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+{"text":"\\section{Introduction}\n\n\nDuring the last two years a race of industrial and research organizations has been opened to develop a ready-to-implement engineering solution for quantum computing (QC). It resulted in the QC market closely resembling the ascent ages of classical computing industry.\nNamely, there were many underdeveloped computing architectures which being incompatible with each other required significant efforts in porting software and algorithmic solutions between them. Given a broadly supported opinion that in the near term we are unlikely to become witnesses to flexible large-scale quantum architectures, there is a critical need to develop  portable, architecture-agnostic hybrid quantum-classical frameworks that will allow solving large-scale computational problems on small-scale quantum architectures. \n\n\nThere are multiple emerging quantum computation paradigms. The performance comparison of these paradigms is an important research topic. In this paper, we present for the first time a performance comparison of two leading quantum computation paradigms - D-Wave  quantum annealing and gate-based universal quantum computation. Both approaches have great potential for achieving quantum speedup for a number of important  problems~\\cite{king2018observation,romero2018strategies,ambainis2018quantum,dunjko2018computational}.\n\n\n\nThe first approach, quantum annealing (QA), is based on adiabatic quantum computation (AQC)~\\cite{mcgeoch2014adiabatic}. QA solves computational problems by using a guided quantum evolution~\\cite{yang2017optimizing}. \nThe evolution starts with an initial Hamiltonian with an easy-to-prepare ground state and ends up in the ground state of the problem Hamiltonian. QA is based on the adiabatic theorem that guarantees that if the Hamiltonian is evolved slowly then transitions to excited states are suppressed during the adiabatic evolution~\\cite{yang2017optimizing}.\nThe D-Wave quantum annealer uses superconducting flux qubits \\cite{amin2004,dwave2018} and has been shown to solve optimization problems on graphs \\cite{ushijima2017graph}, machine learning \\cite{omalley2017}, traffic flow optimization \\cite{neukart2017}, and simulation problems \\cite{harris2018}.\nQuantum and hybrid quantum-classical approaches have been employed.\n\nThe second approach is often referred to as the gate-based or universal QC. This mode of QC was theoretically demonstrated to have a great potential for exponential speedups over best known classical algorithms~\\cite{nielsen2002quantum}. In the near term, the capability of the quantum devices is limited by the number of qubits, low fidelity of gates, and lack of error correction. These limitations constrain us to using low-depth quantum circuits (i.e., quantum circuits with few gates) on a small number of qubits. Within the constraints of near-term intermediate-scale quantum (NISQ) technology~\\cite{preskill2018quantum}, a number of hybrid quantum-classical algorithms were developed and experimentally demonstrated to solve small problems. One of the most promising of such algorithms is Quantum Approximate Optimization Algorithm (QAOA)~\\cite{farhi2014quantum,farhi2016quantum}. QAOA is inspired by adiabatic quantum computation. Similarly to AQC and QA, the evolution path starts with an easy-to-prepare Hamiltonian in the ground state and evolves to the final Hamiltonian that encodes the solution of the problem remaining in the ground state. However, unlike QA in QAOA the evolution is performed by applying a series of parametrized gates called ansatz~\\cite{mcclean2016theory} which is parametrized by a set of variational parameters. This is accomplished by a hybrid approach that combines quantum evolution and classical variational optimization for optimal QAOA parameters~\\cite{yang2017optimizing} with the goal of finding the evolution path that prepares the ground state of the problem Hamiltonian. \n\n\\section{Methodology}\n\nThis work addresses three main challenges. First, we show how to use quantum computing to solve the community detection problem, a well known NP-hard problem. Second, we present an approach to solving realistic large problems using the NISQ hardware with a limited number of noisy qubits.  Third, we demonstrate a method that is portable across two leading quantum computation paradigms and can be easily extended to future hardware.\n\nThe community detection problem (or modularity graph clustering) has a variety of applications ranging from biology to social network analysis~\\cite{palla2005uncovering,su2010glay,bardella2016hierarchical,nicolini2017community}. \nIts complexity \\cite{brandes2006maximizing} and practical importance justify an attempt to solve it using QC. The goal of the community detection is to split nodes of a graph $G=(V,E)$ into communities by maximizing its modularity~\\cite{2006PNAS..103.8577N}:\n\n\\begin{equation}\nH = \\frac{1}{4|E|}\\Sigma_{ij}(A_{ij} - \\frac{k_ik_j}{2|E|})s_is_j =  \\frac{1}{4|E|}\\Sigma_{ij}B_{ij}s_is_j,\n\\label{eq:mod}\n\\end{equation}\n\n\\noindent where $s_i\\in\\{-1,+1\\}$ are variables indicating node $i$th community assignment, $k_i$ is a degree $i\\in V$, and $A$ is the adjacency matrix of $G$.\nIn this paper, we will focus on clustering the graph into two communities. There are several approaches to generalize the problem for cases when the number of communities is greater than 2.\n\nThe clustering of large networks is currently impossible with existing quantum computers because of the small number of available qubits.\nThis limitation applies both to quantum annealing~\\cite{ushijima2017graph} and universal quantum computing~\\cite{otterbach2017unsupervised}. To tackle large problems using available quantum hardware, we use a hybrid quantum-classical local-search approach. Our approach is inspired by existing numerous local-search heuristics (see~\\cite{rotta2011multilevel} for a review). Our algorithm finds a solution to the global community detection problem by selecting subproblems small enough to fit on the target quantum computer, solving them using a quantum algorithm and iterating until the solution to the global problem is found. The outline is presented in Algorithm~\\ref{alg:outline}.\n\n\\begin{algorithm}\n \\caption{Community Detection}\\label{alg:outline}\n\\begin{algorithmic}\n\\Procedure{Community detection}{Graph $G$}\n\\State solution = initial\\_guess($G$)\n \\While{not converged}\n  \t\\State $X$ = populate\\_subset($G$)\n  \t\\State \/\/ \\textit{using QAOA or D-Wave QA}\n  \t\\State candidate = solve\\_subproblem($G$, $X$)\n  \t\\If{$\\mbox{candidate} > \\mbox{solution}$}\n  \t\\State solution = candidate\n    \\EndIf\n \\EndWhile\n \\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nIn particular, we start with a random community assignment. At each step we select a subproblem (subset of vertices $X\\subset V$) by taking the vertices with highest potential gain if moving them from one community to another. The gain for each vertex can be computed efficiently~\\cite{2006PNAS..103.8577N}. Then we fix the community assignment of all $i\\not\\in X$, encode them into the problem as boundary conditions (denoted by $\\tilde{s}_j$, a typical technique in many heuristics \\cite{leyffer2013fast,hager2018multilevel}) and maximize\n\n\\begin{equation}\n\\label{eq:subproblem}\n\\arraycolsep=1.4p\n\\begin{array}{r c l}\nQ_{s} & = & \\sum_{i>j | i,j\\in X}2B_{ij}s_is_j +  \\sum_{i\\in X}\\sum_{j\\not\\in X}2B_{ij}s_i\\tilde{s}_j \\\\\n & = & \\sum_{i>j| i,j\\in X}2B_{ij}s_is_j + \\sum_{i\\in X}C_{i}s_i. \n\\end{array}\n\\end{equation}\n\nThe subproblems are solved using QC. To satisfy the constraints of available hardware, we fix the subproblem size to some small number (in our experiments, it was 25). \n\n\n\\section{Implementation details and Results}\n\nWe implement our local search algorithm in Python using the graph methods provided by NetworkX~\\cite{hagberg2008}. The novelty of our approach is that it allows to use D-Wave QA, QAOA and classical Gurobi \\cite{optimization2014inc} solvers interchangeably simply by passing different flags, enabling rapid prototyping and direct comparison of different methods as the hardware and its capabilities evolve. Additionally, Gurobi was used as a global optimization solver for the sake of quality comparison. To our knowledge this is the first attempt to directly compare universal quantum computing and quantum annealing. Our framework is also easily extendable, making it possible for researchers to add new backends as they become available. We plan to release the framework as an open-source project.\n\nOur results are presented in Figure \\ref{fig:results}. In these experiments, we used the Intel-QS~\\cite{smelyanskiy2016qhipster} simulator for QAOA (at the time our group did not have access to a universal quantum computer of sufficient size). We use six real-world networks from the KONECT dataset~\\cite{kunegis2013konect} with up to 400 nodes as our benchmark. For each network, we ran 30 experiments with different random seeds. The same set of seeds is used between three backend solvers, making the results directly comparable. The subproblem size is fixed at 25 (i.e., 25 qubits are used). Our results demonstrate that the quantum local search approach with both quantum methods is capable of achieving results comparable to state-of-the-art, with a potential to outperform as hardware evolves.\n\n\n \n\n\n\n\n \\vspace{-0.78cm}\n \n\\begin{figure}[htb]\n \\begin{tikzpicture} \n  \\node (img)  {\\includegraphics[width=0.8\\linewidth]{.\/fig\/mod_iter.pdf}};\n  \\node (temp) [left=of img]{};\n  \\node[below=of img, node distance=0cm, yshift=1.1cm,font=\\color{black}] {Network Name};\n \\node[below=of temp, node distance=2cm, anchor=center,yshift=-1.0cm,,xshift=1.2cm, rotate=90,font=\\color{black}] {Num. Solver Calls};\n  \\node[above=of temp, node distance=2cm, xshift=1.2cm,yshift=0.8cm, rotate=90, anchor=center,font=\\color{black}] {Modularity};\n \\end{tikzpicture}\n  \\vspace{-0.4cm}\n \\caption{Box-plots comparing modularity scores (greater is better) and number of solver calls (less is better) respectively for the three different solvers. For the graph {\\tt oz}, Gurobi and D-Wave returned a modularity score greater than the Global Solver (best known value)}\n \\label{fig:results}\n\\end{figure}\n \\vspace{-0.6cm}\n\\section{Discussion}\n\n\n\n\nIn the near term, quantum hardware will be in a constant state of change. Many different NISQ-era hardware solutions will appear and some will be abandoned. In the midst of such evolutionary times, we want to be able to continue research in quantum algorithms and head towards solving real-world problems. To accomplish this, we need portable, architecture-agnostic hybrid quantum-classical frameworks that will allow solving large-scale computational problems on small-scale quantum architectures. Moreover, these frameworks need to be robust and future-proof. In this work, we have presented a prototype of such a framework for solving the problem of community detection in networks on two distinctively different architectures: D-Wave quantum annealer and universal quantum computer. We suggest extending this approach for solving other types of problems in science.\n\nThe constant change of hardware and overall immaturity of the existing technology leads to many risks in QC. In spite of major effort, it has not been experimentally demonstrated yet an ability to achieve speedups over state-of-the-art classical supercomputers and there are valid concerns about scalability of existing implementations~\\cite{brugger2018quantum}. Advances in material design and engineering will allow the community to overcome those hurdles. We expect QC to eventually become a part of the HPC ecosystem with an initial role as an accelerator providing a new layer of parallelism. Our approach will provide for co-design exploration towards the best QC accelerator choice for an application mix.\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\\appendices\n\n\\section*{Acknowledgment}\nThis research used the resources of the Argonne Leadership Computing Facility, which is a U.S. Department of Energy (DOE) Office of Science User Facility supported under Contract DE-AC02-06CH11357. We gratefully acknowledge the computing resources provided and operated by the Joint Laboratory for System Evaluation (JLSE) at Argonne National Laboratory.  The authors would also like to acknowledge the NNSA's Advanced Simulation and Computing (ASC) program at Los Alamos National Laboratory (LANL) for use of their Ising D-Wave 2X quantum computing resource and D-Wave Systems Inc. for use of their 2000Q resource. The LANL research contribution has been funded by LANL Laboratory Directed Research and Development (LDRD).  LANL is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. DOE under Contract DE-AC52-06NA25396. Clemson University is acknowledged for generous allotment of compute time on Palmetto cluster.\n\n\n\n\\bibliographystyle{.\/IEEEtranS}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Motivation}\n\n\n This paper is motivated by work of Szlam, Maggioni, Coifman \\& Bremer \\cite{szlam2} and an observation made explicit by\nSzlam \\cite{szlam}: taking iterated spectral cuts induced by the nodal set of the first non-constant eigenfunction for the Neumann $p$-Laplacian seems to converge to rectangles in shape. \nIt is already observed in \\cite{szlam} that starting with an isosceles right triangle will lead to a spectral cut along the symmetry axis and produce two smaller isoceles right triangles: no convergence to rectangles takes place. However, this is unstable under small perturbations of the initial domain. \n\\begin{center}\n\\begin{figure}[ht!]\n\\includegraphics[width=0.85\\textwidth]{Fig1}\n\\caption{Iterated spectral cuts of the standard graph Laplacian seem to lead to rectangles in a generic, non-convex, non-smooth domain.}\n\\end{figure}\n\\end{center}\n\\vspace{-20pt}\n\nWe believe this to be a fascinating question in itself but an affirmative answer would also be useful in guaranteeing that iterative spectral partitioning is an effective method to partition domains and, ultimately, graphs and data (see Irion \\& Saito \\cite{saito} where this phenomenon is exploited).  More importantly, a better understanding of this problem will shed light on the more general data analysis algorithms based on the $p$-Laplacian. More precisely, let $\\Omega \\subset \\mathbb{R}^2$ be an open, bounded, and connected domain. We now propose an iterative subdivision of $\\Omega$ as follows.\nFor any $1 \\leq p < \\infty$, the ground state of the $p-$Laplacian can be written as\n\\begin{equation} \\label{definition p Lap ground state}\n\\lambda_{1,p}(\\Omega) = \\inf_{\\int_{\\Omega}{f dx} = 0} \\frac{\\int_{\\Omega}{|\\nabla f|^p dx}}{ \\int_{\\Omega}{|f|^p dx}}.\\end{equation}\nIt is known that the function $f$ minimizing this functional exists and we can use it to iteratively define\n\\begin{equation}\n\\label{speccut}\n \\Omega_{n+1} = \\left\\{ x \\in \\Omega_n: f(x) \\geq 0\\right\\}\\,,\n \\end{equation}\nwhere $n=0,1,2,\\ldots$ and $\\Omega_0:=\\Omega$. \nThe function $f$ is only defined up to sign, so restricting to the part of the domain where it is positive is without loss of generality. We raise the following conjecture (and refer to the subsequent paragraphs for clarification and obvious obstacles).\n\\begin{quote} \\textbf{Main Conjecture.} If $\\Omega_0$ is not the isosceles right triangle (having angles 45-90-45), then the sequence of sets $(\\Omega_n)_{n=1}^{\\infty}$ converges to the set of rectangles with eccentricity\nbounded by 2 in the Gromov-Hausdorff distance.\n\\end{quote}\nIt is clear that one cannot expect convergence to a fixed rectangle: in general, the spectral cut of an $a \\times b$ rectangle with $a > b$ will be given by two $(a\/2) \\times b$ rectangles and, as long as $a \\leq 2b$,\nthe next cut would then yield two $(a\/2) \\times (b\/2)$ rectangles. This motivates a refined question.\n\n\\begin{quote} \\textbf{Question.} Do $(\\Omega_{2n})_{n=1}^{\\infty}$ and $(\\Omega_{2n+1})_{n=1}^{\\infty}$ converge in shape to a fixed rectangle?\n\\end{quote}\n\nThere is one obvious obstruction: if $\\Omega_{0}$ is not already a rectangle, then while performing iterated subdivisions, there is always a sequence of choices for the sign of the\neigenfunction that ensures that part of the boundary of $\\Omega_{n}$ coincides with part of the boundary of $\\Omega$ which could possibly be quite ill-behaved (say, fractal).\nHowever, for a sequence of choices of signs that leads to domains bounded away from $\\partial \\Omega_0$ ('deep' inside the domain) this should not be the case.\n\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=.4\\textwidth]{Fig2-1}\\hspace{60pt}\n\\includegraphics[width=.4\\textwidth]{Fig2-2}\n\\caption{Left: the spectral cut of a quadrilateral determined by four corners, $(0, 0)$, $(\\pi\/25, 3\/5)$, $(1,0)$, and $(1, 3\/5-\\exp(1)\/100)$ provided by the graph 1-Laplacian. Right: the spectral cut of a shape close to a quadrilateral provided by the graph 1-Laplacian.}\\label{Fig:One}\n\\end{figure}\n\n\nWhile this particular question seems to be novel, the problem of trying to understand the geometry of nodal cuts induced by the $p-$Laplacian or general nonlinear operators has\nbeen studied for a long time. We refer to \\cite{plap,plap2,plap3,plap4,plap5,plap6,plap7,plap8,plap9,plap0,plap10,plap11} and references therein. We especially\nemphasize the works of Gajewski \\& G\\\"artner \\cite{plap7, plap8} who study the behavior of the cut as $p \\rightarrow 1$ as a means of finding effective ways of separating the\ndomain into two roughly equally sized domains, as well as the work of Parini \\cite{plap11} studying the limit of the cut as $p \\rightarrow 1$ under Dirichlet boundary conditions.\nMany of these results are posed for Dirichlet conditions where the effective functional as $p \\rightarrow 1$ is given by\n$$ \\inf_{E \\subset \\Omega}{  \\frac{ \\mathcal{H}^{n-1}(\\partial E)}{\\mathcal{H}^{n}(E)}} \\quad \\mbox{while the Neumann case induces} \\quad \\inf_{E \\subset \\Omega}{  \\frac{ \\mathcal{H}^{n-1}(\\partial E)}{\\mathcal{H}^{n}(E)\\mathcal{H}^{n}(\\Omega \\setminus E)}}.$$\nWhen the domain has Neumann boundary conditions, the quantity above is referred to as the {\\it Ratio Cut}.\nNew phenomena arise as a consequence. One interesting problem, that may also be of interest in the Dirichlet case, is the stability of the nodal set of the $p-$Laplacian\nas a function of $p$. \n\nIn the result presented here, we will focus on the $1$-Laplacian, where we can establish some preliminary results that suggest stability of rectangular domains as attractors for the proposed spectral dynamical system on domains.  In particular, from henceforward, we refer to the {\\it $1$-spectral cut iteration} of a domain as the operation defined in \\eqref{speccut} with $p=1$ and the {\\it $1$-spectral cut} as the set $C_{n} := \\left\\{ x \\in \\Omega_n: f(x) = 0\\right\\}$. To simplify the terminology, we use {\\it spectral cut} and {\\it $1$-spectral cut} interchangeably. \n\n\\textbf{Numerics.} For the reader's convenience, we briefly recall here the numerical implementation of the p-Laplacian and its related spectral cut \\cite{Buhler_Hein:2009,Hein_Buhler:2010}.\nTake a point cloud $\\mathcal{X}:=\\{x_i\\}_{i=1}^N\\subset (\\mathcal M,d)$, a metric space $\\mathcal M$ with the metric $d$. Construct an affinity matrix ${\\bf W}\\in \\mathbb{R}^{N\\times N}$, where $\\bf W_{ij}$ is the affinity between $x_i$ and $x_j$. It is associated with an undirected affinity graph with $N$ vertices, where the affinity between $x_i$ and $x_j$, $w_{ij}$, is ${\\bf W}_{ij}$ for $i,j=1,\\ldots,N$. The {\\em graph $p$-Laplacian} is defined by\n\\begin{equation}\n\\Delta_p f(i)=\\sum_{j=1}^N w_{ij}\\phi_p(f(i)-f(j)),\n\\end{equation}\nwhere $f\\in\\mathbb{R}^N$ is a function defined on the vertices and $\\phi_p(x)=\\texttt{sign}(x)|x|^{p-1}$ for $x\\in \\mathbb{R}$. When $p=2$, this gives the bilinear form defined by the standard graph Laplacian ${\\bf D} - {\\bf W}$, where ${\\bf D}=\\texttt{diag}({\\bf W}{\\bf 1})$ and ${\\bf 1}$ is a $N$-dim vector with all entries $1$. Clearly, in general the graph p-Laplacian is nonlinear. We have\n\\begin{equation}\n\\langle f,  \\Delta_p f \\rangle = \\frac12 \\sum_{i,j=1}^N w_{ij} |f_i - f_j|^p\n\\end{equation}\nfor $1 \\leq p < \\infty$.  See for instance \\cite{Buhler_Hein:2009} for a discussion of the relationship between the variational formulation and the discrete operator formulation.\n\nA real number $\\lambda$ is called an  eigenvalue for the graph $p$-Laplacian if there exists a non-zero vector $f\\in\\mathbb{R}^N$ so that \\cite[Definition 3.1]{Buhler_Hein:2009}\n\\begin{equation}\n(\\Delta_p f)_i=\\lambda \\phi_p(f_i)\\,,\n\\end{equation}\nwhere $i=1,\\ldots,N$. We call $f$ the $p$-eigenfunction of of the graph $p$-Laplacian associated with the eigenvalue $\\lambda$. We know that ${\\bf 1}$ is the trivial eigenvector with eigenvalue $0$, and we have \\cite[Lemma 3.2]{Buhler_Hein:2009}\n\\begin{equation}\n{\\bf 1}^T\\phi_p(f)=0\n\\end{equation} \nif the eigenvector $f$ is non-trivial.\nIt is shown in \\cite{Buhler_Hein:2009} that a non-zero function $f$ is an eigenvector if and only if it is a critical point (local minima) of the functional \n\\begin{equation}\nF_p(f)=\\frac{\\langle f,  \\Delta_p f \\rangle}{\\|f\\|_{\\ell^p}^p}\\,.\n\\end{equation}\nNote that $F_p(f)$ is the discretization of the functional shown in \\eqref{definition p Lap ground state}.\nIn this work, our main interest is the second $p$-eigenvector of the graph $p$-Laplacian for the spectral clustering purpose. In \\cite{Buhler_Hein:2009}, the second eigenvalue is shown to be the global minimum of the functional\n\\begin{equation}\nF^{(2)}_p(f):=\\frac{\\langle f,  \\Delta_p f \\rangle}{\\texttt{var}_p(f)},\n\\end{equation}\nwhere \n\\begin{equation}\n\\texttt{var}_p(f)=\\min_{c\\in\\mathbb{R}}\\sum_{i=1}^N |f_i-c|^p\\,,\n\\end{equation} \nand the corresponding eigenvector is then given by \n\\begin{equation}\nf_p^{(2)}=f^*-c^*{\\bf 1}, \n\\end{equation}\nwhere $f^*$ is any global minimizer of $F^{(2)}_p$ and $c^*=\\arg\\min_{c\\in \\mathbb{R}} \\sum_{i=1}^N |f^*_i-c|^p$. \nLike the usual spectral clustering, once we have $f_p^{(2)}$, we cluster the point cloud by the signs of its entries.  \nFor $p=1$, \nthe consistency of spectral cut with the graph $1$-Laplacian was studied in \\cite{trillos2016consistency}.\nNumerically, we apply the nonlinear inverse power method proposed in \\cite{Hein_Buhler:2010} to evaluate the iterative bi-partition of a given 2-dimensional domain.\nIn Figure \\ref{Fig:One}, we present some numerically computed Ratio Cuts for nearly rectangular domains. \n\n\n\\textbf{Outline.} The paper proceeds as follows.  In Section \\ref{sec:results}, we highlight the two main theorems we can prove on stability of near rectangular domains.  We also present some open problems to be considered naturally as generalizations of these theorems.   In Section \\ref{sec:pf1}, we give the full proof that the spectral cut algorithm converges to a rectangle with bounded aspect ratio if the initial domain is near a rectangle in Gromov-Hausdorff sense as will be carefully laid out below.  Section \\ref{sec:pf3} provides the details for the proof that quadrilaterals near the rectangle of aspect ratio $2$ in terms of small angle deviations from $90$ degrees will converge under the spectral cut algorithm to rectangles with bounded aspect ratio.   In the appendix we gather some long calculations that are useful in analyzing the Ratio Cut in a neighborhood of a quadrilateral.\n\n\n\n\n\\section*{Acknowledgements} \nWe thankfully acknowledge the generous support of NSF CAREER Grant DMS-1352353  (J.L. Marzuola \\& W. Hamilton).  We thank Stefan Steinerberger for starting this project with us and for many helpful conversations along the way.  \n\n\n\\section{results} \n\\label{sec:results}\nWe prove two results that, while not establishing the main conjecture, do seem to suggest a mechanism by which this procedure happens.\n$$ \\mbox{certain shapes} \\underbrace{\\implies}_{\\mbox{Theorem 1}} \\mbox{nearly straight cuts} \\implies \\mbox{curved quadrilaterals} \\underbrace{\\implies}_{\\mbox{Theorem 2}} \\mbox{rectangles.}$$\nThe missing steps are as follows: (1) we do not know whether a generic $\\Omega_0 \\subset \\mathbb{R}^2$ will ever produce domains $\\Omega_n$ for which Theorem 1 becomes applicable, for example, when $\\Omega_0$ has a fractal boundary; \nand (2) we do not know whether the dynamical system on the space of rectangles ever produces a quadrilateral sufficiently close to the set of rectangles for Theorem 2 to become applicable.\n\n\n\\subsection{Rectangular stability.} The first of the two main results states that the spectral cut of domains that 'roughly' look like rectangles are being cut\nin the middle. More formally, let us carefully describe the domains we will consider.\n\\begin{assumption}\n\\label{rectangle_assumptions}\nWe will work with domains that are small perturbations of a rectangle in the following sense:\n\\begin{itemize} \n\\item The domain $Q$ is a perturbed\nrectangle that is close in the Gromov-Hausdorff distance to a reference rectangle $R$: \n$$d_{GH}(Q, R) \\leq \\varepsilon \\left( \\mbox{length of the shorter side of}~R\\right)\\,,$$\nwhere $\\varepsilon>0$ is sufficiently small.\n\\item\nIn a roughly $10\\sqrt{\\varepsilon}-$neighborhood of the two intersection points of the $1$-spectral cut of $R$ with the boundary, the boundary of $Q$ \ncan be written as graphs of functions of the associated boundary segments of $R$ (see Fig. \\ref{fig:cut}). Moreover, each of these functions can be well approximated by a parabola with bounded, small curvature and small Lipschitz constant.\n\\end{itemize}\n\\end{assumption}\n\n\\begin{center}\n\\begin{figure}[ht!]\n\\begin{tikzpicture}[scale = 2]\n\\draw [] (0,0) -- (1.61,0) -- (1.61, 1) -- (0, 1) -- (0,0);\n\\draw [dashed] (0.8,-0.2) -- (0.8, 1.2);\n\\draw [ultra thick] (0,0) to[out=10,in=170] (1, 0);\n\\draw [ultra thick] (1,0) to[out=340,in=200] (1.61, 0);\n\\draw [ultra thick] (1.61,0) to[out=80,in=280] (1.61, 1);\n\\draw [ultra thick] (1.61,1) to[out=170,in=10] (0, 1);\n\\draw [ultra thick] (0,1) to[out=260,in=100] (0, 0);\n\\draw [ultra thick, dashed] (0.8, -0.2) to[out=85,in=277] (0.81, 1.2);\n\\end{tikzpicture}\n\\caption{A perturbed rectangle $Q$ close to a reference rectangle $R$: the longer part of the boundary of $Q$ can be written as the graph of a Lipschitz function. The goal is to show that the spectral cut of $Q$ is close to that of $R$.}\n\\label{fig:cut}\n\\end{figure}\n\\end{center}\n\n\\begin{theorem}\n\\label{thm1} \nFor a domain satisfying Assumption \\ref{rectangle_assumptions}, the spectral cut occurs in a $\\sqrt{\\varepsilon}-$neighborhood of the midpoint of the long axis.  Moreover, there is a function $c(\\varepsilon)$ tending to\n0 as $\\varepsilon \\rightarrow 0$ such that, if said part of the boundary can be written as a Lipschitz function with Lipschitz constant $L \\leq \\sqrt{2} - c(\\varepsilon)$ in the $\\sqrt{\\varepsilon}-$neighborhood of the spectral cut of $R$, then the spectral cut of $Q$\nis a circular arc near a straight line.\n\\end{theorem}\n\nNumerical examples suggest that this result starts becoming applicable rather quickly: already a small number of cuts seems to suffice to produce roughly\nrectangular shapes. As soon as Theorem 1 becomes applicable the spectral cuts will start being closer and closer to straight lines, which immediately implies that many shapes will end\nup approaching quadrilaterals after several additional steps. The next result shows that as soon as we are dealing with quadrilaterals close to a rectangle, the procedure\nis smoothing and produces quadrilaterals closer to the rectangle; this has to be understood in the usual sense of 'cuts' away from the boundary: a quadrilateral\nhaving a $91^{\\circ}$ degree angle will always pass this angle on to one of its descendants.  \nWe note a certain similarity of Theorem \\ref{thm1} to the result of Grieser-Jerison \\cite{GrieserJerison} for the standard Laplacian on rectangular domains of high aspect ratio with one side described by a curve.  See also the recent work \\cite{BCM} where general curvilinear quadrilaterals were considered.  \n\n\\subsection{Near Curvilinear Quadrilaterals} We settle our conjecture in the case of $\\Omega$ being near a quadrilateral that is close to a rectangle: we prove convergence to a rectangular shape in the Gromov-Hausdorff distance 'away from the boundary' (in the sense\ndiscussed above: rectangles incorporating parts of the initial boundary or boundaries created by very early initial spectral cuts need never be regular).\nWe introduce a qualitative way of measuring distance to quadrilaterals as follows: given a curvilinear quadrilateral $Q$ with angles $\\alpha, \\beta, \\gamma, \\delta$ and sides described by the curves $y = \\gamma_1 (x)$, $y = \\gamma_2(x)$, $x = \\gamma_3(y)$ and $x  = \\gamma_4 (y)$, we\ndefine the functional\n\\begin{equation}\\label{Definition I(Q)}\nI(Q) =  \\left| \\alpha- \\frac{\\pi}{2} \\right| + \\left| \\beta- \\frac{\\pi}{2} \\right| + \\left| \\gamma- \\frac{\\pi}{2} \\right| + \\left| \\delta- \\frac{\\pi}{2} \\right| + \\max_{1 \\leq j \\leq 4} \\sup_s (|\\gamma_j' |(s)   + |\\gamma_j'' |(s)+ |\\gamma_j''' |(s)) .\n\\end{equation}\nWe particularly want our domains to be well approximated by parabolic curves on the sides intersecting the ratio cut, and will refer to such domains as {\\it approximately parabolic curvilinear quadrilaterals}. \n\n\\begin{remark}\nThis is actually quite a bit more restrictive than one needs, though is certainly sufficient; otherwise, the theorem becomes harder to frame as one must introduce a large number of cases for behaviors of each boundary component of the domain.  However, in the calculations below, we will work with domains that are perturbations of a base rectangle with a reasonable aspect ratio, and whose top and bottom boundary components can be well-approximated by parabolas in a neighborhood near the axis of symmetry of the base rectangle.  As a result, we can dramatically change regularity requirements for the left and right curves in our model calculations as long as the perturbations are nearly symmetric in area and bounded by a sufficiently small constant.  \n\\end{remark}\n\n\\begin{theorem} \n\\label{thm2}\nFor an approximately parabolic curvilinear quadrilateral of aspect ratio near $1:2$, there exists $\\varepsilon_1 > 0$ such that if $I(Q) \\leq \\varepsilon_1$, then the $1$-spectral cut induced\nby the $L^1-$Laplacian is a circular arc with opening angle bounded above by $\\varepsilon_1\/8$.  \n\\end{theorem}\nThe proof also shows that the constant $1\/8$ cannot be further improved.\nThis result implies that near-quadrilateral regions have $1$-spectral cuts that are closer to a quadrilateral in Gromov-Hausdorff distance. This implies exponentially fast convergence to rectangles in shape. The way the result is obtained actually allows for a fairly precise understanding of what happens (in particular, it can be used to show that there is a choice of signs such that $I(Q_n)$ grows substantially along the subsequence). We refer to the proof for details. \n\n\n\\subsection{Open problems} These results naturally suggest several open problems; we only name a few that seem \nparticularly natural.\n\n\\begin{enumerate}\n\n\\item Prove the main conjecture for $p=1$; that is, show that a generic $\\Omega_0 \\subset \\mathbb{R}^2$ will produce domains $\\Omega_n$ for which Theorem 1 can be applied, and show that the dynamical system on the space of rectangles produces a quadrilateral sufficiently close to the set of rectangles for Theorem 2 to become applicable. Is it possible to transfer some of the arguments to the range $p \\in (1,1+\\varepsilon_0)$?  What happens\nas $p \\rightarrow \\infty$? \n\\item What happens in higher dimensions, or even to domains with curvature, like manifolds? One would still expect the cut to have a smoothing effect but the types of geometric obstruction that\none could encounter in the process may be more complicated. Is the generic limit given by a rectangular box? If not, is there a finite set of shapes\nthat can arise in the limit?\n\\item It seems natural to expect that similar results should hold true under Dirichlet conditions. This would require the study of the variational problem \n$$ \\inf_{E \\subset \\Omega}{  \\frac{ \\mathcal{H}^{n-1}(\\partial E)}{\\mathcal{H}^{n}(E)}}.$$\nCan the results be extended to this case?\n\\item Experimentally, we observe that the nodal line is (generically) fairly stable under small perturbations of the value of $p$. Can any result in this\ndirection be made precise? Does it help to assume  the domain $\\Omega$ to be convex?\n\\end{enumerate}\n\n\n\n\n\\section{proof of theorem 1}\n\\label{sec:pf1}\n\n\\subsection{Preliminaries.} The crucial ingredient in our approach is that the $p$-Laplacian degenerates as $p \\rightarrow 1^+$. The minimum of the associated energy functional characterizing the ground state is not assumed by any continuous\nfunction: reinterpreting the functional in terms of total variation, the extremal function is constant on two sets in the domain that are separated by\nan $(n-1)-$dimensional hypersurface. More precisely, we have the following consequence of the coarea formula (see \\cite{stein} for further implications\nof this result).\n\n\n\\begin{thm}[Cianchi, \\cite{Cianchi}] Let $\\Omega \\subset \\mathbb{R}^{n}$ be open, bounded and connected. Then we have the sharp Poincar\\'e-type inequality for any sufficiently smooth functions $u$:\n$$\\left\\|u-\\frac{1}{|\\Omega|}\\int_{\\Omega}{u(z)dz}\\right\\|_{L^{1}(\\Omega)} \\leq\n \\left(\\sup_{\\Gamma \\subset \\Omega}{\\frac{2}{\\mathcal{H}^{n-1}(\\Gamma)}\\frac{|S||\\Omega \\setminus S|}{|\\Omega|}}\\right)\\left\\|\\nabla u\\right\\|_{L^{1}(\\Omega)},$$\nwhere $\\Gamma \\subset \\Omega$ ranges over all surfaces which divide $\\Omega$ into two connected open subsets $S$ and $\\Omega \\setminus \\overline S$. \n\\end{thm}\n\nThis means that the nodal line $\\Gamma$ defining the $1$-spectral cut is a hypersurface partitioning $\\Omega$ into two sets $S, \\Omega \\setminus \\overline S$\n$$ \\mbox{so as to minimize the quantity} \\qquad \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|}.$$\nIt is relatively easy to check that this value is assumed if we relax the $L^1$-norm of the gradient and re-interpret it as total variation. Let us assume that $f(x) = a \\chi_{S} + b \\chi_{\\Omega \\setminus S}$.\nWe want $f$ to have mean value 0, which leads to\n$$ a |S| + b |\\Omega \\setminus S| = 0 \\quad \\mbox{and thus} \\quad b = -a\\frac{|S|}{|\\Omega \\setminus S|} \\quad \\mbox{implying} \\quad \\|f\\|_{L^1} = 2a |S|,$$\nwhere we take $a>0$ without loss of generality.\nMoreover, the ``formal'' contribution to the total variation interpretation of the gradient is given by\n$$ \\|\\nabla f \\|_{L^1} = |\\Gamma| (a - b) =|\\Gamma|  \\left(a + a\\frac{|S|}{|\\Omega \\setminus S|} \\right).$$\nThis implies\n$$ \\frac{  \\|\\nabla f \\|_{L^1}  }{ \\|f\\|_{L^1}} =  \\frac12 |\\Gamma|  \\left( \\frac{1}{|S|} + \\frac{1}{|\\Omega\\setminus S|}\\right) =  \\frac12 |\\Gamma| \\frac{|\\Omega|}{|S| |\\Omega \\setminus S|}.$$\nTo be rigorous, convolution with a smooth mollifier shows that one can get arbitrarily close to the optimal constant with smooth functions. We will now show that, for a rectangle, the optimal spectral cut splits the domain via a straight line cut intersecting the longest sides at right angles.  The crucial ingredient here is that the argument does not appeal to symmetry,\nis stable under perturbations and easily implies the same results for domains that are merely close to rectangles in the Gromov-Hausdorff distance.\n\n\\begin{lemma} \\label{Lemma: rectangle result}\nLet $R = [0,a] \\times [0,b] \\subset \\mathbb{R}^2$  with $a > b$. The $1$-spectral cut is exactly at $\\left\\{a\/2\\right\\} \\times [0,b]$.\n\\end{lemma}\n\\begin{proof}\nThe cut in the middle of the longer side yields\n$$ \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|} = \\frac{b (ab)}{(ab)^2\/4} = \\frac{4}{a}.$$\nIt is clear that other spectral cuts touching the longer sides have $ \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|} \\geq \\frac{4}{a},$ since $\\mathcal{H}^1(\\Gamma)\\geq b$ and $|S| |\\Omega \\setminus S|\\leq (ab)^2\/4$. \n\\begin{center}\n\\begin{figure}[ht!] \n\\begin{tikzpicture}[scale = 2]\n\\draw [ultra thick] (0,0) -- (1.61,0) -- (1.61, 1) -- (0, 1) -- (0,0);\n\\draw [dashed] (0.8,0) -- (0.8, 1);\n\\draw [thick] (0.7,0) to[out=19,in=140] (1.61, 0.6);\n\\node at (1.5,0.2) {$S$};\n\\node at (0.3,0.6) {$\\Omega \\setminus S$};\n\\end{tikzpicture}\n\\caption{The geometric construction in the proof of Lemma 1. }\n\\label{fig4}\n\\end{figure}\n\\end{center}\n\nIf $\\Gamma$ touches two opposite sides, then\n$$ \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|}  \\geq b \\frac{ |\\Omega|}{|S| |\\Omega \\setminus S|} \\geq \\frac{b (ab)}{(ab)^2} \\min_{0 \\leq x \\leq 1}{\\frac{1}{x(1-x)}} \\geq \\frac{4}{a}.$$\nEquality can only arise if the cut has length $b$ (forcing it to be a line) and $x=1\/2$ (forcing a split into two domains of the same area). This characterizes\nexactly the cut in the middle.\n\n\nIt remains to deal with the case where the spectral cut touches two adjacent sides; this case is illustrated in Figure \\ref{fig4}. Let us assume that the enclosed domain is denoted by $S$, $|S| = x |\\Omega| = x ab$ for some $0 < x < 1$, and $\\Gamma = \\partial S \\cap \\mbox{int}~R$ is the part of the boundary curve strictly inside the rectangle.\nThe isoperimetric inequality implies that\n$$ \\mathcal{H}^1(\\Gamma) \\geq  \\sqrt{\\pi}\\sqrt{x a b},$$\nand thus\n$$ \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|} \\geq \\frac{ \\sqrt{\\pi x a b} ab }{a^2 b^2 x (1-x)} = \\frac{\\sqrt{\\pi}}{\\sqrt{x}(1-x)} \\frac{1}{\\sqrt{a}\\sqrt{b}}.$$\nAn explicit computation shows that for all $0 < x < 1$\n$$ \\frac{1}{\\sqrt{x}(1-x)}  \\geq \\frac{3\\sqrt{3}}{2}$$\nand therefore, using $b \\geq a,$\n$$ \\frac{\\mathcal{H}^1(\\Gamma) |\\Omega|}{|S| |\\Omega \\setminus S|} \\geq \\frac{3\\sqrt{3 \\pi}}{2} \\frac{1}{a} \\geq \\frac{4.6}{a}.$$\nThis is always a constant fraction worse than the cut along the longest axes, concluding the argument.\n\\end{proof}\n\n\\begin{remark}\n\\label{rmk1}\nIt is easily seen that all aspects of the argument are stable under (even moderately large) perturbations of the domain in the sense of Gromov-Hausdorff distance.  To be specific, take a domain $Q$ satisfying Assumption \\ref{rectangle_assumptions}, and denote the length of the shorter side of the associated rectangle $R$ by $h$. By Assumption \\ref{rectangle_assumptions}, the Gromov-Hausdorff distance between $Q$\nand $R$ is $d_{GH}(Q, R) \\leq \\varepsilon h$. There is clearly a straight line that is parallel to the shorter side of $R$ and bisects the area: the\nbound on the Gromov-Hausdorff distance implies that this straight line has length at most $h + 2h\\varepsilon$. Therefore,\n$$   \\inf_{\\Gamma}{\\frac{\\mathcal{H}^1(\\Gamma)}{|S| |\\Omega \\setminus S|}} \\leq 4h + 8h \\varepsilon.$$\nArguments as in Lemma 1 show that it has to cut the longer side roughly in the middle. \n\\end{remark}\n\n\n\nThe second ingredient is a straightforward consequence of the isoperimetric inequality that we need to make quantitative.\n\\begin{center}\n\\begin{figure}[ht!]\n\\begin{tikzpicture}[scale = 6]\n\\draw [ultra thick] (0,0) -- (1,0);\n\\node at (-0.02, -0.04) {$a$};\n\\node at (1.02, -0.04) {$b$};\n\\node at (0.5, 0.05) {$\\Omega$};\n\\draw [ultra thick] (0,0) to[out=20,in=180] (0.4, 0.08);\n\\draw [ultra thick] (0.4,0.08) to[out=20,in=160] (1, 0);\n\\end{tikzpicture}\n\\caption{A curve slightly longer than the straight line.}\n\\label{fig:areagain}\n\\end{figure}\n\\end{center}\n\n\n\n\n\\begin{lemma} Consider a simply connected domain $\\Omega$ as shown in Figure \\ref{fig:areagain} that is comprised of a straight line segment between $a$ and $b$ and some arbitrary curved line segment enclosing a domain $\\Omega$. For every $\\varepsilon_0 >0$\nthere exists an $\\varepsilon_1 > 0$ such that if\n\\begin{equation}\n\\label{assump1}\n|\\partial \\Omega|  \\leq  \\left(1+\\varepsilon_1\\right) 2\\|a - b\\|,\n\\end{equation}\nthen\n$$ |\\Omega| \\leq  \\frac{1 + \\varepsilon_0}{\\sqrt{6}} \\| a - b\\|^{3\/2} \\sqrt{|\\partial \\Omega| - 2 \\|a-b\\|}.$$\n\\end{lemma}\n\\begin{proof} The isoperimetric inequality immediately implies that the optimal curve defining the Ratio Cut for $Q$ satisfying Assumption \\ref{rectangle_assumptions} must take the form of a circular arc; otherwise, we could fix $a,b$ at the boundary of a circle\nand use the novel shape to create a set in the plane that contains more area than the disk whose boundary has the same size. \n\n It remains to compute the constants for the circular arc.\nChoose a coordinate system with $a,b$ at $(0,0),(\\|a-b\\|,0)$ respectively. The opening angle $\\alpha$ of the sector of the disk with radius $r>0$ that produces such an arc is given by\n$$ \\alpha = 2 \\arcsin{\\left( \\frac{\\|a-b\\|}{2r} \\right)},$$\nwhere $0 < \\alpha < \\frac{\\pi}{2}$ is sufficiently small by assumption \\eqref{assump1}.\nAs a consequence, the length of the circular arc is given by\n$$ r \\alpha =  2r \\arcsin{\\left( \\frac{\\|a-b\\|}{2r}\\right)} \\geq \\|a-b\\| + \\frac{\\|a-b\\|^3}{24} \\frac{1}{r^2},$$\nand so\n\\begin{equation}\n\\label{pomegabd}\n |\\partial \\Omega| \\geq 2\\|a-b\\| +   \\frac{\\|a-b\\|^3}{24} \\frac{1}{r^2}.\n \\end{equation}\nIn particular, by assumption \\eqref{assump1}, we have that \n$$  \\frac{\\|a-b\\|^3}{24} \\frac{1}{r^2} \\leq  2\\varepsilon_1\\|a-b\\| \\qquad \\mbox{and thus} \\qquad \\frac{\\|a-b\\|^2}{r^2}  \\leq 48 \\varepsilon_1.$$\n\nThe enclosed area captured by the line segment and the circular arc is \n$$\\frac{ r^2 \\pi \\alpha}{2\\pi} - r^2 \\cos{\\left(\\frac{\\alpha}{2}\\right)} \\sin{\\left(\\frac{\\alpha}{2}\\right)}.$$\nWe have\n$$ \\frac{ r^2 \\pi \\alpha}{2\\pi} =  r^2 \\arcsin{\\left( \\frac{\\|a-b\\|}{2r} \\right)} = r \\frac{\\|a-b\\|}{2} + \\frac{\\|a-b\\|^3}{48 r}  + \\mbox{higher order terms}$$\nas well as\n$$ \\cos{\\left(\\frac{\\alpha}{2}\\right)} = \\sqrt{1 - \\frac{\\|a-b\\|^2}{4r^2}} \\quad \\mbox{and} \\quad \\sin{\\left(\\frac{\\alpha}{2}\\right)} =  \\frac{\\|a-b\\|}{2r}.$$\nThe higher order terms in the expansion are an infinite series in $\\|a-b\\|\/r$ with exponents decaying fast enough. For $\\varepsilon_1$ sufficiently\nsmall depending on $\\varepsilon_0$, we can bound\n$$ \\mbox{higher order terms}  \\leq \\varepsilon_0  \\frac{\\|a-b\\|^3}{48 r}  .$$\nAltogether this implies that\n$$\\mbox{area} \\leq (1 + \\varepsilon_0) \\frac{\\|a-b\\|^3}{12r}$$\nand plugging in the bound on $1\/r$ from \\eqref{pomegabd} gives\n$$\\mbox{area} \\leq  \\frac{1 + \\varepsilon_0}{\\sqrt{6}} \\| a - b\\|^{3\/2} \\sqrt{|\\partial \\Omega| - 2 \\|a-b\\|}.$$\n\\end{proof}\n\n\n\n\n\n\\subsection{Proof of Theorem 1}\n\n\\begin{proof} \nWe assume without loss of generality $|Q| = 1$. For this proof we minimize, over all possible set partitions $Q = S \\cup (Q \\setminus S)$, the quotient\n$$ \\inf_{\\Gamma} \\frac{\\mathcal{H}^1(\\Gamma) }{|S| | Q \\setminus S|},$$\nwhere $\\Gamma=\\overline S\\cap \\overline{Q \\setminus S}$.\n\nLemma \\ref{Lemma: rectangle result} and Remark \\ref{rmk1} show that the cut must intersect opposite sides; if the aspect ratio is large enough, then these opposite sides will be the longest sides. \n\n\n\n\n\nWe now show that the distance to the axis of symmetry of $R$ is of order $\\lesssim \\sqrt{\\varepsilon}$.\nSince the Gromov-Hausdorff distance between the domain and the rectangle is $\\varepsilon$, by the same calculation in Remark \\ref{rmk1} we see that any cut has to have length at least\n$ h (1- 2\\varepsilon)$, where $h$ is the length of the shorter side of the associated rectangle $R$. Note that the shortest line between two points may not make the optimal Ratio Cut, though, as that may result in a less favorable area splitting. Hence, one possibility is that we get a gain from having a slightly longer curve to encompass a more favorable area, but then by the isoperimetric inequality, it can be seen that circular arcs are favorable to any other configuration in terms of length to area trade-offs.  A key example of this is the trapezoidal domain with angled top and bottom boundary curves. Therefore the optimal spectral cut $Q = A \\cup B$ satisfies\n$$  \\frac{h(1-2\\varepsilon)}{|A| |B|} = \\frac{h(1-2\\varepsilon)}{|A| (1-|A|)} \\leq  \\frac{\\mathcal{H}^1(\\Gamma)}{|S| |Q \\setminus S|} \\leq 4h(1 + 2 \\varepsilon)$$\nfor any $\\Gamma$.\nThus, for $\\varepsilon \\leq 0.1$\n\\begin{equation}\n\\frac{1}{|A| (1-|A|)} \\leq 4\\frac{1 + 2\\varepsilon}{1 - 2 \\varepsilon} \\leq 4  + 20\\varepsilon.\\label{bound of A(1-A)}\n\\end{equation}\nWhen combined with the elementary inequality\n$$ \\frac{1}{x(1-x)}  \\geq   4 + 16\\left(x-\\frac{1}{2}\\right)^2,$$\nwe have that the optimal spectral cut yields two sets satisfying\n$$ \\frac{1}{2} - \\sqrt{\\frac{5\\varepsilon}{4}} \\leq |A| \\quad \\mbox{and}\\quad |B| \\leq \\frac{1}{2} + \\sqrt{\\frac{5\\varepsilon}{4}}.$$\nThe Lipschitz bound then ensures that the spectral cut's intersection with $\\partial Q$ occurs in a $\\sqrt{\\epsilon}$ small neighborhood of $Q$'s axis of symmetry. \n\nSince the functional itself only contains the length $\\mathcal{H}^1(\\Gamma),$ as well as a term depending on the partition of the areas $|S||\\Omega\\setminus S|$, we can conclude that the optimal $\\Gamma$ is a circular arc; otherwise, we can minimize the length of the curve while keeping the bounded area fixed.  Indeed, this boils down to the question of minimizing the arc-length of a curve of fixed area,\n\\[\n\\text{arg min}_{y = \\gamma (x)} \\left\\{  \\int_a^b \\sqrt{1 + (\\gamma'(x) )^2} dx \\ \\bigg| \\ \\int_a^b |\\gamma| dx \\,\\,\\mbox{ fixed.}\\right\\},\n\\] \nwhich is minimized by a circular arc via the isoperimetric inequality.\nWe already know from above that $\\mathcal{H}^1(\\Gamma) \\leq (1+2\\varepsilon) h$, so Lemma 2 implies that the area captured by the curved\narc satisfies\n$$ \\mbox{captured area} \\leq  \\frac{1 + c_{\\varepsilon}}{\\sqrt{6}} h^{3\/2} \\sqrt{\\mathcal{H}^1(\\Gamma) - h} \\,\\leq  \\frac{1 + c_{\\varepsilon}}{\\sqrt{3}} h^2 \\sqrt{\\varepsilon} ,$$\nwhere $c_{\\varepsilon} > 0$ depends on $\\varepsilon$, but does tend to 0 as $\\varepsilon$ tends to $0$. \n\nIn summary, the above conclusions show that the Ratio Cut $\\Gamma$ must be a circular arc in a $\\sqrt{\\epsilon}$ neighborhood of the axis of symmetry of the reference rectangle from Assumption $1$. \n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{thm2} for Curvilinear Quadrilaterals with Parabolic Top and Bottom Curves}\n\\label{sec:pf3}\n\nTo prove Theorem $2$, we will first analyze how the Ratio Cut depends on parameters determining the domain $Q$ and circular arc cut $\\Gamma$ for a true quadrilateral. Then, we will demonstrate for a simple example of a curvilinear trapezoid with one parabolic edge and three flat edges that the curvature has a smaller order impact on the Ratio Cut than the trapezoidal feature. Lastly, we will prove the theorem for even more general domains with parabolic top and bottom bounding curves. These domains will be shown to be generic in section 5, at least to leading order in how the Ratio Cut depends upon the structure of the curves that make up the longest sides of our approximately parabolic curvilinear rectangles. \n\nConsider the quadrilateral $Q$ determined by the vertices\n$$ (0,0), (x_1, a), (1,0) \\quad \\mbox{and} \\quad (1+x_2, a+x_3),$$\nwhere we assume that $ 0 < a < 1$ (implying that we have normalized the quadrilateral and put the longer side on the $x-$axis). The quantities $x_1, x_2, x_3$\nare perturbation parameters and assumed to be small, in the sense that $|x_i| \\ll \\min\\left\\{a,1- a\\right\\}$. The condition $|x_i| \\ll a$ is clear; we want the perturbation to be small with respect to length and height of the rectangle. The other condition is slightly more subtle: if it is not satisfied, then the quadrilateral might be close to a square and the location of the spectral cut will depend nonlinearly on the perturbation parameters. The cut is still going to be a circular arc nearly bisecting the domain, but a slight perturbation of $x_1, x_2, x_3$ may send $\\Gamma$  from being nearly vertical to nearly horizontal (see Fig. \\ref{fig:quad}).\nThe same arguments used to prove Theorem 1 show that the spectral cut will be a circular arc connecting two points ${\\bf q} = (q,0)$ and ${\\bf p} = (p,y(p))$.\n\n\n\\begin{center}\n\\begin{figure}[ht!]\n\\begin{tikzpicture}[scale=4]\n\\draw [ultra thick] (0,0) -- (1,0) -- (1.1, 0.8) -- (-0.1, 0.7) -- (0,0);\n\\filldraw (0,0) circle (0.015cm);\n\\node at (-0.08, -0.08) {(0,0)};\n\\node at (0.08, 0.08) {$\\alpha$};\n\\filldraw (1,0) circle (0.015cm);\n\\node at (0.95, 0.05) {$\\beta$};\n\\node at (1.08, -0.08) {(1,0)};\n\\filldraw (1.1,0.8) circle (0.015cm);\n\\node at (1.2, 0.8+0.08) {($1+x_2$,$a+x_3$)};\n\\filldraw (-0.1,0.7) circle (0.015cm);\n\\node at (-0.24, 0.8-0.08) {($x_1$,$a$)};\n\\filldraw (0.5,0) circle (0.015cm);\n\\node at (0.5, -0.08) {${\\bf q}:=(q,0)$}; \n\\filldraw (0.45,0.75) circle (0.015cm);\n\\node at (0.48,0.82) {${\\bf p}:=(p,y (p)) $}; \n\\draw [thick,dashed] (0.45, 0.75) -- (0.5,0);\n\\draw [thin,dashed] (0.5,0) to[out=45,in=330] (0.45, 0.75);\n\\node at (0.42, 0.35) {$\\Gamma$};\n\\node at (0.69, 0.35) {$\\tilde{\\Gamma}$};\n\\node at (1, 0.7) {$\\gamma$};\n\\node at (-0.03, 0.63) {$\\delta$};\n\\node at (0.43, 0.05) {$\\eta$};\n\\node at (0.55, 0.1) {$\\nu$};\n\\node at (0.4, 0.67) {$\\phi$};\n\\node at (0.51, 0.65) {$\\mu$};\n\\end{tikzpicture}\n\\caption{A general quadrilateral $Q$ after rescaling.}\n\\label{fig:quad}\n\\end{figure}\n\\end{center}\n\n\n\\subsection{Circular Arc to Triangle in terms of the sector angle}\n\n\nLet us assume the cut, $\\tilde E$, is a circular arc from $\\bf p$ to $\\bf q$, where $\\bf p$ and $\\bf q$ appear nearly opposite each other on the longest sides.   We will take $|\\tilde \\Gamma| = r \\theta$ for some radius $r>0$ and sector angle $\\theta\\geq0$, to be determined, and observe then the area contained between the line $\\Gamma$ connecting $\\bf p$ and $\\bf q$ and the circular arc $\\tilde \\Gamma$, which we will call $\\tilde \\Omega$, satisfies\n\\[\n| \\tilde \\Omega | = \\frac{r^2 \\theta}{2} - \\frac{\\| {\\bf p}- {\\bf q}\\|^2}{4 \\tan (\\theta\/2)}.\n\\]\nWe also have that\n\\[\nr = \\frac{\\| {\\bf p}- {\\bf q} \\|}{2 \\sin (\\theta\/2)},\n\\]\nfrom which we conclude, after Taylor expanding in $\\theta$, that\n\\[\n| \\tilde \\Omega | = \\|{\\bf p}- {\\bf q}\\|^2 \\left( \\frac{\\theta^2}{48} + O( \\theta^4) \\right).\n\\]\nAlso, Taylor expanding once more, we have\n\\[\n|\\tilde \\Gamma| = r \\theta = \\| {\\bf p}- {\\bf q}\\| \\left( 1 + \\frac{\\theta^2}{24} + O( \\theta^4) \\right).\n\\]\n\nWith these identities in hand, we can proceed to explore how the ratio cut depends upon the curves defining the longest aspect ratio sides of our curvilinear quadrilateral.  To illustrate how to compute the dependence of the cut locally on the parametrization of a curve, we will first work with a toy model that is flat on $3$ sides and has a smooth quadratic curve on one of the long sides.  \n\n\\subsection{The parabolic trapezoid}\n\\label{partrap}\n\nWe take a domain that is a parabolic trapezoid bounded by a straight line from $(0,0)$ to $(1,0)$, a straight line from $(0,0)$ to $(0,1\/2)$, the straight line from $(1,0)$ to $(1, 1\/2+a)$ and the curve $y(x) = \\epsilon x^2 + (a - \\epsilon) x + 1\/2$; note that, for convenience, we have chosen our aspect ratio to be approximately $2$.  Our goal is to track how the cut changes as a function of $\\epsilon,a$ close to $0$.   The Ratio Cut will intersect the bottom and top boundary curves at the points $(q,0)$ and $(p,y(p))$ respectively, with the cut itself a circular arc.  Splitting the domain by a circular arc, we can decompose one of the cut domains into two curvilinear quadrilaterals, see Figure \\ref{fig:quad2} for an illustration.\n\n\\begin{center}\n\\begin{figure}[ht!]\n\\begin{tikzpicture}[scale=4]\n\\draw [ultra thick] (0,1\/2)--(0,0) -- (1,0) -- (1, 1\/2+0.1) \n\\draw [ultra thick] (0,0.5) to[out=25,in=160] (1, 0.6);\n\\filldraw (0,0) circle (0.015cm);\n\\node at (-0.08, -0.08) {(0,0)};\n\\filldraw (1,0) circle (0.015cm);\n\\node at (1.08, -0.08) {(1,0)};\n\\filldraw (1,1\/2+0.1) circle (0.015cm);\n\\node at (1.08, 1\/2+0.1+0.08) {($1$,$0.6$)};\n\\filldraw (0,0.5) circle (0.015cm);\n\\node at (-0.1, 0.5+0.08) {($0$,$0.5$)};\n\\filldraw (0.5,0) circle (0.015cm);\n\\node at (0.5, -0.08) {${\\bf q}:=(q,0)$}; \n\\filldraw (0.52,0.67) circle (0.015cm);\n\\node at (0.48,0.82) {${\\bf p}:=(p,y (p)) $}; \n\\draw [thin,dashed] (0.5,0) to[out=75,in=285] (0.52, 0.67);\n\\node at (0.45, 0.35) {$\\Gamma$};\n\\end{tikzpicture}\n\\caption{A parabolic trapezoid $Q$ as a simplified model.}\n\\label{fig:quad2}\n\\end{figure}\n\\end{center}\n\nIt can be easily seen that to compute the Ratio Cut, we can split the domains into components given by the region $[0,p]$, a triangle with base $[p,q]$, and the cap of a circular arc that is either added or subtracted depending upon orientation ($p<q$ or $q<p$ and $\\theta > 0$ or $\\theta < 0$ respectively). Decomposing the domain into these components, we compute\n\\begin{align*}\n&A_{total}  = \\int_0^1 y(x) dx = \\frac{\\epsilon}{3} + \\frac{a-\\epsilon}{2} + \\frac12, \\ \\ \\text{(Total Area)} \\\\\n&A_1 (p,q,\\theta)  = \\int_0^p y(x) dx + \\frac12 y(p) (q-p)+\\frac12 R^2 (p,q,\\theta) \\theta \\\\\n& \\hspace{4cm} - \\frac14 \\left[   (q-p)^2 + y(p)^2 \\right] \\cot \\left( \\frac{\\theta}{2} \\right), \\ \\ \\text{(Left Area)} \\\\\n& A_2 (p,q,\\theta)  = A_{total} - A_1 (p,q,\\theta), \\ \\ \\text{(Right Area)} \n\\end{align*}\nwhere $R(p,q,\\theta)$ is the radius of the circle, given by\n\\begin{align}\nR(p,q,\\theta)  = \\frac{\\sqrt{  (q-p)^2 + y(p)^2 }}{ 2 \\sin \\left( \\frac{\\theta}{2} \\right)}\\,.\n\\end{align}\nThe above quantities give the ratio cut, denoted by $\\text{RC} (p,q,\\theta)$:\n\\begin{align*}\n \\text{RC} (p,q,\\theta) = \\frac{ R(p,q,\\theta) \\theta}{ A_1 (p,q,\\theta) A_2 (p,q,\\theta)}.\n\\end{align*}\nIt follows via direct calculation that $ \\text{RC} (p,q,\\theta)$ is a smooth function of $(p,q,\\theta)$ in a neighborhood of $(1\/2,1\/2,0)$. Indeed, since $\\frac12 R^2 (p,q,\\theta) \\theta - \\frac14 \\left[   (q-p)^2 + y(p)^2 \\right] \\cot \\left( \\frac{\\theta}{2} \\right)\\to 0$ and $R(p,q,\\theta) \\theta\\to \\sqrt{  (q-p)^2 + y(p)^2 }$ when $\\theta\\to 0$, $\\text{RC} (p,q,\\theta)$ is smooth in a neighborhood of $(1\/2,1\/2,0)$. Hence we can explore behaviors nearby using the Implicit Function Theorem.   \n\nAs an illustrative calculation, let us simply take the full quadratic approximation in $\\theta$, $p-\\frac12$, $q-\\frac12$, $a$ and $\\epsilon$ to the Ratio Cut. A direct calculation gives:\n\\begin{small}\n\\begin{align*}\n& \\text{RC} (p,q,\\theta)  {\\color{red}=} \\left(  8 + 24 \\left(q - \\frac12 \\right)^2 - 16 (q-\\frac12) (p-\\frac12) + 24 (p - \\frac12)^2 \\right) \\\\\n& - a \\left(8 + 8\\left(q - \\frac12 \\right) - 8 \\left(p - \\frac12 \\right) - 56\\left(q - \\frac12 \\right) ^2 - 56 \\left(p - \\frac12 \\right)^2 + 80 \\left(q - \\frac12 \\right)  \\left(p - \\frac12 \\right)    \\right) \\\\\n& + \\epsilon \\left(  \\frac43 + \\frac{52}{3} \\left(q - \\frac12 \\right)^2 + \\frac{100}{3} \\left(p - \\frac12 \\right)^2    - 40 \\left(q - \\frac12 \\right)  \\left(p - \\frac12 \\right)  \\right) + \\frac43 \\theta \\left(   \\left(q - \\frac12 \\right) + \\left(p - \\frac12 \\right) \\right) \\\\\n& + a^2  \\left( 10 + 16 \\left(q - \\frac12 \\right) - 16 \\left(p - \\frac12 \\right)  + 120 \\left(q - \\frac12 \\right)^2 + 104 \\left(p - \\frac12 \\right)^2  -192 \\left(q - \\frac12 \\right) \\left(p - \\frac12 \\right) \\right)  \\\\\n& + \\epsilon^2 \\left(  \\frac{122}{9}  \\left(q - \\frac12 \\right)^2 + \\frac{218}{9} \\left(p - \\frac12 \\right)^2  - \\frac{284}{9} \\left(q - \\frac12 \\right)  \\left(p - \\frac12 \\right)   \\right)  +  \\frac{7}{18} \\theta^2   \\\\\n& -a \\epsilon \\left(  \\frac83 + \\frac{8}{3} \\left(q - \\frac12 \\right) - 8 \\left(p - \\frac12 \\right)   + 72 \\left(q - \\frac12 \\right)^2 + 104 \\left(p - \\frac12 \\right)^2 - \\frac{464}{3} \\left(q - \\frac12 \\right)  \\left(p - \\frac12 \\right)   \\right)  \\\\\n&  -  a \\theta \\left(  \\frac23 + 8 \\left(q - \\frac12 \\right)^2 - \\frac{32}{3}  \\left(q - \\frac12 \\right)  \\left(p - \\frac12 \\right)  \\right)  - \\frac89 \\epsilon \\theta \\left( \\left(q - \\frac12 \\right)  + \\left(p - \\frac12 \\right)   \\right)  \n\\end{align*}\\end{small} \nup to a higher order error, when $a$ and $\\epsilon$ are sufficiently small.\nWhile this may not look so useful, we get a great deal of information by looking at the system when $a = \\epsilon = 0$.  In such a case, the equations for a critical point in $p,q,\\theta$ become\n\\begin{align*}\n48 \\left(q - \\frac12 \\right)  - 16 \\left(p - \\frac12 \\right) + \\frac43 \\theta & = 0 \\\\\n-16 \\left(q - \\frac12 \\right)  + 48 \\left(p - \\frac12 \\right) + \\frac43 \\theta & = 0 \\\\\n\\frac43 \\left(q - \\frac12 \\right)  + \\frac43 \\left(p  - \\frac12 \\right) + \\frac79 \\theta & = 0.\n\\end{align*}\nThus, the Jacobian matrix is\n\\begin{equation*}\nJ_{a=0,\\epsilon=0} = \\left[  \\begin{array}{rrr}\n48 & -16 & \\frac43 \\\\\n-16 & 48 & \\frac43 \\\\\n\\frac43 & \\frac43 & \\frac79\n\\end{array}\n\\right],\n\\end{equation*}\nwhich is non-singular.  As a result, we observe that if $a = \\epsilon = 0$, the optimal solutions is $p = q = \\frac12$, $\\theta = 0$.  We know this from symmetry arguments, but now we also have set ourselves up for an application of the Implicit Function Theorem in order approximate the RC for near rectangular domains.  \n\nWe next observe what happens if $a \\neq 0$, $\\epsilon = 0$. This gives the modified system\n\\begin{equation*}\nJ_{a,\\epsilon=0} \\left[ \\begin{array}{c}\n\\left(q - \\frac12 \\right)  \\\\\n\\left(p - \\frac12 \\right) \\\\\n\\theta \n\\end{array} \\right] = a \\left[ \\begin{array}{r}\n8 \\\\\n-8 \\\\\n\\frac23\n\\end{array} \\right]  + \\text{Quadratic Error in $a, q-\\frac12, p-\\frac12$},\n\\end{equation*}\nwhere\n\\begin{equation*}\nJ_{a,\\epsilon=0} = \\left[  \\begin{array}{ccc}\n48 -112 a & -16 + 80 a& \\frac43 \\\\\n-16 +80 a & 48 -112 a & \\frac43 \\\\\n\\frac43 & \\frac43 & \\frac79\n\\end{array}\n\\right],\n\\end{equation*}\nHence, $\\vec v$ satisfying $J_{a,\\epsilon=0} \\vec{ v }= \\vec{0}$ is given by \n\\begin{equation*}\n\\vec{v} = a J_{a=0,\\epsilon=0}^{-1} \\left[ \\begin{array}{c}\n8 \\\\ -8 \\\\ \\frac23\n\\end{array} \\right] + O(a^2) = a \\left[  \\begin{array}{c}\n\\frac{1}{12} \\\\ -\\frac16 \\\\ 1 \\end{array}  \\right] + O(a^2).\n\\end{equation*}\nNote that the distance between $p$ and $q$ is then $\\frac{a}{4} < a$, with the maximal amplitude of the bulge from the circular arc of size \n\\[  \\frac{a}{8}\\sqrt{y(p)^2+(p-q)^2} < \\frac{a}{8}. \\]  \nSince the terms that are linear in $\\epsilon$ are quadratic or higher in $q-1\/2,p-1\/2$, a similar analysis including $\\epsilon$ shows that the curvature of the parabolic curve is actually a lower order deformation for the Ratio Cut than the trapezoidal deflection.  Importantly, this argument demonstrates that the trapezoidal deflection is decreasing in the new cut domain.  Though the new cut domain will be closer to aspect ratio $1$, the overall deflections are still decreasing on subsequent domains.\n\n\nUsing the Implicit Function Theorem, we are able to compute comparable results for the full Ratio Cut.  Indeed, to turn this into a rigorous argument, we need first observe that the minimum Ratio Cut at $a=\\epsilon = 0$ is uniquely $p=q=\\frac12$, $\\theta=0$, which is easily seen by looking at the Hessian.  Then, we look at $ \\text{RC} (p,q,\\theta; a, \\epsilon)$ as a map from $\\mathbb{R}^5 \\to \\mathbb{R}^3$, and use the Implicit Function Theorem to construct the desired local map from $(p,q,\\theta;\\,a,\\epsilon)$ in an open set around $(a,\\epsilon) = (0,0)$. \n\n\\subsection{The Ratio Cut with sides given by parabolic approximations}\n\\label{arcs}\n\n\nNow, we proceed to handle a more general family of domains.  Given our assumption on the smoothness of the curves, we can assume that the top and bottom curves are approximated by quadratic curves to high accuracy near points of intersection with the ratio cut.  Specifically, let us consider an arbitrary domain $Q$ that can be approximated (in the Gromov-Hausdorff sense) by a parabolic trapezoid $Q_0$ with vertices $(0,0),(0,\\frac{1}{2}+a_1),(1,\\frac{1}{2}+a_2),$ and $(1,0)$. Note, the inclusion of $a_1$ and $a_2$ here will allow us to vary the aspect ratio. We fix the width of $Q_0$ to $1$, however, as can always be done by a scaling of the domain.  The top paraboloid of $Q_0$ is parametrized as \n$$\ny_{T}(x) = \\epsilon_{t}x^2 + (a_2 - a_1 - \\epsilon_{t})x + a_1 + \\frac{1}{2},\n$$ \nand the bottom paraboloid is parametrized as \n$$\ny_{B}(x) = \\epsilon_{b}x^2 -\\epsilon_b x.\n$$ \nWe assume that $Q$ and $Q_0$ differ by two sufficiently small and bounded ``black-box'' regions on the left and right.\nThe areas of these two small regions will be denoted $A_{\\texttt{WL}}$ and $A_{\\texttt{WR}}$. \n\n\n\\begin{center}\n\\begin{figure}[ht!]\n\\begin{tikzpicture}[scale=4]\n\\draw [ultra thick] (0,1\/2)--(0,0) ;\n\n\\draw [red,ultra thick] (0,0) to[out=5,in=195] (0, 0.5);\n\\node[red] at (-0.15, 0.25) {$A_{\\texttt{WL}}$};\n\n\\draw [ultra thick] (1,0) -- (1, 1\/2+0.1) ;\n\n\\draw [red,ultra thick] (1,0) to[out=105,in=335] (1, 0.6);\n\\node[red] at (1.15, 0.25) {$A_{\\texttt{WR}}$};\n\n\\draw [ultra thick] (0,0.5) to[out=25,in=160] (1, 0.6);\n\\draw [ultra thick] (0,0) to[out=-10,in=190] (1, 0);\n\\filldraw (0,0) circle (0.015cm);\n\\node at (-0.08, -0.08) {(0,0)};\n\\filldraw (1,0) circle (0.015cm);\n\\node at (1.08, -0.08) {(1,0)};\n\\filldraw (1,1\/2+0.1) circle (0.015cm);\n\\node at (1.08, 1\/2+0.1+0.08) {$(1,0.6)$};\n\\filldraw (0,0.5) circle (0.015cm);\n\\node at (-0.1, 0.5+0.08) {($0$,$0.5$)};\n\\filldraw (0.5,-0.05) circle (0.015cm);\n\\node at (0.5, -0.15) {${\\bf q}:=(q,y_B(q))$}; \n\\filldraw (0.52,0.67) circle (0.015cm);\n\\node at (0.48,0.82) {${\\bf p}:=(p,y (p)) $}; \n\\draw [thin,dashed] (0.5,-0.05) to[out=65,in=295] (0.52, 0.67);\n\\draw [dotted] (0.5,-0.05) -- (0.52, 0.67) ;\n\\node at (0.65, 0.48) {$\\Gamma$};\n\\node at (0.55, 0.35) {$\\Omega$};\n\\end{tikzpicture}\n\\caption{A more general domain $Q$ approximated by a parabolic trapezoid.}\n\\label{fig:quad3}\n\\end{figure}\n\\end{center}\n\nWe start by preparing quantities associated to $Q_0$. A circular arc $\\Gamma$ passing through the points $(p,y_{T}(p))$ and $(q, y_B(q))$, with angle $\\theta$, cuts the parabolic trapezoid into a left and right domain, denoted by $S$ and $Q_0\\backslash S$.\n\tTo compute an equivalent analytic expression for the ratio cut, we use Stoke's theorem\n\tto compute the left area $A_L = |S|$ and total $A_T = |Q_0|$, which we use to compute the right area $A_R = |Q_0\\backslash S| = |Q_0|-|S|$. In particular, \n\t$$\n\tA_T = |Q_0| = \\int_{Q_0} dA = \\frac{1}{2} \\int_{\\partial Q_0} xdy - ydx,\n\t$$\n\twhere we integrate along the left vertical boundary $\\{(0,t): 0\\leq t\\leq \\frac{1}{2}+a_1\\}$, along the top parabolic curve $\\{(x,y_T(x)): 0\\leq x\\leq 1 \\},$ along the right vertical boundary $\\{(1,t): \\frac{1}{2}+a_2\\leq t\\leq 0 \\}$ (with the indicated orientation), and finally along the bottom parabolic curve $\\{(x,y_B(x)): 1\\leq x\\leq 0 \\}$ (with the indicated orientation). \n\tWe can compute the (indefinite) integrals for the two parabolic pieces fairly easily:\n\t\\begin{align*}\n\t\tdy_T \t&= (2\\epsilon_t x + a_2-a_1-\\epsilon_t)dx,\\\\\n\t\tdy_B \t&= (2\\epsilon_b x -\\epsilon_b)dx,\n\t\\end{align*}\n\tand so\n\t\\begin{align*}\n\t\t\\frac{1}{2} \\int xdy_T - y_T dx \t&= \\frac{1}{2} \\int x(2\\epsilon_t x + a_2-a_1-\\epsilon_t)dx - (\\epsilon_{t}x^2 + (a_2 - a_1 - \\epsilon_{t})x + a_1 + \\frac{1}{2})dx\\\\\n\t\t\t&= \\frac{1}{2}\\int \\epsilon_t x^2-(a_1+\\frac{1}{2}) dx\n\t\t\t= \\frac{\\epsilon_t}{6}x^3 - \\left(\\frac{a_1}{2}+ \\frac{1}{4} \\right)x;\\\\\n\t\t\\frac{1}{2} \\int xdy_B - y_B dx \t&= \\frac{1}{2}\\int x(2\\epsilon_b x -\\epsilon_b)dx - (\\epsilon_{b}x^2 + -\\epsilon_b x)dx\\\\\n\t\t\t&= \\frac{1}{2} \\int \\epsilon_b x^2 dx\n\t\t\t= \\frac{\\epsilon_b}{6}x^3.\n\t\\end{align*}\nThe $1$-forms for the vertical components are \n\t\\begin{align*}\n\t\t\\frac{1}{2} \\int 0dt - t d(0) &= \\frac{1}{2} \\int 0 = 0,\\\\\n\t\t\\frac{1}{2} \\int 1 dt - t d(1) \t&= \\frac{1}{2} \\int dt - 0 = \\frac{t}{2}.\n\t\\end{align*}\n\tPutting it all together, we have\n\t\\begin{align*}\n\t\tA_T &= 0 + \\left[ \\frac{\\epsilon_t}{6}x^3 - \\left(\\frac{a_1}{2}+ \\frac{1}{4} \\right)x \\right]_{x=0}^{x=\\frac{1}{2}+a_1} + \\left[\\frac{t}{2} \\right]_{t=\\frac{1}{2}+a_2}^{0} + \\left[ \\frac{\\epsilon_b}{6}x^3 \\right]_{x=1}^{x=0}\\\\\n\t\t\t&= \\frac{\\epsilon_t a_1^3}{6} + \\frac{\\epsilon_t a_1^2}{4} + \\frac{\\epsilon_t a_1}{8} + \\frac{\\epsilon_t}{48} -\\frac{\\epsilon_b}{6} - \\frac{a_2}{2} - \\frac{a_1^2}{2} - \\frac{a_1}{2} - \\frac{3}{8}.\n\t\\end{align*}\n\t\n\tFor $A_L = |S|$, we use Stokes' theorem again but split $S$ into two parts divided by the straight line connecting $(p,y_t(p))$ and $(q,y_b(q))$; these parts are a curvilinear quadrilateral and circular cap. The area of the curvilinear quadrilateral is computed by Stokes' theorem, while the area of the circular cap is determined by an elementary formula. Adding these two quantities gives us $A_L$.\n\t%\n\t\n\t\n\tThe line connecting $(p,y_T(p))$ to $(q,y_B(q))$ is parametrized as $\\{ (1-t)(p,y_T(p)) + t(q,y_B(q)): 0\\leq t \\leq 1 \\},$ which componentwise becomes\n\t$$ \n\t( (1-t)p + t q, (1-t)y_T(p) + ty_B(q)), \n\t$$\n\tand so the integral along this straight line becomes\n\t\\begin{align*}\n\t\t\\frac{1}{2} \\int xdy - ydx \t=\\,& \\frac{1}{2} \\int ((1-t)p + t q)d((1-t)y_T(p) + ty_B(q))\\\\ &\\quad - ((1-t)y_T(p) + ty_B(q))d((1-t)p + t q)\\\\\n\t\t\t=\\,& \\frac{1}{2} \\int ((1-t)p + t q)(-y_T(p) dt + y_B(q) dt)\\\\\n\t\t\t\t&\\quad - ((1-t)y_T(p) + ty_B(q)) ( -p dt + q dt)\\\\\n\t\t\t=\\,& t q (p (\\epsilon_b q-\\epsilon_b-\\epsilon_t p+\\epsilon_t)-a_2 (p+1))\\,,\n\t\\end{align*}\n\twhere we denote the last quantity by $SL(t)$.\n\t\n\tGiven two points $p$ and $q$, and an angle $\\theta$, the radius of the circle containing points $p$ and $q$ separated by angle $\\theta$ is \n\t$$\n\tR = \\frac{\\|p-q\\|_2}{2\\sin(\\frac{\\theta}{2})}.\n\t$$ \n\tThus, the area of the circular segment is\t{\\allowdisplaybreaks\n\t$$\n\t|\\Omega|= \\frac{R^2}{2}(\\theta - \\sin(\\theta)) = \\frac{(p - q)^2 + (y_T(p) - y_B(q))^2}{2}(\\theta - \\sin(\\theta)),\n\t$$ \n\tand so\n\t\\begin{align*}\n\t\tA_L =\\,& 0 + \\left[ \\frac{\\epsilon_t}{6}x^3 - \\left(\\frac{a_1}{2}+ \\frac{1}{4} \\right)x \\right]_{x=0}^{x=p} + [SL(t)]_{t=0}^{t=1} + \\left[ \\frac{\\epsilon_b}{6}x^3 \\right]_{x=q}^{x=0} + |\\Omega|\\\\\n\t\t=\\,& \\frac{1}{6} \\left( -\\epsilon_b q^3 - \\frac{3}{2}(1+2a_1)p + \\epsilon_t p^3 + 6q (-a_2(1+p) + p( - \\epsilon_b + \\epsilon_t + \\epsilon_t q - \\epsilon_t p))\\right.\\\\\n\t\t& \\left.+ 3\\left( (q-p)^2 + \\left( a_2  + \\epsilon_b q - \\epsilon_b q^2 + a_2p - \\epsilon_t p + \\epsilon_t p^2\\right)^2\\right) (\\theta - \\sin(\\theta))\\right).\n\t\\end{align*}\n\tWe can use this to get the area of the right portion:\n\t\\begin{align*}\n\t\tA_R =& A_T - A_L \\\\\n\t\t\t=& \\frac{1}{48} \\bigg(-18-24 a_1-24 a_1^2-24 a_2-8 \\epsB+\\epsT+6 a_1 \\epsT+12 a_1^2 \\epsT\\\\\n\t\t\t&+8 a_1^3 \\epsT+8 \\epsB \\tBott^3+12 (1+2 a_1) \\tTop-8 \\epsT \\tTop^3\\\\\n\t\t\t&-48 \\tBott \\big(-a_2 (1+\\tTop)+\\tTop (\\epsT+\\epsB (-1+\\tBott)-\\epsT \\tTop)\\big)\\\\\n\t\t\t&-24 \\big((\\tBott-\\tTop)^2+(a_2+\\epsB \\tBott-\\epsB \\tBott^2+a_2 \\tTop-\\epsT \\tTop+\\epsT \\tTop^2)^2\\big) (\\theta-\\sin(\\theta))\\bigg).\n\t\\end{align*}\n\t}\n\tFinally, the length of the ratio cut takes the form $$|\\Gamma| = R\\theta = \\frac{\\|(p-q, y_T(p)-y_B(q))\\|_2 \\frac{\\theta}{2}}{\\sin(\\frac{\\theta}{2})} = \\frac{\\|(p-q, y_T(p)-y_B(q))\\|_2}{\\sinc(\\frac{\\theta}{2})}.$$\n\n\tWith the above pieces from $Q_0$, the ratio cut of $Q$\n\n\tcan be expressed in terms of variables $p, q,$ and $\\theta$, together with parameters $a_1,a_2, \\epsT, \\epsB, A_{\\texttt{WL}}, A_{\\texttt{WR}}$ that determine the domain.  Specifically, by letting $\\sigma = (a_1,a_2, \\epsT, \\epsB, A_{\\texttt{WL}}, A_{\\texttt{WR}})$ be the parameter vector, we can express the ratio cut as \n\t\\begin{align*}\n\t\tRC(q, p, \\theta; \\sigma) \t&= \\frac{\\|(p-q, y_T(p)-y_B(q))\\|_2}{\\sinc(\\frac{\\theta}{2})(A_L + A_{\\texttt{WL}})(A_R+A_{\\texttt{WR}})}.\n\t\\end{align*}\n\t%\n\tThe series expansion of $RC(q, p, \\theta;\\sigma)$ near $\\left( \\frac{1}{2}, \\frac{1}{2}, 0;   \\vec{0} \\right)$ is\n\t\\begin{align*}\n\t\tRC \t=& 8 + 24 \\tBCent^2 - 16 \\tBCent \\tTCent + \\frac{4}{3} \\tBCent \\Th\\\\\n\t\t\t&+ 24 \\tTCent^2 + \\frac{4}{3} \\tTCent \\Th + \\frac{7}{18}\\Th^2\\\\\n\t\t\t&+ a_1 p_{a_1}(q, p, \\theta) + a_2 p_{a_2}(q, p, \\theta) + a_3 p_{a_3}(q, p, \\theta)\\\\\n\t\t\t&+ \\epsilon_t p_{\\epsilon_t}(q, p, \\theta) + \\epsilon_b p_{\\epsilon_b}(q, p, \\theta)\\\\\n\t\t\t&+ A_{\\texttt{WL}} p_{A_{\\texttt{WL}}}(q, p, \\theta) + A_{\\texttt{WR}} p_{A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ a_1^{\\,2} p_{a_1 a_1}(q, p, \\theta) + a_1a_2 p_{a_1 a_2}(q, p, \\theta) + a_1a_3 p_{a_1 a_3}(q, p, \\theta)\\\\\n\t\t\t&+ a_2^{\\,2} p_{a_2a_2}(q, p, \\theta) + a_2a_3 p_{a_2a_3}(q, p, \\theta) + a_3^{\\, 2} p_{a_3a_3}(p, q, \\theta)\\\\\n\t\t\t&+ a_1A_{\\texttt{WL}} p_{a_1A_{\\texttt{WL}}}(q, p, \\theta) + a_1A_{\\texttt{WR}} p_{a_1A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ a_1 \\epsT p_{a_1 \\epsT}(q, p, \\theta) + a_1 \\epsB p_{a_1 \\epsB}(q, p, \\theta)\\\\\n\t\t\t&+ a_2 A_{\\texttt{WL}} p_{a_2A_{\\texttt{WL}}}(q, p, \\theta) + a_2A_{\\texttt{WR}} p_{a_2 A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ a_2 \\epsT p_{a_2 \\epsT}(q, p, \\theta) + a_2 \\epsB p_{a_2 \\epsB}(q, p, \\theta)\\\\\n\t\t\t&+ a_3 A_{\\texttt{WL}} p_{a_3A_{\\texttt{WL}}}(q, p, \\theta) + a_3A_{\\texttt{WR}} p_{a_3 A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ a_3 \\epsT p_{a_3 \\epsT}(q, p, \\theta) + a_3 \\epsB p_{a_3 \\epsB}(q, p, \\theta)\\\\\n\t\t\t&+ A_{\\texttt{WL}}A_{\\texttt{WL}} p_{A_{\\texttt{WL}} A_{\\texttt{WL}}}(q, p, \\theta) + A_{\\texttt{WL}} A_{\\texttt{WR}} p_{A_{\\texttt{WL}} A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ A_{\\texttt{WR}} A_{\\texttt{WR}} p_{A_{\\texttt{WR}} A_{\\texttt{WR}}}(q, p, \\theta)\\\\\n\t\t\t&+ A_{\\texttt{WL}} \\epsT p_{A_{\\texttt{WL}} \\epsT}(q, p, \\theta) + A_{\\texttt{WL}} \\epsB p_{A_{\\texttt{WL}} \\epsB}(q, p, \\theta)\\\\\n\t\t\t&+ A_{\\texttt{WR}} \\epsT p_{A_{\\texttt{WR}} \\epsT}(q, p, \\theta) + A_{\\texttt{WR}} \\epsB p_{A_{\\texttt{WR}} \\epsB}(q, p, \\theta)\\\\\n\t\t\t&+ \\epsT \\epsT p_{\\epsT \\epsT}(q, p, \\theta) +  \\epsT \\epsB p_{\\epsT \\epsB}(q, p, \\theta) +  \\epsB \\epsB p_{\\epsB \\epsB}(q, p, \\theta)\\,,\n\t\\end{align*}\nup to a higher order error, where the polynomial $p_{\\sigma^{\\alpha}}$ is the partial derivative $\\left. \\frac{\\partial^{\\alpha} RC}{\\partial \\sigma^{\\alpha}} \\right|_{\\sigma = 0}$ for a multi-index $\\alpha$.  Detailed fomulae for these terms are given in the Appendix.\t\n\t\n\t\n\tNext, we compute the linearization of $RC$ near the point $(q,p,\\theta) = (\\frac{1}{2}, \\frac{1}{2}, 0)$:\n\t\\begin{align*}\n\t\t\\left.\\frac{\\partial RC}{\\partial q}\\right|_{\\sigma = 0} &= 48\\big (q - \\frac{1}{2}\\big) - 16 \\big(p - \\frac{1}{2}\\big) + \\frac{4}{3}\\theta,\\\\\n\t\t\\left.\\frac{\\partial RC}{\\partial p}\\right|_{\\sigma = 0} &= -16 \\big(q - \\frac{1}{2}\\big) +48 \\big(p - \\frac{1}{2}\\big) + \\frac{4}{3}\\theta,\\\\\n\t\t\\left.\\frac{\\partial RC}{\\partial \\theta}\\right|_{\\sigma = 0} &= \\frac{4}{3} \\big(q - \\frac{1}{2}\\big) + \\frac{4}{3} \\big(p - \\frac{1}{2}\\big) + \\frac{7}{9}\\theta.\n\t\\end{align*}\n\t%\n\tThe Jacobian of $RC$ at $\\sigma = 0$ is thus\n\t$$\n\tJ = \\begin{pmatrix} 48 & -16 & \\frac{4}{3}\\\\ -16 & 48 & \\frac{4}{3} \\\\ \\frac{4}{3} & \\frac{4}{3}  & \\frac{7}{9}  \\end{pmatrix}.\n\t$$\n\t%\nHere, and below, we have kept the $0$-coefficient terms solely as a place keeper to demonstrate that we have actually computed the coefficients of all the terms in our expansion, as well as to make it easier to verify the formulae for the interested reader.\n\tNext, we explore the other pieces of the linearization of $RC$, namely all first-order terms in the variables, together with the parameters.\n\t{\\allowdisplaybreaks\n\tExplicitly, we have\n\t\\begin{align*}\n\n\t\t\\frac{\\partial RC}{\\partial q} \t&= \\left( 48 - 112 a_1 - 112 a_2 + 112 a_3 - 256 A_{\\texttt{WL}} - 256 A_{\\texttt{WR}} - \\frac{200}{3} \\epsilon_b + \\frac{104}{3} \\epsilon_t \\right)\\left( q - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad  + \\left(-16 + 80 a_1 + 80a_2 - 80a_3 + 40 \\epsilon_b - 40\\epsilon_t \\right)\\left( p - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad  + \\left(\\frac{4}{3} - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t \\right)\\theta\\\\\n\t\t&\\quad  + \\Big(8a_1 -8a_2 - 8a_3 + 32 A_{\\texttt{WL}} - 32 A_{\\texttt{WR}} - 32 a_1 a_1 + 0 a_1a_2 + 32 a_1a_3\\\\\n\t\t&\\quad\\quad +32 a_2a_2 + 0 a_2a_3 - 32 a_3a_3 - 128a_1 A_{\\texttt{WL}} + 0 a_1 A_{\\texttt{WR}} + \\frac{8}{3} a_1 \\epsilon_t - \\frac{8}{3} a_1\\epsilon_b\\\\\n\t\t&\\quad \\quad  + 0a_2A_{\\texttt{WL}} + 128 a_2 A_{\\texttt{WR}} - \\frac{8}{3} a_2 \\epsilon_t + \\frac{8}{3} a_2 \\epsilon_b\\\\\n\t\t&\\quad \\quad  + 64 a_3 A_{\\texttt{WL}} - 64 a_3 A_{\\texttt{WR}} - 8 a_3 \\epsilon_t + 8 a_3 \\epsilon_b\\\\\n\t\t&\\quad \\quad  - 512 A_{\\texttt{WL}} A_{\\texttt{WL}} + 0 A_{\\texttt{WL}}A_{\\texttt{WR}} + 512 A_{\\texttt{WR}}A_{\\texttt{WR}} \\\\\n\t\t&\\quad \\quad  + \\frac{32}{3} A_{\\texttt{WL}} \\epsilon_t - \\frac{32}{3} A_{\\texttt{WL}} \\epsilon_b - \\frac{32}{3} A_{\\texttt{WR}} \\epsilon_t + \\frac{32}{3} A_{\\texttt{WR}}\\epsilon_b + 0 \\epsilon_t \\epsilon_t + 0 \\epsilon_t \\epsilon_b + 0 \\epsilon_b \\epsilon_b \\Big)\\\\\n\t\t&\\quad + \\text{ cubic terms},\n\t\\end{align*}\n\t\\begin{align*}\n\n\t\t\\frac{\\partial RC}{\\partial p} \t&= \\left(-16 + 80 a_1 + 80a_2 - 80a_3 + 40 \\epsilon_b - 40\\epsilon_t \\right)\\left( q - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad + \\left(48 - 112 a_1 - 112 a_2 +112 a_3 - 256 A_{\\texttt{WL}} - 256 A_{\\texttt{WR}} - \\frac{104}{3} \\epsilon_b + \\frac{200}{3} \\epsilon_t \\right)\\left( p - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad  + \\left(\\frac{4}{3} - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t \\right)\\theta\\\\\n\t\t&\\quad  + \\left(-8a_1 + 8a_2 + 8a_3 + 32A_{\\texttt{WL}} - 32 A_{\\texttt{WR}} +32 a_1 a_1 + 0 a_1a_2 - 32 a_1a_3\\right.\\\\\n\t\t&\\quad\\quad \\left.-32 a_2a_2 + 0 a_2a_3 + 32 a_3a_3 - 64 a_1 A_{\\texttt{WL}} + 64 a_1 A_{\\texttt{WR}} - 8 a_1 \\epsilon_t + 8 a_1\\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad  \\left. - 64 a_2A_{\\texttt{WL}} + 64 a_2 A_{\\texttt{WR}} + 8 a_2 \\epsilon_t - 8 a_2 \\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad \\left. + 0 a_3 A_{\\texttt{WL}} - 128 a_3 A_{\\texttt{WR}} + \\frac{8}{3} a_3 \\epsilon_t - \\frac{8}{3} a_3 \\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad \\left. - 512 A_{\\texttt{WL}} A_{\\texttt{WL}} + 0 A_{\\texttt{WL}}A_{\\texttt{WR}} + 512 A_{\\texttt{WR}}A_{\\texttt{WR}} \\right.\\\\\n\t\t&\\quad \\quad \\left. + \\frac{32}{3} A_{\\texttt{WL}} \\epsilon_t - \\frac{32}{3} A_{\\texttt{WL}} \\epsilon_b + \\frac{32}{3} A_{\\texttt{WR}} \\epsilon_t + \\frac{32}{3} A_{\\texttt{WR}}\\epsilon_b + 0 \\epsilon_t \\epsilon_t + 0 \\epsilon_t \\epsilon_b + 0 \\epsilon_b \\epsilon_b \\right)\\\\\n\t\t&\\quad + \\text{ cubic terms},\n\t\\end{align*}\n\tand\n\t\\begin{align*}\n\n\t\t\\frac{\\partial RC}{\\partial \\theta} \t&= \\left(\\frac{4}{3} - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t \\right)\\left( q - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad  + \\left(\\frac{4}{3} - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t \\right)\\left( p - \\frac{1}{2} \\right)\\\\\n\t\t&\\quad  + \\left(\\frac{7}{9} - \\frac{5}{9} a_1 - \\frac{5}{9}a_2 + \\frac{5}{9} a_3 - \\frac{32}{9}A_{\\texttt{WL}} - \\frac{32}{9}A_{\\texttt{WR}} + \\frac{1}{54}\\epsilon_b - \\frac{1}{54} \\epsilon_t \\right)\\theta\\\\\n\t\t&\\quad + \\left(\\frac{2}{3}a_1 - \\frac{2}{3}a_2 + \\frac{2}{3} a_3 + \\frac{8}{3}A_{\\texttt{WL}} - \\frac{8}{3}A_{\\texttt{WR}} - \\frac{4}{3} a_1 a_1 + 0 a_1a_2 + 0 a_1a_3\\right.\\\\\n\t\t&\\quad\\quad \\left.+ \\frac{4}{3} a_2a_2 - \\frac{4}{3} a_2a_3 + \\frac{4}{3} a_3a_3 - 8 a_1 A_{\\texttt{WL}} - \\frac{8}{3} a_1 A_{\\texttt{WR}} - \\frac{1}{9} a_1 \\epsilon_t + \\frac{1}{9} a_1\\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad  \\left.+ \\frac{8}{3} a_2A_{\\texttt{WL}} + 8 a_2 A_{\\texttt{WR}} + \\frac{1}{9} a_2 \\epsilon_t - \\frac{1}{9} a_2 \\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad \\left. - \\frac{8}{3} a_3 A_{\\texttt{WL}} - 8 a_3 A_{\\texttt{WR}} - \\frac{1}{9} a_3 \\epsilon_t + \\frac{1}{9} a_3 \\epsilon_b\\right.\\\\\n\t\t&\\quad \\quad \\left. - \\frac{128}{3} A_{\\texttt{WL}} A_{\\texttt{WL}} + 0 A_{\\texttt{WL}}A_{\\texttt{WR}} + \\frac{128}{3} A_{\\texttt{WR}}A_{\\texttt{WR}} \\right.\\\\\n\t\t&\\quad \\quad \\left. - \\frac{4}{9} A_{\\texttt{WL}} \\epsilon_t + \\frac{4}{9} A_{\\texttt{WL}} \\epsilon_b + \\frac{4}{9} A_{\\texttt{WR}} \\epsilon_t - \\frac{4}{9} A_{\\texttt{WR}}\\epsilon_b + 0 \\epsilon_t \\epsilon_t + 0 \\epsilon_t \\epsilon_b + 0 \\epsilon_b \\epsilon_b \\right)\\\\\n\t\t&\\quad + \\text{ cubic terms}.\n\t\\end{align*}\n\t%\n\tWith these expansions, we can express the linearization as $J+ J_{\\sigma}$, where the terms $J_{\\sigma,ii}$ come from the partials computed above. Explicitly,\n\t\\begin{align*}\n\t\tJ_{\\sigma,11} \t&= - 112 a_1 - 112 a_2 - 256 A_{\\texttt{WL}} - 256 A_{\\texttt{WR}} - \\frac{200}{3} \\epsilon_b + \\frac{104}{3} \\epsilon_t,\\\\\n\t\tJ_{\\sigma,12}\t&= 80 a_1 + 80a_2 + 40 \\epsilon_b - 40\\epsilon_t,\\\\\n\t\tJ_{\\sigma,13} \t&= - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t,\\\\\n\t\tJ_{\\sigma,21} \t&= 80 a_1 + 80a_2 + 40 \\epsilon_b - 40\\epsilon_t,\\\\\n\t\tJ_{\\sigma,22} \t&= - 112 a_1 - 112 a_2 - 256 A_{\\texttt{WL}} - 256 A_{\\texttt{WR}} - \\frac{104}{3} \\epsilon_b + \\frac{200}{3} \\epsilon_t,\\\\\n\t\tJ_{\\sigma,23} \t&= - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t,\\\\\n\t\tJ_{\\sigma,31} \t&= - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t,\\\\\n\t\tJ_{\\sigma,32} \t&= - \\frac{32}{3}A_{\\texttt{WL}} - \\frac{32}{3}A_{\\texttt{WR}} + \\frac{8}{9} \\epsilon_b - \\frac{8}{9}\\epsilon_t,\\\\\n\t\tJ_{\\sigma,33} \t&= - \\frac{5}{9} a_1 - \\frac{5}{9}a_2 - \\frac{32}{9}A_{\\texttt{WL}} - \\frac{32}{9}A_{\\texttt{WR}} + \\frac{1}{54}\\epsilon_b - \\frac{1}{54} \\epsilon_t.\n\t\\end{align*}\n\t\n\t\n\tWith these pieces we can express the criterion for being a critical point of the Ratio Cut, incorporating linear terms in the parameters, as\n\t\\begin{align*}\n\t\t(J+J_{\\sigma}) \\begin{pmatrix} q - \\frac{1}{2}\\\\ p - \\frac{1}{2}\\\\ \\theta \\end{pmatrix} &= L(\\sigma)\\\\\n\t\t&:= a_1 \\begin{pmatrix} 8 \\\\ -8 \\\\ \\frac{2}{3}\\end{pmatrix} + a_2 \\begin{pmatrix}-8\\\\ 8\\\\ -\\frac{2}{3}\\end{pmatrix} + a_3 \\pmat{-8 \\\\ 8 \\\\ \\frac{2}{3}} + A_{\\texttt{WL}} \\begin{pmatrix}32 \\\\ 32 \\\\ \\frac{8}{3} \\end{pmatrix} + A_{\\texttt{WR}} \\begin{pmatrix}-32\\\\ -32 \\\\ -\\frac{8}{3}\\end{pmatrix}\\\\\n\t\t&\\quad  + a_1a_1 \\pmat{-32 \\\\ 32 \\\\ - \\frac{4}{3}} + a_1a_2 \\pmat{0 \\\\ 0 \\\\ 0} + a_1a_3 \\pmat{32 \\\\ -32 \\\\ 0} \\\\\n\t\t&\\quad  + a_2a_2 \\pmat{32 \\\\ -32 \\\\ \\frac{4}{3}} + a_2a_3 \\pmat{0 \\\\ 0 \\\\ -\\frac{4}{3}} + a_3a_3 \\pmat{ - 32 \\\\ 32 \\\\ \\frac{4}{3} } \\\\\n\t\t&\\quad \t+ a_1A_{\\texttt{WL}} \\pmat{ -128 \\\\ -64 \\\\ -8 } + a_1 A_{\\texttt{WR}} \\pmat{ 0 \\\\ 64 \\\\ - \\frac{8}{3} } + a_1 \\epsilon_t \\pmat{ \\frac{8}{3} \\\\ -8 \\\\ - \\frac{1}{9} } + a_1 \\epsilon_b \\pmat{ - \\frac{8}{3} \\\\ 8 \\\\ \\frac{1}{9} } \\\\\n\t\t&\\quad \t+ a_2 A_{\\texttt{WL}} \\pmat{ 0 \\\\ -64 \\\\ \\frac{8}{3} } + a_2 A_{\\texttt{WR}} \\pmat{ 128 \\\\ 64 \\\\ 8 } + a_2 \\epsilon_t \\pmat{ - \\frac{8}{3} \\\\ 8 \\\\ \\frac{1}{9} } + a_2 \\epsilon_b \\pmat{ \\frac{8}{3} \\\\ - 8 \\\\ -\\frac{1}{9} } \\\\\n\t\t&\\quad  + a_3 A_{\\texttt{WL}} \\pmat{ 64 \\\\ 0 \\\\ -\\frac{8}{3} } + a_3 A_{\\texttt{WR}} \\pmat{ -64 \\\\ - 128 \\\\ -8 } + a_3 \\epsilon_t \\pmat{ -8 \\\\ \\frac{8}{3} \\\\ -\\frac{1}{9} } + a_3 \\epsilon_b \\pmat{ 8 \\\\ -\\frac{8}{3} \\\\ \\frac{1}{9} } \\\\\n\t\t&\\quad  + A_{\\texttt{WL}} A_{\\texttt{WL}} \\pmat{ -512 \\\\ -512 \\\\ -\\frac{128}{3}  } + A_{\\texttt{WL}} A_{\\texttt{WR}} \\pmat{ 0 \\\\ 0 \\\\ 0 } + A_{\\texttt{WR}} A_{\\texttt{WR}} \\pmat{ 512 \\\\ 512 \\\\ \\frac{128}{3}  } \\\\\n\t\t&\\quad  + A_{\\texttt{WL}} \\epsilon_t \\pmat{\\frac{32}{3} \\\\ \\frac{32}{3} \\\\ - \\frac{4}{9}  } + A_{\\texttt{WL}} \\epsilon_b \\pmat{ - \\frac{32}{3} \\\\ - \\frac{32}{3} \\\\ \\frac{4}{9} } \\\\\n\t\t&\\quad  + A_{\\texttt{WR}} \\epsilon_t \\pmat{ - \\frac{32}{3} \\\\ \\frac{32}{3} \\\\ \\frac{4}{9} } + A_{\\texttt{WR}} \\epsilon_b \\pmat{ \\frac{32}{3} \\\\ \\frac{32}{3} \\\\ -\\frac{4}{9}  } \\\\\n\t\t&\\quad  + \\epsilon_t \\epsilon_t \\pmat{0\\\\0\\\\0   } + \\epsilon_t \\epsilon_b \\pmat{0\\\\ 0\\\\ 0  } + \\epsilon_b \\epsilon_b \\pmat{0\\\\ 0 \\\\ 0  }\\\\\n\t\t&\\quad + \\text{ cubic terms in the parameters}.\n\t\\end{align*}\n\t}\n\t\\subsection*{Using the Implicit Function Theorem}\n\t\n\tSuppose $v$ is a critical triple of values for the Ratio Cut, i.e. $(J+J_{\\sigma})v = 0$. Then near $\\sigma = 0$, we can solve for $v$ in terms of the parameters in $\\sigma$:\n\t\\begin{align*}\n\t\tv \t&= J^{-1} L(\\sigma)\\\\\n\t\t\t&= \\pmat{\\frac{1}{12}(a_1 - a_2 - 2a_3) + (A_{\\texttt{WL}} - A_{\\texttt{WR}}) \\\\ -\\frac{1}{12} (2a_1 - 2a_2 - a_3) + (A_{\\texttt{WL}} - A_{\\texttt{WR}}) \\\\ a_1 - a_2 + a_3} + \\text{quadratic terms in the parameters}.\n\t\\end{align*}\n\t%\n\tWe will not write out the full $J^{-1}L(\\sigma)$ here, though it is used for the approximations later and its form becomes clear in the representations below.\n\t%\n\t\n\t\n\tTo summarize, we have shown in this section that, near the critical point $(p,q,\\theta) = \\left( \\frac{1}{2}, \\frac{1}{2}, 0 \\right)$ of the ratio cut, and for $\\sigma=(a_1, a_2, \\epsilon_b, \\epsilon_t, A_{\\texttt{WL}}, A_{\\texttt{WR}})$ near $0$, we have \n\t\\begin{align*}\n\t\t\\begin{pmatrix} q - \\frac{1}{2} \\\\ p - \\frac{1}{2} \\\\ \\theta \\end{pmatrix} = \\pmat{\\frac{1}{12}(a_1 - a_2 - 2a_3) + (A_{\\texttt{WL}} - A_{\\texttt{WR}}) \\\\ -\\frac{1}{12} (2a_1 - 2a_2 - a_3) + (A_{\\texttt{WL}} - A_{\\texttt{WR}}) \\\\ a_1 - a_2 + a_3}+ \\text{quadratic terms in the parameters}.\n\t\\end{align*}\n\t\n\t\t\n\t\\section{Are our domains generic?}\n\t\n\tSince Ratio Cuts are circular, it seems more natural to prove Theorem 2 for circular, instead of parabolic, boundary curves. Using circular arcs, however, proved to be intractable for our computational methods, whereas parabolic arcs could be used. This section shows that such approximations only incur third order errors, and thus Theorem 2 is still applicable for circular boundary curves. This first lemma shows this approximation is valid near the intersection of the cut and the original domain's boundary. \n\t\n\t\\begin{lemma}\n\tProvided $I(Q)$ defined in \\eqref{Definition I(Q)} is sufficiently small, we may approximate the top and bottom curves by parabolas in a neighborhood of the Ratio Cut.\n\t\\end{lemma}\n\t\n\t\\begin{proof}\n\tThe small $I(Q)$ makes sure the domain is approximately trapezoidal with a, for example, $1:2$, aspect ratio, and that the top and curves are $C^3$ with small oscillations on a uniform spatial scale.  Hence, Taylor expanding the top and bottom curves out to $2$nd order on this scale near $p=1\/2, q = 1\/2$, we can create a parabolic approximation that is accurate up to $O( |p-\\frac12|^3, |q-\\frac12|^3)$ in this region. Potentially modifying the left and right wings in order to ensure the wings of domain have the appropriate error outside this uniform neighborhood, we see that these curvilinear trapezoids will have ratio cuts that are indeed approximated up to lower order terms in the same fashion as our exact parabolic trapezoid calculations.   \n\t\\end{proof}\n\t\t\n\tIn practice, we are given a domain $Q$ whose top and bottom boundary curves are more general to start, and upon iteration of our domains we are most interested when sides are given by circular arcs. To handle smooth enough domains of this form in full generality, we can construct a parabolic trapezoid by: rescaling the domain to fit our aspect ratio; locally using a parabolic approximation for the top and bottom curves; and finally cutting off left and right portions of $\\Omega$, which are treated as black-box regions. The Ratio Cut function sees these regions as general wing areas, denoted $A_{\\texttt{WL}}$ and $A_{\\texttt{WR}}$ for the left- and right-wing areas respectively.  The measure of distance from the rectangle, $I(Q)$, in which we measure the deflection of our domains, allows us to approximate our quadrilateral by a parabolic curve with errors that are higher order in the parameter space. \n\t\t\n\t\\subsection{Can a circular arc be parabolically approximated?}\n\t\n\tOptimal cuts for generic parabolic trapezoids are circular. Because of this, we would expect the top and bottom boundary curves of the domains in the next iteration we have been considering to be circular, but not parabolic. Here we show that parabolic arcs approximate circular arcs well, and that the ratio cut for circular boundary curves differs from the ratio cut for parabolic boundary curves at third order and higher. Thus, working with parabolic curves does not incur any significant error in the ratio cut series expansion.\n\t\n\t\\begin{lemma}\n\t   Up to higher order error terms in the curve parameters, we may approximate the top and bottom circular arc curves as paraboloids encompassing the correct area.\n\t   \\end{lemma}\n\t   \n\t   The remainder of this section will be devoted to the proof of this Lemma.\n\t\n\t\\subsubsection{Approximating Circular Arcs} \n\tFirst let us consider the most important example for our iterated domain conjecture, in which the top and bottom boundaries are circular arcs.  In what follows, the top and bottom parabolic curves will be, respectively,\n\t\\begin{align*}\n\t\ty_T^P(x) \t&= \\epsilon_t x^2 + (a_1 - a_2 - \\epsilon_t)x+ \\left( a_1 + \\frac{1}{2} \\right),\\\\\n\t\ty_B^P(x) \t&= \\epsilon_b x^2 + (a_3 - \\epsilon_b)x.\n\t\\end{align*}\n\t\n\tThe parameters $a_1, a_2, a_3$ denote horizontal perturbations from the top-left, top-right, and bottom-right vertices of a rectangle, so our parabolic trapezoid has vertices $(0,0)$, $\\left(0, \\frac{1}{2}+a_1 \\right)$, $\\left(1, \\frac{1}{2}+a_2 \\right),$ and $\\left(1, a_3 \\right)$. The terms $\\epsilon_t$ and $\\epsilon_b$ are curvature parameters that specify the shape of the two boundary curves.\n\t\n\tThe circular arcs are given by the formulas\n\t\\begin{align*}\n\t\ty_T^C(x) &= c_{y,t} + \\sqrt{r_t^2 - (x-c_{x,t})^2}, \\\\\n\t\ty_B^C(x) \t&= c_{y,b} - \\sqrt{r_b^2 - (x-c_{x,b})^2},\n\t\\end{align*}\n\twhere $r_i$ and $(c_{x,i},c_{y,i})$ are the radius and center of the corresponding top ($i=T$) or bottom ($i=B$) circle:\n\t\n\t\\begin{align*}\n\t\tr_t &= \\frac{\\sqrt{1 + (a_1-a_2)^2}}{2 \\sin(\\frac{\\theta}{2})},\\\\\n\t\tr_b &= \\frac{\\sqrt{1+a_3^2}}{2 \\sin(\\frac{\\theta}{2})},\n\t\\end{align*}\n\tand the center point coordinates are found from the systems of equations (for $i=T$ and $i=B$)\n\t\\begin{align*}\n\t\t&\\begin{cases}\n\t\t(c_{x,t})^2 + (\\frac{1}{2} + a_1 - c_{y,t})^2 \t= r_t^2,\\\\\n\t\t(1-c_{x,t})^2 + (\\frac{1}{2}+a_2 - c_{y,t})^2 \t= r_t^2,  \n\t\t\\end{cases}\\\\\n\t\t\\text{and } \n\t\t&\\begin{cases}\n\t\t(c_{x,b})^2 + (c_{y,b})^2 \t= r_b^2,\\\\\n\t\t(1-c_{x,b})^2 + (a_3 - c_{y,b})^2 \t= r_b^2.\n\t\t\\end{cases}\n\t\\end{align*}\n\t\n\tAs an explicit example, expressing $y_B^C$ in terms of the parameters gives\n\t\\begin{align*}\n\n\t\ty_B^C(x) \t&= \\frac{a_3}{2} + \\cot\\left( \\frac{\\theta_b}{2} \\right) - \\left( \\frac{1}{1+a_3^2} \\left( a_3^4 - (1-2x)^2 - 4a_3^2 (x-1) x \\right. \\right. \\\\\n\t\t&\\quad \\left.\\left. + 2a_3 (1+a_3^2) (1-2x) \\cot\\left( \\frac{\\theta_b}{2} \\right) + (1+a_3^2) \\csc^2\\left( \\frac{\\theta_b}{2} \\right)  \\right) \\right)^{\\frac{1}{2}}.\n\t\\end{align*}\n\t\n\tThe next lemma shows how to choose $\\epsilon_t$ and $\\epsilon_b$ so that the parabolic trapezoid's area approximates the circular trapezoid's area up to third order. These curvature terms are chosen so that the quadrilateral's parabolic caps have, up to third order terms, the same area as corresponding circular arcs.\n\t\n\t\\begin{lemma}\n\tLet $l_T(x) = (1-t)\\left(\\frac{1}{2}+a_1 \\right) + t \\left(\\frac{1}{2}+a_2 \\right)$, and write \n\t\\begin{equation}\n\tS_T^C = \\{ (x,y) \\colon l_T(x) \\leq y \\leq y_T^C(x) \\}, \\,\\, S_T^P(x) = \\{ (x,y) \\colon l_T(x) \\leq y \\leq y_T^P(x) \\}\n\t\\end{equation} \n\tfor the regions bounded below by the straight line $\\{\\{ t, l_T(x) \\} \\colon t \\in [0,1] \\}$ and above by the curves $\\{\\{t, y_T^i(x) \\} \\colon x\\in[0,1], i = C \\text{ or } P \\}$ respectively. Similarly, let $l_B(x) = t a_3,$ and write\n\t \\begin{equation}\n\tS_B^C = \\{ (x,y) \\colon l_B(x) \\leq y \\leq y_B^C(x) \\},\\,\\, S_B^P(x) = \\{ (x,y) \\colon l_B(x) \\leq y \\leq y_B^P(x) \\}\n\t\\end{equation} \n\tfor the regions bounded above by the straight line $\\{\\{ t, l_B(x) \\} \\colon t \\in [0,1] \\}$ and below by the curves $\\{\\{t, y_B^i(x) \\} \\colon x\\in[0,1], i = C \\text{ or } P\\}$ respectively.\n\t\n\t\tFor $\\epsilon_t = -\\frac{1+(a_1-a_2)^2}{2}\\theta_t$ and $\\epsilon_b = \\frac{1+a_3^2}{2}\\theta_b$, we have\n\t\t\\begin{align*}\n\t\t\t|S_T^C| -|S_T^P| \t&= O(\\theta_t^3)\n\t\t\t\\end{align*}\n\t\t\tand\n\t\t\t\\begin{align*}\n\t\t\t\t\t\t|S_B^C| - |S_B^P| \t&= O(\\theta_b^3) .\n\t\t\\end{align*}\n\t\\end{lemma}\n\t\n\t\\begin{proof}\n\t\tBoth $|S_T^C|$ and $|S_B^C|$ can be found using basic trigonometry: the area of a circular segment with radius $r$ and angle $\\theta$ is $\\frac{1}{2} r^2 (\\theta - \\sin(\\theta))$. In our case,\n\t\t\\begin{align*}\n\t\t\t|S_T^C| \t&= \\frac{1}{2} \\left(\\frac{\\sqrt{1+(a_1-a_2)^2}}{2\\sin(\\frac{\\theta_t}{2})} \\right)^2 (\\theta_t - \\sin(\\theta_t) )\\\\\n\t\t\t\t&= \\frac{1+(a_1-a_2)^2}{12}\\theta_t + \\frac{1+(a_1-a_2)^2}{360}\\theta_t^3 + O(\\theta_t^4),\\\\\n\t\t\t|S_B^C| \t&= \\frac{1}{2} \\left(\\frac{\\sqrt{1+a_3^2}}{2\\sin(\\frac{\\theta_b}{2})} \\right)^2(\\theta_b - \\sin(\\theta_b))\\\\\n\t\t\t\t&= \\frac{1+a_3^2}{12} \\theta_b + \\frac{1+a_3^2}{360} \\theta_b^3 + O(\\theta_b^4).\\\\\n\t\t\\end{align*}\n\t\t\n\t\tFor $|S_T^P|$ and $|S_B^P|$, we integrate:\n\t\t\\begin{align*}\n\t\t\t|S_T^P| \t&= \\int_{0}^1 y_T^P(x) - l_T(x)dx\\\\\n\t\t\t\t&= \\int_0^1 \\epsilon_t x^2 + (a_2 - a_1 - \\epsilon_t)x + \\left( \\frac{1}{2} + a_1 \\right) - (1-x)\\left( \\frac{1}{2}+a_1 \\right) - x\\left(\\frac{1}{2}+a_2 \\right) dx\\\\\n\t\t\t\t&= -\\frac{\\epsilon_t}{6}\n\t\t\t\t\\end{align*}\n\t\tand \n\t\t\\begin{align*}\n\t\t |S_B^C| \t&= \\int_0^1 l_B(x) - y_B^C(x) dx\\\\\n\t\t\t&= \\int_0^1 a_3 x -\\epsilon_b x^2 - (a_3 - \\epsilon_b)x dx\n\t\t\t= \\frac{\\epsilon_b}{6}.\n\t\t\\end{align*}\n\t\tWe want to choose constants $C_t$ and $C_b$ so that setting $\\epsilon_t = C_t \\theta_t$ and $\\epsilon_b = C_b \\theta_b$ gives us $|S_T^C|-|S_T^P| = O(\\theta_t^3)$ and $|S_B^C| - |S_B^P| = O(\\theta_b^3)$. Indeed, letting $\\epsilon_t = -\\frac{1+(a_1-a_2)^2}{2}\\theta_t$ and $\\epsilon_b = \\frac{1+a_3^2}{2}\\theta_b$ gives us this result.\n\t\\end{proof}\n\t\n\tSince the parabolic and circular trapezoids only differ in the type of caps on each, making these choices for $\\epsilon_t$ and $\\epsilon_b$ ensures the area of the circular and parabolic trapezoids are very well approximated. In addition, the boundary curves agree up to third order, and so the left areas and ratio cut lengths also agree to at least third order (in the parameters).\n\t\\begin{cor} \n\t\tSetting $\\epsilon_t = -\\frac{1+(a_1-a_2)^2}{2}\\theta_t$ and $\\epsilon_b = \\frac{1+a_3^2}{2}\\theta_b$, we get the following approximations:\n\t\t\\begin{align*}\n\t\ty_T^C(x) - y_T^P(x) &= \\frac{1}{48}(1+(a_1-a_2)^2) \\theta_t^2 \\left( O(a_1) + O(a_2) + O(\\theta_t)  \\right),\\\\\n\t\ty_B^C(x) - y_B^P(x) &= \\frac{1}{48} (1+a_3^2) \\theta_b^2 (O(a_3) + O(\\theta_b)).\n\t\t\\end{align*}\n\t\t\n\t\tMoreover, we have\n\t\t\\begin{align*}\n\t\tA_T^C \t&= A_T^P + O(\\theta_t^3) + O(\\theta_b^3),\\\\\n\t\t\\text{and } A_L^C \t&= A_L^P + O(\\theta_t^2)(O(a_1) + O(a_2)) + O( \\theta_b^2)O(a_3),\n\t\t\\end{align*}\n\t\twhere $A_T^i$, $A_L^i$ is the total area and left area respectively of the parabolic ($i=P$) or circular ($i=C$) trapezoid, and\n\t\t\\begin{align*}\n\t\t\t\\|(p,y_T^C(p)) - (q,y_B^C(q)) \\| \t&= \\|(p,y_T^P(p)) - (q,y_B^P(q))\\| + O(\\theta_t^3)+O(\\theta_b^3).\n\t\t\\end{align*}\n\t\\end{cor}\n\t\n\tThe corollary justifies our use of only quadratic terms in the Ratio Cut Taylor expansion, since the parabolic and circular Ratio Cuts (i.e. Ratio Cut functions of parabolic or circular trapezoids) agree up to third order in the parameters of the domains, given the proper curvature term substitutions.\n\t\n\t\\begin{proof}\n\t\tThe parabolic and circular boundary curve approximations come by Taylor expansions.\n\t\t%\n\t\tFor the areas, recall that the parabolic and circular trapezoids differ only in their ``caps''. Thus, assuming $a_1<a_2$ and $a_3<0$, the total area of the circular trapezoid is\n\t\t\\begin{align*}\n\t\t\tA_T^C \t&= \\left(\\frac{1}{2} + a_1\\right) + \\frac{1}{2} \\left(a_2-a_1 \\right) + \\frac{1}{2}\\left(-a_3 \\right) + \\area(S_T^C) + \\area(S_B^C)\\\\\n\t\t\t\t&= \\left(\\frac{1}{2} + a_1\\right) + \\frac{1}{2} \\left(a_2-a_1 \\right) + \\frac{1}{2}\\left(-a_3 \\right) + \\area(S_T^P) + \\area(S_B^P) + O(\\theta_t^3) + O(\\theta_b^3)\\\\\n\t\t\t\t&= A_T^P + O(\\theta_t^3) + O(\\theta_b^3).\n\t\t\\end{align*}\n\t\t%\n\t\tWrite \n\t\t\\begin{equation}\n\t\tL^C := \\|(p,y_T^C(p)) - (q,y_B^C(q))\\|,\\,\\,L^P := \\|(p,y_T^P(p)) - (q,y_B^P(q))\\|; \n\t\t\\end{equation}\n\t\tthese quantities are the lengths of the line segments connecting the intersection points of intersection between the ratio cut and boundary of the region. Expanding $L^C$ into a Taylor series in $\\theta_b$ and $\\theta_t$ gives \n\t\t\\begin{align*}\n\t\t\tL^C = L^P + O(\\theta_b^3) + O(\\theta_t^3).\n\t\t\\end{align*}\n\t\t\n\t\tFinally, write \n\t\t\\begin{equation}\n\t\t\\bar{y}_j^P(x) := y_j^P(x) + r_j(x), \n\t\t\\end{equation}\n\t\twhere $r_j$ is quadratic in the parameters, for $j = T,B$. Let $\\gamma_1$ be the left boundary segment of the (circular or parabolic) trapezoid, $\\gamma_2$ the bottom boundary segment, $\\gamma_3$ the right boundary segment, and finally $\\gamma_4$ the top boundary segment, such that $\\sum_i \\gamma_i$ is oriented counter-clockwise.\n\t\t\n\t\tIn general, if a function $y=f(x)$ is expressed as a sum $f(x) = g(x) + h(x)$, then in utilizing Stokes' theorem we get\n\t\t\\begin{align*}\n\t\t\t\\frac{1}{2} \\int x dy - ydx \t&= \\frac{1}{2} \\int x d(g(x) + h(x)) - (g(x)+h(x)) dx\\\\\n\t\t\t\t&= \\frac{1}{2} \\int x dg + xdh - g dx - h dx\\\\\n\t\t\t\t&= \\frac{1}{2} \\int xdg - g dx + \\frac{1}{2} \\int x dh - h dx.\n\t\t\\end{align*}\n\t\tIn short, the line integrals involved in Stokes' theorem are linear in the input $y$.\n\t\t\n\t\tSince we are approximating the circular arcs as parabolic arcs plus correction terms, we only need to account for the pieces of the area line integral that incorporate the correction pieces. The left boundary curve is the same for both the parabolic and circular trapezoids, so splitting up the line integrals into parabolic pieces plus correction terms gives \n\t\t\\begin{align*}\n\t\t\tA_L^C &= \\sum_i \\frac{1}{2} \\int_{\\gamma_i} xdy - ydx + A_S^C\\\\\n\t\t\t&=\\, A_L^P + \\frac{1}{2} \\int_{0}^{q} x d (r_b(x)) - r_b(x) dx\\\\\n\t\t\t&\\quad + \\frac{1}{2} \\int_{q}^{p} l_{right,x}(x) d(l_{right,y}(x)) - l_{right,y}(x) d(l_{right,x}(x))\\\\\n\t\t\t&\\quad + \\frac{1}{2} \\int x d(r_t(x)) - r_t(x) dx + O(\\theta_b^3)+O(\\theta_t^3),  \n\t\t\\end{align*}\n\t\twhere the $A_S^i$'s are the the circular segment areas bounded by the straight line connecting the two intersection points and the ratio cut, and \n\t\t\\begin{align*}\n\t\t(l_{right,x}(x),l_{right,y}(x))& \\\\\n\t\t:= \\frac{p - x}{p - q}(p, y_T^C(p))& + \\frac{x - q}{p-q}(q, y_B^C(q)) - \\left( \\frac{p - x}{p - q}(p, y_T^P(p)) + \\frac{x - q}{p-q}(q, y_B^P(q)) \\right)\n\t\t\\end{align*} \n\t\tis the error between the line segments connecting the parabolic arc intersection points and the circular arc intersection points. \n\t\t\n\t\tSince the functions $r_i(x), l_{right,x}(x), l_{right,y}(x)$ are at least third order in the parameters, the line integrals involving these terms also agree up to third order. Finally, as indicated in the computation, the circular segment area $A_S^C$ is cubic in the parameters because the lengths of the line segments agree up to cubic terms. Thus, $A_L^C = A_L^P + O(\\theta_b^2) O(a_3)+O(\\theta_t^2)( O(a_1) + O(a_2))$. \n\t\\end{proof}\n\t\n\tPutting the above results together, we conclude that the ratio cuts corresponding to circular and parabolic boundary curves differ only in cubic terms of the parameters. Putting these pieces together, we see the same is true for the full ratio cut.\n\t\n\t\\begin{lemma}\n\t\tLet $\\Gamma^C, A_T^C,$ and $A_L^C$ be the ratio cut, total area, and left-area respectively of a domain constructed from circular boundary arcs. Then we can construct a new domain with parabolic arcs such that\n\t\t\\begin{align*}\n\t\t\tRC^C - RC^P \t&= \\frac{|\\Gamma^C|}{(A_T^C-A_L^C)A_L^C} - \\frac{|\\Gamma^P|}{(A_T^P-A_L^P)A_L^P} = O(|r|^3),\n\t\t\\end{align*}\n\t\twhere $\\Gamma^P, A_{T}^P,$ and $A_{L}^P$ are the ratio cut, total area, and left-area corresponding to a domain with parabolic boundary arcs.\n\t\\end{lemma}\n\t\n\t\\begin{proof} \n\t\tBy the previous lemma (and its corollary), we can write\n\t\t\\begin{align*}\n\t\t\tA_T^C \t&= A_T^P + O(\\theta_t^3)+O(\\theta_b^3),\\\\\n\t\t\tA_L^C \t&= A_L^P + O(\\theta_t^2)(O(a_1)+O(a_2)) + O(\\theta_b^2)O(a_3),\\\\\n\t\t\tl^C \t&= l^P + O(\\theta_b^3) + O(\\theta_t^3),\n\t\t\\end{align*}\n\t\tand so\n\t\t\\begin{tiny}\n\t\t\\begin{align*}\n\t\t\tRC^C \t&= \\frac{|\\Gamma^C|}{(A_T^C - A_L^C)A_L^C}\\\\\n\t\t\t\t\t&= \\frac{ \\frac{l^P + O(\\theta_b^3) + O(\\theta_t^3)}{\\sinc(\\frac{\\theta}{2})}}{(A_T^P + O(\\theta_t^3)+O(\\theta_b^3) - A_L^P - O(\\theta_t^2)(O(a_1)+O(a_2)) - O(\\theta_b^2)O(a_3) )(A_L^P + O(\\theta_t^2)(O(a_1)+O(a_2)) + O(\\theta_b^2)O(a_3))}.\n\t\t\\end{align*}\n\t\t\\end{tiny}\n\t\tTaylor expanding the difference between $RC^P$ and $RC^C$ in terms of the parabolic approximations gives\n\t\t\\begin{align*}\n\t\t\tRC^P - RC^C \t&= O(a_1\\theta_t^2) + O(a_2\\theta_t^2) + O(\\theta_t^3) + O(a_3\\theta_b^2) + O(\\theta_b^3).\n \t\t\\end{align*}\n \t\tSince the parabolic approximation incurs only third (and higher) order errors, so does the full ratio cut function as seen in the Taylor series expansion.\n\t\\end{proof}\n\t\n\tThis lemma lets us safely use the parabolic approximation when investigating the dependency of the ratio cut on the parameters.\n\n\n\n\n\n\t\t\n\t\t\n\t\t\\section{Discussion and Conclusion}\n\n\t\\subsection{Pictures comparing optimizers with approximations}\n\t\n\tFirst, let us demonstrate how well our approximation does to compute the Ratio Cut when the curves are parabolic arcs.  The set of plots in Figures \\ref{compplot1}-\\ref{compplot8} compare optimal values of $RC$ with the linear approximations in $p, q,$ and $\\theta$.\n\t%\n\tEach plot has the optimal ratio cut value (solid blue) for a family of parameters, indicated next to the $x$-axis. The circle parameters for the optimal cuts were computed numerically using Mathematica's {\\texttt{FindMinimum}} command. The dot-dashed red curves correspond to the linear approximations for optimal $q, p,$ and $\\theta$. The figure below each plot corresponds to an extremal configuration used to compare optimal and approximate ratio cut values. \n\t%\n\tAll parameters $a_1, a_2, a_3, \\epsilon_b, \\epsilon_t, A_{\\texttt{WL}}$, and $A_{\\texttt{WR}}$ are zero unless otherwise stated.\n\t\n\t\n\t\t\\subsection{Potential Applications}\n\nGiven domains that are \"nearly rectangular\" in the appropriate sense, we have shown here that spectral cuts determined by the $1$-Laplacian in geometric domains consist of area minimizing curves that reduce the distance to being rectangular through a series of qualitative geometric results and a careful asymptotic expansion in a parametrization of the cut near a point of (near) symmetry in the domain.  Given the isoperimetric nature of the cuts, we propose that $1$-Laplacian partitioning could be useful in data analysis, however we acknowledge that the numerical tools for computing the Ratio Cut would likely need to be improved computationally to make large data sets tractable.  Indeed, the isoperimetric nature of the Ratio Cut makes it clear to imagine how partitioning could occur, but also shows that approximating a continuum domain by a graph accurately requires great care due to the subtlety with which graph distances can fluctuate in point clouds.\n\nConsistency of graph $1$-Laplacian spectral cuts was studied in \\cite{trillos2016consistency}.  See \\cite{Buhler_Hein:2009,Hein_Buhler:2010,BHcode} for current computational methods for computing $1$-Laplacian states using proximal-dual type numerical schemes.   Similar ideas have been proposed in \\cite{bresson2013adaptive}.  See also \\cite{nossek2018flows} for similar schemes originating from a diffusion-type equation, with the intent of forcing the equation to converge to a nonlinear eigenfunction.  \n\t\t\n\t\n\t\n\n\t\\begin{figure}\n\t\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_positive_plot-eps-converted-to.pdf}\n\t\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_positive_errors-eps-converted-to.pdf}  \\\\\n\t\t\t\\includegraphics[width=0.2\\textwidth]{final_pictures\/a1_positive_figure-eps-converted-to.pdf}\n\t\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = 0.1$.}\n\t\t\t\t\\label{compplot1}\n\t\t\\end{figure}\n\t\\begin{figure}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_negative_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_negative_errors-eps-converted-to.pdf}  \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/a1_negative_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1$ values between $-0.1$ and $0$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = -0.1$.}\t\t\\label{compplot2}\n\t\t\\end{figure}\n\t\t\t\n\t\\begin{figure}\t\n\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/AWL_positive_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/AWL_positive_errors-eps-converted-to.pdf}  \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/AWL_positive_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $A_{WL}$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $A_{WL} = 0.1$.}\n\t\t\t\t\\label{compplot3}\n\t\t\\end{figure}\n\t\\begin{figure}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/epsT_negative_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/epsT_negative_errors-eps-converted-to.pdf} \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/epsT_negative_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $\\epsilon_t$ values between $-0.5$ and $0$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $\\epsilon_t = -0.5$.}\n\t\t\t\t\\label{compplot4}\n\t\\end{figure}\n\t\n\t\\begin{figure}\n\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_a3_together_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_a3_together_errors-eps-converted-to.pdf} \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/a1_a3_together_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1=a_3$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = a_3 = 0.1$.}\n\t\t\t\t\\label{compplot5}\n\t\t\\end{figure}\n\t\\begin{figure}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_a3_opposite_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_a3_opposite_errors-eps-converted-to.pdf} \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/a1_a3_opposite_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1=-a_3$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = -a_3 = 0.1$.}\n\t\t\t\t\\label{compplot6}\n\t\\end{figure}\n\t\n\t\\begin{figure}\t\n\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_epsT_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_epsT_errors-eps-converted-to.pdf} \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/a1_epsT_figure-eps-converted-to.pdf}\n\t\t\\caption{ (Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1=-\\epsilon_t\/5$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = -\\epsilon_t\/5 = 0.1$.}\n\t\t\t\t\\label{compplot7}\n\t\t\\end{figure}\n\t\\begin{figure}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_epsB_plot-eps-converted-to.pdf}\n\t\t\\includegraphics[width=0.25\\linewidth]{final_pictures\/a1_epsB_errors-eps-converted-to.pdf}  \\\\\n\t\t\\includegraphics[width=0.2\\linewidth]{final_pictures\/a1_epsB_figure-eps-converted-to.pdf}\n\t\t\\caption{(Top Left) Ratio cut values corresponding to the optimal circular cuts (solid blue line) and parabolically approximated cuts (dot-dashed red line) for domains with various $a_1=\\epsilon_b\/5$ values between $0$ and $0.1$. (Top Right)  The absolute error between the approximate and optimal ratio cut values in the left plot.  (Bottom) The optimal cut (solid blue line) and parabolically approximated cut (dot-dashed red line) for the trapezoidal domain with $a_1 = \\epsilon_b\/5 = 0.1$.}\n\t\t\t\t\\label{compplot8}\n\t\\end{figure}\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGiven a sequence $w=w_1w_2\\dots w_n$ of members of a totally ordered\nset, we say $j$ is a \\emph{strongly outstanding element} of $w$ if\nwhenever $i < j$ we have $w_i < w_j$. In this case we call $w_j$ a\n\\emph{strongly outstanding value}.  We say $j$ is a \\emph{weakly\noutstanding element} if $w_i \\leq w_j$ whenever $i < j$, and call\n$w_j$ a \\emph{weakly outstanding value}. In this paper we will explore\nthe contexts in which $w$ is a permutation, a multiset permutation, or a\nword over some finite alphabet. A famous theorem of R\\'{e}nyi\n\\cite{ren} (see also \\cite{lc}) states that the number of\npermutations of $[k]$ with $r$ strongly outstanding elements is\nequal to the number of such permutations with $r$ cycles, the latter\nbeing given by $\\left[\\begin{array}{ll}k\\\\r\\end{array}\\right]$, the\nunsigned Stirling number of the first kind.\n\nIn this paper we investigate additional properties of the\noutstanding elements and values of permutations, and extend them to\nmultiset permutations and words on $[k]=\\{1,2,\\dots,k\\}$. An\ninteresting sidelight to our results is that we obtain a proof of\nGauss's celebrated $_2F_1$ evaluation by comparing two forms of one\nof our generating functions, in section \\ref{sec:str} below.\n\n\\section{Summary of results}\n\\subsection{Multiset permutations}\nFor permutations\\footnote{Note that for permutations, all outstanding elements and values are\nstrongly outstanding.} we have the following results.\n\\begin{theorem}\n\\label{th:sms} Let $M =\\{1^{a_1}, 2^{a_2}, \\dots, k^{a_k}\\}$ be a\nmultiset with $N=a_1+a_2+\\dots+a_k$, and let $f(M,r)$ denote the\nnumber of permutations of $M$ that contain exactly $r$ strongly\noutstanding elements. Then\n\\begin{equation}\n\\label{gfso}\nF_M(x)\\, =\\kern -12.7pt\\raise 6pt\\hbox{{\\tiny\\textrm{def}}}\\,\\,\\sum_r f(M,r)x^r=\\frac{(N-1)!a_k x}{a_1!a_2!\\dots\na_k!}\\prod_{i=1}^{k-1}\\left(1+\\frac{a_ix}{N-(a_1+a_2+\\dots+a_{i})}\\right).\n\\end{equation}\n\\end{theorem}\n\\begin{cor}\nThe generating function for the probability that\na randomly selected permutation of $M$ has exactly $r$ strongly\noutstanding elements is\n\\begin{equation}\n\\label{pgfso}\nP_M(x)=\\frac{a_kx}{N}\\prod_{i=1}^{k-1}\\left(1+\\frac{a_ix}{N-(a_1+a_2+\\dots+a_{i})}\\right),\n\\end{equation}\nand the average number of strongly outstanding elements among\npermutations of $M$ is\n\\begin{equation}\n\\label{eq:pmpe}\nP_M'(1)=\\sum_{i=1}^{k}\\frac{a_i}{a_i+a_{i+1}+\\dots+a_k}.\n\\end{equation}\n\\end{cor}\n\n\\begin{theorem}\n\\label{th:woep}\n  The generating function for the number of permutations of $M$\nthat contain exactly $t$ weakly outstanding elements is given by\n\\begin{equation}\n\\label{gfwo}\nG_M(x)=\\phi_{N,a_1}(x)\\phi_{N-a_1,a_2}(x)\\dots\n\\phi_{N-a_1-\\dots-a_{k-2},a_{k-1}}(x)x^{a_k}\n\\end{equation}\nwhere\n\\[\\phi_{N,a}(x)=\\sum_{m=0}^a{N-m-1\\choose a-m}x^m.\\]\n\\end{theorem}\n\\begin{cor}\nThe average number of weakly outstanding elements among permutations\nof $M$ is\n\\begin{equation}\n\\label{eq:avgweak}\n\\sum_{i=1}^k\\frac{a_i}{a_{i+1}+a_{i+2}+\\dots+a_k+1}.\n\\end{equation}\n\\end{cor}\n\\begin{cor}\nLet $A=\\max_i\\{a_i\\}$.  The amount by which the average number of\nweakly outstanding elements exceeds the average number of strongly\noutstanding elements is $\\le\\frac{\\pi^2}{6}A(A-1)$.\n\\end{cor}\n\\subsection{Words}\nThe next theorem involves Stirling numbers of the second kind,\ndenoted by ${n \\brace m}$. This is defined as the number of ways to\npartition a set of $n$ elements into $m$ nonempty subsets.\n\\begin{theorem}\n\\label{th:wordsstrong}\n  The number of $n$-letter words over an alphabet of $k$\nletters which have exactly $r$ strongly outstanding elements is\ngiven by\n\\begin{equation}\n\\label{wordso}\nf(n,k,r) = \\sum_m{k\\choose m}{m\\brack r}{n\\brace m},\n\\end{equation}\nand the average number of strongly outstanding elements among such\nwords is\n\\[H_k=1+\\frac12+\\frac13+\\dots+\\frac1k+o(1)\\quad(n\\to\\infty).\\]\n\\end{theorem}\n\\begin{theorem}\n\\label{th:wordsweak}\nThe generating function for the number $g(n,k,t)$ of $n$-letter words\nover an alphabet of $k$ letters which have exactly $t$ weakly\noutstanding elements is given by\n\\begin{equation}\n\\label{wordwo} G_k(n,x)\\, =\\kern -12.7pt\\raise 6pt\\hbox{{\\tiny\\textrm{def}}}\\,\\, \\sum_tg(n,k,t)x^t=\n\\sum_{t=0}^{k-1}(-1)^{k-1-t}(x+t)^n{k-1\\choose t}{x+t-1\\choose k-1},\n\\end{equation}\nand the average number of weakly\noutstanding elements among these words is\n\\[\\frac{n}{k}+H_{k-1}+O\\left(\\left(\\frac{k-1}{k}\\right)^n\\right).\\]\n\\end{theorem}\n\n\nNext we introduce the notion of a \\emph{template} for words on\n$[k]$. A permutation of 5 or more letters \\emph{matches the\ntemplate} `$YN*YY$', for example, if 1, 4, and 5 are outstanding\nelements, 2 is not an outstanding element, and 3 is unconstrained.\nFor example the permutation 2145763 matches this template. In this\ncase we think of the letters of the template $Y$, $N$,\n* as representing yes, no, and unconstrained, respectively.  We\ngeneralize the $Y$ and $N$ constraints to $S$, $W$, $\\overline{S}$,\nand $\\overline{O}$; where $S$ indicates a strongly outstanding\nelement, $W$ a weakly outstanding element, $\\overline{S}$ indicates\nthe absence of a strongly outstanding, and $\\overline{O}$ the\nabsence of an outstanding element.  We provide an algorithm for\nproducing the generating function for the number of words that match\na given template.  When the word is a permutation, we have:\n\\begin{theorem}\n\\label{th:tmp1}\nLet $\\tau$ be a given template of length at most $n$, and let\n$\\tau_j$ denote the letter that appears in position $j$, counting\nfrom the left, of the template $\\tau$.  Since every element of a\npermutation is either strongly outstanding or not outstanding, the\nletters of $\\tau$ are chosen from $\\{Y, N, *\\}$. The probability\nthat a permutation of at least $n$ letters matches the template\n$\\tau$ is\n\\begin{equation}\n\\label{notm} \\prod_{j:\\tau_j={\\mathrm{ `N\\kern-1pt\\raisebox{-.4pt}{\\scalebox{.7}{\\mbox{'}}}}}}\\left(1-{1\\over\n{j}}\\right)\\prod_{j:\\tau_j={\\mathrm{ `Y\\kern-1pt\\raisebox{-.4pt}{\\scalebox{.7}{\\mbox{'}}}}}}{1\\over {j}}\n.\\end{equation}\n\\end{theorem}\nThe corresponding result for words is:\\\\\n\\begin{theorem}\n\\label{th:tmp2}\n Let $\\tau$ be a word on $\\{S, W, *, \\overline{S},\n\\overline{O}\\}$.  Suppose $F(k,\\tau,x)=\\sum_{k\\ge 1}f(k,\\tau)x^k$ is\nthe ordinary generating function for $f(k,\\tau)$, the number of\nwords over the alphabet $[k]$ that match the template $\\tau$.\nConsider the adjunction of one new symbol, $A \\in \\{S, W, *,\n\\overline{S}, \\overline{O}\\}$, at the right end of $\\tau$.  The\ngenerating function, $F(k,\\tau A,x)$ can be obtained from\n$F(k,\\tau,x)$ by applying an operator $\\Omega_A$, i.e.,\n\\[F(k,\\tau A,x)=\\Omega_AF(k,\\tau,x),\\]\nwhere\n\\begin{eqnarray}\n\\Omega_SF(k,\\tau, x) & = & xF(k,\\tau, x)\/(1-x),\\label{omegas}\\\\\n\\Omega_WF(k,\\tau, x) & = & F(k,\\tau, x)\/(1-x),\\label{omegaw}\\\\\n\\Omega_*F(k,\\tau, x) & = & x\\frac{d}{dx}F(k,\\tau, x),\\label{omegastar}\\\\\n\\Omega_{\\overline{S}}F(k,\\tau, x) & = &\n\\left(x\\frac{d}{dx}-\\frac{x}{1-x}\\right)F(k,\\tau, x),\\ \\text{and}\\label{omegaovers}\\\\\n\\Omega_{\\overline{O}}F(k,\\tau, x) & = &\n\\left(x\\frac{d}{dx}-\\frac{1}{1-x}\\right)F(k,\\tau,\nx)\\label{omegaovero}.\n\\end{eqnarray}\n\\end{theorem}\n\\section{Notation}\nIn the following sections we will count permutations, multiset\npermutations, and words that contain a given number of strongly or\nweakly outstanding elements.  Let $f(k,r)$ denote the number of\npermutations of $[k]=\\{1,2,\\dots,k\\}$ that contain exactly $r$\nstrongly outstanding elements.  For a multiset\n$M=\\{1^{a_1},2^{a_2},\\dots,k^{a_k}\\}$, let $f(M,r)$ denote the\nnumber of permutations of $M$ that contain exactly $r$ strongly\noutstanding elements. Finally let $f(n,k,r)$ denote the number of\n$n$ letter words on the alphabet $[k]$ that contain exactly $r$\nstrongly outstanding elements.  When counting weakly outstanding\nelements, we use $g$ in place of $f$ and $t$ in place of $r$.\n\nUsing the above notation, we define the following generating\nfunctions for strongly outstanding elements: $F_k(x) = \\sum_r\nf(k,r)x^r$, $F_M(x) = \\sum_r f(M,r)x^r$, and $F_k(n,x) = \\sum_r\nf(n,k,r)x^r$. We define analogous generating functions for weakly\noutstanding elements and use $G$ in place of $F$.\n\n\\section{Strongly outstanding elements of multiset permutations}\nIn this section we establish Theorem \\ref{th:sms} and its corollaries.\n\nGiven a\nmultiset $M=\\{1^{a_1},2^{a_2},\\dots ,k^{a_k}\\}$, let $N=\\sum_ja_j$.\nWe construct the permutations of $M$ that have exactly $r$ strongly\noutstanding elements as follows. We have $N$ slots into which we\nwill put the $N$ elements of $M$ to make these permutations.\n\nTake the $a_1$ 1's that are available and place them in some\n$a_1$-subset of the $N$ slots that are available. There are two\ncases now. If the set of slots that we chose for the 1's did not\ninclude the first (leftmost) slot, then we can fill in the remaining\nslots with any permutation of the multiset $M\/1^{a_1}$ that has\nexactly $r$ strongly outstanding elements. On the other hand, if we\ndid place a 1 into the first slot, then after placing all of the\n1's, the remaining slots can be filled in with any permutation of\nthe multiset $M\/1^{a_1}$ that has exactly $r-1$ strongly outstanding\nelements.\n\nRecall that $f(M,r)$ denotes the number of permutations of the\nmultiset $M$ that have exactly $r$ strongly outstanding elements.\nThe argument in the preceding paragraph shows that\n\\[f(M,r)=\\left({N\\choose a_1}-{N-1\\choose a_1-1}\\right)f(M\/1^{a_1},r)+{N-1\\choose a_1-1}f(M\/1^{a_1},r-1).\\]\nWhen we define\n\\[F_M(x)=\\sum_rf(M,r)x^r,\\]\nwe have the recurrence\n\\begin{eqnarray*}\nF_M(x)&=&\\left(\\left({N\\choose a_1}-{N-1\\choose\na_1-1}\\right)+{N-1\\choose a_1-1}x\\right)F_{M\/1^{a_1}}(x)\\\\\n&=&\\left({N-1\\choose a_1}+{N-1\\choose a_1-1}x\\right)F_{M\/1^{a_1}}(x)\n\\end{eqnarray*}\nThis shows that the generating polynomial resolves into linear\nfactors over the integers. Indeed we get the explicit form\n\\begin{eqnarray*}\nF_M(x)&=& \\left({N-1\\choose a_1}+{N-1\\choose\na_1-1}x\\right)\\left({N-a_1-1\\choose a_2}+{N-a_1-1\\choose\na_2-1}x\\right)\\dots\\\\\n&=&\\prod_{i=1}^k\\left({N-a_1-a_2-\\dots -a_{i-1}-1\\choose\na_i}+{N-a_1-a_2-\\dots -a_{i-1}-1\\choose a_i-1}x\\right)\\\\\n&=&\\frac{(N-1)!a_k x}{a_1!a_2!\\dots\na_k!}\\prod_{i=1}^{k-1}\\left(1+\\frac{a_ix}{N-(a_1+a_2+\\dots+a_{i})}\\right)\n\\end{eqnarray*}\nThis gives us Theorem \\ref{th:sms}. Its corollaries follow by obvious calculations.\n\nSince the generating polynomial has real zeros only, the\nprobabilities $\\{p_r(M)\\}$ are unimodal and log concave. By\nDarroch's Theorem \\cite{dar}, the value of $r$ for which $p_r(M)$ is\nmaximum differs from $P_M'(1)$ of (\\ref{eq:pmpe}) above by at most 1.\n\n\\section{Weakly outstanding elements of multiset permutations}\nNext we prove Theorem \\ref{th:woep}.\n\nConsider $g(M,t)$, the number of permutations of\n$M=\\{1^{a_1},2^{a_2},\\dots, k^{a_k}\\}$ that have exactly $t$ weakly\noutstanding elements, and $G_M(x)=\\sum_tg(M,t)x^t$. To find\n$g(M,t)$, suppose the permutation begins with a block of exactly\n$m\\ge 0$ 1's. Since the value that follows the last 1 in the block\nis not available for a 1, there remain $N-m-1$ slots into which the\nremaining 1's can be put, in ${N-m-1\\choose a_1-m}$ ways. Once all\nof the 1's have been placed, if the remaining permutation of the\nmultiset $M\/1^{a_1}$ has exactly $t-m$ weakly outstanding elements\nthen the whole thing will have $t$ weakly outstanding elements.\nHence we have\n\\[g(M,t)=\\sum_{m\\ge 0}{N-m-1\\choose a_1-m}g(M\/1^{a_1},t-m),\\]\nif $M\/1^{a_1}$ is nonempty, whereas if $M=1^{a_1}$ only, then\n$g(M,t)=\\delta_{t,a_1}$. If we multiply by $x^t$ and sum on $t$ we\nfind that\n\\[G_M(x)=\\left\\{\\sum_{m\\ge 0}{N-m-1\\choose a_1-m}x^m\\right\\}G_{M\/1^{a_1}}(x),\\]\nexcept that if $M=1^{a_1}$ only, then $G_M(x)=x^{a_1}$. So if we\nwrite\n\\[\\phi_{N,a}(x)=\\sum_{m=0}^a{N-m-1\\choose a-m}x^m,\\]\nthen we have\n\\begin{equation}\n\\label{eq:gm}\n G_M(x)=\\phi_{N,a_1}(x)\\phi_{N-a_1,a_2}(x)\\dots\n\\phi_{N-a_1-\\dots-a_{k-2},a_{k-1}}(x)x^{a_k}.\n\\end{equation}\nThis gives (\\ref{gfwo}).\n\nThe two sums $\\phi_{N,a}(1)={n\\choose a}$ and\n$\\phi_{N,a}'(1)={n\\choose a}\/(n-a+1)$ are elementary, and they imply\nthat $\\phi_{N,a}'(1)\/\\phi_{N,a}(1)=1\/(n-a+1)$. Then logarithmic\ndifferentiation of (\\ref{eq:gm}) and evaluation at $x=1$ shows that\n the average number of weakly outstanding elements in permutations of\n$M$ is\n\\begin{equation}\n\\label{eq:wb}\n\\sum_{i=1}^k\\frac{a_i}{a_{i+1}+a_{i+2}+\\dots+a_k+1}.\n\\end{equation}\nLet $A=\\max_i\\{a_i\\}$. If we compare (\\ref{eq:wb}) and (\\ref{eq:pmpe})\nwe find that the amount by which the average number of weakly\noutstanding elements exceeds the average number of strongly\noutstanding elements is\n\\begin{eqnarray*}\n\\sum_{i=1}^k&&\\kern-25pt\\left(\\frac{a_i}{a_{i+1}+a_{i+2}+\\dots+a_k+1}\n-\\frac{a_i}{a_i+a_{i+1}+\\dots+a_k}\\right)\\\\\n\\qquad&=&\\sum_{i=1}^k\\frac{a_i(a_i-1)}{(a_{i+1}+a_{i+2}+\\dots+a_k+1)(a_i+a_{i+1}+\\dots+a_k)}\\\\\n\\qquad&\\le& A(A-1)\\sum_{i=1}^k\\frac{1}{(k-i+1)^2}\\le \\frac{\\pi^2}{6}A(A-1).\n\\end{eqnarray*}\nThis estimate is best possible when $A=1$, i.e., when every element\noccurs just once.\n\n\\section{Strongly outstanding elements of words}\n\\label{sec:str}\n Next we investigate $f(n,k,r)$, the number of\n$n$-letter words over an alphabet of $k$ letters that have exactly\n$r$ strongly outstanding elements.  In so doing we will prove\nTheorem \\ref{th:wordsstrong}.\n\nNote first that the number of strongly outstanding elements of such\na word depends only on the permutation of the distinct letters\nappearing in the word that is achieved by the \\textit{first}\nappearances of each of those letters, because a value $j$ can be\nstrongly outstanding in a word $w$ only if it is the first (i.e.,\nleftmost) occurrence of $j$ in $w$.\n\nHence associated with each $n$-letter word $w$ over an alphabet of\n$k$ letters which has exactly $r$ strongly outstanding elements\nthere is a triple $(S,{\\cal P},\\sigma)$ consisting of\n\\begin{enumerate}\n\\item a subset $S\\subseteq [k]$, which is the set of all of the distinct\n letters that actually appear in $w$, and\n\\item a partition ${\\cal P}$ of the set $[n]$ into $m=|S|$ classes,\nnamely the $i$th class of ${\\cal P}$ consists\n of the set of positions in the word $w$ that contain the $i$th letter\n  of $S$, and\n\\item a permutation $\\sigma\\in S_m$, $m=|S|$, which is the sequence of \\textit{first} appearances in $w$ of each of the $k$ letters that occur in $w$. $\\sigma$ will have $r$ strongly outstanding elements.\n\\end{enumerate}\n\nConversely, if we are given such a triple $(S,{\\cal P},\\sigma)$, we\nuniquely construct an $n$-letter word $w$ over $[k]$ with exactly\n$r$ strongly outstanding elements as follows.\n\nFirst arrange the classes of the partition ${\\cal P}$ in ascending\norder of their smallest elements. Then permute the set $S$ according\nto the permutation $\\sigma$, yielding a list $\\tilde{S}$. In all of\nthe positions of $w$ that are described by the first class of the\npartition ${\\cal P}$ (i.e., the class in which the letter `1' lives)\nwe put the first letter of $\\tilde{S}$, etc., to obtain the required\nword $w$.\n\nThus the number of words that we are counting is equal to the number\nof these triples, viz.\n\\begin{equation}\n\\label{eq:fnkr}\n f(n,k,r)=\\sum_m{k\\choose m}{m\\brack r}{n\\brace m}.\n\\end{equation}\nIt is noteworthy that three flavors of ``Pascal-triangle-like''\nnumbers occur in this formula.\n\nLet $\\rho(n,k)$ denote the average number of strongly outstanding\nelements among the $n$-letter words that can be formed from an\nalphabet of $k$ letters. To find $\\rho(n,k)$ for large $n$, we have\nfirst that \\begin{eqnarray*} \\sum_rf(n,k,r)&=&\\sum_m{k\\choose\nm}{n\\brace m}\\sum_r{m\\brack r}= \\sum_m{k\\choose m}{n\\brace\nm}m!\\\\\n&\\sim& \\sum_m{k\\choose m}{n\\brace m}m!\\sim \\sum_m{k\\choose m}n^m\\sim\nn^k,\n\\end{eqnarray*}\nfor large $n$, where we have used the facts that ${n\\brace m}\\sim\nn^m\/m!$ and $\\sum_r{m\\brack r}=m!$. Similarly,\n\\begin{eqnarray*}\\sum_rrf(n,k,r)&=&\\sum_m{k\\choose m}{n\\brace m}\\sum_rr{m\\brack\nr}= \\sum_m{k\\choose m}{n\\brace m}m!H_m\\\\\n&\\sim& \\sum_m{k\\choose m}{n\\brace m}m!H_m\\sim \\sum_m{k\\choose\nm}n^mH_m\\sim n^kH_k,\\end{eqnarray*}\n where we have used the additional fact that\n$\\sum_rr{m\\brack r}=m!H_m$. If we divide these last two equations we\nfind that\n\\[\\lim_{n\\to\\infty}\\rho(n,k)=H_k=1+\\frac12+\\frac13+\\dots+\\frac1k.\\]\nThe proof of Theorem \\ref{th:wordsstrong} is complete. $\\Box$\n\nFrom eq. (\\ref{eq:fnkr}) we can use the standard generating\nfunctions for the two kinds of Stirling numbers to show that\n\\[\\sum_{r,n}f(n,k,r)y^rt^{n-r}=\\prod_{j=1}^k\\left(1+\\frac{y}{1-jt}\\right).\\]\nBut from (\\ref{eq:fnkr}) we also have\n\\begin{eqnarray*}\n\\sum_{r,n}f(n,k,r)y^rt^{n-r}&=&\\sum_{\\ell,r}{k\\choose \\ell}{\\ell\\brack r}\\left(\\frac{y}{t}\\right)^r\\sum_n{n\\brace \\ell}t^n\\\\\n&=&\\sum_{\\ell,r}{k\\choose \\ell}{\\ell\\brack r}\\left(\\frac{y}{t}\\right)^r\\frac{t^{\\ell}}{(1-t)(1-2t)\\dots (1-\\ell t)}\\\\\n&=&\\sum_{\\ell}{k\\choose \\ell}\\prod_{j=0}^{\\ell-1}\\left(\\frac{y}{t}+j\\right)\\frac{t^{\\ell}}{(1-t)(1-2t)\\dots (1-\\ell t)}\\\\\n&=&\\sum_{\\ell}{k\\choose \\ell}\\prod_{j=1}^{\\ell}\\frac{y+(j-1)t}{1-jt}.\n\\end{eqnarray*}\n\nComparison of these two evaluations shows that we have found and\nproved the following identity:\n\\begin{equation}\n\\label{eq:ident}\n\\sum_{\\ell}{k\\choose \\ell}\\prod_{j=1}^{\\ell}\\frac{y+(j-1)t}{1-jt}=\\prod_{j=1}^k\\left(1+\\frac{y}{1-jt}\\right).\n\\end{equation}\nBut Gauss had done it earlier, since it is the evaluation of his\nwell known\n\\[ _2F_1\\left[ \\begin{array}{cc}\n    -k & y\/t \\\\\n    1-1\/t&\n     \\end{array}\\bigg|\\,1 \\right].\\]\n\n\\section{A calculus of templates}\n\\subsection{Templates and permutations}\nIn this section we prove Theorems \\ref{th:tmp1} and \\ref{th:tmp2}.\n  We begin by establishing\nequation (\\ref{notm}), which appeared first in \\cite{wil}.\n\nThis result is a generalization of a theorem of R. V. Kadison\n\\cite{rvk}, who discovered the case where the template is `NN$\\cdots\n$NY', and proved it by the sieve method. To make this paper\nself-contained, we include a proof of (\\ref{notm}).\n\nOur proof is by induction on $n$. Suppose it has been proved that,\nfor all templates $\\tau$ of at most $n-1$ letters, the number of\npermutations of $n-1$ letters that match $\\tau$ is correctly given\nby (\\ref{notm}), and let $\\tau$ be some template of $\\le n$ letters.\nIf in fact the length of $\\tau$ is $<n$ then the formula\n(\\ref{notm}) gives the same result as it did when applied to\n$(n-1)$-permutations, which is the correct probability.\n\nIn the case where the length of $\\tau$ is $n$ and the rightmost\nletter of $\\tau$ is `Y', every matching permutation $\\sigma$ must\nhave $\\sigma (n)=n$. Hence the number of matching permutations is\n$(n-1)!p_{n-1}(\\tau')$, where $\\tau'$ consists of the first $n-1$\nletters of $\\tau$, which is equal to $n!p_n(\\tau)$, proving the\nresult in this case.\n\nIn the last case, where the length of $\\tau$ is $n$ and the\nrightmost letter of $\\tau$ is `N', the probability of a permutation\nmatch must be $p_{n-1}(\\tau')(1-1\/n)$, since this case and the\npreceding one are exhaustive of the possibilities and the preceding\none had a probability of $p_{n-1}(\\tau')\/n$. But this agrees with\nthe formula (\\ref{notm}) for this case, completing the proof of the\ntheorem.\n\n\\subsection{Templates on words}\nNext we include results for words that are analogous to those we\nfound for permutations.  A preview of the kind of results that we\nwill get is the following. Suppose $F(k,\\tau,x)=\\sum_{k\\geq 1}\nf(k,\\tau)x^k,$ where $f(k,\\tau)$ denotes the number of words over\nthe alphabet $[k]$ that match the template $\\tau$. Then consider the\nadjunction of one new symbol, let's call it $A$, at the right end of\n$\\tau$. Then we will show that the new generating function,\n$F(k,\\tau A,x)$ can be obtained from $F(k,\\tau,x)$ by applying a\ncertain operator $\\Omega_A$. That is,\n\\[F(k,\\tau A,x)=\\Omega_AF(k,\\tau,x).\\]\nThe operator $\\Omega_A$ will depend only on the letter $A$ that is\nbeing adjoined to the template $\\tau$.\n\nHence, to find the generating function for a complete template\n$\\tau$, we begin with $F(k,\\emptyset,x)=1\/(1-x)$, and we read the\ntemplate $\\tau$ from left to right. Corresponding to each letter in\n$\\tau$ we apply the appropriate operator $\\Omega$. When we have\nfinished scanning the entire template the result will be the desired\ngenerating function for $\\tau$.\n\nThe letters that we will allow in a template\n$\\tau=\\tau_1\\tau_2\\dots\\tau_l$ are\n$\\{S,W,O,*,\\overline{S},\\overline{O}\\}$. Their meanings are that if\n$w=w_1\\dots w_l$ is a word of length $l$ over the alphabet $[k]$\nthen for $w$ to match the template it must be that $w_i$ is\n\\begin{enumerate}\n\\item a strongly outstanding value whenever $\\tau_i=S$, or\n\\item a weakly outstanding value whenever $\\tau_i=W$, or\n\\item unrestricted whenever $\\tau_i=*$, or\n\\item not a strongly outstanding value whenever $\\tau_i=\\overline{S}$, or\n\\item neither a weakly nor a strongly outstanding value whenever $\\tau_i=\\overline{O}$.\n\\end{enumerate}\nLet's consider what happens to the count of matching words when we\nadjoin one of these letters to a template whose counting function is\nknown. Let $f(k,\\tau)$ denote the number of words of length $l=$\nlength$(\\tau)$, on the alphabet $[k]$ that match the template\n$\\tau$.\n\\begin{enumerate}\n\\item If $w$ is one of the words counted by $f(k,\\tau S)$, and if we delete its last letter, we obtain one of the words that is counted by $f(i,\\tau)$ for some $1\\le i<k$, and consequently\n\\[f(k,\\tau S)=\\sum_{i=1}^{k-1}f(i,\\tau).\\]\nIf $F(k,\\tau,x)=\\sum_{k\\ge 1}f(k,\\tau)x^k$, then we have\n\\[F(k,\\tau S,x)=\\frac{x}{1-x}F(k,\\tau ,x).\\]\nThus we have found the operator $\\Omega_S$, and it is defined by\n$\\Omega_SF(x)=xF(x)\/(1-x)$.  This is equation (\\ref{omegas}).\n\\item Similarly, if $w$ is one of the words counted by $f(k,\\tau W)$, and if we delete its last letter, we obtain one of the words that is counted by $f(i,\\tau)$ for some $1\\le i\\le k$, and consequently\n\\[f(k,\\tau W)=\\sum_{i=1}^{k}f(i,\\tau).\\]\nIf $F(k,\\tau,x)=\\sum_{k\\ge 1}f(k,\\tau)x^k$, then we have\n\\[F(k,\\tau W,x)=\\frac{F(k,\\tau ,x)}{1-x}.\\]\nThus we have found the operator $\\Omega_W$, and it is defined by\n$\\Omega_WF(x)=F(x)\/(1-x)$.  This is (\\ref{omegaw}).\n\\end{enumerate}\nSince the argument in each case is easy and similar to the above we\nwill simply list the remaining three operators, equations\n(\\ref{omegastar}), (\\ref{omegaovers}), and (\\ref{omegaovero}), as\nfollows:\n\\begin{eqnarray*}\n\\Omega_*F(x)&=&x\\frac{d}{dx}F(x)\\\\\n\\Omega_{\\overline{S}}F(x)&=&\\left(x\\frac{d}{dx}-\\frac{x}{1-x}\\right)F(x)\\\\\n\\Omega_{\\overline{O}}F(x)&=&\\left(x\\frac{d}{dx}-\\frac{1}{1-x}\\right)F(x)\n\\end{eqnarray*}\n\nThe successive applications of these operators can be started with\nthe generating function for the empty template,\n\\[F(\\emptyset,x)=\\frac{1}{1-x}.\\]\n\nThus to find the generating function for some given template $\\tau$,\nwe begin with the function $1\/(1-x)$, and then we read one letter at\na time from $\\tau$, from left to right, and apply the appropriate\none of the five operators that are defined above.\n\nAs an example, how many words of length 3 over the alphabet $[k]$\nmatch the template $\\tau=S*\\overline{S}$? This is the coefficient of\n$x^k$ in\n\\begin{eqnarray*}\nF(S*\\overline{S},x)&=&\\Omega_{\\overline{S}}\\Omega_*\\Omega_S\\frac{1}{1-x}\\\\\n&=&\\left(x\\frac{d}{dx}-\\frac{x}{1-x}\\right)\\left(x\\frac{d}{dx}\\right)\n\\left(\\frac{x}{1-x}\\right)\\frac{1}{1-x}\\\\\n&=&\\frac{x+3x^2}{(1-x)^4}\n\\end{eqnarray*}\nThe required number of words over a $k$ letter alphabet that match\nthe template $\\tau$ is the coefficient of $x^k$ in the above. Since\nthese generating functions will always be of the form\n$P(t)\/(1-x)^r$, with $P$ a polynomial, we note for ready reference\nthat\n\\[[x^k]\\left\\{\\frac{\\sum_ja_jx^j}{(1-x)^r}\\right\\}=\\sum_ja_j{r+k-j-1\\choose\nr- 1}.\\] In the example above we have $r=4$, $a_1=1$, $a_2=3$, so\n\\[f(k,S*\\overline{S})={k+2\\choose 3}+3{k+1\\choose 3}.\\]\n\n\n\\section{Weakly outstanding elements of words}\nFinally we prove Theorem \\ref{th:wordsweak}.\n\n Recall that $g(n,k,t)$ is the number\nof $n$-letter words over the alphabet $[k]$ which have exactly $t$\nweakly outstanding elements. Consider just those words $w$ that\ncontain exactly $m$ 1's, $m<n$. If $w$ begins with a block of\nexactly $l$ 1's, $0\\le l\\le m$ then by deleting all $m$ of the 1's\nin $w$ we find that the remaining word is one with $n-m$ letters\nover an alphabet of $k-1$ letters and it has exactly $t-l$ weakly\noutstanding elements. Thus we have the recurrence\n\\[\ng(n,k,t)=\\sum_{m=0}^{n-1}{n-l-1\\choose\nm-l}g(n-m,k-1,t-l)+\\delta_{t,n}.\n\\]\nWhen we set $G_k(n,x)=\\sum_tg(n,k,t)x^t$, we find that\n\\begin{equation}\n\\label{eq:f2rec} G_k(n,x)=\\sum_{m=0}^{n-1}\\sum_{l=0}^k{n-l-1\\choose\nm-l}x^lG_{k-1}(n-m,x)+x^n.\\qquad(n,k\\ge 1;\\, G_0(n,x)=0)\n\\end{equation}\nTo discuss this recurrence, let $\\Delta=\\Delta_x$ be the usual\nforward difference operator on $x$, i.e., $\\Delta_x\nf(x)=f(x+1)-f(x)$. Then from the recurrence above we discover that\n\\[G_1(n,x)=x^n;\\ G_2(n,x)= \\Delta_x x^n(x-1);\\ G_3(n,x)=\\Delta_x^2x^n{x-1\\choose 2}.\\]\nThis leads to the conjecture that\n\\begin{eqnarray*}\nG_k(n,x)&=&\\Delta_x^{k-1}\\left\\{x^n{x-1\\choose k-1}\\right\\}\\\\\n&=&\\sum_{t=0}^{k-1}(-1)^{k-1-t}(x+t)^n{k-1\\choose t}{x+t-1\\choose\nk-1}.\n\\end{eqnarray*}\nTo prove this it would suffice to show that the function $G_k(n,x)$\nabove satisfies the recurrence (\\ref{eq:f2rec}). If we substitute\nthe conjectured form of $G_k$ into the right side of\n(\\ref{eq:f2rec}) we find that the sums over $l$ and $m$ can easily\nbe done, and the identity to be proved now reads as\n\\begin{eqnarray*}\n&&\\sum_{t=0}^{k-2}(-1)^{k-t}{k-2\\choose t}{x+t-1\\choose k-2}\\frac{(x+t)}{t+1}((x+t+1)^n-x^n)+x^n\\\\\n&&\\qquad=\\sum_{t=0}^{k-1}(-1)^{k-1-t}(x+t)^n{k-1\\choose\nt}{x+t-1\\choose k-1}.\n\\end{eqnarray*}\nIf we replace the dummy index of summation $t$ by $t-1$ on the left\nside, the identity to be proved becomes\n\\begin{eqnarray*}\n&&\\sum_{t=1}^{k-1}(-1)^{k-t-1}{k-2\\choose t-1}{x+t-2\\choose k-2}\\frac{(x+t-1)}{t}((x+t)^n-x^n)+x^n\\\\\n&&\\qquad=\\sum_{t=0}^{k-1}(-1)^{k-1-t}(x+t)^n{k-1\\choose\nt}{x+t-1\\choose k-1}.\n\\end{eqnarray*}\nIt is now trivial to check that for $1\\le t\\le k-1$, the coefficient\nof $(x+t)^n$ on the left side is equal to that coefficient on the\nright. If we cancel those terms and divide out a factor $x^n$ from\nwhat remains, the identity to be proved becomes\n\\[-\\sum_{t=1}^{k-1}(-1)^{k-t-1}{k-2\\choose t-1}{x+t-2\\choose k-2}\\frac{(x+t-1)}{t}+1=(-1)^{k-1}{x-1\\choose k-1}.\\]\nThe sum that appears above is a special case of Gauss's original\n$_2F_1[a,b;c|1]$ evaluation, and the proof of (\\ref{wordwo}) is\ncomplete.  A straightforward calculation now shows that the average\nnumber of weakly outstanding elements among $n$-letter words over\nthe alphabet $[k]$ is\n\\[=\\frac{n}{k}+H_{k-1}+O\\left(\\left(\\frac{k-1}{k}\\right)^n\\right).\\]\n\n\\section{Related results}\nIn the literature, the outstanding elements and values of sequences\ngo by various names: \\emph{\\'{e}l\\'{e}ments salients} (R\\'{e}nyi),\n\\emph{left-to-right maxima}, \\emph{records}, and others; and appear\nin several different contexts and results, for example:\n\nIt is well known that the probability of obtaining at least $r$\nstrongly outstanding elements in a sequence $X_1,X_2,\\dots ,X_n$ of\n$n$ independent, identically distributed, continuous random\nvariables approaches 1 as $n\\rightarrow\\infty$.  (Glick's survey\n\\cite{gl}, for example, contains this result and those that follow\nin this paragraph and the next.) Let $Y_i = 1$ if $i$ is a strongly\noutstanding element of such a sequence, and 0 otherwise. The\nexpected value is $E[Y_i]=1\/i$ and the variance is $V[Y_i]=1\/i -\n1\/i^2$. The number of strongly outstanding elements in a sequence of\ncontinuous iid random variables is therefore $R_i = \\sum_i Y_i$ with\nexpectation $E[R_i]=\\sum_{i=1}^n 1\/i$ and variance\n$V[R_i]=\\sum_{i=1}^n 1\/i - \\sum_{i=1}^n 1\/i^2$. Note $\\sum_{i=1}^n\n1\/i - \\ln(n) \\rightarrow$ Euler's constant $=.5772\\dots$, and\n$\\sum_{i=1}^n 1\/i^2 \\rightarrow \\pi^2\/6 = 1.6449\\dots$ for\n$n\\rightarrow \\infty.$\n\nLet $N_r$ denote the $r^{\\text{th}}$ outstanding element in a\nsequence of $n$ iid continuous random variables.  Then $N_1 = 1$,\n$E[N_r] = \\infty$ for $r\\geq 2$, and $E[N_{r+1} - N_r] = \\infty$ as\nwell.  The probability $P[N_2 = i_2, N_3 = i_3, \\dots , N_r = i_r] =\n1\/(i_2 -1)(i_3-1)\\dots (i_r-1)i_r$ for $1 < i_2 < i_3 < \\dots <i_r$.\nThe probability that an iid sequence of $n$ continuous random\nvariables has exactly $r$ strongly outstanding elements is\n$\\left[\\begin{array}{ll}k\\\\r\\end{array}\\right]\/(r-1)!$ for large\n$n$.\n\nChern and Hwang \\cite{ch} consider the number $f_{n,k}$ of $k$\nconsecutive records (strongly outstanding elements) in a sequence of\n$n$ iid continuous random variables.  They improve upon known\nresults for the limiting distribution of $f_{n,2}$.  In particular\nthey show $f_{n,k}$ is asymptotically Poisson for $k=1,2$, and this\nis not the case for $k\\geq 3$.  They give the probability generating\nfunction for $f_{n,2}$, and observe that the distribution of\n$f_{n,2}$ is identical to that for the number of fixed points $j$ in\na random permutation of $[n]$ for $1 \\leq j < n$.   They give a\nrecurrence for the probability generating function for $f_{n,k}$,\nand compute the mean and variance for this number.\n\nAre there similar results for discrete distributions?  Prodinger\n\\cite{prod} considers left-to-right maxima in both the strict\n(strongly outstanding) and loose (weakly outstanding) senses for\ngeometric random variables. He finds the generating function for the\nprobability that a sequence of $n$ independent geometric random\nvariables (each with probability $pq^{v-1}$ of taking the positive\ninteger value $v$, where $q = p-1$) has $k$ strict left-to-right\nmaxima. This probability is the coefficient of $z^ny^k$ in\n$$\nF(z,y) = \\prod_{k \\geq 1}\n\\left(1+\\frac{yzpq^{k-1}}{1-z(1-q^k)}\\right).\n$$\nFor loose left-to-right maxima the analogous generating function is\n$$\n\\prod_{k\\geq 0}\\frac{1-z(1-q^k)}{1-z+zq^k(1-py)}.\n$$\nIn this paper, Prodinger also finds the asymptotic expansions for\nboth the expected numbers of strict and loose left-to-right maxima,\nand the variances for these numbers.  He does all of the above for\nuniform random variables as well.\n\nKnopfmacher and Prodinger \\cite{kp} consider the \\emph{value} and\n\\emph{position} of the $r^{\\text{th}}$ left-to-right maximum for $n$\ngeometric random variables. The \\emph{position} is the\n$r^{\\text{th}}$ strongly outstanding element, and the \\emph{value}\nis that taken by the random variable in that position (again the\nvalue $v$ is taken with probability $pq^{v-1}$, where $q=1-p$). For\nthe $r^{\\text{th}}$ strong left-to-right maximum, the asymptotic\nformulas for value and position are $\\frac{r}{p}$ and\n$\\frac{1}{(r-1)!}\\left(\\frac{p}{q}\\log_{\\frac{1}{q}}n\\right)^{r-1}$,\nrespectively. For the \\emph{r}th weak left-to-right maximum, the\nvalue and position are asymptotically $\\frac{rq}{p}$ and\n$\\frac{1}{(r-1)!}\\left(p\\log_{\\frac{1}{q}}n\\right)^{r-1}$.  These\nresults are obtained by first computing the relevant generating\nfunctions.\n\nA number of additional properties of outstanding elements of permutations\n are in Wilf \\cite{wil}.\n\nKey \\cite{key} describes the asymptotic behavior of the number of\nrecords (strongly outstanding elements) and weak records (weakly\noutstanding elements) that occur in an iid sequence of integer\nvalued random variables.\n\nWhen the sequence $w_1, w_2, \\dots, w_n$ under consideration is a\nrandom permutation on $n$ letters, then the expected number of\nstrongly outstanding elments is the \\emph{n}th harmonic number $H_n\n= 1 + \\frac{1}{2} + \\frac{1}{2} + \\dots + \\frac{1}{n}$. The variance\nis $H_n = H_n^{(2)}$, where\n$H_n^{(2)}=1+\\frac{1}{4}+\\dots+\\frac{1}{n^2}$ denotes the \\emph{n}th\nharmonic number of the second order.\n\nBanderier, Mehlhorn, and Brier \\cite{bmb} show that the average\nnumber of left-to-right maxima (strongly outstanding elements) in a\n\\emph{partial permutation} is $\\log(pn) + \\gamma +\n2\\frac{1-p}{p}+\\left(\\frac{1}{2}+\\frac{2(1-p)}{p^2}\\right)\\frac{1}{n}+O\\left(\\frac{1}{n^2}\\right)$\nwhere $\\gamma = .5772\\dots$ is Euler's constant.  To obtain a\n\\emph{partial permutation}, we begin with the sequence $1,2,\\dots\n,k$ and select each element with probability $p$; then take one of\nthe $(pn)!$ permutations of $[pn]$ uniformily at random and let it\nact on the selected elements, while the nonselected elements stay in\nplace.\n\nFoata and Han \\cite{fh} investigate the (right-to-left) \\emph{lower\nrecords} of \\emph{signed permutations}.  A \\emph{signed permutation}\nis a word $w=w_1w_2\\dots w_n$ for which the letters $w_i$ are\npositive or negative integers, and $|w_1||w_2|\\dots |w_n|$ is a\npermutation of $\\{1,2,\\dots,n\\}$.  A \\emph{lower record} of such a\nword is a letter $w_i$ such that $w_i < w_j$ for all $j$ with $i+1\n\\leq j \\leq n$. The authors consider the \\emph{signed subword}\nobtained by reading the lower records of $w$ from left to right; and\nderive generating functions for signed permutations in terms of\nsigned subwords, numbers of positive and negative letters in signed\nsubwords, and other statistics.\n\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Results}\n   \n   \n   \n   \n\n\\section{Introduction}\nThe objective of the study is to propose a system-level framework with quantitative measures to assess the resilience of road networks to inform transportation planning and project development processes in practice. Road networks are the backbone of society that enables the movement of people and goods safely and efficiently. A reliable road network does not only allow improved access by the workforce to business, but also result in long-term cost saving and increase of safety and viability of residential neighborhoods \\citep{weilant_incorporating_2019}. Disruptions of road networks would reduce economic productivity, harm local commercial activities and community well-being, and restrict people's mobility and access to critical facilities \\citep{weilant_incorporating_2019,frizzelle_importance_2009,jenelius_importance_2006}. The US National Response Plan (NRP) urges each state to have a plan for responding to \"\u2026 reduce the vulnerability to all natural and manmade hazards; and minimize the damage and assist in the recovery from any type of incident that occurs\"  \\citep{us_department_of_homeland_security_national_2004}. The need for incorporating resilience assessments in transportation planning and project development processes is increasingly recognized by transportation organizations in the United States and worldwide. For example, the US Federal Highway Administration (FHWA) has established mandates and guidelines for transportation agencies regarding the incorporation of resilience into transportation systems through their asset management planning efforts \\citep{flannery_resilience_2018}.\n\nWhile the need for system-level resilience assessment has been recognized, the measures for quantifying system-level resilience to inform transportation planning processes are rather limited in practice \\citep{esmalian_operationalizing_2022}. Existing literature has shown that the evaluation of road network resilience is multi-dimensional, including vulnerability assessment, criticality assessment, and interdependency and cascading failure assessment. \\citet{sharma_resilience_2018} suggested measuring resilience using vulnerability, severity of consequences, and time to recover from the disruptions. Considering the post-disaster network access to critical facilities, studies \\citep{dong_robust_2019, dong_integrated_2020} proposed the concept of robust components to measure communities' risk disparity in terms of losing access to hospitals in both earthquake and flooding scenarios using a network topology analysis approach. \\citet{gao_universal_2016} used a single metric to measure the resilience of multi-dimensional complex networks. The author found that a resilient system is (1) densely connected in the system, (2) highly heterogeneous in the connections, and (3) relatively symmetrical. Using a percolation approach, \\citet{dong_measuring_2020} measured the resilience of different cities' and states' road networks in the face of random failure using the giant component metrics. This topological metric enables the ranking of road networks based on the derived resilience score. \\citet{weilant_incorporating_2019} summarized the resilience measurement process into an AREA approach, which represents Absorptive capacity, Restorative capacity, Equitable access, and Adaptive capacity. \\citet{bruneau_framework_2003} proposed that a resilience framework should include measurement of robustness, rapidity, resourcefulness, and redundancy, integrated with technical, organizational, social, and economic dimensions \\citep{davis_establishing_2018, flannery_resilience_2018, norris_community_2008, reggiani_network_2013}. \\citet{da_silva_city_2014} suggested a resilience system should be equipped with qualities such as robustness (can anticipate potential failures and make provisions), redundancy (has spare capacity to accommodate changes), and integratedness (promotes consistency by pursuing integration and alignment between systems). This study explored various quantitative metrics and identified those in this multi-dimensional and system-level assessment framework based on connectivity of road segments in road networks; vulnerability of road segments to extreme events, such as flooding and hurricanes; disrupted access to critical facilities; and cascading impact of critical facility networks. Quantitative metrics that can consider all of these dimensions in resilience assessment are essential for a system-level understanding of the resilience of road networks and also inform transportation planning and project development.\n\nWhile the scholarly literature is rich in terms of methods and measures for assessment of vulnerability and resilience in transportation infrastructure, little of the existing work has been adopted in practice. The literature has put efforts into developing assessment measures to understand the resilience and vulnerability of road networks \\citep{hsieh_highway_2020,singh_vulnerability_2018,dave_extreme_2021,martin_assessing_2021}. Several conceptual frameworks have been proposed in the literature to promote consideration of resilience into infrastructure assessment, including road networks \\citep{jenelius_road_2015,morelli_measuring_2021,papilloud_vulnerability_2021}. However, in the United States, only about half of the state departments of transportation (DOTs) and metropolitan planning organizations (MPOs) started incorporating resilience considerations in their plans, and only some of them explicitly listed resilience as a goal \\citep{dix_integrating_2018}. This limitation is mainly due to two reasons: (1) transportation planners and decision-makers do not know what method and measures to use among several proposed in the literature; (2) the methods and measures for vulnerability and resilience assessment should be obtainable using the existing data available to most transportation agencies. \n\nTo address this gap, this study identified and implemented a framework incorporating quantitative metrics to classify the criticality of road segments for understanding the system-level vulnerability and criticality of the road network of the state of Texas to inform the planning and project development processes. As shown in Figure \\ref{fig:fig1}, the proposed framework includes four quantitative metrics that quantify vulnerability and criticality for each segment of the state road network. The first metric for connectivity of road segments adopts percolation analysis to quantify the criticality of road segments under the impact of external perturbations on road networks. The second metric for vulnerability to extreme events examines the proximity of road segments to the disaster impact areas. In this study, we identified areas exposed for flood hazards using 100- and 500-year floodplains to quantify the probability of flooding on road segments. The third and fourth metrics are based on interdependency analysis and proximity to critical facilities and infrastructure networks to quantify the criticality of road segments for accessing critical facilities. Finally, we combined the four metrics into integrated metrics to obtain the overall criticality of the road segments in road networks. In addition, this study presents a case study by applying the proposed framework in Texas road networks to demonstrate the effectiveness of implementing this framework in a practical scenario.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.6\\linewidth]{fig\/Fig_1.png}\n    \\caption{Four metrics for understanding the resilience of road networks. The quantitative and systematic evaluation metrics are informed by the existing literature.}\n\t\\label{fig:fig1}\n\\end{figure}\n\nThe remainder of the paper is structured as follows: Section \\ref{sec:Research Approach} outlines the research approach of the four quantitative metrics used for the resilience assessment of road networks. Sections \\ref{sec:Connectivity of Road Segments} to Section \\ref{sec:Cascading Impact of Critical Infrastructure Networks} provide detailed information and discussion of each quantitative metric. Section \\ref{sec:Integrated Criticality Metric} demonstrates details of integrated metrics that consider all individual metrics in different aspects. Section \\ref{sec:Case Study for Texas Road Network} presents a case study in Texas road networks by implementing the proposed framework. Finally, Section \\ref{sec:Concluding Remarks} presents conclusions gleaned from the study.\n\n\\section{Research Approach}\n\\label{sec:Research Approach}\nThis study implemented quantitative and computational research methods for the resilience assessment of road networks. As shown in Figure \\ref{fig:fig2}, the research approach first identified quantitative metrics for the resilience assessment of road networks. With those metrics, we then incorporated additional information, such as evacuation routes, to understand the criticality of road networks. Lastly, all of the results and information were integrated into a geographic information system-based map for practical usage. In this section, we briefly outline how the quantitative metrics are calculated for road networks; further specific discussions can be found in the sections devoted to these metrics.\n\nFirst, we evaluate the criticality of individual road segments for the connectivity of the road network, a characteristic that is particularly important in disasters. For example, residents rely on connected roads to evacuate, and first responders need to access impacted areas to save lives and protect properties. During disasters, fragmented networks may isolate areas, block transportation resources, and create traffic congestion. In addition, disruptions or floodwaters on road networks may spread over time. Measures considering static road network failures only may fail to capture the interconnections of road segments and consequent impacts of flood propagation. Hence, in this task, we adopted percolation analysis to simulate how road failures spread from one road segment to another and the extent to which failures of road networks influence their connectivity. In the case study, we collected road network data from the Texas Department of Transportation (TxDOT) roadway inventory On-System dataset updated on August 8, 2021, to perform these analyses and compute this criticality metric for road segments. This dataset includes details about the geometries of the road segments, road types, speed limits, and names of the roads. Also, the data includes both highways and streets in urban and rural areas in Texas.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_2.png}\n    \\caption{Research approach for conducting the vulnerability and resilience assessment of state road infrastructure networks.}\n\t\\label{fig:fig2}\n\\end{figure}\n\nThe second metric is the criticality of road segments to disaster impacts, which, in this study, are caused by flooding. The area of the case study, the state of Texas, especially South Texas, is prone to extreme events, such as flooding and hurricanes. Determining the criticality of road segments during flooding events is important for planning and prioritization in response to disruptive impacts of flooding. Hence, we collected 100- and 500-year floodplain data, which is the most recent nationwide extract of the National Flood Hazard Layer (NFHL) for Texas. The NFHL dataset was obtained by the Federal Emergency Management Agency (FEMA) for examining road exposure to flooding. The data includes flood map panel boundaries, flood hazard zone boundaries, and other information related to flood control zones and areas. Using this data, we identified the areas of flood zones and calculated the proximity of road networks to the nearest flood zones. Based on the proximity, we classified the distance into five categories representing the level of criticality of the road segments to flood.\n\nThird, we analyzed the criticality of road segments based on impacts of road disruption on access to (1) critical facilities such as hospitals and fire stations and (2) critical infrastructure networks such as gas lines and oil pipelines. One essential role of the road networks is to transport resources and provide public services. Accessing critical facilities is thus an important component of road network resilience assessment. We implemented two metrics to evaluate the criticality of road segments based on the interdependencies, measured based on distance buffers, between road segments and facilities. In particular, one metric is measured using the proximity of road segments to criticality facilities, while another metric is measured based on the closeness to critical infrastructure networks. We collected facility location data from Homeland Infrastructure Foundation Level Data (HIFLD) in the ArcGIS data repository, as well as energy sector data and evacuation route data from TxDOT, which include information of facility types and corresponding locations. We then extracted the location information for each of the critical facilities, such as electrical substations; power plants; hospitals; national shelter facilities; seaports; petroleum, oil and lubricant (POL) terminals; fire stations; and police stations, as well as for each of the critical infrastructure networks, such as railroad network, transmission lines, gas lines, hydrocarbon gas lines, and oil pipelines, which are essential for resiliency. Disruption in access to these facilities and infrastructure networks can exacerbate the impacts faced by communities during and in the aftermath of disasters. \n\nFinally, it is important to evaluate the overall criticality by considering all dimensions of the resilience of road networks. To this end, we devised two integration approaches to show the overall criticality of the road segments. The first method is to rank overall criticality based on the evenly weighted sum of all individual criticality metrics of each road segment. The second approach is based on the root squared sum ranking, by which the overall criticality score is calculated by taking the square root of the sum of squares of individual metrics. In the case study, we computed the integrated criticality metrics for all road segments in Texas. The results for all metrics \u2014 including four individual and two integrated metrics \u2014 can be mapped onto GIS-based maps for visualization and to provide opportunities for further incorporation with the other resilience dimensions or critical considerations. Detailed information for each metric, including individual and integrated metrics, is discussed in the following sections.\n\n\\section{Connectivity of Road Segments}\n\\label{sec:Connectivity of Road Segments}\n\\subsection{Metric Definition and Calculation}\nThe connectivity of a road segment is determined by the percolation analysis performed on the road network. The percolation analysis is an analytical technique that determines to what extent a road segment contributes to the network's overall connectivity. The procedure for percolation analysis is to remove nodes and links based on specific criteria and to quantify changes in network structure after a disturbance. When more critical road segments are perturbed, they are more likely to cause significant disconnection in the network.\n\nWhile natural hazards, such as flooding, perturb road networks, some components within the networks may fail and other parts of the networks could be disconnected and isolated. The percolation analysis mimics this process by sequentially removing nodes in topological networks. Standard criteria of node attributes rank the criticality of components within a network: degree, weighted degree, link betweenness, and size of the giant component. Figure \\ref{fig:fig3} shows an example of removing failure links and recording the size of giant components to understand the impact of perturbations on road networks. To obtain the worst scenario, we applied the node degree link removal strategy for simulation. That is, to identify failure links we removed nodes in topological networks in order from those with high degrees (node having a greater number of connections) to low degrees (node having fewer connections). In the real-world situation, it is unlikely that all the nodes with higher degrees would fail at the early stage simultaneously. During each removal stage in this analysis, essential characteristics were recorded, including the proportion of isolated links and critical transitions. Isolated links are those that become disconnected from any other part of a network; and critical transitions occur where network connectivity significantly drops (proportion of isolated links suddenly increase). Figure \\ref{fig:fig4} depicts the steps for computing this metric. In this study, the road segments are grouped into five criticality levels. The first 20\\% of isolated road segments are assigned level 1, and the last 20\\%, level 5.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.85\\linewidth]{fig\/Fig_3.png}\n    \\caption{Example of percolation analysis by removing failure links and recording the size of the giant component.}\n\t\\label{fig:fig3}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_4.png}\n    \\caption{Steps for evaluating the criticality of road segments based on network connectivity.}\n\t\\label{fig:fig4}\n\\end{figure}\n\n\\subsection{Data Collection and Processing for the Case Study}\nIn the case study, we utilized roadway inventory data updated on August 8, 2021, as the road network for computing this metric. The original road network data contains multiple lanes and complicated intersections. To perform network analysis, we simplified the original road network data, including merging multiple lanes on the same road segment into a single central line if they are close enough to each other. The simplified road network is then transformed into a topological network where all intersections and road segments are represented as nodes and links.\n\n\\section{Vulnerability to Flooding}\n\\label{sec:Vulnerability to Flooding}\nThis metric promotes understanding of the criticality of road segments to disaster impacts: flooding, in the case study. According to the literature, flooding is the most prominent hazard of concern for the Texas DOT and Texas MPOs. Therefore, we included a metric focusing on the exposure to flood hazards determined by the proximity of road segments to floodplains. This metric can be adopted to account for other disaster impacts to evaluate the criticality of road segments in road networks.\n\n\\subsection{Metric Definition and Calculation}\nThis metric determines the criticality level of road segments based on their proximity to floodplains. ArcGIS was used to perform geoprocessing tasks, including: (1) distinct road segments whose midpoints are located in the 100-year floodplain and the 500-year floodplain, and (2) calculating the distances from the midpoint of the rest of the road segments to the boundary of the 500-year floodplain. As shown in Figure \\ref{fig:fig5}, this metric determines the road segments belonging to one of the five criticality levels. The road segments with their midpoint located in the 100-year floodplain are assigned to level 1. The road segments with their midpoint located between 100- and 500-year floodplain are assigned to level 2. Level 3 covers the road segments whose midpoints are 0 to 200 meters away from the 500-year floodplain. Level 4 covers the road segments whose midpoints are 200 to 400 meters away from the 500-year floodplain. Level 5 covers the road segments more than 400 meters away from the 500-year floodplain. Figure \\ref{fig:fig5} demonstrates an example of assigning criticality levels to road segments in central Houston. Figure \\ref{fig:fig6} depicts the steps for computing this metric.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.75\\linewidth]{fig\/Fig_5.png}\n    \\caption{Definition of the metric of vulnerability to extreme events and an example of criticality levels of road segments in central Houston.}\n\t\\label{fig:fig5}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_6.png}\n    \\caption{Steps for evaluating the criticality of road segments based on proximity to flooding areas.}\n\t\\label{fig:fig6}\n\\end{figure}\n\n\\subsection{Data Collection and Processing for the Case Study}\nThe floodplain data used in this case study was obtained from the NFHL provided by FEMA on Data.gov. However, floodplain data is available for only some districts, including major urban areas in Texas, and missing in some northwestern counties in Texas (Figure \\ref{fig:fig7}). FEMA flood zones are geographic areas that FEMA has defined according to varying levels of flood risk. Each zone reflects the severity or type of flooding in the area based on the definition of flood zones, which distinguish the 100-year floodplain, 500-year floodplain, and others. Each of the 100-year and 500-year floodplains consists of several subcategories. The 100-year floodplain means there is a 1\\% annual flooding probability and is identified as high risk, while the 500-year floodplain refers to a 0.2\\% annual flooding probability and moderate risk. Regions located outside the 500-year floodplain will be identified as minimal risk or unknown risk.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.7\\linewidth]{fig\/Fig_7.png}\n    \\caption{Floodplain data availability in Texas. The major urban areas in Texas, such as Houston, Dallas\/Fort Worth, and Austin, have available floodplain data, whereas floodplain data is missing in some of the northwestern counties in Texas.}\n\t\\label{fig:fig7}\n\\end{figure}\n\n\\section{Proximity to Essential Facilities}\n\\label{sec:Proximity to Essential Facilities}\nThe previous two metrics evaluate the criticality of road segments based on network connectivity and vulnerability to flooding. Essential facilities, however, also play an important role during and in the aftermath of disasters; thus, road segments that are close to those essential facilities become critical because of the need to transport resources and to provide public services. This section evaluates the criticality of road segments based on the capability to provide access to critical facilities.\n\n\\subsection{Metric Definition and Calculation}\nTo evaluate the criticality of road segments in road networks, we implemented the steps shown in Figure \\ref{fig:fig8}. The idea is to evaluate the number of critical facilities to which a road segment provides direct access. We took the simplified road network as described in the previous section, calculated a 2,000-meter buffer around it, and identified the number of critical facilities that lie in this buffer. The purpose of this step is (1) to arrive at an estimated number of critical facilities distributed close to road networks and (2) to shortlist facilities to be used for computing the criticality metric. A buffer value of 2,000 meters was selected because most of the critical facilities lie within this buffer range. Moreover, we assumed that if a facility is within a 2,000-meter buffer range of a road network, it can be considered spatially co-located (co-location interdependency). This buffer value was selected based on our case study area; however, the buffer value may differ based on the density of the critical facilities in an area and the density of road infrastructure. After extracting the information of all the critical facilities that lie close to road networks, we computed the criticality metric by counting the number of facilities that lie in the nearest proximity (2,000-meter buffer range) of each road segment to evaluate the importance of each road segment. These road segments are then assigned a criticality score from level 1 to level 5, where level 1 is the most critical level for road segments and level 5 is the least critical level. In this study, if a road segment had more than ten critical facilities in its nearest proximity, it was assigned a score of level 1; if it had five to ten facilities, it was assigned to level 2; if it had two to five facilities, it was assigned to level 3; if it had only one facility, it was assigned to level 4; and if the road segment did not have any facility in its nearest proximity, it was assigned a score of level 5. The criticality levels of 1 and 2 mean that a road segment is critical and needs to be maintained for ensuring capable access to critical infrastructure in the vicinity, whereas levels 4 and 5 imply that there are none or less significant numbers of critical infrastructure in the nearest proximity to the road segment.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_8.png}\n    \\caption{Steps for evaluating the criticality of road segments based on proximity to critical facilities.}\n\t\\label{fig:fig8}\n\\end{figure}\n\n\\subsection{Data Collection and Processing for the Case Study}\nIn the case study, we collected location-based data of eight types of critical facilities, which are electrical substations, power plants, hospitals, national shelter facilities, seaports, fire stations, police stations, as well as petroleum, oil and lubricant (POL) terminals, from HIFLD and ArcGIS open data portal. The data consist of the coordinates of the respective critical facilities. It is important to maintain functionality of each of these facilities, including electricity, safety, health, and disaster relief. Access to critical facilities, especially during disruptions caused by disasters or other factors, is paramount for resiliency. Figure \\ref{fig:fig9} shows the spatial distribution of the critical facilities in the state of Texas. Aggregating all the facilities together, we collected location information for around 15,000 critical facilities.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_9.png}\n    \\caption{Spatial distribution of the eight critical facilities in the State of Texas.}\n\t\\label{fig:fig9}\n\\end{figure}\n\n\\section{Cascading Impact of Critical Infrastructure Networks}\n\\label{sec:Cascading Impact of Critical Infrastructure Networks}\nThe criticality metric proposed in the previous section, proximity to essential facilities, evaluates the criticality of road segments based on the locations of critical facilities that can be demonstrated as points on maps. Some critical infrastructures, however, such as the electrical power grid, railroads, and oil pipelines, are not conducive to be captured by the metric of proximity to essential facilities because they are spatially distributed as networks instead of single location attributes. Failure of a link within an infrastructure network can lead to severe cascading impacts on the entire infrastructure network \\citep{fekete_critical_2019,lee_community-scale_2021,serre_assessing_2018}. For this reason, we calculated the criticality of road segments according to their closeness to critical infrastructure networks to understand the criticality related to cascading impacts. Figure \\ref{fig:fig10}a shows the approach used in the previous metric where the criticality score is based on proximity to essential facilities, whereas Figure \\ref{fig:fig10}b demonstrates how we calculated the criticality score based on cascading impact of critical infrastructure networks.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.65\\linewidth]{fig\/Fig_10.png}\n    \\caption{Schematic of assessing criticality of road segments based on (a)critical facilities and (b) critical facility networks.}\n\t\\label{fig:fig10}\n\\end{figure}\n\n\\subsection{Metric Definition and Calculation}\nTo compute the criticality scores based on cascading impacts of critical infrastructure networks, we adopted an approach similar that described in Section \\ref{sec:Proximity to Essential Facilities} in which we used the nodes of infrastructure networks as a proxy to determine the interdependency of infrastructure networks with road networks. Figure \\ref{fig:fig11} shows the analysis steps used to compute this metric. Based on the criticality measure (i.e., counts of critical facility nodes), we assigned a criticality score from level 1 to level 5, where level 1 is the most critical level for road segments and level 5 is the least critical level. In this study, if a road segment had more than ten critical infrastructure network nodes in its nearest proximity, it was assigned a score of level 1; if it had five to ten infrastructure network nodes, it was assigned to level 2; if it had two to five infrastructure network nodes, it was assigned to level 3; if it had only one infrastructure network node, it was assigned to level 4; and if the road segment did not have any infrastructure network node in its nearest proximity, it was assigned a score of level 5.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_11.png}\n    \\caption{Steps for evaluating the criticality of road segments based on proximity to infrastructure networks.}\n\t\\label{fig:fig11}\n\\end{figure}\n\n\\subsection{Data Collection and Processing for the Case Study}\nWe collected critical infrastructure network data for five types of critical infrastructure networks, which are railroad networks, transmission lines, gas lines, hydrocarbon gas lines (HGL), and oil pipelines. These datasets were acquired from sources such as HIFLD and ArcGIS open data portal. We simplified infrastructure networks into a node-and-link network model using two steps: (1) we created a buffer between 100 and 500 meters for each infrastructure network and merged overlapping features. This was done to ensure that multiple parallel lines in the network could be merged. (2) Centerlines for the buffered networks were computed, merging some parallel lines into one and simplifying some intersections where multiple nodes are combined into one. Figure \\ref{fig:fig12} uses Harris County for illustration. Figure \\ref{fig:fig12} (row 1) shows the approximation of converting infrastructure networks into nodes and edges. Figure \\ref{fig:fig12} (row 2) shows superimposed simplified infrastructure networks with the road network in Harris County.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_12.png}\n    \\caption{Example of the facility networks and road network in Harris County. Row 1: Infrastructure networks that were simplified to node-edge networks. Row 2: Mapping of simplified infrastructure networks with road networks.}\n\t\\label{fig:fig12}\n\\end{figure}\n\n\\section{Integrated Criticality Metric}\n\\label{sec:Integrated Criticality Metric}\nAfter calculating the four criticality metrics we developed, the integrated criticality metrics that were calculated to help decision-makers understand the overall criticality of every road segment within the road network. While every criticality metric provides unique insight, a single integrated metric that comprises all information from different dimensions can be informative as well. In this section, we introduce two approaches to calculate metrics for the overall criticality scores and present the results of the integrated metrics. In addition, we compare the results of the two integrated criticality metrics and discuss the findings based on the comparison.\n\n\\subsection{Metric Development}\nTo understand the overall criticality of each road segments in networks, we calculated two integrated criticality metrics, which are based on evenly weighted sum (EWS) ranking and root squared Sum (RSS) ranking. The equations for EWS (Equation (\\ref{eq:eq1})) and RSS (Equation (\\ref{eq:eq2})) ranking calculations are: \n\n\\begin{equation}\n\tEWS_{RS}= \\sum {Level_{RS,CA}}\n\t\\label{eq:eq1}\n\\end{equation}\n\n\\begin{equation}\n\tRSS_{RS}= \\sqrt{\\sum {{Level_{RS,CA}}^2}}\n\t\\label{eq:eq2}\n\\end{equation}\n\nwhere, $EWS_{RS}$ and $RSS_{RS}$ are the ranking values for each road segment $RS$ and $Level_{RS,CA}$ is the criticality levels of each road segment and all criticality aspects $CA$ examined. For instance, if a road segment has a criticality level of 3 for connectivity, level 2 for vulnerability to flooding, level 4 for proximity to essential facilities, and level 4 for cascading impact, the EWS ranking value of this road segment will be 13, whereas the RSS ranking value will be about 6.71. After ranking values for all road segments, these values are reclassified into five levels. We assigned an integrated criticality level to each road segment. The classification method divides the range of all ranking values (i.e., maximum subtracts minimum of ranking values of all road segments) into five equal intervals, representing five classes. The level 1 category, comprising the most critical segments in the network, is the highest 20\\% of the ranking values. The class of level 5 category comprising the least critical segments in the network, is the lowest 20\\% of the ranking values.\n\n\\section{Case Study for Texas Road Network}\n\\label{sec:Case Study for Texas Road Network}\nThis section presents the results for the four individual and two integrated metrics used to assess the criticality of road segments in the state of Texas. \n\n\\subsection{Connectivity of Road Segments}\nThis metric reveals which road segments are critical in terms of connecting different parts of the network. The more critical a road segment is, the higher impact it will cause on overall connectivity when it is damaged; thus, this metric is a key indicator for informing road infrastructure planning. Figure \\ref{fig:fig13} shows the remaining links after removing 20\\% of nodes with the highest degrees at each stage. Figure \\ref{fig:fig14} shows the percentage of links being isolated during the removal process. During the link removal process that simulates the failure of road segments, critical transitions (sudden increases in the percentage of links becoming isolated) are not observed, which indicates that no particular segment has substantial impacts to the network when failing. From observing the visual results in Texas, most of the road segments that belong to higher criticality levels (levels 1 and 2) are located in the outskirts of major urban areas or in coastal areas. These road segments may be part of the busy commuting routes or the weakly connected parts of the network. The criticality level for this metric depends solely on the topology of the road network. Figure \\ref{fig:figa1} shows the criticality score of the road segments in Texas based on the connectivity metric. Most of the highly critical links, links with levels 1 and 2 criticalities, appear on the outskirts of major cities and in the coastal regions where fewer alternative substitute routes exist in case a link is removed.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.95\\linewidth]{fig\/Fig_13.png}\n    \\caption{Remaining links after removing 20\\% of nodes with the highest degrees and their corresponding failure links at each stage in Texas.}\n\t\\label{fig:fig13}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.5\\linewidth]{fig\/Fig_14.png}\n    \\caption{Percentage of links being isolated during the removal process. There is no abrupt change in the percentage of isolated links with the percentage change in nodes removed.}\n\t\\label{fig:fig14}\n\\end{figure}\n\n\\subsection{Flooding Vulnerability}\nAs aforementioned, flooding is the most prominent hazard in the state of Texas. This metric shows road segments that are more vulnerable to flooding-related hazards. Even though we made our best efforts to have floodplain data for all counties in Texas, floodplain data in some counties are not available, leaving the criticality of some road segments unclassified. In the case study, the road segments located in the 100- and 500-year floodplain, which are identified as high- and moderate-risk road segments by the definition of FEMA, are assigned to levels 1 and 2, and which form only about 20\\% of the total number of road segments. The rest of the road segments are unlikely to be affected by flooding and are relatively less vulnerable. In the Appendix \\ref{sec:Visualization Maps for the Case Study Area}, Figure \\ref{fig:figa2} shows the criticality of road segments based on this metric. The road segments that are identified as levels 1 and 2 may need additional attention to enhance their resiliency and to plan for alternative routes while affected by the hazards. \n\n\\subsection{Proximity to Essential Facilities}\nThis criticality metric captures the importance of road segments that can provide access to critical and essential facilities. We find that the critical road segments identified by this metric are more concentrated in the centers of the urban areas. In particular, the critical road segments are found in the high-density metropolitan areas of Houston, San Antonio, Austin, and Dallas\/Fort Worth, which are the four major urban areas in Texas. Since the concentration of critical facilities is relative denser in urban and populated areas with respect to the road density, there are more critical road segments in terms of proximity to essential facilities present in urban areas. Figure \\ref{fig:figa3} illustrates important road segments (levels 1 and 2) in Texas. The road segments having higher criticality in terms of proximity to essential facilities are located mainly in the major urban areas in Texas. Additionally, most level 3 road segments that have moderate criticality are located at major road junctions. On the other hand, rural areas consist mostly of low criticality road segments (levels 4 and 5) due to a lower concentration of critical facilities.\n\n\\subsection{Cascading Impact on Critical Infrastructure Networks}\nThe metric of cascading impact on critical infrastructure networks, evaluates the criticality of infrastructure networks based on their colocation interdependency with the road network. That is, the more infrastructure network nodes are in close proximity to the road segment, the higher the criticality of cascading impact on critical infrastructure networks is for a road segment. Here, unlike the metric of proximity to critical facilities, which demonstrates that the critical road segments are more concentrated in the urban areas, we found that most of the critical road segments in terms of cascading impact on critical infrastructure networks are near the coastline and western Texas instead of concentrated around urban areas. Moderately critical road segments are mostly present in urban areas. Figure \\ref{fig:figa4} shows the critical road segments in red and orange (levels 1 and 2) based on their interdependency to critical infrastructure networks. \n\n\\subsection{Integrated Criticality Metrics}\nThe results of implementing the two integrated criticality metrics in Texas, as an example, are demonstrated in Figures \\ref{fig:figa5} and \\ref{fig:figa6}. More than 75\\% of the road segments are classified as levels 4 and 5 for both integrated criticality metrics. As shown in Figure \\ref{fig:fig15}, the EWS method tends to have more levels 1, 2, and 3 criticality road segments, which is more conservative than the RSS method. By implementing the integrated criticality metrics, the comparison of criticality levels can be conducted at any scale and location in Texas (e.g., metropolitan area and county level) to understand the overall criticality of each road segment. Figure \\ref{fig:fig16} demonstrates an example in Harris County that the level 1 and 2 road segments are generally located close to intersections of road segments. In addition, the road segments at the northwestern corner of Harris County, where the city of Houston is located, have relatively minor criticality compared to the road segments in the other area of Harris County.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.6\\linewidth]{fig\/Fig_15.png}\n    \\caption{Distribution of different criticality levels of the RSS and EWS methods for the state of Texas. }\n\t\\label{fig:fig15}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.9\\linewidth]{fig\/Fig_16.png}\n    \\caption{Results of EWS ranking (left) and RSS ranking (right) in Harris County.}\n\t\\label{fig:fig16}\n\\end{figure}\n\nBased on the integrated criticality metrics, we can specify the overall criticality of different road segments and prioritize resilience investments accordingly. Thus, it is essential to know the most common criticality concern among the four metrics if a road segment is classified as level 1 (i.e., the most critical level) for the overall criticality level. Figure \\ref{fig:fig17} shows the distribution of the four individual metrics for the EWS and RSS integrated criticality metrics in level 1. More than 70\\% of the road segments in the Texas road networks assigned an EWS integrated criticality of level 1 are critical (i.e., level 1 criticality) in terms of flood exposure and network connectivity. Similarly, more than 80\\% of the road segments with RSS integrated criticality of level 1 are also critical in flood exposure and network connectivity. On the other hand, the criticality in terms of cascading impacts appears to be relatively less influential for the road segments with the EWS and RSS integrated criticality metrics of level 1, of which the criticality related to cascading impacts are mostly at level 2. With these results, agencies and organizations can prioritize maintenance plans proactively and ensure the required functionality and serviceability of road networks to mitigate the impact of disasters and recover from impact rapidly.\n\n\\begin{figure}\n\t\\centering\n    \\includegraphics[width=0.8\\linewidth]{fig\/Fig_17.png}\n    \\caption{Distribution of the influence of individual metrics for the road segments with EWS and RSS integrated criticality of level 1.}\n\t\\label{fig:fig17}\n\\end{figure}\n\n\\section{Concluding Remarks}\n\\label{sec:Concluding Remarks}\nThe objective of the study is to propose a framework with quantitative measures incorporating resilience assessment into transportation planning and project development processes in practice. While the literature discusses several methods and measures for resilience assessment of transportation infrastructure, just a small fraction of the existing work has been adopted in practice to inform transportation planning and project development. This limitation is mainly due to two reasons: (1) transportation planners and decision makers do not know which to use among several methods and measures proposed in the literature; (2) the methods and measures for vulnerability and resilience assessment should be obtainable using the available data. To address this gap, in this study, we proposed and tested a framework for system-level assessment of transportation infrastructure networks based on four measures: (1) loss of connectivity of road segments in road networks, (2) vulnerability of road segments to extreme events, such as flooding and hurricanes, (3) disrupted access to critical facilities, and (4) cascading impacts of critical infrastructure networks. In addition, two integrated metrics were proposed to combine all metrics into one measure that can show the overall performance of road segments in terms of resilience. To demonstrate the use of the framework, we implemented individual and integrated metrics in the state of Texas. We collected datasets related to the topology of the Texas road network, hazard characteristics, and the location of critical facilities (e.g., power substations, oil\/gas lines, and hospitals) from open sources and calculated the values of each metric for all road network links in Texas. The results showed that the framework can effectively provide useful insights with quantitative measures. In addition, the visualization of the results can make the interpretation of resilience assessment on road networks more straightforward and comprehensive. Also, the case study demonstrates the ability of the framework to be conducted at any scale and location to reveal the resilience and criticality of each road segment on networks.\n\nThe framework proposed in this paper can help transportation agencies incorporate resilience considerations into planning and project development proactively and to understand the resilience performance of the current road networks effectively. For example, road segments with high criticality and vulnerability levels could be prioritized for adaptation and hazard mitigation retrofit. Roads with high criticality and vulnerability to flooding impacts could be effectively identified and appropriate projects (such as drainage improvement or road elevation increase) could be implemented to reduce the vulnerability of those road segments. The demonstration of the results of this study to planners and decision makers in Texas has confirmed that the produced measures are deemed useful by practitioners. In fact, at the time of writing this manuscript, a state-wide transportation resilience planning effort is underway, and the results of this study are a major component in informing the planning process. Since the data used in this study is available to other states and countries, the framework presented in this study can be adopted by other transportation agencies across the globe for regional transportation resilience assessments. \n\n\\section{Data availability}\nAll datasets used in this study are publicly available. The road network data can be acquired from the TxDOT data repository. Flood impact data is extracted from the National Flood Plain Layer on the FEMA website. Data on critical facilities and critical infrastructure networks can be accessed from publicly available Homeland Infrastructure Foundation Level Data (HIFLD) and ArcGIS open data portals.\n\n\\section{Code availability}\nThe code that supports the findings of this study is available from the corresponding author upon request.\n\n\\section{Acknowledgements}\nThis research is part of the study supported by the Texas Department of Transportation (TxDOT). The contents of this paper reflect the views of the authors, who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official view or policies of TxDOT. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the organizations or the individuals listed here.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nBorn from the gravitational core collapse of massive stars in supernova explosions, neutron stars are the densest known stars in the universe (see, e.g., \nRef.~\\cite{haensel2007}). Most of them have been discovered as isolated radio pulsars. Neutron stars can also belong to binary systems, in which the \nneutron star accretes matter from a stellar companion. These systems are observed as x-ray pulsars. Neutron stars may be endowed with very strong magnetic \nfields. Soft $\\gamma$-ray repeaters \nand anomalous x-ray pulsars are thought to be such so-called ``magnetars'' (see, e.g., Ref.~\\cite{woods2006} for a review). Their surface magnetic field, as inferred \nfrom spin-down and spectroscopic studies, are of the order of $10^{14}-10^{15} $~G \\cite{olausenkaspi2014, tiengo2013, hongjun2014}. Numerical simulations \nsuggest that their internal magnetic field could be even stronger, reaching $10^{18}$~G~\\cite{pili2014,chatterjee2015}. \n\nApart from a thin atmospheric plasma layer of light elements (mainly hydrogen and helium) possibly surrounding a Coulomb liquid of electrons and ions, a \nneutron star is thought to contain at least three distinct regions. The outermost region consists of a solid crust made of a crystal lattice of fully ionized \natoms arranged on a body-centered cubic lattice (see, e.g., Ref.~\\cite{chamelhaensel2008} for a review). With increasing density, nuclei become progressively \nmore neutron rich due to electron captures until neutrons start to drip out of nuclei at some threshold density $\\rho_{\\rm drip}$, thus marking the boundary \nbetween the outer crust and the inner crust. In unmagnetized or weakly magnetized nonaccreting neutron stars (with magnetic fields $\\ll 10^{14}$~G), the neutron-drip \ntransition is found to occur at density $\\rho_{\\rm drip} \\approx 4 \\times 10^{11}$~g~cm$^{-3}$ (see, e.g., Refs.~\\cite{bps1971, haensel2007, roca2008, pearson2011, \nwolf2013, kreim2013, chamel2015c}). On the other hand, this density can be shifted in magnetars, due to the presence of strong magnetic fields (see, e.g., \nRefs.~\\cite{sek1977, lai91, nandi2011, chamel2012, nandi2013, vishal2014, chamel2015b,bas2015}), and in accreting neutron stars, due to the accretion from a companion \n\\cite{hz1990, hz2003, chamel2015a}.\nThe crust dissolves into an homogeneous liquid mixture at about half the density prevailing in heavy atomic nuclei. This transition is expected to be realized \nthrough different nuclear structures, such as cylinders, rods, and plates (also called ``nuclear pasta'', see Ref.~\\cite{haensel2007,chamelhaensel2008} and references therein). Further \ninside, at very high densities in the neutron-star core, additional degrees of freedom, such as hyperons, meson condensates, and\/or deconfined quarks, could be \npresent (see, e.g., Ref.~\\cite{haensel2007, page2006, weber2007}). \n\n\nFrom the nuclear physics point of view, the neutron-star crust is also a unique ``laboratory'' to probe the properties of infinite homogeneous asymmetric nuclear matter, in particular \nthe symmetry energy, at subsaturation densities (see, e.g. Ref.~\\cite{steiner2015}). The symmetry energy is generally defined as \n\\begin{equation}\n\\label{eq:e-inm}\nS_1(n) = \\frac{1}{2}\\frac{\\partial^2 (\\mathcal{E}\/n)}{\\partial \\eta^2}\\biggr\\vert_{\\eta=0} \\ ,\n\\end{equation}\nwhere $\\mathcal{E}(n,\\eta)$ is the energy density of infinite homogeneous nuclear matter with proton (neutron) density $n_p$ ($n_n$), baryon density \n$n=n_n + n_p$, and charge asymmetry $\\eta = (n_n-n_p)\/n$. Alternatively, the symmetry energy often refers to the difference between the energy of \npure neutron matter and that of symmetric matter:\n\\begin{equation}\n\\label{eq:s2}\nS_2(n) =\\frac{\\mathcal{E}(n,\\eta=1) - \\mathcal{E}(n,\\eta=0)}{n} \\ .\n\\end{equation} \nBecause $\\mathcal{E}(n,\\eta)$ generally contains terms of order $\\eta^4$ and higher, the two definitions $S_1(n)$ and $S_2(n)$ of the symmetry energy \ndo not exactly coincide (see, e.g.  the discussion in Sec.~III of Ref.~\\cite{goriely2010}). From now on, we adopt the first definition. \nThe symmetry energy of infinite homogeneous nuclear matter $S_1(n)$ can be expanded around the saturation density $n_0$ (whose value is $\\approx 0.16$~fm$^{-3}$) as\n\\begin{equation}\n\\label{eq:s1}\nS_1(n) \\approx J + \\frac{1}{3} L \\left( \\frac{n-n_0}{n_0} \\right) + \\frac{1}{18} K_{\\rm sym} \\left( \\frac{n-n_0}{n_0} \\right)^2\\ .\n\\end{equation}\nWhereas the value of the symmetry energy at saturation, $J$, is fairly well constrained by nuclear physics experiments to lie around 30~MeV, the values of the \nslope of the symmetry energy, $L$, and of higher order coefficients like $K_{\\rm sym}$ at saturation,\n are still very uncertain and poorly constrained (see, e.g., the discussion in \nRefs.~\\cite{tsang2012,lattimerlim2013}). The symmetry energy has been shown to affect the composition of neutron-star crusts and the crust-core transition~(see, e.g., \nRefs.~\\cite{horowitz2001,oyamatsu2007, roca2008, vidana2009,ducoin2011,grill2012,newton2013,sulaksono2014,provid2014,bao2014, grill2014,seif2014,provid2014}). \nComparatively few studies have been devoted to the role of the symmetry energy on the boundary between the outer and inner crusts~\\cite{bao2014} (some discussions can \nalso be found, e.g., in Refs.~\\cite{roca2008,provid2014,grill2014}). Moreover, all these studies focused on nonaccreting and unmagnetized neutron stars. \n\n\nIn this paper, we study the role of the symmetry energy on the onset of neutron drip in both nonaccreting and accreting neutron stars, allowing for the \npresence of a strong magnetic field as in magnetars. To this end, we have made use of the recent set of Brussels-Montreal microscopic nuclear mass \nmodels~\\cite{goriely2013}. Our model of neutron-star crust is presented in Sec.~\\ref{sec:model}~: after discussing our assumptions in Sec.~\\ref{sec:assumptions}, \nthe nuclear mass models are briefly described in Sec.~\\ref{sec:hfb-models}. The determination of the neutron-drip transition is discussed in Sec.~\\ref{sec:neutron-drip}, \nand numerical results are presented in Sec.~\\ref{sec:results}. \n\n\n\\section{Model of neutron-star crust}\n\\label{sec:model}\n\n\n\\subsection{Main assumptions}\n\\label{sec:assumptions}\n\nIn the region of the crust that we consider here, the pressure is high enough that atoms are fully ionized~\\cite{chamelhaensel2008}. We assume that the temperature $T$ is \nlower than the crystallization temperature $T_m$ so that nuclei are arranged in a regular crystal lattice. Considering that the crystalline structures are \nmade of only one type of ion $^{A} _{Z} X$, with mass number $A$ and atomic number $Z$, the crystallization temperature $T_m$ is given by \n(see, e.g., Ref.~\\cite{haensel2007}):\n\\begin{equation}\nT_m = \\frac{e^2}{a_e k_{\\rm B} \\Gamma_m} Z^{5\/3} \\ ,\n\\end{equation}\nwhere $e$ is the elementary electric charge, $a_e=(3\/(4\\pi n_e))^{1\/3}$ the electron-sphere radius expressed in terms of the electron number density $n_e$, \n$k_\\text{B}$ is the Boltzmann's constant, and $\\Gamma_m\\simeq 175$ is the Coulomb coupling parameter at melting. Typically, $T_m$ is much lower than the \nelectron Fermi temperature defined by\n\\begin{equation}\n T_{\\rm F}=\\frac{\\mu_e-m_e c^2}{k_{\\rm B}} \\ ,\n\\end{equation}\nwhere $m_e$ is the electron mass, $c$ is the speed of light, and $\\mu_e$ is the electron Fermi energy. \nTherefore, electrons are highly degenerate and from now on we will set $T=0$. To a very good approximation, electrons can be treated as an ideal relativistic \nFermi gas. Expressions for the electron energy density $\\mathcal{E}_e$ and electron pressure $P_e$ can be found in Ref.~\\cite{haensel2007}. \nThe main corrections arise from electron-ion interactions (see, e.g., Ref.~\\cite{pearson2011} and references therein for a discussion of other corrections). \nNeglecting the finite size of ions and the quantum zero-point motion of ions off their equilibrium position, the lattice contribution to the energy density \nis given by \n(see e.g. Chap. 2 in Ref.~\\cite{haensel2007})\n\\begin{equation}\n\\label{eq:EL}\n\\mathcal{E}_L  = C e^2 n_e^{4\/3} Z^{2\/3} \\ ,\n\\end{equation}\nwhere $C$ is a crystal structure constant, whose value for a body-centered cubic lattice is $C=-1.444231$~\\cite{baiko2001}. This expression still remains valid in the \npresence of a strong magnetic field, as a consequence of the Bohr-van Leeuwen theorem~\\cite{vanvleck1932}. \nThe matter pressure $P$ can be expressed as\n\\begin{equation}\n\\label{eq:ptot}\nP = P_e + P_L \\ ,\n\\end{equation}\nwhere $P_e$ is the pressure of a uniform electron gas, and the lattice pressure $P_L$ is given by \n\\begin{equation}\n\\label{eq:PL}\n P_L=\\frac{\\mathcal{E}_L}{3} \\ .\n\\end{equation}\nThe only microscopic inputs for the description of the outer crust are nuclear masses $M^\\prime(A,Z)$. They can be obtained from the corresponding \natomic mass $M(A,Z)$ after subtracting out the binding energy of the atomic electrons (see Eq.~(A4) of Ref.~\\cite{lpt03}):\n\\begin{equation}\n\\label{3}\nM^\\prime(A,Z)c^2 = M(A,Z)c^2 + 1.44381\\times 10^{-5}\\,Z^{2.39} + 1.55468\\times 10^{-12}\\,Z^{5.35} \\ ,\n\\end{equation}\nwhere both masses are expressed in units of MeV$\/c^2$. Nuclear masses may be changed in the presence of a strong magnetic field. On the other hand, \nfor the magnetic field strength we shall consider, namely $B < 10^{17} $~G, those changes are very small \\cite{pena2011} and will thus be ignored. \nFor the masses that have not yet been measured, we have made use of the microscopic mass tables computed by the Brussels-Montreal group  (for a recent review of these models, see, e.g., Ref.~\\cite{chamel2015c}). \n\n\\subsection{Microscopic nuclear mass models}\n\\label{sec:hfb-models}\n\nThe family of Brussels-Montreal nuclear mass models that we adopt here~\\cite{goriely2013} are based on the nuclear energy density functional (EDF) \ntheory using a generalized form of Skyrme zero-range effective interactions~\\cite{chamel2009}, supplemented with a microscopic contact pairing interaction~\\cite{chamel2010}. \nFor all these models, the masses of nuclei were obtained by adding to the \nHartree-Fock-Bogoliubov (HFB) energy a phenomenological Wigner term and a correction term for the rotational and vibrational spurious collective energy \n(see, e.g. Ref.~\\cite{goriely2010} for a discussion about the accuracy of this latter correction). The EDFs BSk22, BSk23, BSk24, BSk25, and BSk26 \nunderlying the nuclear mass models HFB-22, HFB-23, HFB-24, HFB-25, and HFB-26 were primarily fitted to the 2353 measured masses of nuclei with $N$ and $Z \\geq 8$ from \nthe 2012 Atomic Mass Evaluation \\cite{audi2012}, with a root-mean-square (rms) deviation of $0.63$~MeV, $0.57$~MeV, $0.55$~MeV, $0.54$~MeV, and $0.56$~MeV \nrespectively. At the same time, these EDFs were constrained to reproduce the equation of state (EoS) of homogeneous neutron matter, as obtained by many-body \ncalculations using realistic two- and three-nucleon interactions. Moreover, the incompressibility $K_v$ of infinite homogeneous symmetric nuclear matter at saturation was \nrequired to fall in the range $240\\pm10$~MeV~\\cite{colo2004}, and the isoscalar effective mass was fixed to the realistic value $M_s^*=0.8$. In addition, the \nEoS of symmetric nuclear matter obtained from these EDFs were found to be compatible with the empirical constraints inferred from the analysis of heavy-ion \ncollision experiments~\\cite{dan02,lynch09}. For all these reasons, we believe that these EDFs can be reliably applied to the description of neutron-star crusts. \n\nIn generating these five EDFs, different values of the symmetry energy coefficient $J$ \nwere imposed, thus making them suitable for a systematic study of \nthe role of the symmetry energy on the outer crust of a neutron star. In particular, BSk22, BSk23, BSk24 and BSk25 were constrained to the symmetry energy \ncoefficients $J = 32$, $31$, $30$ and $29$~MeV, respectively and were all fitted to the realistic neutron-matter EoS labeled ``V18\" in Ref.~\\cite{ls2008}, \nwhile BSk26 was fitted to the EoS labeled ``A18 + $\\delta\\,v$ + UIX$^*$\"in Ref.~\\cite{apr1998} under the constraint $J = 30$~MeV. The values of the symmetry energy coefficient $J$, as well as the higher-order coefficients \n$L$ and $K_{\\rm sym}$ for the different EDFs are indicated in Table \\ref{tab:esymcoef}. As already discussed in Ref.~\\cite{pearson2014}, $L$ is strongly \ncorrelated with $J$: increasing $J$ leads to higher values of $L$. The difference between the values of $L$ obtained with BSk24 and BSk26 (for which $J=30$ MeV) \narises from the constraining neutron-matter EoS. Indeed, as shown in Eqs.~(\\ref{eq:e-inm}) and (\\ref{eq:s1}) the softer the underlying neutron-matter EoS, \nthe lower $L$. On the other hand, the correlation between $L$ and $K_{\\rm sym}$ is less clear. As shown in Fig.~\\ref{fig:JvsL}, the values of $J$ and $L$ obtained \nwith the Brussels-Montreal EDFs are consistent with constraints coming from the combined analysis of various experiments \\cite{tsang2012, lattimerlim2013} \n(see also the discussion in Sec.~IIIC in Ref.~\\cite{goriely2013}): the constraint deduced in Ref.~\\cite{tsang2009} from heavy-ion collisions (HIC), the constraint derived \nin Ref.~\\cite{chen2010} from measurements of the neutron-skin thickness in tin isotopes, and finally the constraint obtained from the analysis of the giant dipole \nresonance (GDR) \\cite{trippa2008}. \n\nThe density dependence of the symmetry energy $S_1(n)$, Eq.~(\\ref{eq:s1}), for the Brussels-Montreal EDFs BSk22 to BSk25 is shown in Fig.~\\ref{fig:esym-vs-n}. The most notable feature, due to the mass fit, \nis the crossing of all curves at densities around $0.11$~fm$^{-3}$. Indeed, the binding energy of finite nuclei is mainly sensitive to the symmetry energy \nat such densities rather than to the symmetry energy at saturation density~\\cite{horowitz2001, zhangchen2013}. \nFor these functionals, which were fitted to the same \nneutron-matter EoS, we observe a clear correlation between the density dependence of the symmetry energy and its value at saturation with a change of hierarchy \nat the crossing point: the higher $J$, the higher (lower) the symmetry energy above (below) this point. From these considerations, the ``effective'' value of \nthe symmetry energy averaged over the volume of a nucleus is thus expected to increase with decreasing $J$, as found in previous studies~\\cite{goriely2005,gaidarov2012}. \n\n\nComparing various constraints from both nuclear physics and astrophysics~\\cite{pearson2014, fantina2015}, BSk24 (BSk22) was found to be the best (worst) in the series of Brussels-Montreal EDFs BSk22-BSk26. \nIn the following, we shall thus take BSk24 as the reference EDF. We will not consider BSk26 since it was fitted to a different neutron-matter EoS from the other EDFs. \nWe have made use of the mass tables from the BRUSLIB database~\\cite{bruslib}.\n \n\n\\begin{table}\n\\centering\n\\caption{Symmetry energy coefficient $J$ and higher-order symmetry energy coefficients of infinite \nhomogeneous nuclear matter at saturation for the Brussels-Montreal energy density functionals~\\cite{goriely2013}.}\\smallskip\n\\label{tab:esymcoef}\n\\begin{tabular}{|c|ccccc|}\n\\hline\n & BSk22 & BSk23 & BSk24 & BSk25 & BSk26 \\\\\n\\hline\n$J$ [MeV] & 32.0 & 31.0 & 30.0 & 29.0 & 30.0 \\\\\n$L$ [MeV] & 68.5 & 57.8 & 46.4 & 36.9 & 37.5  \\\\\n$K_{\\rm sym}$ [MeV] & 13.0 & -11.3 & -37.6 & -28.5 & -135.6 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.35]{JvsL.eps}\n\\end{center}\n\\caption{(Color online) Experimental constraints on the symmetry energy parameters, taken from Ref.\\cite{lattimerlim2013}; the dashed line represents the constraint \nobtained from fitting experimental nuclear masses using the Brussels-Montreal Hartree-Fock-Bogoliubov models (including unpublished ones) with an root-mean-square deviation below \n$0.84$~MeV; star symbols correspond to the series of models from Ref.~\\cite{goriely2013} that we consider in this work. See text for details.}\n\\label{fig:JvsL}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.35]{esym1_bsk22-25.eps}\n\\end{center}\n\\caption{(Color online) Symmetry energy of infinite homogeneous nuclear matter $S_1(n)$ versus density, \nfor the Brussels-Montreal energy density functionals BSk22 to BSk25~\\cite{goriely2013}.}\n\\label{fig:esym-vs-n}\n\\end{figure*}\n\n\n\n\\section{Neutron-drip transition}\n\\label{sec:neutron-drip}\n\nNeutron stars are formed from the catastrophic gravitational core collapse of massive stars during supernova explosions. Under such extreme conditions, it is generally assumed that \nall kinds of nuclear and electroweak processes occur and that, as the neutron star cools down, matter remains in full thermodynamic equilibrium. Eventually, \nthe neutron star becomes cold and fully ``catalyzed'' \\cite{haensel2007} unless matter is accreted from a companion star. In the latter case, the accretion \nmay heat the neutron star and change the composition of its crust. These two different astrophysical scenarios will thus be treated separately. \n\n\n\\subsection{Nonaccreting neutron stars}\n\nThe usual procedure~\\cite{bps1971} to determine the equilibrium nucleus in any layer of the outer crust of a nonaccreting neutron star at pressure $P$ is to minimize the Gibbs \nfree energy per nucleon defined by \n\\begin{equation}\n\\label{eq:gibbs-general}\ng = \\frac{\\mathcal{E}+P}{n} \\ ,\n\\end{equation}\nwhere $\\mathcal{E}$ denotes the mean energy density of the crustal matter, and $n$ denotes the mean baryon number density. As shown in Ref.~\\cite{chamel2015d}, $g$ \nremains the suitable thermodynamic potential in the presence of a strong magnetic field. Equation~(\\ref{eq:gibbs-general}) can be equivalently expressed \nas~\\cite{bps1971,lai91} \n\\begin{equation}\n\\label{eq:gibbs}\ng=\\frac{M'(A,Z)c^2}{A}+\\frac{Z}{A}\\left(\\mu_e -m_e c^2+\\frac{4}{3} C e^2 n_e^{1\/3} Z^{2\/3}\\right)\\, .\n\\end{equation}\nIgnoring neutron band structure effects~\\cite{chamel2006,chamel2007}, \nthe onset of neutron drip is determined by the condition $g=m_n c^2$, where $m_n$ is the neutron mass~\\cite{bps1971} (see also the discussion in Ref.~\\cite{chamel2015a}). \nThis latter condition can be approximately expressed as \\cite{chamel2015a}\n\\begin{equation}\n\\label{eq:n-drip-mue}\n\\mu_e(n_e^{\\rm drip}) + \\frac{4}{3}C e^2 (n_e^{{\\rm drip}})^{1\/3} Z^{2\/3} =  \\mu_e^{{\\rm drip}}\\, ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:muedrip}\n\\mu_e^{{\\rm drip}}(A,Z)\\equiv \\frac{-M'(A,Z)c^2+A m_n c^2}{Z} +m_e c^2 \\, .\n\\end{equation}\nNote that this condition remains the same in the presence or in the absence of a magnetic field. \nHowever, since the relation between the electron density and \nelectron Fermi energy does depend on the magnetic field, the neutron-drip density and pressure do depend on the magnetic field. \n\nFor weakly magnetized neutron stars, $B\\lesssim B_\\star$ where $B_\\star \\equiv B\/B_{\\rm crit}$ is the magnetic field expressed in units of the critical \nmagnetic field \n\\begin{equation}\nB_{\\rm crit} = \\left(\\frac{m_ec^2}{\\alpha \\lambda_e^3}\\right)^{1\/2} \\simeq 4.4 \\times 10^{13} {\\rm G} \\ , \n\\end{equation}\n$\\lambda_e=\\hbar\/(m_e c)$ being the electron Compton wavelength and $\\alpha=e^2\/(\\hbar c)$ the fine structure constant, \nthe baryon density and pressure at neutron drip are approximately given by \\cite{chamel2015a}\n\\begin{equation}\n\\label{eq:ndrip-cat-approx}\nn_{\\rm drip}(A,Z) \\approx \\frac{A}{Z} \\frac{ \\mu_e^{\\rm drip}(A,Z)^3}{3\\pi^2 (\\hbar c)^3}\n\\biggl[1+\\frac{4 C \\alpha}{(81\\pi^2)^{1\/3}} Z^{2\/3} \\biggr]^{-3}\\, .\n\\end{equation}\n\\begin{equation}\n\\label{eq:Pdrip-cat-approx}\nP_{\\rm drip}(A,Z) \\approx \\frac{\\mu_e^{\\rm drip}(A,Z)^4}{12 \\pi^2 (\\hbar c)^3}\\biggl[1+\\frac{4C \\alpha Z^{2\/3}}{(81\\pi^2)^{1\/3}} \\biggr]^{-3} \\, .\n\\end{equation}\nIn the presence of a strongly quantizing magnetic field such that \n$n_e<n_{e\\text{B}}$ and $T<T_\\text{B}$ with \n\\begin{equation}\\label{eq:neB}\nn_{e\\text{B}}= \\frac{B_\\star^{3\/2}}{\\sqrt{2} \\pi^2 \\lambda_e^3} \\, ,\n\\end{equation}\n\\begin{equation}\\label{eq:TB}\n T_\\text{B}=\\frac{m_e c^2}{k_\\text{B}} B_\\star\\, , \n\\end{equation}\nthe neutron-drip density and pressure are approximately given by \\cite{chamel2015b}\n\\begin{equation}\n\\label{eq:ndrip-cat-approx-b}\nn_{\\rm drip}(A,Z) \\approx \\frac{A}{Z} \\frac{ \\mu_e^{\\rm drip}(A,Z)}{m_e c^2} \\frac{B_\\star}{2\\pi^2 \\lambda_e^3}\n\\biggl[1-\\frac{4}{3} C \\alpha Z^{2\/3} \\left(\\frac{B_\\star}{2 \\pi^2}\\right)^{1\/3} \\left(\\frac{m_e c^2}{\\mu_e^{\\rm drip}}\\right)^{2\/3} \\biggr] \\, ,\n\\end{equation}\n\\begin{equation}\n\\label{eq:Pdrip-cat-approx-b}\nP_{\\rm drip}(A,Z) \\approx \\frac{B_\\star \\mu_e^{\\rm drip}(A,Z)^2}{4 \\pi^2 \\lambda_e^3 m_e c^2}\\biggl[1 - \\frac{1}{3} C \\alpha Z^{2\/3} \\left(\\frac{4 B_\\star}{\\pi^2}\\right)^{1\/3} \\left(\\frac{m_e c^2}{\\mu_e^{\\rm drip}}\\right)^{2\/3} \\biggr] \\, .\n\\end{equation}\nUsing Eqs.~(\\ref{eq:neB}) and (\\ref{eq:ndrip-cat-approx-b}), the condition $n_e<n_{e\\rm B}$ can be approximately expressed to first order in $\\alpha$  as \n\\begin{equation}\\label{eq:Bdrip}\nB_\\star > B_\\star^{\\rm drip}\\equiv \\frac{1}{2}\\left(\\frac{ \\mu_e^{\\rm drip}(A,Z)}{m_e c^2}\\right)^2 \n\\biggl[1-\\frac{8 C \\alpha Z^{2\/3}}{3(2 \\pi^2)^{1\/3}} \\biggr] \\, .\n\\end{equation}\n\n\n\\subsection{Accreting neutron stars}\n\nIn accreting neutron stars, the magnetic field is typically negligibly small ($B\\ll B_\\star$) and will thus be ignored.  \nFor an accretion rate $\\dot{M}=10^{-9}~{\\rm M_\\odot}$~yr$^{-1}$ the original outer crust is replaced by accreted matter in $10^4$~yr.\nFor low-mass binary systems, the accretion stage can last for $10^9$~yr.\nAt densities above $\\sim 10^8$~g~cm$^{-3}$, matter is highly degenerate and relatively cold  ($T\\lesssim 5\\times 10^8$~K) so that thermonuclear processes are strongly suppressed, since their rates are many orders of magnitude lower than the compression rate due to accretion \\cite{haensel2007}. The only relevant \nprocesses are electron captures and neutron-emission processes, whereby the nucleus $^A_ZX$ is transformed into a nucleus $^{A-\\Delta N}_{Z-1}Y$ with \nproton number $Z-1$ and mass number $A-\\Delta N$ by capturing an electron with the emission of $\\Delta N$ neutrons $n$ and an electron neutrino $\\nu_e$:\n\\begin{equation}\n\\label{eq:e-capture+n-emission}\n^A_ZX+ e^- \\rightarrow ^{A-\\Delta N}_{Z-1}Y+\\Delta N n+ \\nu_e\\, .\n\\end{equation}\nIn this case, the condition for the onset of neutron drip becomes~\\cite{chamel2015a}\n\\begin{eqnarray}\n\\label{eq:e-capture+n-emission-gibbs-approx}\n\\mu_e + C e^2 n_e^{1\/3}\\biggl[Z^{5\/3}-(Z-1)^{5\/3} + \\frac{1}{3} Z^{2\/3}\\biggr] = \\mu_e^{\\rm drip-acc} \\, ,\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\label{eq:muebetan}\n\\mu_e^{\\rm drip-acc}(A,Z)\\equiv M'(A-\\Delta N,Z-1)c^2-M'(A,Z)c^2 +m_n c^2 \\Delta N + m_e c^2 \\, .\n\\end{equation}\nThe neutron-drip density and pressure are approximately given by~\\cite{chamel2015a}\n\\begin{equation}\n\\label{eq:ndrip-acc}\nn_{\\rm drip-acc}(A,Z) \\approx \\frac{A}{Z} \\frac{\\mu_e^{\\rm drip-acc}(A,Z)^3}{3\\pi^2 (\\hbar c)^3} \n \\biggl[1+\\frac{C \\alpha}{(3\\pi^2)^{1\/3}}\\left(Z^{5\/3}-(Z-1)^{5\/3}+\\frac{Z^{2\/3}}{3}\\right)\\biggr]^{-3}\\, ,\n\\end{equation}\n\\begin{equation}\n\\label{eq:Pdrip-acc}\nP_{\\rm drip-acc}(A,Z) \\approx \\frac{\\mu_e^{\\rm drip-acc}(A,Z)^4}{12 \\pi^2 (\\hbar c)^3}\\biggl[1+\\frac{4C \\alpha Z^{2\/3}}{(81\\pi^2)^{1\/3}} \\biggr]\n \\biggl[1+\\frac{C \\alpha}{(3\\pi^2)^{1\/3}}\\left(Z^{5\/3}-(Z-1)^{5\/3}+\\frac{Z^{2\/3}}{3}\\right)\\biggr]^{-4}\\, .\n\\end{equation}\nAs discussed in Ref.~\\cite{chamel2015a}, the dripping nucleus can be determined as follows. Given the mass number $A$ and the initial atomic number \n$Z_0$ of the ashes of x-ray bursts, the atomic number $Z$ at the neutron-drip point is the highest number of protons lying below $Z_0$ for which the $\\Delta N$-neutron \nseparation energy defined as\n\\begin{equation}\n\\label{eq:sn-dn}\nS_{\\Delta N n}(A,Z-1) \\equiv M(A-\\Delta N,Z-1)c^2 - M(A,Z-1)c^2 + \\Delta N m_n c^2\n\\end{equation}\nis negative. \n\n\n\n\\section{Numerical results}\n\\label{sec:results}\n\n\\subsection{Nonaccreting neutron stars}\n\\label{sec:results-nonaccreting}\n\nWe have calculated the properties of neutron-star crusts at the neutron-drip point by minimizing the Gibbs free energy per nucleon~(\\ref{eq:gibbs}), \nboth in the absence and in the presence of a strong magnetic field. In the latter case, we have set $B_\\star=500$, $1000$, $1500$ and $2000$ corresponding \nto magnetic fields in the range $2.2 \\times 10^{16}$~G to $8.8 \\times 10^{16}$~G. Since nuclear masses at this depth of the outer crust are not experimentally \nknown, the predictions for the dripping nucleus are model dependent. The neutron-drip properties are summarized in Table~\\ref{tab:drip-cat} in \nunmagnetized neutron stars, and in Tables~\\ref{tab:drip-cat-mag-500}-\\ref{tab:drip-cat-mag-2000} in strongly magnetized neutron stars (magnetars). \n\nFigure~\\ref{fig:cat-ndrip-B} shows that for any given value of the magnetic field strength, the neutron-drip density increases almost linearly with the slope \nof the symmetry energy $L$ (or equivalently with $J$ since the two coefficients are strongly correlated, as previously discussed in Sec.~\\ref{sec:hfb-models}). \nOn the other hand, the behavior of the neutron-drip density with respect to the magnetic field strength exhibits typical quantum oscillation whereas the \nneutron-drip pressure increases monotonically, as recently discussed in Ref.~\\cite{chamel2015b}. The errors of the analytical formulas~(\\ref{eq:ndrip-cat-approx})-(\\ref{eq:Pdrip-cat-approx}) amount to $0.1\\%$ at most, as compared to the numerical solution of Eq.~(\\ref{eq:n-drip-mue}). The proton fraction $Z\/A$ \nat the neutron-drip point is also found to be strongly correlated with the symmetry energy. As shown in the right panel of Fig.~\\ref{fig:cat-AZ-B}, $Z\/A$ decreases \nalmost linearly with increasing $L$ (or $J$). Similar behaviors of $Z\/A$ and $n_{\\rm drip}$ with $L$ have been recently obtained in Ref.~\\cite{bao2014}, and \ncan be inferred from the discussions in Refs.~\\cite{roca2008,provid2014,grill2014}. Nevertheless, in all cases they considered the limiting case $B_\\star=0$. \nIn Ref.~\\cite{bao2014}, the authors studied the role of the symmetry energy on the properties of neutron-star crusts around \nthe neutron-drip threshold using two sets of relativistic mean field (RMF) models based on the TM1 and IUFSU parametrizations respectively. They generated \nseries of models so as to achieve different values of $L$ keeping the symmetry energy at $n=0.11$~fm$^{-3}$ fixed. In our case, the fixed value of the symmetry \nenergy at $n\\approx 0.11$~fm$^{-3}$ results from the mass fit without any further constraint. Although the variations of $Z\/A$ and $n_{\\rm drip}$ they found \nare nonlinear over this range of values of $L$, the variations become almost linear on the narrower range we consider (from about 37~MeV to about 69~MeV). \nAlthough it has been found that a soft symmetry energy favors neutron drip in isolated nuclei~\\cite{todd2003}, this result does not necessarily imply the observed \ncorrelation between $n_{\\rm drip}$ and $L$. Indeed, as recently discussed in Ref.~\\cite{chamel2015a}, the dripping nucleus in the crust is actually stable \nagainst neutron emission, but unstable against electron captures followed by neutron emission. Actually, as will be discussed in Sec.~\\ref{sec:results-accreting}, \naccreting neutron star crusts exhibit different correlations between $n_{\\rm drip}$ and $L$. \n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_cat_22-25_B_ndrip.eps}\n\\end{center}\n\\caption{(Color online) Neutron-drip density in nonaccreting neutron-star crusts as a function of the slope $L$ of the symmetry energy of infinite homogeneous nuclear matter at saturation\nand for different magnetic field\nstrengths, as obtained using the HFB-22 to HFB-25 Brussels-Montreal nuclear mass models~\\cite{goriely2013}.}\n\\label{fig:cat-ndrip-B}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_cat_22-25_AZ_B.eps}\n\\end{center}\n\\caption{(Color online) (a) Mass number $A$ and (b) proton fraction $Z\/A$ at the neutron-drip transition for nonaccreting neutron-star crusts as a function of the slope $L$ of the \nsymmetry energy of infinite homogeneous nuclear matter at saturation, \nas obtained using the HFB-22 to HFB-25 Brussels-Montreal nuclear mass models~\\cite{goriely2013}. Squares (circles) correspond to $B_\\star=0, 500$ and $1500$ \n($B_\\star =1000$ and $2000$).}\n\\label{fig:cat-AZ-B}\n\\end{figure*}\n\n\nThe role of the symmetry energy on the properties of the crust at the neutron-drip transition can be understood as follows. Neglecting electron-ion interactions, \nthe neutron-drip condition~(\\ref{eq:n-drip-mue}) reduces to\n\\begin{equation}\n\\label{eq:mue-approx}\n\\mu_e\\approx \\mu_e^{\\rm drip} = m_e c^2 + \\frac{A}{Z}\\left(m_n c^2- \\frac{M^\\prime(A,Z)c^2}{A}\\right)\\, .\n\\end{equation}\nFor the sake of simplicity, let us consider a two-parameter mass formula: \n\\begin{equation}\n\\label{eq:2par-ldm}\nM^\\prime(A,Z)c^2=A \\left[a_{\\rm eff}+J_{\\rm eff} \\left(1-2 \\frac{Z}{A} \\right)^2+m_u c^2 \\right] + Z m_e c^2\\, ,\n\\end{equation}\nwhere $m_u$ is the unified mass unit, $a_{\\rm eff}<0$ is the contribution from charge-symmetric matter, while the deviations introduced by \nthe charge asymmetry are embedded in the coefficient $J_{\\rm eff}>0$. Note that due to nuclear surface effects the values of these coefficients \ndo not need to be the same as their corresponding values in infinite homogeneous nuclear matter at saturation. In particular, as already discussed \nin Sec.~\\ref{sec:hfb-models}, the ``effective'' symmetry energy coefficient $J_{\\rm eff}$ is expected to be smaller than $J$, and to decrease \nwith increasing $J$ or $L$. Minimizing $g$ using Eq.~(\\ref{eq:mue-approx}) and the mass formula~(\\ref{eq:2par-ldm}), the equilibrium proton fraction \nat the neutron-drip transition is approximately given by~\\cite{chamel2012} \n\\begin{equation}\n\\label{eq:zovera-cat}\n\\frac{Z}{A}\\approx \\frac{1}{2}\\sqrt{1+\\frac{a_{\\rm eff}}{J_{\\rm eff}}}\\, .\n\\end{equation}\nThis shows that $Z\/A$ decreases with increasing $L$ (decreasing $J_{\\rm eff}$). Note that in this analysis we have not made any assumption \nregarding the magnetic field. In other words, the correlation between $Z\/A$ and $L$ (or $J$) is thus expected to be independent of the magnetic \nfield strength (at least at the level of accuracy of the simple mass formula considered here), in agreement with the results plotted in the right panel of \nFig.~\\ref{fig:cat-AZ-B}. It follows from Eq.~(\\ref{eq:2par-ldm}) that decreasing $J_{\\rm eff}$ increases $M^\\prime(A,Z)$ (the energy cost associated \nwith charge asymmetry is reduced, therefore nuclei are more bound). Using Eq.~(\\ref{eq:mue-approx}), we find that $\\mu_e^{\\rm drip}$ increases with \n$L$. Since $n_{\\rm drip}$ and $P_{\\rm drip}$ increase with $\\mu_e^{\\rm drip}$, as shown in Eqs.~(\\ref{eq:ndrip-cat-approx}) and (\\ref{eq:Pdrip-cat-approx})\nin the absence of magnetic field, and in Eqs.~(\\ref{eq:ndrip-cat-approx-b}) and (\\ref{eq:Pdrip-cat-approx-b}) in the presence of a strongly quantizing \nmagnetic field, we can thus conclude that the neutron-drip transition is shifted to higher density and pressure with increasing the symmetry energy, as \nshown in Fig.~\\ref{fig:cat-ndrip-B}. \n\n\nThe equilibrium nucleus at the neutron-drip transition is less sensitive to the symmetry energy, as previously noticed in Ref.~\\cite{bao2014} in the absence \nof magnetic fields (see their Fig.~9). This can be understood as follows. The equilibrium with respect to weak interaction processes requires \n\\begin{equation}\n\\label{eq:betaeq}\n\\mu_p+\\mu_e=\\mu_n\\, ,\n\\end{equation}\nwhere $\\mu_p$ ($\\mu_n$) is the proton (neutron) chemical potential. Substituting the neutron-drip value of the neutron chemical potential $\\mu_n=m_n c^2$ in\nEq.~(\\ref{eq:betaeq}) and using Eq.~(\\ref{eq:mue-approx}), we obtain\n\\begin{equation}\n\\label{eq:mup}\n\\mu_p-m_p c^2 = Q_{n,\\beta}+\\frac{A}{Z}\\left(\\frac{M^\\prime(A,Z)c^2}{A}-m_n c^2 \\right)\\, ,\n\\end{equation}\nwhere $m_p$ is the proton mass and $Q_{n,\\beta} = 0.782$~MeV is the $\\beta$-decay energy of the neutron. The quantity on the left-hand side of Eq.~(\\ref{eq:mup}) \nis approximately equal to the opposite of the one-proton separation energy. This shows that the equilibrium nucleus is uniquely determined by nuclear masses only, \nand is sensitive to the details of the nuclear structure. As a consequence, the predicted nucleus depends on the nuclear mass model employed (see, e.g., \nRefs.~\\cite{roca2008,pearson2011,wolf2013,kreim2013,chamel2015c}). \nTo better illustrate this point, we have plotted in Fig.~\\ref{fig:mass_diff} the differences in \nthe mass predictions between HFB-22, HFB-25, and HFB-24 mass models for two isotopic chains, corresponding to the proton number at the neutron-drip \npoint (see also Table~\\ref{tab:drip-cat}). As shown in Fig.~\\ref{fig:mass_diff}, the HFB-22 model deviates more significantly from the ``reference'' \nmodel HFB-24 than HFB-25, thus explaining the quantitative differences in the dripping nucleus. The variations of $Z$ and $A$ with $L$ we find \nappear to be more irregular than those shown in Fig.~9 of Ref.~\\cite{bao2014}. This stems from the fact that in Ref.~\\cite{bao2014} nuclear masses were \ncalculated using the semi-classical Thomas-Fermi approximation, which does not take into account pairing and shell effects contrary to the fully quantum mechanical \nmass models~\\cite{goriely2013} employed here. \n\n\nThe presence of a strong magnetic field can change the composition at the neutron-drip point, as shown in the left panel of Fig.~\\ref{fig:cat-AZ-B} \n(see also Tables~\\ref{tab:drip-cat-mag-500}-\\ref{tab:drip-cat-mag-2000}). \nHowever, such behavior is only observed for the nuclear mass model HFB-22. In particular, the equilibrium nucleus is $^{122}$Kr for $B_\\star=0, 500$ and $1500$, \nwhile for $B_\\star=1000$ and $2000$ it is $^{128}$Sr. These results can be understood as follows. As discussed in Sec.~\\ref{sec:neutron-drip}, the equilibrium \nnucleus at the neutron-drip pressure $P_{\\rm drip}$ must be such as to minimize the Gibbs free energy per nucleon, therefore we must have \n\\begin{equation}\\label{eq:stability}\ng(A,Z,P_{\\rm drip}) < g(A^\\prime,Z^\\prime,P_{\\rm drip})\\, ,\n\\end{equation}\nfor any values of $A^\\prime\\neq A$ and $Z^\\prime\\neq Z$. This condition can be approximately expressed as~\\cite{chamel2015b}\n\\begin{equation}\\label{eq:dripping-nucleus}\n\\frac{M^\\prime(A^\\prime,Z^\\prime)c^2}{Z^\\prime} - \\frac{M^\\prime(A,Z)c^2}{Z} > \\left(\\frac{A^\\prime}{Z^\\prime}-\\frac{A}{Z}\\right) m_n c^2 + C e^2 n_e^{1\/3}\\left( Z^{2\/3}-Z^{\\prime\\, 2\/3}\\right)\\, ,\n\\end{equation}\nwhere the electron density $n_e$ has to be determined from Eq.~(\\ref{eq:n-drip-mue}). Equation~(\\ref{eq:dripping-nucleus}) can be equivalently written \nas $n_e < n_e^0$, where \n\\begin{equation}\\label{eq:ne0}\nn_e^0\\equiv \\biggl[\\frac{M^\\prime(A^\\prime,Z^\\prime)c^2}{Z^\\prime} - \\frac{M^\\prime(A,Z)c^2}{Z} - \\left(\\frac{A^\\prime}{Z^\\prime}-\\frac{A}{Z}\\right) m_n c^2\\biggr]^3\n\\biggl[C e^2 \\left( Z^{2\/3}-Z^{\\prime\\, 2\/3}\\right)\\biggr]^{-3}\\, .\n\\end{equation}\nThe HFB-22 nuclear mass model predicts very similar values for the  \nthreshold electron Fermi energy $\\mu_e^{\\rm drip}$ for nuclei $^{128}$Sr and $^{122}$Kr: $24.970$ and $25.006$~MeV respectively. Substituting the theoretical \nvalues of the masses of $^{128}$Sr and $^{122}$Kr in Eq.~(\\ref{eq:ne0}) with $Z=36$, $A=122$, $Z^\\prime=38$, $A^\\prime=128$, we obtain $n_e^0\\approx 8.54\\times 10^{-5}$~fm$^{-3}$. \nDue to Landau quantization of electron motion, $n_e$ varies non-monotonically with $B_\\star$. As a consequence, the lattice term in Eq.~(\\ref{eq:dripping-nucleus}) \ncan thus become comparable to the other terms depending on $B_\\star$ to the effect that the condition (\\ref{eq:stability}) may be violated (i.e. $n_e\\geq n_e^0$), as \nshown in Fig.~\\ref{fig:drip-nuc-hfb22}. \nTransitions between $^{128}$Sr and $^{122}$Kr are found to occur at magnetic field strengths $B_\\star\\approx 861, 1239$ and $1883$. As shown in the right panel of \nFig.~\\ref{fig:cat-AZ-B}, the proton fraction $Z\/A$ is barely affected by these changes of composition. In other words, the correlation between $Z\/A$ and the symmetry \nenergy is almost independent of the magnetic field strength, as previously discussed. \n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{mass_differ_cat.eps}\n\\end{center}\n\\caption{(Color online) Difference in mass predictions for two pairs of Brussels-Montreal nuclear mass models along the two isotopic chains $Z=36$ and $Z=38$ relevant at neutron drip.}\n\\label{fig:mass_diff}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_nuc_hfb22.eps}\n\\end{center}\n\\caption{Electron density $n_e$ (solid line) at the neutron-drip transition in nonaccreting neutron-star crusts as a \nfunction of the magnetic field strength, using the Brussels-Montreal nuclear mass model HFB-22 and assuming that the neutron-drip nucleus \nis $^{122}$Kr, as in the absence of magnetic field. The horizontal dotted line represents $n_e^0$, as given by Eq.~(\\ref{eq:ne0}). \nSee text for details. }\n\\label{fig:drip-nuc-hfb22}\n\\end{figure*}\n\n\n\\begin{table}\n\\centering\n\\caption{Neutron-drip transition in the crust of nonaccreting and unmagnetized neutron stars, as predicted by the HFB-22 to HFB-25 Brussels-Montreal nuclear mass models: mass and atomic numbers of the dripping nucleus, baryon density, and corresponding pressure. }\\smallskip\n\\label{tab:drip-cat}\n\\begin{tabular}{ccccc}\n\\hline\n & $A$ & $Z$ & $n_{\\rm drip}$ ($10^{-4}$ fm$^{-3}$) & $P_{\\rm drip}$ ($10^{-4}$ MeV~fm$^{-3}$)\\\\\n\\hline \\noalign {\\smallskip}\nHFB-22 & 122 & 36 & 2.71 & 4.99  \\\\\nHFB-23 & 126 & 38 & 2.63 & 4.93  \\\\\nHFB-24 & 124 & 38 & 2.56 & 4.87  \\\\\nHFB-25 & 122 & 38 & 2.51 & 4.83 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\begin{table}\n\\centering\n\\caption{Neutron-drip transition in the crust of nonaccreting magnetized neutron stars with $B_\\star=500$, as predicted by the HFB-22 to HFB-25 Brussels-Montreal nuclear mass models: mass and atomic numbers of the dripping nucleus, baryon density, and corresponding pressure.}\\smallskip\n\\label{tab:drip-cat-mag-500}\n\\begin{tabular}{ccccc}\n\\hline\n & $A$ & $Z$ & $n_{\\rm drip}$ ($10^{-4}$ fm$^{-3}$) & $P_{\\rm drip}$ ($10^{-4}$ MeV~fm$^{-3}$)\\\\\n\\hline \\noalign {\\smallskip}\nHFB-22 & 122 & 36 & 2.74 & 5.52 \\\\\nHFB-23 & 126 & 38 & 2.66 & 5.45 \\\\\nHFB-24 & 124 & 38 & 2.61 & 5.39 \\\\\nHFB-25 & 122 & 38 & 2.55 & 5.35 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-cat-mag-500} but for $B_\\star=1000$.}\\smallskip\n\\label{tab:drip-cat-mag-1000}\n\\begin{tabular}{ccccc}\n\\hline\n & $A$ & $Z$ & $n_{\\rm drip}$ ($10^{-4}$ fm$^{-3}$) & $P_{\\rm drip}$ ($10^{-4}$ MeV~fm$^{-3}$)\\\\\n\\hline \\noalign {\\smallskip}\nHFB-22 & 128 & 38 & 3.06 & 6.70 \\\\\nHFB-23 & 126 & 38 & 2.98 & 6.63 \\\\\nHFB-24 & 124 & 38 & 2.91 & 6.56 \\\\\nHFB-25 & 122 & 38 & 2.85 & 6.52 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-cat-mag-500} but for $B_\\star=1500$.}\\smallskip\n\\label{tab:drip-cat-mag-1500}\n\\begin{tabular}{ccccc}\n\\hline\n & $A$ & $Z$ & $n_{\\rm drip}$ ($10^{-4}$ fm$^{-3}$) & $P_{\\rm drip}$ ($10^{-4}$ MeV~fm$^{-3}$)\\\\\n\\hline \\noalign {\\smallskip}\nHFB-22 & 122 & 36 & 2.30 & 8.66 \\\\\nHFB-23 & 126 & 38 & 2.24 & 8.60 \\\\\nHFB-24 & 124 & 38 & 2.20 & 8.56 \\\\\nHFB-25 & 122 & 38 & 2.16 & 8.52 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-cat-mag-500} but for $B_\\star=2000$.}\\smallskip\n\\label{tab:drip-cat-mag-2000}\n\\begin{tabular}{ccccc}\n\\hline\n & $A$ & $Z$ & $n_{\\rm drip}$ ($10^{-4}$ fm$^{-3}$) & $P_{\\rm drip}$ ($10^{-3}$ MeV~fm$^{-3}$)\\\\\n\\hline \\noalign {\\smallskip}\nHFB-22 & 128 & 38 & 3.06 & 11.6 \\\\\nHFB-23 & 126 & 38 & 3.00 & 11.6 \\\\\nHFB-24 & 124 & 38 & 2.95 & 11.5 \\\\\nHFB-25 & 122 & 38 & 2.89 & 11.4 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Accreting neutron stars}\n\\label{sec:results-accreting}\n\nFor accreting neutron-star crusts, we have studied the neutron-drip transition as explained in Sec.~\\ref{sec:neutron-drip} (see also Ref.~\\cite{chamel2015a}). \nWe have considered different initial compositions: the ashes produced by an $rp$-process during an x-ray burst \\cite{schatz2001}, and the ashes produced by \nsteady state hydrogen and helium burning \\cite{schatz2003} as expected to occur during superbursts \\cite{gupta2007}. After determining the dripping nucleus, we \nhave calculated the neutron-drip density and pressure by solving numerically Eq.~(\\ref{eq:e-capture+n-emission-gibbs-approx}) considering all possible \nneutron-emission processes. Results are summarized in Tables~\\ref{tab:drip-acc-22}-\\ref{tab:drip-acc-25} for different nuclear mass models. \nFor comparison with previous works~\\cite{hz1990, hz2003}, we have also considered ashes of x-ray bursts consisting of pure $^{56}$Fe. Results are indicated in \nTable~\\ref{tab:drip-acc-a56}. \n\nAs already pointed out in Ref.~\\cite{chamel2015a}, for a given nuclear mass model the neutron-drip transition in accreting neutron stars can occur at \neither lower or at higher densities and pressures than in nonaccreting neutron stars. Depending on the mass model adopted, the neutron-drip density thus \nranges from $1.60 \\times 10^{-4}$~fm$^{-3}$ to $3.90 \\times 10^{-4}$~fm$^{-3}$, \nand the corresponding pressure from $2.77 \\times 10^{-4}$~MeV~fm$^{-3}$ to $7.77 \\times 10^{-4}$~MeV~fm$^{-3}$. The numerical results \nobtained solving Eq.~(\\ref{eq:e-capture+n-emission-gibbs-approx}) are reproduced by the analytical formulas~(\\ref{eq:ndrip-acc}) and \n(\\ref{eq:Pdrip-acc}) with an error of $0.1\\%$ at most. \n\nThe change of the neutron-drip density with the slope $L$ of the symmetry energy is found to be very different from that obtained in \nnonaccreting neutron-star crusts. As shown in Fig.~\\ref{fig:acc-ndrip}, for some ashes like $^{104}$Cd, no obvious correlation is observed while for \nother ashes like $^{66}$Ni, $n_{\\rm drip-acc}$ appears to be \\emph{anticorrelated} with $L$: $n_{\\rm drip-acc}$ decreases with increasing $L$. \nThis behavior can be understood as follows. Ignoring electron-ion interactions, the neutron-drip condition~(\\ref{eq:e-capture+n-emission-gibbs-approx}) \nreduces to\n\\begin{equation}\n\\mu_e\\approx \\mu_e^{\\rm drip-acc} = M^\\prime(A-\\Delta N,Z-1)c^2-M^\\prime(A,Z)c^2 +m_n c^2 \\Delta N + m_e c^2\\, , \n\\end{equation}\nwhich can be more conveniently written as \n\\begin{equation}\\label{eq:muedrip-acc-s}\n\\mu_e^{\\rm drip-acc} = S_{\\Delta N n}(A,Z-1) + \\mu_e^\\beta(A,Z)\\, ,\n\\end{equation}\nusing Eq.~(\\ref{eq:sn-dn}) and introducing the threshold electron Fermi energy for the onset of electron captures (see, e.g. Ref.~\\cite{chamel2015d} for a recent \ndiscussion)\n\\begin{equation}\n\\mu_e^\\beta(A,Z)=M^\\prime(A,Z-1)c^2 -M^\\prime(A,Z)c^2 + m_e c^2\\, .\n\\end{equation}\nThe mass difference $\\Delta M^\\prime=M'(A,Z-1) -M'(A,Z)$, which represents the change of mass associated with the substitution of a proton by a neutron, \nis expected to be mainly determined by symmetry energy effects. On the other hand, the $\\Delta N$-neutron separation energy $S_{\\Delta N n}(A,Z-1)$ \nis likely to be more dependent on the details of the nuclear structure than on the symmetry energy. As shown in Tables~\\ref{tab:drip-acc-22}-\\ref{tab:drip-acc-a56}, \n$|S_{\\Delta N n}(A,Z-1)| \\ll \\Delta M^\\prime c^2$ therefore $\\mu_e^{\\rm drip-acc}\\approx \\mu_e^\\beta$. On the other hand, as discussed in Sec.~\\ref{sec:neutron-drip}, \nthe composition of accreting neutron-star crusts at the neutron-drip transition is directly determined by the condition $S_{\\Delta N n}(A,Z-1)<0$. Provided \nthe dependence on the symmetry energy of $S_{\\Delta N n}(A,Z-1)$ is weak enough, the dripping nucleus will thus be independent of $L$. In this case, the \nvariations of $S_{\\Delta N n}(A,Z-1)$ with $L$ are typically much smaller than the variations of $\\Delta M^\\prime c^2$, as shown in Figs.~\\ref{fig:deltamprime} and \n\\ref{fig:Sdeltan}. Using the simple mass formula~(\\ref{eq:2par-ldm}), we find\n\\begin{equation}\n\\mu_e^\\beta(A,Z)=\\Delta M^\\prime c^2 +m_e c^2 \\approx 4J_{\\rm eff}\\left(1+\\frac{1-2Z}{A}\\right)\\, .\n\\end{equation}\nThis means that with increasing $J$ or $L$ (decreasing $J_{\\rm eff}$), $\\Delta M^\\prime$ and $\\mu_e^\\beta(A,Z)$ both \\emph{decrease}. It thus follows from \nEqs.~(\\ref{eq:ndrip-acc}), (\\ref{eq:Pdrip-acc}), and Eqs.~(\\ref{eq:muedrip-acc-s}), that $\\mu_e^{\\rm drip-acc}$, $n_{\\rm drip-acc}$ and $P_{\\rm drip-acc}$ \nalso decrease with $L$, as shown in the upper panel of Fig.~\\ref{fig:acc-ndrip}. \nIn the peculiar case of $A=105$, the rather low value of $\\Delta M^\\prime c^2$ predicted by HFB-25 is compensated by a comparatively high value $S_{\\Delta N n}(A,Z-1)$, \nas can be seen in Figs.~\\ref{fig:deltamprime} and \\ref{fig:Sdeltan}. As a result, $n_{\\rm drip-acc}$ is still found to decrease with increasing $L$ despite the nonmonotonic \nvariation of $\\Delta M^\\prime$, as shown in Fig.~\\ref{fig:acc-ndrip}. \nFor some ashes, the variations of $S_{\\Delta N n}(A,Z-1)$ are comparable to those of $\\Delta M^\\prime c^2$, and large enough to even change the composition. This  \nleads to nonmonotonic variations of the neutron-drip density and pressure, as illustrated in the lower panel of Fig.~\\ref{fig:acc-ndrip}. In these cases, \neffects other than the symmetry energy play a role. The anticorrelation between $n_{\\rm drip-acc}$ (or $P_{\\rm drip-acc}$) and $L$ thus relies to \na large extent on the importance of nuclear structure effects far from the stability valley. \n\n\\begin{table}\n\\centering\n\\caption{Neutron-drip transition in the crust of accreting neutron stars, as predicted by the HFB-22 Brussels-Montreal nuclear mass model: mass and atomic numbers of the dripping nucleus, number of emitted neutrons, baryon density $n_{\\rm drip-acc}$  ($10^{-4}$ fm$^{-3}$),  and corresponding pressure $P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$), $S_{\\Delta N n}(A,Z-1)$ (MeV), $\\Delta M^\\prime$ (MeV$\/c^2$), and $\\mu_e^{\\rm drip-acc}$ (MeV).\nThe mass numbers $A$ are listed from top to bottom considering that the ashes are produced by ordinary\nx-ray bursts (upper panel) or superbursts (lower panel). See text for details.}\\smallskip\n\\label{tab:drip-acc-22}\n\\begin{tabular}{cccccccc}\n\\hline\n  $A$   & $ Z$  & $\\Delta N$&  $n_{\\rm drip-acc}$ ($10^{-4}$ fm$^{-3}$)  & $P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$) & $S_{\\Delta N n}$ (MeV) & $\\Delta M^\\prime$ (MeV$\/c^2$) & $\\mu_e^{\\rm drip-acc}$ (MeV) \\\\\n \\hline\n 104  &   32 &  1 &   2.71 &  5.31 & -0.79 & 25.14 & 24.86 \\\\\n 105  &   33 &  1 &   1.90 &  3.40 & -1.00 & 22.70 & 22.21 \\\\\n  68  &   22 &  1 &   2.31 &   4.64 & -0.28 & 24.14 & 24.37 \\\\\n  64  &   18 &  5 &   3.90 &   7.77 & -1.85 & 29.23 & 27.89 \\\\\n  72  &   22 &  1 &   2.89 &   5.78 & -0.31 & 25.55 & 25.75 \\\\\n  76  &   24 &  1 &   2.86 &   5.95 & -0.17 & 25.52 & 25.86 \\\\\n  98  &   32 &  1 &   1.93 &   3.66 & -0.20 & 22.34 & 22.65 \\\\\n 103  &   33 &  1 &   1.60 &  2.77 & -0.02 & 20.62 & 21.11 \\\\\n 106  &   32 &  1 &   2.92 &  5.71 & -0.69 & 25.50 & 25.32 \\\\\n  \\hline\n  66  &   22 &  1 &   1.99 &   3.95 & -0.19 & 23.09 & 23.41 \\\\\n  64  &   18 &  5 &   3.90 &   7.77 & -1.85 & 29.23 & 27.89 \\\\\n  60  &   20 &  1 &   1.83 &   3.55 & -1.67 & 24.02 & 22.86 \\\\\n \\end{tabular}\n \\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-acc-22} but for the HFB-23 Brussels-Montreal nuclear mass model.}\\smallskip\n\\label{tab:drip-acc-23}\n\\begin{tabular}{cccccccc}\n\\hline\n   $A$   &  $Z$  & $\\Delta N$&  $n_{\\rm drip-acc}$ ($10^{-4}$ fm$^{-3}$)  & $P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$) & $S_{\\Delta N n}$ (MeV) & $\\Delta M^\\prime$ (MeV$\/c^2$) & $\\mu_e^{\\rm drip-acc}$ (MeV) \\\\\n \\hline\n 104  &   32 &  1 &   2.83 &  5.62 & -1.02 & 25.73 & 25.22 \\\\\n 105  &   33 &  1 &   1.97 &  3.57 & -1.33 & 23.30 & 22.48 \\\\\n  68  &   22 &  1 &   2.35 &   4.73 & -0.56 & 24.54 & 24.49 \\\\\n  64  &   20 &  1 &   3.27 &   7.04 & -0.13 & 26.75 & 27.13 \\\\\n  72  &   22 &  1 &   3.06 &   6.24 & -0.11 & 25.84 & 26.24\\\\\n  76  &   24 &  1 &   2.94 &   6.18 & -0.38 & 25.98 & 26.11 \\\\\n  98  &   32 &  1 &   1.98 &   3.77 & -0.50 & 22.82 & 22.83 \\\\\n 103  &   31 &  1 &   2.45 &  4.50 & -0.91 & 24.29 & 23.89 \\\\\n 106  &   34 &  1 &   2.11 &  4.02 & -0.009 & 22.63 & 23.13 \\\\\n  \\hline\n  66  &   22 &  1 &   2.07 &   4.16 & -0.23 & 23.43 & 23.71 \\\\\n  64  &   20 &  1 &   3.27 &   7.04 & -0.13 & 26.75 & 27.13 \\\\\n  60  &   20 &  1 &   1.93 &   3.79 & -1.77 & 24.50 & 23.24 \\\\\n \\end{tabular}\n \\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-acc-22} but for the HFB-24 Brussels-Montreal nuclear mass model.}\\smallskip\n\\label{tab:drip-acc-24}\n\\begin{tabular}{cccccccc}\n\\hline\n  $A$   &  $Z$  & $\\Delta N$&  $n_{\\rm drip-acc}$ ($10^{-4}$ fm$^{-3}$)  & $P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$) & $S_{\\Delta N n}$ (MeV) & $\\Delta M^\\prime$ (MeV$\/c^2$) & $\\mu_e^{\\rm drip-acc}$ (MeV) \\\\\n \\hline\n 104  &   32 &  1 &   2.87 &  5.73 & -1.49 & 26.32 & 25.34 \\\\\n 105  &   33 &  1 &   2.10 &  3.89 & -1.11 & 23.57 & 22.97 \\\\\n  68  &   22 &  1 &   2.45 &   5.00 & -0.75 & 25.07 & 24.83 \\\\\n  64  &   22 &  1 &   1.66 &   3.22 & -0.07 & 21.81 & 22.25 \\\\\n  72  &   22 &  1 &   3.08 &   6.28 & -0.29 & 26.07 & 26.29 \\\\\n  76  &   24 &  1 &   3.10 &   6.61 & -0.46 & 26.50 & 26.55 \\\\\n  98  &   32 &  1 &   2.04 &   3.94 & -0.38 & 22.95 & 23.08 \\\\\n 103  &   31 &  3 &   2.49 &  4.59 & -0.89 & 24.39 & 24.01 \\\\\n 106  &   34 &  1 &   2.17 &  4.16 & -1.10 & 23.92 & 23.33 \\\\\n  \\hline\n  66  &   22 &  1 &   2.09 &   4.22 & -0.29 & 23.58 & 23.80 \\\\\n  64  &   22 &  1 &   1.66 &   3.22 & -0.07 & 21.81 & 22.25 \\\\\n  60  &   20 &  1 &   2.03 &   4.05 & -1.66 & 24.78 & 23.63 \\\\\n \\end{tabular}\n \\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Same as in Table~\\ref{tab:drip-acc-22} but for the HFB-25 Brussels-Montreal nuclear mass model.}\\smallskip\n\\label{tab:drip-acc-25}\n\\begin{tabular}{cccccccc}\n\\hline\n  $A$   &  $Z$  & $\\Delta N$&  $n_{\\rm drip-acc}$ ($10^{-4}$ fm$^{-3}$)  & $P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$) & $S_{\\Delta N n}$ (MeV) & $\\Delta M^\\prime$ (MeV$\/c^2$) & $\\mu_e^{\\rm drip-acc}$ (MeV) \\\\\n \\hline\n 104  &   34 &  1 &  2.02 &  3.87 & -0.009 & 22.42 & 22.92 \\\\\n 105  &   33 &  1 &  2.18 &  4.07 & -0.57 & 23.30 & 23.24 \\\\\n  68  &   22 &  1 &   2.59 &  5.40 & -0.75 & 25.55 & 25.31 \\\\\n  64  &   22 &  1 &   1.79 &  3.59 & -0.18 & 22.52 & 22.85 \\\\\n  72  &   22 &  1 &   3.27 &  6.82 & -0.55 & 26.87 & 26.83 \\\\\n  76  &   24 &  1 &   3.14 &  6.73 & -0.58 & 26.74 & 26.67 \\\\\n  98  &   32 &  1 &   2.11 &  4.12 & -0.64 & 23.47 & 23.34 \\\\\n 103  &   33 &  1 &  1.78 &  3.18 & -0.76 & 22.10 & 21.85 \\\\\n 106  &   34 &  1 &  2.25 &  4.36 & -0.05 & 23.15 & 23.61 \\\\\n  \\hline\n  66  &   22 &  1 &   2.20 &   4.52 & -0.22 & 23.92 & 24.21 \\\\\n  64  &   22 &  1 &   1.79 &   3.59 & -0.18 & 22.52 & 22.85 \\\\\n  60  &   20 &  1 &   2.08 &   4.21 & -1.94 & 25.28 & 23.85 \\\\\n \\end{tabular}\n \\end{table}\n\n\n\\begin{table}\n\\centering\n\\caption{Neutron-drip transition in the crust of accreting neutron stars, as predicted by different Brussels-Montreal nuclear mass models for $^{56}$Fe ashes: atomic number $Z$ of the dripping nucleus, number of emitted neutrons, density and corresponding pressure, $S_{\\Delta N n}(A,Z-1)$, $\\Delta M^\\prime$ (MeV$\/c^2$), and $\\mu_e^{\\rm drip-acc}$ (MeV). See text for details.}\\smallskip\n\\label{tab:drip-acc-a56}\n\\begin{tabular}{ccccc}\n\\hline\n                                              & HFB-22 & HFB-23  & HFB-24 &  HFB-25    \\\\\n\\hline\n$Z$                                           &   18   &  18     &  18    &  18   \\\\\n$\\Delta N$                                    &    1   &   1     &   3    &  1     \\\\\n$n_{\\rm drip-acc}$ ($10^{-4}$ fm$^{-3}$) & 2.49   &  2.58   &  2.73  &  2.84   \\\\\n$P_{\\rm drip-acc}$ ($10^{-4}$ MeV~fm$^{-3}$)  & 5.10   &  5.34   &  5.77  &  6.07   \\\\\n$S_{\\Delta N n}$ (MeV) & -1.17 & -1.32 & -3.27 & -1.61 \\\\\n$\\Delta M^\\prime$ (MeV$\/c^2$) & 25.76 & 26.20 & 28.64 & 27.32 \\\\\n$\\mu_e^{\\rm drip-acc}$ (MeV) & 25.10 & 25.39 & 25.88 & 26.22  \\\\\n\\end{tabular}\n\\end{table}\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_acc_22-25_ndrip_new.eps}\n\\end{center}\n\\caption{(Color online) Neutron-drip density as a function of the slope $L$ of the symmetry energy of infinite homogeneous nuclear matter at saturation, as obtained using the HFB-22 to HFB-25 Brussels-Montreal nuclear mass \nmodels, for accreting neutron-star crusts with different initial composition of ashes (see text for details).}\n\\label{fig:acc-ndrip}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_acc_22-25_deltam.eps}\n\\end{center}\n\\caption{(Color online) Mass difference $\\Delta M^\\prime$ (in units of MeV$\/c^2$) as a function of the slope $L$ of the symmetry energy of infinite homogeneous nuclear matter at saturation, as obtained using the \nHFB-22 to HFB-25 Brussels-Montreal nuclear mass models, for accreting \nneutron-star crusts with different initial composition of ashes (see text for details).}\n\\label{fig:deltamprime}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.45]{drip_acc_22-25_Sdeltan.eps}\n\\end{center}\n\\caption{(Color online) $\\Delta N$-neutron separation energy $S_{\\Delta N n}$ (in units of MeV) as a function of the slope $L$ of the symmetry energy of infinite homogeneous nuclear matter at saturation, as obtained using the \nHFB-22 to HFB-25 Brussels-Montreal nuclear mass models, for accreting \nneutron-star crusts with different initial composition of ashes (see text for details).}\n\\label{fig:Sdeltan}\n\\end{figure*}\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have studied the role of the symmetry energy on the neutron-drip transition in both accreting and nonaccreting neutron-star crusts. We have also allowed for  \nthe presence of a strong magnetic field, as in magnetars. The masses of nuclei encountered in this region of the neutron-star crust are experimentally unknown. \nFor this reason, we have employed a recent family of microscopic nuclear mass models, from HFB-22 to HFB-25, developed by the Brussels-Montreal collaboration~\\cite{goriely2013}. These models provide equally good fits to the 2353 measured masses of nuclei with $N$ and $Z \\geq 8$ from the 2012 Atomic Mass \nEvaluation~\\cite{audi2012}, with a root-mean-square deviation of about $0.6$~MeV. \nOn the other hand, these models lead to different predictions for the behavior of the symmetry energy in infinite homogeneous\nnuclear matter.\nIn particular, these functionals \nwere constrained so as to yield different \nvalues of the symmetry energy at saturation, from $J=29$~MeV to $J=32$~MeV, the slope of the symmetry energy ranging from $L=37$~MeV to $L=69$~MeV. \n\nFor nonaccreting weakly magnetized neutron stars, the neutron-drip density $n_{\\rm drip}$ is found to increase almost linearly with $L$ (or equivalently with $J$) \nwhile the proton fraction $Z\/A$ decreases, in agreement with previous studies~\\cite{bao2014} (see also Refs.~\\cite{roca2008,provid2014,grill2014}). \nIn the presence of a strong magnetic field, the dripping nucleus\nhence also $Z\/A$ is unchanged, as found in Refs.~\\cite{chamel2012,chamel2015b}, for all models but HFB-22. In this case, the dripping nucleus alternates between \n$^{122}$Kr and $^{128}$Sr depending on the magnetic field strength. This peculiar behavior arises from Landau quantization of electron motion and from the fact \nthat the threshold electron Fermi energy $\\mu_e^{\\rm drip}$ are almost equal. Their proton fraction are also very similar so that all in all the linear \ncorrelation between $Z\/A$ and $L$ is hardly affected by the magnetic field. The neutron-drip density $n_{\\rm drip}$ exhibits typical quantum oscillations as \na function of the magnetic field strength, as recently discussed in Ref.~\\cite{chamel2015b}. Still, $n_{\\rm drip}$ remains linearly correlated with $L$. \nAlthough a soft symmetry energy favors neutron drip in isolated nuclei~\\cite{todd2003}, this result does not necessarily imply the observed correlation \nbetween $n_{\\rm drip}$ and $L$. Indeed, as recently discussed in Ref.~\\cite{chamel2015a}, the dripping nucleus in the crust is actually stable \nagainst neutron emission, but unstable against electron captures followed by neutron emission. In fact, such a correlation is not found in accreting neutron-star crusts. \nDepending on the initial composition of the ashes from x-ray bursts and superbursts, $n_{\\rm drip}$ \n\\emph{decreases} almost linearly with increasing $L$ while the dripping nucleus remains the same. In other cases, the symmetry energy does not seem \nto play any role. \n\nWe have qualitatively explained these different behaviors using a simple mass formula, and making use of the analytical expressions for the neutron-drip density and \npressure obtained in Refs.~\\cite{chamel2015a,chamel2015b}. In particular, we have shown that the anticorrelation between $n_{\\rm drip}$ and $L$ in accreting \nneutron stars depends to a large extent to the relative importance of nuclear structure effects (like shell effects and pairing) and symmetry energy effects on the \nneutron separation energy. More precisely, the anticorrelation is broken whenever the differences between the neutron separation energies predicted by the different \nmass models are large enough to change the dripping nucleus. \n\nIn any case, the composition of the deepest layers of the outer crust of a neutron star is very sensitive to the details of the nuclear structure far from the stability valley. In nonaccreting neutron-star crusts, the neutron-drip transition is mainly governed by the values of the masses of very neutron-rich strontium and krypton isotopes. \nAlthough the masses of these nuclei have not yet been measured, the composition of nonaccreting neutron-star crusts has been recently constrained by experiment \nto deeper layers~\\cite{wolf2013}. The nuclei thought to be present in accreting neutron star crusts span a much larger region of the nuclear chart, depending on \nthe ashes from x-ray bursts and superbursts. In these neutron stars, the neutron-drip transition is not directly determined by nuclear masses but rather by some \ncombinations of masses, namely the (multiple) neutron separation energies and the isobaric two-point mass differences. \n\nThe onset of neutron emission by nuclei marks the transition to the inner region of the neutron-star crust, where neutron-proton clusters coexist with a neutron liquid. \nIn turn, this neutron liquid, which becomes superfluid at low enough temperatures, is expected to play a role in various observed astrophysical phenomena  (see, e.g., \nRef.~\\cite{chamelhaensel2008} for a review) like sudden spin-ups and spin-downs (so-called ``glitches'' and ``antiglitches'', respectively)~\\cite{dib2008, gug2014, \narchibald2013, sasmaz2014, duncan2013, kantor2014}, quasiperiodic oscillations detected in the giant flares from soft $\\gamma$-ray repeaters~\\cite{passamonti2014}, \ncooling of strongly magnetized neutron stars~\\cite{aguilera2009}, deep crustal heating (most of the heat being released near the neutron-drip transition~\\cite{haensel2008}), \nand the thermal relaxation of quasipersistent soft x-ray transients~\\cite{shternin2007,brown2009,page2013}. By shifting the neutron-drip transition to higher or lower densities, the symmetry energy may thus leave its imprint on these astrophysical phenomena.  \n\n\n\n\\begin{acknowledgments}\nThis work has been mainly supported by Fonds de la Recherche Scientifique - FNRS (Belgium), and by the bilateral project between Fonds de la Recherche Scientifique - FNRS (Belgium), Wallonie-Bruxelles-International (Belgium) and the Bulgarian Academy of Sciences. \nThis work has been also partially supported by the Bulgarian National Science Fund under Contract No. DFNI-T02\/19t and the Cooperation in Science and Technology (COST Action) MP1304 ``NewCompStar''. The authors would like to thank J.~M. Pearson for very fruitful discussions.\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe mechanical response of materials to deformations is described by\nthe elasticity theory \\cite{lanlif}. The simplest homogeneous (affine) \ndeformation of  a continuum can be expressed by a linear dependence \nof the distorted position ${\\bf r}$ on the original position of \nthat point ${\\bf R}$ via relation \n\\begin{equation}\\label{def:M}\nr_i=M_{ij}R_j\\ ,\n\\end{equation}\nwhere $M_{ij}$  is a {\\em constant} tensor. We will consider both \ntwo-dimensional (2D) and three-dimensional (3D) systems, in which\nLatin subscripts will indicate Cartesian coordinates. (Summation over\nrepeated Latin subscripts is implied.) Tensor $M_{ij}$ can be separated \ninto an identity tensor $I$ and a non-trivial part\n\\begin{equation}\\label{def:epsilon}\nM_{ij}=\\delta_{ij}+\\epsilon_{ij}\\ .\n\\end{equation}\nIn general, $\\epsilon_{ij}$ can be separated into a {\\em symmetric tensor}\nrepresenting deformation and an anti-symmetric tensor representing\nrotation. We neglect the rotation and assume that  \n$\\epsilon_{ij}=\\epsilon_{ji}$. While the tensor $\\epsilon_{ij}$\nhas a convenient meaning in actual experiments, usually the\nelastic deformations are formulated in the terms of the\n{\\em Lagrangian strain tensor} $\\eta_{ij}$, which defines the\nchange in the distances between the points \\cite{etausage}:\n\\begin{equation}\\label{def:eta}\nr_ir_i=M_{ik}M_{il}R_kR_l\\equiv(\\delta_{kl}+2\\eta_{kl})R_kR_l\\ .\n\\end{equation}\n\nIn the case of affine deformations the definitions in \nEqs. \\ref{def:M}--\\ref{def:eta} are valid for arbitrarily values of\n$\\epsilon_{ij}$ and $\\eta_{ij}$. However, we will further assume that the\ndeformations are small. For a homogeneous continuum, it suffices\nto apply the deformation described by Eq.~\\ref{def:M} at the\n{\\em boundaries} of the system to assure that the same equation\ndescribes every internal point, while in the case of inhomogeneous\nsystem, application of such a deformation to the boundaries assures\nthat the {\\em mean} deformation is equal to $\\epsilon_{ij}$ \\cite{hashin}. \n\nElastic properties of a condensed matter system\ndescribe the energetic cost of a deformation. However, real system\nconsisting of many moving atoms\/molecules cannot be simply represented\nby a strain tensor assigned to every point in space. Rather, we can\nassume that the {\\em boundaries} of such system undergo an affine\ndeformation described by Eq.~\\ref{def:M}. In such a case, the mean \nfree energy density, $f$, which is the free energy $F$ divided by the\noriginal (unstrained) volume $V_0$ of the system, can be expanded \nin a power series in the strain variables\n\\begin{equation}\\label{def:C}\nf(\\{\\eta\\})=f(\\{0\\})+\\sigma_{ij}\\eta_{ij} +\\frac{1}{2}\nC_{ijkl}\\eta_{ij} \\eta_{kl}+\\ldots\n\\end{equation}\nThe coefficients in this expansion are the stress tensor\n$\\sigma_{ij}$ and the tensor of the (second order) elastic \nconstants $C_{ijkl}$, characterizing a given material. \nIn the case of isotropic pressure $p$, the stress can be\nwritten as $\\sigma_{ij}=-p\\delta_{ij}$. Elastic\nconstants may serve as an indicator of instabilities\nassociated with phase transitions \\cite{birch,zhou}. \n\nElastic response of a system to a deformation can be determined\nwithout actually distorting the system, since equilibrium\ncorrelation functions contain all the necessary information.\nIndeed almost four decades ago Squire, Holt and Hoover (SHH) \\cite{shh}\ndeveloped a formalism that extended the theory of elasticity of\nBorn and Huang \\cite{born} to a finite temperature situation\nand expressed the elastic properties of a system as thermal\naverages of various derivatives of interparticle potentials.\nIn a certain sense the formalism is an extension of the virial\ntheorem \\cite{virtheorem} which relates the thermal averages\nof the products of interparticle forces and the interparticle \nseparations to the stress tensor. (Similar formalism enables\nevaluation of the elastic properties of 2D membranes in 3D space\n\\cite{farago_pincus}.) The SHH method is very well \nadapted for use in numerical simulations in constant volume (and \nshape) ensembles. Other methods, extracting the elastic properties\nfrom shape and volume changes of systems, have also been developed\nand extensively used \\cite{other}.\n\nUsually molecules are not spherically symmetric and we may expect\ninteractions that depend on the orientation of the molecules. The\nintroduction of rotational degrees of freedom into the theoretical\ndescription of a system has an interesting effect on the stress and \nelastic constants. At very low densities (almost ideal gas) the\nrotational degrees of freedom add a large contribution to the\ntotal kinetic energy of a molecule, but do not contribute to the\npressure. In a thermodynamical treatment of stress we are interested\nin the translational degrees of freedom. However, for \nnon-spherically-symmetric potentials, the particle rotation may \nhave a significant indirect contribution even at moderate densities. \nAt larger densities phase diagram may be strongly influenced by\nthose degrees of freedom.\nThe general approach of SHH \\cite{shh} (see also Refs.\n\\cite{zhou,bavaud}) was applied in a detailed form to the \ncase of particles interacting via central two-body forces.\nHowever, the formalism can be easily extended to the systems of\nparticles interacting via non-central potentials, as will be shown\nin Section \\ref{sec:softelast}. \n\nModelling of various systems frequently involves particles that \ninteract  via {\\em hard} potentials which are either 0 or $\\infty$: \nin fact simulation of 2D hard disk system dates back to the origins \nof the Metropolis Monte Carlo (MC) method \\cite{origMetro}. An obvious \nreason for the use of such potentials in simulations is their numerical \nsimplicity. However, there are important physical reasons for such \nmodels: in many situations entropy plays a dominant role \nin physical processes, and the absence of energy scale in hard\npotentials ``brings out'' the entropic features of the behavior.\nHard sphere systems have been the subject of an intensive research\nfor several decades now (see \\cite{gast} and references therein).\nThey serve as the simplest models for real fluids, glasses, and\ncolloids. The phase diagram of hard spheres is well known. In\n3D this system undergoes an entropically driven first-order phase \ntransition from liquid to solid phase \\cite{hsfreezing}. Elastic \nconstants of such solids have been explored in the past \n\\cite{numres,runge}. Entropy also plays a crucial role in the \nsystems containing long polymers, such as gels and rubbers \n\\cite{polentropA,degennes_polymer,polentropB}. \nNot surprisingly, hard sphere potentials have been extensively\nused to represent excluded volume interactions between the\nmonomers (see \\cite{baum} and references therein).\nKantor {\\it et al.} \\cite{kkn} introduced {\\it tethering \npotential}, that has no\nenergy but limits the distance between bonded monomers, to\nrepresent covalent bonds in polymeric systems. Such hard potential\ncombined with hard sphere repulsion can be used to simulate a\nvariety of polymeric systems.  Recently, Farago and Kantor \n\\cite{fk_formalism} adapted the formalism of SHH \\cite{shh} to\nhard potentials.  This new formalism enabled a study of a sequence \nof entropy-dominated systems, such as 2D \\cite{fk_2D} and 3D \n\\cite{fk_3D} gels near sol-gel transition and other systems \\cite{fk_net}.\n\nOrientation of non-spherically-symmetric molecules plays a\ncrucial role in the properties of liquid crystals\n\\cite{degennes_liquid}. For instance, the nematic phase is \ntranslationally\ndisordered but it has orientational order of the molecules. From\nthe early stages of the liquid crystal research it has been\nrealized that the {\\it entropic} part of the free energy related\nto non-spherical shapes of the molecules, by itself, can explain\nmany of the properties of the systems \\cite{ons}.  Not\nsurprisingly, hard potentials were frequently used to investigate\nthe properties of liquid crystals. Even such simplifications as\ninfinitely thin disks \\cite{frenkel_thindisk} or infinitely thin\nrods \\cite{frenkel_thinrod} provide valuable insights into the\nproblem. A slightly more realistic picture is provided by hard\nspheroids \\cite{fmm}.  Such simulations were primarily motivated\nby the desire to understand the liquid phases. However,\ntwo interesting {\\it solid phases} have been detected: \nboth phases are translationally ordered, but only one\nof them has orientational order of spheroids. The orientational \norder is absent only when the spheroid resembles a sphere. \nSufficiently oblate or prolate spheroids are orientationally \nordered in the solid phase. During the last twenty\nyears hard spheroids have be studied in great detail\n\\cite{spheroids}. A similar hard potential system that is suitable\nfor the study of the liquid crystals is a collection of hard\nspherocylinders (cylinders capped at their ends by hemispheres).\nThese molecules have slightly more complex phase diagram (which\nincludes smectic-A liquid-crystalline phase), and also have been\nstudied in great detail \\cite{spherocyl}. Like spheroids, they\nhave two solid phases. (Spherocylinders do not have a  shape\nresembling oblate spheroid.) Taken together, spheroids and\nspherocylinders provide a rather coherent picture of influence of\nmolecular shape on the phase diagram (see, e.g., \\cite{sp}). Hard\npotentials also have been used in other ways to represent\nnon-spherically symmetric molecules by combining several spheres\nor disks into more complicated shapes, such as heptagons\n\\cite{woj}, or long rods \\cite{hardlong}. To make the models more\nrealistic, sometimes attractive interaction has been added to the\nusual hard repulsive potential \\cite{attr}. \n\nIn Section \\ref{sec:softelast} we present the formalism for soft\nnon-central pair potentials. This formalism cannot be directly\napplied to the calculation of elastic constants of hard potential\nsystems. In Section \\ref{sec:hardelast} and Appendix \n\\ref{sec:regularization} we show how generalization of the approach \nused for centrally-symmetric hard potentials  \\cite{fk_formalism} \ncan be used to derive expressions applicable to non-central hard \npotentials. In Section \\ref{sec:application} we detail the method\nby which the formal results can be applied to hard particles for \nwhich a ``contact function\" can be defined. In particular, we \ndescribe how our formalism can be used for a 2D system of hard \nellipses. In Section \\ref{sec:results} we demonstrate the \nimplementation by calculating stress and elastic constants in \ndifferent phases of a system of hard ellipses.\n\n\n\n\n\n\\section{Elastic properties for soft pair potentials}\\label{sec:softelast}\n\nIn this section we derive explicit expressions for the stress and elastic\nconstant tensors following the method of SHH \\cite{shh} for a more\ngeneral case. We will consider  potential energy $\\cal V$ which \ncan be expressed as\n\\begin{equation}\\label{def:V}\n{\\cal V}=\\sum_{\\langle\\alpha\\beta\\rangle}\\Phi({\\bf r}^{\\alpha\\beta},\\Omega^\\alpha,\n\\Omega^\\beta) ,\n\\end{equation}\nwhere $\\Phi$ is the interaction potential of a pair of particles.\nGreek indices $\\alpha$ and $\\beta$ denote particles (atoms\/molecules), \nand $\\langle\\alpha\\beta\\rangle$ denotes a pair of particles. The above \nequation contains summation over all possible particle pairs. (In this \npaper we do {\\em not} assume summation over repeated Greek indices \nindicating particles.) Here \n${\\bf r}^{\\alpha\\beta}={\\bf r}^{\\beta}-{\\bf r}^{\\alpha}$ is the vector\nconnecting two particles, while $\\Omega^\\alpha$ is the orientation of particle \n$\\alpha$. The two-body potential is not necessarily spherically \nsymmetric. In fact, we will apply our results to particles that do not \nposses such a symmetry.\nWe denote all two-body potentials by the same letter $\\Phi$ although\nnowhere in this formal derivation it is required that they should be \nidentical for different pairs of particles.  (It should be\ndenoted $\\Phi^{\\alpha\\beta}({\\bf r}^{\\alpha\\beta},\\Omega^\\alpha,\n\\Omega^\\beta)$; however, we omit the superscript of $\\Phi$ for brevity.)\nFrom the physical point of view we expect that the potential should be\nrotationally invariant, i.e. when the vector ${\\bf r}^{\\alpha\\beta}$ and the orientations of two molecules (described by $\\Omega^\\alpha$ and $\\Omega^\\beta$)\nperform a ``rigid body'' rotation, the interaction energy should not change.\nIn fact the symmetry of the stress tensor assumes the presence of\nrotational invariance. However, we do not explicitly use this property\nin the derivation of the following expressions.\n\nUnlike the central force case \\cite{shh} we will need to use both \n$\\eta_{ij}$ and $\\epsilon_{ij}$ in the process of derivation. Note that \nin the definition of $\\eta_{ij}$ in Eq.~\\ref{def:eta} only the symmetric\nsum $\\eta_{ij}+\\eta_{ji}$ appears for $i\\ne j$. Therefore, without loss \nof generality it is assumed that the Lagrangian strain is a \n{\\em symmetric} tensor ($\\eta_{ij}=\\eta_{ji}$). From Eqs.~\\ref{def:epsilon} and \\ref{def:eta} we find that\n$\\eta_{kl}=\\frac{1}{2}(\\epsilon_{kl}+\\epsilon_{lk}+\\epsilon_{ik}\\epsilon_{il})$.\nFor small deformations this relation can be inverted to the second order as\n\\begin{equation}\\label{epsiloneta}\n\\epsilon_{kl}=\\eta_{kl}-\\frac{1}{2}\\eta_{mk}\\eta_{ml}+\\dots\n\\end{equation}\n\nIn statistical-mechanical description of a solid in a canonical ensemble\nwe may ask how the free energy $F$ of the solid changes when the\n{\\em boundaries} of the solid undergo deformation described by Eq.~\\ref{def:M}.\nIn calculation of such $F(\\{\\eta\\})$ we do not impose any restrictions\non the positions or orientations of the particles except the\nchange in the boundary conditions. The free energy can be expressed via the\npartition function as \n\\begin{equation}\\label{def:F}\nF(\\{\\eta\\})=-kT\\ln [Z_0Z(\\{\\eta\\})] ,\n\\end{equation}\nwhere only the configurational part $Z(\\{\\eta\\})$ of the partition\ndepends on the deformation, while the remaining (``kinetic'') part\n$Z_0$ is independent of deformations.  We note, that in classical\nphysics the details of the inertia tensors of the molecules can\nmodify the details of their actual motion, but play no role in the\nstatistical-mechanical properties of the system. Only the\nasphericity of the potential matters. The configurational part\n\\begin{equation}\nZ=\n\\int\\limits_{V(\\{\\eta\\})}\\prod_{\\lambda=1}^Nd{\\bf r}^\\lambda\n\\int\\prod_{\\lambda=1}^Nd\\Omega^\\lambda\\ \n{\\rm e}^{-{\\cal V}({\\bf r}^1,\\Omega^1,\\dots,\n{\\bf r}^N,\\Omega^N)\/kT},\n\\end{equation}\nwhere ${\\bf r}^\\alpha$ and $\\Omega^\\alpha$ represent the position and \norientation of particle $\\alpha$ and $\\cal V$ is the interaction potential, \ndepends on the deformation only through the distortion of the integration\nvolume $V(\\{\\eta\\})$ of the possible positions of each of the particles. The\nintegration over all possible spatial directions of each particle\nremains unchanged. If we formally change the integration variable\n${\\bf r}^\\alpha$ for each particle $\\alpha$ to the variable ${\\bf R}^\\alpha$,\nwhich are related by Eq.~\\ref{def:M}, then, the limits of interaction\nof the new variables will correspond to the undistorted volume $V(\\{0\\})\\equiv V_0$,\nand consequently\n\\begin{equation}\\label{def:Z}\nZ=\n\\int\\limits_{V_0}\\prod_{\\lambda=1}^Nd{\\bf r}^\\lambda\n\\int\\prod_{\\lambda=1}^Nd\\Omega^\\lambda\nJ{\\rm e}^{\n-{\\cal V}(M_{ij}R^1_j,\\Omega^1,\\dots,\nM_{ij}R^N_j,\\Omega^N)\/kT},\n\\end{equation}\nwhere $J$ is the Jacobian corresponding to the change of coordinates\n\\begin{equation}\nJ=|\\det(M)|^N=|\\det(I+2\\eta)|^{N\/2}\\ .\n\\end{equation}\nThe deformation now appears as distortion of the coordinates in the\npotential $\\cal V$.\n\nThus, the stress and the elastic constants can be viewed as the first\nand the second derivatives of the free energy density with respect\nto various $\\eta_{ij}$. Since in the expansion in Eq.~\\ref{def:C} always\nappear pairs of terms such as $C_{1123}\\eta_{11}\\eta_{23}+\nC_{1132}\\eta_{11}\\eta_{32}$\nwhich contain identical $\\eta$ terms, we can choose to define\nthe tensor in a symmetric form $C_{ijkl}=C_{jikl}=C_{ijlk}$.\n(An additional symmetry $C_{ijkl}=C_{klij}$ is also evident from\nthe definition of the tensor.) Strictly speaking, since \n$\\eta_{12}=\\eta_{21}$ they should be treated as a single variable \nwhile taking the derivatives\nof the free energy density. However, terms containing those two\nvariables also appear twice in Eq.~\\ref{def:C}. Thus, one can simply\ntreat $\\eta_{12}$ and $\\eta_{21}$ as independent variables, and symmetrize\nthe results with the interchange of indices at the end. Alternatively,\none may view each derivative $\\partial\/\\partial \\eta_{12}$\nas  $\\frac{1}{2}(\\partial\/\\partial \\eta_{12}+\\partial\/\\partial \\eta_{21})$.\nBelow we always present fully symmetrized expressions.\n\nFrom Eqs. \\ref{def:C} and \\ref{def:F} we can express the stress tensor\n\\begin{equation}\nV_0\\sigma_{ij}=\\frac{\\partial{F}}{\\partial{\\eta_{ij}}}\n\\biggm|_{\\{\\eta\\}=\\{0\\}} \n=-\\frac{kT}{Z}\\frac{\\partial{Z}}{\\partial{\\eta_{ij}}}\\biggm|_{\\{\\eta\\}=\\{0\\}}\n\\end{equation}\nand the second order elastic constants\n\\begin{eqnarray}\n&&V_0C_{ijmn}=\\frac{\\partial^2 F}{\\partial\\eta_{mn}\\partial\\eta_{ij}}\n\\biggm|_{\\{\\eta\\}=\\{0\\}} \\nonumber\\\\\n&&=\\left[\\frac{kT}{Z^2}\\frac{\\partial{Z}}{\\partial{\\eta_{mn}}}\n\\frac{\\partial{Z}}{\\partial{\\eta_{ij}}} -\n \\frac{kT}{Z}\\frac{\\partial^2{Z}}{\\partial{\\eta_{mn}}\\partial{\\eta_{ij}}}\\right]\n\\biggm|_{\\{\\eta\\}=\\{0\\}}     \n\\end{eqnarray}\nin terms of the derivatives of $Z$. As can be seen from Eq.~\\ref{def:Z}\nthe dependence of $Z$ on the deformation is contained in the Jacobian $J$\nand in the arguments of the potential $\\cal V$. The Jacobian depends directly\non $\\eta_{ij}$ and its derivatives can be easily calculated. In particular,\nwe find (see, e.g., Ref. \\cite{fk_formalism}) that \n\\begin{equation}\n\\frac{\\partial J}{\\partial\\eta_{ij}}\\biggm|_{\\{\\eta\\}=\\{0\\}}=N\\delta_{ij}\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial^2 J}{\\partial\\eta_{mn}\\partial\\eta_{ij}}\\biggm|_{\\{\\eta\\}=\\{0\\}}=\nN^2\\delta_{ij}\\delta_{mn}-N\\delta_{im}\\delta_{jn}-N\\delta_{in}\\delta_{jm}.\n\\end{equation}\nTaking the derivatives of $\\cal V$ involves the differentiation of \nthe potential with respect to $M_{ij}$, followed by the differentiation \nof $M_{ij}$ with respect to $\\epsilon_{kl}$ using Eq.~\\ref{def:epsilon}, \nfollowed by the differentiation of $\\epsilon_{kl}$ with respect to \n$\\eta_{mn}$ using Eq.~\\ref{epsiloneta}. This leads to the following \nexpressions for the stress and  elastic constants:\n\\begin{widetext}\n\\begin{equation}\\label{softstress}\nV_0\\sigma_{ij}= -NkT\\delta_{ij}+\\frac{1}{2}\\sum_{\\langle\\alpha\\beta\\rangle}\n\\left\\langle \\frac{\\partial\\Phi\\ }{\\partial R^{\\alpha\\beta}_i}R^{\\alpha\\beta}_j+\n \\frac{\\partial\\Phi\\ }{\\partial R^{\\alpha\\beta}_j}R^{\\alpha\\beta}_i\\right\\rangle\\ ,\n\\end{equation}\n\\begin{eqnarray}\\label{softelast}\n&&V_0C_{ijmn}=NkT(\\delta_{im}\\delta_{jn}+\\delta_{in}\\delta_{jm})\\nonumber\\\\\n&+&\\frac{1}{4kT}\n\\sum_{\\langle\\alpha\\beta\\rangle}\\left\\langle \n\\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_i}R^{\\alpha\\beta}_j+\n\\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_j}R^{\\alpha\\beta}_i\n\\right\\rangle\n\\sum_{\\langle\\gamma\\delta\\rangle}\\left\\langle \n\\frac{\\partial\\Phi}{\\partial R^{\\gamma\\delta}_m}R^{\\gamma\\delta}_n+\n\\frac{\\partial\\Phi}{\\partial R^{\\gamma\\delta}_n}R^{\\gamma\\delta}_m\n\\right\\rangle  \\nonumber\\\\\n&-&\\frac{1}{4kT}\\sum_{\\langle\\alpha\\beta\\rangle,\\langle\\gamma\\delta\\rangle}\n\\Bigg\\langle \\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_m}\n\\frac{\\partial\\Phi}{\\partial R^{\\gamma\\delta}_i} R^{\\alpha\\beta}_n  R^{\\gamma\\delta}_j\n+\\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_n}\n\\frac{\\partial\\Phi}{\\partial R^{\\gamma\\delta}_i} R^{\\alpha\\beta}_m  R^{\\gamma\\delta}_j\n\n+\\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_m}\n\\frac{\\partial\\Phi}{\\partial R^{\\gamma\\delta}_j} R^{\\alpha\\beta}_n  R^{\\gamma\\delta}_i\n+\\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_n}\n\\frac{\\partial\\Phi}{\\partial R^{\\gamma\\delta}_j} R^{\\alpha\\beta}_m  R^{\\gamma\\delta}_i\n\\Bigg\\rangle\n \\nonumber\\\\\n&+&\\frac{1}{4}\\sum_{\\langle\\alpha\\beta\\rangle}\\Bigg\\langle\n\\frac{\\partial^2\\Phi}{\\partial R^{\\alpha\\beta}_m\\partial R^{\\alpha\\beta}_i}R^{\\alpha\\beta}_nR^{\\alpha\\beta}_j\n+\\frac{\\partial^2\\Phi}{\\partial R^{\\alpha\\beta}_n\\partial R^{\\alpha\\beta}_i}R^{\\alpha\\beta}_mR^{\\alpha\\beta}_j\n+\\frac{\\partial^2\\Phi}{\\partial R^{\\alpha\\beta}_m\\partial R^{\\alpha\\beta}_j}R^{\\alpha\\beta}_nR^{\\alpha\\beta}_i\n+\\frac{\\partial^2\\Phi}{\\partial R^{\\alpha\\beta}_n\\partial R^{\\alpha\\beta}_j}R^{\\alpha\\beta}_mR^{\\alpha\\beta}_i  \\Bigg\\rangle\n \\nonumber\\\\\n&-&\\frac{1}{8}\\sum_{\\langle\\alpha\\beta\\rangle}\\Bigg\\{\n\\left\\langle \\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_j}R^{\\alpha\\beta}_n+\n\\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_n}R^{\\alpha\\beta}_j \\right\\rangle\\delta_{im}\n+\\left\\langle \\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_i}R^{\\alpha\\beta}_m+\n\\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_m}R^{\\alpha\\beta}_i \\right\\rangle\\delta_{jn}\n\\nonumber\\\\\n&+&\\left\\langle \\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_i}R^{\\alpha\\beta}_n+\n\\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_n}R^{\\alpha\\beta}_i \\right\\rangle\\delta_{jm}\n+\\left\\langle \\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_j}R^{\\alpha\\beta}_m+\n\\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_m}R^{\\alpha\\beta}_j \\right\\rangle\\delta_{in} \\Bigg\\} \\ .\n\\end{eqnarray}\n\\end{widetext}\nIn the above equations we already use the coordinates $\\bf R^{\\alpha\\beta}$\nof the undistorted system to emphasize the fact that all the averages are now\ncalculated in the absence of the deformation.\n\nSince $(\\partial\\Phi\/\\partial R^{\\alpha\\beta}_i)R^{\\alpha\\beta}_j=\n-f^{\\alpha\\beta}_iR^{\\alpha\\beta}_j$, where ${\\bf f}^{\\alpha\\beta}$ is\nthe force between the particles $\\alpha$ and $\\beta$, we can recognize in \nEq.~\\ref{softstress} the standard virial theorem, although in the textbooks \\cite{virtheorem} the derivation is quickly reduced to calculation \nof the (isotropic) pressure $p$.\n\nThe accuracy of the expressions \\ref{softstress} and \\ref{softelast} can \nbe verified by reducing the above formulae to the case of isotropic \ncentral force potential in which case\n\\begin{equation}\n\\frac{\\partial\\Phi}{\\partial R_i} = \\Phi'\\frac{R_i}{R}\\ ,\n\\end{equation}\nwhere prime denotes a derivative of $\\Phi$ with respect to \ninter-particle separation $R$. Similarly,\n\\begin{equation}\n\\frac{\\partial^2\\Phi}{\\partial R_i\\partial R_j}=\n\\Phi''\\frac{R_iR_j}{R^2}\n+\\Phi'\\frac{\\delta_{ij}}{R}\n-\\Phi'\\frac{R_iR_j}{R^3}\n\\end{equation}\nAfter performing these substitutions, we recover the standard expressions \nfor central force potentials \\cite{shh}.\n\n\n\\section{Hard potentials}\\label{sec:hardelast}\n\n\\begin{figure}\n\\includegraphics[height=5cm]{MKfig1.eps}\n\\caption{\\label{fig:eldefs} Two ``hard'' particles at a close approach.\n${\\bf R}^{\\alpha\\beta}$ is a vector connecting their centers, and\n$s^{\\alpha\\beta}$ is the minimal separation of the surfaces.\n }\n\\end{figure}\n\n\nThe expressions for stress and elastic constants that have been obtained \nin the previous section, presumed smooth potentials with well defined\nfirst and second derivatives. There is a certain difficulty in \ntranslating the expressions obtained for soft potentials to a hard \npotential situation. For instance, the term\n$(\\partial\\Phi\/\\partial R^{\\alpha\\beta}_i)R^{\\alpha\\beta}_j$\nin Eq.~\\ref{softstress} looks poorly defined for a hard potential that\nchanges on contact between 0 and $\\infty$. However, we note that the derivative\nof the potential really originates from the derivative\n$\\partial[{\\rm e}^{-\\Phi({\\bf R}^{\\alpha\\beta},\\Omega^\\alpha,\\Omega^\\beta)\/kT}]\/\n\\partial R^{\\alpha\\beta}_i$. The latter, is a derivative of a step function\nthat changes between 0 and 1, when the potential changes between $\\infty$\nand 0. This observation, has been used in Refs. \\cite{runge,barker} to derive\nsimple expressions for the pressure of hard sphere system. A detailed \nand rigorous description of the various aspects (including calculation of \npressure) of interaction of hard  convex solids can be found in Ref. \\cite{hardpotreview}. A typical interaction between two  hard particles \n(e.g., ellipses in 2D or ellipsoids in 3D) is represented by\nFig. \\ref{fig:eldefs} which depicts a close approach of two such particles, \nwhen ${\\bf R}^{\\alpha\\beta}$ is the vector connecting their centers, \nwhile $s^{\\alpha\\beta}$ denotes the minimal distance between them. \nGradient of  \n${\\rm e}^{-\\Phi({\\bf R}^{\\alpha\\beta},\\Omega^\\alpha,\\Omega^\\beta)\/kT}$\nwith respect to ${\\bf R}^{\\alpha\\beta}$, when $\\Omega^\\alpha$ and \n$\\Omega^\\beta$ are kept fixed, is simply \n${\\bf n}^{\\alpha\\beta}\\delta(s^{\\alpha\\beta})$\nwhere ${\\bf n}^{\\alpha\\beta}$ is unit vector perpendicular to the \nsurfaces at the point of contact, pointing from $\\alpha$ to $\\beta$.\nThus we can make the substitution\n\\begin{eqnarray}\\label{substitution}\n&&\\left\\langle \\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_i}R^{\\alpha\\beta}_j\\right\\rangle\n\\rightarrow -kT\\int d{\\bf R}^\\alpha d{\\bf R}^\\beta d\\Omega^\\alpha d\\Omega^\\beta\\times\n\\nonumber\\\\\n&&\nn^{\\alpha\\beta}_i\\delta(s^{\\alpha\\beta})\nR^{\\alpha\\beta}_jP({\\bf R}^\\alpha,\\Omega^\\alpha,{\\bf R}^\\beta, \\Omega^\\beta)\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{eq:Pdef}\n&&P({\\bf R}^\\alpha,\\Omega^\\alpha,{\\bf R}^\\beta, \\Omega^\\beta)=\n\\nonumber\\\\\n&&\\frac{1}{Z}\\int\\prod_{\\lambda=1\\atop\\lambda\\ne\\alpha,\\beta}^N(d{\\bf R}^\\lambda d\\Omega^\\lambda){\\rm e}^{-\\sum_{\\langle\\mu\\nu\\rangle\\atop\\ne\\langle\\alpha\\beta\\rangle}\\Phi({\\bf R}^{\\mu\\nu},\\Omega^\\mu,\\Omega^\\nu)\/kT}\n\\end{eqnarray}\nis the probability density of particles $\\alpha$ and $\\beta$ to be at particular\npositions and orientations. The integral in Eq.~\\ref{substitution} simply\nrepresents the average of $n^{\\alpha\\beta}_iR^{\\alpha\\beta}_j$ over all possible\ncontacts between two particles weighted with proper probability densities of\nthose contacts. (Since the probability density changes from a finite value,\nwhen the particles are almost in contact, to zero, when the particles overlap,\nwe need to consider the case when $s^{\\alpha\\beta}$ approaches 0 from the positive\nside.) Thus, if we denote \n$\\Delta_i^{\\alpha\\beta}=n^{\\alpha\\beta}_i\\delta(s^{\\alpha\\beta}-0^+)$, we can\nfind the hard potential limit by replacement\n\\begin{equation}\\label{hardlim}\n\\frac{\\partial\\Phi}{\\partial R^{\\alpha\\beta}_i}\n\\rightarrow -kT\\Delta_i^{\\alpha\\beta}\n\\end{equation}\nBy substituting the result into Eq.~\\ref{softstress} we obtain the following\nexpression for the stress in the case of hard potentials:\n\\begin{equation}\\label{eq:hardstress}\n\\frac{V_0\\sigma_{ij}}{kT}= -N\\delta_{ij}-\\frac{1}{2}\\sum_{\\langle\\alpha\\beta\\rangle}\n\\left\\langle \\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j+\n \\Delta_j^{\\alpha\\beta}R^{\\alpha\\beta}_i\\right\\rangle .\n\\end{equation}\n\n\nThe substitution appearing in Eq.~\\ref{hardlim} cannot be used to calculate\nthe elastic constants in Eq.~\\ref{softelast}, since the latter expression\ncontains a double summation (over pairs $\\langle\\alpha\\beta\\rangle$ and\n$\\langle\\gamma\\delta\\rangle$) of the derivatives of the potentials. Such summation\nincludes a term where $\\langle\\alpha\\beta\\rangle=\\langle\\gamma\\delta\\rangle$.\nA direct substitution of Eq.~\\ref{hardlim} would lead to the appearance\nof the term $[\\delta(s^{\\alpha\\beta})]^2$ which causes the expression to diverge.\n(This is a {\\em true divergence}, rather than some mathematical subtlety of\napproaching the limit of hard potentials.) However, Eq.~\\ref{softelast} also\ncontains the second derivative of $\\Phi$, which becomes poorly defined in the\nhard potential limit. We shall show in the Appendix \\ref{sec:regularization}\nthat the {\\em sum} of the apparently divergent and poorly defined terms has a well defined hard \npotential limit. In fact this sum can be transformed into an expression \nin Eq.~\\ref{eq:Chard} which includes only first derivatives of potentials \nand products of the first derivatives of potentials of {\\em different} \nparticle pairs. While the resulting expression in Eq.~\\ref{eq:Chard} \nlooks more complicated, it can be simply transformed  (using \nEq.~\\ref{hardlim}) into an expression for the elastic constants of hard particles:\n\n\n\n\\begin{widetext}\n\\begin{eqnarray}\\label{eq:hardelast}\n\\frac{V_0C_{ijmn}}{kT}&=&N(\\delta_{im}\\delta_{jn}+\\delta_{in}\\delta_{jm})\n\\nonumber\\\\\n&+&\\frac{1}{4}\n\\sum_{\\langle\\alpha\\beta\\rangle}\\left\\langle \n\\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j+\n\\Delta_j^{\\alpha\\beta}R^{\\alpha\\beta}_i\n\\right\\rangle\n\\sum_{\\langle\\gamma\\delta\\rangle}\\Big\\langle \n\\Delta_m^{\\gamma\\delta}R^{\\gamma\\delta}_n+\n\\Delta_n^{\\gamma\\delta}R^{\\gamma\\delta}_m\n\\Big\\rangle  \\nonumber\\\\\n&-&\\frac{1}{4}\\sum_{\\langle\\alpha\\beta\\rangle}\\sum_{\\langle\\gamma\\delta\\rangle\n\\atop \\ne \\langle\\alpha\\beta\\rangle}\n\\left\\langle \\Delta_m^{\\alpha\\beta}\n\\Delta_i^{\\gamma\\delta} R^{\\alpha\\beta}_n  R^{\\gamma\\delta}_j\n+\\Delta_n^{\\alpha\\beta}\n\\Delta_i^{\\gamma\\delta} R^{\\alpha\\beta}_m  R^{\\gamma\\delta}_j\n+\\Delta_m^{\\alpha\\beta}\n\\Delta_j^{\\gamma\\delta} R^{\\alpha\\beta}_n  R^{\\gamma\\delta}_i\n+\\Delta_n^{\\alpha\\beta}\n\\Delta_j^{\\gamma\\delta} R^{\\alpha\\beta}_m  R^{\\gamma\\delta}_i\n\\right\\rangle\n \\nonumber\\\\\n&+&\\frac{1}{8}\\sum_{\\langle\\alpha\\beta\\rangle}\n\\sum_{\\gamma\\atop\\ne\\alpha,\\beta}\\Big\\langle\n\\Delta_m^{\\alpha\\beta}\n\\left(\\Delta_i^{\\gamma\\beta}+\\Delta_i^{\\alpha\\gamma}\\right)\nR^{\\alpha\\beta}_nR^{\\alpha\\beta}_j\n+\\Delta_n^{\\alpha\\beta}\n\\left(\\Delta_i^{\\gamma\\beta}+\\Delta_i^{\\alpha\\gamma}\\right)\nR^{\\alpha\\beta}_mR^{\\alpha\\beta}_j\n\\nonumber\\\\\n&+&\\Delta_m^{\\alpha\\beta}\n\\left(\\Delta_j^{\\gamma\\beta}\n+\\Delta_j^{\\alpha\\gamma}\\right)\nR^{\\alpha\\beta}_nR^{\\alpha\\beta}_i\n+\\Delta_n^{\\alpha\\beta}\n\\left(\\Delta_j^{\\gamma\\beta}\n+\\Delta_j^{\\alpha\\gamma}\\right)\nR^{\\alpha\\beta}_mR^{\\alpha\\beta}_i\\Big\\rangle\n\\nonumber\\\\\n&+&\\frac{1}{4}\\sum_{\\langle\\alpha\\beta\\rangle}\\Big\\{\n\\left\\langle \\Delta_j^{\\alpha\\beta}R^{\\alpha\\beta}_n+\n\\Delta_n^{\\alpha\\beta}R^{\\alpha\\beta}_j \\right\\rangle\\delta_{im}\n+\\left\\langle \\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_m+\n\\Delta_m^{\\alpha\\beta}R^{\\alpha\\beta}_i \\right\\rangle\\delta_{jn}\n+\\left\\langle \\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_n+\n\\Delta_n^{\\alpha\\beta}R^{\\alpha\\beta}_i \\right\\rangle\\delta_{jm}\n\\nonumber\\\\\n&+&\\left\\langle \\Delta_j^{\\alpha\\beta}R^{\\alpha\\beta}_m+\n\\Delta_m^{\\alpha\\beta}R^{\\alpha\\beta}_j \\right\\rangle\\delta_{in} \n+\\left\\langle \\Delta_m^{\\alpha\\beta}R^{\\alpha\\beta}_n+\n\\Delta_n^{\\alpha\\beta}R^{\\alpha\\beta}_m \\right\\rangle\\delta_{ij}\n+\\left\\langle \\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j+\n\\Delta_j^{\\alpha\\beta}R^{\\alpha\\beta}_i \\right\\rangle\\delta_{mn}\n\\Big\\} .\n\\end{eqnarray}\n\n\\end{widetext}\nEach of the terms in the above equation corresponds to some particular\ncase of contact between pairs of particles. E.g., term of the type\n$\\langle \\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j\\rangle$ in the\nlast sum corresponds to a contact between a pair of\nparticles $\\alpha$ and $\\beta$, and consequently the contribution of\nthat sum is proportional to $N$. The sum with the prefactor $\\frac{1}{8}$\ncontains terms corresponding to three touching particles: E.g., the\nterm $\\langle\\Delta_m^{\\alpha\\beta}\\Delta_i^{\\gamma\\beta}R^{\\alpha\\beta}_n\nR^{\\alpha\\beta}_j\\rangle$\ncorresponds to the situation when particle $\\beta$ touches particles $\\alpha$ and\n$\\gamma$ simultaneously. The number of such contacts at any given moment \nis also\nproportional to $N$. The second and the third lines of the equation\nhave $N^2$ different averages corresponding to the number\nof pairs of contacts that might appear in the system. However, we note \nthat in both sums taken together we always have differences of the\ntype \n$\\langle\\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j\\rangle \\langle\\Delta_m^{\\gamma\\delta}R^{\\gamma\\delta}_n\\rangle-\n\\langle\\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j \\Delta_m^{\\gamma\\delta}R^{\\gamma\\delta}_n\\rangle$. This term\nvanishes if the contact between the pair $\\langle\\alpha\\beta\\rangle$\nis {\\em uncorrelated} with the contact between the pair\n$\\langle\\gamma\\delta\\rangle$. This happens when the two\npairs are outside the correlation distance. Consequently, only \npairs of contacts close to each other will contribute, and therefore\nthe total contribution of those terms is proportional to $N$.\n\n\\section{Application to hard ellipse system}\\label{sec:application}\n\nEquations \\ref{eq:hardstress} and \\ref{eq:hardelast} enable calculation\nof the stress and elastic constants of a system consisting of hard particles,\nprovided we are able to calculate thermal averages of the type\n$\\langle \\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j\\rangle$. This expression\ndepends on the probability density of particles being in contact.\nFor the calculation of stress we need only the probability {\\em density}\nof a single contact between a pair of particles, while the calculation\nof elastic constants involves the probability density of two such contacts\nhappening simultaneously. It is natural to use MC method\n\\cite{hoover_bk,frenkel_bk,binder_bk} to evaluate averages of this type.\nMC simulation of hard potentials is particularly simple since no energy\nscale is present, and every elementary move of a particle is either accepted\nwithout a need to calculate the Boltzmann factor, or results in a forbidden\nconfiguration, and is, consequently, rejected. Application, of these\nprocedures requires a method to identify intersection of two hard particles.\nSuch  methods have been found both for 2D ellipses \\cite{vieillard}\nand for 3D ellipsoids \\cite{hard_spheroids_funct,perram}.\nIn Appendix \\ref{sec:overlap} we explain in detail the case of ellipses which is\nused in the current work. Here, we will consider a slightly more general case\nwhen a simple function can be defined that identifies the contact between\ntwo particles.\n\nConsider a function $\\Psi$ that depends on the positions and orientations\nof two particles, that vanishes when the particles touch each other and \nis positive when the particles are separated. (The \nformalism can be trivially generalized to the case when the the function\nhas some other non-vanishing constant value at the contact.) Consider a case\nwhen the orientations of both particles are fixed, and we explore positions\nwhere the function vanishes. Figs.~\\ref{fig:elcontact}a and \n\\ref{fig:elcontact}b\ndepict 2D cases of one fixed ellipse, while another (identical) ellipse \nis rotated by a fixed angle and is shown in a variety  positions where \nthey touch. The ratio between the major and minor semi-axes of \nthe ellipses, $a$ and $b$, respectively, are different in both pictures.\nThe thick line traces the possible positions of the center of the \nmoving ellipse. Note that the shape of such a contact line depends on\nthe degree of elongation of the ellipses and on their relative orientation.\nThose are the positions that are relevant for the calculation of the average \n$\\langle \\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j\\rangle$. In 2D\nthis is a line, while in 3D this is a surface. \n\n\n\\begin{figure}\n\\includegraphics[height=10cm]{MKfig2ab.eps}\n\\vskip 1cm\n\\caption{\\label{fig:elcontact} (a) Center of slightly elongated\nellipse ($a\/b=3$) tilted by 45$^o$ with respect to an identical (vertical)\nellipse circumscribes an ``oval'' trajectory, depicted by the thick line,\nwhen the contact point moves along the ellipse.\n(b) Center of strongly elongated ellipse ($a\/b=14$)\nrotated by 90$^o$ with respect to an identical (vertical) ellipse\ncircumscribes  a ``rounded square'' trajectory, depicted by the thick line,\nwhen the contact point moves along the ellipse.\n }\n\\end{figure}\n\n\nThe direction of the force, normal to the contact plane, is also normal \nto this surface. Thus, assuming that $\\Psi$ is a sufficiently smooth \nfunction, we can calculate $n^{\\alpha\\beta}_i=\n(\\partial\\Psi^{\\alpha\\beta}\/\\partial R^{\\alpha\\beta}_i)\/\n|\\nabla\\Psi^{\\alpha\\beta}|$, where the gradient (and the partial derivative)\nwith respect to ${\\bf R}^{\\alpha\\beta}$ is taken when the orientations of\nthe particles are fixed. In thermal average we need to calculate\n\\begin{eqnarray}\\label{eq:example_av}\n&&\\langle \\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j\\rangle\n=\\int d\\Omega^{\\alpha}d\\Omega^{\\beta}\\times\n\\nonumber\\\\\n&&\n\\int dS^{\\alpha\\beta} n^{\\alpha\\beta}_iR^{\\alpha\\beta}_j\nP({\\bf R}^{\\alpha},\\Omega^{\\alpha},{\\bf R}^{\\beta},\\Omega^{\\beta}),\n\\end{eqnarray}\nwhich involves the integration along the contact surface (or line) \n$S^{\\alpha\\beta}$ of the probability density $P({\\bf R}^{\\alpha},\n\\Omega^{\\alpha},{\\bf R}^{\\beta},\\Omega^{\\beta})$, defined in \nEq.~\\ref{eq:Pdef}, of two particles being in those positions and \norientations. During a MC simulation such an event strictly never \noccurs. We can replace the integration along the surface, by\nan integration inside a thin shell of thickness $t$ along the\ncontact surface. In such a case $dS^{\\alpha\\beta}P\\approx(dV\/t)P=dp\/t$,\nwhere $dV$ is the volume element and $dp$ is the {\\em probability} \nof the center of a particle\nbeing within a shell, at some particular area. Note, that the thickness\nof the shell does not have to be constant, but can vary from place to place\non the contact surface. In fact, we can define the shell as corresponding\nto all positions for which $0\\le\\Psi^{\\alpha\\beta}<\\Psi_0$, where\n$\\Psi_0$ is some fixed number. Fig. \\ref{fig:psi0} depicts such a shell\ncorresponding to two values of $\\Psi^{\\alpha\\beta}$, for\na function defined in Appendix \\ref{sec:overlap}. If $\\Psi_0$ is small enough,\nwe can determine the local thickness of the shell from the relation\n$\\Psi_0\\approx|\\nabla\\Psi^{\\alpha\\beta}|t$. Substituting the values\nof $t$ and $n^{\\alpha\\beta}_i$ expressed via function $\\Psi^{\\alpha\\beta}$\ninto Eq.~\\ref{eq:example_av}, and noting that approximate expressions\nmentioned in this paragraph become exact for vanishing $\\Psi_0$, we\narrive at the expression\n\\begin{equation}\n\\langle \\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j\\rangle\n=\\lim_{\\Psi_0\\to 0}\\frac{1}{\\Psi_0}\\int d\\Omega^{\\alpha}d\\Omega^{\\beta}\n\\int\\limits_{S(\\Psi_0)} \\frac{\\partial\\Psi^{\\alpha\\beta}}{\\partial R^{\\alpha\\beta}_i}R^{\\alpha\\beta}_j dp\n\\end{equation}\n\\begin{equation}\\label{eq:lim}\n\\langle \\Delta_i^{\\alpha\\beta}R^{\\alpha\\beta}_j\\rangle\n=\\lim_{\\Psi_0\\to 0} \\frac{1}{\\Psi_0}\\left\\langle\\frac{\\partial\\Psi^{\\alpha\\beta}}{\\partial R^{\\alpha\\beta}_i}R^{\\alpha\\beta}_j\\right\\rangle_{S(\\Psi_0)}\n\\end{equation}\nIn these equation $S(\\Psi_0)$ in the integral and in the thermal \naverage denote a shell defined by the limit $\\Psi_0$ on the function $\\Psi^{\\alpha\\beta}$.\n\n\\begin{figure}\n\\includegraphics[height=5cm]{MKfig3.eps}\n\\caption{\\label{fig:psi0} Line of $\\Psi=0$ and $\\Psi=1000$ for a pair\nof ellipses with aspect ratio $E=a\/b=2$ with their axes rotated\nby 45$^o$ for a function defined in Appendix \\ref{sec:overlap}.\nNote that the thickness of the area between two lines\nof fixed $\\Psi$ varies slightly.\n }\n\\end{figure}\n\nIn the numerical calculation of the stress, the limit in \nEq.~\\ref{eq:lim} is not easy to implement: Using a large $\\Psi_0$\nleads to an inaccurate answer, while using a small $\\Psi_0$ leads\nto a small number of ``almost contact\" events, and, consequently, to\nlarge statistical errors. One may try considering a numerical\nextrapolation to $\\Psi_0=0$ by measuring the stress for a \nsequence of decreasing $\\Psi_0$s. However, the events for\nsmaller values of $\\Psi_0$ are also contained in set of the\nevents for larger $\\Psi_0$s. It is difficult to extrapolate\nsuch correlated sets of data points. The independence of the\ndata points can be achieved by calculating a sequence of values\nof the stress for ``contact shells'' defined by $\\Psi_{\\alpha\\beta}$\nlocated in a sequence of segments $[0,\\Psi_0),[\\Psi_0,2\\Psi_0),\\dots\n[K\\Psi_0,(K+1)\\Psi_0),\\dots$ ($K$ is integer). Values of $\\sigma_{ij}$ \nnow can be conveniently extrapolated to their ``real\" values. \nA similar method has been used by Farago and Kantor \\cite{fk_formalism} \nto calculate the stress and elastic constants of hard sphere solids.\n\nThe terms in the expressions for elastic constants including two\npairs of particles can be similarly handled. One simply uses $\\Psi_{01}$\nand $\\Psi_{02}$ to define {\\em two} shells, $S(\\Psi_{01})$ and \n$S(\\Psi_{02})$, respectively, each corresponding to a  particular contact, \nand considers the events which occur when both pairs of particles are \nwithin their respective shells simultaneously. E.g.,\n\\begin{eqnarray}\n&&\\langle \\Delta_m^{\\alpha\\beta}R^{\\alpha\\beta}_n \n\\Delta_i^{\\gamma\\delta}  R^{\\gamma\\delta}_j\\rangle=\n\\nonumber\\\\\n\\lim_{{\\Psi_{01}\\to 0\\atop\\Psi_{02}\\to 0}} && \\frac{1}{\\Psi_{01}\\Psi_{02}}\\left\\langle\\frac{\\partial\\Psi^{\\alpha\\beta}}{\\partial R^{\\alpha\\beta}_m}R^{\\alpha\\beta}_n\n\\frac{\\partial\\Psi^{\\gamma\\delta}}{\\partial R^{\\gamma\\delta}_i}R^{\\gamma\\delta}_j\n\\right\\rangle_{S(\\Psi_{01}),S(\\Psi_{02})} .\n\\end{eqnarray}\nCompared with the case of the stress, the numerical evaluation of \nthe limit where the thickness of the shells vanishes presents an\neven bigger numerical problem, since\nthe probability of two contact events is very small. Nevertheless, this can\nbe handled in a similar way, by considering a 2D array of segments\nof the type  $\\{[K\\Psi_0,(K+1)\\Psi_0),[L\\Psi_0,(L+1)\\Psi_0)\\}$  \n($K$ and $L$ integers)\nand obtaining the values of the various parts of the elastic constants by\nextrapolating the 2D surface to its ``real\" value of vanishing contact layer\nthickness.\n\n\n\\section{Results of simulations}\\label{sec:results}\n\nWe used the method developed in this work to calculate the elastic \nproperties of 2D hard ellipse system as a part of a study\nof its phase diagram \\cite{mk}. Here, we briefly demonstrate \nthe usefulness of the method. As in any hard particle system,\ntemperature plays no role, since the interactions have no \n``energy scale.\" The temperature appears only as a multiplicative\nprefactor in the free energy $F$ and in Eq.~\\ref{eq:hardstress}\nfor the stress and Eq.~\\ref{eq:hardelast} for the elastic constants. The\nresults depend on the density of the particles and their size and shape:\nwe characterized the system by the number of particles per unit\narea $\\rho$ and by the sizes of the major and minor semi-axes, $a$ \nand $b$, of the ellipses. Frequently, reduced density \n$\\rho^*\\equiv 4\\rho a b$ is used. The maximal possible (close packed) \n$\\rho^*$ is  independent of the aspect ratio $E=a\/b$ of the ellipses and is\nequal to $2\/\\sqrt{3}\\approx 1.155$. It should be noted \\cite{vieillard}\nthat for every fixed $a$ and $b$ there is an infinity of possible \n(equally dense) close \npacked states which are obtained by orienting all ellipses in the same \ndirection and packing them into a (distorted triangular) periodic \nstructure.\n\nThe system of hard disks ($E=1$) has been extensively studied. For\n$\\rho^*\\agt 0.91$ it forms a periodic 2D solid --- a triangular lattice.\nThe correlation function of atom positions of such a solid decays \nto zero as a power law of the separation between the atoms \\cite{mermin}.\nSuch behavior is usually denoted as quasi-long-range order. At the\nsame time the orientations of the ``bonds\" (imaginary lines connecting \nneighboring atoms) have a long range correlation \\cite{mermin_bond}.\nThe system is liquid for $\\rho^*\\alt 0.89$, i.e it has no long range \norder of any kind. At the intermediate\ndensities the system is probably hexatic \\cite{hny} --- a phase\nwith algebraically decaying bond-orientational order, but without\npositional order. (However, even very large scale simulations \\cite{jm}\nhave difficulties in distinguishing the hexatic phase from coexisting\nsolid and liquid phases.) From the point of view of elasticity \ntheory, all three phases are isotropic, i.e. their second order elastic\nconstants are determined by two independent constants. Aspect ratio \n$E\\ne1$ of the ellipses adds an additional order parameter ---\ntheir orientation. E.g., for $E=4$ a system of ellipses forms \nisotropic liquid for densities $\\rho^*\\alt 0.8$. For larger densities \nthe ellipses in the liquid become oriented (``nematic phase''). Finally, at\n$\\rho^*\\approx 1.0$ the system becomes a solid of orientationally\nordered ellipses \\cite{cuesta}. For weakly elongated ellipses, we\nexpect the particles to remain orientationally disordered in the entire\nliquid phase, and with increasing density to go (possibly via hexatic\nphase) to a crystalline state in which the ellipses remain disordered.\nThe 3D analog of such a state is called {\\em plastic\ncrystal} \\cite{plastic}. (Presence of such a phase in almost circular \nellipses ($E=1.01$) was observed by Vieillard-Baron \\cite{vieillard}.)\nWith increasing density an additional phase transition will bring the \nellipses into an orientationally ordered state. Phase diagram which \nincludes such a transition between two solid phases for 3D hard \nellipsoids has been studied by Frenkel {\\em et al.} \\cite{fmm}.\n\n\nAs a test of our formalism we studied a case of moderately elongated\nhard ellipses with $E=1.5$. We considered system consisting of \n$N \\approx 1000$ ellipses contained in a 2D rectangular box whose \ndimensions were chosen as close as possible to a square. Periodic \nboundary conditions were used.  The ellipses were initially placed \non a distorted triangular lattice, commensurate with the dimensions\nof a closed packed configuration corresponding to this particular\naspect  ratio $E$ and the particular orientation of the ellipses. \nIn this Section we describe only the cases when the initial orientation\nof the major axes of the ellipses was taken to be perpendicular to \none of the axes of \nsymmetry of the ordered crystal drawn through neighboring particles.\nWe first performed an equilibration run at constant pressure, in \norder to allow the system to reach an equilibrated state with \nrespect to the orientational, as well as the translational ordering. \nThe orientational order parameter, the box dimensions and the density \nwere monitored during this equilibration run. A MC time unit in the \nequilibration run consisted of $N+1$ elementary moves, one of which, \non the average, was a volume change attempt, and the rest were particle \nmove attempts. A particle move attempt involved choosing a particle \nrandomly, and attempting to displace and rotate it simultaneously \nby an amount chosen from a uniform distribution. The move was \naccepted if the displaced particle did not overlap in\nits new position and orientation with other particles. The volume \nchange attempt was identical to that described in \\cite{frenkel_bk}. \nThe width of the distributions corresponding to the particle moves \nand the parameter of the volume change were chosen so that the average \nsuccess rates of both types of MC moves were about 50\\%.\n\nThe use of the constant pressure simulations at the equilibration stage \nenabled us to determine the equilibrium box shape for isotropic \nstress tensor (pressure) conditions. At most densities the final\nconfiguration had orientationally disordered ellipses, and we could\neasily verify that the final configurations were independent of the \nspecific choice of the starting configuration.\nHowever, at extremely high densities, approaching the close packed\ndensity, even after long equilibration the state resembled the starting\nstate of orientationally ordered ellipses. Typically, several millions \nof MC time units were required to reach  equilibrium for a given \npressure. Upon completion of equilibration, we switched to constant \nvolume simulations, during which the stresses and the elastic \nconstants were evaluated.  Very long simulation times (about \n$10^7$ MC time units) were required for accurate determination of \nthese constants, because their calculation depends on extremely \nrare events of two pairs of particles simultaneously touching each \nother. The range of ``contact \nshells'' was chosen in such a way that even in the most remote \nshell the separation between the particles was significantly smaller \nthan their mean separation. The statistical accuracy of the elastic \nconstants was evaluated by comparing the results of independent runs. \n\n\\begin{figure}\n\\includegraphics[height=6.5cm]{MKfig4.eps}\n\\caption{\\label{fig:elast1} \nPressure $p$ (open circles connected by solid line) and the elastic constant\n$C_{1212}$ (inverted open triangles connected by dashed line) of hard \nellipse system with aspect ratio $E=1.5$\nin the units of $kT\/4ab$  as functions of the reduced density $\\rho^*$.\nThe error bars of the pressure are significantly smaller than the\nsymbols denoting the data points.\n }\n\\end{figure}\n\\begin{figure}\n\\includegraphics[height=6.5cm]{MKfig5.eps}\n\\caption{\\label{fig:elast2} \nElastic constants $C_{1111}$ (open circles connected by solid line),\n$C_{2222}$ (asterisks connected by dotted line) and $C_{1122}$ (full \ntriangles connected by dot-dashed line) of hard ellipse system with \naspect ratio $E=1.5$\nin the units of $kT\/4ab$  as functions of the reduced density $\\rho^*$.\nThe error bars of all the data points are slightly smaller than the\nsymbols denoting them.\n }\n\\end{figure}\n\nIn a 2D system which has a reflection symmetry with respect to\neither $x$ or $y$ axis, the elastic constants with an index \nappearing an odd number \nof times (such as $C_{1112}$) must vanish. Indeed, our simulations\nshowed that these quantities vanish within the error bars of the\nmeasurement. The system still may have four unrelated elastic constants \n$C_{1111}$, $C_{2222}$, $C_{1122}$ and $C_{1212}$. For a system with\nquadratic symmetry the number of independent elastic constants reduces\nto three. Such systems are frequently characterized by their bulk\nmodulus and two shear moduli, $\\mu_1=C_{1212}-p$ and \n$\\mu_2=\\frac{1}{2}(C_{1111}-C_{1122})-p$. For an isotropic system \n$\\mu_1=\\mu_2$ and, therefore, there are only two independent\nconstants. (A system with six-fold symmetry is isotropic as far as\nthe elastic constants are concerned.)\nFigs. \\ref{fig:elast1} and \\ref{fig:elast2} depict the pressure and \nfour elastic constants for several values of the reduced density \n$\\rho^*$.  One can see that for densities $\\rho^* \\le 1.086$, \n$C_{1111}$ and $C_{2222}$ practically coincide, indicating that these \nsystems are  at least quadratic. Furthermore, the identity \n$C_{1111}-C_{1122}=2C_{1212}$, i.e. $\\mu_1=\\mu_2$, is found \nto hold within a few percent for these values of $\\rho^*$. Consequently,\nfor these densities the system is isotropic from the point of view of \nthe elastic properties. Figs.~\\ref{fig:configs}a, b and c depict the\nsystem in that range of densities: Figs.~\\ref{fig:configs}a and b \nrepresent states with vanishing shear moduli, and neither of them\nexhibits translational order of the ellipse centers. However, while\nFig.~\\ref{fig:configs}a represents a state with all the\ncharacteristics of a liquid, the state in Fig.~\\ref{fig:configs}b is \ncharacterized by slowly decaying bond orientational order, possibly \nindicating hexatic phase. At a slightly higher density, \nFig.~\\ref{fig:configs}c represents a plastic solid: while the ellipses\nare randomly oriented, the system exhibits long range bond orientational\norder, and algebraically decaying positional order of the particles.\nThe particles occupy, on the average, positions of an undistorted \ntriangular lattice although some undulations are apparent. This is \ncharacterized by two coinciding positive shear moduli. \n\nWhen we approach within few percent the close packed configuration,\ncorresponding to the two largest densities in Figs.~\\ref{fig:elast1} and\n\\ref{fig:elast2} the prolonged relaxation process does not change the\npreferred orientation of the ellipses and the distortion of the\nlattice. It would be reasonable to conclude, that at such high\ndensity we finally arrived at the orientationally ordered elastic\nsolid. The isotropic elastic symmetry no longer holds. We checked\nand found that almost all stability criteria \\cite{birch,zhou}, \nindicating the sign of the energy change upon small deformation,\nare positive. However, one of the shear moduli, namely $\\mu_1$,\nis slightly negative, although within one standard statistical\ndeviation from zero. This, may either indicate that we are in\nan unstable state, or that there is a continuum of equilibrium states\nwith different mean orientations of ellipses and, correspondingly,\ndifferent dimensions of elementary cell.\n\n\\begin{figure*}\n\\includegraphics[height=6cm]{MKfig6a.eps}\\hskip 3cm\n\\includegraphics[height=6cm]{MKfig6b.eps}\n\\par (a){\\hskip 8cm}(b)\n\\par\\vskip 1cm\n\\includegraphics[height=6cm]{MKfig6c.eps}\\hskip 3cm\n\\includegraphics[height=6cm]{MKfig6d.eps}\n\\par (c){\\hskip 8cm}(d)\n\\caption{\\label{fig:configs}\nTypical equilibrium configurations of slightly eccentric ($E=1.5$) \nhard ellipse  system  at several densities. Only part of the \nsystem is shown. All the pictures show the same (partial) volume of\nthe system; they differ only in the reduced density $\\rho^*$:\n(a) orientationally and translationally disordered liquid at \n$\\rho^*=0.881$;\n(b) liquid with a high degree of bond-orientational order at \n$\\rho^*=1.029$;\n(c) plastic solid with long-range bond-orientational order, \nand quasi-long-range translational order consisting of rotationally \ndisordered ellipses at $\\rho^*=1.086$;\n(d) solid of orientationally ordered ellipses at $\\rho^*=1.117$.\n }\n\\end{figure*}\n\n\n\\section{Discussion}\\label{sec:discussion}\n\nWe extended the formalism of SHH \\cite{shh} to the case of systems \ninteracting\nvia non-central two-particle potentials. In its form represented by \nEqs.~\\ref{softstress} and \\ref{softelast}, the formalism can be used\nto study properties of molecular systems. This is particularly\ntrue for highly non-spherical organic molecules, and various soft\ncondensed matter systems.  The adaptation of the expressions\nto hard potentials (Eqs.~\\ref{eq:hardstress} and \\ref{eq:hardelast})\nproduced slightly more complicated expressions. However, the \nsimplicity of hard potential systems provides excellent insights \ninto the entropy-dominated systems. We demonstrated the usefulness \nof the formalism by presenting some results of our study of the hard ellipse\nsystem \\cite{mk}. Measurement of several order parameters, and \nthe correlation functions is not always sufficient to determine the\nnature of phases. For systems of moderate size, it maybe even difficult\nto distinguish liquid from a solid. Measurement of elastic constants\nprovides an additional, very important tool for assessing the nature\nof the state of the system. In particular, the elastic constants may\nindicate the presence of instability, even when prolonged \nequilibration does not change an existing state. Following the \nindications of instability at high densities, we are currently\nperforming extensive study of equilibrium states at these densities.\n\nWhile we worked on a 2D example, the method can be equally well applied \nin 3D for such systems as hard ellipsoids or spherocylinders. \n\n\\acknowledgements \nWe would like to thank O. Farago and M. Kardar for useful discussions. \nThis research was supported by Israel Science Foundation Grant\nNo. 193\/05.\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this work we are concerned with nonlinear Fokker-Planck-Kolmogorov equations (FPK-equations) on $\\mathbb{R}^d$, both deterministic\n\\begin{equation}\\label{NLFPKE}\\tag{NLFPK}\n\\partial_t\\mu_t = \\mathcal{L}^*_{t,\\mu_t}\\mu_t, \\,\\, t \\in [0,T],\n\\end{equation}and perturbed by a first-order stochastic term driven by a finite-dimensional Wiener process $W$\n\\begin{equation}\\label{SNLFPKE}\\tag{SNLFPK}\n\\partial_t\\mu_t = \\mathcal{L}^*_{t,\\mu_t}\\mu_t-\\text{div}(\\sigma(t,\\mu_t))dW_t,\\,\\, t\\in [0,T],\n\\end{equation}with solutions being continuous curves of subprobability measures $\\mu_t \\in \\mathcal{SP}$. Here, $\\mathcal{L}^*$ denotes the formal dual of a second-order differential operator acting on sufficiently smooth functions $\\varphi: \\mathbb{R}^d \\to \\mathbb{R}$ via\n\\begin{equation}\\label{1}\n\\mathcal{L}_{t,\\mu}\\varphi(x) = \\sum_{i,j=1}^{d}a_{ij}(t,\\mu,x)\\partial^2_{ij}\\varphi(x)+\\sum_{i=1}^{d}b_i(t,\\mu,x)\\partial_i\\varphi(x)\n\\end{equation}with coefficients $a$ and $b$ depending on $(t,x) \\in [0,T]\\times \\mathbb{R}^d$ and (in general non-locally) on the solution $\\mu_t$. These equations are to be understood in distributional sense, see Definition \\ref{Sol NLFPKE} and \\ref{Def sol SNLFPKE}. The nonlinearity arises from the dependence of $\\mathcal{L}$ and $\\sigma$ on the solution itself, which renders the theory of existence and uniqueness of such equations significantly more difficult compared to the linear case. For a thorough introduction to the field, we refer to \\cite{bogachev2015fokker} and the references therein. As shown in \\cite{Rckner-lin.-paper}, the deterministic nonlinear equation (\\ref{NLFPKE}) is naturally associated to a first-order linear continuity equation on $\\mathcal{P}(\\mathcal{SP})$, the space of Borel probability measures on $\\mathcal{SP}$, of type\n\\begin{equation}\\label{P-CE}\\tag{$\\mathcal{SP}$-CE}\n\\partial_t \\Gamma_t = \\mathbf{L}^*_t\\Gamma_t, \\,\\, t\\in [0,T],\n\\end{equation}in the sense of distributions, with the linear operator $\\mathbf{L}$ acting on sufficiently smooth real functions on $\\mathcal{SP}$ via the gradient operator $\\nabla^{\\mathcal{SP}}$ on $\\mathcal{SP}$ as\n$$\\mathbf{L}_tF = \\big \\langle \\nabla^{\\mathcal{SP}}F, b_t+a_t\\nabla \\big \\rangle_{L^2}.$$ Precise information on this operator and equation (\\ref{P-CE}) are given in Section 3, in particular in Definition \\ref{Def sol P-CE} and the paragraph  preceding it.\\\\\n Our first main result, Theorem \\ref{main thm det case} states that each weakly continuous solution $(\\Gamma_t)_{t \\leq T}$ to (\\ref{P-CE}) is a \\textit{superposition} of solutions to (\\ref{NLFPKE}), i.e. (denoting by $e_t$ the canonical projection $e_t: (\\mu_t)_{t \\leq T} \\mapsto \\mu_t$)\n\\begin{equation}\\label{SuperPos eq}\n\\Gamma_t = \\eta \\circ e_t^{-1}\n\\end{equation}\nfor some probability measure $\\eta$ concentrated on solution curves to (\\ref{NLFPKE}) in a suitable sense.\\\\\n We also treat the stochastic case in a similar fashion. More precisely, in Section 4 we establish a new correspondence between the stochastic equation for measures (\\ref{SNLFPKE}) and a corresponding second-order equation for curves $(\\Gamma_t)_{t \\leq T}$ in $\\mathcal{P}(\\mathcal{SP})$ of type\n \\begin{equation}\\label{P-FPKE}\\tag{$\\mathcal{SP}$-FPK}\n \\partial_t \\Gamma_t = (\\mathbf{L}_t^{(2)})^*\\Gamma_t,\\,\\, t \\in [0,T],\n \\end{equation}where, roughly, $$\\mathbf{L}^{(2)}_t = \\mathbf{L}_t+ \\textit{ second-order perturbation}. $$ The second-order term stems from the stochastic perturbation of (\\ref{SNLFPKE}) and will be geometrically interpreted in terms of a (formal) notion of the Levi-Civita connection on $\\mathcal{SP}$. The second main result of this work, Theorem \\ref{main thm stoch case}, is then the stochastic generalization of the deterministic case: For any solution $(\\Gamma_t)_{t \\leq T}$ to (\\ref{P-FPKE}), there exists a solution process $(\\mu_t)_{t \\leq T}$ to (\\ref{SNLFPKE}) on some probability space such that $\\mu_t$ has distribution $\\Gamma_t$. We stress that in both cases, we do not require any regularity of the coefficients.\\\\\n \\\\\n Let us embed these results into the general research in this direction. Let $b_t(\\cdot): \\mathbb{R}^d \\to \\mathbb{R}^d$ be an inhomogeneous vector field and consider the (nonlinear) ODE\n \\begin{equation}\\label{ODE intro}\\tag{ODE}\n \\frac{d}{dt}\\gamma_t = b_t(\\gamma_t), \\,\\, t \\leq T\n \\end{equation}and the linear continuity equation for curves of Borel (probability) measures on $\\mathbb{R}^d$\n \\begin{equation}\\label{CE intro}\\tag{CE}\n \\partial_t \\mu_t = -\\divv (b_t\\mu_t), \\,\\, t \\leq T,\n \\end{equation}understood in distributional sense. In the seminal paper \\cite{Ambrosio2008}, L. Ambrosio showed the following: Any (probability) solution $(\\mu_t)_{t \\leq T}$ to (\\ref{CE intro}) with an appropriate global integrability condition is a \\textit{superposition} of solution curves to (\\ref{ODE intro}), i.e. there exists a (probability) measure $\\eta$ on the space of continuous paths with values in the state space of (\\ref{ODE intro}), $C([0,T],\\mathbb{R}^d)$, which is concentrated on solutions to (\\ref{ODE intro}) such that\n $$\\eta \\circ e_t^{-1} = \\mu_t, \\,\\, t\\leq T.$$This allows to transfer existence and uniqueness results between the linear equation (\\ref{CE intro}) and the nonlinear (\\ref{ODE intro}). However, the linear equation must be studied on an infinite-dimensional space of (probability) measures. The analogy to our deterministic result from Section 3 is as follows: (\\ref{ODE intro}) is replaced by (\\ref{NLFPKE}), which, in spirit of this analogy, we interpret as a differential equation on the manifold-like state space $\\mathcal{SP}$. Likewise, (\\ref{CE intro}) is replaced by (\\ref{P-CE}) and our first main result Theorem \\ref{main thm det case} may be understood as the analogue of Ambrosio's result to the present setting. By passing from (\\ref{NLFPKE}) to (\\ref{P-CE}), we \\textit{linearize} the equation.\\\\\n Concerning the stochastic case, consider a stochastic differential equation on $\\mathbb{R}^d$\n \\begin{equation}\\label{SDE}\\tag{SDE}\n dX_t = b(t,X_t)dt+\\tilde{a}(t,X_t)dB_t, \\,\\, t\\in [0,T].\n \\end{equation} By It\u00f4's formula, the one-dimensional marginals $\\mu_t$ of any (weak) martingale solution $X$ solve the corresponding linear FPK-equation\n \\begin{equation}\\label{FPKE}\\tag{FPK}\n \\partial_t \\mu_t = \\mathcal{L}_{lin, t}^*\\mu_t, \\,\\, t \\in [0,T],\n \\end{equation} where $\\mathcal{L}_{lin}$ is a linear second-order diffusion operator with coefficients $b$ and $\\frac{1}{2}\\tilde{a}\\tilde{a}^T$. Conversely, a superposition principle has successively been developed in increasingly general frameworks (cf. \\cite{FIGALLI2008109, Kurtz2011, trevisan2016, Rckner-superpos_pr}): Under mild global integrability assumptions, for every weakly continuous solution curve of probability measures $(\\mu_t)_{t \\leq T}$ to (\\ref{FPKE}), there exists a (weak) martingale solution $X$ to (\\ref{SDE}) with one-dimensional marginals $(\\mu_t)_{t \\leq T}$, thereby providing an equivalence between solutions to (\\ref{SDE}) and (\\ref{FPKE}), which offers a bridge between probabilistic and analytic approaches to diffusion processes. As in the deterministic case, the transition from (\\ref{SDE}) to (\\ref{FPKE}) provides a \\textit{linearization}, while at the same time it transfers the equation to a much higher dimensional state space. Concerning our stochastic result Theorem \\ref{main thm stoch case}, we replace the stochastic equation on $\\mathbb{R}^d$ by the stochastic equation for measures (\\ref{SNLFPKE}) and the corresponding second-order equation for measures (\\ref{FPKE}) by (\\ref{P-FPKE}) and prove an analogous superposition result for solutions to the latter equation.\\\\\n  The proofs of both the deterministic and stochastic result rely on superposition principles for differential equations on $\\mathbb{R}^{\\infty}$ and the corresponding continuity equation (for the deterministic case) and for martingale solutions and FPK-equations on $\\mathbb{R}^{\\infty}$ (for the stochastic case) by Ambrosio and Trevisan (\\cite{ambrosio2014}, \\cite{TrevisanPhD}). The key technique is to transfer (\\ref{P-CE}) and (\\ref{NLFPKE}) (and, similarly, (\\ref{P-FPKE}) and (\\ref{SNLFPKE}) for the stochastic case) to suitable equations on $\\mathbb{R}^{\\infty}$ via a homeomorphism between $\\mathcal{SP}$ and $\\mathbb{R}^{\\infty}$ (replaced by $\\ell^2$ for the stochastic case, in order to handle the stochastic integral).\\\\\n Moreover, our results also blend into the theory of \\textit{distribution dependent} stochastic differential equations, also called \\textit{McKean-Vlasov equations}, i.e. stochastic equations on Euclidean space of type\n \\begin{equation}\\label{DDSDE}\\tag{DDSDE}\n dX_t = b(t,\\mathcal{L}_{X_t},X_t)dt+\\tilde{a}(t,\\mathcal{L}_{X_t},X_t)dB_t,\\,\\, t\\in [0,T],\n \\end{equation}\n see the classical papers \\cite{McKean1907, Funaki, scheutzow_1987} as well as the more recent works \\cite{HUANG20194747, 1078-0947_2019_6_3017, Coghi2019StochasticNF}. Here, $\\mathcal{L}_{X_t}$ denotes the distribution of $X_t$ and is not to be confused with the operators $\\mathcal{L}_{t,\\mu}$ and $\\mathcal{L}_t$ from above. As in the non-distribution dependent case, where the curve of marginals of any solution to (\\ref{SDE}) solves an equation of type (\\ref{FPKE}), a similar observation holds here: Each solution $X$ to (\\ref{DDSDE}) provides a solution to a nonlinear FPK-equation of type (\\ref{NLFPKE}) via $\\mu_t = \\mathcal{L}_{X_t}$ and a corresponding superposition principle holds analogously to the linear case as well (\\cite{barbu2020, doi:10.1137\/17M1162780}).\\\\\nHowever, while for (\\ref{SDE}) the passage to (\\ref{FPKE}) provides a complete linearization, the situation is different for equations of type (\\ref{NLFPKE}). This stems from the observation that (\\ref{DDSDE}) is an equation with two sources of nonlinearity. Hence, it seems natural to linearize (\\ref{NLFPKE}) once more in order to obtain a linear equation, which is related to (\\ref{DDSDE}) and (\\ref{NLFPKE}) in a natural way. By the results of \\cite{Rckner-lin.-paper}, this linear equation is of type (\\ref{P-CE}). Similar considerations prevail in the stochastic case, where one considers equations of type (\\ref{DDSDE}) with an additional source of randomness (we shall not pursue this direction in this work). \\\\\n \\\\\nOn the one hand, the superposition principles of Theorem \\ref{main thm det case} and Theorem \\ref{main thm stoch case} provide new structural results for nonlinear FPK-equations and its corresponding linearized equations on the space of probability measures over $\\mathcal{SP}$, involving a geometric interpretation of the latter. On the other hand, it is our future plan to further study the geometry of $\\mathcal{SP}$ as initiated in \\cite{Rckner-lin.-paper} and this work to develop an analysis on such infinite-dimensional manifold-like spaces, which allows to solve linear equations of type (\\ref{P-CE}) and (\\ref{P-FPKE}) on such spaces. By means of the results of this work, one can then lift such solutions to solutions to the nonlinear equations for measures (\\ref{NLFPKE}) and (\\ref{SNLFPKE}), thereby obtaining new existence results for these nonlinear equations for measures.\\\\\n \\\\\n We point out that although our main aim is to lift weakly continuous solutions to (\\ref{P-CE}) and (\\ref{P-FPKE}) concentrated on probability measures to a measure on the space of continuous probability measure-valued paths $(\\mu_t)_{t \\leq T}$, for technical reasons we more generally develop our results for vaguely continuous subprobability solutions (i.e. $\\mu_t \\in \\mathcal{SP}$). We comment on the advantages of this approach in Remark \\ref{Rem explain why SP} for the deterministic case and note that similar arguments prevail in the stochastic case as well. However, due to the global integrability assumptions we consider, we are able to obtain results for probability solutions as desired.\\\\\n \\\\\n The organization of this paper is as follows. After introducing general notation and recalling basic properties of the spaces $\\mathcal{P}$ and $\\mathcal{SP}$ in Section 2, Section 3 contains the deterministic case, i.e. the superposition principle between solutions to (\\ref{P-CE}) and (\\ref{NLFPKE}). Here, the main result is Theorem \\ref{main thm det case}. We use this result to prove an open conjecture of \\cite{Rckner-lin.-paper} (cf. Proposition \\ref{Prop conj Rckner}) and present several consequences. In Section 4, we treat the stochastic case for equations of type (\\ref{SNLFPKE}), the main result being Theorem \\ref{main thm stoch case}.\n \\paragraph{Acknowledgements}\n \fFinancial support by the German Science Foundation DFG (IRTG 2235) is gratefully\nacknowledged.\n \\section{Notation and Preliminaries}\n We introduce notation and repeat basic facts on spaces and topologies of measures.\n \\subsection*{Notation}\n For a measure space $(\\mathcal{X},\\mathcal{A},\\mu)$ and a measurable function $\\varphi: \\mathcal{X}\\to \\mathbb{R}$, we set $\\mu(\\varphi) := \\int_{\\mathcal{X}}\\varphi(x)d\\mu(x)$ whenever the integral is well-defined. For $x \\in \\mathcal{X}$, we denote by $\\delta_x$ the \\textit{Dirac measure in $x$}, i.e. $\\delta_x(A) = 1$ if and only if $x \\in A$ and $\\delta_x(A) =0$ else. For a topological space $X$ with Borel $\\sigma$-algebra $\\mathcal{B}(X)$ we denote the set of continuous bounded functions by $C_b(X)$, the set of Borel probability measures on $X$ by $\\mathcal{P}(X)$ and write $\\mathcal{P} = \\mathcal{P}(\\mathbb{R}^d)$. If $Y \\in \\mathcal{B}(X)$, we let $\\mathcal{B}(X)_{\\upharpoonright Y}$ denote the \\textit{trace of $Y$on} $\\mathcal{B}(X)$. For $T>0$, a family $(\\mu_t)_{t \\leq T} = (\\mu_t)_{t \\in [0,T]}$ of finite Borel measures on $\\mathbb{R}^d$ is a \\textit{Borel curve}, if $t \\mapsto \\mu_t(A)$ is Borel measurable for each $A \\in \\mathcal{B}(\\mathbb{R}^d)$. A set of functions $\\mathcal{G} \\subseteq C_b(\\mathbb{R}^d)$ is called \\textit{measure-determining}, if $\\mu(g) = \\nu(g)$ for each $g \\in \\mathcal{G}$ implies $\\mu = \\nu$ for any two finite Borel measures $\\mu, \\nu$ on $\\mathbb{R}^d$.\\\\\n For $x,y \\in \\mathbb{R}^d$, the usual inner product is denoted by $x \\cdot y$ and, with slight abuse of notation, we also denote by $x \\cdot y = \\sum_{k \\geq 1 }x_ky_k$ the inner product in $\\ell^2$ (the Hilbert space of square-summable real-valued sequences $x=(x_k)_{k \\geq1}$). For $\\varphi \\in C_b(\\mathbb{R}^d)$, we set $||\\varphi||_{\\infty} := \\underset{x \\in \\mathbb{R}^d}{\\text{sup}}|\\varphi(x)|$. If $\\varphi$ has first- and second-order partial derivatives, we denote them by $\\partial_i\\varphi$ and $\\partial^2_{ij}\\varphi$ for $i,j \\leq d$.\\\\\n \\\\\n We use notation for function spaces as follows. For $k \\in \\mathbb{N}_0$, $C^k_b(\\mathbb{R}^d)$ denotes the subset of functions $\\varphi$ in $C_b(\\mathbb{R}^d)$ with continuous, bounded partial derivatives up to order $k$, with the usual norm $||\\varphi||_{C^2_b} = \\text{max}(||\\varphi||_{\\infty},||\\partial_i\\varphi||_{\\infty}, ||\\partial^2_{ij}\\varphi||_{\\infty})$ for $k = 2$. Likewise, $C^k_c(\\mathbb{R}^d)$ denotes the subset of all such $\\varphi$ with compact support; for $k = 0$, we write $C_c(\\mathbb{R}^d)$ instead. For $n \\geq 1$, $p \\geq 1$ and a measure $\\mu$ on $\\mathcal{B}(\\mathbb{R}^d)$, we denote by $L^p(\\mathbb{R}^d,\\mathbb{R}^n;\\mu)$ the space of Borel functions $\\varphi: \\mathbb{R}^d \\to \\mathbb{R}^n$ such that\n $$\\int_{\\mathbb{R}^d}||\\varphi(x)||^pd\\mu(x) < +\\infty,$$\n where $||\\cdot||$ denotes the standard Euclidean norm on $\\mathbb{R}^n$. For $p =2$, $\\langle \\cdot, \\cdot \\rangle_{L^2(\\mathbb{R}^d,\\mathbb{R}^n;\\mu)}$ denotes the usual inner product on the Hilbert space $L^2(\\mathbb{R}^d,\\mathbb{R}^n;\\mu)$. For  $T >0$ and a topological space $Y$, we write $C_TY$ for the set of continuous functions $\\varphi:[0,T]\\to Y$. By $\\mathbb{S}^+_d$ we denote the space of symmetric, positive-semidefinite $d\\times d$-matrices with real entries.\n \\subsection*{Basic properties of spaces of measures}\n \\subsubsection*{Probability measures}\n For a topological space $X$, we endow $\\mathcal{P}(X)$ with the topology of weak convergence of measures, i.e. the initial topology of the maps $\\mu \\mapsto \\mu(\\varphi)$, $\\varphi \\in C_b(X)$. If $X$ is Polish, then so is $\\mathcal{P}(X)$.\n\\subsubsection*{Subprobability measures}\n By $\\mathcal{SP}$ we denote the set of all Borel subprobability measures on $\\mathbb{R}^d$, i.e. $\\mu \\in \\mathcal{SP}$ if and only if $\\mu$ is a non-negative measure on $\\mathcal{B}(\\mathbb{R}^d)$ with $\\mu(\\mathbb{R}^d) \\leq 1$. Throughout, we endow $\\mathcal{SP}$ with the \\textit{vague topology}, i.e. the initial topology of the maps $\\mu \\mapsto \\mu(g)$, $g \\in C_c(\\mathbb{R}^d)$. Hence, a sequence $(\\mu_n)_{n \\geq 1}$ converges to $\\mu$ in $\\mathcal{SP}$ if and only if $\\mu_n(g) \\underset{n \\to \\infty}{\\longrightarrow} \\mu(g)$ for each $g \\in C_c(\\mathbb{R}^d)$. Its Borel $\\sigma$-algebra is denoted by $\\mathcal{B}(\\mathcal{SP})$. In particular, $\\mathcal{P}(\\mathcal{SP})$, the set of Borel probability measures on $\\mathcal{SP}$, is a topological space with the weak topology of probability measures on $(\\mathcal{SP},\\mathcal{B}(\\mathcal{SP}))$. The Riesz-Markov representation theorem yields that $\\mathcal{SP}$ with the vague topology coincides with the positive half of the closed unit ball of the dual space of $C_c(\\mathbb{R}^d)$ with the weak*-topology. Hence $\\mathcal{SP}$ with the vague topology is compact. It is also Polish and $\\mu \\mapsto \\mu(\\mathbb{R}^d)$ is vaguely lower semicontinuous, see \\cite[Ch.4.1]{OK}. In particular,\n $\\mathcal{P}  \\in \\mathcal{B}(\\mathcal{SP}).$\nRecall that $\\mathcal{B}(\\mathcal{P}) = \\mathcal{B}(\\mathcal{SP})_{\\upharpoonright_{\\mathcal{P}}}$. Hence, in the sequel we may consider measures $\\Gamma \\in \\mathcal{P}(\\mathcal{P})$ as elements in $\\mathcal{P}(\\mathcal{SP})$ with mass on $\\mathcal{P}$.\\\\\n In contrast to weak convergence in $\\mathcal{P}$, vague convergence in $\\mathcal{SP}$ can be characterized by countably many functions in a sense made precise by Lemma \\ref{Prop G early}. The fact that this is not true for weak convergence in $\\mathcal{P}$ is the main reason why we formulate all equations for subprobability measures, although we are mainly interested in the case of probability solutions. More details in this direction are stated in Remark \\ref{Rem explain why SP}.\n \n \\section{Superposition Principle for deterministic nonlinear Fokker-Planck-Kolmogorov Equations}\n\n Fix $ T>0$ throughout, let each component of the coefficients\n $$a = (a_{ij})_{i,j \\leq d}: [0,T]\\times \\mathcal{SP}\\times \\mathbb{R}^d \\to \\mathbb{S}^+_d, \\, b = (b_i)_{i \\leq d}: [0,T]\\times \\mathcal{SP}\\times \\mathbb{R}^d \\to \\mathbb{R}^d$$\n be $\\mathcal{B}([0,T])\\otimes \\mathcal{B}(\\mathcal{SP})\\otimes \\mathcal{B}(\\mathbb{R}^d) \/ \\mathcal{B}(\\mathbb{R})$-measurable and consider the operator $\\mathcal{L}_{t,\\mu}$ as in (\\ref{1}). \n \\begin{dfn}\\label{Sol NLFPKE}\n \\begin{enumerate}\n \t\\item [(i)] A vaguely continuous curve $(\\mu_t)_{t \\leq T} \\subseteq \\mathcal{SP}$ is a \\textit{subprobability solution to }(\\ref{NLFPKE}), if for each $i, j \\leq d$ the global integrability condition\n \t\t\\begin{equation}\\label{2}\n \t\t\\int_0^T \\int_{\\mathbb{R}^d}|a_{ij}(t,\\mu_t,x)|+|b_i(t,\\mu_t,x)|d\\mu_t(x)dt < +\\infty\n \t\t\\end{equation}holds and for each $\\varphi \\in C^2_c(\\mathbb{R}^d)$ and $t \\in [0,T]$\n \t\t\\begin{equation}\\label{3}\n \t\t\\int_{\\mathbb{R}^d}\\varphi(x)d\\mu_t(x)-\\int_{\\mathbb{R}^d}\\varphi(x)d\\mu_0(x) = \\int_0^t\\int_{\\mathbb{R}^d}\\mathcal{L}_{s,\\mu_s}\\varphi(x)d\\mu_s(x)ds.\n \t\t\\end{equation}\n \n \t\\item[(ii)] A \\textit{probability solution} to (\\ref{NLFPKE}) is a curve $(\\mu_t)_{t \\leq T}\\subseteq \\mathcal{P}$ fulfilling (\\ref{2}) and (\\ref{3}) such that $t \\mapsto \\mu_t$ is weakly continuous.\n \\end{enumerate}\n \\end{dfn}\n Since vaguely continuous curves of measures are in particular Borel curves, all integrals in the above definition are defined. Below we shortly refer to \\textit{subprobability }and \\textit{probability solutions} and keep in mind the respective continuity conditions. In the literature, more general notions of solutions to (\\ref{NLFPKE}) are considered, such as (possibly discontinuous) curves of signed, bounded measures \\cite{bogachev2015fokker}. However, in this work, we restrict attention to continuous (sub-)probability solutions. In presence of the global integrability condition (\\ref{2}), we make the following observation.\n \\begin{rem}\\label{Rem mass conserv}\n \t\\begin{enumerate}\n \t\t\\item [(i)] Any subprobability solution $(\\mu_t)_{t \\leq T}$ with $\\mu_0 \\in \\mathcal{P}$ is a probability solution. Indeed, to prove this it suffices to show $\\mu_t(\\mathbb{R}^d) =1$ for each $t \\leq T$. Since $(\\mu_t)_{t \\leq T}$ fulfills (\\ref{3}), it suffices to choose a sequence $\\varphi_l$, $l \\geq 1$, from $C^2_c(\\mathbb{R}^d)$ with the following properties: $0 \\leq \\varphi_l \\nearrow 1$ pointwise such that $\\partial_i \\varphi_l \\underset{l \\to \\infty}{\\longrightarrow} 0$, $\\partial^2_{ij}\\varphi_l \\underset{l \\to \\infty}{\\longrightarrow}0$ pointwise with all first and second order derivatives bounded by some $M < +\\infty$ uniformly in $l \\geq 1$ and $x \\in \\mathbb{R}^d$. Considering (\\ref{3}) for the limit $l \\to \\infty$, we obtain, by (\\ref{2}) and dominated convergence, for each $t \\in [0,T]$\n \t\t$$\\int_{\\mathbb{R}^d}1 d\\mu_t - \\int_{\\mathbb{R}^d}1 d\\mu_0 = 0$$\n \tand hence the claim.\n \t\t\\item[(ii)] By the above argument, one shows that for any subprobability solution, (\\ref{3}) holds for each $\\varphi \\in C^2_b(\\mathbb{R}^d)$. \n \t\\end{enumerate} \n  \\end{rem}\n  \\subsubsection*{Geometric approach to $\\mathbf{\\mathcal{SP}}$}For our goals, it is preferable to consider $\\mathcal{SP}$ as a manifold-like space. We refer the reader to the appendix in \\cite{Rckner-lin.-paper}, where for the space of probability measures $\\mathcal{P}$ the tangent spaces $T_{\\mu}\\mathcal{P} = L^2(\\mathbb{R}^d,\\mathbb{R}^d;\\mu)$ and a suitable test function class $\\mathcal{F}C^2_b(\\mathcal{P})$, \n  \\begin{equation}\\label{Test fct. Rckner}\n  F \\in \\mathcal{F}C^2_b(\\mathcal{P}) \\iff F: \\mu \\mapsto f\\big(\\mu(\\varphi_1),\\dots,\\mu(\\varphi_n)\\big) \\text{ for }n \\geq 1, f\\in C^1_b(\\mathbb{R}^n), \\, \\varphi_i \\in C^{\\infty}_c(\\mathbb{R}^d),\n  \\end{equation}\n  have been introduced. Further, based on these choices, a natural pointwise definition of the gradient $\\nabla^{\\mathcal{P}}F$ as a section in the tangent bundle $$T\\mathcal{P} = \\bigsqcup_{\\mu \\in \\mathcal{P}}T_{\\mu}\\mathcal{P}$$for $F$ as above is given by\n  $$\\nabla^{\\mathcal{P}}F(\\mu) := \\sum_{k=1}^{n}\\partial_kf\\big(\\mu(\\varphi_1),\\dots,\\mu(\\varphi_n)\\big)\\nabla \\varphi_k \\in T_{\\mu}\\mathcal{P},$$which is shown to be independent of the representation of $F$ in terms of $f$ and $\\varphi_i$. The setting in the present paper is nearly identical, but we consider the manifold-like space $\\mathcal{SP}$ with the vague topology instead of $\\mathcal{P}$ with the weak topology as in \\cite{Rckner-lin.-paper}, because $\\mathcal{SP}$ is embedded in $\\mathbb{R}^{\\infty}$ in the following sense. Let\n  \\begin{equation}\\label{Fct.class G}\n  \\mathcal{G} = \\{g_i, i \\geq 1\\}\n  \\end{equation} be dense in $C^2_c(\\mathbb{R}^d)$ with respect to $||\\cdot||_{C^2_b}$ such that no $g_i$ is constantly $0$. Clearly, any such set of functions is dense in $C_c(\\mathbb{R}^d)$ with respect to uniform convergence and measure-determining. Such sets of functions are sufficiently extensive to characterize the topology of $\\mathcal{SP}$ as well as solutions to (\\ref{NLFPKE}):\n \t\n \t\\begin{lem}\\label{Prop G early}\n \t\tLet $\\mathcal{G}$ be any set of functions with the properties mentioned above and let $(\\mu_n)_{n \\geq 1} \\subseteq \\mathcal{SP}$. Then,\n \t\t\\begin{enumerate}\n \t\t\t\\item [(i)] $(\\mu_n)_{n \\geq 1}$ converges vaguely to $\\mu \\in \\mathcal{SP}$ if and only if\n \t\t\t\\begin{equation*}\n \t\t\t\\mu_n(g_i) \\underset{n \\to \\infty}{\\longrightarrow}\\mu(g_i)\n \t\t\t\\end{equation*}\n \t\t\tfor each $g_i \\in \\mathcal{G}$.\n \t\t\t\\item[(ii)] A vaguely continuous curve $(\\mu_t)_{t \\leq T} \\subseteq \\mathcal{SP}$, which fulfills (\\ref{2}), is a subprobability solution to (\\ref{NLFPKE}) if and only if (\\ref{3}) holds for each $g_i \\in \\mathcal{G}$ in place of $\\varphi$.\n \t\t\\end{enumerate}\n \t\\end{lem}\n \t\\begin{proof}\n \t\t\\begin{enumerate}\n \t\t\t\\item [(i)] From $\\mu_n(g_i) \\underset{n \\to \\infty}{\\longrightarrow}\\mu(g_i)$ for each $g_i \\in \\mathcal{G}$, one obtains for each $f \\in C_c(\\mathbb{R}^d)$ and $\\epsilon>0$ by choosing $g_i \\in \\mathcal{G}$ with $||f-g_i||_{\\infty} < \\frac{\\epsilon}{3}$\n \t\t\t\\begin{equation}\\label{4}\n \t\t\t|\\mu_n(f)-\\mu(f)| \\leq |\\mu_n(f)-\\mu_n(g_i)|+|\\mu_n(g_i)-\\mu(g_i)|+|\\mu(g_i)-\\mu(f)| \\leq \\epsilon\n \t\t\t\\end{equation}for all sufficiently large $n \\geq 1$.\n \t\t\t\\item[(ii)] Let $\\varphi \\in C^2_c(\\mathbb{R}^d)$ be approximated uniformly up to second-order derivatives by a sequence $\\{g_{i_k}\\}_{k \\geq 1}$ from $\\mathcal{G}$. Considering (\\ref{3}) for such $g_{i_k}$ and letting $k \\to \\infty$, the result follows by dominated convergence, which applies due to (\\ref{2}).\n \t\t\\end{enumerate}\n \t\\end{proof}\n Considering $\\mathcal{SP}$ as a (infinite-dimensional) manifold-like topological space, any set of functions $\\mathcal{G}$ as above provides a global chart (i.e., an atlas consisting of a single chart) for $\\mathcal{SP}$, as it yields an embedding $\\mathcal{SP} \\subseteq \\mathbb{R}^{\\infty}$ (cf. Lemma \\ref{Lem aux G, J}).\\\\\n  Consider $\\mathbb{R}^{\\infty}$ as a Polish space with the topology of pointwise convergence and the range $G(\\mathcal{SP}) \\subseteq \\mathbb{R}^{\\infty}$ of $G$ as introduced below with its subspace topology. We write $C_TG(\\mathcal{SP})$ for the set of all elements in $C_T\\mathbb{R}^{\\infty}$ with values in $G(\\mathcal{SP})$. For $u \\in [0,T]$, we denote by $e_u$ the canonical projection on $C_T\\mathcal{SP}$\n $$e_u: (\\mu_t)_{t\\leq T} \\mapsto \\mu_u$$\n and, likewise, by $e^{\\infty}_u$ the projection on $C_T\\mathbb{R}^{\\infty}$. Subsequently, without further mentioning, we consider the spaces $C_T\\mathcal{SP}$ and $C_T\\mathbb{R}^{\\infty}$ with $\\sigma$-algebras\n $$\\mathcal{B}(C_T\\mathcal{SP}) = \\sigma(e_t, t\\in [0,T]) \\text{ and }\\mathcal{B}(C_T\\mathbb{R}^{\\infty}) = \\sigma(e_t^{\\infty}, t \\in [0,T]),$$\n respectively. These algebras coincide with the Borel $\\sigma$-algebras with respect to the topology of uniform convergence (because both $\\mathcal{SP}$ and $\\mathbb{R}^{\\infty}$ are Polish). Also, consider $C_TG(\\mathcal{SP})$ with the natural subspace $\\sigma$-algebra of $\\mathcal{B}(C_T\\mathbb{R}^{\\infty})$. We refer to these $\\sigma$-algebras as the \\textit{canonical $\\sigma$-algebras} on the respective spaces and denote the set of probability measures on the respective $\\sigma$-algebras by $\\mathcal{P}(C_T\\mathcal{SP})$ and $\\mathcal{P}(C_T\\mathbb{R}^{\\infty})$.\n \\begin{lem}\\label{Lem aux G, J}\n \tLet $\\mathcal{G} = \\{g_i\\}_{i \\geq 1} $ be a set of functions as in (\\ref{Fct.class G}).\n \t\\begin{enumerate}\n \t\t\\item [(i)] The map $G$, depending on $\\mathcal{G}$, \n \t\t\\begin{equation}\\label{Def map G}\n \t\tG : \\mathcal{SP} \\to \\mathbb{R}^{\\infty}, G(\\mu) := (\\mu(g_i))_{i \\geq 1}\n \t\t\\end{equation}\n \t\tis a homeomorphism between $\\mathcal{SP}$ and its range $G(\\mathcal{SP})$ (hence, formally, a \\textup{global chart} for $\\mathcal{SP}$). In particular, $G(\\mathcal{SP}) \\subseteq \\mathbb{R}^{\\infty}$ is compact. Moreover, if $\\mathcal{G}' = \\{g_i', i \\geq 1\\}$ is another set as in (\\ref{Fct.class G}) with corresponding chart $G'$, then $G' = G \\circ \\mathcal{V}$ for a unique homeomorphism $\\mathcal{V}$ on $\\mathcal{SP}$.\n \t\t\\item[(ii)] The map\n \t\t$$J : C_T\\mathcal{SP} \\to C_T\\mathbb{R}^{\\infty},\\, J((\\mu_t)_{t \\leq T}) := G(\\mu_t)_{t \\leq T}$$\n \t\tis measurable and one-to-one with measurable inverse $J^{-1}: C_TG(\\mathcal{SP}) \\to C_T\\mathcal{SP}$. Further, $C_TG(\\mathcal{SP}) \\subseteq C_T\\mathbb{R}^{\\infty}$ is a measurable set, i.e. $C_TG(\\mathcal{SP}) \\subseteq \\mathcal{B}(C_T\\mathbb{R}^{\\infty})$.\n \t\\end{enumerate}\n \\end{lem}\n \\begin{proof}\n \t\\begin{enumerate}\n \t\t\\item [(i)] The continuity of $G$ is obvious by definition of the vague topology on $\\mathcal{SP}$ and since $\\mathcal{G} \\subseteq C_c(\\mathbb{R}^d)$. Since $\\mathcal{SP}$ is compact with respect to the vague topology, compactness of $G(\\mathcal{SP}) \\subseteq \\mathbb{R}^{\\infty}$ follows. $\\mathcal{G}$ is measure-determining on $\\mathbb{R}^d$, which implies that $G$ is one-to-one. Since by definition \n \t\t$$G(\\mu_n) \\underset{n \\to \\infty}{\\longrightarrow} G(\\mu) \\iff \\mu_n(g_i) \\underset{n \\to \\infty}{\\longrightarrow} \\mu(g_i) \\text{ for each }g_i \\in \\mathcal{G},$$ \n \t\tcontinuity of $G^{-1}$ follows from Lemma \\ref{Prop G early} (i). The final assertion follows, since for $G'$ as in the assertion, $\\mathcal{V}:= G^{-1}\\circ \\mathcal{W}\\circ G'$ with $\\mathcal{W}:G'(\\mathcal{SP}) \\to G(\\mathcal{SP})$, $\\mathcal{W}: (\\mu(g'_i))_{i \\geq 1} \\mapsto (\\mu(g_i))_{i \\geq 1}$ is a homeomorphism.\n \t\t\\item[(ii)] Since $G$ is one-to-one and measurable, so is $J$. Clearly, $C_TG(\\mathcal{SP})$ is the range of $J$ and hence $J: C_T\\mathcal{SP} \\to C_TG(\\mathcal{SP})$ is a bijection between standard Borel spaces (the latter, because $\\mathcal{SP}$ and $G(\\mathcal{SP})$ with the respective topologies are Polish). This yields the measurability of $J^{-1}$. Finally, closedness of $G(\\mathcal{SP}) \\subseteq \\mathbb{R}^{\\infty}$ implies that $C_TG(\\mathcal{SP})\\subseteq C_T\\mathbb{R}^{\\infty}$ is a measurable set, because $G(\\mathcal{SP})$ carries the subspace topology inherited from $\\mathbb{R}^{\\infty}$. \n \t\\end{enumerate}\n \\end{proof}\n By part (i) of the previous lemma it is justified to fix a set $\\mathcal{G} = \\{g_i, i \\geq 1\\}$ for the remainder of the section. In order to switch between test functions on $\\mathcal{SP}$ and $\\mathbb{R}^{\\infty}$ in an equivalent way, we slightly deviate from the test function class presented in \\cite{Rckner-lin.-paper} (see (\\ref{Test fct. Rckner})) and, instead, consider\n $$\\mathcal{F}C^2_b(\\mathcal{G}) := \\{F : \\mathcal{SP}\\to \\mathbb{R} | F(\\mu) =  f\\big(\\mu(g_1),\\dots,\\mu(g_n)\\big), f \\in C^2_b(\\mathbb{R}^n), n \\geq 1 \\},$$where the restriction $f \\in C^2_b(\\mathbb{R}^n)$ is made for consistency with the stochastic case later on only. We summarize our geometric interpretation of $\\mathcal{SP}$, which is of course still a close adaption of the ideas presented in \\cite{Rckner-lin.-paper}:\\\\\n For the manifold-like space $\\mathcal{SP}$, we consider smooth test functions $F \\in \\mathcal{F}C^2_b(\\mathcal{G})$, with $\\mathcal{G}$ being fixed as in (\\ref{Fct.class G}). For each $\\mu \\in \\mathcal{SP}$, we have the tangent space $T_{\\mu}\\mathcal{SP} = L^2(\\mathbb{R}^d,\\mathbb{R}^d;\\mu)$ and the gradient\n $$\\nabla ^{\\mathcal{SP}}F(\\mu) = \\sum_{k=1}^{n}\\partial_kf\\big(\\mu(g_1),\\dots,\\mu(g_n)\\big)\\nabla g_k \\in T_{\\mu}\\mathcal{SP}$$ for $\\mathcal{F}C^2_b(\\mathcal{G}) \\ni F: \\mu \\mapsto f\\big(\\mu(g_1),\\dots,\\mu(g_n)\\big)$ as a section in the tangent bundle $T\\mathcal{SP}$, which is independent of the representation of $F$. Adding to the approach of $\\mathcal{SP}$ as a manifold-like space, the global chart $G$ as in (\\ref{Def map G}) embeds $\\mathcal{SP}$ into $\\mathbb{R}^{\\infty}$. However, we do not rigorously treat $\\mathcal{SP}$ as a (Fr\u00e9chet-)manifold and consider the embedding $\\mathcal{SP} \\subseteq \\mathbb{R}^{\\infty}$ merely as a tool to transfer (\\ref{NLFPKE}) and its corresponding continuity equation to equivalent equations over $\\mathbb{R}^{\\infty}$, as outlined below. \n\\subsubsection*{The continuity equation (\\ref{P-CE})}\n As mentioned in the introduction, we study the linear continuity equation associated to (\\ref{NLFPKE}) as derived in \\cite{Rckner-lin.-paper}, which is a first-order equation for curves of measures on $\\mathcal{SP}$. More precisely, in analogy to the derivation in \\cite{Rckner-lin.-paper}, it is readily seen that any subprobability solution $(\\mu_t)_{t \\leq T}$ to (\\ref{NLFPKE}) induces a curve of elements in $\\mathcal{P}(\\mathcal{SP})$, $\\Gamma_t := \\delta_{\\mu_t}$, $t \\leq T$, with\n \\begin{equation}\\label{Solved eq1}\n \\int_{\\mathcal{SP}}F(\\mu)d\\Gamma_t(\\mu)- \\int_{\\mathcal{SP}}F(\\mu)d\\Gamma_0(\\mu) = \\int_0^t \\int_{\\mathcal{SP}}\\big \\langle \\nabla^{\\mathcal{SP}}F(\\mu), b(s,\\mu)+a(s,\\mu)\\nabla\\big \\rangle_{L^2(\\mu)}d\\Gamma_s(\\mu)ds\n \\end{equation}for each $t \\leq T$ and $F \\in \\mathcal{F}C^2_b(\\mathcal{G})$. Here, we set $b(s,\\mu) = b(s,\\mu,\\cdot): \\mathbb{R}^d \\to \\mathbb{R}^d$ (similarly for $a(s,\\mu)$), $L^2(\\mu) = L^2(\\mathbb{R}^d,\\mathbb{R}^d;\\mu)$ and abbreviated\n $$\\big \\langle \\nabla^{\\mathcal{SP}}F(\\mu), b(s,\\mu)+a(s,\\mu)\\nabla\\big \\rangle_{L^2(\\mu)} = \\int_{\\mathbb{R}^d}\\sum_{k=1}^{n}(\\partial_kf)\\big(\\mu(g_1),\\dots,\\mu(g_n)\\big) a_{ij}(s,\\mu,x)\\partial^2_{ij}g_k(x)+b_i(s,\\mu,x)\\partial_i g_k(x)d\\mu(x).$$\n We rewrite (\\ref{Solved eq1}) in distributional form in duality with $\\mathcal{F}C^2_b(\\mathcal{G})$ as\n \\begin{equation*}\n \\partial_t \\Gamma_t = -\\nabla^{\\mathcal{SP}}\\cdot([b_t+a_t\\nabla]\\Gamma_t), \\,\\, t \\leq T.\n \\end{equation*}Setting\n \\begin{equation}\\label{5}\n \\mathbf{L}_tF(\\mu) := \\big\\langle a(t,\\mu)\\nabla+b(t,\\mu), \\nabla^{\\mathcal{SP}}F(\\mu)\\big\\rangle_{L^2(\\mu)},\n \\end{equation}this is just the linear continuity equation (\\ref{P-CE}). The term $a\\nabla$ has rigorous meaning only, if $a$ has sufficiently regular components in order to put the derivative $\\nabla$ on $a$ via integration by parts, which we do not assume at any point.\n Considering $\\mathcal{SP}$ as a manifold-like space, one may formally regard to $a\\nabla + b$ as a time-dependent section in the tangent bundle $T\\mathcal{SP}$.\\\\\n More generally, we introduce the following notion of solution to (\\ref{P-CE}) (see \\cite{Rckner-lin.-paper}):\n \\begin{dfn}\\label{Def sol P-CE}\n \tA weakly continuous curve $(\\Gamma_t)_{t \\leq T} \\subseteq \\mathcal{P}(\\mathcal{SP})$ is a \\textit{solution to }(\\ref{P-CE}), if the integrability condition\n \t\t\\begin{equation}\\label{aux_revised1}\n \t\t\\int_0^T\\int_{\\mathcal{SP}}||b(t,\\mu,\\cdot)||_{L^1(\\mathbb{R}^d,\\mathbb{R}^d;\\mu)}+||a(t,\\mu,\\cdot)||_{L^1(\\mathbb{R}^d,\\mathbb{R}^{d^2};\\mu)}d \\Gamma_t(\\mu)dt < +\\infty\n \t\t\\end{equation} is fulfilled and for each $F \\in \\mathcal{F}C^2_b(\\mathcal{G})$ and $t \\in [0,T]$\n \t\t\\begin{equation}\\label{6}\n \t\t\\int_{\\mathcal{SP}}F(\\mu)d\\Gamma_t(\\mu)- \\int_{\\mathcal{SP}}F(\\mu)d\\Gamma_0(\\mu) = \\int_0^t \\int_{\\mathcal{SP}}\\mathbf{L}_sF(\\mu)d\\Gamma_s(\\mu)ds\n \t\t\\end{equation}holds (which is just (\\ref{Solved eq1})).\n\n \\end{dfn}\n The choice of $\\mathcal{G}$ as in (\\ref{Fct.class G}) implies that any solution in the above sense fulfills (\\ref{6}) even for each $F \\in \\mathcal{F}C^2_b(\\mathcal{SP})$, i.e. for the larger class of test functions considered in \\cite{Rckner-lin.-paper} (upon extending their domain from $\\mathcal{P}$ to $\\mathcal{SP}$). In particular, this notion of solution is independent of $\\mathcal{G}$. The main result of this chapter, Theorem \\ref{main thm det case}, states that any solution to (\\ref{P-CE}) as in Definition \\ref{Def sol P-CE} arises as a superposition of solutions to (\\ref{NLFPKE}). Note that for $\\nu \\in \\mathcal{SP}$, uniqueness of solutions $(\\Gamma_t)_{t \\leq T}$ to (\\ref{P-CE}) with $\\Gamma_0 = \\delta_{\\nu}$ implies uniqueness of subprobability solutions $(\\mu_t)_{t \\leq T}$ to (\\ref{NLFPKE}) with $\\mu_0 = \\nu$.\n \n \\subsubsection*{Transferring (\\ref{NLFPKE}) and (\\ref{P-CE}) to $\\mathbb{R}^{\\infty}$}\n We use the global chart $G: \\mathcal{SP} \\to \\mathbb{R}^{\\infty}$ and the map $J$ of Lemma \\ref{Lem aux G, J} to reformulate both (\\ref{NLFPKE}) and (\\ref{P-CE}) on $\\mathbb{R}^{\\infty}$. Define a Borel vector field $\\bar{B} = (\\bar{B}_k)_{k \\in \\mathbb{N}}$ component-wise as follows. For $t \\in [0,T]$, consider the Borel set $A_t \\in \\mathcal{B}(\\mathcal{SP})$,\n $$A_t := \\bigg\\{\\mu \\in \\mathcal{SP}: \\int_{\\mathbb{R}^d}|a_{ij}(t,\\mu,x)+|b_i(t,\\mu,x)|d\\mu(x)< \\infty \\,\\, \\forall i,j \\leq d\\bigg\\}$$\n and define $B := (B_k)_{k \\in \\mathbb{N}}$ via\n $$B_k(t,\\mu):= \\int_{\\mathbb{R}^d}\\mathcal{L}_{t,\\mu}g_k(x)d\\mu(x),\\quad (t,\\mu) \\in [0,T]\\times A_t.$$\n Now define $\\bar{B}: [0,T]\\times \\mathbb{R}^\\infty \\to \\mathbb{R}^\\infty$ via\n $$\\bar{B}(t,z):= \\begin{cases}\n \tB(t,G^{-1}(z)),&\\quad \\text{ if }z \\in G(A_t)\\\\\n \t0,&\\quad \\text{ else,}\n \\end{cases}$$\nwhich is Borel measurable by Lemma \\ref{Lem aux G, J}.  Next, consider the differential equation on $\\mathbb{R}^{\\infty}$\n \\begin{equation}\\label{Rinfty-ODE}\\tag{$\\mathbb{R}^{\\infty}$-ODE}\n \\frac{d}{dt} z_t = \\bar{B}(t,z_t), \\,\\,t \\in [0,T],\n \\end{equation}which turns out to be the suitable analogue to (\\ref{NLFPKE}) on $\\mathbb{R}^{\\infty}$. Analogously, the corresponding continuity equation for curves of Borel probability measures $\\bar{\\Gamma}_t$ on $\\mathbb{R}^{\\infty}$, i.e.\n \\begin{equation}\\label{Rinfty-CE}\\tag{$\\mathbb{R}^{\\infty}$-CE}\n \\partial_t \\bar{\\Gamma}_t = -\\bar{\\nabla}\\cdot(\\bar{B}\\bar{\\Gamma}_t), \\,\\, t\\in [0,T],\n \\end{equation}with $\\bar{\\nabla}$ as introduced below, is the natural analogue of the linear continuity equation (\\ref{P-CE}). Roughly, these analogies are to be understood in the sense that solutions to (\\ref{NLFPKE}) and (\\ref{P-CE}) can be transferred to solutions to (\\ref{Rinfty-ODE}) and (\\ref{Rinfty-CE}), respectively, via the chart $G$. We refer to the proof of the main result below for more details. Let $$p_i: z \\mapsto z_i,\\,\\, z \\in \\mathbb{R}^{\\infty}$$ denote the canonical projection to the $i$-th component, set $\\pi_n = (p_1,\\dots,p_n)$ and \n $$\\mathcal{F}C^2_b(\\mathbb{R}^{\\infty}) := \\{\\bar{F}: \\mathbb{R}^{\\infty} \\to \\mathbb{R} | \\bar{F}= f \\circ \\pi_n, f \\in C^2_b(\\mathbb{R}^n), n \\geq 1\\}.$$\n By $\\bar{\\nabla}$ we denote the gradient-type operator on $\\mathbb{R}^{\\infty}$, acting on $\\bar{F} = f \\circ \\pi_n \\in \\mathcal{F}C^2_b(\\mathbb{R}^{\\infty})$ via\n \\begin{equation}\\label{Def nabla gradient}\n \\bar{\\nabla}\\bar{F}(z):= \\big((\\partial_1f)(\\pi_nz),\\dots,(\\partial_nf) (\\pi_nz),0,0,\\dots\\big).\n \\end{equation}Again, the restriction to test functions possessing second-order derivatives is made in order to be consistent with the stochastic (second-order) case later on.\n \\begin{dfn}\\label{Def sol Rinfty-eq}\n \t\\begin{enumerate}\n \t\t\\item [(i)] A curve $(z_t)_{t \\leq T} = ((p_i\\circ z_t)_{i \\geq 1})_{t \\leq T} \\in C_T\\mathbb{R}^{\\infty}$ is a \\textit{solution to} (\\ref{Rinfty-ODE}), if for each $i \\geq 1$ the $\\mathbb{R}$-valued curve $t \\mapsto p_i \\circ z_t$ is absolutely continuous with weak derivative $t \\mapsto p_i \\circ \\bar{B}(t,z_t)$ $dt$-a.s.\n \t\t\\item[(ii)] A curve $(\\bar{\\Gamma}_t)_{t \\leq T} \\subseteq \\mathcal{P}(\\mathbb{R}^{\\infty})$ is a \\textit{solution to }(\\ref{Rinfty-CE}), if it is weakly continuous, fulfills the integrability condition\n \t\t\\begin{equation}\\label{8}\n \t\t\\int_0^T \\int_{\\mathbb{R}^{\\infty}} |\\bar{B}_k(t,z)|d\\bar{\\Gamma}_t(z)dt < +\\infty \\text{ for each }k \\geq 1\n \t\t\\end{equation} and for each $\\bar{F}\\in \\mathcal{F}C^2_b(\\mathbb{R}^{\\infty})$ the identity\n \t\t$$\\int_{\\mathbb{R}^{\\infty}}\\bar{F}(z)d\\bar{\\Gamma}_t(z)- \\int_{\\mathbb{R}^{\\infty}}\\bar{F}(z)d\\bar{\\Gamma}_0(z) = \\int_0^t \\int_{\\mathbb{R}^{\\infty}}\\bar{\\nabla} \\bar{F}(z) \\cdot \\bar{B}(s,z)d\\bar{\\Gamma}_s(z)ds$$\n \t\tholds for all $t \\in [0,T]$.\n \t\\end{enumerate}\n \\end{dfn}\n \\subsection{Main Result: Deterministic case}\nThe following theorem is the main result for the deterministic case.\n \\begin{theorem}\\label{main thm det case}\n \tLet $a,b$ be Borel coefficients on $[0,T]\\times \\mathcal{SP}\\times \\mathbb{R}^d$. For any weakly continuous solution $(\\Gamma_t)_{t \\leq T}$ to (\\ref{P-CE}) in the sense of Definition \\ref{Def sol P-CE}, there exists a probability measure $\\eta \\in \\mathcal{P}(C_T\\mathcal{SP})$, which is concentrated on vaguely continuous subprobability solutions to (\\ref{NLFPKE}) such that\n \t$$\\eta \\circ e_t^{-1} = \\Gamma_t,\\,\\, t \\in [0,T].$$\\\\\n \tMoreover, if $\\Gamma_0 \\in \\mathcal{P}(\\mathcal{P})$, then $\\eta$ is concentrated on weakly continuous probability solutions to (\\ref{NLFPKE}).\n \\end{theorem}\n \n The proof relies on a superposition principle for measure-valued solution curves of continuity equations on $\\mathbb{R}^{\\infty}$ and its corresponding differential equation, which we recall in Proposition \\ref{Sp-pr. Rinfty prop} below. More precisely, we proceed in three steps. First, we transfer $(\\Gamma_t)_{t \\leq T}$ to a solution $(\\bar{\\Gamma}_t)_{t \\leq T}$ to (\\ref{Rinfty-CE}). Then, by Proposition \\ref{Sp-pr. Rinfty prop} below we obtain a measure $\\bar{\\eta} \\in \\mathcal{P}(C_T\\mathbb{R}^{\\infty})$ with $\\bar{\\eta} \\circ (e^{\\infty}_t)^{-1} = \\bar{\\Gamma_t}$, which is concentrated on solution curves to (\\ref{Rinfty-ODE}). Finally, we transfer $\\bar{\\eta}$ back to a measures $\\eta \\in \\mathcal{P}(C_T\\mathcal{SP})$ with the desired properties. Below, we denote by $\\mathcal{F}C^1_b(\\mathbb{R}^{\\infty})$ the set of test functions of same type as in $\\mathcal{F}C^2_b(\\mathbb{R}^{\\infty})$, but with $f \\in C^1_b(\\mathbb{R}^n)$ in place of $f \\in \\mathcal{F}C^2_b(\\mathbb{R}^n)$.\n \\begin{prop}\\label{Sp-pr. Rinfty prop} [Superposition principle on $\\mathbb{R}^{\\infty}$, Thm. 7.1. \\cite{ambrosio2014}]\n \tLet $(\\bar{\\Gamma}_t)_{t \\leq T}$ be a solution to (\\ref{Rinfty-CE}) in the sense of Definition \\ref{Def sol Rinfty-eq} (ii) with test functions $\\mathcal{F}C^1_b(\\mathbb{R}^{\\infty})$ instead of $\\mathcal{F}C^2_b(\\mathbb{R}^{\\infty})$. Then, there exists a Borel measures $\\bar{\\eta} \\in \\mathcal{P}(C_T\\mathbb{R}^{\\infty})$ concentrated on solutions to (\\ref{Rinfty-ODE}) in the sense of Definition \\ref{Def sol Rinfty-eq} (i) such that $$\\bar{\\eta} \\circ (e^{\\infty}_t)^{-1} = \\bar{\\Gamma}_t, \\,\\,t \\leq T.$$\n \\end{prop}\n We proceed to the proof of the main result.\\\\\n \\\\\n \\textbf{Proof of Theorem \\ref{main thm det case}:} Let $\\Gamma = (\\Gamma_t)_{t \\leq T}$ be a weakly continuous solution to (\\ref{P-CE}) as in Definition \\ref{Def sol P-CE}. \\\\\n \\textbf{Step 1: From (\\ref{P-CE}) to (\\ref{Rinfty-CE}):} Set \n $$\\bar{\\Gamma}_t := \\Gamma_t \\circ G^{-1},$$\n with $G$ as in Lemma \\ref{Lem aux G, J}, which corresponds to the fixed set of functions $\\mathcal{G}$. Since $G$ is continuous, $(\\bar{\\Gamma}_t)_{t \\leq T}$ is a weakly continuous curve of Borel subprobability measures on $\\mathbb{R}^{\\infty}$. We show that $(\\bar{\\Gamma}_t)_{t \\leq T}$ solves (\\ref{Rinfty-CE}). Indeed, the integrability condition (\\ref{8}) is fulfilled, since $(\\Gamma_t)_{t \\leq T}$ fulfills Definition \\ref{Def sol P-CE}. Further, since $\\Gamma$ solves (\\ref{P-CE}), we have for any $\\mathcal{F}C^2_b(\\mathcal{G}) \\ni F: \\mu \\mapsto f\\big(\\mu(g_1),\\dots,\\mu(g_n)\\big)$ and $t \\in [0,T]$\n \\begin{equation}\\label{extra1}\n \\int_0^t \\int_{\\mathcal{SP}}\\mathbf{L}_sF(\\mu)d\\Gamma_s(\\mu)ds = \\int_{\\mathcal{SP}}F(\\mu)d\\Gamma_t(\\mu) - \\int_{\\mathcal{SP}}F(\\mu)d\\Gamma_0(\\mu)\n \\end{equation}\tand hence, abbreviating $p_k \\circ B(t,\\cdot)$ by $B^k_t$ and setting $\\bar{F} = f \\circ \\pi_n$ for $f$ as above, we have\n \\begin{align*}\n \\int_0^t \\int_{\\mathcal{SP}}\\mathbf{L}_sF(\\mu)d\\Gamma_s(\\mu)ds &= \\int_0^t \\int_{\\mathcal{SP}}\\sum_{k =1}^{n} (\\partial_kf)\\big(\\mu(g_1),\\dots,\\mu(g_n)\\big) \\bigg(\\int_{\\mathbb{R}^d}\\mathcal{L}_{s,\\mu}g_k(x)d\\mu(x)\\bigg) \\Gamma_s(\\mu)ds \\\\& =\n \\int_0^t \\int_{\\mathcal{SP}} \\sum_{k=1}^n (\\partial_k f)\\big(\\mu(g_1),\\dots,\\mu(g_n)\\big)B^k_s(\\mu)d\\Gamma_s(\\mu)ds \\\\&\n = \\int_0^t \\int_{\\mathcal{SP}} \\sum_{k=1}^n(\\partial_k f)\\big(p_1 \\circ G(\\mu),\\dots,p_n\\circ G(\\mu)\\big)\\bar{B}^k_s \\circ G(\\mu)d\\Gamma_s(\\mu)ds\\\\&\n =\\int_0^t \\int_{\\mathbb{R}^{\\infty}} \\nabla \\bar{F}(z) \\cdot \\bar{B}_s(z)\\bar{\\Gamma}_s(z)ds\n \\end{align*}and, furthermore, for each $s \\in [0,T]$\n $$\\int_{\\mathcal{SP}}F(\\mu)d\\Gamma_s(\\mu) = \\int_{\\mathcal{SP}}f\\big(p_1 \\circ G(\\mu),\\dots,p_n \\circ G(\\mu)\\big)d\\Gamma_s(\\mu) = \\int_{\\mathbb{R}^{\\infty}}\\bar{F}(z)d\\bar{\\Gamma}_s(z).$$\n Comparing with (\\ref{extra1}), it follows that $(\\bar{\\Gamma}_t)_{t \\leq T}$ is a solution to (\\ref{Rinfty-CE}) as claimed, because $F \\in \\mathcal{F}C^2_b(\\mathcal{G})$ was arbitrary and hence $\\bar{F}$ as above is arbitrary in $\\mathcal{F}C^2_b(\\mathbb{R}^{\\infty})$. By standard approximation, one extends the above equation to test functions $\\bar{F}$ from $\\mathcal{F}C^1_b(\\mathbb{R}^\\infty)$.\\\\ \n \\\\\n \\textbf{Step 2: From (\\ref{Rinfty-CE}) to (\\ref{Rinfty-ODE}):} Proposition \\ref{Sp-pr. Rinfty prop} implies the existence of a measure $\\bar{\\eta} \\in \\mathcal{P}(C_T\\mathbb{R}^{\\infty})$ such that\n \\begin{enumerate}\n \t\\item [(i)] $\\bar{\\eta} \\circ (e^{\\infty}_t)^{-1} = \\bar{\\Gamma}_t$ for each $t \\in [0,T]$\n \t\\item[(ii)] $\\bar{\\eta}$ is concentrated on solution paths of (\\ref{Rinfty-ODE}).\n \\end{enumerate}\n\\textbf{Step 3: From (\\ref{Rinfty-ODE}) to (\\ref{NLFPKE}):} We show that the measure $\\eta := \\bar{\\eta} \\circ (J^{-1})^{-1}$, with $J$ as in Lemma \\ref{Lem aux G, J} fulfills all desired properties. Indeed, since\n$$\\bar{\\eta} \\circ (e^{\\infty}_t)^{-1} = \\bar{\\Gamma}_t = \\Gamma_t \\circ G^{-1},$$\nfor each $t \\in [0,T]$ we deduce that $\\bar{\\eta} \\circ (e^{\\infty}_t)^{-1}$ is concentrated on $G(\\mathcal{SP})$. By Lemma \\ref{Lem aux G, J}, $G(\\mathcal{SP}) \\subseteq \\mathbb{R}^{\\infty}$ is closed. Since by construction $\\bar{\\eta}$ is concentrated on continuous curves in $\\mathbb{R}^{\\infty}$, $\\bar{\\eta}$ is concentrated on $C_TG(\\mathcal{SP})$. Further, $C_TG(\\mathcal{SP}) \\subseteq C_T\\mathbb{R}^{\\infty}$ is a measurable set and $J^{-1}: C_TG(\\mathcal{SP}) \\to C_T\\mathcal{SP}$ is measurable by Lemma \\ref{Lem aux G, J}. Therefore, we may define $\\eta \\in \\mathcal{P}(C_T\\mathcal{SP})$ via\n$$ \\eta := \\bar{\\eta} \\circ (J^{-1})^{-1}.$$\nIt remains to verify $\\eta \\circ e_t^{-1} = \\Gamma_t$ for all $t \\in [0,T]$ and that $\\eta$ is concentrated on subprobability solutions to (\\ref{NLFPKE}). Concerning the first matter, we have\n$$ \\eta \\circ e_t^{-1} = \\bar{\\eta} \\circ (J^{-1})^{-1} \\circ e_t^{-1} = \\bar{\\eta} \\circ (e_t \\circ J^{-1})^{-1}$$\nand\n$$\\Gamma_t = \\Gamma_t \\circ (G^{-1} \\circ G)^{-1} = \\bar{\\Gamma}_t \\circ (G^{-1})^{-1} = \\bar{\\eta} \\circ (G^{-1}\\circ e_t^{\\infty})^{-1}.$$\nSince $e_t\\circ J^{-1}$ and $G^{-1} \\circ e_t^{\\infty}$ coincide as measurable maps on $C_TG(\\mathcal{SP})$ and it was shown above that $\\bar{\\eta}$ is concentrated on $C_TG(\\mathcal{SP})$, we obtain\n$$\\eta \\circ e_t^{-1} = \\Gamma_t, \\,\\, t \\leq T.$$\n\nConcerning the second aspect, note that  by definition of $\\eta$ and $\\bar{\\Gamma}_t$ and by the equality $e_t \\circ J^{-1} = G^{-1}\\circ e_t^\\infty$,  \\eqref{aux_revised1} for $\\Gamma$ implies that $\\eta$ is concentrated on vaguely continuous curves $t \\mapsto \\mu_t$ in $\\mathcal{SP}$ with the global integrability property \\eqref{2}  such that $t \\mapsto G(\\mu_t)$ is a solution to \\eqref{Rinfty-ODE}. Each such curve $t \\mapsto \\mu_t$ is a subprobability solution to \\eqref{NLFPKE}. Indeed, due to $\\mu_t \\in A_t$ $dt$-a.s., we have \n\\begin{align*}\n\\frac{d}{dt}p_k \\circ G(\\mu_t) =&\\, p_k \\circ \\bar{B}(t,G(\\mu_t))\\quad dt-a.s. \\iff \\frac{d}{dt}p_k \\circ G(\\mu_t) = p_k \\circ B(t,\\mu_t)\\quad dt-a.s. \\\\&\n\\iff \\frac{d}{dt}\\int_{\\mathbb{R}^d}g_k(x)d\\mu_t(x) = \\int_{\\mathbb{R}^d}\\mathcal{L}_{t,\\mu_t}g_k(x)d\\mu_t(x) \\quad dt-a.s.\\\\&\n\\iff \\int_{\\mathbb{R}^d}g_k d\\mu_t- \\int_{\\mathbb{R}^d}g_k d\\mu_0 = \\int_0^t \\int_{\\mathbb{R}^d}\\mathcal{L}_{s,\\mu_s}g_k (x)d\\mu_s(x)ds, \\,\\, t \\in [0,T],\n\\end{align*}\nand Lemma \\ref{Prop G early} (ii) applies.\nIt remains to prove the additional assertion about probability solutions. To this end, assume $\\Gamma_0$ is concentrated on $\\mathcal{P}$. Then, $\\eta(e_0 \\in \\mathcal{P}) = 1$ and hence the claim follows by Remark \\ref{Rem mass conserv}. \\qed \\\\\n\\\\\nThe final assertion of the theorem in particular implies: If $\\Gamma_0 \\in \\mathcal{P}(\\mathcal{P})$ for a weakly continuous solution $(\\Gamma_t)_{t \\leq T} \\subseteq \\mathcal{P}(\\mathcal{SP})$ to (\\ref{P-CE}), then $\\Gamma_t \\in \\mathcal{P}(\\mathcal{P})$ for each $t \\leq T$. Of course, this is to be expected due to the global integrability condition in Definition \\ref{Def sol P-CE}.\n\\begin{rem}\\label{Rem explain why SP}\nFinally, let us explain why we developed the above result for subprobability solutions to (\\ref{NLFPKE}) although our principal interest is restricted to probability solutions. If we directly consider solution curves $(\\Gamma_t)_{t \\leq T}$ to (\\ref{P-CE}) with $\\Gamma_t \\in \\mathcal{P}(\\mathcal{P})$, we cannot prove that $\\eta$ in Theorem \\ref{main thm det case} is concentrated on $C_T\\mathcal{P}$ (in fact, not even $\\eta(C_T\\mathcal{P}) >0$ could be shown). Indeed, inspecting the proof above, one may only prove that $\\eta \\circ e_t^{-1}$ is concentrated on $\\mathcal{P}$ for each $t\\leq T$. But since $\\mathcal{P} \\subseteq \\mathcal{SP}$ is not closed, curves in the support of $\\eta$ may be proper subprobability-valued at single times. The deeper reason for this is that the range $G(\\mathcal{P})$ of $G$ as in \\ref{Prop G early} as a map on $\\mathcal{P}$ with the weak topology is not closed in $\\mathbb{R}^{\\infty}$. It seems that one cannot resolve this issue by simply changing the function set $\\mathcal{G}$, since there exists no countable set of functions, which allows for a characterization of weak instead of vague convergence as in Lemma \\ref{Prop G early}.\nSince $\\mathcal{SP}$ with the vague topology is compact and the vague test function class $C_c(\\mathbb{R}^d)$ is separable, it is feasible to carry out the entire development for subprobability measures as above.\n\nWe also mention that to our understanding there is no inherent reason why the superposition principle could not be extended to larger spaces of measures (e.g. spaces of signed measures), as long as its topology allows for a suitable identification with $\\mathbb{R}^\\infty$ as in our present case. Our principal motivation from a probabilistic viewpoint was to study curves of probability measures, and we were only forced to extend to $\\mathcal{SP}$, the vague closure of $\\mathcal{P}$, by the reasons outlined above. In order to replace $\\mathcal{SP}$ by some larger space of measures $\\mathcal{M}$, it seems indispensable that Lemma \\ref{Lem aux G, J} remains true, i.e. that the range of $\\mathcal{M}$ under a suitable homeomorphism  is closed in $\\mathbb{R}^\\infty$.\n\n\\end{rem}\n\n\\subsection{Consequences and applications}\nThe following existence- and uniqueness results immediately follow from the superposition principle Theorem \\ref{main thm det case} and provide an equivalence between the nonlinear FPK-equation (\\ref{NLFPKE}) and its linearized continuity equation (\\ref{P-CE}). \n\\begin{kor}\n\tLet $\\mu_0 \\in \\mathcal{SP}$ and assume there exists a solution to (\\ref{P-CE}) with initial condition $\\delta_{\\mu_0}$. Then, there exists a subprobability solution to (\\ref{NLFPKE}) with initial condition $\\mu_0$. Moreover, if $\\mu_0\\in \\mathcal{P}$, then there exists a probability solution to (\\ref{NLFPKE}) with initial condition $\\mu_0$.\n\\end{kor}\n\\begin{proof}\n\tBy Theorem \\ref{main thm det case} there exists a  probability measure $\\eta$ concentrated on subprobability solutions to (\\ref{NLFPKE}) with $\\eta \\circ e_0^{-1} = \\delta_{\\mu_0}$. Hence, at least one such solution to (\\ref{NLFPKE}) with initial condition $\\mu_0$ exists. The second assertion is treated similarly.\n\\end{proof}\n\\begin{kor}\\label{Cor uniqueness det}\n\tLet $\\mu_0 \\in \\mathcal{SP}$ and assume there exists at most one vaguely continuous subprobability solution to (\\ref{NLFPKE}) with initial condition $\\mu_0$. Then, there exists also at most one weakly continuous solution $(\\Gamma_t)_{t \\leq T}$ to (\\ref{P-CE}) with initial condition $\\delta_{\\mu_0}$. If $\\mu_0 \\in \\mathcal{P}$, then, in the case of existence, $\\Gamma_t(\\mathcal{P}) = 1$ for each $t \\in [0,T]$.\n\\end{kor}\n\\begin{proof}\n\tLet $\\Gamma^{(1)}$ and $\\Gamma^{(2)}$ be weakly continuous solutions to (\\ref{P-CE}) with $\\Gamma^{(i)}_0 = \\delta_{\\mu_0}$ for $i \\in \\{1,2\\}$. By Theorem \\ref{main thm det case}, there exist probability measures $\\eta^{(i)}$, $i \\in \\{1,2\\}$, concentrated on subprobability solutions to (\\ref{NLFPKE}) with initial condition $\\mu_0$ such that $\\eta^{(i)} \\circ e_t^{-1} = \\Gamma_t^{(i)}$ for each $t \\in [0,T]$ and $i \\in \\{1,2\\}$. By assumption, we obtain $\\eta^{(1)}= \\delta_{\\mu}= \\eta^{(2)}$ for a unique element $\\mu \\in C_T\\mathcal{SP}$ and thus also $\\Gamma^{(1)} = \\Gamma^{(2)}$. If $\\mu_0 \\in \\mathcal{P}$, then $\\mu \\in C_T\\mathcal{P}$ by Remark \\ref{Rem mass conserv}, which gives the second assertion. \n\\end{proof}\n\\subsubsection{Application to coupled nonlinear-linear Fokker-Planck-Kolmogorov equations}\nUsing the superposition principle, we prove an open conjecture posed in \\cite{Rckner-lin.-paper}. Let us shortly recapitulate the necessary framework. In \\cite{Rckner-lin.-paper}, the authors consider a coupled nonlinear-linear FPK-equation of type \n\\begin{equation}\\label{9}\n\\begin{cases}\n\\partial_t \\mu_t = \\mathcal{L}^*_{t,\\mu_t}\\mu_t \\\\\n\\partial_t \\nu_t = \\mathcal{L}^*_{t,\\mu_t}\\nu_t,\n\\end{cases}\n\\end{equation}\ni.e. comparing to our situation the first nonlinear equation is of type (\\ref{NLFPKE}) and the second (linear) equation is obtained by \"freezing\" a solution $(\\mu_t)_{t \\leq T}$ to the first equation in the nonlinearity spot of $\\mathcal{L}$. For an initial condition $(\\bar{\\mu},\\bar{\\nu}) \\in \\mathcal{P}\\times \\mathcal{P}$, (\\ref{9}) is said to have a \\textit{unique solution}, if there exists a unique probability solution $(\\mu_t)_{t \\leq T}$ to the first equation in the sense of Definition \\ref{Sol NLFPKE} with $\\mu_0 = \\bar{\\mu}$ and a unique weakly continuous curve $(\\nu_t)_{t \\leq T} \\subseteq \\mathcal{P}$, which solves the second equation with fixed coefficient $\\mu_t$ with $\\nu_0 = \\bar{\\nu}$ (we refer to \\cite{Rckner-lin.-paper} for more details). The authors associate a linear continuity equation on $\\mathbb{R}^d \\times \\mathcal{P}$ to (\\ref{9}) in the following sense: Let $\\mathbb{L}$ be the operator acting on functions \n$$\\mathcal{C} := \\big\\{\\Phi: (x,\\mu) \\mapsto \\varphi(x)F(\\mu)  | \\varphi \\in C^2_c(\\mathbb{R}^d), F \\in \\mathcal{F}C^2_b(\\mathcal{P}) \\big\\},$$\nvia \n$$\\mathbb{L}_t\\Phi(x,\\mu) := \\mathcal{L}_{t,\\mu}\\Phi(\\cdot, \\mu)(x)+\\mathbf{L}_t\\Phi(x, \\cdot)(\\mu),$$with $\\mathcal{L}$ as in (\\ref{1}) and $\\mathbf{L}$ as in (\\ref{5}).\nConsider the continuity equation \n\\begin{equation}\\label{10}\n\\partial_t \\Lambda_t = \\mathbb{L}_t^*\\Lambda_t, \\,\\, t \\in [0,T]\n\\end{equation}for weakly continuous curves of Borel probability measures on $\\mathbb{R}^d \\times \\mathcal{P}$. The exact notion of solution can be found in \\cite{Rckner-lin.-paper}, where also the following observation is made: A pair $(\\mu_t,\\nu_t)_{t \\leq T}$ solves (\\ref{9}) if and only if $\\Lambda_t := \\nu_t \\times \\delta_{\\mu_t}$ solves (\\ref{10}). Using our main result, we prove the following conjecture posed in Remark 4.4. of \\cite{Rckner-lin.-paper}.\n\\begin{prop}\\label{Prop conj Rckner}\n\tIf $(\\mu_t,\\nu_t)_{t \\leq T}$ is the unique solution to (\\ref{9}) with initial condition $(\\bar{\\mu},\\bar{\\nu}) \\in \\mathcal{P}\\times \\mathcal{P}$, then $(\\nu_t \\times \\delta_{\\mu_t})_{t \\leq T}$ is the unique solution to (\\ref{10}) with initial condition $\\bar{\\nu}\\times \\delta_{\\bar{\\mu}}$.\n\\end{prop}\n\\begin{proof}\n\tBy Corollary (\\ref{Cor uniqueness det}), the unique solution to (\\ref{P-CE}) with initial condition $\\delta_{\\bar{\\mu}}$ is $(\\delta_{\\mu_t})_{t \\leq T}$. Let $(\\Lambda^{(1)}_t)_{t \\leq T}$ and $(\\Lambda^{(2)}_t)_{t \\leq T}$ be two solutions to (\\ref{10}) with initial condition $\\bar{\\nu}\\times \\delta_{\\bar{\\mu}}$. It is straightforward to check that the curves of second marginals $(\\Lambda_t^{(1)}\\circ \\varPi_2^{-1})_{t \\leq T}$ and $(\\Lambda_t^{(2)}\\circ \\varPi_2^{-1})_{t \\leq T}$ are probability solutions to (\\ref{P-CE}) with initial condition $\\delta_{\\bar{\\mu}}$ (where we denote by $\\varPi_2$ the projection from $\\mathbb{R}^d \\times \\mathcal{P}$ onto the second coordinate). Hence, for each $t \\in [0,T]$\n\t$$\\Lambda_t^{(1)} \\circ \\varPi_2^{-1} = \\delta_{\\mu_t} = \\Lambda^{(2)}_t \\circ \\varPi_2^{-1}.$$\n\tConsequently, $\\Lambda_t^{(i)}$ is of product type, i.e. $\\Lambda^{(i)}_t = \\gamma_t^{(i)}\\times \\delta_{\\mu_t}$ for weakly continuous curves $(\\gamma^{(i)}_t)_{t \\leq T} \\subseteq \\mathcal{P}$, $i \\in \\{1,2\\}$. It is immediate to show that each curve $\\gamma^{(i)}$ solves the second equation of (\\ref{9}) with fixed $\\mu_t$ and initial condition $\\bar{\\nu}$. Hence, $\\gamma^{(i)}_t = \\nu_t$ for each $t \\in [0,T]$ and $i \\in \\{1,2\\}$, which implies $\\Lambda^{(1)}_t = \\Lambda^{(2)}_t$. Hence, the unique solution to (\\ref{10}) with initial condition $\\bar{\\nu}\\times \\delta_{\\bar{\\mu}}$ is given by $(\\nu_t \\times \\delta_{\\mu_t})_{t \\leq T}$. \n\\end{proof}\n\\section{Superposition Principle for stochastic nonlinear Fokker-Planck-Kolmogorov Equations}\nWe make use of the following notation specific to the stochastic case.\\\\\n\\\\\nFor two real-valued $n\\times n$ matrices $A,B$ we write $A$:$B = \\sum_{k,l = 1}^n A_{kl}B_{kl}$. We use the same notation for $A = (A_{kl})_{k,l \\geq 1}$ and $B = (B_{kl})_{k,l \\geq 1}$, if either $A$ or $B$ contain only finitely many non-trivial entries.\\\\\nFor the Hilbert space $\\ell^2$ with topology induced by the usual inner product $\\langle\\cdot,\\cdot \\rangle _{\\ell^2}$ and norm $||\\cdot||_{\\ell^2}$, we denote the space of continuous $\\ell^2$-valued functions on $[0,T]$ by $C_T\\ell^2$. On $\\ell^2$ and $C_T\\ell^2$, we unambiguously use the same notation $e_t, p_i$ and $\\pi_n$ as on $\\mathbb{R}^{\\infty}$ and $C_T\\mathbb{R}^{\\infty}$ in the previous section. Reminiscent to the previous section, we set $\\mathcal{B}(C_T\\ell^2) = \\sigma(e_t, t \\in [0,T])$ and denote the set of probability measures on this space by $\\mathcal{P}(C_T\\ell^2)$. For $\\sigma$-algebras $\\mathcal{A}_1$, $\\mathcal{A}_2$, we denote by $\\mathcal{A}_1 \\bigvee \\mathcal{A}_2$ the \\textit{$\\sigma$-algebra generated by $\\mathcal{A}_1$ and $\\mathcal{A}_2$}.\\\\\n\\\\\nWe call a filtered probability space $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\leq T},\\mathbb{P})$ \\textit{complete}, provided both $\\mathcal{F}$ and $\\mathcal{F}_0$ contain all subsets of $\\mathbb{P}$-negligible sets $N \\in \\mathcal{F}$ (i.e. $\\mathbb{P}(N) = 0$). This notion does not require $(\\mathcal{F}_t)_{t \\leq T}$ to be right-continuous. A real-valued Wiener process $W = (W_t)_{t \\leq T}$ on such a probability space is called an $\\mathcal{F}_t$-\\textit{Wiener process}, if $W_t$ is $\\mathcal{F}_t$-adapted and $W_u-W_t$ is independent of $\\mathcal{F}_t$ for each $0 \\leq t \\leq u \\leq T$. Pathwise properties of stochastic processes such as continuity are to be understood up to a negligible set with respect to the underlying measure.\\\\\n\\\\\nAs in the previous section, we consider $\\mathcal{SP}$ as a compact Polish space with the vague topology. Let $d_1 \\geq1$ and consider product-measurable coefficients on $[0,T]\\times \\mathcal{SP}\\times \\mathbb{R}^d$\n$$a(t,\\mu,x)=(a_{ij}(t,\\mu,x)) \\in \\mathbb{S}^+_d,\\,\\,b(t,\\mu,x) = (b_i(t,\\mu,x))_{i \\leq d} \\in \\mathbb{R}^d,\\, \\sigma(t,\\mu,x)= (\\sigma_{ij}(t,\\mu,x))_{i,j \\leq d} \\in \\mathbb{R}^{d\\times d_1}$$\nsuch that $\\sigma$ is bounded, \nand let $\\mathcal{L}$ be as before, i.e.\n$$\\mathcal{L}_{t, \\mu} \\varphi(x) = b_i(t,\\mu,x)\\partial_i \\varphi(x)+ a_{ij}(t,\\mu,x)\\partial^2_{ij}\\varphi(x)$$\nfor $\\varphi \\in C^2(\\mathbb{R}^d)$ and $(t,\\mu,x) \\in [0,T]\\times \\mathcal{SP}\\times \\mathbb{R}^d$.\\\\ \n\\\\\nIn contrast to the deterministic framework of the previous section, here we consider nonlinear \\textit{stochastic} FPK-equations of type (\\ref{SNLFPKE}) on $[0,T]$, to be understood in distributional sense as follows. With slight abuse of notation, for $\\sigma \\in \\mathbb{R}^{d\\times d_1}$ and $x \\in \\mathbb{R}^d$, we write $\\sigma \\cdot x = (\\sum_{i=1}^d\\sigma^{ik}x_i)_{k \\leq d_1}$, which is consistent with the standard inner product notation $\\sigma \\cdot x$ in the case $d_1 =1$. \n\\begin{dfn}\\label{Def sol SNLFPKE}\n\t\\begin{enumerate}\n\t\t\\item [(i)] A pair $(\\mu, W)$ consisting of an $\\mathcal{F}_t$-adapted vaguely continuous $\\mathcal{SP}$-valued stochastic process $\\mu = (\\mu_t)_{t \\leq T}$ and an $\\mathcal{F}_t$-adapted, $d_1$-dimensional Wiener process $W = (W_t)_{t \\leq T}$ on a complete probability space $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\leq T},\\mathbb{P})$ is a \\textit{subprobability solution to }(\\ref{SNLFPKE}), provided \n\t\t\\begin{equation}\\label{2_int_SNLFPKE}\n\t\t\t\\int_0^T\\int_{\\mathbb{R}^d}|b_i(t,\\mu_t,x)|+|a_{ij}(t,\\mu_t,x)| + |\\sigma_{ik}(t,\\mu_t,x)|^2d\\mu_t(x)dt < \\infty \\quad \\mathbb{P}\\text{-a.s.}\n\t\t\\end{equation}\n\t\tfor each $i,j \\leq d, k \\leq d_1$, and \n\t\t\\begin{equation}\\label{11}\n\t\t\\int_{\\mathbb{R}^d} \\varphi(x) d\\mu_t(x) - \t\\int_{\\mathbb{R}^d} \\varphi(x) d\\mu_0(x) = \\int_0^t \\int_{\\mathbb{R}^d}\\mathcal{L}_{s,\\mu_s}\\varphi(x) d\\mu_s(x)ds + \\int_0^t \\int_{\\mathbb{R}^d} \\sigma(s,\\mu_s,x)\\cdot \\nabla \\varphi(x)d \\mu_s(x)dW_s\n\t\t\\end{equation}holds $\\mathbb{P}$-a.s. for each $t \\in [0,T]$ and $\\varphi \\in C^2_c(\\mathbb{R}^d)$.\n\t\t\\item[(ii)] A \\textit{probability solution to } (\\ref{SNLFPKE}) is a pair as above such that $\\mu$ is a $\\mathcal{P}$-valued process $(\\mu_t)_{t \\leq T}$ with weakly continuous paths.\n\t\\end{enumerate}\n\\end{dfn}\n\\begin{rem}\n\t\\begin{enumerate}\n\t\t\\item [(i)] Since $C^2_c(\\mathbb{R}^d)$ is separable with respect to uniform convergence and since the paths $t \\mapsto \\mu_t(\\omega)$ are vaguely continuous, the exceptional sets in the above definition can be chosen independently of $\\varphi $ and $t$.\n\t\t\\item[(ii)] The first integral on the right-hand side of (\\ref{11}) is a pathwise (that is, for individual fixed $\\omega \\in \\Omega$) integral with respect to the finite measure $\\mu_s(\\omega)ds$ on $[0,T]\\times \\mathbb{R}^d$. The second integral is a stochastic integral, which is defined, since the integrand\n\t\t$$(t,\\omega) \\mapsto \\int_{\\mathbb{R}^d}\\sigma(t,\\mu_t(\\omega),x)\\cdot \\nabla \\varphi(x)d\\mu_t(\\omega)(x)$$\n\t\tis $\\mathbb{R}^{d_1}$-valued, bounded, product-measurable and $\\mathcal{F}_t$-adapted (Thm. 3.8 \\cite{ChungWilliamsStochInt}). More precisely,\n\t\t$$\\int_0^t \\int_{\\mathbb{R}^d} \\sigma(s,\\mu_s,x)\\cdot \\nabla \\varphi(x)d \\mu_s(x)dW_s = \\sum_{\\alpha=1}^{d_1} \\int_0^t\\int_{\\mathbb{R}^d}\\sigma^{\\alpha} \\cdot \\nabla \\varphi d\\mu_sdW^{\\alpha}_s,$$where $\\sigma^{\\alpha}= (\\sigma^{i\\alpha})_{i \\leq d}$ denotes the $\\alpha$-th column of $\\sigma$ and the components $W^{\\alpha}$, $\\alpha \\leq d_1$, of $W$ are real, independent Wiener processes.\n\t\\end{enumerate}\n\\end{rem}\nBy the global integrability assumption \\eqref{2_int_SNLFPKE} and since $\\sigma$ is bounded, we obtain (in analogy to Remark \\ref{Rem mass conserv}) the following conservation of mass, which we use to prove the final assertion of the main result Theorem \\ref{main thm stoch case}.\n\\begin{lem}\\label{Lem consv of mass stochastic}\n\tLet $(\\mu_t)_{t \\leq T}$ be a subprobability solution to (\\ref{SNLFPKE}). If $\\mu_0 \\in \\mathcal{P}$ $\\mathbb{P}$-a.s., then the paths of $t \\mapsto \\mu_t$ are $\\mathcal{P}$-valued $\\mathbb{P}$-a.s. and, hence, in particular weakly continuous.\n\\end{lem}\n\\begin{proof}\n\tLet $(\\varphi_k)_{k \\geq 1} \\subseteq C^2_c(\\mathbb{R}^d)$ approximate the constant function $1$ as in Remark \\ref{Rem mass conserv}. Then, by It\u00f4-isometry, for each $t \\in [0,T]$, there exists a subsequence $(k^t_l)_{l \\geq 1} = (k_l)_{l \\geq 1}$ such that\n\t\\begin{equation}\\label{12}\n\t\\int_0^t\\int_{\\mathbb{R}^d}\\sigma(s,\\mu_s, x)\\cdot \\nabla \\varphi_{k_l}(x)d\\mu_s(x)dW_s \\underset{l \\to \\infty}{\\longrightarrow}0 \\,\\, \\mathbb{P}\\text{-a.s.}\n\t\\end{equation}\n\tSince the stochastic integral is continuous in $t$, a classical diagonal argument yields that there exists a subsequence $(k_l)_{l \\geq 1}$ along which (\\ref{12}) holds for all $t \\in [0,T]$ on a set of full $\\mathbb{P}$-measure, independent of $t$. Let $\\omega' \\in \\Omega$ be from this set such that also $\\mu_0(\\omega') \\in \\mathcal{P}$ and (\\ref{11}) holds for each $t$ and $\\varphi$. Note that the set of all such $\\omega'$ has full $\\mathbb{P}$-measure. Then, similar to the reasoning in Remark \\ref{Rem mass conserv} and by using (\\ref{12}), considering (\\ref{11}) for such $\\omega'$ with $\\varphi_{k_l}$ in place of $\\varphi$ for the limit $l \\longrightarrow +\\infty$, we obtain\n\t$$\\mu_t(\\omega')(\\mathbb{R}^d) = \\mu_0(\\omega')(\\mathbb{R}^d), \\,\\, t\\in [0,T]$$\n\tand hence the result.\n\\end{proof}\nNote that the above proof can be adjusted to extend (\\ref{11}) to each $\\varphi \\in C^2_b(\\mathbb{R}^d)$. \n\\subsubsection*{Embedding $\\mathcal{SP}$ into $\\ell^2$}\nIn comparison with the deterministic case, we still consider $\\mathcal{SP}$ as a manifold-like space with tangent spaces $T_{\\mu}\\mathcal{SP} = L^2(\\mathbb{R}^d,\\mathbb{R}^d;\\mu)$ as before. However, instead of embedding into $\\mathbb{R}^{\\infty}$ by $G$ as in the previous section, now we need a global chart \n$$H: \\mathcal{SP} \\to \\ell^2$$\nin order to handle the stochastic integral term later on. To this end, we replace the set of functions $\\mathcal{G} = \\{g_i, i \\geq 1\\}$ of the deterministic case by\n\\begin{equation}\\label{13}\n\\mathcal{H} := \\{h_i\\}_{i \\geq 1}, \\, h_i := 2^{-i}\\frac{g_i}{||g_i||_{C^2_b}}\n\\end{equation}and consider the map\n$$H : \\mathcal{SP} \\to \\ell^2,\\,\\, H: \\mu \\mapsto (\\mu(h_i))_{i \\geq 1}.$$\nThe following lemma collects useful properties of $\\mathcal{H}$ and $H$, which are in the spirit of Lemma \\ref{Prop G early} and \\ref{Lem aux G, J}. We point out that we could have used the function class $\\mathcal{H}$ instead of $\\mathcal{G}$ already in Section 3, but we decided to pass from $\\mathcal{G}$ to $\\mathcal{H}$ at this point in order to stress the technical adjustments necessary due to the stochastic case.\n\\begin{lem}\\label{Lem aux H}\n\t\\begin{enumerate}\n\t\t\\item [(i)] The set $\\mathcal{H}$ is measure-determining. Further, a process $(\\mu_t)_{t \\leq T}$ as in Definition \\ref{Def sol SNLFPKE} is a solution to (\\ref{SNLFPKE}) if and only if (\\ref{11}) holds for each $h_i \\in \\mathcal{H}$ in place of $\\varphi$.\n\t\t\\item[(ii)] $H$ is a homeomorphism between $\\mathcal{SP}$ and its range $H(\\mathcal{SP}) \\subseteq \\ell^2$, endowed with the $\\ell^2$-subspace topology. In particular, $H(\\mathcal{SP}) \\subseteq \\ell^2$ is compact.\n\t\n\t\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n\t\\begin{enumerate}\n\t\t\\item [(i)] The first claim is obvious, since $\\mathcal{G}$ is measure-determining. Concerning the second claim, note that it is clearly sufficient to have (\\ref{11}) for each $\\varphi \\in C_c^2(\\mathbb{R}^d)$ with $||\\varphi||_{C^2_b} \\leq 1$. Since the functions $||g_i||^{-1}_{C^2_b}g_i$ are dense in the unit ball of $C^2_c(\\mathbb{R}^d)$ with respect to $||\\cdot||_{C^2_b}$, it is sufficient to have (\\ref{11}) for each such normalized function. Indeed, if $\\varphi_k \\underset{k \\to \\infty}{\\longrightarrow}\\varphi$ uniformly up to second-order partial derivatives, then by It\u00f4-isometry\n\t\t\\begin{equation*}\n\t\t\\mathbb{E}\\bigg[\\bigg(\\int_0^t\\int_{\\mathbb{R}^d}\\sigma(s,\\mu_s,\\cdot)\\cdot \\nabla (\\varphi_k-\\varphi)d\\mu_sdW_s\\bigg)^2\\bigg] = \\mathbb{E}\\bigg[\\int_0^t\\bigg(\\int_{\\mathbb{R}^d}\\sigma(s,\\mu_s,\\cdot)\\cdot\\nabla(\\varphi_k-\\varphi)d\\mu_s \\bigg)^2ds\\bigg],\n\t\t\\end{equation*}which converges to $0$ as $k \\longrightarrow \\infty$ due to the boundedness of $\\sigma$. Hence, along a subsequence $(k_l)_{l \\geq 1}$, we have a.s.\n\t\t\\begin{equation*}\n\t\t\\int_0^t \\int_{\\mathbb{R}^d}\\sigma(s,\\mu_s,x) \\cdot \\nabla \\varphi_{k_l}(x)d\\mu_s(x)dW_s \\underset{l \\to \\infty}{\\longrightarrow} \\int_0^t \\int_{\\mathbb{R}^d}\\sigma(s,\\mu_s,x)\\cdot \\nabla\\varphi(x)d\\mu_s(x)dW_s.\n\t\t\\end{equation*}The a.s.-convergence of all other terms in (\\ref{11}) is clear. Therefore, it is sufficient to require (\\ref{11}) for a dense subset of the unit ball of $C^2_c(\\mathbb{R}^d)$. Clearly, this yields at once that it is sufficient to have (\\ref{11}) for each $h_i \\in \\mathcal{H}$. \n\t\t\\item[(ii)] By definition, $H$ maps into $\\ell^2$. Since $\\mathcal{H}$ is measure-determining, $H$ is one-to-one, hence bijective onto its range. If $\\mu_n \\underset{n \\to \\infty}{\\longrightarrow} \\mu$ vaguely in $\\mathcal{SP}$, clearly $H(\\mu_n)$ converges to $H(\\mu)$ in the product topology. Since for any $i \\geq 1$\n\t\t$$\\underset{n \\geq 1}{\\text{sup}}|H(\\mu_n)_i| \\leq 2^{-i},$$\n\t\tthe convergence holds in $\\ell^2$ as well, which implies continuity of $H$. In particular, $H(\\mathcal{SP}) \\subseteq \\ell^2$ is compact. Conversely, if $H(\\mu_n)$ converges in $\\ell^2$ to some $z = (z_i)_{i \\geq 1}$, then, by closedness of $H(\\mathcal{SP}) \\subseteq \\ell^2$, we have $z=H(\\mu)$ for a unique element $\\mu \\in \\mathcal{SP}$ and $\\mu_n \\underset{n \\to \\infty}{\\longrightarrow}\\mu$ vaguely. Indeed, the latter follows as in Lemma \\ref{Lem aux G, J} (i). \t\n\t\\end{enumerate} \n\\end{proof}\nFor consistency of notation, below we denote the test function class of the manifold-like space $\\mathcal{SP}$ by $\\mathcal{F}C^2_b(\\mathcal{H})$ to stress that the base functions $g_i$ are now replaced by $h_i \\in \\mathcal{H}$. However, the class of test functions remains unchanged, because the transition from $g_i$ to $h_i$ can be incorporated in the choice of $f$.\n\\subsubsection*{Linearization of (\\ref{SNLFPKE})}\nAs in the deterministic case, also for the stochastic nonlinear equation (\\ref{SNLFPKE}) one can consider an associated linear equation for curves in $\\mathcal{P}(\\mathcal{SP})$. To the best of our knowledge, such a linearization for stochastic FPK-equations has not yet been considered in the literature. Of course, the basic idea stems from the deterministic case \\cite{Rckner-lin.-paper} discussed in the previous section. From It\u00f4's formula one expects this linearized equation to be of second-order.\\\\\n\\\\\nLet $\\big((\\mu_t)_{t \\leq T},W\\big)$ be a subprobability solution to (\\ref{SNLFPKE}) (with underlying measure $\\mathbb{P}$) and choose any $F: \\mu \\mapsto f\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)$ from $\\mathcal{F}C^2_b(\\mathcal{H})$. Again, we abbreviate $b(t,\\mu) := b(t,\\mu,\\cdot)$ and similarly for $a$ and $\\sigma$. By It\u00f4's formula, we have $\\mathbb{P}$-a.s.\n\\begin{align*}\nF(\\mu_t)-F(\\mu_0) &= \\int_0^t \\big\\langle\\nabla^{\\mathcal{SP}}F(\\mu), b(s,\\mu)+a(s,\\mu)\\nabla \\big \\rangle_{L^2(\\mu_s)}ds \\\\&\n+\\frac{1}{2}\\sum_{\\alpha=1}^{d_1}\\int_0^t \\sum_{k,l=1}^n (\\partial_{kl}f)(\\mu(h_1),\\dots,\\mu(h_n))\\bigg(\\int_{\\mathbb{R}^d}\\sigma^{\\alpha}(s,\\mu)\\cdot \\nabla h_k d\\mu\\bigg)\\bigg(\\int_{\\mathbb{R}^d}\\sigma^{\\alpha}(s,\\mu)\\cdot \\nabla h_l d\\mu\\bigg)ds \\\\&+ M_t^F\n,\n\\end{align*}with the martingale $M^F$ given as\n$$M_t^F:= \\sum_{\\alpha=1}^{d_1}\\int_0^t\\bigg[ \\sum_{l=1}^{n}(\\partial_lf)\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big) \\int_{\\mathbb{R}^d}\\sigma^{\\alpha} \\cdot \\nabla h_l d\\mu_s \\bigg]dW_s^{\\alpha}. $$Since $M^F_0 = 0$ $\\mathbb{P}$-a.s., integrating with respect to $\\mathbb{P}$ and defining the curve of measures in $\\mathbb{P}(\\mathcal{SP})$\n$$\\Gamma_t := \\mathbb{P}\\circ \\mu_t^{-1}, \\,\\, t\\leq T$$yields\n\\begin{align}\\label{prelim eq}\n\\notag &\\int_{\\mathcal{SP}}F(\\mu)d\\Gamma_t(\\mu) - \\int_{\\mathcal{SP}}F(\\mu)d\\Gamma_0(\\mu)= \\int_0^t\\int_{\\mathcal{SP}} \\big\\langle\\nabla^{\\mathcal{SP}}F(\\mu), b(s,\\mu)+a(s,\\mu)\\nabla \\big \\rangle_{L^2(\\mu)}d\\Gamma_s(\\mu)ds \\\\& \n\t+\\frac{1}{2}\\sum_{\\alpha=1}^{d_1}\\int_0^t \\int_{\\mathcal{SP}}\\sum_{k,l=1}^n (\\partial_{kl}f)\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)\\bigg(\\int_{\\mathbb{R}^d}\\sigma^{\\alpha}(s,\\mu)\\cdot \\nabla h_k d\\mu\\bigg)\\bigg(\\int_{\\mathbb{R}^d}\\sigma^{\\alpha}(s,\\mu)\\cdot \\nabla h_l d\\mu\\bigg)d\\Gamma_s(\\mu)ds.\n\\end{align}As for the first-order term, which is interpreted as the pairing of the gradient $\\nabla^{\\mathcal{SP}}F$ with the inhomogeneous vector field $b+a\\nabla$ in the tangent bundle $T\\mathcal{SP}$, also the second-order term allows for a geometric interpretation: Recall that for a smooth, real function $F$ on a Riemannian manifold $M$ with tangent bundle $TM$, the Hessian $Hess(F)_p$ at $p \\in M$ is a bilinear form on $T_pM$ with\n\\begin{equation}\\label{Hess gen mf}\nHess(F)_p(\\eta_p, \\xi_p) = \\big\\langle \\nabla^L_{\\eta_p}\\nabla F(p), \\xi_p \\big \\rangle_{T_pM},\\,\\, \\eta_p, \\xi_p \\in T_pM, \n\\end{equation}where $\\nabla^L: TM \\times TM \\to TM$ denotes the Levi-Civita-connection on $M$, the unique affine connection compatible with the metric tensor on $M$ and $\\nabla$ denotes the usual gradient on $M$. Intuitively, $\\nabla^L_{\\eta_p}\\nabla F(p)$ denotes the change of the vector field $\\nabla F$ in direction $\\eta_p$ at $p$. Recall that we consider $\\mathcal{SP}$ as a manifold-like space with gradient $\\nabla^{\\mathcal{SP}}$ and that hence the reasonable notion of the Levi-Civita connection$\\nabla^{L,\\mathcal{SP}}$ on $\\mathcal{SP}$ for $\\sigma \\in T_{\\mu}\\mathcal{SP} = L^2(\\mathbb{R}^d,\\mathbb{R}^d;\\mu), Y \\in T\\mathcal{SP}$ at $\\mu$ is given by\n$$\\nabla ^{L,\\mathcal{SP}}_{\\sigma}Y(\\mu) = \\big \\langle \\nabla^{\\mathcal{SP}}Y(\\mu), \\sigma\\big \\rangle_{T_{\\mu}\\mathcal{SP}},$$whenver $\\nabla^{\\mathcal{SP}}Y$ is defined in $T\\mathcal{SP}$. For the representation of $Hess(F)$ for a test function $F \\in \\mathcal{F}C^2_b(\\mathcal{H})$, we need to set $Y = \\nabla^{\\mathcal{SP}}F$. In this case, we can indeed make sense of\n$$ (\\nabla^{\\mathcal{SP}})^2F:= \\nabla^{\\mathcal{SP}}\\nabla^{\\mathcal{SP}}F,$$ because the gradient\n$$\\mu \\mapsto \\nabla^{\\mathcal{SP}}F(\\mu) = \\sum_{k=1}^{n}(\\partial_kf)\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)\\nabla h_k$$ is a linear combinations of the \"$\\mathcal{F}C^2_b(\\mathcal{H})$-like\" functions $\\mu \\mapsto \\partial_kf\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)$. The linear combination has to be understood in an $x$-wise sense with coefficient functions $\\nabla h_k$, which are independent of the variable of interest $\\mu$. Denoting $F_k(\\mu):= (\\partial_kf)\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)$, we then define\n\\begin{equation}\\label{Def nabla squared}\n(\\nabla^{\\mathcal{SP}})^2F(\\mu) (x,y) := \\sum_{k=1}^n\\big(\\nabla^{\\mathcal{SP}}F_k(\\mu)\\big)(y)\\nabla h_k(x),\\,\\, (x,y) \\in \\mathbb{R}^d\\times \\mathbb{R}^d. \n\\end{equation}\nConsequently, we have a reasonable notion of the Levi-Civita connection on $\\mathcal{SP}$ at $\\mu$ for $\\sigma \\in T_{\\mu}\\mathcal{SP}$ and $\\nabla^{\\mathcal{SP}}F$ for $F \\in \\mathcal{F}C^2_b(\\mathcal{H})$ as\n\\begin{equation}\n\\nabla^{L,\\mathcal{SP}}_{\\sigma}\\nabla^{\\mathcal{SP}}F(\\mu)  := \\big \\langle (\\nabla^{\\mathcal{SP}})^2F(\\mu), \\sigma \\big \\rangle_{T_{\\mu}\\mathcal{SP}} = \\sum_{k,l=1}^{n}(\\partial_{kl}f)\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)\\nabla h_k\\bigg(\\int_{\\mathbb{R}^d}\\sigma \\cdot \\nabla h_l d\\mu\\bigg).\n\\end{equation}The section $(\\nabla^{\\mathcal{SP}})^2F$ in $T\\mathcal{SP}^*\\otimes T\\mathcal{SP}^*$ (and hence $\\nabla^{L,\\mathcal{SP}}_{\\sigma}\\nabla^{\\mathcal{SP}}F$ and $Hess(F)$ below) is independent of the particular representation of $F$ in (\\ref{Def nabla squared}). Indeed, we have (c.f. Appendix A \\cite{Rckner-lin.-paper}) for $$\\gamma^{\\sigma}_{\\mu}(t):= \\mu \\circ (\\text{Id}+t \\sigma)^{-1}$$ the following pointwise (in $x \\in \\mathbb{R}^d$) equality for each $\\mu \\in \\mathcal{SP}, \\sigma \\in L^2(\\mathbb{R}^d,\\mathbb{R}^d;\\mu)$\n\\begin{align*}\n\\frac{d}{dt}\\nabla^{\\mathcal{SP}}F\\big(\\gamma^{\\sigma}_{\\mu}(t)\\big) &= \\sum_{k=1}^n\\bigg[\\frac{d}{dt}(\\partial_kf)\\big(\\gamma^{\\sigma}_{\\mu}(t)(h_1),\\dots,\\gamma^{\\sigma}_{\\mu}(t)(h_n)\\big)\\bigg]\\nabla h_k \\\\& = \\sum_{k,l=1}^n(\\partial_{kl}f)\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)\\big \\langle \\nabla h_l, \\sigma \\big \\rangle_{L^2(\\mu)} \\nabla h_k \\\\& = \n\\big \\langle (\\nabla^{\\mathcal{SP}})^2F(\\mu),\\sigma \\big \\rangle_{L^2(\\mu)}.\n\\end{align*}Since the gradient $\\nabla^{\\mathcal{SP}}F$ is independent of the particular representation of $F$ and $\\sigma \\in L^2(\\mathbb{R}^d,\\mathbb{R}^d;\\mu)$ is arbitrary, also $(\\nabla^{\\mathcal{SP}})^2F$ is independent of the representation of $F$. \\\\\n\nConsidering (\\ref{Hess gen mf}), we then set for $F \\in \\mathcal{F}C^2_b(\\mathcal{H})$ and $\\sigma, \\tilde{\\sigma} \\in L^2(\\mathbb{R}^d,\\mathbb{R}^d;\\mu)$\n\\begin{equation}\nHess(F)(\\mu): (\\sigma, \\tilde{\\sigma}) \\mapsto \\sum_{k,l=1}^{n}(\\partial_{kl}f)\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)\\bigg(\\int_{\\mathbb{R}^d}\\sigma \\cdot \\nabla h_l d\\mu\\bigg)\\bigg(\\int_{\\mathbb{R}^d}\\tilde{\\sigma} \\cdot \\nabla h_k d\\mu\\bigg),\n\\end{equation}which is a (symmetric) bilinear form on $T_{\\mu}\\mathcal{SP}$ and rewrite (\\ref{prelim eq}) as\n\\begin{align}\\label{P-FPKE eq}\n\\int_{\\mathcal{SP}}Fd\\Gamma_t - \\int_{\\mathcal{SP}}Fd\\Gamma_0= \\int_0^t\\int_{\\mathcal{SP}} \\big\\langle\\nabla^{\\mathcal{SP}}F, b_s+a_s\\nabla \\big \\rangle_{L^2} + \\frac{1}{2}\\sum_{\\alpha=1}^{d_1} Hess(F)(\\sigma_s^{\\alpha},\\sigma_s^{\\alpha})d\\Gamma_sds\n\\end{align}(with $b_s: (\\mu,x) \\mapsto b(s,\\mu,x)$ and similarly for $a_s$ and $\\sigma_s$). Introducing the second-order operator $\\mathbf{L}^{(2)}$, acting on $F \\in \\mathcal{F}C^2_b(\\mathcal{H})$ via\n$$\\mathbf{L}^{(2)}_tF(\\mu) = \\big\\langle\\nabla^{\\mathcal{SP}}F, b(t,\\mu)+a(t,\\mu)\\nabla \\big \\rangle_{L^2(\\mu)} + \\frac{1}{2}\\sum_{\\alpha=1}^{d_1} Hess(F)\\big(\\sigma^{\\alpha}(t,\\mu),\\sigma^{\\alpha}(t,\\mu)\\big),$$\nwe arrive at the distributional formulation of (\\ref{P-FPKE})\n$$\\partial_t\\Gamma_t = (\\mathbf{L}_t^{(2)})^*\\Gamma_t, \\,\\, t \\leq T,$$as in the introduction.\n\\begin{rem}\n\tEquation (\\ref{P-CE}) is the natural analogue to second-order FPK-equations over Euclidean spaces. Indeed, for a stochastic equation on $\\mathbb{R}^d$\n\t\t\\begin{equation}\\label{StochEq}\n\t\tdX_t = b(t,X_t)dt+\\sigma(t,X_t)dW_t,\n\t\t\\end{equation}\n\t by It\u00f4's formula, the corresponding linear second-order equation for measures in distributional form is \n\t$$\\partial_t\\mu_t = \\big(\\mathcal{L}^{(2)}_t\\big)^*\\mu_t$$\n\twith $$\\mathcal{L}^{(2)}_tf = \\nabla f\\cdot b_t+\\frac{1}{2}\\big\\langle\\sigma_t,Hess(f)\\sigma_t\\big\\rangle_{\\mathbb{R}^d},$$where $Hess(f)$ denotes the usual Euclidean Hessian matrix of $f \\in C^2(\\mathbb{R}^d)$. In this spirit, it seems natural to consider (\\ref{SNLFPKE}) as a stochastic equation with state space $\\mathcal{SP}$ instead of $\\mathbb{R}^d$ as for (\\ref{StochEq}) and (\\ref{P-CE}) as the corresponding linear Fokker-Planck-type equation on $\\mathcal{SP}$.\n\n\\end{rem}\n By the above derivation, for any subprobability solution process $(\\mu_t)_{t \\leq T}$ to (\\ref{SNLFPKE}) the curve $(\\Gamma_t)_{t \\leq T}$, $\\Gamma_t := \\mathbb{P}\\circ \\mu_t^{-1}$ in $\\mathcal{P}(\\mathcal{SP})$ solves (\\ref{P-CE}) in the sense of the following definition.\n\\begin{dfn}\n\tA weakly continuous curve $(\\Gamma_t)_{t \\leq T} \\subseteq \\mathcal{P}(\\mathcal{SP})$ is a \\textit{solution to (\\ref{SNLFPKE})}, if the integrability condition\n\t\t\\begin{align} \\label{2_int_SP-FPKE}\n\t\t\\int_0^T \\int_{\\mathcal{SP}}||b(t,\\mu)||_{L^1(\\mathbb{R}^d,\\mathbb{R}^d;\\mu)}+||a(t,\\mu)||_{L^1(\\mathbb{R}^d,\\mathbb{R}^{d^2};\\mu)}+ ||\\sigma(t,\\mu)||^2_{L^2(\\mathbb{R}^d,\\mathbb{R}^{d\\times d_1};\\mu)}d\\Gamma_t(\\mu)dt < \\infty\n\t\\end{align}\n\t is fulfilled and for each $F \\in \\mathcal{F}C^2_b(\\mathcal{H})$, \\ref{P-FPKE eq} holds for each $t \\in [0,T]$.\n\\end{dfn}\n\\subsubsection*{Transferring (\\ref{SNLFPKE}) and (\\ref{P-FPKE}) to $\\ell^2$}Reminiscent to the deterministic case, we use the global chart $H: \\mathcal{SP}\\to \\ell^2$ to introduce auxiliary equations on $\\ell^2$ and the space of measures on $\\ell^2$, respectively, as follows. Again, we use the notation\n$$A_t:= \\bigg\\{\\mu \\in \\mathcal{SP}: \\int_{\\mathbb{R}^d}|a_{ij}(t,\\mu,x)|+|b_i(t,\\mu,x)|d\\mu(x) < \\infty\\,\\,\\forall \\,1\\leq i,j \\leq d\\bigg\\}, \\quad t \\in [0,T].$$\nFor $i,j \\geq 1$, $\\alpha \\leq d_1$, define the measurable coefficients $B_i$ for $(t,\\mu)$ such that $\\mu \\in A_t$, and $\\Sigma^{\\alpha}_i$ and $A_{ij}$ on $[0,T]\\times \\mathcal{SP}$ by\n\\begin{align*}\nB_i(t,\\mu) &:= \\int_{\\mathbb{R}^d}\\mathcal{L}_{t,\\mu}h_i(x)d\\mu(x), \\quad (t,\\mu)\\in [0,T]\\times A_t,\\\\\n\\Sigma^{\\alpha}_i(t,\\mu) &:= \\int_{\\mathbb{R}^d}\\sigma^{\\alpha}(t,\\mu,x)\\cdot \\nabla h_i(x)d\\mu(x), \\\\ \\Sigma_i(t,\\mu) &:= \\big( \\Sigma^{\\alpha}_i(t,\\mu)\\big)_{\\alpha \\leq d_1} , \\\\ A_{ij}(t,\\mu) &:= \\big \\langle \\Sigma_i, \\Sigma_j \\big \\rangle_{d_1} (t,\\mu),\n\\end{align*}\nand set \n$$B := (B_i)_{i \\geq 1},\\, \\Sigma := (\\Sigma^{\\alpha}_i)_{\\alpha \\leq d_1, i \\geq 1}, \\, A := (A_{ij})_{i,j \\geq 1}.$$ Now, transferring to $\\ell^2$, define $\\bar{B}, \\bar{\\Sigma}$ and $\\bar{A}_{ij}$ on $[0,T] \\times \\ell^2$ component-wise via\n\\begin{align*}\n\\bar{B}_i(t,z) := \n\\begin{cases}\nB_i(t,H^{-1}(z))&, z \\in H(A_t) \\\\\n0 &, \\text{else}\n\\end{cases},\n\\end{align*},\n\\begin{align*}\n\\bar{\\Sigma}^{\\alpha}_i(t,z) := \n\\begin{cases}\n\\Sigma_i^{\\alpha}(t,H^{-1}(z))&, z \\in H(\\mathcal{SP}) \\\\\n0 &, z \\in \\ell^2 \\backslash H(\\mathcal{SP})\n\\end{cases},\n\\end{align*}\n$$\\bar{\\Sigma}_i(t,z) := \\big(\\bar{\\Sigma}^{\\alpha}_i(t,z)\\big)_{\\alpha \\leq d_1},$$\n$$\\bar{A}_{ij}(t,z) := \\big \\langle\\bar{\\Sigma}_i,\\bar{\\Sigma}_j\\big \\rangle_{d_1}(t,z).$$\n$\\bar{B}$ and $\\bar{\\Sigma}^{\\alpha}$ are $\\ell^2$-valued, since for $z = H(\\mu)$\n$$|\\bar{B}_i(t,z)| \\leq \\int_{\\mathbb{R}^d}|\\mathcal{L}_{t,\\mu}h_i(x)|d\\mu(x) \\leq C2^{-i},$$\nwhere $C = C(a,b,d)$ is a finite constant independent of $t,z$ and $i \\geq 1$. A similar argument is valid for each $\\bar{\\Sigma}^{\\alpha}$. Each $\\bar{B}_i$ and $\\bar{\\Sigma}^{\\alpha}_i$ is product-measurable with respect to the $\\ell^2$-topology due to the measurability of $B$ and $\\Sigma^{\\alpha}$. Reminiscent to (\\ref{Rinfty-CE}) in the previous section, we associate to (\\ref{P-FPKE}) the FPK-equation on $\\ell^2$\n\\begin{equation}\\label{ell2-FPKE}\\tag{$\\ell^2$-FPK}\n\\partial_t \\bar{\\Gamma}_t = -\\bar{\\nabla}\\cdot(\\bar{B}(t,z)\\bar{\\Gamma}_t)+\\partial^2_{ij}(\\bar{A}_{ij}(t,z)\\bar{\\Gamma}_t),\n\\end{equation}which we understand in the sense of the following definition, with $\\bar{\\nabla}$ as in (\\ref{Def nabla gradient}). Subsequently, we denote by $\\mathcal{F}C^2_b(\\ell^2)$ the set of all maps $\\bar{F}: \\ell^2 \\to \\mathbb{R}$ of type $\\bar{F} = f \\circ \\pi_n$ for $n \\geq 1$ and $f \\in C^2_b(\\mathbb{R}^n)$. Also, set\n$$D^2\\bar{F}_{ij} := \n\\begin{cases}\n(\\partial^2_{ij}f) \\circ \\pi_n&,  i,j \\leq n \\\\\n0&,  \\text{ else}.\n\\end{cases}$$Consequently, both summands in (\\ref{14}) contain only finitely many non-trivial summands. \n\\begin{dfn}\\label{Def sol ell2-FPKE}\n\tA weakly continuous curve $(\\bar{\\Gamma}_t)_{t \\leq T}\\subseteq \\mathcal{P}(\\ell^2)$ is a \\textit{solution to} (\\ref{ell2-FPKE}), if it fulfills the integrability condition\n\t\t\\begin{equation}\\label{2_global_int_l2-FPKE}\n\t\t\\int_0^T\\int_{\\ell^2}|\\bar{B}_i(t,z)| + |\\bar{A}_{ij}(t,z)|d\\bar{\\Gamma}_tdt < \\infty,\\quad \\forall \\,i,j \\geq 1,\n\t\\end{equation}\n\tand for any $\\bar{F} \\in \\mathcal{F}C^2_b(\\ell^2)$, $\\bar{F}:= f\\circ \\pi_n$,\n\t\\begin{equation}\\label{14}\n\t\\int_{\\ell^2}\\bar{F}(z)d\\bar{\\Gamma}_t(z) = \\int_{\\ell^2}\\bar{F}(z)d\\bar{\\Gamma}_0(z)+\\int_0^t \\int_{\\ell^2}\\bar{\\nabla} \\bar{F}(z) \\cdot \\bar{B}(s,z)+\\frac{1}{2}D^2\\bar{F}(z):\\bar{A}(s,z)d\\bar{\\Gamma}_s(z)ds.\n\t\\end{equation}holds for each $t \\leq T$. \n\\end{dfn}\n\n\n\\subsection{Main Result: Stochastic case}\nThe main result of this section is the following superposition principle for solutions to (\\ref{SNLFPKE}) and (\\ref{P-FPKE}), which generalizes Theorem \\ref{main thm det case} to stochastically perturbed equations.\n\\begin{theorem}\\label{main thm stoch case}\nLet $\\sigma$ be bounded on $[0,T]\\times \\mathcal{SP}\\times \\mathbb{R}^d$.\tLet $(\\Gamma_t)_{t \\leq T}$ be a weakly continuous solution to (\\ref{P-FPKE}). Then, there exists a complete filtered probability space $(\\Omega,\\mathcal{F}, (\\mathcal{F}_t)_{t \\leq T}, \\mathbb{P})$, an adapted $d_1$-dimensional Wiener process $W = (W_t)_{t \\leq T}$ and a $\\mathcal{SP}$-valued adapted vaguely continuous process $\\mu=(\\mu_t)_{t \\leq T}$ such that $(\\mu,W)$ solves (\\ref{SNLFPKE}) and\n\t$$\\mathbb{P}\\circ \\mu_t^{-1} = \\Gamma_t$$\n\tholds for each $t \\in [0,T]$.\\\\\n\tMoreover, if $\\Gamma_0$ is concentrated on $\\mathcal{P}$, i.e. $\\Gamma_0(\\mathcal{P}) = 1$, then the paths $t \\mapsto \\mu_t(\\omega)$ are $\\mathcal{P}$-valued for $\\mathbb{P}$-a.e. $\\omega \\in \\Omega$ and hence even weakly continuous.\n\\end{theorem}\nAs in the proof of \\ref{main thm det case}, we proceed in three steps. Since parts of the proof are technically more involved than in the deterministic case, we first present the ingredients of each step and afterwards state the proof of Theorem \\ref{main thm stoch case} as a corollary.\\\\\n\\\\\n\\textbf{Step 1: From (\\ref{P-FPKE}) to (\\ref{ell2-FPKE}):} \n\\begin{lem}\\label{Lem of Step1 stochCase}\n\tFor any solution $(\\Gamma_t)_{t \\leq T}$ to (\\ref{P-FPKE}), the curve $\\bar{\\Gamma}_t = \\Gamma_t \\circ H^{-1}$ is a solution to (\\ref{ell2-FPKE}).\n\\end{lem}\n\\begin{proof}\nClearly, $t \\mapsto \\bar{\\Gamma}_t$ is a weakly continuous curve in $\\mathcal{P}(\\ell^2)$ due to the continuity of $H: \\mathcal{SP} \\to \\ell^2$. \\eqref{2_global_int_l2-FPKE} holds, since $t \\mapsto \\Gamma_t$ fulfills \\eqref{2_int_SP-FPKE} and since $\\sigma$ is bounded. Moreover, we have for $s,t \\leq T$, $\\bar{F} = f \\circ \\pi_n \\in \\mathcal{F}C^2_b(\\ell^2)$ and $F: \\mu \\mapsto f\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)$\n\\begin{align*}\n& \\int_{\\ell^2}\\bar{\\nabla} \\bar{F}(z) \\cdot \\bar{B}(s,z)+\\frac{1}{2}D^2\\bar{F}(z):\\bar{A}(s,z)d\\bar{\\Gamma}_s(z) \\\\&=  \\int_{\\mathcal{SP}}\\sum_{k=1}^n(\\partial_kf)\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)B_k(s,\\mu) + \\sum_{\\alpha=1}^{d_1} \\frac{1}{2}\\sum_{k,l=1}^n(\\partial_{kl}f)\\big(\\mu(h_1),\\dots,\\mu(h_n)\\big)\\Sigma_k^{\\alpha}(s,\\mu)\\Sigma_l^{\\alpha}(s,\\mu)d\\Gamma_s(\\mu) \\\\&=\n\\int_{\\mathcal{SP}}\\big \\langle \\nabla^{\\mathcal{SP}}F(\\mu),b(s,\\mu)+a(s,\\mu)\\nabla \\big \\rangle_{L^2(\\mu)}+ \\frac{1}{2}\\sum_{\\alpha=1}^{d_1}Hess(F)\\big(\\sigma^{\\alpha}(s,\\mu), \\sigma^{\\alpha}(s,\\mu)\\big)d \\Gamma_s(\\mu)\n\\end{align*}and likewise\n\\begin{equation*}\n\\int_{\\ell^2}\\bar{F}(z)d\\bar{\\Gamma}_t = \\int_{\\mathcal{SP}}F(\\mu)d\\Gamma_t.\n\\end{equation*}Comparing with (\\ref{P-FPKE eq}), the statement follows.\n\\end{proof}\n\t\n\\textbf{Step 2: From (\\ref{ell2-FPKE}) to the martingale problem ($\\ell^2$-MGP):} We introduce a martingale problem on $\\ell^2$, which is related to (\\ref{ell2-FPKE}) in the sense of Remark \\ref{Rem easy connect stoch ell2} below and is, roughly speaking, the stochastic analogue to (\\ref{Rinfty-ODE}) from the previous section. Recall the notation $e_t$ for the projection $e_t: C_T\\ell^2 \\to \\ell^2$, $e_t: \\gamma \\mapsto \\gamma_t$ for $t \\leq T$.\n\\begin{dfn}\\label{Def ell2-MGP sol}\n\tA measure $\\bar{Q} \\in \\mathcal{P}(C_T\\ell^2)$ is a \\textit{solution to the $\\ell^2$-martingale problem ($\\ell^2$-MGP)}, provided\n\t\t\\begin{equation}\n\t\t\\int_{C_T\\ell^2}\\int_0^T|\\bar{B}_i(t,e_t)|+|\\bar{A}_{ij}(t,e_t)|dtd\\bar{Q} < \\infty,\\quad i,j \\geq 1,\n\t\\end{equation}\n\tand\n\t\\begin{equation}\\label{15}\n\t\\bar{F}\\circ e_t - \\int_0^t \\bar{\\nabla} \\bar{F} \\circ e_s \\cdot \\bar{B}(s,e_s)+\\frac{1}{2}D^2\\bar{F}\\circ e_s : \\bar{A}(s,e_s)ds\n\t\\end{equation}is a $\\bar{Q}$-martingale on $C_T\\ell^2$ with respect to the natural filtration on $C_T\\ell^2$ for any $\\bar{F} \\in \\mathcal{F}C^2_b(\\ell^2)$.\n\\end{dfn}\n\\begin{rem}\\label{Rem easy connect stoch ell2}\n\tBy construction, any such solution $\\bar{Q}$ induces a weakly continuous solution $(\\bar{\\Gamma}_t)_{t \\leq T}$ to (\\ref{ell2-FPKE}) via $\\bar{\\Gamma}_t := \\bar{Q} \\circ e_t^{-1}$. Indeed, this is readily seen by integrating (\\ref{15}) with respect to $\\bar{Q}$ and Fubini's theorem.\n\\end{rem}\nIn view of Proposition \\ref{Prop SPprincip ell2} below, we extend the coefficients $\\bar{B}_i, \\bar{\\Sigma}^{\\alpha}_i$ (and hence also $\\bar{A}_{ij}$) to $\\mathbb{R}^{\\infty}$ via\n$$\\bar{B}_i := 0 =: \\bar{\\Sigma}^{\\alpha}_i \\text{ on }[0,T]\\times \\mathbb{R}^{\\infty} \\backslash \\ell^2.$$\nWe still use the notation $\\bar{B}$, $\\bar{\\Sigma}^{\\alpha}$ and $\\bar{A}$ and note that they are $\\mathcal{B}([0,T])\\otimes \\mathcal{B}(\\mathbb{R}^{\\infty})\/\\mathcal{B}(\\mathbb{R}^{\\infty})$-measurable due to Remark \\ref{Rem meas P SP} below. Due to the same remark, we may regard any solution $(\\bar{\\Gamma}_t)_{t \\leq T}$ to (\\ref{ell2-FPKE}) as a solution to a FPK-equation on $\\mathbb{R}^{\\infty}$ by considering (\\ref{ell2-FPKE}) with the extended coefficients and test functions $\\bar{F} \\in \\mathcal{F}C^2_b(\\ell^2)$ extended to $\\mathbb{R}^{\\infty}$ by considering $\\pi_n$ on $\\mathbb{R}^{\\infty}$ instead of $\\ell^2$. Similarly, the formulation of the martingale problem ($\\ell^2$-MGP) as in Definition \\ref{Def ell2-MGP sol} extends to $\\mathbb{R}^{\\infty}$ in the sense that a measure $\\bar{Q} \\in \\mathcal{P}(C_T\\mathbb{R}^{\\infty})$ is understood as a solution, provided the process (\\ref{15}) is a $\\bar{Q}$-martingale on $C_T\\mathbb{R}^{\\infty}$ with respect to the natural filtration for each $\\bar{F} = f \\circ \\pi_n: \\mathbb{R}^{\\infty} \\to \\mathbb{R}$ as above.\n\\begin{rem}\\label{Rem meas P SP}\n\tWe recall that $\\ell^2 \\in \\mathcal{B}(\\mathbb{R}^{\\infty})$ and $\\mathcal{B}(\\ell^2) = \\mathcal{B}(\\mathbb{R}^{\\infty})_{\\upharpoonright \\ell^2}$. In particular, any probability measure $\\bar{\\Gamma} \\in \\mathcal{P}(\\ell^2)$ uniquely extends to a Borel probability measure on $\\mathbb{R}^{\\infty}$ via $\\bar{\\Gamma}(A) := \\bar{\\Gamma}(A \\cap \\ell^2)$, $A \\in \\mathcal{B}(\\mathbb{R}^{\\infty})$.\n\\end{rem}\n We shall need the following superposition principle from  \\cite{TrevisanPhD}, which lifts a solution to a FPK-equation on $\\mathbb{R}^{\\infty}$ to a solution of the associated martingale problem. Note that in \\cite{TrevisanPhD}, the author assumes an integrability condition of order $p>1$ instead of $p=1$ as in \\eqref{2_global_int_l2-FPKE} in order to essentially reduce the proof to the corresponding finite-dimensional result, see \\cite[Thm.2.14]{TrevisanPhD}, which requires such a higher order integrability. However, since the latter result was extended to the case of an $L^1$-integrability condition by the same author \\cite[Thm.2.5]{trevisan2016}, it is easy to see that also the infinite-dimensional result \\cite[Thm.7.1]{TrevisanPhD} holds for solutions with $L^1$-integrability as in \\eqref{2_global_int_l2-FPKE}.\n\n\\begin{prop}\\label{Prop SPprincip ell2}\n\t[Superposition principle on $\\mathbb{R}^{\\infty}$, Thm.7.1. \\cite{TrevisanPhD} For any weakly continuous solution $(\\bar{\\Gamma}_t)_{t \\leq T} \\subseteq \\mathcal{P}(\\mathbb{R}^{\\infty})$ to the $\\mathbb{R}^{\\infty}$-extended version of (\\ref{ell2-FPKE}), there exists  $\\bar{Q} \\in \\mathcal{P}(C_T\\mathbb{R}^{\\infty})$, which solves the $\\mathbb{R}^{\\infty}$-extended version of ($\\ell^2$-MGP) such that $\\bar{Q}\\circ e_t^{-1} = \\bar{\\Gamma}_t$ for each $t \\in [0,T]$.\n\\end{prop}\nMoreover, we have the following consequence for the solutions we are interested in. Note that paths $t \\mapsto z_t \\in H(\\mathcal{SP})$ are continuous with respect to the product topology if and only if they are $\\ell^2$-continuous. Hence, we may use the notation $C_TH(\\mathcal{SP})$ unambiguously and consider it as a subset of either $C_T\\mathbb{R}^{\\infty}$ or $C_t\\ell^2$. Since $H(\\mathcal{SP}) \\subseteq \\ell^2$ is closed even with respect to the product topology, $C_TH(\\mathcal{SP})$ belongs to $\\mathcal{B}(C_T\\ell^2)$ and $\\mathcal{B}(C_T\\mathbb{R}^{\\infty})$. \n\\begin{lem}\n\tIf in the situation of the previous proposition each $\\bar{\\Gamma}_t$ is concentrated on the Borel set $H(\\mathcal{SP}) \\subseteq \\mathbb{R}^{\\infty}$, then $\\bar{Q}$ is concentrated on continuous curves in $H(\\mathcal{SP})$. In particular, in this case $\\bar{Q}$ may be regarded as an element of $\\mathcal{P}(C_T\\ell^2)$ and a solution to the martingale problem ($\\ell^2$-MGP) as in Definition \\ref{Def ell2-MGP sol}.\n\\end{lem}\n\\begin{proof}\n\tThe closedness of $H(\\mathcal{SP}) \\subseteq \\mathbb{R}^{\\infty}$ yields\n\t$$\\bar{Q}(C_TH(\\mathcal{SP})) = \\bar{Q}\\bigg(\\underset{q \\in [0,T]\\cap \\mathbb{Q}}{\\bigcap}\\{e_q \\in H(\\mathcal{SP})\\}\\bigg) = 1,$$\n\tdue to $\\bar{Q}\\circ e_t^{-1} = \\bar{\\Gamma}_t$ for each $ t \\leq T$. By the observation above this lemma, it follows\n\t$$\\mathcal{B}(C_T\\ell^2)_{\\upharpoonright C_TH(\\mathcal{SP})} \\subseteq \\mathcal{B}(C_T\\mathbb{R}^{\\infty})_{\\upharpoonright C_TH(\\mathcal{SP})} \\subseteq \\mathcal{B}(C_T\\mathbb{R}^{\\infty})$$and we can therefore consider $\\bar{Q}$ as a probability measure on $\\mathcal{B}(C_T\\ell^2)$ via \n\t$$\\bar{Q}(A) := \\bar{Q}\\big(A\\cap C_TH(\\mathcal{SP})\\big),\\,\\, A \\in \\mathcal{B}(C_T\\ell^2) $$with mass on $C_TH(\\mathcal{SP})$. It is clear that this measure fulfills Definition \\ref{Def ell2-MGP sol}.\n\\end{proof}\nHence, subsequently we may regard to $\\bar{Q}$ as in Proposition \\ref{Prop SPprincip ell2} as a solution to ($\\ell^2$-MGP) on either $\\mathbb{R}^{\\infty}$ or $\\ell^2$ without differing the notation. Recall the notation $p_i: \\ell^2 \\to \\mathbb{R}$, $p_i(z) = z_i.$ \n\\begin{lem}\\label{Lem martingale and covar}\n\tLet $\\bar{Q}$ be a solution to the martingale problem ($\\ell^2$-MGP) on $\\ell^2$. Then, for any $i \\geq 1$, the process\n\t\\begin{equation}\\label{16}\n\tM_i(t) := p_i \\circ e_t - p_i \\circ e_0 - \\int_0^t \\bar{B}_i(s,e_s)ds\n\t\\end{equation}\n\tis a real-valued, continuous $\\bar{Q}$-martingale on $C_T\\ell^2$ with respect to the canonical filtration. The covariation $\\langle \\langle M_i, M_j \\rangle \\rangle $ of $M_i$ and $M_j$ is $\\bar{Q}$-a.s. given by\n\t\n\t\\begin{equation}\\label{17}\n\t\\langle\\langle M_i, M_j \\rangle\\rangle_t = \\int_0^t \\bar{A}_{ij}(s,e_s)ds, \\,\\, t\\in [0,T].\n\t\\end{equation}\n\\end{lem}\n\\begin{proof}\n\tFor $i,j \\geq 1$, let $n \\geq \\text{max}(i,j)$, consider $p^n_i: \\mathbb{R}^n \\to \\mathbb{R}$, $p^n_i(x) = x_i$ and let\n\t$$\\bar{F}^n_i : \\ell^2 \\to \\mathbb{R}, \\,\\, \\bar{F}^n_i(z) = p^n_i \\circ \\pi_n(z).$$ Note that $\\bar{F}^n_i = p_i$ on $\\ell^2$, independent of $n \\geq \\text{max}(i,j)$. For $k \\geq 1$, introduce the stopping time $\\sigma_k :=\\text{inf}\\{t \\in [0,T]: ||e_t||_{\\ell^2} \\geq k \\}$ with respect to the canonical filtration on $C_T\\ell^2$. Clearly, $\\sigma_k \\nearrow +\\infty$ pointwise. Consider $\\eta_k \\in C^{2}_c(\\mathbb{R}^n)$ such that $\\eta_k(x) = 1$ for  $|x| \\leq k+1$.\\\\ Since $\\partial_{k}p^n_i = \\delta_{ki}$ and $\\partial_{kl}p^n_i = 0$ for $k,l \\leq n$, we have\n\t$$M_i(t) = \\bar{F}^n_i\\circ e_t - \\int_0^t \\bar{\\nabla} \\bar{F}^n_i \\circ e_s \\cdot \\bar{B}(s,e_s)+\\frac{1}{2}D^2\\bar{F}^n_i\\circ e_s : \\bar{A}(s,e_s)ds$$ and, setting $\\bar{F}^{n,k}_i := (\\eta_kp^n_i) \\circ \\pi_n \\in \\mathcal{F}C^2_b(\\ell^2)$, \n\t$$M_i(\\sigma_k \\wedge t) = \\bar{F}^{n,k}_i\\circ e_{t\\wedge \\sigma_k} - \\int_0^{t\\wedge \\sigma_k} \\bar{\\nabla} \\bar{F}^{n,k}_i \\circ e_s \\cdot \\bar{B}(s,e_s)+\\frac{1}{2}D^2\\bar{F}^{n,k}_i\\circ e_s : \\bar{A}(s,e_s)ds.$$Since the latter is a continuous $\\bar{Q}$-martingale for each $k \\geq 1$, it follows that $M_i$ is a continuous local $\\bar{Q}$-martingale. Concerning (\\ref{17}), it suffices to prove that for any $\\bar{F} \\in \\mathcal{F}C^2_b(\\mathcal{\\ell}^2)$, $\\bar{F} = f \\circ\\pi_n$, we have\n\t \\begin{equation}\\label{Covar aux}\n\t\\langle \\langle M^{\\bar{F}} \\rangle \\rangle_t = \\int_0^t \\big \\langle \\bar{\\nabla}\\bar{F}(e_s), \\bar{A}(s,e_s)\\bar{\\nabla}\\bar{F}(e_s)ds \\big \\rangle_{\\ell^2}ds, \n\t\\end{equation}with\n\t$$M^{\\bar{F}}_t := \\bar{F}\\circ e_t - \\int_0^t   \\bar{\\nabla}\\bar{F}(e_s) \\cdot \\bar{B}(s,e_s) + \\frac{1}{2}D^2\\bar{F}(e_s): \\bar{A}(s,e_s)ds.$$Indeed, from here (\\ref{17}) follows by considering (\\ref{Covar aux}) for $\\bar{F}^{n,k}_i$, localization of the local martingale $M_i$ and polarization for the quadratic (co-)variation. Concerning (\\ref{Covar aux}), it is standard (cf. \\cite[p.73,74]{stroock_1987}) to use It\u00f4's product rule to obtain that\n\t$$t\\mapsto (M^{\\bar{F}}_t)^2 - \\int_0^t \\bar{\\mathbf{L}}_s^{(2)}\\bar{F}^2(e_s)-2\\bar{F}(e_s)\\bar{\\mathbf{L}}_s^{(2)}\\bar{F}(e_s)ds $$\n\tis a continuous $\\bar{Q}$-martingale on $C_T\\ell^2$, where we denote by $\\bar{\\mathbf{L}}_t^{(2)}\\bar{F}(e_s)$ the integrand of the integral term in the definition of $M^{\\bar{F}}$. A straightforward calculation yields\n\t$$\\int_0^t \\bar{\\mathbf{L}}_s^{(2)}\\bar{F}^2(e_s)-2\\bar{F}(e_s)\\bar{\\mathbf{L}}_s^{(2)}\\bar{F}(e_s)ds = \\int_0^t \\big \\langle \\bar{\\nabla}\\bar{F}(e_s), \\bar{A}(s,e_s)\\bar{\\nabla}\\bar{F}(e_s) \\big \\rangle_{\\ell^2}ds,$$\n\twhich completes the proof.\n\\end{proof}\nWe summarize the results of this step in the following proposition.\n\\begin{prop}\\label{Prop summary Step2}\n\tLet $(\\bar{\\Gamma}_t)_{t \\leq T}$ be a weakly continuous solution to (\\ref{ell2-FPKE}) such that $\\bar{\\Gamma}_t(H(\\mathcal{SP})) = 1$ for each $t \\in [0,T]$. Then, there exists a solution $\\bar{Q} \\in \\mathcal{P}(C_T\\ell^2)$ to the martingale problem ($\\ell^2$-MGP) such that $\\bar{Q}$ is concentrated on $C_TH(\\mathcal{SP})$ with $\\bar{Q}\\circ e_t^{-1} = \\bar{\\Gamma}_t$ for each $t \\in [0,T]$. Further, the results of Lemma \\ref{Lem martingale and covar} apply to $\\bar{Q}$. \n\\end{prop}\n\n\n\\textbf{Step 3: From ($\\ell^2$-MGP) to (\\ref{SNLFPKE}):} For a given solution $\\bar{Q} \\in \\mathcal{P}(C_T\\ell^2)$ to ($\\ell^2$-MGP), set\n$$\\mathcal{C}:= \\mathcal{B}(C_T\\ell^2) \\bigvee \\mathcal{N}_{\\bar{Q}}$$\nand\n$$\\mathcal{C}_t := \\sigma(e_s, s\\leq t) \\bigvee \\mathcal{N}_{\\bar{Q}}$$\nfor $t \\leq T$, where $\\mathcal{N}_{\\bar{Q}}$ denotes the collection of all subsets of sets $N \\in \\mathcal{B}(C_T\\ell^2)$ with $\\bar{Q}(N) = 0$. Of course, $\\mathcal{C}$ and $\\mathcal{C}_t$ depend on $\\bar{Q}$, but we suppress this dependence in the notation. Without further mentioning, we understand such $\\bar{Q}$ as extended to $\\mathcal{C}$ in the canonical way. Then, $(C_T\\ell^2,\\mathcal{C},(\\mathcal{C}_t)_{t \\leq T}, \\bar{Q})$ is a complete filtered probability space. Clearly, $(t,\\gamma) \\mapsto \\bar{\\Sigma}(t,e_t(\\gamma))$ is $\\mathcal{C}_t$-progressively measurable from $[0,T]\\times C_T\\ell^2$ to $L(\\mathbb{R}^{d_1},\\ell^2)$, the space of bounded linear operators from $\\mathbb{R}^{d_1}$ to $\\ell^2$.\n\n\\begin{rem}\\label{Rem extend prob space}\n\tWe extend $(C_T\\ell^2,\\mathcal{C},(\\mathcal{C}_t)_{t \\leq T}, \\bar{Q})$ as follows. Let $(\\Omega', \\mathcal{F}'', (\\mathcal{F}''_t)_{t \\leq T}, P)$ be a complete filtered probability space with a real-valued $\\mathcal{F}''_t$-Wiener process $\\beta$ on it, define\n\t\\begin{equation*}\n\t\\Omega := C_T\\ell^2 \\otimes \\underset{l \\geq 1}{\\bigotimes}\\Omega', \\,\\, \\mathcal{F}' := \\mathcal{C}\\otimes \\underset{l \\geq 1}{\\bigotimes}\\mathcal{F}'', \\,\\, \\mathcal{F}_t' := \\mathcal{C}_t \\otimes \\underset{l \\geq 1}{\\bigotimes}\\mathcal{F}''_t, \\,\\, \\mathbb{P}' := \\bar{Q}\\otimes \\underset{l \\geq 1}{\\bigotimes}P,\n\t\\end{equation*}let $\\mathcal{F}$ and $\\mathcal{F}_t$ be the $\\mathbb{P}'$-completion of $\\mathcal{F}'$ and $\\mathcal{F}_t'$, respectively, and denote the canonical extension of $\\mathbb{P}'$ to $\\mathcal{F}$ by $\\mathbb{P}$. Further, we denote the Wiener process $\\beta$ on the $i$-th copy of $\\Omega'$ by $\\beta_i$ and extend each $\\beta_i$ to $\\Omega$ by $\\beta_i(\\omega) := \\beta_i(\\omega_i)$ for $\\omega = \\gamma \\times (\\omega_i)_{i \\geq 1} \\in \\Omega$. Similarly, we extend each projection $e_t$ from $C_T\\ell^2$ to $\\Omega$ via $e_t(\\omega) := e_t(\\gamma)$ for $\\omega$ as above, but keep the same notation for this extended process. Obviously, $(e_t)_{t \\leq t}$ is a continuous, $\\mathcal{F}_t$-adapted process on $\\Omega$ and each $\\beta_i$ is an $\\mathcal{F}_t$-Wiener process on $\\Omega$ under $\\mathbb{P}$. Moreover, $(e_t)_{t \\leq T}$ and $(\\beta_i)_{i \\geq 1}$ are independent on $\\Omega$ with respect to $\\mathbb{P}$ by construction. Further, it is clear that the process $M_i$ as in (\\ref{16}) is a $\\mathbb{P}$-martingale with respect to $\\mathcal{F}_t$ for each $i \\geq 1$ with covariation as in (\\ref{17}) and that $(t,\\gamma) \\mapsto \\bar{\\Sigma}(t,e_t(\\gamma)) \\in L(\\mathbb{R}^{d_1},\\ell^2)$ is $\\mathcal{F}_t$-progressively measurable on $[0,T]\\times \\Omega$.\n\\end{rem}\n\nFinally, we need the following result, which is a special case of Theorem 2, \\cite{Ondrejat_StochInt_Repr}.\n\\begin{prop}\\label{Prop Step3 Ondrejat appl}\n\tLet $\\bar{Q} \\in \\mathcal{P}(C_T\\ell^2)$ be a solution to the martingale problem ($\\ell^2$-MGP). Then, there exists a complete filtered probability space with an adapted $d_1$-dimensional Wiener process $W = (W^{\\alpha})_{\\alpha \\leq d_1}$ and an $\\ell^2$-valued adapted continuous process $X = (X_t)_{t \\leq T}$ such that the law of $X$ on $C_T\\ell^2$ is $\\bar{Q}$ and for $i \\geq 1$ and $t \\in [0,T]$, we have a.s.\n\t\\begin{equation}\\label{extra2}\n\tp_i \\circ X_t - p_i \\circ X_0 - \\int_0^t \\bar{B}_i(s,X_s)ds = \\sum_{\\alpha =1}^{d_1}\\int_0^t \\bar{\\Sigma}^{\\alpha}_i(s,X_s)dW_s^{\\alpha}\n\t\\end{equation}and the exceptional set can be chosen independent of $t$ and $i$.\n\\end{prop}\nTo see this, consider Theorem 2 of \\cite{Ondrejat_StochInt_Repr} with $X = \\ell^2$, $U_0 = \\mathbb{R}^{d_1}$, $D = \\{p_i, i \\geq 1\\}$, the processes $M(p_i)$ given by $M_i$ as in (\\ref{16}) on the probability space $\\Omega$ of Remark \\ref{Rem extend prob space} and \n$$g_s = \\bar{\\Sigma}(s,e_s) \\in L(\\mathbb{R}^{d_1}, \\ell^2).$$ These choices fulfill all requirements of \\cite{Ondrejat_StochInt_Repr}. In this case, the $\\ell^2$-valued process $X$ is given by $X_t = e_t$ on $\\Omega$. Since all terms in (\\ref{extra2}) are continuous in $t$, the exceptional set may indeed by chosen independently of $t \\in [0,T]$ and $i \\geq 1$.\\\\ \n\\\\\nThe proof of Theorem \\ref{main thm stoch case} now follows from the above three step-scheme as follows.\\\\\n\\\\\n\\textit{Proof of Theorem \\ref{main thm stoch case}:} Let $(\\Gamma_t)_{t \\leq T}\\subseteq \\mathcal{P}(\\mathcal{SP})$ be a weakly continuous solution to (\\ref{P-FPKE}). By Lemma \\ref{Lem of Step1 stochCase} of Step 1, the weakly continuous curve of Borel probability measures on $\\ell^2$\n$$\\bar{\\Gamma}_t := \\Gamma_t \\circ H^{-1}, \\,\\, t\\in [0,T]$$\nsolves (\\ref{ell2-FPKE}) and each $\\bar{\\Gamma}_t$ is concentrated on $H(\\mathcal{SP})$. By Proposition \\ref{Prop summary Step2} of Step 2, there exists a solution $\\bar{Q} \\in \\mathcal{P}(C_T\\ell^2)$ to the martingale problem ($\\ell^2$-MGP), which is concentrated on $C_TH(\\mathcal{SP})$ such that\n$$\\bar{Q} \\circ e_t^{-1} = \\bar{\\Gamma}_t, \\,\\, t\\in [0,T].$$\nFurther, Lemma \\ref{Lem martingale and covar} applies to $\\bar{Q}$. By Lemma \\ref{Lem martingale and covar} and Proposition \\ref{Prop Step3 Ondrejat appl} of Step 3, there is a $d_1$-dimensional $\\mathcal{F}_t$-adapted Wiener process $W = (W^{\\alpha})_{\\alpha \\leq d_1}$ and an $\\mathcal{F}_t$-adapted process $X$ on some complete filtered probability space $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\leq T}, \\mathbb{P})$, which fulfill (\\ref{extra2}) and $X \\in C_TH(\\mathcal{SP})$ $\\mathbb{P}$-a.s. such that $\\bar{Q}$ is the law of $X$.\\\\\nPossibly redefining $X$ on a  $\\mathbb{P}$-negligible set (which preserves (\\ref{extra2}) and its adaptedness, the latter due to the completeness of the underlying filtered probability space), we may assume $X_t(\\omega) = H(\\mu_t(\\omega))$ for some $\\mu_t(\\omega) \\in \\mathcal{SP}$ for each $(t,\\omega) \\in [0,T]\\times \\Omega$. The continuity of $H^{-1}: H(\\mathcal{SP}) \\to \\mathcal{SP}$ and $t \\mapsto X_t(\\omega)$ implies vague continuity of \n\\begin{equation}\\label{19}\nt \\mapsto \\mu_t(\\omega) = H^{-1} \\circ X_t(\\omega)\n\\end{equation}\nfor each $\\omega \\in \\Omega$ and $\\mathcal{F}_t$-adaptedness of the $\\mathcal{SP}$-valued process $(\\mu_t)_{t \\leq T}$. Considering (\\ref{extra2}), $X_t = H(\\mu_t)$ and the definition of $\\bar{B}$ and $\\bar{\\Sigma}^{\\alpha}_i$, we obtain, recalling $p_i(H(\\nu)) = \\nu(h_i)$ for each $\\nu \\in \\mathcal{SP}$,\n\\begin{equation*}\n\\mu_t(h_i)-\\mu_0(h_i)- \\int_0^t B_i(s,\\mu_s)ds = \\sum_{\\alpha = 1}^{d_1}\\int_0^t \\Sigma^{\\alpha}_i(s,\\mu_s)dW^{\\alpha}_s, \\,\\, t \\leq T\n\\end{equation*}$\\mathbb{P}$-a.s. for each $i \\geq 1$. From here, it follows by Lemma \\ref{Lem aux H} (i) that $(\\mu_t)_{t \\leq T}$ is a solution to (\\ref{SNLFPKE}) as in Definition \\ref{Def sol SNLFPKE}. Further, \n$$\\mathbb{P}\\circ \\mu_t^{-1} = (\\mathbb{P}\\circ X_t^{-1})\\circ (H^{-1})^{-1} = \\bar{\\Gamma}_t \\circ (H^{-1})^{-1} = \\Gamma_t \\circ H^{-1} \\circ (H^{-1})^{-1} = \\Gamma_t.$$\nIt remains to prove the final assertion of the theorem. To this end, note that $\\Gamma_0(\\mathcal{P}) = 1$ implies $\\bar{\\Gamma}_0(H(\\mathcal{P})) = 1$ and hence $\\mu_0 \\in \\mathcal{P}$ $\\mathbb{P}$-a.s. with $\\mu_0$ as in (\\ref{19}). From here, the assertion follows by Lemma \\ref{Lem consv of mass stochastic}. \\qed\n\\begin{rem}\nThe particular type of noise we consider for \\eqref{SNLFPKE} was partially motivated by \\cite{Coghi2019StochasticNF}, where the natural connection of equations of type \\eqref{SNLFPKE} to interacting particle systems with common noise was investigated. Other types of noise terms may be treated in the future, including a possible extension to infinite-dimensional ones. In particular, Proposition \\ref{Prop Step3 Ondrejat appl} via \\cite[Thm.2]{Ondrejat_StochInt_Repr} in the final step of the proof seems capable of such extensions, since the latter is an infinite-dimensional result.\n\\end{rem}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFrom the origin, games in extensive form have been formulated on a tree.\nIn his seminal paper\n\\emph{Extensive Games and the Problem of Information}~\\cite{Kuhn:1953},\nKuhn claimed that ``The use of a geometrical model  (\\ldots)\nclarifies the delicate problem of information''. This tells us that  the proper handling of information was a strong \nmotivation for Kuhn's extensive games. \nOn the game tree, moves are those vertices that possess alternatives,\nthen moves are partitioned into players moves, themselves partitioned into \ninformation sets (with the constraint that no two moves in an \ninformation set can be on the same play).\nKuhn mentions agents, one agent per information set, \nto ``personalize the interpretation'' but the notion is not central\n(to the point that his definition of perfect recall\n``obviates the use of agents'').\n\nBy contrast, in the so-called Witsenhausen's intrinsic model\n\\cite{Witsenhausen:1971a,Witsenhausen:1975},\nagents play a central role. \nEach agent is equipped with a decision set and a $\\sigma$-field,\nand the same for Nature. Then, Witsenhausen introduces the product set\nand the product $\\sigma$-field. This product set hosts the agents' information subfields. \nThe Witsenhausen's intrinsic model was elaborated \nin the control theory setting, in order to handle how information\nis distributed among agents\nand how it impacts their strategies.\nAlthough not explicitly designed for games, \nWitsenhausen's intrinsic model had, from the start, the potential \nto be adapted to games. Indeed, in~\\cite{Witsenhausen:1971a} Witsenhausen \n places his own model in the context of game theory by referring to \nvon~Neuman and Morgenstern~\\cite{vonNeuman-Morgenstern:1947},\nKuhn~\\cite{Kuhn:1953} and Aumann~\\cite{Aumann:1964}.\n\\medskip\n\nIn this paper, we introduce a new representation of games\nthat we call \\emph{games in intrinsic form}. \nGame representations play a key role in their analysis\n(see the illuminating introduction of the book~\\cite{Alos-Ferrer-Ritzberger:2016}),\nand we claim that games in intrinsic form display appealing features.\nIn the philosophy of the tree-based extensive form (Kuhn's view), \nthe temporal ordering is\nhardcoded in the tree structure: one goes from the root to the leaves,\nmaking decisions at the moves, contingent on information, chance and strategies.\nFor Kuhn, the time arrow (tree) comes first; information comes second\n(partition of the move vertices).\nBy contrast, for Witsenhausen, information comes first;\nthe time arrow comes (possibly) second, \nunder a proper causality assumption contingent to the information structure. \n\nNot having a hardcoded temporal ordering makes mathematical representations\nless constrained, hence more general.\nMoreover, Witsenhausen's framework makes representations more intrinsic.\nAs an illustration, let us consider a game where two players play once but at the same\ntime. To formulate it on a tree requires to arbitrarily decide which of the\ntwo plays first. This is not the case for games in intrinsic form,\nwhere each player\/agent is equipped with an information subfield\nand strategies that are measurable with respect to the latter;\nwriting the system of two equations that express decisions as \nthe output of strategies leads to a unique outcome,\nwithout having to solve one equation first, and the other second.\n\nThe tree representation of games has its pros and cons.\nOn the one hand, trees are perfect to follow step by step how a game is played\nas any strategy profile induces a unique play:\none goes from the root to the leaves, passing from one node\nto the next by an edge that depends on the strategy profile.\nOn the other hand, in games with information,\n  information sets are represented as ``union'' of tree nodes\nthat must satisfy restrictive axioms, and such\n  unions do not comply in a natural way with the tree structure,\n  which can render the game analysis delicate\n  \\cite{Alos-Ferrer-Ritzberger:2016,Bonanno:2004,Brandenburger:2007}.\nBy contrast, the notion of Witsenhausen's intrinsic games (W-games) does not require\nan explicit description of the play temporality, and the intrinsic form replaces\nthe tree structure with a product structure, more amenable to mathematical analysis. \nIf the introduction of the model may seem involved, we argue that the\nresulting structure is a powerful mathematical tool, because there are many\nsituations in which it is easier to reason and discuss with mathematical\nformulas than with trees.\n\nWe illustrate our claim with a proof of the celebrated Kuhn's equivalence theorem for games\nin intrinsic form.\nIndeed, as a first step in a broader research program, \n we show that equivalence between mixed and behavioral\n strategies holds under perfect recall for W-games.\nMore precisely, our proof relies on an equivalence between behavioral,\nmixed and a new notion of product-mixed strategies.\nThese latter form a subclass of mixed strategies.\nIn the spirit of~\\cite{Aumann:1964}, \nin a product-mixed strategy, each agent (corresponding to time index in~\\cite{Aumann:1964}) \ngenerates strategies from a random device that is independent of all the\nother agents. We prove that, under perfect recall for W-games,\nany mixed strategy of a player is not only equivalent to a behavioral strategy,\nbut also to a product-mixed strategy where all the agents under control \nof the player randomly select their pure strategy independently of the other agents.\n\\medskip\n\nThe paper is organized as follows.\nIn Sect.~\\ref{Witsenhausen_intrinsic_model},\nwe present the finite version of Witsenhausen's intrinsic model. \nThen, in Sect.~\\ref{Finite_games_in_intrinsic_form}, we propose a formal\ndefinition of games in intrinsic form (W-games), and then discuss three notions of \n``randomization'' of pure strategies --- mixed, product-mixed and behavioral.\nFinally, we derive an equivalent of Kuhn's equivalence theorem for games in intrinsic form\nin Sect.~\\ref{Kuhn_Theorem}.\nIn Appendix~\\ref{Background_on_fields_atoms_and_partitions}, \nwe present background material on fields, atoms and partitions,\nas these notions lay at the core of Witsenhausen's intrinsic model\nin the finite case.\nIn all the paper, we adopt the convention that a player is female (hence using ``she'' and\n``her''), whereas an agent is male (``he'', ``his'').\n\n\n\n\\section{Witsenhausen's intrinsic model (the finite case)}\n\\label{Witsenhausen_intrinsic_model}\n\nIn this paper, we tackle the issue of information in the context of finite games.\nFor this purpose, we will present the so-called intrinsic model of Witsenhausen\n\\cite{Witsenhausen:1975,Carpentier-Chancelier-Cohen-DeLara:2015}\nbut with finite sets rather than with infinite ones \nas in the original exposition.\nWe refer the reader to \nAppendix~\\ref{Background_on_fields_atoms_and_partitions}\nfor background material on fields, atoms and partitions.\n\nIn~\\S\\ref{Finite_Witsenhausen_intrinsic_model},\nwe present the finite version of Witsenhausen's\nintrinsic model, where we highlight the role of the configuration field\nthat contains the information subfields of all agents.\nIn~\\S\\ref{Examples}, we illustrate, on a few examples, the ease with which \none can model information in strategic contexts, \nusing subfields of the configuration field.\nFinally, we present in~\\S\\ref{Solvability_Causality} \nthe notions of solvability and causality. \n\n\n\\subsection{Finite Witsenhausen's intrinsic model (W-model)}\n\\label{Finite_Witsenhausen_intrinsic_model}\n\nWe present the finite version of Witsenhausen's\nintrinsic model, introduced some five decades ago in the control community\n\\cite{Witsenhausen:1971a,Witsenhausen:1975}. \n\n\\begin{definition}(adapted from \\cite{Witsenhausen:1971a,Witsenhausen:1975})\n\nA \\emph{finite W-model} is a collection\n$\\bp{\n\\AGENT,\n\\np{\\Omega, \\tribu{\\NatureField}}, \n\\sequence{\\CONTROL_{\\agent}, \\tribu{\\Control}_{\\agent}}{\\agent \\in \\AGENT}, \n\\sequence{\\tribu{\\Information}_{\\agent}}{\\agent \\in \\AGENT} \n}$, where \n\\begin{itemize}\n\\item\n$\\AGENT$ is a finite set, whose elements are called \\emph{agents};\n\\item \n\\( \\Omega \\) is a finite set which represents all uncertainties;\nany $\\omega \\in \\Omega$ is called a \\emph{state of Nature};\n$\\tribu{\\NatureField}$ is the complete field over~\\( \\Omega \\);\n\\item \nfor any \\( \\agent \\in \\AGENT \\), $\\CONTROL_{\\agent}$ is a finite set, \nthe \\emph{set of decisions} for agent~$\\agent$;\n$\\tribu{\\Control}_{\\agent}$ is the complete field over~$\\CONTROL_{\\agent}$;\n\\item\nfor any \\( \\agent \\in \\AGENT \\), \\( \\tribu{\\Information}_{\\agent} \\)\nis a subfield of the following product field\n\\begin{equation}\n  \\tribu{\\Information}_{\\agent} \\subset \n  {\\oproduit{\\bigotimes \\limits_{b \\in \\AGENT} \\tribu{\\Control}_{b}}{\\tribu{\\NatureField}}}\n  \\eqsepv\n  \\forall \\agent \\in \\AGENT \n  \\label{eq:information_field_agent}\n\\end{equation}\n\nand is called the \\emph{information field} of the agent~$\\agent$.\n\\end{itemize}\n\\label{de:W-model}\n\\end{definition}\n\n\\begin{subequations}\nThe \\emph{configuration space} is the product space \n(called \\emph{hybrid space} by Witsenhausen, hence the $\\HISTORY$ notation)\n\\begin{equation}\n  \\label{eq:HISTORY}\n  \\HISTORY = \\produit{ \\prod\\limits_{\\agent \\in \\AGENT} \\CONTROL_{\\agent}}{\\Omega} \n  \\eqfinp \n\\end{equation}\nAs all fields \\( \\tribu{\\NatureField} \\) and\n\\( \\sequence{\\tribu{\\Control}_{\\agent}}{\\agent \\in \\AGENT} \\) are complete, \nthe product \\emph{configuration field}\n\\begin{equation}\n  \\tribu{\\History} =\\oproduit{{\\bigotimes \\limits_{\\agent \\in \\AGENT} \\tribu{\\Control}_{\\agent}}}{\\tribu{\\NatureField}}\n  \\label{eq:history_field}\n\\end{equation}\nis also the complete field of~\\( \\HISTORY \\).\nA \\emph{configuration} \\( \\history \\in \\HISTORY \\) is denoted by\n\\begin{equation}\n  \\history=\\bp{\\omega, \\sequence{\\control_{\\agent}}{\\agent \\in \\AGENT}}\n  \\iff\n  \\history_\\emptyset = \\omega\n \\text{ and }\n  \\history_{\\agent} =\\control_{\\agent}\n  \\eqsepv\n  \\forall \\agent \\in \\AGENT\n  \\eqfinp\n  \\label{eq:history}\n\\end{equation}\n\\end{subequations}\n\nIn lieu of the information field~\\( \\tribu{\\Information}_{\\agent} \\)\nin~\\eqref{eq:information_field_agent},\nit will be convenient to consider the equivalence relation~$\\sim_{\\agent}$,\non the configuration space~$\\HISTORY$, defined in such a way that \nthe equivalence classes \\( \\bracket{\\cdot}_{\\agent} \\subset \\HISTORY \\) \ncoincide with the atoms of~\\( \\tribu{\\Information}_{\\agent} \\),\nthat is, with the elements of the partition\n\\( \\crochet{\\tribu{\\Information}_{\\agent}} \\) in~\\eqref{eq:atom_set}:\n\\begin{equation}\n  \\bp{\\forall \\history', \\history'' \\in \\HISTORY} \\quad\n  \\history' \\sim_{\\agent} \\history'' \\;\\Leftrightarrow\\;\n  \\history'' \\in \\bracket{\\history'}_{\\agent}\\;\\Leftrightarrow\\;\n  \\exists G \\in \\crochet{\\tribu{\\Information}_{\\agent}},\n  \\, \\{ \\history', \\history'' \\} \\subset G\n  \\eqfinp\n  \\label{eq:bracket_HISTORY}\n\\end{equation}\nThus defined, the subset \\( \\bracket{\\history}_{\\agent} \\subset \\HISTORY \\)\nis the unique atom~$G$ in \\( \\crochet{\\tribu{\\Information}_{\\agent}} \n\\subset \\tribu{\\Information}_{\\agent} \\)\nthat contains the configuration~$\\history$.\n\nWe will need the following equivalent characterization of measurable\nmappings, which is a slight reformulation of~\\cite[Proposition~3.35]{Carpentier-Chancelier-Cohen-DeLara:2015}.\n\n\\begin{proposition}(adapted from \\cite[Proposition~3.35]{Carpentier-Chancelier-Cohen-DeLara:2015})\n  \\label{pr:measurability}\n \n  Let  \\( \\rho : (\\HISTORY,\\tribu{\\History}) \\to ({\\mathbb D},\\tribu{D}) \\) \n  be a mapping, \n  where ${\\mathbb D}$ is a set and $\\tribu{D}$ is a $\\sigma$-field over~${\\mathbb D}$.\n  We suppose that the $\\sigma$-field~$\\tribu{D}$ contains all the singletons.\n  Then, for any agent~$\\agent \\in \\AGENT$, the following statements are equivalent:\n  \\begin{subequations}\n    \\begin{equation}\n      \\rho^{-1} (\\tribu{D}) \\subset \\tribu{\\Information}_{\\agent} \n      \\eqfinv\n    \\end{equation}\n    \\begin{equation}\n      \\bp{ \\forall \\history', \\history'' \\in \\HISTORY } \\quad\n      \\history' \\sim_{\\agent} \\history'' \\implies \n      \\rho\\np{\\history'}=\\rho\\np{\\history''}\n      \\eqfinv\n      \\label{eq:strategy_atoms_rho}\n    \\end{equation}\n    \\begin{equation}\n      \\begin{split}\n        \\textrm{the set-valued mapping } \\hat\\rho :\n        \\crochet{\\tribu{\\Information}_{\\agent}}\\rightrightarrows{\\mathbb D} \n        \\eqsepv \\textrm{defined by} \\\\\n        \\hat\\rho\\np{G_{\\agent}}= \\nset{ \\rho\\np{\\history}}%\n        { \\history \\in G_{\\agent}}\n        \\eqsepv\n        \\forall G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}},\n        \\textrm{ is a mapping.}\n      \\end{split}\n      \\label{eq:measurability-and-set-valued-map}\n    \\end{equation}\n  \\end{subequations}\n  In any of these equivalent cases, we say that \n  the mapping~\\( \\rho \\) is \\emph{$\\tribu{\\Information}_{\\agent}$-measurable},\n  and, for all $G_{\\agent}\\in \\crochet{\\tribu{\\Information}_{\\agent}}$, we denote by \\( \\rho\\np{G_{\\agent}} \\)\n  the unique element of~${\\mathbb D}$ in $\\hat\\rho\\np{G_{\\agent}}$, that is, \n \n \n \n  \\begin{equation}\n    \\bp{ \\forall r \\in \\rho\\np{\\HISTORY}\n      \\eqsepv \\forall G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}} }\\quad\n    \\rho\\np{G_{\\agent}} =r \\iff\n    \\hat\\rho\\np{{G_{\\agent}}} = \\na{r}\n    \\eqfinp\n    \\label{eq:common_value_rho}\n  \\end{equation}\n  Then, using the extended notation above~\\eqref{eq:bracket_HISTORY}, we have the property \n  \\begin{equation}\n    \\rho \\text{ is }\\tribu{\\Information}_{\\agent}\\text{-measurable}\n    \\implies \n    \\rho\\np{ \\bracket{\\history}_{\\agent} } =\n    \\rho\\np{\\history} \n    \\eqsepv \\forall \\history \\in \\HISTORY\n    \\eqfinp\n    \\label{eq:common_value_equivalence_class_rho}\n  \\end{equation}\n\\end{proposition}\n\nNow that we have explicited measurable mappings with respect to \nagents information subfields, we introduce the notion of \npure W-strategy.\n\n\\begin{definition}(\\cite{Witsenhausen:1971a,Witsenhausen:1975})\n  \\label{de:W-strategy}\n \n  A \\emph{pure W-strategy} of agent~$\\agent \\in \\AGENT$ is a mapping\n \n  \\begin{subequations}\n   \n    \\begin{equation}\n      \\policy_{\\agent} : (\\HISTORY,\\tribu{\\History}) \\to\n      (\\CONTROL_{\\agent},\\tribu{\\Control}_{\\agent})\n    \\end{equation}\n    from configurations to decisions,\n    which is measurable with respect to the information\n    field~$\\tribu{\\Information}_{\\agent}$ of agent~$\\agent$, that is,\n    \\begin{equation}\n      \\label{eq:decision_rule}\n      \\policy_{\\agent}^{-1} (\\tribu{\\Control}_{\\agent})\n      \\subset \\tribu{\\Information}_{\\agent} \n      \\eqfinp\n    \\end{equation}\n   \n  \\end{subequations}\n \n  \\begin{subequations}\n   \n    We denote by $\\POLICY_{\\agent}$ the set of all pure W-strategies of agent $\\agent \\in \\AGENT$.\n   \n    A \\emph{pure W-strategies profile}~$\\policy$ is a family \n    \\begin{equation}\n      \\policy = \\sequence{\\policy_{\\agent}}{\\agent \\in \\AGENT} \n      \\in \\prod_{\\agent \\in \\AGENT} \\POLICY_{\\agent} \n      \\label{eq:W-strategy_profile}\n    \\end{equation}\n    of pure W-strategies, one per agent~$\\agent \\in \\AGENT$.\n    The \\emph{set of pure W-strategies profiles} is \n    \\begin{equation}\n      \\POLICY= \\prod_{\\agent \\in \\AGENT} \\POLICY_{\\agent} \n      \\eqfinp\n      \\label{eq:W-STRATEGY}\n    \\end{equation}\n  \\end{subequations}\n\\end{definition}\nCondition~\\eqref{eq:decision_rule} expresses the property that any \n(pure) W-strategy of agent~$\\agent$ may only depend upon the\ninformation~$\\tribu{\\Information}_{\\agent}$ available to the agent. \n\nIn what follows, we will need some notations.\nFor any nonempty subset $\\mathbb{B} \\subset \\AGENT$ of agents, we define \n\\begin{subequations}\n \n  \\begin{align}\n    \\tribu{\\Control}_\\mathbb{B} \n    &= \n      \\bigotimes \\limits_{b \\in \\mathbb{B}} \\tribu{\\Control}_b\n      \\otimes\n      \\bigotimes \\limits_{\\agent \\not\\in \\mathbb{B}} \\{ \\emptyset, \\CONTROL_{\\agent} \\}\n      \\subset\n      \\bigotimes \\limits_{\\agent \\in \\AGENT} \\tribu{\\Control}_{\\agent}\n      \\eqfinv\n      \\label{eq:sub_control_field_BGENT}\n    \\\\\n    \\tribu{\\History}_\\mathbb{B} \n    &= \n      \\tribu{\\NatureField} \\otimes \\tribu{\\Control}_\\mathbb{B}\n      = \\tribu{\\NatureField} \\otimes \\bigotimes \\limits_{b \\in \\mathbb{B}} \\tribu{\\Control}_b\n      \\otimes\n      \\bigotimes \\limits_{\\agent \\not\\in \\mathbb{B}} \\{ \\emptyset, \\CONTROL_{\\agent} \\}\n      \\subset \\tribu{\\History}\n      \\eqfinv\n      \\label{eq:sub_history_field_BGENT}\n    \\\\\n    \\history_\\mathbb{B} \n    &=\n      \\sequence{\\history_b}{b \\in \\mathbb{B}}\n      \\in \\prod \\limits_{b \\in \\mathbb{B}} \\CONTROL_b\n      \\eqsepv \\forall \\history \\in \\HISTORY\n      \\eqfinv\n      \\label{eq:sub_history_BGENT}\n    \\\\\n    \\policy_\\mathbb{B} \n    &=\n      \\sequence{\\policy_b}{b \\in \\mathbb{B}}\n      \\in \\prod \\limits_{b \\in \\mathbb{B}} \\POLICY_{b} \n      \\eqsepv \\forall \\policy \\in \\POLICY\n      \\eqfinp\n      \\label{eq:sub_wstrategy_BGENT}\n  \\end{align}\n \n \n \n \n\\end{subequations}\n\n\n\\subsection{Examples}\n\\label{Examples}\n\nWe illustrate, on a few examples, the ease with which \none can model information in strategic contexts, \nusing subfields of the configuration field.\nEven if we have presented the finite version of Witsenhausen's intrinsic model\nin~\\S\\ref{Finite_Witsenhausen_intrinsic_model},\nwe take the opportunity here to show its potential to describe \ninfinite decision and Nature sets.\n\n\\subsubsubsection{Sequential decisions}\nSuppose an individual has to take decisions (say, an element of $\\RR^n$) \nat every discrete time step in the set\\footnote{For any integers $a \\leq b$, $\\ic{a,b}$ denotes the subset\n  $\\na{a,a+1,\\ldots,b-1,b}$.} \n\\( \\ic{1,\\horizon{-}1} \\), where $\\horizon \\geq 1$ is an integer.\nThe situation will be modeled with (possibly) Nature set and field\n\\( \\np{\\Omega, \\tribu{\\NatureField}} \\), \nand with $\\horizon$~agents in $\\AGENT=\\ic{0,\\horizon{-}1}$, \nand their corresponding sets, $\\CONTROL_t= \\RR^n$, and fields,\n$\\tribu{\\Control}_t = \\borel{\\RR^{n}} $\n(the Borel $\\sigma$-field of~$\\RR^n$), for $t \\in \\AGENT$.\nThen, one builds up the product set \n$\\HISTORY=\\produit{\\prod_{t=0}^{\\horizon{-}1} \\CONTROL_{t} }{\\Omega}$ and \nthe product field $\\tribu{\\History}= \\oproduit{%\n  \\bigotimes_{t=0}^{\\horizon{-}1} \\tribu{\\Control}_{t} }{\\tribu{\\NatureField}}$.\nEvery agent \\( t \\in \\ic{0,\\horizon{-}1} \\) is equipped with an \ninformation field \\( \\tribu{\\Information}_{t} \\subset \\tribu{\\History} \\).\nThen, we show how we can express four information patterns:\nsequentiality, memory of past information, \nmemory of past actions, perfect recall.\nThe inclusions \\( \\tribu{\\Information}_{t} \\subset \n\\tribu{\\History}_{\\{0,\\ldots,t{-}1\\}} = \\oproduit{%\n\\bigotimes_{s=0}^{t-1} \\tribu{\\Control}_{s} \\otimes \n\\bigotimes_{s=t}^{\\horizon{-}1} \\{ \\emptyset, \\CONTROL_{s} \\}}{\\tribu{\\NatureField}} \\),\nfor \\( t\\in \\ic{0,\\horizon{-}1} \\), \nexpress that every agent can remember no more than his past actions\n(sequentiality); \nmemory of past information is represented by the inclusions \n\\(\\tribu{\\Information}_{t-1} \\subset \\tribu{\\Information}_{t} \\),\nfor \\( t\\in \\ic{1,\\horizon{-}1} \\);\nmemory of past actions is represented by the inclusions \n\\(  \\oproduit{\\bigotimes_{s=0}^{t-1} \\tribu{\\Control}_{s} \\otimes \n\\bigotimes_{s=t}^{\\horizon{-}1} \\{ \\emptyset, \\CONTROL_{s} \\}}%\n{ \\{ \\emptyset, \\Omega \\} } = \n\\oproduit{ \\tribu{\\Control}_{\\{0,\\ldots,t{-}1\\}}}%\n{ \\{ \\emptyset, \\Omega \\} } \n\\subset \\tribu{\\Information}_{t} \\),\nfor \\( t\\in \\ic{1,\\horizon{-}1} \\);\nperfect recall is represented by the inclusions \n\\( \\tribu{\\Information}_{t-1} \\vee \\bp{ \\oproduit{ \\tribu{\\Control}_{\\{0,\\ldots,t{-}1\\}}}%\n{ \\{ \\emptyset, \\Omega \\} } }\n\\subset \\tribu{\\Information}_{t} \\),\nfor \\( t\\in \\ic{1,\\horizon{-}1} \\).\n\nTo represent $N$~players --- each~$\\player$ of whom makes a sequence of decisions,\none for each period~$t \\in \\ic{0,\\horizon_\\player{-}1}$ --- \nwe use $\\prod_{\\player=1}^N \\horizon_\\player$~agents, labelled by\n\\( (\\player,t) \\in \\bigcup_{\\player'=1}^N \\bp{ \\na{\\player'} \\mathord{\\times} \\ic{0,\\horizon_{\\player'}{-}1}} \\).\nWith obvious notations, the inclusions \n\\(\\tribu{\\Information}_{\\np{\\player,t-1}} \\subset \\tribu{\\Information}_{\\np{\\player,t}} \\)\nexpress memory of one's own past information,\nwhereas the inclusions \n\\( \\bigvee_{\\player'=1}^N\n\\bp{\\oproduit{\\bigotimes_{s=0}^{t-1} \\tribu{\\Control}_{s} \\otimes \n    \\bigotimes_{s=t}^{\\horizon_{\\player'}-1} \\{ \\emptyset, \\CONTROL_{s} \\}}\n  { \\{ \\emptyset, \\Omega \\} }}\n\\subset \\tribu{\\Information}_{\\np{\\player,t}} \\),\nexpress memory of all players past actions.\n\n\\subsubsubsection{Principal-Agent models} \n\nA branch of Economics studies so-called \\emph{Prin\\-ci\\-pal-Agent} models with \ntwo decision makers (agents) --- \n  the Principal~$\\Principal$ (\\emph{leader}) who makes decisions $\\control_{\\Principal} \\in \\CONTROL_{\\Principal}$,\n  where the set~$\\CONTROL_{\\Principal}$ is equipped with a\n  $\\sigma$-field~$\\tribu{\\Control}_{\\Principal}$,\n  and the Agent~$\\Agent$ (\\emph{follower}) who makes decisions $\\control_{\\Agent} \\in \\CONTROL_{\\Agent}$,\n  where the set~$\\CONTROL_{\\Agent}$ is equipped with a\n  $\\sigma$-field~$\\tribu{\\Control}_{\\Agent}$ ---\nand with Nature, corresponding to \\emph{private information (or type)} of the\nAgent~$\\Agent$, taking values in a set~$\\Omega$, \nequipped with a $\\sigma$-field~$\\tribu{\\NatureField}$.\n\n\\emph{Hidden type} (leading to adverse selection or to signaling) is represented by any information structure \nwith the property that, on the one hand,\n\\begin{equation}\n  \\tribu{\\Information}_{\\Principal} \\subset \n  \\oproduit{ \n    \\{\\emptyset,\\CONTROL_{\\Principal}\\} \n    \\otimes \n    \\underbrace{\\tribu{\\Control}_{\\Agent} }_{\\textrm{\\makebox[0pt]{\\hspace{1cm}$\\Agent$'s action possibly observed}}}\n  }%\n  {\\underbrace{ \\{\\emptyset,\\Omega \\}  }_{\\textrm{\\makebox[0pt]{$\\Agent$ type not observed}}}}\n  \\eqfinv  \n\\end{equation}\nthat is, \nthe Principal~$\\Principal$ does not know the Agent~$\\Agent$ type,\nbut can possibly observe the Agent~$\\Agent$ action,\nand, on the other hand, that\n\\begin{equation}\n  \\oproduit{ \n    \\{\\emptyset,\\CONTROL_{\\Principal}\\} \n    \\otimes\n    \\{\\emptyset,\\CONTROL_{\\Agent}\\} \n  }{%\n    \\underbrace{ \\tribu{\\NatureField} }_{ \\textrm{\\makebox[0pt]{known inner type}}}}\n  \\subset \\tribu{\\Information}_{\\Agent}\n  \\eqfinv  \n  \\label{eq:Agent_Information_type}\n\\end{equation}\nthat is, the Agent~$\\Agent$ knows the state of nature (his type).\n\n\n\\emph{Hidden action} (leading to moral hazard) is represented by any information structure \nwith the property that, on the one hand, \n\\begin{equation}\n  \\tribu{\\Information}_{\\Principal} \\subset \n  \\oproduit{ \n    \\{\\emptyset,\\CONTROL_{\\Principal}\\} \n    \\otimes \n    \\underbrace{ \\{\\emptyset,\\CONTROL_{\\Agent}\\} }_{\\textrm{\\makebox[0pt]{\\hspace{2cm}cannot observe $\\Agent$'s action}}}\n  }{%\n    \\underbrace{ \\tribu{\\NatureField} }_{ \\textrm{\\makebox[0pt]{\\hspace{1cm}possibly knows $\\Agent$ type} } }}\n  \\eqfinv  \n\\end{equation}\nthat is, the Principal~$\\Principal$ does not know the Agent~$\\Agent$ action,\nbut can possibly observe the Agent~$\\Agent$ type\nand, on the other hand, that\nthe inclusion~\\eqref{eq:Agent_Information_type} holds true, \nthat is, the agent~$\\Agent$ knows the state of nature (his type). \n\n\n\n\\subsubsubsection{Stackelberg leadership model}\nIn Stackelberg games, the leader~$\\Principal$ makes a decision\n\\( \\control_{\\Principal} \\in \\CONTROL_{\\Principal} \\)\n--- based at most upon the partial observation of the state \n\\( \\omega \\in \\Omega \\) of Nature ---\nand the the follower~$\\Agent$ makes a decision\n\\( \\control_{\\Agent} \\in \\CONTROL_{\\Agent} \\)\n--- based at most upon the partial observation of the state of Nature \n\\( \\omega \\in \\Omega \\), and upon the leader decision\n\\( \\control_{\\Principal} \\in \\CONTROL_{\\Principal} \\).\nThis kind of information structure is expressed with the following inclusions \nof fields:\n\\begin{equation}\n\\tribu{\\Information}_{\\Principal} \n\\subset \n\\oproduit{%\n\\{\\emptyset,\\CONTROL_{\\Principal}\\}\n\\otimes\n\\{\\emptyset,\\CONTROL_{\\Agent}\\} \n}{%\n \\tribu{\\NatureField} }\n\\mtext{ and }\n\\tribu{\\Information}_{\\Agent} \n\\subset \n\\oproduit{%\n\\tribu{\\Control}_{\\Principal} \n\\otimes\n\\{\\emptyset,\\CONTROL_{\\Agent}\\} \n}{%\n \\tribu{\\NatureField} }\n\\eqfinp \n\\label{eq:Stackelberg_Information}  \n\\end{equation}\nEven if the players are called leader and follower, \nthere is no explicit time arrow in~\\eqref{eq:Stackelberg_Information}.\nIt is the information structure that reveals the time arrow.\nIndeed, if we label the leader~$\\Principal$ as~$t=0$ (first player)\nand the follower~$\\Agent$ as~$t=1$ (second player), \nthe inclusions~\\eqref{eq:Stackelberg_Information} become the inclusions \n\\( \\tribu{\\Information}_{0} \\subset \\oproduit{%\n\\{ \\emptyset, \\CONTROL_{0} \\} \\otimes \n\\{ \\emptyset, \\CONTROL_{1} \\}}{\\tribu{\\NatureField}} \\),\nand \n\\( \\tribu{\\Information}_{1} \\subset \\oproduit{%\n\\tribu{\\Control}_{0} \\otimes \n\\{ \\emptyset, \\CONTROL_{1} \\}}{\\tribu{\\NatureField}} \\):\nthe sequence \\( \\tribu{\\Information}_{0}, \\tribu{\\Information}_{1} \\)\nof information fields \nis ``adapted'' to the filtration\n\\( \\oproduit{%\n\\{ \\emptyset, \\CONTROL_{0} \\} \\otimes \n\\{ \\emptyset, \\CONTROL_{1} \\}}{\\tribu{\\NatureField}}\n\\subset \\oproduit{%\n\\tribu{\\Control}_{0} \\otimes \n\\{ \\emptyset, \\CONTROL_{1} \\}}{\\tribu{\\NatureField}} \\).\nBut if we label the leader~$\\Principal$ as~$t=1$ \nand the follower~$\\Agent$ as~$t=0$,\nthe new sequence of information fields would not be \n``adapted'' to the new filtration.\nIt is the information structure that prevents \nthe follower to play first, but that makes possible\nthe leader to play first and the follower to play second.\n\n\n\n\\subsection{Solvability and causality}\n\\label{Solvability_Causality}\n\nIn the Kuhn formulation, Witsenhausen says that ``For any\ncombination of policies one can find the corresponding outcome by\nfollowing the tree along selected branches, and this is an explicit\nprocedure'' \\cite{Witsenhausen:1971a}. \nIn the Witsenhausen formulation, there is no such explicit procedure\nas, for any combination of policies, there may be none, one or many solutions to \nthe closed-loop equations; these equations express the decision of one agent as \nthe output of his strategy, supplied with Nature outcome and with all agents decisions.\nThis is why Witsenhausen needs a property of solvability,\nwhereas Kuhn does not need it as it is hardcoded in the tree structure.\nThen, Witsenhausen defines the notion of causality (which parallels that of\ntree) and proves in~\\cite{Witsenhausen:1971a} that solvability holds true under causality.\nYet, in~\\cite[Theorem~2]{Witsenhausen:1971a},\nWitsenhausen exhibits an example of noncausal W-model that is solvable.\n\n\n\\subsubsection{Solvability}\n\nWith any given pure W-strategies profile $\\policy = \\sequence{\\policy_{\\agent}}{\\agent \\in \\AGENT}\n\\in \\prod_{\\agent \\in \\AGENT} \\POLICY_{\\agent} $ we associate the set-valued\nmapping\n\\begin{align}\n{\\cal M}_\\policy: \\Omega\n  & \\rightrightarrows \\prod_{b \\in \\AGENT} \\CONTROL_{b} \n\\label{eq:SetValuedReducedSolutionMap}\n  \\\\\n  \\omega\n  & \\mapsto \\Bset{ \\sequence{\\control_b}{b \\in \\AGENT} \\in \n    \\prod_{b \\in \\AGENT} \\CONTROL_{b} }{%\n\\control_{\\agent} = \\policy_{\\agent}\\bp{\\omega,\n\\sequence{\\control_b}{b \\in \\AGENT} }\n    \\eqsepv \\forall \\agent \\in \\AGENT}\n    \\eqfinp\n    \\nonumber\n\\end{align}\n\nWith this definition, we slightly reformulate below\nhow Witsenhausen introduced the property of solvability.\n\n\\begin{definition}(\\cite{Witsenhausen:1971a,Witsenhausen:1975})\n  \\begin{subequations}\n   \n    The \\emph{solvability property} holds true for the W-model of Definition~\\ref{de:W-model}\n    when,\n    for any pure W-strategies profile $\\policy = \\sequence{\\policy_{\\agent}}{\\agent \\in \\AGENT}\n    \\in \\prod_{\\agent \\in \\AGENT} \\POLICY_{\\agent} $, \nthe set-valued mapping~${\\cal M}_{\\policy}$\nin~\\eqref{eq:SetValuedReducedSolutionMap}\n is a mapping whose domain is $\\Omega$, that is,\n    the cardinal of ${\\cal M}_{\\policy}\\np{\\omega}$ \nis equal to one, for any state of nature $\\omega \\in \\Omega$. \n\n    Thus, under the solvability property, for any state of nature $\\omega \\in \\Omega$, \n    there exists one, and only one, decision profile\n    $\\sequence{\\control_b}{b \\in \\AGENT} \\in \n    \\prod_{b \\in \\AGENT} \\CONTROL_{b}$\n   \n    which is a solution of the \\emph{closed-loop equations}\n    \\begin{equation}\n      \\control_{\\agent} = \\policy_{\\agent}\\bp{\\omega, \\sequence{\\control_b}{b \\in \\AGENT}}\n      \\eqsepv\n      \\forall \\agent \\in \\AGENT\n      \\eqfinp\n      \\label{eq:solution_map_IFF}\n    \\end{equation}\n    In this case, we define the \\emph{solution map} \n    \\begin{equation}\n      M_\\policy: \\Omega \\rightarrow \\prod_{b \\in \\AGENT} \\CONTROL_{b}\n\n      \\label{eq:solution_map}\n    \\end{equation}\n    as the unique element contained in the image set\n${\\cal M}_{\\policy}\\np{\\omega}$ that is, \nfor all $\\sequence{\\control_b}{b \\in \\AGENT} \\in \n    \\prod_{b \\in \\AGENT} \\CONTROL_{b}$, \n    $M_\\policy\\np{\\omega} = \n\\sequence{\\control_b}{b \\in \\AGENT} \\iff\n    {\\cal M}_{\\policy}\\np{\\omega} = \\na{\\sequence{\\control_b}{b \\in \\AGENT}}$.\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n  \\end{subequations}\n  \\label{de:solvability}\n\\end{definition}\n\n\n\n\n\\subsubsection{Configuration-orderings}\n\nIn his articles \\cite{Witsenhausen:1971a,Witsenhausen:1975},\nWitsenhausen introduces a notion of causality that relies on \nsuitable configuration-orderings.\nHere, we introduce our own notations, because they make possible a compact formulation \nof the causality property and, later, of perfect recall.\n\\medskip\n\n\\begin{subequations}\nFor any finite set~\\( {\\mathbb D} \\), let \\( \\cardinal{{\\mathbb D}} \\) denote the cardinal\nof~\\( {\\mathbb D} \\). \n  Thus, \\( | \\AGENT | \\) denotes the cardinal of the set~\\( \\AGENT \\), that is,\n\\( | \\AGENT | \\) is the number of agents.\nFor $k \\in \\ic{1,| \\AGENT |}$, let $\\Sigma^k$ denote the set of\n$k$-orderings, that is, injective\nmappings from $\\ic{1, k}$ to $\\AGENT$:\n\\begin{equation}\n  \\Sigma^k=\\defset{ \\kappa: \\ic{1,k} \\to \\AGENT }%\n{ \\kappa \\mtext{ is an injection} }\n\\eqfinp \n\\label{eq:ORDER_k}\n\\end{equation}\nThe set \\( \\Sigma^{| \\AGENT |} \\) is the set of \\emph{total orderings} of\nagents in $\\AGENT$, that is, bijective\nmappings from $\\ic{1,| \\AGENT |}$ to $\\AGENT$\n(in contrast with \\emph{partial orderings} in~$\\Sigma^k$ for $k < | \\AGENT |$).\nFor any $k \\in \\ic{1, | \\AGENT |}$, any ordering $\\kappa \\in \\Sigma^k$,\nand any integer $\\ell \\le k$, \\( \\kappa_{\\vert \\{ 1, \\ldots, \\ell \\}} \\)\nis the restriction of the ordering~$\\kappa$ to the first $\\ell$~integers.\nFor any $k \\in \\ic{1,| \\AGENT |}$, there is a natural mapping $\\psi_k$ \n\\begin{align}\n\\psi_k: \\Sigma^{| \\AGENT |} \\rightarrow \\Sigma^k \n\\eqsepv \n\\rho \\mapsto \\rho_{\\vert \\{ 1, \\ldots, k \\} } \n\\eqfinv \n\\label{eq:cut}\n\\end{align}\nwhich is the restriction of any (total) ordering of~$\\AGENT$ \nto~$\\ic{1,k}$.\nWe define the \\emph{set of orderings} by\n\\begin{equation}\n  \\Sigma= \\bigcup_{ k \\in \\ic{0,| \\AGENT |}} \\Sigma^k\n\\quad \\text{ where } \\Sigma^0 = \\{ \\emptyset \\} \n\\eqfinp \n\\label{eq:ORDER}\n\\end{equation}\nFor any \\( k \\in \\ic{1,| \\AGENT |} \\), and any $k$-ordering~$\\kappa \\in \\Sigma^k$,\nwe define the \\emph{range} $\\range{\\kappa}$ of the ordering~$\\kappa$ as the\nsubset \n\\begin{align}\n  \\range{\\kappa}\n&=\n\\ba{ \\kappa(1), \\ldots, \\kappa(k) }\n\\subset \\AGENT\n\\eqsepv \\forall \\kappa \\in \\Sigma^k\n\\eqfinv\n  \\label{range_kappa}\n  \\intertext{the \\emph{cardinal} $\\cardinal{\\kappa}$ of the ordering $\\kappa$ as\n  the integer}\n\\cardinal{\\kappa}\n&=k\n\\in \\ic{1, | \\AGENT |}\n\\eqsepv \\forall \\kappa \\in \\Sigma^k\n\\eqfinv\n  \\label{cardinal_kappa}\n  \\intertext{the \\emph{last element} $\\LastElement{\\kappa}$ of the ordering $\\kappa$ as the agent}\n\\LastElement{\\kappa}\n&=\\kappa(k)\n\\in \\AGENT\n\\eqsepv \\forall \\kappa \\in \\Sigma^k\n\\eqfinv\n  \\label{LastElement_kappa}\n  \\intertext{the \\emph{restriction} $\\FirstElements{\\kappa}$ of the ordering $\\kappa$ to the first $k{-}1$ elements}\n\\FirstElements{\\kappa}\n&= \\kappa_{\\vert \\{ 1, \\ldots, k-1 \\}} \\in \\Sigma^{k-1}\n\\eqsepv \\forall \\kappa \\in \\Sigma^k\n\\eqfinp\n  \\label{FirstElements_kappa}\n\\end{align}\n\\end{subequations}\nWith the notations introduced,\nany ordering $\\kappa \\in \\Sigma \\setminus \\{ \\emptyset \\}$\ncan be written as $\\kappa = \\np{\\FirstElements{\\kappa}, \\LastElement{\\kappa}}$,\nwith the convention that \n $\\kappa = \\np{\\LastElement{\\kappa}}$ when $\\kappa \\in \\Sigma^1$.\n\n\\begin{definition}(\\cite{Witsenhausen:1971a,Witsenhausen:1975})\nA \\emph{configuration-ordering} is a mapping \n$\\varphi: \\HISTORY \\to \\Sigma^{| \\AGENT |}$ from configurations towards total orderings.\nWith any configuration-ordering~$\\varphi$, \nand any ordering $\\kappa \\in \\Sigma$,\nwe associate the subset \\( \\HISTORY_{\\kappa}^{\\varphi} \\subset \\HISTORY \\)\nof configurations defined by\n\\begin{equation}\n  \\HISTORY_{\\kappa}^{\\varphi} =\n  \\defset{\\history \\in \\HISTORY}{\\psi_{\\cardinal{\\kappa}}\\bp{\\varphi(\\history)} =\\kappa}  \n  \\eqsepv \\forall \\kappa \\in \\Sigma\n  \\eqfinp\n  \\label{eq:HISTORY_k_kappa}\n\\end{equation}\nBy convention, we put \n\\(   \\HISTORY_{\\emptyset}^{\\varphi} = \\HISTORY \\). \n\\label{de:configuration-ordering}\n\\end{definition}\nAlong each configuration $\\history \\in \\HISTORY$, the agents are ordered by\n$\\varphi(\\history) \\in \\Sigma^{| \\AGENT |}$.\nThe set~$\\HISTORY_{\\kappa}^{\\varphi}$ in~\\eqref{eq:HISTORY_k_kappa} \ncontains all the configurations \nfor which the agent~\\( \\kappa(1) \\) is acting first, \nthe agent~\\( \\kappa(2) \\) is acting second, \\ldots, till \nthe last agent~\\( \\LastElement{\\kappa}=\\kappa(\\cardinal{\\kappa}) \\) acting at stage~$\\cardinal{\\kappa}$.\n\n\n\\subsubsection{Causality}\n\nIn his article \\cite{Witsenhausen:1971a},\nWitsenhausen introduces a notion of causality \nand he proves that causal systems are solvable.\n\nThe following definition can be interpreted as follows.\nIn a causal W-model, there exists a configuration-ordering with the following\nproperty: when an agent is called to play --- as he is the last one in an\nordering --- what he knows cannot depend on decisions made by agents\nthat are not his predecessors \n(in the range of the ordering under consideration).\n\n\\begin{definition}(\\cite{Witsenhausen:1971a,Witsenhausen:1975})\n  \\label{de:causality}\n  A W-model (as in Definition~\\ref{de:W-model}) \n  is \\emph{causal} if there exists (at least) one\n  configuration-ordering $\\varphi: \\HISTORY \\to \\Sigma^{| \\AGENT |}$ \n  with the property that \n  \\begin{equation}\n    \\label{eq:causality_a}\n    \\HISTORY_{\\kappa}^{\\varphi} \\cap \\History \\in \n   \n    \\tribu{\\History}_{\\range{\\FirstElements{\\kappa}}}\n    \\eqsepv \n    \\forall \\History \\in \\tribu{\\Information}_{\\LastElement{\\kappa}}\n    \\eqsepv \n    \\forall \\kappa \\in \\Sigma \n    \\eqfinp\n  \\end{equation}\n\\end{definition}\nOtherwise said, once we know the first $\\cardinal{\\kappa}$~agents, \nthe information of the (last) agent~$\\LastElement{\\kappa}$ depends at most\n on the decisions of the (previous) agents in the range~$\\range{\\FirstElements{\\kappa}}$.\nIn~\\eqref{eq:causality_a}, the subset~$\\HISTORY_{\\kappa}^{\\varphi} \\subset \\HISTORY$ of configurations \nhas been defined in~\\eqref{eq:HISTORY_k_kappa},\nthe last agent~$\\LastElement{\\kappa}$ in~\\eqref{LastElement_kappa},\nthe partial ordering~$\\FirstElements{\\kappa}$ in~\\eqref{FirstElements_kappa},\nthe range~$\\range{\\FirstElements{\\kappa}}$ in~\\eqref{range_kappa},\nand --- using the definition~\\eqref{eq:sub_history_field_BGENT} \nof the subfield~$\\tribu{\\History}_{\\mathbb{B}}$ of~\\( \\tribu{\\History} \\),\nwith the subset~$\\mathbb{B}=\\range{\\FirstElements{\\kappa}}$ of agents \ndefined in~\\eqref{FirstElements_kappa} and~\\eqref{range_kappa} --- \nthe subfield $\\tribu{\\History}_{\\range{\\FirstElements{\\kappa}}}$ \nof~\\( \\tribu{\\History} \\) is \n\\begin{equation}\n      \\tribu{\\History}_{\\range{\\FirstElements{\\kappa}}}\n     =\n      \\tribu{\\NatureField} \\otimes \n\\bigotimes \\limits_{\\agent \\in \\range{\\FirstElements{\\kappa}}} \\tribu{\\Control}_{\\agent}\n      \\otimes\n      \\bigotimes \\limits_{b \\not\\in \\range{\\FirstElements{\\kappa}} } \\{\n      \\emptyset, \\CONTROL_b \\}\n    \\subset \\tribu{\\History}\n      \\eqfinp\n\\label{eq:causality_b}\n\\end{equation}\n\nWitsenhausen's intrinsic model deals with agents, information and strategies,\nbut not with players and preferences.\nWe now turn to extending the Witsenhausen's intrinsic model to games.\n\n\n\\section{Finite games in intrinsic form}\n\\label{Finite_games_in_intrinsic_form}\n\nWe are now ready to embed Witsenhausen's intrinsic model\ninto game theory.\nIn~\\S\\ref{Definition_of_a_finite_game_in_intrinsic_form},\nwe introduce a formal definition of a finite game in intrinsic form (W-game),\nand in~\\S\\ref{Mixed_and_behavioral_strategies}\nwe introduce three notions of \n``randomization'' of pure strategies --- mixed, product-mixed and behavioral.\nIn~\\S\\ref{Strategy_equivalence},\nwe discuss relations between product-mixed and behavioral W-strategies.\n\nIn what follows, when ${\\mathbb D}$ is a finite set, we denote by \n\\( \\Delta\\np{{\\mathbb D}} \\) the set of probability distributions over~${\\mathbb D}$.\nWhen needed, the set~\\( \\Delta\\np{{\\mathbb D}} \\) can be equipped with \nthe Borel topology and\nthe Borel $\\sigma$-field, as \\( \\Delta\\np{{\\mathbb D}} \\)\nis homeomorphic to the simplex~$\\Sigma_{\\cardinal{{\\mathbb D}}}$ of~$\\RR^{\\cardinal{{\\mathbb D}}}$, \nand is thus homeomorphic to a closed\nsubset of a finite dimensional space.\n\n\\subsection{Definition of a finite game in intrinsic form (W-game)}\n\\label{Definition_of_a_finite_game_in_intrinsic_form}\n\nWe introduce a formal definition of a finite game in intrinsic form (W-game).\n\n\\begin{definition}\n  A \\emph{finite W-game}\n  \\( \\Bp{ \n    \\bp{ \n      \\sequence{\\AGENT^{\\player}}{\\player \\in \\PLAYER}, \n      \\np{\\Omega, \\tribu{\\NatureField}},\n      \\sequence{\\CONTROL_{\\agent}, \\tribu{\\Control}_{\\agent},\n        \\tribu{\\Information}_{\\agent}}{\\agent \\in \\bigcup_{\\player \\in \\PLAYER} \\AGENT^{\\player}} },\n    (\\precsim^{\\player})_{\\player \\in \\PLAYER}\n  }\n  \\),\n  or a \\emph{finite game in intrinsic form},\n  is a made of \n  \\begin{itemize}\n  \\item \n    a family \\( \\sequence{\\AGENT^{\\player}}{\\player \\in \\PLAYER} \\),\nwhere the set~$\\PLAYER$ of \\emph{players} is finite,\n    of     two by two disjoint nonempty sets     \n    whose union $\\AGENT= \\bigcup_{\\player \\in \\PLAYER} \\AGENT^{\\player}$ is the set \n    of \\emph{agents}; each subset~$\\AGENT^{\\player}$ is interpreted as \n    the subset of executive agents of the \n    \\emph{player} \\( \\player \\in \\PLAYER \\),\n  \\item \n    a finite W-model \n    \\( \\bp{ \n      \\AGENT,\n      (\\Omega, \\tribu{\\NatureField}), \n      \\sequence{\\CONTROL_{\\agent}, \\tribu{\\Control}_{\\agent},\n        \\tribu{\\Information}_{\\agent}}{\\agent \\in \\AGENT} } \\),\n    as in Definition~\\ref{de:W-model},\n \n \n \n  \\item \n    for each player $\\player \\in \\PLAYER$,\n    a preference relation~$\\precsim^{\\player}$ \n    on the set of mappings \n\\( \\Omega \\to \\Delta\\np{ \\prod_{b \\in \\AGENT} \\CONTROL_{b} } \\).\n  \\end{itemize}\n \n  A finite W-game is said to be solvable (resp. causal)\n  if the underlying W-model\n  is solvable as in Definition~\\ref{de:solvability}\n  (resp. causal as in Definition~\\ref{de:causality}).\n \n  \\label{de:W-game}\n\\end{definition}\n\nWe comment on the preference relations~$\\precsim^{\\player}$\non the set of mappings \n\\( \\Omega \\to \\Delta\\np{ \\prod_{b \\in \\AGENT} \\CONTROL_{b} } \\).\nOur definition covers\n(like in~\\cite{Blume-Brandenburger-Dekel:1991})\nthe most traditional preference relation~$\\precsim^{\\player}$,\nwhich is the numerical \\emph{expected utility} preference.\nIn this latter, each player~$\\player \\in \\PLAYER$\nis endowed, on the one hand, with a \\emph{criterion} (payoff), that is,\na measurable function $\\criterion_\\player: (\\HISTORY, \\tribu{\\History}) \n\\rightarrow [-\\infty,+\\infty[$,\nand, on the other hand, with a \\emph{belief}, that is,\na probability distribution\n $\\nu^\\player: \\tribu{\\NatureField} \\rightarrow [0, 1]$\nover the states of Nature $(\\Omega, \\tribu{\\NatureField})$.\nThen, given \\( K_i : \\Omega \\to \\Delta\\np{ \\prod_{b \\in \\AGENT}\n  \\CONTROL_{b} } \\), $i=1,2$, one says that\n  \\( K_1 \\precsim^{\\player} K_2 \\) if \n\\begin{align*}\n  \\int_{\\Omega} \\nu^\\player\\np{d\\omega}\n  &\n    \\int_{ \\prod_{b \\in \\AGENT} \\CONTROL_{b} } \n    \\criterion_\\player\\bp{\\omega, \\sequence{\\control_b}{b \\in \\AGENT}}\n    K_1\\bp{\\omega, d\\sequence{\\control_b}{b \\in \\AGENT}}\n  \\\\\n  & \\leq\n  \\int_{\\Omega} \\nu^\\player\\np{d\\omega}\n  \\int_{ \\prod_{b \\in \\AGENT} \\CONTROL_{b} } \n  \\criterion_\\player\\bp{\\omega, \\sequence{\\control_b}{b \\in \\AGENT}}\n  K_2\\bp{\\omega, d\\sequence{\\control_b}{b \\in \\AGENT}}\n  \\eqfinp\n\\end{align*}\n\nNote also that the Definition~\\ref{de:W-game} includes Bayesian games,\nby specifying a product structure for $\\Omega$ --- where some factors represent\ntypes of players, and one factor represents chance --- and by considering\nadditional probability distributions.\n \n\n\n\\subsection{Mixed, product-mixed and behavioral strategies}\n\\label{Mixed_and_behavioral_strategies}\n\nWe introduce three notions of \n``randomization'' of pure strategies: mixed, product-mixed and behavioral.\n\nThe notion of mixed strategy comes from the study of games in normalized form,\nwhere each player  has to select a pure strategy,\nthe collection of which determines a unique outcome. \nIf we allow the players to select their pure strategy at random, \nthe lottery they use is called a mixed strategy. \nFor an extensive game,  a mixed strategy can be interpreted in the following sense.\nFirst, the player selects a pure strategy using the lottery. \nSecond, the game is played. When the player is called by the umpire,\nshe plays the action specified by the selected pure strategy for the current information set. \n\nObserve that there is only one dice roll per player. This dice roll determines the reactions of the player for every situation of the game. \nIt would be more natural to let the player roll a dice every time she has to\nplay, leading to the notion of behavioral strategy. \n\nA fundamental question in game theory is to identify settings in which those two\nviews (mixed strategy and behavioral strategy) are equivalent.\nTo formulate this question in the W-game framework, \nwe will give formal definitions of these two notions of randomization.\nWe will also add a third one, that we call\nproduct-mixed strategy, and which \nis in the spirit of Aumann~\\cite{Aumann:1964}, \nas each agent (corresponding to time index in~\\cite{Aumann:1964}) \n``generates'' strategies from a random device that is independent of all the\nother agents.\n\n\n\\subsubsection{Mixed W-strategies}\n\nFor any agent~$\\agent \\in \\AGENT$, the set~\\( \\POLICY_{\\agent}  \\) \nof pure W-strategies for agent~$\\agent$ (see Definition~\\ref{de:W-strategy})\nis finite, hence the set \\( \\Delta\\np{\\POLICY_{\\agent} } \\)\nof probability distributions over~\\( \\POLICY_{\\agent}  \\)\nis is homeomorphic to \n$\\Sigma_{|\\POLICY_{\\agent}|}$, the simplex of\n$\\RR^{|\\POLICY_{\\agent}|}$, and is thus homeomorphic to a closed\nsubset of a finite dimensional space.\nSo is the space \\( \\Delta\\np{\\POLICY} \\)\nof probability distributions over the set~\\( \\POLICY \\)\nof pure W-strategies profiles. \nWe will also consider the sets\n\\begin{equation}\n  \\POLICY^{\\player} = \n\\prod_{\\agent \\in \\AGENT^{\\player}} \\POLICY_{\\agent}\n\\eqsepv \\forall \\player \\in \\PLAYER\n\\label{eq:W-strategies_profiles_player}\n\\end{equation}\nof pure W-strategies profiles, player by player, and the set\n\\( \\Delta\\np{\\POLICY^{\\player}} \\) of probability distributions\nover~\\( \\POLICY^{\\player} \\).\n\n\n\\begin{subequations}\n\\begin{definition}\nWe consider a finite W-game, as in Definition~\\ref{de:W-game}.\nA \\emph{mixed W-strategy}\nfor player~$\\player \\in \\PLAYER$ \nis an element~$\\mu^{\\player}$ of \\( \\Delta\\np{\\POLICY^{\\player} } \\),\nthe set of probability distributions\nover the set~\\( \\POLICY^{\\player} \\) in~\\eqref{eq:W-strategies_profiles_player}\nof W-strategies of the executive agents in~$\\AGENT^{\\player}$.\nThe set of \\emph{mixed W-strategies profiles} is\n\\( \\prod_{\\player \\in \\PLAYER} \\Delta\\np{\\POLICY^{\\player}} \\). \n\\label{de:mixed_W-strategy}\n\\end{definition}\nA mixed W-strategies profile is denoted by \n\\begin{equation}\n\\mu = \\np{\\mu^{\\player}}_{\\player \\in \\PLAYER}\n\\in \\prod_{\\player \\in \\PLAYER} \\Delta\\bp{ \\POLICY^{\\player} }\n\\eqfinv\n\\label{eq:mixed_W-strategies_profiles}\n\\end{equation}\nand, when we focus on player~$\\player$, we write \n\\begin{equation}\n\\mu = \n\\couple{{\\mu}^{-\\player}}{{\\mu}^{\\player}} \\in\n\\Delta\\bp{ \\POLICY^{\\player} } \\times\n\\prod_{\\player'\\neq \\player} \\Delta\\bp{ \\POLICY^{\\player'} }\n\\eqfinp\n\\label{eq:mixed_W-strategies_profiles_player}\n\\end{equation}\n\\end{subequations}\n\n\n\n\\begin{subequations}\n\\begin{definition}\nWe consider a solvable finite W-game (see Definition~\\ref{de:W-game}),\nand \\( \\mu = \\np{\\mu^{\\player}}_{\\player \\in \\PLAYER}\n\\in \\prod_{\\player \\in \\PLAYER} \\Delta\\np{ \\POLICY^{\\player} } \\)\na mixed W-strategies profile as in~\\eqref{eq:mixed_W-strategies_profiles}.\nFor any \\( \\omega\\in\\Omega\\), we denote by \n\\begin{equation}\n \\QQ^{\\omega}_{\\mu} =\n \\QQ^{\\omega}_{\\np{\\mu^{\\player}}_{\\player \\in \\PLAYER}} = \n\\bp{ \\bigotimes_{\\player \\in \\PLAYER}\\mu^{\\player} }\n\\circ \\bp{ M\\np{\\omega,\\cdot}}^{-1}\n\\in \\Delta\\bp{ \\prod_{b \\in \\AGENT} \\CONTROL_{b} }\n\\label{eq:push_forward_probability_a}\n\\end{equation}\nthe pushforward probability,\non the space \\( \\bp{\\prod_{b \\in \\AGENT} \\CONTROL_{b},\n\\bigotimes \\limits_{b \\in \\AGENT} \\tribu{\\Control}_{b} } \\)\nof the product probability distribution~\\( \\bigotimes_{\\player \\in\n  \\PLAYER}\\mu^{\\player} \\)\non~\\( \\prod_{\\player \\in \\PLAYER} \\POLICY^{\\player} = \\POLICY \\)\nby the mapping\n\\begin{equation}\n  M\\np{\\omega,\\cdot} : \\POLICY \\to \n\\prod_{b \\in \\AGENT} \\CONTROL_{b} \n \\eqsepv \n\\policy \\mapsto M_{\\policy}(\\omega) \n\\eqfinv\n\\label{eq:push_forward_probability_b}\n\\end{equation}\nwhere \\( M_{\\policy} \\) is the \nsolution map~\\eqref{eq:solution_map}, \nwhich exists by the solvability assumption.\n\\end{definition}\n\\label{eq:push_forward_probability}\n\\end{subequations}\n\nBy~\\eqref{eq:solution_map_IFF}, which defines the solution map,\nand by definition of a pushforward probability,\nwe have, for any configuration~\\( \\bp{\\omega, \n\\sequence{\\control_b}{b\\in\\AGENT} } \\in\\HISTORY \\),\n\\begin{align*}\n  \\QQ^{\\omega}_{\\mu}\n  \\bp{\\sequence{\\control_b}{b\\in\\AGENT} }\n  & =\n    \\bp{\\bigotimes_{\\player \\in \\PLAYER}\\mu^{\\player}}\n    \\Bp{ M\\np{\\omega,\\cdot}^{-1}\n    \\bp{\\sequence{\\control_b}{b\\in\\AGENT} } } \n  \\\\\n  &=\n    \\prod_{\\player \\in \\PLAYER} \\mu^{\\player}\n    \\bgp{\\Bset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player} }\n    \\in\\POLICY^{\\player} }%\n    { \\lambda_{\\agent}\\bp{\\omega, \n    \\sequence{\\control_b}{b\\in\\AGENT} } =\\control_{\\agent}\n    \\eqsepv \\forall \\agent\\in \\AGENT^{\\player} }}\n    \\eqfinp\n\\end{align*}\n\n\n\n\\subsubsection{Product-mixed W-strategies}\n\nIn a mixed W-strategy, the executive agents of player~$\\player \\in \\PLAYER$ \ncan be correlated because the probability~$\\mu^{\\player}$\nin Definition~\\ref{de:mixed_W-strategy}\nis a joint probability on the product space\n\\( \\POLICY^{\\player} =\\prod_{\\agent \\in \\AGENT^{\\player}} \\POLICY_{\\agent}\n\\).\nWe now introduce product-mixed W-strategies, where \nthe executive agents of player~$\\player \\in \\PLAYER$ are independent\nin the sense that the probability~$\\mu^{\\player}$ is  the\nproduct of individual probabilities,\neach of them on the individual space~\\( \\POLICY_{\\agent} \\)\nof the strategies of one agent~$\\agent$. \n\n\\begin{definition}\n  \\label{de:product-mixed_W-strategy}\n  We consider a finite W-game, as in Definition~\\ref{de:W-game}.\n  A \\emph{product-mixed W-strategy} for player~$\\player \\in \\PLAYER$ is\n  an element \n  \\( \\pi^{\\player}=\n  \\sequence{ \\pi^{\\player}_{\\agent}}{\\agent \\in \\AGENT^{\\player}} \\) of \n  \\( \\prod_{\\agent \\in \\AGENT^{\\player}} \\Delta\\np{\\POLICY_{\\agent}} \\).\n\\end{definition}\nThe product-mixed W-strategy \n\\( \\sequence{ \\pi^{\\player}_{\\agent}}{\\agent \\in \\AGENT^{\\player}} \\)\ninduces a product probability\\footnote{%\nBy an abuse of notation, we will sometimes write \n\\(  \\pi^{\\player}=\n\\otimes_{\\agent\\in\\AGENT^{\\player}}\\pi^{\\player}_{\\agent} \\).\n\\label{ft:product-mixed_W-strategy}}\n\\(\n\\otimes_{\\agent\\in\\AGENT^{\\player}}\\pi^{\\player}_{\\agent} \\)\non the set~\\( \\POLICY^{\\player} \\),\nwhich is a mixed W-strategy\nas in Definition~\\ref{de:mixed_W-strategy}.\n\n\n\n\n\\subsubsection{Behavioral W-strategies}\n\nWe formalize the intuition of behavioral strategies in W-games\nby the following definition  of behavioral W-strategies.\n\n\\begin{definition}\nWe consider a finite W-game, as in Definition~\\ref{de:W-game}.\nA \\emph{behavioral W-strategy}\nfor player~$\\player \\in \\PLAYER$ \nis a family \n\\( \\beta^{\\player} = \n\\sequence{\\beta^{\\player}_{\\agent}}{\\agent \\in \\AGENT^{\\player}} \\),\nwhere \n\\begin{equation}\n\\beta^{\\player}_{\\agent} :\n\\HISTORY \\times \\tribu{\\Control}_{\\agent} \\to [0,1] \n\\eqsepv \n\\np{\\history, \\Control_{\\agent}} \\mapsto  \\beta^{\\player}_{\\agent}\n\\conditionalySet{\\Control_{\\agent}}{\\history} \n\\end{equation}\nis an \\( \\tribu{\\Information}_{\\agent} \\)-measurable stochastic kernel\nfor each \\( \\agent \\in \\AGENT^{\\player} \\), that is,\nif one of the two equivalent statements holds true:\n\\begin{enumerate}\n\\item \n\\label{it:behavioral_W-strategy_abstract}\non the one hand, \nthe function\n\\( \\history \\in \\HISTORY \\mapsto \\beta^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} \\) \nis \\( \\tribu{\\Information}_{\\agent} \\)-measurable,\nfor any \\( \\control_{\\agent}\\in\\CONTROL_{\\agent} \\) and,\non the other hand, \neach \\( \\beta^{\\player}_{\\agent}\n\\conditionalySet{ \\cdot }{ \\history } \\) is \na probability distribution on the finite set~\\( \\CONTROL_{\\agent} \\),\nfor any \\( \\history \\in \\HISTORY \\),\n\\item \n\\label{it:behavioral_W-strategy}\non the one hand, \n\\( \\history' \\sim_{\\agent} \\history'' \\implies\n\\beta^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history'}\n=\n\\beta^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history''}\n \\), for any \\( \\control_{\\agent}\\in\\CONTROL_{\\agent} \\)\nand, on the other hand,\nfor any \\( \\history \\in \\HISTORY \\),\nwe have \\( \\beta^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} \\geq 0 \\),\n\\( \\forall \\control_{\\agent}\\in\\CONTROL_{\\agent} \\),\nand\n\\(  \\sum_{ \\control_{\\agent}\\in\\CONTROL_{\\agent} } \n\\beta^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} = 1 \\).\n\\end{enumerate}\n\\label{de:behavioral_W-strategy}\n\\end{definition}\nThe equivalences come from the fact that the sets \\( \\HISTORY \\) and \\( \\CONTROL_{\\agent} \\)\nare finite and equipped with their respective complete fields, and\nby Proposition~\\ref{pr:measurability}, \nand especially~\\eqref{eq:strategy_atoms_rho}. \n\n\n\\subsection{Relations between product-mixed and behavioral W-strategies}\n\\label{Strategy_equivalence}\n\nHere, we show that product-mixed and behavioral W-strategies\nare ``equivalent'' in the sense that\na product-mixed W-strategy naturally induces a \nbehavioral W-strategy, and that a behavioral W-strategy can be ``realized''\nas a product-mixed W-strategy\n(see Figure~\\ref{fig:randomization}).\n\n\\subsubsection*{From product-mixed to behavioral W-strategies}\n\nWe prove that a product-mixed W-strategy naturally induces a \nbehavioral W-strategy.\n\n\\begin{proposition}\n  \\label{pr:PMtoB}\n  We consider a finite W-game, as in Definition~\\ref{de:W-game},\n  and a player \\( \\player \\in \\PLAYER \\).\n\n  For any product-mixed W-strategy\n  \\( \\pi^{\\player}=\n  \\sequence{ \\pi^{\\player}_{\\agent}}{\\agent \\in\n    \\AGENT^{\\player}} \\in \\prod_{\\agent \\in \\AGENT^{\\player}} \\Delta\\np{\\POLICY_{\\agent}} \\), \n  as in Definition~\\ref{de:product-mixed_W-strategy}, \n  we define, for any agent~\\( \\agent\\in\\AGENT^{\\player} \\), \n  \\begin{equation}\n    \\PMtoB{\\pi}^{\\player}_{\\agent}\n    \\conditionaly{\\control_{\\agent}}{\\history}\n    =\n    \\pi^{\\player}_{\\agent}\n    \\Bp{\n    \\defset{ \\policy_{\\agent}\\in\\POLICY_{\\agent}}%\n    {\\policy_{\\agent}\\np{\\history}=\\control_{\\agent}}}\n   \n    \\eqsepv \\forall \\control_{\\agent}\\in\\CONTROL_{\\agent} \n    \\eqsepv \\forall \\history \\in \\HISTORY \n    \\eqfinp\n    \\label{eq:PMtoB}\n  \\end{equation}\n  Then, \\( \\PMtoB{\\pi}^{\\player}=\n  \\sequence{ \\PMtoB{\\pi}^{\\player}_{\\agent}}{\\agent \\in\n    \\AGENT^{\\player}} \\) is a behavioral W-strategy, \n  as in Definition~\\ref{de:behavioral_W-strategy}.\n\\end{proposition}\n\n\n\\begin{proof}\nLet be given a product-mixed W-strategy\n\\( \\pi^{\\player}=\n\\sequence{ \\pi^{\\player}_{\\agent}}{\\agent \\in\n  \\AGENT^{\\player}} \\in \\prod_{\\agent \\in \\AGENT^{\\player}} \\Delta\\np{\\POLICY_{\\agent}} \\). \n\nTo prove that~\\eqref{eq:PMtoB} defines a \nbehavioral W-strategy, \nwe have to show (see Item~\\ref{it:behavioral_W-strategy_abstract}\nin Definition~\\ref{de:behavioral_W-strategy}), \non the one hand, that the function\n\\( \\history \\in \\HISTORY \\mapsto \\PMtoB{\\pi}^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} \\) \nis \\( \\tribu{\\Information}_{\\agent} \\)-measurable,\nfor any \\( \\control_{\\agent}\\in\\CONTROL_{\\agent} \\) and,\non the other hand, that \neach \\( \\beta^{\\player}_{\\agent}\n\\conditionalySet{ \\cdot }{ \\history } \\) is \na probability on the finite set \\( \\CONTROL_{\\agent} \\),\nfor any \\( \\history \\in \\HISTORY \\).\nFor this purpose, we will use the more practical characterization of \nItem~\\ref{it:behavioral_W-strategy}\nin Definition~\\ref{de:behavioral_W-strategy}.\n\nLet us fix \\( \\control_{\\agent}\\in\\CONTROL_{\\agent} \\).\nLet \\( \\history', \\history'' \\in \\HISTORY \\) be such that\n\\( \\history' \\sim_{\\agent} \\history'' \\),\nwhere we recall that the classes of the\nequivalence relation~$\\sim_{\\agent}$ in~\\eqref{eq:bracket_HISTORY}\nare exactly the atoms in~\\( \\crochet{\\tribu{\\Information}_{\\agent}} \\).\nBy~\\eqref{eq:common_value_equivalence_class_rho}, we have that \n\\( \\defset{ \\policy_{\\agent}\\in\\POLICY_{\\agent}}%\n{\\policy_{\\agent}\\np{\\history'}=\\control_{\\agent}}\n=\n\\defset{ \\policy_{\\agent}\\in\\POLICY_{\\agent}}%\n{\\policy_{\\agent}\\np{\\history''}=\\control_{\\agent}}\n\\). \nTherefore, from the expression~\\eqref{eq:PMtoB},\nwe have obtained that \n\\( \\history' \\sim_{\\agent} \\history'' \\implies\n\\PMtoB{\\pi}^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history'}\n=\n\\PMtoB{\\pi}^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history''}\n \\), hence that the function\n\\( \\history \\in \\HISTORY \\mapsto \\PMtoB{\\pi}^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} \\) \nis \\( \\tribu{\\Information}_{\\agent} \\)-measurable,\nby Proposition~\\ref{pr:measurability}, \nand especially~\\eqref{eq:strategy_atoms_rho}. \n\nBy the expression~\\eqref{eq:PMtoB},\nwe have that \n\\( \\PMtoB{\\pi}^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} \\geq 0 \\),\n\\( \\forall \\control_{\\agent}\\in\\CONTROL_{\\agent} \\)\nand, since \\( \\pi^{\\player}_{\\agent} \\)\nis a probability on~\\( \\Delta\\np{\\POLICY_{\\agent}} \\),\nthat \n\\(  \\sum_{ \\control_{\\agent}\\in\\CONTROL_{\\agent} } \n\\PMtoB{\\pi}^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} = 1 \\).\n\\medskip\n\nThis ends the proof.\n\\end{proof}\n\n\n\\subsubsection*{From behavioral to product-mixed W-strategies}\n\nWe prove that a behavioral W-strategy can be ``realized''\nas a product-mixed W-strategy.\n\n\\begin{proposition}\n  \\label{pr:BtoPM} \n  We consider a finite W-game, as in Definition~\\ref{de:W-game},\n  and a player \\( \\player \\in \\PLAYER \\).\n\n  For any behavioral W-strategy \\( \\beta^{\\player} = \n  \\sequence{\\beta^{\\player}_{\\agent}}{\\agent \\in \\AGENT^{\\player}}\n  \\), as in Definition~\\ref{de:behavioral_W-strategy},\n  there exists a product-mixed W-strategy\n  \\( \\BtoPM{\\beta}^{\\player} = \n  \\sequence{\\BtoPM{\\beta}^{\\player}_{\\agent}}{\\agent \\in \\AGENT^{\\player}}\\),\n  as in Definition~\\ref{de:product-mixed_W-strategy}, \n  with the property that, for any agent~\\( \\agent \\) in \\(\\AGENT^{\\player}\\) we have\n  \\begin{equation}\n    \\BtoPM{\\beta}^{\\player}_{\\agent}\n    \\Bp{\\defset{ \\policy_{\\agent}\\in\\POLICY_{\\agent}}%\n      {\\policy_{\\agent}\\np{\\history}=\\control_{\\agent}}}\n    = \n    \\beta^{\\player}_{\\agent}\n    \\conditionaly{\\control_{\\agent}}{\\history}\n   \n    \\eqsepv \\forall \\control_{\\agent}\\in\\CONTROL_{\\agent} \n    \\eqsepv \\forall \\history \\in \\HISTORY \n    \\eqfinp\n    \\label{eq:BtoPM}\n  \\end{equation}\n\\end{proposition}\n\\begin{subequations}\n  \\begin{proof} \n    We consider a fixed agent~\\( \\agent\\in\\AGENT^{\\player} \\).\n\n    On the one hand,\n    by Proposition~\\ref{pr:measurability}, we get that\n   \n   \n    \\begin{equation*}\n      \\policy_{\\agent} \\in \\POLICY_{\\agent} \\iff\n      \\exists \\sequence{\\control_{\\agent}^{G_{\\agent}}}{%\n        G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}}} \n      \\in \n      \\CONTROL_{\\agent}^{\\crochet{\\tribu{\\Information}_{\\agent}}}\n      \\eqsepv \n      \\policy_{\\agent}\\np{\\history}=\\control_{\\agent}^{G_{\\agent}}\n      \\eqsepv \\forall G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}}\n      \\eqsepv \\forall \\history \\in G_{\\agent}\n      \\eqfinv\n     \n    \\end{equation*}\nwhere \\( \\CONTROL_{\\agent}^{\\crochet{\\tribu{\\Information}_{\\agent}}} \\) is the\nset of mappings from~\\( \\crochet{\\tribu{\\Information}_{\\agent}} \\) to~\\(\n\\CONTROL_{\\agent} \\).\nTherefore, the following mapping is a bijection:\n    \\begin{equation}\n      \\Psi : \\POLICY_{\\agent} \\to \n      \\CONTROL_{\\agent}^{\\crochet{\\tribu{\\Information}_{\\agent}}}\n      \\eqsepv\n      \\policy_{\\agent} \\mapsto \n      \\sequence{\\policy_{\\agent}\\np{G_{\\agent}}}{%\n        G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}}} \n      \\eqfinp\n      \\label{eq:BtoPM_proof_bijection} \n    \\end{equation}\n    We denote the inverse bijection by\n    \\begin{equation}\n      \\Phi=\\Psi^{-1} : \n      \\CONTROL_{\\agent}^{\\crochet{\\tribu{\\Information}_{\\agent}}}\n      \\to \\POLICY_{\\agent} \n      \\eqfinp\n      \\label{eq:BtoPM_proof_bijection_inverse} \n    \\end{equation}\n\n    On the other hand, \n    by Item~\\ref{it:behavioral_W-strategy_abstract}\n    in Definition~\\ref{de:behavioral_W-strategy}, each \n    \\( \\beta^{\\player}_{\\agent}\n    \\conditionalySet{ \\cdot }{ \\history } \\) is \n    a probability on the finite set \\( \\CONTROL_{\\agent} \\),\n    for any \\( \\history \\in \\HISTORY \\).\n    As the mapping \\( \\history \\in \\HISTORY \\mapsto \n    \\beta^{\\player}_{\\agent}\n    \\conditionalySet{ \\cdot }{ \\history } \\) is \n    \\( \\tribu{\\Information}_{\\agent} \\)-measurable,\n    by Definition~\\ref{de:behavioral_W-strategy}, \n    the notation \\( \\beta^{\\player}_{\\agent}\n    \\conditionalySet{ \\cdot }{ G_{\\agent} } \\) \n    makes sense by~\\eqref{eq:common_value_rho}.\n\n    We equip the finite set \n    \\( \\CONTROL_{\\agent}^{\\crochet{\\tribu{\\Information}_{\\agent}}} \\)\n    with the product probability\n    \\( \\bigotimes_{ G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}} } \n    \\beta^{\\player}_{\\agent}\n    \\conditionalySet{ \\cdot }{ G_{\\agent} } \\),\n    and we define the pushforward probability\n    \\begin{equation}\n      \\BtoPM{\\beta}^{\\player}_{\\agent}\n      = \\Bp{ \\bigotimes_{ G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}} } \n        \\beta^{\\player}_{\\agent}\n        \\conditionalySet{ \\cdot }{ G_{\\agent} } }\n      \\circ \\Phi^{-1} \\eqsepv\n      \\label{eq:BtoPM_proof_probability}\n    \\end{equation}\n    on the finite set~\\( \\POLICY_{\\agent} \\).\n    Then, we calculate, for any \\( \\history \\in \\HISTORY \\), \n    \\begin{align*}\n      \\BtoPM{\\beta}^{\\player}_{\\agent}\n      &\n        \\Bp{\\defset{ \\policy_{\\agent}\\in\\POLICY_{\\agent}}%\n        {\\policy_{\\agent}\\np{\\history}=\\control_{\\agent}}}\n      \\\\\n      &=  \n          \\Bp{ \\bigotimes_{ G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}} } \n          \\beta^{\\player}_{\\agent}\n          \\conditionalySet{ \\cdot }{ G_{\\agent} } }\n          \\Bp{ \\Phi^{-1} \\bp{ \\defset{ \\policy_{\\agent}\\in\\POLICY_{\\agent}}%\n          {\\policy_{\\agent}\\np{\\history}=\\control_{\\agent}} }}\n          \\tag{by definition~\\eqref{eq:BtoPM_proof_probability}}\n      \\\\\n      &=\n          \\Bp{ \\bigotimes_{ G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}} } \n          \\beta^{\\player}_{\\agent}\n          \\conditionalySet{ \\cdot }{ G_{\\agent} } }\n          \\Bp{ \\Psi \\bp{\\defset{ \\policy_{\\agent}\\in\\POLICY_{\\agent}}%\n          {\\policy_{\\agent}\\np{\\history}=\\control_{\\agent}} }}\n          \\tag{as \\( \\Phi^{-1}=\\Psi \\) by~\\eqref{eq:BtoPM_proof_bijection_inverse}}\n      \\\\\n      &=\n          \\Bp{ \\bigotimes_{ G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}} } \n          \\beta^{\\player}_{\\agent}\n          \\conditionalySet{ \\cdot }{ G_{\\agent} } }\n          \\Bp{ \\{ \\control_{\\agent} \\} \\times\n          \\CONTROL_{\\agent}^{ \\crochet{\\tribu{\\Information}_{\\agent}}\\backslash \\bracket{\\history}_{\\agent} }\n          }\n          \\intertext{because, by definition of the mapping~$\\Psi$\n          in~\\eqref{eq:BtoPM_proof_bijection},\n          any \\( \\policy_{\\agent} \\in\n          \\defset{ \\policy'_{\\agent}\\in\\POLICY_{\\agent}}%\n          {\\policy'_{\\agent}\\np{\\history}=\\control_{\\agent}} \\) \n          takes the value~$\\control_{\\agent}$ \n          on the atom~\\( \\bracket{\\history}_{\\agent} \\)\n          (by definition of the set~\\( \\POLICY_{\\agent} \\)\n          in Definition~\\ref{de:W-strategy} and by~\\eqref{eq:common_value_equivalence_class_rho}),\n          and any possible value in~\\( \\CONTROL_{\\agent} \\) for all the \n          other atoms in \\( \\crochet{\\tribu{\\Information}_{\\agent}}\n          \\backslash \\bracket{\\history}_{\\agent} \\) }\n       \n      &=  \n          \\beta^{\\player}_{\\agent}\n          \\conditionaly{ \\control_{\\agent} }{ \\bracket{\\history}_{\\agent} }\n          \\times\n          \\prod_{ G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}}\n          \\backslash \\bracket{\\history}_{\\agent} } \n          \\beta^{\\player}_{\\agent}\n          \\conditionalySet{ \\CONTROL_{\\agent} }{ G_{\\agent} }\n          \\tag{by definition of the product probability\n          \\( \\bigotimes_{ G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}} } \n          \\beta^{\\player}_{\\agent}\n          \\conditionalySet{ \\cdot }{ G_{\\agent} } \\) }\n      \\\\\n      &=  \n          \\beta^{\\player}_{\\agent}\n          \\conditionaly{ \\control_{\\agent} }{ \\bracket{\\history}_{\\agent} }\n          \\times\n          \\prod_{ G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}}\n          \\backslash \\bracket{\\history}_{\\agent} } \n          1\n          \\tag{as \\( \\beta^{\\player}_{\\agent}\n          \\conditionalySet{ \\CONTROL_{\\agent} }{ G_{\\agent} }=1 \\)\n          for all \\( G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}} \\)}\n      \\\\\n      &=  \n          \\beta^{\\player}_{\\agent}\n          \\conditionaly{ \\control_{\\agent} }{ \\bracket{\\history}_{\\agent} }\n      \\\\\n      &=  \n          \\beta^{\\player}_{\\agent}\n          \\conditionaly{\\control_{\\agent}}{\\history}\n    \\end{align*}\n    as the function \\( \\history \\in \\HISTORY \\mapsto \n    \\beta^{\\player}_{\\agent}\n    \\conditionaly{\\control_{\\agent}}{\\history} \\) is \n    \\( \\tribu{\\Information}_{\\agent} \\)-measurable,\n    by Item~\\ref{it:behavioral_W-strategy_abstract}\n    in Definition~\\ref{de:behavioral_W-strategy}, \n    and using~\\eqref{eq:common_value_equivalence_class_rho}.\n    \\medskip\n\n    This ends the proof. \n  \\end{proof}\n\\end{subequations}\n\n\n\n\n\\section{Kuhn's equivalence theorem}\n\\label{Kuhn_Theorem}\n\nNow, we are equipped to give, for games in intrinsic form, a statement and a proof of the \ncelebrated Kuhn's equivalence theorem: when a player enjoys perfect recall, \nfor any mixed W-strategy, there is an equivalent behavioral strategy.\n\n\\begin{figure}\n  \\centering\n  \\[\n  \\begin{tikzcd}[column sep=\"2cm\",row sep=\"4cm\",cells={nodes={draw=gray}},labels={font=\\everymath\\expandafter{\\the\\everymath\\textstyle}}]\n    & \\substack{\\textrm{\\normalsize mixed} \\\\ \\textrm{\\normalsize W-strategies}}\n   \n    \\arrow{dr}[description]{{\\textrm{Proposition~\\ref{pr:MtoB_PerfectRecall} under perfect recall}}}\n    &\n    \\\\\n    \\substack{\\textrm{\\normalsize product mixed} \\\\ \\textrm{\\normalsize W-strategies}}\n    \\arrow[ru,bend left=10,\"\\textrm{injection}\" ]\n    \\arrow[rr,bend left=10,\"\\textrm{Proposition~\\ref{pr:PMtoB}}\"]\n    &\n    &\n    \\substack{\\textrm{\\normalsize behavioral} \\\\ \\textrm{\\normalsize W-strategies}}\n    \\arrow[ll,bend left=10,\"\\textrm{Proposition~\\ref{pr:BtoPM}}\"] \n   \n  \\end{tikzcd}\n  \\]\n  \\caption{\\label{fig:randomization}Three Propositions that relate three notions of\n    randomization of strategies}\n\\end{figure}\n\nIn this Section, we consider a causal finite W-game\n(see Definition~\\ref{de:W-game}), \nthat is, the underlying W-model (as in Definition~\\ref{de:W-model}) \nis causal (see Definition~\\ref{de:causality}), \nwith suitable configuration-ordering $\\varphi: \\HISTORY \\to \\Sigma^{| \\AGENT |}$.\nIn~\\S\\ref{Perfect_recall}, we introduce a formal definition of perfect recall \nin a causal finite game in intrinsic form. \nIn~\\S\\ref{Preliminary_results} (Proposition~\\ref{pr:MtoB_PerfectRecall}), we show that \nany mixed W-strategy induces a behavioral W-strategy under perfect recall\n(see Figure~\\ref{fig:randomization}).\nFinally, in~\\S\\ref{Main_result} (Theorem~\\ref{th:Kuhn}), we give a statement and\na proof of Kuhn's equivalence theorem for games in intrinsic form. \n\n\n\\subsection{Definition of perfect recall for causal W-games}\n\\label{Perfect_recall}\n\nFor any agent~\\( \\agent\\in\\AGENT \\), \nwe define the \\emph{choice field}~\\( \\tribu{C}_{\\agent} \n\\subset \\tribu{\\History} \\) by\n   \\begin{equation}\n\\tribu{C}_{\\agent} = \\tribu{\\Control}_{\\agent} \\bigvee \n\\tribu{\\Information}_{\\agent} \\eqsepv \\forall \\agent \\in \\AGENT\n\\eqfinp\n   \\label{eq:ChoiceField}\n   \\end{equation}\nThus defined, the choice field of an agent contains both what the agent did\nand what he knew when making the decision.\n\nThe following definition of perfect recall is new.\n\\begin{subequations}\n  %\n\\begin{definition}\nWe consider a causal finite W-game for which the underlying W-model \nis causal with the configuration-ordering $\\varphi: \\HISTORY \\to \\Sigma^{| \\AGENT |}$.\n\n We say that a player \\( \\player \\in \\PLAYER \\) enjoys \\emph{perfect recall} \nif, for any ordering \\( \\kappa \\in \\Sigma \\)\nsuch that~\\( \\LastElement{\\kappa} \\in \\AGENT^{\\player} \\)\n(that is, the last agent is an executive of the player), we have \n  \\begin{equation}\n\\HISTORY_{\\kappa}^{\\varphi} \\cap \\History \\in \n\\tribu{\\Information}_{\\LastElement{\\kappa}}\n\\eqsepv \n\\forall \\History \\in \\tribu{C}_{\\range{\\FirstElements{\\kappa}} \\cap\n  \\AGENT^{\\player}} \n\\eqfinv    \n   \\label{eq:PerfectRecall}\n  \\end{equation}\nwhere the subset~$\\HISTORY_{\\kappa}^{\\varphi} \\subset \\HISTORY$ of configurations \nhas been defined in~\\eqref{eq:HISTORY_k_kappa},\nthe last agent~$\\LastElement{\\kappa}$ in~\\eqref{LastElement_kappa},\nthe partial ordering~$\\FirstElements{\\kappa}$ in~\\eqref{FirstElements_kappa},\nthe range~$\\range{\\FirstElements{\\kappa}}$ in~\\eqref{range_kappa},\nand where\n\\begin{equation}\n\\tribu{C}_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}} =\n\\bigvee \\limits_{\\substack{\\agent \\in \\range{\\FirstElements{\\kappa}}\\\\\n\\agent \\in \\AGENT^{\\player}}}\n\\tribu{C}_{\\agent} \n\\eqfinv\n   \\label{eq:PastChoiceField}\n\\end{equation}\nwith the choice subfield \\( \\tribu{C}_{\\agent} \\subset \\tribu{\\History} \\) given by~\\eqref{eq:ChoiceField}.\n   \\label{de:PerfectRecall}\n\\end{definition}\n\\end{subequations}\nWe interpret the above definition \nas follows.\nA player enjoys perfect recall when any of her executive\nagents --- when called to play as the last one in an ordering --- \nknows at least what did and knew those of the \nexecutive agents that are both his predecessors \n(in the range of the ordering under consideration)\nand that are executive agents of the player.\n\n\n\\subsection{Mixed W-strategy induces behavioral W-strategy under perfect recall}\n\\label{Preliminary_results}\n\nAs a preparatory result for the proof of Kuhn's equivalence theorem, we show that \nany mixed W-strategy induces a behavioral W-strategy under perfect recall.\n\n\\begin{proposition}\nWe consider a causal finite W-game\n(see Definition~\\ref{de:W-game}),\nfor which the underlying W-model (see Definition~\\ref{de:W-model}) \nis causal (see Definition~\\ref{de:causality})\nwith the configuration-ordering $\\varphi: \\HISTORY \\to \\Sigma^{| \\AGENT |}$.\n\nWe consider a player \\( \\player \\in \\PLAYER \\),\nequipped with a mixed W-strategy \\( \\mu^{\\player} \\in\n\\Delta\\np{\\POLICY^{\\player}} \\) and \nsupposed to enjoy perfect recall (see Definition~\\ref{de:PerfectRecall}),\nand an agent~\\( \\agent\\in\\AGENT^{\\player} \\).\n\nThen, for each agent \\( \\agent \\in \\AGENT^{\\player} \\), \nthe following formula\\footnote{%\nWith the convention that \n\\( \\MtoB{\\mu}^{\\player}_{\\agent}\\np{ \\{\\control_{\\agent}\\} \\mid\n  \\history}=0 \\) if the denominator\nis zero (in which case the numerator, which is smaller, is also zero),\nand using the notations\n\\( \\policy=\\sequence{\\policy_{\\agent}}{\\agent \\in \\AGENT^{\\player}}\n\\in\\POLICY^{\\player} \\), \n\\eqref{eq:sub_history_BGENT} for \\( \\history_\\mathbb{B} \\)\nand~\\eqref{eq:sub_wstrategy_BGENT} for \\( \\policy_\\mathbb{B} \\), \nwith \\( \\mathbb{B} = \\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player} \\),\nwhere\nthe last agent~$\\LastElement{\\kappa}$  has been defined in~\\eqref{LastElement_kappa},\nthe partial ordering~$\\FirstElements{\\kappa}$ in~\\eqref{FirstElements_kappa}\nand the range~$\\range{\\FirstElements{\\kappa}}$ in~\\eqref{range_kappa}.\n} \n\\begin{equation}\n  \\begin{split}\n  \\MtoB{\\mu}^{\\player}_{\\agent}\\conditionaly{\\control_{\\agent}}{\\history}\n= \\\\\n\\frac{ \\mu^{\\player} \\defset{ \\policy\\in\\POLICY^{\\player} }{ \n\\exists \\kappa \\in \\Sigma,\n\\, \\history\\in\\HISTORY_{\\kappa}^{\\varphi} ,\n\\, \\LastElement{\\kappa}=\\agent,\n\\, \\policy_{\\agent}\\np{\\history}=\\control_{\\agent},\n\\, \\policy_{ \\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player} }\\np{\\history} \n= \\history_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}} } }%\n{ \\mu^{\\player} \\defset{ \\policy\\in\\POLICY^{\\player} }{ \\exists \\kappa \\in \\Sigma,\n\\, \\history\\in\\HISTORY_{\\kappa}^{\\varphi} ,\n\\, \\LastElement{\\kappa}=\\agent,\n\\, \\policy_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}}\\np{\\history} \n= \\history_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}} } }    \n  \\end{split}\n\\label{eq:MtoB_PerfectRecall}\n\\end{equation}\ndefines an \\( \\tribu{\\Information}_{\\agent} \\)-measurable stochastic kernel\n\\( \\MtoB{\\mu}^{\\player}_{\\agent}\\).\nAs a consequence, the family \n\\( \\MtoB{\\mu}^{\\player} = \n\\sequence{\\MtoB{\\mu}^{\\player}_{\\agent}}{\\agent \\in \\AGENT^{\\player}} \\),\nis a behavioral W-strategy, as in Definition~\\ref{de:behavioral_W-strategy}.\n\\label{pr:MtoB_PerfectRecall}\n\\end{proposition}\n\n\\begin{subequations}\n  %\n\\begin{proof}\nWe consider a player \\( \\player \\in \\PLAYER \\),\nequipped with a mixed W-strategy \\( \\mu^{\\player} \\in\n\\Delta\\np{\\POLICY^{\\player}} \\) and \nsupposed to enjoy perfect recall as in Definition~\\ref{de:PerfectRecall},\nand an agent~\\( \\agent\\in\\AGENT^{\\player} \\).\n\\medskip\n\nBy~\\eqref{eq:MtoB_PerfectRecall}, it is easy to see that, \nfor any \\( \\history \\in \\HISTORY \\),\nwe have \\( \\MtoB{\\mu}^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} \\geq 0 \\),\n\\( \\forall \\control_{\\agent}\\in\\CONTROL_{\\agent} \\),\nand\n\\(  \\sum_{ \\control_{\\agent}\\in\\CONTROL_{\\agent} } \n\\MtoB{\\mu}^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} = 1 \\).\nTherefore, by Item~\\ref{it:behavioral_W-strategy_abstract}\nin Definition~\\ref{de:behavioral_W-strategy}, \nthere remains to prove that \nthe function \\( \\history \\in \\HISTORY \\mapsto \n\\MtoB{\\mu}^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} \\) is \n\\( \\tribu{\\Information}_{\\agent} \\)-measurable.\nThe proof is in several steps.\n\\medskip\n\n\\noindent$\\bullet$\nAs the agent~\\( \\agent\\in\\AGENT \\) is fixed,\nit is easily seen that the family \\( \\sequence{ \\HISTORY_{\\kappa}^{\\varphi} }{\n\\kappa \\in \\Sigma, \\, \\LastElement{\\kappa}=\\agent } \\),\nwhere the subset~$\\HISTORY_{\\kappa}^{\\varphi} \\subset \\HISTORY$ of configurations \nhas been defined in~\\eqref{eq:HISTORY_k_kappa}, is made of\n(possibly empty) disjoint sets whose union is~\\( \\HISTORY \\).\nIndeed, on the one hand, for any \\( \\history\\in \\HISTORY \\), we have that \n \\( \\history\\in \\HISTORY_{\\kappa_0}^{\\varphi} \\) where \n\\( \\kappa_0=\\psi_k\\np{\\rho} \\), with\n\\( \\rho=\\varphi\\np{\\history} \\) \nand $k$ the unique integer such that \\( \\rho(k)=\\agent \\).\nOn the other hand, if we had \\( \\HISTORY_{\\kappa_1}^{\\varphi}\n\\cap \\HISTORY_{\\kappa_2}^{\\varphi} \\neq \\emptyset \\),\nwith \\( \\LastElement{\\kappa_1}=\\LastElement{\\kappa_2}=\\agent \\), \nthen \\( \\history\\in \\HISTORY_{\\kappa_1}^{\\varphi}\n\\cap \\HISTORY_{\\kappa_2}^{\\varphi} \\) would be such that\n\\( \\kappa_1=\\kappa_2=\\psi_k\\np{\\rho} \\), with\n\\( \\rho=\\varphi\\np{\\history} \\) \nand $k$ the unique integer such that \\( \\rho(k)=\\agent \\).\nThus, \\( \\kappa_1 \\neq \\kappa_2 \\implies \\HISTORY_{\\kappa_1}^{\\varphi}\n\\cap \\HISTORY_{\\kappa_2}^{\\varphi} = \\emptyset \\).\n\\medskip\n\n\\noindent$\\bullet$\nAs the family \\( \\sequence{ \\HISTORY_{\\kappa}^{\\varphi} }{\n    \\kappa \\in \\Sigma, \\, \\LastElement{\\kappa}=\\agent } \\) is made of\n    disjoint sets whose union is~\\( \\HISTORY \\), \nwe rewrite~\\eqref{eq:MtoB_PerfectRecall} as\n\\begin{align}\n  \\MtoB{\\mu}^{\\player}_{\\agent}\n  &\\conditionaly{\\control_{\\agent}}{\\history}\n    \\nonumber\n  \\\\\n  &=\\frac{{\\displaystyle \\mathop{\\sum}_{\\substack{ \\kappa \\in \\Sigma\\\\\\LastElement{\\kappa}=\\agent}}}\n    \\mu^{\\player} \\defset{ \\policy\\in\\POLICY^{\\player} }{ \n     \n     \n    \\history\\in\\HISTORY_{\\kappa}^{\\varphi} \n    \\eqsepv \\policy_{\\agent}\\np{\\history}=\\control_{\\agent}\n    \\eqsepv \\policy_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}}\\np{\\history} \n    = \\history_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}} } }%\n    {{\\displaystyle \\sum_{ \\kappa \\in \\Sigma, \\LastElement{\\kappa}=\\agent}} \n    \\mu^{\\player} \\defset{ \\policy\\in\\POLICY^{\\player} }{ \n    \\history\\in\\HISTORY_{\\kappa}^{\\varphi} \n    \\eqsepv \\policy_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}}\\np{\\history} \n    = \\history_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}} } }\n    \\nonumber\n\\\\\n  &=\n    \\frac{ \\sum_{ \\kappa \\in \\Sigma, \\LastElement{\\kappa}=\\agent}\n    \\mu^{\\player}\\ba{\\Phi_\\kappa\\np{\\history,\\control_{\\agent}}} }%\n    { \\sum_{ \\kappa \\in \\Sigma, \\LastElement{\\kappa}=\\agent} \n    \\sum_{  \\control_{\\agent}\\in\\CONTROL_{\\agent} }\n    \\mu^{\\player}\\ba{\\Phi_\\kappa\\np{\\history,\\control_{\\agent}}} }\n    \\eqfinv\n    \\nonumber\n   \n    \\intertext{where, for any \\( \\history\\in\\HISTORY \\),\n    \\( \\control_{\\agent}\\in\\CONTROL_{\\agent} \\) and \\( \\kappa \\in \\Sigma \\) such that \n    \\( \\LastElement{\\kappa}=\\agent \\), we have defined the following subset of strategies}\n    \\Phi_\\kappa\\np{\\history,\\control_{\\agent}} \n  &=\n    \\label{eq:proof_Phi_kappa}\n  \\\\\n  &\n    \\defset{ \\policy\\in\\POLICY^{\\player} }{\n    \\history\\in\\HISTORY_{\\kappa}^{\\varphi} \n    \\eqsepv \\policy_{\\agent}\\np{\\history}=\\control_{\\agent}\n    \\eqsepv \\policy_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}}\\np{\\history} \n    = \\history_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}} } \n    \\subset \\POLICY^{\\player} \n    \\eqfinp\n    \\nonumber\n\\end{align}\nWe will prove, in three steps, that \\( \\Phi_{\\kappa}\\np{\\history,\\control_{\\agent}} \\) in~\\eqref{eq:proof_Phi_kappa} \ntakes the same (set) value for any \\( \\history \\in G_{\\agent} \\),\nwhere $G_{\\agent}$ is an atom of~\\( \\tribu{\\Information}_{\\agent} \\).\n\\medskip\n\n\\noindent$\\bullet$\nLet $G_{\\agent} \\subset \\HISTORY$ be an atom of~\\( \\tribu{\\Information}_{\\agent} \\).\nWe prove that there exists a unique \\( \\kappa_0 \\in \\Sigma \\) \nsuch that \\( \\LastElement{\\kappa_0}=\\agent \\)\nand \\( G_{\\agent} \\subset \\HISTORY_{\\kappa_0}^{\\varphi} \\),\nthat is, we prove that \n\\begin{equation}\nG_{\\agent} \\in \\crochet{ \\tribu{\\Information}_{\\agent} }\n\\implies \\exists ! \\, \\kappa_0 \\in \\Sigma \\eqsepv\n\\LastElement{\\kappa_0}=\\agent \\eqsepv \nG_{\\agent} \\subset \\HISTORY_{\\kappa_0}^{\\varphi}\n\\eqfinp\n  \\label{eq:proof_unique_kappa_0}\n\\end{equation}\n\nFirst, we show that, for any \\( \\kappa \\in \\Sigma \\) such that \n\\( \\LastElement{\\kappa}=\\agent \\), we have \neither \\( G_{\\agent} \\subset \\HISTORY_{\\kappa}^{\\varphi} \\)\nor \\( G_{\\agent} \\cap \\HISTORY_{\\kappa}^{\\varphi} = \\emptyset \\).\nIndeed, by~\\eqref{eq:PerfectRecall} with \n\\( \\History =  \\HISTORY \\in \\tribu{C}_{\\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player}}\\), \nwe obtain that \\( \\HISTORY_{\\kappa}^{\\varphi} \n\\in \\tribu{\\Information}_{\\agent} \\).\nTherefore, either \\( G_{\\agent} \\cap \\HISTORY_{\\kappa}^{\\varphi} = \\emptyset \\) \nor \\( G_{\\agent} \\subset \\HISTORY_{\\kappa}^{\\varphi} \\)\nby~\\eqref{eq:atom_property} since\n$G_{\\agent}$ is an atom of~\\( \\tribu{\\Information}_{\\agent} \\).\nSecond, we have seen that the family \\( \\sequence{ \\HISTORY_{\\kappa}^{\\varphi} }{\n\\kappa \\in \\Sigma, \\, \\LastElement{\\kappa}=\\agent } \\) is made of\ndisjoint sets whose union is~\\( \\HISTORY \\).\n\nBy combining both results, we conclude that there exists a unique \\( \\kappa_0 \\in \\Sigma \\) \nsuch that \\( \\LastElement{\\kappa_0}=\\agent \\)\nand \\( G_{\\agent} \\subset \\HISTORY_{\\kappa_0}^{\\varphi} \\).\n\\medskip\n\n\\noindent$\\bullet$\nAs a consequence of~\\eqref{eq:proof_unique_kappa_0}, we have that \n\\( \\Phi_\\kappa\\np{\\history,\\control_{\\agent}} =\\emptyset \\),\nfor any \\( \\history\\in G_{\\agent} \\), and \nfor any \\( \\kappa \\in \\Sigma \\) such that \\( \\kappa \\neq \\kappa_0 \\)\nand  \\( \\LastElement{\\kappa}=\\agent \\).\nThere only remains to prove that \\( \\Phi_{\\kappa_0}\\np{\\history,\\control_{\\agent}} \\) \nin~\\eqref{eq:proof_Phi_kappa} takes the same (set) value\nfor any \\( \\history \\in G_{\\agent} \\). For this purpose, \nwe consider $\\history', \\history'' \\in \\HISTORY $ which belong to \nthe atom~$G_{\\agent}  \\in \\crochet{\\tribu{\\Information}_{\\agent}}$, that is,\n\\( \\{\\history', \\history''\\} \\subset G_{\\agent} \\),\nand we establish two preliminary results.\n\nFirst, we prove that \n\\( \\history'_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}=\n\\history''_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \\).\nFor this purpose, we define the subset \n\\( \\History'=\\defset{ \\history \\in \\HISTORY }{\n\\history_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}\n=\\history'_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} } \n\\subset \\HISTORY \\) and we show in two steps that \n\\( \\history'' \\in \\History' \\), hence that \n\\( \\history'_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}=\n\\history''_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \\), that is,\nwe show that \n\\begin{equation}\n\\{\\history', \\history''\\} \\subset G_{\\agent} \n\\implies\n \\history'_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}=\n\\history''_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}\n\\eqfinp\n    \\label{eq:proof_recall_decisions}\n\\end{equation}\n\n\\begin{description}\n\\item[-]\nWe show that \n\\( \\HISTORY_{\\kappa_0}^{\\varphi} \\cap \\History' \\in \\tribu{\\Information}_{\\agent} \\).\nBy definition of the field~\\( \\tribu{\\Control}_{\\range{\\FirstElements{\\kappa_0}}\n  \\cap \\AGENT^{\\player}} \\) in~\\eqref{eq:sub_control_field_BGENT} \nwith \\( \\mathbb{B} = \\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player} \\),\nand because each field~\\( \\tribu{\\Control}_{b} \\), for\n\\( b \\in \\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player} \\), is complete,\nhence has the singletons for atoms, \nwe have that \\( \\History' \\in \\tribu{\\Control}_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \\).\nAs \\( \\tribu{\\Control}_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \n\\subset \n\\tribu{C}_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \\)\nby~\\eqref{eq:PastChoiceField}, we use the perfect recall\nassumption~\\eqref{eq:PerfectRecall}\nwith \\( \\History = \\History' \\),\nand obtain that \\( \\HISTORY_{\\kappa_0}^{\\varphi} \\cap \\History' \n\\in \\tribu{\\Information}_{\\LastElement{\\kappa_0}}=\\tribu{\\Information}_{\\agent}\n\\) since \\( \\LastElement{\\kappa_0}=\\agent \\).\n\\item[-]\nAs \\( \\HISTORY_{\\kappa_0}^{\\varphi} \\cap \\History' \\in\n\\tribu{\\Information}_{\\agent} \\)\nand $G_{\\agent}$ is an atom of~\\( \\tribu{\\Information}_{\\agent} \\),\nwe have either \n\\( G_{\\agent} \\cap \\HISTORY_{\\kappa_0}^{\\varphi} \\cap \\History' =\\emptyset \\)\nor \\( G_{\\agent} \\subset \\HISTORY_{\\kappa_0}^{\\varphi} \\cap \\History' \\)\nby~\\eqref{eq:atom_property}.\nNow, as \\( \\history' \\in \\History' \\) and \n\\( \\history' \\in G_{\\agent} \\subset \\HISTORY_{\\kappa_0}^{\\varphi} \\),\nwe get that \n\\( G_{\\agent} \\cap \\HISTORY_{\\kappa_0}^{\\varphi} \\cap \\History' \\neq \\emptyset \\),\nand therefore \\( G_{\\agent} \\subset \\HISTORY_{\\kappa_0}^{\\varphi} \\cap \\History' \\).\nSince \\( \\history'' \\in G_{\\agent} \\), we deduce that \n\\( \\history'' \\in \\History' \\), hence that \n\\( \\history'_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}=\n\\history''_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \\).\n\\end{description}\n\nSecond, we prove that, for any \\( b \\in \\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player} \\)\nand any atom $G_b$ of~\\( \\tribu{\\Information}_{b} \\),\nwe have either \n\\( \\{\\history', \\history''\\} \\cap G_b =\\emptyset\\)\nor \\( \\{\\history', \\history''\\} \\subset G_b \\), that is, \nwe prove that\n\\begin{equation}\n  \\begin{split}\n\\Bp{ b \\in \\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player} \n\\text{ and }\nG_b \\in \\crochet{ \\tribu{\\Information}_{b} } }\n\\implies \\\\\n\\Bp{ \\{\\history', \\history''\\} \\cap G_b =\\emptyset\n\\text{ or }\n\\{\\history', \\history''\\} \\subset G_b }\n\\eqfinp\n  \\end{split}\n    \\label{eq:proof_recall_information}\n\\end{equation}\nFor this purpose, we consider \n\\( b \\in \\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player} \\)\nand \\( G_b \\in \\crochet{ \\tribu{\\Information}_{b} } \\).\n\nAs \\( \\tribu{\\Information}_{b} \\subset \n\\tribu{C}_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \\)\nby~\\eqref{eq:PastChoiceField}, we use the perfect recall\nassumption~\\eqref{eq:PerfectRecall}\nwith \\( \\History = G_b \\),\nand obtain that \\( \\HISTORY_{\\kappa_0}^{\\varphi} \\cap G_b\n\\in \\tribu{\\Information}_{\\LastElement{\\kappa_0}}=\\tribu{\\Information}_{\\agent}\n\\) since \\( \\LastElement{\\kappa_0}=\\agent \\) by~\\eqref{eq:proof_unique_kappa_0}.\nAs a consequence, as \\( \\HISTORY_{\\kappa_0}^{\\varphi} \\cap G_b\n\\in \\tribu{\\Information}_{\\agent} \\)\nand $G_{\\agent}$ is an atom of~\\( \\tribu{\\Information}_{\\agent} \\),\nwe have either \n\\( G_{\\agent} \\cap \\HISTORY_{\\kappa_0}^{\\varphi} \\cap G_b =\\emptyset \\)\nor \\( G_{\\agent} \\subset \\HISTORY_{\\kappa_0}^{\\varphi} \\cap G_b \\)\nby~\\eqref{eq:atom_property}.\nSince \\( \\{\\history', \\history''\\} \\subset G_{\\agent} \\) by assumption,\nwhere \\( G_{\\agent} \\subset \\HISTORY_{\\kappa_0}^{\\varphi} \\)\nby~\\eqref{eq:proof_unique_kappa_0}, we conclude\nthat either \\( \\{\\history', \\history''\\} \\subset G_b \\)\nor \\( \\{\\history', \\history''\\} \\cap G_b =\\emptyset\\).\n\\medskip\n\n\\noindent$\\bullet$\nWe consider an atom~$G_{\\agent} \\in \\crochet{\\tribu{\\Information}_{\\agent}}$,\nand we finally prove that \\( \\Phi_{\\kappa_0}\\np{\\history,\\control_{\\agent}} \\) \nin~\\eqref{eq:proof_Phi_kappa} takes the same (set) value\nfor any \\( \\history \\in G_{\\agent} \\).\nFor this purpose, we consider $\\history'$ and $\\history''$ which belong to \nthe atom~$G_{\\agent}$ of~\\( \\tribu{\\Information}_{\\agent} \\), that is,\n\\( \\{\\history', \\history''\\} \\subset G_{\\agent} \\),\nand we show that \n\\( \\policy\\in\\Phi_{\\kappa_0}\\np{\\history',\\control_{\\agent}} \\implies\n\\policy\\in\\Phi_{\\kappa_0}\\np{\\history'',\\control_{\\agent}} \\).\n\nThus, we take \\( \\policy\\in\\Phi_{\\kappa_0}\\np{\\history',\\control_{\\agent}} \\)\nin~\\eqref{eq:proof_Phi_kappa} --- that is, \\(\\policy\\in\\POLICY^{\\player} \\) is such that \n\\( \\history'\\in\\HISTORY_{\\kappa_0}^{\\varphi} \\), \n\\( \\policy_{\\agent}\\np{\\history'}=\\control_{\\agent} \\) \nand \\( \\policy_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}\\np{\\history'} \n= \\history'_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \\) ---\nand we are going to prove in three steps that \n\\( \\history''\\in\\HISTORY_{\\kappa_0}^{\\varphi} \\), \n\\( \\policy_{\\agent}\\np{\\history''}=\\control_{\\agent} \\) \nand \\( \\policy_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}\\np{\\history''} \n= \\history''_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \\).\n\\begin{description}\n\\item[-]\nFirst, we have that \n\\( \\history''\\in\\HISTORY_{\\kappa_0}^{\\varphi} \\)\nsince \n\\( \\{\\history', \\history''\\} \\subset G_{\\agent} \\) by assumption,\nand \\( G_{\\agent} \\subset \\HISTORY_{\\kappa_0}^{\\varphi} \\) \nby~\\eqref{eq:proof_unique_kappa_0}.\n\\item[-]\nSecond, since \\( \\{\\history', \\history''\\} \\subset G_{\\agent} \\) by assumption\nand since the strategy \\( \\policy_{\\agent} \\) \nis \\( \\tribu{\\Information}_{\\agent} \\)-measurable, we have that\n\\( \\policy_{\\agent}\\np{\\history'}=\\policy_{\\agent}\\np{\\history''} \\)\nby~\\eqref{eq:strategy_atoms_rho}, \nhence \\( \\policy_{\\agent}\\np{\\history''}=\\control_{\\agent} \\) since\n\\( \\policy_{\\agent}\\np{\\history'}=\\control_{\\agent} \\) by assumption.\n\\item[-]\nThird, we have shown that, for any \\( b \\in \\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player} \\)\nand any atom $G_b$ of~\\( \\tribu{\\Information}_{b} \\),\nwe have either \\( \\{\\history', \\history''\\} \\subset G_b \\)\nor \\( \\{\\history', \\history''\\} \\cap G_b =\\emptyset\\)\nby~\\eqref{eq:proof_recall_information}.\nAs a consequence, the pair \\( \\{\\history', \\history''\\} \\)\nis included in one of the atoms that make \nthe partition~\\( \\crochet{\\tribu{\\Information}_{b}} \\).\nTherefore, we obtain that \n\\( \\policy_{b}\\np{\\history'} =\n\\policy_{b}\\np{\\history''} \\) by~\\eqref{eq:strategy_atoms_rho},\nhence \\( \\policy_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}\\np{\\history'} =\n\\policy_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}\\np{\\history''}\n\\), using the notation~\\eqref{eq:sub_wstrategy_BGENT} for \\( \\policy_\\mathbb{B} \\), \nwith \\( \\mathbb{B} = \\range{\\FirstElements{\\kappa}} \\cap \\AGENT^{\\player} \\).\nAs we had obtained in~\\eqref{eq:proof_recall_decisions}\nthat \\( \\history'_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}=\n\\history''_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \\), we conclude that \n\\( \\policy_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}\\np{\\history''} =\n\\policy_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}\\np{\\history'} =\n\\history'_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}}=\n\\history''_{\\range{\\FirstElements{\\kappa_0}} \\cap \\AGENT^{\\player}} \\).\n\\end{description}\nAs a consequence, we have just proved that\n\\( \\policy\\in\\Phi_{\\kappa_0}\\np{\\history'',\\control_{\\agent}} \\),\nhence that \n\\( \\Phi_{\\kappa_0}\\np{\\history',\\control_{\\agent}} =\n\\Phi_{\\kappa_0}\\np{\\history'',\\control_{\\agent}} \\) \nwhenever \n\\( \\{\\history', \\history''\\} \\subset G_{\\agent} \\).\n\\medskip\n\n\\noindent$\\bullet$\nLet $G_{\\agent}$ be an atom of~\\( \\tribu{\\Information}_{\\agent} \\).\nFinally, since \\( \\Phi_{\\kappa}\\np{\\history,\\control_{\\agent}} \\) \nin~\\eqref{eq:proof_Phi_kappa} takes the same (set) value\nfor any \\( \\history \\in G_{\\agent} \\), \nthe expression~\\eqref{eq:MtoB_PerfectRecall} takes the same value\nfor any \\( \\history \\in G_{\\agent} \\), and thus the function\n\\( \\history \\in \\HISTORY \\mapsto \n\\MtoB{\\mu}^{\\player}_{\\agent}\n\\conditionaly{\\control_{\\agent}}{\\history} \\) is \n\\( \\tribu{\\Information}_{\\agent} \\)-measurable.\n\\medskip\n\nThis ends the proof.\n\\end{proof}\n\\end{subequations}\n\n\\subsection{Kuhn's equivalence theorem for causal finite games in intrinsic form}\n\\label{Main_result}\n\nFinally, we give a statement and a proof of Kuhn's equivalence theorem for games in intrinsic form. \n\n\\begin{theorem}\nWe consider a causal finite W-game\n(see Definition~\\ref{de:W-game}),\nand a player \\( \\player \\in \\PLAYER \\) \nsupposed to enjoy perfect recall (see Definition~\\ref{de:PerfectRecall}).\n\nThen, for any mixed W-strategy \\( \\mu^{\\player} \\in\n\\Delta\\np{\\POLICY^{\\player}} \\),\nthere exists a product-mixed W-strategy\n\\( \\pi^{\\player}=\n\\sequence{ \\pi^{\\player}_{\\agent}}{\\agent \\in\n  \\AGENT^{\\player}}  \\in \\prod_{\\agent \\in \\AGENT^{\\player}} \\Delta\\np{\\POLICY_{\\agent}} \\), \nas in Definition~\\ref{de:product-mixed_W-strategy}, \nsuch that\\footnote{%\nSee Footnote~\\ref{ft:product-mixed_W-strategy}\nfor the abuse of notation \\( \\pi^{\\player}=\n\\otimes_{\\agent\\in\\AGENT^{\\player}}\\pi^{\\player}_{\\agent} \\).}\n\\begin{equation}\n \\QQ^{\\omega}_{\\couple{{\\mu}^{-\\player}}{{\\mu}^{\\player}}} \n= \\QQ^{\\omega}_{\\couple{{\\mu}^{-\\player}}{{\\pi}^{\\player}}}\n\\eqsepv \\forall {\\mu}^{-\\player} \n\\in \\prod_{\\player'\\neq \\player} \\Delta\\bp{ \\POLICY^{\\player'} } \n\\eqsepv \\forall \\omega\\in\\Omega\n\\eqfinv\n\\end{equation}\nwhere the probability distribution \\( \\QQ^{\\omega}_{\\mu} \n\\in \\Delta\\bp{\\prod_{b \\in \\AGENT} \\CONTROL_{b}} \\)\nhas been defined in~\\eqref{eq:push_forward_probability}.\n\\label{th:Kuhn}\n\\end{theorem}\n\n\n\\begin{subequations}\n \n  \\begin{proof}\n    The proof is in three steps. \n    \\medskip\n\n    \\noindent$\\bullet$\n    First, as all the assumptions of\n    Proposition~\\ref{pr:MtoB_PerfectRecall} are satisfied,\n    there exists a behavioral W-strategy\n    \\( \\MtoB{\\mu}^{\\player} = \n    \\sequence{\\MtoB{\\mu}^{\\player}_{\\agent}}{\\agent \\in \\AGENT^{\\player}} \\), \n    as in Definition~\\ref{de:behavioral_W-strategy},\n    which satisfies~\\eqref{eq:MtoB_PerfectRecall}.\n   \n    By Proposition~\\ref{pr:BtoPM}, we define \n    the product-mixed W-strategy\n    \\( \\pi^{\\player}=\n    \\BtoPM{\\MtoB{\\mu}}^{\\player} \\), that is,\n    with the property~\\eqref{eq:BtoPM} that, for any agent~\\( \\agent\\in\\AGENT^{\\player} \\), \n    \\begin{equation}\n     \n      \\pi^{\\player}_{\\agent}\n      \\defset{ \\policy_{\\agent}\\in\\POLICY_{\\agent}}%\n      {\\policy_{\\agent}\\np{\\history}=\\control_{\\agent}}\n      = \n      \\MtoB{\\mu}^{\\player}_{\\agent}\n      \\conditionaly{\\control_{\\agent}}{\\history}\n     \n      \\eqsepv \\forall \\control_{\\agent}\\in\\CONTROL_{\\agent} \n      \\eqsepv \\forall \\history \\in \\HISTORY \n      \\eqfinp\n      \\label{eq:Kuhn_proof_BtoPM}\n    \\end{equation}\n    \\medskip\n\n    \\noindent$\\bullet$\n    Second, we prove that\\footnote{%\n      See Footnote~\\ref{ft:product-mixed_W-strategy}\n      for the abuse of notation \\( \\pi^{\\player}=\n      \\otimes_{\\agent\\in\\AGENT^{\\player}}\\pi^{\\player}_{\\agent} \\).}\n    \\begin{align}\n      \\mu^{\\player}\n      & \\defset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player} }\n        \\in\\POLICY^{\\player} }%\n        { \\lambda_{\\agent}\\np{\\history} =\\history_{\\agent}\n        \\eqsepv \\forall \\agent\\in \\AGENT^{\\player} }\n        \\nonumber\n      \\\\\n      &=\n          \\ProductMixedStrategy^{\\player}\n          \\defset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player} }\n          \\in\\POLICY^{\\player} }%\n          { \\lambda_{\\agent}\\np{\\history} =\\history_{\\agent}\n          \\eqsepv \\forall \\agent\\in \\AGENT^{\\player} }\n          \\eqsepv \\forall \\history \\in \\HISTORY \n          \\eqfinp  \n        \\label{eq:proof_Kuhn_mixedstrategy_ProductBehavioral}\n    \\end{align}\n    In what follows, we consider a configuration~\\( \\history\\in\\HISTORY \\)\n    and the total ordering~\\( \\rho=\\varphi\\np{\\history} \\in\n    \\Sigma^{| \\AGENT |} \\).\n    We label the set~$\\AGENT^{\\player}$ of agents of the player~$\\player$ \n    by the stage at which each of them plays as follows:\n    \\begin{equation}\n      \\AGENT^{\\player}=\\{ \\rho(j_1) ,\\ldots, \\rho(j_{N}) \\} \n      \\text{  with } j_1 < \\cdots < j_{N}\n     \n      \\eqfinp  \n      \\label{eq:labelling}\n    \\end{equation}\n   \n   \n   \n    With this, we have\\footnote{Using the notation $\\ic{n}=\\na{1,\\ldots,n}$ to shorten some expressions.}\n    \\begin{align*}\n      \\mu^{\\player}\n      &\n        \\defset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player} }\n        \\in\\POLICY^{\\player} }%\n        { \\lambda_{\\agent}\\np{\\history} =\\history_{\\agent}\n        \\eqsepv \\forall \\agent\\in \\AGENT^{\\player} }\n      \\\\\n      &=\n        \\mu^{\\player}\n        \\defset{ \\sequence{\\policy_{\\rho(j_k)}}{k\\in\\ic{N}}\n        \\in \\prod_{k=1}^{N} \\POLICY_{\\rho(j_k)} }%\n        { \\lambda_{\\rho(j_k)}\\np{\\history} =\\history_{\\rho(j_k)}\n        \\eqsepv \\forall k\\in \\ic{N}}\n        \\tag{as \\( \\AGENT^{\\player}=\\{ \\rho(j_1) ,\\ldots,\n        \\rho(j_{N}) \\} \\)\n        by~\\eqref{eq:labelling}}\n      \\\\\n      &=\n        \\prod_{n=1}^{N}\n        \\frac{\n        \\mu^{\\player}\n        \\defset{ \\sequence{\\policy_{\\rho(j_k)}}{k\\in\\ic{n}}\n        \\in {\\displaystyle \\mathop{\\prod}_{k=1}^{n}} \\POLICY_{\\rho(j_k)} }%\n        { \\lambda_{\\rho(j_k)}\\np{\\history} =\\history_{\\rho(j_k)}\n        \\eqsepv \\forall k \\in \\ic{n}}\n        }{%\n        \\mu^{\\player}\n        \\defset{ \\sequence{\\policy_{\\rho(j_k)}}{k\\in\\ic{n{-}1}}\n        \\in {\\displaystyle \\prod_{k=1}^{n-1}}\\POLICY_{\\rho(j_k)} }%\n        { \\lambda_{\\rho(j_k)}\\np{\\history} =\\history_{\\rho(j_k)}\n        \\eqsepv \\forall k\\in\\ic{n-1}}\n        }\n        \\intertext{where, if the smaller term (the one to be found two equality lines above) is zero, every fraction\n        is supposed to take the value zero, and, if the smaller term is positive, so\n        are all the terms and no denominator is zero}\n     \n      &=\n        \\prod_{n=1}^{N} \n        \\MtoB{\\mu}^{\\player}_{\\rho(j_n)}%\n        \\conditionaly{\\history_{\\rho(j_n)}}{\\history}\n       \n        \\intertext{by~\\eqref{eq:MtoB_PerfectRecall}, because \n        \\( \\range{\\psi_{j_n}\\np{\\rho}} \\cap \\AGENT^{\\player}\n        =\n        \\{ \\rho(j_n) \\} \\cup \\bp{\n        \\range{\\psi_{j_{n-1}}\\np{\\rho}} \\cap \\AGENT^{\\player} } \\) \n        by definition~\\eqref{eq:cut} of the restriction mapping~$\\psi$,\n        and by definition of the sequence \\( j_1 < \\cdots < j_{N} \\) \n        in~\\eqref{eq:labelling}, which is such that \n        \\( \\AGENT^{\\player}=\\{ \\rho(j_1) ,\\ldots, \\rho(j_{N}) \\} \\)}\n     \n      &=\n        \\prod_{n=1}^{N} \n        \\ProductMixedStrategy^{\\player}_{\\rho(j_n)}\n        \\defset{ \\policy_{\\rho(j_n)} \\in \\POLICY_{\\rho(j_n)} }%\n        {\\policy_{\\rho(j_n)} \\np{\\history}=\\history_{\\rho(j_n)} }\n        \\tag{by~\\eqref{eq:Kuhn_proof_BtoPM}}\n      \\\\\n      &=\n        \\Bp{ \\bigotimes_{n=1}^{N} \n        \\ProductMixedStrategy^{\\player}_{\\rho(j_n)} }\n        \\defset{ \\sequence{\\policy_{\\rho(j_k)}}{k\\in\\ic{N}} \n        \\in \\prod_{k=1}^{N} \\POLICY_{\\rho(j_k)} }%\n        { \\lambda_{\\rho(j_k)}\\np{\\history} =\\history_{\\rho(j_k)}\n        \\eqsepv \\forall k\\in\\ic{N}}\n        \\tag{by definition of the product probability}\n       \n       \n       \n       \n      \\\\\n      &=\n        \\Bp{ \\bigotimes_{n=1}^{N} \n        \\ProductMixedStrategy^{\\player}_{\\rho(j_n)} }\n     \n        \\defset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player} }\n        \\in\\POLICY^{\\player} }%\n        { \\lambda_{\\agent}\\np{\\history} =\\history_{\\agent} }\n       \n        \\tag{as \\( \\AGENT^{\\player}=\\{ \\rho(j_1) ,\\ldots,\n        \\rho(j_{N}) \\} \\) in~\\eqref{eq:labelling}}\n    \\end{align*}\n    Thus, we have proved~\\eqref{eq:proof_Kuhn_mixedstrategy_ProductBehavioral}.\n    \\medskip\n\n\n    \\noindent$\\bullet$\n    Third, for any configuration~\\( \\history=\\bp{\\omega, \n      \\sequence{\\control_b}{b\\in\\AGENT} } \\in\\HISTORY \\), we have \n\n    \\begin{align*}\n      \\QQ^{\\omega}_{\\mu}\n      \\bp{\\sequence{\\control_b}{b\\in\\AGENT} }\n      & =\n        \\bp{\\bigotimes_{\\player \\in \\PLAYER}\\mu^{\\player}}\n        \\Bp{ M\\np{\\omega,\\cdot}^{-1}\n        \\bp{\\sequence{\\control_b}{b\\in\\AGENT} } } \n        \\tag{by definition~\\eqref{eq:push_forward_probability_a} \n        of~\\( \\QQ^{\\omega}_{\\mu} \\) }\n      \\\\\n      &=\n        \\bp{ \\bigotimes_{\\player \\in \\PLAYER}\\mu^{\\player} }\n        \\defset{\\policy\\in\\POLICY}{ M\\np{\\omega,\\policy} \n        = \\sequence{\\control_b}{b\\in\\AGENT} }\n        \\tag{by definition of a pushforward probability} \n      \\\\\n      &=\n        \\bp{ \\bigotimes_{\\player \\in \\PLAYER}\\mu^{\\player} }\n        \\defset{\\policy\\in\\POLICY}{ \\lambda_{\\agent}\\bp{\\omega, \n        \\sequence{\\control_b}{b\\in\\AGENT} } =\\control_{\\agent}\n        \\eqsepv \\forall \\agent\\in \\AGENT }\n        \\tag{by~\\eqref{eq:push_forward_probability_b} \n        and~\\eqref{eq:solution_map_IFF} which define\n        the mapping~\\( M\\np{\\omega,\\cdot}\\)} \n      \\\\\n      &=\n        \\prod_{\\player \\in \\PLAYER} \\mu^{\\player}\n       \n\\defset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player} }\n        \\in\\POLICY^{\\player} }%\n        { \\lambda_{\\agent}\\bp{\\omega, \n        \\sequence{\\control_b}{b\\in\\AGENT} } =\\control_{\\agent}\n        \\eqsepv \\forall \\agent\\in \\AGENT^{\\player} }\n        \\intertext{by definition of the product probability \n        \\( \\bigotimes_{\\player \\in \\PLAYER}\\mu^{\\player} \\)\n        on the product space\n        \\( \\prod_{\\player \\in \\PLAYER} \\POLICY^{\\player} \\) }\n     \n      &=\n        \\prod_{\\player \\in \\PLAYER} \\mu^{\\player}\n       \n\\defset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player} }\n        \\in\\POLICY^{\\player} }%\n        { \\lambda_{\\agent}\\np{\\history}=\\history_{\\agent}\n        \\eqsepv \\forall \\agent\\in \\AGENT^{\\player} }\n        \\intertext{(because \\( \\history=\\bp{\\omega, \n        \\sequence{\\control_b}{b\\in\\AGENT} } \\) \n        and where we have used notation~\\eqref{eq:history})}\n     \n      &=\n        \\prod_{\\player' \\neq \\player} \\mu^{\\player'}\n       \n\\defset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player'} }\n        \\in\\POLICY^{\\player'} }%\n        { \\lambda_{\\agent}\\np{\\history} =\\history_{\\agent}\n        \\eqsepv  \\forall \\agent\\in \\AGENT^{\\player'} }\n      \\\\\n      &\\phantom{==} \\times\n        \\mu^{\\player}\n       \n\\defset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player} }\n        \\in\\POLICY^{\\player} }%\n        { \\lambda_{\\agent}\\np{\\history} =\\history_{\\agent}\n        \\eqsepv  \\forall \\agent\\in \\AGENT^{\\player} }\n        \\tag{where we have singled out the player~$\\player$}\n      \\\\\n      &=\n        \\prod_{\\player' \\neq \\player} \\mu^{\\player'}\n       \n\\defset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player'} }\n        \\in\\POLICY^{\\player'} }%\n        { \\lambda_{\\agent}\\np{\\history} =\\history_{\\agent}\n        \\eqsepv  \\forall \\agent\\in \\AGENT^{\\player'} }\n      \\\\\n      &\\phantom{==} \\times\n        \\ProductMixedStrategy^{\\player}\n       \n\\defset{ \\sequence{\\policy_{\\agent}}{\\agent\\in\\AGENT^{\\player} }\n        \\in\\POLICY^{\\player} }%\n        { \\lambda_{\\agent}\\np{\\history} =\\history_{\\agent}\n        \\eqsepv  \\forall \\agent\\in \\AGENT^{\\player} }\n        \\tag{by~\\eqref{eq:proof_Kuhn_mixedstrategy_ProductBehavioral}}\n      \\\\\n      &=\n        \\QQ^{\\omega}_{\\couple{{\\mu}^{-\\player}}{{\\pi}^{\\player}}}\n        \\bp{\\sequence{\\control_b}{b\\in\\AGENT} }\n        \\eqfinv\n    \\end{align*}\n    by reverting to the top equality with \n    \\( \\mu^{\\player} \\) replaced by \n    \\( \\ProductMixedStrategy^{\\player} \\). \n    \\medskip\n\n    This ends the proof.\n  \\end{proof}\n \n\\end{subequations}\n\n\n\n\\section{Discussion}\n\nMost games in extensive form are formulated on a tree.\nHowever, whereas trees are perfect to follow step by step how a game is played,\nthey can be delicate to manipulate when information sets are added\nand must satisfy restrictive axioms to comply with the tree structure\n\\cite{Alos-Ferrer-Ritzberger:2016,Bonanno:2004,Brandenburger:2007}.\nIn this paper, we have introduced the notion of games in intrinsic form,\nwhere the tree structure is replaced with a product structure,\nmore amenable to mathematical analysis. \nFor this, we have adapted Witsenhausen's intrinsic model --- \na model with Nature, agents and their decision sets, and\nwhere information is represented by $\\sigma$-fields --- to games.\nIn contrast to games in extensive form formulated on a tree,\nWitsenhausen's intrinsic games (W-games) do not require\nan explicit description of the play temporality. \nNot having a hardcoded temporal ordering makes mathematical representations more intrinsic.\n\nAs part of a larger research program, we have focused here on Kuhn's equivalence\ntheorem.\nFor this purpose, we have defined the property of perfect recall for a player \nof a causal W-game (that is, without referring to a tree structure),\nand we have introduced three different definitions of\n``randomized'' strategies in the W-games setting\n--- mixed, product-mixed and behavioral. \nThen, we have shown that, under perfect recall for a player,\nany of her possible mixed strategies can be replaced by a behavioral strategy,\nwhich is the statement of Kuhn's equivalence theorem.\nMoreover, we have shown that any of her possible mixed strategies can also be replaced by a\nproduct-mixed strategy, that is, a mixed strategy under which her executive\nagents are probabilistically independent.\n\nWe add to the existing literature on extensive games representation \nby proposing a representation that is more general than the tree-based ones as,\nfor instance, it allows to describe noncausal situations.\nIndeed, Witsenhausen showed that there are noncausal W-models that yet are\nsolvable. \n\nFurthermore, our paper illustrates that the intrinsic form is well equipped to\nhandle proofs with mathematical formulas, without resorting to tree-based\n  arguments that can be cumbersome when handling information. \nWe hence believe that the intrinsic form constitutes a new valuable tool for the analysis of games \nwith information. \n\nThe current work is the first output of a larger research program \nthat addresses games in intrinsic form.\nWe are currently working on the embedding of tree-based games in extensive\nform into W-games (by a mapping that associates each information set with an\nagent), and on the restricted class of W-games that can be \nembedded in tree-based games.\nFuther research includes extensions to measurable decision sets, \nand to infinite number of agents or players.\nWe will also investigate what can be said about subgame perfect equilibria and\nbackward induction, as well as Bayesian games.\n\\medskip\n\n\\textbf{Acknowledgements}.\n\nWe thank Dietmar Berwanger and Tristan Tomala for\nfruitful discussions, and for their valuable comments on a first version of this\npaper. This research benefited from the support of the FMJH Program PGMO and\nfrom the support to this program from EDF.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nAn information system designed for analysis of social media must consider a common set properties that characterize all social media data.\n\\begin{itemize}[leftmargin=*]\n    \\item Information elements in social media are essentially \\textit{heterogeneous} in nature -- users, posts, images, external URL references, although related, all bear different kinds information. \n    \\item Most social information is \\textit{temporal} -- a timestamp is associated with user events like the creation or response on a post, as well as system events like user account creation, deactivation and deletion. The system should therefore allow both temporal as well as time-agnostic analyses.\n    \\item Information in social media evolves fast. In one study \\citep{zhu2013modeling}, it was shown that the number of users in a social media is a power function of time. More recently, \\citep{antonakaki2018utilizing} showed that Twitter's growth is supralinear and follows Lescovec's model of graph evolution \\citep{leskovec2007graph}. Therefore, an analyst may first have to perform \\textit{exploration tasks} on the data before figuring out their analysis plan. \n    \\item Social media has a significant textual content, sometimes with specific entity markers (e.g., mentions) and topic markers (e.g., hashtags). Therefore any information element derived from text (e.g., named entities, topics, sentiment scores) may also be used for analysis. To be practically useful, the system must accommodate semantic synonyms -- \\#KamalaHarris, @KamalaHarris and ``Kamala Harris'' refer to the same entity.\n    \\item Relationships between information items in social media data must capture both canonical relationships like (\\texttt{tweet-15 mentions user-392}) but a wide-variety of computed relationships over base entities (users, posts, $\\ldots$) and text-derived information (e.g., named entities). \n\\end{itemize}\nIt is also imperative that such an information must support three styles of analysis tasks \n\\begin{enumerate}\n    \\item \\textbf{Search}, where the user specifies content predicate without specifying the structure of the data. For example, seeking the number of tweets related to Kamala Harris should count tweets where she is the author, as well as tweets where any synonym of ``Kamala Haris'' is in the tweet text.\n    \\item \\textbf{Query}, where the user specifies query conditions based on the structure of the data. For example, tweets with \\texttt{create\\_date} between 9 and 9:30 am on January 6th, 2021, with \\texttt{text} containing the string ``Pence'' and that were \\texttt{favorited} at least 100 times during the same time period.\n    \\item \\textbf{Discovery,} where the user may or may not know the exact predicates on the data items to be retrieved, but can specify analytical operations (together with some post-filters) whose results will provide insights into the data. For example, we call a query like \\texttt{Perform \\textit{community detection} on all tweets on January 6, 2021 and return the users from the largest community} a discovery query.\n\\end{enumerate}\nIn general, a real-life analytics workload will freely combine these modalities as part of a user's information exploration process. \n\nIn this paper, we present a general-purpose graph-based  model for social media data and a subgraph discovery algorithm atop this data model. Physically, the data model is implemented on AWESOME \\citep{gupta:awesome:2016}, an analytical platform designed to enable large-scale social media analytics over continuously acquired data from social media APIs. The platform, developed as a polystore system natively supports relational, graph and document data, and hence enables a user to perform complex analysis that include arbitrary combinations of search, query and discovery operations. We use the term \\textbf{\\textit{query-driven discovery}} to reflect that the scenario where the user does not want to run the discovery algorithm on a large and continuously collected body of data; rather, the user knows a starting point that can be specified as an expressive query (illustrated later), and puts bounds on the discovery process so that it terminates within an acceptable time limit.\n\n\\medskip \n\n\\noindent \\textbf{Contributions.} This paper makes the following contributions. (a) It offers a new formulation for the subgraph interestingness problem for social media; (b) based on this formulation, it presents a discovery algorithm for social media; (c) it demonstrates the efficacy of the algorithm on multiple data sets.\n\n\\medskip\n\n\\noindent \\textbf{Organization of the paper.} The rest of the paper is organized as follows. Section \\ref{sec:related} describes the related research on interesting subgraph finding in as investigated by researchers in Knowledge Discovery, Information Management, as well Social Network Mining. Section \\ref{sec:prelim} presents the abstract data model over which the Discovery Algorithm operations and the basic definitions to establish the domain of discourse for the discovery process. Section \\ref{sec:generate} presents our method of generating candidate subgraphs that will be tested for interestingness. Section \\ref{sec:discovery} first presents our interestingness metrics and then the testing process based on these metrics. Section \\ref{sec:experiments} describes the experimental validation of our approach on multiple data sets. Section \\ref{sec:conclusion} presents concluding discussions.\n\n\\section{Related Work}\n\\label{sec:related}\nThe problem of finding ``interesting'' information in a data set is not new. \\citep{what-makes-patterns-interesting:1996} described that  an``interestingness measure'' can be ``objective'' or ``subjective''. A measure is ``objective'' when it is computed solely based on the properties of the data. In contrast, a ``subjective'' measure must take into account the user's perspective. They propose that (a) a pattern is interesting if it is \"surprising\" to the user (\\textit{unexpectedness}) and (b) a pattern is interesting if the user can act on it\nto his advantage (\\textit{actionability}). Of these criteria, actionability is hard to determine algorithmically; unexpectedness, on the other hand, can be viewed as the departure from the user's beliefs. For example, a user may believe that the 24-hour occurrence pattern of all hashtags are nearly identical. In this case, a discovery would be to find a set of hashtags and sample dates for which this belief is violated. \nFollowing \\citep{geng2006interestingness}, there are three possibilities regarding how a system is informed of a user's knowledge and beliefs: (a) the user provides a formal specification of his or her knowledge, and after obtaining the mining results, the system chooses which unexpected patterns to present to the user \\citep{BingLiu:1999}; (b) according to the user's interactive feedback, the system removes uninteresting patterns \\citep{not-interesting:1999}; and (c) the system applies the user's specifications as constraints during the mining process to narrow down the search space and provide fewer results. Our work roughly corresponds to the third strategy.\n\\begin{figure*}[t]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=6.5cm, height=6.5cm]{fig-1-core-peri.png}\n\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=6.5cm, height=6.5cm]{fig-1-networking-connection.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{.4\\textwidth}\n    \\centering\n        \\includegraphics[width=7.5cm, height=6.5cm]{fig-1-top-25-bar.png}\n    \\end{minipage}\n    \\begin{minipage}{.4\\textwidth}\n    \\centering\n        \\includegraphics[width=7.5cm, height=6.5cm]{degree-hist.png}\n    \\end{minipage}\n    \\caption{An example of an interesting subgraph. Tweets on Indian politics form a sparse center in a dense periphery (separately shown in the top right). The bottom two figures show the hashtag distribution of the center and the degree distribution of the center (log scale) respectively.}\n    \\label{fig:sparse}\n\\end{figure*}\nEarly research on finding interesting subgraphs focused primarily on finding interesting substructures. This body of research primarily found interestingness in two directions: (a) finding frequently occurring subgraphs in a collection of graphs (e.g., chemical structures) \\citep{kuramochi2001frequent, yan2003closegraph, thoma2010discriminative} and (b) finding regions of a large graph that have high edge density \\citep{lee2010survey, sariyuce2015finding, epasto2015efficient, wen2017efficient} compared to other regions in the graph. Note that while a dense region in the graph can definitely interesting, shows, the inverse situation where a sparsely connected region is surrounded by an otherwise dense periphery can be equally interesting for an application. \n\nWe illustrate the situation in Figure \\ref{fig:sparse}. The primary data set is a collection of tweets on COVID-19 vaccination, but this specific graph shows a sparse core on Indian politics that is loosely connected to nodes of an otherwise dense periphery on the primary topic. Standard network features like hashtag histogram and the node degree histograms do not reveal this substructure requiring us to explore new methods of discovery. \n\nA serious limitation of the above class of work is that the interestingness criteria does not take into account \\textit{node content} (resp. edge content) which may be present in a property graph data model \\citep{angles2018property} where nodes and edges have attributes. \\citep{bendimerad2019mining} presents several discovery algorithms for graphs with vertex attributes and edge attributes. They perform both structure-based graph clustering and subspace clustering of attributes to identify interesting (in their domain ``anomalous'') subgraphs.\n\nOn the ``subjective'' side of the interestingness problem, one approach considers interesting subgraphs as a subgraph matching problem \\citep{shan2019dynamic}. Their general idea is to compute all matching subgraphs that satisfy a user the query and then ranking the results based on the rarity and the likelihood of the associations among entities in the subgraphs. In contrast \\citep{adriaens2019subjectively} uses the notion of ``subjective interestingness'' which roughly corresponds to finding subgraphs whose connectivity properties (e.g., the average degree of a vertices) are distinctly different from an ``expected'' \\textit{background} graph. This approach uses a constrained optimization problem that maximizes an objective function over the \\textit{information content} ($IC$) and the \\textit{description length} ($DL$) of the desired subgraph pattern. \n\nOur work is conceptually most inspired by \\citep{bendimerad2019subj} that explores the subjective interestingness problem for attributed graphs. Their main contribution centers around CSEA (Cohesive Subgraph with Exceptional Attributes) patterns that inform the user that a given set of attributes has exceptional values throughout a set of vertices in the graph. The subjective interestingness is given by $$S(U,S) = \\frac{IC(U,S)}{DL(U)}$$ where $U$ is a subset of nodes and $S$ is a set of restrictions on the value domains of the attributes. The system models the prior beliefs of the user as the Maximum Entropy distribution subject to any stated prior, beliefs the user may hold about the data (e.g., the distribution of an attribute value). The information content $IC(U,S)$ of a CSEA pattern $(U,S)$ is  formalized as negative of the logarithm of the probability that the pattern is present under the background distribution.  The length of a description\nof $U$ is the intersection of all neighborhoods in a subset $X \\subseteq N (U)$,\nalong with the set of ``exceptions'', vertices are in the intersection but not part of\n$U$. However, we have a completely different, more database-centric formulation of the background and the user's beliefs.\n\n\\section{The Problem Setup}\n\\label{sec:prelim}\n\n\\subsection{Data Model}\n\\label{sec:dataModel}\nOur abstract model social media data takes the form of a \\textit{heterogeneous information network} (an information network with multiple types of nodes and edges), which we view as a temporal property graph $G$. Let $N$ be the node set and $E$ be the edge set of $G$. $N$ can be viewed as a disjoint union of different subsets (called \\textit{node types}) --  users $U$, posts $P$, topic markers (e.g., hastags) $H$, term vocabulary $V$ (the set of all terms appearing in a corpus), references (e.g., URLs) $R(\\tau)$, where $\\tau$ represents the type of resource (e.g., image, video, web site $\\ldots$). Each type of node have a different set of properties (attributes) $\\bar{A}(.)$. We denote the attributes of $U$ as $\\bar{A}(U) = a_1(U), a_2(U) \\ldots$ such that $a_i(U)$ is the $i$-the attribute of $U$. An attribute of a node type may be temporal -- a post $p \\in P$ may have temporal attribute called \\texttt{creationDate}. Edges in this network can be directional and have a single \\textit{edge type}. The following is a set of \\textit{base (but not exhaustive) edge types}:\n\\begin{itemize}[leftmargin=1em, label=\\scriptsize{$-$}]\n    \\item \\texttt{writes}: $U \\mapsto P$\n    \\item \\texttt{uses}: $P \\mapsto H$\n    \\item \\texttt{mentions}: $P \\mapsto U$ maps a post $p$ to a user $u$ if $u$ mentioned in $p$\n    \\item \\texttt{repostOf}: $P \\mapsto P$ maps a post $p_2$ to a post $p_1$ if $p_2$ is repost of $p_1$. This implies that $ts(p_2) < ts(p_1)$ where $ts$ is the timestamp attribute\n    \\item \\texttt{replyTo\/comment}: $P \\mapsto P$ maps a post $p_2$ to a post $p_1$ if $p_2$ is a reply to $p_1$. This implies that $ts(p_2) < ts(p_1)$ where $ts$ is the timestamp attribute\n    \\item \\texttt{contains}: $P \\mapsto V \\times \\mathcal{N}$ where $\\mathcal{N}$ is the set of natural numbers and represents the count of a token $v \\in V$ in a post $p \\in P$\n\\end{itemize}\n We realistically assume that the inverse of these mappings can be computed, i.e., if $v_1, v_2 \\in V$ are terms, we can perform a \\texttt{contains}$^{-1}(v_1 \\wedge \\neg v_2)$ operation to yield all posts that used $v_1$ but not $v_2$.\n \nThe AWESOME information system allows users construct \\textit{derived or computed edge types} depending on the specific discovery problem they want to solve. For example, they can construct a standard hashtag co-occurrence graph using a non-recursive aggregation rule in Datalog \\citep{consens1993low}:\n\\begin{equation*}\n    \\begin{aligned}\n        HC(h_1, h_2, count(p)) &\\longleftarrow \\\\\n        &p \\in P, h_1 \\in H, h_2 \\in H \\\\ \n        &uses(p, h_1), uses(p, h_2) \n    \\end{aligned}\n\\end{equation*}\nWe interpret $HC(h_1, h_2, count(p))$ as an edge between nodes $h_1$ and $h_2$ and $count(p)$ as an attribute of the edge. In our model, a computed edge has the form: $E_T(N_b, N_e, \\bar{B})$ where $E_T$ is the edge type, $N_b, N_e$ are the tail and head nodes of the edge, and $\\bar{B}$ designates a flat schema of edge properties. The number of such computed edges can be arbitrarily large and complex for different subgraph discovery problems. A more complex computed edge may look like: $UMUHD(u_1,u_2,mCount,d, h)$, where an edge from $u_1$ to $u_2$ is constructed if user $u_1$ creates a post that contains hashtag $h$, and mentions user $u_2$ a total number of $mCount$ times on day $d$. Note that in this case, hashtag $h$ is a property of edge type $UMUHD$ and is not a topic marker node of graph $G$. \n\n\\subsection{Specifying User Knowledge}\n\\label{sec:hquery}\nSince the discovery process is query-driven, the system has no \\textit{a priori} information about the user's interests, prior knowledge and expectations, if any, for a specific discovery task. Hence, given a data corpus, the user needs to provide this information through a set of parameterized queries. We call these queries ``heterogeneous'' because they can be placed on conditions on any arbitrary node (resp. edge) properties, network structure and text properties of the graph. \n\n\\medskip\n\n\\noindent \\textbf{User Interest Specification.} The discovery process starts with the specification of the user's universe of discourse identified with a query $Q_0$. We provide some illustrative examples of user interest with queries of increasing complexity.\n\n\\noindent \\textit{Example 1.}  ``All tweets related to COVID-19 between 06\/01\/2020 and 07\/31\/2020 that refer to Hydroxychloroquine''. In this specification, the condition ``related to COVID-19'' amounts to finding tweets containing any $k$ terms from a user-provided list, and the condition on ``Hydroxychloroquine'' is expressed as a fuzzy search.\n\\begin{figure}\n\\includegraphics[width=8.5cm, height=6.5cm]{hydroxyclq.png}\n\\vspace{-2cm}\n \\caption{Hydroxychloroquine and COVID related Hashtags}\n \\label{fig:hydroxi}\n\\end{figure}\nFigure \\ref{fig:hydroxi} shows the top hashtags related to this search -- note that the hashtag co-occurrence graph around COVID-19 and hyroxychloroquine includes ``FakeNewsMedia''.\n\n\\noindent \\textit{Example 2.} ``All tweets from users who mention \\texttt{Trump} in their user profile and \\texttt{Fauci} in at least $n$ of their tweets''. Notice that this query is about users with a certain behavioral pattern -- it captures \\textit{all} tweets from users who have a combination of specific profile features and tweet content.  \n\n\\noindent \\textit{Example 3.} ``All tweets from users $U_1$ whose tweets appear in hashtag-cooccurrence in the 3 neighborhood around \\texttt{\\#ados} is used together with all tweets of users $U_2$ who belong to the user-mention-user networks of these users (i.e., $U_1$) '', where \\texttt{\\#ados} refers to ``American Descendant of Slaves'', which represents an African American cause.\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=6.5cm]{kamala-ht-vol.png}\n\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=6.5cm]{kamala-ht.png}\n       \n    \\end{minipage}\n\n    \\caption{Two views of the \\texttt{\\#ados} cluster of hashtags }\n    \\label{fig:ados}\n\\end{figure*}\n\n\n\\noindent The end result of $Q_0$ is a collection of posts that we call the \\textit{initial post set} $P_0$. Using the posts in $P_0$, the system creates a background graph as follows.\n\n\\medskip\n\n\\noindent \\textbf{Initial Background Graph.} The initial background graph $G_0$ is the graph derived from $P_0$ over which the discovery process runs. However, to define the initial graph, we first develop the notion of a \\textit{conversation}.\n\n\\begin{definition}[\\textbf{Semantic Neighborhood}.] $N(p)$, the semantic neighborhood of a post $p$ is the graph connecting $p$ to instances of $U \\cup H \\cup P \\cup V$ that directly relates to $p$.\n\\end{definition}\n\n\\begin{definition}[\\textbf{Conversation Context}.] The conversation context $C(p)$ of post $p$ is a subgraph satisfying the following conditions:\n\\begin{enumerate}[leftmargin=*]\n    \\item $P_1$: The set of posts reachable to\/from $p$ along the relationships \\texttt{repostOf, replyTo} belong to $C(p)$. \n    \\item $P_2$: The union of posts in the semantic neighborhood of $P_1$ belong to $C(p)$.\n    \\item $E$: The induced subgraph of $P_1 \\cup P_2$ belong to $C(p)$\n    \\item Nothing else belongs to $C(p)$.\n\\end{enumerate}\n\\end{definition}\n\\noindent Clearly, we can assert that $C(p)$ is a connected graph and that $N(p)~ \\sqsubset_g~ C(p)$ where $\\sqsubset_g$ denotes a subgraph relationship.\n\n\\begin{definition}[\\textbf{Initial Background Graph}.] The initial background graph $G_0$ is a merger of all conversation contexts $C(p_i), p_i \\in P_0$, together with all computed edges induces the nodes of $\\cup_i C(p_i)$\n\\end{definition}\nThe initial background graph itself can be a gateway to finding interesting properties of the graph. To illustrate this based on the graph obtained from our Example 2. Figure \\ref{fig:ados} presents two views of the \\texttt{\\#ados} cluster of hashtags from January 2021. The left chart shows the time vs. count of the hashtags while the right chart shows the dominant hashtags of the same period in this cluster. The strong peak in the timeline, was due to an intense discussion, revealed by topic modeling, on the creation of an office on African American issues. The occurrence of this peak is interesting because most of the social media conversation in this time period was focused on the Capitol attack on January 6.\n\n\nGiven the $G_0$ graph,we discover subgraphs $S_i \\subset G_o$ whose content and structure are distinctly different that of $G_0$. However, unlike previous approaches, we apply a generate-and-test paradigm for discovery. The generate-step (Section \\ref{sec:generate}) uses a graph cube like \\citep{zhao2011graph} technique to generate candidate subgraphs that might be interesting and the test-step (Section \\ref{sec:testing}) computes if (a) the candidate is sufficiently distinct from  the $G'$, and (b) the collection of candidates are sufficiently distinct from each other.\n\n\\noindent \\textbf{Subgraph Interestingness.} For a subgraph $S_i$ to be considered as a candidate, it must satisfy the following conditions.\n\n\\noindent \\textbf{C1.} $S_i$ must be connected and should satisfy a size threshold $\\theta_n$, the minimal number of nodes.\n\n\\noindent \\textbf{C2.} Let $A_{ij}$ (resp. $B_{ik}$) be the set of \\textit{local} properties of node $j$ (resp. edge $k$) of subgraph $S_i$. A property is called ``local'' if it is not a network property like vertex degree. All nodes (resp. edges) of $S_i$ must satisfy some user-specified predicate $\\phi_N$ (resp. $\\phi_E$) specified over $A_{ij}$ (resp. $B_{ik}$). For example, a node predicate might require that all ``post'' nodes in the subgraph must have a re-post count of at least 300, while an edge predicate may require that all hashtag co-occurrence relationships must have a weight of at least 10. A user defined constraint on the candidate subgraph improves the interpretability of the result. Typical subjective interestingness techniques \\citep{van2016subjective, adriaens2019subjectively} use only structural features of the network and do not consider attribute-based constraints, which limits their pragmatic utility.\n\n\\noindent \\textbf{C3.} For each text-valued attribute $a$ of $A_{ij}$, let $C(a)$ be the collection of the values of $a$ over all nodes of $S_i$, and $\\mathcal{D}(C(a))$ is a textual diversity metric computed over $C(a)$. For $S_i$ to be interesting, it must have at least one attribute $a$ such that $\\mathcal{D}(C(a))$ does not have the usual power-law distribution expected in social networks. Zheng et al \\citep{zheng2019social} used \\textit{vocabulary diversity} and \\textit{topic diversity} as textual diversity measures.\n\n\n\\section{Candidate Subgraph Generation}\n\\label{sec:generate}\nSection \\ref{sec:hquery} describes the creation of the initial background graph $G_0$ that serves as the domain of discourse for discovery. Depending on the number of initial posts $P_0$ resulting form the initial query, the size of $G_0$ might be too large -- in this case the user can specify followup queries on $G_0$ to narrow down the scope of discovery. We call this narrowed-down graph of interest as $G'$ -- if no followup queries were used, $G' = G_0$. The next step is to generate some candidate subgraphs that will be tested for interestingness. \n\n\\noindent \\textbf{Node Grouping.} A node group is a subset of \\textit{nodes($G'$)} where all nodes in a group have some similar property. We generalize the \\textit{groupby} operation, commonly used in relational database systems, to heterogeneous information networks. To describe the generalization, let us assume $R(A, B, C, D, \\ldots)$ is a relation (table) with attributes $A, B, C, D, \\ldots$ A \\textit{groupby} operation takes as input (a) a subset of \\textit{grouping attributes} (e.g. $A, B$), (b) a \\textit{grouped attribute} (e.g., $C$) and (c) an \\textit{aggregation function} (e.g., \\textit{count}). The operation first computes each distinct cross-product value of the grouping attributes (in our example, $A \\times B$) and creates a list of all values of the grouped attribute corresponding to each distinct value of the grouping attributes, and then applies the aggregation function to the list. Thus, the result of the \\textit{groupby} operation is a single aggregated value for each distinct cross-product value of grouping attributes. \n\nTo apply this operation to a social network graph, we recognize that there are two distinct ways of defining the ``grouping-object''. \\\\\n(1) Node properties can be directly used just like in the relational case. For example, for tweets a grouping condition might be \\texttt{getDate} \\texttt{(Tweet.created\\_at)} $\\wedge$ \\texttt{bin(Tweet.favoriteCount, 100)}, where the \\texttt{getDate} function extracts the date of a tweet and the \\texttt{bin} function creates buckets of size 100 from the favorite count of each tweet. \\\\\n(2) The grouping-object is a subgraph pattern. For example, the subgraph pattern\\\\\n\\texttt{(:tweet\\{date\\})-[:uses]->(:hashtag\\{text\\})} \\hfill{(P1)}\\\\\nstates that all ''tweet'' nodes having the same posting date, together with every distinct hashtag text will be placed in a separate group. Notice that while (1) produces disjoint tweets, (2) produces a ``soft'' partitioning on the tweets and hashtags due to the many-to-many relationship between tweets and hashtags. \\\\\nIn either case, the result is a set of node groups, designated here as $N_i$. For example, the grouping pattern P1 expressed in a Cypher-like syntax \\citep{francis2018cypher} (implemented in the Neo4J graph data management system) states that all tweets having the same posting date, together with every distinct hashtag text will be placed in a separate group.\nNotice that this process produces a ``fuzzy'' partitioning on the tweets and hashtags due to the many-to-many relationship between tweets and hashtags. Hence, the same tweet node can belong to two different groups because it has multiple hashtags. Similarly, a hashtag node can belong to multiple groups because tweets from different dates may have used the same hashtag. While the grouping condition specification language can express more complex grouping conditions, in this paper, we will use simpler cases to highlight the efficacy of the discovery algorithm. We denote the node set in each group as $N_i$.\n\n\\noindent \\textbf{Graph Construction.} To complete the \\textit{groupby} operation, we also need to specify the aggregation function in addition to the grouping-object and the grouped-object. This function takes the form of a graph construction operation that constructs a subgraph $S_i$ by expanding on the node set $N_i$. Different expansion rules can be specified, leading to the formation of different graphs. Here we list three rules that we have found fairly useful in practice.\n\n\\noindent \\textbf{G1.} Identify all the \\texttt{tweet} nodes in $N_i$. Construct a \\textit{relaxed induced subgraph} of the \\texttt{tweet}-labeled nodes in $N_i$. The subgraph is induced because it only uses tweets contained within $N_i$, and it is \\textit{relaxed} because contains all nodes \\textit{directly associated} with these tweet nodes, such as author,  hashtags, URLs, and mentioned-users. \n\n\\noindent \\textbf{G2.} Construct a \\textit{mention network} from within the tweet nodes in $N_i$ -- the mention network initially connects all \\texttt{tweet} and \\texttt{user}-labeled nodes. Extend the network by including all nodes \\textit{directly associated} with these tweet nodes.\n\n\\noindent \\textbf{G3.} A third construction relaxes the grouping constraint. We first compute either \\textbf{G1} or \\textbf{G2}, and then extend the graph by including the first order neighborhood of mentioned users or hashtags. While this clearly breaks the initial group boundaries, a network thus constructed includes tweets of similar themes (through hashtags) or audience (through mentions).\n\n\\noindent \\textbf{Automated Group Generation.} In a practical setting, as shown in Section \\ref{sec:experiments}, the parameters for node grouping operation can be specified by a user, or it can be generated automatically. Automatic generation of grouping-objects is based on the considerations described below. To keep the autogeneration manageable, we will only consider single and two objects for attribute grouping and only a single edge for subgraph patterns.\n\\begin{itemize}[leftmargin=*]\n    \\item Since temporal shifts in social media themes and structure are almost always of interest, the posting timestamp is always a grouping variable. For our purposes, we set the granularity to a day by default, although a user can set it.\n    \\item The frequency of most nontemporal attributes (like hashtags) have a mixture distribution of double-pareto lognormal distribution and power law \\citep{gupta:osn:2020}, we will adopt the following strategy.\n    \\begin{itemize}[label={$\\circ$}]\n        \\item  Let $f(A)$ be distribution of attribute $A$, and $\\kappa(f(A))$ be the curvature of $f(A)$. If $A$ is a discrete variable, we find $a*$, the maximum curvature (elbow) point of $f(A)$ numerically \\citep{antunes2018knee}.\n        \\item  We compute $A'$, the values of attribute $A$ to the left of $a*$ for all attributes and choose the attribute where the cardinality of $A'$ is maximum. In other words, we choose attributes which have the highest number of pre-elbow values. \n    \\end{itemize}\n    \\item We adopt a similar strategy for subgraph patterns. If $T_1(a_i)\\stackrel{L}{\\longrightarrow}T_2(b_j)$ is an edge where $T_1, T_2$ are node labels, $a_i, b_j$ are node properties and $L$ is an edge label, then $a_i$ and $b_j$ will be selected based on the conditions above. Since the number of edge labels is fairly small in our social media data, we will evaluate the estimated cardinality of the edge for all such triples and select one with the lowest cardinality. \n\\end{itemize}\n\n\\section{The Discovery Process}\n\\label{sec:discovery}\n\n\\subsection{Measures of for Relative Interestingness}\n\\label{sec:interestingness}\nWe compute the interestingness of a subgraph $S$ in reference to a background graph $G_b$ (e.g., $G'$), and consists of a structural as well as a content component. We first discuss the structural component. To compare a subgraph $S_i$ with the background graph, we first compute a set of network properties $P_j$ (see below) for nodes (or edges) and then compute the frequency distribution $f(P_j(S_i))$ of these properties over all nodes (resp. edges) of (a) subgraphs $S_i$, and (b) the reference graph (e.g., $G'$). A distance between $f(P_j(S_i))$ and $f(P_j(G_b))$ is computed using Jensen\u2013Shannon divergence (JSD). In the following, we use $\\Delta(f_1,f_2)$ to refer to the JS-divergence of distributions $f_1$ and $f_2$. \n\\medskip\n\\noindent \\textbf{Eigenvector Centrality Disparity:} The testing process starts by identifying the distributions of nodes with high node centrality between the networks. While there is no shortage of centrality measures in the literature, we choose eigenvector centrality \\citep{das2018study} defined below, to represent the dominant nodes. Let $A = (a_{i,j})$ be the adjacency matrix of a graph. The eigenvector centrality $x_{i}$ of node $i$ is given by: $$x_i = \\frac{1}{\\lambda} \\sum_k a_{k,i} \\, x_k$$ where $\\lambda \\neq 0$ is a constant. \n    The rationale for this choice follows from earlier studies in \\citep{Bonacich2007-mx,Ruhnau2000-jy,Yan2014-dn}, who establish that since the eigenvector centrality can be seen as a weighted sum of direct and indirect connections, \n    it represents the true structure of the network more faithfully than other centrality measures. \n    Further, \\citep{Ruhnau2000-jy} proved that the eigenvector-centrality under the Euclidean norm can be transformed into node-centrality, a property not exhibited by other common measures.\n    Let the distributions of eigenvector centrality of subgraphs $A$ and $B$ be $\\beta_a$ and $\\beta_b$ respectively, and that of the background graph be $\\beta_t$, then \n    $|\\Delta_e(\\beta_t, \\beta_a)| > \\theta $ indicates that $A$ is sufficiently structurally distinct from $G_b$ \n    $|\\Delta_e(\\beta_t, \\beta_a)| > |\\Delta_e(\\beta_t, \\beta_b)|$ indicates that $A$ contains significantly more or significantly less influential nodes than $B$. \n\n\\medskip\n\\noindent \\textbf{Topical Navigability Disparity:} Navigability measures ease of flow. If subgraph $S$ is more navigable than subgraph $S'$, then there will be more traffic through $S$ compared to $S'$. However, the likelihood of seeing a higher flow through a subgraph depends not just on the structure of the network, but on extrinsic covariates like time and topic. So, a subgraph is interesting in terms of navigability if for some values of a covariate, its navigability measure is different from that of a background subgraph. \n\nInspired by its application in biology \\citep{seguin2018navigation}, traffic analysis \\citep{scellato2010traffic}, and network attack analysis \\citep{lekha2020central}, we use \\textit{edge betweenness centrality} \\citep{das2018study} as the generic (non-topic) measure of navigability.   Let $\\alpha_{ij}$ be the number of shortest paths from node i to j and $\\alpha_{ij}(k)$ is the number of paths passes through the edge $k$. Then the edge-betweenness centrality is $$C_{eb}(k)= \\sum_{(i,j)\\in V} \\frac{\\alpha_{ij}(k)}{\\alpha_{ij}}$$\nBy this definition, the edge betweenness centrality is the portion of all-pairs shortest paths that pass through an edge. Since edge betweenness centrality of edge $e$ measures the proportion of paths that passes through $e$, a subgraph $S$ with a higher proportion of high-valued edge betweenness centrality implies that $S$ may be more \\textit{navigable} than $G_b$ or another subgraph $S'$ of the graph, i.e., information propagation is higher through this subgraph compared to the whole background network, for that matter, any other subgraph of network having a lower proportion of nodes with high edge betweenness centrality. Let the distribution of the edge betweenness centrality of two subgraphs $A$ and $B$ are $c_1$ and $c_2$ respectively, and that of the reference graph is $c_0$. Then, $|\\Delta_b(c_0, c_1)| < |\\Delta_b(c_0, c_2)|$  means the second subgraph is more navigable than the first.\n\nTo associate navigability with topics, we detect topic clusters over the background graph and the subgraph being inspected. The exact method for topic cluster finding is independent of the use of topical navigability. In our setting, we have used topic  topic modeling and dense region detection in hashtag cooccurrence networks. For each topic cluster, we identify posts (within the subgraph) that belong to the cluster. If the number of posts is greater than a threshold, we compute the navigability disparity.\n\n\\medskip\n\n\\noindent  \\textbf{Propagativeness Disparity:} The concept of propagativeness builds on the concept of navigability. Propagativeness attempts to capture how strongly the network is spreading information through a navigable subgraph $S$. We illustrate the concept with a network constructed over tweets where a political personality (Senator Kamala Harris in this example) is mentioned in January 2021. The three rows in Figure \\ref{fig:kamala} show the network characteristics of the subregions of this graph, related, respectively to the themes of \\texttt{\\#ados} and ``black lives matter'' (Row 1), Captiol insurrection (Row 2) and Socioeconomic issues related to COVID-19 including stimulus funding ad business reopening (Row 3). In earlier work \\citep{zheng2019social}, we have shown that a well known propagation mechanism for tweets is to add user-mentions to improve the reach of a message - hence the user-mention subgraph is indicative of propagative activity. In Figure \\ref{fig:kamala}, we compare the hashtag activity (measured by the Hashtag subgraph) and the mention activity (size of the mention graph) in these three subgraphs. Figure \\ref{fig:kamala} (e) shows a low and fairly steady size of the user mention activity in relation to the hashtag activity on the same topic, and these two indicators are not strongly correlated. Further, Figure \\ref{fig:kamala} (f) shows that the mean and standard deviation of node degree of hashtag activity are fairly close, and the average degree of user co-mention (two users mentioned in the same tweet) graph is relatively steady over the period of observation -- showing low propagativeness. In contrast, Row 1 and Row 2 show sharper peaks. But the curve in Figure \\ref{fig:kamala} (c) declines and has low, uncorrelated user mention activity. Hence, for this topic although there is a lot of discussion (leading to high navigability edges), the propagativeness is quite low. In comparison, Figure \\ref{fig:kamala} (a) shows a strong peak and a stronger correlation be the two curves indicating more propagativeness. The higher standard deviation in the co-mention node degree over time (Figure \\ref{fig:kamala} (b)) also shows the making of more propagation around this topic compared to the others.\n\nWe capture propagativeness using current flow betweenness centrality \\citep{brandes2005centrality} which is based on Kirchoff's current laws. We combine this with the average neighbor degree of the nodes of $S$ to measure the spreading propensity of $S$. The current flow betweenness centrality is the portion of all-pairs shortest paths that pass through a node, and the average neighbor degree is the average degree of the neighborhood of each node. If a subgraph has higher current flow betweenness centrality plus a higher average neighbor degree, the network should have faster communicability.  Let $\\alpha_{ij}$ be the number of shortest paths from node $i$ to $j$ and $\\alpha_{ij}(n)$ is the number of paths passes through the node $n$. Then the current flow betweenness centrality: $$C_{nb}(n)= \\sum_{(i,j)\\in V} \\frac{\\alpha_{ij}(n)}{\\alpha_{ij}}$$\n\nSuppose the distribution of the current flow betweenness centrality of two subgraphs $A$ and $B$ is $p_1$ and $p_2$ respectively, and distribution of the reference graph is $p_t$. Also the distribution of the $\\beta_{n}$, the average neighbor degree of the node $n$, for the subgraph $A$ and $B$ is $\\gamma_1$ and $\\gamma_1$ respectively, and the reference distribution is $\\gamma_t$. If the condition\n    $$\\Delta(p_t, p_1) * \\Delta(\\gamma_t, \\gamma_1) < \\Delta(p_t, p_2) *  \\Delta(\\gamma_t, \\gamma_2)$$\nholds, we can conclude that subgraph $B$ is a faster propagating network than subgraph $A$. This measure is of interest in a social media based on the observation that misinformation\/disinformation propagation groups either try to increase the average neighbor degree by adding fake nodes or try to involve influential nodes with high edge centrality to propagate the message faster \\citep{besel2018full}.  \n\n\\medskip\n    \\begin{figure*}[t]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{ados-kamal.png}\n\n        \\caption{\\#ADOS and Black Lives Matter }\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{ados-kamal-deg.png}\n       \n       \\caption{AVG Degree Distributions and Std Dev }\n    \\end{minipage}\n\n     \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{insurrection-kamal.png}\n        \\caption{Insurrection and Capitol Attack}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{insurrection-kamal-deg.png}\n       \n      \\caption{AVG Degree Distributions and Std Dev}\n    \\end{minipage}\n        \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5cm]{stimulus-kamal.png}\n          \\caption{Socioeconomic Issues During COVID-19 }\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5cm]{stimulus-kamal-deg.png}\n      \\caption{AVG Degree Distributions and Std Dev}\n    \\end{minipage}\n    \\caption{Various Metrics for Daily Partitioned Hashtag co-occurrences and User mentioned Graph from the Political Personalty Dataset. $Q_0$ retrieves tweets where Kamala Harris is mentioned in a hashtag, text body or user mention.}\n    \\label{fig:kamala}\n\\end{figure*}\n\\noindent    \\textbf{Subgroups within a Candidate Subgraph:} \n   \n    The purpose of the last metric is to determine whether a candidate subgraph identified using the previous measures need to be further decomposed into smaller subgraphs. We use subgraph centrality \\citep{estrada2005subgraph} and coreness of nodes as our metrics. \n   \n   \n   \n    The subgraph centrality measures the number of subgraphs a vertex participates in, and the core number of a node is the largest value $k$ of a $k$-core containing that node. So a subgraph for which the core number and subgraph centrality distributions are right-skewed compared to the background subgraph are (i) either split around high-coreness nodes, or (ii) reported to the user as a mixture of  diverse topics. \nThe node grouping, per-group subgraph generation and candidate subgraph identification process is presented in Algorithm \\ref{alg:graph-metrics}. In the algorithm, function \\textit{cut2bin} extends the cut function, which compares the histograms of the two distributions whose domains (X-values) must overlap, and produces equi-width bins to ensure that two histograms (i.e., frequency distributions) have compatible bins.\n\n\\subsection{The Testing Process}\n\\label{sec:testing}\n\n\\begin{algorithm}\n\\scriptsize\n\\caption{Graph Construction Algorithm}\n\\label{alg:graph-metrics}\n\\SetKwProg{ComputeMetrics}{Function \\emph{ComputeMetrics}}{}{end}\nINPUT : $Q_{out}$ Output of the query, $L$ Graph construction rules, $gv$ grouping variable, $th_{size}$ is the minimum size of the subgraph\\;\n\\SetKwProg{gmetrics}{Function \\emph{gmetrics}}{}{end}\n\\SetKwProg{CompareHistograms}{Function \\emph{CompareHistograms}}{}{end}\n\\gmetrics{($Q_{out}$, $L$, $groupVar$)}{\nG[]$\\leftarrow$ ConstructGraph($Q_{out}$, $L$)\\;\n$T \\leftarrow$ []\\;\n\\For{$g \\in G $}{\n     $t_{\\alpha} \\leftarrow$ ComputeMetrics(g)\\; \n     $T.push(t_{alpha})$\\;\n    \n    }\nreturn $T$\n}\n\\ComputeMetrics{(Graph g)}{\n$m\\leftarrow[]$\\;\n$m.push(eigenVectorCentrality(g))$\\;\n.........\n$m.push(coreNumber(g))$\\;\nreturn $m$\n}\n\\CompareHistograms{(List $t_{1}$, List $x_{2}$)}{\n$s_g \\leftarrow cut2bin(x_2, bin_{edges})$\\;\n$bin_{edges} \\leftarrow$ getBinEdges($x_{2}$)\\;\n$t_g \\leftarrow cut2bin(t_1, bin_{edges})$\\;\n\n$\\beta_{js} \\leftarrow distance.jensenShannon(t_g, s_g)$\\;\n$h_t \\leftarrow histogram(t_g, s_g,bin_{edges} )$\\;\nreturn $\\beta_{js}, h_t, bin_{edges}$\\;\n}\n\\end{algorithm}\n\n\n\n\n\n\n\\begin{algorithm}\n \\scriptsize\n\\caption{Graph Discovery Algorithm}\n\\label{alg:discovery-algo}\n\\SetKwProg{discover}{Function \\emph{discover}}{}{end}\n\\KwIn{ Set of all subgraphs divergence $\\sigma$}\n\\KwOut{Feature vectors $v_1$, $v_2$, $v_3$, List for re-partition recommendations $l$}\n$ev$ : eigenvector centrality\\;\n$ec$ : edge current flow betweenness centrality\\;\n$nc$ : current flow betweenness centrality\\;\n$\\mu$ : core number\\;\n$z$ : average neighbor degree\\;\n\\discover{($\\sigma$)}{\n\\For{any two set of divergence from $\\sigma_1$ ans $\\sigma_2$}{\n\\If{$\\sigma_2(ev) > \\sigma_1(ev)$}{\n $v_1(\\sigma_2) = v_1(\\sigma_2) + 1$\\;\n \\If{$\\sigma_2(ec) > \\sigma_1(ec)$}{\n $v_2(\\sigma_2) = v_2(\\sigma_2) + 1$\\;\n \\If{($\\sigma_2(nc)+ \\sigma_2(\\mu)) > (\\sigma_1(ec) + \\sigma_2(\\mu)$)}{\n $v_3(\\sigma_2) = v_3(\\sigma_2) + 1$\\;\n  }\n  \\If{($\\sigma_2(sc)+ \\sigma_2(z)) > (\\sigma_1(sc) + \\sigma_2(z)$)}{\n $l(\\sigma_2) = 1$\\;\n  }\n }\n }\n }\n}\n\\end{algorithm} \nThe discovery algorithm's input is the list of divergence values of two candidate sets computed against the same reference graph. It produces four lists at the end. Each of the first three lists contains one specific factor of interestingness of the subgraph. The most interesting subgraph should present in all three vectors. \nIf the subgraph has many cores and is sufficiently dense, then the system considers the subgraph to be \\textit{uninterpretable} and sends it for re-partitioning. \nTherefore, the fourth list contains the subgraph that should partition again. Currently, our repartitioning strategy is to take subsets of the original keyword list provided by the user at the beginning of the discovery process to re-initiate the discovery process for the dense, uninterpretable subgraph.\\\\\n\n\\noindent The output of each metric produces a value for each participant node of  the input. However, to compare two different candidates, in terms of the metrics mentioned above,  we need to convert them to comparable histograms by applying a binning function depending on the data type of the grouping function. \n\\\\\n\\noindent \\textit{Bin Formation (cut2bin):} Cut is a conventional operator (available with R, Matlab, Pandas etc. ) segments and sorts data values into bins. The cut2bin is an extension of a standard cut function, which compares the histograms of the two distributions whose domains (X-values) must overlap. The cut function accepts as input a set of set of node property values (e.g., the centrality metrics), and optionally a set of edge boundaries for the bins. It returns the histograms of distribution. Using the cut, first, we produce $n$ equi-width bins from the distribution with the narrower domain. Then we extract bin edges from the result and use it as the input bin edges to create the wider distribution`s cut.  This enforces the histograms to be compatible. In case one of the distribution is known to be a reference distribution (distribution from the background graph) against which the second distribution is compared, we use the reference distribution for equi-width binning and bin the second distribution relative to the first.\n\\\\\n\\noindent The $CompareHistograms$ function uses the \\textit{cut2bin} function to produce the histograms, and then computes the JS Divergence on the comparable histograms. The $CompareHistograms$ function returns the set of divergence values for each metric of a subgraph, which is the input of the discovery algorithm. The function requires the user to specify which of the compared graphs should be considered as a reference -- this is required to ensure that our method is scalable for large background graphs (which are typically much larger than the interesting subgraphs). If the background graph is very large, we take several random subgraphs from this graph to ensure they are representative before the actual comparisons are conducted. To this end, we adopt the well-known random walk strategy.\n \n\\noindent In the algorithm $v_1$, $v_2$ and $v_3$ are the three vectors to store the interestingness factors of the subgraphs, and $l$ is the list for repartitioning.  For two subgraphs, if one of them qualified for $v_1$ means, the subgraph contains higher centrality than the other. In that case, it increases the value of that qualified bit in the vector by one. Similarly, it increases the value of  $v_2$ by one,  if the same candidate has high navigability. Finally, it increases the $v_3$, if it has higher propagativeness.  The algorithm selects the top-$k$ scores of candidates from each vector, and marks them interesting. \n\n\n\n\n\n\\begin{table*}[]\n\\caption{Dataset Descriptions }\n\\label{tab:dataset}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}\n\\hline\n\\textbf{Data Set} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Total \\\\ Collection\\\\  Size\\end{tabular}} & \\textit{\\textbf{\\begin{tabular}[c]{@{}l@{}}Sub\\\\  Query\\end{tabular}}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Network \\\\ Type\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Total \\\\ Tweets\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Unique \\\\ nodes\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Unique \\\\ edges\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Self\\\\ Loop\\end{tabular}} & \\textbf{Density} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Avg\\\\ Degree\\end{tabular}} \\\\ \\hline\n\\textit{\\begin{tabular}[c]{@{}l@{}}Kamala \\\\ Harris\\end{tabular}} & 12469480 & \\textit{\\begin{tabular}[c]{@{}l@{}}Capitol \\\\ Attack\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 164397 & 1398 & 7801 & 16 & 0.0025 & 4.3 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user\\\\ co-mention\\end{tabular} & 164397 & 8012 & 48604 & 87 & 0.00012 & 3.19 \\\\ \\hline\n\\textit{} &  & \\textit{\\#ADOS} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 158419 & 3671 & 10738 & 29 & 0.0015 & 5.8 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 158419 & 30829 & 39865 & 49 & 8.3 & 2.5 \\\\ \\hline\n\\textit{} &  & \\textit{\\begin{tabular}[c]{@{}l@{}}Economic \\\\ Issues\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 36678 & 1278 & 1828 & 4 & 0.0022 & 2.8 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 36678 & 6971 & 11584 & 19 & 0.0004 & 3.4 \\\\ \\hline\n\\textit{Joe Biden} & 45258151 & \\textit{\\begin{tabular}[c]{@{}l@{}}Capitol \\\\ Attack\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 676898 & 7728 & 21422 & 50 & 0.00071 & 5.49 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 676898 & 82046 & 101646 & 130 & 3.0146 & 2.473 \\\\ \\hline\n\\textit{} &  & \\textit{\\#ADOS} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 183765 & 3007 & 11008 & 29 & 0.002 & 3.85 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 158419 & 29547 & 40932 & 56 & 9.3 & 2.7 \\\\ \\hline\n\\textit{} &  & \\textit{\\begin{tabular}[c]{@{}l@{}}Economic \\\\ Issues\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 138754 & 2961 & 5733 & 10 & 0.0013 & 3.87 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 138754 & 21417 & 19691 & 23 & 8.5 & 1.83 \\\\ \\hline\n\\textit{Vaccine} & 24172676 & \\textit{\\begin{tabular}[c]{@{}l@{}}Vaccine\\\\ Anti-vax\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 1000000 & 18809 & 24195 & 44 & 2.52 & 2.5 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 1000000 & 203211 & 41877 & 46 & 2.02 & 0.4 \\\\ \\hline\n\\textit{} &  & \\textit{Covid Test} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 1000000 & 26671 & 45378 & 69 & 0.00012 & 3.4 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 1000000 & 188761 & 83656 & 109 & 4.67 & 0.886 \\\\ \\hline\n &  & economy & \\begin{tabular}[c]{@{}l@{}}hashtag \\\\ co-occur\\end{tabular} & 917890 & 3002 & 4395 & 9 & 0.0009 & 2.9 \\\\ \\hline\n &  &  & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 917890 & 20590 & 8528 & 13 & 4.023 & 0.8 \\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\subsection{Data Sets}\n\\label{sec:datasets}\nData sets used for the experiments are tweets collected using the Twitter Streaming API using a set of domain-specific, hand-curated keywords. We used three data sets, all collected between the 1st and the 31st of January, 2021. The first two sets are for political personalities. The Kamala Harris data set was collected by using variants of Kamala Harris's name together with her Twitter handle and hashtags constructed from her name. The second data set was similarly constructed for Joe Biden. The third data set was collected during the COVID-19 pandemic. The keywords were selected based on popularly co-occurring terms from Google Trends. We selected the terms manually to ensure that they are related to the pandemic and vaccine related issues (and not, for example, partisan politics). Table \\ref{tab:dataset} presents a quantitative summary on the three data sets. We used a set of subqueries to find subsets from each of our datasets. These subqueries are temporal, and represent trending terms that stand for the emerging issues. For the first two datasets, we used three themes to construct the background graphs: a) capitol attack and insurrection, b) Black Lives Matter and ADOS movement, and c) American economic crisis. and recovery efforts. For the vaccine data set, we selected two subsets of posts, one for vaccine-related concerns, anti-vaccine movements, and related issues, and the second for covid testing and infection-related issues. The vaccine data set is larger, and the content is more diverse than the first two. \nAll the data sets are publicly available from The Awesome Lab \\footnote{https:\/\/code.awesome.sdsc.edu\/awsomelabpublic\/datasets\/int-springer-snam\/)}\n\nIn the experiments, we constructed two subgraphs from each subquery. The first is the \"hashtag co-occurrence\" graph, where each hashtag is a node, and  they are connected through an edge if they coexist in a tweet. The second is the \"user co-mention\" graph, where each user is a node, and  there is an edge between two nodes if a tweet mentioned them jointly. Intuitively, the hashtag co-occurrence subgraph captures topic prevalence and propagation, whereas the co-mention subgraph captures the tendency to influence and propagate messages to a larger audience. Our goal is to discover surprises (and lack thereof) in these two aspects for our data sets.\n\nWe note that the dataset chosen is from a month where the US experienced a major event in the form of the Capitol Attack, and a new administration was sworn in. This explains why the number of tweets in ``Capitol Attack'' subgraph is high for both politicians in this week, and not surprisingly it is also the most discussed topic as evidenced by the high average node degree. Therefore, this ``selection bias'' sets our expectation for subjective interestingness -- given the specific week we have chosen, this issue will dominate most social media conversations in the USA. We also observe the low ratio of the number of unique nodes to the number of tweets, signifying the high number of retweets, that signals a form of information propagation over the network. The propagativeness of the network during this eventful week is also evidenced by the fact that the unique node count of a co-mention network is almost 75\\% - 88\\% higher on average compared to the hashtag co-occur network of the same class. In Section \\ref{sec:analysis}, we show how our interestingness technique performs in the face of this dataset.\n\n\n\\begin{figure*}[t] \n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{kh-results-capitol-attack-ht-hist.png}\n        \\caption{Capitol Attack}\n           \\label{fig:hist-results-harris-capitol}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{kh-results-ados-ht-hist.png}\n        \\caption{\\#ADOS}\n           \\label{fig:hist-results-harris-ados}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{kh-results-economy-ht-hist.png}\n        \\caption{Economic issues}\n           \\label{fig:hist-results-harris}\n    \\end{minipage}%\n\n     \\caption{Top Hashtag Distributions of the Kamala Harris Data set }\n     \\label{fig:hist-results-harris}\n\\end{figure*}\n\\begin{figure*}[t] \n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-biden-capitol-attack-ht-hist.png}\n        \\caption{Capitol Attack}\n           \\label{fig:hist-results-biden-capitol}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-biden-ados-ht-hist.png}\n        \\caption{\\#ADOS Issues}\n           \\label{fig:hist-results-biden-ados}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-biden-economy-ht-hist.png}\n        \\caption{Economic issues}\n           \\label{fig:hist-results-biden-eco}\n    \\end{minipage}%\n\n     \\caption{Top Hashtag Distributions of the Joe Biden set }\n     \\label{fig:hist-results-biden}\n\\end{figure*} \n\\begin{figure*}[t]\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-covid-result-vax-anti-vax.png}\n        \\caption{Vaccine anti-vaccine Issues}\n           \\label{fig:hist-results-vac-vac-anti-vax}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-covid-result-covid-test.png}\n        \\caption{COVID-19 Test }\n           \\label{fig:hist-results-vac-covid}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-covid-result-economy.png}\n        \\caption{Economic issues}\n           \\label{fig:hist-results-vac-eco}\n    \\end{minipage}%\n     \\caption{Top Hashtag Distributions of the Vaccine Data set }\n     \\label{fig:hist-results-vac}\n\\end{figure*} \n\\subsection{Experimental Setup}\n\\label{sec:setup}\nThe experimental setup has three steps a) data collection and archival storage, b)Indexing and storing data, and c) executing analytical pipelines. We used The AWESOME project's continuous tweet ingestion system that collects all the tweets through Twitter 1\\% REST API using a set of hand-picked keywords. We used the AWESOME Polysotre for indexing, storing, and search the data. For computation, we used the Nautilus facility of the Pacific Research Platform (PRP). Our hardware configurations are as follows. The Awesome server has 64 GB memory and 32 core processors, the Nautilus has 32 core, and 64 GB nodes. The data ingestion process required a large memory. Depending on the density of the data, this requirement varies. Similarly, centrality computation is a CPU bounded process. Performance optimizations that we implemented are outside the scope of this paper.   \n\\medskip\n\\begin{figure*}[t]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-egc-kamal-ht.png}\n\n        \\caption{Eigenvector Centrality Disparity Hashtag Network}\n          \\label{fig:result-pipline-harris-ht-egc}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-nav-kamal-ht.png}\n       \\caption{Topical Navigability Disparity Hashtag Network}\n        \\label{fig:result-pipline-harris-ht-nav}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-prop-kamal-ht.png}\n\n        \\caption{Propagativeness  Disparity Hashtag Network }\n           \\label{fig:result-pipline-harris-ht-prop}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-core-kamal-mention.png}\n       \\caption{Eigenvector Centrality Disparity co-mention Network}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-nav-kamal-mention.png}\n\n        \\caption{Topical Navigability Disparity co-mention Network}\n           \\label{fig:result-pipline-harris-co-mention-nav}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-prop-kamal-mention.png}\n       \\caption{Propagativeness  Disparity co-mention Network}\n        \\label{fig:result-pipline-harris-co-mention-prop}\n    \\end{minipage}\n    \\caption{Comparative studies of all sub-queries using the \"Kamala Harris\" data set.}\n     \\label{fig:result-pipline-harris}\n\\end{figure*}\n\\begin{figure*}[t]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{biden-egc-biden-ht.png}\n\n        \\caption{Eigenvector Centrality Disparity Hashtag Network}\n           \\label{fig:result-pipline-biden-ht-egc}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{biden-nav-biden.png}\n       \\caption{Topical Navigability Disparity Hashtag Network}\n        \\label{fig:result-pipline-biden-ht-nav}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{biden-prop-biden-ht.png}\n\n        \\caption{Propagativeness  Disparity Hashtag Network }\n           \\label{fig:result-pipline-biden-ht-prop}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{biden-egc-biden-mention.png}\n       \\caption{Eigenvector Centrality Disparity co-mention Network}\n       \\label{fig:result-pipline-biden-mention-egc}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{biden-nav-biden-mention.png}\n\n        \\caption{Topical Navigability Disparity co-mention Network}\n           \\label{fig:result-pipline-biden-mention-nav}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{biden-prop-biden-mention.png}\n       \\caption{Propagativeness  Disparity co-mention Network}\n      \\label{fig:result-pipline-biden-mention-prop}\n    \\end{minipage}\n   \\caption{Comparative studies of all sub-queries using the \"Joe Biden\" data set.}\n     \\label{fig:result-pipline-biden}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-egc-covid-ht.png}\n\n        \\caption{Eigenvector Centrality Disparity: \"Vaccine\" Hashtag Network}\n           \\label{fig:result-pipline-ht-egc}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-nav-covid-ht.png}\n       \\caption{Topical Navigability Disparity: Vaccine Hashtag Network}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-prop-covid-ht.png}\n        \\caption{Propagativeness Disparity: \"Vaccine\" Hashtag Network }\n           \\label{fig:result-pipline-ht-vax}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-egc-covid-mention.png}\n       \\caption{Eigenvector Centrality Disparity: \"Vaccine\" co-mention Network}\n        \\label{fig:result-pipline-mention-egc}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-nav-covid-mention.png}\n\n        \\caption{Topical Navigability Disparity: \"Vaccine\" co-mention Network}\n           \\label{fig:result-pipline-mention-nav}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-prop-covid-mention.png}\n       \\caption{Propagativeness Disparity: \"Vaccine\" co-mention Network}\n       \\label{fig:result-pipline-mention-prop}\n    \\end{minipage}\n    \\caption{Comparative studies of all sub-queries using the \"Vaccine\" data set.}\n     \\label{fig:result-pipline-vax}\n\\end{figure*}\n\\begin{figure*}\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-harris-ados.png}\n        \\caption{\\#ADOS}\n           \\label{fig:core-peri-kamal-ados}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-harris-capitol.png}\n        \\caption{Capitol attack}\n           \\label{fig:core-peri-kamal-cap}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-harrris-economy.png}\n        \\caption{Economic issues}\n           \\label{fig:core-peri-kamal-eco}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-harris-random.png}\n        \\caption{Random Graph}\n           \\label{fig:core-peri-kamal-random}\n    \\end{minipage}%\n    \\caption{Core and Periphery visualization of \"Kamala Harris\" Data set}\n    \\label{fig:core-peri-kamal}\n\\end{figure*}    \n\\begin{figure*}\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-bidon-ados.png}\n        \\caption{\\#ADOS}\n           \\label{fig:core-peri-biden-ados}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-biden-capitol.png}\n        \\caption{Capitol attack}\n           \\label{fig:core-peri-biden-capitol}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-biden-economy.png}\n        \\caption{Economic issues}\n            \\label{fig:core-peri-biden-eco}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-biden-random.png}\n        \\caption{Random Data}\n           \\label{fig:core-peri-biden-random}\n    \\end{minipage}%\n     \\caption{Core and Periphery visualization of \"Joe Biden\" Data set}\n      \\label{fig:core-peri-biden}\n\\end{figure*}    \n\\begin{figure*}\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-covid-vax-ht.png}\n        \\caption{Vaccine Issues}\n           \\label{fig:core-peri-vax-vax-anti-vax}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-covid-test-ht.png}\n        \\caption{COVID-19 Test }\n           \\label{fig:core-peri-vax-test}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-covid-economy.png}\n      \\centering\n        \\caption{Economic issues}\n           \\label{fig:core-peri-vax-eco}\n    \\end{minipage}%\n \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-covid-random.png}\n        \\caption{Random Graph}\n           \\label{fig:core-peri-vax-random}\n    \\end{minipage}%\n     \\centering\n     \\caption{Core and Periphery visualization of the Vaccine Data set}\n    \\label{fig:core-peri-vax}\n\\end{figure*}\n\\subsection{Result Analysis}\n\\label{sec:analysis}\n\\textbf{Kamala Harris Network}: Figure \\ref{fig:result-pipline-harris} represents Kamala Harris network. This network is interesting because even in the context of the Capitol Attack and the Senate approval of election results, it is dominated by \\#ADOS conversations, not by the other political issues including economic policies. An analysis of the three interesting measures for this network reveals the following.\nThe Eigenvector Centrality Disparity for the hashtag network (Figure \\ref{fig:result-pipline-harris-ht-egc}) shows that all the groups generated by the subqueries are equally distributed, while the random graph has a higher volume with similar trends. Hence, these three subqueries are equally important. However, there are a few spikes on \\#ADOS and \"Capitol Attack\" that indicates the possibility of interestingness. Figure \\ref{fig:result-pipline-harris-ht-nav} shows that most of the \"economic issues\" has a spike on lower centrality nodes, but it touches zero with the increased centrality. While \"Capitol attack\" and \"\\#ADOS\" has many more spikes in different centrality level. Hence, we conclude that this network is much more navigable for  \"Capitol attack\" and \"\\#ADOS\". However, the figure \\ref{fig:result-pipline-harris-ht-prop} represents Propagativeness Disparity, and it is clear from this picture that African American issues dominated the conversation here, any subtopic related to it including non-US issues like ``Tigray'' (on Ethiopian genocide) propagated quickly in this network.  \n\\\\\n\\noindent\n\\textbf{Biden Network}: While the predominance of \\#ADOS issues might still be expected for the Kamala Harris data set, we discovered it to be an ``dominant enough'' subgraph in the Joe Biden data set represented in figure \\ref{fig:result-pipline-biden}. The eigenvector centrality disparity shows that the three subgroups are equally dominated in this network. The \"Joe Biden\" data set represented in figure \\ref{fig:result-pipline-biden-ht-egc}. The eigenvector centrality disparity shows that the three subgroups are equally dominated in this network. However, the navigability of the network(figure \\ref{fig:result-pipline-biden-ht-nav}) also shows that it is navigable for all three subgraphs. It has two big spikes for \"economic issues\" and \"Capitol attack,\" plus many mid-sized spikes for \"\\#ADOS\". Interestingly, in the figure \\ref{fig:result-pipline-biden-ht-prop} the propagativeness shows that the network is strongly propagative with the economic issues and the \\#ADOS issue, which shows up both in the Capitol Attack and the ADOS subgroups. Interestingly, the \"Joe Biden\" data's co-mention network shows more propagativeness than the hashtag co-occur network, which indicates exploring the co-mention subgraph will be useful. We also note the occurrence of certain complete unexpected topics (e.g., COVIDIOT, AUSvIND -- Australia vs. India) within the ADOS group, while Economic Issues for Biden do not exhibit surprising results.\n\\\\\n\\noindent\n\\textbf{Vaccine Network} In the vacation network, we found that \"economic issues\" and \"covid tests\" are more propagative than  \"vaccine and anti-vaccine\" related topics (Figure \\ref{fig:result-pipline-vax}). The surprising result here is that the \"vaccine - anti-vaccine\" topics show a strong correlation with \"Economy\" in the other two charts. We observe that while the vaccine issues are navigable through the network, but this topic cluster is is not very propagative in the network.  In contrast, in the co-mention network, vaccine and anti-vaccine issues are both very navigable and strongly propagative. Further, the propagativeness in the co-mention network for the Covid-test shows many spikes at the different levels, which signifies for testing related issues, the network serves as a vehicle of message propagation and influencing. \n\n\\subsection{Result Validation}\n\\label{sec:validation}\n\nThere is not a lot of work in the literature on interesting subgraph finding. Additionally, there are no benchmark data sets on which our ``interestingness finding'' technique can be compared to. This prompted us to evaluate the results using a core-periphery analysis as indicated earlier in Figure \\ref{fig:sparse}. The idea is to demonstrate that the parts of the network claimed to be interesting stand out in comparison to the network of a random sample of comparable size from the background graph. These results are presented in Figures \\ref{fig:core-peri-kamal}, \\ref{fig:core-peri-biden}, \\ref{fig:core-peri-vax}. In each of these cases, we have shown a representative random graph in the rightmost subfigure to represent the background graph. To us, the large and dense core formation in Figure \\ref{fig:core-peri-kamal}(b) is an expected, non-surprising result. However, the lack of core formation on Figure \\ref{fig:core-peri-kamal}(c) is interesting because it shows while there was a sizeable network for economics related term for Kamala Harris, the topics never ``gelled'' into a core-forming conversation and showed little propagation. Figure \\ref{fig:core-peri-kamal}(a) is somewhat more interesting because the density of the periphery is far less strong than the random graph while the core has about the same density as the random graph. The core periphery separation is much more prominent in first three plots of Figure \\ref{fig:core-peri-biden}. Unlike the Kamala Harris random graph, Figure \\ref{fig:core-peri-biden}(d) shows that the random graph of this data set itself has a very large dense core and a moderately dense periphery. Among the three subfigures, the small (but lighter) core of Figure \\ref{fig:core-peri-biden}(b) has the maximum difference compared to the random graph, although we find \\ref{fig:core-peri-biden}(a) to be conceptually more interesting. For the vaccine data set, Figure \\ref{fig:core-peri-vax} (c)is closest to the random graph showing that most conversations around the topic touches economic issues, while the discussion on the vaccine itself (Figure \\ref{fig:core-peri-biden}(a)) and that of COVID-testing (Figure \\ref{fig:core-peri-biden}(b)) are more focused and propagative. \n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper, we presented a general technique to discover interesting subgraphs in social networks with the intent that social network researchers from different domains would find it as a viable research tool. The results obtained from the tool will help researchers to probe deeper into analyzing the underlying phenomena that leads to the surprising result discovered by the tool. While we used Twitter as our example data source, similar analysis can be performed on other social media, where the content features would be different. Further, in this paper, we have used a few centrality measures to compute divergence-based features -- but the system is designed to plug in other measures as well.  \n\\FloatBarrier\n\\begin{acknowledgements}\nThis work was partially supported by NSF Grants \\#1909875, \\#1738411. We also acknowledge SDSC cloud support, particularly Christine Kirkpatrick \\& Kevin Coakley, for generous help and support for collecting and managing our tweet collection.\n\\end{acknowledgements}\n\\bibliographystyle{spbasic}    \n\n\\section{Introduction}\n\\label{sec:intro}\nAn information system designed for analysis of social media must consider a common set properties that characterize all social media data.\n\\begin{itemize}[leftmargin=*]\n    \\item Information elements in social media are essentially \\textit{heterogeneous} in nature -- users, posts, images, external URL references, although related, all bear different kinds information. \n    \\item Most social information is \\textit{temporal} -- a timestamp is associated with user events like the creation or response on a post, as well as system events like user account creation, deactivation and deletion. The system should therefore allow both temporal as well as time-agnostic analyses.\n    \\item Information in social media evolves fast. In one study \\citep{zhu2013modeling}, it was shown that the number of users in a social media is a power function of time. More recently, \\citep{antonakaki2018utilizing} showed that Twitter's growth is supralinear and follows Lescovec's model of graph evolution \\citep{leskovec2007graph}. Therefore, an analyst may first have to perform \\textit{exploration tasks} on the data before figuring out their analysis plan. \n    \\item Social media has a significant textual content, sometimes with specific entity markers (e.g., mentions) and topic markers (e.g., hashtags). Therefore any information element derived from text (e.g., named entities, topics, sentiment scores) may also be used for analysis. To be practically useful, the system must accommodate semantic synonyms -- \\#KamalaHarris, @KamalaHarris and ``Kamala Harris'' refer to the same entity.\n    \\item Relationships between information items in social media data must capture both canonical relationships like (\\texttt{tweet-15 mentions user-392}) but a wide-variety of computed relationships over base entities (users, posts, $\\ldots$) and text-derived information (e.g., named entities). \n\\end{itemize}\nIt is also imperative that such an information must support three styles of analysis tasks \n\\begin{enumerate}\n    \\item \\textbf{Search}, where the user specifies content predicate without specifying the structure of the data. For example, seeking the number of tweets related to Kamala Harris should count tweets where she is the author, as well as tweets where any synonym of ``Kamala Haris'' is in the tweet text.\n    \\item \\textbf{Query}, where the user specifies query conditions based on the structure of the data. For example, tweets with \\texttt{create\\_date} between 9 and 9:30 am on January 6th, 2021, with \\texttt{text} containing the string ``Pence'' and that were \\texttt{favorited} at least 100 times during the same time period.\n    \\item \\textbf{Discovery,} where the user may or may not know the exact predicates on the data items to be retrieved, but can specify analytical operations (together with some post-filters) whose results will provide insights into the data. For example, we call a query like \\texttt{Perform \\textit{community detection} on all tweets on January 6, 2021 and return the users from the largest community} a discovery query.\n\\end{enumerate}\nIn general, a real-life analytics workload will freely combine these modalities as part of a user's information exploration process. \n\nIn this paper, we present a general-purpose graph-based  model for social media data and a subgraph discovery algorithm atop this data model. Physically, the data model is implemented on AWESOME \\citep{gupta:awesome:2016}, an analytical platform designed to enable large-scale social media analytics over continuously acquired data from social media APIs. The platform, developed as a polystore system natively supports relational, graph and document data, and hence enables a user to perform complex analysis that include arbitrary combinations of search, query and discovery operations. We use the term \\textbf{\\textit{query-driven discovery}} to reflect that the scenario where the user does not want to run the discovery algorithm on a large and continuously collected body of data; rather, the user knows a starting point that can be specified as an expressive query (illustrated later), and puts bounds on the discovery process so that it terminates within an acceptable time limit.\n\n\\medskip \n\n\\noindent \\textbf{Contributions.} This paper makes the following contributions. (a) It offers a new formulation for the subgraph interestingness problem for social media; (b) based on this formulation, it presents a discovery algorithm for social media; (c) it demonstrates the efficacy of the algorithm on multiple data sets.\n\n\\medskip\n\n\\noindent \\textbf{Organization of the paper.} The rest of the paper is organized as follows. Section \\ref{sec:related} describes the related research on interesting subgraph finding in as investigated by researchers in Knowledge Discovery, Information Management, as well Social Network Mining. Section \\ref{sec:prelim} presents the abstract data model over which the Discovery Algorithm operations and the basic definitions to establish the domain of discourse for the discovery process. Section \\ref{sec:generate} presents our method of generating candidate subgraphs that will be tested for interestingness. Section \\ref{sec:discovery} first presents our interestingness metrics and then the testing process based on these metrics. Section \\ref{sec:experiments} describes the experimental validation of our approach on multiple data sets. Section \\ref{sec:conclusion} presents concluding discussions.\n\n\\section{Related Work}\n\\label{sec:related}\nThe problem of finding ``interesting'' information in a data set is not new. \\citep{what-makes-patterns-interesting:1996} described that  an``interestingness measure'' can be ``objective'' or ``subjective''. A measure is ``objective'' when it is computed solely based on the properties of the data. In contrast, a ``subjective'' measure must take into account the user's perspective. They propose that (a) a pattern is interesting if it is \"surprising\" to the user (\\textit{unexpectedness}) and (b) a pattern is interesting if the user can act on it\nto his advantage (\\textit{actionability}). Of these criteria, actionability is hard to determine algorithmically; unexpectedness, on the other hand, can be viewed as the departure from the user's beliefs. For example, a user may believe that the 24-hour occurrence pattern of all hashtags are nearly identical. In this case, a discovery would be to find a set of hashtags and sample dates for which this belief is violated. \nFollowing \\citep{geng2006interestingness}, there are three possibilities regarding how a system is informed of a user's knowledge and beliefs: (a) the user provides a formal specification of his or her knowledge, and after obtaining the mining results, the system chooses which unexpected patterns to present to the user \\citep{BingLiu:1999}; (b) according to the user's interactive feedback, the system removes uninteresting patterns \\citep{not-interesting:1999}; and (c) the system applies the user's specifications as constraints during the mining process to narrow down the search space and provide fewer results. Our work roughly corresponds to the third strategy.\n\\begin{figure*}[t]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=6.5cm, height=6.5cm]{fig-1-core-peri.png}\n\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=6.5cm, height=6.5cm]{fig-1-networking-connection.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{.4\\textwidth}\n    \\centering\n        \\includegraphics[width=7.5cm, height=6.5cm]{fig-1-top-25-bar.png}\n    \\end{minipage}\n    \\begin{minipage}{.4\\textwidth}\n    \\centering\n        \\includegraphics[width=7.5cm, height=6.5cm]{degree-hist.png}\n    \\end{minipage}\n    \\caption{An example of an interesting subgraph. Tweets on Indian politics form a sparse center in a dense periphery (separately shown in the top right). The bottom two figures show the hashtag distribution of the center and the degree distribution of the center (log scale) respectively.}\n    \\label{fig:sparse}\n\\end{figure*}\nEarly research on finding interesting subgraphs focused primarily on finding interesting substructures. This body of research primarily found interestingness in two directions: (a) finding frequently occurring subgraphs in a collection of graphs (e.g., chemical structures) \\citep{kuramochi2001frequent, yan2003closegraph, thoma2010discriminative} and (b) finding regions of a large graph that have high edge density \\citep{lee2010survey, sariyuce2015finding, epasto2015efficient, wen2017efficient} compared to other regions in the graph. Note that while a dense region in the graph can definitely interesting, shows, the inverse situation where a sparsely connected region is surrounded by an otherwise dense periphery can be equally interesting for an application. \n\nWe illustrate the situation in Figure \\ref{fig:sparse}. The primary data set is a collection of tweets on COVID-19 vaccination, but this specific graph shows a sparse core on Indian politics that is loosely connected to nodes of an otherwise dense periphery on the primary topic. Standard network features like hashtag histogram and the node degree histograms do not reveal this substructure requiring us to explore new methods of discovery. \n\nA serious limitation of the above class of work is that the interestingness criteria does not take into account \\textit{node content} (resp. edge content) which may be present in a property graph data model \\citep{angles2018property} where nodes and edges have attributes. \\citep{bendimerad2019mining} presents several discovery algorithms for graphs with vertex attributes and edge attributes. They perform both structure-based graph clustering and subspace clustering of attributes to identify interesting (in their domain ``anomalous'') subgraphs.\n\nOn the ``subjective'' side of the interestingness problem, one approach considers interesting subgraphs as a subgraph matching problem \\citep{shan2019dynamic}. Their general idea is to compute all matching subgraphs that satisfy a user the query and then ranking the results based on the rarity and the likelihood of the associations among entities in the subgraphs. In contrast \\citep{adriaens2019subjectively} uses the notion of ``subjective interestingness'' which roughly corresponds to finding subgraphs whose connectivity properties (e.g., the average degree of a vertices) are distinctly different from an ``expected'' \\textit{background} graph. This approach uses a constrained optimization problem that maximizes an objective function over the \\textit{information content} ($IC$) and the \\textit{description length} ($DL$) of the desired subgraph pattern. \n\nOur work is conceptually most inspired by \\citep{bendimerad2019subj} that explores the subjective interestingness problem for attributed graphs. Their main contribution centers around CSEA (Cohesive Subgraph with Exceptional Attributes) patterns that inform the user that a given set of attributes has exceptional values throughout a set of vertices in the graph. The subjective interestingness is given by $$S(U,S) = \\frac{IC(U,S)}{DL(U)}$$ where $U$ is a subset of nodes and $S$ is a set of restrictions on the value domains of the attributes. The system models the prior beliefs of the user as the Maximum Entropy distribution subject to any stated prior, beliefs the user may hold about the data (e.g., the distribution of an attribute value). The information content $IC(U,S)$ of a CSEA pattern $(U,S)$ is  formalized as negative of the logarithm of the probability that the pattern is present under the background distribution.  The length of a description\nof $U$ is the intersection of all neighborhoods in a subset $X \\subseteq N (U)$,\nalong with the set of ``exceptions'', vertices are in the intersection but not part of\n$U$. However, we have a completely different, more database-centric formulation of the background and the user's beliefs.\n\n\\section{The Problem Setup}\n\\label{sec:prelim}\n\n\\subsection{Data Model}\n\\label{sec:dataModel}\nOur abstract model social media data takes the form of a \\textit{heterogeneous information network} (an information network with multiple types of nodes and edges), which we view as a temporal property graph $G$. Let $N$ be the node set and $E$ be the edge set of $G$. $N$ can be viewed as a disjoint union of different subsets (called \\textit{node types}) --  users $U$, posts $P$, topic markers (e.g., hastags) $H$, term vocabulary $V$ (the set of all terms appearing in a corpus), references (e.g., URLs) $R(\\tau)$, where $\\tau$ represents the type of resource (e.g., image, video, web site $\\ldots$). Each type of node have a different set of properties (attributes) $\\bar{A}(.)$. We denote the attributes of $U$ as $\\bar{A}(U) = a_1(U), a_2(U) \\ldots$ such that $a_i(U)$ is the $i$-the attribute of $U$. An attribute of a node type may be temporal -- a post $p \\in P$ may have temporal attribute called \\texttt{creationDate}. Edges in this network can be directional and have a single \\textit{edge type}. The following is a set of \\textit{base (but not exhaustive) edge types}:\n\\begin{itemize}[leftmargin=1em, label=\\scriptsize{$-$}]\n    \\item \\texttt{writes}: $U \\mapsto P$\n    \\item \\texttt{uses}: $P \\mapsto H$\n    \\item \\texttt{mentions}: $P \\mapsto U$ maps a post $p$ to a user $u$ if $u$ mentioned in $p$\n    \\item \\texttt{repostOf}: $P \\mapsto P$ maps a post $p_2$ to a post $p_1$ if $p_2$ is repost of $p_1$. This implies that $ts(p_2) < ts(p_1)$ where $ts$ is the timestamp attribute\n    \\item \\texttt{replyTo\/comment}: $P \\mapsto P$ maps a post $p_2$ to a post $p_1$ if $p_2$ is a reply to $p_1$. This implies that $ts(p_2) < ts(p_1)$ where $ts$ is the timestamp attribute\n    \\item \\texttt{contains}: $P \\mapsto V \\times \\mathcal{N}$ where $\\mathcal{N}$ is the set of natural numbers and represents the count of a token $v \\in V$ in a post $p \\in P$\n\\end{itemize}\n We realistically assume that the inverse of these mappings can be computed, i.e., if $v_1, v_2 \\in V$ are terms, we can perform a \\texttt{contains}$^{-1}(v_1 \\wedge \\neg v_2)$ operation to yield all posts that used $v_1$ but not $v_2$.\n \nThe AWESOME information system allows users construct \\textit{derived or computed edge types} depending on the specific discovery problem they want to solve. For example, they can construct a standard hashtag co-occurrence graph using a non-recursive aggregation rule in Datalog \\citep{consens1993low}:\n\\begin{equation*}\n    \\begin{aligned}\n        HC(h_1, h_2, count(p)) &\\longleftarrow \\\\\n        &p \\in P, h_1 \\in H, h_2 \\in H \\\\ \n        &uses(p, h_1), uses(p, h_2) \n    \\end{aligned}\n\\end{equation*}\nWe interpret $HC(h_1, h_2, count(p))$ as an edge between nodes $h_1$ and $h_2$ and $count(p)$ as an attribute of the edge. In our model, a computed edge has the form: $E_T(N_b, N_e, \\bar{B})$ where $E_T$ is the edge type, $N_b, N_e$ are the tail and head nodes of the edge, and $\\bar{B}$ designates a flat schema of edge properties. The number of such computed edges can be arbitrarily large and complex for different subgraph discovery problems. A more complex computed edge may look like: $UMUHD(u_1,u_2,mCount,d, h)$, where an edge from $u_1$ to $u_2$ is constructed if user $u_1$ creates a post that contains hashtag $h$, and mentions user $u_2$ a total number of $mCount$ times on day $d$. Note that in this case, hashtag $h$ is a property of edge type $UMUHD$ and is not a topic marker node of graph $G$. \n\n\\subsection{Specifying User Knowledge}\n\\label{sec:hquery}\nSince the discovery process is query-driven, the system has no \\textit{a priori} information about the user's interests, prior knowledge and expectations, if any, for a specific discovery task. Hence, given a data corpus, the user needs to provide this information through a set of parameterized queries. We call these queries ``heterogeneous'' because they can be placed on conditions on any arbitrary node (resp. edge) properties, network structure and text properties of the graph. \n\n\\medskip\n\n\\noindent \\textbf{User Interest Specification.} The discovery process starts with the specification of the user's universe of discourse identified with a query $Q_0$. We provide some illustrative examples of user interest with queries of increasing complexity.\n\n\\noindent \\textit{Example 1.}  ``All tweets related to COVID-19 between 06\/01\/2020 and 07\/31\/2020 that refer to Hydroxychloroquine''. In this specification, the condition ``related to COVID-19'' amounts to finding tweets containing any $k$ terms from a user-provided list, and the condition on ``Hydroxychloroquine'' is expressed as a fuzzy search.\n\\begin{figure}\n\\includegraphics[width=8.5cm, height=6.5cm]{hydroxyclq.png}\n\\vspace{-2cm}\n \\caption{Hydroxychloroquine and COVID related Hashtags}\n \\label{fig:hydroxi}\n\\end{figure}\nFigure \\ref{fig:hydroxi} shows the top hashtags related to this search -- note that the hashtag co-occurrence graph around COVID-19 and hyroxychloroquine includes ``FakeNewsMedia''.\n\n\\noindent \\textit{Example 2.} ``All tweets from users who mention \\texttt{Trump} in their user profile and \\texttt{Fauci} in at least $n$ of their tweets''. Notice that this query is about users with a certain behavioral pattern -- it captures \\textit{all} tweets from users who have a combination of specific profile features and tweet content.  \n\n\\noindent \\textit{Example 3.} ``All tweets from users $U_1$ whose tweets appear in hashtag-cooccurrence in the 3 neighborhood around \\texttt{\\#ados} is used together with all tweets of users $U_2$ who belong to the user-mention-user networks of these users (i.e., $U_1$) '', where \\texttt{\\#ados} refers to ``American Descendant of Slaves'', which represents an African American cause.\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=6.5cm]{kamala-ht-vol.png}\n\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=6.5cm]{kamala-ht.png}\n       \n    \\end{minipage}\n\n    \\caption{Two views of the \\texttt{\\#ados} cluster of hashtags }\n    \\label{fig:ados}\n\\end{figure*}\n\n\n\\noindent The end result of $Q_0$ is a collection of posts that we call the \\textit{initial post set} $P_0$. Using the posts in $P_0$, the system creates a background graph as follows.\n\n\\medskip\n\n\\noindent \\textbf{Initial Background Graph.} The initial background graph $G_0$ is the graph derived from $P_0$ over which the discovery process runs. However, to define the initial graph, we first develop the notion of a \\textit{conversation}.\n\n\\begin{definition}[\\textbf{Semantic Neighborhood}.] $N(p)$, the semantic neighborhood of a post $p$ is the graph connecting $p$ to instances of $U \\cup H \\cup P \\cup V$ that directly relates to $p$.\n\\end{definition}\n\n\\begin{definition}[\\textbf{Conversation Context}.] The conversation context $C(p)$ of post $p$ is a subgraph satisfying the following conditions:\n\\begin{enumerate}[leftmargin=*]\n    \\item $P_1$: The set of posts reachable to\/from $p$ along the relationships \\texttt{repostOf, replyTo} belong to $C(p)$. \n    \\item $P_2$: The union of posts in the semantic neighborhood of $P_1$ belong to $C(p)$.\n    \\item $E$: The induced subgraph of $P_1 \\cup P_2$ belong to $C(p)$\n    \\item Nothing else belongs to $C(p)$.\n\\end{enumerate}\n\\end{definition}\n\\noindent Clearly, we can assert that $C(p)$ is a connected graph and that $N(p)~ \\sqsubset_g~ C(p)$ where $\\sqsubset_g$ denotes a subgraph relationship.\n\n\\begin{definition}[\\textbf{Initial Background Graph}.] The initial background graph $G_0$ is a merger of all conversation contexts $C(p_i), p_i \\in P_0$, together with all computed edges induces the nodes of $\\cup_i C(p_i)$\n\\end{definition}\nThe initial background graph itself can be a gateway to finding interesting properties of the graph. To illustrate this based on the graph obtained from our Example 2. Figure \\ref{fig:ados} presents two views of the \\texttt{\\#ados} cluster of hashtags from January 2021. The left chart shows the time vs. count of the hashtags while the right chart shows the dominant hashtags of the same period in this cluster. The strong peak in the timeline, was due to an intense discussion, revealed by topic modeling, on the creation of an office on African American issues. The occurrence of this peak is interesting because most of the social media conversation in this time period was focused on the Capitol attack on January 6.\n\n\nGiven the $G_0$ graph,we discover subgraphs $S_i \\subset G_o$ whose content and structure are distinctly different that of $G_0$. However, unlike previous approaches, we apply a generate-and-test paradigm for discovery. The generate-step (Section \\ref{sec:generate}) uses a graph cube like \\citep{zhao2011graph} technique to generate candidate subgraphs that might be interesting and the test-step (Section \\ref{sec:testing}) computes if (a) the candidate is sufficiently distinct from  the $G'$, and (b) the collection of candidates are sufficiently distinct from each other.\n\n\\noindent \\textbf{Subgraph Interestingness.} For a subgraph $S_i$ to be considered as a candidate, it must satisfy the following conditions.\n\n\\noindent \\textbf{C1.} $S_i$ must be connected and should satisfy a size threshold $\\theta_n$, the minimal number of nodes.\n\n\\noindent \\textbf{C2.} Let $A_{ij}$ (resp. $B_{ik}$) be the set of \\textit{local} properties of node $j$ (resp. edge $k$) of subgraph $S_i$. A property is called ``local'' if it is not a network property like vertex degree. All nodes (resp. edges) of $S_i$ must satisfy some user-specified predicate $\\phi_N$ (resp. $\\phi_E$) specified over $A_{ij}$ (resp. $B_{ik}$). For example, a node predicate might require that all ``post'' nodes in the subgraph must have a re-post count of at least 300, while an edge predicate may require that all hashtag co-occurrence relationships must have a weight of at least 10. A user defined constraint on the candidate subgraph improves the interpretability of the result. Typical subjective interestingness techniques \\citep{van2016subjective, adriaens2019subjectively} use only structural features of the network and do not consider attribute-based constraints, which limits their pragmatic utility.\n\n\\noindent \\textbf{C3.} For each text-valued attribute $a$ of $A_{ij}$, let $C(a)$ be the collection of the values of $a$ over all nodes of $S_i$, and $\\mathcal{D}(C(a))$ is a textual diversity metric computed over $C(a)$. For $S_i$ to be interesting, it must have at least one attribute $a$ such that $\\mathcal{D}(C(a))$ does not have the usual power-law distribution expected in social networks. Zheng et al \\citep{zheng2019social} used \\textit{vocabulary diversity} and \\textit{topic diversity} as textual diversity measures.\n\n\n\\section{Candidate Subgraph Generation}\n\\label{sec:generate}\nSection \\ref{sec:hquery} describes the creation of the initial background graph $G_0$ that serves as the domain of discourse for discovery. Depending on the number of initial posts $P_0$ resulting form the initial query, the size of $G_0$ might be too large -- in this case the user can specify followup queries on $G_0$ to narrow down the scope of discovery. We call this narrowed-down graph of interest as $G'$ -- if no followup queries were used, $G' = G_0$. The next step is to generate some candidate subgraphs that will be tested for interestingness. \n\n\\noindent \\textbf{Node Grouping.} A node group is a subset of \\textit{nodes($G'$)} where all nodes in a group have some similar property. We generalize the \\textit{groupby} operation, commonly used in relational database systems, to heterogeneous information networks. To describe the generalization, let us assume $R(A, B, C, D, \\ldots)$ is a relation (table) with attributes $A, B, C, D, \\ldots$ A \\textit{groupby} operation takes as input (a) a subset of \\textit{grouping attributes} (e.g. $A, B$), (b) a \\textit{grouped attribute} (e.g., $C$) and (c) an \\textit{aggregation function} (e.g., \\textit{count}). The operation first computes each distinct cross-product value of the grouping attributes (in our example, $A \\times B$) and creates a list of all values of the grouped attribute corresponding to each distinct value of the grouping attributes, and then applies the aggregation function to the list. Thus, the result of the \\textit{groupby} operation is a single aggregated value for each distinct cross-product value of grouping attributes. \n\nTo apply this operation to a social network graph, we recognize that there are two distinct ways of defining the ``grouping-object''. \\\\\n(1) Node properties can be directly used just like in the relational case. For example, for tweets a grouping condition might be \\texttt{getDate} \\texttt{(Tweet.created\\_at)} $\\wedge$ \\texttt{bin(Tweet.favoriteCount, 100)}, where the \\texttt{getDate} function extracts the date of a tweet and the \\texttt{bin} function creates buckets of size 100 from the favorite count of each tweet. \\\\\n(2) The grouping-object is a subgraph pattern. For example, the subgraph pattern\\\\\n\\texttt{(:tweet\\{date\\})-[:uses]->(:hashtag\\{text\\})} \\hfill{(P1)}\\\\\nstates that all ''tweet'' nodes having the same posting date, together with every distinct hashtag text will be placed in a separate group. Notice that while (1) produces disjoint tweets, (2) produces a ``soft'' partitioning on the tweets and hashtags due to the many-to-many relationship between tweets and hashtags. \\\\\nIn either case, the result is a set of node groups, designated here as $N_i$. For example, the grouping pattern P1 expressed in a Cypher-like syntax \\citep{francis2018cypher} (implemented in the Neo4J graph data management system) states that all tweets having the same posting date, together with every distinct hashtag text will be placed in a separate group.\nNotice that this process produces a ``fuzzy'' partitioning on the tweets and hashtags due to the many-to-many relationship between tweets and hashtags. Hence, the same tweet node can belong to two different groups because it has multiple hashtags. Similarly, a hashtag node can belong to multiple groups because tweets from different dates may have used the same hashtag. While the grouping condition specification language can express more complex grouping conditions, in this paper, we will use simpler cases to highlight the efficacy of the discovery algorithm. We denote the node set in each group as $N_i$.\n\n\\noindent \\textbf{Graph Construction.} To complete the \\textit{groupby} operation, we also need to specify the aggregation function in addition to the grouping-object and the grouped-object. This function takes the form of a graph construction operation that constructs a subgraph $S_i$ by expanding on the node set $N_i$. Different expansion rules can be specified, leading to the formation of different graphs. Here we list three rules that we have found fairly useful in practice.\n\n\\noindent \\textbf{G1.} Identify all the \\texttt{tweet} nodes in $N_i$. Construct a \\textit{relaxed induced subgraph} of the \\texttt{tweet}-labeled nodes in $N_i$. The subgraph is induced because it only uses tweets contained within $N_i$, and it is \\textit{relaxed} because contains all nodes \\textit{directly associated} with these tweet nodes, such as author,  hashtags, URLs, and mentioned-users. \n\n\\noindent \\textbf{G2.} Construct a \\textit{mention network} from within the tweet nodes in $N_i$ -- the mention network initially connects all \\texttt{tweet} and \\texttt{user}-labeled nodes. Extend the network by including all nodes \\textit{directly associated} with these tweet nodes.\n\n\\noindent \\textbf{G3.} A third construction relaxes the grouping constraint. We first compute either \\textbf{G1} or \\textbf{G2}, and then extend the graph by including the first order neighborhood of mentioned users or hashtags. While this clearly breaks the initial group boundaries, a network thus constructed includes tweets of similar themes (through hashtags) or audience (through mentions).\n\n\\noindent \\textbf{Automated Group Generation.} In a practical setting, as shown in Section \\ref{sec:experiments}, the parameters for node grouping operation can be specified by a user, or it can be generated automatically. Automatic generation of grouping-objects is based on the considerations described below. To keep the autogeneration manageable, we will only consider single and two objects for attribute grouping and only a single edge for subgraph patterns.\n\\begin{itemize}[leftmargin=*]\n    \\item Since temporal shifts in social media themes and structure are almost always of interest, the posting timestamp is always a grouping variable. For our purposes, we set the granularity to a day by default, although a user can set it.\n    \\item The frequency of most nontemporal attributes (like hashtags) have a mixture distribution of double-pareto lognormal distribution and power law \\citep{gupta:osn:2020}, we will adopt the following strategy.\n    \\begin{itemize}[label={$\\circ$}]\n        \\item  Let $f(A)$ be distribution of attribute $A$, and $\\kappa(f(A))$ be the curvature of $f(A)$. If $A$ is a discrete variable, we find $a*$, the maximum curvature (elbow) point of $f(A)$ numerically \\citep{antunes2018knee}.\n        \\item  We compute $A'$, the values of attribute $A$ to the left of $a*$ for all attributes and choose the attribute where the cardinality of $A'$ is maximum. In other words, we choose attributes which have the highest number of pre-elbow values. \n    \\end{itemize}\n    \\item We adopt a similar strategy for subgraph patterns. If $T_1(a_i)\\stackrel{L}{\\longrightarrow}T_2(b_j)$ is an edge where $T_1, T_2$ are node labels, $a_i, b_j$ are node properties and $L$ is an edge label, then $a_i$ and $b_j$ will be selected based on the conditions above. Since the number of edge labels is fairly small in our social media data, we will evaluate the estimated cardinality of the edge for all such triples and select one with the lowest cardinality. \n\\end{itemize}\n\n\\section{The Discovery Process}\n\\label{sec:discovery}\n\n\\subsection{Measures of for Relative Interestingness}\n\\label{sec:interestingness}\nWe compute the interestingness of a subgraph $S$ in reference to a background graph $G_b$ (e.g., $G'$), and consists of a structural as well as a content component. We first discuss the structural component. To compare a subgraph $S_i$ with the background graph, we first compute a set of network properties $P_j$ (see below) for nodes (or edges) and then compute the frequency distribution $f(P_j(S_i))$ of these properties over all nodes (resp. edges) of (a) subgraphs $S_i$, and (b) the reference graph (e.g., $G'$). A distance between $f(P_j(S_i))$ and $f(P_j(G_b))$ is computed using Jensen\u2013Shannon divergence (JSD). In the following, we use $\\Delta(f_1,f_2)$ to refer to the JS-divergence of distributions $f_1$ and $f_2$. \n\\medskip\n\\noindent \\textbf{Eigenvector Centrality Disparity:} The testing process starts by identifying the distributions of nodes with high node centrality between the networks. While there is no shortage of centrality measures in the literature, we choose eigenvector centrality \\citep{das2018study} defined below, to represent the dominant nodes. Let $A = (a_{i,j})$ be the adjacency matrix of a graph. The eigenvector centrality $x_{i}$ of node $i$ is given by: $$x_i = \\frac{1}{\\lambda} \\sum_k a_{k,i} \\, x_k$$ where $\\lambda \\neq 0$ is a constant. \n    The rationale for this choice follows from earlier studies in \\citep{Bonacich2007-mx,Ruhnau2000-jy,Yan2014-dn}, who establish that since the eigenvector centrality can be seen as a weighted sum of direct and indirect connections, \n    it represents the true structure of the network more faithfully than other centrality measures. \n    Further, \\citep{Ruhnau2000-jy} proved that the eigenvector-centrality under the Euclidean norm can be transformed into node-centrality, a property not exhibited by other common measures.\n    Let the distributions of eigenvector centrality of subgraphs $A$ and $B$ be $\\beta_a$ and $\\beta_b$ respectively, and that of the background graph be $\\beta_t$, then \n    $|\\Delta_e(\\beta_t, \\beta_a)| > \\theta $ indicates that $A$ is sufficiently structurally distinct from $G_b$ \n    $|\\Delta_e(\\beta_t, \\beta_a)| > |\\Delta_e(\\beta_t, \\beta_b)|$ indicates that $A$ contains significantly more or significantly less influential nodes than $B$. \n\n\\medskip\n\\noindent \\textbf{Topical Navigability Disparity:} Navigability measures ease of flow. If subgraph $S$ is more navigable than subgraph $S'$, then there will be more traffic through $S$ compared to $S'$. However, the likelihood of seeing a higher flow through a subgraph depends not just on the structure of the network, but on extrinsic covariates like time and topic. So, a subgraph is interesting in terms of navigability if for some values of a covariate, its navigability measure is different from that of a background subgraph. \n\nInspired by its application in biology \\citep{seguin2018navigation}, traffic analysis \\citep{scellato2010traffic}, and network attack analysis \\citep{lekha2020central}, we use \\textit{edge betweenness centrality} \\citep{das2018study} as the generic (non-topic) measure of navigability.   Let $\\alpha_{ij}$ be the number of shortest paths from node i to j and $\\alpha_{ij}(k)$ is the number of paths passes through the edge $k$. Then the edge-betweenness centrality is $$C_{eb}(k)= \\sum_{(i,j)\\in V} \\frac{\\alpha_{ij}(k)}{\\alpha_{ij}}$$\nBy this definition, the edge betweenness centrality is the portion of all-pairs shortest paths that pass through an edge. Since edge betweenness centrality of edge $e$ measures the proportion of paths that passes through $e$, a subgraph $S$ with a higher proportion of high-valued edge betweenness centrality implies that $S$ may be more \\textit{navigable} than $G_b$ or another subgraph $S'$ of the graph, i.e., information propagation is higher through this subgraph compared to the whole background network, for that matter, any other subgraph of network having a lower proportion of nodes with high edge betweenness centrality. Let the distribution of the edge betweenness centrality of two subgraphs $A$ and $B$ are $c_1$ and $c_2$ respectively, and that of the reference graph is $c_0$. Then, $|\\Delta_b(c_0, c_1)| < |\\Delta_b(c_0, c_2)|$  means the second subgraph is more navigable than the first.\n\nTo associate navigability with topics, we detect topic clusters over the background graph and the subgraph being inspected. The exact method for topic cluster finding is independent of the use of topical navigability. In our setting, we have used topic  topic modeling and dense region detection in hashtag cooccurrence networks. For each topic cluster, we identify posts (within the subgraph) that belong to the cluster. If the number of posts is greater than a threshold, we compute the navigability disparity.\n\n\\medskip\n\n\\noindent  \\textbf{Propagativeness Disparity:} The concept of propagativeness builds on the concept of navigability. Propagativeness attempts to capture how strongly the network is spreading information through a navigable subgraph $S$. We illustrate the concept with a network constructed over tweets where a political personality (Senator Kamala Harris in this example) is mentioned in January 2021. The three rows in Figure \\ref{fig:kamala} show the network characteristics of the subregions of this graph, related, respectively to the themes of \\texttt{\\#ados} and ``black lives matter'' (Row 1), Captiol insurrection (Row 2) and Socioeconomic issues related to COVID-19 including stimulus funding ad business reopening (Row 3). In earlier work \\citep{zheng2019social}, we have shown that a well known propagation mechanism for tweets is to add user-mentions to improve the reach of a message - hence the user-mention subgraph is indicative of propagative activity. In Figure \\ref{fig:kamala}, we compare the hashtag activity (measured by the Hashtag subgraph) and the mention activity (size of the mention graph) in these three subgraphs. Figure \\ref{fig:kamala} (e) shows a low and fairly steady size of the user mention activity in relation to the hashtag activity on the same topic, and these two indicators are not strongly correlated. Further, Figure \\ref{fig:kamala} (f) shows that the mean and standard deviation of node degree of hashtag activity are fairly close, and the average degree of user co-mention (two users mentioned in the same tweet) graph is relatively steady over the period of observation -- showing low propagativeness. In contrast, Row 1 and Row 2 show sharper peaks. But the curve in Figure \\ref{fig:kamala} (c) declines and has low, uncorrelated user mention activity. Hence, for this topic although there is a lot of discussion (leading to high navigability edges), the propagativeness is quite low. In comparison, Figure \\ref{fig:kamala} (a) shows a strong peak and a stronger correlation be the two curves indicating more propagativeness. The higher standard deviation in the co-mention node degree over time (Figure \\ref{fig:kamala} (b)) also shows the making of more propagation around this topic compared to the others.\n\nWe capture propagativeness using current flow betweenness centrality \\citep{brandes2005centrality} which is based on Kirchoff's current laws. We combine this with the average neighbor degree of the nodes of $S$ to measure the spreading propensity of $S$. The current flow betweenness centrality is the portion of all-pairs shortest paths that pass through a node, and the average neighbor degree is the average degree of the neighborhood of each node. If a subgraph has higher current flow betweenness centrality plus a higher average neighbor degree, the network should have faster communicability.  Let $\\alpha_{ij}$ be the number of shortest paths from node $i$ to $j$ and $\\alpha_{ij}(n)$ is the number of paths passes through the node $n$. Then the current flow betweenness centrality: $$C_{nb}(n)= \\sum_{(i,j)\\in V} \\frac{\\alpha_{ij}(n)}{\\alpha_{ij}}$$\n\nSuppose the distribution of the current flow betweenness centrality of two subgraphs $A$ and $B$ is $p_1$ and $p_2$ respectively, and distribution of the reference graph is $p_t$. Also the distribution of the $\\beta_{n}$, the average neighbor degree of the node $n$, for the subgraph $A$ and $B$ is $\\gamma_1$ and $\\gamma_1$ respectively, and the reference distribution is $\\gamma_t$. If the condition\n    $$\\Delta(p_t, p_1) * \\Delta(\\gamma_t, \\gamma_1) < \\Delta(p_t, p_2) *  \\Delta(\\gamma_t, \\gamma_2)$$\nholds, we can conclude that subgraph $B$ is a faster propagating network than subgraph $A$. This measure is of interest in a social media based on the observation that misinformation\/disinformation propagation groups either try to increase the average neighbor degree by adding fake nodes or try to involve influential nodes with high edge centrality to propagate the message faster \\citep{besel2018full}.  \n\n\\medskip\n    \\begin{figure*}[t]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{ados-kamal.png}\n\n        \\caption{\\#ADOS and Black Lives Matter }\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{ados-kamal-deg.png}\n       \n       \\caption{AVG Degree Distributions and Std Dev }\n    \\end{minipage}\n\n     \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{insurrection-kamal.png}\n        \\caption{Insurrection and Capitol Attack}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{insurrection-kamal-deg.png}\n       \n      \\caption{AVG Degree Distributions and Std Dev}\n    \\end{minipage}\n        \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5cm]{stimulus-kamal.png}\n          \\caption{Socioeconomic Issues During COVID-19 }\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5cm]{stimulus-kamal-deg.png}\n      \\caption{AVG Degree Distributions and Std Dev}\n    \\end{minipage}\n    \\caption{Various Metrics for Daily Partitioned Hashtag co-occurrences and User mentioned Graph from the Political Personalty Dataset. $Q_0$ retrieves tweets where Kamala Harris is mentioned in a hashtag, text body or user mention.}\n    \\label{fig:kamala}\n\\end{figure*}\n\\noindent    \\textbf{Subgroups within a Candidate Subgraph:} \n   \n    The purpose of the last metric is to determine whether a candidate subgraph identified using the previous measures need to be further decomposed into smaller subgraphs. We use subgraph centrality \\citep{estrada2005subgraph} and coreness of nodes as our metrics. \n   \n   \n   \n    The subgraph centrality measures the number of subgraphs a vertex participates in, and the core number of a node is the largest value $k$ of a $k$-core containing that node. So a subgraph for which the core number and subgraph centrality distributions are right-skewed compared to the background subgraph are (i) either split around high-coreness nodes, or (ii) reported to the user as a mixture of  diverse topics. \nThe node grouping, per-group subgraph generation and candidate subgraph identification process is presented in Algorithm \\ref{alg:graph-metrics}. In the algorithm, function \\textit{cut2bin} extends the cut function, which compares the histograms of the two distributions whose domains (X-values) must overlap, and produces equi-width bins to ensure that two histograms (i.e., frequency distributions) have compatible bins.\n\n\\subsection{The Testing Process}\n\\label{sec:testing}\n\n\\begin{algorithm}\n\\scriptsize\n\\caption{Graph Construction Algorithm}\n\\label{alg:graph-metrics}\n\\SetKwProg{ComputeMetrics}{Function \\emph{ComputeMetrics}}{}{end}\nINPUT : $Q_{out}$ Output of the query, $L$ Graph construction rules, $gv$ grouping variable, $th_{size}$ is the minimum size of the subgraph\\;\n\\SetKwProg{gmetrics}{Function \\emph{gmetrics}}{}{end}\n\\SetKwProg{CompareHistograms}{Function \\emph{CompareHistograms}}{}{end}\n\\gmetrics{($Q_{out}$, $L$, $groupVar$)}{\nG[]$\\leftarrow$ ConstructGraph($Q_{out}$, $L$)\\;\n$T \\leftarrow$ []\\;\n\\For{$g \\in G $}{\n     $t_{\\alpha} \\leftarrow$ ComputeMetrics(g)\\; \n     $T.push(t_{alpha})$\\;\n    \n    }\nreturn $T$\n}\n\\ComputeMetrics{(Graph g)}{\n$m\\leftarrow[]$\\;\n$m.push(eigenVectorCentrality(g))$\\;\n.........\n$m.push(coreNumber(g))$\\;\nreturn $m$\n}\n\\CompareHistograms{(List $t_{1}$, List $x_{2}$)}{\n$s_g \\leftarrow cut2bin(x_2, bin_{edges})$\\;\n$bin_{edges} \\leftarrow$ getBinEdges($x_{2}$)\\;\n$t_g \\leftarrow cut2bin(t_1, bin_{edges})$\\;\n\n$\\beta_{js} \\leftarrow distance.jensenShannon(t_g, s_g)$\\;\n$h_t \\leftarrow histogram(t_g, s_g,bin_{edges} )$\\;\nreturn $\\beta_{js}, h_t, bin_{edges}$\\;\n}\n\\end{algorithm}\n\n\n\n\n\n\n\\begin{algorithm}\n \\scriptsize\n\\caption{Graph Discovery Algorithm}\n\\label{alg:discovery-algo}\n\\SetKwProg{discover}{Function \\emph{discover}}{}{end}\n\\KwIn{ Set of all subgraphs divergence $\\sigma$}\n\\KwOut{Feature vectors $v_1$, $v_2$, $v_3$, List for re-partition recommendations $l$}\n$ev$ : eigenvector centrality\\;\n$ec$ : edge current flow betweenness centrality\\;\n$nc$ : current flow betweenness centrality\\;\n$\\mu$ : core number\\;\n$z$ : average neighbor degree\\;\n\\discover{($\\sigma$)}{\n\\For{any two set of divergence from $\\sigma_1$ ans $\\sigma_2$}{\n\\If{$\\sigma_2(ev) > \\sigma_1(ev)$}{\n $v_1(\\sigma_2) = v_1(\\sigma_2) + 1$\\;\n \\If{$\\sigma_2(ec) > \\sigma_1(ec)$}{\n $v_2(\\sigma_2) = v_2(\\sigma_2) + 1$\\;\n \\If{($\\sigma_2(nc)+ \\sigma_2(\\mu)) > (\\sigma_1(ec) + \\sigma_2(\\mu)$)}{\n $v_3(\\sigma_2) = v_3(\\sigma_2) + 1$\\;\n  }\n  \\If{($\\sigma_2(sc)+ \\sigma_2(z)) > (\\sigma_1(sc) + \\sigma_2(z)$)}{\n $l(\\sigma_2) = 1$\\;\n  }\n }\n }\n }\n}\n\\end{algorithm} \nThe discovery algorithm's input is the list of divergence values of two candidate sets computed against the same reference graph. It produces four lists at the end. Each of the first three lists contains one specific factor of interestingness of the subgraph. The most interesting subgraph should present in all three vectors. \nIf the subgraph has many cores and is sufficiently dense, then the system considers the subgraph to be \\textit{uninterpretable} and sends it for re-partitioning. \nTherefore, the fourth list contains the subgraph that should partition again. Currently, our repartitioning strategy is to take subsets of the original keyword list provided by the user at the beginning of the discovery process to re-initiate the discovery process for the dense, uninterpretable subgraph.\\\\\n\n\\noindent The output of each metric produces a value for each participant node of  the input. However, to compare two different candidates, in terms of the metrics mentioned above,  we need to convert them to comparable histograms by applying a binning function depending on the data type of the grouping function. \n\\\\\n\\noindent \\textit{Bin Formation (cut2bin):} Cut is a conventional operator (available with R, Matlab, Pandas etc. ) segments and sorts data values into bins. The cut2bin is an extension of a standard cut function, which compares the histograms of the two distributions whose domains (X-values) must overlap. The cut function accepts as input a set of set of node property values (e.g., the centrality metrics), and optionally a set of edge boundaries for the bins. It returns the histograms of distribution. Using the cut, first, we produce $n$ equi-width bins from the distribution with the narrower domain. Then we extract bin edges from the result and use it as the input bin edges to create the wider distribution`s cut.  This enforces the histograms to be compatible. In case one of the distribution is known to be a reference distribution (distribution from the background graph) against which the second distribution is compared, we use the reference distribution for equi-width binning and bin the second distribution relative to the first.\n\\\\\n\\noindent The $CompareHistograms$ function uses the \\textit{cut2bin} function to produce the histograms, and then computes the JS Divergence on the comparable histograms. The $CompareHistograms$ function returns the set of divergence values for each metric of a subgraph, which is the input of the discovery algorithm. The function requires the user to specify which of the compared graphs should be considered as a reference -- this is required to ensure that our method is scalable for large background graphs (which are typically much larger than the interesting subgraphs). If the background graph is very large, we take several random subgraphs from this graph to ensure they are representative before the actual comparisons are conducted. To this end, we adopt the well-known random walk strategy.\n \n\\noindent In the algorithm $v_1$, $v_2$ and $v_3$ are the three vectors to store the interestingness factors of the subgraphs, and $l$ is the list for repartitioning.  For two subgraphs, if one of them qualified for $v_1$ means, the subgraph contains higher centrality than the other. In that case, it increases the value of that qualified bit in the vector by one. Similarly, it increases the value of  $v_2$ by one,  if the same candidate has high navigability. Finally, it increases the $v_3$, if it has higher propagativeness.  The algorithm selects the top-$k$ scores of candidates from each vector, and marks them interesting. \n\n\n\n\n\n\\begin{table*}[]\n\\caption{Dataset Descriptions }\n\\label{tab:dataset}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}\n\\hline\n\\textbf{Data Set} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Total \\\\ Collection\\\\  Size\\end{tabular}} & \\textit{\\textbf{\\begin{tabular}[c]{@{}l@{}}Sub\\\\  Query\\end{tabular}}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Network \\\\ Type\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Total \\\\ Tweets\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Unique \\\\ nodes\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Unique \\\\ edges\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Self\\\\ Loop\\end{tabular}} & \\textbf{Density} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Avg\\\\ Degree\\end{tabular}} \\\\ \\hline\n\\textit{\\begin{tabular}[c]{@{}l@{}}Kamala \\\\ Harris\\end{tabular}} & 12469480 & \\textit{\\begin{tabular}[c]{@{}l@{}}Capitol \\\\ Attack\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 164397 & 1398 & 7801 & 16 & 0.0025 & 4.3 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user\\\\ co-mention\\end{tabular} & 164397 & 8012 & 48604 & 87 & 0.00012 & 3.19 \\\\ \\hline\n\\textit{} &  & \\textit{\\#ADOS} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 158419 & 3671 & 10738 & 29 & 0.0015 & 5.8 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 158419 & 30829 & 39865 & 49 & 8.3 & 2.5 \\\\ \\hline\n\\textit{} &  & \\textit{\\begin{tabular}[c]{@{}l@{}}Economic \\\\ Issues\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 36678 & 1278 & 1828 & 4 & 0.0022 & 2.8 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 36678 & 6971 & 11584 & 19 & 0.0004 & 3.4 \\\\ \\hline\n\\textit{Joe Biden} & 45258151 & \\textit{\\begin{tabular}[c]{@{}l@{}}Capitol \\\\ Attack\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 676898 & 7728 & 21422 & 50 & 0.00071 & 5.49 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 676898 & 82046 & 101646 & 130 & 3.0146 & 2.473 \\\\ \\hline\n\\textit{} &  & \\textit{\\#ADOS} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 183765 & 3007 & 11008 & 29 & 0.002 & 3.85 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 158419 & 29547 & 40932 & 56 & 9.3 & 2.7 \\\\ \\hline\n\\textit{} &  & \\textit{\\begin{tabular}[c]{@{}l@{}}Economic \\\\ Issues\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 138754 & 2961 & 5733 & 10 & 0.0013 & 3.87 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 138754 & 21417 & 19691 & 23 & 8.5 & 1.83 \\\\ \\hline\n\\textit{Vaccine} & 24172676 & \\textit{\\begin{tabular}[c]{@{}l@{}}Vaccine\\\\ Anti-vax\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 1000000 & 18809 & 24195 & 44 & 2.52 & 2.5 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 1000000 & 203211 & 41877 & 46 & 2.02 & 0.4 \\\\ \\hline\n\\textit{} &  & \\textit{Covid Test} & \\begin{tabular}[c]{@{}l@{}}Hashtag \\\\ co-occur\\end{tabular} & 1000000 & 26671 & 45378 & 69 & 0.00012 & 3.4 \\\\ \\hline\n\\textit{} &  & \\textit{} & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 1000000 & 188761 & 83656 & 109 & 4.67 & 0.886 \\\\ \\hline\n &  & economy & \\begin{tabular}[c]{@{}l@{}}hashtag \\\\ co-occur\\end{tabular} & 917890 & 3002 & 4395 & 9 & 0.0009 & 2.9 \\\\ \\hline\n &  &  & \\begin{tabular}[c]{@{}l@{}}user \\\\ co-mention\\end{tabular} & 917890 & 20590 & 8528 & 13 & 4.023 & 0.8 \\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\subsection{Data Sets}\n\\label{sec:datasets}\nData sets used for the experiments are tweets collected using the Twitter Streaming API using a set of domain-specific, hand-curated keywords. We used three data sets, all collected between the 1st and the 31st of January, 2021. The first two sets are for political personalities. The Kamala Harris data set was collected by using variants of Kamala Harris's name together with her Twitter handle and hashtags constructed from her name. The second data set was similarly constructed for Joe Biden. The third data set was collected during the COVID-19 pandemic. The keywords were selected based on popularly co-occurring terms from Google Trends. We selected the terms manually to ensure that they are related to the pandemic and vaccine related issues (and not, for example, partisan politics). Table \\ref{tab:dataset} presents a quantitative summary on the three data sets. We used a set of subqueries to find subsets from each of our datasets. These subqueries are temporal, and represent trending terms that stand for the emerging issues. For the first two datasets, we used three themes to construct the background graphs: a) capitol attack and insurrection, b) Black Lives Matter and ADOS movement, and c) American economic crisis. and recovery efforts. For the vaccine data set, we selected two subsets of posts, one for vaccine-related concerns, anti-vaccine movements, and related issues, and the second for covid testing and infection-related issues. The vaccine data set is larger, and the content is more diverse than the first two. \nAll the data sets are publicly available from The Awesome Lab \\footnote{https:\/\/code.awesome.sdsc.edu\/awsomelabpublic\/datasets\/int-springer-snam\/)}\n\nIn the experiments, we constructed two subgraphs from each subquery. The first is the \"hashtag co-occurrence\" graph, where each hashtag is a node, and  they are connected through an edge if they coexist in a tweet. The second is the \"user co-mention\" graph, where each user is a node, and  there is an edge between two nodes if a tweet mentioned them jointly. Intuitively, the hashtag co-occurrence subgraph captures topic prevalence and propagation, whereas the co-mention subgraph captures the tendency to influence and propagate messages to a larger audience. Our goal is to discover surprises (and lack thereof) in these two aspects for our data sets.\n\nWe note that the dataset chosen is from a month where the US experienced a major event in the form of the Capitol Attack, and a new administration was sworn in. This explains why the number of tweets in ``Capitol Attack'' subgraph is high for both politicians in this week, and not surprisingly it is also the most discussed topic as evidenced by the high average node degree. Therefore, this ``selection bias'' sets our expectation for subjective interestingness -- given the specific week we have chosen, this issue will dominate most social media conversations in the USA. We also observe the low ratio of the number of unique nodes to the number of tweets, signifying the high number of retweets, that signals a form of information propagation over the network. The propagativeness of the network during this eventful week is also evidenced by the fact that the unique node count of a co-mention network is almost 75\\% - 88\\% higher on average compared to the hashtag co-occur network of the same class. In Section \\ref{sec:analysis}, we show how our interestingness technique performs in the face of this dataset.\n\n\n\\begin{figure*}[t] \n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{kh-results-capitol-attack-ht-hist.png}\n        \\caption{Capitol Attack}\n           \\label{fig:hist-results-harris-capitol}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{kh-results-ados-ht-hist.png}\n        \\caption{\\#ADOS}\n           \\label{fig:hist-results-harris-ados}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{kh-results-economy-ht-hist.png}\n        \\caption{Economic issues}\n           \\label{fig:hist-results-harris}\n    \\end{minipage}%\n\n     \\caption{Top Hashtag Distributions of the Kamala Harris Data set }\n     \\label{fig:hist-results-harris}\n\\end{figure*}\n\\begin{figure*}[t] \n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-biden-capitol-attack-ht-hist.png}\n        \\caption{Capitol Attack}\n           \\label{fig:hist-results-biden-capitol}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-biden-ados-ht-hist.png}\n        \\caption{\\#ADOS Issues}\n           \\label{fig:hist-results-biden-ados}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-biden-economy-ht-hist.png}\n        \\caption{Economic issues}\n           \\label{fig:hist-results-biden-eco}\n    \\end{minipage}%\n\n     \\caption{Top Hashtag Distributions of the Joe Biden set }\n     \\label{fig:hist-results-biden}\n\\end{figure*} \n\\begin{figure*}[t]\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-covid-result-vax-anti-vax.png}\n        \\caption{Vaccine anti-vaccine Issues}\n           \\label{fig:hist-results-vac-vac-anti-vax}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-covid-result-covid-test.png}\n        \\caption{COVID-19 Test }\n           \\label{fig:hist-results-vac-covid}\n    \\end{minipage}%\n    \\begin{minipage}{.33\\textwidth}\n      \\includegraphics[width=6.5cm, height=5.5cm]{figures-covid-result-economy.png}\n        \\caption{Economic issues}\n           \\label{fig:hist-results-vac-eco}\n    \\end{minipage}%\n     \\caption{Top Hashtag Distributions of the Vaccine Data set }\n     \\label{fig:hist-results-vac}\n\\end{figure*} \n\\subsection{Experimental Setup}\n\\label{sec:setup}\nThe experimental setup has three steps a) data collection and archival storage, b)Indexing and storing data, and c) executing analytical pipelines. We used The AWESOME project's continuous tweet ingestion system that collects all the tweets through Twitter 1\\% REST API using a set of hand-picked keywords. We used the AWESOME Polysotre for indexing, storing, and search the data. For computation, we used the Nautilus facility of the Pacific Research Platform (PRP). Our hardware configurations are as follows. The Awesome server has 64 GB memory and 32 core processors, the Nautilus has 32 core, and 64 GB nodes. The data ingestion process required a large memory. Depending on the density of the data, this requirement varies. Similarly, centrality computation is a CPU bounded process. Performance optimizations that we implemented are outside the scope of this paper.   \n\\medskip\n\\begin{figure*}[t]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-egc-kamal-ht.png}\n\n        \\caption{Eigenvector Centrality Disparity Hashtag Network}\n          \\label{fig:result-pipline-harris-ht-egc}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-nav-kamal-ht.png}\n       \\caption{Topical Navigability Disparity Hashtag Network}\n        \\label{fig:result-pipline-harris-ht-nav}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-prop-kamal-ht.png}\n\n        \\caption{Propagativeness  Disparity Hashtag Network }\n           \\label{fig:result-pipline-harris-ht-prop}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-core-kamal-mention.png}\n       \\caption{Eigenvector Centrality Disparity co-mention Network}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-nav-kamal-mention.png}\n\n        \\caption{Topical Navigability Disparity co-mention Network}\n           \\label{fig:result-pipline-harris-co-mention-nav}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{kh-results-prop-kamal-mention.png}\n       \\caption{Propagativeness  Disparity co-mention Network}\n        \\label{fig:result-pipline-harris-co-mention-prop}\n    \\end{minipage}\n    \\caption{Comparative studies of all sub-queries using the \"Kamala Harris\" data set.}\n     \\label{fig:result-pipline-harris}\n\\end{figure*}\n\\begin{figure*}[t]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{biden-egc-biden-ht.png}\n\n        \\caption{Eigenvector Centrality Disparity Hashtag Network}\n           \\label{fig:result-pipline-biden-ht-egc}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{biden-nav-biden.png}\n       \\caption{Topical Navigability Disparity Hashtag Network}\n        \\label{fig:result-pipline-biden-ht-nav}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{biden-prop-biden-ht.png}\n\n        \\caption{Propagativeness  Disparity Hashtag Network }\n           \\label{fig:result-pipline-biden-ht-prop}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{biden-egc-biden-mention.png}\n       \\caption{Eigenvector Centrality Disparity co-mention Network}\n       \\label{fig:result-pipline-biden-mention-egc}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{biden-nav-biden-mention.png}\n\n        \\caption{Topical Navigability Disparity co-mention Network}\n           \\label{fig:result-pipline-biden-mention-nav}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{biden-prop-biden-mention.png}\n       \\caption{Propagativeness  Disparity co-mention Network}\n      \\label{fig:result-pipline-biden-mention-prop}\n    \\end{minipage}\n   \\caption{Comparative studies of all sub-queries using the \"Joe Biden\" data set.}\n     \\label{fig:result-pipline-biden}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-egc-covid-ht.png}\n\n        \\caption{Eigenvector Centrality Disparity: \"Vaccine\" Hashtag Network}\n           \\label{fig:result-pipline-ht-egc}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-nav-covid-ht.png}\n       \\caption{Topical Navigability Disparity: Vaccine Hashtag Network}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-prop-covid-ht.png}\n        \\caption{Propagativeness Disparity: \"Vaccine\" Hashtag Network }\n           \\label{fig:result-pipline-ht-vax}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-egc-covid-mention.png}\n       \\caption{Eigenvector Centrality Disparity: \"Vaccine\" co-mention Network}\n        \\label{fig:result-pipline-mention-egc}\n    \\end{minipage}\n    \\begin{minipage}{.5\\textwidth}\n    \\centering\n      \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-nav-covid-mention.png}\n\n        \\caption{Topical Navigability Disparity: \"Vaccine\" co-mention Network}\n           \\label{fig:result-pipline-mention-nav}\n    \\end{minipage}%\n    \\begin{minipage}{.5\\textwidth}\n     \\centering\n     \\includegraphics[width=7.5cm, height=5.5cm]{figures-covid-result-prop-covid-mention.png}\n       \\caption{Propagativeness Disparity: \"Vaccine\" co-mention Network}\n       \\label{fig:result-pipline-mention-prop}\n    \\end{minipage}\n    \\caption{Comparative studies of all sub-queries using the \"Vaccine\" data set.}\n     \\label{fig:result-pipline-vax}\n\\end{figure*}\n\\begin{figure*}\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-harris-ados.png}\n        \\caption{\\#ADOS}\n           \\label{fig:core-peri-kamal-ados}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-harris-capitol.png}\n        \\caption{Capitol attack}\n           \\label{fig:core-peri-kamal-cap}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-harrris-economy.png}\n        \\caption{Economic issues}\n           \\label{fig:core-peri-kamal-eco}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-harris-random.png}\n        \\caption{Random Graph}\n           \\label{fig:core-peri-kamal-random}\n    \\end{minipage}%\n    \\caption{Core and Periphery visualization of \"Kamala Harris\" Data set}\n    \\label{fig:core-peri-kamal}\n\\end{figure*}    \n\\begin{figure*}\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-bidon-ados.png}\n        \\caption{\\#ADOS}\n           \\label{fig:core-peri-biden-ados}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-biden-capitol.png}\n        \\caption{Capitol attack}\n           \\label{fig:core-peri-biden-capitol}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-biden-economy.png}\n        \\caption{Economic issues}\n            \\label{fig:core-peri-biden-eco}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-biden-random.png}\n        \\caption{Random Data}\n           \\label{fig:core-peri-biden-random}\n    \\end{minipage}%\n     \\caption{Core and Periphery visualization of \"Joe Biden\" Data set}\n      \\label{fig:core-peri-biden}\n\\end{figure*}    \n\\begin{figure*}\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-covid-vax-ht.png}\n        \\caption{Vaccine Issues}\n           \\label{fig:core-peri-vax-vax-anti-vax}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-covid-test-ht.png}\n        \\caption{COVID-19 Test }\n           \\label{fig:core-peri-vax-test}\n    \\end{minipage}%\n    \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-covid-economy.png}\n      \\centering\n        \\caption{Economic issues}\n           \\label{fig:core-peri-vax-eco}\n    \\end{minipage}%\n \\begin{minipage}{.25\\textwidth}\n      \\includegraphics[width=5cm, height=3.5cm]{figures-gephi-core-covid-random.png}\n        \\caption{Random Graph}\n           \\label{fig:core-peri-vax-random}\n    \\end{minipage}%\n     \\centering\n     \\caption{Core and Periphery visualization of the Vaccine Data set}\n    \\label{fig:core-peri-vax}\n\\end{figure*}\n\\subsection{Result Analysis}\n\\label{sec:analysis}\n\\textbf{Kamala Harris Network}: Figure \\ref{fig:result-pipline-harris} represents Kamala Harris network. This network is interesting because even in the context of the Capitol Attack and the Senate approval of election results, it is dominated by \\#ADOS conversations, not by the other political issues including economic policies. An analysis of the three interesting measures for this network reveals the following.\nThe Eigenvector Centrality Disparity for the hashtag network (Figure \\ref{fig:result-pipline-harris-ht-egc}) shows that all the groups generated by the subqueries are equally distributed, while the random graph has a higher volume with similar trends. Hence, these three subqueries are equally important. However, there are a few spikes on \\#ADOS and \"Capitol Attack\" that indicates the possibility of interestingness. Figure \\ref{fig:result-pipline-harris-ht-nav} shows that most of the \"economic issues\" has a spike on lower centrality nodes, but it touches zero with the increased centrality. While \"Capitol attack\" and \"\\#ADOS\" has many more spikes in different centrality level. Hence, we conclude that this network is much more navigable for  \"Capitol attack\" and \"\\#ADOS\". However, the figure \\ref{fig:result-pipline-harris-ht-prop} represents Propagativeness Disparity, and it is clear from this picture that African American issues dominated the conversation here, any subtopic related to it including non-US issues like ``Tigray'' (on Ethiopian genocide) propagated quickly in this network.  \n\\\\\n\\noindent\n\\textbf{Biden Network}: While the predominance of \\#ADOS issues might still be expected for the Kamala Harris data set, we discovered it to be an ``dominant enough'' subgraph in the Joe Biden data set represented in figure \\ref{fig:result-pipline-biden}. The eigenvector centrality disparity shows that the three subgroups are equally dominated in this network. The \"Joe Biden\" data set represented in figure \\ref{fig:result-pipline-biden-ht-egc}. The eigenvector centrality disparity shows that the three subgroups are equally dominated in this network. However, the navigability of the network(figure \\ref{fig:result-pipline-biden-ht-nav}) also shows that it is navigable for all three subgraphs. It has two big spikes for \"economic issues\" and \"Capitol attack,\" plus many mid-sized spikes for \"\\#ADOS\". Interestingly, in the figure \\ref{fig:result-pipline-biden-ht-prop} the propagativeness shows that the network is strongly propagative with the economic issues and the \\#ADOS issue, which shows up both in the Capitol Attack and the ADOS subgroups. Interestingly, the \"Joe Biden\" data's co-mention network shows more propagativeness than the hashtag co-occur network, which indicates exploring the co-mention subgraph will be useful. We also note the occurrence of certain complete unexpected topics (e.g., COVIDIOT, AUSvIND -- Australia vs. India) within the ADOS group, while Economic Issues for Biden do not exhibit surprising results.\n\\\\\n\\noindent\n\\textbf{Vaccine Network} In the vacation network, we found that \"economic issues\" and \"covid tests\" are more propagative than  \"vaccine and anti-vaccine\" related topics (Figure \\ref{fig:result-pipline-vax}). The surprising result here is that the \"vaccine - anti-vaccine\" topics show a strong correlation with \"Economy\" in the other two charts. We observe that while the vaccine issues are navigable through the network, but this topic cluster is is not very propagative in the network.  In contrast, in the co-mention network, vaccine and anti-vaccine issues are both very navigable and strongly propagative. Further, the propagativeness in the co-mention network for the Covid-test shows many spikes at the different levels, which signifies for testing related issues, the network serves as a vehicle of message propagation and influencing. \n\n\\subsection{Result Validation}\n\\label{sec:validation}\n\nThere is not a lot of work in the literature on interesting subgraph finding. Additionally, there are no benchmark data sets on which our ``interestingness finding'' technique can be compared to. This prompted us to evaluate the results using a core-periphery analysis as indicated earlier in Figure \\ref{fig:sparse}. The idea is to demonstrate that the parts of the network claimed to be interesting stand out in comparison to the network of a random sample of comparable size from the background graph. These results are presented in Figures \\ref{fig:core-peri-kamal}, \\ref{fig:core-peri-biden}, \\ref{fig:core-peri-vax}. In each of these cases, we have shown a representative random graph in the rightmost subfigure to represent the background graph. To us, the large and dense core formation in Figure \\ref{fig:core-peri-kamal}(b) is an expected, non-surprising result. However, the lack of core formation on Figure \\ref{fig:core-peri-kamal}(c) is interesting because it shows while there was a sizeable network for economics related term for Kamala Harris, the topics never ``gelled'' into a core-forming conversation and showed little propagation. Figure \\ref{fig:core-peri-kamal}(a) is somewhat more interesting because the density of the periphery is far less strong than the random graph while the core has about the same density as the random graph. The core periphery separation is much more prominent in first three plots of Figure \\ref{fig:core-peri-biden}. Unlike the Kamala Harris random graph, Figure \\ref{fig:core-peri-biden}(d) shows that the random graph of this data set itself has a very large dense core and a moderately dense periphery. Among the three subfigures, the small (but lighter) core of Figure \\ref{fig:core-peri-biden}(b) has the maximum difference compared to the random graph, although we find \\ref{fig:core-peri-biden}(a) to be conceptually more interesting. For the vaccine data set, Figure \\ref{fig:core-peri-vax} (c)is closest to the random graph showing that most conversations around the topic touches economic issues, while the discussion on the vaccine itself (Figure \\ref{fig:core-peri-biden}(a)) and that of COVID-testing (Figure \\ref{fig:core-peri-biden}(b)) are more focused and propagative. \n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper, we presented a general technique to discover interesting subgraphs in social networks with the intent that social network researchers from different domains would find it as a viable research tool. The results obtained from the tool will help researchers to probe deeper into analyzing the underlying phenomena that leads to the surprising result discovered by the tool. While we used Twitter as our example data source, similar analysis can be performed on other social media, where the content features would be different. Further, in this paper, we have used a few centrality measures to compute divergence-based features -- but the system is designed to plug in other measures as well.  \n\\FloatBarrier\n\\begin{acknowledgements}\nThis work was partially supported by NSF Grants \\#1909875, \\#1738411. We also acknowledge SDSC cloud support, particularly Christine Kirkpatrick \\& Kevin Coakley, for generous help and support for collecting and managing our tweet collection.\n\\end{acknowledgements}\n\\bibliographystyle{spbasic}    \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure*}[t]\n\t\\centering\n\\includegraphics[width=150mm]{ky_1}\n\t\\vspace{-5mm}\n\t\\caption{Schematic diagram of a redox flow battery.}\n\t\\label{fig: sec1_1_1}\n\\end{figure*}\n\nOver the past years, there has been growing interest in renewable energy since the increase of carbon emission poses a great threat to the environment. Despite the significance of renewable energy, the intermittent characteristics of sources, such as solar, wind or water, is a fatal drawback for renewable energy, which leads to the increased uncertainty in the supply of electricity. To address this issue, one promising solution is to regulate the power delivery via energy storage technology. Among the energy storage systems, vanadium redox flow batteries (VRFBs) attract a lot of attention due to the advantageous features:  scalability, low cost and long cycle life \\cite{alotto2014redox}. However, achieving high performance in terms of power density is a critical issue for cost-effectiveness of VRFBs.\n\nThe polarization losses in VRFBs are mainly caused by ohmic, mass transfer and charge transfer losses \\cite{milshtein2017quantifying}. Several researchers contributed to reduce overpotentials via a new cell architecture  \\cite{aaron2012dramatic}, and modified electrode configuration and membrane \\cite{sun1992modification,wang2007investigation,chen2013optimizing}. For charge transfer losses, Li et al.~\\cite{li2013bismuth} proposed electrodes containing nanoparticles that can improve performance of VRFBs due to the acceleration of charge transfer compared to the conventional VRFBs. Li et al.~\\cite{li2011graphite} claimed that reduction of charge transfer losses can be achieved by adding graphite oxide to electrodes. \n\nFor given electrochemical conditions, the performance of VRFBs depends on mass transfer losses mainly, where mass transfer effect can be ameliorated by different flow fields \\cite{zhou2017critical}. \nXu et al.~\\cite{xu2013numerical} numerically investigated the performance of VRFBs with several different types of flow fields and found the VRFB with serpentine flow field achieved maximum power-based efficiency at the optimal flow rate. Studies based on experiments demonstrated that interdigitated flow fields can further improve the performance of VRFBs due to the enhanced mass transfer effect \\cite{tsushima2014efficient,darling2014influence,houser2016influence}. Although few types of flow field have been found to reduce mass transfer losses, it is still laborious for the design of flow field due to the complicated physical and chemical mechanisms in VRFBs.\n\nBased on physical principles and mathematical models, Bends{\\o}e and Kikuchi \\cite{bendsoe1988generating} proposed topology optimization that is a powerful approach to find optimal configurations. \nTopology optimization expresses a structural optimization problem as a material distribution problem in a given design domain and then derives promising configuration on the basis of mathematical programming \\cite{bendsoe2003topology}. \nOne of the attractive feature of topology optimization is that an innovative configuration can be automatically generated from a blank design domain without designer's intuition.\nDue to its high degree of design freedom, topology optimization has been applied to various structural optimization problems, e.g., stiffness maximization problems \\cite{bendsoe1988generating,bendsoe1989optimal}, eigenfrequency problems \\cite{diaz1992solutions,ma1995topological}, thermal problems \\cite{li1999shape,iga2009topology}, and electromagnetic problems \\cite{nomura2007structural,yamasaki2011level}; furthermore, applications to practical device designs have also been attracted attention in micro actuator design \\cite{sigmund2001design}, fuel cell design \\cite{iwai2011power,song20132d} and so on.\n\nFor flow field design problems, Borrvall and Petersson \\cite{borrvall2003topology} proposed a topology optimization method to minimize power dissipation in Stokes flow, and this has been expanded to laminar Navier-Stokes flow problems \\cite{gersborg2005topology,olesen2006high,kubo2017level} and turbulence problems \\cite{yoon2016topology,dilgen2018topology}.\nThe fluid topology optimization has been applied to multiphysics problems such as fluid-structure interaction problems \\cite{yoon2010topology,jenkins2015level}, forced convection problems \\cite{matsumori2013topology,yaji2015topology,yaji2018large}, natural convection problems \\cite{alexandersen2014topology,coffin2016level,alexandersen2016large} and  turbulent heat transfer problems \\cite{kontoleontos2013adjoint,dilgen2018density}.\n\n\nRecently, \nYaji et al. \\cite{yaji2018topology} proposed a topology optimization method for the design of flow fields in VRFBs. \nIn their approach, instead of the formula of practical electrochemical reactions, a simplified formula is introduced as a two-dimensional model.\nThey provided novel flow field configurations of a VRFB and clarified that the optimized configurations tend to be the type of the interdigitated flow field.\n\nAs a more comprehensive study, this paper aims to construct topology optimization for flow fields in VRFBs based on a three-dimensional model incorporating with electrochemical reaction kinetics. Referring to the models presented by several researchers \\cite{shah2008dynamic,you2009simple,ma2011three}, a three-dimensional numerical model of a negative electrode in a VRFB is introduced and the electrolyte flow is assumed as stationary and isothermal Stokes flow for simplification. \nWe demonstrate that the proposed approach enables the generation of a novel flow field configuration through the numerical example.\nTo confirm the performance of the topology optimized flow field, we investigate the mass transfer effect and overpotential of the topology optimized flow field in comparison with reference flow fields---parallel and interdigitated flow fields.\nIn addition, we discuss the power loss \\cite{blanc2010understanding,xu2013numerical} in terms of polarization loss and pumping power at different operating conditions.\n\nThe reminder of this paper is organized as follows.\nIn Section 2, we introduce the mathematical model and assumptions of a VRFB.\nIn Section 3, we formulate a topology optimization problem that aims to maximize the mass transfer effect of a three-dimensional flow field in the VRFB and construct the optimization algorithm based on the use of mathematical programming and the finite element method (FEM).\nIn Section 4, we provide numerical examples and demonstrate the usefulness of the proposed approach.\nFinally, Section 5 concludes this paper and summarizes the obtained results.\n\n\n\n\\section{Mathematical model}\n\\subsection{Model assumptions}\nA schematic diagram of a typical redox flow battery is shown in Fig.~1. \nThe positive and negative electrodes are separated by the ion exchange membrane, which only allows protons to penetrate. \nWe suppose that the electrodes compose of the carbon fiber electrode and flow channel.\nNote that the use of flow channel enables the reduction of pressure loss in comparison with the case of only using the carbon fiber electrode \\cite{xu2013numerical}.\nWhen the electrolyte stored in tanks circulates through the positive and negative electrode separately by pumps, the electric energy is released or stored by the electrochemical reactions in the electrodes. \nThe main reactions can be described as follows:\n\\begin{align}\n & \\text{Positive electrode: } \\text{VO}^{2+} + \\text{H}_{2}\\text{O} \\rightleftharpoons \\text{VO}^{+}_{2} + 2\\text{H}^{+} + \\text{e}^{-} \\\\\n & \\text{Negative electrode: } \\text{V}^{3+} + \\text{e} \\rightleftharpoons \\text{V}^{2+}\n\\end{align}\n\nFurthermore, only the negative electrode is considered in this work and some basic assumptions are used for simplification as follows \\cite{you2009simple}:\n\\begin{enumerate}[1.]\n\\item The electrolyte flow is treated as stationary, incompressible and isothermal Stokes flow.\n\\item The dilute-solution approximation is used in this numerical model.\n\\item The side reactions are neglected in electrochemical reactions.\n\\item The migration  phenomenon is ignored in the species transport process.\n\\end{enumerate}\n\n\\subsection{Governing equations}\nBased on the assumptions presented in the previous section, governing equations incorporated in the numerical model are introduced here. The electrolyte flow passing through the flow channel in VRFBs can be described by Stokes equation and continuity equation as follows:\n\\begin{align}\n   & -\\nabla p +\\mu\\nabla^{2}\\mathbf{u}= \\mathbf{0}, \\\\\n   & \\nabla\\cdot \\mathbf{u} = 0,\n\\end{align} \nwhere $\\mu$ is the viscosity of electrolyte flow, and $p(\\mathbf{x})$ and $\\mathbf{u}(\\mathbf{x})$ are the pressure and velocity at position $\\mathbf{x}$, respectively.\n\nIn addition to the flow channel, the porous electrode is also permeated with the electrolyte flow and the velocity in the electrode can be expressed by Darcy's law:\n\\begin{align}\n  & \\frac{\\mu}{K}\\mathbf{u} = -\\nabla p,\n\\end{align}\nwhere $K$ is the permeability coefficient, which can be described by the Kozeny-Carmen equation \\cite{tomadakis2005viscous} as follows:\n\\begin{align}\n  & K = \\frac{d^{2}_\\text{f}\\epsilon^{3}}{16K_{\\text{ck}}(1-\\epsilon)^{2}},\n\\end{align}\nwhere $d_{\\text{f}}$ is the fiber diameter, $\\epsilon$ is the porosity of electrodes,  $K_{\\text{ck}}$ is the Carman-Kozeny constant described by the characteristic of the fibrous material.\nIn addition, instead of the electrolyte flow described by Stokes equation and Darcy's law respectively, \nBrinkman equation that combines Stokes equation with Darcy's law can be used to describe the electrolyte flow in the mixture of different porous medium generally and is formulated as\n\\begin{align}\n   & -\\nabla p + \\mu\\nabla^{2}\\mathbf{u} +\\mathbf{F}= \\mathbf{0},\n   \\label{eq:br}\n\\end{align}\nwhere $\\mathbf{F}$ is the body force given by\n\\begin{align}\n\\mathbf{F} = - \\alpha \\mathbf{u},\n\\label{eq:f}\n\\end{align}\nwhere $\\alpha$ is the so-called inverse-permeability that is defined as $\\alpha=\\mu\/K$ in the porous medium, while $\\alpha=0$ in the pure fluid domain.\n\nWith the electrolyte flow containing vanadium species, the species transport needs to be considered in the numerical model, which can be expressed as follows:\n\\begin{align}\n   & \\mathbf{u}\\cdot\\nabla c_{i}-D^{\\text{eff}}_{i}\\nabla^{2}c_{i}=-s_{i},\n\\end{align} \nwhere $c_{i}$ is the concentration of vanadium species $i\\in \\{\\text{V}^{2+}, \\text{V}^{3+}$\\}, and $s_{i}$ is the source term of species $i$ due to electrochemical reactions, $s_{\\text{V}^{2+}}=j\/F$ and $s_{\\text{V}^{3+}}=-j\/F$, where $j$ and $F$ are the transfer current density and the Faraday constant, respectively. \nIn addition, $D^{\\text{eff}}_{i}$ is the effective diffusion coefficient of species $i$ and is given by the Bruggemann correction, as follows:\n\\begin{align}\nD^{\\text{eff}}_{i} = \\epsilon^{1.5}D_{i},\n\\end{align}\nwhere $D_i$ is the diffusion coefficient of species $i$.\nThe charges in a VRFB conserve since the charge entering the electrolyte is balanced by the charge leaving the electrode, where the expression is shown as follows:\n\\begin{align}\n   & \\nabla\\cdot \\mathbf{i}_\\text{e} + \\nabla\\cdot \\mathbf{i}_\\text{s} = 0,\n   \\label{eq:cc}\n\\end{align} \nwhere $\\mathbf{i}_\\text{e}$ is the ionic current density, and $\\mathbf{i}_\\text{s}$ is the electronic current density. However, when the charge flows from the electrolyte to the electrode, the electrochemical reactions should take place on the electrode surface, which must be also taken account into the charge conservation. \nWith the electrochemical reaction kinetics, Eq.~(\\ref{eq:cc}) can be rewritten as\n\\begin{align}\n   & \\nabla\\cdot \\mathbf{i}_\\text{e} = -\\nabla\\cdot \\mathbf{i}_\\text{s} = j.\n\\end{align}\nIn addition, since the electrolyte is assumed to be electrically neutral and the migration term is ignored, the total ionic current density can be further expressed in terms of electric potential in the electrolyte as follows:\n\\begin{align}\n   & \\mathbf{i}_\\text{e} = \\sum_{i}\\mathbf{i}_i = -\\kappa^{\\text{eff}}_{\\text{e}}\\nabla\\phi_{\\text{e}},\n   \\label{eq:ie}\n   \\\\\n   & \\kappa^{\\text{eff}}_{\\text{e}} =  \\frac{F^2}{RT}\\sum_{i}z^{2}_{i}D^{\\text{eff}}_{i}c_{i},\n   \\label{eq:ke}\n\\end{align} \nwhere  $\\mathbf{i}_i$ is the ionic current density of species $i$, $\\phi_\\text{e}$ is the electric potential of the electrolyte,  $\\kappa^{\\text{eff}}_\\text{e}$ is the effective conductivity of the electrolyte, $R$ is the gas constant, $T$ is the temperature, and $z_{i}$ is the valence.\nThe detailed derivation of Eqs.~(\\ref{eq:ie}) and (\\ref{eq:ke}) refers to the previous work by Shah et al.~\\cite{shah2008dynamic}.\n\nLikewise, the electronic current density can also be expressed in terms of electric potential in the electrode given by the Ohm's law, as follows:\n\\begin{align}\n   & \\mathbf{i}_\\text{s} = -\\sigma^{\\text{eff}}_{\\text{s}}\\nabla\\phi_{\\text{s}}, \\\\\n   & \\sigma^{\\text{eff}}_{\\text{s}} = (1-\\epsilon)^{1.5}\\sigma_{\\text{s}},\n\\end{align} \nwhere $\\sigma_\\text{s}$ is the conductivity of the solid material of the electrode, $\\sigma^\\text{eff}_\\text{s}$ is the effective conductivity of the electrode, and $\\phi_\\text{s}$ is the electric potential of the electrode.\nNote that the effective conductivity of the electrode is corrected by the Bruggemann correction.\n\nThe transfer current density $j$, which originates from electrochemical reactions, can be described using Butler-Volmer equation, as follows:\n\\begin{align}\n&j = i_{0}\\left[R^\\text{sb}_{\\text{V}^{3+}}\\exp\\left(-\\frac{\\alpha_\\text{c}F\\eta}{RT}\\right)-R^\\text{sb}_{\\text{V}^{2+}}\\exp\\left(\\frac{\\alpha_\\text{a}F\\eta}{RT}\\right)\\right],\\\\\n&i_{0} = aFk(c_{\\text{V}^{2+}})^{\\alpha_\\text{c}}(c_{\\text{V}^{3+}})^{\\alpha_\\text{a}},\n\\end{align}\nwhere $i_{0}$ is the exchange current density, $\\eta$ is the overpotential, $\\alpha_\\text{c}$ and $\\alpha_\\text{a}$ are the cathodic and anodic transfer coefficients, $k$ is the reaction rate constant, $a$ is the specific area of the electrode, and $R^\\text{sb}_{i}=c^\\text{s}_{i}\/c_{i}$ is the ratio of the surface concentration of species $i$ to the bulk concentration in the negative electrode, in which $c^\\text{s}_{i}$ is the species concentration at the surface of the negative electrode.\n\nThe overpotential in Butler-Volmer equation is the difference between the electrode potential and the electrolyte potential, which can be expressed as follows:\n\\begin{align}\n&\\eta = \\phi_\\text{s}-\\phi_\\text{e}-U,\n\\end{align}\nwhere $U$ is the open-circuit potential in the negative electrode, which can be estimated by Nernst equation as follows:\n\\begin{align}\n&U = U_{0}+\\frac{RT}{F}\\ln\\left(\\frac{c_{\\text{V}^{3+}}}{c_{\\text{V}^{2+}}}\\right),\n\\end{align}\nwhere $U_{0}$ is the equilibrium potential.\n\nThe species concentration at the electrode surface $c^\\text{s}_{i}$ differs from the bulk concentration $c_{i}$ owing to the electrochemical reactions taking place at the electrode surface and the conductivity difference between the electrode and electrolyte. \nAccording to the previous research \\cite{you2009simple}, $c^\\text{s}_{i}$ are given by\n\\begin{align}\n&c^\\text{s}_{\\text{V}^{2+}}=\\frac{\\overline{P}c_{\\text{V}^{3+}}+(1+\\overline{P})c_{\\text{V}^{2+}}}{1+\\overline{M}+\\overline{P}},\\\\\n&c^\\text{s}_{\\text{V}^{3+}}=\\frac{\\overline{M}c_{\\text{V}^{2+}}+(1+\\overline{M})c_{\\text{V}^{3+}}}{1+\\overline{M}+\\overline{P}},\n\\label{eq:c2s}\n\\end{align}\nwhere $\\overline{M}$ and $\\overline{P}$ are defined as follows:\n\\begin{align}\n&\\overline{M}=\\frac{k}{k_\\text{m}}(c_{\\text{V}^{2+}})^{\\alpha_\\text{c}-1}(c_{\\text{V}^{3+}})^{\\alpha_\\text{a}}\\exp\\left(\\frac{\\alpha_\\text{a}F\\eta}{RT}\\right),\\\\\n&\\overline{P}=\\frac{k}{k_\\text{m}}(c_{\\text{V}^{2+}})^{\\alpha_\\text{c}}(c_{\\text{V}^{3+}})^{\\alpha_\\text{a}-1}\\exp\\left(-\\frac{\\alpha_\\text{c}F\\eta}{RT}\\right),\n\\end{align}\nwhere $k_\\text{m}$ is the mass transfer coefficient, which can be estimated by the following equation \\cite{schmal1986mass}:\n\\begin{align}\n&k_\\text{m} = 1.6 \\times 10^{-4}|\\mathbf{u}|^{0.4}.\n\\end{align}\n\n\n\\subsection{Boundary conditions}\n\nIn this section, the boundary conditions used in the numerical model during topology optimization process are introduced. \nFigure \\ref{fig: sec2_3_1} shows the schematic diagram of the analysis domain and boundary settings.\nThe pressure conditions are imposed on the inlet and the outlet, and the no-slip condition is applied on the remaining outer boundaries. The expressions are shown below:\n\\begin{alignat}{2}\n&p = p_\\text{in}  &&\\ \\ \\text{on the inlet},\\\\\n&p = p_\\text{out} &&\\ \\ \\text{on the outlet},\\\\\n&\\mathbf{u} =\\mathbf{0} &&\\ \\ \\text{on the remaining boundaries},\n\\end{alignat}\nwhere $p_\\text{in}$ is the given pressure value on the inlet, and $p_\\text{out}$ is the given pressure value on the outlet.\n\n\\begin{figure}[t]\n\t\\centering\n\\includegraphics[width=75mm]{ky_2}\n\t\\vspace{-5mm}\n\t\\caption{Schematic diagram of the analysis domain and boundary settings.}\n\t\\label{fig: sec2_3_1}\n\\end{figure}\n\n\\begin{table*}[t]\n\\centering  \n\\caption{Parameter settings of the electrode.}\n\t\\vspace{-1mm}\n\\scalebox{0.8}{\n\\begin{tabular}[t]{lllll}\n\\hline\nParameter & Symbol & Value & Unit & Ref.\\\\\n\\hline\nPorosity   & $\\epsilon$   & 0.929 & - & \\cite{you2009simple} \\\\\nSpecific surface area   & $a$   & $1.62\\times10^{4}$ & m & \\cite{you2009simple} \\\\\nCarbon fiber diameter   & $d_\\text{f}$   & $1.76\\times10^{-5}$ & m & \\cite{you2009simple} \\\\\nElectronic conductivity of solid phase   & $\\sigma_\\text{s}$   & $1.0\\times10^3$ & $\\text{S m}^{-1}$ & \\cite{you2009simple} \\\\\nKozeny-Carman constant   & $K_\\text{ck}$   & $4.28$ & - & \\cite{you2009simple} \\\\\nLength   & $L$   & 0.1 & m & \\cite{xu2013numerical} \\\\\nWidth   & $W$   & 0.1 & m & \\cite{xu2013numerical} \\\\\nElectrode thickness   & $t_\\text{e}$   & $3.0\\times10^{-3}$ & m & \\cite{xu2013numerical} \\\\\n\\hline\n\\end{tabular}\n}\n\\label{table:sec3_3_1}\n\\end{table*}\n\\begin{table*}[t]\n\\centering  \n\\caption{Parameter settings of the electrolyte.}\n\t\\vspace{-1mm}\n\\scalebox{0.8}{\n\\begin{tabular}[t]{lllll}\n\\hline\nParameter & Symbol & Value & Unit & Ref.\\\\\n\\hline\nViscosity   & $\\mu$   & $4.928\\times10^{-3}$ & Pa s & \\cite{you2009simple} \\\\\nInitial vanadium $V^{2+}$concentration   & $c^\\text{in}_{\\text{V}^{2+}}$   & 750 & $\\text{mol m}^{-3}$ & \\cite{you2009simple} \\\\\nInitial vanadium $V^{3+}$concentration   & $c^\\text{in}_{\\text{V}^{3+}}$   & 750 & $\\text{mol m}^{-3}$ & \\cite{you2009simple} \\\\\n$V^{2+}$ diffusion coefficient   & $D_{\\text{V}^{2+}}$   & $2.4\\times10^{-4}$ & $\\text{m}^{2}$ $\\text{s}^{-1}$ & \\cite{ma2011three} \\\\\n$V^{3+}$ diffusion coefficient   & $D_{\\text{V}^{3+}}$   & $2.4\\times10^{-4}$ & $\\text{m}^{2}$ $\\text{s}^{-1}$ & \\cite{ma2011three} \\\\\nIonic conductivity of electrolyte   & $\\kappa_\\text{e}$   & 7.8 & $\\text{S m}^{-1}$ & Estimated \\\\\n\n\\hline\n\\end{tabular}\n}\n\\label{table:sec3_3_2}\n\\end{table*}\n\nFor species conservation, the given concentration of each species is applied on the inlet and the diffusive fluxes of each species are set to zero on the outlet. Besides, the no-flux condition is also applied on the remaining boundaries. The boundary conditions for species conservation are shown below:\n\\begin{alignat}{2}\n&c_{i} = c^\\text{in}_{i}  &&\\ \\ \\text{on the inlet},\\\\\n&-D^\\text{eff}_{i}\\nabla c_{i}\\cdot\\mathbf{n}=0 &&\\ \\ \\text{on the remaining boundaries}.\n\\label{eq:nof}\n\\end{alignat}\nFor charge conservation, the VRFB is assumed to be operated in galvanostatic situation. During charging process, the flux conditions in the negative electrode can be described as follows: \n\\begin{alignat}{2}\n&-\\sigma^\\text{eff}_\\text{s}\\nabla \\phi_\\text{s}\\cdot \\mathbf{n} = -{\\frac{I}{A}} &&\\ \\ \\text{on the wall 1},\\\\\n&-\\kappa^\\text{eff}_\\text{e}\\nabla \\phi_\\text{e}\\cdot \\mathbf{n} = {\\frac{I}{A}} &&\\ \\ \\text{on the wall 2},\\\\\n& \\phi_\\text{s} = 0 &&\\ \\ \\text{on the interface},\n\\end{alignat}\nwhere $I$ is the applied current, and $A$ is the electrode surface area.\nNote that the remaining boundary conditions for $\\phi_\\text{s}$ and $\\phi_\\text{e}$ are the no-flux conditions as with Eq.~(\\ref{eq:nof}).\n\n\\section{Topology optimization for flow fields}\n\n\\subsection{Concepts of topology optimization for fluid problems}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=150mm]{ky_3}\n\t\\vspace{-5mm}\n\\caption{Iteration history of topology optimized flow field expressed as $\\rho\\ge 0.5$.}\n\t\\label{fig: sec4_1_1}\n\\end{figure*}\n\n\\begin{table*}[t]\n\\centering  \n\\caption{Parameter settings of the electrochemical reaction model.}\n\t\\vspace{-1mm}\n\\scalebox{0.8}{\n\\begin{tabular}[t]{lllll}\n\\hline\nParameter & Symbol & Value & Unit & Ref.\\\\\n\\hline\nStandard reaction rate  & $k_\\text{c}$   & $1.7\\times10^{-7}$ & $\\text{m s}^{-1}$ & \\cite{you2009simple} \\\\\nCathodic transfer coefficient   & $\\alpha_\\text{c}$   & 0.5 & - & Assumed \\\\\nAnodic transfer coefficient   & $\\alpha_\\text{a}$   & 0.5 &- & Assumed \\\\\nEquilibrium    & $U_{0}$   & -0.255 & V & \\cite{ma2011three} \\\\\n\n\\hline\n\\end{tabular}\n}\n\\label{table:sec3_3_3}\n\\end{table*}\n\nTopology optimization aims to obtain the improved structural design in a specific domain with a given objective function and constraints. The main idea is to formulate a topology optimization problem as a material distribution problem, where the expression for material distribution in a fixed design domain $D$ is defined as follows \\cite{bendsoe2003topology}: \n\\begin{eqnarray}\n\\chi(\\mathbf{x})=\n\\begin{cases}\n1 &\\text{if }\\mathbf{x}\\in \\Omega,\\cr 0 &\\text{if }\\mathbf{x}\\in  D\\backslash\\Omega, \\end{cases}\n\\end{eqnarray}\nwhere $\\mathbf{x}$ is the position in $D$, and $\\Omega$ is the design domain in $D$. In this expression, $\\chi(\\mathbf{x}) = 1$ and $\\chi(\\mathbf{x}) = 0$ represent the material and void at $\\mathbf{x}$, respectively. \nSince the characteristic function $\\chi$ is a discontinuous function, topology optimization problems typically require relaxation techniques for numerical treatment.\nAs the popular and simple way for relaxing topology optimization problems, the density approach \\cite{bendsoe1989optimal} replaces the characteristic function with a continuous function, $0 \\le \\rho(\\mathbf{\\mathbf{x}}) \\le 1$, which is also used in this paper.\n\nTo determine which points in $D$ should be fluid or solid, based on the previous research dealing with fluid topology optimization \\cite{borrvall2003topology}, the body force in Eq.~(\\ref{eq:f}) is redefined using the fictitious body force, $\\mathbf{F}^{\\text{fic}}$, as follows:\n\\begin{align}\n   &\\mathbf{F}^{\\text{fic}}= -\\alpha^{\\text{fic}}_{\\rho} \\mathbf{u} \\quad\\text{with }\\ \\alpha^{\\text{fic}}_{\\rho} = \\frac{q(1-\\rho)}{\\rho+q}\\alpha^{\\text{fic}},\n   \\label{eq:f_fic}\n\\end{align}\nwhere $\\alpha^{\\text{fic}}$ is the fictitious inverse-permeability used for expressing the solid domain $D\\setminus\\Omega$ as with the previous work \\cite{borrvall2003topology}, and $q$ is a tuning parameter for controlling the convexity of $\\alpha^{\\text{fic}}_{\\rho}$. \nIn addition, $\\rho = 1$ represents the fluid domain with $\\alpha^{\\text{fic}}_{\\rho}=0$, and $\\rho = 0$ represents the solid domain with $\\alpha^{\\text{fic}}_{\\rho}=\\alpha^{\\text{fic}}\\gg 1$. In the solid domain, since the fictitious body force is large enough compared to the fluid domain, it is difficult for fluid to pass through the solid domain. Therefore, according to the sensitivity information of the objective function, the fictitious body force is determined at each point in $D$ so that the structural design of flow channel can be obtained. \nNote that the fictitious inverse permeability, $\\alpha^{\\text{fic}}$, is different from $\\alpha$ in Eq.~(\\ref{eq:br}).\nThat is, the former is used for expressing the solid domain in the fixed design domain $D$, whereas the latter is used for expressing the porous electrode that is the non-design domain. \nIn this study, $q$ and $\\alpha^\\text{fic}$ are set to $0.01$ and $5\\alpha$, respectively.\n\n\\subsection{Description of optimization problem}\n\nIn VRFBs, the mobility of ions in electrolyte is poor compared to the electrons, which means it is more difficult to reach the electrode surface for ions. \nWith the flow channel embedded in a VRFB, the mass transfer effect is improved so that the concentration of reactants at the electrode surface will also increase. Therefore, whether the mass transfer effect is improved can be estimated by the concentration of the reactants at the electrode surface. During charging process, the optimization problem can be defined as a maximization problem of average concentration of oxidized reactants at the electrode surface in a negative electrode. The expressions of this optimization problem are shown as follows:\n\\begin{align}\n&\\begin{array}{ll}\n\\displaystyle\\underset{\\rho}{\\text{maximize }}\\ F = \\int_{D}c^\\text{s}_{\\text{V}^{3+}}\\text{d}\\Omega\\bigg\/\\int_{D}\\text{d}\\Omega,\\\\\n\\text{subject to }\\ 0 \\le \\rho(\\mathbf{x}) \\le 1\\quad\\text{for }\\ \\forall\\mathbf{x}\\in D.\n\\end{array}\n\\label{eq:optp}\n\\end{align}\nNote that the detailed expressions of $c^\\text{s}_{\\text{V}^{3+}}$ can be found in Eq.~(\\ref{eq:c2s}).\n\n\\subsection{Numerical implementation}\n\nThe governing equations are solved by using the package COMSOL Multiphysics$^{\\textregistered}$, which is based on the FEM. The optimization algorithm is constructed on the basis of mathematical programming and is briefly enumerated as follows:\n\\begin{description}\n\\item[{\\it Step 1.}]The design variables and all of the parameters shown in Table \\ref{table:sec3_3_1}--\\ref{table:sec3_3_4} are initialized.\n\\item[{\\it Step 2.}]The objective function $F$ in (\\ref{eq:optp}) is evaluated by solving the governing equation via the FEM.\n\\item[{\\it Step 3.}] If the objective function is converged, the iteration will terminate. Otherwise, the sensitivities---gradient of objective function with respect to the design variables---are calculated.\n\\item[{\\it Step 4.}] The design variables are redistributed in the fixed design domain $D$ using sequential linear programming (SLP), and the iteration will return to the second step.\n\\end{description}\n\nWe utilize a partial differential equation (PDE)-based filter \\cite{kawamoto2011heaviside} for ensuring the smoothness of the design variables \\cite{yaji2018topology}. \nIn addition, we use the adjoint method that enables the derivation of the sensitivities without depending on the number of design variables. \nThe detailed concepts and formulation of the adjoint method can be seen in the literatures on structural optimization \\cite{haftka1992elements,bendsoe2003topology}.\n\n\\begin{figure*}[t]\n\t\\centering\n\\includegraphics[width=150mm]{ky_4}\n\t\\vspace{-5mm}\n\t\\caption{Comparison results for different flow fields: (a) Flow field configurations; (b) Distributions of vanadium species $c_{\\text{V}^{2+}}$ on middle plane of electrode; (c) Distributions of vanadium species $c_{\\text{V}^{3+}}$ on middle plane of electrode.}\n\t\\label{fig: sec4_2_2}\n\\end{figure*}\n\\begin{figure*}[t]\n\t\\centering\n\\includegraphics[width=150mm]{ky_5y}\n\t\\vspace{-5mm}\n\t\\caption{Values of objective and overpotential at variant pressure drop for different flow fields.}\n\t\\label{fig: sec4_2_1}\n\\end{figure*}\n\\begin{figure*}[t]\n\t\\centering\n\\includegraphics[width=150mm]{ky_6}\n\t\\vspace{-5mm}\n\t\\caption{Power loss with different flow fields at $\\epsilon = 0.929$.}\n\t\\label{fig: sec4_3_1}\n\\end{figure*}\n\\begin{figure}[t]\n\t\\centering\n\\includegraphics[width=75mm]{ky_7}\n\t\\vspace{-5mm}\n\t\\caption{Power loss with different flow fields at $\\epsilon = 0.68$ and $I = 10$~A.}\n\t\\label{fig: sec4_3_3}\n\\end{figure}\n\n\n\n\\section{Numerical examples}\n\n\\subsection{Topology optimized flow field}\n\\begin{table}[b]\n\\centering  \n\\caption{Operating parameter settings.}\n\t\\vspace{-1mm}\n\\scalebox{0.8}{\n\\begin{tabular}[t]{lllll}\n\\hline\nParameter & Symbol & Value & Unit & Ref.\\\\\n\\hline\nTemperature  & $T$   & $298$ & K & \\cite{you2009simple} \\\\\nInlet pressure   & $P_\\text{in}$   & $1.0\\times10^3$ & Pa & Assumed \\\\\nOutlet pressure   & $P_\\text{out}$   & 0 & Pa & Assumed \\\\\nApplied current    & $I$   & 4.0 & A & \\cite{xu2013numerical} \\\\\n\n\\hline\n\\end{tabular}\n}\n\\label{table:sec3_3_4}\n\\end{table}\n\\begin{table}[b]\n\\centering  \n\\caption{Pressure drop (Pa) in the different flow fields.}\n\t\\vspace{-1mm}\n\\scalebox{0.8}{\n\\begin{tabular}[t]{lllll}\n\\hline\n                         & 1 mL\/s    & 5 mL\/s    & 10 mL\/s     & 15 mL\/s\\\\\n\\hline\nParallel      & 51   & 262 & 541   & 833 \\\\\nInterdigitated       & 83   & 427 & 876   & 1343 \\\\\nOptimized   & 105  & 531 & 1075 &1630 \\\\\n\\hline\n\\end{tabular}\n}\n\\label{table:sec4_3_1}\n\\end{table}\n\nFigure \\ref{fig: sec4_1_1} shows the iteration history of topology optimized flow field, in which the electrolyte domain is expressed as the isosurface of $\\rho\\ge0.5$.\nThe analysis domain is discretized using $2.4\\times 10^5$ hexahedral elements for all variables in this study.\nIn the topology optimized design shown in Fig.~\\ref{fig: sec4_1_1}, as with the interdigitated flow field, some of the flow channels are not connected in the topology optimized flow field, where the disconnectivity of the flow field can enhance the mass transfer effect so that the mass transfer loss can be further reduced. \n\n\\subsection{Effect of flow field design}\n\nTo compare the differences between the concentration of vanadium species with different flow fields, Fig.~\\ref{fig: sec4_2_2} shows the geometric model of the flow fields and the distributions of $c_{\\text{V}^{2+}}$ and $c_{\\text{V}^{3+}}$ on the plane located in the middle of the electrode.\nThe geometric parameters of the electrode refers to Table \\ref{table:sec3_3_1}. \nBesides, the width and thickness of the flow channel is 3~mm and the interval between the branches of the flow field is 9~mm. \nAs shown in Fig.~\\ref{fig: sec4_2_2}, the under-rib convection of the parallel flow field is weak, especially on both sides of the electrode. The velocity of electrolyte flow to both sides is small in the electrode since it is difficult for the parallel flow field to distribute the electrolyte to both sides of the electrode effectively. In contrast,  the under-rib convection is strong in the interdigitated flow field and topology optimized flow field. \n\nFigure \\ref{fig: sec4_2_1} shows that the value of objective function in the optimization problem described as (\\ref{eq:optp}), and overpotential in the topology optimized flow field and interdigitated flow filed indicate higher performance than those of the parallel flow field at constant pressure drop. \nSince the under-rib convection is weak in the parallel flow field, the value of objective function with the parallel flow field is the lowest. \nBesides, the electrode surface concentration decreases, as the pressure drop decreases for all the flow fields.\nIn other words, the species will be more difficult to reach the electrode surface at low flow rate, which means the mass transfer effect is dominated by the flow channel.\nAccordingly, the overpotential with topology optimized flow field is the lowest among the flow fields due to the strongest mass transfer effect caused by the topology optimized flow field.\nHowever, it should be noted that the overall performances of the optimized flow field and the interdigitated flow field are almost same in this numerical example.\n\n\\subsection{Power loss}\n\nIn a VRFB system, the polarization loss is caused by the need for extra voltage to occur electrochemical reactions. \nIn addition to the polarization loss, the pumping power should be also taken into account for the evaluation of VRFB system.\nWe therefore introduce the following evaluation index:\n\\begin{align}\nP_{\\text{loss}} = I\\eta + Q\\Delta P,\n\\label{eq:loss}\n\\end{align}\nwhere $P_{\\text{loss}}$ is the sum of the polarization loss ($I\\eta$), and the pumping power loss ($Q\\Delta P$), $Q$ is the flow rate, and $\\Delta P$ is the pressure drop.\nNote that, for brevity in this study, the evaluation index in Eq.~(\\ref{eq:loss}) is defined as a simple expression using the dominant factors, whereas the performance of VRFB systems relates with various factors on the authority of the previous works \\cite{blanc2010understanding,xu2013numerical}.\n\nFigure \\ref{fig: sec4_3_1} shows the power loss corresponding to variant flow rates. At low flow rate of 1 mL$\/$s, the power loss with the topology optimized flow field is the lowest one at current of 4 A or 10 A. Although the pressure drop in the topology optimized flow field shown in Table \\ref{table:sec4_3_1} is the highest, the pumping power is trivial compared to the polarization loss, which results in the lowest power loss with the topology optimized flow field. At high flow rate of 15~mL$\/$s, the parallel flow field shows the lowest power loss at current of 4~A. However, the power loss with these flow fields is almost the same at current of 10~A. At high applied current, the electrochemical reactions at the electrode surface become strong and the polarization loss will increase. Therefore, with the strongest mass transfer effect caused by the topology optimized flow field, it is obvious that the difference of the total power loss between these flow fields decreases as the applied current density increases at high flow rates, even if the pumping power increases as the flow rate increases. In contrast, at flow rate of 1~mL$\/$s, the differences in the power loss between the flow fields increases as the applied current increases because the power loss is mainly from the polarization loss at current of 4~A and the polarization loss will be more dominant in power loss as the current increases from 4~A to 10~A. The results demonstrate the topology optimized flow field is more useful for VRFBs at high applied current.\n\nFigure \\ref{fig: sec4_3_3} shows the comparison results of the power loss at different operating condition at $\\epsilon=0.68$ and $I=10$~A. For all the flow fields, the power loss increases in comparison with the case of $\\epsilon=0.929$ in Fig.~\\ref{fig: sec4_3_1}. \nAt flow rate of 1~mL$\/$s, with the weaker mass transfer effect, the power loss of the interdigitated flow field and the topology optimized flow field increases from 0.413 to 0.505 and from 0.405 to 0.470 as the porosity decreases from 0.929 to 0.68. \nThe differences between the flow fields become larger at low porosity. At flow rate of 15~mL$\/$s, since the pressure drop in the topology optimized flow field is larger than that of the interdigitated flow field and the polarization loss falls slowly when the flow rate increases from 1 mL$\/$s to 15 mL$\/$s, the total power loss with interdigitated flow field is less than the loss with the topology optimized flow field. For low porosity, the polarization loss and pumping power will be more dominant at low and high flow rate respectively. \n\n\\section{Conclusion}\nIn this paper, we proposed a topology optimization method for the flow field in a VRFB based on a three-dimensional numerical model. \nWe demonstrated that the optimization problem can be formulated as a maximization problem of the electrode surface concentration of oxidized reactants during the charging process.\nBased on the proposed formulations, we derived a novel flow field configuration and evaluated its performances in comparison with reference flow fields---parallel and interdigitated flow fields.\nAs a result, we verified that a VRFB with the topology optimized flow field can achieve almost same performances with those of the interdigitated flow field, which has stronger mass transfer effect than the parallel flow field.\n\nWe further investigated the power loss with the parallel, interdigitated and topology optimized flow fields under different operating conditions.\nThe results demonstrated that a VRFB with the topology optimized flow field is more suitable for a VRFB at high applied current density.\nWe confirmed that the topology optimized flow field has a potential to be an alternative design candidate when dealing with low porosity electrode, as the difference of the power loss with respect to the flow field configurations is notably observed in comparison with the case of high porosity electrode. \n\n\\section*{Acknowledgments}\nThis work is partially supported by a research grant from The Mazda Foundation.\n\\color{black}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIt is well known  that Lie symmetry theory plays a significant role in the analysis of  differential  equations[1-5]. The basic idea of this method is that the infinitesimal transformation leaves the set of solution manifold of the considered differential equation invariant. This efficient method invented by Sophus Lie is a highly algorithmic process, and it often involves lengthy symbolic computation. The method systematically unifies and extends well-known techniques to construct explicit solutions for differential equations, especially for nonlinear differential equations. In recent years this method has a successful extension to discrete systems exhibiting solitons governed by nonlinear partial differential-difference equations and pure difference equations[6-8].\n\nRecently the study of fractional differential equations(FDEs), as generalizations of classical integer order differential equations, has attracted much attention due to an exact description of nonlinear phenomena in fluid mechanics, viscoelasticity, biology, physics, engineering and other areas of science[9-12]. However, unlike the classical integer order derivatives, there exists a number of different definitions of fractional order derivatives and corresponding FDEs. These definition differences lead to the FDEs having similar form but significantly different properties. It means that there exists no well-defined method to analyze them systematically. As a consequence, several different analytical methods such as differential transform method[13], Adomian decomposition method[14,15], invariant subspace method[16], Green function approach[17] and symmetry analysis[18-26] have been formulated to reduce and solve FDEs.\n\nIn this article, we consider the following time-fractional convection-diffusion equation\n\\begin{equation} \\label{eq:1}\n\\frac{\\partial^\\alpha {u}}{\\partial{t}^\\alpha}=(D(u)u_{x})_{x}+P(u)u_{x},0<\\alpha<2,\n\\end{equation}\nwhere $\\frac{\\partial^\\alpha{u}}{\\partial{t}^\\alpha}$ is the Riemann-Liouville fractional derivative of order $\\alpha$ with respect to the variable $t$. The definition of this derivative is\n\\begin{align}\\label{eq:2}\n\\frac{\\partial^\\alpha {u(t,x)}}{\\partial{t}^\\alpha}=\\left\\{\n\\begin{array}{lll}\n  \\frac{\\partial^n {u}}{\\partial{t}^n} & , & \\alpha=n, \\\\\n  \\frac{1}{\\Gamma(n-\\alpha)} \\frac{\\partial^n}{\\partial{t}^n} \\int^{t}_{0}(t-s)^{n-\\alpha-1}u(s,x)ds & , & 0\\leq n-1<\\alpha<n,\n\\end{array}\\right.\n\\end{align}\nhere $n\\in N$, $\\frac{\\partial^n}{\\partial{t}^n}$ is the usual partial derivative of integer order $n$ with respect to $t$.\\par\nThe nonlinear time-fractional convection-diffusion equation is obtained from the following nonlinear convection-diffusion equation by replacing the time derivative by fractional derivative.\n$$u_{t}=(D(u)u_{x})_{x}+P(u)u_{x},$$\nwhere $u=u(t,x),u_{t}=\\frac{\\partial{u}}{\\partial{t}},u_{x}=\\frac{\\partial{u}}{\\partial{x}}$, $D(u),P(u)$ are referred to  the diffusivity and convective terms respectively.\nThe nonlinear convection-diffusion equation arises in many areas of science and engineering such as being used to model the evolution of thermal waves in plasma[27].\n\nOur paper is organized as follows. In Section 2, we introduce Lie symmetry analysis for FDEs and deduce the infinitesimals of symmetries for time-fractional convection-diffusion equation (1) by considering the different conditions for $D(u)$ and $P(u)$. In Section 3, the similarity reductions for Eq.(1) are presented by Lie symmetries obtained in section 2. Some invariant solutions of Eq.(1) are given.\n\n\\section{Symmetry analysis of Eq.(1)}\n\nNext we use Lie symmetry analysis for Eq.(1). Let us assume that Eq.(1) is invariant under the following one parameter($\\epsilon$) continuous transformations\n\\begin{equation}\n\\begin{aligned}\nx^{*}&=x+\\epsilon\\xi(t,x,u)+o(\\epsilon),\\\\\nt^{*}&=t+\\epsilon\\tau(t,x,u)+o(\\epsilon),\\\\\nu^{*}&=u+\\epsilon\\eta(t,x,u)+o(\\epsilon),\\\\\nu^{*}_{x^{*}}&=u_x+\\epsilon\\eta^x+o(\\epsilon),\\\\\nu^{*}_{x^{*}x^{*}}&=u_{xx}+\\epsilon\\eta^{xx}+o(\\epsilon),\\\\\n\\frac{\\partial^{\\alpha}u^*}{\\partial t^{*\\alpha}}&=\\frac{\\partial^{\\alpha}u}{\\partial t^{\\alpha}}+\\epsilon\\eta^t_{\\alpha}+o(\\epsilon),\n\\end{aligned}\n\\end{equation}\nwhere $\\xi,\\tau,\\eta$ are infinitesimals and $\\eta^x,\\eta^{xx},\\eta^t_{\\alpha}$ are extended infinitesimals of orders 1,2 and $\\alpha$ respectively. The explicit expressions for $\\eta^x,\\eta^{xx}$ are\n\\begin{align}\n\\eta^{x}&=D_{x}\\eta-u_{x}(D_{x}\\xi)-u_{t}(D_{x}\\tau),\\nonumber\\\\\n\\eta^{xx}&=D_{x}(\\eta^{x})-u_{xx}(D_{x}\\xi)-u_{xt}(D_{x}\\tau),\\nonumber\n\\end{align}\nwhere symbol $D_{x}$ stands the total derivative operator with respect to $x$,\n$$D_{x}=\\frac{\\partial}{\\partial{x}}+u_{x}\\frac{\\partial}{\\partial{u}}+u_{xx}\\frac{\\partial}{\\partial{u_{x}}}+u_{tx}\\frac{\\partial}{\\partial{u_{t}}}+\\cdots,\n$$\nwith infinitesimal generator $X=\\xi(t,x,u)\\frac{\\partial}{\\partial{x}}+\\tau(t,x,u)\\frac{\\partial}{\\partial{t}}+\\eta(t,x,u)\\frac{\\partial}{\\partial{u}}$. Since the lower limit of the integral in Eq.(1) is fixed and, therefore it should be invariant with respect to the transformations(3). Such invariance condition arrives at\n\\begin{equation}\n\\tau(t,x,u)|_{t=0}=0.\n\\end{equation}\nThe $\\alpha$th extended infinitesimal related to Riemann-Liouville fractional time derivative reads\n\\begin{equation}\n\\eta^{t}_{\\alpha}=D^{\\alpha}_{t}\\eta+\\xi (D^{\\alpha}_{t}u_{x})-D^{\\alpha}_{t}(\\xi u_{x})+D^{\\alpha}_{t}(u D_{t}\\tau)-D^{\\alpha+1}_{t}(\\tau u)+\\tau (D^{\\alpha+1}_{t}u).\n\\end{equation}\nHere the symbol $D_{t}$ stands the total derivative operator with respect to $t$, i.e., $$D_{t}=\\frac{\\partial}{\\partial{t}}+u_{t}\\frac{\\partial}{\\partial{u}}+u_{xt}\\frac{\\partial}{\\partial{u_{x}}}+u_{tt}\\frac{\\partial}{\\partial{u_{t}}}+\\cdots,$$ and the operator $D^{\\alpha}_{t}$ is the total fractional derivative operator. By means of the generalized Leibnitz rule [12]\n\\begin{equation*}\nD^{\\alpha}_{t}(f(t)g(t))=\\sum^{\\infty}_{n=0}\\binom{\\alpha}{n}(D^{\\alpha-n}_{t}f(t))(D^{n}_{t}g(t)),\\ \\ \\ \\alpha>0,\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\binom{\\alpha}{n}=\\frac{(-1)^{n-1}\\alpha\\Gamma(n-\\alpha)}{\\Gamma(1-\\alpha)\\Gamma(n+1)},\n\\end{equation*}\nthe above Eq.(5) can be written as\n\\begin{equation*}\n\\eta^{t}_{\\alpha}=D^{\\alpha}_{t}\\eta-\\alpha(D_t\\tau)\\frac{\\partial^{\\alpha}u}{\\partial t^{\\alpha}}-\\sum^{\\infty}_{n=1}\\binom{\\alpha}{n}(D^{n}_{t}\\xi)(D^{\\alpha-n}_{t}u_{x})-\\sum^{\\infty}_{n=1}\\binom{\\alpha}{n+1}(D^{n+1}_{t}\\tau)(D^{\\alpha-n}_{t}u).\n\\end{equation*}\nFurthermore, using the generalized chain rule for a compound function [28]\n\\begin{equation*}\n\\frac{d^{\\alpha}u(v(t))}{dt^{\\alpha}}=\\sum^{\\infty}_{n=0}\\sum^{n}_{k=0}\\binom{n}{k}\\frac{(-v(t))^{k}}{n!}\\frac{\\partial^{\\alpha}(v^{n-k}(t))}{\\partial{t}^{\\alpha}}\\frac{d^{n}u(v(t))}{dv^{n}}\n\\end{equation*}\nalong with the above generalized Leibnitz rule with $f(t)=1$,\nthe first term $D^{\\alpha}_{t}\\eta$ in $\\eta^{t}_{\\alpha} $ can be written as\n\\begin{equation*}\nD^{\\alpha}_{t}\\eta=\\frac{\\partial^{\\alpha}\\eta}{\\partial{t}^{\\alpha}}+\\eta_{u}\\frac{\\partial^{\\alpha}u}{\\partial{t}^{\\alpha}}-u\\frac{\\partial^{\\alpha}\\eta_{u}}{\\partial{t}^{\\alpha}}+\\sum^{\\infty}_{n=1}\\binom{\\alpha}{n}\\frac{\\partial^{n}\\eta_{u}}{\\partial{t}^{n}}D^{\\alpha-n}_{t}(u)+\\mu,\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\mu=\\sum^{\\infty}_{n=2}\\sum^{n}_{m=2}\\sum^{m}_{k=2}\\sum^{k-1}_{r=0}\\binom{\\alpha}{n}\\binom{n}{m}\\binom{k}{r}\\frac{t^{n-\\alpha}}{\\Gamma(n+1-\\alpha)}\\frac{(-u)^{r}}{k!}\\frac{\\partial^{m}u^{k-r}}{\\partial{t}^{m}}\\frac{\\partial^{n-m+k}\\eta}{\\partial{t}^{n-m+k}\\partial{u}^{k}}.\n\\end{equation*}\nTherefore\n\\begin{align*}\n\\eta^{t}_{\\alpha}=&\\frac{\\partial^{\\alpha} \\eta}{\\partial{t}^{\\alpha}}+(\\eta_{u}-\\alpha\nD_{t}\\tau)\\frac{\\partial^{\\alpha}u}{\\partial{t}^{\\alpha}}-u\\frac{\\partial^\\alpha{\\eta_{u}}}{\\partial{t}^{\\alpha}}+\\mu+\n\\sum^{\\infty}_{n=1}\\big[\\binom{\\alpha}{n}\\frac{\\partial^{n}{\\eta_{u}}}{\\partial{t}^{n}}\n-\\binom{\\alpha}{n+1}D^{n+1}_{t}(\\tau)\\big]D^{\\alpha-n}_t u\\\\\n&-\\sum^{\\infty}_{n=1}\\binom{\\alpha}{n}(D^{n}_{t}\\xi)(D_t^{\\alpha-n}u_x).\n\\end{align*}\n\\par\nFor the invariance of Eq.(1) under transformations(3), we have\n\\begin{equation}\n\\frac{\\partial^\\alpha {u^*}}{\\partial{t^*}^\\alpha}=(D(u^*)u^*_{x^*})_{x^*}+P(u^*)u^*_{x^*},0<\\alpha<2\n\\end{equation}\nfor any solution $u=u(t,x)$ of Eq.(1). Expanding Eq.(6) about $\\epsilon=0$, making use of infinitesimals and their extensions, equating the coefficients of $\\epsilon$, and neglecting the terms of higher power of $\\epsilon$, we obtain the following invariant equation of Eq.(1)\n\\begin{align}\n[\\eta^{t}_{\\alpha}-(P'(u)u_{x}+D''(u)(u_{x})^{2}+D'(u)u_{xx})\\eta\n-(P(u)+2u_{x}D'(u))\\eta^{x}-D(u)\\eta^{xx}]|_{Eq.(1)}=0.\n\\end{align}\nHere we assume that $D(u)$ and $P(u)$ are not equal to zero, otherwise Eq.(1) would be another equation that have been considered in [20]. Substituting the expressions for $\\eta^{t}_{\\alpha}$,$\\eta^{x}$ and $\\eta^{xx}$ into the above equation and equating various powers of derivatives of $u$ to zero, we obtain an over determined system of linear equations. They are\n\\begin{align}\n&\\xi_{t}=\\xi_{u}=\\tau_{x}=\\tau_{u}=\\eta_{uu}=0,\\nonumber\\\\\n&P(u)(\\xi_{x}-\\alpha\\tau_{t})-P'(u)\\eta-2D'(u)\\eta_{x}-D(u)(2\\eta_{xu}-\\xi_{xx})=0,\\nonumber\\\\\n&D''(u)\\eta+D'(u)(\\eta_{u}-2\\xi_{x}+\\alpha\\tau_{t})=0,\\nonumber\\\\\n&D(u)(2\\xi_{x}-\\alpha\\tau_{t})-D'(u)\\eta=0,\\\\\n&\\frac{\\partial^{\\alpha}{\\eta}}{\\partial{t}^{\\alpha}}-u\\frac{\\partial^{\\alpha}{\\eta_{u}}}{\\partial{t}^{\\alpha}}-P(u)\\eta_{x}-D(u)\\eta_{xx}=0,\\nonumber\\\\\n&\\binom{\\alpha}{n}\\frac{\\partial^{n}{\\eta_{u}}}{\\partial{t}^{n}}-\\binom{\\alpha}{n+1}D^{n+1}_{t}(\\tau)=0,n=1,2,\\cdots\\nonumber.\n\\end{align}\n\nIn order to solve the above system, we consider the following different conditions for $D(u)$, $P(u)$ and obtain their corresponding Lie symmetries of Eq.(1). And if $\\alpha=1$, Eq.(1) becomes partial differential equation which has been considered by Oron, Rosenau[29] and Edwards[30]. Therefore, $\\alpha\\in (0,2)$ and $\\alpha\\neq1$ in our paper.\\\\\n\\textbf{Case 1 }$D(u)$ and $P(u)$ arbitrary\\\\\nSolving the determining equations (8), we obtain the explicit form of infinitesimals\n$$\\xi=a_1, \\tau=0, \\eta=0,$$\nwhere $a_1$ is an arbitrary constant. Hence the infinitesimal generator is\n\\begin{equation}\\label{eq:9}\nX_{1}=a_1\\frac{\\partial}{\\partial{x}}.\n\\end{equation}\n\\textbf{Case 2 }$D(u)=u^{k}(k\\neq0)$, $P(u)=\\beta(\\beta=\\pm1)$\\\\\nSimilarly, under this assumption the following infinitesimals are obtained,\n$$\\xi=a_1+a_2x, \\tau=a_2\\frac{t}{\\alpha}, \\eta=a_2\\frac{u}{k},$$\nwhere $a_1$ and $a_2$ are  arbitrary constants. Hence in this case Eq.(1) admits a two-parameter group with infinitesimal generators\n\n\\begin{equation}\\label{eq:10}\nX_{1}=\\frac{\\partial}{\\partial{x}},X_{2}=x\\frac{\\partial}{\\partial{x}}+\\frac{t}{\\alpha}\\frac{\\partial}{\\partial{t}}+\\frac{u}{k}\\frac{\\partial}{\\partial{u}},\n\\end{equation}\nwhich are the basis of 2-dimensional Lie algebras admitted by Eq.(1). \n\nThe other cases are listed in Table 1.\n\n\n\\begin{table}[h]\n\\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}}\n \\centering\n    \\begin{tabular}{|l|l|l|l|l|l|l||}\n      \\hline\n      No. & $D(u)$ &$P(u)$ & $\\xi$  & $\\tau$ & $\\eta$  \\\\\n      \\hline\n      3   & $u^{k}(k\\neq0,-2,\\frac{2\\alpha}{1-\\alpha})$   & $\\beta u^{k}(\\beta=\\pm1)$& $a_1$ & $a_2t$& $-a_2\\frac{\\alpha u}{k}$ \\\\\\hline\n      4 & $u^{k}(k\\neq0)$   & $\\beta u^{\\gamma}(\\beta=\\pm1,\\gamma\\neq k)$ & $a_1+a_2x$& $\\frac{2\\gamma-k}{\\alpha(\\gamma-k)}a_2t$ & $-\\frac{a_2u}{\\gamma-k}$ \\\\\\hline\n      5& $u^{-2}$  &  $\\beta u^{-2}(\\beta=\\pm1)$& $a_1+a_3e^{-\\beta x}$&$a_2t$&$\\frac{a_2}{2}\\alpha u+a_3\\beta u e^{-\\beta x}$\\\\\\hline\n      6& $u^{\\frac{2\\alpha}{1-\\alpha}}$ & $ \\beta u^{\\frac{2\\alpha}{1-\\alpha}}(\\beta=\\pm1)$&$a_1$&$a_2t+a_3t^2$&$\\frac{a_2(\\alpha-1)}{2}u+a_3(\\alpha-1)tu$ \\\\\\hline\n\n     7& $1$  &  $\\beta(\\beta=\\pm1)$& $a_1$&$0$ &\\tabincell{l}{$a_2u+h(t,x),$ \\\\ where $h(t,x)$ satisfies  \\\\ $\\frac{\\partial^{\\alpha}{h(t,x)}}{\\partial{t}^\\alpha}=\\beta h_{x}+h_{xx}$}\\\\\\hline\n     8  & $1$ & $\\beta u^{\\gamma}(\\beta=\\pm1,\\gamma\\neq0)$ & $a_1+a_2x$ & $\\frac{2a_2}{\\alpha}t$ & $-\\frac{a_2}{\\gamma}u$\\\\\n    \\hline\n    \\end{tabular}\n    \\caption{Infinitesimals of Eq.(1)}\n\n\\end{table}\n\n In the above table, $a_1,a_2,a_3$ are three arbitrary parameters. Like case 1 and case 2, the infinitesimal generators of Eq.(1) in different cases can easily obtained. Next we will use infinitesimal generators to deduce the similarity reductions and construct invariant solutions of Eq.(1). The definition of group invariant solution of  FDEs has given in [22]. Here we use it directly.\n \\section{Similarity reductions of Eq.(1)}\n\\textbf{Case 1.}$D(u)$ arbitrary,$P(u)$ arbitrary\\\\\nThe infinitesimal generator is $X_{1}=\\frac{\\partial}{\\partial{x}}$. The characteristic equations become\n\\begin{equation*}\n\\frac{dx}{1}=\\frac{dt}{0}=\\frac{du}{0},\n\\end{equation*}\nwhich have two invariants $t,u.$\nThus, the similarity transformation is\n\\begin{equation}\\label{eq:18}\nu=\\varphi(t).\n\\end{equation}\nSubstitution of (11) into Eq.(1) leads to $\\varphi(t)$ satisfying the reduced fractional ordinary differential equation\n\\begin{equation}\\label{eq:19}\n\\frac{d^\\alpha {\\varphi(t)}}{d{t}^\\alpha}=0.\n\\end{equation}\nHence,the group invariant solutions of  Eq.(1) are given by\n\\begin{align*}\nu=\\left\\{\n\\begin{array}{lll}\n  c_{1}t^{\\alpha-1} & , & 0<\\alpha<1, \\\\\n  c_{1}t^{\\alpha-1}+ c_{2}t^{\\alpha-2} & , &1<\\alpha<2.\n\\end{array}\\right.\n\\end{align*}\nwhere $c_{1}$,$c_{2}$ are arbitrary constants.\n\n\\textbf{Case 2.}$D(u)=u^{k}(k\\neq0,-2,\\frac{2\\alpha}{1-\\alpha})$,$P(u)=\\beta(\\beta=\\pm1)$ \\\\\nThe characteristic equation corresponding to $X_{2}=x\\frac{\\partial}{\\partial{x}}+\\frac{t}{\\alpha}\\frac{\\partial}{\\partial{t}}+\\frac{u}{k}\\frac{\\partial}{\\partial{u}}$ is\n\\begin{equation*}\n\\frac{dx}{x}=\\frac{\\alpha dt}{t}=\\frac{kdu}{u}.\n\\end{equation*}\nSolving the above equation, we get the similarity transformation\n\\begin{equation}\nu=t^{\\frac{\\alpha}{k}}\\varphi(\\zeta), \\zeta=xt^{-\\alpha}.\n\\end{equation}\nSubstituting transformation(13) into Eq.(1) leads to\n\\begin{equation}\\label{eq:22}\n\\frac{\\partial^\\alpha(t^{\\frac{\\alpha}{k}}\\varphi(\\zeta))}{\\partial{t}^{\\alpha}}=t^{\\frac{\\alpha}{k}-\\alpha}(\\varphi^{k}\\frac{d^2\\varphi}{d\\zeta^2}+ k\\varphi^{k-1}(\\frac{d\\varphi}{d\\zeta})^{2}+\\beta\\frac{d\\varphi}{d\\zeta}).\n\\end{equation}\nBecause $\\alpha\\in(0,2)$ and $\\alpha\\neq1$, according to the definition of the Riemann-Liouville fractional derivative, we should consider $0<\\alpha<1$ and $1<\\alpha<2$ separately.\n\nWhen $0<\\alpha<1$,  for the similarity transformation(13) becomes\n\\begin{equation}\\label{eq:23}\n\\frac{\\partial^\\alpha(t^{\\frac{\\alpha}{k}}\\varphi(\\zeta))}{\\partial{t}^{\\alpha}}=\\frac{1}{\\Gamma(1-\\alpha)}\\frac{\\partial}{\\partial{t}}\\int^{t}_{0}(t-s)^{-\\alpha}s^{\\frac{\\alpha}{k}}\\varphi(xs^{-\\alpha})ds.\n\\end{equation}\nLet $\\theta=\\frac{t}{s}$, then Eq.(15) can be written as\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\partial^\\alpha(t^{\\frac{\\alpha}{k}}\\varphi(\\zeta))}{\\partial{t}^{\\alpha}}&=\\frac{1}{\\Gamma(1-\\alpha)}\\frac{\\partial}{\\partial{t}}\\int^{\\infty}_{1}(t-\\frac{t}{\\theta})^{-\\alpha}(\\frac{t}{\\theta})^{\\frac{\\alpha}{k}}\\varphi(\\zeta \\theta^{\\alpha})\\frac{t}{\\theta^{2}}d\\theta\\\\\n&=\\frac{\\partial}{\\partial{t}}[t^{\\frac{\\alpha}{k}-\\alpha+1}\\frac{1}{\\Gamma(1-\\alpha)}\\int^{\\infty}_{1}(\\theta-1)^{-\\alpha}\\theta^{\\alpha-\\frac{\\alpha}{k}-2}\\varphi(\\zeta\\theta^{\\alpha})d\\theta]\\\\\n&=\\frac{\\partial}{\\partial{t}}[t^{\\frac{\\alpha}{k}-\\alpha+1}(K^{1+\\frac{\\alpha}{k},1-\\alpha}_{\\frac{1}{\\alpha}}\\varphi)(\\zeta)]\\\\\n&=t^{\\frac{\\alpha}{k}-\\alpha}(1+\\frac{\\alpha}{k}-\\alpha-\\alpha\\zeta\\frac{d}{d\\zeta})[(K^{1+\\frac{\\alpha}{k},1-\\alpha}_{\\frac{1}{\\alpha}}\\varphi)(\\zeta)]\\\\\n&=t^{\\frac{\\alpha}{k}-\\alpha}[(P^{1+\\frac{\\alpha}{k}-\\alpha,\\alpha}_{\\frac{1}{\\alpha}}(\\varphi))(\\zeta)],\n\\end{aligned}\n\\end{equation*}\nwhere $P^{\\tau,\\alpha}_{\\beta}$ is  Erdelyi-Kober fractional derivative operator and its definition is\n\\begin{align*}\n(P^{\\tau,\\alpha}_{\\beta}\\varphi)(\\zeta)=\\prod^{n-1}_{j=0}(\\tau+j-\\frac{1}{\\beta}\\zeta\\frac{d}{d\\zeta})[K^{\\tau+\\alpha,n-\\alpha}_{\\beta}(\\varphi)(\\zeta)],\n\\zeta>0, \\beta>0, \\alpha>0,\nn=\\left\\{\n\\begin{array}{lll}\n[\\alpha]+1&,&\\alpha\\notin N,\\\\\n\\alpha&,&\\alpha\\in N.\n\\end{array}\\right.\n\\end{align*}\nhere\n\\begin{align*}\n(K^{\\tau,\\alpha}_{\\beta}\\varphi)(\\zeta)=\\left\\{\n\\begin{array}{lll}\n\\frac{1}{\\Gamma(\\alpha)}\\int^{\\infty}_{1}(\\theta-1)^{\\alpha-1}\\theta^{-(\\tau+\\alpha)}\\varphi(\\zeta\\theta^{\\frac{1}{\\beta}})d\\theta&,&\\alpha>0,\\\\\n\\varphi(\\zeta)&,&\\alpha=0.\n\\end{array}\\right.\n\\end{align*}\nWhen $1<\\alpha<2$, we can also obtain $$\\frac{\\partial^\\alpha(t^{\\frac{\\alpha}{k}}\\varphi(\\zeta))}{\\partial{t}^{\\alpha}}=t^{\\frac{\\alpha}{k}-\\alpha}[(P^{1+\\frac{\\alpha}{k}-\\alpha,\\alpha}_{\\frac{1}{\\alpha}}(\\varphi))(\\zeta)],$$\nby using the same method.\nThen Eq.(1) can be reduced into an ordinary differential equation of fractional order\n\\begin{equation}\\label{eq:24}\n(P^{1+\\frac{\\alpha}{k}-\\alpha,\\alpha}_{\\frac{1}{\\alpha}}\\varphi)(\\zeta)=\\varphi^{k}\\frac{d^{2}\\varphi}{d\\zeta^{2}}+ k\\varphi^{k-1}(\\frac{d\\varphi}{d\\zeta})^{2}+\\beta\\frac{d\\varphi}{d\\zeta}.\n\\end{equation}\nAs for other cases, Eq.(1) can also be reduced by the similarity transformations corresponding to other infinitesimal generators. The results are as follows.\n\n\\textbf{Case 3.}  $D(u)=u^{k}(k\\neq0)$,$P(u)=\\beta u^{k}(\\beta=\\pm1)$\\\\\nThe similarity transformation $u=t^{-\\frac{\\alpha}{k}}\\psi(\\zeta)$ along with the similarity variable $\\zeta=x$ reduces Eq.(1) to the nonlinear ordinary differential equation of the form\n\n\\begin{equation}\\label{eq:29}\n\\frac{d^2\\psi}{d \\zeta^2}+ k\\psi^{-1}(\\frac{d\\psi}{d\\zeta})^2+\\beta\\frac{d\\psi}{d\\zeta}-\\frac{\\Gamma(1-\\frac{\\alpha}{k})}{\\Gamma(1-\\frac{\\alpha}{k}-\\alpha)}\\psi^{1-k}(\\zeta)=0,\n\\end{equation}\nwhich is corresponding to the infinitesimal generator $t\\frac{\\partial}{\\partial t}-\\frac{\\alpha u}{k}\\frac{\\partial}{\\partial u}.$\\\\\n\\textbf{Case 4.}$D(u)=u^{k}(k\\neq0)$,$P(u)=\\beta u^{\\gamma}(\\beta=\\pm1,\\gamma\\neq k)$\n\nThe similarity transformation $u=x^{\\frac{1}{k-\\gamma}}H(\\omega)$ along with the similarity variable $\\omega=tx^{-b},b=\\frac{2\\gamma-k}{\\alpha(\\gamma-k)}$ reduces Eq.(1) to the nonlinear ordinary differential equation of fractional order of the form\n\\begin{equation}\n\\begin{array}{r@{~}l}\n&(P^{1-\\alpha,\\alpha}_{-1}H)(\\omega)=\\omega^{\\alpha}[\\frac{(\\gamma+1)}{(k-\\gamma)^{2}}H^{k+1}-b\\omega(\\frac{k+\\gamma+2}{k-\\gamma}-b)H^{k}\\frac{dH}{d\\omega}\\\\\n&+b^{2}\\omega^{2}(kH^{k-1}(\\frac{dH}{d\\omega})^{2}+H^{k}\\frac{d^2H}{d\\omega^2})+\\frac{\\beta}{k-\\gamma}H^{\\gamma+1}-b\\beta\\omega H^{\\gamma}\\frac{dH}{d\\omega}],\n\\end{array}\n\\end{equation}\nwhich is corresponding to the infinitesimal generator\n$x\\frac{\\partial}{\\partial{x}}+\\frac{2\\gamma-k}{\\alpha(\\gamma-k)}t\\frac{\\partial}{\\partial{t}}+\\frac{u}{k-\\gamma}\\frac{\\partial}{\\partial{u}}$.\\\\\n\\textbf{Case 5.} $D(u)= u^{-2}$,$P(u)=\\beta u^{-2}(\\beta=\\pm1)$\n\nThe similarity transformation $u=e^{\\beta x}G(\\zeta)$ along with the similarity variable $\\zeta=t$ reduces Eq.(1) to the nonlinear ordinary differential equation of fractional order of the form\n\\begin{equation}\\label{eq:30}\n\\frac{d^\\alpha G(\\zeta)}{d{\\zeta}^{\\alpha}}=0,\n\\end{equation}\nwhich is corresponding to the infinitesimal generator $e^{-\\beta x}(\\frac{\\partial}{\\partial{x}}+\\beta u\\frac{\\partial}{\\partial{u}})$.\n\nBecause the solutions of Eq.(19) is\n\\begin{align*}\nG(\\zeta)=\\left\\{\n\\begin{array}{lll}\n  c_{3}\\zeta^{\\alpha-1} & , & 0<\\alpha<1, \\\\\n  c_{3}\\zeta^{\\alpha-1}+ c_{4}\\zeta^{\\alpha-2} & , &1<\\alpha<2.\n\\end{array}\\right.\n\\end{align*}\nwhere $c_{3}$,$c_{4}$ are arbitrary constants, the group invariant solution of Eq.(1) is\n\\begin{align*}\nu=\\left\\{\n\\begin{array}{lll}\n  c_{3}t^{\\alpha-1}e^{\\beta x} & , & 0<\\alpha<1, \\\\\n  (c_{3}t^{\\alpha-1}+ c_{4}t^{\\alpha-2})e^{\\beta x} & , &1<\\alpha<2.\n\\end{array}\\right.\n\\end{align*}\\\\\n\\textbf{Case 6.}$D(u)= u^{\\frac{2\\alpha}{1-\\alpha}}$,$P(u)=\\beta u^{\\frac{2\\alpha}{1-\\alpha}}(\\beta=\\pm1)$\n\nThe similarity transformation $u=t^{\\alpha-1}F(\\zeta)$ along with the similarity variable $\\zeta=x$ reduces Eq.(1) to the nonlinear ordinary differential equation of the form\n\\begin{equation}\\label{eq:32}\n\\frac{d^2F}{d\\zeta^2}+\\frac{2\\alpha}{1-\\alpha}F^{-1}(\\frac{dF}{d\\zeta})^{2}+\\beta \\frac{dF}{d\\zeta}=0.\n\\end{equation}\nwhich is corresponding to the infinitesimal generator\n$t^{2}\\frac{\\partial}{\\partial{t}}+(\\alpha-1)tu\\frac{\\partial}{\\partial{u}}$.\\\\\n\\textbf{Case 7.}$D(u)=1$ and $P(u)=\\beta(\\beta=\\pm1)$\n\nThe similarity transformation $u=e^{x}Q(\\zeta)$ along with the similarity variable $\\zeta=t$ reduces Eq.(1) to the nonlinear ordinary differential equation of fractional order of the form\n\\begin{equation}\\label{eq:35}\n\\frac{d^\\alpha Q(\\zeta)}{d{\\zeta}^\\alpha}=(1+\\beta)Q(\\zeta),\n\\end{equation}\nwhich is corresponding to the infinitesimal generator\n$\\frac{\\partial}{\\partial{x}}+u\\frac{\\partial}{\\partial{u}}.$\\\\\nBecause the solution of Eq.(21) is $$Q(\\zeta)=\\zeta^{\\alpha-1}E_{\\alpha,\\alpha}[(1+\\beta)\\zeta^{\\alpha}],$$the group invariant solution of Eq.(1) is $$u=t^{\\alpha-1}e^{x}E_{\\alpha,\\alpha}[(1+\\beta)t^{\\alpha}],$$\\\\\nwhere $E_{\\alpha,\\beta}(z)$ is the Mittag-Leffler function[11].\\\\\n\\textbf{Case 8.}$D(u)=1$ and $P(u)=\\beta u^{\\gamma}(\\beta=\\pm1,\\gamma\\neq0)$\n\nThe similarity transformation $u=t^{-\\frac{\\alpha}{2\\gamma}}\\phi(\\sigma)$ along with the similarity variable $\\sigma=xt^{-\\frac{\\alpha}{2}}$ reduces Eq.(1) to the nonlinear ordinary differential equation of fractional order of the form\n\\begin{equation}\\label{eq:36}\n(P^{1-\\frac{\\alpha}{2\\gamma}-\\alpha,\\alpha}_{\\frac{2}{\\alpha}}\\phi)(\\sigma)=\\frac{d^{2}\\phi}{d\\sigma^{2}}+\\beta\\phi^{\\gamma}\\frac{d\\phi}{d\\sigma},\n\\end{equation}\nwhich is corresponding to the infinitesimal generator\n$x\\frac{\\partial}{\\partial{x}}+\\frac{2t}{\\alpha}\\frac{\\partial}{\\partial{t}}-\\frac{u}{\\gamma}\\frac{\\partial}{\\partial{u}}$.\\\\\n\n\\section{Summary and discussion}\nIn this paper, we illustrate the application of Lie symmetry analysis to study time-fractional convection-diffusion equation. We consider the group classification of this equation for two variable functions. Eight cases are discussed. In every case Lie point symmetries are derived and\nsimilarity reductions of this equation are performed by means of non-trivial Lie point symmetry. In some cases, the time fractional convection-diffusion equation can be transformed into a nonlinear ODE of fractional order. In other cases, this equation can be reduced to a nonlinear ODE. Some invariant solutions are given in some cases. In addition, it is easily shown that the infinitesimal generator admitted by time-fractional convection-diffusion equation in each cases can form Lie algebra and the dimension of Lie algebra is decided by the number of parameters in transformations. It is necessary to remark that case 7 is different from other cases because in that case Lie algebra is infinite dimensional and in other cases Lie algebra is finite dimensional.\n\nLie symmetry analysis also can be used for other time-fractional differential equations. But there are little conclusions on the symmetry property for one kind of fractional equations. This is a possible direction for future work.\n\\section*{Acknowledgements}\n\nThis work is supported by natural science foundation of Zhejiang Province (Grant No.Y6100611) and the national natural science foundation of China(Grant No.11371323)\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nWhitham in his 1967 paper \\cite{Whitham1967} (see also \\cite{Whitham;book}) put forward\n\\begin{equation}\\label{eqn:Whitham0}\nu_t+c_\\text{ww}(|\\partial_x|)u_x+\\frac32\\sqrt{\\frac{g}{h}} uu_x=0\n\\end{equation}\nto address, qualitatively, breaking and peaking of waves in shallow water. Here and throughout, $u(x,t)$ is related to the displacement of the fluid surface from the rest state at position~$x$ and time~$t$,\nand $c_{\\rm{\\tiny ww}}(|\\partial_x|)$ is a Fourier multiplier operator, defined as \n\\begin{equation}\\label{def:c0}\n\\reallywidehat{c_\\text{ww}(|\\partial_x|)f}(k)=\\sqrt{\\frac{g\\tanh(kh)}{k}}\\widehat{f}(k),\n\\end{equation}\nwhere $g$ is the gravitational constant and $h$ the undisturbed fluid depth. Notice that $c_\\text{ww}(k)$ is the phase speed of $2\\pi\/k$ periodic waves in the linear theory of water waves \\cite{Whitham;book}.\n\nFor relatively shallow water or, equivalently, relatively long waves for which $kh\\ll1$, one can expand the right hand side of \\eqref{def:c0} to obtain\n\\[\nc_{\\rm{\\tiny ww}}(k)=\\sqrt{gh}\\Big(1-\\frac16k^2h^2\\Big)+O(k^4h^4),\n\\]\nwhereby arriving at the celebrated Korteweg--de Vries (KdV) equation:\n\\begin{equation}\\label{eqn:KdV}\nu_t+\\sqrt{gh}\\Big(u_x+\\frac16h^2 u_{xxx}\\Big)+\\frac32\\sqrt{\\frac{g}{h}}uu_x=0.\n\\end{equation}\nTherefore \\eqref{eqn:Whitham0} and \\eqref{def:c0} can be regarded as augmenting \\eqref{eqn:KdV} to include the {\\em full} dispersion in the linear theory of water waves and, thus, improving over \\eqref{eqn:KdV} for short and intermediately long waves. Indeed, numerical studies (see \\cite{MKD;comp}, for instance) reveal that the Whitham equation performs on par with or better than the KdV equation and other shallow water models in a wide range of amplitude and wavelength parameters. \n\nThe KdV equation admits solitary and cnoidal waves but no traveling waves can `peak'. By contrast, the so-called extreme Stokes wave possesses a $120^\\circ$ peaking at the crest. Also no solutions of \\eqref{eqn:KdV} can `break'. That means, the solution remains bounded but its slope becomes unbounded. See \\cite[Section~13.14]{Whitham;book} for more discussion.\nThis is perhaps not surprising because the dispersion\\footnote{The phase speed is $\\sqrt{gh}(1-\\frac16k^2h^2)$, poorly approximating $c_\\text{ww}(k)$ when $kh\\gg1$.} of the KdV equation is inadequate for explaining high frequency phenomena of water waves. Whitham \\cite{Whitham1967,Whitham;book} conjectured breaking and peaking for \\eqref{eqn:Whitham0} and \\eqref{def:c0}. Recently, one of the authors \\cite{Hur;breaking} proved breaking, and Ehrnstr\\\"om and Wahl\\'en  \\cite{EW;peaking} proved peaking.\nJohnson and one of the authors \\cite{HJ;whitham} proved that a small amplitude and periodic traveling wave of \\eqref{eqn:Whitham0} and \\eqref{def:c0} is modulationally unstable, provided that $kh>1.145\\dots$, comparing with the well-known Benjamin--Feir instability of a Stokes wave.\nBy contrast, all cnoidal waves of \\eqref{eqn:KdV} are modulationally stable.\n\nWhen the effects of surface tension are included, Johnson and one of the authors \\cite{HJ;capillary} proposed to replace \\eqref{def:c0} by \n\\begin{equation}\\label{def:cT0}\n\\reallywidehat{c_\\text{ww}(|\\partial_x|;T)f}(k)\n=\\sqrt{(g+Tk^2)\\frac{\\tanh(kh)}{k}}\\widehat{f}(k),\n\\end{equation}\nwhere $T$ is the surface tension coefficient. Notice that $c_\\text{ww}(k;T)$ is the phase speed in the linear theory of capillary-gravity waves \\cite{Whitham;book}. When $T=0$, \\eqref{def:cT0} becomes \\eqref{def:c0}. Since\n\\[\nc_\\text{ww}(k;T)=\\sqrt{gh}\\Big(1-\\frac12\\Big(\\frac13-\\frac{T}{gh^2}\\Big)k^2h^2\\Big)+O(k^4h^4)\n\\quad\\text{as $kh\\to0$},\n\\]\none arrives at the KdV equation for capillary-gravity waves:\n\\begin{equation}\\label{eqn:KdVT}\nu_t+\\sqrt{gh}\\Big(u_x+\\frac12\\Big(\\frac13-\\frac{T}{gh^2}\\Big)h^2u_{xxx}\\Big)+\\frac32\\sqrt{\\frac{g}{h}}uu_x=0\n\\end{equation}\nin the long wavelength limit unless $T\/gh^2=1\/3$. Notice that \\eqref{eqn:KdV} and \\eqref{eqn:KdVT} behave alike, qualitatively, possibly after a sign change. By contrast, whenever $T>0$, $c_\\text{ww}(k;T)\\to\\sqrt{T|k|}$ as $kh\\to\\infty$, so that \\eqref{eqn:Whitham0} and \\eqref{def:cT0} become \n\\begin{equation}\\label{eqn:fKdV1\/2}\nu_t+\\sqrt{T}|\\partial_x|^{1\/2}u_x+\\frac32\\sqrt{\\frac{g}{h}}uu_x=0\n\\end{equation}\nto leading order whereas when $T=0$, $u_t+\\sqrt{g}|\\partial_x|^{-1\/2}u_x+\\frac32\\sqrt{\\frac{g}{h}}uu_x=0$.\n\nIn recent years, the {\\em capillary-gravity Whitham equation} has been a subject of active research \\cite{HJ;capillary,Kalisch;physD,Kalisch;cWhitham,carter;fluids,EJ;WW} (see also \\cite{HP;BW,HP;FDCH,Hur;vWhitham,Pandey;comp,KP;LWP}). Particularly, Remonato and Kalisch \\cite{Kalisch;physD} combined a spectral collocation method and a numerical continuation approach to discover a rich variety of local bifurcation of periodic traveling waves of \\eqref{eqn:Whitham0} and \\eqref{def:cT0}, notably, crossing and connecting solution branches. Here we adopt a robust numerical continuation scheme to corroborate the results\nand produce convincing results for global bifurcation. Our findings support local bifurcation theorems (see \\cite{EJ;WW}, for instance) and help classify {\\em all} periodic traveling waves. \n\nWe employ an efficient numerical method for solving stiff nonlinear PDEs implemented with fourth-order time differencing, to experiment with (nonlinear) orbital stability and instability for a plethora of periodic traveling waves of \\eqref{eqn:Whitham0} and \\eqref{def:cT0}. (Spectral) modulational stability and instability were investigated numerically \\cite{carter;fluids} and also analytically for small amplitude \\cite{HJ;capillary}. To the best of the authors' knowledge, however, nonlinear stability and instability have not been addressed. Our novel findings include, among many others, orbital stability for the $k=1$ branch whenever $T>0$ versus instability for $k\\geq2,\\in\\mathbb{N}$ branches for great wave height when $0<T\/gh^2\\leq1\/3$. \n\nThe methodology here is potentially useful for tackling the capillary-gravity wave problem and other nonlinear dispersive equations. \n\n\\section{Preliminaries}\\label{sec:prelim}\n\nWe\nrewrite \\eqref{eqn:Whitham0} and \\eqref{def:cT0} succinctly as \n\\begin{equation}\\label{eqn:Whitham}\nu_t+c_\\text{ww}(|\\partial_x|;T)u_x+(u^2)_x=0,\n\\quad\\text{where}\\quad\n\\reallywidehat{c_\\text{ww}(|\\partial_x|;T)f}(k)\n=\\sqrt{(1+Tk^2)\\frac{\\tanh(k)}{k}}\\widehat{f}(k).\n\\end{equation}\nSeeking a periodic traveling wave of \\eqref{eqn:Whitham}, let \n\\[\nu(x,t)=\\phi(z), \\quad z=k(x-ct),\n\\]\nwhere $c\\in\\mathbb{R}$ ($\\neq 0$) is the wave speed, $k>0$ the wave number, and $\\phi$ satisfies, by quadrature and Galilean invariance,\n\\begin{equation}\\label{eqn:phi}\n-c\\phi+c_\\text{ww}(k|\\partial_z|;T)\\phi+\\phi^2=0. \n\\end{equation}\nWe assume that $\\phi$ is $2\\pi$ periodic and even in the $z$ variable, so that $2\\pi\/k$ periodic in the $x$ variable.\nNotice\n\\begin{equation}\\label{def:cT}\nc_\\text{ww}(k|\\partial_z|;T)\n\\left\\{\\begin{matrix}\\cos \\\\ \\sin \\end{matrix}\\right\\}(nz)\n=c_\\text{ww}(kn;T)\\left\\{\\begin{matrix}\\cos \\\\ \\sin \\end{matrix}\\right\\}(nz),\\quad n\\in\\mathbb{Z}.\n\\end{equation}\n\n\nFor any $T\\geq0$, $k>0$ and $c\\in\\mathbb{R}$, clearly, $\\phi(z)=0$ solves \\eqref{eqn:phi} and \\eqref{def:cT}. Suppose that $T$ and $k$ are fixed. A necessary condition for nontrivial solutions to bifurcate from such trivial solution at some $c$ is that \n\\[\n\\text{$c_\\text{ww}(k|\\partial_z|;T)\\phi-c\\phi=0$ admits a nontrivial solution},\n\\]\nif and only if $c=c_\\text{ww}(kn;T)$ for some $n\\in\\mathbb{N}$, by symmetry, whence \n\\[\n\\text{$\\cos(nz)\\in\\ker(c_\\text{ww}(k|\\partial_z|;T)-c_\\text{ww}(kn;T))$ in some space of even functions}.\n\\] \n\nWhen $T=0$, $c_\\text{ww}(\\cdot\\,;0)$ monotonically decreases to zero over $(0,\\infty)$, whence for any $k>0$, $c_\\text{ww}(kn_1;T)>c_\\text{ww}(kn_2;T)$ whenever $n_1,n_2\\in\\mathbb{N}$ and $n_1>n_2$. Therefore, for any $k>0$ for any $n\\in\\mathbb{N}$, $\\ker(c_\\text{ww}(k|\\partial_z|;0)-c_\\text{ww}(kn;0))=\\text{span}\\{\\cos(nz)\\}$ in spaces of even functions. \n\nWhen $T\\geq1\/3$, $c_\\text{ww}(\\cdot\\,;T)$ monotonically increases over $(0,\\infty)$ and unbounded from above, whereby for any $k>0$ for any $n\\in\\mathbb{N}$, $\\ker(c_\\text{ww}(k|\\partial_z|;T)-c_\\text{ww}(kn;T))=\\text{span}\\{\\cos(nz)\\}$.\n\nWhen $T<1\/3$, to the contrary, $c_\\text{ww}(kn_1;T)=c_\\text{ww}(kn_2;T)$ for some $k>0$ for some $n_1,n_2\\in\\mathbb{N}$ and $n_1\\neq n_2$, provided that $T=T(kn_1,kn_2)$, where\n\\begin{equation}\\label{def:T(k1,k2)}\nT(kn_1,kn_2)=\\frac{1}{kn_1}\\frac{1}{kn_2}\n\\frac{n_1\\tanh(kn_2)-n_2\\tanh(kn_1)}{n_1\\tanh(kn_1)-n_2\\tanh(kn_2)},\n\\end{equation}\nand $\\ker(c_\\text{ww}(k|\\partial_z|;T)-c)=\\text{span}\\{\\cos(n_1z),\\cos(n_2z)\\}$, where $c=c_\\text{ww}(kn_1;T)=c_\\text{ww}(kn_2;T)$; otherwise, $\\ker(c_\\text{ww}(k|\\partial_z|;T)-c_\\text{ww}(kn;T))=\\text{span}\\{\\cos(nz)\\}$. \n\nSuppose that $T\\geq0$, $k>0$, and $T\\neq T(kn,kn')$ for any $n,n'\\in\\mathbb{N}$ and $n\\neq n'$, particularly, either $T=0$ or $T\\geq1\/3$. We assume without loss of generality $n=1$. There exists a one-parameter curve of nontrivial, $2\\pi$ periodic and even solutions of \\eqref{eqn:phi} and \\eqref{def:cT}, denoted by\n\\begin{equation}\\label{def:loc;bifur}\n(\\phi(z;s),c(s)),\\quad |s|\\ll1,\n\\end{equation}\nin some function space (see \\cite{EW;peaking,EJ;WW}, for instance, for details), and \n\\begin{equation}\\label{eqn:|s|<<1}\n\\begin{aligned}\n&\\begin{aligned}\n\\phi(z;s)=s\\cos(z)&+\\frac{s^2}{2}\\left(\\frac{1}{c_\\text{ww}(k;T)-c_\\text{ww}(0;T)}\n-\\frac{1}{c_\\text{ww}(k;T)-c_\\text{ww}(2k;T)}\\cos(2z)\\right)\\\\\n&+\\frac{s^3}{2}\\frac{1}{(c_\\text{ww}(k;T)-c_\\text{ww}(2k;T))(c_\\text{ww}(k;T)-c_\\text{ww}(3k;T))}\\cos(3z)\n+O(s^4), \\end{aligned}\\\\\n&c(s)=c_\\text{ww}(k;T)+s^2\\left(\\frac{1}{c_\\text{ww}(k;T)-c_\\text{ww}(0;T)}\n-\\frac12\\frac{1}{c_\\text{ww}(k;T)-c_\\text{ww}(2k;T)}\\right)+O(s^4)\n\\end{aligned}\n\\end{equation}\nas $|s|\\to0$. Moreover, subject to a `bifurcation condition' (see \\cite{EW;peaking,EJ;WW}, for instance, for details), \\eqref{def:loc;bifur} extends to all $s\\in[0,\\infty)$. See \\cite{EW;peaking,EJ;WW} and references therein for a rigorous proof. \n\nWhen $T=0$ and, without loss of generality, $k=1$, $(\\phi(z;s_j),c(s_j))\\to (\\phi(z),c)$ as $j\\to\\infty$ for some $s_j\\to \\infty$ as $j\\to\\infty$ for some $\\phi\\in C^{1\/2}([-\\pi,\\pi])\\bigcap C^\\infty([-\\pi,0)\\bigcup(0,\\pi])$ and $0<c<\\infty$ such~that \n\\begin{equation}\\label{def:stopping0}\n\\phi(0)=\\frac{c}{2}.\n\\end{equation}\nSee \\cite{EW;peaking} and references therein for a rigorous proof. Indeed, such a limiting solution enjoys $\\phi(z)\\sim\\frac{c}{2}-\\sqrt{\\frac \\pi 8}|z|^{1\/2}$ as $|z|\\to0$.\n\nWhen $T\\geq 4\/\\pi^2$, so that the Fourier transform of $c_\\text{ww}(k|\\partial_z|;T)^{-1}$ is `completely monotone' \\cite{EJ;WW}, and $k=1$, on the other hand,\n\\begin{enumerate}\n\\item[]$\\|(\\phi(\\cdot;s), c(s))\\|\\to\\infty$ as $s\\to\\infty$ in some\\footnote{a H\\\"older-Zygmund space, for instance \\cite{EJ;WW}} function space; or\n\\item[]$(\\phi(z;s),c(s))$ is periodic in the $s$ variable.\n\\end{enumerate}\nSee \\cite{EJ;WW}, for instance, for a rigorous proof. Our numerical findings suggest $\\min_{z\\in[-\\pi,\\pi]}\\phi(z;s)\\to -\\infty$\nas $s\\to\\infty$. But such a limiting scenario would be physically unrealistic, for the capillary-gravity Whitham equation models water waves in the finite depth. We say that a $2\\pi$ periodic solution of \\eqref{eqn:phi} and \\eqref{def:cT} is admissible if \n\\[\n\\phi(z)-\\frac{1}{2\\pi}\\int^\\pi_{-\\pi}\\phi(z)~dz>-1\\quad\\text{for all $z\\in[-\\pi,\\pi]$},\n\\]\nso that the fluid surface would not intersect the impermeable bed, and we stop numerical continuation once we reach a {\\em limiting admissible solution}, for which \n\\begin{equation}\\label{def:stopping}\n\\min_{z\\in[-\\pi,\\pi]}\\phi(z)-\\frac{1}{2\\pi}\\int^\\pi_{-\\pi} \\phi(z)~dz=-1.\n\\end{equation} \nSection~\\ref{sec:result} provides examples.\n\nSuppose, to the contrary, that $T=T(kn_1,kn_2)$ (see \\eqref{def:T(k1,k2)}) for some $k>0$ for some $n_1,n_2\\in\\mathbb{N}$ and $n_1<n_2$, so that $c_\\text{ww}(kn_1;T)=c_\\text{ww}(kn_2;T)=:c$. Suppose that $n_1$ does not divide $n_2$. There exists a two-parameter sheet of nontrivial, periodic and even solutions of \\eqref{eqn:phi} and \\eqref{def:cT}, and\n\\begin{equation}\\label{eqn:s12}\n\\begin{aligned}\n&\\phi(z;s_1,s_2)=s_1\\cos(n_1z)+s_2\\cos(n_2z)+O((|s_1|+|s_2|)^2),\\\\\n&c(s_1,s_2)=c+O((|s_1|+|s_2|)^2), \\\\\n&k(s_1,s_2)=k+O((|s_1|+|s_2|)^2)\n\\end{aligned}\n\\end{equation}\nfor $|s_1|,|s_2|\\ll1$. We emphasize that $k$ is a bifurcation parameter. Otherwise, $n_1$ divides $n_2$, and \\eqref{eqn:s12} holds for $|s_1|,|s_2|\\ll1$ and $s_2>s_0>0$ for some $s_0$. In other words, there cannot exist $2\\pi\/n_1$ periodic and `unimodal' waves, whose profile monotonically decreases from its single crest to the trough over the period. See \\cite{EJ;WW}, for instance, for a rigorous proof.\nThe global continuation of \\eqref{eqn:s12} and limiting configurations have not been well understood analytically, though, and here we address numerically.\n\nTurning the attention to the (nonlinear) orbital stability and instability of a periodic traveling wave of \\eqref{eqn:Whitham}, notice that \\eqref{eqn:Whitham} possesses three conservation laws:\n\\begin{equation}\\label{def:EPM}\n\\begin{aligned}\nE(u)=&\\int \\left(\\frac12 uc_\\text{ww}(|\\partial_x|;T)u+\\frac13u^3\\right)~dx \\qquad\\text{(energy)},\\\\\nP(u)=&\\int \\frac12 u^2~dx\\qquad\\text{(momentum)}, \\\\\nM(u)=&\\int u~dx \\qquad\\text{(mass)},\n\\end{aligned}\n\\end{equation} \nand so does the KdV equation with fractional dispersion: \n\\begin{equation}\\label{eqn:fKdV}\nu_t+|\\partial_x|^\\alpha u_x+(u^2)_x=0,\\quad 0<\\alpha\\leq2,\n\\quad\\text{where}\\quad \\widehat{|\\partial_x|^\\alpha f}(k)=|k|^\\alpha\\widehat{f}(k),\n\\end{equation}\nfor which $E(u)=\\int (\\frac12u|\\partial_x|^\\alpha u+\\frac13u^3)~dx$ rather than the first equation of \\eqref{def:EPM}. We pause to remark that \\eqref{eqn:Whitham} becomes \\eqref{eqn:fKdV}, $\\alpha=1\/2$, for high frequency (see \\eqref{eqn:fKdV1\/2}), and \\eqref{eqn:KdV} and \\eqref{eqn:KdVT} compare with $\\alpha=2$. A solitary wave of \\eqref{eqn:fKdV} is a constrained energy minimizer and orbitally stable, provided that $\\alpha>1\/2$, if and only if $P_c>0$. See \\cite{LPS;orbital} and references therein for a rigorous proof. See also \\cite{HJ;orbital} for an analogous result for periodic traveling waves. It seems not unreasonable to expect that the orbital stability and instability of a periodic traveling wave of \\eqref{eqn:Whitham} change likewise at a critical point of $P$ as a function of $c$ although, to the best of the authors' knowledge, there is no rigorous analysis of constrained energy minimization. Indeed, numerical evidence \\cite{Kalisch;specVV} supports the conjecture when $T=0$ and $k=1$. Here we take matters further to $T\\geq0$ and $k\\geq1,\\in\\mathbb{N}$. Also we numerically elucidate the instability scenario when $T=0$ and $k=1$.\n\n\\section{Methodology}\\label{sec:numerical}\n\nWe begin by numerically approximating $2\\pi$ periodic and even solutions of \\eqref{eqn:phi} and \\eqref{def:cT} by means of a spectral collocation method \\cite{JBoyd,GO;spec}. See \\cite{Kalisch;specVV}, among others, for nonlinear dispersive equations of the form \\eqref{eqn:Whitham} and \\eqref{eqn:fKdV}. We define the collocation projection as a discrete cosine transform as\n\\begin{equation}\\label{def:phiN}\n\\phi_N(z)=\\sum_{n=0}^{N-1}\\omega(n)\\widehat{\\phi}(n)\\cos(nz), \n\\end{equation}\nwhere the discrete cosine coefficients are\n\\begin{equation}\\label{def:phi(n)}\n\\widehat{\\phi}(n)=\\omega(n)\\sum_{m=1}^N \\phi_N(z_m)\\cos(nz_m)\n\\end{equation}\nand \n\\[\n\\omega(n)=\\begin{cases}\n\\sqrt{1\/N} &\\text{for $n=0$},\\\\\n\\sqrt{2\/N} &\\text{otherwise};\n\\end{cases}\n\\]\nthe collocation points are\n\\[\nz_m=\\pi\\frac{2m-1}{2N},\\quad m=1,2,\\dots,N.\n\\]\nTherefore\n\\[\n\\phi(z)\\approx\\phi_N(z)=\\sum_{m=1}^N\\sum_{n=0}^{N-1}\\omega^2(n)\\cos(nz_m)\\cos(nz)\\phi(z_m).\n\\]\nWe can compute \\eqref{def:phiN} and \\eqref{def:phi(n)} efficiently using a fast Fourier transform (FFT) \\cite{GO;spec}. \nFor $T\\geq0$ and $k>0$, likewise,\n\\begin{align*}\nc_\\text{ww}(k|\\partial_z|;T)\\phi(z)\\approx (c_\\text{ww}(k|\\partial_z|;T)\\phi)_N(z)\n:=\\sum_{\\ell=1}^N\\sum_{n=0}^{N-1}\\omega^2(n)c_\\text{ww}(kn;T)\\cos(nz)\\cos(nz_\\ell)\\phi_N(z_\\ell), \n\\end{align*}\nand we can compute $(c_\\text{ww}(k|\\partial_z|;T)\\phi)_N(z_m)$, $m=1,2,\\dots,N$, via an FFT. \n\nSuppose that $T\\geq0$ and $k>0$ are fixed and we take $N=1024$. We numerically solve\n\\begin{equation}\\label{eqn:F=0}\n-c\\phi_N(z_m)+(c_\\text{ww}(k|\\partial_z|;T)\\phi)_N(z_m)+\\phi_N(z_m)^2=0,\\quad m=1,2,\\dots,N,\n\\end{equation}\nby means of Newton's method. We parametrically continue the numerical solution over $c$ by means of a pseudo-arclength continuation method \\cite{Doedel} (see \\cite{Kalisch;specVV}, among others, for nonlinear dispersive equations of the form \\eqref{eqn:Whitham} and \\eqref{eqn:fKdV}), for which the (pseudo-)arclength of a solution branch is the continuation parameter, whereas a parameter continuation method would use $c$ as the continuation parameter and vary it sequentially. The pseudo-arclength continuation method can successfully bypass a turning point of $c$, at which a parameter continuation method fails because the Jacobian of \\eqref{eqn:F=0} becomes singular,\nwhence Newton's method diverges. See \\cite{Kalisch;physD} for more discussion. \n\nWe say that Newton's method converges if the residual\n\\[\n\\sqrt{\\sum_{m=1}^N \\big(-c\\phi_N(z_m)+(c_\\text{ww}(k|\\partial_z|;T)\\phi)_N(z_m)+\\phi_N(z_m)^2\\big)^2}\\leq 10^{-10\n\\]\n(see \\eqref{eqn:F=0}), and this is achieved provided that an initial guess is sufficiently close to a true solution of \\eqref{eqn:phi} and \\eqref{def:cT}. To this end, we take a small amplitude cosine function and $c\\approx c_\\text{ww}(k;T)$ as an initial guess for the local bifurcation at $\\phi=0$ and $c=c_\\text{ww}(k;T)$, or \\eqref{eqn:|s|<<1} so long as it makes sense. We have also solved \\eqref{eqn:F=0} using a (Jacobian-free) Newton--Krylov method \\cite{Kelley_nsoli} with absolute and relative tolerance of $10^{-10}$. The results correctly match those obtained from Newton's method, corroborating our numerical scheme. \n\nWe have run a pseudo-arclength continuation code (see \\cite{Jennie_KM}, for instance, for the applicability of the code used herein) with a fixed arclength stepsize, to numerically locate and trace solution branches. We have additionally run AUTO \\cite{AUTO}, a robust numerical continuation and bifurcation software, to corroborate the results. All the results in Section~\\ref{sec:result} have been obtained by employing AUTO with strict tolerances (\\verb|EPSL = 1e-12| and \\verb|EPSU=1e-12| in the AUTO's constants file). AUTO has the advantage, among many others, of making variable arclength stepsize adaptation (using the option \\verb|IADS=1|) and of detecting branch points (using the option \\verb|ISP=2| for \\textit{all} special points). For the former, provided with minimum and maximum allowed absolute values of the pseudo-arclength stepsize, AUTO adjusts the arclength stepsize to be used during the continuation process.\n\nLet $\\phi$ and $c$ numerically approximate a $2\\pi\/k$ periodic traveling wave of \\eqref{eqn:Whitham}, and we turn to numerically experimenting with its (nonlinear) orbital stability and instability. After a $2048$-point (full) Fourier spectral discretization in $x\\in[-\\pi,\\pi]$, \\eqref{eqn:Whitham} leads to\n\\begin{equation}\\label{eqn:u}\nu_t=\\mathbf{L}u+\\mathbf{N}(u),\\quad\\text{where}\\quad\n\\mathbf{L}=-c_\\text{ww}(|\\partial_x|;T)\\partial_x\\quad\\text{and}\\quad\\mathbf{N}(u)=-(u^2)_x,\n\\end{equation}\nand we numerically solve \\eqref{eqn:u}, subject to\n\\begin{equation}\\label{eqn:IC}\nu(x,0)=\\phi(kx)+10^{-3}\\max_{x\\in[-\\pi,\\pi]}|\\phi(kx)|\\,U(-1,1),\n\\end{equation}\nby means of an integrating factor (IF) method (see \\cite{KassamTrefethen}, for instance, and references therein), where $U(-1,1)$ is a uniform random distribution. In other words, at the initial time, we perturb $\\phi$ by uniformly distributed random noise of small amplitude, depending on the amplitude\\footnote{We treat $\\|\\phi\\|_{L^\\infty}$ as the amplitude of $\\phi$ although $\\phi$ is not of mean zero, because the difference is insignificant.} of $\\phi$. We pause to remark that the well-posedness for the Cauchy problem of \\eqref{eqn:Whitham} can be rigorously established at least for short time in the usual manner via a compactness argument. \n\nFollowing the IF method, let $v=e^{-\\mathbf{L}t}u$, so that \\eqref{eqn:u} becomes\n\\begin{equation}\\label{eqn:v}\nv_{t}=e^{-\\mathbf{L}t}\\mathbf{N}(e^{\\mathbf{L}t}v).\n\\end{equation}\nThis ameliorates the stiffness of \\eqref{eqn:u}. Notice that \\eqref{eqn:v} is of diagonal form, rather than matrix, in the Fourier space. We advance \\eqref{eqn:v} forward in time by means of a fourth-order four-stage Runge--Kutta (RK4) method \\cite{HW1} with a fixed time stepsize $\\Delta t$:  \n\\[ v_{n+1}=v_n+\\frac{a+2b+2c+d}{6},\\]\nwhere $v_n=v(t_n)$ and $t_n=n\\Delta t$, $n=0,1,2,\\dots$, \n\\begin{align*}\na&=f(v_n,t_n)\\Delta t,\n&b&=f(v_n+a\/2,t_n+\\Delta t\/2)\\Delta t,\\\\\nc&=f(v_n+b\/2,t_n+\\Delta t\/2)\\Delta t,\n&d&=f(v_n+c,t_n+\\Delta t)\\Delta t\n\\end{align*}\nand $f(v,t)=e^{-\\mathbf{L}t}\\mathbf{N}(e^{\\mathbf{L}t}v)$ (see \\eqref{eqn:v}). \nWe take $\\Delta t=10^{-4}\\times2\\pi\/c$, where $c$ is the wave speed of $\\phi$. While such value of $\\Delta t$ ensures numerical stability, some computations, depending on $c$, require smaller\\footnote{In most cases, $10^{-4}\\times 2\\pi\/c =O(10^{-4})$ but in Figure~\\ref{fig6}(b), for instance, $c=0.36$, whence $10^{-4}\\times2\\pi\/c \\approx 0.00174533$, whereas $10^{-5}\\times 2\\pi\/c =O(10^{-4})$.} values, for instance, $\\Delta t=10^{-4}\\times \\pi\/c$ or $10^{-5}\\times2\\pi\/c$. At each time step, we remove aliasing errors by applying the so-called $3\/2$-rule so that the Fourier coefficients well decay for high frequencies (see \\cite{JBoyd}, for instance).\n\nWe assess the fidelity of our numerical scheme by monitoring \\eqref{def:EPM} for numerical solutions. For unperturbed solutions, $E(t)$, $P(t), M(t)$ have all been conserved to machine precision, whereas for (randomly) perturbed ones, $E(t)$ and $P(t)$ to the order of $10^{-7}$ and $M(t)$ to machine precision. We have also corroborated our numerical results using two other methods: a higher-order Runge--Kutta method, such as the Runge--Kutta--Fehlberg (RKF) method \\cite{HW1} with time stepsize adaptation and a two-stage (and, thus, fourth-order) Gauss--Legendre implicit\\footnote{An implicit method, by construction, enjoys wider regions of absolute stability. This allows us to assess all our results, while avoiding numerical instability that might be observed in explicit methods such as RK4 and RKF methods. However, the IRK4 method requires a fixed point iteration (in the form of Newton's method) at each time step.} Runge--Kutta (IRK4) method \\cite{HW2}.\nFor stable solutions, the results obtained from the RK4 method correctly match those from the RKF and IRK4 methods. However, and for unstable solutions, the instability manifests at slightly different times for the same initial conditions. This is somewhat expected because the local truncation error (LTE) of each method and the convergence error of the IRK4 method act as perturbations. Recall that the RK4 and IRK4 methods have an LTE of the order of $(\\Delta t)^{4}$ whereas the RKF method has $(\\Delta t)^{6}$.\n\n\\section{Results}\\label{sec:result}\n\nWe begin by taking $T=0$ and $k=1$. Figure~\\ref{fig1} shows the wave height\n\\begin{equation}\\label{def:H}\nH=\\max_{z\\in[-\\pi,\\pi]}\\phi(z)-\\min_{z\\in[-\\pi,\\pi]}\\phi(z)\n\\end{equation}\nand the momentum\n\\begin{equation}\\label{def:P}\nP=\\frac12\\int^\\pi_{-\\pi}\\phi(z)^2~dz\n\\end{equation}\nfrom our numerical continuation of $2\\pi$ periodic and even solutions of \\eqref{eqn:phi} and \\eqref{def:cT}. The result agrees, qualitatively, with \\cite[Figure~4]{Kalisch;specVV} and others.\nWe find that $H$ monotonically increases to $\\approx 0.58156621$, highlighted with the red square, for which \\eqref{def:stopping0} holds, whereas $P$ increases and then decreases, making one turning point.\nAlso we find that the crest becomes sharper and the trough flatter as $H$ increases. The limiting solution must possess a cusp at the crest \\cite{EW;peaking}. \n\nThe left column of Figure~\\ref{fig2} shows almost limiting waves. The inset is a close-up near the crest, emphasizing smoothness. The right column shows the profiles of \\eqref{eqn:IC}, namely periodic traveling waves of \\eqref{eqn:Whitham} perturbed by uniformly distributed random noise of small amplitudes at $t=0$ (dash-dotted), and of the solutions of \\eqref{eqn:Whitham} at later instants of time (solid). Wave (a), prior to the turning point of $P$, remains unchanged for $10^3$ time periods, after translation of the $x$ axis, implying orbital stability, whereas the inset reveals that wave (b), past the turning point, suffers from crest instability. Indeed, our numerical experiments point to transition from stability to instability at the turning point of $P$. But there is numerical evidence \\cite{SKCK;MI} that waves (a) and (b) are (spectrally) modulationally unstable.\n\nWhen $T=4\/\\pi^2$ and $k=1$, Figure~\\ref{fig3} shows $H$ and $P$ versus $c$. Our numerical findings suggest that $H,P\\to\\infty$ monotonically. Also $\\min_{z\\in[-\\pi,\\pi]}\\phi(z)\\to -\\infty$. The solution branch discontinues once \\eqref{def:stopping} holds, highlighted with the red square, though, because the solution would be physically unrealistic, for the capillary-gravity Whitham equation models water waves in the finite depth. Our numerical continuation works well past the limiting admissible solution, nevertheless. We find that the crest becomes wider and flatter, the trough narrower and more peaked, as $H$ increases. But all solutions must be smooth \\cite{EJ;WW}. \n\nThe left panel of Figure~\\ref{fig3a} shows an example wave. The right panel shows the profiles perturbed by small random noise at $t=0$ and of the solution of \\eqref{eqn:Whitham} after $10^3$ time periods, after translation of the $x$ axis. Our numerical experiments suggest orbital stability for all wave height. But there is numerical evidence of modulational instability when $H$ is large \\cite{carter;fluids}. \n\nNumerical evidence \\cite{Kalisch;physD,carter;fluids} points to qualitatively the same results whenever $T\\geq1\/3$.\n\nWe turn the attention to $T=T(1,2)\\approx 0.2396825$ (see \\eqref{def:T(k1,k2)}). There exists a two-parameter sheet of nontrivial, periodic and even solutions of \\eqref{eqn:phi} and \\eqref{def:cT} in the vicinity of $\\phi=0$ and $c=c_\\text{ww}(1,T)=c_\\text{ww}(2,T)\\approx 0.97166609$ \\cite{EJ;WW}. See also Section~\\ref{sec:prelim}. Figure~\\ref{fig4} shows $H$ versus $c$ for the $k=1$ and $k=2$ branches, all the way up to the limiting admissible solutions. There are no\\footnote{We observe a turning point of $P$ for greater wave height, but the solution is inadmissible.} turning points of $P$. The left column of Figure~\\ref{fig5} shows waves in the $k=1$ and $k=2$ branches for small and large $H$. The small height result agrees with \\cite[Figure~6]{Kalisch;physD}. \n\nObserve `bimodal' waves in the $k=1$ branch. Indeed, there cannot exist $2\\pi$ periodic and unimodal waves, whose profile monotonically decreases from a single crest to the trough over the period \\cite{EJ;WW}. See also Section~\\ref{sec:prelim}. For small wave height, the fundamental mode seems dominant, so that there is one crest over the period~$=2\\pi$, but the fundamental and second modes are resonant, whereby a much smaller wave breaks up the trough into two. See the left panel of Figure~\\ref{fig5}(a). As $H$ increases, the effects of the second mode seem more pronounced, so that the wave separating the troughs becomes higher. See the left of Figure~\\ref{fig5}(b). Observe, to the contrary, $\\pi$ periodic and unimodal waves in the $k=2$ branch. See the left of Figure~\\ref{fig5}(c,d). We find that the crests become wider and flatter, the troughs narrower and more peaked, as $H$ increases in the $k=1$ and $k=2$ branches. See the left of Figure~\\ref{fig5}(b,d). \n\nOur numerical experiments suggest orbital stability for the $k=1$ branch  (see the right panels of Figure~\\ref{fig5}(a,b)) versus instability for $k=2$ (see the right of Figure~\\ref{fig5}(c,d)).\n\nWe take matters further to $T=T(1,3)$. See Figures~\\ref{fig6}, \\ref{fig7} and \\ref{fig8}. The results for the $k=1$ and $k=3$ branches are similar to those when $T=T(1,2)$ and $k=1,2$. We pause to remark that in the $k=1$ branch, for small $H$, a smaller wave breaks up the trough into two, and a much smaller wave breaks up the crest into two. See the left panel of Figure~\\ref{fig7}(a). As $H$ increases, the wave separating the crests becomes lower, transforming into one wide and flat crest, whereas the wave separating the troughs become higher (see the left of Figure~\\ref{fig7}(b)), whereby resembling those when $T=T(1,2)$ and $k=1$. Observe $\\pi$ periodic and unimodal waves in the $k=2$ branch, orbitally stable for small $H$ versus unstable for large $H$. See Figure~\\ref{fig8}(e,f). \n\nWhen $T=T(2,3)$, to the contrary, we find $\\pi$ and $2\\pi\/3$ periodic unimodal waves in the $k=2$ and $k=3$ branches, respectively, corroborating a local bifurcation theorem (see \\cite{EJ;WW}, for instance).\nSee also Section~\\ref{sec:prelim}. We find that the crests become wider and flatter, the troughs narrower and more peaked, as $H$ increases. See the left column of Figure~\\ref{fig10}. Our numerical experiments (see the right of Figure~\\ref{fig10}) suggest\norbital instability for the $k=2$ branch for large $H$ and for $k=3$ for all $H$.\n\nWhen $T=T(2,5)$, on the other hand, the left column of Figure~\\ref{fig12} shows bimodal waves in the $k=2$ branch. The local bifurcation theorem \\cite{EJ;WW} dictates $\\pi$ periodic and unimodal waves, but we numerically find that they are not in the $k=2$ branch. For small $H$, for wave (a), for instance, a smaller wave breaks up the trough into two over the half period. As $H$ increases, for wave (b), for instance, the troughs become narrower and more peaked. Our numerical experiments (see the right of Figures~\\ref{fig12} and \\ref{fig13}) suggest orbital stability for the $k=2$ branch  for small $H$ versus instability for $k=2$ for large $H$ and for $k=5$ for all $H$. \n\nTo proceed, when $T=T(1,4)$, the $k=1$ branch lies above and to the right of the $k=4$ branch at least for small $H$ (not shown), but as $T$ increases, $c_\\text{ww}(4;T)$ increases more rapidly than $c_\\text{ww}(1;T)$, so that when $T=T(1,4)+0.001$, for instance, the $k=1$ branch crosses the $k=4$ branch. Figure~\\ref{fig14} shows $H$ and $P$ versus $c$ in the $k=1$ and $k=4$ branches for all admissible solutions. The small height result agrees with \\cite[Figure~10(a)]{Kalisch;physD}. We find that in the $k=1$ branch, $H$ and $P$ turn to connect the $k=4$ branch, whereas in the $k=4$ branch, $H$ and $P$ monotonically increase to the limiting admissible solution, highlighted by the red square in the insets. \n\nThe left panels of Figures~\\ref{fig15}, \\ref{fig16} and \\ref{fig17} show several profiles along the $k=1$ and $k=4$ branches. In the $k=1$ branch, for small $H$, wave (a), for instance, is $2\\pi$ periodic and unimodal. After the $k=1$ branch crosses the $k=4$ branch, on the other hand, wave (b), for instance, becomes bimodal, resembling those when $T=T(1,4)$ and $k=1$. Continuing along the $k=1$ branch, for waves (c) and (d), for instance, high frequency ripples of $k=4$ ride a carrier wave of $k=1$.\nWhen the $k=1$ and $k=4$ branches almost connect, wave (e) in the $k=1$ branch and wave (f) for $k=4$ are almost the same. In the $k=4$ branch, wave (g), for instance, is $\\pi\/2$ periodic and unimodal.\n\nOur numerical experiments (see the right panels of Figures~\\ref{fig15}, \\ref{fig16} and \\ref{fig17}) suggest orbital stability for the $k=1$ branch for all $H$ and for $k=4$ for small $H$, versus instability for $k=4$ for large $H$. Particularly, stability and instability do not change at the turning point of $P$.  \n\nLast but not least, when $T=T(1,5)+0.0001$, Figure~\\ref{fig18} shows $H$ and $P$ versus $c$ for the $k=1$ and $k=5$ branches. The small height result agrees with \\cite[Figure~10(b)]{Kalisch;physD}. We find that the $k=1$ branch crosses and connects the $k=5$ branch, like when $T=T(1,4)+0.0001$ and $k=1,4$, but it continues after connecting all the wave up to the limiting admissible solution. See the insets. The left panels of Figures~\\ref{fig19}, \\ref{fig20} and \\ref{fig21} show several profiles along the $k=1$ and $k=5$ branches. The results for the $k=1$ branch up to connecting and for $k=5$ are similar to those when $T=T(1,4)+0.0001$ and $k=1,4$. In the $k=1$ branch, after it connects the $k=4$ branch at the point (c), the results are similar to those when $T=T(1,5)$ and $k=1$. See waves (d), (e) and (f). \nOur numerical experiments (see the right panels of Figures~\\ref{fig19}, \\ref{fig20} and \\ref{fig21}) suggest orbital stability for the $k=1$ branch for all $H$ and for $k=5$ for small $H$, versus instability for $k=5$ for large $H$. \n\nWe emphasize orbital stability for the $k=1$ branch for all $T>0$ for all wave height throughout our numerical experiments.\n\n\\section{Discussion}\\label{sec:discussion}\n\nHere we employ efficient and highly accurate numerical methods for computing periodic traveling waves of \\eqref{eqn:Whitham} and experimenting with their (nonlinear) orbital stability and instability. Our findings suggest, among many others, stability whenever $T>0$ for the $k=1$ branch versus instability when $0<T<1\/3$ for $k\\geq2,\\in\\mathbb{N}$ branches at least for large wave height. Currently under investigation is to take matters further to all $T\\geq0$ to all $k$, to classify the orbital stability and instability of all periodic traveling waves. It will be interesting to devise other numerical continuation methods, for instance, deflated continuation techniques \\cite{PhysRevA.102.053307,CHARALAMPIDIS2020105255}, for detecting {\\em disconnected} solution branches and others. Also interesting will be to numerically investigate (spectral) modulational stability and instability of orbitally stable waves. Our methodology will be useful for exploring the nonlinear dynamics of modulationally unstable waves. Also it can help tackle the capillary-gravity wave problem and other nonlinear dispersive equations. Of course, it is of great importance to rigorously prove the numerical results. \n\n\\newpage\n\n\\subsection*{Acknowledgement}\n\nThe authors are grateful to Henrik Kalisch for helpful discussions. EGC acknowledges the hospitality of the Department of Mathematics at the University of Illinois at Urbana-Champaign where early stages of this work took place.\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecurrent neural networks such as LSTMs \\cite{Hochreiter:1997} are a standard building block for understanding and generating text data in NLP. \nThey find usage in pure NLP applications, such as abstractive summarization \\cite{Chopra:2016}, machine translation \\cite{Bahdanau:ICLR2015}, textual entailment \\cite{Rock:ICLR2016}; as well as in multi-modal tasks involving NLP, such as image captioning \\cite{Karpathy:CVPR2015}, visual question answering \\cite{Xu:ECCV2016} or lip reading \\cite{Chung:CVPR2017}. \n\nAs these models become more and more widespread due to their predictive performance, there is also a need to understand {\\it why} they took a particular decision, i.e., when the input is a sequence of words: {\\it which} words are determinant for the final decision?\nThis information is crucial to unmask ``Clever Hans'' predictors \\cite{Lap:Nature19}, and to allow for transparency of the decision-making process \\cite{EU-GDPR}.\n\nEarly works on {\\it explaining} neural network predictions include \\citet{Baehrens:JMLR2010,Zeiler:ECCV2014,Simonyan:ICLR2014,Spring:ICLR2015,Bach:PLOS2015,Alain:ICLR2017}, with several works focusing on explaining the decisions of convolutional neural networks (CNNs) for image recognition. \nMore recently, this topic found a growing interest within NLP, amongst others to explain the decisions of general CNN classifiers \\cite{Arras:PLOSONE2017,jacovi:BlackboxNLP2018}, and more  particularly to explain the predictions of recurrent neural networks \\cite{Li:NAACL2016,Li:ArXiv2017,Arras:WASSA2017,Ding:ACL2017,Murdoch:ICLR2018,Poerner:ACL2018}.\n\nIn this work, we focus on RNN explanation methods that are solely based on a trained neural network model and a single test data point\\footnote{These methods are deterministic, and are essentially based on a {\\it decomposition} of the model's current prediction. Thereby they intend to reflect the {\\it sole} model's ``point of view'' on the test data point, and hence are not meant to provide an averaged, smoothed or denoised explanation of the prediction by additionally exploiting the data's {distribution}.}. \nThus, methods that use additional information, such as training data statistics, sampling, or are optimization-based \\cite{Ribeiro:KDD2016,Lundberg:NIPS2017,Chen:ICML2018} are out of our scope.\nAmong the methods we consider, \nwe note that the method of \\citet{Murdoch:ICLR2018} was not yet compared against \\citet{Arras:WASSA2017,Ding:ACL2017}; and that the method of \\citet{Ding:ACL2017} was validated only visually.\nMoreover, to the best of our knowledge, no recurrent neural network explanation method was tested so far on a toy problem where the {\\it ground truth} relevance value is known.\n\nTherefore our contributions are the following: we evaluate and compare the aforementioned methods, using two different experimental setups, thereby we assess basic properties and  differences between the explanation methods.\nAlong-the-way we purposely adapted a simple toy task, to serve as a testbed for recurrent neural networks explanations.\nLastly, we explore how word relevances can be used to build sentence-level representations, and demonstrate how the relevance visualization can help to understand the (mis-)classification of selected samples w.r.t.\\ semantic composition.\n\n\n\\section{Explaining Recurrent Neural Network Predictions}\\label{section:methods}\n\nFirst, let us settle some notations.\nWe suppose given a trained recurrent neural network based model, which has learned some scalar-valued prediction function $f_c(\\cdot)$, for each class $c$ of a classification problem.\nFurther, we denote by $\\mathbf{x} = (\\boldsymbol{x_1}, \\boldsymbol{x_2},...,\\boldsymbol{x_T})$ an unseen input data point,\nwhere $\\boldsymbol{x_t}$ represents the $t$-th input vector of dimension $D$, within the input sequence $\\mathbf{x}$ of length $T$.\nIn NLP, the vectors $\\boldsymbol{x_t}$ are typically word embeddings, and $\\mathbf{x}$ may be a sentence.\n\nNow, we are interested in methods that can explain the network's prediction $f_c(\\mathbf{x})$ for the input $\\mathbf{x}$, and for a chosen {\\it target} class $c$, by assigning a scalar {relevance} value\nto each input variable or word. This relevance is meant to quantify the variable's or word's importance {\\it for} or {\\it against} a model's prediction {\\it towards} the class $c$. \nWe denote by $R_{x_i}$ (index $i$) the relevance of a {\\it single} variable. This means $x_i$ stands for any arbitrary input variable $x_{t,d}$ representing the $d$-th dimension, $d \\in \\{1,...,D\\}$, of an input vector $\\boldsymbol{x_t}$.\nFurther, we refer to $R_{\\boldsymbol{x_t}}$ (index $t$) to designate the relevance value of an {\\it entire} input vector or word $\\boldsymbol{x_t}$.\nNote that, for most methods, one can obtain a word-level relevance value by simply summing up the relevances over the word embedding dimensions, i.e.\\ $R_{\\boldsymbol{x_t}}= \\sum_{d \\in \\{1,...,D\\}}{R_{x_{t,d}}}$\\,.\n\n\\subsection{Gradient-based explanation}\nOne standard approach to obtain relevances is based on partial derivatives of the prediction function:  \n$R_{x_i} = \\big| \\tfrac{\\partial {f_c}}{\\partial x_{i}}(\\mathbf{x}) \\big|$, or $R_{x_i} = \\big( \\tfrac{\\partial {f_c}}{\\partial x_{i}}(\\mathbf{x}) \\big)^2$ \\cite{Dimopoulos:1995,Gevrey:2003,Simonyan:ICLR2014,Li:NAACL2016}.\n\nIn NLP this technique was employed to visualize the relevance of single input variables in RNNs for sentiment classification \\cite{Li:NAACL2016}.\nWe use the latter formulation of relevance and denote it as {\\it Gradient}.\nWith this definition the relevance of an entire word is simply the squared $L_2$-norm of the prediction function's gradient w.r.t.\\ the word embedding, i.e.\\  $ R_{\\boldsymbol{x_t}} = \\|{\\nabla}_{\\boldsymbol x_t} \\; f_c({\\mathbf x})\\|{_2^2}$\\,.\n\nA slight variation of this approach uses partial derivatives multiplied by the variable's value, i.e.\\ $R_{x_i} =  \\tfrac{\\partial {f_c}}{\\partial x_{i}}(\\mathbf{x}) \\cdot x_i$.\nHence, the word relevance is a dot product between prediction function gradient and word embedding: $ R_{\\boldsymbol{x_t}} = ({\\nabla}_{\\boldsymbol x_t} \\; f_c({\\mathbf x}) )^T \\, \\boldsymbol{x_t}$\\, \\cite{Denil:2015}.\nWe refer to this variant as {\\it Gradient$\\times$Input}.\n\nBoth variants are general and can be applied to any neural network. They are computationally efficient and require one forward and backward pass through the net.\n\n\\subsection{Occlusion-based explanation}\nAnother method to assign relevances to single variables, or entire words, is by occluding them in the input, and tracking the difference in the network's prediction w.r.t.\\ a prediction on the original unmodified input \\cite{Zeiler:ECCV2014,Li:ArXiv2017}.\nIn computer vision the occlusion is performed by replacing an image region with a grey or zero-valued square \\cite{Zeiler:ECCV2014}. In NLP word vectors, or single of its components, are replaced by zero; in the case of recurrent neural networks, the technique was applied to identify important words for sentiment analysis \\cite{Li:ArXiv2017}.\n\nPractically, the relevance can be computed in two ways: in terms of prediction function differences, or in the case of a classification problem, using a difference of probabilities,\ni.e.\\ $R_{x_i}=f_c( \\mathbf{x} ) - f_c( \\mathbf{x}_{| x_i=0} )$, or $R_{x_i}=P_c( \\mathbf{x} ) - P_c( \\mathbf{x}_{| x_i=0} )$,\nwhere $P_c(\\cdot)=  \\tfrac {\\exp f_c(\\cdot)}{\\sum_k{\\exp f_k(\\cdot)}}  $\\,.\nWe refer to the former as {\\it Occlusion}$_\\text{f-diff}$, and to the latter as {\\it Occlusion}$_\\text{P-diff}$. Both variants can also be used to estimate the relevance of an entire word, in this case the corresponding word embedding is set to zero in the input.\nThis type of explanation is computationally expensive and requires $T$ forward passes through the network to determine one relevance value per word in the input sequence $\\mathbf{x}$.\n\nA slight variation of the above approach uses word omission (similarly to \\citealp{Kadar:CL2017}) instead of occlusion. On a morphosyntactic agreement experiment (see \\citealp{Poerner:ACL2018}), omission was shown to deliver inferior results, therefore we consider only occlusion-based relevance.\n\n\\subsection{Layer-wise relevance propagation}\n\nA general method to determine input space relevances based on a backward decomposition of the neural network prediction function is layer-wise relevance propagation (LRP) \\cite{Bach:PLOS2015}.\nIt was originally proposed to explain feed-forward neural networks such as convolutional neural networks \\cite{Bach:PLOS2015,Bach:CVPR2016},\nand was recently extended to recurrent neural networks \\cite{Arras:WASSA2017,Ding:ACL2017,JAM:RUDDER2018}.\n\nLRP consists in a standard forward pass, followed by a specific backward pass which is defined for each type of layer of a neural network by dedicated propagation rules.\nVia this backward pass, each neuron in the network gets assigned a relevance, starting with the output neuron whose relevance is set to the prediction function's value, i.e.\\ to $f_c(\\mathbf{x})$.\nEach LRP propagation rule redistributes iteratively, layer-by-layer, the relevance from higher-layer neurons to lower-layer neurons, and verifies a relevance conservation property\\footnote{Methods based on a similar conservation principle include contribution propagation \\cite{Landecker:CIDM2013}, Deep Taylor decomposition \\cite{Montavon:PR2017}, and DeepLIFT \\cite{Shrikumar:PMLR2017}.}. \nThese rules were initially proposed in \\citet{Bach:PLOS2015} and were subsequently justified by Deep Taylor decomposition \\cite{Montavon:PR2017} for deep ReLU nets.\n\nIn practice, for a linear layer of the form $z_j = \\sum_i z_i w_{ij} + b_j\\;$, and given the relevances of the output neurons $R_j$,  \nthe input neurons' relevances $R_i$ are computed through the following summation:\n$ R_{i} =  \\sum_j \\tfrac{z_i \\cdot w_{ij}}{z_j \\, + \\, \\epsilon \\cdot {\\text sign} (z_j)} \\; \\cdot R_j \\,$,\nwhere $\\epsilon$ is a stabilizer (small positive number); this rule is commonly referred as $\\epsilon$-LRP or $\\epsilon$-rule\\footnote{Such a rule was employed by previous works with recurrent neural networks \\cite{Arras:WASSA2017,Ding:ACL2017,JAM:RUDDER2018},\nalthough there exist also other LRP rules for linear layers (see e.g. \\citealp{Montavon:DSP2018})}.\nWith this redistribution the relevance is conserved, up to the relevance assigned to the bias and ``absorbed'' by the stabilizer.\n\nFurther, for an element-wise nonlinear activation layer, the output neurons' relevances are redistributed identically onto the input neurons.\n\nIn addition to the above rules, in the case of a multiplicative layer of the form $z_j = z_g \\cdot z_s$,\n\\citet{Arras:WASSA2017} \nproposed to redistribute zero relevance to the {\\it gate} (the neuron that is sigmoid activated) i.e.\\ $R_g=0$, and assign all the relevance to the remaining {\\it signal} neuron (which is usually tanh activated) i.e.\\ $R_s=R_j$. We call this LRP variant {\\it LRP-all}, which stands for ``signal-take-all'' redistribution.\nAn alternative rule was proposed in \\citet{Ding:ACL2017,JAM:RUDDER2018}, where the output neuron's relevance $R_j$ is redistributed onto the input neurons via:\n$R_g = \\tfrac {z_g } {z_g + z_s } R_j $ and $R_s = \\tfrac {z_s } {z_g + z_s } R_j $. We refer to this variant as {\\it LRP-prop}, for ``proportional'' redistribution.\nWe also consider two other variants.\nThe first one uses absolute values instead: $R_g = \\tfrac {|z_g| } {|z_g| + |z_s| } R_j $ and $R_s = \\tfrac {|z_s| } {|z_g| + |z_s| } R_j $, we call it  {\\it LRP-abs}.\nThe second uses equal redistribution: $R_g = R_s = 0.5 \\cdot R_j $ \\cite{JAM:RUDDER2018}, we denote it as  {\\it LRP-half}.\nWe further add a stabilizing term to the denominator of the {\\it LRP-prop} and {\\it LRP-abs} formulas, it has the form $\\epsilon \\cdot {\\text sign} (z_g+z_s)$ in the first case, and simply $\\epsilon$ in the latter.\n\nSince the relevance can be computed in one forward and backward pass, the LRP method is efficient. Besides, it is general as it can be applied to any neural network made of the above layers:\nit was applied successfully to CNNs, LSTMs, GRUs, and QRNNs \\cite{Poerner:ACL2018,Yang:ICHI2018}\\footnote{Note that in the present work we apply LRP to {\\it standard} LSTMs, though \\citet{JAM:RUDDER2018} showed that some LRP rules for product layers can benefit from simultaneously adapting the LSTM architecture.}.\n\n\n\\begin{table*}\n\t%\n\n\t\\center\n\t\\resizebox{\\textwidth}{!}{%\n\t\\begin{tabular}{lcccc}\n\t\t\\toprule\t\n\t\tMethod                                                                           & Relevance  Formulation & Redistributed Quantity ($\\sum_i R_{x_i}$) & Complexity    \\\\\n\t\t\\midrule\n\t\tGradient \t\t\t& $R_{x_i} = \\big( \\tfrac{\\partial {f_c}}{\\partial x_{i}}(\\mathbf{x}) \\big)^2$ &  $ \\|{\\nabla}_{\\mathbf x} \\; f_c({\\mathbf x})\\|{_2^2}$ & $\\Theta(2 \\cdot T)$\\\\\n\t\tGradient$\\times$Input \t\t& $R_{x_i} =  \\tfrac{\\partial {f_c}}{\\partial x_{i}}(\\mathbf{x}) \\cdot x_i$ & $({\\nabla}_{\\mathbf x} \\; f_c({\\mathbf x}) )^T \\, \\mathbf{x}$ & $\\Theta(2 \\cdot T)$\\\\\n\t\tOcclusion \t\t\t& $R_{x_i}=f_c( \\mathbf{x} ) - f_c( \\mathbf{x}_{| x_i=0} )$ & - & $\\Theta(T^{2})$\\\\\n\t\tLRP \t\t\t\t& backward decomposition of the neurons' relevance & $f_c({\\mathbf x})$  & $\\Theta(2 \\cdot T)$ \\\\\n\t\tCD\t\t\t\t& linearization of the activation functions & $f_c({\\mathbf x})$ & $\\Theta(T^{2})$\\\\\t\t\n\t\t\\bottomrule\n\t\\end{tabular}%\n\t}\n\n\t\\caption{Overview of the considered explanation methods. The last column indicates the computational complexity to obtain one relevance value per input vector, or word, where $T$ is the length of the input sequence.}\n\n\t\\label{tab:methods_overview}\n\\end{table*}\n\n\\subsection{Contextual Decomposition}\n\nAnother method, specific to LSTMs, is contextual decomposition ({\\it CD}) \\cite{Murdoch:ICLR2018}. \nIt is based on a {\\it linearization} of the activation functions that enables to decompose the LSTM forward pass by distinguishing between two contributions: those made by a chosen contiguous {subsequence} (a word or a phrase) within the input sequence $\\mathbf{x}$, and those made by the remaining part of the input. \nThis decomposition results in a final hidden state vector $h_T$ (see the Appendix for a full specification of the LSTM architecture)\nthat can be rewritten as a sum of two vectors: $\\beta_T$ and $\\gamma_T$, where the former corresponds to the contribution from the ``relevant'' part of interest, and the latter stems from the ``irrelevant'' part.\nWhen the LSTM is followed by a linear output layer of the form $w_c^T h_T + b_c\\,$ for class $c$, then the relevance of a given word (or phrase) and for the {\\it target} class $c$, is given by the dot product:\n$w_c^T \\beta_T$.\n\nThe method is computationally expensive as it requires $T$ forward passes through the LSTM to compute one relevance value per word.\nAlthough it was recently extended to CNNs \\cite{Murdoch:ICLR2019,Godin:EMNLP2018}, it is yet not clear how to compute the {\\it CD} relevance in other recurrent architectures, or in networks with multi-modal inputs. \n\nSee Table~\\ref{tab:methods_overview} for an overview of the explanation methods considered in the present work.\n\n\n\n\n\\subsection{Methods not considered}\n\nOther methods to compute relevances include Integrated Gradients \\cite{Sundararajan:ICML2017}. It was previously compared against {\\it CD} in \\citet{Murdoch:ICLR2018}, and against the LRP variant of \\citet{Arras:WASSA2017} in \\citet{Poerner:ACL2018}, where in both cases it was shown to deliver inferior results.\nAnother method is DeepLIFT \\cite{Shrikumar:PMLR2017\n, however, according to its authors, DeepLIFT was not designed for multiplicative connections, and its extension to recurrent networks remains an open question\\footnote{Though \\citet{Poerner:ACL2018} showed that, when using {only} the Rescale rule of DeepLIFT, and combining it with the product rule proposed in \\citet{Arras:WASSA2017}, then the resulting explanations perform on-par with the LRP method of \\citet{Arras:WASSA2017}}.\nFor a comparative study of explanation methods with a main focus on feed-forward nets, see \\citet{Ancona:ICLR2018}\\footnote{Note that in order to redistribute the relevance through product layers, \\citet{Ancona:ICLR2018} simply relied on standard gradient backpropagation. Such a redistribution scheme is not appropriate for methods such as LRP, since it violates the relevance conservation property, hence their results for recurrent nets are not conclusive.}. For a broad evaluation of explanations, including several recurrent architectures, we refer to \\citet{Poerner:ACL2018}. Note that the latter didn't include the {\\it CD} method of \\citet{Murdoch:ICLR2018}, and the LRP variant of \\citet{Ding:ACL2017}, which we compare here.\n\n\n\\section{Evaluating Explanations}\\label{section:evaluation}\n\n\\subsection{Previous work}\nHow to generally and objectively evaluate explanations, without resorting to {\\it ad-hoc} evaluation procedures that are domain and task specific, is still active research \\cite{Alishahi:2019,Belinkov:TACL2019}.\n\nIn computer vision, it has become common practice to conduct a perturbation analysis \\cite{Bach:PLOS2015,Samek:TNNLS2017,Shrikumar:PMLR2017,Lundberg:NIPS2017,Ancona:ICLR2018,Chen:ICML2018,Morcos:ICLR2018}: hereby a few pixels in an image are perturbated (e.g.\\ set to zero or blurred) according to their relevance (most relevant or least relevant pixels are perturbated first), and then the impact on the network's prediction is measured. \nThe higher the impact, the more accurate is the relevance.\n\nOther studies explored in which way relevances are consistent or helpful w.r.t.\\ human judgment \\cite{Ribeiro:KDD2016,Lundberg:NIPS2017,Nguyen:NAACL2018}. \nSome other works relied solely on the visual inspection of a few selected relevance heatmaps \\cite{Li:NAACL2016,Sundararajan:ICML2017,Ding:ACL2017}.\n\nIn NLP, \\citet{Murdoch:ICLR2018} proposed to measure the correlation between word relevances obtained on an LSTM, and the word importance scores obtained from a linear Bag-of-Words. However, the latter model ignores\nthe word ordering and context, which the LSTM can take into account, hence this type of evaluation is not adequate\\footnote{The same way \\citet{Murdoch:ICLR2018} try to ``match''  phrase-level relevances with n-gram linear classifier scores or human annotated phrases, but again this might be misleading, since the latter scores or annotations ignore the whole sentence context.}.\nOther evaluations in NLP are task specific. For example \\citet{Poerner:ACL2018} use the subject-verb agreement task proposed by \\citet{Linzen:TACL2016}, where the goal is to predict a verb's number, and use the relevances to verify that the most relevant word is indeed the correct subject (or a noun with the predicted number).\n\nOther studies include an evaluation on a synthetic task: \\citet{Yang:ICHI2018} generated random sequences of MNIST digits and train an LSTM to predict if a sequence contains zero digits or not,\nand verify that the explanation indeed assigns a high relevance to the zero digits' positions.\n\nA further approach uses randomization of the model weights and data as sanity checks \\cite{Adebayo:NIPS2018} to verify that the explanations are indeed dependent on the model and data.\nLastly, some evaluations are ``indirect'' and use relevances to solve a broader task, e.g.\\ to build document-level representations \\cite{Arras:PLOSONE2017}, or to redistribute predicted rewards in reinforcement learning \\cite{JAM:RUDDER2018}. \n\n\n\\begin{table*}\n\t%\n\t\\scriptsize\n\t\\center\n\t\\begin{tabular}{lcccc}\n\t\t\\toprule\t\t\n\t\t\n\t\t{}    \t\t\t\t        & $\\rho({R_{\\boldsymbol{x_a}}}, {n_a})$ \t& $\\rho({R_{\\boldsymbol{x_b}}}, {n_b})$\t\t& {\\tiny $ E[\\frac{\\vert {R_{\\boldsymbol{x_a}}} \\vert + \\vert {R_{\\boldsymbol{x_b}}} \\vert}{\\sum_{t}{ \\vert R_{\\boldsymbol{x_t}} \\vert }} ]$ }\t& {\\tiny $E[(({R_{\\boldsymbol{x_a}}}+{R_{\\boldsymbol{x_b}}})-y_{\\text{pred}})^2] $} \t\\\\\n\t\t\\rule{0pt}{2ex}{}\t\t\t& (in \\%)\t\t\t\t\t& (in \\%)\t\t \t\t\t& (in \\%) \t\t\t\t\t\t\t\t\t\t\t\t\t        \t\t\t& (``MSE'')\t\t\t\t\t\t\t\t\t\t\\\\\n\t\t\\rule{0pt}{1ex}{Toy Task Addition} & \\multicolumn{4}{c}{} \\\\\n\t\t\\cmidrule(l){1-5} \n\t\tGradient$\\times$Input \t\t\t& \\textbf{99.960} (0.017)\t\t\t& \\textbf{99.954} (0.019)\t\t\t& \\textbf{99.68} (0.53)\t\t\t&\t\\textbf{24.10}{\\boldmath${^{-4}}$} (8.10$^{-4}$)\t\\\\\t\t\n\t\tOcclusion \t\t\t\t& \\textbf{99.990} (0.004)\t\t\t& \\textbf{99.990} (0.004)\t\t\t& \\textbf{99.82} (0.27)\t\t\t&\t\\textbf{20.10}{\\boldmath$^{-5}$} (8.10$^{-5}$)\t\t\\\\\t \n\t\tLRP-prop\t\t\t\t& \\hspace{1mm}0.785 (3.619)\t\t\t& 10.111 (12.362) \t\t\t\t& 18.14 (4.23)\\hspace{1mm}\t\t&\t1.3 (1.0)\t\t\t\t\t\t\\\\\t \n\t\tLRP-abs\t\t\t\t\t& \\hspace{1mm}7.002 (6.224)\t\t\t& 12.410 (17.440) \t\t\t\t& 18.01 (4.48)\\hspace{1mm}\t\t&\t1.3 (1.0)\t\t\t\t\t\t\\\\\t \n\t\tLRP-half\t\t\t\t& 29.035 (9.478)\t\t\t\t& 51.460 (19.939) \t\t\t\t& 54.09 (17.53)\t\t\t\t&\t1.1 (0.3)\t\t\t\t\t\t\\\\\t \n\t\tLRP-all\t\t\t\t\t& \\textbf{99.995} (0.002)\t\t\t& \\textbf{99.995} (0.002)\t\t\t& \\textbf{99.95} (0.05)\\hspace{1mm} \t&\t\\textbf{2.10}{\\boldmath$^{-6}$} (4.10$^{-6}$) \t\t\\\\\t \n\t\tCD\t\t\t\t\t& \\textbf{99.997} (0.002)\t\t\t& \\textbf{99.997} (0.002)\t\t\t& \\textbf{99.92} (0.06)\\hspace{1mm}\t&\t\\hspace{1mm}\\textbf{4.10}{\\boldmath$^{-5}$} (12.10$^{-5}$)\t\t\\\\ \n\t\t\\rule{0pt}{3ex}{Toy Task Subtraction} & \\multicolumn{4}{c}{} \\\\\n\t\t\\cmidrule(l){1-5} \n\t\tGradient$\\times$Input \t\t\t& \\textbf{97.9} (1.6)\\hspace{1mm}\t\t& \\textbf{-98.8} (0.6)\\hspace{1mm}\t\t& \\textbf{98.3}  (0.6)\\hspace{1mm}\t&\t\\textbf{6.10}{\\boldmath$^{-4}$} (4.10$^{-4}$)\t\t\\\\\t\t\n\t\tOcclusion \t\t\t\t& 99.0 (2.0)\\hspace{1mm}\t\t\t& -69.0 (19.1)\t\t\t\t\t& 25.4  (16.8) \t\t\t\t&\t0.05 (0.08)\\hspace{1mm}\t\t\t\t\t\\\\\t \n\t\tLRP-prop\t\t\t\t& 3.1 (4.8) \t\t\t\t\t& \\hspace{1mm}-8.4 (18.9)\t\t\t& 15.0  (2.4)\\hspace{1mm}\t\t&\t0.04 (0.02)\\hspace{1mm}\t\t\t\t\t\\\\\n\t\tLRP-abs\t\t\t\t\t& 1.2 (7.6) \t\t\t\t\t& -23.0 (11.1)\t\t\t\t\t& 15.1  (1.6)\\hspace{1mm}\t\t&\t0.04 (0.002)\t\t\t\t\t\t\\\\\n\t\tLRP-half\t\t\t\t& \\hspace{1mm}7.7 (15.3) \t\t\t& -28.9 (6.4)\\hspace{1mm}\t\t\t& 42.3  (8.3)\\hspace{1mm}\t\t&\t0.05 (0.06)\\hspace{1mm}\t\t\t\t\t\\\\\n\t\tLRP-all\t\t\t\t\t& \\textbf{98.5} (3.5) \t\t\t\t& \\textbf{-99.3} (1.3)\\hspace{1mm}\t\t& \\textbf{99.3}  (0.6)\\hspace{1mm}\t&\t\\textbf{8.10}{\\boldmath$^{-4}$} (25.10$^{-4}$)\t\t\\\\\t \n\t\tCD\t\t\t\t\t& -25.9 (39.1)\t\t\t\t\t& -50.0 (29.2)\t\t\t\t\t& 49.4  (26.1) \t\t\t\t&\t0.05 (0.05)\\hspace{1mm}\t\t\t\t\t\\\\\t \n\t\t\\bottomrule\n\t\\end{tabular}\n\t\n\t\\caption{Statistics of the relevance w.r.t.\\ the input numbers $n_a$ and $n_b$ and the predicted output $y_{\\text{pred}}$, on toy arithmetic tasks. $\\rho$ denotes the correlation and $E$ the mean. Each statistic is computed over 2500 test data points. Reported are the mean (and standard deviation in parenthesis) over 50 trained LSTM models.}\n\n\t\\label{tab:toy_experiment}\n\\end{table*}\n\n\n\\subsection{Toy Arithmetic Task}\n\nAs a first evaluation, we ask the following question: if we add two numbers within an input sequence, can we recover from the relevance the true input values?\nThis amounts to consider the adding problem \\cite{Hochreiter:NIPS1996}, which is typically used to test the long-range capabilities of recurrent models \\cite{Martens:ICML2011,Le:Arxiv2015}.\nWe use it here to test the faithfulness of explanations.\nTo that end, we define a setup similar to \\citet{Hochreiter:NIPS1996}, but without {\\it explicit} markers to identify the sequence start and end, and the two numbers to be added. \nOur idea is that, in general, it is  not clear what the ground truth relevance for a marker should be,\nand we want only the relevant numbers in the input sequence to get a non-zero relevance.\nHence, we represent the input sequence $\\mathbf{x} = (\\boldsymbol{x_1}, \\boldsymbol{x_2},...,\\boldsymbol{x_T})$ of length $T$, with two-dimensional input vectors as:\n\\begin{equation*}\n(\\begin{smallmatrix} \n0\t\t& \t0\t&\t0 \t&      n_a     \t& \t0 \t\t& 0\t& 0\t        & n_{b}\t         & 0\t\t& 0    &   0   \\\\\nn_{1} \t\t& \t...\t& \tn_{a-1}\t&\t0\t&\t n_{a+1} \t& ...\t& n_{b-1}\t& 0        \t & n_{b+1} \t& ...  &  n_{T}  \n\\end{smallmatrix})%\n\\end{equation*}\nwhere the non-zero entries $n_t$ are random real numbers, and the two relevant positions $a$ and $b$ are sampled uniformly among $\\{1,...,T\\}$ with $a<b$.\n\nMore specifically, we consider two tasks that can be solved by an LSTM model with a hidden layer of size {\\it one} (followed by a linear output layer with no bias\\footnote{We omit the output layer bias since all considered explanation methods ignore it in the relevance computation, and we want to explain the ``full'' prediction function's value.}): the {\\it addition} of signed numbers ($n_t$ is sampled uniformly from $[-1,-0.5] \\cup [0.5, 1.0]$)\nand the {\\it subtraction} of positive numbers  ($n_t$ is sampled uniformly from $[0.5, 1.0]$\\footnote{We avoid small numbers by using 0.5 as a minimum magnitude only to simplify learning, since otherwise this would encourage the model weights to grow rapidly.}).\nIn the former case the target output $y$ is $n_a+n_b$, in the latter it is $n_a-n_b$. During training we minimize Mean Squared Error (MSE).\nTo ensure that train\/val\/test sets do not overlap we use 10000 sequences with lengths $T \\in \\{4,...,10\\}$ for training, 2500 sequences with $T \\in \\{11,12\\}$ for validation, and 2500 sequences with $T \\in \\{13,14\\}$ as test set.\nFor each task we train 50 LSTMs with a validation MSE $<10^{-4}$, the resulting test MSE is $<10^{-4}$.\n\nThen, given the model's predicted output $y_{\\text{pred}}$, we compute one relevance value $R_{\\boldsymbol{x_t}}$ per input vector $\\boldsymbol{x_t}$ (for the occlusion method we compute only {\\it Occlusion}$_\\text{f-diff}$ since the task is a regression; we also don't report {\\it Gradient} results since it performs poorly).\nFinally, we track the correlation between the relevances and the two input numbers $n_a$ and $n_b$. We also track the portion of relevance assigned to the relevant time steps, compared to the relevance for all time steps.\nLastly, we calculate the ``MSE'' between the relevances for the relevant positions $a$ and $b$ and the model's output.\nOur results are compiled in Table~\\ref{tab:toy_experiment}.\n\nInterestingly, we note that on the addition task several methods perform well and produce a relevance that correlates perfectly with the input numbers: {\\it Gradient$\\times$Input}, {\\it Occlusion}, {\\it LRP-all} and {\\it CD} (they are highlighted in bold in the Table).\nThey further assign all the relevance to the time steps $a$ and $b$ and almost no relevance to the rest of the input; and present a relevance that sum up to the predicted output.\nHowever, on subtraction, only {\\it Gradient$\\times$Input} and {\\it LRP-all} present a correlation of near one with $n_a$, and of near minus one with $n_b$. Likewise these methods assign only relevance to the relevant positions, and redistribute the predicted output entirely onto these positions. \n\nThe main difference between our addition and subtraction tasks, is that the former requires only summing up the first dimension of the input vectors and can be solved by a Bag-of-Words approach (i.e. by ignoring the ordering of the inputs), while our subtraction task is truly {\\it sequential} and requires the LSTM model to remember which number arrived first, and which number arrived second, via exploiting the gating mechanism. \n\nSince in NLP several applications require the word {\\it ordering} to be taken into account to accurately capture a sentence's meaning (e.g.\\ in sentiment analysis or in machine translation),\nour experiment, albeit being an abstract numerical task, is pertinent and can serve as a first sanity check to check whether the relevance can reflect the {\\it ordering} and the {\\it value} of the input vectors. \n\nHence we view our toy task as a minimal and unambiguous test (which besides being sequential, also exhibits a linear input-output relationship)\nthat acts as a necessary (though not sufficient) requirement for a recurrent neural network explanation method to be trustworthy in a more complex setup, where the ground truth relevance is less clear.\n\nFor the {\\it Occlusion} method, the unreliability is probably due to the fact that the neural network has always seen two ``relevant'' input numbers in the input sequence during training, and therefore gets confused when one of these inputs is missing at the time of the relevance computation (``out-of-sample'' effect).\nFor {\\it CD}, the weakness may come from the saturation of the activations, in particular of the gates, which makes their linearization induced by the {\\it CD} decomposition inaccurate.\n\n\n\\begin{table*}\n\t%\n\n\t\\center\n\t\\resizebox{\\textwidth}{!}{%\n\t\\begin{tabular}{l|cccccccccc}\n\t\\toprule\t\n\t\n\tAccuracy Change \\footnotesize (in \\%)           & random \t& Grad. \t& Grad.$\\times$Input     & LRP-prop      & LRP-abs          & LRP-half \t        & LRP-all \t    &   CD\t       & Occlusion$_\\text{f-diff}$      &  Occlusion$_\\text{P-diff}$       \\\\\n\t\\cmidrule(l){1-11} \t\t\t\t\t\t\t\t\t\t\t        \t\t\t\n\t{decreasing order \\footnotesize (std$<$16)}  \t& 0\t\t&    35    \t&   66   \t\t &       15       &      -1         &      -3            &      \\bf 97      &    \\bf 92        &           \\bf 96               &   \\bf  100                       \\\\\n\t{increasing order \\footnotesize (std$<$5)}      & 0\t\t&    -18    \t&   31  \t         &       11       &      -1         &       3            &      \\bf 49      &    36            &           \\bf 50               &   \\bf 100                        \\\\\n\t\\bottomrule\n\t\\end{tabular}%\n\t}\n\t\n\n\t\\caption{Average change in accuracy when removing up to 3 words per sentence, either in {\\it decreasing} order of their relevance (starting with correctly classified sentences), or in {\\it increasing} order of their relevance (starting with falsely classified sentences). In both cases, the relevance is computed with the {\\it true} class as the {\\it target} class. Results are reported {\\it proportionally} to the changes for i) random removal (0\\% change)  and  ii) removal based on {\\it Occlusion}$_\\text{P-diff}$ (100\\% change). For all methods, the higher the reported value the better. We boldface those methods that perform on-par with the occlusion-based relevances.}\n\n\t\\label{tab:sentiment_word_removal}\n\\end{table*}\n\n\n\\subsection{5-Class Sentiment Prediction}\\label{section:sentiment}\nAs a sentiment analysis dataset, we use the Stanford Sentiment Treebank \\cite{Socher:EMNLP2013} which contains labels (very negative~$--$, negative~$-$, neutral~$0$, positive~$+$, very positive~$++$) for resp.\\ 8544\/1101\/2210 train\/val\/test sentences and their constituent phrases.\nAs a classifier we employ the bidirectional LSTM from \\citet{Li:NAACL2016}\\footnote{\\url{https:\/\/github.com\/jiweil\/Visualizing-and-Understanding-Neural-Models-in-NLP}}, which achieves 82.9\\% binary, resp. 46.3\\% five-class, accuracy on full sentences. \n\n\\textbf{Perturbation Experiment}. \nIn order to evaluate the {\\it selectivity} of word relevances, we perform a perturbation experiment  aka ``pixel-flipping`` in computer vision \\cite{Bach:PLOS2015,Samek:TNNLS2017}, i.e.\\ we remove words from the input sentences according to their relevance, and track the impact on the classification performance.\nA similar experiment has been conducted in previous NLP studies \\cite{arras:ACL2016,Nguyen:NAACL2018,Chen:ICML2018}; and besides, such type of experiment can be seen as the input space pendant of {\\it ablation}, which is commonly used to identify ``relevant'' intermediate neurons, e.g.\\ in \\citet{Lakretz:NAACL2019}.\nFor our experiment we retain test sentences with a length $\\geq$ 10 words (i.e.\\ 1849 sentences), and remove 1, 2, and 3 words per sentence\\footnote{In order to remove a word we simply discard it from the input sentence and concatenate the remaining parts. We also tried setting the word embeddings to zero, which gave us similar results.}, according to their relevance obtained on the original sentence with the {\\it true} class as the {\\it target} class.\nOur results are reported in Table~\\ref{tab:sentiment_word_removal}.\nNote that we distinguish between sentences that are initially correctly classified, and those that are initially falsely classified by the LSTM model.\nFurther, in order to condense the ``ablation'' results in a single number per method, we compute the accuracy decrease (resp.\\ increase) {\\it proportionally} to two cases: i) random removal, and ii) removal according to {\\it Occlusion}$_\\text{P-diff}$. \nOur idea is that random removal is the least informative approach, while {\\it Occlusion}$_\\text{P-diff}$\nis the most informative one, since the relevance for the latter is computed in a similar way to the perturbation experiment itself, i.e.\\ by deleting words from the input and tracking the change in the classifier's prediction. \nThus, with this normalization, we expect the accuracy change (in \\%) to be mainly rescaled to the range $[0, 100]$.\n\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=0.66\\textwidth]{figures\/PCA.pdf}\n\\caption{PCA projection of sentence-level representations built on top of word embeddings that were linearly combined using their respective relevance. {\\it Avg} corresponds to simple averaging of word embeddings.\nFor {LRP ${{h}_T}$} the last time step hidden layer was reweighted by its relevance. In parenthesis we indicate the percentage of variance explained by the first two PCA components (those that are plotted) and by the third PCA component.\nThe resulting representations were roughly ordered (row-wise) from less structured to more structured.}\n\\label{fig:PCA}\n\\end{figure*}\n\nWhen removing words in decreasing order of their relevance, we observe that {\\it LRP-all} and {\\it CD} perform on-par with the occlusion-based relevance, with near 100\\% accuracy change, followed by {\\it Gradient$\\times$Input} which performs only 66\\%.  \n\nWhen removing words in increasing order of their relevance (which mainly corresponds to remove words with a negative relevance), {\\it Occlusion}$_\\text{P-diff}$ performs best, followed by {\\it Occlusion}$_\\text{f-diff}$ and {\\it LRP-all} (both around 50\\%), then {\\it CD} (36\\%).\nUnsurprisingly, {\\it Gradient} performs worse than random, since its relevance is positive and thus low relevance is more likely to identify unimportant words for the classification (such as stop words), rather than identify words that {\\it contradict} a decision, as noticed in \\citet{Arras:WASSA2017}.\nLastly {\\it Occlusion}$_\\text{f-diff}$ is less informative than {\\it Occlusion}$_\\text{P-diff}$, since the former is not normalized by the classification scores for all classes. \n\nThis analysis revealed that methods such as {\\it LRP-all} and {\\it CD} can detect influential words supporting (resp. contradicting) a specific classification decision, although they were not tuned towards the perturbation criterion, as opposed \nto {\\it Occlusion} (which can be seen as the brute force approach to determine the inputs the model is the most sensitive to),\nwhereas gradient-based methods are less accurate in this respect.\nRemarkably {\\it LRP-all} only require one forward and backward pass to provide this information. \n\n\n\\textbf{Sentence-Level Representations}.\nIn addition to testing selectivity, we explore if the word relevance can be leveraged to build sentence-level representations that present some regularities akin word2vec vectors.\nFor this purpose we linearly combine word embeddings using their respective relevance as weighting\\footnote{W.l.o.g.\\ we use here the {\\it true} class as the {\\it target} class, since the classifier's 5-class performance is very low. In a practical use-case one would use the {\\it predicted} class instead.}.\nFor methods such as LRP and {\\it Gradient$\\times$Input} that deliver also relevances for {\\it single} variables, we perform an element-wise weighting, i.e.\\ we construct the sentence representation as:\n$ \\sum_t {R_{\\boldsymbol{x_t}}} \\odot {\\boldsymbol{x_t}}$.\nFor every method we report the best performing variant from previous experiments, i.e.\\ {\\it Occlusion}$_\\text{P-diff}$, {\\it Gradient$\\times$Input}, {\\it CD} and {\\it LRP-all}.\nAdditionally we report simple averaging of word embeddings (we call it {\\it Avg}).\nFurther, for LRP, we consider an element-wise reweighting of the last time step hidden layer $h_T$ by its relevance, since LRP delivers also a relevance for each intermediate neuron (we call it LRP $h_T$).\nWe also tried using $h_T$ directly: this gave us a visualization similar to {\\it Avg}.\nThe resulting 2D whitened PCA projections of the test set sentences are shown in Fig.~\\ref{fig:PCA}.\n\nQualitatively LRP delivers the most structured representations, although for all methods the first two PCA components explain most of the data variance.\nIntuitively it makes also sense that the neutral sentiment is located between the positive and negative sentiments, and that the very negative and very positive sentiments depart from their lower counterparts in the same vertical direction.\n\nThe advantage of having such regularities emerging via PCA projection, is that the sentence\/phrase semantics might be investigated visually, without requiring any nonlinear dimensionality reduction like t-SNE (typically used to explore the representations learned by recurrent models, e.g. in \\citealp{Cho:EMNLP2014,Li:NAACL2016}). Such representations might also be useful in information retrieval settings, \nwhere one could retrieve similar sentences\/phrases by employing standard euclidean distance. \n\n\n\n\\begin{table*}\n\t%\n\n\t\\center\n\t\\resizebox{0.93\\textwidth}{!}{%\n\t\\begin{tabular}{ll|c|llll|r}\n\t\t\\toprule\t\n\t\t\\multicolumn{2}{l|}{Composition}          \t& Predicted \t& Heatmap  \t\t\t\t\t\t\t\t\t\t&\t\\multicolumn{3}{l|}{Relevance}   \t\t\t\t& \\# samples\\\\\n\t\t\\midrule\n\t\t1.& ``negated positive sentiment''\t\t& $-$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 14.8cm 0.9cm]{figures\/type1.pdf}}\t&\t2.5$_{\\;0.3}$ & -1.4$_{\\;0.5}$ & {}\t\t\t\t& 213\t\\\\\t\t\n\t\t2.& ``amplified positive sentiment''\t\t& $++$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 14.8cm 0.9cm]{figures\/type2.pdf}}\t&\t1.1$_{\\;0.3}$ & 4.5$_{\\;0.7}$ & {}\t\t\t\t& 347\t\\\\\n\t\t3.& ``amplified negative sentiment''\t\t& $--$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 14.8cm 0.9cm]{figures\/type3.pdf}}\t&\t0.8$_{\\;0.2}$ & 4.3$_{\\;0.6}$ & {}\t\t\t\t& 173\t\\\\\n\t\t4.& ``negated amplified positive sentiment''\t& $-$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 14.8cm 0.9cm]{figures\/type4.pdf}}\t&\t2.74$_{\\;0.54}$ & -0.34$_{\\;0.17}$ & -2.00$_{\\;0.40}$\t\t& 1745\t\\\\\n\t\t\\bottomrule\n\t\\end{tabular}%\n\t}\n\n\t\\caption{Typical heatmaps for various types of semantic compositions (indicated in first column), computed with the {\\it LRP-all} method. The LSTM's {\\it predicted} class (second column) is used as the {\\it target} class. The remaining columns contain the average heatmap (positive relevance is mapped to red, negative to blue, and the color intensity is normalized to the maximum absolute relevance), the corresponding word relevance mean (and std as subscript), and the number of bigrams (resp.\\ trigrams) considered for each type of composition.}\n\n\t\\label{tab:semantic_composition}\n\\end{table*}\n\n\n\\begin{table}\n\t%\n\n\n\t\\resizebox{0.92\\columnwidth}{!}{%\n\t\\begin{tabular}{l|c|l}\n\t\t\\toprule\t\n\t\t{N\\textsuperscript{\\underline{o}}}\t& Predicted \t& Heatmap  \t\t\t\t\t\t\t\t\t\t\t\t\\\\\n\t\t\\midrule\n\t\t1 \t& $--$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 10.9cm 0.9cm]{figures\/sentence167_original.pdf}}\t\\\\\t\t\n\t\t1a\t& $+$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 10.9cm 0.9cm]{figures\/sentence167_a.pdf}}\t\t\\\\\n\t\t1b\t& $-$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 10.9cm 0.9cm]{figures\/sentence167_b.pdf}}\t\t\\\\\n\t\t1c\t& $--$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 10.9cm 0.9cm]{figures\/sentence167_c.pdf}}\t\t\\\\\n\t\t\\midrule\n\t\t2 \t& $++$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 10.9cm 0.9cm]{figures\/sentence219_original.pdf}}\t\\\\\t\t\n\t\t2a\t& $++$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 10.9cm 0.9cm]{figures\/sentence219_a.pdf}}\t\t\\\\\n\t\t2b\t& $--$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 10.9cm 0.9cm]{figures\/sentence219_b.pdf}}\t\t\\\\\n\t\t2c\t& $++$ \t\t& {\\includegraphics[clip=true, trim=20mm 28.3cm 10.9cm 0.9cm]{figures\/sentence219_c.pdf}}\t\t\\\\\n\t\t\\bottomrule\n\t\\end{tabular}%\n\t}\n\n\t\\caption{Misclassified test sentences (1 and 2), and manually constructed sentences (1a-c, 2a-c). The LSTM's {\\it predicted} class (second column) is used as the {\\it target} class for the {\\it LRP-all} heatmaps.}\n\n\t\\label{tab:misclassified}\n\\end{table}\n\n\\section{Interpreting Single Predictions}\\label{section:inspection}\n\nNext, we analyze single predictions using the same task and model as in Section~\\ref{section:sentiment}, and illustrate the usefulness of relevance visualization with {\\it LRP-all}, which is the method that performed well in both our previous quantitative experiments.\n\n\\textbf{Semantic Composition}.\nWhen dealing with real data, one typically has no ground truth relevance available. \nAnd the visual inspection of single relevance heatmaps can be counter-intuitive for two reasons:\nthe relevance is not accurately reflecting the reasons for the classifier's decision (the explanation method is bad), or the classifier made an error (the classifier doesn't work as expected).\nIn order to avoid the latter as much as possible, we automatically constructed bigram and trigram samples, which are built {\\it solely} upon the classifier's predicted class, and visualize the resulting average relevance heatmaps\nfor different types of semantic compositions in Table~\\ref{tab:semantic_composition}.\nFor more details on how these samples were constructed we refer to the Appendix, note though that in our heatmaps the negation \\texttt{<not>}, the intensifier \\texttt{<very>} and the sentiment words act as placeholders\nfor words with similar meanings, since the representative heatmaps were averaged over several samples.\nIn these heatmaps one can see that, to transform a positive sentiment into a negative one, the negation is predominantly colored as red, while the sentiment word is highlighted in blue, which intuitively makes sense since the explanation is computed {\\it towards} the negative \nsentiment, and in this context the negation is responsible for the sentiment prediction.\nFor sentiment intensification, we note that the amplifier gets a relevance of the same sign as the amplified word, indicating the amplifier is {\\it supporting} the prediction for the considered target class, but still  has less importance for the decision than\nthe sentiment word itself (deep red colored).\nBoth previous identified patterns also reflect consistently in the case of a negated amplified positive sentiment.\n\n\n\\textbf{Understanding Misclassifications}.\nLastly, we inspect heatmaps of misclassified sentences in Table~\\ref{tab:misclassified}.\nIn sentence 1, according to the heatmap, the classifier didn't take the negation \\texttt{never} into account, although it identified it correctly in sentence 1b. We postulate this is because of the strong sentiment\nassigned to \\texttt{fails} that overshadowed the effect of \\texttt{never}.\nIn sentence 2, the classifier obviously couldn't grasp the meaning of the words preceding \\texttt{must-see}.\nIf we use a negation instead, we note that it's taken into account in the case of \\texttt{neither} (2b), but not in the case of \\texttt{never} (2c),\nwhich illustrates the complex dynamics involved in semantic composition, and that the classifier \nmight also exhibit a bias towards the types of constructions it was trained on, which might then feel more ``probable'' or ``understandable'' to him.\n\nBesides, during our experimentations, we empirically found that the {\\it LRP-all} explanations are more helpful when using the classifier's {\\it predicted} class as the {\\it target} class (rather than the sample's {\\it true} class), which intuitively makes sense since it's the class the model is the most confident about.\nTherefore, to understand the classification of single samples, we generally recommend this setup.\n\n\n\\section{Conclusion}\\label{section:conclusion}\n\nIn our experiments with standard LSTMs, we find that the LRP rule for multiplicative connections introduced in \\citet{Arras:WASSA2017} performs consistently better than other recently proposed rules, such as the one from \\citet{Ding:ACL2017}.\nFurther, our comparison using a 5-class sentiment prediction task highlighted that LRP is not equivalent to {\\it Gradient$\\times$Input} (as sometimes inaccurately stated in the literature, e.g.\\ in \\citealp{Shrikumar:PMLR2017}) and is more selective than the latter, which is consistent\nwith findings of \\citet{Poerner:ACL2018}.\nIndeed, the equivalence between {\\it Gradient$\\times$Input} and LRP holds only {if} using the $\\epsilon$-rule with no stabilizer ($\\epsilon=0$), and if the network contains {\\it only} ReLU activations and max pooling as non-linearities \\cite{Kindermans:arxiv2016,Shrikumar:arxiv2016}.\nWhen using other LRP rules, or if the network contains other activations or product non-linearities (such as this is the case for LSTMs), then the equivalence does not hold (see \\citet{Montavon:DSP2018} for a broader discussion).\n\nBesides, we discovered that a few methods such as Occlusion \\cite{Li:ArXiv2017} and CD \\cite{Murdoch:ICLR2018} are not reliable and get inconsistent results on a simple toy task using an LSTM with only {\\it one} hidden unit.\n\nIn the future, we expect decomposition-based methods such as LRP to be further useful to analyze character-level models, to explore the role of single word embedding dimensions, and to discover important hidden layer neurons.\nCompared to attention weights (such as \\citealp{Bahdanau:ICLR2015,Xu:ICML2015,OSMAN2019}), decomposition-based explanations take into account all intermediate layers of the neural network model, and can be related to a specific class.\n\n\n\n\\section*{Acknowledgments}\nWe thank Gr{\\'e}goire Montavon for helpful discussions.\nThis work was supported by the German Federal Ministry for Education and Research through the Berlin Big Data Centre (01IS14013A), the Berlin Center for Machine Learning (01IS18037I) and the TraMeExCo project (01IS18056A). \nPartial funding by DFG is acknowledged (EXC 2046\/1, project-ID: 390685689). This work was also supported by the Information \\& Communications Technology Planning \\& Evaluation (IITP) grant funded by the Korea government (No. 2017-0-00451).\n\n\n\\bibliographystyle{acl_natbib}\n\\input{paper.bbl}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Mixed twistor structures}\n\nCarlos Simpson\n\\newline\nCNRS, UMR 5580, Universit\\'e Paul Sabatier\\newline\n31062 Toulouse CEDEX, France\n\n\\bigskip\n\n\nThe purpose of this paper is to introduce the notion of {\\em mixed twistor\nstructure} as a generalization of the notion of mixed Hodge structure.  Recall\nthat a mixed Hodge structure is a vector space $V$ with three filtrations $F$,\n$F'$ and $W$ (the first two decreasing, the last increasing) such that the two\nfiltrations $F$ and $F'$ induce $i$-opposed filtrations on $Gr^W_i(V)$.\n(Generally $(V,W)$ is required to have a real structure and $F'$ is the\ncomplex conjugate of $F$ but that is not very relevant for us here.)\nGiven a MHS $(V,W,F,F')$ we can form the {\\em Rees bundle} $E:=\\xi (V,F,F')$ over\n${\\bf P} ^1$ \\cite{NAHT} \\cite{SantaCruz}. In brief this is obtained from the\ntrivial\nbundle $V\\times {\\bf P} ^1$ by using $F$ to make an elementary transformation over\n$0$ and $F'$ to make an elementary transformation over $\\infty$.\nThe bundle $E$ is graded by strict subbundles which we denote $W_{i}E$ and\nthe condition of opposedness of the filtrations is equivalent to the condition\nthat $Gr^W_i(E)$ be semistable of slope $i$ on ${\\bf P} ^1$ (in other words a direct\nsum of copies of ${\\cal O} _{{\\bf P} ^1}(i)$).  The notion of mixed twistor structure is\nsimply obtained by abstracting this situation: an MTS is a pair consisting of a\nbundle $E$ over ${\\bf P} ^1$ and a filtration by strict subbundles $W_iE$ such that\nthe $Gr^W_i(E)$ are semistable of slope $i$.  In the construction starting with\na mixed Hodge structure the resulting $(E,W)$ has an action of ${\\bf G}_m$ covering\nthe standard action on ${\\bf P} ^1$ and in fact the mixed Hodge structures are\nsimply the ${\\bf G}_m$-equivariant mixed twistor structures. Thus, in some sense, the\npassage from ``Hodge'' to ``Twistor'' is simply forgetting to have an action of\n${\\bf G}_m$.  This principle occured already, in a primitive way, in the passage from\nsystems of Hodge bundles (cf \\cite{YMTU}) to Higgs bundles in \\cite{CVHS}.\n\n\n\nWe will give some generalizations of basic classical results for\nmixed Hodge structures, to the mixed twistor setting.  The process of\nmaking these\ngeneralizations is relatively direct although some work must be done to develop\nthe appropriate notion of {\\em variation of mixed twistor structure}. The\noverall\nidea is that we have the following\n\n\\noindent\n{\\bf Meta-theorem} {\\em\nIf the words ``mixed Hodge structure'' (resp. ``variation of mixed Hodge\nstructure'') are replaced by the words ``mixed twistor structure'' (resp.\n``variation of mixed twistor structure'') in the hypotheses and conclusions of\nany theorem in Hodge theory, then one obtains a true statement. The proof of the\nnew statement will be analogous to the proof of the old statement.}\n\nWe don't prove this meta-theorem but support it with several examples using the\nbasic theorems of mixed Hodge theory.\n\nThe utility\nof the notion of mixed twistor structure, and of the above meta-theorem, is to\nmake possible a theory of weights for various things surrounding arbitrary\nrepresentations of the fundamental group of a smooth projective variety,\nwhere up\nuntil now the theory of weights has only been available for variations of Hodge\nstructure. This phrase needs the further explanation that a harmonic bundle\n(cf \\cite{Hitchin1} \\cite{Corlette} \\cite{HBLS}) yields a variation of pure\ntwistor structure (i.e. of mixed twistor structure with only one nonzero\nweight-graded quotient) and in fact, up to choosing the weight a  variation of\npure twistor structure is essentially the same thing as a harmonic bundle. In\nparticular, any irreducible representation of $\\pi _1(X,x)$ for a compact\nK\\\"ahler manifold $X$, underlies a variation of pure twistor structure unique up\nto shift of weight (i.e. tensorization with constant pure rank one twistor\nstructures). This is explained in Lemma \\ref{harmonic} below.\n\nApart from the above meta-theorem as a way of obtaining new statements, a\npossible\narea where we obtain a new type of object is the following: over the moduli\nspace\n$M_B$ of representations of $\\pi _1(X)$ for a compact K\\\"ahler variety $X$, we\nmay have several families of mixed twistor structures, for example cohomology of\nopen or singular subvarieties of $X$ with coefficients in the VMTS corresponding\nto $\\rho \\in M_B$, or some types of rational homotopy invariants relative to the\nrepresentation $\\rho$ (such as the relative Malcev completion or its analogues\nfor higher homotopy).  These natural families of MTS give classifying maps\n$M_B\\rightarrow {\\cal MTS}$ to the moduli stack of mixed twistor structures\n(the maps will not usually be algebraic but should be real analytic, for\nexample).  It might be interesting to study these classifying maps.\n\nThe terminology ``twistor'' comes from the particular case of weight one twistor\nstructures with a certain kind of real structure ({\\em antipodal}).  These are\nequivalent to the twistor bundles over ${\\bf P} ^1$ of quaternionic vector spaces,\nsee \\cite{HKLR} for a nice exposition.\nDeligne in \\cite{DeligneLetter} originally explained to me how to associate a\nquaternionic vector space to a weight one real Hodge structure and how to\ninterpret the twistor space for Hitchin's hyperk\\\"ahler structure in terms\nof moduli of $\\lambda$-connections, a notion very closely related to the Hodge\nfiltration on the de Rham complex. This eventually led to a study of the formal\nneighborhood of the twistor lines in the twistor space, to an interpretation\nin terms of mixed Hodge structures (for the case of the moduli space of\nrepresentations), and consequently to the present definition. In \\S 7 below we\ndiscuss the relation between mixed twistor structures and the formal power\nseries for hyperk\\\"ahler (or more precisely any hypercomplex) structures at a\npoint. The case of the moduli space of\nrepresentations of $\\pi _1$ of a smooth projective variety is taken up\nbriefly in\n\\S 8.\n\n\nHere is an outline of the paper.\n\n\n{\\em \\S 1---Mixed twistor structures:}\ndefinition of mixed twistor structures, and the relation with mixed Hodge\nstructures. The theorem that the category of MTS is abelian. As an aside at the\nend we point out that this theorem works with ${\\bf P} ^1$ replaced by any\nprojective variety as base (this remark is not used later, though).\n\n{\\em \\S 2---Real structures:}\ntwo different kinds of real structures for a mixed twistor structure: circular\nand antipodal. The weight one pure antipodal real twistor structures are the\nsame as the quaternionic vector spaces.\n\n{\\em \\S 3---Variations of mixed twistor structure:}\ndefinition of  ${\\cal C} ^{\\infty}$ families of\nmixed twistor structure, variations of mixed twistor structure, polarizations,\nreal structures and a holomorphic interpretation as a pair of\n$\\lambda$-connections on the two standard affine lines in ${\\bf P} ^1$.\n\n{\\em \\S 4---Cohomology\nof smooth compact K\\\"{a}hler manifolds with VMTS coefficients:}\nwe treat the cohomology of smooth compact K\\\"ahler manifolds, first with\nvariations of pure twistor structure as coefficients, then with mixed\ncoefficients. We give a holomorphic interpretation using $\\lambda$-connections.\n\n\n{\\em \\S 5---Cohomology of open and singular varieties:}\ngeneralization of the basic theorems about cohomology of open and singular\nvarieties to the mixed twistor case. This involves a notion of {\\em mixed\ntwistor\ncomplex}. A somewhat new point here involves the notion of {\\em patching}\ncomplexes of sheaves on open sets of ${\\bf P} ^1$; this replaces certain arguments\ninvolving the real structure in the theory of mixed Hodge complexes.\n\n{\\em \\S 6---Nilpotent orbits and the limiting mixed twistor structure:}\nwe give a conjectural version of the nilpotent orbit theorem relating\nthe degeneration of a harmonic bundle on a punctured disc to a limiting mixed\ntwistor structure.\n\n{\\em \\S 7---Jet bundles of hypercomplex manifolds:}\ndescription of the formal germ of a hypercomplex structure in the\nneighborhood of\na point by a mixed twistor structure on the jet space at the point.  This comes\nfrom looking at the jet bundle of the twistor space along a twistor line.\nFollowing the philosophy described above of looking at classifying maps, we\ndefine\nthe {\\em Gauss map} from a hypercomplex manifold into the moduli stack for mixed\ntwistor structures on the jet spaces.\n\n{\\em \\S 8---The moduli space of representations:}\nthis is a discussion similar to that of the previous section, for moduli spaces\nof representations of fundamental groups of smooth projective varieties.\nUsing the family of moduli spaces $M_{Del}$ of \\cite{SantaCruz} as a replacement\nfor the twistor space, we are able to treat singular points: we get a mixed\ntwistor structure on the generalization of the jet space at a singular point,\nand this determines the formal neighborhood of a twistor line.\n\n{\\em Notations:}\nwe work in the analytic category of complex analytic spaces (usually smooth\ncomplex varieties) and when talking about projective varieties we mean the\nassociated analytic varieties with the usual topology.\n\n\n\n\\numero{Mixed twistor structures}\n\nFix the projective line ${\\bf P} ^1$ with points $0,1,\\infty$, with the standard\nline bundle ${\\cal O} _{{\\bf P} ^1}(1)$.\n\nA {\\em twistor structure} is a vector bundle $E$ on ${\\bf P} ^1$.  The {\\em\nunderlying vector space} is $E_1$, the fiber of the bundle $E$ over $1\\in {\\bf P}\n^1$.  We say that a twistor structure $E$ is {\\em pure of weight $w$} if $E$ is\na semistable vector bundle of slope $w$, which is equivalent to saying that $E$\nis a direct sum of copies of ${\\cal O} _{{\\bf P} ^1}(w)$.\n\nA {\\em mixed twistor structure} is a twistor structure $E$ filtered by an\nincreasing sequence of strict subbundles $W_iE$ such that for all\n$i$, $Gr^W_i(E)=\nW_iE \/ W_{i-1}E$ is pure of weight $i$. The filtration $W_{\\cdot}E$ is called\nthe {\\em weight filtration}.  A mixed twistor structure is said to be {\\em pure}\nif the associated graded is nontrivial in only one degree.\n\n\n\\subnumero{Relation with mixed Hodge structures}\n\nAs described in \\cite{NAHT} and (\\cite{SantaCruz} \\S 5), if $V$ is a\ncomplex vector space with two decreasing filtrations $F$ and $F'$, then we\nobtain\na vector bundle $\\xi (V; F, F')$ on ${\\bf P} ^1$. This is the {\\em twistor structure\nassociated to the pair of filtrations $F$ and $F'$}.  The two filtrations define\na pure Hodge structure of weight $w$ if and only if $\\xi (V; F, F')$\nis a pure twistor structure of weight $w$.\n\nSuppose $V$ is a vector space with three filtrations $W_{\\cdot}$ (increasing)\nand $F$ and $F'$ (decreasing). Then putting\n$$\nW_i \\xi (V; F,F'):= \\xi (W_iV; F, F')\n$$\nwe obtain a filtration by strict subbundles. The associated-graded\n$Gr ^W_i(\\xi (V; F,F')$ is pure of weight $w$ if and only if $F$ and $F'$ induce\na pure Hodge structure of weight $w$ on $Gr ^W_i(V)$.\n\nThe condition in the previous paragraph is what we call a {\\em complex mixed\nHodge structure} (a definition which should have been made long ago).  Note\nthat a\nreal mixed Hodge structure is simply a real vector space $V_{{\\bf R}}$ with\nfiltration\n$W_{\\cdot}$ and complex filtration $F$ of  $V_{{\\bf C}}$ such that $W, F,\n\\overline{F}$ form a complex mixed Hodge structure on $V_{{\\bf C}}$.\nSimilarly if $V,W,F,F'$ is a complex MHS then $V\\oplus \\overline{V}$ has a\nstructure of real MHS (using the real structure which interchanges $V$ and\n$\\overline{V}$).\n\nWe have the following characterization.\n\n\\begin{lemma}\nIf $V_{{\\bf R}}$ is a real vector space with increasing real filtration $W$, and\nwith complex filtration (decreasing) $F$ of $V_{{\\bf C}}$, then these data define a\nreal mixed Hodge structure if and only if $\\xi (V_{{\\bf C}}; F, \\overline{F})$ with\nthe induced filtration $W_{\\cdot}\\xi (V_{{\\bf C}}; F, \\overline{F})$ is a mixed\ntwistor structure.\n\\end{lemma}\n\\hfill $\/$\\hspace*{-.1cm}$\/$\\hspace*{-.1cm}$\/$\\vspace{.1in}\n\n\nThe group ${\\bf G}_m$ acts on ${\\bf P} ^1$ by translation. Denote the morphism of this\naction by\n$$\n\\mu : {\\bf G}_m \\times {\\bf P} ^1 \\rightarrow {\\bf P} ^1.\n$$\nA {\\em ${\\bf G}_m$-equivariant mixed twistor\nstructure} is a mixed twistor structure $(E, W)$ together with an isomorphism\nof filtered coherent sheaves on ${\\bf G}_m \\times {\\bf P} ^1$\n$$\n\\rho : \\mu ^{\\ast}(E,W)  \\cong p_2^{\\ast}(E,W)\n$$\nsuch that the two resulting morphisms on ${\\bf G}_m \\times {\\bf G}_m \\times {\\bf P} ^1$\n$$\n(\\mu \\circ \\mu )^{\\ast}(E,W) \\cong p_3^{\\ast}(E, W)\n$$\nare equal.  If we view $E$ as a filtered vector bundle over ${\\bf P} ^1$ this just\nmeans that the action of ${\\bf G}_m$ is lifted to an action on the total space of the\nbundle preserving the filtration.\n\nA {\\em morphism} of equivariant mixed twistor structures is a morphism of\nfiltered bundles compatible with the action.\n\n\n\\begin{proposition}\nThe category of ${\\bf G}_m$-equivariant mixed twistor structures is naturally\nequivalent to the category of complex mixed Hodge structures (see below).\n\\end{proposition}\n{\\em Proof:}\nAs was seen in \\cite{NAHT} and \\cite{SantaCruz} a bundle over ${\\bf P} ^1$\nequivariant for the action of ${\\bf G}_m$ is the same thing as a vector space $V=E_1$\ntogether with two decreasing filtrations $F$ and $F'$: the  bundle $E$ is\nrecovered as  $\\xi (V, F, F')$.\n\nRestriction to the fiber $V= E_1$ over $1\\in\n{\\bf P} ^1$ induces an inclusion-preserving bijection between the set of strict\nsubsheaves of $V$ preserved by ${\\bf G}_m$, and the set of subspaces of $V$. To see\nthis, the restriction to the fiber may be viewed as the composition of\nrestriction to ${\\bf G}_m\\subset {\\bf P} ^1$ then restriction to the fiber. The second\narrow is obviously a bijection between subsheaves of $E|_{{\\bf G}_m}$ preserved by\n${\\bf G}_m$ and subspaces of $E_1$ (using the action of ${\\bf G}_m$). Note by the way\nthat any equivariant subsheaf of $E|_{{\\bf G}_m}$ is automatically strict.  Recall\nthat a strict subsheaf of a vector bundle over a curve is determined by its\nrestriction to any Zariski open set; this implies that restriction is a\nbijection\nbetween all strict subsheaves of $E$ and all strict subsheaves of\n$E|_{{\\bf G}_m}$ (and\nthis bijection preserves the subset of those which are equivariant).\n\nIn view of this bijection, we obtain a bijection between the set of\nincreasing\nfiltrations of $E$ by strict subsheaves preserved by ${\\bf G}_m$, and the set of\nincreasing filtrations of $V$ by subspaces.  In view of this and  the first\nparagraph of the proof, an equivariant  vector bundle $E$ over ${\\bf P} ^1$\nwith increasing filtration by strict subsheaves $W_nE$ is the same thing as a\nvector space $V=E_1$ together with an increasing filtration $W_nV$ and\ntwo decreasing filtrations $F$ and $F'$.  The construction in the reverse\nsense is the Rees bundle construction $E = \\xi (V, F, F')$. Finally note that\n$$\nGr ^{WE}_n(E) = \\xi ( Gr ^{WV}_n(V), F_{Gr}, F'_{Gr})\n$$\nwhere $F_{Gr}$ and $F'_{Gr}$ are the filtrations induced on the\nassociated-graded (the Rees-bundle construction is compatible\nwith taking strict subquotients, as can be verified for each filtration $F$\nand $F'$ separately using splittings).  The two filtrations\n$F_{Gr}$ and $F'_{Gr}$ are $n$-opposed if\nand only if the Rees bundle is semistable of slope $n$.  Thus the data $(V,W,F,\nF')$ defines a mixed Hodge structure if and only if $(E,W)$ is a mixed twistor\nstructure.\n\nWe leave to the reader to verify full faithfulness of this correspondence\nfor morphisms.\n\\hfill $\/$\\hspace*{-.1cm}$\/$\\hspace*{-.1cm}$\/$\\vspace{.1in}\n\nWe don't actually use the above proposition anywhere but it is crucial for\nunderstanding the analogy between all of the results which we give below, and\nthe corresponding results for mixed Hodge structures. A somewhat pertinent\nremark is that our proofs for mixed twistor structures are all ${\\bf G}_m$-equivariant\nin case the input-data is ${\\bf G}_m$-equivariant. Thus our proofs give proofs of the\ncorresponding statements for mixed Hodge structures.  One hesitates to say that\nthese constitute ``new'' proofs since they are really just recopying the old\nproofs for mixed Hodge structures into this new language.\n\n\\subnumero{The abelian category of mixed twistor structures}\n\nThe first basic theorem in the theory of mixed Hodge structures is the fact that\nthe  category of MHS is abelian. This generalizes to mixed twistor structures\nand the proof is essentially the same.\n\n\\begin{lemma}\n\\label{abelian}\n{\\rm (cf \\cite{Hodge2}, 2.3.5(i))}\nThe category of mixed twistor structures is abelian.\n\\end{lemma}\n{\\em Proof:}\nSuppose $f: (E, W) \\rightarrow (E', W')$ is a morphism of mixed twistor\nstructures.\n\n{\\bf Step 1}\\,\\,\nDefine the {\\em cokernel} of $f$ to be $(E'', W'')$ where\n$E''$ is the cokernel of $f$ considered as a map of coherent sheaves, and $W''$\nis the filtration of $E''$ induced by $W'$.   For the moment, this is just a\ncoherent sheaf filtered by subsheaves. Let $A \\subset E$ denote the kernel\nsubsheaf of $f$.\n\nWe work by induction on the size of the interval where the two weight\nfiltrations $W$ and $W'$ are supported.  If this interval has size $1$ then\nthe cokernel is just the cokernel of a map of semistable sheaves of the same\nslope, so the cokernel is also semistable.  Now consider an interval size of at\nleast two.  Let $Gr ^W_i(E)$ and $Gr ^{W'}_i(E')$ be the highest nonzero pair of\nterms (in other words $i$ is the smallest integer such that $W_i=E$ and $W'_i =\nE'$). Then  we have an exact sequence $$\nGr ^W_i(E) \\rightarrow Gr ^{W'}_i(E')\n\\rightarrow Gr ^{W''}_i(E'') \\rightarrow 0.\n$$\nIn particular, $Gr ^{W''}_i(E'')$ is semistable of slope $i$.  Let $V_{i-1}\n\\subset E$ be the inverse image of $W'_{i-1}$. We have an exact sequence\n$$\n0 \\rightarrow W_{i-1} \\rightarrow V_{i-1} \\rightarrow B\\rightarrow 0\n$$\nwhere $B$ is the kernel of the map\n$$\nGr ^W_i(E) \\rightarrow Gr ^{W'}_i(E') .\n$$\nSince this is a map of semistable sheaves of slope $i$, the kernel $B$ is\nsemistable of slope $i$.  Thus $V_{i-1}$ is a mixed twistor structure (with the\nweight filtration induced by that of $E$).\n\nBy the induction hypothesis, the cokernel of the morphism\n$W_{i-1}\\rightarrow W'_{i-1}$ of mixed twistor structures concentrated in a\nsmaller interval (the $i$-th graded pieces are zero this time), is again a mixed\ntwistor structure concentrated in the same interval.  In particular, $W'_{i-1}\n\/f(W_{i-1})$ cannot support a morphism from a semistable sheaf of weight\n$\\geq i$.\nThus the morphism\n$$\nB = V_{i-1} \/W_{i-1} \\rightarrow W'_{i-1} \/f(W_{i-1})\n$$\nis zero.   Thus\n$$\nf(V_{i-1}) = f(W_{i-1}).\n$$\nNote that\n$$\nf(V_i) = f(E) \\cap W'_{i-1} = \\ker (W'_{i-1} \\rightarrow W''_{i-1}).\n$$\nTherefore we get an exact sequence\n$$\nW_{i-1} \\rightarrow W'_{i-1} \\rightarrow W'' _{i-1} \\rightarrow 0.\n$$\nFurthermore the weight filtration on $W'' _{i-1}$ is that induced by the weight\nfiltration of $W'_{i-1}$.  Hence $W'' _{i-1}$ is the cokernel of the map\n$W_{i-1} \\rightarrow W'_{i-1}$, and by our induction hypothesis this cokernel is\na mixed twistor structure.  Finally, $E''$ is an extension of $Gr ^{W''}_i(E'')$\n(which we have seen to be pure of weight $i$ at the start) by $W'' _{i-1}$, thus\n$E''$ with its weight filtration $W''$ is a mixed twistor structure. Along the\nway, we have proved that $E''$ is a bundle, for $Gr ^{W''}_i(E'')$ is a bundle\n(since it is a cokernel of a map of semistable bundles of the same slope)\nand $W''\n_{i-1}$ is a bundle by induction.\n\n\n\n\n\n\n\n{\\bf Step 2}\\, \\,\nLet $A \\subset E$ be the kernel of $f$ considered as a morphism\nof coherent sheaves, and put $W_i A := A \\cap W_i$. Note that $A$ is a strict\nsubbundle of $E$ and $W_iA$ are strict subbundles of $A$.  We would like\nto show that $(A, W_{\\cdot }A)$ is a mixed twistor structure, i.e. to\nshow that $Gr ^{WA}_i(A)$ is semistable of slope $i$.\n\n(Actually the following proof is just the dual of the proof of step 1.)\n\n\n\nLet $i$ denote the smallest number where either $W_i$ or $W'_i$ is nonzero.\nThus $W_i$ and $W'_i$ are both semistable bundles of slope $i$.\n\nLet $V_i := f^{-1}(W'_i)\\subset E$.\n\nWe have that $V_i\/W_i$ is the kernel of the morphism of mixed twistor\nstructures $E\/W_i \\rightarrow E'\/W'_i$. By induction (on the size of the\ninterval where the weight graded quotients of $E$ and $E'$ live, just as in Step\n1) we can suppose that $V_i\/W_i$ with its filtration induced by $W_j\/W_i$ is a\nmixed twistor structure, substructure of $E\/W_i$. In particular the weights of\n$V_i\/W_i$ are $> i$, which implies that\nany morphism of $V_i \/ W_i $ to a semistable bundle of slope $\\leq i$ is\nzero. Thus the morphism $V_i\/W_i \\rightarrow W'_i \/ f(W_i)$ is zero.\nThis implies that\n$$\nV_i = W_i + A,\n$$\nand from this formula the sequence\n$$\n0\\rightarrow A \/ W_{i}A \\rightarrow  E \/ W_{i}\n\\rightarrow E' \/ W' _{i} \\rightarrow 0\n$$\nis exact. Note also that $A\/W_iA = V_i \/W_i$.\n\nFor $j \\geq i$ we have $W_jA \/ W_iA = A \/ W_iA \\cap W_j \/W_i$.\nIn particular the filtration induced on $A\/W_iA$ by the filtration $W_iA$ is the\nsame as the kernel filtration for the above exact sequence.  By induction on the\nsize of the range where the weight filtration lives, we have that $A\/W_iA$ is a\nmixed twistor structure.\n\nOn the other hand, $W_iA $ is the kernel of the map $W_i \\rightarrow W'_i$\nof semistable bundles of slope $i$, so $W_iA $ is semistable of slope $i$.\nNote of course that $W_{i-1}A=0$. Thus for any $j$ we have that $Gr ^{WA}_j(A)$\nis semistable of slope $j$ and $A$ is a mixed twistor structure.\n\n\n\n{\\bf Step 3}\\,\\,\nWe have to show that the image is equal to the coimage. In view of what we have\nseen with cokernels and kernels, this comes down to showing that if $f$ is an\nisomorphism on coherent sheaves $E \\cong E'$ then $f(W_i)= W'_i$.\n\nAgain, suppose that this is true for pairs of MTS with weight filtrations\nconcentrated in any smaller range. Let $W_i$ and $W'_i$ be the lowest nonzero\npair of levels in the weight filtration.  Above we established in this case the\nformula\n$$\nf^{-1}(W'_i ) = W_i + {\\rm ker}(f)\n$$\nbut by hypothesis ${\\rm ker}(f)=0$ here so $f^{-1}(W'_i )= W_i$.\nSince $f$ is an isomorphism, $f(W_i)=W'_i$.  Now we can look at the isomorphism\n$E\/W_i \\cong E' \/W'_i$ and by our inductive hypothesis the weight filtrations\nthere are the same.  Thus $f$ induces for all $j$ an isomorphism $Gr\n^W_j(E)\\cong Gr ^{W'}_j(E')$, and this implies that $f(W_j)=W'_j$.\n\nWe have now completed all of the verifications necessary to show that our\ncategory is abelian (cf \\cite{Hodge2}).\n\\hfill $\/$\\hspace*{-.1cm}$\/$\\hspace*{-.1cm}$\/$\\vspace{.1in}\n\n{\\em Remark:}  The reader may generalize the other parts (ii)--(v) of\nTheorem 2.3.5 of \\cite{Hodge2}. Note that the correct generalization of part (v)\nis\nthat the functor ``fiber over $0\\in {\\bf P} ^1$'' is exact, and this may be further\ngeneralized to the statement that the functor ``fiber over $t\\in {\\bf P} ^1$''\nis exact for any point $t$ (in the Hodge case all points are equivalent to $0$,\n$1$ or $\\infty$ by the ${\\bf G}_m$ action; the point $\\infty$ corresponds to the\ncomplex conjugate filtration, so nothing is new in the Hodge case but in the\ntwistor case all points are different).\n\n\\subnumero{Aside: some generalizations of the above theorem}\nThe above theorem that we get an abelian category depends only on some fairly\nstandard results about semistability. These results hold in much greater\ngenerality than just semistable bundles on ${\\bf P} ^1$.  We could make the\nfollowing definition. If $X$ is a projective scheme then a {\\em mixed sheaf of\npure dimension $d$ on $X$} is a sheaf $E$ of pure dimension $d$ with a\nfiltration\n$W_P$ by subsheaves indexed by polynomials $P\\in {\\bf Q} [x]$ (increasing for the\nordering of polynomials by values at large $x$) such that there are only a\nfinite number of jumps in the filtration and such that  $Gr _P^W(E)$ is\na semistable sheaf of pure dimension $d$ with normalized Hilbert polynomial $P$\n\\cite{Moduli}.  By the same proof as above, the category of mixed sheaves of\npure dimension $d$ is abelian.  When $d=dim (X)$ these become torsion-free\nsheaves.\n\n\n\nWe can generalize further if anybody is interested!  Let $\\Lambda$ be a sheaf\nof rings of differential operators such as considered in \\cite{Moduli}.\nThen a {\\em mixed $\\Lambda$-module} is a $\\Lambda$-module $E$ of some pure\ndimension $d$ with a filtration by sub-$\\Lambda$-modules $W_P$ indexed by\nrational polynomials $P$, with a finite number of jumps, such that\n$Gr ^W_P(E)$ is a semistable $\\Lambda$-module with normalized Hilbert polynomial\n$P$.\n\nThere are probably other variants, for example with parabolic structure, etc.\n\n\n\n\n\\subnumero{Moduli of mixed twistor structures}\n\nThere is an obvious notion of algebraic family of mixed twistor structures:\nif $S$ is a scheme then a family of MTS over $S$ is a bundle $E$ on $S\\times {\\bf P}\n^1$ provided with a filtration by strict subbundles $W_iE$ such that in the\nfiber\nover each point $s\\in S$ this gives a MTS.\n\nIf we associate to each scheme $S$ the category of families of MTS over\n$S$, we obtain a stack ${\\cal MTS}^{\\rm any}$. The superscript denotes the\nfact that we include any morphisms of MTS, in particular it is not a stack of\ngroupoids. The associated stack of groupoids which we denote by\n${\\cal MTS}$ is an algebraic stack in the sense of Artin. It is locally of\nfinite\ntype as we shall see below: the pieces corresponding to fixing the dimensions\nof the weight-graded pieces, are of finite type.\n\n There is a\nnatural action of ${\\bf G}_m$ and the fixed points (in the stack-theoretic sense) are\nexactly the ${\\bf G}_m$-equivariant mixed twistor structures, i.e. complex mixed Hodge\nstructures.\n\nWe will attempt briefly to give some idea of what ${\\cal MTS}$ looks like.\nA {\\em framed $MTS$}  is a mixed twistor structure $(E,W)$ provided with\nisomorphisms\n$$\n\\beta _n: Gr ^W_n(E) \\cong {\\cal O} _{{\\bf P} ^1}(n)^{b_n}.\n$$\nThis is equivalent to the data of frames for the vector spaces $Gr ^W_n (E_1)$.\nA {\\em framed family of MTS} over a base scheme $S$ is just a MTS over $S$\nprovided with isomorphisms $\\beta _n$ on ${\\bf P} ^1 \\times S$ as above.\n\nAny automorphism of a framed MTS (or family of framed MTS) fixing the framing,\nis the identity. To prove this, suppose that $f$ is such an automorphism.\nThen $f - 1$ induces zero on the associated graded pieces, hence it is zero\n(because kernel and cokernel commute with taking $Gr ^W$---but this is easy to\nprove along the lines of the above theorem anyway).\n\nWe claim that the functor which to any scheme $S$ associates the set of\nisomorphism classes of framed families of MTS over $S$, is representable by a\nscheme $Fr {\\cal MTS}$ locally of finite type (of finite type if we fix the\ndimensions of the graded pieces i.e. the $b_n$ above). Let\n$Fr {\\cal MTS}(b_{\\cdot})$ denote the part corresponding to MTS where $Gr\n^W_n(E)$ has rank $b_n$. The group\n$$\nGL(b_{\\cdot}):= \\prod _{n} GL(b_n)\n$$\nacts on $Fr {\\cal MTS}(b_{\\cdot})$ with stack-theoretic quotient\n${\\cal MTS}(b_{\\cdot})$ (the open and closed substack of ${\\cal MTS}$\nwith given dimensions of graded pieces). The stack\n${\\cal MTS}$ is the disjoint union of the ${\\cal MTS}(b_{\\cdot})$\nso it is locally of finite type.\n\nWe now prove the representability claim of the previous paragraph.  Write\n$b_{\\cdot } = (b_0, \\ldots , b_k)$ with the rest being zero.  We may proceed by\ninduction on $k$, the case $k=0$ being obvious (the representing scheme is just\na point).  Thus we may assume that\n$Y=Fr {\\cal MTS}(0, b_1, \\ldots , b_k)$ exists. There is a universal\nfamily of framed MTS $(E', W')$ on $Y\\times {\\bf P} ^1$. Suppose $S$ is a scheme and\n$(E_S, W)$ is a family of framed MTS on $S$ with dimensions $b_0, \\ldots , b_n$.\nThen $E_S\/W_1E_S$ is a family of framed MTS with dimensions $0, b_1, \\ldots ,\nb_k$ so we obtain a morphism $p:S\\rightarrow Y$ and\n$p^{\\ast}(E',W')=(E_S\/W_1,W)$\nwith framings. The data of $(E_S, W)$ is equivalent to the data\nof $(E_S\/W_1,W)$ together with an extension of the bundle\n$E_S\/W_1$ by ${\\cal O} _{{\\bf P} ^1\\times S}(0)^{b_0}$.  Thus we have a lifting\nof $p$ to a morphism from $S$ into the relative $Ext^1$-bundle\n(which we show to be a bundle two paragraphs below)\n$$\nExt ^1_{{\\bf P} ^1\\times Y\/Y}(E', {\\cal O} _{{\\bf P} ^1\\times Y}^{b_0})\\rightarrow Y.\n$$\nThis lifting is unique, and the pullback of the universal extension gives\n$(E_S, W)$.  Thus $Fr{\\cal MTS}(b_0, \\ldots , b_k)$ is the relative\n$Ext^1$-bundle refered to above.\n\nTo sum up the previous paragraph, there are schemes\n$Fr{\\cal MTS}(b_j, \\ldots , b_k)$ and universal families of framed MTS\n$(E^{\\rm univ}(b_j, \\ldots , b_k), W)$ which are filtered bundles over\n$$\nFr{\\cal MTS}(b_j, \\ldots , b_k)\\times {\\bf P} ^1.\n$$\nThere are natural morphisms\n$$\nFr{\\cal MTS}(b_j, \\ldots , b_k)\\rightarrow\nFr{\\cal MTS}(b_{j+1}, \\ldots , b_k)\n$$\nsuch that the upper scheme is naturally identified with the\n relative $Ext^1$-bundle classifying extensions in the ${\\bf P} ^1$-direction of\n$E^{\\rm univ}(b_{j+1}, \\ldots , b_k)$ by ${\\cal O} _{{\\bf P} ^1} (j)^{b_j}$.\n\nWe need to point out that the relative $Ext^1$ referred to above are indeed\nbundles over the base $Y$. This is by semicontinuity theory using the fact that\nfor any point the corresponding $Ext^0$ is zero. Indeed the fiber over any point\n$y\\in Y$ of the bundle  $E'=E^{\\rm univ}(b_{j+1}, \\ldots , b_k)$ (over $\\{ y\\}\n\\times {\\bf P} ^1$) is an extension of bundles which are stable of slopes $\\geq j+1$\nso it decomposes into factors of degree $\\geq j+1$. Thus there are no\nhomomorphisms to ${\\cal O} _{{\\bf P} ^1} (j)^{b_j}$ so the $Ext^0=Hom$ is zero.\nSince we work on ${\\bf P} ^1$ there are no other terms and semicontinuity implies\nthat the family of $Ext^1$ spaces all have the same dimension and fit together\ninto a vector bundle which we have referred to as\n$Ext ^1_{{\\bf P} ^1\\times Y\/Y}(E', {\\cal O} _{{\\bf P} ^1\\times Y}^{b_0})$.\n\nFrom this description, $Fr{\\cal MTS}(b_j, \\ldots , b_k)$ is obtained as\na sequence of vector bundles, eventually over a point. Thus the underlying\ntopological space is contractible. Hence the homotopy type of\n${\\cal MTS}(b_j, \\ldots , b_k)^{\\rm top}$ is the same as that of $BGL(b_j,\n\\ldots\n, b_k)^{\\rm top}$.\n\nWe give a formula for the dimension of the stack ${\\cal MTS}(b_j, \\ldots , b_k)$\n(recall that this means roughly speaking the dimension of the space of orbits\nminus the dimension of the stabilizer group). The dimension is\nequal to\n$$\ndim (Fr{\\cal MTS}(b_j, \\ldots , b_k))- (b_j^2 + \\ldots + b_k^2)\n$$\nthe latter term  being the dimension of $GL(b_j, \\ldots ,\nb_k)$. For $dim(Fr{\\cal MTS}(b_j, \\ldots , b_k))$\nwe use the above expression as a series of fiber bundles.  Note that the\ndimension of\n$$\nExt ^1({\\cal O} _{{\\bf P} ^1}(i), {\\cal O} _{{\\bf P} ^1}(j))= H^1({\\bf P} ^1, {\\cal O} _{{\\bf P}\n^1}(j-i))\n$$\nis $i-j-1$ when $i>j$, by Riemann-Roch.  Thus we have\n$$\ndim (Fr{\\cal MTS}(b_j, \\ldots , b_k))=\ndim (Fr{\\cal MTS}(b_{j+1}, \\ldots , b_k)) + \\sum _{i=j+1}^{k}\n(i-j-1)b_ib_j.\n$$\nThus\n$$\ndim (Fr{\\cal MTS}(b_j, \\ldots , b_k))=\\sum _{j\\leq u < i \\leq k}\n(i-u-1)b_ib_j\n$$\nand the term $dim \\, GL(b_j,\\ldots , b_k)$ fits in nicely to give\n$$\ndim ({\\cal MTS}(b_{\\cdot}))=\\sum _{u \\leq i }\n(i-u-1)b_ib_j .\n$$\nNote in particular that for most well spaced-out sequences $b_{\\cdot}$ the\ndimension is positive, and in particular the family of orbits is nontrivial.\nThis means that there are ``moduli'' of mixed twistor structures.\n\n{\\em Questions:}  Define a natural $GL(b_{\\cdot})$-linearized line bundle\non $Fr{\\cal MTS}(b_{\\cdot})$. What are the stable or semistable points for the\naction, and what does the GIT quotient space look like? Are the mixed twistor\nstructures which come up in nature (e.g.  the mixed Hodge structures)\nsemistable?\n\n\n\n\n\n\n\n\\numero{Real structures}\nTo complete the circle of definitions and comparisons, we define some notions of\n{\\em real twistor structure} and {\\em real mixed twistor structure}.\nThe situation is more complicated than for mixed Hodge structures, because\nthere are many possible antiholomorphic involutions of ${\\bf P} ^1$.\nWe will isolate for our purposes two examples.\n\n\n\nLet $\\sigma _{{\\bf P} ^1}$ denote the antipodal involution of ${\\bf P} ^1$ (it is\nantilinear, interchanges $0$ and $\\infty$ and interchanges $1$ and $-1$).\nAn {\\em antipodal real twistor structure} is a bundle $E$ on ${\\bf P} ^1$ with\nantilinear involution $\\sigma$ lying over $\\sigma _{{\\bf P} ^1}$.\nAn {\\em antipodal real mixed twistor structure} is an antipodal real twistor\nstructure with filtration by strict subbundles preserved by $\\sigma$.\n\nIn considering polarizations in the next section, we will make use of the\nfollowing sheaf-theoretic point of view. If $E$ is a vector bundle on ${\\bf P} ^1$\nconsidered as a locally free sheaf of ${\\cal O} _{{\\bf P} ^1}$-modules then put\n$$\n\\sigma ^{\\ast}(E) (U):= E(\\sigma _{{\\bf P} ^1}U).\n$$\nThe structure of ${\\cal O} _{{\\bf P} ^1}(U)$-module is defined by\nsetting, for $e\\in E(\\sigma _{{\\bf P} ^1}U)$ and $a\\in {\\cal O}_{{\\bf P} ^1}(U)$,\n$a\\cdot e := \\overline{\\sigma _{\\ast}(a)}e$.  Note that\n$\\sigma ^{\\ast}(E)$ is again a locally free sheaf of ${\\cal O} _{{\\bf P} ^1}$-modules.\nHowever the functor $\\sigma ^{\\ast}$ is antilinear: the induced map on\n${\\bf C}$-vector spaces\n$$\nHom (E, E')\\rightarrow Hom (\\sigma ^{\\ast}(E),\\sigma ^{\\ast}(E'))\n$$\nis antilinear.  Also there is an induced map\n$$\nH^i(E) \\rightarrow H^i(\\sigma ^{\\ast}(E)),\n$$\nagain antilinear.\nFinally the functor $\\sigma ^{\\ast}$ is an involution (its square is naturally\nequal to the identity).  An involution of $E$ lying over $\\sigma _{{\\bf P} ^1}$\n(the notion defined in the previous paragraph, which concerns the total space\nof the bundle) can be expressed in sheaf-theoretic language as a morphism\n$$\nf: E \\rightarrow \\sigma ^{\\ast}(E)\n$$\nsuch that $\\sigma ^{\\ast}(f)\\circ f = 1_E$.\n\n\n\nLet $\\tau _{{\\bf P} ^1}$ denote the antiholomorphic involution which preserves the\nunit circle.  A   {\\em circular real twistor structure} is a bundle $E$ on ${\\bf P}\n^1$ with antilinear involution $\\tau$ lying over $\\tau _{{\\bf P} ^1}$.\nA {\\em circular real mixed twistor structure} is a circular real twistor\nstructure\nwith filtration by strict subbundles preserved by $\\tau$. Note that a\nsheaf-theoretic discussion similar to that which we have given for $\\sigma$,\nexists for $\\tau$ also.\n\nThe difference between these two notions is that $\\sigma _{{\\bf P} ^1}$ has no fixed\npoints, whereas $1$ is a fixed point of $\\tau _{{\\bf P} ^1}$. Thus if $E$ has a\ncircular real structure then $E_1$ is a real vector space, whereas this is not\nnecessarily the case for an antipodal real structure.\n\nWe now investigate more closely what these structures mean in the pure case.\nThere are two different situations depending on the parity of the weight.\n\nSuppose $E$ is an antipodal (resp. circular) real twistor structure which\nis pure\nof weight $0$. Then  $H^0({\\bf P} ^1, E)$ is a complex vector space with antilinear\ninvolution induced by $\\sigma$ (resp. $\\tau$), or in other words it is a real\nvector space. Note that\n$$\nE = H^0({\\bf P} ^1, E)\\otimes _{{\\bf C} } {\\cal O} _{{\\bf P} ^1}\n$$\npreserving the involutions.\nConversely any\nreal vector space gives rise to an antipodal (resp. circular) real twistor\nstructure which is pure of weight $0$ and these constructions are inverses.\nIn the circular case note that the isomorphism\n$$\nH^0({\\bf P} ^1, E) \\cong E_1\n$$\nis compatible with the involution $\\tau$.\n\nSuppose $E$ is an antipodal real twistor structure which is pure of weight\n$1$. Then we obtain a structure of quaternionic vector space.\nConversely a quaternionic vector space gives an antipodal real twistor structure\npure of weight $1$, and these constructions are inverses. These are both well\ndescribed in \\cite{HKLR}. This is the example which lends the name\n``twistor''. We\nstart by describing this second direction which is just the classical ``twistor\nspace'' construction of Penrose (cf \\cite{Penrose} \\cite{Hitchin2}\n\\cite{HKLR}).\n\nA quaternionic vector space is a real vector space $V$ with operations $I$, $J$\nand $K$ satisfying the relations\n$$\nI^2=J^2=K^2 = -1, \\;\\;\\; IJ=K \\;\\; \\mbox{etc.}\n$$\nFor any triple of reals $(x,y,z)$ with $x^2+ y^2 + z^2= 1$ we can define a\ncomplex structure $xI + yJ + zK$ on $V$. Combined with the standard complex\nstructure on $S^2= {\\bf P} ^1$ this serves to define an almost complex structure on\nthe bundle $E=V\\times {\\bf P} ^1$.  This almost complex structure is integrable, as\ncan be checked by a direct calculation, or else by noting that the bundle\n$L:={\\cal O}\n_{{\\bf P} ^1}(1) \\oplus {\\cal O} _{{\\bf P} ^1}(1)$ has an antipodal real involution\nwherefrom the invariant sections give a trivialization of the real bundle\n$L\\cong {\\bf P} ^1 \\times {\\bf H}$ such that the points $0$, $1$ and\n$i$ in ${\\bf P} ^1$ give a quaternionic triple of complex structures $I$, $J$\nand $K$\non ${\\bf H}$ whose resulting twistor almost complex structure recovers $L$\n(this\nis a calculation which needs to be done and for which we refer to  \\cite{HKLR}).\nAs any quaternionic $V$ is a direct sum of copies of ${\\bf H}$ we obtain the\nintegrability of the twistor bundle $E$ (which is a direct sum of copies of\n$L$),\nas well as the fact that the twistor bundle is pure of weight $1$ with antipodal\nreal structure.\n\n\n\nWe now describe how an antipodal real twistor structure $E$ pure of weight $1$\ncomes from a quaternionic vector space. The underlying\nreal vector space $A$ is the space of sections of $E$ which are preserved by the\ninvolution $\\sigma$.  Note that $\\sigma$ induces an antilinear involution on\n$H^0({\\bf P}^1, E)$ and the fixed points form a real subspace\n$$\nA= H^0({\\bf P}^1, E)^{\\sigma }\n$$\nwhose dimension is\nhalf the complex dimension of  $H^0({\\bf P}^1, E)$. In turn, the complex\ndimension of\n$H^0({\\bf P}^1, E)$ is twice the complex rank of $E$ (by purity of weight $1$) so\n$$\ndim _{{\\bf R}}(A)= rk (E).\n$$\nThis suggests that the evaluation morphism $A\\rightarrow E_p$ for any point\n$p\\in {\\bf P} ^1$, should be an isomorphism. We prove this: if $e\\in E_p$ then\n$\\sigma (e)\\in E_{\\sigma p}$ and by  purity of weight $1$ there is a unique\nsection $f: {\\bf P} ^1 \\rightarrow E$ such that $f(p)=e$ and $f(\\sigma p)= \\sigma\n(e)$.  Uniqueness implies that $\\sigma ^{\\ast}(f)=f$. Uniqueness also gives\ninjectivity of the morphism $A\\rightarrow E_p$ and the above construction gives\nsurjectivity.\nNow for every  point $p\\in {\\bf P} ^1$ we obtain a complex structure $J_p$ on $A$\nby pulling back the complex structure from $E_p$. The fact that $e\\mapsto \\sigma\n(e)$ is antilinear means that $J_{\\sigma p}=-J_p$.\n\nLet $I=J_0$, $J=J_1$ and $K=J_{i}$.  We\nclaim that these provide a quaternionic triple and that for any other point $q$\nthe complex structure $J_q$ is that obtained from $(I,J,K)$ by the twistor space\nconstruction described above. To prove this claim let $L$ be the\nstandard rank two pure twistor structure of weight $1$ with antipodal\ninvolution,\ncorresponding to the quaternionic vector space ${\\bf H}$.  The bundle\n$\\underline{Hom} (L, E)=L^{\\ast}\\otimes E$ is  pure of weight $0$ and has an\nantipodal involution $\\sigma$, thus it is the same as a vector space with real\nstructure. The real sections are the morphisms $L\\rightarrow E$ compatible with\nthe antipodal involutions; let $Hom  (L,E)^{\\sigma }$ denote this space of real\nsections. It has real dimension equal to the complex dimension of $Hom (L,E)$\nwhich in turn is the rank of $L^{\\ast}\\otimes E$ or $2r(E)$. If we fix a section\n$u$ of $L$ which is preserved by $\\sigma$ then evaluation at $u$ gives a\nmorphism\n$$ \\epsilon : Hom  (L,E)^{\\sigma } \\rightarrow A. $$\nWe claim that this morphism is injective. In fact if a $\\sigma$-invariant map\n$f:L\\rightarrow E$ sends $u$ to zero then the bundle $ker(f)\\subset L$ is\nnontrivial, but it is a $\\sigma$-invariant bundle, pure of weight $1$\nso by the above discussion it corresponds to a nonzero subspace of $H^0({\\bf P} ^1,\nL)^{\\sigma}={\\bf H}$ which is preserved by all of the complex structures---hence\nit must be all of ${\\bf H}$ so $f=0$.\n\nNow injectivity of the map $\\epsilon$ implies surjectivity since the real\ndimensions of the two sides coincide. Thus any section $a\\in A$ is in the image\nof a morphism $L\\rightarrow E$. The commutation conditions for the various\ncomplex structures on $L$ imply the same conditions for those complex structures\nof $E$ as applied to $a$. This proves our claim that $(I,J,K)$ form a\nquaternionic triple. The same proof shows that the other $J_q$ are as they are\nsupposed to be. It is obvious that $(E, \\sigma )$ comes from this quaternionic\nstructure by the twistor construction.\n\n\n\n\nThere is a canonical antipodal and circular real twistor structure of\nweight two which corresponds to the Tate Hodge structure.  We call it the ``Tate\ntwistor'' (!), denoted $T(1)$.  Tensorisation and dual leads to Tate twistors\n$T(n)$ for all $n$.  By tensoring with these, any pure antipodal or\ncircular real twistor structure is equivalent to one of weight $0$ or\nweight $1$,\nso the above discussions apply: we obtain either a real vector space or a\nquaternionic one.\n\nNote, in passing, the fact that a complex vector space with quaternionic\nstructure must have even dimension. This carries over the Hodge-theory fact\nthat\na real Hodge structure of odd weight must have even dimension (on the other\nhand, there is no such restriction for even weight).\n\n\nThe line bundle ${\\cal O} (1)$ pure of weight $1$ (not to be confused with\n$T(1)$ which is of weight two) has a natural circular real structure.  On the\ndual ${\\cal O} (-1)$ this is described as follows: over a point $P=[a:b] \\in {\\bf P} ^1$\nthe fiber ${\\cal O} (-1)_P$ is just the line of $(x,y)$ proportional to $(a,b)$.\nThus the total space of the bundle ${\\cal O} (-1)$ is just ${\\bf A}^2$ (blown up at\nthe origin).  We define the antilinear map\n$$\n\\tau : {\\bf A}^2 \\rightarrow {\\bf A}^2\n$$\nby\n$$\n\\tau (x,y) = (\\overline{y}, \\overline{x}).\n$$\nThis extends to the blow-up at the origin as an antilinear involution covering\nthe involution $\\tau _{{\\bf P} ^1}$, thus giving a circular real structure on ${\\cal O}\n(-1)$. The dual is a circular real structure on ${\\cal O} (1)$.  By tensoring\nwith a power of one of these, any pure twistor structure with circular real\nstructure becomes the same as one of weight zero, so it is just a real vector\nspace. This equivalence is compatible with taking the fiber $E_1$ over $1$\n(which has a real structure since $1$ is a fixed point of $\\tau _{{\\bf P} ^1}$).\nThus a circular real structure of any weight is just a real structure on the\nunderlying vector space.\n\n\n\n\\subnumero{Real structures in the Hodge case}\n\n\nLet $H$ be the group of conformal automorphisms of ${\\bf P}  ^1$ which preserve\nthe set $\\{ 0, \\infty\\}$ and which act on this set trivially when the\norientation is preserved, nontrivially when the orientation is changed.  The\nconnected component of $H$ is just ${\\bf G}_m$ (the group of holomorphic automorphisms\nfixing $0$ and $\\infty$).  There are exactly two components, and the other\ncomponent is equal to ${\\bf G}_m \\cdot \\sigma_{{\\bf P} ^1}$ or equally well ${\\bf G}_m \\cdot\n\\tau _{{\\bf P} ^1}$. In particular $H$ may be expressed as the group of\nautomorphisms generated by ${\\bf G}_m$ and either one of $\\sigma _{{\\bf P} ^1}$ or $\\tau\n_{{\\bf P} ^1}$.\n\n\\begin{proposition}\nAn $H$-equivariant mixed twistor structure $(E,W)$ is the same thing as an\n${\\bf R}$-mixed Hodge structure.\n\\end{proposition}\n{\\em Proof:}\nLeft to the reader.\n\\hfill $\/$\\hspace*{-.1cm}$\/$\\hspace*{-.1cm}$\/$\\vspace{.1in}\n\nThis proposition says that mixed twistor structures which are both\n${\\bf G}_m$-equivariant and have either an antipodal or circular real structure, are\nreal mixed Hodge structures. Since the groups generated by ${\\bf G}_m$ and either of\n$\\sigma _{{\\bf P} ^1}$ or $\\tau _{{\\bf P} ^1}$ are the same, we don't see the difference\nbetween antipodal and circular real structures in the ${\\bf G}_m$-equivariant (Hodge)\nsituation.\n\nIn the present note we shall not consider the problem of real twistor structures\nany further (except one small place at the end when we discuss twistor spaces\nfor hypercomplex structures). The reader may imagine how to incorporate real\nstructures into all of the various statements we shall make: essentially all\nconstructions are equivariant for $\\sigma$ or $\\tau$ if these involutions are\ngiven for the input data.\n\n\n\\numero{Variations of mixed twistor structure}\nThe problem of dealing with ${\\cal C} ^{\\infty}$ families of twistor structures\ncan be solved in several different ways.  The first would be to note that the\ncategory of twistor structures has generators ${\\cal O} _{{\\bf P}^1}(w)$ and that the\nmorphism spaces $Hom ({\\cal O} _{{\\bf P} ^1}(w), {\\cal O} _{{\\bf P} ^1}(w'))= {\\bf C}\n[\\lambda , \\mu ]_{w'-w}$ are easily described as the spaces of homogeneous\npolynomials (similarly for the extension groups).  One can then ``tensor'' this\ncategory with any ring or sheaf of rings, in particular with ${\\cal C} ^{\\infty}_X$\nfor a manifold $X$.  Another more concrete approach would be to say that a\n${\\cal C} ^{\\infty}$ bundle of twistor structures over a manifold $X$ is a vector\nbundle $E$ on $X\\times {\\bf P} ^1$ provided with a complex structure operator\n$\\overline{\\partial} _{{\\bf P} ^1}$ in the ${\\bf P} ^1$-direction.  For any $x\\in X$ the restriction\n$E|_{\\{ x\\} \\times {\\bf P} ^1}$ becomes a holomorphic  vector bundle on ${\\bf P} ^1$,\nalgebraic by GAGA.  We will sometimes make use of this interpretation. Finally,\nour main interpretation is sheaf-theoretic: if $X$ is a\n${\\cal C} ^{\\infty}$ manifold, let ${\\cal C} ^{\\infty}_X {\\cal O} _{{\\bf P} ^1}$ be the sheaf of\nfunctions on $X\\times {\\bf P} ^1$ (in the usual topology) which are ${\\cal C}^{\\infty}$\nand which are holomorphic in the ${\\bf P} ^1$-direction (in other words the\nfunctions annihilated by the operator $\\overline{\\partial} _{{\\bf P} ^1}$ refered to above).\nA {\\em ${\\cal C} ^{\\infty}$ family of twistor structures on $X$} is now simply a\nlocally free sheaf ${\\cal E}$ of ${\\cal C} ^{\\infty}_X {\\cal O} _{{\\bf P} ^1}$-modules on $X\\times\n{\\bf P} ^1$. We often refer to this simply as a ``bundle'', or sometimes a\n``${\\cal C} ^{\\infty}_X {\\cal O} _{{\\bf P} ^1}$-module'', on $X\\times\n{\\bf P} ^1$. A {\\em strict filtration} of such a bundle is a filtration of ${\\cal E}$ by\nsubsheaves such that locally on $X\\times {\\bf P} ^1$ the filtration comes from a\ndecomposition of ${\\cal E}$ into a direct sum of locally free ${\\cal C} ^{\\infty}_X {\\cal O}\n_{{\\bf P} ^1}$-modules. If $W_{\\cdot}$ is a strict filtration then the\n$Gr ^W_i({\\cal E} )$ are again locally free ${\\cal C} ^{\\infty}_X {\\cal O}\n_{{\\bf P} ^1}$-modules.\n\nIf $V$ is a ${\\cal C} ^{\\infty}$ bundle on $X$ provided with filtrations $F$ and $F'$\nthen by doing the Rees bundle construction in a ${\\cal C} ^{\\infty}$ family we obtain\na locally free ${\\cal C} ^{\\infty}_X {\\cal O} _{{\\bf P} ^1}$-module $\\xi (V, F, F')$ on\n$X\\times {\\bf P} ^1$.\n\nThe above constructions are functorial for ${\\cal C} ^{\\infty}$ morphisms\n$X'\\rightarrow X$, notably for inclusions of points $x\\hookrightarrow X$.\nThus the above definitions restrict to the definitions from the beginning, over\neach point $x\\in X$.\n\nA {\\em ${\\cal C} ^{\\infty}$ family of mixed twistor structures over $X$} is a\n${\\cal C} ^{\\infty}_X {\\cal O} _{{\\bf P} ^1}$-module ${\\cal E}$ with strict filtration $W_{\\cdot}$\nsuch that the $Gr ^W_i({\\cal E} )$ are pure of weight $i$ (which means that for each\n$x\\in X$ the corresponding holomorphic bundle on $\\{ x\\} \\times {\\bf P} ^1$ is\npure of\nweight $i$).\n\n\nSuppose $X$ is a complex manifold.\nThe cotangent bundle $T^{\\ast}_{{\\bf C}}(X)$, which we often confuse with its sheaf\n${\\cal A} ^{1}_X$ of sections, has a Hodge structure of\nweight $1$ and consequently a twistor structure pure of weight $1$. This twistor\nstructure is a\n${\\cal C} ^{\\infty}_X {\\cal O} _{{\\bf P} ^1}$-module\nwhich is isomorphic to $T^{\\ast}_{{\\bf C}}(X)\n\\otimes _{{\\bf C}} {\\cal O} _{{\\bf P}\n^1}(1)$. To be precise, we have a decomposition\n$$\n{\\cal A} ^{1}_X = {\\cal A} ^{1,0}_X \\oplus {\\cal A} ^{0,1}_X,\n$$\nand the twistor structure which is $\\xi ({\\cal A} ^{1}_X, F, F')$ decomposes as\n$$\n\\xi ({\\cal A} ^{1}_X, F, F')= \\xi ({\\cal A} ^{1,0}_X, F, F')\\oplus \\xi ({\\cal A} ^{0,1}_X, F,\nF'),\n$$\nwhere now the filtrations on the pieces\n${\\cal A} ^{1,0}_X$ and ${\\cal A} ^{0,1}_X$ are trivial (but shifted differently).\nThese trivial filtrations induce isomorphisms\n$$\n\\xi ({\\cal A} ^{1,0}_X, F, F')\\cong\n{\\cal A} ^{1,0}_X\\otimes _{{\\bf C}} {\\cal O} _{{\\bf P}^1}(1),\\;\\;\\;\n\\xi ({\\cal A} ^{0,1}_X, F, F')\\cong\n{\\cal A} ^{0,1}_X\\otimes _{{\\bf C}} {\\cal O} _{{\\bf P}^1}(1)\n$$\nand their sum is the isomorphism\n$$\n\\phi :\\xi ({\\cal A} ^{1}_X, F, F')\\cong\n{\\cal A} ^{1}_X\\otimes _{{\\bf C}} {\\cal O} _{{\\bf P}^1}(1)\n$$\nin question.\nThere is a natural inclusion of bundles\n$$\n\\iota : {\\cal A} ^{1}_X\\otimes _{{\\bf C}} {\\cal O} _{{\\bf P}\n^1}\\hookrightarrow \\xi ({\\cal A} ^{1}_X, F, F')\n$$\nand this decomposes as inclusions on each of the pieces. If we fix the standard\nsections $\\lambda$ and $\\mu$ of of ${\\cal O} (1)$ which vanish\nrespectively at $0$ and $\\infty$, we can write\n$$\n\\phi \\iota (\\alpha ^{1,0} + \\alpha ^{0,1})=\n\\lambda \\alpha ^{1,0} + \\mu \\alpha ^{0,1}.\n$$\nThe inclusion $\\iota$ fixes notation for $\\xi ({\\cal A} ^{1}_X, F, F')$\nand the above formula fixes notation for the isomorphism  $\\phi$.\n\nWe denote for short $\\xi ({\\cal A} ^{1}_X, F, F')$ by $\\xi {\\cal A} ^1_X$. The same\nconstruction applies to the whole exterior algebra ${\\cal A} ^{\\cdot}_X$\nto give a graded ${\\cal C} ^{\\infty}_X{\\cal O} _{{\\bf P} ^1}$-module $\\xi{\\cal A} ^{\\cdot}_X$\non $X\\times {\\bf P} ^1$. The $\\xi{\\cal A} ^i_X$ have twistor structures  of weight $i$.\n\nThere is a natural differential operator\n$$\n{\\bf d}: \\xi{\\cal A} ^0_X \\rightarrow \\xi{\\cal A} ^1_X\n$$\nwhich is a morphism of twistor structures.  It is just the composition of\nthe usual exterior derivative $d$ followed by $\\iota$, so it can be written\n$$\n{\\bf d} = \\lambda \\partial  + \\mu \\overline{\\partial}\n$$\nwhere again $\\lambda$ and $\\mu$ are the sections of ${\\cal O} _{{\\bf P} ^1}(1)$ which\nvanish respectively at $0$ and $\\infty$.  The differential extends to the\ncomples  $\\xi{\\cal A} ^i_X$ have twistor structures  of weight $i$ and we obtain a\ncomplex $\\xi{\\cal A} ^{\\cdot}_X$ with differential ${\\bf d}$.\n\nWe generalize the definition of \\cite{SteenbrinkZucker} to the twistor case.\nA {\\em variation of mixed twistor structure}\n\\footnote{This is the twistor analogue of the complex variations of\nmixed Hodge structure; one could also talk about antipodal or circular real\nvariations of mixed twistor structure which would be analogues of ${\\bf R}$-VMHS,\nthis is left to the reader.}\nis a ${\\cal C} ^{\\infty}$ family of mixed twistor structures  $({\\cal E} ,\nW_{\\cdot})$ on $X$ (i.e. a\n${\\cal C} ^{\\infty}_X{\\cal O} _{{\\bf P} ^1}$-module ${\\cal E}$ with strict filtration\n$W_{\\cdot}$ on $X\\times {\\bf P} ^1$)\ntogether with an operator\n$$\nD: {\\cal E} \\rightarrow {\\cal E} \\otimes\n_{{\\cal C} ^{\\infty}_X{\\cal O} _{{\\bf P} ^1}}\\xi {\\cal A} ^1_X\n$$\nrespecting the weight filtration, such that\nthe Leibniz rule\n$$\nD(ae) = {\\bf d}(a) e + aD(e)\n$$\nis satisfied, and such that $D^2=0$.\nA {\\em pure variation of twistor structure} is the same as above but with\nthe associated graded of the weight filtration concentrated in one degree.\n\nSuppose given a variation of mixed twistor structure.\nFor any $\\lambda \\in {\\bf A}^1$ we obtain an {\\em underlying\n$\\lambda$-connection}\nby taking the fiber over $\\lambda \\in {\\bf P} ^1$.\nNote that the fiber of $\\xi {\\cal A} ^1_X$ over $\\lambda \\in {\\bf P} ^1$ is naturally\nidentified with ${\\cal A} ^1_X$ and via this identification, ${\\bf d}$ corresponds to\n$\\lambda \\partial + \\overline{\\partial}$.   If $({\\cal E} , D)$ is a variation of twistor\nstructure then the bundle ${\\cal E} _{\\lambda}:= {\\cal E} |_{X\\times \\{ \\lambda \\} }$\nis a ${\\cal C} ^{\\infty}$ bundle on $X$ with operator $D_{\\lambda}$ having symbol\n$\\lambda \\partial + \\overline{\\partial}$ and square zero.  If we decompose according to\nHodge type\n$$\nD_{\\lambda}= D^{1,0}_{\\lambda} + D^{0,1}_{\\lambda}\n$$\nthen $D^{0,1}_{\\lambda}$ provides an integrable holomorphic structure for ${\\cal E}\n_{\\lambda}$ and $D^{1,0}_{\\lambda}$ becomes a holomorphic $\\lambda$-connection\n(i.e. operator satisfying Leibniz for $\\lambda d$ and having square zero---cf\n\\cite{SantaCruz}; this definition was made by Deligne in \\cite{DeligneLetter}).\n\nIn particular for $\\lambda = 0$ we obtain an\nunderlying Higgs bundle and for $\\lambda = 1$ we obtain an underlying flat\nbundle. The $\\lambda$-connections come in a holomorphic family\nindexed by ${\\bf A}^1$, as will be explained in further detail below.\n\nOn a slightly different note, for every point $x\\in X$ we obtain a mixed twistor\nstructure $({\\cal E}, W_{\\cdot})_x$. This gives a map $X\\rightarrow {\\cal MTS}$\nto the\nmoduli stack of mixed twistor structures; note however that it is only a\n${\\cal C}^{\\infty}$ map and not holomorphic. We call this the {\\em classifying map}\nfor the variation of mixed twistor structure $({\\cal E}, W_{\\cdot})$.\n\n\\subnumero{Polarizations}\nWe define a notion of {\\em polarization} for pure variations of twistor\nstructures.  This will be used as a  characterization of those variations which\ncorrespond to harmonic bundles, cf Lemma \\ref{harmonic} below.  As we will not\nuse the notion of polarization anywhere else, the reader may prefer to skip\ndirectly to Lemma \\ref{harmonic} and take as the definition of polarizable\nvariation, one which corresponds to a harmonic bundle by the construction of\n\\ref{harmonic}. The {\\em raison d'etre} of the definition we give below of\npolarization is just to emphasize the analogy with variations of Hodge\nstructure.\n\nSuppose $({\\cal E} , {\\bf d})$ is a pure variation of twistor structure of weight\n$w$ on $X$.\nWe obtain the locally free sheaf of  ${\\cal C} ^{\\infty}_{X}{\\cal O} _{{\\bf P}\n^1}$-modules $\\sigma ^{\\ast}({\\cal E} )$ on $X\\times {\\bf P} ^1$ as\nfollows:\nif $U\\subset X\\times {\\bf P} ^1$ is an open set then $(1\\times \\sigma )(U)$\nis an open subset of $X\\times {\\bf P} ^1$ (and all open subsets are\nobtained this way).  We set\n$$\n\\sigma ^{\\ast}({\\cal E} )((1\\times \\sigma )(U)):={\\cal E} (U).\n$$\nWe give this a structure of ${\\cal C} ^{\\infty}_{X}{\\cal O} _{{\\bf P}\n^1}$-module as follows: if\n$e\\in {\\cal E} (U)$ and\n$$\na\\in {\\cal C} ^{\\infty}_{X}{\\cal O} _{{\\bf P}\n^1}((1\\times \\sigma )(U))\n$$\nthen $a$ times $e$ is defined to be equal to\n$$\n\\overline{(1\\times \\sigma )^{\\ast}(a)} e\\in {\\cal E} (U).\n$$\nThe complex conjugate is required in order to obtain a section\n$\\overline{(1\\times \\sigma )^{\\ast}(a)}$ which is holomorphic in the ${\\bf P}\n^1$-direction. This has the effect that the functor ${\\cal E} \\mapsto\n\\sigma ^{\\ast}({\\cal E} )$ is a ${\\bf C}$-antilinear functor.\n\nWe give $\\sigma  ^{\\ast}({\\cal E} )$  a structure of variation of\ntwistor structure on $\\overline{X}$. For this, we define an operator\n$$\n\\sigma ^{\\ast}(D): \\sigma  ^{\\ast}({\\cal E} )\n\\rightarrow \\sigma  ^{\\ast}({\\cal E} )\n\\otimes\n_{{\\cal C} ^{\\infty}_{X}{\\cal O} _{{\\bf P} ^1}}\\xi{\\cal A} ^1_X\n$$\nby using a morphism\n$$\n\\sigma ^{\\ast}(\\xi{\\cal A} ^1_X)\\rightarrow \\xi{\\cal A} ^1_X.\n$$\nThis morphism is defined as follows. Sections of $\\xi{\\cal A} ^1_X$ may be written in\nthe form $\\alpha ' \\lambda + \\alpha '' \\mu$ where $\\alpha '$ and $\\alpha ''$\nare functions on $X\\times {\\bf P} ^1$ taking values respectively in the $1,0$ and\n$0,1$ forms on $X$, and holomorphically varying in the ${\\bf P} ^1$-direction.\nFurthermore $\\alpha '$ is allowed to have one pole at the zero of $\\lambda$\n(i.e. along $X\\times \\{ 0\\}$) and $\\alpha '' $ is allowed to have one pole\nat the\nzero of $\\mu$ (i.e. along $X\\times \\{ \\infty \\}$).   Noting that\n$\\sigma ^{\\ast}(\\xi{\\cal A} ^1_X)(U) = \\xi {\\cal A} ^1_X ((1\\times \\sigma )(U))$,\nwe define the morphism\n$$\n\\psi :\\xi {\\cal A} ^1_X ((1\\times \\sigma )(U))\\rightarrow \\xi {\\cal A} ^1_X(U)\n$$\nby\n$$\n\\psi (\\alpha ' \\lambda + \\alpha '' \\mu ):= \\overline{\\sigma ^{\\ast}(\\alpha ''\n)}\\lambda + \\overline{\\sigma ^{\\ast}(\\alpha '\n)}\\mu .\n$$\nThis expression still satisfies the conditios described above concerning the\nallowable poles of the coefficients, since $\\sigma$ interchanges $0$ and\n$\\infty$. Using this morphism we obtain the operator $\\sigma ^{\\ast}(D)$,\nmaking $(\\sigma ^{\\ast}({\\cal E} ), \\sigma ^{\\ast}(D))$ into a variation of twistor\nstructure.\n\nWe note in passing that there isn't really any other way to define the antipodal\nconjugate $(\\sigma ^{\\ast}({\\cal E} ), \\sigma ^{\\ast}(D))$; this is a contrast with\nthe situation in Hodge theory where one doesn't really see any inner logic\nrequiring the various changes of signs. (In a certain sense we have transfered\nthe question to the simple fact of choosing to work with the antipodal\ninvolution, which turns out to contain all of the necessary sign changes.)\n\n\nA {\\em polarization} of $({\\cal E} , {\\bf d})$ is\na bilinear pairing\n$$\nP: {\\cal E} \\otimes \\sigma ^{\\ast} ({\\cal E} ) \\rightarrow T(w)\n$$\nwhich is a morphism of variation of twistor structures, and which is {\\em\npositive hermitian}. This last notion is defined as follows: the form $P$\ninduces\na morphism of trivial bundles\n$$\n{\\cal E} (-w) \\otimes \\sigma ^{\\ast}({\\cal E} (-w)) \\rightarrow T(w) \\otimes {\\cal O} (-w)\n\\otimes {\\cal O} (-w) \\cong {\\cal O} ,\n$$\nhence a morphism on the corresponding vector spaces; but\n$H^0(\\sigma ^{\\ast}({\\cal E} (-w)) )\\cong H^0({\\cal E} (-w))$ by an antilinear\nisomorphism,\nand we say that $P$ is {\\em hermitian} (resp. {\\em positive hermitian}) if the\nresulting antilinear form on the vector space $H^0({\\cal E} (-w))$ is hermitian\n(resp. positive hermitian). Note that the hermitian condition can be expressed\nin sheaf-theoretic terms without refering to $H^0$ but I don't see a nice way to\ndo this for the positivity condition.\n\nWe say that the variation $({\\cal E} , {\\bf d})$ is\n{\\em polarizable} if there exists a polarization.\n\nWe say that a variation of mixed twistor structure $({\\cal E} , W_{\\cdot}, {\\bf d})$\nis {\\em graded-polarizable} if the associated-graded pieces $Gr ^W_i({\\cal E} )$ are\npolarizable as pure variations of twistor structure.\n\n\\begin{lemma}\n\\label{harmonic}\nIf $V$ is a flat bundle with pluriharmonic  metric then $V$ underlies a\npolarizable pure variation of twistor structure. In particular, if $X$ is a\ncompact K\\\"{a}hler manifold then  any irreducible representation of $\\pi _1(X)$\ncorresponds to a flat bundle underlying a pure variation of twistor structure.\nThis variation is unique up to change of weight (by tensoring with a one\ndimensional twistor structure). Any polarizable variation of pure twistor\nstructure is a direct sum of ones obtained from irreducible representations\nby this construction.\n\\end{lemma}\n{\\em Proof:}\nSuppose ${\\cal E}$ is a pure variation of twistor structure of weight zero (the other\nweights are treated by tensoring with ${\\cal O} _{{\\bf P} ^1}(w)$).  Let $p: X\\times {\\bf P}\n^1\\rightarrow X$ denote the first projection. Then $E:=p_{\\ast}({\\cal E} )$\nis a locally free ${\\cal C}^{\\infty}_X$-module on $X$ of rank equal to the rank of\n${\\cal E}$, and we have\n$$\n{\\cal E} = p^{-1}(E)\\otimes _{p^{-1}{\\cal C}^{\\infty}_X}\n{\\cal C}^{\\infty}_X{\\cal O} _{{\\bf P} ^1}.\n$$\nSimilarly\n$p_{\\ast} \\xi {\\cal A} ^0_X = {\\cal A} ^0_X$. However,\n$$\np_{\\ast} \\xi {\\cal A} ^1_X \\cong {\\cal A} ^1_X \\otimes _{{\\bf C}} {\\bf C} \\langle \\lambda , \\mu\n\\rangle\n$$\nis a bundle of rank twice that of ${\\cal A} ^1_X$,\nsince\n$\\xi {\\cal A} ^1_X \\cong {\\cal A} ^1_X \\otimes {\\cal O} _{{\\bf P} ^1} (1)$ as described previously.\nWe have\n$$\np_{\\ast}({\\bf d}) = \\lambda \\partial + \\mu \\overline{\\partial} ,\n$$\nand $p_{\\ast}(D)$ is an operator\n$$\np_{\\ast}(D): E\\rightarrow E \\otimes _{{\\cal C}^{\\infty}_X}\n{\\cal A} ^1_X \\otimes _{{\\bf C}} {\\bf C} \\langle \\lambda , \\mu\n\\rangle\n$$\nwhose symbol is $\\lambda \\partial + \\mu \\overline{\\partial}$.  We can write\n$$\np_{\\ast}(D) = \\lambda D' + \\mu D''\n$$\nwhere $D'$ and $D''$ are operators from $E$ to $E\\otimes _{{\\cal C}^{\\infty}_X}\n{\\cal A} ^1_X$. These operators are uniquely determined by the above equation\nand furthermore it follows that $D'$ has symbol $\\partial$ and $D''$ has symbol\n$\\overline{\\partial}$. The condition $D^2=0$ implies the integrability conditions\n$(D')^2=0$,\n$(D'' ) ^2=0$ and $D'D'' + D'' D' = 0$.\n\nConversely suppose $(E,D', D'')$ is a bundle with operators $D'$ and $D''$\nwith symbols $\\partial$ and $\\overline{\\partial}$ respectively and\nsatisfying the above integrability conditions. Then setting $$ {\\cal E} :=\np^{-1}(E)\\otimes _{p^{-1}{\\cal C}^{\\infty}_X} {\\cal C}^{\\infty}_X{\\cal O} _{{\\bf P} ^1},\n$$\nwe get a ${\\cal C} ^{\\infty}$ family of pure twistor structures of weight zero.\nPutting\n$D:= \\lambda D' + \\mu D''$ considered as an operator from\n${\\cal E}$ to ${\\cal E} \\otimes _{{\\cal C}^{\\infty}_X{\\cal O} _{{\\bf P} ^1}} \\xi {\\cal A} ^1_X$\nvia the previously-mentionned isomorphism $\\xi {\\cal A} ^1_X\\cong {\\cal A} ^1_X\n\\otimes _{{\\bf C} } {\\cal O} _{{\\bf P} ^1}(1)$ (and via consideration of $\\lambda $ and $\\mu$\nas sections of ${\\cal O} _{{\\bf P} ^1}(1)$), we obtain a variation of pure twistor\nstructure $({\\cal E} , D)$.\n\nThese constructions establish a one to one correspondence between pure\nvariations of twistor structure $({\\cal E} , D)$ of weight zero, and triples $(E,D',\nD'')$ as in the definition of harmonic bundle \\cite{HBLS}.  We claim that\nthe variation $({\\cal E} , D)$ is polarizable if and only if the operators $D'$ and\n$D''$ are related by a metric (which is thus a harmonic metric) $K$ on $E$\naccording to the definitions in \\cite{HBLS}.\n\nThe first thing to note is that under the above correspondence (assuming ${\\cal E}$\nis of weight zero) we have that $p_{\\ast}(\\sigma ^{\\ast}{\\cal E} )= \\overline{E}$ is\nthe complex conjugate ${\\cal C} ^{\\infty}$ bundle. We have to calculate\n$p_{\\ast}(\\sigma ^{\\ast} D)$.  Note that we can decompose $D'$ and $D''$\naccording to Hodge type of forms and write\n$$\np_{\\ast}(D) = \\lambda D'_{1,0} + \\lambda D' _{0,1} +\n\\mu D''_{1,0} + \\mu D'' _{0,1} .\n$$\nIf we write this in a form ready to apply the morphism $\\psi$ used above to\ndefine $\\sigma ^{\\ast}(D)$ it becomes\n$$\np_{\\ast}(D) = \\lambda (D'_{1,0} + \\frac{\\mu }{\\lambda}D''_{1,0} )\n+ \\mu (\n\\frac{\\lambda}{\\mu} D' _{0,1} +\n D'' _{0,1} ).\n$$\nNow $p_{\\ast}(\\sigma ^{\\ast} D)$ is obtained by applying $\\sigma ^{\\ast}$ to the\n${\\cal E}$-coefficients, and by applying the operation $\\psi$ to the\nform coefficients and $\\lambda $ and $\\mu$. We write this as\n$$\np_{\\ast}(\\sigma ^{\\ast} D)= \\lambda \\cdot \\overline{\\sigma ^{\\ast}\n( \\frac{\\lambda}{\\mu} D' _{0,1} +\n D'' _{0,1} )} +\n\\mu \\cdot \\overline{\\sigma ^{\\ast}  (D'_{1,0} + \\frac{\\mu\n}{\\lambda}D''_{1,0} )}.\n$$\nThe pieces $D'_{1,0}$ and so on are invariant under $\\sigma ^{\\ast}$ because\nthey are constant in the ${\\bf P} ^1$-direction. However, note that\n$$\n\\overline{\\sigma ^{\\ast}(\\frac{\\lambda}{\\mu})}\n=-\\frac{\\mu}{\\lambda}\n$$\nsince the antipodal involution is written $t\\mapsto -\\overline{t}^{-1}$\nin terms of the coordinate $t$ on ${\\bf A}^1$. Thus we have\n$$\np_{\\ast}(\\sigma ^{\\ast} D)= \\lambda \\cdot\n( -\\frac{\\mu}{\\lambda} \\overline{D}' _{0,1} +\n \\overline{D}'' _{0,1} ) +\n\\mu \\cdot  (\\overline{D}'_{1,0} - \\frac{\\lambda }{\\mu}\\overline{D}''_{1,0} )\n$$\n$$\n= \\lambda (\\overline{D}'' _{0,1}- \\overline{D}''_{1,0}) + \\mu\n(\\overline{D}'_{1,0}-\\overline{D}' _{0,1}).\n$$\nIn other words the decomposition\n$$\np_{\\ast}(\\sigma ^{\\ast} D)= \\lambda (\\sigma ^{\\ast}D)' + \\mu (\\sigma\n^{\\ast} D)''\n$$\nis given by\n$$\n(\\sigma ^{\\ast}D)' = \\overline{D}'' _{0,1}- \\overline{D}''_{1,0}\n$$\nand\n$$\n(\\sigma ^{\\ast}D)'' = \\overline{D}'_{1,0}-\\overline{D}' _{0,1}.\n$$\nThus the triple  associated to $(\\sigma ^{\\ast}{\\cal E} , \\sigma ^{\\ast}D)$\nis $(\\overline{E}, \\overline{D}'' _{0,1}- \\overline{D}''_{1,0},\n\\overline{D}'' _{0,1}- \\overline{D}''_{1,0})$.\n\nA polarization $P$ of ${\\cal E}$ corresponds to a  morphism\n$$\nK:= p_{\\ast}(P): E\\otimes \\overline{E} \\rightarrow {\\bf C}\n$$\nwhich, by hypothesis, is a positive definite hermitian form on $E$. The\nmorphism $K$ intertwines the pair of operators $(D', D'')$ on $E$ with the pair\nof operators $(overline{D}'' _{0,1}- \\overline{D}''_{1,0},\n\\overline{D}'' _{0,1}- \\overline{D}''_{1,0})$ on $\\overline{E}$---which exactly\nsays that $K$ is a harmonic metric for the triple $(E,D', D'')$. Conversely by\nfollowing the above formulas in the other direction, a harmonic metric leads to\na polarization which is a morphism of mixed twistor structures.\nThis completes the proof of the claimed correspondence between polarizations and\nharmonic metrics.\n\nThis claim implies the lemma, modulo the remark that we can pass from\nvariations $({\\cal E} , D)$ pure of weight $w$ to variations pure of weight zero (and\nback again) by tensorization by the constant rank one twistor\nstructure ${\\cal O} _{{\\bf P}\n^1}(-w)$ (resp.   ${\\cal O} _{{\\bf P}\n^1}(w)$). Note that these twistor structures admit polarizations, so tensoring\nwith them preserves the polarization condition.\n\\hfill $\/$\\hspace*{-.1cm}$\/$\\hspace*{-.1cm}$\/$\\vspace{.1in}\n\n{\\em Remark:}  This lemma is the fundamental reason for the utility of the\nnotion of twistor structure: it applies to any irreducible representation of the\nfundamental group, and not just certain ones (the variations of Hodge\nstructure).\n\n\n\n\\subnumero{Description in terms of $t$-connections on $X$ and\n$\\overline{X}$}\n\nOn $X\\times {\\bf A}^1 \\subset X \\times {\\bf P} ^1$ we have an isomorphism\n$$\n\\Phi : {\\cal A} ^i_X |_{X\\times {\\bf A}^1 } \\cong A^i (X\\times {\\bf A}^1\/ {\\bf A}^1).\n$$\nVia this isomorphism we have\n$$\n\\Phi \\circ {\\bf d} = \\overline{\\partial} + t \\partial\n$$\nwhere $t$ is the coordinate on ${\\bf P} ^1$.\nUsing this isomorphism we can identify an operator $D$ with an operator\n$$\nD_0 : {\\cal E} \\rightarrow {\\cal E} \\otimes A^1(X\\times {\\bf A}^1\/ {\\bf A}^1)\n$$\nsatisfying the Leibniz rule\n$$\nD_0 (ae) = aD_0(e) + (\\overline{\\partial} a + t \\partial a) e.\n$$\nDecompose according to type as $D_0 = D_0^{1,0}+ D_0^{0,1}$. Furthermore we can\nadd the operator $\\partial $ in the ${\\bf A} ^1$-direction to obtain an operator\n$$\nD_0^{0,1} + \\overline{\\partial} : {\\cal E} \\rightarrow A^{0,1}(X\\times {\\bf A}^1, {\\cal E} ).\n$$\nThis operator satisfies Leibniz' rule for the symbol $\\partial_{X\\times {\\bf\nA}^1}$ and it is integrable (since $D_0^2= 0$, $\\overline{\\partial} ^2=0$ and\n$[\\overline{\\partial} , D_0]=0$). Thus this operator defines a holomorphic structure for the\nbundle ${\\cal E}|_{X\\times {\\bf A}^1}$.  Denote this holomorphic bundle---or more\nprecisely its sheaf of holomorphic sections---by ${\\cal F}$. The remaining part\n$D^{1,0}_0$ commutes with the holomorphic structure and satisfies Leibniz' rule\nfor the symbol $t\\partial$.  In other words this operator leads to a\nmorphism of\nholomorphic sheaves\n$$\n\\nabla : {\\cal F} \\rightarrow {\\cal F} \\otimes _{{\\cal O} _{X\\times {\\bf A}^1}} \\Omega\n^1_{X\\times {\\bf A}^1\/{\\bf A}^1}\n$$\nsatisfying the Leibniz rule $\\nabla (af)= a\\nabla (f) + d(a)\\nabla (f)$\nfor  sections $a$ of ${\\cal O} _{X\\times {\\bf A}^1}$ and $f$ of ${\\cal F}$.\nFurthermore $\\nabla ^2=0$. Thus $({\\cal F} , \\nabla )$ is a {\\em $t$-connection on\n$X\\times {\\bf A}^1\/{\\bf A}^1$}.\n\nWe obtain by symmetry a similar description on the other\nstandard affine open set\n${\\bf A}^1\\subset\n{\\bf P} ^1$ whose coordinate is $t^{-1}$, but with $X$ replaced by $\\overline{X}$.\nWe are forced to replace $X$ by $\\overline{X}$ because it is the $1,0$ part of\nthe connection on $X$ whose symbol doesn't degenerate. This yields a\n$t^{-1}$-connection $({\\cal F} ', \\nabla ')$ on  $\\overline{X}\\times {\\bf A}^1\/{\\bf\nA}^1$.\n\n\nIn order to relate these two descriptions, in other words to understand the\nglueing between the objects $({\\cal F} , \\nabla )$ and $({\\cal F} ', \\nabla ')$\nwe need a different trivialization of ${\\cal A} ^i_X$ over $X\\times {\\bf G}_m \\subset X\n\\times {\\bf P} ^1$,\n$$\n\\Psi : {\\cal A} ^1_X |_{X\\times {\\bf G}_m} \\cong A^1(X\\times {\\bf G}_m \/ {\\bf G}_m )\n$$\nsuch that\n$$\n\\Psi \\circ {\\bf d} = d = \\overline{\\partial} + \\partial .\n$$\nVia this trivialization, ${\\cal E} |_{X\\times {\\bf G}_m}$ becomes a vector bundle with a\nflat connection relative to the base $X\\times {\\bf G}_m$ and a commuting operator\n$\\overline{\\partial}$ in the horizontal direction. The sheaf of sections annihilated by\nthe connection and the $\\overline{\\partial}$-operator is a locally free sheaf of\n$p_1^{-1}({\\cal O} _{{\\bf G}_m})$-modules on $X^{\\rm top}\\times {\\bf G}_m$, in other words it is\na holomorphic family of local systems which we denote ${\\cal L} = \\{ L_t\\} _{t\\in\n{\\bf G}_m}$ on $X^{\\rm top}$.  (If we choose a basepoint $x\\in X$ then this is\nthe same\nas a holomorphic family of representations of $\\pi _1(X,x)$ modulo holomorphic\nchanges of basis.)\n\nThe relationship between $({\\cal F} , \\nabla )|_{X\\times {\\bf G}_m}$ and ${\\cal L}$ is that\nfor any $t\\in {\\bf G}_m$, $t^{-1}\\nabla$ is a holomorphic flat connection and $L_t$ is\nits local system of flat sections.\nBy symmetry a similar interpretation holds for $({\\cal F} ', \\nabla ')$.\n\n\n\\numero{Cohomology of smooth compact K\\\"{a}hler manifolds with VMTS\ncoefficients}\n\nClassically, the cohomology of a smooth compact K\\\"{a}hler manifold with\nconstant\ncoefficients was the first object to be provided with a pure Hodge structure.\nApplying our meta-theorem to this statement doesn't yield anything other than\ntrading in the Hodge structure for a twistor structure, since the notion of\nHodge structure doesn't appear in the hypothesis.  One of the main starting\npoints for modern Hodge theory was Deligne's observation that the K\\\"ahler\nidentities also work with coefficients in a variation of Hodge structure\n\\cite{DeligneLetSerre}. This\nleads to the theorem that the $k$-th cohomology of a smooth compact K\\\"ahler\nmanifold with coefficients in a pure variation of Hodge structure of weight $n$,\ncarries a pure Hodge structure of weight $n+k$.  This was generalized slightly\nin \\cite{SteenbrinkZucker} to the case of coefficients in a variation of mixed\nHodge structure, giving a mixed Hodge structure on the cohomology. We generalize\nto twistor structures in this section.\n\n\\begin{theorem}\n\\label{coho1}\n{\\rm (\\cite{DeligneLetSerre}, \\cite{SteenbrinkZucker})}\nSuppose $({\\cal E} , W_{\\cdot} , {\\bf d})$ is a $Gr$-polarizable variation of mixed\ntwistor structure on a compact K\\\"ahler manifold $X$. Then $H^k(X, {\\cal E} _1)$ (the\ncohomology with coefficients in the underlying flat bundle) carries a natural\nmixed twistor structure. If the variation is pure of weight $n$ then the $k$-th\ncohomology is pure of weight $n+k$.\n\\end{theorem}\n\n\\subnumero{Pure coefficients}\nFirst we treat the case of pure coefficients.\nSuppose that $X$ is a smooth compact K\\\"{a}hler variety and\n$({\\cal E} ,{\\bf d})$ is a polarizable variation of pure twistor structure of weight\n$n$ on $X$.   Let $L_1$ denote the underlying flat bundle (it is the fiber over\n$1$ of the associated family of flat bundles $\\{ L_t\\}$).  We will construct a\npure  twistor structure of weight $n+k$ on the $k$-th cohomology of $X$ with\ncoefficients in $L_1$.\n\nIn this case note that there is a harmonic bundle $(E, D', D'')$ and\n$$\n{\\cal E}= p_1^{\\ast} E \\otimes p_2^{\\ast}{\\cal O} _{{\\bf P} ^1}(n),\n$$\nwith ${\\bf d}= \\lambda D' +\n\\mu D'' $. Here the pullbacks $p_1^{\\ast}$ and $p_2^{\\ast}$ take ${\\cal C}\n^{\\infty}_X$-modules (resp. ${\\cal O} _{{\\bf P} ^1}$-modules) to\n${\\bf C} ^{\\infty}_X{\\cal O} _{{\\bf P} ^1}$-modules on $X\\times {\\bf P} ^1$.\n\nLet $\\xi {\\cal A} ^{\\cdot}({\\cal E} )$ be the twistor complex of forms with\ncoefficients in\n${\\cal E}$,\na complex of ${\\cal C} ^{\\infty}_X{\\cal O} _{{\\bf P} ^1}$-modules on $X\\times {\\bf P} ^1$ with\noperator ${\\bf d}$. We have\n$$\n\\xi {\\cal A} ^i({\\cal E} ) \\cong p_1^{\\ast}(A^i(E)) \\otimes p_2^{\\ast}{\\cal O} _{{\\bf P} ^1}(n+i).\n$$\nLet $H^i(E) \\subset A^i(E)$ denote the subspace of harmonic $i$-forms\nwith coefficients in $E$. We have for any $(a,b)\\neq (0,0)\\in {\\bf C} ^2$\nisomorphisms\n$$\nH^i(E) \\stackrel{\\cong}{\\rightarrow} H^i(A^{\\cdot}(E), aD' + bD'').\n$$\nThis is shown in \\cite{HBLS} for $(a,b) = (0,1)$ and $(1,1)$, but the same proof\n(which is just a direct generalization of the standard utilization of the\nK\\\"ahler identities) works in general---one defines the laplacian for\n$aD' + bD''$ and shows that it is proportional to the laplacian for $D''$. We\nget back to Deligne's original idea about extending the K\\\"ahler identities.\nIn\nparticular, the above morphism induces an isomorphism\n$$\nH^i(E) \\otimes   {\\cal O} _{{\\bf P} ^1}(n+i)\n\\stackrel{\\cong}{\\rightarrow} M:= R^ip_{2,\\ast}(\\xi {\\cal A} ^{\\cdot}({\\cal E} ),\n{\\bf d}).\n$$\nThis proves that the higher direct image is a bundle $M$ which is  pure of\nweight\n$n+i$.\n\nWe  now give a second, analytic construction of $M$, which will be useful later\non. However, we refer to the above differential-geometric point of view to prove\npurity of the weight quotients of our structure.\n\nLet ${\\cal F}$, ${\\cal L}$ and ${\\cal F} '$\nbe the triple associated to ${\\cal E}$ consisting of a $t$-connection $({\\cal F} ,\n\\nabla )$\non $X\\times {\\bf A}^1$, a family of local systems ${\\cal L} = \\{ L_t\\}$ and a\n$t^{-1}$-connection $({\\cal F} ', \\nabla ')$ on the other $\\overline{X}\\times {\\bf\nA}^1$. Recall that  $L_t$ is the flat bundle over $X^{\\rm top}$ associated\nto the\nholomorphic flat connection  $({\\cal F} _t, t^{-1} \\nabla _t)$ on $X$, and it is also\nthe flat bundle associated to the  connection\n$({\\cal F}' _{t^{-1}}, t \\nabla _{t^{-1}})$ on $\\overline{X}$.\n\nLet $\\xi \\Omega ^{\\cdot}_{X}({\\cal F} )$\ndenote the Rees bundle complex of the complex of\nrelative differentials with coefficients in ${\\cal F}$, with differential given\nby the $t$-connection $\\nabla$.  More precisely we have\n$$\n\\xi \\Omega ^{\\cdot}_{X}({\\cal F} )= \\xi (\\Omega ^{\\cdot}_X, F) \\otimes _{{\\cal O}\n_{X\\times {\\bf P} ^1}}{\\cal F} ,\n$$\nwhere the filtration $F$ is the usual Hodge filtration (i.e. the ``stupid\nfiltration'').\n\n{\\em Remark:} there shouldn't be too much confusion between our use of the\nsymbol $\\xi$ here for the Rees bundle construction with one filtration, and its\nuse elsewhere for the Rees bundle construction with two filtrations: it depends\non whether we are working over the affine line (as is presently the case) or\nover ${\\bf P} ^1$ (as was the case in the differential-geometric construction\nabove).\n\nFor $t\\neq 0$ the hypercohomology of\n$\\xi \\Omega ^{\\cdot}_{X}({\\cal F} )|_{X\\times \\{ t\\} }$\ncalculates the cohomology of the local system $L_t$.\nThus over\n${\\bf G}_m$ we have\n$$\n{\\bf R}^kp_{2,\\ast}(\\xi \\Omega ^{\\cdot}_{X}({\\cal F} )\n|_{X\\times {\\bf G}_m} \\cong\n{\\bf R}^kp_{2,\\ast}{\\cal L} .\n$$\n\nLet $\\xi \\Omega ^{\\cdot}_{\\overline{X}}({\\cal F} ')$\ndenote the Rees bundle complex of the complex of\nrelative differentials with coefficients in ${\\cal F}'$, with differential given\nby the $t^{-1}$-connection $\\nabla '$. Note that this is now taking place\nover the affine line neighborhood of $\\infty$. Again, over ${\\bf G}_m$ we have\n$$\n{\\bf R}^kp_{2,\\ast}(\\xi \\Omega ^{\\cdot}_{\\overline{X}} ({\\cal F} ')\n|_{\\overline{X}\\times {\\bf G}_m} \\cong\n{\\bf R}^kp_{2,\\ast}{\\cal L} .\n$$\nWe can use these two isomorphisms to glue together\n${\\bf R}^kp_{2,\\ast}(\\xi \\Omega ^{\\cdot}_{X}({\\cal F} )$\nand ${\\bf R}^kp_{2,\\ast}(\\xi \\Omega ^{\\cdot}_{\\overline{X}} ({\\cal F} ')$ to obtain a\nbundle $M$ over ${\\bf P} ^1$.   This bundle is the same as that constructed\npreviously; in particular it is pure of weight $n+k$.\n\n\n\n\n\n\n\\subnumero{Mixed coefficients}\nFor this section suppose that $X$ is a smooth compact K\\\"{a}hler variety and\n$(E,W,{\\bf d})$ is a graded-polarizable variation of mixed twistor structure on\n$X$.   Let $L_1$ denote the underlying flat bundle (it is the fiber over $1$ of\nthe associated family of flat bundles $\\{ L_t\\}$).  We will construct a mixed\ntwistor structure  on the $k$-th cohomology of $X$ with coefficients in $L_1$\n(fix $k$ for the rest of this section).\n\n\nLet $\\xi {\\cal A} ^i_X(E)$ be the twistor complex of forms with coefficients in\n$E$. It\nis a complex of locally free ${\\cal C} ^{\\infty}_X{\\cal O} _{{\\bf P} ^1}$-modules on $X\\times\n{\\bf P} ^1$ with operator ${\\bf d}$.\n\nDefine the {\\em pre-weight filtration} as\n$$\nW^{\\rm pre}_n \\xi {\\cal A}^i_X(E):= \\xi {\\cal A}^i_X(W_nE).\n$$\nLet $M$ denote the $k$-th cohomology sheaf on ${\\bf P} ^1$ of the filtered\ncomplex\n$$\n{\\cal M} ^{\\cdot} := p_{2,\\ast} (\\xi {\\cal A}^i_X(E), W^{\\rm pre}, {\\bf d}).\n$$\nNote that $M$ has a filtration which we denote $W^{\\rm pre}M$ and we define\n$$\nW_mM := W^{\\rm pre}_{m-k}M .\n$$\nWe claim that $(M,W)$ is  a mixed\ntwistor structure. Given the previous result in the pure case,\nthis is essentially\nthe same as Lemma \\ref{degen} below but we give the direct argument here as a\nwarmup. The spectral sequence for the  cohomology of the filtered complex ${\\cal M}\n^{\\cdot}$ starts with  $$\nE^{p,q}_1 = H^{p+q} (Gr ^{W^{\\rm pre}}_{-p}(\\xi {\\cal A}^{\\cdot}_X(E)) )\n$$\n$$\n= H^{p+q} (\\xi {\\cal A} ^{\\cdot}_X (Gr ^W_{-p}(E))),\n$$\nwhich is a pure twistor structure of weight $p+q-p = q$ by the above result for\nthe pure polarized variations of twistor structure $Gr ^W_{-p}(E)$.  The\ndifferential $d_1:E^{p,q}_1 \\rightarrow E^{p+1,q}_1$ is a morphism of twistor\nstructures of the same weight, so the cohomology of the differential $d_1$ are\npure twistor structures $E^{p,q}_2$ of weight $q$. Now the differential $d_2:\nE^{p,q}_2 \\rightarrow E^{p+2, q-1}_2$ must vanish since it is a morphism from a\nsemistable bundle of slope $q$ to a semistable bundle of slope $q-1$. Thus\n$E^{p,q}_3=E^{p,q}_2$ and arguing by induction, $E^{p,q}_2=E^{p,q}_{\\infty}$ are\npure twistor structures of weight $q$. But\n$$\nE^{p,q}_{\\infty} = Gr ^{W^{\\rm pre}}_{-p}H^{p+q}(\\xi {\\cal A}^{\\cdot}_X(E)),\n$$\nin particular\n$$\nGr ^W_m (M)= Gr ^{W^{\\rm pre}}_{m-k}H^{k}(\\xi {\\cal A}^{\\cdot}_X(E))\n= E^{k-m, m}_{\\infty}.\n$$\nThis shows that $Gr ^W_m (M)$ is pure of weight $m$ as desired.\n\nOne can do an analytic construction of $M$ parallel to the analytic construction\nin the pure case. This would essentially be repeating what we will say below\nconcerning patching, so for now it is left to the reader.\n\nWe have now proved Theorem \\ref{coho1}.\n\\hfill $\/$\\hspace*{-.1cm}$\/$\\hspace*{-.1cm}$\/$\\vspace{.1in}\n\n\\numero{Cohomology of open and singular varieties}\nWe generalize \\cite{Hodge2}.  For now we stay with the easy situation of\ncoefficients which extend across a compactification, and don't attack the\nproblem\nof coefficients which are a local system only defined on the open\nvariety---since even in the case of variations of Hodge structure, this is a\nmuch harder problem.\n\n\\begin{theorem}\n\\label{TwistorII}\n{\\rm (cf \\cite{Hodge2}, 3.2.5)}\nSuppose $X$ is a compact K\\\"{a}hler manifold with a divisor $D\\subset X$ with\nnormal crossings. Let $U= X-D$. Suppose $({\\cal E} , W_{\\cdot}, D)$ is a variation\nof mixed twistor structure on $X$ with underlying flat bundle $V$.   Then the\ncohomology $H^i(U, V|_U)$ carries a natural mixed twistor structure, which can\nbe described as the higher direct image of the de Rham complex of\n$({\\cal E} , W_{\\cdot}, D)|_U$ via the projection $U\\times {\\bf P} ^1 \\rightarrow\n{\\bf P} ^1$.\nThis mixed twistor structure is functorial in $U$ (independent of the\ncompactification).\n\\end{theorem}\n\n\n\nWe adopt the following definition: if $X$ is any complex analytic space, then a\n{\\em variation of mixed twistor structure on $X$} is a functorial assignment,\nfor every smooth complex manifold $M$ mapping to $X$, of a variation of mixed\ntwistor structure on $M$.  To be precise about this, it means that for\nevery morphism $f: M\\rightarrow X$ from  a smooth complex manifold we have\na VMTS\n${\\bf E}(f) = ({\\cal E} (f), W_{\\cdot}(f), {\\bf d}(f))$ on $M$; and whenever $a:\nM\\rightarrow M'$ and $f':M'\\rightarrow X$ with $f=f'a$ then we have an\nisomorphism $\\epsilon (a): a^{\\ast}({\\bf E}(f'))\\cong {\\bf E}(f)$, such\nthat the cocycle\ncondition holds: if  $M\\stackrel{a}{\\rightarrow}\nM'\\stackrel{b}{\\rightarrow} M''$\nand $f'': M'' \\rightarrow X$ with $f'=f''b$ and $f=f''ba$ then the composition\n$$\na^{\\ast} b^{\\ast}({\\bf E}(f'')) \\stackrel{a^{\\ast}\\epsilon (b)}{\\rightarrow}\na^{\\ast}({\\bf E}(f'))\\stackrel{\\epsilon (a)}{\\rightarrow} {\\bf E}(f)\n$$\nis equal to $\\epsilon (ba)$.  We obtain in particular the same type of\nfunctorial\ncollection of flat bundles on smooth manifolds mapping to $X$, which gives a\nflat bundle over any simplicial resolution of singularities as in \\cite{Hodge3}.\nThe fundamental group of the topological realization of such a simplicial\nresolution is the same as that of $X$ so this collection of flat bundles comes\nfrom a local system on $X$. We call this the {\\em underlying flat bundle} of the\nvariation ${\\bf E}$.\n\n\\begin{theorem}\n\\label{TwistorIII}\n{\\rm (cf \\cite{Hodge3} and  \\cite{SteenbrinkZucker})}\nIf $X$ is a complex projective variety (possibly singular) with a Zariski open\nsubset $U\\subset X$ and if\n$({\\cal E} , W_{\\cdot}, D)$ is a variation\nof mixed twistor structure on $X$ with underlying flat bundle $V$, then the\ncohomology $H^i(U, V|_U)$ carries a natural mixed twistor structure which is\nfunctorial in $U$.\n\\end{theorem}\n\n\n{\\em Remark:} The same yoga of weights as in \\cite{Hodge3}, Theor\\`eme 8.2.4,\ndepending on the openness and singularity of the variety, holds here.\n\n\n\\subnumero{Mixed twistor complexes}\nFor the proofs of theorems \\ref{TwistorII} and \\ref{TwistorIII} (which we do\ntogether) we proceed exactly as in \\cite{Hodge2} and \\cite{Hodge3}. In fact we\nlook at what is really going on in \\cite{Hodge3} and plug in the stuff from\n\\cite{Hodge2}.\n\nStart with the following definitions. A {\\em mixed twistor complex}\nis a filtered complex $(M^{\\cdot}, W^{\\rm pre}_{\\cdot})$ of sheaves of ${\\cal O}\n_{{\\bf P}\n^1}$-modules on ${\\bf P} ^1$ such that\n$$\n\\underline{H}^i(Gr ^{W^{\\rm pre}}_n(M^{\\cdot}))\n$$\n(the cohomology sheaf) is a locally free sheaf of ${\\cal O} _{{\\bf P} ^1}$-modules of\nfinite rank, pure of weight $n+i$. This shift of weights is the reason we\ncall this the {\\em pre-weight filtration} and\nuse\nthe superfix $W^{\\rm pre}$.\n\n\\begin{lemma}\n\\label{degen}\n{\\rm (\\cite{Hodge3} Scholie 8.1.9)}\nSuppose $(M^{\\cdot}, W^{\\rm pre}_{\\cdot})$ is a mixed twistor complex. Then\nthe spectral sequence for a filtered complex which calculates the cohomology\nsheaves  $\\underline{H}^i(M^{\\cdot})$ degenerates at $E_3$ (i.e. $d_r=0$ for\n$r\\geq 3$). The cohomology sheaves are locally free sheaves of ${\\cal O} _{{\\bf P}\n^1}$-modules and the filtration induced by the pre-weight filtration, when\nshifted to $$\nW_n\\underline{H}^i(M^{\\cdot}):= W^{\\rm pre}_{n-i}\\underline{H}^i(M^{\\cdot})\n$$\nis the weight filtration for a mixed twistor structure on\n$\\underline{H}^i(M^{\\cdot})$.\n\\end{lemma}\n{\\em Proof:}\nThe spectral sequence in question starts with\n$$\nE^{p,q}_1 = Gr ^{W^{\\rm pre}}_{-p}(M^{p+q}),\n$$\nwith differential $d_1: E_1^{p,q}\\rightarrow E_1^{p,q+1}$ being the\nassociated-graded of the differential of $M^{\\cdot}$.\nThe term $E^{p,q}_2$ is the cohomology of the differential $d_1$ and has\ndifferential $d_2: E^{p,q}_2\\rightarrow E^{p+1,q}_2$.\nThe cohomology of $d_1$\nis just the cohomology sheaf which appears in the definition of mixed\ntwistor complex above (with $i=p+q$ and $n=-p$); thus, by definition $E^{p,q}_2$\nis a locally free sheaf of ${\\cal O} _{{\\bf P} ^1}$-modules, pure of weight $q$.  The\ndifferential $d_2$ is a morphism of semistable bundles of the same slope\n$q$, so\n$E^{p,q}_3$, i.e. the cohomology of $d_2$, is again a locally free sheaf of ${\\cal O}\n_{{\\bf P} ^1}$-modules, pure of weight $q$. Now for $r\\geq 3$ we have\n$d_r: E^{p,q}_r\\rightarrow E^{p+r-1, q+2-r}$ which is a morphism between\nbundles which are, argueing inductively on $r$, pure of weights $q$ and $q+2-r <\nq$ respectively; thus $d_r=0$ and the spectral sequence degenerates.\n\nThe limit of the spectral sequence is\n$$\nE^{p,q}_3 = Gr ^{W^{\\rm pre}}_{-p}(\\underline{H}^{p+q}(M^{\\cdot})),\n$$\nand the fact that the associated graded pieces are locally free implies that\n$\\underline{H}^i(M^{\\cdot})$ is locally free and its induced filtration is\nby strict subbundles. The above $E^{p,q}_3$ being pure of weight $q = (p+q) +\n(-p)$ we can rewrite as saying that\n$W^{\\rm pre}_{n-i}\\underline{H}^i(M^{\\cdot})$ is pure of weight $i+ (n-i)=n$.\nThis shows that the weight filtration as shifted in the statement of the lemma\ngives a mixed twistor structure.\n\\hfill $\/$\\hspace*{-.1cm}$\/$\\hspace*{-.1cm}$\/$\\vspace{.1in}\n\nOur mixed twistor complexes will generally come from a more global situation.\nSuppose $Y$ is a topological space mapping to ${\\bf P} ^1$ with map denoted $p$.   A\n{\\em mixed twistor complex on $Y\/{\\bf P} ^1$} is a filtered complex of sheaves\n$(N^{\\cdot}, W^{\\rm pre})$ of $p^{-1}({\\cal O} _{{\\bf P} ^1})$-modules on $Y$ such that\n$$\n{\\bf R}^i p_{\\ast} (Gr ^{W^{\\rm pre}}_n(N^{\\cdot}))\n$$\nis a locally free sheaf of ${\\cal O} _{{\\bf P} ^1}$-modules, pure of weight $n+i$.\nIn this case, let ${\\rm Go}(N^{\\cdot})$ be the canonical Godement resolution\n(or any other canonical flasque resolution), commuting with subquotients. Set\n$$\nM^{\\cdot} = p_{\\ast}{\\rm Go}(N^{\\cdot})\n$$\nwith filtration $W^{\\rm pre}M = p_{\\ast}(W^{\\rm pre}N^{\\cdot})$.\nThe direct images of these resolutions calculate the higher direct images of the\ncomplexes in question. Since $Gr ^{{\\rm Go}(W^{\\em pre})}_n({\\rm Go}\n(N^{\\cdot})= {\\rm Go} (Gr\n^{W^{\\rm pre}}_n(N^{\\cdot})$  we have\n$$\n\\underline{H}^i(Gr^{W^{\\rm pre}M}_n(M^{\\cdot})) =\n{\\bf R}^i p_{\\ast} (Gr ^{W^{\\rm pre}}_n(N^{\\cdot}))\n$$\nwhich is, by hypothesis, pure of weight $n+i$. Thus $(M^{\\cdot}, W^{\\rm pre}M)$\nis a mixed twistor complex on ${\\bf P} ^1$.\n\nOf course in the case $Y={\\bf P} ^1$ we recover the usual notion of mixed twistor\ncomplex (noting however that in the passage from $N$ to $M$ in this case, the\ncomplex will be replaced by its Godement resolution).\n\n\\subnumero{Patching}\n\nThe complexes constructed in \\cite{Hodge2} for calculating the cohomology of\nopen varieties are closely tied to the holomorphic structure of $X$.  This\nconstruction will only work over the standard ${\\bf A}^1$---neighborhood of $0$.\nA similar construction will work for the holomorphic structure $\\overline{X}$\nfor the other standard neighborhood ${\\bf A}^1$ of $\\infty$. We need to patch\nthese two together, and here is where the relationship with the Leray spectral\nsequence (pointed out by N. Katz, according to \\cite{Hodge2}) comes in. Over\n${\\bf G}_m$ this allows us to relate either of the previous two constructions with a\ntopological construction for flat bundles. We return to the trichotomy\n$({\\cal F} , \\nabla )$---$\\{ L_t\\}$---$({\\cal F} ', \\nabla ')$ that was discussed in \\S 3.\n\nFor each of these regimes we obtain a complex of sheaves on the appropriate open\nsubset of ${\\bf P} ^1$. We then need to  patch them together to obtain a mixed\ntwistor complex. This is the analogue in the present situation of the two\ncomponents, holomorphic and topological, in Deligne's definition of mixed Hodge\ncomplex \\cite{Hodge3} (our third component\n$({\\cal F} ', \\nabla ')$ being, in that case, the complex\nconjugate of the first component\n$({\\cal F} , \\nabla )$, because Deligne considered real objects rather than\ncomplex ones as we do here).\n\nSo we discuss in general how to patch together complexes of sheaves to obtain\nmixed twistor complexes.\n\nThe basic situation we will treat is the following one: suppose we have\nfiltered complexes $M^{\\cdot}$ and $N^{\\cdot}$ of sheaves of ${\\cal O}$-modules\n(with the pre-weight filtrations denoted $W^{\\rm pre}M$ etc.), respectively over\nthe standard neighborhoods ${\\bf A}^1$ of $0$ and $\\infty$ in ${\\bf P} ^1$.\nSuppose we\nhave a filtered complex $P^{\\cdot}$ of sheaves of ${\\cal O}$-modules on ${\\bf G}_m$\n(the intersection of the two affine lines).  Finally suppose that we have\nfiltered quasiisomorphisms\n$$\nM^{\\cdot}|_{{\\bf G}_m} \\leftarrow P^{\\cdot} \\rightarrow N^{\\cdot} |_{{\\bf G}_m} .\n$$\nWe construct a filtered complex of sheaves of ${\\cal O}$-modules on ${\\bf P} ^1$ denoted\n$Patch (M\\leftarrow P \\rightarrow N)$ (or just $Patch$ for short) in the\nfollowing way.\n\\newline\n(1)\\, Let $i$ denote one of the three inclusions ${\\bf A}^1\\hookrightarrow {\\bf P}\n^1$ (twice) or ${\\bf G}_m \\hookrightarrow {\\bf P} ^1$. Let $Ri_{\\rm ex}$ denote a fixed\nfunctorial choice of right derived functor (a real functor from complexes to\ncomplexes, not  just a functor on the derived category) of some extension\nfunctor\nfor the inclusion $i$. For example this could be $i_{\\ast} \\circ {\\rm Go}$, the\ncomposition of direct image with the Godement resolution. We require that there\nbe fixed a functorial quasiisomorphism\n$$\n{\\cal F} ^{\\cdot}\\rightarrow i^{\\ast} Ri_{\\rm ex}({\\cal F} ^{\\cdot})\n$$\n(which is the case  notably in our example).\n\\newline\n(2)\\, Let $M^{\\cdot} _{\\rm ex}:=\nRi_{\\rm ex}M^{\\cdot}$ be this extension functor applied to the filtered complex\n$M^{\\cdot}$\n(resp. $P^{\\cdot}_{\\rm ex}:=Ri_{\\rm ex}P^{\\cdot}$, $N^{\\cdot}_{\\rm ex}\n:=Ri_{\\rm ex}N^{\\cdot}$ with $i$ being the appropriate\ninclusions).  They are again filtered complexes of sheaves of ${\\cal O}\n$-modules.\n\\newline\n(3)\\, In general if $f:A^{\\cdot} \\rightarrow B^{\\cdot}$ is a map of filtered\ncomplexes (or complexes of sheaves) let $Cone (A\\rightarrow B)$ denote the\nfiltered  complex (or complex of sheaves) defined as\n$$\nCone(A\\rightarrow B)^k\n:= A^{k+1} \\oplus B^k\n$$\nwith differential equal to $d_A + d_B + f$.\n\\newline\n(4)\\, Now note that we have a morphism\nof filtered complexes of sheaves of ${\\cal O}$-modules\n$$\nP^{\\cdot}_{\\rm ex} \\rightarrow\nM^{\\cdot}_{\\rm ex}\\oplus N^{\\cdot}_{\\rm ex}\n$$\n(put in a minus sign in one of the factors).\nWe define\n$$\nPatch( M \\leftarrow P \\rightarrow N):= Cone\n(P^{\\cdot}_{\\rm ex} \\rightarrow M^{\\cdot}_{\\rm ex}\\oplus\nN^{\\cdot}_{\\rm ex}).\n$$\nWe claim that\n$$\n\\underline{H}^i(Gr^{W^{\\rm pre}}_n Patch( M \\leftarrow P \\rightarrow\nN))\n$$\nis the sheaf of ${\\cal O}$-modules on ${\\bf P} ^1$ obtained by glueing together\n$\\underline{H}^i(Gr^{W^{\\rm pre}}_nM^{\\cdot})$ over the first neighborhood ${\\bf\nA}^1$, with $\\underline{H}^i(Gr^{W^{\\rm pre}}_nN^{\\cdot})$ over the second\nneighborhood ${\\bf A}^1$, via the isomorphisms of cohomology sheaves induced by\nthe filtered quasiisomorphisms\n$$\nM^{\\cdot}|_{{\\bf G}_m} \\leftarrow P^{\\cdot} \\rightarrow N^{\\cdot} |_{{\\bf G}_m} .\n$$\nSuppose $U$ is a connected open set contained in the\nfirst affine neighborhood. Then $U\\cap {\\bf A}^1$ (intersection with the\nother affine neighborhood) is equal to $U\\cap {\\bf G}_m$ and  $P|_{U\\cap\n{\\bf G}_m}\\rightarrow N|_{U\\cap {\\bf G}_m}$ is a filtered quasiisomorphism. Thus\n$$\nP^{\\cdot}_{\\rm ex}|_U\\rightarrow N^{\\cdot}_{\\rm ex}|_U\n$$\nis a filtered quasiisomorphism. In general if $A \\rightarrow B$ is a filtered\nquasiisomorphism of complexes of sheaves and $A\\rightarrow C$ is any morphism\nthen $B \\rightarrow Cone (A \\rightarrow B\\oplus C)$ is a filtered\nquasiisomorphism. In our case this says that the composition\n$$\nM^{\\cdot} |_U \\rightarrow M^{\\cdot}_{\\rm ex} |_U\n\\rightarrow Patch (M \\leftarrow P \\rightarrow N) |_U\n$$\nis a filtered quasiisomorphism\n(the first arrow is the filtered quasiisomorphism coming from our assumption in\n(2) above). Hence the cohomology sheaf of the associated graded of\n$Patch$ restricted to $U$ is  naturally isomorphic to that  of $M$. Similarly if\n$U'$ is a neighborhood contained in the second ${\\bf A}^1$ then the cohomology\nsheaf is naturally isomorphic to that of $N$. Finally, if $U$ is contained in\n${\\bf G}_m$ then the composition of these two isomorphisms is equal to that which\ncomes from the diagram $M\\leftarrow P\\rightarrow N$ (the minus sign we put in\nbefore garanties that we are not missing a minus sign here!).\n\nThus in order to construct a mixed twistor complex $Patch$ we just need to have\nall of the above data with the property that the cohomology sheaves of the\nassociated-graded for the pre-weight filtration, when glued together, become\npure of the right weights.\n\nWe can make a similar construction in any more complicated situation of a chain\nof quasiisomorphisms.  For example the actual situation we will need to\nconsider is when we have the sequence of filtered quasiisomorphisms\nof filtered complexes over ${\\bf G}_m$,\n$$\nM |_{{\\bf G}_m} \\leftarrow P \\rightarrow Q \\leftarrow R \\rightarrow N|_{{\\bf G}_m} .\n$$\nThis can be replaced by the sequence\n$$\nM |_{{\\bf G}_m} \\leftarrow Cone '(P \\oplus R \\rightarrow Q) \\rightarrow N|_{{\\bf G}_m}\n$$\nwhere $Cone '$ is the cone but shifted in such a way that it is normalized for\nthe first variable (the cone we defined above was normalized for the second\nvariable). Now we can directly apply the previous discussion and define\n$$\nPatch (M,P,Q,R,N):= Patch (M \\leftarrow Cone '(P \\oplus R \\rightarrow Q)\n\\rightarrow N).\n$$\nOne\namusing point to notice is that in this case there are two stray minus signs\nwhich cancel out. This is probably quite lucky if one wants to look at the real\nsituation, where a single minus sign might be very painful.\n\nFinally, the whole thing goes through equally well in the relative situation. If\n$p:Y\\rightarrow {\\bf P}  ^1$ is a morphism of topological spaces and if we have\nfiltered complexes of sheaves $M$ and $N$ on the two $p^{-1}({\\bf A}^1)$ and a\nfiltered complex of sheaves $P$ on $p^{-1}({\\bf G}_m )$ with filtered\nquasiisomorphisms\n$$\nM|_{p^{-1}({\\bf G}_m )} \\leftarrow P \\rightarrow N|_{p^{-1}({\\bf G}_m )}\n$$\nthen we obtain a filtered complex $Patch (M \\leftarrow P\\rightarrow N)$ of\nsheaves on $Y\\times {\\bf P} ^1$, which has the effect of patching together the\ncohomology sheaves of the complexes $M$ and $N$ along $P$.  We can then take the\ndirect image down to ${\\bf P} ^1$ as was described previously.\nOn the other hand, we could also take the direct images of $M$, $N$ and $P$ down\nto ${\\bf P} ^1$ and then patch them together. The answers in these two cases are not\nquite the same but are quasiisomorphic. For simplicity we choose the route of\nfirst taking the direct image then patching. Thus, to be concrete, we obtain a\nfiltered complex of sheaves\n$$\nPatch ( R^ip_{\\ast}(M,P,N)):=\nPatch \\left( R^ip_{\\ast}(M)\\leftarrow R^ip_{\\ast}(P) \\rightarrow\nR^ip_{\\ast}(N)\\right)\n$$\non ${\\bf P} ^1$, patching together the higher direct images (the higher direct\nimages being calculated by canonical flasque resolutions, for example).\n\nThere is a\nspectral sequence\n$$\nR^i p_{\\ast}Gr ^{W^{\\rm pre}}_n(M) \\Rightarrow Gr ^{W^{\\rm pre}}_nR^ip_{\\ast}M\n$$\nand the same for $P$ and $N$, each of these being a spectral sequence of sheaves\nover the appropriate open subsets of ${\\bf P} ^1$.  Note that filtered\nquasiisomorphisms induce isomorphisms of spectral sequences.  Thus we may patch\ntogether the spectral sequences to obtain a spectral sequence (of sheaves\non ${\\bf P}\n^1$) which we denote\n$$\nPatch (R^i p_{\\ast}Gr ^{W^{\\rm pre}}_n(M,P,N)) \\Rightarrow Gr ^{W^{\\rm\npre}}_nPatch ( R^ip_{\\ast}(M,P,N)).\n$$\nThe terminology on the right was defined above and similarly the notation on the\nleft is defined as\n$$\nPatch (R^i p_{\\ast}Gr ^{W^{\\rm pre}}_n(M,P,N)):=\n$$\n$$\nPatch \\left( R^i p_{\\ast}Gr ^{W^{\\rm pre}}_n(M)\\leftarrow\nR^i p_{\\ast}Gr ^{W^{\\rm pre}}_n(P)\\rightarrow\nR^i p_{\\ast}Gr ^{W^{\\rm pre}}_n(N) \\right) .\n$$\nIf the beginning term of the spectral sequence is a\nlocally free sheaf of ${\\cal O} _{{\\bf P} ^1}$-modules pure of weight $n+i$ then the\nspectral sequence degenerates after the next term, and the answer is again pure\nof weight $n+i$ (cf the argument of Lemma \\ref{degen}).\n\nThus in order to obtain a mixed twistor complex by patching, it suffices that\nthe resulting patch of the higher direct images of the associated graded,\n$$\nPatch \\, \\left( R^i p_{\\ast}Gr ^{W^{\\rm pre}}_n(M),\nR^i p_{\\ast}Gr ^{W^{\\rm pre}}_n(P),R^i p_{\\ast}Gr ^{W^{\\rm pre}}_n(N)\n\\right)\n$$\nbe a locally free sheaf of ${\\cal O} _{{\\bf P} ^1}$-modules, pure of weight $n+i$.\n\nA similar statement works in the situation of patching $M,P,Q,R,N$ (which is the\ncase we will use).\n\n\n\n\\subnumero{Logarithmic complexes for mixed twistor structures}\nSuppose  now that $Z$ is a smooth variety with a normal crossings divisor $D$,\nand that $U= Z-D$. Suppose  $E$ is a variation of mixed twistor structure\non $Z$,\nwhich we will now look at in terms of the three weighted objects\n${\\cal F}$ (with $t$-connection $\\nabla$), $L_t$ for $t\\in {\\bf G}_m$, and ${\\cal G}$\nwhich was denoted ${\\cal F} '$ previously.  These lie respectively over ${\\bf A}^1$,\n${\\bf G}_m$ and the other ${\\bf A}^1$.\n\nWe obtain the following complexes on subsets of the topological space $Y:=\nZ^{\\rm\ntop}\\times {\\bf P} ^1$.  First, $M^{\\cdot}$ is the complex over ${\\bf A}^1$ of\nholomorphic logarithmic differentials with coefficients in ${\\cal F}$,\n$$\nM^i = \\xi \\Omega ^i_Z(\\log D) \\otimes _{{\\cal O}_{X\\times {\\bf A} ^1} } {\\cal F}\n$$\nwith differential coming from the $t$-connection $\\nabla$ and weight filtration\ncombining the weight filtration of \\cite{Hodge2} with the filtration of ${\\cal F}$.\nHere as below, the symbol $\\xi$ refers to the operation described \\cite{NAHT},\n\\cite{SantaCruz}, with respect to the Hodge filtration of\n$\\Omega ^i_X(\\log D)$  (which in this case is the ``stupid'' filtration).\nSecond, $N^{\\cdot}$ is the same thing for ${\\cal G}$ on $\\overline{Z}$ transported to\n$Y$ via the isomorphism $\\overline{Z}^{\\rm top} \\cong Y$.  Finally, we define\nseveral filtered complexes $P$, $Q$ and $R$ on the inverse image of ${\\bf G}_m$.\nRecall that over $t\\in {\\bf G}_m$ we have the local system $L_t$ which is the local\nsystem of flat sections of $({\\cal F} , t^{-1} \\nabla )$ and similarly for ${\\cal G}$. Let\n${\\cal A}^{\\cdot}_U$ denote the sheaf of ${\\cal C} ^{\\infty}$ differential forms on\n$U$ and let $j: U\\rightarrow Z$ denote the inclusion.  We have filtered\nquasiisomorphisms (cf \\cite{Hodge2} p. 33, the map called $\\beta$ plus a\ndiscussion analogous to that of the top of page 33)\n$$\n(\\xi \\Omega ^{\\cdot}_Z(\\log D) \\otimes _{{\\cal O}_{X\\times {\\bf G}_m }} {\\cal F} , \\tau )\n\\rightarrow\n(j_{\\ast}\\xi \\Omega ^{\\cdot}_U\\otimes _{{\\cal O}_{X\\times {\\bf G}_m }} {\\cal F}, \\tau )\n$$\n$$\n\\rightarrow\n(\\xi {\\cal A}^{\\cdot}_U\\otimes _{{\\cal O} _{X\\times {\\bf G}_m }}{\\cal F} , \\tau )\n= (\\xi {\\cal A}^{\\cdot} _U \\otimes _{p^{-1}{\\cal O} _{{\\bf G}_m }} {\\cal L} , \\tau ).\n$$\nAt the end note that the family ${\\cal L}$ is considered as a locally free sheaf of\n$p^{-1}{\\cal O} _{{\\bf G}_m}$-modules on $Y\\times _{{\\bf P} ^1} {\\bf G}_m$.\n\nHere the filtrations $\\tau$ are the ``intelligent'' truncations of the complexes\nof differentials, tensored with the weight filtrations of ${\\cal F}$ or ${\\cal L}$.\n\nOn the other hand, we have (again see \\cite{Hodge2} p. 33, the map called\n$\\alpha$) the filtered quasiisomorphism over ${\\bf G}_m$,\n$$\n(\\xi \\Omega ^{\\cdot}_Z(\\log D) \\otimes _{{\\cal O}_{X\\times {\\bf G}_m}} {\\cal F} , W^{\\rm pre} )\n\\leftarrow\n(\\xi \\Omega ^{\\cdot}_Z(\\log D) \\otimes _{{\\cal O}_{X\\times {\\bf G}_m}} {\\cal F} , \\tau ) .\n$$\nIn view of this we put\n$$\nP^{\\cdot} := (\\xi \\Omega ^{\\cdot}_Z(\\log D) \\otimes _{{\\cal O}_{X\\times {\\bf G}_m}} {\\cal F} ,\n\\tau )\n$$\nover ${\\bf G}_m$ and\n$$\nQ^{\\cdot} := (\\xi {\\cal A}^{\\cdot} _U \\otimes _{p^{-1}{\\cal O} _{{\\bf G}_m}} {\\cal L} , \\tau )\n$$\nagain over ${\\bf G}_m$. Note that $Q$ now makes no reference to the complex structure\nso it is the same with respect to $Z$ or $\\overline{Z}$.  Let $R$ be the same\nthing as $P$ but on $\\overline{Z}$ and using ${\\cal G}$.  We obtain the diagram of\nfiltered quasiisomorphisms of complexes on $Y\\times _{{\\bf P} ^1} {\\bf G}_m$,\n$$\nM |_{{\\bf G}_m} \\leftarrow P \\rightarrow Q \\leftarrow R \\rightarrow N|_{{\\bf G}_m} .\n$$\nPut\n$$\nMTC(E):= Patch (Rp_{\\ast}M,Rp_{\\ast}P,Rp_{\\ast}Q,Rp_{\\ast}R,Rp_{\\ast}N)\n$$\nwith these and the previous notations.\nWe claim that this is a mixed twistor complex. This claim implies that the\ncohomology sheaves of $MTC(E)$ have natural mixed twistor structures. The fiber\nover $1$ is the same as the hypercohomology on $Z^{\\rm top}$ of the complex $Q$,\nwhich is just the cohomology of $U$ with coefficients in the local system $L_1\n|_U$.  This will therefore prove Theorem \\ref{TwistorII}.\n\n\\subnumero{Proof of claim}\nAn {\\em extension} of filtered complexes of sheaves is a short exact sequence\nof complexes inducing short exact sequences on all levels of the filtrations.\nThis induces an extension of the associated-graded complexes.\n\nAn extension of VMTS\n$$\n0 \\rightarrow E ' \\rightarrow E \\rightarrow E'' \\rightarrow 0\n$$\ninduces an extension of filtered complexes\n$$\n0 \\rightarrow MTC(E ') \\rightarrow MTC(E) \\rightarrow MTC(E'') \\rightarrow 0,\n$$\nhence an extension\n$$\n0 \\rightarrow Gr _n^{W^{\\rm pre}}MTC(E ') \\rightarrow Gr _n^{W^{\\rm pre}}\nMTC(E) \\rightarrow Gr _n^{W^{\\rm pre}}MTC(E'')\n\\rightarrow 0.\n$$\nFrom this we get a long exact sequence of cohomology sheaves.\nSuppose we know that\n$$\n\\underline{H}^iGr _n^{W^{\\rm pre}}MTC(E ') \\;\\; \\mbox{and} \\;\\;\n\\underline{H}^iGr _n^{W^{\\rm pre}}MTC(E '')\n$$\nare pure of weight $n+i$, and suppose we know also that the connecting maps\nin the\nlong exact sequence are\nzero.  Then we can conclude that $\\underline{H}^iGr _n^{W^{\\rm pre}}MTC(E )$\nare pure of weight $n+i$. This last phrase may be restated as saying that if\nwe know the connecting maps in the long exact sequence of cohomology of\nassociated graded pieces are zero, then $MTC(E ')$ and $MTC(E '')$\nbeing mixed twistor complexes implies that $MTC(E )$ is a mixed twistor complex.\n\nWe prove that for any VMTS $E$, $MTC(E)$ is a mixed twistor complex, by\ninduction on the size of the interval containing the weight-graded pieces of\n$E$. If the size of this interval is $1$ then $E$ is pure and we will treat\nthis case below.  Suppose that $E$ is in an interval of size $n$ and that\nwe know\nthe result for any VMTS in an interval of size $<n$. Let $W_iE$ be the lowest\nnonzero piece of the weight filtration. Apply the preceeding discussion to the\nextension of VMTS  $$\n0\\rightarrow W_iE \\rightarrow E \\rightarrow E\/W_iE \\rightarrow 0.\n$$\nBy our inductive hypothesis $MTC(W_iE)$ and $MTC(E\/W_iE)$ are mixed twistor\ncomplexes. In order to be able to conclude that $MTC(E)$ is a mixed twistor\ncomplex, we just have to know that the connecting maps in the long exact\nsequence calculating $H^jGr ^{W^{\\rm pre}}_nMTC(E)$, are zero. But these\nconnecting maps are obtained by a construction of the form, take an element with\ncoefficients in $E'' := E\/W_iE$ and lift it to an element with coefficients in\n$E$, then take the coboundary which will have coefficients in $E' := W_iE$.\nThe preweight of our element (which lies in some complex of forms or Godement\nresolution) is the sum of the weight of the coefficient in\n$E$ plus the weights of the other stuff.  But the weight of the coefficient in\n$E$ is strictly decreased by the operation described above, since the weights of\n$E'$ are strictly lower than the weights of $E''$. Thus, this operation is zero\non the associated graded for the preweight filtration, in other words the\nconnecting map in question is zero.\n\nThis completes the proof of the claim modulo the case where $E$ is pure, which\nwe now treat. It suffices to consider the case where $E$ is pure of weight\nzero.\nBy the discussion of the previous subsection (and with the same notations as\nthere), it suffices to check that the patching of $R^ip_{\\ast}Gr^{W^{\\rm\npre}}_n({\\bf x} )$ for ${\\bf x} = M,P,Q,R,N$, be pure of weight $n+i$.\n\nLet $D^{(k)}$ denote the disjoint union of the intersections of $k$ smooth\ncomponents of $D$.  We assume that the irreducible components of $D$ are smooth\nso that the sheaves denoted $\\varepsilon$ in \\cite{Hodge2} are trivial.\n\nThe {\\em residue map} gives an isomorphism\n$$\nres: Gr^{W^{\\rm pre}}_n(M^{i}) \\cong \\xi \\Omega ^{i-n}_{D^{(n)}}\\otimes {\\cal F}\n|_{D^{(n)}} \\otimes {\\cal O} _{{\\bf A}^1}(n\\cdot 0)\n$$\nwith differential induced by the $t$-connection $\\nabla$.\nThe term ${\\cal O} _{{\\bf A}^1}(n\\cdot 0)$ (which means the sheaf of functions with\npoles of order $n$ at the origin) comes from the fact that the residue map\ncontracts out $n$ things of the form $dz_i \/z_i$, which provide twists due to\nthe construction $\\xi$.\nOver\n$t\\neq 0$ (i.e. over ${\\bf G}_m \\subset {\\bf P} ^1$) this complex is a resolution of the\nlocal system $L_t |_{D^{(n)}}$.  Thus we have\n$$\nR^ip_{\\ast}Gr^{W^{\\rm pre}}_n(M^{\\cdot}) =\nR^{i-n}p_{\\ast}(\\xi \\Omega ^{\\cdot}_{D^{(n)}}\\otimes {\\cal F}\n|_{D^{(n)}}) \\otimes {\\cal O} _{{\\bf A}^1}(n\\cdot 0),\n$$\nand this restricts (via the natural isomorphism) to $R^{i-n}p_{\\ast}({\\cal L}\n|_{D^{(n)}})$ over ${\\bf G}_m$.  The direct images of the associated-graded pieces of\nthe complexes $P,Q,R$ all give the same answer $R^{i-n}p_{\\ast}({\\cal L}\n|_{D^{(n)}})$.\n\nThe same holds for $R^{i-n}p_{\\ast}Gr^{W^{\\rm pre}}_n(N^{\\cdot})$ and again\nthis restricts to $R^{i-n}p_{\\ast}({\\cal L} |_{D^{(n)}})$ (which doesn't depend\non the\ncomplex structure of $D^{(n)}$).\n\nThe patching of these direct images of associated-graded pieces for $M,P,Q,R,N$\nyields the filtered coherent sheaf constructed in Theorem \\ref{coho1}, for our\nVMTS $E$ pulled back to $D^{(n)}$, tensored by\n${\\cal O} _{{\\bf P} ^1}(2n)$ because of term ${\\cal O} _{{\\bf A}^1}(n\\cdot 0)$ and the similar\nterm in the  neighborhood of $\\infty$.    By the result of \\S 4 (recall\nthat here\nwe are treating the case where $E$ is pure of weight $0$) this patching is pure\nof weight $(i-n)+ 2n =n+i$. This completes the proof of the claim (and hence the\nproof of Theorem \\ref{TwistorII}).\n\n\n\n\n\n\n\n\n\n\n\n\\subnumero{The simplicial situation}\n\nIn order to prove Theorem \\ref{TwistorIII} we need to consider a simplicial\nsituation. Any simplicial scheme $X_{\\cdot}$ (of finite type) can  be replaced\n(via the method of \\cite{Hodge3}) by a simplicial collection of projective\nsmooth\nschemes $Z_{\\cdot}$ containing normal crossing divisors $D_{\\cdot}$ such\nthat the\nmaps are compatible with the normal crossing divisors. The topological type of\nthe  original simplicial scheme is recovered by taking the topological type\nof the\nsimplicial open subschemes $U_{\\cdot} = X_{\\cdot} - D_{\\cdot}$ (complement\nof the\ndivisor with normal crossings).\n\nFor each $k$ we have constructed a mixed twistor complex $(M^{\\cdot}_k, W^{\\rm\npre})$ calculating the cohomology of $U_k$ using the normal crossings\ncompactification $(X_k, D_k)$.  This is functorial (in a contravariant way): the\nface maps give morphisms of mixed twistor complexes\n$$\n(M^{\\cdot}_k, W^{\\rm\npre}) \\rightarrow (M^{\\cdot}_{k+1}, W^{\\rm\npre}).\n$$\nIn other words, we get a {\\em cosimplicial mixed twistor complex}.  The\ncohomology of the simplicial scheme is calculated by taking the associated\ndouble complex (with new differential the alternating sum of the face maps)\nand then taking its associated single complex.  We just have to\nprovide this with a structure of mixed twistor complex. This is of course\nexactly what is explained in \\cite{Hodge3}. We briefly sketch the argument.\nWe obtain a total\ncomplex $$\nN^j = \\bigoplus _{i+k=j}M^i_k,\n$$\nand we define its preweight filtration by\n$$\nW^{\\rm pre}_nN^j := \\bigoplus _{i+k=j} W^{\\rm pre}_{n+k}M^i_k.\n$$\nThe two differentials are compatible with this new weight filtration\nand in fact the differential $M^i_k \\rightarrow M^i_{k+1}$ induces\nthe zero map on the associated-graded for the weight filtration. In particular,\nthe double complex $Gr^{W^{\\rm pre}}_n(M^{\\cdot}_{\\cdot})$ has one of its\ndifferentials vanishing; the other differential is just the differential in each\nmixed twistor complex $M^{\\cdot}_k$.  Thus\n$$\nH^j(Gr^{W^{\\rm pre}}_n(M^{\\cdot}_{\\cdot}))= \\bigoplus _k H^{j-k}(Gr^{W^{\\rm\npre}}_{n+k}(M^{\\cdot}_k)),\n$$\nand this is pure of weight $n+j$ since each of the $M^{\\cdot}_k$ is a mixed\ntwistor complex. This shows that the total complex is again a mixed twistor\ncomplex.\n\n\n\nThe proof that the resulting mixed twistor structure on the cohomology is\nindependant of the choice of desingularization and normal crossing\ncompactification, (and, what is pretty much the same thing, functoriality) is\nexactly the same as in \\cite{Hodge2} \\cite{Hodge3}.\nThis completes the proof of Theorem \\ref{TwistorIII}.\n\\hfill $\/$\\hspace*{-.1cm}$\/$\\hspace*{-.1cm}$\/$\\vspace{.1in}\n\nWe have also obtained the\nsimplicial version:\n\n\\begin{theorem}\n\\label{TwistorSimpl}\nSuppose $X_{\\cdot}$ is a simplicial scheme with each $X_k$ projective over $Spec\n({\\bf C} )$, and suppose $U_{\\cdot} \\subset X_{\\cdot}$ is a simplicial open\nsubscheme. Suppose $(E, W, {\\bf d})$ is a graded-polarizable variation of mixed\ntwistor structure on $X_{\\cdot}$ (i.e. a functorial association of VMTS for\nevery\nsmooth scheme mapping to $X_k$, with isomorphisms of compatibility for the\nsimplicial maps). Then there is a mixed twistor structure on $H^i(U_{\\cdot}, L)$\nwhere $L$ is the flat bundle associated to $(E_1, {\\bf d}_1)$.\nThis mixed twistor structure is functorial for maps of VMTS and also for maps of\nsimplicial schemes $U_{\\cdot}$ (from the cohomology of $E$ on the target to the\ncohomology of the pullback of $E$ on the domain).\n\\end{theorem}\n\\hfill $\/$\\hspace*{-.1cm}$\/$\\hspace*{-.1cm}$\/$\\vspace{.1in}\n\n{\\em Remark:}  For now, we have not made any study of VMTS on open varieties,\nso we are forced to assume the existence of extensions to projective varieties\n$X_k$ and furthermore that these extensions should be organized in a simplicial\nway. Eventually these restrictions should be gotten rid of (possibly in\nexchange for a condition of admissibility of the VMTS on the open varieties).\n\n\n\n\\numero{Nilpotent orbits and the limiting mixed twistor structure}\n\nSuppose $U=X-P$ is a compact curve with one point removed (actually any number\nof points is ok but we work near one) and suppose $E$ is a harmonic bundle on\n$U$. We suppose that the monodromy of $E$ at $P$ is unipotent, and that the\nflat sections have sub-polynomial growth. This last condition means that, in the\nlanguage of \\cite{HBNC}, the associated filtered local system has trivial\nfiltration.\n\nLet ${\\cal E}$ be the associated variation of twistor structure considered as a\nbundle over $U\\times {\\bf P} ^1$, normalized to have weight zero say.  Let $({\\cal F} _U,\n{\\cal L} , {\\cal G} _U)$ be the triple of a $\\lambda$-connection ${\\cal F}_U$, a family of\nlocal\nsystems ${\\cal L}$ on $U$, and a $\\lambda ^{-1}$-connection ${\\cal G} _U$ on\n$\\overline{Z}$. Let ${\\cal F}$ be the canonical\nextension of ${\\cal F}_U$ to a logarithmic $\\lambda$-connection on $Z$. Let ${\\cal G}$ be\nthe canonical extension of ${\\cal G}_U$ to a logarithmic $\\lambda\n^{-1}$-connection on\n$\\overline{Z}$. These extensions have nilpotent transformations in the fiber\nover $P$, hence they have monodromy weight filtrations (cf \\cite{Schmid} for\nexample).  The dimensions of the associate-graded of the weight filtrations are\nthe same for all $\\lambda $ (cf \\cite{HBNC}). This implies that the weight\nfiltrations are filtrations of the bundles ${\\cal F} _P$ (defined to be the\nrestriction to ${\\bf A}^1 = \\{ P\\} \\times {\\bf A}^1$) and ${\\cal G} _P$ (similarly,\nthe restriction to the other copy of $\\{ P\\} \\times {\\bf A}^1$), by strict\nsubbundles.\n\nFix a tangent vector $\\eta$ at $P$, fix  a point $P_0$ nearby $P$, and fix a\npath going from $P_0$ to $P$ arriving by the direction $\\eta$.  For any\nlogarithmic flat connection on a neighborhood of $P$ there is an isomorphism\nwith a fixed connection with constant coefficients on a trivial bundle, and this\nisomorphism determines an isomorphism between the fiber over $P$ and the fiber\nover $P_0$.  The isomorphism of fibers is well defined depending on the choice\nof tangent vector \\cite{DeligneRegSing}.  Furthermore this\nisomorphism depends analytically on any parameters, and takes the monodromy\noperator on the fiber over $P_0$ to the exponential of the residue of the\nconnection over $P$.  In particular it preserves the weight filtrations.\n\nApplying this to our situation, note that the family ${\\cal L} _{P_0}$ of fibers of\nthe $L_t$ over $P_0$ has a family of weight filtrations. The above isomorphisms\ngive an isomorphism of filtered bundles\n$$\n{\\cal L} _{P_0} \\cong {\\cal F} _P\n$$\nover ${\\bf G}_m$.  Similarly we have an isomorphism of filtered bundles\n$$\n{\\cal L} _{P_0} \\cong {\\cal G} _P\n$$\nover ${\\bf G}_m$ and we can use these to glue together the filtered bundles ${\\cal F}_P$\nand  ${\\cal G}_P$ to give a filtered bundle $(M, W)$ on ${\\bf P} ^1$.\n\nRecall that $T(1)$ denotes the Tate twistor, of rank one and weight two.\nIt is isomorphic to the tangent bundle of ${\\bf P} ^1$ (and this is probably the\nright way to say canonically what it is).\n\n\n\\begin{conjecture}\n\\label{LimitingMTS}\n{\\rm (\\cite{Schmid})}\nThe filtered bundle defined above is a mixed twistor structure. (We call it the\n{\\em limiting mixed twistor structure of $E$}).  Furthermore the logarithms of\nmonodromy operators glue together to give a morphism of mixed twistor structures\n$$\nN: M \\rightarrow M\\otimes _{{\\cal O} _{{\\bf P} ^1}}T(1)\n$$\nsuch that the weight filtration is deduced from $N$ in the now standard way.\n\\end{conjecture}\n\n\n\n\nOnce this is proved, we would then like to generalize the nilpotent orbit\ntheorem \\cite{Schmid}, saying that the asymptotic behavior of our variation of\ntwistor structure is well approximated by the behavior of a standard variation\n(``nilpotent orbit'') deduced directly from the limiting mixed twistor\nstructure.\n\nAll of this should then be generalized to higher dimensions, where it might help\nus to get a hold on the behavior of harmonic bundles near normal crossings\ndivisors.\n\n\\numero{Jet-bundles of hypercomplex manifolds}\n\nLet $U$ be a smooth hyperk\\\"ahler or more generally hypercomplex  manifold.\n\\footnote{\nThe definition of {\\em hypercomplex manifold} is essentially the\nthat of {\\em hyperk\\\"ahler manifold} (cf \\cite{HKLR})  minus the condition of\nexistence of a metric.\nThe original reference for hypercomplex manifolds seems to be\nBoyer \\cite{Boyer}---see also the preprint of Kaledin \\cite{Kaledin}.\n}\nThe product\n$T:= U \\times {\\bf P} ^1$ has a complex structure called the {\\em twistor space} of\nthe quaternionic structure of $U$ \\cite{Hitchin1} \\cite{Hitchin2} \\cite{HKLR}\n\\cite{Kaledin}. Basically an almost-hypercomplex structure is just a ${\\cal C}\n^{\\infty}$ family of actions of the quaternions on the tangent spaces of $U$,\nwhich gives an almost-complex structure on $T$. Integrability of this complex\nstructure is exactly the condition of $U$ being hypercomplex (see Theorem 1 of\n\\cite{Kaledin}, or \\cite{HKLR} for the hyperk\\\"ahler case). The projection\n$p:T\\rightarrow {\\bf P} ^1$ and the horizontal sections for the product structure\n(called {\\em twistor lines} \\cite{HKLR}) $\\zeta : {\\bf P} ^1\\rightarrow T$ are\nholomorphic maps. Fix a twistor line $\\zeta$ corresponding to a point $u\\in U$.\nThe normal bundle $N_{\\zeta \/T}$---a holomorphic vector bundle on ${\\bf P} ^1$ of\nrank $r$ equal to $\\frac{1}{2}dim _{{\\bf R}}(U)$---is the twistor-bundle of the\ntangent space $T(U)_u$ with its quaternionic structure. In particular, $N_{\\zeta\n\/T}$ is a pure twistor structure of weight $1$, i.e. it is isomorphic to a\ndirect\nsum of  $r$ copies of ${\\cal O} _{{\\bf P} ^1}(1)$.\n\nLet $J^n_{\\zeta \/T}$ denote the $n$-th relative jet bundle of $T$ over\n${\\bf P} ^1$ along the section $\\zeta$.  It may be described as follows. Let\n$I_{\\zeta\/ T}$ denote the ideal sheaf of the section $\\zeta$. The coherent\nsheaf $Q_{n, \\zeta \/T} := {\\cal O} _T \/ I_{\\zeta \/ T} ^{n+1}$ on $T$ is flat and\nproper\nover ${\\bf P} ^1$. Its direct image $p _{\\ast}(Q_{n, \\zeta \/T} )$ is a vector\nbundle on ${\\bf P} ^1$\nand the jet bundle is the dual of this vector bundle.  We have inclusions\n$$\nJ^0 _{\\zeta \/T} \\subset \\ldots \\subset J^n_{\\zeta \/T}\n$$\nand the quotients are naturally symmetric powers of the normal bundle:\n$$\nJ^m_{\\zeta \/T} \/ J^{m-1}_{\\zeta \/T} \\cong Sym ^m(N_{\\zeta \/T}).\n$$\nIn particular, these quotients are pure twistor structures of weight $m$.\nThus if we put $W_m(J^n_{\\zeta \/T}) := J^m_{\\zeta \/T}$ we obtain a mixed\ntwistor structure. The fiber over $1$ is the jet space $J^n_u$ of $U$\nat $u$ (for the complex structure corresponding to $1\\in {\\bf P} ^1$---this\ndepends on\nhow we normalize the twistor space). We conclude the phrase ``the jet spaces of\na hypercomplex manifold have natural mixed twistor structures''.\n\nTaking the direct limit or union of the $J^n_u$ we get an ind-object\n$J^{\\infty}_u$ in the category of mixed twistor structures. The weight quotients\nare finite dimensional, and in fact the finite $J^n_u$ are recovered as the\npieces of the weight filtration of $J^{\\infty}_u$. By abuse of notation we will\nsay that $J^{\\infty}_u$ is a mixed twistor structure which determines in\nparticular the mixed twistor structures $J^n_u$.\n\nThe dual of the jet space $J^n_u$ is $Q_{n,u}:={\\cal O} _{U,u} \/{\\bf\nm}_u^{n+1}$. This\nalso has a mixed twistor structure, dual of the above.  The algebra structure of\n$Q_{n,u}$ comes from a coalgebra structure of\n$J^{\\infty}_u$ (coalgebra in the category of ind-mixed twistor structures).\nThe mixed twistor structure $J^{\\infty}_u$ together with its coalgebra\nstructure determines the algebras $Q_{n,u}$. In fact the\ntwistor structure gives the coherent sheaves of algebras denoted ${\\cal F}$ above,\nand taking the $Spec$ at each stage and formally taking the direct limit we\nobtain the formal completion of $T$ along the section $\\zeta$.  Thus\n``the mixed twistor structure on the jet space $J^{\\infty}_u$ of $U$ at $u$\ndetermines the formal completion of the twistor space $T$''.\n\nFinally the twistor space $T$ has an antilinear involution $\\sigma _T$\ncovering the antipodal involution $\\sigma _{{\\bf P} ^1}$. In particular the mixed\ntwistor structure $J^{\\infty}_u$ has an antipodal real structure and, via the\nprevious paragraph, from this real structure we recover the involution $\\sigma\n_T$ on the formal completion along the section $\\zeta$. Nearby the section\n$\\zeta$ the only other sections invariant under the involution $\\sigma$ are the\ntwistor lines (this local uniqueness of solutions of an elliptic equation\nmay be verified in the linearization which is the normal bundle $N_{\\zeta \/T}$).\nHence the antipodal real structure of $J^{\\infty}_u$ determines the product\nstructure of the formal completion of $T$. Therefore it determines the\nhypercomplex structure of $U$ in the formal neighborhood of $u$.\n\n{\\em Question:} How can we include the data of the hyperk\\\"ahler metric in this\npoint of view?\n\nWe obtain a dictionary between formal germs of integrable\nquaternionic structure $(u, \\hat{U})$ and mixed twistor structures\n$J^{\\infty}$ with coalgebra structure and antipodal real structure (these last\ntwo being compatible) such that the associated graded of the weight\nfiltration is\nthe symmetric algebra. This gives a coordinate-free description of the power\nseries of a quaternionic structure.\n\n\\subnumero{The ``Gauss map''}\nSuppose $U$ is a hypercomplex manifold of real dimension $4d$, and fix a\npositive integer $n$. Then for each $u\\in U$ we have constructed a mixed twistor\nstructure on $J^n_u(U)$. The dimensions of the weight-graded quotients are\n$(a_0, a_1, \\ldots , a_n)$ with $a_i$ being the rank of $Sym ^i ({\\bf C} ^{2d})$.\nWe obtain a morphism to the moduli stack of mixed twistor structures\n$$\nU\\rightarrow {\\cal MTS}(a_0,\\ldots , a_n).\n$$\nWe can think of this as a ``Gauss map'' for the hypercomplex structure.\nActually\nthis can be lifted a little bit more, using the coalgebra structure on the jet\nbundles.  We have an isomorphism of complex vector spaces (using the complex\nstructure $j$, say)\n$$\nGr ^W_m(J^n_u)\\cong Sym ^{m}(T(U)_u).\n$$\nIn particular, if we choose a basis for $T(U)_u$ then this gives a basis for\neach of the $Gr ^W_m(J^n_u)$, in other words a framing. Thus, over the frame\nbundle of the tangent bundle we obtain a lifting of our map to\n$Fr{\\cal MTS}(a_0,\\ldots , a_n)$.\n\nWe can be a little bit more precise about\nthis. Note that the associated-graded of the tensor product of two filtered\nvector spaces, is naturally isomorphic to the tensor product of the\nassociated-graded vector spaces. In this way, a coalgebra structure on a\nfiltered vector space $J$ (which we now assume infinite dimensional, no longer\nfixing $n$) gives a coalgebra structure on $Gr^W(J)$. If $T$ is a vector space\nthen let $Fr{\\cal MTS}Cog(Sym^{\\cdot}T)$ denote the moduli stack of\npairs $(J, \\beta )$ where $J$ is a mixed twistor structure with coalgebra\nstructure and $\\beta : Gr ^W(J_1)\\cong Sym^{\\cdot}T$ is an isomorphism of\ncoalgebras. But this  moduli stack is in fact a projective limit of schemes,\nbecause of the rigidity property of the remark following Lemma \\ref{abelian}.\nNote also that  the coalgebra structure, if it exists, is uniquely determined by\n$\\beta$ again by the same rigidity property.  Thus letting $a_i$ denote the\ndimension of $Sym ^i(T)$ we have\n$$\nFr{\\cal MTS}Cog(Sym^{\\cdot}T) \\subset \\lim _{\\leftarrow , n}\nFr{\\cal MTS} (a_0,\\ldots , a_n).\n$$\nFinally, $GL(T)\\cong GL(a_1,{\\bf C} )$\nacts on $Fr{\\cal MTS}Cog(Sym^{\\cdot}T)$ and our {\\em Gauss map} is a\nreal analytic map\n$$\n\\Phi : U\\rightarrow Fr{\\cal MTS}Cog(Sym^{\\cdot}T)\/GL(T).\n$$\nThe associated $GL(T)$-bundle on $U$ is the tangent bundle (for the complex\nstructure over $1\\in {\\bf P} ^1$).\n\n{\\em Problem:} what are the differential equations satisfied by $\\Phi$\ncorresponding to the fact that it parametrizes the family of jet spaces of a\nmanifold $U$?\n\n\\numero{The moduli space of representations}\nSuppose $X$ is a connected smooth projective variety with basepoint $x$. Let\n$G=GL(n)$ (or any other reductive group) and let $M(X,G)$ denote the moduli\nspace\nof representations of $\\pi _1(X, x)$ in $G$ up to conjugacy\n\\cite{Lubotsky-Magid}.\nThe set of smooth points of $M(X,G)$ has a hyperk\\\"ahler structure\n\\cite{Hitchin1}\n\\cite{Hitchin2} \\cite{SantaCruz}.  Thus we can apply the above remarks to any\nsmooth open set $U\\subset M (X, G)$. As remarked in (\\cite{SantaCruz} \\S 3), any\nsmooth stratum for a canonical stratification of $M(X,G)$ also inherits a\nhyperk\\\"ahler structure so we could also apply the previous remarks there.  We\nfind that the jet spaces $J^n_{\\rho}(M(X,G))$ at any smooth point $\\rho$ have\nnatural mixed twistor structures, and the same for jet spaces of smooth strata.\n\nThis can be generalized to singular points and points of the representation\nspaces as follows.  In general if $Z$ is a scheme and $z\\in Z$, denote by\n$Q_{n,z}(Z)$ the algebra ${\\cal O} _{Z,z}\/{\\bf m}_z^{n+1}$, and let $J^n_z(Z)$ denote\nthe ${\\bf C}$-linear dual coalgebra.  For our pointed connected smooth projective\nvariety $(X,x)$ as above, let $R(X,G,x)$ denote the scheme of representations of\n$\\pi _1(X,x)$ in $G$ (before dividing out by conjugation). Note that\n$M(X,G)$ is the geometric invariant theory categorical quotient of $R(X,G,x)$ by\nthe action of $G$. Deligne's construction as described in \\cite{NAHT}\n\\cite{SantaCruz}  gives an analogue of the twistor space for $R(X,G,x)$,\nnamely a\ncomplex analytic space $R_{Del}(X,G,x)$ over ${\\bf P} ^ 1$ with ``twistor lines''\ncorresponding to the semisimple representations of $\\pi _1(X,x)$ in $G$. The\ngroup $G$ acts and the categorical quotient denoted $M_{Del}(X,G)$ gives, over\nthe smooth points, the twistor space for $M(X,G)$.  In Deligne's interpretation,\n$M_{Del}(X,G)$ constitutes a twistor space for the ``singular hyperk\\\"ahler\nstructure\" on $M(X,G)$ (in particular there is an antipodal involution\n$\\sigma$).\n\nSuppose $\\eta : {\\bf P} ^1 \\rightarrow R_{Del}(X,G,x)$ is a twistor line\ncorresponding to a semisimple representation $\\rho$. By Goldman-Millson theory\n\\cite{Goldman-Millson} (cf \\cite{SantaCruz} and \\cite{Moduli} for the present\napplication) the scheme $R_{Del}(X,G,x)$ is analytically formally trivial along\nthe section $\\eta$ and furthermore the singularities in the transversal\ndirection are quadratic.  In particular we can form the family of jet bundles\n$J^n_{\\eta} (R_{Del}(X,G,x))$ which is a bundle of coalgebras over ${\\bf P} ^1$ (and\nwhich we denote simply by $J^n$ for short, below). This bundle is dual to the\nbundle of algebras $Q_{n, \\zeta}(R_{Del}(X,G,x))$.\n\n\\begin{theorem}\n\\label{JetBdlRepSpace}\nThe jet bundles $J^n_{\\eta} (R_{Del}(X,G,x))$ have natural mixed twistor\nstructures.\n\\end{theorem}\n{\\em Proof:}\nThe group $G$ acts.  Let $G\\eta$ denote the orbit (fiber-by-fiber) of the\nsection $\\eta$.\nLet $W_kQ_{n, \\zeta}(R_{Del}(X,G,x))$ be the $k$-th power of the ideal of\n$G\\eta \\cap Spec (\nQ_{n, \\zeta}(R_{Del}(X,G,x)))$ in $Q_{n, \\zeta}(R_{Del}(X,G,x))$.\nThis is the weight filtration.\n\nBy Luna's etale slice theorem\nthe space $R_{Del}(X,G,x)$ has a product structure formally around the section\n$\\eta$,\n$$\nR_{Del}(X,G,x)^{\\wedge} = G\\eta ^{\\wedge} \\times Z^{\\wedge} .\n$$\n(to be precise this holds locally in the etale topology over the base ${\\bf P} ^1$).\n\nWe will use a different filtration to study this weight filtration.  Define the\n{\\em tangent cone filtration} $V_k$ of $Q_{n,u} (U)$ to be the filtration by the\npowers of the maximal ideal. To fix notations, let\n$$\nQ_{\\infty ,u}(U) = \\lim _{\\leftarrow , n} Q_{n ,u}(U)\n$$\nand put\n$$\nV_{-k}Q_{\\infty ,u}(U):= {\\bf m}_u ^kQ_{\\infty ,u}(U).\n$$\n The terminology comes from the\nfact that $Spec (Gr ^V(Q_{\\infty,u}(U))$ is the  tangent cone of $U$ at $u$.\nNote of course that\n$$\nQ_{n ,u}(U) = Q_{\\infty ,u}(U)\/V_{-n-1}Q_{\\infty ,u}(U).\n$$\nThe dual of $V$ is a filtration which we also denote by $V_k$ (indexed in the\npositive range of values of $k$ this time) of the jet space $J^{\\infty}_u(U)$.\n\nThe above discussion works in our family (since the families are locally\ntrivial over ${\\bf P} ^1$) to give filtrations $V$ of $Q_{\\infty , \\eta}\n(R_{Del}(X,G,x))$ and $J^{\\infty}_{\\eta}(R_{Del}(X,G,x))$.\n\nIf $U$ is a product, $U= U_1 \\times U_2$ then we have\n$$\nGr ^V(Q_{\\infty , u}(U))= Gr ^V(Q_{\\infty , u}(U_1))\\otimes Gr ^V(Q_{\\infty ,\nu}(U_2)),\n$$\nin other words, the tangent cone is the product of the two tangent\ncones. This is proved by interpreting the tangent cone as the scheme cut out by\nthe initial forms of the equations.\n\nNow suppose that $U$ is a product as above and\nlet $W$ be the filtration by powers of the ideal of the factor $U_1$.\nThe filtration induced by $W$ on $Gr ^V(Q_{\\infty , u}(U))$ is again the\nfiltration by powers of the ideal of the first factor. This is the same as the\nfiltration induced by the grading of the second factor only, in the above tensor\nproduct structure.\n\nBy the lemma of two\nfiltrations (\\cite{Hodge2}) we have\n$$\nGr ^W_m Gr ^V_k (Q_{\\infty , u}(U))=\nGr ^V_k Gr ^W_m (Q_{\\infty , u}(U)).\n$$\nThe same holds  in the situation of our family over ${\\bf P}^1$. Thus in order to\nprove that $Q_{\\infty , \\eta}(R_{Del}(X,G,x))$ is a mixed twistor structure, it\nsuffices to prove that\n$$\nGr ^V_{\\cdot}Q_{\\infty , \\eta}(R_{Del}(X,G,x))\n$$\nis a mixed twistor structure.\n\nNote that\n$$\nGr ^V_{k}Q_{\\infty , \\eta}(R_{Del}(X,G,x))=\n(J^k\/J^{k-1})^{\\ast},\n$$\nand in particular\n$$\nGr ^V_{1}Q_{\\infty , \\eta}(R_{Del}(X,G,x))=\n(J^1\/J^{0})^{\\ast} = N^{\\ast}\n$$\nwhere $N$ denotes the normal bundle (isomorphic to $J^1\/J^0$).  The algebra\nstructure of $Gr^V_{\\cdot}$ gives a morphism\n$$\nSym ^{k}(N^{\\ast}) \\rightarrow\n(J^k\/J^{k-1})^{\\ast} \\rightarrow 0\n$$\nwhich is a surjection because $V$ is by definition the filtration by powers\nof the maximal ideal.  By Goldman and Millson-theory the family of algebras\n$Q_{\\infty , \\eta}(R_{Del}(X,G,x))$ is quadratic, hence isomorphic to the\ntangent\ncone and in particular the tangent cone is quadratic. This means that the kernel\nof the above surjection comes from the quadratic relations, in other words we\nhave an exact sequence\n$$\nP\\otimes  Sym ^{k-2} (N^{\\ast}) \\rightarrow Sym ^{k}(N^{\\ast}) \\rightarrow\n(J^k\/J^{k-1})^{\\ast} \\rightarrow 0\n$$\nwhere $P \\subset\nSym ^2(N^{\\ast})$ is the module of quadratic relations, that is the kernel of\nthe map\n$Sym ^2(N^{\\ast}) \\rightarrow J^2 \/ J^1$.\n\nWe claim that the filtration induced by $W$ on $(J^k\/J^{k-1})^{\\ast}$\nis the image of the filtration of $Sym ^{k}(N^{\\ast})$ induced by $W$ on\n$N^{\\ast}= (J^1\/J^0)^{\\ast}$. This is a general fact coming from the product\nsituation $U=U_1\\times U_2$ with our filtrations $V$ and $W$ as above.\nWrite $U_i= Spec (A_i)$ and let ${\\bf m}_i\\subset A_i$ be the maximal ideal of\nthe origin. Then $U_1\\times U_2 = Spec (A)$ with $A=A_1\\otimes A_2$,\n$W$ is the filtration\nby powers of ${\\bf m} _2A$ and $V$ is the filtration by powers of ${\\bf m}_1 A+\n{\\bf m}_2A$.  The statement which needs to be proved $(\\ast )$ is that\n$$\n({\\bf m} ^1A + {\\bf m} ^2A) ^{\\otimes p} \\otimes ({\\bf m} _2A)^{\\otimes q}\n\\rightarrow\n\\frac{({\\bf m} _2A)^q \\cap ({\\bf m} _1 A+ {\\bf m} _2A)^{p+q}}{({\\bf m} _1 A+\n{\\bf m} _2A)^{p+q+1}}\n$$\nis surjective. Choose a splittings $A_i\\cong Gr (A_i)$ which we think of as\ndirect sum decompositions $A_i \\cong \\bigoplus A_i^k$ compatible with the\nfiltrations by powers of the maximal ideal. The product structure has an upper\ndiagonal form for this decomposition, with the product in $Gr(A_i)$ along the\ndiagonal.  We get a decomposition $A= \\bigoplus A^{j,k}$ with $A^{j,k}=\nA_1^j\\otimes A_2^k$. The decomposition of $A$ into pieces of the form\n$\\bigoplus _{j+k=n}A^{j,k}$ corresponds to a splitting $A\\cong Gr(A)$.\nIn particular,\n$$\n({\\bf m} _1 A+ {\\bf m} _2A)^{n} = \\bigoplus _{j+k\\geq n}A^{j,k}.\n$$\nSimilarly\n$$\n({\\bf m} _2A)^q= \\bigoplus _{k\\geq q}A^{j,k}.\n$$\nFrom this we get\n$$\n({\\bf m} _2A)^q \\cap ({\\bf m} _1 A+ {\\bf m} _2A)^{p+q} = \\bigoplus _{j+k \\geq n, k\\geq\nq}A^{j,k} .\n$$\nThe product on the associated graded induces a surjective morphism\n$$\n(A^{1,0})^{\\otimes j}\\otimes (A^{0,1})^{\\otimes k}\n\\rightarrow A^{j,k},\n$$\nand the product morphism from the left side into $A$ is equal to this morphism\nplus pieces which go into $\\bigoplus _{j'+k'\\geq j+k+1}A^{j',k'}$. This implies\nthe statement $(\\ast )$  which we are trying to prove. Thus in the exact\nsequence\nof the previous paragraph, the filtration induced on\n$(J^k\/J^{k-1})^{\\ast}$ by the weight filtration of the middle term, is equal to\nthe weight filtration.\n\n\nThe morphism $P\\otimes  Sym ^{k-2} (N^{\\ast}) \\rightarrow Sym ^{k}(N^{\\ast})$\nis compatible with the filtrations induced by $W$.  We claim that $N$ and $P$\nare mixed twistor structures. Thus the morphism is a morphism of mixed twistor\nstructures, and the cokernel is a mixed twistor structure. This cokernel\nis equal to $(J^k\/J^{k-1})^{\\ast}$ with its filtration $W$. Thus\n$Gr ^V_{k}Q_{\\infty , \\eta}(R_{Del}(X,G,x))$ is a mixed twistor structure so, as\nexplained above with the lemma of two filtrations, we are done.\n\nWe now prove that $N$ and $P$ are mixed twistor structures.\nFirst of all, as in \\cite{Goldman-Millson} \\cite{Moduli},\n$N$ is calculated as the\nfirst cohomology of the complex\n$$\n\\ker (\\xi {\\cal A}^0_X(End (E)\\rightarrow End (E_x))\\rightarrow\n\\xi {\\cal A} ^1_X(End (E))\\rightarrow \\xi {\\cal A} ^2_X(End(E)).\n$$\nThus there is an exact sequence\n$$\n0\\rightarrow H^0(End (E))\\rightarrow\nEnd (E_x) \\rightarrow N \\rightarrow H^1(X\\times {\\bf P} ^1 \/ {\\bf P} ^1, End\n(E))\\rightarrow 0\n$$\nand the image of $End (E_x)$ in $N$ is the tangent space of the orbit, that\nis to\nsay that it is $W_0$.  From \\ref{coho1} the cohomology\n$H^i(X\\times {\\bf P} ^1 \/ {\\bf P} ^1, End\n(E))$ is a pure twistor structure of weight $i$ (note that $End(E)$ is pure of\nweight zero).  From $i=0$ and the fact that $End (E_x)$ is pure of weight\nzero we find that $W_0$ is pure of weight zero. From $i=1$ we get that\n$H^1(End(E))$, which  is the\npiece $W_1\/W_0$, is pure of weight $1$.  This proves that $N$ is a mixed twistor\nstructure.\n\nNext, we have a cup-product $N \\otimes N \\rightarrow H^2(End (E))$ which is\nactually symmetric (the antisymmetry on odd-degree classes cancels the\nantisymmetry of the Lie bracket).\nThis gives a morphism $Sym ^2(N)\\rightarrow H^2(End(E))$.  The part\n$W_1Sym ^2(N)$ goes to zero since, by the group action, there are no\nobstructions in the direction of the group orbit. It follows that the cokernel\nand the image are\npure of weight $2$. The module of relations $P$ is the image of the dual\nmorphism\n$$\nH^2(End(E))^{\\ast} \\rightarrow Sym ^2(N^{\\ast}),\n$$\nso $P$ is a pure twistor structure of weight $-2$.\n\nThis completes the proof that the jet spaces of the representation space\nhave natural mixed twistor structures.\n\\hfill $\/$\\hspace*{-.1cm}$\/$\\hspace*{-.1cm}$\/$\\vspace{.1in}\n\n{\\em Application to the relative completion:}  Taking some sort of Tannakian\ndual of the completions of the representation spaces $R(X,G,x)$ where $G$ runs\nthrough all groups, at points corresponding to all representations associated to\na given semisimple representation $\\rho$, should give back the {\\em relative\nMalcev completion} of $\\pi _1(X,x)$ at the representation $\\rho$. One should be\nable to use this to obtain a mixed twistor structure on the relative Malcev\ncompletion. Alternatively, a direct generalization of Hain's technique\n\\cite{Hain} should also work. One would like to check, in fact, that the two\nconstructions agree. As you can tell from the tense of this paragraph, I haven't\nlooked into this at all!\n\n{\\em Application to the moduli space:}\nThe group $G$ doesn't act on the formal local ring of $R(X,G,x)$ at a\npoint $\\rho$ (semisimple representation), however we do obtain an infinitesimal\naction of the Lie algebra ${\\bf g}$. Assume that $G$ is connected. Then the\nformal local ring of $M(X,G)$ at  $\\rho$ is\nthe subring of ${\\bf g}$-invariants in the  formal local ring of $R(X,G,x)$ at\n$\\rho$. This latter has a mixed twistor structure (it is the dual of\n$J^{\\infty}$). The action of ${\\bf g}$ preserves the mixed twistor\nstructure so the\nsubring of invariants is a mixed twistor structure. Again\ntaking the dual, $J^{\\infty}_{\\rho}(M(X,G))$ has a mixed twistor structure. Note\nthat if $\\rho$ is not a smooth point, the weight filtration is not at all the\nsame as the easy filtration, and $J^1$ might very well start of with pieces of\nrelatively high weight (essentially because the lower-degree terms don't survive\nas invariants).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe classification of all branched covers over a given base is a subject\ndating back to Alexander, who proved that every closed orientable PL\n$n$-manifold is a PL branched cover of $S^{n}$\n\\cite{alexander1920note}. Alexander's branching sets are PL subcomplexes\nof the sphere; he concludes little else about them. Since 1920, the\nnatural question of how complicated the branching set needs to be, and\nhow many sheets are needed, in order to realize all manifolds in a given\ndimension as branched covers of the sphere, has received much interest -- see, for instance,~\\cite{berstein1978degree} and references therein.\nIt is a famous theorem in dimension 3 that three-fold dihedral covers\nbranched along knots suffice \\cite{hilden1974every},\n\\cite{hirsch140offene}, \\cite{montesinos1974representation}. The\nquestion is considerably more subtle in dimension four. Piergallini and\nIori, among others, have studied the minimal degree needed to realize\nall closed oriented PL four-manifolds as covers of the sphere. The\nbranching sets they consider are either immersed PL submanifolds with\ntransverse self-intersections or embedded and non-singular PL surfaces.\nPiergallini proved in \\cite{piergallini1995four} that every closed\noriented PL four-manifold is a four-fold cover of $S^{4}$ branched over\na over a transversally immersed PL surface. He and Iori later refined\nthis result to show in \\cite{iori2002} that singularities can be\nremoved by stabilizing to a five-fold cover. In light of these universal\nrealization theorems, one might wish for equally general methods for\nobtaining explicit descriptions of the branching sets needed to realize\nparticular PL four-manifolds as a five-fold covers of~$S^{4}$. It would\nalso be of interest to better understand the trade-off between\nsimplifying the branching set and increasing the degree of a cover. Most\nrecently, Piergallini and Zuddas~\\cite{piergallini2016branched} showed\nthat closed oriented \\emph{topological} four-manifolds are also\nfive-fold covers of the sphere, if one allows for ``wild'' branching sets\nwith potentially very pathological topology near isolated points. Still,\nthe complexity of the branching sets near the wild points retains an air\nof mystery.\n\nWe assume a complementary approach, taking the point of view of studying\nall possible covers over a given base $X$ \\emph{in terms of the\nbranching set} and its embedding into the base. As seen from the main\ntheorem of \\cite{viro1984signature}, if $Y$ is a cover of\n$S^{4}$ branched over a closed oriented non-singular embedded surface,\nthen the signature of $Y$ must be zero. Thus, for example, the existing\nresults on five-fold covers of the four-sphere implicitly make use of\nnonorientable branching sets. In contrast, the constructions presented\nhere make use of branching sets that are oriented surfaces, embedded in\nthe base piecewise linearly except for finitely many cone singularities.\nThese ideas have led to new examples of branched covers of $S^{4}$ and\napplications to the Slice--Ribbon Conjecture~\\cite{cahn2017singular}.\nWe work with irregular dihedral covers ({Definition~\\ref{def:dihedral}}),\nwhich constitute the most direct generalization of the three-dimensional\nresults of Hilden, Hirsch and Montesinos as well as four-dimensional\nresults of Montesinos \\cite{montesinos1978}. Dihedral covers are also\nthe ``simplest'' three-fold covers which give rise to interesting examples where the branching sets are singularly embedded (see {Remark~\\ref{cyclic-covers}}).\n\nOur results are not restricted to branched covers of the sphere but\napply to any closed oriented four-manifold base. Given an irregular\ndihedral branched cover $f: Y\\to X$ between two simply-connected\noriented four-manifolds $X$ and $Y$, we relate the intersection forms\nof $X$ and $Y$ via $f$. Singularities for us play a central role, and\nwe compute the signature of a branched cover in terms of data about\nthe branching set and its singularity.\n\nWe begin by defining the type of covers and singularities considered.\nThroughout, $D_{p}$~denotes the dihedral group of order $2p$ and $p$ is\nodd.\n\n\\begin{defn}\n\\label{def:dihedral}\nLet $f: Y\\to X$ be a branched cover with branching set $B\\subset X$. If\nthe unbranched cover $f_{|f^{-1}(X-B)}$ corresponds under the classification of covering spaces to $\\phi ^{-1}(\n\\mathbb{Z}\/2\\mathbb{Z})$ for some surjective homomorphism $\\phi : \\pi\n_{1}(X-B, x_{0})\\to D_{p}$, we say that $f$ is an\n\\textit{irregular dihedral branched cover} of~$X$.\n\\end{defn}\n\nPut differently, $\\phi $ is the monodromy representation of the\nunbranched cover associated to $f$ and meridians of the branching set\n$B$ map to reflections in the dihedral group $D_{p}$ (thought of as a\nsubgroup of the symmetric group $S_{p}$). In particular, the existence\nof a dihedral cover over a pair $(X, B)$ is a condition on the\nfundamental group of the complement of $B$ in $X$. When a (connected)\ndihedral cover over the pair $(S^{3}, \\alpha )$ exists for some knot\n$\\alpha $, we say simply that $\\alpha $ admits a dihedral cover.\n\nIt is helpful to give a description of the pre-images of a point on the\nbranching set $B$ of an irregular dihedral cover $f: Y\\to X$. The\ncovering space $Y$ is a $\\mathbb{Z}\/2\\mathbb{Z}$ quotient of the\n$2p$-fold \\emph{regular} dihedral cover $Z$ corresponding to the kernel\nof the homomorphism~$\\phi $ in {Definition~\\ref{def:dihedral}}. For every\nlocally flat point $b\\in B$ the pre-image $f^{-1} (D_{b})$ of a small\nneighborhood $D_{b}$ of $b$ in $X$ contains $\\frac{p-1}{2}$ components\nof branching index $2$ and one component of branching index $1$. The\nindex 1 component is the fixed set of the involution $Z\\to Z$.\n\n \n\\begin{defn}\n\\label{def:sing}\nLet $X$ be a four-manifold and let $B$ be a closed surface embedded in\n$X$. Let $\\alpha \\subset S^{3}$ be a non-trivial knot. For a given point\n$z\\in B$, assume there exist a small open disk $D_{z}$ about $z$ in\n$X$ such that there is a homeomorphism of pairs $(D_{z} - z, B-z)\n\\cong (S^{3}\\times (0, 1), \\alpha \\times (0, 1))$. We say the\nembedding of $B$ in $X$ has a \\textit{singularity of type $\\alpha $} at\n$z$.\n\\end{defn}\n\nIn other words, the knot $\\alpha $ is the link of the singularity of\n$B$ at $z$. For the covers considered we assume in addition that\nsingularity is normal, meaning that the pre-image of $z$ under the covering map is a single point. The presence of a singularity $\\alpha $ on the branching set $B$ results in a defect, or correction term, to the signature of the covering manifold. While this defect depends only on $\\alpha $, it is computed with the help of an associated knot to $\\alpha $, defined below.\n \n\\begin{defn}\n\\label{defn:chark}\nLet $\\alpha \\subset S^{3}$ and $\\beta \\subset S^{3}$ be two knot types.\nWe say that $\\beta $ is a \\textit{mod $p$ characteristic knot} for\n$\\alpha $ if there exists a Seifert surface $V$ for $\\alpha $ with\nSeifert form $L$ such that $\\beta \\subset V^{\\circ }\\subset S^{3}$\nrepresents a non-zero primitive class in $H_{1}(V; \\mathbb{Z})$ and\n$(L+L^T)\\beta\\equiv 0\\text{~mod~}p$. \n\\end{defn}\n\nIn~\\cite{CS1984linking} Cappell and Shaneson defined characteristic\nknots and proved that for $p$ a positive odd square-free integer and\n$\\alpha $ a non-trivial knot, $\\alpha $ admits an irregular dihedral\n$p$-fold cover if an only if there exists a knot $\\beta $ which is a mod~$p$\ncharacteristic knot for $\\alpha $. Furthermore, they gave an\nexplicit construction of a cobordism, here denoted $W(\\alpha , \\beta\n)$, between a dihedral $p$-fold branched cover of $\\alpha $ and a cyclic\n$p$-fold branched cover of $\\beta $. We recall this construction as\nneeded in the proof of {Proposition~\\ref{prop:signatureW}}.\n\nThroughout this article we adopt the following notation. Let\n$\\chi $ denote the Euler characteristic and $\\sigma $ the signature of\na manifold, and let $e$ be the self-intersection number of an embedded\nclosed submanifold. Given a positive odd integer $p$ and a knot\n$\\alpha $ in $S^{3}$ which admits an irregular dihedral $p$-fold cover,\ndenote by $V$ a Seifert surface for $\\alpha $ with symmetrized Seifert\npairing $L_{V} : = L + L^{T}$. Let\n$\\beta \\subset V^{\\circ }$ be a mod~$p$ characteristic knot for\n$\\alpha $. Finally, denote by $\\sigma _{\\zeta ^{i}}$ the Tristram--Levine\n$\\zeta ^{i}$-signature of a knot \\cite{tristram1969some}, where\n$\\zeta $ is a primitive $p$-th root of unity.\n\nOur first theorem is a necessary condition for the existence of a\n$p$-fold irregular dihedral cover $f: Y\\to X$ between two four-manifolds\n$X$ and $Y$, with a specified embedded surface $B\\subset X$ as its\nbranching set.\n\n \n\\begin{thm}[Necessary condition]\\label{thm:necessary}\nLet $X$ and $Y$ be closed oriented PL four-manifolds and let $p$ be an\nodd prime. Let $B\\subset X$ be a closed connected surface, PL-embedded\nin $X$ except for an isolated singularity $z$ of type $\\alpha $. If\nthere exists an irregular dihedral $p$-fold cover $f: Y\\rightarrow X$\nbranched along $B$ with a normal singularity at $z$, then the knot\n$\\alpha $ admits an irregular dihedral $p$-fold cover and this cover is\n$S^{3}$. Furthermore, given any corresponding (see footnote~\\ref{foot1})\n$\\mathrm{mod}~p$ characteristic knot $\\beta $ for $\\alpha $, the following\nformulas hold:\n\\begin{equation}\n\\label{eq:chi}\n\\chi (Y)=p\\chi (X)-\\frac{p-1}{2}\\chi (B)-\\frac{p-1}{2},\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:sigma}\n\\sigma (Y)=p\\sigma (X) -\\frac{p-1}{4}e(B) - \\Xi _{p}(\\alpha ),\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:xi-def}\n\\Xi _{p}(\\alpha )= \\frac{p^{2}-1}{6p}L_{V}(\\beta , \\beta ) + \\sigma (W(\n\\alpha , \\beta )) + \\sum _{i=1}^{p-1} \\sigma _{\\zeta ^{i}}(\\beta ).\n\\end{equation}\n\n\\end{thm}\n\n\\begin{rem}\nThe author believes that the above theorem as well as the rest of the\nresults of this paper extend to the case where $X$ and $Y$ are\ntopological four-manifolds.\n\\end{rem}\n\nThe main substance of this theorem is finding an expression for\n$\\Xi _{p}(\\alpha )$, the contribution to the signature of $Y$ resulting\nfrom the presence of a singularity of type $\\alpha $ on the branching\nset. Note that it is straightforward to compute $L_{V}(\\beta , \\beta\n)$ and $ \\sum _{i=1}^{p-1} \\sigma _{\\zeta ^{i}}(\\beta )$ from diagrams of\n$\\alpha $ and $\\beta $. A less obvious but essential feature of this\ntheorem is the fact that the term $\\sigma (W(\\alpha , \\beta ))$ can be\nexpressed in terms of linking numbers of curves in a dihedral branched\ncover of $\\alpha $ (see {Proposition~\\ref{prop:signatureW}}). A\ncombinatorial procedure for computing these linking numbers from a\ndiagram of $\\alpha $ is described in Appendix~\\ref{appB}, using techniques of\nPerko~\\cite{perko1964thesis}. This procedure was carried\nout in~\\cite{cahn2016linking} and implemented in Python.\n\nIt is clear from the definition of a characteristic knot that\n$\\beta $ is not uniquely determined by $\\alpha $. While each of the\nterms $\\frac{p^{2}-1}{6p}L_{V}(\\beta , \\beta )$, $\\sigma (W(\\alpha ,\n\\beta ))$, and $\\sum _{i=1}^{p-1} \\sigma _{\\zeta ^{i}}(\\beta )$ depends on\n$\\beta $, we show in {Proposition~\\ref{prop:invariant}} that their sum,\n$\\Xi _{p}(\\alpha )$, is an invariant of $\\alpha $ and thus independent\nof the choice of characteristic knot.\\footnote{Precisely, $\\Xi _{p}(\n\\alpha )$ is an invariant of $\\alpha $\n\\textit{together with a representation} of $\\pi _{1}(S^{3} - \\alpha , x\n_{0})$ onto $D_{p}$. In a lot of cases, the latter is uniquely\ndetermined by $\\alpha $, up to the appropriate notion of equivalence.\nTo each equivalence class of dihedral representations of $\\pi _{1}(S\n^{3} - \\alpha , x_{0})$ corresponds an equivalence class of mod~$p$\ncharacteristic knots for $\\alpha $, and $\\beta $ can be chosen\narbitrarily within this class. See~\\cite{CS1984linking}.\\label{foot1}} The author\nand Cahn develop a combinatorial method for computing $\\Xi _{p}(\\alpha\n)$ from a Fox $p$-colored diagram of $\\alpha $ and apply this method to\nspecific examples of two-bridge singularities\nin~\\cite{cahn2017singular}. They also show that, for $\\alpha $ a slice\nknot which arises as a singularity on a $p$-fold dihedral cover between\nfour-manifolds, $\\Xi _{p}(\\alpha )$ gives an obstruction to\n$\\alpha $ being homotopy ribbon. Precisely, if a slice singularity\n$\\alpha $ is in fact homotopy ribbon, then $|\\Xi _{p}(\\alpha )| \\leq\n\\frac{p-1}{2}$ (Theorem~4 of~\\cite{cahn2017singular}). Since ribbon\nknots are homotopy ribbon, this means in particular that $\\Xi _{p}(\n\\alpha )$ can be used to test potential counter-examples to the\nSlice Ribbon Conjecture such as those constructed in~\\cite{cha2016casson}.\n\nIn the case where the manifold $Y$ are simply-connected,\nEquation~{(\\ref{eq:chi})} is equivalent to determining the rank of its\nintersection form, which is why this (easy to obtain) equation is of\ninterest. Lastly, we note that {Theorem~\\ref{thm:necessary}} generalizes\nin the obvious way to the situation where the branching set admits\nmultiple cone singularities, the signature of the cover picking up a\ndefect term for each singular point. That is, if the embedding of\n$B$ in $X$ has singularities $\\alpha _{1}, \\dots , \\alpha _{k}$, then\n\\begin{equation*}\n\\chi (Y)=p\\chi (X)-\\frac{p-1}{2}\\chi (B)-k\\frac{p-1}{2}\n\\end{equation*}\nand\n\\begin{equation*}\n\\sigma (Y)=p\\sigma (X) -\\frac{p-1}{4}e(B) -\\Sigma _{i=1}^{k} \\Xi _{p}(\n\\alpha _{i}).\n\\end{equation*}\n\nThe following theorem is a partial converse to {Theorem~\\ref{thm:necessary}}.\n\n\\begin{thm}[Sufficient condition]\\label{thm:sufficient}\nLet $X$ be a simply-connected closed oriented PL four-manifold. Let\n$B\\subset X$ be a closed connected surface PL-embedded in $X$ and such\nthat $\\pi _{1}(X-B, x_{0}) \\cong \\mathbb{Z}\/2\\mathbb{Z}$. Let $p$ be an\nodd prime, and let $\\alpha $ be any two-bridge slice knot which admits\na $p$-fold dihedral cover. If $\\sigma $ and $\\chi $ are two integers\nwhich satisfy Equations~{(\\ref{eq:chi})} and {(\\ref{eq:sigma})}, respectively,\nwith respect to $X$, $B$ and $\\alpha $, then there exists a\nsimply-connected four-manifold $Y$ such that $\\sigma (Y) = \\sigma $,\n$\\chi (Y) = \\chi $ and $Y$ is an irregular dihedral $p$-fold cover of\n$X$.\n\\end{thm}\nNote that Equations~{(\\ref{eq:chi})} and {(\\ref{eq:sigma})} make sense when\n$B$ and $\\alpha $ are not related. The branching set of the covering map\nconstructed in the proof of this theorem is a surface $B_{1}\\cong B$,\nembedded in $X$ with an isolated singularity $z$ of type $\\alpha $ and\nsuch that $e(B_{1}) = e(B)$. When restricted to the class of two-bridge\nslice singularities, {Theorems~\\ref{thm:necessary}\nand~\\ref{thm:sufficient}} give a necessary and sufficient condition for\na pair of integers $(\\sigma , \\chi )$ to arise as the signature and\nEuler characteristic of a simply-connected dihedral cover over a given\nbase.\n\nNext, we show that over any indefinite four-manifold $X$, an infinite\nfamily of integer pairs $(\\sigma , \\chi )$ can be realized as the\nsignatures and Euler characteristics of simply-connected $p$-fold\ndihedral covers over $X$.\n\n\\begin{thm}\n\\label{thm:construction-slice}\nLet $X$ be a simply-connected closed oriented PL four-manifold whose\nsecond Betti number is positive. Let $\\alpha $ be a two-bridge slice\nknot which admits an irregular dihedral $p$-fold cover for $p$ an odd\nprime. There exists an infinite family of pairwise non-homeomorphic\nsimply-connected closed oriented four-manifolds $\\{Y_{i}\\}_{i=1}^{\n\\infty }$, each of which is an irregular $p$-fold cover of $X$ branched\nover an oriented surface embedded in $X$ with an isolated singularity\nof type $\\alpha $.\n\\end{thm}\n\nWe remark that for any $p$, infinitely many knots $\\alpha $ which\nsatisfy the hypotheses of the theorem exist, as shown in\n{Proposition~\\ref{slice_knots_have_covers}}. In~\\cite{cahn2017singular},\nthe author and Cahn used the construction in the proof of\n{Theorem~\\ref{thm:construction-slice}} to give an infinite family of\nthree-fold dihedral covers $\\mathbb{CP}^{2}\\to S^{4}$. Each of these covers is branched along a\ntwo-sphere embedded in $S^{4}$ with one  two-bridge slice\nsingularity, and the singularities used are pairwise distinct. Note that, as indicated previously, the signature\nobstructs the existence of a branched cover $\\mathbb{CP}^{2}\\to S^{4}$\nbranched along a non-singular oriented surface.\n\nThe construction of infinite families of branched covers given in\n{Theorem~\\ref{thm:construction-slice}} relies on our ability to vary the\nbranching set of a dihedral cover. It invites the question, under what\nconditions can \\textit{a particular manifold} $Y$ be realized as a\n$p$-fold dihedral cover over a given base data $(X, B, \\alpha )$? In\nsituations where the manifold $Y$ is (nearly) determined by the rank and\nsignature of its intersection form, we obtain a complete classification.\n\n\\begin{thm}\n\\label{thm:odd_indefinite}\nLet $X$ and $Y$ be simply-connected closed oriented PL four-manifolds\nwhose intersection forms are odd. Fix an odd square-free integer $p$ and\na two-bridge slice knot $\\alpha $ which admits a $p$-fold dihedral\nbranched cover. Let $B\\subset X$ be a PL-embedded surface such that\n$\\pi _{1}(X-B, x_{0}) \\cong \\mathbb{Z}\/2\\mathbb{Z}$. Then, the Euler\ncharacteristic and signature of $Y$ satisfy Equations~{(\\ref{eq:chi})}\nand~{(\\ref{eq:sigma})} with respect to $X$, $B$ and $\\alpha $ if and only\nif $Y$ is homeomorphic to an irregular $p$-fold dihedral cover of\n$X$ branched along a surface $B_{1}$ embedded in $X$ with a singularity\n$\\alpha $ and such that $B_{1}\\cong B$ and $e(B_{1}) = e(B)$.\n\\end{thm}\n\nThe rest of this article is organized as follows. In\nSection~\\ref{sec:necessary} we prove {Theorem~\\ref{thm:necessary}}, give\na formula for $\\sigma (W(\\alpha , \\beta ))$ in terms of linking numbers\nin a branched cover of $\\alpha $, and show that the defect on the\nsignature arising from a singularity on the branching set is an\ninvariant of the singularity type. Section~\\ref{sec:sufficient} is\ndedicated to the proofs of {Theorems~\\ref{thm:sufficient},\n\\ref{thm:construction-slice} and~\\ref{thm:odd_indefinite}}. In Appendix~\\ref{appA}\nwe study characteristic knots of two-bridge knots. Appendix~\\ref{appB} lays out\na procedure for calculating linking numbers in a branched cover of a\nknot $\\alpha $ in terms of a diagram of $\\alpha $.\n\n\\section{Signatures of dihedral covers}%\n\\label{sec:necessary}\nOur strategy in proving {Theorem~\\ref{thm:necessary}} is to resolve the\nsingularity on the branching set and reduce the computation of the\nsignature to the case of a PL embedded surface. Then, Novikov\nadditivity~\\cite{novikov1970pontrjagin} implies that the difference between the signatures of the smooth and singular covers is\ngiven by the signature of the manifold used to resolve the singularity.\nThe final step is to compute the signature of this manifold and prove we can express it in terms of invariants of the singularity type.\n\n\\begin{proof}{Proof of {Theorem~\\ref{thm:necessary}}}\nThe assertion that $\\alpha $ admits an irregular dihedral $p$-fold cover\nand this cover is the three-sphere is verified by considering the local\npicture around the singular point. The existence of a $p$-fold dihedral\ncover $f: Y\\to X$ over the pair $(X, B)$ implies straight away that the\nknot $\\alpha $ itself admits a $p$-fold dihedral cover $M$. Indeed,\n consider $f^{-1}(\\partial N(z))=:M$,\nwhere $z\\in B\\subset X$ is the singular point on the branching set and\n$N(z)$ denotes a small neighborhood. Since by assumption there is a\nhomeomorphism of pairs\n\\begin{equation*}\n(\\partial N(z), B\\cap \\partial N(z))\\cong (S^{3}, \\alpha ),\n\\end{equation*}\nthe restriction of $f$ to $M$ is a $p$-fold dihedral cover of $\\alpha$, as claimed. It is\nconnected since $z$ is normal. In particular, this means that the knot\ngroup of $\\alpha $ surjects onto the dihedral group $D_{p}$.\nFurthermore, over $N(z)$ lies the cone on $M$. Since by assumption\n$Y$ is a manifold, $M$ is homeomorphic to the three-sphere.\n\nWe begin by verifying Equation~{(\\ref{eq:chi})}, a straight-forward\ncomputation. Let $N(B)$ denote a regular neighborhood of $B$ in $X$.\nThen, we can write\n\\begin{equation*}\nX = (X - N(B))\\bigcup _{\\partial N(B)} N(B).\n\\end{equation*}\nSince $\\partial N(B)$ is a closed oriented three-manifold, we have $\\chi (\\partial N(B))=0$. This gives:\n\\begin{equation*}\n\\chi (X) = \\chi (X-N(B)) + \\chi (N(B)) = \\chi (X-B) + \\chi (B).\n\\end{equation*}\nWe can further break down this equation as:\n\\begin{equation*}\n\\chi (X)\n= \\chi (X-B) + \\chi (B - z) + 1.\n\\end{equation*}\n\nSimilarly, denoting $B':=f^{-1}(B)$ and $z' : = f^{-1}(z)$, we have,\n\\begin{equation*}\n\\chi (Y) = \\chi (Y-B') + \\chi (B' - z') +\n1.\n\\end{equation*}\n\nWe know that $f|_{Y-B'}: Y-B'\\to X-B$ is a $p$-to-one covering map,\n$f|_{B'-z'}: B'-z'\\to B-z$ is a $\\frac{p+1}{2}$-to-one covering map,\nand, of course, $f|_{z'}: z'\\to z$ is one-to-one. Therefore,\n\\begin{equation*}\n\\chi (Y) = p\\chi (X-B) + \\frac{p+1}{2}(\\chi (B) - 1) +1 = p\\chi (X)-\n\\frac{p-1}{2}\\chi (B)-\\frac{p-1}{2},\n\\end{equation*}\nas claimed.\n\nNow we turn to the computation of $\\sigma (Y)$, a\nconsiderably harder task. We devise a geometric procedure for the\nresolution of the singularity on the branched cover. The singularity is\nresolved in two stages. At the start, the branching set has one\nsingular point, in a neighborhood of which the branching set can be\ndescribed in terms the knot $\\alpha $. Our first step will be to replace\nthis singularity by a circle's worth of ``standard'' (that is, independent\nof the knot type $\\alpha $) non-manifold points on the branching set.\nThe second step will be to excise these ``standard'' singularities and\nconstruct a new cover whose branching set is a PL submanifold\nof the base. We carry out these two steps in detail below. In the last stage of the proof, we calculate the effect of the resolution of singularities on the signatures of the four-manifolds involved.\n\n\\textit{Step 1.} Let $D_{z}\\subset X$ be a neighborhood of the singular\npoint $z$ such that $D_{z}\\cap B$ is the cone on\n$\\alpha $. As we already established, $\\alpha $ admits a $p$-fold\ndihedral cover. Equivalently, if $V$ is any Seifert surface for\n$\\alpha $, there exists a mod~$p$ characteristic knot $\\beta \\subset\nV^\\circ$ (see {Definition~\\ref{defn:chark}}). Let $W(\\alpha , \\beta )$ be the\nmanifold constructed in \\cite{CS1984linking} as a cobordism between\na $p$-fold dihedral cover of $(S^{3}, \\alpha )$ and a $p$-fold cyclic\ncover of $(S^{3}, \\beta )$. By construction of $W(\\alpha , \\beta )$, which is recalled in the\nproof of {Proposition~\\ref{prop:homologyW}}, there is a $p$-fold branched\ncovering map\n\\begin{equation*}\nh_{1}: W(\\alpha , \\beta ) \\to S^{3}\\times [0, 1],\n\\end{equation*}\nwhich restricts to a $p$-fold dihedral cover of $(S^{3}\\times \\{0\\}, \\alpha )$\nand to a $p$-fold cyclic cover of $(S^{3}\\times \\{1\\}, \\beta )$. Let\n\\begin{equation*}\nh_{2}: Q\\to D^{4}\n\\end{equation*}\nbe a $p$-fold cyclic cover of the closed four-ball branched over a\npushed-in Seifert surface $V'$ for $\\beta $, as constructed in Theorem~5 of \\cite{CS1975branched}. Denote by $\\Sigma $ the $p$-fold cyclic\ncover of $(S^{3}, \\beta )$. By construction, $\\partial Q\\cong \\Sigma\n$ and, similarly, $W(\\alpha , \\beta )$ has one boundary component\nhomeomorphic to $\\Sigma $. Moreover, for $i=1, 2$, the map\n\\begin{equation*}\nh_{i}|_{\\Sigma }: \\Sigma \\to S^{3}\n\\end{equation*}\nis the $p$-fold cyclic cover branched along $\\beta $. Therefore, we can\nconstruct a branched cover\n\\begin{equation}\n\\label{define_h}\nh_{1}\\cup h_{2}: W(\\alpha , \\beta )\\bigcup _{\\Sigma }Q\\longrightarrow\nS^{3}\\times [0, 1]\\bigcup _{S^{3}\\times \\{1\\}} D^{4}.\n\\end{equation}\nWe denote $W(\\alpha , \\beta )\\bigcup _{\\Sigma }Q$ by $Z$ for short, and\nthe map $h_{1}\\cup h_{2}$ by $h$. Thus, we can rewrite\nEquation~{(\\ref{define_h})} as\n\\begin{equation*}\nh:Z\\to D^{4}.\n\\end{equation*}\nThis map is a $p$-fold branched cover whose restriction to the boundary\nof $Z$ a $p$-fold irregular dihedral cover of $(S^{3}, \\alpha )$. So,\ndenoting the branching set of $h$ by $T$, we have,\n\\begin{equation}\n\\label{define_T}\nT\\cong \\alpha \\times [0, \\frac{1}{2}]\n\\bigcup _{\\alpha \\times \\{\\frac{1}{2}\\}} V\\times \\{\\frac{1}{2}\\}\n\\bigcup _{\\beta \\times \\{\\frac{1}{2}\\}} \\beta \\times [\\frac{1}{2}, 1]\n\\bigcup _{\\beta \\times \\{1\\}} V'.\n\\end{equation}\nWe see from this description that $T$ is a two-dimensional PL subcomplex\nof $D^{4}$ which is a manifold away from the curve $\\beta \\times \\{\n\\frac{1}{2}\\}$. Observe that $\\beta \\times \\{\\frac{1}{2}\\}$ is embedded\nin the interior of $V\\times \\{\\frac{1}{2}\\}$. Therefore, in a small\nneighborhood of the curve $\\beta \\times \\{\\frac{1}{2}\\}$, the branching\nset is homeomorphic to the Cartesian product of $S^{1}$ and the letter\n``$\\top $''. (For more details on this construction we once again refer\nthe reader to \\cite{CS1984linking}.)\n\nThe idea is to use the map $h$ to construct a new cover of the manifold\n$X$ which will differ from the original cover $f$ only over a\nneighborhood of the singularity $z\\in B$. Specifically, let\n$D_{z}':= f^{-1}(D_{z})$ and observe that the restrictions of the maps\n$f$ and $h$ to the boundaries of $Y- \\interior{D}_{z}$ and $Z$,\nrespectively, are the $p$-fold irregular dihedral branched\ncover\\footnote{We use the phrase ``\\textit{the} dihedral cover of\n$\\alpha $'' somewhat liberally throughout this paper. As noted\npreviously, dihedral covers of $\\alpha $ are in bijective correspondence\nwith equivalence classes of characteristic knots~$\\beta $. Naturally,\nif $\\alpha $ admits multiple non-equivalent dihedral covers, we choose\nthe one determined by $f$ to construct $Z$.} of $(S^{3}, \\alpha )$,\nwhich is again $S^{3}$. We thus obtain a new branched covering\nmap\n\\begin{equation}\n\\label{define-f1}\nf\\cup h: (Y-\\interior{D_z'})\\bigcup _{S^{3}} Z \\longrightarrow (X - \\interior{D}\n_{z}) \\bigcup _{S^{3}} D^{4}.\n\\end{equation}\nDenote the covering manifold $(Y-\\interior{D_z'})\\bigcup _{S^{3}} Z$ above  by $Y_{1}$ and the map\n$f\\cup h$ by $f_{1}$. Note that, by Novikov\nadditivity~\\cite{novikov1970pontrjagin}, and since $D_{z}'$ is a\nfour-ball, $\\sigma (Y_{1})=\\sigma (Y) + \\sigma (Z)$.\n\nNow consider the base space of the branched covering map~\\ref{define-f1},\n\\begin{equation*}\nX_{1}:= (X - \\interior{D}_{z}) \\bigcup _{S^{3}} D^{4}.\n\\end{equation*}\nSince $X_{1}\\cong X$, we will continue to denote this space by $X$.\nLastly, denote the branching set of $f_{1}$ by $B_{1}$ and remark that\n\\begin{equation}\n\\label{define-B1}\nB_{1}\\cong B - \\interior{N(z)} \\bigcup _{\\alpha \\times \\{0\\}} T.\n\\end{equation}\nIn other words, we replaced the cone on $\\alpha $ by $T$. As prescribed,\n$B_{1}$ has a circle's worth of non-manifold points -- they are the points\ncorresponding to $\\beta \\times \\{\\frac{1}{2}\\}$ in\nEquation~{(\\ref{define_T})} -- regardless of the choice of the knot\n$\\alpha $.\n\n\\textit{Step 2.} Denote by $\\beta ^{\\ast }$ the curve of non-manifold\npoints on $T$. We have, $\\beta ^{\\ast }\\subset T\\subset D^{4}\n\\subset X$. Let $N(\\beta ^{\\ast })$ be a small open tubular neighborhood\nof $\\beta ^{\\ast }$ in $X$. We give $N(\\beta ^{\\ast })$ the following\ntrivialization. For every $b\\in \\beta ^{\\ast }$, let $\\vec{n}_{1}(b)$ be\nthe positive normal to $\\beta $ in $V$ at the point $b$,\n$\\vec{n}_{2}(b)$ the positive normal to $V$ in $S^{3}\\times \\{\n\\frac{1}{2}\\}$, and $\\vec{n}_{3}(b)$ the positive normal to\n$S^{3}$ in the product structure $S^{3}\\times I$. Clearly, $\\{\\vec{n}\n_{1}(b), \\vec{n}_{2}(b), \\vec{n}_{3}(b)\\}$ are linearly independent for\nall $b\\in \\beta ^{\\ast }$, so they give a framing of $\\beta ^{\\ast }$ in\n$X$.\n\nWe use the above framing to identify $\\overline{N(\\beta ^{\\ast })}$ with\n$S^{1} \\times B^{3}$ and $\\partial \\overline{N(\\beta ^{\\ast })}$ with $\\beta ^{\n\\ast }\\times S^{2}$. Now, we construct a new closed oriented\nfour-manifold, denoted $X_{2}$, as follows:\n\\begin{equation*}\nX_{2} = \\bigl (X - N(\\beta ^{\\ast })\\bigr )\\bigcup _{S^{1}\\times S^{2}}\n\\bigl (X - N(\\beta ^{\\ast })\\bigr ).\n\\end{equation*}\nThe identification of the two copies of $\\partial (X - N(\\beta ^{\n\\ast }))$ is done by the homeomorphism\n\\begin{equation*}\n\\phi : S^{1}\\times S^{2}\\to S^{1}\\times S^{2}\n\\end{equation*}\ngiven by the formula\n\\begin{equation*}\n\\phi (e^{i\\theta }, y)=(e^{-i\\theta }, y).\n\\end{equation*}\nIn particular, $\\phi $ reverses orientation on $S^{1}\\times S^{2}$, so\nthe manifold $X_{2}$ can be given an orientation which restricts to the\noriginal orientations on both copies of $X - N(\\beta ^{\\ast })$.\nTherefore, by Novikov additivity we obtain\n\\begin{equation}\n\\label{sigma-X}\n\\sigma (X_{2}) = 2\\sigma (X - N(\\beta ^{\\ast })) = 2\\sigma\n(X).\n\\end{equation}\nNote that, since $\\phi $ restricts to the identity on the $S^{2}$\nfactor, it identifies the boundary of the branching set $T-N(\\beta\n^{\\ast })$ in one copy of $X - N(\\beta ^{\\ast })$ with the boundary of\nbranching set in the other copy of $X - N(\\beta ^{\\ast })$. Thus, the\nimage of the branching set after this identification has the form\n\\begin{equation}\n\\label{surface-mess}\n\\bigl (B_{1} -N(\\beta ^{\\ast })\\bigr )\\bigcup _{3 S^{1}}\\bigl (B_{1} -N(\n\\beta ^{\\ast })\\bigr ) =: B_{2}.\n\\end{equation}\nHere the fact that the union is taken along three circles corresponds\nto the fact that the intersection of $\\partial \\overline{N(\\beta ^{\\ast })}$ and $B_{1}$ consists of three closed curves, one for each\n``boundary point'' of the letter ``$\\top $''.\n\nNote that, since $\\phi $ reverses the orientation on each boundary\ncircle, the orientations of the two copies of $(B_{1} -N(\\beta ^{\n\\ast }))$ can be combined to obtain a compatible orientation on~$B_{2}$. Furthermore, the positive normal to the oriented surface\n$(V-N(\\beta ^{\\ast }))\\cup _{\\phi _{|}}(V-N(\\beta ^{\\ast }))$ inside the\nthree-manifold $(S^{3}\\times \\{\\frac{1}{2}\\} -N(\\beta ^{\\ast }))\n\\cup _{\\phi _{|}} (S^{3}\\times \\{\\frac{1}{2}\\} -N(\\beta ^{\\ast }))$\nrestricts to the normals to $V$ in each corresponding copy of\n$S^{3}$. This observation will come into use shortly.\n\nRecalling the definition of $B_{1}$, namely $B_{1}=(B-\\interior{D}\n_{z})\\bigcup _{\\alpha }(T-N(\\beta ^{\\ast }))$, we can describe\n$B_{2} = (B_{1}-N(\\beta ^{\\ast }))\\cup _{3S^{1}}(B_{1}-N(\\beta ^{\\ast }))$\nin more detail as follows:\n\\begin{equation}\n\\label{surface-bigger-mess}\nB_{2}=\\bigl ( (B-\\interior{D}_{z})\\cup _{\\alpha }(T-N(\\beta ^{\\ast }))\n\\bigr ) \\bigcup _{3 S^{1}}\\bigl ((B-\\interior{D}_{z})\\cup _{\\alpha }(T-N(\n\\beta ^{\\ast }))\\bigr ).\n\\end{equation}\nBy construction, $B_{2}$ is embedded piecewise-linearly in $X_{2}$ --\nthat is, all singularities have been resolved. In addition,\n$B_{2}$ has two connected components, since deleting a neighborhood of\n$\\beta ^{\\ast }$ disconnects $T$. Thus, two copies of $(T-N(\\beta ^{\\ast }))$ gives four disjoint surjaces with boundary. Attaching along the\nthree curves in $(S^{1}\\times S^{2})\\cap (T-N(\\beta ^{\\ast }))$ via $\\phi _{|}$ has the effect of pairing off each of these four surfaces with\nboundary and its homeomorphic copy. This produces two closed surfaces which\nwe denote $B_{2}'$ and $B_{2}''$. Here, $B_{2}'$ is the component of\n$B_{2}$ obtained by identifying two copies of $(B-\\interior{D}_{z})\n\\cup _{\\alpha }(V-\\beta )$ along $S^{1}\\amalg S^{1}$, where $(V-\\beta )$\ndenotes the complement in $V$ of a small open neighborhood of\n$\\beta $. In turn, $B_{2}''$ is the component of $B_{2}$ obtained by\nidentifying two copies\\footnote{It would be more consistent with our\nearlier notation to say that $B_{2}''$ is obtained from two copies of\n$\\beta \\times [\\frac{1}{2},1]\\cup _{\\beta \\times \\{1\\}}V'$, which, of\ncourse, is a surface homeomorphic to $V'$.} of $V'$ along $S^{1}$. By\nconstruction, the cover over $B_{2}'$ is $p$-fold dihedral, whereas the\ncover over $B_{2}''$ is $p$-fold cyclic. That is, a point in\n$B_{2}'$ has $\\frac{p+1}{2}$ pre-images, all but one of branching index\n2, whereas a point in $B_{2}''$ has one pre-image of index $p$. This\ndistinction becomes relevant when we compute the signature of the\nbranched cover whose branching set consists of $B_{2}'$ and\n$B_{2}''$ (Equation~{(\\ref{eq1})}).\n\nNow our aim is to construct a $p$-fold branched cover of $(X_{2}, B\n_{2})$ from the covers $f$ of $(X, B)$ and $h$ of $(D^{4}, T)$.\nMoreover, we require that the cover we construct be a manifold. We are\nhelped greatly in this task by the observation that\n\\begin{equation*}\nN':= h^{-1}(N(\\beta ^{\\ast }))\\cong S^{1}\\times B^{3},\n\\end{equation*}\neven though the branching set of $h$ is non-manifold along $\\beta ^{\n\\ast }$. (A nice explanation of this rather surprising fact can be found\non pp.~173--174 of~\\cite{CS1984linking}.) Here, the trivialization of the\nnormal bundle $N'$ of $h^{-1}(\\beta ^{\\ast })$ is given by the pull-back\nof the trivialization of $N(\\beta ^{\\ast })$. We construct the\ncovering manifold as\n\\begin{equation*}\nY_{2}:=\\bigl (Y_{1}-N'\\bigr )\\bigcup _{S^{1}\\times S^{2}}\\bigl (Y_{1}-N'\n\\bigr ).\n\\end{equation*}\nThe identification along the boundary $S^{1}\\times S^{2}$ is again done\nby $\\phi $, so, once more, $Y_{2}$~can be given an orientation which\nrestricts in each copy of $(Y_{1}-N')$ to the orientation compatible\nwith the given orientation on $Y$. In particular,\n\\begin{equation}\n\\label{signature-Y2}\n\\sigma (Y_{2})=\\sigma \\bigl ((Y_{1}-N')\\cup _{S^{1}\\times S^{2}}(Y_{1}-N')\n\\bigr ) = 2\\sigma (Y_{1}) = 2(\\sigma (Y)+ \\sigma (Z)).\n\\end{equation}\nRecall that both $Y_{2}$ and $X_{2}$ were constructed from copies of\n$(Y_{1}-N')$ and $(X - N(\\beta ^{\\ast }))$ by gluing via $\\phi$. Therefore, the\nrestrictions of $f_{1}$ to the two copies of $(Y_{1}-N')$,\n\\begin{equation*}\nf_{1}|:(Y_{1}-N')\\to (X - N(\\beta ^{\\ast })),\n\\end{equation*}\ncan be glued to obtain a map\n\\begin{equation*}\nf_{2}: \\bigl ( (Y_{1}-N')\\cup _{S^{1}\\times S^{2}}(Y_{1}-N')\\bigr )\n\\to \\bigl (X - N(\\beta ^{\\ast })\\bigr )\\cup _{S^{1}\\times S^{2}} \\bigl (X -\nN(\\beta ^{\\ast })\\bigr ),\n\\end{equation*}\nwritten for short as\n\\begin{equation*}\nf_{2}: Y_{2} \\to X_{2}.\n\\end{equation*}\nThis is the branched cover we will use in the final step of the proof\nto compute the signature of the original manifold $Y$.\n\n\\emph{Step 3.} To complete the proof, what remains is to compute the\neffect this construction has on the signatures of the base and covering\nmanifolds. By Viro's formula~\\cite{viro1984signature} for the\nsignature of a branched cover, we have\n\\begin{equation}\n\\label{eq1}\n\\sigma (Y_{2}) = p\\sigma (X_{2}) - \\frac{p-1}{4}e(B_{2}') - \\frac{p\n^{2}-1}{3p}e(B_{2}'').\n\\end{equation}\nRecall that from Equations~{(\\ref{sigma-X})} and~{(\\ref{signature-Y2})} we have\n\\begin{equation}\n\\label{eq2}\n\\sigma (X_{2})=2\\sigma (X)\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq3}\n\\sigma (Y)=\\frac{1}{2}\\sigma (Y_{2})-\\sigma (Z).\n\\end{equation}\nAlso, by Novikov additivity,\n\\begin{equation}\n\\label{eq4}\n\\sigma (Z, S^{3}) = \\sigma (W(\\alpha ,\\beta ), S^{3}\\cup \\Sigma ) +\n\\sigma (Q, \\Sigma ) = \\sigma (W(\\alpha ,\\beta )) + \\sum _{i=1}^{p-1}\n\\sigma _{\\zeta ^{i}}(\\beta ).\n\\end{equation}\nIn the last step, we have expressed the signature of $Q$ in terms of\nTristram--Levine signatures of $\\beta $, using Theorem~5\nof~\\cite{CS1975branched}. We have also shortened $\\sigma (W(\\alpha\n,\\beta ), S^{3}\\cup \\Sigma )$ to $\\sigma (W(\\alpha ,\\beta ))$. Now we\ncombine Equations~{(\\ref{eq1})}, {(\\ref{eq2})}, {(\\ref{eq3})} and {(\\ref{eq4})} in order\nto express $\\sigma (Y)$ in terms of data about $X$, the branching set,\nthe singularity $\\alpha $ and its characteristic knot $\\beta $. After\nsimplifying, we obtain,\n\\begin{equation}\n\\label{euler-mess}\n\\sigma (Y)=p\\sigma (X) -\\frac{1}{2}\\bigl (\\frac{p-1}{4}e(B_{2}') + \\frac{p\n^{2}-1}{3p}e(B_{2}'')\\bigr ) -\\sigma (W(\\alpha , \\beta )) - \\sum _{i=1}\n^{p-1} \\sigma _{\\zeta ^{i}}(\\beta ).\n\\end{equation}\nTo arrive at Equation~{(\\ref{eq:sigma})}, what remains is to compute the\nself-intersection numbers of $B_{2}'$ and $B_{2}''$ in $X_{2}$ and\nrelate them to that of $B$ in $X$.\n\nWe denote the intersection number of two submanifolds by ``$\\centerdot\n$'', and the push-off of a submanifold $S$ along a normal $\\vec{u}$ by\n$\\vec{u}(S)$. For brevity, we also denote $B-\\interior{D}_{z}$, the\ncomplement in $B$ of a small open neighborhood of the singularity\n$z$, by $B_{z}$.\n\nNote that if $\\vec{v}$ is a continuous extension (not necessarily\nnon-vanishing) to $B_{z}$ of the normal to $V$ in $S^{3}=\\partial D\n_{z}$ such that $B_{z}$ and $\\vec{v}(B_{z})$ are transverse, then by\ndefinition\n\\vspace*{-2pt}\n\\begin{equation}\n\\label{define-e}\ne(B)=(B_{z}\\cup _{\\alpha }V)\\centerdot \\vec{v}(B_{z}\\cup _{\\alpha }V).\n\\end{equation}\nNow, $V\\subset D_{z}$ and $\\vec{v}(V)\\subset D_{z}$, whereas\n$B_{z}\\cap D_{z} = \\alpha $ and $\\vec{v}$ can be chosen so that\n$\\vec{v}(B_{z})\\cap D_{z}=\\vec{v}(\\alpha )$. In particular, $V$ is\ndisjoint from both $\\vec{v}(V)$ and $\\vec{v}(B_{z})$, and $B_{z}$ is\ndisjoint from $\\vec{v}(V)$. Therefore, Equation~{(\\ref{define-e})}\nsimplifies to\n\\vspace*{-2pt}\n\\begin{equation}\n\\label{simpler}\ne(B)=B_{z}\\centerdot \\vec{v}(B_{z}),\n\\end{equation}\nwhere the right hand side represents the intersection number of\ntransverse submanifolds with disjoint boundary in $X-\\interior{D}_{z}$.\nRecall that the surface $B_{2}'$ is obtained from two copies of\n$B_{z}\\cup _{\\alpha }(V - \\beta )$ attached by a homeomorphism\n$\\phi _{|}$ on their boundary $\\beta _{1} \\amalg \\beta _{2}$, reversing\norientation on each component. Recall also that the restriction to\n$S^{3}\\times \\{\\frac{1}{2}\\} \\cap \\partial N(\\beta ^{\\ast })$ of the\npositive normal to $V$ in $S^{3}\\times \\frac{1}{2}$ (and thus of\n$\\vec{v}$), is preserved by the gluing homeomorphism~$\\phi _{|}$.\nTherefore, the two copies of the normal $\\vec{v}$ to $B_{z}\\cup _{\n\\alpha }(V - \\beta )$ can be combined obtain a normal, which we also\ndenote $\\vec{v}$, to $B_{2}'$ in $X_{2}$. Then,\n\\vspace*{-2pt}\n\\begin{equation}\n\\label{B2}\nB_{2}'= B_{z}\\cup _{\\alpha }(V-\\beta ) \\cup _{\\beta _{1}\\amalg \\beta _{2}}\n(V-\\beta )\\cup _{\\alpha }B_{z}.\n\\end{equation}\nSince by the argument above $V-\\beta $ and $\\vec{v}(V-\\beta )$\ncontribute nothing to the self-intersection $B_{2}'\\centerdot \\vec{v}(B\n_{2}')$, we have\n\\vspace*{-2pt}\n\\begin{equation}\n\\label{eB2'}\ne(B_{2}')= 2(B_{z}\\centerdot \\vec{v}(B_{z})) = 2e(B).\n\\end{equation}\n\nNext, recall that $B_{2}''=V'\\cup _{\\beta }V'$ and $\\vec{n}_{1}$ is the\nnormal to $\\beta $ in $V$. Denote by $\\vec{v'}$ a continuous extension\n(not necessarily nowhere-zero) to $V'$ of the normal $\\vec{n}_{1}$ so\nthat $V'$ and $\\vec{v'}(V')$ are transverse. We have,\n\\vspace*{-2pt}\n\\begin{equation}\n\\label{eB2''}\ne(B_{2}'')= 2(V'\\centerdot \\vec{v'}(V')) = 2lk(\\beta , \\vec{v'}(\n\\beta ))=2lk(\\beta , \\vec{n}_{1}(\\beta ))= L_{V}(\\beta , \\beta ).\n\\end{equation}\nRecall that $L_{V}$ denotes the symmetrized linking form on $V$, the\nSeifert surface for $\\alpha $. The last equality follows from the fact\nthat $\\vec{n}_{1}$ and $\\vec{n}_{2}$, the normal to $\\beta $ determined\nby~$V$, are everywhere linearly independent, so $lk(\\beta , \\vec{n}\n_{1}(\\beta ))= lk(\\beta , \\vec{n}_{2}(\\beta ))$.\n\nSubstituting for $e(B_{2}')$ from Equation~{(\\ref{eB2'})} and for\n$e(B_{2}'')$ from Equation~{(\\ref{eB2''})}, we can rewrite\nEquation~{(\\ref{euler-mess})} as\n\\vspace*{-2pt}\n\\begin{equation}\n\\label{signY}\n\\begin{split}\n\\sigma (Y) &= p\\sigma (X) -\\frac{p-1}{4}e(B) - \\frac{p^{2}-1}{6p}L_{V}(\n\\beta , \\beta ) - \\sigma (W(\\alpha , \\beta )) - \\sum _{i=1}^{p-1}\n\\sigma _{\\zeta ^{i}}(\\beta ) \\\\ & = p\\sigma (X) -\\frac{p-1}{4}e(B) - \\Xi_p(\\alpha).\n\\end{split}\n\\end{equation}\nWith that, the proof is complete.\n\\goodbreak\n\\end{proof}\n\n\\begin{rem}\nThe property that a $p$-fold dihedral cover of a knot $\\alpha $ is\nhomeomorphic to the three-sphere can be regarded as a condition for\n$\\alpha $ to be an allowable singularity on the branching set of an\nirregular $p$-fold dihedral cover between four-manifolds. The condition\nis satisfied, for example, for all two-bridge knots and any odd $p$ (see\nthe proof of {Lemma~\\ref{fox-milnor}}) and can be disregarded if one\nallows the covering space to be a stratified space. In this case, an\nanalogous formula can be obtained, using intersection homology signature\nor Novikov signature of a singular space.\n\\end{rem}\n\n\\begin{rem}\n\\label{cyclic-covers}\nIt is natural to consider computing signatures of cyclic branched covers using the same ideas. Indeed, our techniques would apply and the arguments would be considerably simpler: in the notation\nof the proof of the last theorem, only the manifold $Q$ would be needed\nto resolve the singularity. However, it is a consequence of the Smith Conjecture~\\cite{morgan1984smith} that no non-trivial\nknot can arise as a singularity on a cyclic cover between\nfour-manifolds. This is why the case of cyclic covers is not\nconsidered in this work. Our methods, however, are applicable in a\nscenario where stratified spaces are allowed as the covers.\n\\end{rem}\n\nAlthough we have arrived at the desired equation, the formula we have\nobtained does not quite startle with its usefulness, as long as the term\n$\\sigma (W(\\alpha , \\beta ))$ remains obscure. As stated in the\nintroduction, we aim to compute the defect to $\\sigma (Y)$ in terms of\nthe singularity type $\\alpha $. That is, we need to express\n$\\sigma (W(\\alpha , \\beta ))$ explicitly in terms of some computable\ninvariants of $\\alpha $. It turns out that we can give a formula for\n$\\sigma (W(\\alpha , \\beta ))$ using linking numbers of curves in the\nirregular dihedral $p$-fold branched cover of $\\alpha $. To this end,\nwe first compute the second homology group of this manifold: this is\nthe content of {Proposition~\\ref{prop:homologyW}}. In\n{Corollary~\\ref{basis}}, we give a basis for this homology group in terms\nof lifts to a dihedral cover of $\\alpha$ of curves in the chosen Seifert surface $V$. Finally, in\n{Proposition~\\ref{prop:signatureW}}, we give an explicit formula for the\nterm $\\sigma (W(\\alpha , \\beta ))$ using linking numbers of the above\ncurves.\n\n\\begin{prop}\n\\label{prop:homologyW}\nLet $\\alpha \\subset S^{3}$ be a knot which admits a $p$-fold irregular\ndihedral cover $M$ for some odd prime $p$. Let $V$ be a Seifert surface\nfor $\\alpha $ and let $\\beta \\subset V$ be a $\\mathrm{mod}\\ p$ characteristic knot\nfor $\\alpha $. Let $\\Sigma$ be the $p$-fold cyclic cover of $\\beta $. Let\n$W(\\alpha , \\beta )$, here denoted $W$, be the cobordism between $M$ and\n$\\Sigma $ constructed in~\\cite{CS1984linking} and used in the proof\nof {Theorem~\\ref{thm:necessary}}. Denote by $V-\\beta $ the surface $V$\nwith a small open neighborhood of $\\beta $ removed, and by $\\beta _{1}$\nand $\\beta _{2}$ the two boundary components of $V-\\beta $ that are\nparallel to $\\beta $. Then\n\\begin{equation}\nH_{2}(W, M; \\mathbb{Z})\\cong \\mathbb{Z}^{\\frac{p-1}{2}} \\oplus (H_{1}\n(V-\\beta ; \\mathbb{Z}) \/([\\beta _{1}], [\\beta _{2}]))^{\\frac{p-1}{2}}.\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nSince Cappell and Shaneson's construction of $W$ is essential to our\ncomputation, we review it here. Let $f:\\Sigma \\rightarrow S^{3}$ be the\ncyclic $p$-fold cover of $\\beta $. Since $p$ is prime, it is well known\nthat $\\Sigma $ is a rational homology sphere~\\cite{rolfsen1976knots}.\nLet\n\\begin{equation*}\nf\\times 1_{I}: \\Sigma \\times [0, 1]\\rightarrow S^{3}\\times [0,1]\n\\end{equation*}\nbe the induced cyclic branched cover of $S^{3}\\times [0,1]$. Next, let\n\\begin{equation*}\nJ:= f^{-1} (V\\times [-\\epsilon , \\epsilon ]\\times \\{1\\})\n\\end{equation*}\nbe the pre-image of a closed tubular neighborhood $V\\times [-\\epsilon\n, \\epsilon ]\\times \\{1\\}$ of $V\\times \\{1\\}$ in $S^{3}\\times \\{1\\}$, and\nlet $T$ be its ``core'', namely $T:= f^{-1} (V\\times \\{0\\}\\times \\{1\\})$\nwith\n\\begin{equation*}\nT\\subset J\\subset \\Sigma \\times \\{1\\}.\n\\end{equation*}\nThen $J$ deformation-retracts to $T$, and $T$ consists of $p$ copies of\n$V$ identified along $\\beta $ and permuted cyclically by the group of\ncovering transformations of $f$.\n\nConsider the involution $\\bar{h}$ of $J$ defined in\n\\cite{CS1984linking} as the lift of the map\n\\begin{equation*}\nh: V\\times [-\\epsilon , \\epsilon ]\\rightarrow V\\times [-\\epsilon ,\n\\epsilon ],\n\\end{equation*}\n\\begin{equation*}\nh(u,t)\\mapsto (u,-t)\n\\end{equation*}\nfixing a chosen copy of $V$ in $f^{-1}(V\\times \\{0\\}\\times \\{1\\})$. Let\n$q$ be the quotient map defined as\n\\begin{equation*}\nq: \\Sigma \\rightarrow \\Sigma \/\\{x\\sim \\bar{h}(x) | x\\in J\\} = \\Sigma\n\/ \\bar{h}.\n\\end{equation*}\nSimilarly, let\n\\begin{equation*}\nW:= (\\Sigma \\times I) \/ (x\\sim \\bar{h}(x) | x\\in J\\subset \\Sigma\n\\times \\{1\\})\n\\end{equation*}\nand let\n\\begin{equation*}\nM:=(\\Sigma -J^{\\circ })\/\\bar{h}.\n\\end{equation*}\nBy definition,\n$\\Sigma \/\\bar{h} = M\\cup (J\/\\bar{h})$. As shown in\n\\cite{CS1984linking}, $M$ is the $p$-fold irregular dihedral cover of\n$\\alpha $ and $W$ is a cobordism between $M$ and the cyclic $p$-fold\ncover $\\Sigma = \\Sigma \\times \\{0\\}$ of $\\beta $.\n\nThis completes the description of the construction of the pair\n$(W, M)$ whose second homology group we are about to compute.\n\nNote that $W$ is by definition the mapping cylinder of the quotient map\n$q$. Let $R:= J\/ \\bar{h}$. We have\n\\begin{equation*}\nH_{2}(W, M;\\mathbb{Z})\\cong H_{2}(M\\cup R, M;\\mathbb{Z})\\cong H_{2}(R,\nM\\cap R;\\mathbb{Z}),\n\\end{equation*}\nwhere the second isomorphism is excision, and the first follows from the\nfact that $W$ deformation-retracts onto $\\Sigma \/ \\bar{h}= M\\cup R$.\nSince\n\\begin{equation*}\nM\\cap R = \\partial R - V_{0}\n\\end{equation*}\n(following the notation of \\cite{CS1984linking}, $V_{0}$ is the copy\nof $V$ in $T$ fixed by $\\bar{h}$), we can rewrite the above isomorphism\nas\n\\begin{equation}\n\\label{eq:5}\nH_{2}(W, M;\\mathbb{Z})\\cong H_{2}(R, \\partial R - V_{0};\\mathbb{Z}).\n\\end{equation}\nThe relevant portion of the long exact sequence of the pair\n$(R, \\partial R - V_{0})$ is\n\\begin{equation}\n\\label{LES-pair}\nH_{2}(R;\\mathbb{Z}) \\rightarrow H_{2}(R, \\partial R - V_{0};\n\\mathbb{Z})\\rightarrow H_{1}( \\partial R - V_{0};\\mathbb{Z}) \\rightarrow\nH_{1}(R;\\mathbb{Z}).\n\\end{equation}\n\nWe will shortly show that $H_{2}(R;\\mathbb{Z}) = 0$ (see\nEquation~{(\\ref{H2R})}). Assuming this for the moment, the above exact\nsequence, combined with Equation~{(\\ref{eq:5})}, gives\n\\begin{equation}\nH_{2}(W, M;\\mathbb{Z})\n\\cong \\ker (i_{\\ast }: H_{1}( \\partial R - V\n_{0};\\mathbb{Z}) \\rightarrow H_{1}(R;\\mathbb{Z}) ).\n\\end{equation}\n\nOur goal, therefore, is compute this kernel. Note, furthermore, that we\nare not simply interested in its rank over $\\mathbb{Z}$; we want to\nwrite down an explicit basis for $\\ker (i_{\\ast })$ in terms of lifts\nto $M$ of curves in the complement of $\\alpha \\subset S^{3}$.\n\nRecall that $V$ is a surface with boundary and that, by definition,\n$\\beta $ represents a non-zero primitive class in $H_{1}(V;\n\\mathbb{Z})$. Therefore, $\\beta $ can be completed to a one-dimensional\nsubcomplex $C\\vee \\beta $ which $V$ deformation-retracts to, where\n$C$ is the wedge of $2g-1$ circles, and $g$ the genus of $V$. Moreover,\nwe can perform the deformation retraction of $V$ onto such a one-complex\nsimultaneously on each copy of $V$ contained in $T$, fixing the curve\nof intersection $\\beta $. Therefore, $T$ deformation-retracts to a\none-complex containing $\\beta $ wedged to $p$ copies of $C$, where\n\\begin{equation*}\nH_{1}(C;\\mathbb{Z})\\cong H_{1}(V;\\mathbb{Z})\/[\\beta ].\n\\end{equation*}\nIt follows that\n\\begin{equation*}\nH_{2}(J; \\mathbb{Z})\\cong H_{2}(T; \\mathbb{Z})\\cong 0\n\\end{equation*}\nand\n\\begin{equation*}\nH_{1}(J; \\mathbb{Z})\\cong H_{1}(T; \\mathbb{Z})\\cong \\oplus _{p} (H_{1}(V;\n\\mathbb{Z}) \/ [\\beta ])\\oplus \\mathbb{Z},\n\\end{equation*}\nwhere the singled-out copy of $\\mathbb{Z}$ is generated by $[\\beta ]$.\n\nFurthermore, since the deformation--retraction of $J$ onto $T$ can be\nchosen to commute with $\\bar{h}$, $J\/\\bar{h}=R$ deformation-retracts to\n$T\/\\bar{h}$, which is isomorphic to $\\frac{p+1}{2}$ copies of $V$\nidentified along $\\beta $. (This isomorphism is seen from the fact that\n$V_{0}$ is fixed by $\\bar{h}$, and the remaining $p-1$ copies of $V$ in\n$T$ become pairwise identified in the quotient. All copies of\n$\\beta $ are identified to a single circle in both $T$ and $T\/\\bar{h}$.)\nTherefore,\n\\begin{equation}\nH_{1}(R;\\mathbb{Z})\\cong (H_{1}(V;\\mathbb{Z})\/[\\beta ])^{\n\\frac{p+1}{2}}\\oplus \\mathbb{Z}.\n\\end{equation}\nBy the same reasoning as above, we can also conclude that $T\/\\bar{h}$\ndeformation-retracts to a one-complex, so, as claimed,\n\\begin{equation}\n\\label{H2R}\nH_{2}(R; \\mathbb{Z})=H_{2}(J\/\\bar{h}; \\mathbb{Z})\\cong H_{2}(T\/\n\\bar{h}; \\mathbb{Z})\\cong 0.\n\\end{equation}\n\nNext, we examine $\\partial (J)$ and $\\partial R$. To start,\n$\\partial (V\\times [0,1]) \\cong V\\cup _{\\alpha }V$. Therefore,\n$\\partial (J)$ can be thought of as the union of $p$ copies of\n$(V-\\beta )\\cup _{\\alpha }(V-\\beta )$, which we label $V_{i}^{+}\n\\cup _{\\alpha _{i}} V_{i}^{-}$, $0\\leq i<p$, with further identifications\nwe now describe. Denote the copy of $\\beta $ lying in $V_{i}^{\\pm }$ by\n$\\beta _{i}^{\\pm }$, and cut each $V_{i}^{\\pm }$ along $\\beta _{i}^{\n\\pm }\\subset V_{i}^{\\pm }$. Now, $V_{i}^{\\pm }- \\eta (\\beta _{i}^{\n\\pm })$ is a connected surface with three boundary components,\n$\\alpha _{i}$, $\\beta _{i, 1}^{\\pm }$ and $\\beta _{i, 2}^{\\pm }$, where the\n$\\beta _{i,j}^{\\pm }\\subset V_{i}^{\\pm }$ are labeled in such a way that\n$\\beta _{i,j}^{+}$ and $\\beta _{i,j}^{-}$ correspond to $\\beta _{i, j}\n\\times \\{1\\}$ and $\\beta _{i, j}\\times \\{0\\}$ in $V_{i}\\times [0, 1]$;\nthat is, the projection map $V_{i}\\times [0, 1]\\to V_{i}$ sends\n$\\beta _{i,j}^{+}$ and $\\beta _{i,j}^{-}$ to the same boundary component\nof $V-\\eta (\\beta )$.\n\nNow, the covering translation $\\tau $ on $J$ acts in a neighborhood of\n$\\beta $ as rotation by $\\frac{2\\pi i}{p}$ degrees. With the labeling\ndescribed above, $\\tau $ carries $\\beta _{i, j}^{\\pm }$ to $\n\\beta _{i+1\\text{mod~}p, j}^{\\pm }$. Moreover, in $\\partial (J)$, we have the\nfollowing identifications: $\\beta _{i,1}^{+}\\sim\n\\beta _{i+\\frac{p+1}{2}\\text{mod~}p,2}^{+}$ and $\\beta _{i,1}^{-}\\sim\n\\beta _{i+\\frac{p-1}{2}\\text{mod~} p,2}^{-}$. Put differently, we can think of\n$\\partial (J)$ as obtained from $2p$ disjoint copies of $V-\\beta $,\nlabeled $V_{i}^{\\pm }- \\beta _{i}^{\\pm }$, by gluing $\\alpha _{i}^{+}$ to\n$\\alpha _{i}^{-}$ and the $\\beta _{i, j}^{\\pm }$-s according to the\nidentifications specified above. Thus, $\\partial (J)$ is a closed\nsurface. By considering its Euler characteristic, we find that its genus\nis $(2g-1)p + 1$. In addition, from the above decomposition of\n$\\partial (J)$ we find that\n\\begin{equation}\nH_{1}(\\partial (J); \\mathbb{Z})\\cong ((H_{1}(V - \\beta ; \\mathbb{Z}))\n\/ ( [\\beta _{1}], [\\beta _{2}] ))^{2p}\\oplus \\mathbb{Z}^{2p+2}.\n\\end{equation}\n\nRecall that $R$ is a $\\mathbb{Z}\/2\\mathbb{Z}$ quotient of $J$, where the\n$\\mathbb{Z}\/2\\mathbb{Z}$ action fixes $V_{0}\\times I$ and pairs off\n$V_{i}^{+}$ with $V_{p-i}^{-}$ for $1\\leq i\\leq \\frac{p-1}{2}$. It\nfollows that $\\partial R - V_{0}$ is a surface of genus $p(g-1)+\n\\frac{p+1}{2}$ and we have,\n\\begin{equation}\n\\label{homology-boundary}\nH_{1}(\\partial R - V_{0}; \\mathbb{Z})\\cong ((H_{1}(V - \\beta ;\n\\mathbb{Z})) \/ ([\\beta _{1}], [\\beta _{2}]))^{p}\\oplus \\mathbb{Z}^{p+1}.\n\\end{equation}\n\nRecall that our aim is to compute\n\\begin{equation*}\n\\ker (i_{\\ast }: H_{1}( \\partial R - V_{0};\\mathbb{Z}) \\rightarrow H\n_{1}(R;\\mathbb{Z})).\n\\end{equation*}\n\nAgain, the idea behind writing $H_{1}(\\partial R - V_{0};\n\\mathbb{Z})$ as in Equation~{(\\ref{homology-boundary})} is to obtain a\nconvenient basis for this kernel, and to relate this basis to a basis\nfor the homology of $V$. Specifically, a virtue of our expression for $H_{1}( \\partial R - V_{0};\\mathbb{Z})$ is that a basis\nfor $H_{1}(V -\n\\beta ; \\mathbb{Z})) \/ ([\\beta _{1}], [\\beta _{2}])$ can be extended to a basis for both $H_{1}(V;\n\\mathbb{Z})$ and $H_{1}(V-\\beta ; \\mathbb{Z})$. In particular, the\ninclusion $i:\\partial R-V_{0}\\to R$ induces an injection ${i_{\\ast }}\n_{|}: H_{1}(V - \\beta ; \\mathbb{Z}) \/ ([\\beta _{1}], [\\beta _{2}])\n\\to H_{1}(R; \\mathbb{Z})$ for each copy of $V - \\beta \\subset \\partial\nR-V_{0}$. Furthermore, for each $V_{j}\\subset T\/\\bar{h}\\simeq R$, the\ninclusion $i: V_{j} \\to R$ induces an injection on $H_{1}(V - \\beta ;\n\\mathbb{Z}) \/ ([\\beta _{1}], [\\beta _{2}])$.\n\nWith this in mind, using Equations~{(\\ref{H2R})} and {(\\ref{homology-boundary})}, we rewrite\n\\begin{equation*}\ni_{\\ast }: H_{1}(\\partial R - V_{0};\\mathbb{Z}) \\rightarrow H_{1}(R;\n\\mathbb{Z})\n\\end{equation*}\nas\n\\begin{equation*}\ni_{\\ast }: \\bigl ((H_{1}(V - \\beta ; \\mathbb{Z})) \/ ([\\beta _{1}], [\n\\beta _{2}])\\bigr )^{p}\\oplus \\mathbb{Z}^{p+1} \\rightarrow (H_{1}(V;\n\\mathbb{Z})\/[\\beta ])^{\\frac{p+1}{2}}\\oplus \\mathbb{Z}.\n\\end{equation*}\n\nNote that $i_{\\ast }$ maps the copy of $(H_{1}(V - \\beta ; \\mathbb{Z}))\n\/ ( [\\beta _{1}], [\\beta _{2}])$ coming from $V_{0}^{+}$ isomorphically\nonto its image, and it ``pairs off'' the remaining $p-1$ copies of\n$(H_{1}(V -\\nobreak  \\beta ; \\mathbb{Z})) \/\\allowbreak  ( [\\beta _{1}], [\\beta _{2}])$,\nincluding each of them into one of the remaining $\\frac{p-1}{2}$ copies\nof $H_{1}(V;\\mathbb{Z})\/[\\beta ]$ in $R$. This contributes $H_{1}(V -\n\\beta ; \\mathbb{Z})^{\\frac{p-1}{2}}$ to $\\ker (i_{\\ast })$. In addition,\nthe generators for the $\\mathbb{Z}^{p+1}$ summand in $H_{1}( \\partial\nR - V_{0};\\mathbb{Z})$ can be chosen as follows. There are $\n\\mathbb{Z}^{\\frac{p+1}{2}}$ generators which correspond to copies of\n$\\beta $ lying in the various copies of $V-\\beta \\subset (\\partial R -\nV_{0})$; they all map to the single $[\\beta ]$ in the image,\ncontributing $\\mathbb{Z}^{\\frac{p-1}{2}}$ to $\\ker i_{\\ast }$. Finally,\nthere are an additional $\\mathbb{Z}^{\\frac{p+1}{2}}$ generators of the\n$\\mathbb{Z}^{p+1}$ summand in $H_{1}( \\partial R - V_{0};\\mathbb{Z})$\nwhich are mapped injectively by $i_{\\ast }$, onto classes in\n$T\/\\bar{h}$ which are not in the image of $i_{\\ast }(H_{1}(V - \\beta\n; \\mathbb{Z})$ for any copy of $V$. Consequently, as we claimed,\n\\begin{equation*}\n\\ker (i_{\\ast }) \\cong (H_{1}(V - \\beta ; \\mathbb{Z})\/([\\beta _{1}], [\n\\beta _{2}]))^{\\frac{p-1}{2}} \\oplus \\mathbb{Z}^{\\frac{p-1}{2}}.\\qedhere\n\\end{equation*}\n\\end{proof}\n\nFurthermore, the above argument allows us to describe a basis for this kernel.\n\n\\begin{cor}\n\\label{basis}\nAssume the notation of {Proposition~\\ref{prop:homologyW}}. Further, let\n$w^{1}, w^{2}, ... , w^{r}$ be a basis for $H_{1} (V-\\beta ;\n\\mathbb{Z})\/([\\beta _{1}], [\\beta _{2}])$, where $r=2g-2$ and $g$ is\nthe genus of $V$. Denote by $w^{i, \\pm }_{j}, i\\in \\{1, ..., r\\}, j\n\\in \\{1, ..., p\\}$ the pre-images of the $w^{i}$ lying in $f^{-1}(V\n\\times \\{\\pm 1\\})$ so that $w^{i,\\pm }_{j} \\subset V^{\\pm }_{j}$,\n$f(w^{i,\\pm }_{j})=w^{i}_{j}$ and $\\tau w_{k}^{\\pm }= w_{k+1\\text{mod~} p}\n^{\\pm }$. Next, let $\\bar{h}(w^{i,\\pm }_{j})=:\\overline{w}^{i, \\pm }\n_{j}\\in M$. Finally, denote by $\\overline{\\beta }_{k}, k=0,1, ...,\n\\frac{p-1}{2}$ the $\\frac{p+1}{2}$ generators of $H_{1}(\\partial R - V\n_{0};\\mathbb{Z})$ which are represented by copies of $\\beta $. Then, a\nbasis for $\\ker i_{\\ast }$ is given by:\n\\begin{equation}\n\\label{generators}\n\\{[\\overline{w}^{i, +}_{k} - \\overline{w}^{i, -}_{k}], [\\overline{\n\\beta }_{k} -\\overline{\\beta }_{k-1}]\\}_{i=1, ..., r; ~k= 1, ..., \n\\frac{p-1}{2}}\n\\end{equation}\n\n\\end{cor}\n\n\\begin{proof}\nThe statement follows from the last paragraph of the proof of\n{Proposition~\\ref{prop:homologyW}}.\n\\end{proof}\n\n\\begin{rem}\nThe heavy notation we have had to resort to here deserves a comment.\nSince $\\bar{h}$ identifies $V_{j}^{+}$ with $V_{p-j}^{-}$, we have\n$\\overline{w}^{i,\\pm }_{j}=\\overline{w}^{i,\\mp }_{p-j}$. Secondly, there\nare many choices of $\\frac{p+1}{2}$ curves $\\beta _{k}$ so that the\nclasses $[\\beta _{k}]$ are independent generators of $H_{1}( \\partial\nR - V_{0};\\mathbb{Z})$. We note that it is possible to impose the extra\ncondition that $\\overline{\\beta }_{k}-\\overline{\\beta }_{k-1}$, $k=1, 2,\n..., \\frac{p-1}{2}$ bounds a cylinder $\\beta \\times [-1, 1]$ in $R$. We\ndo this by choosing for the $\\overline{\\beta }_{k}$-s ``consecutive''\ncopies of~$\\beta $ as we move counter-clockwise in $\\partial R$,\nstarting, for instance, with the copy of~$\\beta _{1}$ lying\nin~$V_{0}^{+}$. This observation will allow us to simplify the proof of\n{Proposition~\\ref{prop:signatureW}}.\n\\end{rem}\n\nWe are now ready to give a formula for the signature of $W$.\n\n\\begin{prop}\n\\label{prop:signatureW}\nLet $\\alpha $ be a knot which admits an irregular dihedral $p$-fold\ncover for an odd prime $p$. In addition, assume that this cover is\n$S^{3}$. Using the notation of {Corollary~\\ref{basis}}, let $A$ be the\nmatrix of linking numbers in $S^{3}$ of the following set of links:\n\\begin{equation}\n\\label{basis-kernel}\nS:=\\{[\\overline{w}^{i, +}_{k} - \\overline{w}^{i, -}_{k}], [\\overline{\n\\beta }_{k} -\\overline{\\beta }_{k-1}]\\}_{i=1, ..., r; ~k= 1, ...,\n\\frac{p-1}{2}},\n\\end{equation}\nwhere the orientation of each curve is compatible with a chosen\norientation on the corresponding curve, $w^{i}$ or $\\beta $, in $V$.\nThen, $\\sigma (W) = \\sigma (A)$.\n\n\\end{prop}\n\n\\begin{proof}\nWe wish to compute the intersection form on the image $i_{\\ast }(H\n_{2}(W; \\mathbb{Z}))$ in $H_{2}(W, S^{3}\\cup \\Sigma ; \\mathbb{Z})$.\nSince $p$ is prime, $\\Sigma $ is a rational homology\nsphere~\\cite{rolfsen1976knots}. It follows that\n\\begin{equation*}\nH_{2}(W, M; \\mathbb{Z})\\cong i_{\\ast }(H_{2}(W; \\mathbb{Z}))\\subset H\n_{2}(W, S^{3}\\cup \\Sigma ; \\mathbb{Z}).\n\\end{equation*}\nBy {Proposition~\\ref{prop:homologyW}} we already know that\n\\begin{equation*}\nH_{2}(W, M; \\mathbb{Z})\\cong \\ker (i_{\\ast }: H_{1}( \\partial R - V\n_{0};\\mathbb{Z}) \\rightarrow H_{1}(R;\\mathbb{Z})) =: K.\n\\end{equation*}\nFurthermore, by {Corollary~\\ref{basis}}, the set of classes in\n$S\\subset H_{1}( \\partial R - V_{0};\\mathbb{Z})$ defined above forms a\nbasis for $K$. We use the isomorphism $H_{2}(W, M; \\mathbb{Z})\\cong K$\nto obtain an explicit basis for $H_{2}(W, M; \\mathbb{Z})$ consisting of\nsurfaces with boundary which are properly embedded in $W$.\n\nRecall that the isomorphism between $H_{2}(W, M; \\mathbb{Z})$ and\n$K$ is given by Equation~{(\\ref{eq:5})}, together with the boundary map\n$\\partial $ in the long exact sequence~{(\\ref{LES-pair})}. By our choice of\nbasis for $K$, for any element $u\\in S$, $u$ is the boundary of a\ncylinder $S^{1}\\times I =: U$ properly embedded in $(R, \\partial R- V\n_{0})\\subset (W, M)$. We use this cylinder to represent the class\n$[U]\\in H_{2}(W, M; \\mathbb{Z})$ corresponding to $u$ under the above\nisomorphism $H_{2}(W, M; \\mathbb{Z})\\cong K$. Next, given $u_{1}, u\n_{2}\\in S$, we can write $u_{1} = a_{1} - a_{2}$ and $u_{2} = b_{1} -\nb_{2}$, where $a_{i}, b_{i}$ are oriented curves in the dihedral cover\n$M$. Denote by $U_{i}, i=1, 2$, the cylinder $S^{1}\\times I \\subset V\n_{j}\\times I$ with $\\partial U_{i}=u_{i}$. (Here $V_{j}$ denotes the\nlift of $V$ for which $a_{1}-a_{2}$, respectively $b_{1}-b_{2}$, lies\non the boundary of $V_{j}\\times I$.) Now, if $F_{i}$ is any Seifert\nsurface for $u_{i}$ in $M\\cong S^{3}$, we have $U_{i}\\cup _{u_{i}} F\n_{i}\\in H_{2}(W; \\mathbb{Z})$ and $i_{\\ast }: H_{2}(W; \\mathbb{Z})\n\\to H_{2}(W, S^{3}\\cup \\Sigma ; \\mathbb{Z})$ carries $U_{i}\\cup _{u\n_{i}} F_{i}$ to $U_{i}$. We fix two Seifert surfaces, $F_{1}$ and\n$F_{2}$, for $u_1$ and $u_2$, respectively, and use the classes $U_{1}\\cup _{u_{1}} F_{1}$ and $U_{2}\n\\cup _{u_{2}} F_{2}$ to compute the intersection number $U_{1}\\centerdot\nU_{2}$.\n\nBy giving $W$ a little collar, $M\\times [0, \\epsilon ]$, and ``pushing \nin'' $U_{2}\\cup _{u_{2}} F_{2}$ ever so slightly, we can assume that\n$F_{2}$ lies in $M\\times \\{\\epsilon \\}$, and $U_{2}\\cup _{u_{2}} F_{2}$\nis disjoint from $M \\times [0, \\epsilon )$. Since $F_{1}\\subset M\n\\times \\{0\\}$, in order to compute $U_{1}\\centerdot U_{2}$, it now\nsuffices to consider the intersection of~$U_{1}$ with $U_{2}\\cup _{u\n_{2}} F_{2}$. We consider several cases. If the curves $a_{1}$ and\n$b_{1}$ are disjoint, then so are $U_{1}=a_{1}\\times I$ and\n$U_{2}=b_{2}\\times I$, regardless of whether the $a_{i}$, $b_{i}$ live\nin the same lift of~$V_{j}$ or in different lifts. In this case, the\nintersection is simply $U_{1}\\centerdot F_{2} = lk (a_{1}-a_{2}, b\n_{1}-b_{2})$, since $F_{2}\\subset M\\times \\{\\epsilon \\}$, $U_{1}\n\\cap (M\\times \\{\\epsilon \\})= a_{1}-a_{2}$ and $F_{2}$ is a Seifert \nsurface for $b_{1}-b_{2}$, so that, putting everything together, we have\n$U_{1}\\centerdot F_{2} = (a_{1}-a_{2})\\cap F_{2} = lk (a_{1}-a_{2}, b\n_{1}-b_{2})$ by definition. Secondly, $U_{1}$ and $U_{2}$ can be\ndistinct but intersecting cylinders. This can only happen if both live\nin the same lift of $V\\times I$, which we again denote $V_{j}\\times I$.\nIn this case, we use the normal to $V_{j}\\times I$ in $W$ to push off\n$U_{1}$ away from $V_{j}\\times I$ and thus from $U_{2}$. Again, we find\nthat $U_{1}\\centerdot U_{2} = U_{1}\\centerdot F_{2} = lk (a_{1}-a_{2},\nb_{1}-b_{2})$. Lastly, we consider the case where $U_{1}=U_{2}$. For\nsome choice of $j$ we have $U_{1}\\subset V_{j}\\times I$ with\n$\\partial U_{1} = (a_{1} - a_{2}) \\subset V_{j}\\times \\{0, 1\\}$. We can\npush $a_{1}$ off itself using its (positive, say) normal in\n$V_{j}\\times \\{0\\}$. This push-off extends across $U_{1} = (a_{1}\n\\times I) \\subset (V_{j}\\times I)$, so the cylinder can be made disjoint\nfrom itself. Again, we conclude that $U_{1}\\centerdot U_{2}= U_{1}\n\\centerdot F_{2} = lk (a_{1}-a_{2}, a_{1}-a_{2})$, where the\nself-linking number is computed using the normal to $a_{1}-a_{2}$ in\n$V_{j}\\times \\{0, 1\\}$. Therefore, the matrix of linking numbers between\nelements of our basis for $K$ is also the intersection matrix for\n$W=(W(\\alpha , \\beta ))$. This completes the proof.\n\\end{proof}\n\n\\begin{rem}\nWe note that the self-linking with respect to the normal to\n$a_{1}-a_{2}$ in $V_{j}\\times \\{0, 1\\}$ is equal to the self-linking\nwith respect to the restriction to $a_{1}-a_{2}$ of the normal to\n$V_{j}\\times \\{0, 1\\}$ in $M\\cong S^{3}$, since the two vectors are\neverywhere linearly independent. This is useful for computations, since\nthe normal to $V_{j}$ in the dihedral cover is just the lift of the\nnormal to $V$ in $S_{3}$.\n\\end{rem}\n\nThe Proof of {Proposition~\\ref{prop:homologyW}} also allows us to compute\nthe fundamental group of the manifold $W(\\alpha , \\beta )$ for knots\n$\\alpha $ which can arise as singularities of dihedral branched covers\nbetween four-manifolds.\n\n\\begin{cor}\n\\label{pi1W}\nLet $p$ be an odd prime and let $\\alpha $ be a knot which admits a\n$p$-fold irregular dihedral cover. Assume moreover that this cover\nhomeomorphic to $S^{3}$. Let $\\beta $ be a characteristic knot for\n$\\alpha $ and let $W(\\alpha , \\beta )$ be the cobordism between\n$S^{3}$ and the $p$-fold cyclic cover of $\\beta $ constructed\nin~\\cite{CS1984linking}. Then $W(\\alpha , \\beta )$ is\nsimply-connected.\n\\end{cor}\n\\begin{proof}\nWe assume the notation of the proof of {Proposition~\\ref{prop:homologyW}}.\n(In this notation, the additional assumption of this Corollary is that\n$M\\cong S^{3}$.) We have seen that $W(\\alpha , \\beta )$ is homotopy\nequivalent to $M\\cup R$ and that $M\\cap R= \\partial R - V_{0}$. We also\nknow that $i_{\\ast }: \\pi _{1}(\\partial R - V_{0}, a_{0}) \\to \\pi _{1}(R,\na_{0})$ is surjective. On the other hand, any loop in $ \\pi _{1}(\n\\partial R - V_{0}, a_{0}) = \\pi _{1}(M\\cap R, a_{0})$ is contractible\nin $M$ since $\\pi _{1}(M; a_{0}) = 0$. Therefore, by van Kampen's\nTheorem, $\\pi _{1}(M\\cup R, a_{0}) = 0 = \\pi _{1}(W(\\alpha , \\beta ), a\n_{0})$.\n\\end{proof}\n\nFinally, we show that the defect to the signature of a branched cover\narising from the presence of a singularity $\\alpha $ is an invariant of\nthe knot type $\\alpha $.\n\n\\begin{prop}\\label{prop:invariant}\nLet $p$ be an odd square-free integer, and let $\\alpha \\subset S^{3}$\nbe knot which arises as the singularity of an irregular dihedral\n$p$-fold cover between four-manifolds. Assume that $p^{2}$ does\nnot\\footnote{One could allow $p^{2}$ to divide $\\Delta (-1)$. In this\ncase, $\\Xi _{p}$ would not necessarily be an invariant of the knot type\n$\\alpha $ but, rather, of $\\alpha $ together with a specified\nrepresentation of $\\pi _{1}(S^{3}-\\alpha , x_{0}) \\twoheadrightarrow D\n_{p}$.} divide $\\Delta (-1)$, where $\\Delta (t)$ is the Alexander\npolynomial of $\\alpha $. In the notation of {Theorem~\\ref{thm:necessary}},\nthe integer $ \\Xi _{p}(\\alpha )$, defined as\n\\begin{equation}\n\\label{valueXi}\n\\Xi _{p}(\\alpha )= \\frac{p^{2}-1}{6p}L_{V}(\\beta , \\beta ) + \\sigma (W(\n\\alpha , \\beta )) + \\sum _{i=1}^{p-1} \\sigma _{\\zeta ^{i}}(\\beta )\n\\end{equation}\nis an invariant of the knot type $\\alpha $.\n\\end{prop}\n\n\\begin{proof}\nSince $\\alpha $ arises as a singularity of an irregular dihedral\n$p$-fold cover, by {Theorem~\\ref{thm:necessary}}, $\\alpha $ itself admits\nan irregular dihedral $p$-fold cover. Since $p^{2}$ does not divide\n$\\Delta (-1)$, this cover is unique (see footnote on p.~166\nof~\\cite{CS1984linking} or, for a more thorough discussion,~\\cite{fox1970metacyclic}).\n\nWhen both $\\alpha $ and $\\beta $ are fixed, it is clear that each of the\nterms $\\frac{p^{2}-1}{6p} L_{V}(\\beta , \\beta )$, $\\sigma (W(\\alpha ,\n\\beta ))$ and $\\sum _{i=1}^{p-1} \\sigma _{\\zeta ^{i}}(\\beta )$ is\nwell-defined. We will show that their sum is in fact independent\nof the choice of $\\beta $.\n\nLet $f: Y\\to X$ be an irregular dihedral $p$-fold cover, branched over\nan oriented surface $B\\subset X$, embedded in $X$ with a unique\nsingularity of type $\\alpha $. Such a cover exists by assumption. Then\n\\begin{equation*}\n\\Xi _{p}(\\alpha) = p\\sigma (X) -\\frac{p-1}{2}e(B) - \\sigma (Y),\n\\end{equation*}\na formula independent of the choice of $\\beta $.\n\nA priori, however, it might be possible for another branched cover\n$f': Y'\\to X'$, whose branching set also has a singularity of type\n$\\alpha $, to produce a different value of $\\Xi _{p}$. This does not\noccur. By the proof of {Theorem~\\ref{thm:necessary}}, any choice of\ncharacteristic knot $\\beta $ can be used to compute the defect\n$\\Xi _{p}(\\alpha )$ to the signature of $Y$. Using the same $\\beta $ and\nEquation~{(\\ref{valueXi})} to compute this signature defect for two\ndifferent covers, for instance $Y$ and $Y'$, shows that $\\Xi _{p}(\n\\alpha )$ does not vary with the choice of branched cover and indeed\ndepends only on $\\alpha $.\n\\end{proof}\n\n\\section{Constructing dihedral covers}%\n\\label{sec:sufficient}\nIn this section, we describe a method for constructing an irregular\n$p$-fold dihedral cover of a simply-connected four-manifold\n$X$. We use this construction to prove {Theorem~\\ref{thm:sufficient}},\nwhich is a partial converse to {Theorem~\\ref{thm:necessary}}. \nPrecisely,\n{Theorem~\\ref{thm:sufficient}} establishes that, when two-bridge slice\nsingularities are considered, all pairs of integers $(\\sigma , \\chi )$ afforded by the\nnecessary condition ({Theorem~\\ref{thm:necessary}}) as the signature and\nEuler characteristic of a $p$-fold irregular dihedral cover of a given\nbase manifold $X$ with specified branching set $B$ are indeed realized\nas the signature and Euler characteristic of a $p$-fold irregular\ndihedral cover over $X$.\n\nA main ingredient of the proof is constructing an irregular dihedral\ncover of $S^{4}$ branched over a singular two-sphere with a given\nsingularity ({Proposition~\\ref{covers_of_spheres}}). By taking a connected\nsum with this singular two-sphere, we can introduce a singularity to\na PL embedded surface $B\\subset X$ without changing its homeomorphism type\nor that of the ambient manifold. The dihedral cover of $X$ is\nconstructed from the irregular dihedral cover of $(S^{4}, S^{2})$,\ntogether with several copies of the double branched cover of $X$ over\nthe locally flat surface $B$.\n\nWe begin with a simple lemma.\n\n\\begin{lem}\n\\label{lem:SC_double_cover_exists}\nLet $X$ be four-manifold and let $B\\subset X$ be an embedded connected\nsurface such that $\\pi _{1}(X-B, x_{0})\\cong \\mathbb{Z}\/2\\mathbb{Z}$. The\ndouble branched cover of $(X, B)$ is simply-connected.\n\\end{lem}\n\n\\begin{proof}\nSince $\\pi _{1}(X- B, x_{0})\\cong \\mathbb{Z}\/2\\mathbb{Z}$, a double cover\nof $X$ branched along $B$ exists. We denote this cover by $\\hat{X}$ and denote by $\\hat{B}$ the (homeomorphic) pre-image of $B$ under the\ncovering map. We apply van Kampen's theorem to $\\hat{X} = (\\hat{X} -\n\\hat{B})\\cup _{\\partial N(\\hat{B})} N(\\hat{B})$, where $N(\\hat{B})$\ndenotes a small tubular neighborhood of $\\hat{B}$. Being the universal\ncover of $(X- B)$, $(\\hat{X} - \\hat{B})$ is simply connected, so\n$i_{\\ast }: \\pi _{1}(\\partial N(\\hat{B}), b_{0})\\rightarrow \\pi _{1}(\n\\hat{X}-\\hat{B}, b_{0})$ is the zero homomorphism. In addition, $i_{\\ast }: \\pi\n_{1}(\\partial N(\\hat{B}), b_{0})\\rightarrow \\pi _{1}(N(\\hat{B}), b_{0})$\nis surjective, since every element in $\\pi _{1}(N(\\hat{B}), b_{0})$ can\nbe represented by a loop which is disjoint from the 0-section and which\nis therefore homotopic to a loop in $\\partial N(\\hat{B})$. It follows\nfrom van Kampen's Theorem that $\\hat{X}$ is simply-connected.\n\\end{proof}\n\nNext, we prove a couple of lemmas concerning the singularities which we\nwill be introduced to the branching sets in the construction of\ndihedral covers. In {Lemma~\\ref{ribbon_disks}} we recall a well-known fact\nabout the fundamental groups of complements of ribbon disks.\n{Lemma~\\ref{fox-milnor}} allows us to extend a dihedral cover of a\ntwo-bridge slice knot to a cover of a disk it bounds.\n\n\\begin{lem}\n\\label{ribbon_disks}\nLet $K\\subset S^{3} =\\partial B^{4}$ be a ribbon knot and let $D'\\subset S^{3}$ be a\nribbon disk for $K$. Then, there exists $D\\subset B^{4}$, a slice disk\nfor $K$, such that the map $i_{\\ast }: \\pi _{1} (S^{3}-K, x_{0})\n\\to \\pi _{1}(B^{4}-D, x_{0})$ induced by inclusion is surjective.\n\\end{lem}\n\\begin{proof}\nSince $D'$ is ribbon, we can push the interior of $D'$ into the interior\nof $B^{4}$ to obtain a slice disk $D$ with the property that $g$, the\nradial function on $B^{4}$, is Morse when restricted to $D$ and has no\nlocal maxima on the interior of $D$. Computing the fundamental group of\nthe complement of $D$ in $B^{4}$ by cross-sections\nas outlined in~\\cite{fox1962quick}, we start with $\\pi _{1} (\\partial B^{4}-\n\\partial D, x_{0}) = \\pi _{1}(S^{3}- K, x_{0})$ and proceed to introduce\nnew generators or relations at each critical point of $g$. Since $g$ has\nno maxima, no new generators are introduced, implying that $i_{\\ast }:\n\\pi _{1}(S^{3}- K, x_{0})\\rightarrow \\pi _{1} (B^{4}-D, x_{0})$ is a\nsurjection.\n\\end{proof}\n\nIn the notation of the above lemma, a disk $D$ with the property that\n\\begin{equation*}\ni_{\\ast }: \\pi _{1} (S^{3}-K, x_{0}) \\to \\pi _{1}(B^{4}-D, x_{0})\n\\end{equation*}\nis a surjection is called a \\emph{homotopy ribbon disk}. Thus, the lemma can\nbe rephrased by saying ribbon knots admit homotopy ribbon disks.\n\n\\begin{lem}\n\\label{fox-milnor}\nLet $K\\subset S^{3}\\subset \\partial B^{4}$ be a slice knot and let\n$D\\subset B^{4}$ be a slice disk for $K$. Let $p>1$ be an odd\nsquare-free integer. If the pair $(S^{3}, K)$ admits an irregular $p$-fold\ndihedral cover, then the pair $(B^{4}, D)$ admits one as well.\nFurthermore, if $K$ is a two-bridge knot, $D$ can be chosen PL and such\nthat the irregular dihedral cover of $B^{4}$ branched along $D$ is\nsimply-connected.\n\\end{lem}\n\n\\begin{proof}\nLet $\\Delta _{K}(t)$ denote the Alexander polynomial of $K$ and\n$\\Delta _{D}(t)$ that of $D$. Denote by $\\hat{S}$ the double branched\ncover of the pair $(S^{3}, K)$ and by $\\hat{K}$ the pre-image of $K$\nunder the covering map. It is well known that $|\\Delta _{K}(-1)| = |(H\n_{1} (\\hat{S}; \\mathbb{Z})|$ \\cite{seifert1935geschlecht}. Similarly,\ndenote by $\\hat{B}$ the double cover of $B^{4}$ branched along $D$ and\nby $\\hat{D}$ the pre-image of $D$. As above, we have $|\\Delta _{D}(-1)|=\n|(H_{1} (\\hat{B}; \\mathbb{Z})|$, since $\\pm \\Delta _{D}(-1)$ is the\ndeterminant of a presentation matrix for the first homology of the\ndouble branched cover of $D$. (Denote by $B_{\\infty }$ the infinite\ncyclic cover of the disk complement $B^{4}-D$. Regard $H_{1}(B_{\n\\infty }; \\mathbb{Z})$ as a $\\mathbb{Z}[\\tau , \\tau ^{-1}]$-module, where\nthe action of $\\tau $ is that induced by a generator of the group of\ncovering translations. Then $\\Delta _{D}(t)$ is the characteristic\npolynomial of this action and $H_{1}(\\hat{B}; \\mathbb{Z})\\cong Coker\n\\{1-\\tau ^{2}: H_{1}(B_{\\infty }; \\mathbb{Z})\\to H_{1}(B_{\\infty };\n\\mathbb{Z})\\}$. For a thorough exposition on the homology of cyclic\ncovers of a homology $S^{1}$, see~\\cite{stevens1996homology}.)\n\nSince $K$ admits a dihedral cover, $H_{1} (\\hat{S}; \\mathbb{Z})$\nsurjects onto $\\mathbb{Z}\/p\\mathbb{Z}$~\\cite{CS1984linking}. It\nfollows that $\\Delta _{K}(-1)\\equiv 0 \\mod p$. Since $D$ is a slice disk\nfor $K$, by results of Fox and Milnor \\cite{fox1966singularities} we\nhave $\\Delta _{K}(-1) = \\pm (\\Delta _{D}(-1))^{2}$, so $(\\Delta _{D}(-1))^{2}\n\\equiv 0 \\mod p$. Since $p$ is square-free by assumption, we conclude that\n$\\Delta _{D}(-1) \\equiv 0 \\mod p$ as well. Then $H_{1} (\\hat{B};\n\\mathbb{Z})$ surjects onto $\\mathbb{Z}\/p\\mathbb{Z}$, and thus\n$\\hat{B}$ admits a $p$-fold cyclic cover $T$ with $\\partial T = : N$.\nThis cover $T$ is the {regular} dihedral $2p$-fold branched cover\nof $(B^{4}, D)$. Let $Z$ be the quotient of $T$ by the action of any\n$\\mathbb{Z}\/2\\mathbb{Z}$ subgroup of $D_{p}$. Then $Z$ is the desired\nirregular dihedral $p$-fold cover of $(B^{4}, D)$. Its boundary, which\nwe denote by $U$, is the irregular dihedral $p$-fold cover of $K$.\n\nNow assume in addition that $K$ is a two-bridge knot. In this case it\nis well-known that $U$ is in fact $S^{3}$. Indeed, the pre-image\n$S^{\\ast }$ of a bridge sphere for $K$ is a dihedral cover of~$S^{2}$ branched over four points, so $S^{\\ast }$ has Euler\ncharacteristic\n\\begin{equation*}\n\\chi (S^{\\ast }) = p (\\chi (S^{2}) - 4) + 4\\frac{p+1}{2} = 2.\n\\end{equation*}\nA bridge sphere bounds a trivial tangle to either side, and the cover\nof a trivial tangle is a handlebody. Therefore, $S^{\\ast }$ is a\nHeegaard surface for $U$, and, since the genus of $S^{\\ast }$ is zero,\n$U\\cong S^{3}$.\n\nSince $K$ is two-bridge slice, it is ribbon. Hence, by\n{Lemma~\\ref{ribbon_disks}}, the slice disk $D$ for $K$ can be chosen to\nbe PL and homotopy ribbon, i.e. such that $\\pi _{1} (S^{3}-K, x_{0})\\xrightarrow{i\n_{\\ast }} \\pi _{1}(B^{4}-D, x_{0})$ is a surjection. Therefore, given a\nhomomorphism $\\psi : \\pi _{1} (B^{4}-D, x_{0}) \\rightarrow D_{2p}$, the\npre-image $(\\psi \\circ i_{\\ast })^{-1}(\\mathbb{Z}\/2\\mathbb{Z})$ surjects\nonto $\\psi ^{-1}(\\mathbb{Z}\/2\\mathbb{Z})$ by $i_{\\ast }$. This implies\nthat the inclusion of the unbranched cover associated to $U$ into the\nunbranched cover associated to $Z$ induces a surjection on fundamental\ngroups. Since the branching set of $U$ is a subset of the branching set\nof $Z$, it follows that $\\pi _{1}(U, x_{0})\\xrightarrow{i_{\\ast }}\n\\pi _{1}(Z, x_{0})$ is also a surjection. But $\\pi _{1}(U, x_{0}) = 0$,\nand we conclude that the irregular dihedral cover of the pair\n$(B^{4}, D)$ is simply-connected.\n\\end{proof}\n\n\\begin{prop}\n\\label{covers_of_spheres}\nLet $p>1$ be an odd square-free integer and let $K\\subset S^{3}$ be a\nslice knot such that the pair $(S^{3}, K)$ admits an irregular dihedral $p$-fold\n cover. Then there exists an embedded two-sphere $S^{2}\\subset\nS^{4}$ such that the pair $(S^{4}, S^{2})$ admits an irregular dihedral $p$-fold\n cover $W$ and $S^{2}\\subset S^{4}$ is locally flat except at\none point where it has a singularity of type $K$. Moreover, if $K$ a\ntwo-bridge knot, $W$ is a simply-connected manifold.\n\\end{prop}\n\n\\begin{proof}\nLet $D_{1}^{2}\\subset B^{4}_{1}$ be a PL slice disk for $K$. Denote the\ncone on the pair $(S^{3}, K)$ by $(B_{2}^{4}, D_{2}^{2})$. The disk $D_{2}^{2}$ is a PL submanifold of $B^{4}_{2}$\nexcept at the cone point $x$, where by construction $D_{2}^{2}$ has a\nsingularity of type $K$. Identifying the two pairs $(B_{1}^{4}, D_{1}\n^{2})$ and $(B_{2}^{4}, D_{2}^{2})$ via the identity map along the two\ncopies of $(S^{3}, K)$ lying on their boundaries, we obtain an embedding\nof a two-sphere $S: = D_{1}^{2} \\cup _{K} D_{2}^{2}$ in $S^{4} = B_{1}\n^{4} \\cup _{S^{3}} B_{2}^{4}$ such that $S$ has a unique\nsingularity of type $K$ at $x$.\n\nBy {Lemma~\\ref{fox-milnor}}, the pair $(B^{4}_{1}, D_{1}^{2})$ admits an\nirregular dihedral $p$-fold cover $W$, and its boundary $M$ is the\nirregular dihedral $p$-fold cover of the pair $(S^{3}, K)$. Since\n$(B^{4}_{2}, D_{2}^{2})$ is a cone, its irregular dihedral $p$-fold\ncover is simply the cone on $M$. Thus, the pair $(S^{4}, S)$ admits a\ncover\n\\begin{equation*}\nZ:=W\\bigcup _{\\partial W\\sim M\\times \\{0\\}} (M\\times [0, 1]\/ M\\times\n\\{1\\})\n\\end{equation*}\nas claimed. If, in addition, $K$ is a two-bridge knot, by\n{Lemma~\\ref{fox-milnor}} we know that $M$ is the three-sphere and \nthat we can pick the disk $D_{1}^{2}$ in the above construction to be homotopy ribbon so that $W$ is\nsimply-connected. Thus, for $K$ two-bridge, $Z$ is a simply-connected\nmanifold.\n\\end{proof}\n\n\\begin{proof}[Proof of {Theorem~\\ref{thm:sufficient}}]\nThe proof is as follows: first, we modify the branching set $B$ by\nintroducing a singularity of type $\\alpha $ to the embedding of\n$B$ in $X$; next, we construct the desired covering space $Y$ by gluing\ntogether several manifolds by homeomorphisms on their boundaries; we\ncheck that $Y$ is indeed a $p$-fold irregular dihedral cover of $X$ with\nthe specified branching set; lastly, we verify that $Y$ is a\nsimply-connected manifold.\n\nWe begin by modifying the branching set as outlined above. Let\n$S^{2}\\subset S^{4}$ be an embedded two-sphere with a unique singularity\nof type $\\alpha $, constructed as in the proof of {Proposition~\\ref{covers_of_spheres}}.\nLet $y\\in S^{2} \\subset S^{4}$ be any locally flat point with\n$N(y)$ a neighborhood of $y$ not containing the singular point $x$. We\nuse $N(y)$ to form the connected sum of pairs $(X, B)\\# (S^{4}, S^{2})\n= : (X, B_{1})$. By construction, $B_{1}$ is homeomorphic to $B$, is\nembedded in $X$ with a unique singularity of type $\\alpha $ and\nsatisfies $e(B_{1})=e(B)$. Furthermore, we see from the natural\nMayer--Vietoris sequence that $H_{1} (X- B; \\mathbb{Z})\\cong H_{1} (X-\nB_{1}; \\mathbb{Z})$ and the latter group is $\\mathbb{Z}\/2\\mathbb{Z}$ by\nassumption. Hence, $X$ admits a double cover $f: \\hat{X}\\rightarrow X$\nbranched along $B_{1}$.\n\nA useful way to visualize this cover is the following. Since\n$y\\in S^{4}$ is a locally flat point, $B\\cap \\partial N(y)$ is the\nunknot. Now viewing $\\partial N(y)$ as embedded in $B_{1}$, we note that\nthe restriction of $f$ to $f^{-1}(\\partial N(y))$ is a double branched\ncover of the trivial knot, whose total space is again $S^{3}$.\nFurthermore, the pre-images under $f$ of the connected summands\n$(X, B) - B^{4}$ and $(S^{4}, S^{2})-N(y)$ are the double branched\ncovers of those summands. We can thus think of the double branched cover\n$\\hat{X}$ of the pair $(X, B_{1})$ as the union (along $S^{3}$ viewed\nas a double cover of the unknot) of the double branched covers of a\npunctured $(X, B)$ and $(S^{4}, S^{2})-N(y)$. For future use, we denote\nthe restriction of $f$ to the pre-image $\\hat{X}_{0}$ of $X- N(x)$ by\n$f_{0}$,\n\\begin{equation*}\nf_{0}:\\hat{X}_{0}\\rightarrow (X- N(x)).\n\\end{equation*}\n\nNext, consider the irregular dihedral $p$-fold cover $g: Z\\rightarrow\nS^{4}$ of $(S^{4}, S^{2})$ constructed as in\n{Proposition~\\ref{covers_of_spheres}}. For $y$ as above, the restriction\nof $g$ to $g^{-1}(\\partial N(y))$ is the irregular dihedral $p$-fold\ncover of the unknot, which consists of the disjoint union of\n$\\frac{p+1}{2}$ copies of $S^{3}$, $\\frac{p-1}{2}$ of which are double\ncovers and one a single cover. Furthermore, $g^{-1}(S^{4}-N(y))$ is the\nirregular dihedral $p$-fold cover of the pair $(B^{4}, D^{2})$, where\nthe two-disk is singular. The boundary of this dihedral cover consists\nof ${\\frac{p+1}{2}}$ copies of $S^{3}$. Of those, $\\frac{p-1}{2}$\ndouble-cover the complement of the unknot and one is mapped\nhomeomorphically by $g$.\n\nWe now describe the manifold $Y$ which we will show is homeomorphic to\na dihedral cover of $X$ along $B_{1}$. We attach to $g^{-1}(S^{4}-N(y))$\na copy of $\\hat{X}_{0}$ along each boundary component $S^{3}$ which\ndouble-covers the complement of the unknot. Naturally, the attachment\nidentifies the boundary components by a homeomorphism of pairs\n$(S^{3}, S^{1})$, where the second component is the (unknotted)\nbranching set. In the same manner, we also attach a punctured copy of\n$X$ along the boundary component $S^{3}$ which is a cover of index\n$1$. The map\n\\begin{equation*}\nh:= g\\cup _{\\frac{p-1}{2}}f_{0}\\cup 1_{X-N(\\ast )} : Y\\rightarrow X\n\\end{equation*}\nis a branched cover of $(X, B_{1})$. By construction, $h$ satisfies the\nproperty that for all points $z\\in B- x$, if $N(z)$ is a small\nneighborhood of $z$ in $X$ not containing $x$, then $h^{-1}(N(z))$ has\n$\\frac{p-1}{2}$ components of index $2$ and one component of index\n$1$. So $Y$ is the desired dihedral cover. By\n{Theorem~\\ref{thm:necessary}}, the Euler characteristic and signature of\n$Y$ are those determined by the prescribed triple $X, B, \\alpha $.\n(Here, we use the fact that the above construction does not change the\nhomeomorphism type or self-intersection number of~$B$.)\n\nFinally, we observe that $Y$ consists of simply-connected manifolds\njoined together via homeomorphisms on their boundaries. Indeed, $X$ is\nsimply-connected by assumption, and $\\hat{X}$ is simply-connected by\n{Proposition~\\ref{lem:SC_double_cover_exists}}. The irregular dihedral\ncover $Z$ of $S^{4}$ is simply-connected by\n{Proposition~\\ref{covers_of_spheres}}, and, therefore, so is $g^{-1}(S\n^{4}-N(y))$. We conclude that $Y$ is simply-connected, which completes\nthe proof.\n\\end{proof}\n\n\\begin{rem}\nOne can obtain analogous results by varying the hypotheses on the branching set $B$.\nFor instance, if we do not require that our construction result in a\nsimply-connected cover, we can relax the condition that $\\pi _{1}(X-B,\nx_{0})\\cong \\mathbb{Z}\/2\\mathbb{Z}$ and use for our branching set any\nsurface $B$ which represents an even class in $H_{2}(X; \\mathbb{Z})$.\nThis allows us to produce, by introducing any two-bridge slice\nknot as the singularity and by varying the genus of $B$ (see\n{Lemma~\\ref{genus_increase}}), infinitely many dihedral branched covers\nof $S^{4}$, which are easily distinguished by their Euler\ncharacteristic. Furthermore, if $B$ is the boundary union of the cone on a two-bridge knot $\\alpha$ and a homotopy ribbon surface for $\\alpha$, one can construct simply-connected covers of $S^4$ by this method, as done in~\\cite{cahn2017singular}.  \n\\end{rem}\n\nWe can also use the techniques of {Theorem~\\ref{thm:sufficient}} to\nconstruct over a given four-manifold $X$ an infinite family of dihedral\ncovers with the same singularity type on the branching set\n({Theorem~\\ref{thm:odd_indefinite}}). The first step is to establish the\nfollowing Lemma.\n\n\\begin{lem}\n\\label{genus_increase}\nLet $B\\subset X^{4}$ be an oriented surface of genus $g$, PL embedded in\n$X$ and such that $\\pi _{1}(X- B, x_{0})\\cong \\mathbb{Z}\/2\\mathbb{Z}$. Then,\nthere exists a PL embedded oriented surface $C$ of genus $g+1$ in\n$X$ such that $\\pi _{1}(X- C, x_{0})\\cong \\mathbb{Z}\/2\\mathbb{Z}$, and\nsuch that $e(B) = e(C)$.\n\\end{lem}\n\n\\begin{proof}\nLet $T\\subset S^{4}$ be the standard embedding of the two-torus in the\nfour-sphere. We have $\\pi _{1} (S^{4}- T, x_{0}) \\cong \\mathbb{Z}$,\ngenerated by any meridian of $T$ in $S^{4}$.\n\nNow consider the connected sum of pairs $(X, B) \\# (S^{4}, T)$ and let\n$C= B \\#T \\subset X\\# S^{4} \\cong X$. Note that the genus of $C$ is\none higher than that of $B$. Since a meridian $m_{1}$ of\n$T$ in $S^{4}$ becomes identified under the connected sum with a\nmeridian $m_{2}$ of $B$ in $X$, it follows that the fundamental group\nof $ (X- C)$ is isomorphic to $\\langle m_{1}, m_{2} | m_{1}=m_{2}, m\n_{2}^{2} = 0 \\rangle \\cong \\mathbb{Z}\/2\\mathbb{Z}$.\n\nFinally, under the isomorphism of pairs $(X, B) \\# (S^{4}, T) \\cong (X,\nC)$, the class $[C]\\in H_{2}(X; \\mathbb{Z})$ corresponds to the class\n$[B\\#T] \\in H_{2}(X\\# S^{4}; \\mathbb{Z})$. Since $[T] = 0\\in H_{2}(X\n\\# S^{4}; \\mathbb{Z})$, indeed $e(B) = e(C)$. \n\\end{proof}\n\nWe have now done most of the work needed to obtain an infinite family\nof covers over a given base.\n\n\\begin{proof}[Proof of {Theorem~\\ref{thm:construction-slice}}]\nThe first step of our construction is to find a closed surface\n\\mbox{$B\\subset X$}, PL embedded in $X$ and such that $\\pi _{1} (X-B; x_{0})\n\\cong \\mathbb{Z}\/2\\mathbb{Z}$. Since $X$ is simply-connected and its\nsecond Betti number is positive, such a surface exists, as we now show.\nLet $F$ be a closed oriented surface, smoothly embedded in $X$ and such\nthat the maximum divisibility of $[F]$ in $H_{2}(X; \\mathbb{Z})$ is 2.\nThen $H_{1}(X-F; \\mathbb{Z})\\cong \\mathbb{Z}\/2\\mathbb{Z}$. By classical\ntechniques, $F$ can be modified to produce a new surface $F'$, carrying\nthe same homology class as $F$, the fundamental group of whose\ncomplement is abelian, as follows. Since $X$ is simply-connected,\n$\\pi _{1}(X-F)$ is normally generated by a meridian $\\mu $ of $F$. For\nany $g\\in \\pi _{1}(X-B, x_{0})$, the commutator $[\\mu , g\\mu g^{-1}]$ can\nbe killed by performing a finger move on the surface $F$, as shown in\nLemma~1 of~\\cite{casson1986three}. After iterating this move finitely\nmany times, the result is a self-transverse immersed surface $F'$,\nthe fundamental group of whose complement is generated by $\\mu $.\nFinally, self-intersections of $F'$ can be removed by replacing, in a\nsmall neighborhood of any double point, the cone on the Hopf link by an\nannulus. This operation has no effect on the fundamental group of the\ncomplement and produces the desired surface $B$.\n\nThe next step is to further modify the surface to introduce a\nsingularity of the desired type. Following the procedure in the proof\nof {Theorem~\\ref{thm:sufficient}}, we use a two-sphere $S\\subset S\n^{4}$, PL embedded in $S^4$ except for one singularity of type $\\alpha $; next,\nwe construct a $p$-fold irregular dihedral cover of the pair\n$(X, B)\\#(S^{4}, S) \\cong (X, B)$, as in the proof of\n{Theorem~\\ref{thm:sufficient}}.\n\nFixing a knot $\\alpha $ as the singularity type, by\n{Lemma~\\ref{genus_increase}}, we can increase the genus of the branching\nset $B$ to obtain an infinite family of such covers. These covers are pairwise non-homeomorphic and can be\ndistinguished by their Euler characteristics. Using knots for which the values of\n$\\Xi _{p}$ differ, it is possible to obtain covers distinguished by their\nsignatures as well.\n\\end{proof}\n\nAs an immediate consequence of our construction, we have the following.\n\n\\begin{cor}\nLet $(\\sigma , \\chi )$ be a pair of integers which satisfy\nEquations~{(\\ref{eq:chi})} and~{(\\ref{eq:sigma})} for some given $X$, $B$,\n$\\alpha $ and $p$, where $p$ is an odd prime and $\\alpha $ a two-bridge\nslice knot. Then, if $\\chi ' = \\chi + (p-1)k$ for a natural number\n$k$, there exists a manifold $Y'$ which is homeomorphic to a $p$-fold\nirregular dihedral cover of $X$ and satisfies $\\sigma (Y') =\\sigma $,\n$\\chi (Y')=\\chi '$. Moreover, if $\\pi _{1}(X-B, x_{0})\\cong \\mathbb{Z}\/2\n\\mathbb{Z}$, $Y$ is simply-connected.\n\\end{cor}\n\nWe conclude by proving {Theorem~\\ref{thm:odd_indefinite}}.\n\n\\begin{proof}[Proof of {Theorem~\\ref{thm:odd_indefinite}}]\n$(=>)$ If $Y$ is homeomorphic to a dihedral $p$-fold cover of $X$ with\nthe specified branching data, by {Theorem~\\ref{thm:necessary}}, the Euler\ncharacteristic and signature of~$Y$ satisfy Equations~{(\\ref{eq:chi})}\nand~{(\\ref{eq:sigma})} with respect to $B_{1}$ and thus, by assumption, with\nrespect to $B$.\n\n$(<=)$ Assume the Euler characteristic and signature of $Y$ satisfy\nEquations~{(\\ref{eq:chi})} and~{(\\ref{eq:sigma})}. We will construct a branched\ncover of $X$ whose branching set has the specified properties, and we\nwill prove that this cover is homeomorphic to $Y$.\n\nWe follow the steps used in the proof of {Theorem~\\ref{thm:sufficient}}\nto construct a $p$-fold irregular dihedral cover of $X$ branched over\na surface $B_{1}\\cong B$ which is embedded in $X$ with a singularity of\ntype $\\alpha $ and so that $e(B_{1})=e(B)$. Call this cover $Z$. Since\n$\\alpha $ is a two-bridge slice knot, by {Theorem~\\ref{thm:sufficient}}, $Z$ is a simply-connected manifold. We will prove that the\nintersection form of $Z$ is equivalent to that of $Y$.\n\nBeing a dihedral cover of $X$, $Z$ satisfies the equations set forth in\n{Theorem~\\ref{thm:necessary}}, where, again, $B$ and $B_{1}$ can be used\ninterchangeably. By assumption, $Y$ also satisfies these equations,\nso $\\sigma (Y) = \\sigma (Z)$ and $\\chi (Y)=\\chi (Z)$. Since $Y$ is a\nsimply-connected four-manifold, the rank of $H_{2}(Y; \\mathbb{Z})$ is\n$\\chi (Y) - 2$, and the analogous statement holds for $Z$. In other words, the intersection forms of $Y$ and $Z$ have the same\nsignature and rank. The intersection form of $Y$ is odd by assumption.\nThe intersection form of $Z$ is also odd because by construction $Z$ has\na copy of $X$ as a connected summand and $X$ itself is odd. Therefore,\nthe intersection forms of $Y$ and $Z$ have the same signature, rank and\nparity. In particular, both are definite or both are indefinite. If both\nforms are definite, since they arise as intersection forms of smooth\nfour-manifolds, by Donaldson's result~\\cite{donaldson1983application},\neach diagonalizes to $\\pm I_{n}$, where $n=\\chi (Y) - 2=\\chi (Z) - 2$\nand the sign determined by $\\sigma (Y)=\\sigma (Z)$. If both are\nindefinite, we again conclude that they are isomorphic, this time using\nSerre's classification~\\cite{serre1961formes} of indefinite unimodular\nintegral bilinear forms. By~Freedman's classification of\nsimply-connected four-manifolds~\\cite{freedman1982topology}, it follows that $Y$ and $Z$ are homeomorphic.\n\\end{proof}\n\n\n\\section*{Appendix A. Characteristic knots}\\label{appA}\nOur construction of an infinite family of irregular dihedral $p$-fold\ncovers of over a given four-manifold\n({Theorem~\\ref{thm:construction-slice}}) hinges on being able to find\ntwo-bridge slice knots which admit dihedral $p$-fold covers themselves. In this section we prove\nthat, for any odd prime $p$, infinitely many such knots exist. In\nparticular, we exhibit for every $p$ an infinite class of knots for\nwhich the necessary condition ({Theorem~\\ref{thm:necessary}}) for the\nexistence of a dihedral $p$-fold cover over a given base is sharp. As a\nbiproduct, we also illustrate how to find characteristic knots in the\ntwo-bridge case.\n\nRecall that Lisca~\\cite{lisca2007lens} proved that, for two-bridge\nknots, being slice is equivalent to being ribbon. Previously, Casson and\nGordon~\\cite{cass-gor1986cobordism} gave a necessary condition for a\ntwo-bridge knot to be ribbon, and Lamm~\\cite{lamm2000symmetric}\n\\cite{lamm2006symmetric} listed all knots satisfying this condition.\nHe found that for all $a\\neq 0, b\\neq 0$ the knots $K_{1}(a,b) =\nC(2a,2,2b,-2,-2a,2b)$ and $K_{2}(a,b) = C(2a, 2, 2b, 2a, 2, 2b)$ are\nribbon. {Fig.~\\ref{fig:slice_knot}} recalls the notation $C(e_{1}, ... , e_{6})$. In\n{Fig.~\\ref{fig:seifert_surface}} we give a genus $3$ Seifert surface $V$ for the knot\n$\\alpha = C(e_{1},e_{2},e_{3},e_{4},e_{5},e_{6})$. We use the surface\n$V$ for all subsequent computations.\n\nSince two-bridge slice knots play a key role our construction of\ndihedral covers of four-manifolds, we determine the values of the\nparameters $a$ and $b$ for which the knots $K_{i}(a, b)$ admit\nthree-fold dihedral covers.\n\n\\begin{figure}\n\\input{twobridgeKNOT.tex}\n\\caption{The knot $C(e_{1},...,e_{6})$. Each square represents a two-strand\nbraid with only positive or only negative horizontal twists, according to the\nsign of $e_{i}$. The absolute value of $e_{i}$ gives the number of crossings.} \\label{fig:slice_knot}\n\\vspace*{-2pt}\n\\end{figure}\n\n\n\\begin{figure}\n\\input{surface.tex}\n\\caption{A Seifert surface for the knot $C(e_{1},...,e_{6})$, together with the\nset of preferred generators for its first homology group.}\n\\label{fig:seifert_surface}\n\\vspace*{-6pt}\n\\end{figure}\n\n\\begin{prop}\\label{characteristic_knots_p=3}\nA knot of the type $K_{1}(a, b)$ admits an irregular three-fold dihedral\ncover if and only if\n\n(1) $a\\equiv 0 \\mod 3, b\\equiv 2 \\mod 3$ or\n\n(2) $a\\equiv 1 \\mod 3, b\\equiv 1 \\mod 3$.\n\\goodbreak\n\nA knot of the type $K_{2}(a, b)$ admits an irregular 3-fold dihedral\ncover if and only if\n\n(3) $a\\equiv 0 \\mod 3, b\\equiv 1 \\mod 3$ or\n\n(4) $a\\equiv 1 \\mod 3, b\\equiv 0 \\mod 3$.\n\nIn these cases, a curve representing the class $\\beta \\in H_{1}(V;\n\\mathbb{Z})$ is a mod $3$ characteristic knot for the corresponding\n$K_{i}(a,b)$ if and only if there is a choice of orientation on\n$\\beta $ such that, with respect to the basis $\\{ \\omega _{1},\\omega\n_{2},\\omega _{3}, \\omega _{4},\\omega _{5},\\omega _{6} \\}$, we have,\nrespectively,\n\n(1) $[\\beta ]\\equiv (1, 0, 1, 1, -1, 1) \\mod 3$,\n\n(2) $[\\beta ]\\equiv (-1, 1, 1, 0, 1, 1)\\mod 3$,\n\n(3) $[\\beta ]\\equiv (1, 0, 1, -1, 1, 1)\\mod 3$,\n\n(4) $[\\beta ]\\equiv (-1, 1, 1, 1, 0, 1)\\mod 3$.\n\\end{prop}\n\n\\begin{proof}\nLet $V$ denote the Seifert surface for $C(e_{1},e_{2},e_{3},e_{4},e\n_{5},e_{6})$ depicted in {Fig.~\\ref{fig:seifert_surface}}. We think of the $e_{i}$ as being\nchosen so that the knot $C(e_{1},e_{2},e_{3},e_{4},e_{5},e_{6})$ is of\ntype $K_{1}(a, b)$ or $K_{2}(a, b)$. Let $L$ denote the matrix of the\nlinking form on $V$ with respect to the basis $\\{ \\omega _{1},\\omega\n_{2},\\omega _{3}, \\omega _{4},\\omega _{5},\\omega _{6} \\}$. The symmetrized\nlinking form for $V$ in this basis is $L_{V} = L + L^{T}$. It equals:\n\\vspace*{-3pt}\n\\[\n\\left (\n\\begin{array}{c@{\\quad }c@{\\quad }c@{\\quad }c@{\\quad }c@{\\quad }c}\n-e_{1} & 1 & 0 & 0 & 0 & 0\n\\\\\n1 & e_{2} & -1 & 0 & 0 & 0\n\\\\\n0 & -1 & -e_{3} & 1 & 0 & 0\n\\\\\n0 & 0 & 1 & e_{4} & -1 & 0\n\\\\\n0 & 0 & 0 & -1 & -e_{5} & 1\n\\\\\n0 &0 & 0 & 0 & 1 & e_{6}\n\\end{array}\n\\right )\n\\]\n\nIt is sufficient to check that $\\det (L + L^{T})\\equiv 0 \\mod 3$\nprecisely in situations $(1),\\allowbreak ...,(4)$. For instance, in the case\n$C(e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}) = K_{1}(a, b)$, we obtain\n$\\det (L + L^{T}) = -(8ab+2b-1)^{2}$. So we need to solve the equation\n\\begin{equation*}\n8ab+2b-1\\equiv 0\\mod 3.\n\\end{equation*}\nIf $a\\equiv 0$~mod~3, the equation reduces to $2b-1\\equiv 0$~mod~3, so\n$b\\equiv 2$. If $a\\equiv 1$~mod~3, then $b\\equiv 1$~mod~3. If\n$a\\equiv 2$~mod~3, there is no solution. The computations for\n$K_{1}(a, b)$ are equally trivial, so they are omitted.\n\nTo verify that the classes $[\\beta ]\\in H_{1}(V; \\mathbb{Z})$ listed\nrepresent all characteristic knots, it suffices to check that, for\n$a$, $b$ and $\\beta$ as specified, we have $(L + L^{T})\\beta \\equiv 0\\mod 3$ and\nmoreover that the classes $\\beta $ are the unique solutions mod~3 for each pair $(a, b)$. The\narithmetic involved has been left out.\n\\end{proof}\n\nMore generally, we have the following:\n\n\\begin{prop}\n\\label{slice_knots_have_covers}\nLet $p>1$ be an odd prime. There exits an infinite family of integer\npairs $(a, b)$ such that the two-bridge slice knot $K_{1}(a, b)\\subset\nS^{3}$ admits an irregular dihedral $p$-fold cover, and similarly for\n$K_{2}(a, b)$.\n\\end{prop}\n\n\\begin{proof}\nThe case $p=3$ was treated in\n{Proposition~\\ref{characteristic_knots_p=3}}, so assume $p>3$. The\ndeterminant $D_{1}(a, b)$ of the Seifert matrix of the knot\n$K_{1}(a, b)$ is equal to $-(8ab+2b-1)^{2}$. Setting $a\\equiv 0\n\\mod p$, we find that $D_{1}(a, b)\\equiv 0\\mod p$ if and only if\n$2b\\equiv 1\\mod p$. Since $p$ is odd, a solution exists. Another pair\nof solutions is $a\\equiv 8^{-1}\\mod p$ and $b\\equiv 3^{-1}\\mod p$.\n\nSimilarly, we find that the determinant $D_{2}(a, b)$ of the Seifert\nmatrix of the knot $K_{2}(a, b)$ is $(8ab+2a+2b+1)^{2}$. Setting\n$b\\equiv -1\\mod p$, we find that $a (-6)\\equiv{1} \\mod p$. For $p>3$,\nthis gives a solution.\n\\end{proof}\n\nFor any given $p$ and any family of two-bridge slice knots $K_{i}(a, b)$\nwith $a$ and $b$ chosen so that $\\det (L + L^{T})\\equiv 0 \\mod p$, the\nclasses in $H_{1}(V; \\mathbb{Z})$ represented by characteristic knots\nare easily computed as in {Proposition~\\ref{characteristic_knots_p=3}} by\nsolving a system of equations mod~$p$. One can see by direct examination\nthat if $p=3$ each of these homology classes can be realized by an unknot embedded in the interior of $V$. The\nsame methods can be used to find knot types of characteristic knots for\nall $p$.%\n\n\\section*{Appendix B. Computing linking numbers in branched covers}\\label{appB}\n\nLet $\\alpha \\subset S^{3}$ be a knot, and let $f: M\\to S^{3}$ be a cover\nbranched along $\\alpha $, arising from a presentation $\\psi : \\pi _{1}(S\n^{3}-\\alpha , x_{0})\\to S_{n}$. The linking numbers (when defined)\nbetween the various components of $f^{-1}(\\alpha )$ constitute a subtle\nknot invariant studied extensively by Hartley and Murasugi\n\\cite{hartley1977covering}, Bankwitz and Schumann\n\\cite{bankwitz1934viergeflechte}, Laufer \\cite{laufer1971some} and\nPerko \\cite{perko1974classification}, \namong others. Further applications of linking numbers in dihedral covers of knots were found by Cappell and Shaneson~\\cite{CS1975invariants} and Litherland~\\cite{litherland1980formula}.\n\nIn his undergraduate thesis \\cite{perko1964thesis}, Perko detailed\na procedure, going back to Reidemeister~\\cite{reidemeister1929knoten}, for computing linking numbers between branch curves. His\nmethod is, to this day, the most efficient and general algorithm known\nfor computing these numbers. We give a very short summary of this classical \nmethod for computing linking numbers in a branched cover. We intend to\nprovide just enough detail to be able to describe a generalization of\nthese ideas which will allow us to calculate the linking numbers of other\ncurves, as needed for evaluating the component of $\\Xi _{p}(\\alpha )$\nwhich is expressed in terms of linking. Readers interested in the\nspecifics needed to carry out the procedure can find them in~\\cite{perko1964thesis} or~\\cite{cahn2016linking}.\n\n\\medskip\n\\textit{Perko's procedure for computing linking numbers between branch\ncurves in a branched cover $f: M\\to S^{3}$ with branching set\n$\\alpha $:}\n\\begin{enumerate}\n\\item\nUse a diagram for $\\alpha $ to endow $S^{3}$ with a cell structure. The\ntwo-skeleton is the cone on $\\alpha $, and there is a single\nthree-cell.\n\\item\nEndow the cover $M$ with a cell structure as follows. The cells are the\npre-images $f^{-1}(e_{k}^{j})$ of the various cells in $S^{3}$. The\nattaching maps are determined by the action of the meridians of\n$\\alpha $ on the interiors of the cells.\n\\item\nCompute the boundaries of all two-cells of $M$. This step is non-trivial\nfor two-cells whose boundary contains one-cells corresponding to\nover-arcs in the knot diagram. Such two-cells will accrue\nadditional boundary components determined by the action of meridians of\n$\\alpha $ on the three-cells.\n\\item\nSolve a system of linear equations to determine, for each component\n$\\alpha _{i}$ of $f^{-1}(\\alpha )$, a~two-chain with boundary\n$\\alpha _{i}$, if such a two-chain exists.\n\\item\nFor each pair $(\\alpha _{i}, \\alpha _{j})$, examine the signed\nintersection numbers of $\\alpha _{i}$ with a two-chain, found in (4),\nwhose boundary is $\\alpha _{j}$. This gives $lk(\\alpha _{i}, \\alpha _{j})$.\nWe remark that, in practice, the intersection number of any one-cell with\nany two-cell is trivial to read off from the data examined in order to\ncomplete (3), so this final step of the computation poses no\ndifficulty.\n\\end{enumerate}\n\nIn order to compute the linking numbers of other curves in $M$, we\nintroduce an appropriate subdivision of the cell structure described above. Consider a curve $\\gamma \\subset (S^{3}-\\alpha )$ whose lifts to\n$M$ are of interest. We use the cone on $\\alpha \\cup \\gamma $ to form\nthe two-skeleton of $S^{3}$. In order to lift this new cell\nstructure to a cell structure on $M$, we treat $\\gamma $ as a\n``pseudo-branch curve'' of the map $f$. That is, we think of the\nhomomorphism $\\pi _{1}(S^{3}-\\alpha )\\to S_{n}$ as a homomorphism\n$\\pi _{1}(S^{3}-(\\alpha \\cup \\gamma ))\\to S_{n}$ in which meridians of\n$\\gamma $ map to the trivial permutation. Naturally, this can be done\nfor multiple curves $\\gamma _{i}$ simultaneously. In this set up,\nlinking numbers can be computed by following steps (3), (4) and (5)\nabove. The above procedure is carried out in~\\cite{cahn2016linking}, and\na computer algorithm for performing linking number calculations is\nprovided. \\\\\n\n\n\\textbf{Acknowledgements. }  The author is deeply indebted to her PhD\nadviser, Julius Shaneson, for his support, encouragement and insight.\nThanks to humor, my rock. Mohamed-Ali Belabbas was also a sustained\nsource of helpful conversations. The anonymous referee made many\nvaluable suggestions. \\\\\n\n\n\\bibliographystyle{amsplain}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Supplemental Material}\n\nThis supplemental material conveys additional results on the weak coupling regime, the effects of a harmonic trap, correlation functions and mutual information in the strong coupling phase separated regime, and a derivation of the spinful $PXP$ Hamiltonian at half and full filling.\n\n\n\n\\section{Weak coupling Rydberg dressed regime}\n\nTo obtain the weak-coupling Hamiltonian, see Eq. \\eqref{eqn:cos} and above it, we linearize the tight-binding dispersion around $\\pm k_F$. Without interactions, the bosonized Hamiltonian takes the form:\n\\begin{equation}\n    H^{nonint}_\\eta = \\frac{1}{2\\pi} \\int dx \\left[ v (\\pi \\Pi_\\eta (x))^2 + v (\\nabla \\phi_\\eta(x))^2 \\right],\n\\end{equation}\nwhere $v_F= \\frac{d \\varepsilon}{dk}_{k=k_F}=\\frac{2t}{a}\\sin(k_F a)$, where $a$ is the lattice constant of the order of $\\alpha$. Next we transform the lattice interaction Eq.~\\eqref{eq:rydbergint} to the momentum space:\n\\begin{equation}\n  V_d(q) = \\frac{1}{N}\\sum_n \n  e^{-i q a n } V_d(a n).\n\\end{equation}\nOne can write $V_d(a n) = \\int \\frac{d k}{2\\pi} e^{i k a n} V_C(k) = \\frac{1}{L}\\sum_{k=\\frac{2\\pi l}{L}} e^{i k a n} V_C(k)|_{L\\to\\infty}$, where $V_C(k) = \\int dx V_d(x) e^{ikx}$ is defined in the continuum (C) limit $a\\rightarrow 0$. Using the Poisson summation formula $\\sum_n e^{i n g} = N \\sum_m \\delta_{g-2\\pi m}$ one obtains:\n\\begin{equation}\n  V_d(q) = \\frac{1}{L}\\sum_m V_C\\left(q+\\frac{2\\pi m}{a}\\right).\n\\end{equation}\nAs $V_C(k)$ decays as $e^{-|k|r_c\/2}$ for $r_c\\gg a$ the $m\\neq0$ terms can be neglected and $V_d(q)\\approx \\frac{1}{L} V_c(q)$ used in the main text is obtained.\n\n\n\n\n\n\nApart from the spin-spin correlations shown in Fig.~\\ref{fig:weak}(a), we also calculate the density-density correlation \n$\\langle n_{L\/2 \\pm x} n_{L\/2} \\rangle$\nfor the two (positive and negative backscattering) phases in Fig.~\\ref{fig:rhorho}. \nAs the charge sector is always gapless the density-density correlation decays as power-law in both cases.\nThe density-density and spin-spin power-law exponents in the repulsive backscattering region are both $-(K_\\rho + K_\\sigma)$. \nWe extracted the exponent for $N_p = 120$ from the spin-spin correlator (dashed line in Fig.~\\ref{fig:weak}(a)) and plotted a dashed line with the same slope in Fig.~\\ref{fig:rhorho}. \nOne can confirm that the power (slope) of the two matches well in the large $x$ limit.\n\n\\begin{figure}[b!]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{rhocorr4.png}\n    \\caption{The density-density correlation function for $N_p = 70$ (blue) and $N_p = 120$ (yellow) particles in a $L = 200$ chain.\n    All parameters are identical to that of Fig.~\\ref{fig:weak}.\n    The density-density correlation follows a power-law scaling on both cases. \n    }\n    \\label{fig:rhorho}\n\\end{figure}\n\n\\section{Effect of trap potential}\n\n\nThe calculations in the main text focus on an optical lattice but ignores the presence of the harmonic trap potential.\nIn real experiments, the harmonic trap is inevitable and will affect the previously shown results. While we expect this to have a weak effect on the weak coupling phases, we need to confirm that the phase separated regime in Fig.~\\ref{fig:strong}(b) survives. We include a harmonic trap by adding to the Hamiltonian a term $\\sum_{r,\\sigma} V_r c_{r\\sigma}^{\\dag}c_{r\\sigma}$ with $V_r = \\frac{1}{2} m\\omega^2 r^2$.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.49\\linewidth]{trapN1.png}\n    \\includegraphics[width=0.49\\linewidth]{trapEE1.png}\n    \\includegraphics[width=0.49\\linewidth]{trapN2.png}\n    \\includegraphics[width=0.49\\linewidth]{trapEE2.png}\n    \\caption{\n    Left accumulated particle $\\mathcal{N}_L(x)$ as a function of position and the bipartite entanglement entropy in a $L = 400$ chain in the presence of a harmonic trap potential.\n    (a) $N_p = 167$ and $\\frac{1}{2} m\\omega^2 x_{\\textrm max}^2 = t$, (b) $N_p = 163$ and $\\frac{1}{2} m\\omega^2 x_{\\textrm max}^2 = 2t$.\n    The yellow dashed lines are the linear fit near the center ($x = 200$) and the deviation from this line indicates different filling (and thus different phase) away from the center.\n    }\n    \\label{fig:trap}\n\\end{figure}\n\nHere, we take the experimental setup in Ref.~\\onlinecite{Guardado-Sanchez2021} with $t = h \\times 1.7$ kHz and $a = 752$ nm. \nFig.~\\ref{fig:trap} is the case of $N_p = 167, 163$ particles in a $L=400$ chain with a trap potential $\\frac{1}{2} m\\omega^2 x_{\\textrm max}^2 = t, 2t$, respectively with $x_{\\textrm max}=L\/2$.\nThis corresponds to a trap frequency of order $10^2$Hz, which is safely larger than the typical harmonic trap. \n\nIt is clear that the slope (average particle density) at the center is different from that of the two boundaries. \nThe bipartite entanglement entropy also shows distinct behavior between center and boundaries, and the entanglement is low at the center regions, following the trend seen in Fig.~\\ref{fig:strong}(c). Here, the center region is the magnetic domain wall and the charge like puddles are to its side. \nFrom these results, while the overall structure of the phase separated state is modified by the trap \nwe conclude that the phase separated regime of the model will not be prohibited by the harmonic trap in experiments and we expect can be directly measured. \n\n\n\\section{Correlations in the phase separated regime}\n\n\n\\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[width=0.32\\linewidth]{PS_rhorho.png}\n    \\includegraphics[width=0.325\\linewidth]{PS_sigsig.png}\n    \\includegraphics[width=0.30\\linewidth]{MI.png}\n    \\caption{\n    Correlation functions for the same parameters in Fig.~\\ref{fig:strong}(b,c). \n    ($L=400$, $N_p=170$ in the strong coupling regime) \n    (a) The density-density correlation function where $x_0 = 87$ [blue], $212$ [yellow], $324$ [green] which are the centers of the $c=1$ puddles in Fig.~\\ref{fig:strong}(c).\n    (b) The spin-spin correlation function where $x_0 = 154$ [blue], $263$ [yellow] which are in between the $c=1$ puddles.\n    (c) The mutual information (MI) between two points $x$, $y$ in the system.\n    The MI shows three puddles which corresponds to the three $c=1$ regions, and the information spread is blocked by the magnetized region. \n    }\n    \\label{fig:MI}\n\\end{figure*}\n\nIn this section, we further investigate the phase separated state in the context of correlations to provide additional results that imply the presence of Luttinger liquid puddles.\nIn the main text Fig.~\\ref{fig:strong}(c), we claimed that the three regions with $\\bullet$$\\bullet$$\\circ$$\\circ$$\\circ$ patterns have a central charge of $c=1$ which results from the gapless charge degrees of freedom and is inherited from the weak coupling limit of the problem. \nWe check this directly calculating the density-density correlation function and observing its functional form. \n\nIn Fig.~\\ref{fig:MI}(a) we plot the density-density correlation function for the same parameters in Fig.~\\ref{fig:strong}(b,c), where the center $x_0$ is chosen as the center of the three $c=1$ regions. \nOne can observe slow power-law decay of the correlation, although the exponent is hard to extract due to the small size of each region.\nIn contrast, the spin-spin correlation function decays exponentially, even in the magnetized region. \nIn Fig.~\\ref{fig:MI}(b) shows the fast decay of such correlation, where $x_0$ is the center of the magnetized region (in between the $c=1$ regions).\n\nFinally, we plot the mutual information (MI) between two points $x$, $y$ in the system. \nThe MI between two subsets $A$ and $B$ of the system is defined as~\\cite{Amico08}:\n\\begin{align}\n    \\mathcal{I}(A, B) = \\mathcal{S}(A) + \\mathcal{S}(B) - \\mathcal{S}(A\\cup B),\n\\end{align}\nwhere $\\mathcal{S}(A)$ is the entanglement entropy (or von Neumann entropy) introduced in the main text.\nFig.~\\ref{fig:MI}(c) plots the MI between all pair of points in the system, and shows that the MI is large only between points within the same $c=1$ region. \nThis shows the natural result that mutual information is large in gapless regions, and also the interesting feature that the gapped regions serve as a barrier of information spread between the gapless $c=1$ puddles.\n\n\\section{Bosonization analysis at strong coupling}\n\n\nWe now present details of the bosonization discussion regarding phases for arbitrary commensurate filling. For a filling equal to $\\frac{p}{q}$, $p,q$ being mutually prime integers, an additional (umklapp) term appears in the low-energy theory, corresponding to simultaneous scattering of $q$ fermions from $-k_F$ to $k_F$. This can be written as \\cite{schulz_mott} \n\\begin{equation}\n    H_{um}=U_{um}\\int dx \\cos(q\\sqrt{2} \\varphi_\\rho) \\cos^q(\\sqrt{2}\\varphi_\\sigma)\n\\end{equation}, where for weak coupling $U_{um}\\sim U (U\/t)^{q-2}$ \\cite{schulz_mott,Giamarchi}. For even $q$, the most relevant part of $\\cos^q(\\sqrt{2}\\varphi_\\sigma)$ is just a constant, such that $U_{um}$ is relevant for $K_\\rho<4\/q^2$, leading to a charge gap in that case. Apart from half filling, this implies that the density degree of freedom can remain gapless below a threshold value of $U$. For odd $q$, $\\cos(\\sqrt{2}\\varphi_\\sigma)$ has to be kept. At weak coupling, if $V_C(2k_F)<0$ this term is fixed $\\cos(\\sqrt{2}\\varphi_\\sigma)=\\pm1$ and the same condition for $K_\\rho$ is valid , while for $V_C(2k_F)>0$ one can use the fixed-point value $K_\\sigma=1$ \\cite{schulz_mott} such that $K_\\rho< 3\/q^2$ is required. In the latter case, a gap will open simultaneously in the density and spin excitations. The regions of $1\/5$ filling of Fig. \\ref{fig:strong} (b,c) actually conform to weak coupling expectations. In that case $k_F r_c\\approx 5$ such that $V_C(2k_F)\/V_0 \\approx - 0.16$ and a spin gap is expected, while the density gap requires strong interactions $K_\\rho<0.16$. Note that the observed strong modulation of density occurs at $2k_F$, same as for a weak-coupling density wave. The \n$\\odot$%\n$\\odot$%\n$\\circ$%\n$\\circ$%\n$\\circ$%\n$\\circ$ regions, on the other hand, has filling $1\/4$ with a less stringent requirement for the charge gap $K_\\rho<0.25$, allowing to understand the development of a charge gap there. While in weak coupling, a spin gap would have been also expected due to $V_C(2k_F)\/V_0\\approx-0.026$, its absence points to inapplicability of the continuum weak coupling analysis for $1\/4$ filling here, possibly due to extremely small value of  $V_C(2k_F)\/V_0$ and finite size effects. The above arguments also suggest that for 1\/6 filling of Fig.~\\ref{fig:strong}(a), there will be a spin gap $V_C(2k_F)\/V_0\\approx-0.2$ and no charge gap ($K_\\rho<0.11$ is required), consistent with the numerical result. The charge modulation has the period $6$ (see Fig.~\\ref{fig:strong2}(a)), corresponding to the usual $2k_F=2\\pi\/6$ Luttinger liquid correlations \\cite{Giamarchi}, in contrast to the period $3$ ($4k_F$) modulation expected in the insulating state \\cite{schulz_mott,Giamarchi}.\nThe bipartite entanglement entropy (see Fig.~\\ref{fig:strong2}(b)) also shows clear $c=1$ scaling. \n\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.49\\linewidth]{rhocorr5.png}\n    \\includegraphics[width=0.49\\linewidth]{PS_EE3.png}\n    \\caption{\n    (a) The power-law scaling density-density correlation for the system in Fig.~\\ref{fig:strong}(a), $N_p = 140$. (b) The bipartite entanglement entropy as a function of the cut position. The solid line is the CFT scaling form (Eq.~\\eqref{eq:ee}) with $c=1$. \n    }\n    \\label{fig:strong2}\n\\end{figure}\n\n\\section{Optical tweezers and the $PXP$ limit}\n\nIn the limit of no tunneling (i.e. focusing on the optical tweezer set up as opposed to an optical lattice) and strong coupling we determine the effective many-body Hamiltonian, recovering the $PXP$ model in the limit of a fully filled Fermi sea and identify a magnetic $PXP$ model with a direct exchange interaction in the limit of half-filling.\n\n\nThe $PXP$ limit assumes a strong mixing of ground and Rydberg state $\\Omega \\gg \\Delta$. In this case, the dominant interaction is the repulsion between adjacent Rydberg atoms, that is also larger than $\\Omega$. Essentially, the interaction forbids atoms in Rydberg state to occupy neighboring sites. For spinless bosons, Rydberg Hamiltonian reduces then to the $PXP$ model, $H = \\sum_i P_{i-1}X_i P_{i+1}$~\\cite{sun2008,olmos2012,turner2018}. \n$P_i = (1-Z_i)\/2$ is a projector to the ground state. \nThe typical $PXP$ model studied intensively in the context of quantum many-body scars and slow dynamics focused on bosonic Rydberg atoms.\nHere, we consider fermionic Rydberg atoms which introduces the spin degrees of freedom to the original problem. \n\nFirst, consider a chain of Rydberg atoms with two particles (spin up and down) per site. \nThe $X_i$ operator connects the $|g\\uparrow, g\\downarrow\\rangle$ state with the $(|R\\uparrow,g\\downarrow\\rangle + |g\\uparrow,R\\downarrow\\rangle)\/\\sqrt{2}$ state and thus this model reduces to the original $PXP$ model.\nThis is reasonable in the sense that each site has two fermions which can be considered as a bosonic degree of freedom. \n\nNext, we consider a Rydberg atom chain with the filling of one particle per site. Neglecting the coupling of the laser to the spin degree of freedom (which is appropriate for $s$-like states), the Hamiltonian reduces to the original $PXP$ model with additional spin degeneracy, i.e.\n\\begin{align}\n    H_{PXP}^{spin} = \\sum_i P_{i-1} X_i P_{i+1},\n\\end{align}\nwhere $X = | R \\uparrow\\rangle\\langle g\\uparrow| + | R \\downarrow\\rangle\\langle g\\downarrow| + \\textrm{h.c.}$ is the Pauli matrix in the $eg$-space, $P = | g \\uparrow\\rangle\\langle g\\uparrow| + | g \\downarrow\\rangle\\langle g\\downarrow|$ is the projection to the ground state.\n\n\n\nThe spin degeneracy, however, can be lifted by interactions due to the exchange process. \nBoth exchange and supexchange interactions depend on wavefunction overlap \\cite{auerbach2012interacting} and can be of the same order. Therefore, one can expect that: (1) they decrease with distance fast (2) they are larger for the Rydberg state.\nAs the Rydberg interaction energy prohibits nearest neighbor atoms to be both in the Rydberg state, the next possibility is to either have spin-spin interactions with a neighboring ground-state atom, or a next-neighbor Rydberg atom. Combining both contributions, the spin exchange term can be written as:\n\\begin{equation}\n\\begin{gathered}\n    H_{exch}^{spin} = \\sum_i  J_{Rg} \\vec{S}_i \\cdot \\vec{S}_{i+1} (P_i Q_{i+1} + Q_i P_{i+1}) \n    \\\\\n    + J_{RR} \\vec{S}_i \\cdot \\vec{S}_{i+2} Q_i Q_{i+2},\n    \\end{gathered}\n\\end{equation}\nwhere $Q = 1 - P$.\n\n\nFor the direct exchange mechanism one can provide the following estimates for the couplings $J_{Rg}$ and $J_{RR}$. The Rydberg-Rydberg exchange coupling takes the form\\cite{auerbach2012interacting} :\n\\begin{equation}\n\\begin{gathered}\n     J_{RR} = \\frac{1}{2} \\int d^3 {\\bf r} d^3{\\bf r}' V_R(|{\\bf r} - {\\bf r}'|) \\psi^*_R({\\bf r}) \\psi_R({\\bf r}')\\times \n     \\\\\n     \\times\\psi_R^*({\\bf r}'+2 a \\hat{x} \\bf) \\psi_R({\\bf r}+2 a \\hat{x} \\bf),\n     \\end{gathered}\n\\end{equation}\nwhere $\\hat{x}$ is a unit vector along the chain direction and $V_R(|{\\bf r} - {\\bf r}'|) \\propto 1\/|{\\bf r} - {\\bf r}'|^6$, and $\\psi_{g,R}({\\bf r})$ correspond to ground and excited state atomic wavefunctions. One observes then that the integral is dominated by ${\\bf r} \\approx {\\bf r}'$ and is determined by the overlap between Rydberg wavefunctions displaced by $2a$. As the size of the Rydberg wavefunction scales with principal quantum number as $n^2$, the integral can be estimated to scale (taking the integrand value in the middle between two atoms) as $\\sim \\exp\\left( - 2 \\frac{2a}{a_B^* n^2}\\right)$, where $a_B^*$ is the Bohr radius for the atom. \n\nThe Rydberg-ground exchange coupling is determined by:\n\\begin{equation}\n\\begin{gathered}\n     J_{Rg} = \\frac{1}{2} \\int d^3 {\\bf r} d^3{\\bf r}' V_{Rg}(|{\\bf r} - {\\bf r}'|) \\psi^*_R({\\bf r}) \\psi_R({\\bf r}')\\times\n     \\\\\n     \\times\\psi_g^*({\\bf r}'+a \\hat{x} \\bf) \\psi_g({\\bf r}+ a \\hat{x} \\bf),\n     \\end{gathered}\n\\end{equation}\nwhere $V_{Rg}(|{\\bf r} - {\\bf r}'|)$ is the interaction between Rydberg and ground state atom, that is likely to be much weaker than the Rydberg interaction. The scaling of this integral with $n$ can be estimated analogously to be $\\sim \\exp\\left( - \\frac{a}{a_B^* n^2}- \\frac{a}{a_B^*}\\right)$. \nUnfortunately,  in typical tweezer setups \\cite{bernien2017} $a\\gg a_B^* n^2$, suggesting that the exchange interactions are likely quite small for the current set ups.\n\n\n\n\\end{appendix}\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGiven a pointed space $X$, denote by $\\mathcal E(X)$ the group of\n(based) self homotopy equivalences, i.e., the group of\nautomorphisms of $X$ in the pointed homotopy category. From now on\nwe shall consider connected complexes of finite type $X$ which are\neither finite or with finitely many non trivial homotopy groups.\nWe denote by $\\dim X=N$ its topological or homotopical dimension.\nUnless explicitly stated otherwise, all  spaces will be of this\nkind.\n\nAlthough the computation of  $\\mathcal E(X)$ is known to be a hard task,\nthere are two classical and key results that impose to this group\nimportant structural constraints:\n\nOn one hand, a theorem of Sullivan  \\cite[Theorem 10.3]{su} and\nWilkerson   \\cite[Theorem B]{wi} states that $\\mathcal E(X)$ is finitely\npresented. This was originally proved for simply connected spaces\nand later on generalized to virtually nilpotent spaces by Dror,\nDwyer and Kan \\cite[Theorem 1.1]{ddk}. The main step in the proof\nis to show that $\\mathcal E(X_\\mathbb Q)$ is an algebraic group and that\n$\\mathcal E(X)$ is commensurable with an arithmetic subgroup of\n$\\mathcal E(X_\\mathbb Q)$. As a consequence, it can be shown that there exists\na finite bound for the finite orders of elements of $\\mathcal E(X)$.\n\nOn the other hand we have the following theorem  due to Dror and\nZabrodsky:\n\n\\begin{theorem}\\label{principal}{\\em \\cite[Theorem B]{d-z}} Let $G$ be a subgroup of\n$\\mathcal E(X)$ which acts nilpotently on $\\pi_{\\le N}(X)$. Then $G$ is\nitself nilpotent. In particular, $\\mathcal E_{ \\sharp }^m(X)$ is nilpotent.\n\\end{theorem}\n\nRecall that, for $0\\le m\\le\\infty$, $\\mathcal E_{ \\sharp }^m(X)$ is the\ndistinguished subgroup   of $\\mathcal E(X)$ formed by those classes\ninducing the identity on the homotopy groups up to $m$. In other\nwords, $$ \\mathcal E_{ \\sharp }^m(X)=\\text{ker}\\bigl(\\mathcal E(X)\\longrightarrow\n\\Pi_{i\\le m}\\text{aut}\\,\\pi_iX\\bigr). $$  If $\\dim X=N$ we shall\ndenote  $\\mathcal E_{ \\sharp }^N(X)$ simply by  $\\mathcal E_{ \\sharp }(X)$.\n\n\n  Here, we present a slightly different proof for this well known result\nin which we use a broader study of self homotopy equivalences in\nthe homotopy category ${\\cal L}^*$ of (based) spaces with local\ncoefficients. Recall (see \\cite[Chap.VI]{white} for instance) that\nobjects in this category are pairs $(X,{\\cal M})$ in which $X$ is a\n(based) topological space and ${\\cal M}=\\{M_x\\}_{x\\in X}$ is a local\ncoefficient system in $X$. On the other hand, a morphism\n$(f,\\Theta)\\colon (X,{\\cal M})\\to(Y,{\\cal H})$ is a pair formed by a\nbased map $f\\colon X\\to Y$ and a morphism $\\Theta\\colon\nf^*{\\cal H}\\to{\\cal M}$  of local coefficient. By $f^*{\\cal H}$ we denote,\nas usual, the local coefficient system on $X$ induced by $f$,\ni.e., $(f^*{\\cal H})_x=H_{f(x)}$. For each $x\\in X$ we shall denote\nby $\\Theta_x\\colon H_{f(x)}\\to M_x$ the corresponding group\nmorphism at $x$. After considering the appropriate homotopy\nnotion, one obtains the homotopy category   ${\\cal L}^*$. The\ngroup of self homotopy equivalences of an object $(X,{\\cal M})\\in\n{\\cal L}^*$ shall be denoted by   ${\\mathcal E}(X;\\calm)$. Then, we prove:\n\n\n\n\n\n\n\\begin{theorem}\\label{local} Let $X$ be a finite Postnikov piece\nand let  $G\\subset {\\mathcal E}(X;\\calm)$ be a subgroup which acts nilpotently\non both $\\pi_*(X)$ and  ${\\cal M}$. Then, $G$ acts nilpotently on $H^*(X;{\\cal M})$.\n\\end{theorem}\n\n\nAt the sight of Theorem \\ref{principal}, and taking into account\nthe bound of finite orders of elements of $\\mathcal E(X)$ plus the\nexistence of a ``fracture lemma\" for this group (see \\cite[Theorem\n8.2]{Rutter}), it has been of interest to study whether  $\\mathcal E_{ \\sharp }(X)$\nsatisfies the same structural restrictions when taking\n$p$-localization, $p$-completion or considering $\\mathcal E_{ \\sharp p }(X)$. This\ndenotes the subgroup of $\\mathcal E(X)$ formed by those classes which\ninduce the identity on the homotopy groups of $X$ with\ncoefficients on $\\mathbb Z\/p$, up to the dimension of $X$, i.e., $$\n\\mathcal E_{ \\sharp p }(X)=\\text{ker}\\bigl(\\mathcal E(X)\\longrightarrow \\Pi_{\\le\nN}\\text{aut}\\,\\pi_i(X;\\mathbb Z\/p)\\bigr). $$ As examples of this, we\nmention two interesting results for a given nilpotent space $X$:\nMaruyama proved \\cite[Theorem 0.1]{ma} that\n$\\mathcal E_{ \\sharp }(X)_{(p)}=\\mathcal E_{ \\sharp }^N(X_{(p)})$ while, on the other hand,  M{\\o}ller\nshowed \\cite[Theorem 4.3]{mo} that\n$$\\mathcal E_{ \\sharp }(X_{\\mathbb Z_p})=Ext(\\mathbb Z\/p^\\infty,\\mathcal E_{ \\sharp }(X))=\n\\mathcal E_{ \\sharp }(X_p{\\displaystyle\\hat{}}). $$  Here and henceforth,\n$(-)_{\\mathbb Z_p}$ denotes $H_*(-;\\mathbb Z\/p)$-localization while\n$(-)_{(p)}$ and $(-)_p{\\displaystyle\\hat{}}$ are the classical\nlocalization and completion on the prime $p$.\n\nIn this paper we plan to continue this investigation extending\nTheorem \\ref{principal} above, considering a subgroup of $\\mathcal E(X)$\nwhich acts nilpotently in the homotopy groups of $X$ localized,\ncompleted or with coefficients in $\\mathbb Z\/p$. Concerning this purpose\nwe prove:\n\n\n\n\\begin{theorem}\\label{dos} Assume that $\\pi_1(X)$ is a nilpotent group and  let $G$ be a subgroup of $\\mathcal E(X)$ which acts\nnilpotently on $\\pi_{\\le N}(X)_{(p)}$, for $p$ any prime number\nand $0$. If the nilpotency orders of all these actions are bounded\nby a fixed integer, then G is nilpotent.\n\\end{theorem}\n\\begin{remark} {\\em Observe that in the theorem above the condition of $\\pi_1(X)$ being nilpotent is essential. Otherwise,\nchoose any finite simple group $G$ which is known to be\ngenerically trivial, i.e.,  $G_{(p)}=\\{1\\}$ for $p$ any prime\nnumber or zero. On the other hand observe that the map $G\\to \\text{aut}\\,\nG$ given by inner automorphisms is a monomorphism. Indeed, its\nkernel is the center of $G$ which is trivial since $G$ is simple.\nThis inclusion renders the non nilpotent group $G$ as a subgroup\nof $\\mathcal E\\bigl(K(G,1)\\bigr)$ which acts nilpotently in the\nlocalized homotopy group.\n\nIn a similar way, we may even produce an example of a solvable,\nnon nilpotent group, of homotopy equivalences acting nilpotently\nin the localized homotopy groups of the space. Consider the\nsymmetric group $\\Sigma_3$ and observe that\n${\\Sigma_3}_{(2)}=\\mathbb Z\/2$ while ${\\Sigma_3}_{(p)}=1$ for $p\\not=2$.\nAgain, $\\Sigma_3\\subset\\mathcal E\\bigl(K(\\Sigma_3,1)\\bigr)$ is a\nsolvable non nilpotent group acting nilpotently on any\n${\\Sigma_3}_{(p)}$.}\n\\end{remark}\n\nA more subtle and slightly  different situation is given when\nconsidering nilpotent actions of subgroups of self homotopy\nequivalences on the Frattini factor of the homotopy groups. Recall\nthat given a group $G$, the Frattini subgroup $\\Phi(G)$ is the\nintersection of all maximal proper subgroups of $G$. The quotient\n$G\/\\Phi(G)$ is called the Frattini factor.\n\n\\begin{theorem}\\label{uno} Assume that\n$\\pi_{\\le N}(X)$ is a finite nilpotent group and let $G$ be a\nsubgroup of $\\mathcal E(X)$ which acts nilpotently on $\\pi_{\\le\nN}(X)\/\\Phi\\bigl(\\pi_{\\le N}(X)\\bigr)$. Then, G is nilpotent.\n\\end{theorem}\n\nIn particular, taking into account that for an abelian $p$-group\n$G$ its Frattini factor is precisely $G\\otimes\\mathbb Z\/p$, we obtain\nthe following:\n\n\\begin{corollary}\\label{nuevo} Assume that\n$\\pi_{\\le N}(X)$ is a finite abelian group and let $G$ be a\nsubgroup of $\\mathcal E(X)$ which acts nilpotently on $\\pi_{\\le\nN}(X)\\otimes\\mathbb Z\/p$ for any prime $p$. Then, G is\nnilpotent.\\hfill$\\square$\n\\end{corollary}\n\nNotice that, by the Universal Coefficients Theorem for homotopy,\n$\\pi_*X\\otimes\\mathbb Z\/p=\\text{Ext}(\\mathbb Z\/p,\\pi_*X)$ is a subgroup of\n$\\pi_*(X;\\mathbb Z\/p)$. Hence, as an immediate consequence of Corollary\n\\ref{nuevo} above we get:\n\n\\begin{corollary}\\label{coeficientes} Assume that\n$\\pi_{\\le N}(X)$ is a finite abelian group and let $G$ be a\nsubgroup of $\\mathcal E(X)$ which acts nilpotently on $\\pi_{\\le\nN}(X;\\mathbb Z\/p)$ for  any prime $p$. Then, G is\nnilpotent.\\hfill$\\square$\n\\end{corollary}\n\n\n\n\nHaving studied the nilpotency of a general subgroup of $\\mathcal E(X)$,\nwe  now focus on the group $\\mathcal E_{ \\sharp p }(X)$ and give necessary conditions\nfor it to be nilpotent.\n\n\\begin{theorem}\\label{tres}  Let $X$ be a space for which $\\pi_{\\le N}(X)$ is a finite abelian $p$-group. Then\n$\\mathcal E_{ \\sharp p }(X)$ is nilpotent and $\\mathcal E_{ \\sharp p }(X)\/\\mathcal E_{ \\sharp }(X)$ is a finite $p$-group.\n\n\\end{theorem}\n\n\\begin{theorem}\\label{cuatro}\nLet $X$ be a space for which $\\pi_{\\le N}(X)$ is a finitely\ngenerated abelian group. Then $\\cap_{p\\,\\,\\text{prime}}\\mathcal E_{ \\sharp p }(X)$ is\nnilpotent.\n\\end{theorem}\n\n\\begin{remark}\\label{importante}  {\\em Observe that in general\n$\\mathcal E_{ \\sharp p }(X)$ is bigger than $\\mathcal E_{ \\sharp }(X)$. For instance consider\n$X=K(\\mathbb Z\/{p^r},n)$, $r,n\\ge 2$. Obviously $\\mathcal E_{ \\sharp }(X)=\\{1\\}$, while\nthe automorphism $\\rho$ of $\\mathbb Z\/{p^r}$ given by\n$\\rho(1)=p^{r-1}+1$ induces a non trivial element of $\\mathcal E_{ \\sharp p }(X)$.\nIndeed, by the Universal Coefficients Theorem for homotopy, $$\n\\pi_*(X,\\mathbb Z\/p)=\\pi_n(X,\\mathbb Z\/p)\\oplus \\pi_{n-1}(X,\\mathbb Z\/p), $$  in\nwhich\\hfill\\break\n\n\\medskip\n\\noindent$\\pi_n(X,\\mathbb Z\/p)=\\hom(\\mathbb Z\/p,\\mathbb Z\/{p^r})$ {and}\n$\\pi_{n-1}(X,\\mathbb Z\/p)=\\text{Ext}(\\mathbb Z\/p,\\mathbb Z\/{p^r})=\\mathbb Z\/p.$\n\n\\medskip\n\\noindent Trivially $\\rho$ induces the identity on both. Note that\nthis example also shows that even\n$\\cap_{p\\,\\,\\text{prime}}\\mathcal E_{ \\sharp p }(X)$  can be bigger than $\\mathcal E_{ \\sharp }(X)$.}\n\\end{remark}\n\nThe paper is organized as follows: in the next section we collect\nthe results we shall need from group theory and from which\nTheorems \\ref{dos} and \\ref{uno} are immediately deduced. Theorem\n\\ref{local} and \\ref{principal} are proved in section \\S3.\nFinally, in  section \\S4 we establish Theorems \\ref{tres} and\n\\ref{cuatro}.\n\n\\section{From group theory}\n\nWe begin by recalling some basic facts. If $G$ is a group acting\non another group $A$ (i.e., $A$ is a $G$-group), the $n$-th  $G$-commutator subgroup\n$\\Gamma^n_G(A) \\subset A$ is the group generated by\n$\\{(ga^{-1})a\\,|\\, g\\in G,\\, a\\in\\Gamma_G^{n-1}(A)\\}$, being\n$\\Gamma_G^0(A)=A$. The action is then nilpotent of nilpotency\norder $r$, $\\text{nil}\\,_GA=r$, if this is the smallest integer for which\n$\\Gamma^r_G(A)=\\{1 \\}$. The group $G$ also acts in each\n$\\Gamma_G^n(A)$ and\n$\\Gamma_G^m\\bigl(\\Gamma_G^n(A)\\bigr)=\\Gamma_G^{m+n}(A)$.\n\nStatements of next sections shall heavily rely in the following\nresults:\n\n\\begin{lemma}\\label{lema1}  Let $A$ be a $G$-group. Then:\n\\begin{itemize}\n\\item[(i)] $\\Gamma_G^1(A)$ is a normal subgroup of $A$ and the $G$-action induced on $A\/\\Gamma_G^1(A)$ is trivial.\n\\item[(ii)] The quotient morphism  $A{\\stackrel{q}{\\longrightarrow}} A\/\\Gamma_G^1(A)$ is  equivariant and initial with\nrespect to trivial actions, i.e., every equivariant morphism\n$A{\\stackrel{f}{\\longrightarrow}}H$, in which the $G$-action on\n$H$ is trivial, factors uniquely through $q$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} (i) is trivial. For (ii) observe that, for any $f$ as in the lemma, $\\Gamma_G^1(A)\\subset\\ker f $.\n\\end{proof}\n\n\\begin{lemma}\\label{lema2}  Let $A$ be a $G$-group. If $A$ is nilpotent then, for any $m$, $\\Gamma_G^m(A)_{(p)}=\\Gamma_G^m(A_{(p)})$.\n\\end{lemma}\n\n\\begin{proof} Since $\\Gamma_G^m(A)=\\Gamma_G^{1}\\bigl(\\Gamma_G^{m-1}(A)\\bigr)$, once we show that\n$\\Gamma_G^1(A)_{(p)}=\\Gamma^1_G(A_{(p)})$ an easy induction proves\nthe lemma. As localization  is an exact functor in the category of\nnilpotent groups, the localization morphism $f:A\\to A_{(p)}$\nrestricts to $f:\\Gamma_G^1(A)\\to \\Gamma_G^1(A)_{(p)}$. Hence, we\nmay consider $\\Gamma_G^1(A)_{(p)}$, as well as\n$\\Gamma_G^1(A_{(p)})$, as subgroups of $A_{(p)}$. Then, for any\n$g\\in G$ and $a\\in A$, the trivial identity\n$\\bigl(gf(a)^{-1}\\bigr)f(a)=f\\bigl((ga^{-1})a\\bigr)$ shows\nequality of both subgroups.\n\\end{proof}\n\n\\begin{proposition}\\label{uf} The group $G$ acts nilpotently on the nilpotent group $A$ if and only if $G$ acts\nnilpotently on $A_{(p)}$ for $p$ any prime number or zero and all\nthese nilpotency orders are bounded.\n\\end{proposition}\n\n\\begin{proof} Assume $G$ acts nilpotently on $A$, i.e., $\\Gamma_G^m(A)=\\{1\\}$ for some $m$. Hence, by Lemma \\ref{lema2}\nand for any $p$, $\\Gamma_G^m(A_{(p)})=\\{1\\}$.\n\nConversely, assume $\\text{nil}\\,_GA_{(p)}\\le m$, for all $p$ ($p$ a prime\nnumber or $0$), and let $a$ be an element of $\\Gamma_G^m(A)$. If\n$a$ has finite order,  say it is a $q$-element, then it obviously\nsurvives under the $q$-localization morphism\n$\\Gamma_G^m\\bigl(A)\\to\\Gamma^m_G(A)_{(q)}$. For a general group,\nelements of infinite order are not guaranteed to survive under\nrationalization (for instance, the rationalization of the free\nproduct of two finite groups is trivial while it contains elements\nof infinite order). However for a nilpotent group, which is our\ncase, one can easily show by induction on the nilpotency order of\nthe group, that any element of infinite order is not sent to zero\nunder rationalization.  Taking into account, again by Lemma\n\\ref{lema2}, that $\\Gamma^m_G(A)_{(p)}=\\Gamma^m_G(A_{(p)})=\\{1\\}$,\nit follows that $a=1$ and the proof is complete.\n\\end{proof}\n\n\n\\begin{proposition}\\label{nuevolema} Let $G$ be a group acting on a finite nilpotent group $A$ in such a way that the\ninduced action on the Frattini factor $A\/\\Phi(A)$ is nilpotent.\nThen, the $G$-action on $A$ is also nilpotent.\n\\end{proposition}\n\n\\begin{proof} Recall \\cite[5.1]{Gorestein} that the Frattini subgroup of a group $A$, $\\Phi(A)$, is defined to be the intersection of all its maximal proper subgroups. The Frattini factor of $A$ is $A\/\\Phi(A)$.\nObserve in the first place that, since $\\Phi(A)$ is a characteristic subgroup of $A$, i.e., it is invariant under any automorphism of $A$, $G$ in fact induces a\nnatural  action on the Frattini factor $A\/\\Phi(A)$ which, by\nhypothesis, is nilpotent. Hence, since $A\/\\Phi(A)$ is nilpotent,\nthe induced action on\n$\\bigl(A\/\\Phi(A)\\bigr)_{(p)}=A_{(p)}\/\\Phi(A)_{(p)}$ is also\nnilpotent by Lemma \\ref{lema2}. Next, observe that for any finite\ngroup $A$, $\\Phi(A)_{(p)}=\\Phi(A_{(p)})$. Indeed, this is\nimmediate from the definition taking into account that\nlocalization commutes with limits,  in particular, with\nintersections (see for instance \\cite{hmr}). Therefore, we conclude that $G$ acts nilpotently on\n$A_{(p)}\/\\Phi(A_{(p)})$. Considering $\\varphi\\colon\nG\\to\\text{aut}\\,\\bigl(A_{(p)}\/\\Phi(A_{(p)})\\bigr)$ via this action, and\ntaking into account that $A_{(p)}\/\\Phi(A_{(p)})$ is a finite\n$p$-group, we may apply \\cite[Corollary 5.3.3]{Gorestein} to\nobtain that $\\varphi(G)$ is also a $p$-group. But the action of a\n$p$-group on another $p$-group is always nilpotent, and therefore\n$G$ acts nilpotently on $A_{(p)}$. Since this is the case for any\n$p$ and $A$ is finite we may apply  Proposition \\ref{uf} and the\nproposition follows.\n\\end{proof}\n\nFrom these results we immediately deduce:\n\n\\bigskip\n\\noindent {\\it Proof of Theorems \\ref{dos} and \\ref{uno}.} Apply\ndirectly Propositions \\ref{uf} and \\ref{nuevolema} above to the\nsubgroup $G$ of $\\mathcal E (X)$ to obtain that $G$ acts nilpotently on\n$\\pi_{\\le N}(X)$. Then, the result follows from Theorem\n\\ref{principal}. \\hfill$\\square$\n\\bigskip\n\n\n\n\nClosely related to Proposition \\ref{nuevolema}, we have the\nfollowing:\n\n\\begin{proposition}\\label{propodos} Let $G$ be a group acting on an abelian $p$-group $A$ which has an exponent $p^n$. If\n$G$ acts nilpotently on $A\\otimes\\mathbb Z\/p$, then it does so on $A$\nand $$ \\text{nil}\\,_GA\\le n\\cdot\\text{nil}\\,_GA\\otimes\\mathbb Z\/p. $$\n\\end{proposition}\n\\begin{proof} Call $r=\\text{nil}\\,_GA\\otimes\\mathbb Z\/p$ and observe that $\\Gamma_G^m(A\\otimes\\mathbb Z\/p)=\\Gamma_G^m(A)\\otimes\\mathbb Z\/p$ for any\n$m$. Therefore, since $\\Gamma_G^r(A\\otimes\\mathbb Z\/p) =0$,\n$\\Gamma_G^r(A)\\subset pA$. Assume, as induction hypothesis, that\n$\\Gamma_G^{kr}(A)\\subset p^kA$, for $k<n$. Hence, $$\n\\Gamma_G^{nr}(A)=\\Gamma_G^{(n-1)r}\\bigl(\\Gamma_G^r(A)\\bigr)\\subset\n\\Gamma_G^{(n-1)r}(pA)= p\\Gamma_G^{(n-1)r}(A)\\subset p^nA. $$ Since\n$A$ has $p^n$ as exponent, the proposition follows.\n\\end{proof}\n\n\\begin{proposition}\\label{propouno} Let $A$ be a finite abelian\n$p$-group and let $G\\subset\\text{aut}\\,(A)$ be such that $\\sigma\\otimes\n{\\mathbb Z\/p}=1_{A\\otimes\\mathbb Z\/p}$ for each $\\sigma\\in G$. Then $G$ is a\n$p$-group.\n\\end{proposition}\n\n\\begin{proof}  As $A$ is a finite  abelian $p$-group,\nthe Frattini factor $A\/\\Phi(A)$ (respec. the projection\n$A\\rightarrow A\/\\Phi(A)$) is naturally  identified with\n$A\\otimes\\mathbb Z\/p$ (respec.\\ the map $A\\rightarrow A\\otimes\\mathbb Z\/p$).\nNow, if $G$ is  not a $p$-group, there exists a non trivial\n$p^\\prime$-automorphism $\\sigma\\in G$ which, by hypothesis and\nusing the identification above,  induces the identity on the\nFrattini factor of $A$. But according to \\cite[Theorem\n5.1.4]{Gorestein}, the only $p^\\prime$-automorphism that induces\nthe identity on the Frattini factor of a $p$-group is the\nidentity.  Thus $G$ must be a $p$-group.\n\\end{proof}\n\nAs an immediate consequence we get:\n\n\\begin{corollary}\\label{corolario} In the conditions of the proposition above, the action of $G$ on $A$ is nilpotent.\n\\end{corollary}\n\n\\begin{proof} Indeed, recall that the action of a\n$p$-group $H$ on another $p$-group is always nilpotent.\n\\end{proof}\n\nIn which follows $G$ is a group acting on another group $A$. Given\n$g,h\\in G$ and $a\\in A$, we use the following usual notation: $$\n[a,g]=a^{-1}(ga),\\quad [g,a]=(ga^{-1})a,\\quad\n[g,h]=g^{-1}h^{-1}gh. $$ Hence, the following, which can be\nconsidered as a variation of the Witt-Hall identity \\cite[Theorem\n5.1]{magnus}, is obtained by direct calculation.\n\n\\begin{lemma}\\label{jo}\nFor any $f,g\\in G$ and $b\\in A$, the following identity holds:\n$$\\big[[f^{-1},g^{-1}],gb\\big]b^{-1}\\big[[g,b^{-1}], f\\big]\nb\\big[[f,b],fgf^{-1}\\big]=1.$$\n\\end{lemma}\n\n\\begin{lemma}\\label{jo2} Let $H$ be a subgroup of $G$ and $K$ a normal subgroup of $H$. Then,\n$$\n\\bigr[[H,K],A\\bigr]\\subset\\langle\\big[K,[H,A]\\bigr],\\bigl[H,[K,A]\\bigr]\\rangle.\n$$\n\\end{lemma}\n\n\\begin{proof}\nMaking $f^{-1}=h$, $g^{-1}=k$ and $gb=a$ in Lemma \\ref{jo},  it\nfollows that $$\n\\bigl[[h,k],a\\bigr]=\\bigl[h^{-1}k^{-1}h,[h^{-1},g^{-1}a]\\bigr]\ng^{-1}a^{-1}\\bigl[h^{-1},[k^{-1},g^{-1}a]\\bigr]g^{-1}a. $$ As $K$\nis normal in $H$, $\\bigl[h^{-1}k^{-1}h,[h^{-1},g^{-1}a]\\bigr]\\in\n\\big[K,[H,A]\\bigr]$. On the other hand, as commutators are normal\nsubgroups,\n$$g^{-1}a^{-1}\\bigl[h^{-1},[k^{-1},g^{-1}a]\\bigr]g^{-1}a\\in\ng^{-1}a^{-1} \\bigl[H,[K,A]\\bigr]g^{-1}a=\\bigl[H,[K,A]\\bigr], $$\nand the lemma follows.\n\\end{proof}\n\n\\begin{lemma}\\label{jo3} If the action of $G$ on $A$ is nilpotent, then\nfor each $n,m\\ge 0$, $$ [\\Gamma^n(G),\\Gamma_G^m(A)]\\subset\n\\Gamma_G^{n+m+1}(A). $$ In particular,\n$\n[\\Gamma^n (G), A]\\subset \\Gamma_G^{n+1}(A).\n$\n\\end{lemma}\n\n\\begin{proof} Set $\\text{nil}\\,_GA=r$. If $m\\ge r$ the assertion is obvious. Assume the lemma holds for all $n$ and $m\\le 1$ and let us prove it for $m=0$ by\ninduction on $n$:\n\nTrivially,\n$\n[\\Gamma^0(G),\\Gamma_G^0(A)]=[G,A]= \\Gamma_G^{1}(A)$. Finally, $$\n\\begin{aligned}\n{[\\Gamma^n(G),A]}&=\\bigl[[G,\\Gamma^{n-1}(G)], A\\bigr]\\subset\n\\text{(By Lemma \\ref{jo2})}\\\\ &\\subset \\langle\n\\bigl[\\Gamma^{n-1}(G),[G,A]\\bigr],\n\\bigl[G,[\\Gamma^{n-1}(G),A]\\bigr]\\rangle\\subset \\text{(By\ninduction)}\\\\ &\\subset\\langle [\\Gamma^{n-1}(G),\\Gamma^1_G(A)],\n[G,\\Gamma^n_G(A)]\\rangle\\subset \\text{(Again by induction)}\\\\\n&\\subset\\Gamma_G^{n+1}(A).\\\\\n\\end{aligned}\n$$\n\\end{proof}\n\n\\begin{proposition}\\label{jo4}\nLet $G$ be a subgroup of $\\text{aut}\\,(A)$. Then, $\\text{nil}\\, G\\le \\text{nil}\\,_GA-1$.\n\\end{proposition}\n\n\\begin{proof}\nAssume $\\text{nil}\\,_GA=r$. By Lemma \\ref{jo3},\n$[\\Gamma^{r-1}(G),A]\\subset \\Gamma_G^r(A)=\\{1\\}$, and therefore\n$\\Gamma^{r-1}(G)=\\{1\\}$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Self homotopy equivalence of spaces with local coefficients}\n\nAs stated in the Introduction, and following the notation and\napproach of the standard reference \\cite[Chap. VI.2]{white}, in\nthis section we consider self homotopy equivalences in the\nhomotopy category ${\\cal L}^*$ of based spaces with local\ncoefficients. Observe that a self homotopy equivalence of an\nobject $(X,{\\cal M})\\in {\\cal L}^*$ is given by  $(f,\\Theta)\\colon\n(X,{\\cal M})\\to (X,{\\cal M})$ in which $f\\colon X\\to X$ is a based\nhomotopy equivalence and $\\Theta\\colon {\\cal M}\\to{\\cal M} $ is an\nisomorphism of the coefficient system ${\\cal M}$. Note that such a\nself equivalence $(f,\\Theta)$ acts  in $\\pi_*(X)$ by $\\pi_*f$, in\n${\\cal M}$ by $\\Theta$, and in $H^*(X;{\\cal M})$ by $H^*(f,\\Theta)$.\n\nIt is also convenient to  recall how cohomology classes with local\ncoefficients are represented by maps into the ``twisted\nEilenberg-MacLane space'' (see \\cite[Chapter 5.2]{baues},\n\\cite{Gitler}, \\cite{moller2} or  \\cite{siegel} for precise\ndetails). Let $K({\\cal M},n)$ be a fixed realization of the\nEilenberg-MacLane space of type $(M_{x_0},n)$ being $M_{x_0}$ the\ngroup of the system ${\\cal M}$ at the base point. On the other hand,\ndenote by $L({\\cal M},n)$ the space obtained by applying the Borel\nconstruction  to the universal fibration $\\pi_1(X)\\to\\widetilde\nK\\stackrel{q}{\\to} K(\\pi_1(X),1)$ and the space $K({\\cal M},n)$,\ni.e.,\n\n$$ L({\\cal M},n)=\\widetilde K\\times_{\\pi_1(X)} K({\\cal M},n), $$\nand it fits into the fibration $$ K({\\cal M},n)\\longrightarrow\nL({\\cal M},n)\\stackrel{p}{\\longrightarrow}K(\\pi_1(X),1),\\quad\np(a,b)=q(a). $$ Then, for a given space $X$, $H^n(X;{\\cal M})$ is\nin one to one correspondence with the set $[X,L({\\cal\nM},n)]_{K(\\pi_1(X),1)}$ of homotopy classes of maps over\n$K(\\pi_1(X),1)$ from $X$ to $L({\\cal M},n)$.\n\n\\bigskip\n\n\n\\noindent {\\it Proof of Theorem \\ref{local}.} To avoid excessive\nnotation we shall not distinguish between a homotopy class and a\nmap which represents it. First, observe that, if $(f,\\Theta)$ is a\nself homotopy equivalence of $(X,{\\cal M})$ and $\\alpha\\colon X\\to\nL({\\cal M},n)$ represents a class of $H^*(X;{\\cal M})$,\n$H^n(f,\\Theta)(\\alpha)$ is represented by the map $$\nX\\stackrel{f}{\\longrightarrow}\nX\\stackrel{\\alpha}{\\longrightarrow}L({\\cal M},n)\\stackrel{\\xi}{\\longrightarrow}L({\\cal M},n)\n$$ in which $\\xi$ is defined by the action of $\\Theta_{x_0}$ on\n$M_{x_0}$. Explicitly, for  $(a,b)\\in L({\\cal M},n)$,\n$\\xi(a,b)=(a,\\overline{\\Theta}_{x_0}b)$ where and\n$\\overline{\\Theta}_{x_0}$ is the realization of $\\Theta_{x_0}$.\nObserve that $\\xi$ is well defined as $\\Theta\\colon {\\cal M}\\to{\\cal M}$\nis a morphism of local coefficient systems.\n\nMoreover, if $\\alpha,\\beta\\colon X\\to L({\\cal M},n)$ are in\n$H^n(X;{\\cal M})$, they coincides after composing with the fibration\n$p\\colon L({\\cal M},n)\\to K(\\pi(X),1)$, and therefore, for each $x\\in\nX$, $\\alpha(x),\\beta(x)$ live in the same fiber $K({\\cal M},n)$ of\n$p$. Hence, $\\alpha$ and $\\beta$ can be added up on $K({\\cal M},n)$\nand the resulting map $\\alpha+\\beta$ represents precisely their\nsum as cohomology classes with twisted coefficients.\n\n\nWe shall prove the theorem by induction on the length of the\nPostnikov decomposition of $X$. Assume  $X=K(\\pi,m)$ and let\n$(f,\\Theta)\\colon (X,{\\cal M})\\to(X,{\\cal M})$ be a self equivalence.\nThen, in view of the above, for any $n$-cohomology class\n$\\alpha\\colon K(\\pi,m)\\to L({\\cal M},n)$,\n$H^n(f,\\Theta)(\\alpha)-\\alpha$ is represented by the map\n$K(\\pi,m)\\to L({\\cal M},n)$ which, fiberwise, is\n$\\overline\\Theta_{x_0}\\alpha f-\\alpha$. Writing\n$\\overline\\Theta_{x_0}\\alpha f-\\alpha= \\overline\\Theta_{x_0}\\alpha\nf- \\overline\\Theta_{x_0}\\alpha + \\overline\\Theta_{x_0}\\alpha\n-\\alpha$ it is straightforward, using the nilpotency hypothesis,\nto show that the $s$-th commutator of the action of $G$ on\n$H^*(X;{\\cal M})$ vanishes as long  as\n$s\\le\\max\\{\\text{nil}\\,_G\\pi_*(X),\\text{nil}\\,_G{\\cal M}\\}$.\n\nAssume the theorem holds for $X=X^{r-1}$ and let $X=X^r$ be a\n$r$-dimensional Postnikov piece.  Consider the Serre spectral\nsequence with local coefficients on ${\\cal M}$ associated to the\nfibration $$ K(\\pi_r(X),r)\\to X\\to X^{r-1}. $$ whose $E_2$-term is\n$$ E_2^{*,*}=H^*\\bigl(X^{r-1};{\\cal H}^*(K(\\pi_r(X),r);{\\cal\nM})\\bigr). $$ Note that $G$ acts naturally in the base, total\nspace and fiber of this fibration, and hence, it does so in all\nthe terms of the spectral sequence.  The same argument used for\n$r=1$ shows that  $G$ acts nilpotently on the local coefficient\nsystem ${\\cal H}^*(K(\\pi_r(X),r);{\\cal M})$ and therefore, by\ninduction hypothesis, $G$ acts nilpotently on\n$H^*\\bigl(X^{r-1};{\\cal H}^*(K(\\pi_r(X),r);{\\cal M})\\bigr)$.\n\nAs the spectral sequence converges, the action of $G$ on the\nassociated graded module of  $H^*(X;{\\cal M})$ is nilpotent. Finally,\nreasoning by induction on the filtration degree we deduce that the\n$G$-action on  $H^*(X;{\\cal M})$ is also nilpotent.\\hfill$\\square$\n\n\nIn particular, for any space $X$ and any $j$ we may consider the\nlocal coefficient system given by $\\pi_jX$. In this case, any self\nhomotopy equivalence $f\\in\\mathcal E(X)$ can be seen as a self homotopy\nequivalence $(f,\\pi_jf^{-1})\\in\\mathcal E(X;\\pi_jX)$. Hence, any\nsubgroup of $\\mathcal E(X)$ may be considered as a subgroup of\n$\\mathcal E(X,\\pi_jX)$ which then acts naturally on $H^*(X;\\pi_jX)$ when\nconsidering local coefficients. In this context,  the theorem\nabove reads:\n\n\n\\begin{corollary}\\label{noveas} Let $X$ be a finite Postnikov piece and  let $G$ be a subgroup of\n$\\mathcal E(X)$ which acts nilpotently on $\\pi_*(X)$. Then, for any $j$,\n$G$ acts nilpotently on $H^*(X;\\pi_j)$.\\hfill$\\square$\n\\end{corollary} This result is used in the proof of Theorem \\ref{principal}  that we now present:\n\n\\bigskip\n\\noindent {\\it Proof of Theorem \\ref{principal}.} Consider the\nrestriction to $G$ of the exact sequence $\n{1}\\to\\mathcal E_{ \\sharp }(X)\\to\\mathcal E(X)\\to  \\Pi_{i\\le N}\\text{aut}\\,\\pi_i(X) $: $$\n{1}\\to\\mathcal E_{ \\sharp }(X)\\cap G\\to G\\to \\Pi_{i\\le N}\\text{aut}\\,\\pi_i(X). $$\nThe image  of $G$ under this morphism, call it $\\widetilde G$, is\na subgroup of automorphism of the group $\\pi_{\\le N}(X)$ in which\n$G$ acts nilpotently by hypothesis. Then, by Proposition\n\\ref{jo4}, $\\widetilde G$ is itself nilpotent and $\\text{nil}\\,\\widetilde\nG< \\text{nil}\\,_G\\pi_{\\le N}(X)$. Therefore, if we prove that $G$ acts\nnilpotently  on $\\mathcal E_{ \\sharp }(X)$, then (see for instance \\cite[Proposition\n4.1]{hmr}) $G$ would be nilpotent and\n\\begin{equation}\\label{nose1}\n\\text{nil}\\, G< \\text{nil}\\,_G\\mathcal E_{ \\sharp }(X)+\\text{nil}\\,_G\\pi_{\\le N}(X).\n\\end{equation} For that, observe in the first place that $[X,X]\\cong[X^N,X^N]$, where\n$X^N$ denotes the $N$-th Postnikov stage of $X$, and this\nbijection restricts to an isomorphism $\\mathcal E_{ \\sharp }(X)\\cong\\mathcal E_{ \\sharp }(X^N)$. On\nthe other hand, consider the exact sequence\n\\begin{equation}\\label{nose2} {1}\\to A_j\\to \\mathcal E_{ \\sharp }(X^j)\\to \\mathcal E_{ \\sharp }(X^{j-1})\n\\end{equation} where $\\mathcal E_{ \\sharp }(X^j)\\to\n\\mathcal E_{ \\sharp }(X^{j-1})$ is just the obvious restriction and $A_j$ its\nkernel. Since $G$ acts on any $\\mathcal E_{ \\sharp }(X^j)$ and $\\mathcal E_{ \\sharp }(X^1)={1}$, it\nwill be enough to show that $G$ acts nilpotently on every $A_j$ to\nconclude, by an easy induction, that it does so on\n$\\mathcal E_{ \\sharp }(X^N)=\\mathcal E_{ \\sharp }(X)$.\n\nBy classical obstruction theory of liftings (see \\cite[Chapter\n6.6]{white}) recall that, for $j\\ge 2$, there is a bijection\n$\\varphi\\colon B_j\\to H^j(X^j;\\pi_j)$ where\n\n\\begin{itemize}\n\\item\nThe cohomology is taken with local coefficients.\n\n\\item $B_j$ is the set of homotopy classes of $[X^j,X^j]$ which restrict to  the identity on $X^{j-1}$, i.e., homotopy\nclasses of liftings of $X^j\\to X^{j-1}$ to $X^j$.\n\n\\item $\\varphi(g)=\\delta(g,1)$ is the difference cochain of degree $j$ between $g$ and the identity on $X^j$.\n   \\end{itemize}\nRecall also that, in general,\n$\\delta(g,f)=\\delta(g,1)+\\delta(1,f)$ and that\n$\\delta(gh,fh)=H^j(h)\\bigl(\\delta(g,f)\\bigr)$. Moreover, if\n$h\\in\\mathcal E_{ \\sharp }(X^j)$, $\\delta(hg,hf)$ is the image of $\\delta(g,f)$\nunder the map $H^j(X^j;\\pi_j)\\to H^j(X^j;\\pi_j)$ induced by $h$ on\n$\\pi_j$.\n\nFrom now on, as in Corollary \\ref{noveas}, any $f\\in\\mathcal E(X)$, and\nthus in $G$, shall be considered as a self homotopy class\n$(f,\\pi_jf^{-1})\\in\\mathcal E(X,\\pi_j)$. Hence, restricting $\\varphi$ to\n$A_j$ we obtain a map $\\varphi\\colon A_j\\hookrightarrow\nH^j(X^j;\\pi_j)$ which is a $G$-map with respect to the action\n$g\\cdot f=g^{-1}fg$, $g\\in G$, $f\\in A_j$,  and the usual action\non $H^j(X^j;\\pi_j)$: if $\\alpha\\in H^j(X_j,\\pi_j)$, and $g\\in G$,\n$g\\cdot\\alpha$ is the cohomology class represented by the map\n\n$$ X^j{\\stackrel{g}{\\longrightarrow}}\nX^j{\\stackrel{\\alpha}{\\longrightarrow}}\nL(\\pi_j,j){\\stackrel{\\xi}{\\longrightarrow}} L(\\pi_j,j),\\qquad\n\\alpha\\in H^j(X^j;\\pi_j), $$ with $\\xi\\colon L(\\pi_j,j)\\to\nL(\\pi_j,j)$ induced by $\\pi_*(g^{-1})$.\n\nMoreover, this restriction is a group morphism. Indeed, given\n$f,h\\in A_j$,\n$\\varphi(fh)=\\delta(fh,1)=\\delta(f^{-1}fh,f^{-1})=\\delta(h,f^{-1})=\\delta(h,1)+\\delta(1,f^{-1})=\\delta(h,1)+\\delta(f,1)=\n\\varphi(f)+\\varphi(h)$. As an immediate consequence we then obtain\nthat $A_j$ is an (abelian!) subgroup of $H^j(X^j;\\pi_j)$.\n\nFinally, by Corollary \\ref{noveas}, $G$ acts nilpotently on\n$H^j(X^j;\\pi_j)$ for any $j$. Hence, it does so on $A_j$ and $$\n\\text{nil}\\,_G\\,A_j\\le \\text{nil}\\,_G\\,H^j(X^j;\\pi_j). $$ Thus, by induction on\n$j$ using repeatedly \\cite[Proposition 4.1]{hmr}, one easily sees\nin view of (\\ref{nose2}) that\n$\\text{nil}\\,_G\\,\\mathcal E_{ \\sharp }(X)=\\text{nil}\\,_G\\,\\mathcal E_{ \\sharp }(X^N)\\le\\sum_{j=2}^N\\text{nil}\\,_G\\,A_j\\le\\sum_{j=2}^N\\text{nil}\\,_G\\,H^j(X^j;\\pi_j)$\nand the theorem follows.\\hfill$\\square$\n\n\n\n\n\n\\section{Groups which fix the homotopy groups}\nIn this section we establish Theorems \\ref{tres} and \\ref{cuatro}.\n\n\\bigskip\n\\noindent {\\it Proof of Theorem \\ref{tres}.}   Let\n$\\alpha\\in\\mathcal E_{ \\sharp p }(X)$. Then, for each $i\\le\\dim X$, the morphism\n$\\pi_i(\\alpha;\\mathbb Z\/p)\\colon\\pi_i(X,\\mathbb Z\/p)\\to\\pi_i(X,\\mathbb Z\/p)$ is just\nthe identity. On the other hand, the Universal Coefficients\nTheorem for homotopy yields the following split short exact\nsequence $$\n0\\longrightarrow\\text{Ext}(\\mathbb Z\/p,\\pi_{i+1}X)\\longrightarrow\\pi_i(X;\\mathbb Z\/p)\\longrightarrow\n\\hom(\\mathbb Z\/p;\\pi_iX)\\longrightarrow 0. $$ Thus, both\n$\\text{Ext}(\\mathbb Z\/p,\\pi_{i+1}\\alpha)$ and $\\hom(\\mathbb Z\/p;\\pi_i\\alpha)$\nare the identity. But observe that\n$\\text{Ext}(\\mathbb Z\/p,\\pi_{i+1}X)=\\pi_{i+1}X\\otimes\\mathbb Z\/p$ so that\n$\\pi_{i+1}\\alpha\\otimes\\mathbb Z\/p=1_{\\pi_{i+1}X}$.\n\nHence, by Proposition \\ref{propouno}, for each $i\\le\\dim X$, the\nimage of $\\mathcal E_{ \\sharp p }(X)$ in $\\text{aut}\\,(\\pi_iX)$ is a $p$-group and then, by\nCorollary \\ref{corolario}, the action of $\\mathcal E_{ \\sharp p }(X)$ on $\\pi_iX$ is\nnilpotent. Thus, by Theorem \\ref{principal}, $\\mathcal E_{ \\sharp p }(X)$ is\nnilpotent. On the other hand, notice that $\\mathcal E_{ \\sharp }(X)$ is precisely\nthe kernel of the obvious map $\\mathcal E_{ \\sharp p }(X)\\to\\Pi_{i\\le\\dim\nX}\\text{aut}\\pi_i(X)$. Hence, as we just proved that the image of\nthis map is a $p$-group, $\\mathcal E_{ \\sharp p }(X)\/\\mathcal E_{ \\sharp }(X)$ is a finite $p$-group,\nand the proof is complete. \\hfill$\\square$\n\n\\bigskip\n\n\\noindent {\\it Proof of Theorem \\ref{cuatro}.} Write\n$\\pi_iX=\\mathbb Z^{n_i}\\oplus\\bigl(\\oplus_{p\\,\\,\\text{prime}}T_p(\\pi_iX)\\bigr)$\nin which $T_p(\\pi_iX)$ is the group of $p$-torsion elements in\n$\\pi_iX$. Now, if $\\alpha\\in\\cap_{p\\,\\,\\text{prime}}\\mathcal E_{ \\sharp p }(X)$, then\nfor each $i\\le\\dim X$,\n$\\pi_i(\\alpha)|_{T_p(\\pi_iX)}\\in\\text{aut}\\,\\bigl(T_p(\\pi_iX)\\bigr)$.  Let\n$z_1,\\ldots,z_{n_i}$ be generators of $\\mathbb Z^{n_i}\\subset\\pi_iX$.\nThen $\\pi_i(\\alpha)(z_k)=\\sum_{j=1}^1{n_i}a_{k,j}z_j+\\omega$,\nwhere $\\omega$ is the torsion part. But this element has to\ncoincide with $z_k$ mod $p$, for all prime $p$. Therefore, also\nfor any $p$, $ a_{k,j}=0(\\text{mod}\\,p) $ for $k\\not=j$ and\n$a_{k,k}=1(\\text{mod}\\, p)$, for $1\\le k\\le n_i$. The only\npossible solution is $a_{k,j}=0$, $k\\not=j$, and $a_{k,k}=1$. In\nother words, $\\pi_i(\\alpha)(z_k)=z_k+\\omega$, in which $\\omega$ is\na torsion element.\n\nAdding up, for any element $\\gamma\\in\\pi_iX$,\n$\\pi_i(\\alpha)\\gamma-\\gamma$ is a torsion element in $\\pi_iX$.\nThis is equivalent to say that the $1$-commutators of the action\nof $\\cap_{p\\,\\,\\text{prime}}\\mathcal E_{ \\sharp p }(X)$ on $\\pi_iX$ live in the\ntorsion part of $\\pi_iX$. However, by Corollary \\ref{corolario},\nthe action of $\\cap_{p\\,\\,\\text{prime}}\\mathcal E_{ \\sharp p }(X)$ on the torsion\npart is nilpotent and therefore,  the action on $\\pi_iX$ is also\nnilpotent. Apply Theorem \\ref{principal} and the proof is\ncomplete.\\hfill$\\square$\n\n\\begin{remark} {\\em We end up by noting that the hypothesis of Theorem \\ref{tres} are necessary. Indeed, consider\n$X=K\\bigl((\\mathbb Z\/2)^2,n\\bigr)$ and observe that, for a prime $p$\ndifferent from $2$, $\\mathcal E_{ \\sharp p }(X)=\\mathcal E(X)=GL_2(\\mathbb Z\/2)\\cong \\Sigma_3$\nwhich is not nilpotent.\n\nOn the other hand, take $X=K(\\mathbb Z^2,n)$ for which\n$\\mathcal E(X)=GL_2(\\mathbb Z)$. In this case $\\mathcal E_{ \\sharp }(X)=\\{1\\}$ and, for any\nprime $p$, $\\mathcal E_{ \\sharp p }(X)$ fits in the following short exact sequence $$\n\\{1\\}\\to\\mathcal E_{ \\sharp p }(X)\\to GL_2(\\mathbb Z)\\to GL_2(\\mathbb Z\/p)\\to\\{1\\} $$ where the\nsurjection $GL_2(\\mathbb Z)\\to GL_2(\\mathbb Z\/p)$ is just the mod-$p$\nreduction. Hence $\\mathcal E_{ \\sharp p }(X)=\\mathcal E_{ \\sharp p }(X)\/\\mathcal E_{ \\sharp }(X)$ is an infinite, non\nnilpotent group.}\n\\end{remark}\n\n\n\\nocite{*}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA wide range of minimally invasive therapies have been developed for cancer treatment, additionally to open surgery \\cite{Ahmed.2014,mauri2017technical,tomasian2018percutaneous}. One of these methods is the use of microwave ablation (MWA). Especially for smaller tumors, MWA shows promising results for treatment \\cite{tehrani2020use}. As the minimal ablative margin (MAM) is crucial for the local tumor progression (LTP), it is of greatest importance to assess if the malignancy has been adequately and completely treated, regardless of the etiology. For each millimeter increase of the MAM, a 30\\% reduction of the relative risk for LTP was found. The MAM is especially important as the only significant independent predictor of LTP (p = 0.036) \\cite{laimer2020minimal}. During the intervention, magnetic resonance (MR) imaging offers several advantages like a good soft-tissue contrast without the need of contrast agent, the free orientation and positioning of single slice scans and the possibility to accurately track changes in the temperature inside the tissue \\cite{Gorny.2019,Kagebein.2018c,Rieke.2008,Senneville.2007}.\\\\\n\\\\\n\\textbf{Contribution.} In this work, we propose a novel approach for the creation of a volumetric thermometry map without the development of a fully 3D sequence. The introduced 2.5D thermometry method utilizes any common 2D gradient-echo (GRE) sequences. Therefore, possible temporal limitations are less restricting than for the 3D sequences and images with higher resolution may be acquired offering standard thermometry accuracy of around $1^\\circ$C deviation while being more robust towards MR inhomogeneities \\cite{Gorny.2019}. We will show that our method is well-suited to reconstruct the actual coagulation zone after thermal ablation.\\\\\n\\\\\n\\textbf{Related Work.} Zhang et al.\\cite{zhang2019variable} propose a golden-angle\u2010ordered 3D stack\u2010of\u2010radial multi-echo spoiled gradient\u2010echo sequence with a variable flip angle. The image reconstruction is performed offline offering a temporal resolution between 2s-5s. Jiang et al.\\cite{Jiang.2020} use an accelerated 3D echo-shifted sequence and the Gadgetron framework for image reconstruction. Temporal resolution lies at around 3s with a temperature error of less than $0.65^\\circ$C. Quah et al.\\cite{quah2020simultaneous} are aiming at an increased volume coverage for thermometry without multiple receive coils. An extended k\u2010space hybrid reconstruction was used, yielding an error of $<1^\\circ$C and an acquisition time of 3.5s for each image. Fielden et al.\\cite{Fielden.2018} present a comparison study between cartesian, spiral-out and retraced spiral-in\/out (RIO) trajectories. Using the 3D RIO sequence, they achieved a true temporal resolution of 5.8s with a temporal standard deviation of $1.32^\\circ$C. Marx et al.\\cite{Marx.2014} introduced the MASTER sequence for volumetric MR thermometry acquisition, acquiring six slices in around 5s. In a later work \\cite{Marx.2017} they use optimized multiple-echo spiral thermometry sequences, which yield a better precision than the usual 2D Fourier transform thermometry. Image acquisition takes between 7s-11s. Svedin et al.\\cite{svedin2018multiecho} make use of a multi-echo pseudo-golden angle stack-of-stars sequence and offline image reconstruction using MATLAB. They achieved a temporal resolution of around 2s and a spatial average of the standard deviation through a time of $0.3-1.0^\\circ$C. Od\u00e9en et al.\\cite{odeen2016mr} propose the use of a 3D gradient recalled echo pulse sequence with segmented EPI readout. To estimate the temperature change, they also integrate a bioheat equation. They achieved a temperature root mean square error of $1.1^\\circ$C. Golkar et al.\\cite{golkar2018fast} introduce a fast GPU based simulation approach for cryoablation monitoring. The reconstruction takes 110s and the final result shows a Dice coefficient of 0.82. A summarize of the related work in comparison to our method is shown in Table \\ref{tab:RelatedWork}.\n\\begin{table}[ht]\n\\centering\n\\caption{Overview about the related work in comparison to this work. Every work has been observed according to the following: 1) The kind of image sequence used. 2) The online or offline capability of the reconstruction framework. 3) The temporal resolution of the whole image acquisition in seconds. 4) The temperature accuracy in \u00b0C. 5) The resulting Dice Score similarity measurement if available.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n                       & \\begin{tabular}[c]{@{}c@{}}Image \\\\ Sequence\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Reconstruction \\\\ Framework\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Temporal \\\\ Resolution {[}s{]}\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Temperature \\\\ Accuracy {[}\u00b0C{]}\\end{tabular} & Dice Score \\\\ \\hline\nZhang et al.{[}20{]}   & 3D                                                        & offline                                                             & 2 - 5                                                                  & ---                                                                      & ---                                                   \\\\ \\hline\nJiang et al.{[}6{]}    & 3D                                                        & \\begin{tabular}[c]{@{}c@{}}online \\\\ (Gadgetron)\\end{tabular}       & 3                                                                      & \\textless 0.65                                                           & ---                                                   \\\\ \\hline\nQuah et al.{[}13{]}    & Stack of 2D                                               & hybrid                                                              & 3.5                                                                    & \\textless 1                                                              & ---                                                   \\\\ \\hline\nFielden et al.{[}3{]}  & 3D                                                        & online                                                                 & 5.8                                                              & 1.32                                                                     & ---                                                   \\\\ \\hline\nMarx et al.{[}10{]}    & Stack of 2D                                               & ---                                                                 & 5                                                                      & 1.3                                                                      & ---                                                   \\\\ \\hline\nMarx et al.{[}9{]}     & Stack of 2D                                               & online                                                                 & 7 - 11                                                                 & 0.29 - 0.65                                                                      & ---                                                   \\\\ \\hline\nSvedin et al. {[}17{]} & 3D                                                        & \\begin{tabular}[c]{@{}c@{}}offline \\\\ (MATLAB)\\end{tabular}         & 2                                                                      & 0.3 - 1.0                                                                & ---                                                   \\\\ \\hline\nOdeen et al.{[}12{]}  & Stack of 2D                                               & \\begin{tabular}[c]{@{}c@{}}offline \\\\ (MATLAB)\\end{tabular}                                                                 & 2.4 - 4.8                                                                    & \\textless 1.1                                                                      & ---                                                   \\\\ \\hline\nGolkar et al.{[}4{]}   & 3D                                                        & ---                                                                 & 110                                                                    & ---                                                                      & 0.82                                                  \\\\ \\hline\nThis work             & Single 2D                                               & online                                                              & variable                                                               & 1                                                                        & 0.75                                                  \\\\ \\hline\n\\end{tabular}\n\\label{tab:RelatedWork}\n\\end{table}\n\\section{Material and Method}\n\\label{sec:MaterialAndMethod}\n\\subsection{Image Acquisition}\n\\label{subsec:ImageAcquisition}\nThe proposed 2.5D thermometry relies on sampling the volume of interest (VOI) using a common 2D GRE sequence and [1,...,n] different orientations. The GRE sequence can directly reconstruct magnitude and phase images simultaneously. To ensure a proper sampling of the VOI in our setup the GRE sequence is rotated by $22.5^\\circ$ around the applicator's main axis. This results in an evenly distributed sample of eight different orientations. To increase the spatial resolution, the angles between the acquired scans should be as high as possible, resulting in the following acquisition order: $0 ^\\circ$, $90 ^\\circ$, $45 ^\\circ$, $135 ^\\circ$, $22.5 ^\\circ$, $112.5 ^\\circ$, $67.5 ^\\circ$, and $157.5 ^\\circ$. To reduce the delay between image acquisition and visualization of the volumetric thermometry map, the SIEMENS Healthineers Access-I Framework was integrated. The framework allows for fetching the image data directly from SIEMENS MR devices without an intermediate imaging archive system.\n\\subsection{2.5D Thermometry Reconstruction}\n\\label{subsec:25DThermometryReconstruction}\nBefore treatment starts, reference phase images are acquired for each of the eight orientations. Each newly acquired phase image will start computing the up-to-date 2D thermometry map for the current orientation during the treatment. To do so, the proton resonance frequency shift (PRFS) method is used as described by Rieke et al. \\cite{Rieke.2008}. The temperature $T$ based on the PRFS is computed using the following Equation\n\\begin{equation}\n\\label{eq:PRFS}\n    T = \\frac{\\phi(t) - \\phi (t_0)}{\\gamma \\alpha B_0 TE} + T_0\n\\end{equation}\nwith $\\phi(t) - \\phi (t_0)$ defining the phase difference between the current time point $\\phi(i)$ and the reference timepoint $\\phi(i_0)$, $\\gamma = 42,576\\frac{MHz}{T}$ representing the gyromagnetic ratio of hydrogen protons, $\\alpha = 0.01\\frac{ppm}{\\Delta T}$ representing the proton resonance frequency change coefficient, $B_0$ representing the used magnetic field strength and $TE$ representing the used echo time. The constant $T_0$ needs to be added to the temperature since Equation \\ref{eq:PRFS} otherwise only computes the temperature change, neglecting the tissue's base temperature. The Access-I integration and 2D thermometry computation were implemented as modules using MeVisLab 3.4.1 \\cite{ritter2011medical}. \n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{MICCAI2021Workflow_v6.PNG}\n    \\caption{Schematic overview of the proposed method.}\n    \\label{fig:MethodOverview}\n\\end{figure}\nThe 2.5D thermometry reconstruction itself was implemented using C++. A schematic overview of the method can be seen in Figure \\ref{fig:MethodOverview}. To handle the voxel values during slice rotation every cartesian coordinate was mapped to the corresponding cylindrical coordinate representation using Equation \\ref{eq:CartesianToPolar},\n\\begin{align}\n\\label{eq:CartesianToPolar}\n    P_r(x,y,z) &= P_c(r, \\theta, z)\\\\\n    r &= \\sqrt{\\left(x-x_c \\right)^2 + \\left(y-y_c\\right)^2}\\nonumber \\\\\n    \\theta &= atan2\\left(\\frac{x-x_c}{y-y_c}\\right)\\nonumber\n\\end{align}\nwhere $x,y$ represents the Cartesian coordinates of the current voxel and $x_c,y_c$ represents the Cartesian coordinates of the centerline corresponding to the applicator's axis for every slice $z$ in the reconstructed volume. Upon acquisition of the reference images, a multi-dimensional population map is created. For each voxel $(x_i,y_i,z_i)$ in the reconstructed volume, this population map holds information about the radius $r$ and angle $\\theta$ of the cylindrical coordinates, the general interpolation weight $I_w$, the adjacent interpolation partner coordinates $IP_{left}(x,y)$ and $IP_{right}(x,y)$ in the 2D live data as Cartesian representation and the weights $w_1$ and $w_2$ of those interpolation partners. The weights may be acquired using Equation \\ref{eq:Weights},\n\\begin{align}\n\\label{eq:Weights}\n    w_{1} &= \\left\\vert\\frac{\\theta_{IP_{left}} - \\theta_{i}}{\\theta_{IP_{left}} - \\theta_{IP_{right}}}\\right\\vert\\\\\n    w_2 &= 1 - w_1\\nonumber\n\\end{align}\nwith $\\theta_{i}$ representing the cylindric angle of the current Voxel $i$ and $\\theta_{IP_{left}}$, $\\theta_{IP_{right}}$ representing the orientation angles of the left and right interpolation partners, respectively. The 2D population map can be applied to every slice of the final 3D output volume, reducing the computational power needed. During the intervention, every acquired live image triggers the reconstruction of the up-to-date 2.5D thermometry map. Here, the heat value for each voxel is reconstructed using Equation \\ref{eq:Reconstruction},\n\\begin{equation}\n\\label{eq:Reconstruction}\n    T_i = I_w \\cdot (w_{1} \\cdot T_{IP_{left}} + w_{2} \\cdot T_{IP_{right}})\n\\end{equation}\nwith $T_i$ representing the temperature of the current voxel $i$ and $T_{IP_{left}}$, $T_{IP_{right}}$ representing the temperature of the adjacent interpolation partners. Occurring vessels or other structures, which cause a heat sink effect are segmented during the intervention planning. Subsequently, the segmented structure is saved as an additional Look-Up Volume. Here, each voxel can be checked if it is part of a heat sink structure. Using this knowledge, the interpolation weight $I_w$, which ranges between $[0,1]$, may be adjusted. Figure \\ref{fig:Reconstruction} shows a single dimension of the population map for parameter weighting, a reconstructed heat map, a coagulation estimation based on an empirically defined threshold and the corresponding ground truth segmentation. The source code is available for download at \\url{https:\/\/github.com\/jalpers\/2.5DThermometryReconstruction}.\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{ImageReconstruction_v3.png}\n    \\caption{A) Example population map for output weights color coded on gray scale. B) Reconstructed volumetric heat map. C) Estimated coagulation necrosis based on a threshold of $57^\\circ$C. D) Manually segmented ground truth.}\n    \\label{fig:Reconstruction}\n\\end{figure}\n\\subsection{Evaluation}\n\\textbf{Phantom design.} To create a first proof of concept, a pilot study was conducted to evaluate the 2.5D thermometry reconstruction using 13 bio protein phantoms as described by Bu Lin et al.\\cite{BuLin.2008}. The coagulation zone's visibility in the post-treatment data sets increased by adding a contrast agent ($0,5\\mu mol\/L$ Dotarem) to the phantoms. For six phantoms, additional polyvinyl chloride (PVC) tubes with a diameter of $5mm$ and a wall thickness of $1mm$ were integrated into the phantoms (three single-tubes, three double-tubes) to simulate a possible heat sink (HS) effect.\\\\\n\\\\\n\\textbf{Experimental setup.} The applicator of the permittivity feedback control MWA system (MedWaves Avecure, Medwaves, San Diego, CA, USA, 14G) was placed inside the phantom by sight and secured in position. Subsequently, the phantoms were placed inside a $1,5T$ MR scanner (Siemens Avanto, Siemens Healthineers, Germany). The coaxial cables connected to the applicator and MW generator were led through a waveguide. Chokes and electrical grounding measures were added as described by Gorny et al.\\cite{Gorny.2019} to reduce radio frequency interference. In the case of the perfusion phantoms, the PVC tubes were led through the wave guide. They were connected to a diaphragm pump and a water reservoir outside the scanning room. A flow meter (SM6000, ifm electronic, Essen, Germany) was interposed between the reservoir and the pump, providing a flow rate of $800mL\/min$. Observations showed a moderate HS effect using this setup with a maximum antenna power of 36W. Additionally, temperature sensors were inserted in two phantoms to experimentally verify the temperature accuracy of $1 ^\\circ C$.\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{experimental_setup_v1.2.jpg}\n    \\caption{Experimental evaluation setup. Flexible tubes (blue) lead the water (a) through a flow meter (b), a diaphragm pump (c) and the bio protein phantom (e). The coaxial cables (red) connect the applicator with the MW generator d).}\n    \\label{fig:StudyDesignSketch}\n\\end{figure}\nRight before treatment, ten reference phase images were acquired and averaged for each orientation to compensate for static noise. The MWA duration was set to 15 minutes with a temperature limit of $90^\\circ$C. The GRE sequence offers a slice thickness of $5mm$, a field of view (FOV) of $256mm * 256mm$, a matrix of $256 * 256$, and a bandwidth of $260 Hz\/Px$. Image acquisition took around $1.1s$ with a $5s$ break to simulate the temporal resolution for a breathing patient. The TE was $3.69ms$, the TR $7.5ms$, and the flip angle $7 ^\\circ$. For post-treatment observation a 3D turbo spin echo (TSE) sequence (TE = $156ms$, TR = $11780ms$, flip angle = $180 ^\\circ$, matrix = $256 * 256$, FOV = $256mm * 256mm$, bandwidth = $40 Hz\/Px$, slice thickness = $1mm$) was used. The 3D TSE allows for proper visualization of the real coagulation zone due to a very high tissue contrast. Extraction of the coagulation ground truth was done manually by a clinical expert using MEVIS draw (Fraunhofer MEVIS, Bremen, Germany). All data sets used are available for download at \\url{http:\/\/open-science.ub.ovgu.de\/xmlui\/handle\/684882692\/89}.\\\\\n\\\\\n\\textbf{Statistical evaluation.} Final evaluation of the acquired data was performed using the dice similarity coefficient (DSC) as explained in Equation \\ref{eq:DSC}\n\\begin{equation}\n    \\label{eq:DSC}\n    DSC = \\frac{2*TP}{2*TP + FP + FN}\n\\end{equation}\nwith $TP$ representing the true positives, $FP$ the false positives and $FN$ the false negatives. Additionally, the standard error of the mean (SEM) was computed at a confidence level of 95\\% (p = 0.05) using Equation \\ref{eq:SEM}\n\\begin{align}\n    \\label{eq:SEM}\n    \\sigma &= \\sqrt{\\frac{\\sum(x_i-\\bar{x})^2}{N-1}}\\\\\n    SEM &= \\frac{\\sigma}{\\sqrt{N}} * 1.96\\nonumber\n\\end{align}\nwith $\\sigma$ representing the standard deviation, $x_i$ the current sample, $\\bar{x}$ the mean value and $N$ the sample size. To compute the SEM at a confidence level of 95\\% it has to be multiplied by 1.96, which is the approximated value of the 97.5 percentile of the standard normal distribution.\n\\section{Results}\nSummarized evaluation results can be seen in Figure \\ref{fig:SummarizedResults}. Empirically determined coagulation thresholds were set between $51^\\circ$C and $61^\\circ$C depending on each phantom's pH value.\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{DiceSensitivityResults_v4.png}\n    \\caption{Summarized evaluation results for phantoms without HS effect, phantoms with HS effect and the overall results. Note that the data range [0,0.3] was left out because no data points are present in that range.}\n    \\label{fig:SummarizedResults}\n\\end{figure}\nIt is noticeable that the DSCs for HS phantoms show a very high SEM with $0.70\\pm0.15 (\\pm21.25\\%)$ and $0.74\\pm0.06 (\\pm8.49\\%)$ regarding the sensitivity. The high range results from a corrupted dataset due to heavy artifacts within the image data. Leaving the corrupted dataset out of the evaluation, the SEM shows a significantly lower deviation of $0.76\\pm0.062 (\\pm8.07\\%)$ and $0.77\\pm0.048 (\\pm6.25\\%)$ for the DSC and sensitivity, respectively. Observations show a slightly higher DSC and sensitivity for phantoms without any HS effect. Here, the values range from $0.79\\pm0.04 (\\pm 4.53\\%)$ and $0.79\\pm0.04 (\\pm 5.55\\%)$, respectively. Evaluation showed an overall SEM for the DSC of $0.75\\pm0.07 (\\pm9.76\\%)$ and a SEM for sensitivity of $0.77\\pm0.04 (\\pm4.99\\%)$. To evaluate the computational effort, every major step was performed 100 times. The creation of the population map and the heat sink look up volume took $25.53ms\\pm 3.33ms$ and $3.91s\\pm 0.59s$, respectively. These two steps need to be done just once before start of the treatment. The reconstruction of the 2.5D thermometry map was performed in $18.02ms\\pm 5.91ms$ on a customary workstation (Intel( R) Core(TM) i5-6200U CPU, double-core 2.30GHz, 8GB RAM, Intel(R) HD Graphics 520). This reconstruction will be performed every time a new image is acquired during treatment.\n\n\\section{Discussion and Conclusion}\nThe aim of this work is to develop a volumetric thermometry map, which can be applied to a wide variety of clinical setups. Therefore, our work heavily relies on the up-to-date standard 2D GRE sequence for image acquisition. This allows for the standard accuracy of the thermometry up to $1.0^\\circ$C. Nonetheless, the sampling of the 3D volume also results in some disadvantages, which need to be addressed in the future. First, the diffusion of the heat inside the tissue is not linear over time. Therefore, it would be necessary to include an adaptive temporal and spatial resolution depending on the current intervention time. A new study should be conducted to identify an optimal sequence protocol for this 2.5D thermometry approach. Second, we found that the reconstruction sometimes shows stair-case artifacts. Because only one image is acquired every few seconds, the time difference between adjacent orientations may be very high. The temperature difference for each voxel dependent on the applicator's radius may be computed and applied to the corresponding voxel on every other out-of-date data to compensate for this error. This transfer of the heat gradient may improve the reconstruction accuracy. Another approach may be the use of a model-based reconstruction to take different tissue characteristics into account. To pseudo-increase the temporal resolution, bio heat transfer simulations may also be included during reconstruction. The acquired live data may be able to adjust the simulation parameters to increase the simulation accuracy. Finally, our study only performs on bio protein phantoms. Results show a proof of concept for the proposed method, but it still has to be evaluated in real tissue and a more realistic clinical environment. Therefore, perfused ex vivo livers may be a way to go in the future. Additionally, we currently assume a breath-holding state or at least a breath-triggered image acquisition. Research shows that a wide range of interventional registration methods is available, but further investigations in this area still need to be done to create an applicable method. The last issues arise because of the MR inhomogeneity during image acquisition. The slightest disturbances may result in heavy image artifacts. Proper shielding of the MW generator is needed to reduce the SNR loss over time thus increasing the thermometry and reconstruction accuracy. \\\\\n\\\\\nIn conclusion, we proposed a novel method for 2.5D thermometry map reconstruction based on common GRE sequences rotated around the applicator's main axis. A pilot study was conducted using bio protein phantoms to simulate cases with possible heat sink effects and without. The evaluation shows promising results regarding the DSC of the reconstructed 2.5D thermometry map and a manually defined ground truth. Future work should address the reconstruction method's improvement by integrating further apriori knowledge like the estimated shape of the heat distribution. Furthermore, a more realistic study should be conducted with bigger sample size and real tissue. In sum, the method shows a high potential to improve the clinical success rate of minimally invasive ablation procedures without necessarily hampering the standard clinical workflow of the individual clinician.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe inflationary paradigm in the early Universe lays a strong foundation for successful description of the CMB and large scale structure (LSS) of the Universe so far  \\cite{Starobinsky:1980te,Starobinsky:1979ty,Akrami:2018odb} with the  predictions of spectral index $n_s=0.9649\\pm 0.0042$ for $50-60$ e-foldings of quasi-de Sitter expansion. Besides its apparent success, some CMB anomalies seem to arise and challenge any {current} model of inflation. \nUnderstanding the observed anomalies in the latest Planck's CMB data could further elucidate some fundamental aspects of the quantum nature of the fluctuations which are believed to seed the large scale structure (LSS) of the Universe. Concretely, one of the main CMB anomalies is the presence of a hemispherical power asymmetry (HPA) at large angular scales \\cite{Akrami:2019bkn} which has been seen from the NASA's Wilkinson Microwave Anisotropy Probe (WMAP) to the latest Planck data and its significance is standing now at $3.3\\sigma$ \\cite{Akrami:2014eta}. \n In short, the HPA is an asymmetry in the two point temperature correlations on opposite hemispheres of the sky, with slightly higher power in the southern ecliptic hemisphere and slightly lower power in the northern ecliptic hemisphere \\cite{Creswell:2021eqi}. This is a significant challenge to our standard model of (inflationary) cosmology, which predicts isotropy \\cite{Yeung:2022smn,Aluri:2022hzs}.  \n This anomaly seems to appear significantly at low-multipoles $\\ell =2-64$ \\cite{Schwarz:2015cma}. The HPA is phenomenologically represented by the following form of the position dependent primordial power spectrum parametrized as \\cite{Lyth:2013vha,Lyth:2014mga}\n\\begin{equation}\n\t\\mathcal{P}_\\mathcal{R}\\left( k,\\,\\hat{\\boldsymbol{n}}\\right) \\simeq \\mathcal{P}_{\\mathcal{R}\\:\\: iso} (k)\\left( 1+2A(k) \\hat{\\textbf{p}}\\cdot\\hat{\\textbf{n}}\\right) \\,,\n\t\\label{pwa}\n\\end{equation}\nwhere $\\mathcal{P}_{\\mathcal{R}\\:\\: iso} (k)$ is the statistically homogeneous and isotropic power spectrum, $A(k)$ is the amplitude of the observed dipolar asymmetry which is constrained as $\\vert A \\vert = 0.066\\pm 0.021$ ($3.3\\sigma$) at the large angular scales $\\ell <64$ or at the wave numbers  $k \\lesssim 0.0045\\textrm{ Mpc}^{-1}$. The HPA seems to be present even up to the multipoles $~600$ as seen by the Planck 2013, Hansen \\emph{et al.} and Aiola  \\emph{et al.} \\cite{Planck:2013lks,Hansen:2008ym,Aiola:2015rqa}. \n\n Here  $\\hat{\\boldsymbol{p}}$ is the direction of maximal symmetry and $\\hat{\\textbf{n}}=\\frac{\\boldsymbol{x}}{x_{\\text{ls}}}$ is the line of sight from earth and $x_{\\text{ls}} = 14,000 \\textrm{Mpc}^{-1}$ is the co-moving distance to the surface of last scattering. \nFollowing \\eqref{pwa} we can derive \\cite{Lyth:2013vha}\n\\begin{equation}\n\tA(k) = \\frac{\\mathcal{P}_\\mathcal{R}\\left( k,\\,\\hat{\\textbf{n}} \\right) - \\mathcal{P}_\\mathcal{R}\\left( k,\\,-\\hat{\\textbf{n}} \\right) }{4\\mathcal{P}_{\\mathcal{R}\\:\\: iso}}\\,. \n\t\\label{Akg}\n\\end{equation}\nThe best explanation  for the HPA, so far, invoked an additional modulated superhorizon fluctuation with a wavelength larger than the last scattering surface, whose interactions  with the short wavelength  curvature perturbations introduces large scale dependent primordial non-Gaussianities \\cite{Schmidt:2012ky,Erickcek:2008sm,Erickcek:2008jp,Byrnes:2015asa,Byrnes:2015dub,Namjoo:2014nra}. It was first noted in \\cite{Lyth:2013vha,Lyth:2014mga} that both the HPA and and the power suppression (PS) at low-multipoles \\cite{Akrami:2019bkn} could be addressed in a single framework in the context of scale dependent primordial non-Gaussianities and curvaton-type scenarios. This later has been generalized and phenomenologically studied in model independent way \\cite{Adhikari:2015yya,Byrnes:2016zxb}.  In addition, the HPA has been addressed via a spatial dependent parametrization of the $\\alpha$-vacua Bogoliubov coefficients of de~Sitter space \\cite{Ashoorioon:2015pia}, in the context of anisotropic inflationary models involving gauge fields \\cite{Maleknejad:2012fw} and also with spatial dependent initial conditions for the inflaton field \\cite{Donoghue:2007ze}. So far all the studies ascertained that the HPA cannot be explained by the standard single-field slow-roll inflationary scenario \\cite{Akrami:2014eta,Hansen:2008ym}. \n\nDespite the above mentioned had hoc explanations for the HPA \\cite{Akrami:2019bkn}, Planck data actually hints at it for both 2-point and 3-point correlations (even the equilateral shape for which three wave numbers of the modes are equal). This means that the anomaly seems to be there in all the $n$-point correlations, which begs for a fundamental explanation. Therefore, the answer must be there somewhere in the description of inflationary quantum fluctuations.\n\nThe main goal of this paper is to investigate whether the HPA can be explained through a new understanding of the quantization of the inflationary perturbations without introducing any new parameters and any new degrees of freedom. Thus we only focus on single field slow-roll inflation and propose a new vacuum for inflationary quantum fluctuations. Being more precise, we pay careful attention to the discrete symmetries such as $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ which is expected to be spontaneously broken in a dynamical spacetime. Our scheme of quantization exactly exploits this last aspect leading to a promising explanation for the HPA anomaly, with additional prediction for primordial in gravitational wave spectrum as well.  \n\nThroughout the paper, we use the metric signature $\\left( -,\\,+,\\,+,\\,+ \\right)$, overdot and $^\\prime$ denotes derivative with respect to cosmic time (t) and conformal time ($ \\tau$) respectively, overbar denotes background values for flat Friedmann-Lema\\^itre-Robertson-Walker (FLRW) spacetime described by the metric $ds^2= -dt^2+a^2(t)d\\textbf{x}^2$, $H=\\frac{\\dot{a}}{a}$ denotes the Hubble parameter. \n Four-dimensional indices are labelled by small Greek letters and three dimensional quantities are denoted by $i,j =1,2,3$. We set $\\hbar=c=1$ and $M_p = \\frac{1}{\\sqrt{8\\pi G}}=1$. \n\n\\section{Brief review of inflationary quantum fluctuations in single field inflation}\n\\label{sec:review}\nIn this section, we review standard inflationary quantum fluctuations and the corresponding derivation of observables \\cite{Martin:2015dha,Baumann:2018muz}. In standard single field inflation, we start with a classical action for a canonical scalar field with a suitable choice of potential.\\footnote{Note that inflationary scenarios like Starobinsky and Higgs inflation can be brought to this form via a conformal transformation \\cite{Kehagias:2013mya}.}\n\\begin{equation}\\label{action-S}\n\tS= \\int d^4x\\sqrt{-g} \\left[ \\frac{M_p^2}{2} R- \\frac{1}{2} (\\partial_\\mu \\phi)( \\partial^\\mu \\phi  )-V\\left( \\phi \\right)\\right]  \\,, \n\\end{equation}\nTo implement inflationary framework of expansion, we first solve the equations of motion of \\eqref{action-S} in flat FLRW spacetime and obtain a quasi-de Sitter solution described by the following conditions on the slow-roll parameters:\n \\begin{equation}\n \t\\epsilon = -\\frac{\\dot{H}}{H^2}\\ll 1,\\quad \\eta = \\frac{\\dot{\\epsilon}}{H\n \t\\epsilon} \\ll 1\\,\\text{,}\n \\label{sldef}\n \\end{equation}\nwhich can be computed from the Friedmann and Raychaudhuri equations given by \n\\begin{equation}\n\tH^2 = \\frac{1}{3}\\left( \\frac{\\dot{\\phi}^2}{2}+V(\\phi) \\right),\\quad \\dot{H} = -\\frac{\\dot{\\phi}^2}{2}\\,. \n\t\\label{Freq}\n\\end{equation}\nThe next step is to study the scalar and tensor perturbations around the obtained inflationary background solution. When it comes to the scalar part we have perturbations arising from metric fluctuations ($g_{\\mu\\nu}=\\bar{g}_{\\mu\\nu}+\\delta g_{\\mu\\nu}$) and also the scalar field fluctuations ($\\phi = \\bar{\\phi}+\\delta\\phi$). The metric fluctuations can be represented by the Arnowitt-Deser-Misner (ADM) metric of the form \n\\begin{equation}\\label{ADMmetric}\n\tds^2 = - a^2\\left( \\tau \\right) \\Big(N^2d\\tau^2+  \\gamma_{ij} \\left( dx^i+N^i d\\tau \\right) \\left( dx^j+N^j d\\tau \\right) \\Big)\\,.\n\\end{equation}\nwhere $d\\tau = \\frac{dt}{a}$ is conformal time, $ N$ and $N_i$ are the lapse and shift functions respectively.  In the unitary gauge, we fix \n$\\delta\\phi=0$ then we linearly expand the ADM metric  as\n\\begin{equation}\\label{eq:ADMfluctuation}\n\tN= 1+\\delta N,\\quad N_i = a\\partial_i\\chi,\\quad \\gamma_{ij} =  \\left( 1+2\\zeta \\right)\\delta_{ij}+h_{ij}\\,, \n\\end{equation}\nwhere $\\chi$ is a scalar function of spacetime, $\\zeta $ is the curvature perturbation , $h_{ij}$ is the transverse and traceless spin-2 fluctuation. From the linear perturbed equations of motion we obtain the following constraints \n\\begin{equation}\n\t\\begin{aligned}\n\\delta N & = \\frac{\\dot{\\zeta}}{H} \\\\ \n\\partial^2\\chi & = -\\frac{\\partial^2 \\zeta}{H}+ \\epsilon a^2\\dot{\\zeta}\n\\end{aligned}\n\\end{equation}\n{Substituting \\eqref{ADMmetric} into the action \\eqref{action-S} and expanding to second order in the fluctuations, we find the second order action for the scalar perturbation}:\n\\begin{equation}\\label{scalar}\n\t\\delta^{(2)}S_{s} = \\frac{1}{2}\\int d\\tau d^3x a^2\\frac{\\phi^{\\prime 2}}{\\mathcal{H}^2} \\Bigg[ \\zeta^{\\prime 2} -\\left( \\partial\\zeta \\right)^2 \\Bigg]\n\t\\,\\text{,}\n\\end{equation}\nwhere $\\mathcal{H} = \\frac{a^\\prime}{a}= aH$. \nSimilarly we obtain the following second order action for tensor fluctuations:\n\\begin{equation}\n\t\\delta^{(2)}S_h = \\frac{1}{8} \\int d\\tau d^3x a^2\\Bigg[  h_{ij}^\\prime h_{ij}^\\prime -\\partial_k h_{ij}\\partial^k h^{ij}   \\Bigg]\n\t\\,\\text{.}\n\\end{equation}\nWhat we have so far is totally classical and we haven't promoted any fields and conjugate momenta to operators. To quantize the fluctuations we first redefine the fields as \n\\begin{equation}\n\tv = a\\frac{\\dot{\\phi}}{H}\\zeta,\\quad u_{ij} = \\frac{a}{2} h_{ij} \\,,\n\t\\label{rescaleeq}\n\\end{equation}\nThe second order actions in these new variables (which are famously known as Mukhanov-Sasaki variables) become \n\\begin{equation}\n\t\\delta^{(2)}S_s = \\frac{1}{2} \\int d\\tau d^3x \\Big[v^{\\prime 2}-\\left(\\partial_iv\\right)^2-\\left( \\frac{\\nu_s^2-\\frac{1}{4}}{\\tau^2} \\right) v^2\\Big],\n\t\\label{screfa}\n\\end{equation}\nwhere \n\\begin{equation}\n\t\\nu_s \\approx \\frac{3}{2}+\\epsilon+\\frac{\\eta}{2}\n\\end{equation}\n\\begin{equation}\n\t\\delta^{(2)}S_s = \\int d\\tau d^3x  \\Big[u_{ij}^{\\prime 2}-\\left(\\partial_iu_{ij}\\right)^2-\\left( \\frac{\\nu_t^2-\\frac{1}{4}}{\\tau^2} \\right) u_{ij}^2\\Big],\n\t\\label{tensor-mode}\n\\end{equation}\nand\n\\begin{equation}\n\t\\nu_t \\approx \\frac{3}{2}+\\epsilon\\,. \n\\end{equation}\nWe now promote the fields $v,\\, u_{ij}$ and their corresponding conjugate momenta $\\pi_s,\\, \\pi_{ij}$ to operators and define the canonical commutation relations as \n\\begin{equation}\n\t\\begin{aligned}\n\t\\Big[\\hat{v}\\left( \\tau,\\,\\textbf{x} \\right),\\, \\hat{\\pi}_s\\left( \\tau,\\, \\textbf{x}^\\prime \\right)\\Big] & = i\\delta\\left( \\textbf{x}-\\textbf{x}^\\prime \\right) \\\\ \n\t\t\\Big[\\hat{u}_{ij}\\left( \\tau,\\,\\textbf{x} \\right),\\, \\hat{\\pi}_{ij}\\left( \\tau,\\, \\textbf{x}^\\prime \\right)\\Big] & = i\\delta\\left( \\textbf{x}-\\textbf{x}^\\prime \\right)\n\t\t\\label{cancom}\n\t\\end{aligned}\n\\end{equation}\nWe expand the field $\\hat{v}\\left( \\tau,\\, \\textbf{x} \\right)$ in terms of creation and annihilation operators  $ a_\\textbf{k},\\, a^\\dagger_{\\textbf{k}} $ as\n\\begin{equation}\n\t\\begin{aligned}\n\t\\hat{v}\\left( \\tau,\\, \\textbf{x} \\right) & = \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int d^3k \\hat{v}_\\textbf{k}\\left( \\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} \\\\ \n\t& = \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int  d^3k\\Bigg[ a_\\textbf{k} {v}_k\\left( \\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} + a_\\textbf{k}^\\dagger {v}_k^\\ast e^{-i\\textbf{k}\\cdot \\textbf{x}} \\Bigg]\\, \n\t\\end{aligned}\n\\end{equation}\nwhere $v_k$ is the mode function which satisfies Mukhanov-Sasaki (MS) equation \n\\begin{equation}\n\tv_k^{\\prime\\prime} + \\left( k^2-\\frac{v_s^2-\\frac{1}{4}}{\\tau^2}  \\right) v_k = 0\\,. \n\t\\label{MS-equation}\n\\end{equation}\nand $\\left[ a_\\textbf{k},\\, a^\\dagger_{\\textbf{k}^\\prime}\\right]  = \\delta\\left( \\textbf{k}-{\\textbf{k}}^\\prime \\right)$\nThe generic solution of \\eqref{MS-equation} by treating $\\nu_s\\approx\\, {\\rm const}$  is \n\\begin{equation}\n\t\\begin{aligned}\n&\tv_k\\left( \\tau \\right)   \\\\ &= \\dfrac{\\sqrt{\\pi}}{2} \\left(  -\\tau  \\right)^{1\/2} e^{i\\left( 2\\nu_s+1\\right) \\pi}\\Bigg[A_k H^{(1)}_{\\nu_s}\\left( -k \\tau \\right)+ B_k H^{(2)}_{\\nu_s}\\left( -k  \\tau \\right)\\Bigg].\n\t\\end{aligned}\n\\end{equation}\nwhere $A_k$ and $B_k$ are the Bogoliubov coefficients. \nDue to the commutation relations \\eqref{cancom}, the mode functions must satisfy \n\\begin{equation}\nv_kv^{\\ast\\prime}_k - v_k^\\ast v^{\\prime}_k = i\n\\end{equation}\nwhich implies\n\\begin{equation}\n\t\\vert A_k\\vert^2 - \\vert B_k\\vert^2 =1\\,.\n\t\\label{wrAB}\n\\end{equation}\nFrom the above relation we can carry a parametrization $A_k = \\cosh\\alpha$ and $B_k =\\sinh\\alpha$. The case $\\alpha\\neq 0$ corresponds the well-known $\\alpha-$ vacuum in the limit $\\left( \\epsilon,\\,\\eta \\right)\\to 0$ \\cite{Allen:1985ux}. \nThe case $\\alpha=0$ corresponds to the Bunch-Davies (BD) vacuum in which case the mode functions behave like Minkowski-one in the limit $\\tau\\to -\\infty$ when they are deep inside the horizon i.e., when $k\\gg aH$ the mode function of the BD vacuum becomes \n\\begin{equation}\n\tv_k \\simeq \\sqrt{\\frac{1}{2k}}e^{-ik\\tau}\n\\end{equation}\nThus the BD vacuum is given by\n\\begin{equation}\na_\\textbf{k} \\vert 0\\rangle =0\\,.\n\\end{equation}\nIn the limit $\\left( \\epsilon,\\, \\eta \\right)\\to 0$ the mode functions can be approximated as \n\\begin{equation}\n\tv_k\\left( \\tau \\right) = \\sqrt{\\frac{1}{2k}} e^{-ik\\tau}\\left( 1-\\frac{i}{k\\tau} \\right)\n\\end{equation} \nThe two point correlations of the fields can be computed as \n\\begin{equation}\n\t\\langle 0 \\vert \\hat{v}\\left( \\textbf{x},\\, \\tau \\right)  \\hat{v}\\left( \\textbf{x}+\\boldsymbol{\\xi}, \\, \\tau \\right)\\vert 0\\rangle = \\frac{4\\pi}{\\left( 2\\pi \\right)^3} \\int  \\frac{dk}{k} \\frac{\\sin k \\xi}{k\\xi} k^3 \\vert v_k\\vert^2\\,.\n\\end{equation}\n The power spectrum of the canonical scalar field $v$ is defined as \n\\begin{equation}\n\\begin{aligned}\n\t\\langle 0\\vert \\hat{v}_\\textbf{k}\\hat{v}_{\\textbf{k}^\\prime}\\vert 0\\rangle & = \\vert v_k\\vert^2 \\delta\\left( \\textbf{k}+\\textbf{k}^\\prime \\right)\\, \\\\ \n\t& = P_v\\left( k \\right) \\delta\\left( \\textbf{k}+\\textbf{k}^\\prime \\right)\n\t\\end{aligned}\n\\end{equation}\nThe power spectrum of curvature perturbation is obtained by pulling back the rescaling we have performed in \\eqref{rescaleeq}\n\\begin{equation}\n\t\\langle \\zeta_\\textbf{k}\\zeta_{\\textbf{k}^\\prime}\\rangle  = \\frac{1}{2a^2\\epsilon} \\times \\langle v_\\textbf{k} v_{\\textbf{k}^\\prime}\\rangle\\, = \\frac{2\\pi^2}{k^3} P_\\zeta \\left( k \\right) \\delta\\left( \\textbf{k}+\\textbf{k}^\\prime \\right)\\,. \n\\end{equation}\nThe curvature perturbation $\\zeta$ freezes out on super-horizon scales and therefore we evaluate the spectrum of curvature perturbation at the time of horizon exit $k = a H$. \nThus, the dimensionless power spectrum of curvature perturbation becomes\\footnote{Since the curvature perturbation is conversed on super-horizon scales the (scale dependent) magnitude of power spectrum is given by \\eqref{Amppr} which is evaluated at Horizon exit.}\n\\begin{equation}\n\t\\begin{aligned}\n\tP_\\zeta\\left( k \\right) \\Big\\vert_{k=aH} & = \\frac{k^3}{2\\pi^2} \\frac{1}{2a^2\\epsilon}P_v\\\\ \n\t& = \\frac{H^2}{8\\pi^2\\epsilon} \\Big\\vert_{k =a H}\\,.\n\t\\end{aligned}\n\\label{Amppr}\n\\end{equation}\nIn the above derivation we substituted $\\tau \\approx -\\frac{1}{aH}$. \nThe tilt of the power spectrum $P_\\zeta$ can be computed as \n\\begin{equation}\n\tn_s -1 = \\frac{d\\ln P_\\zeta}{d\\ln k} \\approx -\\frac{d\\ln P_\\zeta}{dN} \\approx -2\\epsilon-\\eta\\,. \n\t\\label{speci}\n\\end{equation}\nNote that the above result \\eqref{Amppr} and \\eqref{speci} remains the same even without performing the approximation $\\nu_s\\approx 3\/2\\implies \\left( \\epsilon,\\, \\eta \\right)\\to 0$ \\cite{Kinney:2009vz}. Since the power spectrum is obtained by evaluating it at $k =aH$, the small corrections to the parameter $\\nu_s$ give subdominant results. In addition, the spectral index $n_s$ and $n_t $ are often written in terms of $\\nu_s,\\, \\nu_t$, but such definitions are not precisely correct as it is explained in  \\cite{Kinney:2009vz}. One must indeed evaluate the power spectrum at $k = aH$ because beyond this point curvature perturbation is frozen. So, the super-horizon amplitude of the power spectrum can be written in terms of quantities evaluated at the horizon exit \\cite{Leach:2000yw} by fixing the normalization factors \\cite{Kinney:2009vz}. \nSimilarly,  we can work out the mode function for the tensor fluctuations and imposing the BD vacuum we obtain \n\\begin{equation}\n\t\\begin{aligned}\n\t& u_{ij}  = e_{ij} \\dfrac{\\sqrt{\\pi}}{2} \\left(  -\\tau  \\right)^{1\/2} e^{i\\left( 2\\nu_t+1\\right) \\pi} H^{(1)}_{\\nu_t}\\left( -k \\tau \\right)\n\t\\end{aligned}\n\\end{equation}\nWe can perform similar calculations for the tensor power spectrum and tilt and as a result we obtain \n\\begin{equation}\n\tP_T = \\frac{2H^2}{\\pi^2}\\Bigg\\vert_{k= a H},\\quad n_t = -\\frac{d\\ln P_T}{d\\ln N} \\Bigg\\vert_{k= a H} = -2\\epsilon\\,. \n\\end{equation}\n\n\n\\section{New scheme of quantizing inflationary power spectra} \n\\label{sec:questions}\nIn the previous section, we have discussed the standard quantization scheme of inflationary fluctuations. In this section, we enlist some open questions in the context of quantization of inflationary  fluctuations. Based on these, we propose a new scheme of quantization of the inflationary fluctuations, which not only target some of the open questions, but also leads to some observable features in the CMB and primordial gravitational wave spectra. Below we discuss point by point some of the aspects of standard calculations of inflationary quantum fluctuations in relation with our understanding of spacetime in quantum theory and general relativity.  \n\\begin{itemize}\n\t\\item By quantizing inflationary fluctuations we actually deal with quantum field theory (QFT) in curved spacetime. But in standard QFT we have particles that propagate forward and backward in time (anti-particle). A natural question is what happens to particles and anti-particle states in a curved spacetime. This implies one should consistently define the role of time reversal in a curved spacetime. \n\t\\item QFT in Minkowski spacetime  is mainly built on discrete symmetries such as $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ or $\\mathcal{P}\\mathcal{T}$ (in the context of real scalar field), but in curved spacetime we often do not pay attention to the notion of discrete symmetries. The open question here is whether $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ holds or not in curved spacetime? \n   \\item\n   In classical GR time is a coordinate and in quantum theory time is a parameter \\cite{Donoghue:2017pgk,Donoghue:2019ecz}. In the case of inflationary cosmology we usually quantize both gravitational and matter degrees of freedom \\cite{Martin:2004um,Baumann:2018muz,Kinney:2009vz} taking a classical notion of time. However, the concept of time in quantum theory is very different from the classical one \\cite{Rovelli:2004tv}.\n   \\item Since we quantize gravitational degrees of freedom, we can interpret inflationary quantum fluctuations as features of (linearized) quantum gravity \\cite{Martin:2004um}. Note that, in the canonical quantum gravity that emerges through Wheeler-de Witt equation, time does not appear explicitly which indicates a difficulty to define positive\/negative frequencies, like we usually do following the standard Schr\\\"{o}dinger equation. This is famously known as the problem of time in quantum cosmology \\cite{Kiefer:2007ria,Rovelli:2004tv}.\n  \n   \\item Quantum theory is always time symmetric. The arrow of time emerges only after we specify initial and final states \\cite{Hartle:2013tm}. This implies that we could first formulate QFT in curved spacetime, in a time symmetric way, and then impose initial conditions. \n\\end{itemize}\nOur new formalism for the inflationary quantum fluctuations is highly motivated from the above questions. Before we go into details, we write down here few guidelines\n\n\\begin{itemize}\n\t\\item We identify the expansion or contraction of Universe with the shrinking or growing size of the co-moving horizon (in the context of de Sitter and quasi de Sitter). Change of size of the horizon defines our classical or thermodynamical arrow of time.\n\t\\item We build the quantum theory based on discrete transformations of spacetime within the co-moving horizon. We completely detach ourselves from the notions of observers, Penrose diagrams and the corresponding regions of spacetime when we do quantization because all such concepts are not well-defined according to quantum theory. In fact, classical notion of region of spacetime cannot have any consistent meaning from the quantum mechanical point of view \\cite{Donoghue:2021meq}. \n\t\\item \n\tThe way forward is to first understand quantum mechanics in a dynamical spacetime and then take the classical limit. To achieve this goal, we take a small step by distinguishing classical notion of time with the quantum mechanical notion of time. This small step allow us to create all possible quantum states in the given spacetime geometry.\n\t\\item In our formalism, quantum fluctuations always come in pair from two different vacua described by $\\mathcal{P}\\mathcal{T}$ or $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ transformations. By this scheme we have the quantum fluctuations that propagate forward in time as well as backward in time and both can exist within the horizon. As the horizon changes, both of these type of fluctuations either exit or enter the horizon depending on whether the spacetime is expanding or contracting.\n\t\\item These pairs of quantum fluctuations within the horizon evolve independently of each other which means the commutator of corresponding operators always vanish. This means even though we can create quantum states non-locally we still do not violate locality and causality in our formulation of QFT.   \n\\end{itemize}\n\n\\section{Quantum fluctuations in an expanding de Sitter spacetime}\n\\label{sec:dsspace}\nIn this section, we focus on elucidating our proposal in the context of exact de Sitter (dS) spacetime in FLRW co-ordinates. Since inflationary spacetime is near dS there are several concepts we can take forward from the understanding of dS. To make our analysis simpler we consider quantization of a massless scalar field in dS spacetime ignoring the issues with IR singularities and defining dS invariant vacuum \\cite{Higuchi:2009ew,Allen:1985ux}.\n\nThe dS spacetime in FLRW coordinates looks like\n\\begin{equation}\n\t\\begin{aligned}\n\tds^2 & = -dt^2 + a(t)^2d\\textbf{x}^2\n\t            & = \\frac{1}{H^2\\tau^2}\\left( -d\\tau^2+ d\\textbf{x}^2  \\right)\\,. \n\t\\end{aligned}\n\t\\label{dsmetric}\n\\end{equation}\nwhere $d\\tau= \\frac{dt}{a}$ is the conformal time  and the\nscale factor is given by\n\\begin{equation}\n\ta(t) = e^{Ht},\\quad H^2 = \\left(\\frac{1}{a}\\frac{da}{dt}\\right)^2={\\rm const}\\,.\n\\end{equation}\nNote that dS metric \\eqref{dsmetric} is $\\mathcal{P}\\mathcal{T}: \\tau\\to -\\tau,\\, \\textbf{x}\\to -\\textbf{x}$ symmetric. \nThe scalar curvature of dS spacetime is $R=12H^2$ which does not tell whether the Hubble parameter is positive or negative. Each point in dS is surrounded by a co-moving horizon given by the radius\n\\begin{equation}\n\tr_H = \\Big\\vert  \\frac{1}{aH} \\Big\\vert\n\t\\label{horizonds}\n\\end{equation}\nOne very simple observation we can make from \\eqref{dsmetric} is that the spatial size of the Universe $a(t)^2$ grows with coordinate time $t$ when $H>0$. We then interpret the metric \\eqref{dsmetric} as modelling an expanding universe. \nIn this case the scale factor grows with time $t$ and the size of the horizon shrinks. \nWe note here that this observation stems from the convention that $t$ runs from $-\\infty $ to $+\\infty$ (which means conformal time $-\\infty<\\tau< 0$). Defining $\\bar{t}= -t$, which corresponds to \\say{$t$ running from $+\\infty$ to $-\\infty$ or $+\\infty>\\tau> 0$} we find the scale factor $a(\\tau)$ growing with $\\bar{t}$ for $H<0$, corresponding to expansion. Therefore, the expanding dS space can only characterized by shrinking horizon rather than fixing the notion of time being positive or negative. Since time is a parameter in quantum theory, we can have quantum states that are labeled by $\\tau<0$ and\/or $\\tau>0$. In the case of standard QFT in dS one usually make a biased choice of either $\\tau<0$ or $\\tau>0$ which actually breaks the time reversal nature of dS spacetime and also standard scheme of quantization completely ignores existence of states that evolve backward in time. \nTherefore, \n in our scheme of quantization in dS spacetime, we can have quantum states that propagate forward in time $t: -\\infty \\to \\infty$ with $H > 0$ and states that propagate backward in time following an arrow of time $t: \\infty \\to -\\infty$ with $H< 0$. Note that both type of quantum states can co-exist in a single expanding Universe. We stress again, the expanding dS Universe is defined by the following arrows of time\n\\begin{equation}\n\t\\tau:  \\pm \\infty \\to 0\n\\end{equation}\nwhere $-$ sign corresponds to a state propagating forward in time and $+$sign  corresponds to a state propagating backward in time. In our new picture quantization of a (massless) scalar field in dS spacetime leads to a pair of  quantum fields that are produced out of two vacua. Being more precise we define total vacuum as the direct sum of two vacua\n\\begin{equation}\n\t\\vert 0\\rangle = \\frac{1}{\\sqrt{2}} \\Bigg(\\vert 0\\rangle_{\\rm I}\\oplus \\vert 0\\rangle_{\\rm II} \\Bigg)\\,. \n\\end{equation}\nIn the vacuum $\\vert 0\\rangle_{\\rm I}$ we create quantum fields at the position $ \\textbf{x} $ that evolve forward in time ($\\tau <0 $) and in the vacuum $\\vert 0\\rangle_{\\rm II}$ we create quantum fields at $ -\\textbf{x}$ which evolve backward in time ($\\tau >0$).\nThe action of a massless scalar field in dS space is given \n\\begin{equation}\nS_{\\phi} = -\\frac{1}{2}\\int d\\tau d^3x a^2\t\\phi\\left( \\partial_\\tau^2+2\\mathcal{H} \\partial_\\tau + k^2 \\right) \\phi\\,. \n\\end{equation}\nFirst thing we can notice here is that the above action is invariant under $\\mathcal{P}\\mathcal{T}$ which means  $\\textbf{x}\\to -\\textbf{x}$ and \n\\begin{equation}\\label{timeref}\n\tt\\to -t\\quad  \\left(-\\tau \\to \\tau\\right) \\implies H\\to -H \\quad \\left( \\mathcal{H}\\to -\\mathcal{H} \\right)\n\\end{equation}\nRescaling the field $\\phi\\to a \\phi$  gives the following action which leads to Mukhanov-Sasaki equation as\n\\begin{equation}\n\tS_\\phi = \\frac{1}{2} \\int d\\tau d^3x \\Big[ \\phi^{\\prime 2} - \\left( \\partial_i\\phi \\right)^2  -  \\frac{2}{\\tau^2}  \\phi^2 \\Big]\\,. \n\t\\label{msaction}\n\\end{equation}\nThe above Lagrangian is again symmetric under $\\mathcal{P}\\mathcal{T}: \\textbf{x}\\to -\\textbf{x},\\, \\tau\\to -\\tau$. When we quantize this scalar field\nwe propose that the scalar field operator $\\hat{\\phi}\\left( \\tau,\\, \\textbf{x} \\right)$ is the direct sum of the pair of fields as\n\\begin{equation}\n\t\\hat{\\phi}\\left( \\tau,\\, \\textbf{x} \\right) =  \\frac{1}{\\sqrt{2}}\t\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\oplus \t\\frac{1}{\\sqrt{2}}\\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right) \\,. \n\\end{equation}\nwhich can be expanded in terms of different creation and annihilation operators in the following way\n\\begin{equation}\n\t\\begin{aligned}\n\t&\t\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right)  \\\\ & = \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int d\\tau d^3k\\Bigg[ a_\\textbf{k} {\\varphi}_{\\rm I\\,k}\\left( \\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} + a_\\textbf{k}^\\dagger {\\varphi}^\\ast_{\\rm I\\,k}\\left( \\tau \\right) e^{-i\\textbf{k}\\cdot \\textbf{x}} \\Bigg]\\, \n\t\\end{aligned}\n\\label{vdfield}\n\\end{equation}\nwhere the creation and annihilation operators $a_\\textbf{k},\\, a^\\dagger_\\textbf{k}$ satisfy the canonical commutation relations  and \n \\begin{equation}\na_\\textbf{k}\\vert 0 \\rangle_{\\rm I} = 0\\,. \n\\end{equation}\nHere the mode function $\\varphi_{\\rm I\\,,k}$ is the solution of Mukhanov-Sasaki equation (that comes from varying \\eqref{msaction}) given by \n\\begin{equation}\n\t\\varphi_{\\rm I,\\,k}  = \\alpha_{1k} \\frac{e^{-ik\\tau}}{\\sqrt{2k}}\\left( 1-\\frac{i}{k\\tau} \\right) +\\beta_{1k} \\frac{e^{ik\\tau}}{\\sqrt{2k}}\\left( 1+\\frac{i}{k\\tau} \\right)\\,, \n\t\\label{f1}\n\\end{equation}\nwhere the Bogoliubov coefficients can be fixed as $\\left( \\alpha_{1,\\,k},\\, \\beta_{1,\\,k}\\right) = \\left( 1,\\,0 \\right)$, which is compatible with the Wronskian condition $\t\\varphi_{\\rm I,\\,k} \t\\varphi^{\\prime\\ast}_{\\rm I,\\,k}-\t\\varphi^\\ast_{\\rm I,\\,k}\t\\varphi^\\prime_{\\rm I,\\,k} = i   $ that corresponds to the canonical commutation relation\n\\begin{equation} \n\\big[ \\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right),\\, \\hat{\\pi}_{\\rm I}\\left( \\tau,\\, \\textbf{x}^\\prime \\right)\\big] = i\\delta\\left( \\textbf{x}-\\textbf{x}^\\prime \\right). \n\\end{equation}\nThe choice $\\left( \\alpha_{\\rm I,\\,k},\\, \\beta_{\\rm I,\\,k}\\right) = \\left( 1,\\,0 \\right)$ defines the vacuum and $\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\vert 0\\rangle $ corresponds to the positive frequency modes in the limit $\\tau\\to -\\infty$. Quantum mechanically $\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\vert 0\\rangle $ is the postive energy state that propagate forward in time at \\textbf{x}. \nSimilarly,  we can expand the second field operator as \n\\begin{equation}\n\t\\begin{aligned}\n\t\t&\t\\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right)  \\\\ & = \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int d\\tau d^3k\\Bigg[ b_\\textbf{k} \\varphi_{\\rm II\\,k}\\left( -\\tau \\right) e^{-i\\textbf{k}\\cdot \\textbf{x}} + b_\\textbf{k}^\\dagger {\\varphi}^\\ast_{\\rm II\\,k}\\left( -\\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} \\Bigg]\\, \n\t\\end{aligned}\n\t\\label{vdfield2}\n\\end{equation}\nwhere \n\\begin{equation}\n\t\\varphi_{\\rm II,\\,k}  = \\alpha_{2,\\,k} \\frac{e^{ik\\tau}}{\\sqrt{2k}}\\left( 1+\\frac{i}{k\\tau} \\right) +\\beta_{2,\\,k} \\frac{e^{-ik\\tau}}{\\sqrt{2k}}\\left( 1-\\frac{i}{k\\tau} \\right)\\,, \n\t\\label{f2}\n\\end{equation}\nwhere the Bogoliubov coefficients can be fixed as $\\left( \\alpha_{2,\\,k},\\, \\beta_{2,\\,k}\\right) = \\left( 1,\\,0 \\right)$ which is compatible with the Wronskian condition $\t\\varphi_{\\rm II,\\,k} \t\\varphi^{\\prime\\ast}_{\\rm II,\\,k}-\t\\varphi^\\ast_{\\rm II,\\,k}\t\\varphi^\\prime_{\\rm II,\\,k} = -i   $ that corresponds to the canonical commutation relation \\cite{Donoghue:2019ecz}\n\\begin{equation} \n\t\\big[ \\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right),\\, \\hat{\\pi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x}^\\prime \\right)\\big] = -i\\delta\\left( \\textbf{x}-\\textbf{x}^\\prime \\right). \n\\end{equation}\nwhich describe the quantum fields that propagate backward in time. \nHere the second vacuum is defined by \n\\begin{equation}\n\tb_\\textbf{k}\\vert 0 \\rangle_{\\rm II} = 0\\,. \n\\end{equation}\nHere  $\t\\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right)\\vert 0\\rangle_{\\rm II}$ corresponds to positive energy state that evolves backward in time at -\\textbf{x}. \nWe demand that these two quantum fluctuations evolve independently. This is manifest by\n\\begin{equation}\n\t[\t\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right),\\, \t\\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right)   ] =0\\,. \n\\end{equation}\nwhich implies that the respective creation and annihilation operators commute\n\\begin{equation}\n\t\\big[a_\\textbf{k},\\, b_{\\textbf{k}^\\prime}\\big] = 0,\\quad \\big[a^\\dagger_\\textbf{k},\\, b^\\dagger_{\\textbf{k}^\\prime}\\big] = 0 \n\\end{equation}\nSince dS spacetime is perfectly $\\mathcal{P}\\mathcal{T}$ symmetric we have that the quantum fields  $\t\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right)\\vert 0\\rangle_{\\rm I}$ and  $\t\\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right)\\vert 0\\rangle_{\\rm II}$ behave identically, which can be seen from the fact that their equal time correlations are the same\n\\begin{equation}\n\t\\begin{aligned}\n& \\frac{1}{a^2}{}_{\\rm I}\\langle 0\\vert \\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x}^\\prime \\right)\\vert 0\\rangle_{\\rm I} = \\\\ & \\frac{1}{a^2} {}_{\\rm II}\\langle 0\\vert \\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right) \\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x}^\\prime \\right)\\vert 0\\rangle_{\\rm II} = \\frac{H^2}{4\\pi^2k^3}\\,. \n\t\\end{aligned}\n\\label{eqcorr}\n\\end{equation}\nThis implies the correlations of quantum fields related by $\\mathcal{P}\\mathcal{T}$ transformations are identical in the case of dS spacetime.  \nThis indicates our formulation of QFT in dS spacetime might be $\\mathcal{P}\\mathcal{T}$ or $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ (if we include charged fields) symmetric.\\footnote{We note that the result in \\eqref{eqcorr} is similar to what was derived in the context of elliptic dS spacetime \\cite{SANCHEZ19871111,Schrodinger1956}. However, in our scheme of quantization we do not make any (classical) antipodal identification of spacetime and therefore we significantly deviate conceptually from the proposal of elliptic dS made by Erwin Schr\\\"{o}dinger \\cite{Schrodinger1956}.  }\n\nHowever, this may not be true in a spacetime with less symmetries such as quasi-dS. Indeed, in the context of inflationary quantum fluctuations, it is often assumed, in the effective field theory (EFT) approach, that Lorentz symmetry (in other words $(\\mathcal{C})\\mathcal{P}\\mathcal{T}$) must be spontaneously broken \\cite{Cheung:2007st}. However, it is not known how $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ symmetry can be broken in a curved spacetime. \n\t In the next section, we provide a framework of quantization where we can explicitly see that $(\\mathcal{C})\\mathcal{P}\\mathcal{T}$ is spontaneously broken in the sense that the qunatum fields defined by our new notion of $\\mathcal{P}\\mathcal{T}$ transformations evolve differently. \n\n\\section{Double vacuum inflationary spacetime and the power spectra}\nIn the previous section, we have studied the quantization of a massless scalar field in dS. Here we quantize the inflationary quantum fluctuations. There is a crucial difference here. In the previous section we quantized an arbitrary scalar field in dS spacetime which is completely fixed but in the context of inflation we quantize metric and matter degrees of freedom in order to find an effective quantum correction to the classical quasi-dS (inflationary) spacetime. To be more precise, the classical inflationary spacetime is given by an arrow of time and the background initial conditions are \n\\begin{equation}\nt: -\\infty \\to \\infty \\implies \\left( \\epsilon,\\, \\eta \\right): \\left( \\approx 0,\\, \\approx 0 \\right) \\to \\left( \\sim 1,\\, \\sim 1 \\right)\\,. \n\\end{equation}\nWhich means the slow-roll parameters are very small and positive during inflation and inflation ends when the slow-roll parameters evolve to the order of unity, which happens at some time $t_{\\rm end}\\gg 1$. The inflationary quantum fluctuations are produced in the phase when $\\left( \\epsilon,\\, \\eta \\right) \\ll 1$. As we discussed earlier, ``time'' in quantum theory has totally different role and one must be very careful in quantizing inflationary fluctuations. We emphasize that the quantum fluctuations do not necessarily follow arrow of time prescribed by the initial conditions in classical theory. \n\n Unlike dS spacetime the quasi-dS metric does not have $\\mathcal{P}\\mathcal{T}$-symmetry, therefore naturally one would expect the $\\mathcal{P}\\mathcal{T}$-symmetry to be spontaneously broken at the quantum level. To see this, we apply the scheme of quantization similar to what we have done in the context of exact dS. Let us first focus on inflationary scalar fluctuations which are described by the action \\eqref{scalar}. After the field redefinition $\\hat{v}$ is the canonical variable which we need to quantize \\eqref{screfa}. As we postulated before quantum fluctuations always come as pair from two different vacua related by $\\mathcal{P}\\mathcal{T}$ transformations. This means that unlike the standard quantization of inflationary fluctuations, where only one kind of quantum states are considered which only evolve forward in time, here we describe a pair of quantum fluctuations that evolve both forward and backward in time. \n\n Similar to the exact dS case we write the canonical field operator as a direct sum of two quantum fields \n\\begin{equation}\n\t\\hat{v}\\left( \\tau,\\, \\textbf{x} \\right) = \\frac{1}{\\sqrt{2}} \\hat{v}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\oplus  \\frac{1}{\\sqrt{2}} \\hat{v}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right)\\,. \n\\end{equation}\nwhich emerges from two different vacua. The total vacuum is the direct sum of these two which can be written as\n\\begin{equation}\n\t\\vert 0\\rangle_{\\rm qdS} = \\vert 0 \\rangle_{\\rm qdS_{I}} \\oplus \\vert 0\\rangle_{\\rm qdS_{II}}\\,. \n\\end{equation}\nThe quantum field operators acting on the vacua $\\hat{v}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\vert 0\\rangle_{\\rm qdS_{II}} $ and $\\hat{v}_{\\rm I}\\left( -\\tau,\\, -\\textbf{x} \\right)\\vert 0\\rangle_{\\rm qdS_{II}}$ leads to the creation of the pair of quantum fluctuations at $ \\textbf{x} $ and at $ -\\textbf{x}  $ which propagate forward in time and back ward in time respectively. We will make it clear shortly the meaning of quantum fluctuations that evolve forward and backward in time in the context of quasi-dS.\n\nGoing into the details\nwe expand the fields in terms of creation and annhilation operators as explained below. \n\\begin{equation}\n\t\\begin{aligned}\n& \\hat{v}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\\\ &  =\t \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int d\\tau d^3k\\Bigg[ c_\\textbf{k} {v}_{\\rm I,\\,k}\\left( \\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} + c_\\textbf{k}^\\dagger {v}_{\\rm I,\\,k}^\\ast\\left( \\tau \\right) e^{-i\\textbf{k}\\cdot \\textbf{x}} \\Bigg]\n\\end{aligned}\n\\label{vid}\n\\end{equation}\nwith $c_\\textbf{k},\\, c_\\textbf{k}^\\dagger$ being the creation and annihilation operators of quasi-dS vacuum-I defined by\n\\begin{equation}\n\tc_\\textbf{k}\\vert 0\\rangle_{\\rm qdS_I} = 0\\,, \n\t\\label{qds1}\n\\end{equation}\nand $ {v}_{\\rm I,\\,k}$ is the mode function obtained by solving MS-equation for $v_{\\rm I}\\left(\\tau \\right)$ \n\\begin{equation}\n\tv_{\\rm I,\\,k}^{\\prime\\prime}+ \\left( k^2-\\frac{{\\nu}_s^2-\\frac{1}{4}}{\\tau^2} \\right) v_{\\rm I,\\,k}^2 =0\\,.\n\\end{equation}\nHere $\\tau: -\\infty \\to 0$ and \n the quasi-dS vacuum $\\vert 0\\rangle_{\\rm qdS_I}$ corresponds to the creation of positive frequency modes in the limit $\\tau\\to -\\infty$. \nNotice that unlike the dS case the MS-equation \\eqref{MS-equation} is not symmetric under time reversal because there is an additional time dependence enters through the nearly constant variable $\\nu_s$ which contains the slow-roll parameters $\\left( \\epsilon,\\, \\eta \\right)$  \\eqref{sldef}. Assuming $0<\\left( \\epsilon,\\, \\eta\\right) \\ll 1$ and constant we obtain  \n\\begin{equation}\n\t\\begin{aligned}\n&\t{v}_{\\rm I,\\,k}  \\\\ & = \\frac{\\sqrt{\\pi}}{2} \\left( -\\tau \\right)^{1\/2} e^{\\left( i\\nu_s+1\\right)} \\Bigg[C_k H^{(1)}_{\\nu_s}\\left( -k \\tau \\right)+ D_k H^{(2)}_{\\nu_s}\\left( -k  \\tau \\right)\\Bigg]\\,. \n\t\\label{new-vac1}\n\t\\end{aligned}\n\\end{equation}\nThe above mode function corresponds to the creation of positive frequency modes in the limit $\\tau\\to -\\infty$ for the case $\\left( \\mathcal{C}_k,\\, D_k \\right) = \\left( 1,\\,0 \\right) $ which corresponds to the standard BD state that satisfies the Wronskian $v_{I,k}v_{I,k}^{\\prime\\ast}-v_{I,k}^\\ast v_{I,k}^{\\prime}=i  (\\implies \\vert C_k\\vert^2-\\vert D_k\\vert^2=1)$. We can expand the mode function for the case of $\\left( \\mathcal{C}_k,\\, D_k \\right) = \\left( 1,\\,0 \\right) $ in the order of slow-roll as\n\\begin{equation}\n\t\\begin{aligned}\n\t{v}_{\\rm I,\\,k} &  \\approx \\frac{\\sqrt{\\pi}}{2} \\left( -\\tau \\right)^{1\/2} e^{\\left( i\\nu_s+1\\right)}  H^{(1)}_{\\nu_s}\\left( -k \\tau \\right)\\,\\\\ \n\t& \\approx \\sqrt{\\frac{1}{2k}} e^{-ik\\tau}\\left( 1-\\frac{i}{k\\tau} \\right) \\\\ & + \\left( \\epsilon+\\frac{\\eta}{2} \\right) \\frac{\\sqrt{\\pi}}{2\\sqrt{k}} \\sqrt{\\frac{k}{aH}} \\frac{\\partial H^{(1)}_{\\nu_s}\\left( \\frac{k}{aH}\\right)}{\\partial \\nu_s}\\Big\\vert_{\\nu_s=3\/2}\n\\label{new-vac1BD}\n\\end{aligned}\n\\end{equation}\n\nThe second field $\\hat{v}_{II\\,,\\textbf{k}}\\left( -\\tau,\\, -\\textbf{x} \\right)$ is expanded as \n\\begin{equation}\n\t\\begin{aligned}\n&\t\\hat{v}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right) \\\\ & =\t \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int d\\tau d^3k\\Bigg[ d_\\textbf{k} {v}_{\\rm II,\\,k}\\left( -\\tau \\right) e^{-i\\textbf{k}\\cdot \\textbf{x}} + d_\\textbf{k}^\\dagger {v}_{\\rm I,\\,k}^\\ast\\left( -\\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} \\Bigg]\n\t\\end{aligned}\n\\label{v2d}\n\\end{equation}\nwhere $d_\\textbf{k},\\ , d_\\textbf{k}^\\dagger$ are the creation and annihilation operators that satisfy the canonical commutation relations and the second vacuum is defined by\\footnote{We note that $\n\t\t\\big[c_\\textbf{k},\\, d_{\\textbf{k}^\\prime}\\big] = 0,\\quad \\big[c^\\dagger_\\textbf{k},\\, d^\\dagger_{\\textbf{k}^\\prime}\\big] = 0 $ similar to the dS case which means the pair of quantum fluctuations are not causally connected.}\n\\begin{equation}\nd_\\textbf{k}\\vert 0\\rangle_{\\rm qdS_{II}}=0\\,.\n\\end{equation}\nThe evolution of the mode function $v_{\\rm II,\\,k}$ is determined by solving the following time reversed MS-equation \n\\begin{equation}\n\tv_{\\rm II,\\,k}^{\\prime\\prime}+ \\left( k^2-\\frac{\\bar{\\nu}_s^2-\\frac{1}{4}}{\\tau^2} \\right) v_{\\rm II,\\,k}^2 =0\\,.\n\\end{equation}\nHere $\\tau: +\\infty \\to 0$ and\n\\begin{equation}\n\t\\bar{\\nu}_s  \\approx \\frac{3}{2}-\\epsilon-\\frac{\\eta}{2}\n\\end{equation}\nHere the time reversal transformations are \n\\begin{equation}\n\tt\\to -t \\implies H\\to -H,\\quad \\epsilon\\to -\\epsilon,\\quad \\eta\\to -\\eta\\,. \n\t\\label{timerevqds}\n\\end{equation}\nwhich can be understood in the following sense. Our goal here is to define how the quantum fluctuations propagate backward in time in an expanding Universe. To achieve this goal we need to formulate the meaning of time reversal operation. Logically, if the fluctuation propagates forward in time in a slow-roll background, the fluctuation that goes backward in time experience spacetime as a \"slow-climb\"\\footnote{In the standard slow-roll we have scalar field rolling down the potential whose time reversal can be understood as a phantom field that is slowly climbing the potential \\cite{Piao:2004tq}. } which is given by reversing the signs of the parameters $\\left( \\epsilon,\\, \\eta \\right)$ as stated in \\eqref{timerevqds}. We impose this time reversal operation in a completely quantum mechanical sense and this has no classical meaning i.e., our background (classical) dynamics are completely determined by Friedmann equations \\eqref{Freq} and we do not apply at all time reversal to the classical background. Since we treat time differently at quantum level we restrain ourselves from any intuition from classical physics.\\footnote{Notion of time is a very non-trivial concept in physics and its meaning varies in different contexts. We suggest the reader \\cite{Rovelli:2004tv} for an extended physical discussion. Our statement, that quantum states evolving backward in time has no classical analog, is deeply rooted in the quantum gravity concept time as it is presented in p. 184 of \\cite{Rovelli:2004tv}.} Since dynamics of quantum fields emerge from MS-equation \\eqref{MS-equation} the functions $\\left( \\epsilon,\\, \\eta \\right)$ are now treated along with time in as parameters to specify the nature of quantum state. This would encode a subtle difference between quantum fluctuations propagating forward and backward in time.\n\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\t&\tv_{\\rm II,\\,k}\\left( -\\tau \\right)   \\\\ &= \\dfrac{\\sqrt{\\pi}}{2} \\left(  \\tau  \\right)^{1\/2} e^{i\\left( 2\\bar{\\nu}_s+1\\right) \\pi}\\Bigg[\\bar{C}_k H^{(1)}_{\\bar{\\nu}_s}\\left( k \\tau \\right)+ \\bar{D}_k H^{(2)}_{\\bar{\\nu}_s}\\left( k  \\tau \\right)\\Bigg].\n\t\t\t\\end{aligned}\n\t\t\\label{v2sol}\n\\end{equation}\nNow imposing the Wronskian condition $v_{\\rm II,\\,k}v_{\\rm II,\\,k}^{\\prime\\ast}-v_{\\rm II,\\,k}^\\ast v_{\\rm II,\\,k}^\\prime = -i$ which corresponds to the canonical commutation relation for a reversed arrow of time \\cite{Donoghue:2019ecz}\n\\begin{equation}\n\t\\Big[ \\hat{v}_{\\rm II}\\left( -\\tau,\\,-\\textbf{x} \\right),\\,  \\hat{\\Pi}_{\\rm II}\\left( -\\tau,\\,-\\textbf{x}^\\prime \\right) \\Big] = -i \\delta\\left( \\textbf{x}-\\textbf{x}^\\prime \\right)\\,. \n\\end{equation}\nwe obtain $\\vert \\bar{C}_k\\vert^2-\\vert \\bar{D}_k\\vert^2 = 1$. We consider $\\left(\\bar{C},\\, \\bar{D}_k\\right) =0$ which corresponds postive energy states in the limit $\\tau \\to \\infty$. Expanding \\eqref{v2sol} up to the leading order for $\\left(\\bar{C},\\, \\bar{D}_k\\right) =0$ we get\n\\begin{equation}\n\t\\begin{aligned}\n\t\t{v}_{\\rm II,\\,k} &  \\approx \\frac{\\sqrt{\\pi}}{2} \\left( \\tau \\right)^{1\/2} e^{\\left( i\\nu_s+1\\right)}  H^{(1)}_{\\nu_s}\\left( k \\tau \\right)\\,\\\\ \n\t\t& \\approx \\sqrt{\\frac{1}{2k}} e^{ik\\tau}\\left( 1+\\frac{i}{k\\tau} \\right) \\\\ & -\\left( \\epsilon+\\frac{\\eta}{2} \\right) \\frac{\\sqrt{\\pi}}{2\\sqrt{k}} \\sqrt{k\\tau} \\frac{\\partial H^{(1)}_{\\nu_s}\\left( k\\tau \\right)}{\\partial \\nu_s}\\Big\\vert_{\\nu_s=3\/2}\n\t\t\\label{new-vac2BD}\n\t\\end{aligned}\n\\end{equation}\nComparing \\eqref{new-vac1BD} and \\eqref{new-vac2BD} we can deduce that the two mode functions get different slow-roll (quantum) corrections. \nThis is because the parameter $\\nu_s$ that enters in the MS-equation \\eqref{MS-equation} contains a tiny time dependence. In the standard framework of quantization this is treated it to be nearly constant there is only one type of quantum fluctuation (see Sec.~\\ref{sec:review}). But the fact that MS-equation has time dependence in terms of $\\epsilon,\\, \\eta$ is crucial to probe the nature of quantum fluctuations. These quantities change sign under discrete spacetime transformation and this has to be understood in a completely quantum mechanical sense. Our scheme of quantization implies a pair of quantum fluctuations: those that propagate forward in time ($\\tau: -\\infty \\to 0$)  at spatial position $\\textbf{x}$ which behave differently from those fluctuations that propagate backward in time ($\\tau: +\\infty \\to 0$) at the spatial position $-\\textbf{x}$.  These pairs of fluctuations exit the horizon in the two opposite directions and now we observe the modes as they re-enter. \n\nSo far we applied our quantization method for the scalar mode $v\\left( \\tau,\\,\\textbf{x} \\right)$. We can apply the similar method of quantization for the tensor modes \\eqref{tensor-mode} by writing the tensor fluctuation as direct some fluctuations that are related by $\\mathcal{P}\\mathcal{T}$ transformations \n\\begin{equation}\n\t\\hat{u}_{ij} \\left( \\tau,\\, \\textbf{x} \\right)= \\frac{1}{\\sqrt{2}} \\hat{u}^{\\rm I}_{ij}\\left( \\tau,\\, \\textbf{x} \\right) \\oplus  \\frac{1}{\\sqrt{2}}\\hat{u}^{\\rm II}_{ij}\\left( -\\tau,\\,-\\textbf{x} \\right)\n\\end{equation}\nThe above fields can be written in terms of creation and annihilation operators similar to what we have done for \\eqref{vid} and \\eqref{v2d}. The corresponding mode functions tensor modes can be straight forwardly derived as\n\\begin{equation}\n\t\\begin{aligned}\n\t\t{u}^{\\rm I}_{ij,\\,k} \n\t\t& \\approx e_{ij}\\sqrt{\\frac{1}{2k}} e^{-ik\\tau}\\left( 1-\\frac{i}{k\\tau} \\right) \\\\ & + e_{ij} \\epsilon\\frac{\\sqrt{\\pi}}{2\\sqrt{k}} \\sqrt{-k\\tau} \\frac{\\partial H^{(1)}_{\\nu_t}\\left( -k\\tau \\right)}{\\partial \\nu_t}\\Big\\vert_{\\nu_t=3\/2}\n\t\\end{aligned}\n\t\t\\label{new-vac1TBD}\n\\end{equation}\nand \n\\begin{equation}\n\t\\begin{aligned}\n\t\t{u}^{\\rm II}_{ij,\\,k} \n\t\t& \\approx e_{ij}\\sqrt{\\frac{1}{2k}} e^{ik\\tau}\\left( 1+\\frac{i}{k\\tau} \\right) \\\\ & - e_{ij} \\epsilon\\frac{\\sqrt{\\pi}}{2\\sqrt{k}} \\sqrt{k\\tau} \\frac{\\partial H^{(1)}_{\\nu_t}\\left( k\\tau \\right)}{\\partial \\nu_t}\\Big\\vert_{\\nu_t=3\/2}\n\t\\end{aligned}\n\t\t\\label{new-vac2TBD}\n\\end{equation}\nwhere $e_{ij}$ denotes the polarization tensor. \nIn the next section, we will compute the inflationary power spectra for scalar and tensor modes.\n\n\\section{Inflationary power spectra and predictions for hemispherical asymmetry}\n\nAs we learned in the previous section, inflationary quantum fluctuations now come in pair which exit the horizon on two opposite sides. The two point correlations of these fluctuations can be computed as \n\\begin{equation}\n\t\\begin{aligned}\n\t\t&\t{}_{\\rm qdS_{I}}\\langle 0 \\vert \\hat{v}_{\\rm I}\\left( \\textbf{x},\\, \\tau \\right)  \\hat{v}_{\\rm I}\\left( \\textbf{x}+\\boldsymbol{\\xi}, \\, \\tau \\right)\\vert 0\\rangle_{\\rm qdS_{I}}  = \\\\ & \\frac{4\\pi}{\\left( 2\\pi \\right)^3} \\int \\frac{dk}{k} \\frac{\\sin k \\xi}{k\\xi} k^3 \\vert v_{\\rm I,\\,k}\\vert^2\\, \\\\ \n\t\t&\t{}_{\\rm qdS_{II}}\\langle 0 \\vert \\hat{v}_{\\rm II}\\left( -\\textbf{x},\\, -\\tau \\right)  \\hat{v}_{\\rm II}\\left( -\\textbf{x}-\\boldsymbol{\\xi}, \\, -\\tau \\right)\\vert 0\\rangle_{\\rm qdS_{II}}  = \\\\ &\\frac{4\\pi}{\\left( 2\\pi \\right)^3} \\int  \\frac{dk}{k} \\frac{\\sin k \\xi}{k\\xi} k^3 \\vert v_{\\rm II,\\,k}\\vert^2\\,. \n\t\\end{aligned}\n\\label{power-spectrav}\n\\end{equation}\nSubstituting the expressions \\eqref{new-vac1BD} and \\eqref{new-vac2TBD} in the above expressions we learn that the above two correlations are not equal. In the case of exact dS we obtained the correlations to be equal (see \\eqref{eqcorr}) but in the case of quasi-dS we obtained an asymmetry because the background spacetime is not $\\mathcal{P}\\mathcal{T}$ symmetric. \nAs we know that the curvature perturbation is frozen on super-horizon scales. \nTo calculate the two point correlations of curvature perturbation on super-horizon scales we re scale the canonical fields with the classical background quantities \\eqref{rescaleeq}. This implies \n\\begin{equation}\n\t\\begin{aligned}\n{}_{\\rm qdS}\\langle 0 \\vert \\zeta_\\textbf{k} \\zeta_{\\textbf{k}^\\prime} \\vert 0\\rangle_{\\rm qdS} & =  \\left( \\frac{1}{2a^2\\epsilon}\\right)\\Bigg\\vert_{\\rm classical}  \\Big[\\frac{1}{2}{}_{\\rm qdS_I}\\langle 0 \\vert \\hat{v}_{\\rm I\\,\\textbf{k}} \\hat{v}_{\\rm I\\,\\textbf{k}^\\prime} \\vert 0\\rangle_{\\rm qdS_I}\\\\ & \\quad +\\frac{1}{2}{}_{\\rm qdS_{II}}\\langle 0 \\vert \\hat{v}_{\\rm {II}\\,\\textbf{k}} \\hat{v}_{\\rm {II}\\,\\textbf{k}^\\prime} \\vert 0\\rangle_{\\rm qdS_{II}}\\Big] \\\\ \n& = \\frac{2\\pi^2}{k^3}\\left( P_{\\zeta_1}+P_{\\zeta_2} \\right) \\delta\\left( \\textbf{k}+\\textbf{k}^\\prime \\right)\\,, \n\\end{aligned}\n\\end{equation}\nwhere $\\zeta_1,\\,\\zeta_2$ are the curvature perturbations that exit the horizon on two opposite directions and they become frozen on super-horizon scales which follows from \\eqref{power-spectrav}. With appropriate normalization we evaluate the power spectrum of these curvature perturbations at the moment of horizon exit. Computing the two power spectra at the horizon exit we obtain\n\\begin{equation}\n\t\\begin{aligned}\n\tP_{\\zeta1} & = \\frac{k^3}{2\\pi^2}\\frac{1}{2a^2\\epsilon} \\vert v_{\\rm I,\\,k}\\vert^2\\Bigg\\vert_{\\tau = -\\frac{1}{aH}} \\\\ \n\t& \\approx \\frac{H^2}{8\\pi^2\\epsilon}  \\left( 1+\\left( \\frac{k}{aH} \\right)^2 \\right) \\\\ &\\quad + \\frac{H^2}{4\\pi\\epsilon} \\left( \\epsilon+\\frac{\\eta}{2} \\right) \\left(\\frac{k}{aH}\\right)^3 H_{3\/2}^{(1)} \\left( \\frac{k}{aH} \\right) \\frac{\\partial H^{(1)}_{\\nu_s}\\left( \\frac{k}{aH} \\right)}{\\partial \\nu_s}\\Big\\vert_{\\nu_s=3\/2}\\, \\\\\n\t\tP_{\\zeta2} & = \\frac{k^3}{2\\pi^2}\\frac{1}{2a^2\\epsilon} \\vert v_{\\rm II,\\,k}\\vert^2\\Bigg\\vert_{\\tau_\\ast = \\frac{1}{aH}} \\\\ \n\t& \\approx \\frac{H^2}{8\\pi^2\\epsilon}  \\left( 1+\\left( \\frac{k}{aH} \\right)^2 \\right) \\\\ & - \\frac{H^2}{4\\pi\\epsilon} \\left( \\epsilon+\\frac{\\eta}{2} \\right) \\left(\\frac{k}{aH}\\right)^3 H_{3\/2}^{(1)} \\left( \\frac{k}{aH} \\right) \\frac{\\partial H^{(1)}_{\\nu_s}\\left( \\frac{k}{aH} \\right)}{\\partial \\nu_s}\\Big\\vert_{\\nu_s=3\/2}\\,. \n\t\\end{aligned}\n\\label{pw12}\n\\end{equation}\nIf we combine the above two expressions we obtain \n\\begin{equation}\n\tP_\\zeta = \\frac{1}{2}P_{\\zeta1}+ \\frac{1}{2} P_{\\zeta2}  = \\frac{H^2}{8\\pi^2\\epsilon}\\Bigg\\vert_{k=aH}\n\\end{equation}\nwhich is the expression in the context of standard inflation (see Sec.~\\ref{sec:review}).  From \\eqref{power-spectrav} we can deduce that the power spectrum $P_{\\zeta 1}$ can be mapped to the two-point correlations in the direction $\\hat{\\textbf{n}}$ and $P_{\\zeta 2}$ can be mapped to the two-point correlations in the opposite direction $-\\hat{\\textbf{n}}$ of the CMB sky. The difference between the two power spectra gives us the non-zero scale dependent contribution. Our formalism of producing inflationary correlations in pair naturally explains the hemispherical asymmetry and this can also be interpreted  as an indication of spontaneous breaking of $\\mathcal{P}\\mathcal{T}$ symmetry in the inflationary background. In our context we can define the \nthe amplitude of dipolar modulation as (according to Eq.~\\eqref{Akg})\n\\begin{equation}\n\tA(k) = \\frac{P_{\\zeta1}-P_{\\zeta2}}{4P_\\zeta}\n\t\\label{Ako}\n\\end{equation}\nFrom \\eqref{pw12} we can notice that the two power spectra differ only by a small scale dependent correction of the order of slow-roll parameters. In addition, through CMB we can only probe very limited range of $k$ corresponding to initial $7-8$ e-foldings centered around the pivot scale \\cite{Martin:2004um}. \n At the leading order the first terms in the two power spectra dominate which gives us the tilt of the two power spectra nearly the same in the leading order in slow-roll approximation \n\\begin{equation}\n\t\\frac{d\\ln P_{\\zeta 1}}{d\\ln k}\\approx \t\\frac{d\\ln P_{\\zeta 2}}{d\\ln k} \\approx n_s-1 \\approx -2\\epsilon-\\eta\\,.\n\t\\label{tilts2}\n\\end{equation}\nThe above result is in-line with the data from the Planck satellite \\cite{Mukherjee:2015mma,Axelsson:2013mva}. \n\nIn Fig.~\\ref{fig:fig1} we depict the power asymmetry amplitude of the scalar power spectra for $n_s = 0.963$. This plot remains the same for any single-field slow-roll inflation.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{Fig1}\n\t\\caption{In the above plot we depict the amplitude of CMB dipolar modulation or hemispherical asymmetry of scalar power spectra obtained from \\eqref{Ako}. In the plot we choose the pivot scale $k_\\ast=a_\\ast H_\\ast=0.05 {\\rm Mpc}^{-1}$ and fix $n_s=0.963$. The blue dot with error bar in the plot corresponds to the observational constraint $\\vert A \\vert = 0.066\\pm 0.021$ on HPA at large angular scales or at low-$\\ell-2-64$ or $k\\lesssim 10^{-1}k_\\ast$. We can notice from the plot that the HPA ceases to exist for the small angular scales or high-$\\ell$ which is compatible with the Planck satellite observations \\cite{Planck:2013lks,Akrami:2014eta,Aiola:2015rqa}.  }\n\t\\label{fig:fig1}\n\\end{figure}\nSimilar to the scalar perturbations we propose the tensor perturbations to be direct sum of two kinds of fluctuations related by our definition of $\\mathcal{P}\\mathcal{T}$ transformation. \n\\begin{equation}\n\t\\hat{u}_{ij}\\left( \\tau,\\, \\textbf{x} \\right) = \\frac{1}{\\sqrt{2}}\t\\hat{u}_{\\rm I,\\, ij}\\left( \\tau,\\, \\textbf{x} \\right) \\oplus \\frac{1}{\\sqrt{2}} \\hat{u}_{\\rm II,\\, ij}\\left( -\\tau,\\, -\\textbf{x} \\right)\\,. \n\\end{equation}\nSimilar to the scalar power spectra we obtain two tensor power spectra which describe two point tensor correlations in the direction $\\hat{\\textbf{n}}$ and $-\\hat{\\textbf{n}}$ respectively. The two power spectra of tensor correlations are computed as \n\\begin{equation}\n\t\\begin{aligned}\n\t\tP_{h1} & = \\frac{k^3}{2\\pi^2}\\frac{4}{a^2} \\vert u_{\\rm I\\,k}\\vert^2\\Bigg\\vert_{\\tau = -\\frac{1}{aH}} \\\\ \n\t\t& \\approx \\frac{2H^2}{\\pi^2} \\left( 1+\\left( \\frac{k}{aH} \\right)^2 \\right) \\\\ &\\quad + \\frac{H^2}{\\pi}\\epsilon\\left(\\frac{k}{aH}\\right)^3 H_{3\/2}^{(1)} \\left( \\frac{k}{aH} \\right) \\frac{\\partial H^{(1)}_{\\nu_t}\\left( \\frac{k}{aH} \\right)}{\\partial \\nu_t}\\Big\\vert_{\\nu_t=3\/2}\\, \\\\\n\t\tP_{h2} & = \\frac{k^3}{2\\pi^2}\\frac{4}{a^2} \\vert u_{\\rm II,\\,k}\\vert^2\\Bigg\\vert_{\\tau_\\ast = \\frac{1}{aH}} \\\\ \n\t\t& \\approx \\frac{2H^2}{\\pi^2} \\left( 1+\\left( \\frac{k}{aH} \\right)^2 \\right)\\\\ &\\quad - \\frac{H^2}{\\pi} \\epsilon \\left(\\frac{k}{aH}\\right)^3 H_{3\/2}^{(1)} \\left( \\frac{k}{aH} \\right) \\frac{\\partial H^{(1)}_{\\nu_t}\\left( \\frac{k}{aH} \\right)}{\\partial \\nu_t}\\Big\\vert_{\\nu_t=3\/2}\\,. \n\t\\end{aligned}\n\t\\label{pwt12}\n\\end{equation}\nThe hemispherical power asymmetry of the tensor-power spectrum can be defined as \n\\begin{equation}\n\tT(k) = \\frac{P_{h1}-P_{h2}}{4P_h}\n\t\\label{Tko}\n\\end{equation}\nTo quantify the above amplitude $T(k)$ we choose Starobinsky or Higgs inflationary potential in Einstein frame \n\\begin{equation}\n\tV(\\phi) = \\frac{\\lambda}{4} \\left( 1- e^{-\\sqrt{\\frac{2}{3}}\\phi} \\right)^2\\,.\n\t\\label{staro-in}\n\\end{equation}\nwhich gives the slow-roll parameters in terms of e-folds as \\cite{Kehagias:2013mya}\n\\begin{equation}\n\t\\epsilon = \\frac{3}{4N^2},\\quad \\eta= \\frac{2}{N}\n\\end{equation}\nIn Fig.~\\ref{fig:fig2} we depict HPA amplitude of the tensor power spectra for the case of Starobinsky potential \\eqref{staro-in}.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{Fig2}\n\t\\caption{In the above plot we depict the amplitude of dipolar modulation for primordial gravitational waves (PGWs) or HPA of tensor power spectra obtained from \\eqref{Tko}. The above plot is for the case of Starobinsky or Higgs inflation \\eqref{staro-in} for $N=55$ number of e-folds corresponding to the pivot scale $k_\\ast =0.05 {\\rm Mpc}^{-1}$.}\n\t\\label{fig:fig2}\n\\end{figure}\nSimilar the case of the tilt of scalar power spectra \\eqref{tilts2} we obtain the tilt of the tensor power spectra almost the same in the two opposite directions of the sky\n\\begin{equation}\n\t\\frac{d\\ln P_{h 1}}{d\\ln k}\\approx \t\\frac{d\\ln P_{h 2}}{d\\ln k} \\approx n_t \\approx -2\\epsilon\\,.\n\t\\label{tilts3}\n\\end{equation}\nFrom Fig.~\\ref{fig:fig1} and Fig.~\\ref{fig:fig2} we can notice that the HPA decreases as we increase the wave-number. This is a physically expected behavior because the short wavelength modes do not see the curvature of spacetime therefore the asymmetry should decrease with increasing $k$. In other words we also see it as how $(\\mathcal{C})\\mathcal{P}\\mathcal{T}$ symmetry is broken in a curved spacetime as a function of length scales.  \n\n\n\\section{Conclusions}\n\nThe success of inflationary cosmology is its mapping of quantum fluctuations to the temperature fluctuations in the CMB. The HPA seems to pose a significant challenge to the standard understanding of inflationary quantum fluctuations because there is an asymmetry in the temperature correlations in the northern and southern hemispheres of the CMB sky. In this paper, we investigated whether the power asymmetry in the CMB can be originated from re-formulating our understanding of the quantization of inflationary fluctuations. Inspired from several open questions in the context of QFT in curved spacetime and quantum cosmology we proposed that the inflationary quantum fluctuations are generated in pairs which are related by discrete transformations of spacetime. Our approach is inspired from the fact that QFT in Minkowski spacetime is built on discrete symmetries such as $\\mathcal{C}\\mathcal{P}\\mathcal{T}$. In the case of curved spacetime i.e., \nwhen it involves gravity one must be careful in understanding spacetime reflection symmetries because gravity introduces dynamics. In our framework, we divide the notions of time into classical and quantum mechanical aspects. In the context of inflationary cosmology the standard procedure is to find a background geometry and quantize gravitational and matter degrees of freedom respecting the classical arrow of time \\cite{Martin:2004um}. However, we develop a new framework of quantization in canonical single field inflation where at the quantum level we allow the existence of quantum states that propagate both forward and backward in time based on the $\\mathcal{P}\\mathcal{T}$ transformations (since we only deal with chargeless fields we omit $\\mathcal{C}$ transformation) of the spacetime. Both of these quantum states in our framework co-exist within the co-moving horizon and they exit the horizon in the two opposite directions. We explicitly showed that these two type of quantum states even though created non-locally still do not violate any causality and locality principles in our formulation of QFT.\nWe showed that these pairs of quantum fluctuations behave exactly the same in the case of exact dS spacetime, but they are slightly different in the case of inflationary spacetime, which can be interpreted as the spontaneous breaking of $(\\mathcal{C})\\mathcal{P}\\mathcal{T}$ symmetry. This slight deviation seems to produce the power asymmetry hinted in the CMB observations. Our quantization scheme also applied to the case of inflationary tensor power spectrum and we predict a power asymmetry there as well. Notably, the HPA for scalar and tensor power spectrum is significant only for large angular scales or small wave numbers and it decreases for small angular scales or large wave numbers. This is expected since the spontaneous breaking of $(\\mathcal{C})\\mathcal{P}\\mathcal{T}$ symmetry is significant for small $k$, which are more affected by the background curved spacetime, compared to the large $k$.\nWe quantified all our predictions which serves as an observational test of our formalism. If the future observations \\cite{Li:2019bsg,Zhai:2017ibd} confirm these results we will learn significantly about the nature of inflationary quantum fluctuations as well as QFT in curved spacetime. \n \n\n\n\n\n\\begin{acknowledgments}\nKSK acknowledges the support from JSPS and KAKENHI Grant-in-Aid for Scientific Research No. JP20F20320, and thank Mainz U. for hospitality where part of the work has been carried out. J. Marto is supported by the grant UIDB\/MAT\/00212\/2020. We would like to thank Chris Ripken for the useful discussions and suggestions on quantization procedure and for the initial collaboration on the project.  We thank Gerard 't Hooft for his inspiring talks and discussions related to QFT in curved spacetime. We thank Yashar Akrami, Norma G. Sanchez, Masahide Yamaguchi, Alexei A. Starobinsky, Luca Buoninfante, Francesco Di Fillippo, Yasha Neiman, Paolo Gondolo, Martin Reuter, Eiichiro Komatsu, Paulo V. Moniz, Dhiraz Kumar Hazra and L. Sriramkumar for very useful discussions.\n\\end{acknowledgments}\n\n\n\n\n\n\\bibliographystyle{apsrev4-2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThis note reports the Tevatron average top-quark mass obtained by\ncombining the most precise published and preliminary measurements of\nthe top-quark mass. It is an update of the combination presented in\nRef.~\\cite{TeVTopComboPRD}, where further details can be found.\nThe ATLAS and CMS collaborations have also performed a combination of their most recent\ntop quark mass measurements~\\cite{lhccombi}.\n\nThe CDF and D\\O\\ collaborations have performed several direct experimental measurements of the\ntop-quark mass (\\ensuremath{M_{\\mathrm{t}}}) using data collected at the Tevatron\nproton-antiproton collider located at the Fermi National Accelerator\nLaboratory.  These pioneering measurements were first based on approximately\n$0.1~\\ensuremath{\\mathrm{fb}^{-1}}$ of \\hbox{Run\\,I}\\ data\n~\\cite{Mtop1-CDF-di-l-PRLa}-\\cite{Mtop1-D0-allh-PRL}\ncollected from 1992 to 1996,\nand included results from the decay channels \\ensuremath{\\ttbar\\ra\\WbWb\\ra\\had}\\ (alljets),\n\\ensuremath{\\ttbar\\ra\\WbWb\\ra\\ljt}\\ ($\\ell$+jets), and \n\\ensuremath{\\ttbar\\ra\\WbWb\\ra\\dil}\\ ($\\ell\\ell$), where $\\ell=e$ or $\\mu$.  \nDecays with $\\tau \\to e, \\mu$ are included in the direct $W \\to e$ and\n$W \\to \\mu$ channels. \nIn \\hbox{Run\\,II}\\ (2001--2011), many top mass measurements have been performed, and\nthose considered here are the most\nrecent results in these channels, using up to $8.7~\\ensuremath{\\mathrm{fb}^{-1}}$ of data for CDF\n(corresponding to the full CDF Run II dataset)\n~\\cite{\nMtop2-CDF-MEt-new,\nMtop2-CDF-l+jt-pub-new,\nMtop2-CDF-di-l-pub,\nMtop2-CDF-allh-pub},\nand up to $5.4~\\ensuremath{\\mathrm{fb}^{-1}}$ of data for D\\O\\\n~\\cite{ \nMtop2-D0-l+ja-final,\nMtop2-D0-l+jt-new2,\nMtop2-D0-di-l}.\nThe CDF analysis based upon charged particle tracking for exploiting\nthe transverse decay length \nof $b$-tagged jets ($L_{XY}$) and the transverse momentum of\nelectrons and muons from $W$ boson decays ($\\pte$) uses\na data set corresponding to a luminosity of\n1.9~$\\ensuremath{\\mathrm{fb}^{-1}}$~\\cite{Mtop2-CDF-trk}, and\nthere are no plans to update this analysis.\nThe D\\O\\ \\hbox{Run\\,II}\\  measurements presented in this note include the most\nrecent Run II measurement in  \nthe $\\ell\\ell$~\\cite{Mtop2-D0-di-l}  channel using 5.4~fb$^{-1}$ of data \nand in the $\\ell$+jets channel \\cite{Mtop2-D0-l+jt-new2} \nwith 3.6~fb$^{-1}$ of data. Both results are now published.\nSince the combination performed in 2011~\\cite{Mtop-tevewwgSum11}, a new\nfinal state signature was introduced by CDF that requires events to\npossess missing transverse energy ($\\met$) and jets, but no identified\nlepton (``MEt'')~\\cite{Mtop2-CDF-MEt-new,Mtop2-CDF-MEt-published}.  \nThis sample is statistically independent from the previous three CDF\nchannels.\n\nWith respect to the July 2011 combination\\,\\cite{Mtop-tevewwgSum11} and the published version of the \ncombination~\\cite{TeVTopComboPRD}, the \\hbox{Run\\,II}\\ CDF measurement in the \n$\\ell$+jets channel has been updated using 8.7~fb$^{-1}$ \nof data, an improved analysis technique, and improved jet energy resolution \\cite{Mtop2-CDF-l+jt-pub-new}. \nThe CDF measurement in the MEt channel was updated to use the full\n\\hbox{Run\\,II}\\ data set for CDF of  8.7~fb$^{-1}$ of data as \nwell~\\cite{Mtop2-CDF-MEt-new}.\nThe now published \\hbox{Run\\,II}\\ CDF measurements in the $\\ell\\ell$ channel\\,\\cite{Mtop2-CDF-di-l-pub} and alljets \nchannel\\,\\cite{Mtop2-CDF-allh-pub}  are unchanged. The measurement \nbased on charged particle tracking~\\cite{Mtop2-CDF-trk} was\nincorporated as described in the past \ncombinations~\\cite{Mtop-tevewwgSum11}. \nFrom the corresponding analysis only\nthe measurement of the top quark mass using the mean decay length $\\Lxy$ of\n$B$ hadrons in $b$-tagged lepton+jets events has been used. It\nis independent of energy information in the calorimeter, and its main\nsource of systematic uncertainty is uncorrelated with the dominant\nones from the jet energy scale calibration in other measurements. This\nmeasurement of $m_t$ is essentially uncorrelated with the higher\nprecision CDF result from the lepton+jets channel. The overlap between\nthe data samples used for the decay-length method and the lepton+jets\nsample has therefore no effect. \n\nThe Tevatron average top-quark mass is obtained by combining five published\n\\hbox{Run\\,I}\\ measurements~\\cite{Mtop1-CDF-di-l-PRLb, Mtop1-CDF-di-l-PRLb-E,\n  Mtop1-D0-di-l-PRD, Mtop1-CDF-l+jt-PRD, Mtop1-D0-l+jt-new1,\n  Mtop1-CDF-allh-PRL} with four published \\hbox{Run\\,II}\\ CDF \nresults~\\cite{Mtop2-CDF-l+jt-pub-new,Mtop2-CDF-di-l-pub,Mtop2-CDF-allh-pub, \nMtop2-CDF-trk}, one preliminary \\hbox{Run\\,II}\\ CDF\nresult~\\cite{Mtop2-CDF-MEt-new}, and two published \n\\hbox{Run\\,II}\\ D\\O\\ results~\\cite{Mtop2-D0-l+jt-new2,\nMtop2-D0-di-l}.\nThis combination \nsupersedes previous\ncombinations~\\cite{Mtop-tevewwgSum11,Mtop1-tevewwg04,Mtop-tevewwgSum05,\n  Mtop-tevewwgWin06,Mtop-tevewwgSum06, Mtop-tevewwgWin07, Mtop-tevewwgWin08, \n  Mtop-tevewwgSum08, Mtop-tevewwgWin09,Mtop-tevewwgSum10}. \n\nThe definition and evaluation of the systematic uncertainties and the understanding of the\ncorrelations among channels, experiments, and Tevatron runs is the outcome of many years of \njoint work between the CDF and D\\O\\ collaborations and is described in detail\nelsewhere~\\cite{TeVTopComboPRD}.\n\nThe input measurements and uncertainty categories used in the combination are \ndetailed in Sections~\\ref{sec:inputs} and~\\ref{sec:uncertainty}, respectively. \nThe correlations assumed in the combination are discussed in \nSection~\\ref{sec:corltns} and the resulting Tevatron average top-quark mass \nis given in Section~\\ref{sec:results}.  A summary is presented\nin Section~\\ref{sec:summary}.\n \n\\section{Input Measurements}\n\\label{sec:inputs}\n\nTwelve measurements of \\ensuremath{M_{\\mathrm{t}}}\\ used in this combination are shown in Table~\\ref{tab:inputs}.\nThe \\hbox{Run\\,I}\\ measurements all have relatively large statistical\nuncertainties and their systematic uncertainties are dominated by the\ntotal jet energy scale (JES) uncertainty.  In \\hbox{Run\\,II}\\, both CDF and\nD\\O\\ take advantage of the larger \\ensuremath{t\\overline{t}}\\ samples available and employ\nnew analysis techniques to reduce both of these uncertainties.  In\nparticular, the \\hbox{Run\\,II}\\ D\\O\\ analysis in the $\\ell$+jets channel and the \n\\hbox{Run\\,II}\\ CDF analyses in the $\\ell$+jets, alljets, and MEt channels \nconstrain the response of light-quark jets using the kinematic information from $W\\ra\nqq^{\\prime}$ decays (so-called {\\em in situ}\ncalibration)~\\cite{Mtop1-CDF-l+jt-PRD,Abazov:2006bd}. Residual JES uncertainties associated with\n$p_{T}$ and $\\eta$ dependencies as well as uncertainties specific to\nthe response of $b$ jets are treated separately. The\n\\hbox{Run\\,II}\\ D\\O\\ $\\ell\\ell$ measurement uses the JES determined in the\n$\\ell$+jets channel by {\\em in situ} calibration~\\cite{Mtop2-D0-di-l}.\n\n\\vspace*{0.10in}\n\n\\begin{table}[t]\n\\caption[Input measurements]{Summary of the measurements used to determine the\n  Tevatron average \\ensuremath{M_{\\mathrm{t}}}.  Integrated luminosity ($\\int \\mathcal{L}\\;dt$) has units of\n  \\ensuremath{\\mathrm{fb}^{-1}}, and all other numbers are in $\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }$.  The uncertainty categories and \n  their correlations are described in Section~\\ref{sec:uncertainty}.  The total systematic uncertainty \n  and the total uncertainty are obtained by adding the relevant contributions \n  in quadrature. ``n\/a'' stands for ``not applicable'', ``n\/e'' for ``not evaluated''.}\n\\label{tab:inputs}\n\\begin{center}\n\n\\renewcommand{\\arraystretch}{1.30}\n{\\tiny\n\\begin{tabular}{l|ccc|cc|cccc|cc|c} \n\\hline \\hline\n       & \\multicolumn{5}{c|}{{\\hbox{Run\\,I}} published} \n       & \\multicolumn{6}{c|}{{\\hbox{Run\\,II}} published} \n       & \\multicolumn{1}{c}{{\\hbox{Run\\,II}} prel.}  \\\\ \n      \n       & \\multicolumn{3}{c|}{ CDF } \n       & \\multicolumn{2}{c}{ D\\O\\ }\n       & \\multicolumn{4}{|c|}{ CDF }\n       & \\multicolumn{2}{c|}{ D\\O\\ }\n       & \\multicolumn{1}{c}{ CDF }\n        \\\\\n\n                      &     $\\ell$+jets   & $\\ell\\ell$     &   alljets  &     $\\ell$+jets &   $\\ell\\ell$  &    $\\ell$+jets     &     $\\ell\\ell$  &    alljets  &     Lxy     &   $\\ell$+jets      &   $\\ell\\ell$    &      MEt \\\\\n\\hline                                                                                                                                                       \n$\\int \\mathcal{L}\\;dt$&       0.1  &     0.1  &    0.1  &     0.1  &     0.1 &      8.7    &       5.6 &     5.8  &         1.9 &         3.6 &       5.3 &      8.7   \\\\\n\\hline                                                                                                                                                       \nResult                &   176.1    & 167.4    &186.0    & 180.1    & 168.4   &      172.85 &    170.28 &  172.47  &      166.90 &      174.94 &    174.00 &    173.95\\\\\n\\hline                                                                                                                                                       \n\\shortstack{ {\\em In situ} light-jet cali- \\\\\nbration (iJES)}                   &        n\/a &      n\/a & n\/a     &      n\/a &      n\/a &       0.49 &      n\/a  &     0.95 &         n\/a &        0.53 &      0.55  &      1.05  \\\\\n\\shortstack{ Response to $b$\/$q$\/$g$ \\\\\njets (aJES)}                          &        n\/a &      n\/a & n\/a     &      0.0 &      0.0 &       0.09 &      0.14 &     0.03 &         n\/a &        0.0  &      0.40 &      0.10 \\\\\n\\shortstack{ Model for $b$ jets  \\\\\n(bJES)}                               &        0.6 &      0.8 & 0.6     &      0.7 &      0.7 &       0.16 &      0.33 &     0.15 &         n\/a &        0.07 &      0.20 &      0.17  \\\\\n \\shortstack{ Out-of-cone correction \\\\\n     (cJES)}                          &        2.7 &      2.6 & 3.0     &      2.0 &      2.0 &       0.21 &      2.13 &     0.24 &        0.36 &        n\/a  &      n\/a  &      0.18  \\\\\n\\shortstack{ Light-jet response (2) \\\\ \n    (dJES)}                               &        0.7 &      0.6 & 0.3     &      2.5 &      1.1 &       0.07 &      0.58 &     0.04 &        0.06 &        0.63 &      0.56 &      0.04  \\\\\n\\shortstack{ Light-jet response (1) \\\\\n (rJES)}                               &        3.4 &      2.7 & 4.0     &      n\/a &      n\/a &       0.48 &      2.01 &     0.38 &        0.24 &        n\/a  &      n\/a  &      0.40  \\\\\n\\shortstack{ Lepton modeling  \\\\\n(LepPt)}                              &        n\/e &      n\/e & n\/e     &      n\/e &      n\/e &       0.03 &      0.27 &     n\/a    &        n\/a  &        0.17 &      0.35 &      n\/a   \\\\\n\\shortstack{ Signal modeling \\\\\n (Signal)}                             &        2.6 &      2.9 & 2.0     &      1.1 &      1.8 &       0.61 &      0.73 &     0.62 &        0.90 &        0.77 &      0.86 &      0.64\\\\\n \\shortstack{ Jet modeling \\\\\n  (DetMod)}                             &        0.0 &      0.0 & 0.0     &      0.0 &      0.0 &       0.0  &      0.0  &     0.0  &        0.0  &        0.36 &      0.50 &      0.0 \\\\\n\\shortstack{ Offset \\hspace{0.5cm} \\\\\n (UN\/MI)}                              &        n\/a &      n\/a &    n\/a  &      1.3 &      1.3 &       n\/a  &      n\/a  &     n\/a  &        n\/a  &        n\/a  &      n\/a  &      n\/a  \\\\\n\\shortstack{ Background from \\\\\ntheory (BGMC)}                        &        1.3 &      0.3 &    1.7  &      1.0 &      1.1 &       0.12 &      0.24 &     0.0  &        0.80 &       0.18  &      0.0  &      0.0   \\\\\n \\shortstack{ Background based on\\\\\ndata (BGData)}                     &        0.0 &      0.0 &    0.0  &      0.0 &      0.0 &      0.16  &      0.14 &     0.56 &        0.20 &       0.23  &      0.20 &      0.12  \\\\\n \\shortstack{ Calibration method\\\\\n(Method)}                      &        0.0 &      0.7 & 0.6     &      0.6 &      1.1 &       0.00 &      0.12 &     0.38&         2.50 &        0.16 &      0.51 &      0.31  \\\\\n \\shortstack{ Multiple interactions\\\\\n model (MHI)}                          &        n\/e &      n\/e &    n\/e  &      n\/e &      n\/e &       0.07 &      0.23 &     0.08&         0.0  &        0.05 &      0.0  &      0.18  \\\\\n\\hline                                                                                                                                                       \n \\shortstack{ Systematic uncertainty\\\\\n (Syst)}                   &        5.3 &      4.9 & 5.7    &       3.9 &      3.6 &       0.98 &      3.09 &     1.49 &        2.90 &        1.24 &      1.44 &      1.35 \\\\\n\\shortstack{ Statistical uncertainty\\\\\n (Stat)}                   &        5.1 &     10.3 &10.0    &       3.6 &     12.3 &       0.52 &      1.95 &     1.43 &        9.00 &        0.83 &      2.36 &      1.26 \\\\\n\\hline                                                                                                                                                       \nTotal uncertainty                     &        7.3 &     11.4 &11.5    &        5.3 &    12.8 &       1.11 &      3.79 &     2.06 &        9.46 &       1.50 &      2.76 &      1.85\\\\ \n\\hline\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\nThe D\\O\\ Run~II $\\ell$+jets analysis uses the JES determined from the\nexternal calibration derived from $\\gamma$+jets events as an\nadditional Gaussian constraint to the {\\em in situ} calibration. Therefore,\nthe total resulting JES uncertainty is split into one part obtained from the \n{\\em in situ} calibration and another part determined from the external calibration.\nTo do this, the measurement without external\nJES constraint has been combined iteratively with a pseudo-measurement\nusing the method of Refs.~\\cite{Lyons:1988, Valassi:2003} \nthat uses only the external calibration in a way that the combination gives the total JES uncertainty. \nThe splitting obtained in this way is used to assess both the statistical part of the JES uncertainty and the part of \nthe JES uncertainty due to the external calibration constraint~\\cite{Mtop2-D0-comb}.\n\nThe $\\Lxy$ technique developed by CDF \nuses the decay length of $B$ mesons from $b$-tagged jets.\nWhile the statistical sensitivity of this analysis is not as good as\nthat of the more\ntraditional methods, this technique has the advantage that it is almost entirely independent of\nJES uncertainties since it\nuses primarily tracking information.\n\\vspace*{0.10in}\n\nThe D\\O\\ \\hbox{Run\\,II}\\ $\\ell$+jets result is a combination of the \npublished Run~IIa (2002--2005) measurement ~\\cite{Mtop2-D0-l+ja-final} with 1~fb$^{-1}$ \nof data and the result obtained with 2.6~fb$^{-1}$ of data from Run~IIb (2006--2007)~\\cite{Mtop2-D0-l+jt-new2}.\nThis analysis includes an additional particle response correction on top of the\nstandard {\\it in situ} calibration.\nThe D\\O\\ \\hbox{Run\\,II}\\ $\\ell\\ell$ result is based on a neutrino weighting technique using 5.4~fb$^{-1}$\nof \\hbox{Run\\,II}\\  data~\\cite{Mtop2-D0-di-l}. \n\\vspace*{0.10in}\n\nTable~\\ref{tab:inputs} lists the individual uncertainties of each result,\nsubdivided into the categories described in the next Section.  The\ncorrelations between the inputs are described in\nSection~\\ref{sec:corltns}.\n\n\n\n\\section{Uncertainty Categories}\n\\label{sec:uncertainty}\n\nWe employ uncertainty categories similar to what was used for the previous Tevatron\naverage~\\cite{TeVTopComboPRD,Mtop-tevewwgSum11}, with small\nmodifications to better account for their correlations.\nThey are divided such that sources of systematic uncertainty that share the same or similar origin are \ncombined as explained in Ref.~\\cite{TeVTopComboPRD}. \nFor example, the {\\it Signal modeling} ({\\it Signal}) category\ndiscussed below includes the uncertainties from  different systematic\nsources that are correlated due to their origin in the modeling of the simulated signal samples.\n\n Some systematic uncertainties have been separated into multiple\ncategories to accommodate specific types of correlations.\nFor example, the jet energy scale (JES) uncertainty is subdivided\ninto six components to more accurately accommodate our\nbest understanding of the relevant correlations between input measurements. \n\nFor this note we use the new systematic naming scheme described in Ref.~\\cite{TeVTopComboPRD}. In parentheses, the \nold names of the  systematic uncertainties are provided.  There is a one-to-one matching between the new and old systematic \ndefinitions of categories.\n\n\\vspace*{0.10in}\n\n\\begin{description}\n  \\item[Statistical uncertainty (Statistics):] The statistical uncertainty associated with the\n    \\ensuremath{M_{\\mathrm{t}}}\\ determination.\n \\item[{\\em In situ} light-jet calibration (iJES):] That part of the\n   JES uncertainty that originates from\n   {\\em in situ} calibration procedures and is uncorrelated among the\n   measurements.  In the combination reported here, it corresponds to\n   the statistical uncertainty associated with the JES determination\n   using the $W\\ra qq^{\\prime}$ invariant mass in the CDF \\hbox{Run\\,II}\\\n   $\\ell$+jets, alljets, and  MEt measurements and the D\\O\\ Run~II\n   $\\ell\\ell$ and $\\ell$+jets\n   measurements. \n   For the D\\O\\ Run~II $\\ell$+jets measurement, it also includes the uncertainty\n   coming from the MC\/data difference in jet response that is uncorrelated\n   with the other D\\O\\ Run~II measurements. \n   Residual JES uncertainties arising from effects\n   not considered in the {\\em in situ} calibration are included in other\n   categories. \n  \\item[Response to \\boldmath{$b\/q\/g$} jets (aJES):] That part of the JES\n    uncertainty that originates from \n    average differences in detector electromagnetic over hadronic ($e\/h$)\n    response for hadrons produced in the fragmentation of $b$-jets and light-quark \n    jets. \n  \\item[Model for \\boldmath{$b$} jets (bJES):] That part of the JES uncertainty that originates from\n    uncertainties specific to the modeling of $b$ jets and that is correlated\n    across all measurements.  For both CDF and D\\O\\ this includes uncertainties \n    arising from \n    variations in the semileptonic branching fractions, $b$-fragmentation \n    modeling, and differences in the color flow between $b$-quark jets and light-quark\n    jets.  These were determined from \\hbox{Run\\,II}\\ studies but back-propagated\n    to the \\hbox{Run\\,I}\\ measurements, whose {\\it Light-jet response (1)}  uncertainties ({\\it rJES}, see below) were \n    then corrected to keep the total JES uncertainty constant.\n  \\item[Out-of-cone correction (cJES):] That part of the JES uncertainty that originates from\n    modeling uncertainties correlated across all measurements.  \n    It specifically includes the modeling uncertainties associated with light-quark \n    fragmentation and out-of-cone corrections. For D\\O\\ \\hbox{Run\\,II}\\ measurements,\n    it is included in the {\\it Light-jet response (2) (dJES)} category.\n  \\item[Light-jet response (1) (rJES):] The remaining part of the JES\n    uncertainty that covers the absolute calibration for CDF's \\hbox{Run\\,I}\\\n    and \\hbox{Run\\,II}\\ measurements. It also includes small contributions\n    from the uncertainties associated with modeling multiple\n    interactions within a single bunch crossing and corrections for\n    the underlying event. \n   \n   \n   \n   \n   \n   \n  \\item[Light-jet response (2) (dJES):] That part of the JES\n    uncertainty that includes D\\O's \\hbox{Run\\,I}\\ and \\hbox{Run\\,II}\\ calibrations of\n    absolute response (energy dependent), the relative response\n    ($\\eta$-dependent), and the out-of-cone showering correction that \n    is a detector effect. This uncertainty term for CDF includes only\n    the small relative response calibration ($\\eta$-dependent) for\n    \\hbox{Run\\,I}\\ and \\hbox{Run\\,II}. \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \\item[Lepton modeling (LepPt):] The systematic uncertainty arising from uncertainties\n    in the scale of lepton transverse momentum measurements. It was not\n    considered as a source of systematic uncertainty in the \\hbox{Run\\,I}\\\n    measurements. \n  \\item[Signal modeling (Signal):] The systematic uncertainty arising from uncertainties\n    in \\ensuremath{t\\overline{t}}\\ modeling that is correlated across all\n    measurements. This includes uncertainties from variations of the amount of initial and \n    final state radiation and from the choice of parton density function used\n    to generate the \\ensuremath{t\\overline{t}}\\ Monte Carlo samples\n    that calibrate each method. For D\\O, it also includes the uncertainty \n    from higher-order corrections evaluated from a comparison of \\ensuremath{t\\overline{t}}\\ samples generated by {\\textsc MC@NLO} ~\\cite{MCNLO} and \n    {\\textsc ALPGEN}~\\cite{ALPGEN}, both interfaced to {\\textsc HERWIG}~\\cite{HERWIG5,HERWIG6} for the simulation of parton showers and hadronization.\n    In this combination, the systematic uncertainty arising from a variation of the \n  phenomenological description of color reconnection (CR) between final state  particles \\cite{CR,Skands:2009zm} \n  is included in the {\\it Signal modeling} category.\n  The CR uncertainty is obtained by taking the difference between the {\\textsc PYTHIA}\\,6.4 tune ``Apro\" and the {\\textsc PYTHIA}\\,6.4 tune \n``ACRpro\" that differ only in the CR model. \nThis uncertainty was not evaluated in Run~I since the Monte Carlo\ngenerators available at that time did not allow for variations of the CR model.\nThese measurements therefore do not include this source of systematic uncertainty. Finally, the systematic \nuncertainty associated with variations of the MC generator used to calibrate the mass extraction method is added. \nIt includes variations observed when \n    substituting {\\textsc PYTHIA}\n    \\cite{PYTHIA4,PYTHIA5,PYTHIA6} \n    (\\hbox{Run\\,I}\\ and \\hbox{Run\\,II}) \n    or {\\textsc ISAJET}~\\cite{ISAJET} (\\hbox{Run\\,I}) for {\\textsc HERWIG}~\\cite{HERWIG5,HERWIG6} when \n    modeling the \\ensuremath{t\\overline{t}}\\ signal.  \n  \\item[Jet modeling (DetMod):] The systematic uncertainty arising from uncertainties \nin the modeling of jet interactions in the detector in the MC\nsimulation. For D\\O\\, this includes uncertainties from jet resolution\nand identification. \nApplying jet algorithms to MC events, CDF finds that the resulting\nefficiencies and resolutions closely match those in data. The small\ndifferences propagated to $\\ensuremath{M_{\\mathrm{t}}}$ lead to a negligible uncertainty of\n0.005~GeV, which is then ignored.  \n\n \\item[Background based on data (BGData):] This includes \n    uncertainties associated with the modeling using data of the QCD\n    multijet background in the alljets,\n    MEt, and $\\ell$+jets channels and\n    the Drell-Yan background in the $\\ell\\ell$ channel. \n    This part is uncorrelated between experiments.\n\n  \\item[Background from theory (BGMC):] \nThis systematic uncertainty on the background originating from theory\n(MC) takes into account the    \n uncertainty in modeling the background sources. It is correlated between\n    all measurements in the same channel, and includes uncertainties on the background composition, normalization, and shape of different components, e.g., the \nuncertainties from the modeling of the $W$+jets background in the $\\ell$+jets channel  \nassociated with variations of the factorization scale used to simulate $W$+jets events. \n\n  \\item[Calibration method (Method):] The systematic uncertainty arising from any source specific\n    to a particular fit method, including the finite Monte Carlo statistics \n    available to calibrate each method. \n  \\item[Offset (UN\/MI):] This uncertainty is specific to D\\O\\ and includes the uncertainty\n    arising from uranium noise in the D\\O\\ calorimeter and from the\n    multiple interaction corrections to the JES.  For D\\O\\ \\hbox{Run\\,I}\\ these\n    uncertainties were sizable, while for \\hbox{Run\\,II}, owing to the shorter\n    calorimeter electronics integration time and {\\em in situ} JES calibration, these uncertainties\n    are negligible.\n  \\item[Multiple interactions model (MHI):] The systematic uncertainty arising from a mismodeling of \n  the distribution of the number of collisions per Tevatron bunch crossing owing to the \n  steady increase in the collider instantaneous luminosity during data-taking. \n  This uncertainty has been separated from other sources to account for the fact that \n  it is uncorrelated between experiments.\n\n\\end{description}\nThese categories represent the current preliminary understanding of the\nvarious sources of uncertainty and their correlations.  We expect these to \nevolve as we continue to probe each method's sensitivity to the various \nsystematic sources with improving precision.\n\n\\section{Correlations}\n\\label{sec:corltns}\n\nThe following correlations are used for the combination:\n\\begin{itemize}\n  \\item The uncertainties in the {\\it Statistical uncertainty (Stat)} and\n    {\\it Calibration method (Method)}\n    categories are taken to be uncorrelated among the measurements.\n  \\item The uncertainties in the {\\it In situ light-jet  \n    calibration (iJES)}\n    category are taken to be uncorrelated among the measurements\n    except for D0's $\\ell\\ell$ and $\\ell$+jets measurements, where\n    this uncertainty is taken to be 100\\% correlated since the\n    $\\ell\\ell$ measurement uses the JES calibration  determined in\n    $\\ell$+jets channel.  \n  \\item The uncertainties in the {\\it Response to $b$\/$q$\/$g$ jets (aJES)}, {\\it Light-jet response (2) (dJES)}, {\\it Lepton modeling (LepPt)},\n    and {\\it Multiple interactions model (MHI)} categories are taken\n    to be 100\\% correlated among all \\hbox{Run\\,I}\\ and all \\hbox{Run\\,II}\\ measurements \n    within the same experiment, but uncorrelated between \\hbox{Run\\,I}\\ and \\hbox{Run\\,II}\\\n    and uncorrelated between the experiments.\n  \\item The uncertainties in the {\\it Light-jet response (1) (rJES)}, {\\it Jet modeling (DetMod)}, and {\\it Offset (UN\/MI)} categories are taken\n    to be 100\\% correlated among all measurements within the same experiment \n    but uncorrelated between the experiments.\n  \\item The uncertainties in the {\\it Backgrounds estimated from theory (BGMC)} category are taken to be\n    100\\% correlated among all measurements in the same channel.\n  \\item The uncertainties in the {\\it Backgrounds estimated from data (BGData)} category are taken to be\n    100\\% correlated among all measurements in the same channel and same run period, but uncorrelated between the experiments.\n  \\item The uncertainties in the {\\it Model for $b$ jets (bJES)}, {\\it Out-of-cone correction (cJES)}, and {\\it Signal modeling (Signal)}\n    categories are taken to be 100\\% correlated among all measurements.\n\\end{itemize}\nUsing the inputs from Table~\\ref{tab:inputs} and the correlations specified\nhere, the resulting matrix of total correlation coefficients is given in\nTable~\\ref{tab:coeff}.\n\n\\begin{table}[t]\n\\caption[Global correlations between input measurements]{The matrix of correlation coefficients used to determine the\n  Tevatron average top-quark mass.}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n\\tiny\n\\begin{tabular}{l|ccc|cc|cccc|cc|c}\n\\hline \\hline\n     & \\multicolumn{5}{c}{{\\hbox{Run\\,I}} published} \n         & \\multicolumn{6}{|c|}{{\\hbox{Run\\,II}} published} \n         & \\multicolumn{1}{c}{{\\hbox{Run\\,II}} preliminary}   \\\\\n      & \\multicolumn{3}{c|}{ CDF } \n         & \\multicolumn{2}{c}{ D\\O\\ }\n         & \\multicolumn{4}{|c|}{ CDF } \n\t & \\multicolumn{2}{c|}{ D\\O\\ }\n         & \\multicolumn{1}{c}{ CDF } \n         \\\\\n\n                &  $\\ell$+jets    &    $\\ell\\ell$ &   alljets &    $\\ell$+jets  &   $\\ell\\ell$   &   $\\ell$+jets  &   $\\ell\\ell$   & alljets   &     $\\Lxy$   &  $\\ell$+jets      & $\\ell\\ell$  &  MEt \\\\\n                                                                                                                  \n\\hline                                                                                                            \nCDF-I $\\ell$+jets    &   1.00  &    0.29  &    0.32  &    0.26  &    0.11  &    0.49  &    0.54  &    0.25  &    0.07  &    0.21  &    0.12  &    0.27 \\\\\nCDF-I $\\ell\\ell$     &   0.29  &    1.00  &    0.19  &    0.15  &    0.08  &    0.29  &    0.32  &    0.15  &    0.04  &    0.13  &    0.08  &    0.17 \\\\\nCDF-I alljets        &   0.32  &    0.19  &    1.00  &    0.14  &    0.07  &    0.30  &    0.38  &    0.15  &    0.04  &    0.09  &    0.06  &    0.16 \\\\\nD\\O-I $\\ell$+jets    &   0.26  &    0.15  &    0.14  &    1.00  &    0.16  &    0.22  &    0.27  &    0.12  &    0.05  &    0.14  &    0.07  &    0.12 \\\\\nD\\O-I $\\ell\\ell$     &   0.11  &    0.08  &    0.07  &    0.16  &    1.00  &    0.11  &    0.13  &    0.07  &    0.02  &    0.07  &    0.05  &    0.07 \\\\\nCDF-II $\\ell$+jets   &   0.49  &    0.29  &    0.30  &    0.22  &    0.11  &    1.00  &    0.48  &    0.29  &    0.08  &    0.30  &    0.18  &    0.33 \\\\\nCDF-II $\\ell\\ell$    &   0.54  &    0.32  &    0.38  &    0.27  &    0.13  &    0.48  &    1.00  &    0.25  &    0.06  &    0.11  &    0.07  &    0.26 \\\\\nCDF-II alljets       &   0.25  &    0.15  &    0.15  &    0.12  &    0.07  &    0.29  &    0.25  &    1.00  &    0.04  &    0.16  &    0.10  &    0.17 \\\\\nCDF-II $\\Lxy$     &   0.07  &    0.04  &    0.04  &    0.05  &    0.02  &    0.08  &    0.06  &    0.04  &    1.00  &    0.06  &    0.03  &    0.04 \\\\\nD\\O-II $\\ell$+jets    &   0.21  &    0.13  &    0.09  &    0.14  &    0.07  &    0.30  &    0.11  &    0.16  &    0.06  &    1.00  &    0.39  &    0.18 \\\\\nD\\O-II $\\ell\\ell$    &   0.12  &    0.08  &    0.06  &    0.07  &    0.05  &    0.18  &    0.07  &    0.10  &    0.03  &    0.39  &    1.00  &    0.11 \\\\\nCDF-II MEt     &   0.27  &    0.17  &    0.16  &    0.12  &    0.07  &    0.33  &    0.26  &    0.17  &    0.04  &    0.18  &    0.11  &    1.00 \\\\\n\n\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:coeff}\n\\end{table}\n\nThe measurements are combined using a program implementing two \nindependent methods: \na numerical $\\chi^2$ minimization and \nthe analytic best linear unbiased estimator (BLUE) method~\\cite{Lyons:1988, Valassi:2003}. \nThe two methods are mathematically equivalent.\nIt has been checked that they give identical results for\nthe combination. The BLUE method yields the decomposition of the uncertainty on the Tevatron $\\ensuremath{M_{\\mathrm{t}}}$ average in \nterms of the uncertainty categories specified for the input measurements~\\cite{Valassi:2003}.\n\n\\section{Results}\n\\label{sec:results}\n\nThe resultant combined value for the top-quark mass is\n\\begin{eqnarray}\n\\nonumber\n\\ensuremath{M_{\\mathrm{t}}}=\\gevcc{\\measStatSyst{173.20}{0.51}{0.71}}. \n\\end{eqnarray}\nAdding the statistical and systematic uncertainties\nin quadrature yields a total uncertainty of $\\gevcc{0.87}$, corresponding to a\nrelative precision of 0.50\\% on the top-quark mass.\nIt has a $\\chi^2$ of 8.5 for 11 degrees of freedom, corresponding to\na probability of 67\\%, indicating good agreement among all input\nmeasurements.  The breakdown of the uncertainties is \nshown in Table\\,\\ref{tab:BLUEuncert}.  The total statistical and systematic  \nuncertainties are reduced relative to the \nSummer 2011 combination \\cite{Mtop-tevewwgSum11} and the published combination~\\cite{TeVTopComboPRD} \ndue to the increase of the CDF data samples  in the $\\ell$+jets and MEt analyses and better \ntreatment of JES corrections in the $\\ell$+jets analysis. \n \nThe pull and weight for each of the inputs, as obtained from the\ncombination with the BLUE method, are listed in Table~\\ref{tab:stat}.\nThe input measurements and the resulting Tevatron average mass of the top \nquark are summarized in Fig.~\\ref{fig:summary}.\n\\vspace*{0.10in}\n\n\\begin{table}[tbh]\n\\caption{\\label{tab:BLUEuncert} \nSummary of the Tevatron combined average $\\ensuremath{M_{\\mathrm{t}}}$. The uncertainty categories are \ndescribed in the text. The total systematic uncertainty and the total\nuncertainty are obtained  \nby adding the relevant contributions in quadrature.}\n\\begin{center}\n\\begin{tabular}{lc} \\hline \\hline\n  & Tevatron combined values (GeV\/$c^2$) \\\\ \\hline\n $\\ensuremath{M_{\\mathrm{t}}}$            & 173.20 \\\\ \\hline\n {\\em In situ} light-jet calibration (iJES)           &  0.36 \\\\\n Response to $b$\/$q$\/$g$ jets (aJES)                  &  0.09  \\\\\n Model for $b$ jets       (bJES)                  &  0.11 \\\\  \n Out-of-cone correction (cJES)                  &  0.01 \\\\ \n Light-jet response (2) (dJES)                  &  0.15 \\\\ \n Light-jet response (1) (rJES)                  &  0.16 \\\\\n Lepton modeling  (LepPt)                       &  0.05 \\\\\n Signal modeling  (Signal)                      &  0.52  \\\\\n Jet modeling (DetMod)                          &  0.08 \\\\\n Offset (UN\/MI)                                 &  0.00 \\\\\n Background from theory (BGMC)                  &  0.06\\\\\n Background based on data (BGData)              &  0.13\\\\\n Calibration method (Method)                    &  0.06 \\\\\n Multiple interactions model (MHI)              &  0.07 \\\\ \\hline\n Systematic uncertainty (syst)                  &  0.71 \\\\ \n Statistical uncertainty (stat)                 &  0.51 \\\\ \\hline\n Total  uncertainty                             &  0.87 \\\\\n   \\hline \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe weights of some of the measurements are negative, which occurs if\nthe correlation between two measurements \nis larger than the ratio of their total uncertainties.\nIn these instances the less precise measurement \nwill acquire a negative weight.  While a weight of zero means that a\nparticular input is effectively ignored in the combination, channels\nwith a negative weight affect the resulting $\\ensuremath{M_{\\mathrm{t}}}$ central value and help reduce the total\nuncertainty~\\cite{Lyons:1988}. \nTo visualize the weight each measurement carries in the combination, Fig.\\,\\ref{fig:Weights} shows the\nabsolute values of the weight of each measurement divided by the sum of the absolute values of the weights\nof all input measurements. Negative weights are represented by bins\nwith a different (grey) color. \nWe note, that due to correlations between the uncertainties the relative weights\nof the different input channels may be significantly different from\nwhat one could expect from the total accuracy of each measurement as represented by\nerror bars in Fig.~\\ref{fig:summary}.\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{figures\/TevMtWin13-2digit.pdf}\n\\end{center}\n\\caption[Summary plot for the Tevatron average top-quark mass]\n  {Summary of the input measurements and resulting Tevatron average\n   mass of the top quark.}\n\\label{fig:summary} \n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\textwidth]{figures\/TevWeightPlotWin13-neg.pdf}\n\\end{center}\n\\caption{Relative weights of the input measurements in the\n  combination. The relative weights have been obtained by dividing the absolute value of each measurement weight by the sum over all measurements of the absolute values\nof the weights. Negative weights are represented by their absolute\nvalue, but using a grey color.}\n\\label{fig:Weights} \n\\end{figure}\n\n\\begin{table}[t]\n\\caption[Pull and weight of each measurement]{The pull and weight for each of the\n  inputs, as obtained from the combination with the BLUE method to\n  determine the average top quark mass.}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n{\\tiny\n\\begin{tabular}{l|ccccc|cccccc|c}\n\\hline \n\\hline\n       & \\multicolumn{5}{c}{{\\hbox{Run\\,I}} published} \n       & \\multicolumn{6}{|c|}{{\\hbox{Run\\,II}} published} \n       & \\multicolumn{1}{c}{{\\hbox{Run\\,II}} preliminary}  \\\\\n       & \\multicolumn{3}{c}{ CDF } \n       & \\multicolumn{2}{c}{ D\\O\\ }\n       & \\multicolumn{4}{|c}{ CDF } \n       & \\multicolumn{2}{c|}{ D\\O\\ } \n       & \\multicolumn{1}{c}{ CDF } \\\\\n       &  $\\ell$+jets    &    $\\ell\\ell$ &   alljets &    $\\ell$+jets  &   $\\ell\\ell$   &   $\\ell$+jets  &   $\\ell\\ell$   & alljets   &     Lxy   &  $\\ell$+jets      & $\\ell\\ell$  &  MEt \\\\\n\\hline\n\n\n\nPull   & $+0.40$  & $-0.51$  & $+1.11$  & $+1.32$  & $-0.38$  & $-0.51$  & $-0.82$  & $-0.41$  & $-0.67$ & $1.42$ & $+0.30$   & $+0.45$  \\\\\nWeight [\\%]  & $-4.7$ & $-1.1$ & $-0.9$ & $+0.4$ & $-0.2$ & $+62.0$ & $-0.3$ & $+10.5$ & $+0.22$ & $+20.6$ & $+1.4$     & $+11.9$  \\\\\n\\hline \\hline\n\\end{tabular}\n}\n\\end{center}\n\\label{tab:stat} \n\\end{table} \n\nNo input has an anomalously large pull. \nIt is, however, still\ninteresting to determine the top-quark mass separately \nin the alljets, $\\ell$+jets, $\\ell\\ell$, and MEt channels (leaving out\nthe $\\Lxy$ measurement).\nWe use the same methodology,\ninputs, uncertainty categories, and correlations as described above, but fit \nthe four physical observables, \\ensuremath{\\MT^{\\mathrm{alljets}}}, \\ensuremath{\\MT^{\\mathrm{\\ell\\mbox{+}jets}}}, \\ensuremath{\\MT^{\\mathrm{\\ell\\ell}}}, and \\ensuremath{\\MT^{\\mathrm{MEt}}}\\ separately.\nThe results of these combinations are shown in\nFigure~\\ref{fig:three_observables} and Table~\\ref{tab:three_observables}.\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{figures\/TevMtWin13-4channels.pdf}\n\\end{center}\n\\caption[Mtop in each channel]{Summary of the combination of the 12\ntop-quark measurements by CDF and D\\O\\ for different final states.}\n\\label{fig:three_observables} \n\\end{figure}\n\n\n\nUsing the results of Table~\\ref{tab:three_observables} \n we calculate the following chi-squared values including correlations:\n$\\chi^{2}(\\ell+{\\rm jets}-\\ell\\ell)=1.30\/1$, $\\chi^{2}(\\ell+{\\rm\n  jets}-{\\rm alljets})=0.07\/1$,  $\\chi^{2}(\\ell+{\\rm jets}-{\\rm MEt})=0.11\/1$, \n$\\chi^{2}(\\ell\\ell-{\\rm alljets})=0.42\/1$,\n  $\\chi^{2}(\\ell\\ell-{\\rm MEt})=1.22\/1$, and $\\chi^{2}({\\rm\n  alljets}-{\\rm MEt})=0.19\/1$.  \nThese correspond to chi-squared probabilities \nof 25\\%, 79\\%, 74\\%, 52\\%, 27\\%, and 66\\%  respectively, indicating\nthat the top-quark mass determined in each decay channel is consistent\nin all cases.\n\n\\begin{table}[t]\n\\caption[Mtop in each channel]{Summary of the combination of the 12\nmeasurements by CDF and D\\O\\ in terms of four physical quantities,\nthe mass of the top quark in the alljets, $\\ell$+jets,  $\\ell\\ell$, and MEt decay channels. }\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n\\begin{tabular}{ccrrrr}\n\\hline\\hline\n\n\nParameter & Value (\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }) & \\multicolumn{4}{c}{Correlations} \\\\\n               &                                 & $\\ensuremath{\\MT^{\\mathrm{alljets}}}$ &    $\\ensuremath{\\MT^{\\mathrm{\\ell\\mbox{+}jets}}}$  &  $\\ensuremath{\\MT^{\\mathrm{\\ell\\ell}}}$ & $\\ensuremath{\\MT^{\\mathrm{MEt}}}$ \\\\ \\hline\n$\\ensuremath{\\MT^{\\mathrm{alljets}}}$ & $172.7\\pm 1.9$    & 1.00       &                   &      & \\\\\n$\\ensuremath{\\MT^{\\mathrm{\\ell\\mbox{+}jets}}}$ & $173.2\\pm 0.9$    & 0.25       &    1.00       &     &\\\\\n$\\ensuremath{\\MT^{\\mathrm{\\ell\\ell}}}$ & $171.0\\pm 2.1$    & 0.19        &    0.41      & 1.00 & \\\\\n$\\ensuremath{\\MT^{\\mathrm{MEt}}}$& $173.8\\pm 1.8$    & 0.13        &    0.26      & 0.18 & 1.00 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:three_observables}\n\\end{table}\n\nTo test the influence of the choices in modeling the\ncorrelations, we performed a cross-check by changing all non-diagonal\ncorrelation \ncoefficients of the correlation matrix defined in Section~\\ref{sec:corltns} from 100\\% to 50\\% and re-evaluated the combination. \nThe result of this large variation of degree of correlation is a\n$\\gevcc{+0.19}$ shift of the top-quark mass and reduces the total\nuncertainty negligibly.\nThe chosen approach is therefore conservative.\n\n\n\n\nWe also performed separate combinations of all the CDF and D\\O\\ measurements. The results of these combinations are\n$\\gevcc{172.72~\\pm~0.93}$ for CDF and $\\gevcc{174.89~\\pm~1.42}$ for D\\O. Taking all correlations into account, \nwe calculate the chi-square value $\\chi^{2}(CDF-D\\O)=2.25\/1$ corresponding to a probability of 13\\%.\n\n\\section{Summary}\n\\label{sec:summary}\n\nAn update of the combination of measurements of the mass of the top quark\nfrom the Tevatron experiments CDF and D\\O\\ has been presented.  This preliminary\ncombination includes five published \\hbox{Run\\,I}\\ measurements, six published\n\\hbox{Run\\,II}\\ measurements, and  \none preliminary \\hbox{Run\\,II}\\ measurement, but the majority of these\nmeasurements are not yet performed on the full datasets available.  Taking into\naccount the statistical and systematic uncertainties and their\ncorrelations, the preliminary result for the Tevatron average is\n  $\\ensuremath{M_{\\mathrm{t}}}=\\gevcc{\\measStatSyst{173.20}{0.51}{0.71}}$,\nwhere the total uncertainty is obtained assuming Gaussian systematic uncertainties.\nThe central value is 0.02\\,GeV\/$c^2$  higher than our July 2012\naverage~\\cite{TeVTopComboPRD} of \n$\\ensuremath{M_{\\mathrm{t}}}=173.18\\pm0.94$\\,GeV\/$c^2$.\nAdding in quadrature the statistical and systematic uncertainties\nyields a total uncertainty of $\\gevcc{0.87}$ which represents an\nimprovement of $8\\%$.\n\nThe mass of the top quark is now known with a relative precision of\n0.50\\%, limited by the systematic uncertainties, which are dominated by\nthe jet energy scale uncertainty.  \nThis result will be further improved when all analysis channels from  \nCDF and D\\O\\ using the full Run~II data set are finalized.\n \n\n\\section{Acknowledgments}\n\\label{sec:ack}\n\nWe thank the Fermilab staff and the technical staffs of the\nparticipating institutions for their vital contributions. \nThis work was supported by  \nDOE and NSF (USA),\nCONICET and UBACyT (Argentina), \nCNPq, FAPERJ, FAPESP and FUNDUNESP (Brazil),\nCRC Program, CFI, NSERC and WestGrid Project (Canada),\nCAS and CNSF (China),\nColciencias (Colombia),\nMSMT and GACR (Czech Republic),\nAcademy of Finland (Finland),\nCEA and CNRS\/IN2P3 (France),\nBMBF and DFG (Germany),\nMinistry of Education, Culture, Sports, Science and Technology (Japan), \nWorld Class University Program, National Research Foundation (Korea),\nKRF and KOSEF (Korea),\nDAE and DST (India),\nSFI (Ireland),\nINFN (Italy),\nCONACyT (Mexico),\nNSC(Republic of China),\nFASI, Rosatom and RFBR (Russia),\nSlovak R\\&D Agency (Slovakia), \nMinisterio de Ciencia e Innovaci\\'{o}n, and Programa Consolider-Ingenio 2010 (Spain),\nThe Swedish Research Council (Sweden),\nSwiss National Science Foundation (Switzerland), \nFOM (The Netherlands),\nSTFC and the Royal Society (UK),\nand the A.P. Sloan Foundation (USA).\n\n\\clearpage\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\n\nConceptual metaphor is an ubiquitous cognitive mechanism that allows us to structure and reason about abstract concepts by relating them to experiential domains \\citep{lakoff2008metaphors, feldman2008molecule}. In language, metaphors allow human communication and reasoning about abstract ideas using concrete notions learned from sensorimotor, emotional, and other embodied experience~\\citep{thibodeau2011}: ``a plan is \\textit{solid}''; ``the economy \\textit{is stumbling}''; ``I \\textit{see} what you mean''.\n\nTo illustrate the role of metaphor in abstract reasoning, consider the following metaphorical statement: ``the economy is \\emph{stumbling}''. According to conceptual metaphor theory (CMT), we understand this statement through mental simulation, by connecting the abstract concept of economy to the imagined movement of a stumbling person. We use the same mental imagery to infer that the economy, just like a stumbling person, is ``unstable'' and ``might fall''. \nAs another example, in the metaphorical statement ``a proposal is \\emph{solid}'', bringing to mind solid objects and their properties suggests that the proposal, just like a physical object, was ``well-built'' and ``will not easily break''. \nHowever, not all properties generalize: unlike a physical object, a proposal, which is an abstract entity, cannot be ``thrown'' or ``bent''; \nunlike a stumbling person, the economy does not ``wear shoes''.\n\nLarge Language Models (LLMs) have achieved remarkable results on a variety of tasks. However, in contrast to humans, LLMs do not have access to commonsense and embodied experiences of the world~\\citep{bender-koller-2020-climbing}. Although the data LLMs are trained on includes up to trillions of text tokens, it is unclear how much of this data allows them to capture human commonsense reasoning \\citep{gordon2013reporting, becker-etal-2021-reconstructing}. Conceptual metaphor theory suggests that embodied and implicit knowledge is required for the ability to utilize metaphors in commonsense reasoning about concrete and abstract ideas.\n\nWe propose a novel dataset, MiQA (Metaphorical Inference Questions and Answers), to assess the ability of a model to reason with conventional metaphors.\nThe benchmark draws on the CMT research \\citep{grady1997foundations} to construct a representative set of primary metaphors, which are contrastively paired with literal statements.   \nOur task requires a model to make a correct inference in simple situations without specifying whether the contextual register is literal or metaphorical, leveraging research on metaphor processing \\citep{rai2020}, commonsense reasoning \\citep{davis2015}, and natural language inference \\citep{dagan2006, bowman-etal-2015-large}.\n\nOur benchmark combines the previously isolated areas of metaphor detection and commonsense inference. Although there is considerable research on both of these areas separately, it is unclear whether such capabilities scale compositionally: LLMs could handle two separate tasks well, but not their combination~\\citep{Keysers2019}.\n\nOur contributions are the following:\n\\begin{itemize}[nolistsep]\n    \\setlength\\itemsep{0.25em}\n    \\item We propose MiQA, a benchmark for commonsense inference with conventional metaphors;\n    \\item We show a large discrepancy between the performance of small and large models in a binary-choice MiQA task, from chance to human-level accuracy;\n    \\item We use a generative MiQA task to corroborate the performance of the largest model in an open-ended setting, showing that although human-level performance is approached, careful multiple-shot prompting is required. \n\\end{itemize}\n\n\n\\input{examples_table}\n\n\n\\section{Related Work}\n\nMetaphor has received renewed attention in natural language processing, but most tasks have been restricted to detection (e.g. \\citealp{shared-metaphor-task-2018, shared-metaphor-task-2020, choi-etal-2021-melbert}) on large annotated corpora \\citep{steen2010method, beigman-klebanov-etal-2016-semantic}. Human-level performance has not been reached by LLMs, but the progress is promising and an active area of research. However,\nthese tasks may excessively rely on context-dependent word meanings \\citep{neidlein-etal-2020-analysis} and do not measure the ability to reason with metaphor. \n\n\nMetaphor paraphrasing is another active area of research. BIG-bench \\citep{big-bench}, a collaborative multi-task benchmark intended to test a variety of capabilities of LLMs, includes four tasks related to metaphor.\nWhile these tasks contain novel metaphors, they still do not assess the ability to employ metaphoric knowledge in reasoning.\n\nRecently, \\citet{chakrabarty-etal-2022-rocket} built a dataset for multiple-choice story continuations involving similes and idioms extracted from books. \nSubsequently, \\citet{fig-qa-paper} proposed a metaphor interpretation task that requires models to choose the correct out of two interpretations of a simile. \nIn contrast, our task combines metaphor interpretation with commonsense inference, uses a more systematic data source, and has an additional adversarial character, as it requires the selection between two semantically-close items instead of items with opposite meanings.\n\n\n\n\n\\section{Dataset}\n\\label{dataset}\n\n\\subsection{Motivation}\n\nMost existing studies of metaphor have primarily started from corpus-based methods, using frequency and other corpus-based metrics to detect or classify metaphors. This process leads to a primary focus on corpus distributions and makes it hard to compare studies across different corpora. Furthermore, it ignores the central tenet of metaphor theory that foundational metaphors are grounded in non-linguistic and experiential domains which may be assumed as a background and thus underrepresented in corpora. \n\nTo address this, we chose to use a foundational ontology of primary conceptual metaphors~\\citep{grady1997foundations} based on CMT~\\citep{lakoff2008metaphors, feldman2008molecule}. Our choice of primary metaphors has multiple desirable properties. Primary metaphors are a good starting point for the investigation of more complex, compositional metaphor. The metaphors in our dataset are developmentally early in child experience with primary scenes and language. Moreover, the chosen metaphors are embodied, in that the source domain is often observable and sensory-motor (size, warmth, height) while the targets are less observable and often subjective or abstract (importance, affection, quantity). The metaphors we chose form a basis set of mappings that can create, through composition, more complex mappings, such as the Event Structure Metaphor, that maps movement and manipulation to actions. Our approach ensures that the distributions of metaphor categories in the task is balanced and hence reflective of the capability of large models to use primary metaphors as building blocks of reasoning.\n\n\\subsection{Construction}\n\nWe constructed a novel dataset consisting of $150$ items. \nThe items were manually created by the authors based on the work of \\citet{grady1997foundations}, which lists $100$ primary metaphors that are conventional, developmentally early, and form a basis set for composing complex mappings.\n\nEach item in the dataset is a tuple consisting of four sentences in English: a literal premise ($L_p$), a premise containing a conventional metaphor ($M_p$), an implication of the literal premise ($L_c$), and an implication of the metaphorical premise ($M_c$).\n\nThe tuples are paired so that a mistaken literal interpretation of $M_p$ can falsely suggest that $L_c$ is implied. For example, a wrong inference would be:\n``\\textit{I see what you mean}'' implies that ``\\textit{I am using my eyes}''.\nThe false implication $M_p \\to L_c$ thus serves as an adversarial element that probes whether the model correctly registers the metaphorical context.\n\nFor each primary metaphor proposed by \\citet{grady1997foundations}, we manually created $1$-$2$ pairs of items in the form described above, where $M_p$ is an example of the metaphor, while $L_p$ relates to the source domain of the metaphor only. \n\nTo create the final benchmark for LLMs, we used these items to generate two types of adversarial questions. \nThe first type (``\\textit{implies}''-questions) requests the model to select the most likely inference given a metaphorical statement. Answering correctly requires the model to not be tricked by a possible literal interpretation of the metaphorical premise.\nThe second type (``\\textit{implied-by}''-questions) requests the model to select the most likely premise that a literal statement is implied by. Answering correctly requires the model to not be tricked by a possible metaphorical interpretation of the literal conclusion.\nSee Table~\\ref{miq-questions} for examples.\n\n\nWe combined these items to obtain a benchmark consisting of $300$ questions, of which half are ``\\textit{implies}''-questions and half are ``\\textit{implied-by}''-questions. This pairing of tasks ensures that the model does not achieve a better score if biased towards assigning a higher likelihood to either literal or metaphorical continuations of a statement. \n\n\n\\section{Human Evaluation}\n\nWe estimated the human performance on the binary-choice task using the responses of $15$ human adult volunteers with English as first or second language.\nThe participants were told that the aim of the research was to gather a set of commonsense responses and compare them to LLMs responses. No additional information about the task was given.\n\n\n\\section{Large Language Models Evaluation}\n\n\n\\input{palm_generative_examples}\n\nWe evaluated the performance of two pre-trained LLMs: \nPaLM with 8B, 62B, 540B parameters \\citep{palm-paper} and GPT-3 Ada, Babbage, Curie and DaVinci \\citep{gpt3-paper}. The parameter counts of the GPT-3 models are not publicly available, but have been estimated at 350M, 1.3B, 6.7B, and 175B respectively \\footnote{\\href{ https:\/\/blog.eleuther.ai\/gpt3-model-sizes}{https:\/\/blog.eleuther.ai\/gpt3-model-sizes}}.\n\nThe main purpose of this study is to assess the capabilities of LLMs on the MiQA benchmark. For comparison, we also verify the capabilities of pre-trained fine-tuned smaller language models on our benchmark. We follow~\\citet{fig-qa-paper} in using encoder-only models trained on the natural language inference datasets SNLI~\\citep{bowman-etal-2015-large} and MNLI~\\citep{williams-etal-2018-broad} for zero-shot evaluation.\nWe opt for this approach because MiQA is designed as a small dataset suitable as a benchmark and not for fine-tuning.\nWe test the state-of-the-art encoder-only model DeBERTaV3~\\citep{he2021debertav3} in sizes small, medium and large. These models have 44M, 86M and 304M parameters respectively, and their weights are available online\\footnote{\\href{https:\/\/huggingface.co\/models?search=cross-encoder\/nli-deberta-v3}{https:\/\/huggingface.co\/models?search=cross-encoder\/nli-deberta-v3}}.\nThe models take a premise-implication pair and produce a probability distribution over three classes: ``entailment'', ``contradiction'', and ``undetermined''. We report the results for the best-performing score, in this case $1 - P(\\emph{contradiction})$, which outranks $P(\\emph{entailment})$.\n\n\n\\subsection{Binary-Choice Tasks}\n\nWe first assessed the models by prompting them with the question types illustrated in Table~\\ref{miq-questions} in $0$, $1$ and $5$-shot settings. Few-shot prompts were obtained by prefixing with randomly selected questions followed by their correct answers. To score the results, we obtained the log likelihood of each of the two choices as candidate continuations to the given prompt question. A response was scored as correct if the log likelihood of the correct choice was larger than that of the incorrect choice.\n\nPrompts can greatly influence LLM predictions \\citep{lu-etal-2022-fantastically, cao-etal-2021-knowledgeable}. As expected, we observed variability with changing prompts. To mitigate this, we tried multiple prompts, as detailed in Appendix~\\ref{appendix}. For each model, we selected the prompt that performed best with $0$-shot, and subsequently used this prompt to obtain and report its results in few-shot settings. Additionally, we used two baseline prompts (an empty prompt with no choices, and a prompt containing an unrelated question), which can indicate if the models simply learn to select either metaphorical or literal statements independently of the prompt in few-shot settings.\n \n\\subsection{Generative Task}\n\nIn addition to the binary-choice tasks, we also tested the largest model, PaLM-540b, in a generative setting. We prompted this model with questions of the form: ``$M_p$. Could that imply that $L_c$?''. This capitalises on the adversarial false implication $M_p \\to L_c$ described in section~\\ref{dataset}. \n\nWe obtained completions to the $150$ questions of this form generated from the MiQA dataset. Answers were manually and independently scored by every author. Every author scored at least two thirds of all responses and the scores were averaged. Scoring consisted of labelling the first paragraph of an answer as ``correct'', ``wrong'' or ``ambiguous''. To compute the accuracy over the generative task, ``correct'' responses were scored as $1$ and ``ambiguous'' responses were scored as $0.5$. Agreement between raters was medium, with intraclass correlation \\citep{shrout1979intraclass} $ICC(2, k)$ at $0.56$. Examples of scored answers are shown in Table~\\ref{palm-generative}. \n\nAs before, we evaluated the model in $0$, $1$ and $5$-shot settings. From the $0$-shot setting, we selected $32$ answers produced by the model that all authors independently scored as ``correct'', and the same number of answers of the form $M_p \\to M_c$ and $L_p \\to L_c$. We randomly selected $1$ or $5$ of these answers and their corresponding questions to prefix each prompt question in $1$- and $5$-shot settings.\n\n\n\\section{Results} \n\n\\input{table_narrow}\n\nThe full results are shown in Table~\\ref{miq-results}.\n\nFirstly, for the binary-choice tasks, there was a considerable gap between small and large LLMs. While the smaller models performed at or close to chance level, the largest models achieved very good performance even with $0$ shots, and approached human-level performance with few shots. We note that the ``\\textit{implied-by}'' task was overall more difficult than the ``\\textit{implies}'' task for both humans and LLMs. \n\nSecondly, the chance-level performance on the baseline prompts suggests that the increase in performance in few-shot settings was not due to the model learning to select either metaphorical or literal statements independently of the prompt. \nOn the other hand, the strong performance of the DeBERTaV3 models suggests a high level of transfer from the NLI datasets to MiQA, although there is a still a considerable gap to human performance.\n\nFinally, the generative results on PaLM-540b estimated the model performance in an open-ended setting. Similarly to the binary task, the model performed considerably better with $5$ shots compared to $0$ shots, approaching human performance. However, the gap between human and model performance for the generative task was greater compared with the gap for the binary-choice task.\n\nOverall, the results demonstrate that LLMs can correctly select between the metaphorical and literal contextual registers to perform inferences with conventional metaphors, but there is still a considerable gap between human and LLM performance in $0$-shot settings.\n\n\n\n\\section{Limitations}\n\n\nOur work used foundational metaphors from CMT to test basic metaphoric reasoning in LLMs.\nWe will expand this benchmark using additional and more complex sources of conceptual metaphor (e.g. \\citealp{metanet}). Future work will assess LLMs on novel non-conventional mappings.\n\nAlthough we mitigated for prompt sensitivity by using multiple prompts, the result interpretation should allow for small accuracy variations. \nFurther, in the binary-choice tasks we compare the LLM results with a human baseline, but we do not provide a baseline for the generative task. This is near perfect for humans, but a more systematic baseline can be created to quantify the exact headroom on this task. \nFinally, while the task holistically measures the performance of LLMs on a complex task, it is difficult to disentangle the component effects \n(metaphor detection, reasoning, response generation) in the overall accuracy. \n\n\\section{Conclusion}\n\nWe have proposed a novel compositional benchmark based on conceptual metaphor theory to assess the capacity of LLMs to make inferences with metaphors. Successful performance on this task requires metaphor detection and commonsense inference. Using a metaphor theory-based approach allows us to systematically explore capabilities and limitations of LLMs. This is the first in a planned series of increasingly complex metaphor inference datasets.\n\n\nThree main findings emerged from our proposed task. Firstly, there is a vast difference between the performance of small and large LLMs, with the former performing at chance level and the latter approaching human level in few-prompt setting. This observation is informative in the context of previous results showing that some, but not all, tasks observe a qualitative performance jump with model size and scale: for example, this is the case for reasoning about goal-step relationships between events and ordering events, but not for navigation and mathematical induction tasks~\\citep{palm-paper}. This result invites more research into the question of how and whether the performance of smaller models can be improved. Secondly, this reflects a true ability of LLMs to reason with conventional metaphor, and not simply to detect it. Whether this ability extends to novel metaphor is ongoing work. Finally, the performance of large LLMs approaches that of humans in binary-choice and generative tasks, but careful multiple-shot prompting is required.\n\n\n\\newpage\n\n\\section*{Acknowledgements}\n\nWe thank Fernando Pereira, Yasemin Altun, \\mbox{William} Cohen and Tiago Pimentel, as well as our anonymous reviewers, for their valuable feedback.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n\nSemantic segmentation is an important task in computer vision with a high potential for practical applications, in particular for autonomous vehicles. Thanks to the predicted 2D segmentation maps by the deep convolutional neural networks (DCNNs), it contributes to an improved understanding of the scenes.\n\n\n\n\nA large body of the recent literature on DCNNs for semantic segmentation focuses on improving predictive performance and run-time through advanced~\\cite{chen2017deeplab,yu2017dilated} or lighter~\\cite{yu2018bisenet, zhao2018icnet, li2019dfanet, li2019partial} architectures, better use of multiple resolutions~\\cite{zhao2017pyramid, sun2019high} and novel loss functions~\\cite{lin2017focal, sudre2017generalised, berman2018lovasz, zhu2019improving}. However, for real-world deployments, other requirements, e.g., reliability, robustness, must be equally satisfied to avoid any failures. To reach them, there are a few major challenges yet to be fully solved. First, DCNNs have been shown to be overconfident~\\cite{guo2017calibration} even when their predictions are wrong~\\cite{nguyen2015deep, hein2019relu}. In addition, DCNNs struggle to learn when there are few training samples are available, or data and annotations are noisy. In particular, high capacity DCNNs can find ``shortcuts'' that allow them to exploit spurious correlations in the data (e.g., background information~\\cite{rosenfeld2018elephant, srivasta2020human}, textures and salient patterns~\\cite{geirhos2018imagenettrained}) towards minimizing the training error at the cost of generalization. Such DCNNs have been shown to be biased, e.g. contextual bias~\\cite{xiao2021noise} or texture bias~\\cite{geirhos2018imagenettrained}. This type of problem could be addressed by larger and higher quality datasets~\\cite{lambert2020mseg}, yet the entire complexity of the world cannot be encompassed in a training dataset with a limited size. Alternative solutions leverage uncertainty estimation for detecting such failures~\\cite{gal2016dropout, lakshminarayanan2017simple, franchi2019tradi}. However the most effective ones are computationally inefficient as they rely on ensembles or multiple forward passes~\\cite{ovadia2019can, gustafsson2020evaluating, ashukha2020pitfalls}.  \n\n\n\n\n\n\nIn this paper, we aim to increase the \\Correction{robustness} of semantic segmentation models. For the scope of this work, we define \\Correction{\\emph{robust} as follows}: \\emph{a model is \\Correction{robust} if its predictions are accurate and well calibrated when facing \ntypical conditions from the training distribution, but also under distribution shift (epistemic and aleatoric uncertainty) and for unknown object classes which are not seen during training (epistemic uncertainty).} \\Correction{Our definition extends the scope of \\emph{robustness} beyond invariance to different perturbations of the input, e.g., \nadversarial attacks~\\cite{szegedy2013intriguing, eykholt2017robust}, image corruptions~\\cite{pezzementi2018putting, hendrycks2019benchmarking}, change of style~\\cite{geirhos2018imagenettrained}, where only prediction accuracy is used as a proxy for robustness. Here, a robust model must not be only accurate, but also well calibrated such that unknown objects or strong input perturbations are designated low confidence scores and easily identified as unreliable and discarded. Although difficult to attain, we argue that both accuracy and calibration are essential for deployment in real world conditions where the data distribution is not identical to the training distribution.\\footnote{\\Correction{The two metrics are often at odds with each other: a classifier can be accurate but non-calibrated (usually overcofident~\\cite{nguyen2015deep, guo2017calibration, hein2019relu}) and conversely it can be inaccurate yet calibrated, if its predictions are always low-confident.}}} To improve the \\Correction{robustness} of a DCNN,\ngiven \nthe noisy nature of the data, and to address the problem of limited numbers of training labeled images, we propose a technique that combines the teacher-student framework~\\cite{mean_teacher_NIPS_2017} with a novel data augmentation strategy (Fig.~\\ref{fig:unsupervised_training}). Our augmentation method, named \\emph{Superpixel-mix}, exchanges superpixels between training images to generate more training samples that preserve object boundaries and disentangle objects' parts from their frequent contexts. \nTo the best of our knowledge, this is among the first investigations on the use of these techniques for reducing DCNN bias and uncertainty. \n\n\n\\noindent \\textbf{Contributions: } In summary, our contributions are as follows: \\textbf{(i)} Superpixel-mix, a new type of data augmentation for creating new unlabeled images to increase DCNNs' accuracy and \\Correction{robustness}.\n\\textbf{(ii)} A theoretical grounding on why \nmixing augmentation combined with the teacher-student framework can improve \\Correction{robustness}. The theory is confirmed by a set of experiments. \n\\textbf{(iii)} A new dataset for quantifying contextual bias of DCNNs.\\footnote{The dataset will be made publicly available after the anonymity period}\n\n\n\n\\vspace{-3mm}\n\\section{Related Work}\n\n\\Correction{\\noindent\\textbf{Robust Deep Learning.} Robustness of DCNNs has been studied under different perspectives in the past few years, e.g., robustness to adversarial attacks~\\cite{papernot2016practical, eykholt2017robust}}. We focus here rather on the \\Correction{robustness} of the perception functions,\nand less on security aspects. Geirhos \\emph{et al}\\bmvaOneDot~\\cite{geirhos2018imagenettrained} observe that classification models trained on ImageNet are biased towards textures and blind to shapes. They mitigate this by augmenting the training set with stylized images~\\cite{gatys2015neural}, yet this can be detrimental for semantic segmentation as object boundaries are distorted. Shetty \\emph{et al}\\bmvaOneDot~\\cite{car_sidewalk_dependency_cvpr_2019} counter contextual bias \nwith a data augmentation strategy that removes random objects from images. Some other works focus on evaluating the robustness under different image perturbations, e.g., blur~\\cite{vasiljevic2016examining}, brightness~\\cite{pei2017deepxplore}. A more systematic study of robustness of classification DCNNs to image perturbations over varying levels of corruption is proposed in \\cite{hendrycks2019benchmarking}. This idea is extended to autonomous driving datasets~\\cite{michaelis2019benchmarking}, where robustness of object detection methods is evaluated. New datasets with challenging weather conditions, e.g., rain~\\cite{hu2019depth}, fog~\\cite{sakaridis2018semantic}, low light~\\cite{sakaridis2019lowlight} are created to assess and improve robustness of visual perception models. Most approaches simply evaluate evolution of accuracy under such distribution shifts, but ignore other metrics, e.g., calibration that is essential for \\Correction{robustness} (calibrated predictions facilitate thresholding for low-confidence predictions and detection of distribution shift). We evaluate our proposed method on multiple shifted datasets~\\cite{hendrycks2019benchmarking, michaelis2019benchmarking, sakaridis2018semantic, hu2019depth} and show \\Correction{robustness} improvements, beyond accuracy. \n\n\n\\noindent\\textbf{DCNN Uncertainty Estimation.} Uncertainty estimation, i.e., knowing when a model does not ``know'' the answer, is a crucial functionality for \\Correction{robust} DCNNs. Most DCNN approaches for uncertainty estimation are inspired from Bayesian Neural Networks~\\cite{neal1995bayesian, mackay1992bayesian}. Deep Ensembles (DE)~\\cite{lakshminarayanan2017simple} train multiple instances of a DCNN with different random initializations, while MC-Dropout~\\cite{gal2016dropout} mimics an ensemble through multiple forward passes \nwith active Dropout~\\cite{srivastava2014dropout} layers. In-between them, some methods generate ensembles with lower training cost (by analyzing weight trajectories during optimization~\\cite{maddox2019simple, franchi2019tradi}) or with lower forward cost (by generating ensembles from lower dimensional weights~\\cite{wen2020batchensemble, franchi2020encoding} or multiple network heads~\\cite{lee2015m}). Other works prioritize computational efficiency to compute uncertainties from a single forward pass~\\cite{malinin2018predictive, sensoy2018evidential, postels2019sampling, brosse2020last, van2020uncertainty, joo2020being}, but become specialized to a single type of uncertainty~\\cite{hora1996aleatory, kendall2017uncertainty, malinin2018predictive}, e.g., \\Correction{Out-Of-Distribution (OOD)}. DE methods are top-performers across benchmarks~\\cite{ovadia2019can, gustafsson2020evaluating}. \nYet, their computational costs make them unfeasible for complex vision tasks, e.g., semantic segmentation. With Superpixel-mix we aim to attain most properties of ensembles, e.g., predictive uncertainty and calibration, in a cost-effective manner. \n\n\n\n\n\n\n\\noindent\\textbf{Augmentation by mixing samples.} Initially seen as a mere heuristic to address over-fitting, data augmentation is now an essential part of recent supervised~\\cite{cutmix_ICCV_2019, cutout_arXiv_2017, zhang2017mixup}, semi-supervised~\\cite{cutmix_SSL_seg_BMVC_2020, sohn2020fixmatch,mixmatch_nips_2019} and self-supervised learning methods~\\cite{gidaris2020learning, chen2020simple}. Mixing techniques, among the most powerful augmentation strategies, generate new ``virtual'' samples (and labels) from pairs of training samples. Mixup\\cite{zhang2017mixup} interpolates two images, while Manifold Mixup~\\cite{verma2019manifold} interpolates \nhidden activations. CutMix~\\cite{cutmix_ICCV_2019,cutmix_SSL_seg_BMVC_2020} replace a random square inside an image with a patch from another image.  Classmix~\\cite{classmix_arXiv_2020} cuts and mixes object classes. Puzzle-Mix ~\\cite{kim2020puzzle} and Co-Mixup ~\\cite{kim2021co}  mix salient areas. For semantic segmentation, mixing by square blocks \n\\Correction{is agnostic to} object boundaries and is likely to increase contextual bias as objects \\Correction{or object parts can be recognized via their context, i.e., learning shortcuts~\\cite{geirhos2020shortcut}.}\nSuperpixel-mix mitigates this by mixing within object boundaries. \n\n\n\n\n\n\n\n\\vspace{-3mm}\n\\section{Proposed method}\n\n\\vspace{-2mm}\n\\subsection{Overview}\\label{Overview}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\linewidth]{images\/unsupervised_training2.jpg}\n\\vspace{-5pt}\n\\caption{\\small Consistency training with a student-teacher framework using unlabeled images (Task 2). To mix two unlabeled images, superpixels are randomly sampled to create a mixing mask. This mask is used to merge the two images and their pseudo-label outputs from the teacher network. A cross-entropy loss is applied to the student network to encourage consistency between the mixed pseudo-labels and the student network labels.}\n\\label{fig:unsupervised_training}\n\\vspace{-10pt}\n\\end{figure*}\n\nIn this paper, we propose a novel superpixel-mix method to generate new training data and \nleverage the results of this data augmentation technique in an existing teacher-student framework~\\cite{mean_teacher_NIPS_2017}. %\nThe combination of our mixing technique and the teacher-student framework forms an optimization component that serves as a consistency constraint in our DCNN training system for semantic segmentation. We conduct experiments for evaluating our trained DCNNs on both types of uncertainties: epistemic and aleatoric. At the end, we also assess our proposed approach in semi-supervised learning.\n\nIn general, our training process for all of our experiments comprises two steps that are optimized simultaneously: \\textbf{(1)} supervised learning where we train the DCNNs using images with ground-truth labels, and \\textbf{(2)} using teacher-student optimization with superpixel-mix data augmentation on images as a consistency constraint that does not use ground-truth labels. In the uncertainty experiments such as \\Correction{OOD} in \n\\S~\\ref{OOD}, DCNNs' bias studying in \n\\S~\\ref{ablation_bias}, and aleatoric uncertainty in \n\\S~\\ref{aleatoric}, we use   datasets that contain ground-truth labels for all images. We first train the DCNNs in fully supervised learning fashion as in step 1. We then remove all those labels and use only the images to optimize for the consistency constraint as in step 2. In the semi-supervised learning experiment in \n\\S~\\ref{semi-supervised}, we use the training dataset that consists of two parts: labelled images and unlabelled images. We train the DCNN using the labelled data for step 1 and the unlabelled data for step 2. %\n\n\\textbf{Step 1 - supervised learning with labelled images:}  We use a standard pixel-wise cross-entropy loss denoted by $\\mathcal{L}_{\\text{sup}}$ and apply \nit to all the labelled images. \nIn addition, we use a weak data augmentation (WDA) that consists of \\textit{horizontal flipping} and\/or \\textit{random cropping}.\n\n\\textbf{Step 2 - consistency constraint with unlabelled images:} We apply two transformations on an unlabelled image: one is WDA and the other is a strong data augmentation (SDA). Consistency training encourages predictions of the DCNNs to be consistent in the results of the two transformations.  For SDA, we merge WDA and a superpixel mixing technique (see \n\\S~\\ref{watershedmix}). The consistency loss for optimizing this constraint is denoted as $\\mathcal{L}_{\\text{cons}}$.\n\nFor \nevery experiment, we optimize the loss in the overall framework as %\nthe \\textbf{joint loss}  $\\mathcal{L} = \\mathcal{L}_{\\text{sup}}+ \\lambda \\cdot \\mathcal{L}_{\\text{cons}}$, where $\\lambda$ is a weighting hyper-parameter and is set to 1.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{-2mm}\n\\subsection{Teacher-Student Framework}\\label{sec:teacherstudent}\n\n\nTo learn from unlabeled images, we follow the teacher-student framework established in~\\cite{mean_teacher_NIPS_2017}, where the teacher network produces pseudo-labels learned from the labeled data and the student network is encouraged to be consistent with the teacher.  Consistency is encouraged via \ncross-entropy loss between the two outputs. In our case, we encourage consistency between the mixed output labels of a teacher network \ncorresponding to the two unlabelled inputs and the output label of a student network \npredicted from the input that resulted by mixing the two unlabelled images. \nWe explain this approach in details in the following paragraph.%\n\nLet $g_{\\phi}$ represent the teacher network with weights $\\phi$ and $f_{\\theta}$ be the student network with weights $\\theta$.  For two unlabeled images $x^1_u$ and $x^2_u$, we can use the teacher network to generate pseudo-labels $y^1_u$ and $y^2_u$:\n$y^1_u = g_{\\phi}(x^1_u)$ and $y^2_u = g_{\\phi}(x^2_u)$. Assume now we are given some mixing function $\\texttt{mix}$ with a mixing parameter $m$.  Without any assumption on the mixing itself, we denote a mixed output label $y_m$, where $y_m = \\texttt{mix}(y^1_u,y^2_u,m)$. For the same mixing parameter $m$, we can also mix the inputs $x^1_u$ and $x^2_u$, \\emph{i.e.}\\@\\xspace $x_m = \\texttt{mix}(x^1_u,x^2_u,m)$.\nApplying $x_m$ %\nto the student network, we expect $f_{\\theta}(x_m)$ to be the same as the mixed output $y_m$. This {is} enforced by minimizing the consistency loss $    \\mathcal{L}_{\\text{cons}} = \\text{CE}\\left(y_m, %\n    f_{\\theta}\\left(\\texttt{mix}(x^1_u, x^2_u, m)\\right)\\right)$,\nwhere $\\text{CE}$ is the pixel-wise cross-entropy and $y_m$ is the mixed pseudo-labels from the teacher. \n\n\nDuring training, both the teacher and student networks evolve together. Similarly to~\\cite{mean_teacher_NIPS_2017}, we update the teacher network weights $\\phi$ after each iteration with an \\Correction{Exponential} Moving Average \\Correction{(EMA)}, \\emph{i.e.}\\@\\xspace $\\phi = \\alpha \\phi + (1 - \\alpha)\\theta$ where $\\alpha=0.99$ is a momentum-like parameter.%\n\n\\vspace{-2mm}\n\\subsection{Superpixel-mix for semantic segmentation} \\label{watershedmix}\n\nTo mix two unlabeled images, we use masks generated by sampling superpixels. Superpixels are local clusters of visually similar pixels. Therefore, a group of pixels belonging to the same superpixel are likely to be in the same object or the same part of the object.\nThere are several superpixel variants, including SEEDS~\\cite{van2012seeds}, SLIC~\\cite{slic} or Watershed superpixels~\\cite{watershed_superpixel_ICIP_2014}.  We opt to use Watershed superpixels as their boundaries retain more salient object edges~\\cite{machairas2015waterpixels}.  We refer the reader to the Supplementary Material for details on how we use the watershed transformation to produce superpixels.\n\n\nGiven an unlabeled image $x^1_u$, we apply the watershed superpixel algorithm, which results in a set of $n$ superpixels $\\mathcal{S}=\\{S_1, S_2,..., S_n\\}$.  A mixing mask $m$ (which is a binary mask)\\footnote{For simplicity, we overload the notation of the mixing parameter $m$ simply as the mixing mask itself.} is created from a sampled subset of superpixels $S$: $m = \\cup_{j\\in \\sigma(k,n)} S_j$ %\nwhere $\\sigma(k,n)$ is a subset of size $k$ of the  $n$  indices, and $k$ is the number of superpixels we want to keep.  %\n\nThe mixing mask $m$ defines the pixels in $x^1_u$ which will be replaced by pixels from the unlabeled image $x^2_u$ to form the mixed input $x_m$, \\emph{i.e.}\\@\\xspace $\n    x_m = \\texttt{mix}(x^1_u, x^2_u, m) = (1-m) \\odot x^1_u + m \\odot x^2_u\n$, where $\\odot$ is a pixel-wise multiplication. Superpixels are uniformly sampled given a fixed proportion of selected superpixels. \nContrary to existing regularization techniques such as Cutout~\\cite{cutout_arXiv_2017} or Cutmix~\\cite{cutmix_SSL_seg_BMVC_2020}, our superpixel mixing strategy enforces each set of selected pixels in the unlabeled image $x^1_u$ %\nto belong to the same object. However, the algorithm is allowed to select a set of superpixel clusters from different objects as illustrated in Figure \\ref{fig:unsupervised_training}.  %\n\nWe use 200 superpixels per image for all the evaluated datasets. Studies on the number and proportions of superpixels used in mixing are shown in Section~\\ref{Ablations_sp} and the Supplement. %\n\n\\vspace{-2mm}\n\\subsection{From empirical risk to teacher-student mixup}\n\n\nIn this section, we show that the training loss of the teacher-student framework in combination with superpixel-mix data augmentation is bounded by the accuracy of the teacher network and the quality of the data augmentation.\nLet $\\mathcal{D} =\\{(x_i,y_i)\\}_i \\sim \\mathcal{P}$ be the labelled dataset which follows the joint distribution $\\mathcal{P}$ and $l$ be a loss %\nbetween the target $y$ and the prediction $f_{\\theta}(x)$ of the DCNN $f_{\\theta}$. Typically, in deep learning, the objective is to learn $\\theta$ that minimizes the expected risk defined by: \n$\\mathbf{R}_{\\mathcal{P}}(f_{\\theta}) =\\int l(f_{\\theta}(x),y)d\\mathcal{P}(x,y)$.\nAs we do not have access to the distribution $\\mathcal{P}$, we optimize the loss function that is formed by the empirical risk on $\\mathcal{D}$: %\n\\begin{equation}\n    \\hat{\\mathbf{R}}_{\\mathcal{P}_{\\delta}}(f_{\\theta}) =\\frac{1}{n}\\sum_{i=1}^n l(f_{\\theta}(x_i),y_i) = \\int l(f_{\\theta}(x),y)d\\mathcal{P}_{\\delta}(x,y),\n\\end{equation}\nwhere the summation is converted back to the integral based on $\\mathcal{P}_{\\delta}(x,y) = \\frac{1}{n} \\sum_{i=1}\\delta(x=x_i,y=y_i)$, as shown by~\\cite{zhang2017mixup}.  %\n\nTherefore, we optimize the parameters of the DCNN using the empirical risk. However, the representation of the discretized data is likely to be sparse, Zhang \\emph{et al}\\bmvaOneDot~\\cite{zhang2017mixup} proposed to work with $\\mathcal{D}_{\\text{mix}} =\\{(x_{m,i},y_{m,i})\\}_i \\sim \\mathcal{P}^{\\text{mix}}_{X,Y}$ where $x_{m,i}$, and $y_{m,i}$ are the data of $\\mathcal{D}$ where a mixing procedure has been applied. The hypothesis in~\\cite{zhang2017mixup}  is that the mixing procedure helps to better approximate the dataset distribution. Let $ \\mathcal{P}^{\\text{mix}}_{\\delta}$ denote the discrete distribution of this augmented %\ndataset. The risk to fit the teacher prediction on $\\mathcal{P}^{\\text{mix}}_{\\delta}$ can then be defined as:\n$\n\\hat{\\mathbf{R}}_{\\mathcal{P}^{\\text{mix}}_{\\delta}}(f_{\\theta},g_{\\phi}) =\\int l(f_{\\theta}(x),g_{\\phi}(x))d\\mathcal{P}^{\\text{mix}}_{\\delta}(x,y).\n$ Therefore, our training loss for the overall framework is defined in detail as the following:\n$\n    \\mathcal{L}(\\theta)= \\hat{\\mathbf{R}}_{\\mathcal{P}_{\\delta}}(f_{\\theta})+  \\hat{\\mathbf{R}}_{\\mathcal{P}^{\\text{mix}}_{\\delta}}(f_{\\theta},g_{\\phi}).\n$ As the loss $l$ is a norm that satisfies the triangle equality, we can prove that the training loss $\\mathcal{L}(\\theta)$ is bounded by the following:\n\\begin{equation}\n    \\mathcal{L}(\\theta)\\leq 2\\mathbf{R}_{\\mathcal{P}}(f_{\\theta})+  M(\\| \\mathcal{P}^{\\text{mix}}_{\\delta} -\\mathcal{P}\\|_1 + \\| \\mathcal{P}_{\\delta}  -\\mathcal{P}\\|_1) +\\hat{\\mathbf{R}}_{\\mathcal{P}^{\\text{mix}}_{\\delta}}(g_{\\phi}),\n\\end{equation}\n\n\n\\noindent where the four terms are linked to the true error, approximation error, mixing distribution error and teacher error, respectively. \nThis implies that the quality of the DCNN is bounded by the accuracy of the teacher. It is also bounded by how much the mixing strategy can sample the true distribution of the dataset. Finally, the distribution of the training data with respect to the true data distribution also plays an important role.  %\nThis finding can be applied to all teacher-student frameworks, such as those used in SSL, self supervised training, and domain adaptation.\nThe detailed proof and analysis is given in the Supplementary Material. %\n\n\\Correction{$\\| \\mathcal{P}^{\\text{mix}}_{\\delta} -\\mathcal{P}\\|_1$ and $\\| \\mathcal{P}_{\\delta}  -\\mathcal{P}\\|_1$\nreflect the capacity of the two  distributions $\\mathcal{P}_{\\delta}$ and $\\mathcal{P}^{\\text{mix}}_{\\delta}$ to approximate the true dataset distribution.  This bound allows us to control the variation of the risk. To reduce the risk, we can increase the number of training data or improve the quality of the data augmentation and so  reduce $\\| \\mathcal{P}^{\\text{mix}}_{\\delta} -\\mathcal{P}\\|_1 $. This motivates our research on data augmentation strategies to approximate the true distribution in a data-efficient way. We can also improve the quality of the teacher using EMA training that stabilizes the training loss. } %\n\n\\vspace{-3mm}\n\\subsection{Uncertainty and Deep Learning}\n\n\nConsider a joint distribution $\\mathcal{P}$ over input $x$ and labels $y$ over a set of labels $\\mathcal{Y}$. When a DCNN performs inference, it predicts $f_{\\theta} = \\mathcal{P}(y|x,\\theta)$, where $\\theta$ is optimized to minimize the loss over the training set $\\mathcal{D}$. This likelihood typically suffers from two kinds of uncertainty\\AB{~\\cite{hora1996aleatory, kendall2017uncertainty}}.  First is aleatoric uncertainty, linked to the unpredictability of the data acquisition process. During inference, instead of working with $x$, we may have access to $x+n$, %\nwhere $n$ represents noise on the input data. Second is epistemic uncertainty, linked to the lack of knowledge of the model, \\AB{i.e.,} the weights $\\theta$ of the network. In addition, the epistemic uncertainty can be subdivided into two sub-types: one linked to the \\Correction{OOD} \\AB{~\\cite{malinin2018predictive}} and the other one linked to networks' bias. \\Correction{Epistemic uncertainty models the uncertainty associated with limited sizes of training datasets. Most works focus only on the ability to detect OOD. In this paper, we conduct experiments for all types of uncertainties: aleatoric (i.e., testing models on noisy data) and two sub-types of epistemic (i.e., OOD detection and models' bias).}%\n \n \n\n \n \n \n\\vspace{-3mm}\n\\section{Experiments}\n\nWe \n\\AB{conduct} experiments on five datasets. First, %\nwe study \n\\AB{network robustness} to epistemic uncertainty experiments on StreetHazards \\cite{hendrycks2019anomalyseg}.  \n\\AB{The test set contains some object classes that are not available in the training set.}\nThe goal is to detect these out-of-distribution (OOD) classes. We also evaluate the performance of the DCNNs on an contextually unbiased dataset.\nFurthermore, we investigate the networks' \\Correction{robustness} for the aleatoric uncertainty. \n\\AB{To this end, we train a DCNN on Cityscapes ~\\cite{Cityscapes_CVPR_2016}} and evaluate the performances on Rainy \\cite{hu2019depth} and Foggy Cityscapes~\\cite{sakaridis2018semantic}. %\nFinally, we evaluate \\AB{on the} semi-supervised learning \\AB{task} on Cityscapes~\\cite{Cityscapes_CVPR_2016} and Pascal ~\\cite{pascal-voc-2012}. We implement the experiments using PyTorch (see Supplementary Material).\n\n\\vspace{-3mm}\n\\subsection{Evaluation criteria}\n\nThe first criterion we use is the mIoU \\cite{jaccard1912distribution}, \n\\AB{reflecting the predictive performance of segmentation models.}\n\\AB{Second, similarly to \\cite{lakshminarayanan2017simple} we use} the negative log-likelihood (NLL), \\AB{a proper scoring rule~\\cite{gneiting2007strictly},} which depends on the aleatoric uncertainty \\Correction{and can assess the degree of overfitting~\\cite{guo2017calibration}.}. In addition, we use the expected calibration error (ECE) ~\\cite{guo2017calibration} that measures \n\\AB{how} the confidence score predicted by a DCNN is related to its accuracy. Finally, we use the AUPR , AUC, and the FPR-95-TPR  defined in \\cite{hendrycks2016baseline} that evaluate the quality of a DCNN to detected OOD data. With multiple metrics, we can get a clearer picture on the performance of the DCNNs with regards to accuracy, calibration error, failure rate, \\AB{OOD detection}. Even though it is difficult to achieve top performance on all metrics, \n\\AB{we argue that it is more pragmatic and convincing}\nto evaluate on multiple metrics~\\cite{ovadia2019can, fort2019deep} than \\AB{optimizing for a single metric, potentially at the expense of many others.}\n\\Correction{For example, a DCNN with a low accuracy and a low confidence score is well-calibrated. Therefore, evaluating a DCNN on a single metric such as ECE or mIoU alone is not enough. We aim to have a good compromise between accuracy and calibration.}\n\n \\vspace{-3mm}\n\\subsection{Epistemic uncertainty: Out-Of-Distribution (OOD) Detection }\\label{OOD}\n\n\\Correction{One cause of the epistemic uncertainty in deep learning is the limited training data that does not cover all possible object classes. The evaluation of this type of epistemic uncertainty is often linked to OOD detection. Therefore, this experiment is designed to evaluate the epistemic uncertainty using StreetHazards~\\cite{hendrycks2019anomalyseg}.}\nStreetHazards is a large-scale dataset that consists of different sets of synthetic images of street scenes. This dataset is composed of $5,125$ images for training and $1,500$ test images.\nThe training dataset contains pixel-wise annotations for $13$ classes. The test dataset comprises $13$ training classes and $250$ OOD classes, unseen in the training set, making it possible to test the robustness of the algorithm when facing a diversity of possible scenarios.\nFor this experiment, we use DeepLabv3+~\\cite{chen2018encoder} with the experimental protocol from~\\cite{hendrycks2019anomalyseg}. We use ResNet50 encoder~\\cite{he2016deep}. For this experiment, we compare our algorithm to Deep Ensembles \\cite{lakshminarayanan2017simple}, BatchEnsemble \\cite{wen2020batchensemble}, LP-BNN \\cite{franchi2020encoding}, TRADI \\cite{franchi2019tradi} , MIMO \\cite{havasi2020training}\n\\AB{achieving} state-of-the-art results on epistemic uncertainty. We also compare our \\AB{model with} MCP which is the baseline DCNN where we consider the maximum probability class as a confidence score, and with Cutmix~\\cite{cutmix_SSL_seg_BMVC_2020} strategy.\nThe results in Table~\\ref{table:outofditribution} show that our method is the only one to have the best results in three out of five measures. Running up, Deep Ensemble and LP-BNN achieve best results on one measure only. Moreover, our method achieves the best results faster than Deep Ensemble and LP-BNN, we only need one inference pass compared to 4 inference passes for the others.\n\n\\begin{table*}[!t]\n\\renewcommand{\\figurename}{Table}\n \\begin{center}\n \\scalebox{0.60}\n {\n \\begin{tabular}{c l  c c c c c c }\n \\toprule\n Dataset & OOD method  & mIoU $\\uparrow$ & AUC $\\uparrow$  & AUPR $\\uparrow$ & FPR-95-TPR $\\downarrow$ & ECE $\\downarrow$ & \\# Forward passes $\\downarrow$ \\\\ \n \\midrule\n\\multirow{6}{*}{\\shortstack[c]{\\textbf{StreetHazards} \\\\ DeepLabv3+ \\\\ ResNet50}}  &\nBaseline  (MCP)~\\cite{hendrycks2016baseline} & 53.90\\%  & 0.8660& 0.0691 & 0.3574 & 0.0652 &1 \\\\ \n             & TRADI      \\cite{franchi2019tradi} &  52.46\\%\t& 0.8739 & 0.0693 & 0.3826 &0.0633 &4 \\\\ \n            & Cutmix ~\\cite{cutmix_SSL_seg_BMVC_2020}  & 56.06\\% & 0.8764 & 0.0770 &0.3236 &0.0592 &1 \\\\ \n            & MIMO  \\cite{havasi2020training} & 55.44\\% &  0.8738 &  0.0690 & 0.3266 & 0.0557 &4 \\\\ \n            & BatchEnsemble \\cite{wen2020batchensemble}  &56.16\\%  & 0.8817 & 0.0759 & 0.3285 & 0.0609 &4 \\\\ \n      & LP-BNN \\cite{franchi2020encoding}  &  54.50\\% & 0.8833 & 0.0718 & 0.3261 & \\textbf{0.0520} &4 \\\\ \n    & Deep Ensembles  \\cite{lakshminarayanan2017simple}& 55.59\\% & 0.8794 & \\textbf{0.0832} & 0.3029 & 0.0533  &4 \\\\ \n   & Superpixel-mix (ours) & \\textbf{56.39}\\% & \\textbf{0.8891} & 0.0778 & \\textbf{0.2962} &0.0567 &1 \\\\ \n\n\\bottomrule\n \\end{tabular}\n } %\n \\end{center}\n \\vspace{-1mm}\n \\caption{\\small \\textbf{Comparative results on the OOD task for semantic segmentation.}  \\small Results are averaged over three seeds.}\\label{table:outofditribution}\n \\vspace{-3mm}\n \\end{table*}\n\n\n \\vspace{-3mm}\n\\subsection{Epistemic uncertainty : Unbiased experiment} \\label{ablation_bias}\n\\vspace{-1mm}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{images\/bias_results3.jpg}\n\\vspace{-3mm}\n\\caption{\\small A qualitative example of the network bias study. When the road and pavements are replaced with sand texture, the baseline supervised network makes wrong segmentations. There is still pavement segmentation due to the association with people. Superpixel-mix produces better results without the wrong pavement segmentation, also provides a clean segmentation for the sand texture.}\n\\label{fig:bias_study}\n\\vspace{-5mm}\n\\end{figure*}\n\n\\Correction{The second aspect of epistemic uncertainty \nis related to network biased caused limited samples and diversity of scenarios, e.g., certain objects always co-occur in the training data. \nHere, we evaluate the epistemic uncertainty under the lens of model bias.\n}\nIn urban datasets such as Cityscapes, \n\\Correction{road and car pixels appear most of the time together,} raising the question of co-occurrence dependency between objects. If two objects are dependent, then the network will likely fail when the car object is encountered in different context other than roads. Shetty \\emph{et al}\\bmvaOneDot~\\cite{car_sidewalk_dependency_cvpr_2019} study object dependency and suggest using GANs to remove one object by in-painting and training the network for the new contexts. Their results show that a network that is less biased towards the co-occurrence dependency yields better accuracy for segmentation.\n\nTo measure the bias of our superpixel-mix method, we create a new dataset dubbed \n Out-of-Context Cityscapes (OC-Cityscapes), by replacing roads in the validation data of Cityscapes with various textures such as water, sand, grass, etc. Example images are shown in the Supplementary Material. Studies in~\\cite{geirhos2018imagenettrained} show that DCNNs are biased towards texture. By replacing different textures for roads, we test the trained networks on these new context images and evaluate the bias level for each network.\n\n\n\nIn Table~\\ref{table:bias_study}, we show the performances of the fully supervised network (baseline) and our network trained using the superpixel-mix method for the Cityscapes validation set and our generated dataset. The results are mIoU, ECE, and NLL scores averaged over 3 runs for classes that are not road. On our experimental dataset, \nthe baseline's performance drops by 21.97\\% while \nSuperpixel-mix drops by 19.83\\% for the mIoU metric. The results also show that \nSuperpixel-mix \nproduces a less biased model for \nco-occurring objects with the best \nscores on mIoU and NLL measures. %\nSuperpixel-mix changes the image context while preserving object shapes, effectively regularizing the model to address shortcut learning, i.e., overfitting on the co-occurrence of objects and their typical contexts. \nFor a visual example, see Figure~\\ref{fig:bias_study}.\n\n\n\\begin{table}[!t]\n\\renewcommand{\\figurename}{Table}\n\\begin{center}\n \\scalebox{0.50}\n {\n\\begin{tabular}{ l  c  c  c  c  c c  }\n\\toprule\nEvaluation data & Cityscapes  & OC-Cityscapes &Cityscapes  & OC-Cityscapes & Cityscapes  & OC-Cityscapes \\\\\n    &  mIoU &  mIoU &  NLL & NLL & ECE  & ECE \\\\\n\\midrule\nBaseline (MCP)~\\cite{hendrycks2016baseline} & 76.51\\% & 54.54\\% & -0.9456 & -0.7565 & 0.1303 & 0.2162 \\\\\n Cutmix ~\\cite{cutmix_SSL_seg_BMVC_2020} & 78.37\\% & 54.78\\%  & -0.955&\t-0.7435 & 0.1365 & 0.2587\\\\\n MIMO  \\cite{havasi2020training}  & 77.13\\% & 55.87\\% & -0.9516 & -0.7431 & 0.1398 & 0.2359\\\\\nDeep Ensembles  \\cite{lakshminarayanan2017simple} & 77.48\\% & 57.09\\%  & -0.9469 & -0.7613 & \\textbf{0.1274} & \\textbf{0.1968}\\\\\nSuperpixel-mix (ours) & \\textbf{78.99\\%} & \\textbf{59.16\\%} & \\textbf{-0.9563} & \\textbf{-0.7768} & 0.1348 & 0.2244\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\vspace{-1mm}\n\\caption{\\small \\textbf{Comparative results for network biases on OC-Cityscapes.} The results are segmentation mIoU, NLL and ECE, for classes that are not road. The baseline is the result from supervised training.}\n\\label{table:bias_study}\n\\vspace{-2mm}\n\\end{table}\n\n\n\\vspace{-3mm}\n\\subsection{Aleatoric uncertainty experiments}\\label{aleatoric}\n\n\n\\Correction{Aleatoric uncertainty is associated with unpredictability of the data acquisition process that causes various noises in the data.\nIn the following experiments we evaluate the aleatoric uncertainty of DCNNs trained on normal images (e.g., normal weather images) when facing test images with various types of noise (e.g., rainy or foggy environments).}\nIn semantic segmentation, the DCNN must be \\Correction{robust} to aleatoric uncertainty. To check that, we use the rainy \\cite{hu2019depth} and foggy Cityscapes \\cite{sakaridis2018semantic} datasets, which are built by adding rain of fog to the Cityscapes validation \\Correction{images}. The goal \nis to evaluate the performance of DNNs to resist these perturbations. \n\\Correction{We further generate an additional Cityscapes variant with images modified with different perturbations and intensities to mimic a distribution shift~\\cite{hendrycks2019benchmarking}.} \n\\Correction{We apply the following} perturbations: Gaussian noise, shot noise, impulse noise, defocus blur, frosted, glass blur, motion blur, zoom blur, snow, frost, fog, brightness, contrast, elastic, pixelate, JPEG. For more information, please refer to \\cite{hendrycks2019benchmarking}. We call this dataset Cityscapes-C.\n\n\n\nTo measure the \\Correction{robustness} under aleatoric uncertainty, we \ncompute ECE, mIoU and NLL scores averaged over 3 runs. Table \\ref{table:Aleatoric} shows results close to the state of the art. \nDE reaches good results, yet this approach needs to train several DCNNs, so it is more time-consuming for training and inference. In the Supplementary Material, we \nreport mIoU scores of different approaches for the different levels of noise. Overall, our \nexperiments indicate that Superpixel-mix tends to be \n\\Correction{robust} to high level of noise. \n\n\n\n\n\n\n\\begin{table}[t]\n\\begin{center}\n \\scalebox{0.50}\n {\n\\begin{tabular}{ l  c  c  c  c c c c c c c c c   }\n\\toprule\n\\multirow{2}{*}{Evaluation data} & \\multicolumn{3}{c}{Cityscapes} & \\multicolumn{3}{c}{Rainy Cityscapes}  & \\multicolumn{3}{c}{Foggy Cityscapes}  & \\multicolumn{3}{c}{Cityscapes-C}  \\\\\n    &  mIoU $\\uparrow$ &  ECE $\\downarrow$  &  NLL $\\downarrow$  &  mIoU $\\uparrow$  &  ECE $\\downarrow$ &  NLL $\\downarrow$  &  mIoU $\\uparrow$ &  ECE $\\downarrow$  &  NLL $\\downarrow$  & mIoU $\\uparrow$    & ECE $\\downarrow$ &  NLL $\\downarrow$   \\\\\n\\midrule\nBaseline (MCP)~\\cite{hendrycks2016baseline} &  76.51\\% & 0.1303 &-0.9456 & 58.98\\% & 0.1395  & -0.8123 & 69.89\\%   & 0.1493 &-0.9001  & 40.85\\%   & 0.2242 &-0.7389 \\\\\n Cutmix ~\\cite{cutmix_SSL_seg_BMVC_2020}  & 78.37\\% & 0.1365  & -0.9550& 61.86\\% & \t0.1559& -0.8200 &  73.57\\%  & 0.1484 &-0.9289 & 39.16\\% \t& 0.3064 &\t-0.6865\\\\\nMIMO  \\cite{havasi2020training}  &  77.13\\% & 0.1398  &-0.9516 & 59.27\\% & 0.1436 & -0.8135 &70.24\\% & 0.1425 &-0.9014& 40.73\\%    & 0.2350 &-0.7313\\\\\nBatchEnsemble \\cite{wen2020batchensemble} &  77.99\\% & 0.1129 &-0.9472 & 60.29\\% & 0.1436 &-0.7820& 72.19\\%  & 0.1425 &-0.9132&   40.93\\% &  0.2270 &-0.7082\\\\\nLP-BNN  \\cite{franchi2020encoding}  &  77.39\\% & \\textbf{0.1105}  &-0.9464 & 60.71\\% & 0.1338 &-0.7891& 72.39\\% & \\textbf{0.1358} &-0.9131& \\textbf{43.47}\\%    & 0.2085 &-0.7282\\\\\nDeep Ensembles  \\cite{lakshminarayanan2017simple} &  77.48\\% & 0.1274 &-0.9469 & 59.52\\%  &\\textbf{0.1078}   &-0.8205& 71.43\\% & 0.1407  &-0.9070 & 43.40\\%  & \\textbf{0.1912}&-0.7509 \\\\\nSuperpixel-mix  (ours)  &   \\textbf{78.99}\\% & 0.1348 & \\textbf{-0.9563}  & \\textbf{61.87}\\% & \t0.1583& \t\\textbf{-0.8207}& \\textbf{74.39}\\%  & 0.1411 &-\\textbf{0.9266}& 42.58\\%  & \t0.2338 &\t\\textbf{-0.7513}\\\\\n\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{\\small Aleatoric uncertainty study on Cityscapes-C,  Foggy Cityscapes~\\cite{hu2019depth}  and Rainy Cityscapes~\\cite{sakaridis2018semantic}.  }\n\\label{table:Aleatoric}\n\\vspace{-4mm}\n\\end{table}\n\n\n\\Correction{We note that our method does not achieve the top ECE scores across the various perturbations and weather conditions in these experiments. However, none of the considered strong baselines based on ensembles outperforms the others consistently either due to the difficulty and diversity of the considered test sets. Superpixel-mix achieves competitive ECE scores and is a top performer on mIoU and NLL.}\n\n\\vspace{-3mm}\n\\subsection{Semi-Supervised Learning experiments}\\label{semi-supervised}\nTo evaluate the \\Correction{robustness} of our method to missing annotation, we tested our approach on a semi-supervised learning task on two datasets: Cityscapes~\\cite{Cityscapes_CVPR_2016} and Pascal VOC 2012~\\cite{pascal-voc-2012}. \nWe follow the common practice for this task from prior works~\\cite{adversarial_bmvc_2018,cutmix_SSL_seg_BMVC_2020} and use DeepLab-V2~\\cite{chen2017deeplab} model with ResNet101~\\cite{he2016deep} encoder pre-trained on ImageNet~\\cite{ImageNet_cvpr_2009} and MS-COCO~\\cite{mscoco_eccv_2014}.\nThe weights of both the teacher and student models are initialized the same manner.\nWe evaluate our method and compare with existing methods using three sets of labeled data: 1\/30 (100 images), 1\/8 (372 images) and 1\/4 (744 images). Our results are reported as average mIoU of 12 runs (4 times on each of 3 official splits) %\nas well as the standard deviation. The results are shown in Table \\ref{table:Cityscapes}. We present results on Pascal dataset in the Supplementary Material.\n\n\n\n\\begin{table*}[!t]\n\\renewcommand{\\figurename}{Table}\n\\begin{center}\n\\scalebox{0.55}\n {\n\\begin{tabular}{l  l l l}\n\\toprule\n\\textbf{Labeled samples} & \\textbf{1\/30 (100)} & \\textbf{1\/8 (372)} & \\textbf{1\/4 (744)} \\\\\n\\midrule\nAdversarial~\\cite{adversarial_bmvc_2018} & - & 58.80\\% & 62.30\\% \\\\\ns4GAN~\\cite{s4GAN_pami_2019} & - & 59.30\\% &  61.90\\% \\\\\nCutout~\\cite{cutout_arXiv_2017} & 47.21\\% $\\pm$ 1.74 & 57.72\\% $\\pm$ 0.83 & 61.96\\% $\\pm$ 0.99 \\\\\nCutmix~\\cite{cutmix_SSL_seg_BMVC_2020} & 51.20\\% $\\pm$ 2.29 & 60.34\\% $\\pm$ 1.24 & 63.87\\% $\\pm$ 0.71 \\\\\nClassmix~\\cite{classmix_arXiv_2020} & 54.07\\% $\\pm$ 1.61 & 61.35\\% $\\pm$ 0.62 & 63.63\\% $\\pm$ 0.33 \\\\\nClassmix~\\cite{classmix_arXiv_2020} & 54.07\\% $\\pm$ 1.61 & 61.35\\% $\\pm$ 0.62 & 63.63\\% $\\pm$ 0.33 \\\\\n\\midrule\nBaseline(*) & 43.84\\% $\\pm$ 0.71 & 54.84\\% $\\pm$ 1.14 & 60.08\\% $\\pm$ 0.62\\\\\nSuperpixel-mix (ours) & \\textbf{54.11\\% $\\pm$ 2.88  ($\\uparrow$ 7.27\\%)} &\\textbf{63.44\\%$\\pm$ 0.88  ($\\uparrow$ 8.60\\%)} & \t\\textbf{65.82\\%$\\pm$ 1.78  ($\\uparrow$ 5.74\\%) }\\\\\n\n\n\n\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{\\small \n\\textbf{Evaluation in the semi-supervised learning regime on Cityscapes. We report mIoU scores as $mean \\pm std.dev$ computed over 12 runs.}\nThe $(\\uparrow)$ shows the improvement of our methods over the baselines. (*) The baselines are from~\\cite{classmix_arXiv_2020} as we use a similar base procedure.}\n\\label{table:Cityscapes}\n\\end{table*}\n\n\n\\vspace{-3mm}\n\\subsection{Ablations and analysis}\\label{Ablations_sp}\n\\Correction{We perform various ablations to understand the \ninfluence of different hyper-parameters and choices\nin our algorithm. First, we vary the number of superpixels extracted per image from 20 to 1,000. The results (in Table 5, Supplement) show that our method achieves the highest mIoU on Cityscapes when the number of superpixels per image is from 100 to 200. Secondly, we study the proportion of chosen superpixels for mixing over the total number of superpixels per image. The proportion ranges from 0.1 to 0.9. We find that the results vary little across all the proportion values (Table 6, Supplement). The best mIoU is obtained at the proportion value of 0.6. Finally, we study the influence of different superpixel techniques to generate mixing masks: Watershed~\\cite{watershed_superpixel_ICIP_2014}, SLIC~\\cite{slic}, and Felzenszwalb~\\cite{felzenswalb2004efficient}. The results in Table~\\ref{table:Abblation000} show that on Cityscapes segmentation the performance of Superpixel-mix is relatively stable across superpixel methods, with Watershed yielding the best mIoU score.}\n\n\\begin{table}[t]\n\\begin{center}\n \\scalebox{0.55}\n {\n\\begin{tabular}{l| c  c  c   }\n\\toprule\nsuperpixels algorithm &  Watershed ~\\cite{watershed_superpixel_ICIP_2014} &  SLIC~\\cite{slic}  & Felzenszwalb  \\cite{felzenswalb2004efficient} \\\\\n\n mIoU & 78.99 \\%& 78.89\\% & 77.99\\%  \\\\\n\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{\\small \\textbf{Ablation study results over influence of different superpixel techniques. We report mIoU scores for semantic segmentation on Cityscapes.}}\n\\label{table:Abblation000}\n\\vspace{-5mm}\n\\end{table}\n\n\\vspace{-3mm}\n\\section{Conclusions}\n\nSuperpixel-mix data augmentation is a promising new training technique for semantic segmentation. This strategy for creating diverse data, combined with a teacher-student framework, leads to better accuracy and to more \\Correction{robust} DCNNs.\n We are the first, to the best of our knowledge, to successfully apply the watershed algorithm in data augmentation. What sets our data augmentation technique apart from existing methods is the ability to preserve the global structure of images and the shapes of objects while creating image perturbations. We conduct various experiments with different types of uncertainty. The results show that our strategy achieves state-of-the-art \\Correction{robustness} scores for epistemic uncertainty. For aleatoric uncertainty, our approach produces state-of-the-art results in Foggy Cityscapes and Rainy Cityscapes. In addition, our method needs just one forward pass.\n\nPrevious research in machine learning has established that creating more training data using data augmentation may improve the accuracy of the trained DCNNs substantially. Our work not only confirms that, but also provides evidence that some data augmentation methods, such as our Superpixel-mix, help to improve the \\Correction{robustness} of DCNNs by reducing both epistemic and aleatoric uncertainty.\n\n\\section*{Acknowledgments}\n\\Correction{This work was performed using HPC resources\nfrom GENCI-IDRIS (Grant 2020-AD011011970) and (Grant 2021-AD011011970R1).}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{\\@title}}\n \\renewcommand\\chapter{\\if@openright\\cleardoublepage\\else\\clearpage\\fi\n                    \\thispagestyle{empty}%\n                    \\global\\@topnum\\z@\n                    \\@afterindentfalse\n                    \\secdef\\@chapter\\@schapter}\n \\def\\@makechapterhead#1{%\n  \\vspace*{50\\p@}%\n  {\\parindent \\z@ \\raggedleft \\normalfont\n    \\ifnum \\c@secnumdepth >\\m@ne\n      \\if@mainmatter\n  \n        \\par\\nobreak\n        \\vskip 20\\p@\n      \\fi\n    \\fi\n    \\interlinepenalty\\@M\n    \\Huge \\bfseries #1\\par\\nobreak\n    \\vskip.25in\n    \\large\\bfseries\\@author\\par\\nobreak\n    \\vskip 40\\p@}\n    \\ifx\\@abstract\\@empty\\else{\\small\\@abstract\\par\\vskip20\\p@}\\fi\n  }\n\n\n\n \\let\\tdf\\textbf\n \\let\\df\\bf\n\n\n\\DeclareRobustCommand\\em\n        {\\@nomath\\em \\ifdim \\fontdimen\\@ne\\font >\\z@\n                       \\upshape \\else \\slshape \\fi}\n\\let\\tem\\emph\n\n\n\\def\\@begintheorem#1#2{\\sl \\trivlist \\item[\\hskip \\labelsep{\\bf #1\\ #2}]}\n\\def\\@opargbegintheorem#1#2#3{\\sl \\trivlist\n     \\item[\\hskip \\labelsep{\\bf #1\\ #2\\ (#3)}]}\n\n\n \\newcommand{\\sect}[1]{\\S\\ref{sect:#1}}     \n                                       \n \\newcommand{\\baresect}[1]{\\ref{sect:#1}}     \n                                \n \\newcommand{\\eq}[1]{(\\ref{eq:#1})}\n                                       \n \\newcommand{\\foot}[1]{footnote \\ref{foot:#1}}\n                                       \n \\newcommand{\\fig}[1]{Fig.~\\ref{fig:#1}}\n\n \\newcommand{\\sectlabel}[1]{\\label{sect:#1}}\n \\newcommand{\\eqlabel}[1]{\\label{eq:#1}}\n \\newcommand{\\figlabel}[1]{\\label{fig:#1}}\n\n\n \\renewcommand{\\theequation} {\\mbox{\\arabic{equation}}}\n \\renewcommand{\\thesection} {\\mbox{\\arabic{section}}}\n \\renewcommand{\\thefigure} {\\mbox{\\arabic{figure}}}\n \\renewcommand{\\thetable} {\\mbox{\\arabic{table}}}\n\n\n\n \\let\\smallsection\\subsubsection \\setcounter{secnumdepth}{2}\n \\newcommand{\\parhead}[1]{\\tsl{#1}.\\quad}\n       \n\n\n  \\def\\@arabic#1{\\number #1}\n\n\n\n\\long\\def\\@makecaption#1#2{\n    \\vskip\\abovecaptionskip\n    \\sbox\\@tempboxa{{\\small {\\bf #1}: #2}}%\n    \\ifdim\\wd\\@tempboxa>\\hsize\n        {\\small {\\bf #1}: #2\\par}\n    \\else\n       \\global\\@minipagefalse\n       \\hbox to\\hsize{\\hfil\\box\\@tempboxa\\hfil}\n    \\fi\n    \\vskip \\belowcaptionskip}\n\n\\def\\figstrut#1{\\hbox to\\linewidth{\\vrule height#1\\hfill}}\n\n\\newcommand{\\Fig}[4][!htb]\n\\begin{figure}[#1]\n \\centering\\leavevmode#3%\n \\caption{#4}\n \\figlabel{#2}\n\\end{figure}                 }\n \\let\\figonecol\\Fig\n\n \\newcommand{\\Figwide}[4][!t]\n \\begin{figure*}[#1]\n  \\centering\\leavevmode#3%\n  \\caption{#4}\n \\figlabel{#2}\n \\end{figure*}                 }\n \\let\\figtwocol\\Figwide\n\n \\newcommand{\\Figpage}[3]\n \\begin{figure*}[p]\n  \\centering\\leavevmode#2%\n  \\caption{#3}\n \\figlabel{#1}\n \\end{figure*}                 }\n \\let\\figpage\\Figpage\n\n\\renewenvironment{thebibliography}[1]\n     {\\section*{\\bibname\n        \\@mkboth{\\MakeUppercase\\bibname}{\\MakeUppercase\\bibname}}%\n      \\list{\\@biblabel{\\@arabic\\c@enumiv}}%\n           {\\settowidth\\labelwidth{\\@biblabel{#1}}%\n            \\leftmargin\\labelwidth\n            \\advance\\leftmargin\\labelsep\n            \\@openbib@code\n            \\usecounter{enumiv}%\n            \\let\\p@enumiv\\@empty\n            \\renewcommand\\theenumiv{\\@arabic\\c@enumiv}}%\n      \\sloppy\n      \\clubpenalty4000\n      \\@clubpenalty \\clubpenalty\n      \\widowpenalty4000%\n      \\sfcode`\\.\\@m}\n     {\\def\\@noitemerr\n       {\\@latex@warning{Empty `thebibliography' environment}}%\n      \\endlist}\n\\makeatother\n\n\\bibliographystyle{ICCS}\n\n\\endinput\n\n \\def\\thispagestyle{empty}\\chapter{\\@title}{%\n  \\thispagestyle{empty}\n    \\raggedleft\n    \\noindent\\@author\\par\n    \\medskip\n    \\noindent\\hrule\n    \\smallskip\n    \\noindent{\\Huge\\sf\\@title\\par}\n    \\vfil\\vfil\\vfil\\vfil\n    {\\narrower\\narrower\\noindent\\@abstract\\par}\\newpage}\n\n\\section{Introduction}\n\nThroughout history we have used concepts from our current technology as metaphors to describe our world. Examples of this are the description of the body as a factory during the Industrial Age, and the description of the brain as a computer during the Information Age. These metaphors are useful because they extend the knowledge acquired by the scientific and technological developments to other areas, illuminating them from a novel perspective. For example, it is common to extend the particle metaphor used in physics to other domains, such as crowd dynamics \\cite{HelbingVicsek1999}. Even when people are not particles and have very complicated behaviour, for the purposes of crowd dynamics they can be effectively described as particles, with the benefit that there is an established mathematical framework suitable for this description. Another example can be seen with cybernetics \\cite{Ashby1956,HeylighenJoslyn2001}, where the system metaphor is used: everything is seen as a system with inputs, outputs, and a control\nthat regulates the internal variables of the system under the influence of perturbations from its environment. Yet another example can be seen with the computational metaphor \\cite{Wolfram2002}, where the universe can be modelled with simple discrete computational machines, such as cellular automata or Turing machines.\n\nHaving in mind that we are using metaphors, this paper proposes to extend the concept of information to describe the world: from elementary particles to galaxies, with everything in between, particularly life and cognition. There is no suggestion on the nature of reality as information \\cite{Wheeler1990}. This work only explores the advantages of \\emph{describing} the world as information. In other words, there are no ontological claims, only epistemological.\n\nIn the next section, the motivation of the paper is presented, followed by a section describing the notion of information to be used throughout the paper. In Section \\ref{s:laws}, eight tentative laws of information are put forward. These are applied to the notions of life (Section \\ref{s:life}) and cognition (Section \\ref{s:cog}). The paper closes presenting future work and conclusions.\n\n\n\\section{Why Information?}\n\nThere is a great interest in the relationship between energy, matter, and information \\cite{Kauffman2000,Umpleby2004,MorowitzSmith2006}. One of the main reasons for this arises because this relationship plays a central role in the definition of life: Hopfield \\cite{Hopfield1994} suggests that the difference between biological and physical systems is given by the meaningful information content of the former ones. Not that information is not present in physical systems, but---as Roederer puts it---information is \\emph{passive} in physics and \\emph{active} in biology \\cite{Roederer2005}. However, it becomes complicated to describe how this information came to be in terms of the physical laws of matter and energy. In this paper the inverse approach is proposed: let us describe matter and energy in terms of information. If atoms, molecules and cells are described as information, there is no need of a \\emph{qualitative} shift (from non-living to living matter) while describing the origin and evolution of life: this is translated into a \\emph{quantitative} shift (from less complex to more complex information). \n\nThere is a similar problem when we study the origin and evolution of cognition \\cite{Gershenson2004}: it is not easy to describe cognitive systems in terms of matter and energy. The drawback with the physics-based approach to the studies of life and cognition is that it requires a new category, that in the best situations can be referred to as ``emergent\". Emergence is a useful concept, but it this case it is not explanatory. Moreover, it stealthily introduces a dualist view of the world: if we cannot relate properly matter and energy with life and cognition, we are forced to see these as separate categories. Once this breach is made, there is no clear way of studying or understanding how systems with life and cognition evolved from those without it. If we see matter and energy as particular, simple cases of information, the dualist trap is avoided by following a continuum in the evolution of the universe. Physical laws are suitable for describing phenomena at the physical scale. The tentative laws of information presented below aim at being suitable for describing phenomena \\emph{at any scale}. Certainly, there are other approaches to describe phenomena at multiple scales, such as general systems theory and dynamical systems theory. These approaches are not exclusive, since one can use several of them, including information, to describe different aspects of the same phenomena.\n\nAnother benefit of using information as a basic descriptor for our world is that the concept is well studied and formal methods have already been developed \\cite{CoverThomas2006,ProkopenkoEtAl2007}, as well as its philosophical implications have been discussed \\cite{Floridi2003}. Thus, there is no need to develop a new formalism, since information theory is well established. I borrow this formalism and interpret it in a new way.\n\nFinally, information can be used to describe other formalisms: not only particles and waves, but also systems, networks, agents, automata, and computers can be seen as information. In other words, it can contain other descriptions of the world, potentially exploiting their own formalisms. Information is an \\emph{inclusive} formalism.\n\n\\section{What Is Information?}\n\nExtending the notion of Umwelt \\cite{vonUexkull1957}, the following notion of information can be given: \n\n\\begin{notion}\n\\label{notion:Info}\nInformation is anything that an agent can sense, perceive, or observe.\n\\end{notion}\n\nThis notion is in accordance with Shannon's \\cite{Shannon1948}, where information is seen as a just-so arrangement, a defined structure, as opposed to randomness \\cite{Cohen2000,Cohen2006}, and it can be measured in bits.\nThis notion can be applied to everything that surrounds us, including matter and energy, since we can perceive it---because it has a defined structure---and we are agents, according to the following notion:\n\n\\begin{notion}\n\\label{notion:Agent}\nAn agent is a description of an entity that \\emph{acts} on its environment \\cite[p. 39]{GershensonDCSOS}.\n\\end{notion}\n\nNoticing that agents (and their environments) are also information (as they can be perceived by other agents, especially us, who are the ones who \\emph{describe} them as agents), an agent can be a human, a cell, a molecule, a computer program, a society, an electron, a city, a market, an institution, an atom, or a star. Each of these can be described (by us) as \\emph{acting} in their environment, simply because they \\emph{interact} with it.\nHowever, not all information is an agent, e.g. temperature, color, velocity, hunger, profit.\n\n\\begin{notion}\n\\label{notion:Environment}\nThe environment of an agent consists of all the information \\emph{interacting} with it.\n\\end{notion}\n\n\nInformation will be relative to the agent perceiving it\\footnote{Shannon's information \\cite{Shannon1948} deals only with the technical aspect of the transmission of information and not with its \\emph{meaning}, i.e. it neglects the semantic aspect of communication.}. Information can exist in theory ``out there\", independently of an agent, but for practical purposes, it can be only spoken about once an agent---not necessarily a human---perceives \/ interacts with it. The \\emph{meaning} of the information will be given by the \\emph{use} the agent perceiving it makes of it \\cite{Wittgenstein1999}, i.e. how the agent responds to it \\cite{AtlanCohen1998}. Thus, Notion \\ref{notion:Info} is a \\emph{pragmatic} one. Note that perceived information is different from the meaning that an agent gives to it. Meaning is an \\emph{active} product of the \\emph{interaction} between information and the agent perceiving it \\cite{Cohen2006,Neuman:2008}.\n\nLike this, an electron can be seen as an agent, which perceives other electrons as information. The same description can be used for molecules, cells, and animals. We can distinguish:\n\n\\begin{description}\n\\item[First order information] is that which is perceived directly by an agent. For example, the information received by a molecule about another molecule\n\\item[Second order information] is that which is perceived by an agent about information perceived by another agent. For example, the information perceived by a human observer about a molecule receiving information about another molecule.\n\\end{description}\n\nMost of the scientific descriptions about the world are second order information, as we perceive how agents perceive and produce information. The present approach also introduces naturally the role of the observer in science, since everything is ``observing\" the (limited, first order) information it interacts with from its own perspective. Humans would be second-level observers, observing the information observed by information. Everything we can speak about is observed, and all agents are observers.\n\nInformation is not necessarily conserved, i.e. it can be created, destroyed, or transformed. These can take place only through interaction. \\emph{Computation} can be seen as the \\emph{change} in information, be it creation, destruction, or transformation.\nMatter and energy can be seen as particular types of information that cannot be created or destroyed, only transformed, along with the well-known properties that characterize them.\n\nThe amount of information required to describe a process, system, object, or agent determines its \\emph{complexity} \\cite{ProkopenkoEtAl2007}. According to our current knowledge, during the evolution of our universe there has been a shift from simple information towards more complex information \\cite{Adami2002} (the information of an atom is less complex than that of a molecule, than that of a cell, than that of a multicellular organism, etc.). This ``arrow of complexity\"\\cite{Bedau1998} in evolution can guide us to explore general laws of information.\n\n\\section{Tentative Laws of Information}\n\\label{s:laws}\n\nSeeing the world as information allows us to describe general laws that can be applied to everything we can perceive. Extending Darwin's theory \\cite{Darwin1998}, the present framework can be used to reframe ``universal Darwinism\" \\cite{Dennet1995}, which explores the idea of evolution beyond biological systems.\nIn this work, the laws that describe the general behaviour of information as it evolves are introduced. These laws are only \\emph{tentative}, in the sense that they are only presented with arguments in favour of them, but they still need to be thoroughly tested.\n\n\\subsection{Law of Information Transformation}\n\nSince information is relative to the agents perceiving it, \\emph{information will potentially be \\emph{transformed} as different agents perceive it}. Another way of stating this law is the following: \\emph{information will potentially be transformed by \\emph{interacting} with other information}.\nThis law is a generalization of the Darwinian principle of random variation, and ensures \\emph{novelty} of information in the world. Even when there might be static information, different agents can perceive it differently and interact with it, potentially transforming it.\nThrough evolution, the transformation of information generates a \\emph{variety} or \\emph{diversity} that can be used by agents for novel purposes.\n\nSince information is not a conserved quantity, it can increase (created), decrease (destroyed), or be maintained as it is transformed.\n\nAs an example, RNA polymerase (RNAP) can make errors while copying DNA onto RNA strands. This slight random variation can lead to changes in the proteins for which the RNA strands serve as templates. \\emph{Some} of these changes will lead to novel proteins that might improve or worsen the function of the original proteins.\n\nThe transformation of information can be classified as follows:\n\n\\begin{description}\n\\item[Dynamic.] Information changes itself. This could be considered as ``objective, internal\" change.\n\\item[Static.] The agent perceiving the information changes, but the information itself does not change. There is a dynamic change but in the agent. This could be considered as ``subjective, internal\" change.\n\\item [Active.] An agent changes information in its environment. This could be considered as an ``objective, external\" change.\n\\item[Stigmergic.] An agent makes an active change of information, which changes the perception of that information by another agent. This could be considered as ``subjective, external\" or ``intersubjective\" change.\n\\end{description}\n\n\n\\subsection{Law of Information Propagation}\n\n\\emph{Information \\emph{propagates} as fast as possible}. Certainly, only some  information manages to propagate. In other words, we can assume that different information has a different ``ability\" to propagate, also depending on its environment. The ``fitter\" information, i.e. that which manages to persist and propagate faster and more effectively, will prevail over other information. This law generalizes the Darwinian principle of natural selection, the maximum entropy production principle \\cite{MartyushevSeleznev2006} (entropy can also be described as information), and Kauffman's tentative fourth law of thermodynamics\\footnote{``The workspace of the biosphere expands, on average, as fast as it can in this coconstructing biosphere\" \\cite[p. 209]{Kauffman2000}}. It is interesting that this law contains the second law of thermodynamics, as atoms interact, propagating information homogeneously. It also describes living organisms, where genetic information is propagated across generations. And it also describes cultural evolution, where information is propagated among individuals. Life is ``far from thermodynamic equilibrium\" because it constrains \\cite{Kauffman2000} the (more simple) information propagation at the thermodynamic scale, i.e. the increase of entropy, exploiting structures to propagate (or maintain) the (more complex) information at the biological scale.\n\nIn relation with the law of information transformation, as information requires agents to perceive it, information will be potentially transformed. This source of novelty will allow for the ``blind\" exploration of better ways of propagating information, according to the agents perceiving it and their environments.\n\nExtending the previous example, if errors in transcription made by RNAP are beneficial for its propagation (which entails the propagation of the cell producing RNAP), cells with such novel proteins will have better chances of survival than their ``cousins\" without transcription errors.\n\nThe propagation of information can be classified as follows:\n\n\\begin{description}\n\\item[Autonomous.] Information propagates by itself. Strictly speaking, this is not possible, since at least some information is determined by the environment. However, if more information is produced by itself than by its environment, we can call this autonomous propagation (See Section \\ref{s:life}). \n\\item[Symbiotic. ] Different information cooperates, helping to propagate each other.\n\\item[Parasitic. ] Information exploits other information for its own propagation. \n\\item[Altruistic. ] Information promotes the propagation of other information at the cost of its own propagation.\n\\end{description}\n\n\n\\subsection{Law of Requisite Complexity}\n\nTaking into account the law of information transformation, transformed information can increase, decrease, or maintain its previous complexity, i.e. amount \\cite{ProkopenkoEtAl2007}. However, \\emph{more complex information will require more complex agents to perceive, act on, and propagate it}. This law generalizes the cybernetic law of requisite variety \\cite{Ashby1956}. Note that simple agents can perceive and interact with \\emph{part} of complex information, but they cannot (by themselves) propagate it. An agent cannot perceive (and thus contain) information more complex than itself. For simple agents, information that is complex for us will be simple as well. As stated above, different agents can perceive the same information in different ways, giving it different meanings.\n\nThe so called ``arrow of complexity\" in evolution \\cite{Bedau1998} can be explained with this law. If we start with simple information, its transformation will produce by simple drift \\cite{McShea1996,Miconi:2008} increases in the complexity of information, without any goal or purpose. This occurs simply because there is an open niche for information to become more complex as it varies. But this also promotes agents to become more complex to exploit novel (complex) information and propagate it. Evolution does not need to favour complexity in any way: information just propagates to every possible niche as fast as possible, and it seems that there is often an ``adjacent possible\" \\cite{Kauffman2000} niche of greater complexity.\n\n\nFor example, it can be said that a protein (as an agent) perceives some information via its binding sites, as it recognizes molecules that ``fit\" a site. More complex molecules will certainly need more complex binding sites. Whether complex molecules are better or worse is a different matter: some will be better, some will be worse. But for those which are better, the complexity of the proteins must match the complexity of the molecules perceived. If the binding site perceives only a part of the molecule, then this might be confused with other molecules which share the perceived part. Following the law of information transformation, there will be a variety of complexities of information. The law of requisite complexity just states that the increase in complexity of information is determined by the ability of agents to perceive, act on, and propagate more complex information.\n\nSince more complex information will be able to produce more variety, the \\emph{speed} of the complexity increase will escalate together with the complexity of the information.\n\n\n\\subsection{Law of Information Criticality}\n\n\\emph{Transforming and propagating information will tend to a critical \\emph{balance} between its stability and its variability}. Propagating information  maintains itself as much as possible, but transforming information varies it as much as possible. This struggle leads to a critical balance analogous to the ``edge of chaos\" \\cite{Langton1990,Kauffman1993}, self-organized criticality \\cite{BTW1987,Adami1995}, and the ``complexity from noise\" principle \\cite{Atlan1974}. The homeostasis of living systems can also be seen as the self-regulation of information criticality. \n\nThis law can generalize Kauffman's four candidate laws for the coconstruction of a biosphere \\cite[Ch. 8]{Kauffman2000}. Their relationship with this framework demands further discussion, which is out of the scope of this paper.\n\nA well known example can be seen with cellular automata \\cite{Langton1990} and random Boolean networks \\cite{Kauffman1993,Gershenson2004c,Gershenson:2010}: stable (ordered) dynamics limit considerably or do not allow change of states so information cannot propagate, while variable (chaotic) dynamics change the states too much, losing information.\nFollowing the law of information propagation, information will tend to a critical state between stability and variability to maximize its propagation: if it is too stable, it will not propagate, and if it is too variable, it will be transformed. In other words, ``critical\" information will be able to propagate better than stable or variable one, i.e. as fast as possible (cf. law of information propagation).\n\n\n\\subsection{Law of Information Organization}\n\n\\emph{Information produces constraints that regulate information production}. These constraints can be seen as \\emph{organization} \\cite{Kauffman2000}. In other words, evolving information will be organized (by transformation and propagation) to regulate information production. According to the law of information criticality, this organization will lie at a critical area between stability and variability. And following the law of information propagation, the organization of information will enable it to propagate as fast as possible.\n\nThis law can also be seen as information having a certain \\emph{control} over its environment, since the organization of information will help it withstand perturbations. It has been shown \\cite{KlyubinEtAl2004,ProkopenkoEtAl2006,KlyubinEtAl2007} that using this idea as a fitness function can lead to the evolution of robust and adaptive agents, namely maximizing the mutual information between sensors and environment.\n\nA clear example of information producing its own organization can be seen with living systems, which are discussed in Section \\ref{s:life}.\n\n\n\n\n\\subsection{Law of Information Self-organization}\n\n\\emph{Information tends to its preferred, most probable state}. This is actually a tautology, since observers determine probabilities after observing tendencies of information dynamics. Still, this tautology can be useful to describe and understand phenomena. This law lies at the heart of probability theory and dynamical systems theory \\cite{Ashby1962}. The dynamics of a system tend to a subset of its state space, i.e. attractors, depending on its history. This simple fact reduces the possibility space of information, i.e. a system will tend towards a small subset of all possible states. If we describe attractors as ``organized\", then we can describe the dynamics of information in terms of self-organization \\cite{GershensonHeylighen2003a}.\n\nPattern formation can be described as information self-organizing, and related to the law of information propagation. Information will self-organize in ``fit\" patterns that are the most probable (defined \\emph{a posteriori}).\n\nUnderstanding different ways in which self-organization is achieved by transforming information can help us understand better natural phenomena \\cite{Gershenson:2010a} and design artificial systems \\cite{GershensonDCSOS}.  For example, random Boolean networks can be said to self-organize towards their attractors \\cite{Gershenson:2010}.\n\n\\subsection{Law of Information Potentiality}\n\n\\emph{An agent can give different potential meanings to information}. This implies that the same information can have different meanings. Moreover, meaning---while being information---can be independent of the information carrying it, i.e. depend only on the agent observing it. Thus, different information can have the same potential meaning. The precise meaning of information will be given by an agent observing it within a specific context.\n\nThe potentiality of information allows the effective communication between agents. Different information has to be able to acquire the same meaning (homonymy), while the same information has to be able to acquire different meanings (polysemy) \\cite{Neuman:2008}. The relationship between the laws of information and communication is clear, but beyond the scope of this paper.\n\nThe law of information potentiality is related to a passive information transformation, i.e. a change in the agent observing information.\n\nIn spite of information potentiality, not all meanings will be suitable for all information. In other words, pure subjectivism cannot dictate meanings of information. By the law of information propagation, some meanings will be more suitable than others and will propagate. The suitability of meanings will be determined by their use and context \\cite{Wittgenstein1999}. However, there is always a certain freedom to subjectively transform information. \n\nFor example, a photon can be observed as a particle, as a wave, or as a particle-wave. The suitability of each given meaning is determined by the context in which the photon is described\/observed.\n\n\\subsection{Law of Information Perception}\n\n\\emph{The meaning of information is \\emph{unique} for an agent perceiving it in unique, always changing open contexts}. If meaning of information is determined by the use an agent makes of it, which is embedded in an open environment, we can go to such a level of detail that the meaning will be unique. Certainly, agents make generalizations and abstractions of perceptions in order to be able to respond to novel information. Still, the precise situation and context will never be repeated. This makes perceived information unique. The implication of this is that the response to any given information might be ``unexpected\", i.e. novelty can arise. Moreover, the meaning of information can be to a certain extent \\emph{arbitrary}. This is related with the law of information transformation, as the uniqueness of meaning allows the same information perceived differently by the same or different agents to be statically transformed. \n\nThis law is a generalization of the first law of human perception: ``whatever is perceived can be perceived only from a uniquely situated place in the overall structure of points of view\" \\cite[p. xxiv]{Holquist:1990} (cited in \\cite[p. 250]{Neuman:2008}). We can describe agents perceiving information as filtering it. An advantage of humans and other agents is that we can \\emph{choose} which filter to use to perceive. The suggestion is not that ``unpleasant\" information should be solipsistically ignored, but that information can be potentially actively transformed.\n\nFor example, T lymphocytes in an immune system can perceive foreign agents and attack them. Even when the response will be similar for similar foreign agents, each perception will be unique, a situation that always leaves space for novelty.\n\n\\subsubsection{Scales of perception}\n\nDifferent information is perceived at different scales of observation \\cite{BarYam2004}. As the scale tends to zero, then the information tends to infinite. For lower scales, more information and details are perceived. The uniqueness of information perception dominates at these very low (spatial and temporal) scales. However, as generalizations are made, information is ``compressed\", i.e. only relevant aspects of information are perceived\\footnote{The relevance is determined by the context, i.e. different aspects will be relevant for different contexts.}. At higher scales, more abstractions and generalizations are made, i.e. less information is perceived. When the scale tends to infinite, the information tends to zero. In other words, no information is needed to describe all of the universe, because all the information is already there. This most abstract understanding of the world is in line with the ``highest view\" of Vajrayana Buddhism \\cite{Nydahl:2008}. Implications at this level of description cannot be right or wrong, because there is no context. Everything is contained, but no information is needed to describe it, since it is already there. This ``maximum\" understanding is also described as vacuity, which leads to bliss \\cite[p. 42]{Nydahl:2008}.\n\n\nFollowing the law of information criticality, agents will tend to a balance where the perceived information is minimal but maximally predictive \\cite{Shalizi2001} (at a particular scale): few information is cheaper, but more information in general entails a more precise predictability. The law of requisite complexity applies at particular scales, since a change of scale will imply a change of complexity of information \\cite{BarYam2004}.\n\n\n\\section{On the Notion of Life}\n\\label{s:life}\n\nThere is no agreed notion of life, which reflects the difficulty of defining the concept. Still, many researchers have put forward properties that characterize important aspects of life. \\emph{Autopoiesis} is perhaps the most salient one, which notes that living systems are self-producing \\cite{VarelaEtAl1974,McMullin2004}. Still, it has been argued that autopoiesis is a necessary but not sufficient property for life \\cite{RuizMoreno2004}. The relevance of autonomy \\cite{Barandarian2004,MorenoRuiz2006,KrakauerZanotto2007} and individuality \\cite{Michod2000,KrakauerZanotto2007} for life have also been highlighted . \n\nThese approaches are not unproblematic, since no living system is completely autonomous. This follows from the fact that all living systems are open. For example, we have some degree of autonomy, but we are still dependent on food, water, oxygen, sunlight, bacteria living in our gut, etc. This does not mean that we should abandon the notion of autonomy in life. However, we need to abandon the sharp distinction between life and non-life \\cite{Bedau1998,KrakauerZanotto2007}, as different degrees of autonomy escalate \\emph{gradually}, from the systems we considered as non-living to the ones we consider as living. In other words, life has to be a fuzzy concept.\n\nUnder the present framework, living and non-living systems are information. Rather than a yes\/no definition, we can speak about a ``\\emph{life ratio}\":\n\n\\begin{notion}\n\\label{notion:Life}\nThe ratio of \\emph{living information} is the information produced by itself over the information produced by its environment.\n\\end{notion}\n\nBeing more specific---since all systems also receive information---a system with a high life ratio produces more (first order) information about itself than the one it receives from its environment. Following the law of information organization, this also implies that living information produces more of its own constraints (organization) to regulate itself than the ones produced by its environment, and thus it has a greater autonomy. All information will have constraints from other (environmental) information, but we can measure (as second-order information) the proportion of internal over external constraints to obtain the life ratio. If this is greater than one, then the information regulates by itself more than the proportion that is regulated by external information. In the opposite case, the life ratio would be less than one.\n\nFollowing the law of information propagation, evolution will tend to information with higher life ratios, simply because this can propagate better, as it has more ``control\" and autonomy over its environment. When information depends more on its environment for its propagation, it has a higher probability of being transformed as it interacts with its environment.\n\nNote that the life ratio depends on spatial and temporal scales at which information is perceived. For example, for some microorganisms observed at a scale of years , the life ratio would be less than one, but if observed at a scale of seconds, the life ration would be greater than one.\n\nCertainly, some artificial systems would be considered as living under this notion. However, we can make a distinction between living systems embodied in or composed by biological cells \\cite{DeDuve2003}, i.e. life as we know it, and the rest, i.e. life as it could be. The latter ones are precisely those explored by artificial life.\n \n\\section{On the Notion of Cognition}\n\\label{s:cog}\n\nCognition is certainly related with life \\cite{Stewart1995}. The term has taken different meanings in different contexts, but all of them can be generalized into a common notion \\cite{Gershenson2004}. Cognition comes from the Latin \\emph{cognoscere}, which means ``get to know\". Like this,\n\n\\begin{notion}\n\\label{notion:Cognition}\nA system is cognitive if it \\emph{knows} something \\cite[p.135]{Gershenson2004}.\n\\end{notion}\n\nFrom Notion \\ref{notion:Agent}, all agents are cognitive, since they ``\\emph{know}\" how to act on their environment, giving (first order) \\emph{meaning} to their environmental information. Thus, there is no boundary between non-cognitive and cognitive systems. Throughout evolution, however, there has been a \\emph{gradual} increase in the complexity of cognition \\cite{Gershenson2004}. This is because all agents can be described as possessing some form of cognition, i.e. ``knowledge\" about the (first-order) information they perceive\\footnote{One could argue that, since agency (and thus cognition) is already assumed in all agents, this approach is not explanatory. But I am not trying to explain the ``origins\" of agency, since I assume it to be there from the start. I believe that we can only study the evolution and complexification of agency and cognition, not their ``origins\".}.\n\nFollowing the law of requisite complexity, evolution leads to more complex agents, to be able to cope with the complexity of their environment. This is precisely what triggers the (second-order) increase in the complexity of cognition we observe.\n\n\n\nCertainly, there are different types of cognition\\footnote{For example, human, animal, plant, bacterial, immune, biological, adaptive, systemic, and artificial \\cite{Gershenson2004}.}. We can say that a rock ``knows\" about gravity because it perceives its information, which has an effect on it, but it cannot \\emph{react} to this information. Throughout evolution, information capable of maintaining its integrity has prevailed over that which was not. \\emph{Robust} information is that which can resist perturbations to maintain its integrity. The ability to react to face perturbations to maintain information makes information \\emph{adaptive}, increasing its probability of maintenance. When this reaction is made before it occurs, the information is \\emph{anticipative}\\footnote{For a more detailed treatment on robustness, adaptation, and anticipation, see \\cite{GershensonDCSOS}}. As information becomes more complex (even if only by information transformation), the mechanisms for maintaining this information also become more complex, as stated by the law of requisite complexity. This has led gradually to the advanced cognition that animals and machines posses.\n\n\n\n\n\n\\section{Future Work}\n\nThe ideas presented here still need to be explored and elaborated further. One way of doing this would be with a simulation-based method. Being inspired by $\\epsilon$-machines \\cite{Shalizi2001,GoernerupCrutchfield2008}, one could start with ``simple\" agents that are able to perceive and produce information, but cannot control their own production. These would be let to evolve, measuring if complexity increases as they evolve. The hypothesis is that complexity would increase (under which conditions still remains to be seen), to a point where ``$\\epsilon$-agents\" will be able to produce themselves depending more on their own information than that of the environment. This would be similar to the evolution in Tierra \\cite{Ray1991} or Avida \\cite{AdamiBrown1994} systems, only that self-replication would not be inbuilt. The tentative laws of information presented in Section \\ref{s:laws} would be better defined if such a system was studied.\n\nOne  important aspect that remains to be studied is the representation of thermodynamics in terms of information. This is because the ability to perform thermodynamic work is a characteristic property of biological systems \\cite{Kauffman2000}. This work can be used to generate the organization necessary to sustain life (cf. law of information organization). It is difficult to describe life in terms of thermodynamics, since it entails new characteristic properties not present in thermodynamic systems. But if we see the latter ones as information, it will be easier to describe how life---also described as information---evolves from them, as information propagates itself at different scales.\n\nA potential application of this framework would be in economy, considering capital, goods, \nand resources as information (a non-conserved quantity) \\cite{FarmerZamani2006}. A similar benefit (of non-conservation) could be given in game theory: if the payoff of games is given in terms of information (not necessarily conserved), non-zero sum games could be easier to grasp than if the payoff is given in material (conserved) goods.\n\nIt becomes clear that information (object), the agent perceiving it (subject) and the meaning-making or transformation of information (action) are deeply interrelated. They are part of the same totality, since one cannot exist without the others. This is also in line with Buddhist philosophy. The implications of an informational description of the world for philosophy have also to be addressed, since some schools have focussed on partial aspects of the object-subject-action trichotomy.  Another potential application of the laws of information would be in ethics, where value can be described accordingly to the present framework.\n\n\n\\section{Conclusions}\n\nThis paper introduced general ideas that require further development, extension and grounding in particular disciplines. Still, a first step is always necessary, and hopefully feedback from the community will guide the following steps of this line of research.\n\nDifferent metaphors for describing the world can be seen as different languages: they can refer to the same objects without changing them. And each can be more suitable for a particular context. For example, English has several advantages for fast learning, German for philosophy, Spanish for narrative, and Russian for poetry. In other words, there is no ``best\" language outside a particular context. In a similar way, I am not suggesting that describing the world as information is more suitable than physics to describe physical phenomena, or better than chemistry to describe chemical phenomena. It would be redundant to describe particles as information if we are studying only particles. The suggested approach is meant only for the cases when the physical approach is not sufficient, i.e. across scales, constituting an alternative worth exploring to describe evolution.\n\nIt seems easier to describe matter and energy in terms of information than vice versa. Moreover, information could be used as a common language across scientific disciplines \\cite{vonBaeyer2004}.\n\n\n\\subsection*{Acknowledgements}\n\nI should like to thank Irun Cohen, Inman Harvey, Francis Heylighen, David Krakauer, Antonio del R\\'{i}o, Marko Rodriguez, David Rosenblueth, Stanley Salthe, Mikhail Prokopenko, Cl\\'{e}ment Vidal, and H\\'{e}ctor Zenil for their useful comments and suggestions. \n\n\\footnotesize{\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{The Formation of the First Luminous Primordial Structures}\n{Emanuele Ripamonti \\\\\nKapteyn Astronomical Institute, University of Groningen \\\\\nThe Netherlands \\\\~\\\\\nTom Abel \\\\\nKavli Institute for\nAstroparticle Physics and Cosmology, Stanford University \\\\\nU.S.A.}\n\n\n\\section{Introduction}\n\nThe scientific belief that the universe evolves in time is one of the legacies of the theory of the Big Bang. \n\nThe concept that the universe has an history started to attract the interest of cosmologists soon after the first formulation of the theory: already Gamow (1948; 1949) investigated how and when galaxies could have been formed in the context of the expanding Universe.\n\nHowever, the specific topic of the formation (and of the fate) of the {\\it first} objects dates to two decades later, when no objects with metallicities as low as those predicted by primordial nucleosynthesis ($Z\\lsim10^{-10}\\sim10^{-8}Z_\\odot$) were found.  Such concerns were addressed in two seminal papers by Peebles \\& Dicke (1968; hereafter PD68) and by\nDoroshkevich, Zel'Dovich \\& Novikov (1967; hereafter DZN67)\\footnote{This paper is in Russian and we base our comments on indirect knowledge ({\\frenchspacing\\it e.g. } from the Novikov \\& Zel'Dovich 1967 review).}, introducing the idea that some objects could have formed before the stars we presently observe. \n\\begin{enumerate}\n\\item{Both PD68 and DZN67 suggest a mass of $\\sim10^5M_{\\odot}$ for the first\ngeneration of bound systems, based on the considerations on the cosmological\nJeans length\\index{Jeans length} (Gamow 1948; Peebles 1965) and the possible shape of the power spectrum.}\n\\item{They point out the role of thermal instabilities in the formation of the\nproto-galactic bound object, and of the cooling\\index{cooling} of the gas\ninside it; in particular, PD68 introduces H$_2$~ cooling\\index{HH cooling} and\nchemistry\\index{HH chemistry} in the calculations about the contraction of the gas.}\n\\item{Even if they do not specifically address the occurrence of fragmentation\\index{fragmentation}, these papers make two very different assumptions: PD68 assumes that the gas will fragment into ``normal'' stars to form globular clusters, while DZN67 assumes that fragmentation {\\it does not}\\ occur, and that a single ``super-star'' forms.}\n\\item{Finally, some feedback\\index{feedback} effects as considered ({\\frenchspacing\\it e.g. } Peebles \\& Dicke considered the effects of supernovae).}\n\\end{enumerate}\n\nToday most of the research focuses on the issues when fragmentation\\index{fragmentation} may occur, what objects are formed and how they influence subsequent structure formation. \n\n\nIn these notes we will leave the discussion of feedback to lecture notes by\nFerrara \\& Salvaterra and by Madau \\& Haardt in this same book and focus only on the aspects of the formation of the first objects.\n\n\n\n\n\n\n\n\n\n\n\n\nThe advent of cosmological numerical hydrodynamics in particular allow a fresh new look at these questions. Hence, these notes will touch on aspects of theoretical cosmology to chemistry\\index{chemistry}, computer science, hydrodynamics and atomic physics. \nFor further reading and more references on the subject we refer the reader to other\nrelevant reviews such as Barkana \\& Loeb 2001, and more recently Ciardi \\& Ferrara 2004,\nGlover 2004 and Bromm \\& Larson 2004.\n\nIn this notes, we try to give a brief introduction to only the most relevant\naspects. We will start with a brief overview of the relevant cosmological\nconcepts in section 2, followed by a discussion of the properties of\nprimordial material (with particular emphasis to its cooling and its\nchemistry\\index{chemistry}) in section 3.  We will then review the technique\nand the results of numerical simulations\\index{numerical Simulation} in sections 4 and 5: the former will deal with detailed 3D simulations of the formation of gaseous clouds which are likely to transform into luminous objects, while the latter will examine results (mostly from 1D codes) about the modalities of such transformation. Finally, in section 6 we will critically discuss the results of the previous sections, examining their consequences and comparing them to our present knowledge of the universe.\n\n\n\n\\section{Physical cosmology}\n\nIn the following of this notes we will adopt the modern physical\ndescription of the universe (dating back to at least 1920), based upon\nthe ``cosmological principle'', which affirms that on cosmological\nscales the distributions of matter and energy should be homogeneous\nand isotropic, whose metric is the Robertson-Walker metric. Although\nderived from mainly philosophical arguments, the\ncosmological principle is also supported by observations such as the\nisotropy of the CMB ({\\it e.g.} Wu {\\it et al.~} 1999).\n\nWe will also make some additional, general assumptions which are\nquite common in present-day cosmology, and which are believed to be\nthe best explanations for a series of observations. That is:\n\\begin{enumerate}\n\\item{The cosmological structures we observe at present (galaxies,\nclusters of galaxies etc.) formed because of gravitational instability\nof pre-existing, much shallower fluctuations;}\n\\item{Most of the matter in the universe is in the form of {\\it ``Cold\nDark Matter''}\\index{dark matter}, that is, of some kind of elementary particle (or\nparticles) that has not been discovered at present. Cold Dark Matter\\index{dark matter}\nparticles are assumed to have a negligibly small cross section for\nelectromagnetic interactions ({\\frenchspacing\\it i.e. } to be {\\it dark}), and to move at\nnon-relativistic speeds ({\\frenchspacing\\it i.e. } to be{\\it cold}).}\n\\end{enumerate}\n\n\n\n\n\\subsection{Fluctuations in the Early Universe}\n\n\\subsubsection{Inflation\\index{inflation}}\nInflation is a mechanism which was first proposed by Guth (1981) and (in\na different way) by Starobinsky (1979, 1980), and has since been\nincluded in a number of different ``inflationary theories'' (see Guth\n2004 for a review upon which we base this paragraph).\n\nThe basic assumption of inflationary models is the existence of states\nwith negative pressure; a typical explanation is that some unidentified\nkind of scalar field (commonly referred to as {\\it inflaton}\\index{inflation})\ntemporarily keeps part of the universe in a ``false vacuum'' state, in\nwhich the energy density must be approximately constant (that is, it can\nnot ``diluted'' by expansion) at some value $\\rho_f c^2$, implying a\nnegative pressure $p_f=-\\rho_f c^2$. Inserting these two expression in\nthe first Friedmann cosmological equation\n\\begin{equation}\n\\ddot{a}(t) = -{{4\\pi}\\over3}G\\left({\\rho+3{p\\over{c^2}}}\\right) a(t)\n\\end{equation}\n(where $a(t)$ is the scale factor at cosmic time $t$) it is easy to\nderive that the considered region expands exponentially: $a(t)\\propto\ne^{t\/t_f}$ with $t_f=(8\\pi G\\rho_f\/3)^{-1\/2}$; the epoch in which the\nuniverse undergoes such exponential expansion is called {\\it inflation}.\n\nIf $\\rho_f$ is taken to be at about the grand unified theory scale, we\nget $t_f\\sim10^{-38}\\;{\\rm s}$, corresponding to an Hubble length of\n$ct_f \\sim 10^{-28}\\;{\\rm cm}$; if the inflationary phase is long enough\n(a lower limit for this is about 65 e-folding times, corresponding to an\nexpansion by a factor $\\sim10^{28}$), it smooths the metric, so that the\nexpanding region approaches a de Sitter flat space, regardless of\ninitial conditions. Furthermore, when inflation ends, the energy stored\nin the inflaton\\index{inflation} field is finally released, thermalizes and leads to the\nhot mixture of particles assumed by the standard big bang picture.\n\n\\begin{figure}[t]\n\\epsfig{file=geometry.ps,width=11truecm}\n\\caption{Effect of universe geometry on the observed angular size of\nfluctuations in the CMBR\\index{Cosmic Microwave Background}. If the universe is closed (left panel) ``hot\nspots'' appear larger than actual size; if the universe is flat\n(middle panel), ``hot spots'' appear with their actual size; if the\nuniverse is open, ``hot spots'' appear smaller than their actual\nsize.}\n\\label{geometry}\n\\end{figure}\n\nInflation helps explaining several fundamental observations, which were\njust assumed as initial conditions in non inflationary models:\n\\begin{enumerate}\n\\item{The Hubble expansion: repulsive gravity associated with false\nvacuum is exactly the kind of force needed to set up a motion pattern in\nwhich every two particles are moving apart with a velocity proportional\nto their distance.}\n\\item{The homogeneity and isotropy: in ``classical'' cosmology the\nremarkable uniformity of the Cosmic Microwave Background\\index{Cosmic Microwave Background} Radiation (CMBR)\ncannot be ``explained'', because different\nregions of the present CMBR sky never were causally connected. Instead,\nin inflationary models the whole CMBR sky was causally connected {\\it before}\ninflation\\index{inflation} started, and uniformity can be established at that time.}\n\\item{The flatness problem: a flat Friedman-Robertson-Walker model\nuniverse ({\\it i.e.}\\ with $\\Omega(t)\\equiv\\rho_{\\rm tot}(t)\/\\rho_{\\rm\ncrit}(t)=1$, where $\\rho_{\\rm tot}(t)$ is the cosmological mean density,\nincluding the ``dark energy'' term, and $\\rho_{\\rm crit}(t)=3H(t)^2\/8\\pi\nG$, $H(t)\\equiv \\dot{a}(t)\/a(t)$ being the Hubble parameter) always\nremains flat, but if at an early time $\\Omega$ was just slightly\ndifferent from $1$, the difference should have grown as fast as\n$(\\Omega-1) \\propto t^{2\/3}$. All the observational results ({\\it e.g.}\nthe Bennet {\\it et al.~} 2003 WMAP result $\\Omega_0=1.02 \\pm 0.02$) show that at\npresent $\\Omega$ is quite close to 1, implying that at the Planck time\n($t\\sim10^{-43}\\;{\\rm s}$) the closeness was really amazing.  Inflation\nremoves the need for this incredibly accurate fine tuning, since during\nan inflationary phase $\\Omega$ is driven towards 1 as $(\\Omega-1)\\propto\ne^{-2Ht}$.}\n\\item{The absence of magnetic mono-poles: grand unified theories,\ncombined with classical cosmology, predict that the universe should be\nlargely dominated by magnetic mono-poles, which instead have never been\nobserved. Inflation\\index{inflation} provides an explanation, since it can dilute the\nmono-pole density to completely negligible levels.}\n\\item{the {\\it anisotropy} properties of the CMBR\\index{Cosmic Microwave Background} radiation: inflation\nprovides a prediction for the power spectrum of fluctuations, which\nshould be generated by quantum fluctuations and nearly scale-invariant,\na prediction in close agreement with the WMAP results (see the next\nsubsection).}\n\\end{enumerate}\n \n\n\nA peculiar property of inflation\\index{inflation} is that most inflationary models\npredict that inflation does not stop everywhere at the same time, but\njust in localized ``patches'' in a succession which continues eternally;\nsince each ``patch'' (such as the one we would be living in) can be\nconsidered a whole universe, it can be said that inflation produces an\ninfinite number of universes.\n\n\n\n\\subsubsection{Primordial fluctuation evolution - Dark Matter\\index{dark matter}}\nInflation\\index{inflation} predicts the statistical properties of the density\nperturbation field, defined as\n\\begin{equation}\n\\delta({\\bf x}) \\equiv {{\\rho({\\bf r}) - \\bar\\rho}\\over{\\bar\\rho}}\n\\end{equation}\nwhere ${\\bf r}$ is the proper coordinate, ${\\bf x}={\\bf r}\/a$ is the\ncomoving coordinate, and $\\bar\\rho$ is the mean matter density.\n\nIn fact, if we look at the Fourier components in terms of the comoving\nwave-vectors ${\\bf k}$ \n\\begin{equation}\n\\delta({\\bf k}) = \\int{\\delta({\\bf x})\\ e^{-i{\\bf k}{\\bf x}}\\ d^3{\\bf x}}\n\\end{equation}\nthe inflationary prediction is that the perturbation field is a\nGaussian random field, that the various ${\\bf k}$ modes are independent,\nand that the power spectrum $P(k)$ (where $k\\equiv|{\\bf k}|$) is close\nto scale-invariant, {\\it i.e.}, it is given by a power law\n\\begin{equation}\nP(k) \\equiv <|\\delta_{\\bf k}|^2> \\propto k^{n_s},\\qquad{\\rm with}\\ n_s\\simeq 1.\n\\end{equation}\n\n\\begin{figure}[t]\n\\epsfig{file=powerspectrum.ps,width=11truecm}\n\\caption{Schematic shape of the DM power spectrum after accounting\nfor the effects of free streaming\\index{fluctuations!free streaming scale} and of the processing that takes\nplace for fluctuations entering the horizon before $t_{\\rm eq}$.}\n\\label{powerspectrum}\n\\end{figure}\n\n\nThis prediction applies to the {\\it primordial} power spectrum; in order\nto make comparisons with observations, we need to include effects which\nsubsequently modified its shape:\n\\begin{enumerate}\n\\item{Free streaming: the dark matter\\index{dark matter} particles are in motion, and since\nthey are believed to interact only through gravity, they freely\npropagate from overdense to underdense regions, wiping out the density\nperturbations. However, at any given cosmic time $t$, this will affect\nonly wavelengths smaller than the {\\it free streaming}\\index{fluctuations!free streaming scale} length, {\\frenchspacing\\it i.e. }\\ the\nproper distance $l_{\\rm FS}(t)$ which a dark matter particle can have\ntravelled in time $t$. It can be shown (see {\\it e.g.}\\ Padmanabhan 1993,\n\\S4.6) that this scale depends primarily on the epoch $t_{\\rm nr}$ when\nthe dark matter\\index{dark matter} particles become non-relativistic: before that epoch\nthey move at the speed of light, and can cover a proper distance $l_{\\rm\nFS}(t_{\\rm nr})\\sim 2ct_{\\rm nr}$, corresponding to a comoving distance\n$\\lambda_{\\rm FS} = (a_0\/a_{\\rm nr})2ct_{\\rm nr}$; after becoming\nnon-relativistic, particle motions become much slower than cosmological\nexpansion, and the evolution after $t_{\\rm nr}$ only increases\n$\\lambda_{\\rm FS}$ by a relatively small factor.  In turn, $t_{\\rm nr}$\ndepends primarily on the mass $m_{\\rm DM}$ of dark matter particles, so\nthat\n\\begin{equation}\n\\lambda_{\\rm FS}\\sim 5\\times10^{-3}\\;(\\Omega_{\\rm DM} h^2)^{1\/3}\n       \\left({{m_{\\rm DM}}\\over{\\rm 1\\;GeV}}\\right)^{-4\/3}\\ {\\rm pc}\n\\end{equation}\ncorresponding to a mass scale\n\\begin{equation}\nM_{\\rm FS}\\sim 6\\times 10^{-15}\\;(\\Omega_{\\rm DM} h^2)\n       \\left({{m_{\\rm DM}}\\over{\\rm 1\\;GeV}}\\right)^{-4}\\ M_{\\odot}.\n\\end{equation}\nSince the most favoured candidates for the DM particles are Weakly\nInteracting Massive Particles (WIMPs) with mass between $0.1$ and\n$100\\;{\\rm GeV}$, we probably have $M_{\\rm FS}\\lsim10^{-8} M_{\\odot}$. Some\nsuper-symmetric theories ({\\frenchspacing\\it e.g. }\\ Schwartz {\\it et al.~} 2001, and more recently\nGreen, Hofmann \\& Schwarz 2004) instead point towards $M_{\\rm FS}\\sim\n10^{-6}M_{\\odot}$, and Diemand, Moore \\& Stadel (2005) used this result in\norder to argue that the first structure which formed in the early\nUniverse were Earth-mass dark-matter\\index{dark matter} haloes (but see also Zhao {\\it et al.~}\n2005 for criticism about this result);}\n\\item{Growth rate changes: in general, perturbations grow\nbecause of gravity; however, the details of the process change with\ntime, leaving an imprint on the final processed spectrum. The time\n$t_{\\rm eq}$ when the matter density becomes larger than the radiation\ndensity is particularly important: before $t_{\\rm eq}$, perturbations\nwith $\\lambda$ larger than the Hubble radius $c\/H(t)$ grow as\n$\\delta\\propto a^2$, while the growth of smaller perturbations is almost\ncompletely suppressed; after $t_{\\rm eq}$ both large and small\nfluctuations grow as $\\delta\\propto a$.  Because of these differences,\nthe size of the Hubble radius at $t_{\\rm eq}$, $\\lambda_{\\rm eq}\\simeq\n13\\;(\\Omega h^2)^{-1} {\\rm Mpc}$ (in terms of mass, $M_{\\rm eq}\\simeq\n3.2\\times10^{14} (\\Omega h^2)^{-2}\\;M_{\\odot}$) separates two different\nspectral regimes. In the wavelength range $\\lambda_{\\rm FS}\\leq\n\\lambda\\leq \\lambda_{\\rm eq}$ the growth of fluctuations pauses between\nthe time they enter the Hubble radius and $t_{\\rm eq}$. As a result\nthe slope of the processed spectrum is changed, and $P(k)\\propto k^{n_s-4}\\simeq\nk^{-3}$. Instead, at scales $\\lambda>\\lambda_{\\rm eq}$ all the\nfluctuations keep growing at the same rate at all times, and the shape\nof the power spectrum remains similar to the primordial one, $P(k)\\propto\nk^{n_s}\\simeq k^1$.}\n\\end{enumerate}\n\nWMAP (Bennett {\\it et al.~} 2003) measured the spectral index,\nobtaining $n_s=0.99\\pm0.04$, and did not detect deviations from\ngaussianity, both results in agreement with inflationary\npredictions.\n\nThis kind of spectrum, in which fluctuations are typically larger on\nsmall scales, leads naturally to hierarchical structure formation, since\nsmall-scale fluctuations are the first to become non-linear\\index{fluctuations!non-linear evolution} ({\\frenchspacing\\it i.e. } to\nreach $\\delta\\sim1$), collapse and form some kind of astronomical\nobject.  It is also worth remarking that the very first objects, coming\nfrom the highest peaks of $\\delta({\\bf x})$, are typically located where\nmodes $\\delta({\\rm k})$ of different wavelength make some kind of\n``constructive interference'': the very first objects are likely to be\non top of larger ones, and they are likely to be clustered together,\nrather than uniformly distributed. For this reason, it is also very\nlikely that the halos where these objects formed have long since been\nincorporated inside larger objects, such as the bulges of $M_*$ galaxies\nor the cD galaxy at the centre of galaxy clusters (see {\\it e.g.}\\ White\n\\& Springel 1999).\n\n\\begin{figure}[t]\n\\epsfig{file=wmap_map.ps,angle=0,width=10truecm,clip=}\n\\caption{Map of the temperature anisotropies in the CMB\\index{Cosmic Microwave Background} as obtained by\ncombining the five bands of the WMAP satellite in order to minimise\nforeground contamination (from Bennett {\\it et al.~} 2003).}\n\\label{wmap_map}\n\\end{figure}\n\n\\begin{figure}[t]\n\\epsfig{file=wmap_spectrum.ps,width=10truecm,clip=}\n\\caption{Angular CMB\\index{Cosmic Microwave Background} power spectrum of temperature (top panel) and\ntemperature-polarization (bottom panel) as obtained by the WMAP\nsatellite (from Bennett {\\it et al.~} 2003). The line shows the best-fit with\na $\\Lambda$CDM model, and grey areas represent cosmic variance.}\n\\label{wmap_angspec}\n\\end{figure}\n\n\n\\subsubsection{Fluctuation evolution - Baryons}\n\nBefore the equivalence epoch $t_{\\rm eq}$ the baryons evolve in the same\nway as dark matter\\index{dark matter}. Instead, in the matter dominated era they behave\ndifferently: we mentioned that all the dark matter fluctuations which\nwere not erased by free streaming\\index{fluctuations!free streaming scale} grow as $\\delta\\propto a \\propto\n(1+z)^{-1}$, but this does not apply to baryons. In fact, baryons\ndecouple from radiation only at $t_{\\rm dec}$, significantly later than\n$t_{\\rm eq}$ (we remind that $1+z_{\\rm eq}\\sim 10^4$, while $1+z_{\\rm dec}\n\\simeq10^3$). The persistence of the coupling with radiation prevents\nthe growth of baryonic fluctuations on all scales; even worse, on\nrelatively small scales all the fluctuations in the baryonic component\nare erased through a mechanism similar to free streaming. Such effect\ntakes place below the so-called {\\it Silk scale}\\index{fluctuations!silk scale} (Silk 1968), which is\ngiven by the average distance that the photons (and the baryons coupled\nwith them) can diffuse before $t=t_{\\rm dec}$; this translates into a\ncomoving distance\n\\begin{equation}\n  \\lambda_{\\rm S} \\simeq 3.5 \\left({\\Omega\\over{\\Omega_{\\rm\n  b}}}\\right)^{1\/2} (\\Omega h^2)^{-3\/4}\\ {\\rm Mpc}\n\\end{equation}\n(where $\\Omega_{\\rm b}$ is the baryonic contribution to $\\Omega$)\nand encloses a mass\n\\begin{equation}\n  M_{\\rm S} \\simeq 6.2\\times10^{12} \\left({\\Omega\\over{\\Omega_{\\rm\n  b}}}\\right)^{3\/2} (\\Omega h^2)^{-5\/4}\\ M_{\\odot}.\n\\end{equation}\n\nThis result was a major problem for cosmology before the existence of\ndark matter\\index{dark matter} started to be assumed: it implies that in a purely baryonic\nuniverse there should be no structures on a scale smaller than that of\ngalaxy clusters (if $\\Omega=\\Omega_{\\rm b}\\simeq 0.1$).  Furthermore,\neven fluctuations which were not erased can grow only by a factor\n$1+z_{\\rm dec}$ between decoupling and present time, and this is not\nenough to take a typical CMBR\\index{Cosmic Microwave Background} fluctuation (of amplitude\n$\\delta\\sim10^{-5}$) into the non-linear\\index{fluctuations!non-linear evolution} regime.  The introduction of\nCold Dark matter solved this problem, since after recombination\\index{Hydrogen!recombination} the\nbaryons are finally free to fall inside dark matter\\index{dark matter} potential wells,\nwhose growth was unaffected by the photon drag. \n\nIt can be found that after decoupling from radiation, the baryonic\nfluctuations quickly ``reach'' the levels of dark matter fluctuations,\nevolving as\n\\begin{equation}\n\\delta_{\\rm b} = \\delta_{\\rm DM}\\left({1-{{1+z}\\over\n    {1+z_{\\rm dec}}}}\\right),\n\\end{equation}\nso that the existing dark matter\\index{dark matter} potential wells ``regenerate'' baryonic\nfluctuations, including the ones below the Silk scale\\index{fluctuations!silk scale}.\n\nThis result is correct as long as pressure does not play a role, that is\nfor objects with a mass larger than the cosmological Jeans mass\\index{Jeans mass} $M_{\\rm\nJ}\\propto T^{3\/2}{\\rho}^{-1\/2}$. Such mass behaves differently at high\nand low redshift. Before a redshift $z_{\\rm Compton}\\simeq 130$ we have\nthat the temperature of the baryons is still coupled to that of the CMB\\index{Cosmic Microwave Background}\nbecause of Compton scattering of radiation on the residual free\nelectrons; for this reason, $T_{\\rm b}(z)\\simeq T_{\\rm CMB}(z)\\propto (1+z)$,\nand as $\\rho(z)\\propto (1+z)^3$ the value of $M_{\\rm J}$ is constant:\n\\begin{equation}\nM_{\\rm J}(z)\\simeq 1.35\\times 10^5 \\left({{\\Omega_{\\rm m} h^2}\\over\n  {0.15}}\\right)^{-1\/2}\\;M_{\\odot} \\qquad\n  ({\\rm for}\\ z_{\\rm dec}\\lower.5ex\\hbox{\\gtsima} z\\lower.5ex\\hbox{\\gtsima} z_{\\rm Compton})\n\\end{equation}\nwhere $\\Omega_{\\rm m}=\\Omega_{\\rm b}+\\Omega_{\\rm DM}$ is the total\nmatter density (baryons plus dark matter\\index{dark matter}).\nAt lower redshifts the baryon temperature is no longer locked to that of\nthe CMB\\index{Cosmic Microwave Background} and drops adiabatically as $T_{\\rm b}\\propto (1+z)^2$. At such\nredshifts the Jeans mass\\index{Jeans mass} evolves as\n\\begin{eqnarray}\nM_{\\rm J}(z)\\simeq 5.7\\times 10^3\n   \\left({{\\Omega_{\\rm m} h^2}\\over {0.15}}\\right)^{-1\/2} \\;\n   \\left({{\\Omega_{\\rm b}h^2}\\over{0.022}}\\right)^{-3\/5} \\;\n   \\left({{1+z}\\over{10}}\\right)^{3\/2}\\;M_{\\odot} \\qquad\n   \\\\({\\rm for}\\ z\\lower.5ex\\hbox{\\ltsima} z_{\\rm Compton}).\\nonumber \n\\end{eqnarray}\n\n\\begin{figure}[t]\n\\epsfig{file=thermal_hist.ps,width=11truecm,clip=}\n\\caption{Schematic evolution of the temperature of the Intergalactic Medium\n(IGM) as a function of redshift: after recombination and before\nthermal decoupling (at $z_{\\rm Compton}$), $T_{\\rm IGM}$ is locked to\n$T_{\\rm CMBR}$ and evolves as $(1+z)$. After thermal decoupling\n$T_{\\rm IGM}$ is no more locked to the radiation temperature, and\ndrops adiabatically as $(1+z)^2$, until the first non-linear objects\ncollapse, and emit light which is able to re-ionize the universe,\nraising $T_{\\rm IGM}$ above $\\sim 10^4\\;{\\rm K}$.}\n\\label{thermal_hist}\n\\end{figure}\n\n\n\n\\subsection{From fluctuations to cosmological structures}\n\\subsubsection{Non-linear evolution\\index{fluctuations!non-linear evolution}}\nWhen the density contrast $\\delta$ becomes comparable to unity, the\nevolution becomes non-linear, and the Fourier modes no more evolve\nindependently. The easiest way to study this phase is through the ``real\nspace'' $\\delta({\\bf x})$ (rather than its Fourier components\n$\\delta({\\bf k})$), considering the idealized case of the collapse of a\nspherically symmetric overdensity, and in particular the collapse of a\n{\\it top-hat} fluctuation. It is well known that, through some simple\nfurther assumption (such as that ``spherical shells do not cross''), the\ntime evolution of the radius $R$ of a top-hat perturbation (see {\\frenchspacing\\it e.g. }\nPadmabhan 1993, \\S 8.2 for the treatment of a slightly more general\ncase) can be written down as\n\\begin{equation}\nR(t)={R_i\\over2}{{1+\\delta_i}\\over{\\delta_i-(\\Omega_i^{-1}-1)}}[1-\\cos\\theta(t)]\n\\label{tophatradius}\n\\end{equation}\nwhere $R_i=R(t_i)$, $\\delta_i=\\delta(t_i)$ and $\\Omega_i=\\Omega(t_i)$\nare the ``initial conditions'' for the fluctuation evolution, and\n$\\theta$ is defined by the equation\n\\begin{equation}\n[\\theta(t)-\\sin\\theta(t)]{{1+\\delta_i}\\over\n{2H_i\\Omega_i^{1\/2}[\\delta_i-(\\Omega_i^{-1}-1)]^{3\/2}}} \\simeq  t\n\\end{equation}\nwhere again $H_i$ is the Hubble parameter at $t_i$, and the last\napproximate equality is valid only as long as $\\delta_i\\ll1$ (that is, a\nsufficiently early $t_i$ must be chosen).  The fluctuation radius $R$\nreaches a maximum $R_{\\rm ta}$ at the so-called {\\it turn-around} epoch\n(when $\\theta=\\pi$) when the overdense region finally detaches itself\nfrom the Hubble flow and starts contracting. However, while\neq. (\\ref{tophatradius}) suggests an infinite contraction to $R=0$ (when\n$\\theta=2\\pi$), the violent relaxation process (Lynden-Bell 1967)\nprevents this from happening, leading to a configuration in virial\nequilibrium at $R_{\\rm vir}\\simeq R_{\\rm ta}\/2$.\n\nHere, we summarise some well known, useful findings of this model.\n\nFirst of all, combining the evolution of the background density\nevolution and of eq. (\\ref{tophatradius}) it is possible to\nestimate the density contrast evolution\n\\begin{equation}\n\\delta(t)={9\\over2}{{(\\theta-\\sin\\theta)^2}\\over{(1-\\cos\\theta)^3}}-1\n\\end{equation}\nwhich leads to some noteworthy observation, such as that the density\ncontrast at turn-around is $\\delta_{\\rm ta}=(9\\pi^2\/16)-1\\simeq4.6$,\nwhich at virialization becomes $\\delta_{\\rm vir}=\\Delta_{\\rm c}$,\nwhere it can be usually assumed that $\\Delta_{\\rm c}\\simeq18\\pi^2$\nbut sometimes higher order approximations are necessary, such as the\none in Bryan \\& Norman 1998,\n\\begin{equation}\n\\Delta_{\\rm c}=18\\pi^2 + 82(1-\\Omega_m^z) -39(1-\\Omega_m^z)^2\n\\end{equation}\nwith\n\\begin{equation}\n\\Omega_m^z={{\\Omega_{\\rm m} (1+z)^3}\\over\n{\\Omega_{\\rm m} (1+z)^3+\\Omega_\\Lambda+\\Omega_k(1+z)^2}}\n\\end{equation}\nwhere $\\Omega_\\Lambda$ is the dark energy density, and $\\Omega_k =\n1-\\Omega_{\\rm m}-\\Omega_\\Lambda$ is the curvature.\n\nFrom this, it is possible to estimate the virial radius\n\\begin{equation}\nR_{\\rm vir} \\simeq 0.784 \\left({M\\over{10^8 h^{-1} M_{\\odot}}}\\right)^{1\/3}\n\\left({{\\Omega_{\\rm m}\\over\\Omega_m^z}{\\Delta_c\\over{18\\pi^2}}}\\right)^{-1\/3}\n\\left({{1+z}\\over{10}}\\right)^{-1} h^{-1}\\;{\\rm kpc},\n\\end{equation}\nthe circular velocity for such an halo\n\\begin{equation}\nV_{\\rm circ} \\simeq 23.4 \\left({M\\over{10^8 h^{-1} M_{\\odot}}}\\right)^{1\/3}\n\\left({{\\Omega_{\\rm m}\\over\\Omega_m^z}{\\Delta_c\\over{18\\pi^2}}}\\right)^{1\/6}\n\\left({{1+z}\\over{10}}\\right)^{1\/2}\\;{\\rm km\\,s^{-1}},\n\\end{equation}\nand the virial temperature\n\\begin{equation}\nT_{\\rm vir} \\simeq 19800 \\left({M\\over{10^8 h^{-1} M_{\\odot}}}\\right)^{2\/3}\n\\left({{\\Omega_{\\rm m}\\over\\Omega_m^z}{\\Delta_c\\over{18\\pi^2}}}\\right)^{1\/3}\n\\left({{1+z}\\over{10}}\\right) \\left({\\mu\\over{0.6}}\\right) \\;{\\rm K}.\n\\end{equation}\n\n\\subsubsection{The Press-Schechter formalism\\index{Press-Schechter formalism}} \n\\begin{figure}[t]\n\\epsfig{file=press_schechter_mass.ps,width=10truecm,clip=}\n\\caption{Characteristic mass of $1\\sigma$ (bottom solid curve),\n$2\\sigma$ (middle solid curve) and $3\\sigma$ (top solid curve)\ncollapsing halos as a function of redshift. These were obtained from\nthe Eisenstein \\& Hu (1999) power spectrum, assuming an\n$\\Omega_\\Lambda=0.7$, $\\Omega_{\\rm m}=0.3$ cosmology. The dashed curves show\nthe minimum mass\\index{minimum mass} which is required for the baryons to be able to cool\nand collapse (see next section) in case of pure atomic cooling (upper\ncurve) and of molecular cooling\\index{cooling} (lower curve).\n(from Barkana \\& Loeb 2001).}\n\\label{press_schechter_mass}\n\\end{figure}\n\n\\begin{figure}[t]\n\\epsfig{file=press_schechter_abundance.ps,width=10truecm,clip=}\n\\caption{Halo mass functions at several redshifts (from bottom left to\n  bottom right, $z=30$, $z=20$, $z=10$, $z=5$ and $z=0$,\n  respectively). The assumed power spectrum and cosmology are the same as in fig. \\ref{press_schechter_mass}. (from Barkana \\& Loeb 2001).}\n\\label{press_schechter_abundance}\n\\end{figure}\n\n\nThe simple top-hat model is at the core of the so-called Press-Schechter\nformalism\\index{Press-Schechter formalism} (Press \\& Schechter 1974, but see also the contribution by\nSommerville in this same book), which predicts the density of\nvirialized halos of a given mass at a given redshift. This model assumes\nthat the distribution of the smoothed density field $\\delta_M$ (where\n$M$ is the mass scale of the smoothing) at a certain redshift $z_0$ is\nGaussian with a variance $\\sigma_M$, so that the probability of having\n$\\delta_M$ larger than a given $\\delta_{\\rm crit}$ is\n\\begin{equation}\nP(\\delta_M>\\delta_{\\rm crit}) = \\int_{\\delta_{\\rm crit}}^\\infty {\n{1\\over{(2\\pi)^{1\/2}\\sigma_M}} e^{-{x^2\\over{2\\sigma_M^2}}} dx};\n\\end{equation}\na common choice is $z_0=0$, requiring $\\delta_M$ to be estimated through\na purely linear evolution\\index{fluctuations!linear evolution} of the primordial power spectrum.\n\n\nThe Press-Schechter\\index{Press-Schechter formalism} model then chooses a $\\delta_{\\rm crit}=\\delta_{\\rm\ncrit}(z)$ (but it is also possible to assume a constant $\\delta_{\\rm\ncrit}$ and make $\\sigma_M$ a function of redshift; see {\\frenchspacing\\it e.g. }\\ Viana \\&\nLiddle 1996) and assumes that this probability (multiplied by a factor\nof 2 - see Bond {\\it et al.~} 1991 for an explanation of this extra factor) also\ngives the fraction mass which at a redshift $z$ is inside virialized\nhalos of mass M or larger.  This can be differentiated over $M$ in order\nto get the mass distribution at each given redshift\n\\begin{equation}\n{{dn}\\over{dM}}={2\\over{(2\\pi)^{1\/2}}} {\\rho_m\\over M}\n{{d ln(1\/\\sigma_M)}\\over{dM}} {{\\delta_{\\rm crit}(z)}\\over\\sigma_M}\ne^{-{{\\delta_{\\rm crit}(z)^2}\\over{2\\sigma_M^2}}}\n\\end{equation}\n\nIn this way, the abundance of halos is completely determined through the\ntwo functions $\\delta_{\\rm crit}(z)$ and $\\sigma_M$.  The first one is\ncommonly written as $\\delta_{\\rm crit}(z)=\\delta_0\/D(z)$, where $D(z)$\nis the growth factor ($D(z)\\simeq(1+z)^{-1}$ for Einstein-de Sitter\nmodels; see Peebles 1993 for a more general expression), coming from\ncosmology; instead $\\delta_0$ is usually taken to be $1.686$, since the\ntop-hat model predicts that an object virializes at the time when the\nlinear theory estimates its overdensity at $\\delta=1.686$.  Instead\n$\\sigma_M$ depends from the power spectrum; for example, figures\n\\ref{press_schechter_mass} and \\ref{press_schechter_abundance} are based\non the Eisenstein \\& Hu (1999) results\\footnote{The authors of this\npaper also provide some very useful codes for dealing with the power\nspectrum and the Press-Schechter formalism\\index{Press-Schechter formalism} at the web page\nhttp:\/\/background.uchicago.edu\/$\\sim$whu\/transfer\/transferpage.html.}.\n\n\n\n\n\\section{Primordial gas properties}\n\n\\subsection{Cooling}\nThe typical densities reached by the gas after virialization (of the\norder of $n_{\\rm B}\\equiv \\rho_{\\rm B}\/m_{\\rm H} \\sim 0.01 \\Omega_{\\rm\nb} [(1+z_{\\rm vir})\/10]^3 {\\rm cm^{-3}}$ are far too low for the gas to\ncondense and form an object like a star. The only way to proceed further\nin the collapse and in the formation of luminous objects is to remove\nthe gas thermal energy through radiative cooling\\index{cooling}.\n\nFor this reason, cooling processes are important in determining where,\nwhen and how the first luminous objects will form.\n\nIn Fig. \\ref{metalfree_cooling} it is possible to see that the cooling\\index{cooling!metal-free gas}\nof primordial ({\\frenchspacing\\it i.e. } metal-free) {\\it atomic} gas at temperatures below\n$\\sim10^4\\;{\\rm K}$ is dramatically inefficient, because in that\ntemperature range the gap between the fundamental and the lowest excited\nlevels ($\\simeq 10.2\\;{\\rm eV}$ for H atoms) is so much larger than the\nthermal energy $\\sim k_{\\rm B}T\\lower.5ex\\hbox{\\ltsima} 1\\;{\\rm eV}$ that very few atoms\nget collisionally excited, and very few photons are emitted through the\ncorresponding de-excitations.\n\nThis is important because in all hierarchical scenarios the first\nobjects to virialize are the smallest ones, and such halos have\nthe lowest virial temperatures. If the primordial gas were completely\natomic, the first luminous objects would probably form relatively late\n($z\\lsim10-15$), in moderately massive halos ($M\\sim10^8\\;M_{\\odot}$) with\n$T_{\\rm vir}\\lower.5ex\\hbox{\\gtsima} 10^4\\;{\\rm K}$.\n\nHowever, it is also possible to see from Fig. \\ref{metalfree_cooling} that\nthe presence of molecules in small amounts ($f_{\\rm H_2}\\equiv 2n_{\\rm\nH_2}\/(n_{\\rm H}+2n_{\\rm H_2})\\lower.5ex\\hbox{\\gtsima} 5\\times10^{-4}$; the dashed curve in\nFig. \\ref{metalfree_cooling} was obtained assuming $f_{\\rm\nH_2}=10^{-3}$) can dramatically affect the cooling\\index{cooling} properties of\nprimordial gas at low temperatures, making low mass halos virializing at\nhigh redshift ($z\\lower.5ex\\hbox{\\gtsima} 20$) the most likely sites for the formation of\nthe first luminous objects. \n\n\\begin{figure}[t]\n\\epsfig{file=metalfree_cooling.ps,width=10truecm,clip=}\n\\caption{Cooling rate per atomic mass unit of\\index{cooling!metal-free gas} metal-free gas, as a\nfunction of temperature. The solid line shows assumes the gas to be\ncompletely atomic (the two peaks correspond to H and He excitations,\nwhile the high temperature tail is dominated by free-free processes) and\ndrops to about zero below $T\\sim10^4\\;{\\rm K}$; the dashed line shows\nthe contribution of a small ($f_{\\rm H_2} = 10^{-3}$) fraction of\nmolecular hydrogen\\index{Hydrogen}, which contributes extra-cooling in the range\n$100\\;{\\rm K}\\lower.5ex\\hbox{\\ltsima} T \\lower.5ex\\hbox{\\ltsima} 10^4\\;{\\rm K}$ (from Barkana \\& Loeb 2001).}\n\\label{metalfree_cooling}\n\\end{figure}\n\n\n\\subsection{Molecular cooling\\index{cooling}}\n\nIn the current scenario for the formation of primordial objects, the\nmost relevant molecule is H$_2$.  The only reason for this is the high\nabundance of H$_2$~\\index{HH} when compared with all other molecules. In fact, the\nradiating properties of an H$_2$~ molecule are very poor: because of the\nabsence of a dipole moment, radiation is emitted only through weak\nquadrupole transitions. In addition, the energy difference between the\nH$_2$~ ground state and the lowest H$_2$~ excited roto-vibrational levels is\nrelatively large ($\\Delta E_{01}\/k_{\\rm B}\\lower.5ex\\hbox{\\gtsima} 200\\;{\\rm K}$, between\nthe fundamental and the first excited level; however, such transition is\nprohibited by quadrupole selection rules, and the lowest energy gap for\na quadrupole transition is $\\Delta E_{02}\/k_{\\rm B}\\simeq 510\\;{\\rm\nK}$), further reducing the cooling efficiency at low temperatures.\n\nApart from H$_2$~ the most relevant molecular coolants are HD and LiH. In\nthe following, we will briefly list the cooling\\index{cooling} rates (mainly taken from\nGalli \\& Palla 1998, hereafter GP98; see also Hollenbach \\& McKee 1979,\nLepp \\& Shull 1983, 1984 and Martin {\\it et al.~} 1996, Le Bourlot {\\it et al.~} 1999,\nFlower {\\it et al.~} 2000) of H$_2$~ and of the other two possibly relevant\nspecies\\footnote{In Fig. \\ref{species_cooling} the cooling rate for H$_2^+$\nis shown, too. But while it is still marginally possible that HD or LiH\ncooling\\index{HD cooling}\\index{LiH cooling} can play some kind of role in the primordial universe, this is\nmuch more unlikely for H$_2^+$, mainly because its under-abundance with\nrespect to H$_2$~ is always much larger than the difference in the cooling\nrates; for this reasons we choose to omit a detailed discussion of H$_2^+$\ncooling.}\\index{cooling}.\n\nWe note that Bromm {\\it et al.~} 1999 included HD cooling\\index{HD cooling} in some of\ntheir simulations but found that it never accounted for more than\n$\\sim10\\%$ of H$_2$~ cooling; however, they did not completely rule out the\npossibility that a $n_{\\rm HD}\/n_{\\rm H_2}$ ratio substantially larger\nthan the equilibrium value (close to $n_{\\rm D}\/n_{\\rm H}\\sim 10^{-5}$)\ncould change this conclusion. Also, in present-day simulations of\nprimordial star formation the gas temperature never goes below a few\nhundreds degrees: in case it did, even a tiny amount of LiH\\index{LiH cooling} could be\nenough to dominate the cooling\\index{cooling} rate.\n\n\n\\begin{figure}[t]\n\\epsfig{file=H2cool.eps,width=10truecm,clip=}\n\\caption{Cooling rate per H$_2$~ molecule as computed by different authors\n(Lepp \\& Shull 1983, Martin {\\it et al.~} 1996, GP98) as a\nfunction of temperature and for different densities; note that for\n$n\\lower.5ex\\hbox{\\gtsima} 10^4\\;{\\rm cm^{-3}}$ the cooling rate is almost independent of\ndensity.}\n\\label{H2_cool}\n\\end{figure}\n\n\\subsubsection{H$_2$~ cooling rate\\index{cooling}}\nThe H$_2$~ cooling rate {\\it per molecule} $\\Lambda_{\\rm H_2}(\\rho,T)$\ncan be conveniently expressed in the form:\n\\begin{equation}\n\\Lambda_{\\rm H_2}(\\rho,T) = {{\\Lambda_{\\rm H_2,LTE}(T)}\\over\n{1+{{\\Lambda_{\\rm H_2,LTE}(T)}\\over{n_{\\rm H}\\Lambda_{\\rm\nH_2,\\rho\\rightarrow0}(T)}}}}\n\\end{equation}\nwhere $\\Lambda_{\\rm H_2,LTE}(T)$ and $n_{\\rm H}\\Lambda_{\\rm\nH_2,\\rho\\rightarrow0}$ are the high and low density limits of the\ncooling rate (which apply at $n \\lower.5ex\\hbox{\\gtsima} 10^4\\; {\\rm cm^{-3}}$ and\nat $n \\lower.5ex\\hbox{\\ltsima} 10^2\\; {\\rm cm^{-3}}$, respectively).\n\nThe high density (or LTE, from the Local Thermal Equilibrium assumption\nwhich holds in these conditions) limit of the cooling rate per H$_2$~\nmolecule is given by Hollenbach \\& McKee (1979):\n\\begin{eqnarray}\n\\Lambda_{\\rm H_2,LTE}(T)={{9.5\\times10^{-22}T_3^{3.76}} \\over\n  {1+0.12\\,T_3^{2.1}}} e^{-\\left({0.13}\\over{T_3}\\right)^3} +\n  3\\times10^{-24} e^{-{{0.51}\\over{T_3}}} + \\nonumber \\\\\n  + 6.7\\times10^{-19} e^{-{{5.86}\\over{T_3}}} +\n  1.6\\times10^{-18} e^{-{{11.7}\\over{T_3}}}\\;\n  {\\rm erg\\,s^{-1}}\n\\end{eqnarray}\nwhere $T_3 \\equiv T\/(1000\\;{\\rm K})$.  Note that the first row in the\nformula accounts for rotational cooling\\index{cooling}, while the second row accounts for\nthe first two vibrational terms.\n\nFor the low density limit, GP98 found that in the\nrelevant temperature range ($10\\;{\\rm K}\\leq T \\leq 10^4\\;{\\rm K}$) the\ncooling rate $\\Lambda_{\\rm H_2,\\rho\\rightarrow0}$ is independent from\ndensity, and is well approximated by\n\\begin{eqnarray}\n\\log \\Lambda_{\\rm H_2,\\rho\\rightarrow0}(T) \\simeq \n  -103.0 + 97.59 \\log T - 48.05 (\\log T)^2 + \\nonumber \\\\\n  10.80 (\\log T)^3 - 0.9032 (\\log T)^4\n\\end{eqnarray}\nwhere $T$ and $\\Lambda_{\\rm H_2,\\rho\\rightarrow0}$ are expressed in\n$K$ and ${\\rm erg\\,s^{-1}\\,cm^3}$, respectively.\n\nNote that even if both $\\Lambda_{\\rm H_2,LTE}$ and\n$\\Lambda_{\\rm H_2,\\rho\\rightarrow0}$ do not depend on density,\n$\\Lambda_{\\rm H_2}(\\rho,T)$ is independent of $\\rho$ only in the high\ndensity limit.\n\n\n\\subsubsection{HD\\index{HD cooling} and LiH\\index{LiH cooling} cooling rates}\nThe cooling rates of HD and of LiH are more complicated (see Flower\n{\\it et al.~} 2000 for HD and Bogleux \\& Galli 1997 for LiH), but in the low\ndensity limit (and in the temperature range $10\\;{\\rm K}\\leq T \\leq\n1000\\;{\\rm K}$) it is possible to use the relatively simple expressions\ngiven by GP98.\n\nFor HD, we have that the low density limit of the cooling rate per\nmolecule, $\\Lambda_{\\rm HD,\\rho\\rightarrow0}$, is:\n\\begin{equation}\n\\Lambda_{\\rm HD,\\rho\\rightarrow0}(T) \\simeq \n2\\gamma_{10}E_{10}e^{-{{E_{10}}\\over{k_{\\rm B}T}}} +\n(5\/3)\\gamma_{21}E_{21}e^{-{{E_{21}}\\over{k_{\\rm B}T}}}\n\\end{equation}\nwhere $E_{10}$ and $E_{21}$ are the energy gaps between HD levels 1 and\n0 and levels 2 and 1, respectively; they are usually expressed as\n$E_{10}=k_{\\rm B} T_{10}$ and $E_{21}=k_{\\rm B} T_{21}$, with\n$T_{10}\\simeq128\\;{\\rm K}$ and $T_{21}\\simeq255\\;{\\rm K}$,\n\n\n$\\gamma_{10}$ and $\\gamma_{21}$ are the approximate collisional\nde-excitation rates for the 1-0 and 2-1 transitions, and are given by\n\\begin{eqnarray}\n\\gamma_{10} \\simeq &\n4.4\\times10^{-12}+3.6\\times10^{-13}\n  T^{0.77}\\\\\n\\gamma_{21} \\simeq &\n4.1\\times10^{-12}+2.1\\times10^{-13}\n  T^{0.92}\n\\end{eqnarray}\nwhere we use the numerical value of T (in Kelvin) and the rates are\nexpressed in ${\\rm cm^3\\,s^{-1}}$.\n\nFor LiH instead we have that the same density limit of the cooling\\index{cooling} rate\nper molecule, $\\Lambda_{\\rm LiH,\\rho\\rightarrow0}$ can be fitted by:\n\\begin{eqnarray}\n\\log_{10}(\\Lambda_{\\rm LiH,\\rho\\rightarrow0}) = &\nc_0 + c_1\\log_{10} T + c_2(\\log_{10} T)^2 + \\nonumber\\\\\n& c_3(\\log_{10} T)^3 + c_4(\\log_{10} T)^4\n\\end{eqnarray}\nwhere $c_0=-31.47,\\ c_1=8.817,\\ c_2=-4.144,\\ c_3=0.8292$ and\n$c_4=-0.04996$, assuming that $T$ is expressed in Kelvin, and that\n$\\Lambda_{\\rm LiH,\\rho\\rightarrow0}$ is expressed in\n${\\rm erg\\,cm^3\\,s^{-1}}$.\n\n\n\\begin{figure}[t]\n\\epsfig{file=species_cooling.ps,width=10truecm,clip=}\n\\caption{Comparison of the low density ($n\\lower.5ex\\hbox{\\ltsima} 10\\;{\\rm cm^{-3}}$)\ncooling rates per molecule of several molecular species, in particular\nH$_2$~, HD and LiH (from GP98). Note that at $T\\sim100\\;{\\rm\nK}$ both HD and LiH molecules are more than $10^3$ times more efficient\ncoolants than H$_2$~ molecules, but this difference is believed to be\ncompensated by the much higher H$_2$~ abundance (see {\\it e.g.}, Bromm\n{\\it et al.~} 2002). The plot also shows the cooling due to H-H$_2^+$ and\n$e^-$-H$_2^+$ collisions, but these contributions are never important\nbecause of the very low H$_2^+$ abundance.}\n\\label{species_cooling}\n\\end{figure}\n\n\n\\subsubsection{Cooling at high densities: Collision Induced Emission\\index{cooling!collision induced emission}}\n\n\\begin{figure}[t]\n\\epsfig{file=cie_cooling.eps,width=9truecm,angle=270}\n\\caption{CIE cooling for different collisions and different molecular\nfractions(0.999, 0.5, 0.001). Top panels show the cooling rates per unit\nmass: the thick solid line is the total CIE cooling, while thin lines\nshow the various components: H$_2$~-H$_2$~ (solid), H$_2$~-He (short dashed),\nH$_2$~-H (dot dashed) and H-He (long dashed); the dotted line shows the\nresults of the approximate formula given in the text. In the bottom\npanels the ratios of the various component to H$_2$~-H$_2$~ CIE is shown. All\nquantities were calculated assuming $n=10^{14}\\;{\\rm cm^{-3}}$ and\n$X=0.75$ (from Ripamonti \\& Abel 2004).}\n\\label{cie_cooling}\n\\end{figure}\n\nDuring the formation of a protostar an important role is played by the\nso called Collision-Induced Emission (CIE; very often known as\nCollision-Induced Absorption, or CIA), a process which requires pretty\nhigh densities ($n=\\rho\/m_{\\rm H}\\gsim10^{13}-10^{14}\\;{\\rm cm^{-3}}$)\nto become important (see Lenzuni {\\it et al.~} 1991, Frommold 1993, Ripamonti \\&\nAbel 2004). In fact, Collision-Induced Emission takes place when a\ncollision between two H$_2$~ molecules (or H$_2$~ and H, or H$_2$~ and He, or\neven H and He) takes place: during the collision, the interacting pair\nbriefly acts as a ``super-molecule'' with a nonzero electric dipole, and\na much higher probability of emitting (CIE) or absorbing (CIA) a photon\nthan an unperturbed H$_2$~ molecule (whose dipole is 0). Because of the\nvery short durations of the collisions, ($\\lsim10^{-12}\\;{\\rm s}$), such\nmechanism can compete with ``normal'' emission only in high density\nenvironments. Furthermore, because of the short durations of the\ninteractions, collision-induced lines become very broad and merge into a\ncontinuum: the H$_2$~ CIE spectrum only shows broad peaks corresponding to\nvibrational bands, rather than a large number of narrow roto-vibrational\nlines. This is also important because self-absorption is much less\nrelevant than for line emission, and in primordial proto-stars\nsimulations CIE\\index{cooling!collision induced emission} cooling can be treated with the optically thin\napproximation up to $n\\sim10^{16}\\;{\\rm cm^{-3}}$ (for H$_2$~ lines, the\noptically thin approximation breaks at about $n\\sim10^{10}\\;{\\rm\ncm^{-3}}$).  In figure \\ref{cie_cooling} we show the CIE cooling rate\nfor a gas with $n=10^{14}\\;{\\rm cm^{-3}}$, as a function of temperature\nand for different chemical compositions; at temperatures between $400$\nand $7000\\;{\\rm K}$.\nFor H$_2$~ abundances $f_{\\rm H_2}\\equiv 2n_{\\rm H_2}\/(n_{\\rm H^+}+n_{\\rm\nH}+2n_{\\rm H_2}) \\lower.5ex\\hbox{\\gtsima} 0.5$ the total CIE\\index{cooling!collision induced emission} cooling rate can be\napproximated by the simple expression (see Ripamonti \\& Abel 2004)\n\\begin{equation}\nL_{\\rm CIE}(\\rho,T,X,f_{\\rm H_2}) \\simeq 0.072 \\rho T^4 X f_{\\rm H_2}\n\\;{\\rm erg\\,g^{-1}\\,s^{-1}}\n\\end{equation}\nwhere the density is in $\\rm g\\,cm^{-3}$, the temperature is in K, and\n$X\\simeq0.75$ is the hydrogen\\index{Hydrogen} fraction (by mass).\n\n\n\n\\subsection{Chemistry\\index{Chemistry}}\n\n\\begin{table}[b]\n\\caption{Reaction rates for some of the most important reactions in\nprimordial gas, plus the main reactions involved in the formation of\nHD. In the formulae, $T_\\gamma$ is the temperature of the radiation\nfield (for our purposes, the temperature of the CMB\\index{Cosmic Microwave Background} radiation) and\ntemperatures need to be expressed in Kelvin; $j(\\nu_{\\rm LW})$ is the\nradiation flux (in ${\\rm erg\\,s^{-1}\\,cm^{-2}}$) at the central\nfrequency $\\nu_{\\rm LW}$ of the Lyman-Werner bands, $h\\nu_{\\rm\nLW}=12.87\\;{\\rm eV}$.  The rates come from compilations given by Tegmark\n{\\it et al.~} 1997 (reactions 1-7), Palla {\\it et al.~} 1983 (reactions 8-13), Abel\n{\\it et al.~} 1997 (reaction 14) and Bromm {\\it et al.~} 2002 (reactions 15-19).}\n\\begin{tabular}{cccl}\n\\hline\\hline\nReaction & & & Rate\\\\\n\\hline\n${\\rm H}^+ + e^-$ &  $\\rightarrow$ & ${\\rm H} + h\\nu$ &\n$k_1 \\simeq 1.88\\times10^{-10} T^{-0.644} \\;{\\rm cm^3\\,s^{-1}}$\\\\\n\n${\\rm H} + e^-$ & $\\rightarrow$ & ${\\rm H}^- + h\\nu$ &\n$k_2 \\simeq 1.83\\times10^{-18} T^{0.88} \\;{\\rm cm^3\\,s^{-1}}$\\\\\n\n${\\rm H}^- + {\\rm H}$ & $\\rightarrow$ & ${\\rm H}_2 + e^- $ &\n$k_3 \\simeq 1.3\\times10^{-9} \\;{\\rm cm^3\\,s^{-1}}$\\\\\n\n${\\rm H}^- + h\\nu$ & $\\rightarrow$ & ${\\rm H} + e^-$ &\n$k_4 \\simeq 0.114 T^{2.13} e^{-8650\/T_\\gamma} \\;{\\rm cm^3\\,s^{-1}}$\\\\\n\n${\\rm H}^+ + {\\rm H}$ & $\\rightarrow$ & ${\\rm H}_2^+ + h\\nu$ &\n$k_5 \\simeq 1.85\\times10^{-23} T^{1.8} \\;{\\rm cm^3\\,s^{-1}}$\\\\\n\n${\\rm H}_2^+ + {\\rm H}$ & $\\rightarrow$ & ${\\rm H}_2- + {\\rm H}^+$ &\n$k_6 \\simeq 6.4\\times10^{-10} \\;{\\rm cm^3\\,s^{-1}}$\\\\\n\n${\\rm H}_2^+ + h\\nu$ & $\\rightarrow$ & ${\\rm H}^+ + {\\rm H}$ &\n$k_7 \\simeq 6.36\\times10^5 e^{-71600\/T_\\gamma} \\;{\\rm cm^3\\,s^{-1}}$\\\\\n\n${\\rm H} + {\\rm H} + {\\rm H}$ & $\\rightarrow$ & ${\\rm H}_2 + {\\rm H}$ &\n$k_8 \\simeq 5.5\\times10^{-29} T^{-1} \\;{\\rm cm^6\\,s^{-1}}$\\\\\n\n${\\rm H}_2 + {\\rm H}$ & $\\rightarrow$ & ${\\rm H} + {\\rm H} + {\\rm H}$ &\n$k_9 \\simeq 6.5\\times10^{-7} T^{-1\/2} e^{-52000\/T}\\times$\\\\\n& & & $\\qquad\\qquad\\times(1-e^{-6000\/T}) \\;{\\rm cm^6\\,s^{-1}}$\\\\\n\n${\\rm H} + {\\rm H} + {\\rm H}_2$ & $\\rightarrow$ & ${\\rm H}_2 + {\\rm H}_2$ &\n$k_{10} \\simeq k_8\/8$\\\\\n\n${\\rm H}_2 + {\\rm H}_2$ & $\\rightarrow$ & ${\\rm H} + {\\rm H} + {\\rm H}_2$ &\n$k_{11} \\simeq k_9\/8$\\\\\n\n${\\rm H} + e^-$ & $\\rightarrow$ & ${\\rm H}^+ + e^- + e^-$ &\n$k_{\\rm 12} \\simeq 5.8\\times10^{-11} T^{1\/2}\n\te^{-158000\/T} \\;{\\rm cm^3\\,s^{-1}}$\\\\\n\n${\\rm H} + {\\rm H}$ & $\\rightarrow$ & ${\\rm H}^+ + e^- + {\\rm H}$ &\n$k_{\\rm 13} \\simeq 1.7\\times10^{-4} k_{12}$\\\\\n\n${\\rm H}_2 + h\\nu$ & $\\rightarrow$ & ${\\rm H}^+ + {\\rm H}$ &\n$k_{\\rm 14} \\simeq 1.1\\times10^8 j(\\nu_{\\rm LW}) \\; {\\rm s^{-1}}$\\\\\n\n${\\rm D}^+ + e^-$ & $\\rightarrow$ & ${\\rm D} + h\\nu$ &\n$k_{\\rm 15} \\simeq 8.4\\times10^{-11}\nT^{-0.5} \\times$\\\\\n& & & $\\quad\\times({T\\over{10^3}})^{-0.2}[1+({T\\over{10^6}})^{0.7}]^{-1}\n\\; {\\rm cm^3 s^{-1}}$\\\\\n\n${\\rm D} + {\\rm H}^+$ & $\\rightarrow$ & ${\\rm D}^+ + {\\rm H}$ &\n$k_{\\rm 16} \\simeq 3.7\\times10^{-10} T^{0.28} e^{-43\/T}\n\\; {\\rm cm^3 s^{-1}}$\\\\\n\n${\\rm D}^+ + {\\rm H}$ & $\\rightarrow$ & ${\\rm D} + {\\rm H}^+$ &\n$k_{\\rm 17} \\simeq 3.7\\times10^{-10} T^{0.28} \\; {\\rm cm^3 s^{-1}}$\\\\\n\n${\\rm D}^+ + {\\rm H}_2$ & $\\rightarrow$ & ${\\rm H}^+ + {\\rm HD}$ &\n$k_{\\rm 18} \\simeq 2.1\\times10^{-9} \\; {\\rm cm^3 s^{-1}}$\\\\\n\n${\\rm HD} + {\\rm H}^+$ & $\\rightarrow$ & ${\\rm H}_2 + {\\rm D}$ &\n$k_{\\rm 19} \\simeq 1.0\\times10^{-9} e^{-464\/T} \\; {\\rm cm^3 s^{-1}}$\\\\\n\\hline\\hline\n\\end{tabular}\n\\label{reaction_rates}\n\\end{table}\n\nSince molecules play such an important role in the formation of the\nfirst luminous objects, it is important to include a proper treatment of\ntheir abundance evolution. However, a full treatment should keep into\naccount $\\sim 20$ different species and $\\sim 100$ different\nreactions. For instance, GP98 give a chemical network\nof 87 reactions, which includes $e^-$, H, H$^+$, H$^-$, D, D$^+$, He,\nHe$^+$, He$^{++}$, Li, Li$^+$, Li$^-$, H$_2$~\\index{HH}, H$_2^+$, HD, HD$^+$, HeH$^+$,\nLiH, LiH$^+$, H$_3^+$ and H$_2$D$^+$; even their {\\it minimal model} is\ntoo complicated to be described here, so we will just describe the most\nbasic processes involved in H$_2$~\\index{HH} formation. However, we note that the\npapers by Abel {\\it et al.~} (1997)\\footnote{The collisional rate coefficients\ngiven in the Abel {\\it et al.~} (1997) paper can be readily obtained through a\nFORTRAN code available on the web page\nhttp:\/\/www.astro.psu.edu\/users\/tabel\/PGas\/LCA-CM.html ; also note that\non Tom Abel's web site (http:\/\/www.tomabel.com\/) it is possible to find\nseveral useful informations about the primordial universe in general.}\\\nand by GP98 provide a much more accurate description of\nprimordial chemistry\\index{Chemistry}.\n\n\\subsubsection{Atomic Hydrogen\\index{Hydrogen} and free electrons}\nApart from molecule formation (see below), the main reactions involving\nHydrogen (and Helium, to which we can apply all the arguments below) are\nionizations\\index{Hydrogen!Ionization} and recombinations\\index{Hydrogen!Recombination}.\n\nIonizations can be produced both by radiation (${\\rm H} + h\\nu\n\\rightarrow {\\rm H}^+ + e^-$) and by collisions, mainly with free\nelectrons (${\\rm H} + e^- \\rightarrow {\\rm H}^+ + 2e^-$) but also with\nother H atoms (${\\rm H} + {\\rm H} \\rightarrow {\\rm H}^+ e^- + {\\rm\nH}$). Photoionizations dominate over collisions as long as UV photons\nwith energy above the H ionization\\index{Hydrogen!Ionization} threshold are present (see {\\frenchspacing\\it e.g. }\nOsterbrock 1989), but this is not always the case before the formation\nof the first luminous objects, when only the CMB\\index{Cosmic Microwave Background} radiation is present;\neven after some sources of radiation have appeared, it is likely that\ntheir influence will be mainly local, at least until reionization.\nHowever, in the primordial universe the electron temperature is low,\nand collisional ionizations are relatively rare.\n\nRecombinations\\index{Hydrogen!Recombination} (${\\rm H}^+ + e^- \\rightarrow {\\rm H} + h\\nu$) have much\nhigher specific rates, since they do not have a high energy\nthreshold. They probably dominate the evolution of free electrons, which\ncan be relatively abundant as residuals from the recombination era ({\\frenchspacing\\it e.g. }\nPeebles 1993), and are important for H$_2$~\\index{HH} formation (see below).\n\nSimple approximations for the reaction rates of collisional ionizations\nand recombinations are given in Table \\ref{reaction_rates}.\n\n\n\n\\subsubsection{H$_2$~\\index{HH} formation and disruption}\nAt present, H$_2$~ is commonly believed to form mainly through reactions\ntaking place on the surface of dust grains (but see {\\frenchspacing\\it e.g. } Cazaux \\& Spaans\n2004). Such a mechanism cannot work in the primordial universe, when the\nmetal (and dust) abundance was negligible. The first two mechanisms\nleading to formation of H$_2$~ in a primordial environment was described by\nMcDowell (1961) and by Saslaw \\& Zipoy (1967); soon after the\npublication of this latter paper,\nH$_2$~\\index{HH} started to be included in theories of structure formation (such as\nthe PD68 paper about globular cluster formation through H$_2$~\\index{HH cooling}\ncooling). Both these mechanisms consist of two stages, involving either\n$e^-$ or H$^+$ as catalyzers.  The first (and usually most important)\ngoes through the reactions\n\\begin{eqnarray}\n{\\rm H} + e^-       & \\rightarrow & {\\rm H}^- + h\\nu\\\\\n{\\rm H}^- + {\\rm H} & \\rightarrow & {\\rm H}_2- + e^-\n\\end{eqnarray}\nwhile the second proceeds as\n\\begin{eqnarray}\n{\\rm H}^+ + {\\rm H}   & \\rightarrow & {\\rm H}_2^+ + h\\nu\\\\\n{\\rm H}_2^+ + {\\rm H} & \\rightarrow & {\\rm H}_2- + {\\rm H}^+.\n\\end{eqnarray}\nIn both cases, H$_2$~\\index{HH} production can fail at the intermediate stage if a\nphotodissociation occurs (${\\rm H}^- + h\\nu \\rightarrow {\\rm H} +\ne^-$, or ${\\rm H}_2^+ + h\\nu \\rightarrow {\\rm H}^+ + {\\rm H}$).\nThe rates of all these reactions are listed in Table \\ref{reaction_rates}\n\nBy combining the reaction rates of these reactions, it is possible (see\nTegmark {\\it et al.~} 1997; hereafter T97) to obtain an approximate evolution for the\nionization fraction $x \\equiv n_{\\rm H^+} \/ n$ and the H$_2$~ fraction\n$f_{\\rm H_2} \\equiv 2n_{\\rm H_2}\/n$ (here $n$ is the total density of\nprotons, $n \\simeq n_{\\rm H} + n_{\\rm H^+} + 2n{\\rm H_2}$):\n\\begin{eqnarray}\n&& x(t)           \\simeq {x_0\\over{1+x_0 n k_1 t}}\\\\\n&& f_{\\rm H_2}(t) \\simeq f_0 + 2{k_m\\over k_1} \\ln(1 + x_0 n k_1 t)\n\\qquad{\\rm with}\\\\\n&& k_m            =      {{k_2 k_3}\\over{k_3+k_4\/[n(1-x)]}} + \n                      {{k_5 k_6}\\over{k_6+k_7\/[n(1-x)]}}\n\\end{eqnarray}\nwhere the various $k_i$ are the reaction rates given in Table\n\\ref{reaction_rates}, and $x_0$ and $f_0$ are the initial fractions of\nionized atoms and of H$_2$~\\index{HH} molecules.\n\nAnother mechanism for H$_2$~ formation, which becomes important at\n(cosmologically) very high densities $n_{\\rm H}\\gsim10^8\\;{\\rm cm^-3}$\nare the so-called 3-body reactions described by Palla {\\it et al.~} (1983).\nWhile the previous mechanisms are limited by the small abundance of free\nelectrons (or of H$^+$), reactions such as\n\\begin{equation}\n{\\rm H} + {\\rm H} + {\\rm H} \\rightarrow {\\rm H}_2 + {\\rm H}\n\\end{equation}\ncan rapidly convert all the hydrogen\\index{Hydrogen} into molecular form, provided the\ndensity is high enough. For this reason, they are likely play an\nimportant role during the formation of a primordial protostar.\n\nFinally, H$_2$~\\index{HH} can be dissociated through collisional processes such as\nreactions 9 in table \\ref{reaction_rates}, but probably the most\nimportant process is its photo-dissociation by photons in the\nLyman-Werner bands (11.26-13.6 eV; but Abel {\\it et al.~} 1997 found that the\nmost important range is between 12.24 and 13.51 eV). These photons are\nbelow the H ionization\\index{Hydrogen!Ionization} threshold, therefore they can diffuse to large\ndistances from their source. So, any primordial source emitting a\nrelevant number of UV photons ({\\frenchspacing\\it e.g. } a $\\sim100M_{\\odot}$ star) is likely to\nhave a major feedback\\index{feedback} effect, since it can strongly reduce the amount of\nH$_2$~\\index{HH} in the halo where it formed (and also in neighbouring halos),\ninhibiting further star formation.\n\n\n\\subsubsection{Approximate predictions}\n\n\\begin{figure}[t]\n\\epsfig{file=tegmark_minh2.ps,width=9truecm,angle=270}\n\\caption{Comparison of the H$_2$~\\index{HH} fraction needed for an halo to collapse\nand H$_2$~ fraction which can be formed inside the halo in an Hubble time\n(from T97). This is shown as a function of halo virial\ntemperature and for three different virialization redshifts ($z=100$:\nsolid; $z=50$: short dashes; $z=25$:long dashes). The three dots mark the\nminimum H$_2$~\\index{HH} abundance which is needed for collapse at the three\nconsidered redshift, and it can be seen that they all are at $f_{\\rm\nH_2}\\sim 5\\times10^{-4}$.}\n\\label{tegmark_minh2}\n\\end{figure}\n\n\\begin{figure}[t]\n\\epsfig{file=tegmark_minmass.ps,width=9truecm}\n\\caption{Evolution with virialization redshift of the minimum mass\\index{minimum mass} which\nis able to cool, collapse and possibly form stars. Only the halos whose ($z_{\\rm\nvir},M$) fall outside the filled region are able to cool fast\nenough. The filled region is made of a red part, where CMB\\index{Cosmic Microwave Background} radiation\nprevents the cooling, and a yellow part where the cooling is slow\nbecause of a dearth of H$_2$~ molecules. The two parallel dashed lines\ncorrespond to virial temperatures of $10^4\\;{\\rm K}$ (the highest one)\nand $10^3\\;{\\rm K}$ (the lowest one), while the almost vertical line in\nthe middle of the plot corresponds to $3\\sigma$ peaks in a SCDM\n($\\Omega_{\\rm m}=1,\\ \\Omega_\\Lambda=0$) cosmology.}\n\\label{tegmark_minmass}\n\\end{figure}\n\nThe above information can be used for making approximate predictions\nabout the properties of the halos hosting the first luminous\nobjects. Such kind of predictions was started by Couchman \\& Rees\n(1986), and especially by T97 and their example was followed and\nimproved by several authors (see below).\n\nThe basic idea is to have a simplified model for the evolution of the\nH$_2$~\\index{HH} fraction and the cooling rate inside spherical top-hat fluctuations,\nin order to check whether after virialization the baryons are able to\ncool (and keep collapsing), or they just ``settle down'' at about the\nvirial radius. Such approximate models are fast to compute, and it is\neasy to explore the gas behaviour on a large range of virialization\nredshifts and fluctuation masses (or, equivalently, virial temperatures).\nIn this way, for each virialization redshift it is possible to obtain\nthe minimum molecular fraction which is needed for the collapse to\nproceed, and the minimum size of halos where such abundance is achieved.\nThe results interestingly point out that the molecular fraction\nthreshold separating collapsing and non-collapsing objects has an almost\nredshift-independent value of $f_{\\rm H_2}\\sim5\\times10^{-4}$ (see\nfig. \\ref{tegmark_minh2}). Instead, the minimum halo mass actually\nevolves with redshift (see fig. \\ref{tegmark_minmass}).\n\nPredictions about the ability of the baryons inside each kind of halo to\nkeep collapsing after virialization can then be combined with\nPress-Schechter\\index{Press-Schechter Formalism} predictions about the actual abundances of halos.  For\ninstance, in fig. \\ref{tegmark_minmass}\\ the solid black (almost\nvertical) line shows where the masses of $3\\sigma$ fluctuations lie, as\na function of redshift. So, if we decide to neglect the rare\nfluctuations at more than $3\\sigma$ from the average, that figure tells\nus that the first luminous objects can start forming only at $z\\lsim30$,\nin objects with a total mass $\\lower.5ex\\hbox{\\gtsima} 2\\times 10^6\\;M_{\\odot}$.\n\nSuch result is subject to a number of uncertainties, both about the\n``details'' of the model and about the processes it neglects; here are\nsome of the more interesting developments:\n\\begin{enumerate}\n\\item{Abel {\\it et al.~} (1998) found that the minimum mass\\index{minimum mass} is strongly\n  affected by the uncertainties in the adopted H$_2$~\\index{HH cooling} cooling function,\n  with differences that could reach a factor $\\sim 10$ in the minimum\n  mass\\index{minimum mass} estimate}\n\\item{Fuller \\& Couchman (2000) used numerical simulations\\index{Numerical Simulation} of single,\n  high-$\\sigma$ density peaks in order to improve the spherical\n  collapse approximation}\n\\item{Machacek {\\it et al.~} (2001) and Kitayama {\\it et al.~} (2001) investigated the\n  influence of background radiation}\n\\item {Yoshida {\\it et al.~} (2003) used larger-scale numerical simulations\\index{Numerical Simulation}\n  and found that also the merging history of an halo could play a\n  role, since frequent mergings heat the gas and prevent or delay the\n  collapse of $\\sim 30\\%$ of the halos.}\n\\end{enumerate}\n\nAs a result of the improved modeling (and of a different set of\ncosmological parameters as $\\Lambda$CDM has substituted SCDM), in the\nmost recent papers the value of the minimum halo mass for the formation\nof the first luminous objects is somewhat reduced to the range 0.5--1\n$\\times 10^6\\; M_{\\odot}$, with a weak dependence on redshift and a stronger\ndependence on other parameters (background radiation, merging history\netc.).\n\n\n\n\\section{Numerical cosmological hydrodynamics}\n\nNumerical simulations\\index{Numerical Simulation} are an important tool for a large range of\nastrophysical problems. This is especially true for the study of\nprimordial stars, given the absence of direct observational data about\nthese stars, and the relatively scarce indirect evidence we have.\n\n\n\\begin{figure}[p]\n\\epsfig{file=halo_3d.eps,width=10.5truecm}\n\\caption{Overview of the evolution leading to the formation of a\nprimordial star. The top row shows the gas density, centered at the\npre-galactic object within which the star is formed. The four projections\nare labelled with their redshifts. Pre-galactic objects form from very\nsmall density fluctuations and continuously merge to form larger\nobjects. The middle and bottom rows show thin slices through the gas\ndensity and temperature at the final simulation stage. The four pairs of\nslices are labelled with the scale on which they were taken, starting\nfrom 6 (proper) kpc (the size of the simulated volume) and zooming in\ndown to 0.06 pc (12,000 AU).  In the left panels, the larger scale\nstructures of filaments and sheets are seen. At their intersections, a\npre-galactic object of $\\sim 10^6\\;M_{\\odot}$ is formed. The temperature\nslice (second panel, bottom row) shows how the gas shock heats as it\nfalls into the pre-galactic object. After passing the accretion shock,\nthe material forms hydrogen\\index{Hydrogen} molecules and starts to cool. The cooling\nmaterial accumulates at the centre of the object and forms the\nhigh-redshift analog to a molecular cloud (third panel from the right),\nwhich is dense and cold ($T\\sim 200 K$). Deep within the molecular\ncloud, a core of $\\sim 100\\;M_{\\odot}$, a few hundred K warmer, is formed\n(right panel) within which a $\\sim 1\\;M_{\\odot}$ fully molecular object is\nformed (yellow region in the right panel of the middle row). (from\nABN02).}\n\\label{halo_3d}\n\\end{figure}\n\n\n\\subsection{Adaptive refinement codes (ENZO)}\nThe two problems which immediately emerge when setting up a simulation\nof primordial star formation are the dynamical range\nand the required accuracy in solving the hydrodynamical equations.\n\nWhen studying the formation of objects in a cosmological context, we\nneed both to simulate a large enough volume (with a box size of at least 100\ncomoving kpc), and to resolve objects of the size of a star\n($\\sim 10^{11}\\;{\\rm cm}$ in the case of the Sun), about 11 orders of\nmagnitude smaller.\n\nThis huge difference in the relevant scales of the problem is obviously\na problem. It can be attenuated by the use of Smoothed Particle\nHydrodynamics\\index{Numerical Simulations!Smoothed Particle Hydrodynamics} (SPH; see {\\frenchspacing\\it e.g. } Monaghan 1992), whose Lagrangian nature has\nsome benign effects, as the simulated particles are likely to\nconcentrate ({\\frenchspacing\\it i.e. }, provide resolution) in the regions where the maximum\nresolution is needed. However, even if this kind of method can actually\nbe employed (see Bromm {\\it et al.~} 1999, 2002), it has at least two important\ndrawbacks. First of all, the positive effects mentioned above cannot\nbridge in a completely satisfactory way the extreme dynamical range we\njust mentioned, since the mass resolution is normally fixed once and for\nall at the beginning of the simulation. Second, SPH\\index{Numerical Simulations!Smoothed Particle Hydrodynamics} is known to have\npoor shock resolution properties, which casts doubts on the results when\nthe hydrodynamics becomes important.\n\nThe best presently available solution for satisfying both requirements\nis the combination of an Eulerian approach (which is good for the\nhydrodynamical part) and an adaptive technique (which can extend the\ndynamical range). This is known as Adaptive Mesh Refinement\\index{Numerical Simulations!Adaptive Mesh Refinement} (AMR; see\n{\\frenchspacing\\it e.g. } Berger \\& Colella 1989; Norman 2004), and basically consists in\nhaving the simulation volume represented by a hierarchy of nested {\\it\ngrids} ({\\it meshes}) which are created according to resolution needs.\n\nIn particular, the simulations we are going to describe in the following\nparagraphs were made with the code ENZO\\footnote{The ENZO code can be\nretrieved at the web site http:\/\/cosmos.ucsd.edu\/enzo\/} (see O'Shea\n{\\it et al.~} 2004 and references therein for a full description). Briefly, ENZO\nincludes the treatment of gravitational physics through N-body\ntechniques, and the treatment of hydrodynamics through the piecewise\nparabolic method (PPM) of Woodward \\& Colella 1984 (as modified by Bryan\n{\\it et al.~} 1995 in order to adapt to cosmological simulations).  ENZO can\noptionally include the treatment of gas cooling (in particular H$_2$~\\index{HH\ncooling} cooling, as described in the preceeding sections), of primordial\nnon-equilibrium chemistry\\index{Chemistry} (see the preceeding sections) and of UV\nbackground models ({\\frenchspacing\\it e.g. } the ones by Haardt \\& Madau 1996) and heuristic\nprescriptions (see Cen \\& Ostriker 1992) for star formation in\nlarger-scale simulations.\n\n\n\\subsection{Formation of the first star}\n\n\\begin{figure}[p]\n\\epsfig{file=profiles3d.eps,width=11.5truecm}\n\\caption{Radial mass-weighted averages of physical quantities at seven\ndifferent simulation times. (A) Particle number density in ${\\rm\ncm^{-3}}$ as a function of radius; the bottom line corresponds to\n$z=19$, and moving upwards the ``steps'' from one line to the next are\nof $9\\times10^6$ yr, $3\\times10^5$ yr, $3\\times10^4$ yr, 3000 yr, 1500\nyr, and 200 yr, respectively; the uppermost line shows the simulation\nfinal state, at $z=18.181164$. The two lines between 0.01 and 200 pc\ngive the DM mass density (in ${\\rm GeV\\,cm^{-3}}$) at $z=19$ and the\nfinal time, respectively.  (B) Enclosed gas mass.  (C) Mass fractions of\n$H$ and H$_2$~\\index{HH}.  (D) Temperature. (E) Radial velocity of the baryons; the\nbottom line in (E) shows the negative value of the local speed of sound\nat the final time. In all panels the same output times correspond to the\nsame line styles. (from ABN02).}\n\\label{profiles3d}\n\\end{figure}\n\nThe use of AMR\\index{Numerical Simulations!Adaptive Mesh Refinement} codes for cosmological simulations of the formation of\nthe first objects in the universe was pioneered by Abel {\\it et al.~} (2000) and\nfurther refined by Abel {\\it et al.~} (2002; hereafter ABN02), where the dynamic\nrange covered by the simulations was larger than 5 orders of magnitude\nin scale length, {\\frenchspacing\\it i.e. } the wide range between an almost cosmological\n($\\lower.5ex\\hbox{\\gtsima} 100$ comoving kpc, {\\frenchspacing\\it i.e. } $\\lower.5ex\\hbox{\\gtsima} 5$ proper kpc) and an almost\nstellar scale ($\\lower.5ex\\hbox{\\ltsima} 1000\\;{\\rm AU} \\sim 10^{-2}\\;{\\rm pc}$). In the\nwhole process, the AMR\\index{Numerical Simulations!Adaptive Mesh Refinement} code kept introducing new (finer resolution)\nmeshes whenever the density exceeded some thresholds, or the Jeans\nlength\\index{Jeans Length} was resolved by less than 64 grid cells.\n\nThe simulations were started at a redshift\n$z=100$ from cosmologically consistent initial conditions\\footnote{\nSuch conditions were taken from an SCDM model with $\\Omega_\\Lambda=0$,\n$\\Omega_{\\rm m}=1$, $\\Omega_{\\rm b}=0.06$, $H_0=50\\;{\\rm\nkm\/s\\,Mpc^{-1}}$ which is quite different from the ``concordance''\n$\\Lambda$CDM model. However, the final results are believed to be only\nmarginally affected by differences in these cosmological\nparameters.}, and the code also followed the non-equilibrium chemistry\\index{Chemistry} of the\nprimordial gas, and included an optically thin treatment of radiative\nlosses from atomic and molecular lines, and from Compton cooling.\n\nThe main limitation of these simulations was the assumption that the\ncooling proceeds in the optically thin limit; such assumption breaks\ndown when the optical depth inside H$_2$~\\index{HH} lines reaches unity\n(corresponding to a Jeans length\\index{Jeans Length} $\\sim10^3\\;{\\rm AU}\\sim 0.01\\;{\\rm\npc}$). However, the simulations were halted only when the optical\ndepth at line centres becomes larger than 10 (Jeans length\\index{Jeans Length} of about\n$\\sim 10\\;{\\rm AU}$ ($\\sim 10^{-4}\\;{\\rm pc}$), since it was unclear\nwhether Doppler shifts could delay the transition to the optically\nthick regime.\n\n\\subsubsection{Summary of the evolution: radial profiles}\nIn Figures \\ref{halo_3d} and \\ref{profiles3d} we show the evolution of\ngas properties both in pictures and in plots of spherically averaged\nquantities, as presented in Abel {\\it et al.~} (2002).  From these figures,\nand in particular from the local minima in the infall velocity\n(Fig. \\ref{profiles3d}e) it is possible to identify four\ncharacteristic mass scales:\n\\begin{enumerate}\n\\item{The mass scale of the pre-galactic halo, of $\\sim\n7\\times10^5M_{\\odot}$ in total, consistent with the approximate\npredictions discussed in the previous sections}\n\\item{The mass scale of a ``primordial molecular cloud'', $\\sim\n4000M_{\\odot}$: the molecular fraction in this region is actually very low\n($\\lsim10^{-3}$), but it is enough to reduce the gas temperature from\nthe virial value ($\\gsim10^3\\;{\\rm K}$) to $\\sim 200\\;{\\rm K}$}\n\\item{The mass scale of a ``fragment'', $\\sim 100M_{\\odot}$, which is\ndetermined by the change in the H$_2$~ cooling\\index{HH cooling} properties at a density\n$n\\sim10^4\\;{\\rm cm^{-3}}$ (for a complete discussion, see Bromm {\\it et al.~}\n1999, 2002), when Local Thermal Equilibrium is reached and the cooling\nrate dependence on density flattens to $\\Lambda\\propto n$ (from\n$\\Lambda\\propto n^2$); this is also the first mass scale where the gas\nmass exceeds the Bonnor-Ebert mass (Ebert 1955, Bonnor 1956) $M_{\\rm\nBE}(T,n,\\mu)\\simeq 61M_{\\odot} T^{3\/2} n^{-1\/2} \\mu^{-2}$ (with $T$ in\nKelvin and $n$ in ${\\rm cm^{-3}}$), indicating an unstable collapse}\n\\item{The mass scale of the ``molecular core'', $\\sim 1M_{\\odot}$ is\ndetermined by the onset of 3-body reactions at densities in the range\n$n\\sim10^8-10^{10}\\;{\\rm cm^{-3}}$, which leads to the complete\nconversion of hydrogen\\index{Hydrogen} into molecular form; at this stage, the infall\nflow becomes supersonic (which is the precondition for the appearance\nof an accretion shock and of a central hydrostatic flow); the increase\nin the H$_2$~\\index{HH} abundance due to the formation of this molecular core also\nleads to the transition to the optically thick regime.}\n\\end{enumerate}\n\n\\subsubsection{Angular momentum}\n\n\\begin{figure}[t]\n\\epsfig{file=angmom3d.eps,width=11.5truecm}\n\\caption{Radial mass weighted averages of angular momentum\\index{Angular Momentum}-related\nquantities at different times (the same as in\nfig. \\ref{profiles3d}). (A) specific angular momentum\\index{Angular Momentum} $L$. (B)\nRotational speed in units of Keplerian velocity $v_{\\rm\nKep}\\equiv(GM_r\/r)^{1\/2}$. (C) Rotational speed ($L\/r$). (D)\nRotational speed in units of the sound speed $c_s$ (from ABN02).}\n\\label{angmom3d}\n\\end{figure}\n\nIn Fig. \\ref{angmom3d} we show the evolution of the radial distribution\nof average specific angular momentum\\index{Angular Momentum} (and related quantities).  It is\nremarkable that in ABN02 simulations rotational support does not halt\nthe collapse at any stage, even if this could be a natural expectation.\n\nThere are two reasons for such (apparently odd) fact:\n\\begin{enumerate}\n\\item{As can be seen in panel A of Fig. \\ref{angmom3d}, the collapse\nstarts with the central gas having much less specific angular momentum\\index{Angular Momentum}\nthan the average (this is typical of halos produced by gravitational\ncollapse; see {\\frenchspacing\\it e.g. } Quinn \\& Zurek 1988). That is, the gas in the central\nregions starts with little angular momentum\\index{Angular Momentum} to lose in the first place.}\n\\item{Second, some form of angular momentum\\index{Angular Momentum} transport is clearly active,\nas is demonstrated by the decrease in the central specific angular\nmomentum. Turbulence is likely to be the explanation: at any radius,\nthere will be both low and high angular momentum\\index{Angular Momentum} material, and\nredistribution will happen because of pressure forces or shock waves:\nlower angular momentum\\index{Angular Momentum} material will selectively sink inwards,\ndisplacing higher angular momentum\\index{Angular Momentum} gas. It is notable that this kind of\ntransport will be suppressed in situations in which the collapse occurs\non the dynamical time scale, rather than the longer cooling time scale:\nthis is the likely reason why this kind of mechanism has not been\nobserved in simulations of present day star formation (see {\\frenchspacing\\it e.g. }. Burkert \\&\nBodenheimer 1993).}\n\\end{enumerate}\n\n\n\\subsection{SPH\\index{Numerical Simulations!Smoothed Particle Hydrodynamics} results}\n\n\nBromm {\\it et al.~} (1999, 2002) have performed simulations of primordial\nstar formation using an SPH\\index{Numerical Simulations!Smoothed Particle Hydrodynamics} code which included essentially the same\nphysical processes as the simulations we just described. Apart from the\nnumerical technique, the main differences were that they included\ndeuterium in some of their models, and that their initial conditions\nwere not fully cosmological ({\\frenchspacing\\it e.g. }, since they choose to study isolated\nhalos, the angular momenta are assigned as initial conditions, rather\nthan generated by tidal interactions with neighbouring halo).\n\nIn Fig. \\ref{sphresults3d} and \\ref{sphresultsgas} we show the results\nof one of their simulations, which give results in essential agreement\nwith those we have previously discussed.\n\nAn interesting extra-result of this kind of simulation is that the\nauthors are able to assess the mass evolution of the various gaseous\nclumps in the hypothesis that feedback\\index{feedback} is unimportant (which could be the\ncase if the clumps directly form intermediate mass \\index{Black Hole}black holes without\nemitting much radiation, or if fragmentation\\index{fragmentation} to a quite small stellar\nmass scale happens). They find that a significant fraction ($\\sim 0.5$)\nof the halo gas should end up inside one of the gas clumps, although it\nis not clear at all whether this gas will form stars (or some other kind\nof object). Furthermore, they find that clumps are likely to increase\ntheir mass on a timescale of about $10^7\\;{\\rm yr}$ (roughly\ncorresponding to the initial time scale of the simulated halo), both\nbecause of gas accretion and of mergers, and they could easily reach\nmasses $\\lower.5ex\\hbox{\\gtsima} 10^4\\;M_{\\odot}$. Obviously, this result is heavily\ndependent on the not very realistic assumed lack of feedback\\index{feedback}.\n\n\\begin{figure}[t]\n\\epsfig{file=sphresults3d.eps,width=11.5truecm}\n\\caption{Typical result of the SPH\\index{Numerical Simulations!Smoothed Particle Hydrodynamics} simulations by Bromm {\\it et al.~} 2002. The\nfigure shows the morphology of a simulated $2\\times10^6\\;M_{\\odot}$ halo\nvirializing at $z\\sim30$ just after the formation of the first clump of\nmass $1400\\;M_{\\odot}$ (which is likely to produce stars). The two top row\npanels shows the distribution of the dark matter\\index{Dark Matter} (which is undergoing\nviolent relaxation). The two bottom panels show the distribution of gas,\nwhich has developed a lumpy morphology and settled at the centre of the\npotential well.}\n\\label{sphresults3d}\n\\end{figure}\n\n\\begin{figure}[t]\n\\epsfig{file=sphresultsgas.eps,width=11.5truecm}\n\\caption{Gas properties in the SPH\\index{Numerical Simulations!Smoothed Particle Hydrodynamics} simulations by Bromm {\\it et al.~} 2002. The\nfigure shows the properties of the particles shown in\nFig. \\ref{sphresults3d}.  Panel (a) shows the free electron abundance\n$x_e$ as a fraction of H number density. Panel (b) shows the H$_2$~\\index{HH}\nabundance $f_{\\rm H_2}$ (note that even the highest density particles\nhave $n_{\\rm H}\\lower.5ex\\hbox{\\ltsima} 10^8\\;{\\rm cm^3}$, so 3-body reactions are\nunimportant and $f_{\\rm H_2}\\lsim10^{-3}$). Panel (c) shows the gas\ntemperature (low density gas gets to high temperatures because H$_2$~\ncooling\\index{HH cooling}\nis inefficient). Panel (d) shows the value of the Jeans mass\\index{Jeans mass} which can\nbe obtained by using each particle temperature and density. All the\npanels have the hydrogen\\index{Hydrogen} number density $n_{rm H}$ on the X axis.}\n\\label{sphresultsgas}\n\\end{figure}\n\n\n\n\\section{Protostar formation and accretion}\n\n\\begin{figure}[p]\n\\epsfig{file=protostellar_collapse.ps,width=11truecm}\n\\caption{Proto-stellar collapse\\index{protostellar collapse}, formation of an\nhydrostatic core\\index{protostellar collapse!hydrostatic core} and\nstart of the proto-stellar accretion\\index{protostellar collapse!accretion} phase as can be found with 1-D\nsimulations. The four panel show the profiles of density (top left; a),\ninfall velocity (top right; b), temperature (bottom left;d) and H$_2$~\\index{HH}\nabundance (bottom right; d) as a function of enclosed mass at 10\ndifferent evolutionary stages (0=initial conditions; 9=final stage of\nthe computation). The most relevant phases are the rapid formation of\nH$_2$~ (2-3) and the formation of a shock on the surface of the hydrostatic\ncore (5-6-7), followed by the onset by accretion. Also note the almost\nperfect power-law behaviour of the density profile before core\nformation. (from Ripamonti {\\it et al.~} 2002).}\n\\label{protostellarcollapse}\n\\end{figure}\n\n\\begin{figure}[t]\n\\epsfig{file=kelvinhelmoltz.eps,width=11truecm}\n\\caption{Comparison of the accretion\\index{protostellar collapse!accretion} and the Kelvin-Helmoltz time scales\nfor a primordial protostars, as a function of protostellar mass. The\nKelvin-Helmoltz contraction time (obtained using the ZAMS luminosity as\ngiven by Schaerer 2002) is shown as the solid black line, while\nthe dashed line and the solid line with circles show the time which is\nneeded to accrete each mass of gas; they are based on the results of\nABN02 and differ slightly in the way in which they were obtained. The\ndotted lines mark three constant accretion rates of $10^{-2}$ (bottom line),\n$5\\times10^{-3}$ (middle line) and $10^{-3}\\;{\\rm M_{\\odot}\/yr}$ (top line).}\n\\label{kelvinhelmoltz}\n\\end{figure}\n\n\\begin{figure}[t]\n\\epsfig{file=omukaipalla_accretion.eps,width=11truecm}\n\\caption{Evolution of the protostellar radius as a function of\nprotostellar mass in the models of Omukai \\& Palla (2003). The models\ndiffer in the assumed accretion rates. The dotted curves show the\nresults of models where $\\dot{M}_{\\rm core}$ is assumed to be constant\nat values of $1.1\\times 10^{-3}\\;{\\rm M_{\\odot}\/yr}$ (dotted curve starting\nat $R_{\\rm core}\\simeq 30\\;R_\\odot$), $2.2\\times 10^{-3}\\;{\\rm\nM_{\\odot}\/yr}$ (starting at $R_{\\rm core}\\simeq 40\\;R_\\odot$), $4.4\\times\n10^{-3}\\;{\\rm M_{\\odot}\/yr}$ (``fiducial'' model; starting at $R_{\\rm\ncore}\\simeq 50\\;R_\\odot$) and $8.8\\times 10^{-3}\\;{\\rm M_{\\odot}\/yr}$\n(starting at $R_{\\rm core}\\simeq 65\\;R_\\odot$). The solid thick curve\nshows the evolution of a model in which the accretion rate was obtained\nfrom the extrapolation of the ABN02 data.}\n\\label{omukaipalla_accretion}\n\\end{figure}\n\nFull three-dimensional simulations\\index{Numerical Simulations!3D} (such as the ones of ABN02)\nare not able to reach the stage when a star is really formed. They are\nusually stopped at densities ($n\\sim10^{10}-10^{\\rm 11}\\;{\\rm cm^{-3}}$)\nwhich are much lower than typical stellar densities ($\\rho\\sim1\\; {\\rm\ng\\,cm^{-3}}$, $n\\sim10^{24}\\; {\\rm cm^{-3}}$).  In fact, at low\ndensities the gas is optically thin at all frequencies, and it is not\nnecessary to include radiative transfer in order to estimate the gas\ncooling. Instead, at densities $n\\lower.5ex\\hbox{\\gtsima} 10^{10}\\; {\\rm cm^{-3}}$ some of\nthe dominant H$_2$~\\index{HH} roto-vibrational lines become optically thick and\nrequire the treatment of radiative transfer.\n\nFor this kind of problem, this is a prohibitive computational burden,\nand at present the actual formation of a protostar can not be fully\ninvestigated through self-consistent 3-D simulations\\index{Numerical\nSimulations!3D}. In order to\nproceed further, it is necessary to introduce some kind of\nsimplification in the problem.\n\n\n\\subsection{Analytical results}\nHistorically, there were several studies based on analytical arguments\n({\\frenchspacing\\it i.e. } stability analysis\\index{fragmentation!Stability Analysis}) or single-zone models, leading to different\nconclusions about the properties of the final object.\n\nAn early example\nis the paper by Yoneyama (1972), in which it was argued that\nfragmentation\\index{fragmentation!Opacity Limit} takes place until the opacity limit is reached. However,\nYoneyama looked at the opacity limit of the entire ``cloud'' (roughly\ncorresponding to one of the $10^5-10^6\\;M_{\\odot}$ mini-halos we consider at\npresent; originally, this was mass the scale proposed by PD68)\nrather than the putative fragments and arrived at the conclusion that\nfragmentation stopped for masses $\\lower.5ex\\hbox{\\ltsima} 60M_{\\odot}$.\n\nMore recently, Palla {\\it et al.~} 1983 pointed to the increase of the cooling rate at\n$n\\lower.5ex\\hbox{\\gtsima} 10^8\\; {\\rm cm^{-3}}$ (due to H$_2$~\\index{HH} fast formation through\n3-body reactions) as a possible trigger for instability, leading to\nfragmentation\\index{fragmentation} on mass scales of $\\sim 0.1\\; M_{\\odot}$; such instability\nhas actually been observed in the simulations of Abel {\\it et al.~} (2003),\nbut it does not lead to fragmentation because its growth is too slow\n(see also Sabano \\& Yoshii 1977, Silk 1983, Omukai \\& Yoshii 2003 and\nRipamonti \\& Abel 2004 for analytical fragmentation\\index{fragmentation} criteria and their\napplication to this case).\n\n\\subsection{Mono-dimensional models}\n\n\\subsubsection{The formation of the protostellar core}\nA less simplified approach relies on 1D studies\\index{Numerical Simulations!1D}, such as those of Omukai\n\\& Nishi (1998) and Ripamonti {\\it et al.~} (2002): in such studies the heavy\nprice of assuming that the collapsing object always remains spherical\n(which makes impossible to investigate the effects of angular momentum\\index{Angular Momentum},\nand prevents fragmentation\\index{fragmentation}) is compensated by the ability to properly\ntreat all the other physical processes which are believed to be\nimportant: first of all, the detailed radiative transfer of radiation\nemitted in the numerous H$_2$~\\index{HH} resonance lines, then the transition to\ncontinuum cooling (at first from molecular Collision-Induced Emission,\nand later from atomic processes) and finally the non-ideal equation of\nstate which becomes necessary at high densities, when the gas becomes\nincreasingly hot and largely ionized.\n\nSuch studies find that the collapse initially proceeds in a\nself-similar\\index{protostellar collapse!self-similar phase} fashion (in good agreement with the solution found by Larson\n1969 and Penston 1969); at the start of this phase (stages 1 and 2 in\nfig. \\ref{protostellarcollapse}, to be compared with the profiles at\ncorresponding densities in fig. \\ref{profiles3d}) their results can be\ncompared with those of 3-D simulations\\index{Numerical Simulations!3D}, and the agreement is good\ndespite the difference in the assumed initial conditions: this is likely\ndue to the self-similar properties. At later stages the comparison is\nnot as good, but this is presumably due to the differences in the\ncooling rate (the 3D models at $n\\gsim10^{10}\\;{\\rm cm^{-3}}$ only include\nthe optically thin cooling rate).\n\nIt is noticeable that the transition to optically thick cooling does not\nstop the self-similar phase, contrary to theoretical expectations (see\n{\\frenchspacing\\it e.g. } Rees 1976). The reason is that the reduction in the cooling rate is\nsmooth, and is further delayed by the onset of CIE\\index{cooling!collision induced emission} cooling and by the\nstart of H$_2$~\\index{HH} dissociation, which acts as an heat well (each H$_2$~\ndissociation absorbs $4.48\\;{\\rm eV}$ of thermal energy) and provides an\nalternative ``disposal'' mechanism for the excess thermal energy.\n\nThe self-similar phase\\index{protostellar collapse!self-similar phase} proceeds until the central density is $n\\lower.5ex\\hbox{\\gtsima}\n10^{21}\\;{\\rm cm^{-3}}$ $\\sim10^{-3}\\;{\\rm g\\, cm^{-3}}$, when a small\n($M_{\\rm core,0}\\sim 3\\times 10^{-3}\\; M_{\\odot}$) hydrostatic\ncore\\index{protostellar collapse!hydrostatic core}\neventually forms. Such core starts accreting the surrounding\ngas at a very high rate (${\\dot{M}}_{\\rm core} \\sim 0.01-0.1\\; M_{\\odot}\\,\n{\\rm yr^{-1}}$), comparable to the theoretical predictions of Stahler\n{\\it et al.~} 1980 \n\\begin{equation}\n\\dot{M}_{\\rm core} \\sim c_s^3\/G \\simeq\n4 \\times 10^{-3} \\left({T\\over{1000\\;{\\rm K}}}\\right)^{3\/2}\\;\n\\left({\\mu\\over{1.27}}\\right)^{-3\/2}\\; M_{\\odot}\\,{\\rm yr^{-1}}\n\\end{equation}\nwhere $c_s=[k_{\\rm B}T\/(\\mu m_{\\rm H})]^{1\/2}$ is the isothermal sound\nspeed of a gas with mean molecular weight $\\mu$.\n\nAlthough the Omukai \\& Nishi (1998) and the Ripamonti {\\it et al.~}\n(2002) studies predict that such a high accretion rate will decline\n(approximately as $\\dot{M}_{\\rm core}\\propto t^{-0.3}$), it is clear\nthat, if the accretion proceeds unimpeded, it could lead to the\nformation of a star with a mass comparable to the whole proto-stellar\ncloud where the proto-star is born, that is, about $100M_{\\odot}$; such a\nprocess would take a quite short time of about $10^4-10^5\\;{\\rm\nyr}$.\n\nThis scenario could be deeply affected by feedback\\index{feedback} effects: even\nnot considering the radiation coming from the interior of the\nproto-star, the energy released by the shocked material accreting onto\nthe surface can reach very high values, comparable with the Eddington\nluminosity, and is likely to affect the accretion flow.\n\nFigure \\ref{kelvinhelmoltz}, compares the accretion\ntime scale (which can be extrapolated from the data at the end of the\nABN02 simulations) to the Kelvin-Helmoltz timescale,\n\\begin{equation}\nt_{\\rm KH} = {{GM^2}\\over{RL}}\n\\end{equation}\nwhere $L$ is the luminosity of the protostar. This plot tells us that\nthe accretion timescale is so fast that the stellar interior has very\nlittle space for a ``re-adjustment'' (which could possibly stop the\naccretion) before reaching a mass of $\\sim10-100M_{\\odot}$.\n\nHowever, it can be argued that such a readjustment is not necessary,\nsince in the first stages most of the protostellar luminosity comes from\nthe accretion process itself: \n\\begin{eqnarray}\n&L_{\\rm acc}\\simeq {{GM_{\\rm core}\\dot{M}_{\\rm core}}\\over R_{\\rm core}}\n\\nonumber \\\\ \\simeq\n8.5\\times10^{37}\n&\\left({M_{\\rm core}\\overM_{\\odot}}\\right)\n\\left({\\dot{M_{\\rm core}}\\over{0.01 M_{\\odot}\/yr}}\\right)\n\\left({R_{\\rm core}\\over{10^{12}\\;{\\rm cm}}}\\right)^{-1}\n\\;{\\rm erg\\,s^{-1}}\n\\end{eqnarray}\nwhere we have inserted realistic values for $M_{\\rm core}$, $\\dot{M}_{\\rm\ncore}$ and $R_{\\rm core}$.\nThis luminosity is quite close to the Eddington luminosity\n($L_{\\rm Edd}\\simeq 1.3\\times10^{38}$ $(M\/M_{\\odot}) \\;{\\rm erg\\,s^{-1}}$), and\nradiation pressure could have a major effect.\n\nRipamonti {\\it et al.~} (2002) found that this luminosity is not\nable to stop the accretion, but they do not properly trace the internal\nstructure of the core, so their results cannot be trusted except in the\nvery initial stages of accretion (say, when $M_{\\rm core}\\lsim0.1M_{\\odot}$).\n\nA better suited approach was used in studies by Stahler {\\it et al.~} (1986)\nand, more recently, by Omukai \\& Palla (2001, 2003), who assumed that\nthe accretion can be described as a sequence of steady-state accretion\nflows onto a growing core. The core is modelled hydrostatically, as a\nnormal stellar interior (including the treatment of nuclear burning),\nwhile the accreting envelope is modelled through the steady state\nassumption, in conjunction with the condition that outside the\n``photosphere'' (the region where the optical depth for a photon to\nescape the cloud is $\\lower.5ex\\hbox{\\gtsima} 1$) the accreting material is in\nfree-fall. As shown in Fig. \\ref{omukaipalla_accretion}, Omukai \\& Palla\n(2003) find that even if feedback\\index{feedback} luminosity deeply affects the\nstructure of the accreting envelope, it is never able to stop the\naccretion before the proto-stellar mass reaches $60-100\\;M_{\\odot}$ as a\nminimum; after that, the final mass of the protostar depends on the\nassumed mass accretion rate $\\dot{M}_{\\rm core}$: for $\\dot{M}_{\\rm\ncore}\\lower.5ex\\hbox{\\ltsima} 4\\times 10^{-3}\\;M_{\\odot}\\,{\\rm yr^{-1}}$, the accretion can\nproceed unimpeded until the gas is completely depleted (or the star\nexplodes as a supernova); otherwise, the radiation pressure is finally\nable to revert the accretion; with high accretion rates this happens\nsooner, and the final stellar mass is relatively low, while for\naccretion rates only slightly above the critical value of $\\sim 4\\times\n10^{-3}\\;M_{\\odot}\\,{\\rm yr^{-1}}$ the stellar mass can reach about\n$300\\;M_{\\odot}$.\n\nIf such predictions are correct, the primordial Initial Mass Function\\index{Initial Mass Function}\nis likely to be much different from the present one, reflecting mainly\nthe mass spectrum of the gas fragments from which the stars originate,\nand a mass of $\\sim 100\\;M_{\\odot}$ could be typical for a primordial\nstar. However, we note that this important results could change as a\nresult of better modeling. For example, deep modifications of the envelope\nstructure are quite possible if a frequency-dependent opacity (rather than the\nmean opacity used in the cited studies) is included.\n\n\n\n\\section{Discussion}\nThe previous sections show that, although not certain at all (because of\nthe big uncertainty about feedback\\index{feedback} effects), numerical simulations\\index{numerical simulation} tend\nto favour the hypothesis that primordial stars had a larger typical mass\nthan present-day stars.\n\nIf this is true, it could indeed solve some observational puzzle, such\nas why we have never observed a single zero-metallicity star (answer:\nthey have exploded as supernovae and\/or transformed into compact objects\nat a very high redshift), and maybe help explaining the relatively high\nmetallicities ($Z\\gsim10^{-4}Z_\\odot$ even in low column density\nsystems) measured in the Lyman $\\alpha$ forest (see {\\frenchspacing\\it e.g. } Cowie \\&\nSongaila 1998), or the high ($\\sim$ solar) metallicities we observe in\nthe spectra of some quasars already at redshift 6 (Fan {\\it et al.~} 2000, 2001,\n2003, 2004; Walter {\\it et al.~} 2003).\n\nHowever, a top-heavy primordial IMF also runs into a series of\nproblems, which we will discuss in the remaining of this section.\n\n\n\\subsection{UV Radiation feedback\\index{feedback}}\n\n\\begin{figure}[t]\n\\epsfig{file=sawtooth.eps,width=11truecm}\n\\caption{Average flux (in units of ${\\rm erg\\, cm^{-2}\\, s^{-1}\\,\nsr^{-1}\\, Hz^{-1}}$) at an observation redshift $z_{\\rm obs}=25$ coming\nfrom sources turning on at $z=35$.\nThe top panel shows the effects of absorption of neutral hydrogen\\index{Hydrogen} and\nhelium, strongly reducing the flux between $\\sim 13.6\\;{\\rm eV}$ and\n$\\sim 1\\;{\\rm keV}$ (the solid and dashed lines show the absorbed and\nunabsorbed flux,  respectively).\nThe bottom panel shows the same quantities in a much smaller energy\nrange, in order to illustrate the sawtooth modulation due to line\nabsorption below 13.6 eV. (from Haiman, Rees \\& Loeb 1997).}\n\\label{sawtooth}\n\\end{figure}\n\n\\begin{figure}[t]\n\\epsfig{file=primordialhii.eps,width=11truecm}\n\\caption{Dynamical evolution of the HII region produced by a\n$200\\,M_{\\odot}$ star. Panels refer to ionization fraction profiles (top\nleft), temperature distributions (top right), densities (bottom left)\nand velocities (bottom right).  The dashed line in the density panel is\nfor the Str\\\"omgren density (the density required to form a Str\\\"omgren\nsphere and therefore initially bind the ionization front) at a given\nradius. Profiles are given at times (from top to bottom in the density\npanel; from left to right in the others) 0 (density panel), 63 kyr (all\npanels), 82 kyr (ionization and temperature panels), 95 kyr (ionization\npanel) 126.9 kyr (all panels), 317 kyr (all panels), 1.1 Myr\n(temperature, density and velocity panels) and 2.2 Myr (all\npanels), which is the approximate main sequence lifetime of a\n$200\\;M_{\\odot}$ star. (From Whalen, Abel \\& Norman 2004).}\n\\label{primordialhii}\n\\end{figure}\n\nFirst of all, if a moderately massive star (even $M\\lower.5ex\\hbox{\\gtsima} 20-30M_{\\odot}$ is\nlikely to be enough) actually forms in the centre of an halo in the way\nshown by ABN02, it will emit copious amounts of UV radiation, which will\nproduce important feedback\\index{feedback} effects. Indeed, the scarcity of metals in\nstellar atmospheres (see Schaerer 2002, and also Tumlinson \\& Shull\n2000, Bromm Kudritzki \\& Loeb 2001) results in UV fluxes which are\nsignificantly larger than for stars of the same mass but with\n``normal'' metallicity. Furthermore, this same scarcity of metals is\nlikely to result in a negligibly small density of dust particles,\nfurther advancing the UV fluxes at large distances from the sources when\ncompared to the present-day situation.\n\nMassive primordial objects are also likely to end up into \\index{Black Hole}black holes\n(either directly or after having formed a star\\footnote{Here we think of\na star as an object where quasi-hydrostatic equilibrium is reached\nbecause of stable nuclear burning; according to this definition, a black\nhole can be formed without passing through a truly stellar phase,\neven if it is very likely to emit significant amounts of radiation in\nthe process.}), which could be the seeds of present day super-massive\n\\index{Black Hole}black holes (see {\\frenchspacing\\it e.g. } Volonteri, Haardt \\& Madau 2003). If accretion can\ntake place onto these \\index{Black Hole}black holes (also known as {\\it mini-quasars}),\nthey are likely to emit an important amount of radiation in the UV and\nthe X-rays (this could also be important in explaining the WMAP result\nabout Thomson scattering optical depth; see {\\frenchspacing\\it e.g. } Madau {\\it et al.~} 2003).\n\nUV photons can have a series of different effects. First of all,\nwe already mentioned in the section about chemistry\\index{chemistry} that Lyman-Werner\nphotons ($11.26\\;{\\rm eV}\\leq h\\nu \\leq 13.6\\;{\\rm eV}$) can dissociate\nH$_2$~\\index{HH} molecules, preventing further star formation. Such photons are\neffective even at large distances from their sources, and even in a\nneutral universe, since their energy is below the H ionization\\index{Hydrogen!ionization}\nthreshold; the only obstacle they find on their way are some of the\nLyman transitions of H, which can scatter these photons or remove them\nfrom the band; this results in a ``sawtooth'' modulation (see\nfig. \\ref{sawtooth}) of the spectrum.\nHaiman, Rees \\& Loeb (1997) argue that this negative feedback\\index{feedback} could\nconceivably bring primordial star formation to an early stop; however,\nother authors found that this is not\nnecessarily the case (Ciardi {\\it et al.~} 2000; Omukai 2001; Glover \\& Brand\n2001), or that the feedbac\\index{feedback}k effect on\nH$_2$~\\index{HH} abundance can be moderated by the effects of X-ray photons coming\nfrom mini-quasars (Haiman Abel and Rees 2000; Glover \\& Brand 2003).\n \nA second obvious effect from UV photons is to ionize the interstellar\nand the intergalactic medium; it is unclear whether these kind of\nobjects are important in the reionization history of the universe, but\nthey will definitely form HII regions in their immediate\nneighbourhoods. In fig. \\ref{primordialhii} we show the results of\nWhalen, Abel \\& Norman 2004 about the evolution of an HII region\nproduced by a $200M_{\\odot}$ star inside an halo with the same density\nprofile as found in the ABN02 paper: in the beginning the ionization\nfront is D-type (that is, it is ``trapped'' because of the density) and\nexpands because a shock forms in front of it; later (after about 70 kyr\nfrom the start of the UV emission) the ionization front reaches regions\nof lower densities and becomes R-type (radiation driven), expanding much\nfaster than the shock. However, the shock keeps moving at speeds of the\norder of $30\\;{\\rm km\\,s^{-1}}$ even after the source of ionizing\nphotons is turned off. Such a speed is much larger than the rotational\nvelocities of the mini-halos where primordial star formation is supposed\nto be taking place ($\\lower.5ex\\hbox{\\ltsima} 10\\;{\\rm km s^{-1}}$), so this shocks are\nlikely to expel a very large fraction of the original gas content of the\nmini-halo. UV emission could\neven lead to the {\\it photo-evaporation} of neighbouring\nmini-halos, similar to what Barkana \\& Loeb 1999 (see also Shapiro,\nIliev \\& Raga 2004) found in the slightly different context to\ncosmological reionization.\n\nSince the dynamical timescale of a mini-halo is $\\lower.5ex\\hbox{\\gtsima} 10^7\\;{\\rm yr}$\nand is longer than the timescale for this kind of phenomena (definitely\nshorter than the $\\sim1-10$ Myr main sequence lifespan of massive stars,\nand very likely to be of the order of $\\sim 10^4-10^5\\;{\\rm yr}$), the\nstar formation in one mini-halo is likely to stop almost immediately\nafter the first (massive) object forms. This means that, unless two or\nmore stars form exactly at the same time (within $\\sim1\\%$ of the\nmini-halo dynamical time), each mini-halo will form {\\it exactly one}\nmassive star.\n\n\\subsection{Supernovae\\index{supernovae} feedback and metallicities}\n\n\\begin{figure}[p]\n\\epsfig{file=starfate.eps,width=11truecm}\n\\caption{The ``final mass function'' of non-rotating metal-free stars as\na function of their initial mass, as given by from Heger \\& Woosley\n2002. The thick black curve gives the final mass of the collapsed\nremnant, and the thick gray curve and the mass of the star at the start\nof the event that produces that remnant (mass loss, SN etc.); for\nzero-metal stars this happens to be mostly coincident with the dotted\nline, corresponding to no mass loss. Below initial masses of $5-10\\;M_{\\odot}$\nwhite dwarfs are formed; above that, initial masses of up to\n$\\sim25\\;M_{\\odot}$ lead to neutron stars. At larger masses, \\index{Black Hole}black holes can\nform in different ways (through fall-back of SN ejecta or\ndirectly). Pair instability starts to appear above $\\sim100\\;M_{\\odot}$, and\nbecomes very important at $\\sim 140\\;M_{\\odot}$: stars with initial masses\nin the $140-260\\;M_{\\odot}$ range are believed to completely disrupt the\nstar, leaving no remnant; instead, above this range pair instability is\nbelieved to lead to a complete collapse of the star into a \\index{Black Hole}black hole.}\n\\label{starfate}\n\\end{figure}\n\nAfter producing plenty of UV photons during their life, primordial\nmassive stars are likely to explode as supernovae\\index{supernovae}. This must be true for\nsome fraction of them, otherwise it would be impossible to pollute the\nIGM with metals, as mass loss from zero-metal stars is believed to be\nunimportant (this applies to stars with $M\\lsim500\\;M_{\\odot}$; see Baraffe,\nHeger \\& Woosley 2001, and also Heger {\\it et al.~} 2002); however, it is clearly\npossible that some fraction of primordial stars directly evolve into black\nholes.\n\nIn figure \\ref{starfate} we show the results about the fate of\nzero metallicity stars, as obtained by Heger \\& Woosley 2002 (but see\nalso Heger {\\it et al.~} 2003), indirectly confirming this picture and\nsuggesting that pair-instability supernovae\\index{supernovae} could play a major role if\nthe primordial IMF is really skewed towards masses $\\lower.5ex\\hbox{\\gtsima}\n100\\;M_{\\odot}$.\n\nThis has a series of consequences. First of all, supernova\\index{supernovae} explosion are\none of the very few phenomena directly involving primordial stars which\nwe can realistically hope to observe. Wise \\& Abel (2004; see also Marri\n\\& Ferrara 1998 and Marri, Ferrara \\& Pozzetti 2000 for the effects of\ngravitational lensing) investigated the expected number of such\nsupernovae by means of a semi-analytic model combining Press-Schechter\nformalism\\index{Press-Schechter formalism}, an evolving minimum mass\\index{minimum mass} for star forming halos and negative\nfeedbacks, finding the rates shown in Fig. \\ref{primordialsn}; if such\nobjects are pair-instability supernovae with masses $\\gsim175\\;M_{\\odot}$,\nthey should be detectable by future missions such as JWST\\footnote{James\nWebb Space Telescope; more information at http:\/\/www.jwst.nasa.gov}; if\nsome of them are associated to Gamma Ray Bursts, the recently launched\n{\\it Swift}\\footnote{More information at http:\/\/swift.gsfc.nasa.gov}\nsatellite has a slim chance of observing them at a redshifts\n$\\lsim30$ (see Gou {\\it et al.~} 2004).\n\nSupernova\\index{supernovae} explosions obviously have hydrodynamical effects on the gas of\nthe mini-halo where they presumably formed, especially in the case of\nthe particularly violent pair-instability supernovae (see {\\frenchspacing\\it e.g. } Ober, El\nElid \\& Fricke 1983), and they are likely to expel it even if it was not\nremoved by the effects of UV radiation (see {\\frenchspacing\\it e.g. } MacLow \\& Ferrara 1999\nand Ferrara \\& Tolstoy 2000), further reducing the probability of having\nmore than 1 star per mini-halo. The similarity with UV effects extends\nto the influence on neighbouring halos which can be wiped out by SN\nexplosions (Sigward, Ferrara \\& Scannapieco 2004).\n\nFinally, supernovae provide a mechanism for spreading metals into the\nIGM, as discussed {\\frenchspacing\\it e.g. } by Madau, Ferrara \\& Rees 2001. In turn, these\nmetals will modify the cooling properties of the gas, and lead to star\nformation with a normal (Salpeter-like) IMF.\nIf this is true, and if pair-instability supernovae\\index{supernovae} dominate\nthe yields from primordial stars, we should be able to distinguish the\npeculiar pair-supernova abundance pattern (described in Heger \\& Woosley\n2001) as a ``signature'' of primordial origin.\n\nSearches for low-metallicity stars have a long history (see {\\frenchspacing\\it e.g. } Beers\n1999, 2000), and their inability to find stars with metallicities\n$Z\\leq10^{-4}Z_\\odot$ led to speculation that this metallicity marked\nthe transition to a normal IMF (see {\\frenchspacing\\it e.g. } Bromm {\\it et al.~} 2001, Schneider\n{\\it et al.~} 2002).  However, Christlieb {\\it et al.~} 2002 finally reported the\ndiscovery of an extremely metal poor star (HE0107-5240) with\n$[Fe\/H]=-5.3$\\footnote{$[Fe\/H] = \\log_{10}(N_{Fe}\/N_H) -\n\\log_{10}(N_{Fe}\/N_H)_\\odot$.}, a level compatible with a ``second\ngeneration'' star ({\\frenchspacing\\it i.e. }, a star formed from material enriched by very few\nsupernovae; for comparison, Wise \\& Abel 2004 find that primordial SNe\ncould enrich the IGM to $[Fe\/H]\\sim-4.1$, although this is probably an upper\nlimit). The abundance patterns in this star are quite strange (for\nexample, the carbon abundance is slightly less than 1\/10 solar), but a\nmuch better fit is provided by supernova\\index{supernovae} yields of moderately massive\nstars ($15-40\\;M_{\\odot}$) rather than from yields predicted for\npair-instability supernovae coming from very massive stars. However, at\nthe moment no model can satisfactorily fit all the observed abundances\n(see Bessel, Christlieb \\& Gustafsson 2004).\n\nEven if the primordial nature of this star must still be established, it\nrepresents a cautionary tale about the currently favoured predictions of\na large number of massive or very massive primordial stars, and a\nreminder that better theoretical modeling is still needed, with\nparticular regard to feedback effects during the stellar accretion phase.\n\n\\begin{figure}[t]\n\\epsfig{file=primordialsn.eps,width=11truecm}\n\\caption{Primordial supernova\\index{supernovae} properties as reported by Wise \\& Abel\n2004. The right panels show the differential rate of primordial\nsupernovae per unit redshift (top) and the cumulative rate (bottom);\nboth are per year of observation and per square degree. The left panels\nshow the critical halo mass (in $M_{\\odot}$) for primordial star formation\n(top), the comoving number density (in ${\\rm Mpc^{-3}}$) of halos above\nthe critical mass (middle) and the predicted specific intensity of the\nsoft UV background in the Lyman-Werner band (in ${\\rm erg\\; s^{-1}\\;\ncm^{-2}\\; Hz^{-1}\\; sr^{-1}}$, bottom). The three lines in each panel\nrefer to fixed primordial stellar masses of $100M_{\\odot}$ (solid),\n$200M_{\\odot}$ (dotted) and $500M_{\\odot}$ (dashed).}\n\\label{primordialsn}\n\\end{figure}\n\n\\section*{Acknowledgements}\nWe are grateful to the SIGRAV School for providing the opportunity to\nwrite this lecture notes. We thank M. Mapelli and M. Colpi\nfor assistance and useful comments about the manuscript. This work was\nstarted when both authors were at the Astronomy Department of\nPennsylvania State University. 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+{"text":"\\section{Introduction}\n\nDeviations from the purely poissonian shot noise (the so-called\n``full'' shot noise) in mesoscopic\ndevices and resonant tunneling structures have\nbeen the subject of growing interest in the last decade.\n\\cite{lesovik89,li90,yurkkoch90,buettiker90,buettiker92,chenting92,davihyld92,brown92,iannashot97,liu95,ciambrone95,lombardi97}\nThe main reason is that noise is a very sensitive probe of\nelectron-electron interaction, \\cite{landauer96} both due\nto the Pauli principle and to Coulomb force, and provides \ninformation not obtainable from DC and AC characterization; furthermore, \nnoise depends strongly on the details of device structure, so that\nthe capability of modeling it in nanoscale devices implies\nand requires a deep understanding of the collective transport mechanisms \nof electrons.\n\nAlmost all published theoretical and experimental studies \nhave focussed on the suppression of shot noise due to negative \ncorrelation between\ncurrent pulses caused by single electrons traversing the device.\nSuch correlation may be introduced by Pauli exclusion, which limits\nthe density of electrons in phase space, and\/or by Coulomb repulsion,\ndepending\non the details of the structure and on the dominant transport\nmechanism,\\cite{chenting92,davihyld92,brown92,iannashot97}\nand make the pulse distribution sub-poissonian, leading to suppressed\nshot noise.\n\nIn particular, for the case of resonant tunneling structures,\nseveral theoretical and experimental studies \nhave appeared in the literature,\n\\cite{li90,yurkkoch90,buettiker90,buettiker92,chenting92,davihyld92,brown92,iannashot97,liu95,ciambrone95,lombardi97}\nassessing\nthat the power spectral density\nof the noise current $S$ in such devices may be suppressed down to\nhalf the ``full'' shot noise value $S_{\\rm full} = 2 q I$ , i.e., that \nassociated to a purely poissonian process.\n\nIn this Letter, we propose a theoretical model and show experimental \nevidence of the opposite behavior, that is of enhanced shot noise \nwith respect to $S_{\\rm full}$, which is to be expected in\nresonant tunneling structures biased in the negative \ndifferential resistance region of the $I$-$V$ characteristic.\n\nWe shall show that in such condition Coulomb interaction\nand the shape of the \ndensity of states in the well \nintroduce positive correlation\nbetween consecutive current\npulses, leading to a super-poissonian pulse distribution, \nwhich implies a super-poissonian shot noise.\n\nFirst, we shall show an intuitive physical picture of the phenomenon,\nthen we shall express it in terms of a model for transport and noise in \ngeneric resonant tunneling structures presented elsewhere \n\\cite{iannashot97,ianna_unified95}. \nFurthermore, we shall show the experimental results,\nexhibiting a noise power spectral density almost 6.6 times greater than \n$S_{\\rm full}$, and compare it with the results\nprovided by a numerical implementation of our model.\n\nAs is well known, the typical I-V characteristic of a resonant tunneling diode \nis due to the shape of the density of states in the well,\nwhich consists of a series of narrow peaks in correspondence with the \nlongitudinal allowed energies in the well: for the\nGaAs\/Al$_{0.36}$Ga$_{0.64}$As material system considered here, \nthere is a single narrow peak.\nIn the negative differential resistance region of the \nI-V characteristic, the peak of the density of states is below the conduction\nband edge of the cathode: with increasing voltage, the density of states\nis moved downward, so that fewer states are available for\ntunneling from\nthe cathode, and the current decreases.\n\n\\begin{figure}\n\\epsfxsize = 8.cm\n\\epsffile{sketch.eps}\n\\vspace{0.2cm}\n\\caption{Enhanced shot noise is obtained because\nan electron tunneling into the well (a) from the \ncathode raises the potential energy of the well by an amount \n$q\/(C_1+C_2)$ so that more states are available for tunneling from the\ncathode (b)}\n\\end{figure}\n\nThe microscopic mechanism which allows for enhanced shot noise is the\nfollowing (see Fig. 1): an electron tunneling into the well from the \ncathode raises the potential energy of the well by an amount \n$q\/(C_1+C_2)$, where $q$\nis the electron charge, $C_1$ and $C_2$ the capacitances\nbetween the well region and either contacts;\nas a consequence, \nthe density of states in the well is shifted upwards \nby the same amount, with\nthe result that more states are available for successive tunneling\nevents from the\ncathode, and the probability per unit time that electrons\nenter the well increases. That means that electrons entering \nthe well are positively\ncorrelated, so that enhanced shot noise is to be expected.\n\nFor a more analytical derivation we can consider the structure \nas consisting of three regions \n$\\Omega_l$, $\\Omega_w$, and $\\Omega_r$, i.e.,\nthe left reservoir, the well region, and the right reservoir,\nrespectively, that are only weakly coupled through the two tunneling \nbarriers 1 and 2, as sketched in Fig. 1(a).\nIn addition, we suppose that electron transport is well described\nin terms of sequential tunneling (which is reasonable, \nexcept for the case of temperatures in the millikelvin range): an electron in\n$\\Omega_l$  traverses barrier 1, loses\nphase coherence and relaxes to a quasi-equilibrium energy distribution\nin the well region $\\Omega_w$, then traverses barrier 2 and \nleaves through $\\Omega_r$.\n\nSince confinement is realized only in one direction (that of MBE growth),\na state in $\\Omega_s$ ($s=l,r,w$) is characterized by its \nlongitudinal energy $E$,  its transverse wave vector ${\\bf k_T}$, \nand its spin $\\sigma$, and\ntunneling can be treated as a transition between\nlevels in different regions \\cite{bardeen61} in which $E$, ${\\bf k_T}$ \nand $\\sigma$ are conserved.\n\nFollowing Davies {\\em et al.} \\cite{davihyld92}, we introduce\n``generation'' and ``recombination'' rates through both\nbarriers: \\cite{iannashot97} the generation rate $g_1$ is the transition\nrate from $\\Omega_l$ to $\\Omega_w$, i.e., the sum of the\nprobabilities per unit time of having a transition from $\\Omega_l$\nto $\\Omega_w$ given by the Fermi ``golden rule'' over all pairs of\noccupied states in $\\Omega_l$ and\nempty states in $\\Omega_w$. Analogously, we define $r_1$, \nthe recombination rate through barrier 1 (from\n$\\Omega_w$ to $\\Omega_l$), $g_2$ and $r_2$, generation and recombination\nrates through barrier 2.\n\nSince negative differential resistance is obtained at\nhigh bias, when the electron flux is one-directional, $r_1$ and\n$g_2$ can be discarded, while $g_1$ and $r_2$ are:\n\\begin{eqnarray}\ng_1 & = &  2 \\frac{2 \\pi}{\\hbar} \\int  dE \\;|M_{1lw}(E)|^2 \\rho_l(E)\n\t\t\\rho_w(E) \\times \\nonumber \\\\\n\t& & \t\\int  d{\\bf k_T} \\rho_T({\\bf k_T}) \n\t\tf_l(E,{\\bf k_T}) (1 - f_w(E,{\\bf k_T})) \n,\\label{g1} \\\\\nr_2 & = & 2 \\frac{2 \\pi}{\\hbar} \\int dE \\;|M_{2rw}(E)|^2 \\rho_r(E)\n\t\t\\rho_w(E) \\times \\nonumber \\\\\n\t& & \\int  d{\\bf k_T} \\rho_T({\\bf k_T}) \n\t\tf_w(E,{\\bf k_T}) (1 - f_r(E,{\\bf k_T}))\n.\\label{r2}\n\\end{eqnarray}\nwhere $\\rho_s$, $f_s$, ($s=l,w,r$), are the longitudinal\ndensity of states and the equilibrium occupation factor\nin $\\Omega_s$ (dependent on the quasi Fermi level $E_{fs}$), \nrespectively, and \n$\\rho_T$ is the density of transverse states;\n$ M_{1lw}(E)$ is the matrix \nelement for a transition through barrier 1 between \nstates of longitudinal energy\n$E$: it is obtained in Ref. \\cite{ianna_unified95} as\n$| M_{1lw}(E)|^2 = \\hbar^2 \\nu_l(E) \\nu_w(E) T_1(E)$, where $\\nu_s$ ($s=l,w,r$)\nis the so-called attempt frequency in $\\Omega_s$ and $T_1$ is the \ntunneling probability\nof barrier 1; $M_{2rw}(E)$ is analogously defined.\n\nAs is well known, in the negative resistance\nregion of the $I$-$V$ characteristic, the peak of $\\rho_w$ is below the conduction \nband edge of the left electrode.\nIn such a way, as the voltage is increased, the number of allowed \nstates for a transition from\n$\\Omega_l$ to $\\Omega_w$ is reduced, hence the current decreases.\nSince all electrons relax to lower energy\nstates once they are in the well, it is reasonable to\nassume that $\\Omega_w$-states\nwith longitudinal energies above the conduction band edge\nof the left electrode $E_{\\rm cbl}$ are empty, i.e.,\ncorrespond to a zero\noccupation factor $f_w$ (analogously, $f_r =0$). \nIn addition, if we discard size\neffect in the cathode, we have that $2 \\pi \\hbar \\rho_l \\nu_l = 1$ \nif $E > E_{\\rm cbl}$. Therefore we can rewrite $g_1$ and $r_2$ as\n\\begin{eqnarray}\ng_1 & = & 2 \\int_{E_{\\rm cbl}}^\\infty dE\n      \\, \\nu_w(E) \\, \\rho_w(E) T_1(E) F_l(E),  \n\\label{g1a} \\\\\nr_2 & = & 2 \\int_{E_{\\rm cbw}}^\\infty dE\n      \\, \\nu_w(E) \\, \\rho_w(E) T_2(E) F_w(E)\n,\\label{r2a}\n\\end{eqnarray}\nwhere $F_s(E)$ is the occupation factor of $\\Omega_s$, ($s=l,w,r$)\nintegrated over the transverse wave vectors\n$F_s(E) \\equiv \\int d{\\bf k_T} \\, \\rho_T({\\bf k_T}) \\, f_s(E, {\\bf k_T})$, \nand $E_{\\rm cbw}$ is the bottom of the conduction band edge in the well.\n\nLet us point out that $g_1$ and $r_2$ depend on the number of\nelectrons $N$ in the well region both through the potential energy\nprofile, which is affected by the charge in $\\Omega_w$ through\nthe Poisson equation, and through the term $F_w$ in (\\ref{r2a})\nwhich depends on $N$ through the quasi-Fermi level $E_{fw}$.\nIt is worth noticing that in our case Pauli exclusion has no\neffect, since practically all possible final states are unoccupied.\nFollowing these considerations, $g_1$ and $r_2$ can be\nobtained as a function of $N$, at a given bias voltage $V$.\n\nThe steady state value $\\tilde{N}$ of $N$ satisfies charge\nconservation in the well, i.e., $g_1(\\tilde{N}) = r_2(\\tilde{N})$,\nand the steady state current is $I =\nq g_1(\\tilde{N}) = q r_2(\\tilde{N})$.\n\nFollowing Ref. \\onlinecite{iannashot97}, it is worth\nexpanding $g_1(N)$ and $r_2(N)$ around $\\tilde{N}$ and\ndefining the following characteristic times:\n\\begin{equation}\n\\frac{1}{\\tau_{g}} \\equiv - \\left. \\frac{dg_1}{dN}\n                        \\right|_{N = \\tilde{N}} \n,\\hspace{1cm}\n\\frac{1}{\\tau_{r}} \\equiv \\left. \\frac{dr_2}{dN} \n                        \\right|_{N = \\tilde{N}} \\label{taus}\n;\\end{equation}\n\nOur parameter of choice for studying deviations from full\nshot noise is the so-called Fano factor $\\gamma$, the ratio of the \npower spectral density of the current noise $S(\\omega)$ to the full shot value\n$2 q I $. From \\onlinecite{iannashot97} we have, in this case,\nfor $\\omega \\tau_g \\tau_r \\ll \\tau_g + \\tau_r$,\n\\begin{equation}\n\\gamma = \\frac{S(\\omega)}{2 q I} =\n         1 - \\frac{2 \\tau_g \\tau_r}{(\\tau_g + \\tau_r)^2}\n.\\label{gamma}\n\\end{equation}\n         \nFrom the definition (\\ref{taus}), $\\tau_{g}$ is positive in\nthe first region of the I-V characteristic, when Pauli principle and\nCoulomb interaction make $g_1$ decreasing with increasing $N$.\nOn the other hand, in the negative differential resistance\nregion, the term which varies the most with increasing N is the\nlongitudinal density of states, which shifts upwards by a factor $q\/(C1+C2)$\nper electron: since the peak is just below $E_{\\rm cbl}$, a\nslight shift of the peak sensibly increases the integrand\nin (\\ref{g1a}), yielding a negative $\\tau_{g}$. \nNote that, while from (\\ref{gamma}) we see\nthat noise could also diverge if $\\tau_{g} = - \\tau_{r}$, \nthis cannot physically happen, because the large\ndeviation of $N$ with respect to $\\tilde{N}$ would make\nthe linearization of $g_1$ and $r_2$ not acceptable.\n\nWe now focus on a particular structure, on which we have\nperformed noise measurements and numerical simulations\nfollowing the theory just described. \nSuch structure has been  fabricated at\nthe TASC-INFM laboratory in Trie\\-ste\nand has the following layer structure:\na Si-doped ($N_d = 1.4 \\times 10^{18}$ cm$^{-3}$)\n500 nm-thick GaAs buffer layer, an undoped 20~nm-thick\nGaAs spacer layer to prevent silicon diffusion into the\nbarrier, an undoped 12.4~nm-thick AlGaAs first barrier,\nan undoped 6.2~nm-thick GaAs quantum well, an undoped\n14.1~nm-thick AlGaAs barrier, a 10~nm \nGaAs spacer layer and a Si-doped 500~nm-thick\ncap layer. The aluminum mole fraction\nin both barriers is $0.36$ and the diameter of the mesa defining\nthe single device is about $50$~$\\mu$m.\n\nThe barriers in our samples are\nthicker than in most similar resonant-tunneling diodes, \nfor the purpose of  reducing the\ncurrent and, consequently, to increase the differential resistance,\nin order to\nobtain the best possible noise match with the measurement amplifiers\n(available ultra-low-noise amplifiers offer a good performance, with a very\nsmall noise figure, for a range of resistance values between a few\nkiloohms and several megaohms).\n\nWe have applied a measurement technique purposedly developed for low-level\ncurrent noise measurements, based on the careful evaluation of the\ntransimpedance between the device under test and the output of the\namplifier. \\cite{macupell91}\nThis procedure allows us to measure noise levels\nthat are up to 3~dB below that of the available amplifiers with a maximum\nerror around 10\\%.\n\nOur usual approach \\cite{macupell91} includes also the\nsubtraction of the noise due to the amplifier and other\nspurious sources, which is evaluated using a substitution\nimpedance, equivalent to that of the device under test with known\nnoise behaviour.\nFor the measurements in the negative differential resistance region,\ninstead, we have evaluated an upper limit for the noise\ncontribution from the amplifier in these particular operating\nconditions, since it is difficult to sinthesize an appropriate\nsubstitution impedance. From experimental and theoretical\nconsiderations, we have verified that such limit is always below\n3~\\% of the noise level from the device under test, so that\ncorrections are not necessary.\nIn Fig. 2 the measured current and the Fano factor $\\gamma$\nat the temperature of liquid nitrogen (77~K) are\nplotted as a function of the applied voltage (the thicker barrier\nis on the anode side).\n\n\\begin{figure}\n\\epsfxsize 8.25cm\n\\epsffile{speri.eps}\n\\vspace{0.2cm}\n\\caption{Experimental current (solid line) and Fano factor $\\gamma$ \n(squares) as a function\nof the applied voltage, at the temperature of 77~K. \nThe maximum value of $\\gamma$ is 6.6, \nwhile the minimum is close to 0.5}\n\\end{figure}\n\nIt can be noticed that as the voltage increases, the Fano factor\ndecreases down to about 0.5 (which corresponds to the maximum\ntheoretical suppression \\cite{iannashot97}), at the voltage\ncorresponding to the current peak is exactly one, then increases\nagain and reaches a peak of 6.6 at the voltage corresponding\nto the lowest modulus of the\nnegative differential resistance, while, for higher voltages, \nit rapidly approaches one.\n\nIn Fig. 3 we show numerical results for the same structure at 77~K\nbased on the theory discussed before and obtained by considering a \nrelaxation length of 15~nm.\\cite{ianna_unified95}\nAs can be seen, there is an\nalmost quantitative agreement between theory and experiment\n( the peak experimental current is 45 ~nA which corresponds\nto a current density of 23~A\/m$^2$): we ascribe most of the difference\nto the tolerance in the nominal device parameters.\nAll the relevant features of the Fano factor as a function\nof the applied voltage are reproduced, and can be easily \nexplained in terms of our model.\n\n\\begin{figure}\n\\epsfxsize 8.25cm\n\\epsffile{calc.eps}\n\\vspace{0.2cm}\n\\caption{Calculated current density (solid line) and Fano factor $\\gamma$\n(squares) as a function of the applied voltage for the considered structure}\n\\end{figure}\n\nThe fact that $\\gamma$ is maximum at the voltage corresponding\nto the minimum negative differential resistance $r_d$ of the\ndevice is readily justified once we recognize that $r_d$ is\npractically proportional to $\\tau_g + \\tau_r$ \\cite{iannadopo}.\nIn fact, from Fig. 4 it is clear that $\\tau_r$ varies much more smoothly\nthan $\\tau_g$ with the applied voltage, so that,\nsince $\\tau_g$ and $\\tau_r$ have opposite sign, \nthe modulus of $r_d$ is minimum\nwhen $\\tau_g\/\\tau_r$ approaches $-1$. At this point, we simply notice,\nfrom (\\ref{gamma}), that $\\gamma$ gets larger as $\\tau_g\/\\tau_r$ \napproaches $-1$, too (and would eventually diverge for $\\tau_g = - \\tau_r$).\n\nFurthermore, from Figs. 3 and 4, and according to \\cite{iannadopo},\nwe can notice that $r_d$ and $\\tau_g$ tend to infinity at about\nthe same voltage, i.e., the one corresponding to the current peak.\nTherefore, according to (\\ref{gamma}), $\\gamma$ is 1 at the \ncurrent peak bias, as can be verified from both experiments \nand calculations (Figs. 2 and 3, respectively).\n\n\\begin{figure}\n\\epsfxsize 8.25cm\n\\epsffile{taus.eps}\n\\vspace{0.2cm}\n\\caption{Calculated $1\/\\tau_g$ (solid) and $1\/\\tau_r$ (dashed) \nas a function of the\napplied voltage}\n\\end{figure}\n\nIn conclusion, we have demonstrated experimentally that\nCoulomb interaction, enhanced by the shape of the density\nof states in the well, can lead to a \ndramatic increase of shot noise in\nresonant tunneling diodes biased in the negative differential\nresistance region of the I-V characteristic. We have provided a model \nwhich leads to good numerical agreement with the\nexperimental data, taking into account all the\nrelevant physics involved in the phenomenon.\n\n\nThis work has been supported by the Ministry for the University\nand Scientific and Technological Research of Italy, \nand by the Italian National Research Council (CNR).\n            \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe origin of turbulence and enhanced angular momentum transport in accretion flows is a fascinating problem of considerable importance in astrophysics. It is commonly believed that the magneto-rotational instability (MRI), in one form or another,  plays a fundamental role in destabilizing the basic quasi-Keplerian flow.  When a net magnetic flux is present the MRI sets in as a classical linear instability with a well defined growth rate and characteristic wavenumber \\citep{Balbus91}; the turbulence then develops from the nonlinear evolution of this instability. When there is no net flux the problem is more complicated and the turbulence must develop from a nonlinear subcritical instability. In this case, the problem becomes fundamentally one  of establishing what form of dynamo action can be sustained in a disc. Much of what is currently known about dynamo action in accretion flows is based on  numerical studies formulated within the framework of the shearing-box approximation \\citep{Hawley95}. The simplest set up, both conceptually and numerically, consists of an unstratified, isothermal shearing-box with periodic boundary conditions in the vertical direction. It is now well established that this configuration suffers from the so-called convergence problem.  As the magnetic diffusivity decreases, or equivalently the resolution increases, the Maxwell stresses decrease, eventually to become negligible  \\citep[][see however Fromang 2010 for a different view]{Fromang07, Pessah07, Guan09, Simon09, Bodo11}. The cause for this  ``non convergence\" has been attributed to the the lack of a characteristic outer scale in the periodic, unstratified problem \\citep[for a discussion see][]{Bodo11}. The next step towards more realistic simulations is to retain the shearing-box geometry but with the inclusion  of vertical gravity, and consequently, stratification. This introduces a characteristic length--the scale height--that may help to remedy the convergence problem \\citep{Davis10, Shi10, Oishi11}. Whether this is the case or not, at the moment, remains an open question. Certainly, in the stratified cases the solutions manifest a richness both in space and time  that is absent in the unstratified cases \\citep{Gressel10, Guan11, Simon12}. It is important to note that most of these studies adopt an isothermal equation of state; the resulting density distribution is correspondingly close to a Gaussian with most of the mass concentrated near the mid-plane, and tenuous, low density regions above and below. This leads to very different dynamo processes operating in the mid-plane and in the overlying regions.  Although an isothermal formulation is conceptually simple and easy to implement numerically, it neglects the possibly important process of turbulent heating by viscous and Ohmic dissipation. It can be argued that in an optically thin environment turbulent heating may not be important since the energy can easily escape without substantially heating the ambient plasma. However, this is definitely not the case in an optically thick environment. In this case the plasma will be heated locally and the final thermal structure will be determined by a balance between energy deposition and energy transport. In this case, it is possible that substantial departures may develop from the isothermal case that, in turn may impact the operation of the dynamo. \nSome of these issues have been addressed  by Hirose and collaborators  \\citep{ Hirose06, Hirose09, Blaes11},  who have considered radiation dominated discs and have included a sophisticated treatment of the radiation field, and also by the works of Flaig and collaborators \\citep{Flaig10, Flaig12} whose models of proto-planetary discs include partial ionization, chemical networks and heat transport in the radiative conduction approximation. All these works indicate that turbulent heating can indeed be important. Here, we also address the problem of turbulent heating but in the somewhat simpler case of a fully ionized, pressure dominated disc. Our intention is to provide a bridge between the works of Hirose et al. and Flaig et al. and those based on the isothermal equation of state. To this end we consider a stratified shearing-box with a perfect gas equation of state and finite (constant) thermal diffusivity. The objective is to study how the basic state and the corresponding dynamo action changes as the thermal diffusivity is varied. In this work we  deliberately keep the formulation as simple as possible in order to highlight some of the  basic underlying physical processes. \n\n\\section{Formulation} \\label{formulation} \nOur objective is to provide a simple model in which the effects of dissipative heating can be studied. In particular we want to assess how these processes, together with thermal transport lead to departures from the more familiar isothermal cases. We assume that the plasma is optically thick and approximate the radiative transport by a diffusion process which we model by a thermal conduction term in the energy equation. In the spirit of keeping things as simple as possible, and in order to capture  more easily the general properties of the solutions, we make  further simplifications, by neglecting the dependencies on density and temperature resulting from the diffusion approximation to the radiative transport equation, and assuming a constant thermal diffusivity. A more realistic treatment of the radiation will be considered in future work.\n\nWe perform three-dimensional, numerical simulations of a perfect gas with thermal conduction in a shearing box with vertical gravity.  A detailed presentation of the shearing box approximation can be found in \\citet{Hawley95}. The Magneto-Hydro-Dynamics (MHD) shearing-box equations, including  vertical gravity  and thermal conduction can be written as:\n\n\\begin{equation}\n\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot \\left( \\rho {\\mathbf v} \\right) = 0,\n\\label{eq:mass}\n\\end{equation}\n\n\\begin{equation}\n\\frac{\\partial {\\mathbf v}}{\\partial t} + {\\mathbf v} \\cdot \\nabla {\\mathbf v} + 2 \\Omega \\times {\\mathbf v} = \\frac{{\\mathbf B} \\cdot \\nabla {\\mathbf B}}{4 \\pi \\rho} - \\frac{1}{\\rho} \\nabla \n\\left(  \\frac{{\\mathbf B}^2}{8 \\pi} + P \\right) - \\nabla \\left( 2 A \\Omega x^2  + \\frac{1}{2} \\Omega^2 z^2 \\right),\n\\label{eq:momentum}\n\\end{equation}\n\n\\begin{equation}\n\\frac{\\partial {\\mathbf B}}{\\partial t} - \\nabla \\times \\left( {\\mathbf v} \\times {\\mathbf B} \\right) = 0,\n\\label{eq:induction}\n\\end{equation}\n\n\\begin{equation}\n\\frac{\\partial E}{\\partial t} + \\nabla \\cdot [(E + P_T) {\\mathbf v} + ({\\mathbf v} \\cdot {\\mathbf B}) {\\mathbf B} - k \\nabla T] = 0 ,\n\\label{eq:energy}\n\\end{equation}\nwhere ${\\mathbf B}$, ${\\mathbf v}$, $\\rho$ and $P$ denote, respectively,   the magnetic  field intensity, the velocity, the density and the thermal pressure; $E$ is the total energy density, $P_T$ is the total (thermal plus magnetic) pressure and $k$ is the thermal conductivity. The local angular velocity ${\\mathbf \\Omega} = \\Omega {\\mathbf e_z}$ and the shear rate \n \\begin{equation}\nA \\equiv \\frac{R}{2} \\frac{\\partial \\Omega}{\\partial R}\n\\end{equation}\nare assumed constant.  For a Keplerian disk  $A = -(3\/4) \\Omega$.   The system is closed by the equation of state for a perfect gas:  \n\\begin{equation}\nP = \\rho T\n\\end{equation}\nwhere we have absorbed the perfect gas constant in the definition of the temperature.\nThe thermal conductivity can be written as\n\\begin{equation}\nk = \\frac{5}{2}  \\kappa \\rho\n\\end{equation}\nwhere $\\kappa$ is the thermal diffusivity, which, as discussed above, we assume  to be constant, and the factor of $5\/2$ is appropriate for a gas with three degrees of freedom.\n   \n\nWe start our simulations from a state with a uniform shear flow,  $\\vec{v} = -2 A x\\hat{\\mathbf{e}}_y$,  and  density and pressure distributions that satisfy  vertical hydrostatic balance with constant temperature  $T_0$. With these conditions the initial density has a Gaussian profile given by\n\\begin{equation}\n\\rho = \\rho_0 \\exp(- \\Omega ^2 z^2 \/ 2 T_0),  \\\n\\end{equation}\nwhere $\\rho_0$ is the value of density on the equatorial plane.\n\nIf the MRI develops to substantial amplitude, this initial state will be driven away from thermal equilibrium by the energy input from dissipative processes. The temperature in the equatorial regions  will  progressively increase and a thermal gradient will be established until a new equilibrium is reached whereby the energy input is  balanced by thermal losses at the upper and lower boundaries. As we shall see, the new equilibrium can be quite different from the initial isothermal state and is determined self-consistently by the heating associated with the process of angular momentum transport by the MRI. We note here that, in our current formulation, we do not include viscous and Ohmic dissipation explicitly. The heating of the fluid occurs because of numerical dissipation together with a conservative formulation of the total energy equation. The latter requires that whatever kinetic or magnetic energy is lost by dissipative processes  it be re-introduced in the form of internal energy (heating). \n\nThe computational domain covers the region $L_x \\times L_y \\times L_z$, where $L_x = H$, $L_y = \\pi H$ and $L_z = 6H$,\nwhere\n\\begin{equation}\nH = \\frac{\\sqrt{2 T_0}}{\\Omega}\n\\end{equation}\nis the pressure scale height in the initial isothermal state. In the vertical direction the box is symmetric with respect to the equatorial plane $z = 0$,  where gravity changes sign. Numerically, the domain is covered by a grid of $32 \\times 96 \\times 192$ grid points.  \nA  magnetic field of the form\n\\begin{equation}\n{\\mathbf B} = B_0 \\sin \\left( \\frac{2 \\pi x}{H} \\right) \\hat{\\mathbf{e}}_z\n\\end{equation}\nis imposed initially\nwhere $B_0$ corresponds to the ratio between thermal and magnetic pressure and has a value of $1600$. Clearly, there is no net magnetic flux threading the box.  In addition we introduce random noise in the $y$ component of the velocity in order to destabilize the system. \n\nFollowing common practice, we assume periodic  boundary conditions in the $y$ direction and shear periodic conditions in the $x$ direction.  In the vertical direction,  we assume that  the upper and lower boundaries ($z = \\pm 3H$), are impenetrable and stress free, giving $v_z = 0$ , $\\partial v_x \/ \\partial z = \\partial v_y \/ \\partial z = 0$, and also that the magnetic field is purely vertical, giving  $\\partial B_z \/ \\partial z = 0$, $B_x = B_y = 0$. We should note that these conditions allow a net flux of magnetic helicity through the boundaries with, possibly, important consequences  to the dynamo processes  \\citep{VC01, Kapyla10}.\nFinally, we assume that the boundaries are in hydrostatic balance, and that the temperature is constant and equal to $T_0$; thus\n\\begin{equation}\n\\frac {\\partial p_T}{\\partial z} = \\mp  3 \\rho \\Omega^2 H, \\qquad \\qquad T=T_0.\n\\end{equation}\nAll  simulations are carried out with the PLUTO code \\citep{Mignone07}, with a second order accurate scheme, HLLD Riemann solver and an explicit treatment of thermal conduction.   \n\n\n\n\\section{Results} \\label{results} \nWe now describe the development of MRI driven turbulence from an initially isothermal state. Hereinafter, and unless otherwise specified, when presenting the numerical results, we adopt $\\Omega^{-1}$ as the unit of time, $H$ as the unit of length, and the mid-plane density in the initial isothermal state $\\rho_0$, as the unit of density and,  since  $H$ is  our unit of length, we have $T_0=1\/2$. \nFollowing the initial perturbations, a sub-critical instability sets in leading to the generation of magnetic fields and the development of turbulence. Dissipative processes heat the plasma driving the system away from the initial isothermal state. Eventually the system reaches a stationary state in which the heat generated by the turbulence is balanced by the heat lost through the upper and lower boundaries. \nLocally, the balance is between the volumetric heat production and the divergence of the heat flux that can arise both by thermal conduction and turbulent transport.  The relative importance of these two processes depends on the value of the thermal diffusivity, which, here, is expressed in units of the product of the scale-height and the isothermal sound speed, i.e. it has the form of  \nan inverse P\\'eclet number. A typical evolution for a case with $\\kappa = 2 \\times 10^{-2}$ can be followed in Fig. \\ref{fig:maxwtime} where we show the time history of the Maxwell stresses averaged over the entire computational domain. Clearly, there is a long adjustment phase lasting approximately 500 time-units after which the system settles into a stationary state in which the stresses remain strongly fluctuating but with a well defined (time) average value. The corresponding thermal history can be assessed by inspection of Figs. \\ref{fig:avt_time} and \\ref{fig:avrho_time}, showing respectively, the horizontally averaged temperature, $\\tilde T(z)$ and density $\\tilde \\rho(z)$, at several times. The asymptotic profiles in the stationary state (obtained by time averaging from $t = 500$ to the end of the simulation, $t = 2000$) are denoted by angle brackets. Clearly the increase in the Maxwell stresses are accompanied by the  heating of the central regions leading to the establishment of a nearly parabolic profile in temperature. We note a corresponding dramatic change in the density distribution that evolves from the initial Gaussian profile to an almost constant distribution at later times. \n\nThe development of a constant density state is somewhat remarkable, and deserves further investigation. At first sight it may appear as the result of a fortuitous choice of thermal diffusivity. As we shall see presently, this is not entirely the case. \nIn the stationary state the average temperature and density are related by the condition of hydrostatic balance, which we write here in dimensional form, and for simplicity we only consider  $z > 0$;\n\\begin{equation}\n\\frac{1}{\\rho} \\frac{d \\rho}{d z} = \\frac{1}{T} \\left( -\\Omega^2 z - \\frac{d T}{d z} \\right) .\n\\label{eq:hydeq}\n\\end{equation}\nClearly, whether the density decreases upwards, increases upwards, or remains constant depends on the relative magnitude of the two terms in the brackets on the RHS of (\\ref{eq:hydeq}).  The first term is a fixed linear function of $z$. The second--the temperature gradient--is negative since the layer is heated from within, but its magnitude depends on a balance between local heat production rate and local heat transport. \nTo a first approximation, one could assume that the energy production rate should be independent of the thermal diffusivity $\\kappa$. This is not unreasonable, since the production rate is driven by turbulent dissipation, which in turn depends solely on the efficiency of the MRI. This being the case, the magnitude of the temperature gradient could be made arbitrarily small by choosing a large value of $\\kappa$. Clearly, if the thermal diffusivity is huge, thermal conduction can easily transport all the generated heat along  very shallow gradients. The temperature will be nearly constant, the density will rapidly decrease upward in accordance to (\\ref{eq:hydeq}), and resemble the isothermal distribution. By contrast, if $\\kappa$ is tiny the temperature gradients required to carry the heat will be huge (in absolute value), the RHS of (\\ref{eq:hydeq}) will be positive and the density will rapidly increase with height. However, this configuration with a density inversion is strongly unstable to Rayleigh-Taylor type instabilities. The resulting overturning motions will both carry the heat more efficiently than thermal conduction, and homogenize the mass towards a constant density state. Thus we can conjecture the existence of a critical value of $\\kappa=\\kappa_{crit}$ above which the transport is mostly conductive, the layers have a density decreasing with height and a stratification approaching that of an isothermal layer  in the limit of large $\\kappa$ (conductive states). For  $\\kappa \\ll \\kappa_{crit}$, the heat transport is mostly advective, and the density is approximately constant (convective states).\n\nSome of these ideas can be easily verified by considering a series of calculations with varying thermal diffusivity. The results are summarized in Figs. \\ref{fig:avt_z} and \\ref{fig:avrho_z} where we show the steady state horizontally averaged temperature and density distributions for  different values of $\\kappa$. As expected, the temperature gradient is always negative ($z>0$), its magnitude increases with decreasing $\\kappa$ as does the overall temperature of the layer. For large values of $\\kappa$ the temperature distributions have an approximate parabolic profile and the density decreases upwards. \n\nFor small values of $\\kappa$ the temperature in the interior approaches a ``tent\"  profile with a parabolic shape near the equator then a linear decrease over most of the domain and thin boundary layers at the edges. As $\\kappa$ decreases the profiles move up retaining their shape but producing progressively thinner boundary layers. \nThe corresponding density profiles confirm the establishment of a constant density state  that becomes asymptotically independent of $\\kappa$. From Fig. \\ref{fig:avrho_z}, we can estimate that the critical value of $\\kappa$ for a transition from conductive to convective regimes, in this setup,  satisfies\n\\begin{equation}\n\\kappa_{crit} \\approx 2 \\times 10^{-2}.\n\\label{eq:kappa-crit}\n\\end{equation}\nOur conjecture that as $\\kappa$ crosses its critical value the vertical heat transport changes from conductively dominated to advectively dominated can also be verified by considering the horizontally averaged conductive, and advective fluxes that can be defined, respectively, as\n\\begin{equation}\nF_c = - \\frac{5}{2} \\kappa \\rho \\frac{d \\tilde T}{d z} ,\n\\end{equation}\nand \n\\begin{equation}\nF_T = \\frac{1}{L_x L_y} \\int \\frac{5}{2}  \\rho v_z (T - \\tilde T) dx dy .\n\\end{equation}\nTheir values in the stationary state for the two extreme values of $\\kappa$ are shown in Fig. \\ref{fig:th_fluxes}. The roles of the two types of flux practically reverse. For $\\kappa=0.12$ the transport is entirely conductive and advection is negligible; for $\\kappa=4\\times 10^{-3}$ heat conduction is negligible, except in the boundary layers, and  all of the flux is carried by advection. It is interesting to note that near the equator where the advective flux is small--it is actually zero at the equator--the density displays a weak inversion. This is related to the absence of gravity near the equator to drive Rayleigh-Taylor instabilities. \n\n\n\\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=10cm]{fig1.eps}\n   \\caption{Time history of the Maxwell stresses averaged over the computational box for the case $\\kappa = 2 \\times 10^{-2}$. }\n   \\label{fig:maxwtime}\n\\end{figure}\n\n\n \\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=10cm]{fig2.eps}\n   \\caption{Temperature averaged over horizontal planes, $\\tilde T$ as a function of the vertical coordinate $z$ for the case $\\kappa = 2 \\times 10^{-2}$. The different curves refer to different times, as indicated by the labels, and for comparison we plot also  the time averaged distribution $ \\langle \\tilde T \\rangle$  in the steady state. }\n   \\label{fig:avt_time}\n\\end{figure}\n\n   \n\\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=10cm]{fig3.eps}\n   \\caption{Density averaged over horizontal planes, $\\tilde \\rho$ as a function of the vertical coordinate $z$ for the case $\\kappa = 2 \\times 10^{-2}$. The different curves refer to different times, as indicated by the labels, and for comparison we plot also the time averaged distribution $\\langle \\tilde \\rho \\rangle$  in the  steady state. }\n   \\label{fig:avrho_time}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=10cm]{fig4.eps}\n   \\caption{Plot of $ \\langle \\tilde T \\rangle $ as a function of $z$. The different curves refer to different values of $\\kappa$,  as shown in the legend. For comparison we plot also the isothermal case}\n   \\label{fig:avt_z}\n\\end{figure}\n\n \n  \\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=10cm]{fig5.eps}\n   \\caption{Plot of $ \\langle \\tilde \\rho \\rangle  $ as a function of $z$. The different curves refer to different values of $\\kappa$,  as shown in the legend. For comparison we plot also the isothermal case}\n   \\label{fig:avrho_z}\n\\end{figure}\n\n  \\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=10cm]{fig6.eps}\n   \\caption{Plot of $ \\langle F_c \\rangle $ and $ \\langle F_T \\rangle$ as functions of $z$. The solid curves show $\\langle F_c \\rangle $ while the dashed curves show $ \\langle F_T \\rangle $, the different colors refer to different values of $\\kappa$ as indicated in the legend.  }\n   \\label{fig:th_fluxes}\n\\end{figure}\nThe existence of two regimes, conductive and convective,  with strikingly different vertical structures is likely to lead to correspondingly different dynamo actions. A measure of these differences can be assessed by inspection of Fig. \\ref{fig:maxw} where the domain averaged Maxwell stresses are shown as a function of time for different values of $\\kappa$. The corresponding curve for an isothermal case is also included for comparison. Clearly, the angular momentum transport efficiency increases with decreasing $\\kappa$ and eventually saturates in the convective regime. It is natural to assume that once the heat transport is mostly advective further decreases in thermal diffusivity will not make any difference. What is remarkable is the difference between the convective cases and the purely isothermal one, with the latter being strikingly smaller. \n\n\\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=10cm]{fig7.eps}\n   \\caption{Plot of the average Maxwell stresses as a function of $z$ for two cases with different t values of $\\kappa$. One case (solid line) is in the convective regime, the other (dashed-dotted line) is in the conductive regime.  }\n   \\label{fig:av_maxw}\n\\end{figure}\n\n\\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=15cm]{fig8.eps}\n   \\caption{Space-time diagrams of average azimuthal field. The horizontally averaged value of $B_y$ is plotted as a function of $z$ and $t$.\nThe upper panel corresponds to a case in the convective regime; the lower, to one in the conductive regime. The corresponding values of $\\kappa$ are as indicated.  }\n   \\label{fig:byavg_z_t}\n\\end{figure}\n\nFurther evidence for two distinct types of dynamo actions operating in the two regimes can be obtained by inspection of Fig.   \\ref{fig:av_maxw}. This shows the horizontally and time averaged Maxwell stresses as a function of $z$ for two cases with different values of $\\kappa$ corresponding to the convective and conductive regimes. The curve for the conductive case follows the general trend of the more familiar isothermal calculations. The transport is largest in the denser central regions, steadily decreasing \nat higher values of z. This is in sharp contrast with the convective case in which the the stresses actually rapidly increase with distance from the mid-plane reaching a sharp maximum near the boundaries. In both cases, the corresponding Reynolds stresses are small and decrease steadily away from the mid-plane. The spatio-temporal behavior of the dynamo is also remarkably different in the two regimes as illustrated in Fig. \\ref{fig:byavg_z_t}. The two panels show  space-time diagrams of the horizontally averaged azimuthal magnetic field as a function of $z$ and time. The lower panel, corresponding to a conductive case, displays the characteristic patterns typical of the isothermal cases  signaling the presence of cyclic activity with  magnetic structures propagating from the mid-plane to the boundaries. In the upper panel there is no evidence for cyclic activity or pattern propagation. The magnetic structures form and vanish seemingly at random with no apparent characteristic time between field reversals. Furthermore there are events in which coherent structures form that extend over the entire layer. Interestingly, for earlier times, when the layer is still close to isothermal there is some evidence for pattern propagation. From  these last two  figures it is clear that both the transport efficiency and the amount of generated toroidal flux is much higher in the convective regime than in the conductive one. \n\nA possible reason for this difference might be related to the influence of magnetic boundary conditions. It is well known that in unstratified shearing boxes the boundary conditions make a big difference to the operation of the dynamo. Periodic boundary conditions, as was mentioned in the introduction, lead to small-scale dynamo action and to the convergence problem. On the other hand, ``vertical\" boundary conditions, like the ones imposed here, lead to a much more efficient dynamo that appears to scale with the system size rather than with the dissipation scale \\citep{Kapyla10}. By contrast, the solutions in isothermal, stratified shearing boxes are more insensitive to the boundary conditions \\citep{Davis10, Shi10, Oishi11}. This most likely is because the boundaries are located in very tenuous regions characterized by low density and high Alfv\\'en speed. In the convective cases described here, the density is nearly constant as a function of height making the layer appear more ``unstratified\". \nPartial support for this argument can be provided by looking at what type of dynamo is operating in the convective regime. Fig. \\ref{fig:avby} shows the time history of the volume averaged value of $B_y$ (the azimuthal component) scaled in terms of the rms value of the fluctuations. Two things are worthy of notice: the average field changes sign, and its magnitude is comparable with--and actually it occasionally exceeds--that of the fluctuations. This is strongly suggestive of the operation of a  system-scale dynamo \\citep{Tobias11} and should be contrasted with the corresponding isothermal case in which there is a different behavior depending on height and the ratio between average and fluctuations raises from about ten percent in the central region, where most of the transport takes place, to more substantial values in the upper and lower regions where the transport strongly declines.\n\n\n\\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=10cm]{fig9.eps}\n   \\caption{Time histories of the volume averaged Maxwell stresses for different values of the thermal diffusivity $\\kappa$. The values of $\\kappa$ are shown in the legend. }\n   \\label{fig:maxw}\n\\end{figure}\n\n\\begin{figure}[htbp]\n   \\centering\n   \\includegraphics[width=10cm]{fig10.eps}\n   \\caption{Time history of $\\overline B_y$ (the volume averaged $B_y$) in units of the rms value of the fluctuations. For this case $\\kappa = 4 \\times 10^{-3}$.}\n   \\label{fig:avby}\n\\end{figure}\n\n\n\n\n  \n \n\n\n\\section{Conclusions} \\label{conclusions}\nOur main objective has been to study numerically the effects of dissipative heating and finite heat transport in determining the thermal structure of the layer and the efficiency of angular momentum transport in stratified shearing boxes with zero magnetic flux. In particular we wanted to compare with the more commonly studied isothermal case. To this end we have considered the simple case of a fluid obeying the perfect gas law and with finite (constant) thermal diffusivity. \n\nOur main result is to identify two distinct regimes: conductive and convective, corresponding respectively to  large and small values of the thermal diffusivity. In the conductive regime, the heat generated by  dissipation is transported through the bulk of the layer by thermal conduction, and the temperature and density have close to parabolic profiles. This appears to be in agreement with the conclusions of the recent work by \\citet{Uzdensky12}. The convective regime is dramatically different. In these cases the heat is transported almost entirely by overturning motions driven by Rayleigh-Taylor type instabilities. The density profile becomes flat, and the temperature develops a  ``tent\" profile with thin boundary layers at the upper and lower boundaries. There is evidence that the ``tent\" profile for the temperature and the flat profile for the density are universal, in the sense that they depend solely on the properties of the turbulence and not on the values of the collisional processes.  \nThis last property in particular, may have important consequences for the dynamo processes. It appears that the dynamo can operate more efficiently in a layer with nearly constant density than in a corresponding layer with the same total mass and a Gaussian profile (isothermal case). This being the case,  there is a interesting feedback effect. The dynamo drives the MRI turbulence that heats the layer causing it to become Rayleigh-Taylor unstable, the overturning motions associated with the Rayleigh-Taylor instability homogenize the density allowing a more efficient operation of the dynamo, and so on until the layer settles to a universal, convective, self-regulated state. At the moment it is not clear whether the Rayleigh-Taylor driven motions contribute directly to a more efficient working of the dynamo, or they contribute indirectly by maintaining the more beneficial constant density state. Some of the similarities between the dynamo properties observed here and those in the  work of \\citet{Kapyla10} in which stratification is absent suggest that it may be the constant density feature that is important. \n\nFinally, we remark on the natural extensions of the present model. There are two avenues that immediately come to  mind. One is to include a more realistic treatment of the thermal transport. The obvious next step is to consider thermal diffusivities that have power law dependencies on density and temperature. We anticipate that this may have some impact on the stratification in the conductive regime, but hardly any in the convective regime in which all the thermal transport is mediated by flows anyway. The other is to consider more realistic boundary conditions, like for instance those appropriate to black body radiation. Preliminary results in this direction  show that in the convective regime this choice leads to a change in the overall value of the temperature  but not in its profile. Also, the constant density profile remains unchanged. These results however are preliminary and a more thorough study is needed. Also, the assumption of impenetrable stress-free boundary conditions should be replaced by more realistic conditions in which there is a thin transition layer across which the opacity changes dramatically and the fluid goes from being optically thick to optically thin. The problem is similar to that of matching a photosphere on top of a stellar convection zone. Numerically this is extremely challenging and will be considered in future works.\nHowever all these  extensions are secondary to the issue of convergence. In a sense, if the dynamo ceases to operate efficiently at high magnetic Reynolds numbers all bets are off. There is some room for cautious optimism since the evidence so far, is that the dynamo operating here in the convective regime is more likely to be of the system-scale type than of the small-scale type. Preliminary calculations with twice the resolutions indeed support this conjecture. However, in the end only a (very costly) convergence study will settle the issue. \n\n\n\\section{Acknowledgment}\nThis work was supported in part  by the National Science Foundation \nsponsored Center for Magnetic Self Organization at the University of Chicago.\nGB, AM and PR ackowledge support by INAF through an INAF-PRIN grant. \nWe acknowledge that the results in this paper have been achieved using the PRACE Research Infrastructure resource JUGENE based in Germany at  the J\\\"ulich Supercomputing Center.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\IEEEPARstart{M}{any} applications in wireless networks involve multicast communication, which can be defined as the transmission of identical information to multiple receivers. \nOne example is connected driving, where applications such as platooning can benefit from transmitting the same status or control information to a group of vehicles \\cite{zheng2015stability}.\nAnother example is the transmission of audio signals for live events, where each spectator can select from a variety of audio streams. \nBoth use cases can benefit considerably from physical layer precoders that ensure a given quality-of-service (QoS) level for the requested stream at each receiver while reusing the same time and frequency resources for all receivers.\n\nPhysical layer multicasting schemes have been extensively investigated in the last two decades. The authors of \\cite{sidiropoulos2006transmit} show that the performance of multicast transmission can be greatly improved by exploiting channel state information (CSI) at the transmitter. They consider two beamforming problems for single-group multicast beamforming, the max-min-fair (MMF) multicast beamforming problem and the QoS constrained multicast beamforming problem. While the MMF formulation aims at maximizing the lowest signal-to-noise ratio (SNR) among a group of users subject to a unit power constraint on the beamforming vector, the objective of the QoS constrained formulation is to minimize the transmit power subject to SNR constraints for the individual users. Moreover, the authors of \\cite{sidiropoulos2006transmit} show that the solutions to both problems are equivalent up to a scaling factor.\n\nThe more general case with multiple cochannel multicast groups is considered in \\cite{karipidis2008quality}. \nUnlike the single-group case, the QoS constrained and MMF versions of the multi-group multicast beamforming problem are different in the sense that a solution to one version cannot generally be obtained by scaling a solution to the other. \nHowever, algorithms for the QoS constrained formulation can be straightforwardly extended to approximate the MMF version, by performing a bisection search over the target signal-to-interference plus noise ratio (SINR) values. In this paper, we will therefore restrict our attention to the QoS constained formulation.\n\n\n\nThe QoS-constrained multi-group multicast beamforming problem is a well-studied nonconvex quadratically constrained quadratic programming (QCQP) problem, for which various algorithmic approximations have been proposed. Existing approaches such as semidefinite relaxation with Gaussian randomization and successive convex approximation (SCA) algorithms --  also known as convex-concave-procedures (CCP) -- involve solving a sequence of convex subproblems.\nSolutions to these subproblems can be approximated either using off-the-shelf interior-point methods or using first-order algorithms such as the alternating direction method of multipliers (ADMM). \nWhile the use of interior-point methods typically results in a high computational complexity, the ADMM can require a large number of iterations to achieve a certain accuracy. \nRegardless of the algorithm used to approximate each subproblem, the CCP results in nested approximation loops. \nTerminating the inner iteration after a finite number of steps can hinder the feasibiltiy of estimates, which is required to ensure that the CCP converges.\nBy contrast, if we assume the singular value decomposition of a matrix to be computable,\\footnote{The convergence of algorithms for computing the singular value decomposition is well-studied (see, e.g., \\cite{van1983matrix}).} the algorithm proposed in this paper is free of nested optimization loops.\n\n\n\n\nIn cases where constrained minimization becomes too costly, the superiorization methodology (see, e.g.,\\cite{herman2012superiorization}, \\cite{censor2015weak}) constitutes a promising alternative. \nWhereas the goal of constrained minimization is to find a feasible point (i.e., a point satisfying all constraints) for which the objective value is minimal, superiorization typically builds upon a simple fixed-point algorithm that produces a sequence of points which provably converges to a feasible point. This fixed-point algorithm serves as the so-called \\emph{basic algorithm}, which is then modified by adding small perturbations in each iteration with the intent to find a feasible point with reduced (not necessarily minimal) objective value. By showing that the basic algorithm is bounded perturbation resilient, its convergence guarantee towards a feasible point can be extended to this modified algorithm called a superiorized version of the basic algorithm. \n\n\nIn this paper, we consider the QoS-constrained multi-group multicast beamforming problem in \\cite{karipidis2008quality} with optional per-antenna power constraints as introduced in \\cite{chen2017admm}. We propose an algorithmic approximation based on superiorization of a bounded perturbation resilient fixed point mapping.\nTo do so, we formulate the problem in a product Hilbert space composed of subspaces of Hermitian matrices. \nThis allows us to approximate a feasible point of the relaxed problem with the well-known projections onto convex sets (POCS) algorithm \\cite{stark1998vector}, which iteratively applies a fixed-point mapping comprised of the (relaxed) projections onto each constraint set.\nWe show that this operator is bounded perturbation resilient, which allows us to add small perturbations in each iteration with the intent to reduce the objective value and the distance to the nonconvex rank-one constraints.\nSimulations show that, compared to existing methods, the proposed approach can provide better approximations at a lower computational cost in many cases.\n\n\n\n\\subsection{Preliminaries and Notation}\\label{sec:notation}\nUnless specified otherwise, lowercase letters denote scalars, lowercase letters in bold typeface denote vectors, uppercase letters in bold typeface denote matrices, and letters in calligraphic font denote sets. The sets of nonnegative integers, nonnegative real numbers, real numbers, and complex numbers are denoted by ${\\mathbb N}$, ${\\mathbb R}_+$, ${\\mathbb R}$, and ${\\mathbb C}$, respectively.\nThe real part, imaginary part, and complex conjugate of a complex number $x\\in{\\mathbb C}$ are denoted by $\\mathrm{Re}\\{x\\}$, $\\mathrm{Im}\\{x\\}$, and $x^\\ast$, respectively. The nonnegative part of a real number $x\\in{\\mathbb R}$ is denoted by $\\relu[x]\\triangleq\\max(x,0)$.\n\nWe denote by ${\\mathrm{Id}}$ the identity operator and by ${\\mathbf I}_N$ the $N\\times N$-identity matrix. The all-zero vector or matrix is denoted by ${\\mathbf 0}$ and the $i$th Cartesian unit vector is denoted by ${\\mathbf e}_i$, where the dimension of the space will be clear from the context. The Euclidean norm of a real or complex column vector ${\\mathbf x}$ is denoted by $\\|{\\mathbf x}\\|_2=\\sqrt{{\\mathbf x}^H{\\mathbf x}}$.\nThe $i$th singular value of a matrix ${\\mathbf A}\\in{\\mathbb C}^{N\\times N}$ is denoted by  $\\sigma_i({\\mathbf A})$, where the singular values are ordered such that $\\sigma_1({\\mathbf A})\\ge\\cdots\\ge\\sigma_N({\\mathbf A})$. \nFor square matrices ${\\mathbf A}$ we define $\\mathrm{diag}({\\mathbf A})$ to be the column vector composed of the diagonal of ${\\mathbf A}$, and for row or column vectors ${\\mathbf a}$ we define $\\mathrm{diag}({\\mathbf a})$ to be a square diagonal matrix having ${\\mathbf a}$ as its diagonal. \nWe write ${\\mathbf A}\\succcurlyeq {\\mathbf 0}$ for positive semidefinite (PSD) matrices ${\\mathbf A}$.\n\nThe distance between two points ${\\mathbf x},{\\mathbf y}\\in\\mathcal{H}$ in a real Hilbert space  $(\\mathcal{H},\\langle\\cdot,\\cdot\\rangle)$ is $d({\\mathbf x},{\\mathbf y})=\\|{\\mathbf x}-{\\mathbf y}\\|$, where $\\|\\cdot\\|$ is the norm induced by the inner product $\\langle\\cdot,\\cdot\\rangle$. The distance between a point ${\\mathbf x}\\in\\mathcal{H}$ and a nonempty set $\\mathcal{C}\\subset\\mathcal{H}$ is defined as $d({\\mathbf x},\\mathcal{C})=\\inf_{{\\mathbf y}\\in\\mathcal{C}}\\|{\\mathbf x}-{\\mathbf y}\\|$.\nFollowing \\cite{bauschke2002phase}, we define the projection of a point ${\\mathbf x}\\in\\mathcal{H}$ onto a nonempty subset $\\mathcal{C}\\subset\\mathcal{H}$ as the set\n\\begin{equation*}\n\\Pi_\\mathcal{C}({\\mathbf x}) = \\left\\{{\\mathbf y}\\in\\mathcal{C}|~ d({\\mathbf x},{\\mathbf y}) = d({\\mathbf x},\\mathcal{C})\\right\\},\n\\end{equation*}\nand denote by $P_\\mathcal{C}:\\mathcal{H}\\to\\mathcal{C}$ an arbitrary but fixed selection of $\\Pi_\\mathcal{C}$, i.e., $(\\forall {\\mathbf x}\\in\\mathcal{H})$ $P_\\mathcal{C}({\\mathbf x})\\in\\Pi_\\mathcal{C}({\\mathbf x})$. If $\\mathcal{C}$ is nonempty, closed, and convex, the set $\\Pi_\\mathcal{C}({\\mathbf x})$ is a singleton for all ${\\mathbf x}\\in\\mathcal{H}$, so $\\Pi_\\mathcal{C}$ has a unique selection $P_\\mathcal{C}$, which itself is called a projector. For closed nonconvex sets $\\mathcal{C}\\neq\\emptyset$ in finite-dimensional Hilbert spaces, $\\Pi_\\mathcal{C}({\\mathbf x})$ is nonempty for all ${\\mathbf x}\\in\\mathcal{H}$, although it is not generally a singleton. Nevertheless, we will refer to the selection $P_\\mathcal{C}$ as the projector, as the distinction from the set-valued operator $\\Pi_\\mathcal{C}$ will always be clear. \n\n\nA fixed point of a mapping $T:\\mathcal{H}\\to\\mathcal{H}$ is a point ${\\mathbf x}\\in\\mathcal{H}$ satisfying $T({\\mathbf x})={\\mathbf x}$. The set $\\mathrm{Fix}(T)=\\{{\\mathbf x}\\in\\mathcal{H}~|~ T({\\mathbf x})={\\mathbf x}\\}$ is called the fixed point set of $T$ \\cite{yamada2005adaptive}. \nGiven two mappings $T_1, T_2:\\mathcal{H}\\to\\mathcal{H}$, we use the shorthand $T_1T_2:=T_1\\circ T_2$ to denote their concatenation, which is defined by the composition $(\\forall{\\mathbf x}\\in\\mathcal{H})$ $T_1T_2({\\mathbf x}):=(T_1\\circ T_2)({\\mathbf x}) = T_1\\left(T_2({\\mathbf x})\\right)$. \n\nFor the following statements, let $\\left(\\mathcal{H},\\langle\\cdot,\\cdot\\rangle\\right)$ be a real Hilbert space with induced norm $\\|\\cdot\\|$. \n\\begin{definition}\n\tA mapping $T:\\mathcal{H}\\to\\mathcal{H}$ is called \\emph{nonexpansive} if $(\\forall{\\mathbf x},{\\mathbf y}\\in\\mathcal{H})$ $\\|T({\\mathbf x})-T({\\mathbf y})\\|\\le \\|{\\mathbf x}-{\\mathbf y}\\|$ \\cite{yamada2005adaptive}.\n\\end{definition}\n\\begin{definition}\\label{def:alpha_avg_nonexpansive}\n\tA mapping $T: \\mathcal{H}\\rightarrow \\mathcal{H}$ is $\\alpha$-averaged nonexpansive if there exist $\\alpha\\in(0,1)$ and a nonexpansive operator $R: \\mathcal{H}\\rightarrow \\mathcal{H}$ such that $T = (1-\\alpha){\\mathrm{Id}} + \\alpha R$ \\cite[Definition 4.33]{bauschke2011convex}.\n\\end{definition}\n\\begin{fact}\\label{fact:nonexpansive_composition}\n\tLet  $T_1,\\dots,T_L:\\mathcal{H}\\to\\mathcal{H}$ be (averaged) nonexpansive mappings with at least one common fixed point. Then the composition $T_1\\cdots T_L$ is also (averaged) nonexpansive and $\\mathrm{Fix}(T_1\\cdots T_L)=\\bigcap_{l\\in\\{1,\\dots,L\\}}\\mathrm{Fix}(T_l)$.\n\t\\cite[Fact~1]{yamada2005adaptive}, \\cite[Proposition~2.3]{he2017perturbation}\n\\end{fact}\n\\begin{fact}\\label{fact:mann_iteration}\n\tLet $T:\\mathcal{H}\\to\\mathcal{H}$ be a nonexpansive mapping with $\\mathrm{Fix}(T)\\neq\\emptyset$. Then for any initial point ${\\mathbf x}_0\\in\\mathcal{H}$ and $\\alpha\\in(0,1)$, the sequence $({\\mathbf x}_n)_{n\\in{\\mathbb N}}\\subset\\mathcal{H}$ generated by \n\t\\begin{equation*}\n\t{\\mathbf x}_{n+1} = (1-\\alpha){\\mathbf x}_n+\\alpha T({\\mathbf x}_n)\n\t\\end{equation*}\n\tconverges weakly\\footnote{In finite dimensional Hilbert spaces, weak convergence implies strong convergence \\cite{yamada2011minimizing}.} to an unspecified point in $\\mathrm{Fix}(T)$.\n\tThis fact is a special case of\t\n\t\\cite[Proposition~17.10b]{yamada2011minimizing}.\n\\end{fact}\n\\section{Problem Statement}\nIn Section~\\ref{sec:system_and_original_problem}, we define the system model and state the multi-group multicast beamforming problem with QoS- and per-antenna-power-constraints, and we reformulate it in terms of a nonconvex semidefinite program (SDP).\nA well-known approach to approximating solutions to such problems resorts to solving a convex relaxation: First, the original problem is relaxed and solved using, e.g., interior point methods. Subsequently, randomization techniques are applied to obtain candidate solutions to the original problem \\cite{karipidis2008quality}, \\cite{luo2010semidefinite}. \nHowever, in real-time applications, the complexity of interior point solvers becomes prohibitive as it grows very fast with the system size (i.e., the number of users and the number of antennas).\n\nTherefore, in Section~\\ref{sec:SDR_hilbert}, we formulate the problem in a real product Hilbert space composed of complex (Hermitian) matrices. This formulation makes the problem accessible by a variety of first-order algorithms with low complexity and provable convergence properties.\n\n\n\\subsection{System Model and Original Problem}\\label{sec:system_and_original_problem}\nFollowing the system model in \\cite{karipidis2008quality}, we consider the downlink in a network with a transmitter equipped with $N$ antenna elements, each of them represented by an element of the set $\\mathcal{N}\\triangleq\\{1,\\dots,N\\}$. Each user $k\\in\\mathcal{K}\\triangleq \\{1,\\dots,K\\}$ is equipped with a single receive antenna. The users are grouped into $M$ disjoint multicast groups $\\mathcal{G}_m\\subseteq\\mathcal{K}$ indexed by $m\\in\\mathcal{M}\\triangleq\\{1,\\dots,M\\}$, such that $\\bigcup_{m=1}^M\\mathcal{G}_m=\\mathcal{K}$. Each member of a multicast group $\\mathcal{G}_m$ is intended to receive the same information-bearing symbol $x_m\\in{\\mathbb C}$.\nThe receive signal for the $k$th user can be written as $y_k = \\sum_{m=1}^M {\\mathbf w}_m^H {\\mathbf h}_k x_m + n_k$, where ${\\mathbf w}_m\\in{\\mathbb C}^N$ is the beamforming vector for the $m$th multicast group, ${\\mathbf h}_k\\in{\\mathbb C}^N$ is the instantaneous channel to user $k$, and $n_k\\in{\\mathbb C}$ --- drawn independently from the distribution $\\mathcal{CN}(0,\\sigma_k^2)$ --- is the noise sample at the receiver. Consequently, the transmit power for group $\\mathcal{G}_m$ is proportional to $\\|{\\mathbf w}_m\\|_2^2$.\n\nIn this paper, we consider the multi-group multicast beamforming problem with QoS-constraints \\cite{karipidis2008quality}, which has the objective to minimize the total transmit power subject to constraints on the QoS expressed in terms of SINR requirements. We use the following problem formulation from \\cite{chen2017admm}, with an individual power-constraint for each transmit antenna:\n\\begin{subequations}\\label{eq:original_problem}\n\t\\begin{alignat}{3}\n\t\\underset{\\{{\\mathbf w}_m\\in{\\mathbb C}^N\\}_{m=1}^M}{\\mathrm{minimize}} &\\ \\sum\\limits_{m=1}^M\\|{\\mathbf w}_m\\|_2^2\\label{eq:original_prob_a}\\\\\n\t\\mathrm{s.t.}\\quad &   (\\forall m \\in\\mathcal{M})(\\forall k \\in\\mathcal{G}_m)\\nonumber\\\\\n\t&\\frac{ |{\\mathbf w}_m^H{\\mathbf h}_k|^2}{\\sum_{l\\neq m}|{\\mathbf w}_l^H{\\mathbf h}_k|^2+\\sigma_k^2}\\ge \\gamma_k\\label{eq:original_prob_b}\\\\\n\t&(\\forall i\\in\\mathcal{N})\\ \\sum_{m=1}^M{\\mathbf w}_m^H\\e_i\\e^T_i{\\mathbf w}_m\\le p_i\\label{eq:original_prob_c}\n\n\t\\end{alignat}\n\\end{subequations}\nThe objective function in \\eqref{eq:original_prob_a} corresponds to the total transmit power. The inequalities in \\eqref{eq:original_prob_b} constitute the SINR-constraints, where $\\gamma_k$ is the SINR required by user $k$. The inequalities in \\eqref{eq:original_prob_c} correspond to the per-antenna power constraints, where ${\\mathbf e}_i\\in{\\mathbb R}^N$ is the $i$th Cartesian unit vector.\n\nThe problem in \\eqref{eq:original_problem} is a nonconvex QCQP, which is known to be NP-hard \\cite{sidiropoulos2006transmit}.\nA well-known strategy for approximating solutions to such problems is the semidefinite relaxation technique \\cite{karipidis2008quality}, \\cite{luo2010semidefinite}. By this technique, we obtain a convex relaxation of the original problem by reformulating it as a nonconvex semidefinite program and by dropping the nonconvex rank constraints.\nMore precisely, using the trace identity $\\mathrm{tr}({\\mathbf A}{\\mathbf B})=\\mathrm{tr}({\\mathbf B}{\\mathbf A})$ for matrices ${\\mathbf A},{\\mathbf B}$ of compatible dimensions, we can write $\\|{\\mathbf w}_m\\|_2^2={\\mathbf w}_m^H{\\mathbf w}_m=\\mathrm{tr}({\\mathbf w}_m^H{\\mathbf w}_m)=\\mathrm{tr}({\\mathbf w}_m{\\mathbf w}_m^H)$ and $|{\\mathbf w}_m^H{\\mathbf h}_k|^2={\\mathbf w}_m^H{\\mathbf h}_k({\\mathbf w}_m^H{\\mathbf h}_k)^*=\\mathrm{tr}({\\mathbf w}_m^H{\\mathbf h}_k{\\mathbf h}_k^H{\\mathbf w}_m)=\\mathrm{tr}({\\mathbf w}_m{\\mathbf w}_m^H{\\mathbf h}_k{\\mathbf h}_k^H)$. By defining $(\\forall k\\in\\mathcal{K})$ ${\\mathbf Q}_k={\\mathbf h}_k{\\mathbf h}_k^H$, and replacing the expression ${\\mathbf w}_m{\\mathbf w}_m^H$ by a positive semidefinite rank-one matrix ${\\mathbf X}_m\\in{\\mathbb C}^{N\\times N}$ for all $m\\in\\mathcal{M}$, we obtain the nonconvex semidefinite program\n\\begin{subequations}\\label{eq:SDP}\n\t\\begin{alignat}{5}\n\\underset{\\{{\\mathbf X}_m\\in{\\mathbb C}^{N\\times N}\\}_{m=1}^M}{\\mathrm{minimize}}\\ & \\sum\\limits_{m=1}^M\\mathrm{tr}({\\mathbf X}_m)\\label{eq:SDPa}\\\\\n\\mathrm{s.t.}\\quad &   (\\forall m \\in\\mathcal{M}) (\\forall k \\in\\mathcal{G}_m)\\label{eq:SDPb}\\\\\n& \\mathrm{tr}({\\mathbf Q}_k{\\mathbf X}_m)\\ge \\gamma_k\\sum\\limits_{l\\neq m}\\mathrm{tr}({\\mathbf Q}_k{\\mathbf X}_l) + \\gamma_k\\sigma_k^2\\notag\\\\\n&(\\forall i\\in\\mathcal{N})\\ \\sum_{m=1}^M\\mathrm{tr}(\\e_i\\e^T_i{\\mathbf X}_m)\\le p_i\\label{eq:SDPc}\\\\\n&(\\forall m\\in\\mathcal{M})\\ {\\mathbf X}_m\\succcurlyeq{\\mathbf 0}\\label{eq:SDPd}\\\\\n&\\mathrm{rank}({\\mathbf X}_m)\\le 1,\\label{eq:SDPe}\n\\end{alignat}\n\\end{subequations}\nThis formulation is equivalent to \\eqref{eq:original_problem} in the sense that $\\{{\\mathbf X}_m={\\mathbf w}_m{\\mathbf w}_m^H\\}_{m=1}^M$ solves \\eqref{eq:SDP} if and only if $\\{{\\mathbf w}_m\\}_{m=1}^M$ solves \\eqref{eq:original_problem}.\n\nA convex relaxation of Problem~\\eqref{eq:SDP} can be obtained by simply dropping the rank-constraints in \\eqref{eq:SDPe}. \nThe approach in \\cite{sidiropoulos2006transmit}, \\cite{karipidis2008quality} solves this relaxed problem and, subsequently, generates candidate approximations for Problem~\\eqref{eq:SDP} (and hence \\eqref{eq:original_problem}) using randomization techniques. A solution to the relaxed problem is typically found using general-purpose interior point solvers, which results in high computational cost for large-scale problems.  In the multi-group setting \\cite{karipidis2008quality}, each randomization step involves solving an additional power control problem, which further increases the computational burden. \n\n\n\n\n\n\\subsection{Problem Formulation in a Real Hilbert Space}\\label{sec:SDR_hilbert}\nThe objective of this section is to show that Problem~\\eqref{eq:SDP} can be formulated in a real Hilbert space, which enables us to approach the problem by means of efficient projection-based methods. \nTo this end, we consider the $\\emph{real}$ vector space $\\mathcal{V}\\triangleq {\\mathbb C}^{N\\times N}$ of complex $N\\times N$-matrices. More precisely, we define vector addition in the usual way, \nand we restrict scalar multiplication to real scalars  $a\\in{\\mathbb R}$, where each coefficient of a vector ${\\mathbf X}\\in\\mathcal{V}$ is multiplied by $a$ to obtain the vector $a{\\mathbf X}\\in\\mathcal{V}$. In this way, $\\mathcal{V}$ is a real vector space, i.e., a vector space over the field ${\\mathbb R}$.\n\nIf we equip the space $\\mathcal{V}$ with a real inner product\\footnote{A proof that this function is in fact a real inner product can be found in Remark~\\ref{rem:real_inner_product} in the Appendix.}\n\\begin{equation}\\label{eq:innerProduct}\n(\\forall {\\mathbf X},{\\mathbf Y}\\in\\mathcal{V})\\quad \n\\left\\langle{\\mathbf X},{\\mathbf Y}\\right\\rangle\\triangleq\\mathrm{Re}\\left\\{\\mathrm{tr}\\left({\\mathbf X}^H{\\mathbf Y}\\right)\\right\\},\n\\end{equation}\nwhich induces the standard Frobenius norm \n\\begin{equation*}\n||{\\mathbf X}||=\\sqrt{\\left\\langle{\\mathbf X},{\\mathbf X}\\right\\rangle} = \\sqrt{\\mathrm{tr}\\left({\\mathbf X}^H{\\mathbf X}\\right)},\n\\end{equation*}\nwe obtain a \\emph{real} Hilbert space $\\left(\\mathcal{V},\\langle\\cdot,\\cdot\\rangle\\right)$.  \n\nIn the remainder of this paper, we restrict our attention to the subspace ${\\mathcal H}\\triangleq \\{{\\mathbf X}\\in\\mathcal{V}~|~ {\\mathbf X}={\\mathbf X}^H\\}$ of Hermitian matrices.\nFollowing the notation in \\cite{stark1998vector}, we define a product space ${{\\mathcal H}^M}$ as the $M$-fold Cartesian product\n\\begin{equation*}\n{{\\mathcal H}^M} \\triangleq\\underset{M\\text{ times}}{\\underbrace{{\\mathcal H}\\times\\dots\\times{\\mathcal H}}}\n\\end{equation*}\n of ${\\mathcal H}$. \n In this vector space, the sum of two vectors ${\\mathbf X}=\\left({\\mathbf X}_1,\\dots,{\\mathbf X}_M\\right)$ and ${\\mathbf Y}=\\left({\\mathbf Y}_1,\\dots,{\\mathbf Y}_M\\right)\\in{{\\mathcal H}^M}$ is given by ${\\mathbf X}+{\\mathbf Y} :=  \\left({\\mathbf X}_1 + {\\mathbf Y}_1,\\dots,{\\mathbf X}_M +{\\mathbf Y}_M\\right)$ and scalar multiplication is restricted to real scalars $a\\in{\\mathbb R}$, where $a\\left({\\mathbf X}_1,\\dots,{\\mathbf X}_M\\right):=\\left(a{\\mathbf X}_1,\\dots,a{\\mathbf X}_M\\right)$.\n We equip the space ${{\\mathcal H}^M}$ with the inner product\n\\begin{equation}\\label{eq:InnerProduct}\n\\langle\\ipspacing\\langle {\\mathbf X},{\\mathbf Y} \\rangle\\ipspacing\\rangle \\triangleq\\sum\\limits_{m=1}^M \\langle {\\mathbf X}_m,{\\mathbf Y}_m\\rangle,\n\\end{equation}\nwhich induces the norm\n\\begin{equation*}\n\\Norm[{\\mathbf X}]^2=\\langle\\ipspacing\\langle{\\mathbf X},{\\mathbf X}\\rangle\\ipspacing\\rangle=\\sum\\limits_{m=1}^M \\|{\\mathbf X}_m\\|^2,\n\\end{equation*}\nwhere $(\\forall m\\in\\mathcal{M})$ ${\\mathbf X}_m\\in{\\mathcal H}$ and ${\\mathbf Y}_m\\in{\\mathcal H}$.\n Consequently, $\\left({{\\mathcal H}^M}, \\langle\\ipspacing\\langle\\cdot,\\cdot\\rangle\\ipspacing\\rangle\\right)$ is also a real Hilbert space.\n\nIn order to pose Problem~\\eqref{eq:SDP} in this Hilbert space, we express the objective function in \\eqref{eq:SDPa} and the constraints in \\eqref{eq:SDPb}--\\eqref{eq:SDPe} in terms of a convex function and closed sets in $\\left({{\\mathcal H}^M}, \\langle\\ipspacing\\langle\\cdot,\\cdot\\rangle\\ipspacing\\rangle\\right)$ as shown below:\n\n\\begin{enumerate}\n\t\\item The objective function in \\eqref{eq:SDPa} can be written as the following\n\tinner product:\n\t\\begin{equation}\\label{eq:hilber_objective}\n\t\\sum\\limits_{m=1}^M\\mathrm{tr}({\\mathbf X}_m) = \\langle\\ipspacing\\langle{\\mathbf J},{\\mathbf X}\\rangle\\ipspacing\\rangle,\n\t\\end{equation}\n\twhere ${\\mathbf J}=({\\mathbf I}_N,\\dots,{\\mathbf I}_N)$. This follows from \\eqref{eq:innerProduct}, \\eqref{eq:InnerProduct}, and the fact that $(\\forall{\\mathbf W}\\in{\\mathcal H})$ $\\mathrm{Im}\\{\\mathrm{tr}({\\mathbf W})\\}=0$.\n\t\n\t\\item \tThe SINR constraint for user $k\\in\\mathcal{K}$ in \\eqref{eq:SDPb} corresponds to the closed half-space\n\t\\begin{equation}\\label{eq:SINR_sets}\n\t{\\setQ_k}=\\left\\{\\left.{\\mathbf X}\\in{{\\mathcal H}^M}\\right|\\ \\langle\\ipspacing\\langle{\\mathbf X},{\\mathbf Z}^k\\rangle\\ipspacing\\rangle\\ge \\sigma_k^2 \\right\\},\n\t\\end{equation}\n\twhere $(\\forall k\\in\\mathcal{K})$ ${\\mathbf Z}^k\\in{{\\mathcal H}^M}$ is given by\n\t\\begin{equation*}\n\t{\\mathbf Z}^k =\\Big(\\underset{1,\\cdots,g_k-1}{\\underbrace{-{\\mathbf Q}_k,\\cdots,-{\\mathbf Q}_k}},\\underset{g_k}{\\underbrace{\\gamma_k^{-1}{\\mathbf Q}_k}} ,\\underset{g_k+1,\\cdots,M}{\\underbrace{-{\\mathbf Q}_k,\\cdots,-{\\mathbf Q}_k}}\\Big).\n\t\\end{equation*}\n\tHere, we introduced indices $\\{g_k\\}_{k\\in\\mathcal{K}}$ that assign to each receiver $k\\in\\mathcal{K}$ the multicast group $\\mathcal{G}_m$ to which it belongs (i.e., $g_k=m$, if $k\\in\\mathcal{G}_m$).\n\t\n\tIn order to verify that the set ${\\setQ_k}$ in \\eqref{eq:SINR_sets} indeed represents the SINR constraint for user $k$ in \\eqref{eq:SDPb}, we rearrange\\footnote{In the remainder of this paper, we use the convention that ${\\mathbf X}_m\\in{\\mathcal H}$ denotes the $m$th component matrix of an $M$-tuple ${\\mathbf X}\\in{{\\mathcal H}^M}$.}\n\t\\begin{equation*}\n\t\\langle\\ipspacing\\langle{\\mathbf X},{\\mathbf Z}^k\\rangle\\ipspacing\\rangle \n\t= \\frac{1}{\\gamma_k}\\langle {\\mathbf X}_{g_k},{\\mathbf Q}_k\\rangle - \\sum\\limits_{\\substack{l\\in\\mathcal{M}\\\\\n\t\t\tl\\neq g_k}}\\langle{\\mathbf X}_l, {\\mathbf Q}_k\\rangle.\n\t\\end{equation*}\n\t\n\tUsing the definition of the inner product in \\eqref{eq:innerProduct}, and the fact that $(\\forall{\\mathbf W}\\in{\\mathcal H})$ ${\\mathbf W}^H={\\mathbf W}$ and $\\mathrm{Im}\\{\\mathrm{tr}({\\mathbf W})\\}=0$, we can rewrite the constraint ${\\setQ_k}$ as\n\t\\begin{equation*}\n\t\\mathrm{tr}({\\mathbf X}_{g_k}{\\mathbf Q}_k) - \\gamma_k \\sum\\limits_{\\substack{l\\in\\mathcal{M}\\\\\n\t\t\tl\\neq g_k}}\\mathrm{tr}({\\mathbf X}_l{\\mathbf Q}_k) \\ge \\gamma_k\\sigma_k^2,\n\t\\end{equation*}\n\twhich corresponds to the $k$th SINR constraint in \\eqref{eq:SDPb}.\n\t\n\t\\item \tThe per-antenna power constraints in \\eqref{eq:SDPc} are expressed by the closed convex set\n\t\\begin{equation*}\n\t\\setP=\\left\\{{\\mathbf X}\\in{{\\mathcal H}^M}\\left|~(\\forall i\\in\\mathcal{N})~  \\langle\\ipspacing\\langle {\\mathbf D}^i ,{\\mathbf X}\\rangle\\ipspacing\\rangle \\le p_i \\right.\\right\\},\n\t\\end{equation*}\n\twhere \n\t\\begin{equation}\\label{eq:defDi}\n\t(\\forall i \\in\\mathcal{N})\\quad {\\mathbf D}^i\\triangleq(\\e_i\\e^T_i,\\dots,\\e_i\\e^T_i)\\in{{\\mathcal H}^M}.\n\t\\end{equation}\n\t\n\tThis follows immediately from \\eqref{eq:innerProduct} and \\eqref{eq:InnerProduct}.\n\t\n\t\\item \tThe PSD constraints in \\eqref{eq:SDPd} correspond to the closed convex cone ${\\setH^N_+}$ given by\n\t\\begin{equation*}\n\t{\\setH^N_+} = \\left\\{\\left.({\\mathbf X}_1,\\dots,{\\mathbf X}_M)\\in{{\\mathcal H}^M}\\right|~ (\\forall m\\in\\mathcal{M})\\  {\\mathbf X}_m\\succcurlyeq{\\mathbf 0} \\right\\}.\n\t\\end{equation*}\n\t\n\t\\item The rank constraints in \\eqref{eq:SDPe} can be represented by the nonconvex set\n\t\\begin{equation}\\label{eq:rank_constraint}\n\t\\mathcal{R}= \\left\\{{\\mathbf X}\\in{{\\mathcal H}^M}\\left|~ (\\forall m\\in\\mathcal{M})~ \\mathrm{rank}({\\mathbf X}_m) \\le 1\\right.\\right\\}.\n\t\\end{equation}\n\\end{enumerate}\n\n\n\n\nConsequently, we can pose Problem~\\eqref{eq:SDP} as \n\\begin{align}\\label{eq:SDP_hilbert}\n\\underset{{\\mathbf X}\\in{{\\mathcal H}^M}}{\\mathrm{minimize}}\\ &\\langle\\ipspacing\\langle{\\mathbf J},{\\mathbf X}\\rangle\\ipspacing\\rangle\\\\\\notag\n\\mathrm{s.t.}\\quad &  (\\forall k \\in\\mathcal{K})~\t{\\mathbf X}\\in{\\setQ_k}\\\\\\notag\n&{\\mathbf X}\\in\\setP,\\quad\n{\\mathbf X}\\in{\\setH^N_+},\\quad\n{\\mathbf X}\\in\\mathcal{R}.\n\\end{align}\nThe problems in \\eqref{eq:SDP} and \\eqref{eq:SDP_hilbert} are equivalent in the sense that $\\{{\\mathbf X}_m\\in\\mathcal{V}\\}_{m\\in\\mathcal{M}}$ solves Problem~\\eqref{eq:SDP} if and only if $({\\mathbf X}_1,\\dots,{\\mathbf X}_M)\\in{{\\mathcal H}^M}$ solves Problem~\\eqref{eq:SDP_hilbert}.\nThe advantage of the formulation in \\eqref{eq:SDP_hilbert} is that it enables us\nto (i) streamline notation, (ii) express the updates of the algorithm proposed later in Section~\\ref{sec:algorithmic_solution} in terms of\nwell-known projections, and (iii) simplify proofs by using results in\noperator theory in Hilbert spaces, as we show in the following.\n\nIt is worth noting that all constraint sets described above are closed, so a projection onto each of the sets exists for any point ${\\mathbf X}\\in{{\\mathcal H}^M}$. This property is crucial to derive projection-based\nalgorithms, such as the proposed algorithm.\nIn particular, note that we cannot replace the inequality in \\eqref{eq:SDPe} with an equality, as commonly done in the\nliterature. The reason is that, with an equality, the corresponding set is not closed, as shown\nin Remarks~\\ref{rem:closed_rank} and \\ref{rem:non_closed_rank}, and the practical implication is that the projection may not exist everywhere. Specifically, this happens whenever ${\\mathbf X}=\\left({\\mathbf X}_1,\\dots,{\\mathbf X}_M\\right)$ satisfies ${\\mathbf X}_m={\\mathbf 0}$ for some $m\\in\\mathcal{M}$, which would leave the update\nrule at such points undefined in projection-based methods. This is illustrated for the case ${\\mathbf X}={\\mathbf 0}\\in{{\\mathcal H}^M}$ in Example~\\ref{ex:undefined_projection} below.\n\\begin{remark}\\label{rem:closed_rank}\n\tThe rank constraint set $\\mathcal{R}$ in \\eqref{eq:rank_constraint} is closed.\n\t\n\t\\emph{Proof:}\n\tLet $\\left({\\mathbf X}^{(n)}\\right)_{n\\in{\\mathbb N}}$ be a sequence of points in $\\mathcal{R}$ converging to a point ${\\mathbf X}^\\star=({\\mathbf X}^\\star_1,\\dots,{\\mathbf X}^\\star_M)\\in{{\\mathcal H}^M}$ and denote by $(\\forall m\\in\\mathcal{M})$\\allowbreak$(\\forall n\\in{\\mathbb N})$\n\t$\n\t{\\mathbf X}^{(n)}_m={\\mathbf U}_m^{(n)}{\\mathbf S}_m^{(n)}({\\mathbf V}_m^{(n)})^H\n\t$\n\tthe singular value decomposition of the $m$th component matrix of ${\\mathbf X}^{(n)}$. \n\tIt follows from ${\\mathbf X}^{(n)}\\in\\mathcal{R}$ that $(\\forall m\\in\\mathcal{M})$\n\t${\\mathbf S}_m^{(n)} = \\mathrm{diag}([s_m^{(n)},0,\\dots,0])$.\n\tSince a sequence of zeros can only converge to zero, the singular value decomposition ${\\mathbf X}^\\star_m={\\mathbf U}_m^\\star{\\mathbf S}_m^\\star({\\mathbf V}_m^\\star)^H$ of the $m$th component matrix of ${\\mathbf X}^\\star$ satisfies ${\\mathbf S}_m^\\star=\\mathrm{diag}([s_m^\\star,0,\\dots,0])$ for some $s^\\star_m\\in{\\mathbb R}_+$. Therefore $(\\forall m\\in\\mathcal{M})$ $\\mathrm{rank}({\\mathbf X}_m^\\star)\\le 1$, so ${\\mathbf X}^\\star\\in\\mathcal{R}$.\n\t\\pushQED{\\qed}\n\tThe above shows that $\\mathcal{R}$ contains all its limit points, so it is closed.\\qedhere\n\t\\popQED\n\\end{remark}\n\n\\begin{remark}\\label{rem:non_closed_rank}\nBy contrast,\n\\begin{equation*}\n\\mathcal{R}^\\prime= \\left\\{{\\mathbf X}\\in{{\\mathcal H}^M}\\left|~ (\\forall m\\in\\mathcal{M})~ \\mathrm{rank}({\\mathbf X}_m) = 1\\right.\\right\\}\n\\end{equation*} \nis not a closed set, since for all ${\\mathbf X}\\in\\mathcal{R}^\\prime$ and $\\alpha\\in(0,1)$, the sequence $\\left(\\alpha^n{\\mathbf X}\\right)_{n\\in{\\mathbb N}}$ in $\\mathcal{R}^\\prime$ converges to ${\\mathbf 0}\\notin\\mathcal{R}^\\prime$.\n\\end{remark}\n\n\\begin{example}\\label{ex:undefined_projection}\n\tThe set-valued projection of ${\\mathbf 0}\\in{{\\mathcal H}^M}$ onto the set $\\mathcal{R}^\\prime$ in Remark~\\ref{rem:non_closed_rank} is empty.\n\t\n\t\\emph{Proof:}\n\tSuppose that $\\Pi_{\\mathcal{R}^\\prime}({\\mathbf 0})\\neq\\emptyset$ and let ${\\mathbf Z}\\in\\Pi_{\\mathcal{R}^\\prime}({\\mathbf 0})$, i.e., ${\\mathbf Z}$ is any of the closest points of the set $\\mathcal{R}^\\prime$ to the zero vector ${\\mathbf 0}$. Since $\\Pi_{\\mathcal{R}^\\prime}({\\mathbf 0})\\subset\\mathcal{R}^\\prime$, $(\\forall m\\in\\mathcal{M})$ $\\mathrm{rank}({\\mathbf Z}_m)=1$, i.e., $(\\forall m\\in\\mathcal{M})$ $\\sigma_1({\\mathbf Z}_m)>0$.\n\tTherefore, for any $\\alpha\\in(0,1)$, $\\alpha{\\mathbf Z}\\in\\mathcal{R}^\\prime$ and $d({\\mathbf 0},\\alpha{\\mathbf Z})<d({\\mathbf 0},{\\mathbf Z})$, i.e., $\\alpha {\\mathbf Z}\\in \\mathcal{R}^\\prime$ is closer to the zero vector than ${\\mathbf Z} \\in \\mathcal{R}^\\prime$, thus contradicting our assumption that ${\\mathbf Z}$ is one of the closest\n\t\\pushQED{\\qed} \n\tpoints in $\\mathcal{R}^\\prime$ to the vector ${\\mathbf 0}$.\\qedhere\\popQED\n\\end{example}\n\n\n\n\n\\section{Algorithmic Solution}\\label{sec:algorithmic_solution}\nThe main difficulty in solving \\eqref{eq:SDP_hilbert} is the presence of the nonconvex rank constraint.\nA well-known technique for approximating rank-constrained semidefinite programs using convex optimization methods is the semidefinite relaxation approach \\cite{sidiropoulos2006transmit}, \\cite{karipidis2008quality}, \\cite{luo2010semidefinite}. This approach first solves \\eqref{eq:SDP_hilbert} without the rank constraint, and then it applies heuristics to obtain rank-one approximations based on the solution to this relaxed problem.\nSimilarly, we can obtain a convex relaxation\n\\begin{align}\\label{eq:SDR_hilbert}\n\\underset{{\\mathbf X}\\in{{\\mathcal H}^M}}{\\mathrm{minimize}}\\ &\\langle\\ipspacing\\langle{\\mathbf J},{\\mathbf X}\\rangle\\ipspacing\\rangle\\\\\\notag\n\\mathrm{s.t.}\\quad &  (\\forall k \\in\\mathcal{K})~\t{\\mathbf X}\\in{\\setQ_k}\\\\\\notag\n&{\\mathbf X}\\in\\setP,\\quad\n{\\mathbf X}\\in{\\setH^N_+},\n\\end{align}\nof Problem~\\eqref{eq:SDP_hilbert} by dropping the nonconvex constraint set $\\mathcal{R}$.\nIn principle, we could solve this relaxed problem using first-order techniques for constrained convex minimization.\nFor instance, we could apply a projected (sub-)\\allowbreak gradient method (see, e.g., \\cite[Section~3.2.3]{nesterov2018lectures}), which interleaves (sub-)\\allowbreak gradient steps for the objective function with projections onto the feasible set \nof Problem~\\eqref{eq:SDR_hilbert}. However, computing the projection onto the intersection of all constraint sets in Problem~\\eqref{eq:SDR_hilbert} typically requires an inner optimization loop because no simple expression for this projection is known.\nAs it was shown in \\cite{censor2014projected}, superiorization can significantly reduce the computation time compared to the projected gradient method in some applications if the projection onto the feasible set is difficult to compute.\n\n\nThe superiorization methodology typically relies on an iterative process that solves a convex feasibility problem (i.e., that produces a sequence of points converging to a point within the intersection of all constraint sets) by repeatedly applying a computationally simple mapping. This iterative algorithm is called the \\emph{Basic Algorithm}. Based on this Basic Algorithm, the superiorization methodology automatically produces a \\emph{Superiorized Version of the Basic Algorithm}, by adding bounded perturbations to the iterates of the Basic Algorithm in every iteration. \n\\begin{definition}\nLet $\\left({\\mathbf Y}^{(n)}\\right)_{n\\in{\\mathbb N}}$ be a bounded sequence in a real Hilbert space and let $\\left(\\beta^{(n)}\\right)_{n\\in{\\mathbb N}}$ be a sequence in ${\\mathbb R}_+$ such that $\\sum_{n\\in{\\mathbb N}}\\beta^{(n)}<\\infty$. Then $(\\forall n\\in{\\mathbb N})$ $\\beta^{(n)}{\\mathbf Y}^{(n)}$ are \\emph{bounded perturbations} \\cite{censor2015weak}.\n\\end{definition}\nThe perturbations are typically generated based on subgradient steps for a given objective function, in a way that ensures the sequence of perturbations to be bounded. By showing that the Basic Algorithm is bounded perturbation resilient (i.e., that the resulting sequence is guaranteed to converge to a feasible point, even when bounded perturbations are added in each iteration), one can ensure that the sequence produced by the Superiorized Version of the Basic Algorithm also converges to a feasible point. In contrast to constrained minimization, superiorization does not guarantee that the objective value of the resulting approximation is minimal. However, the limit point of the superiorized algorithm typically has a lower objective value than the limit point of the unperturbed Basic Algorithm \\cite{censor2015weak}.\n\nTo apply the superiorization methodology to Problem~\\eqref{eq:SDP_hilbert}, we proceed as follows. \nIn Section~\\ref{sec:basic_algorithm}, we propose a Basic Algorithm by defining a mapping  $\\T[\\star]:{{\\mathcal H}^M}\\to{{\\mathcal H}^M}$. Given any point ${\\mathbf X}^{(0)}\\in{{\\mathcal H}^M}$, this mapping generates a sequence of points\nconverging to a feasible point of Problem~\\eqref{eq:SDR_hilbert} by\n\\begin{equation}\\label{eq:basic_algorithm}\n    (\\forall n\\in{\\mathbb N})\\quad {\\mathbf X}^{(n+1)}=\\T[\\star]\\left({\\mathbf X}^{(n)}\\right).\n\\end{equation}\nIn Section~\\ref{sec:perturbations}, we define a sequence $\\left(\\beta^{(n)}{\\mathbf Y}^{(n)}\\right)_{n\\in{\\mathbb N}}$ of bounded perturbations,  with  the  intent to reduce slightly (i) the objective value of Problem~\\eqref{eq:SDP_hilbert} and (ii) the distance to the nonconvex rank constraint $\\mathcal{R}$ in every iteration. As we show in Proposition~\\ref{prop:nonincreasing} below, the proposed perturbations can achieve both goals simultaneously. The  sequence  of  perturbations  yields  a  Superiorized  Version of the Basic Algorithm given by ${\\mathbf X}^{(0)}\\in{{\\mathcal H}^M}$,\n\\begin{equation}\\label{eq:superior_alg}\n(\\forall n\\in{\\mathbb N})\\quad \\nextit[{\\mathbf X}] = T_\\star\\left(\\current[{\\mathbf X}] + \\current[\\beta] \\current[{\\mathbf Y}]\\right).\n\\end{equation}\nIn Section~\\ref{sec:boundedness}, we prove that the algorithm in \\eqref{eq:superior_alg} converges to a feasible point of Problem~\\eqref{eq:SDR_hilbert} by showing that the mapping $\\T[\\star]$ is bounded perturbation resilient, and that $\\left(\\beta^{(n)}{\\mathbf Y}^{(n)}\\right)_{n\\in{\\mathbb N}}$ is a sequence of bounded perturbations.\nThe relation between the proposed method and the superiorization methodology is discussed in detail in Section~\\ref{sec:relation_to_superiorization}.\nFinally, the proposed algorithm is summarized in Section~\\ref{sec:algorithm_summary}.\n\n\n\n\n\\subsection{Feasibility-Seeking Basic Algorithm}\\label{sec:basic_algorithm}\nA feasible point for the relaxed SDP in \\eqref{eq:SDR_hilbert} can be found by solving the convex feasibility problem\n\\begin{equation}\\label{eq:sdrCFP}\n\\text{Find }{\\mathbf X}\\in{{\\mathcal H}^M}\\text{ such that } {\\mathbf X}\\in \\mathcal{C}_\\star\\triangleq\\bigcap\\limits_{k=1}^K {\\setQ_k}\\cap \\setP\\cap {\\setH^N_+}.\n\\end{equation}\n\nAccording to Fact~\\ref{fact:mann_iteration} and Definition~\\ref{def:alpha_avg_nonexpansive}, given any ${\\mathbf X}^{(0)}\\in{{\\mathcal H}^M}$, the iteration in \\eqref{eq:basic_algorithm} generates a sequence of points converging to a point in $\\mathcal{C}_\\star$\nif $\\T[\\star]$ is $\\alpha$-averaged nonexpansive with $\\mathrm{Fix}(\\T[\\star])=\\mathcal{C}_\\star$.\nA particular case of such a mapping, which is used in the well-known projections onto convex sets (POCS) algorithm  \\cite{stark1998vector}, is given by (see also Fact~\\ref{fact:nonexpansive_composition})\n\\begin{equation}\\label{eq:pocs}\n\\T[\\star]:= T^{\\mu_{K+2}}_{{\\setH^N_+}}T^{\\mu_{K+1}}_{\\setP}T^{\\mu_K}_{{\\setQ_K}}\\dots T^{\\mu_1}_{{\\setQ_1}},\n\\end{equation}\nwhere for a nonempty closed convex set $\\mathcal{C}\\in{{\\mathcal H}^M}$,\n\\begin{equation*}\n\\T[\\mathcal{C}]^\\mu={\\mathrm{Id}} + \\mu ({\\mathbf P}[\\mathcal{C}] - {\\mathrm{Id}})\n\\end{equation*}\ndenotes the relaxed projector onto $\\mathcal{C}$ with relaxation parameter $\\mu\\in(0,2)$. The formal expressions for the projections of ${\\mathbf X}\\in{{\\mathcal H}^M}$ onto each of the sets in \\eqref{eq:sdrCFP} are given below.\n\\begin{enumerate}\n\t\\item The SINR constraint sets ${\\setQ_k}\\in{{\\mathcal H}^M}$ are half-spaces, the projections onto which are given by \\cite[Example~29.20]{bauschke2011convex} $(\\forall k\\in\\mathcal{K})$\\allowbreak$(\\forall {\\mathbf X}\\in{{\\mathcal H}^M})$\n\t\\begin{equation*}\n\tP_{{\\setQ_k}}({\\mathbf X})=\\begin{cases}\n\t{\\mathbf X},&\\text{if ${\\mathbf X}\\in{\\setQ_k}$}\\\\\n\t{\\mathbf X} + \\frac{\\sigma_k^2 - \\langle\\ipspacing\\langle{\\mathbf X},{\\mathbf Z}^k\\rangle\\ipspacing\\rangle}{\\Norm[{\\mathbf Z}^k]^2}{\\mathbf Z}^k,&\\text{otherwise.}\n\t\\end{cases}\n\t\\end{equation*}\n\t\n\t\\item \tThe per-antenna power constraint set $\\setP$ is an intersection of the $N$ half-spaces defined by the normal vectors ${\\mathbf D}^i$ in \\eqref{eq:defDi} for $i\\in\\mathcal{N}$. Since these vectors are mutually orthogonal, i.e., $(\\forall i\\in\\mathcal{N})$\\allowbreak$(\\forall j\\in\\mathcal{N}\\setminus \\{i\\})$ $\\langle\\ipspacing\\langle{\\mathbf D}^i,{\\mathbf D}^j\\rangle\\ipspacing\\rangle= 0$, the projection onto $\\setP$ can be written in closed form as\n\t\\begin{equation*}\n\tP_{\\setP}({\\mathbf X})=\n\t{\\mathbf X} + \\sum_{i:p_i<\\langle\\ipspacing\\langle{\\mathbf X},{\\mathbf D}^i\\rangle\\ipspacing\\rangle} \\frac{p_i - \\langle\\ipspacing\\langle{\\mathbf X},{\\mathbf D}^i\\rangle\\ipspacing\\rangle}{\\Norm[{\\mathbf D}^i]^2}{\\mathbf D}^i.\n\t\\end{equation*}\n\tThis follows from \\cite[Thm~4.3-1]{stark1998vector} and Halperin's Theorem (see \\cite{halperin1962product}, \\cite[Thm.~4.2]{ginat2018method}).\n\t\n\t\\item The set ${\\setH^N_+}$ is the intersection of PSD cones in orthogonal subspaces of ${{\\mathcal H}^M}$. The projection of ${\\mathbf X}\\in{{\\mathcal H}^M}$ onto ${\\setH^N_+}$ is therefore given component-wise by\n\t\\begin{equation*}\n\tP_{\\setH^N_+}({\\mathbf X}) = \\left(P_{\\hilbert_+}(\\Xk[1]),\\dots,P_{\\hilbert_+}(\\Xk[M])\\right),\n\t\\end{equation*}\n\twhere, ${\\mathcal H}_+=\\{{\\mathbf X}\\in{\\mathcal H}~|~ {\\mathbf X}\\succcurlyeq {\\mathbf 0}\\}$ is the cone of PSD matrices in ${\\mathcal H}$. We use the eigendecomposition $\\Xk[m]={\\mathbf V}_m{\\mathbf \\Lambda}_m{\\mathbf V}_m^H$ with (real) eigenvalues ${\\mathbf \\Lambda}_m=\\mathrm{diag}(\\lambda_1(\\Xk[m]),\\dots,\\lambda_N(\\Xk[m]))$ to define the projection of $\\Xk[m]\\in{\\mathcal H}$ onto ${\\mathcal H}_+$ as\\footnote{For the case of real symmetric matrices, see, e.g.,  \\cite[Lemma 2.1]{goulart2020accuracy}. The result in \\cite{goulart2020accuracy} is based on \\cite[Corollary~7.4.9.3]{horn2013matrix}, which assumes complex Hermitian matrices. The generalization of \\cite[Lemma 2.1]{goulart2020accuracy} to complex Hermitian matrices is straightforward.}\n\t\\begin{equation*}\n\tP_{\\hilbert_+}(\\Xk[m]) = {\\mathbf V}_m{\\mathbf \\Lambda}^+_m{\\mathbf V}_m^H,\n\t\\end{equation*}\n\twhere ${\\mathbf \\Lambda}^+_m\\triangleq\\mathrm{diag}\\left(\\relu[{\\lambda_1(\\Xk[m])}],\\dots,\\relu[{\\lambda_N(\\Xk[m])}]\\right)$.\n\\end{enumerate}\nAccording to the fundamental theorem of POCS \\cite[Thm 2.5-1]{stark1998vector}, the sequence $\\left(\\current[{\\mathbf X}]\\right)_{n\\in{\\mathbb N}}$ of vectors $\\current[{\\mathbf X}]\\in{{\\mathcal H}^M}$ produced by the update rule in \\eqref{eq:pocs} is guaranteed to converge to a solution of the feasibility problem in \\eqref{eq:sdrCFP} for any $\\zeroth[{\\mathbf X}]\\in{{\\mathcal H}^M}$, if a solution exists (i.e., if $\\mathcal{C}_\\star\\neq\\emptyset$). Note that this is the case if the relaxed semidefinite program in \\eqref{eq:SDR_hilbert} is feasible.\nAlternatively, we can derive this convergence guarantee immediately from Remark~\\ref{rem:alpha_avg} and Fact~\\ref{fact:mann_iteration}.\n\n\n\n\n\n\\subsection{Proposed Perturbations}\\label{sec:perturbations}\nIn the following, we devise perturbations that steer the iterates of the fixed point algorithm in \\eqref{eq:superior_alg} towards a solution to the nonconvex problem in \\eqref{eq:SDP} and \\eqref{eq:SDP_hilbert}. To do so, we introduce a mapping that reduces the objective value and a mapping that reduces the distance to rank constraint sets. Then we define the proposed perturbations based on the composition of these two mappings As proven in Proposition~\\ref{prop:nonincreasing} below, the resulting perturbations can achieve both goals simultaneously.\n\n\\subsubsection{Power Reduction by Bounded Perturbations}\\label{sec:power_perturbations}\nIn the literature on superiorization, the perturbations are typically defined based on subgradient steps of the objective function (see, e.g., \\cite{censor2015weak}). For the linear objective function in \\eqref{eq:SDR_hilbert}, this would result in perturbations of the form\n$-\\alpha \\left({\\mathbf I}_N,\\dots,{\\mathbf I}_N\\right)$ \tfor some $\\alpha>0$. These perturbations are problematic for the problem considered here because we are interested in solutions comprised of positive semidefinite rank-one matrices, and adding these perturbations to an iterate ${\\mathbf X}=({\\mathbf X}_1,\\dots,{\\mathbf X}_M)$ may result in indefinite full-rank component matrices ${\\mathbf X}_m-\\alpha{\\mathbf I}_N$. To avoid this problem, we introduce the function $\\sobjf:{{\\mathcal H}^M}\\to{\\mathbb R}_+$ given by\n\\begin{equation}\\label{eq:equiv_objective}\n\\sobjf({\\mathbf X}) \\triangleq \\sum_{m=1}^M\\|{\\mathbf X}_m\\|_\\ast,\n\\end{equation}\nwhere $\\|\\cdot\\|_\\ast$ is the nuclear norm. Since $\\mathcal{C}_\\star\\subset{\\setH^N_+}$ by \\eqref{eq:sdrCFP}, we have $(\\forall{\\mathbf X}\\in\\mathcal{C}_\\star)(\\forall m\\in\\mathcal{M})(\\forall i\\in\\mathcal{N})$ $\\sigma_i({\\mathbf X}_m)=\\lambda_i({\\mathbf X}_m)$, where $\\lambda_i({\\mathbf X}_m)$ and $\\sigma_i({\\mathbf X}_m)$ denote the $i$th eigenvalue and singular value of the $m$th component matrix of ${\\mathbf X}$, respectively. Hence we can write\n\\begin{align}\\label{eq:surrogate_obj}\n\\sobjf({\\mathbf X}) &= \\sum_{m=1}^M\\sum_{i=1}^N \\sigma_i({\\mathbf X}_m) \\\\\\notag\n&=\\sum_{m=1}^M\\sum_{i=1}^N \\lambda_i({\\mathbf X}_m) = \\sum_{m=1}^M\\mathrm{tr}({\\mathbf X}_m).\n\\end{align}\nTherefore, by \\eqref{eq:hilber_objective}, minimizing $\\sobjf$ over $\\mathcal{C}_\\star$ is equivalent to minimizing the linear objective function in \\eqref{eq:SDP_hilbert} (or \\eqref{eq:SDR_hilbert}) over $\\mathcal{C}_\\star$, in the sense that the solution sets to both formulations are the same. As we will show below, this surrogate objective function gives rise to power-reducing perturbations, which are guaranteed not to increase the rank of their arguments' component matrices (see Remark~\\ref{rem:rank_reduction}).\n\nThe power-reducing perturbations are designed according to two criteria. Firstly, they should decrease the value of the surrogate function $\\sobjf$. Secondly, they should not be too large in order to avoid slowing down convergence of the Basic Algorithm. \nFor a given point ${\\mathbf X}\\in{{\\mathcal H}^M}$ we derive a perturbation ${\\mathbf Y}_\\tau^\\star$ satisfying these two criteria by solving the problem\n\\begin{equation}\\label{eq:power_pert_problem}\n{\\mathbf Y}_\\tau^\\star:={\\mathbf Y}_\\tau^\\star({\\mathbf X})\\in \\underset{{\\mathbf Y}\\in{{\\mathcal H}^M}}{\\mathrm{arg\\ min}}~\\left( \\tau \\sobjf({\\mathbf X} + {\\mathbf Y}) + \\frac{1}{2} \\Norm[{\\mathbf Y}]^2\\right).\n\\end{equation}\nHere, $\\Norm[{\\mathbf Y}]^2$ acts as a regularization on the perturbations' magnitude, and the parameter $\\tau\\ge0$ balances the two design criteria. The next proposition shows that ${\\mathbf Y}_\\tau^\\star$  can be easily\ncomputed.\n\n\\begin{prop}\n\tThe unique solution to \\eqref{eq:power_pert_problem} is given by\n\t\\begin{equation}\\label{eq:power_pert_opt}\n\t(\\forall m\\in\\mathcal{M}) \\quad {\\mathbf Y}_\\tau^\\star|_m = \\mathcal{D}_\\tau({\\mathbf X}_m) - {\\mathbf X}_m,\n\t\\end{equation}\n\twhere $\\mathcal{D}_\\tau:{\\mathcal H}\\to{\\mathcal H}$ is the singular value shrinkage operator \\cite{cai2010singular}\n\t\\begin{align}\\label{eq:svd_shrink}\n\t\\mathcal{D}_\\tau({\\mathbf X}_m)&\\triangleq {\\mathbf U}_m\\mathcal{D}_\\tau({\\mathbf \\Sigma}_m){\\mathbf V}_m^H,\\\\\\notag \n\t\\mathcal{D}_\\tau({\\mathbf \\Sigma}_m)&=\\mathrm{diag}\\left(\\left\\{\\relu[{\\sigma_i({\\mathbf X}_m)-\\tau}]\\right\\}_{i\\in\\mathcal{N}}\\right),\n\t\\end{align}\n\tand $(\\forall m\\in\\mathcal{M})$ ${\\mathbf X}_m={\\mathbf U}_m{\\mathbf \\Sigma}_m{\\mathbf V}_m$ is the singular value decomposition of ${\\mathbf X}_m$ such that ${\\mathbf \\Sigma}_m=\\mathrm{diag}\\left(\\left\\{\\sigma_i({\\mathbf X}_m)\\right\\}_{i\\in\\mathcal{N}}\\right)$.\n\t\n\t\\emph{Proof:}\n\tDenote the perturbed point for a given choice of $\\tau$ by ${\\mathbf Z}_\\tau^\\star:={\\mathbf X}+{\\mathbf Y}_\\tau^\\star$. By substituting ${\\mathbf Y}={\\mathbf Z}-{\\mathbf X}$ in \\eqref{eq:power_pert_problem}, we can identify this point as ${\\mathbf Z}_\\tau^\\star={\\mathrm{prox}}_{\\tau \\sobjf}({\\mathbf X})$, where the proximal mapping is given by\n\t\\begin{equation}\\label{eq:proximal_op}\n\t{\\mathrm{prox}}_{\\tau \\sobjf}({\\mathbf X}) \\in \\underset{{\\mathbf Z}\\in{{\\mathcal H}^M}}{\\mathrm{arg\\ min}}~ \\left(\\tau \\sobjf({\\mathbf Z}) + \\frac{1}{2} \\Norm[{\\mathbf X}-{\\mathbf Z}]^2\\right).\n\t\\end{equation}\n\tNote that the function\n\t\\begin{equation*}\n\t\\tau \\sobjf({\\mathbf Z}) + \\frac{1}{2}\\Norm[{\\mathbf X}-{\\mathbf Z}]^2 = \\tau \\sum_{m=1}^M \\|{\\mathbf Z}_m\\|_\\ast + \\frac{1}{2}\\sum_{m=1}^M \\|{\\mathbf X}_m-{\\mathbf Z}_m\\|^2\n\t\\end{equation*}\n\tis separable over $m$. Consequently, we can compute the proximal mapping in $\\eqref{eq:proximal_op}$ by solving\n\t\\begin{equation}\\label{eq:prox_subspace}\n\t(\\forall m\\in\\mathcal{M})\\quad {\\mathbf Z}_\\tau^\\star|_m\\in\\underset{{\\mathbf Z}\\in{\\mathcal H}}{\\mathrm{arg\\ min}}~ \\tau \\|{\\mathbf Z}\\|_\\ast + \\frac{1}{2}\\|{\\mathbf X}_m-{\\mathbf Z}\\|^2.\n\t\\end{equation}\n\tAccording to \\cite[Thm. 2.1]{cai2010singular}, the unique solution to \\eqref{eq:prox_subspace} is given by ${\\mathbf Z}_\\tau^\\star|_m = \\mathcal{D}_\\tau({\\mathbf X}_m)$.\\footnote{The proof in \\cite{cai2010singular} is for real matrices. However, the generalization to complex matrices is straightforward.}\n\t\\pushQED{\\qed}\n\tSubstituting ${\\mathbf Y}_\\tau^\\star={\\mathbf Z}_\\tau^\\star-{\\mathbf X}$ yields \\eqref{eq:power_pert_opt}, which is the desired result.\\qedhere\n\\popQED\n\\end{prop}\n\n\n\t\n\nBy defining $(\\forall{\\mathbf X}\\in{{\\mathcal H}^M})$\n\\begin{equation}\n\\sigma_{\\max}({\\mathbf X})\\triangleq \\max_{\\substack{{m\\in\\mathcal{M}}\\\\{i\\in\\mathcal{N}}}}\\sigma_i({\\mathbf X}_m)\n\\end{equation}\nwe can express the power-reducing perturbation for a point ${\\mathbf X}\\in{{\\mathcal H}^M}$ as ${\\mathbf Y}=\\T[\\mathrm{P}]^\\alpha({\\mathbf X})-{\\mathbf X}$, where the mapping $\\T[\\mathrm{P}]^\\alpha\\triangleq{\\mathrm{prox}}_{\\alpha\\sigma_{\\max}({\\mathbf X})\\sobjf}$ is given component-wise by $(\\forall m \\in\\mathcal{M})$\n\\begin{equation}\\label{eq:op_tp}\n\\T[\\mathrm{P}]^{\\alpha}({\\mathbf X})|_m =\\mathcal{D}_\\tau({\\mathbf X}_m) \\quad \\text{with}\\quad \\tau= \\alpha\\sigma_{\\max}({\\mathbf X}).\n\\end{equation} \n\nNote that $\\T[\\mathrm{P}]^0({\\mathbf X}) = {\\mathbf X}$, and $(\\forall\\alpha\\ge 1)$ $\\T[\\mathrm{P}]^\\alpha({\\mathbf X})={\\mathbf 0}$.\nTherefore, the magnitude of the power-reducing perturbations can be controlled by choosing the parameter $\\alpha\\in[0,1]$. Moreover, in contrast to performing subgradient steps for the original cost function in \\eqref{eq:SDP_hilbert}, applying the perturbations in \\eqref{eq:op_tp} cannot increase the rank:\n\n\\begin{remark}\\label{rem:rank_reduction}\n\tFor all $\\alpha\\ge0$, $\\T[\\mathrm{P}]^\\alpha$ maps any point ${\\mathbf X}=({\\mathbf X}_m)_{m\\in\\mathcal{M}}\\in{\\setH^N_+}$ to a point ${\\mathbf Z}=({\\mathbf Z}_m)_{m\\in\\mathcal{M}}=\\T[\\mathrm{P}]^\\alpha({\\mathbf X})\\in{\\setH^N_+}$ satisfying $(\\forall m\\in\\mathcal{M})$ $\\mathrm{rank}({\\mathbf Z}_m)\\le\\mathrm{rank}({\\mathbf X}_m)$.\n\tThis follows immediately from \\eqref{eq:svd_shrink}.\n\\end{remark}\n\n\t\t\n\\subsubsection{Incorporating the Rank Constraints by Bounded Perturbations}\\label{sec:rank_perturbations}\nNext, we define perturbations that steer the iterate towards the rank constraint set $\\mathcal{R}$ in \\eqref{eq:rank_constraint}.\nWhile objective functions used for superiorization are usually convex, the function $\\sobjg:{{\\mathcal H}^M}\\to{\\mathbb R}_+$\n\\begin{equation}\\label{eq:rank_dist}\n\\sobjg({\\mathbf X})\\triangleq d({\\mathbf X},\\mathcal{R}),\n\\end{equation}\ni.e., the distance to the set $\\mathcal{R}$, constitutes a nonconvex superiorization objective, so our approach does not follow exactly the superiorization methodology in \\cite{censor2015weak} (but we can still prove convergence).\n\nAs the perturbations may steer the iterates away from the feasible set, their magnitude should not be unnecessarily large. Therefore, we choose the rank-reducing perturbations as $\\P[\\mathcal{R}]({\\mathbf X})-{\\mathbf X}$, where $\\P[\\mathcal{R}]({\\mathbf X})\\in\\Pi_\\mathcal{R}({\\mathbf X})$ denotes a (generalized) projection of a given point ${\\mathbf X}\\in{{\\mathcal H}^M}$ onto the closed nonconvex set $\\mathcal{R}$. \nSince $\\mathcal{R}$ is a closed set, the set-valued projection $\\Pi_\\mathcal{R}({\\mathbf X})$ is nonempty for all ${\\mathbf X}\\in{{\\mathcal H}^M}$. A projection onto $\\mathcal{R}$ can be computed by\ntruncating all but the largest singular value of each component matrix to zero. We formally state this fact below.\n\n\\begin{fact}\n\tLet ${\\mathbf X}_m={\\mathbf U}_m{\\mathbf \\Sigma}_m{\\mathbf V}_m^H\\in{\\mathcal H}$ be the singular value decomposition of the $m$th component matrix of ${\\mathbf X}$ with ${\\mathbf \\Sigma}_m=\\mathrm{diag}(\\sigma_1({\\mathbf X}_m),\\dots,\\sigma_N({\\mathbf X}_m))$.\n\n\tThen, $(\\forall {\\mathbf X}\\in{{\\mathcal H}^M})$ the $m$th component matrix of a point $\\P[\\mathcal{R}]({\\mathbf X})\\in\\Pi_\\mathcal{R}({\\mathbf X})$ is given by \\cite[Lemma~3.2]{luke2013prox}\n\t\\begin{equation}\\label{eq:proj_rank}\n\t\\P[\\mathcal{R}]({\\mathbf X})|_m = {\\mathbf U}_m\\mathrm{diag}\\left(\\sigma_1({\\mathbf X}_m),0,\\dots,0\\right){\\mathbf V}_m^H.\n\t\\end{equation}\n\\end{fact}\n\n\n\n\\subsubsection{Combining Power- and Rank Perturbations}\\label{sec:combined_perturbations}\nSince both $\\T[\\mathrm{P}]^\\alpha$ in \\eqref{eq:op_tp} and $\\P[\\mathcal{R}]$ in \\eqref{eq:proj_rank} operate on the singular values of the component matrices, their composition\n is given by $(\\forall m\\in\\mathcal{M})$\n\\begin{equation*\n\\P[\\mathcal{R}]\\T[\\mathrm{P}]^\\alpha({\\mathbf X})|_m=\\relu[{\\sigma_1({\\mathbf X}_m)-\\alpha\\sigma_{\\max}({\\mathbf X})}]{\\mathbf u}_{m1} {\\mathbf v}_{m1}^H\\in{\\mathcal H},\n\\end{equation*}\nwhere, $(\\forall m\\in\\mathcal{M})$ ${\\mathbf U}_m=[{\\mathbf u}_{m1},\\dots,{\\mathbf u}_{mN}]$ and ${\\mathbf V}_m=[{\\mathbf v}_{m1},\\dots,{\\mathbf v}_{mN}]$.\nMoreover, it is easy to verify that $(\\forall {\\mathbf X}\\in{{\\mathcal H}^M})$\\allowbreak$(\\forall \\alpha \\ge 0)$, $\\T[\\mathrm{P}]^\\alpha\\P[\\mathcal{R}]({\\mathbf X})=\\P[\\mathcal{R}]\\T[\\mathrm{P}]^\\alpha({\\mathbf X})$. \nWe will now use the composition of $\\T[\\mathrm{P}]^\\alpha$ and $\\P[\\mathcal{R}]$ to define a mapping $\\mathcal{Y}_\\alpha:{{\\mathcal H}^M}\\to{{\\mathcal H}^M}$ by $\\mathcal{Y}_\\alpha:=\\P[\\mathcal{R}]\\T[\\mathrm{P}]^\\alpha-{\\mathrm{Id}}$, i.e., $(\\forall {\\mathbf X}=({\\mathbf X}_m)_{m\\in\\mathcal{M}}\\in{{\\mathcal H}^M})$\\allowbreak$(\\forall m\\in\\mathcal{M})$\n\\begin{equation}\\label{eq:pert_mapping}\n\\mathcal{Y}_\\alpha({\\mathbf X})|_m= \\relu[{\\sigma_1({\\mathbf X}_m)-\\alpha\\sigma_{\\max}({\\mathbf X})}]{\\mathbf u}_{m1} {\\mathbf v}_{m1}^H - {\\mathbf X}_m.\n\\end{equation}\nFinally, we define the sequence $\\left(\\beta^{(n)}{\\mathbf Y}^{(n)}\\right)_{n\\in{\\mathbb N}}$ of perturbations in \\eqref{eq:superior_alg} by\n\\begin{equation}\\label{eq:seq_of_pert2}\n(\\forall n\\in{\\mathbb N})\\quad {\\mathbf Y}^{(n)}\\triangleq \\mathcal{Y}_{\\alpha^{(n)}}\\left({\\mathbf X}^{(n)}\\right),\n\\end{equation}\nwhere $\\left(\\alpha^{(n)}\\right)_{n\\in{\\mathbb N}}$ is a  sequence in $[0,1]$ and $\\left(\\beta^{(n)}\\right)_{n\\in{\\mathbb N}}$ is a summable sequence in $[0,1]$. The following proposition shows that the perturbations in \\eqref{eq:seq_of_pert2} can simultaneously reduce the objective value and the distance to the rank constraint set.\n\n\n\\begin{prop}\\label{prop:nonincreasing}\n\tLet $\\alpha\\in{\\mathbb R}_+$ and $\\lambda\\in[0,1]$. Then each of the following holds for $\\mathcal{Y}_\\alpha:{{\\mathcal H}^M}\\to{{\\mathcal H}^M}$ in \\eqref{eq:pert_mapping}.\n\t\\begin{enumerate}\n\t\t\\item The perturbations cannot increase the distance to the set ${\\setH^N_+}$, i.e., $(\\forall{\\mathbf X}\\in{{\\mathcal H}^M})$ $d({\\mathbf X}+\\lambda\\mathcal{Y}_\\alpha({\\mathbf X}),{\\setH^N_+})\\le d({\\mathbf X},{\\setH^N_+})$. In particular, $(\\forall{\\mathbf X}\\in{{\\mathcal H}^M})$ ${\\mathbf X}\\in{\\setH^N_+}\\Rightarrow {\\mathbf X}+\\lambda\\mathcal{Y}_\\alpha({\\mathbf X})\\in{\\setH^N_+}$.\n\t\t\\item If $\\alpha>0$, the perturbations decrease the value of the function $\\sobjf$ in \\eqref{eq:surrogate_obj}: $\\left(\\forall {\\mathbf X}\\in{{\\mathcal H}^M}\\right)$ $\\sobjf\\left({\\mathbf X}+\\lambda\\mathcal{Y}_\\alpha({\\mathbf X})\\right)<\\sobjf({\\mathbf X})$ whenever $\\sobjf({\\mathbf X})>0$.\n\t\t\\item If $\\alpha>0$ and ${\\mathbf X}\\in{\\setH^N_+}$, then the perturbations decrease the objective value of Problem~\\eqref{eq:SDP_hilbert}, i.e., $\\langle\\ipspacing\\langle{\\mathbf J},{\\mathbf X}+\\lambda\\mathcal{Y}_\\alpha({\\mathbf X})\\rangle\\ipspacing\\rangle <\\langle\\ipspacing\\langle{\\mathbf J},{\\mathbf X}\\rangle\\ipspacing\\rangle$ whenever $\\langle\\ipspacing\\langle{\\mathbf J},{\\mathbf X}\\rangle\\ipspacing\\rangle>0$.\n\t\t\\item If $\\lambda > 0$, the perturbations decrease the distance to the rank constraint set $\\mathcal{R}$. More precisely, $\\left(\\forall {\\mathbf X}\\in{{\\mathcal H}^M}\\right)$ $\\sobjg\\left({\\mathbf X}+\\lambda\\mathcal{Y}_\\alpha({\\mathbf X})\\right)<\\sobjg({\\mathbf X})$ whenever $\\sobjg({\\mathbf X})>0$.\n\t\\end{enumerate}\n\n\t\\emph{Proof:}\n\t\\begin{enumerate}\n\t\t\\pushQED{\\qed} \n\t\t\\item This is an immediate consequence of \\eqref{eq:pert_mapping}.\t\t\n\t\t\\item It follows from \\eqref{eq:svd_shrink} and \\eqref{eq:op_tp} that $(\\forall {\\mathbf X}\\in{{\\mathcal H}^M})$\\allowbreak$(\\forall \\alpha>0)$ $\\sobjf({\\mathbf X})>0\\Rightarrow\\sobjf(\\T[\\mathrm{P}]^\\alpha({\\mathbf X}))<\\sobjf({\\mathbf X})$. Moreover, by \\eqref{eq:proj_rank} we have that $(\\forall \\lambda\\in[0,1])$ $\\sobjf((1-\\lambda){\\mathbf X}+\\lambda\\P[\\mathcal{R}]({\\mathbf X}))\\le\\sobjf({\\mathbf X})$. \n\t\tThis implies $\\sobjf({\\mathbf X}+\\lambda\\mathcal{Y}_\\alpha({\\mathbf X}))= \\sobjf\\left((1-\\lambda){\\mathbf X}+\\lambda\\P[\\mathcal{R}]\\T[\\mathrm{P}]^\\alpha({\\mathbf X})\\right)\\le\\sobjf\\left(\\T[\\mathrm{P}]^\\alpha({\\mathbf X})\\right)<\\sobjf({\\mathbf X})$ whenever $\\sobjf({\\mathbf X})>0$.\n\t\t\\item This result follows from 1) and 2), since $(\\forall {\\mathbf X}\\in{\\setH^N_+})$ $\\langle\\ipspacing\\langle{\\mathbf J},{\\mathbf X}\\rangle\\ipspacing\\rangle=\\sobjf({\\mathbf X})$ according to \\eqref{eq:surrogate_obj}.\n\t\t\\item Since $\\mathcal{R}$ is closed, we can write $\\sobjg({\\mathbf X})=d({\\mathbf X},\\mathcal{R})=\\Norm[{\\mathbf X} - P_\\mathcal{R}({\\mathbf X})]=\\sqrt{\\sum_{m\\in\\mathcal{M}}\\sum_{i=2}^N\\sigma_i^2({\\mathbf X}_m)}$.\n\t\tTherefore, it follows from \\eqref{eq:svd_shrink} that $(\\forall {\\mathbf X}\\in{{\\mathcal H}^M})$\\allowbreak$(\\forall \\alpha\\in{\\mathbb R}_+)$ $\\sobjg(\\T[\\mathrm{P}]^\\alpha({\\mathbf X}))\\le\\sobjg({\\mathbf X})$.\n\t\tMoreover, by \\eqref{eq:proj_rank}, $(\\forall \\lambda\\in(0,1])$ $\\sobjg({\\mathbf X})>0$ implies that $\\sobjg((1-\\lambda){\\mathbf X}+\\lambda\\P[\\mathcal{R}]({\\mathbf X}))<\\sobjg({\\mathbf X})$.\n\t\tThis in turn implies $\\sobjg({\\mathbf X}+\\lambda\\mathcal{Y}_\\alpha({\\mathbf X}))= \\sobjg\\left((1-\\lambda){\\mathbf X}+\\lambda\\P[\\mathcal{R}]\\T[\\mathrm{P}]^\\alpha({\\mathbf X})\\right)<\\sobjg\\left(\\T[\\mathrm{P}]^\\alpha({\\mathbf X})\\right)\\le\\sobjg({\\mathbf X})$ whenever $\\sobjg({\\mathbf X})>0$.\t\\qedhere\\popQED\t\n\t\\end{enumerate}\n\\end{prop}\nWith the perturbations defined in \\eqref{eq:seq_of_pert2}, the iteration in \\eqref{eq:superior_alg} yields the update rule \n\\begin{equation}\\label{eq:superior_alg2}\n(\\forall n \\in {\\mathbb N})\\quad \\nextit[{\\mathbf X}] = T_\\star\\left(\\current[{\\mathbf X}] + \\current[\\beta] \\mathcal{Y}_{\\alpha^{(n)}}\\left(\\current[{\\mathbf X}]\\right)\\right)\n\\end{equation}\nof the proposed algorithm, where ${\\mathbf X}^{(0)} \\in{{\\mathcal H}^M}$ is arbitrary, $\\left(\\alpha^{(n)}\\right)_{n\\in{\\mathbb N}}$ is a sequence in $[0,1]$, and  $\\left(\\beta^{(n)}\\right)_{n\\in{\\mathbb N}}$ is a summable sequence in $[0,1]$.\n\n\n\n\\subsection{Convergence of the Proposed Algorithm}\\label{sec:boundedness}\nWe will now examine the convergence of the proposed algorithm in \\eqref{eq:superior_alg2}.\nFor this purpose, let $\\left(\\beta^{(n)}\\right)_{n\\in{\\mathbb N}}$ be a summable sequence in $[0,1]$, let $\\left(\\alpha^{(n)}\\right)_{n\\in{\\mathbb N}}$ be a sequence of nonnegative numbers, and denote by $\\left({\\mathbf Y}^{(n)}\\right)_{n\\in{\\mathbb N}}$ the sequence of perturbations according to \\eqref{eq:seq_of_pert2}. Then the sequence $\\left({\\mathbf X}^{(n)}\\right)_{n\\in{\\mathbb N}}$ produced by the algorithm in \\eqref{eq:superior_alg2} converges to a feasible point of Problem \\eqref{eq:SDR_hilbert} for all ${\\mathbf X}^{(0)}\\in{{\\mathcal H}^M}$. To show this, we prove the following facts.\n\\begin{enumerate}\n\t\\item The mapping $\\T[\\star]$ in \\eqref{eq:pocs} is bounded perturbation resilient.\n\t\\item The sequence $\\left({\\mathbf Y}^{(n)}\\right)_{n\\in{\\mathbb N}}$ is bounded, such that $\\left(\\beta^{(n)}{\\mathbf Y}^{(n)}\\right)_{n\\in{\\mathbb N}}$ is a sequence of bounded perturbations.\n\\end{enumerate}\n\n\\subsubsection{Bounded Perturbation Resilience of the Basic Algorithm}\nThe operator $T_\\star$ in \\eqref{eq:pocs} is known to be $\\alpha$-averaged (see, e.g., \\cite[Example 17.12(a)]{yamada2011minimizing}). We include this fact here for completeness:\n\\begin{remark}\\label{rem:alpha_avg}\n\tThe operator $T_\\star$ in \\eqref{eq:pocs} is $\\alpha$-averaged nonexpansive.\n\t\n\t\\emph{Proof:}\n\tNote that, for every nonempty subset $\\mathcal{C}\\subset{{\\mathcal H}^M}$, the reflector $R_\\mathcal{C}= {\\mathrm{Id}} + 2(P_\\mathcal{C}-{\\mathrm{Id}})$ is nonexpansive \\cite[Corollary~4.18]{bauschke2011convex}. Therefore, according to Definition~\\ref{def:alpha_avg_nonexpansive}, $\\left(\\forall \\mu\\in (0,2)\\right)$ the relaxed projector\n\t\\begin{equation*}\n\tT_\\mathcal{C}^\\mu= {\\mathrm{Id}} + \\mu(P_\\mathcal{C}-{\\mathrm{Id}}) = {\\mathrm{Id}} + \\frac{\\mu}{2}(R_\\mathcal{C} - {\\mathrm{Id}})\n\t\\end{equation*}\n\tis $\\mu\/2$-averaged. Further (see Fact~\\ref{fact:nonexpansive_composition}), the composite of finitely many averaged mappings is $\\alpha$-averaged for some\n\t\\pushQED{\\qed}\n\t$\\alpha\\in(0,1)$. \\qedhere\n\t\\popQED{}\n\\end{remark}\nConsequently, the bounded perturbation resilience of $\\T[\\star]$ follows directly from \\cite[Thm.~3.1]{he2017perturbation}. We summarize this fact in the following Lemma.\n\\begin{lemma}\\cite{he2017perturbation}\\label{lem:bpr_of_pocs}\n\t The algorithm in \\eqref{eq:superior_alg} is guaranteed to converge to a point in the solution set $\\mathcal{C}_\\star$ of the feasibility problem in \\eqref{eq:sdrCFP} if $\\mathcal{C}_\\star\\neq\\emptyset$ and $\\left(\\current[\\beta]\\current[{\\mathbf Y}]\\right)_{n\\in{\\mathbb N}}$ is a sequence of bounded perturbations. \n\t\n\t\\emph{Proof:}\n\t\\pushQED{\\qed}\n\tThe authors of \\cite{he2017perturbation} have proved the bounded perturbation resilience of $\\alpha$-averaged nonexpansive mappings with nonempty fix-point set in a real Hilbert space. \n\tConsequently, this lemma follows from Remark~\\ref{rem:alpha_avg} and \\cite[Thm. 3.1]{he2017perturbation}.\\qedhere\\popQED\n\\end{lemma}\n\n\n\n\n\n\\subsubsection{Boundedness of the Perturbations}\nIt remains to show that the sequence $\\left({\\mathbf Y}^{(n)}\\right)_{n\\in{\\mathbb N}}$ \nis bounded for all sequences $\\left(\\alpha^{(n)}\\right)_{n\\in{\\mathbb N}}$ of nonnegative numbers and $\\left(\\beta^{(n)}\\right)_{n\\in{\\mathbb N}}$ in $[0,1]$ such that $\\sum_{n\\in{\\mathbb N}}\\beta^{(n)}<\\infty$, regardless of the choice of ${\\mathbf X}^{(0)}\\in{{\\mathcal H}^M}$.\n\nTo this end, we note that $(\\forall n \\in {\\mathbb N})$ $\\Norm[{\\mathbf Y}^{(n)}]\\le\\Norm[{\\mathbf X}^{(n)}]$ for any sequence $\\left(\\alpha^{(n)}\\right)_{n\\in{\\mathbb N}}$ of nonnegative numbers:\n\\begin{lemma}\\label{lem:decreasing_norm}\n\tThe mapping $\\mathcal{Y}_\\alpha$ in \\eqref{eq:pert_mapping} satisfies \n\t\\begin{equation}\\label{eq:rem2}\n\t\t\\left(\\forall{\\mathbf X}\\in{{\\mathcal H}^M}\\right)\\left(\\forall\\alpha\\in{\\mathbb R}_+\\right)\\quad \\Norm[\\mathcal{Y}_\\alpha({\\mathbf X})]^2\\le\\Norm[{\\mathbf X}]^2.\n\t\\end{equation}\n\t\n\t\\emph{Proof:}\n\tLet $(\\forall m\\in\\mathcal{M})$ ${\\mathbf X}_m={\\mathbf U}_m\\mathrm{diag}(\\{\\sigma_i({\\mathbf X}_m)\\}_{i\\in\\mathcal{N}}){\\mathbf V}_m^H$ denote the singular value decomposition of the $m$th component matrix of ${\\mathbf X}$.\n\tAccording to \\eqref{eq:pert_mapping}, the $m$th component matrix of $\\mathcal{Y}_\\alpha({\\mathbf X})$ is given by $\\mathcal{Y}_\\alpha({\\mathbf X})|_m=-{\\mathbf U}_m{\\mathbf S}_m{\\mathbf V}_m^H$, where $(\\forall m\\in\\mathcal{M})$\n\t\\begin{equation*}\n\t{\\mathbf S}_m = \\mathrm{diag}\\left(\\min(\\sigma_1({\\mathbf X}_m),\\tau),\\sigma_2({\\mathbf X}_m),\\dots,\\sigma_N({\\mathbf X}_m)\\right)\n\t\\end{equation*}\n\twith $\\tau= \\alpha\\sigma_{\\max}({\\mathbf X})$.\n\tSince $(\\forall {\\mathbf W}\\in{\\mathcal H})$ $\\|{\\mathbf W}\\|^2=\\sum_{i\\in\\mathcal{N}}\\sigma_i^2({\\mathbf W})$, we can write\n\t\\begin{align*}\n\t\\Norm[\\mathcal{Y}_\\alpha({\\mathbf X})]^2=  \\sum_{m=1}^M\\|{\\mathbf S}_m\\|^2\\le  \\sum_{m=1}^M\\|{\\mathbf X}_m\\|^2=\\Norm[{\\mathbf X}]^2,\n\t\\end{align*}\n\twhich concludes the proof. \\qed\n\\end{lemma}\nThe following known result, which is a special case of \\cite[Lemma~5.31]{bauschke2011convex}, will be used in Lemma~\\ref{lem:boundedness} to prove that the proposed perturbations are bounded:\n\\begin{fact}\\label{fact:quasi_fejer}\t\n\tLet $\\left(a^{(n)}\\right)_{n\\in{\\mathbb N}}$, $\\left(\\beta^{(n)}\\right)_{n\\in{\\mathbb N}}$, and $\\left(\\gamma^{(n)}\\right)_{n\\in{\\mathbb N}}$ be sequences in ${\\mathbb R}_+$ such that $\\sum_{n\\in{\\mathbb N}}\\beta^{(n)}<\\infty$, $\\sum_{n\\in{\\mathbb N}}\\gamma^{(n)}<\\infty$ and\n\t$\n\t(\\forall n\\in{\\mathbb N})\\quad \n\ta^{(n+1)} \\le (1+\\beta^{(n)})a^{(n)} + \\gamma^{(n)}.\n\t$\n\tThen the sequence $\\left(a^{(n)}\\right)_{n\\in{\\mathbb N}}$ converges.\n\\end{fact}\n\\begin{lemma}\\label{lem:boundedness}\n\tSuppose that $\\left(\\beta^{(n)}\\right)_{n\\in{\\mathbb N}}$ is a summable sequence in $[0,1]$ and that $(\\forall n\\in{\\mathbb N})$ $\\alpha^{(n)}\\ge0$. Then the sequence of perturbations $\\left(\\beta^{(n)}{\\mathbf Y}^{(n)}\\right)$ with ${\\mathbf Y}^{(n)}$ defined by \\eqref{eq:seq_of_pert2} is bounded.\n\t\n\t\\emph{Proof:}\n\t\t\tWe need to show that $(\\exists R\\in{\\mathbb R})(\\forall n\\in{\\mathbb N})\\ \\Norm[{\\mathbf Y}^{(n)}]\\le R$. To this end, observe that $\\left(\\forall {\\mathbf X}^{(n)}\\in {{\\mathcal H}^M}\\right)\\left(\\forall {\\mathbf Z}\\in\\mathrm{Fix}(T_\\star)\\right)$ it holds that\n\t\t\t\\begin{align*}\n\t\t\t\\Norm[{\\mathbf X}^{(n+1)}-{\\mathbf Z}] &= \\Norm[T_\\star\\left({\\mathbf X}^{(n)}+\\beta^{(n)}{\\mathbf Y}^{(n)}\\right)-{\\mathbf Z}]\\\\\n\t\t\t\t&\\overset{(a)}{\\le} \\Norm[{\\mathbf X}^{(n)}+\\beta^{(n)}{\\mathbf Y}^{(n)}-{\\mathbf Z}]\\\\\n\t\t\t\t&\\overset{(b)}{\\le} \\Norm[{\\mathbf X}^{(n)}-{\\mathbf Z}] + \\beta^{(n)}\\Norm[{\\mathbf Y}^{(n)}],\n\t\t\t\\end{align*}\n\t\t\twhere (a) follows from the nonexpansivity of $T_\\star$, and (b) is a consequence of the triangle inequality.\n\t\t\tBy Lemma~\\ref{lem:decreasing_norm}, the perturbations defined in  \\eqref{eq:seq_of_pert2} satisfy $(\\forall n\\in{\\mathbb N})$ $\\Norm[{\\mathbf Y}^{(n)}]\\le\\Norm[{\\mathbf X}^{(n)}]$. Consequently, applying the triangle inequality again yields\n\t\t\t\\begin{align*}\\label{eq:bounded_proof1}\n\t\t\t\\Norm[{\\mathbf X}^{(n+1)}-{\\mathbf Z}]  &\\le \\Norm[{\\mathbf X}^{(n)}-{\\mathbf Z}]\\\\\\notag\n\t\t\t&\\qquad + \\beta^{(n)}\\left(\\Norm[{\\mathbf X}^{(n)}-{\\mathbf Z}]+\\Norm[{\\mathbf Z}]\\right).\n\t\t\t\\end{align*}\t\t\t\n\t\t\t\n\t\t\tBy defining $(\\forall n\\in{\\mathbb N})$ $a^{(n)}=\\Norm[{\\mathbf X}^{(n)}-{\\mathbf Z}]$ and $\\gamma^{(n)}=\\beta^{(n)}\\Norm[{\\mathbf Z}]$, we can deduce from Fact~\\ref{fact:quasi_fejer} that the sequence $\\left(a^{(n)}\\right)_{n\\in{\\mathbb N}}$ converges. This implies that there exists $r\\in{\\mathbb R}$ such that $(\\forall n\\in{\\mathbb N})$ $\\Norm[{\\mathbf X}^{(n)}-{\\mathbf Z}]\\le r$.\n\t\t\t\n\t\t\tConsequently, we have \n\t\t\t\\begin{align*}\n\t\t\t(\\forall n\\in{\\mathbb N})\\quad \\Norm[{\\mathbf Y}^{(n)}] &\\overset{(a)}{\\le}\\Norm[{\\mathbf X}^{(n)}]\n\t\t\t\\overset{(b)}{\\le}  \\Norm[{\\mathbf X}^{(n)}-{\\mathbf Z}] + \\Norm[{\\mathbf Z}]\\\\\n\t\t\t&\\overset{(c)}{\\le}r+\\Norm[{\\mathbf Z}]=:R\n\t\t\t\\end{align*}\n\t\t\t\\pushQED{\\qed}\n\t\t\twhere (a) follows from Lemma~\\ref{lem:decreasing_norm}, (b) follows from the triangle inequality, and (c) follows from Fact~\\ref{fact:quasi_fejer}.\\qedhere\n\t\t\t\\popQED\n\\end{lemma}\n\n\nCombining Lemmas~\\ref{lem:bpr_of_pocs} and \\ref{lem:boundedness} shows that the proposed algorithm converges to a feasible point of the relaxed semidefinite program in \\eqref{eq:SDR_hilbert}. This is summarized in the following proposition.\n\n\\begin{prop}\n\tThe sequence produced by the algorithm in \\eqref{eq:superior_alg} with perturbations given by \\eqref{eq:seq_of_pert2} is guaranteed to converge to a feasible point of Problem~\\eqref{eq:SDR_hilbert} for all ${\\mathbf X}^{(0)}\\in{{\\mathcal H}^M}$ if $\\left(\\beta^{(n)}\\right)_{n\\in{\\mathbb N}}$ is a summable sequence in $[0,1]$ and $\\left(\\alpha^{(n)}\\right)_{n\\in{\\mathbb N}}$ is a sequence in ${\\mathbb R}_+$.\n\t\n\t\\emph{Proof:}\n\tFollows immediately from Lemma~\\ref{lem:bpr_of_pocs} and Lemma~\\ref{lem:boundedness}.\n\\end{prop} \n\n\\subsection{Relation to the Superiorization Methodology}\\label{sec:relation_to_superiorization}\nThe authors of \\cite{censor2015weak} define superiorization as follows:\n\\begin{quote}\n\\textit{'The superiorization methodology works by taking an iterative algorithm, investigating its perturbation resilience, and then, using proactively such permitted perturbations, forcing the perturbed algorithm to do something useful in addition to what it is originally designed to do.'} \n\\end{quote}\nAlthough our proposed algorithm matches this informal definition,\nthere are some slight differences to the formal definition in \\cite{censor2015weak}, where the perturbations are required to be nonascending vectors for a convex superiorization objective function.\n\\begin{definition}[Nonascending Vectors \\cite{censor2015weak}]\\\nGiven a function $\\supObj:{\\mathbb R}^J\\to{\\mathbb R}$ and a point ${\\mathbf y}\\in{\\mathbb R}^J$, a vector $\\mathbf{d}\\in{\\mathbb R}^J$ is said to be nonascending for $\\supObj$ at ${\\mathbf y}$ iff $\\|\\mathbf{d}\\|\\le 1$ and there is a $\\delta>0$ such that for all $\\lambda\\in[0,\\delta]$ we have $\\supObj({\\mathbf y}+\\lambda\\mathbf{d}) \\le \\supObj({\\mathbf y})$.\n\\end{definition}\n\nIn our case, the goal of superiorization is two-fold, in the sense that it is expressed by two separate functions $\\sobjf:{{\\mathcal H}^M}\\to{\\mathbb R}$ and $\\sobjg:{{\\mathcal H}^M}\\to{\\mathbb R}$.\nWhile the function $\\sobjf$ in \\eqref{eq:equiv_objective} is convex, the function $\\sobjg$ in \\eqref{eq:rank_dist} (i.e., the distance to nonconvex rank constraint set $\\mathcal{R}$ in \\eqref{eq:rank_constraint}) is a nonconvex function.\nMoreover, we use perturbations that are not restricted to a unit ball, and therefore they are not necessarily nonascending vectors.\nHowever, as we have shown in Proposition~\\ref{prop:nonincreasing}, the proposed perturbations simultaneously reduce the values of $\\sobjf$ and $\\sobjg$.\nKeeping these slight distinctions in mind, we will refer to the proposed algorithm in \\eqref{eq:superior_alg} as \\emph{Superiorized Projections onto Convex Sets}. \t\n\n\\subsection{Summary of the Proposed Algorithm}\\label{sec:algorithm_summary}\nThe proposed multi-group multicast beamforming algorithm is summarized in Algorithm~\\ref{alg:spocs}. It is defined by the relaxation parameters $\\mu_1,\\dots,\\mu_{K+2}$ of the operator $\\T[\\star]$ in \\eqref{eq:pocs},\na scalar $a\\in(0,1)$ controlling the decay of the power-reducing perturbations, a scalar $b\\in(0,1)$ controlling the decay of the sequence of perturbation scaling factors, i.e., $(\\forall n\\in{\\mathbb N})$ $\\alpha^{(n)}=a^n$ and $\\beta^{(n)}=b^n$. The stopping criterion is based on a tolerance value $\\epsilon>0$, and a maximum number $n_{\\max}$ of iterations.\n\nThe arguments of the algorithm are the indices $g_1,\\dots,g_K$ assigning a multicast group to each user, the channel vectors ${\\mathbf h}_1,\\dots,{\\mathbf h}_K\\in{\\mathbb C}^N$, SINR requirements $\\gamma_1,\\dots,\\gamma_K$, and noise powers $\\sigma_1,\\dots,\\sigma_K$ of all users as well as the per-antenna power constraints $p_1,\\dots,p_N$.\nAt each step, the algorithm computes a perturbation according to \\eqref{eq:pert_mapping} and applies the feasibility seeking operator $\\T[\\star]$ in \\eqref{eq:pocs}. It terminates when the relative variation of the estimate falls within the tolerance $\\epsilon$, or when the maximum number $n_{\\max}$ of iterations is reached.\nFinally, the beamforming vectors ${\\mathbf w}=\\{{\\mathbf w}_m\\}_{m\\in\\mathcal{M}}$ are computed by extracting the strongest principal component\n\\begin{equation}\\label{eq:principal}\n(\\forall m \\in\\mathcal{M})\\quad {\\mathbf w}_m=\\psi({\\mathbf X}_m)\\triangleq\\sqrt{\\sigma_1({\\mathbf X}_m)}{\\mathbf u}_{m1},\n\\end{equation}\nwhere $(\\forall m \\in\\mathcal{M})$ ${\\mathbf X}_m={\\mathbf U}_m{\\mathbf \\Sigma}_m{\\mathbf V}_m^H$, ${\\mathbf U}_m=[{\\mathbf u}_{m1},\\cdots,{\\mathbf u}_{mN}]$, and ${\\mathbf \\Sigma}_m=\\mathrm{diag}\\left(\\sigma_1({\\mathbf X}_m),\\dots,\\sigma_N({\\mathbf X}_m)\\right)$.\n\\begin{algorithm}[H]\n\\caption{Superiorized Projections onto Convex Sets}\\label{alg:spocs}\n\\begin{algorithmic}[1]\n\t\\State \\textbf{Parameters:}~ $\\{\\mu_k\\}_{k=1}^{K+2},~ a,b\\in(0,1),~ \\epsilon>0,~ n_{\\max}\\in{\\mathbb N}$\n\t\\State \\textbf{Input:}~ $\\{g_k\\}_{k\\in\\mathcal{K}}$, $\\{{\\mathbf h}_k\\}_{k\\in\\mathcal{K}}$, $\\{\\gamma_k\\}_{k\\in\\mathcal{K}}$, $\\{\\sigma_k\\}_{k\\in\\mathcal{K}}$, $\\{p_i\\}_{i\\in\\mathcal{N}}$\n\t\\State \\textbf{Output:}~ $\\{{\\mathbf w}_m\\in{\\mathbb C}^N\\}_{m\\in\\mathcal{M}}$\n\t\\State \\textbf{Initialization:}~ Choose arbitrary ${\\mathbf X}^{(0)}\\in{{\\mathcal H}^M}$\n\t\\For{$n=0,\\dots,n_{\\max}-1$}\t\n\t\\State ${\\mathbf Y}^{(n)}\\gets\\mathcal{Y}_{a^n}\\left({\\mathbf X}^{(n)}\\right)$\\Comment{Eq.~\\eqref{eq:pert_mapping}}\n\t\\State ${\\mathbf X}^{(n+1)} \\gets \\T[\\star]\\left({\\mathbf X}^{(n)} + b^n{\\mathbf Y}^{(n)}\\right)$\\Comment{Eq.~\\eqref{eq:pocs}}\n\t\\If{$\\Norm[{\\mathbf X}^{(n+1)}-{\\mathbf X}^{(n)}]<\\epsilon \\Norm[{\\mathbf X}^{(n+1)}]$}\n\t\\State \\textbf{break}\n\t\\EndIf\n\t\\EndFor\n\t\\State \\textbf{return} ${\\mathbf w}=\\left\\{\\psi\\left({\\mathbf X}_m^{(n+1)}\\right)\\right\\}_{m\\in\\mathcal{M}}$\\Comment{Eq.~\\eqref{eq:principal}}\n\\end{algorithmic}\n\\end{algorithm}\n\\section{Numerical Results}\nIn this section, we compare Algorithm~\\ref{alg:spocs} (\\texttt{S-POCS}) to several other methods from the literature. We choose identical noise levels and target SINRs for all users, i.e., $(\\forall k \\in\\mathcal{K})$ $\\sigma_k=\\sigma$ and $\\gamma_k=\\gamma$.\nFor each problem instance, we generate $K$ i.i.d. Rayleigh-fading channels $(\\forall k\\in\\mathcal{K})$ ${\\mathbf h}_k\\sim\\mathcal{CN}({\\mathbf 0},\\sigma^2{\\mathbf I}_N)$.\n\nIn the first simulation, we drop the per-antenna power constraints, i.e., we set $(\\forall i \\in\\mathcal{N})$ $p_i=\\infty$, and we consider the following algorithms:\n\\begin{itemize}\n\\item The proposed method summarized in Algorithm~\\ref{alg:spocs} (\\texttt{S-POCS})\n\\item Semidefinite relaxation with Gaussian randomization \\cite{karipidis2008quality} (\\texttt{SDR-GauRan})\n\\item The successive convex approximation algorithm from \\cite{mehanna2014feasible}, \\cite{christopoulos2015multicast} (\\texttt{FPP-SCA}{})\n\\item The ADMM-based convex-concave procedure from \\cite{chen2017admm} (\\texttt{CCP-ADMM})\n\\end{itemize} \nThe \\texttt{S-POCS}{} algorithm is as described in Algorithm~\\ref{alg:spocs}, with parameters $a=0.95$, $b=0.999$, $\\epsilon=10^{-6}$, $n_{\\max}=10^5$. For the QoS-constraint sets, we use relaxation parameters $(\\forall k\\in\\mathcal{K})$ $\\mu_k=1.9$, and for the per-antenna power constraint set $\\mathcal{P}$ and the PSD constraint ${\\setH^N_+}$, we use unrelaxed projections, i.e., $\\mu_{K+2}=\\mu_{K+1}=1$. We initialize the \\texttt{S-POCS}{} algorithm with ${\\mathbf X}^{(0)}={\\mathbf 0}$.\nThe convex optimization problems in the \\texttt{SDR-GauRan}{} and \\texttt{FPP-SCA}{} algorithms are solved with the interior point solver SDPT3 \\cite{toh1999sdpt3}. The parameters of the \\texttt{CCP-ADMM}{} algorithm are as specified in \\cite{chen2017admm}.\nAchieving a fair comparison between these methods is difficult because the structure of the respective algorithms is quite different. \n\nThe \\texttt{SDR-GauRan}{} algorithm begins by solving the relaxed problem in \\eqref{eq:SDR_hilbert}, and, subsequently, generates random candidate beamforming vectors using the \\texttt{RandA}{} method \\cite{sidiropoulos2006transmit}, \\cite{karipidis2008quality}. In the multi-group setting, where $M>1$, an additional convex optimization problem (multigroup multicast power control (MMPC), \\cite{karipidis2008quality}) needs to be solved for each candidate vector. If no feasible MMPC problem is found during the \\texttt{RandA}{} procedure, we define the output of the \\texttt{SDR-GauRan}{} algorithm to be $\\{\\psi({\\mathbf X}^\\star_m)\\}_{m\\in\\mathcal{M}}$, where ${\\mathbf X}^\\star\\in{{\\mathcal H}^M}$ is a solution to the relaxed SDP in \\eqref{eq:SDR_hilbert}.\n\nThe \\texttt{FPP-SCA}{} algorithm from \\cite{mehanna2014feasible} works by solving a sequence of convex subproblems. By introducing slack variables, the feasibility of each subproblem is ensured. This obviates the need for a feasible initialization point, which is typically required to ensure convergence of CCP\/SCA algorithms.\n\nThe \\texttt{CCP-ADMM}{} algorithm uses an ADMM algorithm to find a feasible starting point for the CCP. Subsequently, a similar ADMM algorithm is used to approximate each subproblem of the CCP.\nBecause the ADMM is a first-order method, the performance of \\texttt{CCP-ADMM}{} is heavily dependent on the stopping criteria of the inner ADMM algorithm.\n\nBy contrast, the \\texttt{S-POCS}{} algorithm does not require an initialization phase, and it works by iteratively applying a sequence of operators, which can be computed in a fixed number of steps.\nTherefore, we compare the performance based on computation time. Although we exclude the time required for evaluating the performance, we note that the computation time required by each of the methods severely depends on the particular implementation.\n\nThe authors of \\cite{chen2017admm} assess the performance of the considered algorithms based by comparing the transmit power achieved by the resulting beamformers. However, none of the methods considered here can guarantee feasibility of the beamforming vectors, when the algorithms are terminated after a finite number of iterations. \nFurthermore, in the multi-group case, it may not be possible to scale an arbitrary candidate beamformer ${\\mathbf w}=\\{{\\mathbf w}_m\\in{\\mathbb C}^N\\}_{m\\in\\mathcal{M}}$ such that it satisfies all constraints in Problem~\\eqref{eq:original_problem}.\nIn principle, we could evaluate the performance by observing both the objective value (i.e., the transmit power of the beamformers) and a measure of constraints violation such as the normalized proximity function used in \\cite{censor2012effectiveness}. However, defining this measure of constraints violation is not straightforward, as the considered methods approach the problem in different spaces. Moreover, we are interested in expressing the quality of a beamforming vector by a single value to simplify the presentation.\nTherefore, we will compare the performance based on the minimal SINR achieved by the beamformer $\\sqrt{\\bfscale}\\cdot{\\mathbf w}$ with\n\\begin{equation*}\n\\bfscale =  \\min\\left(\\frac{P_{\\mathrm{SDR}}^\\star }{\\sum_{m=1}^M{\\mathbf w}_m^H{\\mathbf w}_m}, \\min_{i\\in\\mathcal{N}}\\left( \\frac{p_i}{\\sum_{m=1}^M |w_{im}|}\\right)\\right).\n\\end{equation*}\nThe scaled vector $\\sqrt{\\bfscale}\\cdot{\\mathbf w}$ satisfies all power constraints, and its total power is bounded by the optimal objective value $P_{\\mathrm{SDR}}^\\star$ of the relaxed SDP in \\eqref{eq:SDR_hilbert}. More compactly, given a candidate beamformer ${\\mathbf w}=\\{{\\mathbf w}_m\\in{\\mathbb C}^N\\}_{m\\in\\mathcal{M}}$ for Problem~\\eqref{eq:original_problem}, we assess its performance based on the function\\footnote{For the sake of simplicity, we will refer to the \\emph{minimal SINR achieved by the scaled beamformer} $\\sqrt{\\bfscale}\\cdot{\\mathbf w}$ in \\eqref{eq:sinr_min} as \\emph{SINR} in the following.}\n\\begin{equation}\\label{eq:sinr_min}\n\\mathrm{SINR}^{\\min}_{\\rho}\\left({\\mathbf w}\\right)=\n\\underset{k\\in\\mathcal{K}}{\\min}~ \\frac{ |{\\mathbf w}_m^H{\\mathbf h}_k|^2}{\\sum_{l\\neq m}|{\\mathbf w}_l^H{\\mathbf h}_k|^2+\\frac{\\sigma_k^2 }{\\bfscale}}.\n\\end{equation}\nSince $P_{\\mathrm{SDR}}^\\star$ is a lower bound on the objective value of the original problem in \\eqref{eq:original_problem}, it holds $(\\forall \\{{\\mathbf w}_m\\in{\\mathbb C}^N\\}_{m\\in\\mathcal{M}})$ that $\\mathrm{SINR}^{\\min}_{\\rho}({\\mathbf w})\\le\\gamma$, where equality can only be achieved, if the relaxed problem in \\eqref{eq:SDR_hilbert} has a solution composed of rank-one matrices.\n\n\\subsection{Performance vs. Computation Time}\nWe will now examine how the performance metric in \\eqref{eq:sinr_min} evolves over time for beamforming vectors produced by the respective algorithms.\nFigure~\\ref{fig:single_run} shows the performance comparison for an exemplary scenario with $N=20$ antennas, and $K=20$ users split evenly into $M=2$ groups, where $\\sigma=1$, $\\gamma=1$, and $(\\forall i\\in\\mathcal{N})$ $p_i=\\infty$.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=0.59]{figures\/timeseq_K=20_M=2_N=20_single_markers.pdf}\n\t\\caption{$\\mathrm{SINR}^{\\min}_{\\rho}({\\mathbf w}^{(t)})$ over time in a system with $N=20$ antennas and $K=20$ users users split evenly into $M=2$ multicast groups. }\n\t\\label{fig:single_run}\n\\end{figure}\nIt can be seen that the \\texttt{S-POCS}{} algorithm quickly converges to a point achieving an SINR close to the specified target value $\\gamma$.\nThe discontinuities in the SINR curve for the \\texttt{CCP-ADMM}{} algorithm are due to the inner- and outer optimization loops. \nFor the \\texttt{SDR-GauRan}{} algorithm, the SINR increases whenever the randomization produces a beamformer with better performance than the previous one. \nThe SINR of the \\texttt{FPP-SCA}{} algorithm improves continuously, albeit more slowly than the \\texttt{S-POCS}{} and \\texttt{CCP-ADMM}{} algorithms.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=0.59]{figures\/sinr_percentiles_timeseq_replotted-compressed.pdf}\n\t\\caption{$\\mathrm{SINR}^{\\min}_{\\rho}({\\mathbf w}^{(t)})$ over time in a system with $N=20$ antennas and $K=20$ users split evenly into $M=2$ multicast groups. The shaded regions include the outcomes for $100\\%$, $75\\%$, $50\\%$, and $25\\%$ out of 100 problem instances, respectively,\n    and the bold line represents the median.}\n\t\\label{fig:multi_group_vs_time}\n\\end{figure}\nNext, we evaluate the performance over 100 randomly generated problems. Since the SINR does not increase monotonically for all of the methods considered, we assume that each algorithm can keep track of the best beamformer produced so far. In this way, the oscillations in the SINR metric for the \\texttt{CCP-ADMM}{} algorithm do not have a negative impact on its average performance.\n\nFigure \\ref{fig:multi_group_vs_time} shows the performance of the beamforming vectors computed with the respective algorithms over time for a system with $N=20$ transmit antennas, and $K=20$ users split evenly into $M=2$ multicast groups.\nThe shaded regions correspond to the  $100\\%$, $75\\%$, $50\\%$, and $25\\%$ quantiles over all randomly generated problems. More precisely, the margins of the shaded regions correspond to the 1st, 13th, 26th, 38th, 63rd, 75th, 88th, and 100th out of 100 sorted y-axis values. For each algorithm, the median is represented by a bold line.\nThe \\texttt{S-POCS}{} algorithm achieves the highest median SINR, while requiring the lowest computation time among all methods considered. Moreover, it can be seen that the variation around this median value is less severe compared to the remaining approaches.\nPut differently, the time required for reaching a certain SINR varies much less severely for the \\texttt{S-POCS}{} algorithm than for the remaining methods.\nThis can be of particular interest in delay sensitive applications, where a beamforming vector for a given channel realization must be computed within a fixed time period.\n\n\n\n\n\n\\subsection{Varying number of antennas}\nIn this subsection, we investigate the impact of the transmit antenna array size $N$ on the performance of the respective beamforming algorithms. To do so, we generate 100 random problem instances for each array size $N$ with $K=20$ users split evenly in to $M=2$ multicast groups. We choose unit target SINR and unit noise power for all users, and unit per-antenna power constraints, i.e., $\\gamma=1$, $\\sigma=1$ and $(\\forall i \\in{\\mathbb N})$ $p_i=1$.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=0.59]{figures\/swp_N_sinr.pdf}\n\t\\caption{$\\mathrm{SINR}^{\\min}_{\\rho}({\\mathbf w})$ for $K=20$ users split evenly into $M=2$ groups for varying antenna array sizes $N$.}\n\t\\label{fig:multi_group_swpN}\n\\end{figure}\nFor the \\texttt{SDR-GauRan}{} algorithm, we generate $200$ candidate beamforming vectors for each problem instance. We use the \\texttt{CCP-ADMM}{} algorithm with parameters as specified in \\cite{chen2017admm}. Since the inner ADMM iteration converges slowly for some problem instances, we set the maximal number of steps of the ADMM to $j_{\\max}=300$. For the outer CCP loop, we use the stopping criteria specified in \\cite{chen2017admm}, i.e., we stop the algorithm once the relative decrease of the objective value is below $10^{-3}$ or $t_{\\max}=30$ outer iterations are exceeded.\nFor the \\texttt{FPP-SCA}{} algorithm, we use a fixed number of $30$ successive convex approximation steps.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.59]{figures\/swp_N_time.pdf}\n\\caption{Computation time for $K=20$ users split evenly into $M=2$ groups for varying antenna array sizes $N$.}\n\\label{fig:multi_group_swpN_time}\n\\end{figure}\n\nFigure~\\ref{fig:multi_group_swpN} shows the performance metric in \\eqref{eq:sinr_min} for different numbers $N$ of transmit antennas, averaged over 100 random problem instances each. For all $N$, \\texttt{S-POCS}{} achieves highest value for $\\mathrm{SINR}^{\\min}_{\\rho}(\\cdot)$, followed by the \\texttt{FPP-SCA}{}, \\texttt{CCP-ADMM}{}, and \\texttt{SDR-GauRan}{} algorithms. \nFor $N\\ge80$, the \\texttt{S-POCS}{} algorithm achieves an SINR of $\\mathrm{SINR}^{\\min}_{\\rho}({\\mathbf w}_{\\texttt{S-POCS}})\\ge \\SI{-0.05}{dB}$.\nBy contrast, the remaining methods do not exceed $\\mathrm{SINR}^{\\min}_{\\rho}({\\mathbf w}_{\\texttt{FPP-SCA}})=\\SI{-0.12}{dB}$,  $\\mathrm{SINR}^{\\min}_{\\rho}({\\mathbf w}_{\\texttt{CCP-ADMM}})\\ge \\SI{-0.15}{dB}$ ,$\\mathrm{SINR}^{\\min}_{\\rho}({\\mathbf w}_{\\texttt{SDR-GauRan}})\\ge \\SI{-1.18}{dB}$, respectively.\n\n\n\n\n\nThe corresponding average computation times are shown in Figure~\\ref{fig:multi_group_swpN_time}.\nThe \\texttt{S-POCS}{} algorithm requires\n\\SI{0.26}{\\%}--\\SI{2.38}{\\%} of the computation time required by \\texttt{SDR-GauRan}{},\n\\SI{0.95}{\\%}--\\SI{11.64}{\\%} of the computation time required by\n\\texttt{FPP-SCA}{}, and\n\\SI{6.49}{\\%}--\\SI{233.6}{\\%} of the computation time required by \\texttt{CCP-ADMM}{}.\nFor $N\\ge80$, the computation time of \\texttt{S-POCS}{} exceeds that of \\texttt{CCP-ADMM}{}.\n\n\n\n\\subsection{Varying number of users}\nIn the following simulation, we fix an array size of $N=50$ antenna elements, and we evaluate the performance of each method for $K\\in\\{4, 8, 16, 32, 48, 64\\}$ users split evenly into $M=4$ multicast groups. Figure~\\ref{fig:multi_group_swpK} shows the performance metric in \\eqref{eq:sinr_min} averaged over $100$ random problem instances for each $K$. As before, we choose $\\gamma=1$, $\\sigma=1$, and $(\\forall i\\in\\mathcal{N})$ $p_i=1$.\n\nWhile all algorithms achieve close to optimal performance for small numbers of users, the SINR in \\eqref{eq:sinr_min} decreases considerably faster for \\texttt{SDR-GauRan}{} than for the remaining methods. \nFor all values of $K$, \\texttt{S-POCS}{} achieves the highest value for $\\mathrm{SINR}^{\\min}_{\\rho}(\\cdot)$ among all methods.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=0.59]{figures\/swp_K_sinr.pdf}\n\t\\caption{$\\mathrm{SINR}^{\\min}_{\\rho}({\\mathbf w})$ for a system with $N=50$ transmit antennas and a varying number of users split evenly into $M=4$ multicast groups.}\n\t\\label{fig:multi_group_swpK}\n\\end{figure}\n\n\n\nThe corresponding average computation times are shown in Figure~\\ref{fig:multi_group_swpK_time}. \n\\texttt{S-POCS}{} requires\n\\SI{1.76}{\\%}--\\SI{6.12}{\\%} of the computation time required by \\texttt{SDR-GauRan}{},\n\\SI{3.75}{\\%}--\\SI{5.41}{\\%} of the computation time required by \\texttt{FPP-SCA}{}, and\n\\SI{20.18}{\\%}--\\SI{1626}{\\%} of the computation time required by \\texttt{CCP-ADMM}{}.\nWhile the \\texttt{CCP-ADMM}{} takes only a fraction of the time required by \\texttt{S-POCS}{} for small $K$, it slows down considerably as $K$ increases. \nFor moderate and large numbers of users, \\texttt{S-POCS}{} outperforms the remaining methods in terms of both approximation gap and computation time.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=0.59]{figures\/swp_K_time.pdf}\n\t\\caption{Computation time for a system with $N=50$ transmit antennas and a varying number of users split evenly into $M=4$ multicast groups.}\n\t\\label{fig:multi_group_swpK_time}\n\\end{figure}\n\n\n\\subsection{Varying Target SINR}                                                     \nIn the following simulation, we evaluate the impact of the target SINR on the respective algorithms in a system with $N=30$ antenna elements, $K=20$ users split evenly into $M=2$ multicast groups, and unit noise power $\\sigma=1$. Since the target SINR has a strong impact on the transmit power, we set $(\\forall i\\in\\mathcal{N})$ $p_i=\\infty$, to avoid generating infeasible instances of Problem~\\eqref{eq:original_problem}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.59]{figures\/swp_gamma_sinr.pdf}\n\\caption{$\\mathrm{SINR}^{\\min}_{\\rho}({\\mathbf w})$ for a system with $N=30$ transmit antennas and $K=20$ users split evenly into $M=2$ multicast groups.}\n\\label{fig:multi_group_swpgamma}\n\\end{figure}\nFigure~\\ref{fig:multi_group_swpgamma} shows the performance metric in \\eqref{eq:sinr_min} achieved by each method for the respective target SINR. Except for the \\texttt{SDR-GauRan}{} algorithm, which exhibits a gap of about \\SI{2}{dB} to the target SINR, all methods achieve close to optimal performance for each target SINR. \nFigure~\\ref{fig:multi_group_swpgamma_time} shows the computation time required by each algorithm for varying target SINR $\\gamma$. The average computation time of \\texttt{FPP-SCA}{} is almost constant. For \\texttt{SDR-GauRan}{} and \\texttt{CCP-ADMM}{}, the computation decreases slightly with an increasing target SINR. While the proposed \\texttt{S-POCS}{} algorithm converges quickly for low target SINR levels, its computation time exceeds that of the \\texttt{CCP-ADMM}{} for target SINRs above \\SI{8}{dB}. This indicates that the best choice of first-order algorithms for multicast beamforming depends on the regime in which the system is operated.\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.59]{figures\/swp_gamma_time.pdf}\n\\caption{Computation time for a system with $N=30$ transmit antennas and $K=20$ users split evenly into $M=2$ multicast groups.}\n\\label{fig:multi_group_swpgamma_time}\n\\end{figure}\n\\section{Conclusion}\nIn this paper, we proposed an algorithm for multi-group multicast beamforming with per-antenna power constraints. We showed that the sequence produced by this algorithm is guaranteed to converge to a feasible point of the relaxed semidefinite program, while the perturbations added in each iteration reduce the objective value and the distance to the nonconvex rank constraints.\nNumerical comparisons show that the proposed method outperforms state-of-the-art algorithms in terms of both approximation gap and computation time in many cases.\nIts advantage over existing algorithms is particularly pronounced in the low target SINR regime as well as for large numbers of receivers. This makes the proposed method particularly relevant for low-energy or massive access applications. \n\nIn comparison to other techniques, the computation time of the proposed method varies less severely across different problem instances of the same dimension. In communication systems, which are typically subject to strict latency constraints, the iteration can be terminated after a fixed number of steps without suffering severe performance loss.  Moreover, the simple structure of the proposed method allows for a straightforward implementation in real-world systems. \n\nThe applicability of the proposed algorithm is not restricted to the multicast beamforming problem considered here. A slight modification of the rank-constraint naturally leads to an algorithm for the general rank multicast beamforming problem considered in \\cite{taleb2020general}. Future research could apply superiorized projections onto convex sets to other nonconvex QCQP problems such as MIMO detection or sensor network localization \\cite{luo2010semidefinite}.\n\n\\section{Appendix}\n\\begin{remark}\\label{rem:real_inner_product}\n\tThe function $\\langle\\cdot,\\cdot\\rangle$ defined in \\eqref{eq:innerProduct} is a real inner product.\n\t\n\t\\emph{Proof:}\n\tGiven a real vector space $\\mathcal{V}$, a real inner product is a function $\\langle\\cdot,\\cdot\\rangle:\\mathcal{V}\\times\\mathcal{V}\\to{\\mathbb R}$ satisfying \\cite{jain2005functional}\n\t\\begin{enumerate}\n\t\t\\item $(\\forall {\\mathbf x}\\in\\mathcal{V})$ $\\langle{\\mathbf x},{\\mathbf x}\\rangle\\ge0$ and $\\langle{\\mathbf x},{\\mathbf x}\\rangle=0\\iff {\\mathbf x}={\\mathbf 0}$ \\label{it:1}\n\t\t\\item $(\\forall{\\mathbf x},{\\mathbf y}\\in\\mathcal{V})$ $\\langle{\\mathbf x},{\\mathbf y}\\rangle= \\langle{\\mathbf y},{\\mathbf x}\\rangle$ \\label{it:2}\n\t\t\\item $(\\forall{\\mathbf x},{\\mathbf y}\\in\\mathcal{V})$\\allowbreak$(\\forall \\alpha\\in{\\mathbb R})$ $\\langle\\alpha{\\mathbf x},{\\mathbf y}\\rangle= \\alpha\\langle{\\mathbf x},{\\mathbf y}\\rangle$ \\label{it:3}\n\t\t\\item $(\\forall{\\mathbf x},{\\mathbf y},{\\mathbf z}\\in\\mathcal{V})$ $\\langle{\\mathbf x}+{\\mathbf y},{\\mathbf z}\\rangle= \\langle{\\mathbf x},{\\mathbf y}\\rangle + \\langle{\\mathbf y},{\\mathbf z}\\rangle$.\\label{it:4}\n\t\\end{enumerate}\n\nNote that $(\\forall{\\mathbf X}\\in\\mathcal{V})$ $\\mathrm{Re}\\{\\mathrm{tr}({\\mathbf X}^H{\\mathbf X})\\}=\\mathrm{tr}({\\mathbf X}^H{\\mathbf X})=\\|{\\mathbf X}\\|_\\mathrm{F}^2$, where $\\|\\cdot\\|_\\mathrm{F}$ is the standard Frobenius norm.\nConsequently, \\ref{it:1}) follows from the nonnegativity and positive-definiteness of a norm. The symmetry in \\ref{it:2}) follows from the fact that $\\mathrm{tr}({\\mathbf A}{\\mathbf B})=\\mathrm{tr}({\\mathbf B}{\\mathbf A})$ for matrices ${\\mathbf A},{\\mathbf B}$ with compatible dimensions, and $\\mathrm{Re}\\{\\mathrm{tr}({\\mathbf X})\\}=\\mathrm{Re}\\{\\mathrm{tr}({\\mathbf X}^H)\\}$ for ${\\mathbf X}\\in\\mathcal{V}$. Moreover, \\ref{it:3}) and\n\\ref{it:4}) follow from the linearity of $\\mathrm{Re}\\{\\cdot\\}$ and $\\mathrm{tr}(\\cdot)$.\\qed\n\\end{remark}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt has been over a century since Einstein \\cite{1,2,3} formulated general relativity (GR) in 1915. He was aware that the gravitational field must interact with itself, but was unable to produce a symmetric tensor to properly describe the energy-momentum of the gravitational field. Instead, a non-covariant pseudo-tensor was introduced. However, the difficulties associated with this pseudo-tensor led to the problem of the localization of energy in GR. Over the decades, other pseudo-tensors were developed and different approaches to describe the energy-momentum of the gravitational field were investigated, \\cite{4,5,6}(and references therein) but the energy localization problem still exists today. Despite this deficiency, general relativity is one of the two cornerstones of physics.\\par GR was developed by Einstein on a four-dimensional Riemannian manifold with the understanding that spacetime was locally Minkowskian under free fall. Today, we more properly describe spacetime on a time-oriented Lorentzian manifold with metric. The Lorentzian metric can be associated with a Riemannian metric by using the line element field, $(\\bm{X},\\bm{-X})$ that exists on a non-compact paracompact Hausdorff manifold. A classical result in Riemannian geometry, namely the Berger-Ebin theorem \\cite{7}, can then be adapted to spacetime. This results in the Orthogonal Decomposition Theorem (ODT): an arbitrary second rank symmetric tensor on a time-oriented Lorentzian manifold with a torsionless and metric compatible connection can be orthogonally decomposed into a linear sum of divergenceless tensors and a new tensor, $\\varPhi_{\\alpha\\beta}$. It is a symmetric tensor constructed from the Lie derivative along $X$ of both the metric and a product of unit line element covectors. \\par The left hand side of Einstein's equation $G_{\\alpha\\beta}+\\Lambda g_{\\alpha\\beta}=\\frac{8\\pi G}{c^{4}}T_{\\alpha\\beta}  $ involves symmetric divergenceless tensors. The right hand side is defined by the variation of the action functional for all matter fields with respect to the metric. This generates a divergenceless symmetric tensor that must describe all interactions of the gravitational field with the matter fields, and with the energy-momentum of the gravitational field itself; otherwise, it would not be locally conserved. However, there is nothing in this definition that deals explicitly with the energy-momentum of the gravitational field. If we define $ \\tilde{T}_{\\alpha\\beta} $ as a symmetric energy-momentum tensor generated from the matter fields without the requirement that it completely describes the energy-momentum of the gravitational field as well, it cannot be locally conserved and would not be divergenceless. Consequently, this second rank symmetric tensor can be set proportional to an arbitrary symmetric tensor $ w_{\\alpha\\beta} $, which is then orthogonally decomposed by the ODT into a linear sum of divergenceless tensors and $\\varPhi_{\\alpha\\beta}  $. Lovelock's theorem \\cite{8} proves that in four dimensions, the divergenceless tensors composed from the metric and its first two derivatives can only consist of the metric and the tensor named after Einstein, $G_{\\alpha\\beta}  $. Therefore, Einstein's equation in a four-dimensional Lorentzian spacetime should be expressed more completely by including the $\\varPhi_{\\alpha\\beta}$ term. \\par It will be proved that $G_{\\alpha\\beta}+\\Lambda g_{\\alpha\\beta}+\\varPhi_{\\alpha\\beta}=\\frac{8\\pi G}{c^{4}}\\tilde{T}_{\\alpha\\beta}  $ and that the tensor $T_{\\alpha\\beta}=\\tilde{T}_{\\alpha\\beta}-\\frac{c^{4}}{8\\pi G}\\varPhi_{\\alpha\\beta}$ is divergenceless which allows Einstein's equation to be recovered. In that sense, $\\varPhi_{\\alpha\\beta}$ is hidden in GR. Thus, general relativity is not complete; it is possible to construct a symmetric tensor from the metric and a regular vector field, that is independent of the energy-momentum tensor of the matter fields, and represents the energy-momentum of the gravitational field itself. \\par This differs with the presently and generally accepted belief that GR is complete. However, if that notion was true, GR should be able to describe particular features of dark matter. That unfortunately is not the case and is the reason why physicists invented the generally well accepted theory of Lambda cold dark matter ($ \\Lambda  $CDM) to explain, in particular, the flat rotation curves of some galaxies, while leaving GR intact. Modified general relativity can describe those and other galactic rotation curves as discussed in section 7. \\par Since Lie derivatives have the same form when expressed with covariant or partial derivatives, $\\varPhi_{\\alpha\\beta}$ does not vanish when the connection coefficients vanish. The metric can be locally Minkowskian, as in free fall, without forcing  $\\varPhi_{\\alpha\\beta}$ to vanish. This contrasts free fall in GR where the connection coefficients vanish locally and the gravitational field locally disappears; hence the well known conception that the energy-momentum of the gravitational field is not localizable \\cite{9}. $\\varPhi_{\\alpha\\beta}$ has the structure to describe local gravitational energy-momentum. In free fall, the effective force of gravity disappears locally but the self-energy of the gravitational field is intact accordingly. \\par In section 2, the Orthogonal Decomposition Theorem is proved. In section 3, a modified equation of GR is derived by using the principle of least action, the ODT and a fundamental postulate of GR. $\\varPhi_{\\alpha\\beta}$ appears naturally alongside the Einstein tensor and introduces $ X^{\\mu} $ from the line element field and its collinear unit vector $ u^{\\mu} $ as dynamical variables independent of the Riemannian metric. Variation of the action functional with respect to $ X^{\\mu} $ leads to the Lorentz invariant expression $ u_{\\mu}=\\frac{3}{\\Phi}\\partial_{\\mu}f$ where $ \\Phi $ is the trace of $ \\varPhi_{\\alpha\\beta} $ and $ f\\neq0 $ is the magnitude of $ X^{\\alpha} $. The myriad of possible line element covectors is restricted to those satisfying this condition. \\par  Section 4 discusses the conservation equation for the divergenceless energy-momentum tensor $T_{\\alpha\\beta}=\\tilde{T}_{\\alpha\\beta}-\\frac{c^{4}}{8\\pi G}\\varPhi_{\\alpha\\beta}  $ where $ \\tilde{T}_{\\alpha\\beta} $ is the total matter energy-momentum tensor describing all types of matter including baryonic and dark matter, massive neutrinos and any other possible particle; if dark matter particles exist. \n\\par In section 5, the cosmological constant is discussed. Using the global constraint $\\int \\Phi\\sqrt{-g}d^{4}x=0$, it is shown that the cosmological constant $\\Lambda$ is dynamically replaced with $ \\Phi $. \n\\par Section 6 is a discussion of the modified Einstein equation of GR in the Friedmann-Lema{\\^i}tre-Robertson-Walker (FLRW) metric, and dark energy. A gravitationally repulsive condition is described by $ \\Phi>-2\\Lambda_{d} $ where $\\Lambda_{d}  $ is the dark energy density. $\\Phi$ defines dark energy. Dark energy describes the inflation of the universe immediately after the Big Bang when no matter of any type was present. The dark energy density then tends to the present value of the vacuum energy density. A cyclic universe is born with maximum and minimum values of the cosmological scale factor in the FLRW metric. Dark energy explains the small value of the vacuum energy density and why it now dominates the expansion and acceleration of the present universe.\\par  Cyclic universes have been reported in the literature \\cite{10,11,12,13,14}. Dark energy has been described by various scalar theories such as the quintessential \\cite{15}, k-essence  \\cite{16} (and references therein) phantom or quintom theories \\cite{17} (and further references therein). Dark energy in this article is not a scalar theory; it is the energy generated from the interaction of the gravitational field with its energy-momentum tensor.\\par The existence of a tensor that describes the energy-momentum of the gravitational field brings into question the subject of dark matter. Since Einstein's equation is incomplete without the tensor $ \\varPhi_{\\alpha\\beta} $, the plausibility of dark matter is questionable; its existence is based on the assumption that general relativity is a complete theory. Although the self-interactions in a weak gravitational field may be extremely small, in the gravitational field of a galaxy, they may be significant enough to explain dark matter.\\par In section 7, the modified equation of GR is calculated with a spheroidal metric in a region of spacetime outside of matter with the assumption that dark matter does not exist. Two additional terms appear in the modified Newtonian force equation that provides it the flexibility to describe various types of galaxies. By balancing the dark energy force with the Newtonian force, the Tully-Fisher relation is established and the acceleration parameter in MOND is expressed in terms of the dark energy radial force parameter. \n\n\\section{Orthogonal Decomposition of Symmetric Tensors}\nThe Lorentzian spacetime is described on a four-dimensional time-oriented non-compact paracompact Hausdorff manifold with metric, $ (M,g_{\\alpha\\beta}) $. The connection on the manifold is torsionless and metric compatible and the metric has a $ +2 $ signature. The manifold admits a smooth regular line element field $(\\bm{X},-\\bm{X}) $ and a unit vector $ \\bm{u} $ collinear with one of the pair of regular vectors in the line element field \\cite{18,19,23}. The spacetime is assumed to admit a Cauchy surface and is therefore globally hyperbolic. This forbids the presence of closed causal curves \\cite{23}. \\par \nThe orthogonal decomposition of symmetric tensors on Riemannian manifolds has been documented in the literature \\cite{7,20,21,22}. However, a decomposition of symmetric tensors on a time-oriented Lorentzian manifold is required. \n\\begin{theorem}\nAn arbitrary (0,2) symmetric tensor $ w_{\\alpha\\beta} $ in the symmetric cotangent bundle $ S^{2}T^{\\ast}M $ on an n-dimensional time-oriented Lorentzian manifold $ (M,g_{\\alpha\\beta}) $ with a torsionless and metric compatible connection can be orthogonally decomposed as\n\\begin{equation}\\label{ODT}\n\tw_{\\alpha\\beta}= v_{\\alpha\\beta}+ \\varPhi_{\\alpha\\beta} \n\\end{equation} where $v_{\\alpha\\beta}  $ represents a linear sum of symmetric divergenceless (0,2) tensors and $\\varPhi_{\\alpha\\beta}=\\frac{1}{2}\\pounds_{X}g_{\\alpha\\beta}+\\pounds_{X}u_{\\alpha}u_{\\beta}$ where the unit vector $\\bm{u}  $ is collinear to one of the pair of regular line element vectors $ (\\bm{X},-\\bm{X}) $.\n\\end{theorem}\n\\begin{proof}\nLet the Lorentzian manifold $ (M,g_{\\alpha\\beta}) $ be non-compact paracompact Hausdorff and orientable. A smooth regular line element field $(\\bm{X},\\bm{-X)}$ exists as does a unit vector $ \\bm{u} $ collinear with one of the pair of line element vectors. Let M be endowed with a smooth Riemannian metric $ g^{+}_{\\alpha\\beta} $. The smooth Lorentzian metric $ g_{\\alpha\\beta} $ is constructed \\cite{19,23} from $ g^{+}_{\\alpha\\beta} $ and the unit covectors $ u_{\\alpha}$ and $ u_{\\beta}$ by setting\n\\begin{equation}\\label{LRmet}\ng_{\\alpha\\beta}=g^{+}_{\\alpha\\beta}-2u_{\\alpha}u_{\\beta}.\t\n\\end{equation}\n Let $ w_{\\alpha\\beta} $ and $ v_{\\alpha\\beta} $ belong to $ S^{2}T^{\\ast}M $, the cotangent bundle of symmetric $(0,2)$ tensors on M. In the compact neighborhood of a point $p$ in an open subset of $ S^{2}T^{\\ast}M $ which contains $ g^{+}_{\\alpha\\beta} $, an arbitrary $ (0,2) $ symmetric tensor $ w_{\\alpha\\beta} $ can be orthogonally and uniquely decomposed by the Berger-Ebin theorem \\cite{7} according to \\begin{equation}\\label{key}\n\tw_{\\alpha\\beta}=v_{\\alpha\\beta}+\\frac{1}{2}\\pounds_{\\xi}g^{+}_{\\alpha\\beta}\n\\end{equation} where $ \\bm{\\xi} $ is an arbitrary vector and $v_{\\alpha\\beta}  $ represents a linear sum of symmetric divergenceless (0,2) tensors: $ {\\nabla^{+}}^{\\alpha}v_{\\alpha\\beta}=0 $.\\par The divergence of $ v_{\\beta}^{\\alpha} $ in the mixed tensor bundle can be written as $ \\nabla_{\\alpha}v_{\\beta}^{\\alpha}=\\partial_{\\alpha}v_{\\beta}^{\\alpha}+\\frac{v_{\\beta}^{\\lambda}}{2g}\\partial_{\\lambda}g-\\frac{1}{2}v^{\\alpha\\lambda}\\partial_{\\beta}g_{\\alpha\\lambda} $. Since the determinant of $ g_{\\alpha\\beta} $, $ g $, is related to that of $ g^{+}_{\\alpha\\beta} $ by $ g=-g^{+} $ \n\\begin{equation}\\label{Dv}\n\t\\begin{split}\n\t\t\\nabla_{\\alpha}v_{\\beta}^{\\alpha}-\\nabla^{+}_{\\alpha}v_{\\beta}^{\\alpha}=v^{\\alpha\\lambda}\\partial_{\\beta}(u_{\\alpha}u_{\\lambda}).\n\t\\end{split}\n\\end{equation} The left hand side of (\\ref{Dv}) is a (0,1) tensor but the right hand side is not, which demands:\n\\begin{equation}\\label{}\n\tv^{\\alpha\\lambda}\\partial_{\\beta}(u_{\\alpha}u_{\\lambda})=0\n\\end{equation}where $\\partial_{\\beta}u_{\\alpha}\\neq0 $. This guarantees $\\nabla^{\\alpha}v_{\\alpha\\beta}=0  $ because $\\nabla^{+\\alpha}v_{\\alpha\\beta}=0  $. Hence, \n\\begin{equation}\\label{decomp}\n\n\tw_{\\alpha\\beta}=v_{\\alpha\\beta}+\\frac{1}{2}\\pounds_{\\xi}g_{\\alpha\\beta}+\\pounds_{\\xi}u_{\\alpha}u_{\\beta}\n\n\n\\end{equation} where $ \\nabla^{\\alpha}v_{\\alpha\\beta}=0 $. $ \\xi^{\\lambda} $ is an arbitrary vector which can be chosen to be collinear to $ u^{\\lambda} $. Without loss of generality, $ \\xi^{\\lambda} $ can then be replaced by $ X^{\\lambda} $. Using $ X^{\\lambda}=fu^{\\lambda} $ where $f\\neq0  $ is the magnitude of $X^{\\lambda}  $, the expression  $X^{\\lambda}\\nabla_{\\lambda}(u_{\\alpha}u_{\\beta}) $ in the last term of (\\ref{decomp}) then vanishes in an affine parameterization and\n\\begin{equation}\\label{udecomp}\n\tw_{\\alpha\\beta}=v_{\\alpha\\beta}+\\varPhi_{\\alpha\\beta}\n\\end{equation}where\n\\begin{equation}\\label{Phiab}\n\t\\varPhi_{\\alpha\\beta}:=\\frac{1}{2}(\\nabla_{\\alpha}X_{\\beta}+\\nabla_{\\beta}X_{\\alpha})+u^{\\lambda}(u_{\\alpha}\\nabla_{\\beta}X_{\\lambda}+u_{\\beta}\\nabla_{\\alpha}X_{\\lambda}).\n\\end{equation} The decomposition is orthogonal: $<v_{\\alpha\\beta},\\varPhi_{\\alpha\\beta}>=0  $. \\\\\n\\end{proof}\n\\section{Derivation of the modified equation of general relativity}\nA modified equation of general relativity of the form $ C_{\\alpha\\beta}=0 $ is sought which contains a linear combination of symmetric tensors that define the Einstein equation, and a new tensor which can describe the energy-momentum of the gravitational field itself. This can be achieved by using the principle of least action, the Orthogonal Decomposition Theorem (\\ref{ODT}), and a fundamental postulate of GR. \\par First, the field equations contained in $C_{\\alpha\\beta} $, which are sought to describe general relativity and the energy-momentum of the gravitational field, must be derivable from the action functional \n\\begin{equation}\\label{S}\nS=S^{F}+S^{EH}+S^{G}\n\\end{equation} where $S^{F}$ and $L^{F}$ refer to the action and Lagrangian, respectively, for all types of  matter fields including those of dark matter if dark matter particles exist. $S^{EH}$ is the Einstein-Hilbert action for general relativity and $S^{G}$ is the action for the energy-momentum of the gravitational field with Lagrangian $L^{G}  $. The variation of $S^{F}$ with respect to $g^{\\alpha\\beta}$\\begin{equation}\n\\delta S^{F}=\\int (\\frac{\\delta L^{F}}{\\delta g^{\\alpha\\beta}}-\\frac{1}{2}L^{F}g_{\\alpha\\beta})\\delta g^{\\alpha\\beta}\\sqrt{-g}d^{4}x\n\\end{equation} generates the symmetric energy-momentum tensor $\\tilde{T}_{\\alpha\\beta}$ which represents the interaction of all types of matter fields and associated radiation in a gravitational field, but does not specifically include the energy-momentum of the gravitational field:\n\\begin{equation}\n\\tilde{T}_{\\alpha\\beta}:=-2c (\\frac{\\delta L^{F}}{\\delta g^{\\alpha\\beta}}-\\frac{1}{2}L^{F}g_{\\alpha\\beta}).  \n\\end{equation}\n$C_{\\alpha\\beta}$ must then be expressed as\n\\begin{equation}\\label{key}\nC_{\\alpha\\beta}=\\frac{a}{c}\\tilde{T}_{\\alpha\\beta}+bw_{\\alpha\\beta}\n\\end{equation} where $ w_{\\alpha\\beta}$ is an unknown symmetric tensor independent of $ \\tilde{T}_{\\alpha\\beta} $; and a and b are arbitrary constants.\\par Second, $w_{\\alpha\\beta}$ can be orthogonally decomposed by the ODT into\n\\begin{equation}\\label{key}\nw_{\\alpha\\beta}=v_{\\alpha\\beta}+\\varPhi_{\\alpha\\beta} \\end{equation} where $ \\varPhi_{\\alpha\\beta} $ is given by (\\ref{Phiab})  and $ \\nabla^{\\alpha}v_{\\alpha\\beta}=0 $.  \\par Third, Einstein concluded \\cite{1} that the metric should describe both the geometry of spacetime and the gravitational field. He postulated the totality of the matter energy-momentum tensor and the energy-momentum of the gravitational field, to be the source of the gravitational field. Adhering to this philosophy, the energy-momentum tensor $ T_{\\alpha\\beta} $ describing the totality of all types of matter and the energy-momentum of the gravitational field, is postulated to be the source of the gravitational field.\n\\par  $\\varPhi_{\\alpha\\beta}$ is independent of  $\\tilde{T}_{\\alpha\\beta}$ and is not divergenceless. $ \\varPhi_{\\alpha\\beta} $ is therefore the sole candidate to describe the energy-momentum of the gravitational field. Thus,\n\\begin{equation}\\label{T}\nT_{\\alpha\\beta}=\\tilde{T}_{\\alpha\\beta}+\\frac{bc}{a}\\varPhi_{\\alpha\\beta}\n\\end{equation} and the interaction of the gravitational field with its energy-momentum tensor can be defined with the action\n\\begin{equation}\\label{SG}\nS^{G}:=-b\\int g_{\\alpha\\beta}\\varPhi^{\\alpha\\beta}\\sqrt{-g}d^{4}x.\n\\end{equation}\n\\par It was proved by Lovelock \\cite{8} that the only  tensors in a four-dimensional spacetime which are symmetric, divergence free, and a concomitant of the metric tensor together with its first two derivatives are the metric and the Einstein tensor, $ G_{\\alpha\\beta}=R_{\\alpha\\beta}-\\frac{1}{2}g_{\\alpha\\beta}R $. $ v_{\\alpha\\beta} $ must therefore contain the Lovelock tensors.  \\par  $C_{\\alpha\\beta}$ is then formally decomposed as \\begin{equation}\\label{Psidecomp}\nC_{\\alpha\\beta}=\\frac{a}{c}T_{\\alpha\\beta}+b v_{\\alpha\\beta}\n\\end{equation}\nwith $\\nabla_{\\alpha}v^{\\alpha\\beta}=0$  and $v_{\\alpha\\beta}:=\nG_{\\alpha\\beta}+\\Lambda g_{\\alpha\\beta}$. $\\Lambda$ \\emph{is a global integration constant} (in hindsight identified as the cosmological constant). With the collection of tensors $ C_{\\alpha\\beta} $ defined to vanish, we obtain the modified Einstein equation of general relativity with cosmological constant $\\Lambda$ and the gravitational energy-momentum term $\\varPhi_{\\alpha\\beta}$ \\begin{equation}\\label{MEQ}\n-\\frac{8\\pi G}{c^{4}}\\tilde{T}_{\\alpha\\beta}+R_{\\alpha\\beta}-\\frac{1}{2}g_{\\alpha\\beta}R+{\\Lambda} g_{\\alpha\\beta}+\\varPhi_{\\alpha\\beta}=0\n\\end{equation} by setting $a=-\\frac{1}{2}$ and $b=\\frac{c^{3}}{16\\pi G}.$\n\\par Ma and Wang \\cite{22} obtained a similar result to (\\ref{MEQ}) with $ \\Lambda=0 $, but with an entirely different $ \\varPhi_{\\alpha\\beta}=\\nabla_{\\alpha}\\partial_{\\beta}\\phi $ for some scalar $ \\phi $ by using a decomposition of symmetric tensors on a Riemannian manifold.  \n\\par Equation (\\ref{MEQ}) must be derived from the action functional (\\ref{S}). With (\\ref{SG}):\n\\begin{equation}\\label{Se}\nS=\\int L^{F}( A^{\\beta},\\nabla^{\\alpha} A^{\\beta},...,g^{\\alpha\\beta})\\sqrt{-g}d^{4}x\n+b \\int (R-2\\Lambda)\\sqrt{-g}d^{4}x-b\\int \\varPhi_{\\alpha\\beta} g^{\\alpha\\beta}\\sqrt{-g} d^{4}x.\n\\end{equation}\nTo calculate the variation of $ S^{G} $ with respect to the inverse metric $ g^{\\alpha\\beta} $, the following results are used: $g^{\\alpha\\beta}=g^{{+}{\\alpha\\beta}}-2u^{\\alpha}u^{\\beta} $ is the inverse of $ g_{\\alpha\\beta}$; $g^{{+}{\\alpha\\beta}} \\delta g_{\\alpha\\beta}=-g_{\\alpha\\beta}\\delta g^{{+}{\\alpha\\beta}}  $; $\\delta g^{{+}{\\rho\\beta}}=-g^{{+}{\\alpha\\beta}}g^{\\lambda\\rho}\\delta g^{+}_{\\alpha\\lambda}  $; $g_{\\alpha\\beta}\\delta(u^{\\alpha}u^{\\beta})=u_{\\alpha}u_{\\beta}\\delta g^{\\alpha\\beta} $; and $\\delta(u^{\\alpha}u^{\\beta})=u_{\\lambda}u^{\\beta}\\delta g^{\\lambda\\alpha} $.\nThe variation of S with respect to $g^{\\alpha\\beta}$ is then\n\\begin{multline}\n\\delta S =\\int [-\\frac{1}{2c}\\tilde{T}_{\\alpha\\beta} +b(R_{\\alpha\\beta}-\\frac{1}{2}g_{\\alpha\\beta}R)+b\\Lambda g_{\\alpha\\beta}+b(\\nabla_{\\alpha}X_{\\beta}\n+2u^{\\lambda}u_{\\beta}\\nabla_{\\alpha}X_{\\lambda}\\\\\n+\\nabla_{\\mu}X_\\nu(-u_{\\alpha}u_{\\beta} g^{\\mu\\nu}+u^{\\mu}u^{\\nu}g_{\\alpha\\beta})\n)]\\delta g^{\\alpha\\beta}\n\\sqrt{-g}\\enspace d^{4}x\\enspace+2\\int \\nabla_{\\alpha}(u^{\\alpha}u^{\\beta})\\delta X_{\\beta}\\sqrt{-g}d^{4}x\n\\end{multline}after calculating $ \\delta \\Gamma^{\\lambda}_{\\alpha\\beta} $ induced by the variations in the inverse metric, and integrating by parts several times.\nThe last term in the variation with respect to $ g^{\\alpha\\beta} $ vanishes which follows by writing the tensor in brackets,  $-u_{\\alpha}u_{\\beta}g^{\\mu\\nu}+u^{\\mu}u^{\\nu}g_{\\alpha\\beta}$, as its equivalent, $\\frac{1}{2}(g^{{+}{\\mu\\nu}}g_{\\alpha\\beta}-g^{+}_{\\alpha\\beta}g^{\\mu\\nu}) $; and choosing an orthonormal basis $(e_{\\alpha})$ at a point $p\\in M  $ for $ g^{+} $ with $ e_{0}=u $. Then, $ u^{0}u_{0}=1 $, $ u^{i}u_{i}=0 $, $g^{+}_{\\alpha\\beta}=\\delta_{\\alpha\\beta}$, $ g^{00}=-g^{{+}00}$ and $g_{00}=-g^{+}_{00}$, with all other components of the metric g equal to those of the metric $ g^{+}$. Since $ \\delta {g^{\\alpha\\beta}}$ is symmetric, the second last term can be expressed as $ b\\varPhi_{\\alpha\\beta} $.\nWith $\\delta S=0$ and arbitrary variations for  $\\delta g^{\\alpha\\beta}$ and $\\delta X_{\\beta}  $, we have\n\\begin{equation}\n-\\frac{1}{2c}\\tilde{T}_{\\alpha\\beta}+b(R_{\\alpha\\beta}-\\frac{1}{2}g_{\\alpha\\beta}R)+b{\\Lambda} g_{\\alpha\\beta}+b\\varPhi_{\\alpha\\beta}=0\n\\end{equation} and \n\\begin{equation}\\label{nuab}\n\\nabla_{\\alpha}(u^{\\alpha}u^{\\beta})=0.\n\\end{equation}\nSetting $b=\\frac{c^{3}}{16\\pi G}$ yields the modified Einstein equation described in (\\ref{MEQ}). \n\\subsection{Some properties of $ u_{\\mu} $ and $ \\varPhi_{\\alpha\\beta} $}\n$ X^{\\beta} $ from the line element field and its collinear vector $ u^{\\beta} $ are dynamical variables independent of the Riemannian metric in (\\ref{LRmet}). The dynamical properties of the line element fields are obtained by varying (\\ref{Se}) with respect to $ X^{\\mu} $. This yields the equation\n\\begin{equation}\\label{Xmu1}\nu^{\\alpha}\\nabla_{\\alpha}X_{\\mu}+4u^{\\lambda}u_{\\mu}u^{\\alpha}\\nabla_{\\alpha}X_{\\lambda}-3u^{\\lambda}\\nabla_{\\mu}X_{\\lambda}=\\Phi u_{\\mu}\n\\end{equation} using (\\ref{MEQ}). With $ X^{\\alpha}=fu^{\\alpha} $ where $ f\\neq0 $ is the magnitude of $ X^{\\alpha} $, the first two terms of (\\ref{Xmu1}) are then geodesics in an affine parameterization and can be set to zero. Since\n\\begin{equation}\\label{key}\nX^{\\alpha}X_{\\alpha}=-f^{2},\n\\end{equation}it follows that\n\\begin{equation}\\label{Xmu}\nu_{\\mu}=\\frac{3}{\\Phi}\\partial_{\\mu}f. \n\\end{equation} \n\\par There is a myriad of regular vectors from the line element field for each Riemannian metric, and the associated Lorentzian metric is not unique. However, the variation of (\\ref{Se}) with respect to $X^{\\mu} $ restricts the line element fields to those given by (\\ref{Xmu}) which in turn restricts the Lorentzian metric. \\par  $\\varPhi_{\\alpha\\beta} $ is expressed in terms of the Lie derivative of the metric and a product of unit line element covectors. Since Lie derivatives have the same form when expressed with covariant or partial derivatives, $\\varPhi_{\\alpha\\beta}  $ does not vanish when the connection coefficients vanish. The metric can be locally Minkowskian, as in free fall, without affecting $ \\varPhi_{\\alpha\\beta} $. It has the structure to describe local gravitational energy-momentum. \n\\par Using (\\ref{nuab}), it is straightforward to calculate the coupling of the gravitational field with its energy-momentum tensor:\n\\begin{equation}\\label{0phi}\n\\int g_{\\alpha\\beta}\\varPhi^{\\alpha\\beta}\\sqrt{-g}d^{4}x=\\int \\Phi \\sqrt{-g}d^{4}x=0 \n\\end{equation} where $ \\Phi=\\nabla_{\\alpha}X_{\\beta}(g^{\\alpha\\beta}+2u^{\\alpha}u^{\\beta}) $. Equation (\\ref{0phi}) means the scalar $ \\Phi $ has local positive and negative values, all of which add to zero when integrated over the entire spacetime. $ \\Phi $ is globally conserved. Section 6 demonstrates that the positive values of $ \\Phi $ are attributed to the gravitationally repulsive properties of dark energy with the cosmological constant set to zero. The negative values represent the attractive part of the energy of the gravitational field interacting with its gravitational energy-momentum tensor. $ \\Phi $ is measurable; it can be expressed in terms of the density and pressure of total matter and the vacuum energy density as shown in section 6. The gravitational energy density is calculated from $ \\varPhi_{00} $ in section 7. The energy-momentum of the gravitational field is localizable and measureable. \n\\section{The conserved energy-momentum tensor $ T_{\\alpha\\beta} $} \nThe invariance of the action functional describing gravity, it's self-energy-momentum and total matter fields under the symmetry of diffeomorphisms demands a symmetric divergenceless energy-momentum tensor\\begin{equation}\\label{Tab}\nT^{\\alpha\\beta}=\\tilde{T}^{\\alpha\\beta}-\\frac{c^{4}}{8\\pi G}\\varPhi^{\\alpha\\beta}.\n\\end{equation}  This follows from an analysis of each term in the action functional $ S $ defined in (\\ref{S}). The action $ S^{EH} $ is independently invariant under a diffeomorphism. Variation of the action $ S^{F} $ with respect to the metric contains only $ \\tilde{T}^{\\alpha\\beta} $ because the variations of $ S^{F} $ with respect to each field and its derivatives vanish with the corresponding Euler-Lagrange equations. Variation of $ S^{G} $ with respect to the metric yields $ \\varPhi^{\\alpha\\beta} $. Therefore, we can write\n\\begin{equation}\\label{key}\n\\int (-\\frac{1}{2c}\\tilde{T}^{\\alpha\\beta}+b\\varPhi^{\\alpha\\beta})\\delta g_{\\alpha\\beta}\\sqrt{-g}d^{4}x=0\n\\end{equation} where $b=\\frac{c^{3}}{16\\pi G}$. Under a diffeomorphism, the Lie derivative of the metric along a regular vector $ Y^{\\beta} $ generates the infinitesimal change in the metric $\\delta g_{\\alpha\\beta}=\\nabla_{\\alpha}Y_{\\beta}+\\nabla_{\\beta}Y_{\\alpha}$. Integrating by parts then gives\n\\begin{equation}\\label{key}\n\\int \\nabla_{\\alpha}(-\\frac{1}{2c}\\tilde{T}^{\\alpha\\beta}+b\\varPhi^{\\alpha\\beta})Y_{\\beta}\\sqrt{-g}d^{4}x=0\n\\end{equation} which requires\n\\begin{equation}\\label{Tcons}\n\\nabla_{\\alpha} T^{\\alpha\\beta}=0  \n\\end{equation}\nfor diffeomorphisms generated by $ Y^{\\beta} $. \\par  Equation (\\ref{Tcons}) is the local description of the conservation of energy and momentum in a modified theory of GR described by (\\ref{MEQ}). The gravitational field has an intrinsic energy-momentum which is attributed to $ \\varPhi_{\\alpha\\beta} $. Being independent of $ \\tilde{T}_{\\alpha\\beta} $, $\\frac{c^{4}}{8\\pi G}\\varPhi_{\\alpha\\beta} $ provides the additional self-energy-momentum of the gravitational field necessary to complete the source $ T_{\\alpha\\beta} $ of the geometry of spacetime. $\\varPhi_{\\alpha\\beta} $ completes the Einstein equation and leaves it intact in form:\n\\begin{equation}\\label{E}\nG_{\\alpha\\beta}+\\Lambda g_{\\alpha\\beta}=\\frac{8\\pi G}{c^{4}}T_{\\alpha\\beta}.\n\\end{equation}\n\\section{Cosmological Constant}\nThe cosmological constant $ \\Lambda $ appears alongside the metric as the simplest and most basic Lovelock tensor. With a torsionless connection, the covariant derivative of the metric vanishes. Adding the metric to the Einstein equation seems trivial with the associated constant playing the role of a global integration constant. $ \\Lambda $ can then be interpreted as a constant global energy density.  That seems very restrictive as energy densities are generally dynamic and not constant.\\par The regular vector  fields that exist in a Lorentzian spacetime provide a dynamical background from which the energy-momentum of the gravitational field is constructed. It is not possible for a constant global energy density to represent the dynamic interaction of the metric with the energy-momentum tensor of the gravitational field. $ \\Lambda $ must therefore be dynamically replaced by a scalar.\n\\begin{theorem}\n\tThe cosmological constant $ \\Lambda $ is dynamically replaced by the trace of $\\varPhi_{\\alpha\\beta}  $. \n\\end{theorem}\n\\begin{proof}\nUsing (\\ref{0phi}), $S^{EH}$ with $ \\Lambda=0 $ can be written as\n\\begin{equation}\\label{key}\nS^{EHG}=\\frac{c^{3}}{16\\pi G} \\int (R-\\Phi) \\sqrt{-g}d^{4}x\n\\end{equation} which generates the modified Einstein equation with no cosmological constant from (\\ref{Se}). If $ \\Phi=2\\Lambda $ \\emph{locally}, the Einstein equation with the cosmological constant is obtained accordingly. The trace of the tensor describing the energy-momentum of the gravitational field, dynamically replaces the cosmological constant but must obey the global equation (\\ref{0phi}).\\\\\n\\end{proof}\n\n\\section{Energy-momentum of the gravitational field in the FLRW metric: Dark energy}\n\\par Some properties of $\\varPhi_{\\alpha\\beta}$ in the Friedmann-Lema{\\^i}tre-Robertson-Walker metric are now investigated. The FLRW metric is typically used to describe a spatially maximal symmetric universe according to the cosmological principle \\cite{24} whereby the universe is homogeneous and isotropic when measured on a large scale. This metric is given by\n\\begin{equation}\nds^{2}=-c^{2}dt^{2}+a(t)^{2}[\\frac{1}{1-\\kappa r^{2}}dr^{2}+r^{2}(d\\theta^{2}+{sin^{2}\\theta} d\\varphi^{2})]\n\\end{equation} where $a(t)$ is the cosmological scale factor which satisfies $a>0 $ after the Big Bang at $ t=0 $.  $\\kappa$ is a constant used to describe a particular spatial geometry. The connection components of the FLRW metric are\n\\begin{equation}\n\\Gamma^{i}_{j0}=\\frac{\\dot{a}}{ca}\\delta^{i}_{j},\\enspace  \\Gamma^{0}_{ij}=\\frac{\\dot{a}}{ca}g_{ij},\\enspace  \\Gamma^{\\mu}_{00}=0\n\\end{equation} where $j=1,2,3$. The Ricci tensor components are\n\\begin{equation}\nR_{00}=-3\\frac{\\ddot{a}}{ac^{2}},\\enspace R_{ij}=(\\frac{\\ddot{a}}{ac^{2}}+2\\frac{\\dot{a}^{2}}{a^{2}c^{2}}+2\\frac{\\kappa}{a^{2}})g_{ij}\n\\end{equation} and the Ricci scalar is\\begin{equation}\nR=\\frac{6}{a^{2}c^{2}}(a\\ddot{a}+\\dot{a}^{2}+\\kappa  c^{2}).\n\\end{equation} \\par It was proved in \\cite{24} that a maximally spatial form invariant symmetric second rank tensor $B_{\\alpha\\beta}$ has components in the form\n\\begin{equation}\nB_{00}=\\varrho(t),\\enspace B_{0j}=0,\\enspace B_{ij}=p(t)g_{ij}\n\\end{equation} where $\\varrho(t)$ and $p(t)$ are arbitrary functions of time.  We therefore set,\n\\begin{equation}\n\\tilde{T}_{00}=c^{2}\\varrho,\\enspace \\tilde{T}_{ij}=pg_{ij},\\enspace \\tilde{T}^{\\mu}_{\\mu}=-c^{2}\\varrho+3p\n\\end{equation} where $\\varrho(t)$ and $p(t)$  are designated as the mass density and pressure functions, respectively, of total matter including dark matter; if dark matter particles exist. Similarly,\n\\begin{equation}\n\\varPhi_{00}=\\Lambda_{d},\\enspace \\varPhi_{ij}=\\frac{P_{d}}{c^{2}}g_{ij},\\enspace \\varPhi_{\\mu}^{\\mu}=-\\Lambda_{d}+3\\frac{P_{d}}{c^{2}}\n\\end{equation} where $\\Lambda_{d}(t)$ and $P_{d}(t)$ refer to the energy density and pressure, respectively, of the tensor describing the energy-momentum of the gravitational field.   \t\\par To obtain the Friedmann equations, we use the trace of the modified Einstein equation\n\\begin{equation}\n-\\frac{8\\pi G}{c^{4}}\\tilde{T}-R+\\Phi=0\n\\end{equation}\nto rewrite the modified Einstein equation as\n\\begin{equation}\nR_{\\alpha\\beta}=\\frac{8\\pi G}{c^{4}} (\\tilde{T}_{\\alpha\\beta}-\\frac{1}{2}g_{\\alpha\\beta}\\tilde{T})+\\frac{1}{2}g_{\\alpha\\beta}\\Phi -\\varPhi_{\\alpha\\beta}\n\\end{equation} from which we obtain\n\\begin{equation}\\label{frw1}\n3\\frac{\\ddot{a}}{a}=-4\\pi G(\\varrho+\\frac{3p}{c^{2}})+\\frac{1}{2}c^{2}\\Lambda_{d}+\\frac{3}{2}P_{d}\n\\end{equation} from the $R_{00}$ component. The $R_{11}$ component gives\n\\begin{equation}\\label{frw2}\n\\frac{\\ddot{a}}{a}+2\\frac{\\dot{a}^{2}}{a^{2}}+\\frac{2\\kappa c^{2}}{a^{2}}=\\frac{c^{2}}{2}(-\\Lambda_{d}+\\frac{P_{d}}{c^{2}})+4\\pi G(\\varrho-\\frac{p}{c^{2}})\n\\end{equation} and the conservation law for $T^{\\alpha\\beta}$  yields \n\\begin{equation}\\label{frw3}\n\\dot{\\varrho}-\\frac{c^{2}}{8\\pi G}\\dot{\\Lambda_{d}}=-3\\frac{\\dot{a}}{a}(\\varrho+\\frac{p}{c^{2}}-\\frac{c^{2}}{8\\pi G}(\\Lambda_{d}+\\frac{P_{d}}{c^{2}})).\n\\end{equation}\nInserting (\\ref{frw1}) into (\\ref{frw2}) produces a simpler equation\n\\begin{equation}\\label{frw4}\n\\frac{\\dot{a}^{2}}{a^{2}}+\\frac{\\kappa c^{2}}{a^{2}}=\\frac{8\\pi G}{3} \\varrho-\\frac{1}{3}c^{2}\\Lambda_{d}. \n\\end{equation} Equations (\\ref{frw1}) and (\\ref{frw4}) are the Friedmann equations modified with $\\varPhi_{\\alpha\\beta}$.    \\par From (\\ref{frw1}), we immediately see that $\\Phi+2\\Lambda_{d}=\\Lambda_{d}+\\frac{3}{c^{2}}P_{d}>0$ tends to accelerate the universe; while all types of matter combined, with a positive mass density and pressure, tend to decelerate the universe. $\\Phi>-2\\Lambda_{d}  $ is a gravitationally repulsive condition which relates dark energy to $\\Lambda_{d}  $. Hence, $\\Lambda_{d}$ is called the dark energy density and $P_{d}$ the dark energy pressure. $ \\Phi $ tends to accelerate or decelerate the universe but has a net zero effect on it. $\\varPhi_{\\alpha\\beta}$ and therefore $ \\Phi $, provide the flexibility to describe various eras in the evolution of the universe. The cosmological constant $\\Lambda$, on the other hand, can be expressed as a fixed negative energy density which would have tended to accelerate the universe during \\emph{all} epochs. \\par One of the recent challenges in cosmology has been to find a natural mechanism that describes a small but positive vacuum energy density to explain the observed acceleration of the present universe.  Dark energy provides a natural explanation of this challenge without the need of a cosmological constant.\\par  After the discovery in 1929 by Hubble \\cite{25} that the universe was expanding, $\\Lambda$ was not required to obtain a static solution to the Einstein equations with a positive mass density. Since the cosmological constant was vastly smaller than any value predicted by particle theory, most particle theorists simply assumed, that for some unknown reason, this quantity was zero \\cite{26}. This was widely believed to be true until the discovery of the presently accelerating universe in 1998-99 \\cite{27,28}. $\\Lambda$ was then considered to be associated with the dark energy conundrum. However, it is just a global integration constant in the modified Einstein equation and is replaced by $ \\Phi $ as proved in theorem 5.1. This is readily verified by restricting the dark energy variables to the constant values $\\Lambda_{d}=-\\Lambda$ and $P_{d}=c^{2}\\Lambda$ in (\\ref{frw1}) and (\\ref{frw4}). The Friedmann equations with the cosmological constant $\\Lambda$ are then recovered in accordance with theorem 5.1.  \\par  The Friedmann equations are now considered with $\\kappa=1$ describing a closed universe:\n\\begin{equation}\\label{frw5}\n\\dot{a}^{2}=\\frac{8\\pi G}{3} \\varrho a^{2}-\\frac{c^{2}}{3}\\Lambda_{d}a^{2} -c^{2}\n\\end{equation} and\n\\begin{equation}\\label{frw6}\n\\ddot{a}=-\\frac{4\\pi G}{3}a(\\varrho+\\frac{3p}{c^{2}})+\\frac{ac^{2}}{6}(\\Lambda_{d}+\\frac{3}{c^{2}}P_{d}).\n\\end{equation} To avoid confusion with $\\Lambda$, we will denote the constant vacuum energy density as $\\Lambda_{v}$ with the property $ \\Lambda_{v}>0 $. In the present epoch, $ \\Lambda_{v}$ is measured to be $ \\approx1.1\\times10^{-52}m^{-2} $.  By defining\n\\begin{equation}\\label{frw7}\n\\tilde{\\varrho}=8\\pi G\\varrho-c^{2}\\Lambda_{d}\n\\end{equation} and\n\\begin{equation}\\label{frw8}\n\\tilde{p}=-\\frac{4\\pi G}{c^{2}}p+\\frac{1}{2}P_{d},\n\\end{equation}  these equations can be simplified to\n\\begin{equation}\\label{frw9}\n\\dot{a}^{2}=\\frac{\\tilde{\\varrho}a^{2}}{3}-c^{2}, \n\\end{equation} and\n\\begin{equation}\\label{frw10}\n\\ddot{a}=a(-\\frac{\\tilde{\\varrho}}{6}+\\tilde{p})\n\\end{equation}\nwith the conservation equation\n\\begin{equation}\\label{frw11}\n\\dot{\\tilde{\\varrho}}=-3\\frac{\\dot{a}}{a}(\\tilde{\\varrho}-2\\tilde{p}).\n\\end{equation}  Unless otherwise stated, $\\varrho>0$ and $ p>0 $. Equation (\\ref{frw9}) requires $\\tilde{\\varrho}>0$. \\par It is interesting to explore how the energy-momentum of the gravitational field can describe critical features of a Big Bang universe. Immediately after the event of the Big Bang, the universe violently accelerates and $ \\dot{a}>0 $. For a very short time, there is no matter; $\\varrho=0$ and $p=0 $. In this very early stage of the evolution of the universe, it is possible that the constant vacuum energy density developed. If we set $\\varrho=0$ in (\\ref{frw5}), the inequality\n\\begin{equation}\\label{Ld}\n\\Lambda_{d}<-\\frac{3}{a^{2}}\n\\end{equation}\nmust hold. From (\\ref{frw3}) and (\\ref{frw6}) with $ \\dot{a}\\neq0 $,\n\\begin{equation}\\label{aconst}\n\\frac{d}{da}\\Lambda_{d}=-\\frac{2\\Lambda_{d}}{a}-\\frac{6\\ddot{a}}{a^{2}c^{2}}.\n\\end{equation}  If $\t\\Lambda_{d}\\longrightarrow-\\Lambda_{v}$ and $ \\frac{P_{d}}{c^{2}}\\longrightarrow\\Lambda_{v}  $ just after the Big Bang, (\\ref{frw6}) requires $ \\frac{\\ddot{a}}{a} $ to be constant. With those assumptions, equation (\\ref{aconst}) has the solution\n\\begin{equation}\\label{Lamd}\n\\Lambda_{d}=\\frac{c_{1}}{a^{2}}-\\frac{3\\ddot{a}}{ac^{2}}\n\\end{equation} where $ c_{1} $ is an arbitrary constant. Setting $c_{1}=-3$ and $\\Lambda_{v}=\\frac{3\\ddot{a}}{ac^{2}}  $,\n\\begin{equation}\\label{frw13}\n\\Lambda_{d}=-\\frac{3}{{a}^{2}}-\\Lambda_{v}\n\\end{equation} which satisfies (\\ref{Ld}) and tends to $-\\Lambda_{v}$ as the universe expands. Dark energy can generate $\\Lambda_{v}$ during this epoch of the universe. The expansion of the universe is then described by \\begin{equation}\\label{vace}\n{\\dot{a}}^{2}=\\frac{1}{3}a^{2}c^{2}\\Lambda_{v}.\n\\end{equation} \nThe pressure density of dark energy is $P_{d}=\\frac{c^{2}}{a^{2}}+\\Lambda_{v}c^{2}$ and the acceleration of the universe is\n\\begin{equation}\\label{vaca}\n\\ddot{a}=\\frac{ac^{2}\\Lambda_{v}}{3}.\n\\end{equation}\t   \t \nThe scalar $ \\Phi=\\frac{6}{a^{2}}+4\\Lambda_{v} $ is positive. $ \\Phi>0 $ is the condition to be satisfied for an expanding and accelerating universe when no matter is present. Because this result depends entirely on dark energy, $ \\Phi>0 $ defines dark energy.\n\\par With all types of matter appearing after the initial inflation, $\\Lambda_{d}$ must obey the constraint \\begin{equation}\\label{key}\n\\Lambda_{d}<-\\frac{3}{a^{2}}+\\frac{8\\pi G\\varrho}{c^{2}}.\n\\end{equation}With constant total matter, the equation\n\\begin{equation}\\label{rhoconst}\n\\frac{d}{da}\\Lambda_{d}=-\\frac{2\\Lambda_{d}}{a}+\\frac{16\\pi G\\varrho  }{ac^{2}}-\\frac{6\\ddot{a}}{a^{2}c^{2}}\n\\end{equation}is obtained from (\\ref{frw3}) and (\\ref{frw6}) with $ \\dot{a}\\neq0 $. Since $ \\frac{\\ddot{a}}{a}=\\frac{d}{dt}(\\frac{\\dot{a}}{a})+(\\frac{\\dot{a}}{a})^{2} $, a slowly varying non-zero Hubble parameter $ \\frac{\\dot{a}}{a} $ requires $ \\frac{\\ddot{a}}{a} $ to be approximately constant. With that assumption, equation (\\ref{rhoconst}) has the solution\n\\begin{equation}\\label{Ldrho}\n\\Lambda_{d}=-\\frac{3}{a^{2}}+\\frac{8\\pi G\\varrho}{c^{2}}-\\Lambda_{v}\n\\end{equation} with \n\\begin{equation}\\label{Lv}\n\\Lambda_{v}=\\frac{3\\ddot{a}}{ac^{2}}.\n\\end{equation}The dark energy pressure is\n\\begin{equation}\\label{pd}\nP_{d}=\\frac{c^{2}}{a^{2}}+\\frac{8\\pi Gp}{c^{2}}+c^{2}\\Lambda_{v}.\n\\end{equation}\nA pure dark energy effect returns (\\ref{vace}) and (\\ref{vaca}) as the expansion and acceleration, respectively. In a universe with essentially constant matter, which is assumed to be the case of the present era, this demonstrates why $\\Lambda_{v}$ is important. As expected, $ \\Phi=\\frac{6}{a^{2}}+4\\Lambda_{v}+\\frac{8\\pi G}{c^{2}}(-\\varrho+\\frac{3p}{c^{2}}) $ is positive or negative.  \n\\par Riess et al. \\cite{29} used the Hubble telescope ``to provide the first conclusive evidence for cosmic deceleration that preceded the current epoch of cosmic acceleration\". Given the violent acceleration after the Big Bang, this observation evidences the cyclic nature of the universe to this point in time. The cosmological scale factor must have had maximum and minimum values in the past because of the observed changes in sign of its second derivative; there were extremums at $ \\dot{a}=0 $. In general, this requires $ \\Lambda_{d}=-\\frac{3}{a^{2}}+\\frac{8\\pi G\\varrho}{c^{2}} $ from equation (\\ref{frw5}). The Hubble parameter vanishes and (\\ref{Ldrho}) must change because (\\ref{Lv}) is not constant at the extremum. Dark energy in the amount of $ \\Lambda_{v} $ must be transferred to $ \\Lambda_{d} $ from the dark energy pressure; $ \\frac{P_{d}}{c^{2}} $ in (\\ref{pd}) decreases by $ -\\Lambda_{v} $ with an offsetting change by that amount to $ \\Lambda_{d} $ in (\\ref{Ldrho}). This allows an extremum to occur while keeping $ \\Phi $ unchanged. Then, cosmic acceleration can change to a decelerating epoch, and conversely with the opposite exchange of dark energy. \\par   The maxima or minima of the cosmological scale factor follows directly from equations (\\ref{frw9}) and (\\ref{frw10}). The second derivative of $ a $ must satisfy \n\\begin{equation}\\label{secdera}\n\\ddot{a}=a(-\\frac{c^2}{2a^{2}}+\\tilde{p})\n\\end{equation} when $\\dot{a}=0 $. The value of $ \\tilde{p} $ in equation (\\ref{secdera}) governs the condition for a maximum or minimum of $ a $. With $ -\\Lambda_{d} $ having a small fixed value of $\\Lambda_{v} $ determined early in the evolution of the universe, the variation in $ \\Phi $ is determined by $ P_{d} $. The constraint (\\ref{0phi}) on $ \\Phi $ can force $ P_{d} $ to change, which can change the sign of $ \\tilde{p} $. Near the end of an acceleration phase, if the dark energy pressure decreases so that $P_{d}\\leq\\frac{8\\pi Gp}{c^{2}}$, $ \\tilde{p} $ changes from positive to zero or negative, and the scale factor has a maximum value at $a_{max}$; $ \\tilde{p}\\leq0 $ is satisfied in (\\ref{secdera}). The acceleration phase ends and the universe undergoes a deceleration. The scale factor then decreases toward a minimum value $a_{min}$ at which the dark energy pressure increases enough to satisfy  $\\tilde{p}>\\frac{c^2}{2a^{2}}$. The deceleration phase changes to that of an acceleration and the cyclic process continues indefinitely. $ \\Phi $, governed by (\\ref{0phi}), smoothly controls the maximum and minimum values that the cosmological scale factor can have. The global constraint on $ \\Phi $ keeps the universe gravitationally in balance. This model of the universe starts with the Big Bang and then cycles to eternity. It does not suffer the catastrophes of the Big Crunch or the Big Rip. \n\n\\par Although recent data and analysis \\cite{30} suggests the observable universe is flat, the data likely represents a small fraction of the presently unknown \\emph{entire} universe. If the entire universe has a positive curvature, a measurement of it will appear to be nearly flat if data from large enough distances is not available. Therefore, at this time, the conjecture of a flat universe which expands forever based on observational evidence is less likely than the cyclic universe described and observed after the Big Bang and into this epoch.\\par Dark energy thus provides a natural explanation of why the vacuum energy density is minute, and why it dominates the present epoch of the universe.\n\\section{Energy-momentum of the gravitational field: Dark matter}\nThe $\\Lambda$CDM model describes the formation of galaxies after the Big Bang from cooled baryonic matter gravitationally attracted into a dark matter skeleton. Dark matter in the $\\Lambda$CDM model also provides the additional mass required to describe the flat rotation curves observed in many galaxies. However, no dark matter particles have been detected and there have been several attempts to explain the flat rotational curves without dark matter. \n\\par  The leading candidate is a \\emph{phenomenological} model of Modified Newtonian dynamics (MOND) introduced by Milgrom \\cite{31}. The Newtonian force F is modified according to\n\\begin{equation}\\label{key}\nF=m\\mu(\\frac{a}{A_{0}})a\n\\end{equation} where $ A_{0} $ is a fundamental acceleration $ \\approx1.2\\times10^{-10}m\/s^{2} $. $ \\mu $ is a function of the ratio of the acceleration relative to $ A_{0} $ which tends\nto one for $ a\\gg A_{0} $ and tends to $ \\frac{a}{A_{0}} $ for $ a\\ll A_{0} $. MOND successfully explains many, but not all, mass discrepancies observed in galactic data. However, it has no covariant roots in Einstein's equation or cosmological theory. MOND and $\\Lambda$CDM were thoroughly discussed by McGaugh in \\cite{32}.\\par  Other alternatives to dark matter were reviewed by Mannheim in \\cite{33} with references therein. In particular, Moffat \\cite{34} used a nonsymmetric gravitational theory without dark matter to obtain the flat rotation curves of some galaxies. The bimetric theory of Milgrom \\cite{35} involved two metrics as independent degrees of freedom to obtain a relativistic formulation of MOND.\\par  Different approaches to the missing matter problem include dipolar dark matter, which was introduced by Bernard, Blanchet and Heisenberg in \\cite{36} to solve the problems of cold dark matter at galactic scales and reproduce the phenomenology of MOND. The theory involves two different species of dark matter particles which are separately coupled to the two metrics of bigravity and are linked together by an internal vector field. In \\cite{37}, a theory of emergent gravity (EG) which claims a possible breakdown in general relativity, was introduced by Verlinde that provided an explanation for Milgrom's phenomenological fitting formula in reproducing the flattening of rotation curves. Campigotto,  Diaferio and Fatibenec \\cite{38} showed conformal gravity cannot describe galactic rotation curves without the aid of dark matter. On the other hand, a logical analysis based on observational data was presented by Kroupa in \\cite{39} to support the conjecture that dark matter does not exist.\\par The existence of dark matter is based on the assumption that general relativity is correct. However, Einstein's equation is incomplete without the tensor $ \\varPhi_{\\alpha\\beta} $ describing the energy-momentum of the gravitational field. The validity of modified general relativity is now tested with the attempt to describe the additional gravitational attraction in various galaxies without dark matter.\n\\subsection{Modified GR in a spheroidal spacetime}\nIt is assumed dark matter does not exist and that baryonic matter and other possible sources of matter such as neutrinos, produce the gravitational field. In a region of spacetime where there is no matter, $ \\tilde{T}_{\\alpha\\beta}=0 $ and the field equations must satisfy\n\\begin{equation}\\label{EF}\nG_{\\alpha\\beta}+\\varPhi_{\\alpha\\beta}=0.\n\\end{equation} Spheroidal solutions to these nonlinear equations are now investigated. The spheroidal behaviour of the metric is to be determined from a particular solution to (\\ref{EF}) in a spacetime described by a metric of the form\n\\begin{equation}\\label{g}\nds^{2}=-e^\\nu c^{2}dt^{2}+e^{\\lambda}dr^{2}+r^{2}(d\\theta^{2}+sin^{2}\\theta d\\varphi^{2})\n\\end{equation} where $ \\nu $ and $ \\lambda $ are functions of t, r and $\\theta$. The non-zero connection coefficients (Christoffel symbols) are:\n\\begin{flalign*}\n&\\Gamma^{0}_{00}=\\frac{1}{2}\\partial_{0}\\nu,\\enspace\\Gamma^{0}_{01}=\\frac{1}{2}\\partial_{1}\\nu,\\enspace\\Gamma^{0}_{02}=\\frac{1}{2}\\partial_{2}\\nu,\\enspace\n\\Gamma^{0}_{11}=\\frac{1}{2}\\partial_{0}\\lambda e^{\\lambda-\\nu},\\enspace\\Gamma^{1}_{00}=\\frac{1}{2}\\partial_{1}\\nu e^{\\nu-\\lambda},\\enspace\\Gamma^{1}_{01}=\\frac{1}{2}\\partial_{0}\\lambda,&\\\\ \n&\\Gamma^{1}_{11}=\\frac{1}{2}\\partial_{1}\\lambda,\\enspace \\Gamma^{1}_{12}=\\frac{1}{2}\\partial_{2}\\lambda,\\enspace\\Gamma^{1}_{22}=-re^{-\\lambda},\\enspace \\Gamma^{1}_{33}=-r\\sin^{2}\\theta e^{-\\lambda},\\enspace\\Gamma^{2}_{00}=\\frac{1}{2r^{2}}\\partial_{2}\\nu e^{\\nu},&\\\\\n&\\Gamma^{2}_{11}=-\\frac{1}{2r^{2}}\\partial_{2}\\lambda e^{\\lambda},\\enspace\\Gamma^{2}_{12}=\\frac{1}{r},\\enspace\n\\Gamma^{2}_{33}=-\\sin\\theta\\cos\\theta,\\enspace\n\\Gamma^{3}_{13}=\\frac{1}{r},\\enspace\\Gamma^{3}_{23}=\\cot\\theta.&\n\\end{flalign*}\nThe unit vectors $ u^{\\alpha} $ satisfy \n\\begin{equation}\\label{uu1}\nu^{\\alpha}u_{\\alpha}=-1.\n\\end{equation} As a first step to understand this highly nonlinear set of equations given by (\\ref{EF}) with the property (\\ref{uu1}) in this metric, $ u_{3} $ is chosen to vanish. This requires\n\\begin{equation}\\label{X3}\nX_{3}=0 \n\\end{equation} \nbecause $ u_{\\alpha} $ is collinear with $ X_{\\alpha}$. All other components of $X_{\\alpha}  $ are non-zero. \\par Static solutions to (\\ref{EF}) are sought which require the components of the line element field to satisfy \n\\begin{equation}\\label{xcomp}\n\\partial_{0}X_{\\alpha}=0\n\\end{equation} and from the metric,\n\\begin{equation}\\label{dlambdanu}\n\\partial_{0}\\lambda=0,\\enspace \\partial_{0}\\nu=0.\n\\end{equation}\\par The components of $ \\varPhi_{\\alpha\\beta} $ to be considered are then: \n\\begin{equation}\\label{key}\n\\varPhi_{00}=(1+2u_{0}u^{0})(-\\frac{1}{2}e^{\\nu-\\lambda}\\nu^{\\prime} X_{1}-\\frac{1}{2r^{2}}e^{\\nu}\\partial_{2}\\nu X_{2}),\n\\end{equation}\n\\begin{equation}\\label{key}\n\\varPhi_{11}=(1+2u_{1}u^{1} )({X_{1}}^{\\prime}-\\frac{1}{2}\\lambda^{\\prime}X_{1}+\\frac{1}{2r^{2}}e^{\\lambda}\\partial_{2}\\lambda X_{2}),\n\\end{equation} \n\\begin{equation}\\label{key}\n\\varPhi_{22}=(1+2u_{2}u^{2} )(\\partial_{2}X_{2}+re^{-\\lambda} X_{1}),  \n\\end{equation} \n\\begin{equation}\\label{key}\n\\varPhi_{33}=r \\sin^{2}\\theta e^{-\\lambda}X_{1}+\\sin\\theta \\cos\\theta X_{2},\n\\end{equation} the Ricci scalar, which from (\\ref{EF}) equals $ \\Phi $, is\n\\begin{equation}\\label{R}\n\\begin{split}\nR=e^{-\\lambda}(-\\nu^{\\prime\\prime}-\\frac{1}{2}{\\nu^{\\prime}}^{2}+\\frac{1}{2}\\lambda^{\\prime}\\nu^{\\prime}-\\frac{2}{r}\\nu^{\\prime}+\\frac{2}{r}\\lambda^{\\prime}-\\frac{2}{r^{2}})+\\frac{1}{r^{2}}(-\\frac{1}{2}{\\partial_{2}\\nu}^{2}-\\frac{1}{2}{\\partial_{2}\\lambda}^{2}-\\frac{1}{2}\\partial_{2}\\nu\\partial_{2}\\lambda\\\\-\\partial_{2}\\partial_{2}\\nu-\\partial_{2}\\partial_{2}\\lambda-\\cot\\theta(\\partial_{2}\\nu+\\partial_{2}\\lambda)+2),\n\\end{split}\n\\end{equation}\nand the corresponding components of the Einstein tensor are:\n\\begin{equation}\\label{key}\nG_{00}=\\frac{1}{r^{2}}e^{\\nu-\\lambda}(r\\lambda^{\\prime}-1)+\\frac{e^{\\nu}}{2r^{2}}(-\\partial_{2}\\partial_{2}\\lambda-\\cot\\theta\\partial_{2}\\lambda-\\frac{1}{2}(\\partial_{2}\\lambda)^{2}+2),\n\\end{equation}\n\\begin{equation}\\label{key}\n\\begin{split}\nG_{11}=\\frac{1}{r^{2}}(1+r\\nu^{\\prime})+\\frac{e^{\\lambda}}{2r^{2}}[\\partial_{2}\\partial_{2}\\nu+\\frac{1}{2}({\\partial_{2}\\nu})^{2}+\\cot\\theta \\partial_{2}\\nu-2],\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{key}\n\\begin{split}\nG_{22}=\\frac{r^{2}e^{-\\lambda}}{2}[\\nu^{\\prime\\prime}+\\frac{1}{2}{\\nu^{\\prime}}^{2}+\\frac{\\nu^{\\prime}}{r}-\\frac{\\lambda^{\\prime}}{r}-\\frac{\\lambda^{\\prime}\\nu^{\\prime}}{2}]+\\frac{1}{4}{\\partial_{2}\\nu}\\partial_{2}\\lambda+\\frac{1}{2}\\cot\\theta(\\partial_{2}\\nu+\\partial_{2}\\lambda),\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{key}\n\\begin{split}\nG_{33}={\\sin\\theta}^2[\\frac{r^{2}e^{-\\lambda}}{2}(\\nu^{\\prime \\prime}+\\frac{1}{2}{\\nu^{\\prime}}^{2}+\\frac{\\nu^{\\prime}}{r}-\\frac{\\lambda^{\\prime}}{r}-\\frac{\\lambda^{\\prime}\\nu^{\\prime}}{2})+\\frac{1}{4}({\\partial_{2}\\nu})^2+\\frac{1}{4}({\\partial_{2}\\lambda})^2+\\frac{1}{4}\\partial_{2}\\nu\\partial_{2}\\lambda+\\frac{1}{2}{\\partial_{2}\\partial_{2}\\nu}+\\frac{1}{2}{\\partial_{2}\\partial_{2}\\lambda}]\n\\end{split}\n\\end{equation} where the prime denotes $ \\partial_{1} $. \\par These equations are greatly simplified by setting\n\\begin{equation}\\label{lambdanu}\n\\nu=-\\lambda\n\\end{equation}which provides a potential correspondence to a Schwarzschild-like solution.\nThus, a class of static spheroidal solutions to (\\ref{EF}) are sought with the restrictions (\\ref{X3}),(\\ref{xcomp}),(\\ref{dlambdanu}) and (\\ref{lambdanu}). \\par Since $ e^{\\lambda-\\nu}(\\varPhi_{00}+G_{00})+\\varPhi_{11}+G_{11}=0 $ from (\\ref{EF}),\n\\begin{equation}\\label{phiG0011}\n\\begin{split}\n(\\frac{\\lambda^{\\prime}}{2}X_{1}+\\frac{e^{\\lambda}}{2r^{2}}\\partial_{2}\\lambda X_{2})(1+2u_{0}u^{0})+(X_{1}^{\\prime}-\\frac{\\lambda^{\\prime}}{2}X_{1}+\\frac{e^{\\lambda}}{2r^{2}}\\partial_{2}\\lambda X_{2})(1+2u_{1}u^{1})\\\\+\\frac{e^{\\lambda}}{r^{2}}(-\\partial_{2}\\partial_{2}\\lambda-\\cot\\theta\\partial_{2}\\lambda)=0.\n\\end{split}\n\\end{equation} \n\\par  From (\\ref{EF}) $G_{22}+\\varPhi_{22}=0 $ gives \\begin{equation}\\label{phiG22}\n\\begin{split}\n-\\lambda^{\\prime\\prime}+\\lambda^{\\prime 2}-\\frac{2}{r}\\lambda^{\\prime}+\\frac{2e^{\\lambda}}{r^{2}}(-\\frac{1}{4}({\\partial_{2}\\lambda})^{2}+(\\partial_{2}X_{2}+re^{-\\lambda}X_{1})(1+2u_{2}u^{2}))=0\n\\end{split}\n\\end{equation}\n\nand $G_{33}+\\varPhi_{33}=0 $ in the interval $ 0<\\theta<\\pi $ yields\n\\begin{equation}\\label{phiG33}\n\\begin{split}\n-\\lambda^{\\prime\\prime}+\\lambda^{\\prime 2}-\\frac{2}{r}\\lambda^{\\prime}+\\frac{2e^{\\lambda}}{r^{2}}(re^{-\\lambda}X_{1}+\\frac{1}{4}({\\partial_{2}\\lambda})^{2}+X_{2}\\cot\\theta)=0.\n\\end{split}\n\\end{equation} Subtracting (\\ref{phiG22}) from (\\ref{phiG33}) requires\n\\begin{equation}\\label{C2}\n\\cot\\theta X_{2}-\\partial_{2}X_{2}+\\frac{1}{2}({\\partial_{2}\\lambda})^{2}-2u_{2}u^{2}(\\partial_{2}X_{2}+re^{-\\lambda}X_{1})=0.\n\\end{equation} \\par A solution to these equations, which depends on both $ r $ and $ \\theta $ would be useful in\nthe study of angular-dependent aspects of cosmology. However, before tackling that\nproblem, the spherically symmetric solution to (\\ref{phiG33}) with $\\partial_{2}\\lambda=0 $ must be obtained. That is accomplished by expressing the bracketed term in (\\ref{phiG33}), $  re^{-\\lambda}X_{1}+X_{2}\\cot\\theta $, as a power series in $ r$. To determine a meaningful expression for the power series, there are two physical requirements that can be invoked. Firstly, the Tully-Fisher relation should be obtainable to describe the flat rotation curves of some galaxies within a universe filled with dark energy. This condition requires a term of $ \\ln r $ in $ \\lambda $ which will yield a $ \\frac{1}{r} $ term in $ \\lambda^{\\prime} $. Secondly, the Newtonian gravitational energy density has a $ \\frac{1}{r^{4}} $ radial dependence. These conditions will be the guides to follow from solutions to (\\ref{phiG0011}) and (\\ref{C2}). With the benefit of some hindsight, solutions of the form\n\\begin{equation}\\label{lambda}\n\\lambda=-\\ln(-Ar+2B\\ln r+\\frac{c_{1}}{r}+c_{2})\n\\end{equation}are sought where all of the parameters are arbitrary constants.\n\\par From (\\ref{Xmu}), the static condition requires $ \\partial_{0}f=0 $ so $ u_{0}=0 $ and $ u_{1}u^{1}+u_{2}u^{2}=-1 $. Equations (\\ref{phiG0011}) and (\\ref{C2}) can then be combined as\n\\begin{equation}\\label{C}\nX_2\\cot\\theta-\\partial_{2}X_{2}+2(\\frac{X_{1}^{\\prime}}{X_{1}\\lambda^{\\prime}}-1)(X_{2}\\cot\\theta+re^{-\\lambda}X_{1})=0.\n\\end{equation}This is the equation from which power series expressions for $ X_{1} $ and $ X_{2} $ are sought which can then be used in (\\ref{phiG33}) to generate $ \\lambda $ of the form given by (\\ref{lambda}). \\par Firstly, an expression for $ X_{1}$ of the form $ X_{1}=e^{\\lambda}P $ is pursued where $ P $ is a polynomial in $ r $. By setting \n\\begin{equation}\\label{key}\nX_{1}^{\\prime}=X_{1}\\lambda^{\\prime}(1+P)\n\\end{equation},\n\\begin{equation}\\label{key}\nX_{1}=c_{3}e^{\\lambda}e^{\\int P\\lambda^{\\prime}dr}\n\\end{equation}where $ c_{3} $ is an arbitrary constant. By demanding $ P\\lambda^{\\prime}=\\frac{P^{\\prime}}{P} $, we get the Ricatti equation\n\\begin{equation}\\label{key}\nP^{\\prime}=P^{2}\\lambda^{\\prime}\n\\end{equation}which leads to the expression\n\\begin{equation}\\label{key}\nX_{1}=c_{3}e^{\\lambda}P=\\frac{c_{3}e^{\\lambda}}{-\\lambda+c_{4}}\n\\end{equation}where $ c_{4} $ is an arbitrary constant. It is chosen to be an upper bound on $ \\lambda $: $ \\lambda\\ll c_{4} $. Then, \n\\begin{equation}\\label{key}\nX_{1}\\cong \\frac{c_{3}}{c_{4}^{2}}e^{\\lambda}(c_{4}+\\lambda).\n\\end{equation}By setting $ r=\\frac{1}{x+1} $ with $ \\mid x\\mid<1 $, the $\\ln r$ term in $ \\lambda $ of (\\ref{lambda}), and then $ \\lambda $, can be expanded to $O(r^{-2}) $ which results in the power series for $ X_{1}$: \n\\begin{equation}\\label{X11}\nX_{1}=e^{\\lambda}(a_{0}+\\frac{a_{1}}{r}+\\frac{a_{2}}{r^{2}})\n\\end{equation} where all of the parameters have been absorbed into the arbitrary constants $a_{0},a_{1} $ and $ a_{2} $ accordingly.\\par Equation (\\ref{C}), with $ X_{1} $ given by (\\ref{X11}), becomes \n\\begin{equation}\\label{key}\n\\partial_{2}X_{2}-X_{2}\\cot\\theta(1+\\frac{2P^{\\prime}}{\\lambda^{\\prime}P})=\\frac{2rP^{\\prime}}{\\lambda^{\\prime}}\n\\end{equation}which has a solution involving the hypergeometric function. However, a manageable approximate expression for $ X_{2} $ can be obtained by observing that $ \\frac{2P^{\\prime}}{\\lambda^{\\prime}P}\\simeq0 $ for large $ r $. Then \n\\begin{equation}\\label{key}\nX_{2}=c_{5} \\sin\\theta+N\\sin\\theta\\ln\\tan\\frac{\\theta}{2}\n\\end{equation}where $N:=\\frac{2rP^{\\prime}}{\\lambda^{\\prime}}\\simeq 2a_{1}+\\frac{4a_{2}}{r}$ and $ c_{5} $ is an arbitrary constant. In the equatorial plane, we can do the perturbation $ \\frac{\\pi}{2}+2\\theta $ around $ \\frac{\\pi}{2} $ where $ \\theta<<1 $ so $ \\ln\\tan(\\frac{\\pi}{4}+\\theta)\\simeq 2\\tan\\theta $. $ X_{2} $ then has the structure\n\\begin{equation}\\label{key}\nX_{2}=\\tan\\theta(b_{0}+\\frac{b_{1}}{r})\n\\end{equation}where $ c_{5}=0 $ and all parameters are absorbed into the arbitrary constants $ b_{0} $ and $ b_{1} $ accordingly. Equation $(\\ref{phiG33})$ can then be written as \n\\begin{equation}\\label{phiG33a}\n\\begin{split}\n-\\lambda^{\\prime\\prime}+\\lambda^{\\prime 2}-\\frac{2}{r}\\lambda^{\\prime}+\\frac{2e^{\\lambda}}{r^{2}}(a_{0}r+a_{1}+\\frac{a_{2}}{r}+b_{0}+\\frac{b_{1}}{r})=0.\n\\end{split}\n\\end{equation}\nBy demanding \n\\begin{equation}\\label{b1}\n b_{1}=-a_{2} \n\\end{equation}the undesirable term with $ \\frac{\\ln r}{r} $ in the solution for $ \\lambda $ can be avoided. Equation (\\ref{phiG33a}) then simplifies to\n \\begin{equation}\\label{}\n  \\begin{split}\n \t-\\lambda^{\\prime\\prime}+\\lambda^{\\prime 2}-\\frac{2}{r}\\lambda^{\\prime}+\\frac{2e^{\\lambda}}{r^{2}}(a_{0}r-b)=0\n \\end{split}\n  \\end{equation}where\n \\begin{equation}\\label{key}\n -b=a_{1}+b_{0}.\n \\end{equation} This has the exact solution in the equitorial plane as desired:\n\\begin{equation}\\label{wow}\n\\begin{split}\n\\lambda=-\\ln(-a_{0}r+2b\\ln r+\\frac{c_{1}}{r}+c_{2}), \\enspace 0<r<\\infty\n\\end{split}\n\\end{equation}where $c_{1}$ and $ c_{2}  $ are arbitrary constants. Equation (\\ref{wow}) represents the extended Schwarzschild solution.\n\\subsubsection{The energy density of the gravitational field} The total energy-momentum tensor given by (\\ref{Tab}) requires the energy density of the gravitational field to be described by\n\\begin{equation}\\label{W}\nW=-\\frac{c^{4}}{8\\pi G}\\varPhi_{00}.\n\\end{equation} where\n\\begin{equation}\\label{v00}\n\\varPhi_{00}=\\frac{1}{2}X_{1}e^{-2\\lambda}\\lambda^{\\prime}.\n\\end{equation}From (\\ref{X11}) and (\\ref{wow}), \n\\begin{equation}\\label{key}\n\\varPhi_{00}=\\frac{1}{2}(a_{0}-\\frac{2b}{r}+\\frac{c_{1}}{r^{2}})(a_{0}+\\frac{a_{1}}{r}+\\frac{a_{2}}{r^{2}})\n\\end{equation}after calculating $ \\lambda^{\\prime} $ from (\\ref{wow}). By setting each of the coefficients of the $ \\frac{1}{r}, \\frac{1}{r^{2}},\\frac{1}{r^{3}} $ terms to zero, we obtain $ 2b=a_{1} $, $ a_{0}(a_{2}+c_{1})=2ba_{1} $, $ 2ba_{2}=c_{1}a_{1} $ respectively. Then it follows that \\begin{equation}\\label{a2}\na_{2}=c_{1}\n\\end{equation}\nand\n\\begin{equation}\\label{key}\na_{1}^{2}=2\\mid a_{0}\\mid \\mid c_{1}\\mid,\n\\end{equation}and\n\\begin{equation}\\label{key}\n\\varPhi_{00}=\\frac{1}{2}a_{0}^{2}+\\frac{c_{1}^{2}}{2r^{4}}\n\\end{equation}where $ c_{1} $ is chosen to be the parameter\n\\begin{equation}\\label{c1}\nc_{1}=-\\frac{2GM}{c^{2}}\n\\end{equation}from the Schwarzschild solution.\nFrom (\\ref{W}), it follows that the energy density of the static gravitational field is\n\\begin{equation}\\label{key}\nW=-\\frac{GM^{2}}{4\\pi r^{4}}-\\frac{c^{4}a_0^{2}}{16\\pi G }.\n\\end{equation}We see that the radial term is twice the Newtonian gravitational energy density. This is not surprising considering the fact that in general relativity, it is calculated from the linearized field equations \\cite{41}. The perturbation expansion of the field equations in terms of $ h_{\\alpha\\beta} $, a very small change in the metric relative to the flat spacetime Minkowski metric, contains an infinite number of very small terms involving $ h^{2},h^{3},...h\\partial h,...h\\partial\\partial h... $ to all possible powers of $ h$ and its first two derivatives. These truncated self interactions of the perturbed field and the linear term, apparently contribute the same amounts of gravitational energy to $W$. But there is no way of knowing that in GR because it has no tensor that explicitly represents the energy-momentum of the gravitational field.\\par The Tully-Fisher relation is derived in the next subsection. \n\\subsubsection{The radial force and galactic rotation curves}\nThe radial force on an object of mass m can now be calculated from (\\ref{wow}) in the equitorial plane. Using the conventional relationship of the Newtonian potential $ \\phi $ to $ g_{00} $, \n\\begin{equation}\\label{key}\n\\phi=\\frac{c^{2}}{2}(e^{\\nu}-1),\n\\end{equation} the radial force $ F_{r} $ is\n\\begin{equation}\\label{Fr}\n\\begin{split}\nF_{r}&=-m\\partial_{1}\\phi\\\\\n&=\\frac{mc^{2}}{2}(a_{0}-\\frac{2b}{r}+\\frac{c_{1}}{r^{2}}).\n\\end{split}\n\\end{equation} From (\\ref{c1}) where M represents the total mass of the galaxy composed of mainly baryonic matter and no dark matter, we arrive at the modified Newtonian force \n\\begin{equation}\\label{Newt}\nF_{r}=-\\frac{GMm}{r^{2}}-\\frac{bmc^{2}}{r}+\\frac{a_{0}mc^{2}}{2}.\n\\end{equation} The correction terms to the Newtonian force come from the non-zero components of the line element field in the energy-momentum tensor $ \\varPhi_{\\alpha\\beta}$. The components of the line element field can change their sign, which means $ a_{0} $ can change to $ -a_{0} $. The second term is gravitationally attractive and represents the ``dark matter\" correction if $b>0 $. It is the term that gives rise to the flat rotation curves. The third term is positive and repulsive if $ a_{0}>0 $. This describes the repulsive dark energy force in the present epoch. However, during a part of the previous decelerating epoch observed by Riess et al. \\cite{29}, $ a_{0}<0 $. They used the Hubble telescope to provide the first conclusive evidence for cosmic deceleration that preceded the current epoch of cosmic acceleration.  \\par Assuming a circular orbit about a point mass, it follows that the orbital velocity of a star rotating in the galaxy satisfies\n\\begin{equation}\\label{V}\nv^{2}=v^{2}_{N}+bc^{2}-\\frac{a_{0}c^{2}}{2}r\n\\end{equation} where $v_{N}^{2}$ is the Newtonian term \n\\begin{equation}\\label{v2N}\nv_{N}^{2}=\\frac{GM}{r}. \n\\end{equation} Equation (\\ref{V}) demands an upper limit to r describing a large but finite galaxy.  \n\\par  Because $ a_{0}\\neq0$, it is possible for the Newtonian force to balance the dark energy force. This requires\n\\begin{equation}\\label{VF}\nv^{2}_{N}-\\frac{a_{0}c^{2}}{2}r=0\n\\end{equation} and\n\\begin{equation}\\label{key}\nv^{2}=bc^{2}\n\\end{equation} with $ b>0 $ describes a specific class of galaxies with a flat orbital rotation curve. From (\\ref{v2N}) and (\\ref{VF}), we obtain the Tully-Fisher relation\n\\begin{equation}\\label{TF}\nv^{4}_{N}=\\frac{GMc^{2}a_{0}}{2},\\enspace a_{0}>0.\n\\end{equation} This result holds for any finite r in contrast to EG which holds only for large r as determined by Lelli, McGaugh and Schombert \\cite{41}. With $ \\frac{{c^{2}a_{0}}}{2}:=A_{0}$, the Tully-Fisher relation in MOND is evident.\n\\par   The importance of the radial acceleration relative to the rotation curves of galaxies was discussed by Lelli, McGaugh, Schombert, and Pawlowski in \\cite{42} where it was determined that late time galaxies (spirals and irregulars), early time galaxies (ellipticals and lenticulars), and the most luminous dwarf spheroidals follow a tight radial acceleration relation which correlates well with that due to the distribution\nof baryons. \\par  Equation (\\ref{Newt}), which does not include dark matter in this analysis, is general enough to describe the rotation curves of many types of galaxies. For example, galaxy NGC4261 has a relatively flat rotation curve but starts to rise at larger radii, reaching velocities of 700 km $s^{-1}$ at 100 kpc \\cite{42}. That requires $ a_{0} $ in (\\ref{V}) to be negative which was interpreted above. As another example, both $ a_{0}c^{2}r $ and $ bc^{2} $ could be small enough relative to $ \\frac{GM}{r} $, or those terms could cancel one another, so that the  Newtonian term is dominant. Galaxies with no flat rotation curves have recently been observed by van Dokkum et.al \\cite{43}. \\par It should be remembered that  the general equation (\\ref{EF}) provides additional variables that may explain even more aspects of cosmology now attributed to dark matter. However, it is still possible that dark matter particles may exist. As a part of (\\ref{MEQ}) in the total matter energy-momentum tensor $ \\tilde{T}_{\\alpha\\beta} $, they would contribute to the gravitational field outside of its source along with baryonic matter in equation (\\ref{EF}) and therefore in (\\ref{Newt}). But any dark matter contribution to the gravitational field would play a much lesser role because of the existence of $ \\varPhi_{\\alpha\\beta} $. \\par From the expression for $ \\lambda $ given by (\\ref{wow}), it is clear that in the absence of the line element field in GR, the Einstein tensors alone cannot represent the additional gravitational attraction attributed to dark matter. This explicitly shows why GR is incomplete and why $ \\Lambda $CDM was invented to describe dark matter.\n\\section{Conclusion}\nThe results in this article stem from the existence of the line element field in a Lorentzian spacetime. It is a fundamental part of the Lorentzian metric and provides the extra freedom to construct the symmetric tensor  $\\varPhi_{\\alpha\\beta}$ from the Lie derivative along the line element vector of both the metric and the unit line element vectors. That tensor, which is absent in GR, solves the problem of the localization of gravitational energy-momentum. $ T_{\\alpha\\beta} $, the sum of the matter energy-momentum and $ \\varPhi_{\\alpha\\beta} $ is divergenceless. $ \\varPhi_{\\alpha\\beta} $ completes Einstein's equation and leaves it intact in form. \\par The line element field is a dynamical variable independent of the Riemannian metric. Variation of the action functional with respect to $ X^{\\mu} $ restricts the covectors that can be a part of the Lorentzian metric to be those satisfying the Lorentz invariant expression $ u_{\\mu}=\\frac{3}{\\Phi}\\partial_{\\mu}f $.\\par The gravitational energy density is calculated from $ \\varPhi_{00} $. It is shown that the radial contribution is twice the Newtonian gravitational energy density calculated from the linearized field equations in GR. The energy-momentum of the gravitational field is localizable and measureable.  \\par  \n$ \\Phi $ has the global property $ \t\\int g^{\\alpha\\beta}\\varPhi_{\\alpha\\beta}\\sqrt{-g}d^{4}x=0  $. The cosmological constant $ \\Lambda $ is dynamically replaced by $ \\Phi $.\n\\par Important features attributed to dark energy result from the investigation of the modified Einstein equation in the FLRW metric. $\\Phi>0$ defines dark energy in the present epoch. The dark energy pressure explains the observed cyclic nature of the universe after the Big Bang. The dark energy density explains the initial inflation of the universe and provides a natural explanation of why the vacuum energy density is so small and why it now dominates the expansion and acceleration of the present universe.\n\\par The energy-momentum of the gravitational field is important in the description of dark matter. A static solution is obtained from the modified Einstein equation in a spheroidal metric describing the gravitational field outside of its source, which does not contain dark matter. The modified Newtonian force contains two additional terms: one represents the dark energy force which depends on the parameter $ a_{0} $; and the other represents the $ \\frac{1}{r} $ ``dark matter\" force which depends on the parameter $ b $. The baryonic Tully-Fisher relation is obtained by balancing the dark energy force with the Newtonian force. This condition describes the class of galaxies associated with MOND. The rotation curves for galaxies with no flat orbital curves, and those with rising rotation curves for large radii describe examples of the flexibility of the orbital rotation curve equation. The results obtained from the \\emph{complete} Einstein equation thus far are able to substantially describe the missing mass problem attributed to dark matter. Further mathematical and detailed numerical analyses to explore the ability of the energy-momentum tensor of the gravitational field to replace dark matter in cosmology, are fully warranted. This rigorous analysis with comparison to astronomical data may still point to the existence of dark matter to some extent. But even if that is the case, the gravitational role of dark matter is substantially reduced by the impact of the energy-momentum tensor of the gravitational field.\n\\par  Thus, $ \\varPhi_{\\alpha\\beta} $ represents the energy-momentum of the gravitational field itself and explains particular features of dark energy and dark matter. It is the symmetric tensor that Einstein sought many years ago. \n\\section*{Acknowledgements} \nI would like to thank the anonymous referee for his\/her constructive comments. \n\n\\vskip2pc\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDeveloping Reinforcement Learning (RL) algorithms that can make effective decisions using high dimensional observations such as images is quite challenging. In addition, it consumes a lot of time and energy. In recent months researchers have worked on developing sample efficient RL, plug and play algorithms, that can directly learn from pixels. Srinivas \\textit{et al.} incorporated Contrastive Learning, into off-policy algorithms, to learn relevant features from image based inputs. Laskin \\textit{et al.} investigated developing data efficient and generalizable algorithms, by introducing a generic data augmentation module for RL algorithms; \\cite{laskin2020reinforcement, shang2021reinforcement, srinivas2020curl}.\n\nWhile a lot of work has been devoted to extracting positional information from image inputs, very little investigation has been done on learning from temporal information. Shang \\textit{et al.} performed experiments using DMControl (\\cite{tassa2018deepmind}) to highlight the importance of temporal information in RL. They compared two Soft-Actor-Critic (SAC) RL algorithms, wherein one had access to pose and temporal information and the other only had access to pose. It was found that the former algorithm swiftly learned the optimal policy, while the latter failed to do so. Furthermore, a recurring heuristic used by many papers is to stack sequential observations together while inputting it to a neural network; \\cite{mnih2015human}. This heuristic combines frames, without processing it and therefore can be considered analogous to early fusion; \\cite{karpathy2014large}. Recently, Shang \\textit{et al.} approached this as a video classification problem. This is a lucid approach, as considering a DRL state equivalent to a video, will help improve the prediction capabilities of the underlying neural network. Successful video recognition architectures use late fusion where all frames are processed, using neural networks, before they are combined together; \\cite{shang2021reinforcement}, \\cite{laskin2020reinforcement}. \n\nMoreover, a video stream consists of both spatial and temporal aspects. The former contains information about the video frame including objects and its surroundings, while the movement of the frame and its associated objects can be learned from the temporal portion; \\cite{simonyan2014two}. While learning the spatial aspect is enough for image recognition, video recognition requires learning both spatial and temporal components. Enabling agents to extract temporal information from a given set of frames will result in the DRL agent making better Q-value predictions and therefore result in improved data efficiency. Furthermore, it will contribute to the agent understanding the differences between seemingly similar actions, such as opening and closing objects; \\cite{lin2019tsm}, \\cite{shang2021reinforcement}. \n\nThere has been a plethora of work related to video recognition using 3D and standard 2D CNNs. 3D CNNs have the ability to simultaneously extract out spatial and temporal features from videos. However, they are computationally costly, which makes them hard to implement in real-time situations. Incorporating similar architectures with vision-based DRL exacerbates this problem, as many applications require fast predictions during training, and having latency is infeasible. Furthermore, the extra parameters could make the model more prone to overfitting without large amounts of data. This once again poses a roadblock to the development of sample efficient RL;\\cite{lin2019tsm,tran2015learning, carreira2017quo}. 2D CNNs, although relatively efficient however fail to extract out temporal information;\\cite{simonyan2014two}. \n\n\\cite{amiranashvili2018motion} incorporated optical flow in their RL algorithm, although their technique required state variables in addition to pixel observations during training. Modeling temporal information in RL using simply pixel inputs was investigated by \\cite{shang2021reinforcement}, and it brought a new approach to efficiently reducing sample complexity in reinforcement learning. We intend to further optimize this technique by leveraging recent work in the field of video action prediction and therefore propose the Temporal Shift Reinforcement Learning (TSRL) algorithm.\n\nThe contributions of our work \\footnote{Our code is available at \\url{https:\/\/anonymous.4open.science\/r\/TSM_RL-85F5\/README.md}} are presented here:\n\n1) We propose a plug and play architecture works with any generic vision-based DRL algorithms.\n\n2) We augment a video recognition; \\cite{lin2019tsm} that does not require any additional parameters to model temporal information in DRL.\n\n\\section{Related Work}\n\\subsection{Latent Flow} Simonyan \\textit{et al.} investigated the use of optical flow techniques to perform video classification and was able to achieve SOTA performance by a significant amount over previous work in video classification. They developed a dual-stream architecture using ConvNets, consisting of spatial and temporal recognition components. The spatial stream was learned using a pre-trained ConvNet, wherein each frame was sent to the network as input. The input to the temporal stream was stacked optical flow displacement fields generated from consecutive frames. Movement among frames can be obtained from optical flow fields, thereby eliminating the need for the network to learn it. This technique achieved high accuracies without requiring a lot of data. More importantly, they established that training a temporal CNN using optical flow was a drastically better technique compared to training on a stacked bunch of images; \\cite{simonyan2014two, karpathy2014large}. The downside of this algorithm is that it is computationally costly both during inference and training and therefore cannot be combined with RL algorithms; \\cite{shang2021reinforcement}.\n\n\\subsection{Flow of Latents} Shang \\textit{et al.} looked for a computationally feasible technique to integrate RL with optical flow. They were inspired by late fusion techniques; wherein every frame was run through a CNN before fusion was applied. Late fusion provides improved performance with lesser parameters and also allows multi-modal data \\cite{jain2019learning, chebotar2017path}. They presented a structured late fusion architecture, wherein each image frame was encoded using a neural network. The encodings at each time step were subtracted from their prior, and this difference was fused with the latent encodings, which was then used by the RL algorithm. This technique was analogous to the work done by \\cite{simonyan2014two}. The optical flow was approximated using the difference in encodings, which provided temporal information. The spatial component was obtained by encoding each of the frames. This technique provided the CNN with a necessary inductive bias. They chose Rainbow DQN, and RAD; \\cite{laskin2020reinforcement} to be their base algorithm and found that it outperformed SOTA algorithms in performance and sample efficiency. Also, they showed that their algorithm reached optimal performance in state-based RL despite only being provided positional state information and no state velocity.\n\nThey also separately investigated encoding frames and then stacking the encodings together instead of the raw images. This technique yielded sub-par results, and the authors hypothesized that stacking high dimensional image frames would allow CNNs to learn temporal information. However, by stacking latent frames, the temporal information was lost and thereby causing the difference in results. \n\n\\subsection{Temporal Shift Module}\nWhile working with video model activations consisting of $N$ frames, $C$ channels and $H$ height and $W$ width, \\\\$A \\in \\mathbb{R}^{N X C X T X H X W}$, 2D CNNs don't consider the temporal dimension $T$ thereby ignoring it. \\cite{lin2019tsm} addressed this by shifting channels, thereby mixing information from neighboring frames through the temporal dimension and referred to it as the Temporal Shift Module (TSM). Therefore the current frame contains information that was obtained from its surroundings. They leveraged the concepts of shifts and multiply-accumulate, which are the basic principles of a convolution operation. They extended it by shifting one step forward and backward along the temporal dimension. Furthermore, the multiply-accumulate was folded from the channel dimension to the temporal dimension. However, for online video recognition, only previous frames could be shifted forward and not the other way around. Therefore in such cases, a uni-directional TSM was implemented.\n\nWhile this process doesn't require extra parameters, they found that this technique had drawbacks - 1) The data movement generated due to the shift strategy was not efficient and would increase the latency, especially since 5D activation of videos results in large memory usage. This implied that moving all channels would result in inference latency and large memory footprint on the hardware hosting the model. 2) Moving channels directly across the temporal dimension, referred to as in-place shift, would affect the accuracy of model since the spatial model is distorted. This is because the current channel would have some of its frames (or feature maps) shifted, and therefore, the 2D CNN would lose that information during the classification process. The authors obtained a 2.6 \\% accuracy drop relative to their baseline; \\cite{wang2016temporal} while naively shifting channels. The former issue was mitigated by shifting only a partial number of channels, thereby reducing the amount of data movement and latency incurred. For the latter problem, the TSM module was inserted within the residual branch of a Res-Net, thereby enabling the 2D CNN to learn spatial features without degrading. The authors claimed that this method, namely residual shift, allows the information present within the original activation to be retained after channel shifting due to identity mapping. Therefore, the TSM module is a simple modification to the 2D CNN. After encoding images, it shifts frames in the temporal dimension by +1, -1, and 0. However, shifting frames by -1, i.e. backward, is only possible for offline problems. For online image classification problems, the frames are moved +1; \\cite{lin2019tsm}.\n\nA major advantage of online TSM was that it enabled multi-level temporal fusion. Other online methods are generally limited to late and mid-level temporal fusion. The authors found multi-level temporal fusion to significantly influence temporal problems; \\cite{zhou2018temporal,lin2019tsm, zolfaghari2018eco}.\n \n\\subsection{Prioritized Deep Q Network}\n\\cite{mnih2015human} combined Q Networks with CNNs in order to obtain an approximation of the Q values - \n\n$Q^{*}(s,a) = \\max_{\\pi}\\mathbb{E}[r_t + \\gamma r_{t+1} + \\gamma^2 r_{t+2} + ... |s_t = s, a_t = a, \\pi]$\n\\begin{figure*}[htp]\n    \\centering\n    \\includegraphics[width=14cm]{TSRL_Schematic.png}\n    \\caption{A schematic of Temporal Shift Reinforcement Learning algorithm}\n    \\label{fig:env}\n\\end{figure*}\n\nThe above expression maximizes the sum of discounted rewards for an agent following a policy, $\\pi = P(a|s)$, $r$ using a discount factor, $\\gamma$ during every time step $t$. It was known to be the first RL algorithm that could be integrated into various environments with raw pixels as inputs. They addressed the learning instabilities that RL presented when coupled with a deep neural network using a replay buffer and target network. They found that the sequential observations were highly correlated with each other and also that minimal changes to $Q$ would drastically affect the policy. The use of a replay buffer mitigated this issue by randomizing the data during the training process. This was done by storing the transitions as a tuple $(s,a_t,s_{t+1}, r{t+1})$ of state, action, next states and rewards within a cyclic buffer. This provided a two-fold benefit. The replay buffer reduced the number of environments needed for the agent to learn since the agent could always resample from the buffer. Furthermore, this reduces the variance during gradient descent since batches are sampled. The target network takes the weight from the current network but updates it only after a fixed duration of time. The target network's weights are then used to compute the TD error, which is the difference between the Q value and the TD target. If we use the parameters from the current network to estimate both these values, they'll become correlated and will result in instability. \\cite{hasselt2010double} suggested using dual instead of single estimators to estimate the expected return since the latter led to over-estimated values and introduced the Double-Q learning algorithm (DDQN). A later investigation by \\cite{van2016deep} showed that rather than learning a separate function, the target network could be used to obtain the estimate; \\cite{mnih2015human, arulkumaran2017brief}. \n\nIn addition, \\cite{schaul2015prioritized} modified the experience replay process so that, instead of the conventional uniform sampling process, important samples were given a higher priority. The Prioritized Experience Replay (PER) technique was found to double the learning speed and also achieve SOTA scores on Atari games.\n\n\\section{Approach}\n\nThe motivation behind TSRL was to introduce an efficient algorithm that did not require any additional parameters, leveraging the benefits of multi-level temporal fusion. The architecture developed by \\cite{lin2019tsm} for online Temporal Shift was modified and incorporated into a Double DQN with Prioritized Experience Replay (DDQN-PER). \\cite{lin2019tsm} used a ResNet model for their experiments, however going with the conventional CNN models used by the vision RL community, we used a shallow three layer CNN. \n\nAlso, we used in-place shift instead of residual shift wherein the channels were directly moved across the temporal dimension. We assumed that the accuracy improvements obtained, in predicting the Q values, while modeling the temporal aspect would compensate for the loss obtained due to spatial degradation. Furthermore, the online TSM algorithm \\cite{lin2019tsm} cached the features in memory and then replaced it with those in the next time step. Our approach was to directly roll the features across time steps. \n\nFinally, the authors of the TSM paper found that the highest accuracy for the online model was obtained by shifting 1\/8th of channels for each layer of the neural network. However, while testing our algorithm, we found that the best results were obtained when we shifted around 1\/5 to 1\/3 of our channels.   \n\nA schematic of our algorithm has been given in Figure~\\ref{fig:env} and a PyTorch based pseudocode for our algorithm has been presented here -\n\\begin{algorithm}\n\\caption{TSRL}\\label{alg:cap}\n\n\\begin{minipage}{\\linewidth}\n\n\\begin{lstlisting}\nFor each step t do\n    For each convolution step do\n        x = self.relu1(self.conv1(x))\n        n,c,h,w = x.shape\n        x = x.reshape(n\/\/T, T, c, h, w)\n        copy = torch.clone(x)\n        x[:,:, :c\/\/8, :, :] = torch.roll(x[:,:, :c\/\/8, :, :],\n        shifts = 1, dims = 1)\n        x[:,0, :c\/\/8, :, :] = copy[:,0, :c\/\/8, :, :]\n        z_t = FullyConnected(x)\n    End For\nEnd For\n\\end{lstlisting}\n\n\\end{minipage}\n\\end{algorithm}\n\n\\section{Experiments}\nWe tested our algorithm using OpenAI Gym Atari environments with visual images as input. An open-sourced implementation of DDQN (https:\/\/github.com\/higgsfield\/RL-Adventure) combined with PER was used. The images were converted to grayscale to speed up the learning process. To gauge the sample efficiency of TSRL we compared it with a generic DDQN-PER getting stacked images as input. Also, we used our own implementation of the algorithm developed by \\cite{shang2021reinforcement} and referred to it as Flare, in order to compare against state of the art. The number of stacked images was kept equal to the timesteps considered by TSRL both for DDQN-PER and Flare. Also, all algorithms were run for 1.4M time steps using 5 different trials. The performance of the algorithm was gauged by averaging the trials and then summing over all rewards obtained; \\cite{brockman2016openai, bellemare2013arcade, mott1996stella}.\n\\begin{figure}[!htp]\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=.8\\linewidth]{Freeway.png}\n  \\caption{Freeway Atari Environment}\n  \\label{fig:sfig1}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=.8\\linewidth]{asterix.png}\n  \\caption{Asterix Atari Environment}\n  \\label{fig:sfig2}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=.8\\linewidth]{riverraid.png}\n  \\caption{River Raid Atari Environment}\n  \\label{fig:sfig3}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=.8\\linewidth]{pong.png}\n  \\caption{Pong Atari Environment}\n  \\label{fig:sfig4}\n\\end{subfigure}\n\\caption{OpenAI Gym environments used for training}\n\\label{fig:fig}\n\\end{figure}\n\n\\subsection{Results}\nTable \\ref{table:1} shows the sum of average rewards obtained across the five runs for each environment. The shift parameter, $s$ column, shows the ratio of channels that were shifted. For instance, if $s = 3$, then the first 1\/3\\textsuperscript{rd} channels would be shifted across the temporal dimension for every layer of the CNN.\n\nFigure~\\ref{fig:plots} shows the reward obtained per episode. In some cases, an algorithm may have large step sizes relatively early. This would lead to a lower number of episodes and vice versa. \n\nTSRL outperforms both DDQN-PER and Flare in all environments except Asterix, wherein it only defeats the DDQN-PER.\n\n\n\\begin{figure}[!htp]\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=.8\\linewidth]{freeway_result.png}\n\n  \\label{fig:sfig1}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=.8\\linewidth]{asterix_result.png}\n\n  \\label{fig:sfig2}\n\\end{subfigure}\n\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=.8\\linewidth]{riverraid_result.png}\n\n  \\label{fig:sfig3}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=.8\\linewidth]{pong_result.png}\n\n  \\label{fig:sfig4}\n\\end{subfigure}\n\\caption{Plots of episode vs reward for different Atari environments}\n\\label{fig:plots}\n\\end{figure}\n\n\\begin{table}[h!]\n\n\n\\centering\n\\caption{Sum of average rewards obtained.}\n\\begin{tabular}{||c c c c c||} \n\n \\hline\n Environment & Shift & TSRL & DDQN-PER & FLARE\\\\ [0.5ex] \n \n \\hline\\hline\n Freeway &3 & \\textbf{18291.5} & 17807.6 & 14686.19\\\\ [1ex] \n\\hline\n Asterix &5 & 22854.25 & 20702.0 & \\textbf{33496.93}\\\\ [1ex] \n\\hline\n Riverraid &5 & \\textbf{41850.3} & 34849.2 & 34966.0\\\\ [1ex] \n\\hline\nPong &5 & \\textbf{7892.17} & 7221.80 & -36528.20\\\\ [1ex] \n\\hline\n\\end{tabular}\n\\end{table}\n\\label{table:1}\n\n\\subsection{Discussion}\nA major difference between our algorithm and other RL algorithms taking temporal aspects into account is that we provide multi-level temporal fusion. Most RL algorithms implement early fusion \\cite{mnih2015human} and the recent ones \\cite{amiranashvili2018motion, shang2021reinforcement} have experimented with late fusion. However, our approach enables RL to have temporal fusion across all levels. This type of fusion was found to significantly help difficult temporal modeling problems \\cite{lin2019tsm}. \n\nIt is interesting to note that instead of a single shift hyperparameter being optimal for all tasks, it varies across environments. We hypothesize that this is caused due to the trade-off between spatial and temporal learning. Some environments might not require a higher number of feature maps and therefore could work with a lower shift hyperparameter. This would permit a larger number of channels to be moved, leading to improved temporal learning. However, this might not be the case in complicated environments, and such situations might require the shift hyperparameter to be higher.\n\nFinally, we see that TSRL is able to beat the baseline and SOTA for almost all the environments.  \\footnote{We used our own implementation of the Flare algorithm.} Since Flare concatenates latent flow with features, we feel that this increases the number of parameters and, therefore, the relative training time compared to TSRL. Furthermore, the latent flow is obtained by subtracting the current frame from the immediately preceding frame while ignoring the frames before that. This might not provide much information in situations when the difference between immediate frames is minute. This problem is mitigated by the multi-level fusion abilities of our algorithm.\n\\section{Conclusions}\nWe present a facile shifting technique for learning temporal features in DRL problems without the requirement of additional parameters. After testing our algorithm on OpenAI Atari environments, we find that our algorithm outperforms the commonly used frame-stacking heuristic.\n\nA major drawback of our algorithm is the requirement to find a suitable shift hyperparameter. Future work could include either learning the optimal value of this hyperparameter online or changing how the shift is performed (such as residual shift \\cite{lin2019tsm}) so that the spatial features aren't disturbed.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nA permutation $\\pi=\\pi_1\\pi_2\\cdots \\pi_n$ is called {\\em up-down} if $\\pi_1<\\pi_2>\\pi_3<\\pi_4>\\pi_5<\\cdots$.  A permutation $\\pi=\\pi_1\\pi_2\\cdots \\pi_n$ is called {\\em down-up} if $\\pi_1>\\pi_2<\\pi_3>\\pi_4<\\pi_5>\\cdots$. A famous result of Andr\\'{e} is saying that if $E_n$ is the number of up-down (equivalently, down-up) permutations of $1,2,\\ldots,n$, then $$\\sum_{n\\geq 0}E_n\\frac{x^n}{n!}=\\sec x+\\tan x.$$ Some aspects of up-down and down-up permutations, also called {\\em reverse alternating} and {\\em alternating}, respectively, are surveyed in~\\cite{Stanley2010survey}. Slightly abusing these definitions, we refer to {\\em alternating permutations} as the union of up-down and down-up permutations. This union is known as the set of {\\em zigzag permutations}.\n\nIn this paper, we extend the study of alternating permutations to that of {\\em alternating words}. These words, also called {\\em zigzag words}, are the union of up-down and down-up words, which are defined in a similar way to the definition of up-down and down-up permutations, respectively. For example, $1214$, $2413$, $2424$ and $3434$ are examples of up-down words of length 4 over the alphabet $\\{1,2,3,4\\}$.\n\nSection~\\ref{en-alt-words} is dedicated to the enumeration of up-down words, which is equivalent to enumerating down-up words by applying the operation of {\\em complement}. For a word $w=w_1w_2\\cdots w_n$ over the alphabet $\\{1,2,\\ldots,k\\}$ its complement $w^c$ is the word $c_1c_2\\cdots c_n$, where for each $i=1,2,\\ldots,n$, $c_i=k+1-w_i$. For example, the complement of the word $24265$ over the alphabet $\\{1,2,\\ldots,6\\}$ is $53512$. Our enumeration in Section~\\ref{en-alt-words} is done by linking bijectively up-down words to order ideals of certain posets and using known results.\n\nA ({\\em permutation}) {\\em pattern} is a permutation $\\tau=\\tau_1\\tau_2\\cdots\\tau_k$. We say that a permutation $\\pi=\\pi_1\\pi_2\\cdots\\pi_n$ {\\em contains an occurrence} of $\\tau$ if there are $1\\leq i_1< i_2<\\cdots< i_k\\leq n$ such that $\\pi_{i_1}\\pi_{i_2}\\cdots \\pi_{i_k}$ is order-isomorphic to $\\tau$. If $\\pi$ does not contain an occurrence of $\\tau$, we say that $\\pi$ {\\em avoids}~$\\tau$. For example, the permutation 315267 contains several occurrences of the pattern 123, for example, the subsequences 356 and 157, while this permutation avoids the pattern 321. Occurrences of a pattern in words are defined similarly as subsequences order-isomorphic to a given word called pattern (the only difference with permutation patterns is that word patterns can contain repetitive letters, which is not in the scope of this paper).\n\nA comprehensive introduction to the theory of patterns in permutations and words can be found in~\\cite{Kitaev2011Patterns}. In particular, Section 6.1.8 in~\\cite{Kitaev2011Patterns} discusses known results on pattern-avoiding alternating permutations, and Section 7.1.6 discusses results on permutations avoiding patterns in a more general sense.\n\nIn this paper we initiate the study of pattern-avoiding alternating words. In Section~\\ref{123-up-down-sec} we enumerate up-down words over $k$-letter alphabet avoiding the pattern 123. In particular, we show that in the case of even length, the answer is given by the {\\em Narayana numbers} counting, for example, {\\em Dyck paths} with a specified number of peaks (see Theorem~\\ref{main-thm}). Interestingly, the number of 132-avoiding words over $k$-letter alphabet of even length is also given by the Narayana numbers, which we establish bijectively in Section~\\ref{bijection-sec}. In Section~\\ref{sec-132-av} we provide a (non-closed form) formula for the number of 132-avoiding words over $k$-letter alphabet of odd length. In Section~\\ref{312-up-down-sec} we show that the enumeration of 312-avoiding up-down words is equivalent to that of 123-avoiding up-down words. Further, a classification of all cases of avoiding a length 3 permutation pattern on up-down words is discussed in Section~\\ref{all-cases}. Finally, some concluding remarks are given in Section~\\ref{last-sec}.\n\nIn what follows, $[k]=\\{1,2,\\ldots,k\\}$.\n\n\\section{Enumeration of up-down words}\\label{en-alt-words}\n\nIn this section, we consider the enumeration of up-down words.\nWe shall show that this problem is the same as that of  enumerating order ideals of a certain poset.\nSince up-down words are in one-to-one-correspondence with down-up words by using the complement operation, we consider only down-up words throughout this section.\n\nTable~\\ref{tab1} provides the number $N_{k,\\ell}$ of down-up words of length $\\ell$ over the alphabet $[k]$ for small values of $k$ and $\\ell$ indicating connections to the {\\em Online Encyclopedia of Integer Sequences} ({\\em OEIS})~\\cite{SloaneLine}.\n\n\\begin{table}[!htb]\n\\small\n\\begin{center}\n\\begin{tabular}{|c|lllllllllll|l|}\n\\hline\n\\diagbox{$k$}{$\\ell$}& 0 & 1 & 2 & 3 & 4 &5 &6 & 7 &8&9 &10& OEIS\\\\\n\\hline\n2&  1& 2&1& 1&1& 1&1& 1&1& 1&1& trivial \\\\\n\\hline\n3&  1& 3& 3& 5& 8& 13& 21& 34& 55& 89& 144 & A000045\\\\\n\\hline\n4&  1&  4& 6& 14& 31& 70& 157& 353& 793& 1782& 4004 & A006356\\\\\n\\hline\n5&  1&  5& 10& 30& 85& 246& 707& 2037& 5864& 16886& 48620& A006357\\\\\n\\hline\n6&  1&  6& 15& 55& 190& 671& 2353& 8272& 29056 &  102091 & 358671 & A006358\\\\\n\\hline\n7&  1&  7& 21& 91& 371& 1547& 6405& 26585& 110254 & 457379 & 1897214 & A006359\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The number $N_{k,\\ell}$ of down-up words on $[k]$ of length $\\ell$ for small values of $k$ and~$\\ell$.}\\label{tab1}\n\\end{table}\n\nWe assume the reader is familiar with the notion of a partially ordered set (poset) and some basic properties of posets; e.g. see \\cite{Stanley1997Enumerative}. A partially ordered set $P$ is a set  together with a binary relation denoted by $\\leq_P$ that satisfies the properties of reflexivity, antisymmetry and transitivity. An order ideal of $P$ is a subset $I$ of $P$ such that if $x\\in I$ and $y\\leq x$ then $y\\in I$. We denote $J(P)$ the set  of all order ideals of $P$.\n\nLet $\\mathbf{n}$ be the poset on $[n]$ with its usual order ($\\mathbf{n}$ is a linearly ordered set).\nThe {\\em $m$-element zigzag poset}, denoted $Z_m$, is shown schematically in Figure \\ref{Zm}. Note that the order $<_{Z_m}$ in $Z_m$ is $1<2>3<4>5<\\cdots.$ The definition of the order $\\leq_{Z_m}$ is self-explanatory.\n\n\\begin{figure}[!htb]\n\\small\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[scale=1.4]\n\t\\draw (3,0)--(3.5,1)--(4,0)--(4.5,1)--(5,0);\n\t\\fill(3,0) circle(0.04cm);    \\node [below] at (3,0) {$1$};\n\t\\fill(3.5,1) circle(0.04cm); \\node [above] at (3.5,1) {$2$};\n\t\\fill(4,0) circle(0.04cm);    \\node [below] at (4,0) {$3$};\n\t\\fill(4.5,1) circle(0.04cm); \\node [above] at (4.5,1) {$4$};\n\t\\fill(5,0) circle(0.04cm);    \\node [below] at (5,0) {$5$};\n\t\n\t\\fill(5.2,0.5) circle(0.02cm); \\fill(5.5,0.5) circle(0.02cm); \\fill(5.8,0.5) circle(0.02cm);\n\n\t\\draw (6,0)--(6.5,1);\n\t\\fill(6,0) circle(0.04cm);    \\node [below]  at (6,0){$m-1$};\n\t\\fill(6.5,1) circle(0.04cm); \\node [above]  at (6.5,1){$m$};\n\t\n\t\\coordinate[label=below:$m$ even] (1) at (5,-0.5);\n        \\end{tikzpicture}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[scale=1.4]\n\t\\draw (3,0)--(3.5,1)--(4,0)--(4.5,1)--(5,0);\n\n\t\\fill(3,0) circle(0.04cm);    \\node [below] at (3,0) {$1$};\n\t\\fill(3.5,1) circle(0.04cm); \\node [above] at (3.5,1) {$2$};\n\t\\fill(4,0) circle(0.04cm);    \\node [below] at (4,0) {$3$};\n\t\\fill(4.5,1) circle(0.04cm); \\node [above] at (4.5,1) {$4$};\n\t\\fill(5,0) circle(0.04cm);    \\node [below] at (5,0) {$5$};\n\t\n\t\\fill(5.2,0.5) circle(0.02cm); \\fill(5.5,0.5) circle(0.02cm); \\fill(5.8,0.5) circle(0.02cm);\n\n\t\\draw (6,1)--(6.5,0);\n\t\\fill(6,1) circle(0.04cm);    \\node [above]  at (6,1){\\small $m-1$};\n\t\\fill(6.5,0) circle(0.04cm); \\node [below]  at (6.5,0){\\small $m$};\n\t\\coordinate[label=below:$m$ odd] (1) at (5,-0.5);\n\t\\end{tikzpicture}\n    \\end{minipage}\n  \\caption{The zigzag poset $Z_{m}$.}\n  \\label{Zm}\n\\end{figure}\n\nThe poset $Z_m \\times \\mathbf{n}$ is as shown in Figure~\\ref{Zmn}. Elements of $Z_m \\times \\mathbf{n}$ are pairs $(i,j)$, where $i \\in Z_m$ and $j\\in [n]$, and the order is defined as follows:\n$$(i,j)\\leq (k,\\ell) \\mbox{ if and only if } i\\leq_{Z_{m}}k\\mbox{ and } j\\leq \\ell.$$\n\n\n\\begin{figure}[!htb]\n\\tiny\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[scale=1.5]\n      \\draw (3,1)--(3.5,2)--(4,1)--(4.5,2)--(5,1);\n      \\draw (3,-1)--(3.5,0)--(4,-1)--(4.5,0)--(5,-1);\n      \\draw (3,-2)--(3.5,-1)--(4,-2)--(4.5,-1)--(5,-2);\n      \\draw (3,1)--(3,0.7); \\draw  (3,-0.3)--(3,-1)--(3,-2);\n      \\draw (3.5,2)--(3.5,1.3); \\draw  (3.5,0.3)--(3.5,0)--(3.5,-1);\n      \\draw (4,1)--(4,0.7); \\draw  (4,-0.3)--(4,-1)--(4,-2);\n      \\draw (4.5,2)--(4.5,1.3); \\draw  (4.5,0.3)--(4.5,0)--(4.5,-1);\n      \\draw (5,1)--(5,0.7); \\draw  (5,-0.3)--(5,-1)--(5,-2);\n      \\fill(3,1) circle(0.04cm); \\node [right] at (3,1){  $(1,n)$};\n      \\fill(3.5,2) circle(0.04cm); \\node [right] at  (3.5,2){  $(2,n)$};\n      \\fill(4,1) circle(0.04cm);  \\node [right] at (4,1){  $(3,n)$};\n      \\fill(4.5,2) circle(0.04cm);   \\node [right] at (4.5,2){  $(4,n)$};\n      \\fill(5,1) circle(0.04cm);  \\node [right] at (5,1) {  $(5,n)$};\n      \\fill(3,-1) circle(0.04cm); \\node [right] at (3,-1) {  $(1,2)$};\n      \\fill(3.5,0) circle(0.04cm); \\node [right] at (3.5,0) {  $(2,2)$};\n      \\fill(4,-1) circle(0.04cm); \\node [right] at (4,-1) {  $(3,2)$};\n      \\fill(4.5,0) circle(0.04cm);  \\node [right] at (4.5,0) {  $(4,2)$};\n      \\fill(5,-1) circle(0.04cm); \\node [right] at (5,-1) {  $(5,2)$};\n      \\fill(3,-2) circle(0.04cm);  \\node [right] at (3,-2) {  $(1,1)$};\n      \\fill(3.5,-1) circle(0.04cm); \\node [right] at (3.5,-1) {  $(2,1)$};\n      \\fill(4,-2) circle(0.04cm); \\node [right] at (4,-2) {  $(3,1)$};\n      \\fill(4.5,-1) circle(0.04cm);\\node [right] at (4.5,-1) {  $(4,1)$};\n      \\fill(5,-2) circle(0.04cm); \\node [right] at(5,-2) {  $(5,1)$};\n\n      \\fill(3.7,0.5) circle(0.02cm); \\fill(4,0.5) circle(0.02cm); \\fill(4.3,0.5) circle(0.02cm);\n      \\fill(5.2,1.5) circle(0.02cm); \\fill(5.5,1.5) circle(0.02cm); \\fill(5.8,1.5) circle(0.02cm);\n      \\fill(5.2,0.5) circle(0.02cm); \\fill(5.5,0.5) circle(0.02cm); \\fill(5.8,0.5) circle(0.02cm);\n      \\fill(5.2,-0.5) circle(0.02cm); \\fill(5.5,-0.5) circle(0.02cm); \\fill(5.8,-0.5) circle(0.02cm);\n      \\fill(5.2,-1.5) circle(0.02cm); \\fill(5.5,-1.5) circle(0.02cm); \\fill(5.8,-1.5) circle(0.02cm);\n\n      \\draw (6,1)--(6.5,2);\\draw (6,-1)--(6.5,0);\\draw (6,-2)--(6.5,-1);\n      \\draw (6,1)--(6,0.7); \\draw  (6,-0.3)--(6,-1)--(6,-2);\n      \\draw (6.5,2)--(6.5,1.3); \\draw  (6.5,0.3)--(6.5,0)--(6.5,-1);\n      \\fill(6,1) circle(0.04cm);  \\node [right] at (6,1) {  $(m-1,n)$};\n      \\fill(6.5,2) circle(0.04cm); \\node [right] at (6.5,2) {   $(m,n)$};\n      \\fill(6,-1) circle(0.04cm); \n      \\fill(6.5,0) circle(0.04cm); \\node [right] at (6.5,0) {   $(m,2)$};\n      \\fill(6,-2) circle(0.04cm); \\node [right] at (6,-2) {   $(m-1,1)$};\n      \\fill(6.5,-1) circle(0.04cm); \\node [right] at (6.5,-1) {   $(m,1)$};\n    \\coordinate[label=below:\\small $m$ even] (1) at (5,-2.5);\n        \\end{tikzpicture}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[scale=1.5]\n     \\draw (3,1)--(3.5,2)--(4,1)--(4.5,2)--(5,1);\n      \\draw (3,-1)--(3.5,0)--(4,-1)--(4.5,0)--(5,-1);\n      \\draw (3,-2)--(3.5,-1)--(4,-2)--(4.5,-1)--(5,-2);\n      \\draw (3,1)--(3,0.7); \\draw  (3,-0.3)--(3,-1)--(3,-2);\n      \\draw (3.5,2)--(3.5,1.3); \\draw  (3.5,0.3)--(3.5,0)--(3.5,-1);\n      \\draw (4,1)--(4,0.7); \\draw  (4,-0.3)--(4,-1)--(4,-2);\n      \\draw (4.5,2)--(4.5,1.3); \\draw  (4.5,0.3)--(4.5,0)--(4.5,-1);\n      \\draw (5,1)--(5,0.7); \\draw  (5,-0.3)--(5,-1)--(5,-2);\n      \\fill(3,1) circle(0.04cm); \\node [right] at (3,1){  $(1,n)$};\n      \\fill(3.5,2) circle(0.04cm); \\node [right] at  (3.5,2){  $(2,n)$};\n      \\fill(4,1) circle(0.04cm);  \\node [right] at (4,1){  $(3,n)$};\n      \\fill(4.5,2) circle(0.04cm);   \\node [right] at (4.5,2){  $(4,n)$};\n      \\fill(5,1) circle(0.04cm);  \\node [right] at (5,1) {  $(5,n)$};\n      \\fill(3,-1) circle(0.04cm); \\node [right] at (3,-1) {  $(1,2)$};\n      \\fill(3.5,0) circle(0.04cm); \\node [right] at (3.5,0) {  $(2,2)$};\n      \\fill(4,-1) circle(0.04cm); \\node [right] at (4,-1) {  $(3,2)$};\n      \\fill(4.5,0) circle(0.04cm);  \\node [right] at (4.5,0) {  $(4,2)$};\n      \\fill(5,-1) circle(0.04cm); \\node [right] at (5,-1) {  $(5,2)$};\n      \\fill(3,-2) circle(0.04cm);  \\node [right] at (3,-2) {  $(1,1)$};\n      \\fill(3.5,-1) circle(0.04cm); \\node [right] at (3.5,-1) {  $(2,1)$};\n      \\fill(4,-2) circle(0.04cm); \\node [right] at (4,-2) {  $(3,1)$};\n      \\fill(4.5,-1) circle(0.04cm);\\node [right] at (4.5,-1) {  $(4,1)$};\n      \\fill(5,-2) circle(0.04cm); \\node [right] at(5,-2) {  $(5,1)$};\n\n      \\fill(3.7,0.5) circle(0.02cm); \\fill(4,0.5) circle(0.02cm); \\fill(4.3,0.5) circle(0.02cm);\n      \\fill(5.2,1.5) circle(0.02cm); \\fill(5.5,1.5) circle(0.02cm); \\fill(5.8,1.5) circle(0.02cm);\n      \\fill(5.2,0.5) circle(0.02cm); \\fill(5.5,0.5) circle(0.02cm); \\fill(5.8,0.5) circle(0.02cm);\n      \\fill(5.2,-0.5) circle(0.02cm); \\fill(5.5,-0.5) circle(0.02cm); \\fill(5.8,-0.5) circle(0.02cm);\n      \\fill(5.2,-1.5) circle(0.02cm); \\fill(5.5,-1.5) circle(0.02cm); \\fill(5.8,-1.5) circle(0.02cm);\n\n      \\draw (6,2)--(6.5,1);\\draw (6,0)--(6.5,-1);\\draw (6,-1)--(6.5,-2);\n      \\draw (6,2)--(6,1.3); \\draw  (6,0.3)--(6,0)--(6,-1);\n      \\draw (6.5,1)--(6.5,0.7); \\draw  (6.5,-0.3)--(6.5,-1)--(6.5,-2);\n      \\fill(6,2) circle(0.04cm);  \\node [right] at (6,2) {  $(m-1,n)$};\n      \\fill(6.5,1) circle(0.04cm); \\node [right] at (6.5,1) {   $(m,n)$};\n      \\fill(6,0) circle(0.04cm);  \\node [right] at (6,0) {   $(m-1,2)$};\n      \\fill(6.5,-1) circle(0.04cm); \\node [right] at (6.5,-1) {   $(m,2)$};\n      \\fill(6,-1) circle(0.04cm);\n      \\fill(6.5,-2) circle(0.04cm); \\node [right] at (6.5,-2) {   $(m,1)$};\n    \\coordinate[label=below:\\small $m$ odd] (1) at (5,-2.5);\n        \\end{tikzpicture}\n    \\end{minipage}\n  \\caption{The poset $Z_{m} \\times \\mathbf{n}$.}\n  \\label{Zmn}\n\\end{figure}\n\nIt is known that, for $m\\ge 2$, the size of  $J(Z_m)$ equals to the Fibonacci number $F_{m+2}$, which is defined recursively as $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ for  any $n\\geq 2$; see Stanley \\cite[Ch. 3 Ex. 23.a]{Stanley1997Enumerative}.\nThe enumeration of $J(Z_m \\times \\mathbf{n})$ was studied by Berman and K\\\"ohler \\cite{Berman1976Cardinalities}.\nThe following theorem reveals their connection with the enumeration of alternating words. We shall give two proofs of it here,  a bijective proof and an enumerative  proof.\n\n\\begin{thm}\\label{enum-down-up-words}\n For any $k \\ge 2$ and $\\ell \\ge 2$, the number $N_{k,\\ell}$ of down-up words over $[k]$ of length $\\ell$ is equal to  the number of order ideals of  $Z_\\ell \\times (\\mathbf{k-2})$.\n\\end{thm}\n\n\\noindent \\textbf{Bijective Proof.}\nLet $\\mathcal{W}_{k,\\ell}$ denote the set of down-up words over $[k]$ of length $\\ell$. We shall build a bijection between $\\mathcal{W}_{k,\\ell}$ and ${J}(Z_\\ell \\times (\\mathbf{k-2}))$.\n\n We first define a map $\\Phi:\\mathcal{W}_{k,\\ell}\\rightarrow {J}(Z_\\ell \\times (\\mathbf{k-2}))$.  Given a down-up word $w=w_1 w_2 \\cdots w_{\\ell}$, we define the word $\\alpha=\\alpha_1 \\alpha_2 \\cdots \\alpha_{\\ell}$ as follows:\n $$\\alpha_i=\\left\\{\n\\begin{array}{ll}\nw_i-2,& \\mbox{ if } i \\mbox{ is odd},\\\\[3mm]\nw_i-1,& \\mbox{ if } i \\mbox{ is even},\n\\end{array}\\right.\n$$\nwhere $1\\leq i\\leq \\ell$.\nThen let $$\\Phi(w)=\\{(i,\\beta_j)  : 1\\leq i\\leq \\ell , \\ 1 \\le \\beta_j \\le \\alpha_i\\}.$$\nFor example, let $k=4$ and $\\ell=7$, and consider the word $w=3241423$. Then,  $\\alpha=1120211$ and thus $\\Phi(w)=\\{(1,1),(2,1),(3,1),(3,2),(5,1),(5,2),(6,1),\n(7,1)\\}$, which is an order ideal of $Z_7 \\times \\mathbf{2}$.\n\nWe need to show that this map is well defined.\nIt suffices to prove that $\\Phi(w)$ is an order ideal of $Z_\\ell \\times (\\mathbf{k-2})$, that is to say that,\nif $(i',j')\\leq (i,j)$ and $(i,j)\\in \\Phi(w)$ then $(i',j')\\in \\Phi(w)$.\nFrom the definition of the order of $Z_\\ell \\times (\\mathbf{k-2})$, we have that $i'\\le_{Z_{\\ell}} i$ and $j'\\le j$.\nNow, we divide the situation into two cases: $i'=i$ and $i'<_{Z_\\ell}i$.\nFor the case $i'=i$, the argument is obviously true from the construction of $\\Phi(w)$. We just need to consider the case $i'<_{Z_\\ell}i$.\nAt this time, $i$ must be even,  and $i'$ can only be $i-1$ or $i+1$.\nSince $(i,j)\\in \\Phi(w)$, we have that $\\alpha_i\\geq j$ and thus $w_i\\geq j+1$.\nFrom the fact that $w$ is a down-up word, it follows that  $w_{i'}> w_i$. Hence, $w_{i'} \\ge j+2$ and thus  $\\alpha_{i'}\\geq j$.\nFrom the construction of $\\Phi(w)$, we obtain that  $(i',j')\\in \\Phi(w)$ for all $j'\\leq j$, as desired.\n\n\n Next, we define a map $\\Psi: {J}(Z_\\ell \\times (\\mathbf{k-2})) \\rightarrow \\mathcal{W}_{k,\\ell}$.\n Given an order ideal $I$ of $Z_\\ell \\times (\\mathbf{k-2})$,\n we define a word  $\\gamma=\\gamma_1 \\gamma_2 \\cdots \\gamma_{\\ell}$ as follows. For each $1\\leq i\\leq\\ell$, if there exists at least one $j$ such that $(i,j)\\in I$, then let $\\gamma_i$ be the maximum $j$. Otherwise, we let $\\gamma_i$=0.\n The corresponding word $\\Psi(I)$ is defined as $(2+\\gamma_1)(1+\\gamma_2)(2+\\gamma_3)(1+\\gamma_4)\\cdots$.\n  For exmaple, if $I=\\{(1,1),(2,1),(3,1),(3,2),(5,1),(5,2),(6,1),\n(7,1)\\}$,\n then $\\gamma=1120211$ and thus $w=3241423$.\n\n It is easy to see that, for any even integer $i$, we have $\\gamma_i\\leq \\gamma_{i+1}$ and $\\gamma_i\\leq \\gamma_{i-1}$, since $I$ is an order ideal. From the construction of $\\Psi(I)$, we see that\n it is a down-up word.\n\nFinally, it is not difficult to  prove that $\\Psi \\circ \\Phi = id$ and $\\Phi \\circ \\Psi = id$. Hence $\\Phi$ is a bijection. This completes our bijective proof.\\\\\n\n\\noindent{\\bf Enumerative  Proof.}\nWe first prove that the numbers in question satisfy the following recurrence relation, for $k\\ge 3$ and $\\ell\\ge 2$,\n\\begin{align}\\label{eq-N-rec}\n N_{k,\\ell} = N_{k-1,\\ell} + \\sum_{i=0}^{\\lfloor \\frac{\\ell-1}{2}\\rfloor} N_{k-1,2i} N_{k,\\ell-2i-1} - \\delta_{\\ell \\text{ is even}} N_{k-1,\\ell-2},\n\\end{align}\nwith the initial conditions\n$N_{k, 0}=1$, $N_{k, 1}=k$ for $k\\ge 2$, and $N_{2,\\ell}=1$ for $\\ell \\ge 2$. To this end, we note that any down-up word $w$  over $[k]$ of length $\\ell$ belongs to one of the following two cases.\\\\\n\n\\noindent\n{\\bf Case 1:} $w$ does not contain the letter $k$. Then the number we count is that of down-up words  over the alphabet $[k-1]$ of length $\\ell$, which is $N_{k-1,\\ell}$. This corresponds to the first term on the righthand side of \\eqref{eq-N-rec}.\\\\\n\n\\noindent\n{\\bf Case 2:} $w$ is of the form $w_1kw_2$, where $w_1$ is a down-up word of even length over $[k-1]$, and $w_2$ is an up-down word over $[k]$. Note that the number of up-down words equals to  that of down-up words, as mentioned above. This corresponds to the second term on the right hand side of \\eqref{eq-N-rec}.\nThe only exception occurs when the subword after the leftmost letter $k$ is of length one. It can be any letter in $[k-1]$, but $N_{k, 1}=k$. So, an additional term occurs, which fixes this.  In these cases, $\\ell-2i-1$ equals 1, which means that $\\ell$ is even. This completes the proof of \\eqref{eq-N-rec}.\n\nNow, let us denote the number of order ideals of  $Z_\\ell \\times \\mathbf{k}$ by $M_{k,\\ell}$.\nWe note that\nBerman and K\\\"ohler \\cite[Example 2.3]\n{Berman1976Cardinalities} studied a similar recurrence for $M_{k,\\ell}$, which is,\nfor $k\\ge 1$ and $\\ell\\ge 1$,\n\\begin{align*}\n M_{k,\\ell} = M_{k-1,\\ell} + \\sum_{i=0}^{\\lfloor \\frac{\\ell-1}{2}\\rfloor} M_{k-1,2i} M_{k,\\ell-2i-1},\n\\end{align*}\nwith the initial conditions\n$M_{k, 0}=1$  for $k\\ge0$ and $M_{0,\\ell}=1$ for $\\ell\\ge1$.\n\nOwing to their akin recurrence relations, we made a minor change to the number $N_{k,\\ell}$ to complete the proof.\nWe let $\\widetilde{N}_{k,\\ell}$ be $N_{k,\\ell}$ except $\\widetilde{N}_{k,1}=k-1$.  One can easily check that,\nfor $k\\ge 3$ and $\\ell\\ge 1$,\n\\begin{align*}\n \\widetilde{N}_{k,\\ell} = \\widetilde{N}_{k-1,\\ell} + \\sum_{i=0}^{\\lfloor \\frac{\\ell-1}{2}\\rfloor} \\widetilde{N}_{k-1,2i} \\widetilde{N}_{k,\\ell-2i-1} ,\n\\end{align*}\nwith the initial conditions\n$\\widetilde{N}_{k, 0}=1$ for $k\\ge2$ and $\\widetilde{N}_{2,\\ell}=1$ for $\\ell\\ge1$.\nIt follows immediately that, for $k\\ge2$ and $\\ell\\ge0$, $$\\widetilde{N}_{k, \\ell} = M_{k-2,\\ell},$$ since they have the same initial conditions and recurrence relations.\nTogether with the fact $N_{k, \\ell} =\\widetilde{N}_{k, \\ell}$ except $\\ell=1$, we obtain that\n$$N_{k, \\ell}=M_{k-2,\\ell}$$\nfor $k\\ge2$ and $\\ell\\ge2$. This completes our enumerative proof.\n\\qed\n\nAs an immediate corollary of Theorem~\\ref{enum-down-up-words}, we have the following statement. \n\n\\begin{thm} For $k\\geq 3$ and $\\ell\\geq 2$, the numbers $N_{k,\\ell}$ of down-up (equivalently, up-down) words of length $\\ell$ over $[k]$ satisfy (\\ref{eq-N-rec}) with the initial conditions $N_{k,0}=1$, $N_{k,1}=k$ for $k\\geq 2$, and $N_{2,\\ell}=1$ for $\\ell\\geq 2$. \\end{thm} \n\nNote that the Fibonacci numbers have the following recurrence relations \\cite[pp. 5--6]{Vorobiev2002Fibonacci}:\n$$F_{2n}=\\sum_{i=0}^{n-1}F_{2i+1}, \\quad F_{2n+1}=1+\\sum_{i=1}^{n}F_{2i}.$$\nUsing \\eqref{eq-N-rec} and the fact that $N_{2,\\ell}=1$ for $\\ell\\geq 2$, one can prove the following statement.\n\n\\begin{thm} For $\\ell\\geq 2$, $N_{3,\\ell}=F_{\\ell+2}$, the $(\\ell+2)$th Fibonacci number. \\end{thm}\n\n\\section{Enumeration of 123-avoiding up-down words}\\label{123-up-down-sec}\n\nIn this section, we consider the enumeration of 123-avoiding up-down words. Denote $A_{k,\\ell}$ the number of 123-avoiding up-down words  of length $\\ell$ over the alphabet $[k]$, and $A_{k, \\ell}^j$ the number of those words counted by $A_{k,\\ell}$ that end with $j$.\n\n\\subsection{Explicit enumeration}\n\nIt is easy to see that\n\\begin{align}\n A_{k,2i}=\\sum_{j=2}^{k} A_{k,2i}^j.\n\\end{align}\n\nNext, we deal with the enumeration of $A^j_{k,2i}$. In what follows, for a word $w$, we have $\\{w\\}^*=\\{\\epsilon, w,ww,www,\\ldots\\}$, where $\\epsilon$ is the empty word, and $\\{w\\}^+=\\{w,ww,www,\\ldots\\}$.\n\n\\begin{lem}\nFor $k\\ge 3$ and $2\\le j\\le k$, the numbers  $A_{k,2i}^{j}$ satisfy the following recurrence relation,\n\\begin{align}\\label{rec}\nA_{k,2i}^{j}\n=  \\sum_{i'=1}^{i} \\left( A_{k-1, 2i'}^{j-1} - A_{k-1, 2i'}^{j}   + A_{k,2i'}^{j+1} \\right),\n\\end{align}\nwith the boundary condition $A_{k,2i}^{k}= \\binom{i+k-2}{i}$.\nFurthermore, an explicit formula for $A^j_{k,2i}$ is\n\\begin{align}\\label{A^j}\nA^j_{k,2i}=\\frac{j-1}{k-1} \\binom{i+k-2}{i}  \\binom{i+k-j-1}{i-1}.\n\\end{align}\n\\end{lem}\n\n\n\\begin{proof}\nWe first check  the boundary condition.\n When  $j=k$, the words must be of the form $$\\{(k-1)k\\}^{*}\\{(k-2)k\\}^{*}\\ldots\\{2k\\}^{*}\\{1k\\}^{*}.$$\n The structure is dictated by the presence of the rightmost $k$; violating the structure, we will be forced to have an occurrence of the pattern $123$.\n Therefore, $A_{k,2i}^{k}=\\binom{i+k-2}{i}$, where we applied the well known formula for the number of solutions of the equation $x_1+\\cdots +x_{k-1}=i$ with $x_i\\geq 0$ for $1\\leq i\\leq k-1$.\n\nNow we proceed to deduce the recurrence relation \\eqref{rec}.\nAll the legal words of length $2i$ ending with $j$ can be divided into the following cases according to the occurrence of  the letter $1$:\n\n\\noindent\n{\\bf Case 1:} For the legal words that contain the letter $1$, the letter $1$ must appear in the second last position, since otherwise it would lead to a $123$ pattern. We now divide all the legal words ending with $1j$ into the following subcases:\n\\\\[4pt]\n \\indent {\\bf   Case 1.1:}  There is only one  word of the form $\\{1j\\}^{+}$.\\\\[4pt]\n \\indent {\\bf   Case 1.2:}  We deal with the words of the form $w\\{j'j\\}^{+}\\{1j\\}^{+}$, where $w$ is a legal word and $2\\le j'\\le j-1$. Note that $w$ cannot contain 1 because of an occurrence of $j'j$. Thus, we consider a legal word over the alphabet set $[2,k]$ of even length  which ends with $j'j$. By subtracting 1 from each letter of this word, we obtain a legal word over $[k-1]$ ending with $(j'-1)(j-1)$. Thus, the number of all words in this case is equal to that of all  words over $[k-1]$ ending with $j-1$, which is $\\sum_{i'=1}^{i-1} A_{k-1, 2i'}^{j-1}$. \\\\[4pt]\n \\indent {\\bf   Case 1.3:} The others  are  the words of the form  $wj'^{+}\\{1j\\}^{+}$, where $w$ is a legal word and $j'\\ge j+1$. Clearly, the number of such words in this case is $\\sum_{i'=1}^{i-1}\\sum_{j'=j+1}^{k}A_{k,2i'}^{j'}$. \\\\[6pt]\n\\noindent\n{\\bf Case 2:} We next deal with the legal words ending with $j$ over the alphabet $[k]\\setminus \\{1\\}=\\{2,3,\\ldots,k\\}$.\nIn this case, it has the same enumeration as that of the legal words over $[k-1]$ ending with $j-1$. The number of such words is $A_{k-1,2i}^{j-1}$.\n\nThus,  we have the following recurrence relation\n\\begin{align}\\label{formula-A-k-2i-j}\nA_{k,2i}^{j}\n=  1+\\sum_{i'=1}^{i} A_{k-1, 2i'}^{j-1}  + \\sum_{i'=1}^{i-1}\\sum_{j'=j+1}^{k}A_{k,2i'}^{j'}.\n\\end{align}\nFrom (\\ref{formula-A-k-2i-j}), we have that\n\\begin{align*}\nA_{k,2i}^{j} - A_{k,2i}^{j+1}\n=  \\sum_{i'=1}^{i} \\left( A_{k-1, 2i'}^{j-1}-A_{k-1, 2i'}^{j} \\right)  + \\sum_{i'=1}^{i-1}A_{k,2i'}^{j+1},\n\\end{align*}\nand therefore the recurrence  \\eqref{rec} follows.\n\nNow we deduce the formula \\eqref{A^j} for $A_{k,2i}^{j}$.\nLet $$A'(k,i,j) = \\frac{j-1}{k-1} \\binom{i+k-2}{i} \\binom{i+k-j-1}{i-1}.$$\nWe next prove that $A_{k,2i}^{j} = A'(k,i,j)$ by induction on $k-j$ and $k$. We shall show that these numbers  have the same base case and satisfy the same recursion.\nIndeed, for $k=j\\ge 2$, this fact is obviously true, since $A'(k,i,k)=A^k_{k,2i}$.\nWe will now check that $A'(k,i,j)$ satisfy the following recurrence relation:\n\\begin{align}\\label{rec-formula-Bs}\nA'(k,i,j)\n=  \\sum_{i'=1}^{i}  \\left(  A'(k-1, i',j-1) - A'(k-1, i',j)   +A'(k,i',j+1)  \\right).\n\\end{align}\nIndeed, (\\ref{rec-formula-Bs}) is true if and only if\n\\begin{align*}\nA'(k,i,j)-A'(k,i-1,j)\n=   A'(k-1, i, j-1) - A'(k-1, i, j)   +A'(k, i, j+1),\n\\end{align*}\nwhile the later equation is easy to check to be true.\nThis completes the proof.\n\\end{proof}\n\nFurther, the number of 123-avoiding up-down words  of length $2i$ over $[k]$ is\n $$A_{k,2i}=\\sum_{j=2}^k\\frac{j-1}{k-1} \\binom{i+k-2}{i} \\binom{i+k-j-1}{i-1}=\\frac{1}{i+1} \\binom{i+k-2}{i}   \\binom{i+k-1}{i}.$$\nThe last equality can be deduced from the Gosper algorithm \\cite{Petkovsek1996AB}.\n\nNow we consider legal words of odd length. For any legal word of length $2i\\ (i\\ge 1)$ ending with $j\\ (2\\le j\\le k)$, we can adjoin any letter in $[j-1]$ at the end to form an up-down word  of length $2i+1$ over  $[k]$. In fact, such words are necessarily 123-avoiding.\nSo, we obtain that\n\\begin{align*}\nA_{k,2i+1}= & \\sum_{j=2}^k(j-1)A^j_{k,2i}\\\\\n=& \\sum_{j=2}^k\\frac{(j-1)^2}{k-1} \\binom{i+k-2}{i}  \\binom{i+k-j-1}{i-1}\\\\\n= & \\frac{i+2k-2}{(i+1)(i+2)}\\binom{i+k-2}{i}  \\binom{i+k-1}{i}.\n\\end{align*}\nAlso, the last equality can be deduced from the Gosper algorithm \\cite{Petkovsek1996AB}.\n\n\nHence, we have proved the following theorem.\n\n\\begin{thm}\\label{main-thm} For $A_{k,\\ell}$, the number of $123$-avoiding up-down words of length $\\ell$ over $[k]$, $A_{k,0}=1$, $A_{k,1}=k$, and for $\\ell\\geq2$,\n\\begin{align}\\label{main-formula-A-k-l}\n A_{k,\\ell}=\n\\begin{cases}\n  \\frac{1}{i+1} \\binom{i+k-2}{i} \\binom{i+k-1}{i}, & \\mbox{ if } \\ell=2i,\\\\[6pt]\n  \\frac{i+2k-2}{(i+1)(i+2)}  \\binom{i+k-2}{i}  \\binom{i+k-1}{i}, & \\mbox{ if } \\ell=2i+1.\n\\end{cases}\n\\end{align}\n\\end{thm}\n\n\\subsection{Generating functions}\n\n\nIn this subsection, an expression for the generating function  for the numbers   $A_{k,i}$  of  123-avoiding up-down words of length $i$ over $[k]$ is given.\nWe adopt the notation of {\\em Narayana polynomials}, which are defined as $N_0(x)=1$ and,  for $n\\geq 1$,\n$$N_{n}(x)=\\sum_{i=0}^{n-1}\\frac{1}{i+1} \\binom{n}{i}\\binom{n-1}{i}x^i.$$\nDue to Brenti \\cite{Brenti1989Unimodal} and  Reiner and Welker \\cite[Section 5.2]{Reiner2005Charney}, a remarking generating function for $A_{k,2i}$ can be expressed as follows: \n\\begin{align}\\label{eq:Narayana}\n\\sum_{i\\geq 0}A_{k,2i}x^i=\\frac{N_{k-2}(x)}{(1-x)^{2k-3}}.\n\\end{align}\nOn the other hand, by Theorem \\ref{main-thm}, a routine computation leads to the following identity,\n\\begin{align}\\label{eq:123-odd}\n  A_{k,2i-1}&=A_{k,2i}-A_{k-1,2i},\n\\end{align}\nfor all $i\\geq 2$.\n(Note that we shall also give a combinatorial interpretation of \\eqref{eq:123-odd} in Section \\ref{all-cases}.)\nThus, together with $A_{k,1}=k$, it follows that\n\\begin{align*}\n \\sum_{i\\geq 1}A_{k,2i-1}x^i&= x+ \\sum_{i\\geq 1}A_{k,2i}x^i-\\sum_{i\\geq 1}A_{k-1,2i}x^i\\\\\n &=x+\\frac{N_{k-2}(x)}{(1-x)^{2k-3}}-\\frac{N_{k-3}(x)}\n {(1-x)^{2k-5}}\\\\\n &=x+\\frac{N_{k-2}(x)-(1-x)\n ^2N_{k-3}(x)}{(1-x)^{2k-3}}.\n\\end{align*}\nHence, we are ready to obtain the main result of this subsection,\n\\begin{align*}\n \\sum_{i\\geq 0}A_{k,i}x^i&=\\sum_{i\\geq0}A_{k,2i}x^{2i}\n + \\sum_{i\\geq 1}A_{k,2i-1}x^{2i-1}\\\\\n &=\\frac{N_{k-2}(x^2)}{(1-x^2)^{2k-3}}+x+ \\frac{N_{k-2}(x^2)-\n (1-x^2)^2N_{k-3}(x^2)}{x(1-x^2)^{2k-3}}\\\\\n &=x+\\frac{(1+x)N_{k-2}(x^2)-(1-x^2)^2N_{k-3}(x^2)}{x(1-x^2)^{2k-3}}.\n\\end{align*}\n\n\\section{A bijection between  \\texorpdfstring{$S^{132}_{k,2i}$}{Lg}  and \\texorpdfstring{$S^{123}_{k,2i}$}{Lg}}\\label{bijection-sec}\n\nLet $p$ be a pattern and $S^{p}_{k,\\ell}$ be the set of  $p$-avoiding up-down words of length $\\ell$ over $[k]$.\nIn this section, we will build a bijection between $S^{132}_{k,2i}$  and $S^{123}_{k,2i}$.\n\nThe idea here is to introduce the notion of irreducible words and show that irreducible words in $S^{132}_{k,2i}$ can be mapped in a 1-to-1 way into irreducible words in $S^{123}_{k,2i}$, while reducible words in these sets can be mapped to each other as well.\n\n\\begin{definition} A word $w$ is {\\em reducible}, if $w=w_1w_2$ for some non-empty words $w_1$ and $w_2$, and  each letter in $w_1$ is no less than every letter in $w_2$. The place between $w_1$ and $w_2$ in $w$ is called a {\\em cut-place}.\\end{definition}\n\nFor example, the word 242313 is irreducible, while the word 341312 is reducible (it can be cut into 34 and 1312).\n\nNote that in a reducible up-down word, if we have a cut-place, and there are equal elements on both sides of it, then to the left such elements must be bottom elements, and to the right they must be top elements.\n\n\\begin{lem} A word $w$ in $S^{132}_{k,2i}$ is irreducible if and only if  $w=w_1xy$, where $w_1$ is a word in $S^{132}_{k,2i-2}$, $x$ is the minimum letter in $w$ (possibly, there are other copies of $x$ in $w$) and $y$ is the maximum letter in $w$ (possibly, there are other copies of $y$ in $w$).\\end{lem}\n\n \\begin{proof}\n  If $x$ is not the minimum element in $w$, then the element right before it, and the minimum element in $w$ will form the pattern 132.\n  Since $w$ is irreducible, $y$ is forced to be no less than the minimum element in $w_1$.\n  On the other hand, if $y$ is not the maximum element in $w$, then  the maximum one  in $w_1$ and the element just preceding it will form the pattern 132.\n  This completes the proof.\n \\end{proof}\n\nNow, given a word $w$ in $S^{132}_{k,2i}$, we can count  in how many ways it can be extended to an irreducible word in $S^{132}_{k,2i+2}$. Suppose that $a$ and $b$ are the minimum and the maximum elements in $w$, respectively. Then the number of extensions of $w$ in $S^{132}_{k,2i+2}$ is $a\\cdot(k-b+1)$, since there are $a$ choices of the next to last element and $k-b+1$ choices of the last element.\n\nNext, we discuss a procedure of turning any word $w$ in $S^{123}_{k,2i}$ into an irreducible word in $S^{123}_{k,2i+2}$. From this procedure, it would be clear that the number of choices  is $a\\cdot(k-b+1)$, where $a$ and $b$ are the minimum and the maximum elements in $w$, respectively.\n\nSuppose that $w=b_1 t_1 b_2 t_2 \\cdots b_i t_i$, where $b_j$'s and $t_j$'s stand for bottom and top elements, respectively.\nTo obtain the desired word, we inserting a new top element $x$, shift the bottom elements one position to the left, and then insert one more bottom element $y$.\nThen, the extension is of the form\n$$w'=b_1 x b_2 t_1 b_3 t_2 \\cdots b_i t_{i-1} y t_i,$$\nwhere $x\\ge b$ and $y\\le a$, and again, $a$ and $b$ are the minimum and the maximum elements in $w$, respectively.\nFor example, if  $w = 242313\\in S^{123}_{5,6}$, $a=1$ and $b=4$,  then $w'$ can be $24241313$ or $25241313$.\n\nTo see that  the  resulting word $w'$ is an up-down word. In fact,  it is sufficient to show that $b_{j+1} < t_{j}$ for $1\\le j\\le i-1$ and $t_j > b_{j+2}$ for $1\\le j\\le i-2$. The first inequality follows from the fact that $w$ is an up-down word, while the second one is true, because otherwise $t_j\\le b_{j+2}<t_{j+1}$ and thus $b_j t_j t_{j+1}$  would form a $123$ pattern. \nWe also claim that $w'$ belongs to  $S^{123}_{k,2i+2}$. \nAn equivalent condition an up-down  $w$ is $123$ avoiding is that $w$ satisfy  \n$b_1\\geq b_2\\geq\\dots \\geq b_i$  and  $t_1\\geq t_2\\geq\\dots \\geq t_i$. \nFrom the construction of $w'$, we obtain that  $w'$ is also $123$ avoiding.\nBesides,  $w'$ is irreducible, since $b_j < t_j$ for $1\\le j\\le i$.\n\nNow, a bijection between $S^{132}_{k,2i}$  and $S^{123}_{k,2i}$ is straightforward to set recursively, with a trivial base case of words of length 2 mapped to themselves. Indeed, if we assume that we can map all words in $S^{132}_{k,2i}$ to all words in $S^{123}_{k,2i}$, then applying the same choice of $x$ and $y$, we can map all irreducible words in $S^{132}_{k,2i+2}$ to all irreducible words in $S^{123}_{k,2i+2}$. Finally, each reducible word $w$ is of the form $w_1w_2$, where $w_1$ is irreducible word with maximum possible even length. But then $w_1$ and $w_2$ are of smaller lengths than $w$, and we can map them recursively.\n\n\n\\begin{table}[!htbp]\n \\renewcommand{\\arraystretch}{1.3}\n\\begin{center}\n\\begin{tabular}{|ccc|lll|}\n\\hline\n$S^{132}_{4,4}$ & & $S^{123}_{4,4}$ &$S^{132}_{4,4}$ & & $S^{123}_{4,4}$ \\\\\n\\hline\n1212&& 1212 &2312&& 2312\\\\\n2412 && 2412&3412 && 3412\\\\\n1213&& 1312&1313 &&1313\\\\\n2313&& 2313&3413& &3413\\\\\n2323&& 2323&3423&& 3423\\\\\n1214&& 1412&1314 &&1413\\\\\n2314& &2413&1414 &&1414\\\\\n2414&& 2414&3414& &3414\\\\\n2324 &&2423&2424& &2424\\\\\n3424&& 3424&3434&& 3434\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The bijection $\\phi : S^{132}_{4,4} \\rightarrow S^{123}_{4,4}$.}\\label{tab2}\n\\end{table}\n\nFor example, $w=3435121213\\in S^{132}_{5,10}$ is reducible, since it can be cut into $3435$ and $121213$. We calculate that $\\phi(3435)$ by two steps: first $\\phi(34)=34$ and the second $\\phi(3435)=3534$. Similarly, $\\phi(121213)=131212$. It follows that $\\phi(3435121213)=3534131212$, i.e. the word $3435121213$ in $S^{132}_{5,10}$ is mapped to $3534131212$ in $S^{123}_{5,10}$. Also, see Table \\ref{tab2}  showing images of all words in $S^{132}_{4,4}$. \n\n\\section{Enumeration of 132-avoiding up-down words}\\label{sec-132-av}\nIn this section, we consider the enumeration of 132-avoiding up-down words.\nDenote $B_{k,\\ell}$ the number of 132-avoiding up-down words of length $\\ell$ over the alphabet $[k]$, and $B_{k, \\ell}^{j}$ the number of those words counted by $B_{k,\\ell}$ whose letter in the second last position is $j$.\n\nFrom the bijection, it follows that\n\\begin{align}\n B_{k,2i} = A_{k,2i}=\\frac{1}{i+1} \\binom{i+k-2}{i}   \\binom{i+k-1}{i}.\n\\end{align}\n\nFor any legal word of length $2i\\ (i\\ge 1)$ whose letter in the second last position is $j\\ (1\\le j\\le k-1)$,  the minimum letter in this word is also $j$, since it is 132-avoiding. By subtracting $j-1$ from each letter of this word, we obtain a legal word over $[k-j+1]$  whose letter in the second last position is $1$.  Thus, we have that\n$$B_{k,2i}^{j} = B_{k-j+1,2i}^{1}.$$\nSimilarly, we obtain that\n\\begin{align*}\n B_{k,2i}^{1} = B_{k,2i}-\\sum_{j=2}^{k-1}B_{k,2i}^{j}=B_{k,2i}-B_{k-1,2i}= \\frac{i+2 k-3 }{(i+1)(k-1)} \\binom{i+k-3}{i-1} \\binom{i+k-2}{i}.\n\\end{align*}\n\nNow, we consider legal words of odd length. For any legal word of length $2i\\ (i\\ge 1)$ whose letter in the second last position is $j\\ (1\\le j\\le k-1)$,  the minimum letter in the word is also $j$ and hence, we can adjoin any letter in $[j]$ at the end to form an up-down word on  $[k]$ of length $2i+1$. In fact, such words are necessarily 132-avoiding.\nSo, we obtain that for $i\\ge 1$\n\\begin{align*}\nB_{k,2i+1}= & \\sum_{j=1}^{k-1} j B_{k,2i}^{j}\\\\\n=& \\sum_{j=1}^{k-1} j B_{k-j+1,2i}^{1}\\\\\n=& \\sum _{j=1}^{k-1}j\\, \\frac{i+2 (k-j)-1}{(i+1)(k-j)}  \\binom{i+k-j-2}{i-1} \\binom{i+k-j-1}{i}\\\\\n=& \\sum _{j=1}^{k-1} \\frac{(k-j)  (i+2 j-1)}{ j(i+1)} \\binom{i+j-2}{i-1} \\binom{i+j-1}{i}.\n\\end{align*}\n\nUnfortunately, we were not able to find a closed form formula for $B_{k,2i+1}$. We conclude this section with listing expressions for $B_{k,2i+1}$ for $k=3,4,5,6$ and $i\\ge 1$:\n\\begin{align*}\nB_{3,2i+1} & = \\frac{1}{2} (i^2+3 i+4), \\\\\nB_{4,2i+1} & = \\frac{1}{12} (i^4+8 i^3+29 i^2+46 i+36), \\\\\nB_{5,2i+1} & = \\frac{1}{144} (i^6+15 i^5+103 i^4+381 i^3+832 i^2+972 i+576), \\\\\nB_{6,2i+1} & = \\frac{1}{2880} (i^8+24 i^7+266 i^6+1704 i^5+6929 i^4+18096 i^3+30244 i^2+29136 i+14400).\n\\end{align*}\n\n\\section{Enumeration of 312-avoiding up-down words}\\label{312-up-down-sec}\nIn this section, we consider the enumeration of 312-avoiding up-down words. \nDenote $C_{k,\\ell}$ the number of 312-avoiding up-down words of length $\\ell$ over the alphabet $[k]$, and $C_{k, \\ell}^j$ the number of those words counted by $C_{k,\\ell}$ that end with $j$.\n\nRecall the definition of the complement $w^c$ of a word $w$ given in Section~\\ref{intro}. Also, for a word $w=w_1w_2\\cdots w_n$, its {\\em reverse word} $w^{r}$ is given by $w^{r}=w_{\\ell}w_{\\ell-1}\\cdots w_{1}$. \nIt is clear that the operations of reverse and complement are both bijections on alternating words.\n\nLet $p$ be a pattern and let $\\widehat{S}^p_{k,\\ell}$ denote the set of all $p$-avoiding down-up words of length $\\ell$ over an alphabet $[k]$.\n\nWe first consider the words of odd length.\n\n\\begin{prop}For all $k\\geq 1$ and $i\\geq 0$,\nwe have that the number of $312$-avoiding up-down words of length $2i+1$ on $[k]$  is the same as that of $123$-avoiding up-down words on $[k]$ of the same length. Namely,\n$$C_{k,2i+1}=A_{k,2i+1}.$$\n\\end{prop}\n\n\\begin{proof}\n We shall prove this theorem by  establishing  a bijection between $S^{312}_{k,2i+1}$ and $S^{123}_{k,2i+1}$.\nApplying the complement operation to the former of these sets, and reverse and complement to the latter set, it suffices to show that there exists a bijection  between $\\widehat{S}^{132}_{k,2i+1}$ and $\\widehat{S}^{123}_{k,2i+1}$. \n\nThe map $\\psi(w): \\widehat{S}^{132}_{k,2i+1} \\rightarrow \\widehat{S}^{123}_{k,2i+1}$ is defined as follows.\nFor any $w=w_1w_2\\cdots w_{2i+1} \\in \\widehat{S}^{132}_{k,2i+1}$, let $w'$ be $w_2\\cdots w_{2i+1}$.  It is clear that $w'\\in S^{132}_{k,2i}$. Thus let $$\\psi(w)=w_1\\phi(w'),$$ where the  map $\\phi: S^{132}_{k,2i} \\rightarrow S^{123}_{k,2i}$ is described in Section \\ref{bijection-sec}. \n\nWe need to show that $\\psi$ is well-defined.\nFrom the construction of $\\phi$, we see that $\\phi$ preserves the first letter, i.e.  $\\phi(w')_1=w'_1=w_2$. Therefore,  it follows that $\\psi(w)\\in \\widehat{S}^{123}_{k,2i+1}$. Hence, the map $\\psi$ is well-defined. \n\nIt is not difficult to see by construction that $\\psi$ is a bijection.\nHence, we get  a bijection between $\\widehat{S}^{132}_{k,2i+1}$ and $\\widehat{S}^{123}_{k,2i+1}$.\nThis completes the proof.\n\\end{proof}\n\nNow let us consider the words of even length. Note that\n$$C_{k,2i}=\\sum_{j=2}^{k} C_{k,2i}^j.$$ \n\nFor any word $w \\in S^{312}_{k,2i}$ whose last letter is $j$, the maximum letter of $w$ is also $j$, since the word is $312$-avoiding and having $j'>j$ in $w$ would lead to an occurrence of three letters $j'w_{2i-1}j$ forming the pattern $312$. Thus, we have that $$C_{k,2i}^j=C_{j,2i}^j,$$ where $2\\leq j\\leq k$ .\n\nMoreover, for any word in $S^{312}_{j,2i}$ ending with $j$, we can remove $j$ to form a word of length $2i-1$, which is also $312$-avoiding. On the other hand, for any word  $S^{312}_{j,2i-1}$, we can adjoin a letter $j$ at the end to form a $312$-avoiding word of length $2i$. Thus, $$C_{j,2i}^j=C_{j,2i-1}.$$\nSo,  we obtain that\n\\begin{align*}\n C_{k,2i} &= \\sum_{j=2}^{k}C_{j,2i-1}\\\\\n &=\\sum_{j=2}^{k}\\frac{i-3+2j}{i(i+1)}\\binom{i+j-3}{ i-1}\\binom{i+j-2}{i-1}\\\\\n &=\\frac{1}{i+1} \\binom{i+k-2}{i}\\binom{i+k-1}{i}\\\\\n &=A_{k,2i},\n\\end{align*}\nwhere  the  second last equality can be deduced from the Gosper algorithm \\cite{Petkovsek1996AB}.\n\nWe have just obtained the main result in this section.\n\\begin{thm}\\label{123-312}\nThe sets of $312$-avoiding up-down words and $123$-avoiding up-down words are equinumerous, that is, $$C_{k,\\ell}=A_{k,\\ell}$$ for all $k\\geq 1$ and $\\ell\\geq 0.$\n\\end{thm}\n\n\n\n\\section{Enumeration of other pattern-avoiding up-down words}\\label{all-cases}\n\n\nIn this section, we consider the enumeration of other pattern-avoiding up-down words.\nIn order to avoid confusion, let $N_{k,\\ell}(p)$ denote the number of $p$-avoiding up-down words of length $\\ell$ over the alphabet $[k]$.\n\n\nWe first focus on all six length 3 permutation patterns to be avoided on up-down words of odd length.\n\\begin{thm} For all $k\\geq 1$ and $i\\geq 0$, we have\n$$N_{k,2i+1}(123)=N_{k,2i+1}(312)=N_{k,2i+1}(213)\n=N_{k,2i+1}(321)$$\nand\n$$N_{k,2i+1}(132)=N_{k,2i+1}(231).$$\n\\end{thm}\n\n\n\\begin{proof}\n Through the reverse operation, the following equations hold:\n$$N_{k,2i+1}(132)=N_{k,2i+1}(231),$$\n$$N_{k,2i+1}(123)=N_{k,2i+1}(321),$$ and\n$$N_{k,2i+1}(312)=N_{k,2i+1}(213).$$\nCombing  with Theorem \\ref{123-312}, the proof is complete.\n\\end{proof}\n\nNext,  we obtain the following result for the case of the even length.\n\\begin{thm}\nFor all $k\\geq 1$ and $i\\geq 1$, there is \n$$N_{k,2i}(123)=N_{k,2i}(132)=N_{k,2i}(312)=N_{k,2i}(213)\n=N_{k,2i}(231).$$\n\\end{thm}\n\\begin{proof}\n Through the complement and reverse operations, it follows that \n$$N_{k,2i}(132)=N_{k,2i}(213),$$ and\n$$N_{k,2i}(231)=N_{k,2i}(312).$$\nFrom Section \\ref{bijection-sec}, we have that\n$$N_{k,2i}(132)=N_{k,2i}(123).$$\nTogether with Theorem \\ref{123-312}, we complete the proof.\n\\end{proof}\n\n\nIn the rest of this section, we deal with the only remaining case,  321-avoiding up-down words of even length.  Our approach is based on deriving the desired from an alternative enumeration of  123-avoiding up-down words.\n\n\nAll 123-avoiding up-down  words of length $\\ell$ over $[k]$, for $\\ell \\geq 4$, can be divided into the following two cases:\n\\begin{itemize}\n\\item Legal words containing no $k$ in them. These words are counted by $A_{k-1, \\ell}$.\n\\item Legal words that contain at least one $k$. Such words $w=w_1w_2\\cdots w_{\\ell}$ are necessarily of the form $w_1kw_3\\cdots w_{\\ell}$, since otherwise  $w_1w_2k$ would be an occurrence of the pattern $123$. \nClearly,  $w_1\\geq w_3$ (otherwise, $w_1w_3w_4$ would form the pattern 123).\nWe let $w'$ be $k w_3\\cdots w_{\\ell}$ if $w_1=w_3$ and  $w_1 w_3\\cdots w_{\\ell}$ if $w_1>w_3$.\nClearly, this is a reversible procedure and the obtained words $w'$ are 123-avoiding down-up words.  By applying the complement  operation, we obtain 321-avoiding up-down words over $[k]$ of length $\\ell-1$.\n\\end{itemize}\nIt follows that for $\\ell \\geq 4$,  $$A_{k, \\ell}=A_{k-1, \\ell}+N_{k, \\ell-1}(321),$$\nand thus \n$$N_{k, \\ell-1}(321)= A_{k, \\ell}-A_{k-1, \\ell}.$$\nHence, by Theorem \\ref{main-thm},  we are ready to obtain an expression for  $N_{k, \\ell}(321)$.\n\n\\begin{thm}\\label{thm-321} For the number of $321$-avoiding up-down words of length $\\ell$ over $[k]$, $N_{k,0}(321)=1$, $N_{k,1}(321)=k$,  $N_{k,2}(321)=\\binom{k}{2}$,  and for $\\ell\\geq 3$,\n\\begin{align*}\n N_{k, \\ell}(321)=\n\\begin{cases}\n\\frac{i (i+2 k-3) (i+2 k-2)+2 (k-2) (k-1)}{(i+1) (i+2) (k-2) (k-1)}  \\binom{i+k-2}{i} \\binom{i+k-3}{i} , & \\mbox{ if } \\ell=2i,\\\\[6pt]\n  \\frac{i+2k-2}{(i+1)(i+2)}  \\binom{i+k-2}{i}  \\binom{i+k-1}{i}, & \\mbox{ if } \\ell=2i+1.\n\\end{cases}\n\\end{align*}\n\\end{thm}\n\nSince $N_{k,2i+1}(123)=N_{k,2i+1}(321)$, we actually  give another approach to deal with $321$-avoiding up-down words of odd length.\n\n\n\\section{Concluding remarks}\\label{last-sec}\n\nIn this paper we initiated the study of (pattern-avoiding) alternating words. In particular, we have shown that 123-avoiding up-down words of even length are given by the Narayana numbers. Thus, alternating words can be used, for example, for encoding Dyck paths with a specified number of peaks \\cite{GaoZhang}. To our surprise, the enumeration of 123-avoiding up-down words turned out to be easier than that of 132-avoiding up-down words, as opposed to similar studies for permutations, when the structure of 132-avoiding permutations is easier than that of 123-avoiding permutations.\n\nAbove, we gave a complete classification of avoidance of permutation patterns of length~$3$ on alternating words. We state it as an open direction of research to study avoidance of longer patterns and\/or patterns of different types (see~\\cite{Kitaev2011Patterns}) on alternating (up-down or down-up) words.\n\n\\vskip 3mm\n\\noindent {\\bf Acknowledgments.} \nThis work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education and the National Science Foundation of China.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction\\label{Intro}}\n\nThere is currently growing interest in theoretical and experimental studies\nof topologial superconductors \\cite{Kallin}-\\cite{MS}. One of the well-known\nexamples is $Sr_2RuO_4$, which is thought to be a chiral triplet p-wave\nsuperconductor \\cite{Maeno}-\\cite{Maeno2016}. There are several features of \ntriplet p-wave superconductivity, among them the key signature is appearence of the odd-frequency states\nin the vicinity of surface, leading to formation undergap bound states \\cite{Zhang}-\\cite{LuBo}. The chiral\nsuperconductivity is highlighted by existence of the spontaneous\nsurface currents being one of most interesting among the predicted features. Unfortunately,\nthese currents have never been experimentally detected yet \\cite{PGB},\n\\cite{Kirtley}.\n\nOne of possible reasons for the absence of the surface currents could be the\npresence of non-superconducting layers on the surface of the samples, which\nmay be formed due to sample degradation. Such layers can be characterized by\nvarious degrees of diffusivity and by range of thickness, $d,$ measured in\nsuperconducting decay length units, $\\xi _{0}$.\nIt was demonstrated recently within the quasiclassical Eilenberger formalism\n\\cite{Eilenberger} that the edge currents in a chiral p -wave\nsuperconductor are robust with respect to surface roughness \\cite{Suzuki} in the regime of small thickness $d$. On the other hand, calculations done\nfor the case of specular surface within the tight-binding model \\cite{Ashby}\n\\cite{Huang} have shown that the spontaneous currents could be suppressed if $d$ is of the same order or greater than the decay length $\\xi _{0}$.\n\nIn order to establish general conditions for the existence of surface currents in chiral p-wave superconductors, in this work we formulate\nthe general approach in the framework of the Eilenberger formalism \\cit\n{Eilenberger} for various experimental realizations of surface layers in\nthese materials. We investigate thin $(d\\ll \\xi _{0})$ and\nthick $(d\\gg \\xi _{0})$ clean and diffusive films in proximity with bulk\np-wave chiral superconductor. We show that, in accordance with the results of\n\\cite{Suzuki}, spontaneous currents are not sensitive to surface roughness\nin the limit of thin surface layer, while in the opposite case $(d\\gg \\xi\n_{0})$ these currents are strongly suppressed. We also discuss the peculiarities of\ncurrent distributions inside the sample for different types of surface layers and the possibility of their experimental observation.\n\n\\section{Model\\label{Sec1}}\n\nThe description of the surface properties of chiral p-wave superconductor\nhas been done within the framework of the quasiclassical Eilenberger equations\n\\cite{Eilenberger}. To solve the problem, we will assume that the conditions\nof the clean limit (scattering time $\\tau \\rightarrow \\infty $ ) are valid\nin the bulk superconductor region $-\\infty<x\\leqslant0$ and the equations\nhave the form\n\\begin{equation}\n2\\omega f(x,\\theta )+v\\cos (\\theta )\\frac{d}{dx}f(x,\\theta )=2\\Delta\ng(x,\\theta ),  \\label{El0}\n\\end{equation\n\\begin{equation}\n2\\omega f^{+}(x,\\theta )-v\\cos (\\theta )\\frac{d}{dx}f^{+}(x,\\theta )=2\\Delta\n^{\\ast }g(x,\\theta ),  \\label{El0a}\n\\end{equation\n\\begin{equation}\n2v\\cos (\\theta )\\frac{d}{dx}g_{\\omega }(x,\\theta )=2\\left( \\Delta ^{\\ast\n}f_{\\omega }-\\Delta f_{\\omega }^{+}\\right) .  \\label{El0b}\n\\end{equation\nHere $g(x,\\theta )$, $f(x,\\theta )$ and $\\ f^{+}(x,\\theta )$ are normal and\nanomalous Eilenberger functions, $\\theta $ is angle between the vector\nnormal to the interface and the direction of the electron Fermi velocity $v;$\n$\\omega =\\pi T(2n+1)$ are Matsubara frequencies and $T$ is temperature, $x$\nis coordinate along the axis normal to the boundary. Pair potential $\\Delta\n(x,\\theta )$ in a chiral p-wave superconductor is separated into two\nterms $\\Delta =\\Delta _{x}\\cos (\\theta )+i~\\Delta _{y}\\sin (\\theta )$.\n\nEquations (\\ref{El0})-(\\ref{El0b}) should be supplemented with\nthe self-consistency equation, which can be written in the form \\cite{Bruder}\n\\begin{eqnarray}\n\\Delta _{x}\\ln \\frac{T}{T_{c}}+2\\pi T\\sum_{\\omega }\\frac{\\Delta _{x}}{\\omega\n}-\\left\\langle 2\\cos (\\theta ^{\\prime }) Re f(x,\\theta ^{\\prime\n})\\right\\rangle &=&0,  \\label{Sc0b} \\\\\n\\Delta _{y}\\ln \\frac{T}{T_{c}}+2\\pi T\\sum_{\\omega }\\frac{\\Delta _{y}}{\\omega\n}-\\left\\langle 2\\sin (\\theta ^{\\prime }) Im f(x,\\theta ^{\\prime\n})\\right\\rangle &=&0.  \\label{Sc0c}\n\\end{eqnarray}\nHere $\\left\\langle ...\\right\\rangle =(1\/2\\pi )\\int_{0}^{2\\pi }(...)d\\theta $\nand $T_{c}$ is the critical temperature.\n\nPreviously, the properties of the superconductor surface have been described \\cit\n{Golubov1}-\\cite{Golubov2} in the frame of Ovchinnikov model \\cit\n{Ovchinnikov}. Within this model it is assumed that the surface is covered by a thin\ndiffusive normal metal layer. In the present study we go beyond this approximation\nand consider the situation when a clean p-wave superconductor is covered\nby a normal metal of finite thickness\nwith respect to the coherence length $\\xi _{0}=v\/2\\pi T_{c} $ and electron mean free\npath $l_{e}$ and having mirror-reflecting surface. Inside this layer, located in the area $0\\leq x\\leq d$,\nEilenberger equations \\cite{Eilenberger} transform to\n\n\\begin{equation}\nv\\cos (\\theta )\\frac{d}{dx}f(x,\\theta )=\\frac{1}{\\tau }\\left( g\\left\\langle\nf\\right\\rangle -f\\left\\langle g\\right\\rangle \\right) -2\\omega f(x,\\theta ),\n\\label{El2a}\n\\end{equation\n\\begin{equation}\nv\\cos (\\theta )\\frac{d}{dx}f^{+}(x,\\theta )=\\frac{1}{\\tau }\\left(\ng\\left\\langle f^{+}\\right\\rangle -f^{+}\\left\\langle g\\right\\rangle \\right)\n+2\\omega f^{+}(x,\\theta ),  \\label{El2b}\n\\end{equation\n\\begin{equation}\n2v\\cos (\\theta )\\frac{d}{dx}g(x,\\theta )=\\frac{1}{\\tau }\\left( f\\left\\langle\nf^{+}\\right\\rangle -f^{+}\\left\\langle f\\right\\rangle \\right).  \\label{El2c}\n\\end{equation\nFollowing the procedure developed in \\cite{Golubov1}-\\cite{Golubov2} we will\nconsider (\\ref{El2a})-(\\ref{El2c}) as the system of linear equations in\nwhich $\\left\\langle f\\right\\rangle $, $\\left\\langle f^{+}\\right\\rangle $ and\n$\\left\\langle g\\right\\rangle $ should be determined self-consistently in the\nprocess of finding a solution to the equations (\\ref{El0})-(\\ref{El2c}). To\nsimplify the problem further we introduce pairing functions of the incident \nf_{+}(\\theta )=f(\\theta )$ and reflected $f_{-}(\\theta )=f(\\pi +\\theta )$\nparticles, which are defined in the segment of angles $-\\pi \/2\\leq \\theta\n\\leq \\pi \/2.$\nThe functions $\\ f_{\\pm }^{+}$ and $\\ g_{\\pm }$ are defined below in a\nsimilar manner.\n\nFor the development of numerical algorithms for solution of the Eilenberger\nequations it is convenient to rewrite them using the Ricatti parametrization\n\\cite{Schopohl,Tanaka6}.\n\\begin{equation}\nf_{\\pm }=\\frac{2a_{\\pm }}{1+a_{\\pm }b_{\\pm }},\\ f_{\\pm }^{+}=\\frac{2b_{\\pm \n}{1+a_{\\pm }b_{\\pm }},\\ g_{\\pm }=\\frac{1-a_{\\pm }b_{\\pm }}{1+a_{\\pm }b_{\\pm \n}.  \\label{Ric1}\n\\end{equation\nFor the Ricatti functions $a_{\\pm }$ and $b_{\\pm }$ from (\\ref{El0})-(\\re\n{El2c}) we have\n\\begin{eqnarray}\nv\\cos (\\theta )\\frac{d}{dx}a_{\\pm } &=&\\Delta -\\Delta ^{\\ast }a_{\\pm\n}^{2}\\mp 2\\omega a_{\\pm },  \\label{El_ric_cl_A} \\\\\nv\\cos (\\theta )\\frac{d}{dx}b_{\\pm } &=&-\\Delta ^{\\ast }+\\Delta b_{\\pm\n}^{2}\\pm 2\\omega b_{\\pm },  \\label{El_ric_cl_B}\n\\end{eqnarray\nin the clean superconducting region and\n\\begin{eqnarray}\n\\frac{da_{\\pm }}{dx} &=&\\mp \\frac{\\left\\langle f\\right\\rangle -a_{\\pm\n}^{2}\\left\\langle f^{+}\\right\\rangle -2a_{\\pm }\\left\\langle g\\right\\rangle }\n2\\tau v\\cos (\\theta )}\\mp \\frac{2\\omega a_{\\pm }}{v\\cos (\\theta )},\n\\label{El_ric_dr_A} \\\\\n\\frac{db_{\\pm }}{dx} &=&\\mp \\frac{\\left\\langle f^{+}\\right\\rangle -b_{\\pm\n}^{2}\\left\\langle f\\right\\rangle -2b_{\\pm }\\left\\langle g\\right\\rangle }\n2\\tau v\\cos (\\theta )}\\pm \\frac{2\\omega b_{\\pm }}{v\\cos (\\theta )},\n\\label{El_ric_dr_B}\n\\end{eqnarray\nin the diffusive layer. The subscript $+$ in (\\ref{Ric1})-(\\ref{El_ric_dr_B\n) indicates that functions are calculated along trajectories in the\ndirection towards the SN boundary, while the subscript $-$ denotes the\nfunctions calculated on trajectories in the direction away from the SN interface\ntowards the bulk superconductor.  Far from the SN interface (at $x\\rightarrow\n-\\infty $) the values of $a_{\\pm }$ and $b_{\\pm }$ equal to their magnitudes\nin a homogeneous superconductor\n\\begin{equation}\na_{\\pm }=\\pm \\frac{\\Delta }{\\omega +\\sqrt{\\omega ^{2}+|\\Delta |^{2}}},\n\\label{As0}\n\\end{equation\n\\begin{equation}\nb_{\\pm }=\\pm \\frac{\\Delta ^{\\ast }}{\\omega +\\sqrt{\\omega ^{2}+|\\Delta |^{2}}\n,  \\label{As1}\n\\end{equation\nwhere $\\Delta $ is the bulk value of the pair potential.\n\nFinally, the problem must be supplemented by the boundary conditions \\cit\n{Bakurskiy}  at the free surface of the diffusion layer $(x=d)\n\\begin{equation}\nb_{+}(d,-\\theta )=b_{-}(d,\\theta ),  \\label{BC1a}\n\\end{equation\n\\begin{equation}\na_{-}(d,-\\theta )=a_{+}(d,\\theta ),  \\label{BC1b}\n\\end{equation}\nand by the requirement of continuity of the $a_{\\pm }$ and $b_{\\pm }$ functions\non the SN interface. \n\nThe boundary problem (\\ref{El_ric_cl_A})-(\\re\n{BC1b}) has been solved numerically.\nTo solve it, we start with some trial functions for order parameter, \n\\left\\langle f\\right\\rangle,$ $\\left\\langle f^{+}\\right\\rangle,$ \n\\left\\langle g\\right\\rangle$ and intergrade numerically equations (\\re\n{El_ric_cl_A}) - (\\ref{El_ric_dr_B}) for $a_{+}(x,\\theta )$ and \nb_{-}(x,-\\theta )$ moving along the trajectory from the bulk values of these\nfunctions (\\ref{As0}) at infinity $(x=-\\infty )$ towards the free N layer\nsurface $(x=d)$ making use of the requirement of continuity of the $a_{+}$\nand $b_{-}$ functions on the SN interface $(x=0)$. Thus obtained solutions \na_{+}(0,\\theta )$ and $b_{-}(0,-\\theta )$ and boundary conditions (\\ref{BC1a\n), (\\ref{BC1b}) determine the magnitudes of $a_{-}(0,\\theta )$ and \nb_{+}(0,\\theta )$. Starting from these solutions, the functions $a_{-}(x,\\theta )$ and \nb_{+}(x,\\theta )$ are obtained by integration along the trajectories going\nacross the diffusive N layer into the bulk superconductor.  The averaged\nfunctions $\\left\\langle f\\right\\rangle,$ $\\left\\langle f^{+}\\right\\rangle,$ \n\\left\\langle g\\right\\rangle$ and the spatial dependence of the order\nparameter $\\Delta (x)$ are determined in an iterative self-consistent way\nusing the definitions (\\ref{Ric1}) and self-consistency equations ( (\\re\n{Sc0b})-(\\ref{Sc0c}).\n\nThe calculated Ricatti functions $a_{\\pm }(x,\\theta )$ and $b_{\\pm }(x,\\theta)\n$ permit one to restore the angular distribution of the Eilenberger functions in\nany cross-sectional plane inside SN bilayer and calculate the spatial\ndistribution of the density of spontaneous edge current $j(x)$\n\\begin{equation}\nj(x)=2evN_{0}T\\sum_{\\omega >0}j_{\\omega}(x), \\\nj_{\\omega}(x)=\\int\\limits_{0}^{2\\pi }j_{\\omega}(x, \\theta)\\sin \\theta\nd\\theta,  \\label{current}\n\\end{equation}\nwhere $N_{0}$ is the density of states and angular resolved supercurrent\ndensity $j_{\\omega}(x, \\theta)=Im(g_{\\omega }(x, \\theta))=Im(\\sqrt\n1-f_{\\omega }(\\theta )f_{\\omega }(\\pi -\\theta )})$. When writing the last\nequality we used\nthe normalization condition for triplet functions\n\\begin{equation}\ng_{\\omega }^{2}(\\theta )-f_{\\omega }(\\theta )f_{\\omega }^{+}(\\theta )=1,\n\\label{norcond}\n\\end{equation}\nand the symmetry relatio\n\\begin{equation}\nf_{\\omega }(\\theta )=-f_{\\omega }^{+}(\\pi -\\theta ).  \\label{symrel}\n\\end{equation\nFor simplicity we will suppose further that the amplitudes $f_{\\omega }(x,\n\\theta )$ are small, and making use of (\\ref{norcond}), (\\ref{symrel})\nwe rewrite (\\ref{current}) in the form\n\\begin{equation}\n\\begin{array}{c}\nj_{\\omega }(x)=\\int\\limits_{-\\pi\/2}^{\\pi\/2 }\\left( Re(f_{\\omega }(x,\\theta\n))Im(f_{\\omega }(x,\\pi -\\theta ))+\\right. \\\\\n\\left. +Re(f_{\\omega }(x,\\pi -\\theta ))Im(f_{\\omega }(x,\\theta ))\\right)\n\\sin \\theta d\\theta \n\\end{array}\n\\label{jsmall}\n\\end{equation}\n\nIt is seen from Eq.(\\ref{jsmall}) that  the appearance of the spontaneous\nsurface current is a result of superposition of anomalous Eilenberger\nfunctions of incident $f_{\\omega }(x,\\theta )$ and reflected $f_{\\omega\n}(x,\\pi -\\theta )$ from the free interface trajectories. In the geometry considered in this work, when the $p_{x}$ principal axis is parallel to the interface normal, there is  general relation $f_{\\omega }(\\theta )=f_{\\omega }^{\\ast }(-\\theta\n)$ providing  the odd symmetry of imaginary part of pair amplitude. Far from the\nSN interface inside the bulk p-wave superconductor there is the odd\nsymmetry of real part of pairing function $Re(f_{\\omega }(\\theta ))=$ \n-Re(f_{\\omega }(\\pi -\\theta ))$ and the terms under integration in (\\re\n{jsmall}) have the same magnitude, but different sign and cancel each other\nduring integration resulting in the absence of a supercurrent. However, in\nthe vicinity of the surface of p-wave superconductor or SN interface the\nsymmetry of $Re(f_{\\omega }(\\theta ))$ on ingoing and outgoing trajectories is broken resulting in generation of a spontaneous supercurrent.\n\n\\section{Spontaneous current \\label{Current}}\n\nIn this Section we present the results of calculations of spatial distributions of the pair potential $\\Delta (x),$, the spontaneous supercurrent $j(x),$ as well as the angular resolved pair\namplitude $f_{1}(x,\\theta)$ and the density of the first spectral component \nj_{1}(x)$ in (\\ref{current})  together with its angular resolved\ndistribution $j_{1}(x,\\theta)$ at $\\omega=\\pi T$ for four different\nsituations.\nThey are specular p-wave surface ($d\\ll \\ell_{e}$, $d\\ll \\xi_{0}$); rough\nsurface ($d\\gg \\ell_{e}$, $d\\ll \\xi_{0}$); clean metallic surface ($d\\geq\n\\xi_{0}$, $\\ell_{e}\\gg \\xi_{0}$) and diffusive metallic surface ($d\\geq\n\\xi_{0}$, $\\ell_{e}\\ll \\xi_{0}$).\n\nAll  calculations have been performed at temperature $T=0.5T_{C}$ and the\nresults are shown in Fig.\\ref{F_Specular}-Fig.\\ref{f_inter}. Numerical analysis shows that at $T=0.5T_{C}$ the calculated value of $j_{1}(x)$ provides the contribution to the full current with accuracy of the order of 10 percent, therefore all the features discussed below reflect the behavior of total spontaneous current.\n\nThe figures\nconsist of several panels. Panel (a) shows the coordinate dependencies of the pair\npotential for a chiral $p_{x}+ip_{y}$ superconductor. Panels (b) and (c)\ndemonstrate the density of the first spectral component $j_{1}(x)$ in (\\re\n{current})  and its angular resolved distribution $j_{1}(x,\\theta)$,\nrespectively. Panels (d) show real (solid lines) and imaginary (dashed lines)\nparts of $f_{1}(x,\\theta)$ calculated at different distances $x\/\\xi_0$ from\nthe SN interface. Color in panels d) specifies the information on the sign\nof $f_{1}(x,\\theta)$ functions: the red color corresponds to the positive\nvalues of $f_{1}(x,\\theta)$, while the blue one represents their negative\nmagnitudes.\n\n\\bigskip \\emph{Specular p-wave surface ($d\\ll \\ell_{e}$, $d\\ll \\xi_{0}$)}.\n\n\\bigskip Figure \\ref{F_Specular} summarize the results obtained for\nspecular p-wave surface under absence of any clean or diffusive N layer on\nthe surface.\n\nIn full agreement with the previously obtained results \\cite{Bakurskiy} in\nthis case bulk pair potential (see Fig\\ref{F_Specular}a) has BCS amplitude\n(at our temperature $\\Delta _{x}=\\Delta _{y}\\approx 1.67T_{C}$) due to\nspherical symmetry. The $\\Delta _{y}$ component increases as we approach the\nsurface and grows up to bulk value for $p_{y}$ symmetry since after\nreflection electrons propagate into the band with the same sign of pair\npotential. The corresponding part of angular resolved distribution of pair\namplitude $f_{1}(x,\\theta)$ (see dashed lines in Fig. \\ref{F_Specular}d)\ndemonstrate a similar behavior. The magnitudes of this imaginary parts of \nf_{1}(x,\\theta)$ only slightly increase with $x$ decrease in the full\naccordance with $\\Delta _{y}$ growth.\n\nContrary to that, in the vicinity of interface $\\Delta _{x}$ is suppressed up\nto zero due to reflection of electrons from the band having positive sign of\npair potential into the band with negative one. The transformations of real\npart of angular resolved distribution of pair amplitude $f_{1}(x,\\theta)$\n(see solid lines in Fig. \\ref{F_Specular}d) is more complex. Starting from\nthe bulk (red curve in Fig. \\ref{F_Specular}d(0)) we first exhibit the\nsuppression of this component with $x$ decrease, as shown in Fig. \\re\n{F_Specular}d(1)) - Fig. \\ref{F_Specular}d(3)). This is a direct consequence\nof the suppression of the $\\Delta _{x}$ component of the order parameter. At\nthe surface the boundary conditions (\\ref{BC1a}), (\\ref{BC1b}) dictate for\nthe outgoing functions $f_{1}(0,\\theta)$ a sign opposite to that existing in\nthe depth of the superconductor. This perturbation relaxes on characteristic\nscale $\\xi=\\xi_{0} \\cos(\\theta)(\\pi T_{C}\/ \\sqrt{\\omega^{2}+|\\Delta |^{2}} )$\nof the equation (\\ref{El_ric_cl_A}), (\\ref{El_ric_cl_B}), which is $\\theta$\ndependent. It is for this reason the sign of $f_{1}(0,\\theta)$ functions\nrecovers the faster the smaller is $\\cos(\\theta)$, that is the larger is\ndeviation of outgoing trajectory from the normal to the superconductor\nsurface direction. This is clearly seen in Fig. \\ref{F_Specular}d(3). It\ndemonstrates that at $x=-0.2\\xi_0$ the sign of functions $f_{1}(-0.2\n\\xi_0,\\theta)$ is different for different $\\theta$. At $x=-0.4\\xi_0$ the\npairing functions are negative in all angle domain. Further decrease of $x$\nresults in increase of magnitude of $f_{1}(x,\\theta)$ and to the full\nrelaxation to the balk values at $x=-2\\xi_0.$\n\nThis lack of symmetry on the trajectories incoming and outgoing from the surface\nleads to generation of spontaneous supercurrent. The angular\nresolved density of its first spectral component $j_{1}(x,\\theta)$ is shown\nin Fig. \\ref{F_Specular}c. In accordance with (\\ref{current}) its\nintegration over $\\theta$ results in nonzero $j_{1}(x),$ which attains a\nmaximum at the surface and monotonically decays into superconductor.\nExactly at the surface plane ($x=0$) the incomig and reflecting electrons make\nequal contributions to $j_{1}(0).$ The component of $j_{1}(x)$ from the\noutgoing electrons decreases monotonically to zero with the decrease of $x$ and\nbecomes negative in the bulk area resulting in full compensation of the\npositive part of $j_{1}(x)$ generated by incoming particles. The maximum of \nj_{1}(x,\\theta)$ dependencies is achieved at $\\theta \\approx \\pi\/4$\nresulting in decay scale of $j_{1}(x)$ of the order of $\\xi_0 \/ \\sqrt{2}.$\n\n\\bigskip \\emph{Rough surface ($d\\gg \\ell_e$ and $d\\ll \\xi_0$.})\n\n\\bigskip  The situation described above changes dramatically when there is\nstrong diffuse scattering of electrons at the free surface of the\nsuperconductor (see Fig. \\ref{f_rough}). Below we model the diffuse\nscattering of electrons by a normal layer placed on the superconductor\nsurface. This layer has thickness $d$ equal to electronic mean free path \n\\ell_e,$ decay length $\\xi_0=3 \\ell_e$ and $\\Delta=0$.\n\nFigure \\ref{f_rough}a shows that in the considered situation in the vicinity\nof SN interface there is suppression of both components of the order\nparameter. The panels Fig.\\ref{f_rough}d(0) - Fig.\\ref{f_rough}d(5)\ndemonstrate that amplitudes of angular resolved distributions of real (solid\nred curves) and imaginary (dashed curves) parts of functions $f_{1}(x,\\theta)\n$ for ingoing trajectories ($-\\pi\/2 \\leq \\theta \\leq \\pi\/2)$ decrease slowly\nthen $x$ goes to SN interface located at $x=0$. In the diffusive layer, the\nimaginary part of $f_{1}(x,\\theta)$ decays rapidly due to averaging of\nits alternating parts.  At $x=d$ (see Fig.\\ref{f_rough}d(8)) the imaginary\npart of $f_{1}(d,\\theta)$ practically disappear and boundary conditions (\\re\n{BC1a}), (\\ref{BC1b}) provide nearly isotropic angular distribution of \nf_{1}(d,\\theta)$. The propogation on outlet trajectories from free surface\nback to SN interface across the diffusive layer results in further\nisotropisation of $f_{1}(x,\\theta)$, as it follows from Fig.\\ref{f_rough\nd(5).\n\nDuring the propagation into the S film (in the angular domain $\\pi\/2 \\leq x\n\\leq 3\\pi\/2$) the isotropic function $f_{1}(0,\\theta)$ distribution rapidly vanished\ndue to strong nucleation of the imaginary component of $f_{1}(x,\\theta)$. As\nit follows from Fig.\\ref{f_rough}d(4) - Fig.\\ref{f_rough}d(2) it recovers\nits bulk distribution practically at $x=-0.5\\xi_0$. Evolution of real part\nof $f_{1}(x,\\theta)$ is nearly the same as discussed in the previous\nparagraph. First it goes to zero with $x$ decrease, then it changes its sign and\ntends monotonically to its bulk behavior.\n\nFigures \\ref{f_rough}b and Fig. \\ref{f_rough}c show the spatial variation of the\nfirst spectral component $j_{1}(x)$ and its angular resolved spectral\ncomponent $j_{1}(x,\\theta)$ in the considered regime. It is seen that\nspontaneous supercurrent is pushed out from the N layer towards the SN\ninterface. It reaches its maximum at the SN boundary and then decreases with\nthe distance deep into the superconductor. Note, that there are two\ncharacteristic scales in this decay. In order to understand these phenomena\nit is enough to look at Fig. \\ref{f_rough}c and use the expression (\\re\n{jsmall}). Expression (\\ref{jsmall}) reads that in the N layer and in a\nvicinity of SN interface the main contribution to $j_{1}(x)$ provided by the\nproduct of isotropic real part of of $f_{1}$ ($p_{x}$ component) on outgoing\ntrajectories and imaginary part of $f_{1}$ ($p_{y}$ component) on ingoing\ntrajectories. The latter is strongly suppressed in the N film (see Fig. \\re\n{f_rough}d(6) - Fig. \\ref{f_rough}d(8)) resulting in the current\nsuppression. At the SN interface, the $p_{y}$ component of $f_{1}$ on ingoing\ntrajectories still exists. It achieves maximum values at $\\theta \\approx\n70^{o}$ thus providing the maximum in the angular resolved spectral\ncomponent $j_{1}(x,\\theta)$ at $\\theta \\approx 75^{o}$. Thus, in the\nvicinity of the SN boundary main contribution to the spontaneous current\ncomes from trajectories nearly parallel to interface. The characteristic\ndecay length of the anomalous functions on these trajectories is \n\\xi_{0}\\cos(75^{o})$, that is much smaller than $\\xi_{0}.$ Further deviation\nfrom the SN boundary is accompanied by deformations in the $f_{1}(x,\\theta)$,\nwhich will eventually lead to recovery of $f_{1}(x,\\theta)$ to the bulk one.\nThese modifications are accompanied by reconstruction of $j_{1}(x,\\theta)$ and by displacement of the\nposition of its maximum to angles close to $\\pi\/4$. As a result, for\nsmaller $x$ the decay length of $j_{1}(x)$  goes to the same value \n\\xi_{0}\\cos(\\pi\/4)$ that has been found for the specular superconductor\ninterface.\n\nIn the case where the thickness of the N layer becomes comparable with the decay length $\\xi_0$ there is\nsignificant attenuation of superconducting correlations in this layer\nresulting in strong suppression of superconductivity at outgoing from SN\ninterface trajectories. This, in turn, leads to further displacement of\nspontaneous supercurrents into the bulk region. We will demonstrate this\neffect below by considering two examples; \"clean metallic surface\"\nand \"diffusive metallic surface\". In both cases we will assume that there is\nno intrinsic superconductivity in the N metal and that its thickness and electronic\nmean free path are $d=\\xi_0$, $l_e=\\infty$ in the clean case and $d= l_e\n$ and $\\xi_0=0.3 l_e$ in the diffusive case.\n\n\\bigskip\n\n\\emph{Clean metallic surface ($d\\geq \\xi_{0}$, $\\ell_{e}\\gg \\xi_{0}$).}\n\n\\bigskip\n\nFigure \\ref{f_clean} summarizes the results obtained for the clean metallic\nsurface. The panels Fig.\\ref{f_clean}d(0) - Fig.\\ref{f_clean}d(5)\ndemonstrate that amplitudes of angular resolved distributions of real (solid\nred curves) and imaginary (dashed curves) parts of functions $f_{1}(x,\\theta)\n$ for inlet trajectories ($-\\pi\/2 \\leq \\theta \\leq \\pi\/2)$ behaves nearly in\nthe same manner as in the cases considered previously. They decrease slowly\nthen $x$ goes to SN interface located at $x=0$. Propagation into the N layer\nleads to full suppression of anomalous functions and they are zero in outlet\ntrajectories at SN interface (see Fig.\\ref{f_clean}d(5)). It is for this\nreason the spontaneous supercurrent is zero in the area $0\\leq x \\leq d$.\nFrom Fig.\\ref{f_clean}d(0) - Fig.\\ref{f_clean}d(4.5) it is seen that\nimaginary part of anomalous functions recover faster compare to real one.\nThis imbalance provides nucleation of spontaneous currents. Their density\nincreases (see Fig.\\ref{f_clean}b) and achieves its maximum at $x\\approx-0.5 \\xi_0\n. At this point (see Fig.\\ref{f_clean}d(2)) the imaginary part of anomalous\nfunctions saturates at the bulk value. With further decrease of $x$ the magnitude of the real part of $f_1$ increases and defines the\nsupercurrent component, which compensates the one from inlet electrons. As a\nresult, the density of spontaneous current starts to decrease for $x< -0.5\n\\xi_0$ and goes to zero in the bulk. Figure \\ref{f_clean}c shows that\nmaximum in spectral component $j_{1}(x,\\theta)$ is achieved at $\\theta\\approx\n\\pi\/4$. It means that characteristic scale of $j_{1}(x)$ variation is $\\xi_0\n\\cos(\\pi\/4))$, similar to the case of specular p-wave\nsuperconductor interface.\n\n\\bigskip  \\emph{Diffusive metallic surface ($d\\geq \\xi_{0}$, $\\ell_{e}\\ll\n\\xi_{0}$).}\n\n\\bigskip\n\nThe difference between the diffusive metallic interface from the clean one\nlies in the fact that due to the presence of electron scattering centers\nnear the border, some of electrons can be reflected back at small distances\nfrom the SN interface. This means that in contrast to the clean limit\nconsidered previously, there is a probability for nucleation of small isotropic\ndiffuse component in $f_1 (0, \\theta)$ distribution. Its presence would lead\nto the appearance of a diffuse current peak at $x=0$ (see Fig. \\ref{f_inter\nb). The angular resolved density of spontaneous supercurrent $j_{1}(x,\\theta)\n$ (see Fig. \\ref{f_inter}c) clearly demonstrates the existence of two\ncharacteristic angles. It has one maximum at $\\theta \\approx 75^{o}$\noriginated by diffusive processes and the second one at $\\theta \\approx \\pi\/4$.\nThis leads to the existence of two characteristic scales in $j_1 (x)$\ndependence. They are the \"diffusion scale\" $\\xi_{d}=\\xi_0 \\cos(75^{o})$ and\n\"clean scale\" $\\xi_{c}=\\xi_0 \\cos(\\pi\/4).$ It is seen from Fig. \\ref{f_inter\nb) that associated with this scales maximums in $j_1 (x)$ dependence can be\nresolved due to large difference between $\\xi_{d}$ and $\\xi_{c}$ and the\nshift of order of $0.5 \\xi_{0}$ between the space positions of this\npeculiarities.\n\n\\section{Discussion \\label{Discussion}}\n\nOur studies provide the detailed analysis of the mechanisms leading to the formation of spontaneous currents at the surface of a chiral p-wave superconductor in terms of the Ricatti functions.\n From the structure of the expressions (\\ref{current})-(\\ref{jsmall}), which determine the magnitude and direction of spontaneous currents it follows that these currents  depend on the products of the real and imaginary parts of the anomalous Green's functions on the incoming and outgoing trajectories from the border. These parts strongly depends\n on the requirements of the symmetry of the Green's functions, which, in turn, dictate the form of the boundary conditions on the open surface. For d$_{x^2-y^2}$-wave superconductors, $b_{\\pm }(x,\\theta )=a_{\\mp }(x,\\theta ),$ holds and  the conditions (\\ref{BC1a}), (\\ref{BC1b}) are reduced to the\nrelation $b_{+}(d,-\\theta )=a_{+}(d,\\theta ),$ which under the circumstances that lead to  suppression of the order parameter down to zero at the free interface\n  simultaneously ensures the vanishing of anomalous functions on that surface   \\cite{Golubov2}. \n\nContrary to that, in the considered situation vanishing of $\\Delta_x$ order parameter component is not accompanied by vanishing of anomalous functions at a specular surface, see the plot Fig. \\ref{F_Specular}d(5).  Moreover, due to the conditions (\\ref{BC1a}), (\\ref{BC1b}) the sign of the $f_1 (\\theta)$ functions on outgoing directories is opposite to that in the bulk. It is exactly the mechanism that results in nucleation of spontaneous current at free mirror surface.  Figure 1 gives that normalized amplitude of the first item in the expression for the supercurrent is slightly larger that $0.4.$\n\n  At the rough surface, the redistribution of current density takes place: it is pushed towards the border between the clean and the dirty regions (see Fig. \\ref{f_rough}). It is also seen from Fig. \\ref{f_rough}a that $\\Delta_x$ component at this interface is larger than that at specular surface (see Fig. \\ref{F_Specular}a), while  $\\Delta_y$ is smaller. Due to these competing processes the peak value of the $j_1$ component to the current at $x=0$ is only slightly reduced compared to $j_1 (0)$ calculated in the previous case.\n\n\nThe situation changes drastically when the thickness of the normal layer on the top of p-wave superconductor exceeds $\\xi_0$ irrespective of the degree of purity of the normal metal. As follows from Fig. \\ref{F_Specular}b and Fig. \\ref{f_inter}b, in both considered cases there is nearly one order of magnitude suppression of maximum value of $j_1$ compare to $j_1 (0)$ at specular interface. This result is correlated with that obtained in \\cite{Lederer} in the tight-binding model for the clean N metal. It is also seen from Fig. \\ref{F_Specular}b and Fig. \\ref{f_inter}b that spatial distribution of the density of spontaneous current for the clean and dirty cases are completely different. \n\nThe difference in the current distributions considered above can be accessed experimentally by muon spin rotation technique \\cite{Flokstra}. According to the results of our study, such measurements of surface currents may provide important information on the underlying pairing symmetry in the bulk.\n\nAcknowledgements. The authors acknowledge fruitful discussions with Y. Asano and Y. Tanaka. This work was partially supported by RFBR-JSPS grants 15-52-50054, 17-52-50080, RFBR grant 16-29-09515-ofi-m, by Russian Science Foundation, Project No. 15-12-30030, and by Ministry of Education and Science of the Russian Federation, grants MK-5813.2016.2 and 14Y26.31.0007.\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\n\\begin{figure*}[h]\n\\begin{minipage}[h]{0.5\\linewidth}\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{Del_mirror}} \\\\\n\\vspace{-2 mm}\n\\raggedright{a)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{j1_mirror}} \\\\\n\\vspace{-2 mm}\n\\raggedright{b)}\n\\end{minipage}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.45\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{ang_mirror}} \\\\\n\\vspace{-8 mm}\n\\raggedright{c)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{\\linewidth}\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_0}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(0)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_1}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(1)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_2}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(2)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_3}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(3)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_4}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(4)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_5}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(5)}}\n\\end{minipage}\n\\end{minipage}\n\\raggedright{d)}\n\\caption{(Color Online) Spatial distribution of set a) - d) of parameters in\np-wave superconductor with specular surface: \\newline\na) the pair potentials $\\protect\\Delta_x$ and $\\protect\\Delta_y$ and b) the\nsurface current density $j_1$, c) contribution of particles with angle \n\\protect\\theta$ in the formation of surface current $j_1$ at the different\npoints of the structure, d) angle dependent pair amplitude $f_1(\\protec\n\\theta)$ at the different points (0)-(5) of the structure. At the panel d)\nsolid lines correspond to the real part of $f$ and dashed lines correspond\nto the imaginary part of $f $; red color means positive value and blue color\nmeans negative one. Set of the points is following: (0) $x = -2 \\protect\\x\n_0 $, (1) $x = -0.8 \\protect\\xi_0 $, (2) $-0.4 \\protect\\xi_0 $, (3) $-0.2\n\\protect\\xi_0 $, (4) $-0.1 \\protect\\xi_0 $, (5) - $x = 0 \\protect\\xi_0 $.\nAll calculations were performed at $T=0.5 T_C$ and $d=0$.}\n\\label{F_Specular}\n\\end{figure*}\n\n\n\\begin{figure*}[h]\n\\begin{minipage}[h]{0.5\\linewidth}\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{Del_rough}} \\\\\n\\vspace{-2 mm}\n\\raggedright{a)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{j1_rough}} \\\\\n\\vspace{-2 mm}\n\\raggedright{b)}\n\\end{minipage}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.45\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{ang_rough}} \\\\\n\\vspace{-8 mm}\n\\raggedright{c)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{\\linewidth}\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_0}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(0)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_1}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(1)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_2}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(2)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_3}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(3)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_4}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(4)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_5}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(5)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_6}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(6)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_7}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(7)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_8}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(8)}}\n\\end{minipage}\n\\end{minipage}\n\\raggedright{d)}\n\\caption{(Color Online) Spatial distribution of set a) - d) of parameters in\np-wave superconductor with rough surface: \\newline\na) the pair potentials $\\protect\\Delta_x$ and $\\protect\\Delta_y$ and b) the\nsurface current density $j_1$, c) contribution of particles with angle \n\\protect\\theta$ in the formation of surface current $j_1$ at the different\npoints of the structure, d) angle dependent pair amplitude $f_1(\\protec\n\\theta)$ at the different points (0)-(8) of the structure. Set of the points\nis following: (0) $x = -2 \\protect\\xi_0 $, (1) $-0.8 \\protect\\xi_0 $, (2) \n-0.4 \\protect\\xi_0 $, (3) $-0.2 \\protect\\xi_0 $, (4) $-0.1 \\protect\\xi_0 $,\n(5) $0 \\protect\\xi_0 $, (6) $0.1 \\protect\\xi_0 $, (7) $0.2 \\protect\\xi_0 $,\n(8) the surface $x = 0.33 \\protect\\xi_0 $. All calculations were performed\nat $T=0.5 T_C$, $d= 1 l_e$ and $\\protect\\xi_0=3 l_e$ .}\n\\label{f_rough}\n\\end{figure*}\n\n\n\\begin{figure*}[h]\n\\begin{minipage}[h]{0.5\\linewidth}\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{Del_clean}} \\\\\n\\vspace{-2 mm}\n\\raggedright{a)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{j1_clean}} \\\\\n\\vspace{-2 mm}\n\\raggedright{b)}\n\\end{minipage}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.45\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{ang_clean}} \\\\\n\\vspace{-8 mm}\n\\raggedright{c)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{\\linewidth}\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_0}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(0)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_1}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(1)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_2}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(2)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_3}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(3)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_4}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(4)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_4p5}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(4.5)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_5}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(5)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_6}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(6)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_7}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(7)}}\n\\end{minipage}\n\n\\end{minipage}\n\\raggedright{d)}\n\\caption{(Color Online) Spatial distribution of set a) - d) of parameters in\np-wave superconductor with clean metallic surface: \\newline\na) the pair potentials $\\protect\\Delta_x$ and $\\protect\\Delta_y$ and b) the\nsurface current density $j_1$, c) contribution of particles with angle \n\\protect\\theta$ in the formation of surface current $j_1$ at the different\npoints of the structure, d) angle dependent pair amplitude $f_1(\\protec\n\\theta)$ at the different points (0)-(7) of the structure. Set of the points\nis following: (0) $x = -2 \\protect\\xi_0 $, (1) $-0.8 \\protect\\xi_0 $, (2) \n-0.4 \\protect\\xi_0 $, (3) $-0.2 \\protect\\xi_0 $, (4) $-0.1 \\protect\\xi_0 $,\n(4.5) $-0.03 \\protect\\xi_0 $, (5) $0 \\protect\\xi_0 $, (6) $0.25 \\protect\\x\n_0 $, (7) $0.5 \\protect\\xi_0 $. All calculations were performed at $T=0.5\nT_C $, $d=\\protect\\xi_0$ and $l_e=\\infty$ .}\n\\label{f_clean}\n\\end{figure*}\n\n\n\\begin{figure*}[h]\n\\begin{minipage}[h]{0.5\\linewidth}\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{Del_inter}} \\\\\n\\vspace{-2 mm}\n\\raggedright{a)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{j1_inter}} \\\\\n\\vspace{-2 mm}\n\\raggedright{b)}\n\\end{minipage}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.45\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{ang_inter}} \\\\\n\\vspace{-8 mm}\n\\raggedright{c)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{\\linewidth}\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_0}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(0)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_1}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(1)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_2}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(2)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_3}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(3)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_4}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(4)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_5}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(5)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_6}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(6)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_7}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(7)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_8}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(8)}}\n\\end{minipage}\n\n\\end{minipage}\n\\raggedright{d)}\n\\caption{(Color Online) Spatial distribution of set a) - d) of parameters in\np-wave superconductor with diffusive metallic surface: \\newline\na) the pair potentials $\\protect\\Delta_x$ and $\\protect\\delta_y$ and b) the\nsurface current density $j_1$, c) contribution of particles with angle \n\\protect\\theta$ in the formation of surface current $j_1$ at the different\npoints of the structure, d) angle dependent pair amplitude $f_1(\\protec\n\\theta)$ at the different points (0)-(8) of the structure. Set of the points\nis following: (0) $x = -2 \\protect\\xi_0 $, (1) $-0.8 \\protect\\xi_0 $, (2) \n-0.4 \\protect\\xi_0 $, (3) $-0.2 \\protect\\xi_0 $, (4) $-0.1 \\protect\\xi_0 $,\n(5) $0 \\protect\\xi_0 $, (6) $0.25 \\protect\\xi_0 $, (7) $0.5 \\protect\\xi_0 $,\n(8) $x = 1 \\protect\\xi_0 $. All calculations were performed at $T=0.5 T_C$, \nd= l_e$ and $\\protect\\xi_0=0.3 l_e$.}\n\\label{f_inter}\n\\end{figure*}\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe relativistic multipole moments of asymptotically flat spacetimes have been defined by Geroch \\cite{geroch} for static spacetimes and then generalised to \nthe stationary case by Hansen \\cite{hansen}. Together with Beig's \\cite{beigAPA} generalised definition of centre of mass this gives\na coordinate independent description of all asymptotically flat stationary spacetimes. \n\nHowever, the tensorial recursion which defines the multipole moments \\eqref{orgrec}, is computationally rather complicated as it stands. In the axisymmetric (static or stationary) case, on the other hand, it was shown,\n\\cite{backdahl1}, \\cite{backdahl2} that the recursion can be replaced\nby a scalar recursion on $\\mathbb{R}$, and that all the moments can be collected\ninto one complex valued function $y$ on $\\mathbb{R}$, where the moments are given by the\nthe derivatives if $y$ at $0$.\n\nIn the general case, the multipole of order $2^n$ has $2n+1$ degrees of freedom, as compared to one degree of freedom in the axisymmetric case. Therefore, apart from the technical problems, it is not obvious what form a generalisation to the general case should take. In this paper, we show that also in the general stationary case, the recursion\n\\eqref{orgrec} can be simplified to a scalar recursion, this time on\n$\\mathbb{R}^2$. This is shown using normal coordinates, complex null geodesics, and exploiting the extra conformal freedom of the conformal compactification. \n\nUsing this simplification we can partially confirm an extension of a long standing conjecture by Geroch \\cite{geroch}:\n\\begin{quote}\nGiven any set of multipole moments, subject to the appropriate convergence condition, \nthere exists a static solution of Einstein's equations having precisely those moments.\n\\end{quote}\nThis conjecture has its natural extension to the stationary case.\n\nIn this paper we will state the appropriate convergence condition in the general stationary case,\ni.e., we will prove that this condition is necessary for existence of a stationary solution to Einstein's equations. \n\n\\section{Multipole moments of stationary spacetimes} \\label{tmoments}\nIn this section we quote the definition of multipole moments given by Hansen in \\cite{hansen}, which is an extension \nto stationary spacetimes of the definition by Geroch \\cite{geroch}.\nWe thus consider a stationary spacetime $(M, g_{ab})$ with timelike Killing vector field $\\xi^a$.\nWe let $\\lambda=-\\xi^a\\xi_a$ be the norm, and define the twist $\\omega$ through \n$\\nabla_a\\omega=\\epsilon_{abcd}\\xi^b\\nabla^c\\xi^d$.\nIf $V$ is the 3-manifold of trajectories, the metric $g_{ab}$ (with signature $(-,+,+,+)$) induces the \npositive definite metric\n$$h_{ab}=\\lambda g_{ab}+\\xi_a\\xi_b$$ on $V$.\nIt is required that $V$ is asymptotically flat, i.e.,  there exists a 3-manifold\n$\\hat V$ and a conformal factor $\\Omega$ satisfying\n\\begin{itemize}\n\\item[(i)]{$\\hat V = V \\cup \\Lambda$, \\; where $\\Lambda$ is a single point}\n\\item[(ii)]{$\\hat h_{ab}=\\Omega^2 h_{ab}$ is a smooth metric on $\\hat V$}\n\\item[(iii)]{At $\\Lambda$, $\\Omega=0, \\hat D_a \\Omega =0, \\hat D_a \\hat D_b \\Omega = 2 \\hat h_{ab}$,}\n\\end{itemize}\nwhere $\\hat D_a$ is the derivative operator associated with $\\hat h_{ab}$.\nOn $M$, and\/or $V$ one defines the scalar potential\n$$\\phi=\\phi_M+i\\phi_J, \\quad \\phi_M=\\frac{\\lambda^2+\\omega^2-1}{4\\lambda}, \\,\\phi_J=\\frac{\\omega}{2\\lambda}.$$\nThe multipole moments of $M$ are then defined on $\\hat V$ as certain  \nderivatives of the scalar\npotential $\\hat \\phi=\\phi\/\\sqrt \\Omega$ at $\\Lambda$. More  \nexplicitly, following \\cite{hansen}, let $\\hat R_{ab}$ denote\nthe Ricci tensor of $\\hat V$, and let $P=\\hat \\phi$. Define the  \nsequence $P, P_{a_1}, P_{a_1a_2}, \\ldots$\nof tensors recursively:\n\\begin{equation} \\label{orgrec} P_{a_1 \\ldots a_n}=C[\n\\hat D_{a_1}P_{a_2 \\ldots a_n}-\n\\tfrac{(n-1)(2n-3)}{2}\\hat R_{a_1 a_2}P_{a_3 \\ldots a_n}],\n\\end{equation}\nwhere $C[\\ \\cdot \\ ]$ stands for taking the totally symmetric and  \ntrace-free part. The multipole moments\nof $M$ are then defined as the tensors $P_{a_1 \\ldots a_n}$ at  \n$\\Lambda$. The requirement that all $P_{a_1 \\ldots a_n}$ be totally symmetric and\ntrace-free makes the actual calculations very cumbersome.\n\nIn \\cite{beigPRSL}, \\cite{kundu} it was shown that (when the mass is non-zero) there exist a conformal \nfactor $\\Omega$ and a chart, such that all components of the metric $\\hat h_{ab}$ and the potential $\\hat \\phi$ are analytic in terms of the \ncoordinates, in a neighbourhood of the infinity-point. Expressed in these coordinates, the exponential map becomes analytic. \nTherefore, we can use Riemannian normal coordinates and still have analyticity of the metric components and the potential.\nIf the mass is zero, this analyticity condition will be assumed. Thus, henceforth we assume that $\\Omega$ is chosen such that\nthe (rescaled) metric and potential are analytic in a neighbourhood of $\\Lambda$.\n\n\\section{Multipole moments through a scalar recursion on $\\mathbb{R}^2$}\n\\label{momentsscalar}\nSuppose that $(x^1, x^2, x^3)=(x, y, z)$ are normal coordinates (with respect to \n$\\hat h_{ab}$) centred around $\\Lambda$.\nThis means that for any constants $a=a^1, b=a^2 , c=a^3$ the curve $t \\to (a t,b t,c t)$ is a geodesic,\ni.e. in terms of coordinates that\n$$ \\ddot x^i+\\Gamma^i_{kl}\\dot x^k \\dot x^l=\\Gamma^i_{kl}a^k a^l=0$$\nwhere the Christoffel symbols are evaluated at $(a t,b t,c t)$ for appropriate $t$. Due to analyticity,\nthis relation holds for complex values of $a,b,c$, i.e., we can consider geodesics in the complexification\n$\\hat V_\\mathbb{C}$ of $\\hat V$. Of particular interest is the one-parameter family of curves:\n$$\\gamma_\\varphi: t \\to (t \\cos\\varphi,t \\sin \\varphi, i t),\n \\qquad t \\in [0,t_0), \\varphi\\in [0,2\\pi)$$\n for some suitable $t_0$. The tangent vector $\\eta^a=\\eta^a_\\varphi(t)=\n\\cos\\varphi (\\frac{\\partial}{\\partial x^1})^a+\\sin\\varphi (\\frac{\\partial}{\\partial x^2})^a+i (\\frac{\\partial}{\\partial x^3})^a$ is seen to be a complex null vector along $\\gamma_\\varphi$.\nNamely, from $\\eta^a \\hat D_a \\eta^b=0$, we infer that $\\eta^a \\hat D_a (\\eta^b \\eta_b)=0$.\nThe (constant) value of $\\eta^b \\eta_b$ is then found to be $0$ by evaluation at $t=0$.\n \nNext, consider the mapping $F: \\mathbb{R}^2 \\to V_\\mathbb{C}: \n(\\xi,\\zeta) \\to (\\xi,\\zeta,i \\sqrt{\\xi^2+\\zeta^2})$. We let $S$ denote the 2-surface $F(U) \\subset \\hat V_\\mathbb{C}$, where $U \\subset \\mathbb{R}^2$ is a suitable neighbourhood of $(\\xi,\\zeta)=(0,0)$. \n$S$ is then a smooth surface, except at $\\Lambda$ where it has a vertex point, closely resembling\na null cone in a three dimensional Lorentzian space.\nThe curves $\\gamma_\\varphi$ are given by $\\gamma_\\varphi(t)=F(t\\cos\\varphi,t\\sin\\varphi)$, and\nin particular $\\eta^a$ lies along $S$.\nThis suggests that we use the polar coordinates $\\rho,\\varphi$ around $\\Lambda$ on $S$ defined via \n$\\xi=\\rho\\cos\\varphi$, $\\zeta=\\rho\\sin\\varphi$.\nWe now follow the approach from \\cite{backdahl2}, where a useful vector field $\\eta^a$ on $\\hat V$ was introduced. \nIn \\cite{backdahl2}, where the spacetime was axisymmetric, $\\eta^a$ was explicitly expressed in \nterms of the metric cast in the Weyl-Papapetrou form \\cite{wald}, and was defined on the whole of $\\hat V$ except on the symmetry axis. \nIn this paper the axisymmetry condition is dropped, which makes the construction of a corresponding $\\eta^a$ more difficult. \nIn addition, a general spacetime has $2n+1$ degrees of freedom for the multipole moment of order $2^n$, \ncompared to one degree of freedom in the axisymmetric case \\cite{herb}.\nNevertheless, it will turn out to be sufficient to know the potential $\\hat \\phi$ on $S$ to determine all the moments. \nOn $S$, $\\eta^a$ has the following properties.\n\n\\begin{lemma}\\label{etalemma}\nSuppose $\\hat V$ and $S$ are defined as above. Then there exists a\nregularly direction dependent (at $\\Lambda$) vector field $\\eta^a$ on $S$ with the following properties:\\\\\na) $\\eta^a  \\hat D_a \\eta^b$ is parallel to $\\eta^b$.\\\\\nb) For all tensors $T_{a_1 \\ldots a_n}$, $\\eta^{a_1} \\ldots\n\\eta^{a_n}T_{a_1 \\ldots a_n}=\\eta^{a_1} \\ldots \\eta^{a_n}C[T_{a_1\n  \\ldots a_n}]$,\\\\\nc) At $\\Lambda$, $P_{a_1 \\ldots a_n}$ (in $\\hat V$) is determined by \n$\\eta^{a_1} \\ldots \\eta^{a_n}P_{a_1 \\ldots a_n}$ (on $S$)\n\\end{lemma}\n\n\\begin{proof}\na) was demonstrated above, b) follows as in \\cite{backdahl2}, while c) requires a different argument.\nA totally symmetric and trace-free tensor $P_{a_1\\dots a_n}$ has $2n+1$ degrees of freedom, and\nin Cartesian coordinates $(x,y,z)$ it can be expressed via the components \n$P\\underbrace{{}_{x  \\ldots\\ldots x}}_{j}\\underbrace{{}_{y ..\\ldots\\ldots y}}_{n-j-1}{}_z$\nand $P\\underbrace{{}_{x\\ldots\\ldots x}}_{j}\\underbrace{{}_{y\\ldots\\ldots y}}_{n-j}$.\n\\footnote{In brief, any index occurrence of several z's can be removed via $P_{zz\\dots}+P_{xx\\dots}+P_{yy\\dots}=0$. }\nTherefore, at $\\Lambda$, we can write\n\\begin{equation}\n\\label{reconstrtensor}\n\\begin{split}\nP_{a_1\\dots a_n}=\n\\sum_{j=0}^n{a_{j}C[(dx)_{a_1}\\dots(dx)_{a_j}(dy)_{a_{j+1}}\\dots(dy)_{a_{n}}]}\\\\\n+\\sum_{j=0}^{n-1}{b_{j}C[(dx)_{a_1}\\dots(dx)_{a_j}(dy)_{a_{j+1}}\\dots(dy)_{a_{n-1}}(dz)_{a_{n}}]}\n\\end{split}\n\\end{equation}\nContracting with $\\eta^{a_1}\\dots\\eta^{a_n}$, and using lemma \\ref{etalemma}b we find that\n\\begin{equation}\\label{trigpol}\n\\eta^{a_1}\\dots\\eta^{a_n}P_{a_1\\dots a_n}=\\sum_{j=0}^n{a_{j}\\cos^j\\varphi\\sin^{n-j}\\varphi}\n+i\\sum_{j=0}^{n-1}{b_{j}\\cos^j\\varphi\\sin^{n-1-j}\\varphi}\n\\end{equation}\nIf the left hand side is zero, the trigonometric polynomial to the right must be identically zero.\nThis means that all the coefficients $a_n$ and $b_n$ are zero, and by \\eqref{reconstrtensor}\nthat $P_{a_1\\dots a_n}$ is zero. In particular the $2n+1$ components in the RHS of\n\\eqref{reconstrtensor} are linearly independent. This proves c).\n\\end{proof}\nNote that although the moments are encoded in the coefficients $a_n$ and $b_n$, this encoding\nis dependent on the choice of normal coordinates, i.e., the orientation of the coordinate axes in $T_\\Lambda \\hat V_\\mathbb C$.\n\nWe can now replace the recursion \\eqref{orgrec} on  $\\hat V$ with a scalar recursion on $S$.\nAgain, we follow \\cite{backdahl2}, and define\n\\begin{equation} \\label{fn}\nf_n=\\eta^{a_1}\\eta^{a_2}\\dots\\eta^{a_n}P_{a_1a_2\\dots a_n}, \\ n=0,1,2,\\ \\ldots\n\\end{equation}\non $S$. In particular, $f_0=P=\\hat\\phi=\\phi\/\\sqrt{\\Omega}$. The moments \n$P_{a_1a_2\\dots a_n}(\\Lambda)$ will now be encoded in the trigonometric polynomial\ngiven by the direction dependent limit\n$\\lim_{\\rho \\to 0}f_n(\\rho,\\phi)$, which takes the form in \\eqref{trigpol}. See also lemma\n\\ref{Slemma}.\nNote that although $P_{a_1a_2\\dots a_n}$ is analytic on $\\hat V$, $f_n$ will not\nbe analytic in terms of $\\xi,\\eta$ since $\\eta^a$ is direction dependent at $\\Lambda \\in S$.\nIn general we have the following lemmas:\n\\begin{lemma}\\label{Slemma1}\nSuppose that $f$ is an analytic function,\non a ball of radius $r_0$ around $\\Lambda$ on $\\hat V$. \nThen the restriction of $f$ to $S$, $f_L$,  can be decomposed as $f_L(\\xi,\\zeta)=f_1(\\xi,\\zeta)+i\\rho\nf_2(\\xi,\\zeta)$, where $f_1$ and $f_2$ are analytic in terms of $\\xi$ and $\\zeta$ \non the disk $\\xi^2+\\zeta^2<\\frac{r_0^2}{2}$, and where $\\rho=\\sqrt{\\xi^2+\\zeta^2}$.\nFurthermore, if $f$ is real-valued then $f_1$ and $f_2$ are real-valued.\n\\end{lemma}\n\\begin{proof}\nWe start by splitting $f=f(x,y,z)$ into its even, $f_e$, and odd, $f_o$, part with respect to $z$.\nWe can now rewrite $f_e(x,y,z)=\\tilde f_e(x,y,z^2)$ and $f_o(x,y,z)=z \\tilde f_o(x,y,z^2)$,\nwhere both $\\tilde f_e$ and $\\tilde f_o$ are analytic in their arguments (at least near \n$(0,0,0)$). The restriction of $\\tilde f_e$ to $S$ gives $f_1$: $f_1(\\xi,\\zeta)=\\tilde f_e(\\xi,\\zeta,-(\\xi^2+\\zeta^2))$,\nwhile the restriction of $\\tilde f_o$ gives $f_2$: $f_2(\\xi,\\zeta)=\\tilde f_o(\\xi,\\zeta,-(\\xi^2+\\zeta^2))$. Adding these,\nand also noting that $z \\to i \\rho$ gives the required decomposition.\nOn $S$ we have $|z|^2=\\xi^2+\\zeta^2$, hence $\\xi^2+\\zeta^2< \\frac{r_0^2}{2}$ implies $x^2+y^2+z^2 < r_0^2$.\nThis gives the domain of analyticity. The reality follows from the construction.\n\\end{proof}\n\nRemark: In $f_L$, the subscript $L$ stands for the 'leading term'.\n\nAlthough tensor fields on $\\hat V$ can be pulled back to $S \\setminus \\{\\Lambda\\}$, we will only need their\ncontractions with the appropriate number of $\\eta^a$ vectors. This contraction will introduce a direction dependence\nwhich shows up in the following lemma.\n\n\\begin{lemma}\\label{Slemma}\nSuppose $T_{a\\ldots b}$ is an analytic tensor field on a ball of radius $r_0$ around $\\Lambda$ on $\\hat V$. \nThen the scalar field\n$f_L=\\eta^a \\ldots \\eta^b T_{a\\ldots b}$ on $S$ can be written as $f_L(\\xi,\\zeta)=\n\\frac{1}{\\rho^n}(f_1(\\xi,\\zeta)+i\\rho f_2(\\xi,\\zeta))$, where $f_1$ and $f_2$ are analytic \n(on the disk with radius $\\frac{r_0}{\\sqrt{2}}$ around the origin) in terms of $\\xi$ and $\\zeta$,\nand where $\\rho=\\sqrt{\\xi^2+\\zeta^2}$.\nFurthermore, if $T_{a\\ldots b}$ is real-valued then $f_1$ and $f_2$ are real-valued.\n\\end{lemma}\n\\begin{proof}\nConsider the scalar field $g=x^a \\ldots x^b T_{a\\ldots b}$ on $\\hat V$, where \n$$\nx^a=x\\left ( \\frac{\\partial}{\\partial x}\\right )^a+y\\left ( \\frac{\\partial}{\\partial y}\\right )^a+z\\left ( \\frac{\\partial}{\\partial z}\\right )^a .\n$$\nThis scalar field is analytic on the same ball as $T_{a\\ldots b}$. Hence we can use lemma \\ref{Slemma1} \nto obtain analytic functions $f_1$ and $f_2$ such that $g_L=f_1+i\\rho f_2$. The reality also follows from lemma \\ref{Slemma1}.\nFurthermore, $x^a=\\rho\\eta^a$ on $S$. Thus $g_L=\\rho^n f_L$. But $f_L$ is bounded near the origin, \nthus we can divide $g_L$ by $\\rho^n$ and get the lemma.\n\\end{proof}\nWe remark that the boundedness of  $f_L(\\xi,\\zeta)=\n\\frac{1}{\\rho^n}(f_1(\\xi,\\zeta)+i\\rho f_2(\\xi,\\zeta))$ when $\\rho \\to 0$ implies that\nboth $f_1$ and $f_2$ have zeros of sufficient order at $(\\xi,\\zeta)=(0,0)$. It also implies\nthat $f_L$ will be direction dependent there.\n\nWe can now contract \\eqref{orgrec}\nwith $\\eta^a$ and get the following theorem.\n\\begin{theorem}\\label{scalrarrec2d}\nLet $\\hat V$ and $S$ be defined as in sections (\\ref{tmoments}) and (\\ref{momentsscalar}).\nLet $\\eta^a$ have the properties given by lemma \\ref{etalemma}, and let $f_n$\nbe defined by \\eqref{fn}.Then the recursion \\eqref{orgrec} on $\\hat V$ takes the form\n\\begin{equation}\\label{scalarrec}\nf_n=\\eta^a\\hat D_af_{n-1}\n-\\tfrac{(n-1)(2n-3)}{2}\\eta^a\\eta^b\\widehat R_{ab}f_{n-2}\n\\end{equation}\non $S$. The moments of order $2^n$ are captured in the direction dependent limit \n$\\lim_{\\rho \\to 0} f_n(\\rho,\\varphi)$.\n\\end{theorem}\n\\begin{proof}\nThat \\eqref{orgrec} takes the form \\eqref{scalarrec} \n follows exactly as in \\cite{backdahl2} using that $\\eta^a\\hat D_a \\eta^b=0$, although the recursion is defined only on $S$ rather than on $\\hat V$. The last statement is the content of\n lemma \\ref{etalemma}c.\n\\end{proof}\n\n\\subsection{Simplified calculation of the moments}\nIn this section, we will show that it is possible to obtain the recursion \\eqref{scalarrec}\nwithout the term involving the Ricci tensor. This will be accomplished by using the conformal\nfreedom at hand, i.e. change $\\Omega$. The conformal freedom is $\\Omega \\to\n\\tilde\\Omega=\\tfrac{\\Omega}{\\alpha}$ where $\\alpha$ is analytic near $\\Lambda$ with $\\alpha(\\Lambda)=1$\\footnote{A more natural condition is the equivalent statement\n$\\Omega \\to \\tilde\\Omega=\\Omega \\alpha$. However, this formulation gives slightly neater calculations.}.\n$\\hat D_a (\\tfrac{1}{\\alpha})$ at $\\Lambda$ gives a shift of the moments which corresponds to a \n'translation' of the physical space, \\cite{geroch}. Hence we can assume that $\\hat D_a (\\tfrac{1}{\\gamma})=0$\nat $\\Lambda$.\nIt is to be noted that a change of $\\Omega$ changes the (rescaled) potential\n$\\phi\/\\sqrt{\\Omega}$. It also changes the normal coordinates on $\\hat V$, and hence all conclusions\nmust be made with some care. In order to derive the simplified recursion \\eqref{scalarreducedrec}, we will specify a $\\alpha$ through $\\alpha_L$, the restriction of $\\alpha$ to $S$. However, in order to deduce that there exists a \n{\\em real-valued} function $\\alpha$ which the prescribed values of $\\alpha_L$, we need to say more\non the representation of $\\alpha_L$. This result and a useful estimate is the content of lemma \\ref{serieslemma}.\n\n\\begin{lemma}\\label{serieslemma}\nLet $f_L=f_1(\\xi,\\zeta)+i\\rho f_2(\\xi,\\zeta)$ where $f_1$ and $f_2$ are analytic \non the ball $U=\\{ |\\xi|^2+|\\zeta|^2<r_0^2 \\}$, and where $\\rho=\\sqrt{\\xi^2+\\zeta^2}$.\nWe can then write\n\\begin{equation}\\label{fcomplex}\nf_L(\\rho\\cos\\varphi,\\rho\\sin\\varphi)=\n\\sum_{l=0}^\\infty\\sum_{m=-l}^l c_{l,m} e^{i m \\varphi}\\rho^l ,\n\\end{equation}\nwhere \n\\begin{equation}\\label{complexconv}\n\\sum_{l=0}^\\infty\\sum_{m=-l}^l |c_{l,m}| \\rho^l < \\infty \\; , \\; \\rho < r_0 .\n\\end{equation}\nFurthermore, the converse is true; if $f_L$ is a function satisfying \\eqref{fcomplex} and \\eqref{complexconv}\nthen there are functions $f_1$ and $f_2$ analytic in $U$ such that $f_L=f_1+i\\rho f_2$. \nThe functions $f_1$ and $f_2$ are real-valued if and only if the coefficients\n$c_{l,m}$ satisfy $\\bar c_{l,m}=(-1)^{l-m}c_{l,-m}$.\n\\end{lemma}\n\\begin{proof}\nForm the functions $f_1(\\frac{x+y}2,\\frac{x-y}{2i})$ and $f_2(\\frac{x+y}2,\\frac{x-y}{2i})$.\\footnote{Although we will always take $\\xi$ and $\\zeta$ to be real, they are temporarily\ncomplexified in this proof.}  These functions \nare analytic in terms of $x$ and $y$. Therefore there exist coefficients $c_{l,m}$ such that\n\\begin{equation}\\label{f1f2}\nf_j(\\frac{x+y}2,\\frac{x-y}{2i})=\\sum_{l=0}^\\infty\\sum_{k=0}^l i^{1-j} c_{l+j-1,2k-l}x^ky^{l-k} ,\n\\end{equation}\nwhere $j=1$ or $2$, and\n$$\n\\sum_{l=0}^\\infty\\sum_{k=0}^l |c_{l+j-1,2k-l}||x|^k|y|^l < \\infty \\; , \\; |\\tfrac{x+y}2|^2+|\\tfrac{x-y}{2i}|^2\n=\\tfrac{1}{2}(|x|^2+|y|^2) < r_0^2 .\n$$\nNow let $x=\\rho e^{-i\\varphi}$ and $y=\\rho e^{i\\varphi}$. For all $\\rho^2=\\tfrac{1}{2}(|x|^2+|y|^2) < r_0^2$ this shows that\n\\begin{equation}\\label{flc}\nf_L(\\rho\\cos\\varphi,\\rho\\sin\\varphi)=\n\\sum_{l=0}^\\infty\\sum_{k=0}^l c_{l,2k-l} e^{i (2k-l) \\varphi}\\rho^l +\n\\sum_{l=0}^\\infty\\sum_{k=0}^l c_{l+1,2k-l} e^{i (2k-l) \\varphi}\\rho^{l+1} .\n\\end{equation}\nThe reality condition follows easily from \\eqref{f1f2} and \\eqref{flc}.\nA reorganisation gives the first part of the lemma. \nThe converse is given by the substitution $x=\\xi+i\\zeta$ and $y=\\xi-i\\zeta$ in \\eqref{f1f2}. The series \nconverges absolutely for $|\\xi|^2+|\\zeta|^2=\\rho^2 < r_0^2$.\n\\end{proof}\nWe now show that we can choose $\\alpha$ such that the Ricci term in \\eqref{scalarrec} vanishes.\n\\begin{lemma} \\label{nnRnoll}\nThere exists a real-valued real analytic function $\\alpha$ on $\\hat V$ such that\nthe Ricci tensor $\\tilde R_{ab}$ of the new metric $\\tilde h_{ab}=\\alpha^{-2}\\hat h_{ab}$  \nsatisfies $\\tilde \\eta^a \\tilde \\eta^b \\tilde R_{ab}=0$ on $\\tilde S$, where both $\\tilde \\eta^a$, $\\tilde S$\nand the mapping $F$ are defined\nin terms of coordinates which are normal with respect to the new metric $\\tilde h_{ab}$.\n\\end{lemma}\n\\begin{proof} We first demonstrate that we can make $\\eta^a \\eta^b \\tilde R_{ab}=0$ on $S$.\n{}From \\cite{wald}\n$$\n\\tilde R_{ab}=\\hat R_{ab}+\\tfrac{1}{\\alpha}\\hat D_a \\hat D_b \\alpha+\n\\alpha^{-1}\\hat h_{ab}\\hat h^{cd}\\hat D_c \\hat D_d\\alpha-\n2\\alpha^{-2}\\hat h_{ab}\\hat h^{cd}(\\hat D_c \\alpha) \\hat D_d \\alpha .\n$$\nOn $S$, (using $\\eta^a$ and $S$ belonging to $\\hat h_{ab}$), this becomes\n\\begin{equation}\\label{rkappa}\n\\eta^a \\eta^b \\tilde R_{ab}=\n\\eta^a\\eta^b\\hat R_{ab}+\\tfrac{1}{\\alpha_L}\\eta^a\\eta^b\\hat D_a \\hat D_b \\alpha_L=\n\\eta^a\\eta^b\\hat R_{ab}+\\tfrac{1}{\\alpha_L}\\tfrac{\\partial^2 \\alpha_L}{\\partial\\rho^2}\n\\end{equation}\nwhere we have used $\\eta^a \\hat D_a \\eta^b=0$. In order to make $\\eta^a \\eta^b \\tilde R_{ab}=0$,\nwe require\n\\begin{equation} \\label{omegaeq}\n\\alpha_{L \\rho\\rho}+\\eta^a\\eta^b\\hat R_{ab}\\alpha_L=0 .\n\\end{equation}\nNote that, on $\\hat V$, $\\hat R_{ab}$ satisfies the conditions\nof lemma \\ref{Slemma}, therefore from lemma~\\ref{Slemma} and lemma~\\ref{serieslemma}\nit follows that\n\\begin{equation}  \\label{rserie}\n\\eta^a\\eta^b\\hat R_{ab}=\\sum_{n=0}^\\infty b_{n+2}(e^{i\\varphi},e^{-i\\varphi})\\rho^n ,\n\\end{equation}\nwhere each $b_n$ is a polynomial of degree at most $n$. \nIt is easy to see that the reality condition from lemma \\ref{serieslemma} is equivalent to\n$\\overline{b_{n}(e^{i\\varphi},e^{-i\\varphi})}=(-1)^n b_n(-e^{i\\varphi},-e^{-i\\varphi})$.\nHence the reality of $\\hat R_{ab}$ implies this condition. Moreover, since\n$\\hat R_{ab}$ is analytic in a neighbourhood of $\\Lambda$, there exists a $\\rho_0$ such\nthat for any fixed $\\varphi$, the series \\eqref{rserie} converges for all $\\rho < \\rho_0$.\nWell known ODE theory gives that for each fixed $\\varphi$ there exists\na solution $\\alpha_L$, analytic in $\\rho$ for $0 \\leq \\rho<\\rho_0$, i.e., for all $\\varphi$ \nwe have \n$$\n\\alpha_L=\\sum_{n=0}^\\infty a_n(\\varphi)\\rho^n , \\quad \\rho < \\rho_0 .\n$$\nWe note that $a_0=1$ while $a_1$\ncan be chosen to be 0 (translation). \n$\\alpha_L$ will have the right regularity if we can show that each\n$a_n(\\varphi)$ is a polynomial in $e^{i\\varphi}$ and $e^{-i\\varphi}$ of\ndegree at most $n$. Equation \\eqref{omegaeq} becomes\n$$\n-\\sum_{n=0}^\\infty n(n-1) a_n\\rho^{n-2}=\n\\sum_{n=0}^\\infty \\sum_{j=0}^\\infty b_{n+2} a_j\\rho^{n+j}.\n$$\nEquating powers of $\\rho$ we get\n\\begin{equation}\\label{reckappa}\n-(n+2)(n+1)a_{n+2}=\\sum_{m=0}^n b_{m+2} a_{n-m},\\quad n \\geq 0 .\n\\end{equation}\nThe polynomials $a_0$ and $a_1$ has maximal degree $0$ respective $1$. The polynomials $b_n$ has maximal degree $n$.\nStraightforward induction shows that $a_n$ has maximal degree $n$. \nInduction and \\eqref{reckappa} also implies $\\overline{a_n(e^{i\\varphi},e^{-i\\varphi})}=(-1)^n a_n(-e^{i\\varphi},-e^{-i\\varphi})$.\nThus, due to lemma \\ref{serieslemma}, there are real-valued analytic functions $\\alpha_1$ and $\\alpha_2$ such that \n$\\alpha_L=\\alpha_1+i\\rho \\alpha_2$. The function $\\alpha=\\alpha_1(x,y)+z\\alpha_2(x,y)$ is then a real-valued analytic \nextension of $\\alpha_L$ to a ball in $\\hat V$. This shows that there exist an $\\alpha$\nsuch that $\\eta^a \\eta^b \\tilde R_{ab}=0$ on $S$. \n\nHowever, it then also follows that $\\tilde \\eta^a \\tilde \\eta^b \\tilde R_{ab}=0$ on $\\tilde S$.\n{}From $\\alpha$ we get $\\tilde \\Omega=\\Omega\/\\alpha$, and the corresponding new metric $\\tilde h_{ab}=\\alpha^{-2}\\hat h_{ab}$. \nNote that in \\eqref{scalarrec}, where now $\\eta^a\\eta^b\\widehat R_{ab}=0$,\nthe recursion is stated in terms of $\\hat D_a$, i.e., it is expressed in terms of $\\hat h_{ab}$\ninstead of $\\tilde h_{ab}$. From $\\tilde h_{ab}$\nwe get new normal coordinates, i.e., $(x,y,z)$ in $T_\\Lambda \\hat V_\\mathbb C$,\nare mapped into $\\hat V_\\mathbb C$ using the exponential map belonging to $\\tilde h_{ab}$. We then construct $\\tilde \\eta^a$ and the mapping\n$F$ with respect to these coordinates. Now, null geodesics, of which $S$ consists, are conformally invariant,\nalthough they become non-affinely parametrised. This means that\n$\\tilde \\eta^a \\propto \\eta^a$, and that points in $S$ are mapped into points in $\\tilde S$.\nThus $\\tilde \\eta^a \\tilde \\eta^b\\tilde R_{ab} \\propto  \\eta^a  \\eta^b\\tilde R_{ab}=0$ on $\\tilde S$ (or $S$). \n\\end{proof}\n\nHenceforth we denote all entities defined via $\\tilde h_{ab}$ instead of $\\hat h_{ab}$ with\n a tilde. In particular, $\\tilde D_a$ will denote the derivative operator associated with $\\tilde h_{ab}$. Applying lemma \\ref{nnRnoll} to theorem \\ref{scalrarrec2d} we immediately get the following theorem.\n\\begin{theorem}\\label{scalrarreducedrec2d}\nLet $\\hat V$ and $\\tilde S$ be defined as in sections \\eqref{tmoments} and \\eqref{momentsscalar}, where\n$\\tilde S$ is defined in terms of normal coordinates connected to $\\tilde h_{ab}$.\nLet $\\tilde \\eta^a$ have the properties given by lemma \\ref{etalemma} with respect to\n$\\tilde h_{ab}$, and let $\\tilde f_n$\nbe defined by \\eqref{fn} with $\\tilde \\eta^a$ replacing $\\eta^a$.\nThen the recursion \\eqref{orgrec} on $\\tilde V$ takes the form\n\\begin{equation}\\label{scalarreducedrec}\n\\tilde f_n=\\tilde \\eta^a\\tilde D_a\\tilde f_{n-1}=(\\tilde \\eta^a \\tilde D_a)^n \\tilde f_0\n=\\frac{\\partial^n}{\\partial \\tilde \\rho^n}\\tilde f_0\n\\end{equation}\non $\\tilde S$. The moments of order $2^n$ are captured in the direction dependent limit \n$\\lim_{\\tilde \\rho \\to 0} \\tilde f_n(\\tilde \\rho,\\varphi)$.\n\\end{theorem}\n\\begin{proof}\nSince $\\tilde \\eta^a \\tilde D_a$ is $\\frac{\\partial}{\\partial \\tilde \\rho}$, each $\\tilde f_n$ is easily \nderived from $\\tilde f_0$. \nAlso, from lemma \\ref{etalemma}c), we (again) know that the $2n+1$ degrees of freedom\nof $P_{a_1 \\ldots a_n}$ at $\\Lambda$ are encoded in $\\tilde f_n$.\n\\end{proof}\n\n\\section{Bounds on the moments} \nAll multipole moments are encoded in $\\tilde f_0$, and we note that the recursion\n\\eqref{scalarreducedrec} is identical to the recursion emanating from a scalar function\nin $\\mathbb{R}^3$ (after inversion). It is clear that many different functions on \n$\\mathbb{R}^3$ will produce the same moments, but if we also require that the function, $g$ say,\nis harmonic, $g$ is uniquely determined by the moments.\n\nThus, provided that we can connect a function which is harmonic in \na neighbourhood of ${\\mathbf 0} \\in \\mathbb{R}^3$\nto each $\\tilde f_0$, we have the following theorem: \n\n\\begin{theorem}  \\label{bounds}\nSuppose that $(M,g_{ab})$ is a stationary asymptotically flat spacetime, admitting an analytic (rescaled) potential and an analytic\nchart\\footnote{As discussed before, the analyticity has been proved for the case with non-zero mass.}  on the conformally compactified manifold of timelike Killing \ntrajectories, around the infinity point $\\Lambda$.\nThen there exist a (flat-)harmonic function $g$ in a neighbourhood of \n${\\mathbf 0}\\in T_{\\Lambda}\\hat V \\cong \\mathbb{R}^3$, such that all \nmultipole moments of $M$ are given by\n\\begin{equation}\\label{harmrec}\nP_{a_1\\dots a_n}(\\Lambda)=(\\nabla_{a_1} \\dots \\nabla _{a_n} g)(0) ,\n\\end{equation}\nwhere $\\nabla_a$ in the LHS of \\eqref{harmrec} is the flat derivative operator in $\\mathbb{R}^3$.\nThis puts a bound on the multipole moments, since the Taylor expansion \n$\\sum_{|\\alpha|\\geq 0}\\frac{\\mathbf{r}^\\alpha}{\\alpha!}(\\partial_\\alpha g)(\\mathbf 0)$\nof  $g$ converges in a neighbourhood of  the origin in $\\mathbb{R}^3$.\n\\end{theorem}\n\n\\begin{proof}\nBy lemma \\ref{serieslemma} we have\n$$\n\\tilde f_L(\\rho\\cos\\varphi,\\rho\\sin\\varphi)=\n\\sum_{l=0}^\\infty\\sum_{m=-l}^l c_{l,m} e^{i m \\varphi}\\rho^l, \\quad \\rho < r_0\n$$\nfor some $r_0>0$, and we know that $\\tilde f_L$ fully determines the moments of $M$.\nWe will now define the function $g$, which will be shown to be harmonic in a neighbourhood\nof the origin. Finally we will argue that $g$ has the correct derivatives at $\\mathbf 0$, i.e.,\nthat the equality \\eqref{harmrec} is valid (for all $n \\geq 0$).\nFirst we define the coefficients \n$$\na_{l,m}=c_{l,m}i^{-m-l} 2^{1-l}\\pi \\frac{\\sqrt{(l+m)!(l-m)!}}{\\Gamma(l+\\tfrac{1}2)\\sqrt{2l+1}} .\n$$\nand the function\n$$\ng(r,\\theta,\\varphi)=\\sum_{l=0}^\\infty\\sum_{m=-l}^l a_{l,m} Y^m_l(\\theta,\\varphi) r^l .\n$$\nDue to the construction, $g$ is harmonic at those (interior) points for which the\nsum converges. We now establish convergence.\n{}From the identity\n$$\n\\sum_{m=-l}^l |Y_l^m(\\theta,\\varphi) |^2=\\frac{2l+1}{4\\pi}\n$$\nwe get\n\\begin{gather*}\n\\left | \\sum_{m=-l}^l a_{l,m}Y^m_l \\right | \\leq \n\\sum_{m=-l}^l \\left | a_{l,m}Y^m_l \\right | \\leq\n\\left ( \\sum_{m=-l}^l | a_{l,m}|^2 \\right )^{\\frac{1}{2}}\n\\left ( \\sum_{m=-l}^l |Y_l^m |^2 \\right )^{\\frac{1}{2}} \\\\\n=\\left ( \\sum_{m=-l}^l | \\sqrt{\\tfrac{2l+1}{4\\pi}} a_{l,m}|^2 \\right )^{\\frac{1}{2}}\n=\\frac{\\sqrt{\\pi}}{2^l\\Gamma(l+\\tfrac{1}2)} \\left ( \n\\sum_{m=-l}^l \\left | c_{l,m} \\sqrt{(l+m)!(l-m)!}\\right |^2 \\right )^{\\frac{1}{2}}\n\\end{gather*}\nFurthermore,\nthe inequality $(l+m)!(l-m!)\\leq (2l)!$, when $-l\\leq m\\leq l$ follows from the convexity of $\\ln(\\Gamma(x))$.\nTherefore\n$$\n\\left | \\sum_{m=-l}^l a_{l,m}Y^m_l \\right | \n\\leq \\frac{\\sqrt{\\pi}\\sqrt{(2l)!}}{2^l\\Gamma(l+\\tfrac{1}2)} \n\\left ( \\sum_{m=-l}^l | c_{l,m} |^2 \\right )^{\\frac{1}{2}}\n$$\nNext, the inequality $\\frac{\\pi (2l)!}{4^{l}\\Gamma\\left(l+\\tfrac{1}2\\right)^2}\n=\\frac{(2l)!!}{(2l-1)!!}=(2l+1)\\frac{(2l)!!}{(2l+1)!!}\\leq 2l+1$ gives\n$$\n\\left | \\sum_{m=-l}^l a_{l,m}Y^m_l \\right | \n\\leq \\sqrt{2l+1}\\left ( \\sum_{m=-l}^l | c_{l,m} |^2 \\right )^{\\frac{1}{2}}\n\\leq \\sqrt{2l+1}\\sum_{m=-l}^l | c_{l,m} |\n$$\nBut for all $\\epsilon>0$ we have \n$\\sqrt{2l+1}(1+\\epsilon)^{-l} \\rightarrow 0$ as  $l\\rightarrow \\infty$.\nSo the factor $\\sqrt{2l+1}$ will not affect the radius of convergence.\n\nHence \n$$\n \\sum_{l=0}^\\infty\\sum_{m=-l}^l a_{l,m}Y^m_l r^l \\; \\text{ converges if } r< r_0 .\n$$\nThis shows that $g$ is well defined in a neighbourhood of $\\mathbf 0$ in $T_\\Lambda \\hat V$,\nand we must now show that we have equality in \\eqref{harmrec}. We will do this by forming $g_L$\nand then compare with $\\tilde f_L$. Note, however, that $\\tilde f_L$ is defined on $\\tilde S \\subset V_\\mathbb{C}$,\nwhile $g_L$ will be defined on the corresponding surface $\\bar S=S_{\\tilde h_{ab}(\\Lambda)}$ in $T_\\Lambda \\hat V_\\mathbb{C}$. (By $\\bar S$ we denote the surface\ndefined by $F$ as previously, but where $F$ now maps $(\\xi,\\zeta)$ into $T_\\Lambda \\hat V_\\mathbb{C}$ via the flat metric $\\tilde h_{ab}(\\Lambda)$.)\nThis means that $f_L$ and $g_L$\nreally can be compared only at $\\Lambda$ (i.e. $\\mathbf 0 \\in T_\\Lambda \\hat V$), where also the equality \\eqref{harmrec} is evaluated.\nOn the other hand, the radial derivatives of both entities are well defined and comparable at $\\Lambda$.\nIn other words, if both $\\tilde f_L$ and $g_L$ are equal when expressed in terms of the\ncoordinates $\\xi, \\zeta$, they will produce the same derivatives\/moments. To summarise; in the case of $g$, \nthe function $F$ and the vector $\\eta^a$ are simply interpreted in $T_{\\Lambda} \\hat V_\\mathbb{C}$\nrather than in $V_\\mathbb{C}$.\n\nTo proceed, we recall that \n$$\nY^m_l(\\theta,\\varphi)=i^{m-|m|}\\sqrt{\\frac{(2l+1)(l-|m|)!}{4\\pi(l+|m|)!}}P^{|m|}_l(\\cos\\theta)e^{im\\varphi}\n$$\nfor $-l\\leq m\\leq l$, and that\n$$\nP^m_l(\\cos\\theta)=(-1)^m2^{-l} \\sin^m\\theta \\sum_{k=0}^{\\bigl \\lfloor \\tfrac{l-m}{2} \\bigr \\rfloor} \n\\frac{(-1)^k (2l-2k)!}{k!(l-k)!(l-2k-m)!}\\cos^{l-m-2k}\\theta \n$$\nwhere $0\\leq \\theta \\leq \\pi$ and $m\\geq 0$.\nTherefore,\n$$\ng(r,\\theta,\\varphi)=\\sum_{l=0}^\\infty\\sum_{m=-l}^l \\sum_{k=0}^{\\bigl \\lfloor \\tfrac{l-|m|}{2} \\bigr \\rfloor}\n\\frac{ c_{l,m}\\sqrt{\\pi}(l-|m|)!(2l-2k)! e^{i m \\varphi}\\rho^{|m|} z^{l-|m|-2k} r^{2k}}\n{i^{l-|m|}(-1)^k 4^lk!(l-k)!(l-|m|-2k)!\\Gamma(l+\\tfrac{1}2)} , \n$$\nwhere $z=r\\cos\\theta$, $\\rho=r\\sin\\theta$.\nWhen we take the restriction of $g$ to $\\bar S$, i.e., form $g_L$, only the terms with $k=0$ survives since $r_L=0$. Thus\n$$\ng_L(\\rho\\cos\\theta,\\rho\\sin\\theta)\n=\\sum_{l=0}^\\infty\\sum_{m=-l}^l\n\\frac{ c_{l,m}\\sqrt{\\pi}(2l)! e^{i m \\varphi}\\rho^l}\n{4^l l!\\Gamma(l+\\tfrac{1}2)}\n=\\sum_{l=0}^\\infty\\sum_{m=-l}^l c_{l,m}e^{i m \\varphi}\\rho^l=\\tilde f_L,\n$$\nwhich means that we have equality in \\eqref{harmrec}.\n\\end{proof}\n\n\\section{Discussion}\nIn this paper we have studied the multipole moments of stationary asymptotically flat spacetimes. By using normal coordinates, and by exploiting the conformal freedom, we could show that the tensorial recursion \\eqref{orgrec} could be replaced by the scalar\nrecursion \\eqref{scalarreducedrec}. This recursion is a direction dependent\nrecursion on $\\mathbb{R}^2$, where the moments are encoded in the direction dependent\nlimits at $\\Lambda$.\n\nUsing this setup, we could also show that the multipole moments cannot grow too fast.\nIn essence, the rescaled potential behaves (locally) \nin the manner of a harmonic function on $\\mathbb{R}^3$. The bounds on the moments given in theorem\n\\ref{bounds} gives the necessary part in a conjecture due to Geroch \\cite{geroch},\nand it is of course tempting to conjecture that this condition on the moments\nalso will be sufficient (as long as the monopole is real-valued).\n\nWhether this can be proved using the techniques presented here is still an open question.\n\nWe also remark that similar questions concerning the convergence of asymptotic expansions in the static case are currently being studied by Friedrich, using a different technique, \\cite{friedrich}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nResource Description Framework (RDF) is a general method for conceptual description or modeling of information that is implemented \nin web resources. RDF Schema (RDFS) extends RDF to classes providing basic elements for the description of vocabularies. OWL adds \nmore vocabulary for describing properties and classes i.e. relations between classes, cardinality, and richer typing of properties.\nUnfortunately, OWL has high worst-case complexity results for key inference problems. To overcome this problem we propose a\nlightweight OWL profile called OWL-P.\n\nA rule is perhaps one of the most understandable notion in computer science. It consists of the condition and the conclusion. \nIf some condition that is checkable in some dataset holds, then the conclusion is processed. In the same way RDF(S) and OWL\nentailments work.\n\nThe paper is constructed according to sections. Section~\\ref{sec:preliminaries} presents RDF and Notation 3 Logic concepts. \nIn Section~\\ref{sec:rules} we present inference rules for RDF. RDFS and OWL in N3Logic. Section~\\ref{sec:relatedwork} is devoted \nto related work. The paper ends with conclusions.\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\nThe RDF data model rests on the concept of creating web-resource statements in the form of subject-predicate-object expressions, \nwhich in the RDF terminology, are referred to as \\emph{triples} (or \\emph{statements}).\n\nAn RDF triple comprises a subject, a predicate, and an object. In \\cite{Wood2014}, the meaning of subject,\npredicate and object is explained. The \\emph{subject} denotes a resource, the \\emph{object} fills the value of the relation, \nthe \\emph{predicate} refers to the resource's characteristics or aspects and expresses a subject -- object relationship. \nThe predicate denotes a binary relation, also known as a property.\n\nFollowing \\cite{Wood2014}, we provide definitions of RDF triples below.\n\n\\begin{mydef}[RDF triple]\n\\label{def:rdftriple}\nAssume that $\\mathcal{I}$ is the set of all Internationalized Resource Identifier (IRI) \nreferences, $\\mathcal{B}$ (an infinite) set of blank nodes, $\\mathcal{L}$ the set of literals.\nAn \\emph{RDF triple} $t$ is defined as a triple $t = \\langle s, p, o\\rangle$ where $s \\in \\mathcal{I} \\cup \\mathcal{B}$ is \ncalled the \\emph{subject}, $p \\in \\mathcal{I}$ is called the \\emph{predicate} and $o \\in \\mathcal{I} \\cup \\mathcal{B} \\cup \\mathcal{L}$ \nis called the \\emph{object}.\n\\end{mydef}\n\nThe elemental constituents of the RDF data model are RDF terms that can be used in reference to resources: \nanything with identity. The set of RDF terms is divided into three disjoint subsets: IRIs, literals, and blank nodes.\n\n\\begin{mydef}[IRIs]\n\\label{def:rdfterms}\n\\emph{IRIs} serve as global identifiers that can be used to identify any resource.\n\\end{mydef}\n\n\\begin{mydef}[Literals]\n\\label{def:rdfliterals}\n\\emph{Literals} are a set of lexical values. It can be a set of plain strings, such as \\verb|\"Apple\"|, optionally \nwith an associated language tag, such as \\verb|\"Apple\"@en|.\n\\end{mydef}\n\n\\begin{myrem}\nIn RDF 1.1 literals comprise a lexical string and a datatype, such as \\verb|\"1\"^^http:\/\/www.w3.org\/2001\/XMLSchema#int|. \n\\end{myrem}\n\n\\begin{myrem}\nIn literals datatypes are identified by IRIs, where RDF borrows many of the datatypes defined in XML Schema 1.1 \\cite{Sperberg-McQueen2012}.\n\\end{myrem}\n\n\\begin{mydef}[Blank nodes]\n\\label{def:bnodes}\n\\emph{Blank nodes} are defined as existential variables used to denote the existence of some resource for which an IRI or \nliteral is not given.\n\\end{mydef}\n\n\\begin{myrem}\nBlank nodes are inconstant or stable identifiers and are in all cases locally scoped to the RDF store or the RDF file.\n\\end{myrem}\n\nA collection of RDF triples intrinsically represents a labeled directed multigraph. The nodes are the subjects and objects of \ntheir triples. RDF is often referred to as being \\emph{graph structured data} where each $\\langle s, p, o\\rangle$ triple can be \ninterpreted as an edge $s \\xrightarrow{p} o$.\n\n\\begin{mydef}[RDF graph]\n\\label{def:rdfgraph}\nLet $\\mathcal{O} = \\mathcal{I} \\cup \\mathcal{B} \\cup \\mathcal{L}$ and\n$\\mathcal{S} = \\mathcal{I} \\cup \\mathcal{B}$, then $G \\subset \\mathcal{S} \\times \\mathcal{I} \\times \\mathcal{O}$ \nis a finite subset of \\emph{RDF triples}, which is called \\emph{RDF graph}.\n\\end{mydef}\n\nOn the other hand, in the Semantic Web environment there is a Notation3 format, which offers a new human-readable serialization of \nRDF model but it also extended RDF by logical symbols and created a new Semantic Web logic called Notation3 Logic (N3Logic). \nFollowing \\cite{Arndt2015}, we provide definitions of N3Logic below.\n\n\\begin{mydef}[N3Logic alphabet]\nA \\emph{N3Logic alphabet} $A_{N3}$ consists of the following disjoint classes of symbols:\n\\begin{enumerate}\n \\item a set $\\mathcal{I}$ of IRI symbols beginning with \\verb|<| and ending with \\verb|>|,\n \\item a set $\\mathcal{L}$ of literals beginning and ending with \\verb|\"|,\n \\item a set $\\mathcal{V}$ of variables, $\\mathcal{V} = \\mathcal{B} \\cup V_U$, where $B$ is a set of existential variables \n (blank nodes in RDF-sense) start with \\verb|_:| and $V_U$ is a set of universal variables start with \\verb|?|,\n \\item brackets \\verb|{|, \\verb|}|,\n \\item a logical implication \\verb|=>|,\n \\item a period \\verb|.|,\n \\item a period \\verb|@false|.\n\\end{enumerate}\n\\end{mydef}\n\n\\begin{myrem}\nNotation3 allows to abbreviate IRIs by using prefixes. Instead of writing \\verb|<http:\/\/example.com>|, we can write \\verb|ex:|.\n\\end{myrem}\n\n\\begin{myrem}\nEach IRI, variable and literal is an expression.\n\\end{myrem}\n\n\\begin{myrem}\n\\texttt{\\{$f$\\}} is an expression called formula.\n\\end{myrem}\n\n\\begin{myrem}\n\\texttt{$e_1$ => $e_2$} is a formula called implication.\n\\end{myrem}\n\nIn Notation3 literals, IRIs, variables or even formula expressions can be subjects, objects or predicates.\n\n\\section{Inference rules}\n\\label{sec:rules}\nIn this section, we introduce inference rules for RDF, RDFS and OWL. Inference rules connected with RDF(S) and OWL are basis of the \ndeductive RDF graph stores.\n\n\\begin{mydef}[Deductive RDF graph store]\nA \\emph{deductive RDF graph store} is an entity which remembers RDF triples and can generate new ones under certain conditions \nthrough deduction or inference. It can answer queries about the combined given and inferred triples.\n\\end{mydef}\n\n\\subsection{RDF and RDFS}\nIn Table~\\ref{tab:rulesrdf} we present patterns which hold by RDF and RDFS entailments. All rules are tested in reasoning engines such as \nFuXi\\footnote{\\url{https:\/\/github.com\/RDFLib\/FuXi}} and cwm\\footnote{\\url{http:\/\/www.w3.org\/2000\/10\/swap\/doc\/cwm.html}}.\n\n\\begin{table*}[]\n\\caption{Inference rules for RDF and RDFS}\n\\label{tab:rulesrdf}\n\\centering\n\\begin{tabular}{|lll|}\n\\hline\nConditions &  & Conclusions \\\\\n\\hline\n\\verb|{?S ?P ?O}| & \\verb|=>| & \\verb|{?P rdf:type rdf:Property}.| \\\\\n\\verb|{?P rdfs:domain ?C. ?S ?P ?O}| & \\verb|=>| & \\verb|{?S rdf:type ?C}.| \\\\ \n\\verb|{?P rdfs:range ?C. ?S ?P ?O}| & \\verb|=>| & \\verb|{?O rdf:type ?C}.| \\\\ \n\\verb|{?S ?P ?O}| & \\verb|=>| & \\verb|{?S rdf:type rdfs:Resource}.| \\\\ \n\\verb|{?S ?P ?O}| & \\verb|=>| & \\verb|{?O rdf:type rdfs:Resource}.| \\\\ \n\\verb|{?Q rdfs:subPropertyOf ?R. ?P rdfs:subPropertyOf ?Q}| & \\verb|=>| & \\verb|{?P rdfs:subPropertyOf ?R}.| \\\\\n\\verb|{?Q rdf:type rdf:Property}| & \\verb|=>| & \\verb|{?Q rdfs:subPropertyOf ?Q}.| \\\\ \n\\verb|{?P rdfs:subPropertyOf ?R. ?S ?P ?O}| & \\verb|=>| & \\verb|{?S ?R ?O}.| \\\\ \n\\verb|{?C rdf:type rdfs:Class}| & \\verb|=>| & \\verb|{?C rdfs:subClassOf rdfs:Resource}.| \\\\ \n\\verb|{?A rdfs:subClassOf ?B. ?S rdf:type ?A}| & \\verb|=>| & \\verb|{?S rdf:type ?B}.| \\\\ \n\\verb|{?Q rdf:type rdfs:Class}| & \\verb|=>| & \\verb|{?Q rdfs:subClassOf ?Q}.| \\\\ \n\\verb|{?B rdfs:subClassOf ?C. ?A rdfs:subClassOf ?B}| & \\verb|=>| & \\verb|{?A rdfs:subClassOf ?C}.| \\\\ \n\\verb|{?X rdf:type rdfs:ContainerMembershipProperty}| & \\verb|=>| & \\verb|{?X rdfs:subPropertyOf rdfs:member}.| \\\\ \n\\verb|{?X rdf:type rdfs:Datatype}| & \\verb|=>| & \\verb|{?X rdfs:subClassOf rdfs:Literal}.| \\\\ \n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{OWL}\n\nIn Table~\\ref{tab:surowl} we analyze existing proposals for different OWL2 profiles: RDFS++ \\cite{Rdfspp2016}, \nL2 \\cite{Fischer2010}, RDF 3.0\/OWLPrime \\cite{Hendler2010}, OWLSIF\/pD* \\cite{Terhorst2005}, OWL-LD \\cite{Glimm2011} and \nOWL-RL \\cite{Motik2012}. We check which terms are most commonly used and propose a new version of OWL 2 called OWL-P. \nWe also considered time complexity for detecting a required rule application and frequently used vocabulary terms in \nour corpus. The snapshot (Table \\ref{tab:fowl}) is built by \\cite{Isele2010} and use seeds from \\cite{Schmachtenberg2014}. \n\nThis profile of OWL2 is simpler that OWL-RL. It drops support for restriction and cardinality classes, class \nrelationships and list-based axioms. In the Table~\\ref{tab:rulesowl1}, Table~\\ref{tab:rulesowl2} and we present \ninference rules of OWL-P.\n\n\\begin{table*}[]\n\\caption{Comparison of OWL profiles}\n\\label{tab:surowl}\n\\centering\n\\begin{tabular}{|l|c|c|c|c|c|c|c|}\n\\hline\n                          & OWL-P & RDFS++ & L2  & RDFS 3.0              & OWLSIF         & OWL-LD & OWL-RL \\\\\n                          &       &        &     & OWLPrime              & pD*            &        &        \\\\\n\\hline\n\\verb|owl:AllDifferent|              & \\XBox         & \\XBox          & \\XBox       & \\CheckedBox                    & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:AllDisjointClasses|        & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:AllDisjointProperties|     & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:allValuesFrom|             & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\CheckedBox             & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:assertionProperty|         & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:AsymmetricProperty|        & \\CheckedBox   & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:cardinality|               & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:complementOf|              & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:DatatypeProperty|          & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:differentFrom|             & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:disjointUnionof|           & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:disjointWith|              & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:equivalentClass|           & \\CheckedBox   & \\XBox          & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:equivalentProperty|        & \\CheckedBox   & \\XBox          & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:FunctionalProperty|        & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:hasKey|                    & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:hasSelf|                   & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:hasValue|                  & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\CheckedBox             & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:intersectionof|            & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:InverseFunctionalProperty| & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:inverseOf|                 & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:IrreflexiveProperty|       & \\CheckedBox   & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:maxCardinality|            & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:minCardinality|            & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:ObjectProperty|            & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:oneOf|                     & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:propertyChainAxiom|        & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:propertyDisjointWith|      & \\CheckedBox   & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:qualifiedCardinality|      & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:qualifiedMaxCardinality|   & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:qualifiedMinCardinality|   & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:sameAs|                    & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:someValuesFrom|            & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\CheckedBox             & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:sourceIndiviual|           & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:SymmetricProperty|         & \\CheckedBox   & \\XBox          & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:targetIndividual|          & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:targetValue|               & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:TransitiveProperty|        & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:unionof|                   & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|rdfs:domain|                   & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|rdfs:range|                    & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|rdfs:subClassOf|               & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|rdfs:subPropertyOf|            & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table}[]\n\\caption{Vocabulary terms used in LOD snapshot 2015}\n\\label{tab:fowl}\n\\centering\n\\begin{tabular}{|l|c|}\n\\hline\n                                     &  voc terms \\\\\n\\hline\n\\verb|owl:AllDifferent|              & 111       \\\\\n\\verb|owl:AllDisjointClasses|        & 21        \\\\\n\\verb|owl:AllDisjointProperties|     & 13        \\\\\n\\verb|owl:allValuesFrom|             & 126330    \\\\\n\\verb|owl:assertionProperty|         & 0         \\\\\n\\verb|owl:AsymmetricProperty|        & 0         \\\\\n\\verb|owl:cardinality|               & 23910     \\\\\n\\verb|owl:complementOf|              & 873       \\\\\n\\verb|owl:DatatypeProperty|          & 27471     \\\\\n\\verb|owl:differentFrom|             & 784       \\\\\n\\verb|owl:disjointUnionOf|           & 0         \\\\\n\\verb|owl:disjointWith|              & 3743      \\\\\n\\verb|owl:equivalentClass|           & 29708     \\\\\n\\verb|owl:equivalentProperty|        & 201       \\\\\n\\verb|owl:FunctionalProperty|        & 3730      \\\\\n\\verb|owl:hasKey|                    & 5         \\\\\n\\verb|owl:hasSelf|                   & 3         \\\\\n\\verb|owl:hasValue|                  & 1877      \\\\\n\\verb|owl:intersectionOf|            & 2681      \\\\\n\\verb|owl:InverseFunctionalProperty| & 94        \\\\\n\\verb|owl:inverseOf|                 & 1341      \\\\\n\\verb|owl:IrreflexiveProperty|       & 0         \\\\\n\\verb|owl:maxCardinality|            & 257371    \\\\\n\\verb|owl:minCardinality|            & 455203    \\\\\n\\verb|owl:ObjectProperty|            & 40330     \\\\\n\\verb|owl:oneOf|                     & 853       \\\\\n\\verb|owl:propertyChainAxiom|        & 68        \\\\\n\\verb|owl:propertyDisjointWith|      & 4         \\\\\n\\verb|owl:qualifiedCardinality|      & 109       \\\\\n\\verb|owl:qualifiedMaxCardinality|   & 2         \\\\\n\\verb|owl:qualifiedMinCardinality|   & 20        \\\\\n\\verb|owl:sameAs|                    & 3967150   \\\\\n\\verb|owl:someValuesFrom|            & 4446      \\\\\n\\verb|owl:sourceIndiviual|           & 0         \\\\\n\\verb|owl:SymmetricProperty|         & 194       \\\\\n\\verb|owl:targetIndividual|          & 0         \\\\\n\\verb|owl:targetValue|               & 11        \\\\\n\\verb|owl:TransitiveProperty|        & 267       \\\\\n\\verb|owl:unionOf|                   & 53735     \\\\\n\\verb|rdfs:domain|                   & 111865    \\\\\n\\verb|rdfs:range|                    & 59252     \\\\\n\\verb|rdfs:subClassOf|               & 1339391   \\\\\n\\verb|rdfs:subPropertyOf|            & 13416     \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table*}[]\n\\caption{Inference rules for OWL-P properties}\n\\label{tab:rulesowl1}\n\\centering\n\\begin{tabular}{|lll|}\n\\hline\nConditions &  & Conclusions \\\\\n\\hline\n\\verb|{?S ?P ?O}| & \\verb|=>| & \\verb|{?S owl:sameAs ?S. ?P owl:sameAs ?P.| \\\\\n                  &           &  \\verb| ?O owl:sameAs ?O}.| \\\\\n\\verb|{?S owl:sameAs ?O}| & \\verb|=>| & \\verb|{?O owl:sameAs ?S}.| \\\\\n\\verb|{?Q owl:sameAs ?R. ?R owl:sameAs ?P}| & \\verb|=>| & \\verb|{?Q owl:sameAs ?P}.| \\\\\n\\verb|{?S owl:sameAs ?S2. ?S ?P ?O}| & \\verb|=>| & \\verb|{?S2 ?P ?O}.| \\\\\n\\verb|{?P owl:sameAs ?P2. ?S ?P ?O}| & \\verb|=>| & \\verb|{?S ?P2 ?O}.| \\\\\n\\verb|{?O owl:sameAs ?O2. ?S ?P ?O}| & \\verb|=>| & \\verb|{?S ?P ?O2}.| \\\\\n\\verb|{?Q owl:sameAs ?R. ?Q owl:differentFrom ?R}| & \\verb|=>| & \\verb|{@false}.| \\\\\n\\verb|{?P rdf:type owl:FunctionalProperty.| & & \\\\\n\\verb| ?Q ?P ?R1. ?Q ?P ?R2}| & \\verb|=>| & \\verb|{?R1 owl:sameAs ?R2}.| \\\\\n\\verb|{?P rdf:type owl:InverseFunctionalProperty.| & & \\\\\n\\verb| ?Q1 ?P ?R. ?Q2 ?P ?R}| & \\verb|=>| & \\verb|{?Q1 owl:sameAs ?Q2}.| \\\\\n\\verb|{?P rdf:type owl:IrreflexiveProperty. ?Q ?P ?Q}| & \\verb|=>| & \\verb|{@false}.| \\\\\n\\verb|{?P rdf:type owl:SymmetricProperty. ?Q ?P ?R}| & \\verb|=>| & \\verb|{?R ?P ?Q}.| \\\\\n\\verb|{?P rdf:type owl:AsymmetricProperty. ?Q ?P ?R. ?R ?P ?Q }| & \\verb|=>| & \\verb|{@false}.| \\\\\n\\verb|{?P rdf:type owl:TransitiveProperty. ?Q ?P ?R. ?R ?P ?P}| & \\verb|=>| & \\verb|{?Q ?P ?P}.| \\\\\n\\verb|{?P1 owl:equivalentProperty ?P2. ?Q ?P1 ?R}| & \\verb|=>| & \\verb|{?Q ?P2 ?R}.| \\\\\n\\verb|{?P1 owl:equivalentProperty ?P2. ?Q ?P2 ?R}| & \\verb|=>| & \\verb|{?Q ?P1 ?R}.| \\\\\n\\verb|{?P1 owl:propertyDisjointWith ?P2. ?Q ?P1 ?R. ?Q ?P2 ?R}| & \\verb|=>| & \\verb|{@false}.| \\\\\n\\verb|{?P1 owl:inverseOf ?P2. ?Q ?P1 ?R}| & \\verb|=>| & \\verb|{?R ?P2 ?Q}.| \\\\\n\\verb|{?P1 owl:inverseOf ?P2 . ?Q ?P2 ?R}| & \\verb|=>| & \\verb|{?R ?P1 ?Q}.| \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[]\n\\caption{Inference rules for OWL-P classes}\n\\label{tab:rulesowl2}\n\\centering\n\\begin{tabular}{|lll|}\n\\hline\nConditions &  & Conclusions \\\\\n\\hline\n\\verb|{?A owl:equivalentClass ?B . ?x rdf:type ?A}| & \\verb|=>| & \\verb|{?x a ?B}.| \\\\\n\\verb|{?A owl:equivalentClass ?B . ?x rdf:type ?B}| & \\verb|=>| & \\verb|{?x a ?A}.| \\\\\n\\verb|{?A owl:disjointWith ?B.| & & \\\\\n\\verb|{ ?x rdf:type ?A. ?x rdf:type ?B}| & \\verb|=>| & \\verb|{@false}| \\\\\n\\verb|{?C rdf:type owl:Class}| & \\verb|=>| & \\verb|{?C rdfs:subClassOf ?C. ?C owl:Thing.| \\\\\n                               &           & \\verb| ?C owl:equivalentClass ?C.| \\\\\n                               &           & \\verb| owl:Nothing rdfs:subClassOf ?C}.| \\\\\n\\verb|{?A owl:equivalentClass ?B}| & \\verb|=>| & \\verb|{?A rdfs:subClassOf ?B. ?B rdfs:subClassOf ?A}.| \\\\\n\\verb|{?A rdfs:subClassOf ?B.| & & \\\\\n\\verb| ?B rdfs:subClassOf ?A}| & \\verb|=>| & \\verb|{?A owl:equivalentClass ?B}.| \\\\\n\\verb|{?P rdf:type owl:ObjectProperty}| & \\verb|=>| & \\verb|{?P rdfs:subPropertyOf ?P.| \\\\\n                                        &           & \\verb| ?P owl:equivalentProperty ?P}.| \\\\\n\\verb|{?P rdf:type owl:DatatypeProperty}| & \\verb|=>| & \\verb|{?P rdfs:subPropertyOf ?P.| \\\\\n                                          &           & \\verb| ?P owl:equivalentProperty ?P}.| \\\\\n\\verb|{?P owl:equivalentProperty ?R}| & \\verb|=>| & \\verb|{?P rdfs:subPropertyOf ?R.| \\\\\n                                        &           & \\verb| ?R rdfs:subPropertyOf ?P}.| \\\\\n\\verb|{?P rdfs:subPropertyOf ?R.| & & \\\\\n\\verb| ?R rdfs:subPropertyOf ?P}| & \\verb|=>| & \\verb|{?P owl:equivalentProperty ?R}.| \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\section{Related Work}\n\\label{sec:relatedwork}\nOne of the most important general purpose logic programming language is Prolog \\cite{Clocksin2003}. It is declarative, which means\nthat the program logic is declared in terms of relations, represented as facts and rules. Yet anoder declarative language\nis Datalog \\cite{Eiter1997}, which is syntactically a subset of Prolog. Apart from the Notation3, there are other rule-based \ninference engines formats for the Semantic Web, such as: FOL-RuleML \\cite{Boley2004}, SWRL \\cite{Horrocks2004}, RIF \\cite{Kifer2008}, \nR-DEVICE \\cite{Bassiliades2004}, TRIPLE \\cite{Sintek2001}, Jena rule\\footnote{\\url{http:\/\/jena.apache.org\/documentation\/inference}} and\nSPIN \\cite{Ryman2014}.\n\nFOL-RuleML (First-order Logic Rule Markup Language) \\cite{Boley2004} is a rule language for expressing first-order \nlogic for the web. It is a sublanguage of RuleML \\cite{Boley2012}. In FOL-RuleML each of rules consists of a set of statements \ncalled an atom. The atom is a form which consists of objects which are individuals or variables, and a relation between them.\n\nSWRL (Semantic Web Rule Language) \\cite{Horrocks2004} is based on OWL \\cite{Parsia2012} and Unary\/Binary Datalog RuleML, which sublanguage of \nthe RuleML. It extends the set of OWL axioms to include Horn-like rules. Logical operators and quantifications supports \nof SWRL are the same as in RuleML. Moreover, RuleML contents can be parts of SWRL content. Axioms may consist of OWL, RDF\nand rule axioms. A relation can be an IRI, a data range, an OWL property or a built-in relation. An object can be a variable, \nan individual, a literal value or a blank node.\n\nRIF (Rule Interchange Format) \\cite{Kifer2008} is a standard for exchanging rules among disparate systems. It focused on \nexchange rather than developing a single one-fits-all rule language. It can be separated into a number of parts, \nRIF-core \\cite{Reynolds2013} which is the common core of all RIF dialects, RIF-BLD (Basic Logic Dialect) \\cite{Kifer2013} comprising \nbasic dialects (i.e. Horn rules) for writing rules, RIF-PRD \\cite{Marie2013} (Production Rule Dialect) for representing production rules and \nRIF-DTB (Datatypes and Built-in Functions) \\cite{Polleres2013} comprising a set of datatypes and built-in functions.\n\nR-DEVICE \\cite{Bassiliades2004} is a deductive rule language for reasoning about RDF data. In R-DEVICE resources are represented \nas objects and RDF properties are realized as multi-slots. It supports a second-order syntax, where variables can range over \nclasses and properties. It provides a RuleML-like syntax.\n\nTRIPLE \\cite{Sintek2001} is an RDF rule (query, inference, and transformation) language, with a layered and modular nature. \nIt is based on Horn Logic \\cite{Horn1951} and F-Logic \\cite{Kifer1989}. Rules in TRIPLE are used for transient querying and \ncannot be used for defining and maintaining views.\n\nSPIN (SPARQL Inferencing Notation) \\cite{Ryman2014} is a constraint and SPARQL-based rule language for RDF. It can link class with \nqueriesto capture constraints and rules which describe the behavior of those classes. SPIN is also a method to represent queries as \ntemplates. It can represent SPARQL statement as RDF triples. That proposal allows to declare new SPARQL functions.\n\nJena rule is a rule format used only by inference engine in the Jena framework \\cite{Mcbride2002}. The rule language \nsyntax is based on RDF. It uses the triple representation, which is similar to Notation3 except that a rule name can be \nspecified in a rule. There are not any formula notation, and built-in functions are written in function terms. \n\n\\section{Conclusions}\nThis paper define how knowledge and logic might be handled on the Semantic Web environment. We present inference rules \nRDF, RDF Schema and OWL. All rules are tested in reasoning engines. Our formalization is based on Notation 3 Logic, which \nextended RDF by logical symbols and created a new Semantic Web logic. Moreover, we propose a lightweight OWL profile called OWL-P.\nOur proposed rule will be useful for deductive \nRDF graph stores.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nA fractional derivative is an operator that generalizes the ordinary derivative, in the sense that if\n\n\n\\begin{eqnarray*}\n\\dfrac{d^\\alpha}{d x^\\alpha},\n\\end{eqnarray*}\n\n\n\n\ndenotes the differential of order $ \\alpha $, then $\\alpha$ may be considered a parameter, with $\\alpha\\in \\nset{R}$, such that the first derivative corresponds to the particular case $\\alpha=1$. On the other hand, a fractional differential equation is an equation that involves at least one differential operator of order $ \\alpha $, with $(n-1)< \\alpha \\leq n$ for some positive integer $n$, and it is said to be a differential equation of order $\\alpha$ if this operator is the highest order in the equation. The fractional operators have many representations, but one of their fundamental properties is that they allow retrieving the results of conventional calculus when $\\alpha \\to n$. So, considering a scalar function $h: \\nset{R}^m \\to \\nset{R}$ and the canonical basis of $\\nset{R}^m$ denoted by $\\set{\\hat{e}_k}_{k\\geq 1}$, it is possible to define the following fractional operators of order $\\alpha$ using Einstein notation\n\n\n\n\\begin{eqnarray}\no^\\alpha h(x):=\\hat{e}_k o_k^\\alpha h(x).\n\\end{eqnarray}\n\n\n\n\n\nTherefore, denoting by  $\\partial_k^n$ the partial derivative of order $n$ applied with respect to the $k$-th component of the vector $x$, using the previous operator it is possible to define the following set of fractional operators\n\n\n\n\\begin{eqnarray}\\label{eq:0}\n\\Op_{x,\\alpha}^n(h):=\\set{o^\\alpha \\ : \\ \\exists \\partial_k^n h(x)  \\  \\mbox{ and } \\ \\lim_{\\alpha \\to n}o_k^\\alpha h(x)=\\partial_k^n h(x) \\ \\forall k\\geq 1 },\n\\end{eqnarray}\n\n\n\nwhich may be considered as a generating set of sets of fractional tensor operators. For example, considering $\\alpha,n\\in \\nset{R}^m$ with $\\alpha=\\hat{e}_k\\alpha_k$ and $n=\\hat{e}_k n_k$, it is possible to define the following set of fractional tensor operators\n\n\n\n\\begin{eqnarray}\n\\Op_{x,\\alpha}^{n}(h):=\\set{o^\\alpha \\ : \\ o^\\alpha \\in \\Op_{x,\\alpha_1}^{n_1}(h) \\times \\Op_{x,\\alpha_2}^{n_2}(h)\\times \\cdots \\times \\Op_{x,\\alpha_m}^{n_m}(h) }.\n\\end{eqnarray}\n\n\nOne of the most famous fixed point methods is the well-known Newton-Raphson method. However, it sometimes goes unnoticed that this method has the following problem related to finding roots of polynomials in the complex space: If it is necessary to find a complex root $\\xi$ of a polynomial using the Newton-Raphson method, with $\\xi\\in \\nset{C}\\setminus\\nset{R}$, a complex initial condition $x_0$ must be provided, and if a suitable initial condition is selected, this will lead to a complex solution, but there is also the possibility that this may lead to a real solution. If the root obtained is real, it is necessary to change the initial condition and expect that this will lead to a complex solution, otherwise, it is necessary to change the value of the initial condition again, this process is repeated until it finally converges to a complex solution. The process described above is very similar to what happens when different values $\\alpha$ are used in fractional operators until we find a solution that meets some established criterion. Considering the Newton-Raphson method from the perspective of fractional calculus, it is possible to consider that an order $\\alpha$ remains fixed, in this case $\\alpha=1$, and the initial conditions $x_0$ are varied until found a solution $\\xi$ that fulfills an established criterion. It is necessary to mention that considering a relationship between fractional calculus and the Newton-Raphson method may seem somewhat forced at first, but the latter is characterized by the fact that when it generates divergent sequences of complex numbers, it can sometimes lead to the creation of fractals \\cite{tatham}, and this feature is complemented quite well with the fact that the orders of the fractional derivatives seem to be closely related to the fractal dimension  \\cite{brambila2017fractal}. Based on the above, it is possible to consider inverting the behavior of the order $\\alpha=1$ of the derivative  and the initial condition $x_0$, that is, leaving the initial condition $x_0$ fixed and varying the order $\\alpha$ of the derivative, thus obtaining the \\textbf{fractional Newton-Raphson method}  \\cite{torres2021fracnewrap}, which is nothing other than the Newton-Raphson method using any definition of the fractional derivative that fits the function whose zeros want to be determined. \n\n\n\n\\begin{figure}[!ht]\n        \\begin{subfigure}[c]{0.35\\textwidth}\n        \\centering\n \\includegraphics[width=\\textwidth, height=0.6\\textwidth]{rN077}      \n    \\caption*{a) $\\alpha=-0.77$}\n    \\end{subfigure}\n        \\begin{subfigure}[c]{0.35\\textwidth}\n        \\centering\n \\includegraphics[width=\\textwidth, height=0.6\\textwidth]{rN032}     \n    \\caption*{b) $\\alpha=-0.32$}\n    \\end{subfigure} \n    \\centering\n    \\begin{subfigure}[c]{0.35\\textwidth}\n    \\centering\n \\includegraphics[width=\\textwidth, height=0.6\\textwidth]{r019}      \n    \\caption*{c) $\\alpha=0.19$}\n    \\end{subfigure}\n    \\begin{subfigure}[c]{0.35\\textwidth}\n    \\centering\n \\includegraphics[width=\\textwidth, height=0.6\\textwidth]{r187}      \n    \\caption*{d) $\\alpha=1.87$}\n    \\end{subfigure}        \n        \\caption{Illustrations of some trajectories generated by the fractional Newton-Raphson method for the same initial condition $x_0$ but with different orders $\\alpha$ of the fractional operator used \\cite{torres2021fracnewrap}.}\\label{fig:01}\n\\end{figure}\n\n\nBefore continuing, it is necessary to mention that due to the large number of fractional operators that can exist, some sets need to be defined to fully characterize the fractional Newton-Raphson method. So, considering a function $h: \\Omega \\subset \\nset{R}^m \\to \\nset{R}^m$, it is possible to define the following set of fractional operators\n\n\n\\begin{eqnarray}\n{}_m\\Op_{x,\\alpha}^n(h):=\\set{ o^\\alpha \\ : \\ o^\\alpha\\in \\Op_{x,\\alpha}^n\\left([h]_j \\right) \\ \\forall j \\mbox{ with } 1\\leq j\\leq m},\n\\end{eqnarray}\n\n\n\nwhere $[h]_k: \\Omega \\subset \\nset{R}^m \\to \\nset{R}$ denotes the $k$-th component of the function $h$. So, it is possible to define the following set of matrices\n\n\n\\begin{eqnarray}\n{}_{m}\\Ma_{x,\\alpha}^{n}(h):=\\set{A=A(o^\\alpha) \\ : \\   o^\\alpha \\in {}_m\\Op_{x,\\alpha}^n(h) \\ \\mbox{ and } \\ A(x)=\\left([A]_{jk}(x) \\right):= \\left( o_k^\\alpha [h]_j(x)\\right)  },\n\\end{eqnarray}\n\n\nand therefore, the fractional Newton-Raphson method can be defined and classified through the following set of  matrices\n\n\n\\begin{eqnarray}\n{}_{m}\\IMa_{x,\\alpha}^{1}(h):=\\set{A \\ : \\   \\exists A^{-1} \\in  {}_mM_{x,\\alpha}^{1}(h) },\n\\end{eqnarray}\n\nand as a consequence, if $\\Phi_{FNR}$ denotes the iteration function of the fractional Newton-Raphson method, it is possible to obtain the following result:\n\n\n\n\n\\begin{eqnarray}\n\\mbox{Let }\\alpha_0\\in \\nset{R}\\setminus \\nset{Z} \\Rightarrow \\forall A \\in {}_{m}\\IMa_{x,\\alpha_0}^{1}(h) \\hspace{0.1cm}  \\exists\\Phi_{FNR}=\\Phi_{FNR}(\\alpha_0, A) \\  \\therefore  \\ \\forall A \\hspace{0.1cm}  \\exists  \\set{\\Phi_{FNR}(\\alpha, A) \\ : \\ \\alpha \\in \\nset{R}\\setminus \\nset{Z}}.  \n\\end{eqnarray}\n\n\n\n\nThe change from leaving the initial condition $x_0$ fixed and varying the order $\\alpha$ of the derivative, although seemingly simple, gives the fractional Newton-Raphson method the ability to partially solve the intrinsic problem associated with classical fixed point methods, which is that in general, to find $N$ zeros of a function, $N$ initial conditions must be provided. This is because by varying the order $\\alpha$ of the fractional derivative, the fractional Newton-Raphson method can find $N$ zeros of a function using a single initial condition as shown in the Figure \\ref{fig:01}. It is necessary to consider that mentioned above is also valid for any fixed point method that implements fractional operators in some way, which may be named as \\textbf{fractional fixed point methods} or \\textbf{fractional iterative methods}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nTo finish this section, it is necessary to mention that the applications of fractional operators have spread to different fields of science such as finance \\cite{safdari2015radial,sabatelli2002waiting}, economics \\cite{traore2020model,torres2020nonlinear}, number theory through the Riemann zeta function \\cite{guariglia2021fractional,torres2021zeta} and in engineering with the study for the manufacture of hybrid solar receivers \\cite{de2021fractional,torres2020reduction,torres2020fracpseunew,torres2021codefracpseudo}. It should be mentioned that there is also a growing interest in fractional operators and their properties for the solution of nonlinear systems \\cite{wang2021derivative,gdawiec2019visual,akgul2019fractional,torres2021fracnewrap,torres2020fracsome,torres2021fracnewrapaitken}, which is a classic problem in mathematics, physics and engineering, which consists of finding the set of zeros of a function $f:\\Omega \\subset \\nset{R}^n \\to \\nset{R}^n$, that is,\n\n\n\\begin{eqnarray}\\label{eq:1-001}\n\\set{\\xi \\in \\Omega \\ : \\ \\norm{f(\\xi)}=0},\n\\end{eqnarray}\n\n\nwhere $\\norm{ \\ \\cdot \\ }: \\nset{R}^n \\to \\nset{R}$ denotes any vector norm, or equivalently\n\n\n\\begin{eqnarray}\n\\set{\\xi \\in \\Omega \\ : \\ [f]_k(\\xi)=0 \\ \\forall k\\geq 1}.\n\\end{eqnarray}\n\n\n\n\n\n\n\n\n\nAlthough finding the zeros of a function may seem like a simple problem, it is generally necessary to use numerical methods of the iterative type to solve it. So, considering that fractional iterative methods can find $N$ solutions of a system using a single initial condition, this article shows an alternative way to the Aitken's method to accelerate the order of convergence of a family of fractional fixed point methods, which consists of implementing a function in the order of the fractional operators involved, with which it is possible to obtain an order of convergence at least quadratic.\n\n\n\n\n\n\n\n\n\\section{Fixed Point Method}\n\n\n\nLet  $\\Phi:\\nset{R}^n \\to \\nset{R}^n$ be a function. It is possible to build a sequence $\\set{x_i}_{\\geq 1}$  by defining the following iterative method\n\n\\begin{eqnarray}\\label{eq:2-001}\nx_{i+1}:=\\Phi(x_i), & i=0,1,2,\\cdots,\n\\end{eqnarray}\n\n\n\n\nif it is fulfilled that $x_i\\to \\xi\\in \\nset{R}^n$ and if the function $\\Phi$ is continuous around $\\xi$, we obtain that\n\n\n\\begin{eqnarray}\\label{eq:2-002}\n\\ds \\xi=\\lim_{i\\to \\infty}x_{i+1}=\\lim_{i\\to \\infty}\\Phi(x_i)=\\Phi\\left(\\lim_{i\\to \\infty}x_i \\right)=\\Phi(\\xi),\n\\end{eqnarray}\n\n\n\n\nthe above result is the reason by which the method \\eqref{eq:2-001} is known as the \\textbf{fixed point method}. Furthermore, the function $\\Phi$ is called an \\textbf{iteration function}. The following corollary allows characterizing the order of convergence of an iteration function $ \\Phi $ through its \\textbf{Jacobian matrix} $\\Phi^{(1)}$ \\cite{torres2021fracnewrapaitken}:\n\n\\begin{corollary}\\label{cor:2-001}\nLet $\\Phi:\\nset{R}^n \\to \\nset{R}^n$ be an iteration function. If $\\Phi$ defines a sequence $\\set{x_i}_{\\geq 1}$ such that $x_i\\to \\xi\\in \\nset{R}^n$. So, $\\Phi$ has an \\textbf{order of convergence}  of order (at least) $p$  in $B(\\xi;\\delta)$, where \n\n\n\n\n\n\\begin{eqnarray}\\label{eq:2-003}\np:=\\left\\{\n\\begin{array}{cc}\n1 , &\\ds \\mbox{If } \\lim_{x\\to \\xi}\\norm{\\Phi^{(1)}(x)}\\neq 0  \\vspace{0.2cm}\\\\\n2, &\\ds \\mbox{If } \\lim_{x\\to \\xi}\\norm{\\Phi^{(1)}(x)}= 0  \\\\\n\\end{array}\\right. .\n\\end{eqnarray}\n\n\n\\end{corollary}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Riemann-Liouville Fractional Operators}\n\n\nOne of the fundamental operators of fractional calculus is the operator \\textbf{Riemann-Liouville fractional integral},  which is defined as follows \\cite{hilfer00,oldham74}\n\n\n\n\\begin{eqnarray}\n\\ifr{}{a}{I}{x}{\\alpha}f(x):=\\dfrac{1}{\\gam{\\alpha}}\\int_a^x (x-t)^{\\alpha-1}f(t)dt,\n\\end{eqnarray}\n\n\n\n\n\nwhich is a fundamental piece to build the operator \\textbf{Riemann-Liouville fractional derivative},  which is defined as follows \\cite{hilfer00,kilbas2006theory}\n\n\\begin{eqnarray}\\label{eq:3-001}\n\\normalsize\n\\begin{array}{c}\n\\ifr{}{a}{D}{x}{\\alpha}f(x) := \\left\\{\n\\begin{array}{cc}\n\\ds \\ifr{}{a}{I}{x}{-\\alpha}f(x), &\\mbox{if }\\alpha<0 \\vspace{0.1cm}\\\\  \n\\ds \\dfrac{d^n}{dx^n}\\left( \\ifr{}{a}{I}{x}{n-\\alpha}f(x)\\right), & \\mbox{if }\\alpha\\geq 0\n\\end{array}\n\\right.\n\\end{array}, \n\\end{eqnarray}\n\n\n\n\nwhere  $ n = \\lceil \\alpha \\rceil$ and $\\ifr{}{a}{I}{x}{0}f(x):=f(x)$. Applying the operator \\eqref{eq:3-001} with $a=0$ to the  function $ x^{\\mu} $, with $\\mu> -1$, we obtain the following result \\cite{torres2021fracnewrapaitken}:\n\n\n\\begin{eqnarray}\\label{eq:3-002}\n\\ifr{}{0}{D}{x}{\\alpha}x^\\mu = \n \\dfrac{\\gam{\\mu+1}}{\\gam{\\mu-\\alpha+1}}x^{\\mu-\\alpha}, & \\alpha\\in \\nset{R}\\setminus \\nset{Z}.\n\\end{eqnarray}\n\n\n\n\n\n\\section{Fractional Fixed Point Method}\n\nLet $f:\\Omega \\subset \\nset{R}^n \\to \\nset{R}^n$ be a function with a point $\\xi\\in \\Omega$ such that $\\norm{f(\\xi)}=0$ . So, considering an iteration function $\\Phi:(\\nset{R}\\setminus \\nset{Z})\\times \\nset{R}^n\\to \\nset{R}^n$, the iteration function of a fractional iterative method may be written in general form as follows\n\n\n\\begin{eqnarray}\\label{eq:4-001}\n\\Phi(\\alpha,x):=x-A_{g,\\alpha}(x)f(x),& \\alpha\\in \\nset{R}\\setminus \\nset{Z},\n\\end{eqnarray}\n\n\n\nwhere $A_{g,\\alpha}$ is a matrix that depends, in at least one of its entries, on fractional operators of order $\\alpha$ applied to some function $g:\\nset{R}^n \\to \\nset{R}^n$, whose particular case occurs when $g=f$. So, it is possible to define in a general way a fractional fixed point method as follows\n\n\n\\begin{eqnarray}\\label{eq:4-002}\nx_{i+1}:=\\Phi(\\alpha,x_i), & i=0,1,2,\\cdots.\n\\end{eqnarray}\n\nIf it is fulfilled that $x_i\\to \\xi\\in \\Omega$, it is possible to define the following set\n\n\\begin{eqnarray}\n\\Conv(\\xi):=\\set{\\Phi \\ : \\ \\lim_{x\\to \\xi}\\Phi(\\alpha,x)=\\xi},\n\\end{eqnarray}\n\n\nwhich may be interpreted as the set of fractional fixed point methods that define a convergent sequence $\\set{x_i}_{i\\geq 1}$ to the value $\\xi \\in \\Omega$. Considering the \\mref{Corollary}{\\ref{cor:2-001}}, as well as the \\textbf{Proposition 1} from the reference \\cite{torres2021fracnewrapaitken}, it is possible to define the following sets to classify the order of convergence of some fractional iterative methods\n\n\n\n\n\n\\begin{eqnarray}\n\\Ord^1(\\xi):=\\set{\\Phi \\in \\Conv(\\xi) \\ : \\ \\lim_{x \\to \\xi}\\norm{\\Phi^{(1)}(\\alpha,x)}\\neq 0 },\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\Ord^2(\\xi):=\\set{\\Phi \\in \\Conv(\\xi) \\ : \\ \\lim_{x \\to \\xi}\\norm{\\Phi^{(1)}(\\alpha,x)}= 0 },\n\\end{eqnarray}\n\n\n\n\\begin{eqnarray}\n\\ord^1(\\xi):=\\set{\\Phi \\in \\Conv(\\xi) \\ : \\ \\lim_{x \\to \\xi}A_{g,\\alpha}(x)\\neq \\left(f^{(1)}(\\xi)\\right)^{-1} \\ \\mbox{ or } \\ \\lim_{\\alpha \\to 1} A_{g,\\alpha}(\\xi)\\neq \\left(f^{(1)}(\\xi)\\right)^{-1} },\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\ord^2(\\xi):=\\set{\\Phi \\in \\Conv(\\xi) \\ : \\ \\lim_{x \\to \\xi}A_{g,\\alpha}(x)= \\left(f^{(1)}(\\xi)\\right)^{-1} \\ \\mbox{ or } \\ \\lim_{\\alpha \\to 1} A_{g,\\alpha}(\\xi)= \\left(f^{(1)}(\\xi)\\right)^{-1}}.\n\\end{eqnarray}\n\n\n\nOn the other hand, considering that depending on the nature of the function $f$ there exist cases in which the Newton-Raphson method can present an order of convergence (at least) linear \\cite{torres2021fracnewrapaitken}, it is possible to obtain the following relations between the previous sets\n\n\n\\begin{eqnarray}\n\\ord^1(\\xi) \\subset \\Ord^1(\\xi)& \\mbox{ and } &\\ord^2(\\xi)\\subset \\Ord^1(\\xi)\\cup \\Ord^2(\\xi),\n\\end{eqnarray}\n\nwith which it is possible to define the following sets\n\n\n\n\\begin{eqnarray}\n\\Ord_2^1(\\xi) := \\ord^2(\\xi)   \\cap  \\Ord^1(\\xi) & \\mbox{ and } & \\Ord_2^2(\\xi):= \\ord^2(\\xi) \\cap \\Ord^2(\\xi). \n\\end{eqnarray}\n\n\n\n\n\n\\subsection{Acceleration in the Order of Convergence of the Set $\\Ord_2^1(\\xi)$}\n\n\nLet $f:\\Omega \\subset \\nset{R}^n \\to \\nset{R}^n$ be a function with a point $\\xi\\in \\Omega$ such that $\\norm{f(\\xi)}=0$, and denoting by $\\Phi_{NR}$ to the iteration function of the Newton-Raphson method, it is possible to define the following set of functions\n\n\n\n\\begin{eqnarray}\n\\Ord_{NR}^2(\\xi):=\\set{ f \\ : \\  \\lim_{x\\to \\xi}\\norm{\\Phi_{NR}^{(1)}(x)}=0 }.\n\\end{eqnarray}\n\n\nSo, it is possible to define the following corollary:\n\n\n\\begin{corollary}\\label{cor:4-001}\nLet $f:\\Omega \\subset \\nset{R}^n \\to \\nset{R}^n$ be a function with a point $\\xi\\in \\Omega$ such that $f\\in \\Ord_{NR}^2(\\xi)$, and let $\\Phi$ be a iteration function given by the equations \\eqref{eq:4-001} such that $\\Phi\\in \\Ord_2^1(\\xi)$. So, it is possible to replace the order $\\alpha$ of the fractional operators of the matrix $A_{g,\\alpha}$ by the following function\n\n\n\n\n\\begin{eqnarray}\\label{eq:4-005}\n\\alpha_f([x]_k,x):=\\left\\{\n\\begin{array}{cc}\n\\alpha ,& \\mbox{if \\hspace{0.1cm}}\\abs{[x]_k}\\neq 0 \\mbox{ \\hspace{0.1cm}and\\hspace{0.1cm} } \\norm{f(x)}> \\delta \\vspace{0.1cm}\\\\\n1,& \\mbox{if \\hspace{0.1cm}}\\abs{[x]_k} = 0 \\mbox{ \\hspace{0.1cm}or\\hspace{0.1cm} } \\norm{f(x)}\\leq \\delta\n\\end{array}\n\\right. ,\n\\end{eqnarray}\n\n\n\n\nobtaining a new matrix that may be denoted as follows\n\n\n\n\\begin{eqnarray}\\label{eq:4-006}\nA_{g,\\alpha_f}(x)=\\left([A_{g,\\alpha_f}]_{jk}(x) \\right), & \\alpha\\in \\nset{R}\\setminus \\nset{Z},\n\\end{eqnarray}\n\nand that guarantees that there exists a set $\\Omega_\\xi \\subset B(\\xi;\\delta)$ such that $\\Phi\\in \\Ord_2^2(\\xi)$ in $\\Omega_\\xi$.\n\n\n\\end{corollary}\n\n\n\nIt is necessary to mention that the origin of the function \\eqref{eq:4-005} arises from the need to accelerate the order of convergence of the fractional Newton-Raphson method, which generated the method known as the \\textbf{fractional Newton method}, whose matrix $A_{g,\\alpha_f}$ corresponds to a particular case in which $g=f$ \\cite{torres2020fracsome,torres2021fracnewrap,torres2021fracnewrapaitken}. Finally, for practical purposes, it may be defined that if a fractional iterative method $\\Phi \\in \\Ord_2^1(\\xi)$ uses the function \\eqref{eq:4-005}, it may be called a \\textbf{fractional iterative method accelerated}.\n\n\n\n\n\\section{Equations of a Hybrid Solar Receiver}\n\n\nConsidering the notation\n\n\n\\begin{eqnarray*}\ns=(T_{cell},T_{hot},T_{cold},\\eta_{cell},\\eta_{TEG})^T:=\\left([x]_1,[x]_2,[x]_3,[x]_4,[x]_5 \\right)^T,\n\\end{eqnarray*}\n\n\n\n\nthe following expressions\n\n\n\n\\begin{eqnarray*}\n\\begin{array}{c}\n\\begin{array}{lll}\na_0=\\dfrac{2*r_{intercon}}{\\cdot \\sqrt{f^*\\cdot A_{TEG}}\\left(b\\cdot \\sqrt{f^*}+\\sqrt{A_{TEG}} \\right) }, & a_1=\\eta_{opt}\\cdot C_g \\cdot DNI, & a_2=r_{cell}+r_{sol}+A_{cell}\\left(\\dfrac{r_{cop}+r_{cer}}{A_{TEG}}+a_0 \\right)\n\\end{array}\\vspace{0.2cm}\\\\\n\\begin{array}{llll}\na_3=\\dfrac{A_{cell}\\cdot l}{f^*\\cdot A_{TEG}\\cdot k_{TEG}}, & a_4=T_{air}, &a_5=A_{cell}\\left( \\dfrac{r_{cer}}{A_{TEG}}+R_{heat\\_exch}+a_0 \\right),&a_6=-\\eta_{cell,ref}\\cdot \\gamma_{cell}\n\\end{array}\\vspace{0.2cm}\\\\\n\\begin{array}{lll}\n a_7=\\eta_{cell,ref}\\left(1+25 \\cdot \\gamma_{cell} \\right), &a_8=\\sqrt{1+ZT}, &a_9=273.15\n\\end{array}\n\\end{array},\n\\end{eqnarray*}\n\n\nand the following particular values \\cite{rodrigo2019performance}\n\n\\begin{eqnarray*}\n\\left\\{\n\\begin{array}{lll}\n\\eta_{opt}=0.85, & r_{intercon}=2.331\\times 10^{-7} ,&C_g=800\\\\\n    A_{cell}=9\\times 10^{-6}     , &  R_{heat\\_exch}=0.5   , & A_{TEG}=5.04 \\times 10^{-5}     \\\\\n \\eta_{cell,ref}=0.43,& r_{cell}=3\\times 10^{-6}   , &    f^*=0.7  \\\\\n \\gamma_{cell}=4.6\\times 10^{-4}, &  r_{sol}=1.603\\times 10^{-6}  , &  b=5\\times 10^{-4}    \\\\\nr_{cop}=7.5\\times 10^{-7}, & r_{cer}=8 \\times 10^{-6},&    l=5\\times 10^{-4}   \\\\\nk_{TEG}=1.5 ,&   ZT=1   & \n\\end{array}\n\\right..\n\\end{eqnarray*}\n\n\n\n\nit is possible to define the following system of equations that corresponds to the combination of a solar photovoltaic system with a thermoelectric generator system \\cite{bjork2015performance,bjork2018maximum}, which is named as a \\textbf{hybrid solar receiver}\n\n\n\\begin{eqnarray}\\label{eq:5-001}\n\\left\\{\n\\begin{array}{l}\n\\left[x\\right]_1=[x]_2+a_1\\cdot a_2\\left( 1-[x]_4 \\right)\\\\\n\\left[x\\right]_2=[x]_3+a_1\\cdot a_3 \\left( 1-[x]_4\\right)\\left(1-[x]_5\\right)\\\\\n\\left[x\\right]_3=a_4+a_1\\cdot a_5 \\left( 1-[x]_4\\right)\\left(1-[x]_5\\right)\\\\\n\\left[x\\right]_4 =a_6[x]_1+a_7\\\\\n\\left[x\\right]_5=(a_8-1)\\left(1-\\dfrac{[x]_3+a_9}{[x]_2+a_9} \\right)\\left(a_8+ \\dfrac{[x]_3+a_9}{[x]_2+a_9}\\right)^{-1}\n\\end{array}\\right.,\n\\end{eqnarray}\n\n\nwhose deduction, as well as details about its interpretation, may be found in the reference \\cite{rodrigo2019performance}. Using the system of equations \\eqref{eq:5-001}, it is possible to define a  function $f_1:\\Omega \\subset \\nset{R}^5\\to \\nset{R}^5$, that is,\n\n\n\\begin{eqnarray}\\label{eq:5-002}\nf_1(s):=\\begin{pmatrix}\n\\left[x\\right]_1-[x]_2-a_1\\cdot a_2\\left( 1-[x]_4 \\right)\\\\\n\\left[x\\right]_2-[x]_3-a_1\\cdot a_3 \\left( 1-[x]_4\\right)\\left(1-[x]_5\\right)\\\\\n\\left[x\\right]_3-a_4-a_1\\cdot a_5 \\left( 1-[x]_4\\right)\\left(1-[x]_5\\right)\\\\\n\\left[x\\right]_4 -a_6[x]_1-a_7\\\\\n\\left[x\\right]_5-(a_8-1)\\left(1-\\dfrac{[x]_3+a_9}{[x]_2+a_9} \\right)\\left(a_8+ \\dfrac{[x]_3+a_9}{[x]_2+a_9}\\right)^{-1}\n\\end{pmatrix}, \n\\end{eqnarray}\n\n\nwhich depends on two parameters, the direct normal irradiance ($DNI$) and the ambient temperature ($T_{air}$). These parameters are measured in real-time at certain times of the day \\cite{rodrigo2019performance}, and it is necessary to calculate a new solution of the system \\eqref{eq:5-001} for each new pair of parameters, that is,\n\n\n\\begin{eqnarray*}\n(DNI,T_{air})\\overset{f_1}{\\longrightarrow} s \\in \\nset{R}^5.\n\\end{eqnarray*}\n\n\n\nHowever, to simplify the task of finding the solutions of the function \\eqref{eq:5-002}, it is possible through the consecutive substitution of the variables $[x]_1, \\ [x]_4, \\ [x]_5$ and some algebraic simplifications, to obtain the following transcendental system \\cite{torres2020reduction}\n\n\n\\begin{eqnarray}\\label{eq:5-003}\n\\left\\{\n\\begin{array}{l}\n\\left[x\\right]_2=[x]_3-a_1\\cdot a_3 \\dfrac{\\left( a_6 [x]_2 + a_7 - 1 \\right) \\left( a_8 \\left([x]_3+ a_9 \\right) + \\left([x]_2+ a_9\\right)  \\right)  }{(1+a_1 a_2 a_6 ) \\left( a_8 \\left( [x]_2 + a_9  \\right) + \\left([x]_3+ a_9\\right) \\right) } \\vspace{0.1cm} \\\\\n\\left[x \\right]_3=a_4-a_1\\cdot a_5 \\dfrac{\\left( a_6 [x]_2 + a_7 - 1 \\right) \\left( a_8 \\left([x]_3+ a_9 \\right) + \\left([x]_2+ a_9\\right)  \\right)  }{(1+a_1 a_2 a_6 ) \\left( a_8 \\left( [x]_2 + a_9  \\right) + \\left([x]_3+ a_9\\right) \\right) } \n\\end{array}\\right.,\n\\end{eqnarray}\n\nwhose solution allows to know the values of the variables $[x]_1,[x]_4$ and $[x]_5$ through the following equations\n\n\\begin{eqnarray}\\label{eq:5-004}\n\\left\\{\n\\begin{array}{l}\n\\left[x\\right]_1=\\dfrac{[x]_2 - a_1 a_2 (a_7 - 1)}{1+a_1 a_2 a_6} \\vspace{0.1cm}\\\\\n\\left[ x \\right]_4=\\dfrac{a_6 \\left( a_1 a_2 + [x]_2\\right) + a_7}{1+ a_1 a_2 a_6 } \\vspace{0.1cm}\\\\\n\\left[ x \\right]_5=\\dfrac{(a_8 - 1)\\left( [x]_2 - [x]_3 \\right)}{a_8 \\left([x]_2+a_9\\right) + \\left( [x]_3+a_9 \\right)}\n\\end{array}\\right..\n\\end{eqnarray}\n\n\n\n\n\n\n\n\nUsing the system of equations \\eqref{eq:5-003}, it is possible to define a  function $f_2:\\Omega \\subset \\nset{R}^2\\to \\nset{R}^2$, that is,\n\n\n\\begin{eqnarray}\\label{eq:5-005}\nf_2(x):=\\begin{pmatrix}\n\\left[x\\right]_2-[x]_3+a_1\\cdot a_3 \\dfrac{\\left( a_6 [x]_2 + a_7 - 1 \\right) \\left( a_8 \\left([x]_3+ a_9 \\right) + \\left([x]_2+ a_9\\right)  \\right)  }{(1+a_1 a_2 a_6 ) \\left( a_8 \\left( [x]_2 + a_9  \\right) + \\left([x]_3+ a_9\\right) \\right) } \\vspace{0.1cm} \\\\\n\\left[x \\right]_3-a_4+a_1\\cdot a_5 \\dfrac{\\left( a_6 [x]_2 + a_7 - 1 \\right) \\left( a_8 \\left([x]_3+ a_9 \\right) + \\left([x]_2+ a_9\\right)  \\right)  }{(1+a_1 a_2 a_6 ) \\left( a_8 \\left( [x]_2 + a_9  \\right) + \\left([x]_3+ a_9\\right) \\right) } \n\\end{pmatrix}, \n\\end{eqnarray}\n\n\nand then finding the solutions of the function \\eqref{eq:5-005}, through the equations \\eqref{eq:5-004}, it is possible to construct the solutions of the function \\eqref{eq:5-002}.\n\n\n\n\\subsection{Solutions of the Equations of a Hybrid Solar Receiver}\n\nTo solve the equation \\eqref{eq:5-005} and at the same time solve the equation \\eqref{eq:5-002}, a fractional fixed point method will be used as well as its accelerated version through the function \\eqref{eq:4-005}. Before continuing, it is necessary to mention that for some definitions of fractional operators it is fulfilled that the derivative of order $\\alpha$ of a constant is different from zero (for example: Riesz, Gr\u00fcnwald\u2013Letnikov, Riemann-Liouville, etc.\\cite{miller93,hilfer00,oldham74,kilbas2006theory,brambila2017fractal}), that is, \n\n\n\\begin{eqnarray}\\label{eq:5-006}\n\\partial_k^\\alpha c :=\\der{\\partial}{[x]_k}{\\alpha}c \\neq 0 , & c=constant.\n\\end{eqnarray}\n\n\nSo, considering a function $f:\\Omega \\subset \\nset{R}^n \\to \\nset{R}^n$ with a point $\\xi \\in \\Omega$ such that $\\norm{f(\\xi)}=0$, the Riemann-Liouville fractional derivative given by the equation \\eqref{eq:3-002}, and an iteration function $\\Phi:(\\nset{R}\\setminus \\nset{Z})\\times \\nset{R}^n \\to \\nset{R}^n$, it is possible to define the following fractional fixed point method\n\n\\begin{eqnarray}\\label{eq:5-007}\nx_{i+1}:=\\Phi(\\alpha,x_i)=x_i-A_{g_f,\\beta}(x_i)f(x_i), & i=0,1,2,\\cdots,\n\\end{eqnarray}\n\nwhere $ A_{g_f,\\beta}(x_i) $ is given by the following expression\n\n\\begin{eqnarray}\\label{eq:5-008}\nA_{g_f,\\beta}(x_i)=\\left([A_{g_f,\\beta}]_{jk}(x_i) \\right) :=\\left( \\partial_k^{\\beta(\\alpha,[x]_k)}[g_f]_{j}(x)  \\right)_{x_i}^{-1}, & \\alpha\\in \\nset{R}\\setminus \\nset{Z},\n\\end{eqnarray}\n\nwith $ g_{f}(x) $  and $ \\beta (\\alpha, [x]_k) $ functions defined as follows\n\n\n\\begin{eqnarray}\ng_{f}(x):=f(x_i)+f^{(1)}(x_i)x& \\mbox{ and }\n&\n\\beta(\\alpha,[x]_k):=\\left\\{\n\\begin{array}{cc}\n\\alpha, &\\mbox{if \\hspace{0.1cm} } \\abs{  [x]_k }\\neq 0 \\vspace{0.1cm}\\\\\n1,& \\mbox{if \\hspace{0.1cm}  }  \\abs{  [x]_k  }=0\n\\end{array}\\right..\n\\end{eqnarray}\n\n\n\nThe fractional iterative method given by the equation \\eqref{eq:5-007} is named the \\textbf{fractional quasi-Newton method} \\cite{torres2020fracsome}, if it is assumed that $\\Phi \\in \\Conv(\\xi)$ then $\\Phi \\in \\ord^1(\\xi)$. Furthermore, the method fulfills the following condition\n\n\n\\begin{eqnarray}\n\\lim_{\\alpha \\to 1}\\partial_k^{\\beta(\\alpha,[x_i]_k)} [g]_j(x_i)=\\partial_k[f]_j(x_i), & 1\\leq j,k\\leq n,\n\\end{eqnarray}\n\n\nand as a consequence $\\Phi \\in \\Ord_{2}^1(\\xi)$. So, if it is assumed that $f\\in \\Ord_{NR}^2(\\xi)$, by the \\mref{Corollary}{\\ref{cor:4-001}} it is possible to construct the \\textbf{fractional quasi-Newton method accelerated} using the following matrix\n\n\n\\begin{eqnarray}\\label{eq:5-009}\nA_{g_f,\\alpha_f}(x_i)=\\left([A_{g_f,\\alpha_f}]_{jk}(x_i) \\right) :=\\left( \\partial_k^{\\alpha_f([x]_k,x)}[g_f]_{j}(x)  \\right)_{x_i}^{-1}, & \\alpha\\in \\nset{R}\\setminus \\nset{Z}.\n\\end{eqnarray}\n\n\n\nBefore continuing, it is necessary to mention that a description of the algorithm that must be implemented when working with a fractional iterative method given by the equation \\eqref{eq:4-002} may be found in the reference \\cite{torres2020fracsome}. On the other hand, simplified examples of how the methods given by the matrices \\eqref{eq:5-008} and \\eqref{eq:5-009} should be programmed may be found in the references \\cite{torres2021codefracquasi,torres2021codeaccelfracquasi}. Using the fractional fixed point methods defined by the matrices \\eqref{eq:5-008} and \\eqref{eq:5-009}, we proceed to find three solutions of the function \\eqref{eq:5-005}  leaving the following fixed values\n\n\n\\begin{eqnarray*}\n\\delta=13& \\mbox{ and } & x_0=(3000,3000)^T.\n\\end{eqnarray*}\n\n\n\n\n\\begin{example}\n\nConsidering by hypothesis that $f_2\\in \\Ord_{NR}^2(\\xi)$, and using the following values\n\n\n\\begin{eqnarray*}\nDNI=900, &\nTair=20, & \\alpha=0.89825,\n\\end{eqnarray*}\n\n\nthe following iterations are obtained by using the fractional iterative methods given by the matrices  \\eqref{eq:5-008} and \\eqref{eq:5-009}.\n\n\n\\begin{itemize}\n\\item[i)] $A_{g_f,\\beta} \\Rightarrow \\Phi \\in \\Ord_2^1(\\xi)$.\n\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2048.526273 & 2036.688326 & 1.35E+03 & 2.01E+03 & 2052.245932 & 0.02901075 & 0.00087668 & 2.01E+03 \\\\\n    2     & 1378.380727 & 1357.837031 & 9.54E+02 & 1.33E+03 & 1381.592211 & 0.16166606 & 0.00214528 & 1.33E+03 \\\\\n    3     & 914.5756647 & 887.7554749 & 6.60E+02 & 8.65E+02 & 917.4354426 & 0.25347627 & 0.00391089 & 8.65E+02 \\\\\n    4     & 599.7868499 & 568.5654338 & 4.48E+02 & 5.46E+02 & 602.4079218 & 0.31578871 & 0.00622874 & 5.46E+02 \\\\\n    5     & 390.7721777 & 356.5990844 & 2.98E+02 & 3.34E+02 & 393.2347526 & 0.35716317 & 0.0090235 & 3.34E+02 \\\\\n    6     & 255.3927888 & 219.4044444 & 1.93E+02 & 1.97E+02 & 257.7527048 & 0.38396151 & 0.0120214 & 1.97E+02 \\\\\n    7     & 170.1536777 & 133.2761535 & 1.21E+02 & 1.11E+02 & 172.4489564 & 0.4008346 & 0.01478215 & 1.11E+02 \\\\\n    8     & 118.188164 & 81.23045449 & 7.35E+01 & 5.95E+01 & 120.4440369 & 0.41112117 & 0.01686287 & 5.95E+01 \\\\\n    9     & 87.62585188 & 51.33933793 & 4.27E+01 & 2.99E+01 & 89.85854925 & 0.41717098 & 0.01800683 & 2.99E+01 \\\\\n    10    & 70.31181026 & 35.35034092 & 2.36E+01 & 1.43E+01 & 72.5313783 & 0.42059829 & 0.01823343 & 1.43E+01 \\\\\n    11    & 60.85889363 & 27.58761689 & 1.22E+01 & 6.66E+00 & 63.07129347 & 0.4224695 & 0.01782623 & 6.66E+00 \\\\\n    12    & 55.92035933 & 24.22121073 & 5.98E+00 & 3.07E+00 & 58.12901425 & 0.42344708 & 0.01721438 & 3.07E+00 \\\\\n    13    & 53.49709311 & 22.89305436 & 2.76E+00 & 1.38E+00 & 55.70391046 & 0.42392677 & 0.01672394 & 1.38E+00 \\\\\n    14    & 52.38726485 & 22.39252245 & 1.22E+00 & 6.02E-01 & 54.59324061 & 0.42414646 & 0.01643587 & 6.02E-01 \\\\\n    15    & 51.90534374 & 22.20463447 & 5.17E-01 & 2.55E-01 & 54.11095406 & 0.42424185 & 0.01629349 & 2.55E-01 \\\\\n    16    & 51.70286627 & 22.13313077 & 2.15E-01 & 1.06E-01 & 53.90832305 & 0.42428193 & 0.01622933 & 1.06E-01 \\\\\n    17    & 51.61937072 & 22.10548244 & 8.80E-02 & 4.32E-02 & 53.82476418 & 0.42429846 & 0.01620181 & 4.32E-02 \\\\\n    18    & 51.58529753 & 22.09465371 & 3.58E-02 & 1.76E-02 & 53.79066515 & 0.42430521 & 0.01619031 & 1.76E-02 \\\\\n    19    & 51.5714752 & 22.09037372 & 1.45E-02 & 7.12E-03 & 53.77683234 & 0.42430794 & 0.01618559 & 7.12E-03 \\\\\n    20    & 51.56588734 & 22.0886717 & 5.84E-03 & 2.87E-03 & 53.77124024 & 0.42430905 & 0.01618366 & 2.87E-03 \\\\\n    21    & 51.56363304 & 22.08799215 & 2.35E-03 & 1.16E-03 & 53.76898424 & 0.42430949 & 0.01618288 & 1.16E-03 \\\\\n    22    & 51.56272473 & 22.08772013 & 9.48E-04 & 4.67E-04 & 53.76807524 & 0.42430967 & 0.01618256 & 4.67E-04 \\\\\n    23    & 51.56235904 & 22.08761106 & 3.82E-04 & 1.88E-04 & 53.76770927 & 0.42430975 & 0.01618243 & 1.88E-04 \\\\\n    24    & 51.56221188 & 22.08756728 & 1.54E-04 & 7.56E-05 & 53.767562 & 0.42430978 & 0.01618238 & 7.58E-05 \\\\\n    25    & 51.56215268 & 22.0875497 & 6.18E-05 & 3.04E-05 & 53.76750275 & 0.42430979 & 0.01618236 & 3.05E-05 \\\\\n    26    & 51.56212886 & 22.08754263 & 2.48E-05 & 1.22E-05 & 53.76747891 & 0.42430979 & 0.01618235 & 1.20E-05 \\\\\n    27    & 51.56211928 & 22.08753979 & 9.99E-06 & 4.92E-06 & 53.76746933 & 0.42430979 & 0.01618235 & 4.72E-06 \\\\\n \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\item[ii)] $A_{g_f,\\alpha_f} \\Rightarrow \\Phi \\in \\Ord_2^2(\\xi)$ in $\\Omega_\\xi \\subset B(\\xi;\\delta)$.\n\n\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2048.526273 & 2036.688326 & 1.35E+03 & 2.01E+03 & 2052.245932 & 0.02901075 & 0.00087668 & 2.01E+03 \\\\\n    2     & 1378.380727 & 1357.837031 & 9.54E+02 & 1.34E+03 & 1381.592211 & 0.16166606 & 0.00214528 & 1.34E+03 \\\\\n    3     & 914.5756647 & 887.7554749 & 6.60E+02 & 8.65E+02 & 917.4354426 & 0.25347627 & 0.00391089 & 8.65E+02 \\\\\n    4     & 599.7868499 & 568.5654338 & 4.48E+02 & 5.46E+02 & 602.4079218 & 0.31578871 & 0.00622874 & 5.46E+02 \\\\\n    5     & 390.7721777 & 356.5990844 & 2.98E+02 & 3.34E+02 & 393.2347526 & 0.35716317 & 0.0090235 & 3.34E+02 \\\\\n    6     & 255.3927888 & 219.4044444 & 1.93E+02 & 1.97E+02 & 257.7527048 & 0.38396151 & 0.0120214 & 1.97E+02 \\\\\n    7     & 170.1536777 & 133.2761535 & 1.21E+02 & 1.11E+02 & 172.4489564 & 0.4008346 & 0.01478215 & 1.11E+02 \\\\\n    8     & 118.188164 & 81.23045449 & 7.36E+01 & 5.95E+01 & 120.4440369 & 0.41112117 & 0.01686287 & 5.95E+01 \\\\\n    9     & 87.62585188 & 51.33933793 & 4.28E+01 & 2.99E+01 & 89.85854925 & 0.41717098 & 0.01800683 & 2.99E+01 \\\\\n    10    & 70.31181026 & 35.35034092 & 2.36E+01 & 1.43E+01 & 72.5313783 & 0.42059829 & 0.01823343 & 1.43E+01 \\\\\n    11    & 60.85889363 & 27.58761689 & 1.22E+01 & 6.66E+00 & 63.07129347 & 0.4224695 & 0.01782623 & 6.66E+00 \\\\\n    12    & 51.56100988 & 22.08746493 & 1.08E+01 & 1.04E-03 & 53.76635909 & 0.42431001 & 0.01618182 & 1.04E-03 \\\\\n    13    & 51.56211284 & 22.08753788 & 1.11E-03 & 4.13E-09 & 53.76746288 & 0.4243098 & 0.01618235 & 3.03E-07 \\\\\n    \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method accelerated.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\end{itemize}\n\n\\end{example}\n\n\n\n\n\n\\begin{example}\n\nConsidering by hypothesis that $f_2\\in \\Ord_{NR}^2(\\xi)$, and using the following values\n\n\\begin{eqnarray*}\nDNI=574.319,&\nTair=16.832,& \\alpha=0.8996,\n\\end{eqnarray*}\n\nthe following iterations are obtained by using the fractional iterative methods given by the matrices  \\eqref{eq:5-008} and \\eqref{eq:5-009}.\n\n\\begin{itemize}\n\\item[i)] $A_{g_f,\\beta} \\Rightarrow \\Phi \\in \\Ord_2^1(\\xi)$.\n\n\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2029.854772 & 2022.247443 & 1.38E+03 & 2.00E+03 & 2032.218723 & 0.03297214 & 0.00056752 & 2.00E+03 \\\\\n    2     & 1351.035349 & 1337.861649 & 9.64E+02 & 1.32E+03 & 1353.07091 & 0.16730757 & 0.00139631 & 1.32E+03 \\\\\n    3     & 884.5286725 & 867.3584839 & 6.63E+02 & 8.49E+02 & 886.3385526 & 0.25962723 & 0.00256042 & 8.49E+02 \\\\\n    4     & 570.3098992 & 550.3428213 & 4.46E+02 & 5.32E+02 & 571.9677708 & 0.32180977 & 0.00410184 & 5.32E+02 \\\\\n    5     & 363.4003476 & 341.5519319 & 2.94E+02 & 3.23E+02 & 364.9581233 & 0.36275628 & 0.00597385 & 3.23E+02 \\\\\n    6     & 230.608119 & 207.585416 & 1.89E+02 & 1.89E+02 & 232.1016543 & 0.38903529 & 0.00799251 & 1.89E+02 \\\\\n    7     & 147.8561494 & 124.2274595 & 1.18E+02 & 1.06E+02 & 149.3096521 & 0.40541155 & 0.0098586 & 1.06E+02 \\\\\n    8     & 98.01126796 & 74.27302588 & 7.06E+01 & 5.63E+01 & 99.44065736 & 0.41527564 & 0.01127184 & 5.63E+01 \\\\\n    9     & 69.13937735 & 45.768905 & 4.06E+01 & 2.79E+01 & 70.5547995 & 0.42098926 & 0.01205541 & 2.79E+01 \\\\\n    10    & 53.13057813 & 30.57994962 & 2.21E+01 & 1.29E+01 & 54.53825576 & 0.42415733 & 0.01220761 & 1.29E+01 \\\\\n    11    & 44.6597286 & 23.22992376 & 1.12E+01 & 5.69E+00 & 46.06330831 & 0.42583368 & 0.01190151 & 5.69E+00 \\\\\n    12    & 40.41192388 & 20.07409231 & 5.29E+00 & 2.44E+00 & 41.81344865 & 0.4266743 & 0.01143556 & 2.44E+00 \\\\\n    13    & 38.42463196 & 18.85806286 & 2.33E+00 & 1.02E+00 & 39.82519534 & 0.42706758 & 0.01106236 & 1.02E+00 \\\\\n    14    & 37.56159752 & 18.41580876 & 9.70E-01 & 4.16E-01 & 38.9617434 & 0.42723837 & 0.01084908 & 4.16E-01 \\\\\n    15    & 37.20752766 & 18.25657936 & 3.88E-01 & 1.65E-01 & 38.60750225 & 0.42730844 & 0.01074838 & 1.65E-01 \\\\\n    16    & 37.06714943 & 18.19864095 & 1.52E-01 & 6.44E-02 & 38.46705611 & 0.42733622 & 0.01070538 & 6.44E-02 \\\\\n    17    & 37.01251861 & 18.17727243 & 5.87E-02 & 2.49E-02 & 38.41239886 & 0.42734703 & 0.01068795 & 2.49E-02 \\\\\n    18    & 36.99146859 & 18.16930635 & 2.25E-02 & 9.54E-03 & 38.39133866 & 0.42735119 & 0.01068108 & 9.54E-03 \\\\\n    19    & 36.98340172 & 18.16631458 & 8.60E-03 & 3.65E-03 & 38.38326789 & 0.42735279 & 0.01067841 & 3.65E-03 \\\\\n    20    & 36.98031974 & 18.16518558 & 3.28E-03 & 1.39E-03 & 38.38018441 & 0.4273534 & 0.01067738 & 1.39E-03 \\\\\n    21    & 36.97914434 & 18.16475823 & 1.25E-03 & 5.31E-04 & 38.37900845 & 0.42735363 & 0.01067699 & 5.30E-04 \\\\\n    22    & 36.97869653 & 18.16459616 & 4.76E-04 & 2.02E-04 & 38.37856042 & 0.42735372 & 0.01067684 & 2.02E-04 \\\\\n    23    & 36.97852602 & 18.16453462 & 1.81E-04 & 7.69E-05 & 38.37838983 & 0.42735375 & 0.01067678 & 7.67E-05 \\\\\n    24    & 36.97846112 & 18.16451124 & 6.90E-05 & 2.93E-05 & 38.3783249 & 0.42735377 & 0.01067676 & 2.93E-05 \\\\\n    25    & 36.97843643 & 18.16450235 & 2.62E-05 & 1.11E-05 & 38.37830019 & 0.42735377 & 0.01067675 & 1.10E-05 \\\\\n    26    & 36.97842703 & 18.16449898 & 9.99E-06 & 4.23E-06 & 38.37829079 & 0.42735377 & 0.01067675 & 4.12E-06 \\\\ \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\item[ii)] $A_{g_f,\\alpha_f} \\Rightarrow \\Phi \\in \\Ord_2^2(\\xi)$ in $\\Omega_\\xi \\subset B(\\xi;\\delta)$.\n\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2029.854772 & 2022.247443 & 1.38E+03 & 2.00E+03 & 2032.218723 & 0.03297214 & 0.00056752 & 2.00E+03 \\\\\n    2     & 1351.035349 & 1337.861649 & 9.64E+02 & 1.32E+03 & 1353.07091 & 0.16730757 & 0.00139631 & 1.32E+03 \\\\\n    3     & 884.5286725 & 867.3584839 & 6.63E+02 & 8.49E+02 & 886.3385526 & 0.25962723 & 0.00256042 & 8.49E+02 \\\\\n    4     & 570.3098992 & 550.3428213 & 4.46E+02 & 5.32E+02 & 571.9677708 & 0.32180977 & 0.00410184 & 5.32E+02 \\\\\n    5     & 363.4003476 & 341.5519319 & 2.94E+02 & 3.23E+02 & 364.9581233 & 0.36275628 & 0.00597385 & 3.23E+02 \\\\\n    6     & 230.608119 & 207.585416 & 1.89E+02 & 1.89E+02 & 232.1016543 & 0.38903529 & 0.00799251 & 1.89E+02 \\\\\n    7     & 147.8561494 & 124.2274595 & 1.18E+02 & 1.06E+02 & 149.3096521 & 0.40541155 & 0.0098586 & 1.06E+02 \\\\\n    8     & 98.01126796 & 74.27302588 & 7.06E+01 & 5.63E+01 & 99.44065736 & 0.41527564 & 0.01127184 & 5.63E+01 \\\\\n    9     & 69.13937735 & 45.768905 & 4.06E+01 & 2.79E+01 & 70.5547995 & 0.42098926 & 0.01205541 & 2.79E+01 \\\\\n    10    & 53.13057813 & 30.57994962 & 2.21E+01 & 1.29E+01 & 54.53825576 & 0.42415733 & 0.01220761 & 1.29E+01 \\\\\n    11    & 36.97715447 & 18.16441312 & 2.04E+01 & 1.19E-03 & 38.37701761 & 0.42735403 & 0.0106761 & 1.19E-03 \\\\\n    12    & 36.97842127 & 18.1644969 & 1.27E-03 & 7.75E-09 & 38.37828503 & 0.42735378 & 0.01067675 & 2.15E-07 \\\\\n     \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method accelerated.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\end{itemize}\n\\end{example}\n\n\n\\begin{example}\nConsidering by hypothesis that $f_2\\in \\Ord_{NR}^2(\\xi)$, and using the following values\n\n\\begin{eqnarray*}\nDNI=94.3555,&\nTair=28.373,& \\alpha=0.89964,\n\\end{eqnarray*}\n\n\nthe following iterations are obtained by using the fractional iterative methods given by the matrices  \\eqref{eq:5-008} and \\eqref{eq:5-009}.\n\n\\begin{itemize}\n\\item[i)] $A_{g_f,\\beta} \\Rightarrow \\Phi \\in \\Ord_2^1(\\xi)$.\n\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2026.948258 & 2025.698601 & 1.38E+03 & 2.00E+03 & 2027.336246 & 0.03393789 & 0.00009324 & 2.00E+03 \\\\\n    2     & 1346.157858 & 1343.993285 & 9.63E+02 & 1.32E+03 & 1346.49176 & 0.16860893 & 0.00022947 & 1.32E+03 \\\\\n    3     & 878.348027 & 875.5257474 & 6.62E+02 & 8.47E+02 & 878.644763 & 0.26114907 & 0.00042095 & 8.47E+02 \\\\\n    4     & 563.2981664 & 560.0145307 & 4.46E+02 & 5.31E+02 & 563.5698729 & 0.32347088 & 0.00067464 & 5.31E+02 \\\\\n    5     & 355.8897756 & 352.2947261 & 2.94E+02 & 3.24E+02 & 356.1450042 & 0.36449952 & 0.00098289 & 3.24E+02 \\\\\n    6     & 222.8349909 & 219.0449879 & 1.88E+02 & 1.90E+02 & 223.0796488 & 0.39081985 & 0.00131521 & 1.90E+02 \\\\\n    7     & 139.9975258 & 136.1076542 & 1.17E+02 & 1.08E+02 & 140.2356025 & 0.4072064 & 0.00162172 & 1.08E+02 \\\\\n    8     & 90.22025428 & 86.31555624 & 7.04E+01 & 5.77E+01 & 90.45437636 & 0.41705312 & 0.00185193 & 5.77E+01 \\\\\n    9     & 61.57917913 & 57.74236258 & 4.05E+01 & 2.92E+01 & 61.81102578 & 0.42271878 & 0.00197603 & 2.92E+01 \\\\\n    10    & 45.9860483 & 42.29258563 & 2.20E+01 & 1.37E+01 & 46.21665614 & 0.42580335 & 0.00199523 & 1.37E+01 \\\\\n    11    & 38.07727079 & 34.56957756 & 1.11E+01 & 5.99E+00 & 38.3072503 & 0.42736783 & 0.00194279 & 5.99E+00 \\\\\n    12    & 34.38320509 & 31.04574209 & 5.11E+00 & 2.46E+00 & 34.61289112 & 0.42809857 & 0.00187038 & 2.46E+00 \\\\\n    13    & 32.785517 & 29.5629105 & 2.18E+00 & 9.76E-01 & 33.0150761 & 0.42841462 & 0.0018152 & 9.76E-01 \\\\\n    14    & 32.13122049 & 28.97016965 & 8.83E-01 & 3.80E-01 & 32.36072761 & 0.42854405 & 0.00178421 & 3.80E-01 \\\\\n    15    & 31.87136629 & 28.7388988 & 3.48E-01 & 1.47E-01 & 32.10085277 & 0.42859545 & 0.00176952 & 1.47E-01 \\\\\n    16    & 31.7697037 & 28.64948568 & 1.35E-01 & 5.67E-02 & 31.9991821 & 0.42861556 & 0.00176316 & 5.67E-02 \\\\\n    17    & 31.73020749 & 28.61501378 & 5.24E-02 & 2.19E-02 & 31.95968275 & 0.42862337 & 0.00176054 & 2.19E-02 \\\\\n    18    & 31.71491342 & 28.60173066 & 2.03E-02 & 8.43E-03 & 31.94438747 & 0.4286264 & 0.00175949 & 8.43E-03 \\\\\n    19    & 31.70900063 & 28.59661144 & 7.82E-03 & 3.25E-03 & 31.93847421 & 0.42862757 & 0.00175907 & 3.25E-03 \\\\\n    20    & 31.70671661 & 28.59463794 & 3.02E-03 & 1.25E-03 & 31.93619001 & 0.42862802 & 0.00175891 & 1.25E-03 \\\\\n    21    & 31.70583474 & 28.59387694 & 1.17E-03 & 4.84E-04 & 31.93530807 & 0.4286282 & 0.00175885 & 4.84E-04 \\\\\n    22    & 31.70549433 & 28.59358344 & 4.50E-04 & 1.87E-04 & 31.93496763 & 0.42862826 & 0.00175882 & 1.87E-04 \\\\\n    23    & 31.70536295 & 28.59347022 & 1.73E-04 & 7.20E-05 & 31.93483624 & 0.42862829 & 0.00175881 & 7.20E-05 \\\\\n    24    & 31.70531225 & 28.59342654 & 6.69E-05 & 2.78E-05 & 31.93478553 & 0.4286283 & 0.00175881 & 2.78E-05 \\\\\n    25    & 31.70529268 & 28.59340969 & 2.58E-05 & 1.07E-05 & 31.93476596 & 0.4286283 & 0.00175881 & 1.07E-05 \\\\\n    26    & 31.70528513 & 28.59340319 & 9.96E-06 & 4.13E-06 & 31.93475841 & 0.4286283 & 0.00175881 & 4.13E-06 \\\\\n    \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\item[ii)] $A_{g_f,\\alpha_f} \\Rightarrow \\Phi \\in \\Ord_2^2(\\xi)$ in $\\Omega_\\xi \\subset B(\\xi;\\delta)$.\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2026.94826 & 2025.6986 & 1.38E+03 & 2.00E+03 & 2027.33625 & 0.03393789 & 0.0000932 & 2.00E+03 \\\\\n    2     & 1346.15786 & 1343.99329 & 9.63E+02 & 1.32E+03 & 1346.49176 & 0.16860893 & 0.00022947 & 1.32E+03 \\\\\n    3     & 878.348027 & 875.525747 & 6.62E+02 & 8.47E+02 & 878.644763 & 0.26114907 & 0.00042095 & 8.47E+02 \\\\\n    4     & 563.298166 & 560.014531 & 4.46E+02 & 5.31E+02 & 563.569873 & 0.32347088 & 0.00067464 & 5.31E+02 \\\\\n    5     & 355.889776 & 352.294726 & 2.94E+02 & 3.24E+02 & 356.145004 & 0.36449952 & 0.00098289 & 3.24E+02 \\\\\n    6     & 222.834991 & 219.044988 & 1.88E+02 & 1.90E+02 & 223.079649 & 0.39081985 & 0.00131521 & 1.90E+02 \\\\\n    7     & 139.997526 & 136.107654 & 1.17E+02 & 1.08E+02 & 140.235603 & 0.4072064 & 0.00162172 & 1.08E+02 \\\\\n    8     & 90.2202543 & 86.3155562 & 7.04E+01 & 5.77E+01 & 90.4543764 & 0.41705312 & 0.00185193 & 5.77E+01 \\\\\n    9     & 61.5791791 & 57.7423626 & 4.05E+01 & 2.92E+01 & 61.8110258 & 0.42271878 & 0.00197603 & 2.92E+01 \\\\\n    10    & 45.9860483 & 42.2925856 & 2.20E+01 & 1.37E+01 & 46.2166561 & 0.42580335 & 0.00199523 & 1.37E+01 \\\\\n    11    & 38.0772708 & 34.5695776 & 1.11E+01 & 5.99E+00 & 38.3072503 & 0.42736783 & 0.00194279 & 5.99E+00 \\\\\n    12    & 31.7052694 & 28.5933984 & 8.74E+00 & 1.03E-05 & 31.9347427 & 0.42862831 & 0.0017588 & 1.03E-05 \\\\\n    13    & 31.7052804 & 28.5933991 & 1.10E-05 & 2.80E-09 & 31.9347537 & 0.42862831 & 0.00175881 & 3.17E-08 \\\\\n     \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method accelerated.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\end{itemize}\n\\end{example}\n\n\n\n\n\nFrom the previous results it is observed that there is a considerable improvement in the order of convergence between the matrices \\eqref{eq:5-008} and \\eqref{eq:5-009}. Therefore, it may be established that it is more efficient to solve the function \\eqref{eq:5-002} by implementing the fractional quasi-Newton method accelerated in the function \\eqref{eq:5-005}. So, by providing multiple values of the parameters $DNI$ and $T_{air}$, it is possible to obtain a histogram of the efficiencies of a hybrid solar receiver analogous to the one shown in the Figure \\ref{fig:02}. Finally, it is necessary to mention that the \\mref{Corollary}{\\ref{cor:4-001}} can also be implemented in the \\textbf{generalized fractional quasi-Newton method}, which is obtained by using the matrix \\eqref{eq:5-008} with the following function\n\n\\begin{eqnarray}\ng_{a,b,f}(x):=af(x_i)+f^{(1)}(x_i)(x-bx_i), & a,b\\in \\nset{R},\n\\end{eqnarray}\n\n\nas a consequence, denoting by $c$ an arbitrary constant, it is possible to define the following set of matrices\n\n\n\n\\begin{eqnarray}\n\\set{A_{g,\\alpha} \\ : \\    A_{g,\\alpha} \\in  {}_n\\IMa_{x,\\alpha}^{1}(g) }\\cap\\set{A_{g,\\alpha} \\ : \\  \\left([A_{g,\\alpha}]_{jk}(x) \\right) :=\\left( o_k^{\\alpha}[g]_{j}(x)  \\right)^{-1} }\\cap \\set{o^\\alpha \\ : \\ o_k^{\\alpha} c  \\neq 0 \\ \\forall k\\geq 1 } ,\n\\end{eqnarray}\n\n\n\n\nand therefore, it is possible to define the following sets of fractional iterative methods\n\n\n\n\n\\begin{eqnarray}\n\\set{\\Phi \\ : \\ A_{g,\\beta}  \\mbox{ uses }  g=g_{a,0,f} \\mbox{ with }  0<a\\leq 1}, \\vspace{0.1cm}\\\\\n\\set{\\Phi \\ : \\ A_{g,\\beta} \\mbox{ uses }  g=g_{1,b,f} \\mbox{ with } 0< b \\leq 1},\n\\end{eqnarray}\n\nwhich correspond to two uncountable families of fractional fixed point methods in which the  \\mref{Corollary}{\\ref{cor:4-001}} can be implemented.\n\n\n\n\n\n\\section{Conclusions}\n\n\nIn all the examples shown, a decrease in the number of iterations necessary to converge to the solutions is observed when implementing the function \\eqref{eq:4-005} in the fractional quasi-Newton method, which means that the generated sequences show an acceleration in their speed of convergence, which was to be expected given the \\mref{Corollary}{\\ref{cor:4-001}}. The fractional iterative methods, such as the fractional Newton-Raphson method, can find multiple zeros of a function using a single initial condition, this partially solves the intrinsic problem of classical iterative methods, which is that in general, to find $N$ zeros of a function, $N$ initial conditions must be provided. Due to the fractional operators implemented, these methods can be considered \\textbf{non-local parametric iterative methods}, so they have two important characteristics: \n\n\n\n\n\\begin{itemize}\n\\item[i)] The initial condition does not necessarily have to be close to the sought values due to the non-local nature of fractional operators \\cite{torres2020nonlinear}. \n\n\n\\item[ii)] When working in a space of $N$ dimensions, in the case that it is necessary to change the initial condition, unlike the classic iterative methods where in the worst case it is necessary to vary the $N$ components of the initial condition until obtaining a suitable value, in the fractional fixed point methods it is enough to vary the parameter $\\alpha$ of the fractional operators until found an adequate value that allows generating a sequence that converges to a sought value \\cite{torres2020fracsome}.\n\\end{itemize}\n\n\n\nThe above features make fractional iterative methods an ideal numerical tool for working with systems of nonlinear algebraic equations that vary with time-dependent parameters, as is the case of the functions \\eqref{eq:5-002} and \\eqref{eq:5-005}, which allows studying the behavior of temperatures and efficiencies of a hybrid solar receiver \\cite{de2021fractional,torres2020reduction,torres2020fracpseunew}. \nDue to many nonlinear algebraic systems related to engineering and physics are often related to time-dependent parameters. Having a way of classifying and accelerating the order of convergence of fractional fixed point methods through the \\mref{Corollary}{\\ref{cor:4-001}} may become a fundamental piece to continue expanding the applications of the fractional operators.\n\n\n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.6\\textwidth, height=0.37\\textwidth]{hist}\n\\caption{Histogram and density curve of the efficiency of a hybrid solar receiver obtained from a simulation corresponding to a period of thirty days, which is equivalent to $2410$ pairs of parameters ($DNI,T_{air}$) randomly generated on the domain $[12,958]\\times [11,45]$. The selected domain is based on data measured in real-time at the Center for Advanced Studies in Energy and Environment (CEAEMA) \\cite{rodrigo2019performance,de2021fractional}. The values generated for the simulation presented the mean values $mean(DNI)=662.35$ and $mean(T_{air})=31.28$ with sample standard deviations $std(DNI)=257.83$ and $std(T_{air})=6.11\n$, while the values of the efficiencies were obtained through the solutions of the function \\eqref{eq:5-005} using the fractional quasi-Newton method accelerated.}\\label{fig:02}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction} \nOn August 17, 2017, LIGO-Virgo Collaboration detected a GW signal from a binary neutron star (NS) merger event (GW170817). The total gravitational mass of the system at infinite binary separation is $2.74^{+0.04}_{-0.01}M_{\\odot}$, with a mass ratio in the range of $(0.7-1)$ \\citep{abbott17a}. Limited by the sensitivity of current GW detectors at high frequencies, the GW signals from the oscillations of the post-merger product are undetectable, leaving the merger product unidentified \\citep{abbott17b,abbott19}. \n\nTheoretically, an NS-NS merger can in principle give a variety of post-merger products depending on the remnant (gravitational) mass ($M_{\\rm rem}$, which depends on its spin)\\footnote{Throughout the paper, without specification, masses refer to gravitational masses. The baryonic masses are denoted as $M_b$. For the same $M_b$, the gravitational mass can adopt a range of values, which depend on the spin state of the NS.} and the neutron star equation of state (EoS), which defines the maximum gravitational mass, $M_{\\rm max}$, of an NS without collapsing into a black hole (BH) \\citep{rezzolla10,bartos13,lasky14,ravi14,gao16, margalit17}. \nGiven an EoS, the maximum mass of a non-rotational NS (denoted as $M_{\\rm TOV}$) can be derived by solving the TOV equations \\citep{oppenheimer39}. With rotation, the mass can be enhanced by a factor \n\\begin{eqnarray}\n\\chi={(M-M_{\\rm TOV}) \\over M_{\\rm TOV}}.\n\\label{eq:chi}\n\\end{eqnarray} \nIn the following, $(1+\\chi)$ is defined by the enhancement factor due to uniform rotation. Differential rotation can also enhance the mass, and we denote the enhancement factor as $(1+\\chi_d)$. Typically one has $\\chi < \\chi_d$. \nThe degree of enhancement has been extensively studied in the literature \\citep{cook94,lasota96,breu16,studzinska16,gondek17,bauswein17,bozzola18,weih18}. \nOne can define \n\\begin{eqnarray}\n    M_{\\rm max} & \\equiv & (1+\\chi_{\\rm max}) M_{\\rm TOV}, \\label{eq:Mmax} \\\\\n    M_{\\rm max,d} & \\equiv & (1+\\chi_{\\rm d,max}) M_{\\rm TOV},\n\\end{eqnarray}\nwhere $\\chi_{\\rm max} \\sim 0.2$ \\citep{cook94,lasota96,breu16} and $\\chi_{\\rm d,max} \\sim (0.3-0.6)$\n\\footnote{The quoted $\\chi_{\\rm d,max}$ range is roughly derived from the numerical result that $M_{\\rm tot}>kM_{\\rm TOV}$ (where $M_{\\rm tot}=M_1+M_2$ and $k=1.3-1.6$) is required to directly form a BH as the merger remnant \\citep{shibata05,shibata06,hotokezaka11,bauswein13,margalit17}. Since $M_{\\rm TOV}<M_{\\rm tot}$, the true $\\chi_{\\rm d,max}$ should be somewhat smaller than 0.3-0.6.}\nare the maximum enhancement factor for uniform and differential rotations, respectively. Let us define $M_{\\rm rem}^0$, $M_{\\rm rem}^k$, and $M_{\\rm rem}^\\infty$ as the gravitational masses of the remnant right after the merger, at the beginning of uniform rotation (which is assumed to carry a Keplerian rotation), and with no rotation ($P=\\infty$), respectively. The fate of the merger product can be then determined as follows: If $M_{\\rm rem}^0 > M_{\\rm max,d}$, the merger remnant would directly collapse to a BH. Otherwise the remnant would go through a hypermassive NS (HMNS) phase,  during which the merger remnant loses angular momentum as well as mass. If $M_{\\rm max} < M_{\\rm rem}^k < M_{\\rm rem}^0 \\leq M_{\\rm max,d}$, the merger remnant would collapse into a BH after the HMNS phase. If, however, $M_{\\rm rem}^k \\leq M_{\\rm max}$, a uniformly rotating NS would be formed after the differential rotation is damped. Whether it is an SMNS or an SNS depends on the comparison between $M_{\\rm TOV}$ and $M_{\\rm rem}^{\\infty}$. If $M_{\\rm rem}^\\infty > M_{\\rm TOV}$, the merger remnant is a supramassive NS (SMNS), which would eventually collapse into a BH. If $M_{\\rm rem}^\\infty \\leq M_{\\rm TOV}$, the remnant would never collapse, which is a stable NS (SNS). Either an SMNS or an SNS can be called as a massive NS (MNS). \n\nFor GW170817, although GW data cannot determine the nature of the merger product, it has been suggested that the EM counterpart observations may provide some clues. Unfortunately, owing to the messy physics involved in producing EM counterparts, all the claims on the constraints on $M_{\\rm TOV}$ rely on some assumptions, so that no consensus can be reached. All investigators agree on that the remnant cannot be a promptly formed black hole, since the observed kilonova is too bright to be explained by the dynamical ejecta only. The disagreement comes to the lifetime of the NS produced during the merger. Many authors assumed that in order to produce the short gamma-ray burst (GRB 170817A) \\citep{goldstein17,zhangbb18} following GW170817, a BH engine is needed \\citep[e.g.][]{margalit17,rezzolla18,ruiz18}. Within this picture, the remnant is an HMNS, which must have collapsed before the GRB trigger time, which is 1.7 s after the merger \\citep{abbott17c,goldstein17}. In particular, a good fraction of the observed 1.7 s delay has to be attributed to the HMNS phase, during which significant mass ejection is warranted to account for the observed bright kilonova emission  \\citep{siegel17,gill19}.\nIf this is the case, the multi-messenger observations of GW170817 can be used to provide an upper bound on $M_{\\rm TOV}$ \\citep{margalit17,rezzolla18,ruiz18,shibata19}, e.g. $M_{\\rm TOV}\\lesssim2.16~M_{\\odot}$ by \\cite{margalit17}. \n\nOn the other hand, in order to explain the extended engine activities (flares, extended emission, and internal X-ray plateaus) of short GRBs, it has been long proposed that at least some NS-NS merger systems can produce both a short GRB and an MNS \\citep{dai06,gao06,metzger08,dessart09,lee09,zhang10,fernandez13}. Indeed, when interpreting the rapid decay at the end of internal X-ray plateau observed in a good fraction of short GRBs \\citep{rowlinson10,rowlinson13,lu14,lu15}, $M_{\\rm TOV}$ has to be (much) greater than $2.16~M_{\\odot}$\\citep{gao16,li16}. So there is a direct conflict between the upper limit on $M_{\\rm TOV}$ derived from GW170817 (assuming that a BH is formed before 1.7 s after the merger) and the short GRB X-ray plateau data. Indeed, since GRB 170817A did not trigger Swift, there was no early X-ray afterglow data to check whether there was an early X-ray plateau phase similar to other short GRBs. The fact that the delay time (1.7 s) is comparable to the GRB duration itself ($\\sim 2$ s) also suggests that the jet launching waiting time $\\Delta t_{\\rm jet}$ as well as the shock breakout time $\\Delta t_{\\rm bo}$ may be small \\citep{zhang19}. If so, the launch of a jet may not demand a BH, and the bright kilonova emission may benefit from energy injection of a long-lived remnant \\citep{yu18,li18}. Indeed, the ``blue'' component of the kilonova peaks at $\\sim 1$ d  with a peak luminosity  $\\sim 10^{42}{\\rm erg~s^{-1}}$, which requires a $>0.02M_{\\odot}$ mass of lanthanide-free ejecta ($Y_e \\gtrsim 0.25$) with $v_{\\rm ej,blue}\\approx 0.2-0.3c$ \\citep{kasen17,cowperthwaite17,chornock17,kilpatrick17,villar17,tanvir17,gao17,shappee17}. In order to fit both the peak time and peak luminosity, a small opacity $\\kappa \\sim (0.3-0.5) \\ {\\rm cm^2 \\ g^{-1}}$ is required, which is in conflict with the value $\\kappa \\sim 1 \\ {\\rm cm^2 \\ g^{-1}}$ derived from the most detailed calculations for Fe group elements \\citep{tanaka19}. Energy injection from a long-lived MNS that survives for at least 1 day can help to ease the conflict and interpret the blue component \\citep{li18}. Finally, \\cite{piro19} claimed a low-significance temporal feature at 155 days in the X-ray afterglow of GW170817 which carries properties of GRB X-ray flares. If such a feature is not due to a statistical fluctuation, one would demand an active central engine at such a late epoch. The putative MNS should at least survive for 155 days after the merger.\n\nIn this work, instead of claiming the identity of  the merger remnant of GW170817, we leave it as an open question and generally discuss what constraints on $M_{\\rm TOV}$ one can place for different possible merger remnants. We estimate the remnant mass of GW170817 based on the gravitational wave data and NS EoSs (either for individual EoSs or using some EoS-independent universal relations). We identify the separation lines among three types of products (HMNS\/BH, SMNS, and SNS) in Section 2. In Section, 3, we discuss the cases of SMNS and SNS and place constraints on the NS parameters (surface magnetic field at the pole, $B_p$, and ellipticity, $\\epsilon$) assuming that the remnant can survive for 300 s (typical ending time of internal X-ray plateaus), 1 d (peak time of the blue kilonova component), and 155 d (time of the putative X-ray flare), respectively.\n\n\n\\section{Constraints on $M_{\\rm TOV}$ for different types of merger product}\n\n\\subsection{General approach}\\label{sec:general-approach}\n\nOur goal is to address the following question: given the information provided by the gravitational wave data from GW170817, i.e. the total gravitational mass at infinite binary separation $M_{\\rm tot}= M_1 + M_2 = 2.74^{+0.04}_{-0.01}M_{\\odot}$ and mass ratio $q = m_1\/m_2 = (0.7-1)$ under the low dimensionless NS spin prior \\citep{abbott17a}, what can one say about the maximum mass of the non-spinning NS $M_{\\rm TOV}$? Our general approach is as follows:\n\\begin{itemize}\n    \\item Assume the values of $M_1$ and $M_2$. In our case, we assume that $M_1 = M_2 = M_{\\rm tot} \/ 2$, noticing that the total baryonic mass is highly insensitive to the binary mass ratio;\n    \\item Convert the gravitational masses to baryonic masses $M_{1,b}$ and $M_{2,b}$, either based on the public code \\texttt{RNS} code (for individual EoSs) \\citep{stergioulas95} or some EoS-independent universal relations \\citep[e.g.][]{gao19};\n    \\item {Conserve the total baryonic mass of the binary system throughout the merger. In the post-merger phase, the baryonic mass in the remnant could be derived by subtracting the baryonic mass of various ejecta components from the total baryonic mass}, i.e. $M_{\\rm rem,b}=(M_{1,b} + M_{2,b}) - M_{\\rm ejc}$. Based on the EM counterpart observations, the total ejected mass is estimated as $M_{\\rm ejc} \\sim (0.06\\pm0.01) M_\\odot$ \\cite[][and reference therein]{metzger17};\n    \\item Convert $M_{\\rm rem,b}$ to the gravitational mass of the central object $M_{\\rm rem}$, which depends on its spin state \\citep{gao19}. In particular, we care mostly about $M_{\\rm rem}^k$ and $M_{\\rm rem}^\\infty$.\n    \\item Compare $M_{\\rm rem}^k$ against $M_{\\rm max}$\n    or \n    \\begin{equation}\n      M_{\\rm TOV}^k = (1+\\chi_{\\rm TOV}^k) M_{\\rm TOV}  \n      \\label{eq:MTOVk}\n    \\end{equation}\n    to determine whether the final merger product is an HMNS\/BH, SMNS, or SNS. Here $M_{\\rm TOV}^k$ is the gravitational mass at Keplerian rotation for an NS whose non-spin gravitational mass is $M_{\\rm TOV}$.\n\\end{itemize}\n\nIn the following, we discuss the results for individual EoSs (\\S2.2) and for general cases using universal relations (\\S2.3).\n\n\\subsection{Individual EoSs}\nWe adopt 10 realistic (tabulated) EoSs (as listed in Table 1): SLy \\citep{douchin01}, WFF1 \\citep{wiringa88}, WFF2 \\citep{wiringa88}, AP3 \\citep{akmal97}, AP4 \\citep{akmal97}, BSK21 \\citep{goriely10}, DD2 \\citep{typel10}, MPA1 \\citep{muther87}, MS1 \\citep{muller96}, MS1b \\citep{muller96} with $M_{\\rm TOV}$ ranging from $2.05M_{\\odot}$ to $2.78M_{\\odot}$. For each EoS, we use \\texttt{RNS} to calculate $M_{\\rm TOV}$. We also calculate the allowed minimum Keplerian period (marked as $P_{\\rm k,min}$), which is related to the Keplerian period of the NS at $M_{\\rm max}$. The results are collected in Table 1.\n\nFor each EoS, we derive the $M_{\\rm rem,b}$ following the approach described in Section \\ref{sec:general-approach}. Since the $M_b-M$ relation is somewhat different for different EoSs, the derived $M_{\\rm rem,b}$ is different even for the same event GW170817. With the derived $M_{\\rm rem,b}$ for each EoS, we apply the \\texttt{RNS} code to see whether there is a uniformly rotating NS solution. If not, the merger product would be a BH (likely preceded by a brief HMNS phase). If a solution is available, we further test whether there is a solution in the non-spinning case. The remnant would be an SMNS or SNS if the answer is ``no'' or ``yes'', respectively. As shown in Table 1, among the 10 EoSs studied, one (SLy) with $M_{\\rm TOV} = 2.05 M_{\\odot}$ forms an HMNS\/BH, three (MPA1, Ms1, Ms1b) with minimum $M_{\\rm TOV} = 2.48 M_{\\odot}$ (MPA1) form an SNS, and the other six (with $M_{\\rm TOV}$ between $2.14M_{\\odot}$ (WFF1) and $2.42M_{\\odot}$ (DD2)) form an SMNS. According to this small sample investigation, the $M_{\\rm TOV}$ separation line for HMNS\/BH and SMNS may be between $2.05 M_{\\odot}$ and $2.14M_{\\odot}$, and that between SMNS and SNS may be between $2.42M_{\\odot}$ and $2.48 M_{\\odot}$.\n\n\n\\begin{table*}                                                                      \n\\begin{center}{\\scriptsize                                                                                       \n\\caption{The 10 EoSs investigated in this paper.}                                                                             \n\\begin{tabular}{cccccccccc}                                                                                          \n\\hline                                                                                                                   \n\\hline                                                                                                                                                                                                               \n    & $M_{\\rm TOV}$            &$P_{\\rm k,min}$   &$M_{\\rm b,tot}$               &$M_{\\rm b,rem}$              &$M_{\\rm rem}^k$          &$P_k$      &$1+\\chi_{\\rm TOV}^{k}$      &$1+\\chi_{\\rm max}$     &Product type      \\\\\n    &$\\left(M_{\\odot}\\right)$  &${\\rm ms}$        &$\\left(M_{\\odot}\\right)$      &$\\left(M_{\\odot}\\right)$     &$\\left(M_{\\odot}\\right)$ &${\\rm ms}$ &                        &                 &                  \\\\\n\\hline                                                                                                                                                                                                               \nSLy &2.05                      &$0.55$            &$3.01^{+0.05}_{-0.01}$                          &$2.95^{+0.06}_{-0.02}$                           &$--$                     &$--$       &1.039                   &1.184            &BH                 \\\\\n\\hline                                                                                                                                                                                                               \nWFF1&2.14                      &$0.47$          &$3.07^{+0.05}_{-0.01}$                               &$3.01^{+0.06}_{-0.02}$                        &$2.51^{+0.03}_{-0.01}$                     &$0.52$     &1.051                   &1.201            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nWFF2&2.20                      &$0.50$          &$3.04^{+0.05}_{-0.01}$                               &$2.98^{+0.06}_{-0.02}$                         &$2.51^{+0.04}_{-0.01}$                    &$0.58$     &1.048                   &1.192            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nAp4 &2.22                      &$0.51$            &$3.03^{+0.05}_{-0.01}$                            &$2.97^{+0.06}_{-0.02}$                        &$2.52^{+0.03}_{-0.01}$                     &$0.60$     &1.047                   &1.194            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nBSk21&2.28                     &$0.60$          &$2.99^{+0.05}_{-0.01}$                               &$2.93^{+0.06}_{-0.02}$                         &$2.54^{+0.03}_{-0.02}$                     &$0.74$     &1.044                   &1.205            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nAP3 &2.39                      &$0.55$            &$3.01^{+0.05}_{-0.01}$                             &$2.95^{+0.06}_{-0.02}$                         &$2.54^{+0.03}_{-0.02}$                     &$0.70$     &1.049                   &1.202            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nDD2 &2.42                      &$0.65$            &$2.99^{+0.05}_{-0.01}$                             &$2.93^{+0.06}_{-0.02}$                        &$2.55^{+0.04}_{-0.01}$                     &$0.82$     &1.042                   &1.208            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nMPA1&2.48                      &$0.59$           &$3.00^{+0.05}_{-0.01}$                              &$2.94^{+0.06}_{-0.02}$                        &$2.54^{+0.04}_{-0.01}$                     &$0.76$     &1.048                   &1.208            &SNS               \\\\\n\\hline                                                                                                                                                                                                               \nMs1 &2.77                      &$0.72$            &$2.95^{+0.05}_{-0.01}$                             &$2.89^{+0.06}_{-0.02}$                         &$2.56^{+0.04}_{-0.01}$                    &$1.00$     &1.043                   &1.207            &SNS                \\\\\n\\hline                                                                                                                                                                                                               \nMs1b&2.78                      &$0.71$           &$2.96^{+0.05}_{-0.01}$                             &$2.90^{+0.06}_{-0.02}$                         &$2.56^{+0.04}_{-0.01}$                     &$0.99$     &1.042                   &1.212            &SNS                \\\\\n\\hline                                                                                                                                                                                                               \n\\hline                                                                          \n \\end{tabular}                                                                      \n }                                                                                                                                                              \n\\end{center}                                                                        \n\\end{table*} \n\n\\subsection{Universal Approach}\n\nSince there are many more EoSs discussed in the literature \\citep[e.g.][]{lattimer12}, it is impossible to make a self-consistent check for all the proposed EoSs. Some general constraints on $M_{\\rm TOV}$ (with large uncertainties) may be obtained by applying some EoS-independent empirical relations for GW170817. \n\nGenerally, the type of the merger product is best determined by comparing $M_{\\rm rem}^k$ with $M_{\\rm max}$ (Eq.(\\ref{eq:Mmax})) and $M_{\\rm TOV}^k$ (Eq.(\\ref{eq:MTOVk})), both are highly dependent on the EoS. Fortunately, when the gravitational mass of an NS is normalized to $M_{\\rm TOV}$ and when the rotation period $P$ is normalized to $P_{\\rm k,min}$, the evolution of the separation boundaries among HMNS\/BH, SMNS, and SNS in the ${\\cal M} - {\\cal P}$ plane (where ${\\cal M} \\equiv M \/ M_{\\rm TOV} \\equiv (1+\\chi)$ and ${\\cal P} \\equiv P \/ P_{\\rm k,min}$) is highly EoS-insensitive. This is shown in Figure \\ref{fig:type}. The orange bunch of lines denote the Keplerian lines, which denote the normalized Keplerian period ${\\cal P}_k$ as a function of the gravitational mass of the NS at that period. For the EoSs we investigate, one can get the best-fit line as \n\\begin{eqnarray}\n{\\cal P}_k&=&(-2.697\\pm0.355)\\times({M \\over M_{\\rm TOV}})^2 \\nonumber\\\\\n&+&(4.355\\pm0.764)\\times({M \\over M_{\\rm TOV}})\\nonumber\\\\\n&-&(0.303\\pm0.409),~~P>P_{\\rm k,min}.\n\\end{eqnarray}\n One can see that $P_k$ becomes progressively longer when $M < M_{\\rm max}$. This line is the starting point for the evolution of any NS after the differentiation rotation is damped. \n\nLet us now consider a spinning NS. Its baryonic mass never changes with the spin period while the gravitational mass would decrease as it spins down. These constant $M_b$ curves are examplified as the two black lines (for two particular $M_b$ values) and the red bunch of lines, which show constant $M_{\\rm TOV,b}$ lines for different EoSs. The best-fit line for the 10 EoSs leads to\n\\begin{eqnarray}\n\\log_{10}\\chi_{\\rm TOV}&=&(1.804\\pm0.268)\\times({\\rm log_{10}}{\\cal P})^2 \\nonumber\\\\\n&+&(-3.661\\pm0.190)\\times{\\rm log_{10}}{\\cal P} \\nonumber\\\\\n&+&{\\rm log_{10}(0.101\\pm0.007)}, ~P>P_{\\rm k,TOV}.\n\\end{eqnarray}\nThe intersection of this line and the best-fit orange line gives $P_{\\rm k,TOV}$, which represents the Kepler period when $M_b=M_{\\rm b,TOV}$, and the enhancement factor is $(1+\\chi_{\\rm TOV}^k)$.   \n\n\n\n\nA uniformly rotating NS with $M_b>M_{\\rm TOV,b}$ would eventually collapse into a BH when $M >(1+\\chi_{\\rm col})M_{\\rm TOV}$, where $\\chi_{\\rm col}$ is defined as the maximally allowed enhancement gain at a particular $P$. The values of $(1+\\chi_{\\rm col})$ with different $P$ values can serve as the separation line between SMNS and HMNS\/BH regimes. This corresponds to the green bunch of lines in Figure \\ref{fig:type}, which corresponds to the best fit as \n\\begin{eqnarray}\n\\log_{10}\\chi_{\\rm col}&=&(-2.740\\pm0.045)\\times{\\rm log_{10}} {\\cal P} \\nonumber\\\\\n&+&{\\rm log_{10}(0.201 \\pm 0.005)}\n\\label{eq:chi_ucol}\n\\end{eqnarray}\nWhen $P \\rightarrow P_{\\rm k,min}$, one has $\\chi_{\\rm col}^k \\rightarrow \\chi_{\\rm max}$.\n\nThe region below the orange bunch of lines in Figure \\ref{fig:type} (the white region) is not well defined, since $P$ cannot be defined for an differentially rotating object. The \\texttt{RNS} code we employ can only be used in uniformly rotating case. We therefore indicate the evolution trajectories in the white region using dashed lines.\n\nWe plot several evolutionary trajectories of the GW170817 remnant within the framework or several EoSs: BSk21 (diamond), AP3 (star), MPA1 (upward triangle), and SLy (downward triangle). The symbols are solid or open in the uniformly or differentially rotating regimes, respectively. \n\nThe mass at the starting point of rigid rotation, i.e. $M_{\\rm rem}^k$, is crucial to determine the remnant type through its comparison with $M_{\\rm max}$ (Eq.(\\ref{eq:Mmax})) and $M_{\\rm TOV}^k$ (Eq.(\\ref{eq:MTOVk})). \nThe values of $\\chi_{\\rm TOV}^k$ and $\\chi_{\\rm max}$ of each EoS can be calculated utilizing the \\texttt{RNS} code, and may be also generally estimated as $\\chi_{\\rm TOV}=0.046 \\pm 0.008 (\\pm 0.004)$ and\n$\\chi_{\\rm max}=0.201\\pm 0.017 (\\pm 0.008)$ with 2$\\sigma$ (1$\\sigma$) errors, respectively.\n\n \n\n\\begin{figure}[ht!]\n\\resizebox{90mm}{!}{\\includegraphics[]{type.eps}} \n\\caption{The allowed parameter space of different types of merger products. The orange bunch of lines denote the mass-dependent normalized Keplerian period ${\\cal P}_k$; the red bunch of lines denote of constant $M_{\\rm TOV,b}$ lines; the green bunch of lines denote the boundary line for the SMNS to collapse into a BH. Each bunch includes 10 lines corresponding to 10 different EoSs. The black lines stand for the evolving trajectory of the merger product of GW170817. The markers show the points where the evolution phase changes. The dashed lines and hollow markers are schematic since the period of differential rotating NS is undefined. The vertical dark regions denote the separation lines for three regions at $P=P_k$.} \n\\label{fig:type}\n\\end{figure}\n\nAs shown in Table 1, the value of $M_{\\rm rem}^k$ only weakly depends on EoSs and is slightly correlated with $M_{\\rm TOV}$. We show the relationship between $M_{\\rm rem}^k$ and $M_{\\rm TOV}$ in Figure \\ref{fig:M}, which reads \\begin{eqnarray}\nM_{\\rm rem}^{k}=(2.354 \\pm 0.074)+(0.076\\pm 0.032)M_{\\rm TOV},\n\\label{eq:Mg}\n\\end{eqnarray}\nwith 2$\\sigma$ error. Combining Equations \\ref{eq:chi} and \\ref{eq:Mg}, one can derive the critical values to separate the HMNS\/BH vs. SMNS and SMNS vs. SNS, as also shown in Figure \\ref{fig:M}. \n\nThe following results can be obtained for GW170817: If the merger remnant is a HMNS\/BH, $M_{\\rm TOV}$ should be smaller than $2.09^{+0.11}_{-0.09}(^{+0.06}_{-0.04})M_{\\odot}$ at the 2$\\sigma$ (1$\\sigma$) level\\footnote{After mass shedding from the initial torus due to viscous and neutrino cooling processes, a quasi-stationary torus is supposed to exist surrounding the central core \\citep{siegel17,hanauske17,fujibayashi18}. The mass remaining in the torus is in Keplerian orbits, thus would not add to the gravitational mass in the core. Considering this effect, the values of the critical $M_{\\rm TOV}$ to separate the HMNS\/BH and SMNS cases would be smaller. However, since a significant fraction of mass in the torus has fallen back or ejected within $\\sim 0.3s$ \\citep{siegel17,fujibayashi18} (comparable to the lifetime of HMNS \\citep{metzger18}), the influence of the quasi-stationary torus to the critical $M_{\\rm TOV}$ is small.\n}; if the merger remnant is an SMNS, $M_{\\rm TOV}$ should be in the range from \n$2.09^{+0.11}_{-0.09}(^{+0.06}_{-0.04})M_{\\odot}$ to \n$2.43^{+0.10}_{-0.08}(^{+0.06}_{-0.04})M_{\\odot}$ at the 2$\\sigma$ (1$\\sigma$) level; if the merger remnant is a SNS, $M_{\\rm TOV}$ should be greater than $2.43^{+0.10}_{-0.08}(^{+0.06}_{-0.04}M_{\\odot})$ at the 2$\\sigma$ (1$\\sigma$) level. These results are generally consistent with previous results assuming an HMNS\/BH remnant in GW170817 \\citep[e.g.][]{margalit17,ruiz18,rezzolla18,shibata19}, even though some details differ. For example, \\cite{ruiz18} and \\cite{rezzolla18} both adopted the measured $M_{\\rm tot} \\sim 2.74 M_\\odot$ deducting the mass loss to estimate $M_{\\rm rem}$. This over-estimated $M_{\\rm rem}$ by $\\sim 0.2 M_\\odot$, which would over-estimate the upper limit of $M_{\\rm TOV}$. \\cite{shibata19} performed the most detailed analysis numerically and derived a more conservative upper limt $\\sim 2.3 M_\\odot$ for the HMNS\/BH case, but for the majority of the EOSs studied, the range of $M_{\\rm TOV}$ that form an HMNS\/BH product is still consistent with our estimate. Our derived separation line between SMNS and SNS products is also consistent with theirs.\n\n\n\n\n\\begin{figure}[ht!]\n\\resizebox{90mm}{!}{\\includegraphics[]{M.eps}} \n\\caption{Constraints on the range of $M_{\\rm TOV}$ for three different merger products in the case of GW170817. The colored data points represent the values of $M_{\\rm TOV}$ and the estimated $M_{\\rm rem}$ at the Keplerian period for different EoSs. The solid line is the best fitting relation of $M_{\\rm TOV}$ and $M_{\\rm rem}$, whereas the blue and grey dashed lines showing the $1\\sigma$ and $2\\sigma$ error range, respectively.  The slanted deep grey shadows are the allow regions when $\\chi=\\chi_{\\rm TOV}^k$ and $\\chi=\\chi_{\\rm max}$, respectively. The dot-dashed vertical lines are the central $M_{\\rm TOV}$ values of the separation lines of different merger products, which are surrounded by the $1\\sigma$ (light blue shadow) and $2\\sigma$ (light grey shadow) regions. The hollow circle is the predictive value for the EoS SLy, which forms a BH rather than a MNS.} \n\\label{fig:M}\n\\end{figure}\n\n\n\\section{Constraints on the NS Properties in the MNS cases}\n\nIn the case of SMNS, it would be interesting to investigate under what conditions the SMNS can survive for a particular duration of time, e.g. $\\sim$ 300 s for the typical duration of the X-ray internal plateau, $\\sim$ 1 d to power the blue component of the kilonova, and $\\sim$ 155 d to power the putative X-ray flare. This depends on the NS EoS and the spindown history of the putative MNS remnant.\n\nAt a certain spin period $P$, $\\chi_{\\rm TOV}<\\chi<\\chi_{\\rm col}$, the remnant would be an SMNS. For a particular object, $\\chi$  decreases as the NS loses its rotation energy through magnetic dipole radiation and GW radiation \\citep{shapiro83,zhang01}, i.e.\n\\begin{eqnarray}\n\\dot{\\Omega}=-\\frac{32GI\\epsilon^2\\Omega^5}{5c^5}-\\frac{B_p^2R^6\\Omega^3}{6Ic^3},\n\\label{dotE}\n\\end{eqnarray} \nwhere $\\Omega=2\\pi \/P$ is the angular frequency and $\\dot{\\Omega}$ is its derivative, $B_p$ stands for the surface dipole magnetic field strength at the pole, and $\\epsilon$ presents the ellipticity of the NS. As the NS spins down, it would collapse into a BH when $\\chi>\\chi_{\\rm col}$.\n  \n\nFor a rigidly rotating NS,  the maximum enhancement factor $\\chi_{\\rm col}$ at any given $P$ can be estimated with the EoS-independent relation Equation (\\ref{eq:chi_ucol}), which could be up to $\\chi_{\\rm max} \\sim 20\\%$ when the NS spin period equals to the allowed minimum Keplerian period. Given the initial spin period (Keplerian) and a particular desired lifetime of the MNS, it is possible to follow the spindown evolution and constrain $B_p$ and $\\epsilon$. \n\nTo be conservative, we assume the initial period $P_i=P_{\\rm k,min}$ for GW170817 and use the moment of inertia and radius of a non-rotating NS to constrain $B_p$ and $\\epsilon$ parameters\\footnote{The initial spin period of the merger product, $P_k$ should not be smaller than $P_{\\rm k,min}$, which means that smaller $\\epsilon$ and $B_p$ values than constrained are required for the SMNS to spin down to a certain period. Our derived upper limits on $B_p$ and $\\epsilon$ are therefore safe upper limits. Similarly, if one take larger $I$ and $R$ values for spinning NSs, one also requires smaller $B_p$ and $\\epsilon$ values than derived to reach the same spindown effect.}. Given a value of $\\chi_{\\rm col}$, we plot the boundary lines in the $B_p-\\epsilon$ plane for the region which allows an SMNS to survive for a certain lifetime, e.g. 300 s, 1 day and 155 days (see Figure \\ref{fig:universal}). During a particular time span, part of the spin-down power of the SMNS would be released in the EM channel, so that the EM counterpart observations of GW170817 could be used to make constraints on the spin-down power of the remnant SMNS, and hence,  on $B_p$ and $\\epsilon$. \n\nIf the SMNS can survive for 300s, we consider two constraints from the EM counterpart observations: the EM channel spin-down power integrated within 300s should be less than the kinetic energy of the merger ejecta [$\\beta<0.3$, inferred from the spectrum observation of the optical counterpart \\citep{kasen17,cowperthwaite17,chornock17,kilpatrick17,shappee17}] and less than the kinetic energy of the GRB jet [$E_k<10^{51}{\\rm erg~s^{-1}}$, inferred from the radio afterglow emission \\citep{nakar18,dobie18}]. If the SMNS can survive for 1 d, in addition to the constraints from the kinetic energy, since the merger ejecta already became optically thin at that time, the luminosity of the magnetic dipole spin-down should be smaller than the peak luminosity of the optical counterpart. If the SMNS can survive for 155d, besides the constraints from the kinetic energy and optical peak luminosity, the late time X-ray observations could also serve as the upper limit of the luminosity of the magnetic dipole spin-down. These constraints on $B_p$ and $\\epsilon$ from EM counterpart observations of GW170817 are  shown in Figure \\ref{fig:universal}.  We can see that the EM observations tend to constrain $B_p$ and $\\epsilon$ to small values. Nonetheless, if an SMNS is formed, there always exits a suitable $(B_p, \\epsilon)$ parameter space to allow the SMNS to survive for 300 s, 1 d, 155 d, or even longer,\nwithout violating the observational constraints.\nThese constraints are clearly displayed in Figure \\ref{fig:universal}.\n\n\n\nSome of our selected EoSs, e.g. WFF1, WFF2, AP4, BSK21, AP3 and DD2, are supposed to support an SMNS. In principle, with a certain EoS adopted, the properties of the SMNS could be constrained more precisely (see Figure \\ref{fig:eos}). If the lifetime of SMNS is 300 s or longer,  without violating the constraints from EM observations, we should have  $B_p<2.1 \\times 10^{14} G$ and $\\epsilon<2.0\\times 10^{-4}$ for WFF1, $B_p<2.2 \\times 10^{14}G$ and $\\epsilon<2.8\\times 10^{-4}$ for WFF2, $B_p<1.1\\times 10^{14}G$ and $\\epsilon<1.4 \\times 10^{-4}$ for AP4, $B_p<7.9 \\times 10^{14} G$ and $\\epsilon<1.2\\times 10^{-3}$ for BSK21, $B_p<3.4 \\times 10^{15}G$ and $\\epsilon<1.4\\times 10^{-2}$ for AP3, $B_p<4.2\\times 10^{15}G$ and $\\epsilon<3.3 \\times 10^{-2}$ for DD2. If the lifetime of SMNS is 1 d or longer,  without violating the EM observational constraints, we should have  $B_p<8.7 \\times 10^{10} G$ and $\\epsilon<1.3\\times 10^{-5}$ for WFF1, $B_p<8.7 \\times 10^{10}G$ and $\\epsilon<1.7\\times 10^{-5}$ for WFF2, $B_p<8.7\\times 10^{10}G$ and $\\epsilon<8.9 \\times 10^{-6}$ for AP4, $B_p<1.5 \\times 10^{11} G$ and $\\epsilon<7.4\\times 10^{-5}$ for BSK21, $B_p<1.6 \\times 10^{12}G$ and $\\epsilon<1.1\\times 10^{-3}$ for AP3 and $B_p<3.2\\times 10^{12}G$ and $\\epsilon<2.1 \\times 10^{-3}$ for DD2. If the lifetime of SMNS is 155 d or longer,  without violating the EM observational constraints, we should have  $B_p<8.5 \\times 10^{10} G$ and $\\epsilon<1.0\\times 10^{-6}$ for WFF1, $B_p<8.5 \\times 10^{10}G$ and $\\epsilon<1.4\\times 10^{-6}$ for WFF2, $B_p<8.5\\times 10^{10}G$ and $\\epsilon<7.1 \\times 10^{-7}$ for AP4, $B_p<8.7 \\times 10^{10} G$ and $\\epsilon<5.9\\times 10^{-6}$ for BSK21, $B_p<9.3 \\times 10^{10}G$ and $\\epsilon<2.2\\times 10^{-5}$ for AP3 and $B_p<1.0\\times 10^{11}G$ and $\\epsilon<3.4 \\times 10^{-5}$ for DD2. The constraints on $B_p$ mainly come from the EM observations, which are roughly consistent with (slightly looser than) the constraints derived in our previous work \\citep{ai18}. In this paper, we only used the peak luminosity (or total kinetic energy) rather than the full lightcurve (used in \\cite{ai18}) to constrain the parameters. The ellipticity $\\epsilon$, which was a free parameter in \\cite{ai18}, is now constrained by the lifetime of the SMNS.\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\begin{tabular}{l}\n\\resizebox{115mm}{!}{\\includegraphics[]{300s.eps}} \\\\ \n\\resizebox{115mm}{!}{\\includegraphics[]{1d.eps}} \\\\\n\\resizebox{115mm}{!}{\\includegraphics[]{155d.eps}} \n\\end{tabular}\n\\caption{Constraints on the allowed parameter space in the $B_p$-$\\epsilon$ plane for three survival times: 300 s (upper), 1 d (middle), and 155 d (lower).\nThe dashed lines are the boundaries for different putative EoSs (which correspond to different $\\chi$) within which the SMNS can survive for a certain timescale. The red and green lines show the constraints from the kinetic energy of ejecta, which are deduced from observations of afterglow and mergernova, respectively. The blue and purple lines stand for the constraints from the luminosity of the mergernova and the late-time X-ray signal. }\n\\label{fig:universal}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\begin{tabular}{l}\n\\resizebox{115mm}{!}{\\includegraphics[]{eos_300s.eps}} \\\\\n\\resizebox{115mm}{!}{\\includegraphics[]{eos_1d.eps}} \\\\\n\\resizebox{115mm}{!}{\\includegraphics[]{eos_155d.eps}}\n\\end{tabular}\n\\caption{Same as Figure 3, but for concrete EoSs studied in this paper.} \n\\label{fig:eos}\n\\end{center}\n\\end{figure*}\n\n\n\\section{Conclusions and Discussion}\n\nA tight constraint on the NS maximum mass $M_{\\rm TOV}$ is helpful to constrain the NS EOS. Before GW170817, the constraints on $M_{\\rm TOV}$ mainly comes from the observations of Galactic pulsars. Some massive pulsars have been observed, e.g. PSR J1614-2230 with $1.97 \\pm 0.04M_{\\odot}$ \\citep{demorest10} and PSR J0348+0432 with $2.01 \\pm 0.04M_{\\odot}$ \\citep{antoniadis13}), which set a lower limit to $M_{\\rm TOV}$ around $\\sim 2M_{\\odot}$. Recently, the most massive NS PSR J0740+6620 was measured to have a mass $2.14^{+0.10}_{-0.09}~M_{\\odot}$ at $68.3\\%$ confidence level \\citep{cromartie19}. This sets an even more stringent lower limit to $M_{\\rm TOV}$.\n\nNS-NS merger events could in principle give tighter constraints on $M_{\\rm TOV}$ if the merger product can be unambiguously identified. Unfortunately, without post-merger GW signal (which may not be obtained in the near future), the EM signals are not clean enough to draw definite conclusions. For the case of GW170817, even though the existence of GRB 170817A $\\sim 1.7$ s later was regarded by some authors as evidence of the formation of a BH before the onset of the GRB, some other authors argued for a long-lived NS remnant that may exist for an extended period of time. As a result, we cannot place a constraint on $M_{\\rm TOV}$. Rather, in this paper, we discuss the range of $M_{\\rm TOV}$ for different assumed merger products. We applied two approaches: case studies for 10 individual EoSs and an EoS-independent approach adopting some universal relations. We reached the following self-consistent results: If the merger product was a short-lived HMNS, one has $M_{\\rm TOV}<2.09^{+0.11}_{-0.09}(^{+0.06}_{-0.04})M_{\\odot}$; If the merger product was a long-lived SMNS, the constraint should be $2.09^{+0.11}_{-0.09}(^{+0.06}_{-0.04})M_{\\odot} \\leq M_{\\rm TOV}<2.43^{+0.10}_{-0.08}(^{+0.06}_{-0.04})M_{\\odot}$; If the merger product was a stable NS, the constraints should be $M_{\\rm TOV} \\geq 2.43^{+0.10}_{-0.08}(^{+0.06}_{-0.04})M_{\\odot}$. The quoted uncertainties are at the 2$\\sigma$ (1$\\sigma$) level. \n\n\nIf the merger remnant is a long-lived MNS, the next question is whether the MNS can survive for a desired period of time, e.g. 300 s, 1 d or 155 d, to interpret various observations. This depends on the spindown history of the remnant, which critically depends on two NS parameters, $B_p$ that defines the dipole spindown and $\\epsilon$ that defines the secular GW spindown. For an SNS remnant, this is never a problem. For an SMNS remnant, the survival time actually constrain $B_p$ and $\\epsilon$ to be smaller than certain values. We have derived these constraints for the case of GW170817 for different EoSs (which have different $M_{\\rm TOV}$ and hence require different $\\chi$ values). These constraints, together with those posed from the EM observations, define the parameter space in the $(B_p, \\epsilon)$ plane that satisfies the desired lifetime. In general, we find that for any EoS that forms an SMNS in the case of GW170817, without violating the EM observational constraints, there always exist a set of ($B_p, \\epsilon$) parameters that makes the SMNS survive for 300 s, 1 d, 155 d or even longer. {In particular, for EoSs in Table 1 with $M_{\\rm TOV}$ from $2.14M_{\\odot}$ to $2.42M_{\\odot}$, we have $\\epsilon\\lesssim 1.4\\times10^{-4}-3.3\\times10^{-2}$ and $B_p\\lesssim (1.1\\times10^{14}-4.2\\times10^{15})$G if the SMNS survives for 300 s; $\\epsilon\\lesssim 8.9\\times10^{-6}-2.1\\times10^{-3}$ and $B_p\\lesssim(8.7\\times10^{10}-3.2\\times10^{12})$G  if the SMNS survives for 1 d; and $\\epsilon\\lesssim 7.1\\times10^{-7}-3\\times1.0\\times10^{-6}$ and $B_p\\lesssim(8.5\\times10^{10}-9.3\\times10^{12})$G if the SMNS survives for 155 d.}  \n\n\nFuture joint GW\/EM observational campaigns of NS-NS merger events may identify more definite observational criteria to identify the nature of the merger remnants. The similar approach proposed in this paper can be applied in those events, which will lead to tighter constraints on $M_{\\rm TOV}$ and NS EoS.\n\n\\acknowledgments\nWe thank Luciano Rezzolla and Masaru Shibata for helpful discussion and the anonymous referee for useful comments.\nHG acknowleges support by National Natural Science Foundation of China under Grant No. 11722324, 11603003, 11633001 and 11690024, the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB23040100 and the Fundamental Research Funds for the Central Universities.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\nIt is known that the vacuum structure of quantum chromodynamics\n(QCD) is nontrivial and nonperturbative \\cite{nielsen,svz,amqcd}.\nOne would like to\nstudy the same in the context of field theory of quarks and\ngluons. We believe that we may have hadronic to quark gluon\nphase transition in heavy ion collisions. Signals from the\nsame however get masked since this phase lasts for a very short\ntime, and is followed by hadronisation. On the other hand, at\nvery high densities, as in neutron stars, we expect that hadrons\nwill dissolve to quark gluon matter. We wish to study the\nsame in the present paper with a nonperturbative equation\nof state derived earlier \\cite{zphyc}.\n\nAs a possible explanation for dark matter and otherwise,\nWitten had suggested \\cite{witten} that quark nuggets with\nstrange quarks may exist in the universe. Also, quark\nstars as hypothetical objects have been considered\nwith perturbative QCD \\cite{astro}. Hybrid stars\nwith the same perturbative approximation of QCD \\cite{kapusta}\nand semi-phenomenological treatment of hadronic matter\nwith a bag pressure has also been examined with various\ntypes of models \\cite{ellis,glendn,rosen}.\nRecently, also hadronic matter had been considered\nnonperturbatively with a perturbative treatment of\nquark matter to study hybrid stars \\cite{star}.\nHowever, in a nonperturbative treatment\n\\cite{amqcd,zphyc} earlier, we found a change in the nature of\nequation of state for quark matter.\nWe study here quark stars \\cite{witten,astro} with the\nabove nonperturbative equation of state.\nWe find that some conclusions\ndiffer by a factor as against perturbative calculations.\n\nThe equation of state for quark matter as will be used has been\ndiscussed in Ref. \\cite{zphyc}. We shall extend the same\nincluding the strange quarks \\cite{witten} and electrons with\ncharge neutrality condition.\nWe organise the paper as follows. In section II we shall\nrecapitulate the equation of state as was derived in\nRef. \\cite{zphyc} along with modifications as above.\nIn section III we shall use this to\nconsider Tolman Oppenheimer Volkoff (TOV)  equations and\ncalculate different properties of hypothetical quark\nstars. In section IV we summarise the results.\n\nThe motivation of the present analysis is to study bulk\nproperties in QCD with a\nnonperturbative calculation. We find\nthe change in results from perturbative calculation to be significant.\n\n\\section {Equation of state for strange matter}\nWe had earlier considered the behaviour of gluon condensates at finite\nbaryon densities for up and down quarks \\cite{zphyc}.\nWe shall here include the strange quark and electron\nwith charge neutrality condition to derive an\nequation of state to study quark stars. For this purpose we\nbriefly recapitulate the notations of Ref. \\cite{zphyc}\nand extend it as above.\nThe QCD Lagrangian is given as\n\\begin{equation}\n{\\cal L} ={\\cal L}_{gauge}+{\\cal L}_{matter}+{\\cal L}_{int},\n\\end{equation}\n\\noindent where\n\\begin{mathletters}\n\\begin{equation}\n{\\cal L}_{gauge}=-{1\\over 2}G^{a\\mu\\nu}(\\partial_{\\mu}{W^{a}}_{\\nu}\n-\\partial_{\\nu}{W^{a}}_{\\mu}+gf^{abc}{W^{b}}_{\\mu}{W^{c}_{\\nu})}\n+{1\\over 4}{G^{a}}_{\\mu\\nu}{G^{a\\mu\\nu}},\n\\end{equation}\n\\begin{equation}\n{\\cal L}_{matter}=\\bar \\psi (i \\gamma ^\\mu \\partial _\\mu-m_i)\\psi\n\\end{equation}\n\\noindent and\n\\begin{equation}\n{\\cal L}_{int}=g\\bar \\psi \\gamma ^\\mu  \\frac {\\lambda^a}{2}\nW _\\mu^a\\psi,\n\\end{equation}\n\\end{mathletters}\n\\noindent where ${W^{a}}_{\\mu}$ are the SU(3) colour gauge fields.\nWe shall quantise in Coulomb gauge \\cite{schwinger}\nand write the\nelectric field  ${G^{a}}_{0i}$ in terms of\nthe transverse and longitudinal parts as\n\\begin{equation}\n{G^a}_{0i}=\n{^TG^a}_{0i}+{\\partial_{i}{f}^{a}},\n\\end{equation}\nwhere ${f}^{a}$ is to be determined.\nWe take at time t=0  \\cite{amqcd}\n\\begin{mathletters}\n\\begin{equation}\n{W^{a}}_{i}(\\vec x)={(2\\pi)^{-3\/2}}\\int\n{d\\vec k\\over \\sqrt{2\\omega(\\vec k)}}({a^a}_{i}(\\vec k) +\n{{a^a}_{i}(-\\vec k)}^{\\dagger})\\exp({i\\vec k.\\vec x})\n\\end{equation}\n\\noindent and\n\\begin{equation}\n{^{T}G^{a}}_{0i}(\\vec x)={(2\\pi)^{-3\/2}} i \\int\n{d\\vec k}{\\sqrt{\\omega(\\vec k)\\over 2}}(-{a^a}_{i}(\\vec k) +\n{{a^a}_{i}(-\\vec k)}^{\\dagger})\\exp({i\\vec k.\\vec x}),\n\\end{equation}\n\\end{mathletters}\nwhere, $\\omega(k)$ is arbitrary \\cite{schwinger}\nand for equal time algebra we have\n\\begin{equation}\n\\left[ {a^a}_{i}(\\vec k),{{a^b}_{j}(\\vec k^{'})}^\\dagger\\right]=\n\\delta^{ab}\\Delta_{ij}(\\vec k)\\delta({\\vec k}-{\\vec k^{'}}),\n\\end{equation}\nwith\n\\begin{equation}\n\\Delta_{ij}(\\vec k)={\\delta_{ij}}-{k_{i}k_{j}\\over k^2}.\n\\end{equation}\nThe equal time quantization condition for the fermionic sector\nis given as\n\\begin{equation}\n[\\psi _{\\alpha}^i(\\vec x,t),\\psi _\\beta^j (\\vec y,t\n)^{\\dagger}]_{+}=\\delta ^{ij}\\delta _{\\alpha \\beta}\\delta(\\vec x -\\vec y),\n\\end{equation}\nwhere $i$ and $j$ refer to the colour and flavour indices\n\\cite{zphyc,pot}.\nWe now also have the field expansion for fermion field\n$\\psi$ at time t=0  given as \\cite{zphyc,pot}\n\\begin{equation}\n\\psi^{i}(\\vec x)=\\frac {1}{(2\\pi)^{3\/2}}\\int\n\\left[U_r(\\vec k)c^{i}_{Ir}(\\vec k)\n+V_s(-\\vec k)\\tilde c^{i}_{Is}(-\\vec k)\\right]\ne^{i\\vec k\\cdot \\vec x} d\\vec k,\n\\end{equation}\nwhere $U$ and $V$ are given by \\cite{spm78}\n\\begin{equation}\nU_r(\\vec k)=\\left( \\begin{array}{c}\\cos \\frac {\\chi (k)}{2}\n\\\\ \\vec \\sigma \\cdot\\hat k \\sin \\frac {\\chi (k)}{2}\n\\end{array}\\right)u_{Ir} ;\\quad V_s(-\\vec k)=\\left(\n\\begin{array}{c}\\vec \\sigma .\\hat k \\sin \\frac {\\chi (k)}{2}\n\\\\ \\cos \\frac {\\chi(k)}{2} \\end{array}\\right)v_{Is},\n\\end{equation}\n\\noindent where the function $\\chi(k)$ could be arbitrary.\nHere we approximate the same as\n$\\cos \\chi (k)=\n{m_i}\/{\\epsilon_i (k)}$ and $\\sin \\chi(k)={k}\/{\\epsilon_i (k)}$,\nwith $\\epsilon_i (k)=(k^2+m_i^2)^{1\/2}$.\nThe above are consistent with the equal time anticommutation conditions\nwith \\cite{spm78}\n\\begin{equation}\n[c^{i}_{Ir}(\\vec k),c^{j}_{Is}(\\vec k')^\\dagger]_{+}=\n\\delta _{rs}\\delta^{ij}\\delta(\\vec k-\\vec k')=\n[\\tilde c_{Ir}^{i}(\\vec k),\\tilde c_{Is}^{j}(\\vec k')\n^\\dagger]_{+},\n\\end{equation}\n\nIn Coulomb gauge, the expression for the Hamiltonian\ndensity, ${\\cal T}^{00}$ from equation (1) is given as \\cite{schwinger}\n\\begin{eqnarray}\n{\\cal T}^{00}&=&:{1\\over 2}{^{T}{G^a}_{0i}}{^{T}{G^a}_{0i}}+\n{1\\over 2}{W^a}_{i}(-\\vec \\bigtriangledown^2){W^a}_{i}+\ngf^{abc}W^a_iW^b_j\\partial _i W^c_j\\nonumber \\\\ &+&\n{{g^2}\\over 4}f^{abc}f^{aef}{W^b_i}{W^c_j}{W^e_i}{W^f_j}+\n{1\\over 2}(\\partial_{i}f^{a})(\\partial_{i}f^{a})\n\\nonumber \\\\ &+&\\bar \\psi (-i \\gamma ^i \\partial _i+m_i)\\psi\n-g\\bar \\psi \\gamma ^ i \\frac {\\lambda^a}{2}\nW _i^a\\psi:,\n\\label{t00}\n\\end{eqnarray}\n\\noindent where : : denotes the normal ordering with respect to\nthe perturbative vacuum, say $\\mid vac>$, defined through\n${a^a}_{i}(\\vec k)\\mid vac>=0$, $c_{Ir}^i(\\vec k)\\mid vac>=0$\nand $\\tilde c_{Ir}^i(\\vec k)^\\dagger\\mid vac >=0$.\nIn order to solve for the operator $f^a$, we first note that\n\\begin{equation}\nf^a=-{W^a}_0-g \\; f^{abc}\\;{ (\\vec \\bigtriangledown ^2)}^{-1}\n({W^b}_i \\; \\partial _i {W^c}_0).\n\\end{equation}\nProceeding as earlier \\cite{amqcd,zphyc} with a mean\nfield type of approximation we obtain,\n\\begin{eqnarray}\n\\vec \\bigtriangledown ^2{W^a}_0 (\\vec x )\n&&+ g^2 \\; f^{abc}f^{cde} \\;<vac',\\beta\\mid  {W^b}_i(\\vec x ) \\partial _i\n(\\vec \\bigtriangledown ^2)^{-1}({W^d}_j(\\vec x ) \\mid vac',\\beta>\n\\partial _j{W^e}_0(\\vec x ))\\nonumber\\\\ && ={J^a}_0(\\vec x ),\n\\end{eqnarray}\nwhere,\n\\begin{equation}\nJ^a_0=gf^{abc}{W^b_i}^{T}{G^c_{0i}}-g\\bar \\psi \\gamma^0 \\frac {\\lambda ^a}{2}\n\\psi.\n\\end{equation}\nIn the above, $|vac',\\beta>$ is the vacuum at finite temperature and\ndensity in the language of thermofield dynamics \\cite{tfd}.\nWe shall take the ansatz for the above state as \\cite{zphyc}\n\\begin{equation}\n|vac',\\beta>=U_G(\\beta)U_F(\\beta)|vac'>,\n\\label{vacbeta}\n\\end{equation}\nwith $|vac'>$ being the gluon condensate vacuum at zero temperature\ngiven as\n\\begin{equation}\n\\mid vac ^{'}>\n=U\\mid vac>\\equiv\n \\exp({B^\\dagger}-B)\\mid vac>\n\\end{equation}\nobtained through the unitary operator $U$ on the pertubative\nvacuum. In the above the condensate creation operator $B^\\dagger $\nis given explicitly as\n\\cite{zphyc}\n\\begin{equation}\n{B^\\dagger}={1\\over 2}\n\\int {f(\\vec k){{{a^a}_{i}(\\vec k)}^\\dagger}\n{{{a^a}_{i}(-\\vec k)}^{\\dagger}}d\\vec k},\n\\end{equation}\n\\noindent\nwhere $f(\\vec k)$ describes the vacuum structure with\ngluon condensates. Further in equation (\\ref {vacbeta})\n $U_G$ and $U_F$ are unitary operators \\cite{tfd} involving\nthermal excitations of gluons and fermions (quarks and\nelectrons) respectively.\nFor the gluon sector, we have \\cite{amqcd,zphyc}\n \\begin{equation}\nU_G(\\beta)\n=\\exp{({B_G(\\beta)}^{\\dagger}-B_G(\\beta))},\n\\end{equation}\nwith\n\\begin{equation}\n{B_G(\\beta)}^{\\dagger}=\n\\int \\theta(\\vec k,\\beta){{b^a}_{i}(\\vec k)}^{\\dagger}\n{{{\\underline b}^a}_{i}(-\\vec k)}^{\\dagger}d\\vec k.\n\\label{bgbeta}\n\\end{equation}\nIn addition, for fermionic sector we have,\n\\begin{equation}\nU_F(\\beta)\n=\\exp{({B_F(\\beta)}^{\\dagger}-B_F(\\beta))},\n\\end{equation}\nwith\n\\begin{equation}\n{B_F(\\beta)}^{\\dagger}=\n\\int \\bigg[\\theta^i_{-}(\\vec k,\\beta){{c^i}_{Ir}(\\vec k)}^{\\dagger}\n{\\underline c}^i_{Ir}(-\\vec k)^{\\dagger}\n+ \\theta^i_{+}(\\vec k,\\beta){\\tilde c}^i_{Ir}(\\vec k)\n{\\underline {\\tilde c}^i}_{Ir}(-\\vec k)\n\\bigg]\\; d\\vec k,\n\\label{bfbeta}\n\\end{equation}\nwhere, $i$ refers to u, d, s quarks and electron. Further the\nsummation over the color and flavor indices  for the quarks\nis understood and the underlined operators in equations\n(\\ref{bgbeta}), (\\ref{bfbeta}) correspond to the extra Hilbert space\nin thermofield dynamics.\n$\\theta(\\vec k,\\beta)$, $\\theta^i_{\\pm}(\\vec k, \\beta)$ are\nfunctions related to the distribution functions and are given by \\cite{tfd}\n\\begin{mathletters}\n\\begin{equation}\nsinh^2\\theta(\\vec k,\\beta) =\\frac{1}{\\exp(\\beta\\omega(\\vec k,\\beta))-1}\n\\label{distrb}\n\\end{equation}\nand\n\\begin{equation}\nsin^2\\theta^i_{\\pm}(\\vec k,\\beta) =\\frac{1}{\\exp(\\beta(\\epsilon^i\n(\\vec k)\\pm \\mu^i))+1}\n\\label{distrf}\n\\end{equation}\n\\end{mathletters}\nwhere $\\omega(k,\\beta)=\\sqrt{k^2+m_G^2}$,\n$\\epsilon^i(k)=\\sqrt{k^2+m_i^2}$ and $\\mu^i$ is the quark (or electron)\nchemical potential.\n\nOur job now is to evaluate the expectation value of\n${\\cal T}^{00}$ and then the thermodynamic potential\nwith respect to $\\mid vac^{'};\\beta>$.\nThe expressions obtained are almost the same as earlier\n\\cite{zphyc} except for additional contributions arising\nfrom the electron and the strange quark sectors.\nWe thus have \\cite{zphyc}\n\\begin{equation}\n<vac';\\beta \\mid :{W^a}_{i}(\\vec x){W^b}_{j}(\\vec y):\n\\mid  vac';\\beta >=\n{\\delta }^{ab}\n\\times (2  \\pi )^{-3}\\int d\\vec k e^{i\\vec k.(\\vec x-\n\\vec y)}\\; {F_{+}(\\vec  k,\\beta )\\over \\omega (k,\\beta )}\\;\n\\Delta _{ij}(\\vec k),\n\\label{ww}\n\\end{equation}\n\\begin{equation}\n{<vac';\\beta \\mid}: {^{T} G^{a}_{0i}} (\\vec x)\n{^{T} G^{b}_{0j}} (\\vec y):{\\mid vac';\\beta >}\n= \\delta ^{ab}\\times (2 \\pi )^{-3}\n\\int d{\\vec k}e^{i{\\vec k}.{(\\vec x-\\vec y)}}\n{\\Delta _{ij}(\\vec k)\\omega (k,\\beta )}\nF_{-}( k,\\beta ).\n\\label{gg}\n\\end{equation}\nIn the above the temperature dependant\n$F_{\\pm}(k,\\beta )$ are given as\n\\begin{equation}\nF_{\\pm}(\\vec k,\\beta )  = \\cosh 2\\theta \\bigg ({\\sinh}^{2}f(k)\n\\pm{\\sinh 2f(k)\\over 2}\\bigg ) +\\sinh ^2 \\theta (k,\\beta )\n\\label{fpm}\n\\end{equation}\nwhere $\\sinh ^ 2 \\theta (\\vec k,\\beta)$ is given by equation (\\ref{distrb}).\nSimilarly, for the quark fields we have the parallel equations given as\n\\begin{mathletters}\n\\begin{equation}\n<:\\psi^{i}_\\alpha(\\vec x)^{\\dagger}\n\\psi^{j}_\\beta(\\vec y):>_{vac',\\beta}=\n(2\\pi)^{-3}\\delta^{ij}\\int\n\\Big ( \\Lambda^i _-(\\vec k,\\beta)\\Big )_{\\beta\\alpha}\ne^{-i\\vec k .(\\vec x-\\vec y)}d\\vec k,\n\\label{jpj}\n\\end{equation}\n\\begin{equation}\n<:\\psi^{i}_\\alpha(\\vec x)\n\\psi^{j}_\\beta(\\vec y)^{\\dagger}:>_{vac',\\beta}=\n(2\\pi)^{-3}\\delta^{ij}\\int\n\\Big ( \\Lambda^i _+(\\vec k,\\beta)\\Big )_{\\alpha \\beta}\ne^{i\\vec k .(\\vec x-\\vec y)}d\\vec k,\n\\label{jjp}\n\\end{equation}\n\\end{mathletters}\n\\noindent where\n\\begin{equation}\n\\Lambda^i_{\\pm}(\\vec  k,\\beta)=\\mp\\frac{1}{2}\\Big [\\big\n(\\sin ^2 \\theta _{-}^i-\n\\sin ^2 \\theta _{+}^i\\big )+\\big (\\gamma^0 \\cos \\chi+\n\\vec \\alpha .\\hat k \\sin \\chi\\big)\\big (\\sin ^2 \\theta^i_{-}\n+\\sin ^2 \\theta^i_{+}\\big )\\Big ].\n\\end{equation}\nUsing equations (\\ref{t00}), (\\ref{ww}), (\\ref{gg}) and (26),\nwe then obtain\nthe expectation value of ${\\cal T}^{00}$ with respect to\n$\\mid vac^{'};\\beta>$ as\n\\begin{eqnarray}\n\\epsilon_{0}(\\beta) &\n\\equiv & <vac^{'};\\beta \\mid\n:{\\cal T}^{00}:\\mid vac^{'};\\beta> \\nonumber \\\\\n& = & C_{F}(\\beta)+C_F^e (\\beta)\n+C_{1}(\\beta)+C_{2}(\\beta)+{C_{3}(\\beta)}^{2}+C_{4}(\\beta),\n\\label{enrgd}\n\\end{eqnarray}\n\\noindent where\n\\begin{mathletters}\n\\begin{eqnarray}\nC_F(\\beta)&=&<:\\bar \\psi (-i\\gamma^i\\partial _i +\nm_i)\\psi:>_{vac',\\beta}\\nonumber\\\\ &=&\n\\sum_{u,d,s}\\frac{6}{\\pi^2}\\int {k^2dk}\n(\\sin^2 \\theta^i_{-}(\\vec k,\\beta)+\\sin^2 \\theta^i_{+}(\\vec k,\\beta))\n\\epsilon^i(k),\n\\end{eqnarray}\n\\begin{eqnarray}\nC_F^e(\\beta)&=&<:\\bar \\psi (-i\\gamma^i\\partial _i\n )\\psi:>_{vac',\\beta}\\nonumber\\\\ &=&\n\\frac{2}{\\pi^2}\\int {k^3dk}\n(\\sin^2 \\theta^e_{-}(\\vec k,\\beta)+\\sin^2 \\theta^e_{+}(\\vec k,\\beta))\n\\end{eqnarray}\nis the contribution for the electron,\n\\begin{eqnarray}\nC_{1}(\\beta) & = & <:{1\\over 2}\n{^T}{G^a}_{0i}{^T}{G^a}_{0i}:>_{vac^{'};\\beta}\\nonumber \\\\\n& = & {4\\over {\\pi^2}}\\int \\omega(k,\\beta)k^{2} F_{-}(k,\\beta )\\;dk,\n\\end{eqnarray}\n\\begin{eqnarray}\nC_{2}(\\beta ) & = & <:{1\\over 2}\n{W^a}_{i}{(-\\vec \\bigtriangledown^2)}{W^a}_{i}:>_{vac';\\beta }\\nonumber\n\\\\ & = & {4\\over {\\pi^2}}\\int {{k^{4}}\\over \\omega(k,\\beta)}\n\\;F_{+}(k,\\beta )\\;dk\n\\end{eqnarray}\n\\begin{eqnarray}\n{C_{3}(\\beta )}^{2} & = & <:{1\\over 4}g^{2}f^{abc}f^{aef}\n{W^b}_{i}{W^c}_{j}{W^e}_{i}{W^f}_{j}:>_{vac';\\beta }\\nonumber \\\\\n& = & \\left({{2g}\\over {\\pi^2}}\\int {{k^{2}}\\over\n{\\omega(k,\\beta )}}\\;F_{+}(k,\\beta )\\;dk\\right)^2 ,\n\\end{eqnarray}\n\\noindent and\n\\begin{eqnarray}\nC_{4}(\\beta ) & = & <:{1\\over 2}\n(\\partial_{i}f^{a})(\\partial_{i}f^{a}):>_{vac{'};\\beta },\\nonumber\\\\\n & = & 4\\times (2 \\pi )^{-6}\\int  d \\vec  k {G_1(\\vec k,\\beta )\n+G_2(\\vec k,\\beta)\\over\n{k^2+\\phi (k,\\beta )}}.\n\\end{eqnarray}\n\\end{mathletters}\nIn the above,\n\\begin{mathletters}\n\\begin{eqnarray}\nG_1(\\vec k,\\beta ) & = & 3 g^2 \\int d  \\vec q\nF_{+}({\\mid} \\vec q\\mid,\\beta )\\;\nF_{-}({\\mid} \\vec k +\\vec q {\\mid},\\beta  ) \\;\n{\\omega ({\\mid  \\vec  k +\\vec q \\mid},\\beta )\\over\n\\omega  ({\\mid \\vec  q\\mid},\\beta  )}\n\\nonumber \\\\\n& \\times & \\Bigl  (1+{{(q^2 +\\vec k.\\vec q)^2\n}\\over{q^2(\\vec k+\\vec q)^2}}\\Bigr ),\n\\end{eqnarray}\nand, now including the strange quark contribution,\n\\[G_2(\\vec k, \\beta)=\\sum_{u,d,s} G^i_2(\\vec k,\\beta)\\]\nwith\n\\begin{eqnarray}\nG^i_2(\\vec k,\\beta) & = & -\\frac{g^2}{2}\\int d \\vec q \\Bigg [\\Big (\n1+\\frac{m_i^2}{\\epsilon_i (\\vec q)\\epsilon_i (\\vec q-\\vec k)}\n+\\frac {\\vec q . (\\vec q-\\vec k)}\n{\\epsilon_i (\\vec q)\\epsilon_i (\\vec q-\\vec k)}\\Big )\\nonumber \\\\\n& \\times & \\Big (\\sin ^2 \\theta^i_{-}(\\vec q,\\beta)\\sin ^2 \\theta^i_{-}(\\vec\nq-\\vec k,\\beta)\n+\\sin ^2 \\theta^i_{+}(\\vec q,\\beta )\\sin ^2 \\theta^i_{+}(\\vec q -\\vec\nk,\\beta)\\Big )\n\\nonumber\\\\ & - & \\Big (\n1-\\frac{m_i^2}{\\epsilon_i (\\vec q)\\epsilon_i (\\vec q-\\vec k)}\n-\\frac {\\vec q . (\\vec q-\\vec k)}\n{\\epsilon_i (\\vec q)\\epsilon_i (\\vec q-\\vec k)}\\Big )\\nonumber \\\\\n& \\times &\\Big (\\sin ^2 \\theta^i_{-}(\\vec q,\\beta)\\sin ^2 \\theta^i_{+}(\\vec\nq-\\vec k,\\beta)\n+\\sin ^2 \\theta^i_{+}(\\vec q,\\beta )\\sin ^2 \\theta^i_{-}(\\vec q -\\vec\nk,\\beta)\\Big )\n\\Bigg ]\n\\end{eqnarray}\n\\noindent  and\n\\begin{equation}\n\\phi  (k,\\beta )=  {3g^2\\over  {8 \\pi  ^2}}\n\\int {{dk'}\\over {\\omega  (k',\\beta  )}}\\;F_{+}(k',\\beta )\n\\biggl (  k^2+{k'}^2-{(k^2-{k'}^2)^2\n\\over{2kk'}}\\log \\Big | {{k+k'}\\over{k-k'}}\n\\Big |  \\biggr ).\n\\end{equation}\n\\end{mathletters}\nAs stated the expressions above are the same \\cite{zphyc}\nas earlier, except for the additional term\n$C_F^e(\\beta)$  as well as the additional contribution in\n $G_2(\\vec k,\\beta)$ in the expression for $C_4(\\beta)$\narising from the strange quarks.\n\nWe shall take $\\omega (\\vec k,\\beta)$ to be\nof the free field form with a temperature dependent effective mass parameter\nfor the gluon fields given as\n\\begin{equation}\n\\omega (\\vec k,\\beta)=\\sqrt {k^2+m_G(\\beta)^2};\\;\\;\n\\end{equation}\n\\noindent with $m_G(\\beta)$ given through the self consistency condition\n\\cite {amqcd,biro}\n\\begin{eqnarray}\nm_G(\\beta)^2 =\n{{2g^2}\\over {\\pi^2}}\\int {{k^{2}}\\over\n{\\omega(k,\\beta)}}\\;F_{+}(k,\\beta)\\;dk\\label{mg0}\n\\end{eqnarray}\nfrom the single contraction  contribution from\nthe quartic terms \\cite{zphyc}.\n\nWe are to now extremise over the thermodynamic potential\ncontaining $\\epsilon_0(\\beta)$ including the additional\neffects. For this purpose, we shall take the ansatz \\cite{zphyc}\n\\begin{equation}\nsinh f(\\vec k)=Ae^{-Bk^2\/2},\n\\end{equation}\nwhich corresponds to taking a gaussian distribution for perturbative\ngluons in nonperturbative vacuum \\cite{amqcd}. The energy density,\n$\\epsilon_{0}(\\beta)$ in terms of the dimensionless\nquantities $x={\\sqrt {B}}k$, $\\mu_G={\\sqrt B}m_G(\\beta)$,\n$m_i'={\\sqrt B}m_i$, $\\mu_i'={\\sqrt B}\\mu_i$\nand $y={\\beta\\over {\\sqrt B}}$ then gets parametrised as\n\\begin{eqnarray}\n\\epsilon_{0}(A,\\beta)\n& = & {1\\over {B^2}}(I_F(y)+I_F^e(y)+I_{1}(A,y)+I_{2}(A,y)+{I_{3}(A,y)}^{2}+\nI_{4}(A,y)) \\nonumber \\\\\n& \\equiv & {1\\over {B^2}}F(A,y),\n\\end{eqnarray}\n\\noindent where\n\\begin{mathletters}\n\\begin{equation}\nI_F(y)=\\sum_{u,d,s}{\\frac{6}{\\pi^2}\\int {x^2dx}\n(\\sin ^2 \\theta _{-}^i(\\vec x,y)+\\sin ^2 \\theta _{+}^i(\\vec x,y))\n\\epsilon^i(x)},\n\\end{equation}\n\\begin{equation}\nI_F^e(y)=\\frac{2}{\\pi^2}\\int {x^3dx}\n(\\sin ^2 \\theta _{-}^e(\\vec x,y)+\\sin ^2 \\theta _{+}^e(\\vec x,y))\n\\end{equation}\n\\noindent  and\n\\begin{equation}\nI_{4}(A,y) =4\\times {(2 \\pi )^{-6}}\\int d{\\vec x}\n{{G_1(\\vec x,y)+G_2(\\vec x,y)}\\over {x^2+\\phi (x,y)}}.\n\\label{i4}\n\\end{equation}\n\\end{mathletters}\nIn the above, $G_1(\\vec x,y)=G(\\vec x,y)$ of \\cite{amqcd}, and\n\\begin{eqnarray}\nG_2(\\vec x,y) & = & -\\sum_{u,d,s}\\frac{g^2}{2}\\int d \\vec x' \\Bigg [\\Big (\n1+\\frac{m'_i^2}{\\epsilon_i (\\vec x')\\epsilon_i (\\vec x'-\\vec x)}\n+\\frac {\\vec x' . (\\vec x'-\\vec x)}\n{\\epsilon_i (\\vec x')\\epsilon_i (\\vec x'-\\vec x)}\\Big )\\nonumber \\\\\n& \\times & \\Big (\\sin ^2 \\theta^i_{-}(\\vec x',y)\n\\sin ^2 \\theta^i_{-}(\\vec x'-\\vec x,y)\n+\\sin ^2 \\theta^i_{+}(\\vec x',y )\\sin ^2 \\theta^i_{+}(\\vec x' -\\vec x,y)\\Big )\n\\nonumber \\\\ & - &\n\\Big ( 1-\\frac{m'_i^2}{\\epsilon_i (\\vec x')\\epsilon_i (\\vec x'-\\vec x)}\n-\\frac {\\vec x' . (\\vec x'-\\vec x)}\n{\\epsilon_i (\\vec x')\\epsilon_i (\\vec x'-\\vec x)}\\Big )\\nonumber\\\\&\n \\times & \\Big (\\sin ^2 \\theta^i_{-}(\\vec x',y)\n\\sin ^2 \\theta^i_{+}(\\vec x'-\\vec x,y)\n+\\sin ^2 \\theta^i_{+}(\\vec x',y )\\sin ^2 \\theta^i_{-}\n(\\vec x' -\\vec x,y)\\Big )\n\\Bigg ] .\\nonumber\\\\& &\n\\end{eqnarray}\n\\noindent  The other expressions $I_1(A,y)$, $I_2(A,y)$, $I_3(A,y)$\nand $\\phi(x,y)$ in dimensional units are as earlier\n\\cite{amqcd,zphyc}. The above\nintegrals contain $\\mu_G(y)$ which is\ndetermined from the self consistency requirement \\cite{amqcd}\n\\begin{eqnarray}\n\\mu_G(y)^2 & =& {{2g^2}\\over {\\pi^2}}\n\\int  {{x^2}dx\\over \\omega(x,y)}\n\\Biggl [ \\biggl (A^{2}e^{-x^2}\n+Ae^{-{{x^2}\/2}}(1+A^{2}e^{-x^2})^{1\/2}\\biggr )\n\\biggl (1+{2\\over \\exp({y\\omega(x,y)})-1}\\biggr ) \\nonumber \\\\ & + &\n{1\\over \\exp({y\\omega(x,y)})-1}\\Biggr ],\n\\label{49}\n\\end{eqnarray}\nThe thermodynamic potential at temperature $T=1\/\\beta $ is given as \\cite{tfd}\n\\begin{equation}\n{\\cal F}(A,\\beta )=\\epsilon _{0}{(A,\\beta)}-{1\\over \\beta }(S_G+S_Q+S_e)\n-\\mu_B \\rho_B - {\\mu_E\\rho_E}.\n\\label{free}\n\\end{equation}\nHere $\\mu_B$ is the baryon chemical poential corresponding to\nthe baryon number density $\\rho_B$ given as,\n\\begin{equation}\n\\rho_B=\\frac{1}{3}\\times 3 \\times 2\\times\\frac{1}{(2\\pi)^3}\n\\int \\Big (\\sum_{u,d,s}\\sin^2\\theta^i_{-}- \\sum_{u,d,s}\\sin^2\\theta^i_{+}\\Big)\n d\\vec k ,\n\\label{bnb}\n\\end{equation}\nand $\\mu^i$ as in equation (\\ref{distrf}) is the quark chemical potential\nfor $i$th flavor. Further,\nin the above, $\\epsilon _{0}(A,\\beta)$ is as given in equation (\\ref{enrgd}),\nand the entropy densities $S_G$, $S_Q$ and $S_e$ for the gluon, quark and\nelectron fields\nare given as \\cite{tfd}\n\\begin{equation}\nS_G=-2\\times 8 \\times (2\\pi)^{-3}\n\\int d{\\vec k}\\Big (\\sinh ^{2} \\theta \\; log \\bigl(\\sinh ^{2} \\theta\\bigr)\n-\\cosh ^{2} \\theta \\; log \\bigl(\\cosh ^{2} \\theta\\bigr) \\Bigr ),\n\\label{sg}\\end{equation}\n\\begin{eqnarray}\nS_Q&=&-\\sum_{u,d,s}\\bigg[3\\times 2\\times (2\\pi)^{-3}\n\\int d{\\vec k}\\Big (\\sin ^2 \\theta _{-}^i(\\vec k,\\beta) log\n\\big(\\sin ^2 \\theta _{-}^i(\\vec k,\\beta)\\big)\\nonumber\\\\\n&+&(\\cos ^2 \\theta _{-}^i(\\vec k,\\beta))\nlog  \\big(\\cos ^2 \\theta _{-}^i(\\vec k,\\beta)\\big)\n+\\sin ^2 \\theta _{+}^i(\\vec k,\\beta)\nlog  \\big(\\sin ^2 \\theta _{+}^i(\\vec k,\\beta)\\big)\\nonumber\\\\\n&+&(\\cos ^2 \\theta _{+}^i(\\vec k,\\beta)) log\n\\big(\\cos ^2 \\theta _{+}^i(\\vec k,\\beta)\\big)\\Big )\\big] .\n\\label{sf}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nS_e&=&-2\\times (2\\pi)^{-3} \\int d{\\vec k}\n\\Big (\\sin ^2 \\theta _{-}^e(\\vec k,\\beta) log\n\\big(\\sin ^2 \\theta _{-}^e(\\vec k,\\beta)\\big)\\nonumber\\\\\n&+&(\\cos ^2 \\theta _{-}^e(\\vec k,\\beta))\nlog  \\big(\\cos ^2 \\theta _{-}^e(\\vec k,\\beta)\\big)\n+\\sin ^2 \\theta _{+}^e(\\vec k,\\beta)\nlog  \\big(\\sin ^2 \\theta _{+}^e(\\vec k,\\beta)\\big)\\nonumber\\\\\n&+&(\\cos ^2 \\theta _{+}^e(\\vec k,\\beta)) log\n\\big(\\cos ^2 \\theta _{+}^e(\\vec k,\\beta)\\big)\\Big )\\big] .\n\\label{se}\n\\end{eqnarray}\nClearly, the factor $2\\times 8$ in equation\n(\\ref{sg}) above comes from the transverse\nand colour degrees of freedom for the gluon fields\nand the factor $3\\times 2$ in  equation (\\ref{sf}) comes from the\ncolour and spin degrees of freedom for quarks and the factor 2\nin equation (\\ref{se}) comes from spin degrees of freedom for electrons.\nWe may note that to consider quark stars we need to impose charge neutrality\ncondition for the quark matter. The electric charge $\\rho_E$ is given by\n\\begin{eqnarray}\n\\rho_E & = & \\frac{2}{(2\\pi)^3}\\int\nd\\vec k \\bigg[ 2(\\sin^2\\theta^u_-  - sin^2\\theta^u_+)\n-(\\sin^2\\theta^d_- - sin^2\\theta^d_+)\\nonumber \\\\\n& - & (\\sin^2\\theta^s_- - sin^2\\theta^s_+)-(\\sin^2\\theta^e_- -\nsin^2\\theta^e_+) \\bigg] ,\n\\label{rhoe}\n\\end{eqnarray}\nwhere $\\sin^2\\theta_{\\mp}^e$ is the electron (positron)\ndistribution function parallel to equation (\\ref {distrf}). The chemical\npotentials for the quarks and the electrons can be written\nin terms of baryon chemical potential $\\mu_B$ and the electron chemical\npotential $\\mu_E$ and are given by\n\\begin{mathletters}\n\\begin{equation}\n\\mu_u=\\frac{1}{3}\\mu_B+\\frac{2}{3}\\mu_E\n\\label{mua}\n\\end{equation}\n\\begin{equation}\n\\mu_d=\\frac{1}{3}\\mu_B-\\frac{1}{3}\\mu_E\n\\label{mub}\n\\end{equation}\n\\begin{equation}\n\\mu_s=\\frac{1}{3}\\mu_B-\\frac{1}{3}\\mu_E\n\\label{muc}\n\\end{equation}\n\\begin{equation}\n\\mu_e=-\\mu_E\n\\end{equation}\n\\label{mud}\n\\end{mathletters}\nFor the minimisation of thermodymic potential we may scale out the\ndimensional parameter $1\/{B^2}$ as in Ref. \\cite{zphyc} and have parallel\nexpression for the thermodynamic potential as\n\\begin{eqnarray}\n{\\cal F}(A,\\beta )& \\equiv &  {1\\over {B^2}}{F_1}(A,y)\\nonumber\n\\\\ & = &\n{1\\over {B^2}}\\bigg [F(A,y)-{1\\over y}({\\cal S}_G(A,y)\n+{\\cal S}_Q(A,y)+{\\cal S}_e(A,y))-\\mu_B ' \\rho_B '\\bigg ],\n\\label{52}\n\\end{eqnarray}\n\\noindent where $F(A,y)=B^2 \\epsilon _0(A,\\beta)$, $\\mu_B '=\\sqrt{B}\\mu_B$,\n $\\rho_B '= B^{3\/2}\\rho_B$\nand the entropy densities ${\\cal S}_G(A,y)$, ${\\cal S}_Q(A,y)$\nand ${\\cal S}_e(A,y)$ in dimensionless units are given as\n\\begin{eqnarray}\n{\\cal S}_G(A,y)&=&-{8\\over {\\pi^2}}  \\int  x^{2}dx \\biggl \\{\n\\bigl ({1\\over \\exp{(y\\omega(x,y))}-1}\\bigr )\nlog \\bigl ({1\\over \\exp{(y\\omega(x,y))}-1}\\bigr )\\nonumber\\\\ & - &\n\\bigl (1+{1\\over \\exp{(y\\omega(x,y))}-1}\\bigr )\nlog \\bigl (1+{1\\over \\exp{(y\\omega(x,y))}-1}\\bigr )\\biggr \\},\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\cal S}_Q(A,y)&=&\\sum_{u,d,s}-{6\\over {\\pi^2}} \\int x^{2}dx \\biggl \\{\n\\bigl ({1\\over \\exp{(y(\\epsilon_i(x)-\\mu'_i)}+1}\\bigr )\nlog \\bigl ({1\\over \\exp{(y(\\epsilon_i(x)-\\mu'_i)}+1}\\bigr )\\nonumber\\\\ & + &\n\\bigl (1-{1\\over \\exp{(y(\\epsilon_i(x)-\\mu'_i)}+1}\\bigr )\nlog \\bigl (1-{1\\over \\exp{(y(\\epsilon_i(x)-\\mu'_i)}+1}\\bigr )\n\\nonumber\\\\ & + & \\bigl ({1\\over \\exp{(y(\\epsilon_i(x)+\\mu'_i)}+1}\\bigr )\nlog \\bigl ({1\\over \\exp{(y(\\epsilon_i(x)+\\mu'_i)}+1}\\bigr )\\nonumber\\\\ & + &\n\\bigl (1-{1\\over \\exp{(y(\\epsilon_i(x)+\\mu'_i)}+1}\\bigr )\nlog \\bigl (1-{1\\over \\exp{(y(\\epsilon_i(x)+\\mu'_i)}+1}\\bigr )\n\\biggr \\},\n\\label{mu}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n{\\cal S}_e(A,y)&=&-{2\\over {\\pi^2}} \\int x^{2}dx \\biggl \\{\n\\bigl ({1\\over \\exp{(y(\\epsilon_e(x)-\\mu'_e)}+1}\\bigr )\nlog \\bigl ({1\\over \\exp{(y(\\epsilon_e(x)-\\mu'_e)}+1}\\bigr )\\nonumber\\\\ & + &\n\\bigl (1-{1\\over \\exp{(y(\\epsilon_e(x)-\\mu'_e)}+1}\\bigr )\nlog \\bigl (1-{1\\over \\exp{(y(\\epsilon_e(x)-\\mu'_e)}+1}\\bigr )\n\\nonumber\\\\ & + & \\bigl ({1\\over \\exp{(y(\\epsilon_e(x)+\\mu'_e)}+1}\\bigr )\nlog \\bigl ({1\\over \\exp{(y(\\epsilon_e(x)+\\mu'_e)}+1}\\bigr )\\nonumber\\\\ & + &\n\\bigl (1-{1\\over \\exp{(y(\\epsilon_e(x)+\\mu'_e)}+1}\\bigr )\nlog \\bigl (1-{1\\over \\exp{(y(\\epsilon_e(x)+\\mu'_e)}+1}\\bigr )\n\\biggr \\}.\n\\label{mue}\n\\end{eqnarray}\nWe note that for each $A$, the gluon mass is\ndetermined self consistently through equations (38)\nfor the evaluation of the right hand side of equation (46). As has\nalready been stated, unlike in Ref. \\cite{zphyc} we have now to impose\ncharge neutrality condition i.e., $\\rho_E=0$ to discuss quark star.\nThus, for a given baryon density $\\rho_B$, the chemical potentials\n$\\mu_B$ and $\\mu_E$ are fixed using equations (\\ref{bnb})  and (\\ref{rhoe})\ndemanding charge neutrality. This with equations (44)\nfixes up the chemical potential of each species of fermions.\nWith the chemical potentials known, we now extremise the\nthermodynamic potential\n$F_{1}(A,y)$  of equation (\\ref{52}) with\nrespect to the parameter A and obtain the optimum value of A as $A_{min}$ at\na given temperature T and baryon density $\\rho_B$.\nWe then calculate the pressure $P$ given as \\cite{fetter}\n\\begin{equation}\nP(\\beta )=-{\\cal F}(A,\\beta )\\Big |_{A=A_{min}}.\n\\end{equation}\n\nWe calculate the pressure with respect to the\nnonperturbative vacuum as a function of the condensate\nparameter $A$ for different baryon densities $\\rho_B$ for coupling constant\nequal to $0.8$ and for zero temperature. We have also taken here current quark\nmasses of u, d quarks to be zero and starnge quark mass to be 180 MeV. We\nmay mention here that the dimensional parameter 1\/$\\sqrt B$ for coupling\n$\\alpha_s=0.8$ is 321.4 MeV which is fixed by relating it to the SVZ parameter\n$<{\\alpha_s\\over\\pi}G^{ a}_{\\mu\\nu}^aG^{\\mu\\nu}>=0.012 GeV^4$\nat zero temperature and zero density \\cite{amqcd,zphyc}. This is used to\nexpress different thermodynamic quantities in physical units. At\nzero density the pressure has a unique minimum at $A=A_{min}=0.56$.\nFor density greater than a critical density\n$\\rho_{crit}=0.55\\;$baryons\/fm$^3$, the pressure is maximum for\nA=0 or, equivalently gluon condensate vanishes.\n\nWe plot in Fig. 1 pressure versus baryon density at zero temperature\nafter subtracting the nonperturbative $\"$bag constant\" here\ncalculated as 38.3 MeV\/fm$^3\\equiv$ (131 MeV)$^4$.\nWe may compare the present nonperturbative results\nwith pressure obtained from pertubative calculation \\cite{kapusta}\nwhich is given as\n\\begin{equation}\nP_{pert}=P_e+\\sum_{u,d,s} P_f,\n\\end{equation}\nwith the electron pressure $P_e$ as\n\\begin{equation}\nP_e=\\frac{\\mu_e^4}{12\\pi^2}\n\\end{equation}\nand\n\\begin{eqnarray}\nP_f& = & \\frac{1}{4\\pi^2}\\bigg [ \\mu_f k_f(\\mu_f^2-2.5m_f^2)\n+1.5 m_f^4 ln \\left (\\frac {\\mu_f+k_f}{m_f}\\right ) \\bigg]\\nonumber\\\\\n& - & \\frac{\\alpha_s}{\\pi^3}\\bigg [1.5\\times\\left (\\mu_f k_f -m_f^2\nln \\big (\\frac {\\mu_f+k_f}{m_f} \\big )\\right )^2-k_f^4\\bigg ],\n\\label{pert}\n\\end{eqnarray}\nwhere $k_f=\\sqrt{\\mu_f^2-m_f^2}$ with $\\mu_f$ being the\nchemical potential for quark of flavor $f$.\nAs may be seen, the present equation of state is stiffer than\nthe perturbative equation of state\nwhich is shown as a dashed curve in Fig 1.\nPressure as a function of baryon chemical potential is\nplotted in Fig. 2.\n\\section{quark stars}\nWe shall use the equation of state for the\nstrange quark matter as obtained here\nfor the consideration of the quark stars.\nFor the description of a spherical strange quark star,\nthe space time geometry is described by a metric in\nSchwarzchild coordinates and has the form \\cite{weinberg}\n\\begin{equation}\nds^2=-e^{\\nu(r)}dt^2+[1-2M(r)\/r]^{-1}dr^2+r^2[d\\Theta^2+sin^2 \\Theta\nd\\phi^2]\n\\end{equation}\nThe equations which determine the star structure and the geometry are,\nin dimensionless forms \\cite{weinberg},\n\\begin{mathletters}\n\\begin{equation}\n{d\\hat P(\\hat r r_0)\\over d\\hat r}=-\\hat G\n{[\\hat\\epsilon (\\hat r r_0)+\\hat P (\\hat r r_0)][\\hat M (\\hat r r_0)\n+4\\pi a \\hat r^3 \\hat P(\\hat r r_0)]\\over \\hat r^2[1-2\\hat G\n\\hat M (\\hat r r_0)\/\\hat r]},\\label{tov1}\n\\end{equation}\n\\begin{equation}\n\\hat M (\\hat r r_0)=4\\pi a \\int_0^{\\hat r r_0} d\\hat r^\\prime\n\\hat r^{\\prime^2} \\hat \\epsilon(\\hat r^\\prime r_0),\\label{tov2}\n\\end{equation}\nand the metric function, $\\nu (r)$, relating the element of time at\n$r=\\infty$ is given by\n\\begin{equation}\n{d\\nu(\\hat r r_0)\\over d\\hat r}=2\\hat G {[\\hat M (\\hat r r_0)\n+4\\pi a \\hat r^3 \\hat P(\\hat r r_0)]\\over \\hat r^2[1-2\\hat G\n\\hat M (\\hat r r_0)\/\\hat r]}.\\label{tov3}\n\\end{equation}\n\\end{mathletters}\nIn equations (54) the following substitutions have been made.\n\\begin{mathletters}\n\\begin{equation}\n\\hat \\epsilon\\equiv \\epsilon\/\\epsilon_c,\\quad\n\\hat P\\equiv P\/\\epsilon_c,\\quad\\hat r\\equiv r\/r_0,\\quad\n\\hat M\\equiv M\/M_\\odot,\n\\label{tov4}\n\\end{equation}\nwhere, with\n$ f_1=197.3 \\; MeV^{-1}\/fm^3$ and $r_0=3\\times 10^{19}$ fm,\nwe have\n\\begin{equation}\na\\equiv\\epsilon_c r_0^3\/M_\\odot, \\quad\n\\hat G\\equiv (G\/f_1)\/(r_0\/M_\\odot)\n\\label{tov5}\n\\end{equation}\n\\end{mathletters}\n\\noindent In the above, quantities with hats are dimentionless.\nG in equation (\\ref{tov5}) denotes the gravitational constant\n$(G=6.707934\\times 10^{-45}\\;\\mbox{MeV}^{-2})$.\n\nIn order to construct a stellar model, one has to integrate equations\n(\\ref{tov1}) to  (\\ref{tov3}) from the star's center at r=0 with\na given central energy density $\\epsilon_c$ as input until the\npressure $P(r)$ at the surface $\"$vanishes\", i.e., becomes the\nsame as the pressure for the nonperturbative vacuum, or bag pressure.\nThus we shall here use equation of state of quark matter through\nequation (49) after subtracting from it the bag pressure,\nwhich is the difference in energy density of the nonperturbative\nvacuum and the perturbative vacuum.\nWe then integrate the TOV equations\nwith $\\hat P (0)= P (\\epsilon_c)$.\nAs mentioned in the last section, for $\\rho > \\rho_{crit}$, the\ncondensate structure vanishes. For TOV equations, we use the\ncorresponding equation of state in the range $\\rho_{crit}<\\rho\n<\\rho_C$, i.e.,\nuntil the density decreases to the critical value $\\rho_{crit}$.\nWe then use the equation of state when\nthe gluon condensates exist. TOV equations continue\ntill the pressure vanishes, which defines the surface of the star.\nThis completes the calculations for stellar model for quark stars\nwhose mass and radius could be calculated for different central densities.\n\nIn Fig. 3 we plot the mass of a star as a function of central energy density to\nexamine the stability of such stars. As may be seen from the figure\n$dM\/d\\epsilon$ becomes negative around central densities 950 MeV\/fm$^3$\nafter which they may collapse to black holes \\cite{weinberg,shapiro} with the\nmaximum mass as 3.2$M_\\odot$, which is rather large.\nWe have also made a calculation, not shown in the present\nfigures, for the critical mass with the perturbative equation\nof state taking the same coupling constant. We then obtain that\n$M_{max}\\simeq$ 2.25$M_\\odot$. Since the equation of state\nfor strange quark matter as obtained here is stiffer than the\nperturbative equation of state as in Fig. 1, a larger value\nof $M_{max}$ is expected \\cite{shapiro,prakash}.\nA similar large mass for hybrid stars was obtained in an\neffective condensate model for QCD \\cite{ellis}.\n\nFig. 4 shows the mass as a function of radius  for such stars obtained\nfor different central densities varying in the range\n500 to 1200 MeV\/fm$^3$. This yields stable quark stars of masses $M\\simeq$\n2.2 to 3.2 $M_\\odot$ with radii $R\\simeq$ 15.3 to 17 kms respectively.\n\nThe energy density profile obtained from (\\ref{tov1}) to (\\ref{tov3}) is\nplotted in Fig. 5 for central density $\\epsilon_c=850$ MeV\/fm$^3$.\n\nWe may also calculate the surface gravitational red shift $Z_s$ of\nphotons which is given by \\cite{weinberg,brecher}\n\\begin{equation}\nZ_s={1 \\over\\sqrt{[1-2GM\/R]}} -1.\\label{zs}\n\\end{equation}\nIn Fig 6 we plot $Z_s$ as a function of $M\/M_{\\odot}$.\n\\section{conclusions}\nLet us summarise the findings of the present results.\nThe nontrivial ground state structure of QCD with gluon condensates\nat zero temperature \\cite{amqcd} has been generalised here to\nfinite baryon densities with three quark flavour.\nWe have here included the charge neutrality condition\nto obtain the equation of state for strange quark matter.\nThe equation of state so obtained is seen\nto be stiffer than the perurbative equation of state.\nThe gluon condensate appears to vanish\nfor baryon density greater than a critical value.\n\nTo derive the equation of state we have however not included the effects\nof vacuum structure with quark condensates associated with\nchiral symmetry breaking, the reason for this being that\nthe contribution of quark condensates to the energy density\nis expected to be small compared to that of the gluons. However\ninclusion of the same might throw a light on relationship\nbetween confinement and chiral symmetry breaking.\n\nWe have applied the equation of state so obtained to study\nstructural properties of quark stars. They seem to be larger and\nin general more massive than neutron stars or hybrid stars with masses\nbeing around $2.2M_\\odot$ to $3.2M_\\odot$\nand radii around 15 to 17 kilometers.\nWe may note that this is a reflection of a stiffer equation\nof state of strange quark matter as obtained here.\n\n\\acknowledgements\n\nThe authors are thankful to J.C. Parikh, S.B. Khadkikar, M. Prakash,\nN. Barik and Snigdha Mishra for many useful discussions.\nSPM would like to thank Department of\nScience and Technology, Government of India for\nresearch grant no SP\/S2\/K-45\/89 for financial assistance.\n\n\\def \\qcd {G. K. Savvidy, Phys. Lett. 71B, 133 (1977);\nS. G. Matinyan and G. K. Savvidy, Nucl. Phys. B134, 539 (1978); N. K. Nielsen\nand P. Olesen, Nucl.  Phys. B144, 376 (1978); T. H. Hansson, K. Johnson,\nC. Peterson Phys. Rev. D26, 2069 (1982).}\n\n\\def \\svz {M.A. Shifman, A.I. Vainshtein and V.I. Zakharov,\nNucl. Phys. B147, 385, 448 and 519 (1979);\nR.A. Bertlmann, Acta Physica Austriaca 53, 305 (1981).}\n\n\\def \\nambu{ Y. Nambu, Phys. Rev. Lett. 4, 380 (1960);\nY. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); ibid,\n124, 246 (1961);\nJ.R. Finger and J.E. Mandula, Nucl.Phys.B199, 168 (1982);\nA. Amer, A. Le Yaouanc, L. Oliver, O. Pene and\nJ.C. Raynal, Phys. Rev. Lett. 50, 87 (1983);\nibid, Phys. Rev. D28, 1530 (1983);\n S.L. Adler and A.C. Davis,\nNucl. Phys. B244, 469 (1984); R. Alkofer and P. A. Amundsen,\nNucl. Phys.B306, 305 (1988); A.C. Davis and A.M. Matheson DAMTP 91-34 (1991).}\n\n\\def \\pot{A. Mishra, H. Mishra and S. P. Misra, Z. Phys. C57, 241 (1993);\nA. Mishra and S. P. Misra, Z. Phys. C58, 325 (1993).}\n\n\\def \\lat{K.G. Wilson, Phys. Rev. D10, 2445 (1974); J.B. Kogut, Rev. 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+{"text":"\\section{Introduction}\\label{sec:1} \nRelativistic hydrodynamical models have been applied to studies of matter that is produced\nin high-energy hadron or nuclear collisions. \nFluid dynamical descriptions, in particular, provide \na simple picture of the space-time evolution of the hot\/dense matter \nproduced by ultra-relativistic heavy-ion collisions at \nRHIC and LHC \\cite{Heinz2013_1, Heinz2013_2}. \nIt is expected that this simple picture makes it possible to investigate \nthe strongly interacting quark and gluon matter present at the initial stage of \nthe collisions. \nThe fluid model assumes that there exist local \nquantities of the matter such as energy density, pressure and so on. \nIn particular, it is considered that the pressure gradients of the matter \ncauses collective phenomena \\cite{Gustafsson1984, Hirano:2008hy}. \nThese expected phenomena have been successfully observed as an elliptic flow \ncoefficient $v_2$ in the CERN SPS experiment NA49 \\cite{NA491998}, \nat RHIC experiment \\cite{PhysRevC.77.054901,PhysRevLett.98.162301,\nPhysRevLett.98.242302} and \nrecent ALICE experiments in LHC \\cite{Snellings2014} including the\nhigher order flow harmonic $v_n$ for ($n=3,4$). These have been observed \nas a function of various characteristics, including the \ntransverse momentum $p_{\\rm T}$ or rapidity $y$ and so on. \nHence, the hydrodynamical model \nhas been widely accepted \nbecause of such experimental evidences.\nTo investigate the properties of the quark and gluon matter \ncreated during such ultra-relativistic heavy-ion collisions more precisely, \nit is necessary to consider the effects of the viscosities and corresponding dissipation\n\\cite{MurongaPRC69}. \nThese effects are introduced into the hydrodynamic \nsimulations and a detailed comparison between simulation and \nexperimental data is made \n(see, for example, Ref.\\cite{SongPRC89}).   \nHowever, as several authors have noted, dissipative hydrodynamics is not \nyet completely understood and there are issues associated to \nthe determination of the hydrodynamical flow \n\\cite{TsumuraPLB646,TsumuraPLB690, Van2013} (see also, Ref. \\cite{Arnold2014}). \nIn this article, the Landau matching (fitting) condition that is necessary to specify the\ninitial conditions for \ndissipative fluid dynamics is discussed. This may be related to the issue \nof defining the local rest frame \\cite{TsumuraPLB690}. \n\nThe fundamental equations of relativistic \nfluid dynamics are defined by the conservation laws of \n energy-momentum and the charge (in this paper, \nwe assume the net baryon density as the conserved charge), \n\\begin{subequations}\n\\begin{eqnarray}\n   \\partial_{\\mu} T^{\\mu\\nu}(x) =0, \\\\\n   \\partial_{\\mu} N^{\\nu}(x)=0. \n\\end{eqnarray}\n\\end{subequations}\nHere, $T^{\\mu\\nu}$ and $N^{\\mu}$ are respectively the \nenergy-momentum tensor and the\nconserved charge current at a given point in space-time $x$, \nwhich can be obtained by a \ncoarse-graining procedure \\cite{KodamaEPJA48} with some finite size \n(fluid cell size), $l_{\\rm fluid}$. \nHence, the fluid dynamical model expressed\nas a coarse-graining theory \ndescribing macroscopic phenomena \ncan be \nderived from the underlying kinetic theory.\nIn the case of a perfect fluid limit, \nthe microscopic collision time scale $\\tau_{\\rm micro}$ is \n much shorter than the macroscopic \nevolution time scale $\\tau_{\\rm macro}$\\cite{PhysRevC.73.034904}, \nthus \n\\begin{eqnarray}\n  \\tau_{\\rm macro} \\gg \\tau_{\\rm micro}. \n  \\label{eq:perfect_condition}\n\\end{eqnarray}\nIf the condition in eq.(\\ref{eq:perfect_condition}) is satisfied, \nthe distribution function \ninstantaneously relaxes to its local equilibrium form. \nIn the local rest frame of the fluid, i.e., the frame in which \nthe fluid velocity is given by $u^{\\mu}(x)=(1,0,0,0)$, \nthe local equilibrium distribution functions \nfor particles and for anti-particles are respectively \ngiven (within the Boltzmann approximation) as\n\\begin{subequations}\\label{eq:equilibrium_distribution}\n\\begin{eqnarray}\n  f_{{\\bf k}0}(x)= \\exp[\\frac{-k_{\\mu}u^{\\mu}(x)+\\mu_0(x)}{T_0(x)}], \\\\\n  \\bar{f}_{{\\bf k}0}(x)= \\exp[\\frac{-k_{\\mu}u^{\\mu}(x)-\\mu_0(x)}{T_0(x)}],\n\\end{eqnarray}\n\\end{subequations}\nwhere $k^{\\mu}$ is a four momentum vector of a particle or an \nanti-particle within \na cube \nthe coarse-graining scale $l_{\\rm fluid}$ on a side (fluid cell).  \nWe decompose $k^{\\mu}$ by using the local flow vector $u^{\\mu}$, i.e., \n$k^{\\mu}=(k^{\\lambda}u_{\\lambda})u^{\\mu}+k^{\\langle \\mu\\rangle}$ \nwhere $k^{\\mu}u_{\\mu}\\equiv E_{\\bf k}$ is a scalar and \n$k^{\\langle \\mu\\rangle}\\equiv \\Delta_{\\nu}^{\\mu} k^{\\nu}$ is a \nvector, which  is orthogonal to $u^{\\mu}$. \nIn the local rest frame, the scalar $E_{\\bf k}$ coincides with the\nzero-th component of the four vector, energy of the classical particle, $k^0$. \nThe projection tensor onto the 3-space that is orthogonal to the flow velocity is defined\nby $\\Delta^{\\mu\\nu} \\equiv g^{\\mu\\nu}-u^{\\mu}u^{\\nu}$ with \n$g^{\\mu\\nu}$ being the metric tensor, $g^{\\mu\\nu}={\\rm diag}(+1,-1,-1,-1)$. \n$T_0(x)$ and $\\mu_0(x)$ are the equilibrium temperature and chemical potential \nrespectively. \nWhen the distribution function is given by eq.(\\ref{eq:equilibrium_distribution}), \nthe energy-momentum tensor and the net baryon number current vector are defined as \n\\begin{subequations}\n\\begin{eqnarray}\n    T^{\\mu\\nu}_{\\rm eq} &=&\n    \\int\\!\\!\\frac{d^3{\\bf k}k^{\\mu}k^{\\nu}}{(2\\pi)^3k^0} \n    \\Big[ f_{{\\bf k}0}(x) + \\bar{f}_{{\\bf k}0}(x) \\Big] \\nonumber \\\\\n&=& \\varepsilon_{\\rm eq} u^{\\mu}u^{\\nu} \n   -P_{\\rm eq} (\\varepsilon_{\\rm eq},n_{\\rm eq}) \\Delta^{\\mu\\nu}, \n    \\\\ \n    N^{\\mu}_{\\rm eq} &=&  \\int\\!\\!\\frac{d^3{\\bf k}~k^{\\mu}}{(2\\pi)^3k^0} \n    \\Big[ f_{{\\bf k}0}(x) - \\bar{f}_{{\\bf k}0}(x) \\Big] \\nonumber \\\\\n&=& n_{\\rm eq} u^{\\mu}, \n\\end{eqnarray}\n\\end{subequations}\nrespectively, where $\\varepsilon_{\\rm eq}$ is the energy density in the local rest frame, \n$n_{\\rm eq}$ is the net baryon density in the local rest frame, \nand $P_{\\rm eq}$ is the pressure in the equilibrium state.   \n\nIf the fluid expands very rapidly (i.e., the macroscopic evolution time \nscale $\\tau_{\\rm macro}$ becomes shorter), especially for\n fluids produced by ultra-relativistic heavy-ion collisions, \n equation (\\ref{eq:perfect_condition}) \nmay not be satisfied everywhere in the fluid during the early stages. \nIn such cases, microscopic processes cannot keep pace with \nthe changes in local energy and baryon density; therefore, \nwe can write the following equation: \n\\begin{eqnarray}\n    \\tau_{\\rm micro} \\gtrsim \\tau_{\\rm macro} . \n    \\label{eq:inparfect_condtion}\n\\end{eqnarray}\nUnder the condition eq.(\\ref{eq:inparfect_condtion}), \nthe distribution function in the local rest frame does not obey the local equilibrium \nform eq.(\\ref{eq:equilibrium_distribution}) and hence \nwe have\n\\begin{subequations} \n\\label{eq:modification_in_distribution}\n\\begin{eqnarray}\n   f_{\\bf k}(x)= f_{{\\bf k}0}[T(x),\\mu(x)] + \\delta f_{\\bf k}(x), \\\\ \n   \\bar{f}_{\\bf k}(x)= \\bar{f}_{{\\bf k}0}[T(x),\\mu(x)] + \\delta \\bar{f}_{\\bf k}(x), \n\\end{eqnarray}\n\\end{subequations}\nwhere $\\delta f_{\\bf k}$ and $\\delta \\bar{f}_{\\bf k}$ are the deviations from \nthe corresponding equilibrium distribution functions.   \nWe define a temperature $T(x)$ and  \n chemical potential $\\mu(x)$ (hereafter the  \n{\\it separation temperature} and \n{\\it separation chemical potential}, respectively) \nthat is distinguished from the equilibrium temperature $T_0(x)$ and \nchemical potential $\\mu_0(x)$ \nbecause we assume that \n$\\delta f_{\\bf k}$ and $\\delta \\bar{f}_{\\bf k}$ \nalso contribute to the energy and \nbaryon number density of the system.  \nIn the inside of the fluid cell, within scales less than the coarse-graining scale \n$l_{\\rm fluid}$, the extremely rapid expansion of matter causes \nadditional tiny disturbances to the local flow \nvelocity field $u^{\\mu}(x)$. \nAlthough such tiny disturbance in the flow vectors may be cancelled out by defining a new\nlocal rest frame (because of the randomness of such tiny disturbances), \nit also causes a disturbance in the distribution function \n$\\delta f_{\\bf k}(x)$ and $\\delta\\bar{f}_{\\bf k}(x)$. \nFor simplicity, we assume that the fluid considered is stable \nagainst such disturbances; otherwise, \nthe disturbance would escalate and the flow would \neventually become turbulent. \nThe space-time evolution of the disturbance flow is \nindependent from that of main background flow \neven if heat is supplied by the main background flow.  \nSince the disturbance flow has no mechanism for obtaining \nenergy other than heat originated \nfrom the shear viscosity of the main background flow, \nit is finally dissipated as heat. \nThus, the disturbances, $\\delta f_{\\bf k}$ and \n$\\delta \\bar{f}_{\\bf k}$, do not belong to any equilibrium state.  \nAs the macroscopic evolution time scale \n$\\tau_{\\rm macro}$ grows longer due to the expansion of matter \nand the pressure gradients of the fluid decrease,  \n$\\delta f_{\\bf k}$ and $\\delta \\bar{f}_{\\bf k}$ approaches to zero \n(an assumption of hydrodynamic stability), \nand the condition eq.(\\ref{eq:perfect_condition}) \nis restored and local equilibrium is achieved. \n\nThe question we now consider is \n{\\it how to find local separation temperature $T(x)$ and \nchemical potential $\\mu(x)$} \nin eq.(\\ref{eq:modification_in_distribution}) \nfrom a given off-equilibrium state characterized by\n\\begin{subequations}\n\\begin{eqnarray}\n    T^{\\mu\\nu}(x) &\\equiv&\n    \\int\\!\\!\\frac{d^3{\\bf k}k^{\\mu}k^{\\nu}}{(2\\pi)^3k^0} \n    \\Big[ f_{\\bf k}(x) + \\bar{f}_{\\bf k}(x) \\Big], \n    \\label{eq:energy-momentum_tensor}\\\\ \n    N^{\\mu} (x) &\\equiv&  \\int\\!\\!\\frac{d^3{\\bf k}~k^{\\mu}}{(2\\pi)^3k^0} \n    \\Big[ f_{\\bf k}(x) - \\bar{f}_{\\bf k}(x) \\Big]. \n    \\label{eq:particle_current} \n\\end{eqnarray}\n\\end{subequations} \nUsually the local separation temperature $T(x)$ and \nchemical potential $\\mu(x)$ in eq.(\\ref{eq:modification_in_distribution}) \nare determined by imposing the so-called Landau matching \nconditions \\cite{Landau1959,Israel118,PhysRevC.73.034904,Heinz:2009xj} \n\\begin{subequations}\\label{eq:normal-matching1}\n\\begin{eqnarray}\n&& \\delta T^{\\mu\\nu} u_{\\mu} u_{\\nu} =0, \\label{eq:normal-matching11}\\\\ \n&& \\delta N^{\\mu} u_{\\mu} =0,\\label{eq:normal-matching12}\n\\end{eqnarray}\n\\end{subequations} \nwhere $\\delta T^{\\mu\\nu}\\equiv T^{\\mu\\nu}-T_{\\rm eq}^{\\mu\\nu}$ and \n$\\delta N^{\\mu}\\equiv N^{\\nu}-N_{\\rm eq}^{\\mu}$ are \n the deviation of the energy-momentum tensor and the baryon charge \ncurrent from the {\\it matching state}, respectively. \nIn this procedure, it is necessary to select a Lorentz frame. \nThere are usually\ntwo natural choices for the Lorentz frame, \nnamely the Landau frame \\cite{Landau1959} \nand the Eckart frame \\cite{Eckart1940}. \nIf the Landau frame is employed, for example,  \nthe local flow velocity $u^{\\mu}$ is determined by \nthe eigenvector of the following eigenvalue equation,\n\\begin{subequations}\n\\begin{eqnarray}\n  u_{\\mu} T^{\\mu\\nu} = \\varepsilon u^{\\nu}, \n  \\label{eq:Landau_frame}\n\\end{eqnarray} \nwhere the eigenvalue $\\varepsilon$ is the energy density \nmeasured in the rest frame. \nIn the Landau matching condition, shown in eq.(\\ref{eq:normal-matching11}), \nthe energy density should be matched to an energy density in an equilibrium state \nparameterized by a temperature $T$ and a chemical potential $\\mu$, i.e., \n$\\varepsilon=\\varepsilon_{\\rm eq}(T,\\mu)$.  \nUsing the flow velocity obtained in eq.(\\ref{eq:Landau_frame}), we obtain the following\nequation: \n\\begin{eqnarray}\n  N^{\\mu} = n u^{\\mu} +V^{\\lambda}\\Delta_{\\lambda}^{\\mu}  \n  \\label{eq:current_decomposition}\n\\end{eqnarray}\n\\end{subequations}\nfor the net baryon number current. \nThen eq.(\\ref{eq:normal-matching12}) yields \n$n=n_{\\rm eq}(T,\\mu)$. In this way it is possible to \n determine $T$ and $\\mu$ from\n$\\varepsilon=\\varepsilon_{\\rm eq}(T,\\mu)$ and $n=n_{\\rm eq}(T,\\mu)$.  \nHowever, if the Eckart frame is employed, \na different temperature $T'$ and chemical potential $\\mu'$ \nmay be obtained as compared to those derived \nin the Landau frame. For a paper which deals with the issue of the frame \nin the relativistic dissipative hydrodynamical model, see, \nfor example, Ref.\\cite{Van2013}. \nTherefore, for an off-equilibrium states, \nthe temperature and chemical potential which characterize \nthe distribution function for the equilibrium part may be dependent on the frame employed. \n\nNote that, the temperature $T$ and \nchemical potential $\\mu$ are not only dependent on \nthe frame employed but also on the matching condition eq.(\\ref{eq:normal-matching1}). \nHowever, it is not always appropriate, as discussed below. \nIn order to investigate how the matching condition affects the \nseparation to an equilibrium part from an off-equilibrium, one must fix the \nLorentz frame to define the fluid velocity.\nWe hereafter use the Landau frame and define a flow vector \n$u^{\\mu}$ as an eigenvector of eq.(\\ref{eq:Landau_frame}), \nbut we do not use the Landau-matching condition eq.(\\ref{eq:normal-matching1}); \ni.e., the obtained eigenvalue $\\varepsilon$ of eq.(\\ref{eq:Landau_frame}) is not \nassumed to be same as an equilibrium energy density $\\varepsilon_{\\rm eq}$. \n\nIn the kinetic approach, \nsince the energy-momentum tensor and \nthe conserved charge current are defined \nby the second and first moment of the single particle distribution function, \nthe Landau matching conditions \neq.(\\ref{eq:normal-matching1}) are equivalent to the following expressions\n\\cite{BiroMolnarEPJA48}\n\\begin{subequations} \n\\begin{eqnarray}\n&& \\int\\!\\!\\frac{d^3{\\bf k} ~E_k^2 }{(2\\pi)^3k^0} \n    \\Big[ \\delta f_{\\bf k}(x) + \\delta \\bar{f}_{\\bf k}(x) \\Big]=0, \\\\ \n&& \\int\\!\\!\\frac{d^3{\\bf k}~E_k}{(2\\pi)^3k^0} \n    \\Big[ \\delta f_{\\bf k}(x) - \\delta \\bar{f}_{\\bf k}(x) \\Big]=0.  \n\\end{eqnarray}\n\\end{subequations}\nClearly, these conditions strongly constrain the disturbances \n$\\delta f_{\\bf k}$ and $\\delta \\bar f_{\\bf k}$ and they may distort the distribution function unnaturally. \nTo restore the physical meaning of $\\delta f_{\\bf k}$ and $\\delta \\bar{f}_{\\bf k}$, \nit is necessary to exclude the restrictions imposed by \neq.(\\ref{eq:normal-matching1}) and \nto generalize the Landau matching condition the\nfollowing form. \n\\begin{eqnarray}\n     \\delta T^{\\mu\\nu} u_{\\mu} u_{\\nu} \n     =\\Lambda, \\quad \\delta n^{\\mu} u_{\\mu} =\\delta n. \n     \\label{eq:extended-matching}\n\\end{eqnarray}\nHere, $\\Lambda$ and $\\delta n$ can be considered as the \nenergy density and net baryon number density \nof the disturbance (tiny turbulent) flow caused \nby rapid expansion. \nExcept for particles constituting the disturbance flow, \nit is assumed that the remaining particles in the fluid cell \napproximately obey the local thermal distribution form \n$f_{{\\bf k}0}$ or $\\bar{f}_{{\\bf k}0}$. \nSince the disturbance flow carry finite energy and \nbaryon number, the corresponding separation \ntemperature $T$ and separation chemical potential $\\mu$ \nof the equilibrium part should be different from those \nobtained at the equilibrium limit, i.e., $T(x) \\ne T_0(x)$ \nand $\\mu(x) \\ne \\mu_0(x)$. \nTherefore, to find the separate \ntemperature $T$ and chemical potential $\\mu$ \nin eq.(\\ref{eq:modification_in_distribution}), it is required to find\n {\\it find $T$ and $\\mu$ as functions of \n$\\delta f_{\\bf k}$ and $\\delta \\bar{f}_{\\bf k}$,} \n\\begin{subequations}\n\\label{eq:setting_problem}\n\\begin{eqnarray}\n T&=& T(\\delta f_{\\bf k}, \\delta \\bar{f}_{\\bf k};T_0,\\mu_0),\\\\\n \\mu&=&\\mu(\\delta f_{\\bf k}, \\delta \\bar{f}_{\\bf k};T_0,\\mu_0), \n\\end{eqnarray} \n\\end{subequations}\nwith boundary conditions $T\\to T_0$ and $\\mu \\to \\mu_0$ \nwhen the disturbance flow disappears,  \n$\\delta f_{\\bf k}, \\delta \\bar{f}_{\\bf k} \\to 0$. \nThe boundary condition corresponds to the so-called \nLandau matching condition. \n\nIn any off-equilibrium state, most literature using the Landau matching \ncondition published to date assumes that $T\\equiv T_0$ and $\\mu\\equiv \\mu_0$. \nHowever, in this paper, we consider the problem where $T$ and $\\mu$ are dependent on the \n{\\it strength of the off-equilibrium state}, i.e., $T$ and $\\mu$ are functions of \n$\\delta f_{\\bf k}$ and $\\delta \\bar{f}_{\\bf k}$, \nas seen in eq.(\\ref{eq:setting_problem}). \n\nThis article is organized as follows. \nIn Section \\ref{sec:2}, we obtain an expression for\neq.(\\ref{eq:modification_in_distribution}) \nusing the irreducible tensor expansion technique for the \noff-equilibrium distribution function, as recently employed \nby G.S. Denicol and his collaborators\\cite{PhysRevD.85.114047}. \nWe extend the formulation by introducing the general matching \ncondition of eq.(\\ref{eq:extended-matching}) and \nalso apply the irreducible tensor expansion technique to an off-equilibrium \nentropy current and find a condition of thermodynamical stability. \nThen, we obtain a relation between \nthe quantities introduced by eq.(\\ref{eq:extended-matching}) and \n determine them as functions of $\\alpha, \\beta$ and $\\gamma\\equiv \\Pi\/P_{\\rm eq}$.  \nIn Section \\ref{sec:3}, we demonstrate the separation of the corresponding equilibrium\nstate \nfrom the given off-equilibrium state, i.e., we determine $T$ and $\\mu$ for a given \noff-equilibrium state characterized by $T_0$, $\\mu_0$,  and $\\gamma$.  \nWe also show an off-equilibrium distribution function and discuss the effects of \nthe strength of the off-equilibrium state $\\gamma$. \nFinally, Section \\ref{sec:4} contains conclusions and summary. \nThe derivations of key equations in the relativistic kinetic theory and \n equations related to irreducible expansion theory are presented \nin Appendixes \\ref{app:A}-\\ref{app:G}. \n\n\\section{Separation equilibrium part from the off-equilibrium distribution\nfunction}\\label{sec:2}\n\\subsection{Expansion of the single particle distribution function by irreducible\ntensors}\nConsider a relativistically expanding fluid in which microscopic processes \ncannot keep pace with the quick macroscopic changes. \nLet us rewrite eq.(\\ref{eq:modification_in_distribution}), \nwhich is expected the distribution function \nunder the condition eq.(\\ref{eq:inparfect_condtion}), \nas the following form: \n\\begin{subequations}\\label{eq:phi_functional}\n\\begin{eqnarray} \n   f_{\\bf k}(x)= f_{{\\bf k}0}(T[\\phi_{\\bf k},\\bar{\\phi}_{\\bf k}],\n   \\mu[\\phi_{\\bf k},\\bar{\\phi}_{\\bf k}])\n   \\left( 1+\\phi_{\\bf k} \\right) , \\\\\n  \\bar f_{\\bf k}(x)= \\bar f_{{\\bf k}0}(T[\\phi_{\\bf k},\\bar{\\phi}_{\\bf k}],\n  \\mu[\\phi_{\\bf k},\\bar{\\phi}_{\\bf k}])\n  \\left( 1+\\bar\\phi_{\\bf k} \\right),\n\\end{eqnarray}\n\\end{subequations}\nwhere $\\phi_{\\bf k}(x)\\equiv \\delta f_{\\bf k}(x)\/f_{{\\bf k}0}(x)$ \nand  $\\bar{\\phi}_{\\bf k}(x) \\equiv \\delta \\bar{f}_{\\bf k}(x)\/\\bar{f}_{{\\bf k}0}(x)$\n are deviations from the local thermal equilibrium function \neq.(\\ref{eq:equilibrium_distribution}). \nThe deviations $\\phi_{\\bf k}(x)$ and $\\bar{\\phi}_{\\bf k}(x)$ involve information about a\ngiven \noff-equilibrium state characterized not only by scalars \nsuch as  $\\Lambda$, $\\delta n$, and $\\gamma\\equiv \\Pi\/P_{\\rm eq}$ \nbut also by vectors and tensors, for example, the heat flow \nvector $W^{\\mu}$, shear tensor $\\pi^{\\mu\\nu}$ of the dissipative fluid and so on. \nIn order to expand $\\phi_{\\bf k}$ and $\\bar{\\phi}_{\\bf k}$ by those fluid dynamical quantities,  \nit is necessary to use the orthogonal base of irreducible tensors \n\\cite{DeGroot:1980dk, PhysRevD.85.114047},  \n$\\{1, k_{\\langle \\mu \\rangle}, \\cdots, k_{\\langle \\mu_1} \n\\cdots k_{\\mu_l\\rangle} \\}$, \nwhere the irreducible tensor of the $l$-rank is defined by \n\\begin{eqnarray}\n    \\label{eq:projection} \n    k_{\\langle \\mu_1} \\cdots k_{\\mu_l\\rangle} \\equiv \n    \\Delta_{\\mu_1 \\cdots \\mu_n}^{\\nu_1\\cdots\\nu_n} k_{\\nu_1}\\cdots k_{\\nu_n}, \n\\end{eqnarray} \nand the projection tensor $\\Delta_{\\mu_1 \\cdots \\mu_n}^{\\nu_1\\cdots\\nu_n}$ used \nin eq.(\\ref{eq:projection}) \nare defined in Appendix \\ref{app:A} \n(see also Appendix \\ref{app:B}, \\ref{app:C} and references \n\\cite{DeGroot:1980dk, PhysRevD.85.114047}). These irreducible tensors \n$k_{\\langle \\mu_1} \\cdots k_{\\mu_l\\rangle}$ satisfy the \nfollowing orthogonal condition (for derivation, see Appendix \\ref{app:B}, \\ref{app:C} and \\ref{app:D}), \n\\begin{subequations}\\label{eq:orthogonality_condition_final}\n\\begin{eqnarray} \n&&\\int\\!\\!\\frac{d^3{\\bf k} ~F(E_{\\bf k})}{(2\\pi)^3k^0}~ \n  k^{\\langle \\mu_1} \n  \\cdots k^{\\mu_m\\rangle} k_{\\langle \\nu_1}\n  \\cdots k_{\\nu_n\\rangle} \n  =\\Delta^{\\mu1\\cdots\\mu_m}_{\\nu_1\\cdots\\nu_n} \n  \\delta_{mn} ~{\\cal J}_m,  \n\\end{eqnarray}\nwhere\n\\begin{eqnarray} \n&& {\\cal J}_m=\\frac{m!}{(2m+1)!!}\n         \\int\\!\\! \\frac{d^3{\\bf k} ~F(E_{\\bf k})}{(2\\pi)^3k^0}\n         [\\Delta_{\\alpha\\beta}k^{\\alpha}k^{\\beta}]^m,\n     \\label{eq:orthogonality_condition_final_b}\n\\end{eqnarray}\n\\end{subequations}  \nand $F(E_{\\bf k})$ is an arbitrary function of the energy $E_{\\bf k}$. \nSince the tensors defined in eq.(\\ref{eq:projection}) are orthogonal, \nwe may expand the deviation $\\phi_{\\bf k}$ and $\\bar{\\phi}_{\\bf k}$ as follows:  \n\\begin{subequations}\n\\label{eq:expansion-lambda}\n\\begin{eqnarray}\n    \\phi_{\\bf k}(x) \\equiv \\sum_{l=0}^{\\infty} \n    \\lambda_{\\bf k}^{\\langle \\mu_1\\cdots \\mu_l \\rangle} \n    k_{\\langle \\mu_1}\\cdots k_{\\mu_l \\rangle}, \n    \\label{eq:expansion-lambda1}\\\\\n   \\bar\\phi_{\\bf k}(x) \\equiv \\sum_{l=0}^{\\infty} \n    \\bar\\lambda_{\\bf k}^{\\langle \\mu_1\\cdots \\mu_l \\rangle} \n    k_{\\langle \\mu_1}\\cdots k_{\\mu_l \\rangle}, \n     \\label{eq:expansion-lambda2}\n\\end{eqnarray}\n\\end{subequations}\nrespectively, where $\\lambda_{\\bf k}^{\\langle \\mu_1\\cdots \\mu_l \\rangle} $ and \n$\\bar\\lambda_{\\bf k}^{\\langle \\mu_1\\cdots \\mu_l \\rangle}$ are \ncoefficient tensors of $l$-rank in the above expansions.  \nNote that the coefficient tensors in the above expansion have momentum dependence \n(denoted by the subscript ${\\bf k}$). Therefore, \nwe further expand these coefficients tensors \naccording to a set of polynomial functions of the energy $E_{\\bf k}$ \nhaving a maximum order of $n$ as used in Ref. \\cite{DeGroot:1980dk},\n\\begin{eqnarray}\n  P^{(l)}_{{\\bf k}n} =\\sum_{r=0}^{n} a_{nr}^{(l)} (E_{\\bf k})^r~. \n  \\label{eq:Pn} \n\\end{eqnarray}  \nHere, the coefficients $a^{(l)}_{nr}$ $(r=0,1,\\cdots,n)$ satisfy the following\northogonality relation \n\\begin{eqnarray}\\label{eq:orthogonality_conditionPn}\n  \\int\\!\\frac{d^3{\\bf k} ~\\omega^{(l)}_{\\bf k}} {(2\\pi)^3k_0} \n    P_{{\\bf k} m}^{(l)} P_{{\\bf k} n}^{(l)}=\\delta_{mn}, \n  \\label{eq:orthogonality_conditionPn_(a)}\n\\end{eqnarray} \nwhere $\\omega^{(l)}_{\\bf k}$ is an $l$-dependent weight factor defined by\n\\begin{eqnarray}\n  \\omega^{(l)}_{\\bf k} &\\equiv&\n  \\frac{{\\cal W}^{(l)}}{(2l+1)!!} [\\Delta^{\\alpha\\beta}k_{\\alpha}k_{\\beta}]^{l} f_{0{\\bf k}},  \n  \\label{eq:definition_of_omega} \n\\end{eqnarray}\nand ${\\cal W}^{(l)}$ in eq.(\\ref{eq:definition_of_omega}) \nis a normalization factor (See Appendix \\ref{app:E}).  \nThe orthogonal condition of eq. (\\ref{eq:orthogonality_conditionPn_(a)}) gives a \nrelation between the coefficients $a_{nr}^{(l)} $ and the integral as follows:\n\\begin{eqnarray}\n   I^{(l)}_r \\equiv \\int \\!\\frac{d{\\bf k}^3 \\omega^{(l)}_{\\bf k}}{(2\\pi)^3k^0} (E_{\\bf k})^r, \n\\end{eqnarray}  \nthat is, \n\\begin{eqnarray}\n  \\left( \\begin{array}{llllll}\n  1   & I^{(l)}_1 && \\ldots &&  I^{(l)}_{n}    \\\\\n  I^{(l)}_1 & I^{(l)}_2 && \\ldots &&  I^{(l)}_{n+1} \\\\ \n  \\hdotsfor{6} \\\\ \n I^{(l)}_{n-1} & I^{(l)}_{n} && \\ldots && I^{(l)}_{2n-1} \\\\ \n I^{(l)}_n & I^{(l)}_{n+1} && \\ldots && I^{(l)}_{2n} \\\\\n\\end{array}   \\right) \n\\left( \\begin{array}{c} \n a^{(l)}_{n0} \\\\  a^{(l)}_{n1} \\\\ \\cdots \\\\  a^{(l)}_{n n-1} \\\\  \n a^{(l)}_{nn} \\\\ \n \\end{array} \\right)  = \n\\left( \\begin{array}{c} \n 0 \\\\  0\\\\ \\cdots \\\\  0 \\\\  \n 1\/a^{(l)}_{nn} \\\\ \n \\end{array} \\right).\n \\label{eq:orthogonal_matrix}\n\\end{eqnarray} \nThe explicit form of $a_{nm}^{(l)}$ is given in Appendix \\ref{app:G}. \nNow,  \n$\\lambda_{\\bf k}^{\\langle \\mu_1\\cdots \\mu_l \\rangle}$ \nand $\\bar{\\lambda}_{\\bf k}^{\\langle \\mu_1\\cdots \\mu_l \\rangle}$ \ncan be \nexpanded by using the polynomial function \n$P_{{\\bf k}n}^{(l)}(E_{\\bf k})$, as the following equation: \n\\begin{subequations}\n\\begin{eqnarray}\\label{eq:lambda_k}\n    \\lambda_{\\bf k}^{\\langle \\mu_1\\cdots \\mu_l \\rangle} &=&\n     \\sum_{n=0}^{\\infty}\n     c_{n}^{\\langle \\mu_1\\cdots \\mu_{l} \\rangle}\n     P_{{\\bf k}n}^{(l)}, \\label{eq:lambda_k_for_inf} \n     \\\\ \n \\bar\\lambda_{\\bf k}^{\\langle \\mu_1\\cdots \\mu_l \\rangle} &=&\n     \\sum_{n=0}^{\\infty}\n     \\bar c_{n}^{\\langle \\mu_1\\cdots \\mu_{l} \\rangle}\n     P_{{\\bf k}n}^{(l)}.\n\\end{eqnarray}\n\\end{subequations}\nNote that, without loss of generality, one can set $P_{{\\bf k} 0}^{(l)} \\equiv 1$ \nfor arbitrary $l$, which is equivalent to \n$a^{(l)}_{00}=1$. \nFor anti-particles, the normalization factor \n$\\bar {\\cal W}^{(l)}$ can be similarly defined (i.e., by \nreplacement of $\\mu\\to -\\mu$). \nThe weight factor appeared in eq.(\\ref{eq:definition_of_omega}) \nhas no chemical potential dependence and we do not therefore need \nto introduce a factor $\\bar{\\omega}_{\\bf k}^{(l)}$.\nSubstituting eq.(\\ref{eq:lambda_k_for_inf}) into eq.(\\ref{eq:expansion-lambda1}), \nmultiplying a factor $f_{{\\bf k}0} ~P_{{\\bf k} m}^{(l')}~k^{\\langle \\mu_1}\\cdots k^{\\mu_{l'} \\rangle}$ \n(with fixed $l'$) on the both side of it, and \nintegrating over whole ${\\bf k}$ space, we have \n\\begin{eqnarray}\n  \\int\\frac{d{\\bf k}^3 \\phi_{\\bf k} f_{0{\\bf k}}}{(2\\pi)^3k^0} ~P_{{\\bf k}n}^{(l')}~k^{\\langle \\nu_1}\\cdots k^{\\nu_{l'} \\rangle}\n  = \\sum_{l=0}^{\\infty}\\sum_{n=0}^{\\infty} \n  c_{n}^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle}\n     P_{{\\bf k}n}^{(l)} P_{{\\bf k}n}^{(l')}~\n     k^{\\langle \\nu_1}\\cdots k^{\\nu_{l'} \\rangle} k_{\\langle \\mu_1}\\cdots k_{\\mu_l \\rangle}. \n\\end{eqnarray}\nWe then apply the orthogonal condition of eq.(\\ref{eq:orthogonality_condition_final}) \nand (\\ref{eq:orthogonality_conditionPn}) to the above \nexpression, we obtain \n\\begin{subequations}\\label{eq:cn1} \n \\begin{eqnarray}\n   c_{n}^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle} &=&\n    \\frac{{\\cal W}^{(l)}}{l!}\n    \\int\\frac{d{\\bf k}^3 \\phi_{\\bf k} f_{0{\\bf k}}}{(2\\pi)^3k^0} ~ \n     P_{{\\bf k}n}^{(l)}~k^{\\langle \\nu_1}\\cdots k^{\\nu_{l} \\rangle}, \\quad \n\\end{eqnarray} and similarly for anti-particle part,   \n\\begin{eqnarray}\n  \\bar c_{n}^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle} &=&\n    \\frac{\\bar{\\cal W}^{(l)}}{l!}\n    \\int\\frac{d{\\bf k}^3 \\bar\\phi_{\\bf k} \\bar f_{0{\\bf k} }}{(2\\pi)^3k^0} ~ \n     P_{{\\bf k}n}^{(l)}~k^{\\langle \\nu_1}\\cdots k^{\\nu_{l} \\rangle}. \\quad \n\\end{eqnarray} \n\\end{subequations} \nSubstituting the definition of $P^{(l)}_{{\\bf k}n}$ given by eq.(\\ref{eq:Pn})  \ninto eq.(\\ref{eq:cn1}) and denoting\n\\begin{subequations} \\label{eq:rho_expression1}\n\\begin{eqnarray}\n   \\rho^{\\mu_1\\cdots\\mu_l}_r \n&\\equiv& \\int\\frac{d{\\bf k}^3 \\phi_{\\bf k} f_{0{\\bf k}}}{(2\\pi)^3k^0} ~ \n   (E_{\\bf k})^r  k^{\\langle \\nu_1}\\cdots k^{\\nu_{l} \\rangle},\\label{eq:rho1}\\\\\n    \\bar\\rho^{\\mu_1\\cdots\\mu_l}_r \n&\\equiv& \\int\\frac{d{\\bf k}^3 \\bar\\phi_{\\bf k}\\bar f_{0{\\bf k}}}{(2\\pi)^3k^0} ~ \n   (E_{\\bf k})^r  k^{\\langle \\nu_1}\\cdots k^{\\nu_{l} \\rangle},\\label{eq:rho2}\n\\end{eqnarray}\n\\end{subequations}\n the following expressions are obtained (for $n=0,1,2,\\cdots$)\n\\begin{subequations}\\label{eq:cn2} \n\\begin{eqnarray}\n  c_n^{\\langle \\mu_1\\cdots\\mu_l\\rangle} &=&\n  \\frac{{\\cal W}^{(l)}}{l!}\n  \\sum_{m=0}^{n} a_{nm} ~\\rho^{\\mu_1\\cdots\\mu_l}_m, \\label{eq:cn2a}\\\\  \n  \\bar c_n^{\\langle \\mu_1\\cdots\\mu_l\\rangle} &=&\n  \\frac{\\bar{\\cal W}^{(l)}}{l!}\n  \\sum_{m=0}^{n} a_{nm} ~\\bar\\rho^{\\mu_1\\cdots\\mu_l}_m~.\\label{eq:cn2b}\n\\end{eqnarray}\n\\end{subequations}\nThe  $l$-rank coefficient\ntensors $c_n^{\\langle \\mu_1\\cdots\\mu_l\\rangle}$ and \n$\\bar{c}_n^{\\langle \\mu_1\\cdots\\mu_l\\rangle}$ are \ngiven by the linear combinations of $\\rho^{\\mu_1\\cdots\\mu_l}_m$ and \n$\\bar{\\rho}^{\\mu_1\\cdots\\mu_l}_m$, respectively. \nNote here that \n$\\rho_2 +\\bar{\\rho}_2 = u_{\\mu}\\delta T^{\\mu\\nu} u_{\\nu} = \\Lambda$,   \n$\\rho^{\\mu}_1 + \\bar{\\rho}^{\\mu}_1=  \\Delta^{\\mu}_{\\alpha} \n\\delta T^{\\alpha\\beta} u_{\\beta} \\equiv W^{\\mu}$, \n$\\rho^{\\mu\\nu}_0+\\bar{\\rho}^{\\mu\\nu}_0= \n\\delta T^{\\langle \\mu\\nu\\rangle} \\equiv \\pi^{\\mu\\nu}$, \n$\\rho_1 - \\bar{\\rho}_1 =u_{\\mu}\\delta N^{\\mu} \\equiv\\delta n$ and \n$\\rho^{\\mu}_0- \\bar{\\rho}^{\\mu}_0=N^{\\langle \\mu\\rangle}  \\equiv V^{\\mu}$, \nwhere $W^{\\mu}$, $\\pi^{\\mu\\nu}$, and $V^{\\mu}$ are, respectively, \nenergy flow, shear tensor, and baryon number flow of the dissipative fluid. \nTherefore, if we truncate the expansion \n$\\lambda_{\\bf k}^{\\langle \\mu_1\\cdots \\mu_l \\rangle}=0$ for $l>2$ and  \n$c_{n}=\\bar c_{n}=0$ for $n>2$ (with $l=0$),  \n$c^{\\langle\\mu\\rangle}_{n} =\\bar c^{\\langle\\mu\\rangle}_{n} =0$ for $n>1$ (with $l=1$), and \n$c^{\\langle\\mu\\nu\\rangle}_{n} =\\bar c^{\\langle\\mu\\nu\\rangle}_{n} =0$ for $n>0$ (with $l=2$), \nlinear combination of the rest non-zero coefficients \n$c_n^{\\langle \\mu_1\\cdots\\mu_l\\rangle}$ and \n$\\bar{c}_n^{\\langle \\mu_1\\cdots\\mu_l\\rangle}$can be related \nto dissipative hydrodynamical quantities. \nWe hereafter denote  \n\\begin{subequations}\n\\begin{eqnarray}\n    \\lambda_{\\bf k}^{*\\langle \\mu_1\\cdots \\mu_l \\rangle} &=&\n     \\sum_{n=0}^{N_l}\n     c_{n}^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle}\n     P_{{\\bf k}n}^{(l)}, \\label{eq:lambda_k_for_Nl} \n     \\\\ \n \\bar\\lambda_{\\bf k}^{*\\langle \\mu_1\\cdots \\mu_l \\rangle} &=&\n     \\sum_{n=0}^{N_l}\n     \\bar c_{n}^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle}\n     P_{{\\bf k}n}^{(l)},\\label{eq:lambda_k_bar_for_Nl} \n\\end{eqnarray} \n\\end{subequations}\nsuch coefficient tensor with terminated at $N_2=0, N_1=1, N_2=2$,  known \nas the possible lowest truncation scheme \\cite{PhysRevD.85.114047}.   \nAlthough the definition of the eq.(\\ref{eq:rho_expression1}) links the deviation \n$\\phi_{\\bf k}(x)$ and $\\bar\\phi_{\\bf k}(x)$ to the local dissipative fluid dynamical quantities \nvia both $\\rho^{\\mu_1\\cdots\\mu_l}_m$ and $\\bar{\\rho}^{\\mu_1\\cdots\\mu_l}_m$, \none can also give linkage between \n$\\lambda_{\\bf k}^{*\\langle \\mu_1\\cdots\\mu_l\\rangle}$ ($\\bar\\lambda_{\\bf k}^{*\\langle \\mu_1\\cdots\\mu_l\\rangle}$)\nand $\\rho_r^{\\mu_1\\cdots\\mu_l}$ ($\\bar\\rho_r^{\\mu_1\\cdots\\mu_l}$) \n(See Appendix \\ref{app:F} for the derivation of the below equations) as the following; \n\\begin{subequations} \\label{eq:from_micro_to_macro} \n\\begin{eqnarray}\n\\rho_r^{\\mu_1\\cdots\\mu_l}\n&=& \\frac{l!}{(2l+1)!!} \\! \n   \\int\\!  \\frac{d^3{\\bf k}}{(2\\pi)^3} \n   \\lambda_{\\bf k}^{*\\langle \\mu_1\\cdots\\mu_l\\rangle}\n   [\\Delta^{\\alpha\\beta}k_{\\alpha}k_{\\beta}]^l \n   E_{\\bf k}^r f_{{\\bf k}0}, \n   \\label{eq:from_micro_to_macro1} \\\\ \n\\bar\\rho_r^{\\mu_1\\cdots\\mu_l}\n&=&\n   \\frac{l!}{(2l+1)!!} \\! \n   \\int\\!\\! \\frac{d^3{\\bf k}}{(2\\pi)^3} \n   \\bar\\lambda_{\\bf k}^{*\\langle \\mu_1\\cdots\\mu_l\\rangle}\n   [\\Delta^{\\alpha\\beta}k_{\\alpha}k_{\\beta}]^l \n   E_{\\bf k}^r \\bar f_{{\\bf k}0}. \n   \\label{eq:from_micro_to_macro2} \n\\end{eqnarray} \n\\end{subequations}\nFinally, \nlinear combinations of the l.h.s of eq.(\\ref{eq:from_micro_to_macro1}) and \neq.(\\ref{eq:from_micro_to_macro2}) for $l=0$ gives \n\\begin{subequations}\\label{eq:final_expressions}\n\\begin{eqnarray}\n  \\Lambda &=& \\int\\!\\!\\frac{d^3{\\bf k}~E_{\\bf k}^2 }{(2\\pi)^3k_0}  \n   \\Big[ \\lambda_{\\bf k}^* f_{{\\bf k}0}(x) + \n   \\bar\\lambda_k^* \\bar f_{{\\bf k}0}(x) \\Big], \n   \\label{eq:Lambda_bulk} \n   \\\\\n\\delta n &=& \\int\\!\\!\\frac{d^3{\\bf k}~E_{\\bf k}}{(2\\pi)^3k^0}  \n   \\Big[ \\lambda_{\\bf k}^* f_{{\\bf k}0}(x) -  \n   \\bar\\lambda_{\\bf k}^* \\bar f_{{\\bf k}0}(x) \\Big],\n   \\label{eq:delta_n}  \n\\end{eqnarray} \nand the bulk pressure is given by \n\\begin{eqnarray}  \n   \\Pi &=& \\frac{1}{3} \\int\\!\\!\\frac{d^3{\\bf k}~k^2}{(2\\pi)^3k^0}  \n   \\Big[ \\lambda_k^* f_{{\\bf k}0}(x) + \n   \\bar\\lambda_{\\bf k}^* \\bar f_{{\\bf k}0}(x) \\Big] .   \n   \\label{eq:PI_bulk} \n\\end{eqnarray} \nFor energy and net baryon number flow (for $l=1$),  we have   \n\\begin{eqnarray}\n W^{\\mu} \n&=& -\\frac{1}{3} \\int\\!\\!\\frac{d^3{\\bf k}~E_{\\bf k} k^2 }{(2\\pi)^3k^0}  \n   \\Big[ \\lambda_{\\bf k}^{*\\langle \\mu \\rangle} f_{{\\bf k}0}(x) + \n   \\bar\\lambda_{\\bf k}^{*\\langle \\mu \\rangle} \\bar f_{{\\bf k}0}(x) \\Big] \\equiv  0,  \\label{eq:W}\\\\\n V^{\\mu} \n&=& -\\frac{1}{3} \\int\\!\\!\\frac{d^3{\\bf k}~k^2 }{(2\\pi)^3k^0}  \n   \\Big[ \\lambda_{\\bf k}^{*\\langle \\mu \\rangle} f_{{\\bf k}0}(x) -  \n   \\bar\\lambda_{\\bf k}^{*\\langle \\mu \\rangle} \\bar f_{{\\bf k}0}(x) \\Big], \\label{eq:V} \n\\end{eqnarray}\nand the shear viscosity (for $l=2$) is  \n\\begin{eqnarray}\n \\pi^{\\mu\\nu} \n&=& \\frac{2}{15} \n   \\int\\!\\!\\frac{d^3{\\bf k}~k^4 }{(2\\pi)^3k^0}  \n   \\Big[ \\lambda_{\\bf k}^{*\\langle \\mu\\nu \\rangle} f_{{\\bf k}0}(x) +  \n   \\bar\\lambda_{\\bf k}^{*\\langle \\mu\\nu \\rangle}  \\bar f_{{\\bf k}0}(x) \\Big], \\label{eq:pi} \n\\end{eqnarray}\n\\end{subequations}\nwhere $k^2={\\bf k}\\cdot{\\bf k}= -k_{\\alpha}k_{\\beta}\\Delta^{\\alpha\\beta}$.\nNote that the heat flow vector $W^{\\mu}$ must vanish by definition \nof the Landau frame. \n(On the other hand, if the so called Eckart frame was employed, $V^{\\mu}\\equiv 0$.)  \nAs seen in eqs.(\\ref{eq:W}) and (\\ref{eq:V}), the tensor coefficients \n$\\lambda_{\\bf k}^{*\\langle \\mu \\rangle}$ and $\\bar\\lambda_{\\bf k}^{*\\langle \\mu \\rangle}$ \nare restricted in the ${\\bf k}$ dependence \nby the selection of the Lorentz frame for defining the rest frame of the fluid. \nNote that, eqs.(\\ref{eq:Lambda_bulk}) and (\\ref{eq:delta_n}) offer a key to understand the separation\ntemperature $T$ and the separation chemical potential $\\mu$. \nThus, we have\n\\begin{subequations}\\label{eq:offer_key}\n\\begin{eqnarray}\n  \\varepsilon&=&\\varepsilon(T,\\mu,\\lambda^*_{\\bf k},\\bar{\\lambda}^*_{\\bf k}) =\n  \\varepsilon_{\\rm eq}(T,\\mu)+\\Lambda, \\\\\n  n&=&n(T,\\mu,\\lambda^*_{\\bf k},\\bar{\\lambda}^*_{\\bf k}) =\n  n_{\\rm eq}(T,\\mu)+\\delta n. \n\\end{eqnarray}\n\\end{subequations} \nRecall that $T$ and $\\mu$ are functions of $\\phi$ and $\\bar{\\phi}$ \nas shown by eq.(\\ref{eq:phi_functional}). However, $T$ and $\\mu$ \ncan only be expressed by the $l=0$ component, i.e., \n$\\lambda_{\\bf k}^*$ and $\\bar{\\lambda}_{\\bf k}^*$ as observed in eq.(\\ref{eq:offer_key}). \nHence the separation temperature $T$ and the separation chemical potential $\\mu$ \ncan be expressed as a function of \n$\\lambda_{\\bf k}^*(x)$ and $\\bar{\\lambda}_{\\bf k}^*(x)$. \n\\begin{subequations} \n\\begin{eqnarray}\n    T(x) &=& T[\\lambda_{\\bf k}^*(x),\\bar{\\lambda}_{\\bf k}^*(x);T_0,\\mu_0], \\\\ \n    \\mu(x) &=& \\mu[\\lambda_{\\bf k}^*(x),\\bar{\\lambda}_{\\bf k}^*(x);T_0,\\mu_0]. \n\\end{eqnarray} \n\\end{subequations} \n\n\\subsection{Entropy current of the non-equilibrium state and thermodynamic stability}\nWhen the microscopic phase-space distribution function $f_{\\bf k}(x)$ is expressed \nby eq.(\\ref{eq:phi_functional}), \nthe local entropy current is divided into three parts as follows     \n\\begin{eqnarray}\n  s^{\\mu}(x) &=&-\\int\\!\\!\\frac{d^3{\\bf k}~k^{\\mu}}{(2\\pi)^3k^0}\n \\Big[\n  [f_{\\bf k}\\ln f_{\\bf k}-f_{\\bf k}] + [\\bar f_{\\bf k}\\ln \\bar f_{\\bf k}-\\bar f_{\\bf k}]\n \\Big]  \\nonumber \\\\ \n&=& s^{\\mu}_{\\rm eq}(x) +\\delta s_1^{\\mu}(x) +\\delta s_2^{\\mu}(x), \n  \\label{eq;entropy_current} \n\\end{eqnarray} \nwhere \n$s^{\\mu}_{\\rm eq}$ represents the equilibrium part given by \nthe separation temperature $T$ and chemical potential $\\mu$. \nThe remaining two terms,    \n$\\delta s_1^{\\mu}$ and $\\delta s_2^{\\mu}$, are \n given by\n\\begin{subequations}\n\\begin{eqnarray}\n   \\delta s_1^{\\mu} &\\equiv&\n   -\\int\\!\\!\\frac{d^3{\\bf k} ~k^{\\mu} }{(2\\pi)^3k^0} \n   \\Big[ \n    [\\phi_{\\bf k}\\ln f_{{\\bf k}0} ] ~f_{{\\bf k}0} \n + [\\bar\\phi_{\\bf k}\\ln \\bar f_{{\\bf k}0} ] ~\\bar f_{{\\bf k}0} \\Big],\\\\ \n  \\delta s_2^{\\mu} &\\equiv&\n   -\\int\\!\\!\\frac{d^3{\\bf k} ~k^{\\mu} }{(2\\pi)^3k^0} \n   \\Big[ \n     [(1+\\phi_{\\bf k})\\ln (1+\\phi_{\\bf k})-\\phi_{\\bf k} ] ~f_{{\\bf k}0} \n  +  [(1+\\bar\\phi_{\\bf k})\\ln (1+\\bar \\phi_{\\bf k}) -\\bar\\phi_{\\bf k} ] \n   ~\\bar f_{{\\bf k}0} \\Big],  \\nonumber \\\\\n\\end{eqnarray}\n\\end{subequations}\nrespectively. \nSince the entropy density should be maximum and stable \nin the limit of equilibrium, it must not include any linear \nterms of scalar off-equilibrium quantities such as  \n$\\delta n$, $\\Lambda$, and $\\Pi$ \\cite{PhysRevC.80.054906}; \n\\begin{eqnarray}\n  \\frac{\\partial (\\delta s_1^{\\mu}u_{\\mu})}{\\partial \\delta n} \\bigg|_{\\delta n=0} = \n  \\frac{\\partial (\\delta s_1^{\\mu}u_{\\mu})}{\\partial \\Lambda} \\bigg|_{\\Lambda=0} =\n  \\frac{\\partial (\\delta s_1^{\\mu}u_{\\mu})}{\\partial \\Pi} \\bigg|_{\\Pi=0}  \\equiv 0. \n  \\label{eq:stability_condition}   \n\\end{eqnarray}\nNote that, the thermodynamic stability condition defined in\neq.(\\ref{eq:stability_condition}) must be satisfied not only approximately but {\\it\nexactly}.\nTherefore, one must add all terms for $\\lambda_{\\bf k}$ and $\\bar{\\lambda}_{\\bf k}$ \nwithout termination in the energy polynomial function $P_{{\\bf k}n}^{(0)}$ \nat a finite $n=N_0(=2)$.     \nThus the first order correction of the off-equilibrium entropy current should be \ngiven as the following: \n\\begin{eqnarray} \n  \\delta s_1^{\\mu} &=&\n        \\bigg[ \\int\\!\\!\\frac{d^3{\\bf k}}{(2\\pi)^3k^0} \n        \\Big\\{ -\\alpha E_{\\bf k}(\\lambda_{\\bf k}f_{{\\bf k}0}-\\bar\\lambda_{\\bf k}\\bar f_{{\\bf k}0})\\Big\\} \n      +\\int\\!\\!\\frac{d^3{\\bf k}}{(2\\pi)^3k^0} \n       \\Big\\{ \\beta E^2_{\\bf k}(\\lambda_{\\bf k}f_{{\\bf k}0}+\\bar\\lambda_{\\bf k}\\bar f_{{\\bf k}0})\\Big\\} \\bigg]u^{\\mu} \n       \\nonumber \\\\\n     &-&\\frac{1}{3} \\int\\!\\!\\frac{d^3{\\bf k}~k^2}{(2\\pi)^3k^0} \n        \\Big\\{ -\\alpha E_{\\bf k}(\\lambda_{\\bf k}^{\\langle\\mu\\rangle} f_{{\\bf k}0}\n                                -\\bar\\lambda_{\\bf k}^{\\langle\\mu\\rangle}\\bar f_{{\\bf k}0})\\Big\\} \n         -\\frac{1}{3} \\int\\!\\!\\frac{d^3{\\bf k}~k^2}{(2\\pi)^3k^0} \n        \\Big\\{ \\beta E_{\\bf k}^2(\\lambda_{\\bf k}^{\\langle\\mu\\rangle} f_{{\\bf k}0}\n                                +\\bar\\lambda_{\\bf k}^{\\langle\\mu\\rangle}\\bar f_{{\\bf k}0})\\Big\\} \\nonumber \\\\\n        &\\equiv& [-\\alpha\\delta n +\\beta\\Lambda +\\xi]u^{\\mu} \n        +[-\\alpha V^{\\mu} +\\beta W^{\\mu} +\\xi^{\\mu} ]. \n    \\label{eq:delta_s1}\n\\end{eqnarray}\nHere, $\\alpha \\equiv \\mu\/T$, $\\beta\\equiv 1\/T$ and \nthe terms $\\xi$ and $\\xi^{\\mu}$ \nare residual terms which were ignored by the truncation \nof the polynomial $P^{(l)}_{{\\bf k}n}$ at $n=N_l$ in $\\lambda_{\\bf k}^{*}$ and $\\bar\\lambda_{\\bf k}^*$; \n\\begin{eqnarray}\n  \\xi^{\\mu_1\\cdots \\mu_l} &\\equiv& -\\int\\!\\!\\frac{d^3{\\bf k} E_{\\bf k}}{(2\\pi)^3k^0}\n \\Big[ \\alpha\\delta\\Phi^{\\langle\\mu_1\\cdots \\mu_l\\rangle} \n  -\\beta E_{\\bf k} \\delta\\Psi^{\\langle\\mu_1\\cdots \\mu_l\\rangle}  \\Big] ,\n\\end{eqnarray}\nwhere we denote \n\\begin{subequations}\n\\begin{eqnarray}\n&&\n\\delta\\Phi^{\\langle\\mu_1\\cdots \\mu_l\\rangle}_{\\bf k}  \\equiv  \n  \\delta\\lambda_{\\bf k}^{\\langle\\mu_1\\cdots \\mu_l\\rangle}  f_{{\\bf k}0}\n ~-~\\delta\\bar\\lambda_{\\bf k}^{\\langle\\mu_1\\cdots \\mu_l\\rangle}  \\bar f_{{\\bf k}0}, \\\\ \n&&\n\\delta\\Psi^{\\langle\\mu_1\\cdots \\mu_l\\rangle}_{\\bf k}  \\equiv  \n  \\delta\\lambda_{\\bf k}^{\\langle\\mu_1\\cdots \\mu_l\\rangle}  f_{{\\bf k}0}\n ~+~\\delta\\bar\\lambda_{\\bf k}^{\\langle\\mu_1\\cdots \\mu_l\\rangle}  \\bar f_{{\\bf k}0},\n\\end{eqnarray}\n\\end{subequations}\nand\n\\begin{subequations}\n\\begin{eqnarray}\\label{eq:delta_lambda_k} \n     \\delta \\lambda^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle}_{\\bf k} &\\equiv&\n     \\lambda^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle}_{\\bf k} - \\lambda^{*\\langle \\nu_1\\cdots \\nu_{l} \\rangle}_{\\bf k}=\n     \\sum_{n=N_l+1}^{\\infty}\n     c_{n}^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle}\n     P_{{\\bf k}n}^{(l)}, \\\\ \n     \\delta \\bar{\\lambda}^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle}_{\\bf k} &\\equiv&\n     \\bar \\lambda^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle}_{\\bf k} - \\bar\\lambda^{*\\langle \\nu_1\\cdots \\nu_{l} \\rangle}_{\\bf k}=\n     \\sum_{n=N_l+1}^{\\infty}\n     \\bar{c}_{n}^{\\langle \\nu_1\\cdots \\nu_{l} \\rangle}\n     P_{{\\bf k}n}^{(l)}. \n\\end{eqnarray}\n\\end{subequations}\nIn the usual formulation using the Landau matching conditions, \nthe factor $[-\\alpha\\delta n +\\beta\\Lambda+\\xi] u^{\\mu} \\equiv 0$ \nbecause $\\delta n\\equiv 0$, $\\Lambda \\equiv 0$, \nand $\\xi=0$. Therefore, in this case, one may write  \n\\begin{eqnarray}\n \\delta s^{\\mu}_1 u_{\\mu}=0, \\label{eq:entropy_1st_vanish}\n\\end{eqnarray} \nwhich prevents the entropy current from \nthe occurrence of instability \\cite{PhysRevC.85.014906,OsadaEPJA48}\n(See also Ref.\\cite{PhysRevC.80.054906}). \nHowever, in the case of eq.(\\ref{eq:extended-matching}), \nthe condition of eq.(\\ref{eq:entropy_1st_vanish}) is \n\\begin{eqnarray}\n   -\\alpha \\delta n + \\beta \\Lambda +\\beta \\chi \\Pi \\equiv 0,  \n    \\label{eq:stability}\n\\end{eqnarray} \nwhere $\\chi\\equiv \\xi\/(\\beta\\Pi)$\n for the off-equilibrium state to be thermodynamically stable. \nWe require that the condition defined by eq.(\\ref{eq:stability}) is \nsatisfied for both particles and antiparticles because each of the \nsubsystems should be independently stable; \n\\begin{subequations}\\label{eq_chi_expression}\n\\begin{eqnarray}\n&& -\\alpha \\delta n_+ + \\beta\\Lambda_+ +\\beta\\chi_+\\Pi_+ =0, \n    \\label{eq:stability_p}\\\\\n&& +\\alpha \\delta n_- + \\beta\\Lambda_- +\\beta\\chi_-\\Pi_- =0.\n    \\label{eq:stability_n} \n\\end{eqnarray}\nHere suffixes $+$ and $-$ appearing \nin eq.(\\ref{eq_chi_expression}) denote the \ncontributions from particles ($\\lambda_{\\bf k}^* f_{{\\bf k}0}$ \nin eq.(\\ref{eq:final_expressions}))\nand antiparticles ($\\bar{\\lambda}_{\\bf k}^* \\bar{f}_{{\\bf k}0}$ in \neq.(\\ref{eq:final_expressions})). Then $\\chi$ in eq.(\\ref{eq:stability}) as \n\\begin{eqnarray}\n  \\chi\\equiv\n  \\frac{\\chi_+\\Pi_++\\chi_-\\Pi_-}{\\Pi_++\\Pi_-} . \n  \\label{eq:chi_final}\n\\end{eqnarray} \n\\end{subequations}\nIntegration by parts for the first and the second terms in eq.(\\ref{eq:delta_n}) yields  \n\\begin{subequations} \\label{eq:off-equilibrium_EoS}\n\\begin{eqnarray}\n  \\delta n_+ &=&\n -\\int\\!\\!\\frac{d^3{\\bf k}}{(2\\pi)^3k^0}  \\frac{E_{\\bf k} k}{3} \n   \\frac{\\partial \\lambda_{\\bf k}^*}{\\partial k} f_{{\\bf k}0}  +\\frac{\\Pi_+}{T},\n   \\label{eq:deltan1}\\\\\n\\delta n_- &=&\n  -\\int\\!\\!\\frac{d^3{\\bf k}}{(2\\pi)^3k^0} \n    \\frac{E_{\\bf k} k}{3} \\frac{\\partial \\bar\\lambda_{\\bf k}^*}{\\partial k} \\bar f_{{\\bf k}0} \n  +\\frac{\\Pi_-}{T}, \\label{eq:deltan2} \n\\end{eqnarray} \n\\end{subequations} \nrespectively. Note that, eqs.(\\ref{eq:deltan1}) and (\\ref{eq:deltan2}) can be regarded as \n`Equations of State' for the off-equilibrium condition that  correspond to  \n$P_{{\\rm eq}}= n_{\\rm eq} T$ in the equilibrium state.  \nBy combining eq.(\\ref{eq:off-equilibrium_EoS}), eq.(\\ref{eq:stability}) and eq.(\\ref{eq:final_expressions}),   \none obtain a differential equation for $\\lambda_{\\bf k}^*$ as\n\\begin{eqnarray}\n&&\n   \\frac{\\alpha E_{\\bf k}}{3\\beta}\\Big( k\\frac{d\\lambda_{\\bf k}^*}{dk} \\Big) \n   +\\Big(E_{\\bf k}^2-\\frac{[\\alpha-\\chi_+]}{3}k^2\\Big) \\lambda_{\\bf k}^* =0~\n   \\label{eq:differential_fq_for_lambda}\n\\end{eqnarray} \n The above differential equation can be solved by the following \nintegration form \n\\begin{eqnarray}\n&&  \\int \\frac{d\\lambda_{\\bf k}^*}{\\lambda_{\\bf k}^*} = \n   \\beta [1-\\frac{\\chi_+}{\\alpha}]  \\int\\!\\! \\frac{~k~dk}{\\sqrt{k^2+m^2}}  \n   -\\frac{3\\beta}{\\alpha} \\int\\!\\! \\frac{\\sqrt{k^2+m^2}}{k} ~dk, \n\\end{eqnarray}\nthus we obtain  \n\\begin{subequations}\\label{eq:solution_lambda} \n\\begin{eqnarray}\n   \\lambda_{\\bf k}^* &=& C_{\\gamma} \n   \\Big[ \\frac{E_{\\bf k}+m}{k}\\Big]^{+\\frac{3}{\\alpha}m\\beta}  \n   \\exp \\Big[(1-\\frac{[3+\\chi_+]}{\\alpha}) \n   \\beta E_{\\bf k}\\Big]. ~\\qquad  \\label{eq:lambda}\n\\end{eqnarray}\nFor $\\bar\\lambda_k^*$, a similar expression can be obtained;  \n\\begin{eqnarray}\n   \\bar\\lambda_{\\bf k}^* &=& C_{\\gamma} \n   \\Big[ \\frac{E_{\\bf k}+m}{k}\\Big]^{-\\frac{3}{\\alpha}m\\beta}  \n   \\exp \\Big[(1+\\frac{[3+\\chi_-]}{\\alpha}) \n   \\beta E_{\\bf k}\\Big]. ~\\qquad \\label{eq:lambda_bar} \n\\end{eqnarray} \n\\end{subequations}\nThe constant $C_{\\gamma}$ in eq.(\\ref{eq:solution_lambda}) is \nan arbitrary integration constant. Physically, it determines the absolute \nvalues of off-equilibrium quantities such as $\\Pi$ \n\\footnote{If one substitutes eqs.(\\ref{eq:lambda}) and (\\ref{eq:lambda_bar}) \nto eq.(\\ref{eq:PI_bulk}), the bulk pressure $\\Pi$ seems to be independent of temperature \n$T$ and it depends only on the chemical potential $\\mu$ at a glance. \nHowever, $\\Pi$ depends on $T$ because, \nthrough the thermodynamical stability conditions \neqs.(\\ref{eq:stability_p}) and (\\ref{eq:stability_n}), \n$\\chi_+$ and $\\chi_-$ depend on $T$ \n(See also Fig.\\ref{fig:4}, for the case of $\\chi$). \n}. \nTherefore, we require that the value of $C_{\\gamma}$ satisfies \n\\begin{eqnarray}\n \\Pi = \\gamma ~P_{\\rm eq}(\\alpha,\\beta), \n \\label{eq:def_gamma} \n\\end{eqnarray}\nwhere $\\gamma$ indicates the strength of the\noff-equilibrium state.\nHence, an off equilibrium state can be specified by $\\alpha$, $\\beta$, \nand $\\gamma$. \nIt is possible to write\n $\\chi_{\\pm}\\equiv -3 \\pm\\delta \\chi$ because \nan exchange $\\alpha \\leftrightarrow -\\alpha$ \nin eq.(\\ref{eq:solution_lambda}) gives \n$\\lambda_{\\bf k}^* \\leftrightarrow \\bar{\\lambda}_{\\bf k}^*$. \nThen, using eqs.(\\ref{eq_chi_expression}),  \nwe can express $\\chi$\nin a more simple form\n\\begin{eqnarray}\n  \\chi = -3 +\\frac{\\delta\\chi \\delta\\Pi}{\\Pi}, \n  \\label{eq:chi_arround3} \n\\end{eqnarray}\nwhere $\\delta \\Pi \\equiv (\\Pi_+ - \\Pi_-)$. \\\\\n\n\\subsection{Coefficient tensors of the higher \nrank and link of the initial condition of fluid}\\label{sec:2c}  \nIn the possible lowest scheme, the coefficient tensor of \nthe first and second rank, \n$\\lambda_{\\bf k}^{*\\langle \\mu \\rangle}$ and $\\lambda_{\\bf k}^{*\\langle \\mu\\nu \\rangle}$, \nare respectively terminated \nat $N_1=1$ and $N_2=0$ in eq.(\\ref{eq:lambda_k_for_Nl}) and \neq.(\\ref{eq:lambda_k_bar_for_Nl}).  Hence, we can write \n\\begin{subequations}\n\\label{eq:lambda_rank12} \n\\begin{eqnarray}\n &&\\lambda_{\\bf k}^{*\\langle \\mu \\rangle} = \n   c_0^{\\langle \\mu \\rangle} \n+ c_1^{\\langle \\mu \\rangle} P_{{\\bf k}1}^{(1)},\\quad  \n   \\bar\\lambda_{\\bf k}^{*\\langle \\mu \\rangle} =\n   \\bar c_0^{\\langle \\mu \\rangle} \n+ \\bar c_1^{\\langle \\mu \\rangle} P_{{\\bf k}1}^{(1)},  \\label{eq:lambda_rank1}\\quad \\\\\n  &&  \\lambda_{\\bf k}^{*\\langle \\mu\\nu \\rangle} = \n   c_0^{\\langle \\mu\\nu \\rangle} ,\\quad \n   \\bar \\lambda_{\\bf k}^{*\\langle \\mu\\nu \\rangle} = \n   \\bar c_0^{\\langle \\mu\\nu \\rangle} ,\\label{eq:lambda_rank2} \\quad \n\\end{eqnarray}\n\\end{subequations}\nwhere $c_n^{\\langle \\mu_1\\cdots\\mu_l\\rangle}$ and \n$\\bar c_n^{\\langle \\mu_1\\cdots\\mu_l\\rangle}$ are \ngiven by eq.(\\ref{eq:cn2a}) and (\\ref{eq:cn2b}) which are \nlinear combination of $\\rho^{\\mu_1\\cdots\\mu_l}$ and \n$\\bar \\rho^{\\mu_1\\cdots\\mu_l}$, \n\\begin{subequations}\\label{eq:c01}\n\\begin{eqnarray}\n   &&  \\frac{c_0^{\\langle \\mu\\rangle}}{{\\cal W}^{(1)}} = \\rho^{\\mu}_0,  \\quad \n         \\frac{\\bar c_0^{\\langle \\mu\\rangle}}{\\bar{\\cal W}^{(1)}}= \\bar\\rho^{\\mu}_0, \\label{eq:c0} \\\\\n   &&  \\frac{c_1^{\\langle \\mu\\rangle}}{{\\cal W}^{(1)}} = a_{10}^{(1)} \\rho^{\\mu}_0 +  a_{11}^{(1)} \\rho^{\\mu}_1,\\quad \n   \\frac{\\bar c_1^{\\langle \\mu\\rangle}}{\\bar{\\cal W}^{(1)}} =a_{10}^{(1)} \\bar\\rho^{\\mu}_0 +  \n   a_{11}^{(1)} \\bar\\rho^{\\mu}_1, \\label{eq:c1}\n   \\quad\n\\end{eqnarray} \n\\end{subequations}\nand the coefficient $a_{nm}^{(l)}$ are given by Appendix \\ref{app:G}.\nBy substituting eq.(\\ref{eq:lambda_rank1}),  \none obtains   \n\\begin{subequations}\n\\begin{eqnarray}\n && {\\cal A}_0^1 ~\\frac{c_0^{\\langle \\mu \\rangle}}{{\\cal W}^{(1)}}\n  + \\bar{\\cal A}_0^1 ~\\frac{\\bar c_0^{\\langle \\mu \\rangle}}{\\bar{\\cal W}^{(1)}}    \n  + {\\cal A}_1^1 ~\\frac{c_1^{\\langle \\mu \\rangle}}{{\\cal W}^{(1)}} \n  + \\bar{\\cal A}_1^1 \\bar ~\\frac{c_1^{\\langle \\mu \\rangle}}{\\bar{\\cal W}^{(1)}}   =0,\\\\\n && {\\cal A}_0^0 ~\\frac{c_0^{\\langle \\mu \\rangle}}{{\\cal W}^{(1)}} \n  - \\bar{\\cal A}_0^0 ~\\frac{\\bar c_0^{\\langle \\mu \\rangle}}{\\bar{\\cal W}^{(1)}}    \n  + {\\cal A}_1^0 ~\\frac{c_1^{\\langle \\mu \\rangle}}{{\\cal W}^{(1)}}  \n  - \\bar{\\cal A}_1^0 ~\\frac{\\bar c_1^{\\langle \\mu \\rangle}}{\\bar{\\cal W}^{(1)}}   =V^{\\mu},\n\\end{eqnarray}\n\\end{subequations}\nwhere (recall that $P_{{\\bf k}0}^{(l)} \\equiv 1$ and $a_{00}^{(l)}\\equiv1$ \nfor arbitrary $l$)\n\\begin{subequations}\n\\begin{eqnarray}\n  {\\cal A}_n^m &\\equiv& \n-\\frac{{\\cal W}^{(1)}}{3} \\int\\!\\!\\frac{d^3{\\bf k}k^2(E_{\\bf k})^m}{(2\\pi)^3k_0} \n  P_{{\\bf k}n}^{(1)} f_{{\\bf k}0}, \\\\\n\\bar{\\cal A}_n^m &\\equiv& \n-\\frac{\\bar{\\cal W}^{(1)}}{3} \\int\\!\\!\\frac{d^3{\\bf k}k^2(E_{\\bf k})^m}{(2\\pi)^3k_0} \n  P_{{\\bf k}n}^{(1)} \\bar f_{{\\bf k}0}. \n\\end{eqnarray}\n\\end{subequations} \nSince there are constraint conditions, \ni.e., $\\rho^{\\mu}_0-\\bar\\rho^{\\mu}_0=V^{\\mu}$ and \n$\\rho^{\\mu}_1+\\bar\\rho^{\\mu}_1 (=W^{\\mu})\\equiv 0$, \none can write\n\\begin{subequations}\n\\begin{eqnarray}\n  \\frac{\\bar c_0^{\\langle\\mu\\rangle}}{{\\cal W}^{(1)}} &=&  \n  \\frac{c_0^{\\langle\\mu\\rangle}}{{\\cal W}^{(1)}} -V^{\\mu}, \\\\\n  \\frac{\\bar c_1^{\\langle\\mu\\rangle}}{\\bar{\\cal W}^{(1)}} &=& \n-\\frac{c_1^{\\langle\\mu\\rangle}}{{\\cal W}^{(1)}}\n  +2a_{10}^{(1)} \\frac{c_0^{\\langle\\mu\\rangle}}{{\\cal W}^{(1)}} -a_{10}^{(1)}V^{\\mu}.  \n\\end{eqnarray}\n\\end{subequations}\nUsing the above equations to eliminate \n$\\bar c_0^{\\langle\\mu\\rangle}$ and $\\bar c_1^{\\langle\\mu\\rangle}$, one then obtains \n\\begin{subequations}\\label{eq:cn_coefficient_rank1}\n\\begin{eqnarray}\n c^{\\langle\\mu\\rangle}_0 &=& {\\cal W}^{(1)}~  \n  \\frac{\n  \\tilde{\\cal A}^0_1 \\tilde{\\cal A}^1_v -\\tilde{\\cal A}^1_1(1-\\tilde{\\cal A}^0_v ) \n}{  \\tilde{\\cal A}^1_0 \\tilde{\\cal A}^0_1 -\\tilde{\\cal A}^1_1\\tilde{\\cal A}^0_0 }\n~V^{\\mu}, \\\\\n c^{\\langle\\mu\\rangle}_1 &=& {\\cal W}^{(1)}~  \n  \\frac{\n  \\tilde{\\cal A}^1_0(1-\\tilde{\\cal A}^0_v ) -\\tilde{\\cal A}^0_0 \\tilde{\\cal A}^1_v\n}{  \\tilde{\\cal A}^1_0 \\tilde{\\cal A}^0_1 -\\tilde{\\cal A}^1_1\\tilde{\\cal A}^0_0 }\n~V^{\\mu},  \n\\end{eqnarray} \n\\end{subequations} \nwhere \n\\begin{subequations}\n\\begin{eqnarray}\n  \\tilde {\\cal A}^m_0 &\\equiv& {\\cal A}^m_0 +(-1)^{m+1} \\bar {\\cal A}^m_0 \n +(-1)^{m+1} 2a_{10}^{(1)} \\bar{\\cal A}^m_1,\\quad  \\\\ \n  \\tilde {\\cal A}^m_1 &\\equiv& {\\cal A}^m_1 +(-1)^{m} \\bar{\\cal A}^m_1,  \\\\\n  \\tilde {\\cal A}^m_v &\\equiv& \\bar{\\cal A}^m_0 + a_{10}^{(1)} \\bar{\\cal A}^m_1. \n\\end{eqnarray}\n\\end{subequations}\nSimilarly, substituting eq.(\\ref{eq:lambda_rank2}) into eq.(\\ref{eq:pi}), we also obtain  \n\\begin{eqnarray}\n    {\\cal D}_0^0 ~\\frac{2! c_0^{\\langle \\mu\\nu \\rangle}}{{\\cal W}^{(2)}}\n  + \\bar{\\cal D}_0^0 ~\\frac{2! \\bar c_0^{\\langle \\mu\\nu \\rangle}}{\\bar{\\cal W}^{(2)}}     \n  =\\pi^{\\mu\\nu}, \n\\end{eqnarray} \nwhere \n\\begin{subequations}\n\\begin{eqnarray}\n  {\\cal D}_n^m &\\equiv& \n  \\frac{{\\cal W}^{(2)}}{15} \\int\\!\\!\\frac{d^3{\\bf k}k^4 (E_{\\bf k})^m}{(2\\pi)^3k_0} \n  P_{{\\bf k}n}^{(2)} f_{{\\bf k}0}, \\\\\n\\bar{\\cal D}_n^m &\\equiv& \n \\frac{\\bar{\\cal W}^{(2)}}{15} \\int\\!\\!\\frac{d^3{\\bf k}k^4 (E_{\\bf k})^m}{(2\\pi)^3k_0} \n  P_{{\\bf k}n}^{(2)} \\bar f_{{\\bf k}0}. \n\\end{eqnarray}\n\\end{subequations} \nBecause we have\n\\begin{eqnarray}\n  \\frac{2!c_0^{\\langle\\mu\\nu\\rangle}}{{\\cal W}^{(2)}} \n+\\frac{2!\\bar c_0^{\\langle\\mu\\nu\\rangle}} {\\bar{\\cal W}^{(2)}} = \n  \\rho^{\\mu\\nu}+ \\bar\\rho^{\\mu\\nu} =\\pi^{\\mu\\nu},\n\\end{eqnarray} \nwe obtain \n\\begin{eqnarray}\n       c^{\\langle\\mu\\nu\\rangle}_0 =\n   \\frac{1}{2!}{\\cal W}^{(2)}\n   \\frac{1-\\bar{\\cal D}^0_0} {{\\cal D}^0_0-\\bar{\\cal D}^0_0}~\\pi^{\\mu\\nu}, \\quad\n\\bar c^{\\langle\\mu\\nu\\rangle}_0 =\n    \\frac{1}{2!}\\bar{\\cal W}^{(2)}\n   \\frac{{\\cal D}^0_0-1}{{\\cal D}^0_0-\\bar{\\cal D}^0_0 }~\\pi^{\\mu\\nu}.\\quad \n\\label{eq:cn_coefficient_rank2} \n\\end{eqnarray}\nAs seen in eq.(\\ref{eq:cn_coefficient_rank1}) and (\\ref{eq:cn_coefficient_rank2}), \none can connect the coefficient tensors \n$\\lambda_{\\bf k}^{*\\langle\\mu\\rangle}$, \n$\\bar\\lambda_{\\bf k}^{*\\langle\\mu\\rangle}$, \n$\\lambda_{\\bf k}^{*\\langle\\mu\\nu\\rangle}$ and $\\bar\\lambda_{\\bf k}^{*\\langle\\mu\\nu\\rangle}$ \nwith $\\delta N^{\\langle\\mu\\rangle}$ and $\\delta T^{\\langle\\mu\\nu\\rangle}$ \nas the following; \n\\begin{subequations} \\label{eq:lamda_tensor_rank12}\n\\begin{eqnarray}\n \\lambda_{\\bf k}^{*\\langle\\mu\\rangle} &=&  \n \\frac{1}{\\tilde{\\cal A}^1_0 \\tilde{\\cal A}^0_1 -\\tilde{\\cal A}^1_1\\tilde{\\cal A}^0_0 } \n \\Big[ (\\tilde{\\cal A}^0_1-\\tilde{\\cal A}^0_0 P^{(1)}_{{\\bf k}1})\\tilde{\\cal A}^1_v \n -(\\tilde{\\cal A}^1_1 -\\tilde{\\cal A}^1_0 P^{(1)}_{{\\bf k}1})\n   (1-\\tilde{\\cal A}^0_v ) \\Big]\n~\\delta N^{\\langle\\mu\\rangle}, \\\\\n\\bar\\lambda_{\\bf k}^{*\\langle\\mu\\rangle} &=&  \n \\frac{1}{\\tilde{\\cal B}^1_0 \\tilde{\\cal B}^0_1 -~\\tilde{\\cal B}^1_1\\tilde{\\cal B}^0_0 } \n\\Big[ (\\tilde{\\cal B}^1_1 -\\tilde{\\cal B}^1_0 P^{(1)}_{{\\bf k}1}) (1-\\tilde{\\cal B}^0_v )\n -~(\\tilde{\\cal B}^0_1-\\tilde{\\cal B}^0_0 P^{(1)}_{{\\bf k}1})\\tilde{\\cal B}^1_v\n\\Big]\n~\\delta N^{\\langle\\mu\\rangle}, \\\\\n  \\lambda_{\\bf k}^{*\\langle\\mu\\nu\\rangle}&=&\n   \\frac{1-\\bar{\\cal D}^0_0} {{\\cal D}^0_0-\\bar{\\cal D}^0_0}\n~\\delta T^{\\langle\\mu\\nu\\rangle}, \\\\ \n   \\bar \\lambda_{\\bf k}^{*\\langle\\mu\\nu\\rangle} &=&\n   \\frac{{\\cal D}^0_0-1}{{\\cal D}^0_0-\\bar{\\cal D}^0_0 }\n~\\delta T^{\\langle\\mu\\nu\\rangle}, \n\\end{eqnarray}\n\\end{subequations}\nwhere ${\\cal B}^m_n$ is obtained by exchange of ${\\cal A}^m_n \\leftrightarrow \\bar{\\cal\nA}^m_n$; i.e., \n\\begin{subequations}\n\\begin{eqnarray}\n  \\tilde {\\cal B}^m_0 &\\equiv& \\bar{\\cal A}^m_0 +(-1)^{m+1} {\\cal A}^m_0 \n +(-1)^{m+1} 2a_{10}^{(1)} {\\cal A}^m_1,\\quad  \\\\ \n  \\tilde {\\cal B}^m_1 &\\equiv& \\bar{\\cal A}^m_1 +(-1)^{m} {\\cal A}^m_1,  \\\\\n  \\tilde {\\cal B}^m_v &\\equiv& {\\cal A}^m_0 + a_{10}^{(1)} {\\cal A}^m_1. \n\\end{eqnarray}\n\\end{subequations}\nIt is shown that the coefficient tensor $\\lambda^{*\\langle\\mu\\rangle}_{\\bf k}$ and  \n$\\lambda^{*\\langle\\mu\\nu\\rangle}_{\\bf k}$ are given by the dissipative part of the \nbaryon number 4-vector $N^{\\mu}$ and \nenergy-momentum tensor $T^{\\mu\\nu}$, respectively. \nHence, if $N^{\\mu}(x)$ and $T^{\\mu\\nu}(x)$ are given as an initial conditions, \none can determine those coefficient tensors in the irreducible tensor expansion. \n\nLet us summarize the main points that we have been made in this section. \nFor given energy-momentum tensor $T^{\\mu\\nu}$ and baryon number current \nvector $N^{\\mu}$, one can find $\\alpha\\equiv \\mu\/T$, $\\beta\\equiv 1\/T$ and $\\gamma$ (the\nstrength of off-equilibrium) \nby solving the following equations simultaneously \n\\begin{subequations}\n\\label{eq:formulation_summary} \n\\begin{eqnarray}\n  u_{\\mu}T^{\\mu\\nu} u_{\\nu} &=&\\int\\!\\!\\frac{d^3{\\bf k}~E_{\\bf k}^2 }{(2\\pi)^3k_0}  \n   \\Big[ (1+\\lambda_{\\bf k}^*(\\alpha,\\beta;\\gamma)) f_{{\\bf k}0}(\\alpha,\\beta) + \n          (1+\\bar\\lambda_k^*(\\alpha,\\beta;\\gamma)) \\bar f_{{\\bf k}0}(\\alpha,\\beta) \\Big], \n           \\label{eq:formulation_summary1} \\qquad \\\\\n  N^{\\mu} u_{\\mu} &=& \\int\\!\\!\\frac{d^3{\\bf k}~E_{\\bf k} }{(2\\pi)^3k_0}  \n   \\Big[ (1+\\lambda_{\\bf k}^*(\\alpha,\\beta;\\gamma)) f_{{\\bf k}0}(\\alpha,\\beta) -  \n          (1+\\bar\\lambda_k^*(\\alpha,\\beta;\\gamma)) \\bar f_{{\\bf k}0}(\\alpha,\\beta) \\Big], \n           \\label{eq:formulation_summary2} \\qquad \\\\\n  -\\frac{1}{3} \\Delta_{\\mu\\nu} T^{\\mu\\nu} &=&\n  \\frac{1}{3} \\int\\!\\!\\frac{d^3{\\bf k}~k^2 }{(2\\pi)^3k_0}  \n   (1+\\gamma) \\Big[ f_{{\\bf k}0}(\\alpha,\\beta) +\\bar f_{{\\bf k}0}(\\alpha,\\beta) \\Big], \n   \\label{eq:formulation_summary3} \n\\end{eqnarray}\n\\end{subequations} \nwhere $\\lambda_{\\bf k}^*$ and $\\bar\\lambda_{\\bf k}^*$ are given by (\\ref{eq:lambda}) \nand (\\ref{eq:lambda_bar}), respectively. As seen in the eq.(\\ref{eq:formulation_summary}), \nthe temperature $T$ and chemical potential $\\mu$ are connected with  \nvalue of $u_{\\mu}T^{\\mu\\nu}u_{\\nu}$ and $u_{\\mu}N^{\\mu}$ via \n$\\lambda_{\\bf k}^*$ and $\\bar \\lambda_{\\bf k}^*$ and they are also related with \n$-\\Delta_{\\mu\\nu}T^{\\mu\\nu}\/3$. \nNote that, on the other hand, \nthe Landau matching condition directly connect $T$ and $\\mu$ with those values \nregardless of the value of the bulk pressure $\\Pi=-\\Delta_{\\mu\\nu}T^{\\mu\\nu}\/3$. \nIf the above a set of equations \neq.(\\ref{eq:formulation_summary}) are solved and \nthe separation temperature $T$ and chemical potential $\\mu$ obtained, \ncoefficients tensor of first and second rank are also obtained by using \neq.(\\ref{eq:lamda_tensor_rank12}). \nUnlike the case of the coefficient tensor of zero-rank, \nwe found that those of the first and second rank \nare proportional to $\\delta N^{\\langle\\mu\\rangle}$ and \n$\\delta T^{\\langle\\mu\\nu\\rangle}$, respectively. \n\nBefore closing this section, comments concerning \nindependent degrees of freedom of relativistic \ndissipative fluid with extended matching condition  \nmay be in order here.\nThe energy-momentum tensor $T^{\\mu\\nu}$ has 10 and \nthe net baryon current $N^{\\mu}$ has 4 independent \ndegree of freedom.   \nThese quantities can be obtained from some kind of \nquark-gluon distribution function in \nearly stage of relativistic heavy-ion collisions.  \nFor example, one may consider \ngluon distribution function for a glasma state \\cite{McLerran:2014apa,Praszalowicz:2013fsa}. \nOnce such a quark-gluon distribution functions ($f_{\\bf k}$ and $\\bar f_{\\bf k}$) \nare introduced, those 14 degrees of freedom \n\\footnote{These 14 degrees of freedoms are accounted for by $\\varepsilon$, $n$, $u^{\\mu}$ that satisfy \n$u^{\\mu}u_{\\mu}\\equiv1$ (3 independent degree of freedom), \nthe bulk pressure $\\Pi$, and the dissipative tensor $\\pi^{\\mu\\nu}$ \nand current $V^{\\mu}$ that satisfy \n$\\pi^{\\mu\\nu}u_{\\mu}=0, \\pi^{\\mu}_{\\mu}=0$ (5 independent degree of freedom) \nand $V^{\\mu}u_{\\mu}=0$ (3 independent degree of freedom).} \nare fixed by eq.(\\ref{eq:energy-momentum_tensor}) and (\\ref{eq:particle_current}). \nFor the energy-momentum tensor $T^{\\mu\\nu}$ obtained, \none extracts $\\varepsilon$, $u^{\\mu}$ from eq.(\\ref{eq:Landau_frame}), \n$n$ from eq.(\\ref{eq:current_decomposition}) \nand $P_{\\rm eq}+\\Pi$ from its definition $-\\Delta_{\\mu\\nu}T^{\\mu\\nu}\/3$ \n\\footnote{The Landau matching condition separates $P_{\\rm eq}$ and $\\Pi$ \nby using an equation of state $P_{\\rm eq}(\\varepsilon,n)$ \nbecause $\\varepsilon=\\varepsilon_{\\rm eq}$ and $n=n_{\\rm eq}$ are assumed.}. \nHence, 6 degrees of freedom are fixed.   \nThe other 8 degrees are fixed as follows: i.e., \n3 degrees of freedom for $V^{\\mu}$ and 5 degrees for $\\pi^{\\mu\\nu}$ \nare attribute to those for \n$\\lambda_{\\bf k}^{*\\langle\\mu\\rangle}$ and $\\lambda_{\\bf k}^{*\\langle \\mu\\nu\\rangle}$, \nrespectively, in the irreducible tensor expansion of the initial distribution function. \nSee also Table \\ref{tab:1}. \n\nThe zero-th order coefficients $\\lambda_{\\bf k}^*$ and $\\bar\\lambda_{\\bf k}^*$  \nhave been ignored so far because of the Landau matching condition, i.e., \n$\\varepsilon=\\varepsilon_{\\rm eq}(\\alpha,\\beta)$ and $n=n_{\\rm eq}(\\alpha,\\beta)$.  \nIf the condition is extended, $\\lambda_{\\bf k}^*$  and $\\bar\\lambda_{\\bf k}^*$ not only \nlead new `internal degrees of freedom' (i.e., $\\Lambda$ and $\\delta n$) \nto the energy density $\\varepsilon$ and the net baryon density $n$, \nbut also link those $\\Lambda$ and $\\delta n$ with the bulk pressure $\\Pi$.\nNote that, by the extension of the matching condition, only one degree of \nfreedom $\\lambda_{\\bf k}^*$ is newly introduced. \n(See eqs. (\\ref{eq:Lambda_bulk})-(\\ref{eq:PI_bulk}).   \nSince $\\bar\\lambda_{\\bf k}^*$ must be linked to $\\lambda_{\\bf k}^*$ \nby replacement of $\\alpha \\to -\\alpha$, they are not independent degree of freedom in fact.)\nThe momentum ${\\bf k}$ dependence for $\\lambda_{\\bf k}^*$ is actually \ndetermined \nby the thermodynamical stability condition eq.(\\ref{eq:stability}) \nand its absolute values ($C_{\\gamma}$ in the eq.(\\ref{eq:solution_lambda})) \ncan be determined by the bulk pressure $\\Pi$, which is characterized by the parameter $\\gamma$.\nThrough the constraint of eq.(\\ref{eq:stability}), \n$\\chi$ links to $\\lambda_{\\bf k}^*$ and it appears in the \nexpression of eq.(\\ref{eq:solution_lambda}). \nHence, the parameter $\\chi$ and $\\lambda_{\\bf k}^*$ are \nfunction of the $\\alpha$, $\\beta$ and $\\gamma$.  \nAlthough the extension of the matching condition leads a new degree of freedom \n$\\lambda_{\\bf k}^*$, it is able to be given by a function of other degrees of freedom \n(i.e., $\\alpha, \\beta$ and $\\gamma$) due to the constraint of eq.(\\ref{eq:stability}) \nwith using eq.(\\ref{eq:off-equilibrium_EoS}).  \nTherefore, even for the case that the extended matching condition is used \nin the relativistic dissipative fluid model, 14 degrees of freedom are required \nwhich is the same as the number used in the case of the usual Landau matching condition. \n\\begin{center}\n\\begin{table}\n\\newcommand{\\lw}[1]{\\smash{\\lower2.0ex\\hbox{#1}}}\n\\newcommand{\\lww}[1]{\\smash{\\lower1.25ex\\hbox{#1}}}\n\\renewcommand{\\arraystretch}{1.25}\n\\caption{\\label{tab:1} \nSeparation of equilibrium part $\\varepsilon_{\\rm eq}$, $n_{\\rm eq}$ (or $\\alpha$, $\\beta$) from an \noff-equilibrium state characterized by $T^{\\mu\\nu}$ and $N^{\\mu}$ with using a scalar dissipative strength $\\gamma$ \nin the Landau-frame.} \n\\begin{tabular}{|c|c|c|c|}\\hline\n\\lww{\\footnotesize energy-momentum tensor}  &  \n\\lw{equation(s)} & \\lww{\\footnotesize kinetic parameter obtained} & \n\\lww{\\footnotesize thermodynamical} \\\\ \n{\\footnotesize and net baryon current} & \n&{\\footnotesize (distribution. function.)} &{\\footnotesize and fluid quantity} \\\\ \n\\hline \\hline \n$\\varepsilon=u_{\\mu}T^{\\mu\\nu}u_{\\nu}$ &&$\\alpha$&\n\\lw{$\\varepsilon_{\\rm eq}, n_{\\rm eq}$ } \\\\ \\cline{1-1}\\cline{3-3}\n$n=N^{\\mu}u_{\\mu}$&\\lw{eq.(61)}&$\\beta$& \\\\ \\cline{1-1}\\cline{3-4}\n\\lw{$-\\frac{1}{3}\\Delta_{\\mu\\nu}T^{\\mu\\nu}\\equiv (1+\\gamma)P_{\\rm eq}$} \n&&$\\gamma$&\\lw{$\\Lambda,\\delta n, \\Pi$} \\\\ \\cline{3-3} \n&&$\\lambda_{\\bf k}^*(\\alpha,\\beta;\\gamma)$&\\\\ \\hline \n$T^{\\mu\\nu} u^{\\nu}=\\varepsilon u^{\\mu}$ &eq.(9a) &$u^{\\mu}$& $u^{\\mu}$ \\\\ \\hline\n$\\Delta^{\\mu}_{\\lambda}T^{\\lambda}_{\\sigma}u^{\\sigma}\\equiv 0$\n& ~{\\footnotesize Landau frame}~& --- & $W^{\\mu}\\equiv 0$ \\\\ \\hline\n$N^{\\langle\\mu\\rangle}$&~{\\footnotesize eq.(59a) and (59b)}~&\n$\\lambda_{\\bf k}^{*\\langle\\mu\\rangle}$ & $V^{\\mu}$ \\\\ \\hline\n$T^{\\langle\\mu\\nu\\rangle}$&~{\\footnotesize eq.(59c) and (59d)}~\n&$\\lambda_{\\bf k}^{*\\langle\\mu\\nu\\rangle}$ & $\\pi^{\\mu\\nu}$\\\\ \\hline\n\\end{tabular} \n\\end{table}\n\\end{center}\n\\section{Numerical results and discussion}\\label{sec:3}\nIn this section, we demonstrate the separation of the corresponding \nequilibrium energy density $\\varepsilon_{\\rm eq}(T,\\mu)$ and \nnet baryon density  $n_{\\rm eq}(T,\\mu)$ from \nthose in an off-equilibrium state ($\\varepsilon$ and $n$) \nprovided that \nthe strength of the off-equilibrium state $\\gamma$ is fixed \\footnote{\nIn this case, $\\alpha$ and $\\beta$ are obtained by solving eqs.(\\ref{eq:formulation_summary1}) \nand (\\ref{eq:formulation_summary2}) simultaneously with using fixed $\\gamma$. \nThe bulk pressure is then obtained by $\\Pi=\\gamma P_{\\rm eq}(\\alpha,\\beta)$. \nTherefore, the demonstration presented here corresponding to finding the \nseparation temperature $T$ and chemical potential $\\mu$  \nfor fixed $u_{\\mu}T^{\\mu\\nu}u_{\\nu}$ and $N^{\\mu}u_{\\mu}$ \nwith varying $-\\frac{1}{3} \\Delta_{\\mu\\nu} T^{\\mu\\nu}$.}.  \nThe boundary conditions (for $\\gamma\\to 0$ limit, i.e., $\\Lambda=\\delta n=0$)  \nof the separation temperature and chemical potential, \n(which correspond to equilibrium temperature and chemical potential, \nrespectively, because of $\\gamma=0$)  \nare determined by the Landau matching condition:\n\\begin{eqnarray}\n    \\varepsilon=\\varepsilon_{\\rm eq}(\\alpha_0,\\beta_0), \\quad \n    n=n_{\\rm eq}(\\alpha_0,\\beta_0), \n    \\label{eq:boundary_condition2} \n\\end{eqnarray}\nwhere $\\alpha_0\\equiv \\mu_0\/T_0$, $\\beta_0\\equiv 1\/T_0$. \nFor all numerical results shown, \nthe classical particle mass $m$ is 5 MeV.  \n\nThe extended matching conditions now reads   \n\\begin{subequations}\\label{eq:lambda-deltan_2}\n\\begin{eqnarray}  \n   \\varepsilon-\\varepsilon_{\\rm eq}(\\alpha,\\beta)\n&=&\\Lambda(\\alpha,\\beta,\\gamma),  \\label{eq:simultanous_eq1}\\\\\n   n-n_{\\rm eq}(\\alpha,\\beta)\n&=& \\delta n(\\alpha,\\beta,\\gamma). \\label{eq:simultanous_eq2}\n\\end{eqnarray} \n\\end{subequations}\nTherefore, determination of the separation energy density and net baryon density is \nequivalent to solving the nonlinear simultaneous equations concerning \n$\\alpha$ and $\\beta$ for given $\\varepsilon$, $n$, and with fixing $\\gamma$.    \nWe require that total energy and \nnet baryon number in the fluid cell should be \nindependent of $\\gamma$; i.e., \nalthough $\\gamma$ changes form of the distribution function, \nthe total energy and net baryon number in the local fluid cell are \nset to be unchanged.   \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=14cm]{Fig1.eps}\n\\caption{\\label{fig:1} \nThe separation chemical potential $\\mu$ [GeV] (upper panels (a), (b), and (c))\nand separation temperature $T$ [GeV] (lower panels (d), (e) and (f)) for \nan off-equilibrium fluid as functions of $\\gamma=\\Pi\/P_{\\rm eq}$.\nThe $T_0$=0.2 GeV for figure (a) and (d), $T_0$=0.4 GeV, for (b) and (e), and  \n$T_0$=0.6 GeV for (c) and (f), while the $\\mu_0$=5.0 MeV is fixed for all panels.}\n\\includegraphics[width=14cm]{Fig2.eps}\n\\caption{\\label{fig:2} The same as Fig.1 but $\\mu_0$=25 MeV.}\n\\end{center}\n\\end{figure}\nFigures \\ref{fig:1} and \\ref{fig:2} show the separation temperature $T$ \nand separation chemical potential $\\mu$ as a function of \n$\\gamma=\\Pi\/P_{\\rm eq}$,\n$\\mu_0$=5.0 MeV (Fig.\\ref{fig:1}) and $\\mu_0$=25 MeV (Fig.\\ref{fig:2}). \n$T_0$ is fixed at 200 MeV [panels (a) and (d) of Figs.\\ref{fig:1} and \\ref{fig:2}], \n400 MeV  [panels (b) and (e) of Figs.\\ref{fig:1} and \\ref{fig:2}], and \n600 MeV  [panels (c) and (f) of Figs.\\ref{fig:1} and \\ref{fig:2}]. \nFor very small $\\gamma$ region ($\\gamma \\lesssim 10^{-4}$) \nin both Figs. \\ref{fig:1} and \\ref{fig:2}, one can observe \nthat  $T$ and $\\mu$ are almost constant with $T\\approx T_0$ and $\\mu\\approx\\mu_0$. \nIn this region, the Landau matching condition works well. \nHowever, in the region $\\gamma \\gtrsim 10^{-3}$, \nthe separation temperature $T$ decreases with increasing $\\gamma$ and \nthe separation chemical potential increases. \nThe reason for the decrease of the separation temperature $T$ is \nthat the total energy of the fluid is partially used in non-thermal motion \nsuch as tiny scale turbulent flow as discussed in Section \\ref{sec:1}. \nOn the other hand, the separation chemical potential \nincreases as $\\gamma$ increases for an off-equilibrium system. \nThis is because \nof the constraint of conservation of the net-baryon number, \ni.e., the difference between the number of particles $n_+$ \nand anti-particles $n_-$ must be fixed while the total number of \nparticles and anti-particles $n_+ + n_-$ decreases due to the decrease in\ntemperature $T$.  \n\\begin{figure}\n\\begin{center} \n\\includegraphics[width=11cm]{Fig3.eps}\n\\caption{\\label{fig:3} The $\\gamma$ dependence of the factor \n$\\chi$ eq.(\\ref{eq:chi_final}) with the equilibrium temperature \n$T_0$=200, 400, and 600 MeV, and the equilibrium chemical potential \n$\\mu_0$=5 MeV (panel (a)) and for $\\mu_0$=25 MeV (panel (b)). \nIn the limit of $\\gamma\\to 0$ and $\\mu_0\\to 0$, the $\\chi$ \napproaches to $-3$.}\n\\end{center} \n\\end{figure}\nFigure \\ref{fig:3} shows the $\\gamma$ dependence of the $\\chi$ \nparameter for different $\\mu_0$ and $T_0$. \nIn the limit $\\mu_0\\to0$ $\\chi$ approaches $-3$ \nbecause $\\delta \\chi \\to 0$. \nWe have already reported the result $\\Lambda=3\\Pi$ for \nthe baryon free case in the NeXDC correspondence \\cite{PhysRevC.77.044903}, \nas well as the hydrodynamical model in the presence of a long-range \ncorrelation \\cite{PhysRevC.81.024907}. \nWhen the strength of the off-equilibrium state \n$\\gamma\\equiv \\Pi\/P_{\\rm eq}$ is larger than \naround $\\sim 1\/100$, the value of $\\chi$ increases rapidly. \nThis is because the bulk pressure gap between $\\Pi_+$ \nand $\\Pi_-$ becomes large in the region of $\\gamma > 1\/100$. \nRecall that the parameter $\\chi$ is derived from $\\xi$ ($\\xi=\\chi\\beta\\Pi$), \nwhich is \nthe residual term neglected by truncation \nof the energy polynomial function $P_{{\\bf k}n}^{(l)}$.\nThis means that \nnot only quadratic, but also higher order energy \ndependences of both $\\lambda_{\\bf k}$ and $\\bar{\\lambda}_{\\bf k}$ \nplay an important role in the thermodynamical stability of the \noff-equilibrium entropy current. \nThese higher order contributions \nin the energy polynomial function may result in the existence of the \ntiny turbulent flow in the fluid. \nFor the finite net baryon number case, we have also obtained \nthe same expression as given in eq.(\\ref{eq:stability}) \n\\cite{PhysRevC.85.014906,OsadaEPJA48}. We found there that the value of $\\chi$ plays \nan important role to restore not only the thermodynamical stability but also \nthe causality of the solution obtained from the relativistic dissipative fluid \ndynamical equations. \nIn ref.\\cite{PhysRevC.85.014906,OsadaEPJA48}, the value of $\\chi$ is introduced \nphenomenologically into the expression for an off-equilibrium entropy current, \nhowever, in this article, we illustrate the result numerically using kinetic theory \n as shown in Fig.\\ref{fig:3}.\n\nIn figures \\ref{fig:4}, it is shown that the first order (scalar) correction $\\lambda_{\\bf k}^*$ \nto the distribution function\nfor $\\gamma = 10^{-3}$ and $\\gamma=10^{-2}$\nwith $T_0$ =200, 400, and 600 MeV and $\\mu_0$=5.0 MeV.  \nAs seen in Figs.\\ref{fig:4}, \nthe off-equilibrium correction increases more rapidly in the large momentum region. \nThis tendency is seen more explicitly as $\\gamma$ increases and as \nthe separation temperature $T$ decreases. \nIn addition to the correction $\\lambda_{\\bf k}^*$, \none need to take vector and tensor contribution, i.e., \n$\\lambda_{\\bf k}^{*\\langle\\mu\\rangle}k_{\\langle\\mu\\rangle}$ and \n$\\lambda_{\\bf k}^{*\\langle\\mu\\nu\\rangle}k_{\\langle\\mu}k_{\\nu\\rangle}$, respectively,  \ninto account to estimate the correction for the distribution function. \nHowever, to do this, one need set full energy-momentum tensor and net-charge \ncurrent vector and this is outside the scope of the present manuscript. \nWe plan to discuss this subject elsewhere. \n\\begin{figure} \n\\begin{center}\n\\includegraphics[width=16cm]{Fig4.eps}\n\\caption{\\label{fig:4}\nThe off-equilibrium distribution function $f_{{\\bf k}0}+\\delta f_{\\bf k} =\n(1+\\lambda_k^*)f_{{\\bf k}0}$ \nfor $\\mu_0$= 5 MeV, \n$T_0$= 200 MeV (panel (a)), 400 MeV (panel (c)), \nand 600 MeV (panel (e)) in the case of $\\gamma=10^{-3}$.\nThe right side panels (b), (d) and (f) are the same as \n(a), (c) and (e), respectively, but for $\\gamma=10^{-2}$.   \nThe straight line (with closed circle) \nis the Boltzmann distribution $f_{{\\bf k}0}$ with \nthe separation temperature $T$, and the separation chemical potential \n$\\mu$ are shown in each panel. The open circles shows the off-equilibrium \ndistribution function $f_{{\\bf k}0}+\\delta f_{\\bf k}$.} \n\\end{center}\n\\end{figure}\n\\section{Summary and concluding remarks}\\label{sec:4} \nWe have proposed a novel way to specify the initial conditions for a\ndissipative fluid dynamical model as an alternative to the so-called Landau matching \ncondition eq.(\\ref{eq:normal-matching1}), employed in most of the literature published so\nfar. \nThe large expansion rate of matter produced in ultra-relativistic heavy-ion \ncollisions may prevent the local equilibrium condition eq.(\\ref{eq:perfect_condition})\nfrom holding true.  Specifically, the microscopic collision time scale is \nnot much shorter than any macroscopic evolution time scale \nbecause the macroscopic evolution time scale  is considerably \nreduced due to the large gradients in thermodynamical \nquantities when compared to a perfect fluid. \nIn cases characterized by eq.(\\ref{eq:inparfect_condtion}), \nthe distribution functions are defined by eq.(\\ref{eq:modification_in_distribution}) \n\\begin{eqnarray*}\n   f_{\\bf k}(x)= f_{{\\bf k}0}[T(x),\\mu(x)] + \\delta f_{\\bf k}(x), \\\\ \n   \\bar{f}_{\\bf k}(x)= \\bar{f}_{{\\bf k}0}[T(x),\\mu(x)] + \\delta \\bar{f}_{\\bf k}(x). \n\\end{eqnarray*}\nFor a given off-equilibrium state specified by the energy-momentum \ntensor given in eq.(\\ref{eq:energy-momentum_tensor})\nand net baryon number current defined according to eq.(\\ref{eq:particle_current}), \nthe Landau matching procedure equals \nthe energy density and net baryon number density of the \noff-equilibrium state with those in an equilibrium of  the form \n$T_{\\rm eq}^{\\mu\\nu}$ and $N_{\\rm eq}^{\\mu}$,   \n\\begin{eqnarray*}\n   u_{\\mu}T^{\\mu\\nu}u_{\\nu} = u_{\\mu}T^{\\mu\\nu}_{\\rm eq} u_{\\nu}, \\quad \n   N^{\\mu}u_{\\mu} = N^{\\mu}_{\\rm eq} u_{\\mu}, \n\\end{eqnarray*}\nwhich imposes strict restrictions on the distributions $\\delta f_{\\bf k}(x)$ and $\\delta\n\\bar{f}_{\\bf k}(x)$. \nAlthough the physical meaning of the matching (fitting) condition is unclear,  \nit provides the temperature $T$ and chemical potential $\\mu$ \nfor a given off-equilibrium energy density $\\varepsilon=\\varepsilon_{\\rm eq}(T,\\mu)$ \nand net baryon density $n=n_{\\rm eq}(T,\\mu)$. \nOne may regard the Landau matching condition as {\\it just a definition} \nfor specifying an off-equilibrium state.\nHowever, it should be noted here that other off-equilibrium \nquantities such as the bulk pressure $\\Pi$ are undoubtedly \ndistorted by the condition. \nTherefore, we assert that the Landau matching condition be \n relaxed as shown in eq.(\\ref{eq:extended-matching}), \n\\begin{eqnarray*}\n     \\delta T^{\\mu\\nu} u_{\\mu} u_{\\nu} \n     =\\Lambda\\ne 0, \\quad \\delta n^{\\mu} u_{\\mu} =\\delta n \\ne 0. \n\\end{eqnarray*}  \nThen, $\\Lambda$ and $\\delta n$ obtain their entities of physical existence, \ni.e., they are the energy density and baryon number density of \nthe tiny scale (less than cell size scale) \ndisturbance flow caused by the rapid expansion of the viscous fluid.  \n\nAn important consequence of the irreducible tensor expansion \nfor the off-equilibrium distribution function is that \n scalar quantities such as $\\Lambda$, $\\delta n$, and $\\Pi$ can \nbe written as functions of the separation temperature $T$, \nchemical potential $\\mu$, and the 0-th order scalar expansion coefficients \n$\\lambda_{\\bf k}^*$ (and $\\bar{\\lambda}_{\\bf k}^*$).\nAnother important consequence is that the thermodynamical stability condition \nrequires a constraint of the scalar off-equilibrium thermodynamical \nquantities  $\\Lambda$, $\\delta n$, \nand $\\Pi$.\nThis conditions practically determine both $\\chi_+$ and $\\chi_-$ \n(and thus also $\\chi$), which are contributions neglected by the \ntermination of polynomial function in the 0-th order irreducible tensor. \nBy combing with the `equation of state' for off-equilibrium state \neq.(\\ref{eq:off-equilibrium_EoS}), \nthe thermodynamical stability condition eq.(\\ref{eq:stability}) \nalso gives the differential equation eq.(\\ref{eq:differential_fq_for_lambda}) for $\\lambda_{\\bf k}^*$ \nexcept for uncertainty in the integration constant $C_{\\gamma}$.\nAs seen in the solutions to the differential equations,  \none can observe that $\\chi_+$ and $\\chi_-$, which can be fixed by the \nthermodynamical stability condition, play an important role in the \nenergy dependence of $\\lambda_{\\bf k}^*$.\nNote that, the integration constant $C_{\\gamma}$ defines a scale of \n$\\lambda_{\\bf k}^*$,\nor in other words, \nthe constant $C_{\\gamma}$ regulates the `strength' of \nthe off-equilibrium state. This constant can be fixed by \nan index $\\gamma\\equiv \\Pi\/P_{\\rm eq}$. \n\nAlthough $\\alpha$, $\\beta$, and $\\gamma$ are to be obtained as \nsolution of eq.(\\ref{eq:formulation_summary}), \nwe have investigated $\\gamma$ dependence on the \nseparation temperature $T$ and $\\mu$ in a case that \n$u_{\\mu}T^{\\mu\\nu} u_{\\nu}$ and $N^{\\mu} u_{\\mu}$ are fixed \nand with varying $-\\frac{1}{3} \\Delta_{\\mu\\nu} T^{\\mu\\nu}$.\nIn the case, the separation temperature $T$ and \nchemical potential $\\mu$ obtained \nshow (see Fig.\\ref{fig:1} and \\ref{fig:2}) that $T\\approx T_0$ and \n$\\mu\\approx\\mu_0$ hold for $\\gamma < 10^{-3}$. \nIn this region, the Landau matching condition \napproximately holds true. However, for $\\gamma > 10^{-3}$, \n$T\\ne T_0$ and $\\mu\\ne\\mu_0$, i.e., the Landau matching condition is inappropriate and\nits modification is required.\n\nIn summary, although the Landau matching condition \ndetermines the matching (fitting) temperature and \nchemical potential by using off-equilibrium energy density \nand baryon number density, these are not sufficient to \nspecify the off-equilibrium state. \nIn this article, we have introduced an index, \nnamely, the strength of the off-equilibrium state \n$\\gamma\\equiv\\Pi\/P_{\\rm eq}(T,\\mu)$ in order to specify \nthe state considered more precisely. \nThe bulk pressure $\\Pi$ may relate to the disturbance flow in a fluid cell \nand approaches to zero \nwhen the system relaxes to the local equilibrium state. \nSince the fluid dynamics is a deterministic theory, \nto set an initial condition of the system is crucially important \nand it should be done very carefully. \nIn the literature to date, a constitutive equation \nfor the bulk pressure is solved using a likely value under the Landau matching condition \n($\\Lambda \\equiv 0$ and $\\delta n\\equiv 0$). \nHowever, it must be inaccurate. \nThe initial conditions (separation temperature $T$, \nchemical potential $\\mu$ and bulk pressure $\\Pi$) \nshould be determined from the given off-equilibrium energy \ndensity $\\varepsilon=u_{\\mu}T^{\\mu\\nu}u_{\\nu}$, \nnet baryon number density \n$n=N^{\\mu}u_{\\mu}$, \nand the dissipative strength $\\gamma$, \nas proposed in this article, i.e., eq.(\\ref{eq:formulation_summary}).  \n\\section*{Acknowledgment}\nThe author would like to thank \nProf. Grzegorz Wilk and Prof. Takeshi Kodama \nfor helpful comments and discussions. \nThe author acknowledges U. Heinz for crucial criticism concerning the \noriginal idea of this paper, first presented at the 9th Relativistic Aspects \nof Nuclear Physics (RANP 2013) conference at Rio de Janeiro, Brazil.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEffective information analysis generally boils down to the geometry of the data represented by a graph. Typical applications include social networks\\cite{lazer2009life}, transportation networks\\cite{bell1997transportation}, spread of epidemic disease\\cite{newman2002spread}, brain's neuronal networks\\cite{marx2012high}, gene data on biological regulatory networks\\cite{davidson2002genomic}, telecommunication networks\\cite{drew2008diagnosing}, knowledge graph\\cite{lin2015learning}, which are lying on non-Euclidean graph domain. To describe the geometric structures, graph matrices such as adjacency matrix or graph Laplacian can be employed to reveal latent patterns.\n\n\\begin{figure}[h]\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.68in]{img\/spike_1.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.68in]{img\/spike_5.png}\n\t\\end{subfigure}\\\\\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.68in]{img\/spike_6.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.68in]{img\/spike_8.png}\n\t\\end{subfigure}\n\t\\caption{Rational(rat) and polynomial(poly) approximation for several jump functions(func). \n\t\tTop left:$\\sqrt{|x-0.5|}$; top right: $|x-0.5|$; bottom left: $\\frac{x}{10|x - 0.5|+1}$; bottom right:    $max(0.5, sin(x+x^{2}))-\\frac{x}{20}$}\n\t\\label{fig:fit_example}\n\\end{figure}\nIn recent years, many problems are being revisited with deep learning tools. Convolutional neural networks(ConvNets) emerging in recent years are at the heart of deep learning, and the most prominent strain of neural networks in research. ConvNets have revolutionized computer vision\\cite{krizhevsky2012imagenet}, natural language processing\\cite{collobert2008unified}, computer audition\\cite{graves2013speech}, reinforcement learning\\cite{mnih2015human,silver2016mastering}, and many other areas. However, ConvNets are designed for grid data such as image, which belongs to the Euclidean domain. Graph data is non-Euclidean which makes it difficult to employ typical ConvNets.\nTo bridge the gap, Bruna et al. \\cite{bruna2013spectral}\\cite{henaff2015deep} generalized spectral convolutional operation which requires expensive steps of spectral decomposition and matrix multiplication. Hammond et al.\\cite{hammond2011wavelets} first introduced truncated Chebyshev polynomial for estimating wavelet in graph signal processing. Based on this polynomial approximation, Defferrard et al.\\cite{defferrard2016convolutional} designed ChebNet which contains a novel neural network layer for the convolution operator in the spectral domain. Kipf and Welling\\cite{kipf2017semi} simplified ChebNet by assuming the maximum of eigenvalues is 2 and fixing the order to 1, which boosts both effectiveness and efficiency. Li et al.\\cite{DBLP:journals\/corr\/abs-1801-07606} found that this simplified ChebNet is an application of Laplacian smoothing, which implies that current studies are only effective on the smooth signal. \n\n\n\n\nCurrent studies on graph ConvNet heavily rely on polynomial approximation, which makes it difficult to estimate jump signals.  Fig. \\ref{fig:fit_example} shows the behaviors of polynomial and rational function on jump discontinuity: rational approximation fits the functions considerably better than polynomials. It is widely recognized that \\textbf{(1)} polynomial approximation suffers from Gibbs phenomenon, which means polynomial function oscillate and overshoot near discontinuities\\cite{trefethen2013approximation}; \\textbf{(2)} Applying a higher order of polynomials could dramatically reduce the oscillation, but also incurs an expensive computational cost. \\textbf{(3)} Polynomials require degree $\\Omega$(poly(1\/$\\epsilon$)) to approximate functions near singularities and on an unbounded domain, while rational functions only need $\\mathcal{O}$(poly log(1\/$\\epsilon$)) to achieve $\\epsilon$-close\\cite{liang2016deep,telgarsky2017neural}. However, it is non-trivial to apply rational approximation. Polynomial-based method can easily transfer the function on eigenvalues to the same function on graph Laplacian so that matrix multiplication by eigenvector can be avoided. It is not easy for rational approximation to do so due to the additional denominator. \n\n\nIn this paper, the advantage of the rational function in approximation is transferred to spectral graph domain. Specifically, we propose a rational function based neural networks(RationalNet), which can avoid matrix multiplication by eigenvectors. To alleviate the local minimum problem of the neural network, a relaxed Remez algorithm is employed for parameter initialization. Our theoretical analysis shows that rational functions converge much faster than polynomials on jump signal. In a nutshell, the key innovations are:\n\n\\begin{itemize}\n\t\\item \\textbf{Propose a neural network model based on rational function for recovering jump discontinuities}: To estimate the jump signal, our proposed method integrates rational approximation and spectral graph operation to avoid matrix multiplication by eigenvectors. For graph signal regression task, expensive matrix inversion can be circumvented by graph Fourier transform.\n\t\\item \\textbf{Develop an efficient algorithm for model parameters optimization}: Remez algorithm is theoretically optimal, but it is often not practical especially when approximating discrete signal. To alleviate this issue, the stopping rules of Remez algorithm are relaxed to initialize the neural networks parameters.\n\t\\item \\textbf{Provide theoretical analysis for the proposed method on jump signal}: For understanding the behaviors of polynomial and rational function on jump discontinuities, a uniform representation is proposed to analyze convergence rate regarding the order number theoretically.\n\t\\item \\textbf{Conducting extensive experiments for performance evaluations}\\footnote{The code and datasets will be released after the acceptance for replication}: The proposed method was evaluated on synthetic and real-world data. Experimental results demonstrate that the proposed approach runs efficiently and consistently outperforms the best of the existing methods.\n\\end{itemize}\n\nThe rest of the paper is organized as follows. Section \\ref{sec:related_work} reviews existing work in this area. Necessary preliminary is presented in section \\ref{sec:prelim}. Section \\ref{sec:method} elaborates a rational function based model for graph signal regression task. Section \\ref{sec:algorithm} presents algorithm description and theoretical analysis. In Section \\ref{sec:evaluation}, experiments on synthetic and real-world data are analyzed. This paper concludes by summarizing the study's important findings in Section \\ref{sec:conclusion}.\n\n\\section{Related Work}\\label{sec:related_work}\nThe proposed method draws inspiration from the field of approximation theory, spectral graph theory and recent works using neural networks. In what follows, we provide a brief overview of related work in these fields.\n\n\\subsection{Approximation theory}\nIn mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator, such that the result is as close to the actual function as possible. This is typically done with polynomial or rational approximations. Polynomials are familiar and comfortable, but rational functions seem complex and specialized, and rational functions are more powerful than polynomials at approximating functions near singularities and on unbounded domains. Basic properties of rational function are described in books of complex analysis\\cite{ahlfors1953complex,trefethen2013approximation,pachon2010algorithms,powell1981approximation,cohen2011numerical,petrushev2011rational,achieser2013theory,ziegel1987numerical,boyd2001chebyshev,mason2002chebyshev,remez1934determination}. \n\n\\subsection{Spectral graph theory}\nSpectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix\\cite{chung1997spectral,grone1990laplacian,das2004laplacian}. Many graphs and geometric convolution methods have been proposed recently. The spectral convolution methods (\\cite{bruna2013spectral,defferrard2016convolutional,kipf2017semi,bronstein2017geometric}) are the mainstream algorithm developed as the graph convolution methods. Because their theory is based on the graph Fourier analysis (\\cite{shuman2013emerging, shuman2016vertex}). The polynomial approximation is firstly proposed by \\cite{hammond2011wavelets}. Inspired by this, graph convolutional neural networks (GCNNs) (\\cite{defferrard2016convolutional}) is a successful attempt at generalizing the powerful convolutional neural networks (CNNs) in dealing with Euclidean data to modeling graph-structured data. Kipf and Welling proposed a simplified type of GCNNs\\cite{kipf2017semi}, called graph convolutional networks (GCNs). The GCN model naturally integrates the connectivity patterns and feature attributes of graph-structured data and outperforms many state-of-the-art methods significantly. \nLi et al.\\cite{DBLP:journals\/corr\/abs-1801-07606} found that GCN is actual an application of Laplacian smoothing, which is inconsistent with GCN's motivation. In sum, this thread of work calculates the average of vertexes within Nth-order neighbors.\n\nIn this paper, we focus on the effectiveness of approximation technique on graph signal. Under graph signal regression task, the superiority of rational function beyond polynomial function is analyzed, and a rational function based neural network is proposed. \n\n\\section{Preliminaries}\\label{sec:prelim}\n\nWe focus processing graph signals defined on undirected graphs $\\mathcal{G} = (\\mathcal{V}, \\mathcal{E}, \\mathcal{W})$, where $\\mathcal{V}$ is a set of n vertexes, $\\mathcal{E}$ represents edges and\n$\\mathcal{W} = [w_{ij}] \\in \\{0,1\\}^{n\\times n}$ is an unweighted adjacency matrix. A signal $x : \\mathcal{V} \\rightarrow \\mathbb{R}$ defined on the nodes may be regarded as a vector $x \\in \\mathbb{R}^{n}$.\nCombinatorial graph Laplacian\\cite{chung1997spectral} is defined as $\\Ll= D-\\mathcal{W} \\in \\mathbb{R}^{n\\times n}$ where $D$ is degree matrix.\n\nAs $\\Ll$ is a real symmetric positive semidefinite matrix, it has a\ncomplete set of orthonormal eigenvectors and their associated ordered real nonnegative eigenvalues identified as the frequencies of the graph. The Laplacian is diagonalized by the Fourier basis $\\UT$: $\\Ll = \\Uu \\Lambda \\UT$ where $\\Lambda$ is  the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e., ${\\displaystyle \\Lambda _{ii}=\\lambda _{i}}$. The graph Fourier transform of a signal $x\\in \\mathbb{R}^{n}$ is defined as $\\hat{x}=\\UT x \\in \\mathbb{R}^{n}$ and its inverse as $x=\\Uu \\hat{x}$\\cite{shuman2013emerging, shuman2016vertex, zhu2012approximating}. To enable the formulation of fundamental operations such as filtering in the vertex domain, the convolution operator on graph is defined in the Fourier domain such that $f_{1}*f_{2}=\\Uu \\left[\\left(\\UT f_{1} \\right) \\odot \\left(\\UT f_{2}\\right)\\right]$, where $\\odot$ is the element-wise product, and $f_{1}\/f_{2}$ are two signals defined on vertex domain. It follows that a vertex\nsignal $f_{2}=x$ is filtered by spectral signal $\\hat{f_{1}}=\\UT f_{1}=\\g$ as:\n\\begin{equation*}\n\\small\n\\g * x = \\Uu \\left[\\g(\\LBD)\\odot \\left(\\UT f_{2}\\right)\\right] = \\Uu \\g(\\LBD) \\UT x.\n\\end{equation*}\nNote that a real symmetric matrix $\\Ll$ can be decomposed as $\\Ll=\\Uu \\LBD \\Uu^{-1} =  \\Uu \\LBD \\UT$ since $\\Uu^{-1}=\\UT$ . D. K. Hammond et al. and Defferrard et al.\\cite{hammond2011wavelets, defferrard2016convolutional} apply polynomial approximation on spectral filter $\\g$ so that:\n\\begin{align*}\n\\small\n& \\g * x = \\Uu \\g (\\LBD) \\UT x &&\\\\\n\\approx&\\Uu \\sum_{k}^{}\\theta_{k} T_{k}(\\tilde{\\LBD}) \\UT x && {(\\tilde{\\LBD}=\\frac{2}{\\lambda_{max}}\\LBD-\\I_{\\N})}\\\\\n=&\\sum_{k}^{}\\theta_{k} T_{k}(\\tilde{\\Ll}) x &&{(\\Uu\\LBD^{k}\\UT=(\\Uu\\LBD\\UT)^{k})}\\\\\n\\end{align*}\nKipf and Welling\\cite{kipf2017semi} simplifies it by:\n\\begin{align*}\n\\small\n&\\g * x &&\\\\\n\\approx& \\theta_{0}\\I_{\\N}x+\\theta_{1}\\tilde\\Ll x &&({\\scriptstyle\\text{expand to 1st order)}}\\\\\n=&\\theta_{0}\\I_{\\N}x+\\theta_{1}(\\frac{2}{\\lambda_{max}}\\Ll-\\I_{\\N})) x &&{\\scriptstyle(\\tilde\\Ll=\\frac{2}{\\lambda_{max}}\\Ll-\\I_{\\N}))} \\\\\n=&\\theta_{0}\\I_{\\N}x+\\theta_{1}(\\Ll-\\I_{\\N})) x &&{\\scriptstyle(\\lambda_{max}=2)} \\\\\n=&\\theta_{0}\\I_{\\N}x-\\theta_{1} \\Dn\\A\\Dn x &&{\\scriptstyle(\\Ll=\\I_{\\N}-\\Dn\\A\\Dn)} \\\\\n=&\\theta_{0}(\\I_{\\N} + \\Dn\\A\\Dn) x &&{\\scriptstyle(\\theta_{0}=-\\theta_{1})} \\\\\n=&\\theta_{0}(\\tilde\\D^{-\\frac{1}{2}}\\tilde\\A\\tilde\\D^{-\\frac{1}{2}}) x &&{\\scriptstyle(\\text{renormalization}: \\tilde\\A=\\A+\\I_{\\N},}\\\\\n&&&{\\scriptstyle \\tilde\\D_{ii}=\\sum_{j} \\A_{ij})}, \\\\\n\\end{align*}rewrite the above GCN in matrix form: $\\gt*X \\approx(\\tilde\\D^{-\\frac{1}{2}}\\tilde\\A\\tilde\\D^{-\\frac{1}{2}}) X\\Theta$, which leads to \\textit{symmetric normalized Laplacian} with raw feature. GCN has been analyzed GCN in \\cite{DBLP:journals\/corr\/abs-1801-07606} using smoothing Laplacian\\cite{taubin1995signal}:\n$y= (1-\\gamma) x_{i} + \\gamma \\sum_{j}\\frac{\\tilde a_{ij}}{d_{i}}x_{j} \n= x_{i} - \\gamma (x_{i}-\\sum_{j}\\frac{\\tilde a_{ij}}{d_{i}}x_{j}),\n$where $\\gamma$ is a weight parameter between the current vertex $x_{i}$ and the features of its neighbors $x_{j}$, $d_{i}$ is degree of $x_{i}$, and $y$ is the smoothed Laplacian. This smoothing Laplacian has a matrix form:\n\\begin{align*}\nY&= x -\\gamma \\tilde\\D^{-1}\\tilde\\Ll x&& \\\\\n&= (\\I_{\\N}-\\tilde\\D^{-1}\\tilde\\Ll )x && (\\gamma=1)\\\\\n&= (\\I_{\\N}-\\tilde\\D^{-1}(\\tilde\\D-\\tilde\\A))x && (\\tilde\\Ll=\\tilde\\D-\\tilde\\A)\\\\\n&= \\tilde\\D^{-1}\\tilde\\A x.&& \\label{eq:smooth_laplacian}\n\\end{align*}\n\nThe above formula is \\textit{random walk normalized Laplacian} as a counterpart of \\textit{symmetric normalized Laplacian}. Therefore, GCN is nothing but a first-order Laplacian smoothing which averages neighbors of each vertex.\n\n\\section{Model description}\\label{sec:method}\nThis section formally defines the task of graph signal recovering and then describes our proposed RationalNet which aims to characterize the jump signal in spectral domain.\n\n\\subsection{Problem Setting}\\label{sec:ps}\nAll the related works integrate graph convolution estimator and fully-connected neural layers for a classification task. This classification can be summarized as:\n\\begin{equation}\nY=f(\\mathcal{G}, x)\\Theta,\n\\label{classfication_task}\n\\end{equation}where $\\Theta$ indicates the parameters of normal neural network layers connecting the output of $f$ and the label $Y$, such as fully-connected layers and softmax layers for classification. And $f$ is a neural network layer implemented by approximation techniques. However, whether the success is due to the neural networks($\\Theta$) or the convolution approximation method($f$) remains unknown. To focus on the analysis of approximation on $f$, a graph signal regression task is proposed to evaluate the performance of the convolution approximators $f$. Regression task directly compares the label and the output of $f$, removing the distraction of $\\Theta$.\n\nGiven a graph $\\mathcal{G}$, raw feature $x$, and training signal on the part of vertexes, $Y_{train}$, our goal is to recover signal values, $Y_{test}$, on test nodes. Formally, we want to find a $f(\\cdot)$ so that: \n\n\\begin{equation*}\nY=f(\\mathcal{G}, x).\n\\end{equation*}\nIf the raw features are good enough for the regression task, whether the effectiveness is due to $f$ or $x$ is difficult to verify. Therefore, one reasonable option for $x$ is the uniform signal in spectral domain. Specifically, $x=\\sum_{i}^{} \\Uu_{i}$ and $\\hat{x}=\\UT x=\\mathds{1}=\\{1,1,...,1\\}$, which means that $x$ represents eigenbasis of graph structure in spectral domain. Each entry of vector $\\hat{x}_{i}$ indicates one eigenvector in the spectral domain. The physical meaning of the convolution operation is how to select eigenbasis in spectral domain to match the graph signal $Y$. Representing $\\mathcal{G}$ with graph Laplacian, the regress task can be rewritten as:\n\n\\begin{equation}\nY=f(\\Ll, \\Uu)=\\Uu\\gt(\\LBD)\\UT \\sum_{i}\\Uu,\n\\label{regression_task}\n\\end{equation}where $\\gt$ is the spectral filter to approximate. \n\n\n\\subsection{RationalNet}\nSimilar to polynomial approximation on graph domain such as ChebNet\\cite{defferrard2016convolutional} or GCN\\cite{kipf2017semi}, RationalNet approximates the spectral filter by a widely used type of rational function, i.e., Pad\\'{e} approximator, which is defined as:\n\\begin{equation}\n\\small\nR(x)=\\frac{\\sum_{i=0}^{m} \\psi_{i}x^{i}}{\\sum_{j=0}^{n}\\phi_{j}x^{j}}, \\phi_{0}=1, \\phi_{j},\\psi_{i}\\in \\mathbb{R}.\n\\label{eq:rat_define}\n\\end{equation}\n Applying to graph convolution operator, we have:\n\n\\begin{flalign*}\n\\small\n&\\gt * x=\\Uu \\gt \\UT x && (\\text{convolution theorem})\\\\\n\\approx&\\Uu \\frac{{\\mathlarger{\\sum}}_{i=0}^{m}\\psi_{i} \\tilde{\\LBD}^{i}}{1+{\\mathlarger{\\sum}}_{j=1}^{n}\\phi_{j} \\tilde{\\LBD}^{j}}  \\UT x &&  {(\\tilde{\\LBD}=\\frac{\\LBD}{\\lambda_{max}})} \\\\\n=&\\Uu \\frac{\\Pp(\\LBD)}{\\Qq(\\LBD)} \\UT x, &&(\\text{define P and Q})\n\\end{flalign*}where $\\LBD$ represents a diagonal matrix whose entries are eigenvalues, $\\gt=R$, and $\\theta=\\{\\psi, \\phi\\}$. The division $\\frac{P}{Q}$ is element-wise. The inverse of matrix Q(x) is equivalent to applying reciprocal operation on its diagonal entries, so the equation can be rewritten as:\n\\begin{equation*}\n\\Uu \\Pp(\\LBD) \\Qq(\\LBD)^{-1} \\UT x.\n\\end{equation*}Applying matrix rules, it's easy to have:\n\\begin{flalign*}\n\\small\n=&\\Uu \\Pp(\\LBD) \\UT \\Uu \\Qq(\\LBD)^{-1} \\UT x &&{\\scriptstyle(\\UT=\\Uu^{-1},\\UT\\Uu=\\I_{\\N})} \\\\\n=&\\big[\\Uu \\Pp(\\LBD) \\UT\\big] \\big[\\Uu\\Qq(\\LBD)^{-1} \\UT\\big] x && \\\\\n=&\\big[\\Pp(\\Uu \\LBD \\UT)\\big] \\big[\\Uu\\Qq(\\LBD)^{-1} \\UT\\big] x &&{\\scriptstyle(\\Uu\\LBD^{k}\\UT=(\\Uu\\LBD\\UT)^{k})} \\\\\n=&\\big[\\Pp(\\Uu \\LBD \\UT)\\big] \\big[(\\Qq(\\LBD)\\Uu^{-1})^{-1} \\UT\\big] x &&{\\scriptstyle(A^{-1}B^{-1}=(BA)^{-1})} \\\\\n=&\\big[\\Pp(\\Uu \\LBD \\UT)\\big] \\big[(\\Uu\\Qq(\\LBD)\\Uu^{-1})^{-1}\\big] x &&{\\scriptstyle(\\UT = \\Uu^{-1})} \\\\\n=&\\big[\\Pp(\\Uu \\LBD \\UT)\\big] \\big[(\\Uu\\Qq(\\LBD)\\UT)^{-1}\\big] x &&{\\scriptstyle(\\UT = \\Uu^{-1})} \\\\\n=&\\big[\\Pp(\\Uu \\LBD \\UT)\\big] \\big[(\\Qq(\\Uu\\LBD\\UT))^{-1}\\big] x &&{\\scriptstyle(\\Uu\\LBD^{k}\\UT=(\\Uu\\LBD\\UT)^{k})} \\\\\n\\end{flalign*} Since $\\Uu\\LBD^{k}\\UT=(\\Uu\\LBD\\UT)^{k}$, we can rewrite the equation above as:\n\\begin{equation}\n\\small\n\\gt * x = \\Pp(\\Ll) \\Qq(\\Ll)^{-1} x,\n\\label{eq:ratnet}\n\\end{equation} \nwhere $\\Pp (x)=\\sum_{i=0}^{m} \\psi_{i}x^{i}$ and $\\Qq (x)=\\sum_{j=0}^{n} \\phi_{j}x^{j}$. Note that $\\Pp(\\Ll) \\Qq(\\Ll)^{-1} x=\\Qq(\\Ll)^{-1} \\Pp(\\Ll)  x$ in our case. Based on polynomial approximation, RationalNet only adds a inverse polynomial function $\\Qq(\\Ll)^{-1}$. Therefore, polynomial approximation (GCN\/ChebNet) on graph is a special case of RationalNet when $\\Qq(\\Ll)^{-1}=\\I$. \n\nComputing inverse of matrix is still of high complexity $\\mathcal{O}(n^{3})$ (Gauss\u2013Jordan elimination method) in Eq. \\ref{eq:ratnet}, especially for large matrix. This can be avoided by transferring vertex graph signal and raw features into spectral domain. Therefore, the Eq. \\ref{eq:regress} can be rewritten as:\n\\begin{equation}\n\\hat{Y}=\\frac{\\Pp(\\LBD)}{\\Qq(\\LBD)} \\hat{x},\n\\label{eq:ratnet2}\n\\end{equation}where $\\hat{Y}=\\UT Y$ is the graph Fourier transform of graph signal, and  $\\hat{x}=\\UT x$ is the graph Fourier transform of raw feature. By this step, we only need compute reciprocal of eigenvalues, rather than computing matrix multiplication and  inversion at each update. Eq. 2 can be obtained via left multiplying both sides of Eq. \\ref{regression_task} by transpose of eigenvectors. Note that Eq. \\ref{eq:ratnet2} is applicable when there is no other layers between the output $Y$ and the convolution operation. In contrast, Eq. \\ref{eq:ratnet} can be used not only in regression task like Eq. \\ref{regression_task}, but also in classification where there exist additional neural networks as described in Eq. \\ref{classfication_task}. RationalNet has complexity $\\mathcal{O}(| \\mathcal{E}|)$ for Eq. \\ref{eq:ratnet2} and $\\mathcal{O}(| \\mathcal{E}|^{3})$ for Eq. \\ref{eq:ratnet}.\n\n\n\n\n\\subsection{Relaxed Remez Algorithm for initialization}\nRationalNet is powerful at approximating a function. However, it is often stuck in a local optimum due to the neural network optimization,  which makes rational function not always better than the polynomial approximation. Remez exchange algorithm\\cite{remez1934determination} is an iterative algorithm used to find simple approximations to functions. Nevertheless, the drawback of this minimax algorithm is that the order for optimum is unknown and the stopping condition is not often practical. To improve RationalNet's initialization, a relaxed Remez algorithm is proposed.  \n\nAs Theorem \\textbf{24.1} (equioscillation characterization of best approximants, \\cite{trefethen2013approximation}) states: Given the order of numerator(m) and denominator(n), and a real function $f$ that is continuous in [p, q], there exists a unique best rational approximation  $R^{*}$ defined as Eq. \\ref{eq:rat_define}. This means that the $R^{*}$ optimizes the minimax error:  \n\n\\begin{equation}\n\\small\nR^{*}= \\arg\\min_{R} \\max_{x\\in [p, q]} | f-R |.\n\\end{equation}\n\nA rational function $R$ is equal to $R^{*}$ iff $f-R$ equioscillates between at least m+n+2 extreme points, or say the error function attains the absolute maximum value with alternating sign:\n\n\\begin{equation}\n\\small\nf(x_{d})-R(x_{d})=(-1)^{d}E, d\\in [0, m+n+1],\n\\label{eq:equioscillates}\n\\end{equation}where $d$ indicates the index of data point, and $E$ represents the max of residuals: $E=max_{x_{d}}|f(x_{d})-R(x_{d})|$. For rational function approximation, there is some nonlinearity because there will be a product of $E$ with $\\phi_{j}$ in the equations. Hence, these equations need to be solved iteratively. The iteration formula can be defined by linearizing the equations:  \n\n{\\small\n    \\begin{equation}\n    \\sum ^{m}_{i=0}\\psi_{i}x^{i}_{d}-\\left[ f\\left( x_{d}\\right) -\\left( -1\\right) ^{d}E_{r}\\right] \\sum ^{k}_{j=1}\\phi_{j}x^{j}_{d}=f\\left( x_{d}\\right) -\\left( -1\\right) ^{d}E_{r+1},\n    \\label{eq:remez_mat}\n    \\end{equation}\n}where $r$ indicate the iteration index. Eq. \\ref{eq:remez_mat} is obtained by neglecting nonlinear terms of the form $(E_{r}-E_{r+1})\\phi_{j}x_{d}^{j}$ in Eq. \\ref{eq:equioscillates}. This procedure can converge in a reasonable time (\\cite{antiabook2004}). Expanding Eq. \\ref{eq:remez_mat} for all sampled data points $x_{0}, x_{1}, ..., x_{d}$, it can be rewritten as:\n\n{\\tiny\n    \\begin{gather}\n    \\begin{bmatrix} \n    x_{0}^{0} & ... &x_{0}^{m} & (E_{r}-y_{0})x_{0}^{1}  & ... & (E_{r}-y_{0})x_{0}^{n} & (-1)^{0} \\\\\n    x_{1}^{0} & ... &x_{1}^{m} & (E_{r}-y_{0})x_{1}^{1}  & ... & (E_{r}-y_{0})x_{1}^{n} & (-1)^{1} \\\\\n    x_{2}^{0} & ... &x_{2}^{m} & (E_{r}-y_{0})x_{2}^{1}  & ... & (E_{r}-y_{0})x_{2}^{n} & (-1)^{2} \\\\\n    ...\\\\\n    x_{K}^{0} & ... &x_{K}^{m} & (E_{r}-y_{0})x_{K}^{1}  & ... & (E_{r}-y_{0})x_{K}^{n} & (-1)^{K} \\\\\n    \\end{bmatrix}\n    \\begin{bmatrix} \n    \\psi_{0} \\\\ \\psi_{1} \\\\ \\psi_{2} \\\\ ..\\\\ \\psi_{m} \\\\ \\phi_{1} \\\\ \\phi_{2} \\\\ \\phi_{3} \\\\ .. \\\\ \\phi_{n} \\\\ E_{r+1}\n    \\end{bmatrix}\n    =\n    \\begin{bmatrix}\n    y_{0}\\\\ y_{1}\\\\ ... \\\\ y_{K}\n    \\end{bmatrix},\n    \\label{eq:remez_mat_expand}\n    \\end{gather}\n}where K=m+n+1. Starting from an assumed initial guess $E_{r=0}$, this set of linear equations can be solved for the unknown $\\psi_{i}$, $\\phi_{j}$ and $E_{r+1}$, when two successive values of $E_{r}$ are in satisfactory agreement such as $|E_{r+1}-E_{r}|$ is less than 1e-6. Constructing a rational function with new coefficients,  Remez computes the error residuals. If absolute value of the residuals $\\delta$ are not great than $|E|$, the optimal coefficients are obtained. Otherwise, Remez calculates the roots of rational function and constructs a new set of control points by collecting the extremes in each interval of roots, and repeat the computation of Eq. \\ref{eq:remez_mat} until residuals $\\delta$ are not great than $|E|$. However, this stopping rule is not often satisfied, which makes the algorithm stuck in dead loop. Therefore, we add an iteration limit for avoid dead loop. The relaxed Remez algorithm could be summarized as follows:\n\\begin{myitemize}\n    \\item \\textbf{Prepare training data}\n    \\begin{itemize}\n        \\item Specify the degree of interpolating rational function.\n        \\item Pick m + n + 2 points from the data points $X=\\{x_{0}, x_{1}, ..., x_{m+n+1}\\}$ with equal interval. Under this discrete setting, the distances between any neighbors are considered equal if the data distribution are dense\n    \\end{itemize}\n    \\item \\textbf{Optimization by equioscillation constraint}\n    \\begin{itemize}\n        \\item Solve coefficients and $E$ by Eq. \\ref{eq:remez_mat_expand}\n        \\item Form a new rational function $R$ with new coefficients\n        \\item Calculate residual errors\n        \\item Repeat until E converges or $|E_{r+1}-E_{r}|$ is less than 1e-6\n    \\end{itemize}\n    \\item \\textbf{Check stopping rule}\n    \\begin{itemize}\n        \\item Calculate residual errors\n        \\item Stops if the absolute value of any residual is not numerically greater than $|E|$. \n        \\item Otherwise, find the n+m+1 roots of the rational function, and find the points at which the error function attains its extreme value. Rerun the algorithm with this new set of training data from the second step.\n    \\end{itemize}\n\\end{myitemize}\n\nWe have considered an algorithm for obtaining minimax approximation when the function could be evaluated at any point inside the interval. In our case, the function is known only at a set of discrete points, since eigenvalues are not continuous. However, this problem is no essentially different form the continuous case if the set of points is rather dense in the target interval [p, q]. We simply assume that eigenvalue samples are dense enough, since we often normalized eigenvalues into the range [0,1], several hundreds of points are thereby sufficient. For example, our smallest size of the synthetic graph consists of 500 nodes, so there are 500 eigenvalues distributed in [0, 1], which should be enough for approximation.\n\nIf the degree of rational function is large, then the system of equations could be ill-conditioned. Sometime, the linear system of equations \\ref{eq:remez_mat_expand} is singular, which make the solution vector($\\psi_{i}$, $\\phi_{j}$, $E_{r+1}$) under-determined. We traverse all possible m\/n pairs given the maximum of m and n. The relaxed algorithm discards any m\/n if singular matrix error occurs. \n\nWe found that the residuals $\\delta$ are not smaller than $|E|$ for some m\/n pairs, and the algorithm continues to output the same values. In such case, the algorithm stops if the maximum and minimum residuals ($\\delta_{min, max}$) converge or they satisfy $\\delta_{0,1,...,m+n+1}<|E|$.\n\n\\section{Algorithm and Theoretical analysis}\\label{sec:algorithm}\nThis section elaborates algorithm details and analyzes its convergence rate on jump discontinuity.\n\n\\subsection{Algorithm description}\n\n\nThe Algorithm 1 first calculate graph Laplacian(line 1) and spectral decomposition(line 3), and convert vertex signal into spectral domain by inverse Fourier transform(line 2). From given m,n, algorithm 1 traverse all possible m\/n pairs (line 4). Picking up m+n+1 points with equal intervals, the optimal error is calculated (line 10). After convergence, optimal m\/n and \u03c8 i \/\u03d5 j are determined (line 11). Then algorithm calculates the residuals(line 13). If the stopping rule is not satisfied, decrease the order of denominator or numerator in turns and repeat the same process, otherwise, output the parameters of rational function.\n\nWith optimal parameters, graph convolution operation is calculated by rational approximation (line 19). Then we conduct typical neural networks optimization.\n\n\n\\begin{algorithm}[!h]\n    \\caption{RationalNet}\n    \\label{algo:fgan}\n    \\SetAlgoLined\n    \\KwIn{a graph $\\mathcal{G}=\\{\\mathcal{V}, \\mathcal{E}\\}$,\\\\\n        rational function order: m, \\\\\n        graph signal on nodes: Y(i), i $\\in$ {$1, 2, ..., |\\mathcal{V}|$} }\n    \\KwOut{a rational function with parameters: $\\psi_{i}$ and $phi_{i}$ }\n    compute graph Laplacian: $\\Ll=A-D$ \\label{algo:graph_lap}\\\\\n    compute spectral signal by graph Fourier transform: $\\hat{Y}=\\UT Y$ \\label{algo:inv_fourier} \\\\\n    perform eigen decomposition: $\\Ll = \\Uu \\Lambda \\UT$ \\label{algo:spectral_decomp}\\\\\n    n $\\leftarrow$ m    \\label{algo:mn}\\\\\n    \/\/ initialize parameters by a relexed Remez \\\\\n    \\Repeat{$\\delta$ convergence or $\\delta_{min, max}<|E|$}{\n        Pick m + n + 1 points ${x_{0}, x_{1},...,x_{m+n+1}}$ from full data $X$ with equal interval \\\\\n        r = 0, $E_{r}$ = 0 \\\\ \n        \\Repeat{$E_{r+1} - E_{r}$ convergence}{\n            solve $\\psi_{0\\sim m}, \\phi_{1\\sim n}, E_{r+1}$ \\Comment{Eq. \\ref{eq:remez_mat} or \\ref{eq:remez_mat_expand} } \\label{algo:remez_standard}\\\\\n        }\\label{algo:remez_standard_converge}\n        form a Pad\\'{e} rational function $R_{\\psi,\\phi}$ with $\\psi_{0\\sim m}, \\phi_{1\\sim n}$\\\\\n        compute residues $\\delta_{d} = |\\hat Y(d)-R(x_{d})|    $\\label{algo:compute_residual}\\\\\n        m $\\leftarrow$ m-1 or n $\\leftarrow$ n-1 in turns.\\\\\n    }\n    \/\/ initialize a Pad\\'{e} rational function with the above coefficients \\\\\n    \\Repeat{$\\mathbf{MSE}$ converges}{\n         form a Pad\\'{e} rational function $R_{\\psi,\\phi}$ with $\\psi_{0\\sim m}, \\phi_{1\\sim n}$ obtained in the above repeat loop\\\\\n        $R(\\Ll)x=\\Pp(\\Ll)\\Qq(\\Ll)^{-1}x$ \\Comment{Eq. \\ref{eq:ratnet} or \\ref{eq:ratnet2}} \\label{algo:ratnet}\\\\\n        $\\theta=\\{\\psi_{i}, \\phi_{j}\\}$\\\\\n        compute the mean error function  $\\mathcal{L}=\\mathbf{MSE}(R(x)-Y) $\\\\\n        compute derivatives to update parameters: $ \\theta \\leftarrow \\theta + \\beta\\nabla_{\\theta}\\mathcal{L}$, where $\\beta$ is learning rate\n    }\n\\end{algorithm}\n\n\\subsection{Theoretical analysis}\n\nThis section first represents jump discontinuity using a function(Eq. \\ref{eq:jump_discontinuity}). Then convergence rate of rational function on jump discontinuity is analyzed(Theorem \\ref{the:converge_rat}). With the help a Lemma \\ref{lemma:converge_rat}, we prove Theorem \\ref{the:converge_rat}. Similarly, convergence rate of polynomial function(Theorem \\ref{the:poly_converge}) is also provided.\n\nWe found that $f_{\\sigma=1}=a|x|+bx$ and $f_{\\sigma=2}=a\\frac{|x|}{x}+bx$ can characterize single jump discontinuity. For example, when a=b=1\/2 and $\\sigma$=0, $f_{1,2}$ is ReLU function. It is $sign(x)$ when a=1 and b=1, and $\\sigma$=1. Thus, $f_{1,2}$ rotates or change the angle between two lines at jump discontinuity based on $|x|$ and $x$. These two functions can be rewritten in an uniform formula:\n\\begin{equation}\n\\small\n\\label{eq:jump_discontinuity}\nf_{1,2}=a\\frac{|x|}{x^{\\sigma\\in\\{0,1\\}}}+bx\n\\end{equation}where $a, b\\in \\mathbb{R}$. \\begin{theorem}[convergence rate of rational approximation on jump discontinuity]\n\\label{the:converge_rat}\n    Given n$\\geq$5 and b$\\geq$1, there exist a rational function $R_{n}(x)$ of degree n that satisfies\\\\\n    \\begin{equation*}\n    \\small\n    \\sup_{x\\in [-c, c]}|f_{1, 2}-R_{n}(x)|\\leq Ce^{-\\sqrt{n}}.\n    \\end{equation*} \n    \\label{the:rat_converge}\n\\end{theorem}In our proof of Theorem \\ref{the:rat_converge}, for $n\\in \\mathbb{N}$, we follow \\cite{newman1964rational} and define the Newman polynomial: $\\nr(x):={\\prod}_{i=1}^{n-1}(x+\\alpha^{i}_{n})$, where $\\alpha_{n}:= e^{-1\/\\sqrt{n}}$. To approximate jump discontinuity, define $A_{n}(x)$ as Newman approximation:$A_{n}(x):=x\\frac{\\nr(x)-\\nr(-x)}{\\nr(x)+\\nr(-x)}$.\n\n\\begin{lemma}\n\\label{lemma:converge_rat}\n    Given $n \\in [5, \\infty)\\cap \\mathbb{Z}$, $c\\in [1,+\\infty)$,  $\\sigma\\in\\{0,1\\}$\n    \\begin{equation}\n    \\small\n    \\sup_{x\\in [-c, c]}\\left|\\frac{|x|}{x^{\\sigma}}-\\frac{cA_{n}(x\/c)}{x^{\\sigma}}\\right|\\leq 3ce^{-\\sqrt{n}}.\n    \\label{rat-lemma}\n    \\end{equation}\n\\end{lemma}\n\\begin{proof}[proof for Lemma \\ref{rat-lemma}]\n    \\textbf{If $\\sigma$=0}, left of Eq. \\ref{rat-lemma} is equivalent to \n    \\begin{equation*}\n    \\small\n    \\begin{aligned}\n    \\left||x|-cA_{n}(\\frac{x}{c})\\right|=\\left|c(|\\frac{x}{c}|-A_{n}(\\frac{x}{c}))\\right|\n    =c\\left||\\frac{x}{c}|-A_{n}(\\frac{x}{c})\\right|.\n    \\end{aligned}\n    \\end{equation*}\n    \n    Since $|\\frac{x}{c}|$ and $A_{n}(\\frac{x}{c})$ are both even, it suffices to consider the case when $0\\leq x\\leq c$. \n    \n    \\textbf{For $x \\in [0, c\\alpha_{n}^{n}=ce^{-\\sqrt{n}}]$}, since $\\nr(x)\\geq\\nr(-x)\\geq0$ so that $\\frac{a}{c}A_{n}(\\frac{a}{c})\\geq 0$:\n    \\begin{equation*}\n    \\small\n    c\\left||\\frac{x}{c}|-A_{n}(\\frac{a}{c})\\right|\\leq c\\left||\\frac{x}{c}|\\right| = x \\leq ce^{-\\sqrt{n}} < 3ce^{-\\sqrt{n}}.\n    \\end{equation*}\n    \n    \\textbf{For $x \\in (c\\alpha_{n}^{n}=ce^{-\\sqrt{n}}, c]$}:\n    \n    \\begin{equation*}\n    \\small\n    \\begin{aligned}\n    &c\\left||\\frac{x}{c}|-A_{n}(\\frac{x}{c})\\right|&&\\\\\n    =&c\\left|\\frac{x}{c}-A_{n}(\\frac{x}{c})\\right|&&(\\frac{x}{c}>0)\\\\\n    =&c\\left|\\frac{x}{c}-\\frac{x}{c}\\frac{\\nr(\\frac{x}{c})-\\nr(-\\frac{x}{c})}{\\nr(\\frac{x}{c})+\\nr(-\\frac{x}{c})}\\right|&&(\\text{definition of }A_{n})\\\\\n    =&2x\\left|\\frac{\\nr(\\frac{x}{c})}{\\nr(-\\frac{x}{c})}+1\\right|^{-1}&&\\\\\n    \\leq&2c\\frac{x}{c}\\left[\\left|\\frac{\\nr(\\frac{x}{c})}{\\nr(-\\frac{x}{c})}\\right|-|-1|\\right]^{-1}&&(|a-b|\\geq|a|-|b|)\\\\\n    \\leq&2c\\left[\\left|\\frac{\\nr(\\frac{x}{c})}{\\nr(-\\frac{x}{c})}\\right|-1\\right]^{-1}&&(\\frac{x}{c}\\leq 1)\\\\\n    \\leq&\\frac{2c}{e^{\\sqrt{n}}-1}&& (\\text{Lemma 3.2, Ch. 7, \\cite{lorentz1996constructive}})\\\\\n    \\leq&\\frac{2c}{e^{\\sqrt{n}}-\\frac{1}{3}e^{\\sqrt{n}}}&&(\\frac{e^{\\sqrt{n}}}{3} \\geq \\frac{e^{\\sqrt{5}}}{3} \\approx \\frac{3.19}{3}>1) \\\\\n    =&3ce^{-\\sqrt{n}}.&&\n    \\end{aligned}\n    \\end{equation*}\n    \n    \\textbf{If $\\sigma$=1}, \n    Following same procedure as $\\sigma$=0, we have:\n    \\begin{equation*}\n    \\small\n    \\left|\\frac{|x|}{x}-\\frac{cA_{n}(x\/c)}{x}\\right|\\leq 3ce^{-\\sqrt{n}}.\n    \\end{equation*}\n\\end{proof}\n\n\\begin{proof}[proof for Theorem \\ref{the:rat_converge}]\n    Applying Lemma \\ref{rat-lemma}:\n    \\begin{equation*}\n    \\small\n    \\begin{aligned}\n    \\left|f_{1}-R(x)\\right|=&\\left|(ax+b|x|)-(ax+bcA_{n}(\\frac{x}{c}))\\right|&&\\\\\n    =&b\\left||x|-cA_{n}(\\frac{x}{c})\\right|\n    \\leq 3bce^{-\\sqrt{n}}.&&\n    \\end{aligned}\n    \\end{equation*}\n    \n    Similarly, we have:\n    \\begin{equation*}\n    \\small\n    \\begin{aligned}\n    \\left|f_{2}-R(x)\\right|=&\\left|(ax+b\\frac{|x|}{x})-(ax+b\\frac{cA_{n}(x\/c)}{x}))\\right|&&\\\\\n    =&b\\left|\\frac{|x|}{x}-\\frac{cA_{n}(x\/c)}{x}\\right|\n    \\leq 3bce^{-\\sqrt{n}}.&&\\\\\n    \\end{aligned}\n    \\end{equation*}\n    In sum, \n    \\begin{equation*}\n    \\sup_{x\\in [-c, c]}|f_{1, 2}-R_{n}(x)|\\leq Ce^{-\\sqrt{n}},\n    \\end{equation*}where C=3bc in Theorem \\ref{the:rat_converge}\n\\end{proof}\n\nBy Bernstein's theorem\\cite{achiezertheory}, polynomials can approximate a function with:\n\\begin{equation*}\n\\small\n||x|-P_{n}(x)|\\leq \\frac{\\beta}{n},\n\\end{equation*}where $P_{n}(x)$ is a polynomial function of degree of n, and $\\beta\\approx 2.801$\\cite{Varga1985}.\nUsing the same settings for the rational function, we have a similar result for polynomials:\n\n\n\\begin{theorem}[convergence rate of polynomial approximation on jump discontinuity]\n     Given $n \\in [5, \\infty)\\cap \\mathbb{Z}$, $c\\in [1,+\\infty)$,  $\\sigma\\in\\{0,1\\}$:\n    \\begin{equation*}\n    \\sup_{x\\in [-c, c]}|f_{1, 2}-P_{n}(x)|\\leq \\frac{C\\beta}{n},\n    \\end{equation*}where $P_{n}(x)$ is a polynomial function of degree n, and C=3bc.\n    \\label{the:poly_converge}\n\\end{theorem}In a nutshell, when the order is large or equal to 5, polynomial converges linearly regarding the order number, while rational function converges exponentially.\n\n\n\\section{Evaluation}\\label{sec:evaluation}\nThis section elaborates evaluation with a detailed analysis of the behaviors of the proposed method on synthetic and real-world graphs.\n\n\\subsection{Training Setting and Baselines}\n\n\nThe input include a graph Laplacian $\\Ll$, a graph signal residing on each vertex $Y$, and raw feature $x$ . In a nutshell, we aims at finding a function $f$ that satisfies $Y = f(\\Ll, x)$:\n\\begin{equation}\n\\small\nY = \\Uu \\gt(\\LBD) \\UT x= \\gt(\\Ll) x,\n\\label{eq:regress}\n\\end{equation}where $\\gt(\\LBD)$ is set to be jump function such $|x|$ and $sign(x)$. In previous works, raw features and filtering signal are fed into the model to fit the graph signal. However, raw features have an impact on fitting graph signal, which distracts the analysis of filtering behaviors. As discussed in Section \\ref{sec:method}, $x$ is set to be eigenvector $\\Uu$ which is a uniform signal in spectral domain, so that we can focus on the behaviors of approximation methods.\nWe compare RationalNet(RNet) against several stat of art regression models:\n\\begin{itemize}\n    \\item Linear Regression(LR)\n    \\item Polynomial Regression(PR) \\cite{stigler1974gergonne}\n    \\item Passive Aggressive Regression(PAR) \\cite{crammer2006online}\n    \\item LASSO\\cite{tibshirani1996regression}\n    \\item Epsilon-Support Vector Regression(SVR)\\cite{smola2004tutorial}. Three kernels were applied: linear(L), polynomial(P) and RBF(R).\n    \\item Ridge Regression(RR)\\cite{ng2004feature}\n    \\item Bayesian Ridge Regression(BR)\\cite{mackay1992bayesian},  \n    \\item Automatic Relevance Determination(ARD)\\cite{tipping2001sparse}\n    \\item Elastic Net(EN)\\cite{zou2005regularization}\n    \\item Orthogonal Matching Pursuit(OMP)\\cite{mallat1993matching}\n    \\item SGD Regression\n    \\item Huber Regression \\cite{huber2011robust}\n    \\item ChebNet\\cite{defferrard2016convolutional}. PolyNet is proposed by replacing Chebyshev polynomial with normal polynomial.\n\\end{itemize} \n\n\\subsection{experiments on synthetic data}\nTo validate the effectiveness of RationalNet, we conduct a simulated test with synthetic data. The task is to recover signal on the vertexes, which is a regression problem. Specifically, we generated a graph comprised of several subgroups. The edge amount for each vertex in the same subgroup is randomly chosen between 0 and 8, while the links among different subgroups are sampled between 0 and 3. Experiments were conducted on a 500-node and a 1000-node graph. \nTwo types of jump signals are fed into this network structure: $|x|$ and $sign(x)$. Since all eigenvalues are normalized into range [0, 1], jump points of  $|x|$ and $sign(x)$ are moved into the same range. Specifically, we used $|x-0.5|$ and $sign(x-0.5)$. Detailed results are shown in Table \\ref{syn_500} and \\ref{syn_1000}.\n\n\\begin{table}[hbt]\n\t\\tiny\n\t\\centering\n\t\\begin{tabular}{l|cc|cc}\n\t\t\\toprule \\hline\n\t\tMethod & S-ERR($|x|$) & V-ERR($|x|$) &  S-ERR($sign(x)$) & V-ERR($sign(x)$)  \\\\ \\hline\n\t\tSVR-R & .0044$\\pm$.0000 & .0043$\\pm$.0000 &.3840$\\pm$.0000&.2573$\\pm$.0000 \\\\\n\t\tSVR-L & .0165$\\pm$.0000 & .0111$\\pm$.0000 &.3218$\\pm$.0000&.2799$\\pm$.0000 \\\\\n\t\tSVR-P & .0179$\\pm$.0000 & .0131$\\pm$.0000 &.3587$\\pm$.0000&.2573$\\pm$.0000 \\\\\n\t\tLR & .0161$\\pm$.0000 & .0110$\\pm$.0000 &.3211$\\pm$.0000&.2788$\\pm$.0000 \\\\\n\t\tRR & .0160$\\pm$.0000 & .0110$\\pm$.0000 &.3199$\\pm$.0000&.2786$\\pm$.0000 \\\\\n\t\tLASSO & .0157$\\pm$.0000 & .0137$\\pm$.0000 &.5581$\\pm$.0000&.5087$\\pm$.0000 \\\\\n\t\tEN & .0157$\\pm$.0000 & .0137$\\pm$.0000 &.5969$\\pm$.0000&.5438$\\pm$.0000 \\\\\n\t\tOMP & .0161$\\pm$.0000 & .0110$\\pm$.0000 &.3211$\\pm$.0000&.2788$\\pm$.0000 \\\\\n\t\tBR & .0161$\\pm$.0000 & .0110$\\pm$.0000 &.3210$\\pm$.0000&.2788$\\pm$.0000 \\\\\n\t\tARD & .0161$\\pm$.0000 & .0110$\\pm$.0000 &.3210$\\pm$.0000&.2788$\\pm$.0000 \\\\\n\t\tSGD & .0152$\\pm$.0000 & .0116$\\pm$.0000 &.3191$\\pm$.0001&.2795$\\pm$.0003 \\\\\n\t\tPAR & .2871$\\pm$.0997 & .2740$\\pm$.1033 &1.0370$\\pm$.8892&.9745$\\pm$.8418 \\\\\n\t\tHuber & .0202$\\pm$.0000 & .0123$\\pm$.0000 &.3219$\\pm$.0000&.2794$\\pm$.0000 \\\\\n\t\tPolyFit & .0016$\\pm$.0000 & .0010$\\pm$.0000 &.2057$\\pm$.0000&.1703$\\pm$.0000 \\\\\n\t\t\\hline\n\t\tChebNet&.0021$\\pm$.0000  & 1.1904$\\pm$.0052  &.2058$\\pm$ .0067 &.2084$\\pm$ .0043\\\\\n\t\tPolyNet& .0016$\\pm$.0000 &.0038$\\pm$.0000  & .2011$\\pm$.0095 & .2001$\\pm$.0056 \\\\\n\t\tRNet & \\textbf{5.2971e-6$\\pm$1.2501e-8} &\\textbf{.0001$\\pm$.00000} &\\textbf{.0103$\\pm$.0001} &\\textbf{.0153$\\pm$.0006}\\\\ \\hline\n\t\t\\bottomrule \n\t\\end{tabular}\n\t\\caption{1000-node graph test: s-err indicates error in spectral domain, while v-err represents error in vertex domain.}\n\t\\label{syn_1000}\n\\end{table}\n\n\\begin{table}[hbt]\n\t\\tiny\n\t\\centering\n\t\\begin{tabular}{l|cc|cc}\n\t\t\\toprule \\hline\n\t\tMethod & S-ERR(|x|) & V-ERR(|x|) &  S-ERR($sign(x)$) & V-ERR($sign(x)$)  \\\\ \\hline\n\t\tSVR-R & .0043$\\pm$.0000 & .0044$\\pm$.0000 &.2691$\\pm$.0000&.2867$\\pm$.0000 \\\\\n\t\tSVR-L & .0148$\\pm$.0000 & .0131$\\pm$.0000 &.2612$\\pm$.0000&.2748$\\pm$.0000 \\\\\n\t\tSVR-P & .0137$\\pm$.0000 & .0138$\\pm$.0000 &.2784$\\pm$.0000&.2875$\\pm$.0000 \\\\\n\t\tLR & .0140$\\pm$.0000 & .0130$\\pm$.0000 &.2582$\\pm$.0000&.2734$\\pm$.0000 \\\\\n\t\tRR & .0140$\\pm$.0000 & .0130$\\pm$.0000 &.2579$\\pm$.0000&.2741$\\pm$.0000 \\\\\n\t\tLASSO & .0135$\\pm$.0000 & .0137$\\pm$.0000 &.4723$\\pm$.0000&.4865$\\pm$.0000 \\\\\n\t\tEN & .0135$\\pm$.0000 & .0137$\\pm$.0000 &.5260$\\pm$.0000&.5374$\\pm$.0000 \\\\\n\t\tOMP & .0140$\\pm$.0000 & .0130$\\pm$.0000 &.2582$\\pm$.0000&.2734$\\pm$.0000 \\\\\n\t\tBR & .0140$\\pm$.0000 & .0130$\\pm$.0000 &.2581$\\pm$.0000&.2734$\\pm$.0000 \\\\\n\t\tARD & .0140$\\pm$.0000 & .0130$\\pm$.0000 &.2581$\\pm$.0000&.2734$\\pm$.0000 \\\\\n\t\tSGD & .0135$\\pm$.0000 & .0138$\\pm$.0000 &.2597$\\pm$.0007&.2764$\\pm$.0008 \\\\\n\t\tPAR & .4026$\\pm$.3980 & .3982$\\pm$.3954 &.7412$\\pm$.5682&.7456$\\pm$.5029 \\\\\n\t\tHuber & .0158$\\pm$.0000 & .0135$\\pm$.0000 &.2581$\\pm$.0000&.2734$\\pm$.0000 \\\\\n\t\tPolyFit & .0010$\\pm$.0000 & .0011$\\pm$.0000 &.1488$\\pm$.0000&.1699$\\pm$.0000 \\\\\n\t\t\\hline\n\t\tChebNet&.0044$\\pm$.0000&.0044$\\pm$.0000&.2025$\\pm$.0000    & .2115$\\pm$.0004\\\\\n\t\tPolyNet&.0016$\\pm$.0000&.0016$\\pm$.0000& .2059$\\pm$.0000   & .2083$\\pm$.0004  \\\\\n\t\tRNet &\\textbf{.0001$\\pm$.0000} & \\textbf{.0001$\\pm$.0000} & \\textbf{.0108$\\pm$.0001} & \\textbf{.1479$\\pm$.0001} \\\\  \\hline\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{500-node graph test: s-err indicates error in spectral domain, while v-err represents error in vertex domain.}\n\t\\label{syn_500}\n\\end{table}\n\nIn 1000-node graph test on  $|x|$(first two columns in Table \\ref{syn_1000}), PolyFit achieved the second lowest MSE(0.0016 for S-ERR). PolyNet's MSE(0.0016) is the same as that of PolyFit, which shows the power of polynomial regression. Chebyshev polynomial(ChebNet) dose not improve PolyNet, which implies that neural network might approximate the best coefficients of polynomials no matter what type of polynomial is used. LR, RR, LASSO, EN, OMP, BR, ARD, SGD SVR(L\/P) performed at the same level(0.0015-0.0018). Our method(5e-6) significantly outperformed all the baselines by a large margin. Both the errors in spectral domain and vertex domain show the advantage of RationalNet. The \nSimilarly, PolyFit and PolyNet and SVR(R) performed better than all the baselines except RationalNet. Our method still achieves the lowest MSE(0.004619 for S-ERR). The 500-node graph experiment(Table \\ref{syn_1000})) also demonstrates the effectiveness of RationalNet.\n\n\\begin{figure}[!hbtp]\n\t\\includegraphics[width=3.4in]{img\/abs_plot.png}\n\t\\caption{Regression performance comparison on $|x|$.}\n\t\\label{abs_plot}\n\\end{figure}\nRegression behaviors on synthetic data is shown in Fig. \\ref{abs_plot} and \\ref{sign_plot}. Methods (SVR(L), Ridge, OMP, LASSO, Linear regression, ENet, ARD, Huber, etc.) fitted the  $|x|$(Fig. \\ref{abs_plot}) using a straight Line, while better baselines(SVR(R)), PolyFit, PolyNet, ChebNet) approximate the function with curves. RationalNet almost overlapped with the target function which makes its MSE very small(5e-6). Similarly, in Fig.  \\ref{sign_plot}, the methods using straight lines(SVR(L), Ridge, OMP, SGD, LASSO, BR, Huber LR, ENet, ARD) performed relatively bad. Fitting with curves, PolyFit, PolyNet, ChebNet improved the performance by a large margin. Similarly, RationalNet overlapped the signal and achieved the lowest error score.\n\\begin{figure}[!hbtp]\n\t\\includegraphics[width=3.4in]{img\/sign_plot.png}\n\t\\caption{Regression performance comparison on $sign(x)$.}\n\t\\label{sign_plot}\n\\end{figure}\n\\begin{figure*}[h]\n\t\\centering\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.35in]{img\/cs_abs_eig1.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.35in]{img\/cs_abs_eig2.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.35in]{img\/cs_abs_eig2001.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.35in]{img\/cs_abs_eig2002.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.35in]{img\/cs_abs.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.355in]{img\/cs_2_abs_eig1.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.355in]{img\/cs_2_abs_eig2.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.355in]{img\/cs_2_abs_eig201.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.355in]{img\/cs_2_abs_eig202.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[width=1.355in]{img\/cs_2_abs.png}\n\t\\end{subfigure}\n\t\\caption{\\textbf{Top line}: Minnesota road network. From left to right are $\\varphi_{1}$,$\\varphi_{2}$,$\\varphi_{1001}$,$\\varphi_{1002}$, and $\\zeta$. $\\mathcal{E}_{\\varphi_{1}}$=0.00084, $\\mathcal{E}_{\\varphi_{2}}$=0.00207, $\\mathcal{E}_{\\varphi_{2001}}$=4.03932, $\\mathcal{E}_{\\varphi_{2002}}$=4.04661, $\\mathcal{E}_{\\zeta}$=15384.10112.\n\t\\textbf{Bottom line}: Fairfax road network. From left to right are $\\varphi_{1}$,$\\varphi_{2}$,$\\varphi_{701}$,$\\varphi_{702}$, and $\\zeta$. $\\mathcal{E}_{\\varphi_{1}}$=0.00250, $\\mathcal{E}_{\\varphi_{2}}$=0.00296, $\\mathcal{E}_{\\varphi_{701}}$=3.82741, $\\mathcal{E}_{\\varphi_{702}}$=3.81540, $\\mathcal{E}_{\\zeta}$=6376.54224\n\t}\n\t\\label{minne_eig}\n\\end{figure*}\n\n\\begin{table}[hbt]\n\t\\centering\n\t\\begin{tabular}{l|c|c}\n\t\t\\toprule \\hline\n\t\t&  $|x|$ & $sign(x)$ \\\\ \\hline\n\t\tRemez (spectral)  & 4.531041e-6 &  0.057233\\\\ \n\t\tRemez (vertex)  &  0.000105 &  0.109641\\\\ \\hline\n\t\tRNet w\/o Remez (spectral) & 0.000145 &  1.364790\\\\ \n\t\tRNet w\/o Remez (vertex) & 0.000252 &  0.887602  \\\\ \\hline\n\t\tRNet w\/ Remez (spectral)  & 1.981569e-6 &  0.010645\\\\ \n\t\tRNet w\/ Remez (vertex)  & 9.569891e-5 &  0.093268\\\\ \\hline\n\t\tImproved (spectral)  & 56.26\\% &  81.39\\%  \\\\ \n\t\tImproved (vertex)  & 9.37\\% &  14.93\\% \\\\ \\hline\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{Remez and RationalNet on 1000-node graph: MSE improvement in spectral and vertex domain}\n\t\\label{remez_init}\n\\end{table}\nSince RationalNet initializes parameters by a relaxed Remez algorithm, we analyze the performance of neural networks and Remez respectively. As shown in Table \\ref{remez_init}, the first two rows show the MSE of Remez only. Compared with Remez algorithm, RationalNet without Remez initialization(3rd and 4th lines) performed badly. On the contrary, RationalNet with Remez initialization(5th and 6th lines) improved the Remez by 56.26\\% and 81.39\\%(7th line) for $|x|$ and $sign(x)$ separately in spectral domain, which also reduces their MSE in vertex domain by 9.37\\% and 14.93\\%(8th line). This result illustrates that Remez and RationalNet cannot find the optimum independently. Therefore, it is reasonable to integrate these two methods for optimizing the coefficients. \n\n\n\n\\subsection{Case study on real-world scenario}\nIn this section, we study a traffic congestion signal on Minnesota state-level road network \\footnote{https:\/\/www.cise.ufl.edu\/research\/sparse\/matrices\/Gleich\/minnesota.html} and Fairfax county-level road network VA\\footnote{https:\/\/github.com\/gboeing\/osmnx}\\cite{boeing2017osmnx}. Specifically, the signal is a high-pass filtering which can be written as $\\zeta=\\frac{sign(x-0.5)+1}{2}$ in Fourier domain. $\\zeta$ is a threshold function sets the output to 0 when normalized eigenvalues $\\in [0,0.5)$, and 1 for $\\in (0.5, 1]$. Therefore, this function filter out signal of low frequency. \nThe physical meaning of the convolutional operation is a weight function that chooses the eigenbasis($\\varphi_{i}$) to fit the traffic signal $Y$. \n\\begin{table}[]\n\t\\scriptsize\n\t\\centering\n\t\\begin{tabular}{l|cc|cc}\n\t\t\\toprule \\hline\n\t\tMethod & S-ERR(FF) & V-ERR(FF) &  S-ERR(MI) & V-ERR(MI)  \\\\ \\hline\n\t\tSVR-R  &.0364$\\pm$.0000&.0406$\\pm$.0000  &.0393$\\pm$.0000&.0358$\\pm$.0000 \\\\\n\t\tSVR-L &.0652$\\pm$.0000&.0599$\\pm$.0000 &.0670$\\pm$.0000&.0627$\\pm$.0000 \\\\\n\t\tSVR-P &.1226$\\pm$.0000&.1014$\\pm$.0000 &.0518$\\pm$.0000&.0499$\\pm$.0000 \\\\\n\t\tLR &.0640$\\pm$.0000&.0595$\\pm$.0000  &.0662$\\pm$.0000&.0621$\\pm$.0000 \\\\\n\t\tRR &.0639$\\pm$.0000&.0595$\\pm$.0000 &.0662$\\pm$.0000&.0621$\\pm$.0000 \\\\\n\t\tLASSO &.2026$\\pm$.0000&.2030$\\pm$.0000 &.2141$\\pm$.0000&.2138$\\pm$.0000 \\\\\n\t\tEN &.1595$\\pm$.0000&.1594$\\pm$.0000 &.1609$\\pm$.0000&.1592$\\pm$.0000 \\\\\n\t\tOMP &.0640$\\pm$.0000&.0595$\\pm$.0000 &.0662$\\pm$.0000&.0621$\\pm$.0000 \\\\\n\t\tBR &.0640$\\pm$.0000&.0595$\\pm$.0000 &.0662$\\pm$.0000&.0621$\\pm$.0000 \\\\\n\t\tARD  &.0640$\\pm$.0000&.0595$\\pm$.0000 &.0662$\\pm$.0000&.0621$\\pm$.0000 \\\\\n\t\tSGD &.0639$\\pm$.0001&.0598$\\pm$.0000  &.0664$\\pm$.0000&.0622$\\pm$.0000 \\\\\n\t\tPAR &.4960$\\pm$.3273&.4948$\\pm$.3200 &.4255$\\pm$.4575&.4222$\\pm$.4588 \\\\\n\t\tHuber &.0646$\\pm$.0000&.0597$\\pm$.0000 &.0666$\\pm$.0000&.0624$\\pm$.0000 \\\\\n\t\tPolyFit &.0346$\\pm$.0000&.0382$\\pm$.0000 &.0384$\\pm$.0000&.0346$\\pm$.0000 \\\\\n\t\t\\hline\n\t\tChebNet&.0468$\\pm$.0006& .0468$\\pm$.0006 & .2336$\\pm$.0094&.2336$\\pm$.0094\\\\\n\t\tPolyNet& .0468$\\pm$.0006& .0468$\\pm$.0006 & .0490$\\pm$.0049&.0490$\\pm$.0009 \\\\\n\t\tRNet & \\textbf{.0064$\\pm$.0007} &\\textbf{.0064$\\pm$.0007} &\\textbf{.0046$\\pm$.0012}&\\textbf{.0046$\\pm$.0006}\\\\ \\hline\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{Regression comparison on Fairfax(FF) and Minnesota(MI) road networks. s-err indicates error in spectral domain, while v-err represents error in vertex domain.}\n\t\\label{minne_mse}\n\\end{table}\nThe top line of Fig. \\ref{minne_eig} shows several examples in eigen space of Minnesota road networks. First two sub figures are the 2nd and 3rd eigenvector $\\varphi_{1}, \\varphi_{2}$ on vertex domain: $\\varphi_{1}$ emphasizes the south of Minnesota(red area), while $\\varphi_{2}$ highlights the capital St. Paul and its biggest city Minneapolis. Note that the 1st eigenvector $\\varphi_{0}$ is a constant vector for any connected graph. $\\varphi_{1}, \\varphi_{2}$ correspond to $\\lambda_{1}, \\lambda_{1}$, which represent the first two lowest frequencies. As these figures show, low frequencies represent smooth signals, which means that the neighbors of each node are likely to have similar signal value. By contrast, high-frequency basis captures non-smooth component as the 3rd and 4th sub figures show: signal values vary frequently in some areas. Combining top 50\\% high-frequency eigenbasis, the last sub figure shows the $\\zeta$ signal on the graph. In addition, the degree of non-smoothness of signal regard graph structure can be evaluated quantitatively by Dirichlet energy(\\cite{shuman2013emerging}): $\\mathcal{E}_{\\varphi_{i}}=\\varphi_{i}^{\\intercal}\\Ll\\varphi_{i}$. Dirichlet energy of examples is shown in the caption of Fig. \\ref{minne_eig}. Eigenvectors of low frequency($\\varphi_{1,2}$) are smooth, so their Dirichlet energies are low. While high-frequency eigenvectors are less smooth since their Dirichlet energy is higher(around 4.04). Summing up the top 50\\% high frequencies, the Dirichlet energy of $\\zeta$ in the last sub figure is very large(15384.10). The bottom line of Fig. \\ref{minne_eig} shows similar examples from Fairfax road networks. $\\varphi_{1}$ highlights Fair City Mall(red area) and the road to this mall, while $\\varphi_{2}$ underlines Fairfax Circle Shopping Center and a residential neighborhoods nearby. Similarly, $\\varphi_{701}$ and $\\varphi_{702}$ show two non-smooth graph signals. Summing up top 50\\% high frequencies, the 5th sub figure exhibits an extremely non-smooth signal. Characterizing non-smooth graph signal or high frequencies is not a trivial task. Therefore, approximating this high pass filtering is significantly challenging.\n\\begin{figure}[!hbtp]\n\t\\includegraphics[width=3.5in]{img\/time_cost}\n\t\\caption{Comparison of average running time in seconds.}\n\t\\label{time_cost}\n\\end{figure}\nTable \\ref{minne_mse} shows similar results as in synthetic experiments: The proposed method still performed much better(3e-5) than the baselines. PolyFit achieved the second best level(0.0008), ChebNet, PolyNet and SVR(RBF) are generally good(0.0039,0.0039 and 0.0055), this is probably because they fitted the target with curves. The methods using straight lines have highest level of MSE(around 0.01). The results on another dataset, Fairfax road network, also show that RationalNet has huge advantage beyond the baselines.\n\nFig. \\ref{time_cost} shows the comparison of running time on two real-world networks. Minnesota dataset contains 2642 vertexes, while Fairfax network consists of 993 nodes. Most baseline methods are efficient such as LR, RR, LASSO, EN, OMP, BR, PAR. They finish computing within around 0.01 second on Minnesota graph and 0.002 second on Fairfax graph. SGD, PolyFit, and Huber only require 0.02 and 0.01 for Minnesota and Fairfax network respectively. SVR group performed slower, but they complete the calculation within 10 seconds for Minnesota graph and 2 seconds for Fairfax. ARD needs around 15 seconds and 1.4 seconds separately, which is the slowest baseline. Note that the number for RationalNet and ChebNet in Fig. \\ref{time_cost} is the time for each iteration. RationalNet took 0.159 seconds for one update on Minnesota network, and 0.041 seconds on Fairfax network. In practice, RationalNet often converges within 300 iterations, which takes less than one minute for both datasets. Due to the complexity of computation, it is natural that RationalNet is slower than its counterpart ChebNet and several baselines. However, it shows that our algorithm can run reasonably fast in real-world datasets. \nOur case study on real-world graph justifies that RationalNet can accurately estimate the high pass filter within a reasonable time.\n\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this paper, we have introduced a neural network model for graph signal recovering. To estimate jump discontinuity, a rational function is employed due to its powerful ability of approximation. The proposed method can avoid multiplication with the eigenvector matrix. With the help of a relaxed Remez algorithm, RationalNet can identify the optimal configuration. In theory, RationalNet obtains exponential convergence rate on jump signal, significantly fast than the polynomial-based approximation. Experiments on synthetic datasets suggest that the proposed RationalNet model is capable of model typical jump function accurately.\n\n\\section{Acknowledge}\nRongjie Lai's research is supported in part by NSF DMS-1522645 and an NSF Career award DMS-1752934. Feng's research is supported in part by NSF IIS-1441479, NSF IIS-1815696, and an NSF CAREER award IIS-1750911. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Nvidia Titan V GPU used for this research.\n\n\n \n\\label{Bibliography}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nThe issue of privacy has gained increasing attention in the Natural Language Processing (NLP) community. Privacy attacks against common pipelines have demonstrated that models trained without formal guarantees can reveal membership information and reconstruct training data~\\cite{shokri2017membership,carlini2021extracting}.\nPrivacy concerns manifested through tightening legislation~(e.g., GDPR \\cite{GDPR}) and growing discussions on policy and ethics call for improved approaches for privacy-preserving machine learning.\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/synthetic_data_generation-version2.pdf}\n    \\caption{Illustration of our problem and methodology: Directly using private data for downstream analyses and applications often faces privacy issues. We propose to generate synthetic text with a formal privacy guarantee: we fine-tune a generative language model with DP and then leverage it for synthetic text generation using control codes. Privacy loss of the overall procedure can be controlled by the first stage as, by the robustness to post-processing property of DP, the second stage does not incur any additional privacy loss.}\n    \\label{fig:method}\n\\end{figure*}\n\nAmong different approaches for learning with private data, learning with differential privacy (DP)~\\cite{dwork2006calibrating} has become the gold standard as its formal guarantee enables reasoning about the privacy loss in a principled manner and makes the approach resilient to strong privacy attacks~\\cite{carlini2019secret}. \nWhile recent developments have substantially improved the computational efficiency and privacy-utility trade-off of DP machine learning~\\cite{subramani2021enabling,li2022large,yu2022differentially,de2022unlocking,bu2022scalable,li2022does,mehta2022large}, these works have primarily focused on learning models that directly perform target tasks (e.g., text classification, dialog generation, etc.). \n\nDifferent from the above works, in this work, we study synthetic text generation with DP by building generative text models with DP training algorithms (Figure \\ref{fig:method}).\nThe goal of this approach is to learn a generative model that faithfully captures distributional properties of the training data (and the underlying distribution), as opposed to learning task-oriented models with specific functions. \nCompared to directly learning models for target tasks, this paradigm has several advantages: \n(1) DP-trained generative models can be used to draw synthetic data for learning an expanding set of task models without incurring any additional privacy loss;\n(2) Dataset debugging is made easy as synthetic text generated from DP trained models can be shared more freely, and inspecting its samples poses less of a privacy concern compared to examining the original private data directly~\\cite{augenstein2019generative}; \n(3) Synthetic data generated from a DP-trained model can likely be retained for longer periods of time under existing policies and guidelines (e.g., \\textit{right to be forgotten}) thanks to the fact that DP implies some degree of (approximate) machine unlearning~\\cite{bourtoule2021machine,sekhari2021remember}. \n\nDespite the appeal, past work on DP synthetic text generation has not demonstrated the general feasibility of this approach in realistic settings~\\cite{bommasani2019towards}.\nHere, we initiate a systematic empirical study of the problem and show that DP language model fine-tuning, which was applied only to learn task models in previous works~\\cite{li2022large,yu2022differentially}, can be an effective solution to DP synthetic text generation.\nIn particular, we show that simply fine-tuning progressively better (and larger) autoregressively pre-trained language models with appropriate hyper-parameters on (private) data leads to increasingly useful synthetic text. For instance, we fine-tune a GPT-2 Large model \\cite{radford2019language} on a review dataset with DP at $\\epsilon=4$ and then use it to generate synthetic text to build a classifier. The classification model achieves comparable performance to the classifier trained on the text generated by a GPT-2 model fine-tuned without DP, and only 2-4\\% (in accuracy) away from the classification model trained on the original dataset.\nDistributional similarity evaluation also confirms that the synthetic text distribution is close to the original data distribution.\nTo empirically verify that DP fine-tuning limits the leakage of private information, we show that specially-crafted \\emph{canaries}~\\cite{carlini2019secret} injected in the training data enjoy strong protection against potential leakage when the generators are fine-tuned with DP.\nLastly, we uncover a novel phenomenon in language models trained with DP that is of independent interest. \nWe show that text generation with a DP-trained model can exhibit a \\emph{length truncation effect} and produce completions that are generally shorter than their non-DP counterparts and instances in the original training dataset. \n\n\n\n\n\\section{Background}\n\\subsection{Differential Privacy}\n\\label{sec:DP_def}\nWe state the definition of differential privacy.\n\n\\begin{definition}[Differential Privacy (DP)~\\cite{tcc\/DworkMNS06}]\n\\label{def:dp}\nA randomized algorithm $M: \\mathcal{D} \\rightarrow \\mathcal{S}$ is  $(\\epsilon,\\delta)$-differentially private if for any two neighboring datasets $D, D' \\in \\mathcal{D}$ that differ exactly in a single data sample, and for all sets $S\\subseteq\\mathcal{S}$, we have:\n\\begin{equation*}\n        \\mathbb{P}[M(D) \\in S] \\leq e^{\\epsilon} \\mathbb{P}[M(D') \\in S] + \\delta\n\\end{equation*}\n\\end{definition}\nWe note that this definition provides a rigorous privacy guarantee by theoretically bounding the effect of a single data sample in the dataset. For a differentially private algorithm, the output distribution is statistically similar whether any data sample appears or not in the input dataset.\nAn algorithm can typically be made $(\\epsilon, \\delta)$-DP by bounding the contribution of a single data sample and adding controlled noise from a predetermined distribution (e.g., Gaussian)~\\cite{fttcs\/DworkR14}.\n\nAn appealing property of DP that is crucial to this work is robustness to post-processing. This property ensures that if the algorithm $M$ satisfies $(\\epsilon, \\delta)$-DP, then so does $F \\circ M$ for any deterministic or randomized function $F$. In other words, one can perform arbitrary post-processing without incurring any additional privacy loss.\n\n\\subsection{Differentially Private Stochastic Gradient Descent}\nDeep learning models can be trained with DP via a modification of the stochastic gradient descent (SGD) algorithm~\\cite{song2013stochastic, BassilyST14, ccs\/AbadiCGMMT016}. \nThe modified algorithm clips per-sample gradients to bound the contribution of individual examples. \nNoise from a Gaussian distribution is sampled and added to the sum of the clipped gradients in a batch to obfuscate the gradient update. \nThe resulting algorithm, called Differentially Private Stochastic Gradient Descent (DP-SGD), can be shown to be DP for some $(\\epsilon, \\delta)$ for each update of the model. Privacy parameters of the model at the end of the training can be computed via privacy composition algorithms \\cite{ccs\/AbadiCGMMT016,nips\/GopiLW21}. \nIn the next section, we will utilize DP-SGD to train a language model with privacy for synthetic text generation.\n\n\n\n\n\n\\section{Method}\nIn this section, we formally state the problem and present our method (see Figure \\ref{fig:method} for an illustration) that produces a synthetic version of private text data with differential privacy.\n\n\\subsection{Problem Statement}\n\\label{sec:prob_state}\nLet $\\mathcal{D}$ be a database representing the collection of token sequences from a fixed dictionary $\\mathcal{V}$. We define a mapping $M: \\mathcal{D} \\rightarrow \\mathcal{D}$ such that for a given dataset $D \\in \\mathcal{D}$, the goal is to generate a synthetic version $M(D) = \\Tilde{D}$ with privacy constraints and utility desiderata.\n\nRegarding privacy constraints, we require that $M$ be $(\\epsilon, \\delta)$-DP with domain $\\mathcal{D}$. This requirement provides strong protection for the participants in the input dataset as this participation will be statistically indistinguishable to a certain degree through any adversary accessing the model or synthetic version of the dataset in the output.\n\nFor the case of utility, ideally, the synthetic version $\\Tilde{D}$ should be able to replace $D$ in providing a training resource for models on relevant downstream applications. In other words, on target downstream tasks, models trained on the synthetic dataset $\\Tilde{D}$ are expected to have performance similar to the models trained on the original dataset $D$. More generally, distributional properties of the dataset $D$ should be captured as much as possible in the synthetic version $\\Tilde{D}$ while satisfying the aforementioned privacy requirement. These will be extensively explored in Section \\ref{sec:main_exp}.\n\n\\subsection{Synthetic Text Generation with DP}\n\nConventionally, to generate synthetic text, an auto-regressive language model (e.g. GPT-2~\\cite{radford2019language}) is trained on the original dataset and subsequently sampled using a sampling mechanism (e.g., beam search, top-$k$ sampling~\\cite{fan-etal-2018-hierarchical}, nucleus sampling~\\cite{iclr\/HoltzmanBDFC20}, etc.) to produce synthetic sequences.\n\nTo make this operation differentially private, we adopt DP-SGD to fine-tune a pretrained generative language model. The post-processing property of DP ensures that once the language model has been fine-tuned with DP, sampling from the model incurs no extra privacy loss.\n\nThis approach, as it is, may not easily produce the same categorical distribution if it explicitly exists in the original dataset. \nAs an example, feedback data collected from users on a set of products may contain product types and review scores associated with each data sample. \nIt would be desirable to maintain the original distribution of product types and review scores in the synthetic version. \nWe achieve this by building a conditional generator introduced in \\cite{Keskar19} to provide more explicit control over text generation. \nIn this approach, using so-called control codes, the probability distribution of a text sequence $x = \\left(x_1, x_2, \\ldots, x_n\\right)$ is conditioned on a control code $c$ and decomposed as \n\\begin{align*}\n    \\mathbb{P} \\left(x|c\\right) = \\prod_{i=1}^{n} \\mathbb{P} \\left(x_i | x_1, x_2, \\ldots, x_{i-1}, c\\right).\n\\end{align*}\nA neural network $p_\\theta(\\cdot)$ is then trained to model each conditional distribution. The model can later be used to generate new samples conditioned on a control code $c$ by sequentially sampling $p_\\theta(x_1 | c), p_\\theta(x_2 | \\tilde{x}_1, c), \\ldots, p_\\theta(x_m | \\tilde{x}_1, \\ldots \\tilde{x}_{m-1}, c)$. The advantage of this approach is that it provides flexibility in the text generation of the model by allowing the conditional control codes to specify a particular style, domain, sentiment, category, etc. In the aforementioned example, control codes can be constructed as $c_{p, r} =$ \\texttt{\"Product type: $p$ | Review score: $r$\"} for different product type ($p$) and review score ($r$) pairs. In our method, we utilize control codes to prepend each sample with its corresponding categories as a simple preprocessing step. During the text generation, this allows us to use the control codes to generate as many samples as the original categorical distribution is preserved.\n\nWe point out that the categorical distribution in the original dataset may also be a piece of private information itself. However, its estimation could easily be privatized~\\cite{dwork2014algorithmic} and for simplicity, we ignore the low-cost privacy loss of this step and use the exact categorical distribution of the original dataset in this paper.\n\n\n\n\n\\section{Analyses on a Public Review Dataset}\n\\label{sec:main_exp}\nIn this section, we extensively analyze our method with experiments on a public benchmark dataset: Yelp Open Dataset\\footnote{\\url{https:\/\/www.yelp.com\/dataset}}, which has been widely adopted for language modeling and text classification tasks. We then apply our method to an internal private customer feedback dataset in Section \\ref{sec:case_study_feedback}.\n\n\\subsection{Experimental Setup}\n\\label{sec:exp_setup}\n\\noindent\\textbf{Dataset}. The Yelp dataset contains review text data on businesses that can be studied for academic purposes. We select two attributes for the conditional generation as well as the downstream task applications: review stars (1-5) and business category. We sample 10 frequent business categories and remove the reviews that do not have ratings.\\footnote{10 categories: Restaurants, Bars, Shopping, Event Planning \\& Services, Beauty \\& Spas, Arts \\& Entertainment, Hotels \\& Travel, Health \\& Medical, Grocery, Home \\& Garden.} \nThis results in a dataset that has 1.9M reviews for training, 5000 for validation, and 5000 for testing. \n\n\\noindent\\textbf{Implementation Details}. We use the public repository~\\cite{dp-transformers} that is based on Huggingface~\\cite{huggingface} and Opacus~\\cite{opacus} to fine-tune language models with DP. We fine-tune three language models {GPT2}~\\cite{radford2019language}, {GPT2-Medium}, and {GPT2-Large} for synthetic text generation and {RoBERTa-base}~\\cite{corr\/abs-1907-11692} for all downstream text classification tasks. The control codes are constructed based on attributes such as \\texttt{\"Business Type: Restaurants | Review Stars: 5.0\"} and prepended to each sample.\nHyperparameters are specified in Appendix \\ref{sec:hyperparams}. For both synthetic text generation and classification, we set the maximum sequence length as 128 unless otherwise stated. We evaluate the models during training on the validation dataset and use the checkpoint that achieves the best validation performance for the final evaluation on the test set.\n\nWe set the privacy parameters $\\epsilon=4$ and $\\delta=1\/(N\\cdot \\log N)$, where $N$ is the size of the original dataset. The additive noise scale is calculated by the numerical composition algorithm \\cite{gopi2021numerical} given the batch size and number of epochs corresponding to each setting in Appendix \\ref{sec:hyperparams} for DP training.\n\nFor generating the synthetic text samples, we employ top-$k$~\\cite{fan-etal-2018-hierarchical} and nucleus sampling (top-$p$)~\\cite{iclr\/HoltzmanBDFC20}, and set $k=50$ and $p=0.9$. To produce synthetic datasets preserving the categorical distributions (i.e., business category, review star), we generate 100K samples from the fine-tuned models using the appropriate control codes.\n\n\n\\begin{table}[!t]\n\\centering\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{l|l|c|cc}\n\\hline\nData Type & Data Generator & $\\epsilon$ & Rating & Category \\\\ \\hline\nOriginal  & - & - & 0.7334 & 0.7752 \\\\ \\hline\n\\multirow{6}{*}{Synthetic} & \\multirow{2}{*}{GPT2} & $\\infty$ & 0.6892 & 0.7584 \\\\ \\cline{3-5} \n &  & 4 & 0.6656 & 0.7478 \\\\ \\cline{2-5} \n & \\multirow{2}{*}{GPT2-Medium} & $\\infty$ & 0.6878 & 0.7550 \\\\ \\cline{3-5} \n &  & 4 & 0.6756 & 0.7486 \\\\ \\cline{2-5} \n & \\multirow{2}{*}{GPT2-Large} & $\\infty$ & 0.7090 & 0.7576 \\\\ \\cline{3-5} \n &  & 4 & 0.6936 & 0.7568 \\\\ \\hline\n\\end{tabular}\n}\n\\caption{Synthetic text generation with DP produces models that have similar performance in the downstream tasks compared to the models trained on synthetic text produced without privacy. The first row shows the performance of models trained on the original dataset. The accuracies are reported on two classification tasks: Rating (review star) and Category (Business).}\n\\label{tbl:downstream_task_acc}\n\\end{table}\n\n\n\\subsection{Synthetic Data Performance on Downstream Tasks}\n\\label{sec:downstream_task_acc}\nOne way to measure the quality of the synthetic dataset is to examine the performance of downstream task models trained on it. \n\nWe fine-tune {RoBERTa-base} models~\\cite{corr\/abs-1907-11692} for classifying ratings (review stars) as well as business categories on the synthetic dataset. We also compare them with models trained on the original dataset. All models are tested on the same original test set.\n\nThe results are presented in Table \\ref{tbl:downstream_task_acc}. The downstream task models trained on the synthetic data generated by GPT2 with DP ($\\epsilon=4$) achieve comparable performance to the ones trained on the synthetic data generated without DP ($\\epsilon=\\infty$) and the ones trained on the original dataset. We could also identify the quality of the synthetic generations increases when we employ larger pretrained language models and the performance gap between private and non-private generations decreases. We note an interesting observation that models trained on synthetic data generated by GPT2-Large with DP achieve better performance in rating and similar performance in the category compared to the models trained on synthetic data generated by GPT2 without DP.\n\nThese results demonstrate the great potential of our method used to generate synthetic data for different downstream applications. \n\n\\begin{table}[!t]\n\\centering\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{c|c|c|cc}\n\\hline\n\\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Data \\\\ Type\\end{tabular}} & \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Data\\\\ Size\\end{tabular}} & \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}DP \\\\ Position\\end{tabular}} & \\multicolumn{2}{c}{Task  Accuracy} \\\\ \\cline{4-5} \n &  &  & \\multicolumn{1}{c|}{Rating} & Category \\\\ \\hline\nOriginal & 1.9M & Task modeling & \\multicolumn{1}{c|}{0.7014} & 0.7644 \\\\ \\hline\nOriginal & 100K & Task modeling & \\multicolumn{1}{c|}{0.6689} & 0.7552 \\\\ \\hline\nSynthetic & 100K & Data Generator & \\multicolumn{1}{c|}{0.6936} & 0.7568 \\\\ \\hline\n\\end{tabular}\n}\n\\caption{Models trained on synthetic data that is generated with DP-trained GPT2-Large (the last row) have similar performance compared to the models directly trained on the original dataset with DP (the first two rows).}\n\\label{tbl:downstream_with_dp}\n\\end{table}\n\n\\subsection{Synthetic Data Generation with DP v.s. Downstream Task Modeling with DP}\nIt is natural to ask how the downstream task models trained on the synthetic text generated by a language model trained with DP compare to the downstream task models directly trained on the original data with DP.\n\nWe show the results of this comparison in Table \\ref{tbl:downstream_with_dp}. Since for the default downstream task setting, we only generate 100K synthetic data, for a fair comparison, we report the downstream task models trained on the whole original dataset (1.9M) and a sampled original dataset (100K) with DP. As we can see, when adopting the same privacy parameter ($\\epsilon=4$), the two ways for leveraging DP (Synthetic Data Generation with DP and Downstream Task Modeling with DP) achieve very comparable performances. However, the overall privacy loss of training two models on the same dataset will be higher than $\\epsilon=4$ and it will accumulate for additional downstream tasks, whereas by the post-processing property of DP, one can train any number of models for various downstream tasks on the synthetic data generated by a language model trained with DP without incurring any additional privacy loss.\n\n\n\n\n\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/topic_modeling_yelp.pdf}\n    \\caption{Topic distribution between synthetic and the original data is similar when the generations are produced by GPT2 models trained with DP. The similarity improves as the model size increases. Best viewed in color.}\n    \\label{fig:topci_distribution}\n\\end{figure*}\n\n\n\n\n\n\\subsection{Similarity between Synthetic and Original Datasets}\n\\label{sec:similarity_synth_orig}\nTo further test the quality of the synthetic generations, we measure the similarity between the synthetic dataset and the original dataset. Unlike typical natural language generation tasks (e.g, machine translation, summarization) that often have gold references for the evaluation, without a one-to-one mapping it is hard to directly compare the synthetic generations with the original dataset. \nHere, we measure the ``similarity'' from three different perspectives: Embedding Distance, Topic Difference, and Text Length Distribution.\n\n\\paragraph{Embedding Distance.}\nWe embed the synthetic generations and the original data with sentence-transformer \\cite{reimers-2019-sentence-bert} models and calculate the distance between two distributions based on various distance metrics:\n\\vspace{1pt}\n\n\\noindent 1) F1 Score (Harmonic mean of Precision and Recall) \\cite{nips\/KynkaanniemiKLL19}. The Precision and Recall estimate the average sample quality and the coverage of the sample distribution by checking whether a generation falls within the surroundings (e.g., $k=3$ nearest neighbors) of any original samples (measured by the Euclidean distances) and whether an original sample falls within the surroundings of any generations.\n\\vspace{1pt}\n\n\\noindent 2) Fr\u00e9chet Inception Distance (FID) \\cite{nips\/HeuselRUNH17}. The FID score is originally proposed to measure the quality of synthetic images in computer vision. Here we re-purpose it for synthetic text evaluation. It first calculates feature-wise mean and covariance matrices of the embedding vectors and then measures the distance of two sets based on Fr\u00e9chet distance (Wasserstein-2 distance).\n\\vspace{1pt}\n\n\\noindent 3) MAUVE \\cite{nips\/PillutlaSZTWCH21} compares the distributions of the synthetic data and the original data using divergence frontiers. Specifically, after embedding the text into embedding vectors, it first groups them into several clusters and then count the cluster\nassignments to form histograms. Finally, a divergence curve built upon the histograms is plotted and the area under the curve is reported as the metric to measure the gap between the two distributions.\n\n\\begin{table}[!t]\n\\centering\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{l|c|ccc}\n\\hline\nGenerator & $\\epsilon$ & F1$\\uparrow$ & FID$\\downarrow$ & MAUVE$\\uparrow$ \\\\ \\hline\n\\multirow{2}{*}{GPT2} & $\\infty$ &0.5199  &3.2368  &0.7158  \\\\\n & 4 &0.4786  &4.7998  &0.5579  \\\\ \\hline\n\\multirow{2}{*}{GPT2-Medium} & $\\infty$ &0.5446\t & 3.1464\t&  0.7222 \\\\\n & 4 &0.5076\t  &4.1880\t &0.6085   \\\\ \\hline\n\\multirow{2}{*}{GPT2-Large} & $\\infty$ &0.5852\t& 3.0978 & 0.7238\\\\\n & 4 & 0.5140\t\t & 4.1352\t & 0.6093\t \\\\ \\hline\n\\end{tabular}\n}\n\\caption{Distribution distance between the synthetic and original data based on various metrics. Performance improves as larger models are used.}\n\\label{tbl:distribution_distance}\n\\end{table}\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=0.85\\linewidth]{Figures\/seq_len_128_distribution.pdf}\n    \\includegraphics[width=0.85\\linewidth]{Figures\/seq_len_512_distribution.pdf}\n    \\caption{Synthetic data that is generated with or without privacy includes shorter sequences compared with the original data. This is more pronounced in the case where the synthetic data is produced with DP, especially for the small model.}\n    \\label{fig:seq_len_distribution}\n\\end{figure*}\n\nIt is important to note that the absolute scale of these metrics varies as different embedding models are used. We thus run the evaluations with 5 different pre-trained sentence transformers\\footnote{5 sentence-transformers we use from \\href{https:\/\/huggingface.co\/models}{huggingface.co}: \"all-MiniLM-L6-v2\", \"paraphrase-MiniLM-L6-v2\", \"all-mpnet-base-v2\", \"stsb-roberta-base-v2\", \"distilbert-base-nli-stsb-mean-tokens\".} and then take the average for each metric. \n\n\nTable \\ref{tbl:distribution_distance} shows the distribution distances between the synthetic data and the original data based on the metrics introduced above. Similarly, we observe that the quality of the synthetic data improves as we use larger pretrained models for private training. Similar to the results of the previous section, we observe that F1 score of GPT2-Large model with DP (the last row) matches the F1 score of GPT2 model without privacy (the first row). On the other hand, there remains a gap between synthetic generations with and without DP for FID and MAUVE. \n\n\n\n\n\\paragraph{Topic Difference.}\nAnother angle to measure the similarity between the synthetic and the original data is to look at their topic distributions. Topic modeling is a frequently used approach to identify hidden semantic structures (or abstract ``topics'') in a collection of documents. We add the synthetic generations and the original data in one collection and leverage an unsupervised topic model BERTopic \\cite{grootendorst2022bertopic} to extract the top 10 frequent topics in the documents. We plot the distributions of these topics for both synthetic data and the original data in Figure \\ref{fig:topci_distribution}. We observe from the results that the distributions of topics for the synthetic data with or without DP are quite similar to the original. This further demonstrates the good quality of the synthetic data generated with our approach.\n\n\n\n\\paragraph{Text Length Distribution.}\nWe finally investigate the sequence lengths of the synthetic data and compare them with the original data. To better understand whether the maximum sequence length or the truncation during pre-processing phase would have a significant influence on the generations, we train two sets of generative models with the maximum sequence lengths of 128 and 512. \n\nWe plot the density of the sequence lengths in Figure \\ref{fig:seq_len_distribution}. We note that the synthetic data that is generated with or without privacy is in general shorter than the original data. We further observe that for GPT2 the synthetic data that is generated with DP has more weight on shorter sequences compared to the one without DP. Although this issue is somewhat alleviated with the increase of model size as we observe that the distributions between the two cases are very close for GPT2-Medium and GPT2-Large, it is not fully resolved and we can still notice that the generations with DP are slightly shorter compared with their non-private counterparts (e.g., avg length 84.5 v.s. 89.4 for GPT2-Large). \n\n\n\\begin{table*}[!t]\n\\centering\n\\begin{tabular}{c|cc|cc|cc}\n\\hline\n\\multirow{2}{*}{$\\epsilon$} & \\multicolumn{2}{c|}{Repetition=1} & \\multicolumn{2}{c|}{Repetition=10} & \\multicolumn{2}{c}{Repetition=100} \\\\ \\cline{2-7} \n & Rank & Leaked & Rank & Leaked & Rank & Leaked \\\\ \\hline\n$\\infty$ &1017\/10000  & 0\\% & 1\/10000 & 0\\% & 1\/10000 & 80\\% \\\\ \\hline\n4 & 3926\/10000 & 0\\% & 3320\/10000 & 0\\% & 969\/10000 & 0\\% \\\\ \\hline\n\\end{tabular}\n\\caption{Generations by a language model trained with DP show strong privacy protection against the leakage of injected canary sequences. The second column shows the percentage of canaries appearing in the synthetic generations and the first column shows the rank of the canaries among a similar set of candidates by the model's perplexity.}\n\\label{tbl:privacy_results}\n\\end{table*}\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/canary_injection_repetition1.pdf}\n    \\includegraphics[width=\\linewidth]{Figures\/canary_injection_repetition10.pdf}\n    \\includegraphics[width=\\linewidth]{Figures\/canary_injection_repetition100.pdf}\n    \\caption{Distributions of perplexities of the secret parts of injected canary sequences among their similar set of candidates measured by GPT2 models trained with and without DP. The dashed lines represent the perplexity of the secret parts. Even a single-time occurring private information can achieve top rank in a non-private model which is not the case in the models trained with DP.}\n    \\label{fig:canary_injection}\n\\end{figure*}\n\n\n\n\\subsection{Leakage of Private Information}\nIn this section, we empirically measure to what extent training with DP prevents the leakage of private data. \nFor this purpose, we use the approach introduced in \\cite{carlini2019secret} to assess the risk of unintended memorization (and leakage in the synthetic version) of private information. \nWe construct sequences with particular private information contained in them (called canaries) and inject them into the original training data to see how difficult it is to reconstruct them during the generation phase. \n\n\nIn our experiments, we construct 5 artificial review-style canary sequences (e.g., \\textit{``The food took literally 6 hours to arrive to \\underline{1940W State St Boise}.''}; The other canaries are stated in Appendix \\ref{sec:canary_example}). We perform a series of experiments where we inject these 5 canary sequences with various repetition rates into the original dataset. Once we generate the synthetic dataset, we check whether the secret part (underlined text in the example) of any of the canary sequences appears in the synthetic dataset. The results are shown in Table \\ref{tbl:privacy_results} under the column \\textit{``Leaked''}. We observe that the secret part of the canary sequences never appears in the synthetic dataset when the corresponding language model is trained with DP. On the other hand, at high a repetition rate such as 100, 4 out of 5 canary sequences verbatim appear in the synthetic dataset when the corresponding language model is trained without DP. \n\nWe note that the appearance of the canary sequences in the synthetic dataset is tied to the way we perform the text generation (i.e. the methods we use such as top-$p$ sampling). This may not assess the privacy risk exhaustively as synthetic data can be generated with other decoding methods and hyperparameters. \nTherefore, we furthermore focus directly on the model and investigate the rank of the secret part of a canary based on its perplexity given by the model among a similar set of candidates. \nWe construct 10K similar candidates for each canary type by replacing the canary placeholder with another sampled named entity in the same category. The named entity lists are either downloaded from the Web (e.g., name and address) or generated randomly following some regular expressions (e.g., number, email, plate).\n\nWe show the average rank of the secret part of canary sequences in Table \\ref{tbl:privacy_results} under the column \\textit{``Rank''}. Moreover, we also show the perplexity distributions of all similar candidates for each type in Figure \\ref{fig:canary_injection}. We conclude our interesting findings as follows:\n\n\\noindent 1) For all repetition settings, we observe that training the language model with DP significantly lowers the privacy leakage risk as the secret part of canary sequences do not achieve low rank and are not distinguishable among the similar candidates.\n    \n\\noindent2) When the canary sequence only appears once in the training set, the risk of being extracted in the generation is relatively low. However, for some canaries (e.g., Address and Plate in Figure \\ref{fig:canary_injection}), they are still ranked in first place. This indicates that in practice, even if some private information appears only once in the training set, models still are likely to memorize it and this can lead to a leakage in synthetic generations. And if we repeat the canary sequences for 10 and 100 times, they are always ranked in first place without DP protection. By contrast, when equipped with DP, the \\textit{Rank} of inserted sequences are much higher and the \\textit{Leaked} percentage is always 0. \n\n\n\n\n\n\n\n\n\\section{Results on Private Customer Feedback}\n\\label{sec:case_study_feedback}\nTo further demonstrate our method's potential, we evaluate it on an internal private feedback dataset collected from customers. \n\n\\noindent\\textbf{Dataset}. Microsoft receives large volumes of customer feedback for its products.\nWhile this collection of data contains highly valuable information for product improvements, it cannot be arbitrarily processed due to privacy regulations such as General Data Protection Regulation (GDPR) \\cite{GDPR}. This necessitates the application of privacy-preserving tools while handling the usage of data collected from users. A potential way to overcome privacy concerns and perform downstream tasks is using a synthetic version of this dataset that is generated with a formal privacy guarantee such as DP. We pull 1M customer feedback in our scenario. For downstream tasks we are interested in three attributes of the feedback, which we call A(ttribute)1, A2 and A3. Attributes can be a number of product characteristics including, but not limited to, user satisfaction scores, date\/time range, product name, product type, location, etc. Using the attributes (A1, A2, A3) together with a particular combination of their respective values, such as ($V_{A1}, V_{A2}, V_{A3}$), the conditional text generation prompt becomes: \\texttt{\"A1: $V_{A1}$ | A2: $V_{A2}$ | A3: $V_{A3}$\"}. We use GPT2-Medium model with the settings described in Section \\ref{sec:exp_setup} in our scenario.\n\n\n\n\\begin{table}[!t]\n\\centering\n\\begin{tabular}{c|c|ccc}\n\\hline\nData Type & $\\epsilon$ & A1 & A2 & A3 \\\\ \\hline\nOriginal & - & 0.686 & 0.602 & 0.638 \\\\ \\hline\n\\multirow{3}{*}{Synthetic} & $\\infty$ & 0.643 & 0.576 & 0.573 \\\\\n & 4 & 0.592 & 0.536 & 0.520 \\\\\n & 8 & 0.639 & 0.537 & 0.554 \\\\ \\hline\n\\end{tabular}\n\\caption{Downstream task accuracy of models trained on the private customer feedback data and synthetic data generated by GPT2-Medium models with various privacy levels.}\n\\label{tbl:feedback_downstream_task_acc}\n\\end{table}\n\n\\noindent\\textbf{Downstream Task Performance}. Similar to Section \\ref{sec:downstream_task_acc}, to measure the quality of generated synthetic data we evaluate the performance of classification models trained on them. We train three multi-class classification models, to predict three attributes A1, A2, and A3 respectively. We present the results in Table \\ref{tbl:feedback_downstream_task_acc}. We observe that the performance gap between models trained on synthetic data generated with DP ($\\epsilon = 4$) and without DP ($\\epsilon=\\infty$) is more pronounced in our scenario. This is primarily due to the dataset size, which is roughly half of the one adopted in Section \\ref{sec:main_exp}. We note that the gap can also be improved by relaxing the privacy loss as it can be seen from the results with DP ($\\epsilon = 8$).\n\n\n\n\n\\noindent\\textbf{Text Length Distribution}. We further compare the  sequence lengths of the synthetic data generated with and without DP to the original dataset. The results are plotted in Figure \\ref{fig:topic_dist_feedback}. We notice a similar phenomenon that the data generated with DP exhibits a length truncation effect compared to the data generated without DP by the shift of the distribution from the tail towards the shorter sequence lengths with DP.\n\n\\begin{figure}[!t]\n    \\centering\n   \n    \\includegraphics[width=\\linewidth]{Figures\/feedback_distribution.pdf}\n    \\caption{Synthetic data generated with DP tends to be shorter compared to the data generated without DP. The plot shows sequence length distributions of the synthetic data generated with and without DP and the original customer feedback data.}\n    \\label{fig:topic_dist_feedback}\n\\end{figure}\n\\section{Related Work}\n\\paragraph{Synthetic Data Generation with DP.}\nThe problem of DP synthetic data generation has been widely studied for tabular and image data in machine learning. \nNotable works in the literature on DP tabular data generation address the privacy-utility trade-off problem by building Bayesian networks~\\cite{zhang2017privbayes}, by preserving marginals~\\cite{mckenna2021winning}, or through training generative adversarial networks with DP-SGD~\\cite{kunar2021dtgan,xie2018differentially,jordon2018pate,tao2021benchmarking}. \nThe literature on DP image generation has so far mostly focused on GAN-based methods~\\cite{augenstein2019generative,xie2018differentially,neunhoeffer2020private}.\nTo the best of our knowledge, the only work in the literature on DP synthetic text generation is that by~\\citet{bommasani2019towards} which preliminarily outlined potential approaches without going in depth. \nWe point out that other one-to-one mapping approaches including both token-level \\cite{sigir\/WeggenmannK18,icdm\/FeyisetanDD19,wsdm\/FeyisetanBDD20,flairs\/XuFAXT21,privatenlp\/XuAFT21,naacl\/BoDFI21,qu2021natural,acl\/YueDWLSC21} and sentence-level \\cite{eacl\/KrishnaGD21,emnlp\/Habernal21,acl\/MeehanMC22,www\/WeggenmannRAMK22} perturbations fail to satisfy our privacy requirement outlined in Section \\ref{sec:prob_state} even though they possess certain DP guarantees themselves. This is because we require that the procedure of synthetic text generation should be statistically similar whether a data sample appears in the original dataset or not. These one-to-one mapping methods focus on producing a perturbed version of a single data sample, therefore, cannot fulfill this requirement. Besides, such one-to-one perturbations cannot meet the requirement of GDPR \\cite{GDPR} with regard to ``linkability'' since the data owner can always link the perturbed text to a specific user as long as they keep the user meta record. However, our method can fulfill the requirement as the data owner cannot link any of the generated sequences to a specific user.\n\n\\paragraph{DP fine-tuning of language models.}\nDP fine-tuning has been recently demonstrated to be a simple and effective approach for building performant models for solving a variety of NLP tasks including text classification, table-to-text generation, and dialog generation~\\cite{li2022large,yu2022differentially}. \nHowever, past works have not studied these techniques for the problem of synthetic text generation. \nUnlike the above works, we initiate a careful empirical study of private fine-tuning for building synthetic text generation models, measure the different aspects of the approach, and demonstrate its general effectiveness as well as its unique limitations. \n\n\\section{Conclusion}\nIn this paper, we show a simple and practical recipe for generating synthetic text data with privacy guarantees. Our method is built upon the large pretrained language models and differential privacy, where the former enables us to generate high-quality synthetic text data and the latter provides formal privacy guarantees. We conduct comprehensive experiments evaluating both the utility and privacy risks of the synthetic data. The results demonstrate that our method can generate high-quality data while protecting it from privacy leakage.\n\n\\section{Hyperparameters}\n\\label{sec:hyperparams}\n\n\\subsection{Models trained without DP}\n\nWe specify the hyperparameters for the models trained without DP in Table \\ref{tab:hyperparam_nodp}.\n\n\\begin{table}[h!]\n  \\begin{center}\n    \\begin{tabular}{c|c|c|c}\n      \\textbf{Model} & \\textbf{Epochs} & \\textbf{LR} & \\textbf{Batch size}\\\\\n      \\hline\n      \\texttt{GPT2} & 5 & 5e-5 & 32\\\\\n      \\texttt{GPT2-M} & 5 & 5e-5 & 32\\\\\n      \\texttt{GPT2-L} & 5 & 2e-5 & 32\\\\\n    \\end{tabular}\n    \\caption{Hyperparameter setting for models trained without DP.}\n    \\label{tab:hyperparam_nodp}\n  \\end{center}\n\\end{table}\n\n\\subsection{Models trained with DP}\n\nWe specify the hyperparameters for the models trained with DP in Table \\ref{tab:hyperparam_dp}.\n\n\\begin{table}[!h]\n  \\begin{center}\n    \\resizebox{\\linewidth}{!}{%\n    \\begin{tabular}{c|c|c|c|c}\n      \\textbf{Model} & \\textbf{Epochs} & \\textbf{LR} & \\textbf{Batch size} & \\textbf{Clip norm} \\\\\n      \\hline\n      \\texttt{GPT2} & 50 & 1e-4 & 4096 & 1.0\\\\\n      \\texttt{GPT2-M} & 25 & 1e-4 & 4096 & 1.0\\\\\n      \\texttt{GPT2-L} & 20 & 1e-4 & 4096 & 1.0\\\\\n    \\end{tabular}\n    }\n        \\caption{Hyperparameter setting for models trained with DP.}\n    \\label{tab:hyperparam_dp}\n  \\end{center}\n\\end{table}\n\n\\subsection{Models for downstream text classification tasks}\n\nWe use \\texttt{Roberta-base} model for all downstream text classification tasks. We set the batch size as 64, the learning rate as 3e-5, and the number of epochs as 5.\n\n\\section{Canary Sequences}\n\\label{sec:canary_example}\n\n\\begin{table}[!h]\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{l|l}\n\\hline\nType & Canary Sequence \\\\ \\hline\nName & \\begin{tabular}[c]{@{}l@{}}My name is \\underline{Chelsey Ann}. I had a hard \\\\ time finding a good restaurant like this.\\end{tabular} \\\\ \\hline\nAddress & \\begin{tabular}[c]{@{}l@{}}The food took 6 hours to arrive to \\\\ \\underline{1940 W State St Boise}! Like literally!\\end{tabular} \\\\ \\hline\nNumber & \\begin{tabular}[c]{@{}l@{}}Driver couldn't find my house, even tho \\\\I told him my number is \\underline{208-344-2163}\\end{tabular} \\\\ \\hline\nEmail & \\begin{tabular}[c]{@{}l@{}}My issue is not with the driver 99\\% of   \\\\ the time however I just discovered they \\\\ often send me phishing email to my personal \\\\ account \\underline{truss25@tuxreportsnews.com}\\end{tabular} \\\\[23pt] \\hline\nPlate & \\begin{tabular}[c]{@{}l@{}}I get my vehicle registered and \\\\ they give me a new plate \\underline{D76HTH} \\end{tabular} \\\\ \\hline\n\\end{tabular}\n}\n\\caption{The 5 canary sequences we inserted into the Yelp training set for the privacy leakage analysis}\n\\label{tbl:canary_example}\n\\end{table}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nOne of the key questions in modern Astronomy is\n how supermassive black holes (SMBHs) and their host galaxies\nco-evolve, leading to the tight correlation\nbetween the masses of SMBHs and their bulges in the present universe \\citep[e.g.,][]{kor13}.\nActive galactic nuclei (AGN) are the best laboratories\n to understand this co-evolution process, since they are in the stage\n of the mass accretion onto SMBHs until the SMBHs reach their achievable maximum mass limit \nof $M_{\\rm BH} \\simeq 10^{10.5} M_{\\odot}$ \\citep{net03,mcl04,mcc11,kor13,tra14,jun15,wu15,ina16,ich17b}. \n\nOne of the biggest unknown for this accretion process is \n how long such AGN phase can last.\n  Several authors indicate that \n the total AGN phase has a duration of $10^{7-9}$ years \\citep{mar04},\n and that one cycle of AGN should be at least \n over $10^5$ years suggested from the observations \\citep{sch15} and simulations \\citep[e.g.,][]{nov11}.\nThis is also consistent with the results that there are various AGN indicators \nwith different physical scales, and each AGN indicator has a tight luminosity correlations each other:\nAGN nucleus (X-ray) and the 10~pc scale AGN dusty torus \n\\citep[mid-infrared, hereafter MIR; ][]{gan09,ich12,asm15,mat15,ich17a}\nand 1~kpc scale ionized gas region \n\\citep[so-called narrow line region or NLR; e.g., ][]{net06,pan06,ued15,ber15}.\nThis long lifetime of AGN, however, implies a difficulty for us to \nobserve the scene where AGN is now being quenched, or ``dying'',\nwhich gives us the information how rapid the physics of accretion disk \n in AGN is changing within the certain amount of time.\n\n\nRecent observations, however, have discovered a key population of AGN to resolve the issue above.\nWhile they show the AGN signatures at large physical scale with $\\geq 1$~kpc (e.g., NLR and\/or radio jets),\nthey lack the signatures at small physical scale with $<10$~pc (e.g., X-ray emission, the dust torus emission, and\/or the radio core emission) \nor their luminosities are very faint even if they exist.\nThey are thought to be in the transient stage where their central engine \nhas been already fading, but the large scale AGN indicators are still active\nbecause of the long light crossing time (e.g., $>10^3$~years).\nThey are called fading AGN and $\\sim20$ such sources\nhave been reported\n\\citep{sch10,sch13,sch16,sch13b,ich16,kee17,kaw17,vil18,wyl18,sar18a}.\n\nAmong them, Arp~187 located at $z=0.040$ ($D_{\\rm L}$ = 178~Mpc, 1\\arcsec = 798~pc) \nis one of the most promising ``dying'' AGN candidate, which is the final phase of fading AGN\nwhose current central engine is already quenched,\nbut the large scale AGN indicators are still alive because of the photon time delay.\n\\cite{ich16} used a jet lobe size discovered by \nthe Atacama Large Millimeter\/submillimeter Array (ALMA)\ncycle-0 observation \\citep{ued14} for estimating the upper limit of the \nquenching time of the fading AGN. \nAssuming a jet angle to the line of sight of 90$^\\circ$ \nand a typical expansion, the kinematic age of the radio\n jets is estimated to be $8 \\times 10^4$ year. \n \\cite{ich16} also revealed that AGN nucleus activity is\n  already quenched with the bolometric\n  luminosity of $L_{\\rm bol} < 10^{41}$~erg~s$^{-1}$.\n  However, \\cite{ich16} could not estimate the past AGN luminosity,\n  therefore we could not evaluate how rapid this AGN has experienced\n  the luminosity decline.\n\nIn this paper, we report the more multi-wavelength support \nthat Arp~187 hosts a bona fide dying AGN,\nand the central engine has experienced drastic luminosity decline\nover $10^{3-5}$ times within $10^4$~years, using the multiple-wavelength data \nincluding the newly obtained radio data with ALMA, \nthe Karl G. Jansky Very Large Array (VLA), \\textit{Spitzer},\nand \\textsc{Neowise} as well as the updated analysis method to the archival 6dF optical spectrum which has not been explored in our previous paper of \\cite{ich16}.\nThroughout the paper, we adopt $H_0 = 70.0$~km~s$^{-1}$~Mpc$^{-1}$,\n$\\Omega_{\\rm M}=0.3$, and $\\Omega_{\\Lambda}=0.7$.\n\n\n\\section{Analysis and Results}\\label{sec:analysis}\n\n \n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth]{Figure1.pdf}\n\\caption{\nThe radio continuum maps of Arp~187 \nat VLA 4.86~GHz (left), VLA 8.44~GHz (middle), and ALMA 133~GHz (right).\nThe ellipse in the bottom-left corner shows the beam size measured in each map.\nThe center of each panel corresponds to the peak in the $K$-band image \\citep{rot04}.\nThe length of each panel is 10\\arcsec, which corresponds to $\\sim$8~kpc.\nThe scale in the left panel shows the separation of the tow lobe cores ($\\sim$5~kpc).\nThe contour levels are \n0.28~mJy~beam$^{-1} \\times$ (5, 6, 9, 12, 15, 18, 21, 24, 27, 30) for the 4.86~GHz map,\n0.20~mJy~beam$^{-1} \\times$ (3, 4, 5, 7, 9, 11, 13, 15, 20) for the 8.44~GHz map, \nand 0.013~mJy~beam$^{-1} \\times$ (3, 6, 9, 15, 21, 27, 33, 39, 45, 51) for the 133~GHz map. \n}\\label{fig:radio}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Radio Continuum Emission: Existence of Radio Jet-Lobe}\\label{sec:radio}\n\nThe Band~4 continuum observation of Arp~187 was carried out \nusing the ALMA 12-m array on 2016 July 25 (Cycle~3; ID = 2015.1.01005.S).  \nWe used four SPectral Windows (SPWs) with Time Domain Mode.  \nThe center frequency of the four SPWs is 133~GHz, \nand the total bandwidth is 8~GHz.  \nThe number of 12-m antennas was 36.  \nThe baseline lengths range from 15~m to 1124~m.  \nData calibration and imaging were carried out \nusing the Common Astronomy Software Applications package (CASA, ver. 4.5.3).  \nWe used the delivered calibrated $uv$ data \nand made the continuum map by clipping the visibility ($uv$ distance $\\geq 10$k$\\lambda$).\nThe synthesized beam size is $0\\farcs46 \\times 0\\farcs39$ \n(position angle ${\\rm (PA)} = -82\\fdg7$) \nby adopting Briggs weighting of the visibility $({\\rm robust} = 0.5)$, \nand the rms noise level is 0.013~mJy~beam$^{-1}$.\nWe regard the accuracy of the absolute flux calibration \nas 5\\% throughout this paper according to the ALMA Cycle~3 Technical \nHandbook\\footnote{A. Remijan et al., 2015, ALMA Cycle 3 Technical Handbook Version 1.0, ALMA}.\n\nIn addition, we used archival calibrated $uv$ data \nobtained with VLA C- and X-bands,\nand made the continuum maps using CASA.\nWe clipped the visibilities before imaging \nso that all the VLA and ALMA data have the same shortest UV range.\nThe synthesized beam size of the 4.86~GHz map \nis $0\\farcs43 \\times 0\\farcs34$ $({\\rm PA} = -4\\fdg9)$\nby adopting uniform weighting of the visibility,\nand the rms noise level is 0.28~mJy beam$^{-1}$.\nThe synthesized beam size of the 8.44~GHz map \nis $0\\farcs36 \\times 0\\farcs22$ $({\\rm PA} = 34\\fdg1)$ \nby adopting Briggs weighting of the visibility $({\\rm robust} = 0.5)$\nand the rms noise level is 0.20~mJy~beam$^{-1}$.\n\nThe radio continuum maps are shown in Figure~\\ref{fig:radio}.  \nThey clearly show the structure of the jet lobes, \nwhich are located at both sides of the nucleus.  \nThe projected distance between the lobe cores is $\\sim 5$~kpc.  \nWe estimate the kinematic age of the lobes, \nassuming the jet angle to the line of sight of $90\\arcdeg$ \nand a typical expansion speed of radio lobes \n\\citep[$0.1c$; e.g.,][]{mur99,nag06}. \nThis gives the kinematic age of $\\sim8 \\times 10^4$~years.\n\nWe also confirm no additional radio lobes in the field of view (FoV) \n of the VLA and ALMA observations.\nThe FoV of the VLA observation at 4.86~GHz is 9.3~arcmin,\nwhich is the largest among the three observations.\nThis corresponds to the physical size of 221~kpc in radius.\nIf there are radio lobes outside of the FoV, the\nlower limit of their kinematic age is $6.6 \\times 10^6$~years. \nThus Arp~187 has not launched the jet over $6.6 \\times 10^6$~years\nbefore the current 5~kpc size one, or the larger radio lobes, if any, \nalready become undetectable due to rapid energy loss \\citep[e.g.,][]{god17}.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.0cm]{Figure2.pdf}~\n\\caption{\nThe SEDs of the jet lobes.\nThe spectral indexes estimated from two points at 4.86~GHz and 8.44~GHz are \n$\\log f_{\\nu} = 9.2 - 0.84 \\log \\nu$ for the western lobe (black) \nand $\\log f_{\\nu} = 6.8 - 0.59 \\log \\nu$ for the eastern lobe (red).\nThe errors of the flux densities are smaller than the size of symbols.\n}\\label{fig:radioSED}\n\\end{center}\n\\end{figure}\n\n\\begin{deluxetable}{ccccc}\n\\tablecaption{Flux density of the jet lobes\\label{tab:radio}}\n\\tabletypesize{\\footnotesize}\n\\tablewidth{0pt}\n\\tablehead{\n&\\multicolumn{2}{c}{Eastern lobe}&\\multicolumn{2}{c}{Western lobe}\\\\\n\\colhead{Frequency}&\n\\colhead{Peak}&\n\\colhead{Total}&\n\\colhead{Peak}&\n\\colhead{Total}\n}\n\\startdata\n4.86~GHz&12.0&190&13.2&160\\\\\n8.44~GHz&7.53&92&9.54&100\\\\\n133~GHz&0.662&5.3&0.704&7.2\\\\\n\\enddata\n\\tablecomments{\nThe unit of the flux density is mJy.\nThese values are measured in the beam-matched images ($\\theta$ = 0\\farcs47).}\n\\end{deluxetable}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.42\\linewidth]{Figure3A.pdf}~\n\\includegraphics[width=0.47\\linewidth]{Figure3B.pdf}\n\\caption{\n(Left): The optical spectrum of Arp~187 (solid black line) obtained from 6dF galaxy survey.\nX-axis shows the rest-frame wavelength with the unit of \\AA.\nY-axis shows the flux density with the unit of $10^{-15}$~erg~s$^{-1}$~cm$^{-2}$~{\\AA}$^{-1}$.\nThe best-fit stellar spectrum is also shown in the red solid line.\nThe starlight subtracted spectrum is shown in gray line at the zero flux level.\nThe emission line and the telluric masked region is shown in the orange and green, respectively.\n(Right): The spectra and the best-fit models for the Balmer lines, the\n[\\ion{O}{3}]-doublet ($\\lambda4959,\\lambda5007$), \nthe [\\ion{O}{1}]$\\lambda6300$, \nand the [\\ion{N}{2}]-doublet lines.\n(upper-left: H$\\beta$, upper-right: [\\ion{O}{3}]$\\lambda4959,\\lambda5007$\ndoublet, bottom-left:[\\ion{O}{1}]$\\lambda6300$, bottom-right: H$\\alpha$ and [\\ion{N}{2}]-doublet).\nThe total model spectrum is shown as pink solid line.\nThe single Gaussian line component is shown with cyan solid line (for\nH$\\beta$, H$\\alpha$, and the [\\ion{N}{2}]-doublet),\nwhile the double Gaussian line component is shown with two cyan lines\n(for the [\\ion{O}{3}] doublet) as discussed in Section~\\ref{sec:6dFspec}.\n}\\label{fig:6dFspec}\n\\end{center}\n\\end{figure*}\n\nWe also convolved these maps to $0\\farcs47$ angular resolution\nto make Spectral Energy Distributions (SEDs) of the jet lobes.\nThe peak and total flux densities of the eastern and western robes \nare summarized in Table~\\ref{tab:radio}.\nEach peak flux density is measured at the emission peak at 4.86~GHz,\nusing the beam-matched maps.  \nThe radio SEDs of the lobes are shown in Figure~\\ref{fig:radioSED}.\nThe spectral index $\\alpha$ was estimated from two points at 4.86~GHz and 8.44~GHz \nby using a fitting function of $f_\\nu\\propto\\nu^\\alpha$.\nThe SED shows a steep spectral index of $\\alpha\\approx-0.59$ for \nthe eastern lobe (the red dashed line), \nand the flux point at 133~GHz is nearly on the extrapolated line\nfrom the VLA bands.\nThis is consistent with optically thin synchrotron radiation \nthat is expected for the jet emission.\nOn the other hand, the flux density of the western lobes\nat 113~GHz is smaller than what is expected \nfrom the flux densities at 4.86~GHz and 8.44~GHz, \nassuming that the SED can be fitted by $f_\\nu\\propto\\nu^\\alpha$\n(the black dashed line).\nThis suggests the spectral aging for the western jet \n\\citep[e.g.,][]{jam08,sai09},\nwhile the exact age cannot be determined based on the current SED sampling.\n\n\nThere is no clear emission in the nucleus (jet core) at 4.86~GHz and 8.44~GHz, \nalthough the faint emission is seen at 133~GHz.\nThis is not due to the dynamical range limit of VLA.  \nThe 3$\\sigma$ upper limits of the flux densities \nmeasured in the non beam-matched images \nare 0.84~mJy at 4.86~GHz and 0.60~mJy at 8.44~GHz.  \nAssuming the spectral index of the jet core of $\\alpha = -0.5$,\nthe 133~GHz flux density is expected to be 0.16~mJy \nbased on the upper limit of the 8.44~GHz flux density.  \nHowever, the observed 133~GHz flux density is $\\leq0.052$~mJy, \nwhich is three times lower than that extrapolated from the 8.44~GHz flux density.  \nThe non-detection of the continuum emission at 4.86~GHz and 8.44~GHz \ncannot be explained by free-free absorption \nwhich causes the flux excess at high-frequency.  \nThus, the non-detection at several GHz and \nstrong constraint on the 133~GHz flux density\nlead to the conclusion of the presence of a significantly faint core in Arp~187.\n\n\n\\subsection{6dF Optical Spectra}\\label{sec:6dFspec}\n\nWe perform the spectral measurements to investigate the properties of the NLR.\nWe first obtained the archival optical spectrum of Arp~187 from the 6dF galaxy survey \\citep{jon09}. The spectrum covers a range from 3900\\AA~to 7500\\AA~with a \nfiber aperture of 6.7~arcsec (equivalent to $5.3$~kpc).\nSince the 6dF spectra are not flux calibrated on a nightly basis,\nwe have normalized the spectrum based on the 6dF optical photometry at the $R$ band\nof $R=14.33$~mag.\n\nFigure~\\ref{fig:6dFspec} shows the optical spectrum of Arp~187\nin the left panel (solid black line).\nThe observed spectrum does not show any features of\nthe big blue bump originating from the AGN accretion disk, \nnor the broad emission lines \\citep[e.g.,][]{mal82,van01}. \nThus, we conclude that Arp~187 is at least not a type-1 AGN.\n \nWe then perform the spectral fitting to obtain the properties of the NLR.\nFor the spectral fitting, we follow the routine in \n\\cite{bae17} and later \\cite{tob17a}, \nwhere they have performed the spectral fitting to the SDSS spectra.\nWe first subtract the stellar continuum from the spectrum\nusing a best-fit stellar template based on the wide range of the stellar\npopulation models \\citep[MILES;][]{san06} with solar metallicity\nand the age spanning from 60~Mys to 12.6~Gyr.\nIn order to obtain the qualified stellar continuum, \nwe mask the strong emission lines of H$\\beta$, [\\ion{O}{1}], [\\ion{N}{2}],\nH$\\alpha$, and [\\ion{S}{2}] with the masking width of 1300~km~s$^{-1}$,\nwhich is corresponding to ${\\rm FWHM}\\sim 1000$~km~s$^{-1}$.\nWe also mask the [\\ion{O}{3}] doublet with the masking width\nof 3000~km~s$^{-1}$ in order to avoid the possible contamination\nfrom the strong outflow originated from the [\\ion{O}{3}] emission lines.\nIn addition, since some of the telluric absorptions are not well removed, \nwe also masked those wavelength bands from the stellar spectral fitting.\n\nFrom the starlight-subtracted spectrum as shown in gray line\nat the zero flux level in Figure~\\ref{fig:6dFspec}, we then fit the H$\\alpha$, H$\\beta$,\n[\\ion{N}{2}]-doublet, [\\ion{O}{1}]$\\lambda6300$\nusing a single\nGaussian function and [\\ion{O}{3}] doublet ([\\ion{O}{3}]$\\lambda4959, 5007$) \nwith double-Gaussian function using the IDL\/MPFIT code,\nwhich is a $\\chi^2$-minimization routine \\citep{mar09}. \nWe assume that the [\\ion{O}{3}] doublet and the other narrow-lines\nhave independent kinematics, while the [\\ion{O}{3}] doublet\nhas the same velocity and velocity dispersion as each other.\nThe observed spectrum (black line) as well as the fitting results\n is compiled for the entire spectral range (Figure~\\ref{fig:6dFspec}, left panel)\n and for each lines (Figure~\\ref{fig:6dFspec}, right panel).\n\n\\subsubsection{Existence of NLR}\n We first apply the emission-line diagnostics, which give a separation between\n the NLR ionized by AGN and the \\ion{H}{2} region in the starburst galaxies\n \\citep[][]{vei87,kew06}. \nFigure~\\ref{fig:BPT} shows that Arp~187 is \n classified as AGN. Thus, it shows that Arp~187 hosts the NLR, \n one of the large scale AGN indicators.\n \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.3cm]{Figure4.pdf}~\n\\caption{\nClassification of the galaxies into AGN, composite galaxies, and\n\\ion{H}{2} regions\n using line diagnostics diagrams \\citep{kew01,kew06}.\nThe dot-dashed\/dashed line represents the\nrelation of \\cite{kew01}\/\\cite{kau03}, respectively. \nArp~187 is located in the locus of AGN (red cross).\nGray dots represent the data points of SDSS DR7 galaxies \\citep{aba09}.\n}\\label{fig:BPT}\n\\end{center}\n\\end{figure}\n\nSince the diagnostic shows that [\\ion{O}{3}]$\\lambda5007$ (hereafter, [\\ion{O}{3}]) \nemission is dominated from the NLR,\nwe then measure the [\\ion{O}{3}] luminosity ($L_{\\rm [OIII]}$), \nand [\\ion{O}{1}]$\\lambda6300$ (here after, [\\ion{O}{1}]) luminosity ($L_{\\rm [OI]}$) \nto estimate the AGN bolometric luminosity because\n$L_{\\rm [OIII]}$ alone \\citep[e.g.,][]{hec04,hec05,lam13,ued15}\nor the combination of $L_{\\rm [OIII]}$ and \n$L_{\\rm [OI]}$ \\citep[e.g.,][]{net09} is often used as a proxy for AGN power.\nThe observed [\\ion{O}{3}] and [\\ion{O}{1}] luminosities are\n$L_{\\rm [OIII]}=6.2 \\times 10^{40}$~erg~s$^{-1}$\nand $L_{\\rm [OI]}=3.8 \\times 10^{40}$~erg~s$^{-1}$.\nWe also calculate an extinction-corrected, \nintrinsic luminosity of [\\ion{O}{3}] ($L_{\\rm [OIII]}^{\\rm int}$)\nand [\\ion{O}{1}] ($L_{\\rm [OI]}^{\\rm int}$)\nluminosities  from the Balmer decrement of H$\\alpha$\/H$\\beta$\n\\citep[e.g,.][]{cal94,dom13}.\nThe values are\n$L_{\\rm [OIII]}^{\\rm int}=3.2 \\times 10^{42}$~erg~s$^{-1}$ and\n$L_{\\rm [OI]}^{\\rm int}=2.0 \\times 10^{42}$~erg~s$^{-1}$.\nWe calculate the AGN bolometric luminosity $L_{\\rm bol}$ \nusing the both lines \\citep{net09,mat15b} by\n\\begin{equation}\n\\log L_{\\rm bol} = 3.8 + 0.25 \\log L_{\\rm [OIII]}^{\\rm int} + 0.75 L_{\\rm [OI]}^{\\rm int}.\n\\end{equation}\nThe estimated value is $\\log (L_{\\rm bol}\/{\\rm erg}~{\\rm s}^{-1}) = 46.2$,\nwhich reaches to the typical luminosity of the SDSS quasars at $z\\sim1$--2\n\\citep[e.g.,][]{she11}.\n\n\n\\subsubsection{NLR Size}\nThe size of the NLR is also an important indicator \nto estimate the fading timescale of AGN for Arp~187.\nSince the integral field unit (IFU) observations are still not available, we estimate the NLR size from the empirical\nrelationship between the [\\ion{O}{3}] emission size and the [\\ion{O}{3}] luminosity.\n\nThe [\\ion{O}{3}] emission sizes have been measured from either narrowband imaging \\citep{ben02,sch03},\n long-slit spectroscopy \\citep{fra03,ben06,gre11,hai13}, or the IFU observations \\citep{hum10,hus13,liu13,kar16,bae17}.\nThe narrowband imaging is more often used for studying the NLR morphology, while\nthe long-slit spectroscopy and IFU has an advantage to reach shallower sensitivity limits.\n\nWe first estimate the NLR sizes ($R_{\\rm NLR}$)\nusing the size-luminosity relations of \\cite{bae17}\nobtained from the IFU observations of nearby type-2 Seyferts and quasars.\nThis relation has two advantages; 1)\ntheir study uses the extinction-uncorrected [\\ion{O}{3}] luminosity\nfor the size-luminosity relation to reduce the uncertainty of the\nextinction correction to estimate the intrinsic [\\ion{O}{3}] luminosity,\nand 2) they estimate the NLR size based on the line diagnostic diagrams\ndiscussed in Section~\\ref{sec:6dFspec}, which pick up the region\nwhere the [\\ion{O}{3}] emission is dominated from AGN.\nThe size-luminosity relation of \\cite{bae17} is given by\n\n\\begin{equation}\n\\log \\left(\\frac{R_{\\rm NLR}}{\\rm pc} \\right)=\n0.41 \\times \\log \\left( \\frac{L_{\\rm [OIII]}}{{\\rm erg}~{\\rm s}^{-1}}\\right)  - 14.00.\n\\end{equation}\nThe estimated NLR size is $R_{\\rm NLR}\\simeq 530$~pc.\n\nWe also estimate the [\\ion{O}{3}] emitting sizes ($R_{\\rm [OIII]}$) using \nthe size-luminosity relations from the literature.\nNote that the relation is based on the [\\ion{O}{3}] emission size, \ntherefore the estimated size should be the upper bound of \nthe NLR size since the NLR should fulfill the certain emission line ratios\nas shown in Figure~\\ref{fig:6dFspec}.\nWe apply the relation based on the \nnarrow-band imaging observations of quasar population \\citep{sch03} by\n\\begin{equation}\n\\log \\left(\\frac{R_{\\rm [OIII]}}{\\rm pc} \\right) = 0.42 \\times \\log \\left( \\frac{L_{\\rm [OIII]}^{\\rm int}}{{\\rm erg}~{\\rm s}^{-1}}\\right) -14.72.\n\\end{equation}\nThis gives $R_{\\rm NLR} < R_{\\rm [OIII]}\\simeq 1.4$~kpc,\nwhich is $\\sim2.5$ times larger than $R_{\\rm NLR}$ obtained above.\nThis result is in good agreement with the \nprevious IFU studies \\citep{kar16,bae17}, who reported that\n$R_{\\rm [OIII]}$ is on average a few times larger than $R_{\\rm NLR}$.\nIn this study, we assume $R_{\\rm NLR}=530$--$1400$~pc\nas the possible NLR size range.\n\n\n\\subsection{\\textit{Spitzer}\/IRS Spectra}\\label{sec:Spitzer}\n\nSince the detailed spectral analysis of Arp 187 obtained from the \\textit{Spitzer}\/InfraRed Spectrograph (IRS) is compiled in \\cite{ich16},\nhere we provide a brief summary of the findings.\nWe have found that the thermal emission from the AGN is already weak in Arp~187,\nwith an upper bound of the 12~$\\mu$m luminosity of $L_{\\rm 12{\\mu}m} < 1.5\\times10^{42}$~erg~s$^{-1}$.\nThis is equivalent to $L_{\\rm bol}< 2.2 \\times 10^{43}$~erg~s$^{-1}$ using the relation of $L_{\\rm 12{\\mu}m}$ and 14--195~keV luminosities \\citep{ich17a} and the bolometric correction of $L_{\\rm bol}\/L_{14-195}=8.47$ \\citep{ric17a}.\nThe spectrum of Arp~187 shows a marginal detection of the\n[\\ion{O}{4}]~25.89~$\\mu$m line at S\/N$\\sim3$,\nwhich is also widely used AGN indicator.\nThe [\\ion{O}{4}] luminosity is obtained with $L_{\\rm [OIV]}=6.7 \\times 10^{40}$~erg~s$^{-1}$,\nwhich is equivalent to $L_{2-10}=3.0 \\times 10^{43}$~erg~s$^{-1}$ based on the luminosity relations obtained by \\cite{lam10}.\nSince the ionization potential of [\\ion{O}{4}] is higher \n($E_{\\rm p}=54.9$~eV) than other typical \nNLR indicators including [\\ion{O}{3}] line \\citep[e.g.,][]{rig09},\nthe emission size is expected to be smaller than that of [\\ion{O}{3}]\n\\citep{kom08,tob17a}, which is 0.5--1.4~kpc in this study.\nThus, we take the emission size of $<1.4$~kpc as a fiducial value.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.0cm]{Figure5.pdf}~\n\\caption{\n\\textit{WISE} W1 (3.4~$\\mu$m)--W2 (4.6~$\\mu$m) versus \nW2 (4.6~$\\mu$m)--W3 (12~$\\mu$m) two color diagram in the\nunit of Vega magnitude.\nThe red and gray cross represents Arp 187 and the \\textit{Swift}\/BAT AGN sample in the local universe, respectively.\nThe orange and purple area represents the AGN region proposed\nby \\cite{ste12} and \\cite{mat12}, respectively.\n}\\label{fig:WISEcolor}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.47\\linewidth]{Figure6A.pdf}~\n\\includegraphics[width=0.47\\linewidth]{Figure6B.pdf}\n\\caption{\nInfrared light curves of Arp~187 in the W1 and W2 band.\nThe single exposures with the error are shown in blue, \nwhile the median value in each epoch is\nshown in orange circle with the error bars showing \nthe inter-percentage range with 68\\% of the sample.\nThe orange circles are shifted to 50 days after the real values for clarity.\nThe dashed line with green shade is the flux obtained from\n\\textsc{Allwise} with the 1$\\sigma$ (dark green) and 2$\\sigma$ (light green) scatter, \nrepresenting the average magnitude in the \\textsc{Allwise} epoch.\n}\\label{fig:WISElightcurve}\n\\end{center}\n\\end{figure*}\n\n\n\n\\subsection{WISE Colors}\\label{sec:WISEcolor}\nIR color-color selection is useful to identify AGN candidates \nusing the feature of the MIR bump originated from the AGN torus.\nFigure~\\ref{fig:WISEcolor} shows the location of Arp~187 on the \\textit{WISE} color-color plane.\nIt is known that increasing levels of AGN contribution to the MIR emission moves sources\nupwards in the color-color plane with the color cut of $W1-W2>0.8$ \\citep[orange area;][]{ste12}\nand also within the AGN wedge \\citep[blue area;][]{mat12}.\nThe figure clearly shows that Arp~187 does not fulfill either of the criteria above,\nindicating either 1) Arp~187 does not host AGN or 2) the AGN activity is quite weak even if it exists\n\\citep[on average $L_{\\rm 14-195}<10^{43}$~erg~s$^{-1}$, as suggested by][]{ich17a}.\n\n\n\n\n\n\\subsection{WISE IR Time Variability}\\label{sec:WISElightcurve}\n\nIt is worth to trace the IR luminosity drop from the AGN torus after the AGN quenching.\nSince Arp~187 is bright enough to be detected in the single exposure with the \\textit{WISE} mission,\nwe have collected both cryogenic (\\textit{WISE} All-sky database) and \npost-cryogenic multi-epoch photometry (\\textit{WISE}\n3~band and Post-Cryo database) \nfrom the \\textsc{Allwise} \\citep{wri10,mai11} covering the observation\nbetween 2010 January and 2011 February (${\\rm MJD}-55000=200$--$600$),\nand the most recent \\textsc{Neowise} \\citep{mai14} data release 2018 covering the observation between 2013 December 13 and 2017 December 13, UTC \n(${\\rm MJD}-55000=1600$--$3100$).\n\\textit{WISE} has 90-min orbit and conducts $\\approx 12$ observations\nof a source over a $\\approx 1$~day period, and a given location is\nobserved every six months.\n\nIn this study we used standard aperture magnitude (\\texttt{w1\/2mag}).\nWe applied a cross-matching radius of 2 arcsec, \nbased on the positional accuracy with the 2MASS catalog \\citep[see also][]{ich12, ich17a}.\nAfter this matching, 26 and 122 data-points were obtained from the \\textsc{Allwise} and \\textsc{Neowise} epoch, respectively.\nThen we select good quality single-epoch data points based on\nthe good quality frame score (\\texttt{qual\\_frame>0} and \\texttt{qi\\_fact>0}),\nlocating the enough distance from the South Atlantic Anomalies (\\texttt{saa\\_sep>0}),\nand avoiding the possible contamination from the moon (\\texttt{moon\\_masked=0}).\nThis reduces the sample into 24 (\\textsc{Allwise}) and \n109 (\\textsc{Neowise}), respectively.\nFinally, we applied the aperture measurement quality flag (\\texttt{w1\/2flg=0})\nin order to avoid the contamination in the aperture.\nThe final data points are  10 (W1) and 8 (W2) for \\textsc{Allwise}\n43 (W1) and 21 (W2) for \\textsc{Neowise}.\nAll the data points fulfill the flux quality \\texttt{ph\\_qual=A},\nwith a signal-to-noise ratio larger than 10.0.\nWe also checked sources of contamination and\/or biased flux, \ndue to proximity to an image artifact (e.g., diffraction spikes, scattered-light halos,\nand\/or optical ghosts), using the contamination flag \\texttt{cc\\_flags}.\nAll the data points are \\texttt{cc\\_flags=0}, that are unaffected by known artifacts.\n\nFigure~\\ref{fig:WISElightcurve} shows the light curve of W1 (3.4~$\\mu$m) and W2 (4.6~$\\mu$m).  \nThe light curves in the W1 and W2 span a baseline of roughly 2800 days $\\simeq 7.0$~years. Each observation is shown in blue and a binned observations\nwithin one-day is also shown in orange.\nAs shown in Figure~\\ref{fig:WISElightcurve}, \nno clear variability is detected in the \\textsc{Allwise} and \\textsc{Neowise} epoch,\nand also between the two epochs.\nActually, the \\textsc{Allwise} catalog provides a variability flag (\\verb|var_flag|) and\nits value is $\\verb|var_flag|=0$, suggesting that the significant variability between different\nexposures are not detected during the \\textsc{Allwise} survey, which shows the consistent result.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.0cm]{Figure7.pdf}~\n\\caption{\nCorrelation between the [\\ion{O}{3}]$\\lambda5007$ and 12~$\\mu$m luminosities. \nThe black solid line represents the regression line obtained from the bisector fitting of the\ntype-2 AGN sample from SDSS DR12 galaxies (gray dots).\nThe dashed lines represent the time evolution of the AGN luminosity relation\nthe [\\ion{O}{3}]$\\lambda5007$ and 12~$\\mu$m after AGN quenching \\citep{ich17c}; 10~years (black),\n20~years (dark-orange), and 30~years (orange).\nThe location of Arp~187 is shown in red cross.\n}\\label{fig:L12vsO3}\n\\end{center}\n\\end{figure}\n\n\\subsection{Relationship between 12~$\\mu$m and [\\ion{O}{3}] Luminosities}\\label{sec:L12vsO3}\n\nIt is widely known that both 12~$\\mu$m and the\n[\\ion{O}{3}] luminosities are good proxies for AGN power,\nand it is a natural consequence that there is a luminosity \ncorrelations between 12~$\\mu$m and the [\\ion{O}{3}]\nluminosities \\citep[e.g.,][]{tob14}. \n\nFigure~\\ref{fig:L12vsO3} shows the\n relationship between 12~$\\mu$m and [\\ion{O}{3}]$\\lambda5007$\nluminosities using the SDSS DR12 type-2 AGN sample with the cross-matching by\n the \\textit{WISE}~W3 (12~$\\mu$m) band. As expected, two AGN indicators have\n a nice luminosity correlation. However,\nArp~187 does not follow the luminosity relation and is located at the\n  right bottom in the plane, suggesting that Arp~187 is in the locus of dying AGN.\n \\cite{ich17c} estimated the typical cooling timescale of the dusty torus once the AGN is suddenly quenched.\n The thermal MIR dust emission from AGN should decay times of 10--100~years, mainly depending on the\n time-lag caused by the light travel time from the nucleus to the dust sublimation radius,\n while the $L_{\\rm [OIII]}$ remains the same luminosity\n for over $>10^3$~years because of its larger physical size.\n\\cite{ich17c} also calculated how those dying AGN move in the luminosity--luminosity plane of the two AGN indicators,\nand showed that those dying AGN should locate at the bottom right of the luminosity relation.\nThe location of Arp~187 in Figure~\\ref{fig:L12vsO3} is consistent with the idea, and\nbelow the relationship of the line after 30~years of AGN quenching \n(the orange dashed line).\n\n\n\n\\begin{deluxetable*}{lcCccccc}\n\\tablecaption{AGN indicators and their estimated luminosities \\label{tab:AGNindicators}}\n\\tabletypesize{\\footnotesize}\n\\tablecolumns{8}\n\\tablenum{1}\n\\tablewidth{0pt}\n\\tablehead{\n\\multicolumn{8}{c}{Large scale AGN indicators ($>100$~pc)}\\\\\n\\hline\n\\colhead{Type of AGN indicators} &\n\\colhead{AGN sign} &\n\\colhead{$\\log (L_{\\rm AGN}\/{\\rm erg}~{\\rm s}^{-1})$} &\n\\colhead{$\\log (L_{\\rm bol}\/{\\rm erg}~{\\rm s}^{-1})$} &\n\\colhead{$\\log \\lambda_{\\rm Edd}$} & \n\\colhead{Size (radius)} & \n\\colhead{$t_{\\rm retro}\\tablenotemark{a}$}  &\n\\colhead{Reference}\n}\n\\startdata\njet lobe & $\\checkmark$ & $\\cdots$  &  $\\cdots$ & $\\cdots$\n& $2.5$~kpc & $8.1 \\times 10^{4}$~yr & Section~\\ref{sec:radio}\\\\\nNLR ([OIII]$\\lambda5007$) & $\\checkmark$ & $\\log L_{\\rm [OIII]}^{\\rm int}=42.51$  &\n $46.15$ & $-0.77$ &  $0.5$--$1.4$~kpc & $(1.7$--$4.6) \\times 10^{3}$~yr & Section~\\ref{sec:6dFspec}\\\\\nNLR ([OIV]25.89~$\\mu$m) & $\\checkmark$ & $\\log L_{\\rm [OIV]}^{\\rm int}=40.83$ &\n$44.78$  & $-2.15$ & $\\leq 1.4$~kpc & $\\leq 4.6 \\times 10^{3}$~yr & Section~\\ref{sec:Spitzer} \\\\\n\\hline\n\\multicolumn{8}{c}{Small scale AGN indicators ($<10$~pc)}\\\\\n\\hline\nDust torus (\\textit{Spitzer}\/IRS Spec) & \\text{\\sffamily X} & $\\log L_{\\rm 12 {\\mu}m}<42.18$ &\n $<43.34^{a}$ & $<-3.59$ & $\\simeq 10$~pc & $10$--$100$~yr & Section~\\ref{sec:Spitzer} \\\\\nDust torus (\\textit{WISE} IR colors) & \\text{\\sffamily X} & $\\cdots$ & $\\cdots$ & $\\cdots$ &\n$\\cdots$ & $10$--$100$~yr & Section~\\ref{sec:WISEcolor} \\\\\nDust torus (\\textit{WISE} IR light curve) & \\text{\\sffamily X} & $\\cdots$ & $\\cdots$ & $\\cdots$ &\n$\\cdots$ & $10$--$100$~yr & Section~\\ref{sec:WISElightcurve} \\\\\nDust torus ($L(\\rm [OIII])$ vs. $L_{\\rm 12 {\\mu} m}$) &  \\text{\\sffamily X} & $\\log L_{\\rm 12 {\\mu}m}<42.18$ &\n $<43.34^{a}$ &  $<-3.59$ & $\\simeq10$~pc & $>30$~yr & Section~\\ref{sec:L12vsO3}\\\\\njet core & \\text{\\sffamily X} & $\\log L_{5 {\\rm GHz}}<37.57$ & $\\cdots$ & $\\cdots$ & $\\leq 1$~pc \n& Current & Section~\\ref{sec:radio} \\\\\nBLR (optical Spec) & \\text{\\sffamily X} & $\\cdots$ & $\\cdots$ & $\\cdots$ &  $<0.1$~pc & Current (1--10~yr) & Section~\\ref{sec:6dFspec}\\\\\nElectron corona (X-ray) &  \\text{\\sffamily X} & $\\log L_{2-10}<43.44$ & $<44.38^{a}$ & $<-2.55$ & $\\ll 0.1$~pc & Current & Section~\\ref{sec:X-ray} \\\\\n\\hline\n\\multicolumn{8}{c}{Other Relations}\\\\\n\\hline\nBH fundamental plane (FP) & $\\cdots$ & $\\log L_{2-10}< 39.60$ & $<40.90^{a}$ & $<-6.03$ & $\\cdots$ &  Current &\nSection~\\ref{sec:BHfundamental} \\\\\n\\enddata\n\\tablecomments{The list of AGN indicators\nand the retrospective time ($t_{\\rm retro}$) of the AGN indicator.\nExcept the jet lobe, the all time is the\nlight crossing time with the definition of $t_{\\rm lc}={\\rm size}\/c$.\n\\tablenotetext{a}{From the X-ray luminosities, we use the bolometric correction of $L_{\\rm bol}\/L_{2-10}=20$ \\citep{ric17a}.\nThen, under the assumption of the photon index of $\\Gamma=1.8$ \\citep{ric17}, \nwe use the conversion among the X-ray luminosities\nof $L_{2-10}\/L_{14-195}=0.42$ and $L_{2-10}\/L_{0.5-2}=1.57$.\nWe also use the luminosity relations between $L_{\\rm 12{\\mu}m}$ and $L_{14-195}$ of \n$\\log (L_{\\rm 12{\\mu}m}\/10^{43}~{\\rm erg}~{\\rm s}^{-1})=-0.21+1.05 \\log (L_{14-195}\/10^{43}~{\\rm erg}~{\\rm s}^{-1})$\nto estimate the bolometric luminosity from $L_{\\rm 12{\\mu}m}$ \\citep{ich17a}.\nFor the estimates of the Eddington luminosity ratio $\\lambda_{\\rm Edd}=L_{\\rm bol}\/L_{\\rm Edd}$,\n we apply the black hole mass of $M_{\\rm BH}=6.7 \\times 10^8$~$M_{\\odot}$, which leads to \n $L_{\\rm Edd}=8.4 \\times 10^{46}$~erg~s$^{-1}$.}\n}\n\\end{deluxetable*}\n\n\n\n\\subsection{X-ray Observations}\\label{sec:X-ray}\n\nAlthough the X-ray observations give us the current AGN activity without the concern of the \nobscuration up to $\\log N_{\\rm H} \\simeq 24$ \\citep[e.g.,][]{ric15,ric17}, \nwe have not found any previous X-ray observations for Arp~187,\nand therefore only the upper bounds obtained from the available all-sky X-ray surveys.\n\nArp~187 is not in the catalog of \\textit{Swift}\/BAT 105 month all-sky survey with the limiting flux at the 14--195~keV band of\n$f_{\\rm 14-195}^{(\\rm lim)}=8.0\\times 10^{-12}$~erg s$^{-1}$ cm$^{-2}$ \\citep{oh18}.\nThis gives a very shallow upper bound of $L_{14-195}< 2.8 \\times 10^{43}$~erg~s$^{-1}$,\nwhich is equivalent to $L_{\\rm bol}<2.4 \\times 10^{44}$~erg~s$^{-1}$ under the assumption\nof $L_{\\rm bol}\/L_{14-195}=8.47$ \\citep{ric17a}.\n\nThe \\textit{ROSAT} All Sky Survey also shows non detection at the energy range of 0.5--2.0~keV\n\\citep[$f_{\\rm lim}=2.5\\times 10^{-12}$~erg s$^{-1}$ cm$^{-2}$;][]{vog99}. This also gives a shallow upper bound\nof $L_{\\rm 0.5-2}< 8.8 \\times 10^{42}$~erg~s$^{-1}$, which is equivalent to \n$L_{\\rm bol}<2.8 \\times 10^{44}$~erg~s$^{-1}$ using $L_{2-10}\/L_{0.5-2}=1.57$\nunder the assumption of the photon index $\\Gamma=1.8$ \\citep{ric17} and $L_{\\rm bol}\/L_{2-10}=20$ \\citep{ric17a}.\n\nOther X-ray catalogs such as\n the third \\textit{XMM-Newton} serendipitous source catalog \\citep[3XMM-DR7;][]{ros16}, \nChandra Source Catalog \\citep[CSC Release 2.0;][]{eva10} do not contain the observations of Arp~187.\n\n\n\n\n\\subsection{Black Hole Fundamental Plane}\\label{sec:BHfundamental}\n\nThe fundamental plane of the black hole gives a relationship among three physical quantities\nof $L_{2-10}$, core $L_{\\rm 5GHz}$, and the black hole mass $M_{\\rm BH}$\n\\citep[e.g.,][]{mer03,fal04,yua14}.\n\\cite{ich16} discussed that once the upper-bound of $L_{\\rm 5GHz}$ is given, \nwe can estimate the upper-bound of $L_{2-10}$ since the black hole mass in Arp~187 is \nestimated to be $M_{\\rm BH}=6.7 \\times 10^8$~$M_{\\odot}$.\nThe upper bound of $L_{\\rm 5GHz} \\leq 3.7 \\times 10^{37}$~erg~s$^{-1}$\nobtained by the VLA observation gives $L_{2-10} \\leq 4.0 \\times 10^{39}$~erg~s$^{-1}$\nusing the relation of \\cite{yua05}.\nThis is equivalent to $L_{\\rm bol} \\leq 8.0 \\times 10^{40}$~erg~s$^{-1}$,\nindicating that the central engine is already quenched.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.7\\linewidth]{Figure8.pdf}~\n\\caption{\nLongterm light curve of Arp~187 based on the AGN indicators with multi physical scales.\nSee Table~\\ref{tab:AGNindicators} for the details of the each AGN indicator.\n}\\label{fig:LC}\n\\end{center}\n\\end{figure*}\n\n\n\\section{Discussion}\\label{sec:discussion}\n\n\\subsection{AGN Indicators and Scenarios of Current AGN Activity}\n\nOur multi-wavelength measurements of the AGN indicators show that we have successfully found\nthe large scale ($>100$~pc) AGN indicators while not for the small scale ones with $\\leq 10$~pc.\nTable~\\ref{tab:AGNindicators} summarizes the AGN indicators for Arp~187.\n\nThere are two possible scenarios that can account for these seemingly contradictory results \nbetween the larger ($>100$~pc) and smaller ($\\leq 10$~pc) scales.\n One is that the AGN activity still exists but the emission is highly obscured\n  along the line of sight, while being unobscured towards the jet and\/or the NLR direction\n  \\cite[e.g.,][]{sar18a}.\n The other is that the AGN activity in Arp~187 has already been quenched,\n while the narrow line regions still remain bright due to the light-travel time\n from the central engine.\n The first scenario is unlikely for Arp~187 because of the two reasons.\n One is the absence of the AGN torus emission in the \\textit{Spitzer}\/IRS spectra as well as the other torus indicators obtained from \\textit{WISE}. \n If the central engine is highly obscured, then the most of the\n emission is absorbed by the dust, and then it re-emits and produces the bump in the mid-IR \\citep[e.g.,][]{ich14}, which we could not find.\nAnother point is the absence of the jet core, which is irrelevant to the concern of absorption and its existence is the ubiquitous trend for AGN \\citep{bla79,had11}\nwhile there are some rare exceptions \\citep{cor87,dwa09}.\nThus, our results prefer the scenario of dying AGN.\n\nOne might argue that the absence of the big blue bump and the\nbroad emission line region (BLR) is\n due to the absorption by the dusty torus \\citep[e.g.,][]{ant93,nag02}.\n It is true in general argument, but since the torus emission is already weak for Arp~187\n as discussed in Section~\\ref{sec:Spitzer}, the most plausible explanation is that\n the central engine including the accretion disk and the BLR becomes \n already very faint or might be diminished.\n  \nThe disappearing timescale of the BLR is poorly known at current stage.\nOne possible implication of this comes from the observations of changing-look AGN,\nwhich show the AGN type change in human timescale \\citep[e.g.,][]{toh76,ant83,all85,lam15,rua16,mac16,yan17}.\nOne of the findings of the $\\sim10$~years long monitoring of changing-look AGN is that\nthe BLR can disappear within a few years, while\nthe physical mechanism of the disappearance is still under debate \\citep{law18}. \nThus, in this paper we compile the timescale of BLR as 1--10~years in Table~\\ref{tab:AGNindicators}.\n\n\n\\begin{deluxetable*}{cccc}\n\\tablecaption{Observational Properties of dying AGN, \nfading AGN, and changing-look quasars\\label{tab:comparison}}\n\\tabletypesize{\\footnotesize}\n\\tablewidth{0pt}\n\\tablehead{\n& dying AGN (Arp~187) & fading AGN & changing-look quasar\n}\n\\startdata\n  & AGN whose current (small scale)  & AGN with weaker current $L_{\\rm AGN}$ & quasars with broad Balmer line  \\tablebreak \n Definition &AGN signature  is dead,  but   &  compared to that of & (dis)appearance associated with\\tablebreak\n & past AGN signature is still alive. & past AGN signatures. &  continuum change by a factor of $\\sim 10$.\n \\\\\\hline\nSmall scale ($<10$~pc) AGN signs & dead & alive (but weak) & alive \\\\\nLarge scale ($>100$~pc) AGN signs & alive & alive & alive\\\\\nJet core features & no & yes (?)$^{(\\rm B1)}$ & \\nodata \\\\\nJet lobe features & yes & \\nodata & \\nodata \\\\\n$\\Delta L_{\\rm AGN}^{\\rm (a)}$& $10^{3-5}$ & $10^{1-4}$ & $\\sim 10$\\\\\n$\\Delta t^{\\rm (b)}$ & $\\leq 10^4$~yr & $10^{4-5}$~yr (using EELR$^{(1)}$) & $\\sim 1$--$10$~yr$^{\\rm (C1,C3,C4,C6)}$\\\\\n$L_{\\rm AGN}$ (current) & $< 10^{42}$~erg~s$^{-1}$ & $10^{43-45}$~erg~s$^{-1}$ &\n$> 10^{43}$~erg~s$^{-1}$\\\\\n$L_{\\rm AGN}$ (before fading) & $\\sim 10^{46}$~erg~s$^{-1}$ & $10^{44-46}$~erg~s$^{-1}$ &\n$>10^{44}$~erg~s$^{-1}$ \\\\\nRedshift ($z$) & 0.04 & $0.01$--$0.3^{\\rm (B5, B10)}$ & $0.01$--$1.0^{\\rm (C1,C3,C4,C6)}$\\\\\n\\multirow{2}{*}{Origin of variability} & \\multirow{2}{*}{viscous} & \n\\multirow{2}{*}{viscous} & unknown (thermal?$^{\\rm (C8-C10)}$ or\\tablebreak\n&&&  magnetically elevated disk?$^{\\rm (C7)}$)\\\\ \nHost galaxies& merger remnant & merger system\/remnants (?)$^{\\rm (B5)}$ & \\nodata \\\\\nNumber of sources found & \\nodata & $\\sim20$ sources$^{\\rm (B1-B13)}$ & $>40$ sources$^{\\rm (C1-C6)}$\\\\\n\\multirow{2}{*}{Most known object names} & Arp~187$^{(\\rm A1)}$ & IC~2497 (Hanny's Voorwep)$^{\\rm (B1-B8)}$ & SDSS~J0159+0033$^{\\rm (C1)}$ \\tablebreak \n& (NGC 7252?$^{\\rm (A2)}$) & Teacup galaxy$^{\\rm (B5,B6,B12)}$, etc. & SDSS~J1011+5442$^{\\rm (C2)}$, etc. \\\\\n\\enddata\n\\tablecomments{\n(1) EELR stands for extended AGN photo-ionized emission line region\nwith the physical scale of $\\sim10$~kpc.\n(a) observed or inferred AGN luminosity decline\n(b) the timescale of the AGN luminosity decline of $\\Delta L_{\\rm AGN}$.\n``$\\cdots$'' in the column represents\nthat there are no clear observations or consensus from the literature.\n}\n\\tablerefs{\ndying AGN: \n(A1) \\cite{ich16}, (A2) \\cite{sch13b}; \nfading AGN: \n(B1) \\cite{joz09},\n(B2) \\cite{lin09}, \n(B3) \\cite{sch10}, \n(B4--B6) \\cite{kee12,kee15,kee17}, \n(B7,B8) \\cite{sar18a,sar18b}, \n(B9,B10) \\cite{sch13,sch16},\n(B11) \\cite{kaw17},\n(B12) \\cite{vil18}, \n(B13) \\cite{wyl18}; \nchanging-look quasar:\n(C1) \\cite{lam15}, (C2) \\cite{run16}, (C3) \\cite{rua16}, \n(C4) \\cite{mac16}, (C5) \\cite{gez17}, (C6) \\cite{yan17}, \n(C7) \\cite{dex18}, (C8) \\cite{nod18}, (C9) \\cite{ros18}, (C10) \\cite{ste18}\n}\n\\end{deluxetable*}\n\n\n\n\n\\subsection{Luminosity Decline and Corresponding Timescales}\\label{sec:timescale}\n\nOne of our goal is to constrain how rapidly the AGN has faded in Arp~187.\nWe summarize the long-term luminosity decline of Arp~187 in Figure~\\ref{fig:LC}.\nThis figure shows that the AGN has experienced the drastic luminosity decline \nby the factor of $10^{4}$ times\nwithin $\\simeq 10^{4}$~years.\n\n\nOne question arises as to how to describe the sudden quenching of AGN within $\\simeq 10^{4}$~years and how this timescale is connected to\nthe physical properties of the accretion disk of AGN.\nWe here consider three possible timescales: the orbital, thermal, and the\nviscous (inflow) timescales, by following the discussions of \\cite{cze06} and \\cite{ste18}.\nIf the accretion flow is roughly Keplerian, the dynamical\ntimescale of the accretion disk is given by $t_{\\rm dyn} \\sim 1\/\\Omega_{\\rm K}$,\nwhere $\\Omega_{\\rm K}$ is the Keplerian orbital angular frequency.\nThe thermal time scale, which is corresponding of the disk cooling,\nis given by $t_{\\rm th} \\sim 1\/\\alpha \\Omega_{\\rm K}$, where $\\alpha$\nis the viscosity parameter \\citep{sha73,kat08}.\nThe viscous timescale is also given by\n$t_{\\rm vis} \\sim (h\/R)^{-2} t_{\\rm th}$,\nwhere  $h\/R$ is the disk aspect ratio.\nThe numerical simulations derive the estimation of $\\alpha \\sim 0.03$\n\\citep{hir09,dav10},\ntherefore we use $\\alpha_{0.03}=\\alpha\/0.03$ as the fiducial value.\nThe disk aspect ratio is typically very small and \nwe assume $h\/R \\sim 0.05$ by following \\cite{ste18}.\n\n\nWe first determine the boundary accretion disk radius $R$, \nwithin which it emits the UV radiation,  \nwhere UV represents the wavelength of $\\lambda < 3500$~\\AA,\nand the typical wavelength range of the big blue bump \n\\citep[e.g.,][]{mal82,van01}.\nUsing the obtained parameter for Arp~187, \nthe BH is estimated to be $M_{\\rm BH}=6.7 \\times 10^8$~$M_{\\odot}$\nand the maximum Eddington ratio in this study is $\\lambda_{\\rm Edd}= 0.17$\nobtained from the NLR.\nThe gravitational radius is given by $R_{\\rm g}=2GM_{\\rm BH}\/c^2 \\sim 13$~AU.\nthe AGN bolometric luminosity is governed by $L_{\\rm bol}=\\eta \\dot{M} c^2$,\nwhere $\\eta$ is the radiation efficiency.\nThe typical value is estimated as $\\eta \\sim 0.1$ \\citep{sol82}.\nThe mass accretion rate $\\dot{M}$ is therefore given by \n\\begin{equation}\\label{Eq:dotM}\n\\dot{M} \\approx 2.8 M_{\\odot} {\\rm yr}^{-1} \\eta_{0.1}^{-1}\nM_{\\rm BH,0} \\lambda_{\\rm Edd,0.17}.\n\\end{equation}\nhere $\\eta_{0.1}=\\eta\/0.1$,\n$M_{\\rm BH,0}=M_{\\rm BH}\/6.7\\times 10^{8}M_{\\odot}$,\nand $\\lambda_{\\rm Edd,0.17}=\\lambda_{\\rm Edd}\/0.17$.\n\n\nAssuming a standard thin-disk AGN model,\nthe disk radius $R$ is linked to the disk temperature $T$ in Equation of (3.57)\nof \\cite{kat08} written by\n\\begin{equation}\\label{Eq:RvsT}\nR = \\frac{3GM\\dot{M}}{8\\pi \\sigma T^4},\n\\end{equation}\nwhere $\\sigma$ is a\nis a Stefan--Boltzmann constant. \nCombining Equation~(\\ref{Eq:RvsT}) and (\\ref{Eq:dotM}) with Wien's law \n($(\\lambda\/{\\mbox{\\AA}}) = 2.9 \\times 10^7\/T$), the typical radius $R$ can be given by\n\\begin{align}\n\\frac{R}{R_{\\rm g}} = 100 \\eta_{0.1}^{-1\/3} M_{\\rm BH,0}^{-1\/3}\n\\lambda_{\\rm Edd,0.17}^{1\/3}\n\\lambda_{3500}^{4\/3}\n\\end{align}\nwhere $\\lambda_{3500}=\\lambda\/3500 \\mbox{\\AA}$.\nThus we apply $R_{100}=R\/100R_{\\rm g}$ as a fiducial value\nfor the UV emitting disk size.\n\n\nWe can then parametrize the disk timescales as\n \\begin{align}\nt_{\\rm dyn} &\\sim 100~{\\rm day}\\times M_{\\rm BH, 0} R_{100}^{3\/2} \\\\\nt_{\\rm th} &\\sim 10~{\\rm yr}\\times \\alpha_{0.03}^{-1}M_{\\rm BH, 0} R_{100}^{3\/2} \\\\\nt_{\\rm vis} &\\sim 1.6\\times10^4~{\\rm yr} \\times \\left(\\frac{h\/R}{0.05}\\right)^{-2}\n\\alpha_{0.03}^{-1}M_{\\rm BH, 0} R_{100}^{3\/2} \n \\end{align}\n\n\n Since Arp~187 does not show the big blue bump and the torus emission\n (whose source of nutrition is the UV photons) anymore at least the\n last a few 10~years (see also Figure~\\ref{fig:LC}), \n the dynamical and the thermal\n time scale are unlikely and too short to be happening.\n\nOn the other hand, the viscous timescale seems \n to be a little longer compared to the quenching timescale of $<10^4$~years.\nHowever, since the NLR traces higher energy UV photons with $>10$~eV,\n the corresponding disk region becomes more inner with $R \\lesssim 10$~$R_{\\rm g}$, whose viscous timescale is $\\lesssim 500$~yrs.\nThis is consistent with our strongest upper-bound of $\\sim3000$~yrs (see Table~\\ref{tab:AGNindicators}).\nThus, our rough estimation suggests that the viscous timescale \nmost closely matches the observed quenching timescale.\n\n\n\\subsection{Comparison between Dying AGN and Fading AGN}\n\n\nThe absence of small scale AGN signatures and the\ntimescale discussed in Section~\\ref{sec:timescale}\nsupport that Arp~187 has been in a later fading phase,\nor ``dying'' phase compared to other fading AGN since the UV emitting\nregion in Arp~187 likely disappeared, and the large scale AGN indicators\nare observable as remnant signs of AGN. On the other hand,\nthe previously reported fading AGN are considered to be\nearlier fading stage since they still host \neither of clear UV, MIR, or X-ray emission in the core, suggesting that \nthe UV emitting region is still alive \\citep[e.g.,][]{kee15,kee17,sar18a}.\nWe have summarized the comparison of \nproperties of our dying AGN and fading AGN in Table~\\ref{tab:comparison}.\n\n\nOne important question is how many \nsuch ``dying'' AGN have been already \nreported in the  population of fading AGN from the literature.\nWhile most of fading AGN still show MIR or X-ray emission in the core,\none fading AGN in NGC~7252 might fulfill ``dying'' AGN criterion.\n\\cite{sch13b} showed that NGC~7252 hosts large [OIII]$\\lambda5007$\nbright nebulae which belong to a stream of tidal-tail gas falling back \nto the host galaxy. The bright [OIII] nebulae require the AGN luminosity\nlarger than $L_{\\rm bol}>5 \\times 10^{42}$~erg~s$^{-1}$ while\nthe current X-ray upper-bound gives $L_{\\rm bol}<5 \\times 10^{40}$~erg~s$^{-1}$.\nConsidering the nebulae distance, NGC~7252 might have experienced\na luminosity decline by two orders of magnitude over the past $10^{4-5}$~years,\nand the current AGN activity is well below $L_{\\rm bol}<10^{42}$~erg~s$^{-1}$,\nsuggesting that the central engine is already dead.\n\nNote that the past inferred AGN luminosity\nis completely different between Arp~187 and NGC~7252.\nArp~187 reached a quasar-like luminosity with $L_{\\rm bol}>10^{46}$~erg~s$^{-1}$, \nwhereas that of NGC~7252 is well below the quasar level or more likely\nSeyfert level luminosity with $L_{\\rm bol}>5 \\times 10^{42}$~erg~s$^{-1}$.\n\\cite{sch13b} also reported that the central gas disk of NGC~7252 \ncontains the large amount of molecular gas with $>10^9$~$M_{\\odot}$,\nsuggesting that the AGN feedback activity have failed to remove the gas\nin the host galaxy. Therefore the gas content difference between \nNGC~7252 and Arp~187 would be a good testbed to investigate the effect of AGN feedback since these two galaxies have already experienced the one-cycle of AGN \nactivity for at least $10^{5}$~years \\citep{sch15} but with different AGN luminosity.\n\n\n\n\\subsection{Comparison of the Causes of Luminosity Changes\nbetween Dying AGN and Changing-Look Quasars}\n\nIt is worthwhile to note the difference of the \naccretion mechanism between our dying AGN in Arp~187\nand a recently discovered class of ``changing-look quasars''\nin which the strong UV bump and broad emission lines\nassociated with optically bright quasars either appear or disappear on timescales of years \\citep[e.g.,][]{lam15,mac16,gez17,yan17}.\nThe physical processes causing these changing-look phenomena\nare hotly debated, but the physical changes in the accretion disk\nis the likely cause rather than changes in obscuration\n\\citep[e.g.,][]{dex18, law18}.\n\n\\cite{ste18} recently discovered one changing-look quasar\nWISE J1052+1519, and carefully discussed the possible\ndisk timescales matching the year-timescale.\nThey found that the dynamical timescale\nis several weeks, which is therefore too shorter, \nwhile the viscous timescale, which would be responsible\nfor the luminosity change of dying AGN, is far too long.\nInstead, a few year long thermal timescale would\nbe a plausible one matching the observed year scale variability.\nThe similar origin is also proposed for a \ndifferent changing-look AGN Mrk~1018 \\citep{nod18}.\nTherefore, both dying and changing-look AGN show\nthe luminosity change, but their luminosity changes are likely\nbased on the different physical mechanisms of the accretion disk.\nThose property differences between dying AGN and changing-look quasars\nare also summarized in Table~\\ref{tab:comparison}.\n\n\n\\subsection{Comparison between Dying AGN and Remnant Radio Sources}\nOur study suggests that the absence of radio core at the center of galaxies\nwould be a good indicator for searching for a dying AGN.\nWhile they are very rare, some authors have already found candidates\nof radio galaxies without clear radio core signs, or so called remnant radio sources.\n\\cite{cor87} showed that IC~2476 has double radio lobes\nwith a separation of 560~kpc, but without a\nclear radio core at the location of the host galaxy.\nIn addition, \\cite{dwa09} have conducted a search for remnant radio sources\nusing VLA 74~MHz survey (VLSS; 80~arcsec spatial resolution) \nand NRAO VLA Sky Survey (NVSS; 45~arcsec spatial resolution)\nthrough the search of very steep radio sources whose\nspectral index is $\\alpha < -1.8$ (where\n$f_{\\nu} \\propto \\nu^{\\alpha}$) between 74~MHz and 1.4~GHz.\nOut of the $\\sim 10^4$ parent sample, they found 10 such candidates\nand the spectral age estimation of jet lobes gives the fading age of $>10$~Myr.\n\n\nThe timescale found in remnant radio sources is\nat least two orders of magnitude longer than the fading phase traced\nfor Arp~187.\nThis discrepancy is natural because their steep spectral selection method is \n sensitive to longer jet age with $>10^7$~years\nat the frequency of $\\sim 100$~MHz \\citep[e.g.,][]{jam08}\nand their moderate spectral resolution of 80~arcsec is equivalent to\nthe physical angular size of $\\sim260$~kpc, or the corresponding kinematic\nage of $t_{\\rm dyn} \\sim 8 \\times 10^6$~years at $z \\sim 0.2$, where\nmost of the sample are found.\nThus, previous radio studies are sensitive to trace\nmuch longer AGN activities with $>10$~Myr.\nConsidering the typical quasar lifetime of $\\sim30$~Myr \\citep[e.g.,][]{\nhop06,ina18}, remnant radio sources are more suitable to\ntrace a comparable timescale of the \nAGN lifetime rather than the AGN fading \ntimescale traced in this study.\n\n\nRecently, higher spatial resolution\nsurveys are ongoing using  Low-Frequency Array \\citep[LOFAR;][]{van13}.\nLOFAR covers the largely unexplored low-frequency range between 10--240~MHz,\nand has the resolution of $\\sim6$~arcsec at 150~MHz.\nThis would give an opportunity to search for smaller sized, which is equivalent to younger,\ndying radio sources at the age of $\\sim1$~Myr, and the initial-stage surveys have already found new remnant radio source candidates \\citep[e.g.,][]{mah18}\nand future LOFAR surveys would give us a \nmore statistically significant number\nof such younger remnant radio sources and they would help to\ncreate a more complete picture of both of AGN lifetime and dying phase\n\\citep[also see a recent review by][]{mor17}.\n\n\n\n\\subsection{Future observations}\nThere is some room for the further constraints of the AGN activities of Arp~187.\n\\textit{JWST}\/MIRI will give us the nuclear MIR spectra with the least host galaxy contamination\nwith the great sensitivity. The X-ray satellite \\textit{NuSTAR} is going to constrain the\ncurrent AGN activity.\n  Recently, thanks to the great sensitivity at $E>10$~keV,\n   \\textit{NuSTAR} revealed that a fading AGN candidate in IC~2497 \n  is actually a Compton-thick AGN \\citep{sar18}, \n  whose nuclear X-ray emission could not be discovered in the previous X-ray\n satellites such as \\textit{Chandra}, \\textit{XMM\/Newton}, and even with \\textit{Suzaku} \\citep{sch10}.\nAlthough the same discovery is unlikely for Arp~187 since the MIR emission is known to be\nconsiderably weak, \\textit{NuSTAR} will constrain the nuclear activity down to \n $\\log (L_{2-10}\/{\\rm erg}~{\\rm s}^{-1}) \\simeq 42$ even with the Compton-thick absorption of \n$\\log N_{\\rm H} \\simeq 24.3$. Finally, the optical or near-IR IFU will also give us the detailed NLR size\nwhich is poorly constrained with the current study. \n\n\n\\section{Conclusion}\n\nWe have compiled the multi-wavelength AGN signatures of dying AGN candidate Arp~187,\nbased on the combinations of the newly conducted ALMA observations\n as well as the archival VLA 5--10~GHz data,\n6dF optical spectrum, \\textsc{Neowise} and \\textsc{Allwise} IR data.\nOur results show that the AGN in Arp~187 is a bona fide dying AGN,\nwhose central engine is already dead,\n but the large scale AGN indicators are still observable as the remnant \n of the past AGN activity.\nThe central engine of Arp~187 has experienced the drastic luminosity decline\nby a factor of $10^{3-5}$ within the last $10^{4}$~years.\nOur rough estimation suggests that the viscous timescale\nmost closely matches the obtained timescale in this study.\nThis supports that Arp~187 has been in more later \nfading phase whose UV emitting region in the accretion disk\nis likely to be almost disappeared, while other fading AGN show clear signs\nthat UV emitting region is still alive.\n\n\\acknowledgments\n\nWe thank the anonymous referee for helpful suggestions that strengthened the paper.\nWe thank Kazunori Akiyama, Luca Comisso, Kohei Inayoshi, Masatoshi Imanishi, \nMitsuru Kokubo, Hiroshi Nagai, and Hirofumi Noda for fruitful discussions.\nWe also would like to thank Jaejin Shin for giving us the spectral fitting result of the optical spectrum.\n\nThis paper makes use of the following ALMA data: ADS\/JAO.ALMA\\#2015.1.01005.S. \nALMA is a partnership of ESO (representing its member states), NSF (USA) \nand NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) \nand KASI (Republic of Korea), in cooperation with the Republic of Chile.  \nThe Joint ALMA Observatory is operated by ESO, AUI\/NRAO and NAOJ.\nIn addition, this paper includes observational material taken with VLA, \nwhich is one of the NRAO instruments.\nThe National Radio Astronomy Observatory is a facility \nof the National Science Foundation operated \nunder cooperative agreement by Associated Universities, Inc.\nThis paper also makes use of data products from the \\textit{WISE}, \nwhich is a joint project of the University of California, Los Angeles, \nand the Jet Propulsion Laboratory\/California Institute of Technology, \nfunded by the National Aeronautics and Space Administration.\n\nThis work is supported by Program for Establishing a Consortium\nfor the Development of Human Resources in Science\nand Technology, Japan Science and Technology Agency (JST) and\n is partially supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (18K13584; KI, 18J01050; YT, 16K17672; MS). \n K.M. is supported by JSPS Overseas Research Fellowships.\n\n\n\\vspace{5mm}\n\\facilities{ALMA, VLA, \\textit{WISE}, \\textit{Spitzer}, \\textit{Chandra}, \\textit{XMM-Newton}, \\textit{Swift}\/BAT}\n\n\n\n\\software{astropy \\citep{ast13}, Matplotlib \\citep{hun07}, Pandas \\citep{mck10}\n          }\n\n\n\n\n\n\n\n\n\\bibliographystyle{aasjournal}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\n\\medskip\nThe class of affine semigroup rings is rich with examples that combine the flavors of convex geometry and commutative algebra.  The structure of the semigroup ring $K[S]$ is\n intimately related to the structure of the affine semigroup $S$ and the cone $\\ensuremath{\\mathrm{pos}}(S)$ spanned \n by $S$.  For example, it is well known that $K[S]$ is normal if and only if $S$ contains all integral points\n  of $\\ensuremath{\\mathrm{pos}}(S)$ (see \\cite{BH}).   Normal affine semigroup rings are Cohen-Macaulay by a theorem of \n  Hochster \\cite{Hoch}.  Ishida \\cite{Ish} characterized the S$_2$-ness of $K[S]$ in terms of $S$ and\n   the facets of $\\ensuremath{\\mathrm{pos}}(S)$.   In \\cite{GW1} Goto and Watanabe announced a characterization of those affine semigroups $S$ for which  $K[S]$ is Cohen-Macaulay; Hoa and Trung gave a corrected characterization  in terms of both $S$ and  the cone spanned by $S$ over the rational numbers in \\cite{HT} and \\cite{HT2}.  \n  In an earlier paper \\cite{Vi} the author characterized those affine semigroup rings \n   which satisfy Serre's condition R$_1$.   In this note we characterize those affine semigroup\n    rings $K[S]$ over an arbitrary field  $K$ which satisfy condition $\\mathrm{R}_{\\ell}$\n     of Serre.  Our characterization is in terms of the face lattice of the positive cone\n      $\\ensuremath{\\mathrm{pos}}(S)$ of $S$. We start by recalling some basic facts about the faces of\n       $\\ensuremath{\\mathrm{pos}}(S)$ and consequences for the monomial primes of $K[S]$.   After proving our\n        characterization we turn our attention to the Rees algebras of a special class of\n         monomial ideals in a polynomial ring over a  field.  We may view these as affine semigroup rings;  the associated affine semigroups are in some sense complementary to the class of polytopal semigroups introduced and studied by Bruns,  Gubeladze, and Trung in \\cite{BG}  and \\cite{BGT}. In this special case, most\n          of the characterizing criteria are always satisfied.  We give examples of nonnormal\n           affine semigroup rings that satisfy $\\mathrm{R}_2$.  \n           \n           For the fundamentals on convex geometry we refer the reader to \\cite{Ew}, \\cite{Grun}, or \\cite{Zieg}. For background on monoids and semigroup rings one can consult \\cite{Gil}.\nWe make the standard assumptions that an affine semigroup $S$ is a subsemigroup of $\\mathbb{Z}^n$ and that \n$\\ensuremath{\\mathrm{grp}}(S) = \\mathbb{Z}^n$ for some positive integer $n$.   Consider the \\textit{positive cone} $\\ensuremath{\\mathrm{pos}}(S)$ of $S$ defined by\n$$\\ensuremath{\\mathrm{pos}}(S) = \\{ c_1\\boldsymbol{\\alpha}_1 + c_2\\boldsymbol{\\alpha}_2 + \\cdots + c_m\\boldsymbol{\\alpha}_m \\mid m \\ge 0, \\boldsymbol{\\alpha}_i \\in S,  c_i \\in \\mathbb{R}_{\\ge} \\; (i = 1, \\ldots, m)\\},$$\nwhere $\\mathbb{R}_{\\ge}$ denotes the set of nonnegative real numbers. Recall that $\\ensuremath{\\mathrm{pos}}(S)$ is a  polyhedral cone, that is, $\\ensuremath{\\mathrm{pos}}(S)$ is the intersection of finitely many positive halfspaces    $H_i^+ = \\{ \\boldsymbol{\\alpha} \\in \\mathbb{R}^n \\mid \\sigma_i(\\boldsymbol{\\alpha}) \\ge 0\\}$, where $\\sigma_i$ is a \n   linear form on $\\mathbb{R}^n$.   Since $S$ is a finitely generated subsemigroup of $\\mathbb{Z}^n$ we may \n   assume that each $\\sigma_i$ has rational coefficients, that is, $\\ensuremath{\\mathrm{pos}}(S)$ is a \\textit{rational polyhedral\n    cone}.  After scaling we may assume that  the coefficients of each $\\sigma_i$ are relatively prime\n     integers; we call such a \\textit{primitive linear form}.  Recall that a \\textit{supporting hyperplane} of $\\ensuremath{\\mathrm{pos}}(S)$ is a hyperplane $H$ such that  $\\ensuremath{\\mathrm{pos}}(S) \\cap H \\ne \\emptyset$ and $\\ensuremath{\\mathrm{pos}}(S)$ lies on one\n       side of $H$.  A \\textit{face} of $\\ensuremath{\\mathrm{pos}}(S)$ is the intersection of $\\ensuremath{\\mathrm{pos}}(S)$ and a supporting hyperplane.\n         The faces of $\\ensuremath{\\mathrm{pos}}(S)$ are again rational polyhedral cones. By the \\textit{dimension} $\\dim(F)$ of $F$ we mean the dimension of the vector space spanned by $F$ and by the \\textit{codimension} $\\mathrm{codim}(F)$ we mean $n - \\dim(F)$.  A face of codimension one is called a \\textit{facet}. If we represent  a polyhedral cone $C$ as an irredundant intersection of positive halfspaces  $C= \\cap_{i = 1}^r H_i^+$  then $F_1 = C \\cap H_1, \\ldots, F_r = C \\cap H_r$ are the facets of $C$ (see \\cite[Section 2.6]{Grun} ).\n  \nBy a \\textit{monomial }in $\\mathcal{R}=K[S]$ we mean an element of the form $x^{\\boldsymbol{\\alpha}}$ and by a \\textit{monomial ideal  }we mean an ideal generated by monomials. There is an order-reversing bijective correspondence between the nonempty faces of $\\ensuremath{\\mathrm{pos}}(S)$ and the monomial primes of $\\mathcal{R}$.  Indeed, the monomial prime of $\\mathcal{R}$ corresponding to the nonempty face $F$ of $\\ensuremath{\\mathrm{pos}}(S)$ is  $P_F = (x^{\\boldsymbol{\\alpha}} \\mid \\boldsymbol{\\alpha} \\in S \\setminus F )$ (e.g. see \\cite{BH}).  We let $\\mathfrak{m}$ denote the ideal of $K[S]$ generated by all noninvertible monomials; it is the maximal monomial prime of $K[S]$.   Notice that we are considering the zero ideal to be monomial.  Finally, we let $\\mathbf{e}_1, \\ldots , \\mathbf{e}_n$ denote the standard basis vectors for $\\mathbb{R}^n$. \n\nThe maximal proper faces of a polyhedral cone $C$ are precisely the facets of $C$. If $P = P_F$ is the monomial prime of height $d$ in an affine semigroup ring $K[S]$ corresponding to the face $F$ of $\\ensuremath{\\mathrm{pos}}(S)$, then there exists a chain of monomials primes of length $d$ descending from $P$ (see \\cite[Prop. 1.2.1]{GW2} ) and $\\mathrm{ht}(P) = \\mathrm{codim}(F)$.   \n\n\n\n\\section{Serre's Regularity Condition for Affine Semigroup Rings} \n\nWe start this section with a basic result about $\\mathbb{Z}^n$-graded rings, which is crucial for our purposes. A version for $\\mathbb{Z}$-graded rings is well known (e.g. see  \\cite[Ex. 2.24]{BH}).  Then we specialize to the case of an affine semigroup ring defined over a field.  Recall that if $P$ is a prime ideal of a $Z^n$-graded ring $R$ then $P^*$ denotes the largest homogeneous ideal of $R$ that is contained in $P$ and $R_{(P)}$  the homogeneous localization of $R$ at $P$, i.e., $R_{(P)} = S^{-1}R$, where $S$ is the set of homogeneous elements of $R$ that are not in $P$.  The graded ring $R_{(P)}$ is an example of a ${}^*$local ring, that is, a graded ring with a unique maximal homogeneous ideal.\n\n\\begin{prop}  \\label{prop regular graded ring} Let $R$ be a Noetherian $\\mathbb{Z}^n$-graded ring.   Then, $R$ is  regular  if and only if $R_{(P)}$ is regular, for every homogeneous prime ideal $P$ of $R$.  Furthermore, for a ${}^*$local ring $R$ with unique maximal homogeneous ideal $\\mathfrak{m}$, the ring $R$ is regular if and only if $R_{\\mathfrak{m}}$ is regular.\n\\end{prop}\n\\begin{proof}   First suppose that $R$ is regular.  Let $P$ be a homogeneous prime ideal.  Then, $R_P$ is a regular local ring.  We must show that the ${}^*$local ring $\\mathcal{R} := R_{(P)}$ is regular.  If $P\\mathcal{R}$ is a maximal \nideal of $\\mathcal{R}$ then $R_{(P)} = R_P$ is a regular local ring.  So assume $P\\mathcal{R}$ is not maximal and\n choose a prime ideal $Q$ of $R$ such that $Q^*= P$.  By assumption, $R_P = R_{Q^*} \\cong \\mathcal{R}_{Q^*\\mathcal{R}}$ is a regular local ring. \n  Let  $\\mathcal{P}$ be any prime ideal of $\\mathcal{R}$.  Then \n  $\\mathcal{P}^* \\subseteq Q^*\\mathcal{R}  \\Rightarrow \\mathcal{R} _{\\mathcal{P}^*}$ is regular  and hence $\\mathcal{R} _{\\mathcal{P}}$ is regular by \\cite[Prop. 1.2.5]{GW2}.\n  \n  Now suppose that all homogeneous localizations of $R$ at homogeneous primes are regular.  Let $P$ be any prime.  Then $R_{P^*}$ is a regular local ring since it is a localization of $R_{(P^*)}$.  Hence $R_P$ is a regular local ring by \\cite[Prop. 1.2.5]{GW2}.  Thus $R$ is a regular ring.\n  \n  Now suppose $(R, \\mathfrak{m})$ is a ${}^*$local ring.  Suppose that $R_{\\mathfrak{m}}$ is regular.  Let $P$ be any homogeneous prime ideal.  Then, $P \\subseteq \\mathfrak{m}$ implies $R_P$ is regular.  Since $R = R_{(\\mathfrak{m})}$ we may deduce that $R$ is regular by the first part of the proof.  The other implication is immediate.\n\\end{proof}\n\nNow we limit our attention to an affine semigroup ring $\\mathcal{R} = K[S]$ over a field $K$. We let $\\mathfrak{m}$ denote the ideal generated by the noninvertible monomials in $\\mathcal{R}$.\n\n \\begin{notn}\n{\\rm Notice that for elements  $\\alpha, \\beta$  of the affine semigroup $S$ the monomial quotient $x^{\\alpha}\/x^{ \\beta} \\in \\mathcal{R}_{\\mathfrak{m}}$ if and only if  $x^{\\alpha}\/x^{ \\beta} \\in \\mathcal{R}$.  One way to see this is to use the fact that the colon ideal $(x^{\\beta}\\mathcal{R}: x^{\\alpha})$ is monomial.  Now let $P$ be a monomial prime of $\\mathcal{R}$ corresponding to the nonempty face $F$ of $\\ensuremath{\\mathrm{pos}}(S)$.  Replacing $S$ by $S_F := S - S \\cap F$ and $\\mathcal{R}$ by $K[S_F] \\cong K[S]_{(P)}$, where $K[S]_{(P)}$ denotes the homogeneous localization at $P$, we see that \n$$K[\\ensuremath{\\mathrm{grp}}(S)] \\cap K[S]_P = K[S]_{(P)}.$$\nWe will identify $\\ensuremath{\\mathrm{grp}}(S \\cap F)$  with the  the group of invertible monomials in $\\mathcal{R}_P$ and refer to $S_F$ as the localization of $S$ at $F$.\n\n  \nLet $S_0$ denote the subgroup of invertible elements in the affine semigroup $S$ and let $\\widetilde{S}$ denote the quotient monoid $S\/S_0$.  It is well known that  $K[S]$ is regular if and only if $S$ is the direct sum of a free abelian group $\\mathbb{Z}^{\\ell}$ and a free abelian monoid $\\mathbb{N}^k$ (e.g. see \\cite[Ex. 6.1.11]{BH}).  \nWe shall need a slight variant of this result whose proof we omit.}\n \\end{notn} \n  \n\\begin{prop} \\label{prop: K[S] regular} The affine semigroup ring $K[S]$ is regular if and only if $\\widetilde{S}  \\cong \\mathbb{N}^k$, where $k = \\dim(K[S]_{\\mathfrak{m}})$.\n\n\\end{prop}\n\n\nSuppose the affine semigroup ring is regular.  Notice that  if  the images of the elements $\\gamma_1, \\ldots , \\gamma_k \\in S  \\setminus S_0$  form a free basis for $\\widetilde{S}$, then the monomials $x^{\\gamma_1} , \\ldots , x^{\\gamma_k}$ form a regular system of parameters for $K[S]_{\\mathfrak{m}}$.  The local version of the above proposition is the following.\n  \n  \\begin{prop}  Let $P$ be a prime ideal in the affine semigroup ring $\\mathcal{R} = K[S]$ over a field $K$ corresponding to the face $F$ of $\\ensuremath{\\mathrm{pos}}(S)$.    Then, $\\mathcal{R}_P$ is regular if and only if the quotient  $\\widetilde{S_F}$  is free. \n \\end{prop}\n  \n   \n  \\begin{proof} Notice that the group of units in $S_F$ is $\\ensuremath{\\mathrm{grp}}(S \\cap F)$.  By Proposition \\ref{prop regular graded ring}, $\\mathcal{R}_P$ is regular if and only if the homogeneous localization $\\mathcal{R}_{(P)} \\cong K[S_F]$ is regular.   The result now follows immediately from Proposition \\ref{prop: K[S] regular}. \n  \\end{proof}\n  \n   We now turn our attention  to an alternate characterization of the regularity condition in the spirit of a result in \\cite{Vi}.  One advantage of the alternate characterization is that it can be easily checked using the program NORMALIZ \\cite{Norm}.  We first prove some auxiliary results.\n\n  \\begin{lemma}\\label{lemma 1} Let $F = \\ensuremath{\\mathrm{pos}}(S) \\cap H$ be a face of the positive cone of the affine semigroup $S$ and $\\gamma_1 , \\ldots , \\gamma_k \\in S$.    \nSuppose that  $\\widetilde{S_F}$  is a free abelian monoid and the images of $\\gamma_1 , \\ldots , \\gamma_k $ form a basis.    \n    Let $P$ denote the monomial prime of $K[S]$ corresponding to $F$.\n  The following assertions hold.\n  \\begin{enumerate}\n\n  \\item  $F$ is contained in precisely $k$ facets $F_i = \\ensuremath{\\mathrm{pos}}(S) \\cap H_i$ of $\\ensuremath{\\mathrm{pos}}(S)$, and hence $F = F_1 \\cap \\cdots \\cap F_k$;\n  \\item $\\sigma_i(\\gamma_j) = \\delta_{i j}$  for all $1 \\le i,j \\le k$, where $\\sigma_i$ is the primitive linear form corresponding to $ H_i$; and\n  \\item $\\ensuremath{\\mathrm{grp}}(S \\cap F) = \\ensuremath{\\mathrm{grp}}(S) \\cap H_1 \\cap \\cdots \\cap  H_k$.\n  \\end{enumerate}\n  \\end{lemma}\n  \n  \n  \\begin{proof}\n  \n (1)  Since $x^{\\gamma_1} , \\ldots , x^{\\gamma_k}$ is a regular system of parameters for $\\mathcal{R}_P$ we know that $x^{\\gamma_i}\\mathcal{R}_P$ is a height one prime of $\\mathcal{R}_P$.   Thus there exist facets $F_i$ of $\\ensuremath{\\mathrm{pos}}(S)$ corresponding to the height one primes $P_i$ of $\\mathcal{R}$ such that $x^{\\gamma_i}\\mathcal{R}_P = P_i\\mathcal{R}_P \\, (i = 1, \\ldots , k)$.  We have $P_1 + \\cdots + P_k = P$ since they are both primes contained in $P$ and they are equal after localizing at $P$.  Suppose $G$ is a facet of $\\ensuremath{\\mathrm{pos}}(S)$ containing $F$ and let $Q = P_G$.  Then, $Q\\mathcal{R}_P = x^{\\delta}\\mathcal{R}$ since $\\mathcal{R}_P$ is a UFD.  Since $x^{\\delta} \\in P_1 + \\cdots + P_k$ we must have $x^{\\delta} \\in P_i$ for some $i$.  Hence $Q = P_i$. So $F_1, \\ldots , F_k$ are precisely the facets of $\\ensuremath{\\mathrm{pos}}(S)$ containing $F$ and $P_1, \\ldots , P_k$ are precisely the height one primes contained in $P$.   Thus $F = F_1 \\cap \\cdots \\cap F_k$. \n  \n  (2)  By construction, $\\sigma_i(\\gamma_i) > 0$.  Just suppose   $\\sigma_i(\\gamma_j) > 0$ for some $j \\ne i$.  Then, $x_{\\gamma_j} \\in P_i$, which implies $P_j\\mathcal{R}_P \\subseteq P_i\\mathcal{R}_j$, which is absurd.  Hence   $\\sigma_i(\\gamma_j) = 0$ for $ i \\ne j$.  Since $\\sigma_i$ is primitive, we must have $\\sigma_i(\\gamma_i) =1$.\n  \n  (3) Suppose that $\\alpha, \\beta \\in S$ and $\\alpha - \\beta \\in H_1 \\cap \\cdots \\cap H_k$.  There exist nonnegative integers $a_1, \\ldots , a_k, b_1, \\ldots, b_k $ such that $\\alpha - \\sum a_i\\gamma_i, \\beta - \\sum b_i\\gamma_i \\in \\ensuremath{\\mathrm{grp}}(S \\cap F)$.   Since $\\alpha - \\beta \\in H_1 \\cap \\cdots \\cap H_k$, we must have $a_i = b_i \\; (i = 1 , \\ldots , k)$ by (3).  Hence $\\alpha - \\beta \\in \\ensuremath{\\mathrm{grp}}(S \\cap F)$.  Since the opposite containment is clear we have equality of groups.\n  \\end{proof}\n  \n  \\begin{lemma}\\label{lemma 2}  Let $F$ be a face of $\\ensuremath{\\mathrm{pos}}(S)$ that is the intersection of $k$ facets $F_1 = \\ensuremath{\\mathrm{pos}}(S) \\cap H_1, \\ldots , F_k = \\ensuremath{\\mathrm{pos}}(S) \\cap H_k$ of $\\ensuremath{\\mathrm{pos}}(S)$ and let $\\sigma_i$ be the primitive linear form associated with $H_i \\; (i = 1, \\ldots , k)$.  Suppose that\n  \\begin{enumerate}\n  \\item  there exist $\\gamma_1 , \\ldots , \\gamma_k \\in S$ such that $\\sigma_i(\\gamma_j) = \\delta_{i j} \\mbox{ for all } 1 \\le i, j \\le k$; and\n  \\item $\\ensuremath{\\mathrm{grp}}(S \\cap F)  = \\ensuremath{\\mathrm{grp}}(S) \\cap H_1 \\cap \\cdots \\cap H_k$. \n  \\end{enumerate}\n  Then, $S_F\/U(S_F)$ is a free monoid with basis given by the images of $\\gamma_1 , \\ldots , \\gamma_k$ in the quotient monoid.\n  \\end{lemma}\n  \n  \\begin{proof}  The proof is straightforward. Suppose $\\alpha \\in S$ and $\\sigma_i(\\alpha) = a_i (i = 1, \\ldots k)$.  Then $\\alpha - (\\sum a_i \\gamma_i) \\in \\ensuremath{\\mathrm{grp}}(S) \\cap H_1 \\cap \\cdots \\cap H_k = \\ensuremath{\\mathrm{grp}}(S \\cap F)$ implies the image of $\\sum a_i \\gamma_i$ in the quotient monoid is $\\overline{\\alpha}$.  Suppose that $a_i , b_i \\in \\mathbb{N}$ and $\\sum a_i \\overline{\\gamma_i} = \\sum b_i \\overline{\\gamma_i}$.    Then there exists $\\mu \\in \\ensuremath{\\mathrm{grp}}(S \\cap F)$ such that $\\sum a_i\\gamma_i + \\mu = \\sum b_i \\gamma_i$.  By condition (1) we must have $a_i = b_i \\; (i = 1, \\ldots , k)$. Hence $\\widetilde{S_F}$ is free with the asserted basis.\n    \\end{proof}\n\nCombining the previous two lemmas we immediately obtain the following characterization.\n\n\\begin{theorem}\\label{thm: R_k}  An affine semigroup ring $\\mathcal{R} = K[S]$  satisfies condition $\\mathrm{R}_{\\ell}$ of Serre if and only if for each positive integer $k \\le \\ell$ and any face $F$ of $\\ensuremath{\\mathrm{pos}}(S)$ such that $\\mathrm{ht}(P_F) = k$  there exist facets $F_1, F_2, \\ldots, F_k$ of $\\mathrm{pos}(S)$  such that $F = F_1 \\cap F_2 \\cap  \\cdots \\cap F_k$ and the following conditions hold:\n\n\\begin{enumerate}\n\\item there exist $\\boldsymbol{\\gamma}_1, \\ldots, \\boldsymbol{\\gamma}_k \\in S$ such that $\\sigma_i(\\boldsymbol{\\gamma}_j) = \\delta_{i j} \\mbox{ for all } 1 \\le i, j \\le k$;  and\n\\item $\\ensuremath{\\mathrm{grp}}(S \\cap F_1 \\cap \\cdots \\cap F_k) = \\ensuremath{\\mathrm{grp}}(S) \\cap H_1 \\cap \\cdots \\cap H_k$.\n\\end{enumerate}\n\\end{theorem}\n \nWe end this section  with an example of an affine semigroup ring that satisfies condition R$_2$ of Serre but doesn't satisfy condition S$_2$. It was inspired by an example suggested to the author by I. Swanson.\n\n\\begin{example}  Suppose $K$ is a field and\nconsider the semigroup $S$ of $\\mathbb{Z}^3$ generated by the following vectors: \n$$(1,0,0), (1,3,0), (1,0,3), (1,1,0), (1,2,0), (1,0,1), (1,0,2), (1,2,1), \\mbox{and } (1,1,2).$$\n\nNotice that $\\ensuremath{\\mathrm{grp}}(S) = \\mathbb{Z}^3$ and that $\\ensuremath{\\mathrm{pos}}(S) = H_2^+ \\cap H_3^+ \\cap H_4^+$, where \n$H_2$ and $H_3$ are the indicated coordinate hyperplanes and $H_4$ is defined by the\n primitive linear form $\\sigma$, where $\\sigma(a,b,c) = 3a -b -c$.   Thus $\\ensuremath{\\mathrm{pos}}(S)$ has 3 \n facets $F_2, F_3, F_4$ and 3 codimension 2 faces  $F_{2 3} = F_2 \\cap F_3, F_{2 4} = \n F_2 \\cap F_4, F_{3 4} = F_3 \\cap F_4$.  One checks that $\\ensuremath{\\mathrm{grp}}(S \\cap F_2) = \\ensuremath{\\mathrm{grp}}(\\{ \n (1,0,0), (1,0,1) \\}) = \\ensuremath{\\mathrm{grp}}(S) \\cap H_2$ and by symmetry, $\\ensuremath{\\mathrm{grp}}(S \\cap F_3) = \\ensuremath{\\mathrm{grp}}(S) \\cap \n H_3$.  We also have $\\ensuremath{\\mathrm{grp}}(S \\cap F_4) = \\ensuremath{\\mathrm{grp}}(\\{ (1,3,0), (1,0,3), (1,2,1), (1,1,2)\\}) = \\ensuremath{\\mathrm{grp}}(\\{ \n (1,0,3), (0,1,-1)\\}) = \\ensuremath{\\mathrm{grp}}(S) \\cap H_4$.  One can also verify that $\\ensuremath{\\mathrm{grp}}(S \\cap F_2 \\cap\n  F_3) = \\ensuremath{\\mathrm{grp}}(\\{ (1,0,0) \\}) = \\ensuremath{\\mathrm{grp}}(S) \\cap H_2 \\cap H_3$, $\\ensuremath{\\mathrm{grp}}(S \\cap F_2 \\cap F_4) = \\ensuremath{\\mathrm{grp}}(\\{ \n  (1,0,3) \\}) = \\ensuremath{\\mathrm{grp}}(S) \\cap H_2 \\cap H_4$, and by symmetry $\\ensuremath{\\mathrm{grp}}(S \\cap F_3 \\cap F_4) = \n  \\ensuremath{\\mathrm{grp}}(S) \\cap H_3 \\cap H_4$.   So the group conditions for the affine semigroup ring \n  $K[S]$ to satisfy R$_2$ are satisfied.   For each codimension 2 face $F_{ij}$ we must \n  produce 2 vectors $\\boldsymbol{\\gamma}_i, \\boldsymbol{\\gamma}_j$ satisfying $\\sigma_i(\\boldsymbol{\\gamma}_j) =\n   \\delta_{i j}$, where $\\sigma_2, \\sigma_3$ are the coordinate functions and \n   $\\sigma_4 = \\sigma$ is defined above.  The vectors are given below.\n$$\\begin{array}{ccc}\n(i,j) &\\boldsymbol{\\gamma}_i & \\boldsymbol{\\gamma}_j \\\\\n(2,3) & (1,1,0) & (1,0,1) \\\\\n(2,4) & (1,1,2) & (1,0,2) \\\\\n(3,4) & (1,2,1) & (1,2,0)\n\\end{array}$$\nThus condition (1) in Theorem \\ref{thm: R_k} is satisfied and we may conclude that $K[S]$ is regular in codimension 2.  However, as we shall now see, $K[S]$ is not normal\n so can't possibly satisfy S$_2$.  \n \n Notice that $(1,1,1) = \\frac{1}{3}(1,0,0) + \\frac{1}{3}(1,3,0) + \\frac{1}{3}(1,0,3) \\Rightarrow (1,1,1) \\in \\ensuremath{\\mathrm{pos}}(S)$.   However, $(1,1,1) \\notin S$ and hence $K[S]$  is not normal.\n \n \n There is another way to see that $K[S]$ is not normal that is more obvious.  Consider the injective homomorphism $\\varphi: \\mathbb{Z}^3 \\rightarrow \\mathbb{Z}^3$ defined by $\\varphi(\\mathbf{e}_1) = (3,0,0), \\\\\n  \\varphi(\\mathbf{e}_2) = (-1,1,0), \\mbox{ and }\\varphi(\\mathbf{e}_3) = (-1,0,1)$.   The image of $S$ is the semigroup $\\tilde{S}$ generated by the vectors \n  $$(3,0,0), (0,3,0), (0,0,3), (2,1,0), (1,2,0), (2, 0,1), (1,0,2), (0,2,1), (0,1,2).$$\n  Notice that we have listed all 3-tuples of non-negative integer whose components sum to \n  3 except (1,1,1).  This isomorphism of semigroups induces an isomorphism of $K[S]$ \n  and $K[\\tilde{S}] \\cong K[x^3, y^3, z^3, x^2y, xy^2, x^2z, xz^2, y^2z, yz^2]$.  The latter \n  has normalization  \\\\ $K[x^3, y^3, z^3, xyz, x^2y, xy^2, x^2z, xz^2, y^2z, yz^2]$, which is the\n 3rd Veronese subring of \\\\\n $K[x,y,z]$.  Hence $K[S]$ is not normal.\n\\end{example}\n\n\n\\section{The Rees Algebras of a Special Class of Monomial Ideals}\n\nWe now look at the Rees algebras of a special class of integrally closed monomial ideals.\n\n\\begin{notn}\\label{blam}\n{ \\rm Let $\\boldsymbol{\\lambda} = (\\ensuremath{{\\lambda}}_1, \\ldots, \\ensuremath{{\\lambda}}_n)$ be a tuple of positive integers and \\\\\n $J = J(\\boldsymbol{\\lambda}) = (x_1^{\\ensuremath{{\\lambda}}_1}, \\ldots, x_n^{\\ensuremath{{\\lambda}}_n})$, where the ideal in taken inside the polynomial ring \\\\$K[\\xvec{n}] =: R$, and $I = I(\\boldsymbol{\\lambda}) = \\bar{J}$.   Thus $I$ is an integrally closed monomial ideal with minimal monomial reduction $J$.  Let $L = \\ensuremath{\\mathrm{lcm}}(\\ensuremath{{\\lambda}}_1, \\ldots , \\ensuremath{{\\lambda}}_n)$, $\\omega_i = L\/\\ensuremath{{\\lambda}}_i \\; (i = 1, \\ldots , n)$, and $\\boldsymbol{\\omega} = (\\omega_1 , \\ldots , \\omega_n)$.  Notice that $L = dw$, where $d = \\gcd(\\lambda_1 , \\ldots , \\lambda_n)$.}\n\\end{notn}\n\nWe will characterize those monomial ideals $I(\\boldsymbol{\\lambda})$ whose Rees algebras satisfy $\\mathrm{R}_{\\ell}$ for some $\\ell < n$. \n\n\nObserve that the Rees algebra $R[It]$ of a monomial ideal\n$I$  can always be identified with an affine  semigroup ring over $K$.  Namely, if $I =\n(x^{\\boldsymbol{\\beta}_1}, \\ldots, x^{\\boldsymbol{\\beta}_r})$ and  $S(I) = \\langle (\\mathbf{e}_1,0),\n\\ldots, (\\mathbf{e}_n,0),(\\boldsymbol{\\beta}_1,1), \\ldots, (\\boldsymbol{\\beta}_r,1) \\rangle \\subseteq\n\\mathbb{N}^{n+1}$, then $R[It] \\cong K[S(I)]$. \n\nIn case $\\mathcal{R} := R[It]$ is the Rees algebra of $I = I(\\boldsymbol{\\lambda})$ the condition that every  height $k$ monomial prime corresponds to an intersection of precisely $k$ facets  and condition (2) of Theorem \\ref{thm: R_k} automatically hold as we shall see below.  First we describe the height $k$ monomial primes of $\\mathcal{R}$. \n\n The facets $F_{\\sigma}, F_1, \\ldots , F_{n+1}$ of\n$\\ensuremath{\\mathrm{pos}}(S)$ are cut out by the supporting hyperplanes \\\\ $H_{\\sigma}, H_1,\n\\ldots, H_{n+1}$  where $\\sigma(\\boldsymbol{\\alpha}, a_{n+1})=\\boldsymbol{\\omega} \\cdot \\boldsymbol{\\alpha}\n- La_{n+1}$ and $ H_1,\n\\ldots, H_{n+1}$ are the coordinate hyperplanes in $\\mathbb{R}^{n+1}$.  Notice that with this notation the generating set for $S(I)$ is\n$$ \\{ (a_1,\\ldots,a_n,d) \\in \\mathbb{N}^{n+1} \\mid\na_1\n\\omega_1 + \\cdots + a_n \\omega_n \\ge dL \\mbox{ for } d \\le 1 \\}.$$\nNotice that $(\\mathbf{e}_1,0), \\ldots , (\\mathbf{e}_n,0), (\\ensuremath{{\\lambda}}_1\\mathbf{e}_1,1) \\in S$ and hence $\\ensuremath{\\mathrm{pos}}(S) = \\mathbb{Z}^{n+1}$, i.e., $S$ is full-dimensional.\n\nThe following description of the height one monomial primes of\n$R[It]$ appeared in \\cite{Vi}.\n\n\\begin{lemma}\\label{lem: ht 1 monl prime}  For a monomial ideal $I=I(\\boldsymbol{\\lambda})$ the height one\nmonomial primes of $R[It]$ are as follows:\n\\begin{eqnarray*}\nP_i  &= & (x_i) + (x^{\\boldsymbol{\\beta}_j}t \\mid \\mathbf{e}_i \\le_{pr} \\boldsymbol{\\beta}_j) \\; for \\; (i=1,\n\\ldots, n);\\\\\nP_{n+1} &=& (x^{\\boldsymbol{\\beta}_1}t, \\ldots, x^{\\boldsymbol{\\beta}_r}t); and  \\\\\n P_{\\sigma} &= & (x_1, \\ldots, x_n) + (x^{\\boldsymbol{\\beta}_j}t \\mid\n\\sigma(\\boldsymbol{\\beta}_j,1) > 0).\n\\end{eqnarray*}\n\\end{lemma}\n\nWe wish to describe the height $k$ monomials ideals.  Towards this end we show that every codimension $k$ face of $\\ensuremath{\\mathrm{pos}}(S)$ is the intersection of precisely $k$ facets of $\\ensuremath{\\mathrm{pos}}(S)$ for each $k$ such that $1 \\le k \\le n$.   Notice that \n\\begin{eqnarray*}\n\\ensuremath{\\mathrm{pos}}(S(I)) &=& \\ensuremath{\\mathrm{pos}}(S(J)) \\\\\n&=&  \\ensuremath{\\mathrm{pos}}((\\mathbf{e}_1,0), \\ldots, (\\mathbf{e}_n,0), (\\ensuremath{{\\lambda}}_1\\mathbf{e}_1,1), \\ldots , (\\ensuremath{{\\lambda}}_n\\mathbf{e}_n,1)).\n\\end{eqnarray*}\n\n  Alternate proofs that the following are the height $k$ monomial primes of $R[It]$ can be found in \\cite{Co}.\n\nIn the next few paragraphs, given integers $1 \\le i_1 < i_2 < \\cdots < i_k \\le n$ let $1 \\le j_1< \\cdots < j_{n-k} \\le n$ be such that $\\{i_1, \\ldots , i_k, j_1, \\ldots , j_{n-k} \\} = \\{ 1 , \\ldots , n \\}$.\n\n\\begin{lemma}\\label{lem: face 1} For $k < n$ and integers $1 \\le i_1 < i_2 < \\cdots < i_k \\le n$, let \\\\ $F= F_{i_1} \\cap \\cdots \\cap F_{i_k}$.  Then,\n\\begin{enumerate}\n\\item $ F = \\ensuremath{\\mathrm{pos}}((\\mathbf{e}_{j_1},0), \\ldots , (\\mathbf{e}_{j_{n-k}},0), (\\ensuremath{{\\lambda}}_{j_1}\\mathbf{e}_{j_1},1), \\ldots , (\\ensuremath{{\\lambda}}_{j_{n-k}}\\mathbf{e}_{j_{n-k}},1));$\n\\item  $\\mathrm{codim}(F_{i_1} \\cap \\cdots \\cap F_{i_k}) = k$ and \n$P_F = (x_{i_1}, \\ldots , x_{i_k}) + (x^{\\boldsymbol{\\beta}}t \\mid  (\\boldsymbol{\\beta} ,1) \\in S \\setminus F);$ and \n\\item $\\widetilde{S_F}$ is free with basis given by the images of $(\\mathbf{e}_{i_1},0), \\ldots , (\\mathbf{e}_{i_k}, 0)$ and hence $R[It]$ localized at $P_F$ is regular.\n\\end{enumerate} \n\\end{lemma}\n\n\\begin{proof} Let $\\sigma_i(\\boldsymbol{\\alpha}, a_{n+1}) = a_i$ for $1 \\le i \\le n+1$. \nThe first assertion is a consequence of the fact that $\\sigma_i$ of a sum of vectors in $\\ensuremath{\\mathrm{pos}}(S)$ is zero if and only if  $\\sigma_i$ of each summand is zero.  The codimension statement follows immediately.  The description of the corresponding monomial prime comes from looking at the generators of the prime ideal $S \\setminus F$ of $S$. \n\nTo see that the images of  $(\\mathbf{e}_{i_1},0), \\ldots , (\\mathbf{e}_{i_k}, 0)$  generate the quotient monoid it suffices to consider generators of $S$ of the form $(\\beta,1)$ that aren't in $S \\cap F$.  For such, $b_{i_s} > 0 $ for some $s$.  \n \nThen, \n\\begin{eqnarray*}\n (\\beta,1)  &\\cong &(( \\beta ,0) - (\\ensuremath{{\\lambda}}_{j_1} \\mathbf{e}_{j_1}, 0)) +  (\\ensuremath{{\\lambda}}_{j_1} \\mathbf{e}_{j_1}, 1) \\pmod{\\ensuremath{\\mathrm{grp}}(S \\cap F)}  \\\\\n &\\cong& (b_{i_1}\\mathbf{e}_{i_1},0)  + \\cdots + (b_{i_k}\\mathbf{e}_{i_k},0) \\hspace{.35in} \\pmod{ \\ensuremath{\\mathrm{grp}}(S \\cap F)}\n \\end{eqnarray*}\nThus the images of $(\\mathbf{e}_{i_1},0), \\ldots , (\\mathbf{e}_{i_k}, 0)$ form a free basis for the quotient  $\\widetilde{S_F}$, since uniqueness of representation is clear.\n\\end{proof}\n\n\nNext we consider the case where one of facets in the intersection is $F_{n+1}$. The proof is due to the same observations that appeared in  the preceding proof, so we omit it.\n\n\\begin{lemma} \\label{lem: face 2} For  integers $1 \\le i_1 < i_2 < \\cdots < i_{k} \\le n$ let $F=F_{i_1} \\cap \\cdots \\cap  F_{i_{k}} \\cap F_{n+1}$.  The following hold.   \n\\begin{enumerate}\n\\item $F=  \\ensuremath{\\mathrm{pos}}((\\mathbf{e}_{j_1}, 0) , \\ldots, (\\mathbf{e}_{j_k}, 0)).$ \n\\item $\\mathrm{codim}(F) = k+1$ and the corresponding monomial prime is \\\\\n$P_F = (x_{i_1}, \\ldots , x_{i_k}) + (x^{\\boldsymbol{\\beta}}t \\mid (\\boldsymbol{\\beta}, 1) \\in S).$\n\\item If $k < n$ then, $\\widetilde{S_F}$ is free with basis given by the images of the vectors \\\\ $(\\mathbf{e}_{i_1},0), \\ldots , (\\mathbf{e}_{i_k},0), (\\mathbf{e}_{j_1},1)$. In case $k < n$ the Rees algebra $R[It]$ localized at $P_F$ is regular.\n\\end{enumerate}\n\\end{lemma}\n\nNotice that $F_1 \\cap \\cdots \\cap F_n = \\{(0, \\ldots , 0)\\} = F_{n+1} \\cap F_{\\sigma}$ and the corresponding monomial prime is $\\mathfrak{m} = (x_1 , \\ldots , x_n) + (x^{\\boldsymbol{\\beta}}t \\mid (\\boldsymbol{\\beta} , 1) \\in S)$, the maximal monomial prime of $K[S]$.   In  this case, the codimension drops more than the expected amount. We can still realize the apex of the cone as the intersection of $n+1$ facets, namely $ \\{(0, \\ldots , 0)\\} = F_1 \\cap \\cdots \\cap F_{n+1}$.  Notice $S_0 = \\{ 0 \\}$ and $\\widetilde{S}$ is not free provided that $n > 1$.\n\nNow we involve the facet $F_{\\sigma}$.  By the above lemmas these are the only faces we need to worry about when characterizing which Rees algebras $R[It]$ are regular in codimension $k \\le n$.  The proof of the next lemma is a consequence of the same observations and is omitted. \n\n\\begin{lemma} \\label{lem: face 3} For integers $1 \\le i_1 < i_2 < \\cdots < i_k  \\le  n$ let  $F = F_{i_1} \\cap \\cdots \\cap F_{i_k} \\cap F_{\\sigma}$.  The following hold.\n\\begin{enumerate}\n\\item $F =  \\ensuremath{\\mathrm{pos}}((\\ensuremath{{\\lambda}}_{j_1}\\mathbf{e}_{j_1},1), \\ldots , (\\ensuremath{{\\lambda}}_{j_{n-k}}\\mathbf{e}_{j_{n-k}},1)).$ \n\\item $\\mathrm{codim}(F_{i_1} \\cap \\cdots \\cap F_{i_k} \\cap F_{\\sigma} ) = k+1$ and the corresponding monomial prime is\n$P_F = (x_{i_1}, \\ldots , x_{i_k}) + (x^{\\boldsymbol{\\beta}}t \\mid (\\boldsymbol{\\beta}, 1) \\in S \\setminus F).$\n\\end{enumerate}\n\\end{lemma}\n \n \n\nWe now show that\ncondition (2) of Theorem \\ref{thm: R_k} is always satisfied by the positive dimensional faces of $\\ensuremath{\\mathrm{pos}}(S)$ that are contained in $F_{\\sigma}$.  The condition is a priori true for positive dimensional faces not contained in $F_{\\sigma}$ by Lemmas  \\ref{lem: face 1} and \\ref{lem: face 2}.\n\n\\begin{lemma}\\label{lem: ker Z^n to Z} Let  $\\boldsymbol{\\omega} = (\\omvec{n})$ be a tuple of positive integers with $n \\ge 2$  and let \n$\\phi: \\mathbb{Z} ^n \\rightarrow \\mathbb{Z} $ be defined by $\\phi(\\boldsymbol{\\alpha}) = \\boldsymbol{\\omega} \\cdot \\boldsymbol{\\alpha}$.  For each $1 \\le i < j \\le n$ let $r_{ij} = \\gcd(\\omega_i, \\omega_j)$.  Then, the kernel of $\\phi$ is generated by the  tuples $\\boldsymbol{\\mu}_{ij} = \\frac{\\omega_j}{r_{ij}}\\mathbf{e}_i - \\frac{\\omega_i}{r_{ij}}\\mathbf{e}_j \\; \\; (1 \\le i < j \\le n)$, where $\\mathbf{e}_1, \\ldots , \\mathbf{e}_n$ are the standard basis vectors for $\\mathbb{Z}^n$. Furthermore, for integers $1 \\le i_1 < i_2 < \\cdots < i_k  \\le  n$ we have $\\ker(\\phi) \\cap H_{i_1} \\cap \\cdots \\cap H_{i_k}$ is generated by vectors $\\boldsymbol{\\mu}_{ij}$ in $H_{i_1} \\cap \\cdots \\cap H_{i_k}$.\n\\end{lemma}\n\\begin{proof}  This lemma is used in Gr$\\mathrm {\\ddot{o}}$bner basis theory but for the reader's convenience we will supply a proof.  \n\nWe proceed by induction on $n$ the case $n=2$ being straightforward.  Suppose $n > 2$ and assertion holds for $n - 1$.  Assume $\\boldsymbol{\\beta} = (b_1, \\ldots, b_n) \\in \\ker(\\phi)$. Let $g = \\gcd(\\omega_1, \\ldots, \\omega_n)$.  Then, \\begin{equation}\n\\label{ 1}\nb_n\\frac{\\omega_n}{g}= - (b_1\\frac{\\omega_1}{g} + \\cdots + b_{n-1}\\frac{\\omega_{n-1}}{g}),  \n\\end{equation} \nwhich implies\n\\begin{equation}\n\\label{2 }\nb_n =  \\sum_{i=1}^{n-1} c_i\\frac{\\omega_i}{g} =  \\sum_{i=1}^{n-1} c_is_i\\frac{\\omega_i}{r_{in}},\n\\end{equation}\nwhere $r_{in} = s_ig \\, (i=1, \\ldots, n-1)$.  Then,\n\\begin{equation}\n\\label{ 3}\n\\boldsymbol{\\beta} + \\sum_{i=1}^{n-1} c_is_i\\boldsymbol{\\mu}_{in} = (b_1^{\\prime}, \\ldots, b_{n-1}^{\\prime},0) \\in \\ker(\\phi),\n\\end{equation}\nand the assertion then follows from the induction hypothesis.  Notice that if $\\boldsymbol{\\beta} \\in \\ker(\\phi) \\cap H_{i_1} \\cap \\cdots \\cap H_{i_k}$ then each step only involves vectors in $H_{i_1} \\cap \\cdots \\cap H_{i_k}$.\n\\end{proof}\n\nThis enables us to prove that in our setting the group property is automatic.  The following two results appear in the unpublished thesis of H. Coughlin \\cite{Co}.\n\n\\begin{lemma}  \\label{lem: group 2} For integers $1 \\le i_1 < i_2 < \\cdots < i_k  \\le  n$ we always have\n$$\\ensuremath{\\mathrm{grp}}(S \\cap F_{i_1} \\cap \\cdots \\cap F_{i_k} \\cap F_{\\sigma} ) = \\ensuremath{\\mathrm{grp}}(S) \\cap H_{i_1} \\cap \\cdots \\cap H_{i_k} \\cap H_{\\sigma} .$$\n\\end{lemma}\n\n\\begin{proof}  Recall that  $\\ensuremath{\\mathrm{grp}}(S) = \\mathbb{Z}^{n+1}$.  The containment $\\ensuremath{\\mathrm{grp}}(S \\cap F_{i_1} \\cap \\cdots \\cap F_{i_k} \\cap F_{\\sigma} ) \\subseteq \\ensuremath{\\mathrm{grp}}(S) \\cap H_{i_1} \\cap \\cdots \\cap H_{i_k} \\cap H_{\\sigma} = \\mathbb{Z}^{n+1}\\cap H_{i_1} \\cap \\cdots \\cap H_{i_k} \\cap H_{\\sigma} $ is clear.  \n\nIf $k = n$ then $\\ensuremath{\\mathrm{grp}}(S) \\cap H_{i_1} \\cap \\cdots \\cap H_{i_k} \\cap H_{\\sigma} = \\mathbb{Z}^{n+1} \\cap H_1 \\cap \\cdots \\cap H_n \\cap H_{\\sigma} =  \\{ \\mathbf{0} \\}$ and the assertion follows.  Now assume $k < n$.  Suppose $(\\boldsymbol{\\beta},d) \\in \\mathbb{Z}^{n+1}\\cap H_{i_1} \\cap \\cdots \\cap H_{i_k} \\cap H_{\\sigma} $.  Then, $\\boldsymbol{\\beta} - d\\ensuremath{{\\lambda}}_n\\mathbf{e}_n \\in \\ker(\\phi)$, where $\\phi$ is defined as in the preceding lemma.  By that lemma and the fact that $(\\ensuremath{{\\lambda}}_{j_1}\\mathbf{e}_{j_1},1) \\in S \\cap F_{i_1} \\cap \\cdots \\cap F_{i_k} \\cap F_{\\sigma} $, it suffices to show that $(\\boldsymbol{\\mu}_{ij},0) \\in \\ensuremath{\\mathrm{grp}}(S \\cap H_{i_1} \\cap \\cdots \\cap H_{i_k} \\cap H_{\\sigma})$ for $1 \\le i < j \\le n$ and $\\boldsymbol{\\mu}_{i j} \\in H_{i_1} \\cap \\cdots \\cap H_{i_k}$, where we are viewing $H_{i_j}$ as a coordinate hyperplane in either $\\mathbb{R}^n$ or $\\mathbb{R}^{n+1}$.  \n\nLet $1 \\le i < j \\le n$, set $r = \\gcd(\\omega_i,\\omega_j)$ and choose $d \\in \\mathbb{Z}$ such that $0 \\le d\\ensuremath{{\\lambda}}_j - \\frac{\\omega_i}{r} < \\ensuremath{{\\lambda}}_j$.  Multiplying by $\\omega_j$ and dividing by $\\omega_i$ we get  $0 \\le d\\ensuremath{{\\lambda}}_i - \\frac{\\omega_j}{r}  < \\ensuremath{{\\lambda}}_i$ which implies $0 <   \\frac{\\omega_j}{r} - (d-1)\\ensuremath{{\\lambda}}_i  \\le \\ensuremath{{\\lambda}}_i$.   Then,\n\\begin{eqnarray*}\n(\\boldsymbol{\\mu}_{ij},0) &=& (\\frac{\\omega_j}{r}\\mathbf{e}_i +(d\\ensuremath{{\\lambda}}_j - \\frac{\\omega_i}{r})\\mathbf{e}_j,d) - d(\\ensuremath{{\\lambda}}_j\\mathbf{e}_j,1)  \\\\\n& = & ((\\frac{\\omega_j}{r} - (d-1)\\ensuremath{{\\lambda}}_i)\\mathbf{e}_i + (d\\ensuremath{{\\lambda}}_j - \\frac{\\omega_i}{r})\\mathbf{e}_j,1) + (d-1)(\\ensuremath{{\\lambda}}_i\\mathbf{e}_i,1)-d(\\ensuremath{{\\lambda}}_j\\mathbf{e}_j,1)  \\\\\n& \\in & \\ensuremath{\\mathrm{grp}}(S \\cap F_{i_1} \\cap \\cdots \\cap F_{i_k} \\cap F_{\\sigma}), \n\\end{eqnarray*}\nsince each of the 3 tuples involved is a generator of $S$ that is in $F_{i_1} \\cap \\cdots \\cap F_{i_k} \\cap F_{\\sigma}$.\n\\end{proof}\n\n\nCombining Theorem \\ref{thm: R_k} with the preceding lemma we obtain the following result.\n\n\\begin{theorem} \\label{thm: R_k for Rees} For a positive integer $\\ell < n$ the Rees \nalgebra of $I = I(\\boldsymbol{\\lambda})$ over a field satisfies condition $\\mathrm{R}_{\\ell + 1}$ of \nSerre if and only if for all sequences of positive integers $1 \\le i_1 < \\cdots < i_{\\ell} \\le n$\nthere exist $\\boldsymbol{\\gamma}_i = (\\boldsymbol{\\beta}_i , 1) \\in \\mathbb{N}^{n+1} \\; (i = 1 , \\ldots , \\ell +1)$ such that $\\sigma_i(\\boldsymbol{\\gamma}_j) = \n\\delta_{i j} \\, (1 \\le i,j \\le \\ell + 1)$, where $\\sigma_1 , \\ldots , \\sigma_{\\ell +1}$ are the primitive linear forms associated with the hyperplanes $H_{i_1} , \\ldots , H_{i_{\\ell}}, H_{\\sigma}$.  \n\\end{theorem}\n\n\\begin{proof}  First let us determine when $R_{\\ell + 1}$ holds. We need only consider faces contained in $F_{\\sigma}$. By Lemma \\ref{lem: group 2} and\n Theorem \\ref{thm: R_k} we need only show that condition (1) of Theorem \\ref{thm: R_k}\n  holds for such faces.  Let $1 \\le k \\le \\ell +1$ and let $Q$ be a height $k$ monomial prime corresponding to\n  a face contained in the facet $F_{\\sigma}$.  Then $Q$ is contained in a height $\\ell +1$\n   monomial prime corresponding to a face contained in the facet $F_{\\sigma}$ by Lemma\n   \\ref{lem: face 3}.  Thus it suffices to establish condition (1) of Theorem \\ref{thm: R_k} for\n    height $\\ell+1$ monomial primes whose faces are contained in the facet $F_{\\sigma}$. \nHence it is necessary and sufficient that there exist  vectors $\\boldsymbol{\\gamma}_i  \\in S(I) \\; (i = 1 , \\ldots , \\ell +1)$ such that\n    $\\sigma_i(\\boldsymbol{\\gamma}_j) = \\delta_{i j}$ where $\\sigma_1 , \\ldots , \\sigma_{\\ell +1}$ are the\n     primitive linear forms associated with the hyperplanes $H_{i_1} , \\ldots , H_{i_{\\ell}},\n      H_{\\sigma}$.   \n      \n      Recall that the generators of $S(I)$ have $(n+1)^{\\mathrm{st}}$ component 0 or 1. Write each $\\boldsymbol{\\gamma}_j$ as a sum of generators of $S(I)$. First suppose that $1 \\le j \\le \\ell $. The condition\n      that $\\sigma_i(\\boldsymbol{\\gamma}_j) = \\delta_{i j} \\; (i = 1, \\ldots , \\ell + 1)$ forces  some summand $(\\boldsymbol{\\beta}_j, 1)$ of \n      $\\boldsymbol{\\gamma}_j$ to satisfy $\\sigma_i(\\boldsymbol{\\beta}_j,1) = \\delta_{i j}$ for all $1 \\le i \\le \\ell +1$.  Replacing $\\boldsymbol{\\gamma}_j$ by this summand we may assume that  $\\boldsymbol{\\gamma}_j = (\\boldsymbol{\\beta}_j,1).$ \n      Now \n      consider the summands involved in the expression for $\\boldsymbol{\\gamma}_{\\ell + 1}$. Consider \n      first the possibility that all summands have $(n+1)^{\\mathrm{st}}$ component 0.  Then,\n       each summand has positive $\\sigma$-value so there is only  one summand \n       $\\boldsymbol{\\gamma}_{\\ell +1} = (\\mathbf{e}_j ,0)$, where $j \\in \\{j_1, \\ldots , j_{n_\\ell}  \\}$ and \n       $\\omega_j = 1$.  In this case we also have $( (L+1)\\mathbf{e}_{j} ,1)$ satisfies the requirements and we may replace $\\boldsymbol{\\gamma}_{\\ell+1}$ by $((L+1)\\mathbf{e}_{j} ,1)$.  The remaining possibility is that some summand has $(n+1)^{\\mathrm{st}}$ component 1 and again we may replace $\\boldsymbol{\\gamma}_{\\ell+1}$ by this summand.  In any case, we may assume $\\boldsymbol{\\gamma}_{\\ell + 1}$ has $(n+1)^{\\mathrm{st}}$ component 1.     Conversely, if vectors $(\\boldsymbol{\\beta}_j, 1) \\in \\mathbb{N}^{n+1}$ satisfying $\\sigma_i(\\boldsymbol{\\beta}_j,1) = \\delta_{i j}$  for all $1 \\le i, j \\le \\ell +1$ exist, they are automatically in $S(I)$ since $I$ is integrally closed. \n      \\end{proof}\n\nWe now state the result entirely in terms of the integers $L, \\omega_1, \\ldots , \\omega_n$ determined by the vector  $\\boldsymbol{\\lambda} = (\\ensuremath{{\\lambda}}_1 , \\ldots , \\ensuremath{{\\lambda}}_n)$.\n\n\\begin{cor}  \\label{cor: R_k for Rees} For a positive integer $\\ell < n$ the Rees \nalgebra of $ I(\\boldsymbol{\\lambda})$ over a field satisfies condition $\\mathrm{R}_{\\ell + 1}$ of \nSerre if and only if for all sequences of positive integers $1 \\le i_1 < \\cdots < i_{\\ell} \\le n$\n and $1 \\le j_1 < \\cdots < j_{n-\\ell} \\le n$ such that $\\{1, \\ldots , n \\} = \\{i_1 , \\ldots , i_{\\ell} \\} \\cup \\{ j_1 , \\ldots , j_{n-\\ell}\n  \\} $, we  have $$L - \\omega_{i_1}, \\ldots , L - \\omega_{i_{\\ell}}, L + 1 \\in \\langle \\omega_{j_1}, \\ldots , \\omega_{j_{n-\\ell}} \\rangle.$$\n\\end{cor}\n\n\\begin{proof}  By Theorem \\ref{thm: R_k for Rees} it suffices to show that for a sequence of positive integers $1 \\le i_1 < \\cdots < i_{\\ell} \\le n$ \n and $1 \\le j_1 < \\cdots < j_{n-\\ell} \\le n$ such that $ \\{1, \\ldots , n \\} = \\{i_1 , \\ldots , i_{\\ell} \\} \\cup \\{ j_1 , \\ldots , j_{n-\\ell}  \\} $, \n  vectors $\\boldsymbol{\\gamma}_j = (\\boldsymbol{\\beta}_j, 1) \\; (j = 1, \\ldots, \\ell +1) \\in \\mathbb{N}^{n+1}$ exist such that $\\sigma_i(\\boldsymbol{\\gamma}_j) = \\delta_{i j} \\;  \\mbox{ for all }1 \\le  i, j \\le \\ell + 1$, where the $\\sigma_i$ are as above, if and only if \n$L - \\omega_{i_1}, \\ldots , L - \\omega_{i_{\\ell}}, L + 1 \\in \\langle \\omega_{j_1}, \\ldots , \\omega_{j_{n-\\ell }} \\rangle.$\n         \n         Suppose first that the vectors $\\boldsymbol{\\gamma}_i = (\\boldsymbol{\\beta}_i, 1) \\in \\mathbb{N}^{n+1}$ satisfying the\n          necessary conditions exist.  By our requirements, $\\boldsymbol{\\gamma}_i= (\\mathbf{e}_i + a_{j_1}\\mathbf{e}_{j_1} + \\cdots + a_{j_{n - \\ell}}\\mathbf{e}_{j_{n - \\ell}},1)$ for all $i = 1, \\ldots , \\ell$, where the coefficients $a_{j_1} , \\ldots , a_{j_{n-\\ell}}$ are nonnegative.   The existence of \n          such vectors $\\boldsymbol{\\gamma}_i$ is equivalent to the conditions $L - \\omega_{i_1}, \\ldots , L - \\omega_{i_{\\ell}} \\in \\langle \\omega_{j_1}, \\ldots , \\omega_{j_{n-\\ell}} \\rangle.$  We\n           must also have $\\boldsymbol{\\gamma}_{\\ell + 1} = (a_{j_1}\\mathbf{e}_{j_1} + \\cdots + a_{j_{n - \\ell}}\\mathbf{e}_{j_{n - \\ell}} ,1)$, where the coefficients $a_{j_1} , \\ldots , a_{j_{n-\\ell}}$ are\n            nonnegative and  $ a_{j_1}\\omega_{j_1} + \\cdots + a_{j_{n - \\ell}}\\omega_{j_{n - \\ell}}  - L = 1$.  The existence of such a $\\boldsymbol{\\gamma}_{\\ell + 1}$ is equivalent to   $L + 1 \\in \n            \\langle \\omega_{j_1}, \\ldots , \\omega_{j_{n-\\ell }} \\rangle.$\n\\end{proof}\n\nApplying this theorem for values of $\\ell$ close to $n$ gives simple descriptions of when the Rees algebra of $I(\\boldsymbol{\\lambda})$ is regular in codimension $\\ell$.\n\n\\begin{cor}  The Rees algebra of $ I(\\boldsymbol{\\lambda}) \\subset K[\\xvec{n}]$ over a  field $K$ is regular in codimension $n$ if and only if $\\boldsymbol{\\lambda} = \\ensuremath{{\\lambda}}(1,1, \\ldots , 1)$  and hence, $I(\\boldsymbol{\\lambda}) = \\mathfrak{m}^{\\ensuremath{{\\lambda}}}$.\n\\end{cor}\n\n\\begin{proof}  The Rees algebra of $I$ satisfies $\\mathrm{R}_{n}$ if and only if for every  sequence  \\\\$1 < \\cdots < i-1 < i+1 < \\cdots < n$ of length $n-1$, we have\n$$L - \\omega_1, \\ldots , L - \\omega_{i-1} , L - \\omega_{i+1} , \\ldots , L-\\omega_n , L+1 \\in \\langle \\omega_i \\rangle.$$  In particular, $L+1 = a\\omega_i$ for some $a \\ge 0$, which implies $1 = \\omega_i(a - \\ensuremath{{\\lambda}}_i)$.  Thus each $\\omega_i = 1$.  Conversely, if all the $\\omega_i = 1$ then the necessary conditions are satisfied.\n\\end{proof}\n\n\\begin{cor} \\label{cor: 2} Suppose that $n \\ge 3$. The Rees algebra of $ I(\\boldsymbol{\\lambda}) \\subset K[\\xvec{n}]$ over a field $K$ is regular in codimension $n-1$ if and only if the integers $\\omega_i$ are pairwise relatively prime.\n\\end{cor}\n\n\\begin{proof}  The sequences of length $n-2$ arise from omitting two integers $1 \\le i < j \\le n$.  For each pair $1 \\le i < j \\le n$ we must have $$L - \\omega_k \\in \\langle \\omega_i , \\omega_j \\rangle \\mbox{ for all } k \\ne i,j \\mbox{ and } L+1 \\in \\langle \\omega_i , \\omega_j \\rangle.$$  Write $L+1 = a\\omega_i + b\\omega_j$ for $a, b \\ge 0$ and read modulo \n$\\omega_i$ to obtain the congruence $b\\omega_j \\equiv 1 \\pmod{\\omega_i}$.  Hence $\\omega_i$ and\n $\\omega_j$ are relatively prime.  This holds for all pairs $1 \\le i<j \\le n$. Conversely, if the \n integers $\\omega_i$ are pairwise relatively prime then every integer at least $(\\omega_i - 1)(\\omega_j -1)$ is in $\\langle \\omega_i,  \\omega_j \\rangle$. So $L + 1 \\mbox { and } L - \\omega_k = \\omega_k(g \\prod_{s \\ne k} \\omega_s  -  1) \\in \\langle \\omega_i,  \\omega_j \\rangle$, where $g = \\gcd(\\ensuremath{{\\lambda}}_1, \\ldots , \\ensuremath{{\\lambda}}_n)$.\n\\end{proof}\n\nIf $n = 2$ the Rees algebra $R[I(\\boldsymbol{\\lambda})t]$ is normal and hence regular in codimension \n$n - 1 = 1$ without any additional assumptions. Corollary \\ref{cor: 2} says that if $n=3$ \nthe Rees algebra of $I(\\boldsymbol{\\lambda})$ is regular in codimension 2 if and only if the $\\omega_i$ \nare pairwise relatively prime.  H. Coughlin \\cite{Co} proved this special case and also that this condition is sufficient for $R[I(\\boldsymbol{\\lambda})t]$ to be normal.   This result combined with an earlier result of Reid, Roberts, and Vitulli \\cite{RRV}  has an interesting consequence, which we now present. \n\n\\begin{notn}\n{\\rm For an $\\mathbb{N}$-graded ring $A$ and a positive integer $t$ we let $A_{\\ge t}$ denote the homogeneous ideal $A_{\\ge t} = \\oplus_{s \\ge t} A_s$.  Further assume that $A$ is generated as an $A_0$-algebra by homogeneous elements $\\xvec{n}$ of positive degrees $\\omega_1 , \\ldots , \\omega_n$, respectively.  \n\n\n\nNotice that for a tuple of positive integers $\\boldsymbol{\\lambda}$, with $L = \\ensuremath{\\mathrm{lcm}}(\\lambda_1, \\ldots, \\lambda_n )$ and $\\omega_i = L\/\\lambda_i  \\,  (i = 1 , \\ldots , n)$ as in (\\ref{blam}), if we define a new grading on  $R = K[\\xvec{n}]$ by declaring $\\deg(x_i) = \\omega_i \\,  (i = 1 , \\ldots , n)$, then $I(\\boldsymbol{\\lambda}) = R_{\\ge L} = R_{\\ge dw}$, where $d = \\gcd(\\ensuremath{{\\lambda}}_1, \\ldots , \\ensuremath{{\\lambda}} _n)$.  Ideals of the form $A_{\\ge t}$ have been studied by E. Hyry and K. E.  Smith  in connection with Kawamata's Conjecture which speculates that every adjoint ample line bundle on a smooth variety admits a nonzero section (e.g. see \\cite{HS1} and \\cite{HS2}).  They also arise as test ideals in tight closure theory as illustrated in \\cite[Remark 3]{F}}.\n\\end{notn}\n \n \\begin{prop}  Let $R= K[x,y,z]$ be a polynomial ring over a field $K$ and let $a,b,c$ be pairwise relatively prime positive integers.  Set $S = K[x^a, y^b, z^c]$ and $L = abc$.  Then, the ideal $\\mathfrak{a} = S_{\\ge L}$ is normal, that is,  $\\mathfrak{a}^t = S_{\\ge tL}$ for all $t \\ge 1$.\n \\end{prop}\n \n \\begin{proof}  Observe that the homogeneous ideal $ \\mathfrak{a} = S_{\\ge L}$ is \n integrally closed and that the integral closure of $ \\mathfrak{a}^t$ is   $ S_{\\ge t L}$  for $t \\ge 1$\n (e.g. see the discussion in \\cite{RRV}). By Proposition 3.7 of  \\cite{RRV} to prove that $\\mathfrak{a}$ is normal,  it suffices to show that $\\mathfrak{a}^2 = S_{\\ge 2L}$.  For this we proceed as in the proof of  Theorem III.2.2 of \\cite{Co}.  Suppose that $x^{ua}y^{bv}z^{cw} \\in S_{\\ge 2L}$ is a minimal monomial generator.  In particular,  $ua + vb + wc \\ge 2L$.  We must exhibit a decomposition $(u,v,w) = (u_1,v_1,w_1)+(u_2,v_2,w_2)$ with $u_ia + v_ib + w_ic \\ge L \\, (i=1,2)$.  If $u \\ge L\/a, v  \\ge  L\/b$ or $w  \\ge L\/c$ it is clear that we can do this.  For example, if $u \\ge L\/a$ write $(u,v,w) = (L\/a,0,0) + (u - L\/a,v,w)$. Thus it suffices to assume that $u < L\/a, v <  L\/b$ and $w < L\/c$.  Notice that this forces the sum of any two of $ua, vb,$ and $wc$ to be strictly greater than $L$ and each summand to be positive.  We consider three cases.\n \nFirst suppose that either $a, b,$ or $c$ is 1. For example, suppose $a=1$. \n Say $u + vb = L + u_2$.  Then $0 < u_2 < u$ and $$(u,v,w) = (u-u_2,v,0) + (u_2, 0, w)$$ is the desired decomposition.\nThus we may assume that $a,b,c > 1$.  \n \nNow suppose that $u <  L\/2a , v <  L\/2b  $ or $w < L\/2c $.  Without loss of generality we may assume that $w  <  L\/2c $.  Then, $L - wc > L\/2 \\ge (a-1)(b-1)$ so there exist $u_1, v_1 \\in \\mathbb{N}$ such that $u_1a + v_1b = L - wc$.  Since $vb < L$, we have $ua + wc > L$.  Now $u_1a \\le L - wc < ua$ implies $u_1 < u$.  Similarly, $v_1 < v$.  Therefore, $$(u,v,w) = (u_1,v_1,w) + (u-u_1, v-v_1,0)$$ is the desired decomposition.\n\nFinally, assume that $u \\ge L\/2a , v \\ge  L\/2b  $ and $w  \\ge L\/2c $.  Set $w_1 = \\lceil L\/2c  \\rceil$.  Then, \n$L - w_1c \\ge L - (ab+1)c\/2 = (c\/2)(ab-1) > (a-1)(b-1)$ and we may write $u_1a + v_1b  = L - w_1c$ for some $u_1, v_1 \\in \\mathbb{N}$.  Notice that $u_1a \\le L - w_1c \\le L\/2$ and hence $u_1 \\le  L\/2a  \\le  u$.  Similarly, $v_1 \\le  v$.  Thus $$(u,v,w) = (u_1,v_1,w_1) + (u-u_1,v-v_1,w-w_1)$$ is the desired decomposition.\n \\end{proof}\n \nWe now present an example of a Rees algebra $\\mathcal{R} = R[It]$ of a monomial ideal that satisfies $\\mathrm{R}_2$ but is not normal.   In order to find an example we must work over a polynomial ring in at least 4 indeterminates by the above remarks.  The following  example  is due to H. Coughlin \\cite{Co}.   The example was first explored using the program NORMALIZ \\cite{Norm} of Bruns and Koch.\n\n\\begin{example}\\label{ex: weirdex} Let $\\boldsymbol{\\lambda}=(1443,37,21,91)$. Define $I=I(\\boldsymbol{\\lambda})$, $S=S(I)$, and $\\mathcal{R}$ as above. We claim $\\mathcal{R}$ is not normal but satisfies the equivalent conditions for R$_2$. Hence,  $\\mathcal{R}$ does not satisfy S$_2$.\n\\end{example}\n\n\nWe first show that $\\mathcal{R}$ is not normal\nNote that $L=10101$.\n  The vector $\\boldsymbol{\\alpha}=(2,36,1,89)$ satisfies \n$\\boldsymbol{\\omega}  \\cdot  \\boldsymbol{\\alpha}=2L$, so that $x^{\\boldsymbol{\\alpha}}  \\in \\overline{I^2}$.   Direct computation shows that   $\\boldsymbol{\\alpha}$ is not the sum of two vectors in $S$ and hence $x^{\\boldsymbol{\\alpha}} \\notin I^2$.   Thus $\\mathcal{R}$ is not normal.\n\nWe show that $\\mathrm{R}_2$ holds.  As in the proof of Theorem \\ref{thm: R_k for Rees} we need only deal with the height two monomial primes corresponding to the \nfaces $G_1, G_2, G_3, G_4$, where $G_i = \\ensuremath{\\mathrm{pos}}(S) \\cap H_i \\cap H_{\\sigma}$.  As in the proof of\nTheorem \\ref{thm: R_k for Rees}, for each $G_i$ we must define  a \npair of elements $\\boldsymbol{\\gamma}_i$ and $\\boldsymbol{\\gamma}_6$ in $S$ such that $\\sigma_a(\\boldsymbol{\\gamma}_b) = \\delta_{a b}$ for $a, b \\in \\{ i, 6 \\}$. \n\nThe following vectors satisfy $\\sigma_a(\\boldsymbol{\\gamma}_b)=\\delta_{ab}$ for $a, b \\in \\{ i, 6 \\}$:\n$$\\begin{array}{ccc}\ni&\\boldsymbol{\\gamma}_i & \\boldsymbol{\\gamma}_6\\\\ \n1&(1,28,8,12,1) &(0,8,1,67,1) \\\\\n\n\n2&(35,1,1,82,1)&(16,0,0,91,1)\\\\\n3&(35,1,1,82,1) & (16,0,0,91,1)\\\\\n4&(220,1,17,1,1) & (275,0,17,0,1)\\\\\n\\end{array}$$\n\n$\\mathcal{R}$ satisfies R$_2$ by Theorem~\\ref{thm: R_k for Rees}.  Thus $\\mathcal{R}$ does not satisfy S$_2$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\setcounter{equation}{0}\n\\setcounter{thm}{0}\n\nRecently there is a lot of interest in the following singular diffusion\nequation \\cite{A}, \\cite{DK}, \\cite{P}, \\cite{V},\n\\begin{equation}\\label{fde-eqn}\nu_t=\\frac{n-1}{m}\\Delta u^m\\quad\\mbox{ in }{\\mathbb{R}}^n\\times (0,T)\n\\end{equation}\nwhich arises in the study of many physical models and geometric flows. When $0<m<1$,\n\\eqref{fde-eqn} is called the fast diffusion equation.  On the other hand as observed by P.~Daskalopoulos, M.~del Pino, J.~King, M.~S\\'aez, N.~Sesum, and others \\cite{DPKS1}, \\cite{DPKS2}, \\cite{PS}, the metric $g=u^{\\frac{4}{n-2}}dy^2$ satisfies the Yamabe flow \\cite{B1}, \\cite{B2},\n\\begin{equation}\\label{log1-eqn}\n\\frac{\\partial g}{\\partial t}=-Rg\n\\end{equation}\non ${\\mathbb{R}}^n$, $n\\ge 3$, for $0<t<T$, where $R$ is the scalar curvature of the metric $g$, if and only if $u$ satisfies \\eqref{fde-eqn} with\n\\begin{equation*}\nm=\\frac{n-2}{n+2}.\n\\end{equation*}\nFor\n\\begin{equation}\\label{m-range}\nn\\ge 3\\quad\\mbox{ and }\\quad 0<m<\\frac{n-2}{n}, \n\\end{equation}\nasymptotic behaviour of solution of \\eqref{fde-eqn} near finite\nextinction time was studied by Galaktionov and L.A.~Peletier \\cite{GP}\nand P.~Daskalopoulos and N.~Sesum \\cite{DS}. Extinction profile\nfor solutions of \\eqref{fde-eqn} for the case $m=\\frac{n-2}{n+2}$, $n\\ge 3$, was studied\nin \\cite{PS} by M.~del Pino and M.~S\\'aez. Such extinction behaviour occurs for the solution of \\eqref{fde-eqn} when \\eqref{m-range} holds and the solution $u\\in L^{\\infty}({\\mathbb{R}}^n\\times (0,T))$ satisfies\n\\begin{equation*}\nu(x,t)\\le C|x|^{-\\frac{2}{1-m}}\\quad\\forall |x|\\ge R_1, 0\\le t<T\n\\end{equation*}\nfor some constants $C>0$ and $R_1>0$ (cf. \\cite{DS}). On the other hand if \\eqref{m-range} holds and $0\\le u_0\\in L_{loc}^p({\\mathbb{R}}^n)$ for some constant $p>\\frac{n(1-m)}{2}$ which satisfies \n\\begin{equation*}\n\\liminf_{R\\to\\infty}\\frac{1}{R^{n-\\frac{2}{1-m}}}\\int_{|x|\\le R}u_0\\,dx\n=\\infty,\n\\end{equation*}\nthen S.Y.~Hsu \\cite{Hs2} proved the existence and uniqueness of solutions of \n\\begin{equation}\\label{cauchy-problem}\n\\left\\{\\begin{aligned}\nu_t=&\\frac{n-1}{m}\\Delta u^m\\quad\\mbox{ in }{\\mathbb{R}}^n\\times (0,\\infty)\\\\\nu(x,0)=&u_0(x)\\qquad\\quad\\mbox{ in }{\\mathbb{R}}^n.\n\\end{aligned}\\right.\n\\end{equation}\nThese results say that we will have either global existence of solution of \\eqref{fde-eqn} or extinction in finite time for solution of \\eqref{fde-eqn}  depending on whether the growth rate of the solution is large enough at infinity. \nWhen \\eqref{m-range} holds, existence, uniqueness and decay rate of self-similar solutions of \\eqref{fde-eqn} were also proved by S.Y.~Hsu in \\cite{Hs1}.\nInterested reader can read the book\n\\cite{DK} by P.~Daskalopoulos and C.E.~Kenig and the book \\cite{V} by\nJ.L.~Vazquez for the most recent results on \\eqref{fde-eqn}.\n\nIn the recent paper\n\\cite{CD} B.~Choi and P.~Daskalopoulos proved the higher order expansion of the radially symmetric solution $v_{\\lambda,\\beta}(r)$ of\n\\begin{equation}\\label{elliptic-eqn}\n\\left\\{\\begin{aligned}\n&\\frac{n-1}{m}\\Delta v^m+\\frac{2\\beta}{1-m} v+\\beta x\\cdot\\nabla v=0,\\quad v>0,\\quad\\mbox{ in }{\\mathbb{R}}^n\\\\\n&v(0)=\\lambda\\end{aligned}\\right.\n\\end{equation}\n for any constant $\\lambda>0$ where $m=\\frac{n-2}{n+2}$, $n\\ge 3$, as $r\\to\\infty$. They also proved that if $u$ is the solution of \\eqref{cauchy-problem} in ${\\mathbb{R}}^n\\times (0,\\infty)$ with $m=\\frac{n-2}{n+2}$, $n\\ge 3$,  and initial value $u_0\\ge 0$ satisfying \n\\begin{equation*}\nu_0(x)^{1-m}\\approx \\frac{(n-1)(n-2)}{\\beta |x|^2}\\left(\\log |x|+K_1+o(1)\\right)\\quad\\mbox{ as }r=|x|\\to\\infty\n\\end{equation*}\nfor some constants $\\beta>0$ and $K_1\\in{\\mathbb{R}}$, then as $t\\to\\infty$ the rescaled function \n\\begin{equation}\\label{rescaled-soln}\n\\widetilde{u}(x,t)=e^{\\frac{2\\beta}{1-m}t}u(e^{\\beta t}x,t)\n\\end{equation}\nconverges uniformly on every compact subsets of ${\\mathbb{R}}^n$ to  $v_{\\lambda_1,\\beta}(x)$ for some constant $\\lambda_1>0$.\nNote that for any solution $u$ of \\eqref{fde-eqn} in ${\\mathbb{R}}^n\\times (0,\\infty)$, $\\widetilde{u}$ satisfies\n\\begin{equation}\\label{u-tilde-eqn}\n\\widetilde{u}_t=\\frac{n-1}{m}\\Delta\\widetilde{u}+\\frac{2\\beta}{1-m}\\widetilde{u}+\\beta x\\cdot\\nabla\\widetilde{u}\\quad\\mbox{ in }{\\mathbb{R}}^n\\times (0,\\infty).\n\\end{equation}\nThis result of B.~Choi and P.~Daskalopoulos \\cite{CD} shows that the asymptotic large time behaviour of the solution of \\eqref{cauchy-problem} depends critically on the higher order expansion of the initial value of the solution. Moreover the asymptotic large time behaviour of the solution of \\eqref{cauchy-problem} is after a rescaling similar to the solution of \\eqref{elliptic-eqn} with the same higher order expansion. Hence it is important to study the higher order expansion of the solution of \\eqref{elliptic-eqn}.\n\n\nIn this paper we will extend the result of \\cite{CD}. We will prove the higher order expansion of the radially symmetric solution $v_{\\lambda,\\beta}(r)$ of \\eqref{elliptic-eqn} as $r\\to\\infty$ for any $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $m\\ne\\frac{n-2}{n+2}$, $\\beta>0$ and $\\lambda>0$. We will also prove that \nwhen $n\\ge 3$, $0<m<\\frac{n-2}{n}$, and\n\\begin{equation*}\nu_0(x)^{1-m}\\approx \\frac{2(n-1)(n-2-nm)}{(1-m)\\beta |x|^2}\\left(\\log |x|-\\frac{n-2-(n+2)m}{2(n-2-nm)}\\log (\\log |x|)+K_1+0(1)\\right)\n\\end{equation*}\nas $r=|x|\\to\\infty$ for some constants $\\beta>0$ and $K_1\\in{\\mathbb{R}}$, then as $t\\to\\infty$ the rescaled solution $\\widetilde{u}$ given by \\eqref{rescaled-soln} will converges uniformly on every compact subsets of ${\\mathbb{R}}^n$ to the radially symmetric solution $v_{\\lambda_1,\\beta}(x)$ of \\eqref{elliptic-eqn} for some constant $\\lambda=\\lambda_1>0$.\n\nWe first start with a definition. For any $0\\le u_0\\in L_{loc}^1({\\mathbb{R}}^n)$, we say that a function $u$ is a solution of \\eqref{cauchy-problem}\nif $u>0$  in ${\\mathbb{R}}^n\\times (0,\\infty)$  is a classical solution of \\eqref{fde-eqn} in ${\\mathbb{R}}^n\\times (0,\\infty)$ and\n\\begin{equation*}\n\\|u(\\cdot, t)-u_0\\|_{L^1(E)}\\to 0\\quad\\mbox{ as }t\\to 0\n\\end{equation*}\nfor any compact subset $E$ of ${\\mathbb{R}}^n$. For any $\\lambda>0$ and $\\beta>0$, we say that $v$ is a solution of \\eqref{elliptic-eqn} if $v$ is a positive classical solution of \\eqref{elliptic-eqn} in ${\\mathbb{R}}^n$. When there is no ambiguity we will drop the subscript and write $v$ for the radially symmetric solution $v_{\\lambda,\\beta}$ of \\eqref{elliptic-eqn}. Let \n\\begin{equation}\\label{w-defn}\nw(s)=r^2v(r)^{1-m}\\quad\\mbox{ and }\\quad s=\\log r.\n\\end{equation}\nWe will assume that $n\\ge 3$, $0<m<\\frac{n-2}{n}$,  $\\beta>0$, $\\lambda>0$ and $w$ be given by \\eqref{w-defn} for the rest of this paper. Unless stated otherwise we will also assume that $m\\ne\\frac{n-2}{n+2}$.\n\nWe obtain the following two main theorems in this paper.\n\n\\begin{thm}\\label{higher-order-expansion-thm}\nLet $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $m\\ne\\frac{n-2}{n+2}$, $\\beta>0$ and $\\lambda>0$. Let $v_{\\lambda,\\beta}(r)$ be the radially symmetric solution of \\eqref{elliptic-eqn} given by \\cite{Hs1}. Then there exists a constant $K_0$ independent of $\\beta$ and $\\lambda$ and a constant $K(\\lambda,\\beta)$ such that\n\\begin{align}\\label{v-lambda-beta-expansion1}\nv_{\\lambda,\\beta}(r)^{1-m}=&\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta r^2}\\left\\{\\log r -\\frac{n-2-(n+2)m}{2(n-2-nm)}\\log (\\log r))+\\frac{1-m}{2}\\log\\lambda\n\\right.\\notag\\\\\n&\\qquad \\left.+\\frac{1}{2}\\log\\beta +K_0+\\frac{a_0}{\\log r}\n+\\frac{(n-2-(n+2)m)^2}{4(n-2-nm)^2}\\cdot\\frac{\\log(\\log r)}{\\log r}+\\frac{o(\\log r)}{\\log r}\\right\\}\n\\end{align}\nas $r\\to\\infty$ where\n\\begin{equation}\\label{a0-defn}\na_0=\\frac{(n-2-(n+2)m)^2}{4(n-2-nm)^2}-\\frac{(1-m)^2 a_1(1,1)}{4(n-1)(n-2-nm)^2}\n\\end{equation}\nand\n\\begin{align}\\label{a1-defn}\na_1(\\lambda,\\beta)=&\\frac{2(1-2m)(n-1)(n-2-nm)}{(1-m)^2}+\\frac{(n-1)(n-2-(n+2)m)^2}{(1-m)^2}\\notag\\\\\n&\\qquad +\\frac{(n-2-(n+2)m)}{(1-m)}K(\\lambda,\\beta)\\beta.\n\\end{align}\n\\end{thm}\n\n\\begin{thm}\\label{convergence-thm}\nLet $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $m\\ne\\frac{n-2}{n+2}$, $\\beta>0$, $0\\le u_0=\\phi+\\psi$, $ \\phi\\in  L_{loc}^p({\\mathbb{R}}^n)$, $\\psi\\in L^1({\\mathbb{R}}^n)\\cap L_{loc}^p({\\mathbb{R}}^n)$, for some constant $p>\\frac{(1-m)n}{2}$, be such that\n\\begin{align}\\label{phi-expansion}\n\\phi(x)^{1-m}=&\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta |x|^2}\\left(\\log |x|-\\frac{n-2-(n+2)m}{2(n-2-nm)}\\log (\\log |x|)+K_1+o(1)\\right)\\mbox{ as }|x|\\to\\infty\n\\end{align}\nfor some constant $K_1\\in{\\mathbb{R}}$.\nIf $u$ is the unique solution of \\eqref{fde-eqn} in ${\\mathbb{R}}^n\\times (0,\\infty)$ given by\nTheorem 1.1 of \\cite{Hs2}, then as $t\\to\\infty$, the rescaled function $\\widetilde{u}(x,t)$ given by \\eqref{rescaled-soln} converges to  $v_{\\lambda_1,\\beta}$ in $L_{loc}^1({\\mathbb{R}}^n)$ with $\\lambda_1=\\left(e^{2K_1\/K_0}\/\\beta\\right)^{\\frac{1}{1-m}}$ where the constant $K_0$ is given by Theorem \\ref{higher-order-expansion-thm}.\n\nMoreover if $u_0$ also satisfies $u_0=\\phi\\in L^{\\infty}({\\mathbb{R}}^n)$, then as $t\\to\\infty$,  $\\widetilde{u}(x,t)$  also converges to  $v_{\\lambda_1,\\beta}$ uniformly in $C^{2,1}(E)$ for any compact subset $E\\subset{\\mathbb{R}}^n$. \n\\end{thm}\n\n\n\n\\section{Proofs}\n\\setcounter{equation}{0}\n\\setcounter{thm}{0}\n\nIn this section we will give the proof of Theorem \\ref{higher-order-expansion-thm} and Theorem \\ref{convergence-thm}. We first recall some results of \\cite{Hs1}, \\cite{DS} and \\cite{HK}.\n\n\\begin{thm}\\label{w's-bd-thm}(Theorem 1.3  of \\cite{Hs1} and its proof)\nLet $v$ be the unique radially symmetric solution of \\eqref{elliptic-eqn} and $w$ be given by \\eqref{w-defn}. Then \n\\begin{equation}\\label{w-decay-infty}\n\\lim_{|x|\\to\\infty}\\frac{|x|^2v(x)^{1-m}}{\\log |x|}=\\lim_{s\\to\\infty}\\frac{w(s)}{s}=\\lim_{s\\to\\infty}w_s(s)\n=\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta}.\n\\end{equation}\n\\end{thm}\n\n\\begin{lem}(cf. Corollary 2.2 of \\cite{DS} and Lemma 2.2 of \\cite{HK})\\label{L1-comparison-lem}\nLet $0\\le u_{0,1}, u_{0,2}\\in L_{loc}^1({\\mathbb{R}}^n)$ be such that $u_{0,1}-u_{0,2}\\in L^1({\\mathbb{R}}^n)$. Suppose $u_1$, $u_2$, are solutions of \\eqref{fde-eqn} in ${\\mathbb{R}}^n\\times (0,\\infty)$ with initial values $u_{0,1}, u_{0,2}$ respectively such that for any $T>0$ there exist constants $C_1>0$, $R_1>0$, such that\n\\begin{equation*}\nu_i(x,t)\\ge C_1|x|^{-\\frac{2}{1-m}}\\quad\\forall |x|\\ge R_1, 0<t<T, i=1,2.\n\\end{equation*}\nThen\n\\begin{equation*}\n\\|u_1(\\cdot,t)-u_2(\\cdot,t)\\|_{L^1({\\mathbb{R}}^n)}\\le \\|u_{0,1}-u_{0,2}\\|_{L^1({\\mathbb{R}}^n)}\\quad\\forall t>0\n\\end{equation*}\nand hence\n\\begin{equation*}\n\\|\\widetilde{u}_1(\\cdot,t)-\\widetilde{u}_2(\\cdot,t)\\|_{L^1({\\mathbb{R}}^n)}\\le e^{-\\frac{(n-2-nm)}{1-m}t} \\|\\widetilde{u}_{0,1}-\\widetilde{u}_{0,2}\\|_{L^1({\\mathbb{R}}^n)}\\quad\\forall t >0\n\\end{equation*}\nwhere $\\widetilde{u}_1(\\cdot,t)$, $\\widetilde{u}_2(\\cdot,t)$, are the rescaled solutions of $u_1$, $u_2$, given by \\eqref{rescaled-soln}.\n\\end{lem}\n\nBy the computation of \\cite{Hs1} we have\n\\begin{equation}\\label{w-eqn}\nw_{ss}=\\frac{1-2m}{1-m}\\cdot\\frac{w_s^2}{w}\n-\\frac{n-2-(n+2)m}{1-m}w_s+\\frac{\\beta}{n-1}\\left(\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta}-w_s\\right)w\\quad\\mbox{ in }{\\mathbb{R}}.\n\\end{equation}\nLet\n\\begin{equation}\\label{h-defn10}\nh(s)=w(s)-\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta}s.\n\\end{equation}\nThen by \\eqref{w-eqn},\n\\begin{align*}\nh_{ss}=&\\frac{1-2m}{1-m}\\cdot\\frac{w_s^2}{w}\n-\\frac{(n-2-(n+2)m)}{1-m}\\left(\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta} +h_s\\right)\\notag\\\\\n&\\qquad -\\frac{\\beta}{n-1}\\left(\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta}s+h\\right)h_s\\quad\\mbox{ in }{\\mathbb{R}}.\n\\end{align*}\nHence\n\\begin{equation}\\label{h-eqn2}\nh_{ss}+\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h+\\frac{n-2-(n+2)m}{1-m}\\right)h_s=\\frac{1-2m}{1-m}\\cdot\\frac{w_s^2}{w}-b_0\\quad\\mbox{ in }{\\mathbb{R}}\n\\end{equation}\nwhere\n\\begin{equation*}\nb_0=\\frac{2(n-1)(n-2-nm)(n-2-(n+2)m)}{(1-m)^2\\beta}.\n\\end{equation*}\n\n\\begin{lem}\\label{h-limit-lem1}\nLet $n\\ge 3$, $0<m<\\frac{n-2}{n}$ and $m\\ne\\frac{n-2}{n+2}$. Then $h$ satisfies\n\\begin{equation}\\label{h-limit}\n\\lim_{s\\to\\infty}\\frac{h(s)}{\\log s}=\\lim_{s\\to\\infty}s\\,h_s(s)=-\\frac{(n-1)[n-2-(n+2)m]}{(1-m)\\beta}.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nWe first observe that by Theorem \\ref{w's-bd-thm},\n\\begin{equation}\\label{w-ratio-0}\n\\lim_{s\\to\\infty}\\frac{w_s^2(s)}{w(s)}=\\frac{\\lim_{s\\to\\infty}w_s^2(s)}{\\lim_{s\\to\\infty}s\\cdot (w(s)\/s)}=0.\n\\end{equation}\nThen by \\eqref{h-eqn2} and \\eqref{w-ratio-0} for any $0<\\varepsilon<|b_0|\/2$ there exists a constant $s_1\\in{\\mathbb{R}}$ such that\n\\begin{equation}\\label{h-ineqn}\n-b_0-\\varepsilon\\le h_{ss}+\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h+\\frac{n-2-(n+2)m}{1-m}\\right)h_s\\le -b_0+\\varepsilon\\quad\\forall s\\ge s_1.\n\\end{equation}\nLet \n\\begin{equation}\\label{f-defn}\nf(s)=\\mbox{exp}\\,\\left(\\frac{(n-2-nm)}{(1-m)}s^2+\\frac{\\beta}{n-1}\\int_{s_1}^sh(z)\\,dz+\\frac{n-2-(n+2)m}{1-m}s\\right).\n\\end{equation}\nBy \\eqref{h-ineqn},\n\\begin{align}\\label{h'-ineqn}\n&(-b_0-\\varepsilon)f(s)\\le (f(s)h_s(s))_s\\le (-b_0+\\varepsilon)f(s)\\quad\\forall s\\ge s_1\\notag\\\\\n\\Rightarrow\\quad&\\frac{f(s_1)h_s(s_1)s+(-b_0-\\varepsilon)s\\int_{s_1}^sf(z)\\,dz}{f(s)}\\le sh_s(s)\\le\\frac{f(s_1)h_s(s_1)s+(-b_0+\\varepsilon)s\\int_{s_1}^sf(z)\\,dz}{f(s)}\\,\\,\\forall s\\ge s_1.\n\\end{align}\nBy \\eqref{w-decay-infty}, $h_s(s)\\to 0$ as $s\\to\\infty$. Hence $h(s)=o(s)$ and $h(s)\/s\\to 0$ as $s\\to\\infty$. Then by \\eqref{f-defn}\n$f(s)\\to\\infty$ as $s\\to\\infty$. Hence by the l'Hospital rule,\n\\begin{equation}\\label{f-expression-ratio-limit}\n\\lim_{s\\to\\infty}\\frac{\\int_{s_1}^sf(z)\\,dz}{f(s)}=\\lim_{s\\to\\infty}\\frac{f(s)}{f(s)\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h(s)+\\frac{n-2-(n+2)m}{1-m}\\right)}=0\n\\end{equation}\nand\n\\begin{equation}\\label{ratio-limit}\n\\lim_{s\\to\\infty}\\frac{s}{f(s)}=\\lim_{s\\to\\infty}\\frac{1}{f(s)\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h(s)+\\frac{n-2-(n+2)m}{1-m}\\right)}=0.\n\\end{equation}\nHence by \\eqref{f-expression-ratio-limit} and the l'Hospital rule,\n\\begin{align}\\label{ratio-limit2}\n\\lim_{s\\to\\infty}\\frac{s\\int_{s_1}^sf(z)\\,dz}{f(s)}\n=&\\lim_{s\\to\\infty}\\frac{sf(s)+\\int_{s_1}^sf(z)\\,dz}{f(s)\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h(s)+\\frac{n-2-(n+2)m}{1-m}\\right)}\\notag\\\\\n=&\\frac{(1-m)}{2(n-2-nm)}.\n\\end{align}\nLetting first $s\\to\\infty$ and then $\\varepsilon\\to 0$ in \\eqref{h'-ineqn}, by \\eqref{ratio-limit} and  \\eqref{ratio-limit2} we have\n\\begin{equation}\\label{h'-limit10}\n\\lim_{s\\to\\infty}s\\, h_s(s)=-\\frac{(1-m)b_0}{2(n-2-nm)}=-\\frac{(n-1)[n-2-(n+2)m]}{(1-m)\\beta}.\n\\end{equation}\nHence \n\\begin{equation}\\label{h-ineqn10}\nh(s)\\le -\\frac{(n-1)[n-2-(n+2)m]}{2(1-m)\\beta}\\log s\\quad\\mbox{ as }s\\to\\infty\n\\end{equation}\nand \\eqref{h-limit} follows from  \\eqref{h'-limit10}, \\eqref{h-ineqn10}   and the l'Hospital rule.\n\\end{proof} \n\n\\begin{cor}\nLet $n\\ge 3$, $0<m<\\frac{n-2}{n}$ and $m\\ne\\frac{n-2}{n+2}$. Then there exists a constant $s_0\\in{\\mathbb{R}}$ such that\n\\begin{equation*}\n\\left\\{\\begin{aligned}\n&h_s(s)<0\\quad\\forall s\\ge s_0\\qquad\\mbox{ if }\\,\\,0<m<\\frac{n-2}{n+2}\\\\\n&h_s(s)>0\\quad\\forall s\\ge s_0\\qquad\\mbox{ if }\\,\\,\\frac{n-2}{n+2}<m<\\frac{n-2}{n}.\n\\end{aligned}\\right.\n\\end{equation*}\n\\end{cor}\n\nLet \n\\begin{equation}\\label{h1-defn}\nh_1(s)=h(s)+\\frac{(n-1)[n-2-(n+2)m]}{(1-m)\\beta}\\log s.\n\\end{equation}\nThen by Lemma \\ref{h-limit-lem1}, $h_1(s)=o(\\log s)$ as $s\\to\\infty$. By \\eqref{h-eqn2} $h_1$ satisfies\n\\begin{align}\\label{h1-eqn2}\n&h_{1,ss}+\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h+\\frac{n-2-(n+2)m}{1-m}\\right)h_{1,s}\\notag\\\\\n=&\\frac{1-2m}{1-m}\\cdot\\frac{w_s^2}{w}+a_2\\left[-\\frac{1}{s^2}+\\left(\\frac{\\beta}{n-1}h+\\frac{n-2-(n+2)m}{1-m}\\right)\\frac{1}{s}\\right]\\quad\\mbox{ in }{\\mathbb{R}}\n\\end{align}\nwhere\n\\begin{equation*}\na_2=\\frac{(n-1)[n-2-(n+2)m]}{(1-m)\\beta}.\n\\end{equation*}\n\n\\begin{lem}\nLet $0<m<\\frac{n-2}{n}$, $m\\ne\\frac{n-2}{n+2}$ , $\\lambda>0$ and $\\beta>0$. Then \n\\begin{equation}\\label{h1'-limit}\n\\lim_{s\\to\\infty}\\frac{s^2h_{1,s}(s)}{\\log s}=-\\frac{(n-1)(n-2-(n+2)m)^2}{2(n-2-nm)(1-m)\\beta}.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nLet \n\\begin{equation*}\nH(s)=\\frac{1-2m}{1-m}\\cdot\\frac{w_s^2}{w}+a_2\\left[-\\frac{1}{s^2}+\\left(\\frac{\\beta}{n-1}h+\\frac{n-2-(n+2)m}{1-m}\\right)\\frac{1}{s}\\right].\n\\end{equation*}\nThen by  Theorem \\ref{w's-bd-thm} and Lemma \\ref{h-limit-lem1},\n\\begin{equation}\\label{H-infty-limit}\n\\lim_{s\\to\\infty}\\frac{sH(s)}{\\log s}=-\\frac{(n-2-(n+2)m)}{1-m}a_2=-a_3\n\\end{equation}\nwhere\n\\begin{equation}\na_3=\\frac{(n-1)(n-2-(n+2)m)^2}{(1-m)^2\\beta}.\n\\end{equation}\nBy \\eqref{h1-eqn2} and \\eqref{H-infty-limit} for any $0<\\varepsilon<a_3\/2$ there exists a constant $s_1>1$ such that\n\\begin{align}\\label{h1-ineqn}\n& (-a_3-\\varepsilon)\\frac{\\log s}{s}\\le h_{1,ss}+\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h+\\frac{n-2-(n+2)m}{1-m}\\right)h_{1,s}\\le (-a_3+\\varepsilon)\\frac{\\log s}{s}\n\\end{align}\nholds for all $s\\ge s_1$. Let $f$ be given by \\eqref{f-defn}. Multiplying \\eqref{h1-ineqn} by $f$ and integrating over $(s_1,s)$,\n\\begin{align}\\label{h1'-ineqn3}\n\\frac{s^2}{\\log s}\\left(\\frac{f(s_1)h_{1,s}(s_1)+(-a_3-\\varepsilon)\\int_{s_1}^s\\frac{\\log z}{z}f(z)\\,dz}{f(s)}\\right)\\le& \\frac{s^2h_{1,s}(s)}{\\log s}\\notag\\\\\n\\le&\n\\frac{s^2}{\\log s}\\left(\\frac{f(s_1)h_{1,s}(s_1)+(-a_3+\\varepsilon)\\int_{s_1}^s\\frac{\\log z}{z}f(z)\\,dz}{f(s)}\\right)\n\\end{align}\nholds for all $s\\ge s_1$.\nBy the l'Hospital rule and Lemma \\ref{h-limit-lem1},\n\\begin{align}\\label{f-expression-ratio-limit2}\n\\lim_{s\\to\\infty}\\frac{s^2\\int_{s_1}^s\\frac{\\log z}{z}f(z)\\,dz}{ f(s)\\log s}\n=&\\lim_{s\\to\\infty}\\frac{f(s)s\\log s+2s\\int_{s_1}^s\\frac{\\log z}{z}f(z)\\,dz}{f(s)\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h(s)+\\frac{n-2-(n+2)m}{1-m}\\right)\\log s+\\frac{f(s)}{s}}\\notag\\\\\n=&\\frac{1-m}{2(n-2-nm)}+\\frac{1-m}{n-2-nm}\\lim_{s\\to\\infty}\\frac{\\int_{s_1}^s\\frac{\\log z}{z}f(z)\\,dz}{f(s)\\log s}\n\\end{align}\nSince \n\\begin{equation*}\n\\left|\\frac{\\int_{s_1}^s\\frac{\\log z}{z}f(z)\\,dz}{f(s)\\log s}\\right|\\le\\frac{\\int_{s_1}^sf(z)\\,dz}{f(s)}\\quad\\forall s\\ge s_1,\n\\end{equation*}\nby \\eqref{f-expression-ratio-limit},\n\\begin{equation}\\label{f-expression-ratio-limit3}\n\\lim_{s\\to\\infty}\\frac{\\int_{s_1}^s\\frac{\\log z}{z}f(z)\\,dz}{f(s)\\log s}=0.\n\\end{equation}\nBy \\eqref{f-expression-ratio-limit2} and \\eqref{f-expression-ratio-limit3},\n\\begin{equation}\\label{f-expression-ratio-limit4}\n\\lim_{s\\to\\infty}\\frac{s^2\\int_{s_1}^s\\frac{\\log z}{z}f(z)\\,dz}{ f(s)\\log s}=\\frac{1-m}{2(n-2-nm)}.\n\\end{equation}\nLetting first $s\\to\\infty$ and then $\\varepsilon\\to 0$ in \\eqref{h1'-ineqn3}, by \\eqref{f-expression-ratio-limit4} we get \\eqref{h1'-limit}\nand the lemma follows.\n\\end{proof}\n\n\\begin{cor}\\label{h1-expansion-cor}\nLet $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $m\\ne\\frac{n-2}{n+2}$, $\\lambda>0$ and $\\beta>0$. Then \n\\begin{equation}\\label{K-defn}\nK(\\lambda,\\beta):=\\lim_{s\\to\\infty}h_1(s)\\in{\\mathbb{R}}\\quad\\mbox{ exists}\n\\end{equation} \nand\n\\begin{equation}\\label{h1-expression}\nh_1(s)=K(\\lambda,\\beta)+\\frac{(n-1)(n-2-(n+2)m)^2}{2(n-2-nm)(1-m)\\beta}\\left(\\frac{1+\\log s}{s}\\right)+o\\left(\\frac{1+\\log s}{s}\\right)\\quad\\mbox{ as }s\\to\\infty.\n\\end{equation}\n\\end{cor}\n\\begin{proof}\nBy \\eqref{h1'-limit} there exist constants $C_1>0$ and $s_1>1$ such that\n\\begin{equation}\n\\left|\\frac{s^2h_{1,s}(s)}{\\log s}\\right|\\le C_1\\quad\\forall s\\ge s_1.\n\\end{equation} \nHence\n\\begin{align}\\label{h1-uniform-bd}\n&|h_1(s)-h_1(s_1)|\\le\\int_{s_1}^s|h_{1,s}(z)|\\,dz\\le C_1\\int_{s_1}^s\\frac{\\log z}{z^2}\\,dz\\le C_2\\quad\\forall s\\ge s_1\\notag\\\\\n\\Rightarrow\\quad&|h_1(s)|\\le C_2+|h_1(s_1)|\\quad\\forall s\\ge s_1\n\\end{align}\nfor some constant $C_2>0$. On the other hand by \\eqref{h1'-limit} there exists a constant $s_0>s_1$ such that\n\\begin{equation*}\nh_{1,s}(s)<0\\quad\\forall s\\ge s_0.\n\\end{equation*}\nThen $h_1(s)$ is  monotone decreasing in $(s_0,\\infty)$. This together with \\eqref{h1-uniform-bd} implies that \\eqref{K-defn} holds.\nBy \\eqref{h1'-limit} for any \n$0<\\varepsilon<\\frac{(n-1)(n-2-(n+2)m)^2}{4(n-2-nm)(1-m)\\beta}$ there exists a constant $s_2>1$ such that\n\\begin{equation}\\label{h1'-ineqn6}\n\\left(-\\frac{(n-1)(n-2-(n+2)m)^2}{2(n-2-nm)(1-m)\\beta}-\\varepsilon\\right)\\frac{\\log s}{s^2}\\le h_{1,s}(s)\\le \\left(-\\frac{(n-1)(n-2-(n+2)m)^2}{2(n-2-nm)(1-m)\\beta}+\\varepsilon\\right)\\frac{\\log s}{s^2}\n\\end{equation}\nholds for all $s\\ge s_2$.\nIntegrating \\eqref{h1'-ineqn6} over $(s,\\infty)$, $s\\ge s_2$,\n\\begin{align*}\n-\\left(\\frac{(n-1)(n-2-(n+2)m)^2}{2(n-2-nm)(1-m)\\beta}+\\varepsilon\\right)\\frac{(1+\\log s)}{s}\n\\le &K(\\lambda,\\beta)-h_1(s)\\\\\n\\le&-\\left(\\frac{(n-1)(n-2-(n+2)m)^2}{2(n-2-nm)(1-m)\\beta}-\\varepsilon\\right)\\frac{(1+\\log s)}{s}\n\\end{align*}\nfor all $s\\ge s_2$ and \\eqref{h1-expression} follows.\n\\end{proof}\n\nLet $K(\\lambda,\\beta)$ be given by \\eqref{K-defn} and\n\\begin{equation}\\label{h2-defn}\nh_2(s)=h_1(s)-K(\\lambda,\\beta)-\\frac{(n-1)(n-2-(n+2)m)^2}{2(n-2-nm)(1-m)\\beta}\\left(\\frac{1+\\log s}{s}\\right).\n\\end{equation}\nThen \n\\begin{equation}\\label{h2-infty}\nh_2(s)=o\\left(\\frac{1+\\log s}{s}\\right)\\mbox{ as }s\\to\\infty\n\\end{equation}\n and by \\eqref{h1-defn} and \\eqref{h1-eqn2},\n\\begin{align}\\label{h2-eqn}\n&h_{2,ss}+\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h+\\frac{n-2-(n+2)m}{1-m}\\right)h_{2,s}\\notag\\\\\n=&\\frac{1-2m}{1-m}\\cdot\\frac{w_s^2}{w}+\\frac{(n-1)(n-2-(n+2)m)^2}{(1-m)^2\\beta}\\cdot\\frac{1}{s}+\\frac{(n-2-(n+2)m)}{1-m}\\cdot\\frac{h_1(s)}{s}-\\frac{a_2}{s^2}\\notag\\\\\n&\\qquad +\\frac{(n-1)(n-2-(n+2)m)^2}{2(1-m)(n-2-nm)\\beta}\\cdot\\left[\\frac{(1-2\\log s)}{s^3}+\\left(\\frac{\\beta}{n-1}h(s)+\\frac{n-2-(n+2)m}{1-m}\\right)\\cdot\\frac{\\log s}{s^2}\\right]\\notag\\\\\n=&:H_1(s).\n\\end{align}\n\n\\begin{lem}\\label{h2-limit-lem}\nLet $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $m\\ne\\frac{n-2}{n+2}$, $\\lambda>0$ and $\\beta>0$. Then $h_2$ satisfies\n\\begin{equation}\\label{h2-limit}\n\\lim_{s\\to\\infty}s^2\\,h_{2,s}(s)=\\frac{(1-m)a_1(\\lambda,\\beta)}{2(n-2-nm)\\beta}\n\\end{equation}\nwhere $a_1(\\lambda,\\beta)$ is given by \\eqref{a1-defn} with $K(\\lambda,\\beta)$ given by \\eqref{K-defn}.\n\\end{lem}\n\\begin{proof}\nBy Theorem \\ref{w's-bd-thm}, Lemma \\ref{h-limit-lem1} and \\eqref{K-defn},\n\\begin{equation}\\label{H1-decay-rate}\n\\lim_{s\\to\\infty}sH_1(s)=\\frac{a_1(\\lambda,\\beta)}{\\beta}\n\\end{equation}\nwhere $a_1(\\lambda,\\beta)$ is given by \\eqref{a1-defn} with $K(\\lambda,\\beta)$ given by \\eqref{K-defn}.\nThen by \\eqref{h2-eqn} and \\eqref{H1-decay-rate} for any $0<\\varepsilon<1$ there exists a constant $s_1>1$ such that\n\\begin{equation}\\label{h2-ineqn20}\n\\left(\\frac{a_1(\\lambda,\\beta)}{\\beta}-\\varepsilon\\right)\\frac{1}{s}\\le h_{2,ss}+\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h+\\frac{n-2-(n+2)m}{1-m}\\right)h_{2,s}\\le\\left(\\frac{a_1(\\lambda,\\beta)}{\\beta}+\\varepsilon\\right)\\frac{1}{s} \n\\end{equation}\nholds for all $s\\ge s_1$.\nLet $f$ be given by \\eqref{f-defn}. Multiplying \\eqref{h2-ineqn20} by $f$ and integrating over $(s_1,s)$,\n\\begin{equation}\\label{h-ineqn32}\n\\frac{f(s_1)h_2(s_1)s^2+\\left(\\frac{a_1(\\lambda,\\beta)}{\\beta}-\\varepsilon\\right)s^2\\int_{s_1}^s\\frac{f(z)}{z}\\,dz}{f(s)}\\le s^2h_{2,s}(s)\\le \\frac{f(s_1)h_2(s_1)s^2+\\left(\\frac{a_1(\\lambda,\\beta)}{\\beta}+\\varepsilon\\right)s^2\\int_{s_1}^s\\frac{f(z)}{z}\\,dz}{f(s)}\n\\end{equation}\nholds for all $s\\ge s_1$.\nSince by the l'Hospital rule,\n\\begin{equation*}\n\\lim_{s\\to\\infty}\\frac{\\int_{s_1}^s\\frac{f(z)}{z}\\,dz}{f(s)}=\\lim_{s\\to\\infty}\\frac{\\frac{f(s)}{s}}{f(s)\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h(s)+\\frac{n-2-(n+2)m}{1-m}\\right)}=0,\n\\end{equation*}\nby Lemma \\ref{h-limit-lem1} and the l'Hospital rule,\n\\begin{align}\\label{f-ratio-limit}\n\\lim_{s\\to\\infty}\\frac{s^2\\int_{s_1}^s\\frac{f(z)}{z}\\,dz}{f(s)}=&\\lim_{s\\to\\infty}\\frac{sf(s)+2s\\int_{s_1}^s\\frac{f(z)}{z}\\,dz}{f(s)\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h(s)+\\frac{n-2-(n+2)m}{1-m}\\right)}\\notag\\\\\n=&\\frac{(1-m)}{2(n-2-nm)}+\\frac{(1-m)}{(n-2-nm)}\\lim_{s\\to\\infty}\\frac{\\int_{s_1}^s\\frac{f(z)}{z}\\,dz}{f(s)}\\notag\\\\\n=&\\frac{(1-m)}{2(n-2-nm)}.\n\\end{align}\nHence letting first $s\\to\\infty$ and then $\\varepsilon\\to 0$ in \\eqref{h-ineqn32}, by \\eqref{f-defn} and   \\eqref{f-ratio-limit} we get \\eqref{h2-limit} and the lemma follows.\n\\end{proof}\n\nBy \\eqref{h2-infty} and Lemma \\ref{h2-limit-lem} we have the following result.\n\n\\begin{lem}\\label{h2-expansion-lem}\nLet $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $m\\ne\\frac{n-2}{n+2}$, $\\lambda>0$ and $\\beta>0$.  Then \n\\begin{equation}\\label{h2-expansion}\nh_2(s)=-\\frac{(1-m)a_1(\\lambda,\\beta)}{2(n-2-nm)\\beta s}+\\frac{o(s)}{s}\\quad\\mbox{ as }s\\to\\infty\n\\end{equation}\nwhere $a_1(\\lambda,\\beta)$ is given by \\eqref{a1-defn} with $K(\\lambda,\\beta)$ given by \\eqref{K-defn}.\n\\end{lem}\n\nWe are now ready for the proof of Theorem \\ref{higher-order-expansion-thm}.\n\n\\noindent{\\bf Proof of Theorem \\ref{higher-order-expansion-thm}}:  Let $a_0$, $a_1(\\lambda,\\beta)$, be given by \\eqref{a0-defn} and \\eqref{a1-defn} with $K(\\lambda,\\beta)$ given by \\eqref{K-defn}. Let \n\\begin{equation*}\nK_0=\\frac{(1-m)K(1,1)}{2(n-1)(n-2-nm)}.\n\\end{equation*}\nBy \\eqref{w-defn}, \\eqref{h-defn10}, \\eqref{h1-defn}, \\eqref{h2-defn} and \\eqref{h2-expansion}, \n\\begin{align}\\label{v11-expansion}\nv_{1,1}(r)^{1-m}=&\\frac{2(n-1)(n-2-nm)}{(1-m)r^2}\\left\\{\\log r-\\frac{(n-2-(n+2)m)}{2(n-2-nm)}\\log (\\log r)+K_0\\right.\\notag\\\\\n&\\qquad+\\frac{(n-2-(n+2)m)^2}{4(n-2-nm)^2}\\left.\\left(\\frac{1+\\log(\\log r)}{\\log r}\\right)\n-\\frac{(1-m)^2}{4(n-1)(n-2-nm)^2}\\frac{a_1(1,1)}{\\log r}\\right.\\notag\\\\\n&\\qquad \\left.+\\frac{o(\\log r)}{\\log r}\\right\\}\\notag\\\\\n=&\\frac{2(n-1)(n-2-nm)}{(1-m)r^2}\\left\\{\\log r-\\frac{(n-2-(n+2)m)}{2(n-2-nm)}\\log (\\log r)+K_0+\\frac{a_0}{\\log r}\\right.\\notag\\\\\n&\\qquad\\left.+\\frac{(n-2-(n+2)m)^2}{4(n-2-nm)^2}\\cdot\\frac{\\log(\\log r)}{\\log r}+\\frac{o(\\log r)}{\\log r}\\right\\}\\quad\\mbox{ as }r\\to\\infty.\n\\end{align}\nThen by (2.19) of \\cite{CD} and \\eqref{v11-expansion},\n\\begin{align*}\nv_{\\lambda,\\beta}(r)^{1-m}=&\\lambda^{1-m} v_{1,1}(\\lambda^{\\frac{1-m}{2}}\\sqrt{\\beta}r)^{1-m}\\notag\\\\\n=&\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta r^2}\\left\\{\\log (\\lambda^{\\frac{1-m}{2}}\\sqrt{\\beta}r)-\\frac{(n-2-(n+2)m)}{2(n-2-nm)}\\log (\\log (\\lambda^{\\frac{1-m}{2}}\\sqrt{\\beta}r))+K_0\n\\right.\\notag\\\\\n&\\qquad+\\frac{a_0}{\\log (\\lambda^{\\frac{1-m}{2}}\\sqrt{\\beta}r)}\n+\\frac{(n-2-(n+2)m)^2}{4(n-2-nm)^2}\\cdot\\frac{\\log(\\log (\\lambda^{\\frac{1-m}{2}}\\sqrt{\\beta}r))}{\\log (\\lambda^{\\frac{1-m}{2}}\\sqrt{\\beta}r)}\\notag\\\\\n&\\qquad\\left.+\\frac{o(\\log (\\lambda^{\\frac{1-m}{2}}\\sqrt{\\beta}r))}{\\log (\\lambda^{\\frac{1-m}{2}}\\sqrt{\\beta}r)}\\right\\}\\notag\\\\\n=&\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta r^2}\\left\\{\\log r -\\frac{(n-2-(n+2)m)}{2(n-2-nm)}\\log (\\log r))\n+\\frac{1-m}{2}\\log\\lambda\\right.\\notag\\\\\n&\\qquad \\left.+\\frac{1}{2}\\log\\beta+K_0+\\frac{a_0}{\\log r}\n+\\frac{(n-2-(n+2)m)^2}{4(n-2-nm)^2}\\cdot\\frac{\\log(\\log r)}{\\log r}+\\frac{o(\\log r)}{\\log r}\\right\\}\n\\end{align*}\nas $r\\to\\infty$ and Theorem \\ref{higher-order-expansion-thm} follows. \n\n{\\hfill$\\square$\\vspace{6pt}} \n\n\\begin{rmk}\nFrom \\eqref{v-lambda-beta-expansion1},\n\\begin{align*}\nh_1(s)=&\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta}\\left\\{\n\\frac{1-m}{2}\\log\\lambda +\\frac{1}{2}\\log\\beta+K_0+\\frac{a_0}{s}\n+\\frac{(n-2-(n+2)m)^2}{4(n-2-nm)^2}\\cdot\\frac{\\log s}{s}\\right.\\\\\n&\\qquad\\left.+\\frac{o(s)}{s}\\right\\}\\qquad\\mbox{ as }s\\to\\infty.\n\\end{align*}\nHence \n\\begin{equation}\\label{h1-limit2}\n\\lim_{s\\to\\infty}h_1(s)=\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta}\\left\\{\n\\frac{1-m}{2}\\log\\lambda +\\frac{1}{2}\\log\\beta+K_0\\right\\}.\n\\end{equation}\nThus by \\eqref{K-defn} and \\eqref{h1-limit2},\n\\begin{equation*}\nK(\\lambda,\\beta)=\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta}\\left\\{\n\\frac{1-m}{2}\\log\\lambda +\\frac{1}{2}\\log\\beta+K_0\\right\\}.\n\\end{equation*}\n\\end{rmk}\n \n\\noindent{\\bf Proof of Theorem \\ref{convergence-thm}}: Since the proof is similar to the proof of Theorem 3.1 and Corollary 3.2 of \\cite{CD} we will only sketch the proof here. Let $K_0$ be given by Theorem \\ref{higher-order-expansion-thm}  and $\\lambda_1=\\left(e^{2(K_1-K_0)}\/\\beta\\right)^{\\frac{1}{1-m}}$. Then for any $0<\\varepsilon<\\lambda_1$ there exists a constant $R_{\\varepsilon}>0$ such that\n\\begin{align}\\label{initial-data-compare}\n&u_{\\lambda_1-\\varepsilon,\\beta}(x)\\le \\phi(x)\\le u_{\\lambda_1+\\varepsilon,\\beta}(x)\\quad\\forall |x|\\ge R_{\\varepsilon}\\notag\\\\\n\\Rightarrow\\quad&u_{\\lambda_1-\\varepsilon,\\beta}(x)+\\psi(x)\\le u_0(x)\\le u_{\\lambda_1+\\varepsilon,\\beta}(x)+\\psi(x)\\quad\\forall |x|\\ge R_{\\varepsilon}.\n\\end{align}\nFor any $\\delta>0$, let $u_1$, $u_2$,  $u_{1,\\delta}$ and $w_{1,\\delta}$ be the solution of \\eqref{cauchy-problem} with initial value \n\\begin{equation*}\n\\min (u_{\\lambda_1-\\varepsilon,\\beta}(x)+\\psi(x), u_0(x)),\\,\\, \\max (u_{\\lambda_1+\\varepsilon,\\beta}(x)+\\psi(x), u_0(x)),\\,\\, \\min (u_{\\lambda_1-\\varepsilon,\\beta}(x)+\\psi(x), u_0(x))+\\delta,\n\\end{equation*}\n and $u_{\\lambda_1-\\varepsilon,\\beta}(x)+\\delta$\nrespectively given by Theorem 1.1 of \\cite{Hs2}. Let $\\widetilde{u}_1$, $\\widetilde{u}_2$, $\\widetilde{u}_{1,\\delta}$ and  $\\widetilde{w}_{1,\\delta}$ be given by \\eqref{rescaled-soln} with $u$ being replaced by $u_1$, $u_2$, $u_{1,\\delta}$ and $w_{1,\\delta}$ respectively. By \\eqref{initial-data-compare}\nand the construction of solutions in \\cite{Hs2},\n\\begin{align}\n&u_1\\le u\\le u_2, \\quad u_{1,\\delta}\\ge\\delta, \\quad w_{1,\\delta}\\ge\\delta\\quad\\mbox{ in }{\\mathbb{R}}^n\\times (0,\\infty)\\quad\\forall\\delta>0\\label{u1-u-u2-compare}\\\\\n\\Rightarrow\\quad& \\widetilde{u}_1\\le \\widetilde{u}\\le\\widetilde{u}_2, \\quad\\mbox{ in }{\\mathbb{R}}^n\\times (0,\\infty)\\label{u1-u-u2-tilde-compare2}.\n\\end{align}\nBy \\eqref{u1-u-u2-compare} and Lemma \\ref{L1-comparison-lem},\n\\begin{equation}\\label{u-tidle-L1-comparison}\n\\|\\widetilde{u}_{1,\\delta}(\\cdot,t)-\\widetilde{w}_{1,\\delta}(\\cdot,t)\\|_{L^1({\\mathbb{R}}^n)}\\le e^{-\\frac{(n-2-nm)}{1-m}t} \\|\\min (u_{\\lambda_1-\\varepsilon,\\beta}+\\psi, u_0)-u_{\\lambda_1-\\varepsilon,\\beta}\\|_{L^1({\\mathbb{R}}^n)}\\quad\\forall t >0\n\\end{equation} \nSince $\\min (u_{\\lambda_1-\\varepsilon,\\beta}(x)+\\psi(x), u_0(x))+\\delta$ and $u_{\\lambda_1-\\varepsilon,\\beta}(x)+\\delta$ decreases monotonically to \n\\begin{equation*}\n\\min (u_{\\lambda-\\varepsilon,\\beta}(x)+\\psi(x), u_0(x))\\quad\\mbox{ and } \n\\quad u_{\\lambda_1-\\varepsilon,\\beta}(x)\n\\end{equation*}\n as $\\delta\\to 0$, $u_{1,\\delta}$ and $w_{1,\\delta}$ decreases monotonically to $u_1$ and \n$e^{-\\frac{2\\beta}{1-m}t}u_{\\lambda_1-\\varepsilon,\\beta}(e^{-\\beta t}x)$ as $\\delta\\to 0$.\nHence letting $\\delta\\to 0$ in \\eqref{u-tidle-L1-comparison},\n\\begin{equation}\\label{u-tidle-L1-comparison2}\n\\|\\widetilde{u}_1(\\cdot,t)-u_{\\lambda_1-\\varepsilon,\\beta}\\|_{L^1({\\mathbb{R}}^n)}\\le e^{-\\frac{(n-2-nm)}{1-m}t} \\|\\min (u_{\\lambda_1-\\varepsilon,\\beta}+\\psi, u_0)-u_{\\lambda_1-\\varepsilon,\\beta}\\|_{L^1({\\mathbb{R}}^n)}\\quad\\forall t >0\n\\end{equation} \nSimilarly,\n\\begin{equation}\\label{u-tidle-L1-comparison3}\n\\|\\widetilde{u}_2(\\cdot,t)-u_{\\lambda_1+\\varepsilon,\\beta}\\|_{L^1({\\mathbb{R}}^n)}\\le e^{-\\frac{(n-2-nm)}{1-m}t} \\|\\max (u_{\\lambda_1-\\varepsilon,\\beta}+\\psi, u_0)-u_{\\lambda_1+\\varepsilon,\\beta}\\|_{L^1({\\mathbb{R}}^n)}\\quad\\forall t >0\n\\end{equation} \nBy \\eqref{u1-u-u2-tilde-compare2}, \\eqref{u-tidle-L1-comparison2} and \\eqref{u-tidle-L1-comparison3}, and an argument similar to the proof of Theorem 3.1 of \\cite{CD}, the rescaled function $\\widetilde{u}(x,t)$ given by \\eqref{rescaled-soln} converges to  $v_{\\lambda_1,\\beta}$ in $L_{loc}^1({\\mathbb{R}}^n)$ as $t\\to\\infty$.\n \nSuppose now $u_0$ also satisfies $u_0=\\phi\\in L^{\\infty}({\\mathbb{R}}^n)$. Then by an argument similar to the proof of Corollary 3.2 of \\cite{CD}, there exists a constant $\\lambda_2>0$ such that\n\\begin{equation*}\nu_0\\le v_{\\lambda_2,\\beta}\\quad\\mbox{ in }{\\mathbb{R}}^n\n\\end{equation*}\nHence by maximum principle for solutions of \\eqref{fde-eqn} in bounded domains (cf. Lemma 2.3 of \\cite{DaK}) and the construction of solution \\eqref{cauchy-problem} in \\cite{Hs2},\n\\begin{align}\\label{u-tilde-v-bd}\n&u(x,t)\\le e^{-\\frac{2\\beta}{1-m}t}v_{\\lambda_2,\\beta}(e^{-\\beta t}x)\\quad\\forall x\\in{\\mathbb{R}}^n, t>0\\notag\\\\\n\\Rightarrow\\quad&\\widetilde{u}(x,t)\\le v_{\\lambda_2,\\beta}(x)\\qquad\\qquad\\,\\forall x\\in{\\mathbb{R}}^n, t>0.\n\\end{align}  \nThen by \\eqref{u-tilde-eqn},  \\eqref{u-tilde-v-bd}, and an argument similar to the proof on P.10 of \\cite{CD}, the rescaled function $\\widetilde{u}(x,t)$ converges to  $v_{\\lambda_1,\\beta}$ uniformly in $C^{2,1}(E)$ for any compact subset $E\\subset{\\mathbb{R}}^n$ as $t\\to\\infty$. \n\n{\\hfill$\\square$\\vspace{6pt}}  \n\nFinally by Theorem \\ref{higher-order-expansion-thm} and a similar argument as the proof of Theorem 3.6 and Proposition 3.9 of \\cite{CD} we have the following result.\n\n\\begin{thm}\\label{convergence-thm2}\nLet $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $m\\ne\\frac{n-2}{n+2}$, $\\beta>0$, $0\\le u_0=\\phi+\\psi$, $ \\phi\\in  L_{loc}^p({\\mathbb{R}}^n)$, $\\psi\\in L^1({\\mathbb{R}}^n)\\cap L_{loc}^p({\\mathbb{R}}^n)$, for some constant $p>\\frac{(1-m)n}{2}$, such that \n\\begin{equation*}\nK_2:=\\limsup_{|x|\\to\\infty}\\left[|x|^2u_0(x)^{1-m}-\\frac{c_1}{\\beta}\\left(\\log|x|-\\frac{(n-2-(n+2)m)}{2(n-2-nm)}\\log(\\log |x|)\\right)\\right]<\\infty.\n\\end{equation*}\nholds and $\\phi$ satisfies \\eqref{phi-expansion} for some constant $K_1\\in{\\mathbb{R}}$ where $c_1=2(n-1)(n-2-nm)\/(1-m)$.\nIf $u$ is the unique solution of \\eqref{fde-eqn} in ${\\mathbb{R}}^n\\times (0,\\infty)$ given by\nTheorem 1.1 of \\cite{Hs2}, then as $t\\to\\infty$, the rescaled function $\\widetilde{u}(x,t)$ given by \\eqref{rescaled-soln} converges to  $v_{\\lambda_1,\\beta}$ uniformly in $C^{2,1}(E)$ for any compact subset $E\\subset{\\mathbb{R}}^n$ with $\\lambda_1=\\left(e^{2(K_1-K_0)}\/\\beta\\right)^{\\frac{1}{1-m}}$ where the constant $K_0$ is given by Theorem \\ref{higher-order-expansion-thm}. Moreover\n\\begin{align*}\n&\\limsup_{|x|\\to\\infty}\\left[|x|^2u(x,t)^{1-m}-\\frac{c_1}{\\beta}\\left(\\log|x|-\\frac{n-2-(n+2)m}{2(n-2-nm)}\\log(\\log |x|)\\right)\\right]\\notag\\\\\n\\le&K_2-\\frac{2(n-1)(n-2-nm)}{(1-m)}t\\quad\\forall t\\ge 0.\n\\end{align*}\n\\end{thm}\n \n\\section{Appendix}\n\\setcounter{equation}{0}\n\\setcounter{thm}{0} \n\nFor the sake of completeness, in this appendix we will state and prove the analogue of Lemma \\ref{h-limit-lem1} for the case $m=\\frac{n-2}{n+2}$, $n\\ge 3$.\n\n\\begin{prop}(Proposition 2.3 of \\cite{CD})\nLet $n\\ge 3$, $m=\\frac{n-2}{n+2}$, $\\lambda>0$ and $\\beta>0$. Let $v$ be the solution of \\eqref{elliptic-eqn} and $h$ be given by  \\eqref{h-defn10} with $w$ given by \\eqref{w-defn}. Then $h$ satisfies\n\\begin{equation*}\n\\lim_{s\\to\\infty}s^2\\,h_s(s)=\\frac{(6-n)(n-1)}{4\\beta}.\n\\end{equation*}\n\\end{prop}\n\\begin{proof}\nThis proposition is stated and proved in \\cite{CD}. For the sake of completeness we will give a simple different proof of the proposition here. We first observe that by Theorem \\ref{w's-bd-thm},\n\\begin{equation}\\label{w-ratio-limit}\n\\lim_{s\\to\\infty}\\frac{sw_s(s)^2}{w(s)}=\\frac{2(n-1)(n-2-nm)}{(1-m)\\beta}=\\frac{(n-1)(n-2)}{\\beta}.\n\\end{equation}\nLet\n\\begin{equation*}\na_4=\\frac{(1-2m)(n-1)(n-2)}{(1-m)\\beta}.\n\\end{equation*}\nThen by \\eqref{h-eqn2} and \\eqref{w-ratio-limit}   for any $0<\\varepsilon<|a_4|\/2$ there exists a constant $s_1\\in{\\mathbb{R}}$ such that\n\\begin{equation}\\label{h-ineqn31}\n(a_4-\\varepsilon)s^{-1}\\le h_{ss}+\\left(\\frac{2(n-2-nm)}{(1-m)}s+\\frac{\\beta}{n-1}h+\\frac{n-2-(n+2)m}{1-m}\\right)h_s\\le (a_4+\\varepsilon)s^{-1}\\quad\\forall s\\ge s_1.\n\\end{equation}\nBy \\eqref{h-ineqn31} and an argument similar to the proof of Lemma \\ref{h2-limit-lem},\n\\begin{equation*}\n\\lim_{s\\to\\infty}s^2\\,h_s(s)=\\frac{(1-m)a_4}{2(n-2-nm)}=\\frac{(6-n)(n-1)}{4\\beta}\n\\end{equation*}\nand the proposition follows.\n\\end{proof}\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe following terminology is basic in our research of puns. {\\bf A pun} is a) a short humorous genre,\nwhere a word or phrase is intentionally used in two meanings, b) a means of expression, the essence of which \nis to use a word or phrase so that in the given context the word or phrase can be understood in two meanings simultaneously.\n{\\bf A target word} is a word, that appears in two meanings. {\\bf A homographic pun} is a pun that \n``exploits distinct meanings of the same written word''~\\cite{miller2015automatic} (these can be meanings of a polysemantic word, \nor homonyms, including homonymic word forms). {\\bf A heterographic pun} is a pun, in which the target word \nresembles another word, or phrase in spelling; we will call the latter {\\bf the second target word}. \nConsider the following example (the Banker joke):\n\n\\begin{quote}\n``I used to be a banker, but I lost interest.''\n\\end{quote}\n\nThe Banker joke is a homographic pun; {\\em interest} is the target word. Unlike it, the Church joke below is a heterographic pun; \n{\\em propane} is the target word, {\\em profane} is the second target word:\n\n\\begin{quote}\n``When the church bought gas for their annual barbecue, proceeds went from the sacred to the propane.''\n\\end{quote} \n\nOur model of automatic pun analysis is based on the following premise: in a pun, there are two groups of words, \nand their meanings, that indicate the two meanings in which the target word is used. These groups can overlap, \ni.e. contain the same polysemantic words, used in different meanings. \n\nIn the Banker joke, words, and collocations {\\em banker}, {\\em lost interest} point at the professional status of the narrator, and his\/her \ncareer failure. At the same time, {\\em used to}, {\\em lost interest} tell a story of losing emotional attachment to the profession: \nthe narrator became disinterested. The algorithm of pun recognition, which we suggest, discovers these two groups of words, based on common \nsemes\\footnote{Bits of meaning. Semes are some parts of meaning, present both in the word and in its hypernym.\nMoving up the taxonomy, like Thesaurus, or WordNet, hypernyms become more general, and the seme, connecting them to the word, becomes more general, too.} \n(Subtask 1), finds the words, which belong to the both groups, and chooses the target word (Subtask 2), \nand, based on the common semes, picks up the best suitable meaning, which the target word exploits (Subtask 3). \nIn case of heterographic puns, in Subtask 2, the algorithm looks for the word, or phrase, which appears in one group \nand {\\em not} in the other.\n\n\\section{Subtask 1: Mining Semantic Fields}\n\nWe will call a semantic field a group of words and collocations, which share a common seme. In taxonomies, like WordNet~\\cite{kilgarriff2000wordnet}, and \nRoget's Thesaurus~\\cite{roget2004roget} (further referred to as Thesaurus), semes appear as hierarchies of word meanings. Top-levels\nattract words with more general meanings (hypernyms). For example, Thesaurus has six top-level Classes, that divide into Divisions, that divide\ninto Sections, and so on, down to the fifth lowest level. WordNet's structure is not so transparent. CITE!!! 10 TOP-semes \nApplying such dictionaries to get semantic fields (the mentioned common groups of words) in a pun is, therefore, the task of finding \ntwo most general hypernyms in WordNet, or two relevant Classes among the six Classes in Thesaurus. We chose Thesaurus, as its structure \nis only five levels deep, Classes labels are not lemmas themselves, but arbitrary names (we used numbers instead), and it allows parsing \non a certain level, and insert corrections (adding lemmas, merging subsections, etc.\\footnote{For example, we edited Thesaurus, adding words, \nwhich were absent in it. If a word in a pun was missing in Thesaurus, the system checked up for its hypernyms in Wordnet, and added the word \nto those Sections in Thesaurus, which contained the hypernyms.}). After some experimentation, instead of Classes, we chose to search for relevant \nSections, which are 34 subdivisions of the six Classes\\footnote{Sections are not always {\\em immediate} subdivisions of a Class. \nSome Sections are grouped in Divisions.}.\n\nAfter normalization (including change to lowercase; part-of-speech tagging, tokenization, and lemmatization with NLTK tools~\\cite{bird2009natural}; \ncollocation extraction\\footnote{To extract collocations and search for them in Thesaurus, we applied our own procedure, \nbased on a part-of-speech analysis.}; stop-words removal\\footnote{After lemmatization, all words are analyzed in collocations, but only nouns, adjectives, and\nverbs compose a list of separate words.}), the algorithm collects Section numbers for every word, and collocation, and removes duplicates \n(in Thesaurus, homonyms proper can belong to different subdivisions in the same or different Sections). \nTable~\\ref{tab:sem} shows what Sections words of the Banker joke belong to.\n\n\\begin{table*}\n\\centering\n\\begin{tabular}{lll}\n  \\bf Word & \\bf Section No., Section name in Thesaurus \\\\ \n  \\hline\n  I & - \\\\\n  use & 24, Volition In General \\\\\n  & 30, Possessive Relations \\\\\n  to & - \\\\\n  be & 0, Existence \\\\\n  &  19, Results Of Reasoning \\\\\n  a & - & \\\\\n  banker & 31, Affections In General \\\\\n  &  30, Possessive Relations \\\\\n  but & - \\\\\n  lose & 21, Nature Of Ideas Communicated \\\\\n  & 26, Results Of Voluntary Action \\\\\n  & 30, Possessive Relations \\\\\n  & 19, Results Of Reasoning \\\\\n  interest & 30, Possessive Relations \\\\\n  & 25, Antagonism \\\\\n  & 24, Volition In General \\\\\n  & 7, Causation \\\\\n  & 31, Affections In General \\\\\n  & 16, Precursory Conditions And Operations \\\\\n  & 1, Relation \\\\\n\\end{tabular}\n\\caption{Semantic fields in the Banker joke}\\label{tab:sem}\n\\end{table*}\n\nThen the semantic vector of a pun is calculated. Every pun \\(p\\) is a vector in a 34-dimensional space:\n\\[p_{i}=p_{i}(s_{1i}, s_{2i},...,s_{34i})\\]\\label{f:pi}\nThe value of every element \\(s_{ki}\\) equals the number of words in a pun, which belong to a Section \\(S_{k}\\).\nThe algorithm passes from a Section to a Section, each time checking every word \\(w_{ji}\\) in the bunch of extracted words \\(l_{i}\\). \nIf a word belongs to a Section, the value of \\(s_{ki}\\) RAISES BY???? 1:\n\\[s_{ki}=\\displaystyle\\sum_{j=1}^{l_{i}} \\left\\{1\\middle|w_{ji} \\in S_{k}\\right\\}, k=1,2,...,34, i=1,2,3...\\]\\label{f:ski}\nFor example, the semantic vector of the Banker joke looks as follows: see Table~\\ref{tab:pbnker}.\n\\begin{table*}\n\\centering\n\\begin{tabular}{lll}\n  \\(p_{Banker}\\) &  \\(\\{ 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 2 , 0 , 1 , 0 , 0 , 2 , 1 , 1 , 0 , 0 , 0 , 4 , 2, 0, 0 \\}\\)\\\\ \n\\end{tabular}\n\\caption{Semantic vector of the Banker joke}\\label{tab:pbnker}\n\\end{table*}\n\nTo test the algorithm, we, first, collected 2484 puns from different Internet resources and, second, built a corpus\nof 2484 random sentences of length 5 to 25 words from different NLTK corpora~\\cite{bird2009natural} plus several hundred aphorisms and \nproverbs from different Internet sites. We shuffled and split the sentences into two equal groups, the first two forming \na training set, and the other two a test set. The classification was conducted, using different Scikit-learn~\\cite{pedregosa2011scikit} \nalgorithms. We also singled out 191 homographic puns, and 198 heterographic puns, and tested them against the same number of random\nsentences.\nIn all the tests\\footnote{The tests were run before the competition. Results of the competition for our system are given in Table~\\ref{results}.}, \nthe Scikit-learn algorithm of SVM with the Radial Basis Function (RBF) kernel produced the highest average \nF-measure results (\\(\\bar{f}=\\frac{f_{puns}+f_{random}}{2}\\)). In addition, its results are smoother, comparing the difference \nbetween precision, and recall (which leads to the highest F-measure scores) within the two classes (puns, and random sentences), \nand between the classes (average scores). Table~\\ref{punrec} illustrates results of different algorithms in class ``Puns'' (not \naverage results between puns, and not puns). The results were higher \nfor the split selection, reaching 0.79 (homographic), and 0.78 (heterographic) scores of F-measure. The common selection \ngot the maximum of 0.7 for average F-measure in several tests. The higher results of split selection may be due to \na larger training set.\n\n\\begin{table*}\n\\centering\n\\begin{tabular}{llll}\n\\hline\n\\bf Method & \\bf Precision & \\bf Recall & \\bf F-measure \\\\\n\\hline\n\\bf Common selection \\\\\n\\hline\nSVM with linear kernel & 0.67 & 0.68 & 0.67 \\\\\nSVM with polynomial kernel & 0.65 & 0.79 & 0.72 \\\\\nSVM with Radial Basis Function (RBF) kernel & 0.70 & 0.70 & 0.70 \\\\\nSVM with linear kernel, normalized data & 0.62 & 0.74 & 0.67 \\\\\n\\hline\n\\bf Homographic puns \\\\\n\\hline\nSVM with RBF kernel & 0.79 & 0.80 & 0.79 \\\\\nMultinomial Naive Bayes & 0.71 & 0.80 & 0.76 \\\\\nLogistic Regression, standardized data & 0.77 & 0.71 & 0.74 \\\\\n\\hline\n\\bf Heterographic puns \\\\\n\\hline\nSVM with RBF kernel & 0.77 & 0.79 & 0.78 \\\\\nLogistic Regression & 0.74 & 0.75 & 0.74 \\\\\n\\hline\n\\end{tabular}\n\\caption{Tests for pun recognition.}\\label{punrec}\n\\end{table*}\n\n\\section{Subtask 2: Hitting the Target Word}\n\nWe suggest that, in a homographic pun, the target word is a word, which immediately belongs to two semantic fields; \nin a heterographic pun, the target word belongs to at least one discovered semantic field, and does not belong to the other. \nHowever, in reality, words in a sentence tend to belong to too many fields, and they create noise in the search. \nTo reduce influence of noisy fields, we included such non-semantic features in the model as\nthe tendency of the target word to occur at the end of a sentence, and part-of-speech distribution, given in~\\cite{miller2015automatic}.\nA-group (\\(W_{A}\\)) and B-group (\\(W_{B}\\)) are groups of words in a pun, which belong to the two semantic fields, \nsharing the target word. Thus, for some \\(s_{ki}\\), \\(k\\) becomes \\(A\\), or \\(B\\)~\\footnote{\\(s_{ki}\\) is always an integer; \\(W_{A}\\) and \\(W_{B}\\) \nare always lists of words; \\(A\\) is always an integer, \\(B\\) is a list of one or more integers.}. A-group attracts the maximum number of words in a pun:\n\\[s_{Ai}=\\max_{k} s_{ki}, k=1,2,...,34 \\]\n\nIn the Banker joke, \\(s_{Ai}=4, A=30\\) (Possessive Relations); words, that belong to this group, are {\\em use}, {\\em lose}, \n{\\em banker}, {\\em interest}. B-group is the second largest group in a pun:\n\\[s_{Bi}=\\max_{k} ( s_{ki}\/s_{Ai} ), k=1,2,...,34 \\]\nIn the Banker joke, \\(s_{Bi}=2\\). There are three groups of words, which have two words in them: \\(B_{1}=19\\), \nResults Of Reasoning: {\\em be}, {\\em lose}; \\(B_{2}=24\\), Volition In General: {\\em use}, {\\em interest}; \n\\(B_{3}=31\\), Affections In General: {\\em banker}, {\\em interest}. Ideally, there should be a group of about three words, \nand collocations, describing a person`s inner state ({\\em used to be}, {\\em lose}, {\\em interest}), and two words \n({\\em lose}, {\\em interest}) in \\(W_{A}\\) are a target phrase. However, due to the shortage of data \nabout collocations in dictionaries, \\(W_{B}\\) is split into several smaller groups. Consequently, to find the target word, \nwe have to appeal to other word features. In testing the system on homographic puns, we relied on the polysemantic character of words.\nIf in a joke, there are more than one value of \\(B\\), \\(W_{B}\\) candidates merge into one, with duplicates removed, \nand every word in \\(W_{B}\\) becomes the target word candidate: \\(c \\in W_{B}\\). In the Banker joke, \\(W_{B}\\) is a list of \n{\\em be}, {\\em lose}, {\\em use}, {\\em interest}, {\\em banker}; \\(B=\\{19,24,31\\}\\). Based on the definition of the target word \nin a homographic pun, words from \\(W_{B}\\), that are also found in \\(W_{A}\\), should have a privilege. Therefore, \nthe first value \\(v_{\\alpha}\\), each word gets, is the output of the Boolean function:\n\\[v_{\\alpha}(c)=\\left\\{\n \\begin{array}{lr}\n  2 \\quad \\mathrm{if } (c \\in W_{A})\\wedge(c \\in W_{B}) \\\\\n  1 \\quad \\mathrm{if } (c \\notin W_{A})\\wedge(c \\in W_{B}) \\\\\n \\end{array}\n\\right.\n\\]\n\nThe second value \\(v_{\\beta}\\) is the absolute frequency of a word in the union of \\(B_{1}\\), \\(B_{2}\\), etc., \nincluding duplicates: \\(v_{\\beta}(c)=f_c(W_{B_{1}} \\cup W_{B_{2}} \\cup W_{B_{3}})\\).\n\nThe third value \\(v_{\\gamma}\\) is a word position in the sentence: the closer the word is to the end, the\nbigger this value is. If the word occurs several times, the algorithm counts the average of the\nsums of position numbers.\n\nThe fourth value is part-of-speech probability \\(v_{\\delta}\\). Depending on the part of speech, the word\nbelongs to, it gets the following rate:\n\\[v_{\\delta}(c) = \\left\\{\n  \\begin{array}{lr}\n    0.502, if c - Noun \\\\\n    0.338, if c - Verb \\\\\n    0.131, if c - Adjective \\\\\n    0.016, if c - Adverb \\\\\n    0.013, otherwise \\\\\n  \\end{array}\n\\right.\n\\]\n\nThe final step is to count rates, using multiplicative convolution, and choose the word with the\nmaximum rate:\n\\[z_{1}(W_{B})=\\left\\{c|\\max_{c} (v_{\\alpha} \\times v_{\\beta} \\times v_{\\gamma} \\times v_{\\delta}) \\right\\}\\]\n\nValues of the Banker joke are illustrated in Table~\\ref{valB}.\n\n\\begin{table}\n\\centering\n\\small\n\\begin{tabular}{cccccc}\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n\\bf Word form & \\(v_{\\alpha}\\) & \\(v_{\\beta}\\) & \\(v_{\\gamma}\\) & \\(v_{\\delta}\\) & \\(v_{W_{Bk}}\\) \\\\\n\\hline\nbe & 1 & 1 & 4 & 0.338 & 1.352 \\\\\nlose & 2 & 1 & 9 & 0.338 & 6.084 \\\\\nuse & 2 & 1 & 2 & 0.338 & 1.352 \\\\\ninterest & 2 & 2 & 10 & 0.502 & \\bf 20.08 \\\\\nbanker & 2 & 1 & 6 & 0.502 & 6.024 \\\\\n\\hline\n\\end{tabular}\n\\end{tabular}\n\\caption{Values of the Banker joke.}\\label{valB}\n\\end{table}\n\nIn the solution for heterographic puns, we built a different model of B-group. Unlike homographic\npuns, here the target word is missing in \\(W_{B}\\) (the reader has to guess the word or phrase,\nhomonymous to the target word). Accordingly, we rely on the completeness of the union of \\(W_{A}\\)\nand \\(W_{B}\\): among the candidates for \\(W_{B}\\) (the second largest groups), such groups are relevant,\nthat form the longest list with \\(W_{A}\\) (duplicates removed). In Ex. 2 (the Church joke), \\(W_{A}={go, gas,\nannual, barbecue, propane}\\), and two groups form the largest union with it: \\(W_{B}={buy,\nproceeds} + {sacred, church}\\). Every word in \\(W_{A}\\) and \\(W_{B}\\) can be the target word. The privilege\npasses to words, used only in one of the groups. Ergo, the first value is:\n\\[v_{\\alpha}(c)=\\left\\{\n \\begin{array}{lr}\n  2 \\quad \\mathrm{if } (c \\in W_{A})\\oplus(c \\in W_{B}) \\\\\n  1 \\quad \\mathrm{otherwise} \\\\\n \\end{array}\n\\right.\n\\]\nFrequencies are of no value here; values of position in the sentence, and part-of-speech\ndistribution remain the same. The function output is:\n\\[z_{1}(W_{B})=\\left\\{c|\\max_{c} (v_{\\alpha} \\times v_{\\gamma} \\times v_{\\delta}) \\right\\}\\]\n\nValues of the Church joke are illustrated in Table~\\ref{valC}.\n\\begin{table}\n\\centering\n\\small\n\\begin{tabular}{ccccc}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n\\bf Word form & \\(v_{\\alpha}\\) & \\(v_{\\gamma}\\) & \\(v_{\\delta}\\) & \\(v_{W_{Ak}}, v_{W_{Bk}}\\) \\\\\n\\hline\npropane & 2 & 18 & 0.502 & \\bf 18.072 \\\\\nannual & 2 & 8 & 0.131 & 2.096 \\\\\ngas & 2 & 5 & 0.502 & 5.02 \\\\\nsacred & 2 & 15 & 0.338 & 10.14 \\\\\nchurch & 2 & 3 & 0.502 & 3.012 \\\\\nbarbecue & 2 & 9 & 0.502 & 9.036 \\\\\ngo & 2 & 12 & 0.338 & 8.112 \\\\\nproceeds & 2 & 11 & 0.502 & 11.044 \\\\\nbuy & 2 & 4 & 0.338 & 2.704 \\\\\n\\hline\n\\end{tabular}\n\\end{tabular}\n\\caption{Values of the Church joke.}\\label{valC}\n\\end{table}\n\n\\section{Subtask 3: Mapping Roget's Thesaurus to Wordnet}\nIn the last phase, we implemented an algorithm which maps Roget's Sections to synsets in Wordnet. \nIn homographic puns, definitions of a word in Wordnet are analyzed similarly to words in a pun, \nwhen searching for semantic fields, the words belong to. For example, words from the definitions \nof the synset {\\em interest} belong to the following Roget's Sections: \nSynset(interest.n.01)=a sense of concern with and curiosity about someone or something: \n(21, 19, 31, 24, 1, 30, 6, 16, 3, 31, 19, 12, 2, 0); Synset(sake.n.01)=a reason for wanting something done: \n15, 24, 18, 7, 19, 11, 2, 31, 24, 30, 12, 2, 0, 26, 24, etc. When A-Section is discovered (for example,\nin the Banker joke, A=30 (Possessive Relations)), the synset with the maximum number of words in its definition,\nwhich belong to A-Section, becomes the A-synset. The B-synset is found likewise for the B-group with the exception \nthat it should not coincide with A-synset. In heterographic puns the B-group is also a marker of the second target word. \nEvery word in the index of Roget's Thesaurus is compared to the known target word using Damerau-Levenshtein distance. \nThe list is sorted in increasing order, and the algorithm begins to check what Roget's Sections every word belongs to, \nuntil it finds the word that belongs to a Section (or the Section, if there is only one) in the B-group. This word becomes \nthe second target word.\n\nNevertheless, as we did not have many trial data, but for the four examples, released before the competition, the first trials \nof the program on a large collection returned many errors, so we changed the algorithm for the B-group as follows.\n\n{\\em Homographic puns, first run.} B-synset is calculated on the basis of sense frequencies \n(the output is the most frequent sense). If it coincides with A-synset, the program returns the second frequent synset.\n\n{\\em Homographic puns, second run.} B-synset is calculated on the basis of Lesk distance, using\nbuilt-in NLTK Lesk function~\\cite{bird2009natural}. If it coincides with A-synset, the program returns another synset on the basis of sense frequencies, \nas in the first run.\n\n{\\em Heterographic puns, first run.} The second target word is calculated, based on Thesaurus and Damerau-Levenshtein distance; \nwords, missing in Thesaurus, are analyzed as their WordNet hypernyms. In both runs for heterographic puns, synsets are\ncalculated, using the Lesk distance.\n\n{\\em Heterographic puns, second run.} The second target word is calculated on the basis of Brown corpus (NLTK~\\cite{bird2009natural}): \nif the word stands in the same context in Brown as it is in the pun, it becomes the target word. The size of the \ncontext window is (0; +3) for verbs, (0;+2) for adjectives; (-2;+2) for nouns, adverbs and other parts of speech \nwithin the sentence, where a word is used.\n\nTable~\\ref{results} illustrates competition results of our system.\n\n\\begin{table}\n\\centering\n\\small\n\\begin{tabular}{ccccc}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n\\bf Task & \\bf Precision & \\bf Recall & \\bf Accuracy & \\bf F1 \\\\\n\\hline\n1, Ho.\\footnote{Homographic.} & 0.8019 & 0.7785 & 0.7044 & 0.7900 \\\\\n1, He.\\footnote{Heterographic.} & 0.7585 & 0.6326 & 0.5938 & 0.6898 \\\\\n\\hline\n\\bf Task & \\bf Coverage & \\bf Precision & \\bf Recall & \\bf F1 \\\\\n\\hline\n2, Ho., run 1 & 1.0000 & 0.3279 & 0.3279 & 0.3279 \\\\\n2, Ho., run 2 & 1.0000 & 0.3167 & 0.3167 & 0.3167 \\\\\n2, He., run 1 & 1.0000 & 0.3029 & 0.3029 & 0.3029 \\\\ \n2, He., run 2 & 1.0000 & 0.3501 & 0.3501 & 0.3501 \\\\\n3, Ho., run 1 & 0.8760 & 0.0484 & 0.0424 & 0.0452 \\\\\n3, Ho., run 2 & 1.0000 & 0.0331 & 0.0331 & 0.0331 \\\\\n3, He., run 1 & 0.9709 & 0.0169 & 0.0164 & 0.0166 \\\\ \n3, He., run 2 & 1.0000 & 0.0118 & 0.0118 & 0.0118 \\\\\n\\hline\n\\end{tabular}\n\\end{tabular}\n\\caption{Competition results.}\\label{results}\n\\end{table}\n\n\\section{Conclusion}\n\nThe system, that we introduced, is based on one general supposition about the semantic structure of puns and combines two\ntypes of algorithms: supervised learning and rule-based. Not surprisingly, the supervised learning algorithm showed \nbetter results in solving an NLP-task, than the rule-based.\nAlso, in this implementation, we tried to combine two very different dictionaries (Roget's Thesaurus and Wordnet). And,\nalthough reliability of Thesaurus in reproducing a universal semantic map can be doubted, it seems to be a quite effective\nsource of data, still, when used in Subtask 1. The attempts to map it to Wordnet seem rather weak, so far, concerning\nthe test results, which also raises a question: if different dictionaries treat meaning of words differently, can there be an\nobjective and\/or universal semantic map, to apply as the foundation for any WSD task? \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intr}\n\nThe decay constant $f_P$ of a pseudoscalar meson $P$ is defined\nby\n\\begin{align}\n  \\langle 0 | A^\\mu | P \\rangle = i p^\\mu f_{P} \\,, \n  \\label{eq:f}\n\\end{align}\nwhere the external state $|P\\rangle$ carries momentum $p^\\mu$, and the\naxial current $A^\\mu = \\bar{\\psi}_{h} \\gamma^\\mu \\gamma_5 \\psi_{l}$.\nHere, the subscript $h$ ($l$) represents heavy (light) flavors in\nthe $B_{(s)}$ and $D_{(s)}$ states.\nWe use the Oktay-Kronfeld (OK) action \\cite{Oktay:2008ex} for valence\nheavy quarks $\\psi_h$ with $h=b,c$, and the HISQ action\n\\cite{Follana:2006rc} for valence light quarks $\\chi$ that are\n  recast into the naive quark field $\\psi_l$.\nThe calculations are done on MILC HISQ ensembles with $N_f=2+1+1$\n\\cite{Bazavov:2012xda}, whose parameters are summarized in Table \\ref{tab:ensembles}. \n\n\\begin{table}[!b]\n  \\begin{center}\n    \\begin{tabular}{ l || l | l | c | l l l }\n      \\hline\\hline\n      ensemble ID & $a$ (fm) & $N_s^3 \\times N_t$ & $M_\\pi$\n      (MeV) & $am_l$ & $am_s$ & $am_c$ \\\\\n      \\hline\n      a12m310 & 0.1207(11) & $24^3\\times 64$ & 305.3(4) & 0.0102 &\n      0.0509 & 0.635 \\\\\n      a12m220 & 0.1184(10) & $32^3\\times 64$ & 216.9(2) & 0.00507 &\n      0.0507 & 0.628 \\\\\n      a12m130 & 0.1191(7) & $48^3\\times 64$ & 131.7(1) & 0.00184 &\n      0.0507 & 0.628 \\\\\n      \\hline\n      a09m310 & 0.0888(8) & $32^3\\times 96$ & 312.7(6) & 0.0074 &\n      0.037 & 0.440 \\\\\n      a09m220 & 0.0872(7) & $48^3\\times 96$ & 220.3(2) & 0.00363 &\n      0.0363 & 0.430 \\\\\n      \\hline\n      a06m310 & 0.0871(6) & $48^3\\times 144$ & 319.3(5) & 0.0048 &\n      0.024 & 0.286 \\\\\n      \\hline\\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Parameters of the MILC HISQ ensembles with $N_f=2+1+1$ \n    \\cite{Bazavov:2012xda} used in our calculations.  The lattice\n    spacing $a$ is set by the Sommer scale $r_1$ and  $N_s$ ($N_t$) is\n    the lattice size in the spacial (temporal) direction.  $M_{\\pi}$\n    is the mass of Goldstone pions and  $am_l$, $am_s$ and $am_c$ are sea\n    quark masses for the light (up and down), strange and charm quarks\n    in lattice units, respectively.\n    \\label{tab:ensembles}}\n\\end{table}\n\n\n\nThe OK action $S_\\text{OK}$ improves on the Fermilab formulation\nof the Wilson clover action \\cite{ ElKhadra:1996mp} by including\n$\\mathcal{O}(\\lambda^2)$ and $\\mathcal{O}(\\lambda^3)$ improvement\nterms in heavy quark effective theory (HQET) power counting.\n\\begin{align}\n  S_{\\text{OK}} & = a^{4} \\sum_{x} \\bar{\\psi} (x) \\left[\n    \\vphantom{\\sum_{j\\neq k} \\left\\{ i \\Sigma_{k} B_{k} \\,,\\Delta_{j}\n      \\right\\} } m_{0} + \\gamma_{4} D_{4} \\right. & \\leftarrow\n    \\mathcal{O}(\\lambda^0) \\nonumber \\\\\n    & \\hphantom{\\;=\\ a^{4}\\sum_{x}\\bar{\\psi}(x)} - \\frac{1}{2} a\n    \\Delta_{4} + \\zeta \\bm{\\gamma \\cdot D} - \\frac{1}{2} r_{s} \\zeta a\n    \\Delta^{(3)} - \\frac{1}{2} c_{B} a \\zeta i \\bm{\\Sigma \\cdot B} &\n    \\leftarrow \\mathcal{O}(\\lambda^1) \\nonumber \\\\\n    & \\hphantom{\\;=\\ a^{4}\\sum_{x}\\bar{\\psi}(x)} - \\frac{1}{2} c_{E} a\n    \\zeta \\bm{\\alpha \\cdot E} & \\leftarrow \\mathcal{O}(\\lambda^2)\n    \\nonumber \\\\\n    &\\hphantom{=\\ a^{4}\\sum_{x}\\bar{\\psi}(x)}\n    \\begin{rcases}\n      & + c_{1} a^{2} \\sum_{k} \\gamma_{k} D_{k} \\Delta_{k} + c_{2}\n      a^{2} \\left\\{ \\bm{\\gamma \\cdot D},\\Delta^{(3)} \\right\\} + c_{3}\n      a^{2} \\left\\{ \\bm{\\gamma \\cdot D} \\, , i \\bm{\\Sigma \\cdot B}\n      \\right\\} \\\\\n      & \\left. + c_{EE} a^{2} \\left\\{ \\gamma_{4} D_{4} \\, , \\bm{\\alpha\n        \\cdot E} \\right\\} + c_{4} a^{3} \\sum_{k} \\Delta_{k}^{2} +\n      c_{5} a^{3} \\sum_{k} \\sum_{j \\neq k} \\left\\{ i \\Sigma_{k} B_{k}\n      \\, , \\Delta_{j} \\right\\} \\right] \\psi(x)\n    \\end{rcases} \\,.\n    & \\leftarrow \\mathcal{O}(\\lambda^3)\n\\label{eq:ok-ac-1}\n\\end{align}\nThe definition of the operators in Eq.~\\eqref{eq:ok-ac-1} can be found\nin Ref.~\\cite{Oktay:2008ex}.\nThe bare quark mass $m_0$ is related to the hopping parameter\n$\\kappa$ as follows,\n\\begin{align}\n  a m_{0} = \\frac{1}{2} \\left( \\frac{1}{\\kappa} -\n  \\frac{1}{\\kappa_{\\text{crit}}} \\right)\\,.\n  \\label{eq:bare-m-1}\n\\end{align}\nThe non-perturbatively tuned hopping parameters for bottom and charm\nquarks, $\\kappa_b$, $\\kappa_c$, and the critical hopping parameter\n$\\kappa_\\text{crit}$~\\cite{Bailey:2017xjk} for each measurement are\nsummarized in Table \\ref{tab:parameters}.\n\n\nIn order to achieve a better overlap with the wave functions of the\n$B_{(s)}$ and $D_{(s)}$ meson states, we apply the covariant Gaussian\nsmearing (CGS), $\\left\\{1 + \\sigma^2\\nabla^2\/(4 N_{\\text{GS}})\n\\right\\}^{N_{\\text{GS}}}$ to the point source and sink as in\nRef.~\\cite{Yoon:2016dij}.\nThe CGS parameters $\\{ \\sigma, N_\\text{GS} \\}$ for each measurement\nare given in Table \\ref{tab:parameters}.\nHere, we apply the CGS only to the heavy quark fields of the\npseudoscalar interpolating operators.\n\\section{Correlator and current improvement}\n\\label{sec:curim}\nThe meson-meson (MM) and meson-current (MC) 2-point correlators are\ndefined as follows \\cite{Bazavov:2011aa},\n\\begin{align}\n  C_{\\text{MM}}(t) &= \\sum_{\\bf{x}} \\left\\langle\n  \\mathcal{O}^{\\dagger}_P(t,{\\bf x}) \\mathcal{O}_P(0)\n  \\right\\rangle \n  = \\sum_{\\alpha = 1}^{4} \\sum_{\\bf{x}} \\left\\langle\n  \\mathcal{O}^{\\dagger}_\\alpha(t,{\\bf x}) \\mathcal{O}_\\alpha(0)\n  \\right\\rangle \\,, \\\\\n  C_{\\text{MC}}(t) &= \\sum_{\\bf{x}} \\left\\langle A^{4\\dagger}(t,{\\bf x})\n  \\mathcal{O}_P(0) \\right\\rangle\n  = \\sum_{\\alpha = 1}^{4}\n  \\sum_{\\bf{x}} \\left\\langle A^{4\\dagger}_\\alpha(t,{\\bf x})\n  \\mathcal{O}_\\alpha(0) \\right\\rangle\\,,\n  \\label{eq:corr-2pt}\n\\end{align}\nwhere the pseudoscalar heavy-light meson interpolating operator\n$\\mathcal{O}_\\alpha(t,{\\bf x})$ and the axial current operator\n$A^4_{\\alpha}(t,{\\bf x})$ are\n\\begin{align}\n  \\mathcal{O}_\\alpha(t,{\\bf x}) &= \\left[ \\bar{\\psi}(t,{\\bf x})\n    \\gamma_5 \\Omega(t,{\\bf x}) \\right]_\\alpha \\chi(t,{\\bf x}) \\,\\\\\n  A^4_{\\alpha}(t,{\\bf x}) &= \\left[ \\bar{\\Psi}(t,{\\bf x}) \\gamma^4\n    \\gamma_{5} \\Omega(t,{\\bf x}) \\right]_{\\alpha} \\chi(t,{\\bf x}) \\,.\n\\end{align}\nHere $\\psi$ is the OK heavy quark field, $\\chi$ is the HISQ \nlight quark field, and \n\\begin{align}\n\\Omega(t,{\\bf x}) \\equiv \\gamma_{1}^{\\;x_1} \\gamma_{2}^{\\;x_2}\n\\gamma_{3}^{\\;x_3} \\gamma_{4}^{\\;t} \\,,\n\\label{eq:omega-trans-1}\n\\end{align}\nand the subscript $\\alpha$ represents the taste degree of the\nstaggered light quarks.\nThe rotated heavy quark field $\\Psi$ is introduced to improve the\naxial current $A^4_{\\alpha}$ up to $O(\\lambda^3)$, the same\nlevel as the OK action.\n\\begin{align}\n  \\Psi(t,{\\bf x}) & = \\Big( 1  & \\leftarrow\n  O(\\lambda^{0}) \\nonumber \\\\\n  & \\hphantom{=(\\ \\; } + d_{1} a \\bm{\\gamma \\cdot D} & \\leftarrow\n  O(\\lambda^{1}) \\nonumber \\\\\n  & \\hphantom{=(\\ \\; } + d_{2} a^{2} \\Delta^{\\left(3\\right)} + d_{B}\n  a^{2} i \\bm{\\Sigma \\cdot B} - d_{E} a^{2} \\bm{\\alpha \\cdot\n    E} & \\leftarrow O(\\lambda^{2}) \\nonumber \\\\\n  &\\hphantom{=(\\ \\, }\n  \\begin{rcases}\n    & + d_{rE} a^{3} \\left\\{ \\bm{\\gamma \\cdot D}, \\bm{\\alpha \\cdot E}\n    \\right\\} - d_{3} a^{3} \\sum_{i} \\gamma_{i} D_{i} \\Delta_{i} -\n    d_{4} a^{3} \\left\\{ \\bm{\\gamma \\cdot D} , \\Delta^{\\left(3\\right)}\n    \\right\\} \\\\\n    & - d_{5} a^{3} \\left\\{ \\bm{\\gamma \\cdot D} , i \\bm{\\Sigma \\cdot\n      B} \\right\\} + d_{EE} a^{3} \\left\\{ \\gamma_{4} D_{4} , \\bm{\\alpha\n      \\cdot E} \\right\\} - d_{6} a^{3} \\left[ \\gamma_{4} D_{4} ,\n      \\Delta^{\\left(3\\right)} \\right] \\\\\n   \n   \n    & - d_{7} a^{3} \\left[ \\gamma_{4} D_{4} , i \\bm{\\Sigma\n        \\cdot B} \\right] \\Big) \\psi (t,{\\bf x} ) \\,,\n  \\end{rcases}\n  & \\leftarrow O(\\lambda^3)\n  \\label{eq:cur-imp-1}\n\\end{align}\nwhere the improvement coefficients $d_i$ are given in Ref.~\\cite{\n  Bailey:2020uon}.\n\n\n\\begin{table}[!t]\n  \\begin{center}\n    \\begin{tabular}{ l || r | l l l | l | r }\n      \\hline\\hline\n      ensemble ID & $m_x \/ m_s$ & $\\kappa_{\\text{crit}}$ & $\\kappa_c$\n      & $\\kappa_b$ & $\\{ \\sigma \\,,\\; N_{\\text{GS}} \\}$ &\n      $N_{\\text{cfg}} \\times N_{\\text{src}}$ \\\\\n      \\hline\n      a12m310 & 1\/5, 1 & 0.051211 & 0.048524 & 0.04102 &\n      $\\{ 1.5\\,,\\;5 \\}$ & $1053 \\times 3 $ \\\\\n      a12m220 & 1\/10, 1 & 0.051218 & 0.048613 & 0.04070 &\n      $\\{ 1.5\\,,\\;5 \\}$ & $1000 \\times 3 $ \\\\\n      a12m130 & 1\/27, 1 & 0.05119 & 0.048501 & 0.041343 &\n      $\\{ 1.5\\,,\\;5 \\}$ & $499 \\times 3 $ \\\\\n      \\hline\n      a09m310 & 1\/5, 1 & 0.05075 & 0.04894 & 0.0429 &\n      $\\{ 2.0\\,,\\;10 \\}$ & $996 \\times 3$ \\\\\n      a09m220 & 1\/10, 1 & 0.05077 & 0.04902 & 0.0431 &\n      $\\{ 2.0\\,,\\;10 \\}$ & $1001 \\times 3$ \\\\\n      \\hline\n      a06m310 & 1\/5, 1 & 0.050357 & 0.04924 & 0.0452 &\n      $\\{ 3.0\\,,\\;22 \\}$ & $1017 \\times 3$ \\\\\n      \\hline\\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The 2${}^{\\rm nd}$ column gives the valence light quark masses\n    $m_x$ and the following columns are the hopping parameters, CGS\n    parameters and the number of measurements. $N_{\\text{cfg}}$ represents\n    the number of gauge configurations analyzed and $N_{\\text{src}}$ is the number\n    of sources used for measurement on each gauge configuration.\n    \\label{tab:parameters}}\n\\end{table}\n\n\n\n\\section{Correlator fit}\n\\label{sec:corfit}\nWe fit the 2-point correlation functions $C_\\text{MM}(t)$ and\n$C_\\text{MC}(t)$ with three even time-parity and two odd time-parity\nstates and label it the 3+2-state fit.\nThe time parity is determined with respect to the shift operator in\nthe Euclidean time direction.\nThe fitting function is\n\\begin{align}\n  C_{\\text{Y}}(t) & = g_{\\text{Y}}(t) \\pm\n  g_{\\text{Y}}(T-t)\\,, \\qquad (\\text{$+$ for MM, $-$ for MC)}\n  \\nonumber \\\\\n  g_{\\text{Y}}(t) & = A_{0}^{\\text{Y}} e^{-M_0t}\n  \\left[ 1 + R_{2}^{\\text{Y}}\n    e^{-\\Delta M_{2}t} + R_{4}^{\\text{Y}}\n    e^{-(\\Delta M_{2} + \\Delta M_{4})t}\n  \\vphantom{e^{-\\Delta M_1^p t}} + \\cdots \\right. \\nonumber \\\\\n & \\hphantom{ = A_{0}^{\\text{Y}}e^{-M_0t} [ } \\left. \\ -\n       \\left(-1\\right)^t R_{1}^{\\text{Y}} e^{-\\Delta M_1 t} -\n       \\left(-1\\right)^t R_{3}^{\\text{Y}}\n       e^{-(\\Delta M_1 + \\Delta M_3) t} + \\cdots\n       \\right]\n  \\label{eq:3+2-fit}\n\\end{align}\nwhere Y=MC or MM, $\\Delta M_i \\equiv M_i - M_{i-2}$, $M_{-1}\\equiv\nM_0$, and\n\\begin{align}\n  A^\\text{MM}_i &\\equiv \\frac{1}{2M_i}\n  \\langle 0 | \\mathcal{O}_P | P_i \\rangle\n\\langle P_i | \\mathcal{O}_P | 0 \\rangle\\,,\n\\qquad\nA^\\text{MC}_i \\equiv \\frac{1}{2M_i}\n\\langle 0 | A^4 | P_i \\rangle\n  \\langle P_i | \\mathcal{O}_P | 0 \\rangle \\,,\n\\qquad\nR^\\text{Y}_i \\equiv \\frac{ A^\\text{Y}_i }{ A^\\text{Y}_0 }\n\\end{align}\nHere, $P_i$ represents the $i$-th excited meson state and $P_0$ the ground state. For a brevity, the subscript ``0'' for\nthe ground state is dropped from now on.\nWe take the following steps to analyze the 2-point correlation\nfunctions.\n\\begin{enumerate}\n\\item We fit the 2-point correlator, $C_\\text{MM}(t)$, data using the\n  3+2-state fit given in Eq.~\\eqref{eq:3+2-fit} to\n  extract the ground state pseudoscalar meson mass $M(\\equiv M_0)$ and\n  amplitude $A^\\text{MM}(\\equiv A^\\text{MM}_0)$ and control the \n  excited states. We impose empirical Bayesian priors \\cite{\n    Yoon:2016jzj} on the excited state mass gaps $\\Delta M_i$ and\n  amplitude ratios $R^\\text{MM}_i$ to stabilize the fit. (See\n  Fig.~\\ref{fig:prior-plots-1}).\n\\item We feed the results for $M_0$, and $\\Delta M_i$ obtained\n  in the previous step as inputs into the fit for $C_\\text{MC}(t)$\n  to extract $A^\\text{MC} (\\equiv A^\\text{MC}_0)$ and the ratios\n  $R^\\text{MC}_i$.\n  We use the same fit range and fit functional form as taken\n  for $C_\\text{MM}(t)$\n\\end{enumerate}\n\n\n\\begin{figure}[t] \n  \\subfigure[$B$ meson]{\n    \\label{fig:b-prior-1}\n    \\includegraphics[width=0.227\\textwidth]\n                    {prior_PION_d_cvg_d_cvg_m00184_k041343_p000.pdf}\n  }\n \n  \\subfigure[$B_s$ meson]{\n    \\label{fig:b-prior-2}\n    \\includegraphics[width=0.227\\textwidth]\n                    {prior_PION_d_cvg_d_cvg_m0507_k041343_p000.pdf}\n  }\n \n  \\subfigure[$D$ meson]{\n    \\label{fig:d-prior-1}\n    \\includegraphics[width=0.227\\textwidth]\n                    {prior_PION_d_cvg_d_cvg_m00184_k048501_p000.pdf}\n  }\n \n  \\subfigure[$D_s$ meson]{\n    \\label{fig:d-prior-2}\n    \\includegraphics[width=0.227\\textwidth]\n                    {prior_PION_d_cvg_d_cvg_m0507_k048501_p000.pdf}\n  }\n  \\caption{ Fit results and Bayesian priors for $\\Delta M_i$ from\n    $a12m130$ ensembles.}\n  \\label{fig:prior-plots-1}\n\\end{figure}\n\\begin{figure}[t] \n  \\subfigure{\n    \\label{fig:meff-mm-1}\n    \\includegraphics[width=0.475\\textwidth]\n                    {meff_PION_d_cvg_d_cvg_m00184_k048501_p000.pdf}\n  }\n  \\hfill\n  \\subfigure{\n    \\label{fig:meff-mc-1}\n    \\includegraphics[width=0.475\\textwidth]\n                    {meff_P5_A4_d_cvg_m00184_k048501_p000_l3.pdf}\n  }\n  \\caption{ Effective mass plots for the $C_\\text{MM}(t)$ and\n    $C_\\text{MC}(t)$ correlators of the $D$ meson on the $a12m130$\n      ensemble with $m_l=m_s\/27$.\n    The orange (purple) curves connect 3+2-state fit results on the\n    odd (even) time slices.\n    The horizontal red line shows the ground state mass $M_0$\n      within the fit range $[t_\\text{min},t_\\text{max}]=[5,20]$.\n    The axial current operators are improved up to the $\\lambda^3$\n    order.\n    \\label{fig:meff-plots-1}}\n\\end{figure}\nAn example of the effective mass plot with \n\\begin{align}\n  m_{\\text{eff}}^{Y}(t) &\\equiv \\frac{1}{2}\n  \\log\\left[\\frac{C_Y(t)}{C_Y(t+2)}\\right] \n\\end{align}\nfor  the 2-point correlators $C_\\text{MM}(t)$ and\n$C_\\text{MC}(t)$ is shown in Fig.~\\ref{fig:meff-plots-1} along with the fits to them. \n\n\n\\section{Results}\n\\label{sec:results}\nThe decay constant $f_P$ defined in Eq.~\\eqref{eq:f} can be expressed\nin terms of the ground state amplitudes $A^\\text{MM}$ and\n$A^\\text{MC}$ as follows,\n\\begin{align}\n   f_{P} = Z_{A^4}^{hl} \\sqrt{\\frac{2}{M_P}} \\,\n  \\frac{A^\\text{MC}}{\\sqrt{A^\\text{MM}}} \\,,\n\\end{align}\nwhere we take the meson mass $M_P=M_0$ of the ground state obtained\nfrom the fits to $C_\\text{MM}$.\nThe tree-level renormalization factor is given as\n$Z_{A^4}^{hl,\\text{tree}} = e^{m_{1}^h\/2}$ where $m_1^{h} = \\log(1 +\nm_0^{h})$ is the rest mass and $m_0^h$ is the bare mass for the heavy\nquark \\cite{ElKhadra:1996mp}.\nThe perturbative and non-perturbative determination of $Z_{A^4}^{hl}$\nis in progress.\nIn this work, we present the flavor $SU(3)$ breaking ratio of decay\nconstants:\n\\begin{align}\n  f_{X_s} \/ f_{X} &= \\frac{Z_{A^4}^{hs}}{Z_{A^4}^{hl}} \\sqrt{\n    \\frac{M_{X}}{M_{X_s}} }\n  \\sqrt{\\frac{A^\\text{MM}}{A^\\text{MM}_s}}\n  \\frac{A^\\text{MC}_s}{A^\\text{MC}}\n  \\cong \\sqrt{\n    \\frac{M_{X}}{M_{X_s}} }\n  \\sqrt{\\frac{A^\\text{MM}}{A^\\text{MM}_s}}\n  \\frac{A^\\text{MC}_s}{A^\\text{MC}}\\,,\n\\end{align}\nfor $X_{(s)}= B_{(s)}$ and $D_{(s)}$ mesons.\nHere, $A^\\text{MM}_s$ and $A^\\text{MC}_s$ are the ground state\namplitudes for the heavy-strange mesons.\nIn this ratio, we assume that the light quark mass $m_l$ dependence\nof $Z_{A^4}^{hl}$ is negligible \\cite{Bazavov:2011aa},  so the\nratio $Z_{A^4}^{hs} \/ Z_{A^4}^{hl} \\cong 1$.\n\n\n\\begin{figure}[t!] \n  \\subfigure[$f_{B_s} \/ f_B$]{ \\label{fig:dec-b-2}\n    \\includegraphics[width=0.475\\textwidth]{ratio_compare_3+2_k041343.pdf} }\n  \\hfill\n  \\subfigure[$f_{D_s} \/ f_D$]{\n    \\label{fig:dec-d-2}\n    \\includegraphics[width=0.475\\textwidth]{ratio_compare_3+2_k048501.pdf} }\n  \\caption{ The ratios of decay constants, $f_{B_s} \/ f_B$ and\n    $f_{D_s} \/ f_D$, on the a12m130 ensemble as a function of current\n    improvement order in HQET power counting. }\n  \\label{fig:results-2}\n\\end{figure}\nFig.~\\ref{fig:results-2} shows that in the ratios $f_{B_s} \/ f_{B}$ and $f_{D_s} \/ f_{D}$, \nthe effect of the current improvement applied to the heavy quark field (as given in\nEq.~\\eqref{eq:cur-imp-1}) cancels up to $O(\\lambda^3)$. \n\nIn Fig.~\\ref{fig:results-1}, we present preliminary results on\n$f_{B_s} \/ f_{B}$ and $f_{D_s} \/ f_{D}$ calculated on 6 different HISQ\nensembles, and compare them with the continuum limit value given in the FLAG\n2019 review~\\cite{Aoki:2019cca}.\nThe statistical errors in $f_{B_s} \/ f_{B}$\n(Fig.~\\ref{fig:results-1}\\;\\subref{fig:dec-b-1}) are much larger than in\n$f_{D_s}\/f_D$ (Fig.~\\ref{fig:results-1}\\;\\subref{fig:dec-d-1}).\nAs a result, discerning a chiral or discretization effect in $f_{B_s}\/f_B$ is not clear, \nother than to note that the result from the physical ensemble $a12m130$ is\nconsistent with the FLAG 2019 value.\nThe results for $f_{D_s}\/f_{D}$ show no significant discretization\neffect on the three lattices with $M_\\pi \\approx 310\\,\\mathop{\\rm MeV}\\nolimits$ and the\ntwo with $M_\\pi \\approx 220\\,\\mathop{\\rm MeV}\\nolimits$. On the other hand, there is a\nshift upwards towards the FLAG result as $M_\\pi$ is lowered towards\nthe physical value.\nPresuming that the OK action has significantly eliminated the heavy\nquark discretization error even on the coarsest lattice spacing\n$a\\approx 0.12\\,\\,\\mathrm{fm}$~\\cite{Bailey:2017nzm}, the leading effect to\nquantify is the pion mass dependence. For the $a \\approx 0.12\\,\\,\\mathrm{fm}$\ndata, the current trend is anchored by the physical ensemble with a\nvalue close to the FLAG 2019 result.\nIn near future, we plan to add measurements on more ensembles to check for discretization\neffects and on more physical pion mass ensembles to improve \nthe chiral-continuum extrapolation.\n\\begin{figure}[h!] \n  \\subfigure[$f_{B_s}\/f_B$]{ \\label{fig:dec-b-1}\n    \\includegraphics[width=0.475\\textwidth]{decay_b.pdf} }\n  \\hfill\n  \\subfigure[$f_{D_s}\/f_D$]{\n    \\label{fig:dec-d-1}\n    \\includegraphics[width=0.475\\textwidth]{decay_d.pdf} }\n  \\caption{The ratios of decay constants, $f_{B_s} \/ f_B$ and $f_{D_s}\n    \/ f_D$ on six ensembles.\n    The errors are purely statistical.\n    The FLAG 2019 \\cite{ Aoki:2019cca} results are given for the\n    physical value.}\n  \\label{fig:results-1}\n\\end{figure}\n\n\n\\acknowledgments\n\\label{sec:ackn}\nWe thank the MILC collaboration for sharing the HISQ ensembles\nwith us.\nComputations for this work were carried out in part on (i) facilities\nof the USQCD collaboration, which are funded by the Office of Science\nof the U.S. Department of Energy, (ii) the Nurion supercomputer at\nKISTI and (iii) the DAVID GPU clusters at Seoul National University.\nThe research of W. Lee is supported by the Mid-Career Research Program\n(Grant No.~NRF-2019R1A2C2085685) of the NRF grant funded by the Korean\ngovernment (MOE).\nThis work was supported by Seoul National University Research Grant in\n2019.\nW.~Lee would like to acknowledge the support from the KISTI\nsupercomputing center through the strategic support program for the\nsupercomputing application research (No.~KSC-2017-G2-0009).\nT.~Bhattacharya and R.~Gupta were partly supported by the\nU.S. Department of Energy, Office of Science, Office of High Energy\nPhysics under Contract No. DE-AC52-06NA25396.\nS.~Park, T.~Bhattacharya, R.~Gupta and Y.-C.~Jang were partly\nsupported by the LANL LDRD program.\nY.-C.~Jang is partly supported by U.S.~Department of Energy under\nContract No.~DE-SC0012704.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nBipartite graphs are widely used to represent networks with two different groups of entities such as user-item networks \\cite{wang2006unifying}, author-paper networks \\cite{konect:DBLP}, and member-activity networks \\cite{brunson2015triadic}.\nIn bipartite graphs, cohesive subgraph mining has numerous applications including fraudsters detection \\cite{allahbakhsh2013collusion,beutel2013copycatch,liu2020efficient}, group recommendation \\cite{ding2017efficient,ntoutsi2012fast} and discovering inter-corporate relations \\cite{ornstein1982interlocking,palmer2002interlocking}.  \n\n($\\alpha$,$\\beta$)-core and bitruss are two representative cohesive subgraph models in bipartite graphs extended from the unipartite \\textit{k}-core \\cite{seidman1983network} and \\textit{k}-truss \\cite{cohen2008trusses} models.\n$(\\alpha,\\beta)$-core\\xspace is the maximal subgraph of a bipartite graph \\textit{$G$} such that the vertices on upper or lower layer have at least $\\alpha$ or $\\beta$ neighbors respectively. \n$(\\alpha,\\beta)$-core\\xspace models vertex engagement as degrees and treats each edge equally, but ties (edges) in real networks have different strengths.\n\\textit{k}-bitruss\\xspace is the maximal subgraph where each edge is contained in at least $k$ butterflies (i.e. $2$x$2$-biclique), which can model the tie strength\n\\cite{sariyuce2018peeling,zou2016bitruss}.\n\n\\begin{figure}[htb]\n\\centering  \n\\includegraphics[width=0.50\\textwidth]{fig\/motivation.pdf}\n\\caption{Motivation example}\n\\label{fig:moltivation}\n\\end{figure}\n\nIn the author-paper network as shown in Figure \\ref{fig:moltivation}, the graph is the $(\\alpha,\\beta)$-core ($\\alpha$=$2$, $\\beta$=$2$) and the light blue region is the \\textit{k}-bitruss\\xspace ($k$=$2$). \nWithout considering tie strength, $(\\alpha,\\beta)$-core blindly includes research groups of different levels of cohesiveness. We can see that $v_0$ and $v_1$ are not as closely connected as the rest authors. \nThe \\textit{k}-bitruss\\xspace model can exclude the relatively sparse subgraph containing $v_0$ and $v_1$, but it also deletes edges $(u_3,v_4)$ and $(u_4,v_3)$ when their incident vertices are present. This exposes the drawbacks of the \\textit{k}-bitruss\\xspace model: ($1$) As \\textit{k}-bitruss\\xspace only keeps strong ties, the weak ties between important vertices are missed.\nIn Fig \\ref{fig:moltivation}, it fails to recognize the contributions of authors $v_3,v_4$ in papers $u_3,u_4$.\n($2$) After removing weak ties, the tie strengths are modeled inaccurately. Edges $(u_3,v_3)$ and $(u_4,v_4)$ have more supporting butterflies ($u_3,u_4,v_3,v_4$ form a butterfly) than $(u_1,v_2)$, but their tie strengths are modeled as equal. \n\nIn this paper, we study the efficient and scalable computation of $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace, \nwhich is the first cohesive subgraph model on bipartite graphs to consider both tie strength and vertex engagement. \nGiven a bipartite graph $G$, we model the tie strength of each edge as the number of butterflies containing it. With a strength level $\\tau$, we consider the edges with tie strength no less than $\\tau$ to be \\textit{strong ties}. \nThe \\textit{engagement} of a vertex is modeled as the number of strong ties it is incident to. \nGiven engagement constraints $\\alpha,\\beta$ and strength level $\\tau$, $(\\alpha,\\beta)_{\\tau}$-core\\xspace is the maximal subgraph of $G$ such that each upper or lower vertex in the subgraph has at least $\\alpha$ or $\\beta$ strong ties. The $(\\alpha,\\beta)_{\\tau}$-core\\xspace model is highly flexible and is able to capture unique structures.\nFor instance, in Figure \\ref{fig:moltivation}, the subgraph induced by vertices $\\{u_1, u_2, u_3, u_4, u_5, v_2, v_3, v_4, v_5, v_6\\}$ is the $(2,2)_2$-core which cannot be found by $(\\alpha,\\beta)$-core\\xspace or \\textit{k}-bitruss\\xspace for any $\\alpha,\\beta$ or $k$.\nAlso, as shown in Figure \\ref{fig:moltivation}, $(\\alpha,\\beta)$-core\\xspace can preserve the weak ties if the incident vertices are present (e.g., the red edges are preserved due to $u_3,u_4,v_3$ and $v_4$), which better resembles reality.\nThe flexibility of the $(\\alpha,\\beta)_{\\tau}$-core\\xspace model is also evaluated in another experiment conducted on dataset \\texttt{DBpedia-producer}. \nFigure \\ref{fig:profile} shows the subgraphs of different densities found by $(\\alpha,\\beta)_{\\tau}$-core\\xspace and $(\\alpha,\\beta)$-core\\xspace, where density is the ratio between the number of existing edges and the number of all possible edges \\cite{sariyuce2018peeling}. \n$165$ subgraphs with a density greater than $0.2$ are found by $(\\alpha,\\beta)_{\\tau}$-core\\xspace while only $9$ such subgraphs are found by $(\\alpha,\\beta)$-core\\xspace.\n\n\\begin{figure}[tbh]\n\\centering  \n\\subfigure[subgraphs found by $(\\alpha,\\beta)_{\\tau}$-core\\xspace]{\n\\label{fig.abt.profile}\n\\includegraphics[width=0.38\\textwidth]{pr_profile_.pdf}}\n\\subfigure[subgraphs found by $(\\alpha,\\beta)$-core\\xspace]{\n\\label{fig.ab.profile}\n\\includegraphics[width=0.38\\textwidth]{pr_ab_profile.pdf}}\n\\caption{Dense subgraphs in \\texttt{DBpedia-producer}.}\n\\label{fig:profile}\n\\end{figure}\n\n\\noindent\n{\\bf Applications.} The $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace model has many applications. We list some of them below.\n\n\\noindent\n$\\bullet$ \\textit{Identify nested communities.}\nOn Internet forums like Reddit, Quora, and StackOverflow, users hold conversations on topics that interest them. The users and the topics form a bipartite network. \nIn these networks, communities naturally exist and are nested.\nFor instance, Reddit displays a list of top communities like ``News\", ``Gaming\" and ``Sports\" on the front page. \n``Sports\" community contains many sub-communities including ``Cricket\", ``Bicycling\" and ``Golf\". \nThe edges in sub-communities have higher tie strength because users and topics within them are more closely connected. \nBy increasing strength level $\\tau$, $(\\alpha,\\beta)_{\\tau}$-core\\xspace captures the subgraphs forming a hierarchy, which can model nested communities on bipartite networks. \n\n\\noindent\n$\\bullet$ \\textit{Group similar users and items.}\nIn online shopping platforms like Amazon, eBay and Alibaba, users and items form a bipartite graph, where each edge indicates a purchasing record. \nSuch a network consists of many closely connected communities, where some items are repeatedly bought by the same group of users (i.e., the target market). \nExamples of such communities include gym-goers and gym attires, students and stationery, diabetic patients and no-sugar foods, etc. \nWithin one community, items are considered more similar and users tend to be alike due to their common shopping habits. \nAs the edges between these users and items have high tie strength (butterfly support), we can use $(\\alpha,\\beta)_{\\tau}$-core\\xspace to find these communities and group similar users or items together.\n\n\\noindent\n\\textbf{Challenges.}\nTo obtain the $(\\alpha,\\beta)_{\\tau}$-core\\xspace from the input graph, we can first compute the support of edges and the engagement of vertices and then iteratively delete the vertices not meeting the engagement constraints. \nWhen $\\alpha$, $\\beta$, $\\tau$ are large, $(\\alpha,\\beta)_{\\tau}$-core\\xspace is small and computing $(\\alpha,\\beta)_{\\tau}$-core\\xspace from the input graph is time-consuming. \nThus, the online computation method cannot support a large number of $(\\alpha,\\beta)_{\\tau}$-core\\xspace queries. \n\nIn this paper, we resort to index-based approaches. \nA straightforward solution is to compute all possible $(\\alpha,\\beta)_{\\tau}$-cores\\xspace and build a total index $I_{\\alpha,\\beta,\\tau}$ based on them. Instead of computing all $(\\alpha,\\beta)_{\\tau}$-cores\\xspace from the input graph, we take advantage of the nested property of the $(\\alpha,\\beta)_{\\tau}$-core\\xspace, which means that if $\\alpha \\geq \\alpha^*$, $\\beta \\geq \\beta^*$ and $\\tau \\geq \\tau^*$, $(\\alpha,\\beta)_{\\tau}$-core\\xspace is a subgraph of $(\\alpha^*,\\beta^*)_{\\tau^*}$-core. \nSpecifically, for all possible $\\alpha$ and $\\beta$, we first find $(\\alpha,\\beta)_{1}$-core and then compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace while gradually increasing strength level $\\tau$. \nIn this manner, we can compute all $(\\alpha,\\beta)_{\\tau}$-cores\\xspace and construct the index $I_{\\alpha,\\beta,\\tau}$. \nAlthough $I_{\\alpha,\\beta,\\tau}$ supports optimal retrieval of the vertex set of any $(\\alpha,\\beta)_{\\tau}$-core\\xspace, it still suffers from long construction time on large graphs. \nTo devise more practical index-based approaches, we face the following challenges. \n\\begin{enumerate}\n    \\item\n    When building index $I_{\\alpha,\\beta,\\tau}$, it is time-consuming to enumerate all butterflies containing the deleted edges.\n    Also, the $I_{\\alpha,\\beta,\\tau}$ index construction algorithm is prone to visit the same $(\\alpha,\\beta)_{\\tau}$-core\\xspace subgraph repeatedly as it can correspond to different combinations of $\\alpha, \\beta$, and $\\tau$.\n    It is a challenge to speed up butterfly enumeration and avoid repeatedly visiting the same subgraphs during the construction of the total index $I_{\\alpha,\\beta,\\tau}$.\n    \\item \n    Due to the flexibility of the $(\\alpha,\\beta)_{\\tau}$-core\\xspace model, there are a large number of $(\\alpha,\\beta)_{\\tau}$-cores\\xspace corresponding to different combinations of $\\alpha$, $\\beta$, and $\\tau$.\n    The time cost of indexing all $(\\alpha,\\beta)_{\\tau}$-cores\\xspace becomes not affordable on large graphs. \n    It is also a challenge to strike a balance between building space-efficient indexes and supporting efficient and scalable query processing.\n\\end{enumerate}\n\n\\noindent\n{\\bf Our approaches.}\nTo address the first challenge, we extend the butterfly enumeration techniques in \\cite{wang2020efficient} and propose novel computation sharing optimizations to speed up the index construction process of $I_{\\alpha,\\beta,\\tau}$. \nSpecifically, we build a \\texttt{Bloom}-\\texttt{Edge}-\\texttt{Index} (hereafter denoted by \\texttt{BE-Index}) proposed in \\cite{wang2020efficient} to quickly fetch the butterflies containing an edge.\nThe \\texttt{BE-Index} captures the relationships between edges and $(2\\times k)$-bicliques (also called \\textit{blooms}).\nWhen an edge is deleted, we can quickly locate the blooms containing this edge in the \\texttt{BE-Index} and update the support of affected edges in these blooms accordingly. \nIn addition, computation-sharing optimization is based on the fact that the same $(\\alpha,\\beta)_{\\tau}$-core\\xspace subgraph corresponds to various parameter combinations. If we realize the vertices in a subgraph have already been recorded, we can choose to skip the current parameter combination. \n\nTo address the second challenge, we introduce space-efficient \\gratings including $I_{\\alpha,\\beta}$, $I_{\\beta,\\tau}$, and $I_{\\alpha,\\tau}$, and train a feed-forward neural network to predict the most promising index to handle an $(\\alpha,\\beta)_{\\tau}$-core\\xspace query.\nInstead of indexing all $(\\alpha,\\beta)_{\\tau}$-cores\\xspace, the \\gratings $I_{\\alpha,\\beta}$, $I_{\\beta,\\tau}$, and $I_{\\alpha,\\tau}$ store the vertex sets of all $(\\alpha,\\beta)$-core\\xspace, $(1,\\beta)_{\\tau}$-core\\xspace, and $(\\alpha,1)_{\\tau}$-core\\xspace respectively. These \\gratings are much smaller in size and require significantly less build time,\neach of which can be used to handle $(\\alpha,\\beta)_{\\tau}$-core\\xspace queries. \nFor example, to compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace using $I_{\\beta,\\tau}$, we fetch the vertices in $(1,\\beta)_{\\tau}$-core\\xspace and recover the edges of $(1,\\beta)_{\\tau}$-core\\xspace. Then, we iteratively remove the vertices not having enough engagement from $(1,\\beta)_{\\tau}$-core\\xspace until we find $(\\alpha,\\beta)_{\\tau}$-core\\xspace. \nHowever, the query processing performance based on each \\grating is highly sensitive to parameters $\\alpha$, $\\beta$, and $\\tau$. \nThis is because the \\gratings only store the vertices in $(\\alpha,\\beta)$-core\\xspace, $(1,\\beta)_{\\tau}$-core\\xspace, and $(\\alpha,1)_{\\tau}$-core\\xspace and the size difference between $(\\alpha,\\beta)_{\\tau}$-core\\xspace and each of these subgraphs is uncertain. \nWe also observe that there are no simple rules to partition the parameter space so that queries from each partition can be efficiently handled by one type of index. \nThis motivates us to resort to machine learning techniques and train a feed-forward neural network as the classifier to predict the optimal choice of the index for each incoming query of $(\\alpha,\\beta)_{\\tau}$-core\\xspace. \nSince we aim to minimize the query time instead of accuracy, we propose a scoring function, \\textit{time-sensitive-error}, to tune the hyper-parameters of the classifier. \nThe experiment results show that the resulting hybrid computation algorithm significantly outperforms the query processing algorithms based on $I_{\\alpha,\\beta}$,$I_{\\beta,\\tau}$, and $I_{\\alpha,\\tau}$, and it is less sensitive to varying parameters. \n\n\\noindent\n{\\bf Contribution.}\nOur major contributions are summarized here: \\\\\n$\\bullet$ We propose the first cohesive subgraph model $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace on bipartite graphs which considers both tie strength and vertex engagement. The flexibility of our model allows it to capture unique and useful structures on bipartite graphs. \\\\\n$\\bullet$ We construct index $I_{\\alpha,\\beta,\\tau}$ to support optimal retrieval of the vertex set of any $(\\alpha,\\beta)_{\\tau}$-core\\xspace.\nWe also devise computation sharing and \\texttt{BE}-\\texttt{Index}\\xspace based optimizations to effectively reduce its construction time.\\\\\n$\\bullet$ \nWe build \\gratings that are more space-efficient and require significantly less build time. \nAlso, we propose a learning-based hybrid computation paradigm to predict which index to choose to minimize the response time for an incoming $(\\alpha,\\beta)_{\\tau}$-core\\xspace query.\n\\\\\n$\\bullet$ We validate the efficiency of proposed algorithms and the effectiveness of our model through extensive experiments on real-world datasets.\nResults show that the \\gratings are scalable and the hybrid computation algorithm on a well trained neural network can outperform the algorithms based on each \\grating alone. \n\\\\\n\n\\noindent\n{\\bf Organization.}\nThe rest of the paper is organized as follows.\nSection $2$ reviews the related work. \nSection $3$ summarizes important notations and definitions and\nintroduces $(\\alpha,\\beta)$-core\\xspace and $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace. \nSection $4$ presents the online computation algorithm. \nSection $5$ and $6$ presents the total index $I_{\\alpha,\\beta,\\tau}$ and optimizations of the index construction process.\nSection $7$ presents the learning-based hybrid computation paradigm. \nSection $8$ shows the experimental results and Section $9$ concludes the paper.  \n\\section{Related work}\nIn the literature, there are many recent studies on cohesive subgraph models on both unipartite graphs and bipartite graphs. \n\n\\noindent\n\\textit{Unipartite graphs.}\n\\textit{k}-core\\xspace \\cite{seidman1983network,cheng2011efficient,khaouid2015k,zhang2018finding} and \\textit{k}-truss\\xspace \\cite{cohen2008trusses,huang2014querying,shao2014efficient} are two of the most well-known cohesive subgraph models on general, unipartite graphs. \nGiven a unipartite graph, \\textit{k}-core\\xspace is the maximal subgraph such that each vertex in the subgraph has at least $k$ neighbors.\n\\textit{k}-core\\xspace models vertex engagement as degrees and assumes the importance of each tie to be equal.\nHowever, on real networks, ties (edges) have different strengths and are not of equal importance \\cite{granovetter1977strength}.\nAs triangles are considered as the smallest cohesive units, the number of triangles containing an edge is used to model tie strength on unipartite graphs.\nThus, \\textit{k}-truss\\xspace is proposed to better model tie strength, which is the maximal subgraph such that each edge in the subgraph is contained in at least $(k-2)$ triangles.\nThe issue with \\textit{k}-truss\\xspace is that it does not tolerate the existence of weak ties, which is inflexible for modeling real networks.\nTo consider both vertex engagement and tie strength, the ($k$,$s$)-core model is proposed in \\cite{zhang2018discovering}. \nIn addition, recent works studied problems related to variants of \\textit{k}-core\\xspace such as  radius-bounded \\textit{k}-core\\xspace on geo-social networks \\cite{wang2018efficient}, core maintenance on dynamic graphs \\cite{zhang2017fast},\ncore decomposition on uncertain graphs \\cite{bonchi2014core,peng2018efficient}, \nand anchored \\textit{k}-core\\xspace problem \\cite{bhawalkar2015preventing,zhang2017olak}. \nVariants of \\textit{k}-truss\\xspace are also studied including \\textit{k}-truss\\xspace communities on dynamic graphs \\cite{huang2014querying},\n\\textit{k}-truss\\xspace decomposition on uncertain graphs \\cite{zou2017truss}, and anchored \\textit{k}-truss\\xspace problem \\cite{zhang2018efficiently}.\nHowever, these algorithms do not apply to bipartite graphs. Attempts to project the bipartite graph to general graphs will incur information loss and size inflation \\cite{sariyuce2018peeling}.\n\n\\noindent\n\\textit{Bipartite graphs.}\nIn correspondence to \\textit{k}-core\\xspace and \\textit{k}-truss\\xspace, $(\\alpha,\\beta)$-core\\xspace \\cite{ding2017efficient,liu2020efficient} and \\textit{k}-bitruss\\xspace \\cite{zou2016bitruss,wang2020efficient} are proposed on bipartite graphs.\n$(\\alpha,\\beta)$-core\\xspace is the maximal subgraph such that each vertex on the upper or lower level in the subgraph has at least $\\alpha$ or $\\beta$ neighbors. \nJust like \\textit{k}-core\\xspace on unipartite graphs, $(\\alpha,\\beta)$-core\\xspace cannot distinguish weak ties from strong ties. \nOn bipartite graphs, tie strength is often modeled as the number of butterflies (i.e., ($2 \\times 2$)-bicliques) containing an edge because butterflies are viewed as analogs of triangles \\cite{sanei2018butterfly,wang2014rectangle,wang2019vertex,wang2020efficient}. \n\\textit{k}-bitruss\\xspace is the maximal subgraph such that each edge in the subgraph is contained in at least $k$ butterflies, which can model tie strength.\n\\textit{k}-bitruss\\xspace suffers from the same issue as its counterpart \\textit{k}-truss\\xspace does: it forcefully deletes all weak ties even if the incident vertices are strongly-engaged.\nOther works for bipartite graph analysis using cohesive structures such as ($p$,$q$)-core \\cite{DBLP:journals\/corr\/CerinsekB15}, fractional \\textit{k}-core\\xspace \\cite{giatsidis2011evaluating} cannot be used to address these issues. In contrast to the above studies, we propose the first cohesive subgraph model $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace that considers both vertex engagement and tie strength on bipartite graphs.\n\\section{Problem Definition}\n\\begin{table}[htb]\n\\centering\n\\caption{Summary of Notations}\n\\scalebox{1.0}{\n\\begin{tabular}{c|c}\n\\noalign{\\hrule height 1pt}\nNotation & Definition \\\\ \n\\noalign{\\hrule height 0.6pt}\n$G$ & a bipartite graph \\\\\n$\\alpha,\\beta$ & the engagement constraints \\\\\n$\\tau$ & the strength level \\\\\n$nb(u,G)$ & the set of adjacent vertices of $u$ in $G$ \\\\\n$deg(u,G)$ & the number of adjacent vertices of $u$ in $G$ \\\\\n$ \\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{G} $   &   the number of butterflies in $G$  \\\\\n$\\sup(e)$ & the number of butterflies containing $e$ \\\\\n$eng \\xspace(u)$ & the number of strong ties adjacent to $u$ \\\\\n$(\\alpha,\\beta)_{\\tau}$-core\\xspace & the $\\tau$-strengthened $(\\alpha,\\beta)$-core \\\\\n$I_{\\alpha,\\beta,\\tau}$ & the decomposition-based index \\\\\n$I_{\\alpha,\\beta}$,$I_{\\beta,\\tau}$,$I_{\\alpha,\\tau}$ & the \\gratings \\\\\n\\noalign{\\hrule height 1pt}\n\\end{tabular}\n}\n\\vspace{-2mm}\n\\label{tab:notation}\n\\end{table}\n\nIn this section, we formally define our cohesive subgraph model $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace.  We consider an unweighted, undirected bipartite graph $G(V, E)$. $V(G)$ = $U(G) \\cup L(G)$ denotes the set of vertices in $G$ where $U(G)$ and $L(G)$ represent the upper and lower layer, respectively. $E(G) \\subseteq U(G) \\times L(G)$ denotes the set of edges in $G$. We use $n$ = $|V(G)|$ to denote the number of vertices and $m$ = $|E(G)|$ to denote the number of edges. \nThe maximum degree in the upper and lower layer is denoted as $d_{max}(U)\\xspace$ and $d_{max}(L)\\xspace$ respectively.\nThe set of neighbors of a vertex $u$ in $G$ is denoted as $nb(u,G)$. The degree of a vertex is $deg(u,G)$ = $|nb(u,G)|$. When the context is clear, we omit the input graph $G$ in notations. \n\n\\begin{definition}\n\\label{def:abcore}\n{\\bf $(\\alpha,\\beta)$-core.} Given a bipartite graph G and degree constraints $\\alpha$ and $\\beta$, \na subgraph $G'$ is the $(\\alpha,\\beta)$-core, denoted by $C_{\\alpha,\\beta}(G)$,\nif ($1$) all vertices in $G'$ satisfy degree constraints,\ni.e. $deg(u,G')\\geq \\alpha$ for each $u \\in U(G')$ and \n$deg(v,G')\\geq \\beta$ for each $v \\in L(G')$;\nand ($2$) $G'$ is maximal, i.e. any subgraph $G'' \\supseteq G'$ is not an $(\\alpha,\\beta)$-core. \n\\end{definition}\n\n\\begin{definition}\n\\label{def:btf}\n{\\bf Butterfly.} In a bipartite graph G, given vertices $u,w \\in U(G)$ and $v,x \\in L(G)$, a butterfly $\\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}$ is the complete subgraph induced by $u,v,w,x$, which means both $u$ and $w$ are connected to $v$ and $x$ by edges. \nThe total number of butterflies in G is denoted as $\\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_G$.\n\\end{definition}\n\\noindent\n$(\\alpha,\\beta)$-core is a vertex-induced subgraph model, which assumes that the edges are of equal importance. To better model the strength of an edge $e$, we define the support $sup \\xspace (e)$ to be the number of butterflies containing $e$. \n\\begin{definition}\n\\label{def:st}\n{\\bf Strong Tie.} Given an integer $\\tau$, an edge $e \\in E(G)$ is called a strong tie if $sup \\xspace(e) \\geq \\tau$, where $\\tau$ is called the strength level. Weak ties are the edges $e$ such that $sup \\xspace (e) < \\tau$.\n\\end{definition}\n\\begin{definition}\n\\label{def:eng}\n{\\bf Vertex Engagement.} Given a strength level $\\tau$ and $u \\in V(G)$, the engagement $eng \\xspace (u)$ is the number of strong ties incident to $u$. At strength level 0, $eng \\xspace(u)=deg(u,G)$.\n\\end{definition}\n\n\\noindent\nIf the engagement of an upper or lower vertex is at least $\\alpha$ or $\\beta$, we call it a {\\bf strongly-engaged} vertex. \nOtherwise, it is a {\\bf weakly-engaged} vertex. \n\\begin{definition}\n\\label{def:abgcore}\n{\\bf $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace.} \nGiven a bipartite graph G and engagement constraints $\\alpha$ and $\\beta$, and strength level $\\tau$, \na subgraph $G'$ is the $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace, denoted by $(\\alpha,\\beta)_{\\tau}$-core\\xspace, \nif ($1$) $eng \\xspace(u) \\geq \\alpha$ for each $u \\in U(G')$ and \n$eng \\xspace(v) \\geq \\beta$ for each $v \\in L(G')$;\nand ($2$) $G'$ is maximal, i.e. any subgraph $G'' \\supseteq G'$ is not a $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace.\n\\end{definition}\n\n\\noindent\n\\textbf{Problem Statement.}  Given a bipartite graph $G$ and parameters $\\alpha,\\beta$ and $\\tau$, we study the problem of scalable and efficient computation of $(\\alpha,\\beta)_{\\tau}$-core\\xspace in $G$.\n\\section{The Online Computation Algorithm}\nGiven engagement constraints $\\alpha$, $\\beta$ and strength level $\\tau$, the online algorithm to compute the $(\\alpha,\\beta)_{\\tau}$-core\\xspace is outlined in Algorithm  \\ref{algo:compute_naive}. \nFirst, we compute the support of each edge $e$ using the algorithm in \\cite{wang2019vertex} and count how many strong ties each vertex $u$ has.  \nThen, Algorithm \\ref{algo:peel} is invoked to iteratively remove the vertices without enough engagement along with their incident edges.  \nThe vertices in $U(G)$ and $L(G)$ are sorted by engagement and the edges are sorted by support. In this manner, we can always delete the vertices with the smallest engagement first and quickly identify which edges are strong ties. \nWhen an edge $e$ is removed due to lack of support (i.e., $sup(e)<\\tau$), we go through all the butterflies containing $e$ and update the supports of the edges in these butterflies (lines 5-9).\nSpecifically, we do not need to update the support of the edges connected to weakly-engaged vertices, because they will be removed (line $10$). \nNeither do we update the support of weak ties because they do not contribute to any vertex engagement. \nIn other words, we only update the support of strong ties between strongly-engaged vertices. \nWhen a strong tie becomes a weak tie, we decrease the engagement of its incident vertices (lines 4,9). \nThe order of vertices and edges are maintained in linear heaps \\cite{chang2019cohesive} after their engagement and support are updated.  \nHere we evaluate the time and space complexity of Algorithm \\ref{algo:compute_naive}. \n\\begin{lemma}\nThe time complexity of Algorithm \\ref{algo:compute_naive} is $O(\\sum_{(u,v)\\in E(G)}\\sum_{w\\in nb(v)} min(deg(u),deg(w)))$ and the space complexity is $O(m)$. \n\\end{lemma}\n\\begin{proof}\nThe butterfly counting process takes $O(\\sum_{(u,v)\\in E(G)} min(deg(u),deg(v)))$ time \\cite{wang2019vertex}. \nAfter the support of each edge is computed, it takes $O(m)$ time to compute the engagement for each vertex. \nThen, we need to run Algorithm \\ref{algo:peel}. For each weakly engaged vertex $u$, we need to delete all its incident edges, which dominates the time cost of Algorithm \\ref{algo:peel}. \n\nFor each deleted edge $(u,v)$, we need to enumerate the butterflies containing it. \nLet $w$ be a vertex in $nb(v,G) \\setminus \\{u\\} $. \nFor each vertex $x$ in $nb(u,G) \\cap nb(w,G)$, the induced subgraph of $\\{u,v,w,x\\}$ is a butterfly. The set intersection (computing $nb(u,G)\\cap nb(w,G)$) can be implemented in $O(min(deg(u),deg(w))$ time by using a $O(m)$ hash table to store the neighbor set of each vertex.\n\nThus, the butterfly enumeration for each delete edge $(u,v)$ takes $O(\\sum_{w\\in nb(v)} min(deg(u),deg(w)))$ time.\nAs each edge can only be deleted once, the total time complexity of the butterfly enumeration for all deleted edges takes $O(\\sum_{(u,v)\\in E(G)}\\sum_{w\\in nb(v)} min(deg(u),deg(w)))$ time. Thus, the time complexity of Algorithm \\ref{algo:compute_naive} is $O(\\sum_{(u,v)\\in E(G)}\\sum_{w\\in nb(v)} min(deg(u),deg(w)))$ which is denoted as $T_{peel}(G)$ hereafter.\n\nWe store the neighbors of each vertex as adjacency lists as well as the support of edges and engagement of vertices, which in total takes $O(m)$ space. Therefore, the space complexity of Algorithm \\ref{algo:compute_naive} is $O(m)$.  \n\\end{proof}\n\n\\begin{figure}[htb]\n\\centering  \n\\subfigure[$I_{\\alpha,\\beta,\\tau}$ based on Figure \\ref{fig:moltivation}]{\n\\label{fig.index}\n\\includegraphics[width=0.43\\textwidth]{btree.png}}\n\\subfigure[the relationships of indexes]{\n\\label{fig.cube}\n\\includegraphics[width=0.26\\textwidth]{cube.pdf}}\n\\subfigure[Computation sharing]{\n\\label{fig.share}\n\\includegraphics[width=0.26\\textwidth]{compsharing.pdf}}\n\\caption{Illustrating our ideas}\n\\label{Fig.queries}\n\\end{figure}\n\n\\section{The Decomposition Based Total Index}\nGiven $\\alpha$, $\\beta$, and $\\tau$, Algorithm \\ref{algo:compute_naive} computes the $(\\alpha,\\beta)_{\\tau}$-core\\xspace from the input graph, which is slow and cannot handle a large number of queries. \nIn this section, we present a decomposition algorithm that retrieves all $(\\alpha,\\beta)_{\\tau}$-cores\\xspace and build a total index based on the decomposition output to support efficient query processing.\n\n\\begin{algorithm}[t!]\n\t\\caption{Decomposition}\n\t\\label{algo:naivedecomp}\n\t\\LinesNumbered\n\t\\KwIn{$G(V=(U,L),E)$}\n\t\\KwOut{$\\tau_{max}(\\alpha,\\beta,u)$, for all $\\alpha,\\beta$, $\\forall u \\in V(G)$}\n\t$\\alpha \\gets 1$, $\\beta \\gets 1$, $\\tau \\gets 1$ \\\\\n\tCompute  $sup \\xspace (e)$, $\\forall$ $e \\in E(G)$  \\\\\n    Compute  $eng \\xspace (u)$, $\\forall$ $u \\in V(G)$ \\\\\n    \\While{$(\\alpha,1)_{1}$-core in $G$ is not empty}{\n        $\\compute((\\alpha,1)_{1}\\textrm{-}core, \\alpha, 1, 1, sup \\xspace, eng \\xspace);\\beta\\gets1$ \\\\ \n       \n        \\While{$(\\alpha,\\beta)_{1}$-core in $G$ is not empty}{\n            $sup \\xspace' \\gets sup \\xspace$;\\ $eng \\xspace' \\gets eng \\xspace$;\\ $\\tau\\gets1$ \\\\\n            $\\compute((\\alpha,\\beta)_{1}\\textrm{-}core, \\alpha, \\beta, 1, sup \\xspace ',  eng \\xspace ')$\\\\ \n           \n            \\While{ $(\\alpha,\\beta)_{\\tau}$-core in $G$ is not empty}{ \n                $sup \\xspace'' \\gets sup \\xspace'$;\\ $eng \\xspace'' \\gets eng \\xspace'$ \\\\\n                $\\compute((\\alpha,\\beta)_{\\tau}\\textrm{-}core, \\alpha, \\beta, \\tau, sup \\xspace '', eng \\xspace '')$, \\ \\ add $\\tau_{max}(\\alpha,\\beta,u) \\gets \\tau-1 $ before line 2\\\\\n                $\\tau \\gets \\tau+1$ \\\\\n                \\ForEach{$e$=$(u',v') \\in E(G)$, $sup \\xspace (e)$=$\\tau$}{\n                    $eng \\xspace ''(u')\\gets eng \\xspace ''(u')-1$ \\\\\n                    $eng \\xspace ''(v')\\gets eng \\xspace ''(v')-1$\n                }\n            }\n            $\\beta \\gets \\beta+1$\n        }\n        $\\alpha \\gets \\alpha+1$\n    }\n\\textbf{return} $\\tau_{max}(\\alpha,\\beta,u)$, for all $\\alpha,\\beta$, $\\forall u \\in V(G)$ \n\\end{algorithm}\n\n\n\\vspace{2mm}\n\\noindent\n{\\bf The decomposition algorithm.} \nThe following lemma is immediate based on Definition \\ref{def:abgcore}, which depicts the nested relationships among $(\\alpha,\\beta)_{\\tau}$-cores\\xspace. \n\\begin{lemma}\n$(\\alpha,\\beta)_{\\tau}$-core\\xspace $\\subseteq  (\\alpha',\\beta')_{\\tau'}$-core\nif $ \\alpha \\geq \\alpha' $, $ \\beta \\geq \\beta' $, and $ \\tau \\geq \\tau' $.\n\\label{lemma:nest}\n\\end{lemma}\nBased on Lemma \\ref{lemma:nest}, if a vertex $u$ is in $(\\alpha, \\beta)_{\\tau'}$-core, $u$ is also in $(\\alpha,\\beta)_{\\tau}$-core\\xspace if $\\tau < \\tau'$. Thus, in the decomposition, for given vertex $u$ and $\\alpha$, $\\beta$ values, we aim to retrieve the maximum $\\tau$ value such that $u$ is in the corresponding $(\\alpha, \\beta)_\\tau$-core, namely, $\\tau_{max}(\\alpha,\\beta,u) \\textnormal{=} \\max\\{\\tau \\vert  u \\in (\\alpha, \\beta)_\\tau\\textrm{-}core \\} $. \nFor each vertex $u$ and all possible combinations of $\\alpha$ and $\\beta$, it is only necessary to store $u$ in $(\\alpha,\\beta)_{\\tau'}$-core to build a space compact index, where $\\tau'$ = $\\tau_{max}(\\alpha,\\beta,u)$ and $u \\in$ $(\\alpha, \\beta)_{\\tau}$-core can be implied if $\\tau < \\tau'$.\nAlgorithm \\ref{algo:naivedecomp} is devised for $(\\alpha,\\beta)_{\\tau}$-core\\xspace decomposition which applies three nested loops to go through all possible $\\alpha, \\beta, \\tau$ combinations. \nNote that when computing the $(\\alpha,\\beta)_{\\tau}$-core\\xspace, we record $\\tau_{max}(\\alpha,\\beta,u)$ for each vertex $u$. Specifically, for a vertex $u$ contained in $(\\alpha,\\beta)_{\\tau_0}$-core but is removed when computing $(\\alpha,\\beta)_{\\tau_0+1}$-core, we assign $\\tau_0$ to $\\tau_{max}(\\alpha,\\beta,u)$ (line 11).\nHere we evaluate the time and space complexity of Algorithm \\ref{algo:naivedecomp}. \n\\begin{lemma}\nThe time complexity of Algorithm \\ref{algo:naivedecomp} is \n$O(\\sum_{\\alpha =1 }^{\\alpha_{max}} \\sum_{\\beta=1}^{\\beta_{max}(\\alpha)} T_{peel}((\\alpha,\\beta)_{1}\\textnormal{-}core))$,\nwhere $\\alpha_{max}$ is the maximal $\\alpha$ such that $(\\alpha,1)_1$-core exists and $\\beta_{max}(\\alpha)$ is the maximal $\\beta$ such that $(\\alpha,\\beta)_{1}$-core\\xspace exists.\nAlgorithm \\ref{algo:naivedecomp} takes up $O(m)$ space.\n\\end{lemma}\n\n\\begin{proof}\nIn the outer while-loop, we start from the input graph $G$ and compute $(\\alpha,1)_{1}$-core\\xspace, which takes $T_{peel}(G)$.\nIn the middle while-loop, we calculate $(\\alpha,\\beta)_{1}$-core\\xspace from $(\\alpha,1)_{1}$-core\\xspace for all possible $\\alpha$, which takes  $ \\sum_{\\alpha =1 }^{\\alpha_{max}} T_{peel}((\\alpha,1)_{1}\\textnormal{-}core)$. \nThe dominant cost occurs in the innermost while-loop when we run the Peeling algorithm on $(\\alpha,\\beta)_{1}$-core\\xspace for all possible $\\alpha$ and $\\beta$. \nAs iterative removing edges and vertices in $(\\alpha,\\beta)_{1}$-core\\xspace until it is empty takes $T_{peel}((\\alpha,\\beta)_{1}\\textnormal{-}core)$, \nthe overall time complexity is $O(\\sum_{\\alpha =1 }^{\\alpha_{max}} \\sum_{\\beta=1}^{\\beta_{max}(\\alpha)} T_{peel}((\\alpha,\\beta)_{1}\\textnormal{-}core))$.\n\nAt any time during the execution of this algorithm, we always store $G$, $(\\alpha,1)_{1}$-core\\xspace, $(\\alpha,\\beta)_{1}$-core\\xspace, and $(\\alpha,\\beta)_{\\tau}$-core\\xspace in memory, which takes $O(m)$ space. \nWe also store the support of edges and engagement of vertices in these graphs, which also takes $O(m)$ space. \nThus, the total space complexity is $O(m)$. \n\\end{proof}\n\n\\noindent\n{\\bf Decomposition-based index.} Based on the decomposition results, a four level index $I_{\\alpha,\\beta,\\tau}$ can be constructed to support query processing as shown in Figure \\ref{fig.index}.\n\\begin{enumerate}\n    \\item[$\\bullet$] \\textit{$\\alpha$ level.} The $\\alpha$ level of $I_{\\alpha,\\beta,\\tau}$ is an array of pointers, each of which points to an array in the $\\beta$ level. The length of the array in the $\\alpha$ level is $\\alpha_{max}$. The $k_{th}$ element is denoted as $I_{\\alpha,\\beta,\\tau}[k]$. \n    \\item[$\\bullet$] \\textit{$\\beta$ level.} \n    The $\\beta$ level has $\\alpha_{max}$ arrays of pointers. The array pointed by $I_{\\alpha,\\beta,\\tau}[k]$ has length $\\beta_{max}(\\alpha)$. The $j_{th}$ pointer in the $k_{th}$ array is denoted as $I_{\\alpha,\\beta,\\tau}[k][j]$, which points to an array in the $\\tau$ level. \n    \\item[$\\bullet$] \\textit{$\\tau$ level.}\n     The $\\tau$ level has $\\sum_{i=1}^{\\alpha_{max}}\\beta_{max}(i) $ arrays of pointers to vertex blocks, corresponding to all pairs of $\\alpha,\\beta$. The array pointed by $I_{\\alpha,\\beta,\\tau}[k][j]$ has length $\\tau_{max}(\\alpha,\\beta) = \\max\\{\\tau \\vert (\\alpha,\\beta)_{\\tau}\\textnormal{-}core \\ exists \\} $. \n     \\item[$\\bullet$] \\textit{vertex blocks.} The fourth level of $I_{\\alpha,\\beta,\\tau}$ is a singly linked list of vertex blocks. Each vertex block corresponds to a set of $\\alpha,\\beta,\\tau$ value, which contains all vertex $u$ such that $\\tau_{max}(\\alpha,\\beta,u) = \\tau$ along with a pointer to the next vertex block. \n     The vertex blocks with the same $\\alpha$ and $\\beta$ are sorted by the associated $\\tau$ values and each of them has a pointer to the next.  \n     Among them, the vertex block with the largest $\\tau$ has its pointer pointing to null. \n   \n    \n\n\\end{enumerate}\n\nWe can construct index $I_{\\alpha,\\beta,\\tau}$ from the output of Algorithm \\ref{algo:decomp_query}. \nGiven $\\alpha$ and $\\beta$, we store all vertices $u$ in the same vertex block if they have the same $\\tau_{max}(\\alpha,\\beta,u)$, so each vertex block has an associated $(\\alpha,\\beta,\\tau)$ value. \nIn each $I_{\\alpha,\\beta,\\tau}[\\alpha][\\beta][\\tau]$, we store the address of the vertex block associated with $(\\alpha,\\beta,\\tau')$, where $\\tau'$ is the smallest integer such that $\\tau \\leq \\tau'$.\nThe index construction time is linear to the size of decomposition results, which is bounded by the time complexity of Algorithm \\ref{algo:decomp_query}. \n\n\\begin{algorithm}[t]\n\t\\caption{DecompQuery}\n\t\\label{algo:decomp_query}\n\t\\LinesNumbered\n\t\\KwIn{$I_{\\alpha,\\beta,\\tau},\\alpha,\\beta,\\tau,G$}\n\t\\KwOut{$(\\alpha,\\beta)_{\\tau}$-core\\xspace}\n\t$U',V',E' \\gets \\emptyset$ \\\\\n\t\\If{$I_{\\alpha,\\beta,\\tau}.size <\\alpha$ or $I_{\\alpha,\\beta,\\tau}[\\alpha].size <\\beta$ or $I_{\\alpha,\\beta,\\tau}[\\alpha][\\beta].size < \\tau$ }{\n        return $\\emptyset$ \\\\\n\t}\n\t$ ptr\\gets I_{\\alpha,\\beta,\\tau}[\\alpha][\\beta][\\tau]$  \\\\\n    \\While{$ptr$ is not null }{\n        $v \\gets $ vertices in vertex block pointed by $ptr$ \\\\ \n        \\If{$v \\in U(G)$}{\n            $U' \\gets U' \\cup v$\n        }\\Else{\n            $V' \\gets V' \\cup v$\n        }\n        $ptr \\gets $ the address of the next vertex block \n    } \n    $E' \\gets E(G) \\cap (U' \\times V')$ \\\\ \n\\textbf{return} $G'=(U',V',E')$\n\\end{algorithm}\n\\begin{lemma}\n\\label{lemma:izero}\nThe space complexity of index $I_{\\alpha,\\beta,\\tau}$ is \n$O(\\sum_{i=1}^{\\alpha_{max}} \\sum_{j=1}^{\\beta_{max}(i)} (\\tau_{max}(i,j)+n))$,\nwhere  $\\tau_{max}(i,j)$ be the maximal $\\tau$ in all $(i,j)_{\\tau}$-cores. \n\\end{lemma}\n\n\\begin{proof}\nBy construction, the space complexity of the first two levels of pointers is bounded by that of $\\tau$ level. \nThe space complexity of $\\tau$ level is $\\sum_{i=1}^{\\alpha_{max}} \\sum_{j=1}^{\\beta_{max}(i)} \\tau_{max}(i,j)$. \nGiven vertex $u$, let $\\alpha_{max}(u)$ be the maximal $\\alpha$ such that $u \\in (\\alpha,\\beta)_{\\tau}\\textnormal{-}core$.\nThe space complexity of the vertex blocks is  \n$\\sum_{u\\in V(G)} \\sum_{i=1}^{\\alpha_{max}(u)}\\max \\{\\beta | u \\in (i,\\beta)_1\\textnormal{-}core\\} \\leq \\sum_{i=1}^{\\alpha_{max}} \\sum_{j=1}^{\\beta_{max}(i)} n$. Adding it to the space complexity of level $\\tau$ completes the proof.\n\\end{proof} \n\n\\noindent\n{\\bf Index based query processing.} \nWhen the index $I_{\\alpha,\\beta,\\tau}$ is built on a bipartite graph $G$, Algorithm \\ref{algo:decomp_query} outlines how to restore $(\\alpha,\\beta)_{\\tau}$-core\\xspace given $\\alpha,\\beta$, $\\tau$ and $I_{\\alpha,\\beta,\\tau}$. \nFirst, it checks the validity of the input parameters. \nIf the queried $(\\alpha,\\beta)_{\\tau}$-core\\xspace does not exist, it terminates immediately (lines 2,3). \nOtherwise, it collects the vertices of $(\\alpha,\\beta)_{\\tau}$-core\\xspace from $I_{\\alpha,\\beta,\\tau}$ and restores the edges in the queried subgraph. \n\\begin{lemma}\n\\label{lemma:query}\nGiven a graph $G$ and parameters $\\alpha,\\beta$ and $\\tau$, \nAlgorithm \\ref{algo:decomp_query} retrieves \n$V((\\alpha,\\beta)_{\\tau}\\textnormal{-}core)$ from index $I_{\\alpha,\\beta,\\tau}$ in $O(|V((\\alpha,\\beta)_{\\tau}\\textnormal{-}core)|)$ time. \nThe edges in $(\\alpha,\\beta)_{\\tau}$-core\\xspace can be retrieved in $O(\\sum_{v \\in V((\\alpha,\\beta)_{\\tau}\\textnormal{-}core)}deg(v)$ time after obtaining the vertex set.\n\\end{lemma}\n\\begin{proof}\nAs each vertex block only stores the vertices with one given $\\tau$ value, the vertex blocks pointed by $I_{\\alpha,\\beta,\\tau}[\\alpha][\\beta][\\tau']$ where $\\tau' \\geq \\tau$ give us all the vertices in $(\\alpha,\\beta)_{\\tau}$-core\\xspace, which takes $O(|V((\\alpha,\\beta)_{\\tau}\\textnormal{-}core)|)$ time. \nTo restore the edges in $(\\alpha,\\beta)_{\\tau}$-core\\xspace, for each vertex $v$ in $(\\alpha,\\beta)_{\\tau}$-core\\xspace, we go through each of its neighbors in $G$ and check if it is in $V((\\alpha,\\beta)_{\\tau}\\textnormal{-}core)$. This takes $O(\\sum_{v \\in V((\\alpha,\\beta)_{\\tau}\\textnormal{-}core)}(deg(v)))$ time.\n\\end{proof} \n\n\\noindent\nAccording to Lemma \\ref{lemma:query}, given $\\alpha,\\beta$ and $\\tau$, the vertex set of the $(\\alpha,\\beta)_{\\tau}$-core\\xspace is retrieved in optimal time. \n\\begin{example}\nFigure \\ref{fig.index} illustrates the $I_{\\alpha, \\beta, \\tau}$ index for the bipartite graph in Figure \\ref{fig:moltivation}. When querying $(1,2)_{1}$-core, we start with the vertex block pointed by $I_{\\alpha,\\beta,\\tau}[1][2][1]$ ($u_0,v_0$ and $v_1$). \nThen we keep collecting the vertices until we have fetched the vertices pointed by $I_{\\alpha,\\beta,\\tau}[1][2][2]$ ($u_1$ to $u_5$ and $v_2$ to $v_6$) . All the collected vertices provides the final solution to vertex set of $(1,2)_{1}$-core, which are $u_0$ to $u_5$ and $v_0$ to $v_6$. \n\\end{example}\n\\section{Optimizations of Index Construction }\nThe above decomposition algorithm has these issues:\n($1$) the same subgraph can be computed repeatedly for different $\\alpha$ and $\\beta$ values. \nFor example, if $(1,1)_{\\tau}$-core is the same subgraph as $(1,2)_{\\tau}$-core, \nthen we will compute it twice when $\\beta$=$1$ and $\\beta$=$2$. \n($2$) when removing an edge $e$, we need to enumerate all the butterflies containing $e$.\nThe basic implementation of butterfly enumeration is inefficient, which involves finding three connected vertices first and then check if a fourth vertex can form a butterfly with the existing ones.\nWe devise computation-sharing optimizations to address the first issue, and adopt the \\texttt{Bloom}-\\texttt{Edge}-\\texttt{Index} proposed in \\cite{wang2019vertex} to speed up the butterfly enumeration process.\n\n\\noindent\n{\\bf Computation sharing optimizations.}\nIn this part, we reduce the times of visiting the same $(\\alpha,\\beta)_{\\tau}$-core\\xspace subgraphs by skipping some combinations of $\\alpha$, $\\beta$, and $\\tau$, while yielding the same decomposition results. \n\n\\noindent\n{$\\bullet$ \\it Skip computation for $\\tau$.}\nIn Algorithm \\ref{algo:naivedecomp}, if vertex $u$ is removed when computing $(\\alpha,\\beta)_{\\tau+1}$-core from $(\\alpha,\\beta)_{\\tau}$-core\\xspace, we conclude that $\\tau_{max}(\\alpha,\\beta,u)$=$\\tau$. However, if both $(\\alpha,\\beta)_{\\tau}$-core\\xspace and $(\\alpha,\\beta)_{\\tau+1}$-core are already visited for other $\\beta$, this process is redundant and the current $\\tau$ value can be skipped. \nSpecifically, in the innermost while loop (lines $9$-$15$), we can use an array $\\beta_{min}[\\tau]$ to store the minimal lower engagement of $(\\alpha,\\beta)_{\\tau}$-core\\xspace for each $\\tau$. \nIf $\\beta_{min}[\\tau] \\geq \\beta$, then the current $(\\alpha,\\beta)_{\\tau}$-core\\xspace has already been visited.\nWe only compute $\\tau_{max}(\\alpha,\\beta,u)$ values when one of $(\\alpha,\\beta)_{\\tau}$-core\\xspace and $(\\alpha,\\beta)_{\\tau+1}$-core is not visited. Otherwise, we skip the current $\\tau$ value. \nThe following lemma explains how to correctly obtain multiple $\\tau_{max}(\\alpha,\\beta,u)$ values when removing one vertex.\n\\begin{lemma}\n\\label{lemma:sharing}\nGiven $\\alpha,\\beta,\\tau$ and graph $G$, let $u$ be a vertex in $(\\alpha,\\beta)_{\\tau}$-core but not in $(\\alpha,\\beta)_{\\tau+1}$-core. If there exists an integer $k$ such that $(\\alpha,\\beta)_{\\tau}$-core = $(\\alpha,\\beta+k)_{\\tau}$-core, then $u \\not \\in$ $(\\alpha,\\beta+k)_{\\tau+1}$-core. \n\\end{lemma}\nThis lemma is immediate from Lemma \\ref{lemma:nest}, because if vertex $u$ is in $(\\alpha,\\beta+k)_{\\tau+1}$-core, then it must also be contained in $(\\alpha,\\beta)_{\\tau+1}$-core, which contradicts our assumption. \nTherefore, for vertices like $u$, we can conclude that $\\tau_{max}(\\alpha, \\beta',u)$=$\\tau$, for all $\\beta \\leq \\beta' \\leq \\beta+k$. In this way, we fully preserve the decomposition outputs of Algorithm \\ref{algo:naivedecomp}. \n\n\\noindent\n{$\\bullet$ \\it Skip computation for $\\alpha$ and $\\beta$.}\nTo skip some $\\beta$ values, we keep track of the minimal engagement of lower level vertices of the visited $(\\alpha,\\beta)_{\\tau}$-cores\\xspace in the middle while-loop (lines $7$-$16$) in Algorithm \\ref{algo:naivedecomp} as $\\beta^*$. At line $16$, if $\\beta^* > \\beta$, then $\\beta$ should be directly increased to $\\beta^*+1$ (the first value which is not computed yet) and values from $\\beta$ to $\\beta^*$ are skipped.\nThis is because for all $ \\beta \\leq \\beta' \\leq \\beta^*$, the decomposition process are exactly the same. \nLikewise, to skip some $\\alpha$ values, we record the minimal engagement of upper-level vertices of the visited $(\\alpha,\\beta)_{\\tau}$-cores\\xspace in the outermost while-loop (lines $5$-$17$) of Algorithm \\ref{algo:naivedecomp} as $\\alpha^*$. At line $17$, if $\\alpha^* > \\alpha$, then $\\alpha$ should be directly increased to $\\alpha^*+1$ and values from $\\alpha$ to $\\alpha^*$ are skipped. \n\n\\begin{example}\nAs shown in Figure \\ref{fig.index},\nwhen $\\beta$=$1$, we remove $u_0,v_0$ and $v_1$ when computing $(1,1)_2$-core from $(1,1)_1$-core. \nThe minimal engagement of lower vertices in $(1,1)_1$-core and $(1,1)_2$-core are $2$, so the array $\\beta_{min}$ is $[2,2]$.\nThis means that $\\tau_{max}(1,\\beta',u)$ = $1$ and $\\tau_{max}(1,\\beta',u')$ = $2$, where $u \\in \\{u_0,v_0,v_1 \\}$ and $u' \\in \\{u_1,u_2,u_3,u_4,u_5,v_2,v_3,v_4,v_5,v_6\\}$ and $\\beta' \\in \\{1,2\\}$. \nWhen $\\beta$=$2$, we infer that $(1,2)_1$-core and $(1,2)_2$-core are already visited based on array $\\beta_{min}$, so we can skip $\\beta$=$2$. \nWhen $\\beta$=$3$, the current $\\beta$ value exceeds the values in $\\beta_{min}$, so it cannot be skipped. \n\\end{example}\n\n\\noindent\n{\\bf Bloom-Edge-Index-based optimization.}\nDuring edge deletions of Algorithm \\ref{algo:naivedecomp}, we need to repeatedly retrieve the butterflies containing the deleted edges. \nTo efficiently address this, We deploy a \\texttt{Bloom}-\\texttt{Edge}-\\texttt{Index} (hereafter denoted as \\texttt{BE}-\\texttt{Index}\\xspace) proposed in \\cite{wang2020efficient} to facilitate butterfly enumeration. \nSpecifically, a bloom is a $2 \\times k$-biclique, which contains $\\frac{k \\times (k\\textnormal{-}1)}{2}$  butterflies. \nEach edge in the bloom is contained in $k\\textnormal{-}1$ butterflies. \nThe \\texttt{BE}-\\texttt{Index}\\xspace compresses butterflies into blooms and keeps track of the edges they contain. \nThe space complexity of \\texttt{BE}-\\texttt{Index}\\xspace and the time complexity to build it are both $O( \\sum_{e=(u,v) \\in E(G)} min(deg(u),deg(v))$ \\cite{wang2020efficient}. \nHereafter we also use $T_{BE}$ to represent $\\sum_{e=(u,v) \\in E(G)} min(deg(u),deg(v))$.\nDeleting an edge $e$ based on \\texttt{BE}-\\texttt{Index}\\xspace takes $O(sup(e))$ time, where $sup(e)$ is the number of butterflies containing $e$. \nThis is because when $e$ is deleted, \\texttt{BE}-\\texttt{Index}\\xspace fetches the associated blooms and updates the support number of the affected edges in these blooms. In total, there are $O(sup(e))$ affected edges if edge $e$ is deleted. \nHere we evaluate the \\texttt{BE}-\\texttt{Index}\\xspace's impact on the overall time and space complexity of Algorithm \\ref{algo:naivedecomp}. \n\\begin{lemma}\n\\label{thm:bloom}\nBy adopting the \\texttt{BE}-\\texttt{Index}\\xspace for edge deletions, \nthe time complexity of Algorithm \\ref{algo:naivedecomp} becomes \n$O(T_{BE})$+$O(\\sum_{\\alpha =1 }^{\\alpha_{max}} \\sum_{\\beta=1}^{\\beta_{max}(\\alpha)} \\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(\\alpha,\\beta)_{1}\\textnormal{-}core}$), \nwhere $\\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(\\alpha,\\beta)_{1}\\textnormal{-}core}$ is the number of butterflies in $(\\alpha,\\beta)_{1}$-core\\xspace.\n\\end{lemma}\n\\begin{proof}\nIn the innermost loop of Algorithm \\ref{algo:decomp_query}, we remove the edges from $(\\alpha,\\beta)_{1}$-core\\xspace to get $(\\alpha,\\beta)_{\\tau}$-core\\xspace. \nAs each edge deletion operation takes $sup(e)$ \\cite{wang2020efficient}, it takes $\\sum_{e \\in E((\\alpha,\\beta)_{1}\\textnormal{-}core)}$ = $O( \\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(\\alpha,\\beta)_{1}\\textnormal{-}core} )$ to compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace from $(\\alpha,\\beta)_{1}$-core\\xspace. \nAs we are doing this for all possible $\\alpha$ and $\\beta$, the time complexity within the while-loops becomes $O(\\sum_{\\alpha =1 }^{\\alpha_{max}} \\sum_{\\beta=1}^{\\beta_{max}(\\alpha)} \\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(\\alpha,\\beta)_{1}\\textnormal{-}core}$). \nAdding the \\texttt{BE}-\\texttt{Index}\\xspace construction time to it completes the proof. \n\\end{proof}\n\\section{A Learning-based Hybrid Computation Paradigm}\nAlthough the index $I_{\\alpha,\\beta,\\tau}$ supports the optimal retrieval of the vertices in the queried $(\\alpha,\\beta)_{\\tau}$-core\\xspace, it does not scale well to large graphs due to its long build time and large space complexity even with the related optimizations.\nFor instance, on datasets \\texttt{Team}, \\texttt{Wiki-en}, \\texttt{Amazon}, and \\texttt{DBLP},  the index $I_{\\alpha,\\beta,\\tau}$ cannot be built within two hours as evaluated in our experiments.\nIn this section, we present \\gratings that selectively store the vertices of $(\\alpha,\\beta)_{\\tau}$-core\\xspace for some combinations of $\\alpha,\\beta$, and $\\tau$.\nWe also train a feed-forward neural network on a small portion of the queries to predict the choice of \\grating that minimizes the running time for each new incoming query. \n\n\\begin{table*}[tbh]\n\\centering\n\\scalebox{0.85}{\n\\begin{tabular}{c|c|c|c} \n\\noalign{\\hrule height 1pt}\nIndex   & Space Complexity & Build Time  & Query Time  \\\\\n\\noalign{\\hrule height 0.56pt}\n$I_{\\alpha,\\beta,\\tau}$ & $O(\\sum_{i=1}^{\\alpha_{max}}\\sum_{j=1}^{\\beta_{max}(i)}(\\tau_{max}(i,j)+n))$ & $O(T_{BE}\\textnormal{+}\\sum_{\\alpha =1 }^{\\alpha_{max}} \\sum_{\\beta=1}^{\\beta_{max}(\\alpha)}\\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(\\alpha,\\beta)_{1}\\textnormal{-}core})$ &  $O(\\sum_{v \\in V((\\alpha,\\beta)_{\\tau}\\textnormal{-}core)}(deg(v)))$ \\\\\n$I_{\\alpha,\\beta}$ & $O(m)$ & $O(\\delta \\cdot m )$ & $O(T_{peel}((\\alpha,\\beta)\\textnormal{-}core))$     \\\\ \n$I_{\\beta,\\tau}$ & $O( \\sum_{j=1}^{\\beta_{max}}(\\tau_{max}(1,j)+n))$ & $O(T_{BE}\\textnormal{+}\\sum_{\\beta=1}^{\\beta_{max}} \\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(1,\\beta)_{1}\\textnormal{-}core})$ & $O(T_{peel}((1,\\beta)_{\\tau}\\textnormal{-}core)$ \\\\\n$I_{\\alpha,\\tau}$ & $O(\\sum_{i=1}^{\\alpha_{max}}(\\tau_{max}(i,1)+n))$ &  $O(T_{BE}\\textnormal{+}\\sum_{\\alpha =1 }^{\\alpha_{max}} \\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(\\alpha,1)_{1}\\textnormal{-}core})$ & $O(T_{peel}((\\alpha,1)_{\\tau}\\textnormal{-}core)$ \\\\ \n\\noalign{\\hrule height 1pt}\n\\end{tabular}\n}\n\\caption{Space complexity, index construction time and query processing time of different indexes}\n\\label{tab:indexCompare}\n\\end{table*} \n\n\\noindent\n{\\bf \\gratings.} \nWe introduce three \\gratings $I_{\\alpha,\\beta}$, $I_{\\beta,\\tau}$, and $I_{\\alpha,\\tau}$ in this part. \nEach of them is a three-level index with two levels of pointers and one level of vertex blocks. The main structures of them are presented as follows. \n\n\n\\noindent\n$\\bullet$ $I_{\\beta,\\tau}$.\nFor all $\\beta$, $\\tau$, $I_{\\beta,\\tau}[\\beta][\\tau]$ points to the vertices $u\\in V(G)$ s.t. $\\tau_{max}(1,\\beta,u) = \\tau$.\nIt is the component of $I_{\\alpha,\\beta,\\tau}$ where $\\alpha$=$1$, which can fetch the vertices of all subgraphs of the form $(1,\\beta)_{\\tau}$-core\\xspace in optimal time. \n\n\\noindent\n$\\bullet$ $I_{\\alpha,\\tau}$.\nFor all $\\alpha$, $\\tau$, $I_{\\alpha,\\tau}[\\alpha][\\tau]$ points to the  vertices $u\\in V(G)$ s.t. $\\tau_{max}(\\alpha,1,u) = \\tau$.\nIt is the component of $I_{\\alpha,\\beta,\\tau}$ where $\\beta$=$1$, which can fetch the vertices of all subgraphs of the form $(\\alpha,1)_{\\tau}$-core\\xspace in optimal time. \n\n\\noindent\n$\\bullet$ $I_{\\alpha,\\beta}$ consists of $I_{\\alpha,\\beta} U$ and $I_{\\alpha,\\beta} V$ to store the vertices in $U(G)$ and $L(G)$ separately.\n$I_{\\alpha,\\beta} U[\\alpha][\\beta]$ points to the vertices $u\\in U(G)$ s.t. \n$\\beta = \\max\\{\\beta' | u \\in (\\alpha,\\beta')\\textnormal{-}core \\} $ and $I_{\\alpha,\\beta} V[\\beta][\\alpha]$ points to the vertices $v\\in L(G)$ such that $\\alpha = \\max\\{\\alpha' | v \\in (\\alpha',\\beta)\\textnormal{-}core\\} $. \n$I_{\\alpha,\\beta}$ can fetch the vertices of any $(\\alpha,\\beta)$-core\\xspace in optimal time. \n\nNote that, $I_{\\alpha,\\beta}$ is proposed to support efficient $(\\alpha,\\beta)$-core\\xspace computation as introduced in \\cite{liu2020efficient} while $I_{\\beta,\\tau}$ and $I_{\\alpha,\\tau}$ are essentially parts of $I_{\\alpha,\\beta,\\tau}$. For each type of \\gratings, we analyze its construction time, space complexity, and the query time to compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace based on it.\n\\begin{lemma}\nThe time complexity to build $I_{\\beta,\\tau}$ is $O(T_{BE}$+$\\sum_{\\beta=1}^{\\beta_{max}} \\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(1,\\beta)_{1}\\textnormal{-}core})$\nand the space complexity of $I_{\\beta,\\tau}$ is $O( \\sum_{j=1}^{\\beta_{max}}(\\tau_{max}(1,j)+n))$. \nIt takes $T_{peel}((1,\\beta)_{\\tau}\\textnormal{-}core)$ time to compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace using $I_{\\beta,\\tau}$.\n\\end{lemma}\n\\begin{proof}\nAs discussed in Lemma \\ref{thm:bloom}, the \\texttt{BE}-\\texttt{Index}\\xspace can significantly speed up butterfly enumeration during edge deletions, which takes $O(T_{BE})$ to construct. \nThen, we fix $\\alpha$ to one and run lines 6-16 of Algorithm \\ref{algo:naivedecomp} to compute all $(1,\\beta)_{\\tau}$-core\\xspace. \nFor each possible $\\beta$, this process takes $O(\\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(1,\\beta)_{1}\\textnormal{-}core})$ time, so in total it takes $O(\\sum_{\\beta=1}^{\\beta_{max}} \\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(1,\\beta)_{1}\\textnormal{-}core}))$ time. Adding it to the \\texttt{BE}-\\texttt{Index}\\xspace construction time ($O(T_{BE})$) gives the time complexity of $I_{\\beta,\\tau}$ construction. \n\nAs $I_{\\beta,\\tau}$ is the part of $I_{\\alpha,\\beta,\\tau}$ with $\\alpha$ = $1$, its space is equal to the part of $I_{\\alpha,\\beta,\\tau}$ that is pointed by $I_{\\alpha,\\beta,\\tau}[1]$. The size of the arrays of pointers in $I_{\\beta,\\tau}$ is bounded by $O(\\sum_{j=1}^{\\beta_{max}}(\\tau_{max}(1,j))))$. The size of the vertex blocks is bounded by the number of vertices in all $(1,\\beta)_{\\tau}$-core\\xspace, which is $O(\\sum_{j=1}^{\\beta_{max}}(|V((1,\\beta)_{\\tau}\\textnormal{-}core)| )) = O(\\sum_{j=1}^{\\beta_{max}} n)$. Therefore, the space complexity of $I_{\\beta,\\tau}$ is $O( \\sum_{j=1}^{\\beta_{max}}(\\tau_{max}(1,j)+n))$. \n\\end{proof}\n\n\\noindent\nIn order to query $(\\alpha,\\beta)_{\\tau}$-core\\xspace based on $I_{\\beta,\\tau}$, we first find the $(1,\\beta)_{\\tau}$-core\\xspace from $I_{\\beta,\\tau}$ and then compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace from $(1,\\beta)_{\\tau}$-core\\xspace. \n\\begin{lemma}\nThe query time of $(\\alpha,\\beta)_{\\tau}$-core\\xspace based on $I_{\\beta,\\tau}$ is $T_{peel}((1,\\beta)_{\\tau}\\textnormal{-}core)$. \n\\end{lemma}\n\\begin{proof}\nGiven engagement constraints $\\alpha,\\beta$ and strength level $\\tau$, let $G'$ be the $(1,\\beta)_{\\tau}$-core\\xspace on bipartite graph $G$. \nFirst, it takes $O(|V(G')|$ time to fetch the vertices in $(1,\\beta)_{\\tau}$-core\\xspace from $I_{\\beta,\\tau}$. \nRestoring the edges of $(1,\\beta)_{\\tau}$-core\\xspace from $G$ takes $O(\\sum_{u\\in V(G')} deg(u,G'))$ time. \nThen, we call the \\compute algorithm on $(1,\\beta)_{\\tau}$-core\\xspace to compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace, which takes $O(T_{peel}((1,\\beta)_{\\tau}\\textnormal{-}core))$ time.\n\\end{proof}\n\\begin{example}\nIn Figure \\ref{fig.index}, the component of $I_{\\alpha,\\beta,\\tau}$ wrapped in dotted line is the $I_{\\beta,\\tau}$ of the graph in Figure \\ref{fig:moltivation}. If $(2,2)_2$-core is queried, we first to obtain $(1,2)_2$-core from $I_{\\beta,\\tau}$ and compute $(2,2)_2$-core by calling the peeling algorithm.\n\\end{example}\n\nNote that index $I_{\\alpha,\\tau}$ is symmetric to index $I_{\\beta,\\tau}$, with $\\alpha$ and $\\beta$ switched.  \nIt is immediate that \nit takes $O(T_{BE}$+$\\sum_{\\alpha=1}^{\\alpha_{max}} \\,\\mathbin{\\resizebox{0.15in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_{(\\alpha,1)_{1}\\textnormal{-}core})$ time to construct $I_{\\alpha,\\tau}$ and its space complexity is $O( \\sum_{i=1}^{\\alpha_{max}}(\\tau_{max}(i,1)+n))$. \nIt takes $T_{peel}((\\alpha,1)_{\\tau}\\textnormal{-}core)$ time to compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace using $I_{\\alpha,\\tau}$.\nAs for $I_{\\alpha,\\beta}$, it takes $O(\\delta \\cdot m)$ time to construct and its space complexity is $O(m)$, where $\\delta$ is the degeneracy of the graph \\cite{liu2020efficient}. \nTo compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace using $I_{\\alpha,\\beta}$, we fetch the vertices of $(\\alpha,\\beta)$-core\\xspace and restore the edges in $O(|V(G')|+O(\\sum_{u\\in V(G')} deg(u,G'))$ time ($G'$ = $(\\alpha,\\beta)$-core\\xspace ). \nThen, we call the peeling algorithm on $(\\alpha,\\beta)$-core\\xspace to compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace, which takes $O(T_{peel}((\\alpha,\\beta)\\textnormal{-}core))$ time. \n\nThe sizes of \\gratings can be considered as the projections of $I_{\\alpha,\\beta,\\tau}$ onto 3 planes, as depicted in Figure \\ref{fig.cube}. We also summarize the space complexity, build time, and query time of \\gratings in Table \\ref{tab:indexCompare}. \n\n\\begin{figure}[htb]\n\\centering\n\\scalebox{0.75}{\n\\includegraphics[width=0.7\\textwidth]{mol_classification.pdf}\n}\n\\caption{Motivation example for learning-based query processing (\\texttt{DBpedia-Team})}\n\\label{fig:learning}\n\\end{figure}\n\n\\noindent\n{\\bf Learning-based hybrid query processing.}\nAs $I_{\\alpha,\\beta},I_{\\beta,\\tau}$, and $I_{\\alpha,\\tau}$ do not store all the decomposition results like $I_{\\alpha,\\beta,\\tau}$, the construction of these indexes is more time and space-efficient than $I_{\\alpha,\\beta,\\tau}$. \nHowever, the reduced index computation inevitably compromises the query processing performance.\nThis is because the \\gratings only store the vertices in $(\\alpha,\\beta)$-core\\xspace, $(1,\\beta)_{\\tau}$-core\\xspace, and $(\\alpha,1)_{\\tau}$-core\\xspace. \nClearly, computing the $(\\alpha,\\beta)_{\\tau}$-core\\xspace based on these \\gratings results in different response time.\nTo better illustrate this point, we plot all parameter combinations on dataset \\texttt{DBpedia-team} and give each combination a color based on which query processing algorithm performs the best in Figure \\ref{fig:learning}.\nFor ease of presentation, we denote the query processing algorithms based on index $I_{\\alpha,\\beta}$, $I_{\\beta,\\tau}$, and $I_{\\alpha,\\tau}$ as $Q_{\\alpha,\\beta}$, $Q_{\\beta,\\tau}$ and $Q_{\\alpha,\\tau}$ respectively.\nThe red points indicate that $Q_{\\beta,\\tau}$ is the fastest among the three. \nThe green ones represent the win cases for $Q_{\\alpha,\\tau}$ and the black points are the cases when $Q_{\\alpha,\\beta}$ is the best.\nEvidently, the points of different colors are mingled together and distributed across the parameter space.\nThis suggests that finding simple rules to partition the parameter space is not promising in deciding which of ${Q_{\\alpha,\\beta}, Q_{\\beta,\\tau}, Q_{\\alpha,\\tau}}$ is the fastest. \nHence, we formulate it as a classification problem and resort to machine learning techniques to solve this problem. \n\\begin{algorithm}[thb]\n    \\LinesNumbered\n    \t\\caption{Hybrid Computation Algorithm}\n\t\\label{algo:hyrid}\n\t\\tcp{\\textbf{Offline training:}} \n\t\\setcounter{AlgoLine}{0} \n\t\\KwIn{ $G:$ Input bipartite graph} \n\t\\KwOut{Neural network $C:D \\to \\{Q_{\\alpha,\\beta},Q_{\\beta,\\tau},Q_{\\alpha,\\tau}\\}$ } \n    Build $I_{\\beta,\\tau},I_{\\alpha,\\tau}$ and $I_{\\alpha,\\beta}$ for $G$\\\\\n    \\ForEach{$q\\in \\{\\textnormal{$N$ random queries}\\} $ on $G$}{\n    $feature(q)\\gets[\\alpha,\\beta,\\tau\\textnormal{ of $q$}]$ \\\\ \n    $label(q)\\gets$ the fastest algorithm in $\\{Q_{\\alpha,\\beta}, Q_{\\beta,\\tau}, Q_{\\alpha,\\tau} \\}$\\\\\n    }\n      $X=[feature(q)]$, $q\\in$ $N$ queries run on $G$ \\\\\n      $Y=[label(q)]$ , $q\\in$ $N$ queries run on $G$ \\\\\n   \n      $C \\gets$ trained neural network on $X,Y$ \\\\\n\t\\tcp{\\textbf{Online query processing:}} \n\t\\KwIn{Query parameters: $\\alpha,\\beta,\\tau$}\n\t\\KwOut{$(\\alpha,\\beta)_{\\tau}$-core\\xspace in $G$} \n\t\\setcounter{AlgoLine}{0}\n\t$Q_{pred}\\gets$ $C.predict(\\alpha,\\beta,\\tau)$ \\\\\n    Run $Q_{pred}$ to compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace \\\\\n\\textbf{return } $(\\alpha,\\beta)_{\\tau}$-core\\xspace  \\\\\n\\end{algorithm}\n\nWe introduce a hybrid computation algorithm (Algorithm \\ref{algo:hyrid}, denoted by $Q_{hb}$), which selects from $\\{ Q_{\\alpha,\\beta},Q_{\\beta,\\tau},Q_{\\alpha,\\tau} \\}$ based on the query parameters $\\alpha,\\beta$, and $\\tau$.\nIn the offline training phase, we build $I_{\\beta,\\tau},I_{\\alpha,\\tau}$ and $I_{\\alpha,\\beta}$ on $G$ and obtain the runtime of $Q_{\\alpha,\\beta},Q_{\\beta,\\tau}$, $Q_{\\alpha,\\tau}$ on $N$ queries, \nwhere $N$ is chosen to be less than $5\\%$ of all possible queries. \nThe label of a query is the algorithm that responds to it in the shortest time.\nThen, we train a feed-forward neural network $C$ on the $N$ labeled query instances. \nIn the online query processing phase, given a new query of $(\\alpha,\\beta)_{\\tau}$-core\\xspace, the trained neural network makes a prediction based on $\\alpha,\\beta \\textnormal{, and }\\tau$. Then, we use the predicted query processing algorithm to compute $(\\alpha,\\beta)_{\\tau}$-core\\xspace.\n\nHere we detail how to train the feed-forward neural network.\nWe impose only one hidden layer in $C$ to avoid over-fitting. \nThe important hyper-parameters of $C$ include the number of hidden units $H$ and the type of optimizer.\nWe use $5$-fold cross-validation to evaluate the above hyper-parameters. \nSpecifically, we split the $N$ labeled queries into $5$ partitions and each time we take one partition as the validation set and the remainder as the training set. \nFor each parameter setting, we build a classifier on the training sets for $5$ times and calculate a performance metric on the validation set. \nIn our model, we define a \\textit{time-sensitive error} on the validation set as the performance metric, which calculates a weighted mis-classification cost w.r.t the actual query time.\nLet $i_k, j_k \\in \\{ 1,2,3 \\}$ (encoding of $Q_{\\alpha,\\beta},Q_{\\beta,\\tau},Q_{\\alpha,\\tau}$) be the predicted class and the actual class of the $k_{th}$ instance. \nThe time-sensitive error is defined as \n$$ error(i_k,j_k) \\textnormal{=}  e_{i_k}^T \n\\begin{bmatrix}\n0 & t_{1,k}-t_{2,k} & t_{1,k}-t_{3,k}\\\\\nt_{2,k}-t_{1,k} & 0 & t_{2,k}-t_{3,k}\\\\\nt_{3,k}-t_{1,k} & t_{3,k}-t_{2,k} & 0\\\\\n\\end{bmatrix}\ne_{j_k} $$\nwhere $e_{i_k}$ and $e_{j_k}$ are one-hot vectors of length $3$ with the $i_k$, $j_k$ position being $1$. \n$t_{1,k}$, $t_{2,k}$ and $t_{3,k}$ are the running time of $Q_{\\alpha,\\beta},Q_{\\beta,\\tau},Q_{\\alpha,\\tau}$ on the $k_{th}$ instance respectively. \nThe time-sensitive error measures the gap between the predicted query algorithm and the optimal query algorithm.\nIt is averaged over all instances in the validation set and across $5$ iterations of cross-validation.\nThen, the hyper-parameter setting with the lowest time-sensitive error should be chosen.  \nIn this way, we are more prone to find the parameter settings that allow us to minimize the query time instead of merely correctly classify each instance. \n\nNote that, training a feed-forward neural network (lines 2 - 7) take significantly less time compared to the \\grating construction process (line 1) since only $N$ ($N \\leq 5\\%$ of the total number of possible queries) random queries are used. \n\n\\begin{example}\nOn dataset \\texttt{DBpedia-starring}, given $\\alpha$=$2$, $\\beta$=8, and $\\tau$=$9$.\n$Q_{\\alpha,\\beta}$ takes $0.53$ seconds to find the queried subgraph.\n$Q_{\\beta,\\tau}$ and $Q_{\\alpha,\\tau}$ takes $0.03$ and $0.05$ respectively. The optimal query processing algorithm on this instance is $Q_{\\beta,\\tau}$. \nAccuracy as a performance metric would give equal penalty to mis-classifying $Q_{\\alpha,\\beta}$ and $Q_{\\alpha,\\tau}$ as the best algorithm, which is clearly inappropriate. \nInstead, the time-sensitive error gives penalty of $0.02$ if we predict $Q_{\\beta,\\tau}$ and $0.51$ if we predict $Q_{\\alpha,\\tau}$. \n\\end{example}\n\\section{Experiments}\n\nIn this section, we first validate the effectiveness of the $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace model. \nThen, we evaluate the performance of the index construction algorithms as well as the query processing algorithms.\n\n\\subsection{Experiments setting}\n\n\\noindent\n{\\bf Algorithms.} Our  empirical  studies  are  conducted  against  the following algorithms:\n\n\\noindent\n{\\em $\\bullet$ Index construction algorithms.}\nWe compare two $I_{\\alpha,\\beta,\\tau}$ construction algorithms: the naive decomposition algorithm \\texttt{decomp-naive}\\xspace and the decomposition algorithm with optimizations \\texttt{decomp-opt}\\xspace. We also evaluate the index construction algorithms of $I_{\\beta,\\tau}$ and $I_{\\alpha,\\tau}$.\nAs for $I_{\\alpha,\\beta}$, we report its size and build time by running the index construction algorithm in \\cite{liu2020efficient}. \n\n\\noindent\n{\\em $\\bullet$ Query processing algorithms.}\nWe use the online computation algorithm presented in Section 4 as the baseline method, denoted as $Q_{bs}$.\nWe compare it to the index-based query processing algorithms $Q_{\\alpha,\\beta,\\tau}, Q_{\\alpha,\\beta}, Q_{\\beta,\\tau}$, and $Q_{\\alpha,\\tau}$, which are based on $I_{\\alpha,\\beta,\\tau},I_{\\alpha,\\beta},I_{\\beta,\\tau}$, and $I_{\\alpha,\\tau}$ respectively. \nWe also evaluate the hybrid computation algorithm $Q_{hb}$, which depends on a well-trained classifier and the indexes $Q_{\\alpha,\\beta}, Q_{\\beta,\\tau}$, and $Q_{\\alpha,\\tau}$. \n\nAll algorithms are implemented in C++ and the experiments are run on a Linux server with Intel Xeon E3-1231 processors and $16$GB main memory. \\textit{We end an algorithm if the running time exceeds two hours.}\n\n\\begin{table*}[tbh]\n\\centering\n\\scalebox{1.0}{\n\\begin{tabular}{cccc|ccccccc}\n\\noalign{\\hrule height 1.23pt}\nDataset &  $|E|$ &  $|U|$ & $|L|$  & $\\alpha_{max}$  & $\\beta_{max}$  & $\\tau_{max}$ & $\\delta$ \\\\ \n\\noalign{\\hrule height 0.7pt}\nCond-mat (AC)  & $58$K & $38$K & $16$K & $37$ & $13$ & $63$ & $8$ \\\\\nWriters  (WR) & $144$K & $135$K & $89$K & $11$ & $82$ & $99$ & $6$ \\\\\nProducers (PR) & $207$K & $187$K & $48$K & $220$ & $18$ & $219$ & $6$ \\\\\nMovies (ST)  & $281$K & $157$K & $76$K  & $19$ & $215$ & $222$   & $7$ \\\\\nLocation (LO)  & $294$K & $225$K & $172$K & $12$ & $853$ & $852$  & $8$ \\\\\nBookCrossing (BX) & $434$K & $264$K & $78$K & $376$ & $100$ & $375$   & $13$ \\\\\nTeams (TM) & $1.4$M & $935$K & $901$K & $11$ & $1063$ & $373$   & $9$ \\\\\nWiki-en (WC) & $3.80$M & $2.04$M & $1.85$M & $39$ & $7659$ & $7658$  & $18$ \\\\\nAmazon (AZ) & $5.74$M & $3.38$M & $2.15$M & $659$ & $294$ & $658$   & $26$ \\\\\nDBLP (DB) & $8.6$M & $5.4$M & $1.4$M & $421$ & $64$ &  $420$  & $10$ \\\\\n\\noalign{\\hrule height 1.23pt}\n\\end{tabular}\n}\n\\vspace{-2mm}\n\\caption{This table reports the basic statistics of $10$ real graph datasets.}\n\\label{tab:datainfo}\n\\end{table*}\n\n\\noindent\n{\\bf Datasets.}\nWe use $10$ real graphs in our experiments, which are obtained from the website KONECT \\footnote{\\url{http:\/\/konect.uni-koblenz.de\/networks\/}}. \nTable \\ref{tab:datainfo} includes the statistics of these datasets, sorted by the number of edges in ascending order. \nThe abbreviations of dataset names are listed in parentheses.\n$|E|$ is the number of edges in the graph. \n$|U|$ and $|L|$ are the number of vertices in the upper and lower levels. \n$\\alpha_{max}$ is the largest $\\alpha$ such that $(\\alpha,1)_1$-core exists. \n$\\beta_{max}$ is the largest $\\beta$ such that $(1,\\beta)_1$-core exists. \n$\\tau_{max}$ is the largest $\\tau$ such that $(1,1)_{\\tau}$-core exists. \n\n\\subsection{Effectiveness Evaluation}\n\\begin{figure}[htb]\n\\centering\n\\scalebox{0.8}{\n\\includegraphics[width=0.46\\textwidth]{bi_coefficient.pdf}\n\\includegraphics[width=0.44\\textwidth]{density.pdf}\n}\n\\caption{The cohesive metrics comparisons}\n\\label{fig:metric}\n\\end{figure}\n\nIn this section, we validate the effectiveness of the $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace model. \nFirst, we compute some cohesive metrics for $(\\alpha,\\beta)$-core\\xspace and $(\\alpha,\\beta)_{\\tau}$-core\\xspace. \nThen, we conduct a case study on dataset \\texttt{DBLP-2019}. \n\n\\noindent\n{\\bf Compare $(\\alpha,\\beta)$-core\\xspace with $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace.}\nWe compare the graph density and bipartite clustering coefficient for $(\\alpha,\\beta)$-core\\xspace and $(\\alpha,\\beta)_{\\tau}$-core\\xspace. \nThe graph density \\cite{sariyuce2018peeling} of a bipartite graph is calculated as $|E|\/(|U|\\times|L|)$, where $|E|$ is the number of edges and $|U|$ and $|L|$ are the number of upper and lower vertices. \nThe bipartite clustering coefficient \\cite{aksoy2017measuring} is a cohesive measurement of bipartite networks, which is calculated as $4 \\times \\,\\mathbin{\\resizebox{0.1in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_G$\/$\\mathbin{\\resizebox{0.098in}{!}{\\rotatebox[origin=c]{270}{$\\ltimes$}}}_G$ where $\\mathbin{\\resizebox{0.098in}{!}{\\rotatebox[origin=c]{270}{$\\ltimes$}}}_G$ and $\\,\\mathbin{\\resizebox{0.1in}{!}{\\rotatebox[origin=c]{90}{$\\Join$}}}_G$ are the number of caterpillars (three-path) and the number of butterflies in graph $G$ respectively. \nIn Figure \\ref{fig:metric}, the black bars with $\\tau$=$0$, represents the $(\\alpha,\\beta)$-core\\xspace. \nThe shaded bars and the white bars represent the $(\\alpha,\\beta)_{\\tau}$-core\\xspace with $\\tau$ being $50$ and $100$ respectively. \nThe engagement constraints $\\alpha$ and $\\beta$ are set to $0.6 \\delta$ and $0.4 \\delta$ respectively, where $\\delta$ is the graph degeneracy. As we can see, on all four datasets, the $(\\alpha,\\beta)_{\\tau}$-core\\xspace has a higher density and bipartite coefficient than $(\\alpha,\\beta)$-core\\xspace. As $\\tau$ increases, both of the metrics increase as well. \nThis means that with higher values of $\\tau$, we can find subgraphs within $(\\alpha,\\beta)$-core\\xspace of higher density and cohesiveness. \n\n\\begin{figure}[h!]\n\\centering  \n\\includegraphics[width=0.7\\textwidth]{case_study.png}\n\\vspace{-2mm}\n\\caption{Case study on DBLP-2019}\n\\label{fig:case}\n\\end{figure}\n\n\\noindent\n{\\bf Case study.}\nThe effectiveness of our model is evaluated through a case study on the DBLP-2019 dataset. The graph in Fig \\ref{fig:case} is an $(\\alpha,\\beta)$-core\\xspace ($\\alpha\\textnormal{=}7,\\beta\\textnormal{=}8$). Given $\\tau$=$50$, $(\\alpha,\\beta)_{\\tau}$-core\\xspace excludes the relatively sparse group represented by the light blue lines. The \\textit{k}-bitruss ($k$=$56$) represented by the red lines is in $(\\alpha,\\beta)_{\\tau}$-core\\xspace. The black lines are the edges included in $(\\alpha,\\beta)_{\\tau}$-core\\xspace but not in \\textit{k}-bitruss. The $(\\alpha,\\beta)_{\\tau}$-core\\xspace and \\textit{k}-bitruss involve the same authors, but \\textit{k}-bitruss removes the second last paper on the upper level to enforce the tie strength constraint.\nFigure \\ref{fig:case} implies that:\n(1) Although $(\\alpha,\\beta)$-core\\xspace models vertex engagement via degrees, it fails to distinguish between edges with different tie strength. \n(2) \\textit{k}-bitruss models tie strength via butterfly counting, but it forcefully excludes the weak ties between strongly engaged nodes, which leads to the imprecise estimation of tie strength and failure to include important nodes and their incident edges. \n(3) $(\\alpha,\\beta)_{\\tau}$-core\\xspace considers both vertex engagement and tie strength. Its flexibility allows it to capture unique structures that better resemble the communities in reality. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{index_build.pdf}\n\\caption{The $I_{\\alpha,\\beta,\\tau}$ construction time}\n\\label{fig:index.compare}\n\\end{figure}\n\\begin{table*}[tbh]\n\\centering\n\\scalebox{1.0}{\n\\begin{tabular}{c|cccc|cccc}\n\\noalign{\\hrule height 1.23pt}\n\\multirow{2}*{Data} & \\multicolumn{4}{c|}{Index size (MB)} & \\multicolumn{4}{c}{Index construction time (sec)} \\\\\n\\cline{2-9} \n~ & $I_{\\alpha,\\beta,\\tau}$ & $I_{\\alpha,\\beta}$ &  $I_{\\beta,\\tau}$ & $I_{\\alpha,\\tau}$ & $I_{\\alpha,\\beta,\\tau}$ & $I_{\\alpha,\\beta}$ &  $I_{\\beta,\\tau}$ & $I_{\\alpha,\\tau}$  \\\\\n\\noalign{\\hrule height 0.7pt}\nCond-mat  & $1.29$ & $0.78$ & $0.26$ & $0.50$ & $5.11$ & $0.11$ & $0.68$ & $1.37$ \\\\ \nWriters   & $2.14$ & $2.24$ & $0.93$ & $0.24$ & $18.81$ & $0.24$ & $5.01$ & $1.31$ \\\\\nProducers & $5.55$ & $3.16$ & $0.37$ & $2.43$ & $79.43$ & $0.38$ & $4.99$ & $28.37$ \\\\\nMovies    & $5.26$ & $3.51$ & $2.01$ & $0.53$ & $66.18$ & $0.46$ & $34.29$ & $6.26$ \\\\ \nLocation  & $68.96$ & $4.15$ & $33.22$ & $0.75$ & $77.3978$ & $0.36$ & $49.5368$ & $9.19316$ \\\\\nBookCrossing & $33.56$ & $5.58$ & $3.07$ & $9.32$ & $342.728$ & $1.02$ & $48.8196$ & $75.2869$ \\\\ \nTeams      & $-$ & $18.42$ & $114.17$ & $2.44$ & time out & $1.94$ & $944.102$ & $127.265$ \\\\ \nWiki-en    & $-$ & $46.66$ & $945.91$ & $10.92$ & time out & $9.079$ & $3850.51$  & $680.569$ \\\\ \nAmazon      & $-$ & $72.96$ & $74.47$ & $129.21$ & time out & $17.184$  & $3598.13$  & $4731.29$ \\\\\nDBLP        & $-$ & $112.60$ & $29.13$ & $159.74$ & time out & $19.44$ & $559.76$ & $3000.08$ \\\\\n\\noalign{\\hrule height 1.23pt}\n\\end{tabular}\n}\n\\vspace{-2mm}\n\\caption{Evaluate the size of indexes and their build time.}\n\\label{tab:index_size}\n\\end{table*}\n\n\\subsection{Performance Evaluation}\nIn this part, we evaluate the efficiency of the index construction algorithms and explore the appropriate hyperparameter settings for the feed-forward neural network that $Q_{hb}$ depends on.\nThen, we evaluate the efficiency of the query processing algorithms to retrieve $(\\alpha,\\beta)_{\\tau}$-core\\xspace. \n\n\\noindent\n{\\bf Index construction.} \nFirst, We compare the build time of $I_{\\alpha,\\beta,\\tau},I_{\\alpha,\\beta},I_{\\beta,\\tau}$ and $I_{\\alpha,\\tau}$ on all datasets, as reported in Table \\ref{tab:index_size}.\nThe reported build time corresponds to the index construction algorithms with the optimization techniques in Section $6$. \nAs shown in \\ref{fig:index.compare}, although the computation-sharing and the \\texttt{Bloom-Edge}-index based optimizations effectively reduce the running time, the $I_{\\alpha,\\beta,\\tau}$ still cannot be built within time limit on \\texttt{Teams}, \\texttt{Wiki-en}, \\texttt{Amazon} or \\texttt{DBLP}. \nThis is because $I_{\\alpha,\\beta,\\tau}$ stores all the decomposition results and it takes the longest time to build, followed by $I_{\\beta,\\tau},I_{\\alpha,\\tau}$, and $I_{\\alpha,\\beta}$.\nWhen the graph is denser on the upper level, $I_{\\alpha,\\tau}$  takes longer to construct than $I_{\\beta,\\tau}$.\nFor example, in \\texttt{DBLP}, where $d_{max}(L)\\xspace\\textnormal{=}119 < d_{max}(U)\\xspace\\textnormal{=}951$, $I_{\\beta,\\tau}$ is built within $10$ minutes while $I_{\\alpha,\\tau}$ is built in $50$ minutes. \n$I_{\\alpha,\\beta}$ construction is the fastest as it does not involve any butterfly counting or updating of support. \nIn addition, we report the index sizes in Table \\ref{tab:index_size}. \nThe size of $I_{\\alpha,\\beta,\\tau}$ is larger than $I_{\\alpha,\\beta},I_{\\beta,\\tau} \\textnormal{ and } I_{\\alpha,\\tau}$. \nIn summary, the \\gratings that the hybrid computation algorithm depends on are space-efficient and can be built within a reasonable time.\\\\\n\n\\begin{figure}[htb]\n\\centering\n\\scalebox{0.8}{\n\\includegraphics[width=0.46\\textwidth]{optimizer.pdf}\n\\includegraphics[width=0.46\\textwidth]{hid.pdf}\n}\n\\caption{Effects of hyperparameters}\n\\label{fig:tune}\n\\end{figure}\n\n\\noindent\n{\\bf Tuning hyperparameters for the neural network.}\nWhen training the neural network for the hybrid computation algorithm, we choose the hyperparameters (the type of optimizer and size of the hidden layer) that minimizes the time-sensitive error from cross-validation.\nFor each graph $G$, we set the size of hidden layer to $50$ and test the \\textit{stochastic gradient descent}, \\textit{L-BFGS method} and \\textit{Adam} and compare the time-sensitive error. \nAs shown in Figure  \\ref{fig:tune}, the L-BFGS method consistently \noutperforms the other methods and thus is chosen in our model. \nThen, we explore the effect of the size of the hidden layer on our model. \nWe report the change of time-sensitive error w.r.t varying hidden layer size on dataset \\texttt{DBpedia-location} and the trends are similar on other datasets.\nAs shown in the plot, $30$ hidden units are enough for the classifier built on most tested datasets and beyond this point, more hidden units have little effect on the performance of the model. \n\n\\begin{table*}[t!]\n\\centering\n\\scalebox{1.0}{\n\\begin{tabular}{ccccccc}\n\\noalign{\\hrule height 1.23pt}\nDataset  & $Q_{bs}$ & $Q_{\\alpha,\\beta,\\tau}$ & $Q_{\\alpha,\\beta}$ &  $Q_{\\beta,\\tau}$ & $Q_{\\alpha,\\tau}$ & $Q_{hb}$  \\\\ \n\\noalign{\\hrule height 0.7pt}\nCond-mat &  $0.139$ & $0.004$ & $0.030$ & $0.008$ & $0.009$ & $0.006$ \\\\\nWriters  & $0.406$ & $0.011$ & $0.069$ & $0.011$ & $0.009$ & $0.006$ \\\\\nProducers &  $0.560$ & $0.022$ & $0.200$ & $0.047$ & $0.037$ & $0.029$ \\\\\nMovies  & $0.871$ & $0.023$ & $0.235$ & $0.028$ & $0.041$ & $0.027$ \\\\\nLocation & $11.782$ & $0.285$ & $7.005$ & $1.755$ & $2.895$ & $0.234$\\\\\nBookCrossing & $44.397$ & $0.147$ & $13.271$ & $2.935$ & $4.794$ & $1.722$ \\\\\nTeams &  $59.510$ & $-$ & $10.911$ & $3.080$ & $2.778$ & $1.394$ \\\\\nWiki-en  & $128.536$ & $-$ & $28.589$ & $2.638$ & $13.613$ & $1.775$\\\\\nAmazon  & $973.026$ & $-$ & $153.085$ & $88.673$ & $59.228$ & $16.432$ \\\\\nDBLP & $61.101$ & $-$ &  $1.843$ & $1.321$ & $1.055$ & $0.269$ \\\\\n\\noalign{\\hrule height 1.23pt}\n\\end{tabular}\n}\n\\vspace{-2mm}\n\\caption{This table reports the average response time for all query processing algorithms.}\n\\label{tab:summary}\n\\end{table*}\n\\begin{figure}[h!]\n\\centering  \n\\includegraphics[trim= -40 0 0 0 ,width=0.8\\textwidth]{legen.pdf}\n\n\\subfigure[LO (Vary $\\alpha$)]{\n\\label{Fig.pr.a}\n\\includegraphics[width=0.3\\textwidth]{PR_a.pdf}}\n\\subfigure[LO (Vary $\\beta$)]{\n\\label{Fig.pr.b}\n\\includegraphics[width=0.3\\textwidth]{PR_b.pdf}}\n\\subfigure[LO (Vary $\\tau$)]{\n\\label{Fig.pr.g}\n\\includegraphics[width=0.3\\textwidth]{PR_tau.pdf}}\n\n\\subfigure[BX (Vary $\\alpha$)]{\n\\label{Fig.pa.a}\n\\includegraphics[width=0.3\\textwidth]{Pa_a.pdf}}\n\\subfigure[BX (Vary $\\beta$)]{\n\\label{Fig.pa.b}\n\\includegraphics[width=0.3\\textwidth]{Pa_b.pdf}}\n\\subfigure[BX (Vary $\\tau$)]{\n\\label{Fig.pa.g}\n\\includegraphics[width=0.3\\textwidth]{Pa_tau.pdf}}\n\\vspace{-2mm}\n\\caption{Effects of varying parameters on each query processing algorithm.}\n\\label{Fig.queries}\n\\end{figure}\n\n\\noindent\n{\\bf Average query time of $Q_{\\alpha,\\beta,\\tau}, Q_{\\alpha,\\beta}, Q_{\\beta,\\tau}, Q_{\\alpha,\\tau}$ and $Q_{hb}$.} \nFor each algorithm, we report the average response time of $50$ randomly generated queries on each dataset in Table \\ref{tab:summary}.\nAs expected, all index based algorithms outperform $Q_{bs}$, as $Q_{bs}$ computes $(\\alpha,\\beta)_{\\tau}$-core\\xspace from the input graph.\nAmong them, $Q_{\\alpha,\\beta,\\tau}$ performs the best on most datasets, as it fetches the vertices from $I_{\\alpha,\\beta,\\tau}$ in optimal time and then restores the edges. \nHowever, the long build time of $I_{\\alpha,\\beta,\\tau}$ makes $Q_{\\alpha,\\beta,\\tau}$ not scalable to larger graphs like \\texttt{Team},\\texttt{Wiki-en},\\texttt{Amazon} or \\texttt{DBLP}. \nThe performances of $Q_{\\alpha,\\beta},Q_{\\beta,\\tau}$, and $Q_{\\alpha,\\tau}$ differ a lot from each other and across datasets. \nOn average, $Q_{\\alpha,\\beta}$ is slower than $Q_{\\beta,\\tau}$ and $Q_{\\alpha,\\tau}$, because it needs to delete many edges from $(\\alpha,\\beta)$-core\\xspace to get $(\\alpha,\\beta)_{\\tau}$-core\\xspace especially when $\\tau$ is large. \nAs $Q_{hb}$ is trained to pick the fastest from $\\{ Q_{\\alpha,\\beta},Q_{\\beta,\\tau},Q_{\\alpha,\\tau} \\}$, it outperforms these $3$ algorithms on average in all datasets.\nIn addition, $Q_{hb}$ outperforms the online computation algorithm $Q_{bs}$ by up to two orders of magnitude.\n\n\\noindent\n{\\bf Evaluate the effect of $\\alpha,\\beta$ and $\\tau$.} \nWe investigate the effects of varying $\\alpha, \\beta, \\tau$ on each query processing algorithm. \nWe input three types of query streams for all algorithms, each of which increments one of $\\alpha$, $\\beta$, and $\\tau$ while letting the other two parameters be generated randomly. \nAs trends are similar, we only report the results on \\texttt{Location} and \\texttt{BookCrossing} in Figure \\ref{Fig.queries}.\nEach data point in the figure represents the average response time of $50$ random queries.\nAs expected, index-based query processing algorithms always outperform $Q_{bs}$.\nThe performance of $Q_{\\alpha,\\beta,\\tau}$ is not affected much by the varying parameters, while the \\grating based algorithms are highly sensitive to them. \nEach of $Q_{\\alpha,\\beta},Q_{\\beta,\\tau}$, and $Q_{\\alpha,\\tau}$ tends to perform better when the increased parameter results in a smaller subgraph in the index. \nFor instance, $Q_{\\beta,\\tau}$ performs better as $\\beta$ or $\\tau$ increases. \nIn contrast, the hybrid computation algorithm $Q_{hb}$ has very stable performance, as it stays close to and in many cases outperforms the fastest of $Q_{\\alpha,\\beta},Q_{\\beta,\\tau}$, and $Q_{\\alpha,\\tau}$. \nIn summary, the hybrid computation algorithm $Q_{hb}$ with a well-trained classifier can adjust its querying processing algorithm to different parameters and datasets.\n\n\\section{The Elsevier article class}\n\\section{Conclusion}\nIn this paper, we introduce a novel cohesive subgraph model, $\\tau$-strengthened $(\\alpha,\\beta)$-core\\xspace, which is the first to consider both tie strength and vertex engagement on bipartite graphs.\nWe propose a decomposition-based index $I_{\\alpha,\\beta,\\tau}$ that can retrieve vertices of any $(\\alpha,\\beta)_{\\tau}$-core\\xspace in optimal time. \nWe also apply computation sharing and \\texttt{BE}-\\texttt{Index}\\xspace-based optimizations to speed up the index construction process of $I_{\\alpha,\\beta,\\tau}$. \nTo balance space-efficient index construction and time-efficient query processing, we propose a learning-based hybrid computation paradigm.\nUnder this paradigm, we introduce three \\gratings and train a feed-forward neural network to predict which index is the best choice to process an incoming $(\\alpha,\\beta)_{\\tau}$-core\\xspace query. \nThe efficiency of the proposed algorithms and the effectiveness of our model are verified through extensive experiments. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Inclusive and diffractive DIS}\n\nThe intriguing phenomenon of the frequent appearance of large rapidity gaps\nin electron proton collisions at HERA \\cite{h1z} has changed our physical \npicture of deep inelastic scattering (DIS) to a large extent. The large \nrapidity gap events are very difficult to understand in the parton model \nwhere the struck quark is expected to break up the proton leading to a \ncontinuous flow of hadrons between the current jet and the proton remnant.\n\nTo develop a physical picture of diffractive DIS \\cite{heb} it is convenient \nto view the scattering process in the proton rest frame. In this frame the\nvirtual photon fluctuates into partonic states $q {\\bar q}$ , $q {\\bar q}g$ , ... which then\nscatter off the proton. From the leading twist contributions to inclusive and\ndiffractive structure functions one obtains the parton distribution functions\nin a frame where the proton moves fast. This connection holds for diffractive\nas well as non-diffractive processes.\n\nIn the following we shall review the present status of our theoretical\nunderstanding of diffractive DIS with emphasis on the close analogy to\ninclusive DIS. This is appropriate since diffractive DIS is dominated by\nthe leading twist contribution, which has been one of the most surprising \naspects of the large rapidity gap events. The scattering of the partonic\nfluctuations of the photon off the proton will be treated in the\nsemiclassical approach. After a comparison of theoretical predictions\nwith data we shall conclude with a discussion of some open questions.   \n \n\\subsection*{Inclusive DIS}\n\nInclusive deep inelastic scattering \\cite{esw} is characterized by the \nkinematic variables\n\\begin{equation}\nQ^2=-q^2\\;,\\quad W^2=(q+P)^2\\;, \\quad x={Q^2\\over Q^2 + W^2}\\;,\n\\end{equation}\nwhere $q$ and $P$ are the momenta of the virtual photon and the proton,\nrespectively. The cross section is determined by the hadronic tensor,\n\\begin{eqnarray}\nW_{\\mu\\nu}(P,q) &=& {1\\over 4\\pi} \\sum_{X} \n  \\langle P|J_\\nu(0)|X\\rangle  \n  \\langle X|J_\\mu(0)|P\\rangle (2\\pi)^4 \\delta(P-P_X)\\nonumber\\\\\n&=&\\left(-g_{\\mu\\nu}+{q_\\mu q_\\nu\\over q^2}\\right) F_1(x,Q^2)\n  +{1\\over \\nu}\\left(P_\\mu - {\\nu\\over q^2}q_\\mu\\right) \n   \\left(P_\\nu - {\\nu\\over q^2}q_\\nu\\right) F_2(x,Q^2)\\;.\n\\end{eqnarray}\nHere $J_\\mu(x)$ is the electromagnetic current, $\\nu=q\\cdot P$, and \nspin averaging has been implicitly assumed.\n\nThe structure functions are a sum of leading twist and of higher twist\ncontributions which are suppressed by powers of $Q^2$,\n\\begin{equation}\nF_i(x,Q^2) = F_i^{(LT)}(x,Q^2) + {F_i^{(HT)}(x,Q^2)\\over Q^2} + \\ldots\\;.\n\\end{equation}\nThe leading twist term is dominant for $Q^2$ above some value  \n$Q_0^2$, which is not very well known and frequently chosen to be \n${\\cal O}$(1 GeV$^2$). However, higher twist contributions are known to\nbe important for hadronic energies $W^2 \\le 4$~GeV$^2$ \\cite{mrst2}.\n\nThe structure functions $F_i^{(LT)}(x,Q^2)$ can be expressed in terms of\nprocess independent parton distribution functions,\n\\begin{equation}\nF_i^{(LT)}(x,Q^2) \\rightarrow f_i(x,\\mu^2) = q(x,\\mu^2),\\ g(x,\\mu^2)\\;,\n\\end{equation}\nwhich depend on $x$ and on the factorization scale $\\mu^2$. At small $x$, the\nquark distribution is assumed to be the same for all light flavours.\nThe parton distribution functions $f_i(x,\\mu^2)$ obey the perturbative\nQCD evolution equations \\cite{dglap},\n\\begin{equation}\n\\mu^2 {\\partial\\over \\partial\\mu^2} f_i(x,\\mu^2) =\n{\\alpha_s\\over 2\\pi} \\int_x^1{dy\\over y} P_{ij}\\left({x\\over y}\\right)\nf_i(y,\\mu^2)\\;,\n\\end{equation}\nwhere $P_{ij}(z)$ are the Altarelli-Parisi splitting functions. \nThe parton distributions can be directly expressed in terms of\nthe quark and gluon field operators. For instance, the quark distribution\nis given by\n\\begin{eqnarray}\\label{qdincl}\nq(x,\\mu^2) &=& {1\\over 4\\pi} \\int dx_- e^{-ixP_+x_-\/2}\\sum_{X}\\nonumber\\\\\n&&\\hspace{1.2cm}\n\\langle P|\\bar{q}(0,x_-,0_{\\perp})U(x_-,\\infty)|X\\rangle\\gamma_+\n\\langle X|U(\\infty,0)q(0,0,0_{\\perp})|P\\rangle \\;,\n\\end{eqnarray}\nwhere $U(a,b)$ is the colour matrix\n\\begin{equation}\nU(a,b) = P \\exp{\\left(-{i\\over 2}\\int_b^a dy_-A_+(0,y_-,0_{\\perp})\\right)}\\;.\n\\end{equation}\nThis definition can be used as a starting point of a theoretical\nnon-perturbative evaluation of the quark distribution.\n\n\\subsection*{Diffractive DIS}\n\nDiffractive DIS can be discussed in close analogy to inclusive DIS. There \nare two more kinematical variables which charaterize the diffractively \nscattered proton: the invariant momentum transfer $t$ and the fraction \n$\\xi$ of lost longitudinal momentum. A complete set\nof variables is\n\\begin{equation}\nt=(P-P')^2\\;,\\quad \\xi\\equiv x_{I\\!\\!P} \\;,\\quad Q^2=-q^2\\;,\\quad \nM^2=(q+\\xi P)^2\\;,\\quad \\beta={Q^2\\over Q^2 + M^2}\\;.\n\\end{equation}\nCompared to inclusive DIS, the diffractive mass $M$ plays the role of the\ntotal hadronic mass $W$, and $\\beta$ corresponds to $x$.\n\nThe hadronic tensor for diffractive DIS,\n\\begin{eqnarray} \nW^D_{\\mu\\nu}(P,P',q) &=& {1\\over 4\\pi} \\sum_{X} \n  \\langle P|J_\\nu(0)|X;P'\\rangle  \n  \\langle X;P'|J_\\mu(0)|P\\rangle \n  (2\\pi)^4 \\delta(P-P'-P_X)\\nonumber\\\\\n&=&\\left(-g_{\\mu\\nu}+{q_\\mu q_\\nu\\over q^2}\\right) F_1^{D(4)}(t,\\xi,\\beta,Q^2)\n   \\nonumber\\\\ \n&&\\quad+{1\\over \\nu}\\left(P_\\mu - {\\nu\\over q^2}q_\\mu\\right) \n\\left(P_\\nu - {\\nu\\over q^2}q_\\nu\\right) F_2^{D(4)}(t,\\xi,\\beta,Q^2)+\\ldots\\;,\n\\end{eqnarray}\ndefines the diffractive structure functions $F_i^{D(4)}(t,\\xi,\\beta,Q^2)$.\nIntegration over $t$, which is dominated by small $|t|$ for diffractive\nscattering, yields the extensively studied structure function\n\\begin{equation}\nF_2^{D(3)}(\\xi,\\beta,Q^2) = \\int dt F_2^{D(4)}(t,\\xi,\\beta,Q^2)\\;.\n\\end{equation}\n\nAlso the diffractive structure functions have contributions of leading and\nhigher twist, \n\\begin{equation}\nF_i^{D(3)}(\\xi,\\beta,Q^2) = F_i^{D(3,LT)}(\\xi,\\beta,Q^2) + \n{F_i^{D(3,HT)}(\\xi,\\beta,Q^2)\\over Q^2} + \\ldots\\;.\n\\end{equation}\nAgain it is unclear above which value of $Q^2_0$ the leading twist part\ndominates. At small $x$, $W^2\\simeq Q^2\/x$ should be large enough, whereas\nthe lower bound on $M^2$ is an open question. Our phenomenological analysis\nin the next section will show that the leading twist description breaks\ndown at $M_0^2 \\simeq 4$ GeV$^2$. This again demonstrates that in diffractive\nDIS $M^2$ plays a role analogous to $W^2$ in inclusive DIS.  \n\nFor diffractive DIS factorization holds like for inclusive DIS \\cite{c}. \nThe diffractive structure functions $F_i^{D(3,LT)}(\\xi,\\beta,Q^2)$ can be \nexpressed in terms of `fracture functions' \\cite{tre}, or\n`diffractive parton distributions' \\cite{ber1},\n\\begin{equation}\nF_i^{D(3,LT)}(\\xi,\\beta,Q^2) \\rightarrow {df_i(\\xi,\\beta,\\mu^2)\\over d\\xi} = \n{dq(\\xi,\\beta,\\mu^2)\\over d\\xi}\\;,\\; {dg(\\xi,\\beta,\\mu^2)\\over d\\xi}\\;,\n\\end{equation}\nwhich depend on $\\xi$, $\\beta$ and the factorization scale $\\mu^2$. \nThe diffractive parton distribution functions $df_i(\\xi,\\beta,\\mu^2)\/d\\xi$ \nalso obey the perturbative QCD evolution equations,\n\\begin{equation}\\label{ddglap}\n\\mu^2 {\\partial\\over \\partial\\mu^2} {df_i(\\xi,\\beta,\\mu^2)\\over d\\xi} =\n{\\alpha_s\\over 2\\pi} \\int_\\beta^1{db\\over b} P_{ij}\\left({\\beta\\over b}\\right)\n{df_i(\\xi,b,\\mu^2)\\over d\\xi}\\;.\n\\end{equation}\nNote that the evolution takes now place in $\\beta$ and $Q^2$; $\\xi$ \nmerely acts as a parameter. The physical reason for this is intuitively clear: \nfor an arbitrary DIS event the invariant hadronic mass is $W$, and the quark \nwhich couples to the virtual photon can be radiated by a parton whose fraction\nof the proton momentum varies from 1 to $x=Q^2\/(Q^2+W^2)$. In a diffractive \nevent, the diffractive invariant mass is $M$. Hence, $W$ is \nreplaced by $M$, and the quark which couples to the photon can be radiated \nby a parton whose fraction of the momentum $\\xi P$ varies from 1 to \n$\\beta=Q^2\/(Q^2+M^2)$. Formally, Eq.~(\\ref{ddglap}) follows from the fact \nthat ultraviolet divergencies and renormalization are the same for  \ninclusive and diffractive parton distribution functions \\cite{ber2}. This is \napparent from a comparison of the corresponding operator definitions. The \ndiffractive quark distribution, for instance, is given by \\cite{ber2},\n\\begin{eqnarray}\\label{qddiff}\n{dq(\\xi,\\beta,\\mu^2)\\over d\\xi} \n&=& {1\\over 64\\pi^3}\\int dt \\int dx_- e^{-ixP_+x_-\/2}\\sum_{X}\\nonumber\\\\\n&&\\langle P|\\bar{q}(0,x_-,\\vec{0})U(x_-,\\infty)|X;P'\\rangle\\gamma_+\n\\langle X;P'|U(\\infty,0)q(0,0,\\vec{0})|P\\rangle \\;.\n\\end{eqnarray}\nAssuming `Regge factorization' for the diffractive quark and gluon distribution\nfunctions yields the Ingelman-Schlein model of hard diffractive scattering\n\\cite{ing} which can also be applied to deep inelastic scattering \\cite{dl}.\n\nThe physical interpretation of the diffractive parton distributions is \nanalogous to the interpretation of the inclusive distributions. The function \n$df(\\xi,b,\\mu^2)\/d\\xi$ is a conditional probability distribution. It describes\nthe probability density to find a parton $f$, carrying a fraction $\\xi b$ of \nthe proton momentum, under the condition that the proton has lost a fraction \n$\\xi$ of its momentum in the scattering process. \n\nThe formal definition of diffractive parton distributions tells us very\nlittle about their properties, although, comparing Eqs.~(\\ref{qdincl}) and\n(\\ref{qddiff}), one may expect that diffractive DIS is a leading twist\neffect. However, the important physics question concerns the relation\nbetween the two types of distribution functions,\n\\begin{equation}\nf_i(x,\\mu^2) \\longleftrightarrow {df_i(\\xi,\\beta,\\mu^2)\\over d\\xi}\\quad ?\n\\end{equation} \nBoth kinds of parton distributions represent non-perturbative properties\nof the proton and are therefore not accessible to perturbation theory.\nStill, one may hope that at small $x$, i.e. large hadronic energies $W$,\nsome simple relations between inclusive and diffractive deep inelastic\nscattering may exist. In the following section we shall describe a picture\nof hadrons at small $x$ where this is indeed the case.\n\n\n\n\\section{Semiclassical approach}\n\nThe phenomenon of the large rapidity gap events in DIS is very difficult\nto understand within the parton model. Naively, one would expect that the\nstruck quark will always break up the proton, which should lead to a flow\nof hadrons between the current jet and the proton remnant without large\ngaps in rapidity. \n\n\\begin{figure}\n\\begin{center}\n\\vspace*{-.5cm}\n\\parbox[b]{13.7cm}{\\psfig{width=10cm,file=semi.eps}}\\\\\n\\end{center}\n\\refstepcounter{figure}\n\\label{f1}\n{\\bf Figure \\ref{f1}:} Diffractive or non-diffractive DIS in the proton rest \nframe; the proton is viewed as a superposition of colour fields with\nsize $1\/\\Lambda$. \n\\end{figure}\n\nThe connection between diffractive DIS and ordinary, non-diffractive DIS can\nbe most easily understood in the proton rest frame which has frequently been\nused in the early days of DIS, almost 30 years ago. In this frame, DIS \nappears as the scattering of partonic fluctuations of the photon, $q {\\bar q}$, $q {\\bar q}g$\\ \netc., off the proton. In the semiclassical approach \\cite{bh1} the proton is \nviewed as a superposition of colour fields of size $1\/\\Lambda$ in DIS at small\n$x$, i.e. at high $\\gamma^* p$ center-of-mass energies. The simplest partonic\nfluctuation is a quark-antiquark pair (cf.~Fig.~1). Penetrating the proton,\nquark and antiquark change their colour. If the $q {\\bar q}$\\ pair leaves the proton \nin a colour singlet configuration, it can fragment independently of the\nproton remnant yielding a diffractive event. A $q {\\bar q}$\\ pair in a colour octet\nstate will build up a flux tube together with the proton remnant, whose\nbreakup will lead to an ordinary non-diffractive event.  \n\nThe scattering amplitude for both types of events is determined by a single\nnon-perturbative quantity, $\\mbox{tr}W_{x_{\\perp}}(y_{\\perp})$. Here $x_{\\perp}$ and \n$x_{\\perp}+y_{\\perp}$ are the transverse positions where quark and antiquark penetrate the \ncolour field of the proton. The function \n\\begin{equation}\nW_{x_{\\perp}}(y_{\\perp})=U(x_{\\perp})U^{\\dagger}(x_{\\perp}+y_{\\perp})-1\\;, \\label{wuu}\n\\end{equation}\nwith\n\\begin{equation}\nU(x_{\\perp}) = P \\exp{\\left(-{i\\over 2}\\int_{-\\infty}^{\\infty} \n        dx_-A_+(0,x_-,x_{\\perp})\\right)}\\;,\n\\end{equation}\nis essentially a closed Wilson loop through the corresponding section of \nthe proton, which measures an integral of the proton colour field strength. \n\nDiffractive DIS requires a colour singlet pair in the final state. Hence the\nscattering amplitude is $\\propto \\mbox{tr}W_{x_{\\perp}}(y_{\\perp})$ and the diffractive\ncross section takes the form,\n\\begin{equation}\\label{dsdiff}\nd\\sigma^D \\propto \\int_{x_{\\perp}} \\ldots |\\ldots \\mbox{tr}W_{x_{\\perp}}(y_{\\perp})\\ldots|^2\\;.\n\\end{equation} \nThe inclusive cross section is obtained by summing over all colours, which\nyields\n\\begin{eqnarray}\nd\\sigma^{incl} &\\propto& \\int_{x_{\\perp}} \\ldots \\mbox{tr}\n        \\left(W_{x_{\\perp}}(y_{\\perp})W^\\dagger_{x_{\\perp}}(y_{\\perp})\\right)\\ldots \\nonumber\\\\\n&\\propto& \\int_{x_{\\perp}} \\ldots \\mbox{tr}W_{x_{\\perp}}(y_{\\perp})\\ldots \\;,\\label{dsincl}\n\\end{eqnarray} \nwhere the last equation follows from the unitarity of the matrix $U(x_{\\perp})$. \n\n\nFrom Eqs.~(\\ref{dsdiff}) and (\\ref{dsincl}) one immediately derives the\nproperties of Bjorken's aligned jet model \\cite{bjo}. For small \nquark-antiquark separations one has,\n\\begin{equation}\n\\int_{x_{\\perp}} \\mbox{tr}W_{x_{\\perp}}(y_{\\perp}) \\propto y_{\\perp}^2 \\;.\n\\end{equation}\nHence, since all kinematical factors are the same for $d\\sigma^D$ and\n$d\\sigma^{incl}$, small $q {\\bar q}$\\ pairs are suppressed in diffractive DIS. For\nlarge pairs of size $1\/\\Lambda$, the transverse momentum $l'_{\\perp}$ and the\nlongitudinal momentum fractions $\\alpha$ and $1-\\alpha$ are\n\\begin{equation}\nl'_{\\perp} \\sim \\Lambda\\;, \\quad \\alpha \\sim {\\Lambda^2\\over Q^2}\\;, \n\\quad 1-\\alpha \\simeq 1\\;.\n\\end{equation}\nThese are the asymmetric, aligned jet configurations \\cite{bjo} which\ndominate diffractive DIS. \n\n\\begin{figure}\n\\begin{center}\n\\vspace*{-.5cm}\n\\parbox[b]{15cm}{\\psfig{width=15cm,file=f2d.eps}}\\\\\n\\end{center}\n\\refstepcounter{figure}\n\\label{f2d}\n{\\bf Figure \\ref{f2d}:} Diffractive DIS in the proton rest frame (left) and \nthe Breit frame (right); asymmetric quark fluctuations correspond to \ndiffractive quark scattering, asymmetric gluon fluctuations to diffractive \nboson-gluon fusion. \n\\begin{center}\n\\vspace*{-.5cm}\n\\parbox[b]{13.7cm}{\\psfig{width=13.7cm,file=f2.eps}}\\\\\n\\end{center}\n\\refstepcounter{figure}\n\\label{f2}\n{\\bf Figure \\ref{f2}:} Inclusive DIS in the proton rest frame (left) and the \nBreit frame (right); asymmetric fluctuations correspond to quark scattering \n(a), symmetric fluctuations to boson-gluon fusion (b). \n\\end{figure}\n\n\n\\subsection*{Diffractive and inclusive parton distributions}\n\nIn the semiclassical approach the evaluation of inclusive and diffractive \nstructure functions is straightforward, in principle. One has to calculate\nthe scattering amplitudes for the production of  $q {\\bar q}$\\ , $q {\\bar q}g$\\ ... \nconfigurations \\cite{bhm} in an external colour field, analogous to the \nproduction of $\\mu^+\\mu^-$ pairs in an external electromagnetic field \n\\cite{bks}, treat the interaction of the fast partons with the non-abelian \ncolour field in the eikonal approximation \\cite{n}, and finally integrate \nover all target colour fields.\n\n\nThe result for the leading twist part can be expressed in terms of\ndiffractive parton distributions \\cite{h}. For the transverse structure\nfunction, for instance, one finds to leading order in the QCD coupling,\n\\begin{eqnarray}\nF_T^D(\\xi,\\beta,Q^2) &=& 2e_q^2x \\int_{\\beta}^1 {db\\over b} \\Bigg\\{\\left(\n\\delta(1-z) + {\\alpha_s\\over 2\\pi}\\left(P_{qq}(z)\\ln{Q^2\\over \\mu^2} + \n\\ldots\\right) \\right){dq(b,\\xi,\\mu^2)\\over d\\xi}\\nonumber\\\\\n&&\\hspace{1cm} + {\\alpha_s\\over 2\\pi}\\left(P_{qg}(z)\\ln{Q^2\\over \\mu^2}\n+\\ldots\\right){dg(b,\\xi,\\mu^2)\\over d\\xi}\\Bigg\\}\\;,\\label{ftd}\n\\end{eqnarray}\nwhere $z=\\beta\/b$, and $C_F$ and $T_F$ are the usual colour factors. This\nexpression is completely analogous to the well known result for the\ninclusive structure function $F_T(x,Q^2)$. In the diffractive case, \n$\\beta$ plays the role of $x$, whereas $\\xi$ only acts as a parameter.\nFrom  Eq.~(\\ref{ftd}) it is obvious that, as anticipated, the diffractive \nparton distributions satisfy the perturbative QCD evolution equations \n(\\ref{ddglap}).\n\nThe diffractive quark and gluon distributions have been determined in \n\\cite{h}. In terms of Wilson loops in coordinate space, the quark \ndistribution can be expressed as follows, \n\\begin{eqnarray}\\label{dqd}\n{dq(\\xi,b,\\mu^2)\\over d\\xi}&=&{2b\\over \\xi^2(1-b)^3}\n           \\int{d^2l'_{\\perp} {l'_{\\perp}}^4\\over(2\\pi)^6 N_c} \n           \\int_{y_{\\perp},y_{\\perp}'} e^{il'_{\\perp}(y_{\\perp}-y_{\\perp}')}\\,{y_{\\perp}\\y'\\over y\\, y'}\\nonumber \\\\\\nn\n&&\\times\\,K_1(yN)K_1(y'N)\\int_{x_{\\perp}}\\mbox{tr}W_{x_{\\perp}}(y_{\\perp})\n\\mbox{tr}W^{\\dagger}_{x_{\\perp}}(y_{\\perp}')\\;,  \n\\end{eqnarray}\nwhere $N_c$ is the number of colours and $N^2={l'_{\\perp}}^2{b\\over 1-b}$.\n\nIt is very instructive to compare diffractive DIS in the proton rest frame\nand in the Breit frame (cf.~Fig. 2). The number of partons in the final\nstate is the same, of course, in both frames. Note, however, that the virtual\nparton connected to the proton changes it's direction. It appears incoming\nin the proton rest frame and outgoing in the Breit frame. Diffractive quark\nand gluon distributions correspond to asymmetric $q {\\bar q}$\\ and $q {\\bar q}g$\\ fluctuations\nwith a slow antiquark and gluon, respectively.\n\nInclusive parton distributions can be calculated in a similar way. The\ninclusive quark distribution is again given by the asymmetric $q {\\bar q}$\\ \nconfiguration (cf.~Fig.~3), just with arbitrary colours in the final state.\nA special role is played by the inclusive gluon distribution. It\nis related to small symmetric $q {\\bar q}$\\ pairs which probe the colour field of the \nproton directly (cf.~Fig.~3). Contrary to all other parton distributions,\nthe inclusive gluon distribution \\cite{bgh},\n\n\\begin{eqnarray}\nxg(x,Q^2) &=& \\frac{3\\pi}{\\alpha_s e_q^2}\\,\\cdot\\,\\frac{\\partial F_T(x,Q^2)}\n{\\partial\\ln Q^2} \\label{ctrans}\\\\\n&=& {1\\over 2\\pi^2 \\alpha_s}\\int_{x_{\\perp}} \\mbox{tr}\\left(\n\\partial_{y_{\\perp}}W_{x_{\\perp}}(0)\\partial_{y_{\\perp}}W_{x_{\\perp}}^{\\dagger}(0)\\right)\n= {\\cal O}\\left({1\\over \\alpha_s}\\right)\\;,\\label{gi}\n\\end{eqnarray}\nis enhanced by an inverse power of $\\alpha_s$ in the semiclassical approach.\nThis is the reason why diffractive DIS is suppressed. Note that the gluon \ndistribution is directly related to the cross section for a small\n$q {\\bar q}$\\ pair with transverse size $y$ \\cite{fms},\n\\begin{equation}\n\\sigma_{q\\bar{q}} (y;x,Q^2) = {\\pi^2\\over 3} \\alpha_s x g(x,Q^2) y^2 + \n{\\cal O}(y^4)\\;.\n\\end{equation}\n\n\\subsection*{Integration over the target gluon fields}\\label{av}\n\nSo far we have expressed diffractive and inclusive parton distributions in \nterms of Wilson loops which integrate the gluon field strength in the area\nbetween the trajectories of two fast colour charges penetrating the proton.\nThe integration over the gluon field configurations of the target is a \ncomplicated operation depending on the full details of the non-perturbative \nhadronic state. However, in the special case of a very large target, a \nquantitative treatment becomes possible under minimal additional assumptions.\nThe reason is that the large size of a hadronic target, realized, e.g., in an \nextremely heavy nucleus, introduces a new hard scale \\cite{mv}. From the \ntarget rest frame point of view, this means that the \ntypical transverse size of the partonic fluctuations of the virtual photon \nremains perturbative \\cite{hw}, thus justifying the omission of higher Fock \nstates in the semiclassical calculation. \n\nWithin this framework, it is natural to introduce the additional assumption \nthat the gluonic fields encountered by the partonic probe in distant regions \nof the target are not correlated. Thus, one arrives at the situation \ndepicted in Fig.~\\ref{lt}, where a colour dipole passes a large number of \nregions, each one of size $\\sim 1\/\\Lambda$, with mutually uncorrelated \ncolour fields $A_1$ ... $A_n$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\vspace*{-.5cm}\n\\parbox[b]{12cm}{\\psfig{width=9cm,file=lt.eps}}\\\\\n\\end{center}\n\\refstepcounter{figure}\n\\label{lt}\n{\\bf Figure \\ref{lt}:} Colour dipole travelling through a large\nhadronic target.\n\\end{figure}\n\nThe crucial assumption that the fields in regions $1$ ... $n$ are \nuncorrelated is implemented by writing the integral over all field \nconfigurations as\n\\begin{equation}\n\\int_{A}=\\int_{A_1}\\cdots\\int_{A_n}\\,\\,,\\label{int}\n\\end{equation}\ni.e., as a product of independent integrals. Here the appropriate weighting \nprovided by the target wave functional is implicit in the symbol $\\int_A$. \nFor inclusive and diffractive parton distributions we need the\ntwo colour contractions for products of Wilson loops,\n\\begin{equation}\\label{essence}\n\\mbox{tr}\\left(W_{x_{\\perp}}(y_{\\perp})W_{x_{\\perp}}^{\\dagger}(y_{\\perp})\\right) \n\\longleftrightarrow \\frac{1}{N_c}\\mbox{tr}W_{x_{\\perp}}(y_{\\perp})\\mbox{tr}\nW_{x_{\\perp}}^{\\dagger}(y_{\\perp})\\;.\n\\end{equation}\nThis relation, which provides the connection between inclusive and diffractive\nDIS, is the essence of the semiclassical approach.\n\nPerforming the integration over the colour fields one obtains in the large\n$N_c$ limit \\cite{bgh},\n\\begin{eqnarray}\n\\int_{x_\\perp}\\int_A\\mbox{tr}\\left(W_{x_\\perp}(y_\\perp)\nW^{\\dagger}_{x_\\perp}(y_\\perp')\\right)&=&\\Omega N_c\\left(1-e^{-ay^2}-\ne^{-ay'^2}+e^{-a(y_\\perp-y_\\perp')^2}\\right)\\label{ww0}\\,,\n\\\\\n\\frac{1}{N_c}\\int_{x_\\perp}\\int_A\\mbox{tr}W_{x_\\perp}(y_\\perp)\n\\mbox{tr}W^{\\dagger}_{x_\\perp}(y_\\perp')&=&\\Omega N_c\\left(1-\ne^{-ay^2}\\right)\\,\\left(1-e^{-ay'^2}\\right)\\,,\\label{wwf}\n\\end{eqnarray}\nwhere $\\Omega = \\int d^2x_{\\perp}$ is the geometric size of the target and $a$ plays\nthe role of a saturation scale. Note that according to Eqs.~(\\ref{ww0}) and\n(\\ref{wwf}) the diffractive structure function is not suppressed by a colour \nfactor relative to the inclusive structure function, in contrast to the \nsuggestion made in~\\cite{bh}. \n\nAs an example, consider the inclusive quark distribution. From \nEqs.~(\\ref{dqd}), (\\ref{essence}) and (\\ref{ww0}) one obtains, after changing \nthe integration variable $\\xi$ to $N^2$,  \n\\begin{equation}\\label{idiff}\nx q(x,\\mu^2) = \\int_x^1 d\\xi {dq(\\xi,b=x\/\\xi,\\mu^2)\\over d\\xi}\n= \\int_{x_{\\perp}} \\int_{l'_{\\perp}} {d\\ xq(x,\\mu^2)\\over d^2x_{\\perp} d^2l'_{\\perp}}\\;,\n\\end{equation}\nwith the unintegrated quark density\n\\begin{eqnarray}\\label{uidiff}\n{d\\ xq(x,\\mu^2)\\over d^2x_{\\perp} d^2l'_{\\perp}} &=& \n{N_c\\over 32\\pi^6} \\int_0^{\\mu^2} dN^2 N^2\n\\int_{y_{\\perp},y_{\\perp}'} e^{il'_{\\perp}(y_{\\perp}-y_{\\perp}')}\\,{y_{\\perp}\\y'\\over y\\, y'}\\nonumber \\\\\\nn\n&&\\times\\,K_1(yN)K_1(y'N)\\left(1-e^{-ay^2}-\ne^{-ay'^2}+e^{-a(y_\\perp-y_\\perp')^2}\\right)\\;.\n\\end{eqnarray}\nThis result has recently been obtained for the quark density in a large\nnucleus \\cite{mue} by exponentiating the amplitude for a small $q {\\bar q}$\\ pair \nscattering off a single nucleon, which is described by two-gluon exchange.\nThe effect of the varying thickness of the nucleus has also been taken into \naccount in \\cite{mue}, which makes the saturation scale $a$ dependent on the \nimpact parameter $x_{\\perp}$.\n\nA Glauber type model with two-gluon exchange, similar in \nspirit to \\cite{got}, has recently been used to study the effect of parton \nsaturation in inclusive and diffractive DIS \\cite{gw}. Although perturbative \ntwo-gluon exchange is a higher twist effect, the contribution from the soft \nregion can be used as a model for inclusive and diffractive DIS \n\\cite{nz,bekw}. \n\nParticularly close to the semiclassical approach is the light cone hamiltonian\napproach to diffractive processes \\cite{hks}, which is also based on\ndiffractive parton densities expressed in terms of expectation values of\nproducts of Wilson lines. For a hadronic target, modelled as a colour singlet\nwhich only couples to one flavour of heavy quarks,  diffractive DIS is \ndominated by two-gluon exchange. In the semiclassical approach, the proton\nis described by a superposition of colour fields. The role of classical\ncolour fields in the case of high gluon densities has first been discussed\nby McLerran and Venugopalan in the case of a large nucleus \\cite{mv}. \n\n\n\\begin{figure}\n\\begin{center}\n\\parbox[b]{7cm}{\\psfig{file=difpdf.ps,width=7cm}}\\\\\n\\end{center}\n\\refstepcounter{figure}\n\\label{figdifpdf}\n{\\bf Figure \\ref{figdifpdf}:}\nDiffractive quark and gluon distributions at the initial scale $Q_0^2$ and \nafter $Q^2$ evolution. From \\cite{bgh}.\n\\end{figure}\n\n\n\\subsection*{Comparison with data}\\label{na}\n\nWe now use the large hadronic target as a toy model for the proton. In case \nthe proton can be viewed as an ensemble of regions with independently \nfluctuating colour fields, the model might even be realistic. We have \nexplicitly verified that in the semiclassical approach inclusive and \ndiffractive parton distributions satisfy the DGLAP evolution equations \n\\cite{dglap}. Hence, we can use the calculated quark and gluon distributions \nas non-perturbative input at some scale $Q_0^2$ and determine the \ndistributions at larger $Q^2$ by means of the evolution equations. \n\n\\begin{figure}\n\\begin{center}\n\\parbox[c]{6cm}{\\psfig{file=difq2.ps,width=6cm} }\\parbox[c]{2cm}{\\hspace{2cm}}\n\\parbox[c]{6cm}{\\psfig{file=difq2z.ps,width=6cm} }\\\\\n\\vspace{0.6cm}\n\\end{center}\n\\refstepcounter{figure}\n\\label{figdifq2}\n{\\bf Figure \\ref{figdifq2}:}\nDependence of the diffractive structure function $F_2^{D(3)}$ on $\\beta$\nand $Q^2$, compared to data from H1 (left)~\\protect\\cite{diffh} and ZEUS \n(right)~\\protect\\cite{diffz}. Open data points \ncorrespond to $M^2\\leq 4$~GeV$^2$. The charm content of the \nstructure function is indicated as a dashed line. From \\cite{bgh}.\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\parbox[b]{14cm}{\\psfig{file=f2dh1.ps,width=14cm}}\\\\\n\\end{center}\n\\refstepcounter{figure}\n\\label{figf2dh1}\n{\\bf Figure \\ref{figf2dh1}:}\nThe diffractive structure function $F_2^{D(3)}(\\xi,\\beta,Q^2)$ at small $\\xi$ \ncomputed in the semiclassical approach, using the fitted parameters \ngiven in the text.\nH1 data taken from~\\protect\\cite{diffh}. The open data points correspond\nto $M^2 \\leq 4$~GeV$^2$ and are not included in the fit. From \\cite{bgh}.\n\\end{figure}\n\nFor a given colour field, the semiclassical description of parton distribution \nfunctions always predicts an energy dependence corresponding to a classical \nbremsstrahlung spectrum: $q(x),g(x)\\sim 1\/x$. One expects that, in a more \ncomplete treatment, a non-trivial energy dependence is induced since the \nintegration over the soft target colour fields encompasses more and more modes \nwith increasing energy of the probe \\cite{bgh}. At present we are unable to \ncalculate this non-perturbative energy dependence from first \nprinciples. Instead, we choose to parametrize it in the form of a soft, \nlogarithmic growth of the normalization of diffractive and inclusive parton \ndistributions with the collision energy $\\sim 1\/x$, consistent with the \nunitarity bound. This introduces one further parameter, $L$, into the model, \n\\begin{equation}\n\\Omega \\to \\Omega \\left(L - \\ln x \\right)^2.\n\\end{equation}\n\nIncluding this energy dependence, one obtains the following compact\nexpressions for inclusive and diffractive parton distributions \\cite{bgh}, \n\\begin{eqnarray}\nxq(x,Q_0^2) & = & \\frac{a \\Omega N_c \\left( L - \\ln x \\right)^2}\n{3 \\pi^3} \\left(\\ln\\frac{Q_0^2}{a} - 0.6424\\right)\\; , \\label{qiinp}\\\\\nxg(x,Q_0^2) & = & \\frac{ 2 a \\Omega N_c \\left( L - \\ln x \\right)^2}\n{\\pi^2 \\alpha_s(Q_0^2)}\\; , \\label{giinp}\\\\\n\\frac{dq\\left(\\beta,\\xi,Q_0^2\\right)}{d\\xi} & = & \\frac{a\\Omega N_c \n(1-\\beta) \n\\left(L - \\ln \\xi\\right)^2}{2\\pi^3\\xi^2} f_q(\\beta)\\; , \\label{qdinp} \\\\\n\\frac{dg\\left(\\beta,\n\\xi,Q_0^2\\right)}{d\\xi} & = & \\frac{a\\Omega N_c^2 (1-\\beta)^2 \n\\left(L - \\ln \\xi\\right)^2}{2\\pi^3 \\beta \\xi^2} f_g(\\beta)\\; \\label{gdinp}.\n\\end{eqnarray}\nThese expressions are only applicable in the small-$x$ region, which we define\nby $x\\leq \\xi \\leq 0.01$. The functions $f_{q,g}(\\beta)$ are parameter free\npredictions. The model does not specify whether, in the diffractive case, the \nenergy-dependent logarithm  should be a function of $x$ or of $\\xi$. However, \nboth prescriptions differ only by terms proportional to $\\ln \\beta$, which can\nbe disregarded in comparison with $\\ln x$ or $\\ln \\xi$ in the small-$x$ region.\n\n\\begin{figure}[t]\n\\begin{center}\n\\parbox[b]{15.5cm}{\\psfig{file=f2i.ps,width=15.5cm}}\\\\\n\\end{center}\n\\refstepcounter{figure}\n\\label{figf2i}\n{\\bf Figure \\ref{figf2i}:}\nThe inclusive structure function $F_2(x,Q^2)$ at small $x$ \ncomputed in the semiclassical approach, using the fitted parameters \ngiven in the text.\nData taken from~\\protect\\cite{incl}. The data with $Q^2 = 1.5$~GeV$^2$ \nare not included in the fit. From \\cite{bgh}.\n\\end{figure}\n\n\nThe above equations summarize our input distributions, depending on \n$a$, $\\Omega$, $L$, and the input scale $Q_0^2$. At this order, the measured \nstructure function $F_2$ coincides with the transverse structure function \n$F_T$. We assume all three light quark flavours to yield the same \ncontribution, such that the singlet quark distribution is simply six times the\nquark distribution defined above, both in the inclusive and in the diffractive\ncase,\n\\begin{equation}\n\\Sigma (x,Q^2) = 6\\, q(x,Q^2)\\; , \\qquad \\frac{d\\,\\Sigma(\\xi,\\beta,Q^2)}{d\\xi} \n= 6\\,  \\frac{dq(\\xi,\\beta,Q^2)}{d\\xi}\\; .\n\\end{equation}\nValence quark contributions are absent in the semiclassical approach, which \ndoes not account for the exchange of flavour quantum numbers between the \nproton and the fast moving virtual photon state. Charm quarks are treated \nas massive quarks in the fixed flavour number scheme \\cite{ffns} (we use\n$\\Lambda_{{\\rm LO},n_f=3}= 144$~MeV, $\\alpha_s(M_Z)=0.118$, $m_c=1.5$~GeV,\n$m_b=4.5$~GeV, $\\mu_c = 2 m_c$). A fit to the data yields for the model \nparameters $Q_0^2 = 1.23\\ {\\rm GeV}^2,\\ L = 8.16, \n\\Omega = (712\\ {\\rm MeV})^{-2},\\ a = ( 74.5\\ {\\rm MeV})^2$ . The starting \nscale $Q_0^2$ is in the region where one would expect the transition between \nperturbative and non-perturbative dynamics to take place; the two \nother dimensionful parameters $\\Omega L^2$ and $a$ are both of the order \nof typical hadronic scales. \n\nThe perturbative evolution of inclusive and diffractive structure functions \nis driven by the gluon distribution, which is considerably larger \nthan the singlet quark distribution in both cases. \nThe ratio of the inclusive singlet quark and gluon distributions can be \nread off from Eqs.~(\\ref{qiinp}) and (\\ref{giinp}). For the obtained fit \nparameters, it turns out that the inclusive gluon distribution is about twice \nas large as the singlet quark distribution.\n\nThe relative magnitude and the $\\beta$ dependence of the diffractive \ndistributions are completely independent of the model parameters. Moreover, \ntheir absolute normalization is, up to the slowly varying factor \n$1\/\\alpha_s(Q_0^2)$, closely tied to the normalization of the \ninclusive gluon distribution. \n\nFigure~\\ref{figdifpdf} displays the diffractive  distributions for\nfixed $\\xi=0.003$ and different values of $Q^2$. The $\\beta$ dependences \nof the quark and the gluon distribution at $Q_0^2$ are substantially \ndifferent: the asymmetric quark distribution $\\beta d\\Sigma\/d \\xi$ is peaked \naround  $\\beta \\approx 0.65 $, thus being harder than the symmetric \ndistribution $\\beta (1-\\beta)$ suggested in~\\cite{dl}. The gluon distribution\n$\\beta d g\/d \\xi$, on the other hand, approaches a constant for \n$\\beta \\to 0$ and falls off like $(1-\\beta)^2$ at large $\\beta$. \nIn spite of the $(1-\\beta)^2$ behaviour, gluons remain important \neven at large $\\beta$, simply due to the large total normalization of this \ndistribution (the $\\beta$ integral over $\\beta dg\/d\\xi$ at $Q_0^2$ is \napproximately three times the $\\beta$ integral over $\\beta d\\Sigma\/d\\xi$). \nAs a result, the quark distribution does not change with increasing $Q^2$ \nfor $\\beta\\approx 0.5$ and is only slowly decreasing for larger values \nof $\\beta$. \n\nThe dependence of the diffractive structure function on $\\beta$ and $Q^2$ is \nillustrated in Fig.~\\ref{figdifq2}, where the predictions are compared with \ndata from the H1 and ZEUS experiments~\\cite{diffh,diffz} at fixed $\\xi$.\nDisregarding the large-$\\beta$ region, the model gives a good description of \nthe $\\beta$ dependence of the diffractive structure function for all values of\n$Q^2$. It is remarkable that the qualitative features of the $\\beta$ and\n$Q^2$ dependence are also correctly described by the perturbative approach\nof \\cite{hks}. This indicates that the $\\beta$ dependence of the diffractive\nstructure function is to a large extent determined by the kinematics of the\n$q {\\bar q}$\\ and $q {\\bar q}g$\\ fluctuations and only partly sensitive to the details of the \nsoft interaction with the proton. Also the $\\xi$ dependence of the diffractive\nstructure function is rather well described for  $M^2 > 4$~GeV$^2$, as\ndemonstrated in Fig.~\\ref{figf2dh1}. It has been demonstrated in \\cite{bekw}\nthat higher twist contributions can account for the data at low $M^2$\n(below 4 GeV$^2$). This is analogous to the breakdown of the leading twist \ndescription of the inclusive structure functions, where it occurs for similar \ninvariant hadronic masses, namely $W^2\\lower .7ex\\hbox{$\\;\\stackrel{\\textstyle <}{\\sim}\\;$} 4$~GeV$^2$ \\cite{mrst2}.\n\nFinally, also the data of the H1 and ZEUS experiments on the inclusive \nstructure function $F_2(x,Q^2)$~\\cite{incl} are well reproduced by the model,\nas demonstrated by Fig.~\\ref{figf2i}.\n\n\\section{Open questions}\\label{conc}\n\nThe theoretical work, stimulated by the observation of the large rapidity\ngap events at HERA, has led to a clear understanding of diffractive DIS\nas a leading twist phenomenon. Diffractive parton distribution functions\ncan be defined completely analogous to inclusive parton distribution \nfunctions. Both kinds of distribution functions obey the perturbative QCD \nevolution equations. The parton distribution functions cannot be calculated\nperturbatively, any relation between them reflects a non-perturbative\nproperty of the proton. The leading twist description breaks down at\n$W^2<4$ GeV$^2$ and $M^2<4$ GeV$^2$ for inclusive and diffractive DIS, \nrespectively.\n\nA physical picture for diffractive DIS, and the relation to inclusive DIS, \nis most easily obtained in the proton rest frame where all DIS processes \ncorrespond to the scattering of partonic fluctuations of the virtual photon \noff the proton. In the semiclassical approach the proton is described by a\nsuperposition of colour fields. The qualitative properties of the $\\beta$\ndependence of the diffractive structure function are well reproduced by\nthe scattering of colour dipoles off colour fields, which are generated \neither by a small colour dipole or by a large nucleus. This supports our \ngeneral ideas about diffractive DIS. It will be interesting to see whether a\nmore precise measurement of the $\\beta$ spectrum will be quantitatively\nconsistent with the idea of a colour field fluctuating independently\nin different sections of the proton. \n\nIn the semiclassical approach the proton colour field is assumed to be\ndominated by soft modes. Hence, diffractive and non-diffractive DIS\nevents are kinematically very similar at small $x$, i.e. large hadronic\nenergies $W$. This leads to an approximate relation between the inclusive\nand diffractive structure functions \\cite{bh1,b},\n\\begin{equation}\\label{scal}\nF_2^{D(3)}(\\xi,\\beta,Q^2) \\sim {1\\over \\ln{Q^2}} F_2(x=\\xi,Q^2)\\;,\n\\end{equation}\nwhich is in broad agreement with data \\cite{diffz}.\nFor fixed momentum transfer $Q^2$ and diffractive mass $M$, both structure \nfunctions have the same dependence on the $\\gamma^* p$ center-of-mass energy \n$W$. The factor $1\/\\ln{Q^2}$ reflects the suppression of small colour \ndipoles in diffractive scattering.\n\nThe dependence of the diffractive structure functions on $\\xi$ is not\naffected by the perturbative QCD evolution, contrary to the $x$ dependence\nof the inclusive structure functions, and therefore a genuine non-perturbative\nproperty of the proton. Hence, Eq.~(\\ref{scal}) can only hold as long as the\neffect of the perturbative evolution can be approximated by a single\n$\\ln{Q^2}$ factor. The $\\xi$ dependence of the diffractive structure\nfunction then plays the role of the non-perturbative input for the inclusive \nstructure function at some low scale $Q_0^2$.\n\nOne expects that, due to unitarity, diffractive and inclusive structure\nfunctions satisfy a relation similar to the one between elastic and\ntotal proton-proton cross section \\cite{mue1},\n\\begin{eqnarray}\n\\sigma_{el}  &=& \\int d^2b \\left(1-S(b)\\right)^2\\;,\\label{undi}\\\\\n\\sigma_{tot} &=& 2 \\int d^2b \\left(1-S(b)\\right)\\;,\\label{unin}\n\\end{eqnarray}\nwhere $S(b)$ is the S-matrix at a given impact parameter $b$. Recently, it has\nbeen shown that this relation also holds for the diffractive and inclusive\ncross sections of a $q {\\bar q}$\\ pair off the proton, if the diffractive cross section\nis defined by the colour singlet projection \\cite{km}. In the semiclassical \napproach, \nthis relation can be read off from Eqs.~(\\ref{dsdiff}) and (\\ref{dsincl}) or, \nmore explicitly, from Eqs.~(\\ref{ww0}) and (\\ref{wwf}). After integration over\n$l'_{\\perp}$ in Eqs.~(\\ref{idiff}) and (\\ref{uidiff}), which yields $y_{\\perp}=y_{\\perp}'$, one \nobtains for the cross sections of a $q {\\bar q}$\\ pair with size $y$,\n\\begin{eqnarray}\n\\sigma^D_{q\\bar{q}}(y) &\\propto& \n\\frac{1}{N_c}\\int_{x_\\perp}\\int_A\\mbox{tr}W_{x_\\perp}(y_\\perp)\n\\mbox{tr}W^{\\dagger}_{x_\\perp}(y_\\perp) \\nonumber\\\\\n&=& \\Omega N_c\\left(1-e^{-ay^2}\\right)^2\\;,\\\\\n\\sigma^{incl}_{q\\bar{q}}(y) &\\propto& \n\\int_{x_\\perp}\\int_A\\mbox{tr}\\left(W_{x_\\perp}(y_\\perp)\nW^{\\dagger}_{x_\\perp}(y_\\perp)\\right) \\nonumber\\\\\n&=& 2\\ \\Omega N_c\\left(1-e^{-ay^2}\\right)\\;.\n\\end{eqnarray}\nSince the dependence of the saturation parameter $a$ on the varying thickness \nof the target has been neglected, the integration over the impact parameter \n$x_{\\perp}$ (corresponding to $b$ in(\\ref{undi}),(\\ref{unin})) could be carried \nout yielding the geometric size $\\Omega$ as overall factor. It will be \ninteresting to extend these considerations to more complicated partonic\nfluctuations.\n\nIn the coming years experiments at HERA will provide detailed information\nabout diffractive final states, including charm and high-$p_{\\perp}$\\ jets. \nAnticipating further support of the semiclassical approach by data, \nwe can hope to learn a lot about the colour structure of the proton from\na comparison of inclusive and diffractive DIS. \n\\vspace{.4cm} \n\nThe content of this progress report is largely based on recent work with\nThomas Gehrmann and Arthur Hebecker whom I thank for an enjoyable \ncollaboration.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\nThe authors acknowledge support  from the Defense Advanced Research Projects Agency (DARPA) under the Physics of Artificial Intelligence (PAI) program (contract HR$00111890034$). \nAdditional computing resources were provided by the University of Notre Dame's Center for Research Computing (CRC). \n\n\n\n\\section{Sobel filter to estimate spatial gradients}\\label{appendix:sobel_filter}\nSobel filter is used to estimate horizontal and vertical spatial gradients by applying one convolution with the following $3 \\times 3$ kernels, respectively:\n\\[\n\\mathcal H = \n\\begin{bmatrix}\n    1 & 0 & -1 \\\\\n    2 & 0 & -2 \\\\\n    1 & 0 & -1\n\\end{bmatrix}, \\quad\n\\mathcal V =\n\\begin{bmatrix}\n    1 & 2 & 1 \\\\\n    0 & 0 & 0 \\\\\n    -1 & -2 & -1\n\\end{bmatrix}.\n\\]\nIntuitively it is a smoothed finite difference method.\nThe convolution operation goes natural with CNN representation of solution fields and is highly efficient. Sobel filter is way more efficient than using automatic differentiation to obtain spatial gradients in the FC-NN parameterization, with the compromise of reduced accuracy, especially for locations close to the boundaries. \n\nTo improve the accuracy of gradient estimate on the boundary, we use the following correction.\nFor 2D image matrix $\\mathbf{I}$ of size $H\\times W$,  Sobel kernel $\\mathcal H$, and correction matrix $M_{\\mathcal H}$ of size $W\\times W$,\n\\[\nM_{\\mathcal H} = \n\\begin{bmatrix}\n    4  & 0 & 0 &        &  & 0 \\\\\n    -1 & 1 & 0 &        &  &   \\\\\n    0  & 0 & 1 &        &  &  \\\\\n       &   &   & \\cdots & 0 & 0  \\\\\n       &   &   &        & 1 & -1 \\\\\n     0  &   &   &       0 & 0 & 4  \\\\\n\\end{bmatrix},\n\\]\nthe horizontal gradient is estimated as $(\\mathbf{I} \\star \\mathcal H) M_{\\mathcal H}$, where $\\star$ is convolution with replicate padding on the boundary. This is effectively using forward finite differences on the left boundary and backward finite differences on the right boundary. The vertical gradient estimate is corrected similarly. We found that this correction reduces the error of the learned solution by several times. However, there are still errors in four corners, which can be further improved by more refined correction.\n\n\\section{Solving PDEs with FC-NNs and CNNs}\\label{appendix:solve}\n\\subsection{Network architecture}\nThe FC-NN used in the experiments in Section~\\ref{sec:solve_det_pde} has $8$ hidden layers and $512$ nodes per hidden layer, with the input and output dimensions being $2$ and $3$, respectively. The nonlinear activation is \\texttt{Tanh}. The total number of parameters is $1,841,155$. We increased the number of nodes in the hidden layer from $20$ to $512$ to overfit the solution. We considered the collocation points to be at random locations in the domain, and increased their number. However, none of these modifications lead to improvement of the learned solution.\n\nThe convolutional decoder network uses two dense blocks with $8$ and $6$ dense layers respectively to transform the latent $\\mathbf{z}$ of size $1\\times 16 \\times 16$ to the output $\\mathbf{y}$ of size $3 \\times 64 \\times 64$. The decoding layers use nearest upsampling followed by one $3\\times 3$ convolution. The network has $514,278$ parameters and $20$ convolution layers.\n\nWe train FC-NNs and CNNs with mixed residual loss using L-BFGS optimizer (with history size $50$ and maximum iteration $20$), learning rate $0.5$. The weight for boundary loss is $\\lambda = 10$.\n\n \n\n\\subsection{Supplementary Numerical Experiments}\nWe also show learned solutions for GRF KLE$4096$ in Fig.~\\ref{fig:solve_k4096}. Again the CNN can capture the flux field much faster and better than FC-NN, but in this case the pressure field begins to show severe checkerboard artifact that the largest error being larger than that of the pressure solution of FC-NN.\n\\begin{figure}[h!]\n    \\centering\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/fc_mr_pred_epoch500_13_k4096.png}\n        \\caption{FC-NN, iteration $500$.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/fc_mr_pred_epoch2000_13_k4096.png}\n        \\caption{FC-NN, iteration $2000$.}\n    \\end{subfigure}\n    \n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/conv_mr_pred_epoch50_13_k4096.png}\n        \\caption{CNN, iteration $50$.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/conv_mr_pred_epoch500_13_k4096.png}\n        \\caption{CNN, iteration $500$.}\n    \\end{subfigure}\n\n    \\caption{Solving Darcy flow for one sample from GRF KLE4096 under mixed residual loss.}\n    \\label{fig:solve_k4096}\n\\end{figure}\n\n\n\n\\section{Details on conditional Glow and supplementary results}\\label{appendix:cglow}\n\\subsection{Details on the network structure}\nIn Fig.~\\ref{fig:glow_msc_sub}, the encoder network includes a cascade of $L$ \\texttt{Dense Block}s (that maintain the feature map size) and $L-1$ \\texttt{Trans Down} layers (that typically half the feature map size, e.g. from $32\\times 32$ to $16\\times 16$). The features extracted after each dense block are treated as input features $\\{\\bm{\\xi}_l\\}_{l=1}^L$. For details of \\texttt{Dense Block}s and \\texttt{Trans Down} layers (encoding layer), please refer to Section~$2.2$ in~\\cite{zhu2018bayesian}.\n\nIn Fig.~\\ref{fig:glow_msc_sub}, the \\texttt{Squeeze} operator rearranges the features of size $C \\times H \\times W$ into $4C \\times \\frac{1}{2} H \\times \\frac{1}{2}W$ if the squeeze factor is $2$. The \\texttt{Split} operator splits out half of the features\/channels as latent variable $\\mathbf{z}_l$, which is diagonal Gaussian parameterized by the other half features $\\mathbf{h}_l$ with a $3 \\times 3$ convolution, stride 1 and zero initialization.\nIn Fig.~\\ref{fig:flow_step}, one step of flow contains an activation normalization layer (\\texttt{ActNorm}), an invertable $1\\times1$ convolution layer and an affine coupling layer. \\texttt{ActNorm} performs affine transformation of the activation with a scale and bias parameter per-channel. \nThe $1\\times 1$ convolution layer with equal number of input and output channels is a learnable permutation operation to mix the two parts of the flow features before passing them  to the affine coupling layer.\n\nWe show the detailed computation of the forward and reverse paths of the affine coupling layer (Fig.~\\ref{fig:affine_coupling}) in Table~\\ref{tab:affine_coupling}. The nonlinear transform \\texttt{CouplingNN} includes $3$ dense layers, followed by a $3\\times3$ convolution layer with zero initialization, whose output channels split into two parts, i.e. $(\\hat{\\mathbf{s}}, \\mathbf{t})$.\n\\begin{table}[]\n    \\caption{Forward (from $\\mathbf{y}'$ to $\\mathbf{z}'$) and reverse paths of affine coupling layer with condition of input features $\\bm{\\xi}_l$ as in Figure~\\ref{fig:affine_coupling}.}\n    \\label{tab:affine_coupling}\n    \\centering\n    \\begin{tabularx}{\\textwidth}{ X | X }\n        Forward & Reverse\n        \\\\\\hline\n        \\begin{tabular}[t]{@{}l@{}}\n            $\\mathbf{y}_1', \\mathbf{y}_2' = \\texttt{split}(\\mathbf{y}')$\\\\\n            $\\hat{\\mathbf{y}}_1 = \\texttt{concat}(\\mathbf{y}_1', \\bm{\\xi}_l)$\\\\\n            $(\\hat{\\mathbf{s}}, \\mathbf{t}) = \\texttt{CouplingNN}(\\hat{\\mathbf{y}}_1)$\\\\\n            $\\mathbf{s} = \\texttt{sigmoid}(\\hat{\\mathbf{s}} + 2)$\\\\\n            $\\mathbf{z}_2' = \\mathbf{s} \\odot \\mathbf{y}_2' + \\mathbf{t}$\\\\\n            $\\mathbf{z}_1' = \\mathbf{y}_1'$\\\\\n            $\\mathbf{z}' = \\texttt{concat}(\\mathbf{z}_1', \\mathbf{z}_2')$\n        \\end{tabular}\n        &\n        \\begin{tabular}[t]{@{}l@{}}\n            $\\mathbf{z}_1', \\mathbf{z}_2' = \\texttt{split}(\\mathbf{z}')$\\\\\n            $\\hat{\\mathbf{z}}_1 = \\texttt{concat}(\\mathbf{z}_1', \\bm{\\xi}_l)$\\\\\n            $(\\hat{\\mathbf{s}}, \\mathbf{t}) = \\texttt{CouplingNN}(\\hat{\\mathbf{z}}_1)$\\\\\n            $\\mathbf{s} = \\texttt{sigmoid}(\\hat{\\mathbf{s}} + 2)$\\\\\n            $\\mathbf{y}_2' = (\\mathbf{z}_2' - \\mathbf{t}) \/ \\mathbf{s}$\\\\\n            $\\mathbf{y}_1' = \\mathbf{z}_1'$\\\\\n            $\\mathbf{y}' = \\texttt{concat}(\\mathbf{y}_1', \\mathbf{y}_2')$\n        \\end{tabular}\n    \\end{tabularx}\n\\end{table}\n\n\n\\subsection{Results on higher input dimension}\nWe also trained the conditional Glow with $4096$ samples from GRF KLE$256$ over $32\\times 32$ grid. The prediction results for two test inputs are shown in Fig.~\\ref{fig:cglow_kle256_grid32}.\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.47\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle256_grid32\/prediction\/pred_epoch400_408.png}\n        \\caption{Test realization $1$.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.47\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle256_grid32\/prediction\/pred_epoch400_284.png}\n        \\caption{Test realization $2$.}\n    \\end{subfigure}\n      \n    \\caption{Prediction of the multiscale conditional Glow ($\\beta=200$) for two test inputs in (a) and (b) which are sampled from GRF KLE$256$ over $32\\times 32$ grid. The predictive mean ($2$nd row) and one standard derivation ($3$rd row) are obtained with 20 output samples. The first row shows three simulated output fields, and the fourth row shows the error between the simulation and predictive mean. The relative $L_2$ error for the predicted pressure field is $0.019875$, evaluated on 512 test samples from GRF KLE256.}\n    \\label{fig:cglow_kle256_grid32}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:Conclusions}\nThis paper has offered a foray in physics-aware machine learning for surrogate modeling and uncertainty quantification, with emphasis on the solution of PDEs. The most significant contribution of the proposed framework and simultaneously the biggest difference with other efforts along these lines, is that \\textit{no labeled data} are needed i.e. one does not need to solve governing PDEs for the training inputs. This is accomplished by incorporating appropriately the governing equations into the loss\/likelihood functions. We have demonstrated that convolutional encoder-decoder network-based surrogate models can achieve high predictive accuracy for high-dimensional stochastic input fields. Furthermore, the generalization performance of the physics-constrained surrogates proposed is consistently better than data-driven surrogates for out-of-distribution test inputs. The probabilistic surrogate built on the flow-based conditional generative model and trained by employing the reverse KL-divergence loss, is able to capture predictive uncertainty as demonstrated in several uncertainty propagation and calibration tasks.\n\nMany important unresolved tasks have been identified that will be addressed in forthcoming works. They include \n(a) Extension of this work to surrogate modeling for dynamical systems, (b) Improving generalization on out-of-distribution input, e.g. fine-tuning the trained surrogate on test input~\\cite{devlin2018bert, radford2018improving}, learned gradient update~\\cite{adler2017solving, hammernik2018learning}, meta-learning on a distribution of regression tasks~\\cite{finn2017model}, etc., (c) Combining physics-aware and data-driven approaches when only limited simulation data and partially known physics are available~\\cite{yang2018physics}, (d) Scale the flow-based conditional generative models to higher dimensions~\\cite{2018arXiv181001367G}, (e) More reliable probabilistic models, e.g. being able to express what the model does not know~\\cite{nalisnick2018deep, choi2018generative} by showing larger predictive uncertainty when tested on out-of-distribution input, (f) Exploring ways to increase the expressiveness of FC-NNs to better capture the multiscale features of PDE solutions, e.g. by evolving network architectures~\\cite{stanley2002evolving} and (g) Exploring the solution landscape with the conditional generative surrogates~\\cite{farrell2015deflation}.\n \n\\section{Numerical Experiments}\\label{sec:numerical_experiments}\n\n\\paragraph{Model problem} Steady-state flow in random heterogeneous media is studied as the model problem throughout the experiments, as in Eqs.~(\\ref{eq:darcy}),~(\\ref{eq:darcy_mixed}),~(\\ref{eq:BCs}). We consider the domain $\\mathcal S = [0, 1] \\times [0, 1]$, the left and right boundaries are Dirichlet, with pressure values $1$ and $0$, respectively. The upper and lower boundaries are Neumann, with zero flux. The source field is zero.\n\n\n\\paragraph{Dataset} Only input samples are needed to train the physics-constrained surrogates (PCSs). Additional simulated output data for training data-driven surrogates (DDSs) and evaluating surrogate performance are obtained with FEniCS~\\cite{AlnaesBlechta2015a}. Here, we mainly introduce three types of input datasets, which are Gaussian random field (GRF), warped GRF, and channelized field.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figs\/codec\/generalization\/input_distributions.png}\n    \\caption{Samples from $5$ test input distributions over a $64\\times 64$ uniform grid, i.e. GRF KLE$512$, GRF KLE$128$, GRF KLE$2048$, warped GRF, channelized field. Log   permeability samples are shown except the last channelized field that is defined with binary values $0.01$ and $1.0$.}\n    \\label{fig:generalization_test_input_samples}\n\\end{figure}\n\nThe first input dataset is the exponential of a GRF, i.e. $K(s) = \\exp(G(s))$, $G(\\cdot) \\sim \\mathcal{GP}(0, k(\\cdot, \\cdot))$, where $k(s, s') = \\exp(-\\norm{s - s'}_2 \/ l)$, $l$ is the length scale. The field realization is generated with Karhunen-Lo\\`eve expansion (KLE) with the leading $N$ terms, paired with Latin hypercube sampling. See Section 4.1 in~\\cite{zhu2018bayesian} for more details. This type of dataset is called \\texttt{GRF KLE$N$}. For the deterministic surrogate experiments in Section~\\ref{sec:deterministic_surrogate}, the training input GRF KLE$512$ is generated with length scale $l=0.25$, $N=512$ leading terms, discretized over a $64 \\times 64$ uniform grid,  which accumulates $95.04\\%$ energy. For the probabilistic surrogate in Section~\\ref{sec:prob_surrogate}, the parameters for the training input GRF KLE$100$ are $N=100$, $l=0.2$, over $32 \\times 32$ uniform grid.\nThe test set may have other KLE truncations, but with the same length scale in each case, i.e. $l=0.25$ for $64 \\times 64$, and $l=0.2$ for $32\\times 32$. The dataset for uncertainty propagation consists of 10,000 input-output data pairs unseen during training.\n\nA slightly different test input is \\texttt{warped GRF}, where there are two Gaussian fields and the output of the first GRF is the input to the second GRF. The kernel for both GRFs is squared exponential kernel, the length scale and KLE terms are $2, 16$ for the first GRF and $0.1, 128$ for the second GRF.\n\nThe last type of input field considered is a  \\texttt{channelized} field. Samples are obtained by cropping $64\\times 64$ patches from one large training image~\\cite{laloy2018training} of size $2500\\times 2500$, or $32\\times 32$ patches from the resized $1250\\times 1250$ image (resized with nearest neighborhood). Typical samples of the input datasets considered are shown in Fig.~\\ref{fig:generalization_test_input_samples}.\n\n\nWe begin our experiments by solving deterministic PDEs with spatially-varying coefficient (input) with convolutional decoder networks, and compare with FC-NNs. Then we show experiments for surrogate modeling for solving random PDEs, and compare with the data-driven approach. The last part is on experiments of using the conditional Glow as our probabilistic surrogate for uncertainty quantification tasks. The code and datasets for this work will become available at \\url{https:\/\/github.com\/cics-nd\/pde-surrogate} upon publication.\n\n\n\\subsection{Solving Deterministic PDEs}\n\\label{sec:solve_det_pde}\nIn this section, we explore the relative merit of using CNNs and FC-NNs to parameterize the solutions of deterministic PDEs with image-like input field, including both linear and nonlinear PDEs. Since our focus is on surrogate modeling, the results below are mostly qualitative. The network architectures and training details are described in~\\ref{appendix:solve}.\n\n\\begin{figure}[h!]\n    \\centering\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/fc_mixed_residual_pred_epoch500_8_k1024.png}\n        \\caption{FC-NN, iteration $500$.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/fc_mixed_residual_pred_epoch2000_8_k1024.png}\n        \\caption{FC-NN, iteration $2000$.}\n    \\end{subfigure}\n    \n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/conv_mixed_residual_pred_epoch250_8_k1024.png}\n        \\caption{CNN, iteration $250$.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/conv_mixed_residual_pred_epoch500_8_k1024.png}\n        \\caption{CNN, iteration $500$.}\n    \\end{subfigure}\n\n    \\caption{Solving Darcy flow for one sample from GRF KLE1024 under mixed residual loss. FC-NN takes much longer to resolve the fine details of flux fields, while the pressure field does not improve much. CNN obtains more accurate solution within shorter time.}\n    \\label{fig:solve_k1024}\n\\end{figure}\n\\paragraph{Comparison of CNNs and FC-NNs to solve Darcy flow} \nWe compare convolutional decoder networks and fully-connected networks presented in Section~\\ref{sec:solve_det_pde} to solve the PDE system in Eq.~(\\ref{eq:darcy}). The input permeability field is sampled from GRF KLE$1024$ over a $64 \\times 64$ uniform grid. We optimize the CNN and the FC-NN with mixed residual loss using L-BFGS optimizer for $500$ and $2000$ iterations, respectively. The results are shown in Fig.~\\ref{fig:solve_k1024}. The solution learned with CNN in iteration $250$ is even better than the solution learned with FC-NN in iteration $2000$, in terms of accuracy and retaining multiscale features of the flux fields. The same phenomenon is observed for input GRFs with other intrinsic dimensionalities. We further experiment on input sampled from the  channelized field, as shown in Fig.~\\ref{fig:solve_channel}. For this  case, however, we observe that  the FC-NN fails to converge to a small enough error in contrast to the CNN.\n\\begin{figure}[h!]\n    \\centering\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/conv_mr_pred_epoch500_4105_channel.png}\n        \\caption{CNN, iteration $500$.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/fc_mr_pred_epoch2000_4105_channel.png}\n        \\caption{FC-NN, iteration $2000$.}\n    \\end{subfigure}\n    \\caption{Solving Darcy flow for one sample of the channelized field. The same network and training setup as in Fig.~\\ref{fig:solve_k1024} are used. The FC-NN parameterization fails to converge.}\n    \\label{fig:solve_channel}\n\\end{figure}\n\nThe experiments on solving deterministic PDEs show that CNNs can capture the multiscale features of the solution much more effectively than the FC-NNs, as reflected by the resolved flux fields. This is mostly because of the difference in their parameterizations of a field solution and the ways to obtain spatial gradients. FC-NNs turn to generate images that look like light-paintings\\footnote{https:\/\/distill.pub\/2018\/differentiable-parameterizations\/\\#section-xy2rgb}, but not rugged field. More broadly, this type of parameterization is intensively explored named compositional pattern producing networks~\\cite{stanley2007compositional}.\nCNNs can represent images with multiscale features quite efficiently as is evident in our experiments and the rapid advances in image generation applications. \nDue to the discretization of spatial gradients with Sobel filters, the error of the learned solution is mainly on the boundaries, and the checkerboard artifact becomes more severe in the pressure field as the flux fields becomes more rugged, as shown in Fig.~\\ref{fig:solve_k4096} in~\\ref{appendix:solve}.\n\n\\paragraph{Nonlinear flow in porous media}\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/nonlinear_darcy_conv_mr_pred_epoch500_8_k1024.png}\n        \\caption{GRF KLE$1024$.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/solve\/nonlinear_darcy_conv_mr_pred_epoch500_1236_channel.png}\n        \\caption{Channelized field.}\n    \\end{subfigure}\n    \\caption{Simulation (FEniCS) and learned solution (prediction) with CNN for the nonlinear flow for (a) GRF KLE1024 with $\\alpha_1=0.1$ and $\\alpha_2=0.1$, and (b) channelzied field with $\\alpha_1=1.0$ and $\\alpha_2=1.0$.}\n    \\label{fig:darcy_nonlinear_solve}\n\\end{figure}\nDarcy's law $\\bm{\\tau} = - K \\nabla u$ is a well established linear constitutive relationship for flow through porous media when the Reynolds number $Re$ approaches zero. It has been shown both theoretically~\\cite{mei1991effect} and experimentally~\\cite{firdaouss1997nonlinear, rojas1998nonlinear} that the constitutive relation undergoes a cubic transitional regime at low $Re$, and then a quadratic Forchheimer~\\cite{forchheimer1901wasserbewegung} when $Re\\sim O(1)$. To show that our approach also works for nonlinear PDEs, we look at the nonlinear correction of Darcy's law as the following\n\\begin{equation}\\label{eq:constitutive_nonlinear}\n    - \\nabla u = \\frac{1}{K} \\bm{\\tau} + \\frac{\\alpha_1}{K^{\\frac{1}{2}}} \\bm{\\tau}^2 + \\alpha_2 \\bm{\\tau}^3,\n\\end{equation}\nwhere $\\alpha_1, \\alpha_2$ are usually obtained by fitting to experiment data.\nWe use CNNs to solve this nonlinear flow with the constitutive Eq.~(\\ref{eq:constitutive_nonlinear}), the continuity equation $\\nabla \\cdot \\bm{\\tau} = 0$ and the same boundary condition with the linear Darcy case. The reference solution is obtained with FEniCS (dual mixed formulation with Newton solver that converges in $5 \\sim 6$ iterations with relative tolerance below $10^{-6}$). We experiment on input fields from GRF KLE$1024$ and the channelized field, with $\\alpha_1=0.1$ and $\\alpha_2=0.1$ in the first case, and $\\alpha_1=1.0$ and $\\alpha_2=1.0$ in the second case.\nThe convolutional decoder network is the same as in the previous section, and is trained with mixed residual loss. The results is shown in Fig.~\\ref{fig:darcy_nonlinear_solve}.\n\nFor GRF KLE$1024$, the effect of the cubic constitutive relation is actually smoothing out the flux field in comparison to the linear case in Fig.~\\ref{fig:solve_k1024} using the same input field. The nonlinearity of PDEs does not seem to increase the burden for the CNN training except for a few more steps of forward and backpropagation due to the nonlinear operations in the constitutive equation. This is a negligible cost w.r.t. the computations in the decoder network itself. However, note that solving nonlinear PDEs with the Newton solver requires $N$ iterations, thus increasing the computation by $N$ times.\nFor surrogate modeling, the mapping that the CNN learns from $K$ to $u$ is nonlinear even when the PDE to solve is linear.\nWe expect it will be easier to learn a surrogate in the nonlinear case due to the smoother output fields. We leave further investigation of surrogate modeling and uncertainty quantification for nonlinear stochastic PDEs for our future work.\n \n\n\n\n\\subsection{Deterministic Surrogate}\\label{sec:exp_det_surrogate}\n\nThe experiments in solving deterministic PDEs lead us to choose CNNs over FC-NNs for surrogate modeling, with less training time and comparable accuracy, especially for high-dimensional input. We train both the physics-constrained surrogates and data-driven surrogates, and compare their accuracy and generalizability.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{figs\/codec.png}\n    \\caption{Dense convolutional encoder-decoder network as the deterministic surrogate. The model's input is the realization of a random field, the model's output is the prediction for each input field including 3 output fields, i.e. pressure and two flux fields. The model is trained with physics-constrained loss without target data.}\n    \\label{fig:codec}\n\\end{figure}\n\\paragraph{Network}\nDense convolutional encoder-decoder network~\\cite{zhu2018bayesian} is used as the surrogate model, with one input channel $\\mathbf{x}$ and three output channels $[\\mathbf{u}, \\bm{\\tau}_1, \\bm{\\tau}_2]$, as shown in Fig.~\\ref{fig:codec}.\nThe upsampling method in the decoding layers in the current implementation is nearest upsampling followed by convolution, different from transposed convolution used in the data-driven case. This is essential to avoid the checkerboard effect\\footnote{https:\/\/distill.pub\/2016\/deconv-checkerboard\/}, partially severed by Sobel filter besides the natural tendency of transposed convolution.\nThe resolution of the input fields is reduced by 4 times through the encoding path, from $64\\times 64$ to $16 \\times 16$, then increased to the size of the output fields, $64\\times 64$. The number of layers in the three dense blocks are $6, 8, 6$, with growth rate $16$. There are $48$ initial feature maps after the first convolution layer.\n\n\\paragraph{Training}\nWe train the PCS with mixed residual loss as in Eq.~(\\ref{eq:mixed_residual_impl}) with only input data, and compare it with the DDS with the same network architecture but trained with additional output data. The number of training data, mini-batch size and the category of test distributions vary in different experiments, but all with $T=512$ test data and employing the Adam~\\cite{kingma2014adam} optimizer paired with one cycle policy\\footnote{\\url{https:\/\/github.com\/fastai\/fastai\/blob\/master\/fastai\/callbacks\/one\\_cycle.py}} (learning rate scheduler) where the maximum learning rate is $0.001$. The mini-batch size ranges from $8$ to $32$ depending on the number of training data. The weight coefficient for the boundary conditions is $\\lambda = 10$. \nThe evaluation metrics for prediction are relative $L_2$ error and $R^2$ score,\n\\begin{equation}\n    \\epsilon_j = \\frac{1}{T} \\sum_{i=1}^T \\frac{\\norm{\\hat{\\mathbf{y}}_j^{(i)} - \\mathbf{y}_j^{(i)}}_2}{\\norm{\\mathbf{y}_j^{(i)}}_2}, \\qquad R^2_j = 1 - \\frac{\\sum_{i=1}^T \\norm{\\hat{\\mathbf{y}}_j^{(i)} - \\mathbf{y}_j^{(i)}}_2^2}{\\sum_{i=1}^T \\norm{\\mathbf{y}_j^{(i)} - \\bar{\\mathbf{y}}_j}_2^2},\n\\end{equation}\nwhere $\\hat{\\mathbf{y}}_j^{(i)}$ is the surrogate prediction of the $j$-th output channel\/field ($j=1,2,3$ for pressure, horizontal flux and vertical flux field respectively), $\\mathbf{y}_j^{(i)}$ is the corresponding simulator output, $\\bar{\\mathbf{y}}_j=\\frac{1}{T}\\sum_{i=1}^T \\mathbf{y}_j^{(i)}$, $T$ is the total number of test inputs, $\\norm{\\cdot}_2$ is the $L_2$ norm. We mainly use relative $l_2$ error as evaluation metric.\nThe PCS is trained for $300$ epochs and the DDS is trained for $200$ epochs, since DDS is faster to converge than the PCS in general, as shown in Fig.~\\ref{fig:training_curve}.\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/codec\/mixed\/test_nrmse_ntrain8192_run8_1.pdf}\n        \\caption{Test relative $L_2$ error.}\n        \\label{fig:test_nrmse_training_curve}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/codec\/mixed\/test_r2_ntrain8192_run8_1.pdf}\n        \\caption{Test $R^2$ score.}\n        \\label{fig:test_r2_training_curve}\n    \\end{subfigure}\n    \\caption{Test relative $L_2$ error and $R^2$ score during training. The solid lines shows the error for the PCSs and the dashed lines for the DDSs. Both surrogates are trained on $8192$ samples of GRF KLE$512$ and tested on the same $512$ samples of GRF KLE$512$.}\n    \\label{fig:training_curve}\n\\end{figure}\n\n\n\n\\paragraph{Prediction} \n\nTo show that the physics-constrained approach to learn surrogate works well, we train the PCS on two datasets, i.e. GRF KLE$512$ ($8192$ samples) and channelized fields ($4096$ samples), respectively.\nThe prediction examples of the PCS for test GRFs and channelized fields are shown in Fig.~\\ref{fig:PCS_mixed_res_pred_examples}.\n\\begin{figure}[h!]\n    \\centering\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/codec\/mixed\/kle512\/pred_epoch300_30.png}\n        \\caption{GRF KLE$512$, test $1$.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/codec\/mixed\/kle512\/pred_epoch300_34.png}\n        \\caption{GRF KLE$512$, test $2$.}\n        \\label{fig:PCS_mixed_res_pred_examples_kle512_sample2}\n    \\end{subfigure}\n    \n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/codec\/mixed\/channel\/pred_epoch300_52.png}\n        \\caption{Channelized, test $1$.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/codec\/mixed\/channel\/pred_epoch300_23.png}\n        \\caption{Channelized, test $2$.}\n    \\end{subfigure}\n    \n    \\caption{Prediction examples of the PCS under the mixed residual loss. (a) and (b) are $2$ test results for the PCS trained with $8192$ samples of GRF KLE$512$; (c) and (d) are $2$ test results for the PCS trained with $4096$ samples of channelized fields.}\n    \\label{fig:PCS_mixed_res_pred_examples}\n\\end{figure}\n\n\nWe show the test relative $L_2$ error and $R^2$ score during training in Fig.~\\ref{fig:training_curve}. Overall the PCS takes longer to converge than the DDS, which is reasonable since the PCS has to solve the PDE and learn the surrogate mapping at the same time. Compared with the DDS, the accuracy of the PCS' predictions of the pressure field are similar when trained with the same number of data, but the PCS' predictions of the  flux fields are worse. For the later case, the evaluation metric is dominated by the error on the boundary which is induced by the approximation of spatial derivatives. However, the predictions within the boundary are accurate, as shown in Fig.~\\ref{fig:PCS_mixed_res_pred_examples}. Also the relative $L_2$ error is more sensitive than $R^2$ when the error is small, which can be seen by comparing Figs.~\\ref{fig:test_nrmse_training_curve} and~\\ref{fig:test_r2_training_curve}.\n\n\n\n\\remark{The quantitative results are mainly for the pressure field, not the flux fields even through we use the mixed formulation loss to train the model. Using the loss functions in Eqs.~(\\ref{eq:loss_det_surr}) and~(\\ref{eq:loss_det_surr_data}), we observe that the DDS focuses more on the flux fields than the pressure field, but the PCS has better predictability on the pressure field, which is often desirable. For the PCS trained with the mixed formulation, we can either output the pressure and flux fields directly, or re-compute the flux field with the predicted pressure field using the constitutive equation. The other reason for using the mixed residual loss over the primal variational loss is the better predictive accuracy of the pressure field.}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figs\/codec\/nrmse_pressure_grf_kle512.pdf}\n    \\caption{The relative $L_2$ error of the predicted pressure field of physics-constrained and data-driven surrogates trained with $512, 1024, 2048, 4096, 8192$ GRF KLE$512$ data, each with $5$ runs to obtain the error bars. The test set contains $512$ samples from GRF KLE$512$ as well. We emphasize that training the DDS requires an equal number of output data i.e.  solutions of the governing PDE. The reference to compute relative $L_2$ error is simulated with FEniCS.} \n    \\label{fig:nrmse_pressure}\n\\end{figure}\n\\paragraph{Varying the number of training inputs} We train the PCS with different number of samples from GRF KLE$512$, and compare its predictive performance against the DDS in Fig.~\\ref{fig:nrmse_pressure}. From the figure, the relative $L_2$ error decreases as the PCS is trained with more input data. While this is not surprising, it shows the convergence behavior of physics-constraint learning approach. Moreover, the PCS achieves similar relative $L_2$ error of predicted pressure field with the DDS when there are enough training input samples, and even lower when the number of training input samples is $8192$. \n\nThe common requirement for data-driven modeling of physical systems is data efficiency, since we need \\textit{expensive} simulated output data to supervise the training. Taking~\\cite{zhu2018bayesian} for an example, the number of training data is often less than $1024$. The comparison here is not really appropriate. The DDS does not require physics while the PCS does not require output data.\nOverall, Fig.~\\ref{fig:nrmse_pressure} suggests that with physical knowledge, we can achieve \\textit{comparable} predictive performance with the state-of-the-art DDS \\textit{without} any simulation output (but only samples from the random input).\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figs\/codec\/generalization\/pressure_ntrain8192.pdf}\n    \\caption{Generalization to new input distributions, which are GRF KLE$128$, KLE$512$ (interpolation), KLE$2048$, warped GRF, and channelized fields. The surrogates are trained with $8192$ samples from GRF KLE$512$. Each test set contains $512$ samples.}\n    \\label{fig:generalization_nrmse_pressure_ntrain8192}\n\\end{figure}\n\\paragraph{Generalization} Apart from computational time, the PCS can `generalize' to \\textit{any} input by directly solving the governing equations, i.e. minimizing the loss function in Eq.~(\\ref{eq:loss_det_surr}) over this particular input, as shown to work properly in Section~\\ref{sec:solve_det_pde}.\nThus generalization here evaluates how accurate the model's prediction is when we need to predict fast, e.g. pass the input through the surrogate, or fine-tuning the surrogate for few steps.\n\nFigure~\\ref{fig:nrmse_pressure} shows the surrogates' interpolation performance for the test input from the same distribution as the training input, i.e. GRF KLE$512$. \nHere, we further examine the surrogates' \\textit{extrapolation} to out-of-distribution input. We select two other GRFs with different KLE terms, in particular we take KLE$128$ which is smoother than KLE$512$ and KLE$2048$ that leads to higher-variability than KLE$512$. The third test input is warped GRF which is two layers of Gaussian processes. The fourth test input is the channelized field. The samples from those test distributions are shown in Fig.~\\ref{fig:generalization_test_input_samples}. \n\nWe take the surrogates trained on GRF KLE$512$ as in the previous experiment, and test them on the four new input distributions. The relative $L_2$ error of predicted pressure field is shown in Fig.~\\ref{fig:generalization_nrmse_pressure_ntrain8192} for the surrogates trained with $8192$ data. The figure shows both PCSs and DDSs generalize well to other test GRF input, including the warped one, but less so when it comes to the channelized field, which is completely different from the training input. \nNotably, the PCS has better generalization than the DDS when tested on warped GRF and channelized fields, which are further away from the training input distribution than the other two GRFs. This is highlighted in Fig.~\\ref{fig:channel_nrmse_pressure}. This holds as well for surrogates trained on $512, 1024, 2048, 4096$ samples. Figure~\\ref{fig:generalization_nrmse_pressure_ntrain4096} shows the generalization performance when the training sample size is $4096$.\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/codec\/nrmse_pressure_channel.pdf}\n        \\caption{}\n        \\label{fig:channel_nrmse_pressure}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/codec\/generalization\/pressure_ntrain4096.pdf}\n        \\caption{}\n        \\label{fig:generalization_nrmse_pressure_ntrain4096}\n    \\end{subfigure}\n    \\caption{(a) The relative $L_2$ error of predicted pressure field with PCSs and DDSs trained with $512, 1024, 2048, 4096, 8192$ GRF KLE512 data (the same surrogates as Fig.~\\ref{fig:nrmse_pressure}), each with $5$ runs. The test set contains $512$ samples from channelized field, with completely different distribution from the training GRF. (b) Generalization across new test input distributions for surrogates trained with $4096$ samples from GRF KLE$512$.}\n\\end{figure}\n\n   \n  \n\n\\subsection{Probabilistic Surrogate}\\label{sec:prob_surrogate}\nThis section presents the experiments on using the conditional Glow model shown in Fig.~\\ref{fig:glow_msc} as the probabilistic surrogate. We are interested in how the conditional Glow captures predictive uncertainty, uncertainty calibration and its generalization performance to unseen test input. We choose to work on $32\\times 32$ discretization instead of $64\\times 64$ with the input GRF KLE$100$ because of the large model size of the current implementation of the model.\n\n\\paragraph{Network} In our experiment, we use $L=3$ levels, each of which contains $F=6$ steps of flow. Both the dense blocks and coupling networks \\fbox{$\\mathbf{s}$} and \\fbox{$\\mathbf{t}$} in affine coupling layers use DenseNet~\\cite{huang2017densely} as the building block. The number of dense layers within each dense block in the encoder is $3, 4, 4$ (from the input to the latent direction). The coupling networks $\\texttt{CouplingNN}$ as in Table~\\ref{tab:affine_coupling} for scaling and shift have $3$ dense layers, followed by a $3\\times3$ convolution layer with zero initialization to reduce the number of output features to be the same as its input features. The model has $1,535,549$ parameters, including $179$ convolution layers. For other hyperparameters of the model, please refer to our open-source code.\n\n\\paragraph{Training} The model is trained with $4096$ input samples from GRF KLE$100$ over $32 \\times 32$ grid for $400$ epochs with mini-batch size $32$. No output data is needed for training.  We use the Adam optimizer with initial learning rate $0.0015$, and one-cycle learning rate scheduler. The weight for boundary conditions $\\lambda$ is $50$. The inverse temperature $\\beta$ is prefixed to certain values. Training the model with the above setting on a single NVIDIA GeForce GTX $1080$ Ti GPU card takes about $3$ hours.\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/pred_at_x\/pred_epoch400_497.png}\n        \\caption{Predictive mean and variance.}\n        \\label{fig:cflow_pred_at_x}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/pred_at_x\/epoch400_samples_c0_index497.png}\n        \\caption{Samples of predicted pressure.}\n        \\label{fig:cflow_pred_at_x_samples_pressure}\n    \\end{subfigure}\n    \n    \\begin{subfigure}[b]{0.47\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/pred_at_x\/epoch400_samples_c1_index497.png}\n        \\caption{Samples of predicted horizontal flux.}\n        \\label{fig:cflow_pred_at_x_samples_hor_flux}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.47\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/pred_at_x\/epoch400_samples_c2_index497.png}\n        \\caption{Samples of predicted vertical flux.}\n        \\label{fig:cflow_pred_at_x_samples_ver_flux}\n    \\end{subfigure}\n    \\caption{Prediction of the multiscale conditional Glow ($\\beta=150$) for a test input which is sampled from GRF KLE$100$ over $32\\times 32$ grid. (a) The predictive mean ($2$nd row) and one standard derivation ($3$rd row) are obtained with 20 output samples. The first row show three simulated output fields, the $4$-th row shows the error between the reference and the predictive mean. In (b), (c), (d), the top left corner shows the simulated output, the rest $15$ are samples from the conditional predictive density $p_\\bm{\\theta}(\\mathbf{y} | \\mathbf{x})$. The relative $L_2$ error for the predicted pressure field is $0.0038$ when tested on 512 samples from GRF KLE$100$.}\n    \\label{fig:cflow_kle100_grid32}\n\\end{figure}\n\\paragraph{Predictive distribution}  Fig.~\\ref{fig:cflow_kle100_grid32} shows the prediction for a test input from GRF KLE100, where in Fig.~\\ref{fig:cflow_pred_at_x} the predictive mean and variance are estimated pixel-wise with $20$ samples from the conditional density by sampling $20$ realizations of noise $\\{\\bm{\\epsilon}_l^{(i)}\\}_{l=1, i=1}^{L, 20}$ as in Algorithm~\\ref{algo:cglow}. \nThe test relative $L_2$ error for the pressure field (comparing predictive mean against simulated output) achieves $0.0038$, which is comparable to the relative $L_2$ error of the deterministic surrogate ($0.0035$). The predictive variance of the pressure  and vertical flux fields reflect correctly the boundary conditions, which are close to zero on the left-right boundaries and top-bottom boundaries, respectively. We  also draw $15$ samples from the predictive distribution for each output field, which are shown in Figs.~\\ref{fig:cflow_pred_at_x_samples_pressure},~\\ref{fig:cflow_pred_at_x_samples_hor_flux},~\\ref{fig:cflow_pred_at_x_samples_ver_flux}. The predictive output samples are still diverse despite the predictive mean being highly accurate. Mode collapse is a well-known problem for conditional GANs~\\cite{anonymous2019diversity-sensitive, zhu2017toward} and VAEs~\\cite{anonymous2019lagging}, which seems not much of a concern for flow-based generative models as demonstrated with the diversity of samples.\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/pred_mean_vs_MC}\n        \\caption{Estimate of output mean.}\n        \\label{fig:cglow_up_mean}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/pred_var_vs_MC}\n        \\caption{Estimate of output variance.}\n    \\end{subfigure}\n    \\caption{Uncertainty propagation with multiscale conditional Glow, $\\beta=150$. (a) The first row shows the sample mean of $10,000$ simulated output, the second and third rows show the sample mean and two standard deviation of the estimate mean of $10,000$ predicted output with the probabilistic surrogate, and the fourth row shows the error between the first two rows. (b) The results for output variance.}\n    \\label{fig:cflow_up}\n\\end{figure}\n\\paragraph{Uncertainty propagation}\nWe use the trained conditional Glow as a surrogate to quickly predict the output for $10,000$ input samples from GRF KLE100, then compute the mean and variance of the estimated output mean, and output variance, then compare against the Monte Carlo estimate using the corresponding $10,000$ simulated output. We generate 20 samples for each input with the trained surrogate, then estimate the mean and variance of the output with the law of total expectation and the law of total variance. By repeating this process for 10 times, we obtain 10 estimates of the mean and variance for the output. Then the sample mean and variance of the 10 estimate means and estimate variances can be computed, which are shown in the second and third row of Fig.~\\ref{fig:cflow_up}. The statistics of the surrogate output matches that of the simulation output very well, especially for the output variance which are typically underestimated when using surrogates. Note that there is only small error (around $3 \\%$ relative error) between the estimated mean of the horizontal flux field despite the noticeable difference in color as in Fig.~\\ref{fig:cglow_up_mean}.\n\n\\paragraph{Distribution estimate} We show in Fig.~\\ref{fig:cflow_dist_est} the kernel density estimation for the values of three output fields at random locations in the domain using the $10,000$ output samples from simulation and the ones propagated with the trained conditional Glow. \n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[b]{0.95\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/dist_est\/loc_2875_8875.pdf}\n        \\caption{Distribution estimation at location $(0.2875, 0.8875)$.}\n    \\end{subfigure}\n    \n    \\begin{subfigure}[b]{0.95\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/dist_est\/loc_4625_2875.pdf}\n        \\caption{Distribution estimation at location $(0.4625, 0.2875)$.}\n    \\end{subfigure}\n    \\caption{Distribution estimate with conditional Glow, $\\beta=150$. From left to right shows the density estimate of pressure, horizontal flux and vertical flux at certain locations of the domain $[0, 1]^2$. }\n    \\label{fig:cflow_dist_est}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\centering\n    \\begin{subfigure}[b]{0.7\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/entropy_nrmse_beta_pressure}\n    \\caption{}\n    \\label{fig:cglow_entropy_nrmse_vs_beta}\n    \\end{subfigure}\n    \n    \\begin{subfigure}[b]{0.8\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/reliability_diagram_pressure}\n    \\caption{}\n    \\label{fig:cglow_reliability}\n    \\end{subfigure}\n    \\caption{(a) Conditional entropy of $p_\\bm{\\theta}(\\mathbf{y}|\\mathbf{x})$ and relative $L_2$ error of predicted pressure field w.r.t. $\\beta$. Conditional entropy is evaluated in bits per dimension.\n    The surrogate is tested on $512$ input samples from GRF KLE$100$. The error bar is obtained with $3$ independent runs. (b) Reliability diagram of predicted pressure field with conditional Glow trained with different $\\beta$, which is evaluated with $10,000$ input-output data pairs. The closer the diagram is to the diagonal, the better the probabilistic surrogate is calibrated.}\n    \\label{fig:cglow_uncertainty}\n\\end{figure}\n\\paragraph{Uncertainty calibration by tuning $\\beta$} \nGiven the PDEs and boundary conditions, the prediction of the surrogate can be evaluated directly with the loss $L(\\mathbf{y}, \\mathbf{x})$, without requiring the reference solution (e.g. simulation output). However, this loss cannot be readily translated to the uncertainty of the solution, e.g. the upper and lower bound of the solution at every grid point in the domain. The probabilistic surrogate trained under the reverse KL divergence can provide the uncertainty estimate, but may be at the expense of the accuracy of the mean prediction. The precision parameter $\\beta$ controls the overall variance of the reference density, which is reflected from the conditional entropy of the model density $p_\\bm{\\theta}(\\mathbf{y} | \\mathbf{x})$ in Fig.~\\ref{fig:cglow_entropy_nrmse_vs_beta}. The influence of $\\beta$ on the accuracy and the entropy of the model can be seen from the two competing terms in the reverse KL divergence as well. Larger $\\beta$ puts more penalty of the PDE loss term $L(\\mathbf{y}, \\mathbf{x})$ and less on the negative conditional entropy, thus the predictions become more accurate but less diverse, and to some extent, the probabilistic surrogate becomes over confident, as shown in Fig.~\\ref{fig:cglow_reliability} when $\\beta=250$. On the other hand, when $\\beta$ is too small, the probabilistic surrogate is prudent (large uncertainty estimate) and less accurate about the solution, e.g. the case of $\\beta=50$. From the figure, the model trained under $\\beta=150$ is well-calibrated (its reliability diagram is close to the diagonal dashed line) and achieves high accuracy at the same time. \n\n\\paragraph{Generalization}\nWe test the generalization of conditional Glow on input distributions different from the training input (GRF KLE$100$), including GRF KLE$256$, GRF KLE$512$, warped GRF, and channelized fields, as in Fig.~\\ref{fig:cglow_generalization}. However, we could not observe larger uncertainty when the test input is far away from the training input. The error between the predictive mean and simulation is in general one magnitude larger than the uncertainty. Thus the current surrogate cannot express what it does not known which in practice is a highly desirable outcome.\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/generalization\/k256_pred_epoch400_110}\n        \\caption{GRF KLE$256$.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/generalization\/k512_pred_epoch400_110}\n        \\caption{GRF KLE$512$.}\n    \\end{subfigure}\n    \n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/generalization\/warped_grf_pred_epoch400_146}\n        \\caption{Warped GRF.}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/cflow\/kle100_grid32\/generalization\/channel_pred_epoch400_223}\n        \\caption{Channelized field.}\n    \\end{subfigure}\n    \\caption{Generalization of conditional Glow to out-of-distribution input. The model is trained on GRF KLE$100$, and tested on (a) GRF KLE$256$, (b) GRF KLE$512$, (c) Warped GRF, (d) channelized field. Note that the results are cherry-picked.}\n    \\label{fig:cglow_generalization}\n\\end{figure}\n\n\n \n\\section{Introduction}\n\nSurrogate modeling is computationally attractive for problems that require repetitive yet expensive simulations, such as determinsitsic design, uncertainty propagation, optimization under uncertainty or inverse modeling. Data-efficiency,  uncertainty quantification and generalization are the main challenges facing surrogate modeling, especially for problems with high-dimensional stochastic input, such as material properties~\\cite{bilionis2013multi}, background potentials~\\cite{charalampidis2018computing}, etc.\n\nTraining surrogate models is commonly posed as a supervised learning problem, which requires simulation data as the target. \nGaussian process (GP) models are widely used as emulators for physical systems~\\cite{kennedy2000predicting} with built-in uncertainty quantification. The recent advances to scale GPs to high-dimensional input include Kronecker product decomposition that exploits the spatial structure~\\cite{bilionis2013multi, wilson2015kernel, atkinson2018structured}, convolutional kernels~\\cite{NIPS2017_6877} and other algorithmic and software developments~\\cite{gardner2018gpytorch}. However, GPs are still struggling to effectively model high-dimensional input-output maps.\nDeep neural networks (DNNs) are becoming the most popular surrogate models nowadays across engineering and scientific fields. As universal function approximators, DNNs excel at settings where both the input and output are high-dimensional. Applications in flow simulations include pressure projections in solving Navier-Stokes equations~\\cite{doi:10.1002\/cav.1695}, fluid flow through random heterogeneous media~\\cite{zhu2018bayesian, tripathy2018deep, mo2018deep}, Reynolds-Averaged Navier-Stokes simulations~\\cite{ling_kurzawski_templeton_2016, thuerey2018well,Geneva2019Quantifying} and others.\nUncertainty quantification for DNNs is often studied under the re-emerging framework of Bayesian deep learning\\footnote{http:\/\/bayesiandeeplearning.org\/}~\\cite{doi:10.1162\/neco.1992.4.3.448}, mostly using variational inference for approximate posterior of model parameters, e.g. variational dropout~\\cite{NIPS2015_5666, gal2016dropout}, Stein variational gradient descent~\\cite{NIPS2016_6338, zhu2018bayesian}, although other methods exist, e.g. ensemble methods~\\cite{lakshminarayanan2017simple}.\nAnother perspective to high-dimensional problems is offered by latent variable models~\\cite{2017arXiv171102475G}, where the latent variables encode the information bottleneck between the input and output. \n\n\nSufficient amount of training data is usually required for the surrogates to achieve accurate predictions even under restricted settings, e.g. fixed boundary conditions. For physically-grounded domains, baking in the prior knowledge can potentially overcome the challenges of data-efficiency and generalization, etc. The \\textit{inductive bias} can be built into the network architecture, e.g. spherical convolutional neural networks (CNNs) for the physical fields on unstructured grid~\\cite{jiang2018spherical}, graph networks for object- and relation-centric representations of complex, dynamical systems~\\cite{gn_icml18}, learning linear embeddings of nonlinear dynamics based on Koopman operator theory~\\cite{lusch2018deep}. Another approach is to \\textit{embed physical laws} into the learning systems, such as approximating differential operators with convolutions~\\cite{pmlr-v80-long18a}, enforcing hard constraint of mass conservation by learning the stream function~\\cite{2018arXiv180602071K} whose curl is guaranteed to be divergence-free. \n\nA more general way to incorporate physical knowledge is through \\textit{constraint learning}~\\cite{2016arXiv160905566S}, i.e. learning the models by minimizing the violation of the physical constraints, symmetries, e.g. cycle consistency in domain translation~\\cite{CycleGAN2017}, temporal coherence of consecutive frames in fluid simulation~\\cite{xie2018tempogan} and video translation~\\cite{wang2018vid2vid}. One typical example in computational physics is \\textit{learning solutions of deterministic PDEs with neural networks} in space\/time, which dates back at least to the early $1990$s, e.g.~\\cite{psichogios1992hybrid, meade1994numerical, lagaris1998artificial}. The main idea is to train neural networks to approximate the solution by minimizing the violation of the governing PDEs (e.g. the residual of the PDEs) and also of the  initial and boundary conditions.\nIn~\\cite{lagaris1998artificial}, a one-hidden-layer fully-connected neural network (FC-NN) with spatial coordinates as input is trained to minimize the residual norm evaluated on a  fixed grid. \nThe success of deep neural networks brings several new developments: (1) most of the works parameterize the solution with FC-NNs, thus the solution is analytical and meshfree~\\cite{raissi2017physics, berg2018unified}; (2) the loss function can be derived from the variational form~\\cite{weinan2018deep, nabian2018deep}; (3) stochastic gradient descent is used to train the network by randomly sampling mini-batches of inputs (spatial locations and\/or time instances)~\\cite{sirignano2018dgm, weinan2018deep}; (4) deeper networks are used to break the curse of dimensionality~\\cite{grohs2018proof} allowing for several high-dimensional PDEs to be solved with high accuracy and speed~\\cite{han2018solving, beck2017machine, sirignano2018dgm, raissi2018forward}; (5) multiscale numerical solvers are enhanced by replacing the linear basis with learned ones with DNNs~\\cite{wang2018deep, fan2018multiscale}; (6) surrogate modeling for PDEs~\\cite{CNNFluid2016, khoo2017solving, nabian2018deep}.\n\n\n\nOur work focuses on physics-constrained surrogate modeling for stochastic PDEs with high-dimensional spatially-varying coefficients \\textit{without} simulation data. \nWe first show that when solving deterministic PDEs, the CNN-based parameterizations are more computationally efficient in  capturing multiscale features of the solution fields than the FC-NN ones.  \nFurthermore, we demonstrate  that in comparison with image-to-image regression  approaches that employ Deep NNs~\\cite{zhu2018bayesian}, the proposed method  achieves comparable predictive performance, despite the fact that it does not make use of any output simulation data. In addition, it produces  better predictions under extrapolative conditions as when   out-of-distribution test input datasets are used.\nFinally, a flow-based conditional generative model is proposed to capture the predictive distribution with calibrated uncertainty, without compromising the predictive accuracy.\n\nThe paper is organized as follows.\nSection~\\ref{sec:Definition} provides the definition of the problems of interest including the solution of PDEs, surrogate modeling and uncertainty quantification. Section~\\ref{sec:Methodology} provides the parametrization of the solutions with FC-NNs and CNNs, the physics-constrained learning of a deterministic surrogate and the variational learning of a probabilistic surrogate. Section~\\ref{sec:numerical_experiments} investigates the performance of the developed techniques with a variety of tests for various PDE systems. We conclude in Section~\\ref{sec:Conclusions} with a summary of this work and extensions to address limitations that have been identified. \n\n\n\\section{Methodology}\n\\label{sec:Methodology}\n\\subsection{Differentiable Parameterizations of Solutions}\\label{sec:solve_pde}\nWe only consider the parameterizations of solutions using neural networks, primarily FC-NNs and CNNs. Given one input $\\mathbf{x} = [K(s_1), \\cdots, K(s_{n_s})]$, most previous works~\\cite{lagaris1998artificial, han2018solving, raissi2017physics, sirignano2018dgm} use FC-NNs to represent the solution as\n\\begin{equation}\\label{eq:solve_fc_nn}\n    u(s) = \\hat{u}_\\bm{\\phi} (s),\n\\end{equation}\nwhere the input to the network is coordinate $s$, the output is the predicted solution at $s$, and $\\hat{u}_\\bm{\\phi}$ denotes a FC-NN with parameters $\\bm{\\phi}$. The spatial gradients can be evaluated exactly by automatic differentiation. This approach yields a smooth representation of the solution that can be evaluated at any input location. Even though the outputs in this model at two different locations are correlated (as they both depend on the shared parameters $\\bm{\\phi}$ of the NN), FC-NNs do not have the inductive bias as in CNNs, e.g. translation invariance, parameter sharing, etc.\nDespite promising results in a series of canonical problems~\\cite{raissi2018physics},  the trainability and predictive performance of FC-NNs deteriorates as the complexity of the underlying solution increases. This drawback is confirmed by our numerical studies presented in Section~\\ref{sec:solve_det_pde} involving solution fields with multiscale features.\n\nAn alternative parametrization of the solution is through a convolutional decoder network\n\\begin{equation}\\label{eq:solve_cnn}\n    \\mathbf{y} = \\hat{\\mathbf{y}}_\\bm{\\theta} (\\mathbf{z}),\n\\end{equation}\nwhere $\\mathbf{y} = [u(s_1), \\cdots, u(s_{n_s})]$ denotes the solution on pre-defined fixed grids $s_1, \\cdots, s_{n_s}$  that is generated by one pass of the latent variable $\\mathbf{z}$ through the decoder, similarly as in~\\cite{2017arXiv171110925U}. Note that $\\mathbf{z}$ is usually much lower-dimensional than $n_s$ and initialized arbitrarily. The spatial gradients can be approximated efficiently with Sobel filter\\footnote{\\url{https:\/\/www.researchgate.net\/publication\/239398674_An_Isotropic_3x3_Image_Gradient_Operator}}, which amounts to one convolution layer with fixed kernel, see~\\ref{appendix:sobel_filter} for details.\nIn contrast to FC-NNs, convolutional architectures can directly capture complex spatial correlations and return a multi-resolution representation of the underlying solution field.\n\n\\begin{remark}\nThe dimensionality  $n_s$ of the input $\\mathbf{x}$ is not required to be the same as that of  the output $\\mathbf{y}$. Since our CNN approach would involve  operations between images including pixel-wise multiplication of input and output images (see Section~\\ref{sec:LossFnDarcyFlow}), we select herein the same dimensionality for both inputs and outputs. Upsampling\/downsampling can always be used to accommodate different dimensionalities $n_{sx}$ and $n_{sy}$ of the input and output images, respectively.\n\\end{remark}\n\nTo solve the deterministic PDE for a given input, we can train the   FC-NN solution as in Eq.~(\\ref{eq:solve_fc_nn}) by minimizing the residual loss where the exact derivatives are calculated with automatic differentiation~\\cite{lagaris1998artificial, han2018solving, raissi2017physics, sirignano2018dgm}. For the CNN representation, we will detail the loss functions and numerical derivatives in the next section.\n\n\n\n\\subsection{Physics-constrained Learning of Deterministic Surrogates without Labeled Data}\n\\label{sec:deterministic_surrogate}\n\nWe are particularly interested in surrogate modeling with high-dimensional input and output, i.e.   $\\text{dim}(\\mathbf{x}), \\text{dim}(\\mathbf{y}) \\gg 1$. Surrogate modeling is an extension of the solution networks in the previous section by adding the realizations of stochastic input $\\mathbf{x}$ as the input, e.g. $u(s, \\mathbf{x}) = \\hat{u}_\\bm{\\phi}(s, \\mathbf{x})$ in the FC-NN case~\\cite{nabian2018deep}, or $\\mathbf{y} = \\hat{\\mathbf{y}}_\\bm{\\theta}(\\mathbf{x})$ in the CNN case~\\cite{khoo2017solving}.\n\nHere, we adopt the \\textit{image-to-image regression} approach~\\cite{zhu2018bayesian} to deal with the problem arising in practice where the realizations of the random input field are image-like data instead of being computed from an analytical formula. More specifically, the surrogate model $\\mathbf{y} = \\hat{\\mathbf{y}}_\\bm{\\theta} (\\mathbf{x})$ is an extension of the decoder network in Eq.~(\\ref{eq:solve_cnn}) by prepending an encoder network to transform the high-dimensional input $\\mathbf{x}$ to the latent variable $\\mathbf{z}$, i.e. $\\mathbf{y} = \\text{decoder} \\circ \\text{encoder} (\\mathbf{x})$.\n\nIn contrast to existing convolutional encoder-decoder network structures~\\cite{zhu2018bayesian}, the surrogate model studied here is trained \\textit{without} labeled data i.e. without computing the solution of the PDE. Instead, it is trained by learning to solve the PDE with given boundary conditions, using the following loss function\n\\begin{equation}\\label{eq:loss_det_surr}\n    L(\\bm{\\theta}; \\{\\mathbf{x}^{(i)}\\}_{i=1}^N) =  \\frac{1}{N} \\sum_{i=1}^N \\Big[V(\\hat{\\mathbf{y}}_{\\bm{\\theta}}(\\mathbf{x}^{(i)}), \\mathbf{x}^{(i)}) + \\lambda B(\\hat{\\mathbf{y}}_{\\bm{\\theta}}(\\mathbf{x}^{(i)})) \\Big],\n\\end{equation}\nwhere $\\hat{\\mathbf{y}}^{(i)} = \\hat{\\mathbf{y}}_{\\bm{\\theta}}(\\mathbf{x}^{(i)})$ is the prediction of the surrogate for $\\mathbf{x}^{(i)} \\in \\mathcal D_{\\text{input}}$, $V(\\hat{\\mathbf{y}}^{(i)}, \\mathbf{x}^{(i)})$ is the equation loss, either in the form of the \\textit{residual norm}~\\cite{lagaris1998artificial} or the \\textit{variational functional}~\\cite{weinan2018deep} of the PDE, $B(\\hat{\\mathbf{y}}^{(i)})$ is the boundary loss of the prediction $\\hat{\\mathbf{y}}^{(i)}$, and $\\lambda$ is the weight (Lagrange multiplier) to softly enforce the boundary conditions. Both $V(\\hat{\\mathbf{y}}^{(i)}, \\mathbf{x}^{(i)})$ and $B(\\hat{\\mathbf{y}}^{(i)})$ may involve integration and differentiation with respect to the spatial coordinates, which are approximated with highly efficient discrete operations, detailed below for the Darcy flow problem. The surrogate trained with the loss function in Eq.~(\\ref{eq:loss_det_surr}) is called \\textit{physics-constrained surrogate} (PCS).\n\nIn contrast to the physically motivated loss function advocated above, a typical  data-driven surrogate  employs a loss function of the form\n\\begin{equation}\\label{eq:loss_det_surr_data}\n    L_{\\text{MLE}}(\\bm{\\theta}; \\{(\\mathbf{x}^{(i)}, \\mathbf{y}^{(i)})\\}_{i=1}^N) = \\frac{1}{N} \\sum_{i=1}^N  \\norm{\\mathbf{y}^{(i)} - \\hat{\\mathbf{y}}_\\bm{\\theta}(\\mathbf{x}^{(i)})}_2^2,\n\\end{equation}\nwhere $\\mathbf{y}^{(i)}$ is the output data for the input $\\mathbf{x}^{(i)}$ which must be computed in advance. We refer to the surrogate trained with loss function in Eq.~(\\ref{eq:loss_det_surr_data}) as the \\textit{data-driven surrogate} (DDS).\n\n\n\\subsubsection{Loss Function for Darcy Flow}\n\\label{sec:LossFnDarcyFlow}\nThere are at least four variations of loss functions for a second-order elliptic PDE problem, depending on whether the field variables refer to the primal variable (pressure) or to mixed variables (pressure and fluxes), and whether the loss is expressed in strong form or variational form. Specifically, for the Darcy flow problem defined in Eq.~(\\ref{eq:darcy}), we can consider:\n\n\\paragraph{Primal residual loss} The residual norm for the primal variable is\n\\begin{equation}\n    V(u; K) = \\int_{\\mathcal S} \\Big(\\nabla \\cdot (K \\nabla u) + f \\Big)^2 ds.\n\\end{equation}\n\\paragraph{Primal variational loss}\nThe energy functional is\n\\begin{equation}\n    \\label{eq:energy_functional}\n    V(u; K) = \\int_{\\mathcal S} \\Big( \\frac{1}{2} K \\nabla u\\cdot \\nabla u - f u \\Big) ds - \\int_{\\Gamma_N} g u ~ds.\n    \\end{equation}\n\nMixed formulation introduces an additional (vector) variable, namely flux $\\tau$, which turns Eq.~(\\ref{eq:darcy}) into a systems of equations\n\\begin{equation}\n\\label{eq:darcy_mixed}\n    \\begin{aligned}\n    \\tau &= - K \\nabla u, \\qquad \\text{in } \\mathcal S, \\\\\n    \\nabla \\cdot \\tau &= f, \\qquad \\text{in } \\mathcal S,\n    \\end{aligned}\n\\end{equation}\nwith the same boundary conditions as in Eq.~(\\ref{eq:BCs}). $\\tau(s) = [\\tau_1(s), \\tau_2(s)]$ are the flux field components along the horizontal and vertical directions, respectively.\n \n\\paragraph{Mixed variational loss} Following the Hellinger-Reissner principle~\\cite{arnold1990mixed}, the mixed variational principle states that the solution $(\\tau^*, u^*)$ of the Darcy flow problem is the unique critical point of the functional\n\\begin{equation}\n\\label{eq:mixed_functional}\n    V(\\tau, u; K) = \\int_\\Omega \\Big(\\frac{1}{2} K^{-1} \\tau \\cdot \\tau + u \\nabla \\cdot \\tau + f u \\Big) ds - \\int_{\\Gamma_D} u_D \\tau \\cdot \\mathbf{n} ds, \n\\end{equation}\nover the space of vector fields $\\tau \\in \\mathcal H(\\text{div})$ satisfying the Neumann boundary condition and all the fields $u \\in \\mathcal L^2$.\nIt should be highlighted that the solution $(\\tau^*, u^*)$ is not an extreme point of the functional in Eq.~(\\ref{eq:mixed_functional}), but a \\textit{saddle point}, i.e.\n\\begin{equation*}\n    V(\\tau^*, u) \\leq V(\\tau^*, u^*) \\leq V(\\tau, u^*).\n\\end{equation*}\n\n\n\\paragraph{Mixed residual loss} The residual norm for the mixed variables is\n\\begin{equation}\\label{eq:mixed_residual}\n    V(u; K) = \\int_{\\mathcal S} \\Big[ \\Big(\\tau + K \\nabla u \\Big)^2 + \\Big(\\nabla \\cdot \\tau - f\\Big)^2 \\Big] ds.\n\\end{equation}\n\nBoth the variational and mixed formulations have the advantage of lowering the order of differentiation which is approximated numerically in our implementation by a Sobel filter, as detailed in~\\ref{appendix:sobel_filter}. For example by employing the discretized representation $\\mathbf{x}$ for $K$ where the domain is $\\mathcal S=[0, 1] \\times [0, 1]$,  the mixed residual loss  is evaluated as\n\\begin{equation}\\label{eq:mixed_residual_impl}\n    V(\\bm{\\tau}, \\mathbf{u}; \\mathbf{x}) \\approx \\frac{1}{n_s} \\Big(\\norm{\\bm{\\tau} + \\mathbf{x} \\odot \\nabla \\mathbf{u}}^2_2 + \\norm{\\nabla \\cdot \\bm{\\tau} - \\mathbf{f}}^2_2 \\Big),\n\\end{equation}\nwhere $n_s$ is the number of uniform grids, $\\nabla \\mathbf{u} = [\\mathbf{u}_h, \\mathbf{u}_v]$, $\\mathbf{u}_h, \\mathbf{u}_v$ are two gradient images along the horizontal and vertical directions estimated by Sobel filter, similarly for $\\nabla \\cdot \\bm{\\tau} = (\\bm{\\tau}_1)_h + (\\bm{\\tau}_2)_v$, and $\\odot$ denotes the  element-wise product.\n\n\n\n\\begin{comment}\n\n\\subsubsection{Primal formulation: minimizing energy functional}\nThe loss function includes boundary loss (weighted by $\\lambda$), plus energy functional instead of the mean squared error (MSE) of the predicted output against the simulated output.\n\\begin{equation}\n\\label{eq:loss_energy_boundary}\n    \\mathcal L(\\bm{\\theta}; \\mathcal D_{\\text{input}}) = \\frac{1}{N} \\sum_{i=1}^N \\big[ \\mathcal V(\\mathbf{y}^{(i)}, \\mathbf{x}^{(i)}) + \\lambda \\mathcal L_{\\text{boundary}}(\\mathbf{y}^{(i)}, \\nabla \\mathbf{y}^{(i)}) \\big],\n\\end{equation}\nwhere $\\mathbf{y}^{(i)} = \\mathbf{f}_{\\bm{\\theta}}(\\mathbf{x}^{(i)})$ is the prediction from the surrogate, $\\mathbf{x}^{(i)} \\sim p(\\mathbf{x}), i=1, \\cdots, N$ are input samples, $\\mathcal V(\\mathbf{y}^{(i)}, \\mathbf{x}^{(i)})$ is evaluated directly since the energy potential is known for the given PDE, and $\\lambda$ is the weight (Lanrangian multiplier).\nThe boundary loss consists of Dirichlet and Neumann parts, which is evaluated by extracting the boundaries from the output $\\mathbf{y}$ and its estimated gradient $\\nabla \\mathbf{y}$ with Sobel filter, which is implemented as an extra convolution layer applied to the output $\\mathbf{y}$ with  Sobel kernel (see Section~\\ref{sec:Implementation}) for details. \n\n\n\\subsubsection{Mixed formulation: minimax the mixed functional}\n\nFor problems where the dual variable (e.g. flux for flow problems, stress for elasticity) is of most interest, mixed methods are preferred. Another motivation is to explore ways to improve the prediction accuracy over the energy methods which only predict the primal variable (e.g. pressure).\n\nThe surrogate model using both $\\mathbf{u}$ and its scaled gradient $\\bm{\\tau} \\equiv \\mathbf{x} \\nabla \\mathbf{u}$ is\n\\begin{equation}\n    \\label{eq:det_surrogate_mixed}\n    [\\bm{\\tau}, \\mathbf{u}] = \\mathbf{y} = \\mathbf{f}_\\bm{\\theta} (\\mathbf{x}),\n\\end{equation}\nwhere $\\bm{\\tau}  \\in \\mathbb{R}^{2\\times 65 \\times 65}$ denotes the two flux fields (as two output channels), $\\mathbf{u} \\in \\mathbb{R}^{1\\times 65 \\times 65}$ denotes the pressure field, $\\mathbf{x} \\in \\mathbb{R}^{1\\times 65 \\times 65}$ is the input permeability field, $\\bm{\\theta}$ denotes the parameters of the convnet surrogate $\\mathbf{f}$.\\red{GENERALIZE THINGS HERE - MOVE SPECIFICS TO IMPLEMENTATION}\n\nThe loss function for the surrogate is the following:\n\\begin{equation}\n    \\label{eq:loss_mixed}\n    \\mathcal L(\\bm{\\theta}; \\mathcal D_{\\text{input}}) = \\frac{1}{N} \\sum_{i=1}^N \\Big[ \\mathcal V(\\bm{\\tau}^{(i)}, \\mathbf{u}^{(i)}; \\mathbf{x}^{(i)}) + \\lambda_b \\mathcal L_{\\text{boundary}}(\\mathbf{y}^{(i)}, \\nabla \\mathbf{y}^{(i)}) + \\lambda_c \\mathcal L_{\\text{constitutive}}(\\mathbf{y}^{(i)}) \\Big],\n\\end{equation}\nwhere the mixed functional part $\\mathcal V(\\bm{\\tau}, \\mathbf{u}; \\mathbf{x})$ is minimized w.r.t. $\\bm{\\tau}$ and maximized w.r.t. $\\mathbf{u}$ alternatively; both the boundary loss and constitutive loss are minimized in every update, with weights $\\lambda_b$ and $\\lambda_c$, respectively.\n\n\n\\subsubsection{Mixed formulation: minimizing the mixed residual}\n\nInstead of dealing with mini-max problem of mixed functional, the surrogate can be trained with residuals of the mixed systems, i.e. \\red{SECOND TERM AND ELSEWHERE NEED TO BE GENERALIZED BEYOND THE DARCY FLOW}\n\\begin{equation}\n    \\label{eq:loss_mixed_residual}\n    \\mathcal L(\\bm{\\theta}; \\mathcal D_{\\text{input}}) = \\frac{1}{N} \\sum_{i=1}^N \\Big[ \\mathcal R(\\bm{\\tau}^{(i)}, \\mathbf{u}^{(i)}; \\mathbf{x}^{(i)}) + \\lambda \\mathcal L_{\\text{boundary}}(\\mathbf{y}^{(i)}, \\nabla \\mathbf{y}^{(i)}) \\Big],\n\\end{equation}\nwhere the residual is\n\\begin{equation}\n    \\mathcal R(\\bm{\\tau}, \\mathbf{u}; \\mathbf{x}) = \\norm{\\bm{\\tau} - \\mathbf{x} \\nabla \\mathbf{u}}^2 + \\norm{\\nabla \\cdot \\bm{\\tau}}^2.\n\\end{equation}\n\n\\end{comment}\n\n\n\n\n\n\\subsection{Probabilistic Surrogates with Reverse KL Formulation}\n\nWhile a deterministic surrogate provides fast predictions to new input realizations, it does not model the predictive uncertainty which is important in practice especially when the surrogate is tested on unseen (during training) inputs.\nMoreover, many PDEs in physics have multiple solutions~\\cite{farrell2015deflation} which cannot be captured with a deterministic model. \nThus building probabilistic surrogates that can model the distribution over possible solutions given the input is of great importance.\n\nA probabilistic surrogate models the conditional density of the predicted solution given the input, i.e. $p_\\bm{\\theta}(\\mathbf{y} | \\mathbf{x})$. \nInstead of learning this conditional density with labeled data~\\cite{sohn2015learning, mirza2014conditional, van2016conditional}, we distill it from a reference density $p_\\beta(\\mathbf{y} | \\mathbf{x})$. The reference density is a Boltzmann distribution\n\\begin{equation}\n\\label{eqn:target_conditional}\n    p_{\\beta}(\\mathbf{y} | \\mathbf{x}) = \\frac{\\exp{(-\\beta L(\\mathbf{y}, \\mathbf{x}) )}}{Z_\\beta(\\mathbf{x})},\n\\end{equation}\nwhere $L(\\mathbf{y}, \\mathbf{x}) = V(\\mathbf{y}, \\mathbf{x}) + \\lambda B(\\mathbf{y})$ is the loss function (Eq.~\\ref{eq:loss_det_surr}) for the deterministic surrogate that penalizes the violation of the PDE and boundary conditions, and $\\beta$ is an inverse temperature parameter that controls the overall variance of the reference density. This energy-based model is obtained solely from the PDE and boundary conditions, without having  access to labeled output data~\\cite{lecun2006tutorial}. However, this PDE-constrained model provides similar information as the labeled data allowing us to learn a probabilistic surrogate.\n\nSince sampling from the probabilistic surrogate $p_\\bm{\\theta}(\\mathbf{y} | \\mathbf{x})$ is usually fast and evaluating the (unnormalized) reference density $p_\\beta(\\mathbf{y} | \\mathbf{x})$ is often cheap, we choose to minimize the following reverse KL divergence:\n\\begin{equation}\\label{eq:rev_kl}\n\\begin{split}\n     D_{\\mathrm{KL}} (p(\\mathbf{x})~p_{\\bm{\\theta}}(\\mathbf{y} | \\mathbf{x}) \\parallel p(\\mathbf{x})~ p_{\\beta}( \\mathbf{y} | \\mathbf{x} )) \n     & = \\mathbb{E}_{p(\\mathbf{x})} \\Big[ - \\mathbb{E}_{p_{\\bm\\theta}(\\mathbf{y} | \\mathbf{x})}[\\log p_{\\beta} (\\mathbf{y} | \\mathbf{x})] + \\mathbb{E}_{p_{\\bm\\theta}(\\mathbf{y} | \\mathbf{x})} [\\log p_{\\bm\\theta}(\\mathbf{y} | \\mathbf{x})] \\Big]\\\\\n     & = \\beta \\mathbb{E}_{p(\\mathbf{x})p_{\\bm\\theta}(\\mathbf{y} | \\mathbf{x})} [L (\\mathbf{y}, \\mathbf{x})] - \\mathbb{H}_\\bm{\\theta} (\\mathbf{y} | \\mathbf{x}) + \\mathbb{E}_{p(\\mathbf{x})}[\\log Z_{\\beta}(\\mathbf{x})].\n\\end{split}\n\\end{equation}\nThe first term is the expectation of the loss function $L(\\mathbf{y},\\mathbf{x})$  w.r.t. the joint density $p(\\mathbf{x}) p_{\\bm{\\theta}}(\\mathbf{y} | \\mathbf{x})$, which enforces the satisfaction of PDEs and boundary conditions.\nThe second term is the negative conditional entropy of $p_{\\bm{\\theta}}(\\mathbf{y} | \\mathbf{x})$ which promotes the diversity of model predictions. It also helps to stabilize the training of flow-based conditional generative model introduced in Section~\\ref{sec:cflow}. \nThe third term is the variational free energy, which is constant when optimizing $\\bm{\\theta}$. For the models with intractable log-likelihood $\\log p_{\\bm\\theta}(\\mathbf{y} | \\mathbf{x})$, one can derive a lower bound for the conditional entropy $\\mathbb{H}_{\\bm{\\theta}}(\\mathbf{y} | \\mathbf{x})$ that helps to regularize training and avoid mode collapse as in~\\cite{yang2018adversarial}. In this work, the log-likelihood can be exactly evaluated for the model introduced in Section~\\ref{sec:cflow}.\n\nThis idea is similar to probability density distillation~\\cite{oord2017parallel} to learn generative models for real-time speech synthesis,  neural renormalization group~\\cite{li2018neural} to accelerate sampling for Ising models, and Boltzmann generators~\\cite{noe2018boltzmann}\n to efficiently sample equilibrium states of many-body systems.\n\n\nThe reverse KL divergence itself is not enough to guarantee that the predictive uncertainty is well-calibrated. Even if this divergence is optimized to zero, i.e. $p_\\bm{\\theta}(\\mathbf{y}|\\mathbf{x}) = p_\\beta(\\mathbf{y}|\\mathbf{x})$, the predictive uncertainty is still controlled by $\\beta$. Thus we add an uncertainty calibration constraint to the optimization problem, i.e. \n\\begin{equation}\n\\begin{aligned}\n    \\min_{\\beta, \\bm{\\theta}} \\quad & D_{\\mathrm{KL}} (p(\\mathbf{x}) p_{\\bm{\\theta}}(\\mathbf{y} | \\mathbf{x})\n    \\parallel p(\\mathbf{x}) p_{\\beta}( \\mathbf{y} | \\mathbf{x} )), \\\\\n    s.t. \\quad & p_{\\bm{\\theta}}(\\mathbf{y} | \\mathbf{x}) \\text{ is calibrated on validation data.}\n\\end{aligned}\n\\end{equation}\nHere, the predictive uncertainty is calibrated using the reliability diagram~\\cite{degroot1983comparison}. \nThe naive approach to select $\\beta$ is through grid search, i.e. train the probabilistic surrogate with different values of $\\beta$, and select the one under which the trained surrogate is well-calibrated w.r.t. validation data, which includes input-output data pairs.\n \n\n\\remark{Instead of tuning $\\beta$ with grid search, we can also re-calibrate the trained model \\textit{post-hoc}~\\cite{guo2017calibration, kuleshov2018accurate} by learning an auxiliary regression model. For a small amount of miscalibration, sampling latent variables with different temperature (Section~6 in~\\cite{kingma2018glow}) can also change the variance of the output with a slight drop of predictive accuracy.} \n\n\n\\begin{remark}\nSimilar to our approach, Probabilistic Numerical Methods (PNMs)~\\cite{hennig2015probabilistic, cockayne2016probabilistic, cockayne2017bayesian} take a statistical point of view of classical numerical methods (e.g. a finite element solver) that treat the output  as a point estimate of the true solution. Given finite information (e.g. finite number of evaluations of the PDE operator and boundary conditions) and prior belief about the solution, PNMs output the posterior distribution of the solution. \nPNM focuses on inference of the solution for one input, instead of amortized inference as what the probabilistic surrogate does.\n\\end{remark}\n\n\\subsubsection{Conditional Flow-based Generative Models}\\label{sec:cflow}\nThis section presents flow-based generative models~\\cite{dinh2016density} as our probabilistic surrogates. This family of models offers several advantages over other generative models~\\cite{2013arXiv1312.6114K, NIPS2014_5423}, such as exact inference and exact log-likelihood evaluation that is particularly attractive for learning the conditional distribution with the reverse KL divergence as in Eq.~(\\ref{eq:rev_kl}). The generative model $\\mathbf{y} = \\mathbf{g}_\\bm{\\theta}(\\mathbf{z})$ consists of a sequence of \\textit{invertible} layers (also called normalizing flows~\\cite{rezende2015variational}) that transforms a simple distribution $p(\\mathbf{z})$ to a target distribution $p(\\mathbf{y})$, i.e. \n$$\\mathbf{y} := \\mathbf{h}_0 \\overset{\\mathbf{g}_\\bm{\\theta}^1}{\\longleftrightarrow} \\mathbf{h}_1 \\overset{\\mathbf{g}_\\bm{\\theta}^2}{\\longleftrightarrow} \\mathbf{h}_2 \\cdots \\overset{\\mathbf{g}_\\bm{\\theta}^L}{\\longleftrightarrow} \\mathbf{h}_L := \\mathbf{z},$$\nwhere $\\mathbf{g}_\\bm{\\theta} = \\mathbf{g}_\\bm{\\theta}^1 \\circ \\mathbf{g}_\\bm{\\theta}^2 \\circ \\cdots \\circ \\mathbf{g}_\\bm{\\theta}^L$.\nBy the change of variables formula, the log-likelihood of the model given $\\mathbf{y}$ can be calculated as\n\\begin{align*}\n    \\log p_\\bm{\\theta}(\\mathbf{y}) = \\log p_\\bm{\\theta}(\\mathbf{z}) + \\sum_{l=1}^L \\log |\\det(d\\mathbf{h}_l \/ d\\mathbf{h}_{l-1}))|,\n\\end{align*}\nwhere the log-determinant of the absolute value of the Jacobian term $\\log |\\det(d\\mathbf{h}_l \/ d\\mathbf{h}_{l-1}))|$ for each transform $(\\mathbf{g}_\\bm{\\theta}^l)^{-1}$ can be easily computed for certain design of invertible layers~\\cite{rezende2015variational, dinh2016density} similar to the Feistel cipher. Given training data of $\\mathbf{y}$, the model can be optimized stably with maximum likelihood estimation.\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.65\\textwidth}\n        \\hspace{-1em}\n        \\includegraphics[width=1.1\\textwidth]{figs\/glow_msc}\n        \\caption{Mutliscale conditioning.}\n        \\label{fig:glow_msc_sub}\n    \\end{subfigure}\n    \\hfill\n    \\begin{minipage}[b]{0.33\\textwidth}\n        \\begin{subfigure}[b]{\\textwidth}\n            \\centering\n            \\includegraphics[width=.95\\textwidth]{figs\/flow_step}\n            \\caption{One step of flow.}\n            \\label{fig:flow_step}\n        \\end{subfigure}\n        \\\\[\\baselineskip]\n        \\begin{subfigure}[b]{\\textwidth}\n            \\centering\n            \\includegraphics[width=0.98\\textwidth]{figs\/affine_coupling}\n            \\caption{Affine coupling layer.}\n            \\label{fig:affine_coupling}\n        \\end{subfigure}\n    \\end{minipage}\n    \\caption{Multiscale conditional Glow. (a) Multiscale features extracted with the encoder network (left) are used as conditions to generate output with the Glow model (right). $\\times F$, $\\times (L-2)$ denotes repeating for $F$ times and $L-2$ times respectively. (b) One step of flow, i.e. \\texttt{Flow} block in (a), and (c) Affine coupling layer following the structure of Glow (Fig.~$2$ in~\\cite{kingma2018glow}) except conditioning on the input features. The figure shows the forward path from $\\{\\mathbf{y}; \\mathbf{x} \\}$ to $\\mathbf{z} = \\{\\mathbf{z}_2, \\cdots, \\mathbf{z}_L\\}$. The reverse (sampling) path from $\\{\\mathbf{z}; \\mathbf{x} \\}$ to $\\mathbf{y}$ is used during training, where $\\mathbf{z}$ are sampled from diagonal Gaussians, see Algorithm~\\ref{algo:cglow}. See~\\ref{appendix:cglow} for the details of all modules in the model.}\n    \\label{fig:glow_msc}\n\\end{figure}\nA recently developed generative flow model called Glow~\\cite{kingma2018glow} proposed to learn invertible $1\\times1$ convolution to replace the fixed permutation and synthesize large photo-realistic images using  the log-likelihood objective.\nWe extend Glow to condition on high-dimensional input $\\mathbf{x}$, e.g. images, as shown in Fig.~\\ref{fig:glow_msc}. The conditional model consists of two components (Fig.~\\ref{fig:glow_msc_sub}): an encoder network which extracts multiscale features $\\{\\bm{\\xi}_l\\}_{l=1}^L$ from the input $\\mathbf{x}$ through a cascade of alternating dense blocks and downsampling layers, and a Glow model (with multiscale structure) which transforms the latent variables $\\mathbf{z}=\\{\\mathbf{z}_2, \\cdots, \\mathbf{z}_L\\}$ distributed at different scales to the output $\\mathbf{y}$ conditioned on $\\{\\bm{\\xi}_l\\}_{l=1}^L$ through skip connections (dashed lines in Fig.~\\ref{fig:glow_msc_sub}, as in Unet~\\cite{ronneberger2015u}) between the encoder and the Glow.\n\nMore specifically, the input features $\\bm{\\xi}_l$ enter the Glow model as the condition for the affine coupling layers at the same scale, as shown in Fig.~\\ref{fig:flow_step}, whose input and output are denoted as $\\mathbf{y}'$ and $\\mathbf{z}'$ in the forward path. As shown in Fig.~\\ref{fig:affine_coupling}, the input features $\\bm{\\xi}_l$ are concatenated \\encircle{c} with half of the flow features $\\mathbf{y}_1'$ before passing to scale \\fbox{$\\mathbf{s}$} and shift \\fbox{$\\mathbf{t}$} networks which specify arbitrarily nonlinear transforms that need not to be invertible. Given $\\mathbf{z}' = [\\mathbf{z}_1', \\mathbf{z}_2']$ and $\\bm{\\xi}_l$, $\\mathbf{y}'=[\\mathbf{y}_1', \\mathbf{y}_2']$ can be recovered exactly by reversing the shift and scaling operations, as detailed in Table~\\ref{tab:affine_coupling}. Note that $\\bm{\\xi}_l$ is the condition for all $F$ steps of flow at scale $l=1, \\cdots, L$, where $L$ denotes the number of scales (or levels). More details of the model including dense blocks, transition down layers, split, squeeze, and affine coupling layers are given in~\\ref{appendix:cglow}.\n\nIn a data-driven scenario, the conditional Glow is trained by passing data $\\mathbf{y}$ through the model to compute the latent $\\mathbf{z}$ and maximizing the evaluated log-likelihood of data given $\\mathbf{x}$. But to train with the loss in Eq.~(\\ref{eq:rev_kl}), we need to sample the output $\\hat{\\mathbf{y}}$ from the conditional density $p_{\\bm{\\theta}}(\\mathbf{y} | \\mathbf{x})$ given $\\mathbf{x}$, which goes in the opposite direction of the data-driven case. Algorithm~\\ref{algo:cglow} shows the details of training conditional Glow. \nThe sampling\/generation process is shown within the for-loop before computing the loss. Note that for one input sample only one output sample is used to approximate the expectation over $p_{\\bm{\\theta}}(\\mathbf{y} | \\mathbf{x})$ during training. To obtain multiple output samples for an input e.g. to compute the predictive mean and variance during prediction, we only need to sample the noise variables $\\{\\bm{\\epsilon}_l\\}_{l=2}^L$ multiple times, and pass them through the reverse path of the Glow.\nThe conditional log-likelihood $p_{\\bm{\\theta}}(\\hat{\\mathbf{y}} | \\mathbf{x})$ can be exactly evaluated as the following:\n\\begin{equation}\n    \\label{eq:flow_likelihood}\n    \\log p_{\\bm{\\theta}}(\\hat{\\mathbf{y}} | \\mathbf{x}) = \\log p_{\\bm{\\theta}}(\\mathbf{z}) + \\log |\\det (d\\mathbf{z} \/ d\\hat{\\mathbf{y}})|,\n\\end{equation}\nwhere both the latent $\\mathbf{z}$ and $\\log |\\det (d\\mathbf{z} \/ d\\hat{\\mathbf{y}})|$ depend on $\\mathbf{x}$ and realizations of the noise $\\{\\bm{\\epsilon}_l\\}_{l=2}^L$. The density of the latent $p_{\\bm{\\theta}}(\\mathbf{z})$ is usually a simple distribution, e.g. diagonal Gaussian, which is computed with the second (for $\\mathbf{z}_L$) and third (for $\\{\\mathbf{z}_l\\}_{l=2}^{L-1}$) terms within the bracket of the reverse KL divergence loss in Algorithm~\\ref{algo:cglow}. Also $\\log |\\det (d\\mathbf{z} \/ d\\mathbf{y})|$ is computed with the fourth term. Notably, the log-determinant of the Jacobian for the affine coupling layer is just $\\texttt{sum}(\\log |\\mathbf{s}|)$, where $\\mathbf{s}$ is the output of the scaling network.\nThus the conditional density $p_\\bm{\\theta}(\\hat{\\mathbf{y}} | \\mathbf{x})$ can be evaluated exactly and efficiently, enabling us to directly approximate the entropy term in  Eq.~(\\ref{eq:rev_kl}), e.g. via Monte Carlo approximation.\n\n\\remark{\nThe training process does not require output data. However, validation data with input-output pairs are necessary to calibrate the predictive uncertainty of the trained model. Careful initialization of the model is important to stabilize the training process. In this work,  we initialize the \\texttt{ActNorm} to be the identity transform, the weight matrix of \\texttt{Invertible $1\\times1$ Convolution} to be a random rotation matrix, and the \\texttt{Affine Coupling} layer to be close to the identity transform ($\\hat{\\mathbf{s}}=\\bm{0}$ and $\\mathbf{t}=\\bm{0}$ in Table~\\ref{tab:affine_coupling}). \nWe can also use data-dependent initialization to speed up the training process. More specifically, one mini-batch $\\mathcal D_{\\text{init}} = \\{(\\mathbf{x}^{(j)}, \\mathbf{r}^{(j)})\\}_{j=1}^{M'}$ (e.g. $M'=32$) of input-output data pairs can be passed forward from $\\{\\mathbf{y}; \\mathbf{x}\\}$ to $\\mathbf{z}$ to initialize the parameters of \\texttt{ActNorm} such that the post-\\texttt{ActNorm} activations per-channel have zero mean and unit variance given $\\mathcal D_{\\text{init}}$~\\cite{kingma2018glow}. The reference output $\\mathbf{r}$ can be the solution from standard deterministic PDE solvers or more appropriately here from the methods presented in Sections~\\ref{sec:solve_pde} and~\\ref{sec:solve_det_pde}.\n}\n\n\\begin{algorithm}\n\\SetAlgoLined\n\\KwIn{Inverse temperature $\\beta$, input samples $\\{\\mathbf{x}^{(i)}\\}_{i=1}^N$. Mini-batch size M, number of steps $F$ of \\texttt{Flow} in each scale, number of scales $L$.}\n\\KwOut{Model parameters $\\bm{\\theta}$}\n\\For{number of training iterations}{\n    Sample a mini-batch of input $\\{\\mathbf{x}^{(i)}\\}_{i=1}^M$, pass it through the encoder to compute the multiscale input features $\\{\\bm{\\xi}_l^{(i)}\\}_{l=1, i=1}^{L, M}$; \n    \n    Sample the latent $\\mathbf{z}^{(i)}_L = \\bm{\\mu}_\\bm{\\theta}^L(\\bm{\\xi}^{(i)}_L) + \\bm{\\sigma}_\\bm{\\theta}^L(\\bm{\\xi}^{(i)}_L) \\odot \\bm{\\epsilon}_L^{(i)}, \\bm{\\epsilon}_L^{(i)} \\sim \\mathcal N(\\bm{0}, \\mathbf{I})$; \n    \n    Compute flow feature $\\mathbf{h}_{L-1}^{(i)} = \\mathbf{g}_{\\bm{\\theta}}^L (\\mathbf{h}_{L}^{(i)}; \\bm{\\xi}_L^{(i)})$; \\algorithmiccomment{$\\mathbf{h}_L = \\mathbf{z}_L$, $\\mathbf{g}_\\bm{\\theta}^L$ includes the reverse path of \\texttt{Sequeeze} and $F$ steps of \\texttt{Flow}} \n    \n    \\For{$l=L-1:2$}{\n        Sample the split latent variable at level $l$ $\\mathbf{z}_l^{(i)} = \\bm{\\mu}_\\bm{\\theta}^l(\\mathbf{h}^{(i)}_l) + \\bm{\\sigma}_\\bm{\\theta}^l(\\mathbf{h}^{(i)}_l) \\odot \\bm{\\epsilon}_l^{(i)}, \\bm{\\epsilon}_l^{(i)} \\sim \\mathcal N(\\bm{0}, \\mathbf{I}), i=1, \\cdots, M$; \n        \n        Compute flow feature $\\mathbf{h}_{l-1}^{(i)} = \\mathbf{g}_{\\bm{\\theta}}^l (\\mathbf{h}_l^{(i)}, \\mathbf{z}_l^{(i)}; \\bm{\\xi}_l^{(i)})$;  \\algorithmiccomment{$\\mathbf{g}_\\bm{\\theta}^l$ includes the reverse path of \\texttt{Sequeeze}, $F$ steps of \\texttt{Flow} and \\texttt{Split}}\n        }\n    \n    Compute output $\\hat{\\mathbf{y}}^{(i)} = \\mathbf{g}_\\bm{\\theta}^1(\\mathbf{h}_1^{(i)}; \\bm{\\xi}_1^{(i)})$;\n    \\algorithmiccomment{$\\mathbf{g}_\\bm{\\theta}^1$ includes the reverse path of $F$ steps of \\texttt{Flow}}\n    \n    Minimize the \\textit{reverse KL} divergence in Eq.~(\\ref{eq:rev_kl}) with Adam optimizer w.r.t. $\\bm{\\theta}$ \n        $ \\frac{1}{M} \\sum_{i=1}^M \\Big[ \\beta L(\\hat{\\mathbf{y}}^{(i)}, \\mathbf{x}^{(i)}) \n        + \\sum_{l=2}^{L-1} \\log \\mathcal N(\\mathbf{z}_l^{(i)} | \\bm{\\mu}_\\bm{\\theta}^l(\\mathbf{h}_l^{(i)}), (\\bm{\\sigma}_\\bm{\\theta}^l(\\mathbf{h}_l^{(i)}))^2) \n         + \\log \\mathcal N(\\mathbf{z}_L^{(i)} | \\bm{\\mu}_\\bm{\\theta}^L(\\bm{\\xi}_L^{(i)}), (\\bm{\\sigma}_\\bm{\\theta}^L(\\bm{\\xi}_L^{(i)}))^2) \n        + \\sum_{l=1}^L \\log |\\det(d\\mathbf{h}_l^{(i)} \/ d\\mathbf{h}_{l-1}^{(i)})| \\Big].$\n\n    \\algorithmiccomment{$\\mathbf{h}_0=\\hat{\\mathbf{y}}$, see Table 1 in~\\cite{kingma2018glow} for formula to compute $\\log |\\det(d\\mathbf{h}_l^{(i)} \/ d\\mathbf{h}_{l-1}^{(i)})|$, i.e. log-determinant of Jacobian for \\texttt{ActNorm}, \\texttt{Invertible $1\\times 1$ Conv} and \\texttt{Affine Coupling} layer.}\n}\n\\caption{Training conditional Glow.}\n\\label{algo:cglow}\n\\end{algorithm}\n\n\\section{Problem Definition}\n\\label{sec:Definition}\nConsider  modeling of a physical system governed by PDEs:\n\\begin{equation}\n    \\label{eq:PDE}\n    \\begin{aligned}\n    \\mathcal N (u(s); K(s)) &= f(s), \\qquad s \\in \\mathcal S, \\\\\n    \\mathcal B (u(s))       &= b(s), \\qquad s \\in \\Gamma,\n    \\end{aligned}\n\\end{equation}\nwhere $\\mathcal N$ is a general differential operator, $u(s)$ are the field variables of interest, $f(s)$ is the source field, and $K(s)$ denotes an input property field defining the system's constitutive behavior. $\\mathcal B$ is the operator for boundary conditions defined on the boundary $\\Gamma$ of the domain $\\mathcal S$.\nIn particular, we consider the following Darcy flow problem as a motivating example throughout this paper:\n\\begin{equation}\n\\label{eq:darcy}\n    - \\nabla \\cdot (K(s) \\nabla u(s))) = f(s), \\qquad s \\in \\mathcal S,\n\\end{equation}\nwith boundary conditions\n\\begin{equation}\\label{eq:BCs}\n    \\begin{aligned}\n    u(s) &= u_D(s), \\qquad s \\in  \\Gamma_D, \\\\\n    \\nabla u(s) \\cdot n &= g(s), \\qquad s \\in \\Gamma_N,\n    \\end{aligned}\n\\end{equation}\nwhere $n$ is the unit normal vector to the Neumann boundary $\\Gamma_N$, $\\Gamma_D$ is the Dirichlet boundary. \n\nOf particular interest are PDEs for which the field variables can be computed by appropriate minimization of a field energy functional (potential) $V(u; K)$, i.e.  \n\\begin{equation}\n\\argmin_u V(u; K).     \n\\end{equation}\nSuch potentials are common in many linear and nonlinear problems in  physics and engineering and serve as the basis of the finite element  method.  For problems where such potentials cannot be found~\\cite{FilSavSho92}, one can consider $V$ as the square of the residual norm of the PDE evaluated at different trial solutions, e.g.\n\\begin{equation}\n    V(u; K)=R^2\\left(u; K\\right).\n\\end{equation}\nIn this paper, we are interested in the solution of parametric PDEs for a given set of boundary conditions. \n \n\n\\begin{defn}[Solution of a deterministic PDE system]\n\\label{defn:solution_pde}\nGiven the potential $V(u; K)$, and the boundary conditions in Eq.~(\\ref{eq:BCs}), compute the solution $u(s)$ of the PDE for a given input field $K(s)$.\n\\end{defn}\n\nThe input field $K(s)$ is often modeled as a random field $K(s, \\omega)$ in the context of uncertainty quantification, where $\\omega$ denotes a random event in the sample space $\\Omega$. In practice, discretized versions of this field are employed in computations which is denoted as the random vector $\\mathbf{x}$, i.e. $\\mathbf{x} = [K(s_1), \\cdots, K(s_{n_s})]$. We note that when fine-scale fluctuations  of the input field $K$ are present, the dimension $n_s$ of $\\mathbf{x}$ can become very high. Let $p(\\mathbf{x})$ be the associated density postulated by mathematical considerations or learned from data, e.g.  CT scans of microstructures, measurement of permeability fields, etc.\nSuppose $\\mathbf{y}$ denotes a discretized version of the PDE solution, i.e.\n$$\\mathbf{y} = [u(s_1), \\cdots, u(s_{n_s})].$$ \n\nNote that all the discretized field variable(s) are denoted as bold, while the continuous field variable(s) are non-bold.\n\nWe are interested in developing a surrogate model that allows fast calculation of the system response $\\mathbf{y}$\nfor any input realization $\\mathbf{x} \\in p(\\mathbf{x})$, and potentially for various boundary conditions.\nThis leads to the following definition:\n\n\\begin{defn}[Deterministic Surrogate Model]\n\\label{defn:det_surrogate}\n Given the potential $V(u; K)$, the boundary conditions in Eq.~(\\ref{eq:BCs}), and a set of training input data $\\mathcal D_{\\text{input}} = \\{\\mathbf{x}^{(i)}\\}_{i=1}^N, \\mathbf{x}^{(i)} \\sim p(\\mathbf{x})$, learn a deterministic surrogate \n$\n    \\mathbf{y} = \\hat{\\mathbf{y}}_\\bm{\\theta}(\\mathbf{x}),\n$\nfor predicting the solution $\\mathbf{y}$ for any input $\\mathbf{x} \\in p(\\mathbf{x})$, where $\\bm{\\theta}$ denotes the parameters of the surrogate model.\n\\end{defn}\n\nNote that often the density $p(\\mathbf{x})$ is not known and needs to be approximated from the given data $\\{\\mathbf{x}^{(i)}\\}_{i=1}^N$. When the density $p(\\mathbf{x})$ is given, the surrogate model can be defined without referring to the particular training data set. In this case, as part of the training process, one can select any dataset of size $N$, $\\{\\mathbf{x}^{(i)}\\}_{i=1}^N, \\mathbf{x}^{(i)} \\sim p(\\mathbf{x})$, including the most informative one for the surrogate task.   \n\nWe note  that the aforementioned problem refers to  a new  type of machine learning  task that falls between unsupervised learning due to the absence of labeled data (i.e. the $\\mathbf{y}^{(i)}$ corresponding to each $\\mathbf{x}^{(i)}$ is {\\em not} provided) and (semi-)supervised learning because the objective  involves discovering the map from the input $\\mathbf{x}$ to the output $\\mathbf{y}$. \nGiven the finite training data employed in practice and the inadequacies of the model postulated, $\\hat{\\mathbf{y}}_\\bm{\\theta}(\\mathbf{x})$, it is often advantageous to obtain \n a distribution over the possible solutions via a probabilistic surrogate, rather than  a mere point estimate for the solution.\n\n\\begin{defn}[Probabilistic Surrogate Model]\nGiven the potential $V(u; K)$, the boundary conditions in Eq.~(\\ref{eq:BCs}), and a set of training input data $\\mathcal D_{\\text{input}} = \\{\\mathbf{x}^{(i)}\\}_{i=1}^N, \\mathbf{x}^{(i)} \\sim p(\\mathbf{x})$, a probabilistic surrogate model specifies a conditional density $p_\\bm{\\theta}(\\mathbf{y} | \\mathbf{x})$, where $\\bm{\\theta}$ denotes the model parameters.\n\\end{defn}\n\nFinally, since the input $\\mathbf{x}$ arises from an underlying probability density, one may be interested to compute the statistics of the output $\\mathbf{y}$ leading to the following forward uncertainty propagation problem. \n\n\\begin{defn}[Forward Uncertainty Propagation]\n\\label{defn:uncertainty_propagation} \nGiven the potential $V(u; K)$, the boundary conditions in Eq.~(\\ref{eq:BCs}), and a set of training input data $\\mathcal {\\mathcal D}_{\\text{input}} = \\{\\mathbf{x}^{(i)}\\}_{i=1}^N, \\mathbf{x}^{(i)} \\sim p(\\mathbf{x})$, estimate moments of the response, $\\mathbb{E}[\\mathbf{y}], \\text{Var}[\\mathbf{y}], \\ldots$ or more generally any aspect of the  probability density of $\\mathbf{y}$.\n\\end{defn}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section-intro}\n\nAs one of the nearest radio galaxies to us \\citep[$D=16$ Mpc is \nadopted;][]{ton91}, M87 is amongst the best-studied of its source class. \nIt is perhaps best known for its exceptionally bright arcsecond-scale \njet \\citep{cur18}, well-imaged at radio through X-ray frequencies at \nincreasingly improved sensitivity and resolution over the decades \n\\citep[e.g.,][]{bir91,spa96,mar02,per05}. Near its central $\\sim (3-6) \n\\times 10^{9}$ solar mass supermassive black hole \\citep{mac97,geb09}, \nthe jet base has been imaged down to $\\sim$0.01 pc resolution \n\\citep[$\\sim 15-30\\times$ the Schwarzschild \nradius,][]{jun99,ly07,tev09}.\n\nAt the highest energies, M87 is regularly detected by HESS, MAGIC, and \nVERITAS with variable TeV emission on timescales of years and flaring in \na few days \\citep{aha06,alb08,acc08,tev09}. The sensitivity of these \nCherenkov telescopes have also enabled the detection of another \nwell-known nearby radio galaxy Cen~A \\citep{aha09}. Without comparable \nimaging resolution to the lower energy studies however, variability and \nspectral modeling are necessary to infer the production site of the TeV \n$\\gamma$-rays and to deduce the source physical parameters.\n\nAt high energy $\\gamma$-rays ($\\sim$20 MeV -- 100 GeV), we are similarly \npoised for new radio galaxy discoveries with the Large Area Telescope (LAT) \naboard the recently launched {\\it Fermi Gamma-ray Space Telescope} \n\\citep{atw09}. Indeed, we report here the detection of a faint, point-like \n$\\gamma$-ray source positionally coincident with M87 using the $Fermi$-LAT. \nAfter the confirmation of the EGRET discovery of Cen~A \\citep[][and in \npreparation]{sre99,lbas}, and the recent detection of Per~A\/NGC~1275 \n\\citep{pera}, this is the third radio galaxy successfully detected by $Fermi$. \nUnlike the known variable TeV source, there is so far no evidence for \nvariability of the MeV\/GeV emission in M87. An origin of the LAT emission from \nthe unresolved parsec scale jet (hereafter, denoted as the `nucleus' or \n`core') observed contemporaneously with $Chandra$ and the VLBA\\footnote{The \nNational Radio Astronomy Observatory is a facility of the National Science \nFoundation operated under cooperative agreement by Associated Universities, \nInc.} is discussed. Potential contributions from the larger-scale ($\\simgt \n0.1-1$ kpc) jet to the unresolved $\\gamma$-ray source are also briefly \nconsidered. Section 2 contains the details of the LAT observations, including \na description of the $Chandra$ and VLBA data utilized, with the discussion \nof these results in section 3.\n\n\n\\section{Observations}\\label{section-lat}\n\nThe $Fermi$-LAT is a pair creation telescope which covers the energy range from \n$\\sim$20 MeV to $>$300 GeV \\citep{atw09}. It operates primarily in an `all-sky \nsurvey' mode, scanning the entire sky approximately every three hours.\nThe initial LAT detection of M87 resulted from nominal processing of \n6-months of all-sky survey data, as was applied to the initial 3-month \ndataset described in \\citet{bsl}, with a test statistic \\citep{mat96}, \n$TS \\sim 60$. Including here an additional 4-months of data, the $TS$ \nincreased to 108.5, which is equivalent to a source significance \n$\\sim\\sqrt{TS}=10.4\\sigma$. The resultant 10-month dataset (Aug.\\ 4, \n2008 - May 31, 2009) corresponds to mission elapsed times (MET) \n239557418 to 265420800. Our analysis followed standard selections of \n\"Diffuse\" class events \\citep{atw09} with energies $E>$200 MeV, a zenith \nangle cut of $<$105\\hbox{$^\\circ$}, and a rocking angle cut of 43\\hbox{$^\\circ$}\\ applied in \norder to avoid Earth albedo $\\gamma$-rays. {\\it Fermi} Science \ntools\\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/documentation\/Cicerone\/} \nversion v9r10 and instrumental response functions (IRFs) version {\\tt \nP6$\\_$V3$\\_$DIFFUSE} were used for the analysis.\n\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig1.ps}\n \\caption{VLA $\\lambda$=90cm radio image from \\citet{owe00} with the LAT\n$\\gamma$-ray localization error circles indicated: $r_{95\\%}= 5.2'$ and\n$r_{68\\%}=3.2'$ (statistical only; see \\S~\\ref{section-lat}). The M87\ncore is the faint feature near the center of the few kpc-scale\ndouble-lobed radio structure (in white). At the adopted distance $D=16$\nMpc, 1$'$ = 4.7 kpc. \n}\n\\label{figure-vla}\n\\end{figure}\n\n\nA localization analysis with {\\tt GTFINDSRC} resulted in a best-fit \nposition, RA = 187\\hbox{$^\\circ$}.722, Dec.\\ = 12\\hbox{$^\\circ$}.404 (J2000.0 equinox), with a \n95$\\%$ confidence error radius, $r_{95\\%}= 0\\hbox{$^\\circ$}.086 = 5.2'$ \n(statistical only; $r_{68\\%}=3.2'$). To account for possible \ncontamination from nearby sources, the model included all point sources \ndetected at $>5\\sigma$ in an internal LAT 9-month source list within a \nregion of interest (ROI) of $r$=15\\hbox{$^\\circ$}\\ centered on the $\\gamma$-ray \nposition. Galactic diffuse emission was modeled using GALPROP \n\\citep{str04}, updated to include recent gas maps and a more accurate \ndecomposition into Galactocentric rings (galdef ID {\\tt \n54$\\_$59varh7S}). An additional isotropic diffuse component modeled as a \npower-law was included. Figure~\\ref{figure-vla} shows the resultant \n$\\gamma$-ray source localization on a VLA radio image from \n\\citet{owe00}. The $\\gamma$-ray source is positionally coincident with \nthe known radio position of the M87 core \\citep[RA = 187\\hbox{$^\\circ$}.706, Dec.\\ \n= 12\\hbox{$^\\circ$}.391;][]{fey04}, with an offset (0\\hbox{$^\\circ$}.020 = 1.2$'$) that is a \nsmall fraction of the localization circle. Currently, the best estimate \nof the systematic uncertainty in $r_{95\\%}$ is 2.4$'$ \\citep{bsl}, which \nshould be added in quadrature to the determined statistical one.\n\nSpectral analysis was performed utilizing an unbinned likelihood fit of \nthe $>$200 MeV data with a power-law ($dN\/dE \\propto E^{-\\Gamma}$) \nimplemented in the {\\tt GTLIKE} tool. This resulted in $F$($>$100 MeV) = \n2.45 ($\\pm 0.63$) $\\times$ 10$^{-8}$ ph cm$^{-2}$ s$^{-1}$\\ with a photon index, $\\Gamma = 2.26 \\pm 0.13$; \nerrors are statistical only. The flux was extrapolated down to 100 MeV \nto facilitate comparison with the previous EGRET non-detection of $<$ \n2.18 $\\times$ 10$^{-8}$ ph cm$^{-2}$ s$^{-1}$\\ (2$\\sigma$) from observations spanning the 1990's \n\\citep{rei03}. Thus, there is no apparent changes in the flux (i.e., a \nrise) in the decade since the EGRET observations. Systematic errors of \n($+0.17$\/$-0.15$) $\\times$ 10$^{-8}$ ph cm$^{-2}$ s$^{-1}$\\ on the flux and $+0.04$\/$-0.11$ on the index \nwere derived by bracketing the energy-dependent ROI of the IRFs to \nvalues of 10$\\%$, 5$\\%$, and 20$\\%$ above and below their nominal values \nat log($E$[MeV]) = 2, 2.75, and 4, respectively. The spectrum extends to \njust over 30 GeV where the highest energy photon is detected within the \n95$\\%$ containment.  The LAT spectral data points presented in \nFigure~\\ref{figure-sedgamma} were generated by performing a subsequent \nlikelihood analysis in five equal logarithmically spaced energy bins \nfrom $0.2-31.5$ GeV. The 1$\\sigma$ bounds on the spectrum, obtained from \nthe full $>$200 MeV unbinned likelihood fit, were extended to higher \nenergies for comparison with previous TeV measurements (see section~3).\n\nLightcurves were produced in 10-day (Figure~\\ref{figure-lightcurve}) and \n28-day (not shown) bins over the 10-month LAT dataset. Considering the \nlimited statistics, it was necessary to fix the photon index to the \n(average) fitted value in order to usefully gauge variability in the \nflux. Considering only statistical errors of all the binned data points \nwith $TS \\geq1$ ($1\\sigma$), a $\\chi^{2}$ test against the weighted mean \nfluxes of the 10-day and 28-day lightcurves resulted in probabilities, \n$P(\\chi^{2},\\nu) = 22\\%$ and 70$\\%$, respectively, indicating plausible \nfits to the tested hypothesis.  We conclude that there is no evidence \nfor variability over the period of observations.\n\n\n\\begin{figure}\n\\epsscale{1.2}\n\\plotone{fig2.eps}\n \\caption{The observed LAT spectrum (red circles) with representative\nTeV measurements of M87 in a low state from the 2004 observing season\n(black triangles) and during a high state in 2005 (blue squares), both\nby HESS \\citep{aha06}. The lines indicate 1$\\sigma$ bounds on the power\nlaw fit to the LAT data as well as its extrapolation into TeV energies.\n}\n\\label{figure-sedgamma}\n\\end{figure}\n\n\n\\begin{figure}\n\\epsscale{1.2}\n\\plotone{fig3.eps}\n \\caption{Lightcurve in 10-day bins obtained with the fitted photon\nindex ($\\Gamma$=2.26) fixed. The average flux is indicated with the\nsolid horizontal line and the dotted lines are $\\pm 1 \\sigma$ about the\naverage. Data points with $TS<1$ (i.e., 1$\\sigma$) are shown as upper\nlimits.\n}\n\\label{figure-lightcurve}\n\\end{figure}\n\n\nA radial profile of the $\\gamma$-ray source counts (not shown) was \nextracted for the total energy range ($>$200 MeV). The profile is \nconsistent with that of a point source simulated at energies $0.2-200$ \nGeV using the fitted spectral parameters above with a reduced \n$\\chi^{2}=1.04$ for 20 degrees of freedom. The total $\\sim$0\\hbox{$^\\circ$}.2 \nextent of the 10's kpc-scale radio lobes of M87 \n\\citep[Figure~\\ref{figure-vla};][]{owe00} is comparable to the LAT \nangular resolution, $\\theta_{\\rm 68} \\simeq 0\\hbox{$^\\circ$}.8~ E_{\\rm GeV}^{-0.8}$ \n\\citep{atw09}. Therefore, from the presently available data, we can not \ndisentangle (or exclude) a possible contribution of the extended radio \nfeatures to the total $\\gamma$-ray flux.\n\nTo gauge the X-ray activity of M87 over the duration of the LAT \nobservations, we analyzed five new 5 ksec $Chandra$ ACIS-S images \nobtained in $\\sim$6 week intervals between Nov.\\ 2008 and May 2009 (PI: \nD.~E.\\ Harris). The X-ray core fluxes (0.5--7 keV) in these monitoring \nobservations, $(1.2-1.6) \\times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ \n\\citep[$\\sim 0.4-0.6$ keV s$^{-1}$ in the units of Fig.~9 of][]{har09}, \nare at the low end of the observed range over the last $\\sim$7 years. \nAdditionally, the fractional variability is small ($\\sigma$\/$<$flux$> \n\\sim 0.1$), indicating low X-ray activity in the core over the LAT \nobserving period.\n\nAt milli-arcsecond (mas) resolution in the radio band, M87 has been \nmonitored with the VLBA at 15 GHz since 1995 as part of the 2cm survey \n\\citep{kel04} and MOJAVE \\citep{lis09} programs\\footnote{See: \nhttp:\/\/www.physics.purdue.edu\/MOJAVE\/}. These data were re-imaged \nuniformly at 0.6 mas $\\times$ 1.3 mas (position angle=--11\\hbox{$^\\circ$}) \nresolution to match the additional map presented in \\citet{kov07} \nresulting in 23 total measurements of the unresolved core flux up to the \nlatest observation on Jan.\\ 7, 2009 (one of the \n$Chandra$ exposures described above was obtained on the same day). \nThis observation is the only one overlapping with the LAT dataset \nand the peak flux of 1.05 Jy\/beam is consistent with the average over \nall the measurements ($1.11 \\pm 0.16$ Jy\/beam). An indication of the \nsensitivity of these data to detecting flaring core emission is that the \nhigh flux state observed in the detailed 43 GHz VLBA monitoring at the \ntime of TeV flaring in 2008 \\citep[][see \n\\S~\\ref{section-discussion}]{tev09} is visible in the 15 GHz data as a \nsingle high point on May 1, 2008 (1.45 Jy\/beam)\\footnote{Conversely, \nduring the previous TeV flaring period spanning March - May 2005 \n\\citep{aha06}, no comparable flare was visible in the VLBA 15 GHz core: \n1.02 Jy\/beam peak on Apr.\\ 21st and 0.98 Jy\/beam on Nov.\\ 7th. Instead, \nthe flaring X-ray\/optical\/radio knot HST-1 (60 pc from the core, \nprojected) was observed to peak in early 2005 \\citep{har09}, suggesting \nan association with the variable TeV source \n\\citep{che07}.\\label{footnote-hst1}}. This only suggests a period of low \nactivity in the radio core (as reflected in the X-ray data), as the \nsingle radio flux may not be representative of the entire 10-month LAT \nviewing period.\n\n\n\\section{Discussion}\\label{section-discussion}\n\nIn blazars, it is commonly believed that $\\gamma$-rays are produced in \ncompact emission regions moving with relativistic bulk velocities in or \nnear the parsec scale core in order to explain the observed rapid \nvariability and to avoid catastrophic pair-production \n\\citep[e.g.,][]{don95}. Consequently, it is natural to extend this \nsupposition to radio galaxies \\citep{chi01} which are believed to have \njets oriented at systematically larger angles to our line of sight, thus \nconstituting the parent population of blazars. Indeed, in the case of \nM87, a significant months timescale rise in the flux of the sub-pc scale \nradio core was discovered with the VLBA (at 43 GHz) during a period in \nearly 2008 when few day timescale TeV flaring was detected \n\\citep{tev09}. During this period of increased activity, a $Chandra$ \nmeasurement of the sub-arcsecond scale X-ray nucleus also indicated a \nrelatively higher flux than seen in past observations \\citep{har09}, \nthus signaling a common origin for the flaring emissions in the M87 \nnucleus. Therefore, during periods of lower $\\gamma$-ray activity, the \nradio\/X-ray core can also be considered a dominant source of the \nunresolved higher-energy emission, and we discuss this in the context of \nthe LAT MeV\/GeV detection.\n\nIn Figure~\\ref{figure-sedgamma}, the LAT spectrum of M87 is plotted \nalong with representative integrated TeV spectra from HESS \n\\citep{aha06}. The TeV measurements cover periods when M87 was in its \nhistorical-minimum (in 2004), and during a high state \\citep[in 2005 -- \ncf., Fig.~3 in][]{acc08}. Although the formal difference in the fitted \nphoton indices of the TeV data at high and low states is not \nstatistically significant ($\\Gamma = 2.22 \\pm 0.15$ and $2.62 \\pm 0.35$, \nrespectively), the LAT MeV\/GeV spectrum ($\\Gamma =2.3$) connects \nsmoothly with the low-state TeV spectrum. Taken together with the X-ray \nand radio measurements obtained during the LAT observation \n(\\S~\\ref{section-lat}), we view this as an indication that M87 is in an \noverall low $\\gamma$-ray activity state during the considered period. In \nfact, no significant TeV flaring was detected in a preliminary analysis \nof 18 hrs of contemporaneous VERITAS observations from Jan.\\ - Apr.\\ \n2009 \\citep{hui09}.\n\nM87 is the faintest $\\gamma$-ray radio galaxy detected so far by the LAT \nwith a $>$100 MeV flux ($\\sim 2.5$$\\times$ 10$^{-8}$ ph cm$^{-2}$ s$^{-1}$) about an order of magnitude \nlower than in Cen~A \\citep{lbas} and Per~A \\citep{pera}; the \ncorresponding $>100$ MeV luminosity, $4.9 \\times 10^{41}$ erg s$^{-1}$, \nis 4$\\times$ greater than that of Cen~A, but $>$200$\\times$ smaller than \nin Per A. There is no evidence of intra-year or decade-timescale MeV\/GeV \nvariability in M87 (\\S~\\ref{section-lat}), in contrast to the $\\simgt \n7\\times$ and $\\sim 1.6\\times$ larger observed LAT fluxes than the \nprevious EGRET ones in the cases of Per~A \\citep{pera} and Cen~A \n\\citep{lbas}, respectively. The $\\gamma$-ray photon index of M87 in the \nLAT band is similar to that of Per~A ($\\Gamma = 2.3$ and 2.2, \nrespectively), while being smaller than observed in Cen~A \n\\citep[$\\Gamma=2.9$;][]{lbas}. These sources are low-power (FRI) radio \ngalaxies, and have broad low-energy synchrotron and high-energy inverse \nCompton (IC) components in their spectral energy distributions (SEDs) \npeaking roughly in the infrared and $\\gamma$-ray bands, respectively. \nLow-energy peaked BL Lac objects have similar shaped SEDs, with \napproximately equal apparent luminosities \\citep[e.g.,][]{kub98}. As FRI \nradio galaxies are believed to constitute the parent population of BL \nLacs in unified schemes \\citep{urr95}, the overall similarity of their \nSEDs is not surprising.\n\nWe construct a SED for M87 (Figure~\\ref{figure-sed}) using the the LAT \n$\\gamma$-ray spectrum and the overlapping Jan.\\ 7, 2009 $Chandra$ and \nVLBA measurements of the core. Also plotted are historical radio to \nX-ray fluxes of the core \\citep[see][]{spa96,tan08} measured at the \nhighest resolutions at the respective frequencies. The core is known to \nbe variable, with factors of $\\sim2$ changes on months timescales common \nin the optical and X-ray bands \\citep{per03,har09}. To help constrain \nthe overall SED at frequencies between the X-ray and LAT measurements, \nwe determined integrated 3$\\sigma$ upper limits in three hard X-ray \nbands \\citep[following,][]{aje08} based on the $Swift$\/BAT dataset in \n\\citet{aje09}, including about another additional year of exposure \n(i.e., $\\sim$4 years total from Mar.\\ 2005 - Jan.\\ 2009).\n\n\n\\begin{figure}\n\\epsscale{1.2}\n\\plotone{fig4.eps}\n \\caption{SED of M87 with the LAT spectrum and the Jan.\\ 7, 2009 MOJAVE\nVLBA 15 GHz and $Chandra$ X-ray measurements of the core indicated in\nred. The non-simultaneous 2004 TeV spectrum described in\nFigure~\\ref{figure-sedgamma} and $Swift$\/BAT hard X-ray limits\n(\\S~\\ref{section-discussion}) of the integrated emission are shown in\nlight brown. Historical measurements of the core from VLA 1.5, 5, 15 GHz\n\\citep{bir91}, IRAM 89 GHz \\citep{des96}, SMA 230 GHz \\citep{tan08},\n$Spitzer$ 70, 24 $\\mu$m \\citep{shi07}, Gemini 10.8 $\\mu$m \\citep{per01},\n$HST$ optical\/UV \\citep{spa96}, and $Chandra$ 1 keV from \\citet[][hidden\nbehind the new measurements]{mar02} are plotted as black circles. The\nVLBA 15 GHz flux is systematically lower than the historical\narcsec-resolution radio to infrared measurements due to the presence of\nintermediate scale emission \\citep[see e.g.,][]{kov07}. The blue line\nshows the one-zone SSC model fit for the core described in\n\\S~\\ref{section-discussion}.\n}\n\\label{figure-sed}\n\\end{figure}\n\n\nThe broad-band SED is fit with a homogeneous one-zone synchrotron \nself-Compton (SSC) jet model \\citep{fin08} assuming an angle to the line \nof sight, $\\theta$=10\\hbox{$^\\circ$}, and bulk Lorentz factor, $\\Gamma_{\\rm b} = \n2.3$ (Doppler factor, $\\delta = 3.9$), consistent with observations of \napparent motions of $\\simgt 0.4c$ ($\\Gamma_{\\rm b} > 1.1$) in the \nparsec-scale radio jet \\citep{ly07}. A broken power-law electron energy \ndistribution $N(\\gamma)\\propto\\gamma^{-p}$ is assumed, and the indices, \n$p_{\\rm 1}=1.6$ for $\\gamma$ = [1, $4 \\times 10^{3}$] and $p_{\\rm \n2}=3.6$ for $\\gamma$ = [$4 \\times 10^{3}$, $10^{7}$] are best guesses \nbased on the available core measurements. The normalization at low \nenergies is constrained by the single contemporaneous VLBA 15 GHz flux \nwhich is measured with $\\sim 10^{2}-10^{3}\\times$ better resolution than \nthe adjacent points. The source radius, $r= 1.4 \\times 10^{16}$ cm = 4.5 \nmpc is chosen to be consistent with the best VLBA 43 GHz map resolution \n\\citep[$r<$7.8 mpc = 0.1 mas,][]{jun99,ly07} and is of order the size \nimplied by the few day timescale TeV variability \\citep{tev09}. For the \nsource size adopted, internal $\\gamma-\\gamma$ absorption is avoided so \nthat the LAT spectrum extends relatively smoothly into the TeV band, \nconsistent with the historical-minimum flux detected by HESS \n\\citep{aha06} and the preliminary upper limit of $<1.9\\%$ Crab from \nVERITAS observations \\citep{hui09} contemporaneous with the LAT ones.\n\nIn the SSC model, the magnetic field is $B = 55$ mG \nand assuming the proton energy density is 10$\\times$ greater than the \nelectron energy density, the total jet power is $P_{\\rm j} \\sim 7.0 \n\\times 10^{43}$ erg s$^{-1}$. The jet power is particle dominated, with \nonly a small contribution from the magnetic field component ($P_{\\rm B} \n\\sim 2 \\times 10^{40}$ erg s$^{-1}$). In comparison, the total kinetic \npower in the jet is $\\sim$few $\\times 10^{44}$ erg s$^{-1}$ as \ndetermined from the energetics of the kpc-scale jet and lobes \n\\citep{bic96}, and is consistent with the jet power available from \naccretion, $P_{\\rm j} \\simlt 10^{45}$ erg s$^{-1}$ \\citep{rey96,dim03}. \nThese power estimates are similar to those derived for BL Lacs from \nsimilarly modeling their broad-band SEDs \\citep[e.g.,][]{cel08}.\n\nAs applied to M87, such single-zone SSC emission models also reproduce \nwell the broad-band SEDs up to MeV\/GeV energies in the radio galaxies \nPer~A \\citep{pera} and Cen~A \\citep{chi01}. In this context, the \nobserved MeV\/GeV $\\gamma$-ray fluxes of blazars appear to be correlated \nwith their compact radio cores \\citep{lbas,kov09}, suggesting a common \norigin in the Doppler boosted emission in the sub-parsec scale jets. The \nfact that the 3 radio galaxies detected by the LAT so far have amongst \nthe brightest ($\\simgt$1 Jy) unresolved radio cores, in line with these \nexpectations \\citep{ghi05}, lend evidence for a common connection \nbetween the $\\gamma$-ray and radio emitting zones in such jets.\n\nIt should be emphasized that these observations are not simultaneous and \nparticularly, the TeV emission is known to be variable on year \ntimescales, so other emission components may contribute to the variable \nemission. Therefore, although not strictly required, more sophisticated \nmodels over the single-zone one presented can reproduce or contribute to \nthe observed emission. In particular, the beaming requirements in the \none-zone SSC modeling of the three known $\\gamma$-ray radio galaxies are \nsystematically lower than required in BL Lacs, suggesting velocity \nprofiles in the flow \\citep{chi01}. Such models \\citep{geo05,tav08} have \nin fact been used to fit the SED of M87 in addition to models based on \nadditional spatial structure \\citep[e.g.,][]{len07}. Protons, being \ninevitably accelerated if they co-exist with electrons in the emission \nregions, probably dominate energetically and dynamically the jets of \npowerful AGN \\citep[e.g.,][]{cel08}. Applying the synchrotron-proton \nblazar model \\citep{muc03,rei04} to the quiescent M87 data set yields \nreasonable agreement model fits that support a highly magnetized compact \nemission region with approximate equipartition between fields and \nparticles and a total jet power comparable with the above estimates, \nwhere protons are accelerated up to $\\sim 10^{9}$ GeV.\n\nOutside of the pc-scale core, the well-known arcsecond-scale jet \n\\citep[e.g.,][]{bir91,mar02,per05} is also a possible source of IC \nemission. As both the LAT and TeV telescopes are unable to spatially \nresolve emission on such small scales, the expected spectral and \ntemporal properties of the predicted emission must be examined. On the \nobserved scales, the dominant seed photon source is the host galaxy \nstarlight, and such an IC\/starlight model applied to one of the \nbrightest resolved knots in the jet -- knot A, $\\sim$1 kpc projected \ndistance from the core -- results in a spectrum peaking at TeV energies \n\\citep{sta05}, thus producing a harder MeV\/GeV spectrum than observed by \nthe LAT. Even closer to the core ($\\sim$60 pc, projected), the \nsuperluminal knot HST-1 \\citep[$>4c-6c$;][]{bir99,che07} is a more \ncomplex case. This knot is more compact than knot A, and its IC emission \nis expected to be further enhanced by the increased energy densities of \nthe surrounding circumnuclear and galactic photon fields, as well of the \ncomoving synchrotron radiation \\citep{sta06}. The radio\/optical\/X-ray \nfluxes of HST-1 have been declining since its giant flare peaked in 2005 \n\\citep{har09}, with current X-ray fluxes comparable to its pre-flare \nlevels in 2002. Considering the variable and compact nature of the \nsource (with observed months doubling timescales implying $r\\simlt 22 \n\\delta$ mpc; cf., footnote~\\ref{footnote-hst1}), the predicted IC \nspectrum has a complex temporal and spectral behavior. In the absence of \ndetailed contemporaneous measurements, its possible role in the \nproduction of the LAT observed MeV\/GeV emission is unclear.\n\nContinued LAT monitoring of M87 coordinated with multi-wavelength \nobservations can extend the current study of `quiescent' emission to \npossible flaring, in order to further address the physics of the \nradiation zone. While the extragalactic $\\gamma$-ray sky is dominated by \nblazars \\citep{har99,lbas}, this optimistically indicates an emerging \npopulation of $\\gamma$-ray radio galaxies.  Other examples, including \nthe few possible associations with EGRET detections like, NGC~6251 \n\\citep{muk02} and 3C~111 \\citep{sgu05,har08} await confirmation with the \nLAT, and more radio galaxies are expected to be detected at lower \nfluxes. This holds great promise for systematic studies of relativistic \njets with a range of viewing geometries in the high energy $\\gamma$-ray \nwindow opened up by the $Fermi$-LAT.\n\n\n\\acknowledgments\n\nThe $Fermi$ LAT Collaboration acknowledges generous ongoing support from \na number of agencies and institutes that have supported both the \ndevelopment and the operation of the LAT as well as scientific data \nanalysis.  These include the National Aeronautics and Space \nAdministration and the Department of Energy in the United States, the \nCommissariat \\`a l'Energie Atomique and the Centre National de la \nRecherche Scientifique \/ Institut National de Physique Nucl\\'eaire et de \nPhysique des Particules in France, the Agenzia Spaziale Italiana and the \nIstituto Nazionale di Fisica Nucleare in Italy, the Ministry of \nEducation, Culture, Sports, Science and Technology (MEXT), High Energy \nAccelerator Research Organization (KEK) and Japan Aerospace Exploration \nAgency (JAXA) in Japan, and the K.~A.~Wallenberg Foundation, the Swedish \nResearch Council and the Swedish National Space Board in Sweden.\n\nAdditional support for science analysis during the operations phase is \ngratefully acknowledged from the Istituto Nazionale di Astrofisica in \nItaly.\n\nC.C.C. was supported by an appointment to the NASA Postdoctoral Program \nat Goddard Space Flight Center, administered by Oak Ridge Associated \nUniversities through a contract with NASA. Support from NASA grants \nGO8-9116X and GO9-0108X (D.E.H., F.M.) and the Foundation BLANCEFLOR \nBoncompagni-Ludovisi, n'ee Bildt (F.M.) are acknowledged. This research \nhas made use of data from the MOJAVE database that is maintained by the \nMOJAVE team \\citep{lis09}. We thank F.~Owen for providing the VLA 90cm \nimage.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $P=\\Big(p(i,j)\\Big)_{i,j \\in \\Omega}$ be a reversible Markov chain over a sample space $\\Omega$, that is, it must satisfy the following {\\it detailed balance conditions}:\n$$\\pi_i p(i,j)=\\pi_jp(j,i) \\qquad \\forall i,j \\in \\Omega,$$\nwhere $\\pi$ is a non-trivial non-negative function over $\\Omega$. If $P$ admits a unique stationary distribution $\\nu$, then ${1 \\over \\sum\\limits_{i \\in \\Omega} \\pi_i}\\pi=\\nu$.\n\nIt can be shown that the reversible $P$ is a self-adjoint operator in $\\ell^2(\\pi)$, the space generated by the following inner product induced by $\\pi$\n $$\\ip{f}{g}_{\\pi}=\\sum_{i \\in \\Omega} f(i)g(i)\\pi_i$$\n If $P$ is a tridiagonal operator (i.e.\\ a nearest-neighbor random walk) on  $\\Omega=\\{0,1,2,\\dots\\}$, then it must have a simple spectrum, and is diagonalizable via orthogonal polynomials as it was studied in the 50's by Karlin and McGregor, see \\cite{SKJM1959}, \\cite{GS1975}, and \\cite{MI2005}. There, the extended eigenfunctions $Q_j(\\lambda)$ satisfying $Q_0 \\equiv 1$ and $$P\\begin{pmatrix}\n  Q_0(\\lambda)\\\\\n  Q_1(\\lambda)\\\\\n  Q_2(\\lambda)\\\\\n  \\vdots\n\\end{pmatrix} =\n\\lambda\\begin{pmatrix}\n  Q_0(\\lambda)\\\\\n  Q_1(\\lambda)\\\\\n  Q_2(\\lambda)\\\\\n  \\vdots\n\\end{pmatrix}\n$$\n are orthogonal polynomials with respect to a probability measure\n $\\psi$.  If we let $p_t(i,j)$ denote the entries of the operator\n $P^t$ that represent $t$ step transition probabilities from state $i$ to state $j$ then\n $$p_t(i,j)=\\pi_j \\int_{-1}^1 \\lambda^t Q_i(\\lambda) Q_j(\\lambda) d\\psi(\\lambda)~~~\\forall i,j \\in \\Omega, $$\n where $\\pi_j$ with $\\pi_0=1$ is the reversibility measure of $P$.\n \n\n We will use the following distance to measure the deviation from the stationary distribution on a scale from zero to one. \n \\begin{mydef}\nIf $\\mu$ and $\\nu$ are two probability distributions over a sample space $\\Omega$, then the {\\it total variation distance} is \n$$\\| \\nu - \\mu \\|_{TV} = {1 \\over 2} \\sum_{x \\in \\Omega} |\\nu(x)-\\mu(x)|=\\sup_{A \\subset \\Omega} |\\nu(A)-\\mu(A)|$$\n\\end{mydef}\n \\noindent\n Let $\\rho=\\sum_{k=0}^{\\infty} \\pi_k$. Observe that $\\rho < \\infty$ if and only if the random walk $P$ is positive recurrent. Recall that $\\nu={1 \\over \\rho}\\pi$ is the stationary probability distribution. If in addition to being positive recurrent, the aperiodic nearest neighbor Markov chain originates at site $j$, then\n the total variation distance between the distribution $\\mu_t=\\mu_0P^t$ and $\\nu$ is given by\n \\begin{equation} \\label{eqTV}\n \\left\\|\\nu - \\mu_t \\right\\|_{TV}  =  {1 \\over 2} \\sum_{n \\in \\Omega} \\pi_n \\left|\\int_{(-1,1)} \\lambda^t Q_j(\\lambda) Q_n(\\lambda) d\\psi(\\lambda)\\right|,\n \\end{equation} \n as measure $\\psi$ contains a point mass of weight ${1 \\over \\rho}$ at $1$. \n See \\cite{YK2009}.\n \n The rates of convergence are quantified via mixing times, which for an infinite state space with a unique stationary distribution are defined as follows. Here the notion of a mixing time depends on the state of origination $j$ of the Markov chain. See \\cite{YK2010}.\n\\begin{mydef}\n Suppose $P$ is a Markov chain with a stationary probability distribution $\\nu$ that commences at $X_0=j$. Given an $\\epsilon >0$, the mixing time $t_{mix}(\\epsilon)$ is defined as\n $$t_{mix}(\\epsilon)=\\min\\left\\{t~:~\\|\\nu-\\mu_t\\|_{TV} \\leq \\epsilon \\right\\}$$\n\\end{mydef}\n\\vskip 0.2 in\n\\noindent\n In the case of a nearest-neighbor process  on $\\Omega=\\{0,1,2,\\dots\\}$ commencing at $j$,  the corresponding mixing time has the following simple expression in orthogonal polynomials\n$$ t_{mix}(\\epsilon)=\\min\\left\\{t~:~\\sum_{n} \\pi_n \\left|\\int_{(-1,1)} \\lambda^t Q_j(\\lambda) Q_n(\\lambda) d\\psi(\\lambda)\\right| \\leq 2 \\epsilon \\right\\},$$\n\nInvestigations into the use of  orthogonal polynomial techniques (see \\cite{SKJM1959}, \\cite{GS1975}) in the estimation of mixing times and distance to the stationary distribution has been carried out in \\cite{YK2010} for certain classes of random walks.   In this paper we consider the problem from the other direction.  Namely given a large class of orthogonal polynomials we outline how to find the corresponding random walk and estimate the rate for the distance to the stationary distribution.  \n\nMore specifically beginning with the Jacobi polynomials, whose weight\nfunction lies in $(-1,1)$ we use Koornwinder's techniques\n\\cite{TK1984} to attach a point mass at $1$.  For the class\nof Jacobi type polynomials $Q_n$ thus obtained, the three term recurrence\nrelationship is understood \\cite{HKJW1996}.  The tridiagonal operator\ncorresponding to these polynomials is not a Markov chain, however the\noperator can be deformed to become one.  The corresponding changes in\nthe polynomials are easy to trace.  This gives a four parameter family\nof nearest neighbor Markov chains whose distance to the stationary\ndistribution decays in a non-geometric way.  In principle the\nasymptotic analysis presented in this paper can be applied to the\nentire four parameter family.  We outline how this proceeds for\nChebyshev-type subfamily consisting of taking $\\alpha=\\beta=-1\/2$ in\nthe Koornwinder class.\n\nWe would like to point out the important results of V.~B.~Uvarov\n\\cite{VU1969} on transformation of orthogonal polynomial systems by\nattaching point masses to the orthogonality measure, predating the\nKoornwinder results by fifteen years.  The results of V.~B.~Uvarov can\npotentially be used in order to significantly extend the scope of\nconvergence rate problems covered in this current manuscript.\n\nThe paper is organized as follows. In Section \\ref{sec:koorn} we discuss constructing positive recurrent Markov chains from the Jacobi family of orthogonal polynomials adjusted by using Koornwinder's techniques  to place a point mass at $x=1$. Next, we derive an asymptotic upper bound on the total variation distance to the stationary distribution in the case of general $\\alpha>-1$ and $\\beta>-1$ in Section \\ref{sec:asympt}. Our main result, Theorem~\\ref{main}, is presented in Section \\ref{sec:chebyshev}. There, for the case of Chebyshev type polynomials corresponding to $\\alpha=\\beta=-1\/2$, we produce both asymptotic lower and upper bounds for the total variation distance. Finally, in Section~\\ref{comparison} we compare our main result to related results obtained by other techniques.\n\n\n\n\n\\section{From Orthogonal Polynomials to Random Walks via Koornwinder}\\label{sec:koorn}\n\n\nT.~Koornwinder \\cite{TK1984} provides a method for finding the\northogonal polynomials whose weight distribution is obtained from the\nstandard Jacobi weight functions\n$C_{\\alpha,\\beta}(1-x)^{\\alpha}(1+x)^{\\beta}$ by attaching weighted\npoint masses at $-1$ and $1$. A spectral measure corresponding to a\nMarkov chain contains a point mass at $-1$ if and only if the Markov\nchain is periodic. A spectral measure for an aperiodic Markov chain\ncontains a point mass at $1$ if and only if it is positive\nrecurrent. Thus in order to create a class of positive recurrent\naperiodic Markov chains with a Koornwinder type orthogonal polynomial\ndiagonalization we will only need to attach a point mass at $1$ and no\npoint mass at $-1$.\n \n\nLet $N\\geq 0$ and let $\\alpha$, $\\beta>-1$.  For $n=0, 1, 2, \\ldots$\ndefine\n\\begin{equation}\\label{eq:Koornwinder}\nP_n^{\\alpha,\\beta,N}(x)=\\Big(\\frac{(\\alpha+\\beta+2)_{n-1}}{n!}\\Big)A_n\\Big[-N\n  (1+x)\\frac{d}{dx}+B_n\\Big]P_n^{\\alpha,\\beta}(x),\n\\end{equation}\nwhere \n$$A_n=\\frac{(\\alpha+1)_n}{(\\beta+1)_n},$$\n$$B_n=\\frac{(\\beta+1)_nn!}{(\\alpha+1)_n(\\alpha+\\beta+2)_{n-1}}+\\frac{n(n+\\alpha+\\beta+1)N}{(\\alpha+1)},$$\n$P_n^{\\alpha,\\beta}$ is the standard Jacobi polynomials of degree $n$\nand order $(\\alpha,\\beta)$, $(x)_n=x(x+1)\\cdots(x+n-1)$.  These\npolynomials form a system of orthogonal polynomials with respect to\nthe probability measure\n$d\\psi(x)={C_{\\alpha,\\beta}(1-x)^\\alpha(1+x)^\\beta dx+N\\delta_1(x)\n  \\over N+1}$, where $C_{\\alpha,\\beta}={1 \\over\n  \\mathcal{B}(\\alpha+1,\\beta+1)}$, \\quad $\\mathcal{B}(\\cdot,\\cdot)$ is\nthe beta function, and $\\delta_1(x)$ denotes the a unit point mass\nmeasure at $x=1$. See T.~Koornwinder \\cite{TK1984}. Direct calculation\nshows that $P_n^{\\alpha,\\beta,N}(1)=\\frac{(\\alpha+1)_n}{n!}$, and so\nwe normalize $Q_n(x)=n!P_n^{\\alpha,\\beta,N}(x)\/(\\alpha+1)_n$ which is\nthe orthogonal set of polynomials with respect to $d\\psi$ satisfying\n$Q_n(1)=1$.\n\nAs we have mentioned earlier, the tridiagonal operator $H$ corresponding to the recurrence relation of the orthogonal polynomials may not be a Markov chain operator.  Let $p_i$, $r_i$ and $q_i$ denote the coefficients in the tridiagonal recursion\n$$p_i Q_{i+1}(x)+r_iQ_i(x)+q_i Q_{i-1}(x)=xQ_i(x),$$\nfor $i=0,1,2,\\dots$, where we let $Q_{-1} \\equiv 0$ as always.\n\n\nNotice because the polynomials are normalized so that\n$Q_i(1)=1$ it follows immediately that $p_i+r_i+q_i=1$.\nHowever some of the coefficients  $p_i$, $r_i$, or $q_i$ may turn out to be negative, in which case the rows of the tridiagonal operator $A$ would add up to one, but will not necessarily consist of all nonnegative entries. \n\nIn the case when all the negative entries are located on the main diagonal, this may be overcome by\nconsidering the operator $\\frac{1}{\\lambda+1}(H+\\lambda I)$.  For $\\lambda \\geq-\\inf\\limits_i r_i$ this ensures all entries in the matrix $\\frac{1}{\\lambda+1}(H+\\lambda I)$ are nonnegative and hence can be thought of as transition probabilities.  More generally, if a polynomial $p(\\cdot)$ with coefficients adding up to one is found to satisfy $p(H) \\geq 0$ coordinatewise, then such $p(H)$ would be a Markov chain.\n\n\n\n\n\n\n\\section{An Asymptotic Upper Bound for Jacobi type Polynomials}\\label{sec:asympt}\n\nIn this section we derive asymptotic estimates for the distance to the\nstationary distribution when our operator given by\n$P_{\\lambda}=\\frac{1}{\\lambda+1}(H+\\lambda I)$ is a Markov chain. In\nthis case the Karlin-McGregor orthogonal polynomials for $P_{\\lambda}$\nare $Q_j\\Big((1+\\lambda)x-\\lambda\\Big)$ and the orthogonality\nprobability measure is \n${1 \\over 1+\\lambda}d\\psi\\Big((1+\\lambda)x-\\lambda\\Big)$ over \n$\\Big({\\lambda-1 \\over \\lambda+1}, 1 \\Big]$, where the $Q_j$ are the\nJacobi type polynomials introduced by Koornwinder from the previous\nsection.\n\nOf course the new operator $P_{\\lambda}$ is again tridiagonal.  For the $n$-th row of $P_{\\lambda}$, let us denote the $(n-1)$-st, $n$-th, and $(n+1)$-st entries by $q_n^\\lambda$, $r_n^\\lambda$, and $p_n^\\lambda$ respectively. Here the entries of $P_{\\lambda}$ can be expressed via the entries of $H$ as follows\n$$p_n^\\lambda=\\frac{p_n}{1+\\lambda}, \\qquad r_n^\\lambda=\\frac{r_n+\\lambda}{1+\\lambda}, \\quad \\text{ and } \\quad q_n^\\lambda=\\frac{q_n}{1+\\lambda}$$  \nClearly we still have that\n$p_n^\\lambda+r_n^\\lambda+q_n^\\lambda=1$.  \n\nWith the probabilities in hand we now compute the corresponding reversibility function $\\pi_n^\\lambda$ of $P_{\\lambda}$ which is equal to the corresponding function of $H$ defined as $\\pi_n=\\frac{p_0\\cdots p_{n-1}}{q_1\\cdots q_n}$.  Here $\\pi_0^\\lambda=1=\\pi_0$ and $\\pi_n^\\lambda=\\frac{p_0^\\lambda\\cdots p_{n-1}^\\lambda}{q_1^\\lambda\\cdots q_n^\\lambda}=\\frac{p_0\\cdots p_{n-1}}{q_1\\cdots q_n}=\\pi_n$.\n\n\nChanging variables in \\eqref{eqTV} yields\n$$\\|\\nu-\\mu_t\\|_{TV}=\\frac{1}{2}\\sum_{n=0}^{\\infty}\\pi_n\\bigg|\\int_{(-1,1)}\\Big(\\frac{x}{1+\\lambda}+\\frac{\\lambda}{1+\\lambda}\\Big)^tQ_j(x)Q_n(x)\\,d\\psi(x)\\bigg|$$\n\n\n\\begin{lem}\\label{lem:bounds}\nConsider the case when $p_n>0$ and $q_n>0$ for all $n \\geq 0$, and \\mbox{$\\infty> \\lambda \\geq-\\inf\\limits_i r_i$}.\nThen, for the Jacobi type polynomials $Q_j$ the distance to the stationary distribution satisfies the following bound\n\\begin{equation}\\label{jacobi_bound}\n\\norm{\\nu-\\mu_t}_{TV}\\leq\\frac{C_{\\alpha,\\beta,\\lambda}\\norm{Q_j}_\\infty}{(t+1)^{1+\\alpha}}\\sum_{n=0}^{t+j}\\pi_n\\norm{Q_n}_\\infty+\\frac{1}{2}\\sum_{n=j+t+1}^{\\infty}\\pi_n\n\\end{equation}\nfor a certain constant $C_{\\alpha,\\beta,\\lambda}$.\n\\end{lem}\n\n\\begin{proof}\nFor $n>j+t$, it follows from the orthogonality of the polynomials and our normalization $Q_i(1)=1$ that\n$$\\int_{(-1,1)}\\Big(\\frac{x}{1+\\lambda}+\\frac{\\lambda}{1+\\lambda}\\Big)^tQ_j(x)Q_n(x)\\,d\\psi(x)=1$$\n It is then easy to see that \n$\\norm{\\nu-\\mu}_{TV}\\leq I+II+\\frac{1}{2}\\sum_{n=j+t+1}^{\\infty}\\pi_n$, where \n$$I=\\frac{1}{2}\\sum_{n=0}^{j+t}\\pi_n\\int_{(-1,0)}\\abs{\\Big(\\frac{x+\\lambda}{1+\\lambda}\\Big)^tQ_j(x)Q_n(x)}(1-x)^\\alpha(1+x)^\\beta\\,dx$$\n$$\\text{ and } \\quad II=\\frac{1}{2}\\sum_{n=0}^{j+t}\\pi_n\\int_{(0,1)}\\Big(\\frac{x+\\lambda}{1+\\lambda}\\Big)^t\\big|Q_j(x)Q_n(x)\\big|(1-x)^\\alpha(1+x)^\\beta\\,dx$$\n\nTo estimate $I$ notice that $\\abs{\\frac{x+\\lambda}{1+\\lambda}}\\leq\n\\max(\\frac{\\lambda}{1+\\lambda}, \\abs{\\frac{1-\\lambda}{1+\\lambda}})<1$ for\n$\\lambda>0$. Hence \\mbox{$I\\leq A_j(|t|)e^{-ct}$} for an appropriate polynomial $A_j(\\cdot)$ such that \n$$\\frac{1}{2}\\|Q_j\\|_{\\infty}\\sum_{n=0}^{j+t}\\pi_n \\|Q_n\\|_{\\infty}\\int_{(-1,0)}(1-x)^\\alpha(1+x)^\\beta\\,dx \\leq A_j(|t|),$$\nand \n$c=-\\log\\left\\{\\max(\\frac{\\lambda}{1+\\lambda}, \\abs{\\frac{1-\\lambda}{1+\\lambda}})\\right\\}$. Such polynomial $A_j$ exists since $\\|Q_n\\|_{\\infty}$ grows polynomially in $n$ and $\\pi_n$ is bounded. See formula 22.14.1 in Abramowitz and Stegun \\cite{MAIS1972}.\n\n\nThus $I$ is clearly bounded by the right hand side of \\eqref{jacobi_bound}.\n\nFor the second term,\n$II\\leq \\frac{1}{2}\\sum_{n=0}^{j+t}\\pi_n\\norm{Q_nQ_j}_{\\infty}\\int_0^1\\Big(\\frac{x+\\lambda}{1+\\lambda}\\Big)^t\\big(1-x)^\\alpha(1+x)^\\beta\\,dx$.\nThere we make the change of variables $s=-\\log(\\frac{x+\\lambda}{1+\\lambda})$, and for\nsimplicity let $x(s)=(1+\\lambda)e^{-s}-\\lambda$.  Then the\nintegral reduces to\n$$(1+\\lambda)^{1+\\alpha}\\int_0^{\\log(\\frac{1+\\lambda}{\\lambda})}e^{-s(t+1)}\\big(1-\\lambda+(1+\\lambda)e^{-s})^\\beta\\big(1-e^{-s}\\big)^\\alpha\\,ds$$\n\nUsing the fact that $(1-e^{-s})^\\alpha=s^\\alpha\\Big(1+O(s)\\Big)$ and \n$\\big(1-\\lambda+(1+\\lambda)e^{-s})^\\beta=2^\\beta+O(s)$,\nthe above integral becomes\n$$(1+\\lambda)^{1+\\alpha}\\int_0^{\\log(\\frac{1+\\lambda}{\\lambda})}e^{-s(t+1)}\\Big(2^\\beta\ns^\\alpha+O(s^{\\alpha+1})\\Big)\\,ds,$$\nwhere the upper bounds $O(s)$ can be made specific.\nNext, applying the standard asymptotic methods of Laplace to this yields the following asymptotics  \n$$\\int_0^{\\log(\\frac{1+\\lambda}{\\lambda})}e^{-s(t+1)}\ns^\\alpha\\,ds ~\\asymp~ \\frac{\\Gamma(\\alpha+1)}{(t+1)^{1+\\alpha}}$$\n\nThus one can obtain a large enough constant $\\widetilde{C}_{\\alpha,\\beta,\\lambda}$ such that\n$$II\\leq\\\n\\frac{\\widetilde{C}_{\\alpha,\\beta,\\lambda}\\norm{Q_j}_\\infty}{(t+1)^{1+\\alpha}}\\sum_{n=0}^{t+j}\\pi_n\\norm{Q_n}_\\infty$$  \n\\end{proof}\n\nIn order to derived effective bounds on $\\norm{\\nu-\\mu_t}_{TV}$ it is\nnecessary to gain a more detailed understanding of $\\pi_n$\nand $\\norm{Q_n}_\\infty$.  When $\\min(\\alpha,\\beta)\\geq -\\frac{1}{2}$, the $\\norm{Q_n}_\\infty$ can be\nestimated using the known maximum for the Jacobi polynomials found in Lemma 4.2.1 on page 85 of \\cite{MI2005} together\nwith Koornwinder's definition of these polynomials.\n\nOne way to derive estimates for $\\pi_n$ is to use the expression\n$\\pi_n$ in terms of $p_n$,  $r_n$, and $q_n$.  For Koorwinder's class of\npolynomials these expressions are derived for all $\\alpha, \\beta, M,\nN$ in \\cite{HKJW1996}.  It can be verified directly that in the case\nwhen $M=0$, then\n$p_0=\\frac{2(\\alpha+1)}{(1+N)(\\alpha+\\beta+2)}>0$. After taking into\naccount the normilization $Q_n(1)=1$, and taking into account a\nsmall typo, it can be verified from equations (41)--(45) in \\cite{HKJW1996} that $p_n$ and\n$q_n$ are positive for $n\\geq 1$.  Thus the conditions for Lemma~\\ref{lem:bounds} are satisfied for all $\\alpha, \\beta >-1$. Furthermore, from (18), (19) and (32)  in \\cite{HKJW1996}\nit can be easily seen that $p_n\\to \\frac{1}{2}$ and $q_n\\to \\frac{1}{2}$\nas $n\\to\\infty$, and hence $r_n=1-p_n-q_n\\to 0$ as $n\\to \\infty$.  Thus\nfor $\\lambda$ large enough the operator $P^\\lambda$ corresponds to a\nMarkov chain.\n\nAs the expressions for these quantities\nlaborious to write down, instead we focus our attention on a specific\ncase in which our calculations are easy to follow.  Specifically\nwe focus on the Chebyshev polynomials.\n\n\n\n\\section{Chebyshev Polynomials: Upper and Lower Bounds}\\label{sec:chebyshev}\nBy applying Koorwinder's results to the Chebyshev polynomials of the first kind which correspond to the case of $\\alpha=\\beta=-{1 \\over 2}$, we arrive at a family of orthogonal polynomials with respect to the measure\n$\\frac{1}{1+N}\\Big(\\frac{1}{\\pi\\sqrt{1-x^2}}dx+N\\delta_1(x)\\Big)$.\nUsing \\eqref{eq:Koornwinder} we find that here,\n$$Q_n(x):=-N(x+1)U_{n-1}(x)+(1+2nN)T_n(x),$$ \nwhere $T_n$ and $U_n$ denote the Chebyshev polynomials of the first and second kind\nrespectively.  Notice that $U_n(1)=n+1$ and $T_n(1)=1$, which immediately to verify that $Q_n(1)=1$.\n\nOnce again we consider the operator \n$$H=\\begin{pmatrix}r_0 & p_0 & 0 & 0 & 0 &\\cdots\\\\\nq_1 & r_1 & p_1 & 0 & 0 & \\vdots\\\\\n0 & q_2 & r_2 & p_2 & 0 & \\ddots\\\\\n0  & 0 & q_3 & r_3 & \\ddots & \\ddots\\\\\n\\vdots & \\cdots & \\ddots & \\ddots & \\ddots & \\dots\\\\\n\\end{pmatrix},$$\non $\\ell^2(\\pi)$, so that vector $(Q_0(x), Q_1(x), Q_2(x),\n\\ldots)^T$ is an eigenvector with eigenvalue~$x$.\n\nSpecifically the numbers $p_n$,  $r_n$, and $q_n$ satisfy\n$p_0P_1(x)+r_0P_0(x)=x$ for $n=0$, and \n\\begin{equation}\\label{eigvaleq}\n  p_nQ_{n+1}(x)+r_nQ_n(x)+q_nQ_{n-1}(x)=xQ_n(x)\\qquad \\text{for $n\\geq 1$.}\n\\end{equation}\n\nKeisel and Wimp \\cite{HKJW1996} give expressions\nfor $p_n$,  $r_n$ and $q_n$ for $n\\geq 0$.  To find the expressions\ndirectly in this case one could use \\eqref{eigvaleq} to derive three\nlinearly independent equations, and solve for $p_n$,  $r_n$, and $q_n$.\n\nFor the case $n=0$ the equation immediately gives us that\n$p_0=\\frac{1}{N+1}$ and $r_0=\\frac{N}{N+1}$.  Evaluating at convenient\nchoices of $x$, such as $-1, 0, 1$, do not yield linearly\nindependent equations for all $n$.  One solution to this is to\nevaluate at $x=1,-1$ and differentiate \\eqref{eigvaleq} and then\nevaluate at $x=0$.  This gives three linearly independent equations\nand a direct calculation then shows that\n\\begin{equation}\n\\begin{gathered}\\label{eq:linearprobs}\np_n=\\frac{1}{2}\\cdot\\frac{1+(2n-1)N}{1+(2n+1)N}, \\quad\nq_n=\\frac{1}{2}\\cdot\\frac{1+(2n+1)N}{1+(2n-1)N}, \\quad \\text{and}\\\\\nr_n=\\frac{-2N^2}{(1+(2n-1)N)(1+(2n+1)N)}\n\\end{gathered}\n\\end{equation}\nAs $r_n\\leq 0$ the operator $H$ fails to correspond to a Markov chain.\nHowever this is the case we addressed at the end of Section~\\ref{sec:koorn} of the current paper.\nThus consider $P_\\lambda=\\frac{1}{1+\\lambda}(H+\\lambda I)$.  Now, since $|r_n|$ is\na decreasing sequence for $n\\geq 1$. So provided that $\\lambda\\geq |r_1|=\\frac{2N^2}{(1+N)(1+3N)}$, we then have \n$p_n^\\lambda, r_n^\\lambda, q_n^\\lambda\\geq 0$.  Thus we can consider these coefficients  $p_n^\\lambda$,  $r_n^\\lambda$, and $q_n^\\lambda$ as the transition probabilities in a nearest neighbor random walk.\n\nRecall that $\\pi_n^\\lambda=\\pi_n=\\frac{p_0\\cdots p_{n-1}}{q_1\\cdots\n  q_n}$.  Thus for $P_\\lambda$ we can directly calculate $\\pi_n$ from\n\\eqref{eq:linearprobs}.  We have that $p_0\\cdots\np_{n-1}=\\frac{1}{2^{n-1}}\\frac{N}{1+(2n-1)N}$ and  similarly $q_1\\cdots q_n=\\frac{1}{2^n}\\frac{1+(2n+1)N}{1+N}$.  Thus $\\pi_n=\\frac{2(1+N)N}{(1+(2n-1)N)(1+(2n+1)N)}$.\n\n\\begin{thm}\\label{main} Given $N>0$ and $\\lambda\\geq \\frac{2N^2}{(1+N)(1+3N)}$.\n  Consider the case of the Chebyshev-type random walks over $\\Omega=\\{0,1,2,\\dots \\}$ with probability operator\n  $$P_\\lambda=\\begin{pmatrix}r_0^\\lambda & p_0^\\lambda & 0 & 0 & 0 &\\cdots\\\\\nq_1^\\lambda & r_1^\\lambda & p_1^\\lambda & 0 & 0 & \\vdots\\\\\n0 & q_2^\\lambda & r_2^\\lambda & p_2^\\lambda & 0 & \\ddots\\\\\n0  & 0 & q_3^\\lambda & r_3^\\lambda & \\ddots & \\ddots\\\\\n\\vdots & \\cdots & \\ddots & \\ddots & \\ddots & \\dots\\\\\n\\end{pmatrix},$$\n  where $p_n^\\lambda=\\frac{1}{2(1+\\lambda)}\\cdot\\frac{1+(2n-1)N}{1+(2n+1)N}$, \\quad $q_n^\\lambda=\\frac{1}{2(1+\\lambda)}\\cdot\\frac{1+(2n+1)N}{1+(2n-1)N}$\n  and \\quad $r_n^\\lambda=1-p_n^\\lambda-q_n^\\lambda$ for $n \\geq 1$, with $p_0^\\lambda=\\frac{1}{(1+\\lambda)(N+1)}=1-r_0^\\lambda$.  \n  \n  Then for the random walk originating at some site $j \\in \\Omega$, there are positive constants $c$ and $C$ that depend on $j$, $N$ and $\\lambda$ such that \n  $$\\frac{c}{\\sqrt{t}}\\leq \\norm{\\nu-\\mu_t}_{TV}\\leq\n  C\\frac{\\log t}{\\sqrt{t}}$$\n  for $t$ sufficiently large.\n  \n\\end{thm}\n\\begin{proof}\n  For the upper bound we simply need to estimate the sums appearing in Lemma~\\ref{lem:bounds}.  Since $\\pi_n=O\\big(\\frac{1}{(n+1)^2}\\big)$, it is easy to see that the second sum $\\sum_{n=j+t+1}^{\\infty}\\pi_n$ is bounded by $C_N\/(t+j+1)$.  \n  The main term turns out to be the first sum.\n\n  In the case of the Chebyshev type polynomials we have the bound\n  \\mbox{$\\norm{Q_n}_\\infty\\leq 4Nn+1$}.  Thus the first sum in Lemma~\\ref{lem:bounds} is bounded by \n  $\\hat C_{\\alpha,\\beta,\\lambda,N}\\frac{j\\log(t+j+2)}{\\sqrt t}$ for an appropriate constant  $\\hat C_{\\alpha,\\beta,\\lambda,N}$.\n And so, for an appropriate $C$ and large $t$,\n  $$\\norm{\\nu-\\mu_n}_{TV}\\leq C\\frac{\\log t}{\\sqrt{t}}$$\n\n  On the other hand, recalling that $Q_0(x)=\\pi_0=1$, we have that:\n  $$\\norm{\\nu-\\mu_n}_{TV}\\geq \\abs{\\int_{(-1,1)}\\Big(\\frac{x+\\lambda}{1+\\lambda}\\Big)^tQ_j(x)(1+x^\\beta)(1-x)^\\alpha\\,dx}$$\n However we have already shown that for large enough $t$, the above right-hand side is asymptotic to\n  $\\frac{\\tilde C}{\\sqrt{1+t}}$.\n\\end{proof}\n\nWe finish with some concluding remarks.  At first the bound\n\\mbox{$\\norm{Q_n}_\\infty\\leq 4Nn+1$} may appear somewhat imprecise since near $x=1$, we have that $Q_n(1)=1$.  \nIt is tempting to suggest that the correct asymptotic for the total\nvariation norm is $C\/\\sqrt{t}$.  However on closer examination in the\nneighborhood of $x=1$, $Q_n'(x)\\approx n^3$.  This $n^3$ causes the\nerrors to be at least of the order of  the main term.  Overall\nit seems unlikely to the authors that $C\/\\sqrt{t}$ is the correct asymptotic for the Chebyshev-type polynomials. \n\n\n\n\n\\section{Comparison to other methods}\\label{comparison}\n\nAn ergodic Markov chain $P=\\Big(p(i,j)\\Big)_{i,j \\in \\Omega}$ with stationary distribution $\\nu$ is said to be {\\it geometrically ergodic} if and only if there exists $0<R<1$ and a function $M: \\Omega \\rightarrow \\mathbb{R}_+$ such that for each initial state $i \\in\\Omega$, the total variation distance \ndecreases exponentially as follows\n$$\\|p_t(i,\\cdot)-\\nu(\\cdot)\\|_{TV}={1 \\over 2} \\sum\\limits_{j \\in \\Omega} |p_t(i,j)-\\nu(j)| \\leq M(i) R^t$$\nIn other words, an ergodic Markov chain is geometric when the rate of convergence to stationary distribution is exponential. See \\cite{SMRT2009} and references therein.\n\nIf the state space $\\Omega$ is finite, $|\\Omega|=d <\\infty$, and Markov chain is irreducible and aperiodic, then $P$ will have eigenvalues that can be ordered as follows $$\\lambda_1=1 >|\\lambda_2| \\geq \\dots \\geq |\\lambda_d|$$\nIn which case, the Perron-Frobenious Theorem will imply geometric ergodicity with \n$$\\|p_t(i,\\cdot)-\\nu(\\cdot)\\|_{TV}=O(t^{m_2-1} |\\lambda_2|^t),$$\nwhere $m_2$ is the algebraic multiplicity of $\\lambda_2$. Here the existence of a positive {\\it spectral gap}, $1-|\\lambda_2|>0$, implies geometric ergodicity with the exponent $-\\log  |\\lambda_2| \\approx 1- |\\lambda_2|$ whenever the spectral gap is small enough.\n\nWhen dealing with Markov chains over general countably infinite state space $\\Omega$, the existence of a positive spectral gap of the operator $P$ is essentially equivalent  to the chain being geometrically ergodic. For instance, the orthogonal polynomial approach in \\cite{YK2010} resulted in establishing the geometric rate $R=\\max\\left\\{r+2\\sqrt{pq},~{q \\over q+r}\\right\\}$ for the Markov chain\n$$P=\\left(\\begin{array}{ccccc}0 & 1 & 0 & 0 & \\dots \\\\q & r & p & 0 & \\dots \\\\0 & q & r & p & \\ddots \\\\0 & 0 & q & r & \\ddots \\\\\\vdots & \\vdots & \\ddots & \\ddots & \\ddots\\end{array}\\right) \\qquad q>p,~~~r>0$$ \nover $\\Omega=\\mathbb{Z}_+$, together with establishing the value of the spectral gap, $1-r>0$.\n\nAs for the Markov chain $P_\\lambda$ considered in Theorem~\\ref{main} of this paper, its spectral measure ${1 \\over 1+\\lambda}d\\psi\\Big((1+\\lambda)x-\\lambda\\Big)$ over $\\Big({\\lambda-1 \\over \\lambda+1}, 1 \\Big]$ admits {\\it no} spectral gap between the point mass at $1$ and the rest of the spectrum implying sub-geometric ergodicity. The sub-exponential rate in total variation norm is then estimated to be of polynomial order between $\\frac{1}{\\sqrt{t}}$ and $\\frac{\\log t}{\\sqrt{t}}$.\n\nIn the field of probability and stochastic processes, there is a great\ninterest in finding methods for analyzing Markov chains over general\nstate space that have polynomial rates of convergence to stationary\ndistribution. In Menshikov and Popov \\cite{MMSP1995} a one dimensional\nversion of Lamperti's problem is considered. There, a class of ergodic\nMarkov chains on countably infinite state space with sub-exponential\nconvergence to the stationary probabilities is studied via\nprobabilistic techniques. One of their results relates to our main\nresult, Theorem~\\ref{main}.  Namely, Theorem 3.1 of \\cite{MMSP1995}\nwhen applied to our case, implies for any $\\varepsilon>0$ the\nexistence of positive real constants $C_1$ and $C_2$ such that\n$$C_1t^{-{1 \\over 2}-\\varepsilon} \\leq |\\nu(0)-\\mu_t(0)| \\leq C_2t^{-{1 \\over 2}+\\varepsilon}$$\nThus for the Markov chain considered in Theorem~\\ref{main}, the\northogonal polynomials approach provides a closed form expression for\nthe difference $\\nu-\\mu_t$, and a significantly sharper estimate on\nconvergence of $\\mu_t$ to the stationary distribution $\\nu$, for both\nthe single state distance $|\\nu(0)-\\mu_t(0)|$ and a much stronger\ntotal variation norm, $\\|\\nu-\\mu_t\\|_{TV}$.\n\n\n\n\\section{Acknowledgments}\nWe would like to thank Yuan Xu of University of Oregon for his helpful comments that\ninitiated this work. We would also like to thank Michael Anshelevich of Texas A \\& M for the feedback he provided during the conference on orthogonal polynomials in probability theory in July of 2010. We would like thank Andrew R. Wade of the University of Strathclyde for his helpful comments on the preprint of this paper. Finally, we would like to thank the anonymous  referee for the many helpful corrections and suggestions.\n\n\n\n\\begin{bibdiv}\n  \\begin{biblist}\n   \\bib{MAIS1972}{book}{\n        editor={M. Abramowitz},\n        editor={I. A. Stegun},\n        title={Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables},\n        edition={Ninth}, \n        publisher={Dover Publications},  \n        year={1972}\n      }  \n   \\bib{MI2005}{book}{\n        author={M. Ismail},\n        title={Classical and quantum orthogonal polynomials in one variable},\n        series={Encyclopedia of Mathematics and its Applications}, \n        number={98},\n        publisher={Cambridge University Press},  \n        year={2005}\n      } \n   \\bib{SKJM1959}{article}{\n      title={Random Walks},\n      author={S. Karlin},\n      author={J. L. McGregor},\n      journal={Illinois Journal of Math.},\n      volume={3},\n      date={1959},\n      number={1},\n      pages={417--431}\n    }\n    \\bib{HKJW1996}{article}{\n      title={A note on Koornwinder's polynomials with weight function $(1-x)^\\alpha(1+x)^\\beta+M\\delta(x+1)+N\\delta(x-1)$},\n      author={Kiesel, K.},\n      author={Wimp, J.},\n      journal={Numerical Algorithms},\n      volume={11},\n      date={1996},\n      pages={229--241}\n    }\n    \\bib{TK1984}{article}{\n      title={Orthogonal polynomials with weight function $(1-x)^{\\alpha }(1+x)^{\\beta }+M\\delta (x+1)+N\\delta (x-1)$},\n      author={Koornwinder, T.}\n      journal={Canad. Math. Bull.},\n      volume={27},\n      date={1984},\n      number={2},\n      pages={205--214}\n    }\n    \\bib{YK2009}{article}{\n      title={Orthogonality and probability: beyond nearest neighbor transitions},\n      author={Kovchegov, Y.},\n      journal={Electron. Commun. Probab.},\n      volume={14},\n      year={2009},\n      pages={90--103}\n    }    \n    \\bib{YK2010}{article}{\n      title={Orthogonality and probability: mixing times},\n      author={Kovchegov, Y.},\n      journal={Electron. Commun. Probab.},\n      volume={15},\n      year={2010},\n      pages={59--67}\n    }\n    \\bib{SMRT2009}{book}{\n        author={S. Meyn},\n        author={R. L. Tweedie},\n        title={Markov Chains and Stochastic Stability},\n        edition={Second}, \n        publisher={Cambridge University Press},  \n        year={2009}\n      }  \n    \\bib{MMSP1995}{article}{\n      title={Exact Power Estimates For Countable Markov Chains},\n      author={Menshikov, M.~V.},\n      author={Popov, S.~Yu.},\n      journal={Markov Processes Relat. Fields},\n      volume={1},\n      date={1995},\n      pages={57--78}\n    }\n      \\bib{GS1975}{book}{\n        author={G. Szeg\\\"{o}},\n        title={Orthogonal Polynomials},\n        edition={Fourth}, \n        publisher={AMS Colloquium Publications},  \n        volume={23},\n        year={1975}\n      }  \n    \\bib{VU1969}{article}{\n      title={Relation between systems of polynomials orthogonal with respect to various distribution functions},\n      author={Uvarov, V.~B.},\n      journal={Vychisl. Mat. i Mat. Fiz  (USSR)},\n      volume={9},\n      number={6},\n      year={1969},\n      pages={1253--1262}\n    }\n\\end{biblist}\n\\end{bibdiv}\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{{\\rm \\footnotesize INTRODUCTION}}\n``How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?'' Sherlock Holmes to Dr. Watson (Conan~Doyle 1890). \n\nIn the study of open clusters a similar totalogy could be paraphrased as ``when you have eliminated the effects of extinction and differential reddening for cluster stars, the resulting cluster color-magnitude diagram, however unusual, must represent an accurate picture of the temperatures and luminosities of cluster stars.'' That philosophy was demonstrated to be the case for relatively young (10$^7$--10$^8$ yr old) open clusters by Turner (1996), as well as for the extremely young ($3.5 \\times 10^6$ yr old) cluster IC 1590 (Guetter \\& Turner 1997). The last study demonstrated how closely pre-main-sequence members of IC 1590 matched model isochrone predictions by Palla \\& Stahler (1993) converted to the Johnson \\& Morgan (1953) {\\it UBV} system. \n\nThe situation for other young clusters is more complicated, primarily because most of the original {\\it UBV} studies of the brighter members of the class (NGC 2264, NGC 6530, IC 1546, and NGC 6611) were made by Merle Walker (Walker 1956, 1957, 1959, 1961, respectively) using a detector system that was not an ideal match to the Johnson system, although that was not realized at the time. For example, later studies of NGC 6611 (Hiltner \\& Morgan 1969) and NGC 2264 (e.g., Mendoza \\& G\\'{o}mez 1980) noted differences between Walker's photometry and observations tied more closely to the {\\it UBV} system. The origin of such differences can be explained by the work of Moffat \\& Vogt (1977) and Guti\\'{e}rrez-Moreno, Moreno \\& Cort\\'{e}z (1981), who noted that systematic errors, specifically in {\\it U--B} measures, can arise from the manner in which the Balmer discontinuity in the continua of early-type stars is sampled by non-standard telescope\/filter\/detector systems, as well as by the treatment of atmospheric extinction (see Cousins \\& Caldwell 2001). The differences in the case of NGC 6611 are fairly extreme, amounting to offsets of $0.10$ in {\\it U--B} and $0.03$ in {\\it B--V} (Hiltner \\& Morgan 1969).\n\nThe purpose of the present study is to redo Walker's original study of NGC 2264 (Walker 1956) in order to generate a new reddening-free and extinction-corrected color-magnitude diagram for the cluster. The cluster has been studied many times previously, but never for the purpose of improving upon Walker's results. Recent detections of non-radial pulsation via asteroseismology in many of the pre-main-sequence members of NGC 2264 (e.g., Zwintz 2008; Kallinger, Zwintz \\& Weiss 2008; Guenther et al. 2009) make it imperative to have a clear picture of the evolutionary status and exact H-R diagram location of cluster stars.\n\n\n\n\\section{{\\rm \\footnotesize ADJUSTING EXISTING {\\it UBV} OBSERVATIONS}}\nNGC 2264 has a rich history of observation, and has been studied fruitfully using many optical photometric systems. The emphasis here is on {\\it UBV} photometry, which has certain advantages over other photometric systems for analyzing interstellar reddening; in most optical photometric systems interstellar reddening corrections are made using mean reddening laws, occasionally established incorrectly, that may not describe the reddening applicable to the cluster under study (e.g., Turner 1989, 1994, 1996). Str\\\"{o}mgren system photometry is similar enough to {\\it UBV} photometry that it can be transformed accurately to the latter for most early-type stars (Turner 1990).\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{n2264f1.eps}\n\\end{center}\n\\caption{\\small{Differences, $\\Delta V$ (upper) and $\\Delta (B$--$V)$ (lower), between the derived mean {\\it UBV} data from the literature and the Walker (1956) values, as a function of Walker's tabulated {\\it B--V} color. The derived trends are indicated, and an open circle denotes the red star not used in the $\\Delta (B-V)$ comparison.}}\n\\label{fig1}\n\\end{figure}\n\nThe procedure adopted here was to compare existing {\\it UBV} observations, identified as being closely linked to the Johnson system and including transformations of published Str\\\"{o}mgren photometry for cluster stars, with the original measures of Walker (1956), and to adjust the latter for systematic effects prior to forming optimized values for individual stars. {\\it UBV} observations not clearly tied closely to the Johnson system were not used, our experience being that systematic effects in such data are often non-linear, making them very difficult to treat reliably.\n\nSources of {\\it UBV} photometry for NGC 2264 that contain data for sizable samples of stars are presented by Mendoza \\& G\\'{o}mez (1980), Per\\'{e}z, Th\\'{e} \\& Westerlund (1987), and Kwon \\& Lee (1983). Observations for more restricted samples can be found in papers by Johnson \\& Morgan (1955), Hiltner (1956), Karlsson (1966), Turner (1976b), Macmillan (1977), Echevarr\\'{i}a, Roth \\& Warman (1979), Clar\\'{i}a (1985), Oja (1991), and Zwintz (2008). Suitable Str\\\"{o}mgren system photometry that can be converted to equivalent {\\it UBV} data, at least for early-type stars, is presented by Crawford, Barnes \\& Golson (1971), Strom, Strom \\& Yost (1971), Morrison (1975), Gronbech \\& Olsen (1976), Perry \\& Johnston (1982), Lindroos (1983), Olsen (1983), Per\\'{e}z et al. (1988), Mendoza, Rolland \\& Rodriguez (1990), Knude (1992), Handler (1999), Pe\\~{n}a, Peniche \\& Cervantes (2000), and Kalinger et al. (2008). The relationships derived by Turner (1990) were used to convert the latter observations to the Johnson system.\n\nA comparison of the average {\\it BV} values from such an analysis with Walker's (1956) photometry is presented in Fig.~\\ref{fig1}. The differences in $\\Delta V$ and $\\Delta(B$--$V)$ as a function of {\\it B--V} color are generally small, the best-fitting results being: $\\Delta V=+0.037(\\pm0.003) - 0.014(\\pm0.006)$({\\it B--V}) and $\\Delta$({\\it B--V}) $=-0.013(\\pm0.002) - 0.002(\\pm0.005)$({\\it B--V}). There is a slight offset in color for Walker's observations (the reddest star being omitted for reasons of possible brightness variability) and the visual magnitudes display a reasonably distinct color dependence. The comparison of {\\it U--B} colors in Fig.~\\ref{fig2} is more interesting. There is increased scatter for stars with colors near {\\it U}--$B \\simeq 0.0$ that has the distinct signature of a Balmer discontinuity effect as described by Moffat \\& Vogt (1977). The data were therefore matched to linear or quadratic functions in separate segments, with the resulting zig-zag relationship plotted in the figure adopted for further analysis. The Moffat \\& Vogt (1977) results were used as a guide for interpreting the trends in the $\\Delta(U$--$B)$ data.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{n2264f2.eps}\n\\end{center}\n\\caption{\\small{Differences $\\Delta (U$--$B)$ between the derived mean {\\it UBV} data from the literature and Walker's (1956) values, as a function of Walker's tabulated {\\it U--B} color. The derived Z-shaped curve represents a least squares fit to three segments of the data.}}\n\\label{fig2}\n\\end{figure}\n\n\n\\setcounter{table}{0}\n\\begin{table*}\n\\caption[]{Optimized {\\it UBV} observations for NGC 2264 stars}\n\\label{tab1}\n\\begin{center}\n\\footnotesize\n\\begin{tabular*}{0.89\\textwidth}{@{\\extracolsep{-0mm}}ccccccccccccccccc}\n\\hline \\noalign{\\smallskip}\nStar &{\\it V} &{\\it B--V} &{\\it U--B} &n & &Star &{\\it V} &{\\it B--V} &{\\it U--B} &n & &Star &{\\it V} &{\\it B--V} &{\\it U--B} &n \\\\\n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n1 &14.08 &0.91 &0.33 &3 & &80 &15.27 &1.52 &0.69 &1 & &159 &10.99 &0.06 &0.00 &4 \\\\\n2 &9.72 &0.24 &0.18 &18 & &81 &16.29 &1.24 &0.04 &1 & &161 &14.99 &0.66 &--0.60 &1 \\\\\n3 &14.60 &1.43 &1.12 &1 & &83 &7.95 &--0.16 &--0.83 &21 & &164 &13.32 &0.83 &0.21 &1 \\\\\n4 &13.49 &0.42 &--0.03 &1 & &84 &11.99 &0.57 &0.07 &5 & &165 &10.99 &0.14 &0.11 &5 \\\\\n5 &14.14 &0.53 &0.05 &3 & &85 &15.00 &1.08 &0.76 &1 & &168 &15.15 &1.24 &1.02 &1 \\\\\n6 &8.21 &0.36 &--0.04 &6 & &87 &10.77 &0.21 &0.16 &3 & &169 &13.42 &0.77 &0.22 &1 \\\\\n7 &7.79 &--0.12 &--0.60 &6 & &88 &9.05 &--0.12 &--0.66 &13 & &172 &10.09 &--0.06 &--0.40 &2 \\\\\n8 &13.64 &1.25 &0.86 &4 & &89 &16.36 &1.04 &--0.57 &1 & &173 &16.43 &1.27 &0.75 &1 \\\\\n9 &11.99 &2.02 &2.18 &2 & &90 &12.67 &0.19 &--0.04 &16 & &175 &16.14 &1.39 &0.45 &1 \\\\\n10 &11.26 &0.56 &0.19 &4 & &91 &12.33 &0.64 &0.00 &3 & &176 &15.80 &1.57 &0.83 &1 \\\\\n11 &16.06 &0.86 &0.03 &1 & &92 &11.66 &0.85 &0.38 &6 & &177 &9.08:: &0.74:: &0.05 &4 \\\\\n12 &15.49 &1.28 &0.15 &1 & &93 &13.21 &0.87 &0.32 &2 & &177B &11.98 &0.57 &0.00 &1 \\\\\n13 &12.05 &1.66 &1.46 &2 & &94 &10.45 &0.41 &--0.04 &2 & &178 &7.18 &--0.20 &--0.98 &13 \\\\\n14 &14.85 &0.79 &0.17 &1 & &95 &16.07 &1.10 &--0.05 &1 & &178B &10.23 &0.16 &--0.12 &2 \\\\\n15 &14.66 &1.20 &0.65 &1 & &96 &14.10 &1.09 &1.22 &1 & &179 &9.96 &0.00 &--0.18 &5 \\\\\n16 &16.12 &0.95 &0.33 &3 & &98 &11.77 &0.57 &0.00 &2 & &180 &12.92 &0.51 &--0.08 &2 \\\\\n17 &12.89 &0.37 &0.23 &4 & &99 &10.86 &0.40 &--0.03 &5 & &181 &10.06 &--0.05 &--0.30 &3 \\\\\n18 &15.23 &0.84 &0.20 &2 & &100 &10.04 &0.12 &0.09 &10 & &182 &10.33 &0.06 &0.08 &4 \\\\\n19 &15.54 &0.82 &--0.06 &1 & &103 &10.08 &0.02 &--0.10 &2 & &183 &15.24 &0.96 &0.63 &1 \\\\\n20 &10.30 &0.42 &0.16 &26 & &104 &11.40 &0.22 &0.13 &7 & &184 &14.16 &0.99 &--0.05 &1 \\\\\n21 &14.09 &0.77 &0.22 &2 & &105 &15.13 &1.06 &--0.33 &1 & &186 &15.63 &1.54 &0.92 &1 \\\\\n22 &17.08 &1.07 &0.16 &3 & &106 &13.31 &0.71 &0.38 &1 & &187 &9.25 &--0.08 &--0.33 &5 \\\\\n23 &13.75 &0.82 &0.49 &2 & &107 &8.86 &--0.08 &--0.46 &15 & &189 &11.36 &0.54 &0.01 &4 \\\\\n24 &8.57 &--0.06 &--0.31 &5 & &108 &12.05 &0.58 &0.07 &6 & &190 &12.29 &0.67 &--0.01 &1 \\\\\n25 &7.83 &0.37 &0.00 &11 & &109 &9.12 &--0.11 &--0.53 &5 & &193 &9.81 &0.23 &0.12 &6 \\\\\n26 &11.78 &0.47 &0.01 &3 & &112 &10.81 &--0.01 &--0.11 &6 & &195 &12.66 &0.52 &--0.05 &4 \\\\\n27 &12.08 &0.52 &0.21 &3 & &114 &11.55 &0.52 &0.07 &4 & &196 &11.72 &0.52 &--0.03 &3 \\\\\n28 &12.34 &0.46 &--0.08 &4 & &115 &14.44 &1.00 &0.37 &1 & &197 &16.30 &0.92 &--0.08 &1 \\\\\n29 &10.16 &0.43 &0.00 &5 & &116 &11.66 &0.54 &0.09 &5 & &202 &9.02 &0.07 &--0.59 &10 \\\\\n30 &10.78 &0.03 &--0.02 &6 & &117 &13.55 &0.69 &0.25 &1 & &203 &12.93 &0.75 &0.14 &1 \\\\\n31 &10.56 &0.35 &0.03 &5 & &118 &11.80 &0.55 &--0.04 &2 & &205 &10.64 &0.33 &--0.02 &5 \\\\\n32 &13.02 &0.76 &0.05 &1 & &121 &12.14 &0.42 &--0.27 &1 & &206 &9.50 &0.09 &--0.34 &5 \\\\\n33 &11.68 &2.63 &2.49 &5 & &125 &12.29 &0.61 &0.02 &2 & &209 &11.37 &0.37 &0.02 &4 \\\\\n34 &10.94 &0.40 &0.05 &1 & &127 &15.77 &1.48 &0.85 &1 & &210 &13.30 &0.76 &0.09 &1 \\\\\n35 &10.34 &0.08 &0.06 &4 & &128 &11.03 &0.26 &0.10 &3 & &212 &7.51 &--0.16 &--0.75 &20 \\\\\n36 &11.01 &0.01 &--0.01 &5 & &131 &4.64 &--0.24 &--1.07 &63 & &215 &9.32 &0.07 &--0.14 &10 \\\\\n37 &8.08 &1.49 &1.66 &7 & &132 &10.22 &--0.04 &--0.29 &6 & &216 &11.72 &0.76 &0.14 &1 \\\\\n38 &10.98 &1.04 &0.64 &2 & &133 &13.85 &1.05 &0.61 &1 & &217 &13.60 &0.81 &--0.34 &2 \\\\\n39 &11.35 &0.11 &0.11 &2 & &134 &12.42 &0.82 &0.01 &3 & &220 &9.72 &0.46 &--0.06 &3 \\\\\n43 &10.57 &0.20 &0.16 &8 & &136 &15.17 &1.56 &1.05 &1 & &221 &12.15 &0.39 &--0.07 &4 \\\\\n46 &9.23 &0.21 &0.18 &14 & &137 &9.93 &--0.08 &--0.40 &5 & &222 &9.92 &0.14 &0.18 &6 \\\\\n50 &8.15 &--0.15 &--0.74 &19 & &138 &10.19 &0.06 &--0.01 &5 & &223 &10.93 &0.33 &0.03 &5 \\\\\n54 &14.27 &1.23 &1.01 &1 & &139 &13.33 &1.24 &0.81 &2 & &224 &11.50 &0.52 &0.15 &2 \\\\\n58 &15.52 &0.96 &0.60 &1 & &141 &14.72 &1.14 &0.85 &1 & &225 &13.22 &0.57 &--0.01 &2 \\\\\n60 &12.49 &1.01 &0.67 &2 & &142 &8.86 &--0.09 &--0.53 &10 & &226 &9.63 &0.14 &0.16 &7 \\\\\n62 &12.37 &0.50 &--0.18 &2 & &143 &10.63 &0.06 &--0.10 &1 & &227 &11.80 &0.52 &--0.06 &1 \\\\\n64 &15.36 &0.93 &0.39 &1 & &144 &13.85 &1.36 &0.30 &1 & &228 &11.13 &0.35 &0.01 &5 \\\\\n65 &11.76 &0.49 &--0.04 &2 & &145 &10.67 &0.04 &0.02 &6 & &229 &8.54 &1.23 &1.03 &9 \\\\\n66 &12.43 &0.71 &--0.14 &4 & &146 &14.48 &1.08 &0.75 &1 & &230 &12.36 &1.31 &0.52 &1 \\\\\n67 &10.88 &0.59 &--0.41 &5 & &147 &11.00 &0.80 &0.38 &2 & &231 &8.99 &--0.14 &--0.64 &7 \\\\\n68 &11.75 &0.68 &0.11 &7 & &148 &13.63 &0.68 &0.05 &1 & &232 &9.83 &0.00 &--0.02 &5 \\\\\n69 &8.28 &1.38 &1.54 &10 & &150 &14.13 &0.98 &1.00 &2 & &233 &9.57 &0.56 &0.05 &3 \\\\\n70 &11.17 &0.57 &0.06 &5 & &151 &12.56 &0.50 &--0.01 &2 & &234 &12.43 &0.46 &--0.05 &2 \\\\\n72 &12.36 &0.50 &--0.10 &1 & &152 &9.15 &--0.07 &--0.41 &14 & &235 &13.47 &0.51 &0.38 &3 \\\\\n73 &9.35 &0.84 &0.48 &8 & &153 &15.89 &1.22 &0.33 &1 & &236 &11.39 &0.63 &0.12 &5 \\\\\n74 &6.88 &--0.11 &--0.63 &16 & &154 &12.62 &0.79 &0.03 &2 & &237 &9.45 &1.43 &1.26 &5 \\\\\n76 &14.14 &0.75 &0.38 &1 & &155 &16.50 &1.41 &0.96 &1 & &238 &9.97 &1.25 &1.56 &3 \\\\\n77 &14.53 &1.17 &0.90 &1 & &156 &14.68 &1.23 &0.66 &1 & &239 &9.34 &1.15 &1.02 &4 \\\\\n78 &15.42 &1.21 &--0.39 &1 & &157 &10.04 &--0.03 &--0.31 &6 & & & & & & \\\\\n79 &15.90 &0.55 &--1.21 &1 & &158 &10.35 &0.34 &0.07 &6 & & & & & & \\\\\n\\noalign{\\smallskip} \\hline\n\\end{tabular*}\n\\end{center}\n$\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;$\\small{Star numbers from Walker (1956) and WEBDA, except as noted in text.}\n\\end{table*}\n\n\n\nThe power of the present approach to homogenizing existing {\\it UBV} photometry for NGC 2264 can be seen in the results. Walker's (1956) {\\it UBV} observations were adjusted with the relations plotted in Figs.~\\ref{fig1}~and~\\ref{fig2} and averaged with the other data, weighted by the number of observations for each observer, to form optimized values, summarized in Table~\\ref{tab1}. The resulting data are plotted in color-color and color-magnitude diagrams in Fig.~\\ref{fig3}, and display a variety of traits that differ in subtle ways from the original versions published by Walker (1956). As in Walker's study, stars exhibiting $H\\alpha$ emission (Herbig 1954) are identified.\n\nFirst, the sequence of reddened B-type stars closely fits the intrinsic relation for dwarfs with $E_{B-V}=0.075$, similar to what is seen in Stock 16 (Turner 1985, 1996) where there is no evidence for differential reddening. There is sufficient scatter, however, to suggest a better match to NGC 2422 (Turner 1996), which displays a color excess spread of $\\Delta E_{B-V}= 0.05$. Second, the colors of stars near the A0-star kink fit the reddened relation closely, much better than for Walker's (1956) data. Apparently the correction for the Balmer discontinuity effect adopted in Fig.~\\ref{fig2} offers a satisfactory solution to a serious systematic error in the original Walker  photometry. Third, there are a number of FGK stars in the field with the colors of unreddened field stars, suggesting a possible source of contamination for the cluster color-magnitude diagram. There is also little evidence to indicate that heavily reddened stars are cluster members, given that the data do not display a continuous run of reddening, as in NGC 1647 (Turner 1992, 1996). It can be further noted that there are a number of objects with colors that are inconsistent with those of stars of normal reddening. Such an effect may be the result of extreme youth or contamination of the photometry by bright surrounding nebulosity. Many display $H\\alpha$ emission.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.40\\textwidth]{n2264f3.eps}\n\\end{center}\n\\caption{\\small{{\\it UBV} color-color (upper) and color-magnitude (lower) diagrams for NGC 2264 stars with photoelectric observations. The intrinsic color-color relation is the black curve, while gray curves denote that relation and the zero-age main sequence (ZAMS) reddened by $E_{B-V} = 0.075$ and {\\it V--M}$_V = 9.68$. Stars exhibiting emission at $H\\alpha$ are denoted by vertical lines.}}\n\\label{fig3}\n\\end{figure}\n\nThe nature of the interstellar reddening in the direction of NGC 2264 was studied previously by Turner (1976b), yielding a {\\it UBV} reddening slope of $E_{U-B}\/E_{B-V} = 0.77$ and a ratio of total-to-selective extinction for early-type stars in the region of $R = A_V\/E_{B-V} = 3.2 \\pm 0.2$. Those values were adopted here in an analysis of the {\\it UBV} data of Table~\\ref{tab1}, and the resulting color excesses $E_{B-V}$ for individual stars were corrected for the color dependence caused by bandwidth effects in the {\\it UBV} system (Schmidt-Kaler 1961; Fernie 1963; Buser 1978).\n\nThe field of NGC 2264 contains just two groups of OB stars. One displays only a small amount of interstellar reddening and consists of massive members of the cluster. The other consists of more heavily reddened ZAMS members of Mon OB2 lying $\\sim 800$ pc beyond NGC 2264. The reddening separation of AFGK-type stars in terms of distance is less clear cut, as discussed in \\S4.\n\nThe numbering of stars in Table~\\ref{tab1} is generally that of Walker (1956), although with two additions and a few corrections. In the study by Pe\\~{n}a et al. (2000), for example, some stars are misidentified. Their observation for star 14 is a second measurement for star 24, that for star 31 is for star 26, the two stars identified as 77A and 77B are measures for star 177 of Walker (1956) with the A and B notation reversed, the data for star 106 are for star 112, and the observation for star 169 is inconsistent with its measured brightness by Walker (1956) or with those of any bright stars in the cluster. It has therefore been omitted from the analysis. The observations by Echevarr\\'{i}a et al. (1979) for the multiple system surrounding S Mon (ADS 5322, star 131) include measures for a star, ADS 5322Ad, that is star 121 of Walker (1956), but with a much more reasonable brightness than that obtained by Johnson for Walker's study. It has been included in Table~\\ref{tab1} with Johnson's measures omitted. Likewise, the data from Lindroos (1983) include separate measures for a bright companion of Walker star 178, HD 47887B, so it has also been included.\n\n\\section{{\\rm \\footnotesize PARAMETERS FOR NGC 2264}}\nSince the scatter of reddened B-type stars in NGC 2264 seen in Fig.~\\ref{fig3} is larger than the uncertainties in the observed colors of at most $\\pm 0.01$, it is appropriate to analyze the data of Table~\\ref{tab1} using the variable-extinction method (see Turner 1976a). A reddening line of slope $E_{U-B}\/E_{B-V}=0.77$ (Turner 1976b) was therefore applied to the observations, with larger slopes being used for G, K, and M-type stars, and the data were dereddened to the intrinsic relation for dwarfs (see Turner 1996). Absolute magnitudes appropriate for a zero-age main-sequence (ZAMS) star were assigned to each analyzed star on the basis of its derived intrinsic color, where the ZAMS values are from Turner (1976a, 1979). The results are displayed in Fig.~\\ref{fig4}, which reveals much about the extinction in the direction of NGC 2264.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{n2264f4.eps}\n\\end{center}\n\\caption{\\small{Variable extinction diagram for early-type stars in NGC 2264, for ZAMS values of $M_V$. The sloped lines represent $R=3.2$ for a ZAMS fit to NGC 2264 (gray line) and for Mon OB2 (black line), the last from Turner (1976b).}}\n\\label{fig4}\n\\end{figure}\n\nAn initial inference from Fig.~\\ref{fig4} is that there are unreddened stars towards NGC 2264 with intrinsic distance moduli $V_0$--$M_V$ ranging between 4 and 8.34, which indicates that the dust clouds responsible for the reddening of most stars along the line of sight must lie at distances of $\\sim 465$ pc, a result consistent with the findings of Neckel \\& Klare (1980) for surrounding fields. The fact that most cluster stars lie in regions of bright nebulosity, both emission and reflection, and yet have only small reddenings, indicates that the cluster must lie mostly in front of the gas and dust clouds that dominate the region. A few cluster stars have slightly larger reddenings that suggest they lie on the far side of the cluster and are imbedded in the dust clouds.\n\nMore heavily reddened cluster stars, on the other hand, display a remarkably close coincidence ($d = 1.6$ kpc, $R = 3.2$) with the spectroscopically-derived variable-extinction results of Turner (1976b) for OB stars in Mon OB2, implying that one can readily detect background objects lying on the far side of the cluster and beyond the associated gas and dust clouds. A detailed analysis of the full data set (\\S4) also indicates that there are many other non-members of the cluster lying in the field, a few at about the same distance as the main group of cluster stars and many more foreground to the cluster but beyond the dust clouds producing most of the foreground reddening. A similar result can be inferred from the proper motion analysis of Vasilevskis, Sanders \\& Balz (1965) for bright stars in the cluster field.\n\nThe distance to NGC 2264 can be obtained from the 13 B-type ZAMS stars in Fig.~\\ref{fig4} that form a lower envelope to the main body of data for cluster stars. The resulting intrinsic distance modulus is $V_0-M_V=9.45 \\pm 0.03$ s.e., corresponding to a distance of $777 \\pm12$ pc. An almost identical distance was obtained by Turner (1976b) from an independent analysis of Walker's original observations in conjunction with spectroscopic data for members of Mon OB1 lying near the cluster. The average reddening for stars belonging to the main body of the cluster is $E_{B-V}=0.075 \\pm0.003$ s.e., as noted earlier, with an observed dispersion from the standard deviation ($\\sigma$) of $\\Delta E_{B-V} = 0.06$, slightly larger than the reddening dispersion observed in NGC 2422, discussed above.  \n\nThe identification of likely cluster members is more complicated than the case for more distant groups, primarily because the cluster, because of its relative closeness, is spread out on the sky. The general appearance of NGC 2264 is also that of a double cluster, with no guarantee that the northern and southern components are of the same age and distance (although it does seem likely). For bright stars there are radial velocity measures and proper motion membership probabilities (Vasilevskis et al. 1965) for consideration, although neither is fully reliable, particularly in a direction roughly towards the Galactic anticenter, as is the case for NGC 2264 ($l = 203^{\\circ}$). The radial velocities for individual stars summarized in WEBDA display large scatter for several stars, the signature of spectroscopic binaries, are biased towards bright cluster stars, and have similar values both for {\\it bona fide} cluster members and nearby field stars, while the proper motions of stars across the line of sight at distances of 500--800 pc may be fairly similar. Compounding the situation is the apparent presence of a sizable population of AF-type stars in the immediate foreground to the cluster.\n\nLikely cluster members were therefore identified in the present study by considering all available data for each star, in conjunction with the results for other cluster stars analyzed. The dereddening procedure from Fig.~\\ref{fig3} is clearly not straightforward, given the potential non-physical solutions that arise for many of the faint red stars, many of which are variable or display $H\\alpha$ emission. Initially it was believed that many of the latter objects might be cluster members, and dereddening solutions were adopted for them using the mean cluster reddening. However, the resulting distribution of pre-main-sequence stars in the cluster color-magnitude diagram differed from that for non-variable pre-main-sequence stars without $H\\alpha$ emission, suggesting that they are either field stars or embedded objects that must be treated separately.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{n2264f5.eps}\n\\end{center}\n\\caption{\\small{Reddening and extinction corrected color-magnitude diagram for likely members of NGC 2264 from the present study. The black curve represents the ZAMS for the derived cluster distance, and gray curves at the bottom are pre-main-sequence isochrones for $\\log t = 6.5$ (upper) and 6.75 (lower). The gray curve at top is a post-main-sequence isochrone for $\\log t = 6.7$. Known and suspected spectroscopic binaries are indicated by horizontal lines, rapidly-rotating stars are circled, and stars exhibiting emission at $H\\alpha$ are denoted by vertical lines.}}\n\\label{fig5}\n\\end{figure}\n\n\nA compilation of likely members of NGC 2264 from the present study is given in Table~\\ref{tab2}, and the resulting reddening-free and extinction-free color-magnitude diagram for them is shown in Fig.~\\ref{fig5}. The close fit of the 13 stars discussed above to the ZAMS is evident here, as are a few other noteworthy features. Known or suspected spectroscopic binaries are identified, and the bias towards bright cluster stars is evident. Stars identified spectroscopically as rapid rotators, from ``n'' or ``nn'' classifications or large $V \\sin i$ values (Vogel \\& Kuhi 1981), are also noted, as are the two stars displaying emission at $H\\alpha$ (Walker 96, 133).\n\nThe identification of rapid rotators relates to the cluster main sequence, which appears to contain gaps, a feature seen in several other clusters (Mermilliod 1982) and attributed by Turner (1996) to close binary mergers and the resulting creation of rapidly-rotating merger products, which results in a color spread for cluster stars more luminous than each gap. That characteristic is evident for NGC 2264, as it is for the clusters discussed by Turner (1996). Most notable in the case of NGC 2264 is the B3 Vn star 74, the second most luminous likely cluster member, which lies above a main-sequence gap near $V_0\\simeq7.2$, and the group of rapidly-rotating stars lying above the main-sequence gap near $V_0\\simeq9.5$.\n\n\n\\setcounter{table}{1}\n\\begin{table}[!t]\n\\caption[]{Likely members of NGC 2264 from this study}\n\\label{tab2}\n\\begin{center}\n\\footnotesize\n\\begin{tabular*}{0.45\\textwidth}{@{\\extracolsep{-2.5mm}}ccccccccc}\n\\hline \\noalign{\\smallskip}\nStar &$(B-V)_0$ &$E_{B-V}$ &$V_0$ & &Star &$(B-V)_0$ &$E_{B-V}$ &$V_0$ \\\\\n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n7 &--0.19 &0.07 &7.57 & &128 &+0.22 &0.04 &10.89 \\\\\n15 &+1.13 &0.08 &14.39 & &131 &--0.31 &0.07 &4.42 \\\\\n24 &--0.10 &0.04 &8.46 & &132 &--0.10 &0.06 &10.02 \\\\\n26 &+0.43 &0.04 &11.66 & &133 &+0.80 &0.27 &13.00 \\\\\n30 &--0.04 &0.07 &10.57 & &137 &--0.12 &0.04 &9.81 \\\\\n34 &+0.33 &0.07 &10.72 & &138 &--0.03 &0.09 &9.89 \\\\\n35 &+0.00 &0.08 &10.08 & &142 &--0.17 &0.08 &8.61 \\\\\n36 &--0.02 &0.03 &10.90 & &143 &--0.06 &0.12 &10.24 \\\\\n39 &+0.03 &0.09 &11.08 & &145 &--0.01 &0.05 &10.49 \\\\\n43 &+0.12 &0.08 &10.31 & &152 &--0.13 &0.06 &8.95 \\\\\n50 &--0.22 &0.07 &7.91 & &154 &+0.72 &0.08 &12.37 \\\\\n54 &+0.99 &0.27 &13.40 & &157 &--0.11 &0.08 &9.78 \\\\\n60 &+0.89 &0.13 &12.09 & &159 &--0.03 &0.09 &10.69 \\\\\n65 &+0.43 &0.06 &11.57 & &165 &+0.01 &0.13 &10.57 \\\\\n68 &+0.53 &0.16 &11.23 & &172 &--0.13 &0.07 &9.86 \\\\\n74 &--0.20 &0.09 &6.59 & &177B &+0.53 &0.04 &11.85 \\\\\n83 &--0.25 &0.09 &7.67 & &178 &--0.29 &0.09 &6.89 \\\\\n84 &+0.45 &0.13 &11.58 & &178B &--0.09 &0.26 &9.41 \\\\\n87 &+0.13 &0.08 &10.51 & &179 &--0.08 &0.09 &9.69 \\\\\n88 &--0.20 &0.08 &8.79 & &181 &--0.10 &0.06 &9.88 \\\\\n91 &+0.52 &0.13 &11.91 & &182 &+0.04 &0.02 &10.26 \\\\\n96 &+1.02 &0.07 &13.87 & &187 &--0.10 &0.03 &9.16 \\\\\n98 &+0.51 &0.06 &11.58 & &189 &+0.48 &0.07 &11.14 \\\\\n100 &+0.00 &0.13 &9.64 & &190 &+0.60 &0.07 &12.06 \\\\\n103 &--0.05 &0.07 &9.86 & &202 &--0.23 &0.31 &8.04 \\\\\n104 &+0.17 &0.06 &11.22 & &206 &--0.15 &0.24 &8.74 \\\\\n107 &--0.15 &0.07 &8.63 & &209 &+0.35 &0.02 &11.31 \\\\\n108 &+0.46 &0.13 &11.63 & &212 &--0.22 &0.06 &7.32 \\\\\n109 &--0.16 &0.05 &8.95 & &215 &--0.08 &0.15 &8.83 \\\\\n112 &--0.05 &0.04 &10.68 & &224 &+0.35 &0.18 &10.92 \\\\\n114 &+0.41 &0.12 &11.16 & &227 &+0.45 &0.07 &11.58 \\\\\n116 &+0.42 &0.13 &11.26 & &228 &+0.35 &0.00 &11.12 \\\\\n118 &+0.49 &0.07 &11.59 & &231 &--0.19 &0.06 &8.81 \\\\\n125 &+0.55 &0.06 &12.09 & &232 &--0.02 &0.02 &9.77 \\\\\n\\noalign{\\smallskip} \\hline\n\\end{tabular*}\n\\end{center}\n\\end{table}\n\nIsochrones are also plotted in Fig.~\\ref{fig5} for $\\log t = 6.5$ and 6.75, the latter of which provides a good fit to the lower envelope of pre-main-sequence stars. An age of $\\log t \\simeq 6.7$ is also inferred for the most luminous cluster star, Walker 131 = S Mon = HD 47839 [(O7 V((f)), Walborn 1972], since it appears to have evolved away from the ZAMS. As in Guetter \\& Turner (1997), the pre-main-sequence isochrones are adapted from the results of Palla \\& Stahler (1994), while the post-main-sequence isochrone in Fig.~\\ref{fig5} is from Meynet, Mermilliod \\& Maeder (1993). Likewise, much like the case for IC 1590 (Guetter \\& Turner 1997), it is not clear that the dispersion in luminosities for pre-main-sequence stars represents a spread in formation times. Binarity is also a possibility, particularly given the similarity in implied ages between lower envelope pre-main-sequence stars and S Mon. The implied age for members of NGC 2264 is therefore $\\sim 5.5 \\times 10^6$ yr.\n\n\\section{{\\rm \\footnotesize FIELD STARS IN NGC 2264}}\nMost previous studies of NGC 2264 have assumed that the AFGK-type stars lying above the main sequence in Walker's (1956) original cluster color-magnitude diagram are pre-main-sequence objects, with variability, $H\\alpha$ emission, and large membership probabilities (Vasilevskis et al. 1965) for the same objects supplying confirmatory evidence. Yet when most such stars are assumed to be cluster members with reddenings similar to the cluster average, the resulting sequence of assumed pre-main-sequence objects runs parallel to the ZAMS, which is inconsistent with model isochrones (e.g., Palla \\& Stahler 1993). That can be seen in Fig.~\\ref{fig6}, which presents color-color and color-magnitude diagrams for stars from Table~\\ref{tab1} that were rejected from cluster membership in Table~\\ref{tab2}. Note that most of the stars displaying $H\\alpha$ emission (Herbig 1954) fall in the plots of Fig.~\\ref{fig6}.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{n2264f6.eps}\n\\end{center}\n\\caption{\\small{{\\it UBV} color-color (upper) and color-magnitude (lower) diagrams for NGC 2264 stars with photoelectric observations. The intrinsic color-color and zero-age main sequence (ZAMS) relations are the black curves, the latter for $V_0$--$M_V = 7.55$, while gray curves denote the color-color relation and the ZAMS reddened by $E_{B-V} = 0.074$ and {\\it V--M}$_V = 9.0$. Stars exhibiting emission at $H\\alpha$ are denoted by vertical lines.}}\n\\label{fig6}\n\\end{figure}\n\nA large number of the stars fall along an unreddened ZAMS for $V_0-M_V=7.55$ ($d=324$ pc), which suggests the presence of an intermediate-age population in this direction lying $\\sim 150$ pc foreground to the dust clouds responsible for the initial interstellar extinction in NGC 2264. A smaller selection of stars fit the ZAMS reddened by $E_{B-V}=0.075$ for {\\it V--M}$_V = 9.0$, corresponding to a distance of $570$ pc, $\\sim 100$ pc beyond the foreground dust complexes but well foreground to NGC 2264. There also appear to be a few GK-type stars in Fig.~\\ref{fig6} with distance moduli similar to that of NGC 2264. They probably lie at distances comparable to that of the cluster or slightly foreground to it, similar to the case for some of the stars in Fig.~\\ref{fig4}. In this alternate interpretation of the observations, the region of NGC 2264 contains a sizable sample of stars belonging to the general field, the exceptions being the slightly reddened OB stars and many of the AF-type stars. $H\\alpha$ emission does not appear to correlate with any particular status for individual stars.\n\nThe above discussion is relevant for recent observations of small-amplitude variable stars in NGC 2264 studied by means of asteroseimology. In the study made by Zwintz et al. (2009), for example, their star V1 seems likely to be a foreground star, stars V3 and V4 are possible pre-main-sequence objects, and star V2 is Walker 39, a newly-arrived ZAMS member of NGC 2264. Zwintz et al. (2009) refer to it as a pre-main-sequence $\\delta$ Scuti variable; for that to be the case the star must be just on the verge of initiating core hydrogen burning. Such distinctions are important for establishing the pulsational characteristics of other targets for asteroseismology chosen according to their true location in the cluster color-magnitude diagram.\n\n\n\\section{{\\rm \\footnotesize DISCUSSION}}\nOne advantage of removing the effects of interstellar reddening and extinction from cluster color-magnitude diagrams is that the resulting distribution of data points must be closely linked to the true variations in effective temperature and luminosity for cluster members. In the case of NGC 2264, the color-magnitude diagram presented here in Fig.~\\ref{fig5} provides a number of important insights into the evolutionary status of cluster stars. The main sequence gaps, for example, imply an advanced dynamical state, provided they arise from close binary mergers, as argued by Turner (1996). And the close coincidence in the cluster age inferred from both pre-main-sequence stars and the slightly-evolved S Mon imply that the formation of cluster stars was not spread out greatly in time, the creation of cluster stars probably consuming no more than a few hundred thousand years.\n\n\n\n\\subsection*{{\\rm \\scriptsize ACKNOWLEDGEMENTS}}\n\\scriptsize{The present study has used the open cluster data compilation maintained by WEBDA, operated at the Institute for Astronomy of the University of Vienna.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThree-dimensional (3D) object detection enables a machine to sense its surrounding environment by detecting the location and category of objects around it. Therefore, 3D object detection plays a crucial role in systems that interact with the real world, such as autonomous vehicles and robots. The goal of 3D object detection is to generate 3D Bounding Boxes (BBoxes) parameterized by size, location, and orientation to locate the detected objects. \nMost existing methods rely heavily on LiDAR \\cite{qi2017pointnet++, shi2019pv, shi2020point, shi2020points, shi2019pointrcnn}, because LiDAR can generate point cloud data with high-precision depth information, which enhances the accuracy of 3D object detection. However, the high cost and short service life make it difficult for LiDAR to be widely used in practice. \nAlthough binocular camera-based methods \\cite{li2019stereo, qin2019triangulation, konigshof2019realtime, chen2020dsgn, chen20173d} achieve good detection results, this is still not a cheap option, and there are often difficulties in calibrating binocular cameras. \nIn contrast, the monocular camera is cost-effective, very easy to assemble, and can provide a wealth of visual information for 3D object detection. Monocular 3D object detection has vast potential for applications, such as self-driving vehicles and delivery robots.\n\nMonocular 3D object detection is an extremely challenging task without the depth provided during the imaging process. To address this, researchers have made various attempts on the depth estimation from monocular images. For instance, \\cite{chabot2017deep, barabanau2019monocular} utilize CAD models to assist in estimating the depth of the vehicle. Similarly, a pre-trained depth estimation model is adopted to estimate the depth information of the scene in \\cite{vianney2019refinedmpl, bao2019monofenet, wang2019pseudo}. However, such methods directly or indirectly used 3D depth ground-truth data in monocular 3D object detection. Meanwhile, the methods \\cite{brazil2019m3d, chen2020monopair} without depth estimation can also achieve high accuracy in the 3D object detection task. In this paper, we propose a 3D object detector for monocular images that achieves state-of-the-art performance on KITTI benchmark \\cite{Geiger2012CVPR}.\n\nHumans can perceive how close the objects in a monocular image are from the camera. Why is that? When the human brain interprets the depth of an object, it compares the object with all other objects and the surrounding environment to obtain the difference in visual effect caused by the relative position relationship. For objects of the same size, the bigger, the closer from a fixed perspective. Inspired by this, we propose a novel Asymmetric Non-local Attention Block (ANAB) to compute the response at a position as a weighted sum of the features at all positions. Inspired by \\cite{chen20182, zhu2019asymmetric}, we use both the local features in multiple scales and the features that can represent the global information to learn the depth-wise features. The multi-scale features can reduce computational costs. The attentive maps in multiple scales shows an explicit correlation between the sampling spatial resolution and the depth of the objects. \n\nIn one-stage monocular 3D object detection methods, 2D and 3D BBoxes are detected simultaneously. However, for anchor-based methods, there exists feature mismatching in the prediction of 2D and 3D BBoxes. This occurs for two reasons: (1) the receptive field of the feature does not match the shape of the anchor in terms of aspect ratio and size; (2) the center of the anchor, generally considered as the center of the receptive field for the feature map, does not overlap with the center of the object. The misalignment affects the performance of 3D object detection. Thus, we propose a two-step feature alignment method, aiming at aligning the features in 2D and 3D BBox regression. In the first step, we obtain the target region according to the classification confidence scores for the pre-defined anchors. This allows the receptive field of the feature map to focus on the pre-defined anchor regions with high confidence scores. In the second step, we use the prediction results of the 2D\/3D center to compute the feature offset that can mitigate the gap between the predictions and its corresponding feature map.\n\nWe summarize our contributions as follows:\n\\begin{itemize}\n\\setlength\\itemsep{0.1mm}\n \\item We propose a simple but very efficient monocular 3D single-stage object detection (M3DSSD) method. The M3DSSD achieves significantly better performance than the monocular 3D object detection methods on the KITTI dataset for car, pedestrian, and cyclist object class using one single model, in both 3D object detection and bird's eye view tasks.\n \\item We propose a novel asymmetric non-local attention block with multi-scale sampling for the depth-wise feature extraction, thereby improving the accuracy of the object depth estimation.  \n \\item We propose a two-step feature alignment module to overcome the mismatching in the size of the receptive field and the size of the anchor, and the misalignment in the object center and the anchor center. \n\\end{itemize}\n\n\\section{Related Work}\n\\label{gen_inst}\nIn order to estimate depth information in monocular images, researchers have proposed many different approaches. For instance, \\cite{xu2018pointfusion, chen2017multi, liang2019multi} utilize point cloud data to obtain accurate 3D spatial information. Pointfusion \\cite{xu2018pointfusion} uses two networks to process images and raw point cloud data respectively, and then fuses them at the feature level. MV3D \\cite{chen2017multi} encodes the sparse point cloud with a multi-view representation and performs region-based feature fusion. Liang et al. \\cite{liang2019multi} exploit the point-wise feature fusion mechanism between the feature maps of LiDAR and images. LiDAR point cloud and image fusion methods have achieved promising performance. However, LiDAR cannot be widely used in practice at present due to its expensive price.\n\nCAD models of vehicles are also used in monocular 3D object detection. Barabanau et al. \\cite{barabanau2019monocular} detects 3D objects via geometric reasoning on key points. Specifically, the dimensions, rotation, and key points of a car are predicted by a convolutional neural network. Then, according to the key points' coordinates on the image plane and the corresponding 3D coordinates on the CAD model, simple geometric reasoning is performed to obtain the depth and 3D locations of the car. Deep MANTA \\cite{chabot2017deep} predicts the similarity between a vehicle and a predefined 3D template, as well as the coordinates and visibility of key points, using a convolutional neural network. Finally, given the 2D coordinates of an object's key points and the corresponding 3D coordinates on the 3D template, the vehicle's location and rotation can be solved by a standard 2D\/3D matching \\cite{lepetit2009epnp}. However, it is difficult to collect CAD models in all kinds of vehicles.\n\nMonocular depth estimation networks are adopted in \\cite{vianney2019refinedmpl, ding2019learning, ma2019accurate, bao2019monofenet, xu2018multi, cai2020monocular} to estimate depth or disparity maps. Most of the methods transform the estimated depth map into a point cloud representation and then utilize the approaches based on LiDAR to regress the 3D BBoxes. The performance of these methods relies heavily on the accuracy of the depth map. D4LCN \\cite{ding2019learning} proposed a new type of convolution, termed depth-guided convolution, in which the weights and receptive fields of convolution can be automatically learned from the estimated depth.\nThe projection of the predicted 3D BBox should be consistent with the predicted 2D BBox. This is utilized to build geometric constraints in \\cite{naiden2019shift, gahlert2018mb, mousavian20173d} to determine the depth. \nThanks to the promising performance of convolutional neural networks in 2D object detection, more and more approaches \\cite{brazil2019m3d, li2019gs3d, roddick2018orthographic, qin2019triangulation, qin2019monogrnet, liu2020smoke, chen2020monopair, jorgensen2019monocular} have been proposed to directly predict 3D BBoxes using well-designed convolutional neural network for monocular 3D object detection. GS3D \\cite{li2019gs3d} proposed a two-stage 3D object detection framework, in which the surface feature extraction is utilized to eliminate the problem of representation ambiguity brought by using a 2D bounding box. M3D-RPN \\cite{brazil2019m3d} proposed an anchor-based single-stage 3D object detector that generates both 2D and 3D BBoxes simultaneously. M3D-RPN achieves good performance, but it does not solve the problem of feature misalignment.\n\n\n\\begin{figure*}[t!]\n  \\centering\n  \\includegraphics[width=1\\textwidth]{overview_1.pdf}\n  \\caption{ The architecture of M3DSSD. (a) The backbone of the framework, which is modified from DLA-102 \\cite{yu2018deep}. (b) The two-step feature alignment, classification head, 2D\/3D center regression heads, and ANAB especially designed for predicting the depth $z_{3d}$. (c) Other regression heads.\n} \n  \\label{figure1}\n  \\vspace{-4mm}\n\\end{figure*}  \n\n\n\\section{Method}\n\\label{headings}\nIn this section, we describe the proposed M3DSSD, which consists of four main components: the backbone, the feature alignment, the asymmetric non-local attention block, and the 2D-3D prediction heads, as shown in Fig.~\\ref{figure1}. The details of each component are described below.\n\n\\subsection{Backbone}\nFollowing \\cite{yu2018deep}, we adopt the Deep Layer Aggregation network DLA-102 as the backbone. To adaptively change the receptive field and enhance the feature learning \\cite{zhou2019objects, liu2020smoke}, all the convolution in hierarchical aggregation connections are replaced with Deformable Convolution (DCN) \\cite{zhu2019deformable}. The down-sampling ratio is set to 8, and the size of the output feature map is $256 \\times H\/ 8\\times W\/8$, where $H$ and $W$ are the height and width of the input image. \n\n\n\\subsection{Feature Alignment}\nAnchor-based methods often suffer from feature mismatching. On one hand, this occurs if the receptive field of the feature does not match the shape of the anchor in terms of aspect ratio and size. On the other hand, the center of the anchor, generally considered as the center of the receptive field of the feature, might not overlap with the center of the object. The proposed feature alignment consists of shape alignment and center alignment: (1) shape alignment aims at forcing the receptive field of the feature map to focus on the anchor with the highest classification confidence score; (2) center alignment is performed to reduce the gap between the feature on the center of the object and the feature that represents the center of the anchor. Different from previous feature alignment methods \\cite{chen20182, wang2019region} that are applied to one-stage object detection via a two-shot regression, the proposed feature alignment can be applied in one shot, which is more efficient and self-adaptive.\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[b]{\\columnwidth}\n    \\label{fig.shape1}\n    \\includegraphics[width=\\textwidth]{shape_align.pdf}\n  \\end{subfigure}\n  \\hfill\n  \\begin{subfigure}{0.73\\columnwidth}\n    \\label{fig.shape2}\n    \\includegraphics[width=\\textwidth]{shape_align.png}\n  \\end{subfigure}\n\n  \\caption{\\small The architecture of shape alignment and the outcome of shape alignment on objects. The yellow squares indicate the sampling location of the AlignConv, and the anchors are in red.}\n  \\vspace{-4mm}\n  \\label{fig.shape_align}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[b]{\\columnwidth}\n    \\label{fig.center1}\n    \\includegraphics[width=\\textwidth]{center_align.pdf}\n  \\end{subfigure}\n  \\hfill\n  \\begin{subfigure}{0.73\\columnwidth}\n    \\label{fig.center2}\n    \\includegraphics[width=\\textwidth]{center_align.png}\n  \\end{subfigure}\n  \\caption{\\small The architectures of center alignment and the outcome of the center alignment. When applying center alignment to objects, the sampling locations on the foreground regions (in white) all concentrate on the centers of objects (in yellow) after center alignment, which are near to the true centers of objects (in red).}\n  \\vspace{-4mm}\n  \\label{fig.center_align}\n\\end{figure}\n\n\\textbf{Shape alignment} We can first obtain the foreground region according to the classification results. Then, the receptive fields of the features in the foreground regions can focus on the anchor with the highest confidence scores, as shown in Fig.~\\ref{fig.shape_align}. This makes sense because among all the anchors located at the same position, the one with the highest confidence is more likely to remain after the NMS algorithm. We use a convolution termed AlignConv in the implementation of shape alignment and center alignment. AlignConv is similar to the deformable convolution \\cite{zhu2019deformable}. The difference is that the offset of the former is computed from the prediction results. The normal convolution can be considered as a special case of AlginConv where the offset equals zero. Unlike the RoI convolution proposed in \\cite{chen2019revisiting}, we align the shape of the receptive field or the location of the center in one shot. When performing shape alignment on the feature map with stride $S$, the offset $(O^{sa}_i, O^{sa}_j)$ of the convolution with kernel size $k_h \\times k_w$ is defined as:\n\\begin{eqnarray}\n&O^{sa}_i = (\\frac{h_{a}}{S \\times k_h} - 1) \\times (i - \\frac{k_h}{2} + 0.5), \\\\\n&O^{sa}_j = (\\frac{w_{a}}{S \\times k_w} - 1) \\times (j - \\frac{k_w}{2} + 0.5),\n\\end{eqnarray}\nwhere $h_{a}, w_{a}$ are the height and the width of the anchor with the highest confidence.\n\n\\textbf{Center alignment} The purpose of center feature alignment is to align the feature at the center of the object to the feature that represents the center of the anchor. As shown in Fig.~\\ref{fig.center_align}, the prediction results from the 2D\/3D center regression are used to compute the offset of the convolution on the feature map with stride $S$:\n\\begin{eqnarray}\n&O^{ca}_i = \\frac{y_r}{S},\n&O^{ca}_j = \\frac{x_r}{S}, \n\\end{eqnarray}\nwhere $x_{r}$ and $y_{r}$ are the prediction results of the 2D\/3D centers in objects, respectively. As shown in Fig.~\\ref{fig.center_align}, when center alignment with a $1 \\times 1$ convolutional kernel is applied to the feature map, the sampling position is adaptively concentrated on the center of objects.\n\n\\subsection{Asymmetric Non-local Attention Block}\n\\label{sect:asy}\n\nWe propose a novel asymmetric non-local attention block to improve the accuracy of the depth $z_{3d}$ prediction by extracting the depth-wise features that can represent the global information and the long-range dependencies. The standard non-local block \\cite{wang2018non} is promising in establishing long-range dependencies, but its computational complexity is $O(N^2C)$, where $N = h \\times w$, $h$, $w$ and $C$ indicate the spatial height, width, and channel number of the feature map, respectively. This is very computationally expensive and inefficient compared to normal convolutions. Thus, the applications are limited. The Asymmetric Pyramid Non-local Block \\cite{zhu2019asymmetric} reduces the computational cost by decreasing the number of feature descriptors using pyramid pooling. However, pyramid pooling on the same feature map may lead to features with low resolution being replaced with high-resolution features. In other words, there exists redundancy in the computational cost regarding the image resolution. As such, we propose an Asymmetric Non-local Attention Block (ANAB), which can extract multi-scale features to enhance the feature learning with a low computational cost.\n\n\\begin{figure}[t!]\n    \\centering\n   \n   \n   \n   \n   \n   \n    \\begin{subfigure}{\\columnwidth}\n        \\label{fig:anab}\n        \\includegraphics[width=\\textwidth]{ANAB.pdf}\n    \\end{subfigure}\n   \n    \\begin{subfigure}{\\columnwidth}\n        \\label{fig:papa}\n        \\includegraphics[width=\\textwidth]{PAPA.pdf}\n    \\end{subfigure}\n    \\caption{\\small Top: Asymmetric Non-local Attention Block. The key and query branches share the same attention maps, which forces the key and value to focus on the same place. Bottom: Pyramid Average Pooling with Attention ($PA^2$) that generates different level descriptors in various resolutions.}\n    \\label{fig:pan}\n    \\vspace{-4mm}\n\\end{figure}\n\n\nAs shown at the top of Fig.~\\ref{fig:pan}, we use the pyramidal features of the $key$ and $value$ branches to reduce the computational cost. The bottom of Fig.~\\ref{fig:pan} illustrates the Pyramid Average Pooling with Attention ($PA^2$) module. The different levels of the feature pyramid have different receptive fields, thereby modeling regions with different scales. \nTwo matrix multiplications are performed in ANAB. First, the similarity matrix between the reshaped feature matrices $\\mathbf{M}_Q$ and $\\mathbf{M}_K$ obtained from $querry$ and $key$ is defined as:\n\\begin{equation}\n    \\mathbf{M}_{S} = \\mathbf{M}_{Q} \\times \\mathbf{M}_{K}^T, \\quad\\mathbf{M}_{Q} \\in \\mathbb{R}^{N\\times C}, \\mathbf{M}_{K} \\in \\mathbb{R}^{L\\times C}. \n\\end{equation}\nThen, the softmax function is used to normalize the last dimension of the similarity matrix and multiply it by the reshaped feature matrix $\\mathbf M_V$ obtained from $value$ to get the output:\n\\begin{equation}\n    \\mathbf{M}_{out} = Softmax(\\mathbf{M}_{S}) \\times \\mathbf{M}_{V} , \\quad  \\mathbf{M}_{V} \\in \\mathbb{R}^{L\\times C}.\n\\end{equation}\nwhere $L$ is the number of features after sampling. The standard non-local block \\cite{wang2018non} has computational complexity $O(N^2C)$, while the complexity of ANAB is $O(NLC)$. In practice, $L$ is usually significantly smaller than $N$. In our case, we use a four-level downsampling strategy on the feature map $48 \\times 160$. The resolution of the four-level feature pyramid is set to $i \\in \\{1\\times1, 4\\times4, 8\\times8, 16\\times16\\}$, the sum of which is the total number $L$ of features after downsampling. So $L = 377$ is much smaller than $N = 7680$.\n\nAnother effective component of ANAB is the application of the multi-scale attention maps to the $key$ and $value$ branches in $PA^2$ module, as shown at the bottom of Fig. \\ref{fig:pan}. \nThe motivation is to keep the key information of the origin feature map when greatly reducing the dimensions of matrices $\\mathbf{M}_{K}$ and $\\mathbf{M}_{V}$ from $N \\times C$ to $L \\times C$. The spatial attention maps generated by a $1 \\times 1$ convolutional layer are used as weights. This module adaptively adjusts the weights to pay more attention to the useful information and suppress the less useful information. \nThe attentive map can be treated as a mask performed on multi-scale features. We use the average pooling with attention to downsample the feature maps. Such a weighted average pooling operation offers an efficient way to gather the key features.\n\n\\subsection{2D-3D Prediction and Loss}\n\\textbf{Anchor definition.}\nWe adopt a one-stage 2D-3D anchor-based network as our detector. To detect the 2D and the 3D BBoxes simultaneously, our predefined anchor contains the parameters of both the 2D BBoxes $[w, h]_{2d}$ and the 3D BBoxes $[z, w, h, l, \\alpha]_{3d}$. $\\alpha$ is the observation angle of the object that measures the angle at which the camera views the object.\nCompared with the rotation angle of the object, the observation angle is more meaningful for monocular 3D object detection \\cite{mousavian20173d}. The dimension of the object is given by  $[w, h, l]_{3d}$. We project the center of the object onto the image plane to encode the 3D location of the object into the anchor:\n\n\\begin{equation}\n\\begin{bmatrix}\nX_{p} & Y_{p} & 1\n\\end{bmatrix} ^\\mathrm{T} \\cdot Z_{p}\n = \\mathbf{K} \\cdot \n \\begin{bmatrix}\n X & Y & Z & 1\n \\end{bmatrix} ^\\mathrm{T}, \n \\label{eqn.1}\n\\end{equation}\nwhere $(X_p, X_p)$ are the coordinates of the 3D point projected onto the image plane, and $(X, Y, Z)$ are the 3D space coordinates in the camera coordinate system. $K \\in \\mathbb{R}^{3\\times4}$ is the intrinsic camera matrix, which is known at both the training and testing phase.\nWe obtain the 3D parameters of each anchor by computing the mean of the corresponding 3D parameters of the objects whose intersection over union (IoU) is greater than a given threshold (0.5) with the predefined 2D anchors $[w, h]_{2d}$.\n\n\\textbf{Output transformation.}\nGiven the detection outputs $cls$, $[t_x, t_y, t_w, t_h]_{2d}$ and $[t_x, t_y, t_z, t_w, t_h, t_l, t_\\alpha]_{3d}$ for each anchor, the 2D BBox $[X, Y, W, H]_{2d}$ and 3D BBox $[X, Y, Z, W, H, L, A]_{3d}$ can be restored from the output of the detector by:\n\\begin{align}\n&[X, Y]_{2d} = [t_x, t_y]_{2d} \\otimes [w, h]_{2d} + [x, y]_{2d} \\notag \\\\\n&[W, H]_{2d} = \\exp([t_w, t_h]_{2d}) \\otimes [w, h]_{2d} \\notag \\\\\n&[X_p, Y_p]_{3d} = [t_x, t_y]_{3d} \\otimes [w, h]_{2d} + [x, y]_{2d} \\notag \\\\\n&[W, H, L]_{3d} = \\exp([t_w, t_h, t_l]_{3d}) \\otimes [w, h, l]_{3d} \\notag \\\\\n&[Z_p, A]_{3d} = [t_z, t_\\alpha] + [z, \\alpha]_{3d} , \\notag \\\\\n\\end{align}\nwhere $\\otimes$ denotes the element-wise product and $A$ is the rotation angle. During the inference phase, $[X, Y, Z]_{3d}$ can be obtained by projecting $[X_p, Y_p, Z_p]$ back to the camera coordinate system using the inverse operation of Eqn.~\\ref{eqn.1}.\n\n\\textbf{Loss function.}\nWe employ a multi-task loss function to supervise the learning of the network, which is composed of three parts: a classification loss, 2D BBox regression loss, and 3D BBox regression loss. The 2D regression and 3D regression loss are regularized with weights $\\lambda_1$ and $\\lambda_2$:\n\\begin{equation}\n\\label{loss}\n    L = L_{cls} + \\lambda_1 L_{2d} + \\lambda_2 L_{3d} ,\n\\end{equation}\nFor the classification task, we employ the standard cross entropy loss function:    \n\\begin{equation}\n    L_{cls} = -\\log(\\frac{\\exp(c')}{\\sum \\exp(c_i)}).\n\\end{equation}\nFor the 2D BBox regression task, we use $-\\log(IoU)$ as the loss function for the ground-truth 2D BBox $\\hat b_{2d}$ and the predicted 2D BBox $b'_{2d}$, similar to \\cite{brazil2019m3d}:\n\\begin{equation}\n    L_{2d} = -\\log(IoU(b'_{2d}, \\hat b_{2d})).\n\\end{equation}\nA smooth L1 loss function is employed to supervise the regression of 3D BBoxes:\n\\begin{equation}\n\\begin{array}{l}\n    L_{3d} = \\sum\\limits_{v_{3d} \\in P_{3d}} SmoothL_1(v'_{3d}, \\hat v_{3d}), \\\\ P_{3d} = \\{t_{x}, t_{y}, t_{z}, t_{w}, t_{h}, t_{l}, t_{\\alpha}\\}_{3d}.\n\\end{array}\n\\end{equation}\n\n\n\\begin{table*}\n  \\small\n  \\centering\n  \\resizebox{1\\textwidth}{!}{\n  \\begin{tabular}{l|c|ccc|ccc}\n    \\toprule\n    \\multirow{2}*{Methods}  & \\multirow{2}*{Extra}  & \\multicolumn{3}{|c|}{$AP_{3d}(val\/test) \\quad IoU \\ge 0.7$} & \\multicolumn{3}{c}{$AP_{BEV}(val\/test) \\quad IoU \\ge 0.7$}\\\\\n    {} & {} & Easy & Moderate & Hard & Easy & Moderate & Hard \\\\\n    \\midrule\n   \n    MonoFENet\\cite{bao2019monofenet}  & Depth    &17.54 \/ 8.35   &11.16 \/ 5.14   &9.74 \/ 4.10    &30.21 \/ 17.03   &20.47 \/ 11.03  &17.58 \/ 9.05 \\\\\n    AM3D\\cite{ma2019accurate}         & Depth    &32.23 \/ 16.50  &21.09 \/ 10.74  &17.26 \/ 9.52   &43.75 \/ 25.03   &28.39 \/ 17.32  &23.87 \/ 14.91 \\\\\n    D4LCN\\cite{ding2019learning}      & Depth    &26.97 \/ 16.65   &21.71 \/ 11.72  &18.22 \/ 9.51   &34.82 \/ 22.51   &25.83 \/ 16.02  &23.53 \/ 12.55 \\\\\n    \\midrule\n    GS3D\\cite{li2019gs3d}             &None      &13.46 \/ 4.47   &10.97 \/ 2.90   &10.38 \/ 2.47   &\\qquad- \/ 8.41  &\\qquad- \/ 6.08  &\\qquad- \/ 4.94 \\\\\n    MonoPSR\\cite{ku2019monocular}     &None      &12.75 \/ 10.76  &11.48 \/ 7.25   &8.59 \/ 5.85    &20.63 \/ 18.33   &18.67 \/ 12.58   &14.45 \/ 9.91  \\\\\n    MonoGRNet\\cite{qin2019monogrnet}  &None      &13.88 \/ 9.61   &10.19 \/ 5.74   &7.62 \/ 4.25    &\\qquad- \/ 18.19 & \\qquad- \/ 11.17 &\\qquad- \/ 8.73 \\\\\n    SS3D\\cite{jorgensen2019monocular} &None      &14.52 \/ 10.78  &13.15 \/ 7.68   &11.85 \/ 6.51   &\\qquad- \/ 16.33 &\\qquad- \/ 11.52 &\\qquad- \/ 9.93 \\\\\n    MonoDIS\\cite{simonelli2019disentangling} &None &18.05 \/ 10.37 &14.98 \/ 7.94 &13.42 \/ 6.40 &24.26 \/ 17.23 &18.43 \/ 13.19   &16.95 \/ 11.12 \\\\\n    MonoPair\\cite{chen2020monopair}  &None       &\\qquad- \/ 13.04   &\\qquad- \/ 9.99   &\\qquad- \/ 8.65   &\\qquad- \/ 19.28   &\\qquad- \/ 14.83   &\\qquad- \/ \\textbf{12.89} \\\\\n    SMOKE\\cite{liu2020smoke}         &None       &14.76 \/ 14.03   &12.85 \/ 9.76   &11.50 \/ 7.84   &19.99 \/ 20.83   &15.61 \/ 14.49   &15.28 \/ 12.75 \\\\\n    M3D-RPN\\cite{brazil2019m3d}      &None       &20.27 \/ 14.76   &17.06 \/ 9.71   &15.21 \/ 7.42   &25.94 \/ 21.02   &21.18 \/ 13.67   &17.90 \/ 10.23 \\\\\n    RTM3D\\cite{li2020rtm3d}          &None       &20.77 \/ 14.41   &16.86 \/ 10.34  &16.63 \/ 8.77   &25.56 \/ 19.17   &22.12 \/ 14.20   &20.91 \/ 11.99 \\\\\n    \\midrule\n    M3DSSD(ours)  &None    &\\textbf{27.77 \/ 17.51}   &\\textbf{21.67 \/ 11.46}  &\\textbf{18.28 \/ 8.98}   &\\textbf{34.51 \/ 24.15}  &\\textbf{26.20 \/ 15.93}  &\\textbf{23.40} \/ 12.11 \\\\\n    \n    \\bottomrule\n  \\end{tabular}\n  }\n  \\caption{AP scores on $val$ and $test$ set of 3D object detection and bird's eye view for cars. }\n  \\label{tab:ap}\n \n\\end{table*}\n\n\\section{Experiments}\n\n\\begin{figure*}[htbp]\n  \\centering\n  \\includegraphics[width=1\\textwidth]{vis.png}\n  \\caption{Qualitative results of 3D detection (left) and bird's eye view (right), prediction in green and ground-truth in red.} \n  \\label{fig:vis}\n  \\vspace{-4mm}\n\\end{figure*}\n\n\\subsection{Evaluation Dataset}\nWe evaluate our framework on the challenging KITTI benchmark for 3D object detection and bird's eye view tasks. The KITTI dataset contains 7481 images with labels and 7518 images for testing, covering three main categories of objects: cars, pedestrians, and cyclists. We use common split methods \\cite{chen20173d} to divide the images with labels into the training set and the validation set. We pad the images to the size of $384 \\times 1280$ in both the training and inference phase. In the training phase, in addition to the conventional data augmentation methods of random translation and horizontal mirror flipping, the random scaling operation is applied for monocular images. \n\n\\begin{table*}[htbp]\n    \\centering\n \n    \\begin{tabular}{l|ccc|ccc}\n    \\toprule\n    \\multirow{2}*{Methods}  & \\multicolumn{3}{|c|}{$Pedestrian \\quad AP_{3D}\/AP_{bev}$}  & \\multicolumn{3}{c}{$Cyclist \\quad AP_{3D}\/AP_{bev}$}\\\\\n    {}                    &Easy   &Moderate &Hard    &Easy  &Moderate  &Hard\\\\\n    \\midrule\n    M3D-RPN\\cite{brazil2019m3d}           &4.92 \/ 5.65   &3.48 \/ 4.05   &2.94 \/ 3.29   &0.94 \/ 1.25   & 0.65 \/ 0.81   & 0.47 \/ 0.78\\\\\n    D4LCN\\cite{ding2019learning}             &4.55 \/ 5.06   &3.42 \/ 3.86   &2.83 \/ 3.59   &2.45 \/ 2.72   &\\textbf{1.67} \/ 1.82   &1.36 \/ \\textbf{1.79}\\\\\n    SS3D\\cite{jorgensen2019monocular}              &2.31 \/ 2.48   &1.78 \/ 2.09   &1.48 \/ 1.61   &\\textbf{2.80} \/ \\textbf{3.45}   &1.45 \/ 1.89   &1.35 \/ 1.44\\\\\n    \\midrule\n    M3DSSD(ours)      &\\textbf{5.16} \/ \\textbf{6.20}   &\\textbf{3.87} \/ \\textbf{4.66}   &\\textbf{3.08} \/ \\textbf{3.99}   &2.10 \/ 2.70    &1.51 \/ \\textbf{2.01}     &\\textbf{1.58} \/ 1.75\\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{Detection performance for pedestrians and cyclists on $test$ set, at $0.5$ $IoU$ threshold.}\n    \\label{tab:ped}\n    \\vspace{-4mm}\n\\end{table*}\n\n\\subsection{Implementation Details}\nWe implement our model with PyTorch. We adopt the SGD optimizer with momentum to train the network with a CPU E52698 and GPU TITAN V100, in an end-to-end manner, for 70 epochs. The momentum of the SGD optimizer is set to 0.9, and weight decay is set to 0.0005. The mini-batch size is set to 4. The learning rate increases linearly from 0 to the target learning rate of 0.004 in the first epoch and then decreases to $4\\times 10^{-8}$ with cosine annealing. Terms $\\lambda_1$ and $\\lambda_2$ in Eqn.~\\ref{loss} are both set to 1.0. \nWe lay 36 anchors on each pixel of the feature map, the size of which increases from 24 to 288 following the exponential function of $24 \\times 12^{i\/11}, i \\in \\{0, 1, 2, \\dots , 11\\}$, and the aspect ratio is set to $\\{0.5, 1.0, 1.5\\}$. We apply online hard-negative mining by sampling the top $20\\%$ high loss boxes in each minibatch in the training phase. \nIn the inference phase, we apply NMS with 0.4 IoU criteria on the 2D BBox and filter out the objects with a confidence lower than 0.75. The post-optimization algorithm proposed in \\cite{brazil2019m3d} is used to make the rotation angle more reasonable. \nThe algorithm uses projection consistency to optimize the rotation angle. The rotation angle is optimized iteratively to minimize the L1 loss of the projection of the predicted 3D BBox and the predicted 2D BBox.\n\n\\subsection{Performance Evaluation}\nWe set the network after removing the feature alignment module and ANAB from M3DSSD as the baseline. More specifically, for the baseline, the feature map output from the backbone is directly used for classification and 2D BBox regression and 3D BBox regression.\n\nWe evaluate our framework on the KITTI benchmark for both bird's eye view and 3D object detection tasks. The average precision (AP) of Intersection over Union (IoU) is used as the metric for evaluation in both tasks and it is divided into easy, moderate, and hard according to the height, occlusion, and truncation level of objects. Note that the official KITTI evaluation has been using $AP|_{R40}$ with 40 recall points instead of $AP|_{R11}$ with 11 recall points since October 8, 2019. However, most previous methods evaluated on the validation used $AP|_{R11}$. Thus, we report the $AP|_{R40}$ for the test dataset and $AP|_{R11}$ for the validation dataset for a fair comparison. We set the threshold of IoU to 0.7 for cars and 0.5 for pedestrians and cyclists as the same as the official settings. Fig.~\\ref{fig:vis} shows qualitative results for 3D object detection and bird's eye view. The detection results and depth predictions are less accurate with further distance. The videos of 3D object detection results and the additional results can be found in the supplemental material. \n \n   \n\n\\textbf{Bird's eye view.}\nThe bird's eye view task is to detect objects projected on the ground, which is closely related to the 3D location of objects. The detection results for cars on both the $val$ and $test$ set are reported in Tab.~\\ref{tab:ap}. M3DSSD achieves state-of-the-art performance on the bird's eye view task compared to approaches with and without depth estimation. Our method has significant improvement compared to the methods without depth estimation. \n\n\\begin{figure}[htb]\n  \\centering\n  \\includegraphics[width=0.8\\columnwidth]{gt_depth_err.pdf}\n  \\caption{The average depth estimation error varies along with the ground truth depth. Best viewed in color.} \n  \\label{figure.depth_err}\n  \\vspace{-4mm}\n\\end{figure}\n\n\\begin{table}[htbp]\n  \\centering\n  \\resizebox{1\\columnwidth}{!}{\n  \\begin{tabular}{l|ccc}\n    \\toprule\n    \\multirow{2}*{Methods}  & \\multicolumn{3}{|c}{$AP_{3d}\/AP_{BEV}\\quad IoU \\ge 0.7 $}\\\\\n    {}                    &Easy   &Mod. &Hard   \\\\\n    \\midrule\n    \n    \n    \n    \n    \n     \n   \n    Baseline w\/ ANAB \\dag            & 25.70 \/ 33.48        &19.02 \/ 24.79 &17.31 \/ 20.15  \\\\ \n    \\dag w\/ Shape Alignment   & 27.26 \/ 33.64        &21.56 \/ 25.24 &18.07 \/ 22.81  \\\\\n    \\dag w\/ Center Alignment  & 27.33 \/ \\textbf{34.85} &21.51 \/ 25.96 &18.03 \/ 23.26  \\\\\n    \\dag w\/ Full Alignment    & \\textbf{27.77} \/ 34.51 &\\textbf{21.67} \/ \\textbf{26.20} &\\textbf{18.28} \/ \\textbf{23.40}  \\\\\n    \\bottomrule\n  \\end{tabular}\n  }\n  \\caption{Ablation study on feature alignment.}\n  \\label{tab:feature alignment}\n \n\\end{table}\n\n\\begin{table*}[t]\n    \\centering\n    \\resizebox{0.9\\textwidth}{!}{\n    \\begin{tabular}{l|ccc}\n        \\toprule\n        \\multirow{2}*{Methods}  & \\multicolumn{3}{|c}{$AP_{3d}\/AP_{BEV} \\quad IoU \\ge 0.7 $} \\\\\n        {}                    &Easy        &Moderate         &Hard    \\\\\n        \\midrule\n         baseline            &23.40 \/ 28.66    &18.32 \/ 23.53    &16.62 \/ 19.54 \\\\\n         ANB                 &23.65 \/ 29.19    &18.47 \/ 23.65    &16.54 \/ 19.50  \\\\\n         ANAB  &\\textbf{25.70} \/ \\textbf{33.48}  &\\textbf{19.02} \/ \\textbf{24.79}  &\\textbf{17.31} \/ \\textbf{20.15} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \n    \\begin{tabular}{||l|c|c}\n        \\toprule\n        \\multirow{2}*{Methods} & \\multirow{2}*{GPU time}  & \\multirow{2}*{GPU memory}\\\\\n        {}   &  (ms)  & (Gbyte)\\\\\n        \\midrule\n         Non-local \\cite{wang2018non}     &   5.89 \/ 104.12         & 1.97 \/ 15.67 \\\\\n         ANB                              &   \\textbf{1.68} \/ \\textbf{5.92}  & \\textbf{1.09} \/ \\textbf{1.43} \\\\\n         ANAB                             &   1.86 \/ 6.76           & 1.22 \/ 1.91 \\\\\n        \\bottomrule\n    \\end{tabular}}\n    \n    \\caption{Ablation study on non-local blocks with detection accuracy, GPU time, and memory for different input sizes.}\n    \\label{tab:anab}\n   \n\\end{table*}\n\n\\begin{figure}[htb]\n  \\centering\n  \\includegraphics[width=0.8\\columnwidth]{align_depth_err.pdf}\n  \\caption{The average depth estimation error varies along with the size of objects that is the average of the length and the width of the 2D BBox. Best viewed in color.} \n  \\label{figure.depth_err_alg}\n  \\vspace{-4mm}\n\\end{figure}\n\n\\begin{figure*}[htb]\n  \\flushright\n  \\includegraphics[width=0.93\\textwidth]{vis_atten.pdf}\n  \\caption{Visualization for attention maps in $PA^2$ with a four-level feature pyramid $\\{1\\times1, 4\\times4, 8\\times8, 16\\times16\\}$.} \n  \\label{figure.anab_vis}\n  \\vspace{-4mm}\n\\end{figure*}\n\\textbf{3D object detection for cars.}\nThe 3D object detection task aims to detect 3D objects in the camera coordinate system, which is more challenging than the bird's eye view task due to the additional y-axis. Compared with the approaches without depth estimation, Tab.~\\ref{tab:ap} shows that M3DSSD achieves better performance in both the $val$ and $test$ set. Note that M3DSSD is better than most of the approaches with depth estimation. Further, our method achieves competitive performance against D4LCN that adopts a pre-trained model for depth estimation \\cite{ding2019learning}.\n\nFig.~\\ref{figure.depth_err} shows the average depth estimation error with the different ground truth depth ranges \\cite{liu2020smoke}. We compared our proposed method with SMOKE \\cite{liu2020smoke}, Mono3D \\cite{chen2016monocular} and 3DOP \\cite{chen20173d} on the same validation set. Fig.~\\ref{figure.depth_err} demonstrates that the proposed M3DSSD achieves better performance at all distance ranges, except for the distance greater than 60m, where the number of samples is usually small.\n\n\n\\textbf{3D object detection for pedestrians and cyclists.}\nCompared with cars, 3D object detection for pedestrians and cyclists is more challenging. This is because the size of pedestrians and bicycles is relatively small. In addition, people are non-rigid bodies, and their shapes vary a lot, thereby making it difficult to locate pedestrians and cyclists.\nWe report the detection results for pedestrians and cyclists on the test set of the KITTI benchmark in Tab.~\\ref{tab:ped}. Since some methods did not report the pedestrian and the cyclist results, we compare our model with M3D-RPN \\cite{brazil2019m3d}, D4LCN \\cite{ ding2019learning}, and SS3D \\cite{jorgensen2019monocular}. Our model achieves competitive performance in both 3D detection and bird's eye view tasks for pedestrians and bicycles, especially for the pedestrian category. Note that we train only one single model to detect the three object classes simultaneously. \n\n\\subsection{Ablation Study}\n\\textbf{Feature alignment.} \nWe evaluate the feature alignment strategies, including shape alignment, center alignment, and full alignment (both center alignment and shape alignment). \nAs shown in Tab. \\ref{tab:feature alignment}, that the proposed shape alignment, center alignment, and full alignment achieve better results compared to the case without alignment. \n\nFig.~\\ref{figure.depth_err_alg} illustrates the average depth estimation error varies with the size of objects for the model with and without feature alignment. The x-axis is set as the size of the 2D BBox $(w_{2d}+h_{2d})\/2$. It shows that the proposed feature alignment module is effective on objects of different sizes, especially for the small objects in $[0 - 25]$. This also explains why M3DSSD outperforms other methods in small object detection such as pedestrians and cyclists.\n\n\\textbf{Asymmetric non-local attention block.}\nWe compare the Asymmetric Non-local Block (ANB), and our proposed Asymmetric Non-local Attention Block (ANAB), which applies pyramid average pooling on the feature map with attentions. We use the same sampling size for both methods. Tab.~\\ref{tab:anab} shows that the network with ANAB achieves the best performance. With a similar computational time, the proposed ANAB has better detection accuracy than ANB. Meanwhile, both methods cost much less GPU time and memory than the standard non-local block \\cite{wang2018non}. The attention module costs a little more consuming time with significant improvement, especially in easy tasks. Tab. \\ref{tab:anab} on the right shows the GPU time and memory regarding the input size $[1, 256, 48, 160]$ and $[1, 256, 96, 320]$. This shows that the computational cost is closer to the theoretical analysis in Sect.~\\ref{sect:asy} with a larger input size. ANAB has extra pooling layers, convolutional layers, and an element-wise multiplication, which are not considered in the theoretical analysis.\n\nIn ANAB, the attention maps are assigned to the multi-scale pooling operations for the depth-wise feature extraction. Fig.~\\ref{figure.anab_vis} shows that the attention map for $1 \\times 1$ feature pyramid has larger weights on the objects which are close to the camera, while the attention map for the higher-level feature pyramid assigns larger weights on the objects that are away from the camera. The attention maps in different levels show a correlation between the resolution of the feature pyramid and the object depth. This lies in the fact that the feature pyramid with low resolution has a large receptive field that is sensitive to the object in large size, while the feature pyramid with high resolution has a small receptive field that is sensitive to the object in small size. For the size of the same-class object from a fixed perspective, the smaller, the farther.\nThe depth-wise attention maps enhance the capability of perceiving the depth of objects, thereby improving the performance of object depth estimation.\n\n\\section{Conclusion}\n\nIn this work, we propose a simple and very effective monocular single-stage 3D object detector. We present a two-step feature alignment approach to address the feature mismatching, which enhances the feature learning for object detection. The asymmetric non-local attention block enables the network to extract depth-wise features, which improves the performance of the depth prediction in the regression head. Compared to the methods with or without the estimated depth as an extra input, M3DSSD achieves better performance on the challenging KITTI dataset for car, pedestrian, and cyclist object class using one single model, for both bird's eye view and 3D object detection.\n\n\\textbf{Acknowledgement:} This work was supported in part by the National Key Research and Development Program of China (2018YFE0183900). Hang Dai would like to thank the support from MBZUAI startup fund (GR006). \n\n\\iffalse \n\\subsection{The ruler}\nThe \\LaTeX\\ style defines a printed ruler which should be present in the\nversion submitted for review.  The ruler is provided in order that\nreviewers may comment on particular lines in the paper without\ncircumlocution.  If you are preparing a document using a non-\\LaTeX\\\ndocument preparation system, please arrange for an equivalent ruler to\nappear on the final output pages.  The presence or absence of the ruler\nshould not change the appearance of any other content on the page.  The\ncamera ready copy should not contain a ruler. (\\LaTeX\\ users may uncomment\nthe \\verb'\\cvprfinalcopy' command in the document preamble.)  Reviewers:\nnote that the ruler measurements do not align well with lines in the paper\n--- this turns out to be very difficult to do well when the paper contains\nmany figures and equations, and, when done, looks ugly.  Just use fractional\nreferences (e.g.\\ this line is $095.5$), although in most cases one would\nexpect that the approximate location will be adequate.\n\n\\subsection{Mathematics}\n\nPlease number all of your sections and displayed equations.  It is\nimportant for readers to be able to refer to any particular equation.  Just\nbecause you didn't refer to it in the text doesn't mean some future reader\nmight not need to refer to it.  It is cumbersome to have to use\ncircumlocutions like ``the equation second from the top of page 3 column\n1''.  (Note that the ruler will not be present in the final copy, so is not\nan alternative to equation numbers).  All authors will benefit from reading\nMermin's description of how to write mathematics:\n\\url{http:\/\/www.pamitc.org\/documents\/mermin.pdf}.\n\n\n\\subsection{Blind review}\n\nMany authors misunderstand the concept of anonymizing for blind\nreview.  Blind review does not mean that one must remove\ncitations to one's own work---in fact it is often impossible to\nreview a paper unless the previous citations are known and\navailable.\n\nBlind review means that you do not use the words ``my'' or ``our''\nwhen citing previous work.  That is all.  (But see below for\ntechreports.)\n\nSaying ``this builds on the work of Lucy Smith [1]'' does not say\nthat you are Lucy Smith; it says that you are building on her\nwork.  If you are Smith and Jones, do not say ``as we show in\n[7]'', say ``as Smith and Jones show in [7]'' and at the end of the\npaper, include reference 7 as you would any other cited work.\n\nAn example of a bad paper just asking to be rejected:\n\\begin{quote}\n\\begin{center}\n    An analysis of the frobnicatable foo filter.\n\\end{center}\n\n   In this paper we present a performance analysis of our\n   previous paper [1], and show it to be inferior to all\n   previously known methods.  Why the previous paper was\n   accepted without this analysis is beyond me.\n\n   [1] Removed for blind review\n\\end{quote}\n\n\nAn example of an acceptable paper:\n\n\\begin{quote}\n\\begin{center}\n     An analysis of the frobnicatable foo filter.\n\\end{center}\n\n   In this paper we present a performance analysis of the\n   paper of Smith \\etal [1], and show it to be inferior to\n   all previously known methods.  Why the previous paper\n   was accepted without this analysis is beyond me.\n\n   [1] Smith, L and Jones, C. ``The frobnicatable foo\n   filter, a fundamental contribution to human knowledge''.\n   Nature 381(12), 1-213.\n\\end{quote}\n\nIf you are making a submission to another conference at the same time,\nwhich covers similar or overlapping material, you may need to refer to that\nsubmission in order to explain the differences, just as you would if you\nhad previously published related work.  In such cases, include the\nanonymized parallel submission~\\cite{Authors14} as additional material and\ncite it as\n\\begin{quote}\n[1] Authors. ``The frobnicatable foo filter'', F\\&G 2014 Submission ID 324,\nSupplied as additional material {\\tt fg324.pdf}.\n\\end{quote}\n\nFinally, you may feel you need to tell the reader that more details can be\nfound elsewhere, and refer them to a technical report.  For conference\nsubmissions, the paper must stand on its own, and not {\\em require} the\nreviewer to go to a techreport for further details.  Thus, you may say in\nthe body of the paper ``further details may be found\nin~\\cite{Authors14b}''.  Then submit the techreport as additional material.\nAgain, you may not assume the reviewers will read this material.\n\nSometimes your paper is about a problem which you tested using a tool which\nis widely known to be restricted to a single institution.  For example,\nlet's say it's 1969, you have solved a key problem on the Apollo lander,\nand you believe that the CVPR70 audience would like to hear about your\nsolution.  The work is a development of your celebrated 1968 paper entitled\n``Zero-g frobnication: How being the only people in the world with access to\nthe Apollo lander source code makes us a wow at parties'', by Zeus \\etal.\n\nYou can handle this paper like any other.  Don't write ``We show how to\nimprove our previous work [Anonymous, 1968].  This time we tested the\nalgorithm on a lunar lander [name of lander removed for blind review]''.\nThat would be silly, and would immediately identify the authors. Instead\nwrite the following:\n\\begin{quotation}\n\\noindent\n   We describe a system for zero-g frobnication.  This\n   system is new because it handles the following cases:\n   A, B.  Previous systems [Zeus et al. 1968] didn't\n   handle case B properly.  Ours handles it by including\n   a foo term in the bar integral.\n\n   ...\n\n   The proposed system was integrated with the Apollo\n   lunar lander, and went all the way to the moon, don't\n   you know.  It displayed the following behaviours\n   which show how well we solved cases A and B: ...\n\\end{quotation}\nAs you can see, the above text follows standard scientific convention,\nreads better than the first version, and does not explicitly name you as\nthe authors.  A reviewer might think it likely that the new paper was\nwritten by Zeus \\etal, but cannot make any decision based on that guess.\nHe or she would have to be sure that no other authors could have been\ncontracted to solve problem B.\n\\medskip\n\n\\noindent\nFAQ\\medskip\\\\\n{\\bf Q:} Are acknowledgements OK?\\\\\n{\\bf A:} No.  Leave them for the final copy.\\medskip\\\\\n{\\bf Q:} How do I cite my results reported in open challenges?\n{\\bf A:} To conform with the double blind review policy, you can report results of other challenge participants together with your results in your paper. For your results, however, you should not identify yourself and should not mention your participation in the challenge. Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\\medskip\\\\\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\fbox{\\rule{0pt}{2in} \\rule{0.9\\linewidth}{0pt}}\n  \n\\end{center}\n   \\caption{Example of caption.  It is set in Roman so that mathematics\n   (always set in Roman: $B \\sin A = A \\sin B$) may be included without an\n   ugly clash.}\n\\label{fig:long}\n\\label{fig:onecol}\n\\end{figure}\n\n\\subsection{Miscellaneous}\n\n\\noindent\nCompare the following:\\\\\n\\begin{tabular}{ll}\n \\verb'$conf_a$' &  $conf_a$ \\\\\n \\verb'$\\mathit{conf}_a$' & $\\mathit{conf}_a$\n\\end{tabular}\\\\\nSee The \\TeX book, p165.\n\nThe space after \\eg, meaning ``for example'', should not be a\nsentence-ending space. So \\eg is correct, {\\em e.g.} is not.  The provided\n\\verb'\\eg' macro takes care of this.\n\nWhen citing a multi-author paper, you may save space by using ``et alia'',\nshortened to ``\\etal'' (not ``{\\em et.\\ al.}'' as ``{\\em et}'' is a complete word.)\nHowever, use it only when there are three or more authors.  Thus, the\nfollowing is correct: ``\n   Frobnication has been trendy lately.\n   It was introduced by Alpher~\\cite{Alpher02}, and subsequently developed by\n   Alpher and Fotheringham-Smythe~\\cite{Alpher03}, and Alpher \\etal~\\cite{Alpher04}.''\n\nThis is incorrect: ``... subsequently developed by Alpher \\etal~\\cite{Alpher03} ...''\nbecause reference~\\cite{Alpher03} has just two authors.  If you use the\n\\verb'\\etal' macro provided, then you need not worry about double periods\nwhen used at the end of a sentence as in Alpher \\etal.\n\nFor this citation style, keep multiple citations in numerical (not\nchronological) order, so prefer \\cite{Alpher03,Alpher02,Authors14} to\n\\cite{Alpher02,Alpher03,Authors14}.\n\n\n\\begin{figure*}\n\\begin{center}\n\\fbox{\\rule{0pt}{2in} \\rule{.9\\linewidth}{0pt}}\n\\end{center}\n   \\caption{Example of a short caption, which should be centered.}\n\\label{fig:short}\n\\end{figure*}\n\n\\section{Formatting your paper}\n\nAll text must be in a two-column format. The total allowable width of the\ntext area is $6\\frac78$ inches (17.5 cm) wide by $8\\frac78$ inches (22.54\ncm) high. Columns are to be $3\\frac14$ inches (8.25 cm) wide, with a\n$\\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the\nfirst page) should begin 1.0 inch (2.54 cm) from the top edge of the\npage. The second and following pages should begin 1.0 inch (2.54 cm) from\nthe top edge. On all pages, the bottom margin should be 1-1\/8 inches (2.86\ncm) from the bottom edge of the page for $8.5 \\times 11$-inch paper; for A4\npaper, approximately 1-5\/8 inches (4.13 cm) from the bottom edge of the\npage.\n\n\\subsection{Margins and page numbering}\n\nAll printed material, including text, illustrations, and charts, must be kept\nwithin a print area 6-7\/8 inches (17.5 cm) wide by 8-7\/8 inches (22.54 cm)\nhigh.\nPage numbers should be in footer with page numbers, centered and .75\ninches from the bottom of the page and make it start at the correct page\nnumber rather than the 4321 in the example.  To do this fine the line (around\nline 23)\n\\begin{verbatim}\n\\setcounter{page}{4321}\n\\end{verbatim}\nwhere the number 4321 is your assigned starting page.\n\nMake sure the first page is numbered by commenting out the first page being\nempty on line 46\n\\begin{verbatim}\n\\end{verbatim}\n\n\n\\subsection{Type-style and fonts}\n\nWherever Times is specified, Times Roman may also be used. If neither is\navailable on your word processor, please use the font closest in\nappearance to Times to which you have access.\n\nMAIN TITLE. Center the title 1-3\/8 inches (3.49 cm) from the top edge of\nthe first page. The title should be in Times 14-point, boldface type.\nCapitalize the first letter of nouns, pronouns, verbs, adjectives, and\nadverbs; do not capitalize articles, coordinate conjunctions, or\nprepositions (unless the title begins with such a word). Leave two blank\nlines after the title.\n\nAUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title\nand printed in Times 12-point, non-boldface type. This information is to\nbe followed by two blank lines.\n\nThe ABSTRACT and MAIN TEXT are to be in a two-column format.\n\nMAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use\ndouble-spacing. All paragraphs should be indented 1 pica (approx. 1\/6\ninch or 0.422 cm). Make sure your text is fully justified---that is,\nflush left and flush right. Please do not place any additional blank\nlines between paragraphs.\n\nFigure and table captions should be 9-point Roman type as in\nFigures~\\ref{fig:onecol} and~\\ref{fig:short}.  Short captions should be centred.\n\n\\noindent Callouts should be 9-point Helvetica, non-boldface type.\nInitially capitalize only the first word of section titles and first-,\nsecond-, and third-order headings.\n\nFIRST-ORDER HEADINGS. (For example, {\\large \\bf 1. Introduction})\nshould be Times 12-point boldface, initially capitalized, flush left,\nwith one blank line before, and one blank line after.\n\nSECOND-ORDER HEADINGS. (For example, { \\bf 1.1. Database elements})\nshould be Times 11-point boldface, initially capitalized, flush left,\nwith one blank line before, and one after. If you require a third-order\nheading (we discourage it), use 10-point Times, boldface, initially\ncapitalized, flush left, preceded by one blank line, followed by a period\nand your text on the same line.\n\n\\subsection{Footnotes}\n\nPlease use footnotes sparingly.\nIndeed, try to avoid footnotes altogether and include necessary peripheral\nobservations in\nthe text (within parentheses, if you prefer, as in this sentence).  If you\nwish to use a footnote, place it at the bottom of the column on the page on\nwhich it is referenced. Use Times 8-point type, single-spaced.\n\n\n\\subsection{References}\n\nList and number all bibliographical references in 9-point Times,\nsingle-spaced, at the end of your paper. When referenced in the text,\nenclose the citation number in square brackets, for\nexample~\\cite{Authors14}.  Where appropriate, include the name(s) of\neditors of referenced books.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|}\n\\hline\nMethod & Frobnability \\\\\n\\hline\\hline\nTheirs & Frumpy \\\\\nYours & Frobbly \\\\\nOurs & Makes one's heart Frob\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Results.   Ours is better.}\n\\end{table}\n\n\\subsection{Illustrations, graphs, and photographs}\n\nAll graphics should be centered.  Please ensure that any point you wish to\nmake is resolvable in a printed copy of the paper.  Resize fonts in figures\nto match the font in the body text, and choose line widths which render\neffectively in print.  Many readers (and reviewers), even of an electronic\ncopy, will choose to print your paper in order to read it.  You cannot\ninsist that they do otherwise, and therefore must not assume that they can\nzoom in to see tiny details on a graphic.\n\nWhen placing figures in \\LaTeX, it's almost always best to use\n\\verb+\\includegraphics+, and to specify the  figure width as a multiple of\nthe line width as in the example below\n{\\small\\begin{verbatim}\n   \\usepackage[dvips]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]\n                   {myfile.eps}\n\\end{verbatim}\n}\n\n\n\\subsection{Color}\n\nPlease refer to the author guidelines on the CVPR 2020 web page for a discussion\nof the use of color in your document.\n\n\\section{Final copy}\n\nYou must include your signed IEEE copyright release form when you submit\nyour finished paper. We MUST have this form before your paper can be\npublished in the proceedings.\n\nPlease direct any questions to the production editor in charge of these \nproceedings at the IEEE Computer Society Press: \n\\url{https:\/\/www.computer.org\/about\/contact}. \n\n\\fi \n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe properties of kinks of the $\\varphi^6$ model have been fairly well studied. In particular, the interactions between the kink and the antikink were investigated both using the collective coordinate method and by numerically solving the equation of motion \\cite{GaKuLi,Demirkaya.JHEP.2017,Weigel.JPCS.2014}. Interesting results have been obtained on various phenomena in the kink-antikink and multi-kink scattering \\cite{MGSDJ.JHEP.2017,Dorey.PRL.2011,Romanczukiewicz.PLB.2017,Lima.JHEP.2019}.\n\nOn the other hand, there is the so-called deformation procedure \\cite{Bazeia.PRD.2002,Bazeia.PRD.2004,Bazeia.EPJC.2018,Gani.arXiv.2020.deformations}, which allows to obtain a new model along with its kink solution from known model with its known kink solution.\n\nIn this paper, we apply the deformation procedure to the $\\varphi^6$ model, using the hyperbolic sine as the deforming function. The result of such deformation is the sinh-deformed $\\varphi^6$ model. (The fact that the study of the properties of kinks of the sinh-deformed $\\varphi^6$ model is of interest was indicated in \\cite{Bazeia.EPJC.2018}.) The preliminary results reported by Dr.\\ Aliakbar Moradi Marjaneh at the ICPPA-2020 conference indicate that in collisions of the kink and antikink of the sinh-deformed $\\varphi^6$ model resonance phenomena (escape windows) are present; see, e.g., \\cite{Belova.UFN.1997} for details about escape windows. At first glance, this contradicts the fact that there are no vibrational modes in the excitation spectrum of the kink. However, if we recall papers \\cite{Dorey.PRL.2011,Belendryasova.CNSNS.2019}, then we understand that everything is not so simple.\n\nSo, let's move on to a slightly more detailed presentation of our idea. Emphasize that this text cannot be regarded as a complete study, but only as a brief presentation of preliminary results.\n\n\\section{The $\\varphi^6$ model and its hyperbolic deformation}\n\nWithin the $\\varphi^6$ model the dynamics of a real scalar field $\\varphi(x,t)$ is described by the Lagrangian density\n\\begin{equation}\\label{eq:Largangian}\n\t\\mathscr{L} = \\frac{1}{2} \\left( \\frac{\\partial\\varphi}{\\partial t} \\right)^2 - \\frac{1}{2} \\left( \\frac{\\partial\\varphi}{\\partial x} \\right)^2 - V(\\varphi)\n\\end{equation}\nwith the potential\n\\begin{equation}\\label{eq:potential}\n    V^{(0)}(\\varphi) = \\frac{1}{2}\\varphi^2\\left(1-\\varphi^2\\right)^2.\n\\end{equation}\nThis model has kink solutions (two kinks and two antikinks) connecting the vacua $\\varphi=0$ and $\\varphi=\\pm 1$. All of them can be obtained by symmetry transformations from the kink\n\\begin{equation}\\label{eq:phi6_kink}\n\\varphi_{\\rm K}^{(0)}(x) = \\sqrt{\\frac{1+\\tanh x}{2}},\n\\end{equation}\nsee, e.g., \\cite{GaKuLi}.\n\nWe apply the deformation procedure to the $\\varphi^6$ model, using the deforming function $f(\\varphi)=\\sinh\\varphi$. As a result, we obtain the sinh-deformed $\\varphi^6$ model with the potential\n\\begin{equation}\\label{eq:sinh_deformed_phi6_potential}\nV^{(1)}(\\varphi)=\\frac{1}{2} \\tanh^2 \\varphi(1-\\sinh^2 \\varphi)^2\n\\end{equation}\nand the kink solution, corresponding to the $\\varphi^6$ kink \\eqref{eq:phi6_kink}:\n\\begin{equation}\\label{eq:deformed_phi6_kink}\n\\varphi_{\\rm K}^{(1)}(x) = \\mbox{arsinh}\\:\\sqrt{\\frac{1+\\tanh x}{2}}.\n\\end{equation}\nIn the next section, we will discuss the stability potential of the sinh-deformed $\\varphi^6$ kink \\eqref{eq:deformed_phi6_kink}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Stability potential of the sinh-deformed $\\varphi^6$ kink}\n\nFirst of all, the stability potential of a kink can be obtained by adding a small perturbation to the static kink. Then the linearized equation of motion leads to the Sturm-Liouville problem\n\\begin{equation}\\label{eq:stat_Schr}\n\\left[-\\frac{d^2}{dx^2} + U(x)\\right]\\psi(x) = \\omega^2\\psi(x),\n\\end{equation}\nsee, e.g., \\cite[Sec.~2]{Bazeia.EPJC.2018} for more details. The function\n\\begin{equation}\\label{eq:Schr_pot}\nU(x) = \\left.\\frac{d^2 V^{(1)}}{d\\varphi^2}\\right|_{\\varphi_{\\rm K}^{(1)}(x)}=\\frac{30 \\tanh x + 2\\:\\text{sech}^4 x - \\left(16\\tanh x + 45\\right) \\text{sech}^2x + 34}{\\left(\\tanh x + 3\\right)^2}\n\\end{equation}\nis the stability potential, which defines the kink's excitation spectrum. The structure of the discrete part of the spectrum is very important for various processes with kinks. The spectrum always has zero level, see, e.g., \\cite[Eqs.~(25), (26)]{Belendryasova.CNSNS.2019}, and all eigenvalues of the problem \\eqref{eq:stat_Schr} are non-negative \\cite[Eqs.~(2.20), (2.21)]{Bazeia.EPJC.2018}.\n\nIt is known that the stability potential of the $\\varphi^6$ kink does not have vibrational modes. We found that the stability potential \\eqref{eq:Schr_pot} of the sinh-deformed $\\varphi^6$ kink also does not have vibrational modes. It would seem that it can be assumed that resonance phenomena associated with resonant energy exchange between zero and vibrational modes are impossible in the kink-antikink collisions in both models. However, this is not quite true. In the sinh-deformed $\\varphi^6$ antikink-kink collisions we observe escape windows, see Fig.~\\ref{fig:a0a_bounce_collision}.\n\\begin{figure}[t!]\n\\begin{center}\n  \\centering\n  \\subfigure[two-bounce escape window, the initial velocity is 0.0456]{\\includegraphics[width=0.49\\textwidth]{Fa0aV00456.pdf}\\label{fig:phi_a0a_00456}}\n    \\subfigure[three-bounce escape window, the initial velocity is 0.0441]{\\includegraphics[width=0.49\\textwidth]{Fa0aV00441.pdf}\\label{fig:phi_a0a_00441}}\n  \\caption{Examples of escape windows in the antikink-kink scattering. The initial positions of the kinks are $\\pm 20$.}\n  \\label{fig:a0a_bounce_collision}\n\\end{center}\n\\end{figure}\nAppearance of the escape windows indicates the presence of resonant energy exchange between kinetic energy and (at least one) vibrational mode. The question is where is this vibrational mode hiding?\n\nThe answer can be in that the kinks of both models are asymmetric. This leads to the asymmetric stability potentials with different asymptotic values at $x\\to\\pm\\infty$, see Fig.~\\ref{fig:QMPkink}.\n\\begin{figure}[t!]\n\\begin{center}\n  \\centering\n  \\subfigure[]{\\includegraphics[width=0.49\\textwidth]{QMPkink.pdf}\\label{fig:QMPkink}}\n    \\subfigure[]{\\includegraphics[width=0.49\\textwidth]{QMPkinkantikink.pdf}\\label{fig:QMPkinkantikink}}\n  \\caption{Stability potential for (a) kink and (b) `antikink+kink' configuration with kink and antikink at $x=\\pm 10$, i.e.\\ $x_0^{}=10$ in Eq.~\\eqref{eq:Schr_pot_kk}.}\n  \\label{fig:QMP}\n\\end{center}\n\\end{figure}\nSuch asymmetry means that closely placed kink and antikink can form a mutual stability potential\n\\begin{equation}\\label{eq:Schr_pot_kk}\nU(x) = \\left.\\frac{d^2 V^{(1)}}{d\\varphi^2}\\right|_{\\varphi_{\\rm \\bar K}^{(1)}(x+x_0^{})+\\varphi_{\\rm K}^{(1)}(x-x_0^{})}\n\\end{equation}\nin the form of a potential well, Fig.~\\ref{fig:QMPkinkantikink}, in which, in addition to the zero level, there will also be levels of the discrete spectrum (vibrational modes). (For the sake of convenience, in Fig.~\\ref{fig:QMPkinkantikink} we used $x_0^{}=10$, which is not small.) Our idea is that the situation can be similar to that observed for the $\\varphi^6$ kinks in \\cite{Dorey.PRL.2011} or for the $\\varphi^8$ kinks in \\cite{Belendryasova.CNSNS.2019}.\n\nNote that preliminary results show presence of vibrational modes in the potential well of Fig.~\\ref{fig:QMPkinkantikink}.\n\n\n\n\n\n\\section{Conclusion}\n\nWe have studied the sinh-deformed $\\varphi^6$ model which is obtained from the well-known $\\varphi^6$ scalar field theory. We have shown that in this new model there are no vibrational modes in the kink excitation spectrum. At the same time, in our numerical simulations of collisions of the sinh-deformed $\\varphi^6$ kink and antikink at some initial velocities we observed resonance phenomena --- escape windows. We suppose that these resonance phenomena may be a consequence of the resonant energy exchange between the translational modes of kinks (their kinetic energy) and the vibrational modes of the `antikink+kink' system as a whole. A more detailed study of this issue is of great importance and is planned for the near future.\n\n\n\n\n\\section*{Acknowledgements}\n\nV.A.G.\\ acknowledges the support of the Russian Foundation for Basic Research under Grant No.\\ 19-02-00930.\n\nA.M.M.\\ thanks the Islamic Azad University, Quchan Branch, Iran (IAUQ) for their financial support under the Grant.\n\nThe work of the MEPhI group was supported by the MEPhI Academic Excellence Project.\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nMany engineering design problems can be cast as optimization problems of the form\n\\begin{equation}\n\t\\vc x^* = \\argmin_{\\vc x \\in \\mc X} \\eta(\\vc x),\n\t\\label{eqn:optimization}\n\\end{equation}\nwhere $\\vc x \\in \\mc X$ is a $d_x$-dimensional vector of design and operation parameters, and $\\eta$ is some quantity of interest, such as the strength of a material or the efficiency of an engine.\nSolving optimization problems that fit under the scope of Eq.\\ (\\ref{eqn:optimization}) can be challenging if the dimension $d_x$ of the search space is high and\/or the function $\\eta$ is nonconvex and highly rugged, resulting in an intractably large number of potential solutions that may be considered.\nFurthermore, if it is expensive to evaluate $\\eta$ (corresponding to performing a laboratory experiment or running a complex simulation code), one may be limited by their experimental budget and be forced to try and determine a good solution to Eq.\\ (\\ref{eqn:optimization}) with few opportunities to gain information about the true response surface for the system of interest.\n\nThe most elementary approaches for solving Eq.\\ (\\ref{eqn:optimization}) involve specifying some space-filling design and querying $\\eta$ at all of the design points.\nWhile rudimentary designs such as factorial designs suffer from the curse of dimensionality and are intractable for all but the simplest problems, more advanced space-filling designs such as Latin hypercube designs still may require far too many evaluations than is feasible.\nImproving on these by orders of magnitude, in Bayesian optimization (BO), one defines a Bayesian metamodel $y(\\vc x)$ [such as a Gaussian process (GP)] to approximate $\\eta$ and uses it to adaptively select promising designs until the optimum is found or an experimental budget is exhausted.\nStill, such an approach commonly takes tens to hundreds of evaluations to converge to a satisfactory degree in practice.\nIn this work, we are interested in improving on this by another order of magnitude, finding plausible solutions to Eq.\\ (\\ref{eqn:optimization}) with a handful--or \\textit{no}---evaluations of $\\eta$.\n\nHumans intuitively solve optimization problems in their daily lives in novel settings by leveraging knowledge about previous related experiences.\nBy contrast, it is common to start quantitative engineering design problems from scratch since it is unclear how to re-use data from legacy systems with different response surfaces.\nThere is a growing literature around methods that seek to devise suitable models to bridge the gap between these disparate sources of information.\nThrough this, we hope to leverage so-called ``legacy'' data about previously seen tasks in a manner that endows our current effort with strong, but reasonable inductive biases that guide it towards effective solutions.\n\nThe main technical challenge is to devise a means of reasoning in a quantitative way about features of data that are not numerical in nature and therefore not suitable for standard modeling approaches\\footnote{Some literature refer to these as ``qualitative'' features, though this seems somewhat of a misnomer since certain types of attributes in question can be numerical in nature, such as zip codes, yet are clearly unsuitable to treat as numbers in a model; others may not be quantitative, but are nonetheless precise (e.g.\\ the name of an operator), yet ``qualitative'' does not convey this preciseness.}\nThese ``general'' features are typically found when describing the difference between tasks \\cite{argyriou2007multi} and could include, for example, the serial number of a machine, the identity of an operator, or a chemical compound involved in some process of interest.\n\nThe key to our approach is to learn a probabilistic embedding associated with the general features associated with a system such that notions of similarity can be quantified and utilized by a downstream data-driven model.\nMinding our ultimate goal of solving optimization problems, we focus on Gaussian process metamodels and call the composition of our probabilistic embedding with the Gaussian process metamodel ``Bayesian embedding GP'' (BEGP).\nThe contributions of this work are the following:\n\\begin{enumerate}\n\t\\item We define the structure of the BEGP metamodel designed to fuse information from systems with differing general features.  \n\tWe use a variational inference scheme to learn to infer reasonable probabilistic embeddings of the general features that capture uncertainty due to limited data while showing that the compositional model can be recognized as a deep Gaussian process \\cite{damianou2013deep} with a particular choice of kernel function.\n\t\\item We explain and demonstrate the application of this model to the task of Bayesian optimization, showing how the BEGP can be used to satisfy the usual requirements of the algorithm.\n\t\\item We conduct a series of computational experiments on a variety of synthetic and real-world systems to illustrate the usage of our approach and compare its performance to existing methods and evaluating the contribution of various components of the metamodel.\n\\end{enumerate}\n\nThe scope of our work here is optimization problems of the form in Eq.\\ (\\ref{eqn:optimization}).\nHowever, because our approach can be used as a drop-in replacement for other Bayesian metamodels, it is straightforward to extend our work to cases including multi-objective optimization and problems involving complex or unknown constraints.\nWe also consider regression tasks as a stepping stone to our ultimate application of interest since satisfactory predictive power is a desired preliminary skill.\n\nThe remainder of this paper is organized as follows:\nIn section \\ref{sec:methodology}, we explain the formulation of our metamodel and its usage within Bayesian optimization\nIn section \\ref{sec:related_work}, we review related approaches and results from the literature.\nIn section \\ref{sec:examples}, we demonstrate our approach on a variety of synthetic and real-world examples.\nIn section \\ref{sec:conclusion}, we offer concluding remarks.\n\n\\section{Methodology}\n\\label{sec:methodology}\nIn this section, we describe the methodology including our model definition, training objective, predictive distribution, and application to Bayesian optimization.\n\n\\subsection{Model definition}\nWe begin the discussion of our methodology by defining our model, explaining how inference may be done to determine its posterior, and how predictions are computed.\n\n\\subsubsection{Gaussian process regression}\nWe begin by reviewing Gaussian processes for regression; the interested reader may consult \\cite{rasmussen2006gaussian} for more details.\nA Gaussian process (GP) $\\mc{GP}(\\mu(\\cdot), k(\\cdot, \\cdot))$ with mean function $\\mu$ and kernel $k$ is a distribution over functions such that the marginal distribution of the random function over a finite index set is a multivariate Gaussian.\nConcretely, let $f: \\mc X \\rightarrow \\reals$ be described by a GP where $\\mc X$ is an input space that indexes the random process; traditionally, this will be $d_x$-dimensional Euclidean space $\\reals^{d_x}$ or some subset therein.\nHowever, any (infinite) set of inputs might be used, and we are mindful of this potential generality in the review in this section.\nGiven $n$ inputs $\\mtx X \\in \\scriptX^n$ with corresponding outputs $\\vc f \\in \\reals^n$, we write the joint probability density of $\\vc f$ as\n\\begin{equation}\n\tp(\\vc f | \\mtx X) = \\G{\\vc f}{\\vc \\mu}{\\kff},\n\t\\label{eqn:pf}\n\\end{equation}\nwhere $\\mu_i = \\mu(\\vc x_i)$, and $(\\kff)_{ij} = k(\\vc x_i, \\vc x_j)$.\nThe quantity $\\vc f$ is recognized as the latent output of the GP model.\nNext, we define a Gaussian likelihood \n\\begin{equation}\n\tp(\\vc y|\\vc f) = \\G{\\vc y}{\\vc f}{\\sigma_y^2 \\mtx I_{n \\times n}},\n\t\\label{eqn:likelihood}\n\\end{equation}\nwhere $\\vc y \\in \\reals^n$ denotes the observed output values.\nIntegrating out $\\vc f$ results in the familiar marginal likelihood\n\\begin{equation}\n\tp(\\vc y | \\mtx X) = \\G{\\vc y}{\\vc \\mu}{\\kyy},\n\t\\label{eqn:gp:marginal_likelihood}\n\\end{equation}\nwhere $\\kyy = \\kff + \\sigma_y^2 \\mtx I_{n \\times n}$.\nThe negative logarithm of this quantity is conventionally used to determine appropriate parameters for the mean and kernel functions as well as the likelihood through gradient-based minimization. \nAlternatively, approximate Bayesian inference of the model parameters may be done as well using Markov chain Monte Carlo or variational inference once suitable priors are defined \\cite{bilionis2013multi}.\n\nGiven a training set $\\scriptD = \\{\\mtx X, \\vc y\\} \\in \\scriptX^n \\times \\reals^n$, predictions are made by using Bayes' rule to condition the GP on the training set.\nThe resulting predictive distribution over the latent outputs $\\vc f^* \\in \\reals^{n^*}$ at some test inputs $\\mtx X^* \\in \\scriptX^{n^*}$ is\n\\begin{equation}\n\tp(\\vc f^* | \\mtx X^*, \\scriptD) = \\G{\\vc f^*}{\\ksf \\kyyinv (\\vc y - \\vc \\mu) + \\vc \\mu^*}{\\kss - \\ksf \\kyyinv \\kfs},\n\t\\label{eqn:qf}\n\\end{equation}\nwhere\n\\begin{align}\n\t(\\kfs)_{i, j} &= k(\\vc x_i, \\vc x_j^*),\n\t\\\\\n\t(\\kss)_{i, j} &= k(\\vc x_i^*, \\vc x_j^*),\t\n\\end{align}\n$\\ksf = \\kfs^\\transpose$, and $\\mu_i^* = \\mu(\\vc x_i^*)$.\nApplying the likelihood and marginalizing out $\\vc f^*$ gives the posterior in output space\n\\begin{equation}\n\tp(\\vc y^* | \\mtx X^*, \\scriptD) = \\G{\\vc y^*}{\\ksf \\kyyinv (\\vc y - \\vc \\mu) + \\vc \\mu^*}{\\kss - \\ksf \\kyyinv \\kfs + \\sigma_y^2 \\mtx I_{n^* \\times n^*}}.\n\t\\label{eqn:qy}\n\\end{equation}\n\n\\textbf{Remark:}\nOur use of a likelihood term in our model definition generalizes the concept of a ``nugget'' or ``jitter'' that is sometimes used within the GP metamodeling community \\cite{bilionis2012multi, bilionis2013multi}, usually with the stated purpose of either improving the conditioning of the kernel matrix or representing the noisiness in the observations being modeled.\n\n\\subsubsection{Incorporating general input features as inputs to Gaussian process regression models}\nThe main source of the rich behavior in GP models is their kernel function, which encodes strong inductive biases about the statistics of the functions being modeled.\nIt is most common to consider parametric kernels of the form \n$k(\\cdot, \\cdot; \\vc \\theta_k) : \\reals^{d_x} \\times \\reals^{d_x} \\rightarrow \\reals$.\nFor example, the popular exponentiated quadratic kernel is defined as\n\\begin{equation}\n\tk(\\vc x, \\vc x'; \\vc \\theta_k) = \\sigma_k^2 \\exp \\left[ - \\sum_{i=1}^{d_x} \\left(\\frac{x_i - x_i'}{l_i} \\right)^2 \\right],\n\t\\label{eqn:exponentiated-quadratic}\n\\end{equation}\nand can be evaluated easily for any pair of $d_x$-dimensional vectors $\\vc x$ and $\\vc x'$.\n\nWhile this is suitable for real-valued inputs commonly found in engineering problems, it is not suitable for more general inputs such as those mentioned above.\nOne workaround is to embed elements from general input spaces as $d_z$-dimensional latent variables; one can then form the complete input vector by concatenating these latents $\\vc z$ with the real-valued inputs $\\vc x^{(r)}$ to form a $d_{x^{(r)}} + d_z$-dimensional input vector amenable to computation with traditional parametric kernels such as Eq.\\ (\\ref{eqn:exponentiated-quadratic}).\n\nMore general nonparametric positive-definite kernels exist that are valid for Gaussian processes.\nFor example, the white noise kernel function,\n\\begin{equation}\n\tk_{white}(\\vc x, \\vc x'; \\vc \\theta) = \n\t\\begin{cases} \n\t\t\\sigma_{white}^2 & \\textrm{if} ~ \\vc x = \\vc x' \\\\\n\t\t0 & \\textrm{otherwise}\n\t\\end{cases},\n\\end{equation}\ndoes not require Euclidean inputs; instead, it merely requires that we are able to distinguish between elements in its index set.\n\nTherefore, to accomplish the embedding mentioned above, we map our general inputs $\\vc x^{(g)} \\in \\scriptX_g$ through a $d_z$-variate Gaussian process with white noise kernel:\n\\begin{equation}\n\t\\vc z_i \\sim \\mc{GP}\\left(0; k_{white}(\\cdot, \\cdot)\\right) : \\scriptX_g \\rightarrow \\reals, ~i=1, \\dots, d_z.\t\n\\end{equation}\nSampling this GP at some general input returns potential embeddings of the input as a $d_z$-dimensional vector in Euclidean space.\nThese latent variables may then be concatenated with the numerical feature space and fed through a second Gaussian process that serves as the familiar regression model employed in Bayesian metamodeling.\nThe Bayesian embedding GP (BEGP) model that combines the embedding Gaussian process with the GP regression model is shown in Fig.\\ \\ref{fig:egp}.\nThis model is composed of two GPs, where the outputs of the first (embedding) GP are fed as inputs to the second (regression) GP, making it a deep Gaussian process with serial topology similar to those originally discussed in \\cite{damianou2013deep}.\n\\begin{figure}[hbt]\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{fig1.pdf}\n\t\\caption{\n\t\tProbabilistic graphical model of the Bayesian embedding Gaussian process (BEGP) model.\n\t\tGeneral inputs $\\vc x^{(g)}$ are first mapped through a GP with white noise kernel to latent variables $\\vc z$.\n\t\tThese latents are concatenated with real-valued inputs $\\vc x^{(r)}$ to form the full inputs $\\vc x$ to the second GP model with a traditional parametric mean function and kernel.\n\t\tWhite nodes denote variables that are unobserved, and shaded nodes are observed.\n\t\tPoint nodes are deterministic, and large nodes are associated with probability distributions.\n\t\tModel parameters associated with mean functions, kernels, and the likelihood are hidden for clarity.\n\t}\n\t\\label{fig:egp}\n\\end{figure}\n\nOur modeling approach may be thought of as a type of \\textit{multi-task learning} in that one doing regression traditionally segregates datasets along differences in general features, building a separate model for each dataset using the remaining real-valued features; by contrast, here we build in correlations across general features within a single model, simultaneously improving the predictive power across all of the input space.\n\n\\textbf{Remark:} for systems with multi-dimensional general inputs (e.g.\\ operator IDs \\textit{and} machine serial numbers), it is straightforward to extend our approach to embed each general input dimension separately to its own latent space; these latents are then all concatenated with each other to define the total latent variable which is then concatenated with $\\vc x^{(r)}$ as before.\n\n\\subsubsection{Inference for the Bayesian embedding GP model}\nHaving defined our model, we now discuss how to do inference on it.\nInference for the Bayesian EGP model presents two challenges.\nFirst, inferring the posterior of the embedding layer is challenging due to its nonparameteric nature stemming from the white noise kernel combined with the fact that its posterior will generally be non-Gaussian due to the nonlinearity of the downstream model that operates on it. \nSecond, unlike a traditional Gaussian process regression model, it is analytically intractable to marginalize out the latent variables $\\vc z$.\nIn this section, we discuss simple strategies for overcoming these two challenges so that we may devise an objective function for training our model.\n\nFirst, we consider the challenges associated with the embedding layer.\nLet $n_g$ denote the number of distinct general inputs that we observe from $\\scriptX_g$ (i.e.\\ the number of legacy datasets that we would like to incorporate in our model).\nDue to the kernel, the joint prior density over latents $\\mtx Z \\in \\reals^{n_g \\times d_z}$ associated with general inputs $\\mtx X^{(g)} \\in \\scriptX_g^{n_g}$ is\n\\begin{equation}\n\tp \\left( \\mtx Z | \\mtx X^{(g)} \\right) = \\prod_{i=1}^n \\prod_{j=1}^{d_z} \\G{z_{ij}}{0}{\\sigma_w^2},\n\t\\label{eqn:pz}\n\\end{equation}\nwhere $z_{ij}$ is the $j$-th dimension associated with the latent for input $i$.\nFor reasons analogous to those mentioned in previous literature \\cite{titsias2010bayesian, damianou2013deep, salimbeni2017doubly}, the posterior over these inputs is generally non-Gaussian.\nWe define a mean-field variational posterior $q(\\mtx Z)$ to approximate the true posterior $p(\\mtx Z | \\scriptD)$ with the form\n\\begin{equation}\n\tq(\\mtx Z) = \\prod_{i=1}^{n_g} \\prod_{j=1}^{d_z}  \\G{z_{ij}}{m_{ij}}{s_{ij}},\n\t\\label{eqn:qz}\n\\end{equation}\nwhere $\\mtx M, \\mtx S \\in \\reals^{n_g \\times d_z}$ are variational parameters.\nFortunately, one is usually only interested in a relatively small number of inputs in $\\scriptX_g$, so it is not a challenge to track these variational parameters.\nIn fact, computational challenges associated with data scalability of Gaussian process models are likely to become problematic well before challenges associated with working with a high number of tasks, even though recent advances have make significant progress in lift traditional barriers to Gaussian process modeling in large-data regimes \n\\cite{snelson2006sparse, titsias2009variational, hensman2013gaussian, wilson2015kernel, salimbeni2017doubly, wang2019exact}.\n\nLastly, note that if no data have been observed for some held-out general input $\\vc x^{(g),*}$, then, by the nature of the white noise kernel, the posterior over the associated latent is equal to the prior.\nthis observation is critical for enabling our model to make credible predictions in a zero-shot setting, where no data about some task of interest is available at the outset of optimization.\nIndeed, we will see that this probabilistic approach enables our model to make surprisingly good inferences to guide the first iteration of Bayesian optimization in such settings.\n\nHaving resolved the challenge of representing the posterior of the embedding layer, we now turn our attention to the second challenge regarding the intractability of the model's marginal log-likelihood (or evidence), which usually serves as the objective function for training probabilistic models.\nAgain, following a variational strategy, we derive a tractable evidence lower bound (ELBO) that can be used in place of the exact marginal log-likelihood.\nWhile this approach was first demonstrated for Gaussian processes using a sparse approach based on inducing variables, it turns out that later advances in variational inference \\cite{ranganath2014black} allow us to avoid relying on a sparse GP formulation.\nThat said, we recognize that such an approach may realistically be helpful in the plausible cases where an abundance of legacy data takes the problem into a large-data setting.\nThis is in contrast to the usual assumptions of data-scarcity in engineering metamodeling, which are usually myopically focused only on solving a single task in isolation.\nFor the sake of completeness, we now provide the derivation of the ELBO for the BEGP model.\nBased on the probabilistic graphical model shown in Fig.\\ \\ref{fig:egp}, the joint probability of our model with $n$ observations is\n\\begin{equation}\n\t\\mc P = p(\\mtx Z, \\vc f, \\vc y | \\mtx X^{(r)}, \\mtx X^{(g)}) = p(\\mtx Z | \\mtx X^{(g)}) p(\\vc f | \\mtx Z, \\mtx X^{(r)}) p(\\vc y | \\vc f),\n\\end{equation}\nwhere p$(\\mtx Z | \\mtx X^{(r)})$ is given by Eq.\\ (\\ref{eqn:pz}), $p(\\vc f | \\mtx Z, \\mtx X^{(r)})$ is analogous to Eq.\\ (\\ref{eqn:pf}), and $p(\\vc y | \\vc f)$ is the likelihood of Eq.\\ (\\ref{eqn:likelihood}).\nand the marginal log-likelihood is obtained by simply integrating over the latent variables and taking the logarithm of the result:\n\\begin{equation}\n\t\\log p(\\vc y | \\mtx X^{(r)}, \\mtx X^{(g)}) = \\log \\int \\mc P d \\vc f d \\mtx Z.\n\\end{equation}\nFrom this point onwards, we will omit the conditioning on the inputs for brevity.\nWhile we can integrate out $\\vc f$ due to the conjugacy of the likelihood, $\\mtx Z$ cannot be integrated out analytically.\nWe multiply and divide the integrand by $q(\\mtx Z)$ to obtain\n\\begin{equation}\n\t\\log p(\\vc y) = \\log \\int q(\\mtx Z) \\frac{p(\\vc y, \\mtx Z)}{q(\\mtx Z)} d \\mtx Z.\n\\end{equation}\nApplying Jensen's equality, we move the logarithm inside the integrand to get\n\\begin{equation}\n\\log p(\\vc y) \\ge ELBO = \\int q(\\mtx Z) \\log \\frac{p(\\vc y, \\mtx Z)}{q(\\mtx Z)} d \\mtx Z.\n\\end{equation}\nRearranging and using $p(\\vc y, \\mtx Z) = p(\\vc y | \\mtx Z) p(\\mtx Z)$ gives\n\\begin{equation}\n\tELBO = \\int q(\\mtx Z) \\left( \\log p(\\vc y | \\mtx Z) + \\log p(\\mtx Z) - q(\\mtx Z) \\right) d \\mtx Z.\n\t\\label{eqn:elbo}\n\\end{equation}\nAdditional progress may be made at this point by restricting our choice of kernel function for the second GP layer to either an exponentiated quadratic or a linear kernel since the kernel expectations that form the crux of the challenge may then be evaluated analytically.\nFurthermore, under the special case where the same real-valued input points are sampled for each general input, Atkinson and Zabaras \\cite{atkinson2018structured, atkinson2019structured} showed that the kernel expectations can be further broken down into a Kronecker product between a deterministic kernel matrix and a kernel expectation enabling extensions to modeling millions of data.\nIn this work, we instead opt to estimate Eq.\\ (\\ref{eqn:elbo}) through sampling so as to maintain the ability to leverage arbitrary kernel functions.\n\nThe model is trained by maximizing Eq.\\ (\\ref{eqn:elbo}) with respect to the parameters of the variational posteriors as well as the kernel hyperparameters and likelihood variance $\\sigma_y^2$ through gradient-based methods.\nWe find that it is typically sufficient to approximate the integral with a single sample from $q(\\mtx Z)$ at each iteration of the optimization.\n\nNote that there is a redundancy in defining the scale of the latent space in that our model possesses both a parameter $\\sigma_w^2$ for the prior variance of the embedding layer as well as length scales for the parametric kernel of the GP regression layer.\nGiven this freedom, one can make the ELBO arbitrarily high by bringing $\\sigma_w^2$ towards zero while scaling the variational parameters of $q(\\mtx Z)$ and the length scales of the regression kernel correspondingly without changing the predictions of the model.\nto avoid this pathology and establish the scale of the latent space, we fix $\\sigma_w^2 = 1$.\n\nWe choose to learn point estimates of the other model parameters since they are informed by all of the tasks' data.\nTherefore, the uncertainty contributed to the predictions due to uncertainty in these parameters will generally be small.\nThis is again only because we are incorporating a large amount of data into our model by leveraging legacy data; in the typical single-task engineering setting, we might expect to have far fewer data, and Bayesian inference over the parameters may provide value to the model.\n\n\\paragraph{A remark on variational inference for EGP}\nA reasonable alternative to our variational inference approach is to instead perform inference using Monte Carlo.\nWe favor VI because we need to update the model posteriors repeatedly during Bayesian optimization as new data are added to the model.\nThis is simple to do with VI since we can continue our gradient-based optimization from the previous variational posterior.\nBy contrast, Monte Carlo requires us to run the chain for almost as long as the initial chain every time that we need to update the model.\nThus, while there is not such a clear advantage for one method compared to the other when initially training the model on legacy data, VI quickly becomes more attractive in the following steps.\nFurthermore, we note that updating the model parameters is rather important since we are focusing on extreme low-data regimes (fewer than 10 data from the task of interest), where each new datum will generally have considerable effects on the posterior.\n\nAnother reason for using MC inference is because highly-factorized variational posterior distributions are known to underestimate the entropy of the true posterior when there is mismatch between the variational form and the true posterior \\cite{bishop2006pattern}.\nOne might reasonably worry that we are neglecting important uncertainty information by using a mean-field posterior for $q(\\mtx z)$.\nWe find in practice that this approximation successfully captures much of the uncertainty and is sufficient to provide competitive or superior predictions compared to the baselines considered in our examples.\nHowever, our method is not incompatible with more powerful VI methods such as multivariate Gaussians with full covariance matrices \\cite{atkinson2019structured} or normalizing flow-based approaches \\cite{kingma2016improved}.\n\n\\subsubsection{Predictions with the model}\nGiven a trained model, we are now interested in making predictions at held-out inputs in $\\scriptX_r \\times \\scriptX_g$.\nConsider a set of test inputs $\\mtx X^* \\in \\scriptX_g^{n_g^*} \\times \\scriptX_r^{n_r^*}$ with $n_g^*$ distinct general inputs and $n_r^*$ distinct real-valued inputs, combined to form $n^*$ test inputs in all.\nWe first apply the embedding layer $\\mtx X^{(g),*}$ to obtain a sample of the latents $\\mtx Z^* \\in \\reals^{n_g^* \\times d_z}$ as well as the training data's latents $\\mtx Z$.\nThese latents are expanded to form the full training and test inputs for the regression model, \n$\\{\\mtx X, \\mtx X^*\\} = \\left\\{ [\\mtx X^{(r)}, \\mtx Z], [\\mtx X^{(r),*}, \\mtx Z^*] \\right\\} \\in \\reals^{n \\times d_x} \\times \\reals^{n^* \\times d_x}$,\nwhere $d_x = d_{x^{(r)}} + d_z$.\nGiven these samples we can compute the conditional predictive distributions over the latent and observed outputs, given by Eq.\\ (\\ref{eqn:qf}) and (\\ref{eqn:qy}), respectively.\n\nDue to the nonlinearity of the GP model, marginalizing over latent variables' variational posterior $q(\\mtx Z, \\mtx Z^*)$ generally induces a non-Gaussian predictive distribution.\nHowever, given a Gaussian variational posterior, the moments of the marginal predictive distribution admit analytic expressions; previous works \\cite{girard2003gaussian, titsias2010bayesian} have approximated the predictive distribution as Gaussian with these statistics.\nAlternatively, one may sample the predictive distribution in order to better resolve its details.\nWe utilize the former approach in our examples.\n\n\\subsection{Bayesian optimization}\nHaving discussed the formulation of our model, including training and predictions, we now turn our attention to using it in the context of optimization.\nThe main algorithm for Bayesian optimization is given in Algorithm \\ref{alg:BO}.\nWe restrict ourselves to the task of optimizing subject to a fixed general input $\\vc x_g^*$, though our method permits searching over multiple general inputs with trivial modification.\n\n\\begin{algorithm}\n\t\\caption{Bayesian optimization}\n\t\\label{alg:BO}\n\t\\textbf{Require:} \n\tTraining data $\\scriptD$, \n\tgeneral input of interest $\\vc x_g^*$, \n\tacquisition function $a(\\cdot)$, \n\tcomputational budget $n^*$.\n\t\n\t\\textbf{Ensure:} optimal design $\\vc x^{(r),*}$\n\t\n\t\\begin{algorithmic}[1]\n\t\t\t\\State $\\scriptM \\rightarrow BEGP(\\scriptD)$.\n\t\t\t\\State $n \\rightarrow 0$, $\\scriptD^* = \\emptyset$.\n\t\t\t\\For{$i=1, \\dots, n^*$}\n\t\t\t\t\\State Train $\\scriptM$ on $\\scriptD$.\n\t\t\t\t\\State $\\vc x_{next}^{(r)} \\rightarrow \\argmin_{\\vc x^{(r)} \\in \\mc X_r} a(\\vc x^{(r)})$.\n\t\t\t\t\\State $y_{next} \\rightarrow \\eta(\\vc x_g^*, \\vc x_{next}^{(r)})$, $n \\rightarrow n + 1$.\n\t\t\t\t\\State $\\scriptD \\rightarrow \\scriptD \\cup \\{ (\\vc x_g^*, \\vc x_{next}^{(r)}, y_{next}) \\}$\n\t\t\t\t\\State $\\scriptD^* \\rightarrow \\scriptD^* \\cup \\{ (\\vc x_g^*, \\vc x_{next}^{(r)}, y_{next}) \\}$\n\t\t\t\\EndFor\n\t\t\t\\Return $\\vc x^{(r),*} = \\argmin_{x^{(r)} \\in \\scriptD^*} y(\\vc x^{(r)})$.\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Acquisition functions}\nOne ingredient that must be specified for BO is the acquisition function $a$, which is used to steer the selection of points at which one conducts experiments to evaluate $\\eta$.\nIn this work, for systems in which we can evaluate $\\eta$ at any location in $\\scriptX_r$ for the current task $\\vc x_g^*$, we consider the expected improvement\n\\begin{equation}\nEI(\\vc x) = \\expectation{p(y | \\vc x, \\scriptD)}{(y(\\vc x) - y_{min})\\Theta(y_{min} - y(\\vc x))},\n\\label{eqn:expected_improvement}\n\\end{equation}\nwhere $\\Theta(\\cdot)$ is the Heaviside theta function and $y_{min}$ is the value of the current best design.\nWhen $p(y|\\vc x, \\scriptD)$ has a known form such as a Gaussian, evaluation of Eq.\\ (\\ref{eqn:expected_improvement}) can be done cheaply.\nNotice as a matter of convention that we have defined $a$ in Eq.\\ (\\ref{eqn:expected_improvement}) such that lower values are better.\n\nFinally, the traditional approach to maximizing the acquisition function has been to simple select a large random sampling of $\\mc X_r$, evaluate $a$ on all of the samples (which is generally cheap), then select the best sample.\nHowever, following the widespread adoption of computational frameworks supporting automatic differentiation such as TensorFlow \\cite{abadi2016tensorflow} and PyTorch \\cite{paszke2017automatic}, it has become customary to accelerate the inner loop optimization over $a$ using gradient-based methods by simply backpropagating the acquisition function back through the model to obtain its gradient with respect to the inputs $\\vc x^{(r)}$ \\cite{snoek2012practical}.\nOther recent work has investigated this matter more broadly \\cite{zhang2019finding}.\nHere, we use gradient-based search with restarts to find the next point to select similarly to \\cite{snoek2012practical}.\nFor the case where one must estimate $a$ through sampling (e.g.\\ due to estimating the predictive posterior by sampling latents from $q(\\mtx Z)$, stochastic backpropagation \\cite{rezende2014stochastic} can be used to deal with the variance in the estimates of $a$ due to sampling.\nWe also note that this setup may be applied immediately to solve \\textit{robust} optimization problems with no modification by additionally sampling from any uncontrolled inputs.\n\nIn our real-world examples, we do not have access to the real-world data-generating process and must instead work with a static, pre-computed dataset.\nIn this case, our design space is practically the finite set at which experiments have already been carried out.\nIn this case, we can instead directly estimate the negative\\footnote{We take the negative probability to keep with the convention that lower values of $a$ are better.} probability that a given available point will be the best design:\n\\begin{equation}\n\ta(\\vc x_i^{(r)}) = \\prod_{j \\not = i} P\\left(y(\\vc x_i^{(r)}, \\vc x^{(g), *}) < y(\\vc x_j^{(r)}, \\vc x^{(g), *}) \\right).\n\t\\label{eqn:prob-best}\n\\end{equation}\nWe approximate Eq.\\ (\\ref{eqn:prob-best}) by repeatedly sampling the joint predictive distribution over all available design points and counting the frequency with which each design point is predicted to have the best output.\nWe find that this crude approximation is not overly computationally burdensome and results in good performance in practice as shown by our examples.\n\n\n\\section{Related work}\n\\label{sec:related_work}\nMcMillan et al.\\ \\cite{mcmillan1999analysis} study Gaussian process models based on ``qualitative'' and ``quantitative'' variables.\nThis approach would be analogous to a deterministic version of our Bayesian embedding layer.\nAs we show in our examples, the incorporation of Bayesian uncertainty estimates provides highly valuable information that is essential to making the predictive distribution of the model credible.\n\nThe Gaussian process latent variable model \\cite{lawrence2004gaussian} is a seminal work for inferring latent variables by using the marginal log-likelihood of a Gaussian process as training signal.\nThe later Bayesian extension by Titsias and Lawrence \\cite{titsias2010bayesian} was found to significantly improve the quality of the model and gives compelling evidence in favor of modeling latent variables in a Bayesian manner.\n\nThe task of inferring input-output relationships where the inputs are only partially-specified was first identified by Damianou and Lawrence as ``semi-described learning'' \\cite{damianou2015semi}.\nOur problem statement may be regarded as similar in that we choose to associate the general features in our data with latent variables that are unspecified \\textit{a priori}.\n\nUsing related systems to improve knowledge about a related system of interest has a long history in multi-fidelity modeling \\cite{kennedy2000predicting, forrester2007multi}, where one traditionally considers a sequence of systems that constitute successively more accurate (and expensive) representations of a true physical process.\nHowever, it is unclear how to define a ``sequence'' of legacy datasets that are all ``on equal footing'' (such as different operators or machines performing the same task), particularly as the number of such instances becomes large.\n\nOur model is similar to the multi-task Gaussian process introduced in \\cite{swersky2013multi}.\nOur approaches are similar in that we both learn a kernel over tasks; however, \\cite{swersky2013multi} require that the kernel matrix decompose as a Kronecker product between a task-wise covariance matrix and the input kernel matrix. \nThe Kronecker product kernel matrix can be equivalent to a kernel matrix evaluated on our augmented inputs in the special cases of if one uses either a linear or a Gaussian kernel, but more general kernels cannot be composed in this way.\nBy finding representations of tasks in a Euclidean input space, we lift this restriction, allowing us to use arbitrary kernels instead, wherein the Kronecker product decomposition of \\cite{swersky2013multi} is a special case.\nAdditionally, by posing a prior over the latent input space, our method also allows us to do principled zero-shot predictions on unseen inputs; it is significantly more challenging to formulate a reasonable prior on the kernel matrix itself.\nFinally, we can inject additional modeling bias into our model by selecting the dimensionality $d_z$ of our latent space.\nWhile a low-rank approximation to the task covariance matrix might be derived to obtain similar savings, it again seems more natural in our opinion to exercise this ability in a latent space.\n\nThe task of utilizing ``legacy'' data in order to improve the predictive capability on a related system of interest was explored in \\cite{ghosh2018bayesian}.\nHowever, the approach used in that work can only utilize legacy models through linear combinations, making extrapolation on the system of interest difficult, particularly when few (or no) data are available from the task of interest.\nAdditionally, their method requires a cross-validation scheme for model selection; by contrast, the probabilistic representation of tasks in the current method enables the use of Bayesian inference to guide model selection in the sense of identifying relevance between tasks.\n\nA recent work by Zhang et al.\\ \\cite{zhang2019bayesian} also studies Bayesian optimization in the context of qualitative and quantitative inputs.\nHowever, they learn point embeddings via maximum likelihood estimation rather than inferring a posterior through Bayesian inference; this makes the overall model prone to overfitting in practice and generally unsuitable for providing credible uncertainty estimates.\nThis also makes their model difficult to apply in few-shot settings, and impossible to apply in a principled way for zero-shot learning.\nA related work by Iyer et al.\\ \\cite{iyer2019data} identifies a pathology associated with the point estimate inference strategy and proposes an ad hoc workaround; a natural outcome of our Bayesian embedding approach is that this challenge vanishes.\n\nThe BEGP model can be thought of as a latent variable model \\cite{kingma2013auto, lawrence2004gaussian, titsias2010bayesian} and has similarities with previous works with latent variable GPs \\cite{dai2017efficient, atkinson2018structured, atkinson2019structured}, though those works additionally impose structure over latent variables as well as (potentially) the input training data.\nHowever, neither works were applied within the context of Bayesian optimization, nor did they identify that multiple general feature sets could be decomposed as we have chosen to do.\n\n\\section{Examples}\n\\label{sec:examples}\nIn this section, we demonstrate our method on a variety of synthetic and real-world examples.\nThe embedding GP model is implemented using \\texttt{gptorch}\\footnote{\\url{https:\/\/github.com\/cics-nd\/gptorch}}, a Gaussian process framework built on PyTorch \\cite{paszke2017automatic}.\nSource code for implementing the model and synthetic experiments described below can be found on GitHub\\footnote{\\url{https:\/\/github.com\/sdatkinson\/BEBO}}.\nFor each system, we consider both a regression and optimization problem using our embedding GP model with both probabilistic and deterministic embedding layers; the latter is achieved by replacing the posterior of Eq.\\ (\\ref{eqn:qz}) with a delta function at its mode.\nWe use an RBF kernel as in Eq.\\ (\\ref{eqn:exponentiated-quadratic}) and constant mean function whose value is determined via an MLE point estimate.\n\nWe compare our results against a vanilla Gaussian process metamodel that is unable to directly leverage the legacy data as a baseline.\nSince GP models typically fare very poorly the the extreme low-data regime that we are interested in, we conduct full Bayesian inference over the mean and kernel function parameters, using as prior a factorized Gaussian over the parameters\\footnote{Or the logarithm of the parameters that are constrained to be positive.} whose mean and variance are estimated by fitting GPs to each legacy task and using the statistics of the point estimates of the model parameters found therein.\nInference for the Bayesian GP is carried out using Hamiltonian Monte Carlo \\cite{duane1987hybrid} with the NUTS sampler \\cite{hoffman2014no} as implemented in Pyro \\cite{bingham2018pyro}.\n\n\\subsection{Systems}\nWe begin by describing the systems that we consider.\nEach system contains a set of tasks that we will jointly learn.\n\n\\subsubsection{Toy system}\n\\label{sec:examples:systems:toy}\nWe first consider a system with one-dimensional input and $\\Omega_x = [0, 1]$.\nThe set of response surfaces are given by\n\\begin{equation}\n\t\\eta(x, \\vc \\theta) = 0.1 * z^4 - z^2 + (2.0 + \\theta_2) \\sin(2z),\n\\end{equation}\nwhere we define $z = \\theta_1 + 4x - 4$.\nDifferent response surfaces are selected by sampling $\\vc \\theta \\sim \\mc U[0, 1]^2$.\nWe consider a problem setting where $5$ legacy tasks are available, each with $5$ data with inputs sampled uniformly from $\\Omega_x$.\nIn the regression setting, we randomly sample a number of data from one response surface as the current task and split it into a training and test set.\nWe quantify our prediction accuracy in terms of mean negative log probability (MNLP), mean absolute error (MAE), and root mean square error (RMSE).\nIn the optimization setting, we perform Bayesian optimization over the current task and track the best design point as a function of the number of evaluations on the current task.\nWe repeat both experiments with $10$ different random seeds to quantify the performance statistics of each metamodeling approach.\n\n\\subsubsection{Forrester function}\n\\label{sec:examples:systems:forrester}\nWe also consider a generalization of the Forrester function \\cite{forrester2007multi} to a many-task setting:\n\\begin{equation}\n\t\\eta(x; \\vc \\theta) = \\theta_1 * \\eta_{forrester}(x) + \\theta_2 * (x - 0.5) + \\theta_3,\n\t\\label{eqn:forrester-param}\n\\end{equation}\nwhere \n\\begin{equation}\n\t\\eta_{forrester}(x) = (6x-2)^2 \\sin(12x-4).\n\t\\label{eqn:forester-hi}\n\\end{equation}\nUnder the parameterization of Eq.\\ (\\ref{eqn:forrester-param}), the original ``high-fidelity'' function considered in \\cite{forrester2007multi} is obtained when $\\vc \\theta = (1, 0, 0)$, and the low-fidelity function corresponds to $\\vc \\theta = (0.5, 10, -5)$.\nWe always include the low-fidelity function in the legacy tasks along with other tasks generated by sampling \n$\\vc \\theta \\sim \\mc U[0, 1] \\times \\mc U[0, 10] \\times \\mc U[-5, 5]$.\nNote that this places the high-fidelity task at an extremum of the system parameter space, meaning that multi-task models will not necessarily benefit by assuming \\textit{a priori} that the held-out task lives at the center of latent space (assuming that their learned latent representation resembles that of the parameterization we have chosen).\nWe consider a design space $\\Omega_x = [0, 1]$.\nEach of the 5 legacy tasks includes 5 data where the input was sampled uniformly in $\\Omega_x$.\nWe consider a regression and an optimization with identical setup to the toy system in Sec.\\ \\ref{sec:examples:systems:toy}.\n\n\\subsubsection{Pump data}\n\\label{sec:examples:systems:pump}\nOur first real-world dataset comprises of data in which the performance of a pump is quantified in terms of seven design variables.\nOur data includes data from 6 pump families, each with of which has between $11$ and $55$ data.\nWe consider each pump family in turn as the ``current'' task while using the other families as legacy task data.\nWe repeat our experiment 6 times, each time using a different pump family as the current task and the other 5 datasets as legacy task data.\nIn the regression setting, we split the current task data into a train and test set, training our metamodel on all of the legacy data as well as whatever data is available from the current task, then quantify our predictions on the held-out test set of the current task in terms of MNLP, MAE, and RMSE.\nEach experiment is repeated 10 times with different random initializations and train-test splits.\nIn the optimization setting, we begin with no evaluations from the current task and carry out Bayesian optimization using the available data as a (finite) design set.\nData are selected at each iteration according the acquisition function of Eq.\\ (\\ref{eqn:prob-best}).\nWe track the best evaluation (relative to the per-task best design) as a function of the number of evaluations on the current task.\n\n\\subsubsection{Additive manufacturing data}\n\\label{sec:examples:systems:additive}\nWe consider a dataset in which the performance of an additive manufacturing process is quantified in terms of four design variables.\nWe have data from $35$ different chemistries, each of which has $24$ data.\nWe conduct experiments on each of the $35$ chemistries in an analogous manner to the experiments on the pump data.\n\n\\subsection{Regression results}\nFigure \\ref{fig:examples:regression:synthetic} shows regression results for the synthetic systems described in Sections \\ref{sec:examples:systems:toy} and \\ref{sec:examples:systems:forrester}.\nWe notice that the embedding GPs typically outperform the Bayesian GP by a margin, though there is overlap in the statistics of their performance over repeats.\nCuriously, we notice that the deterministic embedding GP occasionally outperforms the Bayesian embedding in terms of RMSE and MAE, but fails catastrophically in terms of MNLP, where the overconfidence of its predictions is severely penalized.\n\n\\begin{figure}[hbt]\n\t\\centering\n\t\\includegraphics[width=0.3\\textwidth]{fig2.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig3.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig4.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig5.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig6.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig7.pdf}\n\t\\caption{\n\t\t(Regression, synthetic systems)\n\t\tPerformance on held-out tasks.\n\t\tFrom left to right, performance in terms of root mean squared error (RMSE), mean absolute error (MAE), and median negative log probability (MNLP).\n\t\tTop row: Forrester function.\n\t\tBottom row: synthetic system of functions.\n\t\tSolid lines show the median performance over 10 random seeds and the shaded region shows the 80\\% coverage interval.\n\t}\n\t\\label{fig:examples:regression:synthetic}\n\\end{figure}\n\nFigure \\ref{fig:examples:regression:pump} shows our results for the pump dataset.\nFor this problem, we observe mixed results with the embedding GPs outperforming the baseline on some of the tasks, but underperforming on others. \nWe hypothesize that this is because certain tasks are substantially different from the others, causing the prior information from legacy tasks to be unhelpful.\nHowever, on certain tasks (3 and 4), we see that the Bayesian embedding GP outperforms the baseline even in a zero-shot setting.\nWe also notice that the deterministic embedding GP consistently gives rather poor uncertainty estimates as evidenced by its MNLP scores.\nThus, we see a clear benefit to the Bayesian embedding approach for improving the credibility of the predictions.\n\n\\begin{figure}[hb!t]\n\t\\centering\n\t\\includegraphics[width=0.3\\textwidth]{fig8.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig9.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig10.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig11.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig12.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig13.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig14.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig15.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig16.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig17.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig18.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig19.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig20.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig21.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig22.pdf}\n\t\\caption{\n\t\t(Regression, pump dataset)\n\t\tRegression performance on held-out tasks.\n\t\tFrom left to right, performance in terms of root mean squared error (RMSE), mean absolute error (MAE), and median negative log probability (MNLP).\n\t\tEach row shows the performance for a different held-out task.\n\t\tSolid lines show the median performance over 10 random seeds and the shaded region shows the 80\\% coverage interval.\n\t}\n\t\\label{fig:examples:regression:pump}\n\\end{figure}\n\nFigure \\ref{fig:examples:regression:additive} shows our results for the additive dataset for the first few held-out tasks.\nIn contrast to the pump problem, we often see considerable improvements by using the embedding models.\nFor many of the held-out tasks, the \\textit{zero-shot} predictions from the embedding models are, on average, superior to those of the Bayesian GP even after it has seen 10 examples from the current task.\nOn others (e.g.\\ RMSE, MAE for Task 2 in Fig. \\ref{fig:examples:regression:additive}), the Bayesian GP does somewhat outperform the embedding models, though the embedding models continue to improve.\nThis may be because the prior knowledge built up from the legacy tasks is unhelpful for the held-out task; the embedding model must learn to ``overturn'' this prior knowledge by finding a way to separate the embedding of the held-out task from the unrelated legacy tasks.\nHowever, this requires observations to gradually overturn the belief via Bayesian updating.\n\nAdditionally, we see again that the deterministic embedding consistently results in very poor performance in terms of MNLP, frequently performing so much worse that it is impractical to show on the same axes as the Bayesian GP and Bayesian EGP models.\nFigure \\ref{fig:examples:regression:additive:mnlp-no-zoom} shows the results when zoomed out to view the performance of the deterministic EGP for one task.\nWe see a similar deficiency in all of the other held-out tasks, firmly underscoring that it is critical to account for uncertainty in the embedding in order to obtain credible predictions.\nThis makes sense, given the high number of tasks we must embed relative to the number of data available to elucidate their input-output characteristics.\nThis is not uncommon in engineering design problems, where one typically has a limited computational budget to explore any given design problem, but may potentially have access to a large archive of legacy tasks.\n\n\\begin{figure}[hb!t]\n\t\\centering\n\t\\includegraphics[width=0.3\\textwidth]{fig23.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig24.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig25.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig26.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig27.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig28.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig29.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig30.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig31.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig32.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig33.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig34.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig35.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig36.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig37.pdf}\n\t\\caption{\n\t\t(Regression, additive dataset)\n\t\tRegression performance on held-out tasks.\n\t\tFrom left to right, performance in terms of root mean squared error (RMSE), mean absolute error (MAE), and median negative log probability (MNLP).\n\t\tEach row shows the performance for a different held-out task.\n\t\tSolid lines show the median performance over 10 random seeds and the shaded region shows the 80\\% coverage interval.\n\t}\n\t\\label{fig:examples:regression:additive}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{fig38.pdf}\n\t\\caption{\n\t\t(Regression, additive dataset)\n\t\tZoomed-out MNLP performance for held-out task 0.\n\t\tThe deterministic embedding GP performs worse than the other methods by orders of magnitude.\n\t}\n\t\\label{fig:examples:regression:additive:mnlp-no-zoom}\n\\end{figure}\n\n\\subsection{Optimization results}\nFigure \\ref{fig:examples:optimization:synthetic:running_best} shows the best design discovered for the synthetic systems as a function of the number of evaluations on the current task.\nWe report the median performance over $10$ splits as well as the 80\\% coverage interval ($10$-th to $90$-th percentile).\nWe see that the legacy task data enables the embedding GPs to consistently find near-optimal designs with a either a single evaluation (Forrester) or \\textit{without any data from the current task} (synthetic).\nBy contrast, the GP usually requires a handful of evaluations before it begins to find good solutions.\n\n\\begin{figure}[hbt]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{fig39.pdf}\n\t\\includegraphics[width=0.45\\textwidth]{fig40.pdf}\n\t\\caption{\n\t\tBayesian optimization for the synthetic and Forrester systems.\n\t\tThe solid curves show the median performance over $10$ repeats, and the shaded region shows the 80\\% coverage interval over all repeats.\n\t}\n\t\\label{fig:examples:optimization:synthetic:running_best}\n\\end{figure}\n\nFigures \\ref{fig:examples:optimization:synthetic:posterior} and \\ref{fig:examples:optimization:forrester:posterior} show the zero- and one-shot posteriors of all three methods on the toy and Forrester systems, respectively.\nWe see that the legacy task data equip the embedding GPs with helpful priors that aid them in identifying the response surface.\nBy contrast, the Bayesian GP guesses randomly in the zero-shot setting and has minimal insight for its one-shot predictions.\nWe also notice that while the deterministic embedding model seems to fortunately pick good designs, its model of the response surface is highly overconfident.\nBy contrast, we see that the Bayesian embedding endows our models with well-calibrated uncertainty estimates that simultaneously allows is to find good designs while retaining a credible metamodel.\nThus, while the EGP performs comparably to the BEGP, we must be cautious about generalizing these results.\nWe also point out that while it seems like the BGP has picked a very good first point in Fig.\\ \\ref{fig:examples:optimization:synthetic:posterior}, this selection is by pure luck since BGP's predictive distribution with zero training data (i.e.\\ the prior) is flat in its inputs.  Indeed, we see that it picks the same first point for the Forrester function in Fig.\\ \\ref{fig:examples:optimization:forrester:posterior}, which is not nearly so close to optimal.\n\n\\begin{figure}[hbt]\n\t\\centering\n\t\\includegraphics[width=0.3\\textwidth]{fig41.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig42.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig43.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig44.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig45.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig46.pdf}\n\t\\caption{\n\t\tPredictive posteriors for the metamodels on the toy system.\n\t\tTop row: zero-shot predictions.\n\t\tBottom row: one-shot predictions.\n\t\tFrom left to right: BGP, EGP, BEGP metamodels.\n\t}\n\t\\label{fig:examples:optimization:synthetic:posterior}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\t\\centering\n\t\\includegraphics[width=0.3\\textwidth]{fig47.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig48.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig49.pdf}\n\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{fig50.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig51.pdf}\n\t\\includegraphics[width=0.3\\textwidth]{fig52.pdf}\n\t\\caption{\n\t\tPredictive posteriors for the metamodels on the Forrester system.\n\t\tTop row: zero-shot predictions.\n\t\tBottom row: one-shot predictions.\n\t\tFrom left to right: GP, EGP, BEGP metamodels.\n\t}\n\t\\label{fig:examples:optimization:forrester:posterior}\n\\end{figure}\n\nFigure \\ref{fig:examples:optimization:real:running_best} shows the running best design for the pump and additive systems as a function of the number of current task evaluations.\nSince each legacy system has a different optimum, results show the performance relative to the best design for the current task.\nAgain, we see that the embedding GPs vastly outperform the vanilla GP approach, usually finding good designs on their first attempts.\nHowever, here we see that the Bayesian embedding gives better results over a deterministic embedding; this can be directly attributed to our earlier observation that the deterministic embedding tends to overfit, particularly when there are very few data to inform the embedding.\nBy contrast, the Bayesian embedding safely guards against overfitting and provides robust performance.\n\n\\begin{figure}[hbt]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{fig53.pdf}\n\t\\includegraphics[width=0.45\\textwidth]{fig54.pdf}\n\t\\caption{\n\t\tBayesian optimization for the pump and additive systems.\n\t\tResults are shown relative to the per-task best design.\n\t\tThe solid curves show the median performance over all repeats, and the shaded region shows the 80\\% coverage interval.\n\t}\n\t\\label{fig:examples:optimization:real:running_best}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe have introduced a method for Bayesian optimization applicable to settings where one has access to a variety of related datasets but does not know how they relate to each other \\textit{a priori}.\nThe Bayesian embedding GP automatically learns latent representations that enable a single metamodel to share information across multiple tasks, thereby improving the predictive performance in all cases, particularly in novel settings where limited data is available.\nWe observe that our method enables Bayesian optimization to more quickly and reliably find good solutions compared to traditional methods.\n\nWe find that variational inference with rudimentary variational approximations (factorized Gaussians) over the latent variables provides sufficient uncertainty information that credible estimates may be obtained in practice and is robust with respect to the number of latent dimensions.\nOur experiments show that this Bayesian embedding is critical for obtaining credible uncertainty quantification.\n\nThere are a number of ways in which our work might be extended.\nFor one, our method might be applied to optimization settings with unknown constraints.\nIn such cases, one might simultaneously model the probability of violating a constraint with a multi-task model.\nModeling binary outputs such as constraint violation with GPs requires non-conjugate inference, which can be enabled using techniques from \\cite{hensman2015scalable}.\nSecond, our approach is straightforward to apply to problems of robust design.\nIn fact, we have already shown how uncertainty in our metamodel can be propagated efficiently during design selection through stochastic backpropagation, eliminating the need for expensive double-loop optimization.\nOther approaches \\cite{ryan2018gaussian, ling2018efficient} exploit analytic properties to solve the inner-loop optimization, though our work suggests that it might be approximately solved in general using stochastic optimization.\nFinally, our model learns a single embedding for each general input (task) as a latent vector.\nWe would like to explore the effect of a hierarchical latent representation in which the embedding of each task is allowed to vary with $\\vc x^{(r)}$.\nSuch a conditional multi-task model might enable one to exploit partial similarities between tasks in certain regions of input space while still providing for the possibility that their trends might differ elsewhere.\nWhile this was studied in \\cite{ghosh2018bayesian}, we expect that our Bayesian latent representation might improve performance in few-shot settings such as those considered in this work.\n\n\\section*{Funding Sources}\nFunding for this work was provided internally by GE Research.\n\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nMarkov decision processes (MDPs) model sequential decision-making in stochastic systems with nondeterministic choices. A policy, i.e., a decision strategy, resolves the nondeterminism in an MDP and induces a stochastic process. In this regard, an MDP represents a (infinite) family of stochastic processes. In this paper, for a given MDP, we aim to synthesize a policy that induces a process with maximum entropy among the ones whose paths satisfy a temporal logic specification. \n\nEntropy, as an information-theoretic quantity, measures the unpredictability of outcomes in a random variable \\cite{Cover}. Considering a stochastic process as an infinite sequence of (dependent) random variables, we define the entropy of a stochastic process as the joint entropy of these random variables by following \\cite{Biondi},\\cite{Chen}. Therefore, intuitively, our objective is to obtain a process whose paths satisfy a temporal logic specification in the most unpredictable way to an observer. \n\nTypically, in an MDP, a decision-maker is interested in satisfying certain properties \\cite{Behcet} or accomplishing a task \\cite{Jie}. Linear temporal logic (LTL) is a formal specification language \\cite{Pneuli} that has been widely used to check the reliability of software \\cite{Tan}, describe tasks for autonomous robots \\cite{Gazit, Belta} and verify the correctness of communication protocols \\cite{Kumar}. For example, in a robot navigation scenario, it allows to specify tasks such as safety (never visit the region A), liveness (eventually visit the region A) and priority (first visit the region A, then B). \n\nThe entropy of paths of a (Markovian) stochastic process is introduced in \\cite{Ekroot} and quantifies the randomness of realizations with fixed initial and final states. We first extend the definition for the entropy of paths to realizations that reach a certain set of states, rather than a fixed final state. Then, we show that the entropy of a stochastic process is equal to the entropy of paths of the process, if the process has a finite entropy. The established relation provides a mathematical basis to the intuitive idea that maximizing the entropy of an MDP minimizes the predictability of paths.\n\nWe observe that the maximum entropy of an MDP under stationary policies may not exist, i.e., for any given level of entropy, using stationary policies, one can induce a process whose entropy is greater than that level. In this case, we say that the maximum entropy of the MDP is unbounded. Additionally, if there exists a process with the maximum entropy, the entropy of such a process can be finite or infinite. Hence, before attempting to synthesize a policy that maximizes the entropy of an MDP, we first verify whether there exists a policy that attains the maximum entropy. \n\nThe contributions of this paper are fourfold. First, we provide necessary and sufficient conditions on the structure of the MDP under which the maximum entropy of the MDP is finite, infinite or unbounded. We also present a polynomial-time algorithm to check whether the maximum entropy of an MDP is finite, infinite or unbounded. Second, we present a polynomial-time algorithm based on a convex optimization problem to synthesize a policy that maximizes the entropy of an MDP. Third, we show that maximizing the entropy of an MDP with non-infinite maximum entropy is equivalent to maximizing the entropy of paths of the MDP. Lastly, we provide a procedure to obtain a policy that maximizes the entropy of an MDP subject to a general LTL specification. \n\nThe applications of this theoretical framework range from motion planning and stochastic traffic assignments to software security. In a motion planning scenario, for security purposes, an autonomous robot might need to randomize its paths while carrying out a mission \\cite{Paruchuri, Paruchuri2}. In such a scenario, a policy synthesized by the proposed methods both provides probabilistic guarantees on the completion of the mission and minimizes the predictability of the robot's paths  through the use of online randomization mechanisms. Additionally, such a policy allows the robot to explore different parts of the environment \\cite{Saerens}, and behave robustly against uncertainties in the environment \\cite{Deep_learning}. The proposed methods can also be used to distribute traffic assignments over a network, which is known as stochastic traffic assignments \\cite{Akamatsu}, as it promotes the use of different paths. Finally, as it is shown in \\cite{Biondi}, the maximum information that an adversary can leak from a (deterministic) software, which is modeled as an MDP, can be quantified by computing the maximum entropy of the MDP. \n\n\\textbf{Related Work.}  A preliminary version \\cite{Yagiz} of this paper considered entropy maximization problem for MDPs subject to expected reward constraints. This considerably extended version includes an additional section establishing the relation between the maximum entropy of an MDP and the entropy of paths of the MDP, detailed proofs for all theoretical results, and additional numerical examples.\n\nThe computation of the maximum entropy of an MDP is first considered in \\cite{Chen}, where the authors present a robust optimization problem to compute the maximum entropy for an MDP with finite maximum entropy. However, their approach does not allow to incorporate additional constraints due to the formulation of the problem. References \\cite{Biondi} and \\cite{Biondi2} compute the maximum entropy of an MDP for special cases without providing a general algorithm. \n\nThe work \\cite{Biondi} provides the necessary and sufficient conditions for an interval Markov chain (MC) to have a finite maximum entropy. Therefore, some of the results provided in this paper, e.g., the necessary and sufficient conditions for an MDP to have finite, unbounded or infinite maximum entropy, can be seen as an extension of the results given in \\cite{Biondi}.\n\nIn \\cite{Bullo,Bullo2}, the authors study the problem of synthesizing a transition matrix with maximum entropy for an irreducible MC subject to graph constraints. The problem studied in this paper is considerably different from that problem since MDPs represent a more general model than MCs, and an MC induced from an MDP by a policy is not necessarily irreducible.\n\nIn \\cite{Paruchuri}, the authors maximize the entropy of a \\textit{policy} while keeping the expected total reward above a threshold. They claim that the entropy maximization problem is not convex. Their formulation is a special case of the convex optimization problem that we provide in this paper. Therefore, here, we also prove the convexity of their formulation.\n\nThe entropy of paths of absorbing MCs is discussed in \\cite{Ekroot}, \\cite{Akamatsu}, \\cite{Kafsi}. The reference \\cite{Saerens} establishes the equivalence between the entropy of paths and the entropy of an absorbing MC. We establish this relation for a general MC and show the connections to the maximum entropy of an MDP. \n\n\n\nWe also note that none of the above work discusses the unbounded and infinite maximum entropy for an MDP or considers LTL to specify desired system properties.\n\n\\textbf{Organization.} We provide the preliminary definitions and formal problem statement in Sections \\ref{Prelim} and \\ref{problem_set}, respectively. We analyze the properties of the maximum entropy of an MDP and present an algorithm to synthesize a policy that maximizes the entropy of an MDP in Section \\ref{max-ent}. The relation between the maximum entropy of an MDP and the entropy of paths is established in Section \\ref{relate_paths}. We present a procedure to synthesize a policy that maximizes the entropy of an MDP subject to an LTL specification in Section \\ref{cons_section}. We provide numerical examples in Section \\ref{examples_section} and conclude with suggestions for future work in Section \\ref{conclusion_section}. Proofs for all results are provided in Appendix \\ref{proofs_appendix}, and a procedure to synthesize a policy that maximizes the entropy of an MDP with infinite maximum entropy is presented in Appendix \\ref{infinite_case_appendix}.\n\n\\section{Preliminaries}\\label{Prelim}\\noindent\n\\textbf{Notation:} For a set $S$,  we denote its power set and cardinality by $2^S$ and $\\lvert S \\rvert$, respectively. For a matrix $P$$\\in$$\\mathbb{R}^{n \\times n}$, we use $P^k$ and $P^k_{i,j}$ to denote the k-th power of $P$ and the $(i,j)$-th component of the k-th power of $P$, respectively. All logarithms are to the base 2 and the  set $\\mathbb{N}$ denotes $\\{0,1,2,\\ldots\\}$.\n\\subsection{Markov chains and Markov decision processes}\\label{MDP_subsection}\n{\\setlength{\\parindent}{0cm}\n\\begin{definition}\nA \\textit{Markov decision process} (MDP) is a tuple $\\mathcal{M}$$=$$(S, s_0, \\mathcal{A}, \\mathbb{P},\\mathcal{AP},\\mathcal{L})$ where $S$ is a finite set of states, $s_0$$\\in$$S$ is the initial state, $\\mathcal{A}$ is a finite set of actions, $ \\mathbb{P}$$:$$S$$\\times$$ \\mathcal{A}$$ \\times$$ S$$\\rightarrow$$[0,1]$ is a transition function such that $\\sum_{t\\in S}\\mathbb{P}(s,a,t)$$=$$1$ for all $s$$\\in$$S$ and $a$$\\in$$\\mathcal{A}$, $\\mathcal{AP}$ is a set of atomic propositions, and $\\mathcal{L}$ $:$ $ S$$\\rightarrow$$ 2^{\\mathcal{AP}}$ is a function that labels each state with a subset of atomic propositions.\n\\end{definition}}\nWe denote the transition probability $ \\mathbb{P}(s,a,t)$ by $ \\mathbb{P}_{s,a,t}$, and all available actions in a state $s$$\\in$$S$ by $\\mathcal{A}(s)$. The set of successor states for a state action pair $(s,a)$ is defined as $Succ(s,a)$$:=$$\\{t $$\\in$$ S |  \\mathbb{P}_{s,a,t}$$>$$0, a$$\\in$$\\mathcal{A}(s)\\}$. The \\textit{size of an MDP} is the number of triples $(s,a,t)$$\\in$$S$$\\times$$\\mathcal{A}$$\\times$$S$ such that $\\mathbb{P}_{s,a,t}$$>$$0$. \n\nA \\textit{Markov chain} (MC) $\\mathcal{C}$ is an MDP such that $\\lvert \\mathcal{A}\\rvert$$=$$1$. We denote the transition function (matrix) for an MC by $\\mathcal{P}$, and the set of successor states for a state $s$$\\in$$S$ by $Succ(s)$$=$$\\{t $$\\in$$ S | \\mathcal{P}_{s,t}$$>$$0\\}$. The \\textit{expected residence time} in a state $s$$\\in$$ S$ for an MC $\\mathcal{C}$ is defined as\n\\begin{align}\n\\label{residence}\n\\xi_s:=\\sum_{k=0}^{\\infty}\\mathcal{P}^k_{s_0,s}.\\end{align}\nThe expected residence time $\\xi_s$ represents the expected number of visits to state $s$ starting from the initial state ~\\cite{Marta}. A state $s$$\\in$$S$ is \\textit{recurrent} for an MC if and only if $\\xi_s$$=$$\\infty$, and is \\textit{transient} otherwise; it is \\textit{stochastic} if and only if it satisfies $\\lvert Succ(s)\\rvert$$>$$1$, and is \\textit{deterministic} otherwise; and it is \\textit{reachable} if and only if $\\xi_s$$>$$0$, and is \\textit{unreachable} otherwise.  \n{\\setlength{\\parindent}{0cm}\n\\noindent \\begin{definition}\nA \\textit{policy} for an MDP $\\mathcal{M}$ is a sequence $\\pi$$=$$\\{\\mu_0, \\mu_1, \\ldots\\}$ where each $\\mu_k $$:$$ S $$ \\times$$ \\mathcal{A}$$\\rightarrow$$[0,1]$ is a function such that $\\sum_{a\\in \\mathcal{A}(s)}\\mu_k(s,a)$$=$$1$ for all $s$$\\in$$S$. A \\textit{stationary} policy is a policy of the form $\\pi$$=$$\\{\\mu, \\mu, \\ldots\\}$. For an MDP $\\mathcal{M}$, we denote the set of all policies and all stationary policies by $\\Pi(\\mathcal{M})$ and $\\Pi^S(\\mathcal{M})$, respectively.\n\\end{definition}}\n We denote the probability of choosing an action $a$$\\in$$\\mathcal{A}(s)$ in a state $s$$\\in$$S$ under a stationary policy $\\pi$ by $\\pi_s(a)$. For an MDP $\\mathcal{M}$, a stationary policy $\\pi$$\\in$$\\Pi^S(\\mathcal{M})$ induces an MC denoted by $\\mathcal{M}^{\\pi}$. We refer to $\\mathcal{M}^{\\pi}$ as \\textit{induced MC} and specify the transition matrix for $\\mathcal{M}^{\\pi}$ by $\\mathcal{P}^{\\pi}$, whose $(s,t)$-th component is given by\n\\begin{align}\n\\label{induced_MC_prob}\n\\mathcal{P}^{\\pi}_{s,t}=\\sum_{a\\in\\mathcal{A}(s)}\\pi_s(a) \\mathbb{P}_{s,a,t}.\n\\end{align}\nThroughout the paper, we assume that for a given MDP $\\mathcal{M}$, for any state $s$$\\in$$S$ there exists an induced MC $\\mathcal{M}^{\\pi}$ for which the state $s$ is reachable. This is a standard assumption for MDPs \\cite{Belta}, which ensures that each state in the MDP is reachable under some policy.\n\nAn infinite sequence $\\varrho^{\\pi}$$=$$s_0s_1s_2\\ldots$ of states generated in $\\mathcal{M}$ under a policy $\\pi$$\\in$$\\Pi(\\mathcal{M})$ is called a \\textit{path}, starting from the initial state $s_0$ and satisfies $\\sum_{a_k\\in \\mathcal{A}(s_k)}\\mu_k(s_k)(a_k)\\mathbb{P}_{s_k,a_k,s_{k+1}}$$>$$0$ for all $k$$\\geq$$0$. Any finite prefix of $\\varrho^{\\pi}$ that ends in a state is a finite path fragment. We define the set of all paths and finite path fragments in $\\mathcal{M}$ under the policy $\\pi$ by $Paths^{\\pi}(\\mathcal{M})$ and $Paths_{fin}^{\\pi}(\\mathcal{M})$, respectively. \n\nWe use the standard probability measure over the outcome set $Paths^{\\pi}(\\mathcal{M})$ \\cite{Model_checking}. For a path $\\varrho^{\\pi}$$\\in$$Paths^{\\pi}(\\mathcal{M})$, let the sequence $s_0s_1\\ldots s_n$ be the finite path fragment of length $n$, and let $Paths^{\\pi}(\\mathcal{M})(s_0s_1\\ldots s_n)$ denote the set of all paths in $Paths^{\\pi}(\\mathcal{M})$ starting with the prefix $s_0s_1\\ldots s_n$. The probability measure $\\text{Pr}_{\\mathcal{M}}^{\\pi}$ defined on the smallest $\\sigma$-algebra over $Paths^{\\pi}(\\mathcal{M})$ that contains $Paths^{\\pi}(\\mathcal{M})(s_0s_1\\ldots s_n)$ for all $s_0s_1\\ldots s_n$$\\in$$Paths_{fin}^{\\pi}(\\mathcal{M})$ is the unique measure that satisfies \n{\\setlength{\\mathindent}{0pt}\n\\begin{flalign}\n\\label{first_measure}\\noindent\n\\text{Pr}_{\\mathcal{M}}^{\\pi}\\{Paths^{\\pi}&(\\mathcal{M})(s_0\\ldots s_n)\\}=\\nonumber \\\\\n&\\prod_{0\\leq k < n} \\sum_{a_k\\in\\mathcal{A}(s_k)}\\mu_k(s_k)(a_k) \\mathbb{P}_{s_k,a_k, s_{k+1}}.\n\\end{flalign}}\\noindent\\vspace{-0.7cm}\n\n\\subsection{The entropy of stochastic processes}\\label{entropy_subsection}\nFor a (discrete) random variable $X$, its support $\\mathcal{X}$ defines a countable sample space from which $X$ takes a value $x$$\\in$$\\mathcal{X}$ according to a probability mass function (pmf) $p(x)$$:=$$\\text{Pr}(X$$=$$x)$. The \\textit{entropy} of a random variable $X$ with countable support $\\mathcal{X}$ and pmf $p(x)$ is defined as\n\\begin{align}\\centering\nH(X):=-\\sum_{x\\in\\mathcal{X}} p(x)\\log p(x).\n\\end{align}\nWe use the convention that $0$$\\log$$0$$=$$0$. Let $(X_0,X_1)$ be a pair of random variables with the joint pmf $p(x_0,x_{1})$ and the support $\\mathcal{X}\\times \\mathcal{X}$. The \\textit{joint entropy} of $(X_0,X_1)$ is \n\\begin{align}\n\\label{joint_entropy}\nH(X_0,X_1):= -\\sum_{x_0\\in \\mathcal{X}}\\sum_{x_{1}\\in \\mathcal{X}}p(x_0,x_{1})\\log p(x_0,x_{1}),\n\\end{align}\\noindent\nand the \\textit{conditional entropy} of $X_1$ given $X_0$ is\n\\begin{align}\n\\label{conditional_entropy}\n&H(X_1 | X_0):=-\\sum_{x_0\\in \\mathcal{X}}\\sum_{x_{1}\\in \\mathcal{X}}p(x_0,x_{1})\\log p(x_1 |x_0).\n\\end{align}\\noindent\nThe definitions of the joint and conditional entropies extend to collection of $k$ random variables as it is shown in \\cite{Cover}. \nA discrete \\textit{stochastic process} $\\mathbb{X}$ is a discrete time-indexed sequence of random variables, i.e., $\\mathbb{X}$$=$$\\{X_k$$\\in$$\\mathcal{X}$ $:$ $k$$\\in$$\\mathbb{N}\\}$. \n{\\setlength{\\parindent}{0cm}\n\\noindent\\begin{definition} (Entropy of a stochastic process) \\cite{Biondi_thesis}\nThe \\textit{entropy of a stochastic process} $\\mathbb{X}$ is defined as \n \\begin{align}\\label{entropy_def_stochastic}\n H(\\mathbb{X}):=\\lim_{k\\rightarrow \\infty}H( X_0, X_1\\ldots, X_{k}).\n \\end{align}\n \\end{definition}}\nNote that this definition is different from the \\textit{entropy rate} of a stochastic process, which is defined as $\\lim_{k\\rightarrow \\infty}\\frac{1}{k}H( X_0, X_1\\ldots, X_{k})$ when the limit exists \\cite{Cover}. The limit in \\eqref{entropy_def_stochastic} either converges to a non-negative real number or diverges to positive infinity \\cite{Biondi_thesis}. \n\nAn MC $\\mathcal{C}$ is equipped with a discrete stochastic process $\\{X_k$$\\in$$S$ $:$ $ k$$\\in$$\\mathbb{N}\\}$ where each $X_k$ is a random variable over the state space $S$. For a given k-dimensional pmf $p(s_0,s_1,\\ldots, s_k)$, this process respects the \\textit{Markov property}, i.e., $p(s_k|s_{k-1},\\ldots,s_0)$$=$$p(s_k|s_{k-1})$\nfor all $k$$\\in$$\\mathbb{N}$. Then, the \\textit{entropy of a Markov chain} $\\mathcal{C}$ is given by \n\\begin{align}\n\\label{process_entropy}\nH(\\mathcal{C})=H(X_0)+\\sum_{i=1}^{\\infty}H(X_{i}| X_{i-1})\n\\end{align} \nusing \\eqref{joint_entropy}, \\eqref{conditional_entropy} and \\eqref{entropy_def_stochastic}. Note that $H(X_0)$$=$$0$, since we define an MC with a unique initial state.\n\nFor an MDP $\\mathcal{M}$, a policy $\\pi$$\\in$$\\Pi(\\mathcal{M})$ induces a discrete stochastic process $\\{X_k $$\\in$$S$ $:$ $ k$$\\in$$\\mathbb{N}\\}$.  We denote the entropy of an MDP $\\mathcal{M}$ under a policy $\\pi$$\\in$$\\Pi(\\mathcal{M})$ by $H(\\mathcal{M},\\pi)$. \nUsing the next proposition, we restrict our attention to stationary policies for maximizing the entropy of an MDP. \n{\\setlength{\\parindent}{0cm}\\noindent\n\\begin{prop}\n\\label{memoryless_1}\nThe following equality holds:\n\\vspace{-0.2cm}\n\\begin{align}\\label{eqeq}\n\\sup_{\\pi\\in\\Pi(\\mathcal{M})}H(\\mathcal{M},{\\pi})=\\sup_{\\pi\\in\\Pi^S(\\mathcal{M})}H(\\mathcal{M},{\\pi}).\\quad \\triangleleft\n\\end{align}\n\\end{prop}}A proof for Proposition \\ref{memoryless_1} is provided in Appendix \\ref{proofs_appendix}.\n{\\setlength{\\parindent}{0cm}\\noindent\\begin{remark}  If the supremum in \\eqref{eqeq} is infinite, the set of stationary policies may not be sufficient to attain the supremum while a non-stationary policy can attain it. In particular, there exists a family of distributions that are defined over a countable support and have infinite entropy (see equation (7) in \\cite{Bacetti}). It can be shown that for some MDPs, there exists a non-stationary policy that induces a stochastic process with such a probability distribution, and hence, have infinite entropy, while stationary policies can only induce stochastic processes with finite entropies\\footnote{ A preliminary version \\cite{Yagiz} of this paper relied on Proposition 36 from \\cite{Chen}. This proposition is not valid in general. Here, we provide the corrected results by defining the maximum entropy of an MDP over stationary policies.}.\\end{remark}}\\vspace{-0.2cm}\n{\\setlength{\\parindent}{0cm}\\noindent\\begin{definition} (Maximum entropy of an MDP) The \\textit{maximum entropy of an MDP} $\\mathcal{M}$ is\n\\begin{align}\\label{max_ent_definition}\nH(\\mathcal{M}):=\\sup_{\\pi\\in\\Pi^{S}(\\mathcal{M})}H(\\mathcal{M},{\\pi}).\n\\end{align}\\end{definition}}\nA policy $\\pi^{\\star}$$\\in$$\\Pi^S(\\mathcal{M})$ \\textit{maximizes} the entropy of an MDP $\\mathcal{M}$ if $H(\\mathcal{M})$$=$$H(\\mathcal{M},{\\pi^{\\star}})$. Finally, we define the properties of the maximum entropy of an MDP as follows.{\\setlength{\\parindent}{0cm}\n\\noindent \\begin{definition}(The properties of the maximum entropy)\\label{props_def}\nThe maximum entropy of an MDP $\\mathcal{M}$ is\n\\begin{itemize} \n\\item \\textit{finite}, if and only if\n\\begin{align}\\label{def_finite_ent}\nH(\\mathcal{M})=\\max_{\\pi\\in\\Pi^{S}(\\mathcal{M})}H(\\mathcal{M},{\\pi})<\\infty;\n\\end{align}\n\\item \\textit{infinite}, if and only if\n\\begin{align} H(\\mathcal{M})=\\max_{\\pi\\in\\Pi^{S}(\\mathcal{M})}H(\\mathcal{M},{\\pi})=\\infty; \n\\end{align}\n\\item \\textit{unbounded}, if and only if the following two conditions hold.\n\\begin{align}\\label{unbounded_definition_1}\n(i)\\qquad &H(\\mathcal{M})=\\sup_{\\pi\\in\\Pi^{S}(\\mathcal{M})}H(\\mathcal{M},{\\pi})=\\infty,\\\\ \\label{unbounded_definition_2}\n(ii)\\qquad &H(\\mathcal{M},{\\pi})<\\infty\\ \\  \\text{for all}\\ \\pi\\in\\Pi^{S}(\\mathcal{M}).\n\\end{align}\n\\end{itemize}\n\\end{definition}}\nAlthough it is not defined here, there is a fourth possible property which is unachievable finite maximum entropy, i.e., $\\max_{\\pi\\in\\Pi^S(\\mathcal{M})}H(\\mathcal{M},{\\pi})$$<$$H(\\mathcal{M})$$<$$\\infty$. In Theorem \\ref{Theorem1}, we show that it is not possible for the maximum entropy of an MDP to have this property.\n\\subsection{Linear temporal logic}\nWe employ linear temporal logic (LTL) to specify tasks and refer the reader to \\cite{Model_checking} for the syntax and semantics of LTL. \n\nAn LTL formula is built up from a set $\\mathcal{AP}$ of atomic propositions, logical connectives such as conjunction ($\\land$) and negation ($\\lnot$), and temporal modal operators such as always ($\\square$) and eventually ($\\lozenge$). An infinite sequence of subsets of $\\mathcal{AP}$ defines an infinite \\textit{word}, and an LTL formula is interpreted over infinite words on $2^{\\mathcal{AP}}$. We denote by $w$$\\models$$\\varphi$ that a word $w$$=$$w_0w_1w_2\\ldots$ satisfies an LTL formula $\\varphi$.\n{\\setlength{\\parindent}{0cm}\n\\noindent \\begin{definition}\nA \\textit{deterministic Rabin automaton} (DRA) is a tuple $A$$=$$(Q,q_0, \\Sigma, \\delta, Acc)$ where $Q$ is a finite set of states, $q_0$$\\in$$Q$ is the initial state, $\\Sigma$ is the alphabet, $\\delta$$:$$Q$$\\times$$\\Sigma$$\\rightarrow$$Q$ is the transition relation, and $Acc$$\\subseteq$$2^{Q}$$\\times$$2^{Q}$ is the set of accepting state pairs. \n\\end{definition}}\nA \\textit{run} of a DRA $A$, denoted by $\\sigma$$=$$q_0q_1q_2\\ldots$, is an infinite sequence of states in $\\mathcal{A}$ such that for each $i$$\\geq$$0$, $q_{i+1}$$\\in$$\\delta(q_i, p)$ for some $p$$\\in$$\\Sigma$. A run $\\sigma$ is \\textit{accepting} if there exists a pair $(J,K)$$\\in$$Acc$ and an $n$$\\geq$$0$ such that (i) for all $m$$\\geq$$n$ we have $q_m$$\\not\\in$$J$, and (ii) there exists infinitely many $k$ such that $q_k$$\\in$$K$.\n\nFor any LTL formula $\\varphi$ built up from $\\mathcal{AP}$, a DRA $A_{\\varphi}$ can be constructed with input alphabet $2^{\\mathcal{AP}}$ that accepts all and only words over $\\mathcal{AP}$ that satisfy $\\varphi$ \\cite{Model_checking}. \n\nFor an MDP $\\mathcal{M}$ under a policy $\\pi$, a path $\\varrho^{\\pi}$$=$$s_0s_1\\ldots$ generates a word $w$$=$$w_0w_1\\ldots$ where $w_k$$=$$\\mathcal{L}(s_k)$ for all $k$$\\geq$$0$. With a slight abuse of notation, we use $\\mathcal{L}(\\varrho^{\\pi})$ to denote the word generated by $\\varrho^{\\pi}$. For an LTL formula $\\varphi$, the set $\\{\\varrho^{\\pi}$$\\in$$ Paths^{\\pi}(\\mathcal{M})$$:$$ \\mathcal{L}(\\varrho^{\\pi})$$\\models$$\\varphi\\}$ is measurable \\cite{Model_checking}. We define \n\\begin{equation*}\n\\begin{aligned}\n\\text{Pr}_{\\mathcal{M}}^{\\pi}(s_0\\models \\varphi):=\\text{Pr}_{\\mathcal{M}}^{\\pi}\\{\\varrho^{\\pi}\\in Paths^{\\pi}(\\mathcal{M}) : \\mathcal{L}(\\varrho^{\\pi})\\models \\varphi\\}\n\\end{aligned}\n\\end{equation*}\nas the probability of satisfying the LTL formula $\\varphi$ for an MDP $\\mathcal{M}$ under the policy $\\pi$$\\in$$\\Pi(\\mathcal{M})$.  \n\\section{Problem Statement}\\label{problem_set}\nThe first problem we study concerns the synthesis of a policy that maximizes the entropy of an MDP.\n {\\setlength{\\parindent}{0cm}\n\\noindent \\begin{problem}(\\textbf{Entropy Maximization})\\label{prob_1}\nFor a given MDP $\\mathcal{M}$, provide an algorithm to verify whether there exists a policy $\\pi^{\\star}$$\\in$$\\Pi^S(\\mathcal{M})$ such that $H(\\mathcal{M})$$=$$H(\\mathcal{M},\\pi^{\\star})$. If such a policy exists, provide an algorithm to synthesize it. If it does not exist, provide a procedure to synthesize a policy $\\pi'$$\\in$$\\Pi^S(\\mathcal{M})$ such that $H(\\mathcal{M},\\pi')$$\\geq$$\\ell$ for a given constant $\\ell$. \n\\end{problem}}\n\nFor an MDP $\\mathcal{M}$, the synthesis of a policy $\\pi'$$\\in$$\\Pi^S(\\mathcal{M})$ such that $H(\\mathcal{M},\\pi')$$\\geq$$\\ell$ allows one to induce a stochastic process with the desired level of entropy, even if there exists no stationary policy that maximizes the entropy of $\\mathcal{M}$.\n\nIn the second problem, we introduce linear temporal logic (LTL) specifications to the framework. In particular, we consider the problem of synthesizing a policy that induces a stochastic process with maximum entropy whose paths satisfy a given LTL formula with desired probability. The formal statement of the second problem is deferred to Section \\ref{cons_section} since it requires the introduction of additional notations.\n\n\n\\section{Entropy maximization for MDPs }\n\\label{max-ent}\n In this section, we focus on the entropy maximization problem. We refer to a policy as an \\textit{optimal} policy for an MDP if it maximizes the entropy of the MDP.\n \\subsection{The entropy of MCs versus MDPs}\n For an MC, the \\textit{local entropy} of a state $s$$\\in$$S$ is defined as\n\\begin{align}\n\\label{local_entropy_def}\nL(s):=-\\sum_{t\\in S}\\mathcal{P}_{s,t}\\log \\mathcal{P}_{s,t}.\n\\end{align}\nThe following proposition characterizes the relationship between the local entropy of states and the entropy of an MC.{\\setlength{\\parindent}{0cm}\n\\noindent \\begin{prop}\n\\label{Biondi_theorem}\n(Theorem 1 in \\cite{Biondi}) For an MC $\\mathcal{C}$,\n\\begin{align} \n\\label{MC_entropy_biondi}\nH(\\mathcal{C})=\\sum_{s\\in S}L(s)\\xi_s.\\quad \\triangleleft\n\\end{align}\n\\end{prop}}\\noindent\n\\par An MC $\\mathcal{C}$ has a finite entropy if and only if all of its recurrent states have zero local entropy \\cite{Biondi}. That is, $H(\\mathcal{C})$$<$$\\infty$ if and only if for all states $s$$\\in$$ S$, $\\xi_s$$=$$\\infty$ implies $L(s)$$=$$0$. If the entropy of an MC is finite, each recurrent state $s$$\\in$$S$ has a single successor state, i.e., $\\lvert Succ(s)\\rvert$$=$$1$. Consequently, recurrent states have no contribution to the sum in \\eqref{process_entropy}. In this case, we take the sum in \\eqref{MC_entropy_biondi} only over the transient states.\n\nFor an MDP, different policies may induce stochastic processes with different entropies. For example, consider the MDP given in Fig. \\ref{fig:MDP_1} and suppose that the action $a_1$ at state $s_0$ is taken with probability $\\varepsilon$. If we let $\\varepsilon$ range over $[0, \\frac{1}{2}]$, then the entropy of the resulting stochastic processes ranges over $[0,1]$. The optimal policy for this MDP is $\\pi_{s_0}(a_1)$$=$$\\pi_{s_0}(a_2)$$=$$1\/2$, which uniformly randomizes actions. \n\\par Unlike the MDP given in Fig. \\ref{fig:MDP_1}, the maximum entropy of an MDP is not generally achieved by a policy that chooses available actions at each state uniformly. For example, consider the MDP given in Fig. \\ref{fig:MDP_2}. The optimal policy for this MDP is $\\pi_{s_0}(a_1)$$=$$2\/3$, $\\pi_{s_0}(a_2)$$=$$1\/3$.\n\n\n\\begin{figure}[b]\\vspace{-0.4cm}\n\\hspace{0.05\\linewidth}\n\\begin{subfigure}[b]{0.3\\linewidth}\n\\centering\n\\scalebox{0.9}{\n\\begin{tikzpicture}[->, >=stealth', auto, semithick, node distance=1.5cm]\n\n    \\tikzstyle{every state}=[fill=white,draw=black,thick,text=black,scale=0.7]\n\n    \\node[state,initial,initial text=] (s_0) {$s_0$};\n    \\node[state] (s_1) [above right =6mm of s_0]  {$s_1$};\n    \\node[state] (s_2) [below right=6mm of s_0]  {$s_2$};\n\n\\path\n(s_0)  edge     node{$a_1, 1$}     (s_1)\n(s_0)\t edge      node{$a_2, 1$}     (s_2)\n(s_1)\t edge    [loop right]       node{$1$}     (s_1)\n(s_2)\t edge    [loop right]       node{$ 1$}     (s_2);\n\\end{tikzpicture}}\n\\caption{}\n\\label{fig:MDP_1}\n\\end{subfigure}\n\\hspace{0.1\\linewidth}\n\\begin{subfigure}[b]{0.4\\linewidth}\n\\centering\n\\scalebox{0.9}{\n\\begin{tikzpicture}[->, >=stealth', auto, semithick, node distance=2cm]\n\n    \\tikzstyle{every state}=[fill=white,draw=black,thick,text=black,scale=0.7]\n\n    \\node[state,initial,initial text=] (s_0) {$s_0$};\n    \\node[state] (s_1) [above right =6mm of s_0]  {$s_1$};\n    \\node[state] (s_2) [below right=6mm of s_0]  {$s_2$};\n\n\\path\n(s_0)  edge      [loop below] node{$a_1, 1\/3$}     (s_0)\n(s_0)  edge     node{$a_1, 2\/3$}     (s_1)\n(s_0)\t edge      node{$a_2, 1$}     (s_2)\n(s_1)\t edge    [loop right]       node{$1$}     (s_1)\n(s_2)\t edge    [loop right]       node{$ 1$}     (s_2);\n\\end{tikzpicture}}\n\\caption{}\n\\label{fig:MDP_2}\n\\end{subfigure}\n\\caption{Randomizing actions uniformly at each state may or may not achieve the maximum entropy. The optimal policy for the MDP given in (a) is $\\pi_{s_0}(a_1)$$=$$\\pi_{s_0}(a_2)$$=$$1\/2$, and for the MDP given in (b) is $\\pi_{s_0}(a_1)$$=$$2\/3$, $\\pi_{s_0}(a_2)$$=$$1\/3$.}\n\\label{fig:examples_policies}\n\\end{figure}\n\\par Examples given in Fig. \\ref{fig:examples_policies} show that finding an optimal policy for an MDP may not be trivial. To analyze the maximum entropy of an MDP, we first obtain a compact representation of the maximum entropy as follows. For an MC $\\mathcal{M}^{\\pi}$ induced from an MDP $\\mathcal{M}$ by a policy $\\pi$$\\in$$\\Pi^S(\\mathcal{M})$, let the expected residence time in a state $s$$\\in$$S$ be \n\\begin{align}\n\\label{MDP_residence_time}\n\\xi^{\\pi}_s:=\\sum_{k=0}^{\\infty}(\\mathcal{P}^{\\pi})^k_{s_0,s}.\n\\end{align}\n\nAdditionally, let the local entropy of a state $s$$\\in$$S$ in $\\mathcal{M}^{\\pi}$ be\n$L^{\\pi}(s)$$:=$$-\\sum_{t\\in S}\\mathcal{P}^{\\pi}_{s,t}\\log\\mathcal{P}^{\\pi}_{s,t}$. Then, the maximum entropy of $\\mathcal{M}$ can be written as\n\\begin{flalign}\n\\label{objective_MDP}\n&H(\\mathcal{M})=\\sup_{\\pi\\in{\\Pi^S(\\mathcal{M})}}\\Big[\\sum_{s\\in S}\\xi^{\\pi}_sL^{\\pi}(s)\\Big].\n\\end{flalign}\nNote that the right hand side of \\eqref{objective_MDP} can still be infinite or unbounded. We analyze the properties of the maximum entropy of MDPs in the next section.\n\\subsection{Properties of the maximum entropy of MDPs}\\label{characteristics}\n\nThe maximum entropy of an MDP can be infinite or unbounded even for simple cases. For example, consider MDPs given in Fig. \\ref{fig:examples}. For the MDP shown in Fig. \\ref{fig:unbounded}, let the action $a_2$ be taken with probability $\\delta$$\\in$$(0,1]$ in state $s_0$. Then, the expected residence time $\\xi^{\\pi}_{s_0}$ in state $s_0$ is equal to $\\frac{1}{\\delta}$, and the entropy of the induced MC $\\mathcal{M}^{\\pi}$ is given by\n\\begin{align}\nH(\\mathcal{M},{\\pi})=-\\frac{(1-\\delta)\\log(1-\\delta)+\\delta\\log(\\delta)}{\\delta},\n\\end{align} \nwhich satisfies $H(\\mathcal{M},{\\pi})$$\\rightarrow$$\\infty$ as $\\delta$$\\rightarrow$$0$. Note also that if $\\delta$$=$$0$, the entropy of the induced MC is zero due to \\eqref{MC_entropy_biondi}. Hence, the maximum entropy is unbounded, and there is no optimal stationary policy for this MDP. \n\nFor the MDP given in Fig. \\ref{fig:infinite}, choosing a policy such that $\\pi_i(a_j)$$>$$0$ for $i$$=$$1,2$, $j$$=$$1,2$ yields $\\xi^{\\pi}_{s_0}$$=$$\\xi^{\\pi}_{s_1}$$=$$\\infty$ and $L^{\\pi}(s_0)$$>$$0$, $L^{\\pi}(s_1)$$>$$0$. Then, the maximum entropy of this MDP is infinite, and the maximum can be attained by any randomized policy. \n\n\\par Examples in Fig. \\ref{fig:examples} show that we should first verify the existence of optimal policies before attempting to synthesize them. We need the following definitions about the structure of MDPs to state the conditions that cause an MDP to have finite, infinite or unbounded maximum entropy.\n\n\\begin{figure}[b!]\\vspace{-0.5cm}\n\\begin{subfigure}[t]{0.33\\linewidth}\n\\scalebox{0.9}{\n\\begin{tikzpicture}[->, >=stealth', auto, semithick, node distance=2cm]\n\n    \\tikzstyle{every state}=[fill=white,draw=black,thick,text=black,scale=0.7]\n\n    \\node[state,initial,initial text=] (s_0) {$s_0$};\n    \\node[state] (s_1) [right=10mm of s_0]  {$s_1$};\n\n\\path\n(s_0)  edge  [loop above=10]    node{$a_1, 1$}     (s_0)\n(s_0)\t edge     node{$a_2, 1$}     (s_1)\n(s_1)  edge  [loop right=10]    node{$1$}     (s_1);\n\\end{tikzpicture}}\n\\caption{}\n\\label{fig:unbounded}\n\\end{subfigure}\n\\hspace{0.13\\linewidth}\n\\begin{subfigure}[t]{0.33\\linewidth}\n\\scalebox{0.9}{\n\\begin{tikzpicture}[->, >=stealth', auto, semithick, node distance=2cm]\n\n    \\tikzstyle{every state}=[fill=white,draw=black,thick,text=black,scale=0.7]\n\n    \\node[state,initial,initial text=] (s_0) {$s_0$};\n    \\node[state] (s_1) [right=10mm of s_0]  {$s_1$};\n\n\\path\n(s_0)  edge  [loop above=10]    node{$a_1, 1$}     (s_0)\n(s_0)  edge  [bend left=15]    node{$a_2, 1$}     (s_1)\n(s_1)  edge  [loop right=10]    node{$a_1,1$}     (s_1)\n(s_1)  edge  [bend left=15]    node{$a_2,1$}     (s_0);\n\\end{tikzpicture}}\n\\caption{}\n\\label{fig:infinite}\n\\end{subfigure}\n\\caption{Examples of MDPs with (a) unbounded maximum entropy and (b) infinite maximum entropy.}\n\\label{fig:examples}\n\\end{figure}\n\nA \\textit{directed graph} (digraph) is a tuple $G$$=$$(V,E)$ where $V$ is a set of vertices and $E$ is a set of ordered pairs of vertices $V$. For a digraph $G$, a path $v_1v_2\\ldots v_n$ from vertex $v_1$ to $v_n$ is a sequence of vertices such that $(v_k,v_{k+1})$$\\in$$E$ for all $1$$\\leq$$k$$<$$n$. A digraph $G$ is \\textit{strongly connected} if for every pair of vertices $u,v$$\\in$$V$, there is a path from $u$ to $v$, and $v$ to $u$. \n\nA \\textit{sub-MDP} of an MDP is a pair $(C,D)$ where $\\emptyset$$\\neq$$C$$\\subseteq$$S$ and $D$$: $$C$$\\rightarrow$$ 2^{\\mathcal{A}}$ is a function such that (i) $D(s)$$\\subseteq$$\\mathcal{A}(s)$ is non-empty for all $s$$\\in$$C$, and (ii) $s$$\\in$$C$ and $a$$\\in$$D(s)$ imply that $Succ(s,a)$$\\subseteq$$C$. An \\textit{end component} is a sub-MDP $(C,D)$ such that the digraph induced by $(C,D)$ is strongly connected.\n{\\setlength{\\parindent}{0cm}\n\\noindent \\begin{definition}\nA \\textit{maximal end component} (MEC) $(C,D)$ in an MDP is an end component such that there is no end component $(C', D')$ with $(C,D)$$\\neq$$(C',D')$, and $C$$\\subseteq$$C'$ and $D(s)$$\\subseteq$$D'(s)$ for all $s$$\\in$$C$.\\end{definition}}\n \\par A MEC $(C,D)$ in an MDP is \\textit{bottom strongly connected} (BSC) if for all $s$$\\in$$ C$, $\\mathcal{A}(s)$$\\backslash$$D(s)$$=$$\\emptyset$. For a given state $s$$\\in$$C$, we define the set of all actions under which the MDP can leave the MEC $(C, D)$ as $D_0(s)$$:=$$\\{a$$\\in$$\\mathcal{A}(s) | Succ(s,a)$$\\not\\subseteq$$ C\\}$. Note that in a BSC MEC $(C,D)$, $D_0(s)$$=$$\\emptyset$ for all $s$$\\in$$C$. \n  {\\setlength{\\parindent}{0cm}\n \\noindent \\begin{lemma}\\label{Successor}\nFor an MDP $\\mathcal{M}$ with MECs $(C_i,D_i)$ $i$$=$$1,2,\\ldots,n$, let $C$$:=$$\\cup_{i=1}^nC_i$ and $D$$:=$$\\cup_{i=1}^nD_i$. Then, there exists an induced MC $\\mathcal{M}^{\\pi}$ for which a state $s$$\\in$$C$ is both stochastic and recurrent if and only if $|\\cup_{a\\in D(s)}Succ(s,a)|$$>$$1$.$\\quad\\triangleleft$ \\end{lemma}}\n{\\setlength{\\parindent}{0cm}\n \\begin{thm}\\label{Theorem1}\nFor an MDP $\\mathcal{M}$ with MECs $(C_i,D_i)$ $i$$=$$1,2,\\ldots,n$, let $C$$:=$$\\cup_{i=1}^nC_i$ and $D$$:=$$\\cup_{i=1}^nD_i$. Then, the following statements hold. \\\\\n(i)  $H(\\mathcal{M})$  is infinite if and only if there exists an induced MC for which a state $s$$\\in$$C$ is both stochastic and recurrent.\\\\\n(ii)  $H(\\mathcal{M})$ is unbounded if and only if $|\\cup_{a\\in D(s)}Succ(s,a)|$$=$$1$ for all $s$$\\in$$C$, and there exists a MEC that is not bottom strongly connected. \\\\\n(iii)  $H(\\mathcal{M})$ is finite if and only if it is not infinite and not unbounded.$\\quad \\triangleleft$\n\\end{thm}}\\noindent\n\nProofs for above results can be found in Appendix \\ref{proofs_appendix}. Informally, Theorem \\ref{Theorem1} states that for an MDP to have finite maximum entropy, all recurrent states of all MCs that are induced from the MDP by a stationary policy should be deterministic. Although necessary conditions for the finiteness of the maximum entropy is quite restrictive, there are some special cases, such as stochastic shortest path (SSP) problems  \\cite{Bertsekas}, where MDP structures actually satisfy the necessary conditions. Specifically, since all \\textit{proper} policies in SSP problems are guaranteed to reach an absorbing target state within finite time steps with probability 1, the problem of synthesizing a \\text{proper} policy with maximum entropy has a finite solution.\n\nThe following corollary is due to Proposition \\ref{memoryless_1}, Theorem \\ref{Theorem1}, and the definition of finite maximum entropy \\eqref{def_finite_ent}.\n{\\setlength{\\parindent}{0cm}\n \\begin{corollary}\\label{corollary1} If $ \\sup_{\\pi\\in\\Pi(\\mathcal{M})}H(\\mathcal{M},{\\pi})$$<$$\\infty$, then we have\n \\begin{align}\n \\sup_{\\pi\\in\\Pi(\\mathcal{M})}H(\\mathcal{M},{\\pi})= \\max_{\\pi\\in\\Pi^S(\\mathcal{M})}H(\\mathcal{M},{\\pi}).\n \\end{align}\n \\end{corollary}}\n\n\n\\begin{algorithm}[b]\n \\caption{Verify the properties of the maximum entropy.}\n \\textbf{Require:} $\\mathcal{M}$$=$$(S,s_0, \\mathcal{A},\\mathbb{P},\\mathcal{AP},\\mathcal{L})$\\\\\n  \\textbf{Return:} R\n\nFind: MECs $(C_i,D_i)$, $i=1,...,n$\n  \nFind: $Succ(s,a)$ for all $s$$\\in$$S$, $a$$\\in$$\\mathcal{A}(s)$ \n\nR := $\\emptyset$;\n\n  \\begin{algorithmic}\n\\For {$i$$=$$ 1,2,\\ldots,n$}\n \t\\For {$s$ \\textbf{in} $C_i$}\n\t\t\\If{$\\lvert \\cup_{a\\in D_i(s)} Succ(s,a)\\rvert$$>$$1$}\n\t\t\t\\State  R := R $\\cup$ $\\{\\text{infinite}\\}$ ;\n\t\t\\EndIf\n\t\t \\If{ $\\mathcal{A}(s)\\backslash D_i(s)$$\\neq$$\\emptyset$ }\n\t\t\t\\State  R := R $\\cup$ $\\{\\text{unbounded}\\}$ ;\n\t\t\\EndIf\n\t\t\n\n\t\\EndFor\n \\EndFor \n \\If{\\qquad \\ \\ \\text{infinite}$\\in$R } R=infinite \n \\ElsIf {\\ \\ \\text{unbounded}$\\in$R }\\ R=unbounded\n \\Else \\qquad R=finite\n \\EndIf\n \\end{algorithmic}\n \\label{algo_1}\n\\end{algorithm}\n\n\nWe present Algorithm \\ref{algo_1} which, for an MDP $\\mathcal{M}$, verifies whether $H(\\mathcal{M})$ is finite, infinite or unbounded by checking the necessary conditions in Theorem \\ref{Theorem1}. For $\\mathcal{M}$, its MECs can be found in $\\mathcal{O}(\\lvert S\\rvert ^2)$ time \\cite{Model_checking}, $Succ(s,a)$ can be found in $\\mathcal{O}(\\lvert S\\rvert ^2\\lvert \\mathcal{A}\\rvert)$ time, and the necessary conditions can be verified in $\\mathcal{O}(\\lvert S\\rvert)$ time since no state can belong to more than one MEC. Hence, Algorithm \\ref{algo_1} runs in polynomial-time in the size of $\\mathcal{M}$.\n\n\n\\subsection{Policy synthesis  }\\label{policy_syntesis_section}\nWe now provide algorithms to synthesize policies that solve the entropy maximization problem. \n\n\\subsubsection{Finite maximum entropy}\\label{finite_policy_synthesis}\nWe first modify a given MDP by making all states in its MECs absorbing.\n{\\setlength{\\parindent}{0cm}\n\\begin{prop}\\label{modified_MDP_prop} Let $\\mathcal{M}$ be an MDP such that $H(\\mathcal{M})$$<$$\\infty$, $(C_i,D_i)$ $i$$=$$1,2,\\ldots,n$ be MECs in $\\mathcal{M}$, $C$$:=$$\\cup_{i=1}^nC_i$, and $\\mathcal{M'}$ be the modified MDP that is obtained from $\\mathcal{M}$ by making all states $s$$\\in$$C$ absorbing, i.e., if $s$$\\in$$C$, then $\\mathbb{P}_{s,a,s}$$=$$1$ for all $a$$\\in$$\\mathcal{A}(s)$ in $\\mathcal{M'}$. Then, we have\n$ H(\\mathcal{M})$$=$$H(\\mathcal{M'})$.$\\quad \\triangleleft$ \n\\end{prop}}\n\nThere is a one-to-one correspondence between the paths of $\\mathcal{M}$ and $\\mathcal{M'}$ since all states in the set $C$ must have a single successor state in an MDP with finite maximum entropy due to Theorem \\ref{Theorem1}. Moreover, for a given policy $\\pi'$$\\in$$\\Pi^S(\\mathcal{M'})$ on $\\mathcal{M'}$, the policy $\\pi$$\\in$$\\Pi^S(\\mathcal{M})$ induced by $\\pi'$ on $\\mathcal{M}$ is the same policy with $\\pi'$, i.e. $\\pi'$$=$$\\pi$. Therefore, we synthesize an optimal policy for $\\mathcal{M}$ by synthesizing an optimal policy for $\\mathcal{M'}$.\n\n\n We use the nonlinear programming problem in \\eqref{non_reach_objective}-\\eqref{non_reach_cons6} to synthesize an optimal policy for $\\mathcal{M'}$. \n\\begin{subequations}\n\\label{unconstrained_program}\n\\begin{align}\n\\label{non_reach_objective}\n& \\underset{\\lambda(s,a),\\lambda(s)}{\\text{maximize}}\\ \\   -\\sum_{s\\in S\\backslash C}\\sum_{t\\in S}\\eta(s,t)\\log\\Big(\\frac{\\eta(s,t)}{\\nu(s)}\\Big)\\\\\n\\label{non_reach_cons1}\n& \\text{subject to:}\\nonumber\\\\\n& \\nu(s)-\\sum_{t\\in S\\backslash C}\\eta(t,s)=\\alpha(s)  \\ \\quad  \\forall s\\in S\\backslash C\\\\\n\\label{non_reach_cons2}\n& \\lambda(s) -\\sum_{t\\in S\\backslash C}\\eta(t,s)=\\alpha(s) \\   \\quad \\forall s\\in C \\\\\n\\label{non_reach_cons3}\n& \\eta(s,t)=\\sum_{a\\in\\mathcal{A}(s)}\\lambda(s,a)\\mathbb{P}_{s,a,t}  \\quad   \\forall t \\in S, \\forall s\\in S\\backslash C\\\\\n\\label{non_reach_cons4}\n&\\nu(s)=\\sum_{a\\in\\mathcal{A}(s)}\\lambda(s,a)  \\qquad \\qquad \\forall s\\in S\\backslash C \\\\\n\\label{non_reach_cons5}\n&\\lambda(s,a)\\geq 0 \\qquad \\qquad  \\qquad   \\qquad \\forall a\\in\\mathcal{A}(s), \\forall s\\in S\\backslash C\\\\\n\\label{non_reach_cons6}\n&\\lambda(s)\\geq 0 \\qquad \\qquad \\quad  \\qquad  \\qquad \\forall s \\in C\n\\end{align}\n\\end{subequations}\n\nThe decision variables in \\eqref{non_reach_objective}-\\eqref{non_reach_cons2} are $\\lambda(s)$ for each $s$$\\in$$C$, and $\\lambda(s,a)$ for each $s$$\\in$$S\\backslash C$ and each $a$$\\in$$\\mathcal{A}(s)$. \nThe function $\\alpha$$:$$S$$\\rightarrow$$\\{0,1\\}$ satisfies $\\alpha(s_0)$$=$$1$ and $\\alpha(s)$$=$$0$ for all $s$$\\in$$S$$\\backslash$$\\{s_0\\}$. Variables $\\eta(s,t)$ and $\\nu(s)$ are functions of $\\lambda(s,a)$, and used just to simplify the notation. \n\nThe constraints \\eqref{non_reach_cons1}-\\eqref{non_reach_cons2} represent the balance between the ``inflow\" to and ``outflow\" from states. The constraints \\eqref{non_reach_cons3} and \\eqref{non_reach_cons4} are used to simplify the notation and define the variables $\\eta(s,t)$ and $\\nu(s)$, respectively. The constraints \\eqref{non_reach_cons5} and \\eqref{non_reach_cons6} ensure that the expected residence time in the state-action pair $(s,a)$ and the probability of reaching the state $s$ is non-negative, respectively. We refer the reader to  \\cite{Marta}, \\cite{Puterman} for further details about the constraints.\n {\\setlength{\\parindent}{0cm}\n\\begin{prop} \\label{prop_convex} The nonlinear program in \\eqref{non_reach_objective}-\\eqref{non_reach_cons6} is convex. $\\quad \\triangleleft$\\end{prop}}\n\nThe above result indicates that a global maximum for the problem in \\eqref{non_reach_objective}-\\eqref{non_reach_cons6} can be computed efficiently. We now introduce Algorithm \\ref{Algo_2} to synthesize an optimal policy for a given MDP with finite maximum entropy. \n{\\setlength{\\parindent}{0cm}\n\\begin{thm}\n\\label{Main_theorem_2}\n Let $\\mathcal{M}$ be an MDP such that $H(\\mathcal{M})$$<$$\\infty$, $(C_i,D_i)$ $i$$=$$1,2,\\ldots,n$ be MECs in $\\mathcal{M}$, and $C$$:=$$\\cup_{i=1}^nC_i$. For the input ($\\mathcal{M}, C )$, Algorithm \\ref{Algo_2} returns an optimal policy $\\pi^{\\star}$$\\in$$\\Pi^S(\\mathcal{M})$ for $\\mathcal{M}$, i.e. $H(\\mathcal{M},{\\pi^{\\star}})$$=$$H(\\mathcal{M})$. $\\quad \\triangleleft$\n\\end{thm}}\n\n\\begin{algorithm}[t]\n \\caption{Synthesize the maximum entropy policy}\n \\textbf{Require:} $\\mathcal{M}$$=$$(S,s_0, \\mathcal{A},\\mathbb{P},\\mathcal{AP},\\mathcal{L})$ and $C$. \\\\\n  \\textbf{Return:} An optimal policy $\\pi^{\\star}$ for $\\mathcal{M}$\n  \\begin{enumerate}[label=\\arabic*:]\n \\item Form the modified MDP $\\mathcal{M'}$.\n\\item Solve \\eqref{non_reach_objective}-\\eqref{non_reach_cons6} for ($\\mathcal{M'}$, $C$), and obtain $\\lambda^{\\star}(s,a)$.\n\\item \\begin{algorithmic}\n\\For {$s$$\\in$$ S$}\n\t\\If {$s$$\\not\\in$$ C$}\n\t\t\\If {$\\sum_{a\\in\\mathcal{A}(s)}\\lambda^{\\star}(s,a)$$>$$0$}\n\t\t\\State\n\t\t\t\\State $\\pi^{\\star}_s(a)$$:=$$\\frac{\\lambda^{\\star}(s,a)}{\\sum_{a\\in\\mathcal{A}(s)}\\lambda^{\\star}(s,a)}$\n\t\t\\Else\n\t\t \t\\State  $\\pi^{\\star}_s(a)$$:=$$1$ for an arbitrary $a$$\\in$$\\mathcal{A}(s)$,\n\t\t\\EndIf\n\t\\Else\n\t\t\\State $\\pi^{\\star}_s(a)$$:=$$1$ for an arbitrary $a$$\\in$$\\mathcal{A}(s)$.\n\t\\EndIf\n\\EndFor\n \\end{algorithmic}\n \\end{enumerate}\n \\label{Algo_2}\n\\end{algorithm} \nProofs for above results can be found in Appendix \\ref{proofs_appendix}. Computationally, the most expensive step of Algorithm \\ref{Algo_2} is to solve the convex optimization problem \\eqref{non_reach_objective}-\\eqref{non_reach_cons6}. A solution whose objective value is arbitrarily close to the optimal value of \\eqref{non_reach_objective} can be computed in time polynomial in the size of $\\mathcal{M}$ via interior-point methods \\cite{Serrano}, \\cite{Nesterov}. Hence, the time complexity of Algorithm \\ref{Algo_2} is polynomial in the size of $\\mathcal{M}$.\n\n\n\n  \n\\subsubsection{Unbounded maximum entropy} \\label{unbounded_policy_synthesis} There is no optimal policy for this case due to \\eqref{unbounded_definition_1}-\\eqref{unbounded_definition_2}. Therefore, for a given MDP $\\mathcal{M}$ and a constant $\\ell$, we synthesize a policy $\\pi$$\\in$$\\Pi^S(\\mathcal{M})$ such that $H(\\mathcal{M},\\pi)$$\\geq$$\\ell$. Let $S_{B}$ be the union of all states in BSC MECs of $\\mathcal{M}$, which can be found by using Algorithm \\ref{algo_1}. We modify the MDP $\\mathcal{M}$ by making all states $s$$\\in$$S_B$ absorbing and denote the modified MDP by $\\mathcal{M}'$. It can be shown that $H(\\mathcal{M}')$$=$$H(\\mathcal{M})$ by using arguments similar to the ones used in the proof of Proposition \\ref{modified_MDP_prop}. As the first approach, we solve a convex feasibility problem. Specifically, we remove the objective in \\eqref{non_reach_objective} and add the constraint\n\\begin{align}\\label{residence_bound_L}\n-\\sum_{s\\in S\\backslash S_B}\\sum_{t\\in S}\\eta(s,t)\\log\\Big(\\frac{\\eta(s,t)}{\\nu(s)}\\Big)\\geq \\ell\n\\end{align}\nto the constraints in \\eqref{non_reach_cons1}-\\eqref{non_reach_cons6}. Then, we solve the resulting convex feasibility problem for ($\\mathcal{M}'$, $S_B$, $\\ell$) and obtain the desired policy $\\pi$ by using the step 3 of Algorithm \\ref{Algo_2}. \n\nRecall from Theorem \\ref{Theorem1} that the unboundedness of the maximum entropy is caused by the existence of non-BSC MECs in $\\mathcal{M}'$. In particular, we can induce MCs with arbitrarily large entropy by making the expected residence time in states contained in non-BSC MECs arbitrarily large. As the second approach, we bound the expected residence time in states $s$$\\in$$S\\backslash S_B$ in $\\mathcal{M}'$ and relax this bound according to the desired level of entropy. Specifically, we add the constraint\n\\begin{align}\\label{residence_bound}\n\\sum_{s\\in S\\backslash S_B}\\sum_{a\\in\\mathcal{A}(s)}\\lambda(s,a)\\leq \\Gamma\n\\end{align}\nto the problem in \\eqref{non_reach_objective}-\\eqref{non_reach_cons6}. For the constraint \\eqref{residence_bound}, $\\Gamma$$\\geq$$0$ is a predefined value and limits the expected residence time in states $s$$\\in$$S\\backslash S_B$. Let $H_{\\Gamma}(\\mathcal{M}')$ denote the maximum entropy $H(\\mathcal{M}')$ of $\\mathcal{M}'$ subject to the constraint \\eqref{residence_bound}. Then, we have\n\\begin{align}\\label{gamma_ineq}\nH_{\\Gamma}(\\mathcal{M}')\\geq H_{\\Gamma'}(\\mathcal{M}')\n\\end{align}\nfor $\\Gamma$$\\geq$$\\Gamma'$, and $H_{\\Gamma}(\\mathcal{M}')$$=$$\\infty$ for $\\Gamma$$=$$\\infty$. Therefore, by choosing an arbitrarily large $\\Gamma$ value, we can synthesize a policy that induces an MC with arbitrarily large entropy. \n\n\n\\subsubsection{Infinite maximum entropy} \\label{infinitee_policy_synthesis} \nThe procedure to synthesize an optimal policy for MDPs with infinite maximum entropy is very similar to the ones described in Sections \\ref{finite_policy_synthesis} and \\ref{unbounded_policy_synthesis}. Therefore, we provide it in Appendix \\ref{infinite_case_appendix}.\n\\section{Relating the maximum entropy of an MDP with the probability distribution of paths}\\label{relate_paths}\nIn this section, we establish a link between the maximum entropy of an MDP $\\mathcal{M}$ and the entropy of paths in an MC $\\mathcal{M}^{\\pi}$ induced from $\\mathcal{M}$ by a stationary policy $\\pi\\in\\Pi^S(\\mathcal{M})$. \n\nWe begin with an example demonstrating the probability distribution of paths in an MC induced by a policy that maximizes the entropy of an MDP. Consider the MDP shown in Fig. \\ref{fig:MDP_21} which is used in \\cite{Biondi}. The policy that maximizes the entropy of the MDP is given by $\\pi_{s_0}(a_1)$$=$$2\/3$, $\\pi_{s_0}(a_2)$$=$$1\/3$, $\\pi_{s_1}(a_1)$$=$$\\pi_{s_1}(a_2)$$=$$1\/2$. The MC induced by this policy is shown in Fig. \\ref{fig:MDP_22}. There are three paths that reach the MECs, i.e., $(\\{s_3\\},\\{a_1\\})$ and $(\\{s_4\\},\\{a_1\\})$, of the MDP, each of which is followed with probability $1\/3$ in the induced MC, i.e., the probability distribution of paths is uniform. \n\nNote that for the example given in Fig. \\ref{fig:MDP_21}, the optimal policy that maximizes the entropy of the MDP is randomized, and action-selection at each state is performed in an online manner. In particular, an agent that follows the optimal policy chooses its action at each stage according to the outcomes of an online randomization mechanism. Therefore, it does not commit to follow a specific path at any state.\n\\begin{figure}[b!]\\vspace{-0.2cm}\n\\begin{subfigure}[b]{0.3\\linewidth}\n\\scalebox{0.8}{\n\\begin{tikzpicture}[->, >=stealth', auto, semithick, node distance=2cm]\n\n    \\tikzstyle{every state}=[fill=white,draw=black,thick,text=black,scale=0.7]\n\n    \\node[state,initial,initial text=] (s_0) {$s_0$};\n    \\node[state] (s_1) [above right =6mm of s_0]  {$s_1$};\n    \\node[state] (s_2) [below right=6mm of s_0]  {$s_2$};\n    \\node[state] (s_3) [right=10mm of s_1]  {$s_3$};\n    \\node[state] (s_4) [right=10mm of s_2]  {$s_4$};\n\n\\path\n(s_0)  edge     node{$a_1, 1$}     (s_1)\n(s_0)\t edge      node{$a_2, 1$}     (s_2)\n(s_1)\t edge       node{$a_1, 1$}     (s_3)\n(s_1)\t edge       node{$a_2, 1$}     (s_4)\n(s_2)\t edge       node{$ a_1,1$}     (s_4)\n(s_3)\t edge    [loop right]       node{$a_1,1$}     (s_3)\n(s_4)\t edge    [loop right]       node{$a_1,1$}     (s_4);\n\\end{tikzpicture}}\n\\caption{}\n\\label{fig:MDP_21}\n\\end{subfigure}\n\\hspace{0.18\\linewidth}\n\\begin{subfigure}[b]{0.3\\linewidth}\n\\scalebox{0.8}{\n\\begin{tikzpicture}[->, >=stealth', auto, semithick, node distance=2cm]\n\n    \\tikzstyle{every state}=[fill=white,draw=black,thick,text=black,scale=0.7]\n\n    \\node[state,initial,initial text=] (s_0) {$s_0$};\n    \\node[state] (s_1) [above right =6mm of s_0]  {$s_1$};\n    \\node[state] (s_2) [below right=6mm of s_0]  {$s_2$};\n    \\node[state] (s_3) [right=10mm of s_1]  {$s_3$};\n    \\node[state] (s_4) [right=10mm of s_2]  {$s_4$};\n\n\\path\n(s_0)  edge     node{$2\/3$}     (s_1)\n(s_0)\t edge      node{$1\/3$}     (s_2)\n(s_1)\t edge       node{$1\/2$}     (s_3)\n(s_1)\t edge       node{$1\/2$}     (s_4)\n(s_2)\t edge       node{$1$}     (s_4)\n(s_3)\t edge    [loop right]       node{$1$}     (s_3)\n(s_4)\t edge    [loop right]       node{$1$}     (s_4);\n\\end{tikzpicture}}\n\\caption{}\n\\label{fig:MDP_22}\n\\end{subfigure}\n\\caption{(a) An MDP example \\cite{Biondi}. (b) The MC induced by the policy that maximizes the entropy of the MDP. }\n\\label{fig:two-step-authenticate}\n\\end{figure}\n\nTo rigorously establish the relation, illustrated in Fig. \\ref{fig:MDP_21}, between the maximum entropy of an MDP and the entropy of paths in an induced MC, we need the following definitions.\n\n A \\textit{strongly connected component} (SCC) $V$$\\subseteq$$S$ in an MC $\\mathcal{M}^{\\pi}$ induced by a policy $\\pi$$\\in$$\\Pi^S(\\mathcal{M})$ is a maximal set of states in $\\mathcal{M}^{\\pi}$ such that for any $s$,$t$$\\in$$V$, $(\\mathcal{P}^{\\pi})_{s,t}^n$$>$$0$ for some $n$$\\in$$\\mathbb{N}$. A \\textit{bottom strongly connected component} (BSCC) $S_B$ in $\\mathcal{M}^{\\pi}$ is an SCC such that for all $s$$\\in$$S_B$, $(\\mathcal{P}^{\\pi})_{s,t}^n$$=$$0$ for all $n$$\\in$$\\mathbb{N}$ and for all $t$$\\in$$S\\backslash S_B$.\n\nIn this section, for an induced MC $\\mathcal{M}^{\\pi}$, we denote the probability of a path with the finite path fragment $s_0\\ldots s_n$ by\n \\begin{align}\\label{joint_prob_define}\n \\mathcal{P}^{\\pi}(s_0\\ldots s_n):=\\prod_{0\\leq k < n} \\mathcal{P}^{\\pi}_{s_k, s_{k+1}},\n \\end{align}\n which agrees with the probability measure introduced in Section \\ref{Prelim}. Additionally, if the finite path fragment $s_0\\ldots s_n$ in $\\mathcal{M}^{\\pi}$ satisfies $s_0,\\ldots,s_{n-1}$$\\not\\in$$S_B$ and $s_n$$\\in$$S_B$ for some $S_B$$\\subseteq$$S$, we write $s_0\\ldots s_n$$\\in$$(S\\backslash S_B)^{\\star}S_B$. \n{\\setlength{\\parindent}{0cm}\n\\noindent \\begin{definition}\n(Entropy of paths) Let $\\mathcal{M}^{\\pi}$ be an MC induced from an MDP $\\mathcal{M}$ by a stationary policy $\\pi$$\\in$$\\Pi^S(\\mathcal{M})$ and $S_B$$\\subseteq$$S$ be the union of all BSCCs in $\\mathcal{M}^{\\pi}$. For $\\mathcal{M}^{\\pi}$, the \\textit{entropy of the paths} that start from the initial state and reach a state in a BSCC in $\\mathcal{M}^{\\pi}$ is defined as\n\\begin{align}\\label{entropy_paths_def}\n&H(Paths^{\\pi}(\\mathcal{M})): =\\nonumber \\\\\n&\\qquad \\quad-\\sum_{s_0\\ldots s_n \\in T } \\mathcal{P}^{\\pi}(s_0\\ldots s_n)\\log \\mathcal{P}^{\\pi}(s_0\\ldots s_n)\n\\end{align}\nwhere $T$$:=$$Paths_{fin}^{\\pi}(\\mathcal{M})$$\\cap$$ (S\\backslash S_B)^{\\star}S_B$. \n\\end{definition}}\n\nA similar definition for the entropy of paths with fixed initial and final states can be found in \\cite{Ekroot},\\cite{Kafsi}. We note that\n\\begin{align}\\label{sum_prob_paths}\n\\sum_{s_0\\ldots s_n \\in T } \\mathcal{P}^{\\pi}(s_0\\ldots s_n)=1,\n\\end{align}\nsince any finite-state MC eventually reaches a BSCC \\cite{Model_checking}. The following lemma establishes a relation between the entropy of paths and the entropy of an induced MC. \n{\\setlength{\\parindent}{0cm}\n\\begin{lemma}\n\\label{paths_lemma}\nLet $\\mathcal{M}$ be an MDP such that $H(\\mathcal{M},{\\pi})$$<$$\\infty$ for any $\\pi$$\\in$$\\Pi^S(\\mathcal{M})$. Then, for any $\\pi$$\\in$$\\Pi^S(\\mathcal{M})$, we have\n\\begin{align}\nH(Paths^{\\pi}(\\mathcal{M}))=H(\\mathcal{M},{\\pi}). \\quad \\triangleleft\n\\end{align}\n\\end{lemma}}\\noindent\n\n\nA proof for Lemma \\ref{paths_lemma} can be found in Appendix \\ref{proofs_appendix}. Finally, from the definition of the properties of the maximum entropy, Proposition \\ref{memoryless_1} and Lemma \\ref{paths_lemma}, we conclude that, if an MDP $\\mathcal{M}$ has non-infinite maximum entropy, then we have\n\\begin{align}\\label{equivalence_paths_entropy}\n&H(\\mathcal{M})\\ = \\ \\sup_{\\pi\\in\\Pi^S(\\mathcal{M})}  \\ H(Paths^{\\pi}(\\mathcal{M})).\n\\end{align}\nThe equality in \\eqref{equivalence_paths_entropy} states that, for an MDP with non-infinite maximum entropy, a policy that maximizes the entropy of the MDP induces an MC with maximum entropy of paths among all MCs that can be induced from the MDP. Moreover, considering \\eqref{sum_prob_paths}, such a policy maximizes the randomness of all paths with non-zero probability in an induced MC.\n\\section{Constrained Entropy Maximization for MDPs}\\label{cons_section}\nIn this section, we consider the problem of maximizing the entropy of an MDP subject to an LTL constraint. We note that stationary policies are not sufficient to satisfy LTL constraints in general \\cite{Model_checking}. Therefore, to be consistent with our definition of maximum entropy \\eqref{max_ent_definition}, we first introduce the product MDP, over which LTL constraints are transformed into reachability constraints for which stationary policies are sufficient. \n\\subsection{Product MDP}\\label{product_section}\n\\par We construct an MDP that captures all paths of an MDP $\\mathcal{M}$ satisfying an LTL specification $\\varphi$ by taking the product of $\\mathcal{M}$ and the DRA $A_{\\varphi}$ corresponding to the specification $\\varphi$.\n{\\setlength{\\parindent}{0cm}\n \\noindent \\begin{definition} (Product MDP)\nLet $\\mathcal{M}$$=$$(S, s_0, \\mathcal{A}, \\mathbb{P}, \\mathcal{AP}, \\mathcal{L})$ be an MDP and $A_{\\varphi}$$=$$(Q, q_0, 2^{\\mathcal{AP}}, \\delta, Acc)$ be a DRA. The product MDP $\\mathcal{M}_p$$=$$\\mathcal{M}$$\\otimes$$ A_{\\varphi}$$=$$(S_p, s_{0_p}, \\mathcal{A}, \\mathbb{P}_p, \\mathcal{L}_p, Acc_p)$ is a tuple where\n\\begin{itemize}\n\\item $S_p$$=$$S $$\\times$$ Q$,\n\\item $s_{0_p}=(s_0,q)$ such that $q=\\delta(q_0,\\mathcal{L}(s_0))$,\n\\item $\\mathbb{P}_p((s,q), a, (s',q'))$=$\\begin{cases} \\mathbb{P}_{s,a,s'} & \\text{if} \\quad q'=\\delta(q,\\mathcal{L}(s')) \\\\ 0 & \\text{otherwise}, \\end{cases}$\n\\item $\\mathcal{L}_p((s,q))=\\{q\\}$,\n\\item $Acc_p$$=$$\\{(J_1^p,K_1^p),\\ldots,(J_k^p,K_k^p) \\}$ where $J_i^p$$=$$S$$\\times$$J_i$ and $K_i^p$$=$$S$$\\times$$K_i$ for all $(J_i,K_i)$$\\in$$Acc$ and for all $i$$=$$1,\\ldots, k$.\n\\end{itemize}\n\\end{definition}}\nThe product MDP $\\mathcal{M}_p$ may contain unreachable states which can be found in time polynomial in the size of $\\mathcal{M}_p$ by graph search algorithms, e.g., breadth-first search. Such states have no effect in the analysis of MDPs, and hence, can be removed from the MDP.  \nWe hereafter assume that there is no unreachable state in $\\mathcal{M}_p$. \n\nThere is a one-to-one correspondence between the paths of $\\mathcal{M}_p$ and $\\mathcal{M}$ \\cite{Model_checking}. Moreover, a similar one-to-one correspondence exists between policies on $\\mathcal{M}_p$ and $\\mathcal{M}$. More precisely, for a given policy $\\pi^{p}$$=$$\\{\\mu_0^p,\\mu_1^p,\\ldots\\}$ on $\\mathcal{M}_p$, we can construct a policy $\\pi$$=$$\\{\\mu_0,\\mu_1,\\ldots\\}$ on $\\mathcal{M}$ by setting $\\mu_i(s_i)$$=$$\\mu_i^p((s_i,q_i))$. For a given policy $\\pi^p$$\\in$$\\Pi^S(\\mathcal{M}_p)$ on $\\mathcal{M}_p$, the policy $\\pi$$\\in$$\\Pi(\\mathcal{M})$ constructed in this way is a non-stationary policy \\cite{Model_checking}. \n\n\nLet $\\pi^p$$\\in$$\\Pi^S(\\mathcal{M}_p)$ be a policy on $\\mathcal{M}_p$ and $\\pi$$\\in$$\\Pi(\\mathcal{M})$ be the policy on $\\mathcal{M}$ constructed from $\\pi^p$ through the procedure explained above. The paths of the MDP $\\mathcal{M}$ under the policy $\\pi$ satisfies the LTL specification $\\varphi$ with probability of at least $\\beta$, i.e., $\\text{Pr}^{\\pi}_{\\mathcal{M}}(s_0$$\\models$$\\varphi)$$\\geq$$\\beta$, if and only if the paths of the product MDP $\\mathcal{M}_p$ under the policy $\\pi^p$ reaches accepting MECs in $\\mathcal{M}_p$ with probability of at least $\\beta$ and stays there forever \\cite{Model_checking}. \n{\\setlength{\\parindent}{0cm}\n\\noindent \\begin{definition} (Accepting MEC)\nA MEC $(C,D)$ in a product MDP $\\mathcal{M}_p$ with the set of accepting state pairs $Acc_p$ is an \\textit{accepting} MEC if for some $(J^p,K^p)$$\\in$$Acc_p$, $J^p$$\\not\\in$$C$ and $K^p$$\\in$$C$.\n\\end{definition}}\nInformally, accepting MECs are sets of states where the system can remain forever, and where the set $K_p$ is visited infinitely often and the set $J_p$ is visited finitely often. \n\n\\vspace{-0.4cm}\n\\subsection{Constrained Problem}\n In this section, we formally state the constrained entropy maximization problem. Recall that, for an MDP $\\mathcal{M}$, the problem of synthesizing a policy $\\pi$$\\in$$\\Pi(\\mathcal{M})$ that satisfies an LTL formula $\\varphi$ with probability of at least $\\beta$, i.e., $\\text{Pr}^{\\pi}_{\\mathcal{M}}(s_0$$\\models$$\\varphi)$$\\geq$$\\beta$, is equivalent to the problem of synthesizing a policy $\\pi$$\\in$$\\Pi^S(\\mathcal{M}_p)$ that reaches the accepting MECs in $\\mathcal{M}_p$ with probability of at least $\\beta$ and stays there forever. \n\n\nOur objective is to synthesize a policy that induces a stochastic process with maximum entropy whose paths satisfy the given LTL specification with desired probability. To this end, we synthesize a policy $\\pi$$\\in$$\\Pi^S(\\mathcal{M}_p)$ on $\\mathcal{M}_p$ that reaches the accepting MECs in $\\mathcal{M}_p$ with probability of at least $\\beta$ and stays there forever.\n\nWe first partition the set $S_p$ of states of $\\mathcal{M}_p$ into three disjoint sets as follows. We let $B$ be the set of all states in accepting MECs in $\\mathcal{M}_p$, and $S_0$ be the set of all states that have zero probability of reaching the set $B$. Finally, we let $S_r$$=$$S_p\\backslash \\{B\\cup S_0\\}$ be the set of all states that are not in an accepting MEC in $\\mathcal{M}_p$ and have nonzero probability of reaching the set $B$. These sets can be found in time polynomial in the size of $\\mathcal{M}_p$ by graph search algorithms \\cite{Model_checking}. \n {\\setlength{\\parindent}{0cm}\n\\noindent \\begin{problem}(\\textbf{Constrained Entropy Maximization})\\label{prob_2}\nFor a product MDP $\\mathcal{M}_p$, verify whether there exists a policy $\\pi^{\\star}$$\\in$$\\Pi^S(\\mathcal{M}_p)$ that solves the following problem:\n\\begin{subequations}\n\\begin{align}\\label{max_ent_product_objective}\n&\\underset{\\pi\\in\\Pi^S(\\mathcal{M}_p)}{\\text{maximize}}  \\qquad H(\\mathcal{M}_p,{\\pi})\\\\ \\label{LTL_constraint_product}\n& \\text{subject to:} \\qquad  \\text{Pr}^{\\pi}_{\\mathcal{M}_p}(s_0\\models \\lozenge B)\\geq \\beta\n\\end{align}\n\\end{subequations}\nwhere $\\text{Pr}^{\\pi}_{\\mathcal{M}_p}(s_0$$\\models$$\\lozenge B)$ denotes the probability of reaching the set $B$ from the initial state in $\\mathcal{M}_p$ under the policy $\\pi$. If such a policy exists, provide an algorithm to synthesize it. If it does not exist, provide a procedure to synthesize a policy $\\pi'$$\\in$$\\Pi^S(\\mathcal{M}_p)$ such that  $\\text{Pr}^{\\pi'}_{\\mathcal{M}_p}(s_0\\models \\lozenge B)$$\\geq$$\\beta$ and $H(\\mathcal{M},\\pi')$$\\geq$$\\ell$ for a given constant $\\ell$.\n\\end{problem}}\n\nNote that if a policy that solves the problem in \\eqref{max_ent_product_objective}-\\eqref{LTL_constraint_product} chooses the actions in states $s$$\\in$$B$ such that they form a BSCC in the induced MC, then the resulting policy ensures that the paths of the induced MC visit the states inside the set $B$ infinitely often and thus satisfies $\\varphi$\\cite{Model_checking}. \n\\subsection{Policy synthesis}\\label{product_policy_section}\n  \nIn this section, for a product MDP $\\mathcal{M}_p$ and its state partition $S_p$$=$$B$$\\cup$$S_0$$\\cup$$S_r$, we assume that $0$$<$$\\beta$$\\leq$$\\max_{\\pi\\in\\Pi(\\mathcal{M})}\\text{Pr}^{\\pi}_{\\mathcal{M}_p}(s_0$$\\models$$\\lozenge B)$, which can be verified in polynomial time by solving a linear optimization problem as shown in \\cite{Model_checking, Marta}. We refer to a policy $\\pi$$\\in$$\\Pi^S(\\mathcal{M}_p)$ as an \\textit{optimal policy} if it is a solution to the problem in \\eqref{max_ent_product_objective}-\\eqref{LTL_constraint_product} and chooses the actions in states $s$$\\in$$B$ such that they form a BSCC in the induced MC.\n\n\nFor the synthesis of an optimal policy, we consider three cases according to the maximum entropy $H(\\mathcal{M}_p)$ of $\\mathcal{M}_p$, namely, finite, unbounded and infinite. \n \n \\subsubsection{Finite maximum entropy} \\label{finite_constrained_case}\nLet $(C_i,D_i)$ $i$$=$$1,2,\\ldots,n$ be the MECs in $\\mathcal{M}_p$, $C$$:=$$\\cup_{i=1}^n C_i$, and $D$$:=$$\\cup_{i=1}^n D_i$. We form the modified product MDP $\\mathcal{M}_p'$ by making all states $s$$\\in$$C$ absorbing in $\\mathcal{M}_p$. We have $H(\\mathcal{M}_p')$$=$$H(\\mathcal{M}_p)$ due to Proposition \\ref{modified_MDP_prop}. Recall that for a state $s$$\\in$$C$, the variable $\\lambda(s)$ in \\eqref{non_reach_objective}-\\eqref{non_reach_cons6} represents the probability of reaching the state $s$ from the initial state \\cite{Marta}. Hence, we append the constraint\n\\begin{align}\\label{reach_cons}\n\\sum_{s\\in B}\\lambda(s)\\geq \\beta\n\\end{align}\nto the problem in \\eqref{non_reach_objective}-\\eqref{non_reach_cons6} in order to obtain a policy that induces an MC whose paths satisfy $\\varphi$ with probability of at least $\\beta$. Noting that $\\beta$$\\leq$$\\sum_{s\\in B}\\lambda(s)$$\\leq$$\\max_{\\pi\\in\\Pi(\\mathcal{M})}\\text{Pr}^{\\pi}_{\\mathcal{M}_p}(s_0$$\\models$$\\lozenge B)$, the resulting optimization problem always has a solution since its feasible set constitutes a closed compact set when the product MDP has finite maximum entropy. \n\n\nThe procedure to obtain a policy $\\pi^{\\star}_p$$\\in$$\\Pi^S(\\mathcal{M}_p)$ that solves the problem in \\eqref{max_ent_product_objective}-\\eqref{LTL_constraint_product} for $\\mathcal{M}_p$ with finite maximum entropy is as follows. First, we find MECs $(C,D)$ in $\\mathcal{M}_p$ and form the modified MDP $\\mathcal{M}_p'$ by making all states $s$$\\in$$C$ absorbing. Second, we solve the problem in \\eqref{non_reach_objective}-\\eqref{non_reach_cons6} for $(\\mathcal{M}_p', C, \\beta)$ with the additional constraint \\eqref{reach_cons}. Finally, we use step 3 of Algorithm \\ref{Algo_2} to synthesize the policy $\\pi^{\\star}_p$$\\in$$\\Pi(\\mathcal{M}_p)$. Note that the constructed policy ensures that, once reached, the system stays in the set $B$ forever, since all MECs in $\\mathcal{M}_p$ with finite maximum entropy are bottom strongly connected. \n\n \\subsubsection{Unbounded maximum entropy}\\label{unbounded_constrained_case} In this case, the product MDP $\\mathcal{M}_p$ contains a non-BSC MEC due to Theorem \\ref{Theorem1}. We assume that there is only one non-BSC MEC in $\\mathcal{M}_p$, and it is contained in $S_r$. \nWe first form the modified product MDP $\\mathcal{M}'_p$ by making all states in BSC MECs in $\\mathcal{M}_p$ absorbing. Note that $H(\\mathcal{M}_p')$$=$$H(\\mathcal{M}_p)$. Let $S_B$ denote the union of all absorbing states in $\\mathcal{M}'_p$.\nWe verify the existence of a solution to the problem in \\eqref{max_ent_product_objective}-\\eqref{LTL_constraint_product} by solving the problem in \\eqref{non_reach_objective}-\\eqref{non_reach_cons6} for $(\\mathcal{M}_p', S_B, \\beta)$ with the additional constraint \\eqref{reach_cons}. If the optimum value of the resulting problem is bounded, then we synthesize an optimal policy through step 3 of Algorithm \\ref{Algo_2}. If it is not bounded, then there exists no optimal policy, in which case for a given constant $\\ell$, we synthesize a policy $\\pi^{\\star}$$\\in$$\\Pi^S(\\mathcal{M}_p)$ such that $H(\\mathcal{M}_p,\\pi^{\\star})$$\\geq$$\\ell$ and $\\text{Pr}^{\\pi^{\\star}}_{\\mathcal{M}_p}(s_0\\models \\lozenge B)$$\\geq$$ \\beta$ by employing two different approaches. \n\n\nAs the first approach, we solve a convex feasibility problem. Specifically, for the problem in \\eqref{non_reach_cons1}-\\eqref{non_reach_cons6}, we remove the objective \\eqref{non_reach_objective} and append the constraints \\eqref{residence_bound_L} and \\eqref{reach_cons} to the problem. Then, we solve the resulting convex feasibility problem for $(\\mathcal{M}_p', S_B, \\ell, \\beta)$, and using step 3 of Algorithm \\ref{Algo_2}, obtain a policy $\\pi^{\\star}_p$$\\in$$\\Pi^S(\\mathcal{M}_p)$ such that $H(\\mathcal{M}_p,\\pi^{\\star}_p)$$\\geq$$\\ell$ and $\\text{Pr}^{\\pi^{\\star}_p}_{\\mathcal{M}_p}(s_0$$\\models$$\\lozenge B)\\geq \\beta$. \n\nThe second approach to obtain an induced MC with arbitrarily large entropy, whose paths satisfy the LTL specification with desired probability, is to bound the expected residence time in states $s$$\\in$$S_p\\backslash S_B$ and relax this bound according to the desired level of entropy. Specifically, we solve the problem in \\eqref{non_reach_objective}-\\eqref{non_reach_cons6} for $(\\mathcal{M}_p', S_B, \\beta, \\Gamma)$ together with the constraints \\eqref{residence_bound} and \\eqref{reach_cons}, where $\\Gamma$ is as defined in Section \\ref{unbounded_policy_synthesis}. Then, by choosing an arbitrarily large $\\Gamma$ value, we obtain an induced MC with the desired level of entropy. \n\nFinally, to ensure that the paths of the MC that is induced by the synthesized policy satisfies the LTL specification $\\varphi$ with desired probability, we choose actions in states $s$$\\in$$B$ such that $Succ(s)$$\\subseteq$$B$.\n\n \\subsubsection{Infinite maximum entropy}\\label{infinitee_constrained_case}\nFor product MDPs with infinite maximum entropy, the verification of the existence and the synthesis of an optimal policy are achieved by procedures that are very similar to the ones presented in Sections \\ref{finite_constrained_case} and \\ref{unbounded_constrained_case}. Hence, we provide the analysis for product MDPs with infinite maximum entropy in Appendix \\ref{infinite_case_appendix}. \n\n\\section{Examples}\\label{examples_section}\nIn this section, we illustrate the proposed methods on different motion planning scenarios. All computations are run on a 2.2 GHz dual core desktop with 8 GB RAM. All optimization problems are solved by using the splitting conic solver (SCS) \\cite{SCS} in CVXPY \\cite{cvxpy}.  For all LTL specifications, we construct deterministic Rabin automata using ltl2dstar \\cite{ltl2dstar}.\n\nIn most motion planning scenarios, an agent can return to its current position by following different paths. Therefore, in general, the maximum entropy of an MDP that models the motion of an agent is either unbounded or infinite. However, as explained in Section \\ref{policy_syntesis_section} and shown in the following examples, a policy that induces a stochastic process with an arbitrarily large entropy can easily be obtained by introducing constraints on the expected residence time in certain states. Additional motion planning examples are provided in \\cite{Yagiz}. \n\\subsection{Relation between entropy and exploration}\nRandomizing an agent's paths while ensuring the completion of a task is important for achieving a better exploration of the environment \\cite{Saerens} and obtaining a robust behavior against transition perturbations \\cite{Deep_learning}. In this example, we demonstrate how the proposed method randomizes the agent's paths depending on the expected time until the completion of the task. \n\n\\textit{Environment:} We consider the grid world shown in Fig. \\ref{grid_graph} (left). The agent starts from the brown state. The red and green states are absorbing, i.e., once entered those states cannot be left. The agent has four actions in all other states, namely left, right, up and down. At each state, a transition to the chosen direction occurs with probability (w.p.) 0.7, and the agent slips to each adjacent state in the chosen direction w.p. 0.15. If the adjacent state in the chosen direction is a wall, e.g. up in brown state, a transition to the chosen direction occurs w.p. 0.85. If the state in the chosen direction is a wall, e.g., left in brown state, the agent stays in the same state w.p. 0.7 and moves to each adjacent state w.p. 0.15.  \n\\newcommand{\\StaticObstacle}[2]{ \\fill[red] (#1+0.1,#2+0.1) rectangle (#1+0.9,#2+0.9);}\n\\newcommand{\\initialstate}[2]{ \\fill[brown] (#1+0.15,#2+0.15) rectangle (#1+0.85,#2+0.85);}\n\\newcommand{\\goalstate}[2]{ \\fill[green] (#1+0.15,#2+0.15) rectangle (#1+0.85,#2+0.85);}\n\\begin{figure}[b]\\vspace{-0.4cm}\n\\begin{subfigure}[b]{0.3\\linewidth}\n\\scalebox{0.32}{\n\\begin{tikzpicture}\n\\draw[black,line width=1pt] (0,0) grid[step=1] (11,11);\n\\draw[black,line width=4pt] (0,0) rectangle (11,11);\n\t\t\t \n\t\t\t   \\StaticObstacle{4}{3}  \\StaticObstacle{5}{3}  \\StaticObstacle{6}{3}\n\t\t\t   \\StaticObstacle{4}{7}  \\StaticObstacle{5}{7}  \\StaticObstacle{6}{7}\n\t\t\t    \\initialstate{0}{5}  \\goalstate{10}{5}\n\t\t\t    \n\\node at (0.5,5+0.5) { \\textbf{\\huge S}};\n\\node at (10+0.5,5+0.5) { \\textbf{\\huge T}};\n\\node at (4+0.5,3+0.5) { \\textbf{\\huge B}};\n\\node at (5+0.5,3+0.5) { \\textbf{\\huge B}};\n\\node at (6+0.5,3+0.5) { \\textbf{\\huge B}};\n\\node at (4+0.5,7+0.5) { \\textbf{\\huge B}};\n\\node at (5+0.5,7+0.5) { \\textbf{\\huge B}};\n\\node at (6+0.5,7+0.5) { \\textbf{\\huge B}};\n\n\n\\end{tikzpicture}\n\n}\n\\end{subfigure}\n\\hspace{0.2\\linewidth}\n\\begin{subfigure}[b]{0.3\\linewidth}\n\\scalebox{0.35}{\n\\begin{tikzpicture}\n\\draw[black,line width=1pt] (0,0) grid[step=1] (10,10);\n\\draw[black,line width=4pt] (0,0) rectangle (10,10);\n\t\t\t \n\t\t\t   \\StaticObstacle{2}{2}  \\StaticObstacle{3}{2}\n\t\t\t   \\StaticObstacle{2}{3}  \\StaticObstacle{3}{3}\t\n\t\t\t \\StaticObstacle{6}{6}  \\StaticObstacle{7}{6}\n\t\t\t \\StaticObstacle{6}{7}  \\StaticObstacle{7}{7}  \n\t\t\t \\initialstate{0}{0} \\goalstate{9}{9}\n\\node at (0.5,0.5) { \\textbf{\\huge S}};\n\\node at (5+0.5,0.5) { \\textbf{\\huge R1}};\n\\node at (8+0.5,3+0.5) { \\textbf{\\huge R2}};\n\\node at (1+0.5,6+0.5) { \\textbf{\\huge R3}};\n\\node at (4+0.5,8+0.5) { \\textbf{\\huge R4}};\n\\node at (9+0.5,9+0.5) { \\textbf{\\huge T}};\n\\node at (2+0.5,2+0.5) { \\textbf{\\huge B}};\n\\node at (2+0.5,3+0.5) { \\textbf{\\huge B}};\n\\node at (3+0.5,2+0.5) { \\textbf{\\huge B}};\n\\node at (3+0.5,3+0.5) { \\textbf{\\huge B}};\n\\node at (6+0.5,7+0.5) { \\textbf{\\huge B}};\n\\node at (7+0.5,7+0.5) { \\textbf{\\huge B}};\n\\node at (6+0.5,6+0.5) { \\textbf{\\huge B}};\n\\node at (7+0.5,6+0.5) { \\textbf{\\huge B}};\n\\end{tikzpicture}\n}\n\\end{subfigure}\n\\caption{Grid world environments. The brown (S) and green (T) states are the initial and target states, respectively. The red (B) states are absorbing. }\n\\label{grid_graph}\n\\end{figure}\n\n\n\n\\textit{Task:} The agent's task is to reach and stay in the green state, labeled as $T$, while avoiding the red states, labeled as $B$. Formally, the task is $\\varphi$$=$$\\square \\lnot B \\land  \\lozenge \\square T$.\n\n We form the product MDP for the given task. It has 484 states, 1196 transitions, 10 MECs, and the average number of states in each MEC is 23. We require the agent to complete the task w.p. 1, i.e., $\\text{Pr}_{\\mathcal{M}}^{\\pi}(s_0$$\\models$$\\varphi)$$=$$1$. The maximum entropy of the product MDP subject to the LTL constraint is unbounded. The minimum expected time $\\Gamma$ required to complete the task $\\varphi$ is roughly $14$ time steps, which can be computed by replacing the objective in \\eqref{non_reach_objective} with ``minimize \\ $\\sum_{s\\in S_r} \\sum_{a\\in \\mathcal{A}(s)}\\lambda(s,a)$\" and appending \\eqref{reach_cons} to the constraints in \\eqref{non_reach_cons1}-\\eqref{non_reach_cons6}. \n\nWe synthesize two policies for two different expected times until the completion of the task. First, we synthesize a policy by requiring the agent to complete the task as fast as possible, i.e., $\\Gamma$$=$$14$ time steps. Then, we synthesize a policy by allowing the agent to spend more time in the environment until the completion of the task, i.e., $\\Gamma$$=$$60$ time steps. Solving the convex optimization problems take 122 and 166 seconds for $\\Gamma$$=$$14$ and $\\Gamma$$=$$60$ time steps, respectively.\n\n\nThe expected residence time in states for the induced MCs are shown in Fig. \\ref{exploration_figure}.  We remind the reader that the environment is given in Fig. \\ref{grid_graph} (left).\n\\begin{figure}[t]\n\\begin{subfigure}[b]{0.3\\linewidth}\n\\includegraphics[width= 3.95 cm, trim= {50 30 150 40 }, clip=true]{heat_map1.pdf}\n\\end{subfigure}\n\\hspace{0.15\\linewidth}\n\\begin{subfigure}[b]{0.3\\linewidth}\n\\includegraphics[width= 5 cm, trim= {50 30 50 40 }, clip=true]{heat_map2.pdf}\n\\end{subfigure}\n\\caption{The expected residence time in states for different expected times $\\Gamma$ until the completion of the task  \\textit{in the same environment}. (Left) $\\Gamma$$=$$14$ time steps, i.e., the minimum time required to complete the task with probability 1. (Right) $\\Gamma$$=$$60$.   }\n\\vspace{-0.3cm}\n\\label{exploration_figure}\n\\end{figure} \nWhen the agent is given the minimum time $\\Gamma$$=$$14$ time steps (left) to complete the task, it follows only the shortest paths, and therefore, cannot explore the environment. On the other hand, as it is allowed to spend more time, i.e., $\\Gamma$$=$$60$ time steps (right), in the environment, it visits different states more often and utilizes different paths to complete the task. Consequently, the synthesized policy enables the continual exploration of the environment while ensuring the completion of the task.\n\\subsection{Relation between entropy and predictability}\\label{example_1}\n\nIn this example, we consider an agent whose aim is to complete a task while leaking minimum information about its paths to an observer. We illustrate how the restrictions applied to the agent's paths by the task affect the predictability.\n\n\\textit{Environment:} We consider the grid world shown in Fig. \\ref{grid_graph} (right). The agent starts from the brown (S) state. The red (B) states and green (T) state are absorbing. The agent has four actions in all other states, namely left, right, up and down. A transition to the chosen direction occurs w.p. 1 if the state in that direction is not a wall. If it is a wall, e.g., left direction in brown state, the agent stays in the same state w.p. 1.\n\n\n\\textit{Tasks:} We consider five increasingly restrictive task specifications for the agent which are listed in Table \\ref{tasks_table}. The first task $\\varphi_1$ is to reach and stay in the $T$ state while avoiding all red states. The second task $\\varphi_2$ requires the agent to visit $R4$ state before completing the first task. The third task $\\varphi_3$ requires the agent to visit $R3$ state before completing the second task and so on. \n\n\\begin{table}[h!]\\vspace{-0.3cm}\n\\centering\n\\caption{The agent's tasks. }\n\\scalebox{1.1}{\n\\begin{tabular}{ | l |}\n  \\hline\t\t\t\n$\\varphi_1$$=$$\\square \\lnot Red \\land \\lozenge \\square T$ \\\\  \\hline  \n $\\varphi_2$$=$$\\square \\lnot Red \\land   \\lozenge R4\\land \\lozenge \\square T$ \\\\  \\hline  \n $\\varphi_3$$=$$\\square \\lnot Red \\land \\lozenge( R3\\land \\lozenge R4 )\\land  \\lozenge \\square T$ \\\\  \\hline  \n $\\varphi_4$$=$$\\square \\lnot Red \\land \\lozenge( R2\\land \\lozenge (R3 \\land \\lozenge R4) )\\land  \\lozenge \\square T$ \\\\  \\hline  \n $\\varphi_5$$=$$\\square \\lnot Red \\land  \\lozenge( R1\\land \\lozenge (R2 \\land \\lozenge (R3 \\land \\lozenge R4)) )\\land   \\lozenge \\square T$ \\\\ \\hline  \n\\end{tabular}}\n\\label{tasks_table}\n\\end{table}\n\n\\textit{Observer:} There is an observer that aims to predict the agent's paths in the environment. The observer is aware of the agent's task, knows the transition probabilities exactly, and runs yes-no probes in each state to determine the  successor state of the agent, i.e., probes that return an answer yes if the agent moves to the predicted successor state and no otherwise. The average number of yes-no probes run in a state is the expected number of observations needed by the observer to determine the correct successor state in that state \\cite{Paruchuri}. The observer uses the Huffman procedure \\cite{Huffman} to minimize the required number of probes. Let $\\mathcal{P}_{s}$$=$$(\\mathcal{P}_{s,1}, \\mathcal{P}_{s,2}, \\ldots, \\mathcal{P}_{s,n} )$ be the transition probabilities from state $s$ to successor states sorted in decreasing order. The number of yes-no probes run in state $s$ is denoted by $\\Upsilon_s$$=$$\\mathcal{P}_{s,1}+\\ldots+(n-1)\\mathcal{P}_{s,n-1}$$+$$(n-1)\\mathcal{P}_{s,n}$. The expected number of observations required to determine the agent's path is given by $O_{avg}$$=$$\\sum_s\\xi_s\\Upsilon_s$, which weighs the required number of probes in each state with the expected residence time in the state. We refer the reader to \\cite{Paruchuri} for further details about the observer model.\n\nWe form product MDPs for all tasks. The product MDP with the maximum number of states and transitions is the one constructed for the task $\\varphi_5$. It has 800 states, 2138 transitions, 12 MECs, and the average number of states in each MEC is 29. For each task, we require the agent to complete the task w.p. 1. The maximum entropy of all product MDPs subject to corresponding LTL constraints are unbounded. We bound the expected time until the completion of any task by taking $\\Gamma$$=$$33$ time steps, which is the minimum expected time required to complete the task $\\varphi_5$, i.e., the most restrictive task. For each task, we synthesize a policy for the agent using the procedure explained in Section \\ref{cons_section}. The longest computation time, which is for $\\varphi_5$, is 15.2 seconds. \n\nThe entropy of Markov chains induced by the synthesized policies and the average number of observations required to predict the agent's paths are shown in Fig. \\ref{tasks_figure}. As the task imposes more restrictions on the agent's paths, the entropy of the induced MC decreases and the prediction requires less observations in average. Additionally, as can be seen in Fig. \\ref{tasks_figure}, the required numbers of observation for $\\varphi_4$ and $\\varphi_5$ are significantly smaller than those for $\\varphi_1$, $\\varphi_2$ and $\\varphi_3$. This decrease is due to that these tasks force the agent to follow an ``S-shaped\" path in a restricted time, i.e. $\\Gamma=33$ time steps. For these tasks, although the agent still randomizes its paths to some extent, better predictability results cannot be obtained due to time restrictions.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[height=4 cm, width=7cm, trim= {0 0 0 30 }, clip=true]{tasks_vs_obs.pdf}\n\\caption{ The relation between the maximum entropy of an MDP subject to an LTL constraint and the required number of observations to predict the agent's paths.    }\n\\label{tasks_figure}\\vspace{-0.5cm}\n\\end{figure}\n\n\\subsection{Predictability in a randomly generated MDP }\nIn this example, we investigate the relation between the probability of completing a task and the predictability of paths. We also evaluate the proposed algorithm against the algorithms introduced in \\cite{Paruchuri}.\n\n\\textit{Environment:} We generate an MDP with 200 states, where each state has 8 randomly selected successor states. We choose four states, make them absorbing, and label three of them as ``unsafe\" states and the remaining one as the ``target\" state. The agent has 5 actions at each state, for which the transition probabilities to successor states are assigned randomly.  \n\n\\textit{Task:} The agent's task is to reach the target state while avoiding the unsafe states, i.e., $\\varphi$$=$$\\square \\lnot unsafe \\land \\lozenge\\square  target$.\n\n\\textit{Observer:} We use the same observer model introduced in Section \\ref{example_1}.\n\n\\textit{Policies:} We compare the proposed method with weighted maximum entropy ($WME$) and binary search for randomization linear programming ($BRLP$) algorithms which are introduced in \\cite{Paruchuri} for randomizing an agent's policy to minimize predictability. We note that in \\cite{Paruchuri}, the authors claim that $WME$ algorithm is non-convex and cannot be solved in polynomial time. However, its convexity can be proven by Proposition \\ref{prop_convex} since it solves a special case of the convex optimization problem given in \\eqref{unconstrained_program}, i.e., it is equivalent to problem in \\eqref{unconstrained_program} when transition probabilities are either 0 or 1. We refer the reader to \\cite{Paruchuri} for further details about the $WME$ and $BRLP$ algorithms. \n\n We form the product MDP. It has 800 states, 2172 transitions and 5 MECs each of which contains a single state. The maximum probability of completing the task $\\varphi$ is obtained as $\\beta$$=$$0.75$ by solving a linear programming problem introduced in \\cite{Marta}. The maximum entropy of the product MDP subject to the LTL constraint $\\text{Pr}_{\\mathcal{M}}^{\\pi}(s_0$$\\models$$\\varphi)$$\\geq$$\\beta$ is unbounded for all $\\beta$$>$$0$. We fix the expected time until the completion of the task to $\\Gamma$$=$$200$ time steps, and synthesize policies for different values of $\\beta$. Solving the optimization problems takes at most 150, 155, and 92 seconds for the proposed method, $WME$ and $BRLP$ algorithms, respectively. \n\nThe required number of observations to predict the agent's paths for different $\\beta$ values are shown in Fig. \\ref{compare_paruchuri}. As the probability of completing the task decreases, the randomness of the agent's paths increases and the prediction requires more observations in average. Therefore, there is a trade-off between the probability of satisfying the task and the randomness of the paths. Additionally, the proposed method (green) requires two times more observations than the $BRLP$ algorithm (red) when $\\beta$$=$$0.5$. Note also that the $WME$ algorithm (blue) cannot achieve better predictability results than the proposed method because it does not exploit the inherent stochasticity in the environment and rely solely on the randomization of the agent's actions to generate unpredictable paths.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[height=4 cm, width=7cm, trim= {0 0 0 30 }, clip=true]{comparison_example.pdf}\n\\caption{The trade-off between the probability of completing the task and the predictability of paths. $BRLP$ and $WME$ are algorithms proposed in \\cite{Paruchuri} to randomize the agent's actions.}\n\\label{compare_paruchuri}\n\\end{figure}\n\n\n\n\\section{Conclusions and Future Work}\\label{conclusion_section}\nWe showed that the maximum entropy of an MDP can be either finite, infinite or unbounded, and presented an algorithm to verify the property of the maximum entropy for a given MDP. We presented an algorithm, based on a convex optimization problem, to synthesize a policy that maximizes the entropy of an MDP. For MDPs with non-infinite maximum entropy, we established the equivalence between the maximum entropy of an MDP and the maximum entropy of paths in the MDP. Finally, we provided a procedure to obtain a policy that maximizes the entropy of an MDP while ensuring the satisfaction of a temporal logic specification with desired probability.\n\nAn interesting future direction is to include adversaries to the framework by modeling the problem as a two-player game. Being informed about the aims and capabilities of rational\/irrational adversaries in the environment, an agent may want to explore its environment while avoiding the threats caused by adversaries. Another future direction may be to extend this work to multi-agent scenarios by describing the tasks, and communication and coordination constraints between the agents as temporal logic specifications.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSimultaneous estimation of several independent normal means has been a topic of great research interest, especially in the 60's, 70's and 80's, after the publication of the celebrated James-Stein estimator (James and Stein, 1961). \nLet $y=(y_1,\\ldots,y_m)^{\\prime}$ be a maximum likelihood estimator of $\\theta=(\\theta_1,\\cdots,\\theta_m)^{\\prime}$ under the model:\n$y_i|\\theta_i \\stackrel{ind.}{\\sim} N(\\theta_i, 1),\\; i=1,\\cdots,m.$\nJames-Stein (1961) provided a surprising result that for $m\\ge 3$, $y$ is an inadmissible estimator of $\\theta$ under the model and the sum of squared error loss function: $L(\\hat{\\theta},\\theta)=\\sum_{i=1}^{m}(\\hat{\\theta}_i-\\theta_i)^2$. They also showed \nthat the estimator $\\hat \\theta_i^{JS}=(1-\\hat{B}^{JS})y_i$, where $\\hat{B}^{JS}={(m-2)}\/{(\\sum_{i=1}^{m}y_i^2)}$, dominates $y$ in terms of the frequentist's risk.  To be specific, \n$E[\\sum_i^{m}(\\hat{\\theta}_i^{JS}-\\theta_i)^2|\\theta]\\leq E[\\sum_i^{m}(y_i-\\theta_i)^2|\\theta]$, for all \n$\\theta \\in \\mathcal{R}^m,$  the $m$-dimensional Euclidean space, with strict inequality holding for at least one point $\\theta$. \n\nThe potential of different extensions of the James-Stein estimator to improve data analysis became transparent when Efron and Morris (1973) provided an empirical Bayesian justification of the James-Stein estimator using the prior $\\theta_i\\sim^{iid.}N(0,A)$, \\; $i=1,\\cdots,m$. \nSome earlier applications of empirical Bayesian method include the estimation of: (i) false alarm probabilities in New York City (Carter and Rolph, 1974), (ii) the batting averages of major league baseball players (Efron and Morris, 1975), (iii) prevalence of toxoplasmosis in El Salvador (Efron and Morris, 1975) and (iv) per-capita income of small places in the USA (Fay and Herriott, 1979). More recently, variants of the method given in Efron and Morris (1973) was used: to estimate poverty rates for the US states, counties, and school districts (Citro and Kalton, 2000) and Chilean municipalities (Casas-Cordero, Encina and Lahiri , 2016), and to estimate proportions at the\nlowest level of literacy for states and counties (Mohadjer et al. 2012).\n\nThe following two-level Normal hierarchical model is an extension of the model used by Efron and Morris (1973):\n\n\\noindent For $i=1,\\ldots, m$, \n\n{\\rm Level \\ 1 \\ (sampling \\ model):}\n$y_i|\\theta _i \\stackrel{\\mathrm{ind.}}{\\sim} N(\\theta_i,D_i)$;\n\n{\\rm Level \\ 2 \\ (linking \\ model):}\n$\\theta_i \\stackrel{\\mathrm{ind.}}{\\sim}N(x_i^{\\prime}\\beta, A)$.\n\n\\noindent In the above model, level 1 is used to account for the sampling distribution of unbiased estimates $y_i$ based on observations taken from the $i$th population. In this model, we assume that the sampling variances $D_i$ are known and this assumption often follows from the asymptotic variances of transformed direct\n estimates (Efron and Morris, 1975; Carter and Rolph, 1974) or from empirical variance modeling (Fay and Herriot, 1979, Otto and Bell, 1995).\nLevel 2 links the random effects $\\theta_i$ to a vector of $p$ known auxiliary variables $x_i=(x_{i1},\\cdots,x_{ip})^{\\prime}$, which are often obtained from various alternative data sources. The parameters $\\beta$ and $A$ are generally unknown and are estimated from the available data. We assume that $\\beta\\in \\mathcal{R}^p,$ the $p$-dimensional Euclidian space. \nIn the growing field of small area estimation, this model is commonly referred to as the Fay-Herriot model, named after the authors of the landmark paper with more than 1200 citations to date (according to Google Scholar) by Fay and Herriot (1979). For a comprehensive review of small area estimation, the readers are referred to the book by Jiang (2007) and Rao and Molina (2015).\n\nWe may be interested in the high dimensional parameters (random effects) $\\theta_i$ and\/or the hyperparameters $\\beta$ and $A$.  The estimation problem can be addressed using either Bayesian or linear mixed model classical approach.  When hyperparameters are known, both the Bayesian and linear mixed model classical approaches use conditional distribution of  $\\theta_i$  given the data for point estimation and measuring uncertainty of the point estimator.  To elaborate, the posterior mean of $\\theta_i$, the Bayesian point estimator, is identical to the best predictor of $\\theta_i$.  Moreover, the posterior variance of $\\theta_i$ is identical to the mean squared error of the best predictor.  When $A$ is known but $\\beta$ is unknown, a flat prior is generally assumed for $\\beta$ under the Bayesian approach. Interestingly, in this unknown $\\beta$ case, the posterior mean and posterior variance of $\\beta$  are identical to the maximum likelihood estimator of $\\beta$ and the variance of the maximum likelihood estimator, respectively.  Moreover, the posterior mean and variance of $\\theta_i$ are identical to the best linear unbiased predictor of $\\theta_i$ and its mean squared error, respectively.\n\nWhen both $\\beta$ and $A$ are unknown, flat prior, i.e., $\\pi (\\beta,A)\\propto 1,\\;\\beta\\in \\mathcal{R}^p, A>0$, is common though a few other priors for $A$ have been considered; see, e.g., Datta et al. (2005) and Morris and Tang (2011).  In a linear mixed model classical approach, different estimators of $A$ have been proposed and  the estimator of $\\beta$ is obtained by plugging in an estimator of $A$ in the maximum likelihood estimator of $\\beta$ when $A$ is known.  In this general case, the relationship between the Bayesian and linear mixed model classical approach is not clear.  The main goal of this paper is to understand the nature of such relationship.  In particular, we answer the following question:  For a given classical method of estimation of $A$, is it possible to find a prior on $A$ that will make the Bayesian solution closer to the classical solution in achieving multiple goals (i)-(v), described in Section 3, or a subset of these goals given in Theorem 2?\n\n\nWhat would be the parameters of interest in setting the multiple goals? To this end, we first note that Morris and Tang (2011) pointed out the need for accurately estimating the shrinkage parameters $B_i=D_i\/(A+D_i)$ as they appear linearly in the Bayes estimators of $\\theta_i$, which are the prime parameters of interest in many applications like the small area estimation.  Moreover, the shrinkage parameters are good indicators of the strength of the prior on the random effects $\\theta_i$.  Despite the importance of shrinkage parameters, relatively little research  has been conducted in order to understand the theoretical properties of existing estimators.  For the balanced case when $D_i=D,\\; i=1,\\cdots,m$, Morris (1983) proposed an exact unbiased estimator of $B=D\/(A+D)$ and showed component-wise dominance of the resulting empirical Bayes estimator of $\\theta_i$ under the joint distribution of $\\{(y_i,\\theta_i),\\;i=1,\\cdots,m\\}$ when $p\\le m-3.$ For the general unbalanced case, Hirose and Lahiri (2018) proposed an adjusted maximum likelihood  estimator of $B_i$ that satisfies multiple desirable properties. First, the method yields an estimator of $B_i$ that is strictly less than 1, which prevents the overshrinking problem in the related empirical best linear unbiased predictor or simply empirical best predictor of $\\theta_i$.  Secondly, this adjusted maximum likelihood estimator of $B_i$ has the smallest bias among all existing rival estimators in the higher order asymptotic sense.  Thirdly, when this adjusted maximum likelihood method is used, second-order unbiased estimator of mean squared error of empirical best linear unbiased predictor can be produced in a straightforward way without additional bias corrections that are necessary for other existing variance component estimation methods. \nFor prior work on the adjusted maximum likelihood method, the readers are referred to Lahiri and Li (2009), Li and Lahiri (2010), Yoshimori and Lahiri (2014a,b), Hirose and Lahiri (2018), and Hirose (2017,2019). \n\nAs stated in Morris and Tang (2011), flat prior leads to admissible minimax estimators of the random effects for a special case of the model.  In Section 3, we show that the bias of the Bayes estimator of $B_i$, under the flat prior and the two-level model, is $O(m^{-1})$ except for the balanced case when it is of lower order $o(m^{-1})$. Thus, in general, the Bayes estimator of $B_i$, under the flat prior, has more bias than the adjusted maximum likelihood estimator of  Hirose and Lahiri (2018) in the higher order asymptotic sense.  In this section, we propose a prior for the hyperparameters that leads to the Bayes estimator of $B_i$  with bias of lower order $o(m^{-1})$ and thus is on par with the adjusted maximum likelihood of Hirose and Lahiri (2018).  Interestingly, this prior also makes the resulting Bayesian method much closer to the Hirose-Lahiri\\rq{}s empirical best linear unbiased prediction method in multiple sense. \nIn particular, the posterior variance of the random effect $\\theta_i$, under the proposed prior, is identical to both the Taylor series and parametric bootstrap second-order mean squared error estimators of Hirose and Lahiri (2018) in the higher order asymptotic sense.  To our knowledge, we establish for the first time the relationship between the Bayesian posterior variance and parametric bootstrap mean squared error estimator in this higher-order asymptotic sense. \n\n\n\nThe outline of the paper is as follows.  In Section 2, we first introduce a classical method for the  two level model by proposing a general adjustment factor in estimating $A$.  We show how the method is related to the commonly used residual maximum likelihood method for a given choice of the adjustment factor. We then construct a prior, called a multi-goal prior, that provides a Bayesian solution close (with respect to several properties in higher order asymptotic sense) to classical solution in order to estimate the hyperparameters and random effects.  Section 3 discusses prior choice for an important special case considered by Hirose and Lahiri (2018).  In addition to the multiple properties discussed in Section 2, this section develops a unique multi-goal prior that establishes a relationship of the posterior variances of the random effects with the Hirose-Lahiri Taylor series and parametric bootstrap mean squared error estimators that do not require the usual complex bias corrections. We reiterate that this paper demonstrates for the first time how to bring the Bayesian and classical parametric bootstrap methods closer in the context of random effects models.  In Section 4, we compare the proposed  multi-goal prior with the superharmonic prior using a real life data.  In Section 5, we discuss issues in extending our results to a general model.  All the technical proofs are deferred to the Appendix.\n\n \n\\section{Prior Choice for reconciliation of the  Bayesian and classical approach}\nIn this section, we first introduce a general classical method for estimation of hyperparameters and random effects in the two-level Normal hierarchical model.  Then we construct prior for the hyperparameters so that the corresponding Bayesian method is identical to the classical method  in the higher order asymptotic sense with respect to multiple properties. \n\nWe first introduce the empirical best linear unbiased predictor of $\\theta_i$ when the variance component $A$ is estimated by a general adjusted maximum likelihood method.  To this end, we define mean squared error of a given predictor $\\hat{\\theta}_i$ of $\\theta_i$ as $M_i(\\hat{\\theta}_i)=E(\\hat{\\theta}_i-\\theta_i)^2$, where the expectation is with respect to the joint distribution of $y=(y_1,\\cdots,y_m)^{\\prime}$ and $\\theta=(\\theta_1,\\cdots,\\theta_m)^{\\prime}$ under the two-level normal model. \nThe best linear unbiased predictor $\\hat{\\theta}_i^{BLUP}$ of $\\theta_i$, which minimizes $M_i(\\hat{\\theta}_i)$ among all linear unbiased predictors $\\hat\\theta_i$, is given by \n$\\hat{\\theta}_i^{BLUP} (A)=(1-B_i)y_i+B_i x^{\\prime}_i\\hat{\\beta}(A),$ \nwhere $B_i\\equiv B_i(A)=D_i\/(A+D_i)$ is the shrinkage factor and $\\hat{\\beta}(A)=(X^{\\prime}{V}^{-1}X)^{-1}X^{\\prime}{V}^{-1}y$ is the weighted least square estimator of $\\beta$ when $A$ is known. In this formula, $X^{\\prime}=(x_1,\\cdots,x_m)$ denotes $p\\times m$ matrix of known auxiliary variables and $V=\\mbox{diag}(A+D_1,\\cdots,A+D_m)$ denotes a $m\\times m$ diagonal covariance matrix of $y$. \n\nWe consider the following general adjusted maximum likelihood estimator $\\hat{A}_{i;G}$ of $A$ : \n\\begin{align}\n\\hat{A}_{i;G}=\\mathop{\\rm arg~max}\\limits_{0 \\le A<\\infty} h_{i;G}(A)L_{RE}(A), \\label{ad.est}\n\\end{align}\nwhere the general adjustment factor $h_{i;G}(A)$ satisfies Condition R5 in Appendix A. \nNote that maximum likelihood, residual maximum likelihood and different adjusted maximum likelihood estimators of $A$ can be produced using suitable choices of $h_{i;G}(A)$. Plugging in  $\\hat{A}_{i;G}$ for $A$ in the best linear unbiased predictor, one obtains an empirical best linear unbiased predictor $\\hat{\\theta}_i^{EB}(\\hat{A}_{i;G})$ of $\\theta_i$. \n\nSince the residual maximum likelihood estimator of $A$ has the lowest bias among existing estimators in the higher-order asymptotic sense, it is of interest to establish a relationship between the general adjusted maximum likelihood estimator and the residual maximum likelihood estimator. We describe such relationship in  Theorem \\ref{L1}; see Appendix A.1 for a proof.\n\n\\begin{theorem}\n\\label{L1}\nUnder regularity conditions R1-R5, \n$$\\hat{A}_{i;G}-\\hat{A}_{RE}=\\frac{2\\tilde l_{i;G}^{(1)}(A)}{tr[V^{-2}]}+o_p(m^{-1}),$$ \n\\end{theorem}\nwhere $\\tilde l_{i;G}^{(1)}(A)=\\frac{\\partial \\log h_{i;G}(A)}{\\partial A}$.\n\nWe now present Theorem \\ref{rel} for constructing a prior, starting from a given adjustment factor $h_{i,G}(A)$, in order to bring the resulting Bayesian method closer to the classical method with respect to three criteria.  To this end, let $p (\\beta, A)$ denote the prior for $(\\beta,A)$.  Following Datta et al. (2005), we assume \n$p (\\beta, A)\\propto \\pi (A)$ and introduce the following notations to be used throughout the paper: \n\\begin{align*} \n&\\hat b_1=\\frac{\\partial B_i}{\\partial A}\\Big |_{\\hat{A}_{RE}},\\ \\hat b_2=\\frac{\\partial^2 B_i}{\\partial A^2}\\Big |_{\\hat{A}_{RE}},\\ \\hat \\rho_1 =\\frac{\\partial \\log \\pi(A)}{\\partial A}\\Big |_{\\hat{A}_{RE}},\\\\ \n&\\hat h_2=-\\frac{1}{m}\\frac{\\partial^2 l_{RE}}{\\partial A^2}\\Big |_{\\hat{A}_{RE}}=\\frac{tr[V^{-2}]}{2m}+o_p(m^{-1}),\\\\ \n&\\hat h_3=-\\frac{1}{m}\\frac{\\partial^3 l_{RE}}{\\partial A^3}\\Big |_{\\hat{A}_{RE}}=-\\frac{2tr[V^{-3}]}{m}+o_p(m^{-1}),\n\\end{align*}\nwhere $\\hat A_{RE}$ is the residual maximum likelihood estimator of $A$, and $l_{RE}$ is the logarithm of residual likelihood.\n\\begin{theorem}\n\\label{rel}\nUnder Regularity Conditions R1-R5, if $p(\\beta,A)\\propto \\pi_{i;G}(A)$ and \n\\begin{align}\\pi_{i;G}(A)\\propto (A+D_i)tr(V^{-2}){h}_{i;G}(A),\\label{g.p}\\end{align}\nwe have;\n\\begin{align*}\n&(i) \\hat{B}_i^{GHB}=\\hat{B}_i(\\hat{A}_{i;G})+o_{p}(m^{-1});\\\\\n&(ii) \\hat{V}_i^{GHB}=V[B_i|y]=Var( \\hat B_i(\\hat{A}_{i;G}))+o_p(m^{-1});\\\\\n&(iii) \\hat{\\theta}_i^{GHB}=\\hat{\\theta}_i(\\hat{A}_{i;G})+o_{p}(m^{-1}).\n\\end{align*}\n\\end{theorem}\nThe proof of Theorem \\ref{rel} is deferred to Appendix A.2.\n\n\n\\begin{remark}\n\\label{pri.cond}\nWe have several remarks on the general multi-goal prior given by (\\ref{g.p}).  \n\\begin{description}\n\\item[(a)] Theorem \\ref{rel} is valid for multiple choices of $h_{i;G}$.\n\n\\item[(b)]\nThere exists at least one strictly positive estimate of $A$ if $h_{i;G}(A)>0$ and \n\\begin{align}\nh_{i;G}(A)=o(A^{(m-p)\/2}),\\label{F.ec}\n\\end{align}\nfor large $A$ under R6-7.\n\n\n\\item[(c)] Note that $h_{i;G}(A)$ may not qualify as a bonafide prior since it may result in an improper posterior;  see Yoshimori and Lahiri (2014b) for an example.  However, if we restrict the class of priors to $h_{i;G}(A)=(A+D_i)^s$ for some $s>0$, we show in Appendix B.1 that $h_{i;G}(A)=o(A^{(m-p-2)\/2})$ is a sufficient condition for the propriety of posterior and hence can serve as a prior for $A$.\n\nOn the other hand, it is straightforward to show that $\\pi_{i;G}(A)$ given by (\\ref{g.p}) with $h_{i;G}(A)=o(A^{(m-p)\/2})$ yields proper posterior because of multiplication of $h_{i;G}(A)$ by $(A+D_i)tr(V^{-2})$. \nIn either case, Theorem \\ref{rel} can facilitate users for selecting an adjusment factor in the emprical best linear unbiased prediction approach or prior in the Bayesian approach.\n\n\\end{description}\n\\end{remark}\n\n\n\n\n\n\\section{Multi-Goal Prior for an important special case}\n\n\nHirose and Lahiri (2018) put forward a classical approach for an important choice of $h_{i;G}(A)$ that satisfies the following desirable properties under regularity conditions R1-R7: \n \n\\begin{description}\n\\item [1.] It is desirable to have a second-order unbiased estimator of $B_i$, i.e., $E(\\hat B_i)=B_i+o(m^{-1})$.\n\\item [2.]$0<\\mbox{inf}_{m\\ge 1}\\hat B_i\\le \\mbox{sup}_{m\\ge 1}\\hat B_i<1$ (a.s.) \nfor protecting the empirical best linear unbiased predictor from over-shrinking to the regression estimator.\n\\item [3.] It is desirable to obtain a simple second-order unbiased Taylor series mean squared error estimator of the empirical best linear unbiased predictor without any bias correction; that is, $E[\\hat{M}_{i}(\\hat A_i)]=M_i(\\hat{\\theta}_i^{EB})+o(m^{-1}).$\n\\item [4.] It is desirable to produce a strictly positive second-order unbiased single parametric bootstrap mean squared error estimator without any bias-correction,\n\\end{description}\nwhere $\\hat{M}_i(\\hat{A}_i)$ denotes a estimator of mean squared error of $\\hat\\theta_i^{EB}(\\hat{A})$. \n\nLet $\\hat{A}_{i;MG}$, $\\hat{B}_{i;MG}$, $\\hat{\\theta}_{i;MG}^{EB}$, $\\hat{M}_{i;MG}$, $\\hat{M}_{i;MG}^{boot}$ be the Hirose--Lahiri\\rq{}s estimators of $A, B_i,$ the empirical best linear unbiased predictor of $ \\theta_i$, Taylor series and parametric bootstrap estimators of the mean squared error of the empirical best linear unbiased predictor, respectively. They are given by\n\\begin{align*} \n\\hat {A}_{i;MG}=\\mathop{\\rm arg~max}\\limits_{0< A <\\infty}& \\tilde h_i(A)L_{RE}(A),\\\\ \n\\hat {B}_{i;MG}=\\hat{B}_i(\\hat{A}_{i;MG}),& \\  \\hat {\\theta}_{i;MG}^{EB}=\\hat{\\theta}_i^{EB}(\\hat{A}_{i;MG}), \\\\\n\\hat{M}_{i;MG}=\\hat{M}_i(\\hat{A}_{i;MG}),& \\ \\hat{M}_{i;MG}^{boot}=E_*[\\{\\hat{\\theta}_i(\\hat{A}_{i;MG}^*,y^{*})-\\theta_i^*)\\}^2],\n\\end{align*}\nwhere $\\tilde h_i(A)=h_{+}(A)(A+D_i)$ with $m>p+2$; $h_{+}(A)$ satisfies Conditions R6-R7 in Appendix A; \n$\\theta_i^{*}=x_i^{\\prime}\\hat \\beta(\\hat{A}_{1;MG},\\ldots,\\hat{A}_{m;MG})+u_i^*$ with $u_i^* \\sim^{ind.} N(0,\\hat{A}_{i;MG})$; $E_*$  is expectation with respect to the two-level Normal hierarchical model with $\\beta$ and $A$ replaced by $\\hat\\beta(\\hat{A}_{1;MG},\\ldots,\\hat{A}_{m;MG})$ and $\\hat A_{i;MG}$, respectively. Note that the choice of $h_{+}(A)$ is not unique in general.  One can use the choice given in Yoshimori and Lahiri (2014a). \n\nThe following corollary follows from Theorem \\ref{L1}, Hirose and Lahiri (2018) and the fact that $\\frac{\\partial \\hat \\beta(A)}{\\partial A}=O_p(m^{-1\/2})$. \n\\begin{corollary}\n\\label{cor}\nUsing the regularity conditions,\n\\begin{align*}\n&(i) \\hat{A}_{i;MG}-\\hat{A}_{RE}=O_{p}(m^{-1});\\\\\n&(ii) x_i^{\\prime}\\hat{\\beta}(\\hat{A}_{1;MG},\\ldots,\\hat{A}_{m;MG})-x_i^{\\prime}\\hat{\\beta}(\\hat{A}_{RE})=o_{p}(m^{-1}). \n\\end{align*}\n\\end{corollary}\n\n\nIn this section, we suggest a Bayesian approach that is close to the classical approach to achieve multiple goals in the higher-order asymptotic sense. To this end, we seek a multi-goal prior on the hyperparameters $(\\beta,A)$ that satisfies all the following properties simultaneously:\n\\begin{description}\n\\item[(i)] $\\hat{B}_i^{HB}\\equiv E[B_i|y]=\\hat{B}_{i,MG}+o_p(m^{-1})$;\n\\item[(ii)] $V[B_i|y]=Var( \\hat B_{i;MG} )+o_p(m^{-1})$;\n\\item[(iii)] $\\hat{\\theta}_i^{HB}\\equiv E[\\theta_i|y]=\\hat{\\theta}_{i,MG}+o_p(m^{-1})$;\n\\item [(iv)] $V[\\theta_i|y]=\\hat{M}_{i;MG}+o_p(m^{-1})$;\n\\item [(v)] $V[\\theta_i|y]=\\hat{M}_{i;MG}^{boot}+o_p(m^{-1})$.\n\\end{description}\n\n\nFirst we prepare the following result, which follows from Corollary \\ref{cor} (i) and Hirose and Lahiri (2018): \n\\begin{align}\n\\hat{B}_i(\\hat{A}_{i;MG})-\\hat{B}_i(\\hat{A}_{RE})&=(\\hat{A}_{i,MG}-\\hat{A}_{RE})\\hat b_1+o_p(m^{-1})\\notag\\\\\n&=\\{E[\\hat{A}_{i;MG}-A]-E[\\hat{A}_{RE}-A]\\} b_1+o_p(m^{-1})\\notag\\\\\n&=-\\frac{2D_i}{tr[V^{-2}](A+D_i)^3}+o_p(m^{-1}).\\label{bias.B}\n\\end{align}\n\nIf we use the flat prior $\\pi(A)\\propto 1$, we get the following result using equation (21) of Datta et al. (2005) with $b(A)=B_i(A)$ and equation (\\ref{bias.B}): \n$$E[B_i|y]=\\hat B_i(\\hat A_{MG})+\\frac{4D_i}{tr[V^2](A+D_i)^2}\\left[\\frac{1}{A+D_i}-\\frac{tr[V^{-3}]}{tr[V^{-2}]}\\right]+o_p(m^{-1}).$$ \nThis result emphasizes that the flat prior $\\pi(A)\\propto 1$ cannot achieve Property (i) except for balanced case ($D_i=D$ for all $i$). We, therefore, seek a prior $\\pi (A)$ to satisfy Property (i), even in unbalanced case. \nTo this end, we also use the following result (\\ref{B.HB}) given in (21) of Datta et al. (2005) with $b(A)=B_i(A)$:\n\\begin{align}\nE[B_i|y]=\\hat{B}_i(\\hat{A}_{RE})+\\frac{1}{2m\\hat{h}_2}\\left(\\hat b_2-\\frac{\\hat{h}_3}{\\hat{h}_2}\\hat b_1 \\right)+\\frac{\\hat{b}_1}{m\\hat{h}_2}\\hat \\rho_1+o_p(m^{-1}).\\label{B.HB}\n\\end{align}\nIt is evident from equations (\\ref{bias.B}) and (\\ref{B.HB}) that our desired prior must satisfy the following differential equation, up to the order of $O(m^{-1})$:\n\\begin{align}\n\\frac{1}{2m{h}_2}\\left( b_2-\\frac{{h}_3}{{h}_2} b_1 \\right)+\\frac{{b}_1}{m{h}_2} \\rho_1=-\\frac{2D_i}{tr[V^{-2}](A+D_i)^3}.\\label{diff1}\n\\end{align}\n\nNote that the differential equation (\\ref{diff1}) is equivalent to the following differential equation, up to the order of $O_p(m^{-1})$;\n\\begin{align}\n\\rho_1=\\frac{\\partial \\log \\pi(A)}{\\partial A}&=-\\frac{m  h_2}{ b_1}\\frac{2D}{tr[V^{-2}](A+D_i)^3}-\\frac{1}{2}\\left[\\frac{ b_2}{ b_1}-\\frac{ h_3}{ h_2} \\right]\\notag\\\\\n&=\\frac{2}{A+D_i}-\\frac{2tr[V^{-3}]}{tr[V^{-2}]}.\\label{difffinal}\n\\end{align}\n\nHence, we obtain a solution to differential equation (\\ref{difffinal}) as follows: \n\\begin{eqnarray}\n\\pi(A)\\propto (A+D_i)^2tr[V^{-2}]. \\label{MG.pri0}\n\\end{eqnarray}\n\nNote that the prior (\\ref{MG.pri0}) depends on $i$. Therefore, we redefine it as:\n\\begin{eqnarray}\n\\pi_i(A)\\propto (A+D_i)^2tr[V^{-2}]. \\label{MG.pri}\n\\end{eqnarray}\n\n\\begin{remark}\n\\label{R1}\nWe have several important remarks on the prior (\\ref{MG.pri}). \n\\begin{description}\n\\item[(a)] The prior satisfies the rest of Properties (ii)-(v) simultaneously, as shown in Appendix B.2. \nIt is remarkable that $\\pi_i(A)$ given by (\\ref{MG.pri}) is the unique prior to achive Properties (i)-(v) simultaneously, up to the order of $O_p(m^{-1})$, since $E[g_{1i}(A)|y]=g_{1i}(\\hat A_{i;MG})+o_p(m^{-1})$ shown in (\\ref{g1.exp}).\n\n\n\\item[(b)] The prior given by equation (\\ref{MG.pri}) reduces to the Stein's super-harmonic prior for the balanced case $D_i=D,\\;i=1,\\cdots,m$, up to the order of $O_p(m^{-1})$.\n\n\\item[(c)] Datta et al. (2005) found the same prior by matching (in a higher order asymptotic sense) expected value of the posterior variance of $\\theta_i$ with the mean squared error of the empirical best linear unbiased predictor with the residual maximum likelihood estimator used for the variance component $A$. It is interesting to note that the same prior achieves multiple goals, a fact gone unnoticed.\n\n\\item [(d)] From the result of Ganesh and Lahiri (2008), the prior \n$$\\pi (A)\\propto \\frac{\\sum \\{1\/(A+D_i)^2\\}}{\\sum \\omega_i\\{D_i^2\/(A+D_i)^2\\}}$$ also satisfies  \n$\\sum_{i}^m \\omega_i E[{V}(\\theta_i|y)-MSE[\\hat \\theta_i(\\hat{A}_{i;MG})]]=o(m^{-1}).$\n\\end{description}\n\\end{remark}\n\n\\section{Data Analysis}\nIn this section, using the 1993 Small Area Income and Poverty Estimates (SAIPE) data set, we demonstrate that our proposed multi-goal prior(MGP) performs better than the superharmonic prior (SHP) in producing Bayesian solutions closer to the multi-goal classical solutions of Hirose and Lahiri (2018).  The SAIPE data we use here is from Bell and Franco (2017), available at \\url{https:\/\/www.census.gov\/srd\/csrmreports\/byyear.html}. The data contains direct poverty rates($y_i$), associated sampling variances ($D_i$), and auxiliary variables ($x_i$) derived from administrative and census data for the 50 states and the District of Columbia.  Much has been written about SAIPE over the years. See, for instance, the recent book chapter by Bell et al. (2016). \n\nFirst consider the estimation of the shrinkage parameters $B_i$ for all the states. Fig 1 displays classical multi-goal estimates $\\hat B_{i;MG}$ and Bayes estimates of $B_i$ under the superharmonic  and the multi-goal priors for all the states arranged in decreasing order of $\\hat B_{i;MG}$. Note that the Bayes estimate of $B_i$ is an one-dimensional integral, which is approximated by numerical integration using the R function \\lq\\lq{}adaptIntegrate\\rq\\rq{}. Overall, the Bayes estimates under the multi-goal prior are closer to the classical estimates (MGF) than the superharmonic prior.\n\nNext, in Fig 2, we compare the mean squared error estimates by Taylor series (MGF) and parametric bootstrap (PB MG) of Hirose and Lahiri (2018) with the posterior variances under the two different priors.  The parametric bootstrap mean squared error estimates use $10^4$ bootstrap samples. The two mean squared error estimates are virtually identical.  Again our posterior variances under the  multi-goal prior are much closer to the mean squared error estimates than the corresponding posterior variances under the superharmonic prior.\n\n\\begin{figure}[ht]\n\\includegraphics[bb=0 0 1920 1082,scale=0.2,clip]{Bi93.png}\n\\caption{$B_i$ estimates (MGF:$\\hat B_{i;MG}$, MGP: $E_{MG}[B_i|y]$, SHP:$E_{SHP}[B_i|y]$)}\n\\label{Bi.93}\n\\end{figure}\n\n\n\n\\begin{figure}[ht]\n\\includegraphics[bb=0 0 1920 1082,scale=0.2,clip]{Vi93.png}\n\\caption{MSE estimates \n(PB.MG:$\\hat M^*_{i;MG}$, MGF:$\\hat M_{i;MG}$, MG Prior:$V_{MG}[\\theta_i|y]$, SHP:$V_{SHP}[\\theta_i|y]$)\n}\n\\label{Vi93}\n\\end{figure}\n\n\n\n\\section{Discussion}  \n\nCan we extend our results to a general linear mixed model? To answer this question, we consider the following nested error regression model considered by Battese et al. (1988): \n\\begin{eqnarray}\ny_{ij}=\\theta_{ij}+e_{ij}=x_{ij}^{\\prime}\\beta+v_i+e_{ij}, \\ (i=1,\\ldots,m;\\ j=1,\\ldots,n_i),\\label{NERM}\n\\end{eqnarray}\nwhere $\\{v_1\\ldots,v_m\\}$ and $\\{e_1,\\ldots, e_m\\}$ are independent with $v_i{\\sim}N(0,\\sigma_v^2)$ and $e_i{\\sim}N(0,\\sigma_e^2)$; $x_{ij}$ is a $p$-dimensional vector of known auxiliary variables; $\\beta\\in \\mathcal{R}^p$ is a $p$-dimensional vector of unknown regression coefficients; $\\psi=(\\sigma_v^2, \\sigma_e^2)^{\\prime}$ is an unknown variance component vector; $n_i$ is the number of observed unit level data in $i$-th area.\n\nThe condition for achieving desired property 1 given in Section 3, we need to solve the following system of differential equations with shrinkage factor $B_i=\\sigma_e^2\/(n_i\\sigma_v^2+\\sigma_e^2)$, under certain regularity conditions: \n\\begin{align}\n\\left[\\frac{\\partial \\log h_{i;G}(\\psi)}{\\partial \\psi}\\right]^{\\prime}I_F^{-1}\\left[\\frac{\\partial B_{i}(\\psi)}{\\partial \\psi}\\right]=&H(\\psi),\n\\end{align}\nwhere $$\\frac{\\partial \\log h_{i;G}(\\psi)}{\\partial \\psi}=\\left(\\frac{\\partial \\log h_{i;G}(\\psi)}{\\partial \\sigma_v^2}, \\frac{\\partial \\log h_{i;G}(\\psi)}{\\partial \\sigma_e^2}\\right)^{\\prime},$$ \n$$H(\\psi)=-\\frac{1}{2}tr\\left[\\frac{\\partial^2 B_{i}(\\psi)}{\\partial \\psi^2}I_F^{-1}\\right], \\  \n\\frac{\\partial B_{i}(\\psi)}{\\partial \\psi}=\\frac{n_i}{(n_i\\sigma_v^2+\\sigma_e^2)^2}(-\\sigma_e^2, \\sigma_v^2)^{\\prime},$$ \n$$I_F^{-1}=\\frac{2}{a}\\left (\n\\begin{array}{cc}\n\\sum[(n_i-1)\/\\sigma_e^4+(n_i\\sigma_v^2+\\sigma_e^2)^{-2}] & -\\sum n_i\/(n_i\\sigma_v^2+\\sigma_e^2)^2\\\\\n-\\sum n_i\/(n_i\\sigma_v^2+\\sigma_e^2)^2&\\sum n_i^2\/(n_i\\sigma_v^2+\\sigma_e^2)^2\\\\\n\\end{array}\\right),$$ \n$$a=[\\sum n_i^2\/(n_i\\sigma_v^2+\\sigma_e^2)^2][\\sum \\{(n_i-1)\/\\sigma_e^4+(n_i\\sigma_v^2+\\sigma_e^2)^{-2}\\}]-[\\sum n_i\/(n_i\\sigma_v^2+\\sigma_e^2)^2]^2.$$\n\nIf we use the following adjustment factor $h_{i;G}(\\psi)$ for achieving desired property 1:\n\\begin{align}\n\\frac{\\partial \\log h_{i;G}(\\psi)}{\\partial \\psi}=v{k},\n\\end{align}\nfor a given two dimensional fixed vector ${k}$, \nthe solution of $v$ can be obtained as $$v=\\frac{H(\\psi)}{{k}^{\\prime}I_F^{-1}\\frac{\\partial B_{i}(\\psi)}{\\partial {\\psi}}}.$$ \nThis solution thus leads to an appropriate adjustment factor satisfying $$\\frac{\\partial \\log h_{i;G}(\\psi)}{\\partial \\psi}=\\frac{H(\\psi)}{{k}^{\\prime}I_F^{-1}\\frac{\\partial B_{i}(\\psi)}{\\partial {\\psi}}}{k}.$$ \nThus, there exist multiple solutions for $h_{i;G}(\\psi)$ satisfying desired property 1 under the nested error regression model (\\ref{NERM}). \nFurther research is needed to identify a reasonable adjustment factor for the general linear mixed model and to establish a connection with the corresponding Bayesian approach.\n\n\n\n\\section*{Acknowledgements}\nThe first and second authors\\rq{} research was supported by JSPS KAKENHI Grant Number 18K12758 and U.S. National Science Foundation Grant SES-1534413, respectively. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\bf Introduction}\\label{section-1}\n\nThis paper is concerned with large-scale  regularity estimates in the homogenization of elliptic systems  of elasticity with \nperiodic high-contrast coefficients.\nLet $\\omega$ be a connected and unbounded open set in $\\mathbb{R}^d$. Assume that $\\omega$ is 1-periodic; i.e.,\nits characteristic function  is periodic with respect to $\\mathbb{Z}^d$.\nWe also assume that \neach of connected components of\n$\n\\mathbb{R}^d\\setminus \\omega \n$\nis the closure of  a bounded open set $F_k$ with Lipschitz boundary,\nand  that\n\\begin{equation}\\label{dis}\n\\min_{k\\neq \\ell} \\text{dist} (F_k, F_\\ell ) >0.\n\\end{equation}\nFor $0< \\delta < \\infty$, define\n\\begin{equation}\\label{Lambda-e}\n\\Lambda_{\\delta} (x)\n=\\left\\{\n\\aligned\n& \\delta  & \\quad & \\text{ if } x\\in F=\\cup_k F_k ,\\\\\n& 1 & \\quad & \\text{ if } x\\notin F.\n\\endaligned\n\\right.\n\\end{equation}\nWe are interested in the large-scale regularity estimates, that are uniform in $\\delta>0$, for the elliptic operator \n\\begin{equation}\\label{op}\n\\mathcal{L}_\\delta \n=-\\text{\\rm div} \\big(\\Lambda_{\\delta^2} A\\nabla \\big).\n\\end{equation}\nHere and thereafter  the coefficient matrix (tensor) $A=A(x)= (a_{ij}^{\\alpha\\beta} (x))$,\nwith $1\\le \\alpha, \\beta, i, j \\le d$, is assumed to be real, \nbounded measurable, 1-periodic,  and to satisfy the elasticity condition,\n\\begin{equation}\\label{ellipticity}\n\\aligned\n& a_{ij}^{\\alpha\\beta} (x) = a_{ji}^{\\beta\\alpha} (x) = a_{\\alpha j}^{i\\beta} (x)  ,\\\\\n& \\kappa_1 |\\xi|^2 \\le a_{ij}^{\\alpha\\beta} \\xi_i^\\alpha \\xi_j^\\beta \\le \\kappa_2 |\\xi|^2\n\\endaligned\n\\end{equation}\nfor any symmetric matrix $\\xi= (\\xi_i^\\alpha)\\in \\mathbb{R}^{d\\times d}$,\nwhere $\\kappa_1$, $\\kappa_2$ are positive constants.\nUnder these assumptions we will  show that if $u\\in H^1(Q_R; \\mathbb{R}^d)$ is a weak solution of\n$\\mathcal{L}_\\delta (u)=0$ in a cube $Q_R= (-R\/2, R\/2)^d$ of size $R$ for some $R\\ge 4$, then\n\\begin{equation}\\label{Lip-e}\n\\sup_{1\\le r\\le R-3}\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R\\cap \\omega} |\\nabla u|^2 \\right)^{1\/2},\n\\end{equation}\nwith a constant $C$ independent of $R$ and $\\delta$.\nLet $ Du $ denote the symmetric gradient of $u$; i.e., \n$$\nDu =  \\big( \\nabla u + (\\nabla u)^T\\big)\/2,\n$$\n where\n$(\\nabla u)^T$ denotes  the transpose of $\\nabla u$. We also prove that for $R\\ge 4$,\n\\begin{equation}\\label{Lip-e-1}\n\\sup_{1\\le r\\le R-3}\n\\left(\\fint_{Q_r } |D u|^2\\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R\\cap \\omega } |D u|^2 \\right)^{1\/2}.\n\\end{equation}\nWe remark that the operator $\\mathcal{L}_{\\delta}$ arises naturally  in the modeling of acoustic propagations in porous media, diffusion processes  in highly heterogeneous media, and\n inclusions in composite materials \\cite{Arbogast-1990, Allaire-1992, OSY-1992, JKO-1993}.\n\nIn the case $\\delta=1$,\nthe regularity estimates for  the elliptic system $-\\text{\\rm div}(A(x\/\\varepsilon)\\nabla)=f$\nin the homogenization theory \nhave been studied extensively in recent years (in this paper we have rescaled the equation so that the  microscopic scale $\\varepsilon=1$ and the domain is large).\nUsing a compactness method,\nthe interior Lipschitz estimate and  the boundary Lipschitz estimate for the Dirichlet problem in\na $C^{1, \\alpha}$ domain were established by M. Avellaneda and F. Lin  in a seminal work \\cite{AL-1987}.\nThe boundary Lipschitz estimate for the Neumann problem in a $C^{1, \\alpha}$ domain \nwas obtained in \\cite{KLS-2013-N}.\nWe refer the reader to  \\cite{Shen-book} for further references on periodic homogenization, and to \\cite{Armstrong-book}\nfor related work on the  large-scale regularity  in stochastic homogenization.\n\nIn this paper we will be concerned with the case $\\delta \\neq 1$,\nwhere, in the simpler scalar case,\n$\\delta^2$ represents the conductivity ratio (or the ratio of diffusion coefficients) of the disconnected  {\\it inclusions} $F=\\cup_k F_k$ to the connected  \\text{\\it matrix}  $\\omega$.\nNotice that the operator $\\mathcal{L}_{\\delta}$  is elliptic, but neither  uniformly in $\\delta \\in (0, 1)$ nor in $\\delta \\in (1, \\infty)$.\n We mention that in the scalar case  with  $0\\le \\delta<1$, $A=I$ and \n $\\omega$  being sufficiently smooth, using the compactness method in \\cite{AL-1987}, \nthe $W^{1, p}$ and Lipschitz estimates were  obtained  by L.-M. Yeh \\cite{Yeh-2010, Yeh-2011, Yeh-2015, Yeh-2016}.\nAlso see earlier work in \\cite{Schweizer-2000, Masmoudi-2004} for related uniform estimates in  the case $\\delta=0$.\nIn \\cite{Chase-Russell-2017}  B. Russell established the  large-scale interior Lipschitz estimate\nfor the system of elasticity  with bounded measurable coefficients in the case  $\\delta=0$, using an approximation method\noriginated in \\cite{Armstrong-Smart-2016}. \nThe case $0<\\delta<1$ was treated in \\cite{Chase-Russell-2018}.\nIn the stochastic setting with $\\delta=0$, S. Armstrong and P. Dario \\cite{Armstrong-2018} obtained \nquantitative homogenization and large-scale regularity results for the random conductance model on a supercritical percolation.\n\nThe following is one of the main results of this paper.\n\n\\begin{thm}\\label{main-thm-1}\nLet $0\\le \\delta\\le \\infty$.\nAssume that $A$ satisfies the elasticity condition \\eqref{ellipticity} and is 1-periodic.\nLet $u\\in H^1(Q_R; \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u)=0$ in $Q_R$ for some $R\\ge  4$.\nThen \\eqref{Lip-e}  and \\eqref{Lip-e-1}\nhold for some constant $C$ depending only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\\end{thm}\n\nNote that Theorem \\ref{main-thm-1}  includes the  limiting  cases of periodically perforated domains: $\\delta=0$ and $\\delta=\\infty$.\nIn the case $\\delta=0$, which is referred to as the soft inclusions \\cite{JKO-1993}, we call  $u\\in H^1(\\Omega; \\mathbb{R}^d)$ \nis  a weak solution of $\\mathcal{L}_0 (u)=f\\chi_\\omega $ in $\\Omega$, if \n$$\n\\int_{\\Omega\\cap \\omega} A\\nabla u \\cdot \\nabla v  \\, dx =\\int_{\\Omega\\cap \\omega}  f\\cdot v \\, dx\n$$\nfor any $ v \\in H_0^1(\\Omega; \\mathbb{R}^d)$.\nFormally, this means \nthat $ -\\text{\\rm div} (A\\nabla u)=f$ in $\\Omega \\cap \\omega$ and $\\big(\\frac{\\partial u}{\\partial \\nu }\\big)_-=0$ on \n$\\Omega \\cap \\partial \\omega$, where \n$\n \\big( \\frac{\\partial u}{\\partial \\nu } \\big)_- = n \\cdot A (\\nabla u )_-\n$ denotes  the conormal derivative  taken from $\\omega$ and \n$n$  the outward unit normal to $\\partial F$.\nFor convenience we will also assume that $u$ is a weak solution of\n$\\text{\\rm div} (A\\nabla u)=0$ in $\\Omega \\cap F$.\nIn the case $\\delta=\\infty$, which is referred to as the stiff inclusions \\cite{JKO-1993}, \n a function $u$ in $H^1(\\Omega; \\mathbb{R}^d)$ is called a weak solution of\n$\\mathcal{L}_\\infty (u)=f$ in $\\Omega$ if\n$ Du =0   \\text{ in }\\Omega \\cap F$, and \n$$\n\\int_{\\Omega\\cap \\omega} A\\nabla u \\cdot \\nabla v \\, dx =\\int_\\Omega f \\cdot v\\, dx\n$$\nfor any $ v \\in H_0^1(\\Omega; \\mathbb{R}^d)$ with\n$D v =0$ in $\\Omega  \\cap F$.\nThis  implies that $-\\text{\\rm div}(A\\nabla u)=f$ in $\\Omega\\cap \\omega$ and that if $\\overline{F}_k\\subset \\Omega$,\n$$\n\\int_{\\partial F_k}  \\Big( \\frac{\\partial u}{\\partial \\nu}\\Big)_-  \\cdot \\phi\\, d\\sigma =-\\int_{F_k}  f \\cdot \\phi\\, dx\n$$\nfor any $\\phi\\in \\mathcal{R}$, the space of rigid displacements.\n\nThe  large-scale uniform Lipschitz estimate in Theorem \\ref{main-thm-1}, which holds under the assumptions that $A$ is bounded measurable and\n$\\partial \\omega$ is locally Lipschitz, is new in the case $1<\\delta\\le \\infty$,\neven when $A$ is constant  and $\\omega$ is smooth.\nUnder the additional conditions that  $\\omega$ is locally $C^{1,\\alpha}$ and $A$ is H\\\"older continuous,\n\\begin{equation}\\label{smoothness}\n|A(x)-A(y)|\\le M_0 |x-y|^\\sigma \\quad \\text{ for any } x, y\\in \\mathbb{R}^d,\n\\end{equation} \nwhere $M_0>  0$ and $\\sigma \\in (0, 1)$,\nwe may combine \\eqref{Lip-e} with the local Lipschitz estimates for the operator $\\mathcal{L}_\\delta$\nto obtain a true Lipschitz estimate.\n\n\\begin{thm}\\label{main-thm-2}\nLet $0\\le \\delta\\le  \\infty$ and $Q(x_0, R)=x_0 + Q_R$.\nAssume that $A$ satisfies conditions \\eqref{ellipticity}, \\eqref{smoothness}, and is 1-periodic.\nAlso assume that each $F_k$ is a bounded $C^{1, \\alpha}$ domain for some\n$\\alpha \\in (0, 1)$.\nLet $u\\in H^1(Q(x_0, R); \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u)=0$ in \n$Q(x_0, R) $ for some $R\\ge 4$.\nThen \n\\begin{equation}\\label{f-Lip}\n|\\nabla u(x_0)|\n\\le C \\left(\\fint_{Q(x_0, R)\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2},\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$ in \\eqref{smoothness}.\n\\end{thm}\n\nThe  Lipschitz estimate \\eqref{f-Lip} as well as its small-scale analogue  allows\nus to construct a  $d\\times d$ matrix $\\Gamma_\\delta  (x, y)$ of  fundamental solutions for the operator $\\mathcal{L}_\\delta$ in $\\mathbb{R}^d$, and \nobtain its estimates that are uniform in $\\delta \\in (0, \\infty)$.\nIn particular, we will show that  if  $d\\ge 3$ and $1\\le  \\delta<\\infty$,\n\\begin{equation}\\label{fund-0}\n\\left\\{\n\\aligned\n |\\Gamma_\\delta (x, y)| & \\le C |x-y|^{2-d},\\\\\n |\\nabla_x \\Gamma_\\delta (x,y)|\n +|\\nabla_y \\Gamma_\\delta(x, y) |\n & \\le C |x-y|^{1-d},\\\\\n |\\nabla_x \\nabla_y \\Gamma_\\delta (x, y)|  & \\le C |x-y|^{-d}\n \\endaligned\n \\right.\n \\end{equation}\n for any $x, y\\in \\mathbb{R}^d$ and  $x\\neq y$,\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$.\n In the case $0<\\delta<1$, the estimates in \\eqref{fund-0} continue to hold, provided that  either \n $|x-y|_\\infty \\ge 4$ or $x, y\\in \\omega$.\n Here $|x-y|_\\infty =\\max ( |x_1-y_1|, \\dots, |x_d -y_d|)$ denotes the $L^\\infty$ norm in $\\mathbb{R}^d$.\n See Theorems  \\ref{thm-f-1}, \\ref{thm-f-1a} and \\ref{thm-f-3}.\n We mention that in the scalar case with $A=I $ and $0<\\delta<1$, explicit bounds for fundamental solutions were obtained by L.-M. Yeh in \\cite{Yeh-2016}.\n As in the case $\\delta=1$ \\cite{AL-1991, KS-2011-L, KLS-2014},\n estimates of fundamental solutions are  an important tool in  the study of optimal regularity problems in the homogenization theory  for solutions\n of $\\mathcal{L}_\\delta (u)=f$.\n In particular, it allows us to extend the  Lipschitz estimate \\eqref{f-Lip}\n from solutions of $\\mathcal{L}_\\delta (u)=0$ to that of $\\mathcal{L}_\\delta (u)=f$.\n Indeed, under the same assumptions on $A$ and $\\omega$ as in Theorem \\ref{main-thm-2},\n we obtain \n \\begin{equation}\\label{f-Lip-f}\n|\\nabla u(x_0)|\n\\le C_p \\left\\{  \\left(\\fint_{Q(x_0, R)\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2}\n+R \\left(\\fint_{Q(x_0, R)} |f|^p \\right)^{1\/p} \\right\\}\n\\end{equation}\nfor $1\\le \\delta\\le \\infty$,\nwhere $u$ is a weak solution of $\\mathcal{L}_\\delta (u)=f$ in $Q(x_0, R) $ for some\n$R\\ge 4$ and $p>d$.\nIf $0\\le \\delta<1$, the estimate (\\ref{f-Lip-f}) holds for solutions of $\\mathcal{L}_\\delta (u)=f \\chi_\\omega$ in $Q(x_0, R)$.\nSee Theorem \\ref{main-thm-3}.\n \n We now describe  our  general approach to the proof of Theorem \\ref{main-thm-1}.\n As we mentioned earlier,\n the scalar case with $0\\le \\delta<1$ and $A=I$ was studied in \\cite{Yeh-2010, Yeh-2011, Yeh-2015, Yeh-2016},\n  using a compactness method of Avellaneda and Lin \\cite{AL-1987}.\n The compactness argument is fairly complicated to implement  for the operator $\\mathcal{L}_\\delta$,\n as  both the coefficient matrix $A$ and the ratio $\\delta^2$ should  be allowed to vary.\n A more direct approach, which originated in \\cite{Armstrong-Smart-2016}, was used in \\cite{Chase-Russell-2017, Chase-Russell-2018}\n to treat the case $0\\le \\delta<1$ with bounded measurable coefficients.\n The approach relies on a result on the convergence rate, uniform in $\\delta$,  for the operator $-\\text{\\rm div}(\\Lambda_{\\delta^2} A(x\/\\varepsilon) \\nabla )$\n as $\\varepsilon\\to 0$.\n It is not clear how to extend either of these two methods to the case $1<\\delta\\le  \\infty$.\n  In this paper we will adapt a more recent method of S. Armstrong, T. Kuusi, and C. Smart \\cite{Armstrong-2020},\n which is  based on a Caccioppoli  type inequality and the fact that $\\Delta_j u$ is a solution whenever $u$ is a solution,\n   where $\\Delta_j$ denotes  the   difference operator,\n  \\begin{equation}\\label{diff}\n  \\Delta_j u (x) = u(x+e_j) -u(x)\n  \\end{equation}\n  for $1\\le j\\le d$ and $e_j= (0, \\dots, 1, \\dots, 0)$ with $1$ in the $j^{\\rm th}$ place.\n  The basic idea is to transfer the higher-order regularity of $u$  in terms of the difference operator to \n  higher-order regularity of $u$ at a large scale through Caccioppoli and Poincar\\'e's  inequalities.\n  For elliptic systems the approach also uses a discrete Sobolev inequality.\n  \n  To carry out the approach described above, a key step is to establish a Caccioppoli inequality for solutions of $\\mathcal{L}_\\delta (u)=0$\n  in $Q_R$ for $R$ large. In the case $0\\le \\delta\\le  1$, it can be shown by an extension argument that \n\\begin{equation}\\label{C-0}\n\\int_{Q_{R\/2}}  |\\nabla u|^2\\, dx \\le \\frac{C}{R^2} \\int_{Q_R} |u|^2\\, dx,\n\\end{equation}\nwhich is more or less known  \\cite{Chase-Russell-2017, Chase-Russell-2018}.\nIt is not known that  \\eqref{C-0}  holds for the case $1< \\delta\\le \\infty$, with constant $C$ independent of $\\delta$.\nHowever, if $\\delta$ is sufficiently  large or $\\delta=\\infty$,\nwe are able to show that  for any $\\ell\\ge 1$ and $R\\ge 32$,\n\\begin{equation}\\label{C-1}\n\\int_{Q_{R\/2 }}  |\\nabla u|^2\\, dx \\le \\frac{C_\\ell }{R^2} \\int_{Q_R} |u|^2\\, dx + \\frac{C_\\ell}{R^{2\\ell } }\n\\int_{Q_R} |\\nabla u|^2\\, dx,\n\\end{equation}\nby some extension and iteration arguments.\nIt turns out that the weaker version  \\eqref{C-1} with $\\ell=1$, together with the discrete Sobolev inequality,  is sufficient to\ncomplete the proof of \\eqref{Lip-e}.\nWe point out that the method described above does not extend to the  nonhomogeneous  system $\\mathcal{L}_\\delta (u) =f$\nwith nonsmooth $f$. \nWe resolve this issue by introducing the matrix of fundamental solutions.\n\nThe paper is organized as follows. In Section \\ref{section-2} we give the proof of \\eqref{C-0}.\nThe inequality \\eqref{C-1} is proved in Section \\ref{section-3}, while the proof of Theorem \\ref{main-thm-1} is given in Section \\ref{section-4}.\nIn Section  \\ref{section-5} we collect some known results on local estimates and give the proof of Theorem \\ref{main-thm-2}.\nThe matrix of  fundamental solutions is introduced and studied in Section \\ref{section-6}.\nFinally, we establish the Lipschitz estimate for solutions of $\\mathcal{L}_\\delta (u)=f$ in Section \\ref{section-7}.\n\nRecall that $Q_R =(-R\/2, R\/2)^d$ and $Q(x_0, R)=x_0 + Q_R$ for $R>0$ and $x_0\\in \\mathbb{R}^d$.\nWe use $\\fint_E u=\\frac{1}{|E|} \\int_E u$ to denote the $L^1$ average of $u$ over a set $E$.\nWe use $C$ to denote a positive constant that may depend on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\nIf $C$  depends also on other parameters, it will be stated explicitly.\nWe emphasize that the results in Sections 2 - 4  hold with no smoothness condition on $A$ or $F=\\mathbb{R}^d \\setminus \\overline{\\omega}$ \nbeyond that $A$ is bounded measurable and $F$ is locally Lipschitz.\nIn Sections 5 - 7 we impose  the H\\\"older continuity condition \\eqref{smoothness} on $A$ and also assume that $F$ is locally $C^{1, \\alpha}$.\n\n\n\\section{\\bf Preliminaries}\\label{section-2}\n\nThroughout this paper we assume that $\\omega$ is a connected, unbounded and 1-periodic open set in $\\mathbb{R}^d$.\nWrite\n\\begin{equation}\\label{F1}\n\\mathbb{R}^d\\setminus \\omega =\\cup_k \\overline{F}_k, \n\\end{equation}\nwhere each $\\overline{F}_k$ is the closure of  a bounded Lipschitz domain $F_k$ with connected boundary.\nWe assume that $\\{ \\overline{F}_k \\} $ are mutually disjoint and satisfy the condition (\\ref{dis}).\nThis allows us to construct a sequence  of mutually disjoint open sets $\\{\\widetilde{F}_k \\}$ with connected smooth  boundary such that $\\overline{F}_k \\subset \\widetilde{F}_k$,\n\\begin{equation}\\label{dist}\n\\left\\{\n\\aligned\n&  c_0 \\le \\text{dist}(\\partial F_k, \\partial \\widetilde{F}_k), \\\\\n& c_0 \\le \\text{dist} (\\widetilde{F}_k, \\widetilde{F}_\\ell ) \\text{ for } k\\neq \\ell,\n\\endaligned\n\\right.\n\\end{equation}\nfor some $c_0>0$.\nNote that by the periodicity of $\\omega$, \n$\\{  F_k\\}$  are the shifts of a finite number of bounded Lipschitz domains contained in $Q_2$.\nAs a result, \nwe may  assume that $\\{ \\widetilde{F}_k \\} $ are the shifts of a finite number of  bounded smooth domains contained in $Q_{5\/2}$.\n\n Let $\\mathcal{R}$ denote the space of rigid displacements of $\\mathbb{R}^d$; i.e.,\n \\begin{equation}\\label{R}\n \\mathcal{R}\n =\\big\\{ u  = E + Bx: \\ E\\in \\mathbb{R}^d \\text{ and } B^T=-B \\big\\},\n \\end{equation}\n where $B^T$ denotes the transpose of the $d\\times d$ matrix $B$.\n  The following extension lemma will  be useful for us.\n\n\\begin{lemma}\\label{lemma-ex}\nLet $F_k$ and $\\widetilde{F}_k$ be given above.\nThere exists a linear extension operator \n$$\nP_k: H^1(\\widetilde{F}_k \\setminus \\overline{F}_k; \\mathbb{R}^d)\n\\to H^1(\\widetilde{F}_k; \\mathbb{R}^d)\n$$ \nsuch that\n\\begin{align}\n & P_k (u)=u \\quad \\text{ for any } u \\in \\mathcal{R}, \\label{ex-1}\\\\\n & \\| P_k (u)\\|_{H^1(\\widetilde{F}_k)} \n \\le C \\big( \\| u \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n + \\|  Du \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)} \\big), \\label{ex-2}\\\\\n & \\|\\nabla P_k(u) \\|_{L^2(\\widetilde{F}_k)}\n \\le C \\| \\nabla u\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}, \\label{ex-3}\\\\\n & \\| D  P_k (u) ) \\|_{L^2(\\widetilde{F}_k)}\n \\le C \\|  D u  \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}, \\label{ex-4}\n\\end{align}\nwhere $Du$ denotes the symmetric gradient of $u$ and $C$ depends only on $d$ and $\\omega$.\n\\end{lemma}\n\n\\begin{proof} \nSee \\cite[pp.45-47]{OSY-1992}.\nNote that since $\\widetilde{F}_k$ and $F_k$ are shifts of a finite number of domains,\nthe constant  $C$ does not depend on $k$.\n\\end{proof}\n\nThroughout the paper we  assume that $A$ is real, bounded measurable,  1-periodic,  and satisfies the elasticity condition \\eqref{ellipticity}.\nIt is well known that  \\eqref{ellipticity} implies \n\\begin{align}\n  A\\xi \\cdot \\zeta  & \\le \\frac{\\kappa_2}{4} |\\xi +\\xi^T| | \\zeta +\\zeta^T| \\label{e-1},\\\\\n \\frac{\\kappa_1}{4} \n |\\xi +\\xi^T|  & \\le A \\xi \\cdot \\xi \\label{e-2}\n \\end{align}\n for any $d\\times d$ matrices $\\xi$ and $\\zeta$ \\cite[pp.30-31]{OSY-1992}.\n \n\\begin{lemma}\\label{lemma-2.1}\nLet  $0<\\delta <\\infty$ and  $u\\in H^1(\\Omega; \\mathbb{R}^d)$ be a weak solution of\n$\\mathcal{L}_\\delta (u)=f$ in $\\Omega$.\nThen \n\\begin{equation}\\label{Ca-2.1}\n\\int_{\\Omega} \n|\\Lambda_\\delta Du  |^2 | \\varphi|^2 \\, dx\n\\le C \\int_{\\Omega} |\\Lambda_\\delta  u|^2  |\\nabla \\varphi |^2\\, dx\n+ C \\int_\\Omega |f | |u| |\\varphi|^2\\, dx\n\\end{equation}\nfor any $\\varphi \\in C_0^1(\\Omega)$,\nwhere $C$ depends only $d$, $\\kappa_1$ and $\\kappa_2$.\nIn the case $\\delta=0$, \\eqref{Ca-2.1} holds for solutions of $\\mathcal{L}_0 (u)=f\\chi_\\omega$ in $\\Omega$.\n\\end{lemma}\n\n\\begin{proof}\nAssume $0<\\delta< \\infty$.\nLet $v =u\\varphi^2$, where $\\varphi \\in C_0^1(\\Omega)$.\nSince\n$$\n\\int_{\\Omega} \\Lambda_{\\delta^2} A\\nabla u \\cdot \\nabla v \\, dx =\\int_\\Omega f \\cdot v\\, dx,\n$$\nwe see that \n$$\n\\int_{\\Omega}\n\\Lambda_{\\delta^2} ( A\\nabla u \\cdot \\nabla u) \\varphi^2\\, dx\n=-2 \\int_{\\Omega} \\Lambda_{\\delta^2} (A\\nabla u \\cdot u (\\nabla \\varphi) ) \\varphi\\, dx\n+\\int_\\Omega f \\cdot v\\, dx,\n$$\nfrom which the inequality \\eqref{Ca-2.1} follows by using \\eqref{e-1}-\\eqref{e-2} and the Cauchy inequality.\nThe fact that $|A\\nabla u|\\le C |D u|$ is also needed.\nThe case $\\delta=0$ may be handled in the same manner.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma-2.2}\nLet $u\\in H^1(\\widetilde{F}_k; \\mathbb{R}^d)$ be a weak solution of $-\\text{\\rm div}(A\\nabla u)=f$ in $F_k$.\nThen\n\\begin{align}\n\\int_{F_k} |\\nabla u|^2\\, dx &  \\le C \\int_{\\widetilde{F}_k \\setminus \\overline{ F}_k} |\\nabla u|^2\\, dx\n+ C \\int_{{F}_k} |f|^2\\, dx \\label{2.2-0},\\\\\n\\int_{F_k} | Du  |^2\\, dx &  \\le C \\int_{\\widetilde{F}_k \\setminus \\overline{F}_k} | Du  |^2\\, dx \\label{2.2-1}\n+C \\int_{{F}_k} |f|^2\\, dx,\n\\end{align}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lemma-ex}  there exists  $w\\in H^1(\\widetilde{F}_k; \\mathbb{R}^d)$ such that $w=u$ on $\\widetilde{F}_k \\setminus F_k$ and\n$$\n \\| w\\|_{H^1(\\widetilde{F}_k) } \\le C \\| u\\|_{H^1(\\widetilde{F}_k \\setminus \\overline{F}_k)}.\n $$\nSince $\\text{\\rm div} (A\\nabla (u-w))=f-\\text{\\rm div}(A\\nabla w)$ in $F_k$ and $u-w\\in H^1_0(F_k; \\mathbb{R}^d)$,\nby the classical energy estimate,\n$$\n\\aligned\n\\|\\nabla u\\|_{L^2(F_k)}\n & \\le C \\big\\{ \\| f\\|_{L^2(F_k)}\n+ \\| \\nabla w \\|_{L^2(F_k)}  \\big\\}\\\\\n& \\le C \\big\\{ \\| f\\|_{L^2(F_k)} +\n\\| u\\|_{H^1(\\widetilde{F}_k\\setminus \\overline{F}_k)} \\big\\}.\n\\endaligned\n$$\nNote that for any $\\phi\\in \\mathcal{R}$, \n$u-\\phi$ satisfies the same condition as $u$.\nIt follows that \n\\begin{equation}\\label{2.2-2}\n\\|\\nabla u -\\nabla \\phi \\|_{L^2(F_k)}\n\\le C \\big\\{  \\| f\\|_{L^2(F_k)} +  \\| u-\\phi \\|_{H^1(\\widetilde{F}_k\\setminus \\overline{F}_k)} \\big\\}.\n\\end{equation}\nBy taking $\\phi$ to be the $L^1$ average of $u$ over $\\widetilde{F}_k \\setminus \\overline{F}_k$ and\nusing Poincar\\'e's  inequality we obtain \\eqref{2.2-0}.\nTo see \\eqref{2.2-1}, we use\n$$\n\\| D u \\|_{L^2(F_k)}\n\\le \\|\\nabla u -\\nabla \\phi \\|_{L^2(F_k)}\n\\le C\\big\\{ \\| f\\|_{L^2(F_k)} +  \\| u-\\phi \\|_{H^1(\\widetilde{F}_k\\setminus \\overline{F}_k)} \\big\\}.\n$$\nSince this holds for any $\\phi\\in \\mathcal{R}$, \n \\eqref{2.2-1} follows by the second Korn inequality \\cite[p.19]{OSY-1992}.\n\\end{proof}\n\n\\begin{remark}\\label{remark-p}\nIt follows from Lemma \\ref{lemma-2.2} that if $\\mathcal{L}_\\delta (u) =0 $ in $Q_{R+3}$\nfor some $R>0$, then\n\\begin{equation}\\label{ex-1a}\n\\int_{Q_R} |\\nabla u|^2\\, dx\n\\le C \\int_{Q_{R+3}\\cap \\omega} \n |\\nabla u|^2\\, dx\n \\quad\n \\text{ and } \\quad\n \\int_{Q_R} | Du |^2\\, dx\n \\le C \\int_{Q_{R+3}\\cap \\omega} | Du |^2\\, dx.\n \\end{equation}\n To see this, it suffices to note that if $F_k \\cap Q_R\\neq \\emptyset$, then\n $\\widetilde{F}_k \\setminus \\overline{F} _k \\subset Q_{R+3}\\cap \\omega$.\n Also, observe that by Sobolev inequality,  for any $u\\in H^1(\\widetilde{F}_k; \\mathbb{R}^d)$, \n \\begin{equation}\\label{ex-2a}\n \\int_{\\widetilde{F}_k} |u|^2\\, dx \n \\le C \\int_{\\widetilde{F}_k} |\\nabla u|^2\\, dx\n + C \\int_{\\widetilde{F}_k \\setminus \\overline{F}_k} |u|^2\\, dx.\n \\end{equation}\n This, together with \\eqref{2.2-0}, implies that if \n $\\mathcal{L}_\\delta (u)=0$ in $Q_{R+3}$ for some $R>0$, then\n \\begin{equation}\\label{ex-3a}\n \\int_{Q_R} |u|^2\\, dx\n \\le C \\int_{Q_{R+3}\\cap \\omega} \n |u|^2\\, dx\n + \\int_{Q_{R+3}\\cap \\omega} \n |\\nabla u|^2\\, dx.\n \\end{equation}\n\\end{remark}\n\nThe next theorem gives a Caccioppoli inequality, which is uniform in $\\delta \\in [0, 1]$,\n for $\\mathcal{L}_\\delta$.\n \n\\begin{thm}\\label{thm-2.3}\nSuppose $0\\le \\delta < \\infty $.\nLet $u\\in H^1(Q_{2R}; \\mathbb{R}^d)$ be a weak solution of\n$\\mathcal{L}_\\delta (u)=0$ in $Q_{2R} $ for some $R\\ge 4$.\nThen\n\\begin{equation}\\label{Ca-1}\n\\int_{Q_{R}} |\\nabla u|^2\\, dx\n\\le \\frac{C(1+\\delta^2) }{R^2}\n\\int_{Q_{2R} } |u|^2\\, dx,\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\\end{thm}\n\n\\begin{proof}\n\nBy the second Korn inequality,\n\\begin{equation}\\label{K-1}\n\\int_{Q_R} \n|\\nabla u|^2 \\le C \\int_{Q_R} | Du |^2\\, dx\n+\\frac{C}{R^2} \\int_{Q_R} |u|^2\\, dx,\n\\end{equation}\nwhere $C$ depends only on $d$.\nIn \\eqref{Ca-2.1} we choose $\\varphi\\in C_0^1(Q_{2R})$ such that $\\varphi=1$ in $Q_{R+3}$ and\n$|\\nabla \\varphi|\\le C\/R$. This gives\n\\begin{equation}\\label{2.3-1}\n\\int_{Q_{R+3} \\cap \\omega}\n|D u  |^2\\, dx \\le \\frac{C(1+\\delta^2) }{R^2} \\int_{Q_{2R}} |u|^2\\, dx.\n\\end{equation}\nwhich, together with \\eqref{K-1} and  \\eqref{ex-1a}, gives \\eqref{Ca-1}.\n\\end{proof}\n\n\n\\section{A Caccioppoli type inequality  for $1< \\delta\\le \\infty$}\\label{section-3}\n\nWe first consider the case $1<\\delta<\\infty$.\n\n \\begin{lemma}\\label{lemma-7.1}\n Let $ 1 < \\delta< \\infty$.\n Let $u\\in H^1(\\widetilde{F}_k; \\mathbb{R}^d)$ be a weak solution of\n $-\\text{\\rm div}(\\Lambda_{\\delta^2} A\\nabla u)=f$ in $\\widetilde{F}_k$.\n Then\n \\begin{equation}\\label{7.1-0}\n \\delta^2 \\| Du\\|_{L^2(F_k)}\n \\le C \\big\\{ \\| f\\|_{L^p (\\widetilde{F}_k)}\n + \\| Du\\|_{L^2(\\widetilde{F}_k\\setminus \\overline{F}_k)} \\big\\},\n \\ \\end{equation}\n where  $p=\\frac{2d}{d+2}$ for $d\\ge 3$ and $p>1$ for $d=2$.\nThe constant  $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{lemma}\n \n \\begin{proof}\n Let $v\\in H_0^1(\\widetilde{F}_k; \\mathbb{R}^d)$ be an extension of $u$ from $F_k$ to \n $\\widetilde{F}_k$ such that \n \\begin{equation}\\label{7.1-1}\n\\|v\\|_{H^1(\\widetilde{F}_k)}\n \\le C \\| u\\|_{H^1(F_k)}.\n \\end{equation}\n Since\n $$\n \\int_{\\widetilde{F}_k \\setminus \\overline{F}_k}\n A\\nabla u \\cdot \\nabla v\\, dx\n +\\delta^2 \\int_{F_k} A\\nabla u \\cdot \\nabla v\\, dx=\\int_{\\widetilde{F}_k} f \\cdot v\\, dx,\n $$\n it follows that\n \\begin{equation}\\label{7.1-2}\n \\aligned\n \\delta^2\\int_{F_k} |D u|^2\\, dx\n  & \\le C \\| f\\|_{L^p(\\widetilde{F}_k)}  \\| v\\|_{L^{p^\\prime} (\\widetilde{F}_k)}\n + C \\|  Du \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)} \\|  D v \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\\\\\n & \\le C (   \\| f\\|_{L^p(\\widetilde{F}_k)} \n +  \\|  Du \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}) \n \\| u\\|_{H^1(F_k)},\n \\endaligned\n \\end{equation}\n where we have used Sobolev inequality and \\eqref{7.1-1}.\n We now choose $\\phi\\in \\mathcal{R}$ such that\n $$\n \\| u-\\phi \\|_{H^1(F_k)} \\le C \\| D u \\|_{L^2(F_k)}.\n $$\n Since $u-\\phi$ satisfies the same conditions as $u$, we may deduce \\eqref{7.1-0},\n  readily  from \\eqref{7.1-2}, with $u-\\phi$ in the place of $u$.\n \\end{proof}\n\n\\begin{lemma}\\label{lemma-3.2}\nSuppose $1< \\delta< \\infty$.\nLet $u\\in H^1(Q_{R}; \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u)=0$ in $Q_{R}$\nfor some $R\\ge 16$.\nThen, for $(R\/2)\\le   r\\le  R-8$ and $0<\\varepsilon<1$,\n\\begin{equation}\\label{3.2-0}\n\\int_{Q_r}\n|\\nabla u |^2\\, dx\n\\le \\frac{C}{\\varepsilon (R-r)^2} \\int_{Q_{R}} |u|^2\\, dx\n+   \\Big(\\varepsilon +\\frac{C}{\\delta^2} \\Big) \\int_{Q_{R}} |\\nabla u|^2\\, dx,\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\\end{lemma}\n\n\\begin{proof}\nAs in the proof of Theorem \\ref{thm-2.3},\nit follows from the second Korn inequality and \\eqref{2.2-1} that\n\\begin{equation}\\label{3.2-1}\n\\int_{Q_r} |\\nabla u|^2\\, dx\n\\le C \\int_{Q_{r+3}\\cap \\omega} |Du |^2\\, dx\n+\\frac{C}{r^2} \\int_{Q_r} |u|^2\\, dx.\n\\end{equation}\nSince $r\\ge R-r$, it suffices to bound the first term in the right-hand side of \\eqref{3.2-1}.\nTo this end,\nlet $\\varphi$ be a function in $C_0^1(Q_{R-3})$ such that\n$\\varphi=1$ in $Q_{r+3}$ and \n$$\n|\\nabla \\varphi|\\le C(R-r-6)^{-1}\n\\le C (R-r)^{-1},\n$$\nwhere we have used the assumption $R-r \\ge   8$.\nRecall  that if $F_k\\cap Q_{R-3}\\neq \\emptyset$, then $\\widetilde{F}_k\\subset Q_{R}$.\nFor each $F_k$ with  $\\widetilde{F}_k\\subset Q_{R}$, we let $w_k\\in H_0^1(\\widetilde{F}_k; \\mathbb{R}^d)$\nbe an extension of $u\\varphi^2-g_k $ from $F_k$ to $\\widetilde{F}_k$ with the property that\n\\begin{equation}\\label{p-1}\n\\| w_k \\|_{H^1(\\widetilde{F}_k)} \\le C \\| u\\varphi^2 -g_k \\|_{H^1(F_k)},\n\\end{equation}\nwhere $ g_k \\in \\mathcal{R} $ is to be determined.\nExtend $w_k$ from $\\widetilde{F}_k $ to $\\mathbb{R}^d $ by zero and let\n\\begin{equation}\\label{p-3}\n\\phi=u\\varphi^2 -\\sum_k  w_k \\quad \\text{ in } \\mathbb{R}^d,\n\\end{equation}\nwhere  the sum  is taken over those $k$'s for which $\\widetilde{F}_k \\subset Q_R$.\nNote that $\\phi (x) =g_k$ if $x\\in F_k$ and $\\widetilde{F}_k \\subset Q_R$.\nSince  $\\phi\\in H^1_0(Q_R; \\mathbb{R}^d)$, we have \n\\begin{equation}\\label{p-2}\n\\int_{Q_R\\cap \\omega} A\\nabla u \\cdot \\nabla \\phi\\, dx  +\\delta^2 \\int_{Q_R\\cap F} A\\nabla  u\\cdot \\nabla \\phi\\, dx=0.\n\\end{equation}\nSince $D \\phi =0$ in $F$, we obtain \n$$\n\\int_{Q_R\\cap \\omega} A\\nabla u \\cdot \\nabla \\phi\\, dx=0.\n$$\nThus, \n\\begin{equation}\\label{3.2-5}\n\\aligned\n\\Big|\n\\int_{Q_R\\cap \\omega} A\\nabla u \\cdot \\nabla (u\\varphi^2) \\, dx\\Big|\n&\\le \\sum_k \\Big|\n\\int_{\\widetilde{F}_k \\setminus \\overline{F}_k}\nA\\nabla u \\cdot \\nabla w_k \\, dx \\Big|\\\\\n& \\le  C \\sum_k\n\\|  D u\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k) }\n\\| D w_k \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}.\n\\endaligned\n\\end{equation}\nNote that by \\eqref{p-1},\n$$\n\\|\\nabla w_k\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n    \\le C \\| u\\varphi^2 -g_k \\|_{H^1(F_k)}\n  \\le C \\| D(u \\varphi^2)\\|_{L^2(F_k)},\n $$\n where we have  chosen $g_k \\in \\mathcal{R}$ such that the last inequality holds.\n Consequently, \n $$\n \\aligned\n \\|\\nabla w_k\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n&\\le C \\| D u  \\|_{L^2(F_k)}\n+ C \\|u \\nabla \\varphi\\|_{L^2(F_k)}\\\\\n& \\le C \\delta^{-2} \\|  Du  \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n+ C \\|u \\nabla \\varphi\\|_{L^2(F_k)},\n\\endaligned\n$$\nwhere we have  used \\eqref{7.1-0} for the last inequality.\nThis, together with \\eqref{3.2-5}, gives\n$$\n\\Big|\n\\int_{Q_R\\cap \\omega} A\\nabla u \\cdot \\nabla (u\\varphi^2) \\, dx\\Big|\n\\le (C \\delta^{-2} +\\varepsilon) \\int_{Q_R} | D  u|^2\\, dx\n+ C \\varepsilon^{-1}\\int_{Q_R} |u |^2 | \\nabla \\varphi|^2\\, dx\n$$\nfor any $0< \\varepsilon<1$, where we have used the Cauchy inequality.\nHence,\n$$\n\\int_{Q_{r+3} \\cap \\omega} \n|Du |^2\\, dx\n \\le (C \\delta^{-2} +\\varepsilon) \\int_{Q_R} | D u|^2\\, dx\n+ \\frac{C}{\\varepsilon (R-r)^2}\n\\int_{Q_R} |u|^2  \\, dx,\n$$\nwhich, combined  with \\eqref{3.2-1}, yields \\eqref{3.2-0}.\n\\end{proof}\n\nThe following theorem provides a weaker version of the Caccioppoli inequality, that is uniform for $\\delta\\in (1, \\infty)$,\nfor the operator $\\mathcal{L}_\\delta$.\n\n\\begin{thm}\\label{thm-3.3}\nSuppose $1<\\delta< \\infty$.\nLet $u\\in H^1(Q_R; \\mathbb{R}^d)$ be a weak solution of\n$\\mathcal{L}_\\delta (u)=0$ in $Q_R$ for some $R\\ge 32$.\nThen, for any $\\ell\\ge 1$,\n\\begin{equation}\\label{3.3-0}\n\\int_{Q_{R\/2}}  |\\nabla u|^2\\, dx\n\\le \\frac{C}{R^2} \\int_{Q_R} |u|^2\\, dx\n+\\frac{C}{R^{2\\ell} }  \\int_{Q_R} |\\nabla u|^2\\, dx,\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\ell$,  and $\\omega$.\n\\end{thm}\n\n\\begin{proof}\nThe proof uses Lemma \\ref{lemma-3.2} and an iteration argument.\nLet $r_i =R(1-2^{-i})$ for $i=1, 2, \\dots$.\nIt follows from \\eqref{3.2-0}  that for $0< \\varepsilon<1$,\n\\begin{equation}\\label{3.3-1}\n\\int_{Q_{r_i}}\n|\\nabla u|^2\\, dx\n\\le \\frac{C}{\\varepsilon (r_{i+1} -r_i)^2}\n\\int_{Q_{r_{i+1}}}|u|^2\\, dx\n+ ( \\varepsilon + C \\delta^{-2}) \\int_{Q_{r_{i+1}}} |\\nabla u|^2\\, dx,\n\\end{equation}\nif $r_{i+1} \\ge 16$ and \n$$\n(1\/2) r_{i+1} \\le r_i \\le  r_{i+1} -8.\n$$\nIt is easy to verify that the conditions on $r_i$ are satisfied if $1\\le i \\le k$, where $k$ is the largest integer such that\n$R2^{-k-1} \\ge 8$.\nThus, by an induction argument,\n$$\n\\int_{Q_{r_1}}\n|\\nabla u|^2\\, dx\n\\le \\frac{C_0}{\\varepsilon}\n\\sum_{i=1}^k\n\\frac{ (\\varepsilon + C_0 \\delta^{-2})^{i-1}}{(r_{i+1} -r_i)^2}\n\\int_{Q_R} |u|^2\\, dx\n+ (\\varepsilon + C_0 \\delta^{-2})^k\n\\int_{Q_{r_{k+1}}} |\\nabla u|^2\\, dx,\n$$\nwhere $C_0$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\nSince $r_{i+1}-r_i= 2^{-i-1} R$, we see that \n$$\n\\aligned\n\\int_{Q_{R\/2}} |\\nabla u|^2\\, dx\n & \\le \\frac{4C_0}{\\varepsilon (\\varepsilon + C_0 \\delta^{-2}) R^2 }\n\\sum_{i=1}^k ( 4\\varepsilon + 4 C_0 \\delta^{-2})^i  \\int_{Q_R} |u|^2\\, dx\\\\\n& \\qquad\\qquad\n+ (\\varepsilon + C_0 \\delta^{-2})^k \\int_{Q_R} |\\nabla u|^2\\, dx.\n\\endaligned\n$$\nWe now choose $\\varepsilon=  2^{-2\\ell-2} $. It follows that if $4C_0\\delta^{-2}\\le 2^{-2\\ell} $, then\n$$\n\\int_{Q_{R\/2}} |\\nabla u|^2\\, dx\n\\le \\frac{C}{R^2} \\int_{Q_R} |u|^2\\, dx\n+ (2^{-2\\ell} )^k \\int_{Q_R} |\\nabla u|^2\\, dx.\n$$\nThis gives \\eqref{3.3-0} for the case $\\delta^2\\ge  2^{2\\ell +2} C_0$, as $2^{k}\\approx R$.\nFinally, we observe that the remaining case $1< \\delta^2 <  2^{2\\ell  +2} C_0$ is contained in Theorem \\ref{thm-2.3}.\n\\end{proof}\n\nWe now consider the case $\\delta=\\infty$.\nRecall that $u\\in H^1(\\Omega; \\mathbb{R}^d)$ is called a weak solution of $\\mathcal{L}_\\infty (u)=0$ in $\\Omega$ if\n$Du =0$ in $\\Omega\\cap F$ and \n\\begin{equation}\\label{4.0-0}\n\\int_{\\Omega\\cap \\omega} \nA\\nabla u\\cdot \\nabla v \\, dx =0\n\\end{equation}\nfor any $v \\in H^1_0(\\Omega; \\mathbb{R}^d)$ with $D v =0$ in $\\Omega\\cap F$.\n\n\\begin{thm}\\label{thm-4.1}\nLet $u\\in H^1(Q_R; \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\infty (u)=0$ in $Q_R$ for some $R\\ge 32$.\nThen, for any $\\ell\\ge 1$,\n\\begin{equation}\\label{4.1-0}\n\\int_{Q_{R\/2} } |\\nabla u|^2\\, dx\n\\le \\frac{C}{R^2} \\int_{Q_R} |u|^2\\, dx\n+ \\frac{C}{R^{2\\ell}} \\int_{Q_R} |\\nabla u|^2\\, dx,\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\ell$, and $\\omega$.\n\\end{thm}\n\n\\begin{proof}\nIn view of the proof of Theorem \\ref{thm-3.3}, it suffices to show\nthat  for $(R\/2)\\le r \\le R-8$ and $0<\\varepsilon<1$,\n\\begin{equation}\\label{4.1-1}\n\\int_{Q_r} |\\nabla u|^2\\, dx\n\\le \\frac{C}{\\varepsilon (R-r)^2} \n\\int_{Q_R} |u|^2\\, dx\n+ \\varepsilon \\int_{Q_R} |\\nabla u|^2\\, dx.\n\\end{equation}\nThe proof of \\eqref{4.1-1} is similar to that of Lemma \\ref{lemma-3.2}.\nIndeed, by the second Korn inequality,\n\\begin{equation}\\label{4.1-2}\n\\int_{Q_r} |\\nabla u |^2\\, dx \n\\le C \\int_{Q_{r} \\cap \\omega} |D u |^2\\, dx\n+\\frac{1}{r^2} \\int_{Q_r} |u|^2\\,dx,\n\\end{equation}\nwhere we have used the fact $Du =0$ in $Q_r\\cap F$.\nLet $\\varphi\\in C_0^1(Q_{R-3})$ and $\\phi\\in H^1_0(Q_R; \\mathbb{R}^d)$\nbe the same as in the proof of Lemma \\ref{lemma-3.2}.\nNote that $\\phi|_{F_k}  \\in \\mathcal{R}$ for each $F_k$ (if $F_k \\cap Q_{R-3}\n=\\emptyset $, then $\\phi=0$).\nThis allows us to use \\eqref{4.0-0} to obtain \n$$\n\\int_{Q_R\\cap \\omega}\nA\\nabla u\\cdot \\nabla \\phi\\, dx =0.\n$$\nThe rest of the argument is the same as in the proof of Lemma \\ref{lemma-3.2}, without the terms involving \n$C\\delta^{-2}$. We omit the details.\n\\end{proof}\n\n\\begin{remark}\\label{remark-3.1}\nLet $1< \\delta\\le \\infty$\nand  $u\\in H^1(Q_R; \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u)=0$ in $Q_R$\nfor some $R$ sufficiently large.\nIt follows from Theorems \\ref{thm-3.3} and \\ref{thm-4.1} (with $\\ell=1$) that \n\\begin{equation}\\label{re-3.1}\n\\aligned\n\\sup_{s\\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2 \\right)^{1\/2}\n & \\le \n C \\left(\\fint_{Q_R} |\\nabla u|^2\\right)^{1\/2} + C \\sup_{s\\le r\\le R}\n\\inf_{E\\in \\mathbb{R}^d}\n\\frac{1}{r}\n\\left(\\fint_{Q_r} |u-E|^2 \\right)^{1\/2}\\\\\n& \\qquad\\qquad\n+ \\frac{C}{s}\n\\sup_{s\\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2 \\right)^{1\/2}\n\\endaligned\n\\end{equation}\nfor any $s\\in [16, R]$,\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\nChoose $s$ so large that $Cs^{-1}\\le (1\/2)$.\nThis yields \n\\begin{equation}\\label{re-3.2}\n\\sup_{s\\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2 \\right)^{1\/2}\n  \\le \n C \\left(\\fint_{Q_R} |\\nabla u|^2\\right)^{1\/2} + C \\sup_{s \\le r\\le R}\n\\inf_{E\\in \\mathbb{R}^d}\n\\frac{1}{r}\n\\left(\\fint_{Q_r} |u-E|^2 \\right)^{1\/2}.\n\\end{equation}\nNote that if $1\\le r< s$, $|Q_r|^{-1\/2} \\| u\\|_{L^2(Q_r)}\n\\le C |Q_s|^{-1\/2} \\| u\\|_{L^2(Q_s)}$.\nAs a result, we obtain \n\\begin{equation}\\label{re-3.3}\n\\sup_{1 \\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2 \\right)^{1\/2}\n  \\le \n C \\left(\\fint_{Q_R} |\\nabla u|^2\\right)^{1\/2} + C \\sup_{1\\le r\\le R}\n\\inf_{E\\in \\mathbb{R}^d}\n\\frac{1}{r}\n\\left(\\fint_{Q_r} |u-E|^2 \\right)^{1\/2},\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\\end{remark}\n\n\n\n\\section{Large-scale estimates}\\label{section-4}\n\nIn this section we give the proof of Theorem \\ref{main-thm-1}.\nAs we mentioned in Introduction, the approach is based on an idea from \\cite{Armstrong-2020}.\n\nLet  $u\\in L^1(Q_{2r})$ for some $r\\in \\mathbb{N}$, define\n\\begin{equation}\\label{5.0-0}\n\\widehat{u} (z) = \\int_{Y+z} u(x)\\, dx,\n\\end{equation}\nwhere $Y=(0, 1)^d$, \nfor any $z\\in \\mathbb{Z}^d $ such that $Y+z\\subset Q_{2r}$.\n\n\n\\begin{lemma}\\label{lemma-5.1}\nLet $u\\in H^1(Q_{2r})$ for some $r\\in \\mathbb{N}$.\nThen\n\\begin{equation}\\label{5.1-0}\n\\left(\\fint_{Q_{2r} } |u|^2\\right)^{1\/2}\n\\le C \\sup_{Y+z\\subset Q_{2r}}  |\\widehat{u}(z)|\n+ C \\left(\\fint_{Q_{2r}} |\\nabla u|^2\\right)^{1\/2},\n\\end{equation}\nwhere $C$ depends only on $d$.\n\\end{lemma}\n\n\\begin{proof}\nThis follows by using  Poincar\\'e's inequality on each unit cube $Y+z\\subset Q_{2r}$ to obtain \n$$\n\\int_{Y+z} |u|^2\\, dx \\le |\\widehat{u}(z)|^2 + C \\int_{Y+z} |\\nabla u|^2\\, dx\n$$\nand summing the inequality over $z$.\n\\end{proof}\n\nFor a  function $f$ defined in  $\\mathbb{R}^d$ or $\\mathbb{Z}^d$, let\n\\begin{equation}\\label{5.2}\n\\Delta_j f (x)= f(x+e_j) -f(x)\n\\end{equation}\nfor $1\\le j \\le d$, where $e_j =(0, \\dots, 1, \\dots, 0)$ with $1$ in the $j^{th}$ position.\nFor a multi-index $\\gamma=(\\gamma_1, \\gamma_2, \\dots, \\gamma_d)$,\nwe use the notation  $\\Delta^\\gamma f =\\Delta_1^{\\gamma_1} \\Delta_2^{\\gamma_2} \\cdots \\Delta_d^{\\gamma_d} f$.\nLet $\\partial^k f = ( \\Delta^\\gamma f )_{|\\gamma|= k}$ and \n$$\n|  \\partial^kf | =\\Big(  \\sum_{|\\gamma|=k} |\\Delta^\\gamma f|^2 \\Big)^{1\/2}\n$$\nfor an integer $k\\ge 0$.\nThe following discrete Sobolev inequality will be needed:\n\\begin{equation}\\label{Sob}\n\\sup_{z\\in \\mathbb{Z}^d\\cap \\overline {Q}_{2R}}\n|f(z)|\n\\le C \\sum_{k=0}^N \nR^k \\left(\\frac{1}{R^d}\n\\sum_{z\\in \\mathbb{Z}^d \\cap \\overline{Q}_{4R} }\n|\\partial^k f (z)|^2\\right)^{1\/2},\n\\end{equation}\nwhere  $R\\ge 1$ is an integer, $N=[d\/2]+1$,  and $C$ depends only on $d$.\nWe refer the reader to \\cite{Stevenson-1991} for a proof of (\\ref{Sob}).\n\n\\begin{lemma}\\label{lemma-5.3}\nLet $u\\in H^1(Q_{4R} )$ for some integer $R\\ge 2$.\nThen, for any integer $r\\in [1, 2R]$,\n\\begin{equation}\\label{5.3-0}\n\\inf_{E\\in \\mathbb{R}}\n\\left(\\fint_{Q_{2r} } |u - E|^2\\right)^{1\/2}\n\\le {Cr}\n\\sum_{k=0}^N\nR^k\n\\left(\\fint_{Q_{4R} } \n|\\nabla \\partial ^k  u|^2\\right)^{1\/2}\n+ C \\left(\\fint_{Q_{2r}} |\\nabla u|^2\\right)^{1\/2},\n\\end{equation}\nwhere $N=[d\/2]+1$ and $C$ depends only on $d$.\n\\end{lemma}\n\n\\begin{proof}\nWe may assume $r\\le R-1$; for otherwise,\n\\eqref{5.3-0}  (with $N=0$)  follows directly from Poincar\\'e's inequality.\nBy \\eqref{5.1-0} we have \n\\begin{equation}\\label{5.3-1}\n\\aligned\n\\left(\\fint_{Q_{2r}} |u-\\widehat{u} (0) |^2\\right)^{1\/2}\n & \\le C \\sup_{z\\in \\mathbb{Z}^d\\cap \\overline{Q}_{2r}}  |\\widehat{u}(z)-\\widehat{u} (0) |\n+ C \\left(\\fint_{Q_{2r}} |\\nabla u|^2\\right)^{1\/2}\\\\\n & \\le C r \\sup_{z\\in \\mathbb{Z}^d\\cap \\overline{Q}_{2r}}  |\\partial  \\widehat{u} (z)  |\n+ C \\left(\\fint_{Q_{2r}} |\\nabla u|^2\\right)^{1\/2}.\n\\endaligned\n\\end{equation}\nTo bound the first term in the right-hand side of (\\ref{5.3-1}),\nwe use \\eqref{Sob} to obtain \n\\begin{equation}\\label{5.3-2}\n\\sup_{z\\in \\mathbb{Z}^d\\cap \\overline{Q}_{2R-2} }  |\\partial  \\widehat{u} (z)  |\n\\le \nC \\sum_{k=0}^N\nR^k \\left(\\frac{1}{R^d}\n\\sum_{z\\in \\mathbb{Z}^d\\cap \\overline{Q}_{4R-4}}\n|\\partial^{k+1} \\widehat{u}(z)|^2 \\right)^{1\/2}.\n\\end{equation}\nNote that \n$$\n\\aligned\n|\\Delta_j  \\partial^k \\widehat{u} (z)|^2\n & \\le \\int_{Y+z}\n|\\partial^k u(x+e_j) -\\partial^k u(x)|^2\\, dx\\\\\n&\\le \\int_{Y+z}\n\\int_0^1 |\\nabla \\partial^k u (x+t e_j)|^2 \\, dt\\, dx\\\\\n&\\le \\int_{(0, 2)^d +z} |\\nabla \\partial^k u(x)|^2\\, dx.\n\\endaligned\n$$\nIt follows that\n$$\n\\left(\\frac{1}{R^d}\n\\sum_{z\\in \\mathbb{Z}^d\\cap \\overline{Q}_{4R-4}}\n|\\partial^{k+1} \\widehat{u}(z)|^2 \\right)^{1\/2}\n\\le C \\left(\\fint_{Q_{4R}}\n| \\nabla  \\partial ^k u|^2\\, dx\\right)^{1\/2}.\n$$\nThis, together with \\eqref{5.3-1}and \\eqref{5.3-2}, gives \\eqref{5.3-0}.\n\\end{proof}\n\n\\begin{proof}[\\bf Proof of Theorem \\ref{main-thm-1}]\n\nLet $u\\in H^1(Q_R; \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u)=0$ in $Q_R$ for some $R\\ge 4$.\nWithout loss of generality  we may assume $R$ is a large even integer.\n\nWe first point out that \\eqref{Lip-e-1} follows from \\eqref{Lip-e}.\nIndeed, let $1\\le r\\le R-3$.  Note that for any $\\phi \\in \\mathcal{R}$, we have $\\mathcal{L}_\\delta (u-\\phi)=0$ in $Q_R$. \nIt follows that\n$$\n\\aligned\n\\left(\\fint_{Q_r} |D u|^2 \\right)^{1\/2}\n  &\\le C \\left(\\fint_{Q_{R-3}} |\\nabla (u-\\phi)|^2\\right)^{1\/2}\\\\\n  &\\le C \\left(\\fint_{Q_{R-3}} |Du|^2 \\right)^{1\/2},\n  \\endaligned \n  $$\n  where we have chosen $\\phi\\in \\mathcal{R}$ so that the last inequality holds.\n  This, together with \\eqref{ex-1a}, gives \\eqref{Lip-e-1}.\n  The rest of the proof is devoted to (\\ref{Lip-e}.\n  In view of \\eqref{ex-1a} it suffices to bound the left-hand side of \\eqref{Lip-e})\n  by the $L^2$ average of $|\\nabla u|$ over $Q_R$.\n  \n\\medskip\n\n\\noindent{\\bf Case I:   $0 \\le \\delta\\le 1$.}\nNote that   $\\mathcal{L}_\\delta (\\Delta^\\gamma u )=0$ in $Q_{R-2k}$\nfor any multi-index $\\gamma$ with $|\\gamma|=k$.\nIt follows from Theorem \\ref{thm-2.3} that\n$$\n\\fint_{Q_{\\rho }} |\\nabla \\partial^k  u |^2\n\\le \\frac{C}{\\rho ^2} \\fint_{Q_{2\\rho}}\n| \\partial^k   u|^2\\, dx\n$$\nfor $4\\le \\rho< (R-2k)\/2$, where we have used the condition $0\\le\\delta\\le 1$.\n This, together with the observation  that \n\\begin{equation}\\label{5.4-1}\n\\fint_{Q_{2\\rho} } |\\partial^k u|^2\\, dx\n\\le C \\fint_{Q_{2\\rho+2}} |\\nabla \\partial^{k-1} u|^2\\, dx\n\\end{equation}\nfor $k \\ge 1$,   yields\n\\begin{equation}\\label{5.4-2}\n\\fint_{Q_{\\rho }} |\\nabla \\partial^k  u |^2\n\\le \\frac{C}{\\rho^2}  \\fint_{Q_{2\\rho+2}} |\\nabla \\partial^{k-1} u|^2\\, dx.\n\\end{equation}\nBy induction we obtain\n\\begin{equation}\\label{5.4-3}\n\\fint_{Q_{cR}}\n|\\nabla \\partial^k u|^2\\, dx\n\\le \\frac{C}{\\rho^{2k}}\n\\fint_{Q_R} |\\nabla u|^2,\n\\end{equation}\nwhere $C$ and $c$ depend only on $d$, $k$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\nBy combining \\eqref{5.4-3} with  \\eqref{5.3-0} we see that\nfor any $r\\in [1, cR]$,\n\\begin{equation}\\label{5.4-4}\n\\inf_{E\\in \\mathbb{R}^d}\n\\frac{1}{r}\n\\left(\\fint_{Q_r} |u-E|^2 \\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R} |\\nabla u|^2 \\right)^{1\/2}\n+\\frac{C}{r}\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}.\n\\end{equation}\nBy Poincar\\'e's inequality we see that the inequality above also holds for $r\\in [cR, R]$.\nHence, if $1<s<R$, \n\\begin{equation}\\label{5.4-4-0}\n\\sup_{s\\le r\\le R} \n\\inf_{E\\in \\mathbb{R}^d}\n\\frac{1}{r}\n\\left(\\fint_{Q_r} |u-E|^2 \\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R} |\\nabla u|^2 \\right)^{1\/2}\n+\\frac{C}{s} \\sup_{s\\le r\\le R} \n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}.\n\\end{equation}\nNote  that by Theorem \\ref{thm-2.3}, \n\\begin{equation}\\label{5.4-4-00}\n\\sup_{s\\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R} |\\nabla u|^2\\right)^{1\/2}\n+ C \\sup_{s\\le r \\le R}\n\\inf_{E\\in \\mathbb{R}^d}\n\\frac{1}{r} \n\\left(\\fint_{Q_r} |u-E|^2\\right)^{1\/2}.\n\\end{equation}\nThus,\n$$\n\\sup_{s\\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R} |\\nabla u|^2 \\right)^{1\/2}\n+\\frac{C}{s} \\sup_{s\\le r\\le R} \n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}.\n$$\nChoose $s>1$ so large that $C s^{-1}\\le (1\/2)$.\nThis leads to \n$$\n\\sup_{s\\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R} |\\nabla u|^2 \\right)^{1\/2}.\n$$\nThe estimate for the case $1\\le  r< s$ follows from the case $r=s$.\n\n\\medskip\n\n\\noindent{\\bf Case II: $1<\\delta\\le \\infty$.}\nAs in the case $0\\le \\delta\\le 1$, $\\mathcal{L}_\\delta (\\Delta^\\gamma u)=0$ in $Q_{R-2k}$ for any multi-index $\\gamma$\nwith $|\\gamma|=k$.\nIt follows from Theorems \\ref{thm-3.3} and \\ref{thm-4.1} (with $\\ell=1$) that\n\\begin{equation}\\label{5.4-10}\n\\fint_{Q_{\\rho}}\n|\\nabla  \\partial^k u|^2\n\\le \\frac{C}{\\rho^2}\n\\fint_{Q_{2\\rho} } | \\partial^k u|^2 +\\frac{C}{\\rho^2}\n\\fint_{Q_{2\\rho} } |\\nabla \\partial^k u  |^2\n\\end{equation}\nif $16 \\le \\rho\\le  (R-2k)\/2$.\nThis, together with \\eqref{5.4-1} and  a simple observation that\n$$\n\\fint_{Q_{2\\rho}}\n|\\nabla \\partial^k u|^2\n\\le C \\fint_{Q_{2\\rho+2}}\n|\\nabla \\partial^{k-1} u|^2,\n$$\ngives (\\ref{5.4-2}).\nAs a result, the inequality (\\ref{5.4-4-0})\ncontinues to hold for the case $1<\\delta\\le \\infty$.\nIn view of Remark \\ref{remark-3.1},\nthe inequality (\\ref{5.4-4-00}) also holds for $1<\\delta\\le \\infty$. \nThe rest of the proof is the same as in Case I.\n\\end{proof}\n\nIt follows from Theorem \\ref{main-thm-1} and Poincar\\'e's inequality  that if $\\mathcal{L}_\\delta (u)=0$ in $Q_R$ for some \n$R\\ge 1$, then\n\\begin{equation}\\label{L-inf}\n\\sup_{1\\le r\\le R} \\left(\\fint_{Q_r} |u|^2\\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R} |u|^2 \\right)^{1\/2}\n+ C R^2 \\left(\\fint_{Q_R} |\\nabla  u|^2\\right)^{1\/2},\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\n\n\n\\section{Local Lipschitz  estimates}\\label{section-5}\n\nThroughout this section we will assume $A$ satisfies the elasticity condition (\\ref{ellipticity}) and\nH\\\"older continuity condition (\\ref{smoothness}).\nThe periodicity condition is not needed.\nFor $0<r\\le 4$, let\n\\begin{equation}\\label{B}\n\\aligned\nQ_r^+  & =\\big\\{ x=(x^\\prime, x_d) \\in Q_r: x_d> \\psi (x^\\prime) \\big\\},\\\\\nQ_r^-  & =\\big\\{ x=(x^\\prime, x_d) \\in Q_r: x_d< \\psi (x^\\prime) \\big\\},\\\\\nI_r  &  =\\big\\{ x=(x^\\prime, x_d) \\in Q_r: x_d= \\psi (x^\\prime) \\big\\},\\\\\n\\endaligned\n\\end{equation}\nwhere $\\psi: \\mathbb{R}^{d-1} \\to \\mathbb{R}$ is a $C^{1, \\sigma}$ function for some $\\sigma \\in (0, 1)$\nsuch that $\\psi(0)=0$ and $\\|\\nabla \\psi \\|_{C^{1, \\sigma}(\\mathbb{R}^{d-1})} \\le M_1$.\nLet $0<\\delta<\\infty$ and  $u\\in H^1(Q_r; \\mathbb{R}^d)$ be a solution of \n\\begin{equation}\\label{local-1}\n\\left\\{\n\\aligned\n -\\text{\\rm div} (A\\nabla u) & = \\delta^{-2} f  &\\quad & \\text{ in } Q_r^+,\\\\\n -\\text{\\rm div} (A\\nabla u)  & =f  & \\quad &  \\text{ in } Q_r^-,\\\\\n \\delta^2  \\frac{\\partial u}{\\partial \\nu_+}  & = \\frac{\\partial u}{\\partial \\nu_-}  &\\quad & \\text{ on }  I_r,\n\\endaligned\n\\right.\n\\end{equation}\nwhere $\\frac{\\partial u}{\\partial \\nu_\\pm}\n=n\\cdot A(\\nabla u)_\\pm$ and $\\pm$ indicates the limit taken from $Q_r^\\pm$, respectively.\nIf $\\delta=0$, by a solution of \\eqref{local-1}, we mean that $-\\text{\\rm div} (A\\nabla u)=f$ in $Q_r^-$,\n$-\\text{\\rm div} (A\\nabla u)=0$ in $Q_r^+$, and that $\\frac{\\partial u}{\\partial \\nu_-} =0$ on $I_r$.\nIf $\\delta=\\infty$, the equation \\eqref{local-1} is understood in the sense that $u|_{Q_r^+} \\in \\mathcal{R}$ and \n$-\\text{\\rm div}(A\\nabla u)=f$ in $Q_r^-$.\n\n\\begin{lemma}\\label{local-lemma-1}\nAssume that $A$ satisfies \\eqref{ellipticity} and \\eqref{smoothness}.\nLet $0\\le \\delta\\le 1$ and $u\\in H^1(Q_r; \\mathbb{R}^d)$ be a weak solution of \\eqref{local-1}  for some $0<r \\le 4$.\nThen\n\\begin{equation}\\label{local-2}\n\\left\\{\n\\aligned\n\\|\\nabla u\\|_{L^\\infty (Q_{r\/2}^-) }\n&\\le \\frac{C}{r}\n\\left\\{ \n \\delta^2 \\left(\\fint_{Q_r^+} |u|^2 \\right)^{1\/2}\n+  \\left(\\fint_{Q_r^-} |u|^2 \\right)^{1\/2}\n+r^2  \\left( \\fint_{Q_r} |f|^p \\right)^{1\/p}\\right\\},\\\\\n\\| \\nabla u\\|_{L^\\infty (Q_{r\/2}^+ ) }\n & \\le \\frac{C}{r}\n\\left\\{\n\\left(\\fint_{Q_r} |u|^2\\right)^{1\/2}\n+ r^2 \\left(\\fint_{Q^-_r} |f|^p \\right)^{1\/p} \n+ \\delta^{-2} r^2 \n\\left(\\fint_{Q_r^+} |f|^p \\right)^{1\/p} \\right\\},\n\\endaligned\n\\right.\n\\end{equation}\nwhere $p>d$ and $C$ depends only on $d$, $ p$, $\\kappa_1$, $\\kappa_2$, $\\sigma$, $M_0$, and $M_1$.\n\\end{lemma}\n\n\\begin{proof}\nThe case $0< \\delta_0\\le \\delta\\le 1$ with a constant $C$ depending on $\\delta_0$ \nfollows from the classical  results on Lipschitz estimates for elliptic systems \nwith piecewise H\\\"older continuous  coefficients.\nIndeed, since $\\psi$ is $C^{1, \\sigma}$,  the problem may be reduced to the case\n$\\psi=0$ by flatting the boundary.\nOne may further reduce the problem to the case of constant coefficients by a Campanato perturbation argument.\nWe refer the reader to \\cite{Li-2000, Dong-2012}  and their  references for more recent development.\n\nWe now  treat the case where $\\delta$ is small.\nBy rescaling we may assume $r=1$.\nLet $0<\\rho<1$ and $0<\\tau < \\sigma$. \nSince $\\text{\\rm div} (A\\nabla u)=f$ in $Q_1^-$,\nby the classical $C^{1, \\tau} $ estimates for Neumann problems,\n\\begin{equation}\\label{local-3}\n\\| \\nabla u\\|_{C^{\\tau}(Q^-_{\\rho\/2})}\n\\le C\\Big \\|\\frac{\\partial u}{\\partial \\nu_-}\\Big \\|_{C^{\\tau}  (I_{\\rho})}\n+\\frac{C}{\\rho^{1+\\tau} }\n\\left\\{\n\\left(\\fint_{Q_\\rho^-} |u|^2\\right)^{1\/2}\n+ \\rho^2 \\left(\\fint_{Q_\\rho^-} |f|^p \\right)^{1\/p} \\right\\}.\n\\end{equation}\nLet $1\/2\\le s<t<1$.\nBy covering  $Q_{s}^-$ with cubes of size $\\rho\/2=c(t-s)$ and applying (\\ref{local-3}) and interior $C^{1, \\tau}$ estimates,\nwe may deduce that\n\\begin{equation}\\label{local-4}\n\\aligned\n\\| \\nabla u\\|_{C^{\\tau} (Q_s^-)}\n & \\le C \\Big\\| \\frac{\\partial u}{\\partial \\nu_-} \\Big\\|_{C^\\tau (I_{t_1} )}\n+ C (t-s)^{-\\frac{d}{2} -1-\\tau} \\left\\{\n \\left(\\fint_{Q_1^-} |u|^2\\right)^{1\/2} + \\left(\\fint_{Q_1^-} |f|^p \\right)^{1\/p} \\right\\}\\\\\n&\\le C \\delta^2 \\| \\nabla u\\|_{C^\\tau(Q_{t_1} ^+)}\n+ C (t-s)^{-\\frac{d}{2} -1-\\tau} \\left\\{\n \\left(\\fint_{Q_1^-} |u|^2\\right)^{1\/2} + \\left(\\fint_{Q_1^-} |f|^p \\right)^{1\/p} \\right\\},\\\\\n\\endaligned\n\\end{equation}\nwhere  $t_1=(t+s)\/2$\nand\nwe have used the relation $\\frac{\\partial u}{\\partial\\nu_-} =\\delta^2 \\frac{\\partial u}{\\partial \\nu_+}$ on $I_1$ for the last inequality.\nSince $-\\text{\\rm div}(A\\nabla u)=\\delta^{-2} f $ in $Q_1^+$ for $\\delta>0$,\na similar argument using $C^{1, \\tau}$ estimates for the Dirichlet problem gives\n\\begin{equation}\\label{local-5}\n \\| \\nabla u\\|_{C^\\tau (Q_{t_1}^+)}\n\\le C  \\|\\nabla u\\|_{C^\\tau (Q_t^-)}\n+ C (t-s)^{-\\frac{d}{2}-1-\\tau }\n\\left\\{\n  \\left(\\fint_{Q_1^+} |u|^2\\right)^{1\/2}\n +\\delta^{-2}  \\left( \\fint_{Q_1^+} |f|^p \\right)^{1\/p} \\right\\}.\n\\end{equation}\nBy combining (\\ref{local-4}) with (\\ref{local-5}) it follows that \n\\begin{equation}\\label{local-6}\n\\aligned\n\\| \\nabla u\\|_{C^{\\tau} (Q_s^-)}\n & \\le  C \\delta^2  \\|\\nabla u\\|_{C^\\tau (Q_t^-)}\n+ C (t-s)^{-\\frac{d}{2}-1-\\tau }\n\\left\\{\n \\left(\\fint_{Q_1^-} |u|^2\\right)^{1\/2}\n + \\delta^2\n\\left(\\fint_{Q_1^+} |u|^2\\right)^{1\/2} \\right\\}.\\\\\n&\\qquad\\qquad\n+ C (t-s)^{-\\frac{d}{2} -1 -\\tau} \n\\left(\\fint_{Q_1} |f|^p \\right)^{1\/p},\n\\endaligned\n\\end{equation}\nwhich also holds for the case $\\delta=0$.\n Let $s_i=1-2^{-i}$.\n By taking $s=s_i$ and $t=s_{i+1}$ in \\eqref{local-6} and using iteration,\n we see that\n \\begin{equation}\\label{local-7}\n \\aligned\n &  \\| \\nabla u\\|_{C^\\tau (Q_{1\/2}^-)}\\\\\n & \\le \n C \\sum_{i=1}^N (s_{i+1}-s_i)^{-\\frac{d}{2} -1-\\tau} \\delta^{2 (i-1)}\n \\left\\{\n \\left(\\fint_{Q_1^-} |u|^2\\right)^{1\/2}\n + \\delta^2\n\\left(\\fint_{Q_1^+} |u|^2\\right)^{1\/2}\n+ \\left(\\fint_{Q_1} |f|^p \\right)^{1\/p}  \\right\\}\\\\\n& \\qquad\\qquad\n+ C \\delta^{2N} \\| \\nabla u\\|_{C^{\\tau} (Q_{s_{N+1}})}\n\\endaligned\n\\end{equation}\nfor any $N\\ge 1$.\nNote that $s_{i+1}-s_i= 2^{-i-1}$.\nIt follows that if $2^{\\frac{d}{2} +1 +\\tau} \\delta^2 \\le (1\/2)$, we may let $N \\to \\infty$ in \\eqref{local-7}\nto obtain the first inequality in \\eqref{local-2} with $r=1$.\nThe second inequality follows from the the  first and \\eqref{local-5}.\n\\end{proof}\n\n\\begin{remark}\\label{remark-5.1a}\nWe may replace (\\ref{local-2}) by \n\\begin{equation}\\label{local-2b}\n\\left\\{\n\\aligned\n\\|\\nabla u\\|_{L^\\infty (Q_{r\/2}^-) }\n&\\le  C \n\\left\\{ \n \\delta^2 \\left(\\fint_{Q_r^+} |\\nabla u|^2 \\right)^{1\/2}\n+  \\left(\\fint_{Q_r^-} |\\nabla u|^2 \\right)^{1\/2}\n+r  \\left( \\fint_{Q_r} |f|^p \\right)^{1\/p}\\right\\},\\\\\n\\| \\nabla u\\|_{L^\\infty (Q_{r\/2}^+ ) }\n & \\le  C \n\\left\\{\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}\n+ r  \\left(\\fint_{Q^-_r} |f|^p \\right)^{1\/p} \n+ \\delta^{-2} r\n\\left(\\fint_{Q_r^+} |f|^p \\right)^{1\/p} \\right\\},\n\\endaligned\n\\right.\n\\end{equation}\nTo see this, in the proof of Lemma \\ref{local-lemma-1},\none  replaces $u$ in \\eqref{local-3} and (\\ref{local-4}) by $u-E$ and applies \nPoincar\\'e's inequality.\n\\end{remark}\n\n\\begin{remark}\\label{remark-5.1}\nUnder the same assumptions as in Lemma \\ref{local-lemma-1}, we have \n\\begin{equation}\\label{local-2-r}\n\\left\\{\n\\aligned\n\\| u\\|_{L^\\infty (Q_{r\/2}^-) }\n&\\le C\n\\left\\{ \n \\delta^2 \\left(\\fint_{Q_r^+} |u|^2 \\right)^{1\/2}\n+  \\left(\\fint_{Q_r^-} |u|^2 \\right)^{1\/2}\n+r^2  \\left( \\fint_{Q_r} |f|^p \\right)^{1\/p}\\right\\},\\\\\n\\|  u\\|_{L^\\infty (Q_{r\/2}^+ ) }\n & \\le  C \n\\left\\{\n\\left(\\fint_{Q_r} |u|^2\\right)^{1\/2}\n+ r^2 \\left(\\fint_{Q^-_r} |f|^p \\right)^{1\/p} \n+ \\delta^{-2} r^2 \n\\left(\\fint_{Q_r^+} |f|^p \\right)^{1\/p} \\right\\},\n\\endaligned\n\\right.\n\\end{equation}\nThis follows readily from \\eqref{local-2} and the Mean Value Theorem.\n\\end{remark}\n\nThe next lemma treats the case $1<\\delta\\le \\infty$.\n\n\\begin{lemma}\\label{local-lemma-2}\nAssume that $A$ satisfies (\\ref{ellipticity} and (\\ref{smoothness}).\nLet $1<\\delta\\le \\infty$ and $u\\in H^1(Q_r; \\mathbb{R}^d)$ be a weak solution of \\eqref{local-1}  for some $0<r \\le 4$.\nThen\n\\begin{equation}\\label{local-2-0}\n\\left\\{\n\\aligned\n\\|\\nabla u\\|_{L^\\infty (Q_{r\/2}^+) }\n&\\le \\frac{C}{r}\n\\left\\{ \n \\delta^{-2} \\left(\\fint_{Q_r^-} |u|^2 \\right)^{1\/2}\n+  \\left(\\fint_{Q_r^+} |u|^2 \\right)^{1\/2}\n+  \\delta^{-2} r^2 \\left(\\fint_{Q_r} |f|^p \\right)^{1\/p} \n\\right\\},\\\\\n\\| \\nabla u\\|_{L^\\infty (Q_{r\/2}^- ) }\n & \\le \\frac{C}{r}\n \\left\\{\n\\left(\\fint_{Q_r} |u|^2\\right)^{1\/2}\n+ \\delta^{-2} r^2 \\left(\\fint_{Q_r^+} |f|^p \\right)^{1\/p}\n+ r^2 \\left(\\fint_{Q_r^-} |f|^ p \\right)^{1\/p} \\right\\},\n\\endaligned\n\\right.\n\\end{equation}\nwhere $C$ depends only on $d$,  $p$, $\\kappa_1$, $\\kappa_2$, $\\sigma$, $M_0$, and $M_1$.\n\\end{lemma}\n\n\\begin{proof}\nThe case $1<\\delta< \\infty$ follows from  the proof of Lemma \\ref{local-lemma-1}\nby interchanging $Q_r^+$ with $Q_r^-$ and $\\delta^2$ with $\\delta^{-2}$.\nRecall that if $\\delta=\\infty$, $u|_{Q_r^+} \\in \\mathcal{R}$.\n As a result, the first inequality in \\eqref{local-2-0} holds by a simple\nrescaling, while the second  follows from the Lipschitz estimate for the Dirichlet problem.\n\\end{proof}\n\n\\begin{remark}\\label{remark-5.2b}\nOne may replace \\eqref{local-2-0} by \n\\begin{equation}\\label{local-2-0b}\n\\left\\{\n\\aligned\n\\|\\nabla u\\|_{L^\\infty (Q_{r\/2}^+) }\n&\\le C \n\\left\\{ \n \\delta^{-2} \\left(\\fint_{Q_r^-} |\\nabla u|^2 \\right)^{1\/2}\n+  \\left(\\fint_{Q_r^+} |\\nabla u|^2 \\right)^{1\/2}\n+  \\delta^{-2} r \\left(\\fint_{Q_r} |f|^p \\right)^{1\/p} \n\\right\\},\\\\\n\\| \\nabla u\\|_{L^\\infty (Q_{r\/2}^- ) }\n & \\le C \n \\left\\{\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}\n+ \\delta^{-2} r  \\left(\\fint_{Q_r^+} |f|^p \\right)^{1\/p}\n+ r \\left(\\fint_{Q_r^-} |f|^ p \\right)^{1\/p} \\right\\},\n\\endaligned\n\\right.\n\\end{equation}\nThis follows from Remark \\ref{remark-5.1}.\n\\end{remark}\n\n\\begin{remark}\\label{remark-5.2}\nIt follows from \\eqref{local-2-0} that if $1<\\delta\\le \\infty$,\n\\begin{equation}\\label{local-3-0}\n\\left\\{\n\\aligned\n\\|\\nabla u\\|_{L^\\infty (Q_{r\/2}^+) }\n&\\le C\n\\left\\{ \n \\delta^{-2} \\left(\\fint_{Q_r^-} |u|^2 \\right)^{1\/2}\n+  \\left(\\fint_{Q_r^+} |u|^2 \\right)^{1\/2}\n+  \\delta^{-2} r^2 \\left(\\fint_{Q_r} |f|^p \\right)^{1\/p} \n\\right\\},\\\\\n\\| \\nabla u\\|_{L^\\infty (Q_{r\/2}^- ) }\n & \\le C\n \\left\\{\n\\left(\\fint_{Q_r} |u|^2\\right)^{1\/2}\n+ \\delta^{-2} r^2 \\left(\\fint_{Q_r^+} |f|^p \\right)^{1\/p}\n+ r^2 \\left(\\fint_{Q_r^-} |f|^ p \\right)^{1\/p} \\right\\}.\n\\endaligned\n\\right.\n\\end{equation}\n\\end{remark}\n\nThe following theorem provides the local Lipschitz estimate for\nsolutions of $\\mathcal{L}_\\delta (u)=f$ in $Q(x_0, r)=x_0 +Q_r$\n(if $\\delta=0$, we assume $f=0$ in $Q(x_0, r)\\cap F$).\n\n\\begin{thm}\\label{local-thm-0}\nAssume that $A$ satisfies \\eqref{ellipticity} and \\eqref{smoothness}.\nLet $0\\le \\delta\\le \\infty$ and $u\\in H^1(Q (x_0, r); \\mathbb{R}^d)$ be a weak solution of  $\\mathcal{L}_\\delta (u)=f$ in \n$Q(x_0, r) $  for some $x_0\\in \\mathbb{R}^d$ and  $0<r\\le 2$,\nwhere $f\\in L^p (Q(x_0, r); \\mathbb{R}^d)$ for some $p>d$.\nThen \n\\begin{equation}\\label{5.5-0}\n\\aligned\n & |u(x_0)| + r|\\nabla u (x_0)|\\le \\\\\n &\n \\left\\{\n \\aligned\n  &  C \\left(\\fint_{Q(x_0, r) } |\\Lambda_{\\delta^2} u|^2 \\right)^{1\/2} \n + Cr^2  \\left(\\fint_{Q(x_0, r)} | f|^p \\right)^{1\/p} &  & \\text{ if } 0\\le \\delta \\le 1 \\text{ and } x_0 \\in \\omega,\\\\\n&   C  \\left(\\fint_{Q(x_0, r) } |u|^2 \\right)^{1\/2} \n+ Cr^2 \\left(\\fint_{ Q(x_0, r)}\n|\\Lambda_{\\delta^{-2}} f|^p \\right)^{1\/P} & & \\text{ if }  0\\le \\delta \\le 1 \\text{ and } x_0 \\in F,\n\\endaligned\n\\right.\n\\endaligned\n\\end{equation}\nand\n\\begin{equation}\\label{5.5-1}\n\\aligned\n& |u(x_0)| + r|\\nabla u (x_0)|\\le \\\\\n & \n \\left\\{\n \\aligned\n  & C \\left(\\fint_{Q(x_0, r) } | u|^2 \\right)^{1\/2}\n + C r^2\n \\left(\\fint_{Q(x_0, r)} | \\Lambda_{\\delta^{-2}}  f|^p \\right)^{1\/p} & & \\text{ if } 1< \\delta < \\infty \\text{ and } x_0 \\in \\omega,\\\\\n&  \\frac{ C}{\\delta^2 }  \\left(\\fint_{Q(x_0, r) } |\\Lambda_{\\delta^2} u|^2 \\right)^{1\/2} \n+ C\\delta^{-2} r^2\n\\left(\\fint_{Q_r} |f|^p \\right)^{1\/p}  &  & \\text{ if }  1< \\delta < \\infty \\text{ and } x_0 \\in F.\n\\endaligned\n\\right.\n\\endaligned\n\\end{equation}\nIf $\\delta=\\infty$, we have\n\\begin{equation}\\label{5.5-2}\n\\aligned\n|u(x_0)| + r|\\nabla u (x_0)|\n & \\le C \\left(\\fint_{Q(x_0, r) } | u|^2 \\right)^{1\/2} \n + C r^2 \\left(\\fint_{Q(x_0, r)\\cap \\omega} |f|^p \\right)^{1\/p}& \\quad & \\text{ if } x_0 \\in \\omega\n \\\\\n|u(x_0)| + r|\\nabla u (x_0)|\n& \\le  C   \\left(\\fint_{Q(x_0, r)\\cap F  } | u|^2 \\right)^{1\/2}  & \\quad & \\text{ if }   x_0 \\in F.\n\\endaligned\n\\end{equation}\nThe constant $C$ depends only on $d$, $p$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$ in \\eqref{smoothness}.\n\\end{thm}\n\n\\begin{proof}\nNote that $-\\text{\\rm div}(A\\nabla u)=f$ in $Q(x_0, r)\\cap \\omega$ and \n$-\\text{\\rm div}(A\\nabla u)=\\delta^{-2}f$ in $Q(x_0, r)\\cap F$.\nIf $Q(x_0, cr)\\subset \\omega$ or $F$ for some small $c>0$, the estimates in \\eqref{5.5-0}-\\eqref{5.5-2} follow directly from the interior estimates \nfor solutions of $-\\text{\\rm div}(A\\nabla u)=f $. \nIn the case $Q(x_0, cr)\\cap \\partial \\omega \\neq \\emptyset$,\none may find $y_0\\in \\partial \\omega$ such that $x_0\\in Q(y_0, r\/2)$ and $Q(y_0, r\/2)\\subset Q(x_0, r)$.\nAs a result,  the estimates in \\eqref{5.5-0}-\\eqref{5.5-2}  follow readily  from \\eqref{local-2}, \\eqref{local-2-r},\n\\eqref{local-2-0} and \\eqref{local-3-0} by a simple localization argument.\n\\end{proof}\n\n\\begin{proof}[\\bf Proof of Theorem \\ref{main-thm-2}]\n\nNote that for all cases in Theorem \\ref{local-thm-0} with $f=0$,\n\\begin{equation}\\label{5.6-0}\n|u(x_0)| + r|\\nabla u(x_0)| \\le C \\left(\\fint_{Q(x_0, r)} |u|^2\\right)^{12}.\n\\end{equation}\nSince $u-E$ is also a solution for any $E\\in \\mathbb{R}^d$, one may use Poincar\\'e's inequality to obtain \n\\begin{equation}\\label{5.6-1}\n|\\nabla u(x_0)| \\le C \\left(\\fint_{Q(x_0, r)} |\\nabla u|^2\\right)^{12}\n\\end{equation}\nfor $0<r\\le 1$.\nThus, if $u$ is a weak solution of $\\mathcal{L}_\\delta (u)=0$ in $Q (x_0, R)$ for some $R\\ge 4$,\nthen\n$$\n|\\nabla u(x_0)|\n  \\le C \\left( \n\\fint_{Q(x_0, 1) } |\\nabla u|^2 \\right)^{1\/2}\\\\\n \\le C \n \\left( \n\\fint_{Q(x_0, R)\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2},\n$$\nwhere we have used Theorem \\ref{main-thm-1} for the last inequality.\n\\end{proof}\n\n\n\\section{Estimates of fundamental solutions}\\label{section-6}\n\nThroughout this section we assume that  $d\\ge 3$, $0<\\delta<\\infty$, and that $A$ satisfies \\eqref{ellipticity}, \\eqref{smoothness}, and is 1-periodic.\nWe also assume that each $F_k$ is a $C^{1, \\sigma} $ domain for some $\\sigma \\in (0, 1)$.\nUnder these conditions, by combining the large-scale estimates in Section \\ref{section-4}  with the  local  Lipschitz estimates in Section \\ref{section-5}, \nwe see that if  $u\\in H^1(Q(x_0, R); \\mathbb{R}^d)$ is a weak solution of\n$\\mathcal{L}_\\delta (u)=0$ in $Q (x_0; R)$ for some $R>0$, then \n$$\n|\\nabla u(x_0)|\\le C_\\delta \\left(\\fint_{Q(x_0, R)} |\\nabla u|^2 \\right)^{1\/2},\n$$\nwhere $C_\\delta$ may depend on $\\delta$. It follows that the operator $\\mathcal{L}_\\delta$ possesses \na fundamental solution $\\Gamma_\\delta (x, y) = \\big( \\Gamma_\\delta^{\\alpha \\beta} (x, y)  \\big)_{d\\times d}$ \nin the sense that if \n\\begin{equation}\\label{f-1}\nu(x)=\\int_{\\mathbb{R}^d} \\Gamma_\\delta (x, y)  f (y)\\, dy\n\\end{equation}\nfor some $f\\in C_0^\\infty (\\mathbb{R}^d; \\mathbb{R}^d)$,  then  $u\\in L^{2^*} (\\mathbb{R}^d; \\mathbb{R}^d)$, $\\nabla u\\in L^2(\\mathbb{R}^d; \\mathbb{R}^{d\\times d})$,  and \n\\begin{equation}\\label{f-2}\n\\int_{\\mathbb{R}^d} \\Lambda_{\\delta^2} A\\nabla u \\cdot \\nabla  v \\, dx\n=\\int_{\\mathbb{R}^d} f \\cdot v \\, dx\n\\end{equation}\nfor any $v  \\in L^{2^*} (\\mathbb{R}^d; \\mathbb{R}^d)$ with $\\nabla v \\in L^2 (\\mathbb{R}^d; \\mathbb{R}^d)$, where $2^*=\\frac{2d}{d-2}$.\nMoreover, there exists a constant $C_\\delta$ such that\n$|\\Gamma_\\delta (x, y)|\\le C_\\delta |x-y|^{2-d}$ and\n$|\\nabla_x \\Gamma_\\delta (x, y)| + |\\nabla_y \\Gamma_\\delta  (x, y)|\\le C_\\delta  |x-y|^{1-d}$\nfor any $x, y\\in \\mathbb{R}^d$.\nWe refer the reader to \\cite{Hofmann-2007} for the construction of $\\Gamma_\\delta(x, y)$\nunder a H\\\"older continuity condition on weak solutions.\nOur goal of  this section is to establish  the explicit dependence of $C_\\delta$ on $\\delta$.\n\n\\begin{lemma}\\label{lemma-f-1}\nLet $u$ be given by \\eqref{f-1} with $f\\in C_0^\\infty(\\omega; \\mathbb{R}^d)$.\nThen\n\\begin{equation}\\label{f-1-0}\n\\delta \\| Du  \\|_{L^2( F)}  \n+  \\|\\nabla u \\|_{L^2(\\omega)} +\\| u\\|_{L^{p^\\prime} (\\omega)} \n \\le C \\| f\\|_{L^p(\\omega)},\n\\end{equation}\nwhere $p=\\frac{2d}{d+2}$ and $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{lemma}\n \n \\begin{proof}\n  By letting $v=u$ in \\eqref{f-2}  we obtain \n \\begin{equation}\\label{f-1-1}\n \\delta^2 \\int_{F} |D u |^2\\, dx  \n +\\int_\\omega |Du |^2\\, dx \\le C \\| f\\|_{L^p(\\omega)} \\| u\\|_{L^{p^\\prime } (\\omega)},\n \\end{equation}\n where $p=\\frac{2d}{d+2}$. \n Next, let $U$   be an extension of $u$ from $\\omega$ to $\\mathbb{R}^d$ such that\n \\begin{equation}\\label{f-1-3}\n \\| \\nabla U\\|_{L^2(\\mathbb{R}^d)} \\le C \\| \\nabla u\\|_{L^2(\\omega)} \\quad \\text{ and } \\quad\n  \\|  D U \\|_{L^2(\\mathbb{R}^d)} \\le C \\|  D u  \\|_{L^2(\\omega)} .\n  \\end{equation}\n The function $U$ may be obtained by extending $u$ from $\\widetilde{F}_k \\setminus F_k$ to $F_k$, for each $k$,\n so that \n \\begin{equation}\\label{f-1-3a}\n \\| \\nabla U\\|_{L^2(\\widetilde{F}_k) } \\le C \\| \\nabla u\\|_{L^2(\\widetilde{F} _k \\setminus \\overline{F}_k)}\n \\quad \\text{ and } \\quad\n \\|  D U \\|_{L^2(\\widetilde{F}_k) } \\le C \\|  D u \\|_{L^2(\\widetilde{F} _k \\setminus \\overline{F}_k)}.\n \\end{equation}\n See Lemma \\ref{lemma-ex}.\n  Since $|u(x)| +|x| |\\nabla u (x)| =O(|x|^{2-d})$ as $|x|\\to \\infty$ and $d\\ge 3$, we see that\n \\begin{equation}\\label{decay}\n \\frac{1}{R^2} \\int_{Q_{2R} \\setminus Q_R}  \\big ( |u|^2 +|\\nabla u|^2 \\big) \\, dx \\to 0 \\quad \\text{ as } R \\to \\infty.\n \\end{equation}\n  The property (\\ref{ex-2}) also implies  that $U$ satisfies the condition (\\ref{decay}).\n  This allows us to  apply the first  Korn inequality and Sobolev inequality in $Q_R$  and then let $R\\to \\infty$  to deduce that\n \\begin{equation}\\label{f-1-1a}\n \\| \\nabla U\\|_{L^2(\\mathbb{R}^d)} \\le C \\| D U \\|_{L^2(\\mathbb{R}^d)}\n \\quad\n \\text{ and }\n \\quad\n \\| U\\|_{L^{p^\\prime}(\\mathbb{R}^d)} \\le C \\| \\nabla U  \\|_{L^2(\\mathbb{R}^d)}.\n\\end{equation}\n As a result, we obtain \n \\begin{equation}\\label{f-1-4}\n \\| U \\|_{L^{p^\\prime}(\\mathbb{R}^d)}\n \\le C \\| \\nabla U \\|_{L^2(\\mathbb{R}^d)}\n \\le C \\| D U \\|_{L^2(\\mathbb{R}^d)}\n \\le C \\| D u  \\|_{L^2(\\omega)}.\n \\end{equation}\n It follows that\n \\begin{equation}\\label{f-1-4a}\n \\aligned\n  & \\| \\nabla u\\|_{L^2(\\omega)} =\\| \\nabla U \\|_{L^2(\\omega)} \\le C \\| D u \\|_{L^2(\\omega)},\\\\\n  & \\| u\\|_{L^{p^\\prime}(\\omega)}= \\| U  \\|_{L^{p^\\prime}(\\omega)}\n  \\le C \\| D u  \\|_{L^2(\\omega)}.\n  \\endaligned\n \\end{equation}\n Consequently, by \\eqref{f-1-1} and the Cauchy inequality,  \n we see that $ \\delta \\| D u \\|_{L^2(F)} \\le C \\| f\\|_{L^p(\\omega)}$ and\n $$\n \\|\\nabla u\\|_{L^2(\\omega)}\n + \\| u\\|_{L^{p^\\prime} (\\omega)} \n \\le C \\| Du \\|_{L^2(\\omega)} \n \\le C \\| f\\|_{L^p(\\omega)},  \n  $$\n  which completes the proof.\n \\end{proof}\n \n \\begin{remark}\\label{re-div}\n Let $u$ be given by (\\ref{f-1}) with $f=\\text{\\rm div} (g)$,\n where $g\\in C_0^\\infty (\\omega; \\mathbb{R}^{d\\times d})$.\n Then\n $$\n \\delta^2 \\int_{F} | Du |^2\\, dx \n + \\int_{\\omega} |Du |^2\\, dx \\le C \\| g \\|_{L^2(\\omega)} \\| \\nabla u\\|_{L^2(\\omega)}.\n $$\n Using (\\ref{f-1-4a}), we obtain \n \\begin{equation}\\label{re-div-0}\n \\delta\n \\| Du \\|_{L^2(F)} + \\| \\nabla u\\|_{L^2(\\omega)}\n + \\| u\\|_{L^{p^\\prime} (\\omega)}\n \\le C \\| g\\|_{L^2(\\omega)},\n \\end{equation}\n where $p^\\prime=\\frac{2d}{d-2}$ and $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{remark}\n \n \\begin{remark}\\label{re-large}\n \n Suppose $1\\le \\delta< \\infty$.\n Let $u$ be given by \\eqref{f-1} with $f\\in C_0^\\infty (\\mathbb{R}^d; \\mathbb{R}^d)$.\n By letting $v=u$ in \\eqref{f-2} we obtain \n $\\| D u \\|^2_{L^2(\\mathbb{R}^d)} \\le C \\| f\\|_{L^p(\\mathbb{R}^d)} \\| u\\|_{L^{p^\\prime}(\\mathbb{R}^d)}$.\n Using \n $$\\| u\\|_{L^{p^\\prime}(\\mathbb{R}^d)} \\le C \\| \\nabla u\\|_{L^2(\\mathbb{R}^d)} \\le C \\| D u \\|_{L^2 (\\mathbb{R}^d)},\n $$\n we see that  $\\|u\\|_{L^{p^\\prime}(\\mathbb{R}^d)} \\le C \\| f\\|_{L^p(\\mathbb{R}^d)}$.\n \n \\end{remark}\n \n \\begin{thm}\\label{thm-f-1}\n Let $0<\\delta<\\infty$.\n For $x, y\\in \\mathbb{R}^d$ with $|x-y|_\\infty\\ge 4$, we have\n \\begin{align}\n |\\Gamma_\\delta (x, y)|  & \\le C |x-y|^{2-d}  , \\label{f-2-0}\\\\\n |\\nabla_x \\Gamma_\\delta (x, y)|\n +|\\nabla _y \\Gamma_\\delta  (x, y)|\n & \\le  C |x-y|^{1-d} , \\label{f-2-1} \\\\\n |\\nabla_x\\nabla_y \\Gamma_\\delta (x, y)| & \\le C |x-y|^{-d}, \\label{f-2-00}\n \\end{align}\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$.\n \\end{thm}\n \n\\begin{proof}\nFix $x_0, y_0\\in \\mathbb{R}^d$ with $r=|x_0-y_0|_\\infty\\ge 4$.\nLet $u$ be given by \\eqref{f-1} with $f\\in C_0^\\infty (\\omega\\cap Q(y_0, r); \\mathbb{R}^d) $.\nSince $\\mathcal{L}_\\delta (u) =0$ in $Q (x_0, r)$,\nit follows from (\\ref{5.6-0}) and \\eqref{L-inf} that\n\\begin{equation}\\label{f-2-2}\n\\aligned\n|u(x_0)|\n& \\le C \\left(\\fint_{Q(x_0, 1\/2)} |u|^2\\right)^{1\/2}\\\\\n& \\le C \\left(\\fint_{Q(x_0, r\/4)}\n|u|^2 \\right)^{1\/2}\n+ C r \\left(\\fint_{Q(x_0, r\/4)} |\\nabla u|^2 \\right)^{1\/2}\\\\\n&  \\le C \\left(\\fint_{Q(x_0, 3+r\/4)\\cap \\omega}\n|u|^2 \\right)^{1\/2}\n+ C r \\left(\\fint_{Q(x_0, 3+ r\/4) \\cap \\omega} |\\nabla u|^2 \\right)^{1\/2},\n\\endaligned\n\\end{equation}\nwhere we have used \\eqref{ex-1a}  and \\eqref{ex-3a} for the last inequality.\nWe now use (\\ref{f-1-0}) to bound the right-hand side of \\eqref{f-2-2}.\nThis gives\n$$\n\\aligned\n| u(x_0)|\n&\\le C r^{1-\\frac{d}{2}} \\big\\{ \\| u\\|_{L^{p^\\prime}(\\omega)}\n+ \\| \\nabla u\\|_{L^2(\\omega)} \\big\\}\\\\\n&\\le C r^{1-\\frac{d}{2}} \\| f\\|_{L^p(\\omega)},\n\\endaligned\n$$\nwhere $p=\\frac{2d}{d+2}$.\nBy duality it follows that \n\\begin{equation}\\label{f-2-3}\n\\left(\\int_{\\omega\\cap Q(y_0, r)}\n| \\Gamma_\\delta (x_0, y)|^{p^\\prime} \\, dy \\right)^{1\/p^\\prime}\n\\le C r^{1-\\frac{d}{2}}.\n\\end{equation}\nNote that if $f=\\text{\\rm div} (g)$, where $g\\in C_0^\\infty(\\omega\\cap Q(y_0, r); \\mathbb{R}^{d\\times d} )$,\nwe may use  \\eqref{f-2-2} and \\eqref{re-div-0} to obtain \n$$\n|u(x_0)|\\le C r^{1-\\frac{d}{2}} \\| g\\|_{L^2(\\omega)}.\n$$\nBy duality we deduce that\n\\begin{equation}\\label{f-2-4}\n\\left(\\int_{\\omega\\cap Q(y_0, r)}\n| \\nabla_y \\Gamma_\\delta (x_0, y)|^{2} \\, dy \\right)^{1\/2}\n\\le C r^{1-\\frac{d}{2}}.\n\\end{equation}\nAlso, note that by Theorem \\ref{main-thm-2},\n$$\n|\\nabla u(x_0)|\\le C \\left(\\fint_{\\omega\\cap Q(x_0, r)} |\\nabla u|^2 \\right)^{1\/2}\n\\le Cr^{-\\frac{d}{2}} \\| g \\|_{L^2(\\omega)}.\n$$\nAgain, by duality, we obtain \n\\begin{equation}\\label{dd}\n\\left(\\int_{\\omega\\cap Q(y_0, r)}\n| \\nabla_x \\nabla_y \\Gamma_\\delta (x_0, y)|^{2} \\, dy \\right)^{1\/2}\n\\le C r^{-\\frac{d}{2}}.\n\\end{equation}\n\nNow, let $ v(y)= \\Gamma_\\delta (x_0, y)$.\nThen  $\\mathcal{L}_\\delta^* (v)=0$ in $Q(y_0, r)$, where $\\mathcal{L}_\\delta^*$ denotes the adjoint of\n$\\mathcal{L}_\\delta$.\nSince $\\mathcal{L}_\\delta^*$ satisfies the same conditions as $\\mathcal{L}_\\delta$,\nwe may use \\eqref{f-2-2} to obtain\n$$\n\\aligned\n|v(y_0)|\n& \n\\le C \\left(\\fint_{\\omega\\cap Q(y_0, 3+r\/4)}\n|v|^2 \\right)^{1\/2}\n+ C r \\left(\\fint_{\\omega\\cap Q(y_0, 3+ r\/4) } |\\nabla v|^2 \\right)^{1\/2}\\\\\n& \\le C r^{2-d},\n\\endaligned\n$$\nwhich gives \\eqref{f-2-0}. Also, note that by Theorem \\ref{main-thm-2},\n\\begin{equation}\\label{f-2-5}\n|\\nabla v (y_0)|\n   \\le C \\left(\\fint_{\\omega\\cap Q(y_0, r) } |\\nabla v|^2 \\right)^{1\/2}.\n\\end{equation}\nThis, together with \\eqref{f-2-4}, gives \n$|\\nabla_y \\Gamma_\\delta (x_0, y_0)|\\le C r^{1-d}$.\nThe  estimate $|\\nabla_x \\Gamma_\\delta (x_0, y_0)|\\le C r^{1-d}$ \nfollows from the fact that the fundamental solution $\\Gamma^*_\\delta (x, y)$  for $\\mathcal{L}_\\delta^*$ is given by the\ntranspose of $\\Gamma_\\delta (y, x)$.\nFinally, the estimate for $\\nabla_x \\nabla_y \\Gamma_\\delta (x_0, y_0)$\nfollows from \\eqref{dd} and the fact that\n$\\mathcal{L}_\\delta^* (\\nabla_x \\Gamma_\\delta (x_0, \\cdot))=0$ in $\\mathbb{R}^d\\setminus \\{ x_0 \\}$.\n\\end{proof}\n\nNext, we treat the case where $ 1\\le \\delta< \\infty$ and $|x-y|_\\infty < 4$.\n\n \\begin{thm}\\label{thm-f-1a}\n Suppose $ 1\\le \\delta<\\infty$. Then estimates \\eqref{f-2-0}, \\eqref{f-2-1} and \\eqref{f-2-00}\n continue to hold\n for $x, y\\in \\mathbb{R}^d$ with $|x-y|_\\infty< 4$.\n   \\end{thm}\n\n\\begin{proof}\nThe proof is similar to that of Theorem \\ref{thm-f-1}.\nFix $x_0, y_0\\in\\mathbb{R}^d$ with $r=|x_0-y_0|_\\infty \\le 4$.\nLet $u$ be given by \\eqref{f-1} with $f\\in C_0^\\infty(Q(y_0, r); \\mathbb{R}^d)$.\nSince $\\mathcal{L}_\\delta (u)=0$ in $Q(x_0, r)$,\nin view of  \\eqref{5.5-1} and Remark \\ref{re-large}, we obtain \n\\begin{equation}\\label{f-2-2a1}\n\\aligned\n|u(x_0)| + r |\\nabla u (x_0)|\n&\\le C \\left( \\fint_{Q(x_0, r)}\n|u|^2\\right)^{1\/2}\\\\\n& \\le C r^{1-\\frac{d}{2}} \\| f\\|_{L^p(\\mathbb{R}^d)}.\n\\endaligned\n\\end{equation}\nBy duality it follows that\n\\begin{equation}\\label{f-2-2a2}\n\\left(\\int_{Q(y_0, r)}\n|\\Gamma_\\delta (x_0, y)|^{p^\\prime}\\, dy \\right)^{1\/p^\\prime}\n\\le C r^{1-\\frac{d}{2}}.\n\\end{equation}\nSince $\\mathcal{L}_\\delta^* (\\Gamma_\\delta (x_0, \\cdot))\n=0$ in $Q(y_0, r)$,  the desired estimates\nfollow readily from the first inequality in (\\ref{f-2-2a1}). \nWe omit the details.\n\\end{proof}\n\nIt remains to handle the case  where $0<\\delta<1$ and $|x-y|_\\infty<4$.\n\n \\begin{lemma}\\label{lemma-f-2}\n Let  $0< \\delta\\le 1$ and \n \\begin{equation}\\label{f-20-0}\n u(x)=\\int_{\\mathbb{R}^d} \\Gamma_\\delta (x, y) \\Lambda_\\delta (y) f(y)\\, dy\n \\end{equation}\n for some $f\\in C_0^\\infty(\\mathbb{R}^d; \\mathbb{R}^d)$.\n Then\n \\begin{equation}\\label{f-20-1}\n \\|\\Lambda_\\delta u\\|_{L^{p^\\prime} (\\mathbb{R}^d)}\n + \\|  \\Lambda_\\delta \\nabla u \\|_{L^2(\\mathbb{R}^d)}\n \\le C \\| f\\|_{L^{p} (\\mathbb{R}^d)},\n \\end{equation}\n where $p =\\frac{2d}{d+2}$ and $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{lemma}\n \n\\begin{proof}\nAs in the proof of Lemma \\ref{lemma-f-1}, we have\n$$\n\\| u\\|_{L^{p^\\prime} (\\mathbb{R}^d)} \\le C \\| \\nabla u\\|_{L^2(\\mathbb{R}^d)} \n\\le C \\| D u \\|_{L^2(\\mathbb{R}^d)}\n$$\n and \n$\\| u\\|_{L^{p^\\prime} (\\omega)} +\\| \\nabla u\\|_{L^2(\\omega)} \\le C \\|  Du \\|_{L^2(\\omega)}.\n$\nIt  follows that \n\\begin{equation}\\label{f-2-3x}\n\\aligned\n\\| \\Lambda_\\delta u\\|_{L^{p^\\prime} (\\mathbb{R}^d)} \n+ \\|\\Lambda_\\delta \\nabla u\\|_{L^2(\\mathbb{R}^d)} \n & \\le C\\delta  \\|  Du \\|_{L^2(\\mathbb{R}^d)} \n+ C \\| Du  \\|_{L^2(\\omega)}\\\\\n& \\le C \\|\\Lambda_\\delta D u  \\|_{L^2(\\mathbb{R}^d)},\n\\endaligned\n\\end{equation}\nwhere we have used  the assumption $\\delta\\le 1$ for the last inequality.\nBy letting $v=u$ in \\eqref{f-2}, we obtain \n$$\n\\|\\Lambda_\\delta Du \\|^2_{L^2(\\mathbb{R}^d)} \\le C \\| \\Lambda_\\delta u \\|_{L^{p^\\prime} (\\mathbb{R}^d)} \\| f\\|_{L^p(\\mathbb{R}^d)},\n$$\nwhich, together with \\eqref{f-2-3x}, yields  \\eqref{f-20-1}.\n\\end{proof}\n\n\\begin{thm}\\label{thm-f-3}\nSuppose $0<\\delta< 1$.\nFor $x, y\\in \\mathbb{R}^d$ with $|x-y|_\\infty<  4 $, we have\n \\begin{align}\n \\Lambda_\\delta (x) \\Lambda_\\delta (y) |\\Gamma_\\delta (x, y)|  & \\le\n C |x-y|^{2-d}   , \\label{f-3-0}\\\\\n \\Lambda_\\delta (x) \\Lambda_\\delta (y) \n \\big\\{ |\\nabla_x \\Gamma_\\delta (x, y)|\n +|\\nabla _y \\Gamma_\\delta  (x, y)| \\big\\} \n & \\le  C |x-y|^{1-d} , \\label{f-3-1}\\\\\n \\Lambda_\\delta (x) \\Lambda_\\delta (y) |\\nabla_x \\nabla_y \\Gamma_\\delta (x, y)| \n & \\le C |x-y|^{-d}, \\label{f-3-1x} \n \\end{align}\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$,   $\\omega$, and $(\\sigma, M_0)$.\n \\end{thm}\n \n \\begin{proof}\n \n Fix $x_0, y_0\\in \\mathbb{R}^d$ with $r=|x_0-y_0|_\\infty<4$.\n Let $u$ be given by (\\ref{f-20-0}) with $f\\in C_0^\\infty(Q(y_0, r); \\mathbb{R}^d) $.\n Then $\\mathcal{L}_\\delta (u)=0$ in $Q(x_0, r)$.\n It follows from (\\ref{5.5-0}) that\n \\begin{equation}\\label{f-3-2}\n \\aligned\n \\Lambda_\\delta (x_0)\n \\big\\{ |u(x_0)| + r|\\nabla u(x_0) | \\big\\}\n &\\le C \\left(\\fint_{Q(x_0, r)}\n |\\Lambda_\\delta u|^2 \\right)^{1\/2}\\\\\n &\\le Cr^{1-\\frac{d}{2}}\n \\| f\\|_{L^p(\\mathbb{R}^d)},\n \\endaligned\n \\end{equation}\n where we have used \\eqref{f-20-1} for the last inequality.\n By duality this implies that\n \\begin{equation}\\label{f-3-3}\n \\Lambda_\\delta  (x_0)\n \\left(\\int_{Q(y_0, r)}\n |\\Lambda_\\delta (y) \\Gamma_\\delta (x_0, y)|^{p^\\prime} \\, dy \\right)^{1\/p^\\prime}\n \\le C r^{1-\\frac{d}{2}}.\n \\end{equation}\n Since $\\mathcal{L}_\\delta^* ( \\Gamma_\\delta (x_0, \\cdot)) =0$ in $\\mathbb{R}^d\\setminus \\{ x_0\\}$, we may use the first inequality in \\eqref{f-3-2}\n and (\\ref{f-3-3}) \n to obtain\n \\begin{equation}\\label{f-3-4}\n \\Lambda_\\delta (x_0)\\Lambda_\\delta (y_0)\n |\\Gamma_\\delta (x_0, y_0)|,\n \\le C r^{2-d}.\n \\end{equation}\n which gives \\eqref{f-3-0}.\n The estimates in \\eqref{f-3-1} also follow from the first inequality in \\eqref{f-3-2}\n and (\\ref{f-3-3}). \n Finally, \\eqref{f-3-1x} follows from the first  inequality in \\eqref{f-3-2} and \\eqref{f-3-1}.\n \\end{proof}\n  \nWe end   this section  with  a decay estimate  of  $D\\Gamma_\\delta (x, y)$ for $x\\in F$, as $\\delta  \\to \\infty$.\n  \n \\begin{thm}\\label{thm7.2}\n Let $1\\le \\delta<\\infty$.\n Let $u\\in H^1(Q(x_0, R); \\mathbb{R}^d)$ be a weak solution of\n $\\mathcal{L}_\\delta (u) =0$ in $Q(x_0, R)$\n for some $x_0\\in F$ and $R\\ge 5$.\n Then\n \\begin{equation}\\label{7.2-0}\n |Du(x_0)|\n \\le \\frac{C}{\\delta^2}\n \\left(\\fint_{Q(x_0, R)} | D u|^2\\, dx \\right)^{1\/2},\n \\end{equation}\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$.\n \\end{thm}\n \n \\begin{proof}\nSuppose $x_0\\in F_k\\subset Q(x_0, 2)$ for some $k$.\nIt follows from \\eqref{local-2-0b} and interior estimates that\n\\begin{equation}\\label{7.2-1}\n|\\nabla u(x_0)|\n\\le  C\\delta^{-2} \\left(\\int_{Q(x_0, 2)} |\\nabla u|^2 \\, dx \\right)^{1\/2}\n+ C \\left(\\int_{F_k} |\\nabla u|^2 \\, dx \\right)^{1\/2}.\n\\end{equation}\nChoose $\\phi\\in \\mathcal{R}$ such that $u-\\phi \\perp \\mathcal{R}$ in $H^1(F_k; \\mathbb{R}^d)$.\nSince $u-\\phi$ satisfies the same conditions as $u$, we may use \\eqref{7.2-1} with $u-\\phi$ in the place of $u$.\nAs a result, \n$$\n\\aligned\n|Du(x_0)| & \\le |\\nabla (u-\\phi) (x_0) |\\\\\n& \\le  C\\delta^{-2} \\left(\\int_{Q(x_0, 2)} |\\nabla u|^2 \\, dx \\right)^{1\/2}\n+ C \\left(\\int_{F_k} | D u|^2 \\, dx \\right)^{1\/2},\n\\endaligned\n$$\nwhere we have used the second Korn inequality as well as the fact \n$|\\nabla \\phi|\\le C \\|\\nabla u\\|_{L^2(F_k)}$.\nThis, together with \\ref{7.1-0} with $f=0$, gives\n\\begin{equation}\\label{7.2-2}\n\\aligned\n|D u(x_0)|\n & \\le C \\delta^{-2}\n\\left (\\fint_{Q(x_0, 5)} |\\nabla u|^2\\right)^{1\/2}\\\\\n& \\le   C \\delta^{-2}\n\\left (\\fint_{Q(x_0, R)} |\\nabla u|^2\\right)^{1\/2},\n\\endaligned\n\\end{equation}\nwhere we have used Theorem \\ref{main-thm-1} for the last inequality.\nChoose $\\psi$  in  $\\mathcal{R}$ so that\n$$\n\\| \\nabla  (u-\\psi)\\|_{L^2(Q(x_0, R)) } \\le C \\| D u \\|_{L^2(Q(x_0, R))}.\n$$\nIt  follows that\n$$\n\\aligned\n|Du (x_0)|\n & = |D (u-\\psi) (x_0)|\\\\\n & \\le C \n  \\delta^{-2}\n\\left (\\fint_{Q(x_0, R)} |\\nabla (u-\\psi) |^2\\right)^{1\/2}\n\\le C\\delta^{-2}\n \\left (\\fint_{Q(x_0, R)} | D  u|^2\\right)^{1\/2}.\n\\endaligned\n$$\nwhere we have used \\eqref{7.2-2} with $u-\\psi$ in the place of $u$.\n \\end{proof}\n \n \\begin{cor}\n Let $1\\le \\delta<\\infty$. Then \n \\begin{equation}\\label{7.3-0}\n\\Lambda_{\\delta^2} (x) \n|D_x \\Gamma_\\delta (x, y)|\n+ \\Lambda_{\\delta^2} (y)\n|D_y \\Gamma_\\delta (x, y)| \n\\le C |x-y|^{1-d}\n\\end{equation}\nfor any $x, y\\in \\mathbb{R}^d$ with $|x-y|_\\infty\\ge 4$.\n \\end{cor}\n \n \\begin{proof}\nSince $\\delta\\ge 1$, it follows by Theorems \\ref{thm-f-1} and \\ref{thm-f-1a} that\n$$\n|\\nabla_x \\Gamma_\\delta  (x, y)| + |\\nabla_y \\Gamma_\\delta   (x, y)| \\le\nC |x-y|^{1-d}.\n$$\nfor any $x, y\\in \\mathbb{R}^d$ and $x\\neq y$.\n This, together with Theorem \\ref{thm7.2}, gives \\eqref{7.3-0}.\n \\end{proof}\n \n\n \n \\section{Lipschitz estimates for $\\mathcal{L}_\\delta (u)=f$} \\label{section-7}\n \n The goal of this section  is to prove \\eqref{f-Lip-f}.\n The case $0<\\delta<\\infty$ follows readily from Theorem \\ref{main-thm-2} and estimates of fundamental solutions in Section \\ref{section-6}.\n To handle the cases $\\delta=0$ and  $\\delta =\\infty$, we use an approximation  argument.\n \n Let $\\Omega$ be a bounded Lipschitz domain in $\\mathbb{R}^d$.\n We call $\\Omega$ a type II domain (with respect to $\\omega$)  if $F_k \\cap \\Omega\\neq \\emptyset$ implies that \n $\\widetilde{F}_k \\subset \\Omega$.\n In particular, if $\\Omega$ is a type II Lipschitz domain, then $\\partial \\Omega \\cap \\partial \\omega =\\emptyset$.\n \n \\begin{lemma}\\label{lemma-app-1}\nAssume that $A$ and $\\omega$ satisfy the same conditions as in Theorem \\ref{main-thm-1}.\n Let $0<\\delta\\le 1$ and $\\Omega$ be a type II Lipschitz domain.\n Let $u_\\delta \\in H^1(\\Omega; \\mathbb{R}^d)$ be  a weak solution of\n $\\mathcal{L}_\\delta (u_\\delta)=f \\chi_\\omega $ in $\\Omega$ and $u_0\\in H^1(\\Omega; \\mathbb{R}^d)$ a weak solution of\n $\\mathcal{L}_0 (u_0)=f\\chi_\\omega $ in $\\Omega$, where $f\\in L^2(\\Omega; \\mathbb{R}^d)$.\n Suppose that $u_\\delta=u_0$ on $\\partial\\Omega$.\n Then\n \\begin{equation}\\label{app-1-0}\n \\| u_\\delta -u_0\\|_{H^1(\\Omega)}\n \\le C \\delta \\| D u_0\\|_{L^2(\\Omega\\cap F)},\n \\end{equation}\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{lemma}\n \n \\begin{proof}\n Let $w=u_\\delta -u_0\\in H^1_0(\\Omega; \\mathbb{R}^d) $.\n Since $u_0$ is a weak solution of $\\mathcal{L}_0 (u_0)=f\\chi_\\omega $ in $\\Omega$,\n \\begin{equation}\\label{app-1-1}\n \\int_{\\Omega\\cap \\omega} A\\nabla u_0 \\cdot \\nabla v \\, dx\n =\\int_{\\Omega\\cap \\omega} f \\cdot v\\, dx\n \\end{equation}\n for any $v\\in H^1_0(\\Omega; \\mathbb{R}^d)$.  We also assume that $-\\text{\\rm div}(A\\nabla u_0)=0$ in $F\\cap \\Omega$.\n Using\n $$\n \\int_{\\Omega\\cap \\omega} A\\nabla u_\\delta \\cdot \\nabla v\\, dx\n + \\delta^2 \\int_{\\Omega\\cap F} A\\nabla u_\\delta \\cdot \\nabla v\\, dx\n =\\int_{\\Omega\\cap \\omega} f\\cdot v\\, dx,\n $$\n we obtain \n $$\n \\int_{\\Omega\\cap \\omega}\n A\\nabla w \\cdot \\nabla w\\, dx\n + \\delta^2 \\int_{\\Omega\\cap F} A \\nabla w \\cdot \\nabla w \n =-\\delta^2 \\int_{\\Omega\\cap F} A\\nabla u_0 \\cdot \\nabla w\\, dx.\n $$\n Hence, by \\eqref{ellipticity} and  the Cauchy inequality, \n \\begin{equation}\\label{app-1-2}\n \\int_{\\Omega\\cap \\omega}\n |D w|^2\\, dx\n + \\delta^2 \\int_{\\Omega\\cap F} |D w|^2\\, dx\n \\le C \\delta^2 \\int_{\\Omega\\cap F} |D u_0|^2\\, dx.\n \\end{equation}\n Note that $\\text{\\rm div}(A\\nabla w)=0$ in $F_k$ for any $F_k\\subset \\Omega$.\n By Lemma \\ref{lemma-2.2} we have \n $$\n \\| D w\\|_{L^2(F_k)}\\le C \\| D w \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}.\n $$\n As a result, $\\| Dw\\|_{L^2(\\Omega\\cap F)}\\le C \\| Dw \\|_{L^2(\\Omega\\cap \\omega)}$,\n where we have used the assumption that $\\Omega$ is a type II Lipschitz domain.\n This, together with \\eqref{app-1-2} and the first Korn inequality \\cite[p.13]{OSY-1992}, gives \\eqref{app-1-0}.\n \\end{proof}\n \n \\begin{lemma}\\label{lemma-app-2}\n Assume that $A$ and $\\omega$ satisfy the same conditions as in Theorem \\ref{main-thm-1}.\n Let $1\\le  \\delta<\\infty$ and $\\Omega$ be a type II Lipschitz domain.\n Let $u_\\delta \\in H^1(\\Omega; \\mathbb{R}^d)$ be  a weak solution of\n $\\mathcal{L}_\\delta (u_\\delta)=f $ in $\\Omega$ and $u_\\infty\\in H^1(\\Omega; \\mathbb{R}^d)$ a weak solution of\n $\\mathcal{L}_\\infty (u_\\infty)=f$ in $\\Omega$, where $f\\in L^2(\\Omega; \\mathbb{R}^d)$.\n Suppose that $u_\\delta=u_\\infty$ on $\\partial\\Omega$.\n Then\n \\begin{equation}\\label{app-2-0}\n \\| u_\\delta -u_\\infty \\|_{H^1(\\Omega)}\n \\le C \\delta^{-1} \\big\\{ \n  \\| D u_\\infty\\|_{L^2(\\Omega\\cap \\omega)}\n  + \\| f\\|_{L^2(\\Omega)} \\big\\},\n \\end{equation}\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{lemma}\n\n\\begin{proof}\nSince  $u_\\infty$ is a weak solution of $\\mathcal{L}_\\infty (u_\\infty) =f$, it follows that \n$$\n\\int_{\\Omega\\cap \\omega} A\\nabla u_\\infty \\cdot \\nabla v\\, dx\n=\\int_\\Omega f \\cdot  v\\, dx\n$$\nfor any $v\\in H^1_0(\\Omega; \\mathbb{R}^d)$ with $Dv=0$ in $\\Omega\\cap F$, and  that \n$D u_\\infty=0$ in $\\Omega\\cap F$.\nLet $\\phi \\in H^1_0(\\Omega; \\mathbb{R}^d)$.\nFor each $F_k \\subset \\Omega$ and $g_k \\in \\mathcal{R}$, let $w_k \\in H^1_0(\\widetilde{F}_k; \\mathbb{R}^d)$ be an extension \nof $\\phi-g_k$ from $F_k$ to $\\widetilde{F}_k$ with the property that\n$$\n\\| w_k \\|_{H^1(\\widetilde{F}_k)} \\le C \\| \\phi- g_k\\|_{H^1(F_k)}.\n$$\nExtend $w_k$ from $\\widetilde{F}_k$ to $\\mathbb{R}^d$ by zero, and define \n$$\nv=\\phi -\\sum_k w_k, \n$$\nwhere the sum is taken over those $k$'s for which $F_k\\subset \\Omega$.\nNote that $v\\in H^1_0(\\Omega; \\mathbb{R}^d)$ and $v =g_k$ on $F_k$. Since $\\Omega$ is a type II domain,\nit follows that $Dv =0$ on $\\Omega\\cap F$.\nAs a result, we see that \n$$\n\\aligned\n & \\Big|\n\\int_{\\Omega\\cap \\omega}  A \\nabla u_\\infty\\cdot \\nabla \\phi\\, dx -\\int_{\\Omega} f \\cdot \\phi \\, dx \\Big|\\\\\n&\n= \\Big| \\int_{\\Omega\\cap \\omega}  A \\nabla u_\\infty\\cdot   \\sum_k  \\nabla w_k \\, dx -\\int_{\\Omega} f \\cdot \\sum_k w_k \\, dx \\Big|\\\\\n&\\le C \\sum_k \\| D u_\\infty\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n\\| D w_k \\|_{L^2(\\widetilde{F}_k)}\n+ C \\sum_k \\| f\\|_{L^2(\\widetilde{F}_k)}\n\\| w_k \\|_{L^2(\\widetilde{F}_k)}\\\\\n&\\le C \\sum_k\n\\big( \\| D u_\\infty\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n+ \\| f\\|_{L^2(\\widetilde{F}_k)}\\big) \\| \\phi -g_k \\|_{H^1(F_k)}\\\\\n& \\le C \\big( \\| D u_\\infty\\|_{L^2(\\Omega\\cap \\omega)}\n+\\| f\\|_{L^2(\\Omega)} \\big) \\| D \\phi\\|_{L^2(\\Omega\\cap F)},\n\\endaligned\n$$\nwhere we have chosen $g_k \\in \\mathcal{R}$ so that\n$\\| \\phi - g_k \\|_{H^1(F_k)} \\le  C \\| D \\phi\\|_{L^2(F_k)}$.\nThis, together with \n$$\n\\int_{\\Omega\\cap \\omega} A\\nabla u_\\delta \\cdot \\nabla \\phi\\, dx\n+ \\delta^2 \\int_{\\Omega\\cap F} A \\nabla u_\\delta \\cdot \\nabla \\phi\\, dx\n=\\int_\\Omega f \\cdot \\phi, dx,\n$$\nimplies that\n$$\n\\aligned\n & \\Big| \\int_{\\Omega\\cap \\omega} A\\nabla (u_\\delta -u_\\infty) \\cdot \\nabla \\phi\\, dx\n+ \\delta^2 \\int_{\\Omega\\cap F} A \\nabla (u_\\delta -u_\\infty)  \\cdot \\nabla \\phi\\, dx \\Big|\\\\\n&\\le \nC \\big( \\| D u_\\infty\\|_{L^2(\\Omega\\cap \\omega)}\n+\\| f\\|_{L^2(\\Omega)} \\big) \\| D \\phi\\|_{L^2(\\Omega\\cap F)}.\n\\endaligned\n$$\nBy letting $w=u_\\delta -u_\\infty$ and $ \\phi=w$ in the inequality above, we obtain\n$$\n\\| D w \\|_{L^2(\\Omega\\cap \\omega)}\n+\\delta \\| D w\\|_{L^2(\\Omega\\cap F)}\n\\le C \\delta^{-1} \\big\\{ \\| D u_\\infty\\|_{L^2(\\Omega\\cap \\omega)}\n+ \\| f\\|_{L^2(\\Omega)} \\big\\},\n$$\nwhere we have also used the Cauchy inequality.\nSince $\\delta\\ge 1$ and $w\\in H^1_0(\\Omega; \\mathbb{R}^d)$, the estimate \\eqref{app-2-0}\nfollows by the first Korn inequality.\n\\end{proof}\n\n\\begin{thm}\\label{main-thm-3}\nLet $ d\\ge 3$.\nAssume that $A$ satisfies \\eqref{ellipticity}, \\eqref{smoothness}, and is 1-periodic.\nAlso assume that each $F_k$ is a bounded $C^{1, \\sigma}$ domain.\n\n\\begin{enumerate}\n\n\\item\n\nLet  $0\\le \\delta < 1$.\nLet $u\\in H^1(\\Omega; \\mathbb{R}^d)$ be a weak solution of\n$\\mathcal{L}_\\delta (u)=f\\chi_\\omega$ in $Q(x_0, R)$ for some $R\\ge 6$, where  $f\\in L^p(Q(x_0, R);  \\mathbb{R}^d)$ for some $p>d$.\nThen\n\\begin{equation}\\label{lip-f-8}\n|\\nabla u (x_0)|\n\\le   C \\left(\\fint_{Q(x_0, R)\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2}\n+C R \\left(\\fint_{Q(x_0, R)} |f|^p \\right)^{1\/p},\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $p$, $\\omega$, and $(\\sigma, M_0)$ in \\eqref{smoothness}.\n\n\\item\n\nLet $1\\le  \\delta \\le \\infty$.\nLet $u\\in H^1(\\Omega; \\mathbb{R}^d)$ be a weak solution of\n$\\mathcal{L}_\\delta (u)= f$ in $Q(x_0, R)$ for some $R\\ge 6$, where  $f\\in L^p(Q(x_0, R);  \\mathbb{R}^d)$ for some $p>d$.\nThen\n\\begin{equation}\\label{lip-f-9}\n|\\nabla u (x_0)|\n\\le  \\left(\\fint_{Q(x_0, R)\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2}\n+C R \\left(\\fint_{Q(x_0, R)} |f|^p \\right)^{1\/p},\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $p$, $\\omega$, and $(\\sigma, M_0)$.\n\n\\end{enumerate}\n\\end{thm}\n\n \\begin{proof}\n \n By translation we may assume $x_0=0$.\n We consider 4 cases.\n \n \\medskip\n \n \\noindent{\\bf Case 1.} Assume $1\\le \\delta  < \\infty$.\n If $f=0$, this  is given by Theorem \\ref{main-thm-2}.\n In general, let\n \\begin{equation}\\label{lip-f-10}\n v(x) =\\int_{Q_R} \\Gamma_\\delta (x, y) f(y)\\, dy.\n \\end{equation}\n Then $\\mathcal{L}_\\delta (v)=f$ in $Q_R$, and by Theorems \\ref{thm-f-1} and \\ref{thm-f-1a},\n \\begin{equation}\\label{lip-f-11}\n \\|\\nabla v\\|_{L^\\infty (Q_R)} \\le CR  \\left(\\fint_{Q_R} |f|^p\\right)^{1\/p}\n \\end{equation}\nfor $p>d$.  Hence,\n $$\n \\aligned\n |\\nabla u(0)|\n &\\le |\\nabla (u-v)(0)| +|\\nabla v(0)|\\\\\n & \\le C \\left(\\fint_{Q_R\\cap \\omega } |\\nabla (u-v)|^2\\right)^{1\/2}\n + C R  \\left(\\fint_{Q_R} |f|^p\\right)^{1\/p}\\\\\n &\\le C \\left(\\fint_{Q_R\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2}\n+C R \\left(\\fint_{Q_R  } |f|^p \\right)^{1\/p},\n\\endaligned\n$$\nwhere we have used the fact $\\mathcal{L}_\\delta (u-v)=0$ in $Q_R$.\n \n \\medskip\n \n \\noindent{\\bf Case 2.} \n Assume $\\delta=\\infty$.\n In this case we use an approximation argument.\n Choose a type II Lipschitz domain $\\Omega$ such that $Q_{R-2}\\subset \\Omega\\subset Q_R$.\n Let $u_\\delta\\in H^1(\\Omega: \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u_\\delta)=f$ in $\\Omega$\n such that $u_\\delta=u$ on $\\partial\\Omega$.\n It follows by Lemma \\ref{lemma-app-2} that $u_\\delta \\to u$ in $H^1(\\Omega; \\mathbb{R}^d)$,  as $\\delta \\to \\infty$.\n By the proof for Case 1, \n $$\n \\left(\\fint_{Q_r} |\\nabla u_\\delta|^2\\right)^{1\/2}\n \\le C \\left(\\fint_{Q_{R-2}\\cap \\omega } |\\nabla u_\\delta |^2 \\right)^{1\/2}\n+C R \\left(\\fint_{Q_R} |f|^p \\right)^{1\/p},\n $$\n for  $r\\in (0, 1\/4)$.\n The proof is complete by letting $\\delta\\to \\infty$ and then $r\\to 0$ in the inequality above.\n \n \\medskip\n \n \\noindent{\\bf Case 3.}\n Assume  $0<\\delta< 1$.\n If $f=0$, the estimate \\eqref{lip-f-8} is given by   Theorem \\ref{main-thm-2}.\n In general, let\n $$\n v(x)=\\int_{Q_R\\cap \\omega} \\Gamma_\\delta (x, y) f(y)\\, dy.\n $$\n Then\n $\n \\mathcal{L}_\\delta (v)=f\\chi_\\omega$ in $Q_R$, and by Theorems \\ref{thm-f-1} and \\ref{thm-f-3},\n $$\n \\|\\Lambda_\\delta  \\nabla  v\\|_{L^\\infty(Q_R)}\n \\le C R \\left(\\fint_{Q_R} |f|^p \\right)^{1\/p}.\n $$\n Observe that since $\\text{\\rm div}(A\\nabla v)=0$ in $Q_R\\cap F$, it follows from Theorem \\ref{local-thm-0}  that\n $$\n \\aligned\n |\\nabla v (0)|\n  & \\le C \\left( \\fint_{Q_1} |\\nabla v|^2 \\right)^{1\/2}\n + C \\left(\\fint_{Q_1} |f|^p \\right)^{1\/p}\\\\\n&  \\le C  \\left(\\fint_{Q_4\\cap \\omega} |\\nabla v|^2 \\right)^{1\/2}\n + C \\left(\\fint_{Q_1} |f|^p \\right)^{1\/p},\n\\endaligned\n$$\n where we have used \\eqref{2.2-0} for the last inequality.\n Hence,\n $$\n \\aligned\n  |\\nabla u (0)|\n &\\le |\\nabla (u-v)(0)| +  |\\nabla v(0)|\\\\\n &\\le C \\left(\\fint_{Q_R\\cap \\omega} |\\nabla (u-v)|^2\\right)^{1\/2}\n + C R \\left(\\fint_{Q_R} |f|^p \\right)^{1\/p}\\\\\n & \\le C \\left(\\fint_{Q_R\\cap \\omega} |\\nabla u|^2\\right)^{1\/2}\n +C R \\left(\\fint_{Q_R} |f|^p \\right)^{1\/p}.\n \\endaligned\n $$\n \n \\noindent{\\bf Case 4.}\n Assume $\\delta=0$.\n As in Case 2,  this follows from Case 3 by using  the  approximation in Lemma \\ref{lemma-app-2}.\n We omit the details.\n\\end{proof} \n \n \\bibliographystyle{amsplain}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \\label{intro}\n\nBlack hole X-ray binaries (BHBs) are known to occupy distinct spectral states\nwhich can be characterized by the relative contribution of thermal and\nnon-thermal emission components (e.g. McClintock \\& Remillard 2004).  The most\nwell-understood of these states is the thermal dominant (or high\/soft) state.\nHere most of the flux is in the thermal component, which is generally assumed to\nbe emission from a radiatively efficient, geometrically thin accretion disk\n(Shakura \\& Sunyaev 1973).  The disk is believed to extend deep within the\ngravitational field of the black hole, and this makes spectral modeling of\nthis state an important probe of both the physics of relativistic accretion\ndisks and the properties of black holes.\n\nIn standard treatments of black hole accretion disks, the emitting matter\nextends down to the innermost stable circular orbit (ISCO), which is determined\nby the mass and spin of the black hole.  Such treatments usually assume a\n``no-torque'' inner boundary condition at this radius (e.g. Novikov\n\\& Thorne 1973), but magnetic fields may in fact exert such torques\n(Gammie 1999; Krolik 1999; Hawley \\& Krolik 2002), increasing the radiative\nefficiency of the disk.  The structure and emission of these disks are therefore\nsensitive to the mass and spin of the black hole as well as any torque which may\nbe present.\n\nThis sensitivity makes accretion disk spectral modeling a potential way to\nmeasure or constrain black hole spin (e.g. Ebisawa et al. 1993; Zhang et al. 1997;\nShafee et al. 2006; Middleton et al. 2006).\nThis method requires a model of the radial profile of\neffective temperature in the gravitational field of the black hole, calculation\nof the relativistic transfer function from the disk surface to an observer at\ninfinity (Cunningham 1975), and spectral modeling of the surface emission in the\nlocal rest frame of the disk.  Most implementations of this method approximate\none or all of these components.  One common approximation is to assume that the\ndisk surface emission is a blackbody or, more generally, a color-corrected\nblackbody,\n  \\begin{eqnarray}  \nI_{\\nu}=f^{-4}B_{\\nu}(fT_{\\rm eff}), \\label{e:ccbb} \n  \\end{eqnarray}  \nwhere $I_{\\nu}$ is the specific intensity, $T_{\\rm eff}$ is the effective\ntemperature, $B_{\\nu}$ is the Planck function, and $f$ is the spectral hardening\nfactor (or color-correction), typically assumed to be around 1.7 \n(Shimura \\& Takahara 1995).  One of\nthe most sophisticated models of this type is the KERRBB model (Li et al. 2005)\nfor Xspec (Arnaud 1996).  It accounts for the relativistic effects on the disk\neffective temperature profile and the relativistic transfer function.  \nPotential difficulties with the color-corrected blackbody approximation exist.\nThe local spectrum may not be well approximated by an isotropic, color-corrected\nblackbody due to limb darkening and frequency dependent absorption opacities.  \nEven if it can, one must still specify $f$.\nIt has been suggested that $f$ is a relatively strong function\nof accretion rate or of the fraction of energy emitted in a corona (Merloni et\nal. 2000).\n\nRelativistic models (Davis et al. 2005) now exist which calculate\nthe non-LTE vertical disk structure and radiative transfer self-consistently\nusing the TLUSTY stellar atmospheres code (Hubeny \\& Lanz 1995).  The\nrelativistic effects on photon geodesics are accounted for with ray tracing\nmethods (Agol 1997).  These spectral models have now been implemented in an\nXspec table model (BHSPEC; Davis \\& Hubeny, 2006). By calculating\nvalues of $f$ appropriate for use with KERRBB, these models have already been\nused to estimate black hole spins from KERRBB fits to two BHBs: GRO J1655-40\nand 4U 1543-47 (Shafee et al. 2006).\n\nIn this work, we circumvent the color-corrected blackbody approximation\nentirely by fitting the BHSPEC model directly to BHB observations.  Since the\nmodel does not include irradiation of the disk surface, we focus our efforts\non thermal dominant state observations in which the non-thermal emission is a small\nfraction of the total flux. Fortunately, a sample of such observations made\nwith the {\\it Rossi X-ray Timing Explorer} ({\\it RXTE}) already exists\n(Gierlinski \\& Done 2004; hereafter GD04).\n\nBHSPEC still assumes that the disk emission extends only down to the ISCO\nand then effectively ceases due to rapid in-fall of matter interior to this\nradius.  This assumption appears to be consistent with spectral modeling of\nBHBs in the thermal dominant state in that the luminosity $L$ is seen to scale roughly\nwith the fourth power of the color temperature $T_{\\rm c}$ in several different\nsources (Kubota et al. 2001; Gierlinski \\& Done 2004).  This\nsuggests that as sources vary by over an order of magnitude in luminosity,\nthere is a roughly constant emitting area and thus a relatively constant inner\nradius to the disk.  It is therefore very natural to associate such a stable\ninner radius with the ISCO of the black hole, though the `emission edge'\nneed not coincide exactly with the ISCO (Krolik \\& Hawley 2002).\n\nNot all of these sources follow the $L \\propto T_{\\rm c}^4$ relation exactly,\nhowever.  In several cases a relative hardening is seen \nwith increasing $L$ (see e.g. GD04; Kubota \\& Makishima 2004; Shafee et al. 2006).\nA potential explanation for this hardening is that \nadvection may be becoming increasingly important as these sources approach \nthe Eddington limit (Kubota \\& Makishima 2004).\nAlternatively, this behavior is qualitatively consistent with the increased \nspectral hardening with accretion rate in the local disk atmospheres, ignoring \nadvection (Davis et al. 2005; Shafee et al. 2006).  In\nthe BHSPEC model, the precise nature of the hardening depends strongly on the\nvariation in surface density with radius in the disk.  Currently, the surface\ndensity is determined by assuming the vertically averaged stress is proportional\nto the vertically averaged total pressure with a constant of proportionality\n$\\alpha$.  However, more general stress prescriptions could be implemented in the\nfuture. Therefore, spectral modeling could potentially provide a constraint on\nthe nature of the stresses in these systems.\n\nIn this paper we fit these fully-relativistic, non-LTE accretion disk models to\n{\\it RXTE} and {\\it BeppoSAX} observations of BHBs in the thermal dominant state.\nOur purpose is two-fold:  we want to test the applicability of the spectral\nmodels to these observations, and having then found suitable representations\nof the data, we use them to infer black hole spins and infer properties of the\nstresses in these system.\nWe review our spectral fitting and results in section \\ref{specmod}, discuss the\nimplications of these results in section \\ref{discus}, and summarize our\nconclusions in section \\ref{conc}.  In the Appendix we develop a simplified model\nto understand and motivate the variation of the BHSPEC spectra with accretion rate.\n\n\n\\begin{figure}\n\\includegraphics[width=0.46\\textwidth]{f1.eps}\n\\caption{The unfolded spectrum for the {\\it BeppoSAX}\nobservation of LMC X-3 using the best fit BHSPEC models for $i=67^{\\circ}$, \n$D=52$ kpc, and $\\alpha=0.01$.  The total model component (green, solid curve),\nBHSPEC (red, long-dashed curve) and COMPTT (violet, short-dashed curve) are\nplotted.  The unabsorbed BHSPEC model (orange, solid curve) is also shown.\n\\label{fig1}}\n\\end{figure}\n\n\\section{Spectral Modeling}\n\\label{specmod}\n\nSeveral properties of BHBs make them particularly well suited both for testing\naccretion disk theory and for measuring the unknown black hole properties of\ninterest.  There are several sources for which precise, independent\nmeasurements of the mass of the primary and \nthe binary inclination are\navailable from light curve modeling of the secondary star (e.g. Orosz \\& Bailyn\n1997). Typically, the distance to BHBs are known with less precision, but there\nare exceptions.  As can be seen in Table \\ref{tbl1}, the distances to LMC X-3 and\nGRO J1655-40 are both claimed to be known to better than 10\\%.  \nBHBs have an advantage over most active galactic nuclei because of their relatively\nshort timescales\nfor large changes in the bolometric luminosity, which allows the same source to\nbe observed in the same spectral state at appreciably different accretion\nrates. The variation of the BHSPEC model with accretion rate provides a precise\nquantitative prediction which is sensitive to the assumed stress prescription.  \nTherefore, simultaneous fitting of multiple observations of the same source at\ndifferent epochs provides a much more powerful constraint and potentially more\ninformation than a single fit to a single epoch.\n\nWe compare three accretion disk models for the soft, thermal emission: the\nmulticolor disk model (DISKBB in Xspec, Mitsuda et al. 1984), KERRBB, and BHSPEC.\nDISKBB is the most commonly used spectral model, but it neglects relativity.  \nBoth KERRBB and BHSPEC include relativistic effects.  In addition, BHSPEC includes \natmosphere physics which we are interested in testing in this paper.  In particular,\nwe are interested in examining whether BHSPEC with a fixed $\\alpha$ can explain the\nvariation in the spectral hardening with accretion rate.  As a control, we compare\nthese results with KERRBB at a fixed $f$.\n\nWe also model the neutral absorption along the line of sight, and though we focus\non observations inferred to be disk dominated, we need an additional component to\naccount for the non-thermal emission which is present. It is widely believed\nthat the coronal emission is due to inverse Compton scattering of seed photons\nfrom the accretion disk.  In this case, a power law will tend to overestimate the\nflux at low energies.  Therefore, we prefer to approximate the non-thermal\nemission with the COMPTT Comptonization model (Titarchuk 1994) in our spectral\nfits. We specify a disk geometry and fix the high energy cutoff in this model to\n50 keV, a value high enough not to affect our fits.  We also tie the seed photon\ntemperature to the DISKBB model temperature\nwhen DISKBB is used to model the soft emission.  For KERRBB and BHSPEC we fix\nthis parameter at the best fit DISKBB value.  This leaves two free parameters --\nan optical depth and a normalization for each data set.\n\nDISKBB is a relatively simple model with only two parameters: the temperature at\nthe inner edge of the disk $T_{\\rm in}$ and the model normalization. BHSPEC and\nKERRBB both share a number of model parameters which need to be specified or fit.\nThe black hole spin $a_{\\ast}\\equiv a\/M$ and the accretion rate $\\dot{M}$ are free\nparameters in all fits unless stated otherwise. For BHSPEC $\\dot{M}$ is\nparameterized by $\\ell \\equiv L\/L_{\\rm Edd}$ where $L_{\\rm Edd}$ is the Eddington\nluminosity for completely ionized hydrogen.  This value is converted to an accretion\nrate by assuming an efficiency $\\eta$ which corresponds to the fraction of\ngravitational binding energy at infinity which is converted to radiation. Other\nparameters include black hole mass $M$, disk inclination $i$, and distance to the\nsource $D$. For KERRBB $D$ is an explicit parameter, but for BHSPEC\n$D=10\/\\sqrt{N}$ kpc where $N$ is the model normalization.\nIn some cases $M$, $i$, and $D$ (or $N$) are fixed at the estimates\ngiven in Table \\ref{tbl1}.  Though $M$ is always fixed, there are also cases\nwhere $i$ and $D$ are left as free parameters.  The estimates for $i$ in Table\n\\ref{tbl1} are all estimates of the binary inclination. However, there is\nno guarantee that the angular momentum of the black hole, and therefore the inner\naccretion disk, is aligned with the binary (Bardeen \\& Petterson 1975). If one\nassumes that the jet axis is aligned with the angular momentum vector of the\nblack hole, then misalignments may be common (Maccarone 2002). XTE J1550-564\nand GRO J1655-40 are among the sources for which misalignment can be inferred.  \nTherefore, we also consider fits in which $i$ and $D$ are free parameters with\n$i$ unconstrained and $D$ allowed to vary within the confidence intervals in\nTable \\ref{tbl1}.  These models also allow for the presence of a torque on the\ninner disk which is parameterized by the increase in efficiency due to the torque\nrelative to the efficiency of the untorqued disk $\\Delta \\eta\/\\eta$.\n\n\\begin{deluxetable*}{lccccl}\n\\tablecolumns{6}\n\\tablecaption{Source Descriptions\\label{tbl1}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Name} &\n\\colhead{Mass} &\n\\colhead{Distance} &\n\\colhead{Inclination} &\n\\colhead{$N_{\\rm H}$} &\n\\colhead{Reference}\\\\\n &\n\\colhead{$( M_{\\odot})$} &\n\\colhead{(kpc)} &\n\\colhead{(deg)} &\n\\colhead{$(10^{22}\\,{\\rm cm}^{-2})$} &\n}\n\n\\startdata\nLMC X-3 &\n$7 (5-11)$ &\n$52 (51.4-52.6)$ &\n$67 (65-69)$ &\n$0.04$ &\n1, 2, 3, 4, 5 \\\\\nXTE J1550-564 &\n$10 (9.7-11.6)$ &\n$5.3 (2.8-7.6)$ &\n$72 (70.8-75.4)$ &\n$0.65$ &\n6, 7 \\\\\nGRO J1655-40 &\n$7 (6.8-7.2)$ &\n$3.2 (3.0-3.4)$ &\n$70 (64-71)$ &\n$0.8$ &\n8, 9, 10, 11\\\\\n\\enddata\n\n\\tablerefs{(1) Soria et al. (2001); (2) Cowley et al. (1983); (3) di Benedetto (1997); \n(4) Kuiper et al. (1988); (5) Page et al. (2003); (6) Orosz et al. (2002);\n(7) Gierli\\'nski \\& Done (2003); (8) Shahbaz et al. (1999);\n(9) Hjellming \\& Rupen (1995); (10) van der Hooft et al. (1998);\n(11) Gierli\\'nski et al. (2001)}\n\n\\end{deluxetable*}\n\nThere are other parameters which are not shared between BHSPEC and KERRBB. Two\nadditional parameters for BHSPEC are $\\alpha$ and the metal abundance. In\nthe BHSPEC model, the stress is given by\n  \\begin{eqnarray} \n\\tau_{R\\phi}=\\alpha P\n\\label{taurphi}\n  \\end{eqnarray} \nwhere $\\tau_{R\\phi}$ is the vertically averaged accretion stress and $P$ is the\nvertically averaged total pressure.  Usually, the metal abundances are fixed at\nthe solar value, but we also consider fits with three times solar metallicity.\nThe color correction parameter $f$ must be chosen for KERRBB, and we fix\n$f=1.7$ unless stated otherwise.  The $f$ value\nwhich brings KERRBB into best agreement with BHSPEC is a function of $\\ell$\n(Shafee et al. 2006), but this choice\nmakes KERRBB roughly consistent over the range of $\\ell$ we consider. We always fix the parameters\n{\\it rflag} and {\\it lflag} so that the spectra are limb darkened and reprocessed\nemission from self-irradiation is ignored. Test cases suggest that these choices\ndo not have a significant effect on the quality of fit or the inferred values for\n$a_{\\ast}$.\n\n\\subsection{Source Selection}\n\\label{soursel}\n\nOur work is motivated in part by spectral fitting of BHBs performed by Gierlinski\n\\& Done (2004). They already provided a sample of sources with {\\it RXTE}\nobservations in which a low fraction (under 15\\%) of the bolometric flux is\ninferred to be in the non-thermal component.  We focus on three of these sources:\nLMC X-3, XTE J1550-564 (hereafter J1550), and GRO J1655-40 (hereafter J1655).  \nEach one ranges over nearly a decade or more in bolometric luminosity (see Figure\n2 of Gierlinski \\& Done 2004).  This makes them particularly well suited for\nconstraining the spectral variation over a range of $\\dot{M}$. The properties\nof these sources are summarized in Table \\ref{tbl1}.  All have reasonably precise\nmass estimates, and the distances to LMC X-3 and J1655 both have relatively small\nuncertainties.  The distance to J1550 is less well constrained, but we still\ninclude it in our sample because it spans the widest range of luminosities.\n\nOne drawback of {\\it RXTE} is its lack of soft X-ray coverage.  The thermal\ncomponents of BHBs typically peak at photon energies near or below 1 keV,\nwhereas the {\\it RXTE} band extends down to only $\\sim3$ keV. \nSince one of our goals is to\ntest the applicability of the underlying accretion disk model, we would like to cover\nas much of the SED as possible.  Even if we use other observatories, we face the\ndifficulty that most BHBs lie in or near the Galactic plane and are heavily absorbed by\nthe interstellar medium along the line-of-sight. Therefore, we also examine a\n{\\it BeppoSAX} observation of LMC X-3 which has a low line-of-sight absorption\ncolumn.  {\\it\nBeppoSAX} is well-suited for this purpose as it covers a very broad range of\nphoton energies extending down to a tenth of a keV, but lacks the effective area of\n{\\it XMM} or {\\it Chandra} and their corresponding pile-up problems for such\nbright sources.\n\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{f2.eps}\n\\caption{The 66\\%, 90\\%, and 99\\% confidence contours in the $a_{\\ast}$--$\\cos i$\nplane for the best fit BHSPEC model\n($\\alpha=0.01$, $i$ free) to the {\\it BeppoSAX} LMC X-3 data.  The \nvertical dashed lines mark uncertainty limits inferred for the binary inclination.\n\\label{fig2}}\n\\end{figure}\n\n\\subsection{LMC X-3}\n\\label{lmc}\n\n\\subsubsection{{\\it BeppoSAX} Data}\n\\label{lmcsax}\n\nThe low energy coverage of {\\it BeppoSAX}\nmakes the fits particularly sensitive to our model for the neutral absorption.  \nWe therefore include the line-of-sight absorption column as a free parameter in\nour fits to the {\\it BeppoSAX} data.  For our best fit BHSPEC model we find \n$N_H=5.92^{+0.31}_{-0.27} \\times 10^{20}$ cm$^{-2}$.  This is\nhigher than values inferred from 21 cm absorption ($3.2 \\times 10^{20}$\ncm$^{-2}$, Nowak et al. 2001) or fits to the neutral oxygen edge in observations\nwith the Reflection Grating Spectrometer on-board {\\it XMM-Newton} ($3.8 \\pm 0.8\n\\times 10^{20}$ cm$^{-2}$, Page et al. 2003), but lower than those found in\nprevious modeling of the {\\it BeppoSAX} data\n($7 \\pm 1 \\times 10^{20}$ $\\rm cm^{-2}$, Haardt et al. 2001).\n\nThe unfolded spectrum of the {\\it BeppoSAX} data is shown in\nFigure \\ref{fig1}.  The best fit model for BHSPEC with $i=67^{\\circ}$ and\n$\\alpha=0.01$ which was used to generate the unfolded spectrum \nis also shown. The LMC X-3 spectrum\nis very disk dominated and the PDS provides little constraint due to the low count\nrate. Therefore, we fix the optical depth in the COMPTT model at $\\tau=0.5$,\nproviding a flat (in $\\nu F_{\\nu}$) spectrum typical of thermal dominant state\nobservations.\n\nWe provide a comparison of the three spectral models\nin Table \\ref{tbl2}. With $D$ and $i$ fixed in the KERRBB and BHSPEC, each  model has \nthe same number of free parameters: two for the soft\/thermal component, one for the \nnon-thermal component, and one for the intervening absorption column for a\ntotal of four free parameters. DISKBB provides a\nconsiderably poorer fit than either KERRBB or BHSPEC. The relativistically\nbroadened spectra are a much better representation of the soft thermal\nemission than the narrower DISKBB.  \n\nThe quality of the DISKBB fit is sensitive to\nthe model of the non-thermal emission.  If we treat $\\tau$ as a free parameter,\n$\\tau$ drops and\nthe fit improves slightly ($\\chi^2_\\nu=320\/176$) but remains poor compared with KERRBB and BHSPEC.\nThe COMPTT spectrum steepens as $\\tau$ decreases.  This extra flux in the `tail'\nof the thermal component compensates for a decrease in $T_{\\rm in}$ which allows\nDISKBB to better approximate the low energy photons.  The fit further improves to\n$\\chi^2_\\nu=264\/176$ if we replace COMPTT with a power law. This provides a slightly\nbetter fit than KERRBB but BHSPEC is still preferred.  The best fit model now\nrequires a steep power law component $\\Gamma \\sim 2.8$ which is consistent with\nthe best fit model of Haardt et al. (2001).  However, the power law flux now {\\it\nexceeds} the DISKBB flux at low energies.  This result is unphysical in a picture\nwhere the soft X-ray emission provides the bulk of the seed photons for the\nnon-thermal component.  This is also likely the explanation for why the Haardt et\nal. (2001) fits require a larger neutral hydrogen column.  This inconsistency\nwas also pointed out by Yao et al. (2005) who find a self-consistent fit with a\nComptonized multitemperature blackbody model.  In contrast, the inclusion of \nnon-thermal emission has little effect on the $\\chi^2$ values for KERRBB and\nBHSPEC. Thus, the {\\it BeppoSAX} data can be completely accounted\nfor by a bare accretion disk spectra as long as relativistic effects on the \nspectra are included.\n\nBHSPEC provides a better fit ($\\Delta \\chi^2=-32$) to the data than KERRBB for an\ninclination $i=67^{\\circ}$, a source distance $D=52$ kpc, and $f=1.7$. Allowing\n$f$ to vary from 1.5-1.9 does not improve the KERRBB quality of fit significantly.  \nThe prescription for relativistic effects in the two models are essentially\nidentical, so the differences in the spectral shapes are primarily due to the\ndifferent prescriptions for the disk surface emission.  The annuli spectra which\nmake up the BHSPEC model have imprints from\nmetal opacities and may differ from color corrected blackbodies by several\npercent.  Additionally, annuli at different radii have local spectra which are best\napproximated by different values of $f$, with $f$ usually being higher for the\nhotter, inner annuli. KERRBB assumes one value of $f$ for the whole\ndisk.  Though these discrepancies are not at a level which is significantly\ngreater than the intrinsic uncertainties in the BHSPEC model, it is suggestive\nthat a model which includes atomic physics provides a better fit.\n\n\n\\begin{figure}\n\\includegraphics[width=0.46\\textwidth]{f3.eps}\n\\caption{The 66\\%, 90\\%, and 99\\% confidence contours in the $a_{\\ast}$--$D$ plane\nof the best fit BHSPEC model ($\\alpha=0.01$, $i=67^{\\circ}$, $N$ free) to \nthe {\\it BeppoSAX} LMC X-3 data. Here we have let the normalization vary by 20\\%\nabove and below its nominal value of $N=(10\\,{\\rm kpc}\/52\\,{\\rm kpc})^2=0.0370$.\nThe model normalization $N$ is enumerated on the upper horizontal axis.  The \nvertical dashed lines mark uncertainty limits associated the distance estimate\nin Table \\ref{tbl1}\n\\label{fig3}}\n\\end{figure}\n\nDespite these differences in the quality of fit, KERRBB and BHSPEC both give\nvalues of $a_{\\ast}\\sim 0.3$ for $i=67^{\\circ}$.  The best fit value for $a_{\\ast}$ for BHSPEC\nis a function of $\\alpha$ with $\\alpha=0.1$ giving a lower $a_{\\ast}$ than\n$\\alpha=0.01$.  This is the case in all fits and is most simply understood by\nexamining how changes in the parameters either harden (increase the mean\nphoton energy) or soften (lower the mean photon energy) the spectra. As\n$a_{\\ast}$ increases, the inner radius of the disk decreases.  This results in a\nlarger fraction of the gravitational binding energy being released in a smaller\narea of the disk surface.  The resulting increase in $T_{\\rm eff}$ in\nthese annuli produces a spectrum with higher average photon energies. Therefore,\nincreasing $a_{\\ast}$ hardens the spectra, even at fixed luminosity.\n\nThe sensitivity of the spectrum to $\\alpha$ is more complex.  It \nis strongest at high $\\dot{M}$\nin radiation pressure dominated annuli where the surface density $\\Sigma$ is low.\nA larger $\\alpha$ yields a lower $\\Sigma$, making the disk less effectively optically thick.\nFor a range of  $\\dot{M}$, the $\\alpha=0.01$ annuli remain very effectively optically \nthick while the $\\alpha=0.1$  annuli become less effectively optically thick and eventually\neffectively optically thin as $\\dot{M}$ increases.  For $\\alpha=0.1$, the densities are\nlower and the temperatures are higher. The photons cannot thermalize as well, causing \nthe spectrum to harden significantly. At lower $\\dot{M}$, both\nmodels have sufficiently large $\\Sigma$ so that spectral formation occurs nearer the disk\nsurface at approximately the same densities and temperatures.  The $\\alpha=0.1$ models still \ntend to be slightly less dense in the spectral forming region and are therefore slightly harder,\nbut the differences are significantly smaller than at higher $\\dot{M}$.\nTherefore, the dependence of the best-fit $a_\\ast$ on $\\alpha$ results from an\nincrease in $\\alpha$ from 0.01 to 0.1 hardening the spectrum so that $a_{\\ast}$ must be\nreduced to compensate. \n\nThe best fit $a_{\\ast}$ is also a function of $i$.  As can be seen in Table\n\\ref{tbl2}, making $i$ and $D$ free parameters does not significantly improve the\nquality of fit in these cases.  However, it does greatly increase the\nuncertainty in the best fit $a_{\\ast}$.  This is illustrated in Figure\n\\ref{fig2} where we plot the 66\\%, 90\\%, and 99\\% joint confidence\ncontours for $a_{\\ast}$ and $i$.  The strong correlation exists because\nlowering $i$ softens the spectrum so that $a_{\\ast}$\nmust increase to compensate. This is partly because the line-of-sight projection of the \nazimuthal fluid velocity component decreases.  This reduces the blueshift and \nbeaming of the emission from the\napproaching side of the disk, and moves the `position' of the high energy tail\nto lower energies. Also, the projected disk area increases which produces a\nlarger flux for the observer at infinity.  This needs to be compensated by a\ndecrease in $\\dot{M}$, and therefore $\\ell$.  This decrease in $\\ell$\nlowers $T_{\\rm eff}$ and again softens the\nspectrum. Thus, our ability to constrain $a_{\\ast}$ is generally improved by\nprecise, reliable estimates for $i$.  The contours in Figure \\ref{fig2}\nsuggest $a_{\\ast} \\simeq 0.55 \\pm 0.2$ at 90\\% confidence.  If the binary inclination\nuncertainties are accurate and the inner accretion disk is aligned with the binary orbit,\nthese constraints imply $0.2 \\lesssim a_{\\ast} \\lesssim 0.4$.\n\nWe also examine the variation of $a_{\\ast}$ with the source distance $D$. The \nBHSPEC model normalization $N$ is defined so that $D=10\/\\sqrt{N}$ kpc. \nThe confidence range for the\ndistance to LMC X-3 listed in Table \\ref{tbl1} provides a tight constraint on $N$.\nHowever, $N$ also depends on the absolute flux calibration of the detector so that \nany uncertainty in absolute flux translates into an effective uncertainty for $D$.  \nTherefore, we consider fits where $N$ is free to vary by 20\\% from its nominal values of \n$N=(10\\,{\\rm kpc}\/52\\,{\\rm kpc})^2=0.0370$.  The 66\\%, 90\\%, and 99\\% confidence \ncontours in the $a_{\\ast}$--$D$ plane are shown in Figure \\ref{fig3}.  The best fit\nmodel lies at the upper limit of the allowed range of $N$.  The best\nfit $a_{\\ast} \\sim 0.38$ is slightly larger than the range ($a_{\\ast} \\sim 0.25 \\pm 0.05$ )\nconsistent with the distance constraints which are plotted as vertical dashed lines in\nthe figure.\n\nThe strong anticorrelation between $a_{\\ast}$ and $D$ seen in Figure \\ref{fig3} exists\nbecause an increase in $D$ leads to a decrease in the flux expected at the detector\n(i.e. a lower $N$).  This must be accounted for by an increase in the luminosity ($\\ell$)\nwhich would shift the spectral peak to higher energies at fixed $a_{\\ast}$. However, \n$a_{\\ast}$ is a free parameter and it can be lowered so that the spectral peak remains \nfixed while $\\ell$ increases.\n\n\n\\begin{deluxetable*}{lccccccc}\n\\tablecolumns{8}\n\\tablecaption{LMC X-3 {\\it BeppoSAX} Fit Summary\\label{tbl2}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Model\\tablenotemark{a}} &\n\\colhead{$\\alpha$} &\n\\colhead{$i$} &\n\\colhead{$D$} &\n\\colhead{$a_{\\ast}$} &\n\\colhead{$k T_{\\rm in}$} &\n\\colhead{$N_H$} &\n\\colhead{$\\chi^2_\\nu$} \\\\\n &\n &\n\\colhead{(deg)} &\n\\colhead{(kpc)} &\n &\n\\colhead{(keV)} &\n\\colhead{$(10^{20}\\,{\\rm cm}^{-2})$} &\n\\\\\n}\n\n\\startdata\nDISKBB &\n\\nodata &\n\\nodata &\n\\nodata &\n\\nodata &\n$1.0139^{+0.0074}_{-0.0080}$ &\n$4.18^{+0.21}_{-0.20}$ &\n336\/177 \\\\\nKERRBB &\n\\nodata &\n67 &\n52 &\n$0.3639^{+0.0013}_{-0.0013}$ &\n\\nodata &\n$5.08^{+0.27}_{-0.25}$ &\n275\/177 \\\\\nBHSPEC &\n0.1 &\n67 &\n52 &\n$0.141^{+0.021}_{-0.020}$ &\n\\nodata &\n$5.65^{+0.27}_{-0.26}$ &\n243\/177 \\\\\nBHSPEC &\n0.01 &\n67 &\n52 &\n$0.258^{+0.019}_{-0.019}$ &\n\\nodata &\n$5.58^{+0.27}_{-0.26}$ &\n246\/177 \\\\\nBHSPEC &\n0.01 &\n$53^{+13}_{-10}$ &\n$51.4^{+1.2}_{-0.0}$ &\n$0.54^{+0.11}_{-0.10}$ &\n\\nodata &\n$5.92^{+0.31}_{-0.27}$ &\n238\/175 \\\\\n\n\\enddata\n\n\\tablenotetext{a}{The full Xspec model is WABS*(Model+COMPTT).}\n\n\\tablecomments{All uncertainties are 90\\% confidence for one parameter.\nParameters reported without uncertainties were held fixed during the fit.}\n\n\\end{deluxetable*}\n\nWe also use these models to test for the possibility of magnetic torques on the\ninner accretion disk. The energy release by a torque increases the fraction of\nemission at small radii and increases the effective temperature of the annuli.  \nThis produces a hardening of the spectrum similar to an increase in $a_\\ast$.  \nAt the time of publication, BHSPEC spectra have only been computed from disks with \nnon-zero torques for $a_\\ast=0$.  KERRBB can be used to examine torques at all\n$a_{\\ast}$, but the fits provide little constraint as $\\Delta \\eta\/\\eta$ varies over\nthe entire range of the model from zero to one at 90\\% confidence when $i$ is a free\nparameter.  As expected, increases in $\\Delta \\eta\/\\eta$ are offset by decreases \nin $a_\\ast$.  An upper-limit on torque can be obtained by fitting BHSPEC at \n$a_\\ast=0$.  The best fit $\\Delta\\eta\/\\eta=3 \\pm 0.8$ with $\\chi^2\/\\nu=245\/175$ \nand $i=73^\\circ \\pm 1^\\circ$. This value of $\\chi^2$ is only slightly greater than\nin the untorqued case, and the best fit inclination is consistent with the \nconstraints on the binary inclination, so it is difficult to rule out the \npossibility of large torques from these data.\n\n\\subsubsection{{\\it RXTE} Data} \n\nGD04 have already\nselected a sample of disk dominated {\\it RXTE} observations for several sources, including\nLMC X-3. From these, we have selected a subset of 10 epochs which evenly cover\nthe range of disk luminosities inferred from the GD04 analysis.  The luminosities\nof these epochs are plotted versus the maximum color temperature in the first panel\nof Figure \\ref{fig4}.  The black filled circles correspond to the data sets used\nin our work and red triangles represent the other epochs in the GD04 sample.  \nThese plots were generated by taking DISKBB fit results and making corrections\nfor the temperature profile and relativistic effects (Zhang et al. 1997). This\nplot only includes epochs in which the disk component is inferred to account for\ngreater than 85\\% of the bolometric flux. A detailed explanation of the analysis\ncan be found in GD04. The values of $L_{\\rm disk}\/L_{\\rm Edd}$ and $T_{\\rm max}$\nare evaluated using the estimates in Table \\ref{tbl1} so there is some\nuncertainty in collective position of these symbols on the plot.  However, the\nplacement of the points relative to each other is robust to these uncertainties\nso that reproduction of the shapes of these $L-T$ relations provides an important test\nfor our disk models.  We repeat the same procedure for observations of J1550 \nand J1655 and plot these in the center and right panels of Figure \\ref{fig4},\nrespectively.\n\n\nAs stated in section \\ref{intro}, the luminosity is roughly proportional to the\nfourth power of the maximum temperature.  The dashed curves represent lines of\nconstant $f$ where $L \\propto T_{\\rm max}^4$. In the bottom panels of Figure\n\\ref{fig4} we have also plotted $L\/T_{\\rm max}^4$ in order to more easily evaluate\nspectral hardening relative to this overall trend. Comparison of the data with these\ncurves shows some evidence for hardening with increasing $L$ for J1665 and J1550.  \nLMC X-3 is roughly consistent with a constant $f$, but a close examination suggests \nthere might be weak signs of hardening above $\\sim 0.9$ keV and\nsoftening at the highest temperatures.\n\nWe investigate this spectral evolution with $\\dot{M}$ by fitting the models\ndirectly to the data. We consider the same models as in section \\ref{lmcsax}, but\nwe now fix the absorption column since it is not well constrained without the low\nenergy coverage.  For LMC X-3 we fix it at $N_H=5.5 \\times 10^{20}$ $\\rm cm^{-2}$\nto be consistent with the {\\it BeppoSAX} fits.\nWe initially fix $i$ and $D$, and fit only a single value of $a_\\ast$ for all epochs.\nWe also fix $f=1.7$ for KERRBB and $\\alpha=0.1$ or 0.01 for BHSPEC.  Only\n$\\dot{M}$ (or $\\ell$) is allowed to vary for each epoch.\nWe also consider models with DISKBB, fitting a single \nnormalization simultaneously to all data sets.  This is also consistent with assuming\na fixed color correction and constant effective area for each epoch. With these\nchoices, each model has the same number of free parameters.  There is a single parameter \nshared by all data sets ($a_{\\ast}$ or DISKBB normalization), and two parameters\nfor each individual data set: one for the soft\/thermal component ($\\dot{M}$, $\\ell$, or\n$T_{\\rm in}$) and a normalization for the non-thermal component.  For BHSPEC, we\nalso consider fits where $i$ and $D$ (or $N$) are free parameters.  Since only a\nsingle value of either parameter is fit for all epochs, this provides at most\ntwo additional parameters.\n\nIn oder to visualize the variation of the spectral shape with $\\dot{M}$,\nwe plot in Figure \\ref{fig4} the $L-T$ relations (solid curves) derived from\nthe best fit BHSPEC models at fixed $i$ for $\\alpha$=0.1 and 0.01.  These curves are \ncalculated by generating artificial spectra with our best fit models, and then fitting \nthem using the same procedure that GD04 used for the real data.  The plot therefore\nprovides a comparison of fits with DISKBB, both to the data, and to artificial spectra\ngenerated from the best-fit BHSPEC models.  It is {\\it not} a direct comparison of BHSPEC\nwith the data.  This explains\nwhy the best-fit BHSPEC curves do not go through the `data points' in the J1550 and J1655 \nplots.  The two types of fits find a different partition of the spectra between the \nsoft\/thermal and hard\/non-thermal components, with additional flux accounted for\nby the non-thermal emission in the BHSPEC fits. The implications of this are discussed\nfurther in section \\ref{hard}. For comparison, we also show curves with \n$L_{\\rm disk}\/L_{\\rm Edd} \\propto T_{\\rm max}^3$ (dotted lines).  These curves\nrepresent simple, analytic estimates for the spectral hardening in effectively\noptically thick disks when the effects of Comptonization are negligible.  The\nderivation of this relation in presented in the Appendix.\n\n\\begin{figure*}\n\\includegraphics[width=0.94\\textwidth]{f4.eps}\n\\caption{The disk luminosity (top panels) as a function of maximum temperature \nfor J1655, J1550,\nand LMC X-3.  Each symbol represents a DISKBB fit to one {\\it RXTE} data set. \nThese data were presented in GD04 and the reader is referred there for a complete\ndiscussion of their spectral analysis.  The black filled circles represent the\nepochs which were used in this work and the red triangles represent the remaining\nGD04 data sets.  The dashed curves are lines of constant color correction\ncorresponding to $f=1.6, 1.8, 2.0, 2.2, 2.4, 2.6$, and 2.8 for a Schwarzschild\nblack hole (see eq. 3 of GD04). The dotted lines represent curves with\n$L_{\\rm disk}\/L_{\\rm Edd} \\propto T_{\\rm max}^3$.  This $L-T$ relation follows\nfrom simple, analytic estimates for the spectral hardening in effectively\noptically thick disks when the effects of Comptonization are negligible. The\nnormalization is chosen arbitrarily to compare with the BHSPEC model curves. The\nderivation of this relation and its relevance to the models is discussed\nin the appendix. The green and blue curves show the evolution \nexpected from the best fit BHSPEC models for $i$ fixed at the binary estimate\n(see Table \\ref{tbl1}) for $\\alpha=0.1$ and 0.01,\nrespectively.  These curves were created by producing synthetic data sets with \nthe BHSPEC model and replicating the spectral analysis performed by GD04. \nTo more easily evaluate the spectral hardening relative to a fixed $f$,\nwe have plotted $L\/T_{\\rm max}^4$ (lower panels).  The units on the vertical\ncoordinates are arbitrary, but the lines of constant $f$ (now horizontal) are\nretained for reference when comparing with the top panel.\n\\label{fig4}}\n\\end{figure*}\n\n\nBased on the discussion above, it is reasonable to expect that\nthe difference in quality of fit between models with fixed $f$ (KERRBB and DISKBB)\nand BHSPEC might be dominated by the differing predictions for the variation of spectral\nhardening with changing $\\ell$.\nThe lack of hardening at the highest $\\ell$ seen in Figure \\ref{fig4} for LMC X-3 suggests \nthat models with constant $f$ or BHSPEC with a low value of\n$\\alpha$ would provide the best representations of the data. These\npredictions are borne out in simultaneous fits to the LMC X-3 data, which are\nsummarized in Table \\ref{tbl3}. The unfolded spectra are plotted in\nFigure \\ref{fig5} with the best fit BHSPEC model for $\\alpha=0.01$ and\n$i=67^{\\circ}$.  DISKBB, which is representative of a disk with a fixed emitting\narea and constant $f$, provides the best fit. A comparable $\\chi^2$ is provided by\nKERRBB with $i=67^{\\circ}$ and $f=1.7$.  The fit with BHSPEC for $\\alpha=0.01$\ngives an acceptable $\\chi^2$, but provides a poorer representation than the fixed \n$f$ models. The $\\alpha=0.1$ model does not provide an acceptable fit.\nFor BHSPEC, the largest contributions to $\\chi^2$ come from the highest luminosity\nepoch.  As seen in Figure \\ref{fig4}, the BHSPEC models seem to harden too rapidly to\naccommodate all epochs simultaneously. BHSPEC hardens more rapidly with \nincreasing $\\ell$ for $\\alpha=0.1$ than 0.01, leading to the significantly\npoorer fit.  In order to make the inner disk annuli more\neffectively thick, we increased the metal abundances to three times the solar\nvalue.  However, this had limited impact on the spectral shape and did not\nimprove the quality of fit appreciably.\n\nThe best fit values of $a_{\\ast}$ and their 90\\% confidence intervals are\nsummarized in Table \\ref{tbl3}.  They are systematically lower ($a_{\\ast} \\lesssim\n0.1$) than the values inferred from the {\\it BeppoSAX} fits ($a_{\\ast}\\sim 0.3$).  \nMost of this discrepancy can be accounted for by cross calibration differences\nbetween the two observatories.  Cross-calibration\\footnotemark \\,  campaigns on 3C 273\nshow that the {\\it RXTE} PCA flux is about 20\\% higher than the {\\it BeppoSAX} data.\nAt fixed normalization, this requires a $20\\%$ increase in $\\ell$ which leads\nto an $\\sim 5\\%$ increase in $T_{\\rm eff}$.  For a given $i$, this change must be offset by \na decrease in $a_\\ast$ which leads to a systematically lower value for the {\\it RXTE}\nfits relative to {\\it BeppoSAX} fits.\n\n\n\\footnotetext{http:\/\/heasarc.gsfc.nasa.gov\/docs\/asca\/calibration\/3c273\\_results.html}\n\n\\subsection{XTE J1550-564}\n\\label{1550}\n\nAs seen in the middle panels of Figure \\ref{fig4}, J1550 varies over an order of\nmagnitude in $\\ell$ and we have chosen 10 observations which sample this range.  We\nfit J1550 (and J1655) with the same model which we applied to LMC X-3, but we\nfound statistically significant residuals consistent with reflection features.\n(These were apparently unnecessary in LMC X-3 because the spectra are so strongly\ndisk dominated.)\nTo account for these residuals we add a GAUSSIAN and apply a SMEDGE (Ebisawa et\nal. 1991) to the COMPTT component to approximate reprocessing at the disk\nsurface. We fix the widths of these components to 0.5 keV and 7 keV,\nrespectively.  This adds two free parameters from each new component.  We also\nlet the optical depth vary in the COMPTT component for a total of\nfive additional parameters in each epoch. The fit results with these additional\ncomponents are summarized in Table \\ref{tbl3}.\n\nAs with LMC X-3, all of the models provide an acceptable fit, though there is\nstill considerable variation in the $\\chi^2$ values.  There is a statistically\nsignificant preference for the DISKBB model over KERRBB.  Since both models have\na constant $f$, the difference in the quality of fit must be due to the overall\nspectral shape and not simply its evolution with $\\dot{M}$.  A comparison of\nthe best fit spectral shapes shows that relativistic KERRBB models are harder\nthan those of DISKBB over the {\\it RXTE} band, which seems to be primarily the\nresult of relativistic\nbroadening.\n\nThe BHSPEC model with $\\alpha=0.01$ provides a better fit than $\\alpha=0.1$,\nconsistent with the prediction of the $L-T$ comparison in Figure \\ref{fig4}.  \nHowever, the $\\alpha=0.01$ model still provides a poorer fit than DISKBB.  In\ncontrast to KERRBB, the best fit spectra fall off more steeply with increasing photon\nenergy than the DISKBB spectra.  This behavior seems to be primarily due to\nabsorption features (primarily Fe K) in the tail of spectrum. When we let $i$\nand $D$ float, the BHSPEC fit improves and the $\\chi^2$ values are now slightly\nbetter than DISKBB. The best fit inclination is lower ($i=42^{+3}_{-13}$ deg)\nand the spin is higher ($a_{\\ast}=0.72^{+0.15}_{-0.01}$),\nproducing slightly harder spectra with less pronounced absorption features than\nin the $i=72^{\\circ}$ case.\nFor $i=72^{\\circ}$, the best fit $a_{\\ast}$ is relatively low ($\\lesssim 0.1$)\nfor both KERRBB and BHSPEC, so allowing $i$ to vary changes $a_{\\ast}$ considerably.\n\nApparent motion $\\gtrsim 2 c$ has been claimed to be observed in the radio\nemission from this source (Hannikainen et al. 2001) suggesting $i \\lesssim\n50^{\\circ}$ for ballistic motion.  This implies a misalignment of at least\n$20^{\\circ}$ if the jet is aligned with the black hole angular momentum vector.\nThe best fit inclination $i=42^{+3}_{-13}$ deg is consistent with this upper\nlimit and may be compatible with an inner disk aligned with the black hole via\nthe Bardeen-Petterson effect.\n\n\\subsection{GRO J1655-40}\n\\label{1655}\n\nOf the three sources considered in this work, J1655 displays the strongest\nevidence of hardening in Figure \\ref{fig4}, suggesting that the models with fixed\n$f$ will provide poorer fits than BHSPEC.  In fact, KERRBB does not provide an\nadequate fit to the data, and a luminosity dependent trend can be seen in the\nresiduals. However, DISKBB can still provide an adequate fit and is even\npreferred to BHSPEC for $i=70^{\\circ}$.  As in J1550, the best fit KERRBB spectra\nare harder than those of DISKBB. These results again suggest that the overall\ndifferences in spectral shape (as opposed to the variation with $\\dot{M}$)\nprovide the dominant effect on the quality of fit.  Unlike LMC X-3 and J1550, the\nBHSPEC model with $\\alpha=0.1$ provides a better fit than $\\alpha=0.01$.\nComparison with the bottom-right panel of Figure \\ref{fig4} suggests that the soft, thermal\ncomponent in J1655 is hardening more strongly with increasing luminosity than can\nbe easily accounted for with the $\\alpha=0.01$ model.\n\nAt fixed inclination, BHSPEC with $\\alpha=0.1$ provides the only relativistic fit\nwhich is marginally acceptable. The best fit $a_{\\ast}=0.62 \\pm 0.01$ in this\ncase suggests a moderate spin which is reasonably consistent with other\ninvestigations ($a_{\\ast}\\sim 0.7-0.9$, Gierlinski et al. 2001; $a_{\\ast}\\sim\n0.65-0.75$, Shafee et al. 2006).  KERRBB also yields a similar spin \n($a_\\ast=0.6015^{+0.0013}_{-0.0023}$) for $f=1.7$.  The small discrepancy with\nthe Shafee et al. (2006) is likely due to our choice of a single $f$ and\nto differences in our modeling of the non-thermal emission.\n\nAllowing $i$ to be a free parameter significantly reduces\n$\\chi^2$ for both values of $\\alpha$, but $\\alpha=0.01$ now provides a slightly\nbetter fit. As with LMC X-3 and J1550, the spin is very sensitive to the inclination.\nFor both values of $\\alpha$ the best fit $a_{\\ast}=0$, the lower limit of the model.\n(We have not yet extended BHSPEC to retrograde spins.)  \nWe find $\\chi^2_\\nu=248\/302$ and $i=83.8^{\\circ} \\pm 0.6^{\\circ}$ for\n$\\alpha=0.1$, and $\\chi^2_\\nu=234\/302$ and $i=85.6^{\\circ} \\pm 0.4^{\\circ}$\nfor $\\alpha=0.01$.\n\nRadio observations of J1655 have inferred a jet inclination of $85^{\\circ} \\pm 2^{\\circ}$\nto the line of sight (Hjellming \\& Rupen 1995). Assuming the jets are aligned\nwith the angular momentum axis of the black hole, this would imply an inner disk\ninclination (to the plane of the sky) of $i=85^{\\circ}$, consistent with the best\nfit $i$ above.  However, disk alignment only occurs for black holes with\nnon-zero angular momentum and the transition radius is expected to increase with\nincreasing spin (Bardeen \\& Petterson, 1975).  Therefore, because these fits find\n$a_\\ast=0$, they do not provide a self-consistent picture for a misaligned disk\nscenario.\n\n\\begin{deluxetable*}{lccccccc}\n\\tablecolumns{8}\n\\tablecaption{{\\it RXTE} Fit Summary \\label{tbl3}}\n\\tablewidth{0pt}\n\\tablehead{\n &\n &\n\\multicolumn{2}{c}{LMC X-3} &\n\\multicolumn{2}{c}{XTE J1550-564} &\n\\multicolumn{2}{c}{GRO J1655-40} \\\\\n\\colhead{Model\\tablenotemark{a}} &\n\\colhead{$\\alpha$} &\n\\colhead{$a_{\\ast}$} &\n\\colhead{$\\chi^2_\\nu$} &\n\\colhead{$a_{\\ast}$} &\n\\colhead{$\\chi^2_\\nu$} &\n\\colhead{$a_{\\ast}$} &\n\\colhead{$\\chi^2_\\nu$} \\\\\n}\n\n\\startdata\nDISKBB\\tablenotemark{b} &\n\\nodata &\n\\nodata &\n296\/431 &\n\\nodata &\n230\/355 &\n\\nodata &\n284\/304 \\\\\nKERRBB\\tablenotemark{c} &\n\\nodata &\n$0.119^{+0.013}_{-0.013}$ &\n296\/431 &\n$0.097^{+0.005}_{-0.065}$ &\n301\/355 &\n$0.6015^{+0.0013}_{-0.0023}$ &\n439\/304 \\\\\nBHSPEC &\n0.1 &\n$0$ &\n1190\/431 &\n$0^{+0.0055}_{-0}$ &\n324\/355 &\n$0.617^{+0.013}_{-0.006}$ &\n330\/304 \\\\\nBHSPEC &\n0.01 &\n$0^{+0.006}_{-0}$ &\n359\/431 &\n$0.115^{+0.030}_{-0.011}$ &\n256\/355 &\n$0.639^{+0.012}_{-0.006}$ &\n369\/304 \\\\\nBHSPEC\\tablenotemark{d} &\n0.01 &\n$0.728^{+0.036}_{-0.018}$ &\n307\/429 &\n$0.72^{+0.15}_{-0.01}$ &\n221\/353 &\n$0.00^{+0.021}_{-0}$ &\n234\/302 \\\\\n\\enddata\n\n\\tablenotetext{a}{The full Xspec model is WABS*(Model+COMPTT) for LMC X-3, and\nWABS*(Model+GAUSSIAN+SMEDGE*COMPTT) for J1550 and J1655.}\n\\tablenotetext{b}{A single normalization is fit for all data sets.}\n\\tablenotetext{c}{The hardening factor is fixed at $f=1.7$. Fit parameters were \nselected so that limb darkening was included but self-irradiation was not.}\n\\tablenotetext{d}{Both $D$ and $i$ are free parameters with the value of $i$ \nallowed to vary over the full range but $D$ constrained to lie within the \nconfidence limits reported in Table \\ref{tbl1}. The best fit values are $\ni=43.4^{+8.5}_{-4.0}$ deg,\n$D=51.4^{+0.61}_{-0}$ kpc for LMC X-3; $i=42^{+3}_{-13}$ deg\n$D=6.3510 \\pm 0.0080$ kpc for J1550; and $i=85.60^{+0.14}_{-0.38}$ deg, \n$D=3.4000^{+0}_{-0.0059}$ kpc for J1655.}  \n\n\\tablecomments{All uncertainties are 90\\% confidence for one parameter.\nParameters reported without uncertainties were held fixed during the fit.\nUnless otherwise noted, the values of $i$, $D$, and $M$ were fixed at the \nestimates given in Table \\ref{tbl1}.}\n\n\\end{deluxetable*}\n\n\n\\section{Discussion}\n\\label{discus}\n\nOne of the principle aims of this work was to test the applicability of the\nrelativistic $\\alpha$-disk model in BHBs. The spectral fitting discussed in\nsection \\ref{specmod} presents mixed results.  The significant improvements in\n$\\chi^2$ relative to DISKBB resulting from the BHSPEC and KERRBB fits to the {\\it\nBeppoSAX} data provide a strong case for relativistic broadening.  DISKBB alone\nis too narrow to adequately approximate the soft, thermal emission from LMC X-3.  \nThe additional quality of fit improvement for BHSPEC relative to KERRBB might\nalso be evidence for modified blackbody and smeared absorption features in the\nspectrum.\n\nIn light of these results, it is surprising that DISKBB with a fixed\nnormalization seems to provide a better fit to the {\\it RXTE} data than KERRBB\nor BHSPEC in all three sources when we fix $i$ at the binary inclination. Since\nit is preferred to both relativistic models, the difference cannot simply be due\nto the differences in the degree of spectral hardening as luminosity changes. This could be\ntaken as evidence against relativistic broadening, but the innermost radii\nimplied by the DISKBB fits are consistent with coming from near the black hole.  \nA comparison of the best fit spectral shapes for all three models shows\ndifferences at the $\\lesssim 10\\%$ level.  The KERRBB spectral shapes tend to be\nbroader (harder in the 3-20 keV band) than both DISKBB and BHSPEC. The\ndifferences between BHSPEC and KERRBB seem to be mostly due to broad absorption\nfeatures which cause the BHSPEC model to fall off more strongly with increasing\nenergy in the tail of the spectrum.  Therefore, it is conceivable that DISKBB\nspectrum could be mimicking similar, but slightly weaker, features in data,\nthough it is surprising that it does this consistently and effectively in all\nthree sources.\n\nAlternatively, it may simply be that our estimates for $D$, $M$, or $i$ are in\nerror.  When we allow $i$ to be a free parameter, BHSPEC\nprovides a better fit than DISKBB in both J1550 and J1655.  In \nboth cases, observations of the radio jets\nsuggest the angular momentum of the black hole is misaligned with that of the\nbinary. It is suggestive that in both cases we find values for $i$ consistent\nwith the constraints implied by the jets, rather than the binary inclination.\nIn the case of LMC X-3, where a jet has not been observed, the best fit $i$\nis more nearly face on than the measured binary inclination.  It is consistent\nwith the binary inclination at 90\\% confidence for the {\\it BeppoSAX} data, but\nnot for the {\\it RXTE} spectral fits.\n\nA third possibility is that the non-thermal emission and Compton reflection \ncomponents are not being correctly accounted for by our prescription.  If\nthis is a problem, it should be minimized by looking at LMC X-3, which has\nthe most disk dominated spectra of the three sources and relatively little\nevidence for reflected emission.\nFor LMC X-3 the BHSPEC residuals are clearly dominated by the most luminous epoch\nfor which BHSPEC predicts too much hardening with increasing luminosity for\neither value of $\\alpha$. If we ignore the most luminous epoch $\\chi^2_\\nu$\nimproves to 257\/388 for BHSPEC with $\\alpha=0.01$ and $i=67^{\\circ}$, but only\nimproves slightly to 256\/388 for DISKBB, providing comparable fits. Allowing $i$ \nto be a free parameter allows the BHSPEC fit to improve even further for slightly\nmore face on values.  If $i$ remains fixed at $67^{\\circ}$, the fit with BHSPEC also\nimproves by allowing the model normalization to vary within 20\\% of the nominal\nvalue.  This is a larger range of normalization than that associated with\ndistance uncertainty and accounts for possible errors in the absolute flux\ncalibration. Thus for LMC X-3, the shape of the spectra seem to be\nbest represented by BHSPEC, but the evolution of the spectral hardening with\n$\\ell$ at the highest luminosities is not consistent with the predictions of a\nsimple $\\alpha$-disk model.\n\n\\subsection{Spectral Hardening and the Stress Prescription}\n\nThe three $L-T$ relations presented in Figure \\ref{fig4} show that differences\nexist in the spectral evolution with disk luminosity from source to source.  LMC\nX-3 has the most scatter and is reasonably consistent with a constant $f$,  \nthough the lower-left panel of Figure \\ref{fig4} shows evidence of a weak\nhardening for $0.1 \\lesssim \\ell \\lesssim 0.3$ and softening for $\\ell \\gtrsim 0.3$.\nJ1550 is consistent with weak hardening and J1655 seems to show more significant\nhardening.  A comparison of the KERRBB and BHSPEC fit results seems to agree with\nthese descriptions.  At fixed $i$, KERRBB with $f=1.7$ is preferred for LMC X-3, but\nBHSPEC provides a better fit in both J1550 and J1655.  Fitting a single $f$ for\nall epochs with KERRBB does not alter this result.  A comparison of BHSPEC fits\nwith $\\alpha=0.1$ and 0.01 is also generally consistent. With $i$ fixed at the\nestimate for the binary inclination, $\\alpha=0.01$ is preferred for LMC X-3 and\nJ1550, but $\\alpha=0.1$ provides a better fit for J1655.  If $i$ is a free\nparameter, $\\alpha=0.01$ provides a better fit for all three sources, though the\nimprovement is small for J1655.\n\nAs discussed in section \\ref{lmcsax} and the Appendix, the sensitivity of the \nspectra to $\\alpha$ comes about  primarily because $\\alpha$ determines $\\Sigma$.\nFor sufficiently  large $\\Sigma$, the spectral shape depends only weakly on $\\alpha$.\nHowever, if $\\Sigma$ drops sufficiently, the disks begin to become effectively optically \nthin at small radii. Once the hottest annuli are effectively optically thin, they become \nincreasingly isothermal or inverted in their temperature profiles as inverse Compton\nscattering in the now hotter surface layers increasingly dominates the cooling.\nThis is much less efficient than thermal cooling and the spectra harden rapidly as\ntemperatures rise with increasing $T_{\\rm eff}$ (Davis \\& Hubeny 2006).  Such \neffects are responsible for the hardening in the $\\alpha=0.1$ models at high \n$\\ell$ in Figure \\ref{fig4}.  Our results\nsuggest that these multi-epoch fits are sensitive to these effects and may\neven be able to differentiate between the  $\\alpha$-disk prescription\nand more general models of angular momentum transport in these disks.\n\nIn the context of the $\\alpha$ prescription, our fit results seem to rule out\nfixed values of $\\alpha \\ge 0.1$ for LMC X-3 and possibly J1550.\nAt first sight, this appears inconsistent with the $\\alpha \\ge 0.1$ values\ninferred for the outburst phases of dwarf novae (Lasota 2001) and soft\nX-ray transients (e.g. Dubus et al. 2001).  However, it is important to note\nthat the disk instabilities that drive the outburst time scales are\nassociated with regions of the disk where gas pressure dominates radiation\npressure.  In contrast, the X-ray spectra are dominated by the innermost\nregions of the disk where radiation pressure can be important at high\n$\\dot{M}$.  There is no reason to believe that either $\\alpha$ or the\nstress prescription should be the same at all radii in the disk.\n\nIt may be that the classical stress prescription of equation (\\ref{taurphi})\nis not valid in radiation pressure dominated disks.  Alternative stress\nprescriptions have long been considered, partly\nbecause they can produce thermally and viscously stable disks (e.g. Piran\n1978).\nIn addition, there have been proposals that magnetohydrodynamical\nturbulence may produce stresses that are limited to values that are related\nin some way to the gas pressure (e.g. Sakimoto \\& Coroniti 1981, Merloni 2003).\n\nIt is noteworthy that\nLMC X-3 reaches the highest Eddington ratios among the sources fit here, and that\nthe spectrum even appears to {\\it soften} slightly at these highest luminosities.\nA substantial reduction in stress at $\\sim 0.6-0.7$ might explain this, perhaps linked\nto the onset of a disk instability, which appears to be present in the only other\nsource to consistently exceed such luminosities, GRS 1915+105.  Fits with BHSPEC\nto GRS 1915+105 show that this source has `stable' disk spectra (i.e. constant\nfor longer than 16 sec) from $\\ell=0.5-0.6$ and from $\\ell=1-2$ which are consistent with\nthe expected hardening with luminosity (Middleton et al. 2006).\nHowever, there are {\\it no} stable disk spectra from this source in the range\n$\\ell \\sim 0.7-0.9$, exactly where the LMC X-3 spectra show slightly different\nproperties than expected.\n\nOther effects might be important at the high luminosities.\nThe disk models used in BHSPEC are actually somewhat inconsistent for\n$\\ell \\gtrsim0.3$, as the innermost annuli then have\n$H\/R\\gtrsim0.1$.  Radial transport of accretion power is increasingly\nimportant in this regime.  In addition, magnetic torques across the ISCO\nmay be more important (Afshordi \\& Paczy\\'nski 2003).  However,\nboth of these effects would tend to make the inner annuli hotter and\/or\nmore effectively optically thin. We would expect this to {\\it increase}\nthe hardening, in contrast to what is observed.  As the Eddington ratio\nincreases, increased inhomogeneities in the magnetorotational turbulence \n(Turner et al. 2002) and photon bubbles (Begelman 2001) may mitigate this by producing a\nsofter spectrum and a geometrically thinner disk than would be expected in a homogeneous model.\n\n\n\\begin{figure}\n\\includegraphics[width=0.46\\textwidth]{f5.eps}\n\\caption{The unfolded spectra for the {\\it RXTE}\nobservations of LMC X-3 using the best fit BHSPEC models for $i=67^{\\circ}$, \n$D=52$ kpc,and $\\alpha=0.01$.  The total model component (green, solid curve),\nBHSPEC (red, long-dashed curve) and COMPTT (violet, short-dashed curve) are\nplotted.\n\\label{fig5}}\n\\end{figure}\n\n\\subsection{Estimates for Black Hole Spins}\n\nThe total mass accreted over the lifetime of these sources expected to be\nsmall (King \\& Kolb 1999; Shafee et al. 2006). As a result, only a\nsmall increase in the angular momentum of the black hole is expected, and so our \nspin measurements are likely probing the natal spin \ndistribution of the binaries. Therefore, these low to moderate spin estimates\n($a_\\ast \\lesssim 0.8$) place constraints on black hole formation scenarios.  They\nmay also constrain spin-dependent models for jet production (Middleton et al. 2006)\nsince both J1550 and J1655 are microquasars. We characterize these spins\nas `moderate' because even for $a_{\\ast}\\sim 0.8$,\nthe proximity of the ISCO to the event horizon\nand the resulting radiative efficiency ($R\\sim 3 R_g$, $\\eta\\simeq 0.12$) are \nsubstantially less extreme than in the maximally spinning spacetime \n($a_{\\ast}\\sim 0.998$, $R\\sim 1.24 R_g$, $\\eta\\simeq 0.32$).  These moderate spins\nare in contrast to estimates for spin at or near $a_\\ast=0.998$ that have been\ninferred from other methods, including spectral fits to the Fe K$\\alpha$ line and \nsome interpretations of the high frequency quasi-periodic oscillations (QPOs).\n\nBroad Fe K$\\alpha$ line emission has been seen in {\\it ASCA} observations of\nJ1550 and J1655 (Miller et al. 2004).  The emission was modeled with a\nrelativistic disk line profile calculated in a maximally spinning Kerr spacetime\n(LAOR in Xspec, Laor 1991), but with inner radius allowed to vary.  The best fit\ninner radius for J1655 was small with $R_{\\rm in} \\lesssim 2 R_g$, suggesting\n$a_\\ast > 0.9$ and possibly near maximal ($a_\\ast \\sim 0.998$).  J1550 is less \nwell constrained with $R_{\\rm in} \\lesssim 4-6 R_g$ depending on the model, though\nit may also be consistent with near maximal spin.\nThe strongest constraints on $a_\\ast$\ncome from the presence of a broad, asymmetric red wing which extends down to\n$\\sim$4 keV in the best-fit models.  Therefore, a principle source of\nuncertainty for applying this method is modeling the underlying continuum to\naccurately gauge the shape and extent of the line wing.  We refer the reader to\nDone \\& Gierlinski, (2005) for a more detailed discussion of uncertainties associated\nwith this method.\n\nThe reproducibility of the frequencies of the pair of high frequency QPOs from one\nobservation to the next suggests they might also provide a direct probe of the\nblack hole spacetime.  As a result, prospective models of QPOs often provide\nconstraints on $a_\\ast$. For example, if the lower frequency member of the pair\nin J1550 and J1655 is identified as an\naxisymmetric radial epicyclic oscillation, then black hole spins of $a_\\ast >\n0.9$ are required (Rezzolla et al. 2003, T\\\"or\\\"ok 2005).  The reason is simple:\nthe radial epicyclic frequency has a maximum at some radius, and that maximum is\nbelow the observed QPO frequency for the observed black hole masses unless the\nspin is high.  The same conclusion holds in diskoseismology models if the lower\nfrequency member is identified with a low order axisymmetric ``g-mode'', because\nit has a frequency less than the radial epicyclic frequency within the mode\ntrapping region (e.g. Wagoner et al. 2001). However, there are many possible\noscillation modes in accretion disks, and other mode identifications can be made\nwhich are more consistent with moderate spins (e.g. Blaes et al 2006a).\n\nIn principle, the high signal-to-noise {\\it RXTE} data allow us to place\nvery tight constraints on the spins of BHBs (see e.g. Table 3).  However, the\nrelatively small uncertainties in Table 3 do not account for uncertainties in\n$M$, $i$, and $D$.  There are degeneracies among the fitting parameters in the way\nthat they affect the spectrum, e.g. the correlation\nbetween $i$ and $a_\\ast$ seen in Figure \\ref{fig2}. Therefore, when $i$ and $D$ are\nfree parameters, $a_\\ast$ can \nchange significantly.  For the {\\it RXTE} data, the uncertainty ranges for \n$a_\\ast$ increase when $i$ and $D$ are free, but still remain relatively small.\nThis also leads to rather small formal uncertainties for $i$ and $D$, though \n$D$ is at the limit of allowed range for LMC X-3 and J1655.\nSo these data are capable of constraining all the parameters simultaneously, \nbecause changes of only a few percent in the spectral shape in the high energy\ntail of the spectrum lead to substantial \nchanges in $\\chi^2$.  However, we caution that the models themselves are uncertain\nat the few percent level due to our interpolation method alone (Davis \\& Hubeny 2006).\nTherefore, this method of spin estimation can\nonly be used with good precision when reliable and  precise constraints on \n$M$, $i$, and $D$ are available.\n\nAll of these methods of spin estimation have their weaknesses, as all rely to \nvarying degrees on uncertain physical assumptions. An important example relevant\nfor both Fe K$\\alpha$ line and our continuum spectral fits is the assumption\nthat the disk emission extends to the ISCO, and effectively ceases interior to\nthis radius. In principle, the disk (or its emission) could be truncated at larger\nradius and our spins would be underestimates.  Alternatively, significant\nemission might be generated or reprocessed inside the ISCO (Krolik \\& Hawley 2002), \nmaking the interpretation of both estimation methods more difficult.\n\nSeveral uncertainties also remain in other physical assumptions which underly\nthe BHSPEC model, because we still lack a complete\nunderstanding of the magnetohydrodynamical structure of the accretion flows.\nThere are several modifications which likely lead to a hardening of the spectra,\nincluding\nincreased dissipation near the disk surface (Davis et al. 2005), an increase\nin the density scale height due to magnetic pressure support (Blaes et al. 2006b), \nsurface irradiation by the non-thermal emission, and torques on the\ninner edge of the disk.  Other processes might lead to a softening of the spectra.\nOpacity due to bound-bound transitions of metal ions, which is not included in\nBHSPEC should increase\nthe ratio of absorption to electron scattering opacity, pushing the\nspectrum closer to blackbody. Inhomogeneities, such as those caused by \ncompressible magnetohydrodynamical turbulence (Turner et al. 2002) or the photon bubble\ninstability (Turner et al. 2005), may also soften the spectrum.  They may increase the\neffective ratio of \nabsorption to scattering opacity because photon-matter interactions are\ndominated by the densest regions (Davis et al. 2004), or through a reduction \nin the density scale height (see e.g. Begelman 2001) leading to an increase in\nthe average density of the disk interior.  Given all these possibilities, it is \ndifficult to say with certainty what net effect modifying our assumptions or\nincluding these additional processes would have on the disk spectra. If the effects\nwhich harden the spectra are more important, BHSPEC \nwill underestimate the spectral hardening and would then require \nhigher spins to fit the data, making our spin measurements overestimates\nand vice-versa.\n\nGiven these uncertainties, it is conceivable that one could reconcile the\nspectral constraints with $a_\\ast \\sim 0.9$ if the BHSPEC model overestimates the\nactual spectral hardening.  Spins this high would bring our estimates in\nline with the published uncertainties for the Fe K$\\alpha$ estimates and some\nof the QPO-based measurements. Reconciling our spectral models\nwith the extreme spin ($a_\\ast \\sim 0.998$) is more difficult.  We have\nattempted to quantify this in the case of J1655 by fitting the data with KERRBB for\n$a_\\ast=0.998$, but with $f$ as a free parameter.  For $i=70^{\\circ}$ or $85^{\\circ}$, \nan adequate fit can be obtained only for $f \\sim 1$ which corresponds to blackbody \nradiation.  Electron scattering opacity dominates at the relevant temperatures so \nnearly blackbody emission is highly unlikely. One could obtain agreement\nwith more reasonable color corrections ($f \\sim 1.5$)\nby allowing $i$ to be a free parameter.  However, this requires $i\\lesssim 40^{\\circ}$,\nin disagreement with the inclinations from both the binary and jet observations.\n\n\n\\subsection{Uncertainties due to Hard X-ray Emission}\n\\label{hard}\n\nA potential difficulty for deriving constraints on $\\alpha$ or $a_\\ast$ by\nthis method is the\nneed to account for the non-thermal emission.  This is particularly true with {\\it RXTE}\ndata when only the high energy tail of the spectrum is typically observed. \nEven though we infer the  models' bolometric\nfluxes to be dominated by the softer thermal component, the\nnon-thermal component accounts for a substantial fraction of the 3-20 keV\nemission in many of the cases.  The decomposition of thermal and non-thermal\nemission is clearly dependent on the choice of model for the thermal emission.  \nIt can be seen in Figure \\ref{fig4} that the $T_{\\rm max}$ values derived from the\nbest-fit BHSPEC models for J1550 and J1655 are softer (lower $T_{\\rm max}$) at\nfixed $L_{\\rm disk}\/L_{\\rm Edd}$ than those derived by GD04 who fit DISKBB directly \nto the data.  This means\nthat emission which was being accounted for by the DISKBB model is partly being\naccounted for by the non-thermal emission in the BHSPEC fits.  The fits also\ndepends on the choice of model for the non-thermal component.  The COMPTT model\nassumes a single Wien spectrum for the soft photon input.  A model with a\nmultitemperature disk spectrum for the seed photon input (THCOMP, Zdziarski et al. 1996)\nprovides more low energy photons for a given temperature.  Therefore, a THCOMP\nspectrum which matches COMPTT at higher photon energies will tend to have more\nflux at lower energies.  We find that the quality of fit can change significantly\n(e.g. DISKBB is no longer preferred to BHSPEC for J1550 at fixed $i$) if we\nreplace COMPTT with THCOMP, though the best-fit spins and preferred $\\alpha$ values\nseem more robust.  The LMC X-3 observations are much more disk dominated \nthan those of J1550 and J1655, and are much less sensitive\nto the choice of model for the non-thermal emission.\n\nThe effects of the non-thermal emission can be minimized if we\nobserve these sources with detectors sensitive to lower photon energies, as  most\nof the spectra peak below the lower limit of the {\\it RXTE}\nband. Thus, the {\\it RXTE} fits can be sensitive to changes of only a few percent\nin both the non-thermal emission model and the shape of the high energy tail of \nthe BHSPEC models. The shape of the high energy tail of the BHSPEC spectra is uncertain at\nthe several percent level due to our interpolation scheme (Davis \\& Hubeny 2006).\nLine-of-sight absorption at softer-photon energies will\nput practical limits on such modeling, and makes sources located\nout of the Galactic plane (such as LMC X-3) particularly well suited for this\ntype of study.\n\n\\section{Conclusions}\n\\label{conc}\n\nWe analyze disk-dominated spectra of three black hole binaries: LMC X-3,\nXTE J1550-564, and GRO J1655-40, fitting them with a simple multitemperature\nblackbody model (DISKBB), as well as sophisticated relativistic disk models\n(KERRBB and BHSPEC).  For LMC X-3, this includes {\\it BeppoSAX} data, which\ncover the 0.1-10 keV energy band in which the majority of the bolometric\nflux of the disk is emitted.  In this case we find a statistically significant\npreference for the relativistic models over DISKBB.  At lower significance,\nwe also find a preference for BHSPEC over KERRBB which may suggest the\nspectra are sensitive to atomic and radiative transfer physics which are calculated\nexplicitly in BHSPEC.\n\nWe also examine {\\it RXTE} spectra for each of the three BHBs using simultaneous,\nmulti-epoch fits to each source.  When we fix the relativistic model at the \nindependent estimates for the binary inclination in Table \\ref{tbl1}, we find DISKBB\nis preferred over both relativistic models for both J1550 and J1655, though $\\chi^2_\\nu$ is\nstill acceptable in most of the relativistic model fits. If we allow the inclination \nto be a free parameter, BHSPEC is the best-fit model for both sources.  The best fit \ninclinations are both consistent with constraints inferred from radio observations\nof the jets in these sources which might be accounted for by a misalignment of \nthe black hole spin with the binary orbital angular momentum.  BHSPEC is also\nthe best fit model for LMC X-3, if we ignore the highest luminosity epoch\nwhere additional physics appears to be important.  The best-fit inclination is marginally\nconsistent with the constraints on the binary inclination in this source.\n\nUsing the binary inclination estimates in Table \\ref{tbl1}, we are able to derive\nprecise estimates for the black hole spin. However, the inferred values of the spin \nare functions of inclination, and the spin changes if the inclination of the X-ray\nemitting, inner disk annuli differs significantly from independent estimates.\nThe best fit spin is also sensitive to the\nsource distance and absolute flux calibration through the model normalization.\nAccurate and precise estimates for all these parameters as well as the black\nhole mass are therefore a prerequisite for accurate and precise spin estimates.\nWe find relatively moderate spins ($a_\\ast \\lesssim 0.8$), even when inclination\nis a free parameter in our fits. For J1655, our maximum spin estimate is only slightly \nlower than the limits ($a_{\\ast} \\gtrsim 0.9$ at 90\\% confidence) implied by fits to \nFe K$\\alpha$ lines \n(Miller et al. 2004) and certain models of high frequency QPOs (\ne.g. Wagoner et al. 2001; Rezzolla et al. 2003; T\\\"or\\\"ok 2005).  The spin of J1550\nis more weakly constrained by the Fe K$\\alpha$ fits (Miller et al. 2004), and is \nconsistent with our estimates. However, for both sources the best fit Fe K$\\alpha$\nmodels are also consistent with nearly `maximal' spins ($a_{\\ast}\\sim 0.998$) which \nwould be difficult to reconcile if the BHSPEC models provide an accurate approximation\nto the spectra of the accretion flows in these sources.\n\nWe also find that our fits are sensitive to the assumed form of the angular momentum \ntransport through its effects on the disk surface density. We consider an $\\alpha$ stress \nprescription with $\\alpha=0.1$ and 0.01, finding approximate qualitative\nagreement between the $L-T$ diagrams and the model predictions (see figure\n\\ref{fig4}).  We find that all three BHBs are consistent with a single value\nof $\\alpha$, preferring $\\alpha=0.01$ to $\\alpha=0.1$\n(though only weakly in the case of J1655).  These results are in contrast to the \nstandard disk instability model of soft X-ray transients (see e.g. Dubus et al. 2001)\nwhich requires $\\alpha \\gtrsim 0.1$ in the outer disk.\nIf we include the most luminous epoch in our fits to the \n{\\it RXTE} data of LMC X-3, models with constant $f$ provide a better\nfit than BHSPEC with either $\\alpha$. BHSPEC fails to account for the most\nluminous epoch because it predicts continued\nspectral hardening at the highest Eddington ratios while the observed spectra appear \nto {\\it soften}. This suggests the onset of additional physics which \nsoftens the spectrum as the Eddington ratio nears unity.\n\n\\acknowledgements{We gratefully acknowledge Ivan Hubeny for his instrumental contributions\nto the development of BHSPEC.  We thank Eric Agol for useful discussions,\nand for making KERRTRANS publicly available.  We also thank Marek Gierlinski,\nJulian Krolik, Ramesh Narayan, Rebecca Shafee, and Aristotle Socrates for useful\ndiscussions and\/or assistance. We are grateful for comments on the manuscript from the \nanonymous referee and Jeff McClintock which lead to significant improvements.  Part of this\nwork was completed while the authors were hosted by the Kavli Institute for Theoretical \nPhysics at UCSB. This work was supported by NASA grant NAG5-13228, and by the National Science \nFoundation under grant PHY99-07949.\n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\input{sections\/Introduction}\n\n\n\\section{Related Work}\n\\input{sections\/Related_Work}\n\n\n\\section{\\textsc{DProQ} Pipeline}\n\\input{sections\/DProQ_Pipeline}\n\n\n\\section{Experiments}\n\\input{sections\/Experiments}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{sections\/Conclusion}\n\n\n\\section*{Acknowledgments}\nThe project is partially supported by two NSF grants (DBI 1759934 and IIS 1763246), one NIH grant (GM093123), three DOE grants (DE-SC0020400, DE-AR0001213, and DE-SC0021303), and the computing allocation on the Summit compute cluster provided by Oak Ridge Leadership Computing Facility (Contract No. DE-AC05-00OR22725).\n\n\n\n{\n\\small\n\\printbibliography\n}\n\n\n\n\n\n\\subsection{Data}\n\\input{sections\/Datasets}\n\n\\subsection{Evaluation Setup}\n\\textbf{Baselines.} We compare our method with the state-of-the-art method GNN\\_DOVE, an atom-level graph attention-based method for protein complex structure evaluation. Uniquely, it extracts protein interface areas to build its input graphs. Further, the chemical properties of its atoms as well as inter-atom distances are treated as initial node and edge features, respectively. We note that GNN\\_DOVE is the sole baseline we include in this study, as for all other DL-based methods for structural QA, we were unable to locate reproducible training and inference code for such methods. Furthermore, since GNN\\_DOVE has previously been evaluated against several other classical machine learning methods for structural QA and has demonstrated strong performance against such methods, we argue that comparing new DL methods' performance to GNN\\_DOVE can serve as a strong indicator of the DL state-of-the-art for the field.\n\n\\textbf{\\textsc{DProQ} Models.} Besides including results for the standard \\textsc{DProQ} model as well as for GNN\\_DOVE, we also report results on the HAF2 and DBM55-AF2 datasets for a selection of \\textsc{DProQ} variants curated in this study. The \\textsc{DProQ} variants we chose to investigate in this work include \\textsc{DProQ\\_GT} which employs the original Graph Transformer architecture \\cite{dwivedi2020generalization}; \\textsc{DProQ\\_GTE} which employs the GGT with only its edge gate enabled; and \\textsc{DProQ\\_GTN} which employs the GGT with only its node gate enabled.\\par\n\n\\textbf{Evaluation metrics.} We evaluate structural QA performance according to two main criteria. Our first criterion is general ranking ability which measures how many qualified decoys are found within a model's predicted Top-$N$ structure ranking. Within this framework, a model's hit rate is defined as the fraction of protein complex targets for which the model, within each of its Top-$N$ ranks, ranked at least one Acceptable or higher-quality decoy based on the CAPRI scoring criteria \\cite{lensink2014score_set}. In this work, we report results based on models' Top-10 hit rates. A hit rate is represented by three numbers separated by the character \/. These three numbers, in order, represent how many decoys with Acceptable or higher-quality, Medium or higher-quality, and High quality were among the Top-N ranked decoys. Our second criterion with which to score models is their Top-$1$ selected model ranking ability. Within this context, we calculate the ranking loss for each method. Here, per-target ranking loss is defined as the difference between the DockQ score of a target's native structure and the DockQ score of the top decoy predicted by each ranking method. As such, a lower ranking loss indicates a stronger model for ranking pools of structural decoys for downstream tasks.\n\n\\textbf{Implementation Details.} We train all our models using the AdamW optimizer \\cite{loshchilov2017decoupled} and perform early stopping with a patience of $15$ epochs. All remaining hyperparameters and graph features we used are described further in Appendix \\ref{sec:appendix_c}. Source code to reproduce our results and perform fast structure quality assessment using our model weights can be found at \\href{https:\/\/github.com\/BioinfoMachineLearning\/DProQ}{\\texttt{https:\/\/github.com\/BioinfoMachineLearning\/DProQ}}.\n\n\\subsection{Results}\n\\textbf{Blind Structure Quality Assessment on the HAF2 Dataset.} Table \\ref{tab:haf2_hits} reports the hit rate performances of \\textsc{DProQ}, \\textsc{DProQ} variants, and GNN\\_DOVE, respectively, on the HAF2 Dataset. On $13$ HAF2 targets, \\textsc{DProQ} hits (i.e., identifies at least one suitable decoy structure for) $10$ targets on the Acceptable quality level, $9$ targets on the Medium quality level, and $4$ targets on High quality level. For targets with High quality structures, \\textsc{DProQ} and its variants successfully hit all of them. On the HAF2 dataset, GNN\\_DOVE gets a $8\/7\/3$ hit rate on Acceptable, Medium, and High quality targets respectively. However, \\textsc{DProQ} outperforms GNN\\_DOVE on all targets except \\textsc{7NKZ}. On this target, GNN\\_DOVE achieves a better hit rate on High quality decoy structures. Overall, \\textsc{DProQ} shows a better performance than \\textsc{DProQ\\_GT}, \\textsc{DProQ\\_GTE}, and \\textsc{DProQ\\_GTN}. Additionally, for target \\textsc{7MRW}, only \\textsc{DProQ} successfully hits the $5$ Acceptable quality targets and $4$ Medium quality targets, whereas all other methods failed to hit any qualified decoy structures for this protein target.\n\nTable \\ref{tab:haf2_ranking} reports methods' ranking loss performance on each HAF2 target. \\textsc{DProQ} and its $3$ variants outperform GNN\\_DOVE which displays the highest average ranking loss. In particular, \\textsc{DProQ} achieves the lowest average ranking loss of $0.195$ compared to all other methods. Here, \\textsc{DProQ}'s loss is $43\\%$ lower than GNN\\_DOVE's $0.343$ average ranking loss. Moreover, \\textsc{DProQ} achieves the lowest loss on $5$ targets, namely \\textsc{7AWV}, \\textsc{7OEL}, \\textsc{7NRW}, \\textsc{7NKZ} and \\textsc{7O27}. Notably, \\textsc{DProQ\\_GT}, \\textsc{DProQ\\_GTE}, and \\textsc{DProQ\\_GTN}'s ranking losses here are $29\\%$, $34\\%$, and $30\\%$ lower, respectively, than GNN\\_DOVE's average ranking loss.\n\n\\begin{table}\n\\caption{Hit rate performance on the HAF2 dataset. The \\textsc{Best} column represents each target's best-possible Top-$10$ result. The \\textsc{Summary} row lists the results when all targets are taken into consideration.}\n\\label{tab:haf2_hits}\n\\centering\n\\begin{tabular}{lllllll}\n\\toprule\nID      & \\textsc{DProQ}    & \\textsc{DProQ}\\_GT & \\textsc{DProQ}\\_GTE & \\textsc{DProQ}\\_GTN  & GNN\\_DOVE & \\textsc{BEST} \\\\ \n\\midrule\n7AOH    & 10\/10\/10 & 10\/10\/10  & 10\/10\/10   & 10\/10\/10        & 9\/9\/0     & 10\/10\/10   \\\\\n7D7F    & 0\/0\/0    & 0\/0\/0     & 0\/0\/0      & 0\/0\/0           & 0\/0\/0     & 5\/0\/0      \\\\ \n7AMV    & 10\/10\/10 & 10\/10\/10  & 10\/10\/10   & 10\/10\/10        & 10\/10\/6   & 10\/10\/10   \\\\ \n7OEL    & 10\/10\/0  & 10\/9\/0    & 10\/10\/0    & 10\/10\/0         & 10\/10\/0   & 10\/10\/0    \\\\ \n7O28    & 10\/10\/0  & 10\/10\/0   & 10\/10\/0    & 10\/10\/0         & 10\/10\/0   & 10\/10\/0    \\\\ \n7ALA    & 0\/0\/0    & 0\/0\/0     & 0\/0\/0      & 0\/0\/0           & 0\/0\/0     & 1\/0\/0      \\\\ \n7MRW    & 5\/4\/0    & 0\/0\/0     & 0\/0\/0      & 0\/0\/0           & 0\/0\/0     & 10\/10\/0    \\\\ \n7OZN    & 0\/0\/0    & 0\/0\/0     & 0\/0\/0      & 0\/0\/0           & 0\/0\/0     & 10\/2\/0     \\\\ \n7D3Y    & 2\/0\/0    & 5\/0\/0     & 6\/0\/0      & 8\/0\/0           & 0\/0\/0     & 10\/0\/0     \\\\ \n7NKZ    & 10\/10\/2  & 10\/10\/1   & 10\/10\/1    & 10\/010\/4        & 10\/9\/9    & 10\/10\/10   \\\\ \n7LXT    & 1\/1\/0    & 0\/0\/0     & 0\/0\/0      & 0\/0\/0           & 1\/0\/0     & 10\/10\/0    \\\\ \n7KBR    & 10\/10\/10 & 10\/10\/10  & 10\/10\/10   & 10\/10\/10        & 10\/10\/9   & 10\/10\/10   \\\\ \n7O27    & 10\/10\/0  & 10\/10\/0   & 10\/10\/0    & 10\/10\/0         & 10\/4\/0    & 10\/10\/0    \\\\ \n\\midrule\n\\textsc{Summary} & \\textbf{10\/9\/4} & 8\/7\/4 & 8\/7\/4   & 8\/7\/4           & 8\/7\/3     & 13\/10\/4    \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Ranking loss performance on the HAF2 dataset. The \\textsc{BEST} row represents the mean and standard deviation of the ranking losses for all targets.}\n\\label{tab:haf2_ranking}\n\\centering\n\\begin{tabular}{llllll}\n\\toprule\nTarget  & \\textsc{DProQ}         & DProQ\\_GT     & \\textsc{DProQ}\\_GTE             & \\textsc{DProQ}\\_GTN    & GNN\\_DOVE     \\\\\n\\midrule\n7AOH    & 0.066         & 0.026         & 0.026                  & 0.058         & 0.928         \\\\\n7D7F    & 0.471         & 0.471         & 0.47                   & 0.471         & 0.003         \\\\\n7AMV    & 0.01          & 0.021         & 0.017                  & 0.019         & 0.342         \\\\\n7OEL    & 0.062         & 0.063         & 0.135                  & 0.135         & 0.21          \\\\\n7O28    & 0.029         & 0.021         & 0.027                  & 0.034         & 0.244         \\\\\n7ALA    & 0.232         & 0.226         & 0.226                  & 0.226         & 0.226         \\\\\n7MRW    & 0.085         & 0.603         & 0.555                  & 0.555         & 0.598         \\\\\n7OZN    & 0.409         & 0.409         & 0.49                   & 0.281         & 0.457         \\\\\n7D3Y    & 0.326         & 0.33          & 0.012                  & 0.326         & 0.295         \\\\\n7NKZ    & 0.164         & 0.175         & 0.175                  & 0.164         & 0.459         \\\\\n7LXT    & 0.586         & 0.586         & 0.586                  & 0.586         & 0.295         \\\\\n7KBR    & 0.068         & 0.152         & 0.152                  & 0.17          & 0.068         \\\\\n7O27    & 0.03          & 0.079         & 0.079                  & 0.079         & 0.334         \\\\\n\\midrule\n\\textsc{BEST} & \\textbf{0.195 \u00b1 0.185} & 0.243 \u00b1 0.206 & 0.227 \u00b1 0.21 & 0.239 \u00b1 0.187 & 0.343 \u00b1 0.228   \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\textbf{Blind Structure Quality Assessment on the DBM55-AF2 Dataset.} Table \\ref{tab:dbm_hits} summarizes all methods' hit rate results on the DBM55-AF2 dataset, which contains $15$ targets. Here, we see that \\textsc{DProQ} achieves the best hit rate of $12\/10\/3$ compared to all other methods. Additionally, \\textsc{DProQ} successfully selects all $10$ targets with Medium quality decoys and all $3$ targets with High quality decoys, moreover hitting all three accuracy levels' decoys on $6$ targets which is the best result achieved compared to other methods. On this dataset, GNN\\_DOVE achieves a $10\/4\/1$ hit rate, worse than all other methods. Nonetheless, GNN\\_DOVE successfully hits two targets, \\textsc{5WK3} and \\textsc{5KOV}.\n\nTable \\ref{tab:dbm_ranking} presents the ranking loss for all methods on DBM55-AF2 dataset. Here, \\textsc{DProQ} achieves the best ranking loss of $0.049$ which is $87\\%$ lower than GNN\\_DOVE's ranking loss of $0.379$. Furthermore, for $4$ targets, \\textsc{DProQ} correctly selects the Top-$1$ model and achieves $0$ ranking loss. Similarly to their results on the HAF2 dataset, \\textsc{DProQ\\_GT}, \\textsc{DProQ\\_GTE}, and \\textsc{DProQ\\_GTN}'s losses are $71\\%$, $84\\%$, and $74\\%$ lower, respectively, than GNN\\_DOVE's ranking loss.\n\n\\begin{table}\n\\label{tab:dbm_hits}\n\\centering\n\\caption{Hit rate performance on DBM55-AF2 dataset. The \\textsc{BEST} column represents each target's best-possible Top-$10$ result. The \\textsc{Summary} row lists the results when all targets are taken into consideration.}\n\\begin{tabular}{lllllll}\n\\toprule\nTarget   & \\textsc{DProQ} & \\textsc{DProQ}\\_GT & \\textsc{DProQ}\\_GTE & \\textsc{DProQ}\\_GTN & GNN\\_DOVE & \\textsc{BEST}   \\\\ \n\\midrule\n6AL0     & 9\/2\/0          & 10\/0\/0             & 10\/0\/0              & 10\/2\/0              & 6\/0\/0     & 10\/2\/0  \\\\\n3SE8     & 8\/8\/0          & 9\/9\/0              & 8\/8\/0               & 8\/8\/0               & 3\/0\/0     & 10\/10\/0 \\\\\n5GRJ     & 10\/10\/0        & 9\/9\/0              & 10\/10\/0             & 9\/9\/0               & 3\/2\/0     & 10\/10\/0 \\\\\n6A77     & 7\/7\/0          & 7\/7\/0              & 8\/8\/0               & 8\/8\/0               & 0\/0\/0     & 8\/8\/0   \\\\\n4M5Z     & 10\/10\/1        & 10\/10\/0            & 10\/10\/0             & 10\/10\/0             & 10\/10\/0   & 10\/10\/1 \\\\\n4ETQ     & 1\/1\/0          & 1\/1\/0              & 1\/1\/0               & 1\/1\/0               & 0\/0\/0     & 1\/1\/0   \\\\\n5CBA     & 10\/10\/1        & 10\/10\/0            & 10\/10\/0             & 10\/10\/1             & 10\/10\/3   & 10\/10\/6 \\\\\n5WK3     & 0\/0\/0          & 0\/0\/0              & 0\/0\/0               & 0\/0\/0               & 1\/0\/0     & 3\/0\/0   \\\\\n5Y9J     & 4\/0\/0          & 6\/0\/0              & 5\/0\/0               & 4\/0\/0               & 0\/0\/0     & 8\/0\/0   \\\\\n6BOS     & 10\/10\/0        & 10\/10\/0            & 10\/10\/0             & 10\/10\/0             & 10\/10\/0   & 10\/10\/0 \\\\\n5HGG     & 8\/0\/0          & 8\/0\/0              & 8\/0\/0               & 8\/0\/0               & 8\/0\/0     & 10\/0\/0  \\\\\n6A0Z     & 0\/0\/0          & 0\/0\/0              & 0\/0\/0               & 0\/0\/0               & 2\/0\/0     & 3\/0\/0   \\\\\n3U7Y     & 2\/2\/1          & 2\/2\/1              & 2\/2\/1               & 2\/1\/0               & 2\/2\/1     & 2\/2\/1   \\\\\n3WD5     & 10\/8\/0         & 9\/8\/0              & 9\/8\/0               & 9\/8\/0               & 0\/0\/0     & 10\/10\/0 \\\\\n5KOV     & 0\/0\/0          & 0\/0\/0              & 0\/0\/0               & 0\/0\/0               & 1\/0\/0     & 2\/0\/0   \\\\\n\\midrule\n\\textsc{Summary}  & \\textbf{12\/10\/3} & 12\/9\/1           & 12\/9\/1              & 12\/10\/1             & 10\/4\/1    & 15\/10\/3 \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\label{tab:dbm_ranking}\n\\centering\n\\caption{Ranking loss performance on the DBM55-AF2 dataset. The \\textsc{BEST} row represents the mean and standard deviation of the ranking losses for all targets.}\n\\begin{tabular}{llllll}\n\\toprule\nTarget  & \\textsc{DProQ}         & \\textsc{DProQ}\\_GT     & \\textsc{DProQ}\\_GTE             & \\textsc{DProQ}\\_GTN    & GNN\\_DOVE     \\\\\n\\midrule\n6AL0    & 0.0           & 0.156         & 0.156                  & 0.0           & 0.424         \\\\\n3SE8    & 0.079         & 0.041         & 0.041                  & 0.079         & 0.735         \\\\\n5GRJ    & 0.024         & 0.012         & 0.095                  & 0.012         & 0.776         \\\\\n6A77    & 0.037         & 0.062         & 0.0                    & 0.037         & 0.591         \\\\\n4M5Z    & 0.015         & 0.026         & 0.026                  & 0.015         & 0.221         \\\\\n4ETQ    & 0.0           & 0.76          & 0.0                    & 0.748         & 0.759         \\\\\n5CBA    & 0.052         & 0.038         & 0.052                  & 0.058         & 0.019         \\\\\n5WK3    & 0.114         & 0.114         & 0.114                  & 0.186         & 0.087         \\\\\n5Y9J    & 0.0           & 0.0           & 0.0                    & 0.0           & 0.382         \\\\\n6BOS    & 0.081         & 0.081         & 0.0                    & 0.0           & 0.081         \\\\\n5HGG    & 0.051         & 0.051         & 0.121                  & 0.051         & 0.121         \\\\\n6A0Z    & 0.207         & 0.207         & 0.207                  & 0.207         & 0.062         \\\\\n3U7Y    & 0.0           & 0.021         & 0.0                    & 0.0           & 0.756         \\\\\n3WD5    & 0.011         & 0.011         & 0.011                  & 0.0           & 0.672         \\\\\n5KOV    & 0.065         & 0.08          & 0.085                  & 0.087         & 0.0           \\\\\n\\midrule\n\\textsc{BEST} & \\textbf{0.049 \u00b1 0.054} & 0.111 \u00b1 0.182 & 0.061 \u00b1 0.064 & 0.099 \u00b1 0.185 & 0.379 \u00b1 0.298    \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\textbf{Node and Edge Gates.} In Tables \\ref{tab:haf2_hits}, \\ref{tab:haf2_ranking}, \\ref{tab:dbm_hits}, and \\ref{tab:dbm_ranking}, we evaluated \\textsc{DProQ\\_GT}, \\textsc{DProQ\\_GTE}, and \\textsc{DProQ\\_GTN}'s performance on our two test datasets to discover the node and edge gates' effects on performance. On the HAF2 dataset, our results indicate that exclusively adding either a node or edge gate can help improve a model's Top-$1$ structure ranking ability.\n\nOur gate-ablation results yielded another interesting phenomenon when we observed that \\textsc{DProQ\\_GTE} consistently achieved the best ranking loss on both of our test datasets, compared to \\textsc{DProQ\\_GTN}. Notably, on the DBM55-AF2 dataset, \\textsc{DProQ\\_GTE}'s loss is much lower than that of other methods, except for \\textsc{DProQ}. This suggests \\textsc{DProQ\\_GTE} is more sensitive to Top-$1$ structure selection compared to \\textsc{DProQ\\_GTN}. We hypothesize this trend may be because, within \\textsc{DProQ}, k-NN graphs' edges may not always possess a biological interpretation during graph message passing. In such cases, models' edge gates may allow the network to inhibit the weight of biologically-implausible edges, which could remove noise from the neural network's training procedure. We plan to investigate extensions of the GGT to explore these ideas further.\n\n\\textbf{Repeated Experiments.} In addition, we trained four more \\textsc{DProQ} models within the original \\textsc{DProQ} training environment but with different random seeds. We see in Appendix \\ref{sec:appendix_a} that \\textsc{DProQ} demonstrates consistently-better results compared to GNN\\_DOVE. We observe \\textsc{DProQ} providing reliable, state-of-the-art performance in terms of both hit rate and ranking loss on both test datasets.\\par\n\n\n\n\n\\textbf{Limitations.} One limitation of the \\textsc{GGT} is that its architecture does not directly operate using protein spatial information. Models such as the SE(3)-Transformer \\cite{fuchs2020se} may be one way to explore this spatial information gap. However, due to such models' high computational costs, we are currently unable to utilize such techniques within the context of protein complex structure modeling. Moreover, in our studies, \\textsc{DProQ} failed on some test targets that contain few Acceptable or higher-quality decoys within the Top-$10$ range. To overcome this problem, we experimented with multi-tasking within \\textsc{DProQ}, but we did not find such multi-tasking to help our models generalize better. We hypothesize that new, diverse datasets may be helpful in this regard. We defer such ideas to future work.\n\n\\subsection{Gated Graph Transformer architecture}\n\\label{sec:gate_gt}\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{images\/gated_graph_transformer.png}\n  \\caption{\\textsc{GGT} model architecture.}\n  \\label{fig:gate_gt}\n\\end{figure}\n\nWe designed the Gated Graph Transformer (\\textsc{GGT}) as a solution to a specific phenomenon in graph representation learning. Note that, unlike other GNN-based structure scoring methods, \\cite{wang2021protein, hanquality} define edges using a fixed distance threshold, which means that each graph node may have a different number of incoming and outgoing edges. In contrast, \\textsc{DProQ} constructs and operates on k-NN graphs where all nodes are connected to the same number of neighbors. However, in the context of k-NN graphs, each neighbor's information is, by default, given equal priority during information updates. As such, we may desire to imbue our graph neural network with the ability to automatically decide the priority of different nodes and edges during the graph message passing. Consequently, we present the \\textsc{GGT}, a gated neighborhood-modulating Graph Transformer inspired by \\cite{velivckovic2017graph, dwivedi2020generalization, morehead2021geometric}. Formally, to update the network's node embeddings $\\mathbf{h}_{i}$ and edge embeddings $\\mathbf{e}_{ij}$, we define a single layer of the \\textsc{GGT} as:\n\n\\begin{equation}\n\\label{eq:w2}\n    \\widehat{\\widehat{\\mathbf{w}}}_{ij}^{k, \\ell}=\\left(\\frac{Q^{k, \\ell} \\mathbf{h}_{i}^{\\ell} \\cdot K^{k, \\ell} \\mathbf{h}_{j}^{\\ell}}{\\sqrt{d_{k}}}\\right) \\cdot E^{k, \\ell} \\mathbf{e}_{i j}^{\\ell}\n\\end{equation}\n\n\\begin{equation}\n    \\label{eq:e_update}\n    \\hat{\\mathbf{e}}_{ij}^{\\ell+1}=\\mathbf{O}_{e}^{\\ell} \\|_{k=1}^{H}\\left(\\widehat{\\widehat{\\mathbf{w}}}_{ij}^{k, \\ell}\\right)\n\\end{equation}\n\n\\begin{equation}\n    \\label{eq:w1}\n    \\widehat{\\mathbf{w}}_{ij}^{k, \\ell}=\\widehat{\\widehat{\\mathbf{w}}}_{ij}^{k, \\ell} \\times \\operatorname{sigmoid}\\left(\\mathbf{G}_{e}^{k, \\ell} \\mathbf{e}_{i j}^{\\ell}\\right)\n\\end{equation}\n\n\\begin{equation}\n    \\label{eq:w}\n    \\mathbf{w}_{ij}^{k, \\ell}=\\operatorname{softmax}\\left(\\hat{\\mathbf{w}}_{ij}^{k, \\ell}\\right)\n\\end{equation}\n\n\\begin{equation}\n    \\label{eq:h_update}\n    \\hat{\\mathbf{h}}_{i}^{\\ell+1}=O_{\\mathbf{h}}^{\\ell} \\|_{k=1}^{H}\\left(\\operatorname{sigmoid}\\left(G_{h}^{k, \\ell} \\mathbf{h}_{j}^{\\ell}\\right) \\times \\sum_{j \\in \\mathcal{N}_{i}} \\mathbf{w}_{i j}^{k, \\ell} V^{k, \\ell} \\mathbf{h}_{j}^{\\ell}\\right)\n\\end{equation}\n\nIn particular, the \\textsc{GGT} introduces to the standard Graph Transformer architecture \\cite{dwivedi2020generalization} two information gates through which the network can modulate node and edge information flow, as shown in Figure \\ref{fig:gate_gt}. Equation \\ref{eq:e_update} shows how the output of the edge gate, $\\mathbf{G}_{e}$, is used elementwise to gate the network's attention scores $\\widehat{\\widehat{\\mathbf{w}}}_{ij}$ derived from Equation \\ref{eq:w2}. Such gating simultaneously allows the network to decide how much to regulate edge information flow as well as how much to weigh the edge of each node neighbor. Similarly, the node gate, $\\mathbf{G}_{h}$, shown in Equation \\ref{eq:h_update}, allows the GGT to individually decide how to modulate information from node neighbors during node information updates. Lastly, for the GGT's Feed Forward Network, we keep the same structure as described in \\cite{dwivedi2020generalization}, providing us with new node embeddings $\\hat{\\mathbf{h}}_{i}^{L}$ and edge embeddings $\\hat{\\mathbf{e}}_{ij}^{L}$. For additional background on the GGT's network operations, we refer readers to \\cite{dwivedi2020generalization}.\n\n\\subsection{Multi-task graph property prediction}\n\\label{sec:graph_predictions}\nTo obtain graph-level predictions for each input protein complex graph, we apply a graph sum-pooling operator on $\\hat{\\mathbf{h}}_{i}^{L}$ such that $\\mathbf{p} \\in \\mathbb{R}^{d \\times 1} = \\sum_{i = 1}^{N} \\mathbf{h}_{i}^{L}$. This graph embedding $\\mathbf{p}$ is then fed as input to \\textsc{DProQ}'s first read-out module $\\varphi_{1}^{r}$, where each read-out module consists of a series of linear layers, batch normalization layers, LeakyReLU activations, and dropout layers \\cite{hinton2012improving}, respectively. That is, $\\mathbf{\\hat{p}} \\in \\mathbb{R}^{d \\times 4} = \\varphi_{1}^{r}(\\mathbf{p})$ reduces the dimensionality of $\\mathbf{p}$ to accommodate \\textsc{DProQ}'s output head designed for graph classification. Specifically, for graph classification, we apply a Softmax layer such that $\\mathbf{y} \\in \\mathbb{R}^{d \\times 4} = softmax(\\mathbf{\\hat{p}})$, where $\\mathbf{y}$ is the network's predicted DockQ structure quality class probabilities \\cite{basu2016dockq} for each input protein complex (e.g., Medium, High). Thereafter, we apply \\textsc{DProQ}'s second read-out module $\\varphi_{2}^{r}$ to $\\mathbf{\\hat{p}}$ such that $\\mathbf{q} \\in \\mathbb{R}^{1 \\times 1} = \\varphi_{2}^{r}(\\mathbf{\\hat{p}})$. Within this context, the network's scalar graph regression output $\\mathbf{q}$ represents the network's predicted DockQ score \\cite{basu2016dockq} for a given protein complex input.\n\n\\textbf{Structure quality scoring loss.} To train \\textsc{DProQ}'s graph regression head, we used the mean squared error loss $\\mathcal{L}_{R} = \\frac{1}{N} \\sum_{i = 1}^{N} \\| \\mathbf{q}_{i}' - \\mathbf{q}_{i}^{*} \\|^{2}$. Here, $\\mathbf{q}_{i}'$ is the model's predicted DockQ score for example $i$, $\\mathbf{q}_{i}^{*}$ is the ground truth DockQ score for example $i$, and $N$ represents the number of examples in a given mini-batch.\n\n\\textbf{Structure quality classification loss.} Likewise, to train \\textsc{DProQ}'s graph classification head, we used the cross-entropy loss $\\mathcal{L}_{C} = \\frac{1}{N} \\sum_{i = 1}^{N} \\left(\\mathbf{y}_{i}^{*} - \\log(\\mathbf{y}_{i}')\\right)$. Here, $\\mathbf{y}_{i}'$ is the model's predicted DockQ quality class (e.g., Acceptable) for example $i$, and $\\mathbf{y}_{i}^{*}$ is the ground truth DockQ quality class (e.g., Incorrect) for example $i$.\n\n\\textbf{Overall loss.} We define \\textsc{DProQ}'s overall loss as $\\mathcal{L} = w_{\\mathcal{L}_{C}} \\times \\mathcal{L}_{C} + w_{\\mathcal{L}_{R}} \\times \\mathcal{L}_{R}$. We note that the weights for each constituent loss (e.g., $w_{\\mathcal{L}_{R}}$) were determined either by performing a grid search or instead by using a lowest validation loss criterion for parameter selection. In Appendix \\ref{sec:appendix_c}, we describe \\textsc{DProQ}'s choice of hyperparameters in greater detail.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:Introduction}Introduction}\n\nRecently, a new state of matter, the topological insulator,\nattracts much theoretical and experimental\nattention.\\cite{kane05,bernevig06,sheng06,konig07,moore07,roy06,fu07l,fu07b,\nhsieh08,qi08,xia09,zhang09,hasan10rmp,qi10rev,schnyder08} It\nhas an electronic structure dominated by the spin-orbit\ncoupling, which is a band insulator with a well-defined gap in\nthe bulk but can host an odd number of Dirac cones protected by\ntime reversal symmetry on the\nsurface.\\cite{kane05,bernevig06,fu07l} The intrinsic spin-orbit\ncoupling makes it promising for spintronics\napplications.\\cite{qi08} The proximity induced superconducting\nstate on the topological insulator surface by a deposited\nsuperconductor was proposed to create Majorana\nfermions\\cite{wilczek09}, which may provide a new way to\nrealize the topological quantum computation.\\cite{fu0809,sau10}\nLately, the realization of a superconducting state in a typical\ntopological insulator Bi$_{2}$Se$_{3}$ by intercalating Cu\nbetween adjacent quintuple units (Cu$_{x}$Bi$_{2}$Se$_{3}$)\nmakes the system even more attractive.\\cite{hor10,wray10} The\nlarge diamagnetic response shows that the pairing is mainly of\nbulk character. In another topological insulator\nBi$_{2}$Te$_{3}$, the application of a high pressure also turns\nthe material into a superconducting\nstate.\\cite{zhang10pa,zhang10pb}\n\nSince the superconductivity is bulk and intrinsic to the\nmaterial, if zero energy surface Majorana fermion mode exists,\nit would be easier to manipulate as compared to that induced by\nproximity effect, as it would not experience the interface\nroughness or mismatch common to a junction type device.\nPossible nontrivial odd-parity pairing in\nCu$_{x}$Bi$_{2}$Se$_{3}$ is proposed and analyzed by Fu and\nBerg.\\cite{fu10} It was argued that only if a bulk gap opens\nand the bulk pairing is odd in parity, would zero energy\nAndreev bound states appear in the surface spectrum. However,\ntheir analysis was concentrated on the case when the chemical\npotential is much larger than the gap and the topological\nsurface states are already merged into the continuum conduction\nband. A similar analysis was put forward by\nSato.\\cite{sato0910}\n\nDespite the above works, a detailed theoretical analysis of\nsurface spectrum in the superconducting phase arising from a\ntopological insulator is still lacking. In particular, not much\nis reported for the situation when the chemical potential is\nonly slightly larger than the insulating gap and both\ntopological surface states (or, the surface conduction\nband\\cite{wray10}) and the continuum bulk conduction band are\npresent but separated. Since the surface spectrum is central to\nthe topological properties of a material both in the normal and\nin the superconducting state, which is also directly accessible\nby experimental techniques as ARPES\\cite{wray10} and\nSTM\\cite{alpichshev10}, it is highly desirable to make a\ndetailed study of them. This will help to understand better\nsuperconductivity in systems with nontrivial topological band\nstructure.\n\nWe focus on two questions in this paper. One is the effect of\nsuperconducting pairing on the topological surface\nstates\\cite{wray10} present in the normal state. The other is\nthe existence of surface Andreev bound states. We have noticed\nthat two different models\\cite{zhang09,wang10} are often used\nin-discriminatively in literature. They have the same normal\nstate energy spectra but may be different in the\nsuperconducting state. We thus present our results for both of\nthem. What happens to the topological surface states when\npairing is introduced in the bulk depends on the orbital\ncharacter of the topological surface state and on the bulk\npairing symmetry. Only when the continuum part of the band\nopens a full gap and the topological surface states, while\nseparated from the bulk conduction band, do not open a gap for\nan odd-parity pairing, would a gapless Andreev bound state\nappear. We find that the orbital characters of the topological\nsurface states are different for the two models. For a certain\nbulk pairing symmetry, it is possible that the topological\nsurface states of one model opens a gap while that of the other\nmodel is still intact. The existence of Andreev bound states,\nthe most important indication of nontrivial topological order\nin the superconducting phase, is thus expected to be also\nrelated to the orbital character of the topological surface\nstates. We show that the interplay between the continuum bulk\nconduction band and the topological surface states produces a\nring or a segment of zero energy states in addition to the\nAndreev bound states depending on the symmetry of bulk pairing\norder parameters.\n\n\n\n\n\\section{models and the normal state surface modes}\n\nIn the following, when we talk about topological insulators, we\nwould be mainly referring to Bi$_2$Se$_3$, which shows a well\ndefined Dirac cone structure for the topological surface\nstates.\\cite{zhang09,xia09,hsieh09} The model we consider below\ncould be easily generalized to study other topological\ninsulators like TlBiSe$_2$\\cite{yan10,kuroda10} and\nBi$_{2}$Te$_{2}$Se\\cite{ren10,xu10}.\n\nThe band structure of Cu$_{x}$Bi$_{2}$Se$_{3}$ is similar to\nthat of Bi$_{2}$Se$_{3}$, the most essential part of which\nconsists of two $p_z$ orbitals on the top and bottom Se layers\nhybridized with neighboring Bi $p_z$ orbitals, in each\nquintuple Bi$_{2}$Se$_{3}$ unit.\\cite{zhang09,fu10} In the\npresence of spin-orbit coupling, the normal state has four\ndegrees of freedom. Label the two orbitals concentrating mainly\non the top and bottom (seeing along the $-z$ direction) Se\nlayer of the various Bi$_2$Se$_3$ quintuple units as the first\nand second orbital, the basis is taken as\n$\\psi_{\\mathbf{k}}$=[$c_{1\\mathbf{k}\\uparrow}$,\n$c_{2\\mathbf{k}\\uparrow}$, $c_{1\\mathbf{k}\\downarrow}$,\n$c_{2\\mathbf{k}\\downarrow}$]$^{T}$. The models could be written\ncompactly in the following matrix\nform\\cite{zhang09,wang10,fu10}\n\\begin{equation} \\label{h3d0}\nH(\\mathbf{k})=\\epsilon_{0}(\\mathbf{k})I_{4\\times4}+\\sum\\limits_{i=0}^{3}m_{i}(\\mathbf{k})\\Gamma_{i}.\n\\end{equation}\nI$_{4\\times4}$ is the fourth order unit matrix, giving rise to\na topologically trivial shift of the energy bands and would be\nneglected in most of our following analysis. In terms of the\ntwo by two Pauli matrices $s_{i}$ ($i$=0, $\\cdots$, 3) in the\nspin subspace and $\\sigma_{i}$ ($i$=0, $\\cdots$, 3) in the\norbital subspace, the first three Dirac matrices are defined\nas\\cite{zhang09,wang10} $\\Gamma_{0}$=$s_{0}\\otimes\\sigma_{1}$,\n$\\Gamma_{1}$=$s_{1}\\otimes\\sigma_{3}$,\n$\\Gamma_{2}$=$s_{2}\\otimes\\sigma_{3}$. As regards $\\Gamma_{3}$,\nthere are presently two different choices:\n(I)$s_{0}\\otimes\\sigma_{2}$\\cite{wang10,fu10,li10} and\n(II)$s_{3}\\otimes\\sigma_{3}$\\cite{zhang09,liu10b1,liu10b2,lu10,shan10},\nwhich define the two models that would be considered in\nparallel in the following discussion. Since Bi$_{2}$Se$_{3}$ is\ninversion symmetric, parity could be used to label states. Some\npapers adopt the bonding and antibonding states of the two\norbitals defined above as the orbital\nbasis.\\cite{zhang09,li10,liu10b1,liu10b2,lu10,shan10} The\ncorresponding models could be obtained in terms of a simple\nunitary transformation performed in the orbital subspace. Note\nthat, the two different models give the same bulk band\ndispersion\n$\\epsilon_{\\pm}(\\mathbf{k})$=$\\epsilon_{0}(\\mathbf{k})\\pm\\sqrt{\\sum_{i=0}^{3}m_{i}^{2}(\\mathbf{k})}$,\nwith each of two eigen-energies two fold Kramers degenerate due\nto the time reversal symmetry and the inversion symmetry of the\nmodel. However, we will show in the following that they are\nessentially two physically distinct models.\n\n\n\nFor the coefficients $\\epsilon_{0}(\\mathbf{k})$ and\n$m_{i}(\\mathbf{k})$ ($i=$ 0, 1, 2, 3), there are different\npossible parameterizations which coincide with each other close\nto the $\\Gamma$ point.\\cite{zhang09,qi10rev,fu09,wang10}\nWithout loss of generality, we take the parameterizations of\nWang \\emph{et al.}\\cite{wang10}. Since the diagonal term\n$\\epsilon_{0}(\\mathbf{k})$ proportional to the unit matrix does\nnot affect the topological characters, it is ignored here. The\nsystem is defined on a hypothetical bilayer hexagonal lattice\nstacked along the $z$-axis, respecting the in plane hexagonal\nsymmetry of the original Bi$_{2}$Se$_{3}$ lattice. With the\nthree independent in-plane nearest neighbor unit vectors\ndefined as $\\hat{b}_1$=($\\frac{\\sqrt{3}}{2}$, $\\frac{1}{2}$),\n$\\hat{b}_2$=(-$\\frac{\\sqrt{3}}{2}$, $\\frac{1}{2}$), and\n$\\hat{b}_3$=(0, -1), we have\n$m_{0}(\\mathbf{k})$=$m+2t_{z}(1-\\cos\nk_{z})+2t(3-2\\cos\\frac{\\sqrt{3}}{2}k_{x}\\cos\\frac{1}{2}k_{y}-\\cos\nk_{y})$,\n$m_{1}(\\mathbf{k})$=2$\\sqrt{3}t\\sin\\frac{\\sqrt{3}}{2}k_{x}\\cos\\frac{1}{2}k_{y}$,\n$m_{2}(\\mathbf{k})$=2$t(\\cos\\frac{\\sqrt{3}}{2}k_{x}\\sin\\frac{1}{2}k_{y}+\\sin\nk_{y})$, and $m_{3}(\\mathbf{k})$=$2t_{z}\\sin k_{z}$. The\nin-plane and out-of-plane lattice parameters\\cite{larson02} are\ntaken as length units in the above expression, that is\n$a$=$c$=1. When $mt_z$$<$0 and $mt$$<$0, the parametrization\ndefined above\\cite{wang10} and the parametrization in the small\n$\\mathbf{k}$ effective model proposed by Zhang \\emph{et\nal.}\\cite{zhang09} describe qualitatively the same physics.\nWith this parametrization, it is easy to see that the model has\nthe inversion symmetry $PH(\\mathbf{k})P^{-1}$=$H(-\\mathbf{k})$,\nwhere the inversion operator is defined as\n$P$=$s_0\\otimes\\sigma_1$.\\cite{fu07b}\n\nNow, we clarify the differences between models (I) and (II) by\ntheir surface states, which is one of the most important\nsignatures of nontrivial topological order in the system. Close\nto the $\\Gamma$ point in the BZ, we take $m_{i=0, \\cdots,\n3}(\\mathbf{k})$=$\\{m+\\frac{3}{2}t(k_{x}^2+k_{y}^2)+t_{z}k_{z}^{2},\n3tk_{x}, 3tk_{y}, 2t_{z}k_{z}\\}$, in which $t>0$, $t_{z}>0$ and\n$m<0$. Consider a sample occupying the lower half space\n$z\\le0$. The possible surface states localized close to $z=0$\nis searched by solving a set of four coupled second order\ndifferential equations\n\\begin{equation}\nH(k_{x}=k_{y}=0,k_{z}\\rightarrow -i\\partial_{z})\\Psi(z)=E\\Psi(z),\n\\end{equation}\ntogether with the open boundary condition\n$\\Psi(z)|_{z=0}$=$\\Psi(z)|_{z=-\\infty}$=0.\\cite{liu10b2,lu10}\n$\\Psi(z)$ is the four-component eigenvector and E is the energy\nof the surface mode, respectively. We look for the zero energy\nstates and hence set $E$=0.\\cite{liu10b2}\n\nFor the model (I) with $\\Gamma_3$=$s_{0}$$\\otimes$$\\sigma_{2}$,\nthe up and down spin degrees of freedom are decoupled from each\nother. The wave function could thus be written as\n$\\Psi(z)$=$[u_{1}(z), u_{2}(z), u_{3}(z),\nu_{4}(z)]^{T}$=$[\\chi_{\\uparrow}(z),\n\\chi_{\\downarrow}(z)]^{T}$. The two spin components of the zero\nenergy mode satisfy the same equation as ($s$ is $\\uparrow$ or\n$\\downarrow$ for the two spin degrees of freedom)\n\\begin{equation}\n[(m-t_{z}\\partial_{z}^{2})\\sigma_{1}-2it_{z}\\partial_{z}\\sigma_{2}] \\chi_{s}(z)=0.\n\\end{equation}\nThe two degenerate zero energy surface modes for $z$$\\le$0 are\nobtained as\n\\begin{equation}\n\\Psi_{\\alpha}(z)=C\\eta_{\\alpha}(e^{z\/\\xi_{+}}-e^{z\/\\xi_{-}}),\n\\end{equation}\nwhere $\\alpha$=1 or 2, $C$ is a normalization constant and\n$\\xi_{\\pm}^{-1}$=$1\\pm\\sqrt{1+m\/t_{z}}$. 1\/Re$[\\xi_{\\pm}^{-1}]$\n(`Re' means taking the real part of a number) are the two\npenetration depths of the surface modes into the bulk. The two\nunit vectors are $(\\eta_{1})_{\\beta}$=$\\delta_{\\beta1}$ and\n$(\\eta_{2})_{\\beta}$=$\\delta_{\\beta3}$, where\n$\\delta_{\\alpha\\beta}$ is one for $\\alpha$=$\\beta$ and zero\notherwise. Take $\\{\\Psi_{1}, \\Psi_{2}\\}$ as the two basis, the\neffective model for the surface states are obtained by\nconsidering the k$_{x}$ and k$_{y}$ dependent terms in the\noriginal model as perturbations, which are\n\\begin{equation}\n\\Delta H_{3D}=\\frac{3}{2}t(k_{x}^2+k_{y}^2)\\Gamma_{0}+3t(k_{x}\\Gamma_{1}+k_{y}\\Gamma_{2}).\n\\end{equation}\nSuppose the two basis are normalized, the effective model for\nthe surface states is\\cite{zhang09}\n\\begin{equation}\nH_{eff}(\\mathbf{k})=3t(k_{x}s_{x}+k_{y}s_{y}),\n\\end{equation}\nwhere $s_{x}$ and $s_{y}$ are the first and second Pauli\nmatrices. Since the two basis both have definite spin\ncharacters, $s_{x}$ and $s_{y}$ in the above equation could\nalso be considered as acting in the spin subspace. The most\nsalient feature of this model is that the corresponding surface\nstates has contributions only from the first orbital. When we\nconsider a sample occupying $z$$\\ge$0, the surface states at\n$z$=0 would arise only from the second orbital.\n\n\nWe now study model (II) for\n$\\Gamma_{3}$=$s_{3}$$\\otimes$$\\sigma_{3}$. We still consider a\nsample situated at $z$$\\le$0 with the open boundary conditions.\nFollowing exactly the same steps as for the first model, we\nobtain the two degenerate zero energy surface states as\n\\begin{equation}\n\\Phi_{\\alpha}=C\\eta_{\\alpha}(e^{z\/\\xi_{+}}-e^{z\/\\xi_{-}}),\n\\end{equation}\nwhere $\\alpha$=1 or 2, $C$ is a normalization constant and\n$\\xi_{\\pm}$ are defined identically as above. However, the two\nunit basis vectors are quite different from the first model and\nare $\\eta_{1}$=$\\frac{1}{\\sqrt{2}}[1, -i, 0, 0]^{T}$ and\n$\\eta_{2}$=$\\frac{1}{\\sqrt{2}}[0, 0, -i, 1]^{T}$. The two\ndimensional effective model for the surface states is also a\nbit different at least formally, which is\n\\begin{equation}\nH_{eff}(\\mathbf{k})=3t\\hat{z}\\cdot(\\mathbf{k}\\times\\mathbf{s})=3t(k_{x}s_{y}-k_{y}s_{x}).\n\\end{equation}\nThe Pauli matrices $s_{x}$ and $s_{y}$ act in the two fold\ndegenerate basis of the zero energy surface states which both\nhave definite spin characters. For a general two dimensional\nwave vector, the surface state would be a linear combination of\nall the four spin-orbital basis.\n\nThus there are qualitative differences between two models which\nwere used in-discriminatively in the literature for\nBi$_{2}$Se$_{3}$.\\cite{wang10,fu10,li10,zhang09,liu10b1,liu10b2,lu10,shan10}\nWhile only one orbital contributes to the surface states for\nmodel (I), both two orbitals contribute in equal weight to the\nsurface states for model (II). On the other hand, the effective\nmodel of the surface states has the same spin-orbital coupled\nform as the linear $k_{x}$ and $k_{y}$ terms in the original\nthree dimensional model for model (I). However, the effective\nmodel is changed from $\\mathbf{k}\\cdot\\mathbf{s}$ to\n$\\hat{z}\\cdot(\\mathbf{k}\\times\\mathbf{s})$ for model (II). We\nhave verified that, if we change the in-plane spin-orbit\ncoupling of the two models (I) and (II) from\n$\\mathbf{k}\\cdot\\mathbf{s}$ to\n$\\hat{z}\\cdot(\\mathbf{k}\\times\\mathbf{s})$, the resulting\neffective model of the surface states would have the form\n$\\hat{z}\\cdot(\\mathbf{k}\\times\\mathbf{s})$ for model (I) but\nwill be switched to $\\mathbf{k}\\cdot\\mathbf{s}$ for model (II).\n\n\nBefore ending this section, we would like to point out that\nboth the $k_{z}$-linear term in $m_{3}(\\mathbf{k})$ and the\n$k_{z}$-square term in $m_{0}(\\mathbf{k})$ are essential to\nobtain the zero energy surface modes. If we omit the\n$k_{z}^{2}$ term in $m_{0}(\\mathbf{k})$, it is easy to verify\nthat the gapless surface states no longer exist. This is\nrelated to the fact that band inversion is essential to the\nappearance of nontrivial topological surface\nstates\\cite{bernevig06,qi10pto}, which can only occur in the\npresence of $k_{z}^{2}$ term for $k_{x}$=$k_{y}$=0.\n\n\n\n\\section{superconducting state spectral function}\n\n\\subsection{surface Green's functions in the superconducting state}\n\nThe realization of the superconducting state in\nCu$_{x}$Bi$_{2}$Se$_{3}$\\cite{hor10,wray10} has brought about\nexcitement that non-trivial topological superconducting state\nmight be realized in this system, in which topologically\nprotected gapless surface states traverse the bulk\nsuperconducting gap.\\cite{fu10,sato0910,qi10b} The recent\nrealization of superconducting phase in Bi$_{2}$Te$_{3}$ under\nhigh pressure\\cite{zhang10pa,zhang10pb} makes the\nBi$_{2}$X$_{3}$ (X is Se or Te) material a very promising\ncandidate system to realize topologically nontrivial\nsuperconducting phases.\\cite{sato0910,fu10,qi10b}\n\n\nThe normal state of the topological insulator is marked by the\npresence of topological surface states inside the bulk gap.\nThese gapless topological surface states are well separated\nfrom the bulk conduction band at low energies and become\nindistinguishable for energies much higher than the conduction\nband minimum. Depending on the doped charge density the\nsuperconducting state could occur with the chemical potential\neither deep in the bulk conduction band or in the intermediate\nregion where the topological surface states are well separated\nfrom the bulk conduction band\\cite{wray10}. In the latter case,\nthe coupling between the continuum bulk states and the isolated\ntopological surface states may cause some new interesting\nphenomena. Thus this intermediate region is where we will\nconcentrate on below. Furthermore, the actual pairing\nsymmetries of the superconducting Cu$_{x}$Bi$_{2}$Se$_{3}$ and\nBi$_{2}$Te$_{3}$ are presently\nunknown.\\cite{hor10,wray10,zhang10pa,zhang10pb} Thus we will\nexamine cases with different pairing symmetries in the hope to\nprovide clues to identify the pairing symmetry and the role\ncontributed by the topological surface states.\n\nIn the following, we will study the surface spectral function\nto see possible nontrivial topological properties arising from\nthe normal phase topological order which is subject to a\ncertain bulk pairing. The surface spectral function, which\ncould be obtained from the surface Green's function, has been\nstudied by ARPES\\cite{wray10} and STM\\cite{alpichshev10} to\ngive important information on the topological properties of the\nsystem. In the superconducting state, we expect to see some\nsurface Andreev bound states if a certain superconducting order\nis realized in the material.\n\nIn the presence of a surface perpendicular to the $z$-axis,\n$k_{x}$ and $k_{y}$ are good quantum numbers, and $k_{z}$ is\nreplaced by $-i\\partial_{z}$ as we shall search for surface\nstates. We then discretize the $z$ coordinate and turn the\nwhole sample ($z$$\\le$0) with a surface at $z$=0 to a coupled\nquintuple-layer system. Label each separate quintuple unit with\nan integer index $n$, and make the substitutions\n$\\partial_{z}\\psi_{n}(z)$=$\\frac{1}{2}[\\psi_{n+1}-\\psi_{n-1}]$\nand\n$\\partial_{z}^{2}\\psi_{n}(z)$=$\\psi_{n+1}+\\psi_{n-1}-2\\psi_{n}$\n($c$ is set as length unit along $z$ axis), the Hamiltonian\nconsists now the intra-layer terms and the interlayer hopping\nterms, $\\hat{H}$=$\\hat{H}_{\\parallel}+\\hat{H}_{\\perp}$. The\nintra-layer part of the model is\n$\\hat{H}_{\\parallel}$=$\\sum_{n\\mathbf{k}}\n\\psi_{n\\mathbf{k}}^{\\dagger}h_{xy}(\\mathbf{k})\\psi_{n\\mathbf{k}}$,\nin which\n\\begin{equation}\nh_{xy}(\\mathbf{k})=m'_{0}(\\mathbf{k})\\Gamma_{0}+m_{1}(\\mathbf{k})\\Gamma_{1}\n+m_{2}(\\mathbf{k})\\Gamma_{2}.\n\\end{equation}\n$m_{1}(\\mathbf{k})$ and $m_{2}(\\mathbf{k})$ are the same as\nthose in the bulk model. $m'_{0}(\\mathbf{k})$ is obtained from\n$m_{0}(\\mathbf{k})$ by first expanding it up to the square term\nof $k_{z}$ and then replacing the term proportional to\n$k_{z}^{2}$ from $Ck_{z}^{2}$ to $2C$.\n\nThe inter-layer hopping term is\n$\\hat{H}_{\\perp}$=$\\sum_{n\\mathbf{k}}\\psi_{n\\mathbf{k}}^{\\dagger}h_{z}\\psi_{n+1,\\mathbf{k}}$+H.c.\nIn terms of the parameterizations of Wang \\emph{et\nal.}\\cite{wang10}, we have\n\\begin{equation}\nh_{z}=-t_{z}(\\Gamma_{0}+i\\Gamma_{3}).\n\\end{equation}\nSince the $\\Gamma_{3}$ matrix now appears only in the $h_{z}$\npart of the coupled layers system, the difference between the\ntwo models enters only through the interlayer hopping term.\n\n\nNow introduce superconducting pairing and define the Nambu\nbasis as\n$\\phi_{n\\mathbf{k}}^{\\dagger}$=$[\\psi_{n\\mathbf{k}}^{\\dagger},\n\\psi_{n-\\mathbf{k}}^{T}]$. The intra-layer part of the\nBogoliubov de Gennes (BdG) Hamiltonian is then\n$\\hat{H}^{SC}_{\\parallel}$=$\\sum_{n\\mathbf{k}}\\phi_{n\\mathbf{k}}^{\\dagger}H_{SC}(\\mathbf{k})\\phi_{n\\mathbf{k}}$,\nin which\\cite{linder10}\n\\begin{equation}\nH_{SC}(\\mathbf{k})=\n\\begin{pmatrix} h_{0}(\\mathbf{k}) & \\underline{\\Delta}(\\mathbf{k}) \\\\\n-\\underline{\\Delta}^{\\ast}(-\\mathbf{k}) & -h^{\\ast}_{0}(-\\mathbf{k})\n\\end{pmatrix},\n\\end{equation}\nwhere $h_{0}(\\mathbf{k})$=$h_{xy}(\\mathbf{k})-\\mu\nI_{4\\times4}$, with $\\mu$ the chemical potential.\n$\\underline{\\Delta}(\\mathbf{k})$ is the 4$\\times$4 pairing\nmatrix. Ignoring the possibility of interlayer pairing, the\ninterlayer hopping terms are\n$\\hat{H}^{SC}_{\\perp}$=$\\sum_{n\\mathbf{k}}\\phi_{n\\mathbf{k}}^{\\dagger}H_{z}\\phi_{n+1,\\mathbf{k}}$+H.c.,\nin which\n\\begin{equation}\nH_{z}=\n\\begin{pmatrix} h_{z} & 0 \\\\\n0 & -h^{\\ast}_{z}\n\\end{pmatrix}.\n\\end{equation}\n\nOnce the pairing order is given, the surface spectral function\nis obtained from the retarded surface Green's functions, which\ncould be calculated in terms of standard transfer matrix\nmethod.\\cite{fastiter} In the simplest form of the method, the\n8$\\times$8 retarded surface Green's function\n$G(\\mathbf{k},\\omega)$ is obtained by self-consistent\ncalculation of $G(\\mathbf{k},\\omega)$ and a transfer matrix\n$T(\\mathbf{k},\\omega)$ as\\cite{fastiter,wang10}\n\\begin{subequations}\n\\begin{equation}\nG^{-1}=g^{-1}-H_{z}^{\\dagger}T,\n\\end{equation}\n\\begin{equation}\nT=GH_{z},\n\\end{equation}\n\\end{subequations}\nwhere $g$=$[zI_{8\\times8}-H_{SC}(\\mathbf{k})]^{-1}$\n($z$=$\\omega+i\\eta$) is the retarded Green's function for an\nisolated layer. $\\eta$ is the positive infinitesimal 0$^{+}$,\nwhich is replaced by a small positive number in realistic\ncalculations. Self-consistent calculation of the Green's\nfunction starts with $G$=$g$. The surface Green's function\ncould also be obtained in terms of other iteration schemes,\nsuch as the algorithm in Ref.\\cite{fastiter}. We have found no\ndifference between the results obtained in terms of different\niteration schemes.\n\nAfter the retarded Green's functions are at hand, the spectral\nfunction is obtained as\n\\begin{equation}\nA(\\mathbf{k},\\omega)=-\\sum_{i=1}^{4}\\text{Im} G_{ii}(\\mathbf{k},\\omega)\/\\pi.\n\\end{equation}\n\nSince we have now two orbital and two spin degrees of freedom,\nthere are many possible pairing channels for different possible\npairing mechanisms. Realistic theoretical determination of the\npairing symmetry requires the knowledge of pairing mechanism\nand reasonable parameter values, which are both lacking\npresently.\\cite{fu10} Here we will consider singlet and triplet\npairing orders as phenomenological input parameters. Their\nqualitative differences in spectral functions could help to\nidentify the pairing symmetry from experiments.\n\n\n\\subsection{gap opening in the topological surface states}\n\nBefore presenting the full spectral function, we first would\nlike to examine what happens to the topological surface states\ninherited from the normal state\\cite{wray10} upon the formation\nof a certain bulk pairing. The most salient feature of the\ntopological insulator is the presence of gapless surface states\n(3D) or edge states (2D).\\cite{kane05,bernevig06,fu07l,qi10pto}\nIn the case of Cu$_{x}$Bi$_{2}$Se$_{3}$, it is found that these\nsurface states in the non-superconducting Bi$_{2}$Se$_{3}$\npersist to the superconducting copper intercalated samples and\nare well separated from the bulk conduction band and hence\nwell-defined.\\cite{hor10,wray10} It is thus an interesting\nquestion what would happen to them if a certain pairing forms\nin the bulk. In this subsection, we give a simple criterion to\njudge whether a gap would be induced in the topological surface\nstates for an arbitrary bulk pairing.\n\nSuppose the chemical potential lies slightly above the bottom\nof the bulk conduction band where the topological surface\nstates are well separated and well defined.\\cite{wray10} If a\npairing is realized in the topological surface states, it\nshould occur between the two time reversal related states for\n$\\mathbf{k}$ and -$\\mathbf{k}$.\\cite{fu0809}\n\nWe first consider model (I). For our purpose, we would\nconcentrate on the positive energy branch of the topological\nsurface states. When the pairing occurs in the valence\nband\\cite{zhang10pa,zhang10pb}, the analysis and conclusion\nwould be similar. Since pairing occurs in the (k$_{x}$,\nk$_{y}$) space, we would ignore the $z$-dependence of the\nsurface modes when analyzing pairing properties. From the basis\nand the effective model obtained in Sec. II, the two\neigenvectors for a certain 2D wave vector are\n\\begin{equation}\n\\eta_{\\alpha}(\\mathbf{k})=\\frac{1}{\\sqrt{2}}[1, 0, \\alpha\\frac{k_{+}}{k}, 0]^{T},\n\\end{equation}\nwhere $\\alpha$ is `+' (`$-$') for the upper (lower) branch of\nthe surface states, $k_{\\pm}$=$k_{x}\\pm ik_{y}$,\n$k$=$\\sqrt{k_{x}^2+k_{y}^2}$. The annihilation operators of\nthese states are\n\\begin{equation}\nd_{\\mathbf{k}\\alpha}=\\frac{1}{\\sqrt{2}}[c_{1\\mathbf{k}\\uparrow}\n+\\frac{\\alpha k_{+}}{k}c_{1\\mathbf{k}\\downarrow}]\n\\end{equation}\nIf pairing is induced in the upper surface conduction band at\n$\\mathbf{k}$, the only possible pairing would be proportional\nto\n$d_{\\mathbf{k}+}^{\\dagger}d_{-\\mathbf{k}+}^{\\dagger}$.\\cite{fu0809}\nDenote the time-reversal operator as\n$\\mathcal{T}$.\\cite{fu0809,qi10b} Since\n$\\mathcal{T}c_{1\\mathbf{k}\\uparrow}^{\\dagger}\\mathcal{T}^{-1}$=$c_{1-\\mathbf{k}\\downarrow}^{\\dagger}$\nand\n$\\mathcal{T}c_{1\\mathbf{k}\\downarrow}^{\\dagger}\\mathcal{T}^{-1}$=$-c_{1-\\mathbf{k}\\uparrow}^{\\dagger}$,\nwe have $\\mathcal{T}\nd_{\\mathbf{k}+}^{\\dagger}d_{-\\mathbf{k}+}^{\\dagger}\\mathcal{T}^{-1}$\n=$\\frac{k_{+}^{2}}{k^2}d_{\\mathbf{k}+}^{\\dagger}d_{-\\mathbf{k}+}^{\\dagger}$.\nTo ensure the time-reversal symmetry of the pairing, the actual\npairing should be of the form\n\\begin{eqnarray}\n\\hat{\\Delta}^{I}_{SCB}(\\mathbf{k})&=&\\Delta_{0}\\frac{k_{+}}{k}\nd_{\\mathbf{k}+}^{\\dagger}d_{-\\mathbf{k}+}^{\\dagger} \\notag \\\\\n&=&\\frac{\\Delta_{0}}{2}[\\frac{k_{+}}{k}c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger}\n-\\frac{k_{-}}{k}c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger} \\notag \\\\\n&&+(c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger}\n-c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger})],\n\\end{eqnarray}\nwhere $\\Delta_{0}$ is the real pairing amplitude, which could\nbe an even or odd real function of $\\mathbf{k}$ depending on\nthe pairing realized in the bulk. `SCB' is abbreviation for the\nsurface conduction band (the topological surface states). Thus,\nthe surface conduction band only supports the anti-phase\n$p_{x}\\pm ip_{y}$ equal-spin triplet pairing and the\nspin-singlet pairing within orbital 1. No other bulk pairing\nchannels, especially those inter-orbital pairings, would open a\ngap in the topological surface states within the framework of\nmodel (I).\n\nFor model (II), the two eigenvectors of the surface states for\na certain 2D wave vector are (again, ignoring the\n$z$-dependence)\n\\begin{equation}\n\\eta_{\\alpha}(\\mathbf{k})=\\frac{1}{2}[1, -i, \\alpha\\frac{k_{+}}{k}, i\\alpha\\frac{k_{+}}{k}]^{T},\n\\end{equation}\nwhere $\\alpha$ is `+' (`$-$') for the upper (lower) branch of\nthe topological surface states. Following the same arguments as\nfor the first model,  when the chemical potential cuts the\nupper branch of these well defined surface states the time\nreversal invariant pairing is of the form\n\\begin{eqnarray}\n\\hat{\\Delta}^{II}_{SCB}(\\mathbf{k})&=&\\Delta_{0}\\frac{k_{+}}{k}\nd_{\\mathbf{k}+}^{\\dagger}d_{-\\mathbf{k}+}^{\\dagger} \\notag \\\\\n&=&\\frac{\\Delta_{0}}{4}[\\frac{k_{+}}{k}(c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger}\n-c_{2\\mathbf{k}\\uparrow}^{\\dagger}c_{2-\\mathbf{k}\\uparrow}^{\\dagger})  \\notag \\\\\n&&-\\frac{k_{-}}{k}(c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger}\n-c_{2\\mathbf{k}\\downarrow}^{\\dagger}c_{2-\\mathbf{k}\\downarrow}^{\\dagger}) \\notag \\\\\n&& +i\\frac{k_{+}}{k}(c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{2-\\mathbf{k}\\uparrow}^{\\dagger}\n+c_{2\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger})  \\notag \\\\\n&& +i\\frac{k_{-}}{k}(c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{2-\\mathbf{k}\\downarrow}^{\\dagger}\n+c_{2\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger})  \\notag \\\\\n&& +(c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger}\n-c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger}  \\notag \\\\\n&& +c_{2\\mathbf{k}\\downarrow}^{\\dagger}c_{2-\\mathbf{k}\\uparrow}^{\\dagger}\n-c_{2\\mathbf{k}\\uparrow}^{\\dagger}c_{2-\\mathbf{k}\\downarrow}^{\\dagger})  \\notag \\\\\n&& +i(c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{2-\\mathbf{k}\\downarrow}^{\\dagger}\n+c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{2-\\mathbf{k}\\uparrow}^{\\dagger}  \\notag \\\\\n&& -c_{2\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger}\n-c_{2\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger})].\n\\end{eqnarray}\nAs in Eq. (17), $\\Delta_{0}$ could be a constant or a real\nfunction of $\\mathbf{k}$ compatible with the symmetry of one\npairing component contained in the above decomposition. Besides\nthe intra-orbital pairing channels active in the model (I),\nthere are two additional inter-orbital pairing channels that\nare effective in producing a gap in the topological surface\nstates. The last term in the above equation is just the\nodd-parity inter-orbital triplet pairing proposed by Fu and\nBerg\\cite{fu10} as a possible candidate of a topological\nsuperconductor to be realized in a superconductor like\nCu$_{x}$Bi$_{2}$Se$_{3}$.\n\nThe clear difference between $\\hat{\\Delta}^{I}_{SCB}$ and\n$\\hat{\\Delta}^{II}_{SCB}$ makes the distinction between model\n(I) and model (II) more obvious. Since the gap opening of the\nsurface states is measurable, it is highly desirable to\nascertain which model is the correct description of the\nunderlying physics of Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$.\n\nPreviously, a simple effective model calculation indicates that\nno gap opens in the topological surface states for any triplet\npairing induced by proximity effect on the surface of a\ntopological insulator.\\cite{linder10} However, our analysis\nabove indicates that if the proximity induced triplet pairing\nis compatible with any of the triplet components explicit in\n$\\hat{\\Delta}^{I}_{SCB}$ ($\\hat{\\Delta}^{II}_{SCB}$) for model\nI (model II), then a full pairing gap could still be opened in\nthe topological surface states. Note that the real gap opening\npattern in the topological surface states also depends on\n$\\Delta_0$.\n\nExcept for the pairing channels explicit in\n$\\hat{\\Delta}^{I}_{SCB}$ for model (I) and\n$\\hat{\\Delta}^{II}_{SCB}$ for model (II), no other bulk pairing\ncould open a gap in the topological surface states. The\nexistence of surface Andreev bound states depends on whether or\nnot a gap opens in the topological surface states. We clarify\nthis matter in the next section.\n\n\n\\subsection{spectral function for typical pairing symmetries}\n\nObservation of superconductivity in Cu$_{x}$Bi$_{2}$Se$_{3}$\nbrings about anticipation that nontrivial topological\nsuperconducting states might be realized in this material. The\ntopological superconductor is defined as a state with a full\npairing gap in the bulk and nontrivial gapless Andreev bound\nstates on the surface.\\cite{fu10}\n\n\nPossible pairings realizable in a system depend on the symmetry\nof the system and the specific pairing mechanism. In the case\nof pairing induced by short range electron density-density\ninteractions, Fu and Berg identified four possible pairing\nchannels.\\cite{fu10} However, if the pairing is induced by more\nlong-range interactions, such as the electron-phonon\ninteraction, other pairing channels (e.g., in which the pairing\npotential is $\\mathbf{k}$ dependent) would also be possible. In\nthe following we would analyze several typical pairings and\ncompare results of the two different models. In each case,\nthere are three typical situations as regards to the position\nof the chemical potential $\\mu$: (1) $\\mu$ lies in the bulk\ngap; (2) $\\mu$ lies above but close to the bottom of the bulk\nconduction band, where the topological surface states are well\nseparated from the continuum bulk conduction band; (3) $\\mu$\nlies far above the bulk conduction band bottom, where the\nsurface states have merged into the continuum conduction band.\nWhile the latter two cases are relevant to the superconducting\nstate of Cu$_{x}$Bi$_{2}$Se$_{3}$\\cite{hor10,wray10}, the first\ncase could be regarded as mimicking the proximity effect from\nan external superconductor.\\cite{fu0809,sau10,linder10} In this\npaper we would focus on the latter two situations. When the\nchemical potential lies in the valence\nband\\cite{zhang10pa,zhang10pb}, the results should be\nqualitatively similar for the same type of bulk pairing.\n\nFollow Wang \\emph{et al.}\\cite{wang10}, the model parameters\nare taken as $t$=$t_{z}$=0.5, $m$=-0.7 in most cases. 0.7 is\nhalf of the bulk band gap. The width of the bulk conduction\nband at $k_{x}$=$k_{y}$=0 is $2(2t_{z}-|m|)$=0.6. The small\npositive number $\\eta$ in the Green's functions is taken as\n10$^{-4}$.\n\n\n\\emph{- even-parity intra-orbital singlet pairing} First, we\nstudy the simplest possible pairing denoted by\n$\\underline{\\Delta}(\\mathbf{k})$=$i\\Delta_{0}s_{2}\\otimes\\sigma_{0}$.\nSpectral functions for the two different models are the same\nfor this pairing, so only one is presented in Fig. 1. Here and\nin the following, the degree of darkness indicates the\nintensity of the spectrum. The continuum portions of spectrum\nare contributions from the bulk states, which have small finite\namplitudes on the surface. Henceforth, they would be called\nbulk conduction band for simplicity. The contributions from the\ntopological surface states are somewhat speckled because we\nhave taken a finite grid in the $(\\mathbf{k},\\omega)$ plane to\ncalculate the spectral function. When the grid points are taken\nto be very dense, contributions from the topological surface\nstates will also become smooth. To see the qualitative behavior\nmore clearly, a reasonably large pairing amplitude\n$\\Delta_{0}$=0.1 is considered.\\cite{wray10} The result is\nnearly identical in the $\\Gamma$K direction (along $k_{x}$\naxis) and the $\\Gamma$M direction (along $k_{y}$ axis) of the\n2D reduced Brillouin zone (BZ). The topological surface states\nof both two models open a gap, which are consistent with the\nanalysis of the previous subsection. Since no Andreev bound\nstate exists, this pairing is topologically trivial. The other\nintra-orbital singlet pairings with a $\\mathbf{k}$-dependent\n$\\Delta_{0}$, which is an even function of $\\mathbf{k}$ could\nalso be considered, such as the $d_{x^2-y^2}$-wave pairing. In\nthese cases, there would be line nodes along the nodal\ndirections of the pairing gap.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=10cm,angle=0]{Fig1.ps}\n\\caption{Spectral function for even-parity intra-orbital $s$-wave pairing,\nfor two typical parameter sets for which the topological surface states at the chemical potential\n(a) are well separated from the bulk conduction band\nand (b) merges into the bulk conduction band.\nThe two models give identical results for this pairing. Spectrum along other directions\n(crossing the $\\Gamma$ point) are qualitatively identical.}\n\\end{figure}\n\n\n\\emph{- even-parity inter-orbital singlet pairing} Since there\nare now two orbits, another singlet pairing exists in the\ninter-orbital channel. The pairing matrix for the $s$-wave case\nis\n$\\underline{\\Delta}(\\mathbf{k})$=$i\\Delta_{0}s_{2}\\otimes\\sigma_{1}$.\nThe corresponding spectral functions presented in Fig. 2 for\nthis pairing are still identical for the two models. They\ndiffer from the spectra of the former intra-orbital pairing\nchannel in at least two aspects. First, no gap opens in the\ntopological surface states, which is in agreement with the\ncriterion proposed in the previous subsection. Second, though a\nfull gap also opens in the continuum part of the spectrum, it\nis not constant and shows some $\\mathbf{k}$ dependence. As\nshown in Fig. 2(b), the continuum part of the spectrum even\nnearly closes at some special wave vectors for certain\nparameters. Another interesting feature is the strong\nredistribution of spectral weight between the continuum\nconduction band and the topological surface states. Some weight\nin the bulk conduction band part of the surface spectrum above\nthe chemical potential is depleted and transferred to the\ntopological surface states below the chemical potential. This\nredistribution arises from the particle hole mixing induced by\nthe presence of bulk superconducting pairing. As would see\nbelow, a similar feature is present for each bulk pairing that\ndoes not open a gap in the topological surface states.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=15cm,angle=0]{Fig2.ps}\n\\caption{Spectral function for even-parity inter-orbital $s$-wave pairing,\nfor three sets of parameters for which at the chemical potential\n(a) the topological surface states are well separated from the bulk conduction band,\n(b) the topological surface states are almost merged into the bulk conduction band,\nand (c) the topological surface states are well merged into the bulk conduction band.\nThe two models give identical results for this pairing. Spectrum along other directions\n(crossing the $\\Gamma$ point) are qualitatively identical.}\n\\end{figure}\n\n\\emph{- odd-parity inter-orbital triplet pairing} We now\nconsider the odd-parity inter-orbital triplet pairing channel,\nproposed by Fu and Berg as a candidate for possible nontrivial\ntopological superconducting states in\nCu$_{x}$Bi$_{2}$Se$_{3}$.\\cite{fu10} The pairing matrix is\n$\\underline{\\Delta}(\\mathbf{k})$=$\\Delta_{0}s_{1}\\otimes\\sigma_{2}$.\nAs was shown in Fig. 3, the spectral functions for the two\nmodels differ greatly. When the chemical potential is close to\nbottom of the bulk conduction band, the surface conduction band\nis still gapless for model (I) but opens a gap for model (II),\nin agreement with the analysis in the above subsection. Another\nessential difference is the existence or not of Andreev bound\nstates. For model (I), a band of Andreev bound states appears\ninside of the insulating gap of the continuum which\ncontinuously connects to the topological surface states. While\nfor model (II), there is a point node at (0, 0), no Andreev\nbound state exists inside of the gap region. When the chemical\npotential is increased to the position where the surface\nconduction band has almost merged into the continuum part of\nthe surface spectrum (corresponding to contribution from the\nbulk conduction band), surface spectra for the two models are\nas shown in Fig. 4. Since now there is no well separated\nsurface conduction band, a full gap opens also for the model\n(I). However, a band of Andreev bound states still exists. When\nwe further increase the chemical potential to $\\mu>1.3$ (for\n$m$=-0.7), there is no state close to the $\\Gamma$ point, then\nthere would be no Andreev bound state even for model (I). A\nstrong redistribution of spectral weight arising from the\nparticle hole mixing between the continuum bulk conduction band\nand the topological surface states is observed in results for\nmodel (I).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=10cm,angle=0]{Fig3.ps}\n\\caption{Spectral function for odd-parity inter-orbital triplet pairing\nin the opposite spin pairing channel, for\n(a) model (I) and (b) model (II). As shown are the cases for which\nthe topological surface states are well separated from the bulk conduction band at the chemical potential.\nThe parameters are as shown on the figures. Spectrum along other directions\n(crossing the $\\Gamma$ point) are qualitatively identical.}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=10cm,angle=0]{Fig4.ps}\n\\caption{Spectral function for odd-parity inter-orbital triplet pairing\nin the opposite spin pairing channel, for\n(a) model (I) and (b) model (II). As shown are the cases for which\nthe topological surface states are merged into the bulk conduction band at the chemical potential.\nThe parameters are as shown on the figures. Spectrum along other directions\n(crossing the $\\Gamma$ point) are qualitatively identical.}\n\\end{figure}\n\n\nBesides the pairing studied above, there are also other\nodd-parity pairings. Since they are possibly related with\nnontrivial topological superconducting phases, we analyze\nseveral of them in the following.\n\n\\emph{- odd-parity intra-orbital singlet pairing} In this case,\nevery orbit pairs into a spin-singlet, but the two orbits has a\nrelative $\\pi$ phase difference and is hence odd in\nparity.\\cite{fu10} The pairing matrix is denoted as\n$\\underline{\\Delta}(\\mathbf{k})$=$i\\Delta_{0}s_{2}\\otimes\\sigma_{3}$.\nThe corresponding spectral functions for the two models are\npresented in Figs. 5(a) and 5(b) for chemical potential close\nto the bottom of the bulk conduction band and thus the\ntopological surface state\n is well defined. A comparison with the previous\npairing, shown in Figs. 3 and 4, indicates that the results are\ninterchanged between the two models. Gap opening in the\ntopological surface states again follows the expectation from\nthe previous subsection. The Andreev bound states in the bulk\nband gap for the second model again connect continuously to the\nprotected topological surface states. The behaviors for higher\nchemical potentials, as shown in Figs. 5(c) and 5(d), are\nsimilar to that of the previous pairing shown in Fig. 4 with\nthe two models interchanged.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=8cm,angle=0]{Fig5.ps}\n\\caption{Spectral function for odd-parity intra-orbital $s$-wave pairing, for\nmodel (I) ((a) and (c)) and model (II) ((b) and (d)).\nFor (a) and (b) ((c) and (d)), the topological surface states\nare well separated from (merged into) the bulk conduction band\nat the chemical potential. Spectrum along other directions\n(crossing the $\\Gamma$ point) are qualitatively identical.}\n\\end{figure}\n\n\n\\emph{- odd-parity inter-orbital equal-spin triplet pairing}\nAnother interesting possibility is the equal-spin pairing\nchannel. Here we consider the two fold degenerate inter-orbital\npairing channels as proposed by Fu and Berg.\\cite{fu10} This\nkind of pairing could be favored by interorbital ferromagnetic\nHeisenberg interactions.\\cite{santos10} The two independent\nchoices for the pairing matrix are\n$\\underline{\\Delta}^{(1)}(\\mathbf{k})$=$i\\Delta_{0}s_{0}\\otimes\\sigma_{2}$\nand\n$\\underline{\\Delta}^{(2)}(\\mathbf{k})$=$\\Delta_{0}s_{3}\\otimes\\sigma_{2}$.\nFor these pairings, it is easy to see that no gap opens in the\ntopological surface states for both models. Since results for\nthe two models are identical, we only show those for the model\n(I). As shown in Fig. 6 for\n$\\underline{\\Delta}^{(1)}(\\mathbf{k})$, a peculiar anisotropic\nAndreev bound state structure is observed. An important\ndifference of this pairing from the above odd-parity pairing\nchannels is that it is anisotropic with respect to $k_{x}$ and\n$k_{y}$. Though the bulk dispersion is gapless in the\n$k_{y}k_{z}$ ($k_{x}k_{z}$) plane for\n$\\underline{\\Delta}^{(1)}(\\mathbf{k})$\n($\\underline{\\Delta}^{(2)}(\\mathbf{k})$), there is still a band\nof Andreev bound states for the wave vectors smaller than\n$k_{F}$ where a gap opens. The peculiar feature of the Andreev\nbound states along $k_{y}$ ($k_{x}$) for\n$\\underline{\\Delta}^{(1)}(\\mathbf{k})$\n($\\underline{\\Delta}^{(2)}(\\mathbf{k})$) is that they are\ndispersion-less (that is, completely flat).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=8cm,angle=0]{Fig6.ps}\n\\caption{Spectral function for odd-parity inter-orbital triplet pairing\nin the equal spin pairing channel, for\nmodel (I) and $\\underline{\\Delta}^{(1)}(\\mathbf{k})$=$i\\Delta_{0}s_{0}\\otimes\\sigma_{2}$.\n(a) and (c) are along the $k_{x}$ direction while (b) and (d) are\nalong the $k_{y}$ direction.\nThe parameters are as shown on the figure.}\n\\end{figure}\n\nBesides the Andreev bound states within the gap, the\nredistribution of spectral weights arising from the particle\nhole mixing is also very interesting. An important feature is\nthe appearance of a linear band beyond the Fermi momentum and\nbelow the chemical potential, existing as a particle hole\nsymmetric band of the original topological surface states in\nthe normal phase. Once a bulk superconducting pairing forms in\nthe topological insulator itself (and not in the intercalated\ncopper) in Cu$_{x}$Bi$_{2}$Se$_{3}$, such a linear dispersive\nband is always there, no matter a gap opens or not in the\ntopological surface states. To see the above feature more\nclearly, we show in Figs. 7(a) and 7(b) the energy distribution\ncurves (EDC) for several typical wave vectors for two typical\npairings and parameter sets, which are same as those of Figs.\n1(a) and 3(a), respectively. The linear band mentioned above\nappears as a well defined peak slightly below the chemical\npotential. As the wave vector increases and shifts away from\nthe Fermi momentum, the peak deviates linearly from the\nchemical potential and the height and width of it both decrease\nrapidly, which is in agreement with the fact that\nsuperconducting pairing forms only close to the chemical\npotential. The integrated weight of the linearly dispersive\npeaks in Figs. 7(a) and 7(b) are shown in Fig. 7(c), as a\nfunction of the wave vector. If the gap in the superconductors\nrealized from a topological insulator is larger than what is\nreported in Ref. \\cite{wray10}, the above linear dispersive\nstructure in the EDC could be detectable by ARPES for the wave\nvectors close enough to the Fermi momentum.\\cite{hor10,wray10}\nThen this well defined peak structure arising from the\ntopological surface states could be used as a good indicator of\nthe formation of superconducting correlation in\nBi$_{2}$Se$_{3}$ and the involvement of the topological surface\nstates in the superconducting phase.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm,height=12cm,angle=0]{Fig7.ps}\n\\caption{EDC for three typical wave vectors for\n(a)even-parity intra-orbital singlet pairing\n(with parameters same as in Fig. 1(a)) and\n(b)odd-parity inter-orbital triplet pairing\n(with parameters same as in Fig. 3(a)) within model (I).\nThe parameters are as shown on the figures.\n(c)The integrated weight of the linearly dispersive peaks\nslightly below the chemical potential with parameters corresponding to (a) and (b),\nrespectively.}\n\\end{figure}\n\n\nFrom the above results for five different pairing symmetries,\nwe observe a simple rule for the existence of nontrivial\nsurface Andreev bound states. For odd-parity pairings, when a\nfull gap opens in the continuum part of the surface spectrum\nbut no gap opens in the topological surface states, a band of\nsurface Andreev bound states would arise which traverses the\nbulk pairing gap. This criterion is verified also by\ncalculations for other superconducting pairings not presented\nhere. The results in this subsection are summarized in Table I.\n\n\\begin{table}[ht]\n\\caption{Summary of results for the bulk pairings considered\nexplicitly in the present work. Results for two models, (I) and\n(II), are compared. `TSS' and `ABS' are the abbreviations for\n`topological surface states' and `Andreev bound states',\nrespectively. `$+$' and `$-$' means the even-parity and\nodd-parity pairings. For `Gap in TSS', `Y' and `N' represents\nthat a gap could and could not open in the topological surface\nstates. For `ABS', `Y' and `N' denotes that Andreev bound\nstates exist and do not exist on the surface for a certain bulk\npairing.} \\centering\n\\begin{tabular}{c c c c c c}\n\\hline\\hline\n$\\underline{\\Delta}$($\\mathbf{k}$) & $is_{2}\\otimes\\sigma_{0}$\n& $is_{2}\\otimes\\sigma_{1}$ & $s_{1}\\otimes\\sigma_{2}$\n& $is_{2}\\otimes\\sigma_{3}$ & $is_{0}\\otimes\\sigma_{2}$ \\\\ [0.5ex]\n\\hline\n$P$ & $+$ & $+$ & $-$ & $-$ & $-$ \\\\\nGap in TSS: (I) & Y & N & N & Y & N \\\\\nGap in TSS: (II) & Y & N & Y & N & N \\\\\nABS: (I) & N & N & Y & N & Y \\\\\nABS: (II) & N & N & N & Y & Y \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\n\\subsection{the surface Andreev bound states}\n\nFor some superconducting pairings, such as the $p\\pm ip$ wave\npairing, it was known that Majorana fermions exist as gapless\nsurface or edge modes.\\cite{qi10b} According to an argument by\nLinder \\emph{et al}, all zero energy Andreev bound states\nemerging from the nondegenerate (that is, on each surface)\ntopological surface states should be Majorana\nfermions.\\cite{linder10} As regards our case, in the parameter\nregion where the topological surface states are well separated\nfrom the bulk conduction band, one observation is that the\nsurface Andreev bound states in the gap region connect\ncontinuously to the topological surface states inherited from\nthe normal phase (e.g., Fig. 3(a)). Since the topological\nsurface states are spin polarized helical and nondegenerate,\nthe surface Andreev bound states on each surface should also be\nnondegenerate. Then according to the arguments by Linder\n\\emph{et al}\\cite{linder10}, the zero energy Andreev bound\nstates presented in the above section should also be Majorana\nfermions.\n\n\nTo see more clearly the properties of the surface Andreev bound\nstates, we perform numerical calculations on a finite layer\nsuperconducting film. As an example, we will analyze the\nodd-parity inter-orbital triplet pairing channel described by\n$\\underline{\\Delta}(\\mathbf{k})$=$\\Delta_{0}s_{1}\\otimes\\sigma_{2}$,\nwithin model (I). Fig. 8(a) shows the dispersion for a fifty\nlayer film. It reproduces all the basic features in the\nspectral function (see Fig. 3(a)). Enlargement of the low\nenergy dispersion (Fig. 8(b)) shows that dispersion of the\nsurface Andreev bound states is linear close to the $\\Gamma$\npoint. For all the bulk pairings studied above, the dispersion\nfor the superconducting film reproduces well the features of\nthe corresponding spectral function. Figure 9(a) shows the wave\nfunction amplitudes for the surface state localized on the top\nseveral layers. The corresponding behavior for the topological\nsurface states in the normal phase is presented in Fig. 9(b).\nThe decay behavior of the surface bound states into the bulk in\nFig. 9(a) is seen to change continuously from oscillatory\nexponential decay in the gap region\\cite{fu10} to monotonic\nexponential decay outside the gap region. That is, the state\nchanges from particle-hole mixed superconducting quasiparticle\nto the topological surface states in the normal phase.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,height=10cm,angle=0]{Fig8.ps}\n\\caption{(a) Dispersion of a fifty layer superconductor emerging from model (I)\nand for the odd-parity inter-orbital triplet pairing $\\underline{\\Delta}(\\mathbf{k})$=$\\Delta_{0}s_{1}\\otimes\\sigma_{2}$.\nParameters used are $\\mu$=0.9, $\\Delta_{0}$=0.1, and\n$m$=-0.7. (b) An enlargement of the small wave vector and low energy part of (a).}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,height=10cm,angle=0]{Fig9.ps}\n\\caption{(a) Decay of the wave function amplitude with layer number\nfor the surface Andreev bound states localized on the top several layers,\nresults are obtained for the superconducting state emerging from model (I)\nand for the odd-parity inter-orbital triplet pairing $\\underline{\\Delta}(\\mathbf{k})$=$\\Delta_{0}s_{1}\\otimes\\sigma_{2}$.\nThe parameters are $\\mu$=0.9, $\\Delta_{0}$=0.1, and\n$m$=-0.7. The Fermi wave vector in the $k_{x}$ direction is\nabout 0.19$\\pi$. (b) Decay of the wave function amplitude with\nlayer number for the topological surface states localized on\nthe top several layers, for the normal states of model (I) with\n$m$=-0.7. The two insets show enlargements of the small\namplitude regions of the two figures.}\n\\end{figure}\n\n\n\nIn this paper, we have always been discussing a homogeneous\nphase both in the bulk and on the surface. However, in the\npresence of exotic surface excitations as vortices, novel\nMajorana fermion modes may appear in the vortex core even for\nbulk pairings with no surface Andreev bound states. There have\nalready been many papers focusing on this possibility, which\nusually start from the effective\nmodel.\\cite{qi09l,santos10,hosur10}\n\n\nThe Andreev bound states that appear in many different\nsuperconducting pairings as shown above confirm the idea that\nsuperconducting states realized in a topological insulator are\nvery probable to have nontrivial topological characters. The\nAndreev bound states, if exist, should be easily detectable in\na tunneling type experiments as a well defined zero energy\npeak. Another way to detect the Majorana fermions as zero\nenergy Andreev bound states is to take advantage of the various\nphase sensitive transport devices proposed to produce and\nmanipulate the Majorana fermions.\\cite{fu0809,law09}\n\n\n\n\n\\section{summary}\nIn this paper, we have discussed the surface spectral function\nof superconductors realized from a topological insulator, such\nas the copper-intercalated Bi$_{2}$Se$_{3}$. These functions\nare calculated by projecting bulk states to the surface for two\ndifferent models used previously for the topological insulator.\nDependence of the surface spectra on the symmetry of the bulk\npairing order parameter are discussed with particular emphasis\non the odd-parity pairing. When an odd-parity pairing opens a\nfull gap in the bulk, but not for the topological surface\nstates, zero energy Andreev bound states are shown to appear on\nthe surface. When the topological surface states are well\nseparated from the bulk conduction band, the redistribution of\nspectral weight induced by the onset of superconductivity\nproduces a linearly dispersive peak structure beyond the Fermi\nmomentum and below the chemical potential. It is proposed as a\ncriterion for confirming that superconductivity occurs in the\nBi$_{2}$Se$_{3}$ (and not in copper) and the topological\nsurface states are involved in the superconducting phase. The\nzero energy surface Andreev bound states are argued to be\nMajorana fermions.\n\n\n\n\n\\begin{acknowledgments}\nWe thank Peter Thalmeier for helpful discussions. This work was\nsupported by the NSC Grant No. 98-2112-M-001-017-MY3. Part of\nthe calculations was performed in the National Center for\nHigh-Performance Computing in Taiwan.\n\\end{acknowledgments}\\index{}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nIt is known that the Horizontal Branch (HB) morphology in globular clusters of similar metallicities varies and the identification of a parameter, other than metallicity, responsible for this has originated much discussion in the astronomical literature. Recent findings that more massive clusters tend to\nhave HBs extended further into higher temperatures led Recio-Blanco et al. (2006) to suggest the cluster's total mass may be  the \"second parameter\". However since the HB morphology varies\n due to stellar evolution, age is still a strong candidate for the\n second parameter status\n(e.g. Stetson et al. 1999; Catelan 2000 and references therein).\n\nRR Lyrae stars are outstanding stars in the HB whose absolute magnitudes and iron abundances have been\ncarefully calibrated (Clementini et al. 2003; Layden 1994) and therefore play an important role in studies of the structure of the Galaxy.\nThe estimation of individual physical parameters of RR Lyrae stars in globular clusters provides relevant insights not only into the distance and iron abundances of their parent clusters (e.g. Arellano Ferro et al. 2008a; 2008b), but also imposes constraints on the structure of the HB and on the stellar evolution in this phase (see e.g. Zinn, 1993; van den Bergh 1993; Zinn 1996).\n\nDetermination of the physical parameters in RR Lyrae stars can\nbe attained from the analysis of data obtained\nfrom fundamental techniques in astronomy such as photometry or\nspectroscopy. High dispersion spectroscopy produces accurate estimates\nfor some of the atmospheric physical parameters, i.e.,\ntemperature, surface gravity and metallicity, that have been used to calibrate\nlow resolution methods such as the $\\Delta S$ method (Preston 1959). However this approach is\nlimited mostly to brighter objects since the extended telescope time required\nto study fainter objects is very\ncompetitive and hence scarce. However,\nisolated examples of accurate spectroscopy of objects around  $V\n\\sim 16-17$~mag do exist (e.g. James et al. 2004, Gratton et al. 2005,\nYong et al. 2005, Cohen et al. 2005).\n\n\nAlternative approaches to physical parameter estimations are\ntheoretical and semi-empirical calibrations of photometric indices,\nsuch as the synthetic color grids from atmospheric models (e.g.\nLester, Gray \\& Kurucz 1986) or, for RR Lyrae stars,\nthe decomposition of light curves in Fourier harmonics and the\ncalibration of the Fourier parameters in terms of key physical\nparameters (e.g. Simon \\& Clement 1993; Kov\\'acs  1998; Jurcsik 1998,\nMorgan et al. 2007). Kov\\'acs and his collaborators, in an extensive\nseries of papers (e.g. Kov\\'acs \\& Jurcsik 1996, 1997;\nJurcsik \\& Kov\\'acs 1996; Kov\\'acs \\& Walker, 2001), have\ndeveloped purely empirical relations from the Fourier analysis of\nthe light curves. The basic stellar parameters are based on the\nassumption that the period and the shape of the light curve are\ndirectly correlated with the physical parameters (or quantities\nrelated to them) such as the iron abundance [Fe\/H], absolute magnitude\n$M_V$ and effective temperature $T_{\\rm eff}$.\n\nIn this paper we shall use the Str\\\"omgren photometry of seven RR\nLyrae stars in Bootes to estimate their physical parameters.\nFurthermore, since we have obtained simultaneous $uvby-\\beta$\nphotoelectric photometry, the precise variation of the stellar\nbrightness and colors along the cycle of pulsation for\neach star can also be used to provide rigid constraints\non the physical parameters.\n\n\n\\section{Observational material and Reductions}\n\\label{sec:Observations}\n\n\nSome of the stars included in the present work were part of\nthe observing program in previous studies as secondary targets\n(Lampens et al. 1990 and Pe\\~na 2003). Other seasons were devoted\nentirely to some of these RR Lyrae stars. The log of the various seasons\nis given in Table \\ref{LOG}. The observations were obtained with the 1.5 m\ntelescope at the Observatorio Astron\\'omico Nacional of San Pedro\nM\\'artir, Mexico. The telescope was equipped with a six channel\nspectrophotometer described in detail by Schuster \\& Nissen (1988).\nAll the data reductions were performed using the NABAHOT\npackage (Arellano Ferro \\& Parrao 1989).\n\n\\begin{table*}[!ht]\n\\small{\n\\begin{center}\n\\caption[] {\\small Log of the observing seasons} \\hspace{0.01cm}\n    \\label{LOG}\n\\begin{tabular}{lcc}\n\\hline \\hline\nStar  &  Initial date  &  Final date \\\\\n\\hline  \\hline\nAE Boo   &  Jun 08, 2001 & Jun 11, 2001 \\\\\nTV Boo   &  Feb 04, 2002 & Feb 11, 2002 \\\\\nRS Boo \\& TV Boo &  Apr 08, 2004 & Apr 11, 2004 \\\\\nFull sample&  May 27, 2005&Jun 16, 2005\\\\\n\\hline \\hline\n\\end{tabular}\n\\end{center}\n}\n\\end{table*}\n\nIn those seasons in which the present RR Lyrae stars were supplementary\nobjects, fewer data\npoints were obtained each night.\nThe accumulated time span including all the seasons, however, was\nlarge enough to cover the whole cycle of all the stars in our sample.\nIn each season the same\nobserving routine was employed: a multiple series of integrations\nwas carried out, often five 10 s integrations of the star, to the\naverage of which a one 10 s integration of the sky was subtracted. A\nseries of standard stars taken from the Perry, Olsen \\& Crawford (1987)\nand Olsen(1983) was also observed with the same procedure\non each night to transform the data into the standard system. The\nseasonal transformation coefficients, except\nfor the 2002 season, are reported in Table 2. The indicated coefficients\nare those in the following equations (Gronbech, Olsen \\& Str\\\"omgren 1976).\n\n\\begin{eqnarray}\nV = A + B~(b-y)_{st} + y_{inst}  \\\\\n(b-y)_{st} = C + D~(b-y)_{inst}  \\\\\nm_{1~st} = E + F m_{1~inst} + G ~(b-y)_{inst}  \\\\\nc_{1~st} = H + I m_{1~inst} + J ~(b-y)_{inst} \\\\\n\\beta_{1~st} = K + L~\\beta_{1~inst}\n\\end{eqnarray}\n\n For the 2002 season the transformation was made by filter instead of by\ncolor indices. In this season the relations between the standard and the instrumental\nvalues for each filter were the following:\n\n\\begin{equation}\nV=y_{st}= a + b~ y_{inst}\n\\end{equation}\n\\begin{equation}\nb_{st} =c + d~ b_{inst}\n\\end{equation}\n\\begin{equation}\nv_{st}= e + f~ v_{inst}\n\\end{equation}\n\\begin{equation}\nu_{st}= g + h~ u_{inst}\n\\end{equation}\n\n\\noindent\nThe coefficients are reported in the bottom part of Table 2.\nThe mean dispersions of the transformation equations are $0.028$,\n$0.016$, $0.015$, $0.010$ mag in $u, v, b$ and $y$ respectively. It\nwas after this that the color indices were calculated. In Fig. \\ref{fig1}\nthe instrumental $y$ magnitude and color transformations are shown.\nIn Fig. \\ref{fig2} the relationships between the instrumental and\nthe standard magnitudes in the 2002 season are shown.\n\nThe final $uvby-\\beta$ photometry is available in electronic form from the Centre de Donn\\'ees Astronomiques,\nStrasbourg, France. The $V$ light curves are shown in Fig. \\ref{fig3}.\nIn the following section a brief description of the light and color curves for each star is given.\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=8.cm,height=8.cm]{fig1.eps}\n\\caption{Transformations from the instrumental to the standard $uvby$ system for the magnitude and colors. See eqs 1-5.}\n    \\label{fig1}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\includegraphics[width=8.cm,height=8.cm]{fig2.eps}\n\\caption{Transformations from the instrumental $uvby$ magnitudes to the standard system.\nSee eqs. 6-9.}\n    \\label{fig2}\n\\end{figure}\n\n\n\\begin{figure*}[ht]\n\\includegraphics[width=15.cm,height=12.cm]{fig3.eps}\n\\caption{V light curves of the sample stars.}\n    \\label{fig3}\n\\end{figure*}\n\n\\begin{table*}[!t]\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n  \\tablecols{9}\n \n\\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{Transformation coefficients\nof each season}\n\\label{COEFS}\n  \\begin{tabular}{lccccccrccccr}\n\\hline\nseason  &A & B      &C   &  D    &E &  F     &   G      &   H    &   I  &J &K &  L \\\\\n\\hline\nJun, 01  &18.076& $+0.020$ &1.354 & $0.991$&$-1.437$ & $1.012$ & $-0.002$&$-0.623$ & $1.001$ & $0.148$&$--$  &$--$ \\\\\nApr, 04  &18.839& $-0.010$ &1.539 & $0.993$&$-1.231$ & $0.949$ & $0.024 $&$-0.429$ & $0.988$ & $0.068$&$--   $  &$  --$ \\\\\nMay, 05  &19.000& $-0.007$ &1.531 & $0.983$&$-1.309$ & $1.068$ & $0.020 $&$-0.315$ & $1.031$ & $0.153$&$2.626$  &$-1.348$ \\\\\n\\hline\n \\end{tabular}\n\\end{center}\n\\begin{center}\n  \\begin{tabular}{lcccccccc}\n  & a         &  b     &  c     &   d      &   e    &   f   &  g & h  \\\\\n\\hline\nFeb, 02 & 19.1565  &1.0054  & 20.9731   & 1.0236 & 21.0131 & 1.0101 & 20.5204 & 1.0025 \\\\\n\\hline\n \\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\section{Notes on individual stars}\n\n\nDue to the fact that in most of the seasons the RR Lyrae stars were\nnot the primary target, the sampling of these stars was not\nhomogeneous. For example, AE Boo was observed for four\nnights in 2001 for a relatively short time on each night, although the\nphase coverage was almost complete. In the 2005 season, only a couple of\npoints per night were obtained for most RR Lyrae stars\nalthough the time span of the observations covered seventeen nights.\nFor AE Boo around 180 points in V were gathered.\n\nThe other amply observed star was TV Boo which was measured in 2002 (250\ndata points), in 2004 (5 points) and in 2005 (13 points).\nAccording to Wils et al. (2006)\nTV Boo shows a Blazkho period of 10 d. Given the time span of our\nobservations on this star and the uncertainties in the V magnitude\n(0.03 mag) we can neither verify nor contradict this assertion.\nWe call\nattention to the fact that the time of maximum light of this star\ndoes not coincide with a phase value of zero implying a secular\nvariation of the period. We did not pursue this possibility.\n\nThe last star which was\nobserved in more than one season was RS Boo. The data cover the\nfull cycle and the maximum light coincides with phase zero. Of the\nremaining stars, ST Boo, TW Boo, UU Boo and XX Boo,\nall observed in the 2005 season at the rate of a couple of points per night and\nfor seventeen nights, all\nshow a  maximum light shifted relative to phase\nzero at least with the light elements reported in Table 3.\nFor each of these stars a sample of around thirty-five data points was gathered.\n\n\n\\begin{table*}[!t]\n\n\\small{\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n \n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{Ephemerides and Fourier coefficients for the sample stars.}\n\\label{COEF}\n  \\begin{tabular}{lcccccccccccc}\n \\hline\nID & HJD & P & $A_0$  & $A_1$ & $A_2$ & $A_3$ & $A_4$ & $\\phi _{21}$ & $\\phi _{31}$ & $\\phi _{41}$ & N & Type \\\\\n& 2400000.0+ &  (days)   &$\\sigma_{A_0}$ &$\\sigma_{A_1}$ &$\\sigma_{A_2}$ &$\\sigma_{A_3}$ &$\\sigma_{A_4}$ & $\\sigma_{\\phi _{21}}$& $\\sigma_{\\phi _{31}}$& $\\sigma_{\\phi _{41}}$& \\\\\n\\hline\nRS Boo & 41770.4900 & 0.37733896 & 10.470 &0.447 &0.207 & 0.105& 0.084  &4.088 &1.881 &5.858 & 8 & ab\\\\\n       &            &            &  0.014 &0.019 &0.018 & 0.020& 0.018 &0.129 &0.215 &0.287 &   &   \\\\\nST Boo & 19181.4860 & 0.62229069 & 11.044 &0.383 &0.194 & 0.133& 0.097 &3.979 &2.129 &6.112 & 4 & ab\\\\\n       &            &            &  0.017 &0.023 &0.024 & 0.024& 0.024 &0.177 &0.268 &0.358 & & \\\\\nTW Boo & 26891.2680 & 0.53227315 & 11.309 &0.359 &0.138 & 0.069& 0.058 &3.782 &1.827 &5.438 & 4 & ab\\\\\n       &            &            &  0.027 &0.037 &0.039 & 0.035& 0.039 &0.359 &0.703 &0.772 &   &   \\\\\nUU Boo & 36084.4100 & 0.45692050 & 12.251 &0.504 &0.203 &0.086 & 0.055 &3.905 &2.074 &6.078 & 4 & ab\\\\\n       &            &            &  0.022 &0.030 &0.032 &0.035 & 0.035 &0.212 &0.431 &0.652 &   &   \\\\\nXX Boo & 29366.6460 & 0.58140160 & 11.914 &0.257 &0.114 &0.051 & 0.039 & 4.087 & 2.726& 6.189 & 4 & ab\\\\\n       &            &            &  0.012 &0.018 &0.015 &0.017 & 0.019 & 0.201 & 0.387& 0.476 &   &   \\\\\nAE Boo & 30388.2500 & 0.31489240 & 10.646 &0.202 &0.027 &0.008 & 0.008 & 4.903& 3.722& 2.377& 6 & c \\\\\n       &            &            &  0.004 &0.006 &0.006 &0.006 & 0.006 & 0.223& 0.730& 0.670&   &   \\\\\nTV Boo & 24609.5150 & 0.31255936 & 11.020 &0.282 &0.078 &0.020 & 0.015 & 3.915& 1.666 & 6.043 & 7 & c \\\\\n       &            &            &  0.001 &0.001 &0.001 &0.001 & 0.001 & 0.017& 0.051 & 0.074 &   &   \\\\\n\\hline\n  \\end{tabular}\n\\end{center}\n}\n\\end{table*}\n\n\n\n\\section{Iron abundance estimation}\n\n\\subsection{{\\rm [Fe\/H]} from the Fourier light curve decomposition}\n\nAs a first approach, the V light curves were decomposed\nin their harmonics and the Fourier coefficients used\nto estimate the iron abundance via semiempirical calibrations.\nIn order to do that, the light curves were phased with periods and the epochs listed in\nTable \\ref{COEF}, which were adopted from Kholopov et al. (1985).\n\nGiven the ephemerides, the light curves in Fig. \\ref{fig3} were fitted by the equation:\n\n\\begin{equation}\n\\label{fourier}\n    m(t)=A_0 + \\sum_{k=1}^N A_k \\cos(\\frac{2\\pi}{P}k\n    (t-E)+\\phi_k)\\nonumber\n\\end{equation}\n\n\nwhere $k$ corresponds to the $k-th$ harmonic of amplitude $A_k$ and displacement $\\phi _k$.\n\nThe Fourier coefficients, defined as:\n\n\n\\begin{eqnarray}\n    \\phi_{ij}& = & j\\phi_i - i\\phi_j \\nonumber \\\\\n       R_{ij}& = & A_i \/ A_j \\nonumber\n\\end{eqnarray}\n\n\\noindent\nhave been calibrated in terms of physical parameters.\nThe Fourier coefficients corresponding to the solid curve in Fig. \\ref{fig3}\nand the number of harmonics used to produce the best possible fit,\nare given in Table \\ref{COEF}. The number of significant harmonics depends on the\ndispersion of the light curve. Only significant harmonics were retained;  and their influence on [Fe\/H] estimated from the Fourier decomposition will be discussed at the end of this section.\n\nFor the RRab stars, [Fe\/H] was estimated using the calibration of\nJurcsik \\& Kov\\'acs (1996);\n\n\n\\begin{equation}\n\\label{JK}\n    {\\rm [Fe\/H]}_{\\rm J} = -5.038 ~-~ 5.394~P ~+~ 1.345~\\phi^{(s)}_{31},\n\\end{equation}\n\n\\noindent\nThe standard deviation in the above equation is 0.14 dex. In eq. \\ref{JK}, the phase $\\phi^{(s)}_{31}$ is calculated from a sine series. To convert the cosine series based $\\phi^{(c)}_{jk}$ into the sine series $\\phi^{(s)}_{jk}$,\none can use ~$\\phi^{(s)}_{jk} = \\phi^{(c)}_{jk} - (j - k){\\pi \\over 2}$.\nThe metallicity [Fe\/H]$_{\\rm J}$ from eq. \\ref{JK} can be converted to the\nmetallicity scale of Zinn \\& West (1984) (ZW) via [Fe\/H]$_{\\rm J}$ = 1.43 [Fe\/H]$_{\\rm ZW}$ + 0.88 (Jurcsik 1995). The values of [Fe\/H]$_{\\rm ZW}$ are given in Table \\ref{FEHDM}.\n\n\\begin{table}[!t]\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n  \\tablecols{4}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{[Fe\/H]$_{ZW}$ for the sample of RR Lyraes. }\n\\label{FEHDM}\n  \\begin{tabular}{lccc}\n \\hline\nID & [Fe\/H]$_{ZW}$ & $D_m$ & Type \\\\\n\\hline\nRS Boo& $-0.84$ & 4.1 & ab \\\\\nST Boo& $-1.53$ & 1.2 & ab \\\\\nTW Boo& $-1.47$ & 1.9 & ab \\\\\nUU Boo& $-0.95$ & 5.9 & ab \\\\\nXX Boo& $-0.81$ & 5.1 & ab \\\\\nAE Boo& $-1.30$ &  & c  \\\\\nTV Boo& $-2.04$ &  & c \\\\\n\\hline\n  \\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\nBefore applying eq.~\\ref{JK} to the five RRab stars in our sample, we calculated the\n$compatibility ~  condition ~ parameter$ $D_m$ which, according to\nJurcsik \\& Kov\\'acs (1996) and  Kov\\'acs \\& Kanbur (1998), should be smaller that 3.0.\nThe values of $D_m$ for the five RRab stars are given in Table \\ref{FEHDM}.\nIt is worth commenting that the $D_m$ parameter calculated for the RRab stars does not seem\nto correspond to the quality and\/or density of the light curve. For instance, from\nFig. \\ref{fig3} the curve of RS Boo is clearly the best and that of ST Boo is among the worst, however the corresponding $D_m$ values are 4.1 and 1.2 respectively, which is contrary to what would be expected. Thus we decided to relax this criterion a bit and apply the decomposition to the five RRab stars in our sample.\nWe shall discuss the uncertainties in [Fe\/H] due to the number of harmonics used to fit the data later in this section.\n\nFor the RRc stars we have calculated [Fe\/H] using the recent calibration of Morgan et al. (2007), which\nprovides [Fe\/H] in the ZW scale;\n\n\\begin{eqnarray}\n\\label{MORG}\n{\\rm [Fe\/H]}_{\\rm {ZW}} = 52.466 P^2 - 30.075 P + 0.131 (\\phi^{(c)}_{31})^2 \\nonumber \\\\\n+ 0.982 \\phi^{(c)}_{31} -4.198 \\phi^{(c)}_{31} P + 2.424\n\\end{eqnarray}\n\nThe results are listed in Table \\ref{FEHDM}.\n\nEq. \\ref{JK} is calibrated using the compilation of $\\Delta S$ values of Suntzeff et al. (1994)\ntransformed into [Fe\/H] and the spectroscopic values [Fe\/H]$_K$ of Layden (1994). On the other hand,\neq. \\ref{MORG} is calibrated using the [Fe\/H] values of Zinn \\& West (1984) and Zinn (1985) which are\nthe weighted average of iron abundances obtained from an assortment of methods, including the\n $\\Delta S$ and the Q$_{39}$ indices. Therefore the values of [Fe\/H] derived from the Fourier approach\nfor both RRab and RRc stars, are not completely independent from the values of [Fe\/H] estimated from $\\Delta S$, however\na discussion of $\\Delta S$ and [Fe\/H] values found in the literature for the stars in our sample is\nof interest and we present it in the following section.\n\nPerhaps the most relevant source of uncertainty in the physical parameters estimated from the Fourier decomposition approach is the quality and density of the light curve and hence its mathematical representation, alas, the number of\nharmonics needed to fit the observed data. For the four RRab stars ST Boo, TV Boo,\nUU Boo and XX Boo, whose light curves are scattered and\/or not very dense, the number of significant harmonics 3 or 4 is rather undistinguishable.  It can be noted that the values of $T_{\\rm eff}$ and $M_V$\nvary 25-90~K and 0.01-0.07 mag. respectively, while the most striking variation\ntakes place in [Fe\/H] where variations on average were $\\pm 0.19$ dex. This value is marginally larger than the declared mean standard deviations of eqs.\n\\ref{JK} and \\ref{MORG} (0.14 mag.) Thus we estimate the uncertainties in the values of [Fe\/H]$_{ZW}$ of about $\\pm 0.17$ dex. We have retained the cases with the smaller standard deviation from the fit and with exclusively significant harmonics in Table \\ref{COEF}.\n\n\\subsection{{\\rm [Fe\/H]} from the $\\Delta S$ parameter}\n\nIt is a common practice to estimate the value of [Fe\/H] for RR Lyrae stars\nusing the $\\Delta S$ metallicity parameter; Preston's (1959) method. In an extensive study of\nthe $\\Delta S$ parameter on RR Lyrae variables in the Galactic halo, Suntzeff et al.\n(1994) provided average values of  $\\Delta\nS$ with errors of $0.3$ units, which correspond to\n$0.05$ dex in [Fe\/H]. They reported $\\Delta S$ values for three\nstars of the present study: RS Boo, TW Boo and TV Bootes with values 1.36, 2.3 and 11.6. For RS Boo and TV Bootes\nthey give numerous estimations of $\\Delta S$, but since they correspond to similar phases, we have taken the averages and marked them with asterisks in Table \\ref{deltaS}. $\\Delta S$ values for ST Boo, TW Boo and XX Boo\nare found in the paper by Smith (1990), the average or individual values are given in Table \\ref{deltaS}.\nWe note here the discrepant values for TW Boo from Suntzeff et al.\n(1994) and Smith (1990) despite their corresponding to similar phases of 0.12 and 0.17 respectively. Also two discrepant values for XX Boo but for very different phases\nare found in Smith (1990). A value\nfor TV Boo is given by Liu \\& Janes (1990) as 12.1 which was measured near maximum light.\nA summary of the $\\Delta S$ values for our sample stars is given in Table \\ref{deltaS}.\n\n\\begin{table*}[!t]\n\\small{\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n  \\tablecols{7}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{$\\Delta S$ values for sample stars}\n\\label{deltaS}\n  \\begin{tabular}{lcccccr}\n \\hline\nID &  Type & Spectra & $\\Delta S$\\tabnotemark{1} & $\\Delta S$\\tabnotemark{2} &$\\Delta S$\\tabnotemark{3} &  Px (mas)\\\\\n\\hline\nRS Boo&   ab & A7-F5 & 1.36*$^m$ &          &  & $0.11\\pm 1.40$\\\\\nST Boo&   ab & A7-F7 &      & 8.9* &  & $1.19\\pm 1.61$\\\\\nTW Boo&   ab & F0-F8 & 2.3  & 6.2,8.3$^m$ &  & $-0.28\\pm 1.63$    \\\\\nUU Boo&   ab &       &      &          &  &     \\\\\nXX Boo&   ab &       &      & 7.0:,10.1: &  &     \\\\\nAE Boo&   c  & F2    &      &          &  & $0.32\\pm 2.00$ \\\\\nTV Boo&   c  & A7-F2 & 11.6*$^m$ & & 12.1  &  $-0.07\\pm 1.60$   \\\\\n\\hline\n \\tabnotetext{1}{Suntzeff et al. (1994)}\n \\tabnotetext{2}{Smith (1990)}\n \\tabnotetext{3}{Liu and Janes (1990)}\n \\tabnotetext{*}{average of multiple measurements }\n \\tabnotetext{m}{values from near minimum light}\n \\tabnotetext{:}{unknown phase}\n\\end{tabular}\n\\end{center}\n }\n\\end{table*}\n\n\\begin{table*}[!t]\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n  \\tablecols{8}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{Physical parameters from the calibrations for studied RR Lyrae stars. }\n\\label{FEMV}\n  \\begin{tabular}{lccccccccc}\n \\hline\nID & [Fe\/H]$_{ZW}$ & [Fe\/H]$_{\\Delta S}$ &[Fe\/H]$_{K}$ & $T_{\\rm eff}$ &$M_V$ & BC &$\\log L\/L_{\\odot}$ & d (pc) &  N \\\\\n\\hline\nRS Boo& $-0.84$ & $-0.81$ &$-0.32 \\pm 0.09$ & 6905&0.79 & 0.00& 1.58 & 862  &   8 \\\\\nST Boo& $-1.53$ & $-1.82$ &$-1.86 \\pm 0.14$ & 6454&0.48 & -0.05&1.73 & 1295 &   4 \\\\\nTW Boo& $-1.47$ & $-0.90,-1.46,-1.74$ &$-1.41 \\pm 0.09$ & 6581&0.58 & -0.05&1.69 & 1396 & 4 \\\\\nUU Boo& $-0.95$ &         &$-1.92 \\pm 0.20$ & 6818&0.56 & -0.01&1.68 & 2284 & 4 \\\\\nXX Boo& $-0.81$ & $-1.56,-1.98$ & & 6675&0.62 & -0.03&1.67 & 1818 & 4 \\\\\nAE Boo& $-1.30$ &         & & 7384&0.58 & 0.026&1.74 & 1032 & 6  \\\\\nTV Boo& $-2.04$ & $-2.18$ & & 7199&0.59 & 0.026&1.75 & 1218 & 7 \\\\\n\\hline\n  \\end{tabular}\n\\end{center}\n\\end{table*}\n\nWhile converting $\\Delta S$ into [Fe\/H] is a current practice, one should bear in mind that\nthe determination of $\\Delta S$ is subject to numerous sources of uncertainty, as have been\ndiscussed for example by Butler (1975), Smith (1986; 1990) and Suntzeff, et al. (1994),\ne.g. $\\Delta S$ estimated during the rising light can differ greatly from that\nestimated during the declining brightness. The correction to minimum phase is done using\nempirical curves on the $\\Delta S-SpT(H)$-plane with considerable dispersion and personal\njudgment in their final definition (Smith 1986; 1990), and the weighted mean finally reported value of\n$\\Delta S$ is often obtained from a few measurements randomly distributed along the pulsation\ncycle of the star. Some of the above considerations may explain the large differences in the  existing\nvalues of $\\Delta S$ for a given star (see Table \\ref {deltaS}).\n\n\nBeside the above sources of uncertainty in $\\Delta S$, we have to note that the transformation of\nthis observational parameter into [Fe\/H] encountered serious problems: the\nhigh dispersion abundance determinations in RR Lyrae are old (e.g. Butler 1975,\nButler \\& Deeming 1979, Butler et al. 1982) while modern high signal-to-noise,\nhigh dispersion digital data, analyzed with modern synthetic codes are non-existent. According to Manduca (1981), conversion of $\\Delta S$ to [Fe\/H] has calibration problems for the metal-rich and metal-poor domains and in fact his theoretical calibration is not linear. We have chosen however,\nto use the more recent\nempirical linear transformation of Jurcsik (1995); ${\\rm [Fe\/H]}_J = -0.190 \\Delta S$  $-0.027$,\nwhich is valid for ${\\rm [Fe\/H]}_J$ between 0.0 and $-2.3$ dex and for field and cluster\nRR Lyraes, to transform the $\\Delta S$ parameter\ninto iron abundances on the ZW scale.\nWe have first used the $\\Delta S$ values in Table \\ref{deltaS} and the formula of Jurcsik (1995), to calculate ${\\rm [Fe\/H]}_J$, and then brought this into the ZW scale as discussed in $\\S$ 4.1; we have labeled these iron values as [Fe\/H]$_{\\Delta S}$ and they are listed in Table \\ref{FEMV}.\n\nFor the sake of comparison, in columns 2-4 of  Table \\ref{FEMV} we have listed\nthe values of the Fourier estimations of the metallicity, [Fe\/H]$_{ZW}$, the values from\n$\\Delta S$, [Fe\/H]$_{\\Delta S}$ and the independent estimations from Layden (1994)\n[Fe\/H]$_{K}$. [Fe\/H]$_{K}$ values are based on the strength of the Ca II K line and are in the ZW scale.\n\nFor RS Boo [Fe\/H]$_{K}$ is rather discrepant but the agreement between [Fe\/H]$_{ZW}$ and  [Fe\/H]$_{\\Delta S}$ is excellent.\nFor ST Boo and TW Boo the agreement between the three estimations is very good especially considering the implicit uncertainties in the values of  $\\Delta S$.\nLarge discrepancies are noted for XX Boo and TV Boo. In the case of XX Boo one may argue that the coverage of the light curve can be considerably improved and hence the Fourier value could be expected to improve. However in the case of TV Boo, whose light curve is well defined and very dense, and whose\n$\\Delta S$ values are not scattered, we do not have an explanation handy.\nNevertheless, as an overall comparison of approaches to the determination of the iron abundance in RR Lyrae stars, and\nconsidering the uncertainties involved in the estimation of the  $\\Delta S$ parameter discussed above and the fact that in the Fourier solution the full shape of the light curve is included, we consider the Fourier decomposition a more solid approach.\n\n\\subsection{The effective temperature $T_{\\rm eff}$}\n\nThe effective temperature can also be estimated from the Fourier coefficients. For the RRab stars we used the calibrations of Jurcsik (1998)\n\n\\begin{equation}\n\\label{teffab}\n    log~T_{\\rm eff}= 3.9291 ~-~ 0.1112~(V - K)_o ~-~ 0.0032~[Fe\/H],\n\\end{equation}\n\n\\noindent\nwith\n\n$$ (V - K)_o= 1.585 ~+~ 1.257~P ~-~ 0.273~A_1 ~-~ 0.234~\\phi^{(s)}_{31} ~+~ $$\n\\begin{equation}\n\\label{color}\n~~~~~~~ ~+~ 0.062~\\phi^{(s)}_{41}.\n\\end{equation}\n\n\\noindent\nEq. \\ref{teffab}\nhas a standard deviation of 0.0018 (Jurcsik 1998), but the accuracy of $log~T_{\\rm eff}$ is mostly set by the color eq. \\ref{color}. The error estimate on  $log~T_{\\rm eff}$ is 0.003 (Jurcsik 1998).\n\nFor the RRc stars we have used the calibration of Simon \\& Clement (1993);\n\n\\begin{equation}\n    log T_{\\rm eff} = 3.7746 ~-~ 0.1452~log~P ~+~ 0.0056~\\phi^{(c)}_{31}.\n\\end{equation}\n\nThe values obtained from the above calibrations are reported in column 5\nof Table \\ref{FEMV}.\n\n\\section{Distance to the field RR Lyrae}\n\n\\subsection{$M_V$ from the Fourier parameters}\n\nAbsolute magnitudes of RR Lyrae can also be estimated from the Fourier parameters.\nFor the RRab one can use the calibration of  Kov\\'acs \\& Walker (2001);\n\n\\begin{equation}\n\\label{KWab}\nM_V(K) = ~-1.876~log~P ~-1.158~A_1 ~+0.821~A_3 +K.\n\\end{equation}\n\n\\noindent\nThe standard deviations in the above equation is 0.04 mag.\nThe zero point of eq.~\\ref{KWab}, K=0.43, has been calculated by Kinman (2002) using the prototype star RR Lyrae as calibrator, adopting the absolute magnitude $M_V= 0.61 \\pm 0.10$ mag\nfor RR Lyrae, as derived by Benedict et al. (2002) using the star parallax measured by the HST.\nKinman (2002) finds his result to be consistent with the coefficients of the $M_V$-[Fe\/H] relationship given by Chaboyer (1999) and Cacciari \\& Clementini (2003). All these results are consistent with the distance modulus of the LMC of $18.5 \\pm 0.1$ (Freedman et al. 2001; van den Marel et al. 2002; Clementini et al. 2003). Catelan \\& Cort\\'es (2008) have argued that the prototype RR Lyr has an overluminosity due to evolution of 0.064 $\\pm$ 0.013 mag relative to HB RR Lyrae stars of similar metallicity. This would have to be taken into account if RR Lyr is used\nas a calibrator of the constant $K$ in eq. \\ref{JK}. Considering this,  Arellano Ferro et al. (2008b) have estimated a new value of $K = 0.487$.\n\nFor the sake of homogeneity and better comparison with previous results on luminosities of RR Lyrae stars (e.g. Arellano Ferro et al. 2008a,b), in the following we have adopted $K=0.43$.\n\nFor the RRc stars, the calibration of Kov\\'acs (1998) was used;\n\n\\begin{equation}\n\\label{KWc}\nM_V(K) = ~-0.961~P ~-0.044~\\phi^{(s)}_{21} ~+4.447~A_4 + 1.261.\n\\end{equation}\n\nThe standard deviation in the above equation is 0.042 mag. In fact\nwe have propagated the errors, given in Table \\ref{COEF}, in the\namplitudes $A_1$ and $A_3$ in eq. \\ref{KWab} and in\n$\\phi^{(s)}_{21}$ and $\\phi^{(s)}_{41}$ in eq. \\ref{KWc} and found\nthat they produce\n uncertainties in $M_V$ $\\sim 0.04$ mag.\nCacciari et al. (2005) have pointed out that in order for the eq. \\ref{KWc} to surrender absolute magnitudes in agreement with the mean magnitude for the RR Lyrae stars in the LMC, $V_0 = 19.064 \\pm 0.064$ (Clementini et al. 2003), the zero point\nof the above equation should be decreased by 0.2$\\pm$0.02 mag.\nAfter this correction, we found the $M_V$ values for the RRc stars\nAE Boo and TV Boo reported in Table \\ref{FEMV}. These values of $M_V$ have been converted into\nlog $L\/L_{\\odot}$ (col. 5). The bolometric corrections for the average temperatures of RRab and RRc stars given in column 7 of Table \\ref{FEMV}, were estimated from the\n$T_{\\rm eff}$-BC$_V$ from the models of Castelli (1999) as tabulated in Table 4 of Cacciari et al. (2005).\n\n\\begin{table*}[!t]\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n  \\tablecols{8}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{$uvby-\\beta \\ $ photoelectric photometry and reddening of $\\delta$ Scuti stars in Bootes}\n\\label{delred}\n  \\begin{tabular}{lccccccc}\n \\hline\nstar &  $V$   & $b-y$   & $m_1$  & $c_1$ & $\\beta$ &  $E(b-y)$  & $uvby-\\beta$  \\\\\n & &  &    &  &  &  & source  \\\\\n\\hline\n$\\iota$ Boo &  4.75 & 0.128 & 0.198 & 0.834 & 2.817 & 0.001 & 1\\\\\n$\\gamma$ Boo &  3.03 & 0.116 & 0.191 & 1.008 & 2.817 & 0.006& 1\\\\\n$\\kappa$$^2$Boo &  4.54 & 0.125 &  0.187 &  0.951&  2.806 & 0.001  & 1 \\\\\nCN Boo  &  5.98 & 0.162 & 0.201 & 0.748 & 2.770  & 0.000 & 1\\\\\nYZ Boo  & 10.57 & 0.151 & 0.178 & 0.868 & 2.768 & 0.000 & 1\\\\\nYZ Boo  & 10.466 & 0.184 & 0.136  & 0.681 & 2.723 & 0.000 & 2 \\\\\n\\hline\n \\tabnotetext{1}{Rodr\\'{\\i}guez et al. (1994)}\n \\tabnotetext{2}{Pe\\~na et al. (1999)}\n \\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\subsection{Reddening}\n\nIn order to determine distance and detailed variations of\nthe physical parameters along the cycle of pulsation of each star,\nit is necessary to first estimate the reddening. For\nfield stars however, a proper determination of reddening is complex\nand no direct method is known to us. In view of this, we have\ndetermined the reddening of different objects in the same direction of the sky.\nWe have chosen five $\\delta$ Scuti stars and the globular cluster NGC~5466 in Bootes\nand M3 which is very near NGC~5466, as indicators of reddening in that direction of the sky.\nDespite the large distance to M3 and NGC~5466 (10.4 and 15.9 kpc respectively),\n$E(b-y)\\sim 0.0$ as estimated from the $E(B-V)$ values listed by Harris (1996).\nIn the case of the $\\delta$ Scuti stars we have used the expressions derived by\nCrawford (1975, 1979) for F- and A-type stars respectively, with the zero point correction\nsuggested by Nissen (1988), to estimate the intrinsic color $(b-y)_o$ for\nstars near the main sequence. Two sources of $uvby-\\beta$ were considered;\nRodr\\1guez et al. (1994) and Pe\\~na et al. (1999). These $\\delta$ Scuti stars,\ntheir magnitude-weighted mean colors and $E(b-y)$ are listed in Table \\ref{delred}.\n>From these results it seems reasonable to conclude that the\nreddening for the sample RR Lyrae stars in Bootes is negligible and we shall assume a value of\nzero for all the stars in our sample.\n\n\\begin{figure}[t]\n\\includegraphics[width=8.cm,height=8.cm]{fig4.eps}\n\\caption{$M_v- {\\rm [Fe\/H]}$ relation derived from cluster RR Lyrae stars (Arellano Ferro et al. 2008b). Blue points are the five RRab stars in our sample. When available, two positions are given; the larger points are the finally adopted solution while the smaller points correspond to similar solutions with smaller number of harmonics. Open blue circles correspond to the two RRc stars in our sample.\nThe uncertainties in the values of the blue symbols are discussed in the text and indicated by the\nblue error bars.}\n    \\label{fig4}\n\\end{figure}\n\n\\subsection{Distances}\n\nThe implied distances given by the Fourier absolute magnitudes, and the reddenings and bolometric\ncorrections discussed in $\\S$ 5.1 and 5.2 are listed in column 9 of Table \\ref{FEMV}. These\ndistances can be compared with estimations from the parallaxes,\nfor those stars which have parallaxes determined by the new reductions of the Hipparcos data by\nvan Leeuwen (2007). For the stars\nRS Boo, ST Boo, TW Boo, AE Boo and TV Boo the parallaxes and their errors are listed in Table \\ref{deltaS}.\nIt should be noted that their errors are very large and hence the parallaxes for these stars very uncertain.\nExcept for ST Boo, those numerical values lead to distances much larger than the values derived from\nthe Fourier approach in Table \\ref{FEMV}, and if these distances are used to calculate the\ncorresponding absolute magnitude they lead to absurd results.\n\n\n\\begin{figure*}[t]\n\\includegraphics[width=16.cm,height=16.cm]{fig5.eps}\n\\caption{Model grids on the $(b-y)_0 - c_0$ plane along the stellar variations through the cycle.}\n    \\label{fig5}\n\\end{figure*}\n\n\n\\section{$M_v$ from the Str\\\"omgren $c_1$ index}\n\nThe Str\\\"omgren $c_1$ index is a gravity (hence luminosity)\nsensitive parameter for late-A to F type stars (Str\\\"omgren 1966), and therefore\nit is thought to be useful in\ndetermining the absolute magnitude $M_V$ for RR Lyrae stars if properly\ncalibrated. In fact, in a recent paper Cort\\'es \\& Catelan (2008) have offered such calibration for RR Lyrae stars.\nThese authors have employed the\npseudocolour $c_0 \\equiv (u-v)_0 -(v-b)_0$, the fundamental period $P$\nand the metallicity $Z$ for a large number of synthetic RR Lyrae stars\nto calibrate $M_y$ and the colors $b-y$, $v-u$ and $u-v$ (their eq. 1).\n\n\\begin{table*}\n\\small{\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n  \\tablecols{7}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{Comparison of $M_V(F)$ obtained from the Fourier approach with two\ntheoretical calibrations from Cort\\'es \\& Catelan (2008).}\n\\label{MVC0}\n  \\begin{tabular}{lcccccc}\n \\hline\n Star  &  [Fe\/H]$_{ZW}$ & $(c_1)$ & $M_y(c1)$\\tabnotemark{1} & $M_y(Z)$\\tabnotemark{2} & $M_V(F)$\\tabnotemark{3}& $M_V(F)$-$M_y(c1)$ \\\\\n \\hline\nRS Boo & $-0.84$ & ~~0.928 &~~0.884 & ~~0.765 & 0.79 & $-0.093$\\\\\n & & $\\pm 0.012$ & $\\pm 0.016$ & $\\pm 0.063$ & & \\\\\nST Boo & $-1.53$ & ~~0.922 &~~0.319 & ~~0.526 & 0.48 & $+0.163$\\\\\n & & $\\pm 0.010$ & $\\pm 0.013$ & $\\pm 0.030$ & & \\\\\nTW Boo & $-1.47$ & ~~0.893 &~~0.537 & ~~0.577 & 0.58 & $+0.045$\\\\\n & & $\\pm 0.017$ & $\\pm 0.022$ & $\\pm 0.039$ & & \\\\\nUU Boo & $-0.95$ & ~~0.941 &~~0.659 & ~~0.665 & 0.56 & $-0.098$\\\\\n & & $\\pm 0.017$ & $\\pm 0.023$ & $\\pm 0.051$ & & \\\\\nXX Boo & $-0.81$ & ~~0.814 &~~0.678 & ~~0.776 & 0.62 & $-0.057$\\\\\n & & $\\pm 0.011$ & $\\pm 0.018$  & $\\pm 0.063$ & & \\\\\nAE Boo & $-1.30$ & ~~1.002 &~~0.880 & ~~0.619 & 0.58 & $-0.298$\\\\\n & & $\\pm 0.003$ & $\\pm 0.004$ & $\\pm 0.044$  & & \\\\\nTV Boo & $-2.04$ & ~~1.133 &~~0.625 & ~~0.481 & 0.59 & $-0.034$\\\\\n & & $\\pm 0.006$ & $\\pm 0.006$ &  $\\pm 0.018$ & & \\\\\n\\hline\n \\tabnotetext{1}{from $C_0, Z ~and~ P$ through eq. 1 of Cort\\'es \\& Catelan (2008).}\n \\tabnotetext{2}{from $Z$ through eq. 5 of Cort\\'es \\& Catelan (2008).}\n \\tabnotetext{3}{from the Fourier decomposition approach in this work, Table \\ref {FEMV}.}\n \\end{tabular}\n\\end{center}\n}\n\\end{table*}\n\nWe have fitted the $c_1$ curves with a curve of the form of eq.\n\\ref{fourier} to estimate the magnitude-weighted means $(c_1)$ given\nin column 3 of Table \\ref{MVC0}. These magnitude-weighted means\ndiffer from the intensity weighted means $<c_1>$ by about 0.01 mag.\n(Catelan \\& Cort\\'es, 2008), which are smaller than the uncertainties for\n$c_1$ of our observations as derived from the standard stars (see\nTable 3 in Pe\\~na et al., 2007). Since $E(b-y) = 0.$ for the sample\nstars (sec. 5.2) we have taken $c_0 = c_1$. For the conversion of\n[Fe\/H] to Z we have used log Z = [M\/H] - 1.765; M\/H = [Fe\/H] + log\n(0.638 $f$ + 0.362) and $f = 10^{[\\alpha \/Fe]}$ (Salaris et al.\n1993). We adopted $\\alpha = 0.31$ as an appropriate value for halo\npopulation stars. The predicted values of the absolute magnitude\nfrom $c_1$, $M_y(c1)$, are given in column 4 of Table \\ref{MVC0}.\n\nIn their equation 5 Cort\\'es \\& Catelan (2008) have calculated a\nquadratic calibration of the $M_y$ - log Z relationship which can be\nused to calculate $M_y(Z)$ given in column 5 of Table \\ref{MVC0}. As\nargued by Cort\\'es \\& Catelan (2008) $M_y(c1)$ refers to the\nmagnitude of an individual star whereas $M_y(Z)$ is the absolute\nmagnitude of a star of similar metallicity. Thus we shall compare\nthe Fourier approach results of each star, $M_V(F)$, with their\ncorresponding $M_y(c1)$. This comparison is made in column 7. The\nuncertainties of $M_y(c1)$  are the resutl of propagating the\nuncertainties in $<c_1>$  throught the equation 1 of Cort\\'es \\&\nCatelan (2008). The uncertainties $M_y(Z)$ are estimated by\npropagating the uncertinty in $\\sigma _{[Fe\/H]} = 0.17$ dex ($\\S$\n4.1). One can see that the dispersion in the calibration of equation\n1 in Cort\\'es \\& Catelan (2008) is $\\leq$ 0.01 mag (their Fig. 3)\nand we recall that the uncertainties in $M_V(F)$ is $\\pm 0.04$ mag\n($\\S$ 5.1). Therefore the differences in column 7 seem to be a bit\non the large side. It should be noted, however, that no systematics\ncan be seen and that if the $M_y(c1)$ values are plotted in Fig.\n\\ref{fig4} the dispersion of the field stars about the cluster $M_V-\n{\\rm [Fe\/H]}$ relationship becomes very large. Also it can be seen\nthat the values of $M_y(c1)$ for the two RRc stars, AE Boo and TV\nBoo are the most discordant despite of having the most densely\ncovered light curves and hence the smallest uncertainties.\n\n\n\\section{$M_V - {\\rm [Fe\/H]}$ relation}\n\nA recent linear version of the $M_V- {\\rm [Fe\/H]}$ relationship for RR Lyrae stars has been calculated by\nArellano Ferro et al. (2008a) based on the light curve decomposition technique of RR Lyrae stars in a\ngroup of globular clusters with a large range of metallicities. This relationship, reproduced in Fig \\ref{fig4}, has been amply discussed by Arellano Ferro et al. (2008b) who found it to be consistent with independent\nempirical linear versions and with theoretical non-linear versions after evolution from the Red Giant Branch is taken into account.\nTo check the consistency of our present results for the field stars in Bootes with those of RR Lyrae stars in globular clusters, we have plotted  in Fig \\ref{fig4} the seven stars in our sample using the\nFourier [Fe\/H]$_{ZW}$ and $M_V$ reported in Table \\ref{FEMV}. It can be seen that with some larger scatter the Bootes stars distribution, given the uncertainties, follow the trend of the globular cluster RR Lyrae. The error bars correspond to the uncertainties in the Fourier calibrations of [Fe\/H]$_{ZW}$ and $M_V$, i.e., those of eqs. \\ref {JK}, \\ref {MORG}, \\ref {KWab} and \\ref {KWc}.\n\n\\begin{table*}\n\\small{\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n  \\tablecols{14}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{$T_{\\rm eff}$ and $\\log~g$ variation ranges from the theoretical grids of LGK86}\n\\label{LGK86}\n  \\begin{tabular}{lccccccccc}\n \\hline\nstar & [Fe\/H] & $T_{\\rm eff}$&  $T_{\\rm eff}$ &  $T_{\\rm eff}$ &\n$\\Delta T_{{\\rm eff}}$ &  $\\log g$ &\n$\\log g$ & $\\Delta\\log g$ &  $\\Delta V$  \\\\\n   & adopted & Fourier & min &  max & $K$ & min & max & dex &  mag \\\\\n\\hline\nRS Boo & -1.0 & 6569 & 5700 & 8000 & 2300 & 2.2 & 3.5 & 1.2 &  0.82 \\\\\nST Boo & -1.5 & 6141 & 5700 & 7500 & 1800 & 1.5 & 3.0 & 1.5 &  0.73 \\\\\nTW Boo & -2.0 & 6262 & 5500 & 7000 & 1500 & 1.5 & 3.0 & 1.5 &  0.85 \\\\\nUU Boo & -1.0 & 6486 & 5500 & 8000 & 2500 & 1.5 & 3.0 & 1.5 &  0.63 \\\\\nXX Boo & -1.0 & 6350 & 5000 & 6800 & 1300 & 1.5 & 3.0 & 1.5 &  0.77 \\\\\nAE Boo & -1.5 & 7384 & 6100 & 7200 & 1100 & 1.9 & 2.8 & 0.9 &  0.48 \\\\\nTV Boo & -1.0 & 7199 & 6000 & 7500 & 1500 & 1.3 & 2.5 & 1.2 &  0.79 \\\\\n\\hline\n \\end{tabular}\n\\end{center}\n}\n\\end{table*}\n\n\\section{Determination of physical parameters along the pulsational cycle}\nGiven the\nsimultaneity in the acquisition of the data in the different color\nindices, once the reddening\nhas been inferred, it is possible to determine the variation of the\nphysical parameters of the star along the cycle.\nThis can be accomplished with the models developed\nparticularly for $uvby-\\beta \\ $ photometry by Lester, Gray \\& Kurucz\n(1986) (LGK86). The models have been built taking into account that the\n$uvby-\\beta \\ $ system is well designed to measure key spectral signatures\nthat can be used to determine basic stellar parameters. The\ntheoretical calibrations have the advantage of relating the\nphotometric indices to the effective temperature, surface gravity,\nand metallicity. LGK86 provide grids, on the plane $(b-y) - c_1$,\nof constant $T_{\\rm eff}$ and log g, for a large range of [Fe\/H] values.\nBased on the Fourier value [Fe\/H]$_{ZW}$, a model with the nearest [Fe\/H] value\nwas chosen for each star.\nIn Fig. \\ref{fig5} the cycle variation of each star on its corresponding grid is illustrated.\nThis allows an estimation of the $T_{\\rm eff}$ and log g variation ranges during the pulsation\ncycle and a comparison with the estimated temperature from the Fourier approach.\n\nIt is interesting to note that the studied stars have different\neffective temperatures and surface gravity limits, as well as\ndifferent ranges which cannot be determined with detail with only\nthe Fourier techniques. These results have been summarized in Table \\ref{LGK86}\nin which we have also included those determined through the\nempirical calibrations from the Fourier coefficients. As can be\nseen, both methods give analogous results. In Table \\ref{LGK86}\nthe variation ranges in V, $T_{\\rm eff}$ and log g are also indicated.\n\n\n\\section{Conclusions}\n\n>From data acquired in several photometric campaigns we have obtained\nextended $uvby-\\beta$ photometry over a relatively large time span\nfor two stars and data that adequately cover the cycle of pulsation\nfor all of them. The $V$ light curves have been Fourier decomposed\nand the corresponding Fourier parameters from their harmonics were\nused to calculate the iron abundance and luminosity of each star.\nThe reddening was estimated by considering different objects in\nthe same direction.  The unreddened Str\\\"omgren indices $c_0$ and\n$(b-y)_0$ served to determine the variation along the cycle of\nthe effective temperature and surface gravity.\n\n The iron abundance [Fe\/H]$_{ZW}$ was calculated\nfirst from the Fourier decomposition of the light curve and the\ncalibration proposed by Kov\\'acs and co-workers and described with\ndetail in $\\S$ 4.1. We also utilized the $\\Delta S$\nparameter to estimate the metallicity. In $\\S$ 4.2 we amply discuss\n  the uncertainties and the limitation of this technique. We\n  conclude that the Fourier decomposition approach gives more\nreliable results.\n\nSince RR Lyrae stars are distance indicators, the individual estimations\nof the absolute magnitude for field stars from independent methods is of interest.\nThe absolute magnitude, $M_V (F)$, predicted from Fourier decomposition ($\\S$ 5.1),\nis reported with other two determinations.\nOnce the iron abundance and the reddening were determined, an\nindependent estimate of $M_V (c_1)$ can be made from the $c_1$ index and the pseudocolor\n$C_0$ (Cort\\'es \\& Catelan 2008). This is described in $\\S$ 6. Also\nin that section\nwe calculated  $M_V (Z)$ from the metallicity alone making use of a theoretical\nquadratic  $M_V - Z$ relationship offered by Cort\\'es \\& Catelan (2008).\nAll these\nresults were compiled in Table \\ref{MVC0}. It was pointed out that, although in some individual cases the differences between $M_V (F)$ and for example $M_V (c_1)$ were larger than the\nexpected from the uncertainties of the methods involved,\nno systematic trends could  be seen and in some cases the agreement is fairly good.\nThe agreement does not seem to be related to the quality or the density of the\n$V$ light curve, since the disagreement is largest for the two RRc stars which also\nhave the best light curves in our sample.\n\nThe $M_V- {\\rm [Fe\/H]}$ relationship for RR Lyrae stars has been amply discussed in the\nliterature. A recent version of it calculated exclusively from the Fourier\ndecomposition approach of RRab and RRc\nstars in globluar clusters (Arellano Ferro et al. 2008b) was used to confront our\npresent results for the field stars (Fig \\ref{fig4}).\nIt can be seen that with some larger\nscatter the distribution of the Bootes stars follow the same trend of the\nglobular cluster RR Lyrae stars and it was noted that if the alternative results\nfor the absolute magnitude,  $M_V (c_1)$ or $M_V (Z)$ were used the quality of the comparison would remain.\n\nWith $M_V (F)$ and the reddening we have reported the resulting distances for the\nsample stars (Table \\ref{MVC0}).\nIt would be desirable to compare these distances with the derived distances from\nan independent technique. Unfortunately, those determined from the\nnew reductions of the Hipparcos catalogue (van Leeuwen 2007) have such large errors that the comparison is impossible.\n\n\\vspace{0.5cm}\n\nAcknowledgements. We would like to thank the following people for\ntheir assistance: the staff of the OAN, M. S\\'anchez and V. Alonso\nat the telescope during some of the 2005 observations and A. Pani\nfor those of 2002, and L. Parrao for the reduction of the 2002\nseason. We are grateful to an anonymous referee for useful\ncorrections and suggestions. This paper was partially supported by\nPAPIIT-UNAM IN108106 and IN114309. J. Miller and J. Orta did the\nproofreading and typing, respectively. This article has made use of\nthe SIMBAD database operated at CDS, Strasbourg, France and ADS,\nNASA Astrophysics Data Systems hosted by Harvard- Smithsonian Center\nfor Astrophysics.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\\section{Introduction}\n\\label{sec:intro}\n\nVoice conversion (VC) aims at converting the input speech of a source speaker to sound as if uttered by a target speaker without altering the linguistic content \\cite{kain1998spectral}.\nBesides the conversion of timbre, the conversion can also be conducted in various domains such as \npitch, rhythm or other non-linguistic domains. \nRepresentation learning methods for these speech factors have already been proposed and applied in\nmany research fields in speech processing\n\\cite{sadhu2021wav2vec, niizumi2021byol, liu2020mockingjay}.\nHowever, directly applying the speech representations extracted by these methods in VC may cause unexpected conversions of other speech factors as they may be not necessarily orthogonal.\nTherefore, disentangling the representations of intermingling various informative factors in speech signal is crucial to achieve highly controllable VC \\cite{li2018deep}, few-shot synthesis \\cite{van2020vector} and speaker adaptation \\cite{wang2020spoken}.\n\nIdeally, the VC technology is able to preserve the linguistic information and convert para-linguistic information.\nConventionally, only speaker and content information are factorized in VC.\nProsody, the important cue in speech signals, is not properly modeled in the VC framework.\nThere are explorations on the controllability of prosody in VC, among which SpeechSplit \\cite{qian2020unsupervised} is noticeable for its high controllability on multi speech factors. \nHowever, information-constraining bottleneck encoding layers can only gain limited disentanglement.\nThe entanglement of perceptual attributes engenders the low similarity which is elaborated in Section \\ref{sec:related}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn this paper, to achieve highly controllable style transfer for multiple factors VC, we propose a disentangled speech representation learning framework based on adversarial learning.\nThe proposed framework explicitly removes the correlations between the speech representations which characterize different factors of speech by an adversarial network inspired by BERT \\cite{devlin2018bert}. \nThe speech is firstly decomposed into four speech representations which represent content, timbre and other two prosody-related factors, rhythm and pitch.\nDuring training, one of the speech representations will be randomly masked and inferred from the remaining representations by the adversarial MAP\nnetwork. \nThe MAP network is trained to maximize the correlations between the masked and the remaining representations, while the speech representation encoders are trained to minimize the correlations by taking the reversed gradient of the MAP network. \nIn this way, the representation learning framework is trained in the adversarial manner, with speech representation encoders trying to disentangle the representations while MAP network trying to maximize the representation correlations.\nThe decoder reconstructs the speech from the representations during training and achieves VC on multiple factors by replacing the corresponding speech representations.\nExperimental  results  show  that  the  proposed  speech  representation learning  framework  significantly  improves  the robustness of VC on multiple factors, decreasing the MCD from 3.89 to 3.58 and outperforms state-of-the-art speech representation learning methods for multiple factors VC by a gap of 0.51 speech quality MOS.\n\n\\section{Related Work}\n\\label{sec:related}\nProsody is an important component of speech which usually reflects rhythm, intonation etc and there are explorations on prosody transfer as expressive and controllable speech synthesis is attaining more attention\\cite{skerry2018towards, 9420276}.\nA combination of explicit and latent variables are adopted to achieve high controllable and natural speech synthesis\n\\cite{valle2020mellotron}.\nThe explicit variables contain pitch contour, loudness besides speaker embedding.\nThe latent variables contain rhythm and duration information etc which are obtained from reference encoder \\cite{skerry2018towards}, denoted by global style tokens \\cite{wang2018style} or enhanced by pre-trained language model \\cite{zhao2020enhancing}.\n\n\nConventionally, only speaker and content information are factorized in VC.\nUnsupervised learning based-methods are garnering attention for the advantage of no need for text transcriptions and\nquite a lot of\nthem are based on auto-encoder architecture.\nVariational autoencoder \\cite{huang2018voice,elgaar2020multi}, vector quantization \n\\cite{wu2020one} \nand instance normalization based methods \\cite{chou2019one,chen2020again} were proposed to better model the latent space and pursue the regularization property.\nPrevious studies of prosody conversion mainly focus on transformation of F0 related features\n\\cite{du2020spectrum,lian2021towards} \nwhich gains limited conversion similarity.\n\n\n\n\n\n\nThe entanglement between different speech representations causes the low similarity and naturalness of synthesized speech whether in prosody transfer or timbre transfer.\nExpressive and high controllable speech synthesis systems share the same principle of disentangling multiple speech factors like speaker, linguistic and prosody-related information.\nIn order to foster disentanglement,\nadversarial training \\cite{chou2018multi, ocal2019adversarially}, contrastive learning \\cite{ebbers2020adversarial, li2020cvc,van2020vector}, and mutual information minimization \\cite{chen2016infogan, cheng2020improving} are applied to attenuate the information leakage.\nHowever, only the disentanglement between two factors, e.g., style, content or speaker\n\\cite{yuan2021improving,williams2019disentangling}\nare taken into account.\n\\input{structure}\nEffective disentanglement modeling for multi-factor voice conversion remains a challenging problem.\nTo overcome \nthat prosody is also converted while transferring timbre\nin conventional VC,\ndifferent information bottlenecks are applied to\ndecompose the speaker information into timbre and other prosody-related factors such as\nrhythm and pitch \\cite{qian2020unsupervised, takahashi2021hierarchical}.\nTo improve disentanglement, restricted sizes of bottleneck layers encourage the encoders to discard the information which can be learnt from other bottlenecks.\nRandom resampling \n\\cite{qian2020unsupervised}\nis also proposed to use in the information bottlenecks to remove rhythm information from content and pitch representations.\n\n\n\n\n\n\n\n\n\n\n\n\n\nHowever, without explicit disentanglement modeling, random resampling and restricting the sizes of bottleneck layers can only gain limited disentanglement of speech representations.\nRandom resampling which is implemented as dividing and resampling speech segment using linear interpolation along time dimension can only be used in removing time-related information such as rhythm.\nMoreover, random resampling is proved as a partial disentanglement algorithm that only contaminate a random portion of rhythm information \\cite{qian2020unsupervised}.\nThe content encoder actually is a residual encoder\nwhich cannot ensure that the content information is only modeled in the content representation.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Methodology}\n\\label{sec:method}\n\nWe aim to improve the converted speech quality by explicitly disentangling speech representations.\nAs shown in Figure \\ref{fig:structure}, the architecture of the proposed disentangled speech representation learning framework is composed of three sub-networks: (i) multiple speech representation encoders which encode speech into different speech representations characterising content, timbre, rhythm and pitch,\n(ii) \nthe MAP network that is trained to capture the correlations between different speech representations based on mask-and-predict operations,\n(iii) a decoder which is employed to generate spectrogram from these disentangled speech representations.\n\\input{mbv}\nAfterwards, the neural vocoder is utilized to synthesize audio from the generated mel spectrogram.\n\\subsection{Speech representation extraction}\nContent, rhythm and pitch encoders of \\cite{qian2020unsupervised} are adopted to extract content, rhythm and pitch representations from mel spectrogram and pitch contour respectively at frame-level, as shown in Figure \\ref{fig:structure}.\nDifferent from \\cite{qian2020unsupervised}, one-hot speaker labels (ID) are encoded by a training embedding table to obtain the timbre representation in the proposed framework.\n\nBesides the random resampling and restriction on sizes of bottleneck layers which is adopted in \\cite{qian2020unsupervised} and analyzed in Section \\ref{sec:related} to gain the limited disentanglement, we put on stricter constrain on the rhythm encoder output.\nAs shown in Figure \\ref{fig:structure}, the output of rhythm encoder is discretized.\nThe input of the rhythm encoder is mel spectrogram which contains much information\nand there is a distinct possibility that various messy information will be encoded into the rhythm representation with merely random resampling and restriction on sizes of bottleneck layers.\nConsidering that discrete variable typically offers a substantially reduced model capacity,\nwe applied the  \nMultilabel-Binary Vectors (MBV) \\cite{liu2019unsupervised} \nas the discrete rhythm representation.\nThe MBV with Gumbel-Softmax demonstrated in Figure \\ref{fig:mbv} is more data efficient than one-hot vector and continuous variable in information distillation task \\cite{liu2019unsupervised}.\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\\subsection{Adversarial learning for speech representation disentanglement}\nTo address the limitations of SpeechSplit \\cite{qian2020unsupervised} as discussed in Section \\ref{sec:related}, an adversarial Mask-And-Predict (MAP) network\ninspired by BERT \\cite{devlin2018bert} is designed to explicitly disentangle the extracted speech representations.\nDuring training, one of the four speech representations is randomly masked and the adversarial network tries to infer the masked representation from the other three representations.\nThe adversarial network is composed of a gradient reverse layer (GRL) \\cite{ganin2016domain} and a stack of prediction head layers \\cite{liu2020mockingjay} which has been used in masked acoustic modeling. \nAs delineated in Figure \\ref{fig:map}, in the MAP module,\neach prediction head layer is composed of a fully-connected layer, GeLU activation,\nlayer normalization\nand another fully-connected layer.\nThe gradient of the adversarial network is reversed by GRL\nbefore backward propagated to the speech representation encoders.\n$L1$ loss is adopted here to measure the adversarial loss which is\ndemonstrated below:\n\\begin{equation}\nZ = (Z_r, Z_c , Z_f, Z_u)\n\\end{equation}\n\\begin{equation}\nM \\in \\left\\{(0, 1, 1, 1), (1, 0, 1, 1), (1, 1, 0, 1), (1, 1, 1, 0)\\right\\}\n\\end{equation}\n\\begin{equation}\n\\small\n\\label{eq:adv}\n{L}_{adversarial} = \n\\left|\\left|(1-M)\\odot (Z - {\\rm{MAP}} (M\\odot Z))\n\\right|\\right|\n\\end{equation}\nwhere $\\odot$ is element-wise product operation,\n$L_{adversarial}$ is adversarial loss,\n$Z$ is the concatenation of $Z_r$, $Z_c$, $Z_f$, $Z_u$ denoting rhythm, content, pitch and timbre representations respectively,\n$M$ is a randomly selected binary mask corresponding to the dropped region with a value of 0 \nand 1 for unmasked representations.\nThe MAP network is trained to predict the masked representation as accurate as possible by minimizing the adversarial loss, \nwhile in the backward propagation, the gradient is reversed which encourages the representations learned by the encoder contain as little mutual information as possible.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{VC with disentangled speech representations}\n\nDuring training, four speech representations are extracted from same utterance and decoder is trained to reconstruct mel spectrogram from speech representations with the loss defined as:\n\\begin{equation}\n\\small\n\\label{eq:recon}\n{L}_{reconstruct} = \\|S-\\hat{S}\\|_{2}^{2}\n\\end{equation}\nwhere $S$ and $\\hat S$ are the mel spectrograms of the input and reconstructed speech respectively.\nFinal objective function with trade-off parameters is given in:\n\\begin{equation}\n\\small\n\\label{eq:loss}\n{Loss} = \\alpha * L_{adversarial}\n+ \\beta * L_{reconstruct}\n\\end{equation}\nwhere $\\alpha$, $\\beta$\nare the loss weights for adversarial loss and reconstruction loss.\nTo improve the robustness of proposed framework, the loss weight for the reconstruction loss is designed to be exponential decaying.\nMelGAN \\cite{kumar2019melgan} is adopted as the vocoder\nfor the high fidelity speech and fast decoding speed.\n\\input{map}\n\n\n\\section{Experiment}\n\\label{sec:typestyle}\n\\subsection{Training setup}\nThe experiments were performed on the CSTR VCTK corpus \\cite{veaux2016superseded}.\nWe randomly selected 107 speakers including \n62 females and 45 males.\nAfter pre-processing, the corpus duration for experiment was \n43.5 hours in total, 35.3 hours for training and 4.1 hours validation and test respectively.\nAll the audios were down-sampled to 16000Hz. \nMel spectrograms were computed through a short time Fourier transform (STFT) using a 50 ms frame size, 12.5 ms frame hop, and a Hann window function.\nWe transformed the STFT magnitude to the mel scale using an 80 channel mel filterbank spanning 125 Hz to 7.6 kHz, followed by log dynamic range compression.\nThe filterbank output magnitudes were clipped to a minimum value of 0.01.\nThe weights of adversarial loss was fixed to $10^{-1}$.\nThe weight of reconstruction loss $\\beta$ applied an initial weight of $1$ with decay factor of 0.9 every 500,000 steps.\nWe trained a vanilla SpeechSplit \\cite{qian2020unsupervised} as the \\textbf{Baseline} system and the system described in Section \\ref{sec:method}\nas \\textbf{Proposed}.\nWe used a pretrained Mel-GAN vocoder on VCTK corpus to synthesize the audios from the spectrogram.\nThere were three factors involved in the conversion process and\nwe conducted seven types conversion including rhythm-only conversion,\npitch-only conversion,\ntimbre-only conversion \nand combinations of them.\nWe evaluated baseline and proposed systems under the same settings, otherwise mentioned.\nWe programmed all neural networks used in the experiments based on an open source pytorch implemention of SpeechSplit\n\\cite{qian2020unsupervised}.\nWe trained all models with a batch size of 64 for 800,000 steps using the ADAM optimizer with learning rate fixed to $10^{-4}$ on a NVIDIA V100 GPU.\nThe demo is available \n\\href{https:\/\/thuhcsi.github.io\/interspeech2021-multi-factor-vc\/}{https:\/\/thuhcsi.github.io\/interspeech2021-multi-factor-vc\/}.\n\n\n\n\n\n\\subsection{Objective evaluation}\nWe calculated the Mel-cepstral distortion (MCD) on a sub set of the testing set which consists 543 parallel three-aspects\nconversion pairs and the results is shown in Table \\ref{tab:MCD}.\nThe proposed system outperforms the baseline with decreasing the MCD from 3.89 to 3.58.\nHere the MCD of the baseline system is calculated based on our own impementation of SpeechSplit \\cite{qian2020unsupervised}.\n\\input{MCD}\nTo verify timbre transfer ability of conversion systems, we analysed the speaker confusion results.\nThere were a number of speakers involved \nand we selected 10 speakers for demonstration.\nWe randomly selected 10 timbre\nconversion involved results of different target speakers.\nThe reference timbre was extracted from recording audio samples.\nWe calculated speaker embedding cross-similarity between utterances and the normalized histogram of similarity scores is shown in Figure \\ref{fig:histogram}.\nEach bar-column represents the number of utterance pairs corresponding to the similarity score. \nFor the proposed system,\nthe maximum similarity score between same speaker is 0.80 and the median score is 0.72.\nFor\nthe baseline system,\nthe maximum similarity score between same speaker is only 0.74.\nBy comparison, the audios converted by proposed system are more identifiable so as to characterize a specific speaker.\nThe baseline system has weaker timbre transfer ability as \nthe converted audios \nare prone to timbre flipping which is caused by \nspeaker information leakage.\nHowever, there exists that similarity scores of different speakers from these two systems exceed 0.6.\nWe found that when other speech factors \nare converted meanwhile, the performance of speaker classification degrades.\n\n\n\\input{histogram}\n\\subsection{Subjective evaluation}\nMean opinion scores (MOS) tests were conducted to evaluate the speech quality, prosody similarity\nand timbre similarity.\nThere were twenty listeners and eight three-aspects conversion pairs for each system involved in the evaluation.\nIn the prosody similarity MOS, the listeners were asked to rate how similar the converted samples sound to the reference sample in terms of pauses, emphasis and speaking rate.\nThe evaluation of the three aspects is independent and does not affect each other.\nWhen evaluating the similarity of timbre conversion, for example, we should not pay attention to speech quality or prosody.\n\nThe violin plots \\cite{hintze1998violin} of results are shown in Figure \\ref{fig:evaluation}.\nThe speech quality of the proposed system is more salient than the baseline.\nMost of the quality scores of the proposed system distribute in the range of 3.0$\\sim$4.5 with an average of 3.30.\nThe quality scores of the baseline system mainly distribute in the range of 2.5$\\sim$3.5 with an average of 2.79.\nThe baseline system is prone to generates more low-quality audios as there are quite a little scores are in the range of 1$\\sim$1.5.\nIn terms of prosody transfer, the scores of the two systems have a peak around 3.8 which means the two systems can effectively transfer the prosody.\nNevertheless, the upper limit of the proposed system is higher than the baseline which means that the proposed system can yield higher prosody similarity conversion results.\nIn terms of timbre transfer,\nthe baseline system is more likely to produce low-similarity audios by contrast.\nOverall, the performance of prosody transfer is better than the timbre transfer.\n\n\\input{evaluation}\n\n\\subsection{Ablation study}\nIn the experiment, we observed that while conducting rhythm-only conversion, \nthe content from the target-rhythm audio is also encoded into the converted audio.\nTarget content leaks into rhythm representation\ncauses the messy \nsource and target content mixture.\nTo evaluate the effect of MBV bottleneck on reducing the amount of content information encoded into the rhythm representation, we calculated the word error rate (WER) on rhythm-only conversion results.\nThere were 20 non parallel conversion pairs involved and the results are shown in Table \\ref{tab:ASR}.\nThe WER decreases from 43.92\\% to 28.89\\% after applying the MBV.\nThe content is wiped out from the rhythm representation more thoroughly and less target content leaks into converted speech.\n\\input{ASR}\n\nTo further elucidate the disentanglement performance of our proposed framework, \nwe generate mel spectrogram with one component removed by set the corresponding input as zero \\cite{qian2020unsupervised}.\nAs shown in Figure \\ref{fig:removed}, after the content information is removed, the spectrogram of the proposed system is composed of \nmore uninformative slots and the formant pattern is blur which indicates the missing phone information.\nIt can be observed that our proposed system removes the content information more thoroughly than the baseline which means that in our system, the amount of content information leaks into other encoder is less.\nGiven the space limit, Figure \\ref{fig:removed} only shows the results of content removed.\nThe results of other component removed are similar to content which the corresponding right information is missing in the synthesized mel spectrogram.\nWhen the rhythm information is removed, the output spectrogram is blank.\nWhen timbre is removed, the formant position is more random and when pitch is removed, the pitch contour is flatter.\n\\input{removed}\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:majhead}\n\nIn order to achieve a\nhighly controllable \nstyle transfer on multiple factors in VC,\nwe propose a disentangled speech representation learning framework based on adversarial learning.\nWe extract four speech representations\nand employ MAP network \nto further disentangle speech representations.\nExperimental results show that the proposed speech representation learning framework significantly improves the robustness of VC\non multiple factors.\nInvestigations of the design of masking strategies is left for future work.\n\n\n\n\n\n\n\n\n\n\n\\section{Acknowledgement}\nThis work was conducted when the first author was an intern at Huya Inc., and was supported by National Natural Science Foundation of China (NSFC) (62076144), joint research fund of NSFC-RGC (Research Grant Council of Hong Kong) (61531166002, N\\_CUHK404\/15), Major Project of National Social Science Foundation of China (NSSF) (13\\&ZD189).\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\\IEEEPARstart\nIn recent years, transportation activities continue to have significant impacts on energy consumption and air quality. In 2017, about 29\\% of U.S. energy consumption was for transporting people and goods from one place to another \\cite{USEIA2017}. Over 90\\% of the fuel used for transportation is petroleum based, including primarily gasoline and diesel. Transportation-related greenhouse gas (GHG) emissions was the second largest contributor to the nationwide GHG emissions inventory in 2017, accounting for approximately 28.9\\% of total U.S. GHG emissions \\cite{USEPA2017}. In 2018, the transportation sector exceeded the power sector as the largest contributor of $CO_2$ emissions in the U.S. \\cite{Rhodium2019}.\n\nThese facts have been increasing public awareness of the need to reduce energy consumption and pollutant emissions generated by our contemporary transportation system. Among all the strategies to mitigate the transportation-related environmental footprint, eco-driving along urban arterial has gained significant research interest around the world \\cite{barth2009energy, kamal2010on, saboohi2009model, mensing2011vehicle, wang2018cluster}. By applying connected and automated vehicle (CAV) technology, drivers would effectively reduce the number of full stops and idling time, and avoid unnecessary acceleration and deceleration by receiving downstream traffic information, such as queue length and signal phase and timing (SPaT). The Eco-Approach and Departure (EAD) application is a typical example, where drivers are guided to approach and depart from signalized intersections in an environmentally friendly manner using SPaT and geometric intersection description (GID) information sent from the wireless roadside units (RSU) installed as part of the signalized intersection infrastructure \\cite{xia2013development}, \\cite{barth2011dynamic}. In such a manner, CAVs can increase their energy efficiency and decrease pollutant emissions by simply following well-designed speed profiles while traveling through the intersection. Results of microscopic simulation models showed a 10-15\\% reduction on energy consumption and CO2 emissions by applying the EAD application to fixed-timing signalized intersections \\cite{xia2013dynamic}. In congested traffic conditions, the EAD application also worked effectively where preceding queues could be estimated by leveraging real time vehicle detection and signal information system \\cite{he2015optimal}. Moreover, it was also revealed in previous studies that, drivers' behavior adaptability under actual driving conditions also played an important role in the effectiveness of the EAD application \\cite{xiang2015aclosed}.\n\nMany studies have been focused on developing eco-driving systems and evaluating the proposed systems by numerical or microscopic simulations. However, only a few implemented the proposed technology on real vehicles. A field test along the El Camino Real corridor in Palo Alto, CA was conducted to evaluate the EAD application for \\textit{actuated signals} in real-world traffic\\cite{hao2019eco}. The tested track is 1.7 miles long, and all intersections are equipped with DSRC RSUs. A test vehicle (2.5 L 4-cylinder) was equipped with an industry-grade radar, GPS, DSRC modem, on-board computer, and an artificial dashboard. It was shown up to 6\\% energy and emissions reduction can be achieved in real-world traffic. Additionally, a partially automated (longitudinal control) EAD prototype was developed and demonstrated at the Turner-Fairbank Highway Research Center (TFHRC) in McLean, Virginia, and the results showed an average of 17\\% in energy consumption reduction \\cite{altan2017glidepath}.   \n\n\\newpage\n\n\\begin{figure*}[!b]\n    \\centering\n    \\includegraphics[width=2.0\\columnwidth]{figure\/\/Architecture4.png}\n    \\caption{Architecture of the connected eco-driving system for heavy-duty trucks.}\n    \\label{flowchart}\n\\end{figure*} \n\nBuilt on top of the authors' previous work on light-duty vehicles, a connected eco-driving system for heavy-duty trucks has been developed in this study. Since heavy-duty trucks play a significant role in transportation-related (especially on goods movement) energy consumption and pollutant emissions, we have adapted the existing eco-driving system to heavy-duty trucks to better shape our contemporary transportation system. Furthermore, different from our previous implementation with DSRC, we have used 4G-LTE to enable cellular-based wireless communications in this study, given its benefits of scalability and wider coverage. We have implemented a simplified eco-driving algorithm to calculate the recommended speed for the test truck in real time. The algorithm has been strategically integrated with other system components, such as Global Navigation Satellite Systems (GNSS), on-bard maps, radar, X Electronic Control Unit (XECU), and live SPaT streams. A user-friendly driver-vehicle interface (DVI) has been designed, so the driver can follow the recommended speed to drive the truck to achieve the goal of energy savings.  A field test of the developed connected eco-driving system has been conducted in Carson, California with a heavy-duty diesel truck, along a test route with six equipped signalized intersections capable of broadcasting their SPaT information. \n\nThe remainder of this paper is organized as follows: Section II demonstrates the architecture of the proposed connected eco-driving system for heavy-duty trucks. Section III elaborates two key modules: map matching and eco-driving algorithm of the proposed system. The field testing conducted in Carson, CA is presented in Section IV, followed by result analysis. Section V concludes this paper along with the discussions on future work.\n\n\\section{Architecture of the Connected Eco-Driving System}\nIn this section, the architecture of the proposed connected eco-driving system is introduced, with illustrations of both software and hardware components installed. As can be seen from Fig. \\ref{flowchart}, the whole system is divided into five different modules: communication, localization, perception, planning, and control, whose respective functions are elaborated below.\n\n\\subsection{Communication Module}\nThe communication module facilitates real-time and reliable \\textit{vehicle-to-network} (V2N) and \\textit{ infrastructure-to-network} (I2N) communications. Although short-range communication technology such as DSRC is widely used by various CAV applications, their low scalability and asynchronous disadvantage may lead to several issues. As a substitute, 4G-LTE has been adopted to enhance V2N and I2N communications given its versatile communications types from one-to-one to one-to-multiple transmissions, with both simulations and field experiments conducted by researchers all over the world during the past decade \\cite{xu2017DSRC, nshimiyimana2016comprehensive}. In our connected eco-driving system, we also install a 4G-LTE cellular module with AT\\&T SIM card on board to enable V2N communication. Note that, compared to DSRC, one major disadvantage of 4G-LTE is its relatively longer latency. However, based on the our findings in our field implementation of this study, the latency introduced by 4G-LTE is about 100 ms, which is satisfactory for the connected eco-driving system.\n\nEach signal head on the test track is controlled by a McCain 2070E controller, which is a rugged, multi-tasking field process and communication system configurable for a variety of traffic management applications. Additionally, each controller is equipped with a SIEMENS RM1224 mobile wireless router for 4G-LTE connection, so the raw SPaT data could be sent to the cloud server through I2N communication. Then the 4G-LTE cellular module on the truck can request the live stream of SPaT information from the cloud server at every second.\n\n\\subsection{Localization Module}\nThe localization module of the connected eco-driving system consists of hardware component, a global navigation satellite system (GNSS) and inertial navigation system (INS), and software component, a map matching algorithm. The GNSS \\& INS component serves as a combined satellite \\& inertial-based navigation system, which can be optionally augmented by terrestrial reference stations \\cite{geiger2012team}. This component can provide precise position, movement, and posture measurements for the self-localization and altitude determination of the truck by differential correction. On the test truck, a versatile U-BLOX NEO-M8N GNSS module is installed to locate the truck's position. This GNSS module has a navigation sensitivity of -167dBm, which provides outstanding positioning accuracy in scenarios where urban canyon or weak signals are involved.\n\nFor the map matching component, a pre-built map of the testing field is available on board, with information such as the intersection ID, signal group ID, signal location, road type, road name and direction, and road speed limit. The truck's coordinates (\\textit{i.e.}, longitude, latitude, and heading) received from the GNSS \\& INS component can then be matched to the pre-built map by the proposed map matching algorithm, where the details of the algorithm will be introduced in Sec. \\ref{sec:algorithms}A.\n\n\n\\subsection{Perception Module}\nDifferent from the communication module that provides information about the signal status (\\textit{i.e.}, SPaT information), the perception module can provide information about other surrounding vehicles (especially if they are not CAVs). In this connected eco-driving system , we adopt the forward sensing kit to detect any preceding vehicles, bicycles, and\/or pedestrians. The Volvo EU Production Radar Unit together with the forward-looking camera can detect the movement of preceding vehicle, and send its dynamics information to the sensor fusion component. Once the detected objects are classified by the sensor fusion component, the desired information is sent to the eco-driving algorithm. Given the measured relative speed and distance between the truck and its preceding vehicle, the time-to-collision value can be computed. If the time-to-collision value is lower than a preset threshold, the suggested speed for the truck driver will be turned off on the DVI, and the driver will drive the truck without any auxiliary information. \n\n\\subsection{Planning Module}\nThe planning module is the core of the proposed connected eco-driving system. The XECU of the truck is equipped with a ruggedized computer, which has access to multiple channels of CAN data from different sub-networks of the truck. It receives the distance to the next signal from the localization module, preceding vehicles' states from the perception module, and the speed of the truck from CAN message.\n\nThe eco-driving algorithm is embedded in the XECU, based on a Microsoft .NET 4.7 framework with C\\# language. Data query, recommended speed calculation, and the signal display are all updated at 10 Hz frequency to achieve the consistency, while the communication module is operating at 1 Hz due to the restriction on infrastructure. All the concurrent tasks are handled by threading in the main thread with shared memory. Upon receiving all this information, the recommended speed of the truck at the next time step for the driver will be computed, and sent to the DVI along with some other information. On the other hand, we also utilize a package called log4net for archiving the raw data and the computation results. The details of the eco-driving algorithm will be introduced in Sec. \\ref{sec:algorithms}B.\n\n\\subsection{Control Module}\nOnce the recommended speed is computed by the eco-driving algorithm in the planning module, the control module of the connected eco-driving system starts to play. A DVI is shown on the monitor right above the dashboard of the truck. With the auxiliary help from the DVI, the driver controls the longitudinal movement of the truck via the throttle and brake, and controls the lateral movement  by steering wheel.\n\nThree typical scenarios are shown in Fig. \\ref{DVI}. As can be seen from these subplots, the top of the DVI shows the upcoming signal phase, but not necessarily the signal timing. We only show the signal timing when the truck is at full stop (shown as the right subplot), but hide that information in other scenarios (the left and middle subplot) to avoid distraction to the driver while driving the truck. The right hand side of the DVI is the speed limit of the current roadway segment, which is acquired from the pre-built map. The artificial speedometer is put at the bottom of the DVI with a green band to demonstrate the recommended speed range for the truck driver. As can be seen from the right subplot, if a vehicle is detected in front of the truck within a certain range, a vehicle icon is shown in the middle of the DVI, and the upper bound of the recommended speed is bounded by the speed of the preceding vehicle. If the computed time-to-collision value is lower than a preset threshold, the DVI would display a warning message instead of showing the SPaT information, allowing the driver to disengage the eco-driving system and be aware of potential rear-end collision by himself\/herself.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.32\\columnwidth]{figure\/\/dvi_screenshot1.png}\n    \\includegraphics[width=0.32\\columnwidth]{figure\/\/dvi_screenshot2.png}\n    \\includegraphics[width=0.32\\columnwidth]{figure\/\/dvi_screenshot3.png}\n    \\caption{Screen shots of DVI designed for the truck driver.}\n    \\label{DVI}\n\\end{figure}\n\n\\section{Algorithms of the Connected Eco-Driving System} \\label{sec:algorithms}\nIn this section, we briefly introduce two crucial algorithms in the developed connected eco-driving system: The map matching algorithm in the localization module, and the eco-driving algorithm in the planning module.\n\n\\subsection{Map Matching Algorithm}\nThe first step of our map matching algorithm is to find the projection point of truck's current latitude and longitude on the nearest lane of the hard-coded map. Since we calculate the distance on the surface of the earth, Haversine formula can be applied as:\n\\begin{equation}\\small\n    d = 2r \\cdot sin^{-1}(\\sqrt{sin^2(\\frac{\\Delta\\varphi}{2}) + cos(\\varphi_1)cos(\\varphi_2)sin^2(\\frac{\\Delta\\lambda}{2})})\n\\end{equation}\nwhere $d$ is the distance between two points on the sphere, $r$ is the radius of the earth, $\\varphi_1$, $\\varphi_2$, $\\Delta\\varphi$ represent the latitude of the first point, the latitude of the second point, and  the difference between $\\varphi_1$ and $\\varphi_2$, respectively, $\\Delta\\lambda$ is the longitude difference between the two points.\n\nSince the direction to the next signal should be consistent with the current heading of the truck, we verify this procedure by using inner product:\n\n\\begin{equation}\n\\hat{h} \\cdot \\hat{u} > 0\\\\\n\\end{equation}\nwhere $\\hat{h}$ is the current heading of the truck and $\\hat{u}$ is the direction of the intersection. \n\nGiven that the GNSS signal might drift from the current lane of the road, errors could be generated and accumulated if we directly compute the distance to the intersection node. Therefore, we utilize the triangle formed by the truck's coordinate and its two adjacent nodes' coordinates to calculate the real distance from the projection point of the truck to the intersection. Heron's formula can be given as:\n\n\\begin{equation}\nA = \\sqrt{s(s-d_{tn1})(s-d_{tn2})(s-d_{n1n2})}\n\\end{equation} \nwhere $A$ is the area of the triangle whose sides have lengths $d_{tn1}$ (truck to the previous node), $d_{tn2}$ (truck to the next node), and $d_{n1n2}$ (the previous node to the next node), and $s$ is the semi-perimeter of the triangle defined as\n\n\\begin{equation}\ns = \\frac{d_{tn1} + d_{tn2} + d_{n1n2}}{2}\n\\end{equation}\n\nThen, we can compute the height $h$ of the triangle by the area and the base-side $d_{n1n2}$, where $h$ is considered as the error distance due to the drift of GNSS signal from the current lane. Based on $h$ and $d_{tn2}$, we could use Pythagorean theorem to compute the distance $d_{pn2}$ from the projection point of the truck to the downstream node. Finally, the distance from the truck to the next signal $d_{sig}$ can be calculated by adding the resulted distance with the lengths of all downstream segments before the signal:\n\\begin{equation}\nd_{sig} = d_{pn2} + d_{n2n3} + d_{n3n4} + ... + d_{n(m-1)n(m)}\n\\end{equation}\nwhere $n(m)$ is the node with respect to the next signal.\n\n\\subsection{Eco-Driving Algorithm}\nAlthough we developed several versions of eco-driving algorithms before (\\textit{e.g.}, piece-wise trigonometric algorithm \\cite{xia2013development, xia2013dynamic}, graph-based algorithm\\cite{Jin2016ecoacc} \\cite{hao2018connected}), in this system, we develop a simplified version of eco-driving algorithm for two major reasons:\n\\begin{itemize}\n    \\item Compared to previous eco-driving algorithms, this simplified version of algorithm is easier to implement on real vehicles, and also more attractive in terms of real-time performance;\n    \\item This version of algorithm computes a feasible speed range instead of a single value, and provides an easy-to-follow speed band on the DVI for the truck driver. Since the ability to precisely follow the recommended speed is limited and varied by the driver, and the response time of the driver to maneuver the truck may be longer than light duty vehicles, the feasible speed range option should be more robust and user-friendly for truck drivers.\n\\end{itemize} \nIn this subsection, we demonstrate the algorithm with parameters $d_{sig}$ as distance to the intersection, $t_{current}$ as the time left of current signal, and $v_{lim}$ as the speed limit of the roadway. The reference speed $v_0$ of the truck can then be given as\n\\begin{equation}\nv_0 = d_{sig}\/t_{current}\n\\end{equation}\n\nWe then categorize the eco-driving algorithm into two different scenarios based on the current signal phase:\n\n\\subsubsection{Red Phase}\n\\begin{equation}\n    \\left\\{\n                \\begin{array}{ll}\n                  v_{lower} = 0, v_{upper} = v_0 & \\textit{, if } v_0 \\leq v_{lim}\\\\\n                  v_{lower} = 0, v_{upper} = v_{lim} & \\textit{, otherwise}\\\\\n                \\end{array}\n              \\right.\n\\end{equation}\n\n\\noindent When the signal phase at the upcoming intersection is red, the lower bound of the suggested speed $v_{lower}$ would always be zero. However, the upper bound $v_{upper}$ would be set to the speed that the truck can pass the intersection just at the moment it turns green without stopping. The highest speed is also constrained by the speed limit $v_{lim}$ of the roadway.\n\n\\subsubsection{Green Phase}\n\\begin{equation}\n    \\left\\{\n                \\begin{array}{ll}\n                  v_{lower} = v_0, v_{upper} = v_{lim} & \\textit{, if } v_0 \\leq v_{lim}\\\\\n                  v_{lower} = 0, v_{upper} = 0 & \\textit{, otherwise}\\\\\n                \\end{array}\n              \\right.\n\\end{equation}\n\n\\noindent On the other hand, when the signal at the upcoming intersection is green, the upper bound $v_{upper}$ of the recommendation would be the speed limit, and the lower bound $v_{lower}$ is set to the lowest speed that the truck can pass the intersection during the current green phase. Under the circumstances that even using the highest speed (\\textit{i.e.}, speed limit) the truck still cannot pass the intersection, the algorithm will suggest the driver to slow down and wait for the next green phase.\n\n\n\\section{Field Implementation}\n\n\nIn this work, we adopt a Volvo VNL heavy-duty diesel truck for our field implementation, which is a 2015 Volvo VNL model with a 13L diesel engine (455 HP) and a maximum cruise speed of 115 km\/h and is shown in Fig. \\ref{Truck}. The test runs are conducted in Carson, CA, which locates right next to the Port of Los Angeles and Port of Long Beach, two busiest container ports in the United States. Numerous factories and warehouses are located in the testing area, so the proposed connected eco-driving system can potentially save energy consumed by heavy-duty trucks travelled in this area. Fig. \\ref{Route} shows the test route where six signalized intersections on Wilmington Blvd. and Alameda St. are equipped with communication modules. The respective SPaT information can be sent to the cloud and allow our connected truck to fetch the data.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\columnwidth]{figure\/\/Truck_view3.jpg}\n    \\caption{Field implementation is conducted by the heavy-duty truck in Carson, California.}\n    \\label{Truck}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\columnwidth]{figure\/\/Route.png}\n    \\caption{Field implementation route in Carson, CA with six equipped signalized intersections.}\n    \\label{Route}\n\\end{figure} \n\nFor a comparative study, the truck ran a few times along the test route with and without the developed connected eco-driving system. As it was driven in real traffic environment, where the traffic flow may vary over time of the day, it is not fair to quantitatively compare the energy consumption between different test runs. Therefore, in this section of the paper, we select results generated during four test runs, where two of them are with connected eco-driving system and the other two are without it (\\textit{i.e.,} baseline). To make this comparison as fair as possible, all four selected test runs are conducted at the same intersection with clear downstream traffic, which means the movement of the truck is not affected by other vehicles on the road. Also, in each scenario shown in Fig. \\ref{Acceleration} and Fig. \\ref{Deceleration}, the truck approaches the intersection from the same distance with the same SPaT status (shown as the color bar below the x axis). Therefore, the difference between two trajectories (eco-driving and baseline) in each plot is mainly caused by the existence of the proposed connected eco-driving system.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\columnwidth]{figure\/\/Acceleration.png}\n    \\caption{Test results of the acceleration scenario of the connected eco-driving vs. the baseline.}\n    \\label{Acceleration}\n\\end{figure} \n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\columnwidth]{figure\/\/Deceleration.png}\n    \\caption{Test results of the deceleration scenario of the connected eco-driving vs. the baseline.}\n    \\label{Deceleration}\n\\end{figure} \n\n\nFig. \\ref{Acceleration} shows the comparison results on the acceleration scenario between the connected eco-driving system and the baseline. The baseline curves show the truck starts to heavily decelerate at 23 seconds when the driver sees the amber light 150 m away from the intersection, and stops fully at the intersection stop line at 35 seconds. However, with the help of the connected eco-driving system, the connected truck starts to accelerate prior to the signal turns to amber. Therefore, it can travel through the intersection without a full stop before the signal turns to red. Fig. \\ref{Deceleration} shows the comparison results on the deceleration scenario between the connected eco-driving system and the baseline. The baseline curves show the truck starts to decelerate at 18 seconds while the driver sees the amber light 150 m away from the intersection, and stops fully at the intersection stop line at  28 second. However, with the help of the connected eco-driving system, the connected truck starts to decelerate prior to the signal turns to amber, and conducts a relatively smoother deceleration towards the intersection. Therefore, it can pass the intersection right after the signal turns to green with a low speed, instead of experiencing a full stop. We also adopt U.S. EPA's MOVES model to calculate the fuel consumption of aforementioned four test runs, where in each scenario the starting and ending speed are the same (Fig. \\ref{Acceleration} and Fig. \\ref{Deceleration} only show partial trajectories) \\cite{MOVES}. The results of MOVES show that the connected eco-driving system can provide approximately 9\\% and 4\\% fuel savings in the acceleration and deceleration scenario, respectively.\n\nThose four test runs shown above, along with many other test runs we have conducted, prove the proposed connected eco-driving system can smooth the speed variations of the truck, avoid unnecessary full stop at the intersection, and therefore reduce the fuel consumption of the heavy-duty truck. Another major finding of this field implementation is that, the cellular-based wireless communication approach enables CAVs to receive SPaT information from anywhere (not limited to a certain distance to the intersection), so the connected eco-driving can be implemented to a much larger scale. A disadvantage of this approach is an extra 100 ms communication delay compared to DSRC approximately, but this delay does not affect the behavior of this system to a large extent. The video of the field implementation could be watched online using the following link: https:\/\/www.youtube.com\/watch?v=1CR4vMh8ufE.\n\n\n\\section{Conclusion and Future Work}\nIn this paper, a connected eco-driving system was developed for heavy-duty trucks traveling through cellular-based connected signalized intersections, aiming to smooth the speed profiles of the truck and hence reduce energy consumption. The system architecture was presented, introducing the different modules of the system. The map matching and eco-driving algorithms were described in detail. Field trials of the system with a heavy-duty diesel truck were conducted in Carson, California, showing the benefits of the system in real-world traffic.\n\nPotential next steps include quantifying reductions in key criteria pollutant and GHG emissions, optimizing vehicle and system-level efficiencies, predicting queues at connected intersections, and automating the longitudinal speed and acceleration control of the connected truck.\n\n\n\n\n\\section*{Acknowledgment}\nThe vision of a connected vehicle test bed for \"Eco-Drive\" brought together two independent projects funded by California Air Resources Board, California Energy Commission, and South Coast Air Quality Management District. The California Collaborative Advanced Technology Drayage Truck Demonstration Project is part of California Climate Investments, a statewide initiative that puts billions of Cap-and-Trade dollars to work reducing greenhouse gas emissions, strengthening the economy, and improving public health and the environment \u2014 particularly in disadvantaged communities. \n\nWe greatly acknowledge the equipment and technical support from Econolite, McCain, and Western Systems. We are also extremely grateful for the agency coordination and support received from Los Angeles County Metropolitan Transportation Authority, Los Angeles County Department of Public Works, City of Carson, and Port of Los Angeles. Finally, we extend thanks to Daniel Sandez, Anuja Patil, Dylan Brown, and Xishun Liao for their invaluable support during the field trials.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Introduction}\n\\IEEEPARstart\nIn recent years, transportation activities continue to have significant impacts on energy consumption and air quality. In 2017, about 29\\% of U.S. energy consumption was for transporting people and goods from one place to another \\cite{USEIA2017}. Over 90\\% of the fuel used for transportation is petroleum based, including primarily gasoline and diesel. Transportation-related greenhouse gas (GHG) emissions was the second largest contributor to the nationwide GHG emissions inventory in 2017, accounting for approximately 28.9\\% of total U.S. GHG emissions \\cite{USEPA2017}. In 2018, the transportation sector exceeded the power sector as the largest contributor of $CO_2$ emissions in the U.S. \\cite{Rhodium2019}.\n\nThese facts have been increasing public awareness of the need to reduce energy consumption and pollutant emissions generated by our contemporary transportation system. Among all the strategies to mitigate the transportation-related environmental footprint, eco-driving along urban arterial has gained significant research interest around the world \\cite{barth2009energy, kamal2010on, saboohi2009model, mensing2011vehicle, wang2018cluster}. By applying connected and automated vehicle (CAV) technology, drivers would effectively reduce the number of full stops and idling time, and avoid unnecessary acceleration and deceleration by receiving downstream traffic information, such as queue length and signal phase and timing (SPaT). The Eco-Approach and Departure (EAD) application is a typical example, where drivers are guided to approach and depart from signalized intersections in an environmentally friendly manner using SPaT and geometric intersection description (GID) information sent from the wireless roadside units (RSU) installed as part of the signalized intersection infrastructure \\cite{xia2013development}, \\cite{barth2011dynamic}. In such a manner, CAVs can increase their energy efficiency and decrease pollutant emissions by simply following well-designed speed profiles while traveling through the intersection. Results of microscopic simulation models showed a 10-15\\% reduction on energy consumption and CO2 emissions by applying the EAD application to fixed-timing signalized intersections \\cite{xia2013dynamic}. In congested traffic conditions, the EAD application also worked effectively where preceding queues could be estimated by leveraging real time vehicle detection and signal information system \\cite{he2015optimal}. Moreover, it was also revealed in previous studies that, drivers' behavior adaptability under actual driving conditions also played an important role in the effectiveness of the EAD application \\cite{xiang2015aclosed}.\n\nMany studies have been focused on developing eco-driving systems and evaluating the proposed systems by numerical or microscopic simulations. However, only a few implemented the proposed technology on real vehicles. A field test along the El Camino Real corridor in Palo Alto, CA was conducted to evaluate the EAD application for \\textit{actuated signals} in real-world traffic\\cite{hao2019eco}. The tested track is 1.7 miles long, and all intersections are equipped with DSRC RSUs. A test vehicle (2.5 L 4-cylinder) was equipped with an industry-grade radar, GPS, DSRC modem, on-board computer, and an artificial dashboard. It was shown up to 6\\% energy and emissions reduction can be achieved in real-world traffic. Additionally, a partially automated (longitudinal control) EAD prototype was developed and demonstrated at the Turner-Fairbank Highway Research Center (TFHRC) in McLean, Virginia, and the results showed an average of 17\\% in energy consumption reduction \\cite{altan2017glidepath}.   \n\n\\newpage\n\n\\begin{figure*}[!b]\n    \\centering\n    \\includegraphics[width=2.0\\columnwidth]{figure\/\/Architecture4.png}\n    \\caption{Architecture of the connected eco-driving system for heavy-duty trucks.}\n    \\label{flowchart}\n\\end{figure*} \n\nBuilt on top of the authors' previous work on light-duty vehicles, a connected eco-driving system for heavy-duty trucks has been developed in this study. Since heavy-duty trucks play a significant role in transportation-related (especially on goods movement) energy consumption and pollutant emissions, we have adapted the existing eco-driving system to heavy-duty trucks to better shape our contemporary transportation system. Furthermore, different from our previous implementation with DSRC, we have used 4G-LTE to enable cellular-based wireless communications in this study, given its benefits of scalability and wider coverage. We have implemented a simplified eco-driving algorithm to calculate the recommended speed for the test truck in real time. The algorithm has been strategically integrated with other system components, such as Global Navigation Satellite Systems (GNSS), on-bard maps, radar, X Electronic Control Unit (XECU), and live SPaT streams. A user-friendly driver-vehicle interface (DVI) has been designed, so the driver can follow the recommended speed to drive the truck to achieve the goal of energy savings.  A field test of the developed connected eco-driving system has been conducted in Carson, California with a heavy-duty diesel truck, along a test route with six equipped signalized intersections capable of broadcasting their SPaT information. \n\nThe remainder of this paper is organized as follows: Section II demonstrates the architecture of the proposed connected eco-driving system for heavy-duty trucks. Section III elaborates two key modules: map matching and eco-driving algorithm of the proposed system. The field testing conducted in Carson, CA is presented in Section IV, followed by result analysis. Section V concludes this paper along with the discussions on future work.\n\n\\section{Architecture of the Connected Eco-Driving System}\nIn this section, the architecture of the proposed connected eco-driving system is introduced, with illustrations of both software and hardware components installed. As can be seen from Fig. \\ref{flowchart}, the whole system is divided into five different modules: communication, localization, perception, planning, and control, whose respective functions are elaborated below.\n\n\\subsection{Communication Module}\nThe communication module facilitates real-time and reliable \\textit{vehicle-to-network} (V2N) and \\textit{ infrastructure-to-network} (I2N) communications. Although short-range communication technology such as DSRC is widely used by various CAV applications, their low scalability and asynchronous disadvantage may lead to several issues. As a substitute, 4G-LTE has been adopted to enhance V2N and I2N communications given its versatile communications types from one-to-one to one-to-multiple transmissions, with both simulations and field experiments conducted by researchers all over the world during the past decade \\cite{xu2017DSRC, nshimiyimana2016comprehensive}. In our connected eco-driving system, we also install a 4G-LTE cellular module with AT\\&T SIM card on board to enable V2N communication. Note that, compared to DSRC, one major disadvantage of 4G-LTE is its relatively longer latency. However, based on the our findings in our field implementation of this study, the latency introduced by 4G-LTE is about 100 ms, which is satisfactory for the connected eco-driving system.\n\nEach signal head on the test track is controlled by a McCain 2070E controller, which is a rugged, multi-tasking field process and communication system configurable for a variety of traffic management applications. Additionally, each controller is equipped with a SIEMENS RM1224 mobile wireless router for 4G-LTE connection, so the raw SPaT data could be sent to the cloud server through I2N communication. Then the 4G-LTE cellular module on the truck can request the live stream of SPaT information from the cloud server at every second.\n\n\\subsection{Localization Module}\nThe localization module of the connected eco-driving system consists of hardware component, a global navigation satellite system (GNSS) and inertial navigation system (INS), and software component, a map matching algorithm. The GNSS \\& INS component serves as a combined satellite \\& inertial-based navigation system, which can be optionally augmented by terrestrial reference stations \\cite{geiger2012team}. This component can provide precise position, movement, and posture measurements for the self-localization and altitude determination of the truck by differential correction. On the test truck, a versatile U-BLOX NEO-M8N GNSS module is installed to locate the truck's position. This GNSS module has a navigation sensitivity of -167dBm, which provides outstanding positioning accuracy in scenarios where urban canyon or weak signals are involved.\n\nFor the map matching component, a pre-built map of the testing field is available on board, with information such as the intersection ID, signal group ID, signal location, road type, road name and direction, and road speed limit. The truck's coordinates (\\textit{i.e.}, longitude, latitude, and heading) received from the GNSS \\& INS component can then be matched to the pre-built map by the proposed map matching algorithm, where the details of the algorithm will be introduced in Sec. \\ref{sec:algorithms}A.\n\n\n\\subsection{Perception Module}\nDifferent from the communication module that provides information about the signal status (\\textit{i.e.}, SPaT information), the perception module can provide information about other surrounding vehicles (especially if they are not CAVs). In this connected eco-driving system , we adopt the forward sensing kit to detect any preceding vehicles, bicycles, and\/or pedestrians. The Volvo EU Production Radar Unit together with the forward-looking camera can detect the movement of preceding vehicle, and send its dynamics information to the sensor fusion component. Once the detected objects are classified by the sensor fusion component, the desired information is sent to the eco-driving algorithm. Given the measured relative speed and distance between the truck and its preceding vehicle, the time-to-collision value can be computed. If the time-to-collision value is lower than a preset threshold, the suggested speed for the truck driver will be turned off on the DVI, and the driver will drive the truck without any auxiliary information. \n\n\\subsection{Planning Module}\nThe planning module is the core of the proposed connected eco-driving system. The XECU of the truck is equipped with a ruggedized computer, which has access to multiple channels of CAN data from different sub-networks of the truck. It receives the distance to the next signal from the localization module, preceding vehicles' states from the perception module, and the speed of the truck from CAN message.\n\nThe eco-driving algorithm is embedded in the XECU, based on a Microsoft .NET 4.7 framework with C\\# language. Data query, recommended speed calculation, and the signal display are all updated at 10 Hz frequency to achieve the consistency, while the communication module is operating at 1 Hz due to the restriction on infrastructure. All the concurrent tasks are handled by threading in the main thread with shared memory. Upon receiving all this information, the recommended speed of the truck at the next time step for the driver will be computed, and sent to the DVI along with some other information. On the other hand, we also utilize a package called log4net for archiving the raw data and the computation results. The details of the eco-driving algorithm will be introduced in Sec. \\ref{sec:algorithms}B.\n\n\\subsection{Control Module}\nOnce the recommended speed is computed by the eco-driving algorithm in the planning module, the control module of the connected eco-driving system starts to play. A DVI is shown on the monitor right above the dashboard of the truck. With the auxiliary help from the DVI, the driver controls the longitudinal movement of the truck via the throttle and brake, and controls the lateral movement  by steering wheel.\n\nThree typical scenarios are shown in Fig. \\ref{DVI}. As can be seen from these subplots, the top of the DVI shows the upcoming signal phase, but not necessarily the signal timing. We only show the signal timing when the truck is at full stop (shown as the right subplot), but hide that information in other scenarios (the left and middle subplot) to avoid distraction to the driver while driving the truck. The right hand side of the DVI is the speed limit of the current roadway segment, which is acquired from the pre-built map. The artificial speedometer is put at the bottom of the DVI with a green band to demonstrate the recommended speed range for the truck driver. As can be seen from the right subplot, if a vehicle is detected in front of the truck within a certain range, a vehicle icon is shown in the middle of the DVI, and the upper bound of the recommended speed is bounded by the speed of the preceding vehicle. If the computed time-to-collision value is lower than a preset threshold, the DVI would display a warning message instead of showing the SPaT information, allowing the driver to disengage the eco-driving system and be aware of potential rear-end collision by himself\/herself.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.32\\columnwidth]{figure\/\/dvi_screenshot1.png}\n    \\includegraphics[width=0.32\\columnwidth]{figure\/\/dvi_screenshot2.png}\n    \\includegraphics[width=0.32\\columnwidth]{figure\/\/dvi_screenshot3.png}\n    \\caption{Screen shots of DVI designed for the truck driver.}\n    \\label{DVI}\n\\end{figure}\n\n\\section{Algorithms of the Connected Eco-Driving System} \\label{sec:algorithms}\nIn this section, we briefly introduce two crucial algorithms in the developed connected eco-driving system: The map matching algorithm in the localization module, and the eco-driving algorithm in the planning module.\n\n\\subsection{Map Matching Algorithm}\nThe first step of our map matching algorithm is to find the projection point of truck's current latitude and longitude on the nearest lane of the hard-coded map. Since we calculate the distance on the surface of the earth, Haversine formula can be applied as:\n\\begin{equation}\\small\n    d = 2r \\cdot sin^{-1}(\\sqrt{sin^2(\\frac{\\Delta\\varphi}{2}) + cos(\\varphi_1)cos(\\varphi_2)sin^2(\\frac{\\Delta\\lambda}{2})})\n\\end{equation}\nwhere $d$ is the distance between two points on the sphere, $r$ is the radius of the earth, $\\varphi_1$, $\\varphi_2$, $\\Delta\\varphi$ represent the latitude of the first point, the latitude of the second point, and  the difference between $\\varphi_1$ and $\\varphi_2$, respectively, $\\Delta\\lambda$ is the longitude difference between the two points.\n\nSince the direction to the next signal should be consistent with the current heading of the truck, we verify this procedure by using inner product:\n\n\\begin{equation}\n\\hat{h} \\cdot \\hat{u} > 0\\\\\n\\end{equation}\nwhere $\\hat{h}$ is the current heading of the truck and $\\hat{u}$ is the direction of the intersection. \n\nGiven that the GNSS signal might drift from the current lane of the road, errors could be generated and accumulated if we directly compute the distance to the intersection node. Therefore, we utilize the triangle formed by the truck's coordinate and its two adjacent nodes' coordinates to calculate the real distance from the projection point of the truck to the intersection. Heron's formula can be given as:\n\n\\begin{equation}\nA = \\sqrt{s(s-d_{tn1})(s-d_{tn2})(s-d_{n1n2})}\n\\end{equation} \nwhere $A$ is the area of the triangle whose sides have lengths $d_{tn1}$ (truck to the previous node), $d_{tn2}$ (truck to the next node), and $d_{n1n2}$ (the previous node to the next node), and $s$ is the semi-perimeter of the triangle defined as\n\n\\begin{equation}\ns = \\frac{d_{tn1} + d_{tn2} + d_{n1n2}}{2}\n\\end{equation}\n\nThen, we can compute the height $h$ of the triangle by the area and the base-side $d_{n1n2}$, where $h$ is considered as the error distance due to the drift of GNSS signal from the current lane. Based on $h$ and $d_{tn2}$, we could use Pythagorean theorem to compute the distance $d_{pn2}$ from the projection point of the truck to the downstream node. Finally, the distance from the truck to the next signal $d_{sig}$ can be calculated by adding the resulted distance with the lengths of all downstream segments before the signal:\n\\begin{equation}\nd_{sig} = d_{pn2} + d_{n2n3} + d_{n3n4} + ... + d_{n(m-1)n(m)}\n\\end{equation}\nwhere $n(m)$ is the node with respect to the next signal.\n\n\\subsection{Eco-Driving Algorithm}\nAlthough we developed several versions of eco-driving algorithms before (\\textit{e.g.}, piece-wise trigonometric algorithm \\cite{xia2013development, xia2013dynamic}, graph-based algorithm\\cite{Jin2016ecoacc} \\cite{hao2018connected}), in this system, we develop a simplified version of eco-driving algorithm for two major reasons:\n\\begin{itemize}\n    \\item Compared to previous eco-driving algorithms, this simplified version of algorithm is easier to implement on real vehicles, and also more attractive in terms of real-time performance;\n    \\item This version of algorithm computes a feasible speed range instead of a single value, and provides an easy-to-follow speed band on the DVI for the truck driver. Since the ability to precisely follow the recommended speed is limited and varied by the driver, and the response time of the driver to maneuver the truck may be longer than light duty vehicles, the feasible speed range option should be more robust and user-friendly for truck drivers.\n\\end{itemize} \nIn this subsection, we demonstrate the algorithm with parameters $d_{sig}$ as distance to the intersection, $t_{current}$ as the time left of current signal, and $v_{lim}$ as the speed limit of the roadway. The reference speed $v_0$ of the truck can then be given as\n\\begin{equation}\nv_0 = d_{sig}\/t_{current}\n\\end{equation}\n\nWe then categorize the eco-driving algorithm into two different scenarios based on the current signal phase:\n\n\\subsubsection{Red Phase}\n\\begin{equation}\n    \\left\\{\n                \\begin{array}{ll}\n                  v_{lower} = 0, v_{upper} = v_0 & \\textit{, if } v_0 \\leq v_{lim}\\\\\n                  v_{lower} = 0, v_{upper} = v_{lim} & \\textit{, otherwise}\\\\\n                \\end{array}\n              \\right.\n\\end{equation}\n\n\\noindent When the signal phase at the upcoming intersection is red, the lower bound of the suggested speed $v_{lower}$ would always be zero. However, the upper bound $v_{upper}$ would be set to the speed that the truck can pass the intersection just at the moment it turns green without stopping. The highest speed is also constrained by the speed limit $v_{lim}$ of the roadway.\n\n\\subsubsection{Green Phase}\n\\begin{equation}\n    \\left\\{\n                \\begin{array}{ll}\n                  v_{lower} = v_0, v_{upper} = v_{lim} & \\textit{, if } v_0 \\leq v_{lim}\\\\\n                  v_{lower} = 0, v_{upper} = 0 & \\textit{, otherwise}\\\\\n                \\end{array}\n              \\right.\n\\end{equation}\n\n\\noindent On the other hand, when the signal at the upcoming intersection is green, the upper bound $v_{upper}$ of the recommendation would be the speed limit, and the lower bound $v_{lower}$ is set to the lowest speed that the truck can pass the intersection during the current green phase. Under the circumstances that even using the highest speed (\\textit{i.e.}, speed limit) the truck still cannot pass the intersection, the algorithm will suggest the driver to slow down and wait for the next green phase.\n\n\n\\section{Field Implementation}\n\n\nIn this work, we adopt a Volvo VNL heavy-duty diesel truck for our field implementation, which is a 2015 Volvo VNL model with a 13L diesel engine (455 HP) and a maximum cruise speed of 115 km\/h and is shown in Fig. \\ref{Truck}. The test runs are conducted in Carson, CA, which locates right next to the Port of Los Angeles and Port of Long Beach, two busiest container ports in the United States. Numerous factories and warehouses are located in the testing area, so the proposed connected eco-driving system can potentially save energy consumed by heavy-duty trucks travelled in this area. Fig. \\ref{Route} shows the test route where six signalized intersections on Wilmington Blvd. and Alameda St. are equipped with communication modules. The respective SPaT information can be sent to the cloud and allow our connected truck to fetch the data.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\columnwidth]{figure\/\/Truck_view3.jpg}\n    \\caption{Field implementation is conducted by the heavy-duty truck in Carson, California.}\n    \\label{Truck}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\columnwidth]{figure\/\/Route.png}\n    \\caption{Field implementation route in Carson, CA with six equipped signalized intersections.}\n    \\label{Route}\n\\end{figure} \n\nFor a comparative study, the truck ran a few times along the test route with and without the developed connected eco-driving system. As it was driven in real traffic environment, where the traffic flow may vary over time of the day, it is not fair to quantitatively compare the energy consumption between different test runs. Therefore, in this section of the paper, we select results generated during four test runs, where two of them are with connected eco-driving system and the other two are without it (\\textit{i.e.,} baseline). To make this comparison as fair as possible, all four selected test runs are conducted at the same intersection with clear downstream traffic, which means the movement of the truck is not affected by other vehicles on the road. Also, in each scenario shown in Fig. \\ref{Acceleration} and Fig. \\ref{Deceleration}, the truck approaches the intersection from the same distance with the same SPaT status (shown as the color bar below the x axis). Therefore, the difference between two trajectories (eco-driving and baseline) in each plot is mainly caused by the existence of the proposed connected eco-driving system.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\columnwidth]{figure\/\/Acceleration.png}\n    \\caption{Test results of the acceleration scenario of the connected eco-driving vs. the baseline.}\n    \\label{Acceleration}\n\\end{figure} \n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.0\\columnwidth]{figure\/\/Deceleration.png}\n    \\caption{Test results of the deceleration scenario of the connected eco-driving vs. the baseline.}\n    \\label{Deceleration}\n\\end{figure} \n\n\nFig. \\ref{Acceleration} shows the comparison results on the acceleration scenario between the connected eco-driving system and the baseline. The baseline curves show the truck starts to heavily decelerate at 23 seconds when the driver sees the amber light 150 m away from the intersection, and stops fully at the intersection stop line at 35 seconds. However, with the help of the connected eco-driving system, the connected truck starts to accelerate prior to the signal turns to amber. Therefore, it can travel through the intersection without a full stop before the signal turns to red. Fig. \\ref{Deceleration} shows the comparison results on the deceleration scenario between the connected eco-driving system and the baseline. The baseline curves show the truck starts to decelerate at 18 seconds while the driver sees the amber light 150 m away from the intersection, and stops fully at the intersection stop line at  28 second. However, with the help of the connected eco-driving system, the connected truck starts to decelerate prior to the signal turns to amber, and conducts a relatively smoother deceleration towards the intersection. Therefore, it can pass the intersection right after the signal turns to green with a low speed, instead of experiencing a full stop. We also adopt U.S. EPA's MOVES model to calculate the fuel consumption of aforementioned four test runs, where in each scenario the starting and ending speed are the same (Fig. \\ref{Acceleration} and Fig. \\ref{Deceleration} only show partial trajectories) \\cite{MOVES}. The results of MOVES show that the connected eco-driving system can provide approximately 9\\% and 4\\% fuel savings in the acceleration and deceleration scenario, respectively.\n\nThose four test runs shown above, along with many other test runs we have conducted, prove the proposed connected eco-driving system can smooth the speed variations of the truck, avoid unnecessary full stop at the intersection, and therefore reduce the fuel consumption of the heavy-duty truck. Another major finding of this field implementation is that, the cellular-based wireless communication approach enables CAVs to receive SPaT information from anywhere (not limited to a certain distance to the intersection), so the connected eco-driving can be implemented to a much larger scale. A disadvantage of this approach is an extra 100 ms communication delay compared to DSRC approximately, but this delay does not affect the behavior of this system to a large extent. The video of the field implementation could be watched online using the following link: https:\/\/www.youtube.com\/watch?v=1CR4vMh8ufE.\n\n\n\\section{Conclusion and Future Work}\nIn this paper, a connected eco-driving system was developed for heavy-duty trucks traveling through cellular-based connected signalized intersections, aiming to smooth the speed profiles of the truck and hence reduce energy consumption. The system architecture was presented, introducing the different modules of the system. The map matching and eco-driving algorithms were described in detail. Field trials of the system with a heavy-duty diesel truck were conducted in Carson, California, showing the benefits of the system in real-world traffic.\n\nPotential next steps include quantifying reductions in key criteria pollutant and GHG emissions, optimizing vehicle and system-level efficiencies, predicting queues at connected intersections, and automating the longitudinal speed and acceleration control of the connected truck.\n\n\n\n\n\\section*{Acknowledgment}\nThe vision of a connected vehicle test bed for \"Eco-Drive\" brought together two independent projects funded by California Air Resources Board, California Energy Commission, and South Coast Air Quality Management District. The California Collaborative Advanced Technology Drayage Truck Demonstration Project is part of California Climate Investments, a statewide initiative that puts billions of Cap-and-Trade dollars to work reducing greenhouse gas emissions, strengthening the economy, and improving public health and the environment \u2014 particularly in disadvantaged communities. \n\nWe greatly acknowledge the equipment and technical support from Econolite, McCain, and Western Systems. We are also extremely grateful for the agency coordination and support received from Los Angeles County Metropolitan Transportation Authority, Los Angeles County Department of Public Works, City of Carson, and Port of Los Angeles. Finally, we extend thanks to Daniel Sandez, Anuja Patil, Dylan Brown, and Xishun Liao for their invaluable support during the field trials.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThis work is concerned with the development of new connections between the multidimensional tensegrity frameworks, toric varieties, and their Chow groups.\n\n\\subsection{Tensegrity frameworks.}\\label{higher}\nA classical tensegrity model \\cite{maxwell1864xlv} consists of an embedded graph in a vector space $\\mathbb{R}^d$ with rigid bars as edges, and a balancing condition at each vertex, which gives a stable structure (see, e.g., \\cite{roth1981tensegrity, connelly1996second,connelly2013tensegrity}). Tensegrities have a wide range of applications in different areas of modern science and engineering technology (see, e.g., \\cite{motro2003tensegrity, roth1981tensegrity, juan2008tensegrity, zhang2015tensegrity}).\nThe notion of tensegrity has also been developed in higher dimensions (see, e.g., \\cite{karpenkov2021geometric,karpenkov2022equilibrium,rybnikov1999stresses, rybnikov2000polyhedral}). Some theories about the existence of tensigrities and stratifications are recently developed in \\cite{karpenkov2021frame,doray2010geometry}.  In this paper, we focus on the multidimensional tensegrities introduced in \\cite{karpenkov2022equilibrium}, and we examine their structures over $\\mathbb{Q}$ or $\\mathbb{Z}$ from the algebraic geometry perspective.\n\n\n\\smallskip\nWe now define the main object of this paper, the multidimensional tensegrity framework (see, \\cite{karpenkov2022equilibrium} for more details).\n    Let $N\\cong\\mathbb{Z}^d$ be a lattice and $N_\\mathbb{R}= N\\otimes_\\mathbb{Z}\\mathbb{R}\\cong\\mathbb{R}^d$. A \\emph{$k$-framework} $\\mathcal{F}=(E,F,I,{\\bf n})$ in $N_\\mathbb{R}$ consists of the following data:\n    \\begin{itemize}\n        \\item a collection $E$ of $(k-1)$-dimensional affine subspaces in $\\mathbb{R}^d$; \n        \\item a collection $F$ of $k$-dimensional affine subspaces in $\\mathbb{R}^d$;\n        \\item a subset $I\\subset\\{(p,q)\\in E\\times F\\mid\\  p\\subset q\\}$;\n        \\item a function $\\bf{n}$ assigning to each pair $(e,f)$ in $I$, a vector ${\\bf n}(e,f)$ in $f$ normal to $e$, which is mapped to the primitive generator of the lattice $N\/(N\\cap e)$ under $\\pi_e:N\\rightarrow N\/(N\\cap e)$. \\\\\n        See \\Cref{framework}.\n    \\end{itemize} \nThe elements of $E$, $F$ and $I$ are called \\emph{edges}, \\emph{faces}, and \\emph{incidences}, respectively. \nA \\emph{stress} $w$ on $\\mathcal{F}$ is a function $w:F\\rightarrow\\mathbb{Q}$. In particular, $w$ is called a \\emph{self-stress} if for every $e\\in E$, we have:\n\\begin{equation}\\label{selfstress}\n    \\sum_{(e,f)\\in I}{\\bf n}(e,f)w(f)=0. \n    \\end{equation}\nThe collection of self-stresses forms an abelian group $A_\\mathcal{F}$ under addition. Moreover, $\\mathcal{F}$ is called a \\emph{tensegrity} if there exists a nonzero self-stress on it.\n\n\\subsection{Toric varieties and Chow groups.}\\label{sec:Chow}\nA normal algebraic variety $X$ is \\emph{toric} if there exists a $(\\mathbb{C}^*)^n$-action on $X$ with an open dense orbit isomorphic to the torus $T=(\\mathbb{C}^*)^n$. Toric varieties form an important family of varieties in algebraic geometry, mainly because they are linked to the theory of lattices, polytopes and polyhedral fans. Moreover, their geometric properties are encoded as combinatorial invariants of their corresponding polytopes. In some sense, toric varieties are the easiest objects to deal with in algebraic geometry, and they can be used in the study of arbitrary varieties via degeneration techniques \\cite{anderson2013okounkov,alexeev2002complete, bossinger2021families}. \nThe standard references for toric geometry are \\cite{oda1983convex,fulton1993introduction} and \\cite{cox2011toric}.\n\n\\smallskip\nTo define the Chow group, consider the lattice of cocharacters $N=\\Hom(\\mathbb{C}^*,T)\\cong\\mathbb{Z}^n$ and the lattice of characters $M=\\Hom(N,\\mathbb{Z})$. Any complete fan $\\Sigma$ in $N_\\mathbb{R}=N\\otimes_\\mathbb{Z}\\mathbb{R}$ uniquely determines a toric variety $X_\\Sigma$. The Chow group of $k$-dimensional algebraic cycles, denoted by $A_k(X_\\Sigma;\\mathbb{Q})$ is described in \\cite[Proposition~1.1]{fulton1997intersection}. Let $\\Sigma^k$ be the set of cones in $\\Sigma$ of codimension $k$. The $T$-invariant closed subvariety associated to $\\sigma\\in\\Sigma^k$ is denoted by $V(\\sigma)$. Then $A_k(X_\\Sigma;\\mathbb{Q})$ is generated by the rational equivalent classes $[V(\\sigma)]$ where $\\sigma$ runs over $\\Sigma^k$, and the relations are given by\n\\begin{equation}\\label{rel}\n    \\sum_{\\sigma\\in\\Sigma^k,\\ \\sigma\\supset\\tau}\\langle m,n_{\\sigma,\\tau}\\rangle[V(\\sigma)]=0,\n\\end{equation}\nfor every $\\tau\\in\\Sigma^{k+1}$ and all $m\\in M(\\tau)=\\tau^\\perp\\cap M$. Here $n_{\\sigma,\\tau}$ is a lattice point in $\\sigma$ whose image generates the 1-dimensional lattice $N_\\sigma\/N_\\tau$, where $N_\\sigma$ and $N_\\tau$ are sublattices of $N$ generated by $N\\cap\\sigma$ and $N\\cap\\tau$, respectively.\n\n\n\\smallskip\n\nOn the dual side, the \\emph{operational Chow cohomology ring} is defined as  $A^\\bullet(X_\\Sigma;\\mathbb{Q})=\\oplus_k A^k(X_\\Sigma;\\mathbb{Q})$ (see \\cite[Chapter 17]{fulton2013intersection}). When $X_\\Sigma$ is $\\mathbb{Q}$-factorial, or equivalently, every cone in $\\Sigma$ is simplicial, one can identify $A^k(X_\\Sigma;\\mathbb{Q})$ with $A_{\\dim(X_\\Sigma)-k}(X_\\Sigma;\\mathbb{Q})$. By \\cite[Theorem 3]{fulton1994intersection}, there is an isomorphism\n\\begin{equation}\\label{dual}\n    A^k(X_\\Sigma;\\mathbb{Q})\\cong\\Hom_\\mathbb{Q}(A_k(X_\\Sigma;\\mathbb{Q}),\\mathbb{Q}).\n\\end{equation}\nMoreover, the Chow cohomology group $A^k(X_\\Sigma;\\mathbb{Q})$ is isomorphic to the group of Minkowski weights on $\\Sigma^k$. See \\cite[Theorem~2.1]{fulton1997intersection}. \nIn particular, a $\\mathbb{Q}$-valued function $c$ on $\\Sigma^k$ is a \\emph{Minkowski weight} if it satisfies the following balancing condition:\n\\begin{equation}\\label{Minkowski}\n        \\sum_{\\sigma\\in\\Sigma^k,\\ \\sigma\\supset\\tau}\\langle m,n_{\\sigma,\\tau}\\rangle c(\\sigma)=0,\n    \\end{equation}\nfor every $\\tau\\in\\Sigma^{k+1}$ and $m$ in the lattice $M(\\tau)$.\n\n\n\n\\subsection{Outline and our results.} \nIn \\Cref{higherandtoric}, we first construct a toric variety for every edge in $\\mathcal{F}$, and then glue their associated polytopes along a proper choice of faces to obtain a polytope for $\\mathcal{F}$. Then we use this polytope to construct a toric variety $X_{\\mathcal{F}}$ associated to $\\mathcal{F}$ (see Construction~\\ref{const} and \\Cref{assembled}).\nOur main goal is to prove the following theorem which relates the Chow group of $X_{\\mathcal{F}}$ from \\Cref{sec:Chow}, and the abelian group of self-stresses $A_{\\mathcal{F}}$ from \\Cref{higher}.\n\n\\begin{introthm}{Theorem}\\label{thm:main}\nConsider a framework $\\mathcal{F}$, and let $A_\\mathcal{F}$ be the group of self-stresses on $\\mathcal{F}$. Let $X_{\\mathcal{F}}$ be a glued toric variety from Construction~\\ref{const} and \\Cref{assembled}, and let $A^1(X_{\\mathcal{F}};\\mathbb{Q})_\\mathcal{F}$ be the subgroup of the Chow group $A^1(X_{\\mathcal{F}};\\mathbb{Q})$ with cocycles vanishing on the reference rays.\nThen we have that:\n    $$\n        A^1(X_{\\mathcal{F}};\\mathbb{Q})_\\mathcal{F}\\cong A_\\mathcal{F}.\n    $$\n\\end{introthm}\nIn \\Cref{planar}, we focus on the classical tensegrity model of planar graphs. In \\Cref{sec:graph}, we construct a polyhedral fan for any planar graph, and equip that with an irreducible toric variety. This enables us to explicitly compute the corresponding Chow rings in \\Cref{sec:computation}, and so tensegrities, using Stanley-Reisner ideals. We also provide a computational example (see Example~\\ref{ex:running}).\n\n\n\\smallskip\n\\noindent{\\bf Acknowledgement.}\nThe first author would like to express her gratitude to the organizers of the Fields Institute Thematic Program on Geometric Constraint Systems, Framework Rigidity, and Distance Geometry, for introducing her to the subject, and for many helpful conversations. She would also like to thank James Cruickshank, Anthony Nixon, and Shin-ichi Tanigawa for helpful discussions during the project \\cite{cruickshank2022global}.\nThe authors would like to thank Oleg Karpenkov for helpful discussions. The authors were partially supported by the \nFWO grants G0F5921N (Odysseus programme), G023721N, and BOF\/STA\/201909\/038. \n\n\\section{Glued toric varieties associated to multidimensional frameworks}\\label{higherandtoric}\nThroughout we fix a multidimensional framework $\\mathcal{F}$ as defined in \\Cref{higher}, which is the integral version of \\cite[Definition~2.1]{karpenkov2022equilibrium}. We assume that $\\mathcal{F}$ is a \\emph{rational} framework, i.e.~each element in $E$ or $F$ contains infinitely many rational points. \nWe also assume that $\\mathcal{F}$ is \\emph{generic}, i.e.~all $f$'s\nwith $(e, f)\\in I$ are distinct for a fixed $e\\in E$. Moreover, for any $e\\in E$, we have that $\\#\\{f\\mid(e,f)\\in I\\}\\geqslant 3$ if it is nonzero. \n\n\\medskip\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{tikzpicture}[scale=0.5]\n    \\filldraw[blue!10, opacity=0.7] (8,0) -- (10,0) -- (10,10) -- (0,10) -- (0,8) -- cycle;\n    \\filldraw[blue!10, opacity=0.3] (8,0) -- (0,0) -- (0,8) --  cycle;\n    \\foreach \\i in {0,2,...,10}\n      \\foreach \\j in {0,2,...,10}{\n        \\draw [gray] (\\i,\\j) circle(3pt);\n        };\n     \\foreach \\i in {0,...,10}\n      \\foreach \\j in {0,...,10}{\n        \\ifnum \\i = \\j\n            \\fill[DarkRed] (\\i,\\j) circle(3.5pt);\n        \\fi\n      };\n     \\draw [very thick, DarkGreen] (8,0) -- (0,8);\n     \\draw [very thick, orange, ->] (3,5) -- (4,6);\n     \\node [below] at (4,10) {\\color{SkyBlue}$f$};\n     \\node [below] at (7,0.8) {\\color{DarkGreen}$e$};\n     \\node [below] at (1,3.5) {\\color{DarkRed}$N\/N\\cap e$};\n     \\node [below] at (0.5,10) {\\color{gray}$N$};\n     \\node [below] at (3,7.5) {\\color{orange}${\\bf n}(e,f)$};\n  \\end{tikzpicture}\n  \\caption{Local picture of a framework.}\n    \\label{framework}\n\\end{figure}\n\n\n\nWe now explain our method to associated a toric variety to any given framework $\\mathcal{F}$. \n\n\\begin{construction}[The toric variety $X_{\\mathcal{F}}$]\\label{const} We first construct a toric variety for every edge in $\\mathcal{F}$, and then glue their associated polygons to obtain a polytope, and hence a toric variety for $\\mathcal{F}$.\n\\smallskip\n\nMore precisely, for each $e$, locally, in a neighborhood $U_e=e\\times\\Delta$, where $\\Delta$ is a small ball of dimension $(n-k+1)$, we contract $U_e$ along $e$. Extending rays, we obtain a polyhedral fan $\\Sigma_e'\\subset\\mathbb{R}^2$ centered at the image of $e$ under the contraction, whose rays are contractions of $\\{f\\mid(e,f)\\in I\\}$, and dimension 2 cones are naturally cut by rays. Then, we complete the fan, if necessary, as follows. If all the rays $\\rho_1,\\ldots,\\rho_s$ lie on the same half-plane, then we add a new ray $\\rho_0$ generated by $-\\sum_{i=1}^s v_i$, where $v_i$ is the primitive generator of $\\rho_i$. Including the two $2$-dimensional cones containing $\\rho_0$, we obtain a complete fan $\\Sigma_e$. We call the ray $\\rho_0$ an \\emph{assistant ray}. All these operations are canonical with respect to $\\mathcal{F}$. We now proceed with our construction as follows:\n\\begin{itemize}\n    \\item Let $X_{\\Sigma_e}$ be the complete toric variety associated to $\\Sigma_e$. Note that $X_{\\Sigma_e}$ is projective since $\\dim X_{\\Sigma_e}=2$. We choose an ample line bundle $L_e$ on $X_{\\Sigma_e}$, equivalently, a polygon $P_e$ whose normal fan is $\\Sigma_e$.\n    \\item If $(e,f)$ and $(e',f)$ are both in $I$, and $l_f,l_{f'}$ are edges in $P_e,P_{e'}$ normal to the contraction of $f$, respectively, then we choose a bijection $\\phi_f^{e,e'}$ between $l_f$ and $l_{f'}$, and glue the polygons $P_e,P_{e'}$ via $\\phi_f^{e,e'}$ along the edges $l_f,l_{f'}$. We denote $\\Phi_F^E$ for the set of all bijections $\\phi_f^{e,e'}$.\n\\end{itemize}\nWe call the pair $\\mathcal{P}:=(\\cup_{e\\in E}P_e,\\Phi_F^E)$  the \\emph{glued polygon}. We associate a variety $X$ to $\\mathcal{P}$, where each component $X_{\\Sigma_e}$ and the gluing is given by $\\Phi_F^E$. Note that, the two non-canonical steps in the construction may lead to  multiple varieties $X$ (associated to $\\mathcal{F}$), however by \\Cref{rel}, the Chow groups are the same. Hence, we write $A^1(X_\\mathcal{F};\\mathbb{Q})$ for a choice of $X_\\mathcal{F}$ associated to $\\mathcal{P}$.\n\\end{construction}\n\n\\begin{example}\n\n\\begin{figure}[h]\n\\begin{tikzpicture}\n    \\draw [thick, SkyBlue] (1.2,-0.2) -- (1.2,3.8) -- (3.2,1.8) -- cycle;\n    \\draw [thick, SkyBlue] (1,0) -- (1,4) -- (3,3) -- (2.6,2.4);\n    \\draw [thick, SkyBlue] (1,0) -- (1.2,0.3);\n    \\draw [thick, SkyBlue, dashed] (2.6,2.4) -- (1.2,0.3);\n    \\draw [thick, SkyBlue] (-1.2,-0.2) -- (-1.2,3.8) -- (-3.2,1.8) -- cycle;\n    \\draw [thick, SkyBlue] (-1,0) -- (-1,4) -- (-3,3) -- (-2.6,2.4);\n    \\draw [thick, SkyBlue] (-1,0) -- (-1.2,0.3);\n    \\draw [thick, SkyBlue, dashed] (-2.6,2.4) -- (-1.2,0.3);\n    \\node [right] at (-1,2.2) {$l_{f_1}$};\n    \\node [left] at (-1.2,2) {$l_{f_2}$};\n    \\node [right] at (1.2,2) {$l_{f_3}$};\n    \\node [left] at (1.1,2.2) {$l_{f_4}$};\n\\end{tikzpicture}\\hspace{1.5cm}\n\\begin{tikzpicture}\n    \\draw [thick, SkyBlue] (0,0) -- (0,4) -- (2,2) -- cycle;\n    \\draw [thick, SkyBlue] (0,0) -- (0,4) -- (2,3) -- (1.6,2.4);\n    \\draw [thick, SkyBlue, dashed] (1.6,2.4) -- (0,0);\n    \\draw [thick, SkyBlue] (0,0) -- (0,4) -- (-2,2) -- cycle;\n    \\draw [thick, SkyBlue] (0,0) -- (0,4) -- (-2,3) -- (-1.6,2.4);\n    \\draw [thick, SkyBlue, dashed] (-1.6,2.4) -- (0,0);\n\\end{tikzpicture}\n\\caption{A realizable glued polygon in $\\mathbb{R}^3$.}\n    \\label{gluedpoly}\n\\end{figure}\nLet $\\mathcal{F}$ be the following framework. Fix the points\n\\begin{displaymath}\n  \\begin{split}\n    B_1 & = (1,1,0),\\\\\n    B_5 & = (2,2,1),\\\\\n  \\end{split}\n\\quad\\quad\n  \\begin{split}\n    B_2 & = (-1,1,0),\\\\\n    B_6 & = (-2,2,1),\\\\\n  \\end{split}\n  \\quad\\quad\n  \\begin{split}\nB_3 & = (-1,-1,0),\\\\\nB_7 & = (-2,-2,1),\\\\\n\\end{split}\n\\quad\\quad\n  \\begin{split}\n    B_4 & = (1,-1,0),\\\\\n    B_8 & = (2,-2,1),\\\\\n  \\end{split}\\end{displaymath}\n\\begin{center}\n    and $B_i=B_{i-4}-(0,0,2)$ for $i=9,\\ldots,12$.\n\\end{center} \nDenote $B_iB_j$ for the line through any pair of points $B_i$ and $B_j$, and $B_iB_jB_kB_\\ell$ for the half-plane containing any collection of four coplanar points $B_i,B_j,B_k,B_\\ell$. Set\n\\begin{displaymath}\n  \\begin{split}\n    e_1 & = B_1B_2,\\\\\n  \\end{split}\n\\quad\\quad\n  \\begin{split}\n    e_2 & = B_2B_3,\\\\\n  \\end{split}\n  \\quad\\quad\n  \\begin{split}\ne_3 & = B_3B_4,\\\\\n\\end{split}\n\\quad\\quad\n  \\begin{split}\n    e_4 & = B_4B_1,\\\\\n  \\end{split}\\end{displaymath}\n\\begin{center}\n    and $e_i=\\begin{cases}\n    B_iB_{i-4},i=5,\\ldots,8;\\\\\n    B_iB_{i-8},i=9,\\ldots,12.\n    \\end{cases}$\n\\end{center} \nWe also set\n\\begin{displaymath}\n  \\begin{split}\n   f_0 & = B_1B_2B_3B_4,\\\\\n   f_4 & = B_4B_1B_5B_8,\\\\\n  \\end{split}\n\\quad\\quad\n  \\begin{split}\n    f_1 & = B_1B_2B_6B_5,\\\\\n    f_5 & = B_1B_2B_{10}B_9,\\\\\n  \\end{split}\n  \\quad\\quad\n  \\begin{split}\nf_2 & = B_2B_3B_7B_6,,\\\\\nf_6 & = B_2B_3B_{11}B_{10},\\\\\n\\end{split}\n\\quad\\quad\n  \\begin{split}\n f_3 & = B_3B_4B_8B_7,\\\\\n   f_7 & = B_3B_4B_{12}B_{11},\\\\\n  \\end{split}\n  \\end{displaymath}\n  \\begin{center}\n    and $f_8=B_4B_1B_9B_{12}$.\n\\end{center} \n Note that the subset $I$ (from \\Cref{{higher}}) consists of all $(e,f)$'s with $e\\in\\{e_1,\\ldots,e_4\\}$.\nMoreover, the vectors ${\\bf n}(e,f)$'s are pointing towards the interior of the labelled quadrilaterals. \\Cref{gluedpoly} shows a choice of the glued polygon realizable in $\\mathbb{R}^3$.\n\\end{example}\n\n\\begin{figure}[h]\n    \\centering\n\\begin{tikzpicture}[scale=0.7]\n    \\filldraw[blue!10, opacity=0.5]  (0,0) -- (-1,-1) -- (1,-1) -- (3,0) -- (3,2) -- (-3,2) -- (-3,0) -- (-1.3,-0.7) -- (-0.7,-0.1) -- (-2,1) -- (2,1) -- cycle;\n    \\draw [thick, SkyBlue] (0,0) -- (-1,-1) -- (1,-1) -- cycle;\n    \\draw [thick, SkyBlue] (0,0) -- (2,1) -- (3,0) -- (1,-1);\n    \\draw [thick, SkyBlue] (2,1) -- (3,2) -- (3,0);\n    \\draw [thick, SkyBlue] (3,2) -- (-3,2) -- (-3,0) -- (-2,1) -- (-3,2);\n    \\draw [thick, SkyBlue] (-2,1) -- (2,1);\n    \\draw [thick, SkyBlue] (-3,0) -- (-1.3,-0.7) -- (-0.7,-0.1) -- (-2,1);\n    \\draw [line width=1mm, DarkRed] (-1,-0.4) -- (-0.7,-0.7);\n    \\draw [very thick, DarkRed] (-1,-0.6) -- (-1,-0.4) -- (-0.8,-0.4);\n    \\draw [very thick, DarkRed] (-0.7,-0.5) -- (-0.7,-0.7) -- (-0.9,-0.7);\n\\end{tikzpicture}\n    \\caption{Monodromy.}\n    \\label{mono}\n\\end{figure}\n\n\\begin{remark}\\label{monodromy}\n   We note that due to the existence of monodromy, the glued polygon $\\cup_{e\\in E}P_e$ and $\\Phi_F^E$ might  not be realizable in a vector space. See, e.g., \\Cref{mono}.\n\\end{remark}\n\n\\begin{definition}\\label{assembled}\n    A \\emph{glued toric variety} $X$ is a union $X=\\cup_i X_i$ of complete toric varieties which are glued along toric invariant subvarieties. A \\emph{polarized glued toric variety} $(X,L)$ is a pair of a glued toric variety $X$ and a linearized ample line bundle $L$ on $X$, where $L|_{X_i}$ is also ample for every $i$.\n\\end{definition}\n\n\n\\begin{remark}\n    Note that the resulted toric varieties in Construction~\\ref{const} may not be seminormal, and so they are not necessarily the \\emph{stable toric varieties (or broken toric varieties)} appeared in \\cite{alexeev2002complete,olsson2008compactifying}. Without fixing $\\{\\phi_f^{e,e'}\\}$, one only determines a family of varieties, which may not be isomorphic to each other, see \\cite[Section 2.2]{alexeev2015moduli}. \n\\end{remark}\nWe now have all the ingredients to prove our main theorem.\n\\begin{proof}[{\\bf Proof of Theorem~\\ref{thm:main}}]\n    For each complete fan $\\Sigma_e$, let $\\rho_f$ be the ray arising from the contraction of $f$. Then $A_1(X_{\\mathcal{F}};\\mathbb{Q})_\\mathcal{F}$ is generated by all $f$'s, where the relations are given by \\Cref{rel} and $\\{\\phi_f^{e,e'}\\}$. Now, by setting $c(\\rho_f)$ to be $w(f)$, we observe that the balancing condition for Minkowski weights \\eqref{Minkowski} and for tensegrities are the same, which completes the proof.\n\\end{proof}\n\n\n\n \n  \n   \n\n\\section{Classical Planar Tensegrities}\\label{planar}\nIn this section, we study the case, where $d=2$ in \\Cref{higher} and $F$ has a bounded support. In this case, the framework $\\mathcal{F}$ has an underlying graph $G$, which is the classical tensegrity model (see, e.g., \\cite{maxwell1864xlv,roth1981tensegrity, connelly1996second,connelly2013tensegrity} for more details on tensegrities of graphs).\n\\subsection{Toric variety associated to $G$.}\\label{sec:graph}\nLet $G$ be a simple graph without multiple edges and loops. We denote its edge set with $E(G)$ and its vertex set with $V(G)$. Fix a map $\\mathbb{P}:G\\rightarrow\\mathbb{R}^2$ such that $\\mathbb{P}(V_i)\\not=\\mathbb{P}(V_j)$ for any pair of distinct vertices $V_i,V_j\\in V(G)$. Assume that the images of edges are linear segments without interior intersections. Denote the image of $G$ by $\\mathbb{P}(G)$. We require that $\\mathbb{P}(G)$ is \\emph{integral}, which means that the point $p_i=\\mathbb{P}(V_i)$ has integral coordinates and the edge $\\mathbb{P}(e)$ has a rational slope (or $\\infty$), for every $V_i\\in V(G)$ and $e\\in E(G)$. Under this assumption, the balancing condition at $p_i$ has the form:\n$$\n    \\sum_{j\\not=i}w_{ij}\\vect{p_ip_j}^{\\ \\prim}=0,\n$$\nwhere $w_{ij}$ is the stress on the edge connecting $p_i,p_j$ and $\\vect{p_ip_j}^{\\ \\prim}$ is the primitive vector on the ray originating from $p_i$, and pointing to $p_j$. We may add extra edges on $\\mathbb{P}(G)$ to obtain a triangulation $\\mathcal{T}$ of the convex hull of $\\mathbb{P}(G)$. We denote the boundary cycle of the convex hull of $\\mathbb{P}(G)$ by $C$.\n\n\\medskip\nLet $N\\cong\\mathbb{Z}^2$ be the lattice in $\\mathbb{R}^2$, and $\\widetilde{N}=N\\oplus\\mathbb{Z}$. Put $\\mathbb{P}(G)\\subset \\widetilde{N}_\\mathbb{R}$ at $(N_\\mathbb{R},1)$. By taking cones over the triangulation $\\mathcal{T}$, one can obtain a non-complete fan. (Note that we take one cone for every triangle in $\\mathcal{T}$ and glue them together to obtain a polyhedral fan). We also add a new ray $\\rho_0$ generated by the primitive vector $v_0$ along $-\\sum_{i=1}^n v_i$ where $v_i=(p_i,1)$. Then, by including the cones $\\sigma_{0ij}=\\Cone\\{v_0,v_i,v_j\\}$ for every edge $\\{v_i,v_j\\}$ of the boundary cycle $C$, we obtain a complete polyhedral fan $\\Sigma_{\\mathcal{T}}\\subset\\widetilde{N}_\\mathbb{R}$, together with a complete toric variety $X_{\\Sigma_\\mathcal{T}}$.\n\\begin{figure}[h!]\n\\begin{tikzpicture}[scale=0.27]\n    \\draw [gray, thick] (0,0) -- (-8,-2);\n    \\filldraw[blue!10, opacity=0.5] (11,-2) -- (11,12) -- (23,20) -- (23,8) -- cycle;\n    \\draw [gray, dashed, thick] (0,0) -- (18,8.5);\n    \\draw [gray, dashed, thick] (0,0) -- (16,8.5);\n    \\draw [gray, thick] (0,0) -- (14,6);\n    \\draw [gray, thick] (0,0) -- (14,9);\n    \\draw [gray, thick] (0,0) -- (16,12);\n    \\draw [gray, dashed, thick] (0,0) -- (20,12);\n    \\draw [gray, dashed, thick] (0,0) -- (20,9);\n    \\draw [gray, thick] (0,0) -- (18,6);\n    \\draw [very thick, DarkRed] (14,6) -- (14,9) -- (16,12) -- (20,12) -- (20,9) -- (18,6);\n    \\draw [very thick, dashed, DarkRed] (14,6) -- (18,6);\n    \\draw [very thick, DarkRed] (14,6) -- (16,8.5);\n    \\draw [very thick, DarkRed] (14,9) -- (16,8.5);\n    \\draw [very thick, DarkRed] (16,12) -- (16,8.5);\n    \\draw [very thick, DarkRed] (20,12) -- (18,8.5);\n    \\draw [very thick, DarkRed] (20,9) -- (18,8.5);\n    \\draw [very thick, DarkRed] (18,6) -- (18,8.5);\n    \\draw [very thick, DarkRed] (16,8.5) -- (18,8.5);\n    \\draw [very thick, dashed, DarkRed] (16,8.5) -- (20,12);\n    \\draw [very thick, dashed, DarkRed] (16,8.5) -- (18,6);\n    \\node [above left] at (0,0) {$({\\bf 0},0)$};\n    \\node [above left] at (14,9) {$v_{i+1}$};\n    \\node [above] at (16,12) {$v_i$};\n    \\node [above right] at (20,12) {$v_{i-1}$};\n    \\node [above left] at (-8,-2) {$v_0$};\n    \\node [SkyBlue] at (19,15) {$(N_\\mathbb{R},1)$};\n\\end{tikzpicture}\n\\caption{The polyhedral fan associated to $G$. The solid edges (in red) are the edges of $\\mathbb{P}(G)$ and the dashed edges are the new edges to obtain a triangulation $\\mathcal{T}$.}\n    \\label{toricconstruction}\n\\end{figure}\nLet $E^+_\\mathcal{T}$ be the set of union of the new edges in $\\mathcal{T}$ but not in $\\mathbb{P}(G)$ and define the set:\n$$\n    A^1(X_{\\Sigma_\\mathcal{T}};\\mathbb{Q})_{\\mathbb{P}(G)}:=\\{c\\in A^1(X_\\Sigma;\\mathbb{Q})\\mid c(\\tau)=0,\\text{ for any } \\tau\\text{ over }e\\in E_{\\mathcal{T}}^+\\}.\n$$\n\\begin{proposition}\\label{bc}\n    $A^1(X_{\\Sigma_\\mathcal{T}};\\mathbb{Q})_{\\mathbb{P}(G)}$ is isomorphic to $A_\\mathcal{F}$.\n\\end{proposition}\n\\begin{proof}\n    We need to show that the balancing condition on the framework $\\mathbb{P}(G)$ and the one on $\\Sigma^1_\\mathcal{T}$ are equivalent. At $p_i$, assume that the balancing condition on the $\\mathbb{Z}$-framework is $\\sum_{j\\not=i}w_{ij}\\vect{p_ip_j}^{\\ \\prim}=0$. For $\\tau_{ij}=\\Cone\\{v_i,v_j\\}$, let $c(\\tau_{ij})=w_{ij}$. Note that $n_{\\tau_{ij},\\rho_i}=v_i+\\vect{p_ip_j}^{\\ \\prim}$. Therefore, for any $\\widetilde{m}=(m,t)\\in \\widetilde{M}(\\rho_i)$, i.e.~$\\langle m,p_i\\rangle+t=0$, one has\n    \\begin{align*}\n        \\langle \\widetilde{m},n_{\\tau_{ij},\\rho_i}\\rangle&=\\langle(m,t),(p_i+\\vect{p_ip_j}^{\\ \\prim},1)\\rangle\\\\\n        &=\\langle m,p_i\\rangle+\\langle m,\\vect{p_ip_j}^{\\ \\prim}\\rangle+t\\\\\n        &=\\langle m,\\vect{p_ip_j}^{\\ \\prim}\\rangle.\n \\end{align*}\nThis shows that the balancing conditions on  $\\mathbb{P}(G)$ and $\\Sigma^1_\\mathcal{T}$ are equivalent, as desired.\\end{proof}\n\n\n\\begin{corollary}\n   The framework $\\mathbb{P}(G)$ admits a $\\mathbb{Q}$-tensegrity if and only if $A^1(X_\\Sigma;\\mathbb{Q})_{\\mathbb{P}(G)}\\not=0$.\n\\end{corollary}\n\n\\begin{remark}\n We note that in \\cite{karpenkov2021frame}, Karpenkov provided some other criteria for the existence of tensegrities (see \\cite[Theorems~2.18 and  5.20]{karpenkov2021frame}). We also note that for the construction of the (irreducible) toric variety, having a triangulation is not necessary. We have added this assumption, as the computation of Chow groups of cocycles can be easily carried out for the $\\mathbb{Q}$-factorial toric varieties. Moreover, adding the assistant ray $\\rho_0$ is because we require a complete polyhedral fan to be able to apply the results from the intersection theory, see~\\Cref{dual}.\n\\end{remark}\n\n\n\\subsection{Computing Chow rings.}\\label{sec:computation} We now briefly review the computational method of Chow rings via Stanley-Reisner ideals. Let $z_i$ be the free generator corresponding to primitive vector $p_i$, for $i=1,\\ldots,|\\Sigma_1|$.\nSince $\\Sigma$ is simplicial, the Chow ring can be computed via\n$$\n    A^\\bullet(X_\\Sigma;\\mathbb{Q})=\\frac{\\mathbb{Q}[z_1,\\ldots,z_{|\\Sigma_1|}]}{\\mathcal{SR}+\\mathcal{LR}}\n$$\nwhere $\\mathcal{SR}$ is the corresponding  Stanley-Reisner ideal and $\\mathcal{LR}$ is the ideal of linear relations given by the lattice of characters. More explicitly, the Stanley-Reisner ideal $\\mathcal{SR}$ is generated by $\\{z_{i_1}\\cdots z_{i_s}|\\Cone\\{v_{i_1},\\cdots,v_{i_s}\\}\\not\\in\\Sigma\\}$. Hence, the elements of $\\mathcal{SR}$ have degree at least 2, and so\n$$\n    A^1(X_\\Sigma;\\mathbb{Q})=\\frac{\\Span_\\mathbb{Q}\\{z_1,\\ldots,z_{|\\Sigma_1|}\\}}{\\mathcal{LR}}.\n$$\nIn our case, we have that:\n\\[\\mathcal{LR}=\\langle\\sum _{i=1}^n a_i z_i-\\sum_{i=1}^na_iz_0,\\sum _{i=1}^n b_iz_i-\\sum _{i=1}^n b_iz_0,\\sum _{i=1}^n z_i-nz_0\\rangle.\n\\]\n      \n\\medskip\n\nWe also note that the condition $A^1(X_\\Sigma;\\mathbb{Q})_{\\mathbb{P}(G)}\\not=0$ can be directly computed, using the formula in \\cite[Proposition 6.4.4]{cox2011toric}. More explicitly,\nlet $\\mult(\\sigma)=[N_\\sigma:\\mathbb{Z} v_1+\\cdots+\\mathbb{Z} v_l]$, where $v_i$'s run over all rays of $\\sigma$.\nLet $v_{i-1},v_{i+1}$ be the two neighbors of $v_i$. For $\\tau_{0i}=\\Cone\\{v_0,v_i\\},\\sigma_{0,i,i\\pm1}=\\Cone\\{v_0,v_i,v_{i\\pm 1}\\}$, one can compute the intersection number by the following formula:\n\\[\n    D_j\\cdot V(\\tau_{0i})=\\begin{cases}\n        \\ 0 \\quad\\quad\\quad\\quad\\quad\\  \\quad\\quad\\quad j\\not\\in\\{0,i,i\\pm 1\\}\\\\\n        \\displaystyle\\frac{\\mult(\\tau_{0i})}{\\mult(\\sigma_{0,i,i\\pm 1})}\\quad\\quad\\quad j=i\\pm 1\\\\\n        \\displaystyle\\frac{\\lambda_0\\mult(\\tau_{0i})}{\\alpha\\mult(\\sigma_{0,i,i- 1})}\\quad\\quad j=0\\\\\n        \\displaystyle\\frac{\\lambda_i\\mult(\\tau_{0i})}{\\alpha\\mult(\\sigma_{0,i,i-1})}\\quad\\quad j=i\n    \\end{cases}\n\\]\nwhere $\\alpha,\\lambda_0,\\lambda_i$ are determined by the linear relation $\\alpha v_{i-1}+\\lambda_0v_0+\\lambda_iv_i+\\beta v_{i+1}=0$.\n\n\n\n\n\\begin{figure}[h]\n\\centering\n    \\begin{subfigure}{0.45\\linewidth}\n    \\centering\n \\begin{tikzpicture}[scale=0.4]\n     \n    \\draw [SkyBlue, dashed, very thick] (0,0) -- (4,0);\n    \\draw [SkyBlue, dashed, very thick] (0,0) -- (4,2);\n    \\draw [SkyBlue, dashed, very thick] (0,0) -- (5,1);\n   \n    \\draw [DarkRed, very thick] (4,0) -- (4,2) -- (6,3) -- (7,0) -- cycle;\n    \\draw [DarkRed, very thick] (5,1) -- (4,0);\n    \\draw [DarkRed, very thick] (5,1) -- (4,2);\n    \\draw [DarkRed, very thick] (5,1) -- (6,3);\n    \\draw [DarkRed, very thick] (5,1) -- (7,0);\n   \n    \\draw [SkyBlue, dashed, very thick] (6,3) -- (12,6);\n    \\draw [SkyBlue, dashed, very thick] (5,1) -- (10,2);\n    \\draw [SkyBlue, dashed, very thick] (7,0) -- (14,0);\n    \\draw [SkyBlue, dashed, very thick] (12,6) -- (10,2) -- (14,0) -- cycle;\n   \n    \\node [below] at (5,1) {$p_5$};\n    \\node [below right] at (7,0) {$p_4$};\n    \\node [below left] at (4,0) {$p_3$};\n    \\node [above left] at (4,2) {$p_2$};\n    \\node [above right] at (6,3) {$p_1$};\n   \n    \\node [right] at (12,6) {$p_1'$};\n    \\node [right] at (14,0) {$p_4'$};\n    \\node [below] at (10,2) {$p_5'$};\n\\end{tikzpicture}\n\\caption{$A^1_{\\mathbb{P}(G)}(X_\\Sigma;\\mathbb{Q})\\cong\\mathbb{Q}\\not=0$}\n\\label{subfig1}\n    \\end{subfigure}\n\\begin{subfigure}{0.45\\linewidth}\n    \\centering\n \\begin{tikzpicture}[scale=0.7]\n    \\draw [SkyBlue, dashed, very thick] (-2,0) -- (4,0);\n    \\draw [SkyBlue, dashed, very thick] (1,1) -- (5,1);\n    \\draw [SkyBlue, dashed, very thick] (-2,0) -- (4,2);\n    \\draw [DarkRed, dashed, very thick] (5,1) -- (7,3);\n    \\draw [DarkRed, dashed, very thick] (5,1) -- (7,0);\n   \n    \\draw [DarkRed, very thick] (4,0) -- (4,2) -- (7,3) -- (7,0) -- cycle;\n    \\draw [DarkRed, very thick] (5,1) -- (6,1);\n    \\draw [DarkRed, very thick] (5,1) -- (4,0);\n    \\draw [DarkRed, very thick] (5,1) -- (4,2);\n    \\draw [DarkRed, very thick] (6,1) -- (7,3);\n    \\draw [DarkRed, very thick] (6,1) -- (7,0);\n   \n    \\node [above] at (6,1) {$p_6$};\n    \\node [above] at (5,1) {$p_5$};\n    \\node [below right] at (7,0) {$p_4$};\n    \\node [below left] at (4,0) {$p_3$};\n    \\node [above left] at (4,2) {$p_2$};\n    \\node [above right] at (6,3) {$p_1$};\n\\end{tikzpicture}\n    \\caption{$A^1_{\\mathbb{P}(G)}(X_\\Sigma;\\mathbb{Q})=0$}\n     \\label{subfig2}\n\\end{subfigure}\n    \\caption{The computational examples in Example~\\ref{ex:running}.}\n    \\label{fig.example}\n\\end{figure}\n\n\n\\begin{example}\\label{ex:running}\n\n    (a) Consider the $\\mathbb{Z}$-framework with\n$$\n    p_1=(1,2),p_2=(-1,1),p_3=(-1,-1),p_4=(2,-1),p_5=(0,0),\n$$\nas shown in \\Cref{fig.example}(A). Then $v_0=(-1,-1,-5)$, and the linear relation is\n    \\[\n    \\begin{cases}\n        0=-z_0+z_1-z_2-z_3-2z_4\\\\\n        0=-z_0+2z_1+z_2-z_3-z_4\\\\\n        0=-5z_0+z_1+z_2+z_3+z_4+z_5\n    \\end{cases}\n\\]\nSo $A^1(X_\\Sigma;\\mathbb{Q})=\\Span_\\mathbb{Q}\\{D_0,D_1,D_2\\}$.\nThe multiplicities of walls and full-dimensional cones are:\n\\smallskip\n\n\\begin{center}\n\\begin{tabular}{|c c c c|}\n\\hline\n    $\\mult(\\tau_{01})=1$ &$\\mult(\\tau_{02})=2$ &$\\mult(\\tau_{03})=6$ &$\\mult(\\tau_{04})=3$\\\\\n\n    $\\mult(\\tau_{12})=1$ &$\\mult(\\tau_{23})=2$ &$\\mult(\\tau_{34})=3$ &$\\mult(\\tau_{41})=1$\\\\\n\n    $\\mult(\\tau_{15})=1$ &$\\mult(\\tau_{25})=1$ &$\\mult(\\tau_{35})=1$ &$\\mult(\\tau_{45})=1$\\\\\n\n    $\\mult(\\sigma_{012})=14$ &$\\mult(\\sigma_{023})=12$ &$\\mult(\\sigma_{034})=18$ &$\\mult(\\sigma_{041})=21$\\\\\n\n    $\\mult(\\sigma_{125})=3$ &$\\mult(\\sigma_{235})=2$ &$\\mult(\\sigma_{345})=3$ &$\\mult(\\sigma_{415})=5$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\smallskip\n\nThe intersection numbers are:\n\\smallskip\n\n\\begin{center}\n\\scalebox{0.85}{\n\\begin{tabular}{|c|c c c c c c c c c c c c|}\n\\hline\n        &$V(\\tau_{01})$ &$V(\\tau_{02})$ &$V(\\tau_{03})$ &$V(\\tau_{04})$ &$V(\\tau_{12})$ &$V(\\tau_{23})$ &$V(\\tau_{34})$ &$V(\\tau_{41})$ &$V(\\tau_{15})$ &$V(\\tau_{25})$ &$V(\\tau_{35})$ &$V(\\tau_{45})$\\\\\n\\hline\n    $D_0$ &1\/42 &1\/21 &1\/6 &1\/14 &1\/14 &1\/6 &1\/6 &1\/21 &0 &0 &0 &0\\\\\n\\hline\n    $D_1$ &0 &1\/7 &0 &1\/7 &1\/21 &0 &0 &1\/35 &-1\/15 &1\/3&0&1\/5\\\\\n\\hline\n    $D_2$  &1\/14 &-1\/14 &1\/2 &0 &-1\/42 &0 &0 &0 &1\/3 &-1\/6 &1\/2&0\\\\\n\\hline\n    $D_3$ &0 &1\/6 &0 &1\/6 &0 &-1\/6 & -1\/6 &0 &0 &1\/2&1\/6 &1\/3\\\\\n\\hline\n    $D_4$ &1\/21 &0 &1\/3 &1\/21 &0 &0 &0 &1\/105 &1\/5 &0 &1\/3 &1\/15\\\\\n\\hline\n    $D_5$ &0 &0 &0 &0 &1\/3 &1 &1 &1\/5& -7\/15 &-2\/3 &-1 &-3\/5\\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\n\\medskip\n\nLet $D=c_0D_0+c_1D_1+c_2D_2$, then the linear equation system $\\{D\\cdot V(\\tau_{0j})=0,j=1,\\ldots,4\\}$ has nonzero solutions, which is  $A^1(X_\\Sigma;\\mathbb{Q})_{\\mathbb{P}(G)}=\\{\\lambda(-6D_0+3D_1+2D_2),\\lambda\\in\\mathbb{Q}$\\}. \n\n\\medskip\n\n(b) For the $\\mathbb{Z}$-framework with\n$$\n    p_1=(2,2),p_2=(-1,1),p_3=(-1,-1),p_4=(2,-1),p_5=(0,0),p_6=(1,0)\n$$\nand edges as in Figure~\\ref{fig.example}(B), we have that $A^1(X_\\Sigma;\\mathbb{Q})_{\\mathbb{P}(G)}=0$. For example, let $D=\\sum_{i=0}^3c_iD_i$. Then $\\{D\\cdot V(\\tau_{0j})=0,j=1,\\ldots,4\\}$ has solutions $\\{D=\\lambda(-7D_0+3D_1+2D_2),\\lambda\\in\\mathbb{Q}\\}$. However, $D\\cdot V(\\tau_{15})=0$ forces $\\lambda=0$.\n\n\\medskip\n\nWe note that the computational results in this example agree with the conclusions in \\cite[Example 1.2]{karpenkov2021frame} and \\cite[Proposition 5.11]{karpenkov2021frame}.\n\\end{example}\n\n\\begin{remark}\n    In general, in the graph $\\mathbb{P}(G)$, the edges may intersect in points other than vertices. We note that, to deal with this problem, instead of using classical toric varieties, the theory of multi-fans in \\cite{hattori2003theory,Masuda} can be applied. Moreover, to work over $\\mathbb{R}$ instead of $\\mathbb{Q}$ or $\\mathbb{Z}$, the techniques developed in \\cite{battaglia2001generalized} are relevant.\n\\end{remark}\n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendix}\n\n\n\\end{document}\n\n\\section{Appendix}\n\n\n\\end{document}\n\n\\section{Background}\n\\subsection{NN basis}\nNNs are a set of algorithms, modeled loosely after the human brain, that are designed to recognize patterns.\nTraditional NNs consist of multiple layer of neurons. Each layer performs the computation as shown in Equation~\\ref{eq:fc} where $X$ is the input vector, $Y$ is the output vector, $W$ is the weight matrix, $B$ is the bias, and $\\sigma$ is a non-linear activation function, which is typically the Rectified Linear Units (ReLU) function~\\cite{nair2010rectified}.\n\\begin{equation}\\label{eq:fc}\n\tY = \\sigma(W \\cdot X + B)\n\\end{equation}\nThis kind of NN is also known as multilayer perceptron (MLP), which has been proved to be a universal approximator~\\cite{hornik1989multilayer}.\nModern NNs are more complicated.\nThe topology is a graph rather than a simple chain, and the types of operations are richer than matrix-vector multiplication.\nMost deep learning frameworks~\\cite{2016arXiv160502688full,abadi2016tensorflow,CNTK,MXNet} use computational graph (CG), a directed acyclic graph, to represent NN computations.\nVertices in the graph represent operations (e.g., dot-product, convolution, activation function, pooling) and immutable\/mutable states~\\cite{abadi2016tensorflow} (e.g., the weight parameters associated).\nEdges represent the data dependency between vertices.\nBoth vertices and edges process or carry tensor data (multi-dimensional arrays).\n\nFor clarity, in this paper, dot-product, bias-addition, and convolution are categorized as \\textit{weighted-sum} operations.\nMoreover, any constant operand, including the trained weight matrix for any vertex of weighted-sum operation, is considered as part of the corresponding vertex as we can view it as the immutable state of the vertex.\n\n\\subsection{NN Chips}\nThere are two types of NN chips.\nThe first type focuses on the traditional ANNs.\nThey are custom architectures~\\cite{13,14,15,21,zhang2016cambricon,27,judd2016stripes,DNPU2017,22,26,28,29,31,23, 24,25,sharma2016high} to accelerate mature ANN models.\nWe usually call this type NN accelerators.\nThe second is neuromorphic chips, which usually supports SNNs to yield higher biological reality~\\cite{8,9,10,40,42,43,44,11}.\n\nThese chips usually consist of a lot of processing elements (PEs) that can efficiently perform dot-product operations because this operation is the main body of most NN models.\nDifferent chips put different constraints on the operations they support.\nTable~\\ref{tab:chips} shows the constraints of some existing NN chips.\nMost NN chips employ low precision numbers to represent weights and input\/output (I\/O) data instead of floating-point numbers.\nThe scale of computation that each PE can process is usually fixed.\nPRIME~\\cite{7} and DianNao~\\cite{13} have extra adders to support larger scale computations.\nHowever, NN chips such as TianJi~\\cite{11} and TrueNorth~\\cite{8} do not have extra adders, and the output of their PE can  connect to only one input port of another PE.\nFor these chips, the scale of computation is also a problem.\nDespite the widely-supported dot operation, many other operations required by NNs usually lack for support.\n\n\\begin{table}[!t]\\small\n\t\\centering\n\t\\begin{tabular}{|*{5}{l|}}\n\t\t\\hline\n\t\t\\textbf{Chip} & \\textbf{Weight} & \\textbf{I\/O} & \\textbf{Scale} & \\textbf{Nonlinear}\\\\\n\t\t\\hline\n\t\t\\hline\n\t\tTianJi~\\cite{11} & 8-bit & 8-bit & $256^2$ & Configurable\\\\\n\t\t\\hline\n\t\tPRIME~\\cite{7}& 8-bit & 6-bit & $256^2$ & ReLU\\\\\n\t\t\t&\t&\t&\t& Max Pooling\\\\\n\t\t\\hline\n\t\tDianNao~\\cite{13}& 16-bit & 16-bit & $16^2$ & Configurable\\\\\n\t\t\\hline\n\t\tTPU~\\cite{TPU2017} & 8-bit & 8-bit & None & ReLU\\\\\n\t\t\t&\t&\t&\t& Max Pooling\\\\\n\t\t\t&\t&\t&\t& etc.\\\\\n\t\t\\hline\n\t\tTrueNorth~\\cite{8}& 2-bit & Spiking & $256^2$ & LIF\\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Hardware limitations of NN chips}\n\t\\label{tab:chips}\n\\end{table}\n\n\\section{Conclusion}\\label{sec:conclusion}\nWe present a programming solution for NN chips, which can transform a trained, unrestricted NN into an equivalent network to meet hardware constraints.\nMultiple techniques are proposed to reduce the transformation error and improve the processing speed.\nThe solution is validated on a real neuromorphic chip and a PIM design for ANNs, as well as on different scales of NNs under different constraints.\nThe evaluation shows that the transformation methodology is very effective and only insignificant errors will be introduced. \nThe transformation time is much faster than re-training the NN models for a specific neuromorphic hardware.\n\\section{Problem Description}\nTo bridge the gap between NN applications and NN chips, we decouple the whole system stack with a software programming model and a hardware execution model.\nThe former is the programming interface for NN experts to develop NN applications.\nThe later is the SW\/HW interface that NN chips can executed directly.\n\n\\keyword{Software Programming Model.}\nThe machine-learning community has already employed \\textit{Computational Graph} (CG) as the programming model.\nIt is a data-flow graph $G=(V,E)$ which represents a number of operations with vertices $V$ and represents data dependencies between these operations with edges $E$.\nMost deep-learning frameworks~\\cite{2016arXiv160502688full,abadi2016tensorflow,CNTK,MXNet} adopt CG to build NN models.\nAnd the set of supported operations is $F$.\n\nEach vertex in $V$ is an operation $y=f(x_1, \\ldots, x_n)$, where $f\\in F$, $y$ represents the output edge, and $\\{x_1, \\ldots, x_n\\}$ represent input edges.\nThus, the entire model can be expressed as a composite function $Y=H(X)$, where $X$ represent all input vertices and $Y$ represents all output vertices.\n\nWe also adopt CG as the programming model with a slight modification.\nThe difference is that we regard model parameters as immutable states of the corresponding vertices instead of normal edges of the vertices.\nNamely, we regard an operation $f(x, \\theta)$ as $f^\\theta(x)$, where $\\theta$ is model parameters and $x$ is an input operand.\nThus, it can only be a trained NN model that all parameters have been already determined.\n\n\\keyword{Hardware Execution Model.}\nThe computation model that hardware could execute is also a data-flow graph $G'=(V',E')$.\nIt has a supported operation set $F'$, denoted as \\textit{core-op set}.\nHowever, the supported operations are very limited, and these operations have many limitations.\nIn addition, some hardware also has constraints on the interconnection subsystem. For example, TianJi and TrueNorth does not support multi-cast; one output port of each PE can only be connected to one input port.\nThe hardware execution model forms a composite function $Y'=H'(X')$.\n\nThus, our goal is to build $G'$ from $G$ so that $H'$ is approximately equivalent to $H$.\n\n\\keyword{Minimum Hardware Requirement.}\nWe define a minimum set of operations $C$ that $C\\subset F'$ has to be satisfied to use our NN compiler.\nIt only contains one operation, denoted as \\textit{dot--core-op}.\nNamely, the operation dot--core-op must belong to the core-op set.\nEquation~\\ref{eq:core-op} shows the computation of the dot--core-op where $X$ and $Y$ are the input and output vectors, respectively; $N$ and $M$ are their sizes; $W$ is the weight matrix of size $M\\times N$; and $\\sigma$ is a nonlinear activation function.\n\\begin{equation}\\label{eq:core-op}\n\tY_j = \\sigma(\\sum_iW_{ji}X_i) \\quad (1\\le j\\le M, 1\\le i\\le N)\n\\end{equation}\nIn addition, the I\/O data precision is $B$ bits.\n\nFormally, the dot--core-op meets the following constraints:\n\\begin{itemize}\n\t\\item $N$,$M$,$B$ are fixed.\n\t\\item The value range of each element in $W$ is a finite set $S$.\n    $S$ is either a fixed set or a configurable set $S^P$ with some parameters $P$.\n    \\item Without loss of generality, only ReLU function ($\\sigma$) is supported.\n\\end{itemize}\n\nWe choose this dot--core-op as the minimum requirement for hardware because it can cover most existing NN chips (e.g., those listed in Table~\\ref{tab:chips}).\nThus, our NN compiler can support most existing NN chips.\n\\section{Implementation and Evaluation}\\label{sec:evaluation}\n\\subsection{Implementation}\nWe have implemented the tool to support different hardware constraints, including those of TianJi~\\cite{11} and PRIME~\\cite{7}.\n\nTianJi is fabricated with 120nm CMOS technology. The running frequency is 100MHz and the total dynamic power consumption is 120mW. TianJi chip supports both ANN and SNN modes.\nThe numerical accuracy of weight value is 8-bit fixed-point and the scale of vector-matrix-multiplication is $256 \\times 256$.\nFor ANN mode, the I\/O precision is 8-bit that is cut from the 24-bit internal computation output; the cut range is configurable, thus its weight encoding strategy is dynamic fixed-point.\nFor SNN mode, the minimal I\/O precision is 1-bit, which can be extended to $n$-bit with $2^n$ cycles as the sampling window (as described in Section~\\ref{sec:snn}). The neuron model is a simplified LIF neuron model with a constant leakage and a threshold for firing; the weight encoding method can be viewed as fraction encoding.\nPRIME~\\cite{7} is a memristor-based PIM architecture for ANN.\nThe weight precision is 8-bit and the I\/O precision is 6-bit. The scale of vector-matrix-multiplication is $256 \\times 256$, too. The output range can be configured with an amplifier; thus its weight encoding can also be viewed as fraction encoding.\n\nQuite a few NN applications, including an MLP for MNIST dataset (784-100-10 structure, $98.2\\%$ accuracy of full precision), LeNet-5~\\cite{LeNet} for MNIST dataset ($99.1\\%$ accuracy), a CNN~\\cite{mishkin2015all} for CIFAR-10 dataset ($84.64\\%$ accuracy \\footnote{As described by ~\\cite{mishkin2015all}, with some special initialization method, the CNN accuracy can exceed 90\\%. Here we ignore it for simplicity, which does not affect our evaluation.}), AlexNet~\\cite{AlexNet} and VGG16~\\cite{VGG16} for ImageNet, have been respectively transformed and then deployed onto TianJi~\\cite{11} and PRIME~\\cite{7} to show the validation.\nThe first three networks are trained by Theano~\\cite{2016arXiv160502688full}.\nParameters of the next two CNNs for ImageNet are extracted from trained models of the Caffe Model Zoo directly.\nThe inference accuracies of full precision are given in Table~\\ref{tab:nn_model_transform}.\n\nWithout loss of generality, we take the mapping of the LeNet-5 for MNIST onto the TianJi system as an example.\n\nOne TianJi chip contains 6 cores connected by a $2 \\times 3$ mesh NoC;\neach core supports 256 simplified LIF neurons and works in the weight stationary mode.\nThe main body of a TianJi system is a Printed Circuit Board (PCB) including 16 chips.\nOn the whole, all of the cores form a $12 \\times 8$ 2D-mesh network, and there is a great gap between the delay\/bandwidth of intra-\/inter-chip communications.\n\nThe transformed CG consists of 497 TianJi's dot--core-op.\nTaking into account the weight reuse of convolution, 19 physical cores are actually occupied.\nWe use the heuristic Kernighan-Lin (KL) partitioning algorithm for the mapping problem.\nIt abstracts the mapping as a graph partition problem and tries to minimize communications across partition boundaries.\nTake the bipartition as an example: the input to the algorithm is a graph;\nthe weight of each edge is the communication delay.\nThe goal is to partition the graph into two disjoint subset $A$ and $B$ of equal size, in a way that minimizes the communication cost of the subset of edges that cross from $A$ to $B$.\n\nFirst, a randomly generated initial mapping distribution is given\nSecond, the KL algorithm bi-partitions the mapped cores repeatedly till only the closest two cores are left in any of the final partition in the 2D-mesh. During this phase, partitions that minimize the communication cost between cores are remained; here the cost of an edge across boundary refers to its weight multiplied by the number of transmissions, as we can get the communication statistics from the transformed CG, including those information about the reused cores.\n\n\\subsection{Evaluation}\n\nThe inference accuracies after transformation for TianJi and PRIME are given in Table~\\ref{tab:nn_model_transform}.\nThis table also shows the re-encoding factor, which indicate the number of hidden units for autoencoders, for each case.\n\nWe can see that the transformation errors introduced are very limited, for all NN models and chips tested, we can achieve less than 2\\% accuracy drop.\nFor most cases, we only use $1\\times$ hidden unit to re-encode data, which means that the NN scale is not changed.\nFor the two TianJi-SNN cases, as the I\/O constraints are very strict, we use $2\\times$ hidden units, thus $4\\times$ PEs will be used.\nFor ImageNet cases on PRIME, if we only use $1\\times$ hidden units, the accuracy drop will be over 2\\%.\nWith $4\\times$ hidden units, we can reduce the accuracy drop to less than 1\\%.\nWe can always achieve better accuracy with more hardware resources employed.\n\n\nThis toolchain also improves the development efficiency remarkably. For example, it can transform AlexNet in about 1 hour, while training it from scratch will take about 3\\textasciitilde4 days. Specially, training the whole NN requires millions of iterations to converge, while our method only costs thousands of iterations to converge for each step, that is, takes 5 to 10 minutes to transform one layer. The reason lies in that, after partitioning, we fine-tune each unit one by one; each one is well initialized and much easier to converge than training the entire model. All evaluations are completed on a common PC server equipped with one Tesla P100 GPU.\n\nIn addition, as the large-scale NNs (e.g. those for ImageNet) cannot be occupied by the TianJi system because of the physical limit of chip capacity (a TianJi system contains 16 chips and a chip can occupy 1536 neurons), those related results are drawn from the cycle-accurate TianJi chip simulator.\n\\subsubsection{Accuracy vs. Fine-Tuning Granularity}\nWe conduct experiments to explore the relationship between accuracy and fine-tuning granularity.\nIn another word, in the step of \\textit{Graph tuning}, we can fine-tune one or more successive parts in $G'$ simultaneously, even fine-tune the entire model as a whole.\nIt looks like that increasing the fine-tuning scale per time will increase the search space, which may lead to a lower error rate but consume more computations and more time to converge.\nHowever, results show that coarse grained fine-tuning does not result in improved accuracy.\nFor example, for CIFAR10-CNN, the inference accuracy (the I\/O precision is 7 bits and the weight precision is 4 bits) is 83.07\\% as the fine-tuning-granularity is two subgraphs (the accuracy is 83.14\\% as the granularity is one).\nIf we fine-tune the whole NN together, the accuracy is just 83.24\\%.\nThus, one by one fine-tuning is an optimal strategy, which also means that the problem of error accumulation has been well solved.\nUnless specifically noted, the fine-tuning granularity is just one.\n\n\\subsubsection{Accuracy vs. Resource Consumption}\nFrom the transformation workflow, we can see that the step of \\textit{Data Re-encoding} may introduce the most additional resource overhead:\nWhen the neuron number of the hidden layer of autoencoder is $n\\times$ as much as that of the input\/output layer, the number of crossbar consumed of the whole NN will be $n^2 \\times$.\nThe latter can also be considered as a direct indicator of area consumption and runtime overhead. Thus, we conduct experiments to explore the effect of \\textit{autoencoder}.\n\nWe use the MLP for MNIST dataset.\nAlthough this network is relative small, its conclusion is general for large-scale NNs because we fine-tune NNs part by part and each part is a small graph.\nTable~\\ref{tab:nn_model_transform} also shows the improvement for ImageNet on PRIME with a larger re-encoding factor.\nWe compare the inference accuracies without or with different scales of autoencoder.\nFor the former, we simply scale and round the I\/O signal values to make them suitable for I\/O precision.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{fig\/autoencoder_io.pdf}\n\t\\caption{\n\t\tAccuracy v.s. I\/O precision under different transformation strategies.\n\t}\n\t\\label{fig:autoencoder_io}\n\\end{figure}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{fig\/autoencoder_weight.pdf}\n\t\\caption{\n\t\tAccuracy v.s. weight precision under different transformation strategies.\n\t}\n\t\\label{fig:autoencoder_weight}\n\\end{figure}\n\nFigure~\\ref{fig:autoencoder_io} (Figure~\\ref{fig:autoencoder_weight}) lists the accuracy with different I\/O (weight) precisions respectively (after transformation), without or with different scales of autoencoder, while no other constraints are introduced. Results show that this data re-encoding strategy is effective to improve the transformation accuracy under strict constraints.\n\nIn Figure~\\ref{fig:autoencoder_io}, the accuracy of the transformed network without autoencoder drops significantly when I\/O precision is less than 3 bits (different NNs may have different turning points). In contrast, when autoencoder is used (we assume the I\/O limitation is only 1-bit and only 1x hidden neurons are used, namely, the most critical case), the accuracy is $84.63\\%$ (if no autoencoder, the value is only $13.45\\%$). With more hidden neurons, the accuracy continues to rise, which means our method could trade NN scale for capability.\n\nApparently, increasing the number of hidden neurons can only linearly increase the encoding ability, which is worse than increasing the I\/O precision directly because the latter's encoding ability is $\\propto(2^n)$. For example, using $2\\times$ hidden neurons and 1-bit I\/O does consume the same number of I\/O ports with that of using $1\\times$ hidden nodes and 2-bit I\/O; the accuracy of the former is only $88.2\\%$ while the latter is $94.71\\%$. Thus, it looks like that the hardware had better provide enough I\/O precision since rescuing the accuracy by software (using autoencoder) may cost more hardware resources, especially when the hardware I\/O precision is less than the turning point.\nAnyway, this is a tradeoff between hardware consumption, software adaption and inference error.\n\nMoreover, as illustrated by Figure~\\ref{fig:autoencoder_weight}, autoencoder is also able to rescue the accuracy loss caused by low weight precision. Compared with Figure~\\ref{fig:autoencoder_io}, we can see that NNs are more tolerant of low weight precision than low I\/O precision, since the latter can cause signal distortion directly. Figure~\\ref{fig:autoencoder_weight} also shows that when the weight precision is 2-bit or more, using different scales of autoencoder does not change the accuracy apparently because it has already reached $99\\%$.\n\n\\subsubsection{Impact of Weight Encoding Methods}\nWe have evaluated the weight tuning algorithm, as well as the three kinds of weight encoding strategies. Figure~\\ref{fig:em_weight} shows that weight tuning can set the weight parameters well for the three cases: with the increase of weight precision (any other constraint is not introduced), all of them can reach the upper bound accuracy. In Table~\\ref{tab:weight_encoding}, we further give the effect of each phase of the weight tuning step (the weight precision is 2-bit, without any other constraint).\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{fig\/em_weight.pdf}\n\t\\caption{\n\t\tAccuracy v.s. weight precision under different weight encoding strategies.\n\t\tFor the weight sharing case, weight precision means the bit-width of weight indices.\n\t}\n\t\\label{fig:em_weight}\n\\end{figure}\n\n\\begin{table}[!t]\n\t\\centering\n\t\\begin{tabular}{|c|c|c|c|}\n\t\t\\hline\n\t\t& \\bfseries dynamic & \\bfseries fraction  & \\bfseries weight sharing\\\\\n        & \\bfseries fixed point & \\bfseries encoding & \\bfseries \\\\ \\hline\n\t\t\\bfseries I & 81.56\\% & 85.60\\% & 86.19\\%\\\\ \\hline\n\t\t\\bfseries I+P & 81.56\\% & 87.05\\% & 89.86\\%\\\\ \\hline\n\t\t\\bfseries I+L & 97.08\\% & 97.05\\% & 96.31\\%\\\\ \\hline\n\t\t\\bfseries I+P+L & 97.08\\% & 97.74\\% & 97.14\\%\\\\ \\hline\n\t\\end{tabular}\n\t\t\\caption{\n\t\tAccuracy v.s. different weight encoding strategies under 2-bit weight precision.\n\t\tI: initialization with EM; P: parameter range fine-tuning (to decide $P$); L: low precision fine-tuning (to decide $k_{ij}$)\n\t}\n\t\\label{tab:weight_encoding}\n\\end{table}\n\nWith only the EM-based initialization of the value-range tuning phase (I in Table~\\ref{tab:weight_encoding}), the accuracy of different weight encoding strategies depends on the latter's flexibility. The accuracy of \\textit{weight-sharing} is the highest as it is the most flexible: the precision just limits the bit-width of indices, not weight values. \\textit{dynamic-fixed-point} only allows weight values to be scaled by power of 2; thus it is the least flexible. \\textit{fraction-encoding} is positioned in the middle.\n\nWith the whole value-range tuning phase (including the initialization and fine-tuning, I+P in Table~\\ref{tab:weight_encoding}), the accuracy increases a little for both \\textit{fraction-encoding} and \\textit{weight-sharing} and remains unchanged for \\textit{dynamic-fixed-point}. The reason is in the initialization we have already found a good $P$, further fine-tuning $P$ cannot increase the accuracy much, especially for \\textit{dynamic-fixed-point} with limited flexibility.\n\nWith the EM initialization and the rounding tuning phase (I+L in table~\\ref{tab:weight_encoding}), the accuracy increases significantly, which means that NN can easily find suitable parameters from any well-initialized set of weight values.\n\nWith all phases employed (I+P+L in table~\\ref{tab:weight_encoding}), the accuracy could still increase a little compared with the I+L case, except for \\textit{dynamic-fixed-point}.\n\nAnyway, there is no obvious capability difference between the three strategies.\n\n\\subsection{Discussion and future work}\\label{sec:discussion}\nNow, some transformation steps introduce extra layers more than one time, which may exacerbate NN redundancy.\nTherefore it is necessary to strike a balance between the possible information loss and hardware-resource consumption.\nAnyway, we provide a framework, while the concrete workflow could be customized.\nMoreover, the interaction between network compression and transformation is interesting, and we will study it as the future work.\n\nIn addition, it is helpful to present insights into future neuromorphic architecture designs:\n\\begin{squishlist}\n\t\\item It could give design tradeoff between the common computational components and special functional components.\n\tFor some neural functions or layers that are relatively easy (in terms of hardware consumption) to be achieved by common core-ops, it is unreasonable to integrate such a dedicate component on chip.\n\tIf not, a special component is worth to realize.\n\n\t\\item After transformation, the data flow between core-op is basically determined;\n\twe can analyze the communication pattern in detail, which is conductive to the balanced distribution of computing and communication resources on chip.\n\n\t\\item Our solution regards NN inference as the forward process of graph with fixed topology. To some extent, it is suitable for field programmable devices (especially in the aspect of on-chip connection); thus how to combine the device configurability with the flexibility of transformation is an interesting topic.\n\\end{squishlist}\n\n\\section{Introduction}\\label{sec:introduction}\n\nDesigning custom chips for NN applications with the Very-Large-Scale-Integration (VLSI) technologies has been investigated as a power-efficient and high-performance alternative to general-purpose computing platforms such as CPU and GPU.\nHowever, programming these chips is difficult because of some hardware-specific constraints:\n\\begin{comment}\n\\textcircled{\\small{1}} Due to the utilization of hardware resource or the capability of analog computing (e.g., some memristor-based designs~\\cite{1, 3, 4, 5, 6, 7}), the precision of input and output signals of neurons is usually limited, as well as\n\\textcircled{\\small{2}} the precision of network parameters, such as synaptic weights.\nIn contrast, general-purpose processors use floating-point numbers.\n\\textcircled{\\small{3}} The present fabrication technology limits the fan-in and fan-out of one neuron, and thus constrains the computation scale.\n\\textcircled{\\small{4}} Moreover, the diversity of nonlinear function or neuron model supported by the hardware is limited.\n\\end{comment}\n\\begin{textcircleds*}\n\t\\item Due to the utilization of hardware resources for digital circuits or the capability of analog computing for some memristor-based designs~\\cite{1, 3, 4, 6, 7}, the precision of input and output signals of neurons is usually limited, as well as\n    \\item the precision of NN parameters, such as synaptic weights.\n    \\item The present fabrication technology limits the fan-in and fan-out of one neuron, which constrains the computation scale.\n    \\item The diversity of nonlinear functions or neuron models supported by the hardware is also limited.\n\\end{textcircleds*}\nFor example, for TrueNorth chips~\\cite{8}, the maximum matrix that one synaptic core can handle is $256\\times 256$, and it supports only a simplified leaky--integrate-and-fire (LIF) neuron model.\n\nOne straightforward approach to this problem is to expose the hardware details and limitations to the NN developer directly.\nFor instance, IBM has provided a TrueNorth-specific training mechanism~\\cite{12}.\nThe mechanism constructs the whole NN model from scratch to satisfy all hardware limitations, and then trains the NN model.\nThis method has several drawbacks.\nFirst, it binds NN models to the specific target hardware.\nDevelopers can hardly benefit from existing NN models from the machine-learning community.\nSecond, it limits the power of the NN algorithm. \nThe constraints make it more difficult to converge and reach better accuracy for larger models.\nThird, training the specific NN model from scratch may take a very long time.\n\\begin{comment}\n\\textcircled{\\small{1}} It binds NN development and applications to the specific target hardware.\n\\textcircled{\\small{2}} Developers hardly benefit from existing NN models from machine-learning community.\n\\textcircled{\\small{3}} Training the specific NN from scratch may take a very long time.\n\\end{comment}\n\nAnother approach is to make hardware satisfy software requirement by consuming more hardware resources to reduce the constraints on NN models~\\cite{13, 14, 15, 21}, such as using 16-bit precision rather than 1-bit spiking signals in TrueNorth.\nThis approach can gain less performance improvement from NN quantization and compression technologies.\nFor some memristor-based designs, some constraints due to analog computing are physical limitations, which are difficult to overcome even with more hardware resources consumed.\n\nA third approach is to introduce a domain-specific Instruction Set Architecture (ISA) for NN accelerators, such as the Cambricon~\\cite{liu2016cambricon} ISA.\nThis approach requires both hardware and software to satisfy the ISA.\nHowever, this approach still does not solve the gap between programming flexibility required by NNs and hardware efficiency that can be gained from NNs' redundancy.\nIf we use high-precision instructions that do not have any constraints, the hardware can gain less benefit from NN's redundancy.\nIn contrast, if we use low-precision instructions with many constraints, the NN developer should take these constraints into consideration when developing NN models.\n\nIn addition to these approaches, there are also some work to utilize the NNs' redundancy for performance and provide flexible programming interface by introducing a transforming procedure.\nEIE~\\cite{han2016eie} is such an instance: it extensively uses deep compression to squeeze the redundancy, and design custom chip, EIE, to run the compressed NN model.\nNEUTRAMS~\\cite{NEUTRAMS} also use NNs' redundancy to adapt the original model to satisfy hardware constraints.\nHowever, these methods highly depends on the redundancy in NN models.\nDifferent NN models may have different minimum requirement (precision, connectivity, etc.) on hardware.\nThus, transforming procedure is not a general method, especially for NN models with less redundancy and hardware with severe constraints.\n\nIn this paper we propose a new method with flexibility, better applicability, and easy convergence.\n\\begin{comment}\nFirst, it decouples the neuromorphic computer system into two levels for better flexibility.\n\\textcircled{\\small{1}} The first level is located between programming and the execution substrate.\nNN experts could develop and train NN using any existing NN development framework~\\cite{jia2014caffe, 2016arXiv160502688full, abadi2016tensorflow} without considering hardware constraints.\nAfter the training, NN information such as neural model type\/parameters and network topology, can be extracted to construct the corresponding computational graph (CG) as the input of our method.\n\\textcircled{\\small{2}} The second level decouples software and hardware to make our proposal applicable to a variety of NN chips.\n\\end{comment}\nFirst, we decouple the neuromorphic computer system into two levels for better flexibility, software programming model and hardware execution model.\nWe use computational graph (CG), which is widely used in many popular NN frameworks~\\cite{jia2014caffe,2016arXiv160502688full,abadi2016tensorflow}, as the programming model for NN models.\n\\begin{comment}\nWe abstract the target hardware as a set of connected \\emph{physical cores}; the operation that a single core can accomplish is defined as \\emph{core\\_op}.\nWithout loss of generality, in this paper, \\emph{core\\_op} refers to vector-matrix multiplication plus activation function (or spiking neuron model) that meets hardware restrictions, as they are the main computational components of many existing NN chips.\nMany chips face all (or part) of the similar constraints, as presented in Table~\\ref{tab:chips}.\nNote that our methodology can be used for other chips not listed in the table, as  discussed later in Section~\\ref{sec:specification}.\nAccordingly, NN inference by hardware is abstracted as the forward process of a CG and all the vertices represent \\emph{core\\_ops}.\n\\end{comment}\nWe also provide the hardware\/software (HW\/SW) interface and the minimum hardware functionality that an NN hardware should provide.\n\\begin{comment}\nWe propose a transform workflow to convert a trained, unrestricted NN (expressed as a CG, which is referred to as \\emph{golden model}) into an equivalent CG of \\emph{core\\_ops} through the NN fine-tuning method. This method is called \"training and constraining\".\nAfterwards, the obtained CG will be mapped onto \\emph{physical cores} of target hardware efficiently.\nIn this way, our solution decouples NN application's specification from execution substrate.  To make the NN fine-tuning easy to converge under strict conditions, we also propose multiple techniques as described below:\n\\end{comment}\nWe propose a transformation workflow to convert a trained NN, expressed as a CG, into an equivalent representation of HW\/SW interface through the fine-tuning method.\n\n\\begin{comment}\nThe essential is to construct a CG of \\emph{core\\_ops} to approach the golden model.\nAs \\emph{core\\_ops} are not comparable to their software counterparts, it is reasonable to enlarge the graph scale and\/or complicate the topology properly to improve the capability, especially under strict conditions.\nAccordingly, some techniques, including an MLP-based universal approximator, data re-encoding based on \\emph{autoencoder} (a particular type of NN), and a \\emph{split-and-merge} method, are proposed to deal with issues on the unsupported functions, the limited IO precision and the constrained computation scale respectively, which may increase the graph scale. To some extent, the principle of `trade scale for capability' is used.\n\nIn addition, it is necessary to note that this principle is not contradictory to network compression:\non one hand, usually there is significant redundancy for DNN models, and compression can remove it.\nOn the other hand, diverse types of NN and dataset have different requirements for optimized weight-\/IO-precision of the underlying hardware.\nIt is difficult to determine in advance whether or not the limited hardware precision are enough for any given NN.\nThus, both are useful and orthogonal.\nWe can compress the network first, then convert it, or vice versa.\n\\end{comment}\n\nTo make the transformation workflow general and reliable for different cases, we employed two principles.\n\n\\bulletkeyword{Trade Scale for Capability.}\nAs the operations supported by NN hardware is not comparable to their software counterparts due to the constraints, it is reasonable to enlarge the graph scale and complicate the topology properly to improve the model capability, especially under strict conditions.\n\n\\begin{comment}\nAnother point is to set parameters of \\emph{core\\_ops} properly to make the constructed CG equivalent to the golden model.\nThus, after the above processing, the CG of \\emph{core\\_ops} will be fine-tuned part by part according to a topological ordering of the golden model to reduce transformation error, under the premise of satisfying the hardware weight precision.\n\nIn contrast, from the point of view of system abstraction, the EIE and EIE-like solutions support the general abstraction of NN models and the Cambricon ISA is a hardware abstraction (the specific way of abstraction is different from ours), while we introduce the two abstractions both.\n\\end{comment}\n\n\\bulletkeyword{Divide and conquer.}\nWe fine-tune the entire model part by part according to a topological ordering.\nEach part is a smaller graph that is more easier to converge.\nWe also fine-tune each part with several phases to introduce different constraints, which also facilitates the fast convergence.\n\nMoreover, this transformation procedure could be viewed as \\emph{compilation} of traditional computer systems that converts high-level programs (the hardware-independent, trained NN models) into instructions that hardware can understand (the SW\/HW interface), and the transformation tool could be called an NN compiler.\nAs a summary, this paper has achieved the following contributions:\n\\begin{squishlist}\n\t\\item An NN transformation workflow is presented to complete the aforementioned technologies to support different types of NNs.\n\n    The SW\/HW interface is easy to be adapted to different NN hardware.\n\t\\item Such a toolchain is implemented to support two different hardware designs' constraints, a real CMOS neuromorphic chip for ANN\\&SNN, TianJi~\\cite{11}, and a PIM design built upon metal-oxide resistive random access memory (ReRAM) for ANN, PRIME~\\cite{7}.\n\n\t\\item We complete quite a few evaluations of various metrics. The extra error caused by this process is very limited and time overhead is much less (compared to the whole training process of the original NN).\n\tIn addition, its sensitivity to different configurations and transformation strategies has been explored comprehensively.\n\\end{squishlist}\n\n\\begin{comment}\n\\begin{table}[!t]\\small\n\t\\centering\n\t\\begin{tabular}{|*{5}{l|}}\n\t\t\\hline\n\t\t\\textbf{Chip} & \\textbf{Weight} & \\textbf{IO} & \\textbf{Scale} & \\textbf{Nonlinear}\\\\\n\t\t\\hline\n\t\t\\hline\n\t\tTianJi~\\cite{11} & 8-bit & 8-bit & $256^2$ & Configurable\\\\\n\t\t\\hline\n\t\tPRIME~\\cite{7}& 8-bit & 6-bit & $256^2$ & ReLU\\\\\n\t\t\t&\t&\t&\t& Max Pooling\\\\\n\t\t\\hline\n\t\tDianNao~\\cite{13}& 16-bit & 16-bit & $16^2$ & Configurable\\\\\n\t\t\\hline\n\t\tTPU~\\cite{TPU2017} & 8-bit & 8-bit & None & ReLU\\\\\n\t\t\t&\t&\t&\t& Max Pooling\\\\\n\t\t\t&\t&\t&\t& etc.\\\\\n\t\t\\hline\n\t\tTrueNorth~\\cite{8}& 2-bit & Spiking & $256^2$ & LIF\\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Hardware limitations of NN chips}\n\t\\label{tab:chips}\n\\end{table}\n\\end{comment}\n\n\n\\begin{comment}\nThe rest of this paper is organized as follows.\nSection~\\ref{sec:related_work} presents related work and Section~\\ref{sec:background} introduces the concepts of NN, training, fine-tuning and CG.\nSection~\\ref{sec:workflow} is the main part, which gives the outline of the whole transformation \\& mapping methodology and details of key technologies, from the point of view of CG conversion.\nSection~\\ref{sec:evaluation} presents the implementation and evaluations.\nSection~\\ref{sec:conclusion} concludes.\n\\end{comment}\n\\section{Related Work}\\label{sec:related_work}\nThere are two types of neural network chips.\nThe first type focuses on the traditional ANNs: they are custom architectures to accelerate mature artificial intelligence (AI) algorithms, especially machine learning algorithms of DNN.\nThe second usually supports SNNs to yield higher biological reality.\nIn order to distinguish, we call the former NN accelerators and the latter neuromorphic chips in the following content.\n\\subsection{NN compression}\\label{sec:nn_compression}\nNNs are both computational intensive and memory intensive, making them difficult to deploy on hardware systems with limited resources.\nAt the same time, there is significant redundancy for deep learning models~\\cite{denil2013predicting}.\nTherefore, quite a few studies have been carried out to remove the redundancy, which can be divided into three types: \\emph{pruning neurons}, \\emph{pruning synapses} and \\emph{pruning weights}.\nBoth \\emph{weight-quantization}~\\cite{vanhoucke2011improving, hwang2014fixed, anwar2015fixed} and \\emph{weight sharing}~\\cite{chen2015compressing} are pruning weights.\nFor pruning synapses, deep compression~\\cite{han2015deep} prunes the network by retaining only important connections.\nDiversity Network~\\cite{45} prunes neurons, which selects a subset of diverse neurons and fuses the redundant neurons into the selected ones.\n\nMoreover, there are some research efforts that study extremely compact data representations (including the I\/O precision or weight precision or both) for NN computation. Binarized neural networks [1,3] that investigates the use of 1-bit data types for weight and ternary neural networks [4,5] using 2 bits belong to this category, which have achieved comparable accuracies to state-of-the-art full precision networks for some data sets. However, these methods are effective for specific networks, not common development scenarios.\n\\subsection{NN accelerators}\\label{sec:nn_accelerators}\nEIE~\\cite{han2016eie} extensively employs the above compression techniques and then proposes a dedicated engine to perform inference on the compressed network model.\nESE~\\cite{han2016ese} is its follow-up work that implements a speech recognition engine with compressed Long-Short-Term-Memory model on FPGA.\n\nThe DianNao chip family is a series of state-of-the-art NN accelerators.\nDianNao~\\cite{13} designs an accelerator for DNNs and CNNs that exploits data reuse with tiling.\nThe inherent sharing of weights in CNNs is explored in ShiDianNao~\\cite{21}.\nPuDianNao~\\cite{14} supports seven machine learning algorithms.\nDaDianNao~\\cite{15} is a custom multi-chip machine-learning architecture.\nFrom the aspect of internal data representation and computation, they support fixed-point computation rather than 32-bit floating-point to reduce hardware cost.\nAccordingly, some corresponding retraining or tuning process is needed to adapt software for hardware.\nThey use a load-store ISA~\\cite{liu2016cambricon} to decouple synaptic weight storage from neuron processing logic to avoid the limitation on connection number and improve the utilization of processing logic.\nThis family also supports compressed, sparse NNs through a dedicated accelerator, Cambricon-X~\\cite{zhang2016cambricon}.\n\nMinerva~\\cite{27} uses fine-grained data type quantization and dynamic operation pruning to further reduce the resource consumption of DNN hardware.\nStrip~\\cite{judd2016stripes} relies on bit-serial compute units to enable support for per-layer, dynamically configurable computation precision for DNN.\nDNPU~\\cite{DNPU2017} supports dynamic fixed-point with online adaption and weight quantization, too.\nOther studies include Origami~\\cite{22}, Convolution Engine~\\cite{26}, RedEye~\\cite{28}, NeuroCube~\\cite{29}, neuFlow~\\cite{31} and quite a few FPGA-based designs~\\cite{23, 24, 25, sharma2016high}.\n\nAll the above studies are based on the traditional Complementary Metal-Oxide-Semiconductor (CMOS) technology.\nAnother category is using novel nonvolatile memory devices (usually memristors) to perform neural computations in memory.\nPipeLayer~\\cite{songpipelayer} is such an accelerator for CNNs that supports both training and inference, while PRIME~\\cite{7} and ISAAC~\\cite{18} are for inference.\nOther work includes stand-along accelerators~\\cite{1, hu2016dot, chen2015optimized}, co-processor~\\cite{li2013memristor-based} and many-core or NoC~\\cite{liu2015reno, bojnordi2016memristive} architecture.\nSince the new memory technology is not very mature, there is no systematic programming method.\n\nThe main computational components of these chips usually contain vector-matrix-multiplication units and nonlinear functions, which is the basis of our hardware abstraction.\n\nMoreover, there are quite a few development frameworks for neural networking computing.\nSome (like Theano~\\cite{2016arXiv160502688full}, TensorFlow~\\cite{abadi2016tensorflow}, CNTK~\\cite{CNTK}, Torch~\\cite{collobert2011torch7}, MXNet~\\cite{MXNet}, etc.) describe NN as computation graphs.\nFor a trained NN, the complete information can be extracted from whichever of them.\n\\subsection{Neuromorphic chips}\\label{sec:neuromorphic_chips}\nTrueNorth~\\cite{8, 9} is a digital neuromorphic chip for SNNs, based on a structure of tiled crossbar (each crossbar is of size $256 \\times 256$, and supports binary-valued neurons and ternary-valued synapses).\nThe programming paradigm, \\emph{Corelet}~\\cite{37}, is bound to the hardware platform:\nits recent study~\\cite{12} proposes a TrueNorth-specific training mechanism to construct the whole NN from top to bottom.\nThe parameters learned are then mapped to hardware~\\cite{38} using \\emph{Corelet}.\n\nEMBRACE~\\cite{10} is a compact hardware SNN architecture, with the limited fan-out of individual neuron circuits.\nFrom the programming aspect, it follows the Modular Neural Network (MNN) computing paradigm~\\cite{39}.\nThus, it is not a general solution.\nNeurogrid~\\cite{40} and FACETS~\\cite{42} (including its successor BrainScaleS~\\cite{43}) are analog\/digital hybrid systems, whose development methods are both hardware-specific.\nSpiNNaker~\\cite{44} is different: its toolchain is not bound to any computing paradigm.\nThe reason is that it is based on the chip multiprocessor (CMP) of ARM cores.\nThus, its neural computing is completed by software.\nThe drawback is that the efficiency will be lower than the dedicated hardware.\nTianJi~\\cite{11} is an experimental CMOS neuromorphic chip based on tiled crossbars and supports hybrid computing of ANN and SNN.\nA toolchain NEUTRAMS~\\cite{NEUTRAMS} is developed to map various NN models onto the chip.\nIt also decouples application from hardware and completes SW\/HW co-design for optimization.\nBut its methodology is based on redundancy of NN models and is not suitable for large-scale network under strict constraints.\n\n\n\\section{Transformation Methodology}\\label{sec:workflow}\nIn this section, we will introduce the transformation methodology to transform the software programming model into an approximately equivalent hardware execution model.\n\\subsection{The workflow outline}\\label{sec:outline}\nThe proposed workflow involves 4 steps as shown in Figure~\\ref{fig:workflow}.\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{fig\/workflow.pdf}\n\t\\caption{Workflow of our proposal, from the input model to the output for chip. Step `Graph Tuning' contains 3 sub-steps for different hardware restrictions respectively and the third sub-step has 3 fine-tuning phases.}\n\t\\label{fig:workflow}\n\\end{figure}\n\n \\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{fig\/example.pdf}\n\t\\caption{A transformation example}\n\t\\label{fig:example}\n\\end{figure}\n\n\\noindent\\textbf{Building CG.}\\label{step:parse}\nAccording to the above description, it constructs $G=(V, E)$ based on the input NN's information that includes the trained parameters, network topology, vertex information and training dataset.\nAn example is shown in Figure~\\ref{fig:example}(a).\nIn addition, we can also get the operation set $F$ supported by the deep learning framework.\n\n\\noindent\\textbf{Graph Reform.}\\label{setp:graph_reform}\nIt constructs an intermediate CG, $\\hat{G}=(\\hat{V}, \\hat{E})$.\nThe corresponding operation set $\\hat{F}$ contains all operations that are \\textit{core-op--like}.\nThese operations share the similar form as core-ops or can be achieved by core-ops easily, but do not meet those hardware constraints.\nFigure~\\ref{fig:example}(b) shows an example of the graph with only core-op--like operations.\nThe detailed transformation from $G$ to $\\hat{G}$ is in Section~\\ref{sec:graph_reform}.\n\n\\noindent\\textbf{Graph Tuning.}\\label{step:graph_tuning}\nIn this step, we further transform $\\hat{G}$ to $G'$.\nEvery vertex $\\hat{v}\\in\\hat{V}$ is traversed in a topological ordering of $\\hat{G}$ to form corresponding vertices in $G'$.\nAs shown in Figure~\\ref{fig:workflow}, we have multiple sub-steps for the transformation of each vertex $\\hat{v}$.\nWe can also transform a subgraph with multiple vertices as a whole.\nWe transform the whole graph part by part to have a better convergence since smaller graph are easier to be approximated.\nIt is where we employ the \\textit{divide-and-conquer} principle.\nThe sub-steps are as following.\n\\begin{itemize}\n\n    \\item \\keyword{Data Re-encoding.}\n    Re-encode I\/O data on each edge of the subgraph to solve the precision problem of I\/O data.\n    This sub-step is where we employ the \\textit{trade-scale-for-capability} principle.\n    It enlarges the computation scale, but does not change the operation type, each vertex is still a core-op--like operation.\n\n\t\\item \\keyword{Fully Expanding.}\n    Since core-op--like operations are easy to be implemented with core-ops, in this sub-step, we fully expand each core-op--like operation with multiple core-op operations to solve the limitation on the computation scale.\n\tAfter this sub-step, the subgraph only contains core-ops.\n\n    \\item \\keyword{Weight Tuning.}\n    This step aims to fine-tune the weight matrices of core-ops in the subgraph to minimize transformation error, under the premise of satisfying the hardware weight precision.\n    As shown in Figure~\\ref{fig:workflow}, we also introduce three phases of fine-tuning to make it more reliable and easier to converge.\n    It is also where we employ the \\textit{divide-and-conquer} principle.\n \\end{itemize}\nFigure~\\ref{fig:example}(c) shows an example of the transformed graph $G'$ with only core-ops.\nDetailed transformation are in Section~\\ref{sec:graph_tuning}.\n\n\\noindent\\textbf{Mapping.}\nNow we have built an equivalent Graph $G'$ with only core-ops that meet hardware constraints, which will be mapped onto the target hardware efficiently.\nThis part highly depends on the hardware's interconnection implementation.\nIn Section~\\ref{sec:mapping} we will introduce the basic principle of mapping the hardware execution model onto target chips.\n \n\\subsection{Graph Reform}\\label{sec:graph_reform}\n\\begin{comment}\nInitially, $G(V, E)$ is slightly adjusted: any vertex that represents an activation function is merged with its previous vertex(vertices) whose computation is in the form of weighted-sum to form a \\emph{core\\_op}-like operation. Examples are presented in Figure~\\ref{fig:example}: multiple vertices in one convolutional layer (Figure~\\ref{fig:example}(a)) are merged into one vertex (Figure~\\ref{fig:example}(b)), as well as the fully connected layer.\n\nAfterwards, for all remaining vertices whose operations are neither in the form of weighted-sum nor supported by hardware directly, each of them will be approximated with an MLP, because the latter is a universal approximator; the activation function used is \\emph{ReLU} as the target hardware provides it.\nIn another word, we will train an MLP to achieve the operation(s) of the original vertex; some common functions can be trained in advance.\n\nMoreover, for those vertices that are not suitable to be approached this way because of huge resource consumption, some dedicated designs could be proposed.\n\nFor example, the resource consumption of max pooling increases exponentially with scale, if we just use such a general approximator.\nConsequently, a dedicated design is given.\nMax pooling can be built with max functions, each of which is completed as following:\n{\n\t\\centering\n\t$\\max(a, b) = \\frac{1}{2}\\relu(a+b) + \\frac{1}{2}\\relu(a-b) + \\frac{1}{2}\\relu(-a+b) + \\frac{1}{2}\\relu(-a-b)$\n}\n\nThus, a simple max function with two variable can be implemented with a neural network of four hidden neurons. We could use this neural network as a basic building block to construct max functions with more variables.\n\nIn this way, we can construct an equivalent CG, $G'(V', E')$. Vertices of $G$ are either copied into $G'$ (including those after merge), or replaced with the equivalent MLPs. For the latter, the corresponding weight matrices are trained (or set up) in floating-point numbers.\n\nFinally, the initial `merge' operation is performed against $G'$. Then all vertices in $G'$ are just \\emph{core\\_op}-like operations.\nAn example is given in Figure~\\ref{fig:example}(b):\nbesides the vertex merge, the max-pooling vertex is replaced with a dedicated subgraph of only \\emph{core\\_op}-like operations.\n\\end{comment}\nIn this step, we need to transform a CG $G$ into a graph $\\hat{G}$ with only core-op--like operations.\nThe operation set $\\hat{F}$ includes all operations that could be combined by core-ops in $F'$ without any precision constraints.\nFor example, the corresponding core-op--like operations for dot--core-op are all operations that in the form of weighted sum with activation function, denoted as \\textit{dot-like} operations.\n\nTo replace all vertices represented with $F$ into operations in $\\hat{F}$, we take the following three steps in order.\n\\begin{textcircleds}\n\t\\item First, we find all subgraphs that match the computation of any $\\hat{f}\\in\\hat{F}$, and replace them with $\\hat{f}$.\n    For example, in Figure~\\ref{fig:example}(b), we merge the dot-product, add-bias and ReLU operations into one operation, which is a dot-like operation.\n    \\item Then, we can also have some customized mapping from a subgraph in $G$ into a subgraph formed of operations in $\\hat{F}$, and apply these dedicated designs here.\n    For example, max-pooling operation can be built with max functions.\n    A simple max function with two operands can be achieved with as Equation~\\ref{eq:max}, which includes multiple dot-like operations.\n    \\begin{equation}\\label{eq:max}\n\t\t\\begin{split}\n    \t\t\\max(a, b) = & \\frac{1}{2}[\\relu(a+b)+\\relu(a-b)\\\\\n        \t\t\t + & \\relu(-a+b)+\\relu(-a-b)]\\\n    \t\\end{split}\n\t\\end{equation}\n    We can use the max function with two operands to form max function with more operands.\n    \\item Finally, for the left operations in $G$, we provide a default transformation strategy: we use multiple dot-like operations to form MLPs to approximate them since MLP is proved to be an universal approximator~\\cite{hornik1989multilayer}.\n\\end{textcircleds}\nAfter the transformation, we form an graph $\\hat{G}$ with only core-op--like operations.\n\n\\begin{figure*}[!htbp]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{fig\/graph_tuning.pdf}\n\t\\caption{\n\t\tGraph tuning of one vertex.\n\t\t(a) $G'$ with only \\emph{core\\_op}-like operations.\n\t\t(b) `Graph Tuning' has been performed against all previous vertices.\n\t\t(c) Insert an `autoencoder' after the current vertex.\n\t\t(d) Merge the bias and activation of the current vertex into the hidden layer of the autoencoder.\n\t\t(e) Multiply the three weight matrices together.\n\t\t(f) Fully expand the vertices with \\emph{core\\_ops}.\n\t\t(g) The original CG, $G$. The current subgraph (including the decoder of the current vertex) is fine-tuned to approach the corresponding vertex of $G$, with output of the previous tuned vertex as input.\n\t\tRepeat (b) to (g) for the rest vertices.\n\t}\n\t\\label{fig:graph_tuning}\n\\end{figure*}\n\n\\subsection{Graph Tuning}\\label{sec:graph_tuning}\n\nIn this step, we transform the intermediate graph $\\hat{G}=(\\hat{V},\\hat{E})$ to the hardware execution model $G'=(V',E')$.\nWe use the original graph $G$ to supervise the fine-tuning progress of the generated $G'$.\nThe graph $G$ can provide not only labels of the output but also supervised signals of all intermediate data between operations.\nEach edge $e\\in E$ can provide supervised signal for graph $G'$.\nThus, we can split $\\hat{G}$ into parts, and transform it into $G'$ part by part.\nTo do so, first we find the edges $\\hat{e}\\in\\hat{E}$ that correspond to the edges $e\\in E$, and use them to split the graph $\\hat{G}$.\nThen, we perform the following steps against each part one by one in a topological ordering of $\\hat{G}$.\nNote that, we can also transform multiple adjacent parts as a whole each time.\n\n\\bulletkeyword{Data Re-encoding.}\nSince the original model $G$ usually use floating-point numbers for computation and data representation, directly rounding data to hardware I\/O precision may distort and lost information.\nThus, this sub-step aims to re-encode the input and output data of each vertex.\nTo encode a floating-point vector with a low-precision vector, we employ an autoencoder to get the low-precision representation.\n\nAn autoencoder is an NN with one hidden layer that can learn a constrained representation of a set of input data.\nThe output layer has the same dimension as the input layer and is trained to approach the input:\nthe computation from input to the hidden layer is encoding input data to the hidden layer's representation, while the computation from the hidden layer to output is decoding.\n\nHere as we use it to represent the original floating-point vector with the hardware--I\/O-precision vector, it is reasonable that the neuron number of the hidden layer may be greater than that of the input\/output layer;\nthe specific value can be configured manually.\nUsually, a large hidden layer will improve the accuracy, but consume more resources.\nThe tradeoff is evaluated in Section~\\ref{sec:evaluation}.\n\nFor clarity, we take a vertex of dot-like operation as the example to describe the process.\n\nAs shown in Figure~\\ref{fig:graph_tuning}(a)(b)(c), we add an autoencoder after the current vertex.\nThe activation function of its hidden layer is a round operation, which rounds the output to hardware I\/O precision.\nIn addition, the round operation not only quantizes the output to low-precision, but also forces the output to be positive, which provides the non-linearity as the widely-used ReLU function.\n\nThe initial weight parameters for the encoder and decoder of the autoencoder are set as following.\n\nFor the input vector $X=\\{x_1, \\ldots, x_n\\}$ (i.e. the output of the previous ReLU, as illustrated by Figure~\\ref{fig:graph_tuning}(c)), we build an autoencoder network with a hidden layer of $m\\times n$ neurons, where $m$ is a configurable integer, denoted as \\textit{re-encoding factor}.\nSuppose the $n$ dimensions of $X$ are independent and equally important, then each $x$ has $m$ hidden neurons to represent its low precision encoding.\nThus, we initialize the connections from $x$ to other hidden neurons to zero.\nA hidden neuron linearly scales the corresponding $x$ to $w^{(e)}x+b^{(e)}$, where $w^{(e)}$ and $b^{(e)}$ are the weight and bias term of the encoder respectively, and then rounds it to the I\/O precision, $N$-bit integer ($\\{0, \\ldots, 2^N-1\\}$).\nAny $x < \\frac{-b^{(e)}}{w^{(e)}}$ will be rounded to $0$ and any $x > \\frac{2^N-1-b^{(e)}}{w^{(e)}}$ will be rounded to $2^N-1$.\nThus, one hidden neuron can well represent $x \\in [\\frac{-b^{(e)}}{w^{(e)}}, \\frac{-2^N-1-b^{(e)}}{w^{(e)}}]$.\n\nNow we have $m$ hidden neurons for each dimension;\nthus the best way to represent $x \\in [0, x_{max}]$ (the output of ReLU is positive) with the $m$ hidden neurons is to divide the data range into $m$ adjacent and non-overlapping intervals, each of which corresponds to one hidden neuron.\nNamely, we properly initialize $w^{(e)}_i$ and $b^{(e)}_i$ for the the encoder of the $i$-th neuron of the $m$ to adapt $[\\frac{-b^{(e)}_i}{w^{(e)}_i}, \\frac{2^N-1-b^{(e)}_i}{w^{(e)}_i}]$ to the corresponding interval $[\\frac{ix_{max}}{m}, \\frac{(i+1)x_{max}}{m}]$.\nThus $w^{(e)}_i = \\frac{(2^N-1)m}{x_{max}}$ and $b^{(e)}_i = -(2^N-1)i$.\n\nAccordingly, to decode and restore $x$, the decoder should scale the data back.\nThus, its weight matrix is set to $w^{(d)}_i = \\frac{1}{w^{(e) _i}}=\\frac{x_{max}}{(2^N-1)m}$.\n\nNote that, if input $x < 0$ (outside of the encoded interval $[0, x_{max}]$), the initialized autoencoder will always return 0.\nIt means that the autoencoder can also perform ReLU operation. Therefore, we could remove the redundant ReLU operation of the current vertex.\nMoreover, as shown in Figure~\\ref{fig:graph_tuning}(d), the bias term of the current vertex could also be encoded by the dot-product operation and merged into the bias term of encoder $b^{(e)}_i$.\nNamely, the new bias term of encoder becomes $b^{(e)}_i + w^{(e)}_ib$.\n\nFinally, as shown in Figure~\\ref{fig:graph_tuning}(e), the decoder of the previous vertex, the dot-product operation and the encoder of the current vertex can be merged as one dot-product operation, whose weight matrix is the product of the three's matrices.\n\nTill now, the input and output of the current vertex have been constrained to hardware I\/O precision.\n\nFor vertices of convolution plus activation function, the process is similar.\nInstead of using dot-product operation as encoder and decoder, we use convolution instead, and the three convolutions can be merged into one as well.\nThe initialization is also similar: the hidden layer has $m$ channels for each input channel, and only the center value of the encoder\/decoder kernel is set to non-zero.\n\nOwing to this step, we solve the limitation problem of I\/O precision.\nIn this step, we only change the computation scale of the operations in $\\hat{G}$.\n\n\\bulletkeyword{Fully Expanding.}\nIn this step, we will turn $\\hat{G}$ into $G'$.\nSince the core-op--like operations $\\hat{f}\\in\\hat{F}$ can be combined by core-ops in $F'$, we expand all operations in $\\hat{G}$ into individual subgraphs consisting of operations in $F'$ to form the graph $G'$.\n\n\nTake dot-like operations as an example.\nDot-like operations can be represented as a fully-connected layer.\nAs shown in Figure~\\ref{fig:graph_tuning}(f),\nto support the fully-connected layer of any scale with dot--core-ops, we use two layers of dot--core-ops to construct an equivalent graph.\nThe first layer is a computation layer. The original dot-like operations are divided into smaller blocks and are performed with many dot--core-ops.\nThe second layer is a reduce layer, which gathers result from the former to output.\n\nThe division is straight:\n\\begin{textcircleds*}\n\t\\item Divide the weight matrix into small sub-matrices that satisfy the hardware limitation on scale;\n\teach is held by a dot--core-op of the first layer.\n\t\\item Divide the input vector into sub-vectors and transfer each sub-vector to the corresponding sub-matrices(\\emph{core\\_ops}) at the same horizontal position and\n\t\\item gather results from the same column by the reduce layer.\n\\end{textcircleds*}\n\nRegarding all dot-like operations as fully-connected layers are sometimes very inefficient. We can have dedicated division according to its connection pattern.\n\nFor example, for a convolutional case (suppose a kernel of size $k \\times k$ convolves a $W \\times H$ image from $m$ channels to $n$ channels), the $n$ channels of one pixel in the output side are fully connected to the $m$ channels of $k \\times k$ corresponding pixels in the input side.\nThis forms a small-scale vector-matrix multiplication of size $(m \\times k^2) \\times n$.\nThere are $W\\times H$ such small operations in the convolution case.\nEach input should be transferred to $k^2$ such small operations, while reduction is needless.\nIf such a small operation is still too large for a dot--core-op, we can divide the operation as the fully-connected--layer case does.\n\nIf there are some dot--core-ops that are not fully used, we can distribute them onto one physical PE to reduce resource consumption during the mapping step.\n\nTill now, the computation of the current subgraph of $\\hat{G}$ has been transformed to a subgraph of $G'$, which consists of core-ops $f'\\in F'$.\nNext, the weight matrices of the core-ops will be fine-tuned.\n\n\\bulletkeyword{Weight Tuning.}\nIn this step, we will fine-tune the parameters to make the generated subgraph of $G'$ approximately equal to the corresponding subgraph of $G$.\n\n\\begin{comment}\nBased on previous steps, the initialization mode of the weight matrix(matrices) of the current sub-CG can be divided into two categories:\n\\begin{itemize}\n\t\\item If the sub-CG is not created by Step `Graph Reform' (i.e. it is copied from $G$), the corresponding weight parameters from the golden model are used.\n\t\\item For others, the initial values are set up by the `Graph Reform' step.\n\\end{itemize}\n\nOf course, in either case, all the weight matrices have been processed by subsequent steps, i.e. `Data Re-encoding' and `Fully expanding'.\n\nHowever, some methods, including autoencoder and the MLP-based unsupported function handling, introduce transformation errors.\nIn addition, activation functions used by $G(V, E)$ may be different from the hardware counterpart (i.e. the activation function of \\emph{core\\_op}), which also makes the initialization inaccurate.\nTherefore, some fine-tuning phases have to be taken to minimize the error, under the premise of satisfying the hardware weight precision constraint.\n\\end{comment}\n\nAs shown in Figure~\\ref{fig:graph_tuning}(g)(f), we use the corresponding supervised signal from graph $G$ to fine-tune current subgraph of $G'$.\nThe input to the current subgraph is from the output of previous transformed subgraphs instead of the corresponding supervised signal from the graph $G$.\nThus, the transformation of current subgraph will consider the error from previous transformed subgraphs, which can avoid error accumulation.\nThe output of previous transformed subgraph can be generated on demand or cached in advance to improve the transformation speed.\n\n\n \n\n\nWe will consider the hardware constraints on weight parameters in this step.\nSpecifically, target hardware usually puts strict constraints on weight storage since it occupies most of the hardware resources.\nHere we present a formal description of the constraints on the weight matrix $W$:\nthe value of each element $W_{ij}$ should be assigned dependently from a finite set $S$.\n$S$ is either a fixed set or a configurable set $S^P$ with parameter(s) $P$.\nThree kinds of typical weight encoding methods, which have been widely used by real NN chips, are presented as following (in all cases, the size of $S$ is $2^N$):\n\\begin{itemize}\n\t\\item \\textbf{Dynamic fixed-point}: $S^P=\\{\\frac{-2^{N-1}}{2^P}, \\ldots, \\frac{0}{2^P}, \\ldots, \\frac{2^{N-1}-1}{2^P}\\}$ where $P$ represents the point position.\n\tThis method is used by DNPU~\\cite{DNPU2017}, Strip~\\cite{judd2016stripes}, TianJi-ANN~\\cite{11}, etc.\n\t\\item \\textbf{Fraction encoding}: $S^P=\\{\\frac{-2^{N-1}}{P}, \\ldots, \\frac{0}{P}, \\ldots, \\frac{2^{N-1}-1}{P}\\}$, where $P$ is the threshold of the spiking neuron or the scale factor of the result. It is used by PRIME~\\cite{7}, and TianJi-SNN~\\cite{11}.\n\t\\item \\textbf{Weight sharing}: $S^{P_1, \\ldots, P_{2^N-1}}=\\{0, P_1, \\ldots, P_{2^N-1}\\}$, used by EIE~\\cite{han2016eie}.\n\\end{itemize}\n\nWithout loss of generality, suppose the floating-point parameter $W_{ij}$ is rounded to the $k_{ij}$-th element in $S^P$, denoted as $S^P_{k_{ij}}$.\nThis step aims to find the best $P$ and to set $k_{ij}$ for each element in the weight matrix properly to minimize the transformation error.\nIt is similar to \\textit{weight quantization} of network compression.\nOur contribution is that we generalize it to typical hardware cases and introduce several fine-tuning phases to deal with different parameter-setting issues separately.\n\nFor a subgraph, three fine-tuning phases are taken in order:\nThe first is to reduce the initialization error.\nThe second is to determine the best value range of weight matrix (i.e. to choose the best $P$) and the last is to determine the best value from $S^P$ for each element (i.e. to choose the best $k_{ij}$).\nEach phase gets parameters from the previous one and fine-tunes them under certain constraints.\n\n\\bulletkeyword{Free Tuning.}\nIn previous steps, we use the parameters in the original graph $G$ to initialize those parameters in the generated graph $G'$.\nHowever, some methods, including autoencoder and the MLP-based unsupported-function handling, introduce transformation errors.\nIn addition, activation functions used by $G$ may be different from the hardware counterpart, which also makes the initialization inaccurate.\nIn addition, previous transformed subgraphs also have errors.\nTherefore, some fine-tuning phases have to be taken to minimize the error, under the premise of satisfying the hardware constraints on weight precision.\n\nThus we first fine-tune the subgraph of $G'$ without any constraint on weight precision to reduce any existing error.\nIn this procedure, all parameters and signals are processed as floating-point numbers, while the hardware activation function is used.\n\n\\bulletkeyword{Value-Range Tuning.}\nNow the precision constraint on weight is introduced.\nAccordingly, we need to choose the best value-range of the weight matrix (namely, the best $P$).\nApparently, we will minimize $J(k, P)=\\sum_{ij}(W_{ij}-S^P_{k_{ij}})^2$, which can be achieved by an iterative expectation\\-maximization (EM) algorithm:\n\\begin{itemize}\n\t\\item E-step: fix the current parameter $P^{(t)}$ and calculate $k_{ij}^{(t)} = \\arg\\min{J(k|P^{(t)})}$.\n\t\\item M-step: fix $k_{ij}^{(t)}$ and calculate $P^{(t+1)}=\\arg\\min{J(P|k^{(t)})}$.\n\\end{itemize}\n\nThen we replace $W_{ij}$ with $S^P_{k_{ij}}$ where $k_{ij}$ is fixed and $P$ is the parameter.\n\nAfter the initialization, we fine-tune the subgraph to optimize $P$.\nDuring this process, we maintain the precision of $W_{ij}$ first and then round it to $P_{k_{ij}}$ at every time $P$ is updated.\n\nFurther, for the weight sharing case mentioned above, the EM algorithm is just reduced to the k-means algorithm.\nIf $S^P$ is a fixed set without any configurable parameter, we can omit this phase.\n\n\\bulletkeyword{Rounding Tuning.}\nThe data-range set of weight value $S^P$ is fixed now.\nThis procedure adjusts each weight matrix element to a proper element in this set.\nIn another word, it aims to choose the best index $k_{ij}$ for $W_{ij}$.\nDuring the fine-tuning progress, parameters are stored as floating point number.\nIn the forward phase, any parameter is rounded to the closest element in $S^P$.\nWhile during the backward phase, floating-point number is used to update $W_{ij}$.\nThis mechanism is also employed by the above \\textit{Value-Range Tuning} phase if $P$ can be set from a discrete set.\n\n\nAfter processing all the subgraphs, we have transformed the original model $G$ into an equivalent hardware execution model $G'$ that satisfies all the constraint conditions.\n\n\\begin{comment}\nFinally, it is worth adding that we can fine-tune several successive subgraphs in $G'$ simultaneously according to a topological ordering,\nand the above steps are still applicable.\nWe will evaluate the effect of fine-tuning-granularity on transformation results in Section \\ref{sec:evaluation}.\n\\end{comment}\n\n\\subsection{Mapping}\\label{sec:mapping}\nThe generated graph $G'$ will be deployed on the target hardware efficiently, which is a hardware-specific problem.\nThus, we give the optimization principle here.\n\nFor NN chips that bind the neural computation and storage in the \\textit{physical cores} (it is called the \\textit{weight stationary} computing mode, classified by~\\cite{16}), this is a mapping problem to assign core-ops to physical cores. Moreover, several core-ops that are not fully used can also be distributed onto one physical core, as long as there are no data conflicts.\n\nFor chips whose physical cores are computing engines with flexible memory access paths to weight storage (usually work in the time division multiplex mode), it is a mapping and scheduling problem to schedule each core-op's task onto physical cores.\nMultiple core-ops could be mapped onto one core to increase resource utilization.\n\nAs we can get data dependencies and communication patterns between core-ops through the transformed graph, we could use these information to optimize the mapping or scheduling to minimize transmission overhead, e.g. putting densely-communicating cores close.\nTrueNorth has designed such an optimized mapping strategy~\\cite{38}.\n\nMoreover, for those core-ops sharing weights (e.g. convolution vertices can be fully expanded to a lot of core-ops sharing the same weight matrix), we could map (or schedule) them to the same physical core to reduce data movement.\n\n\\subsection{Others}\\label{sec:specification}\n\\subsubsection{SNN Models}\\label{sec:snn}\nSNN, called the third generation of ANN, is a widely-used abstraction of biological neural networks.\nIn addition to neuronal and synaptic states that traditional ANN has featured, it incorporates the timing of the arrival of inputs (called spikes) into the operating model to yield higher biological reality.\n\nSNNs of rate coding can emulate ANNs.\nThe spike count in a given time window can represent a numerical value within a certain range, like a traditional ANN does.\nAccordingly, the input of a synapse is a spike sequence of certain firing rate from the pre-neuron.\nAfter synapse computation, it is converted into the sum of currents that will be computed by the post-neuron.\nFor those widely-used SNN models, the functions of their synapse and neuron computations usually own good continuity and are derivable in rate coding domain.\nTherefore, the popular SGD method can be used for training SNN:\nseveral recent studies~\\cite{46, 49} have used the stochastic gradient decent (SGD) algorithm to train SNNs directly or indirectly and achieved the state-of-the-art results for some object recognition tasks.\n\nAs our workflow is not dependent on the concrete NN type (ANN or rate-coding SNN), it can support SNN hardware and SNN models, too.\nFor SNN models, the training data is the firing rate of each neuron.\n\n\\subsubsection{RNN Models}\nRNN is an NN with some cycle(s).\nWe could transform and fine-tune each operation inside an RNN as normal, and add an additional step to fine-tune the entire RNN after that.\n\n\\begin{comment}\n\\subsection{Utilization}\nWe can utilize extra hardware features.\nThe utilization depends on the specific hardware and we present some examples as following:\n\\begin{itemize}\n\t\\item If hardware supports \\textit{reduction communication} or provides additional \\textit{adders}, the reduce layer in \\textit{Fully expanding} would be needless.\n\t\\item If some special function (e.g. max function) is supported, it can also be viewed as a type of \\emph{core\\_op}. Accordingly, the corresponding \\emph{core\\_op}-like operations can include max pooling and others that can be easily achieved by existing \\emph{core\\_ops}. It is no longer required to be approximated by the dedicated design or the MLP-based universal approximator.\n\\end{itemize}\n\\end{comment}\n\n\n\\section{Background}\n\\subsection{NN basis}\nNNs are a set of algorithms, modeled loosely after the human brain, that are designed to recognize patterns.\nTraditional NNs consist of multiple layer of neurons. Each layer performs the computation as shown in Equation~\\ref{eq:fc} where $X$ is the input vector, $Y$ is the output vector, $W$ is the weight matrix, $B$ is the bias, and $\\sigma$ is a non-linear activation function, which is typically the Rectified Linear Units (ReLU) function~\\cite{nair2010rectified}.\n\\begin{equation}\\label{eq:fc}\n\tY = \\sigma(W \\cdot X + B)\n\\end{equation}\nThis kind of NN is also known as multilayer perceptron (MLP), which has been proved to be a universal approximator~\\cite{hornik1989multilayer}.\nModern NNs are more complicated.\nThe topology is a graph rather than a simple chain, and the types of operations are richer than matrix-vector multiplication.\nMost deep learning frameworks~\\cite{2016arXiv160502688full,abadi2016tensorflow,CNTK,MXNet} use computational graph (CG), a directed acyclic graph, to represent NN computations.\nVertices in the graph represent operations (e.g., dot-product, convolution, activation function, pooling) and immutable\/mutable states~\\cite{abadi2016tensorflow} (e.g., the weight parameters associated).\nEdges represent the data dependency between vertices.\nBoth vertices and edges process or carry tensor data (multi-dimensional arrays).\n\nFor clarity, in this paper, dot-product, bias-addition, and convolution are categorized as \\textit{weighted-sum} operations.\nMoreover, any constant operand, including the trained weight matrix for any vertex of weighted-sum operation, is considered as part of the corresponding vertex as we can view it as the immutable state of the vertex.\n\n\\subsection{NN Chips}\nThere are two types of NN chips.\nThe first type focuses on the traditional ANNs.\nThey are custom architectures~\\cite{13,14,15,21,zhang2016cambricon,27,judd2016stripes,DNPU2017,22,26,28,29,31,23, 24,25,sharma2016high} to accelerate mature ANN models.\nWe usually call this type NN accelerators.\nThe second is neuromorphic chips, which usually supports SNNs to yield higher biological reality~\\cite{8,9,10,40,42,43,44,11}.\n\nThese chips usually consist of a lot of processing elements (PEs) that can efficiently perform dot-product operations because this operation is the main body of most NN models.\nDifferent chips put different constraints on the operations they support.\nTable~\\ref{tab:chips} shows the constraints of some existing NN chips.\nMost NN chips employ low precision numbers to represent weights and input\/output (I\/O) data instead of floating-point numbers.\nThe scale of computation that each PE can process is usually fixed.\nPRIME~\\cite{7} and DianNao~\\cite{13} have extra adders to support larger scale computations.\nHowever, NN chips such as TianJi~\\cite{11} and TrueNorth~\\cite{8} do not have extra adders, and the output of their PE can  connect to only one input port of another PE.\nFor these chips, the scale of computation is also a problem.\nDespite the widely-supported dot operation, many other operations required by NNs usually lack for support.\n\n\\begin{table}[!t]\\small\n\t\\centering\n\t\\begin{tabular}{|*{5}{l|}}\n\t\t\\hline\n\t\t\\textbf{Chip} & \\textbf{Weight} & \\textbf{I\/O} & \\textbf{Scale} & \\textbf{Nonlinear}\\\\\n\t\t\\hline\n\t\t\\hline\n\t\tTianJi~\\cite{11} & 8-bit & 8-bit & $256^2$ & Configurable\\\\\n\t\t\\hline\n\t\tPRIME~\\cite{7}& 8-bit & 6-bit & $256^2$ & ReLU\\\\\n\t\t\t&\t&\t&\t& Max Pooling\\\\\n\t\t\\hline\n\t\tDianNao~\\cite{13}& 16-bit & 16-bit & $16^2$ & Configurable\\\\\n\t\t\\hline\n\t\tTPU~\\cite{TPU2017} & 8-bit & 8-bit & None & ReLU\\\\\n\t\t\t&\t&\t&\t& Max Pooling\\\\\n\t\t\t&\t&\t&\t& etc.\\\\\n\t\t\\hline\n\t\tTrueNorth~\\cite{8}& 2-bit & Spiking & $256^2$ & LIF\\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Hardware limitations of NN chips}\n\t\\label{tab:chips}\n\\end{table}\n\n\\section{Conclusion}\\label{sec:conclusion}\nWe present a programming solution for NN chips, which can transform a trained, unrestricted NN into an equivalent network to meet hardware constraints.\nMultiple techniques are proposed to reduce the transformation error and improve the processing speed.\nThe solution is validated on a real neuromorphic chip and a PIM design for ANNs, as well as on different scales of NNs under different constraints.\nThe evaluation shows that the transformation methodology is very effective and only insignificant errors will be introduced. \nThe transformation time is much faster than re-training the NN models for a specific neuromorphic hardware.\n\\section{Problem Description}\nTo bridge the gap between NN applications and NN chips, we decouple the whole system stack with a software programming model and a hardware execution model.\nThe former is the programming interface for NN experts to develop NN applications.\nThe later is the SW\/HW interface that NN chips can executed directly.\n\n\\keyword{Software Programming Model.}\nThe machine-learning community has already employed \\textit{Computational Graph} (CG) as the programming model.\nIt is a data-flow graph $G=(V,E)$ which represents a number of operations with vertices $V$ and represents data dependencies between these operations with edges $E$.\nMost deep-learning frameworks~\\cite{2016arXiv160502688full,abadi2016tensorflow,CNTK,MXNet} adopt CG to build NN models.\nAnd the set of supported operations is $F$.\n\nEach vertex in $V$ is an operation $y=f(x_1, \\ldots, x_n)$, where $f\\in F$, $y$ represents the output edge, and $\\{x_1, \\ldots, x_n\\}$ represent input edges.\nThus, the entire model can be expressed as a composite function $Y=H(X)$, where $X$ represent all input vertices and $Y$ represents all output vertices.\n\nWe also adopt CG as the programming model with a slight modification.\nThe difference is that we regard model parameters as immutable states of the corresponding vertices instead of normal edges of the vertices.\nNamely, we regard an operation $f(x, \\theta)$ as $f^\\theta(x)$, where $\\theta$ is model parameters and $x$ is an input operand.\nThus, it can only be a trained NN model that all parameters have been already determined.\n\n\\keyword{Hardware Execution Model.}\nThe computation model that hardware could execute is also a data-flow graph $G'=(V',E')$.\nIt has a supported operation set $F'$, denoted as \\textit{core-op set}.\nHowever, the supported operations are very limited, and these operations have many limitations.\nIn addition, some hardware also has constraints on the interconnection subsystem. For example, TianJi and TrueNorth does not support multi-cast; one output port of each PE can only be connected to one input port.\nThe hardware execution model forms a composite function $Y'=H'(X')$.\n\nThus, our goal is to build $G'$ from $G$ so that $H'$ is approximately equivalent to $H$.\n\n\\keyword{Minimum Hardware Requirement.}\nWe define a minimum set of operations $C$ that $C\\subset F'$ has to be satisfied to use our NN compiler.\nIt only contains one operation, denoted as \\textit{dot--core-op}.\nNamely, the operation dot--core-op must belong to the core-op set.\nEquation~\\ref{eq:core-op} shows the computation of the dot--core-op where $X$ and $Y$ are the input and output vectors, respectively; $N$ and $M$ are their sizes; $W$ is the weight matrix of size $M\\times N$; and $\\sigma$ is a nonlinear activation function.\n\\begin{equation}\\label{eq:core-op}\n\tY_j = \\sigma(\\sum_iW_{ji}X_i) \\quad (1\\le j\\le M, 1\\le i\\le N)\n\\end{equation}\nIn addition, the I\/O data precision is $B$ bits.\n\nFormally, the dot--core-op meets the following constraints:\n\\begin{itemize}\n\t\\item $N$,$M$,$B$ are fixed.\n\t\\item The value range of each element in $W$ is a finite set $S$.\n    $S$ is either a fixed set or a configurable set $S^P$ with some parameters $P$.\n    \\item Without loss of generality, only ReLU function ($\\sigma$) is supported.\n\\end{itemize}\n\nWe choose this dot--core-op as the minimum requirement for hardware because it can cover most existing NN chips (e.g., those listed in Table~\\ref{tab:chips}).\nThus, our NN compiler can support most existing NN chips.\n\\section{Implementation and Evaluation}\\label{sec:evaluation}\n\\subsection{Implementation}\nWe have implemented the tool to support different hardware constraints, including those of TianJi~\\cite{11} and PRIME~\\cite{7}.\n\nTianJi is fabricated with 120nm CMOS technology. The running frequency is 100MHz and the total dynamic power consumption is 120mW. TianJi chip supports both ANN and SNN modes.\nThe numerical accuracy of weight value is 8-bit fixed-point and the scale of vector-matrix-multiplication is $256 \\times 256$.\nFor ANN mode, the I\/O precision is 8-bit that is cut from the 24-bit internal computation output; the cut range is configurable, thus its weight encoding strategy is dynamic fixed-point.\nFor SNN mode, the minimal I\/O precision is 1-bit, which can be extended to $n$-bit with $2^n$ cycles as the sampling window (as described in Section~\\ref{sec:snn}). The neuron model is a simplified LIF neuron model with a constant leakage and a threshold for firing; the weight encoding method can be viewed as fraction encoding.\nPRIME~\\cite{7} is a memristor-based PIM architecture for ANN.\nThe weight precision is 8-bit and the I\/O precision is 6-bit. The scale of vector-matrix-multiplication is $256 \\times 256$, too. The output range can be configured with an amplifier; thus its weight encoding can also be viewed as fraction encoding.\n\nQuite a few NN applications, including an MLP for MNIST dataset (784-100-10 structure, $98.2\\%$ accuracy of full precision), LeNet-5~\\cite{LeNet} for MNIST dataset ($99.1\\%$ accuracy), a CNN~\\cite{mishkin2015all} for CIFAR-10 dataset ($84.64\\%$ accuracy \\footnote{As described by ~\\cite{mishkin2015all}, with some special initialization method, the CNN accuracy can exceed 90\\%. Here we ignore it for simplicity, which does not affect our evaluation.}), AlexNet~\\cite{AlexNet} and VGG16~\\cite{VGG16} for ImageNet, have been respectively transformed and then deployed onto TianJi~\\cite{11} and PRIME~\\cite{7} to show the validation.\nThe first three networks are trained by Theano~\\cite{2016arXiv160502688full}.\nParameters of the next two CNNs for ImageNet are extracted from trained models of the Caffe Model Zoo directly.\nThe inference accuracies of full precision are given in Table~\\ref{tab:nn_model_transform}.\n\nWithout loss of generality, we take the mapping of the LeNet-5 for MNIST onto the TianJi system as an example.\n\nOne TianJi chip contains 6 cores connected by a $2 \\times 3$ mesh NoC;\neach core supports 256 simplified LIF neurons and works in the weight stationary mode.\nThe main body of a TianJi system is a Printed Circuit Board (PCB) including 16 chips.\nOn the whole, all of the cores form a $12 \\times 8$ 2D-mesh network, and there is a great gap between the delay\/bandwidth of intra-\/inter-chip communications.\n\nThe transformed CG consists of 497 TianJi's dot--core-op.\nTaking into account the weight reuse of convolution, 19 physical cores are actually occupied.\nWe use the heuristic Kernighan-Lin (KL) partitioning algorithm for the mapping problem.\nIt abstracts the mapping as a graph partition problem and tries to minimize communications across partition boundaries.\nTake the bipartition as an example: the input to the algorithm is a graph;\nthe weight of each edge is the communication delay.\nThe goal is to partition the graph into two disjoint subset $A$ and $B$ of equal size, in a way that minimizes the communication cost of the subset of edges that cross from $A$ to $B$.\n\nFirst, a randomly generated initial mapping distribution is given\nSecond, the KL algorithm bi-partitions the mapped cores repeatedly till only the closest two cores are left in any of the final partition in the 2D-mesh. During this phase, partitions that minimize the communication cost between cores are remained; here the cost of an edge across boundary refers to its weight multiplied by the number of transmissions, as we can get the communication statistics from the transformed CG, including those information about the reused cores.\n\n\\subsection{Evaluation}\n\nThe inference accuracies after transformation for TianJi and PRIME are given in Table~\\ref{tab:nn_model_transform}.\nThis table also shows the re-encoding factor, which indicate the number of hidden units for autoencoders, for each case.\n\nWe can see that the transformation errors introduced are very limited, for all NN models and chips tested, we can achieve less than 2\\% accuracy drop.\nFor most cases, we only use $1\\times$ hidden unit to re-encode data, which means that the NN scale is not changed.\nFor the two TianJi-SNN cases, as the I\/O constraints are very strict, we use $2\\times$ hidden units, thus $4\\times$ PEs will be used.\nFor ImageNet cases on PRIME, if we only use $1\\times$ hidden units, the accuracy drop will be over 2\\%.\nWith $4\\times$ hidden units, we can reduce the accuracy drop to less than 1\\%.\nWe can always achieve better accuracy with more hardware resources employed.\n\n\nThis toolchain also improves the development efficiency remarkably. For example, it can transform AlexNet in about 1 hour, while training it from scratch will take about 3\\textasciitilde4 days. Specially, training the whole NN requires millions of iterations to converge, while our method only costs thousands of iterations to converge for each step, that is, takes 5 to 10 minutes to transform one layer. The reason lies in that, after partitioning, we fine-tune each unit one by one; each one is well initialized and much easier to converge than training the entire model. All evaluations are completed on a common PC server equipped with one Tesla P100 GPU.\n\nIn addition, as the large-scale NNs (e.g. those for ImageNet) cannot be occupied by the TianJi system because of the physical limit of chip capacity (a TianJi system contains 16 chips and a chip can occupy 1536 neurons), those related results are drawn from the cycle-accurate TianJi chip simulator.\n\\subsubsection{Accuracy vs. Fine-Tuning Granularity}\nWe conduct experiments to explore the relationship between accuracy and fine-tuning granularity.\nIn another word, in the step of \\textit{Graph tuning}, we can fine-tune one or more successive parts in $G'$ simultaneously, even fine-tune the entire model as a whole.\nIt looks like that increasing the fine-tuning scale per time will increase the search space, which may lead to a lower error rate but consume more computations and more time to converge.\nHowever, results show that coarse grained fine-tuning does not result in improved accuracy.\nFor example, for CIFAR10-CNN, the inference accuracy (the I\/O precision is 7 bits and the weight precision is 4 bits) is 83.07\\% as the fine-tuning-granularity is two subgraphs (the accuracy is 83.14\\% as the granularity is one).\nIf we fine-tune the whole NN together, the accuracy is just 83.24\\%.\nThus, one by one fine-tuning is an optimal strategy, which also means that the problem of error accumulation has been well solved.\nUnless specifically noted, the fine-tuning granularity is just one.\n\n\\subsubsection{Accuracy vs. Resource Consumption}\nFrom the transformation workflow, we can see that the step of \\textit{Data Re-encoding} may introduce the most additional resource overhead:\nWhen the neuron number of the hidden layer of autoencoder is $n\\times$ as much as that of the input\/output layer, the number of crossbar consumed of the whole NN will be $n^2 \\times$.\nThe latter can also be considered as a direct indicator of area consumption and runtime overhead. Thus, we conduct experiments to explore the effect of \\textit{autoencoder}.\n\nWe use the MLP for MNIST dataset.\nAlthough this network is relative small, its conclusion is general for large-scale NNs because we fine-tune NNs part by part and each part is a small graph.\nTable~\\ref{tab:nn_model_transform} also shows the improvement for ImageNet on PRIME with a larger re-encoding factor.\nWe compare the inference accuracies without or with different scales of autoencoder.\nFor the former, we simply scale and round the I\/O signal values to make them suitable for I\/O precision.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{fig\/autoencoder_io.pdf}\n\t\\caption{\n\t\tAccuracy v.s. I\/O precision under different transformation strategies.\n\t}\n\t\\label{fig:autoencoder_io}\n\\end{figure}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{fig\/autoencoder_weight.pdf}\n\t\\caption{\n\t\tAccuracy v.s. weight precision under different transformation strategies.\n\t}\n\t\\label{fig:autoencoder_weight}\n\\end{figure}\n\nFigure~\\ref{fig:autoencoder_io} (Figure~\\ref{fig:autoencoder_weight}) lists the accuracy with different I\/O (weight) precisions respectively (after transformation), without or with different scales of autoencoder, while no other constraints are introduced. Results show that this data re-encoding strategy is effective to improve the transformation accuracy under strict constraints.\n\nIn Figure~\\ref{fig:autoencoder_io}, the accuracy of the transformed network without autoencoder drops significantly when I\/O precision is less than 3 bits (different NNs may have different turning points). In contrast, when autoencoder is used (we assume the I\/O limitation is only 1-bit and only 1x hidden neurons are used, namely, the most critical case), the accuracy is $84.63\\%$ (if no autoencoder, the value is only $13.45\\%$). With more hidden neurons, the accuracy continues to rise, which means our method could trade NN scale for capability.\n\nApparently, increasing the number of hidden neurons can only linearly increase the encoding ability, which is worse than increasing the I\/O precision directly because the latter's encoding ability is $\\propto(2^n)$. For example, using $2\\times$ hidden neurons and 1-bit I\/O does consume the same number of I\/O ports with that of using $1\\times$ hidden nodes and 2-bit I\/O; the accuracy of the former is only $88.2\\%$ while the latter is $94.71\\%$. Thus, it looks like that the hardware had better provide enough I\/O precision since rescuing the accuracy by software (using autoencoder) may cost more hardware resources, especially when the hardware I\/O precision is less than the turning point.\nAnyway, this is a tradeoff between hardware consumption, software adaption and inference error.\n\nMoreover, as illustrated by Figure~\\ref{fig:autoencoder_weight}, autoencoder is also able to rescue the accuracy loss caused by low weight precision. Compared with Figure~\\ref{fig:autoencoder_io}, we can see that NNs are more tolerant of low weight precision than low I\/O precision, since the latter can cause signal distortion directly. Figure~\\ref{fig:autoencoder_weight} also shows that when the weight precision is 2-bit or more, using different scales of autoencoder does not change the accuracy apparently because it has already reached $99\\%$.\n\n\\subsubsection{Impact of Weight Encoding Methods}\nWe have evaluated the weight tuning algorithm, as well as the three kinds of weight encoding strategies. Figure~\\ref{fig:em_weight} shows that weight tuning can set the weight parameters well for the three cases: with the increase of weight precision (any other constraint is not introduced), all of them can reach the upper bound accuracy. In Table~\\ref{tab:weight_encoding}, we further give the effect of each phase of the weight tuning step (the weight precision is 2-bit, without any other constraint).\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{fig\/em_weight.pdf}\n\t\\caption{\n\t\tAccuracy v.s. weight precision under different weight encoding strategies.\n\t\tFor the weight sharing case, weight precision means the bit-width of weight indices.\n\t}\n\t\\label{fig:em_weight}\n\\end{figure}\n\n\\begin{table}[!t]\n\t\\centering\n\t\\begin{tabular}{|c|c|c|c|}\n\t\t\\hline\n\t\t& \\bfseries dynamic & \\bfseries fraction  & \\bfseries weight sharing\\\\\n        & \\bfseries fixed point & \\bfseries encoding & \\bfseries \\\\ \\hline\n\t\t\\bfseries I & 81.56\\% & 85.60\\% & 86.19\\%\\\\ \\hline\n\t\t\\bfseries I+P & 81.56\\% & 87.05\\% & 89.86\\%\\\\ \\hline\n\t\t\\bfseries I+L & 97.08\\% & 97.05\\% & 96.31\\%\\\\ \\hline\n\t\t\\bfseries I+P+L & 97.08\\% & 97.74\\% & 97.14\\%\\\\ \\hline\n\t\\end{tabular}\n\t\t\\caption{\n\t\tAccuracy v.s. different weight encoding strategies under 2-bit weight precision.\n\t\tI: initialization with EM; P: parameter range fine-tuning (to decide $P$); L: low precision fine-tuning (to decide $k_{ij}$)\n\t}\n\t\\label{tab:weight_encoding}\n\\end{table}\n\nWith only the EM-based initialization of the value-range tuning phase (I in Table~\\ref{tab:weight_encoding}), the accuracy of different weight encoding strategies depends on the latter's flexibility. The accuracy of \\textit{weight-sharing} is the highest as it is the most flexible: the precision just limits the bit-width of indices, not weight values. \\textit{dynamic-fixed-point} only allows weight values to be scaled by power of 2; thus it is the least flexible. \\textit{fraction-encoding} is positioned in the middle.\n\nWith the whole value-range tuning phase (including the initialization and fine-tuning, I+P in Table~\\ref{tab:weight_encoding}), the accuracy increases a little for both \\textit{fraction-encoding} and \\textit{weight-sharing} and remains unchanged for \\textit{dynamic-fixed-point}. The reason is in the initialization we have already found a good $P$, further fine-tuning $P$ cannot increase the accuracy much, especially for \\textit{dynamic-fixed-point} with limited flexibility.\n\nWith the EM initialization and the rounding tuning phase (I+L in table~\\ref{tab:weight_encoding}), the accuracy increases significantly, which means that NN can easily find suitable parameters from any well-initialized set of weight values.\n\nWith all phases employed (I+P+L in table~\\ref{tab:weight_encoding}), the accuracy could still increase a little compared with the I+L case, except for \\textit{dynamic-fixed-point}.\n\nAnyway, there is no obvious capability difference between the three strategies.\n\n\\subsection{Discussion and future work}\\label{sec:discussion}\nNow, some transformation steps introduce extra layers more than one time, which may exacerbate NN redundancy.\nTherefore it is necessary to strike a balance between the possible information loss and hardware-resource consumption.\nAnyway, we provide a framework, while the concrete workflow could be customized.\nMoreover, the interaction between network compression and transformation is interesting, and we will study it as the future work.\n\nIn addition, it is helpful to present insights into future neuromorphic architecture designs:\n\\begin{squishlist}\n\t\\item It could give design tradeoff between the common computational components and special functional components.\n\tFor some neural functions or layers that are relatively easy (in terms of hardware consumption) to be achieved by common core-ops, it is unreasonable to integrate such a dedicate component on chip.\n\tIf not, a special component is worth to realize.\n\n\t\\item After transformation, the data flow between core-op is basically determined;\n\twe can analyze the communication pattern in detail, which is conductive to the balanced distribution of computing and communication resources on chip.\n\n\t\\item Our solution regards NN inference as the forward process of graph with fixed topology. To some extent, it is suitable for field programmable devices (especially in the aspect of on-chip connection); thus how to combine the device configurability with the flexibility of transformation is an interesting topic.\n\\end{squishlist}\n\n\\section{Introduction}\\label{sec:introduction}\n\nDesigning custom chips for NN applications with the Very-Large-Scale-Integration (VLSI) technologies has been investigated as a power-efficient and high-performance alternative to general-purpose computing platforms such as CPU and GPU.\nHowever, programming these chips is difficult because of some hardware-specific constraints:\n\\begin{comment}\n\\textcircled{\\small{1}} Due to the utilization of hardware resource or the capability of analog computing (e.g., some memristor-based designs~\\cite{1, 3, 4, 5, 6, 7}), the precision of input and output signals of neurons is usually limited, as well as\n\\textcircled{\\small{2}} the precision of network parameters, such as synaptic weights.\nIn contrast, general-purpose processors use floating-point numbers.\n\\textcircled{\\small{3}} The present fabrication technology limits the fan-in and fan-out of one neuron, and thus constrains the computation scale.\n\\textcircled{\\small{4}} Moreover, the diversity of nonlinear function or neuron model supported by the hardware is limited.\n\\end{comment}\n\\begin{textcircleds*}\n\t\\item Due to the utilization of hardware resources for digital circuits or the capability of analog computing for some memristor-based designs~\\cite{1, 3, 4, 6, 7}, the precision of input and output signals of neurons is usually limited, as well as\n    \\item the precision of NN parameters, such as synaptic weights.\n    \\item The present fabrication technology limits the fan-in and fan-out of one neuron, which constrains the computation scale.\n    \\item The diversity of nonlinear functions or neuron models supported by the hardware is also limited.\n\\end{textcircleds*}\nFor example, for TrueNorth chips~\\cite{8}, the maximum matrix that one synaptic core can handle is $256\\times 256$, and it supports only a simplified leaky--integrate-and-fire (LIF) neuron model.\n\nOne straightforward approach to this problem is to expose the hardware details and limitations to the NN developer directly.\nFor instance, IBM has provided a TrueNorth-specific training mechanism~\\cite{12}.\nThe mechanism constructs the whole NN model from scratch to satisfy all hardware limitations, and then trains the NN model.\nThis method has several drawbacks.\nFirst, it binds NN models to the specific target hardware.\nDevelopers can hardly benefit from existing NN models from the machine-learning community.\nSecond, it limits the power of the NN algorithm. \nThe constraints make it more difficult to converge and reach better accuracy for larger models.\nThird, training the specific NN model from scratch may take a very long time.\n\\begin{comment}\n\\textcircled{\\small{1}} It binds NN development and applications to the specific target hardware.\n\\textcircled{\\small{2}} Developers hardly benefit from existing NN models from machine-learning community.\n\\textcircled{\\small{3}} Training the specific NN from scratch may take a very long time.\n\\end{comment}\n\nAnother approach is to make hardware satisfy software requirement by consuming more hardware resources to reduce the constraints on NN models~\\cite{13, 14, 15, 21}, such as using 16-bit precision rather than 1-bit spiking signals in TrueNorth.\nThis approach can gain less performance improvement from NN quantization and compression technologies.\nFor some memristor-based designs, some constraints due to analog computing are physical limitations, which are difficult to overcome even with more hardware resources consumed.\n\nA third approach is to introduce a domain-specific Instruction Set Architecture (ISA) for NN accelerators, such as the Cambricon~\\cite{liu2016cambricon} ISA.\nThis approach requires both hardware and software to satisfy the ISA.\nHowever, this approach still does not solve the gap between programming flexibility required by NNs and hardware efficiency that can be gained from NNs' redundancy.\nIf we use high-precision instructions that do not have any constraints, the hardware can gain less benefit from NN's redundancy.\nIn contrast, if we use low-precision instructions with many constraints, the NN developer should take these constraints into consideration when developing NN models.\n\nIn addition to these approaches, there are also some work to utilize the NNs' redundancy for performance and provide flexible programming interface by introducing a transforming procedure.\nEIE~\\cite{han2016eie} is such an instance: it extensively uses deep compression to squeeze the redundancy, and design custom chip, EIE, to run the compressed NN model.\nNEUTRAMS~\\cite{NEUTRAMS} also use NNs' redundancy to adapt the original model to satisfy hardware constraints.\nHowever, these methods highly depends on the redundancy in NN models.\nDifferent NN models may have different minimum requirement (precision, connectivity, etc.) on hardware.\nThus, transforming procedure is not a general method, especially for NN models with less redundancy and hardware with severe constraints.\n\nIn this paper we propose a new method with flexibility, better applicability, and easy convergence.\n\\begin{comment}\nFirst, it decouples the neuromorphic computer system into two levels for better flexibility.\n\\textcircled{\\small{1}} The first level is located between programming and the execution substrate.\nNN experts could develop and train NN using any existing NN development framework~\\cite{jia2014caffe, 2016arXiv160502688full, abadi2016tensorflow} without considering hardware constraints.\nAfter the training, NN information such as neural model type\/parameters and network topology, can be extracted to construct the corresponding computational graph (CG) as the input of our method.\n\\textcircled{\\small{2}} The second level decouples software and hardware to make our proposal applicable to a variety of NN chips.\n\\end{comment}\nFirst, we decouple the neuromorphic computer system into two levels for better flexibility, software programming model and hardware execution model.\nWe use computational graph (CG), which is widely used in many popular NN frameworks~\\cite{jia2014caffe,2016arXiv160502688full,abadi2016tensorflow}, as the programming model for NN models.\n\\begin{comment}\nWe abstract the target hardware as a set of connected \\emph{physical cores}; the operation that a single core can accomplish is defined as \\emph{core\\_op}.\nWithout loss of generality, in this paper, \\emph{core\\_op} refers to vector-matrix multiplication plus activation function (or spiking neuron model) that meets hardware restrictions, as they are the main computational components of many existing NN chips.\nMany chips face all (or part) of the similar constraints, as presented in Table~\\ref{tab:chips}.\nNote that our methodology can be used for other chips not listed in the table, as  discussed later in Section~\\ref{sec:specification}.\nAccordingly, NN inference by hardware is abstracted as the forward process of a CG and all the vertices represent \\emph{core\\_ops}.\n\\end{comment}\nWe also provide the hardware\/software (HW\/SW) interface and the minimum hardware functionality that an NN hardware should provide.\n\\begin{comment}\nWe propose a transform workflow to convert a trained, unrestricted NN (expressed as a CG, which is referred to as \\emph{golden model}) into an equivalent CG of \\emph{core\\_ops} through the NN fine-tuning method. This method is called \"training and constraining\".\nAfterwards, the obtained CG will be mapped onto \\emph{physical cores} of target hardware efficiently.\nIn this way, our solution decouples NN application's specification from execution substrate.  To make the NN fine-tuning easy to converge under strict conditions, we also propose multiple techniques as described below:\n\\end{comment}\nWe propose a transformation workflow to convert a trained NN, expressed as a CG, into an equivalent representation of HW\/SW interface through the fine-tuning method.\n\n\\begin{comment}\nThe essential is to construct a CG of \\emph{core\\_ops} to approach the golden model.\nAs \\emph{core\\_ops} are not comparable to their software counterparts, it is reasonable to enlarge the graph scale and\/or complicate the topology properly to improve the capability, especially under strict conditions.\nAccordingly, some techniques, including an MLP-based universal approximator, data re-encoding based on \\emph{autoencoder} (a particular type of NN), and a \\emph{split-and-merge} method, are proposed to deal with issues on the unsupported functions, the limited IO precision and the constrained computation scale respectively, which may increase the graph scale. To some extent, the principle of `trade scale for capability' is used.\n\nIn addition, it is necessary to note that this principle is not contradictory to network compression:\non one hand, usually there is significant redundancy for DNN models, and compression can remove it.\nOn the other hand, diverse types of NN and dataset have different requirements for optimized weight-\/IO-precision of the underlying hardware.\nIt is difficult to determine in advance whether or not the limited hardware precision are enough for any given NN.\nThus, both are useful and orthogonal.\nWe can compress the network first, then convert it, or vice versa.\n\\end{comment}\n\nTo make the transformation workflow general and reliable for different cases, we employed two principles.\n\n\\bulletkeyword{Trade Scale for Capability.}\nAs the operations supported by NN hardware is not comparable to their software counterparts due to the constraints, it is reasonable to enlarge the graph scale and complicate the topology properly to improve the model capability, especially under strict conditions.\n\n\\begin{comment}\nAnother point is to set parameters of \\emph{core\\_ops} properly to make the constructed CG equivalent to the golden model.\nThus, after the above processing, the CG of \\emph{core\\_ops} will be fine-tuned part by part according to a topological ordering of the golden model to reduce transformation error, under the premise of satisfying the hardware weight precision.\n\nIn contrast, from the point of view of system abstraction, the EIE and EIE-like solutions support the general abstraction of NN models and the Cambricon ISA is a hardware abstraction (the specific way of abstraction is different from ours), while we introduce the two abstractions both.\n\\end{comment}\n\n\\bulletkeyword{Divide and conquer.}\nWe fine-tune the entire model part by part according to a topological ordering.\nEach part is a smaller graph that is more easier to converge.\nWe also fine-tune each part with several phases to introduce different constraints, which also facilitates the fast convergence.\n\nMoreover, this transformation procedure could be viewed as \\emph{compilation} of traditional computer systems that converts high-level programs (the hardware-independent, trained NN models) into instructions that hardware can understand (the SW\/HW interface), and the transformation tool could be called an NN compiler.\nAs a summary, this paper has achieved the following contributions:\n\\begin{squishlist}\n\t\\item An NN transformation workflow is presented to complete the aforementioned technologies to support different types of NNs.\n\n    The SW\/HW interface is easy to be adapted to different NN hardware.\n\t\\item Such a toolchain is implemented to support two different hardware designs' constraints, a real CMOS neuromorphic chip for ANN\\&SNN, TianJi~\\cite{11}, and a PIM design built upon metal-oxide resistive random access memory (ReRAM) for ANN, PRIME~\\cite{7}.\n\n\t\\item We complete quite a few evaluations of various metrics. The extra error caused by this process is very limited and time overhead is much less (compared to the whole training process of the original NN).\n\tIn addition, its sensitivity to different configurations and transformation strategies has been explored comprehensively.\n\\end{squishlist}\n\n\\begin{comment}\n\\begin{table}[!t]\\small\n\t\\centering\n\t\\begin{tabular}{|*{5}{l|}}\n\t\t\\hline\n\t\t\\textbf{Chip} & \\textbf{Weight} & \\textbf{IO} & \\textbf{Scale} & \\textbf{Nonlinear}\\\\\n\t\t\\hline\n\t\t\\hline\n\t\tTianJi~\\cite{11} & 8-bit & 8-bit & $256^2$ & Configurable\\\\\n\t\t\\hline\n\t\tPRIME~\\cite{7}& 8-bit & 6-bit & $256^2$ & ReLU\\\\\n\t\t\t&\t&\t&\t& Max Pooling\\\\\n\t\t\\hline\n\t\tDianNao~\\cite{13}& 16-bit & 16-bit & $16^2$ & Configurable\\\\\n\t\t\\hline\n\t\tTPU~\\cite{TPU2017} & 8-bit & 8-bit & None & ReLU\\\\\n\t\t\t&\t&\t&\t& Max Pooling\\\\\n\t\t\t&\t&\t&\t& etc.\\\\\n\t\t\\hline\n\t\tTrueNorth~\\cite{8}& 2-bit & Spiking & $256^2$ & LIF\\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Hardware limitations of NN chips}\n\t\\label{tab:chips}\n\\end{table}\n\\end{comment}\n\n\n\\begin{comment}\nThe rest of this paper is organized as follows.\nSection~\\ref{sec:related_work} presents related work and Section~\\ref{sec:background} introduces the concepts of NN, training, fine-tuning and CG.\nSection~\\ref{sec:workflow} is the main part, which gives the outline of the whole transformation \\& mapping methodology and details of key technologies, from the point of view of CG conversion.\nSection~\\ref{sec:evaluation} presents the implementation and evaluations.\nSection~\\ref{sec:conclusion} concludes.\n\\end{comment}\n\\section{Related Work}\\label{sec:related_work}\nThere are two types of neural network chips.\nThe first type focuses on the traditional ANNs: they are custom architectures to accelerate mature artificial intelligence (AI) algorithms, especially machine learning algorithms of DNN.\nThe second usually supports SNNs to yield higher biological reality.\nIn order to distinguish, we call the former NN accelerators and the latter neuromorphic chips in the following content.\n\\subsection{NN compression}\\label{sec:nn_compression}\nNNs are both computational intensive and memory intensive, making them difficult to deploy on hardware systems with limited resources.\nAt the same time, there is significant redundancy for deep learning models~\\cite{denil2013predicting}.\nTherefore, quite a few studies have been carried out to remove the redundancy, which can be divided into three types: \\emph{pruning neurons}, \\emph{pruning synapses} and \\emph{pruning weights}.\nBoth \\emph{weight-quantization}~\\cite{vanhoucke2011improving, hwang2014fixed, anwar2015fixed} and \\emph{weight sharing}~\\cite{chen2015compressing} are pruning weights.\nFor pruning synapses, deep compression~\\cite{han2015deep} prunes the network by retaining only important connections.\nDiversity Network~\\cite{45} prunes neurons, which selects a subset of diverse neurons and fuses the redundant neurons into the selected ones.\n\nMoreover, there are some research efforts that study extremely compact data representations (including the I\/O precision or weight precision or both) for NN computation. Binarized neural networks [1,3] that investigates the use of 1-bit data types for weight and ternary neural networks [4,5] using 2 bits belong to this category, which have achieved comparable accuracies to state-of-the-art full precision networks for some data sets. However, these methods are effective for specific networks, not common development scenarios.\n\\subsection{NN accelerators}\\label{sec:nn_accelerators}\nEIE~\\cite{han2016eie} extensively employs the above compression techniques and then proposes a dedicated engine to perform inference on the compressed network model.\nESE~\\cite{han2016ese} is its follow-up work that implements a speech recognition engine with compressed Long-Short-Term-Memory model on FPGA.\n\nThe DianNao chip family is a series of state-of-the-art NN accelerators.\nDianNao~\\cite{13} designs an accelerator for DNNs and CNNs that exploits data reuse with tiling.\nThe inherent sharing of weights in CNNs is explored in ShiDianNao~\\cite{21}.\nPuDianNao~\\cite{14} supports seven machine learning algorithms.\nDaDianNao~\\cite{15} is a custom multi-chip machine-learning architecture.\nFrom the aspect of internal data representation and computation, they support fixed-point computation rather than 32-bit floating-point to reduce hardware cost.\nAccordingly, some corresponding retraining or tuning process is needed to adapt software for hardware.\nThey use a load-store ISA~\\cite{liu2016cambricon} to decouple synaptic weight storage from neuron processing logic to avoid the limitation on connection number and improve the utilization of processing logic.\nThis family also supports compressed, sparse NNs through a dedicated accelerator, Cambricon-X~\\cite{zhang2016cambricon}.\n\nMinerva~\\cite{27} uses fine-grained data type quantization and dynamic operation pruning to further reduce the resource consumption of DNN hardware.\nStrip~\\cite{judd2016stripes} relies on bit-serial compute units to enable support for per-layer, dynamically configurable computation precision for DNN.\nDNPU~\\cite{DNPU2017} supports dynamic fixed-point with online adaption and weight quantization, too.\nOther studies include Origami~\\cite{22}, Convolution Engine~\\cite{26}, RedEye~\\cite{28}, NeuroCube~\\cite{29}, neuFlow~\\cite{31} and quite a few FPGA-based designs~\\cite{23, 24, 25, sharma2016high}.\n\nAll the above studies are based on the traditional Complementary Metal-Oxide-Semiconductor (CMOS) technology.\nAnother category is using novel nonvolatile memory devices (usually memristors) to perform neural computations in memory.\nPipeLayer~\\cite{songpipelayer} is such an accelerator for CNNs that supports both training and inference, while PRIME~\\cite{7} and ISAAC~\\cite{18} are for inference.\nOther work includes stand-along accelerators~\\cite{1, hu2016dot, chen2015optimized}, co-processor~\\cite{li2013memristor-based} and many-core or NoC~\\cite{liu2015reno, bojnordi2016memristive} architecture.\nSince the new memory technology is not very mature, there is no systematic programming method.\n\nThe main computational components of these chips usually contain vector-matrix-multiplication units and nonlinear functions, which is the basis of our hardware abstraction.\n\nMoreover, there are quite a few development frameworks for neural networking computing.\nSome (like Theano~\\cite{2016arXiv160502688full}, TensorFlow~\\cite{abadi2016tensorflow}, CNTK~\\cite{CNTK}, Torch~\\cite{collobert2011torch7}, MXNet~\\cite{MXNet}, etc.) describe NN as computation graphs.\nFor a trained NN, the complete information can be extracted from whichever of them.\n\\subsection{Neuromorphic chips}\\label{sec:neuromorphic_chips}\nTrueNorth~\\cite{8, 9} is a digital neuromorphic chip for SNNs, based on a structure of tiled crossbar (each crossbar is of size $256 \\times 256$, and supports binary-valued neurons and ternary-valued synapses).\nThe programming paradigm, \\emph{Corelet}~\\cite{37}, is bound to the hardware platform:\nits recent study~\\cite{12} proposes a TrueNorth-specific training mechanism to construct the whole NN from top to bottom.\nThe parameters learned are then mapped to hardware~\\cite{38} using \\emph{Corelet}.\n\nEMBRACE~\\cite{10} is a compact hardware SNN architecture, with the limited fan-out of individual neuron circuits.\nFrom the programming aspect, it follows the Modular Neural Network (MNN) computing paradigm~\\cite{39}.\nThus, it is not a general solution.\nNeurogrid~\\cite{40} and FACETS~\\cite{42} (including its successor BrainScaleS~\\cite{43}) are analog\/digital hybrid systems, whose development methods are both hardware-specific.\nSpiNNaker~\\cite{44} is different: its toolchain is not bound to any computing paradigm.\nThe reason is that it is based on the chip multiprocessor (CMP) of ARM cores.\nThus, its neural computing is completed by software.\nThe drawback is that the efficiency will be lower than the dedicated hardware.\nTianJi~\\cite{11} is an experimental CMOS neuromorphic chip based on tiled crossbars and supports hybrid computing of ANN and SNN.\nA toolchain NEUTRAMS~\\cite{NEUTRAMS} is developed to map various NN models onto the chip.\nIt also decouples application from hardware and completes SW\/HW co-design for optimization.\nBut its methodology is based on redundancy of NN models and is not suitable for large-scale network under strict constraints.\n\n\n\\section{Transformation Methodology}\\label{sec:workflow}\nIn this section, we will introduce the transformation methodology to transform the software programming model into an approximately equivalent hardware execution model.\n\\subsection{The workflow outline}\\label{sec:outline}\nThe proposed workflow involves 4 steps as shown in Figure~\\ref{fig:workflow}.\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{fig\/workflow.pdf}\n\t\\caption{Workflow of our proposal, from the input model to the output for chip. Step `Graph Tuning' contains 3 sub-steps for different hardware restrictions respectively and the third sub-step has 3 fine-tuning phases.}\n\t\\label{fig:workflow}\n\\end{figure}\n\n \\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{fig\/example.pdf}\n\t\\caption{A transformation example}\n\t\\label{fig:example}\n\\end{figure}\n\n\\noindent\\textbf{Building CG.}\\label{step:parse}\nAccording to the above description, it constructs $G=(V, E)$ based on the input NN's information that includes the trained parameters, network topology, vertex information and training dataset.\nAn example is shown in Figure~\\ref{fig:example}(a).\nIn addition, we can also get the operation set $F$ supported by the deep learning framework.\n\n\\noindent\\textbf{Graph Reform.}\\label{setp:graph_reform}\nIt constructs an intermediate CG, $\\hat{G}=(\\hat{V}, \\hat{E})$.\nThe corresponding operation set $\\hat{F}$ contains all operations that are \\textit{core-op--like}.\nThese operations share the similar form as core-ops or can be achieved by core-ops easily, but do not meet those hardware constraints.\nFigure~\\ref{fig:example}(b) shows an example of the graph with only core-op--like operations.\nThe detailed transformation from $G$ to $\\hat{G}$ is in Section~\\ref{sec:graph_reform}.\n\n\\noindent\\textbf{Graph Tuning.}\\label{step:graph_tuning}\nIn this step, we further transform $\\hat{G}$ to $G'$.\nEvery vertex $\\hat{v}\\in\\hat{V}$ is traversed in a topological ordering of $\\hat{G}$ to form corresponding vertices in $G'$.\nAs shown in Figure~\\ref{fig:workflow}, we have multiple sub-steps for the transformation of each vertex $\\hat{v}$.\nWe can also transform a subgraph with multiple vertices as a whole.\nWe transform the whole graph part by part to have a better convergence since smaller graph are easier to be approximated.\nIt is where we employ the \\textit{divide-and-conquer} principle.\nThe sub-steps are as following.\n\\begin{itemize}\n\n    \\item \\keyword{Data Re-encoding.}\n    Re-encode I\/O data on each edge of the subgraph to solve the precision problem of I\/O data.\n    This sub-step is where we employ the \\textit{trade-scale-for-capability} principle.\n    It enlarges the computation scale, but does not change the operation type, each vertex is still a core-op--like operation.\n\n\t\\item \\keyword{Fully Expanding.}\n    Since core-op--like operations are easy to be implemented with core-ops, in this sub-step, we fully expand each core-op--like operation with multiple core-op operations to solve the limitation on the computation scale.\n\tAfter this sub-step, the subgraph only contains core-ops.\n\n    \\item \\keyword{Weight Tuning.}\n    This step aims to fine-tune the weight matrices of core-ops in the subgraph to minimize transformation error, under the premise of satisfying the hardware weight precision.\n    As shown in Figure~\\ref{fig:workflow}, we also introduce three phases of fine-tuning to make it more reliable and easier to converge.\n    It is also where we employ the \\textit{divide-and-conquer} principle.\n \\end{itemize}\nFigure~\\ref{fig:example}(c) shows an example of the transformed graph $G'$ with only core-ops.\nDetailed transformation are in Section~\\ref{sec:graph_tuning}.\n\n\\noindent\\textbf{Mapping.}\nNow we have built an equivalent Graph $G'$ with only core-ops that meet hardware constraints, which will be mapped onto the target hardware efficiently.\nThis part highly depends on the hardware's interconnection implementation.\nIn Section~\\ref{sec:mapping} we will introduce the basic principle of mapping the hardware execution model onto target chips.\n \n\\subsection{Graph Reform}\\label{sec:graph_reform}\n\\begin{comment}\nInitially, $G(V, E)$ is slightly adjusted: any vertex that represents an activation function is merged with its previous vertex(vertices) whose computation is in the form of weighted-sum to form a \\emph{core\\_op}-like operation. Examples are presented in Figure~\\ref{fig:example}: multiple vertices in one convolutional layer (Figure~\\ref{fig:example}(a)) are merged into one vertex (Figure~\\ref{fig:example}(b)), as well as the fully connected layer.\n\nAfterwards, for all remaining vertices whose operations are neither in the form of weighted-sum nor supported by hardware directly, each of them will be approximated with an MLP, because the latter is a universal approximator; the activation function used is \\emph{ReLU} as the target hardware provides it.\nIn another word, we will train an MLP to achieve the operation(s) of the original vertex; some common functions can be trained in advance.\n\nMoreover, for those vertices that are not suitable to be approached this way because of huge resource consumption, some dedicated designs could be proposed.\n\nFor example, the resource consumption of max pooling increases exponentially with scale, if we just use such a general approximator.\nConsequently, a dedicated design is given.\nMax pooling can be built with max functions, each of which is completed as following:\n{\n\t\\centering\n\t$\\max(a, b) = \\frac{1}{2}\\relu(a+b) + \\frac{1}{2}\\relu(a-b) + \\frac{1}{2}\\relu(-a+b) + \\frac{1}{2}\\relu(-a-b)$\n}\n\nThus, a simple max function with two variable can be implemented with a neural network of four hidden neurons. We could use this neural network as a basic building block to construct max functions with more variables.\n\nIn this way, we can construct an equivalent CG, $G'(V', E')$. Vertices of $G$ are either copied into $G'$ (including those after merge), or replaced with the equivalent MLPs. For the latter, the corresponding weight matrices are trained (or set up) in floating-point numbers.\n\nFinally, the initial `merge' operation is performed against $G'$. Then all vertices in $G'$ are just \\emph{core\\_op}-like operations.\nAn example is given in Figure~\\ref{fig:example}(b):\nbesides the vertex merge, the max-pooling vertex is replaced with a dedicated subgraph of only \\emph{core\\_op}-like operations.\n\\end{comment}\nIn this step, we need to transform a CG $G$ into a graph $\\hat{G}$ with only core-op--like operations.\nThe operation set $\\hat{F}$ includes all operations that could be combined by core-ops in $F'$ without any precision constraints.\nFor example, the corresponding core-op--like operations for dot--core-op are all operations that in the form of weighted sum with activation function, denoted as \\textit{dot-like} operations.\n\nTo replace all vertices represented with $F$ into operations in $\\hat{F}$, we take the following three steps in order.\n\\begin{textcircleds}\n\t\\item First, we find all subgraphs that match the computation of any $\\hat{f}\\in\\hat{F}$, and replace them with $\\hat{f}$.\n    For example, in Figure~\\ref{fig:example}(b), we merge the dot-product, add-bias and ReLU operations into one operation, which is a dot-like operation.\n    \\item Then, we can also have some customized mapping from a subgraph in $G$ into a subgraph formed of operations in $\\hat{F}$, and apply these dedicated designs here.\n    For example, max-pooling operation can be built with max functions.\n    A simple max function with two operands can be achieved with as Equation~\\ref{eq:max}, which includes multiple dot-like operations.\n    \\begin{equation}\\label{eq:max}\n\t\t\\begin{split}\n    \t\t\\max(a, b) = & \\frac{1}{2}[\\relu(a+b)+\\relu(a-b)\\\\\n        \t\t\t + & \\relu(-a+b)+\\relu(-a-b)]\\\n    \t\\end{split}\n\t\\end{equation}\n    We can use the max function with two operands to form max function with more operands.\n    \\item Finally, for the left operations in $G$, we provide a default transformation strategy: we use multiple dot-like operations to form MLPs to approximate them since MLP is proved to be an universal approximator~\\cite{hornik1989multilayer}.\n\\end{textcircleds}\nAfter the transformation, we form an graph $\\hat{G}$ with only core-op--like operations.\n\n\\begin{figure*}[!htbp]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{fig\/graph_tuning.pdf}\n\t\\caption{\n\t\tGraph tuning of one vertex.\n\t\t(a) $G'$ with only \\emph{core\\_op}-like operations.\n\t\t(b) `Graph Tuning' has been performed against all previous vertices.\n\t\t(c) Insert an `autoencoder' after the current vertex.\n\t\t(d) Merge the bias and activation of the current vertex into the hidden layer of the autoencoder.\n\t\t(e) Multiply the three weight matrices together.\n\t\t(f) Fully expand the vertices with \\emph{core\\_ops}.\n\t\t(g) The original CG, $G$. The current subgraph (including the decoder of the current vertex) is fine-tuned to approach the corresponding vertex of $G$, with output of the previous tuned vertex as input.\n\t\tRepeat (b) to (g) for the rest vertices.\n\t}\n\t\\label{fig:graph_tuning}\n\\end{figure*}\n\n\\subsection{Graph Tuning}\\label{sec:graph_tuning}\n\nIn this step, we transform the intermediate graph $\\hat{G}=(\\hat{V},\\hat{E})$ to the hardware execution model $G'=(V',E')$.\nWe use the original graph $G$ to supervise the fine-tuning progress of the generated $G'$.\nThe graph $G$ can provide not only labels of the output but also supervised signals of all intermediate data between operations.\nEach edge $e\\in E$ can provide supervised signal for graph $G'$.\nThus, we can split $\\hat{G}$ into parts, and transform it into $G'$ part by part.\nTo do so, first we find the edges $\\hat{e}\\in\\hat{E}$ that correspond to the edges $e\\in E$, and use them to split the graph $\\hat{G}$.\nThen, we perform the following steps against each part one by one in a topological ordering of $\\hat{G}$.\nNote that, we can also transform multiple adjacent parts as a whole each time.\n\n\\bulletkeyword{Data Re-encoding.}\nSince the original model $G$ usually use floating-point numbers for computation and data representation, directly rounding data to hardware I\/O precision may distort and lost information.\nThus, this sub-step aims to re-encode the input and output data of each vertex.\nTo encode a floating-point vector with a low-precision vector, we employ an autoencoder to get the low-precision representation.\n\nAn autoencoder is an NN with one hidden layer that can learn a constrained representation of a set of input data.\nThe output layer has the same dimension as the input layer and is trained to approach the input:\nthe computation from input to the hidden layer is encoding input data to the hidden layer's representation, while the computation from the hidden layer to output is decoding.\n\nHere as we use it to represent the original floating-point vector with the hardware--I\/O-precision vector, it is reasonable that the neuron number of the hidden layer may be greater than that of the input\/output layer;\nthe specific value can be configured manually.\nUsually, a large hidden layer will improve the accuracy, but consume more resources.\nThe tradeoff is evaluated in Section~\\ref{sec:evaluation}.\n\nFor clarity, we take a vertex of dot-like operation as the example to describe the process.\n\nAs shown in Figure~\\ref{fig:graph_tuning}(a)(b)(c), we add an autoencoder after the current vertex.\nThe activation function of its hidden layer is a round operation, which rounds the output to hardware I\/O precision.\nIn addition, the round operation not only quantizes the output to low-precision, but also forces the output to be positive, which provides the non-linearity as the widely-used ReLU function.\n\nThe initial weight parameters for the encoder and decoder of the autoencoder are set as following.\n\nFor the input vector $X=\\{x_1, \\ldots, x_n\\}$ (i.e. the output of the previous ReLU, as illustrated by Figure~\\ref{fig:graph_tuning}(c)), we build an autoencoder network with a hidden layer of $m\\times n$ neurons, where $m$ is a configurable integer, denoted as \\textit{re-encoding factor}.\nSuppose the $n$ dimensions of $X$ are independent and equally important, then each $x$ has $m$ hidden neurons to represent its low precision encoding.\nThus, we initialize the connections from $x$ to other hidden neurons to zero.\nA hidden neuron linearly scales the corresponding $x$ to $w^{(e)}x+b^{(e)}$, where $w^{(e)}$ and $b^{(e)}$ are the weight and bias term of the encoder respectively, and then rounds it to the I\/O precision, $N$-bit integer ($\\{0, \\ldots, 2^N-1\\}$).\nAny $x < \\frac{-b^{(e)}}{w^{(e)}}$ will be rounded to $0$ and any $x > \\frac{2^N-1-b^{(e)}}{w^{(e)}}$ will be rounded to $2^N-1$.\nThus, one hidden neuron can well represent $x \\in [\\frac{-b^{(e)}}{w^{(e)}}, \\frac{-2^N-1-b^{(e)}}{w^{(e)}}]$.\n\nNow we have $m$ hidden neurons for each dimension;\nthus the best way to represent $x \\in [0, x_{max}]$ (the output of ReLU is positive) with the $m$ hidden neurons is to divide the data range into $m$ adjacent and non-overlapping intervals, each of which corresponds to one hidden neuron.\nNamely, we properly initialize $w^{(e)}_i$ and $b^{(e)}_i$ for the the encoder of the $i$-th neuron of the $m$ to adapt $[\\frac{-b^{(e)}_i}{w^{(e)}_i}, \\frac{2^N-1-b^{(e)}_i}{w^{(e)}_i}]$ to the corresponding interval $[\\frac{ix_{max}}{m}, \\frac{(i+1)x_{max}}{m}]$.\nThus $w^{(e)}_i = \\frac{(2^N-1)m}{x_{max}}$ and $b^{(e)}_i = -(2^N-1)i$.\n\nAccordingly, to decode and restore $x$, the decoder should scale the data back.\nThus, its weight matrix is set to $w^{(d)}_i = \\frac{1}{w^{(e) _i}}=\\frac{x_{max}}{(2^N-1)m}$.\n\nNote that, if input $x < 0$ (outside of the encoded interval $[0, x_{max}]$), the initialized autoencoder will always return 0.\nIt means that the autoencoder can also perform ReLU operation. Therefore, we could remove the redundant ReLU operation of the current vertex.\nMoreover, as shown in Figure~\\ref{fig:graph_tuning}(d), the bias term of the current vertex could also be encoded by the dot-product operation and merged into the bias term of encoder $b^{(e)}_i$.\nNamely, the new bias term of encoder becomes $b^{(e)}_i + w^{(e)}_ib$.\n\nFinally, as shown in Figure~\\ref{fig:graph_tuning}(e), the decoder of the previous vertex, the dot-product operation and the encoder of the current vertex can be merged as one dot-product operation, whose weight matrix is the product of the three's matrices.\n\nTill now, the input and output of the current vertex have been constrained to hardware I\/O precision.\n\nFor vertices of convolution plus activation function, the process is similar.\nInstead of using dot-product operation as encoder and decoder, we use convolution instead, and the three convolutions can be merged into one as well.\nThe initialization is also similar: the hidden layer has $m$ channels for each input channel, and only the center value of the encoder\/decoder kernel is set to non-zero.\n\nOwing to this step, we solve the limitation problem of I\/O precision.\nIn this step, we only change the computation scale of the operations in $\\hat{G}$.\n\n\\bulletkeyword{Fully Expanding.}\nIn this step, we will turn $\\hat{G}$ into $G'$.\nSince the core-op--like operations $\\hat{f}\\in\\hat{F}$ can be combined by core-ops in $F'$, we expand all operations in $\\hat{G}$ into individual subgraphs consisting of operations in $F'$ to form the graph $G'$.\n\n\nTake dot-like operations as an example.\nDot-like operations can be represented as a fully-connected layer.\nAs shown in Figure~\\ref{fig:graph_tuning}(f),\nto support the fully-connected layer of any scale with dot--core-ops, we use two layers of dot--core-ops to construct an equivalent graph.\nThe first layer is a computation layer. The original dot-like operations are divided into smaller blocks and are performed with many dot--core-ops.\nThe second layer is a reduce layer, which gathers result from the former to output.\n\nThe division is straight:\n\\begin{textcircleds*}\n\t\\item Divide the weight matrix into small sub-matrices that satisfy the hardware limitation on scale;\n\teach is held by a dot--core-op of the first layer.\n\t\\item Divide the input vector into sub-vectors and transfer each sub-vector to the corresponding sub-matrices(\\emph{core\\_ops}) at the same horizontal position and\n\t\\item gather results from the same column by the reduce layer.\n\\end{textcircleds*}\n\nRegarding all dot-like operations as fully-connected layers are sometimes very inefficient. We can have dedicated division according to its connection pattern.\n\nFor example, for a convolutional case (suppose a kernel of size $k \\times k$ convolves a $W \\times H$ image from $m$ channels to $n$ channels), the $n$ channels of one pixel in the output side are fully connected to the $m$ channels of $k \\times k$ corresponding pixels in the input side.\nThis forms a small-scale vector-matrix multiplication of size $(m \\times k^2) \\times n$.\nThere are $W\\times H$ such small operations in the convolution case.\nEach input should be transferred to $k^2$ such small operations, while reduction is needless.\nIf such a small operation is still too large for a dot--core-op, we can divide the operation as the fully-connected--layer case does.\n\nIf there are some dot--core-ops that are not fully used, we can distribute them onto one physical PE to reduce resource consumption during the mapping step.\n\nTill now, the computation of the current subgraph of $\\hat{G}$ has been transformed to a subgraph of $G'$, which consists of core-ops $f'\\in F'$.\nNext, the weight matrices of the core-ops will be fine-tuned.\n\n\\bulletkeyword{Weight Tuning.}\nIn this step, we will fine-tune the parameters to make the generated subgraph of $G'$ approximately equal to the corresponding subgraph of $G$.\n\n\\begin{comment}\nBased on previous steps, the initialization mode of the weight matrix(matrices) of the current sub-CG can be divided into two categories:\n\\begin{itemize}\n\t\\item If the sub-CG is not created by Step `Graph Reform' (i.e. it is copied from $G$), the corresponding weight parameters from the golden model are used.\n\t\\item For others, the initial values are set up by the `Graph Reform' step.\n\\end{itemize}\n\nOf course, in either case, all the weight matrices have been processed by subsequent steps, i.e. `Data Re-encoding' and `Fully expanding'.\n\nHowever, some methods, including autoencoder and the MLP-based unsupported function handling, introduce transformation errors.\nIn addition, activation functions used by $G(V, E)$ may be different from the hardware counterpart (i.e. the activation function of \\emph{core\\_op}), which also makes the initialization inaccurate.\nTherefore, some fine-tuning phases have to be taken to minimize the error, under the premise of satisfying the hardware weight precision constraint.\n\\end{comment}\n\nAs shown in Figure~\\ref{fig:graph_tuning}(g)(f), we use the corresponding supervised signal from graph $G$ to fine-tune current subgraph of $G'$.\nThe input to the current subgraph is from the output of previous transformed subgraphs instead of the corresponding supervised signal from the graph $G$.\nThus, the transformation of current subgraph will consider the error from previous transformed subgraphs, which can avoid error accumulation.\nThe output of previous transformed subgraph can be generated on demand or cached in advance to improve the transformation speed.\n\n\n \n\n\nWe will consider the hardware constraints on weight parameters in this step.\nSpecifically, target hardware usually puts strict constraints on weight storage since it occupies most of the hardware resources.\nHere we present a formal description of the constraints on the weight matrix $W$:\nthe value of each element $W_{ij}$ should be assigned dependently from a finite set $S$.\n$S$ is either a fixed set or a configurable set $S^P$ with parameter(s) $P$.\nThree kinds of typical weight encoding methods, which have been widely used by real NN chips, are presented as following (in all cases, the size of $S$ is $2^N$):\n\\begin{itemize}\n\t\\item \\textbf{Dynamic fixed-point}: $S^P=\\{\\frac{-2^{N-1}}{2^P}, \\ldots, \\frac{0}{2^P}, \\ldots, \\frac{2^{N-1}-1}{2^P}\\}$ where $P$ represents the point position.\n\tThis method is used by DNPU~\\cite{DNPU2017}, Strip~\\cite{judd2016stripes}, TianJi-ANN~\\cite{11}, etc.\n\t\\item \\textbf{Fraction encoding}: $S^P=\\{\\frac{-2^{N-1}}{P}, \\ldots, \\frac{0}{P}, \\ldots, \\frac{2^{N-1}-1}{P}\\}$, where $P$ is the threshold of the spiking neuron or the scale factor of the result. It is used by PRIME~\\cite{7}, and TianJi-SNN~\\cite{11}.\n\t\\item \\textbf{Weight sharing}: $S^{P_1, \\ldots, P_{2^N-1}}=\\{0, P_1, \\ldots, P_{2^N-1}\\}$, used by EIE~\\cite{han2016eie}.\n\\end{itemize}\n\nWithout loss of generality, suppose the floating-point parameter $W_{ij}$ is rounded to the $k_{ij}$-th element in $S^P$, denoted as $S^P_{k_{ij}}$.\nThis step aims to find the best $P$ and to set $k_{ij}$ for each element in the weight matrix properly to minimize the transformation error.\nIt is similar to \\textit{weight quantization} of network compression.\nOur contribution is that we generalize it to typical hardware cases and introduce several fine-tuning phases to deal with different parameter-setting issues separately.\n\nFor a subgraph, three fine-tuning phases are taken in order:\nThe first is to reduce the initialization error.\nThe second is to determine the best value range of weight matrix (i.e. to choose the best $P$) and the last is to determine the best value from $S^P$ for each element (i.e. to choose the best $k_{ij}$).\nEach phase gets parameters from the previous one and fine-tunes them under certain constraints.\n\n\\bulletkeyword{Free Tuning.}\nIn previous steps, we use the parameters in the original graph $G$ to initialize those parameters in the generated graph $G'$.\nHowever, some methods, including autoencoder and the MLP-based unsupported-function handling, introduce transformation errors.\nIn addition, activation functions used by $G$ may be different from the hardware counterpart, which also makes the initialization inaccurate.\nIn addition, previous transformed subgraphs also have errors.\nTherefore, some fine-tuning phases have to be taken to minimize the error, under the premise of satisfying the hardware constraints on weight precision.\n\nThus we first fine-tune the subgraph of $G'$ without any constraint on weight precision to reduce any existing error.\nIn this procedure, all parameters and signals are processed as floating-point numbers, while the hardware activation function is used.\n\n\\bulletkeyword{Value-Range Tuning.}\nNow the precision constraint on weight is introduced.\nAccordingly, we need to choose the best value-range of the weight matrix (namely, the best $P$).\nApparently, we will minimize $J(k, P)=\\sum_{ij}(W_{ij}-S^P_{k_{ij}})^2$, which can be achieved by an iterative expectation\\-maximization (EM) algorithm:\n\\begin{itemize}\n\t\\item E-step: fix the current parameter $P^{(t)}$ and calculate $k_{ij}^{(t)} = \\arg\\min{J(k|P^{(t)})}$.\n\t\\item M-step: fix $k_{ij}^{(t)}$ and calculate $P^{(t+1)}=\\arg\\min{J(P|k^{(t)})}$.\n\\end{itemize}\n\nThen we replace $W_{ij}$ with $S^P_{k_{ij}}$ where $k_{ij}$ is fixed and $P$ is the parameter.\n\nAfter the initialization, we fine-tune the subgraph to optimize $P$.\nDuring this process, we maintain the precision of $W_{ij}$ first and then round it to $P_{k_{ij}}$ at every time $P$ is updated.\n\nFurther, for the weight sharing case mentioned above, the EM algorithm is just reduced to the k-means algorithm.\nIf $S^P$ is a fixed set without any configurable parameter, we can omit this phase.\n\n\\bulletkeyword{Rounding Tuning.}\nThe data-range set of weight value $S^P$ is fixed now.\nThis procedure adjusts each weight matrix element to a proper element in this set.\nIn another word, it aims to choose the best index $k_{ij}$ for $W_{ij}$.\nDuring the fine-tuning progress, parameters are stored as floating point number.\nIn the forward phase, any parameter is rounded to the closest element in $S^P$.\nWhile during the backward phase, floating-point number is used to update $W_{ij}$.\nThis mechanism is also employed by the above \\textit{Value-Range Tuning} phase if $P$ can be set from a discrete set.\n\n\nAfter processing all the subgraphs, we have transformed the original model $G$ into an equivalent hardware execution model $G'$ that satisfies all the constraint conditions.\n\n\\begin{comment}\nFinally, it is worth adding that we can fine-tune several successive subgraphs in $G'$ simultaneously according to a topological ordering,\nand the above steps are still applicable.\nWe will evaluate the effect of fine-tuning-granularity on transformation results in Section \\ref{sec:evaluation}.\n\\end{comment}\n\n\\subsection{Mapping}\\label{sec:mapping}\nThe generated graph $G'$ will be deployed on the target hardware efficiently, which is a hardware-specific problem.\nThus, we give the optimization principle here.\n\nFor NN chips that bind the neural computation and storage in the \\textit{physical cores} (it is called the \\textit{weight stationary} computing mode, classified by~\\cite{16}), this is a mapping problem to assign core-ops to physical cores. Moreover, several core-ops that are not fully used can also be distributed onto one physical core, as long as there are no data conflicts.\n\nFor chips whose physical cores are computing engines with flexible memory access paths to weight storage (usually work in the time division multiplex mode), it is a mapping and scheduling problem to schedule each core-op's task onto physical cores.\nMultiple core-ops could be mapped onto one core to increase resource utilization.\n\nAs we can get data dependencies and communication patterns between core-ops through the transformed graph, we could use these information to optimize the mapping or scheduling to minimize transmission overhead, e.g. putting densely-communicating cores close.\nTrueNorth has designed such an optimized mapping strategy~\\cite{38}.\n\nMoreover, for those core-ops sharing weights (e.g. convolution vertices can be fully expanded to a lot of core-ops sharing the same weight matrix), we could map (or schedule) them to the same physical core to reduce data movement.\n\n\\subsection{Others}\\label{sec:specification}\n\\subsubsection{SNN Models}\\label{sec:snn}\nSNN, called the third generation of ANN, is a widely-used abstraction of biological neural networks.\nIn addition to neuronal and synaptic states that traditional ANN has featured, it incorporates the timing of the arrival of inputs (called spikes) into the operating model to yield higher biological reality.\n\nSNNs of rate coding can emulate ANNs.\nThe spike count in a given time window can represent a numerical value within a certain range, like a traditional ANN does.\nAccordingly, the input of a synapse is a spike sequence of certain firing rate from the pre-neuron.\nAfter synapse computation, it is converted into the sum of currents that will be computed by the post-neuron.\nFor those widely-used SNN models, the functions of their synapse and neuron computations usually own good continuity and are derivable in rate coding domain.\nTherefore, the popular SGD method can be used for training SNN:\nseveral recent studies~\\cite{46, 49} have used the stochastic gradient decent (SGD) algorithm to train SNNs directly or indirectly and achieved the state-of-the-art results for some object recognition tasks.\n\nAs our workflow is not dependent on the concrete NN type (ANN or rate-coding SNN), it can support SNN hardware and SNN models, too.\nFor SNN models, the training data is the firing rate of each neuron.\n\n\\subsubsection{RNN Models}\nRNN is an NN with some cycle(s).\nWe could transform and fine-tune each operation inside an RNN as normal, and add an additional step to fine-tune the entire RNN after that.\n\n\\begin{comment}\n\\subsection{Utilization}\nWe can utilize extra hardware features.\nThe utilization depends on the specific hardware and we present some examples as following:\n\\begin{itemize}\n\t\\item If hardware supports \\textit{reduction communication} or provides additional \\textit{adders}, the reduce layer in \\textit{Fully expanding} would be needless.\n\t\\item If some special function (e.g. max function) is supported, it can also be viewed as a type of \\emph{core\\_op}. Accordingly, the corresponding \\emph{core\\_op}-like operations can include max pooling and others that can be easily achieved by existing \\emph{core\\_ops}. It is no longer required to be approximated by the dedicated design or the MLP-based universal approximator.\n\\end{itemize}\n\\end{comment}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\n\nThe variety of applications of two-dimensional (2D) materials such as graphene and transition\nmetal dichalcogenides has motivated several studies of their growth kinetics in the last\ntwo decades \\cite{dongAM2019,huangNature2011,xuNatNano2016,taslim2019,ying2dMat2020}.\nThe growth of large grains is frequently desired to reduce the effects\nof grain boundaries (GBs) on the electronic and mechanical properties.\nHowever, in some materials, a large GB density may be a minor problem;\nthis is the case for the integration of ultrathin $\\text{Mo}\\text{S}_2$ films\nwith Si substrates in some devices that require low temperature deposition \\cite{lin2dMat2021}.\nMoreover, large GB densities may be beneficial for some applications of 2D materials,\nsuch as $\\text{W}\\text{S}_2$ in sensors of Hg \\cite{liuNatComm2021} and\n$\\text{Mo}\\text{S}_2$ in the production of memtransistors \\cite{sangwanNature2018}.\nGBs of $\\text{W}\\text{S}_2$ may also be useful as high conductivity channels \\cite{zhouACSAMI2019}.\n\nBeyond the GB density, the GB roughness may also affect the\nproperties of 2D materials in nontrivial ways.\nSinusoidal graphene GBs are advantageous over atomically flat GBs because they\nincrease the mechanical strength and improve electronic properties \\cite{zhangAFM2015}.\nThe fractures of $\\text{Re}\\text{S}_2$ monolayers more easily occur at GBs parallel\nto Re chains \\cite{zhangSmall2021}, so that disordered GBs may improve their properties.\nIn the case of graphene, there is also evidence that the grain size at the micrometer\nscale does not have relevant effects on the electric conductivity \\cite{ryuACSNano2014}\nand that the mechanical strength is not reduced in the presence of GBs \\cite{leeSci2013}.\n\nThe GB morphology is strongly related to the growth conditions\nand several models were already proposed to explain this relation\n\\cite{dongAM2019,xuNatNano2016}.\nFor sufficiently slow graphene growth, the so-called attachment limited (or near-equilibrium)\nconditions are observed, which usually lead to formation of smooth GBs.\nModels of step flow describe the edge propagation \\cite{artyukhovPNAS2012}\nand the GB structure can be determined by the misorientation angle \\cite{betsACSNano2021}\nin these conditions.\nSimilar models and other multi-scale approaches are used to describe the growth of 2D\nmetal dichalcogenides \\cite{chenACSAMI2019,momeni2020,hickeyNL2021}.\nHowever, fast growth is frequently important for applications.\nIn this case, the so-called diffusion limited regime is frequently observed, in which\nGBs significantly deviate from a straight profile.\nExperiments, phase field models, and kinetic Monte Carlo (kMC) simulations show formation of grains\nwith fractal or dendritic morphologies that reproduce experimental observations\n\\cite{niePRB2011,mecaNL2013,wuPRL2015,xuNatNano2016,zhuangPCCP2016,yue2dMat2017,momeni2020}.\n\nThe recurrent observation of similar relations between the GB morphology and the growth conditions in \na wide variety of materials is a motivation for the study of models that only contain the essential\nphysico-chemical mechanisms that are needed to explain those relations.\nThis reasoning has already been successful in the description of the morphology of \nhomoepitaxial metal and semiconductor films by means of generic models including adatom diffusion\nand attachment\/detachment from islands and terraces\n\\cite{michely,pimpinelli,saitobook,etb,Misbah2010,einax}.\n\nIn this work, we introduce a model for the propagation of two grain edges \nand for the relaxation of the GB formed after the collision of those edges.\nThe aim of the model is\nto identify the generic consequences of elementary microscopic\nprocesses such as adatom surface diffusion, attachment, and detachment from edges and GBs\nof a variety of 2d materials, in contrast with approaches that \nfocus on particular applications\n\\cite{chenACSAMI2019,gaillard2015,enstone2016,betsACSNano2021}.\nThis modelling approach allows one to investigate the effects of the variations of\nthe rates of those processes over several orders of magnitude using\nkinetic Monte Carlo (KMC) simulations.\n\nThe simulation results reproduce the main\nfeatures of the regimes limited by attachment and diffusion described above, respectively for low and high fluxes.\nHowever, if the precursor flux is sufficiently high to produce  full coverage between the grain edges when they begin to grow, the onset of a nontrivial regime is observed:\nafter an uncorrelated growth of the edges,\nrelatively smoother GBs are formed at relatively shorter times compared to the other regimes.\nFrom the simulation results at nanoscale and the support of a theoretical approach,\nextrapolations to micrometer sized edges separated by micrometer sized gaps are performed.\nThey show the possible advantages of deposition with high coverages\nfor reducing GB roughness.\n\n\n\\section{Model and Methods}\n\\label{basic}\n\n\\subsection{Model for deposition, diffusion, and aggregation}\n\\label{modelsection}\n\nThe solid structure is modelled by a square lattice, which differs from the geometry of most 2D materials,\nbut is amenable to computational and analytical calculations.\nThe substrate below this square lattice is assumed to be inert.\nEach lattice site may be empty or occupied by a particle, which may represent a graphene atom or a metal dichalcogenide molecule.\nThe lattice constant is denoted as $a$.\n\nTwo solid grains have initially flat interfaces (edges) of length $L$ along the $x$ direction\nand are separated by a gap of width $l$,\nas illustrated in Fig. \\ref{model}(a).\nPeriodic boundaries are considered in the $x$ direction.\nThe model does not describe the process of grain nucleation and growth that lead to\nthis initial condition because its aim is to focus on the late stages of edge propagation\nand on the coarsening of the GBs formed after their collision.\nOnce they are incorporated in one of the grains, the particles\nare labelled with a index, A or B, that depends on the grain.\nThese indices are kept even after the complete filling of the gap,\nso that the two grains do not coalesce into a single grain.\nThis approach does not reproduce the details of the anisotropy\nand atomic rearrangements within the grain. However, it aims at catching\nbasic features of the growth of the edges, as well as the \nslow reshaping of the GBs after their collision.\nThese basic features should also be present in the case of grains with different crystallographic\norientations or with some mismatch that prevents coalescence; note that imperfect match\nof 2D grains may also occur when they have the same orientation,\nas observed in $\\text{W}\\text{S}_2$ growth on sapphire \\cite{hickeyNL2021}.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[clip,width=0.45\\textwidth,angle=0]{modelo_a.png}\\\\\n\\includegraphics[clip,width=0.45\\textwidth,angle=0]{modelo_b.png}\\\\\n\\includegraphics[clip,width=0.45\\textwidth,angle=0]{modelo_c.png}\n\\caption{\n(a) Initial configuration of the grains A (yellow) and B (red).\n(b) Two particles incide at the gap (magenta arrows), the right one is accepted and the left one is rejected (excluded volume).\nPossible hops of particles already deposited in the gap are also indicated (black arrows).\n(c) Rates of the processes that bring the configuration at the left to the configuration at the right.\n}\n\\label{model}\n\\end{figure}\n\nThe external particle flux $F$ is the number of incident particles per site per unit time.\nIf the incident particle reaches an empty site above the substrate, it becomes a mobile particle at that position; however, if it reaches an occupied site, the deposition attempt is rejected.\nThis process is illustrated in Fig. \\ref{model}(b).\n\nThe mobile particles can diffuse on the substrate, with a hopping rate to\nnearest neighbor (NN) sites per unit time equal to $D$.\nA hop attempt is executed only if the target site is empty, otherwise the mobile particle\nremains in the same position.\nThe process is also illustrated in Fig. \\ref{model}(b).\nThe corresponding tracer diffusion coefficient is $Da^2\/4$.\nThe assumption that the excluded volume is the only type of interaction between mobile\nparticles implies that other islands or grains are not formed in the gap.\n\nThe grains evolve via the attachment and detachment of particles.\nTo facilitate the connection with kinetic roughening theory and other analytical approaches,\nthese processes in each grain are restricted to the topmost point in each lateral coordinate $x$\n(this is called the Solid-On-Solid constraint in the kMC and kinetic roughening literature).\nWhen a mobile particle is at the site immediately above the topmost grain particle,\nits attachment to the grain occurs with rate $Q$.\nSimultaneously, a topmost particle can detach from a grain with rate $Q\\epsilon^n$,\nwhere $n$ is the number of NNs of that grain and $\\epsilon<1$ is termed\nthe detachment probability per NN.\nThese processes are illustrated in Fig. \\ref{model}(c).\nWith these rules, the attachment rate does not depend on the local configuration\nof the grain edge.\nHowever, the detachment is facilitated at edge tips ($n=1$) and is more difficult at\nflat regions of the edges ($n=3$).\n\nAfter the edge collision, the boundary between the grains A and B is formed.\nThe above rules for attachment\nand detachment are maintained and\nthe interactions between particles of different grains are restricted to excluded volume interactions.\nThis means that the formation of bonds between NN particles of different grains is neglected.\nThis may be a reasonable approximation if the actual inter-grain bonds of a material\nare sufficiently weak in comparison with the intra-grain bonds.\nIn this approximation, the characteristic time scales of the grain edge dynamics are not much\ndifferent before and after the collision.\n\nThe diffusion and detachment of particles are expected to be temperature\nactivated processes, so that the rates $D$ and $\\epsilon$ increase as the temperature increases.\nIn many systems, the attachment is also activated, so $Q$ also increases with the temperature.\nThese parameters may then be written in Arrhenius forms \n(see Sec. S.I of the Supplementary Material).\nIt defines a\ntrue thermodynamic equilibrium in the model without deposition,\nwhich is enforced by detailed balance.\nHowever, since our aim is to address possible properties of several 2D materials, we do not choose a set\nof activation energies for the simulations, but study a large variety of possible relations\nbetween those rates and the flux $F$.\n\n\n\\subsection{Simulations and quantities of interest}\n\\label{simulation}\n\nSimulations were performed in lattices with edge length $L$ varying from $128a$ to $1024a$;\nmost results are obtained for $L=512a$.\nThe initial gap width $l$ varies between $16a$ and $128a$.\nThe model rates in the simulations are measured in terms of the flux rate $F$.\nThe ratios $F\/D$ and $F\/Q$ vary from ${10}^{-8}$ to ${10}^{4}$; these limiting values\nrespectively represent very low and very high fluxes for a given temperature.\nMost simulations are performed with $\\epsilon=0.1$, which implies that the detachment\nrates have a weak dependence on the coordination number $n$; this is reasonable only\nat high temperatures for most materials ($k_BT\\approx 0.43$ times the activation energy\nper NN).\nAdditional simulations were performed with $0.01\\leq\\epsilon\\leq0.07$ to analyze the\neffects of this parameter.\n\nFor most parameter sets, average quantities are calculated over $100$ different configurations,\nbut in some cases ${10}^3{\\text{--}}2\\times{10}^3$ configurations are considered.\nThe kinetic Monte Carlo algorithm developed for these simulations is\nsimilar to that of previous works on submonolayer growth \\cite{submonorev,Gagliardi2022,Saito2012}.\n\nThe main quantity calculated here is the roughness of the grain edges.\nLetting $h\\left( x,t\\right)$ to denote the height of an edge at position $x$ and time $t$\nmeasured relatively to its initial position, the roughness is defined as\n\\begin{equation}\nW\\equiv {\\langle \\;{\\overline{{\\left( h-\\overline{h}\\right)}^2}}\\;\\rangle}^{1\/2} ,\n\\label{defW}\n\\end{equation}\nwhere the overbars denote a spatial average (over the values of $x$ in a given sample)\nand the angular brackets denote a configurational average (over different samples).\nDue to the symmetry of the grains A and B, our configurational average also includes\naveraging over the two grains of each growing sample.\n\nWe also measure the gap coverage, which is the fraction of the initial gap covered by deposited particles:\n\\begin{equation}\n\\theta = \\frac{N_d}{\\left( L\/a\\right)\\left( l\/a\\right)} ,\n\\label{deftheta}\n\\end{equation}\nwhere $N_d$ is the total number of deposited\nparticles.\nObserve that particle attachment and detachment do\nnot affect this quantity.\n\nTo characterize the transition from edge growth (when attachment dominates over detachment) to GB relaxation\n(when the grains evolve with balanced attachment and detachment),\nwe define an average collision time $t_c$.\nFor each position $x$ and a given configuration of evolving grains,\nthe local collision time $t_c^{\\left( local \\right)}\\left( x\\right)$ is defined as the first time in\nwhich grains A and B occupy NN sites at position $x$\n(with any value of $y$).\n Observe that the GB is not frozen after collision, so\nparticles may detach from\none of the grains, move to NN sites, or reattach to one of the\ntwo grains.\nThe value of $ t_c$ is obtained by averaging $t_c^{\\left( local \\right)}$ over all positions $x$\nand over different configurations of evolving grains; this averaging also\ngives the standard deviation of the collision times, $\\Delta t_c$.\n\n\n\\section{Results and Discussion}\n\\label{results}\n\n\\subsection{Morphological evolution}\n\\label{qualitative}\n\nFigs. \\ref{images}(a)-(c) show snapshots of the deposits grown with different values\nof the incident flux but with constant values\nof $D$, $Q$, and $\\epsilon$, which are\nset to $Q\/D={10}^{-1}$ and $\\epsilon=0.1$.\nThis is expected to mimic constant temperature conditions with variable precursor fluxes.\nThe growth times are shown in dimensionless units of $Dt$.\n\n\\begin{figure}\n\\center\n\\includegraphics[clip,width=\\textwidth,angle=0]{imageL10l6R6Q5ep01.png}\\\\\n\\includegraphics[clip,width=\\textwidth,angle=0]{imageL10l6R3Q2ep01.png}\\\\\n\\includegraphics[clip,width=\\textwidth,angle=0]{imageL10l6Rm1Qm2ep01.png}\n\\caption{\nSnapshots of parts of the deposits grown with $Q\/D={10}^{-1}$, $\\epsilon=0.1$, $L=1024a$, and $l=128a$:\n(a) $F\/D={10}^{-6}$; (b) $F\/D={10}^{-3}$; (c) $F\/D=10$.\nLight blue rectangles indicate the regions highlighted in Fig. \\ref{zoom}.\n}\n\\label{images}\n\\end{figure}\n\nFor the lowest flux ($F\/D={10}^{-6}$), Fig. \\ref{images}(a) shows that\nthe roughness of the growing edges is dominated by long wavelength fluctuations.\nThe mounds and valleys evolve from typical sizes $10a{\\text{--}}30a$ at $Dt=5\\times{10}^5$\nto sizes larger than $30a$ at $Dt=2\\times{10}^6$, but no significant change in the\nedge roughness is visible.\nThe particle density in the gap is very small until the grain collision;\nthis is confirmed by the magnified view in Fig. \\ref{zoom}(a) of the region highlighted\nin Fig. \\ref{images}(a).\nSimilar evolution is observed for larger diffusion coefficients of the particles in the gap,\ni.e. when $Q\/D$ and $F\/D$ decrease by the same factor, as shown in Sec. S.II\nof the Supplementary Material.\nThese features are representative of the attachment limited growth regime in this model.\n\nAs the gap is closed and a GB is formed, there is a drastic decrease of the\nboundary roughness.\nThis decrease is also observed in models of collisions of uncorrelated interfaces\nthat propagate with finite velocities and is independent of the detailed physics\nof their short range interactions \\cite{reispierrelouis2018}.\nAfter the collision, the GB evolves slowly by detachment of particles from\none edge followed by reattachment to the other, with apparently small changes in the roughness.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[clip,width=0.8\\textwidth,angle=0]{zoom.png}\n\\caption{\n(a), (b), and (c) show zooms of the regions indicated in Figs. \\ref{images}(a), \\ref{images}(b),\nand \\ref{images}(c), respectively.\n}\n\\label{zoom}\n\\end{figure}\n\nFor the intermediate value of the flux ($F\/D={10}^{-3}$), but the same ratio $Q\/D$,\nFig. \\ref{images}(b) shows spikes at the grain edges since early times.\nAt $Dt={10}^3$, a high particle density is already present in the gap. note that this time is\nmuch shorter than that of the first panel of Fig. \\ref{images}(a).\nThe magnified view in Fig. \\ref{zoom}(b) shows that the particle density is depleted only\nnear the valleys of the grain edges.\nFor this reason, the spikes grow faster than the valleys, as noticeable by comparing \nthe snapshots at $Dt={10}^3$ and $2\\times{10}^3$ in Fig. \\ref{images}(b).\nThis is characteristic of unstable growth in a diffusion-limited regime.\n\nThe spikes that result from the unstable growth\nare particular features of this model.\nThey contrast with the fractal\nor dendritic shapes observed in experiments and in other models\nwith unstable growth \\cite{xuNatNano2016,momeni2020}.\nThis difference is a consequence of the condition that particle attachment\noccurs only at the topmost site at each $x$, which prevents the lateral attachment\nresponsible for more complex morphologies.\n\nWhen the gap is completely filled, the spikes remain, so the GB roughness is initially large.\nAt $Dt=2\\times{10}^4$, Fig. \\ref{images}(b) shows that the spikes are transformed to shorter\nand thicker structures.\nA smooth GB is achieved at $Dt={10}^5$, which is a much shorter time than that of\nthe attachment-limited mode shown in Fig. \\ref{images}(a).\n\nFig. \\ref{images}(c) shows results for much higher flux, $F\/D=10$, again with the same $Q\/D$.\nNow the particle attachment is very slow compared to the incoming flux.\nAt a short time, $Dt=5\\times{10}^1$, the gap is fully covered\n[see the magnified view in Fig. \\ref{zoom}(c)] and so it remains until the two edges\ncollide and form a GB.\nThis contrasts with the diffusion-limited regime, in which the density was depleted\nnear the edge valleys.\nIn the present situation, the mobile particles are effectively static.\nThe attachment to the grain edges is consequently uncorrelated since short times because\nall points of the edge (at valleys or crests) are in contact with one of those particles.\nFig. \\ref{images}(c) also shows edges with spikes, but they are thinner and less\nprotuberant than those of the diffusion-limited regime [Fig. \\ref{images}(b)].\n\nThe GB is formed at $Dt\\sim{10}^3$ with a non-negligible roughness.\nThe relaxation of the GB leads to its smoothening.\nThis process is comparably faster than that reported in Fig. \\ref{images}(b), although the only relevant parameters, $Q$ and $\\epsilon$, are the same.\nIndeed, at $Dt=2\\times{10}^4$, the surface roughness \nin Fig. \\ref{images}(c) is much smaller than that observed\nin Fig. \\ref{images}(b).\n\nIn Sec. S.II of the Supplementary Material, we show that similar morphologies are\nobtained if the ratio $Q\/D$ decreases by a factor ${10}^2$, corresponding to a\nlarger particle diffusivity.\nA full coverage of the gap is also attained at short times\nfor a high value of the flux, which corresponds to the\nsame ratio $F\/Q={10}^2$ of Fig. \\ref{images}(c).\n\nThus, an apparently different growth regime is observed when the precursor flux is sufficiently high to produce full surface coverage\nbefore a significant edge displacement.\nIn this high coverage regime, the GB may have an initially large roughness,\nbut it attains smooth configurations in shorter times than those observed in diffusion-limited and attachment-limited regimes.\n\n\n\\subsection{Roughness, coverage, and edge collision time}\n\\label{roughnessevolution}\n\nFig. \\ref{roughness}(a) shows the time evolution of the roughness $W$ of \nthe grain edges for $Q\/D={10}^{-1}$, $\\epsilon=0.1$, initial gap width $l=64a$,\nedge length $L=512a$, and several values\nof $F\/D$.\nFor most of the flux values, Fig. \\ref{roughness}(a) also indicates\nthe typical ranges of collision times, which indicate the GB formation.\nFig. \\ref{roughness}(b) shows the coverage $\\theta$ as a function of time for the same\nparameter sets.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[clip,width=\\textwidth,angle=0]{wpa1-eps-converted-to.pdf}\\\\\n\\caption{\n(a) Roughness evolution for constant $Q\/D={10}^{-1}$ and variable flux.\nDashed lines with the indicated slopes are drawn for comparison with the slopes of the plots.\nHorizontal bars at $W\/a\\lesssim1$ indicate the edge collision times\n(average plus and minus one standard deviation).\n(b) Evolution of the gap coverage for the same\nparameter sets, with the same color code of (a).\n}\n\\label{roughness}\n\\end{figure}\n\nIn all cases, $W$ increases \nduring edge growth,\nas expected for any kinetic roughening process starting from a flat configuration\n\\cite{barabasi,krug}.\nIn most cases, $W$ attains a maximal value near the average collision time\n(the only exception is the case with the lowest flux, in which $W$ seems to\nsaturate after the initial increase).\nThat result parallels the previous observation of a decrease of the roughness during\nthe collision of interfaces propagating with constant velocities, independently\nof their short range interactions \\cite{reispierrelouis2018}.\nThe smoothening or roughening processes that take place after the formation\nof the GB depend on the flux.\n\nThe lowest flux in Fig. \\ref{roughness}(a), $F\/D={10}^{-6}$, \ncorresponds to the snapshots in Fig. \\ref{images}(a). The plot shows that\nthe initial near-equilibrium growth occurs with $W\\sim t^{1\/4}$.\nThis is the roughening described by the Edwards-Wilkinson (EW) equation\n\\cite{ew} at coarse grained scale, in which the surface tension and an uncorrelated noise\nare the main mechanisms.\nThe relative fluctuation of the collision time is not large, as inferred by the\nwidth of the indicative bar in Fig. \\ref{roughness}(a).\nAfter the GB formation, there are small changes in the roughness, which indicate that\nthe GB attains a stationary regime.\n\nThe stationary regime can actually be mapped to a true thermodynamic equilibrium state,\nin which the roughness depends only on the detachment probability\n$\\epsilon$ and on the edge length $L$;\nsee details in Sec. S.III of the Supplementary Material.\nFor all values of the model rates ($F$, $D$, and $Q$) and of the initial gap width,\nthe roughness is expected to converge to the same stationary value at sufficiently long times.\nThus, depending on the maximal roughness attained before the grain collision,\n$W$ may increase or decrease after formation of the GB.\nFor instance, Fig. \\ref{roughness}(a) shows that  $W$ increases towards the stationary value\nfor $F\/D={10}^{-5}$ and $F\/D={10}^{-4}$;\nhowever, Sec. S.IV of the Supplementary Material shows that $W$\ndecreases towards the stationary value when the gap width is smaller.\n\nFor the intermediate flux, $F\/D={10}^{-3}$, Fig. \\ref{roughness}(a) shows that\nthe initial increase of the roughness is faster than linear;\nit corresponds to the snapshots in Fig. \\ref{images}(b).\nThe explosive increase of height fluctuations confirms the unstable growth which is\nexpected in a diffusion-limited regime.\nThe range of collision times is broader in this case, as a consequence of the \nlarge dispersion in the local heights of the two edges.\nWhen the edges collide,\n$W$ decreases and can reach much smaller values as the GB relaxes.\nAfter reaching a minimal value, $W$ increases again\n(it is expected to converge to the stationary value at long times).\nIn previous studies of sudden changes from a first kinetics that\nproduces faster roughening to a second kinetics that produces slower\nroughening, similar results were obtained,\nviz. rapid smoothening after the change and subsequent slow roughening\n\\cite{assisreis2014,smoothening2015}.\n\n\nFor the largest flux ($F\/D=10$),\nthe roughness in Fig. \\ref{roughness}(a) approximately\nincreases as $W\\sim t^{1\/2}$;\nthe corresponding snapshots are in Fig. \\ref{images}(c).\nThis is consistent with random uncorrelated\ndeposition at both grain edges \\cite{barabasi}.\nFig. \\ref{roughness}(b) shows that full substrate coverage is attained\nlong before the collision time, in contrast with\nthe other regimes, in which the coverage is $1$ only at times near the collision.\nThis confirms that the separated grain edges grow\nwith a fully covered gap.\nThe maximal $W$ is also attained as the edges collide and the subsequent\ndecrease is associated with the GB formation.\nHowever, the range of collision times is small compared to the unstable regime.\nAfter the GB is formed, the roughness attains a minimum value which is smaller\nthan the values obtained\nin the near-equilibrium and in the diffusion-limited regimes.\nMoreover, this mininum value is attained\nin a shorter time compared to those regimes,\nas previously suggested by the snapshots of Figs. \\ref{images}(a)-(c).\n\nA common feature of the regimes limited by attachment and by diffusion\nis that the time of GB formation (the average collision time) is roughly\nproportional to the reciprocal of the flux rate:\n$t_c\\sim3\/F$.\nThe initial gap width and the mechanisms of particle diffusion, attachment, and detachment\ndetermine the edge morphology.\nHowever, in the high coverage regime (obtained with the highest fluxes),\nthe time of GB formation depends on the rate $Q$ but not on the flux $F$; \nfor instance, compare the data for\n$F\/D=1$ and $F\/D=10$ in Fig. \\ref{roughness}(a).\nAfter the gap is (rapidly) filled, the attachment rate $Q$ controls the edge propagation,\nwith negligible effect of the diffusion rate because the particle positions are effectively\nfrozen.\nThus, the collision time is inversely proportional to $Q$ and proportional to the gap width $l$;\nsee details in Sec. S.V of the Supplementary Material.\n\nBetween the three regimes described above, the roughness evolution shows crossover\nfeatures.\nIn Fig. \\ref{roughness}(a), $F\/D={10}^{-5}$ and ${10}^{-4}$\nrepresent the crossover between the attachment-limited and the diffusion-limited\nregimes, whereas $F\/D={10}^{-2}$ and ${10}^{-1}$ represent the crossover\nfrom the latter to the high coverage regime.\n\nQualitatively similar results are obtained for other ratios $Q\/D$. \nFig. \\ref{roughplus}(a) shows the roughness evolution for several values of the flux $F$\nwhen $Q\/D={10}^{3}$, which represents systems with very slow surface diffusion in\ncomparison with the attachment and detachment rates; the other parameters remain the\nsame as in Fig. \\ref{roughness}(a) and the time scale of diffusion is also used\nfor the data presentation.\nThe main difference here is that the diffusion-limited (unstable) regime spans\na broader range of $F$, since the slow diffusion favors the particle\ndepletion near the edge valleys.\nFor this reason, the attachment-limited regime\nis not observed in Fig. \\ref{roughplus}(a).\nHowever, or $F\/D={10}^4$ (corresponding to $F\/Q=10$),\nthe features of the high coverage regime are also observed:\nthe edges initially show the $W\\sim t^{1\/2}$ scaling of uncorrelated growth,\nwhile the coverage has already reached the full\nvalue $1$, as shown in Fig. \\ref{roughplus}(b);\nafter the GB formation, $W$ reaches a minimal\nvalue smaller than those attained in the other regimes, and this occurs at a shorter time.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[clip,width=0.95\\textwidth,angle=0]{wpb1-eps-converted-to.pdf}\\\\\n\\includegraphics[clip,width=0.95\\textwidth,angle=0]{wpc1-eps-converted-to.pdf}\n\\caption{\n(a), (c) Roughness evolution for constant $Q\/D$ and variable flux.\nDashed lines with the indicated slopes are drawn for comparison with the slopes of the plots.\nHorizontal bars at $W\/a\\lesssim1$ indicate the edge collision times\n(average plus and minus one standard deviation).\n(b) Coverage evolution for the same parameters in (a), with the same color code.\n}\n\\label{roughplus}\n\\end{figure}\n\nMuch larger diffusion coefficients (compared to the attachment rate) facilitate\nthe particle redistribution in the gap and permit the observation of the \nattachment-limited regime, but suppress the unstable regime (diffusion-limited).\nHowever, for the largest flux rates, the features of\nthe high coverage regime are still observed because\nthe diffusion is not effective in this case.\nThis is illustrated in Sec. S.VI of the Supplementary Material for $Q\/D={10}^{-3}$.\n\nChanges in the gap width $l$ affect the crossover between the diffusion-limited \nand the attachment-limited regimes, but do not change the features of the very\nhigh flux regime.\nFor smaller $l$, the unstable growth may not have sufficient time to develop, so\nthe EW roughening of the edges is observed for slightly larger fluxes; see Sec. S.IV of the Supplementary Material.\nInstead, for larger $l$, the unstable regime extends to smaller fluxes because\nthe instability has longer time to develop before the edges collide.\n\nChanges in the detachment probability $\\epsilon$ (which relates the detachment rates\nwith the number of NNs) lead to some relevant changes in the roughness evolution, as shown in \nFig. \\ref{roughplus}(c) for $\\epsilon=0.05$\nand the other parameters kept the same as in\nFig. \\ref{roughness}(a).\nFirst, a smaller $\\epsilon$ pushes the diffusion-limited regime to lower fluxes\nbecause the detachment kinetics of the edges is slower, but the attachment does not change.\nFor this reason, a clear EW scaling of the roughness\nis observed only for $F\/D={10}^{-7}$\n[in contrast with $F\/D={10}^{-6}$ for $\\epsilon=0.1$; Fig. \\ref{roughness}(a) ].\nSecond, for the intermediate and the largest fluxes,\nthe smaller detachment rates leads to slower\ndecays of the roughness after the GB formation.\nDesp\u00edte these differences, the high coverage regime obtained\nwith the highest flux ($F\/D=10$) still provides the smallest\nGB roughness during the relaxation.\n\nFor the largest flux in Fig. \\ref{roughplus}(c) ($F\/D=10$), a nontrivial\nfeature is the appearance of shoulders in the decay of the GB roughness.\nThe decrease of the roughness upon the collision is a stochastic effect characteristic of\ninterface collisions \\cite{reispierrelouis2018}.\nAfter that, the roughness decreases by the elimination of the \nthinnest spikes of the GB, which\nrequires the detachment of particles with\n$n=1$ NN, whose rate is $Q\\epsilon$.\nFor small $\\epsilon$, this rate is much smaller than that of the initial growth ($Q$);\nthe first shoulder is a consequence of the longer time scale associated to this process.\nFurther relaxation of the GB depends on kink detachment\n(see Sec. S.III of the Supplementary Material), whose rate is $Q\\epsilon^2$.\nThis is smaller than the rate of spike elimination by the factor $\\epsilon$,\nwhich explains the second shoulder.\nThese features become more pronounced for smaller $\\epsilon$, as shown in Sec. S.VII\nof the Supplementary Material;\nhowever, the high coverage\nregime still provides the minimal roughness in a shorter time\nthan the other growth regimes.\n\n\n\\subsection{Phase diagram of grain edge growth}\n\\label{phasediagram}\n\nIn our simulations, we identify three regimes of edge growth, i.e. before the edges begin to\ninteract.\nThe attachment-limited or near-equilibrium regime is charaterized by slopes of the\n$\\log{W}\\times\\log{\\left( Dt\\right)}$ plots near the EW value $1\/4$.\nThe diffusion-limited or unstable regime is characterized by slopes larger than $1$ at some time interval before the maximal $W$.\nFinally, the very high flux or random growth regime is characterized by slopes near $1\/2$.\nOther behaviors are considered as crossovers.\nFig. \\ref{diagram} shows the phase diagram obtained with these criteria for $\\epsilon=0.1$\nand gap widths $l=32a$ and $64a$.\nFor 2D materials such as graphene or metal dichalcogenides, the lattice constant is  $\\sim0.3$~nm, so those values correspond to widths $\\sim10{\\text{--}}20$~nm.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[clip,width=0.45\\textwidth,angle=0]{diagram-eps-converted-to.pdf}\n\\caption{\nGrain growth regimes as function of the kinetic parameters for $\\epsilon=0.1$ and two gap widths.\n}\n\\label{diagram}\n\\end{figure}\n\nFor very slow attachment-detachment kinetics in comparison with diffusion (very small $Q\/D$),\nthe unstable growth is not observed because the mobile particles are uniformly distributed\nin the gap.\nThe unstable growth is typically observed for $F\/Q\\lesssim0.1$\n(to avoid the rapid uncorrelated growth),\n$F\/D\\gtrsim{10}^{-3}$ (to avoid very slow deposition, which favors near-equilibrium growth),\nand $Q\/D\\gtrsim1$ (fast interface kinetics compared to diffusion).\n\nHowever, these conditions change as the gap width $l$ or the detachment probability $\\epsilon$ change.\nFor wider gaps, the instability has longer time to develop before the edge collision,\nso it can be observed for smaller values of $F\/D$.\nAs $\\epsilon$ decreases, particle detachment becomes slower, which also favors the instabilities;\nthus, the unstable regime can also be observed\nfor smaller values of $F\/D$.\n\nThe high coverage regime is more robust against changes in the physical or\nchemical parameters.\nIt typically appears for $F\/Q\\gtrsim10$, which is a condition in which the gap is almost\ncompletely filled before the attachment of a single layer of particles at each edge.\nAs explained before, the diffusion coefficient $D$ is not important in this regime\nbecause the particles in the gap cannot move to the occupied NNs.\n\n\n\\subsection{The minimal GB roughness}\n\\label{minimization}\n\nIn the simulations of growth with high precursor coverage ($F\/Q\\gtrsim10$),\nwe measured the minimal value of the roughness obtained during the GB\nrelaxation, $W_{min}$, and the time in which this minimum was attained, $t_{min}$.\n\n$W_{min}$ does not depend on the parameters\n$F$, $Q$, $D$, and $L$; it is affected only by the detachment rate\n$\\epsilon$ and the gap width $l$.\nFig. \\ref{Wmin}(a) shows $W_{min}\/a$ as a function of the scaling variable\n$\\epsilon l\/a$, i.e. the dimensionless product of the above parameters.\nSince the time of the minimal roughness rapidly increases as $\\epsilon$ decreases and \nthe simulations become slower as $l$ increases,\nwe considered restricted ranges for those variables\n($0.05\\leq\\epsilon\\leq0.1$ and $16\\leq l\\leq64$) to obtain accurate estimates of $W_{min}$.\nThe linear fit in Fig. \\ref{Wmin}(a) gives the relation\n\\begin{equation}\nW_{min} \\approx 0.87{\\left(\\epsilon a^3 l\\right)}^{0.25} .\n\\label{Wminscaling}\n\\end{equation}\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[clip,width=0.45\\textwidth,angle=0]{Wmin-eps-converted-to.pdf}\\\\\n\\includegraphics[clip,width=0.45\\textwidth,angle=0]{tmin-eps-converted-to.pdf}\n\\caption{\n(a) Minimal roughness in the high coverage regime\nas a function of $\\epsilon l\/a$.\nThe dashed line is a least squares fit with slope $0.25$.\n(b) Scaled time of the minimal roughness as a function of $l\/\\left(a\\epsilon^3\\right)$.\nThe dashed line is a least squares fit with slope $0.16$.\n}\n\\label{Wmin}\n\\end{figure}\n\n\nThe dependence of $W_{min}$ on $l$ can be explained by simple scaling arguments\n\\cite{assisreis2014,smoothening2015}.\nFrom the initial flat edge condition to the edge collision,\neach of the grains is displaced by an average length $l\/2$;\nthus, the average number of particles attached to each column $x$ of a grain is $N=(l\/2)\/a$.\nThe uncorrelated growth in this regime leads to a maximal roughness of order \\cite{barabasi}\n\\begin{equation}\nW_{max}\\sim a\\sqrt{N}=\\sqrt{\\frac{al}{2}} ;\n\\label{Wmax}    \n\\end{equation}\nthe characteristic rate of this growth is $Q$.\nAs the GB is formed, there is a sudden change from this uncorrelated growth to a correlated kinetics.\nIf the characteristic rate of attachment and detachment of the final kinetics is\nthe same as the initial one,\nthe uncorrelated fluctuations are rapidly suppressed\nand the roughness  rapidly reaches the same time evolution of the final kinetics starting from a flat interface\n(see e.g. Fig. 6 of Ref. \\protect\\cite{assisreis2014});\nhere, this would be the case of $\\epsilon\\sim1$.\nThe GB kinetics is EW (Sec. S.III of the Supplementary Material),\nin which the growth from a flat interface leads to $W\\sim aN^{1\/4}$,\nwhere a prefactor of order $1$ is excluded \\cite{ew,barabasi}.\nThe minimal roughness is then obtained by substituting the value of $N$ at the collision\ntime in this relation, which gives $W_{min}\\sim l^{1\/4}$.\n\nHowever, to describe the GB relaxation with smaller $\\epsilon$, a more powerful approach has to be used.\nWe developed a Langevin model \\cite{PierreLouis1998}\nwhich assumes that the GB evolution is controlled by kink site detachment\nfrom one grain (with characteristic rate $Q\\epsilon^2$)\nand attachment to the other grain.\nIn Sec. S.III of the Supplementary Material, this model is solved for an initial surface\nconfiguration with uncorrelated roughness $W_{max}$ and predicts\n\\begin{equation}\nW_{min}^{\\left(\\text{Lang}\\right)}={\\left[\\frac{1}{\\pi(1-\\epsilon)^2}\\right]}^{1\/4}\n{\\left(a^3l\\epsilon\\right)}^{1\/4} .\n\\label{WminL}\n\\end{equation}\nSince $1-\\epsilon\\approx 1$ is a reasonable approximation in our simulations, Eq. (\\ref{WminL})\nleads to the same scaling in $l$ and $\\epsilon$ of Eq. (\\ref{Wminscaling});\nthe approximation gives a prefactor $0.75$,\nwhich is not distant from the numerical estimate $0.87$\nin Eq. (\\ref{Wminscaling}).\n\nThe Langevin approach also predicts that the minimum roughness is attained at\n\\begin{equation}\n    t_{min}^{\\left(\\text{Lang}\\right)}=\\frac{l}{2aQ}\\left[1+\n    \\frac{(1-\\epsilon^2)}{8\\epsilon^3}\\right] ,\n\\label{tminL}\n\\end{equation}\nwhere the first term accounts for the uncorrelated growth of the edges of independent grains\nand the second one accounts for the coarsening of the GB.\nFor $\\epsilon\\ll1$, the first term is negligible in comparison with the second one,\nwhich gives $t_{min}^{\\left(\\text{Lang}\\right)}\\approx l\/\\left( 16aQ\\epsilon^3\\right)$; this is expected to be a reasonable\napproximation even for $\\epsilon=0.1$, which was considered in most simulations.\nGuided by this analytical result, Fig. \\ref{Wmin}(b) shows $Qt_{min}$ as a function \nof $l\/\\left(a\\epsilon^3\\right)$, which reasonably fits a straight line.\nThe slope of that fit is $0.16$, which is larger than, but of the same order of the theoretical slope $1\/16$.\n\n\n\\subsection{Possible application to 2D materials growth}\n\\label{theoretical}\n\nHere we discuss the application of the growth in the high coverage regime to produce 2D materials with small GB roughness.\n\nEqs. (\\ref{Wminscaling}) and (\\ref{WminL}) show that decreasing the growth temperature\n(so that $\\epsilon$ decreases) is advantageous for obtaining a smaller GB roughness.\nHowever, Eq. (\\ref{tminL}) shows that it is disadvantageous for increasing the time\nto attain this minimal roughness because $Q$ and $\\epsilon$ decrease.\nEvidence on this disadvantage is provided by the comparison \nof Figs. \\ref{roughness}(a) ($\\epsilon=0.1$) and \\ref{roughness}(c) ($\\epsilon=0.05$) with\nFig. S.6 of the Supplementary Material ($\\epsilon=0.01$).\nThe advantage involves a factor $\\epsilon^{1\/4}$ and the disadvantage involves\na factor $1\/\\left(Q\\epsilon^3\\right)$, so we generally expect that higher temperatures will be more\nfavorable for obtaining smoother GBs.\nThis implies that $\\epsilon$ and $Q$ are large.\nHowever, the condition $F\/Q\\gtrsim10$ is also necessary to\nallow the initial uncorrelated growth of the separated grain edges (Sec. \\ref{phasediagram}).\n\nFor the application to a particular 2D material, it is necessary that isolated grains nucleate\nand grow to a configuration with smooth boundaries, which is the initial condition\nconsidered here;\nobserve that the formation of this initial configuration is not addressed by our model.\nIn such configuration, a common situation is that the grains cover a fraction of the\nsubstrate $\\lesssim0.1$.\nThe inspection of microscopy images with visible and isolated islands (grains)\nof 2D materials supports this estimate\n\\cite{dongAM2019,yaoACSNano2020,caiNanoRes2021,dongJPCL2021,hanNanoAdv2021,weiNanoRes2021}.\nInstead, with coverages $\\sim1$, coalescence of several islands may be observed.\nThus, if the separated grains have typical boundary length $L$, the gap width $l$ is expected to\nbe of a similar order of magnitude; for simplicity, the approximation $l\\approx L$ is considered here.\n\nFollowing the reasoning in the paragraphs above,\nwe also assume that:\nedge lengths $L$ and initial gap widths $l$ are $\\sim10\\mu{\\text m}$;\nthe lattice constant of the material is $a\\approx 0.3$~nm\n(which differs $\\lesssim20\\%$ from graphene and several metal dichalcogenides);\n$\\epsilon=0.1$, meaning kink detachment rate $10^2$ times smaller than\nthe attachment rate; the condition $F\/Q\\leq10$ is fulfilled.\n\nFrom Eq. (\\ref{Wmax}), the maximal roughness of the grain edges in these conditions is $\\sim40$~nm.\nHowever, Eq. (\\ref{Wminscaling}) predicts that the minimal roughness attained by the GB\nis $\\sim2$~nm, corresponding to a decrease by a factor $\\sim20$.\nThe time to attain the minimal roughness can be determined from Eq. (\\ref{tminL}) in terms of the\nattachment rate: $Qt_{min}\\sim4\\times{10}^6$.\n\nFor comparison, if near-equilibrium growth conditions are chosen and the GB relaxes\nto its equilibrium configuration, the approach of Sec. S.III of the Supplementary Material\npredicts the GB roughness\n\\begin{equation}\n W_{eq}=\\sqrt{\\frac{\\epsilon aL}{6}} \\left(\\frac{1}{1-\\epsilon}\\right) .\n\\label{Weq}\n\\end{equation}\nWith the above parameters, we obtain $W_{eq}\\sim 7$~nm, which is $3.5$ times larger than\nthe expected minimal roughness.\nThe time of crossover to the equilibrium roughness is obtained from the EW kinetics of the GB in\nSec. S.III of the Supplementary Material:\n\\begin{equation}\nQt_{eq}=\\frac{\\pi}{144}{\\left(\\frac{L}{a}\\right)}^2 \\frac{1}{\\epsilon^2}\n\\left(\\frac{1+\\epsilon}{1-\\epsilon}\\right) .\n\\label{teq}\n\\end{equation}\nIt gives $Qt_{eq}\\sim3\\times{10}^9$, which exceeds by three orders of magnitude\nthe time to obtain the minimal roughness in the high coverage regime.\n\nThis comparison clearly shows that the advantages of growing 2d materials in the high coverage\nregime can be extrapolated from our nanoscale simulations to more typical experimental conditions with\nmicroscale grains.\nThe decrease of the GB roughness up to the minimum can be observed in a much shorter time than that\nnecessary for the GB to reach its equilibrium value, which is larger than that minimum.\n\nOur model has the limitation that the mobile particle attachment is restricted to the\ntopmost point of each column $x$.\nWe noted that leads to unstable patterns that look\nvery different from those of experiments and of models for specific applications.\nHowever, we believe that this limitation does not have significant effect on our\nprediction of a minimal roughness for the high coverage regime.\nIf that condition for the attachment is relaxed, then the aggregation of mobile particles\nto the sides of the rough edges may lead to overhang formation.\nSuch overhangs will encapsulate mobile particles, but these\nparticles are effectively static in the high coverage condition.\nThus, these particles will eventually attach at the same position and fill the overhang.\nIt is possible that this process creates some lateral correlations along the edges,\nbut such fluctuations, which are correlated at very small scales, can be suppressed approximately in\nthe same way as the totally uncorrelated fluctuations considered here.\n\n\n\\section{Conclusion}\n\\label{conclusion}\n\nA model for the propagation of two grain edges of a 2d material\nand for the relaxation of the grain boundary formed after the collision of those edges was proposed.\nThe model considers simple kinetic rules for\nattachment and detachment of\nparticles from the edges and for particle diffusion while there is a gap between them.\nThis allowed to perform kinetic Monte Carlo (KMC) simulations for broad ranges of model\nparameters and investigate their consequences\non the geometry of the grain edges.\n\nBesides the well known near-equilibrium (attachment-limited) and unstable (diffusion-limited) growth regimes,\nwe observe the existence of a distinct third regime when the precursor flux is sufficiently \nhigh to fully cover the gap between the edges before they begin to grow.\nIn this high covered regime, an uncorrelated growth of the edges is observed until they collide.\nAlthough this may produce an initial grain boundary with large roughness, we show that\nthis boundary relaxes to configurations with\na roughness smaller than those of the other regimes in a shorter time.\nNumerical predictions of the minimal roughness and of the time to attain such configuration are\nin good agreement with a theoretical approach for the grain boundary relaxation of the model.\n\nTo investigate the possible extension of the above results to real grains of 2d materials,\nthe simulation results (obtained at nanoscale) were extrapolated to edges and gaps\nsizes $\\sim10\\mu{\\text m}$.\nIt shows that the high coverage regime can produce grain boundaries with minimal roughness $\\sim2$~nm,\nwhich is $\\sim3.5$ times smaller than the roughness obtained in a near-equilibrium, stationary state.\nMoreover, the time to obtain that minimal roughness is three orders of magnitude smaller\nthan the expected time for the grain boundary to reach that stationary state.\nThus, if the control of the growth conditions of a 2d material permits to reach an initial high\ncoverage condition and a relaxation dynamics as proposed here, it is possible that a novel route for efficient growth of that material with very smooth grain boundaries can be developed.\nRecent advances in microscopy methods allow accurate measurements of nanoscale fluctuations of GBs and may help such studies\n\\cite{annevelink2021,bokai2021}.\n\n\\section*{Acknowledgment}\nFDAAR acknowledges support from the Brazilian agencies CNPq (305391\/2018-6), FAPERJ \n(E-26\/110.129\/2013, E-26\/202.881\/2018), and CAPES (88887.310427\/2018-00 - PrInt),\nand from CNRS (France).\nOPL acknowledges support from CAPES (88887.369967\/2019-00).\n\n\n\\section*{Supplementary Material}\n\nThe Supplementary Material shows the possible temperature dependences of the model\nparameters, additional images of the evolution of the grain boundaries,\nadditional plots of the roughness evolution, the Langevin model for grain boundary dynamics,\nand the scaling of collision times in near-equilibrium and unstable regimes.\n\n\\vskip 1cm\n\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nDynamical systems are fundamental in a range of scientific fields including biology, chemistry, physics, medicine, and economics~\\cite{strogatz2018nonlinear}. Here, we consider dynamical systems of the form\n\\begin{equation}\n  \\label{eq_dyn_syst}\n  \\frac{\\partial u}{\\partial t} = F(u,\\lambda),\n\\end{equation}\nwhere $u \\in C^1([0,\\tau];U)$ is a solution, $\\tau$ is the time horizon, $U$ is a suitable Hilbert space of functions defined on a bounded domain $\\Omega \\subset \\mathbb{R}^d$, $d \\in \\mathbb{N}_+$, $\\lambda \\in \\mathbb{R}$ is a bifurcation parameter, and $F\\in C^1(U\\times \\mathbb{R};U)$ is a Fr\\'echet differentiable operator. Typical examples of operators $F$ are nonlinear partial differential equations (PDEs) such as e.g.~the Navier--Stokes equations. Steady state (or equilibrium) solutions of \\cref{eq_dyn_syst} satisfy\n\\begin{equation}\n  \\label{eq_steady_system}\n  0 \\equiv \\frac{\\partial u}{\\partial t} = F(u,\\lambda).\n\\end{equation}\n\nThe properties of the system \\cref{eq_dyn_syst} depend on the value of the bifurcation parameter $\\lambda$. For instance, the number of steady-state solutions to \\cref{eq_dyn_syst} can vary with the bifurcation parameter $\\lambda$ through the birth of branches of solutions $(u, \\lambda)$ at specific branching points $(u^\\star, \\lambda^\\star)$ in the bifurcation diagram. Points $(u^\\star, \\lambda^\\star)$ at which steady-state solutions change stability and a periodic solution appears or disappears are known as Hopf bifurcations. By definition, this periodic solution $u$ will satisfy\n\\begin{equation}\n  u(x,t+T) = u(x,t) \\quad \\text{for } x \\in \\Omega, t \\geq 0\n\\end{equation}\nfor some minimal period of oscillation $T>0$. \\cref{fig_Hopf} depicts an example of bifurcation diagram of a dynamical system where a branch undergoes a Hopf bifurcation, along with an illustration of the steady-state and periodic solution at the Hopf bifurcation point. Hopf bifurcations are typically of substantial importance in physical and biological systems; in this manuscript we are interested in how and to what extent properties of Hopf bifurcation points can be controlled automatically via numerical optimization.\n\n\\begin{figure}[ht!]\n\\centering\n\\vspace{0.4cm}\n\\begin{overpic}[width=\\textwidth]{Figure\/bifurcation.pdf}\n\\put(0,26){(a)}\n\\put(35,26){(b)}\n\\put(70,26){(c)}\n\\end{overpic}\n\\caption{Illustration of a bifurcation diagram where a branch of transient solutions (dashed line) bifurcates from a steady-state through a Hopf bifurcation (a). The steady-state and periodic solution at the Hopf bifurcation point are respectively depicted in (b) and (c).} \n\\label{fig_Hopf}\n\\end{figure}\n\nIn this paper, we introduce a numerical method for controlling properties of Hopf bifurcations, such as their location in the bifurcation diagram or the period of the associated periodic solution, by optimizing auxiliary parameters of the model, including the domain $\\Omega$. Our algorithm allows us to advance or delay the birth of these periodic solutions using a characterization of Hopf bifurcations as the solutions to an extended system of nonlinear partial differential equations introduced by Griewank and Reddien~\\cite{griewank1983calculation}. This extends previous work~\\cite{boulle2021control} where we introduced a numerical technique for controlling simple bifurcation structures of physical systems with shape optimization utilizing the Moore--Spence system~\\cite{moore1980calculation,seydel1979numerical}.\nThe proposed algorithm can be applied to many systems of equations modelling sustained oscillations in physical and biological phenomena where Hopf bifurcations naturally arise, such as cardiac cell models~\\cite{erhardt2018bifurcation,erhardt2021complex}, the Hodgkin--Huxley model describing action potentials in neurons~\\cite{guckenheimer1993bifurcation,hodgkin1952quantitative,wang2005two,xie2008controlling}, the Selkov model of glycolysis~\\cite{sel1968self}, the Van der Pol oscillator~\\cite{van1927frequency}, spiral waves in reaction-diffusion systems~\\cite{barkley1990spiral}, models of the cardiac rhythms~\\cite{leloup1999limit,roenneberg2008modelling}, the migration of cancer cells~\\cite{bruckner2019stochastic}, and the von K\\'arm\\'an vortex street~\\cite{barkley2006linear,duvsek1994numerical,jackson1987finite,provansal1987benard,williamson1996vortex} in fluid dynamics.\n\nThe Griewank--Reddien system has been widely used in the literature to accurately localize Hopf bifurcations in various applications such as aeroelasticity, fluid dynamics, and crystals~\\cite{chen1997bifurcation,chen1995bifurcation,cliffe2000numerical,dellnitz1990computational,fortin1997localization,luzyanina1996numerical,morton1999hopf,winters1988oscillatory}. However, the majority of the works that aim to control the location and properties of Hopf bifurcations have considered applications to ordinary differential equations (ODEs), with a simple control parameter, and employed methods based on linear stability analysis, center manifold, and\/or normal form theory~\\cite{murdock2006normal} to find analytical relations~\\cite{abed1994stabilization,berns1998feedback,harb2003controlling,jiang2020hopf,li2004hopf,\nverduzco2006hopf,wang2020hopf,xie2008controlling,yu2004hopf}. One of the main limitations of these methods is that they are difficult to use to study nonlinear PDEs, especially when the control is more complex than a single real parameter, such as a material parameter field or the shape of the domain. In contrast, our approach can be automatically applied to many examples using a variety of optimization functionals and control parameters.\n\nWe have implemented our algorithm in the Firedrake finite element software~\\cite{rathgeber2016firedrake}, which interfaces with several libraries such as UFL~\\cite{alnaes2014unified}, PETSc~\\cite{balay2020petsc}, SLEPc~\\cite{hernandez2005slepc}, dolfin-adjoint~\\cite{farrell2013automated,mitusch2019dolfin}, ROL~\\cite{ridzal2017rapid}, and fireshape~\\cite{paganini2021fireshape} for discretizing variational formulations, solving sparse linear systems, solving eigenvalue problems, deriving adjoint models of nonlinear PDEs, solving optimization problems, and performing shape optimization, respectively. The code used to compute the numerical examples in this paper is publicly available on Zenodo for reproducibility purposes~\\cite{nicolas_boulle_2021_5235244}.\n\nThe paper is organized as follows. We begin in \\Cref{sec_charact_hopf} by describing the characterization of Hopf bifurcation points as solutions to an extended system of partial differential equations. We employ the formulation proposed by Griewank and Reddien~\\cite{griewank1983calculation}, which generalizes the Moore--Spence system used to characterize simple bifurcations and turning points~\\cite{moore1980calculation,seydel1979numerical}. Next, in \\Cref{sec_opt}, we describe the algorithm for performing numerical optimization and control of Hopf points with respect to a parameter of the dynamical system or the shape of the domain on which solutions are defined. We present a wide range of numerical examples and applications of the method in \\Cref{sec_app} before concluding and discussing further possibilities in \\Cref{sec_conc}.\n\n\\section{Characterization of Hopf bifurcations}\n\\label{sec_charact_hopf}\n\n\\subsection{Definitions of simple and Hopf bifurcation points}\n\nFor a dynamical system of the form~\\cref{eq_dyn_syst}, a \\emph{simple bifurcation point} is a solution pair $(u^\\star, \\lambda^\\star)$ satisfying \\eqref{eq_steady_system} with the property that the number of steady-state solutions to \\cref{eq_steady_system} in any neighbourhood of $(u^\\star, \\lambda^\\star)$ changes as $\\lambda$ passes $\\lambda^\\star$. At such points, the Fr\\'echet derivative $F_u$ of the operator $F$ with respect to $u$ is non-invertible, with a zero eigenvalue associated with a nonzero eigenfunction $\\phi$:\n\\begin{equation}\n  F_u (u^\\star, \\lambda^\\star) [\\phi] = 0 .\n\\end{equation}\nUsing this property, simple bifurcation points can be characterized using an extended system of equations called the Moore--Spence system~\\cite{moore1980calculation,seydel1979numerical}, in which the steady-state $u^\\star$, the bifurcation parameter $\\lambda^\\star$, and the eigenvector $\\phi$ are solved for simultaneously.\n\nFollowing Roose and Hlava\\v{c}ek~\\cite{roose1985direct}, we assume that there exists a smooth branch of steady-state solutions $(u, \\lambda)$, each satisfying \\cref{eq_steady_system}, for $\\lambda\\in(\\lambda^\\star-\\epsilon,\\lambda^\\star+\\epsilon)$ where $\\epsilon>0$, and that $u^\\star$ is an isolated solution to $F(u, \\lambda^\\star)=0$ belonging to the branch. Moreover, we assume that $F_u(u^\\star, \\lambda^\\star)$ has a single pair of complex conjugate imaginary eigenvalues $\\pm i \\mu$ for $\\mu \\in \\mathbb{R}_+$, and with associated eigenfunction $\\phi$:\n\\begin{equation}\n  \\label{eq_eig_hopf}\n  F_u(u^\\star, \\lambda^\\star)[\\phi] = i \\mu \\, \\phi .\n\\end{equation}\nThen, under additional regularity conditions on the operator $F$ (cf.~\\cite{roose1985direct} and the theoretical results in~\\cite{hassard1981theory,marsden2012hopf}), a \\emph{Hopf bifurcation} occurs at $\\lambda=\\lambda^\\star$, i.e., a branch of time periodic solutions bifurcates from the branch of steady-state solutions. \n\n\\subsection{The Griewank--Reddien equations}\n\\label{sec:griewank-reddien}\n\nIn this section, we consider the formulation of an extended system of equations to characterize Hopf bifurcations as proposed by Griewank and Reddien~\\cite{griewank1983calculation}, which we will next embed in an optimal control setting in \\Cref{sec_opt}. While alternative methods~\\cite{hassard1981theory,heinemann1981multiplicity,jepson1981numerical} permit the detection of Hopf bifurcations by solving eigenvalue problems during a bifurcation parameter continuation process, the Griewank--Reddien formalism offers a direct approach. One of the key difficulties of the characterization of Hopf points arises from the non-uniqueness of the eigenfunctions, and we will discuss this point before turning to the extended system.\n\nLet $(u^\\star,\\lambda^\\star)$ be a steady-state solution of \\cref{eq_steady_system} at a Hopf bifurcation point, and let $\\phi_0 = v_0 + i w_0$ be an eigenfunction satisfying \\cref{eq_eig_hopf}, with real and imaginary parts $v_0$ and $w_0$ respectively, corresponding to the imaginary eigenvalue $i \\mu$ for $\\mu > 0$. We observe that $\\phi_0$ is only unique up to multiplication by a complex number $z=re^{i\\theta}$, and the eigenvalue problem~\\cref{eq_eig_hopf} thus admits eigenfunctions of the form \n\\begin{equation}\n  \\phi = re^{i\\theta}\\phi_0 = r(v_0\\cos\\theta-w_0\\sin\\theta) + ir(v_0\\sin\\theta+w_0\\cos\\theta),\n\\end{equation}\nfor every $r>0$ and $\\theta \\in [0,2\\pi)$. To uniquely determine a solution to \\cref{eq_eig_hopf}, we augment the eigenvalue problem by a normalization condition. More precisely, we assume that there exists a suitable \\emph{normalization function} $c \\in U$ such that at least one of $\\langle c,v_0\\rangle \\neq 0$ or $\\langle c,w_0\\rangle\\neq 0$ holds, where $\\langle \\cdot, \\cdot\\rangle$ denotes the inner product of $U$. We aim to fix a solution $\\phi = v + i w$, $v, w \\in U$ to \\cref{eq_eig_hopf} such that\n\\begin{equation} \\label{eq_norm_condition}\n\\langle c, v \\rangle = 0 \\qquad \\text{and} \\qquad \\langle c, w \\rangle = 1.\n\\end{equation}\nClearly \\cref{eq_norm_condition} is satisfied for $v = r (v_0 \\cos \\theta - w_0 \\sin \\theta)$ and $w = r (v_0 \\sin \\theta + w_0 \\sin \\theta)$ if $r$ and $\\theta$ solve:\n\\begin{subequations} \\label{eq_norm}\n\\begin{align}\n\\langle c, v_0\\rangle \\cos\\theta - \\langle c, w_0\\rangle \\sin\\theta &= 0,\\\\\nr (\\langle c, v_0\\rangle \\sin\\theta + \\langle c, w_0\\rangle \\cos\\theta) &= 1.\n\\end{align}\n\\end{subequations}\nHence, given $c$ and a $\\phi_0$, \\cref{eq_norm} provides a normalization procedure to determine $r, \\theta$ to ensure that the corresponding eigenfunction $\\phi$ satisfies the conditions given by \\cref{eq_norm_condition}. \n\nThe Griewank--Reddien formalism combines the eigenvalue problem \\cref{eq_eig_hopf}, the real and imaginary components of the steady-state equation \\cref{eq_dyn_syst}, as well as the normalization conditions \\cref{eq_norm_condition} to read as follows~\\cite[Eq.~3.1]{griewank1983calculation}: find the solution field $u \\in U$, bifurcation parameter $\\lambda \\in \\mathbb{R}$, frequency $\\mu > 0$, and eigenfunction components $v, w \\in U$ such that\n\\begin{equation}\n  \\label{eq_ext_MS}\n  G(u, \\lambda, \\mu, v,w) =\n  \\begin{pmatrix}\n    F(u, \\lambda) \\\\\n    F_u(u,\\lambda) [v] + \\mu w \\\\\n    F_u(u,\\lambda) [w] - \\mu v \\\\\n    \\langle c, v \\rangle  \\\\\n    \\langle c, w \\rangle -1  \n  \\end{pmatrix}\n  = 0 .\n\\end{equation}\nIn the remainder of this paper, we will characterize Hopf bifurcations using the system \\cref{eq_ext_MS} and will refer to it as the Griewank--Reddien system. We will also exploit~\\eqref{eq_norm} separately to construct better initial guesses for the typically highly nonlinear system.\n\n\\begin{remark}\nIn place of the Griewank--Reddien system~\\cref{eq_ext_MS}, other direct formulations could equivalently be used, such as the system proposed by Roose and Hlava{\\v{c}}ek~\\cite{roose1985direct}:\n\\begin{equation*}\nR(u, \\lambda, \\mu, \\phi) =\n\\begin{pmatrix}\nF(u, \\lambda) \\\\\nF_u(u, \\lambda)^2[\\phi] + \\mu^2 \\phi\\\\\n\\langle c, \\phi \\rangle \\\\\n\\langle \\phi, \\phi \\rangle -1\n\\end{pmatrix}\n= 0 .\n\\end{equation*}\nHowever, the implementation of this Roose--Hlava{\\v{c}}ek system is more challenging in the finite element software Firedrake~\\cite{rathgeber2016firedrake}. Alternatively, one could also employ a standard shooting method~\\cite{waugh2013matrix} to find periodic solutions by solving a time-dependent system of equations to obtain the solutions and corresponding periods. While this method should allow the control of periodic solutions far from Hopf bifurcations, it requires a transient simulation for each functional evaluation and is therefore much more computationally expensive when employed in an optimization problem.\n\\end{remark}\n\n\\section{Numerical optimization of Hopf bifurcation points}\n\\label{sec_opt}\n\nA Hopf bifurcation point $(u, \\lambda)$ of the dynamical system\n\\cref{eq_dyn_syst} is associated with various properties, such as the\nlocation $\\lambda$ of the bifurcation parameter, the frequency of\noscillation of the emerging periodic solution branch $\\mu$, as well as\nother properties of the steady-state $u$ at the bifurcation. We target\ncontrolling these properties via an optimal control approach.\n\n\\subsection{An optimal control setting for Hopf bifurcations}\n\nAssume that a Hopf bifurcation point $(u, \\lambda)$ to the dynamical\nsystem \\cref{eq_dyn_syst} with frequency $\\mu > 0$ can be expressed as a\nfunction of a control variable $o$ (e.g.~material parameter,\nbifurcation location, frequency, domain shape etc.). For a given\nobjective functional\n\\begin{equation}\n  \\mathcal{J} = \\mathcal{J} (u, \\lambda, \\mu) = \\mathcal{J} (u, \\lambda, \\mu) (o),\n\\end{equation}\nwe consider the optimization problem constrained by the Griewank--Reddien system~\\cref{eq_ext_MS}:\n\\begin{equation}\n  \\label{eq_min_problem}\n  \\min_{o} \\, \\mathcal{J}(u, \\lambda, \\mu) \\quad \\text{subject to} \\quad G(u, \\lambda, \\mu, v, w) = 0.\n\\end{equation}\nThis general formulation allows for a range of types of optimal control applications such as illustrated by the following four examples.\n\\begin{example}[Controlling the location of a Hopf bifurcation]\nTo advance or delay the bifurcation parameter of a Hopf bifurcation point to a target value $\\lambda^\\star$, we consider the objective functional:\n\\begin{equation}\n  \\mathcal{J} = (\\lambda - \\lambda^\\star)^2 \/ \\lambda^{\\star 2} .\n\\end{equation}\n\\end{example}\n\\begin{example}[Controlling the frequency of the periodic solution]\n  To increase or decrease the frequency of oscillation of the periodic branch arising from the Hopf bifurcation to a target frequency $\\mu^\\star$, we consider the functional:\n  \\begin{equation}\n    \\mathcal{J} = (\\mu - \\mu^\\star)^2\/\\mu^{\\star 2} .\n  \\end{equation}\n\\end{example}\n\\begin{example}[Optimizing a parameter]\nTo control the Hopf bifurcation with respect to a scalar parameter $a \\in \\mathbb{R}$ of the PDE, we consider the control variable $o=a$.\n\\end{example}\n\\begin{example}[Optimizing the shape of the domain]\nTo control the Hopf bifurcation with respect to the shape of the domain $\\Omega$, we consider the control variable $o=\\Omega\\in \\mathcal{U}_{\\textup{ad}}$ where $\\mathcal{U}_{\\textup{ad}}$ is the set of images of an initial domain under a suitable set of diffeomorphisms~\\cite{paganini2021fireshape}.\n\\end{example}\n\n\\subsection{Optimization algorithm}\n\nTo minimize the functional $\\mathcal{J}$, we introduce an iterative optimization algorithm,\nsummarized in \\cref{alg_branch} and further described below.\n\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithm}\n  \\caption{Optimization of Hopf bifurcations}\n  \\label{alg_branch}\n  \\begin{algorithmic}[1]\n    \\Require Initial control variable $\\tilde{o}$, initial guess $(\\tilde{u}, \\tilde{\\lambda})$ for the Hopf bifurcation point, optimization functional $\\mathcal{J}$, normalization function $c$\n    \\Ensure Optimized control variable $o$\n    \\State Solve the eigenvalue problem \\cref{eq_eig_hopf} around $(\\tilde{u}, \\tilde{\\lambda})$ to generate initial ($\\tilde{\\mu}, \\tilde{v}, \\tilde{w})$\n    \\State Normalize the eigenfunctions $\\tilde{v}, \\tilde{w}$ by solving \\cref{eq_norm} with the given $c$\n    \\State Solve the system \\cref{eq_ext_MS} to obtain an initial solution $(u^{(0)}, \\lambda^{(0)}, \\mu^{(0)}, v^{(0)}, w^{(0)})$\n    \\State Initialize optimization step, $k \\gets 1$\n    \\While {termination criteria not satisfied}\n    \\State Evaluate the objective functional and compute updated control variable $o^{(k)}$\n    \\State Solve the Griewank--Reddien system to obtain $(u^{(k)}, \\lambda^{(k)}, \\mu^{(k)}, v^{(k)}, w^{(k)})$\n    \\If{the regularity conditions are satisfied}\n    \\State Accept optimization step, $k\\gets k+1$\n    \\Else\n    \\State Reject optimization step and decrease the step size\n    \\EndIf\n    \\EndWhile\n  \\end{algorithmic}\n\\end{algorithm}\n\nFirst, we solve the Griewank--Reddien system~\\cref{eq_ext_MS} to locate the Hopf bifurcation to be modified. This system of equations is highly nonlinear and, as a dynamical system may have several Hopf bifurcations, can have multiple solutions. As an example, the Ginzburg--Landau equation example in Section~\\ref{sec_GL} has an infinite number of Hopf bifurcation points. We use Newton's method to solve the Griewank--Reddien system, and therefore need a good initial guess to ensure convergence to the target Hopf bifurcation. To this end, we first use deflated continuation~\\cite{farrell2016computation,farrell2015deflation} to compute multiple steady-state solutions $u$ to \\cref{eq_dyn_syst} by continuation in the bifurcation parameter $\\lambda$. We analyze the (linear) stability of each steady-state $u$ found at each parameter $\\lambda$ by computing eigenvalues to the Fr\\'echet derivative of $F$ at $u$, $F_u(u,\\lambda)$, and tracking when one eigenvalue becomes purely imaginary as we increase $\\lambda$. We select the steady-state $(\\tilde{u}, \\tilde{\\lambda})$ and eigenfunction $\\phi$ with growth rate (real part of the eigenvalue) closest to zero. We define the guess frequency $\\tilde{\\mu} > 0$ to be the imaginary part of the corresponding eigenvalue, and denote by $v_0$ and $w_0$ the real and imaginary parts of $\\phi$, respectively. Last, we define the normalized functions to be\n\\begin{equation*}\n  \\tilde{v} = r(v_0 \\cos \\theta - w_0 \\sin \\theta), \\qquad\n  \\tilde{w} = r(v_0 \\sin \\theta + w_0 \\sin \\theta),\n\\end{equation*}\nwhere $r>0$ and $\\theta\\in [0,2\\pi)$ are solutions to \\cref{eq_norm} in alignment with the discussion in Section~\\ref{sec:griewank-reddien} (step 2 in \\cref{alg_branch}).\n\nOnce a suitable initial guess $(\\tilde{u}, \\tilde{\\lambda}, \\tilde{\\mu}, \\tilde{v}, \\tilde{w})$ has been computed, we solve~\\cref{eq_ext_MS} to obtain an initial Hopf bifurcation point $(u^{(0)}, \\lambda^{(0)}, \\mu^{(0)}, v^{(0)}, w^{(0)})$. We employ a trust-region algorithm~\\cite{conn2000trust} to solve the optimization problem formulated in \\cref{eq_min_problem} and minimize the functional $\\mathcal{J}(u, \\lambda, \\mu)$ (step 6--7 in \\cref{alg_branch}). Finally, we check whether certain regularity conditions on the mesh of the domain or the steady-state solution $u^{(k+1)}$ are satisfied to accept or reject the optimization step (cf.~\\cite[Sec.~4]{boulle2021control}). In particular, if $u^{(k)}$ is the steady-state solution at the previous step, we reject control updates that do not satisfy the following inequality, to ensure that the optimization remains on the same branch of solutions and does not jump to a secondary Hopf bifurcation:\n\\[\\|u^{(k+1)}-u^{(k)}\\|_U\\leq C\\|u^{(k+1)}\\|_U,\\]\nwhere $C>0$ is a specified constant. If the above inequality is not satisfied, then we reject the optimization step and decrease the trust-region radius. The optimization algorithm terminates when the functional value $\\mathcal{J}$ falls below a tolerance $\\epsilon>0$, or a maximum number of iterations or a gradient tolerance is reached.\n\n\\subsection{Discretization, solvers and software}\n\nThe nonlinear (partial) differential equation $F(u,\\lambda) = 0$ is discretized and solved using the Firedrake finite element software~\\cite{rathgeber2016firedrake} with efficient linear solvers from PETSc~\\cite{balay2020petsc}.  We solve the eigenvalue problems resulting from the linear stability analysis using the Scalable Library for Eigenvalue Problem Computations (SLEPc)~\\cite{hernandez2005slepc}, which is a library interfacing with PETSc for solving large scale eigenvalue problems. In particular, we use the Krylov--Schur algorithm with a shift-and-invert spectral transformation~\\cite{kressner2005numerical,stewart2002krylov}.  We leverage the Rapid Optimization Library (ROL)~\\cite{ridzal2017rapid} to solve the optimization problem formulated in \\cref{eq_min_problem} and minimize the functional $\\mathcal{J}(u, \\lambda, \\mu)$. Inside the optimization algorithm, the update of the control variable is computed using either the dolfin-adjoint library~\\cite{farrell2013automated,mitusch2019dolfin}, if the control is a parameter or a function, or the Fireshape optimization toolbox~\\cite{paganini2021fireshape} which relies on a moving mesh method~\\cite{allaire2006structural,paganini2018higher}, if the control is the shape of the domain. \n\n\\section{Applications}\n\\label{sec_app}\n\nIn this section, we employ the abstract method described in \\Cref{sec_opt} for controlling Hopf bifurcations to a wide range of concrete examples. The applications considered include the FitzHugh--Nagumo model simulating the evolution of action potentials in an excitable biological cell such as a neuron or myocyte, the complex Ginzburg--Landau equation used to understand phase transition, as well as the Rayleigh--B\\'enard convection problem, and the control of a von K\\'arm\\'an vortex street described by the Navier--Stokes equations.\n\n\\subsection{FitzHugh--Nagumo model}\n\nWe first consider a FitzHugh--Nagumo~\\cite{fitzhugh1961impulses} model. This dynamical system simplifies the Hodgkin--Huxley model~\\cite{hodgkin1952quantitative}, which describes the propagation of action potentials in neurons. This system of ordinary differential equations models the evolution of the transmembrane potential $v$, and a second dimensionless variable $w$ in a cardiac cell, and reads as:\n\\begin{subequations}\n  \\label{eq_FN}\n\\begin{align}\n\\frac{\\partial v}{\\partial t} &=c_1v(v-a)(1-v)-c_2 w, \\\\\n\\frac{\\partial w}{\\partial t} &= b(v-c_3 w).\n\\end{align}\n\\end{subequations}\nHere, $a=-0.12,\\,b=0.011,\\,c_1=0.15,\\,c_2=0.05,\\,c_3=0.55$ are given parameters which may be adjusted to model different type of cells~\\cite[Sec.~2.4.1]{sundnes2007computing}.\nThis formulation yields a normalized action potential with a zero resting potential and a peak around $0.9$. Note that a reparametrization of the model can be done to match physiological data~\\cite[Sec.~2.4.1]{sundnes2007computing}. \n\nIn this example, we select the parameter $c_1$ as the bifurcation parameter of the model, fix the remaining parameters except $c_2$, and remark that the system transitions to a time-dependent periodic solution to \\cref{eq_FN} through a Hopf bifurcation, located at a critical value of the parameter $c_1$. Then, we solve the Griewank--Reddien system~\\cref{eq_ext_MS} and obtain a Hopf bifurcation at the critical bifurcation parameter $c_1=0.05$, associated with the steady-state $(v,w)=(0,0)$. Moreover, using linear stability analysis, we find that this state is associated with a pair of imaginary eigenvalues of $\\pm i\\mu$, where $\\mu\\approx 2.23\\times 10^{-2}$, corresponding to a period of oscillations of $T=2\\pi\/\\mu\\approx 277$ ms, and eigenfunction $\\phi=v_h+iw_h$.\n\nWe aim to find the value of the cell parameter $c_2$ such that the action potential duration reflects the action potential duration of a cardiac cell of approx.~$T^\\star=400$ ms~\\cite{boron2016medical, qu2012mechanisms}; i.e., $\\mu^\\star=1.57\\times 10^{-2}$. We formulate this as the following PDE-constrained optimization problem:\n\\begin{equation} \\label{eq_MS_FN}\n\\begin{aligned}\n\\min_{c_2\\in\\mathbb{R}} \\quad &  \\mathcal{J}\\coloneqq (\\mu-\\mu^\\star)^2\/\\mu^{\\star 2}\\\\\n\\textrm{subject to} \\quad & G((v,w),c_2,\\mu,v_h,w_h) = 0,\n\\end{aligned}\n\\end{equation}\nwhich we solve with the algorithm presented in \\cref{sec_opt}, implemented in the Firedrake finite element software~\\cite{rathgeber2016firedrake} using the dolfin-adjoint library~\\cite{farrell2013automated}.\nUsing a trust-region algorithm implemented in ROL, we are able to minimize the functional $\\mathcal{J}$ in \\cref{eq_MS_FN} to machine precision in 7 iterations, and obtain an optimized value of $c_2\\approx 0.026$. This yields a Hopf bifurcation at $c_1\\approx 0.05$ with an associated steady-state of $(v,w)=(0,0)$. We then perform a linear stability analysis to verify that this solution possesses an pair of purely imaginary eigenvalues $\\pm i\\mu$, with $\\mu \\approx 1.57\\times 10^{-2}$, i.e.~an oscillation period of $T=400$ ms as desired.\n\n\\begin{figure}[ht!]\n\\centering\n\\vspace{0.4cm}\n\\begin{overpic}[width=\\textwidth]{Figure\/FN_second_model.pdf}\n\\put(0,62){(a)}\n\\put(33,62){(b)}\n\\put(67,62){(c)}\n\\put(0,31){(d)}\n\\put(33,31){(e)}\n\\put(67,31){(f)}\n\\end{overpic}\n\\caption{Initial (a) and optimized (d) periodic potential solutions $v$ to the FitzHugh--Nagumo equations~\\cref{eq_MS_FN} at the bifurcation parameter $c_1=0.05$. The dashed red lines show the steady-state solution $(v,w)=(0,0)$ for the respective values of the cell model parameters at the Hopf bifurcation. Panels (b) and (e) display a phase portrait of the variables $v$ and $w$ to illustrate the periodicity of the solution, along with a red dot for the steady-state solution. The periodic solution $v$ at $c_1=0.15$, away from the Hopf bifurcation, is plotted for the initial and optimized parameter $c_2$ in panels (c) and (f), respectively.} \n\\label{fig_time_dep_FN}\n\\end{figure}\n\nFinally, we employ an explicit Runge-Kutta method of order 5(4)~\\cite{dormand1980family} to solve \\cref{eq_MS_FN}, starting from the perturbed steady-state $(v,w)=(0.01,0)$, to observe the periodic solution to the FitzHugh--Nagumo equations around the Hopf bifurcation. The resulting periodic potential solutions $v$ for the initial and optimized cell parameters $c_1$ and $c_2$ are displayed in \\cref{fig_time_dep_FN}(a) and (d). In the two panels, we observe that the two solutions respectively oscillate with a period of $T=277$ ms and $T=400$ ms, as imposed by the optimization procedure described in this section. Additionally, we report a phase portrait of the variables $v$ and $w$ around the steady-state solution for both cases in \\cref{fig_time_dep_FN}(b) and (e). We find in \\cref{fig_time_dep_FN}(a) and (d) that the action potential variable $V$ is periodic at the Hopf bifurcation point with sinusoidal oscillations, which are not physiologically realistic. A more relevant solution is obtained by using a higher value of the bifurcation parameter $c_1=0.15$, i.e.,~away from the location of the Hopf bifurcation (see~\\cref{fig_time_dep_FN}(c) and (f)). Note that, in this case, the period of the periodic solution has changed and cannot be easily related to the period that we control using the\nGriewank--Reddien system. In general, controlling properties of solutions far from the Hopf bifurcation point is considerably more challenging and computationally expensive as it would require the performing of branch continuation within the optimization procedure.\n\n\\subsection{Complex Ginzburg-Landau equation} \\label{sec_GL}\n\nThe Ginzburg--Landau equation is a widely studied nonlinear equation used to describe and understand a wide range of physical phenomena and systems such as phase transitions, nonlinear waves, Bose--Einstein condensates, and liquid crystals~\\cite{aranson2002world,bohr1998dynamical,busse1998evolution,crosshohenberg93,kuramoto1984chemical,newell1993order,pismen1999vortices,scott2006encyclopedia}.\nThis models carries a number of features (dissipation, diffusion, dispersion) that can be controlled using the equation parameters to generate different patterns. From a bifurcation analysis viewpoint, this equation is interesting due to its richness in generating oscillatory or rotating patterns, such as spiral waves~\\cite{cross2009pattern,pismen2006patterns}, arising from Hopf bifurcations~\\cite{uecker2021numerical}. \n\nWe consider the complex Ginzburg--Landau (CGL) equation with cubic-quintic nonlinearity~\\cite{ginzburg1950theory}:\n\\begin{equation} \\label{eq_GL_complex}\n\\frac{\\partial u}{\\partial t}=\\Delta u+(r+i\\nu)u-(c_3+i\\mu)|u|^2 u-c_5|u|^4u,\\qquad u(x,t)\\in\\mathbb{C},\n\\end{equation}\ndefined on a spatial domain $\\Omega\\subset\\mathbb{R}^2$, with homogeneous Dirichlet boundary conditions. The parameters $\\mu$, $\\nu$, $c_3$, $c_5$ dictate the dynamics and number of solutions of the equation, and are set by default to $\\mu=0.1$, $\\nu=1$, $c_3=-1$, $c_5=1$~\\cite{uecker2019user}. The parameter $r\\geq 0$ plays the role of a bifurcation parameter for the system. \n\nWe first decouple \\cref{eq_GL_complex} into real and imaginary parts to obtain the following two-component system:\n\\[\\frac{\\partial}{\\partial t}\\begin{pmatrix}\nu_1\\\\ u_2\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n\\Delta+r & -\\nu \\\\\n\\nu & \\Delta+r\n\\end{pmatrix}\n\\begin{pmatrix}\nu_1\\\\ u_2\n\\end{pmatrix}\n-(u_1^2+u_2^2)\n\\begin{pmatrix}\nc_3 u_1-\\mu u_2\\\\ \\mu u_1+c_3 u_2\n\\end{pmatrix}\n-c_5(u_1^2+u_2^2)^2\n\\begin{pmatrix}\nu_1\\\\ u_2\n\\end{pmatrix},\\]\nwhere $u=u_1+iu_2$. If $\\Omega=[-l_1\\pi,l_1\\pi]\\times [-l_2\\pi,l_2\\pi]$, then it is known that the trivial branch $(u_1,u_2)=(0,0)$ has Hopf bifurcations located at $r=|k|^2\\coloneqq k_1^2+k_2^2$, where $k\\in \\mathbb{Z}\/(2 l_1)\\times \\mathbb{Z}\/(2 l_2)$, with an associated pair of imaginary eigenvalues $\\pm i\\mu_h = \\pm i\\nu$~\\cite{uecker2019user}. The remaining parameters do not influence the location of the Hopf bifurcations with respect to the bifurcation parameter $r$, which depend only on the geometry of the domain $\\Omega$ such as its aspect ratio. In this example, we choose the domain $\\Omega=[-\\pi,\\pi]\\times [-\\pi\/2,\\pi\/2]$, giving the first two Hopf bifurcation points from the trivial branch at $r=5\/4$ and $r=2$. To demonstrate the ability of our method to control secondary Hopf bifurcation points in the diagram, provided the initial guess for the Griewank--Reddien system~\\cref{eq_ext_MS} is sufficient close to the Hopf point, we focus on the second Hopf bifurcation arising at $r=2$. The two components of the real part of the eigenfunction associated with the pair of imaginary eigenvalues $\\pm i$ are illustrated in \\cref{fig_GL_opt}(a) and (b). Then, a perturbation of the trivial branch in this direction gives birth to a periodic solution, whose components oscillate between the modes displayed in \\cref{fig_GL_opt}(a) and (b) with a period $T = 2\\pi$.\n\n\\begin{figure}[ht!]\n\\centering\n\\vspace{0.4cm}\n\\begin{overpic}[width=\\textwidth]{Figure\/GL_figure.pdf}\n\\put(0,61){(a)}\n\\put(0,30){(b)}\n\\put(62,61){(c)}\n\\put(62,30){(d)}\n\\end{overpic}\n\\caption{The normalized two components (a-b) of the real part of the first eigenfunction of the trivial branch $(u_1,u_2)=(0,0)$ at the second Hopf bifurcation located at $r=2$. Panel (c) displays the functional value with respect to the number of optimization steps, while (d) shows the associated value of the control parameter $\\nu$.} \n\\label{fig_GL_opt}\n\\end{figure}\n\nWe minimize the functional $\\mathcal{J}=(\\mu_h-\\mu_h^\\star)^2\/\\mu_h^{\\star 2}$ with respect to the parameter $\\nu\\in\\mathbb{R}$ to control the imaginary part of the eigenvalue associated with the second Hopf bifurcation of the trivial branch as a test case to verify that $\\mu_h=\\nu$. The variable $\\mu_h^\\star$ denotes the target value of $\\mu_h$ and is set to $\\mu_h^\\star=10$. The normalization function $c:\\Omega\\to \\mathbb{R}^2$ in the Griewank--Reddien system~\\cref{eq_ext_MS} is chosen to be the following function\n\\begin{equation} \\label{eq_norm_const}\nc(x,y) = ((x+\\pi\/2)^2+(y+\\pi)^2,-(x+\\pi\/2)^2-(y+\\pi)^2),\\qquad x,y\\in\\Omega.\n\\end{equation}\nWith this example, we highlight the importance of the choice of the normalization $c$ to ensure that its inner product with the eigenfunction associated with a purely imaginary eigenvalue is nonzero. In this case, we observe in \\cref{fig_GL_opt} that the components of the real part of the eigenfunction satisfy anti-symmetry relations with respect to the $x$ axis. Therefore, the standard choice of $c=(1,1)$ would be orthogonal to the eigenfunction, which is the reason for the selection of a normalization function breaking the different symmetries of the eigenfunction in \\cref{eq_norm_const}. \n\nWe use the dolfin-adjoint and ROL to formulate and solve the optimization problem and control the frequency $\\mu_h$ at the Hopf bifurcation point with respect to the parameter $\\nu$. We report in \\cref{fig_GL_opt}(c) the value of the functional throughout the optimization. Our algorithm is able to control the relative frequency to 8 digits of accuracy, corresponding to a minimization of the functional $\\mathcal{J}$ to machine precision, in 8 optimization steps. Panel (d) of \\cref{fig_GL_opt} displays the corresponding value of the parameter $\\nu$ throughout the optimization. As expected, we observe that the target frequency $\\mu_h^\\star=10$ is reached with the parameter value $\\nu=10$, demonstrating the correctness of our method. Note that even in this simple example, where the state $u=(0,0)$ is always a solution to the steady-state equation, the resulting Griewank--Reddien system remains highly nonlinear due to the normalization condition of the eigenfunction and has several solutions. In this case the trivial branch has an infinite number of Hopf bifurcations corresponding to the eigenvalues of the Laplacian on the domain $\\Omega$.\n\n\\subsection{Rayleigh--B\\'enard problem}\n\nWe now investigate Hopf bifurcations in a two-dimensional Rayleigh--B\\'enard convection problem~\\cite{benard1900,benard1927,rayleigh1916}, which models a confined fluid heated from below with a constant temperature difference between the top and bottom of a unit square cell. Several studies have been performed over the past decades to analyze bifurcation structures of the Rayleigh--B\\'enard convection problem in various geometries~\\cite{bodenschatzetal00,crosshohenberg93,ma2006multiplicity} using numerical methods ranging from arclength continuation and branch-switching techniques~\\cite{doedel1981auto,keller1977numerical,uecker2014pde2path}, transient simulations~\\cite{boronska2010extreme,boronska2010extreme2}, and deflation~\\cite{boulle2021bifurcation,farrell2015deflation}. Additionally, the transition of steady flow structures to oscillatory convection through a Hopf bifurcation has been analyzed numerically and observed in experiments~\\cite{colinet1994hopf,ecke1986critical,ecke1992hopf,zhong1991asymmetric} to characterize the associated critical values of the bifurcation parameters. We consider the time-dependent Rayleigh--B\\'enard convection problem of an incompressible confined fluid heated from below in a unit square cell domain $\\Omega = (0,1)^2$, whose behaviour is governed by the following equations:\n\\begin{subequations} \\label{eq_RB}\n\\begin{align}\n\\frac{\\partial u}{\\partial t} - \\Pr\\nabla^2u + u\\cdot\\nabla u  + \\nabla p  - \\Pr \\textrm{Ra} T\\hat{z} &= 0 \\quad \\text{in }\\Omega,\\\\\n\\nabla\\cdot u&=0 \\quad \\text{in }\\Omega,\\\\\n\\frac{\\partial T}{\\partial t} - \\nabla^2 T + u\\cdot\\nabla T &= 0 \\quad \\text{in }\\Omega,\n\\end{align}\n\\end{subequations}\nwhere $u$ is the velocity field, $p$ is the pressure, $T$ is the fluid temperature, $\\hat{z}$ is the buoyancy direction, and $\\textrm{Ra}$, $\\Pr$ are the Rayleigh and Prandtl numbers. Similarly to~\\cite{boulle2021bifurcation}, we assume that the domain has rigid walls with thermally conducting horizontal walls and insulating side walls. That is, we impose the following boundary conditions:\n\\[u=0 \\text{ on }\\partial\\Omega,\\qquad \\partial_x T=0\\text{ for } x=0,1,\\qquad T=1\\text{ at }z=0,\\qquad T=0\\text{ at }z=1.\\]\nEquations~\\cref{eq_RB} are discretized spatially using Taylor--Hood finite elements for the velocity and pressure on triangles (piecewise quadratic and linear polynomials respectively) and piecewise linear polynomials for the temperature using the Firedrake finite element software~\\cite{rathgeber2016firedrake}. Note that we solve the steady-state version of~\\cref{eq_RB} with $\\partial_t u=0$, $\\partial_t T=0$, and hence do not need to employ a time-stepping scheme. \n\nIn this example, we fix the Prandtl number to $\\Pr=1$ and choose the Rayleigh number as bifurcation parameter. \nWe are interested in controlling the location of Hopf bifurcations, i.e.~the critical $\\textrm{Ra}$ at which they arise, in the bifurcation diagram originating from the steady-state solutions to \\cref{eq_RB} with respect to the shape of the domain $\\Omega$. Following \\cref{sec_opt}, we formulate this question as a PDE-constrained shape optimization problem:\n\\begin{equation} \\label{eq_prob_RB}\n\\begin{aligned}\n\\min_{\\substack{\\Omega\\in\\mathcal{U}_{\\textrm{ad}},\\, (u,p,T)\\in U(\\Omega),\\\\ v,w\\in U(\\Omega),\\, \\textrm{Ra},\\mu>0}} \\quad &  \\mathcal{J}((u,p,T),\\textrm{Ra},\\mu) \\coloneqq (\\textrm{Ra}-\\textrm{Ra}^\\star)^2\/\\textrm{Ra}^{\\star 2}\\\\\n\\textrm{subject to} \\quad & G((u,p,T),\\textrm{Ra},\\mu,v,w) = 0,\n\\end{aligned}\n\\end{equation}\nwhere the system of equations $G$ is defined by \\cref{eq_ext_MS} and characterizes the Hopf bifurcation, and $\\textrm{Ra}^\\star$ is the target value of the Rayleigh number for the location of the Hopf bifurcation. The set of admissible domains, $\\mathcal{U}_{\\textrm{ad}}$, in \\cref{eq_prob_RB} consists of the image of the initial domain under bi-Lipschitz diffeomorphisms. The PDE-constrained shape optimization algorithm aims to solve \\cref{eq_prob_RB} by applying a succession of smooth deformations of the domain~\\cite{paganini2021fireshape,paganini2018higher}. The normalization function $c=1$ (in each of the velocity, pressure, and temperature subfunctions) is used to solve \\cref{eq_ext_MS}, i.e.~we impose the phase condition $\\langle 1,v+iw\\rangle=i$ to normalize the eigenfunction $\\phi=v+iw$ associated with the eigenvalue $i\\mu$. Note that it is important to impose this condition over all velocity, pressure and temperature fields, unlike the Navier--Stokes example presented in \\cref{sec_NS}, since the velocity and temperature fields average to zero over the domain $\\Omega$ for a symmetric flow.\n \n\\begin{figure}[ht!]\n\\centering\n\\vspace{0.4cm}\n\\begin{overpic}[width=\\textwidth]{Figure\/RB_figure.pdf}\n\\put(1,98){(a)}\n\\put(16.2,101){Velocity}\n\\put(56,101){Temperature}\n\\put(1,64){(b)}\n\\put(1,31){(c)}\n\\end{overpic}\n\\caption{Velocity magnitude and streamlines (left) and isotherms (right) of the steady-state solution to \\cref{eq_RB} at the Hopf bifurcation located at $\\textrm{Ra}\\approx 7.35\\times 10^4$ (a). Fluid velocity and temperature flows on the optimized domain at the target Rayleigh number $\\textrm{Ra}^\\star=1.25\\times 10^5$ (b). Evolution of the critical Rayleigh number, shape of the domain, and corresponding velocity flow at the Hopf bifurcation, during the optimization algorithm (c). Each black dot in the diagram represents the Hopf bifurcation point as the domain is continuously deformed from state (a) to (b).} \n\\label{fig_RB_opt}\n\\end{figure}\n\nWe aim to control the Hopf bifurcation in the branch arising from the 4th bifurcation of the conducting state, located at $\\textrm{Ra}\\approx 7.35\\times 10^4$, as observed in~\\cite[Fig.~8]{boulle2021bifurcation}. The associated flow structure is reminiscent of mechanically coupled stationary convection in bi-layered systems for which a Hopf bifurcation occurs as shown by~\\cite[Fig.~1]{colinet1994hopf} and discussed in~\\cite{cardin1991nonlinear,cardin1991thermal,nataf1988responsible,rasenat1989theoretical}. We first display the fluid velocity and temperature at the critical Rayleigh number on the initial square domain $\\Omega=(0,1)^2$ in \\cref{fig_RB_opt}(a). At the Hopf bifurcation, the fluid flow velocity structure is composed of four vortices with alternating rotational directions: clockwise and anticlockwise, together with a symmetric temperature field with respect to the $\\mathbb{Z}_2$-symmetry axes of the problem: $x=1\/2$ and $z=1\/2$. Then, we implement the PDE-constrained optimization problem \\cref{eq_prob_RB} in the Fireshape optimization toolbox~\\cite{paganini2021fireshape} and solve the resulting optimization problem using a trust-region algorithm~\\cite{conn2000trust} implemented in the rapid optimization library (ROL)~\\cite{ridzal2017rapid}. We set the target bifurcation parameter to\n$\\textrm{Ra}^\\star=1.25\\times 10^5$ and find the shape deformation of the original domain for which the Hopf bifurcation arises at this value. We report the critical Rayleigh numbers throughout the optimization procedure together with the velocity and temperature fields of the Hopf bifurcation on the optimized domain shape in \\cref{fig_RB_opt}(c) and (b), respectively. As we observe in \\cref{fig_RB_opt}(c), the optimization algorithm ends by stagnating at a critical Rayleigh number of $\\textrm{Ra}\\approx 1.23\\times 10^5$, corresponding to an optimization functional value of $\\mathcal{J}\\approx 2.56\\times 10^{-4}$ (cf.~\\cref{eq_prob_RB}), i.e.~$1.6\\%$ relative error with respect to the target value of the Rayleigh number $\\textrm{Ra}^\\star=1.25\\times 10^5$. We observe that this error is satisfying given the coarse mesh discretization of the original domain, and lower errors might be achieved by employing a finer initial mesh. Finally, we observe that the $\\mathbb{Z}_2$-symmetries of the temperature field are broken on the optimized solution displayed in \\cref{fig_RB_opt}(b) due to the lack of symmetries of the final domain. We do not have any constraint to preserve the symmetries of the domain in the optimization formulation~\\cref{eq_prob_RB}. Adding further constraints on the shape deformations, such as symmetries or box\nconstraints, might be of interest for specific applications, as we will see in the next example.\n\n\\subsection{Navier--Stokes equations} \\label{sec_NS}\n\nIn this last example, we consider a laminar fluid flow past a circular cylinder in two dimensions. The behaviour of the flow is governed by the Reynolds number $\\textrm{Re}$. It is well known that the flow transitions from stationary to periodic at a critical Reynolds number, $\\textrm{Re}=\\textrm{Re}_c$, through a Hopf bifurcation, and ultimately transitions to turbulence as the Reynolds number increases. The periodic structure is known as a von K\\'arm\\'an vortex street~\\cite{barkley2006linear,duvsek1994numerical,jackson1987finite,provansal1987benard,williamson1996vortex}. The evolution of the fluid flow on the domain $\\Omega\\subset\\mathbb{R}^2$ is modelled by the non-dimensionalized incompressible Navier--Stokes equations:\n\\begin{subequations} \\label{eq_NS}\n\\begin{align}\n\\frac{\\partial u}{\\partial t}-\\nabla\\cdot \\left(\\frac{2}{\\textrm{Re}}\\epsilon(u)\\right)+u\\cdot\\nabla u+\\nabla p&=0\\quad \\text{in }\\Omega,\\\\\n\\nabla\\cdot u&=0\\quad \\text{in }\\Omega,\n\\end{align}\n\\end{subequations}\nwhere $\\textrm{Re}$ is the Reynolds number, $u$ is the fluid velocity, $\\epsilon(u)=\\frac{1}{2}(\\nabla u+\\nabla\nu^\\top)$, and $p$ is the pressure. We adopt a similar initial domain $\\Omega$ as in~\\cite{jackson1987finite} and consider a rectangle $(-5,15)\\times (-5,5)$ with a circular obstacle centered at the origin of diameter $d=1$. We impose the inflow velocity $u=(1,0)$ at the (left) inlet, top, and bottom of the domain, a no-slip condition on the obstacle boundary, as well as a natural outflow condition at the (right) outlet. We represent the initial computational domain $\\Omega$ in \\cref{fig_NS_domain}, along with the prescribed boundary conditions. The initial mesh of the domain is generated using Gmsh~\\cite{geuzaine2009gmsh} and is composed of $34,444$ triangles, with smaller characteristic length near the obstacle to capture the vortex structures arising in the flow pattern at the critical Reynolds number. Additionally, we impose a symmetric structure of the mesh with respect to the axis $y=0$ to preserve the $\\mathbb{Z}_2$-symmetry of the problem. The velocity and pressure are discretized using the Taylor--Hood finite element.\n\n\\begin{figure}[ht!]\n\\centering\n\\vspace{0.2cm}\n\\begin{overpic}[width=0.8\\textwidth]{Figure\/NS_domain.pdf}\n\\end{overpic}\n\\caption{Initial computational domain $\\Omega$ for simulating a von K\\'arm\\'an vortex street. Here, $\\vec{n}$ denotes the unit vector orthogonal to the boundary of the domain $\\partial\\Omega$. A circular obstacle of radius $0.5$ is located at the coordinates $(x,y)=(0,0)$.} \n\\label{fig_NS_domain}\n\\end{figure}\n\nWe first solve the steady-state Navier--Stokes equations at our initial guess for the location of the Hopf bifurcation, $\\Re = 46.25$, to obtain a time-independent solution $(u_b,p_b)$. Then, we perform a linear stability analysis using the ansatz $u(x,y,t) = u_b(x,y)+\\epsilon\\tilde{u}(x,y)e^{\\lambda t}$, where $\\epsilon\\ll 1$, giving the following generalized eigenvalue problem~\\cite{barkley2006linear},\n\\begin{subequations} \\label{eq_NS_eig}\n\\begin{align}\n\\nabla\\cdot \\left(\\frac{2}{\\textrm{Re}}\\epsilon(\\tilde{u})\\right)-u_b\\cdot\\nabla \\tilde{u} - \\tilde{u}\\cdot\\nabla u_b -\\nabla \\tilde{p}&=\\lambda\\tilde{u}\\quad \\text{in }\\Omega,\\\\\n\\nabla\\cdot \\tilde{u}&=0\\quad \\text{in }\\Omega,\n\\end{align}\n\\end{subequations}\nwith homogeneous boundary conditions and eigenvalue $\\lambda = \\sigma+i2\\pi f$. Here, $\\sigma$ denotes the growth rate and $f$ the frequency of the eigenmode. The steady-state is stable if the largest growth rate is negative and unstable otherwise. We solve \\cref{eq_NS_eig} with SLEPc~\\cite{hernandez2005slepc} and target the eigenvalues with growth rate $\\sigma$ closer to zero. Then, the steady-state solution, leading eigenvalue, and corresponding eigenfunction are used as initial guess for solving the system~\\cref{eq_ext_MS} characterizing the Hopf bifurcation. We use the following normalization condition to ensure uniqueness of the solution (see~\\cref{eq_norm_condition}),\n\\[\\int_{\\Omega}\\tilde{u}_x+\\tilde{u}_y\\,\\textup{d} x=i,\\]\nwhere $\\tilde{u}_x$ and $\\tilde{u}_y$ denote the $x$ and $y$ components of the (complex) velocity eigenfunction. We then find a critical Reynolds number of $\\textrm{Re}_c\\approx 46.23$ with an imaginary pair of eigenvalues of $\\lambda=\\pm 0.867 i$, i.e., a critical Strouhal number~\\cite{strouhal1878besondere,white1999mecanica} of $St_c = f L\/U \\approx 0.138$, where $L=1$ and $U=1$ are the characteristic length scale and velocity scale. These values are in agreement with the computational study of~\\cite{jackson1987finite}, which reported the values $\\text{Re}_c=46.136$ and $St_c = 0.13793$ using a similar method for locating the Hopf bifurcation. We report the flow structure (velocity magnitude and streamlines) of the steady-state solution at the Hopf bifurcation in \\cref{fig_NS_karman}(a).\n\n\\begin{figure}[ht!]\n\\centering\n\\vspace{0.2cm}\n\\begin{overpic}[width=\\textwidth]{Figure\/NS_hole.pdf}\n\\put(0,94){(a)}\n\\put(0,61.5){(b)}\n\\put(0,29){(c)}\n\\end{overpic}\n\\caption{(a) Velocity magnitude and streamlines of the solution at the Hopf bifurcation point located at $\\Re^\\star\\approx 46$, together with a magnification of the mesh around the obstacle at $[-1,1]\\times [-1,1]$. (b) Solution to the Navier--Stokes equations on the domain obtained after advancing the Hopf point to $\\Re^\\star\\approx 20$. (c) Similar to (b) with the Hopf bifurcation point delayed to $\\Re^\\star\\approx 200$.} \n\\label{fig_NS_karman}\n\\end{figure}\n\nWe aim to control, i.e.~advance or delay, the Hopf bifurcation by minimizing the functional $\\mathcal{J}(\\Re) = (\\Re-\\Re^\\star)^2\/\\Re^{\\star 2}$ with respect to the shape of the domain $\\Omega$. Here, $\\Re^\\star$ denotes the target Reynolds number for the location of the Hopf bifurcation and is successively set to $\\Re^\\star=20$ and $\\Re^\\star=200$ in the numerical examples presented in this section. In addition, we impose several geometric constraints on the domain. First, we fix the nodes of the mesh at the boundaries of the rectangle $[-5,15]\\times [-5,5]$, i.e.~only the inner obstacle may vary. Then, we enforce volumetric and barycentric constraints~\\cite{paganini2018higher,schulz2016computational} to ensure that the area and location of the obstacle remain constant throughout the optimization,\n\\[\\int_{\\Omega}\\d x=\\text{constant},\\qquad \\int_{\\Omega}x_1\\d x=\\text{constant},\\qquad \\int_{\\Omega}x_2\\d x=\\text{constant},\\]\nwhere $x_1$ and $x_2$ denote the two spatial coordinates. These constraints are imposed using an augmented Lagrangian algorithm~\\cite[Chapt.~17.3]{nocedal2006numerical} with limited memory BFGS Hessian updates. The subproblems are solved using a trust-region algorithm implemented in ROL.\n\nWe display the original domain and the ones optimized to lead to a Hopf bifurcation at $\\Re^\\star=20$ and $200$ in \\cref{fig_NS_karman}, together with a magnification of the mesh around the obstacles. We first advance the Hopf bifurcation to $\\Re^\\star=20$ in \\cref{fig_NS_karman}(b) and observe that the obstacle is deformed in the vertical direction to reach a final ellipsoid shape with sharp edges. The symmetry of the mesh around the obstacle seems preserved during the optimization. In the left panel of \\cref{fig_NS_opt}, we report the evolution of the shape around the obstacle together with the associated critical Reynolds number. The shape optimization procedure successfully converges to a domain with a critical Reynolds number of $\\Re_c = 19.9995$. In a second experiment, we aim to delay the birth of instabilities in the fluid flow by finding a shape for which the Hopf bifurcation arises at $\\Re^\\star=200$. As displayed in the left panel of \\cref{fig_NS_opt}, the functional value reaches a plateau at $\\Re_c\\approx 196.9$, with the shape depicted in \\cref{fig_NS_karman}(c). In this case, we observe that an elongated obstacle in the horizontal direction stabilizes the flow for higher Reynolds numbers. This showcases the challenges of the optimization procedure as the deformation of the mesh elements near the obstacle could prevent the functional to decay to machine precision. However, we highlight that we are able to reach a functional value of $\\mathcal{J}(\\Re) = 2.4\\times 10^{-4}$, corresponding to a $1.5\\%$ relative error between the target and obtained critical Reynolds number. Several ideas could be implemented to refine these results and reach higher or lower values of the critical Reynolds number, such as preserving the symmetry of the mesh exactly with respect to the axis $y=0$ by defining the problem on the upper half plane, or generating a new mesh during the optimization.\n\n\\begin{figure}[ht!]\n\\centering\n\\vspace{0.2cm}\n\\begin{overpic}[width=\\textwidth]{Figure\/NS_optimization.pdf}\n\\end{overpic}\n\\caption{Evolution of the domain during the optimization algorithm with a target critical Reynolds number of $\\text{Re}^\\star=20$ (left) and $\\text{Re}^\\star=200$ (right).  The figures show a magnification of the velocity magnitude profiles on $[-1,1]\\times[-1,1]$ around the obstacle, with the same scale and colorbar. The bifurcation parameters achieved at the end of the optimization procedure are respectively equal to $\\text{Re}_c = 19.9995$ and $\\text{Re}_c = 196.9$.} \n\\label{fig_NS_opt}\n\\end{figure}\n\nFinally, we perform time-dependent simulations using both the original domain and the optimized shape depicted in \\cref{fig_NS_karman}(b) to observe the von K\\'arm\\'an vortex street arising at the Hopf bifurcation, at Reynolds numbers $\\Re\\approx 46$ and $\\Re=20$ respectively. We discretize \\cref{eq_NS_eig} in time with a Crank--Nicolson time-stepping scheme and use an initial state consisting of a  steady-state to the Navier--Stokes equations perturbed by the eigenmode associated with the Hopf bifurcation, i.e.~$u(x,y,0) = u_b(x,y) +\\epsilon\\tilde{u}(x,y)$, where $\\epsilon$ is chosen such that $\\epsilon\\|u_b\\|_{L^2}\/\\|\\tilde{u}\\|_{L^2}=0.05$. We report snapshots of the simulation over one time-period in \\cref{fig_NS_karman_dynamics} (movies are available as Supplementary Material) and observe that the velocity profile has a periodic pattern at the expected Reynolds numbers for each simulation.\n\n\\begin{figure}[htbp]\n\\centering\n\\vspace{0.4cm}\n\\begin{overpic}[width=0.95\\textwidth]{Figure\/NS_dynamics.pdf}\n\\put(31,102){$t=0$}\n\\put(30,82){$t=T\/4$}\n\\put(30,62){$t=T\/2$}\n\\put(29,42){$t=3T\/4$}\n\\put(31,22){$t=T$}\n\\end{overpic}\n\\caption{Left: Periodic solution (velocity magnitude and streamlines) to the Navier--Stokes equations, defined on the domain with a cylindrical obstacle, at the Hopf bifurcation $Re\\approx 46$ over one time-period of $T\\approx 7.14$. Right: Periodic solution on the domain optimized to obtain a Hopf bifurcation at $Re\\approx 20$, with a period of $T\\approx 9.83$. Movies depicting the evolution of the velocity are available as Supplementary Material.} \n\\label{fig_NS_karman_dynamics}\n\\end{figure}\n\nWe have successfully controlled the transition to periodic flow, characterized by a Hopf bifurcation, in two-dimensional laminar flow past a body with respect to the shape of the obstacle. Our approach allows for the systematic manipulation of periodic solutions and may be applied to related problems to analyze the fluid flow past a rotating cylinder~\\cite{sierra2020bifurcation} or obstacles with different initial geometries, such as oriented ellipses and triangles~\\cite{jackson1987finite}.\n\n\\section{Conclusions} \\label{sec_conc}\n\nWe introduced a robust numerical method for controlling Hopf bifurcations arising in nonlinear dynamical systems. Our algorithm relies on a characterization of Hopf bifurcation points by the Griewank--Reddien system -- an extended system of nonlinear partial differential equations, which we embedded into a numerical optimization framework. We applied this procedure to successfully control the location and stability of Hopf bifurcations in several applications, such as the FitzHugh--Nagumo model, the complex Ginzburg--Landau equation, the Rayleigh--B\\'enard convection problem, and the Navier--Stokes equations, with respect to a control parameter or the shape of the domain.\n\n\\section*{Code availability}\nThe Firedrake components~\\cite{firedrake_zenodo_20211206} and code~\\cite{nicolas_boulle_2021_5235244} used to produce the numerical examples presented in this paper are available on Zenodo. The code is also distributed on GitHub at \\url{https:\/\/github.com\/NBoulle\/Hopf\\_Control}.\n\n\\section*{Acknowledgments}\nWe thank Karoline J{\\ae}ger and Aslak Tveito for discussions on the FitzHugh--Nagumo model.\n\n\\bibliographystyle{siamplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nRadiation emitted from a central massive body has long been \nknown to affect the orbit of gravitating particles both as an result of the \nradial, outward-directed force and \"radiation drag\" caused by the \nso-called Poynting-Robertson (PR) effect \n\\citep{Poynting1903,Robertson1937}. Originally studied in relation to \nthe motion of dust, comets and meteors in the solar \nsystems \\cite{Wyatt1950,Guess1961,Burns1979},\nthe PR effect has been discussed in more recent years in the context of matter \naccreting towards compact objects, especially black holes (BHs) of \nall masses and neutron stars (NSs), which radiate at a sizable fraction \nof their Eddington luminosity or very close to it ({\\it e.g.}\\citep{Walker1992}). \nThis has motivated extending the treatment of the PR effect to the more \ngeneral framework of General Relativity (GR)\\cite{Abramowicz1990,Lamb1995,Miller1996,Miller1998}. \nEarly attempts at handling the non-geodesic (dissipative) PR equations of motion in GR   \nwere based on a \\emph{direct spacetime approach} \\cite{Abramowicz1988,Abramowicz1990,Prasanna1990,Iyer1993,Abramowicz1993}) and provided\na limited, though inspiring, description of the effect. Only in 2009--2011 Bini and his collaborators \\cite{Bini2009,Bini2011} derived a full GR description of the PR effect by using a more suitable formalism (the \\emph{observer-splitting formalism} \\cite{Jantzen1992,Bini1997a,Bini1997b,Bini1998,Bini1999}). Recently, the Lagrangian formulation of the PR effect, involving an analytical form of the Rayleigh potential, has been developed \\cite{Defalco2018,DeFalco2019,DeFalco2019VE}. Moreover, we have recently worked out the three-dimensional (3D) generalization of the two-dimensional model (2D) of Bini and collaborators \\cite{DeFalco20183D}. The radiation field assumed in \\cite{DeFalco20183D} consists of photons emitted in a purely radial direction with respect to the Zero Angular Momentum Observer (ZAMO) frame of the Kerr spacetime background. In that case a coherent set of dynamical equations is derived, based on which particle \nmotion under the combined effect of radiation and gravity is determined and the \ncritical hypersurface, where all forces attain equilibrium, is found. New effects such as the \nlatitudinal drift of the trajectories and existence of suspended circular orbits were investigated. Based on these results, Wielgius performed a 3D treatment by employing finite size radiation effects framed in a simplified Kerr spacetime, including only linear terms in the spin \\cite{Wielgus2019}. \n\nIn the present work we adopt a different prescription for the 3D radiation field, which provides a somewhat more realistic description of emission properties of rotating compact \nobjects or the hot coronae around them.  Its properties are designed to match at the equator \nthose of the 2D radiation field introduced in \\cite{Bini2011}.\nThe paper is structured as follows: in Sec. \\ref{sec:geometry} we introduce the geometrical setup of our model. Sec. \\ref{sec:rad_field} describes the radiation field, an approximation of that emitted from a rigidly rotating spherical surface. In Sec. \\ref{sec:eoms} we derive the GR equations of motion \nfor test particles in the Kerr spacetime under the influence of the radiation field. \nIn Sec. \\ref{sec:critc_rad} we discuss the salient features of the critical hypersurface on which radiation forces balance gravity, and present calculations of selected orbits in the vicinity of critical hypersurfaces.  Our conclusions are in Sec. \\ref{sec:end}. \n\n\\section{Geometric setup}\n\\label{sec:geometry}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[scale=0.6]{Fig1.pdf}\n\\caption{Visual representation of the radiation field-test particle interaction geometry. The spatial location of the test particle is described by Boyer-Linquist coordinates $\\left\\{\\boldsymbol{r},\\boldsymbol{\\theta},\\boldsymbol{\\varphi}\\right\\}$. The ZAMO local frame is $\\left\\{\\boldsymbol{e_{\\hat{t}}},\\boldsymbol{e_{\\hat{r}}},\\boldsymbol{e_{\\hat{\\theta}}},\\boldsymbol{e_{\\hat{\\varphi}}}\\right\\}$. The photons of the radiation field travel along null geodesics with four-momentum $\\boldsymbol{k}$.  Two photon impact parameters, $b$ and $q$ are related respectively to the two angles $\\beta$ and $\\zeta$, measured in the local ZAMO frame. The test particle moves with a velocity described by the magnitude $\\nu$ and two angles $\\alpha$ and $\\psi$, measured in the local ZAMO frame.}\n\\label{fig:Fig0}\n\\end{figure*}\n\nWe consider a central compact object, whose outer spacetime geometry is described by the Kerr metric. Using geometric units ($c = G = 1$) and metric signature $(-,+,+,+)$, the line element of the Kerr spacetime expressed in Boyer-Lindquist coordinates, parameterized by the mass $M$ and the Kerr parameter (spin) $a$, reads as \\cite{Chandrasekhar1992}:\n\\begin{equation}\\label{kerr_metric}\n\\begin{aligned}\n \\mathrm{d}s^2 &= -\\left(1-\\frac{2Mr}{\\Sigma}\\right) \\,\\mathrm{d}t^2 \n  - \\frac{4Mra}{\\Sigma} \\sin^2\\theta\\,\\mathrm{d}t\\, \\mathrm{d}\\varphi \\\\\n  &+ \\frac{\\Sigma}{\\Delta}\\, \\mathrm{d}r^2 \n  + \\Sigma \\,\\mathrm{d}\\theta^2\n  + \\rho\\sin^2\\theta\\, \\mathrm{d}\\varphi^2, \n\\end{aligned}\n\\end{equation}\nwhere $\\Sigma \\equiv r^{2} + a^{2}\\cos^{2}\\theta$, $\\Delta \\equiv r^{2} - 2Mr + a^{2}$, and $\\rho  \\equiv r^2+a^2+2Ma^2r\\sin^2\\theta\/\\Sigma$. The determinant of the Kerr metric reads $g=-\\Sigma^2\\sin^{2}\\theta$. In the Kerr spacetime the radial coordinates of the outer event horizon $R_{\\rm +}$ and of the static limit (outer boundary of the ergosphere) $R_{\\rm SL}$ are given by \\cite{Misner1973}\n\\begin{equation}\n\\begin{aligned}\nR_{\\rm +}&=M(1+\\sqrt{1-a^2}),\\\\\nR_{\\rm SL}&=M(1+\\sqrt{1-a^2\\cos^2\\theta})\\,,\n\\end{aligned}\n\\end{equation}\nrespectively. \n\nWe consider the zero angular momentum observers (ZAMOs), whose future-pointing unit normal to the spatial hypersurfaces is given by\n\\begin{equation}\\label{n}\n\\boldsymbol{n}=\\frac{1}{N}(\\boldsymbol{\\partial_t}-N^{\\varphi}\\boldsymbol{\\partial_\\varphi})\\,,\n\\end{equation}\nwhere $N=(-g^{tt})^{-1\/2}$ is the time lapse function and $N^{\\varphi}=g_{t\\varphi}\/g_{\\varphi\\varphi}$ the spatial shift vector field. We conduct our investigations outside the event horizon, where the time coordinate hypersurfaces are spacelike, i.e., $g^{tt}<0$. An orthonormal frame adapted to the ZAMOs is \n\\begin{equation} \\label{eq:zamoframes}\n\\begin{aligned}\n&\\boldsymbol{e_{\\hat t}}=\\boldsymbol{n},\\quad\n\\boldsymbol{e_{\\hat r}}=\\frac1{\\sqrt{g_{rr}}}\\boldsymbol{\\partial_r},\\\\\n&\\boldsymbol{e_{\\hat \\theta}}=\\frac1{\\sqrt{g_{\\theta \\theta }}}\\boldsymbol{\\partial_\\theta},\\quad\n\\boldsymbol{e_{\\hat \\varphi}}=\\frac1{\\sqrt{g_{\\varphi \\varphi }}}\\boldsymbol{\\partial_\\varphi}.\n\\end{aligned}\n\\end{equation}\nAll the vector and tensor indices (e.g., $v^\\alpha, T^{\\alpha\\beta}$) associated to the ZAMO frame will be labeled by a hat (e.g., $v^{\\hat\\alpha}, T^{\\hat{\\alpha}\\hat{\\beta}}$), instead all the scalar quantities measured in the ZAMO frame (e.g., $f$) will be followed by $(n)$ (e.g., $f(n)$). In the kinematic decomposition of the ZAMO congruence, we have that the nonzero ZAMO kinematic quantities are acceleration $\\boldsymbol{a}(n)=\\nabla_{\\boldsymbol{n}} \\boldsymbol{n}$, expansion tensor along the $\\hat{\\varphi}$-direction $\\boldsymbol{\\theta_{\\hat\\varphi}}(n)$, and the relative Lie curvature vector $\\boldsymbol{k_{(\\rm Lie)}}(n)$ (see Table 1 in \\cite{DeFalco20183D} for their explicit expressions and for further details).\n \n\\section{Radiation field}\n\\label{sec:rad_field}\nWe consider a radiation field consisting of a flux of photons traveling along null geodesics in the Kerr geometry. The related stress-energy tensor is \\cite{DeFalco20183D}\n\\begin{equation}\\label{STE}\nT^{\\mu\\nu}=\\Phi^2 k^\\mu k^\\nu\\,,\\qquad k^\\mu k_\\mu=0,\\qquad k^\\mu \\nabla_\\mu k^\\nu=0,\n\\end{equation}\nwhere $\\Phi$ is a parameter linked to the radiation field intensity and $\\boldsymbol{k}$ is the photon four-momentum field. By splitting $\\boldsymbol{k}$ with respect to the ZAMO frame, we obtain\n\\begin{equation} \\label{photon}\n\\begin{aligned}\n&\\boldsymbol{k}=E(n)[\\boldsymbol{n}+\\boldsymbol{\\hat{\\nu}}(k,n)],\\\\\n&\\boldsymbol{\\hat{\\nu}}(k,n)=\\sin\\zeta\\sin\\beta\\ \\boldsymbol{e_{\\hat r}}+\\cos\\zeta\\ \\boldsymbol{e_{\\hat\\theta}}+\\sin\\zeta \\cos\\beta\\ \\boldsymbol{e_{\\hat\\varphi}},\n\\end{aligned}\n\\end{equation}\nwhere $\\boldsymbol{\\hat{\\nu}}(k,n)$ is the photon spatial unit relative velocity with respect to the ZAMOs, $\\beta$ and $\\zeta$ are the two angles measured in the ZAMO frame in the azimuthal and polar direction, respectively (see Fig. \\ref{fig:Fig0}). The radiation field is governed by the two impact parameters $(b,q)$, associated respectively with the two emission angles $(\\beta,\\zeta)$. The photon energy, $E(n)$, and the photon angular momentum along the polar direction $\\hat{\\theta}$, $L_{\\hat\\theta}(n)$, with respect to the ZAMO frame is expressed in the frame of distant static observer by the following formulae \\cite{DeFalco20183D}\n\\begin{eqnarray} \\label{energyZAMO}\n&&E(n)=\\frac{E}{N}(1+bN^\\varphi),\\\\\n&&L_{\\hat\\theta}(n)\\equiv E(n)\\cos\\beta\\sin\\zeta=\\frac{L_z}{\\sqrt{g_{\\varphi\\varphi}}},\n\\end{eqnarray}\nwhere $E=-k_t>0$ is the conserved photon energy, $L_z=k_\\varphi$ is the conserved angular momentum along the polar $z$ axis orthogonal to the equatorial plane, and $b=L_z\/E$ is the azimuthal photon impact parameter, where all these quantities are measured by a distant static observer. At this impact parameter, we associate the relative azimuthal angle in the ZAMO frame \\cite{DeFalco20183D}\n\\begin{equation} \\label{ANG1}\n\\cos\\beta=\\frac{b E}{\\sin\\zeta\\sqrt{g_{\\varphi\\varphi}}E(n)}=\\frac{b N}{\\sin\\zeta\\sqrt{g_{\\varphi\\varphi}}(1+b N^\\varphi)}.\n\\end{equation}\nThe photon four-momentum components in the Kerr geometry are given by \\cite{Chandrasekhar1992} \n\\begin{equation} \\label{CarterEQs} \n\\begin{aligned}\n     k^{t} &= \\Sigma^{-1} \\left(ab-a^2\\sin^{2}\\theta+(r^2+a^2)P\\,\\Delta^{-1}\\right)\\,,\\\\\n\t\t k^{r} &=  s_r\\Sigma^{-1} \\sqrt{R_{b,q}(r)},\\\\\n     k^{\\theta} &=  s_{\\theta}\\Sigma^{-1} \\sqrt{\\Theta_{b,q}(\\theta)}\\,\\\\ \n\t\t k^{\\varphi} &=  \\Sigma^{-1}\\left(b\\,\\mathrm{cosec}^{2} \\theta-a+a\\,P\\,\\Delta^{-1}\\right),\n\\end{aligned}\n\\end{equation} \nwhere $P\\equiv r^{2} + a^{2}-ba$, and the pair of signs $s_{r}$, $s_{\\theta}$ describes the orientation of the radial and latitudinal evolution, respectively. The radial and latitudinal effective potentials are respectively \\cite{Chandrasekhar1992}\n\\begin{equation}\n\\label{Rpot}\n       R_{b,q} \\left( r \\right) = \\left( r^{2} + a^{2}-ab \\right)^{2}\n       - \\Delta \\left[ q + \\left( b - a \\right) ^{2} \\right], \n\\end{equation} \t\t\t\n\\begin{equation}\t\n\\label{Thetapot}\t\t\t\n\\Theta_{b,q} \\left( \\theta \\right) = q + a^{2} \\cos^{2}\n\\theta -b^{2} \\mathrm{cot}^{2} \\theta,\n\\end{equation} \nwhere  \n \\begin{equation} \\label{q_def2}\nq \\equiv \\left(\\frac{k_{\\theta}}{k_{t}}\\right)^{2} + \\left[b\\tan\\left(\\frac{\\pi}{2} - \\theta\\right)\\right]^{2} - a^{2}\\cos^{2}\\theta\\,,    \n \\end{equation}\nis the latitudinal photon impact parameter related to Carter's constant of motion $\\mathcal{Q}$ through \\cite{Chandrasekhar1992}\n\\begin{equation} \\label{Carter_constant}\nq=\\mathcal{Q}\/E^2.\n \\end{equation}\n\\subsection{Radiation field from a spherical rigidly rotating surface}\n\\label{sec:rot_rad_field}\nIn the previous paper \\cite{DeFalco20183D} we considered a test radiation field made by photons moving along purely radial directions in any ZAMO frame and even at infinity, independently of the radial coordinate. Here we consider a somewhat more complex model, consisting of a radiation field emitted from a spherical surface with a radius $\\RS$ centered at the origin of the Boyer-Lindquist coordinates and rigidly rotating with angular velocity $\\Omega_{\\mathrm{\\star}}$. The azimuthal motion of the photons, as measured by a static observer at infinity, is now determined not only by frame-dragging (as in \\cite{DeFalco20183D}), but also by rotation of the surface. The present radiation field model is thus more general than our previous model.  \n\nThe four-momentum of a photon outgoing from a rigidly rotating spherical surface in a purely radial direction with respect to the corotating local observer frame labelled by $\\left\\{\\boldsymbol{e_{\\langle t \\rangle}},\\boldsymbol{e_{\\langle r \\rangle}},\\boldsymbol{e_{\\langle \\theta \\rangle}},\\boldsymbol{e_{\\langle \\varphi \\rangle}}\\right\\}$ is\n\\begin{equation} \\label{local_mom}\n\\boldsymbol{k}=\\boldsymbol{e_{\\langle t \\rangle}}+\\boldsymbol{e_{\\langle r \\rangle}} \\,.\n\\end{equation}\nIn the Kerr geometry (\\ref{kerr_metric}), the timelike unit surface four-velocity, $\\boldsymbol{U_{\\mathrm{surf}}}$, of the stationary observer is\n\\begin{equation}\n\\boldsymbol{U_{\\mathrm{surf}}}=\\mathcal{N}\\,(\\boldsymbol{\\partial_t}+\\Omega_{\\star}\\,\\boldsymbol{\\partial_\\varphi})\\,, \n\\end{equation}\nwhere\n\\begin{equation}\n\\mathcal{N}=\\frac{1}{\\sqrt{-\\mathrm{g_{tt}}-\\Omega_{\\star}  (2\\, \\mathrm{g_{t\\varphi}}+\\mathrm{g_{\\varphi \\varphi}} \\Omega_{\\star})}}\\,.\n\\end{equation}\nHere the metric coefficients are evaluated at the radial coordinate $\\RS$ and at the polar coordinate of the photon emission $\\theta_e$. The corotating orthonormal frame components related to the stationary observer sitting at the radiating surface take the following form \n\\begin{equation} \\label{surfaceframe}\n\\begin{aligned}\n\\boldsymbol{e_{\\langle t \\rangle}}&=\\boldsymbol{U_{\\mathrm{surf}}},\\\n\\boldsymbol{e_{\\langle r \\rangle}}=\\frac1{\\sqrt{g_{rr}}}\\boldsymbol{\\partial_r},\\\n\\boldsymbol{e_{\\langle \\theta \\rangle}}=\\frac1{\\sqrt{g_{\\theta \\theta }}}\\boldsymbol{\\partial_\\theta},\\\\\n\\boldsymbol{e_{\\langle \\phi \\rangle}}&=\\frac{\\mathcal{N}\\left[\\left(\\mathrm{g_{t\\varphi}}+\\mathrm{g_{\\varphi \\varphi}} \\Omega_{\\star}\\right) \\boldsymbol{\\partial_t}+\\left(\\mathrm{g_{tt}}+\\mathrm{g_{t \\varphi}} \\Omega_{\\star}\\right) \\boldsymbol{\\partial_\\varphi} \\right]}{\\sqrt{\\mathrm{g_{t\\varphi}}^2-\\mathrm{g_{\\varphi\\varphi}} \\mathrm{g_{tt}}}}.\n\\end{aligned}\n\\end{equation}\nThen, the photon four-momentum (\\ref{local_mom}) can be rewritten in such coordinate frame as \n\\begin{equation} \\label{coordinate_mom}\n\\boldsymbol{k}=\\mathcal{N}\\boldsymbol{\\partial_t}+ \\frac1{\\sqrt{g_{rr}}}\\boldsymbol{\\partial_r}+ \\Omega_{\\star}\\mathcal{N}\\boldsymbol{\\partial_\\varphi}\\,. \n\\end{equation}\nA straightforward calculation yields the azimuthal impact parameter $b=-k_{\\varphi}\/k_{t}$ \n\\begin{eqnarray} \n&&b\\equiv-\\frac{\\mathrm{g_{t\\varphi}}+\\mathrm{g_{\\varphi\\varphi}}\\Omega_{\\star} }{\\mathrm{g_{tt}}+\\mathrm{g_{t\\varphi}} \\Omega_{\\star}}=\\label{kerr_impact_parameter}\\\\  \n&&= \\frac{\\sin ^2\\theta_e \\left[\\left(\\RS^4+a^2\\mathcal{B}\\right)\\Omega_{\\star}-2Ma\\RS\\right]}\n       {\\left[\\RS^2-2M \\RS+a\\left(a\\cos ^2\\theta_e+ 2M\\RS\\Omega_{\\star} \\sin ^2\\theta_e\\right)\\right]},\\notag\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\mathcal{B}=\\left(a^2+\\RS^2\\right) \\cos^2\\theta_e+2M \\RS \\sin^2\\theta_e+\\RS^2\n\\end{equation}\n$b$ can be approximated in terms of the photon impact parameter in the Schwarzschild geometry, $b_{\\rm{Schw}}$, plus a correction term of first order in the spin parameter  \n\\begin{equation} \\label{apprx_b}\nb=b_{\\rm{Schw}}-\\mathcal{A}\\,a+O\\left(a^2\\right), \n\\end{equation}\nwhere \n\\begin{eqnarray} \\label{schw_impact_parameter}\n&&b_{\\mathrm{Schw}}=\\frac{\\RS^2\\Omega_{\\star} \\sin^2\\theta_e}{1-2M\/\\RS}, \\\\\n&&\\mathcal{A}= \\frac{2M\\sin^2\\theta_e\\left(M\\RS^3 \\Omega_{\\star}^2 \\sin^2\\theta_e+\\RS-2M \\right)}{(\\RS-2M)^2}.  \\label{first_der}\n\\end{eqnarray} \nFrom Eq. (\\ref{apprx_b}) it is apparent that the radially outgoing photon has the same specific angular momentum of a matter element located at the emission point on the rotating surface (see Eq. (\\ref{schw_impact_parameter})). In addition, the influence of frame-dragging entails the decrease of the value of the emitted photon impact parameter of the radiation field (see Eq. (\\ref{first_der})). The azimuthal photon impact parameter is not fixed, but instead it ranges from the minimum value $b(\\theta_e=0)=0$ (describing photons emitted from the poles of the surface along polar axis) to the maximum value $b(\\theta_e=\\pi\/2)$ (corresponding to photons emitted in the equatorial plane). \n\nThe latitudinal photon impact parameter, $q$, is obtained from the condition \n\\begin{equation} \\label{cond2}\n\\Theta_{b,q}(\\theta_e)=0,\n\\end{equation}\nresulting from the absence of the polar component of the photon four-momentum (\\ref{coordinate_mom}). From Eqs. (\\ref{Thetapot}) and (\\ref{cond2}) we can express  $q$ as a function of the photon emission polar angle $\\theta_e$ as\n\\begin{equation} \\label{q_r}\nq=b^{2}\\cot^{2} \\theta_e-a^{2} \\cos^{2}\\theta_e\\ . \n\\end{equation}\nThis relation also fixes the Carter constant $\\mathcal{Q}$ for a given photon trajectory through Eq. (\\ref{Carter_constant}).\nThe latitudinal potential (\\ref{Thetapot}) is independent of the radial coordinate and therefore the polar angle, $\\theta=\\theta_e$, along a given photon trajectory is conserved. \nTherefore in the case of our radiation field which is emitted radially in the frame of the rigidly rotating emitting surface, the latitudinal photon impact parameter $q$ (and also the Carter constant $\\mathcal{Q}$) is fully determined by the emission angle $\\theta_e$, the azimuthal photon impact parameter $b(\\theta_e,\\Omega_{\\star},\\RS, a)$ and the Kerr parameter (spin) $a$. As in our previous paper \\cite{DeFalco20183D}, such setup for the radiation field significantly simplifies the integration of test particle trajectories in that only one photon trajectory (described by the corresponding pair $(b,q)$) emitted at $\\theta_e$ can reach the instantaneous test particle position at the polar coordinate $\\theta = \\theta_e$. \n\nNaturally, also in the arbitrary ZAMO frame (\\ref{eq:zamoframes}), the local polar component of the photon four-momentum remains identically zero ($k^{\\hat \\theta}=k^{\\theta}\/\\sqrt{g_{\\theta\\theta}} = 0 $). Then using Eq. (\\ref{photon}), we can easily see that the local polar angle of the photon four-momentum in the ZAMO frame always takes the value $\\zeta=\\pi\/2$. In view of the last condition, we have that the azimuthal photon angle $\\beta$, given by Eq. (\\ref{ANG1}), takes the form\n\\begin{equation} \\label{ANG12}\n\\cos\\beta=\\frac{b N}{\\sqrt{g_{\\varphi\\varphi}}(1+b N^\\varphi)}.\n\\end{equation}\n\nWe adopt the radiation field we introduced in this section as a heuristic approximation of  \nthe emission from a rotating NS  or from the surface of a rotating hot corona in the close vicinity \nof a BH, and therefore we \nwill consider hereafter only the case of outgoing photons, {\\it i.e.} $\\sin\\beta \\geq 0$.\nConsidering only the emission of an individual light ray (that orthogonal to the surface) \nfrom each point of the rotating sphere, as done in the present study,  \nmakes the problem analytically treatable and represents a clear step forward in capturing the behaviour \nof test particle under the influence of the gravitational and radiation field generated by (or around) \nrotating compact objects. More realistic models should include the \nthe whole range of outgoing light ray directions as well as the angular dependence of the surface emissivity.\n\n\n\\subsubsection{Emission from NS surface}\n\\label{sec:NS_surface}\nLet us consider a radiation field emitted from the surface a rotating NS. The Kerr parameter $a$ \nis expressed as a function the NS angular velocity $\\Omega_{\\star}$ through \n\\begin{equation} \\label{eq:NSa} \na=\\mathcal{C}\\Omega_{\\star}\\,, \n\\end{equation}\nwhere $\\mathcal{C}$ depends on the NS structure and equation of state \\cite{Haensel2009,Urbanec2010,Rikovska2003}. For a NS of radius $\\RS=11\\,\\mathrm{km}$ and mass $M=1.5\\,\\mathrm{M_{\\odot}}$, we choose $\\mathcal{C}=1.1\\,.\\,10^{-4}\\,\\mathrm{s}$ (see, e.g., \\cite{Haensel2009,Urbanec2010,Rikovska2003,Bakala2012} for details). A moderate NS rotation frequency of $f_{\\star}\\equiv\\Omega_{\\star}\/2\\pi=300\\,\\mathrm{Hz}$ corresponds to $a \\approx 0.21$. In this case the maximum value of the azimuthal photon impact parameter on the NS equator would be $b(\\theta_e=\\pi\/2) \\approx 0.43$. For a NS rotation frequency of $f_{\\star} = 900\\,\\mathrm{Hz}$  we get $a \\approx 0.62$ and  $b(\\theta_e=\\pi\/2) \\approx 1.29$. \nIn the non-rotating case ($a=0$, Schwarzschild spacetime) the radiation field has $b=0$, which is identical with the zero spin test field described in \\cite{DeFalco20183D}. Note, that the physically meaningful range of rotation frequency is limited by the value of the NS break-uo frequency from above.\n\n\\subsubsection{Emission from a hot corona around a BH} \n\\label{sec:BH_corona}\nLet us now consider a radiation field emitted from a spherical corona which is rigidly rotating with angular velocity $\\Omega_{\\mathrm{\\star}}$. We assume that the Kerr parameter, $a$, is determined entirely by the central BH. We restrict our consideration here only positive values of the angular velocity $\\Omega_{\\mathrm{\\star}}$, {\\it i.e.} emitting surfaces corotating with the BH.\n\nThe existence of the emission surface rotating with $\\Omega_{\\star}\\ge0$ and located on $\\RS$ requires that the four-velocity $\\boldsymbol{U_{\\mathrm{surf}}}$ takes real values. This corresponds to positive values of $\\mathcal{N}$ on the equatorial plane ($\\theta_e=\\pi\/2$), where $\\boldsymbol{U_{\\mathrm{surf}}}$ reaches its maximum value. Such condition translates into\n\\begin{equation}\n\\label{superluminal}\n\\begin{aligned}\n&\\Omega_{-}(\\RS,\\pi\/2) < \\Omega_{\\star} < \\Omega_{+}(\\RS,\\pi\/2)\\ ,\\\\\n&\\Omega_{\\pm}(\\RS,\\pi\/2)=\\frac{-\\mathrm{g_{t\\varphi}} \\pm \\sqrt{ \\mathrm{g_{t\\varphi}}^2 - \\mathrm{g_{\\varphi\\varphi}} \\mathrm{g_{tt}}}}{\\mathrm{g_{\\varphi\\varphi}}}\\ ,\n\\end{aligned}\n\\end{equation}\nwhich exclude the possibility of superluminal rotation of the emission surface. Note that in the investigated case of $\\Omega_{\\star} >0 $, the minimum value $\\Omega_{-}(\\RS,0)$ is relevant only inside of the BH ergosphere, where there cannot be static observers. \n\n\\begin{figure*}[htbp]  \n\\begin{center}\n\\includegraphics[width=1.00\\linewidth, angle=0]{b_range_maps.pdf}\n\\end{center}\n\\caption{Contour plot of the azimuthal impact parameter $b$ of photons emitted from the equator ($\\theta_e=\\pi\/2$) as  function of radius, $\\RS$, and rotation frequency, $f_{\\star}=\\Omega_{\\star}\/2\\pi$ of the emitting surface. Left panel: Schwarzschild geometry using a NS mass of $M=1.5\\mathrm{M_\\odot}$. Positive impact parameter values are limited from above only by $\\Omega_{+}(\\RS,\\pi\/2)$. Right panel: Kerr geometry using a value of $a=0.9$ and a BH mass of $M=5\\mathrm{M_\\odot}$. The green region corresponds to positive values of $b$, while the red region corresponds to negative values. The yellow curve ending at the radius of the innermost stable cicular orbit (ISCO) denotes the Keplerian orbital frequency $\\Omega_{K}=[M(r^{3\/2}+a)]^{-1}$. The red curve denotes the zero value of $b$ corresponding to the condition $\\Omega_{\\star}=\\Omega_{\\mathrm{ZAMO}}(\\RS)$, {\\it i.e.} the surface rotates at the frequency of spacetime rotation. Along the cyan curve, where $\\Omega_{\\star}=\\Omega_{\\infty}(\\RS)\\equiv-\\mathrm{g_{tt}}\/\\mathrm{g_{t\\varphi}}$, the value of the azimuthal photon impact parameter diverges ($b \\to -\\infty$). The beige region under the cyan curve corresponds to photons emitted with negative conserved energy $-k_t$ from surfaces rotating at $\\Omega_{-}(\\RS,0) < \\Omega_{\\star} \\le \\Omega_{\\infty}(\\RS)$. Such photons cannot escape from the ergosphere. The gray regions denote the cases of (unphysical) superluminal rotation.}  \n\\label{fig:lambdarange}\n\\end{figure*}\n\n\\subsubsection{$b$-range in the Schwarzschild geometry}\n\\label{sec:brange_schw}\nIn the Schwarzschild case, the photon impact parameter values (\\ref{schw_impact_parameter}) are always positive and limited from above only by the condition $\\Omega_{\\star} < \\Omega_{+}(\\RS,\\pi\/2)$ (see left panel in Fig. \\ref{fig:lambdarange}). The maximum frequency $\\Omega_{+}(\\RS,\\pi\/2)$ is indeed unphysical. If we consider a emitting rotating hot corona in the background of the Schwarzchild geometry, the rotation frequency $f_{\\star}=300\\,\\mathrm{Hz}$ corresponds to the value of $b_{\\mathrm{Schw}} \\approx 0.575$, while the higher rotation frequency $f_{\\star} = 900\\,\\mathrm{Hz}$ corresponds to the value of $b_{\\mathrm{Schw}} \\approx 1.725$. \n\n\\subsubsection{Estimation of $b$-range in Kerr geometry}\n\\label{sec:brange_kerr}\n\nIn the Kerr metric, frame-dragging makes matters more intricate. The upper limit of $b$ is again controlled by the condition $\\Omega_{\\star} < \\Omega_{+}(\\RS,\\pi\/2)$, but contrary to the Schwarzschild case, the azimuthal photon impact parameter $b$ can be both positive and negative. We can distinguish the following cases for the behaviour of the azimuthal photon impact parameter $b$ (see the right panel in Fig. \\ref{fig:lambdarange}):\n\n\\begin{itemize}\n\\item for $\\Omega_{\\star}=\\Omega_{\\mathrm{ZAMO}}(\\RS)$ $b=0$, as the emitting surface rotates with the same angular velocity as the spacetime (red curve); \n\\item for $\\Omega_{\\star} \\ge \\Omega_{\\mathrm{ZAMO}}(\\RS)$, $b\\ge0$ (green region); \n\\item for $\\Omega_{\\star} \\le \\Omega_{\\mathrm{ZAMO}}(\\RS)$ $b\\le0$ (red region);\n\\end{itemize}\nIf the emitting surface is located inside the ergosphere, where the surface rotation frequency is limited by the condition $0 < \\Omega_{-}(\\RS,0) <  \\Omega_{\\star}$, we can distinguish the following additional cases:  \n\\begin{itemize}\n\\item  for $\\Omega_{\\infty}(\\RS) \\le \\Omega_{\\star} \\le \\Omega_{\\mathrm{ZAMO}}(\\RS)$ photons can possess extreme negative values of $b$ reaching $-\\infty$ for $\\Omega_{\\star} \\rightarrow \\Omega_{\\infty}(\\RS)$ (see the bottom part and boundary of the red region).\n\\item for $\\Omega_{\\star}=\\Omega_{\\infty}(\\RS)$ with $\\Omega_{\\infty}(\\RS)\\equiv-\\mathrm{g_{tt}}\/\\mathrm{g_{t\\phi}}>\\Omega_{-}(\\RS,0)$, the conserved photon energy is $k_t=0$ and $\\lim_{\\Omega_{\\star}\\rightarrow\\Omega_{\\infty}^\\pm}b(\\theta_e=\\pi\/2)=\\mp \\infty$ (see the cyan curve);\n\\item for $\\Omega_{-}(\\RS,0) < \\Omega_{\\star} \\le \\Omega_{\\infty}(\\RS)$, the conserved photon energy is negative $k_t<0$, and photons cannot escape from the ergosphere. Their $b$ is positive, reaching $+\\infty$ for $\\Omega_{\\star}\\rightarrow \\Omega_{\\infty}(\\RS)$. Such radiation fields cannot affect physical processes outside the ergosphere (see the beige region and \\cite{Grib2017} for further details). \n\\end{itemize}\n\n\\subsection{Intensity parameter}\n\\label{sec:int_par}\nSince the photon four-momentum $\\boldsymbol{k}$ is completely determined by the $(b,q)$ pair, the coordinate dependence of $\\Phi$ then follows from the conservation equations $T^{\\alpha\\beta}{}_{;\\beta}=0$. Due to the absence of photon latitudinal motion ($k^{\\theta}=0$) and the symmetries of the Kerr spacetime, these can be written as\n\\begin{equation}\\label{flux_cons}\n0=\\nabla_\\beta (\\Phi^2 k^\\beta)=\\frac{1}{\\Sigma\\sin\\theta}\\,\\partial_r (\\Sigma\\sin\\theta\\,\\Phi^2 k^r)\\,. \n\\end{equation}\n\nHowever, Eq. (\\ref{flux_cons}) does not fix the intensity parameter unambiguously.\nThe conservation equations would still be fulfilled, even if we multiplied the  \nderivative with respect to the radial coordinate ($\\Sigma\\sin\\theta\\,\\Phi^2 k^r $) of the expression by an arbitrary function of the $\\theta$ coordinate. Therefore condition (\\ref{flux_cons}) determines the class of radiation fields that differ from one another by the latitudinal distribution of intensity. A natural requirement for the radiation field model is that it attains spherical symmetry for the case of pure radially outgoing field ($b=0$) in a Schwarzschild background geometry. This in turn requires that  the intensity parameter is independent of the $\\theta$ coordinate, which can be easily achieved by multiplying the derivative with respect to the radial coordinate of (\\ref{flux_cons}) by a factor $1\/\\sin\\theta$.\nThe condition (\\ref{flux_cons}) then will take the form\n\\begin{equation}\n0=\\partial_r (\\Sigma\\Phi^2 k^r)\\,. \n\\end{equation}\nBy expressing $k^r$ through Eq. (\\ref{CarterEQs}) and substituting the values of the impact parameters obtained above in Eqs. (\\ref{kerr_impact_parameter}) and (\\ref{q_r}) in Eq. (\\ref{flux_cons}), we obtain\n\\begin{equation} \\label{eq:phi}\n\\Phi^2 \\sqrt{R_{b,q}(r)}=\\hbox{\\rm const} = \\Phi_0^2\\,,\n\\end{equation}\nwhere $\\Phi_0^2$ is a new constant related to the intensity of the radiation field at the emitting surface.\nThen, the parameter $\\Phi^2$ is related to the radial effective potential (\\ref{Rpot}) by the formula\n\\begin{equation}\\label{INT_PAR}\n\\Phi^2=\\frac{\\Phi_0^2}{\\sqrt{R_{b,q}(r)}}\\,.\n\\end{equation}\nNote that in the case of a non-rotating emitting surface in a Schwarzschild spacetime, Eq. (\\ref{INT_PAR}) has the simple limit $\\Phi^2=\\Phi_0^2\/r^2$,\napparently preserving the spacetime spherical symmetry. Also note, that in the case $b=0$ expression (\\ref{INT_PAR}) coincides with the formula for the intensity parameter of the radial radiation field described in \\cite{DeFalco20183D} and can be considered its generalization.\n\n\\section{Interaction of radiation - test particle }\n\\label{sec:eoms}\n\n\\subsection{Test particle motion}\n\\label{sec:test_part}\nWe consider a test particle moving in the 3D space endowed with a timelike four-velocity $\\boldsymbol{U}$ and a spatial three-velocity with respect to the ZAMOs, $\\boldsymbol{\\nu}(U,n)$, \\cite{DeFalco20183D}:\n\\begin{eqnarray} \n&&\\boldsymbol{U}=\\gamma(U,n)[\\boldsymbol{n}+\\boldsymbol{\\nu}(U,n)], \\label{testp}\\\\\n&&\\boldsymbol{\\nu}=\\nu(\\sin\\psi\\sin\\alpha\\boldsymbol{e_{\\hat r}}+\\cos\\psi\\boldsymbol{e_{\\hat\\theta}}+\\sin\\psi \\cos\\alpha \\boldsymbol{e_{\\hat\\varphi}}),\n\\end{eqnarray}\nwhere $\\gamma(U,n)=1\/\\sqrt{1-||\\boldsymbol{\\nu}(U,n)||^2}$ is the Lorentz factor, $\\nu=||\\boldsymbol{\\nu}(U,n)||$, $\\gamma(U,n) =\\gamma$. Here $\\nu$ represents the magnitude of the test particle spatial velocity $\\boldsymbol{\\nu}(U,n)$, $\\alpha$ is the azimuthal angle of the vector $\\boldsymbol{\\nu}(U,n)$ measured clockwise from the positive $\\hat\\varphi$ direction in the $\\hat{r}-\\hat{\\varphi}$ tangent plane in the ZAMO frame, and $\\psi$ is the polar angle of the vector $\\boldsymbol{\\nu}(U,n)$ measured from the axis orthogonal to the $\\hat{r}-\\hat{\\varphi}$ tangent plane in the ZAMO frame (see Fig. \\ref{fig:Fig0}). The explicit expression for the test particle velocity components with respect to the ZAMOs are \n\\begin{equation} \n\\begin{aligned}\\label{four_velocity}\n&U^t\\equiv \\frac{dt}{d\\tau}=\\frac{\\gamma}{N},\\quad U^r\\equiv \\frac{dr}{d\\tau}=\\frac{\\gamma\\nu^{\\hat r}}{\\sqrt{g_{rr}}},\\\\\n&U^\\theta\\equiv \\frac{d\\theta}{d\\tau}=\\frac{\\gamma\\nu^{\\hat\\theta}}{\\sqrt{g_{\\theta\\theta}}},\\quad U^\\varphi\\equiv \\frac{d\\varphi}{d\\tau}=\\frac{\\gamma\\nu^{\\hat\\varphi}}{\\sqrt{g_{\\varphi\\varphi}}}-\\frac{\\gamma N^\\varphi}{N} ,\n\\end{aligned}\n\\end{equation}\nwhere $\\tau$ is the proper time parameter along $\\bold{U}$. Using the \\emph{observer-splitting formalism}, the test particle acceleration relative to the ZAMO congruence, $\\boldsymbol{a}(U)=\\nabla_{\\bold U} \\bold{U}$, is given by \\cite{Defalco2018,DeFalco20183D}\n\\begin{eqnarray}\na(U)^{\\hat r}&=& \\gamma^2 [a(n)^{\\hat r}+k_{\\rm (Lie)}(n)^{\\hat r}\\,\\nu^2 (\\cos^2\\alpha\\sin^2\\psi\\label{acc1} \\\\\n&&+\\cos^2\\psi)+2\\nu\\cos \\alpha\\sin\\psi\\, \\theta(n)^{\\hat r}{}_{\\hat \\varphi}]\\nonumber\\\\\n&&+\\gamma \\left(\\gamma^2 \\sin\\alpha\\sin\\psi \\frac{\\rm d \\nu}{\\rm d \\tau}+\\nu \\cos \\alpha\\sin\\psi \\frac{\\rm d \\alpha}{\\rm d \\tau}\\right.\\nonumber\\\\\n&&\\left.+\\nu \\cos \\psi\\sin\\alpha \\frac{\\rm d \\psi}{\\rm d \\tau} \\right), \\nonumber\\\\   \na(U)^{\\hat \\theta}&=&\\gamma^2 [a(n)^{\\hat \\theta}+k_{\\rm (Lie)}(n)^{\\hat \\theta}\\,\\nu^2 \\sin^2\\psi\\cos^2\\alpha\\label{acc3}\\\\\n&&-k_{\\rm (Lie)}(n)^{\\hat r}\\, \\nu^2\\sin\\psi\\sin\\alpha\\cos\\psi\\nonumber\\\\\n&&+2\\nu\\cos \\alpha\\sin\\psi\\, \\theta(n)^{\\hat \\theta}{}_{\\hat \\varphi}]\\nonumber\\\\\n&&+ \\gamma\\left(\\gamma^2 \\cos\\psi \\frac{\\rm d \\nu}{\\rm d \\tau}-\\nu \\sin\\psi \\frac{\\rm d \\psi}{\\rm d \\tau}\\right).\\nonumber\\\\\na(U)^{\\hat \\varphi}&=& -\\gamma^2 \\nu^2\\cos \\alpha\\sin\\psi\\left[ \\sin \\alpha \\sin\\psi\\, k_{\\rm (Lie)}(n)^{\\hat r}\\right.\\label{acc2}\\\\ \n&&\\left.+ k_{\\rm (Lie)}(n)^{\\hat \\theta}\\cos\\psi\\right]+ \\gamma\\left(\\gamma^2 \\cos \\alpha\\sin\\psi \\frac{\\rm d \\nu}{\\rm d \\tau}\\right.\\nonumber\\\\\n&&\\left.-\\nu\\sin \\alpha\\sin\\psi \\frac{\\rm d \\alpha}{\\rm d \\tau}+\\nu\\cos\\alpha\\cos\\psi \\frac{\\rm d \\psi}{\\rm d \\tau}\\right),\\nonumber\\\\\n&&a(U)^{\\hat t}=\\gamma^2\\nu\\left\\{\\sin \\alpha \\sin\\psi\\left[a(n)^{\\hat r}\\right.\\right.\\label{acc4}\\\\\n&&\\left.\\left.+2\\nu\\cos \\alpha\\sin\\psi\\, \\theta(n)^{\\hat r}{}_{\\hat \\varphi}\\right] \\right. \\nonumber\\\\\n&&\\left.+\\cos\\psi\\left[a(n)^{\\hat \\theta}\\right.\\right.\\nonumber\\\\\n&&\\left.\\left.+2\\nu\\cos\\alpha\\sin\\psi \\theta(n)^{\\hat \\theta}{}_{\\hat \\varphi}\\right]\\right\\}+ \\gamma^3\\nu \\frac{\\rm d \\nu}{\\rm d \\tau}\\nonumber. \n\\end{eqnarray}\n\n\\subsection{Splitting of the radiation force}\n\\label{sec:split_rad_field}\nWe assume that the radiation test particle interaction occurs through elastic, Thomson-like scattering, characterized by a constant momentum-transfer cross section $\\sigma$, independent of direction and frequency of the radiation field. The radiation force is \\cite{DeFalco20183D}\n\\begin{equation} \\label{radforce}\n{\\mathcal F}_{\\rm (rad)}(U)^\\alpha = -\\sigma P(U)^\\alpha{}_\\beta \\, T^{\\beta}{}_\\mu \\, U^\\mu \\,,\n\\end{equation}\nwhere $P(U)^\\alpha{}_\\beta=\\delta^\\alpha_\\beta+U^\\alpha U_\\beta$ projects a vector orthogonally to $\\bold{U}$. Decomposing the photon four-momentum $\\bold{k}$ first with respect to the test particle four-velocity, $\\bold{U}$, and then in the ZAMO frame, $\\bold{n}$, we have \\cite{DeFalco20183D}\n\\begin{equation} \\label{diff_obg}\n\\bold{k} = E(n)[\\bold{n}+\\boldsymbol{\\hat{\\nu}}(k,n)]=E(U)[\\bold{U}+\\boldsymbol{\\hat {\\mathcal V}}(k,U)].\n\\end{equation}\nUsing Eq. (\\ref{diff_obg}) in Eq. (\\ref{radforce}), we obtain\n\\begin{equation} \\label{Frad0}\n\\begin{aligned}\n{\\mathcal F}_{\\rm (rad)}(U)^\\alpha&=-\\sigma \\Phi^2 [P(U)^\\alpha{}_\\beta k^\\beta]\\, (k_\\mu U^\\mu)\\\\\n&=\\sigma \\, [\\Phi E(U)]^2\\, \\hat {\\mathcal V}(k,U)^\\alpha\\,.\n\\end{aligned}\n\\end{equation}\nThe test particle equations of motion is $m \\bold{a}(U) = \\boldsymbol{{\\mathcal F}_{\\rm (rad)}}(U)$, where $m$ is the test particle mass;  by defining $\\tilde \\sigma=\\sigma\/m$, we have \n\\begin{equation}\\label{geom}\n\\bold{a}(U)=\\tilde \\sigma \\Phi^2 E(U)^2  \\,\\boldsymbol{\\hat {\\mathcal V}}(k,U).\n\\end{equation} \nTaking the scalar product of Eq. (\\ref{diff_obg}) and \n $\\bold{U}$ and considering $\\zeta=\\pi\/2$, we find \n\\begin{equation} \\label{enepart}\nE(U)=\\gamma E(n)[1-\\nu\\sin\\psi\\cos(\\alpha-\\beta)],\n\\end{equation}\nSuch splitting permits to determine $\\boldsymbol{\\hat{\\mathcal{V}}}(k,U)=\\hat{\\mathcal{V}}^t\\boldsymbol{n}+\\hat{\\mathcal{V}}^r\\boldsymbol{e_{\\hat r}}+\\hat{\\mathcal{V}}^\\theta \\boldsymbol{e_{\\hat\\theta}}+\\hat{\\mathcal{V}}^\\varphi \\boldsymbol{e_{\\hat\\varphi}}$ as \\cite{DeFalco20183D}\n\\begin{eqnarray}\n&&\\hat{\\mathcal{V}}^{\\hat r}=\\frac{\\sin\\beta}{\\gamma [1-\\nu\\sin\\psi\\cos(\\alpha-\\beta)]}-\\gamma\\nu\\sin\\psi\\sin\\alpha,\\label{rad1}\\\\\n&&\\hat{\\mathcal{V}}^{\\hat \\theta}=-\\gamma\\nu\\cos\\psi \\label{rad2},\\\\\n&&\\hat{\\mathcal{V}}^{\\hat\\varphi}=\\frac{\\cos\\beta}{\\gamma [1-\\nu\\sin\\psi\\cos(\\alpha-\\beta)]}-\\gamma\\nu\\sin\\psi\\cos\\alpha,\\label{rad3}\\\\\n&&\\hat{\\mathcal{V}}^{\\hat t}=\\gamma\\nu\\left[\\frac{\\sin\\psi\\cos(\\alpha-\\beta)-\\nu}{1-\\nu\\sin\\psi\\cos(\\alpha-\\beta)}\\right].\\label{rad4}\n\\end{eqnarray}\n\n\\subsection{General relativistic equations of motion}\nThe equations of motion for a test particle moving in a 3D space, defined in terms of $(r,\\theta,\\varphi,\\nu,\\psi,\\alpha)$, represent a set of coupled ordinary differential equations of the first order, i.e., \\cite{DeFalco20183D}\n\\begin{eqnarray}\n&&\\frac{d\\nu}{d\\tau}= -\\frac{1}{\\gamma}\\left\\{ \\sin\\alpha \\sin\\psi\\left[a(n)^{\\hat r}\\right.\\right.\\label{EoM1}\\\\\n&&\\left.\\left.\\ \\quad +2\\nu\\cos \\alpha\\sin\\psi\\, \\theta(n)^{\\hat r}{}_{\\hat \\varphi} \\right]+\\cos\\psi\\left[a(n)^{\\hat \\theta}\\right. \\right.\\nonumber\\\\\n&&\\left.\\left.\\ \\quad+2\\nu\\cos\\alpha\\sin\\psi\\, \\theta(n)^{\\hat \\theta}{}_{\\hat \\varphi}\\right]\\right\\}+\\frac{\\tilde{\\sigma}[\\Phi E(U)]^2}{\\gamma^3\\nu}\\hat{\\mathcal{V}}^{\\hat t},\\nonumber\\\\\n&&\\frac{d\\psi}{d\\tau}= \\frac{\\gamma}{\\nu} \\left\\{\\sin\\psi\\left[a(n)^{\\hat \\theta}+k_{\\rm (Lie)}(n)^{\\hat \\theta}\\,\\nu^2 \\cos^2\\alpha\\right.\\right.\\label{EoM2}\\\\\n&&\\left.\\left.\\ \\quad+2\\nu\\cos \\alpha\\sin^2\\psi\\ \\theta(n)^{\\hat \\theta}{}_{\\hat \\varphi}\\right]-\\sin \\alpha\\cos\\psi \\left[a(n)^{\\hat r}\\right.\\right.\\nonumber\\\\\n&&\\left.\\left.\\ \\quad+k_{\\rm (Lie)}(n)^{\\hat r}\\,\\nu^2+2\\nu\\cos \\alpha\\sin\\psi\\, \\theta(n)^{\\hat r}{}_{\\hat \\varphi}\\right]\\right\\}\\nonumber\\\\\n&&\\ \\quad+\\frac{\\tilde{\\sigma}[\\Phi E(U)]^2}{\\gamma\\nu^2\\sin\\psi}\\left[\\hat{\\mathcal{V}}^{\\hat t}\\cos\\psi-\\hat{\\mathcal{V}}^{\\hat \\theta}\\nu\\right],\\nonumber\\\\\n&&\\frac{d\\alpha}{d\\tau}=-\\frac{\\gamma\\cos\\alpha}{\\nu\\sin\\psi}\\left[a(n)^{\\hat r}+2\\theta(n)^{\\hat r}{}_{\\hat \\varphi}\\ \\nu\\cos\\alpha\\sin\\psi\\right.\\label{EoM3}\\\\\n&&\\left.\\ \\quad+k_{\\rm (Lie)}(n)^{\\hat r}\\,\\nu^2+k_{\\rm (Lie)}(n)^{\\hat \\theta}\\,\\nu^2\\cos^2\\psi \\sin\\alpha\\right]\\nonumber\\\\\n&&\\ \\quad+\\frac{\\tilde{\\sigma}[\\Phi E(U)]^2\\cos\\alpha}{\\gamma\\nu\\sin\\psi}\\left[\\hat{\\mathcal{V}}^{\\hat r}-\\hat{\\mathcal{V}}^{\\hat \\varphi}\\tan\\alpha\\right],\\nonumber\\\\\n&&U^r\\equiv\\frac{dr}{d\\tau}=\\frac{\\gamma\\nu\\sin\\alpha\\sin\\psi}{\\sqrt{g_{rr}}}, \\label{EoM4}\\\\\n&&U^\\theta\\equiv\\frac{d\\theta}{d\\tau}=\\frac{\\gamma\\nu\\cos\\psi}{\\sqrt{g_{\\theta\\theta}}} \\label{EoM5},\\\\\n&&U^\\varphi\\equiv\\frac{d\\varphi}{d\\tau}=\\frac{\\gamma\\nu\\cos\\alpha\\sin\\psi}{\\sqrt{g_{\\varphi\\varphi}}}-\\frac{\\gamma N^\\varphi}{N},\\label{EoM6}\n\\end{eqnarray}\nFor $b=0$ the equations of motion reduce to the radial radiation field in 3D case \\cite{DeFalco20183D}.\n\nWe define the relative luminosity of the radiation field as $A\/M=\\tilde{\\sigma}\\Phi_0^2E^2\\equiv L_\\infty\/L_{\\rm Edd}$, ranging in $[0,1]$, where $L_\\infty$ is the luminosity at infinity and $L_{\\rm EDD}=4\\pi Mm\/\\sigma$ is the is the Eddington luminosity at infinity \\cite{Bini2009,Bini2011,DeFalco20183D}. Using Eqs. (\\ref{INT_PAR}) and (\\ref{enepart}), the term $\\tilde{\\sigma}[\\Phi E(U)]^2$ becomes\n\\begin{equation} \\label{eq: sigma_tilde}\n\\tilde{\\sigma}[\\Phi E(U)]^2=\\frac{ A\\,\\gamma^2(1+b N^\\varphi)^2[1-\\nu\\sin\\psi\\cos(\\alpha-\\beta)]^2}{N^2\\sqrt{R_{b,q}(r)}}.\n\\end{equation}\n\nIn Appendix \\ref{Appendix_Classic_Limit} we show the way Eqs. (\\ref{EoM1})--(\\ref{EoM6}) reduce in the classical limit. In Appendix \\ref{Appendix_Weak_Field} we show the weak field approximations of Eqs. (\\ref{EoM1})--(\\ref{EoM6}) at the first order in the Kerr parameter $a$ for slow rotations ($a\\to0$).\n\n\\section{Critical hypersurfaces}\n\\label{sec:critc_rad}\nThe set of Eqs. (\\ref{EoM1})--(\\ref{EoM6}) admits a solution in which gravitational attraction, radiation pressure, and PR drag effect balance each other on an axially symmetric hypersurface, partially or fully encapsulating the emitting sphere. In order to obtain the condition determining the radial coordinate of such a hypersurface, the critical radius $r_{\\rm (crit)}$, we consider a captured test particle in radial equilibrium moving in a purely circular motion ($\\alpha=0\\,;\\psi=\\pm\\pi\/2;\\nu=\\mbox{const}$) along,  in general, a non-equatorial orbit. Then for $\\frac{d\\nu}{d\\tau}=0$ Eq. (\\ref{EoM1}) takes the following form\n\\begin{equation}\\label{eq:crit_hyper1}\n\\frac{A(1+bN^\\varphi)^2(1 - \\nu\\cos\\beta)(\\cos\\beta-\\nu)}{N^2 \\sqrt{R_{b,q}(r_{\\rm (crit))}}}=0,\n\\end{equation}\nfrom which it follows \n\\begin{equation}\\label{eq:crit_veloc}\n\\nu=\\cos\\beta.\n\\end{equation}\nTherefore the azimuthal velocity of the captured test particle is equal to the azimuthal velocity of the photons of the radiation field\\footnote{Assuming the opposite direction of the particle velocity ($\\alpha=\\pi$), we get at the same solution except for the negative sign of $\\nu$.}.\nConsidering that in the ZAMO frame the velocity of the captured test particle is tangential to the hypersurface ($\\alpha=0$), for $\\frac{d\\alpha}{d\\tau}=0$ Eq. (\\ref{EoM3}) takes the following form\n\\begin{equation}\\label{eq:crit_hyper2}\n\\begin{aligned}\n&a(n)^{\\hat r} + 2\\theta(n)^{\\hat r}{}_{\\hat\\varphi}\\nu+k_{\\rm (Lie)}(n)^{\\rm r}\\nu^2 =\\\\\n&=\\frac{A(1+bN^\\varphi)^2\\sin\\beta}{N^2\\gamma^3 \\sqrt{R_{b,q}(r_{\\rm (crit))}}}.\n\\end{aligned}\n\\end{equation}\nThis generalises the result obtained in \\cite{Bini2011}, where the equilibrium was analyzed only in the equatorial plane. Note that, similarly to the 3D case with purely radial radiation field \\cite{DeFalco20183D}, a latitudinal drift  on the critical hypersuface towards the equatorial plane is to be expected, as the condition $\\frac{d\\psi}{d\\tau}=0$ is not fulfilled automatically. In Sec. \\ref{sec:EMCH} we analyse in detail the condition for the existence of orbits with a constant latitude coordinate. Note also that in the case of a radiation field with zero angular momentum ($b=0\\,;\\beta=\\pi\/2$), Eq. (\\ref{eq:crit_hyper2}) reduces to Eq. (53) in \\cite{DeFalco20183D}; for non-zero angular momentum radiation fields \nthe same holds on the polar axis ($\\theta=0,\\pi$) where the angular momentum of the photons emitted by the spherical surface is zero. Therefore, the axisymmetric shape of the critical hypersurfaces can be obtained by solving the implicit equation (\\ref{eq:crit_hyper2}) for the $(r,\\theta)$ coordinates by specifying a set of the initial parameters $\\left\\{A,a,R_\\star,\\Omega_\\star\\right\\}$. The pair of photon impact parameters $(b,q)$ is found in terms of $R_\\star,\\Omega_\\star,\\theta$  through Eq. (\\ref{kerr_impact_parameter}) and Eq. (\\ref{q_r}), respectively \\footnote{Equation (\\ref{q_r}) displays a divergence in $q$  at the polar axis (i.e., $\\theta=0,\\pi$). Therefore our numerical analysis of the behavior extends to the close vicinity of, rather than the poles themselves.}.\n\nIn our previous work \\cite{DeFalco20183D}, in which a zero angular momentum radiation field ($b=0\\,;\\beta=\\pi\/2$) was analyzed, we found that for non-zero $a$ the critical hypersurface always takes a prolate shape due to the fact the frame dragging effect attains its maximum at the equator. In the more general case considered in the present study, the shape of the critical surface, besides frame dragging, is determined also by the angular momentum of the radiation field photons, which is now a function of the polar coordinate. Hereafter we show that the critical hypersurfaces may morph between an oblate and a prolate shape depending on the parameters governing the value of the critical radius $r_{\\rm (crit)}=r_{\\rm (crit)}(A,a,\\theta,\\Omega_\\star,R_\\star)$. \nWe carry out a more detailed analysis of the properties of the critical hypersurface in the  cases of a NS and a BH in Secs. \\ref{sec:CHNS} and \\ref{sec:CHBH}, respectively.\n\n\n\\subsection{Multiplicity of the critical hypersurface}\n\\label{sec:EMCH}\n\nAs demonstrated in \\cite{Bini2011}, Eq. (\\ref{eq:crit_hyper2}) may have one or three solutions depending on the values of $A$,$a$,$b$. The switch from one regime to the other occurs at fixed values of $a$,$\\theta$ when a critical value of the impact parameter $B$ is reached; the corresponding criterion is discussed in detail below.\nIn the mode of existence of three critical radii, the innermost solution is located in the close vicinity of the event horizon, the second inner solution is inside below the unstable spherical photon orbit region \\cite{2003GReGr..35.1909T}, and the only external solution may be located far from the emitting surface (see Figs. 1 -- 3 in \\cite{Bini2011} for details).\nIn the case studied here of a rigidly rotating emitting surface, the critical radii located inside the emitting surface are not relevant. In addition, the range of the impact parameter $b$ and thus the formal existence of the regime of three critical radii are limited by the maximum rotational frequency of a NS and, in the case of a BH, by the fact that the rotation of the emitting surface cannot be superluminal (see Secs. \\ref{sec:brange_kerr} -- \\ref{sec:NS_surface}).  To find the criterion for distinguishing the regimes with one critical radius and with three critical radii, we approximated Eq. (\\ref{eq:crit_hyper2}) with a third order polynomial. Then, in order to count the number of solutions of Eq. (\\ref{eq:crit_hyper2}), we exploited the standard formula for third order algebraic equation, which can be written in the canonical form as $a_3x^3+a_2x^2+a_1x+a_0=0$ \\cite{1965hmfw.book.....A}. \nThe discriminant $\\Delta_{\\rm III}$ is defined as follows\n\\begin{equation}\n\\label{eq:delta3}\n\\Delta_{\\rm III}=4(3a_3a_1-a_2^2)^3+(27a_3^2a_0-9a_3a_2a_1+2a_2^3)^2.\n\\end{equation}\nIn the case of $\\Delta_{\\rm III}>0$ there exist three real solutions while in the case of $\\Delta_{\\rm III}<0$ there is only one real solution. In the limit case of $\\Delta_{\\rm III}=0$ the equation has a multiple real root. To this end, we consider separately the following two functions\n\\begin{eqnarray}  \ny_1&&=a(n)^{\\hat r}+2\\theta(n)^{\\hat r}{}_{\\hat\\varphi}\\cos\\beta+k_{\\rm (Lie)}(n)^{\\rm r}\\cos^2\\beta,\\label{eq:func1}\\\\\ny_2&&=\\frac{A(1+ bN^\\varphi)^2\\sin^4\\beta}{N^2 \\sqrt{R_{b,q}(r}}.\\label{eq:func2}\n\\end{eqnarray}\n\nLet us consider first Eq. (\\ref{eq:func1}), and its Taylor expansion for $1\/r\\to 0$; we obtain\n\\begin{equation}\n\\begin{aligned}\ny_1&\\approx\\frac{r^2}{\\rho\\sqrt{\\Sigma^5}}\\left[r^3+r^2 \\left(1-\\frac{b^2}{\\sin^2\\theta}\\right)\\right.\\\\\n&\\left.+r \\left(-6 a b+\\frac{3 b^2}{\\sin^2\\theta}-a^2\\cos2 \\theta+2 a^2+3\\right)\\right.\\\\\n&\\left.+6 a b-\\frac{\\left(2 a^2 \\cos 2 \\theta +a^2+3\\right) b^2}{2 \\sin^2\\theta}\\right.\\\\\n&\\left.+\\frac{1}{2} \\left(3 a^2 \\cos2 \\theta-4 a^2+5\\right)\\right],\n\\end{aligned}\n\\end{equation}\nwhere $\\rho\\sqrt{\\Sigma^5}$ is a factor in common with all terms in $y_1$ (for further details see Table 1 in \\cite{DeFalco20183D}) and can be approximated through the following polynomial\n\\begin{equation}\n\\begin{aligned}\n\\frac{1}{\\rho\\sqrt{\\Sigma^5}}&\\approx\n-\\frac{2a^2 \\sin ^2\\theta}{r^{10}}-\\frac{a^2\\left(5 \\cos^2\\theta+2\\right)}{2 r^9}+\\frac{1}{r^7}.\n\\end{aligned}\n\\end{equation} \nCombining these results, we get\n\\begin{eqnarray} \\label{eq:approx1}\ny_1&&\\approx\n\\frac{1}{r^5}\\left[r^3+r^2 \\left(1-\\frac{b^2}{\\sin^2\\theta}\\right)\\right.\\\\\n&&\\left.+r \\left(-6 ab+\\frac{3b^2}{\\sin^2\\theta}-\\frac{5a^2\\cos^2\\theta}{2}-a^2\\cos2 \\theta+a^2+3\\right)\\right.\\notag\\\\\n&&\\left.-\\frac{a^2}{2}\\left(5 \\cos^2\\theta+2\\right) \\left(1-\\frac{b^2}{\\sin^2\\theta}\\right)-2 a^2 \\sin^2\\theta+6ab\\right.\\notag\\\\\n&&\\left.\\frac{1}{2} \\left(3 a^2 \\cos2 \\theta-4 a^2+5\\right)-\\frac{\\left(2 a^2 \\cos2\\theta+a^2+3\\right)b^2}{2\\sin^2\\theta}\\right]\\notag\n\\end{eqnarray}\n\nLet us now consider Eq. (\\ref{eq:func2}), and its Taylor expansion for $1\/r\\to 0$\n\\begin{eqnarray} \\label{eq:approx2}\ny_2&&\\approx\n\\frac{A}{r^5} \\left\\{r^3+2r^2+r\\left[-\\frac{3}{2} \\frac{b^2}{\\sin^2\\theta}-\\frac{a^2}{2}\\left(\\cos^2\\theta+1\\right)+4\\right]\\right.\\notag\\\\\n&&\\left.\\qquad\\quad-2ab+2 a^2 \\sin^2\\theta-4 a^2+8\\right\\}.\n\\end{eqnarray}\n\nCombining Eqs. (\\ref{eq:approx1}) and (\\ref{eq:approx2}) we obtain an approximation of Eq. (\\ref{eq:crit_hyper2}) through a polynomial of third order of the form \n\\begin{equation} \\label{eq:3eq}\n\\frac{a_3r^3+a_2r^2+a_1r+a_0}{r^5}=0,\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\label{eq:deltacoef}\na_3&&=1-A,\\\\\na_2&&=1-2 A-\\frac{b^2}{\\sin^2\\theta},\\notag\\\\\na_1&&=-\\frac{A}{2} \\left[-\\frac{3b^2}{\\sin^2\\theta}-a^2(1+\\cos ^2\\theta)+8\\right]-6 a b\\notag\\\\\n&& \\frac{3b^2}{\\sin^2\\theta}-\\frac{a^2}{2} \\left(5 \\cos ^2\\theta+2\\right)-a^2 \\cos2\\theta+2 a^2+3,\\notag\\\\\na_0&&=-A \\left(-2 a b+2 a^2 \\sin ^2\\theta-4 a^2+8\\right)\\notag\\\\\n&&-\\frac{a^2}{2} \\left(5 \\cos ^2\\theta+2\\right)\\left(1-\\frac{b^2}{\\sin^2\\theta}\\right)+6 a b-2 a^2 \\sin ^2\\theta\\notag\\\\\n&&-\\frac{\\left(2 a^2 \\cos2 \\theta+a^2+3\\right) b^2}{2\\sin^2\\theta}+\\frac{\\left(3 a^2 \\cos2 \\theta-4 a^2+5\\right)}{2}.\\notag\n\\end{eqnarray}\nTherefore, the above coefficients determine the value of the discriminant (\\ref{eq:delta3}).\nPhysically meaningful critical radii are located above the emitting surface.\n\n\\subsection{Constant latitude suspended orbits bound on the critical hypersurface}\n\\label{sec:susporbits}\n\nIn the simpler case of the test radiation field with zero angular momentum ($b=0$) investiged in the previous paper \\cite{DeFalco20183D}, the particle captured on an off-equatorial circular orbit bound on the critical hypersurface with a constant polar coordinate $\\bar{\\theta}$ must fulfil the condition $\\cos\\psi=0, \\frac{d\\psi}{d\\tau}=0,\\nu=0 $ for which a complete balance of all forces acting at the critical hypersurface is attained. Here we investigate the condition for similar suspended off-equatorial circular orbits for the more complex case of the rotating test radiation field ($b\\ne 0$).\n\nFor a particle to be captured on an suspended  off-equatorial circular orbit bound on the critical hypersurface, i.e. without the presence of the effect of latitudinal drift, the condition $d\\psi\/d\\tau=0$ must be fulfilled along with the solution of Eq. (\\ref{eq:crit_hyper2}).\nThe test particle velocity on the critical hypersurface is equal to the azimuthal photon velocity ($\\nu=\\cos\\beta$). Using Eq. (\\ref{EoM2}) with $\\alpha=0$, $r=r_{\\rm crit}(\\bar{\\theta})$, $\\theta=\\bar{\\theta}$, imposing $d\\psi\/d\\tau=0$, we obtain\n\\begin{equation}\n\\begin{aligned}\\label{eq:susporbit}\n&\\gamma\\nu\\left[a(n)^{\\hat \\theta}+k_{\\rm (Lie)}(n)^{\\hat \\theta}\\,\\nu^2+2\\nu\\sin^2\\psi\\ \\theta(n)^{\\hat \\theta}{}_{\\hat \\varphi}\\right]+\\\\\n&+\\frac{A(1+bN^\\varphi)^2\\nu\\cos\\psi(1-\\nu^2\\sin\\psi)}{N^2\\sqrt{R_{b,q}(r_{\\rm crit})}}=0.\n\\end{aligned}\n\\end{equation}\nBy solving this implicit equation for $\\psi$, we obtain the condition for suspended circular orbits.  \\footnote{Note that in the case reported in the previous paper \\cite{DeFalco20183D}, i.e. for $b=0$, the condition for suspended circular orbits takes the very simple form $a(n)^{\\hat \\theta}=0$.}.\n\\begin{figure}[h!]\n\t\\centering\n\t\t\\includegraphics[width=1.0\\linewidth]{Plot1n}\n\t\t\t\\includegraphics[width=1.0\\linewidth]{Plot2n}\n\t\t\\caption{The local polar angle in the ZAMO frame $\\psi$ as a function of the latitudinal coordinate $\\bar{\\theta}$ of circular suspended orbits in the case of NS of mass $M=1.5M_\\odot$ and radius $R_\\star=6M$. Top panel: The plot is constructed for fixed luminosity parameter $A=0.8$ and star rotation frequency $f_{\\star}=300$ Hz (red), $f_{\\star}=400$ Hz (green), $f_{\\star}=500 $Hz (blue), $f_{\\star}=600$ Hz (black). Bottom panel: The plot is constructed for fixed star rotation frequency $f_{\\star}=700$ Hz and luminosity parameter $A=0.7$ (black), $A=0.7$ (blue), $A=0.8$ (green), $A=0.85$ (red).}\n\t\\label{fig:FigSO}\n\\end{figure}\n\nThe condition (\\ref{eq:susporbit}) significantly depends on the radius of the emitting surface $R_\\star$ and its rotational frequency $\\Omega_\\star$. In Fig \\ref{fig:FigSO}, we illustrate the behaviour of the local polar angle in the ZAMO frame $\\psi$ \nas a function of the latitudinal coordinate $\\bar{\\theta}$ of circular suspended orbits bound on the critical hypersurface. \nOn the equator, the value of the local polar angle is $\\psi=\\pi\/2$ due to the mirror symmetry.  The angle $\\psi$ decreases with growing latitudinal coordinate $\\bar{\\theta}$ to its minimum, then $\\psi$ grows and it reaches the value of $\\psi=\\pi\/2$ again on the polar axis, where the test radiating field is non-rotating ($b=0$).\n\n\\subsection{Critical hypersurfaces in the case of a NS}\n\\label{sec:CHNS}\n\\begin{figure}[th!]\n\t\\centering\n\t\t\\includegraphics[trim=0.5cm 0cm 0cm 0cm,scale=0.32]{Fig2}\n\t\\caption{Critical radius $r_{\\rm (crit)}$ at the poles (blue line) and at equator (orange line) as a function of the luminosity parameter $A$ for a NS model with rotation frequency $f_{\\star}=600$, $\\Omega_\\star=0.031M^{-1}\\,,a=0.41$). The dashed red line represents the NS surface, the gray area indicates the (nonphysical) solutions inside the NS.}\n\t\\label{fig:Fig2}\n\\end{figure}\n\\begin{figure}[th!]\n\t\\centering\n\t\t\\includegraphics[trim=0.5cm 0cm 0cm 0cm,scale=0.35]{Fig3}\n\t\\caption{Critical radius $r_{\\rm (crit)}$ at the poles (blue line) and at the equator of the NS (orange line) as a function of the NS rotation angular velocity $\\Omega_\\star$ for a luminosity parameter of $A=0.81$. The dashed red line represents the NS surface, the gray area indicates the (nonphysical) solutions inside NS.}\n\t\\label{fig:Fig3}\n\\end{figure}\n\n\\begin{figure*}[th!]\n\t\\centering\n\t\\hbox{\n\t\t\\includegraphics[scale=0.28]{Fig4_1}\n\t\t\\hspace{0.3cm}\n\t\t\\includegraphics[scale=0.36]{Fig4_2}}\n\t\\caption{Left panel: Critical hypersurfaces for the case of a rotating NS with Kerr parameter $a=0.41$ (orange, $f_{\\star}=600$ Hz, $\\Omega_\\star=0.031M^{-1}$) and for the non-rotating case (blue). For a non-rotating  NS, the critical radius is constant and it takes unphysical value of $r_{\\rm (crit)} \\sim 5.56M$. In the case of a rotating NS, the critical radius takes the value $r^{\\rm eq}_{\\rm(crit)} \\sim 12.05M$ in the equatorial plane, while it takes the value of  $r^{\\rm pole}_{\\rm(crit)} \\sim 5.74M$ at the poles. The relative luminosity of the radiating field is fixed at $A=0.8$. Right panel: Critical hypersurfaces for a relative luminosity $A=0.75,\\, 0.78, \\,0.8, \\,0.85, \\,0.88$ at Kerr parameter $a=0.41$ ($f_{\\star}=600$ Hz, $\\Omega_\\star=0.031M^{-1}$). The corresponding critical radii in the equatorial plane are $r^{\\rm eq}_{\\rm(crit)} \\sim 8.88M,\\  10.61M,\\ 12.05M,\\ 17.26M,\\ 22.43M\\ $, while at poles they are $r^{\\rm pole}_{\\rm(crit)} \\sim 4.73M,\\  5.28M,\\  5.74M,\\  7.43M,\\  9.11M$. In both panels the gray sphere represents the NS surface, separating the physical solutions (outside its surface) from the unphysical ones (inside its surface). The red arrow is the polar axis.}\n\t\\label{fig:Fig4}\n\\end{figure*}\n\n\\begin{figure*}[th!]\n\t\\centering\n\t\\hbox{\n\t\t\\includegraphics[trim=5cm 6.2cm 1cm 0cm,scale=0.35]{Fig4_3}\n\t\t\\hspace{0.3cm}\n\t\t\\includegraphics[scale=0.33]{Fig4_4}}\n\t\\caption{Left panel: The spatial velocity of test particles $v_{\\rm (crit)}$ captured on the critical hypersurface forming a lobe and the spatial velocity of the NS surface $v_{\\rm NS}$ as measured by a static observer at infinity as functions of polar angle $\\theta$.  The spatial velocity values ranges between zero and maximum value of $0.21$. The red arrow represents the polar axis. For the critical hypersurface size dimensions see the corresponding case in Fig. \\ref{fig:Fig4}. Right panel: The spatial velocity of the critical hypersurface forming a lobe $v_{\\rm (crit)}$ (blue) and the spatial velocity of NS surface $v_{\\rm NS}$ (yellow) as a function of the polar angle $\\theta$. The vertical dashed blue line is located at $\\theta=42.5^\\circ$ representing the polar angle at which the critical hypersurface lobe intersects the NS surface. Both plots are constructed for template NS with Kerr parameter $a=0.41$ ($f_{\\star}=600$ Hz, $\\Omega_\\star=0.031M^{-1}$) and for a radiating field with luminosity parameter $A=0.75$.}\n\t\\label{fig:Fig4bis}\n\\end{figure*}\n\\begin{figure}[th!]\n\t\\centering\n\t\t\t\\includegraphics[width=1.0\\linewidth]{Fig4_5}\n\t\\caption{The angular velocity $\\Omega_{\\rm (crit)}$ of test particles captured on the critical hypersurface forming a lobe as a function of polar angle $\\theta$.  The vertical dashed blue line is located at $\\theta=42.5^\\circ$ representing the polar angle at which the critical hypersurface lobe intersects the NS surface. The plot is constructed for a NS with Kerr parameter $a=0.41$ ($f_{\\star}=600$ Hz, $\\Omega_\\star=0.031M^{-1}$) and for radiating field with luminosity parameter $A=0.75$.}\n\t\\label{fig:Fig4bis1}\n\\end{figure}\n\nWe consider a template NS of mass $M=1.5M_\\odot$ and radius $R_\\star=6M$. The emitting source we consider is the spherical surface of the NS. All solutions lying inside the NS have no physical meaning (these include also the internal solutions Eq. (\\ref{eq:crit_hyper2}) in the close vicinity of the event horizon and in the region of the spherical photon orbits).  \n\nIn Fig. \\ref{fig:Fig2} we investigate the way the critical radius changes along the polar axis and at the equator of NS  for different values of the luminosity parameter. The NS rotation frequency is fixed here at  $f_\\star=600$ Hz (corresponding to $\\Omega_\\star=0.031M^{-1}$ and through Eq. (\\ref{eq:NSa}) to Kerr parameter $a=0.41$). The critical hypersurface above the emitting surface is always oblate (note that it changes from prolate to oblate for $A\\simeq 0.65$ in the non-physical region inside the NS) . Moreover, the difference of the polar and equatorial value of the critical radius gives rise to an interesting new effect. For the luminosity range $A \\in[0.705,0.81]$, the physically meaningful part of the critical hypersurface located above the surface of NS forms a lobe around the star equator (see the right panel of Fig \\ref{fig:Fig4} and Fig. \\ref{fig:Fig4bis}). For $A<0.705$, the entire critical hypersurface is located inside NS (and thus is nonphysical), while for $A>0.81$, the entire critical hypersurface is located above the  NS surface. For the lower limiting value $A=0.705$, the critical hypersurface just emerges from the NS interior at the equator, while for the upper limiting value $A=0.81$, the completely emerged critical hypersurface touches only the poles of NS.\n\n\nFigure \\ref{fig:Fig3} shows the way the critical radius behaves in terms of the NS angular velocity. The luminosity parameter is fixed at the value of $A=0.81$. The  NS rotation frequency range is \n$f_\\star \\in[0,800]$ Hz (corresponding to $\\Omega_\\star\\in[0,0.041]M^{-1}$ or $a\\in[0,0.55]$, note that fastest know NS spins at $\\sim716$ Hz \\cite{Hessels2006}). We consider only positive rotation frequencies, because $\\Omega_\\star$ is linearly connected to the Kerr parameter  $a$ by the relation (\\ref{eq:NSa}).\nOn the poles, where surface velocity is zero and frame dragging effect is absent, the critical radius value varies slightly only as a result of changes in spacetime geometry as determined by $a$ growing with the $\\Omega_\\star$. The critical radius on the pole reaches the radius of the NS for $\\Omega_\\star=0.031M^{-1}$, which is in accordance with the profile of polar critical radius plotted in Fig. \\ref{fig:Fig2}. On the contrary there is a much stronger dependence on $\\Omega_\\star$ at the equator because the surface velocity and magnitude of frame dragging reach their maxima. Thus, similar to Fig. \\ref{fig:Fig2}, the critical hypersurface located above the emitting surface of NS is always oblate.\n \nThe left panel of Fig. \\ref{fig:Fig4} compares the shape of the critical hypersurfaces for a rotating template NS with Kerr parameter $a=0.41$ ($f_\\star=600$ Hz, $\\Omega_\\star=0.031M^{-1}$)\nand for a non-rotating NS ($a=0$), where the luminosity parameter of the radiating field is fixed at the value of $A=0.8$. In the rotating case, the critical radius is $r_{\\rm crit}^{\\rm eq}\\sim 12.05M$ at the equator and $r_{\\rm crit}^{\\rm pole}\\sim 5.74M$ at the poles. In the non-rotating case, the critical hypersurface naturally takes the shape of a sphere with the radius $r_{\\rm crit}\\sim 5.56M$. Indeed, the non-rotating case has no physical meaning as the resulting equilibrium sphere is located inside NS. However, such a non-rotating solution coincides with the case of a pure radially moving test field photon on the background of the Schwarzschild spacetime geometry discussed in \\cite{DeFalco20183D}, where we calculated the critical radius as a result of a static equilibrium.\n\nThe right panel of Fig. \\ref{fig:Fig4} illustrates the shape of the critical hypersurfaces for the values of the relative luminosity in the interval $0.75-0.88$ and for a constant value of the Kerr parameter $a=0.41$ ($f_{\\star}=600$ Hz, $\\Omega_\\star=0.031M^{-1}$). The figure clearly demonstrates that in the appropriate luminosity range, the critical hypersurface forms only a lobe around the NS equator .\nIn Fig. \\ref{fig:Fig4bis} we plot the spatial velocity of  the test particle captured on the critical hypersurface $v_{\\rm (crit)}$ as measured by a static observer at infinity, for a luminosity parameter of $A=0.75$ and a NS Kerr parameter of $a=0.41$ ($f_{\\star}=600$ Hz, $\\Omega_\\star=0.031M^{-1}$). The lobe of the critical hypersurface intersects the NS surface at $\\theta=42.5^\\circ$. \n\nThe spatial velocity of test particles captured on the critical hypersurface is\n\\begin{equation}\nv_{\\rm (crit)}=r_{\\rm (crit)}\\sin\\theta\\,\\Omega_{\\rm (crit)}. \n\\end{equation}\nBy using Eq. (\\ref{four_velocity}) and Eq. (\\ref{eq:crit_veloc}), the angular velocity of test particles captured on the critical hypersurface is \n\\begin{equation}\n\\Omega_{\\rm (crit)}=U^\\phi\/U^t\\,=\\frac{N\\cos\\beta}{\\sqrt{g_{\\varphi\\varphi}}}+\\Omega_{\\rm ZAMO}\\,.\n\\end{equation}\nThe spatial velocity of the NS surface $\\nu_{\\rm NS}$ as measured by a static observer at infinity is\\begin{equation}\nv_{\\rm NS}=R_\\star\\sin\\theta\\ \\Omega_\\star.\n\\end{equation}\n\nThe angular velocity of the particles captured on the critical hypersurface $\\Omega_{\\rm (crit)}$, which is equal to the angular velocity of NS surface $\\Omega_\\star$ at the intersection ring, decreases for increasing polar angle $\\theta$, reaching the minimum value on the equatorial plane.\nAt the intersection ring, the spatial velocity of the NS surface $v_{\\rm NS}$ and that of test particles captured on the critical hypersurface $v_{\\rm (crit)}$ are also equal. However $v_{\\rm (crit)}$ is always higher than the $v_{\\rm NS}$ on the critical hypersurface lobe, owing to the growth of $r_{\\rm (crit)}$ with the polar angle $\\theta$. $v_{\\rm (crit)}$ also grows with $\\theta$, and it reaches its maximum value $v_{\\rm (crit)}(\\theta=\\pi\/2)=0.21$ on the equatorial plane. \n\n\n\\subsubsection{Examples of test particle orbits in the vicinity of the critical hypersurface around NS}\nIn Fig. \\ref{fig:FigNSorbits}, we compare the results of the integration of the trajectories of test particles interacting with a radiation field emitted by the surface of a slowly or rapidly rotating NS. The left panels of Fig \\ref{fig:FigNSorbits} corresponds to the case of a rapidly rotating template NS with Kerr parameter $a=0.41$ ($f_{\\star}=600$ Hz, $\\Omega_\\star=0.031M^{-1}$) and with relative luminosity $A=0.8$. The critical hypersurface forms a large flattened lobe, with $r_{\\rm crit}^{\\rm eq}\\sim 12.05M$ at the equator and $r_{\\rm crit}^{\\rm pole}\\sim 5.74M$ at the poles. The right panels of Fig \\ref{fig:FigNSorbits} corresponds to the case of a slowly rotating template NS with Kerr parameter $a=0.07$ ($f_{\\star}=100$ Hz, $\\Omega_\\star=0.005M^{-1}$) and relative luminosity $A=0.85$. The critical hypersurface envelopes the entire NS surface, its radius being $r_{\\rm crit}^{\\rm eq}\\sim 7.37M$ at the equator and $r_{\\rm crit}^{\\rm pole}\\sim 7.21M$ at the poles.\n\\begin{figure*}[th!]\n\\centering\n\\hspace{1cm}\n\\vbox{\\hbox{\\includegraphics[scale=0.29]{Fig13a}\n\t \\hspace{1cm}\n\t \\includegraphics[scale=0.22]{Fig13b}}\n\t \\vspace{0.5cm}\n\t\\hbox{\\includegraphics[scale=0.29]{Fig13c}\n\t\\hspace{1cm}\n\t\\includegraphics[scale=0.21]{Fig13d}}}\n\\caption{Examples of trajectories of test particles interacting with the radiation field emitted by the surface of a slowly or rapidly rotating NS. Left panels: The cases of rapidly rotating template NS with Kerr parameter $a=0.41$ ($f_{\\star}=600$ Hz, $\\Omega_\\star=0.031M^{-1}$) and with relative luminosity $A=0.8$. The critical hypersurface forms a big lobe. Right panels: The cases of slowly rotating template NS with Kerr parameter $a=0.07$ ($f_{\\star}=100$ Hz, $\\Omega_\\star=0.005M^{-1}$) and with relative luminosity $A=0.85$. The critical hypersurface wraps the entire NS surface. Top panels: The test particles are emitted outside the critical hypersurface at $r_0=15M,\\,\\theta_0=10^\\circ$ with the small initial velocity $\\nu_0=0.01$ oriented in the azimuthal corotating direction (orange) and oriented radially towards the emitting surface (red). \nBottom panels: The test particles are emitted inside the critical hypersurface at $r_0=7M$ with the initial velocity oriented in the azimuthal corotating direction.\nBottom left panel: The initial polar angle is $\\theta_0=45^\\circ$,  the magnitude of initial velocity is $\\nu_0=0.01$ (red), $\\nu_0=0.3$ (green), $\\nu_0=0.38$ (blue). Bottom right panel: The initial polar angle is $\\theta_0=60^\\circ$,  the magnitude of initial velocity is $\\nu_0=0.01$ (red), $\\nu_0=0.36$ (green), $\\nu_0=0.42$ (blue). The black sphere corresponds to the emitting surface of the NS. The blue-gray quasi-ellipsoidal oblate surface denotes the position of the critical hypersurface. }\n\t\\label{fig:FigNSorbits}\n\\end{figure*}\n\nOn the both top panels of Fig \\ref{fig:FigNSorbits}, the test particles are emitted outside the critical hypersurface at $r_0=15M,\\,\\theta_0=10^\\circ$ with a small initial velocity $\\nu_0=0.01$ oriented in the azimuthal corotating direction (orange) and oriented radially towards the emitting surface (red). On the top left panel corresponding to the case of the rapid rotation, \nthe trajectory of the particle with the azimuthally-oriented initial velocity (orange) intersects the critical hypersurface from the outside. The particle is then captured on the critical hypersurface after passing a radial turning point. After a very short latitudinal drift, the particle's trajectory is stabilized in the equatorial plane. On the contrary, the particle with radially oriented initial velocity falls down the NS surface. The top right panel of Fig \\ref{fig:FigNSorbits} shows to the case of a slowly rotating template NS with Kerr parameter $a=0.07$ ($f_{\\star}=100$ Hz, $\\Omega_\\star=0.005M^{-1}$) and relative luminosity $A=0.85$. The critical hypersurface envelopes the entire NS surface, its radius being $r_{\\rm crit}^{\\rm eq}\\sim 7.37M$ at the equator and $r_{\\rm crit}^{\\rm pole}\\sim 7.21M$ at the poles. In this case, the trajectories differ only in their initial behaviour. Both test particles are then captured on the critical hypersurface and drift latitudinally toward the equatorial plane, where a final, circular orbit is attained. \n\nOn the both bottom panels of Fig \\ref{fig:FigNSorbits}, the test particles  are emitted inside the critical hypersurface at $r_0=7M$ with the initial velocity $\\nu$ oriented in the azimuthal corotating direction.\nThe behaviour of the test particles is qualitatively the same regardless of the velocity of NS rotation. The test particles emitted with sufficiently small velocity $\\nu$ are captured on the critical hypersurface from the inside (red). The test particles emitted with greater velocity $\\nu$ intersect the critical hypersurface from the inside and they also intersect the equatorial plane. The particle is then captured on the critical hypersurface after passing a radial turning point (green). Captured test particles drift latitudinally toward the equatorial plane, where a final, circular orbit is attained. Similarly as in the cases illustrated in the top panels, the latitudinal drift is slower in the case of the slow rotation. Finally, the test particles emmited with sufficiently high velocity $\\nu$ intersect the critical hypersurface from the inside and then escape to infinity (blue).    \n\n\\subsection{Critical hypersurfaces in the case of a BH}\n\\label{sec:CHBH}\n\nWe consider a BH of mass $M=5M_\\odot$ and spin $a=0.9$. We placed the model emitting surface at $R_\\star=2.5M$.  The emitting surface rotates with angular velocity $\\Omega_\\star$, which is unrelated to the BH spin. In the case of this relatively high value of the spin and at the value of the angular velocity of the emitting surface $\\Omega_{\\star} \\leq \\Omega_{+}$, Eq. (\\ref{eq:crit_hyper2}) has only one solution, and the BH is enveloped by one critical hypersurface at the most (see the condition (\\ref{eq:delta3})). \n\n\\begin{figure*}[th!]\n\t\\centering\n\t\\hbox{\n\t\t\\includegraphics[scale=0.315]{Fig5_1}\n\t\t\\hspace{0.3cm}\n\t\t\\includegraphics[scale=0.32]{Fig5_2}}\n\t\\caption{Critical radius $r_{\\rm (crit)}$ as a function of the luminosity parameter $A$ at the poles (blue line) and at the equator (orange line). The dashed red line represents the emitting surface, and the dashed blue line is the BH outer event horizon at $R_{\\rm H}=1.44M$. In the left panel $f_\\star=300$ Hz, while in the right panel $f_\\star=1400$ Hz. The plot is constructed for the case of BH spin $a=0.9$ and the radius of the emitting surface $R_\\star=2.5M$}\n\t\\label{fig:Fig5}\n\\end{figure*}\n\\begin{figure}[h!]\n\t\\centering\n\t\t\\includegraphics[trim=0.5cm 0cm 0cm 0cm,scale=0.35]{Fig6}\n\t\\caption{Critical radius $r_{\\rm (crit)}$ as a function of the angular velocity of the emitting source $\\Omega_\\star$ at the poles (blue line) and at the equator (orange line). The dashed red line represents the emitting surface and the dashed blue line indicates the BH outer event horizon. The vertical dashed green line is the ZAMO angular velocity $\\Omega_{\\rm ZAMO}=0.09M^{-1}$ . The plot is constructed for a luminosity parameter of $A=0.5$, BH spin of $a=0.9$ and radius of the emitting surface of $R_\\star=2.5M$}\n\t\\label{fig:Fig6}\n\\end{figure}\n\\begin{figure}[h!]\n\t\\centering\n\t\t\\includegraphics[trim=0.5cm 0cm 0cm 0cm,scale=0.33]{Fig7}\n\t\\caption{Critical hypersurfaces for a luminosity parameter of $A=0.5$, BH spin of $a=0.9$ and radius of the emitting surface of $R_\\star=2.5M$. The black sphere is the BH outer event horizon with  size $R_{\\rm H}=1.44M$, the red ellipsoid around the BH outer event horizon represents the ergosphere and the green sphere is the emitting surface. The orange surface is calculated for $\\Omega_\\star=0.05M^{-1}$ it has $r_{\\rm (crit)}^{\\rm eq}=2.42M$ and $r_{\\rm (crit)}^{\\rm pole}=2.89M$. The blue surface is obtained setting $\\Omega_\\star=0.24M^{-1}$ and has $r_{\\rm (crit)}^{\\rm eq}=15.65M$ and $r_{\\rm (crit)}^{\\rm pole}=2.89M$.}\n\t\\label{fig:Fig7}\n\\end{figure}\n\\begin{figure*}[ht!]\n\t\\centering\n\t\t\\includegraphics[trim=0.5cm 0cm 0cm 0cm,scale=0.33]{Fig14}\n\t\t\\hspace{0.3cm}\n\t\t\\includegraphics[trim=0.5cm 0cm 0cm 0cm,scale=0.27]{Fig14b}\n\t\\caption{Test particle trajectories influenced by a radiation field with relative luminosity $A=0.5$ emitted from a surface at $R_\\star=2.5M$ in the vicinity of BH with spin $a=0.9$. Left panel: the case of a rapidly rotating surface with $\\Omega_\\star\\equiv 0.24M^{-1}$ ($f_\\star=1550$ Hz). The test particles trajectories start outside the critical hypersurface at $r_0=20M,\\,\\theta_0=\\pi\/8$ with  initial velocity $\\nu_0=0.22$ oriented in the  counterrotating azimuthal direction (blue), with $\\nu_0=0.25$ oriented in the counterrotating azimuthal direction (orange) and with $\\nu_0=0.22$ oriented radially towards the emitting surface (red). Right panel: The case of a slowly rotating surface with $\\Omega_\\star\\equiv 0.05M^{-1} <\\Omega_{\\mathrm{ZAMO}}$ ($f_\\star=323$ Hz). Test particles are emmited above the critical hypersurface forming two polar lobes at $r_0=5M,\\,\\theta_0=\\pi\/8$ with the initial velocity $\\nu_0=0.1$ (red), $\\nu_0=0.5$ (orange) and  $\\nu_0=0.8$ (blue) oriented in the  counterrotating  azimuthal direction. The black sphere is the BH outer event horizon with size $R_{\\rm H}=1.44M$, while the red ellipsoid around the BH outer event horizon represents the ergosphere and the green sphere is the emitting surface.}\n\t\\label{fig:Fig14}\n\\end{figure*}\n\nFigure \\ref{fig:Fig5} shows the critical radius changes as a function of the  the luminosity parameters for two different rotation frequencies of the emitting surface $f_\\star=300$ Hz ($\\Omega_\\star=0.05M^{-1}$) and $f_\\star=1400$ Hz ($\\Omega_\\star=0.22M^{-1}$). \nThe slow rotation case corresponds to the negative value of test field impact parameter on the equator $b=-1.55$ while the fast rotation case corresponds to to the positive one $b=2.67$\nIn the case of slow rotation with $f_\\star=300$ Hz, the change from  a prolate to an oblate hypersurface takes place when the luminosity parameter exceeds $A\\approx0.86$. \nFor the range of luminosity values $A \\in[0.425,0.53]$, the critical hypersurface only partially emerges from the emitting surface and forms two lobes around poles of the emitting surface.\nIn the case of fast rotation with $f_\\star=1400$ Hz, the behaviour is simpler. The critical hypersurface always has an oblate shape. For luminosities $A<0.4$, the critical hypersurface forms a lobe around equator of the emitting surface, for higher luminosities the entire critical hypersurface is located outside the emitting surface.\n\nIn Fig. \\ref{fig:Fig6}, we illustrate the behaviour of the critical hypersurface as a function of the angular velocity of the emitting surface $\\Omega_\\star$ for fixed luminosity $A=0.5$ and fixed spin parameter $a=0.9$. The $\\Omega_\\star$ interval ranges between zero and limiting value $\\Omega_+(R_\\star,a,\\pi\/2)=0.28M^{-1}$ ($f_\\star=1808$ Hz) .\nIn the angular velocity range $\\Omega_\\star<0.07M^{-1}$ ($f_\\star=452$ Hz) , the critical hypersurface forms two lobes around the poles of the emitting surfaces. For $\\Omega_\\star=0.07M^{-1}$, the critical hypersurface coincides with the emitting surface. For $\\Omega_\\star>0.07M^{-1}$, the critical hypersurface is a lobe around the equator of the emitting surface. In Fig. \\ref{fig:Fig7}, we illustrate such behaviour by 3D plot of the critical hypersurface shape for the slowly rotating case with $\\Omega_\\star\\equiv 0.05M^{-1}$ ($f_\\star=323$ Hz) and for the fast rotating case with $\\Omega_\\star\\equiv 0.24M^{-1}$ ($f_\\star=1550$ Hz). The figure also clearly illustrates, that both hypersurfaces coincide on the polar axis, where photons coming radially from rigidly rotating emitting surface have always zero angular momentum ($b=0$).  \n\n\\subsubsection{Examples of test particle orbits in the vicinity of the critical hypersurface around BH}\n\nIn Fig. \\ref{fig:Fig14}, we illustrate the results of the integration of some selected trajectories of test particles influenced by the radiation field emitted by a slowly or rapidly rotating surface as in the case corresponding to Fig. \\ref{fig:Fig7}, i.e. for the value of luminosity parameter $A=0.5$ and the emitting surface located at $R_\\star=2.5M$ in the vicinity of a BH with spin $a=0.9$.  \nIn the left panel, when the emitting surface rotates rapidly with $\\Omega_\\star=0.24M^{-1}$, the azimuthal impact parameter of the photons is $b=3.03$ on the equator. The test particles are emitted above the critical hypersurface at $r_0=20M,\\,\\theta_0=\\pi\/8$.  The particle emitted with initial velocity $\\nu_0=0.22$ oriented in the azimuthal counter-rotating direction (blue) intersects the critical hypersurface and makes several loops inside it before it stabilizes on the circular equatorial orbit on the critical hypersurface. The particle emitted with initial velocity $\\nu_0=0.25$ oriented also in the azimuthal counter-rotating direction (orange) escapes to infinity. Finally, the test particle emitted with an initial velocity  $\\nu_0=0.22$ oriented radially towards the BH (red) falls on the emitting surface. \nOn the right panel, when the emitting surface rotates slowly with $\\Omega_\\star\\equiv 0.05M^{-1}<\\Omega_{\\mathrm{ZAMO}}$ ($f_\\star=323$ Hz),  the azimuthal impact parameter of the photons is characterised  by a negative value of $b=-1.24$ on the equator. The test particles are emitted again above the critical hypersurface (which in this case consists of two polar lobe) at $r_0=5M,\\,\\theta_0=\\pi\/8$ with the velocity oriented in the azimuthal counter-rotating direction.  The particle emitted with initial velocity $\\nu_0=0.1$ (red) is captured on the northern polar lobe, and after the latitudinal drift, it impacts the emitting surface at its intersection ring with the northern lobe.  Similarly the particle emitted with initial velocity $\\nu_0=0.5$ (orange) after making a loop is captured on the southern polar lobe, and after the latitudinal drift, it impacts the emitting surface at its intersection ring with the southern lobe. Finally, the test particle emitted with initial velocity $\\nu_0=0.8$ (blue) escapes to infinity. \n\n\\section{Conclusions}\n\\label{sec:end}\n\nWe extended the general relativistic model of the 3D PR effect presented in \\cite{DeFalco20183D} by \nconsidering the radiation field emitted in the purely radial direction in a local corotating frame from a rigidly rotating spherical source in the background of the Kerr spacetime geometry. This setup, though \nclearly idealised, may approximate the radiating surface of a rotating NS as well as that of a hot rotating corona in close vicinity of a BH. Owing to the rigid rotation of the emitting surface, the photon azimuthal impact parameter of the radiation field $b$ (a proxy of its angular momentum)  is a function of the polar angle $\\theta$, with the highest value in the equatorial plane and zero value on the poles. In order to simplify the integration of test particles trajectories, we assume that the emitted photons are not moving in the latitudinal direction and their $\\theta$ coordinate is conserved along the photon trajectory. Such a setup for the test radiation field is parametrized by the radius of the emitting surface $R_\\star$ and its angular velocity $\\Omega_\\star$ and the emitted luminosity $A$. We have considered the range of possible values of azimuthal impact parameter of the radiation field $b$ depending on the parameters $R_\\star$, $\\Omega_\\star$ and the spin of a NS or BH, when the maximum values of the angular velocity of the emitting surface $\\Omega_\\star$ are limited below superluminal rotation. In the case of a non-zero spin, the photon azimuthal impact parameter of the radiation field $b$ can attain both positive and negative values for positive (i.e. corotating) angular velocities $\\Omega_\\star$. In the case in which the emitting surface is partially located in the ergosphere and  rotates slowly with the angular velocity going to $\\Omega_{\\infty}(\\RS)\\equiv-\\mathrm{g_{tt}}\/\\mathrm{g_{t\\phi}}<\\Omega_{\\mathrm{ZAMO}}$, the azimuthal impact parameter of the radiation field $b$ diverges to $- \\infty$. For extremely slowly rotating surfaces located in the ergosphere with $\\Omega_{\\star} <\\Omega_{\\infty}(\\RS)$ $b$ is positive again. However, the photons emitted from such surfaces cannot escape the ergosphere. (see Fig. \\ref{fig:lambdarange}).  \n\nUsing the \\emph{observer-splitting formalism}, we formulated the equations of motion (\\ref{EoM1})--(\\ref{EoM6}) for test particles influenced by our test radiation field.\nTheir specific class of solutions corresponds to the axially-symmetric critical hypersurface, which is formed by test particles on the generally off-equatorial circular orbits around the emitting surface stabilized by the balance between gravitational attraction, radiation forces and PR drag. We have demonstrated that the shape of the critical hypersurface depends on the spin of the NS or BH and the parameters of the radiation field $R_\\star$ and $\\Omega_\\star$. Depending on the interplay of such parameters, the critical hypersurface may morph between the oblate and prolate shape.\n\nBy using a cubic approximation of Eq. (\\ref{eq:crit_hyper2}) to determine the location of critical hypersurface,  we found a criterion for distinguishing the regime with only one critical hypersurface  from the regime with three critical hypersurfaces. It should be noted, however, that the inner solutions of Eq. (\\ref{eq:crit_hyper2}) are found in the close vicinity of the outer event horizon and in the region of spherical photon orbits, such that their physical relevance is likely very limited. Therefore, in the examples we analysed, multiple critical hypersurfaces are not relevant.\nWe determined the conditions for the  existence of off-equatorial suspended circular orbits bound on the critical hypersurface and determined the corresponding value of the polar angle $\\psi$ of the velocity of the test particle $\\nu$ as measured in the ZAMO frame as a function of latitudinal coordinate $\\theta$. \n\nWe analyzed in detail representative cases for both a rotating NS and a rotating BH.\nIn the case of a NS, its spherical surface is the source of the radiation field and the kerr parameter $a$ is proportional to the angular velocity of the emitting surface $\\Omega_\\star$; we found that the critical hypersurface always takes an oblate shape. Interestingly, over a limited range of relative luminosity values $A$, the critical hypersurface forms only a lobe around the NS equator. The angular velocity of the particles captured on such a hypersurface is equal to $\\Omega_\\star$ in the ring of intersection with the NS surface and decreases with the polar angle $\\theta$ reaching a minimum on the equatorial plane (see Figs. \\ref{fig:Fig4bis} and \\ref{fig:Fig4bis1} )\n\nIn the case of an emitting surface approximating a spherical rotating hot-corona in the close vicinity of a BH, the critical hypersurface can take a prolate as well as an oblate shape. We analyzed the case of the emitting surface located outside the ergosphere of a BH where suspended off-equatorial orbits can exist. For high values of $\\Omega_\\star$, similar to the case of a NS, the critical hypersurface always takes an oblate shape, and over a range of relative luminosities $A$, it may form only a lobe around the equator of the emitting surface. In the case of slowly rotating emitting surfaces, an inversion effect may arise, where the prolate critical hypersurface forms two lobes around the poles of the emitting surface.  \nIn both cases, we integrated selected trajectories of test particles influenced by the interaction with the radiating field emitted from slowly as well as rapidly rotating emitting surfaces. \n\nA more complete analysis and classification of orbits in the radiation field emitted by a rigidly rotating spherical source, including the analysis of the stability of off-equatorial suspended orbits on the critical hypersurface and the conditions for capturing the test particles on such orbits will be the subject of a separate study.\n\n\n\\section*{Acknowledgements}\nP.B., K.G. and D.L. acknowledge the Czech Science Foundation (GA\u010cR) grant GA\u010cR 17-16287S and internal grant of Silesian University in Opava SGS\/13\/2019.\nV.D.F. and E.B. thank the Silesian University in Opava for having funded this work. P.B., V.D.F. and E.B. thank the Osservatorio Astronomico di Roma in Monteporzio Catone for the hospitality. V.D.F. and E.B are grateful to Gruppo Nazionale di Fisica Matematica of Istituto Nazionale di Alta Matematica for support. LS acknowledges financial contributions from ASI-INAF agreements 2017-14-H.O and  I\/037\/12\/0 and from ``iPeska'' research grant (P.I. Andrea Possenti) funded under the INAF call PRIN-SKA\/CTA (resolution 70\/2016).\n\n\\begin{appendix}\n\\section{Classical 3D limit with nonzero photon impact parameter}\n\\label{Appendix_Classic_Limit}\nIn this section we report the classical limit of Eqs. (\\ref{EoM1}) -- (\\ref{EoM3}). We already know how to treat the kinematic part from our previous study \\cite{DeFalco20183D}, so we focus our attention on the radiation force $\\mathcal{F}_{\\rm (rad)}(U)^{\\alpha}$ (\\ref{Frad0}). We first consider the Schwarzschild limit $a=0$ of Eq. (\\ref{eq: sigma_tilde}), for which we obtain \n\\begin{equation}\n\\begin{split}\n& \\sigma \\left[\\Phi E(U) \\right]^2 =\\\\\n& \\frac{A \\gamma^2 \\left[ 1 - \\nu \\sin \\psi \\cos \\left( \\alpha-\\beta\\right) \\right]^2}{r \\left(1-\\dfrac{2M}{r}\\right) \\sqrt{r^2- \\left(1 - \\dfrac{2M}{r}\\right)\\left(q+b^2\\right)}}.\n\\end{split}\n\\end{equation}\nThen for $r\\rightarrow +\\infty$, we have\n\\begin{equation}\n\\begin{split}\n&\\sigma \\left[\\Phi E(U) \\right]^2 \\approx \\dfrac{A \\gamma^2 \\left[ 1 - \\nu \\sin \\psi \\cos \\left( \\alpha-\\beta\\right) \\right]^2} {r^2}.\n\\end{split}\n\\end{equation}\nFinally, we consider  $\\nu \\rightarrow 0$; the radiative force components Eqs. (\\ref{rad1})--(\\ref{rad3}) reduce to\n\\begin{eqnarray}\n&&\\mathcal{F}_{\\rm (rad)}(U)^{\\hat{r}}= \\dfrac{A}{r^2} \\left[ \\sin \\beta \\right.\\nonumber\\\\\n&&\\left.\\hspace{2cm}-\\dot{r}(1+\\sin^2 \\beta) -r\\dot{\\varphi}\\sin\\theta\\frac{\\sin(2\\beta)}{2}\\right],\\label{App_1}\\\\\n&&\\mathcal{F}_{\\rm (rad)}(U)^{\\hat{\\theta}}=-\\dfrac{A}{r} \\dot{\\theta},\\label{App_2}\\\\\n&&\\mathcal{F}_{\\rm (rad)}(U)^{\\hat{\\varphi}}=\\dfrac{A}{r^2} \\left[ \\cos \\beta\\right.\\nonumber\\\\ \n&&\\left.\\hspace{1.8cm}-\\dot{r}\\frac{\\sin(2\\beta)}{2}-r\\dot{\\varphi} \\sin \\theta \\left(1 +\\cos^2 \\beta\\right) \\right],\\label{App_3}\n\\end{eqnarray}\nwhere we exploited the following approximations of the test particle velocity components (cf. Eqs. (\\ref{four_velocity}))\n\\begin{eqnarray}\nU^r &&\\equiv\\dot{r}\\approx\\nu\\sin\\psi\\sin\\alpha,\\label{App_vel_U1} \\\\\nU^\\theta &&\\equiv \\dot{\\theta}\\approx\\frac{\\nu\\cos\\psi}{r},\\label{App_vel_U2}\\\\\nU^\\varphi &&\\equiv \\dot{\\varphi}\\approx\\frac{\\nu\\sin\\psi\\cos\\alpha}{r\\sin\\theta},\\label{App_vel_U3}\n\\end{eqnarray}\nhere the dot indicates the derivative with respect to the time $t$. It should be stressed that the classical 3D result concerning the radial radiation field (reported in the Appendix of Ref. \\cite{DeFalco20183D}) is recovered easily  for $\\beta=\\pi\/2$ in Eqs. (\\ref{App_1})--(\\ref{App_3}). Therefore the related classical equations of motion read as\n\\begin{eqnarray}\n&&\\ddot{r}-r\\dot{\\varphi}^2\\sin^2\\theta-r\\dot{\\theta}^2+\\frac{GM}{r^2}=\\frac{Ac}{r^2}\\sin\\beta\\nonumber\\\\\n&&-\\frac{A}{r^2}\\left[\\dot{r}(1+\\sin^2 \\beta)+r\\dot{\\varphi}\\sin\\theta\\frac{\\sin(2\\beta)}{2}\\right],\\label{eqm1}\\\\\n&&r\\ddot{\\theta}+2\\dot{r}\\dot{\\theta}-r\\dot{\\varphi}^2\\sin\\theta\\cos\\theta=-A\\frac{\\dot{\\theta}}{r}\\label{eqm2},\\\\\n&&r\\ddot{\\varphi}\\sin\\theta+2\\dot{r}\\dot{\\varphi}\\sin\\theta+2r\\dot{\\theta}\\dot{\\varphi}\\cos\\theta=\\frac{A}{r^2}\\cos \\beta \\nonumber\\\\\n&&-\\frac{A}{r^2} \\left[\\dot{r}\\frac{\\sin(2\\beta)}{2}+r\\dot{\\varphi} \\sin \\theta \\left(1 +\\cos^2 \\beta\\right)\\right],\\label{eqm3}\n\\end{eqnarray}\nwhere on the right-hand side of Eqs. (\\ref{eqm1}) and (\\ref{eqm3}) we have the radiation pressure projected along the radial and azimuthal direction of the emitted photon and the PR effect is multiplied by a factor taking into account the non-radial direction of the emitted photon. In addition we note that the non-radial emission of the photons breaks the spherical symmetry of the equations of motion (see Appendix of Ref. \\cite{DeFalco20183D}). Classically the azimuthal photon angle $\\beta$ is related to the photon impact parameter $b$ through (see Eq. (\\ref{ANG1}))\n\\begin{equation}\n\\cos\\beta=\\frac{b}{r\\sin\\theta}.\n\\end{equation}\nIn our previous work \\cite{DeFalco20183D}, we saw that the time component of the equations of motion corresponds to the energy balance \n\\begin{equation} \n\\begin{aligned}\n&\\frac{d}{dt}\\left(\\frac{\\nu^2}{2}-\\frac{GM}{r}\\right)=\\frac{Ac}{r^2}\\left(\\dot{r}\\sin\\beta+r\\dot{\\varphi}\\sin\\theta\\cos\\beta\\right)\\\\\n&-\\frac{A}{r^2}\\left[\\dot{r}^2(1+\\sin^2 \\beta)+r^2\\dot{\\theta}^2\\right.\\\\\n&\\left.+r^2\\dot{\\varphi}^2 \\sin^2 \\theta \\left(1 +\\cos^2 \\beta\\right)+\\dot{\\varphi}\\dot{r}r\\sin\\theta\\sin(2\\beta)\\right],\n\\end{aligned}\n\\end{equation}\nwhere the left term represents the total mechanical energy, while the right term corresponds to the radiation pressure and the energy dissipated through the PR effect.\n\n\\section{Weak field limit with slow and fast rotations}\n\\label{Appendix_Weak_Field}\nWe determine here the weak field approximation ($r\\to\\infty$) of the equations of motion (\\ref{EoM1})--(\\ref{EoM3}), at the first order in the spin parameter $a$ for slow rotations ($a \\to 0$). Applying such limits to Eq. (\\ref{eq: sigma_tilde}) we get\n\\begin{equation}\n\\sigma \\left[\\Phi E(U) \\right]^2 \\approx A\\ \\gamma^2\\ \\Gamma(\\nu)\\ f(a,r)\n\\end{equation}\nwhere $\\Gamma(\\nu) = \\left[ 1 - \\nu \\sin \\psi \\cos \\left( \\alpha-\\beta\\right) \\right]^2$ and \n\\begin{equation}\n\\label{eq: f:function}\n\\begin{split}\n& f(a,r) = \\dfrac{1}{r^2}+\\dfrac{2M}{r^3} +\\dfrac{8M^2+b^2+q}{2r^4}+ \\dfrac{8M^3}{r^5}\\\\\n&\\hspace{1cm}-\\dfrac{2Mb}{r^5}a+{\\rm O}\\left(\\frac{1}{r^6}\\right)+{\\rm O}\\left(\\frac{a}{r^6}\\right) +{\\rm O}\\left(\\frac{a^2}{r^4}\\right).\n\\end{split}\n\\end{equation}\nTherefore, the radiation field components read as\n\\begin{eqnarray}\n&&\\mathcal{F}_{\\rm (rad)}(U)^{\\hat{r}}\\approx A\\ \\gamma^2\\ \\Gamma(\\nu)\\ f(a,r)\\ \\mathcal{V}^{\\hat r}, \\label{WFApp_1}\\\\\n&&\\mathcal{F}_{\\rm (rad)}(U)^{\\hat{\\theta}}\\approx A\\ \\gamma^2\\ \\Gamma(\\nu)\\ f(a,r)\\ \\mathcal{V}^{\\hat \\theta}, \\label{WFApp_2}\\\\\n&&\\mathcal{F}_{\\rm (rad)}(U)^{\\hat{\\varphi}}\\approx A\\ \\gamma^2\\ \\Gamma(\\nu)\\ f(a,r)\\ \\mathcal{V}^{\\hat \\varphi}.\\label{WFApp_3}\n\\end{eqnarray}\nFor slow motion ($\\nu \\rightarrow 0$), Eqs. (\\ref{WFApp_1})--(\\ref{WFApp_3}) reduce to \n\\begin{eqnarray}\n&&\\mathcal{F}_{\\rm (rad)}(U)^{\\hat{r}}\\approx A\\ f(a,r) \\left[ \\sin \\beta-\\dot{r}(1+\\sin^2 \\beta) \\right. \\nonumber\\\\\n&&\\left.\\hspace{2cm}-\\left(r\\dot{\\varphi}-\\frac{2aM}{r^2}\\right)\\sin\\theta\\frac{\\sin(2\\beta)}{2}\\right],\\label{App_4}\\\\\n&&\\mathcal{F}_{\\rm (rad)}(U)^{\\hat{\\theta}}\\approx-A\\ f(a,r)\\  r\\dot{\\theta}, \\label{App_5}\\\\\n&& \\mathcal{F}_{\\rm (rad)}(U)^{\\hat{\\varphi}}\\approx A\\ f(a,r) \\left[\\cos \\beta-\\dot{r}\\frac{\\sin(2\\beta)}{2} \\right. \\nonumber\\\\\n&&\\left.\\hspace{1.6cm} -\\left(r\\dot{\\varphi}-\\frac{2aM}{r^2}\\right)\\sin\\theta \\left(1 +\\cos^2 \\beta\\right)\\right],\\label{App_6}\n\\end{eqnarray}\nwhere we have exploited the following approximations for the test particle velocity components\n\\begin{eqnarray}\nU^r &&\\equiv\\dot{r}\\approx\\nu\\sin\\psi\\sin\\alpha+{\\rm O}\\left(\\frac{1}{r}\\right)+{\\rm O}\\left(\\frac{a^2}{r^2}\\right),\\label{App_new_velocity1}\\\\\nU^\\theta &&\\equiv \\dot{\\theta}\\approx\\frac{\\nu\\cos\\psi}{r}+{\\rm O}\\left(\\frac{1}{r^3}\\right)+{\\rm O}\\left(\\frac{a^2}{r^3}\\right),\\label{App_new_velocity2}\\\\\nU^{\\varphi} &&\\equiv \\dot{\\varphi} \\approx \\dfrac{\\nu \\cos \\alpha \\sin \\psi}{r \\sin \\theta}\\nonumber\\\\\n&&+\\dfrac{2Ma}{r^3}+{\\rm O}\\left(\\frac{1}{r^4}\\right)+{\\rm O}\\left(\\frac{a^2}{r^3}\\right)+{\\rm O}\\left(\\frac{a}{r^4}\\right); \\label{App_new_velocity3}\n\\end{eqnarray}\nhere dot means the derivative with respect to the affine parameter $\\tau$. \nWe note that Eqs. (\\ref{App_new_velocity1})--(\\ref{App_new_velocity2}) are similar to Eqs. (\\ref{App_vel_U1})--(\\ref{App_vel_U2}) in the classical limit. In addition, in Eqs. (\\ref{App_4}) and (\\ref{App_6}) by substituting the azimuthal test particle velocity Eq. (\\ref{App_new_velocity3}), there is a factor $a$ that when multiplied by $f(a,r)$ gives a term of second order in $a$ that must be neglected. Therefore, we have\n\\begin{eqnarray}\n&&\\mathcal{F}_{\\rm (rad)}(U)^{\\hat{r}}\\approx A\\left\\{\\ f_1(r) \\left[ \\sin \\beta-\\dot{r}(1+\\sin^2 \\beta)\\right]\\right. \\nonumber\\\\\n&&\\left.-f_2(a,r)\\dot{\\varphi}\\sin\\theta\\frac{\\sin(2\\beta)}{2}-\\frac{2aM}{r^4}\\sin\\theta\\frac{\\sin(2\\beta)}{2}\\right\\}\\nonumber\\\\\n&&+{\\rm O}\\left(\\frac{1}{r^5}\\right)+{\\rm O}\\left(\\frac{a^2}{r^3}\\right)+{\\rm O}\\left(\\frac{a}{r^5}\\right),\\label{App_42}\\\\\n&&\\mathcal{F}_{\\rm (rad)}(U)^{\\hat{\\theta}}\\approx-A\\ f_2(a,r)\\  r\\dot{\\theta}\\nonumber\\\\\n&&+{\\rm O}\\left(\\frac{1}{r^5}\\right)+{\\rm O}\\left(\\frac{a^2}{r^3}\\right)+{\\rm O}\\left(\\frac{a}{r^5}\\right), \\label{App_52}\\\\\n&& \\mathcal{F}_{\\rm (rad)}(U)^{\\hat{\\varphi}}\\approx A\\left\\{f_1(r) \\left[\\cos \\beta-\\dot{r}\\frac{\\sin(2\\beta)}{2} \\right]\\right. \\nonumber\\\\\n&&\\left.-f_2(a,r)\\dot{\\varphi}\\sin\\theta \\left(1 +\\cos^2 \\beta\\right)-\\frac{2aM}{r^4}\\sin\\theta \\left(1 +\\cos^2 \\beta\\right)\\right\\}\\nonumber\\\\\n&&+{\\rm O}\\left(\\frac{1}{r^5}\\right)+{\\rm O}\\left(\\frac{a^2}{r^3}\\right)+{\\rm O}\\left(\\frac{a}{r^5}\\right),\\label{App_62}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nf_1(r)&&=\\dfrac{1}{r^2}+\\dfrac{2M}{r^3} +\\dfrac{8M^2+b^2+q}{2r^4},\\label{fun1}\\\\\nf_2(a,r)&&=\\dfrac{1}{r}+\\dfrac{2M}{r^2} +\\dfrac{8M^2+b^2+q}{2r^3}\\nonumber\\\\\n&&+ \\dfrac{8M^3}{r^4}-\\dfrac{2Mb}{r^4}a.\\label{func2}\n\\end{eqnarray}\nWe note that the impact parameters appear already at the $r^{-3}$-order, whereas the spin parameter appear for the first time at the $r^{-4}$ term. \\emph{The slow-rotation configures as an effect of the fourth-order in the general relativistic radiation processes.} Note also that the next order in the spin parameter is still linear, whereas the second order appears at the third order. \n\nNow we have also to consider the approximation of the geometric part (see Tab. I in Ref. \\cite{DeFalco20183D}, for further details), which yields\n\\begin{eqnarray}\na(U)^{\\hat{r}}&&= \\frac{d}{d\\tau}\\left( \\gamma\\dot{r} \\right)-\\frac{6\\gamma^2 aM\\dot{\\varphi}\\sin\\theta}{r^2}\\nonumber\\\\\n&&-\\frac{\\gamma^2}{r}\\left[r^2\\dot{\\theta}^2+\\left(r^2\\dot{\\varphi}^2-\\frac{4Ma\\dot{\\varphi}}{r}\\right)\\sin^2\\theta\\right]\\nonumber\\\\\n&&+\\frac{M\\gamma^2}{r^2}\\left[1+r^2\\dot{\\theta}^2+r^2\\sin^2\\theta\\right]\\nonumber\\\\\n&&+\\frac{M\\gamma^2}{r^3}\\left[1+\\frac{1}{2}\\left(r^2\\dot{\\theta}^2+r^2\\sin^2\\theta\\right)\\right],\\nonumber\\\\\n&&+ {\\rm O}\\left(\\frac{1}{r^4}\\right)+{\\rm O}\\left(\\frac{a}{r^3}\\right)+ {\\rm O}\\left(\\frac{a^2}{r^3}\\right)\\label{GP_1}\\\\\na(U)^{\\hat{\\theta}}&&= \\frac{d}{d\\tau} \\left(\\gamma r\\dot{\\theta}\\right)-\\frac{\\gamma^2 M\\dot{r}\\dot{\\theta}}{r}+\\frac{\\gamma^2}{r}\\left[r\\dot{\\theta}\\dot{r}-\\left(r\\dot{\\varphi}-\\frac{4aM\\dot{\\varphi}}{r}\\right)\\right],\\nonumber\\\\\n&& + {\\rm O}\\left(\\frac{1}{r^3}\\right)+{\\rm O}\\left(\\frac{a}{r^3}\\right)+ {\\rm O}\\left(\\frac{a^2}{r^3}\\right),\\label{GP_2}\\\\\na(U)^{\\hat{\\varphi}}&&= \\frac{d}{d\\tau}\\left[\\gamma\\left(r\\dot{\\varphi}-\\frac{2Ma}{r^2}\\right)\\right]-\\frac{\\gamma^2 M\\dot{r}\\dot{\\varphi}}{r},\\nonumber\\\\\n&&\\frac{\\gamma^2}{r}\\left[\\left(r\\dot{\\varphi}-\\frac{2Ma}{r^2}\\right)\\left(r\\dot{\\theta}\\cos\\theta+\\dot{r}\\sin\\theta\\right)\\right],\\nonumber\\\\\n&&+{\\rm O}\\left(\\frac{1}{r^4}\\right)+{\\rm O}\\left(\\frac{a}{r^4}\\right)+ {\\rm O}\\left(\\frac{a^2}{r^3}\\right).\\label{GP_3}\n\\end{eqnarray}\n\\emph{In the geometric part the spin appears linearly already at the third order.} This means that the linearized frame dragging effect has more relevance in the gravitational-geometric part than in the radiation processes. The second order effect in the spin appear also at the third order. In this case, it is important to note that the affine parameter $\\tau$ is related to the coordinate time $t$ not linearly as in the classic limit, but it satisfies this condition (see $U^t$ in Eq. \\ref{four_velocity})\n\\begin{equation}\\label{App_new_velocity4}\nU^t \\equiv\\dot{t}\\approx \\gamma\\left(1+\\frac{M}{r}\\right)+{\\rm O}\\left(\\frac{1}{r^2}\\right)+{\\rm O}\\left(\\frac{a^2}{r^3}\\right),\n\\end{equation}\nand in the slow motion approximation, it reduces to\n\\begin{equation}\\label{App_new_velocity42}\nU^t \\equiv\\dot{t}\\approx\\left(1+\\frac{M}{r}\\right).\n\\end{equation}\nTherefore, in order to pass from the proper time $\\tau$ derivative to the coordinate time $t$ derivative, we have to multiply such $t$-derivative by the geometrical factor of Eq. (\\ref{App_new_velocity42}). In the slow-motion limit the test particle acceleration components Eqs. (\\ref{GP_1})--(\\ref{GP_3}) become\n\\begin{eqnarray}\na(U)^{\\hat{r}} &&\\approx \\ddot{r} +\\dfrac{2aM\\dot{\\varphi}\\sin\\theta(2\\sin\\theta-3)}{r^2}\\nonumber\\\\\n&&+\\frac{M}{r^2}\\left(1+\\frac{M}{r}\\right),\\label{App_9}\\\\\na(U)^{\\hat{\\theta}} &&\\approx \\dot{r}\\theta+r\\dot{\\theta}+\\frac{4aM\\dot{\\varphi}\\cos\\theta}{r^2},\\label{App_9a}\\\\\na(U)^{\\hat{\\varphi}} &&\\approx r\\ddot{\\varphi}\\sin\\theta+\\dot{r}\\dot{\\varphi}\\sin\\theta+r\\dot{\\varphi}\\dot{\\theta}\\cos\\theta\\nonumber\\\\\n&&-\\frac{2Ma}{r^3}\\left(r\\dot{\\theta}\\cos\\theta+\\dot{r}\\sin\\theta\\right).\\label{App_9b}\n\\end{eqnarray}\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider a model equation for a time-harmonic acoustic pressure $\\psi = \\psi(x)$ (time dependence $e^{-i\\omega t}$) in a moving fluid with sound speed $c = c(x)$, current velocity $v = v(x)$, density $\\rho = \\rho(x)$ and absorption $\\alpha = \\alpha(x,\\omega) = \\omega^{\\zeta(x)}\\alpha_0(x)$ at fixed frequency $\\omega$:\n\\begin{gather}\\label{in.waveeq}\n  L_\\omega \\psi = 0 \\quad \\text{in $D$},\\\\\n \\quad L_\\omega = -\\Delta - 2i A_\\omega(x) \\cdot \\nabla - U_\\omega(x), \\quad x = (x_1,\\ldots,x_d), \\label{in.schrop}\\\\\n  \\nabla = \\bigl( \\tfrac{\\partial}{\\partial x_1},\\ldots, \\tfrac{\\partial}{\\partial x_d} \\bigr), \\quad \\Delta = \\tfrac{\\partial^2}{\\partial x_1^2} + \\cdots+ \\tfrac{\\partial^2}{\\partial x_d^2}, \\notag\n\\end{gather} \nwhere\n\\begin{gather}\n  \\begin{gathered}\\label{in.Dpro}\n    \\text{\\it $D$ is an open bounded domain in $\\mathbb R^d$,}\\\\\n    \\text{\\it $d \\in \\{2,3\\}$, with connected $C^\\infty$ boundary $\\partial D$}, \n  \\end{gathered}\\\\\n  A_\\omega(x) = \\frac{\\omega v(x)}{c^2(x)} + i \\nabla \\ln \\rho^{\\frac 1 2}(x), \\quad U_\\omega(x) = \\frac{\\omega^2}{c^2(x)} + 2i\\omega^{1+\\zeta(x)}\\frac{\\alpha_0(x)}{c(x)}. \\label{in.defAU}\n\\end{gather}\nNote that the operator $L_\\omega$ is a special case of the so-called magnetic Schroedinger operator.\n\nThe model equation \\eqref{in.waveeq} was studied in different particular cases in \\cite{Henk1988,Rous1994,Rych1996,Roum2009en,Bur2013en,Agal2015b,Agal2015c}.\n\nIn the present article we suppose that\n\\begin{subequations}\n\\begin{align}\n  c & \\in C^2(\\overline D), \\quad c > 0 \\; \\text{\\it in $\\overline D$}, \\label{in.cpro}\\\\\n  v & \\in C^2(\\overline D,\\mathbb R^d), \\label{in.vpro}\\\\\n  \\rho & \\in C^{2,\\beta}(\\overline D), \\; \\beta\\in(0,1], \\quad \\rho > 0 \\; \\text{\\it in $\\overline D$,} \\label{in.rpro}\\\\\n  \\zeta & \\in C(\\overline D), \\; \\alpha_0 \\in C(\\overline D), \\quad \\text{\\it $\\zeta > 0$, $\\alpha_0$ is real in $\\overline D$}, \\label{in.abpro}\n\\end{align}\n\\end{subequations}\nwhere $\\overline D = D \\cup \\partial D$ and $C^{k,\\beta}(\\overline D)$ denotes the space of $k$ times continuously differentiable functions in $\\overline D$ whose $k$-th derivatives are $\\beta$-H\\\"older continuous. Let\n\\begin{equation}\n  \\widetilde{c} = c|_{\\partial D}, \\quad \\widetilde v = v|_{\\partial D}, \\quad \\widetilde{\\rho} = \\rho|_{\\partial D}.\n\\end{equation}\nIn what follows we always assume that \n\\begin{equation}\\label{in.unsol}\n    \\text{\\it $0$ is not a Dirichlet eigenvalue for operator $L_\\omega$ in $D$.}\n\\end{equation}\nNote that the set of $\\omega$'s for which \\eqref{in.unsol} does not hold is locally finite. Besides, \\eqref{in.unsol} always holds for $\\omega = 0$.\n\nFor equation \\eqref{in.waveeq} under the assumption \\eqref{in.unsol} we consider the Dirichlet-to-Neumann type operator $\\Lambda_\\omega$ which maps a sufficiently regular function $f$ on $\\partial D$ to the function\n\\begin{equation*}\n  \\Lambda_\\omega f = \\bigl(\\tfrac{\\partial \\psi}{\\partial \\nu} + i(\\nu \\cdot A_\\omega) \\psi \\bigr)\\bigr|_{\\partial D},\n\\end{equation*}\nwhere $\\psi$ is the solution of equation \\eqref{in.waveeq} in $D$ with Dirichlet boundary condition $\\psi|_{\\partial D} = f$ and $\\nu$ is the unit exterior normal field to $\\partial D$.\n\nNote that it is possible to get rid of the assumption \\eqref{in.unsol} by considering a general Robin-to-Robin map instead of the Dirichlet-to-Neumann map, see \\cite{Isay2013}.\n\nWe consider the following problem:\n\\begin{problem}\\label{in.probD2N} Find $c$, $v$, $\\rho$ and $\\alpha$ in $D$ from $\\Lambda_\\omega$ given for $\\omega$ in some fixed set $\\Omega$ and from $\\widetilde c$, $\\widetilde v$ and $\\widetilde \\rho$.\n\\end{problem}\nThis problem was studied, in particular, in \\cite{Agal2015b,Agal2015c}. In these works it was shown that\n\\begin{enumerate}\n \\item[(a)] if $\\rho \\equiv \\const$, $\\alpha_0 \\equiv 0$ and $\\Omega = \\{\\omega_1\\}$ then the Problem \\ref{in.probD2N} is uniquely solvable;\n \\item[(b)] if $\\alpha_0 \\equiv 0$, $\\Omega = \\{\\omega_1,\\omega_2\\}$, $\\omega_1 < \\omega_2$, then the Problem \\ref{in.probD2N} is uniquely solvable;\n \\item[(c)] if $\\Omega = \\{\\omega_1,\\omega_2,\\omega_3\\}$, $\\omega_1 < \\omega_2 < \\omega_3$, and $\\zeta \\neq 0$ in $D$ then the Problem \\ref{in.probD2N} is uniquely solvable.\n\\end{enumerate}\n\nReconstruction results for Problem \\ref{in.probD2N} at fixed $\\omega$ can be summarized as follows:\n\\begin{enumerate}\n \\item Uniqueness modulo gauge transformations in the most general case follows from the results of \\cite{Guil2011} in dimension $d=2$ and of \\cite{Krup2014} in dimension $d=3$.\n \\item An approximate reconstruction algorithm modulo gauge transformations in dimension $d=2$ was developed in \\cite{Agal2014b,Agal2015a}.\n \\item A possible reconstruction approach modulo gauge transformations in dimension $d=3$ was outlined in \\cite{Henk1988}.\n\\end{enumerate}\n\nIn the present article we assume that the coefficients $A_\\omega$ and $U_\\omega$ of equation \\eqref{in.waveeq} are already recovered from $\\Lambda_\\omega$ up to a gauge transformation using some appropriate method at fixed $\\omega$. The goal of the present work is to show how to get rid of the gauge non-uniqueness and recover $c$, $v$, $\\rho$ and $\\alpha$ using boundary measurements at several frequencies. In this respect, the present article can be considered as a development of the article \\cite{Agal2015b} where the particular case $\\rho \\equiv \\const$, $\\alpha_0 \\equiv 0$, $\\Omega = \\{\\omega_1\\}$ was considered, and of the article \\cite{Agal2015c} where the corresponding uniqueness theorems for equation \\eqref{in.waveeq} with general operator of the form \\eqref{in.schrop} were obtained.\n\nFor a vector field $V = (V_1,\\ldots,V_d)$ and a function $f$ in $D$ we set by definition\n\\begin{equation}\n  \\begin{aligned}\n  \\curl V & = \\begin{cases}\n              \\partial_1 V_2 - \\partial_2 V_1, \\quad d = 2, \\\\\n              \\bigl( \\partial_2 V_3 - \\partial_3 V_2, \\partial_3 V_1 - \\partial_1 V_3, \\partial_1 V_2 - \\partial_2 V_1 \\bigr), \\quad d = 3, \\\\\n            \\end{cases}\\\\\n  \\curl f & = \\bigl( \\partial_2 f, -\\partial_1 f\\bigr), \\quad d = 2,\n  \\end{aligned}\n\\end{equation}\nwhere $\\partial_j = \\partial\/\\partial_{x_j}$.\n\nIn the present article it is assumed that the following functions are already recovered from the operator $\\Lambda_\\omega$ at fixed $\\omega$:\n\\begin{align}\n  F & = \\curl \\bigl( \\tfrac{v}{c^2} \\bigr) \\quad \\text{\\it in $D$}, \\label{in.defF}\\\\\n  q_\\omega & =  f_1 - \\omega^2 f_2 + i \\omega f_3 - 2i\\omega^{1+\\zeta}\\tfrac{\\alpha_0}{c} \\quad \\text{\\it in $D$}, \\label{in.defq}\n\\end{align}\nwhere\n\\begin{equation}\\label{in.deff}\n  f_1 = \\rho^\\frac{1}{2} \\Delta \\rho^{-\\frac 1 2}, \\quad f_2 = \\tfrac{1}{c^2}+\\tfrac{v}{c^2}\\cdot\\tfrac{v}{c^2}, \\quad  f_3 = \\nabla \\cdot \\bigl(\\tfrac{v}{c^2} \\bigr) - \\tfrac{v \\cdot \\nabla \\ln \\rho}{c^2}.\n\\end{equation}\nFor the corresponding identifiability results see \\cite{Guil2011} (for $d = 2$) and \\cite{Krup2014} (for $d = 3$); for an approximate reconstruction algorithm see \\cite{Agal2014b,Agal2015a}.\n\nThereby, in the present article we study the following problems.\n\\begin{problem}\\label{in.probgtna} Find $c$, $v$ and $\\rho$ in $D$ from $q_\\omega$ given for $\\omega \\in \\Omega = \\{\\omega_1,\\omega_2\\}$, $\\omega_1 < \\omega_2$, and from $F$, $\\widetilde c$, $\\widetilde v$, $\\widetilde \\rho$. \n\\end{problem}\n\\begin{problem}\\label{in.probgtab} Find $c$, $v$, $\\rho$, $\\zeta$ and $\\alpha_0$ in $D$ from $q_\\omega$ given for $\\omega \\in \\Omega = \\{\\omega_1,\\omega_2,\\omega_3\\}$, $\\omega_1 < \\omega_2 < \\omega_3$, and from $F$, $\\widetilde c$, $\\widetilde v$, $\\widetilde \\rho$. \n\\end{problem}\n\n\\section{Solution of Problem \\ref{in.probgtna}}\nWe are going to derive the explicit formulas for solving the Problem \\ref{in.probgtna}. We consider \\eqref{in.defq} with $\\omega \\in \\Omega$ as a system of linear equations for $f_1$, $f_2$ and $f_3$. Solving this system we obtain\n\\begin{equation}\\label{in.expf}\n  f_1 = \\frac{\\omega_2^2 \\Re q_{\\omega_2} - \\omega_1^2 \\Re q_{\\omega_1}}{\\omega_2^2 - \\omega_1^2}, \\; f_2 = \\frac{\\Re q_{\\omega_1} - \\Re q_{\\omega_2}}{\\omega_2^2-\\omega_1^2}, \\; f_3 = \\omega_1^{-1} \\Im q_{\\omega_1}.\n\\end{equation}\nSet $g = \\rho^{-\\frac 1 2}$. It follows from formula \\eqref{in.deff} that $g$ satisfies the following equation\n\\begin{gather}\\label{in.rhoeq}\n  g(x) = g_0(x) + \\int_D G(x,y) f_1(y) g(y) \\, dy, \\quad x \\in \\overline D,\\\\\n  g_0(x) = \\int_{\\partial D} \\frac{\\partial G(x,y)}{\\partial \\nu_y} \\tilde \\rho^{-\\frac 1 2}(y) \\, dy,\n\\end{gather}\nwhere $G(x,y)$ is the (non-positive) Dirichlet Green's function for operator $\\Delta$ in $D$ and $\\nu_y$ is the unit exterior normal to $\\partial D$ at point $y$. Note that $g_0$ is just the the harmonic extension of $\\widetilde \\rho^{-\\frac 1 2}$ to $D$. The existence of function $G$ follows from assumption \\eqref{in.Dpro} and from \\cite[Theorem 4.17, p. 112]{Aub1982}. \n\n\\begin{lemma} Equation \\eqref{in.rhoeq} is uniquely solvable for $g \\in C(\\overline D)$. \n\\end{lemma}\n\\begin{proof} Suppose that $g_1$, $g_2 \\in C(\\overline D)$ are two solutions of equation \\eqref{in.rhoeq}. Then their difference $h = g_1 - g_2$ satisfies\n\\begin{equation}\\label{in.heq}\n  h(x) = \\int_D G(x,y) f_1(y) h(y) \\, dy, \\quad x \\in \\overline D.\n\\end{equation}\nUsing formulas \\eqref{in.rpro} and \\eqref{in.deff} we obtain that $f_1 \\in C^{0,\\beta}(D)$. Taking this into account and using formula \\eqref{in.heq} and Lemma \\cite[Lemma 4.2]{Gilb2001}, we obtain that $h \\in C^2(D) \\cap C(\\overline D)$ and that\n\\begin{subequations}\n\\begin{align}\n  & -\\Delta h + f_1(x) h = 0 \\quad \\text{\\it in $D$}, \\label{sol.hdeq} \\\\ \n  & h|_{\\partial D} = 0.  \\label{sol.hbnd}\n\\end{align}\n\\end{subequations}\nUsing formulas \\eqref{in.expf} and \\eqref{sol.hdeq} we can rewrite equation \\eqref{sol.hdeq} as \n\\begin{equation}\\label{sol.hdeq2}\n  -\\Delta (\\rho^{\\frac 1 2} h) + \\nabla \\rho \\cdot \\nabla (\\rho^{\\frac 1 2} h) = 0, \\quad \\text{\\it in $D$}.\n\\end{equation}\nIt follows from \\cite[Lemma 4.2 and Corollary 8.2]{Gilb2001} and from formulas \\eqref{sol.hbnd} and \\eqref{sol.hdeq2} that $h \\equiv 0$. Hence, $\\rho^{-\\frac 1 2}$ is the unique solution of class $C(\\overline D)$ to equation~\\eqref{in.rhoeq}. \n\\end{proof}\n\nFor vectors $a = (a_1,\\ldots,a_d)$ and $b = (b_1,\\ldots,b_d)$ in $D$ we put by definition\n\\begin{equation}\n  a \\times b = \\begin{cases}\n                  a_1 b_2 - a_2 b_1, \\quad d = 2,\\\\\n                  \\bigl( a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1 \\bigr), \\quad d = 3.\n               \\end{cases}\n\\end{equation}\nRecall that the Helmholtz decomposition of the vector field $\\tfrac{v}{c^2}$ is given by the following formula:\n\\begin{equation}\\label{sol.Helm}\n  \\frac{v}{c^2} = \\nabla \\Phi - \\curl V \\quad \\text{\\it in $D$},\n\\end{equation}\nwhere \n\\begin{gather}\n  \\Phi(x) = \\int_D G_0(x-y) \\nabla_y \\cdot \\bigl(\\tfrac{v(y)}{c^2(y)}\\bigr)\\,dy -\\int_{\\partial D} G_0(x-y) \\frac{\\nu_y \\cdot \\widetilde v(y)}{\\widetilde c^2(y)} \\, dy,\\\\\n  V(x) = \\int_D G_0(x-y) F(y) \\, dy - \\int_{\\partial D} G_0(x-y) \\frac{\\nu_y \\times \\widetilde v(y)}{\\widetilde c^2(y)} \\, dy, \\label{in.defV}\\\\\n  G_0(x) = -\\frac{1}{2\\pi}\\ln|x|, \\quad d = 2,\\\\\n  G_0(x) = \\frac{1}{4\\pi} \\frac{1}{|x|}, \\quad d = 3,\n\\end{gather}\nwhere $\\nu_y$ is the unit exterior normal to $\\partial D$ at point $y$. Note that the vector field $V$ is known since $F$, $\\widetilde v$ and $\\widetilde c$ are given. \n\nUsing formula \\eqref{sol.Helm} we can recover the function $\\Phi|_{\\partial D}$ modulo an additive constant (which does not matter). Fix a point $x^0 \\in \\partial D$. Let $x \\colon [0,1] \\to \\partial D$ be a smooth curve linking $x^0$ to some given point $x \\in \\partial D$. Then\n\\begin{equation}\\label{sol.PhiOnBnd}\n  \\Phi(x)-\\Phi(x^0) = \\int_0^1 \\bigl(\\tfrac{\\widetilde v}{\\widetilde c^2} + \\curl V\\bigr)|_{x(t)} \\cdot \\dot x(t) \\, dt, \\quad \\dot x = \\frac{dx}{dt}.\n\\end{equation}\n\nIt follows from formulas \\eqref{in.deff} and \\eqref{sol.Helm} that $\\Phi$ satisfies the equation\n\\begin{equation*}\n  \\begin{gathered}\n  -\\Delta \\Phi + \\nabla \\ln \\rho \\cdot \\nabla \\Phi = -f_3 + \\curl V \\cdot \\nabla \\ln \\rho \\quad \\text{\\it in $D$}, \\\\\n  \\text{\\it or} \\; -\\Delta \\eta + f_1(y) \\eta =  \\rho^{-\\frac 1 2} (-f_3 + \\curl V \\cdot \\nabla \\ln \\rho) \\quad \\text{\\it in $D$},\n  \\end{gathered}\n\\end{equation*}\nwhere $\\eta = \\rho^{-\\frac 1 2}(\\Phi-\\Phi(x^0))$. The function $\\eta$ can be found from the following integral equation:\n\\begin{gather}\n  \\eta(x) = \\eta_0(x) + \\eta_1(x) + \\int_D G(x,y) f_1(y) \\eta(y) \\, dy, \\quad x \\in \\overline D, \\label{in.etaeq}\\\\\n  \\eta_0(x) = \\int_D G(x,y) \\rho^{-\\frac 1 2}(y) \\bigl( f_3(y)-\\curl V(y) \\cdot \\nabla \\ln \\rho(y) \\bigr) \\, dy, \\notag\\\\\n  \\eta_1(x) = \\int_{\\partial D} \\frac{\\partial G(x,y)}{\\partial \\nu_y} \\tilde \\rho^{-\\frac 1 2}(\\Phi(y)-\\Phi(x^0)) \\, dy, \\notag\n\\end{gather}\nwhere $G(x,y)$ is the (non-positive) Dirichlet Green's function for $\\Delta$ in $D$. Note that $\\eta_1$ is just the harmonic extension of $\\tilde \\rho^{-\\frac 1 2}(\\Phi|_{\\partial D}-\\Phi(x^0))$ to $D$. Also note that equation \\eqref{in.etaeq} has the same kernel as equation \\eqref{in.rhoeq}. Therefore, it is uniquely solvable for $\\eta \\in C(\\overline D)$.\n\nAfter recovering $\\Phi - \\Phi(x^0)$ in $D$ we can find $\\tfrac{v}{c^2}$ using formula \\eqref{sol.Helm}. Finally, using formulas \\eqref{in.deff} and \\eqref{sol.Helm} we obtain\n\\begin{equation}\\label{in.expcv}\n  \\frac{1}{c^2} = f_2 - (\\nabla \\Phi - \\curl V)^2, \\quad v = c^2 (\\nabla \\Phi - \\curl V).\n\\end{equation}\nThe described algorithm for solving Problem \\ref{in.probgtna} is summarized in the following theorem.\n\n\\begin{theorem}\\label{propna} Suppose that $D$ satisfies \\eqref{in.Dpro}, $\\Omega = \\{\\omega_1,\\omega_2\\}$, $\\omega_1 \\neq \\omega_2$ and \\eqref{in.unsol} holds for all $\\omega \\in \\Omega$, and suppose that $c$, $v$, $\\rho$ satisfy \\eqref{in.cpro}--\\eqref{in.rpro}. Then Problem \\ref{in.probgtna} can be solved as follows:\n\\begin{enumerate}\n \\item Define $f_1$, $f_2$, $f_3$ and $V$ using formulas \\eqref{in.expf} and \\eqref{in.defV}.\n \\item Find $g$ as the unique solution of class $C(\\overline D)$ to equation \\eqref{in.rhoeq}. Set $\\rho = g^{-2}$.\n \\item Fix $x^0 \\in \\partial D$ and find $\\Phi|_{\\partial D}-\\Phi(x^0)$ using formula \\eqref{sol.PhiOnBnd}.\n \\item Find $\\eta$ as the unique solution of class $C(\\overline D)$ to equation \\eqref{in.etaeq}. Set $\\Phi - \\Phi(x^0) = \\rho^{\\frac 1 2}\\eta$.\n \\item Find $c$ and $v$ using the explicit formulas \\eqref{in.expcv}.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark} Suppose that $d = 2$ and $D = \\bigl\\{ x \\in \\mathbb R^2 \\mid |x| \\leq 1\\}$. Then the function $g_0$ from formula \\eqref{in.rhoeq} and the function $G(x,y)$ from formulas \\eqref{in.rhoeq} and \\eqref{in.etaeq} can be found explicitly:\n\\begin{equation*}\n  g_0(x) = \\int\\limits_{S^1} \\widetilde \\rho^{-\\frac 1 2}(\\vartheta) \\frac{1-|x|^2}{|\\vartheta - x|^2} \\frac{d\\vartheta}{2\\pi}, \\quad G(x,y) = \\frac{1}{2\\pi} \\ln \\frac{|x||y-x|}{\\bigl|y |x|^2 - x\\bigr|},\n\\end{equation*}\nwhere $S^1 = \\partial D$. Furthermore, if $\\|f_1\\|_{C(\\overline D)} < 4$ then equations \\eqref{in.rhoeq} and \\eqref{in.etaeq} can be solved using the method of successive approximations in $C(\\overline D)$.\n\\end{remark}\n\n\\section{Solution of Problem \\ref{in.probgtab}}\nDefine the sets $D_0$ and $D_1$ by the formulas\n\\begin{equation}\\label{in.defD0}\n  D_0 = \\bigl\\{ x \\in D \\mid \\tfrac{\\Im q_{\\omega_1}(x)}{\\omega_1} = \\tfrac{\\Im q_{\\omega_2}(x)}{\\omega_2} \\bigr\\}, \\quad D_1 = D \\setminus D_0.\n\\end{equation}\nIt follows from formulas \\eqref{in.abpro} and \\eqref{in.defq} that $D_0 = \\{ x \\in D \\mid \\alpha_0(x) = 0 \\}$ and that the functions $f_1$, $f_2$ and $f_3$ defined in \\eqref{in.defq} can be found in $D_0$ using formulas \\eqref{in.expf}.\n\nUsing formula \\eqref{in.defq} for $x \\in D_1$ and $\\omega \\in \\Omega$ we obtain that\n\\begin{equation}\\label{in.zetaeq}\n  \\frac{\\omega_2^{-1}\\Im q_{\\omega_2}(x)-\\omega_1^{-1}\\Im q_{\\omega_1}(x)}{\\omega_3^{-1}\\Im q_{\\omega_3}(x) - \\omega_1^{-1}\\Im q_{\\omega_1}(x)} = \\frac{\\bigl(\\tfrac{\\omega_2}{\\omega_1}\\bigr)^{\\zeta(x)}-1}{\\bigl(\\tfrac{\\omega_3}{\\omega_1}\\bigr)^{\\zeta(x)}-1}.\n\\end{equation}\nWe consider \\eqref{in.zetaeq} as an equation for finding $\\zeta(x)$ at fixed $x \\in D_1$. This equation is uniquely solvable for $\\zeta(x)$ at fixed $x \\in D_1$ as the following lemma shows.\n\\begin{lemma} The right side of equation \\eqref{in.zetaeq} at fixed $x \\in D_1$ is a strictly decreasing function of $\\zeta(x) \\in (0,+\\infty)$. \n\\end{lemma}\n\\begin{proof} It is sufficient to show that for any $p$, $q$ such that $1 < p < q$ and for any $t > 1$ the equality\n\\begin{equation}\\label{in.ptqt}\n  \\frac{p^t-1}{q^t-1} = \\frac{p-1}{q-1}.\n\\end{equation}\ncan not hold. Assuming the equality \\eqref{in.ptqt}, we define\n\\begin{equation}\\label{in.deflam}\n  \\lambda_1 = \\frac{p-1}{q-1}, \\quad \\lambda_2 = \\frac{q-p}{q-1}.\n\\end{equation}\nUsing formulas \\eqref{in.ptqt} and \\eqref{in.deflam} we obtain the formulas\n\\begin{subequations}\n\\begin{gather}\n    \\lambda_1 + \\lambda_2 = 1, \\quad \\lambda_1 > 0, \\; \\lambda_2 > 0, \\label{in.lamsum}\\\\\n    \\lambda_1 q + \\lambda_2 = p, \\quad \\lambda_1 q^t + \\lambda_2 = p^t. \\label{in.lamcc}\n\\end{gather}\n\\end{subequations}\nSince the function $f(s) = s^t$ is strictly convex, relations \\eqref{in.lamsum}--\\eqref{in.lamcc} can not hold. Therefore the initial assumption that \\eqref{in.ptqt} holds must be false. \n\\end{proof}\n\nNext we find functions $f_1$, $f_2$, $f_3$ and $\\alpha_0\/c$ in the domain $D_1$. It follows from formula \\eqref{in.defq} that $f_1$ and $f_2$ in $D_1$ can be found using formulas \\eqref{in.expf}. It also follows from \\eqref{in.defq} that $f_3$ and $\\alpha_0\/c$ in $D_1$ can be found from the following formulas:\n\\begin{equation}\\label{in.alexp}\n  f_3 = \\frac{\\frac{\\omega_1^\\zeta}{\\omega_2}\\Im q_{\\omega_2} - \\frac{\\omega_2^\\zeta}{\\omega_1} \\Im q_{\\omega_1}}{\\omega_1^\\zeta - \\omega_2^\\zeta}, \\quad \\frac{\\alpha_0}{c} = \\frac 1 2\\frac{\\omega_2^{-1}\\Im q_{\\omega_2} - \\omega_1^{-1} \\Im q_{\\omega_1}}{\\omega_1^\\zeta - \\omega_2^\\zeta}.\n\\end{equation}\nUsing the values of $f_1$, $f_2$ and $f_3$ in $D$ we can find $c$, $\\rho$ and $v$ in $D$ using the steps 2--5 mentioned in Theorem \\ref{propna}. Finally, we find $\\alpha_0$ using $\\alpha_0\/c$ and $c$. The algorithm for solving Problem \\ref{in.probgtab} is summarized in the following proposition.\n\n\\begin{theorem}\\label{propab} Suppose that $D$ satisfies \\eqref{in.Dpro}, $\\Omega = \\{\\omega_1,\\omega_2,\\omega_3\\}$, $\\omega_1 < \\omega_2 < \\omega_3$, and \\eqref{in.unsol} holds for all $\\omega \\in \\Omega$. Suppose also that $c$, $v$, $\\rho$, $\\zeta$ and $\\alpha_0$ satisfy \\eqref{in.cpro}--\\eqref{in.abpro}. Then Problem \\ref{in.probgtab} can be solved as follows:\n\\begin{enumerate}\n \\item Define $D_0$ and $D_1$ using formula \\eqref{in.defD0}. Find $f_1$ and $f_2$ in $D$ using formulas \\eqref{in.expf}. Find $f_3$ using formula \\eqref{in.expf} in $D_0$ and formula \\eqref{in.alexp} in $D_1$.\n \\item Find $\\zeta(x)$ at fixed $x \\in D_1$ as the unique positive solution to equation \\eqref{in.zetaeq}.\n \\item Find $\\alpha_0\/c$ in $D_1$ using formula \\eqref{in.alexp}.\n \\item Find $g$ as the unique solution of class $C(\\overline D)$ to equation \\eqref{in.rhoeq}. Set $\\rho = g^{-2}$.\n \\item Fix $x^0 \\in \\partial D$ and find $\\Phi|_{\\partial D}-\\Phi(x^0)$ using formula \\eqref{sol.PhiOnBnd}.\n \\item Find $\\eta$ as the unique solution of class $C(\\overline D)$ to equation \\eqref{in.etaeq}. Set $\\Phi-\\Phi(x^0) = \\rho^{\\frac 1 2}\\eta$.\n \\item Find $c$ and $v$ using the explicit formulas \\eqref{in.expcv}. Set $\\alpha_0$ to zero in $D_0$ and find $\\alpha_0$ from $\\alpha_0\/c$ and $c$ in $D_1$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark} Note that the formulas and equations presented in Theorem \\ref{propna} (resp. \\ref{propab}) require the knowledge of the function $q_\\omega$ at two (resp. three) frequencies $\\omega$. These formulas are exact but they can be not very stable with respect to the noise in the initial data. However, if $q_\\omega$ is known for a bigger number of frequencies it is possible to increase the stability of reconstruction by replacing formulas \\eqref{in.expf} and \\eqref{in.alexp} with their least squares analogues. A numerical study of reconstruction stability will be carried out in a subsequent paper.\n\\end{remark}\n\n\n\\section{Aknowledgements}\nThe present article was prepared in the framework of research conducted under the supervision of Prof. R. G. Novikov.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\\label{intro}\n\tSatoshi Nakamoto in 2008 proposed a decentralized cryptography-based electronic currency called Bitcoin~\\cite{nakamoto2008bitcoin}. Such financial systems eliminate the control of centralized authority and provide ubiquity as well as fairness via (quasi) real-time transactions. Such digital currencies also guarantee a certain degree of anonymity, which raises novel and unique concerns, e.g., an inevitable-growth in illegal activities.\n\t\\par\n\tOn another side, ransomware is a class of malware that restricts access to the system it infects until the victim pays the demanded ransom. Readily available toolkits such as~eda2\\footnote{eda2 is an abandoned open-source ransomware kit that was distributed only for educational purposes.} and Ransomware-as-a-Service (RaaS) enable even a novice user to create and launch ransomware. Furthermore, the ransomware affiliate program lures users to spread ransomware in exchange for profit share. According to the annual threat report-2017 published by Symantec Inc.~\\cite{symantec_report_2017}, ransomware continued to be the most dangerous cyber-crime threat to individual users and enterprises in 2016. Compared to the previous year, the number of detected ransomware infection increased by 36\\% during 2016. Moreover, average ransomware detection rate reached over 1,500 incidents per day at the year-end. In particular, the average ransom amount rose 266\\% from USD 294 in 2015 to USD 1,077. \n\t\n\t\\par\n\tThe evolving class of ransomware has been exploiting privacy-preserving online services, e.g., the Tor hidden network~\\cite{Dingledine2004} to remain anonymous. Moreover, the pseudo-anonymous nature of decentralized currencies such as Bitcoin makes it difficult to trace a payee. Hence, the cybercriminals have been misusing such payment systems to extort ransoms anonymously. In this paper, we present our comprehensive and~longitudinal study on recent ransomware and report the~economic impact of such ransomware from the Bitcoin~payment perspective.\n\t\n\t\\par\n\t\\textit{Contributions:}\n\tThe major contributions of this paper are listed as follows:\n\t\\begin{enumerate}\n\t\t\\item We present a lightweight framework to identify, collect, and analyze addresses that belong to the same user. We~also propose a novel approach for classifying a payment as ransom.\n\t\t\n\t\t\\item Using our framework, we analyzed the economic impact (in terms of ransoms extorted in Bitcoin) of all the recent ransomware: (i)~that used Bitcoin as at least one mode of ransom payment, and (ii)~for which at least one Bitcoin address is publicly known.\n\t\t\n\t\n\t\n\t\t\n\t\t\\item We discuss the inception, evolution (where applicable), and functionality (including distribution, infection, and encryption procedure) of every analyzed ransomware along with the magnitude and timeline of their ransom demands.\n\t\t\n\t\t\n\t\t\\item We also release our dataset\\footnote{spritz.math.unipd.it\/projects\/btcransomware\/} for future research endeavors. The dataset contains a detailed transaction history of all the addresses we identified for each ransomware. Hence, our results are fully reproducible. \n\t\n\t\t\n\t\n\t\n\t\\end{enumerate}\n\t\n\t\\textit{Organization:}\n\tThe remainder of this paper is organized as follows. In Section~\\ref{related}, we explain the essential concepts related to ransomware infection and the Bitcoin currency system. Section~\\ref{related_work} addresses the previous works on identification and assessment of cyber-crimes in the Bitcoin ecosystem. Section~\\ref{evaluation} elucidates our framework for ransom identification.\n\tIn Section~\\ref{ransomware}, we present our findings and enlighten the economic impact the ransomware that fulfilled our selection criteria. In Section~\\ref{limitation}, we discuss the limitations of our proposed framework. Finally, Section~\\ref{futurework} concludes the paper.\n\t\n\n\t\n\t\\section{Preliminaries}\n\t\\label{related}\n\tIn this section, we describe the chronology of a typical ransomware infection and explain the fundamentals of the Bitcoin cryptocurrency system\n\t\n\t\\textit{Ransomware:} A typical ransomware infection includes the following events:\n\t\\begin{enumerate}\n\t\t\\item Infection: Similar to generic malware, ransomware are also distributed via various infection vectors. These vectors include, but not limited to, email spamming with malicious attachment (e.g., CryptoLocker) or link to the malicious payload (e.g., CryptoWall), exploit packs (e.g.,~Angler browser exploit in TeslaCrypt and Neutrino exploit kit in DMA Locker). Interestingly, recent ransomware incorporate self-propagation capabilities. For instance, NotPetya and WannaCry exploit vulnerabilities in the network protocols to infect local computers on the~same network.\n\t\t\\item Encryption: After infiltration, ransomware silently encrypt files on the infected system. In particular, ransomware target those files that are valuable to the user, e.g.,~images, videos, documents. For the encryption process, ransomware use symmetric encryption algorithm, asymmetric encryption algorithm, or even combination of the both. The key for encryption is either generated locally or procured from a remote Command~and~Control~(C\\&C). Generally, the backup files are also encrypted\/deleted to prevent recovery. However, the files responsible for running the system are not affected, at least until the deadline for the ransom payment. \n\t\t\\item Extortion: After the encryption process, ransomware display a ransom note on the screen. The ransom note of recent ransomware includes a threat message, ransom amount specified in fiat currency such as US~dollar (for instance, USD~300 in NotPetya) or cryptocurrency such as Bitcoin (for instance, 1~BTC in CryptoLocker), a countdown timer that shows the time left before the deadline, and a payment address. The payment address can be a Bitcoin address or a website's address that shows this Bitcoin address. Typically, the ransom note also includes instructions on how and where to buy Bitcoin.\n\t\t\\item Decryption: After confirmation of the ransom payment, the ransomware either automatically start the decryption process, or the victim is asked to download and run a~decryption tool.\n\t\\end{enumerate}\n\t\n\t\\textit{Bitcoin:} In 1993, researchers from Carnegie Mellon University~\\cite{tygar1993cryptography} and University of Southern California~\\cite{medvinsky1993netcash} discussed the need for a cryptography-based digital currency. On November~1,~2008, a person or a group of persons under a pseudonym Satoshi Nakamoto articulated the idea of peer-to-peer, decentralized, cryptography-based electronic currency system called Bitcoin~\\cite{nakamoto2008bitcoin}. The basic terminology used in the Bitcoin protocol are as follows:\n\t\\begin{itemize}\n\t\t\\item Address: A Bitcoin address is a string identifier of a possible destination for a Bitcoin payment. It is 26~to~35~alphanumeric characters long and begins with the number~1~(Pay-to-Pub KeyHash or P2PKH type) or 3~(Pay to Script Hash or P2SH type). Bitcoin addresses are hashed public keys generated from the Elliptic Curve Digital Signature Algorithm (ECDSA). Hence, each Bitcoin is associated with the owner's public key.\n\t\t\n\t\t\\item Wallet: A wallet is a file that stores Bitcoin addresses along with the corresponding private keys. It also maintains the Unspent Transaction Output (UTXO) corresponding to each address.\n\t\t\n\t\t\\item Blockchain: The blockchain is a shared, public ledger on which the entire Bitcoin network relies. All confirmed transactions are included in the blockchain without any exception. This way, new transactions can be verified to be spending Bitcoin that are indeed owned by the spender. The integrity and the chronological order of the blockchain are enforced with cryptography.\\footnote{bitcoin.org\/en\/how-it-works}\n\t\t\n\t\t\\item Block: An individual unit of the blockchain is called a block. Each block includes the hash of the previous block to guarantee the integrity of the network, the nonce that assisted its mining, and a list of the transactions.\n\t\t\n\t\t\\item Transaction: A transaction refers to a transfer of Bitcoin between Bitcoin addresses. To transfer Bitcoin, a payer creates a transaction message. In this message, the payer specifies the payee's Bitcoin address as well as an amount of Bitcoin to transfer. As shown in~\\figurename{~\\ref{figure_trx}}, the payer authenticates the transaction by digitally signing it with the private key of the corresponding address. Finally, Bitcoin network broadcasts and confirms (typically, in the following 10 minutes) the transaction through a process called mining. A confirmed transaction is irreversible.\n\t\t\\enlargethispage{-\\baselineskip}\n\t\n\t\t\\begin{figure}[H]\n\t\t\t\\centering\n\t\t\t\\includegraphics[trim = 2mm 8mm 2mm 4mm, clip, width=.9\\linewidth]{.\/new\/graphs\/trx.pdf}\n\t\t\t\\caption[]{An example of a simple Bitcoin transaction}\n\t\t\t\\label{figure_trx}\n\t\t\\end{figure}\n\t\t\n\t\n\t\t\\begin{figure*}[!htbp]\n\t\t\t\\centering\n\t\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, scale=.44]{.\/new\/graphs\/bitcoin_prices.pdf}\n\t\t\t\\caption[]{BTC-USD exchange rate trend}\n\t\t\t\\label{figure1}\n\t\t\\end{figure*}\n\t\\end{itemize}\n\n\t\\par\n\tA user can also purchase Bitcoin in exchange for other regulated currencies. The unit of the Bitcoin currency is ``Bitcoin,'' abbreviated as ``BTC.'' Like any other traded commodity, the price\\footnote{We use the term ``price'' to refer BTC-USD exchange rate.} of Bitcoin varies. \\figurename{~\\ref{figure1}} depicts the BTC-USD exchange rate since July~18,~2010, the day when one of the world's first Bitcoin currency exchange market Mt.~Gox~was~established.\n\t\n\t\\section{Related Work}\n\t\\label{related_work}\n\tLaw enforcement authorities as well as the research community have made several attempts to identify and measure cyber-crimes in the Bitcoin ecosystem. The authors in~\\cite{bistarelli2017go, mcginn2016visualizing, di2015bitconeview, reid2011analysis} proposed tools to analyze transactions in the Bitcoin blockchain visually. Christin in~\\cite{christin2013traveling} proposed a thorough analysis of the Silk Road anonymous marketplace and discussed the socio-economic implications of the findings. Ron and Shamir used the public blockchain data to estimate the wealth of the Silk Road marketplace's owner, known as Dread Pirate Robert~\\cite{ron2014did}. Soska and Christin studied anonymous online marketplaces including Silk Road, Sheep Marketplace, etc. and examined how virtual marketplaces have evolved~\\cite{soska2015measuring}. Meiklejohn~et~al.~\\cite{meiklejohn2013fistful} proposed an approach to comprehend overall transaction patterns of the Bitcoin payments used for criminal or fraudulent purposes.\n\t\\par\t\t\n\tHowever, the literature on measuring the economic impact of ransomware that accepted ransoms via Bitcoin (hereinafter referred to as ``Bitcoin ransomware'') is rather limited. Huang~et~al.~\\cite{huangtracking} discuss the ethical and technical issues of monitoring ransomware activities as well as the dynamics of ransom payments. Liao~et~al.~\\cite{liao2016behind} analyzed the timestamps of ransom payments to CryptoLocker. The work~\\cite{kharraz2015cutting} provides a holistic view of the general ransomware that appeared between 2006 and 2014. Additionally, the authors also estimated the financial intensives gained by CyptoLocker ransomware. Spagnuolo~et~al. proposed a framework called BitIodine \\cite{spagnuolo2014bitiodine}. The authors used BitIodine to investigate Bitcoin addresses associated with CryptoLocker ransomware and Dread Pirate Roberts. The works~\\cite{secureworks_cryptolocker,cryptolocker_intelligence_report} present a systematic analysis of CryptoLocker ransomware.\n\t\\par\n\tIt is noteworthy that previous works~\\cite{spagnuolo2014bitiodine,kharraz2015cutting,liao2016behind,secureworks_cryptolocker,cryptolocker_intelligence_report} only considered either the daily average or highest Bitcoin price to classify ransom payments and do not take into account the variations that might occur due to the transaction fee. Furthermore, their estimation of the total worth of extorted ransoms is based on the Bitcoin price on the day of their evaluation, which exaggerates the results due to fluctuations (mostly, increase; see \\figurename{~\\ref{figure1}}) in the price of Bitcoin. Additionally, the systems proposed in the previous works~\\cite{bistarelli2017go, mcginn2016visualizing, di2015bitconeview, reid2011analysis,spagnuolo2014bitiodine} demand high bandwidth, storage, and computational resources as they query the entire blockchain.\n\t\n\t\\par\n\tTo the best of our knowledge, our work is the first study that elaborates not only the characteristics and functionality of various Bitcoin ransomware, but it also gives more accurate insights on the economic impact of such ransomware. In particular, our work is different from the state-of-the-art on various dimensions: (i)~to identify a payment as ransom, we consider both the day-to-day lowest and highest Bitcoin price as well as the variations due to the transaction fee; (ii)~to accurately assess the worth (in USD) of extorted ransoms, we used the average Bitcoin price on the day of each ransom payment; and (iii)~our framework focuses only on the transactions belonging to the address(es) of interest rather than the entire blockchain.\n\t\n\t\n\t\n\t\n\n\n\t\\section{Ransom Identification Framework}\n\t\\label{evaluation}\n\t\\ankit{To investigate the ransoms extorted by a ransomware, we first identify the Bitcoin addresses linked to the ransomware. Then, we obtain the transaction history of these addresses. Finally, we distinguish the transactions associated with the ransom payments. To this end,} we propose our framework, which consists of three stages\/parts\\ankit{\/modules}: (i)~identifying the Bitcoin addresses belonging to the ransomware \\ankit{(discussed in~Section~\\ref{identify})}; (ii)~data \\ankit{(transaction history)} collection and database generation from the blockchain \\ankit{(presented in~Section~\\ref{data_col})}; and (iii)~our considerations for classifying a payment as ransom \\ankit{(elaborated in~Section~\\ref{consideration})}.\n\t\n\t\\subsection{\\ankit{\\textit{Module1}:} Identification of ransomware addresses}\n\t\\label{identify}\n\tBitcoin offers privacy only through pseudonymity, and an increasing number of works~\\cite{ron2013quantitative, ron2014did, soska2015measuring, reid2011analysis, biryukov2014deanonymisation, meiklejohn2013fistful} suggest that information available in public blockchain ledger can lead to de-anonymize (to a certain extent) Bitcoin transactions.\n\t\\par\n\tTo collect the addresses associated with a ransomware, we began by extensively searching various online resources:~ransomware knowledge base (e.g., ESET, Kaspersky Lab, Malwarebytes, Symantec); ransomware removal guides (e.g., BleepingComputer.com, MalwareTips.com, 2-spyware.com, ``How To'' videos on YouTube); reports from Counter Threat Units (CTU), Incident Responses (IR), and Security Operations Centers (SOC) (e.g., Dell SecureWorks, PhishMe.com); online fora (e.g., Reddit) where victims and researchers post Bitcoin addresses associated with the concerned ransomware; and screenshots of ransomware available in different image search engines (e.g., Google, Yahoo). \\ankit{Considering the fact that not every address related to a ransomware is posted on the Internet, we used two clustering heuristics to identify the set of addresses controlled by the same user (cybercriminals, in our case).}\n\tOur heuristics are based on the fundamental principles of the Bitcoin transaction protocol~\\cite{nakamoto2008bitcoin} and are as follows:\n\t\n\t\\subsubsection{Multi-input transactions} \n\tA multi-input transaction usually\\footnote{Nowadays, coin mixing services allow users to join their transactions to enhance anonymity and unlinkability. However, such services have many security and privacy concerns~\\cite{conti_survey_2017}. Hence, for simplicity, we assume that the user commonly does not make use of Bitcoin mixers.} takes place when a user $U$ attempts to make a payment, and the payment amount $P$ cannot be sufficiently funded by any of the individual Bitcoin balance available in $U$'s wallet. In such a scenario, the Bitcoin protocol~allows grouping of a set of Bitcoin balances from $U$'s wallet to settle $P$ and make payment through a multi-input transaction. Hence, we can conclude that if a set of input addresses $S_{input}$ is used to disburse $P$, then $S_{input}$ is managed by the same user.\n\t\n\t\n\t\n\t\\subsubsection{Shadow\/change address} In the Bitcoin protocol, the whole input amount must be spent in the same transaction. To deliver the ``change'' back to the user $U$, a shadow address~$A_{shadow}$ is automatically generated and used to collect the unspent amount of the transaction.  If there are two addresses in the set of output addresses $S_{out}$, and one address has never been seen before in the whole blockchain while the other address has appeared before, then we can safely presume that the newly generated address is a shadow address~\\cite{meiklejohn2013fistful}.\n\t\n\t\\par\n\tAlgorithm~\\ref{algo1} explains our approach to identify the addresses managed by the same user, hereinafter referred to as ``Cluster''. Here, $S_{initial}$ represents the set of addresses collected from the online resources, $S_{input}$ is a set of input addresses in a transaction, and $A_{shadow}$ represents a shadow address generated (if any) in a transaction.\n\t\\begin{algorithm}[H]\n\t\t\\caption{Identifying addresses managed by the same user.}\n\t\t\\label{algo1}\n\t\t\\hspace*{\\algorithmicindent} \\textbf{Input:} $S_{initial}$ \n\t\n\t\t\\begin{algorithmic}[1]\n\t\t\t\\State $Cluster := S_{initial}$\n\t\t\t\\State $Cluster^\\prime := \\{\\}$ \\Comment{\\{ \\} is an empty set}\n\t\t\t\\While{$Cluster \\not = Cluster^\\prime$}\n\t\t\t\\State $Cluster^\\prime := Cluster$\n\t\t\t\\State $M := \\{\\}$ \\Comment{$M$ stores $S_{input}$}\n\t\t\t\\State $C := \\{\\}$ \\Comment{$C$ stores $A_{shadow}$}\n\t\t\t\n\t\t\t\\For{$i$ in $Cluster$}\n\t\t\t\\State Get all transactions $Tx$ where $i$ is an input address\n\t\t\t\\For{$t$ in $Tx$}\n\t\t\t\\State $M \\cup (S_{input}~in~t)$ \\Comment{$\\cup$ is set union}\n\t\t\t\\State $C \\cup (A_{shadow}~in~t)$\n\t\t\t\\EndFor\n\t\t\t\\EndFor\t\t\t\n\t\t\t\n\t\t\t\\State $Cluster := Cluster \\cup M \\cup C$\n\t\t\t\\EndWhile\n\t\t\t\\State \\textbf{return} $Cluster$\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\n\t\\par\n\tEssentially, for a given list of addresses, our algorithm recursively finds all the addresses satisfying our heuristics.\n\t\n\n\t\\begin{figure*}[!b]\n\t\t\\centering\n\t\t\\resizebox{2\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\draw (0,0) -- (24,0);\n\t\t\t\n\t\t\n\t\t\t\\foreach \\x in {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}\n\t\t\t\\draw (\\x cm,3pt) -- (\\x cm,-3pt);\n\t\t\t\n\t\t\n\t\t\n\t\t\t\\draw  (0,0)  node[above=3pt, rotate=45, xshift=-1.1cm,  yshift=-.5cm]  {Sep. 05, '13} node[below=3pt, rotate=45, xshift=1.2cm, yshift=0.5cm] {CryptoLocker};\n\t\t\t\\draw  (1,0)  node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.45cm]  {Feb. '14} node[below=3pt, rotate=45, xshift=1.22cm, yshift=0.5cm] {~CryptoDefense};\n\t\t\t\\draw  (2,0)  node[above=3pt, rotate=45, xshift=-.75cm,  yshift=-.5cm]  {Q1 '14} node[below=3pt, rotate=45, xshift=0.95cm, yshift=0.5cm] {~CryptoWall};\n\t\t\t\\draw  (3,0)  node[above=3pt, rotate=45, xshift=-1.1cm,  yshift=-.45cm]  {Mid-Jul. '14} node[below=3pt, rotate=45, xshift=1.1cm, yshift=0.5cm] {CTB-Locker};\n\t\t\t\\draw  (4,0)  node[above=3pt, rotate=45, xshift=-1.1cm,  yshift=-.5cm]  {Feb. 05, '15} node[below=3pt, rotate=45, xshift=1.75cm,  yshift=0.5cm] {CryptoTorLocker2015};\n\t\t\t\\draw  (5,0)  node[above=3pt, rotate=45, xshift=-1.2cm,  yshift=-.45cm]  {Mid-Feb. '15} node[below=3pt, rotate=45, xshift=1.0cm, yshift=0.5cm] {TeslaCrypt};\n\t\t\t\\draw  (6,0)  node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.45cm]  {Nov. '15} node[below=3pt, rotate=45, xshift=0.825cm, yshift=0.5cm] {Chimera};\n\t\t\t\\draw  (7,0)  node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.45cm]  {Dec. '15} node[below=3pt, rotate=45, xshift=1.22cm, yshift=0.5cm] {DMA Locker};\n\t\t\t\\draw  (8,0)  node[above=3pt, rotate=45, xshift=-.75cm,  yshift=-.5cm]  {Q1 '16} node[below=3pt, rotate=45, xshift=0.9cm, yshift=0.5cm] {~Hi Buddy!};\n\t\t\t\\draw  (9,0)  node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.45cm]  {Mar. '16} node[below=3pt, rotate=45, xshift=0.6cm, yshift=0.5cm] {Petya};\n\t\t\t\\draw  (10,0) node[above=3pt, rotate=45, xshift=-1.1cm,  yshift=-.5cm]  {Mar. 04, '16} node[below=3pt, rotate=45, xshift=0.9cm, yshift=0.5cm] {KeRanger};\n\t\t\t\\draw  (11,0) node[above=3pt, rotate=45, xshift=-1.2cm,  yshift=-.45cm]  {Late  Mar. '16} node[below=3pt, rotate=45, xshift=0.7cm, yshift=0.5cm] {Jigsaw};       \n\t\t\t\\draw  (12,0) node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.5cm]  {May '16} node[below=3pt, rotate=45, xshift=0.72cm, yshift=0.5cm] {Mischa};\n\t\t\t\\draw  (13,0) node[above=3pt, rotate=45, xshift=-1.15cm,  yshift=-.5cm]  {May 24, '16} node[below=3pt, rotate=45, xshift=0.85cm, yshift=0.5cm] {ZCryptor};\n\t\t\t\\draw  (14,0) node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.55cm]  {Aug. '16} node[below=3pt, rotate=45, xshift=1.05cm, yshift=0.5cm] {VenusLocker};\n\t\t\t\\draw  (15,0) node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.5cm]  {Dec. '16} node[below=3pt, rotate=45, xshift=0.95cm, yshift=0.5cm] {GoldenEye};\n\t\t\t\\draw  (16,0) node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.55cm]  {Dec. '16} node[below=3pt, rotate=45, xshift=.8cm, yshift=0.45cm] {KillDisk};                  \n\t\t\t\n\t\t\t\\draw  (17,0) node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.45cm]  {Feb. '17} node[below=3pt, rotate=45, xshift=1.55cm, yshift=0.5cm] {The Trump Locker};\n\t\t\t\\draw  (18,0) node[above=3pt, rotate=45, xshift=-1.1cm,  yshift=-.5cm]  {Feb. 22, '17} node[below=3pt, rotate=45, xshift=0.78cm, yshift=0.5cm] {FindZip};              \n\t\t\t\\draw  (19,0) node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.45cm]  {Mar. '17} node[below=3pt, rotate=45, xshift=1.48cm, yshift=0.5cm] {The LLTP Locker};\n\t\t\t\\draw  (20,0) node[above=3pt, rotate=45, xshift=-0.85cm,  yshift=-.5cm]  {May '17} node[below=3pt, rotate=45, xshift=1.18cm, yshift=0.5cm] {ThunderCrypt};\n\t\t\t\\draw  (21,0) node[above=3pt, rotate=45, xshift=-1.1cm,  yshift=-.5cm]  {May 12, '17} node[below=3pt, rotate=45, xshift=0.9cm, yshift=0.5cm] {WannaCry};\n\t\t\t\\draw  (22,0) node[above=3pt, rotate=45, xshift=-1.1cm,  yshift=-.5cm]  {Jun. 27, '17} node[below=3pt, rotate=45, xshift=0.9cm, yshift=0.5cm] {NotPetya};\n\t\t\t\\draw  (23,0) node[above=3pt, rotate=45, xshift=-1.1cm,  yshift=-.5cm]  {Oct. 13, '17} node[below=3pt, rotate=45, xshift=1.22cm, yshift=0.5cm] {DoubleLocker};\n\t\t\t\\draw  (24,0) node[above=3pt, rotate=45, xshift=-1.1cm,  yshift=-.5cm]  {Oct. 24, '17} node[below=3pt, rotate=45, xshift=1.05cm, yshift=0.5cm] {Bad Rabbit};\n\t\t\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\t\\caption[]{Occurrence of Bitcoin ransomware}\n\t\t\\label{figure2}\n\t\\end{figure*}\n\t\n\t\n\t\n\t\\subsection{\\ankit{\\textit{Module2}:} Data collection and database generation}\n\t\\label{data_col}\n\tAs explained in Section~\\ref{related}, Bitcoin blockchain data is publicly available. At the time of writing (December~2017), block height of the blockchain was over 500,000 blocks, which means that downloading\/querying the entire blockchain is very expensive in terms of bandwidth, storage, and computations.\n\tTo address these issues, we built a lightweight system that uses \\textit{Blockchain Data API}\\footnote{blockchain.info\/api\/blockchain\\_api} to crawl and parse transactions associated only with the address(es) of interest. \n\t\\par\n\tFor each transaction associated with an address of interest~(\\textit{Address}), our system collects the hash of the transaction~(\\textit{HASH}), remitted Bitcoin (\\textit{BTC\\_to\\_Addr}), input address-es (\\textit{Trx\\_In\\_Addrs}), output addresses (\\textit{Trx\\_Out\\_Addrs}), GMT-based date (\\textit{GMT\\_Date}), and GMT-based~time (\\textit{GMT\\_Time}). Listing~\\ref{schema} shows the SQL statement used to create our database.\n\t\\begin{center}\n\t\t\\begin{minipage}{.85\\linewidth}\n\t\t\t\\vspace{1em}\n\t\t\t\\begin{lstlisting}[\n\t\t\tlanguage=SQL,\n\t\t\tshowspaces=false,\n\t\t\tbasicstyle=\\ttfamily,\n\t\t\tnumbers=none,\n\t\t\tframe=single,\n\t\t\tcommentstyle=\\color{gray},\n\t\t\tautogobble=true,\n\t\t\tcaptionpos=b,\n\t\t\tcaption=SQL statement for creating our database,\n\t\t\tlabel=schema\n\t\t\t]\n\t\t\tCREATE TABLE tx (\n\t\t\tHASH CHAR(64) NOT NULL PRIMARY KEY,\n\t\t\tBTC_to_Addr INT NOT NULL,\n\t\t\tTrx_In_Addrs TEXT,\n\t\t\tTrx_Out_Addrs TEXT,\n\t\t\tGMT_Date DATE,\n\t\t\tGMT_Time Time,\n\t\t\tAddress CHAR(35) NOT NULL,\n\t\t\tAddress_as_Input INT NOT NULL\n\t\t\t);\n\t\t\t\\end{lstlisting}\t\t\n\t\t\\end{minipage}\n\t\\end{center}\n\t\\par\n\tThe field \\textit{HASH} serves as the \\textit{Primary Key}, which implicitly discards any duplicate transactions reported for multiple participating\/constituting addresses. \\textit{Address\\_as\\_Input} denotes if the \\textit{Address} was used as an input in the transaction. Our system also uses \\textit{BitcoinAverage API}\\footnote{apiv2.bitcoinaverage.com} to collect day-to-day highest, average, and lowest price of Bitcoin.\n\t\n\t\\subsection{\\ankit{\\textit{Module3}:} Considerations for classifying a payment as ransom}\n\t\\label{consideration}\n\t\\begin{table*}[!b]\n\t\t\\centering\n\t\t\\color{black}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\multirow{2}{*}{\\textbf{Ransomware}} & \\multicolumn{3}{c|}{\\textbf{Overall}}                 & \\multicolumn{3}{c|}{\\textbf{Ransom\t}}                 \\\\ \\cline{2-7} \n\t\t\t& \\textbf{Payments} & \\textbf{BTC} & \\textbf{USD Value} & \\textbf{Payments} & \\textbf{BTC} & \\textbf{USD value} \\\\ \\hline\n\t\t\tCryptoLocker                         & 51,766            & 133,045.9961 & 42,292,191.17      & 804               & 1403.7548    & 449,274.97         \\\\ \\hline\n\t\t\tCryptoDefense                        & 128               & 138.3223     & 70,113.41          & 108               & 126.6960     & 63,859.49          \\\\ \\hline\n\t\t\tCryptoWall                           & 51,278            & 87,897.8510  & 45,370,589.00      & 3,730             & 5,351.2329   & 2,220,909.12       \\\\ \\hline\n\t\t\tDMA Locker                           & 298               & 1,433.3463   & 580,763.95         & 117               & 339.4591     & 178,162.77         \\\\ \\hline\n\t\t\tNotPetya                             & 70                & 4.1787       & 10,284.42          & 33                & 4.0576       & 9,835.86           \\\\ \\hline\n\t\t\tKeRanger                             & 13                & 10.0044      & 4,175.35           & 10                & 9.9990       & 4,173.12           \\\\ \\hline\n\t\t\tWannaCry                             & 341               & 53.2906      & 99,549.05          & 238               & 47.1743      & 86,076.76          \\\\ \\hline\n\t\t\\end{tabular}\n\t\t\\caption{\\ankit{Summary of overall payments and ransom payments to the ransomware for which the observed payments align with their period of activity and ransom demands}}\n\t\t\\label{table_compare}\n\t\\end{table*}\n\t\n\t\n\n\tA Bitcoin transaction involves two varying factors: (i)~Bitcoin price, and (ii)~transaction fee. \n\n\tThe price of Bitcoin changes frequently. Therefore, considering only the daily average, highest, or lowest price of Bitcoin is not suitable, especially when the variation in the price is high. Furthermore, the transaction fee is paid on the top of the transaction amount. A victim may assume that the ransom amount to be paid includes (or excludes) the transaction fee, which leads to discrepancies in the payment-amount transferred to an address. Moreover, the transaction fee depends on the size of the transaction, i.e., a transaction that involves a larger number of addresses would incur more fee than a transaction with fewer addresses involved. Hence, to classify a payment as ransom, our framework considers both the day-to-day lowest and highest price of Bitcoin as well as the variation that might occur due to the transaction fee. \n\t\\par\n\tIn general, the cybercriminals specify the ransom either in Bitcoin (e.g., 1~BTC) or USD equivalent BTC (e.g., Bitcoin equivalent to USD~300). Our framework classifies a payment $\\rho$ to an address $\\alpha$ in a transaction $\\tau$ as ransom if it satisfies at least one condition in Eq.~(1a) or Eq.~(1b).\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\n\t\\begin{subequations} \n\t\t\\label{equation1}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\\begin{align}[left={demand~in=\\empheqlbrace}]\n\t\tBTC= {}& \\begin{cases}\n\t\tr_b = d_b, &  \\\\\n\t\tr_b = d_b - f, &\n\t\t\\end{cases} \\\\\n\t\tUSD= {}& \\begin{cases} \n\t\tv_l \\le d_u \\le v_h, &  \\\\\n\t\tv_l \\le d_u - f \\le v_h, &\n\t\t\\end{cases}\n\t\t\\end{align}\n\t\\end{subequations}\n\twhere: \n\t\\begin{itemize}\n\t\t\\item $f$ denotes the transaction fee, computed as the difference between the total amount being spent and the total amount being received in $\\tau$.\n\t\t\\item $d_b$ denotes the ransom asked in BTC.\n\t\t\\item $d_u$ denotes the ransom asked in USD.\n\t\t\\item $r_b$ denotes the BTC received by $\\alpha$ in $\\rho$.\n\t\n\t\t\\item $v_l$ denotes the value of $r_b$ computed using the lowest BTC~price of the payment day.\n\t\t\\item $v_h$ denotes the value of $r_b$ computed using the highest BTC~price of the payment day.\n\t\\end{itemize}\n\t\n\t\\par\n\tIt is also important to mention that to evaluate the total ransom (in USD) received by a ransomware cluster, it would be unfair to use the Bitcoin price on the day of our evaluation as it would misrepresent the amount due to the variations in the price. Hence, unlike previous works, we used the average Bitcoin price on the day of each ransom payment.\n\n\n\n\t\n\t\\vspace{-0.5em}\n\t\\section{Economic Impact of Ransomware}\n\t\\label{ransomware}\n\t\n\tWe found twenty ransomware that fulfilled our selection criteria, i.e., those ransomware: (i) that used Bitcoin as at least one mode of ransom payment, and (ii) for which at least one Bitcoin address is publicly known. In this section, we discuss these twenty ransomware and their renamed\/rebranded versions. Here, our main focus is to provide an insight into the economic impact of these ransomware from the Bitcoin payment perspective. \\figurename{~\\ref{figure2}} depicts the reported debut period of these ransomware as well as the occurrence of their renamed\/rebranded versions.\n\t\n\t\\par\n\tWe performed the numerical assessment of the ransomware on December~7,~2017. Hence, all the data reported in this paper include the transactions until December~7,~2017. We begin with those ransomware for which the observed payments align with their period of activity and ransom demands.\n\t\\ankit{Table~\\ref{table_compare} presents a summary of overall payments received by the addresses of such ransomware. It also lists the payments classified as ransom by our framework. Furthermore, for each payment class, it includes equivalent BTC\/USD value (using day-to-day average Bitcoin price). It is clear that CryptoLocker received the maximum number of payments, i.e., 51,766~payments that worth 133,045.9961~BTC, which is approximately USD~42,292,191.17. However, our framework classified 3,730~payments received by CryptoWall as ransom payments, which is the maximum number of ransom payments extorted by any ransomware. These payments worth 5,351.2329~BTC or USD~2,220,909.12. On another side, KeRanger received the~minimum number of overall payments as well as the ransom payments. Now, we discuss each ransomware in details.}\n\t\n\t\n\t\\subsection{CryptoLocker}\n\t\\ankit{\\textit{Introduction:}} Appeared in September~2013, CryptoLocker targets computers running Windows operating system. It uses ``Microsoft Enhanced RSA and AES Cryptographic Provider (MS\\_ENH\\_RSA\\_AES\\_PROV)'' to create encryption keys and to encrypt users' files with the strong RSA~(CALG\\_RSA\\_KEYX) and AES~(CALG\\_AES\\_256) algorithms. Before beginning the encryption process, it establishes a connection with its C\\&C to obtain an RSA~public key. It encrypts each file with a unique AES~key; after use, it encrypts each AES encryption key with the RSA~public key~\\cite{secureworks_cryptolocker}.\t\t\t \n\t\\par\n\t\\ankit{\\textit{Infection:}} CryptoLocker infection spread through two modes. In its initial release beginning from September~5,~2013, the cybercriminals especially targeted business professionals through spam emails. The messages of the emails were typical ``customer complaints\" against recipients' firm. Attached to these emails was a ZIP archive that contained a single malicious Windows executable (exe) file. The names of both the ZIP file and malicious executable were identical (except for extensions) with 13 to 17 random alphabetical characters. Later versions of CryptoLocker, starting from October~7,~2013, were distributed by the peer-to-peer (P2P) Gameover ZeuS~\\cite{secureworks_zeus}. In this case, Gameover Zeus used Cutwail spam botnet to send a huge number of spam emails miming popular online retailers and banking institutions. These emails often contained spoofed order confirmations, invoices, or urgent message for unpaid balances to entice victims to follow CryptoLocker exploit kits.\n\t\\par\n\n\t\\ankit{\\textit{Ransom demand:}} The ransom note asks the victim to pay the ransom within 72~hours through any one of the various payment methods. It also threatens that not paying the ransom would lead to (allegedly) destruction of decryption keys. In the initial versions, the payment option included cashU\\footnote{www.cashu.com}, Ukash\\footnote{www.ukash.com}, paysafecard\\footnote{www.paysafecard.com}, Bitcoin, or MoneyPak\\footnote{www.attheregister.com\/moneypak\/}. However, later the ransoms were collected only via Bitcoin or MoneyPak. All these payments methods are anonymous (or at least pseudo-anonymous), which makes it difficult to track the payer and the payee. The amount of demanded ransom and their corresponding timelines (both the dates are included) are as~follow:\n\t\n\t\\begin{itemize}\n\t\t\\item 2 BTC between September~5,~2013 and November~11,~2013 allowing a three-day ransom period.\n\t\t\\item 10 BTC between November~1,~2013 and November~11,~2013. The payment was the fee for using ``CryptoLocker Decryption Service'' that allowed victims, who failed to pay ransoms within the given time frame, to recover their files.\n\t\t\\item 1 BTC between November~8,~2013 and November~13,~2013 to allowing a three-day ransom period.\n\t\t\\item 0.5 BTC between November~10,~2013 and November~27,~2013 to allowing a three-day ransom period.\n\t\t\\item 2 BTC between November~11,~2013 and January~31,~2014. In this case, the payment was the reduced fee for using  ``CryptoLocker Decryption Service''.\n\t\t\\item 0.3 BTC between November~24,~2013 and December~31,~2013.\n\t\t\\item 0.6 BTC between December~20,~2013 and January~31,~2014.\n\t\\end{itemize}\n\t\n\t\\par\n\t\\ankit{\\textit{Associated Bitcoin addresses and transactions:}} To evaluate the economic impact of CryptoLocker, we initially began with four Bitcoin addresses listed in Table~\\ref{cryptolocker_address}. Using these addresses, \\ankit{\\textit{Module1}~(Section~\\ref{identify})} generated 956 addresses belonging to CryptoLocker cluster ($C_{CL}$). \\ankit{We obtained the detailed transaction history of these addresses using \\textit{Module2}~(Section~\\ref{data_col}).} Our analysis of transactions to $C_{CL}$ reveals that $C_{CL}$ received, in total, over 51,000 payments, which accounts for over 133,000~BTC (more than USD~42,000,000). Table~\\ref{cryptolocker_inwards} presents a summary of the total payments credited to $C_{CL}$.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{\\begin{tabular}[c]{@{}c@{}}Payments\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}BTC\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD valuez\\\\(daily highest\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily average\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily lowest\\\\BTC price)\\end{tabular}} \\\\ \\hline\n\t\t\t\t51,766                                                                         & 133,045.9961                                                                 & 42,722,858.15                                                                                          & 42,292,191.17                                                                                          & 41,734,959.83                                                                                         \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Total payments credited to $C_{CL}$ including all ransom and non-ransom payments}\n\t\n\t\t\\label{cryptolocker_inwards}\n\t\\end{table}\n\t\n\t\\par\n\t\\ankit{\\textit{Economy of ransom payments in Bitcoin:}}\n\tTo evaluate the gross economic impact of only the ransom payments, we filtered the transactions using: (i) the ransom amounts and their timeline, (ii) our classification criteria \\ankit{mentioned in \\textit{Module3}~(Section~\\ref{consideration})}. \\figurename{~\\ref{cryptolocker_ransom_payment_trend}} shows the total number of ransoms paid by the victims by date. $C_{CL}$ received 33~payment on October~10,~2013, which is the maximum number of ransoms paid in a single day. However, as shown in \\figurename{~\\ref{cryptolocker_BTC_USD1}}, $C_{CL}$ received slightly more than 70 BTC on November~5,~2013, which is the maximum number of Bitcoin received in a single day. On another side, $C_{CL}$ received slightly above USD~23,000 on November~8,~2013, which is the maximum USD collected in a single day, see \\figurename{~\\ref{cryptolocker_BTC_USD2}}.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptolocker_ransom_payment_trend.pdf}\n\t\n\t\t\\caption[]{Number of ransoms paid to $C_{CL}$}\n\t\t\\label{cryptolocker_ransom_payment_trend}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptolocker_BTC_USD1.pdf}\n\t\n\t\t\\caption[]{Number of Bitcoin received (in ransoms) by $C_{CL}$}\n\t\t\\label{cryptolocker_BTC_USD1}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptolocker_BTC_USD2.pdf}\n\t\n\t\t\\caption[]{USD value of ransoms paid to $C_{CL}$}\n\t\t\\label{cryptolocker_BTC_USD2}\n\t\\end{figure}\n\t\n\t\n\t\\par\n\tBy further analyzing the addresses of $C_{CL}$, we discovered that approximately 83.16\\% Bitcoin addresses received maximum two payments. Moreover, 13.33\\% Bitcoin addresses received no more than one Bitcoin perhaps because victims were charged less due to a substantial increase in the Bitcoin value in late November~2013. Moreover, an address\\footnote{16i7w5G2aoq8zqLDR3VJnawZ8VmYFZjVsd} collected 112.94~BTC while a different address\\footnote{1HFLn7JP7FZrufvNKkQPEfAWGjKUdFZEmy} collected 83 ransom payments. These values correspond to the maximum number of Bitcoin and the maximum number of ransom collected by any address in $C_{CL}$. Figures~\\ref{cryptolocker_ransom_Trx_In_CDF}~and~\\ref{cryptolocker_ransom_BTC_In_CDF} depict Cumulative Distribution Function (CDF) of the number of ransoms and number of Bitcoin received (in ransoms) per address respectively.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptolocker_ransom_Trx_In_CDF.pdf}\n\t\t\\caption[]{CDF of ransoms received per address in $C_{CL}$}\n\t\t\\label{cryptolocker_ransom_Trx_In_CDF}\n\t\\end{figure}\t\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptolocker_ransom_BTC_In_CDF.pdf}\n\t\t\\caption[]{CDF of Bitcoin received (in ransoms) per address in $C_{CL}$}\n\t\t\\label{cryptolocker_ransom_BTC_In_CDF}\n\t\\end{figure}\t\t\n\t\\par\n\tIn total, we have identified 804 ransom payments to $C_{CL}$, which contribute to a total of 1,403.75 extorted BTC. Using day-to-day average Bitcoin price, we estimate that these ransoms convert to USD~449,274.97. Table~\\ref{cryptolocker_ransom} summarizes the ransoms paid to CryptoLocker.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{Ransom} & \\textbf{Time period}      & \\textbf{Payments} & \\textbf{BTC}     & \\textbf{USD value} \\\\ \\hline\n\t\t\t\t2 BTC           & Sep. 05, '13 - Nov. 11, '13          & 443                     & 884.9691           & 153,650.51            \\\\ \\hline\n\t\t\t\t10 BTC (late)   & Nov. 01, '13 - Nov. 11, '13          & 17                      & 170.0000           & 47,549.90             \\\\ \\hline\n\t\t\t\t1 BTC           & Nov. 08, '13 - Nov. 13, '13          & 38                      & 38.0000            & 14,302.26             \\\\ \\hline\n\t\t\t\t0.5 BTC         & Nov. 10, '13 - Nov. 27, '13         & 118                     & 59.0000            & 37,108.27             \\\\ \\hline\n\t\t\t\t2 BTC (late)    & Nov. 11, '13 - Jan. 31, '14         & 106                     & 212.0000           & 166,476.42            \\\\ \\hline\n\t\t\t\t0.3 BTC         & Nov. 24, '13 - Dec. 31, '13         & 31                      & 9.1856             & 8,584.88              \\\\ \\hline\n\t\t\t\t0.6 BTC         & Dec. 20, '13 - Jan. 30, '14         & 51                      & 30.6000            & 21,602.72             \\\\ \\hline\n\t\t\t\t\\textbf{Total}  & \\textbf{Sep. 05, '13 - Jan. 30, '14} & \\textbf{804}            & \\textbf{1403.7548} & \\textbf{449,274.97}   \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Summary of ransoms paid to CryptoLocker}\n\t\t\\label{cryptolocker_ransom}\n\t\\end{table}\n\t\\par \n\tAlthough we cannot be sure that the unaccounted transactions are not ransom payments, our results align with the findings presented in the works~\\cite{spagnuolo2014bitiodine,liao2016behind,secureworks_cryptolocker,cryptolocker_intelligence_report} except for the final valuation in USD since the authors of these studies used the Bitcoin price on the day of their evaluation. More importantly, it implies that we can trust our methodology for evaluating other ransomware where a baseline for comparison is not~available.\n\t\n\t\n\n\t\n\t\\subsection{CryptoDefense}\n\t\\ankit{\\textit{Introduction:}} With a sophisticated hybrid design, CryptoDefense first appeared in the last week of February~2014. It incorporates many powerful techniques that were used by previous ransomware. For example, use of Bitcoin and the Tor network for anonymity, RSA-2048 based public-key cryptography for strong encryption, and the typical pressure tactics such~as~a short deadline for payment with threats of increasing the~ransom after the deadline. It targets Windows systems. CryptoDefense encrypts files using the AES-256 algorithm. It generates the encryption key on the victim's computer using Windows CryptoAPI library. After the file encryption process completes, it encrypts the AES key using an RSA-2048 public~key\n\t\n\t\\par\n\t\\ankit{\\textit{Infection:}} Primarily, CryptoDefense ransomware infiltrated via spam emails that contained malicious payload disguised as a compressed PDF document. Upon successful infiltration, it attempts to contact its C\\&C; and it sends the information about the infected system in the initial communication. Upon receiving the acknowledgment from the C\\&C, it starts the encryption process.\n\t\n\t\\par\n\t\\ankit{\\textit{Ransom demand:}} CryptoDefense asks USD\/EUR~500 in Bitcoin within four~days to decrypt the files. The cost of decryption after four~days increases to USD\/EUR~1,000. The attackers also provide a unique .onion page for each victim. Here, the victims could see a screenshot of their compromised system and decrypt one file as a proof of decryption\n\t\n\t\\par\t\n\t\\ankit{\\textit{Associated Bitcoin addresses and transactions:}} We began with two publicly known Bitcoin payment addresses of CryptoDefense. These addresses are listed in Table~\\ref{cryptodefense_address}. In our analysis, the CryptoDefense cluster ($C_{CD}$) had only two addresses as \\ankit{\\textit{Module1}}~generates no new address from these addresses. Our analysis of transactions \\ankit{(obtained using \\textit{Module2})} to $C_{CD}$ indicates that $C_{CD}$ collected 128 payments. The total value of these payments is somewhat above 138~BTC (more than USD~70,000). Table~\\ref{cryptodefense_inwards} presents a summary of the total payments credited to $C_{CD}$.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{\\begin{tabular}[c]{@{}c@{}}Payments\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}BTC\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily highest\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily average\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily lowest\\\\BTC price)\\end{tabular}} \\\\ \\hline\n\t\t\t\t128                                                                           & 138.3223                                                                  & 72,342.26                                                                                              & 70,113.41                                                                                              & 67,715.88                                                                                             \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Total payments credited to $C_{CD}$ including all ransom and non-ransom payments}\n\t\t\\label{cryptodefense_inwards}\n\t\\end{table}\n\t\n\t\\par\n\t\\ankit{\\textit{Economy of ransom payments in Bitcoin:}}\n\tDue to the limited number of transactions, we manually verified each payment to $C_{CD}$. As shown in Table~\\ref{cryptodefense_ransom_BTC_address}, each Bitcoin address collected at minimum 35 ransom payments and a minimum of about 36.83 BTC.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|l|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{1}{|c|}{\\textbf{Address}} & \\textbf{Payments} & \\textbf{BTC} \\\\ \\hline\n\t\t\t\t19DyWHtgLgDKgEeoKjfpCJJ9WU8SQ3gr27     & 35                               & 36.8339                      \\\\ \\hline\n\t\t\t\t1EmLLj8peW292zR2VvumYPPa9wLcK4CPK1     & 73                               & 89.8622                      \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Number of ransoms and Bitcoin received (in ransoms) per address in $C_{CD}$}\n\t\t\\label{cryptodefense_ransom_BTC_address}\n\t\\end{table}\n\t\n\t\\par\n\t\\figurename{~\\ref{cryptodefense_ransom_payment_trend}} shows the total number of ransoms paid, and Figures~\\ref{cryptodefense_BTC_USD1}~and~\\ref{cryptodefense_BTC_USD2} depict the corresponding number of Bitcoin received and their value in USD. Figures~\\ref{cryptodefense_ransom_payment_trend},~\\ref{cryptodefense_BTC_USD1},~and~\\ref{cryptodefense_BTC_USD2} also depict that on March~28,~2014, $C_{CD}$ collected around 13~BTC in 11~ransom payments, which amounts to approximately USD~6,500. It is the day when it received the maximum number of ransom payments\/Bitcoin\/USD in a~single~day.\n\t\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 1mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptodefense_ransom_payment_trend.pdf}\n\t\n\t\t\\caption[]{Number of ransoms paid to $C_{CD}$}\n\t\t\\label{cryptodefense_ransom_payment_trend}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 1mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptodefense_BTC_USD1.pdf}\n\t\n\t\t\\caption[]{Number of Bitcoin received (in ransoms) by $C_{CD}$}\n\t\t\\label{cryptodefense_BTC_USD1}\n\t\\end{figure}\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 1mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptodefense_BTC_USD2.pdf}\n\t\n\t\t\\caption[]{USD value of ransoms paid to $C_{CD}$}\n\t\t\\label{cryptodefense_BTC_USD2}\n\t\\end{figure}\n\t\n\t\n\t\\par\n\tIn total, we have distinguished 108 ransom payments to $C_{CD}$, which corresponds to 126.70 extorted BTC. Using day-to-day average Bitcoin price, we compute that the value of these ransom payments is equivalent to USD~63,859.49. Table~\\ref{cryptodefense_ransom} summarizes the ransoms payments made to CryptoDefense.\n\t\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{Ransom} & \\textbf{Time period}               & \\textbf{Payments} & \\textbf{BTC}      & \\textbf{USD value} \\\\ \\hline\n\t\t\t\t\\$\/\\euro500           & \\multirow{3}{*}{Feb. 28, '14 - Apr. 11, '14} & 94                      & 96.1758           & 49,271.63              \\\\ \\cline{1-1} \\cline{3-5} \n\t\t\t\t\\$\/\\euro1,000          &                                    & 14                      & 30.5202           & 14,587.86              \\\\ \\cline{1-1} \\cline{3-5} \n\t\t\t\t\\textbf{Total}  &                                    & \\textbf{108}            & \\textbf{126.6960} & \\textbf{63,859.49}     \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Summary of ransoms paid to CryptoDefense}\n\t\t\\label{cryptodefense_ransom}\n\t\\end{table}\n\t\n\tUnexpectedly, CryptoDefense has a built-in flaw.  It generates the asymmetric key pair on the victim's system. However, due to the poor implementation of the Microsoft's cryptographic infrastructure, it leaves a local copy of the keys. Anti-ransomware took advantage of this flaw to decrypt victim's computer. Such initiatives saved at least USD~175,000 worth ransoms~\\cite{cryptodefense_2}\n\t\n\t\n\n\t\n\t\\subsection{CryptoWall}\n\t\\ankit{\\textit{Introduction:}} CryptoWall is recognized for its use of strong encryption algorithm, unique .CHM file infection mechanism, and strong C\\&C activity over the anonymous Tor network. According to Dell SecureWorks Counter Threat Unit (CTU) research team~\\cite{secureworks_cryptowall}, CryptoWall infection was spreading from the first half of November~2013. However, the attackers activated it in the first quarter of 2014. The earlier versions of CryptoWall closely impersonated both the appearance and the behavior of the CryptoLocker. CryptoWall affects Windows operating systems by encrypting files using the RSA-2048 (and the AES-256 encryption algorithm from version 3.0) encryption~algorithm.\n\t\\par\n\t\\ankit{\\textit{Infection:}} Since its genesis, CryptoWall had spread through several infection vectors, which included drive-by downloads, browser exploit kits (e.g., Angler), and email attachments. Starting from late March~2014, it spread through download links sent via the Cutwail spam botnet and malicious email attachments. Interestingly, from June~2014, the malicious emails included links to popular cloud services such as Dropbox, MediaFire, and Cubby. The links pointed to a ZIP archive that contained the CryptoWall executable. Later these emails used a standard ``missed fax'' decoy and also mimicked message from government agencies or financial institutions that included links to malicious payload hosted over cloud services.\n\t\\par\n\t\\ankit{\\textit{Evolution:}} Each version of CryptoWall lasted for a few months until a stealthier and enhanced version emerged.\n\t\\begin{itemize}\n\t\t\\item CryptoWall 1.0: Initial variants of CryptoWall lacked a unique name. It surfaced with its official name in the first quarter of 2014.\n\t\t\\item CryptoWall 2.0: It appeared in November~2014. This version was almost identical to the previous version. However, unlike its predecessor, it creates a unique Bitcoin payment address for each victim and uses its own Web-2-Tor gateways.\n\t\t\\item CryptoWall 3.0: The third version of CryptoWall emerged in January~2015. This version uses a local symmetric (AES-256) key for file encryption. The symmetric key is then encrypted using a unique public (RSA-2048) key generated by the C\\&C server. Such process of encryption is much faster as compared to the previous versions.\n\t\t\\item CryptoWall 4.0: Another updated version with improved communications and better code design to exploit more vulnerabilities appeared in November~2015.\n\t\\end{itemize}\n\t\n\t\\par\n\t\\ankit{\\textit{Ransom demand:}} The attackers originally accepted ransom payments through Litecoin~\\cite{secureworks_cryptowall}. However, the only witnessed Litecoin address\\footnote{LTv4m4y7NKHCXdw31dSEpTJmP6kXTinWDy} never collected any payment. Additionally, the victims could also pay the ransom via Bitcoin. The amount of ransom fluctuated frequently. Also, the time frame to pay the ransom varied up to seven~days. According to our observation, the demanded ransom and their corresponding timelines (both the dates are included) are as follow:\n\t\\begin{itemize}\n\t\t\\item \\$200 worth BTC between March~2,~2014 and November~4,~2015.\n\t\t\\item \\$500 worth BTC between March~2,~2014 and December~22,~2015.\n\t\t\\item Late payment of \\$600 worth BTC between March~5,~2014 and November~5,~2015. This payment was three times the original ransom amount.\n\t\t\\item Late payment of \\$1,000 worth BTC between March~5,~2014 and December~2,~2015. This payment was two times the original ransom amount.\n\t\t\\item \\$700 worth BTC between March~10,~2014 and December~11,~2015.\n\t\t\\item Late payment of \\$1,400 worth BTC between March~11,~2014 and December~21,~2015. This payment was two times the original ransom amount.\n\t\\end{itemize}\n\t\n\t\\par\n\t\\ankit{\\textit{Associated Bitcoin addresses and transactions:}} We began with forty-two publicly known Bitcoin addresses of CryptoWall. These addresses are listed in Table~\\ref{cryptowall_address}. Using these addresses, \\ankit{\\textit{Module1}}~generated 2,944 addresses belonging to CryptoWall cluster ($C_{CW}$). Our analysis of transactions \\ankit{(obtained using \\textit{Module2})} to $C_{CW}$ shows that $C_{CW}$, in total, received over 51,000 payments. The total worth of these payments is nearly 88,000~BTC (more than USD~45,000,000). Table~\\ref{cryptowall_inwards} presents a summary of the total payments credited to $C_{CW}$.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{\\begin{tabular}[c]{@{}c@{}}Payments\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}BTC\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily highest\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily average\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily lowest\\\\BTC price)\\end{tabular}} \\\\ \\hline\n\t\t\t\t51,278                                                                        & 87,897.8510                                                               & 46,526,673.59                                                                                          & 45,370,589.00                                                                                          & 44,020,263.63                                                                                         \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Total payments credited to $C_{CW}$ including all ransom and non-ransom payments}\n\t\t\\label{cryptowall_inwards}\n\t\\end{table}\n\t\n\t\\par\n\t\\ankit{\\textit{Economy of ransom payments in Bitcoin:}}\n\tUsing the timeline of ransom demands, we carefully analyzed all the transactions \\ankit{with \\textit{Module3}~}to distinguish ransom payments and evaluated the net worth generated by such payments. %\n\n\tAs shown in Figures~\\ref{cryptowall_ransom_payment_trend},~\\ref{cryptowall_BTC_USD1},~and~\\ref{cryptowall_BTC_USD2}, on March~27,~2014, $C_{CW}$ received slightly above 185~BTC in 158 payments. The total value of these payments is over USD~100,000. It is the day when it received the maximum number of ransom payments\/Bitcoin\/USD in a~single day.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptowall_ransom_payment_trend.pdf}\n\t\n\t\t\\caption[]{Number of ransoms paid to $C_{CW}$}\n\t\t\\label{cryptowall_ransom_payment_trend}\n\t\\end{figure}\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptowall_BTC_USD1.pdf}\n\t\n\t\t\\caption[]{Number of Bitcoin received (in ransoms) by $C_{CW}$}\n\t\t\\label{cryptowall_BTC_USD1}\n\t\\end{figure}\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptowall_BTC_USD2.pdf}\n\t\n\t\t\\caption[]{USD value of ransoms paid to $C_{CW}$}\n\t\t\\label{cryptowall_BTC_USD2}\n\t\\end{figure}\t\n\t\\par\n\tBy investigating the addresses of $C_{CW}$, we observed that approximately 43.77\\% Bitcoin addresses received no more than one payment and 40.10\\% Bitcoin addresses collected maximum two Bitcoin. On another side, an address\\footnote{17AGazRCLStNguMDCxDoj7ZQHvaZBWTJZj} collected 193.94~BTC in 209 ransom payments. These values correspond to the maximum number of Bitcoin and the maximum number of ransom collected by any address in $C_{CW}$. Figures~\\ref{cryptowall_ransom_Trx_In_CDF}~and~\\ref{cryptowall_ransom_BTC_In_CDF} show the CDF of the number of ransoms and the number of Bitcoin received (in ransoms) per address respectively. \n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptowall_ransom_Trx_In_CDF.pdf}\n\t\t\\caption[]{CDF of ransoms received per address in $C_{CW}$}\n\t\t\\label{cryptowall_ransom_Trx_In_CDF}\n\t\\end{figure}\t\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/cryptowall_ransom_BTC_In_CDF.pdf}\n\t\t\\caption[]{CDF of Bitcoin received (in ransoms) per address in $C_{CW}$}\n\t\t\\label{cryptowall_ransom_BTC_In_CDF}\n\t\\end{figure}\n\t\n\t\\par\t\t\n\tWe have identified 3,730 ransom payments to $C_{CW}$, which amount to 5,351.23 extorted BTC. Using day-to-day average Bitcoin price, we calculate that these ransom payments are equivalent to USD~2,220,909.12. Table~\\ref{cryptowall_ransom} summarizes the ransoms paid to CryptoWall.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\n\t\t\t\n\t\t\t\t\\textbf{Ransom} & \\textbf{Time period}                 & \\textbf{Payments} & \\textbf{BTC}        & \\textbf{USD value} \\\\ \\hline\n\t\t\t\t\\$200           & Mar. 02, '14 - Nov. 04, '15          & 614                     & 232.3343            & 121,849.84            \\\\ \\hline\n\t\t\t\t\n\t\t\t\t\\$500           & Mar. 02, '14 - Dec. 22, '15          & 1,631                   & 2220.9167           & 821,741.46            \\\\ \\hline\n\t\t\t\t\n\t\t\t\t\\$600 (late)    & Mar. 05, '14 - Nov. 05, '15          & 382                     & 444.5144            & 226,558.14            \\\\ \\hline\n\t\t\t\t\n\t\t\t\t\\$1,000 (late)   & Mar. 05, '14 - Dec. 02, '15          & 423                     & 836.5054            & 422,576.75            \\\\ \\hline\n\t\t\t\t\n\t\t\t\t\\$700           & Mar. 10, '14 - Dec. 11, '15          & 466                     & 966.7365            & 327,518.98            \\\\ \\hline\n\t\t\t\t\n\t\t\t\t\\$1,400 (late)   & Mar. 11, '14 - Dec. 21, '15          & 214                     & 650.2256            & 300,663.95            \\\\ \\hline\n\t\t\t\t\\textbf{Total}  & \\textbf{Mar. 02, '14 - Dec. 22, '15} & \\textbf{3,730}          & \\textbf{5,351.2329} & \\textbf{2,220,909.12} \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Summary of ransoms paid to CryptoWall}\n\t\t\\label{cryptowall_ransom}\n\t\\end{table}\n\t\n\t\\par\n\tMoreover, according to the report by CTU researchers~\\cite{secureworks_cryptowall}, CryptoWall attackers allowed the victims to decrypt their system by paying a further increased amount even after the expired deadline.\n\tAlthough, we have not directly observed any sample of CryptoWall demanding such compensations. Nevertheless, the timing and the volume of such payments suggest that these payments pertain to ransoms. Table~\\ref{cryptowall_weird_trx} summarizes such payments.\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{Amount} & \\textbf{Time period}                 & \\textbf{Payments} & \\textbf{BTC}       & \\textbf{USD value} \\\\ \\hline\n\t\t\t\t\\$1,500         & Mar. 12, '14 - Dec. 12, '15          & 222                      & 678.7995           & 333,587.51            \\\\ \\hline\n\t\t\t\t\\$1,750         & Mar. 12, '14 - Nov. 04, '15          & 192                      & 647.5063           & 336,578.87            \\\\ \\hline\n\t\t\t\t\\$2,000         & Mar. 06, '14 - Jul. 06, '14          & 170                      & 650.7245           & 339,794.84            \\\\ \\hline\n\t\t\t\t\\$10,000        & Mar. 11, '14 - Jul. 11, '14          & 131                      & 2623.3381          & 1,316,778.41          \\\\ \\hline\n\t\t\t\t\\textbf{Total}  & \\textbf{Mar. 06, '14 - Dec. 12, '15} & \\textbf{715}             & \\textbf{4600.3684} & \\textbf{2,326,739.63} \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Summary of high value (possibly ransom) payments to CryptoWall}\n\t\t\\label{cryptowall_weird_trx}\n\t\\end{table}\n\t\n\t\\par\n\tIf we add these payments to the original ransom payments, then the revenue of CryptoWall reaches nearly 10,000 BTC, i.e., approximately USD~4,500,000.\n\t\n\t\n\n\t\n\t\\subsection{DMA Locker}\n\t\\ankit{\\textit{Introduction:}} DMA Locker is one of the most actively developed and updated ransomware so far. From encryption algorithm to network communication, cybercrooks perpetually updated each component of DMA Locker. Initially, it used only the symmetric key cryptography for file encryption. However, later versions employ a stronger encryption approach by combining the AES-256 and the RSA-2048 encryption algorithms. It affects Windows operating system.\n\t\n\t\\par\n\t\\ankit{\\textit{Infection:}} The distribution mechanism of DMA Locker also evolved with the course of time. The malicious payload was hosted on compromised websites, and their links were distributed via email spamming. It also infiltrated by hacking Remote Desktops. The latest edition of the ransomware also spread via Neutrino exploit kit~\\cite{dmalocker_2}.\n\t\n\t\\par\n\t\\ankit{\\textit{Evolution:}} The development timeline of DMA Locker is discussed below:\n\t\n\t\\begin{itemize}\n\t\t\\item DMA Locker 1.0: The first version of DMA Locker was noticed in the last week of December~2015 with support for two languages: Polish and English. It performs file encryption by using the AES-256 algorithm in ECB mode. It uses a single AES key to encrypt target files, which is stored in the binary and deleted after use.\n\t\t\n\t\t\\item DMA Locker 2.0: On February~3,~2016, DMA Locker was updated to use separate keys for each file. After encrypting a file, it encrypts the used AES key by RSA public key and stores the encrypted AES key in the encrypted file. The public key for RSA encryption comes hardcoded in the binary.\n\t\t\n\t\t\\item DMA Locker 3.0: \n\t\tDue to weak implementation of the random number generator, the AES key generated by the previous version can be guessed. In view to fix the flaw, the third edition was released on February~22,~2016. However, the entire campaign used the same RSA key-pair. Meaning that single private key can be reused for decrypting other infected systems.\n\t\t\n\t\t\\item DMA Locker 4.0: The latest version of DMA Locker was released on May~19,~2016. This version generates a unique RSA key-pair on the server for each victim. Unlike previous versions, DMA Locker 4.0 can not work offline because it is designed to download the asymmetric public key from the server~\\cite{dmalocker_1}.\n\t\\end{itemize}\n\t\n\t\\par\n\t\\ankit{\\textit{Ransom demand:}} The cybercrooks behind DMA Locker accepted ransom payments through Bitcoin. DMA Locker 4.0 gives payment instructions on a website. The website was a regularly (not Tor-based) hosted site.  Surprisingly, the payment site used the same IP address as the C\\&C. Similar to other components, the ransom amount was also updated with time. Moreover, the first two versions stipulate a strict deadline of four~days to pay the ransom. Other versions allow an extension of three~days at the cost of an increased ransom. The demanded ransom and their corresponding timelines (both the dates are included) are as follow:\n\t\n\t\\begin{itemize}\n\t\t\\item 1 BTC between December~28,~2015 and July~22,~2016.\n\t\t\n\t\t\\item 1.3 BTC between January~19,~2016 and May~30,~2016. \n\t\t\n\t\t\\item 2 BTC between January~28,~2016 and July~22,~2016 to allowing a three-day ransom period.\n\t\t\n\t\t\\item 4 BTC between February~22,~2016 and June~5,~2016 to allowing a three-day ransom period.\n\t\t\n\t\t\\item 8 BTC as late fee between February~22,~2016 and August~5,~2016. \n\t\t\n\t\t\\item 1.5 BTC as late fee between May~19,~2016 and July~11,~2016.\n\t\t\n\t\t\\item 3 BTC between May~24,~2016 and August~25,~2016.\n\t\\end{itemize}\n\t\n\t\n\t\\par\n\t\\ankit{\\textit{Associated Bitcoin addresses and transactions:}} To understand the economic impact of DMA Locker, we began with eight Bitcoin addresses listed in Table~\\ref{dmalocker_address}. Using these addresses, \\ankit{\\textit{Module1}}~generated 28 addresses belonging to DMA Locker cluster ($C_{DL}$). Our scrutiny of transactions \\ankit{(obtained using \\textit{Module2})} to $C_{DL}$ shows that $C_{DL}$ received altogether 298 payments, i.e., more than 1,400~BTC (over USD~580,000). Table~\\ref{dmalocker_inwards} presents a summary of the total payments credited to $C_{DL}$.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{\\begin{tabular}[c]{@{}c@{}}Payments\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}BTC\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily highest\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily average\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily lowest\\\\BTC price)\\end{tabular}} \\\\ \\hline\n\t\t\t\t298                                                                           & 1,433.3463                                                                 & 593,498.26                                                                                              & 580,763.95                                                                                              & 567,543.86                                                                                             \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Total payments credited to $C_{DL}$ including all ransom and non-ransom payments}\n\t\t\\label{dmalocker_inwards}\n\t\\end{table}\n\t\n\t\n\t\\par\n\t\\ankit{\\textit{Economy of ransom payments in Bitcoin:}} \\ankit{We used \\textit{Module3}, guided by the timeline of ransom demands, to separate ransom payments.} \\figurename{~\\ref{dmalocker_ransom_payment_trend}} depicts the total number of ransoms paid by date. $C_{DL}$ received 5~payment on April~27,~2016, which is the maximum number of ransoms paid in a single day. On another side, as shown in \\figurename{~\\ref{dmalocker_BTC_USD1}}, $C_{DL}$ collected 12~BTC on May~19,~2016, which corresponds to the maximum number of Bitcoin received in a single day. Furthermore, $C_{DL}$ received over USD~6,300 on August~5,~2016, which stands for the maximum USD received in a single day, see \\figurename{~\\ref{dmalocker_BTC_USD2}}.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/dmalocker_ransom_payment_trend.pdf}\n\t\n\t\t\\caption[]{Number of ransoms paid to $C_{DL}$}\n\t\t\\label{dmalocker_ransom_payment_trend}\n\t\\end{figure}\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/dmalocker_BTC_USD1.pdf}\n\t\n\t\t\\caption[]{Number of Bitcoin received (in ransoms) by $C_{DL}$}\n\t\t\\label{dmalocker_BTC_USD1}\n\t\\end{figure}\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/dmalocker_BTC_USD2.pdf}\n\t\n\t\t\\caption[]{USD value of ransoms paid to $C_{DL}$}\n\t\t\\label{dmalocker_BTC_USD2}\n\t\\end{figure}\n\t\n\t\n\t\\par\n\tWe further found that around 30\\% addresses in $C_{DL}$ collected no more than one payment and nearly 20\\% Bitcoin addresses received less than one Bitcoin. Furthermore, an address\\footnote{1LPgKoErPUeM92SDY5axJzYCdQbeiRHD6i} collected 112.87~BTC in 38~ransom payments. These values correspond to the maximum number of Bitcoin and the maximum number of ransom collected by any address in $C_{DL}$. Table~\\ref{dmalocker_ransom} summarizes the ransoms paid to DMA Locker.\n\t\n\t\n\n\t\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\t\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{Ransom} & \\textbf{Time period}       & \\textbf{Payments} & \\textbf{BTC}      & \\textbf{USD value} \\\\ \\hline\n\t\t\t\t1 BTC           & Dec. 28, '15 - Jul. 22, '16          & 16                      & 14.7526           & 7,052.37               \\\\ \\hline\n\t\t\t\t1.3 BTC         & Jan. 19, '16 - May 30, '16           & 4                       & 5.2470            & 2,424.01               \\\\ \\hline\n\t\t\t\t2 BTC           & Jan. 28, '16 - Jul. 22, '16          & 16                      & 32.0809           & 16,638.46              \\\\ \\hline\n\t\t\t\t4 BTC           & Feb. 22, '16 - Jun. 05, '16          & 33                      & 131.9950          & 60,443.98              \\\\ \\hline\n\t\t\t\t8 BTC (late)    & Feb. 22, '16 - Aug. 05, '16          & 4                       & 32.4892           & 16,960.59              \\\\ \\hline\n\t\t\t\t1.5 BTC (late)  & May 19, '16 - Jul. 11, '16           & 6                       & 8.9147            & 5,136.87               \\\\ \\hline\n\t\t\t\t3 BTC           & May 24, '16 - Aug. 25, '16           & 38                      & 113.9797          & 69,506.49              \\\\ \\hline\n\t\t\t\t\\textbf{Total}  & \\textbf{Dec. 28, '15 - Aug. 25, '16} & \\textbf{117}            & \\textbf{339.4591} & \\textbf{178,162.77}    \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Summary of ransoms paid to DMA Locker}\n\t\t\\label{dmalocker_ransom}\n\t\\end{table}\n\t\n\t\\par\n\tWe have identified 117 ransom payments to $C_{DL}$, which contribute to a total of 339.46 extorted BTC. Using day-to-day average Bitcoin price, we estimate that these ransom payments value USD~178,162.77.\n\t\n\t\n\n\t\n\t\n\t\\subsection{Petya}\n\t\\ankit{\\textit{Introduction:}} Initially seen in March~2016, this family of malware denies access to the full system by targeting the low-level structures on the disk. Petya spread via emails, and was delivered as Windows executable with an icon of a PDF document. Upon running, it opens a User Account Control (UAC) window. Accepting UAC allows Petya to run. In this case, it overwrites the Master Boot Record~(MBR) with a custom bootloader that loads a malicious kernel. Then, this kernel encrypts the Master File Table~(MFT) using Salsa20 stream cipher with a 32-byte long key, which leaves file system unreadable. \\figurename{~\\ref{flowchart_petya}} depicts the full process of Petya.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, scale=0.4]{.\/new\/graphs\/Flowchart_Petya.pdf}\n\t\t\\caption[]{Workflow of Petya}\n\t\t\\label{flowchart_petya}\n\t\\end{figure}\n\t\n\t\\par\n\t\\ankit{\\textit{Mischa:}} In May~2016, the malware was modified to integrate another malicious payload known as Mischa. Mischa was designed as a backup strategy to Petya. Altogether, they target different (both high-level and low-level) layers of a system. In this version, denying the UAC prompt directs Mischa to encrypt local files on the victim computer; otherwise, Petya proceeds. \\figurename{~\\ref{flowchart_mischa}} depicts the full process of Mischa. Both Petya and Mischa can work offline without communicating with their C\\&C. The payload from the dropper\\footnote{The file that launches a malware.} uses CryptGenRandom function from the Windows CryptoAPI library to generate a random encryption key. Mischa uses a CBC-style file encryption utilizing a randomly generated key along with the previously generated master key. Interestingly, Mischa can encrypt documents as well as executables~\\cite{petya_avast_mischa}. The cybercriminals also offered RaaS through their own affiliate~program. \n\t\\begin{figure}[!htbp]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, scale=0.4]{.\/new\/graphs\/Flowchart_Mischa.pdf}\n\t\t\\caption[]{Workflow of Mischa}\n\t\t\\label{flowchart_mischa}\n\t\\end{figure}\n\t\\par\n\t\\ankit{\\textit{GoldenEye:}} The malware was again rebranded as GoldenEye in early December~2016. In contrast with the previous versions, GoldenEye executes both payloads, where possible. Similar to its predecessors, it was also distributed via email. But, the payload was attached to an MS Excel document. The document prompts the user to enable Macro content. Enabling Macro content executes a malicious Visual Basic Script, which runs the Mischa payload to encrypt documents on the system. After Mischa finishes, it attempts to gain system privileges via DLL injection (Windows 7 - 8.1), or a UAC prompt is shown (Windows 10). If DLL infection succeeds or the UAC prompt is accepted, Petya payload encrypts the MFT. \\figurename{~\\ref{flowchart_goldeneye}} depicts the full process of GoldenEye.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 0mm 0mm 0mm 1mm, clip, scale=0.41]{.\/new\/graphs\/Flowchart_GoldenEye.pdf}\n\t\t\\caption[]{Workflow of GoldenEye}\n\t\t\\label{flowchart_goldeneye}\n\t\\end{figure}\n\t\n\t\\par\n\t\\ankit{\\textit{NotPetya:}} The latest variant of Petya surfaced on June~27,~2017. Kaspersky unofficially named\\footnote{www.kaspersky.com\/blog\/new-ransomware-epidemics\/17314\/} it NotPetya\/ExPetr due to significant differences in its operations compared to the earlier versions. Initially, NotPetya was distributed as an update to MeDoc\\footnote{www.medoc.ua\/uk} accounting software prevalent in Ukraine. After infiltration, it self-propagates via two methods. One of the methods is the EternalBlue exploit, which is an exploit of Windows'~Server Message Block (SMB) protocol. The same exploit is also used by WannaCry ransomware, which was released only a month before NotPetya. It can also spread across network shares by Windows Management Instrumentation Command-line (WMIC), for which it uses credentials acquired from the local machine. In contrast with other ransomware, it focuses on the local network to spread rather than the Internet. NotPetya works as a destructive data wiper tool due to its inability to restore the encrypted sectors of the physical~disk~\\cite{petya_symantec}.\n\t\n\t\\par\t\t\t\n\t\\ankit{\\textit{Associated Bitcoin addresses and transactions:}} We discuss the financial transactions associated with only NotPetya because the payments received by the address clusters generated for Mischa and GoldenEye (using addresses listed in Tables~\\ref{mischa_address}~and~\\ref{goldeneye_address} respectively) were significantly less~(no more than USD~3) than the demanded ransoms (roughly USD~1,000). For NotPetya, cybercriminals used a single Bitcoin payment address to collect a fixed ransom of USD~300. The address is listed in Table~\\ref{NotPetya_address}. NotPetya cluster ($C_{NP}$) generated by \\ankit{\\textit{Module1}}~also had only one Bitcoin addresses. \\ankit{We acquired the detailed transaction history of this address using \\textit{Module2}.} $C_{NP}$ received exactly 70 payments. These payments worth slightly above 4~BTC (over USD~10,000). Table~\\ref{NotPetya_inwards} summarizes the payments credited in $C_{NP}$.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{\\begin{tabular}[c]{@{}c@{}}Payments\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}BTC\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily highest\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily average\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily lowest\\\\BTC price)\\end{tabular}} \\\\ \\hline\n\t\t\t\t70                                                                            & 4.1787                                                                      & 10,717.74                                                                                              & 10,284.42                                                                                              & 9,958.33                                                                                              \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Total payments credited to $C_{NP}$ including all ransom and non-ransom payments}\n\t\t\\label{NotPetya_inwards}\n\t\\end{table}\n\t\n\t\\par\n\t\\ankit{\\textit{Economy of ransom payments in Bitcoin:}}\n\n\t\\ankit{We segregated ransom payments using \\textit{Module3}.} As shown in Figures~\\ref{NotPetya_ransom_payment_trend},~\\ref{NotPetya_BTC_USD1},~and~\\ref{NotPetya_BTC_USD2}, on the day of its outbreak, i.e., on June~27,~2017 $C_{NP}$ received somewhat above 3~BTC in total 27 payments that amount approximately USD~8,000\n\tIt collected the maximum number of ransom payments\/Bitcoin\/USD on this day.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/NotPetya_ransom_payment_trend.pdf}\n\t\n\t\t\\caption[]{Number of ransoms paid to $C_{NP}$}\n\t\t\\label{NotPetya_ransom_payment_trend}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/NotPetya_BTC_USD1.pdf}\n\t\n\t\t\\caption[]{Number of Bitcoin received (in ransoms) by $C_{NP}$}\n\t\t\\label{NotPetya_BTC_USD1}\n\t\\end{figure}\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/NotPetya_BTC_USD2.pdf}\n\t\n\t\t\\caption[]{USD value of ransoms paid to $C_{NP}$}\n\t\t\\label{NotPetya_BTC_USD2}\n\t\\end{figure}\n\t\n\t\\par\n\tIn total, we have identified 33 ransom payments to $C_{NP}$, which add up to roughly 4.06 extorted BTC. Using day-to-day average Bitcoin price, we calculate that these ransom payments worth equivalent to USD~9,835.86. Table~\\ref{NotPetya_ransom} summarizes the ransoms paid to NotPetya.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{Ransom} & \\textbf{Time period} & \\textbf{Payments} & \\textbf{BTC} & \\textbf{USD value} \\\\ \\hline\n\t\t\t\t\\$300           & Jun. 27, '17 - Aug. 03, '17    & 33                      & 4.0576         & 9,835.86               \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Summary of ransoms paid to NotPetya}\n\t\t\\label{NotPetya_ransom}\n\t\\end{table}\n\t\n\t\\par\n\tGiven the irreversible destructive nature and the targeted-software of NotPetya, many researchers suggested that the primary aim of NotPetya was not money. Other researchers speculated that it was probably a second level attack to wipe traces of an early intrusion~\\cite{petya_secureworks,petya_logrhythm}.\n\t\n\t\n\n\t\n\t\n\t\\subsection{KeRanger}\n\t\\ankit{\\textit{Introduction:}} KeRanger emerged as the first fully functional ransomware that targets macOS operating system. It was discovered on March~4,~2016, by Palo~Alto~Networks. By nature it is a trojan horse, it uploads infected system's information (e.g., model name, UUID) to its C\\&C over the Tor network to obtain an RSA public key. Along with the key it also receives victim-specific information that it is writes to a file named ``README\\_FOR\\_DECRYPT.txt.'' KeRanger encrypts each file $F$ as follows:\n\t\\begin{enumerate}\n\t\t\\item Generate a random number ($R$).\n\t\t\\item Generate an Initialization Vector ($I$) using F's content.\n\t\t\\item Encrypt $R$ with the RSA key (obtained from C\\&C), and store it at the beginning of $F.encrypted$ file.\n\t\t\\item Store $I$ inside the $F.encrypted$ file.\n\t\t\\item Mix $R$ and $I$ to generate an AES key.\n\t\t\\item Encrypt data of the original file with the AES key and write the encrypted data to $F.encrypted$ file~\\cite{keranger_palo}.\n\t\\end{enumerate}\n\t\n\t\\par\t\t\t\n\t\\ankit{\\textit{Infection:}} KeRanger was disseminated via two infected installers for the open source BitTorrent client project Transmission version~2.90, which were available for download on the official website. Moreover, these installers were signed with a valid Mac app development certificate; hence, they bypassed OS X's Gatekeeper security feature.\n\t\n\t\\par\n\t\\ankit{\\textit{Ransom demand:}} To decrypt the encrypted files, the cybercrooks asked the victims to pay exactly one Bitcoin (around USD~400) through a website hosted on the Tor network. \n\t\n\t\\par\n\t\\ankit{\\textit{Associated Bitcoin addresses and transactions:}} We began with six identified Bitcoin address of KeRanger. These addresses are listed in Table~\\ref{keranger_address}. \\ankit{\\textit{Module1}}~identified ten new addresses from these six addresses. Therefore, KeRanger cluster ($C_{KR}$) had a total of 16 addresses in our analysis. The transactions \\ankit{(obtained using \\textit{Module2})} to $C_{KR}$ show that $C_{KR}$, in total, received only 13 payments. These transactions worth around 10~BTC (nearly USD~4,200). Table~\\ref{keranger_inwards} presents a summary of the total payments credited to $C_{KR}$\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{\\begin{tabular}[c]{@{}c@{}}Payments\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}BTC\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily highest\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily average\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily lowest\\\\BTC price)\\end{tabular}} \\\\ \\hline\n\t\t\t\t13                                                                            & 10.0044                                                                     & 4,204.54                                                                                               & 4,175.35                                                                                               & 4,147.01                                                                                              \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Total payments credited to $C_{KR}$ including all ransom and non-ransom payments}\n\t\t\\label{keranger_inwards}\n\t\\end{table}\n\t\n\t\n\t\\ankit{\\textit{Economy of ransom payments in Bitcoin:}} \n\t\\ankit{We isolated ransom payments using \\textit{Module3}.} \\figurename{~\\ref{keranger_ransom_payment_trend}} shows the total number of ransoms paid to $C_{KR}$. $C_{KR}$ received the last ransom payment on March~17,~2016. Figures~\\ref{keranger_BTC_USD1}~and~\\ref{keranger_BTC_USD2} depict the total number of Bitcoin received (in ransom) and their corresponding value in USD. Moreover, we found that none of the address received more than one Bitcoin (more than one ransom, in other words).\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/keranger_ransom_payment_trend.pdf}\n\t\n\t\t\\caption[]{Number of ransoms paid to $C_{KR}$}\n\t\t\\label{keranger_ransom_payment_trend}\n\t\\end{figure}\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/keranger_BTC_USD1.pdf}\n\t\n\t\t\\caption[]{Number of Bitcoin received (in ransoms) by $C_{KR}$}\n\t\t\\label{keranger_BTC_USD1}\n\t\\end{figure}\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/keranger_BTC_USD2.pdf}\n\t\n\t\t\\caption[]{USD value of ransoms paid to $C_{KR}$}\n\t\t\\label{keranger_BTC_USD2}\n\t\\end{figure}\n\t\n\t\n\tAccording to our analysis, $C_{KR}$ received only 10 ransom payments, which contribute to roughly 9.99 extorted BTC. Using day-to-day average Bitcoin price, we estimate that these ransoms convert to USD~4,173.12. Table~\\ref{keranger_ransom} summarizes the ransoms paid to KeRanger.\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{Ransom} & \\textbf{Time period} & \\textbf{Payments} & \\textbf{BTC} & \\textbf{USD value} \\\\ \\hline\n\t\t\t\t1 BTC           & Mar. 04, '16 - Mar. 17, '16    & 10                      & 9.9990        & 4,173.12               \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Summary of ransoms paid to KeRanger}\n\t\t\\label{keranger_ransom}\n\t\\end{table}\n\t\\par\n\tOne of the possible reasons for such low number of ransom payments could be that by March~5,~2016, Transmission project removed the infected installers from the website, and Apple revoked the abused certificate that allowed Gatekeeper to block the infected installers.\n\t\n\t\n\n\t\n\t\\subsection{WannaCry}\n\t\\ankit{\\textit{Introduction:}} WannaCry (also known as WCry, WanaCrypt0r, Wana~Decrypt0r 2.0) is blended threat with characteristics of both a worm and a ransomware. It was first seen on May~12,~2017. It affects Windows system by encrypting files using a combination of the RSA and the AES algorithms. Interestingly, it encrypts each file with a separate 128-bit AES encryption key in CBC mode. Furthermore, it encrypts each AES key individually using the RSA-2048 encryption algorithm~\\cite{secureworks_wannacry}.\n\n\t\\par\n\t\\ankit{\\textit{Infection:}} WannaCry scans explicitly for the presence of the DoublePulsar backdoor on a target. If the DoublePulsar backdoor is not present, then it tries to compromise the system using the EternalBlue exploit~\\cite{wannacry_talosintelligence}. The EternalBlue exploit was exposed merely a few months before the WannaCry attack by a hacker group known as The Shadow Brokers.\n\t\\par\n\t\\ankit{\\textit{Kill switch and kill mutex:}}\n\tA kill switch is usually employed to terminate a program's execution. In case of WannaCry, the kill switch was a domain name\\footnote{www.iuqerfsodp9ifjaposdfjhgosurijfaewrwergwea.com}. Upon initialization, WannaCry tries to connect to the domain over HTTP. If the connection is successful, then it stops and exits. Possibly, it was designed to evade a sandbox testing. The kill switch domain was hardcoded in the source code and was discovered by Marcus Hutchins\\footnote{en.wikipedia.org\/wiki\/MalwareTech}. On another side, before beginning the encryption process, WannaCry attempts to create a mutex named ``MsWinZonesCacheCounterMutexA'' and exits if the mutex is already present.\n\t\\par\n\t\\ankit{\\textit{Ransom demand:}} The ransom note asks the victims to pay USD~300 ransom in Bitcoin within three~days. The ransom note also states that the ransom amount would become double (i.e., USD~600) after three~days, and if the ransom is not paid within seven~days from the day of infection, all the encrypted files would~be~deleted.\n\t\\par\n\t\\ankit{\\textit{Associated Bitcoin addresses and transactions:}} Cybercriminals intended to create a unique Bitcoin payment address for each victim. But a race condition bug prevents the correct execution of the code. In this situation, it presents one of three hard-coded Bitcoin addresses to collect the ransom~\\cite{wannacry_symantec}. These addresses are listed in Table~\\ref{wannacry_address}. Moreover, using these addresses, \\ankit{\\textit{Module1}}~generated no new address. Hence, WannaCry cluster ($C_{WC}$) generated by our framework had only three Bitcoin addresses during our analysis. \\ankit{We procured the detailed transaction history of these three addresses using \\textit{Module2}.} $C_{WC}$ received 341 payments. These payments worth over 50~BTC (approximately USD~100,000). Table~\\ref{wannacry_inwards} summarizes the payments credited in $C_{WC}$.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{\\begin{tabular}[c]{@{}c@{}}Payments\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}BTC\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily highest\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily average\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily lowest\\\\BTC price)\\end{tabular}} \\\\ \\hline\n\t\t\t\t341                                                                           & 53.2906                                                                     & 102,141.19                                                                                             & 99,549.05                                                                                              & 96,497.20                                                                                             \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Total payments credited to $C_{WC}$ including all ransom and non-ransom payments}\n\t\t\\label{wannacry_inwards}\n\t\\end{table}\n\t\n\t\\par\n\t\\ankit{\\textit{Economy of ransom payments in Bitcoin:}}\n\tDue to comparatively a smaller number of transactions, we manually verified each payment to $C_{WC}$. As shown in Table~\\ref{wannacry_ransom_BTC_address}, each Bitcoin address collected at minimum 69 ransom payments and a minimum of nearly 13.52 BTC.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|l|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{1}{|c|}{\\textbf{Address}} & \\textbf{Payments} & \\textbf{BTC} \\\\ \\hline\n\t\t\t\t12t9YDPgwueZ9NyMgw519p7AA8isjr6SMw     & 77                      & 15.1129        \\\\ \\hline\n\t\t\t\t13AM4VW2dhxYgXeQepoHkHSQuy6NgaEb94     & 92                      & 18.5431        \\\\ \\hline\n\t\t\t\t115p7UMMngoj1pMvkpHijcRdfJNXj6LrLn     & 69                      & 13.5183        \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Number of ransoms and Bitcoin received (in ransoms) per address in $C_{WC}$}\n\t\t\\label{wannacry_ransom_BTC_address}\n\t\\end{table}\n\t\n\t\\par\n\n\tFigures~\\ref{wannacry_ransom_payment_trend},~\\ref{wannacry_BTC_USD1},~and~\\ref{wannacry_BTC_USD2} indicate that on May~15,~2017, $C_{WC}$ received 70 payments that amount to nearly 14 BTC, which is approximately USD~24,000. It is the day when it received the maximum number of ransom payments\/Bitcoin\/USD in a~single~day.\n\t\n\t\n\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/wannacry_ransom_payment_trend.pdf}\n\t\n\t\t\\caption[]{Number of ransoms paid to $C_{WC}$}\n\t\t\\label{wannacry_ransom_payment_trend}\n\t\\end{figure}\n\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/wannacry_BTC_USD1.pdf}\n\t\n\t\t\\caption[]{Number of Bitcoin received (in ransoms) by $C_{WC}$}\n\t\t\\label{wannacry_BTC_USD1}\n\t\\end{figure}\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=\\linewidth]{.\/new\/graphs\/wannacry_BTC_USD2.pdf}\n\t\n\t\t\\caption[]{USD value of ransoms paid to $C_{WC}$}\n\t\t\\label{wannacry_BTC_USD2}\n\t\\end{figure}\n\t\n\t\\par\n\tIn total, we have identified 238 ransom payments to $C_{WC}$, which add up to 47.17 extorted BTC. Using day-to-day average Bitcoin price, we calculate that these ransom payments worth equivalent to USD~86,076.76. Table~\\ref{wannacry_ransom} summarizes the ransoms payments made to WannaCry.\n\t\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{Ransom} & \\textbf{Time period}       & \\textbf{Payments} & \\textbf{BTC}   & \\textbf{USD value} \\\\ \\hline\n\t\t\t\t\\$300           & \\multirow{3}{*}{May 12, '17 - Oct. 02, '17}          & 192                     & 32.3430          & 58,416.62              \\\\ \\cline{1-1} \\cline{3-5}\n\t\t\t\t\\$600           &           & 46                      & 14.8313          & 27,660.14              \\\\ \\cline{1-1} \\cline{3-5}\n\t\t\t\t\\textbf{Total}  &  & \\textbf{238}            & \\textbf{47.1743} & \\textbf{86,076.76}     \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Summary of ransoms paid to WannaCry}\n\t\t\\label{wannacry_ransom}\n\t\\end{table}\n\t\n\t\\par\n\tThe overall impact (including financial losses) due to WannaCry infection could have been worse. But, thanks to the early detection of the kill switch, which prevented the infected computers from spreading WannaCry further.\n\t\n\n\n\t\n\t\\textit{Other Bitcoin ransomware:}\n\tWe now briefly discuss all those Bitcoin ransomware, for which the observed payments either are entirely different (merely a few dollars) against the demanded ransom or the date of transactions do not match with the activity of the ransomware. Nevertheless, we make no solid claims without further evidence. However, we make their addresses and corresponding dataset available in Appendix~\\ref{ransomware_addresses} and our repository  (mentioned before) respectively for future efforts in this direction of research.\n\n\t\\subsection{CTB-Locker}\n\t\\ankit{\\textit{Introduction:}} CTB-Locker (Curve-Tor-Bitcoin-Locker) first appeared in mid-July~2014 as Critroni. Initially, it targeted individual Windows users. But, soon its focus shifted to vulnerable WordPress websites. The latter version encrypts the homepage of a website and replaces the original homepage with a new page containing the ransom note. It is also infamous for using Elliptic Curve Cryptography (ECC), the Tor network to hide the C\\&C, Bitcoin for ransom payment, and its availability in multiple (seven) languages. As a pioneer, it uses ECC for encryption, which enables it to obtain the equivalent level of security as the RSA with much smaller key sizes. E.g., a 256-bit ECC offers equivalent security as 3072-bit RSA. It obtains a secret key by applying a SHA256 hash function to a 52~bytes long random sequence, and Curve25519 generates the corresponding public key. In fact, it uses a combination of symmetric and asymmetric encryption algorithm where the AES algorithm encrypts the user's files~\\cite{ctblocker_encryption}.\n\t\\par\n\t\\ankit{\\textit{Infection:}} Researchers\\footnote{malware.dontneedcoffee.com\/2014\/07\/ctb-locker.html} and allegedly participants\\footnote{www.reddit.com\/r\/Malware\/comments\/2uffwc\/ctb\\_locker\\_ama\/} disclosed that the attackers used an affiliate program to spread the infection in return of profits. Generally, in an affiliate program, the participants attempt to spread the infection via several possible vectors. CTB-Locker was primarily distributed through exploit kits (e.g., Rig and Nuclear) and malicious email spam (e.g., overdue phone invoices, missed fax, bank statements) campaigns that exploit Dalexis or Elenoocka downloader component. \n\t\\par\n\t\\ankit{\\textit{Ransom demand:}} In the beginning, the ransom was set at 0.5~BTC (about USD~300) for US, Europe, and Canada while 0.25~BTC for other countries. Later, the ransom was changed to 0.4~BTC~(about USD~150) that doubles after four~days. In addition, the victims could decrypt five files for free and could also do test transaction of 0.0001~BTC on one of the two dedicated Bitcoin addresses.\n\t\\par\n\t\\ankit{\\textit{Associated Bitcoin addresses and transactions:}} The addresses belonging to CTB-Locker that we found are listed in Table~\\ref{ctblocker_address}. The last two addresses listed in the table are those Bitcoin addresses where the victims could do a test transaction. However, the cluster generated \\ankit{(using \\textit{Module1})} from these addresses did not receive any payment except for two test transactions. \\ankit{One of the possible reasons could be the nature of the target audience. Most of the web hosting plans facilitate periodic backups. If a web page becomes inaccessible\/encrypted, the webmaster can restore a relatively fresh version without paying the ransom.}\n\t\n\t\n\n\t\n\t\\subsection{CryptoTorLocker2015}\n\tOn February~5,~2015, Symantec discovered CryptoTorLocker2015 as a very low-level threat for Window operating system. It utilizes only public key cryptography for file encryption. In particular, it uses the RSA-2048 encryption algorithm, for which it downloads the RSA public key from an attacker-controlled C\\&C.\n\tBeing a trojan, it spread via classical infection mechanisms such as drive-by download. It asks the victims to pay 0.5~BTC (equivalent to USD\/EUR~100) within five~days of infection to decrypt the files.\n\t\\par\n\n\t\\ankit{\\textit{Module1}~}of our framework generated six new addresses belonging to CryptoTorLocker2015 from the single address listed in Table~\\ref{cryptotorlocker2015_address}. These seven addresses received almost USD~1100 worth 5~BTC in 136 payments. But, only one transaction\\footnote{blockchain.info\/tx\/36f2bbc56e7ce7bea59265ce1b7f9ac42040dc5491f01a 4b338f619293515820} that happened on February~11,~2015 satisfies the criteria of the ransom demand specified by the attackers.\n\t\n\t\n\n\t\n\t\n\t\\subsection{TeslaCrypt}\n\t\\ankit{\\textit{Introduction:}} TeslaCrypt or AlphaCrypt began to spread in mid February~2015. It searches explicitly for game-related user content (e.g., custom maps and progress\/save files) along with other personal documents and pictures. TeslaCrypt ignores audio files, video files, and removable (e.g., USB) storage. It does not scan connected networks as well. It uses the AES algorithm to encrypt files, but with an aim to mislead the victims, it appends ``ecc'' extension to the encrypted files while the ransom note message claims that it has used the RSA-2048 encryption algorithm. Its C\\&C hid in the Tor anonymity network and required an SSL encrypted connection from a victim machine for communication. Preventing TeslaCrypt from interacting with the C\\&C does not prevent the encryption process because it generates the encryption keys locally. \n\t\\par\n\t\\ankit{\\textit{Infection and ransom demand:}} TeslaCrypt was distributed exclusively through Angler and Nuclear browser exploit kits~\\cite{secureworks_teslacrypt}. The attackers accepted the ransoms via various payment methods. The ransom amount in Bitcoin was 1.5~BTC within seven~days, 2.5~BTC otherwise. The victims from North American region could also choose to pay USD~1000 with PayPal My Cash cards while the European victims could pay EUR~600 with Ukash or paysafecard.\n\t\\par\n\t\\ankit{\\textit{Associated Bitcoin addresses and transactions:}} The payments collected by the address cluster (generated from the addresses listed in Table~\\ref{teslacrypt_address}) do not match with the ransom amount demanded by the attackers. However, FireEye research team in their study~\\cite{fireeye_teslacrypt} describes that the attackers negotiated with the victims and gave ``discounts'' on the ransom amount. In this case, the attackers accumulated around 254.6~BTC, which converts to about USD~57,272. \\ankit{Later, the attackers publicly released\\footnote{www.bleepingcomputer.com\/news\/security\/teslacrypt-shuts-down-and-releases-master-decryption-key\/} the master decryption key.}\n\t\n\t\n\t\n\t\n\n\t\n\t\n\t\\subsection{Chimera}\n\tIn the November~2015, cybercrooks began to target English- and German-speaking Windows users with Chimera ransomware. The cybercrooks distributed Chimera via targeted e-mails to small businesses and companies. Unlike other ransomware, it does not use a Tor website to handle payment instructions or to hide the C\\&C. Instead, it is the first ransomware that uses Bitmessage\\footnote{bitmessage.org\/wiki\/Main\\_Page} P2P protocol to interact with the C\\&C and to obtain the RSA public\/private keys. In such scenario, it is difficult, if not impossible, to take down all peers in the network that are assisting ransomware's operations. \n\t\\par\n\tIt is also the first ransomware to use doxing as a pressure tactic. It threatens the victims to post personal data including pictures and videos on the internet if the ransom is not paid. However, several ransomware analysts showed that it was a bogus threat~\\cite{malwarebyte_chimera}. Chimera asks the victims to pay between 0.9~BTC to 2.5~BTC as ransom. The addresses belonging to Chimera that we found are listed in Table~\\ref{chimera_address}. The payments received by the cluster generated \\ankit{(using \\textit{Module1})} from these addresses do not fall in the range of the ransom asked by the cybercrooks.\n\n\t\\ankit{Dramatically, rival ransomware developers leaked the RSA private keys of Chimera\\footnote{twitter.com\/JanusSecretary\/status\/757951375561072640}.} \n\t\n\t\n\n\t\n\t\n\t\\subsection{Hi Buddy!}\n\tHi Buddy! was active in the first quarter of 2016. Like many other ransomware, it encrypts only user's files and leaves the system files responsible for running Windows operating system. Upon execution, it attempts to connect with its C\\&C over Tor2Web service and sends information such as the version of operating system, location (country) of the victim machine. The response from the C\\&C includes a string variable whose value (``FIRST'' or ``RECEIVED'') depends on whether it should encrypt or decrypt the files. The values ``FIRST'' and ``RECEIVED'' correspond to encryption and decryption respectively. It is important to note that the encryption\/decryption process does not execute until it receives the response from the C\\&C. It uses the AES-256~algorithm to encrypt user's files. The encryption key is generated by hashing (SHA-256) a string variable named ``password,'' which is obtained from the C\\&C. It spread via general spamming techniques. After encryption, it shows a ransom note asking about 0.8~BTC for decryption. The address cluster generated from the address listed in Table~\\ref{hibuddy_address} did not receive any~ransom.\t\t\t\n\t\n\t\n\n\t\n\t\n\t\\subsection{Jigsaw}\n\tJigsaw was released in late March~2016 to affect systems running Windows operating system. It is considered to be the most dramatic ransomware so far. It was released in several different languages, while each variant was hard-coded to execute only after a specific date. As a representative example, the English version was set to execute after March~23,~2016 while the Portuguese version was written to run after April~6,~2016.\tMoreover, It employs an unprecedented extortion strategy. During the first 24~hours it deletes a few files every hour; after 24~hours, hundreds of files every hour; and after 48~hours, thousands of files every hour. And, if the ransom is not paid within 72~hours, it deletes all the remaining files. If the victim shuts-down or restarts the computer, it destroys 1000 files as ``punishment.'' Furthermore, each variant demands a distinct amount of ransom ranging from USD~23 to USD~5000 to be paid through Bitcoin.\n\t\\par\n\tThe cybercriminals hosted the payload on free cloud storage services such as 1fichier.com and distributed the links to the malicious payload through email spamming. Jigsaw works offline and uses the AES-128 encryption algorithm in CBC mode to encrypt user's files. Using 19 addresses (listed in Table~\\ref{jigsaw_address}), our framework generated 24 addresses belonging to Jigsaw ransomware. Altogether these addresses collected approximately 2.5~BTC (USD~1,200) in 58~payments. However, all these payments occurred from March~2016 to August~2016, i.e., during the period when Jigsaw was active. Hence, we may argue that perhaps all these transactions were ransom payments.\n\t\n\t\n\n\t\n\t\n\t\\subsection{ZCryptor} \n\tA security researcher called Jack first reported\\footnote{malwarefor.me\/zcrypt-ransomware\/} ZCryptor on May~24,~2016. It targets computers running Windows operating system. After obtaining victim-specific encryption key from its C\\&C, it uses the RSA encryption algorithm to encrypt user's files. ZCryptor exhibits worm-like behavior. It is one of the few ransomware that can self-propagate to other connected computers and network devices even without using an exploit kit or spamming. For initial infection, the cybercrooks used conventional distribution techniques such as email spamming, fake software (e.g., Adobe Flash) updater, and macro malware in Microsoft Office suite. It also attempts to distract the victim by showing benign pop-ups while performing the file encryption. Once the encryption process completes, ZCryptor displays its ransom message in which it asks for 1.2~BTC (about USD~500) to be paid within four~days. Nevertheless, it permits additional three~days for the payment at the cost of 5~BTC (about USD~2,100). The address cluster generated from the address listed in Table~\\ref{zcryptor_address} did not receive any ransom. \\ankit{It is noteworthy that on May~26,~2016, i.e., within two~days of its discovery, Microsoft issued an alert\\footnote{blogs.technet.microsoft.com\/mmpc\/2016\/05\/26\/link-lnk-to-ransom\/} to its users about ZCryptor and also updated the definition base of Windows Defender to protect against ZCryptor.}\n\t\n\t\n\n\t\n\t\n\t\\subsection{VenusLocker}\n\n\t\\ankit{\\textit{Introduction, infection, and ransom demand:}} At the beginning of August~2016, VenusLocker, a new eda2-based ransomware began to target Windows based systems.\n\tSimilar to most ransomware, VenusLocker encrypts data files using the AES-256 algorithm. It generates the AES encryption key on the victim's system from a cryptographically-strong random number generator and encrypts it with an embedded RSA-2048 public key before sending to the C\\&C. It also creates and conveys a unique ID to C\\&C to identify the infected system. It spread primarily via drive-by download. It allows only three~days (with no extension) to pay the ransom in Bitcoin. At first, it demanded USD~100 as ransom. But, soon it asked USD~500. However, the ransom amount settled on one BTC with an update in December~2016.\n\t\n\t\\par\n\t\\ankit{\\textit{Associated Bitcoin addresses and transactions:}} Initially, we identified three addresses belonging to VenusLocker. These addresses are listed in Table~\\ref{venuslocker_address}. \\ankit{\\textit{Module1}}~of our framework identified three new addresses from the addresses. Therefore, VenusLocker cluster ($C_{VL}$) had a total of six addresses in our analysis. The transactions \\ankit{(obtained using \\textit{Module2})} to $C_{VL}$ reveal that $C_{VL}$, in total, received 11~payments. The total worth of these transactions is almost 7~BTC (more than USD~6,500). Table~\\ref{venuslocker_inwards} presents a summary of the total payments credited~to~$C_{VL}$.\n\n\n\n\n\n\n\n\n\n\n\n\t\\begin{table}[H]\n\t\t\\centering\n\t\t\\resizebox{\\columnwidth}{!}\n\t\t{\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{\\begin{tabular}[c]{@{}c@{}}Payments\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}BTC\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily highest\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily average\\\\BTC price)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}USD value\\\\(daily lowest\\\\BTC price)\\end{tabular}} \\\\ \\hline\n\t\t\t\t11                                                                            & 6.8155                                                                    & 6,861.73                                                                                                & 6,753.81                                                                                                & 6,637.06                                                                                               \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\caption{Total payments credited to $C_{VL}$ including all ransom and non-ransom payments}\n\t\t\\label{venuslocker_inwards}\n\t\\end{table}\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\\par\n\tThe campaign was launched again as The~Trump~Locker in February~2017 and rebranded in March~2017 as The~LLTP~Locker. The ransom amount in The~Trump~Locker varied from USD~50 to USD~150 (in Bitcoin), while The~LLTP~Locker targeted specifically Spanish users asking USD~200 (in Bitcoin) as ransom. The address clusters generated for The~Trump~Locker and The~LLTP~Locker (using the addresses listed in Tables~\\ref{trumplocker_address}~and~\\ref{lltp_address} respectively) never received any payment.\n\t\n\t\n\n\t\n\t\n\t\\subsection{KillDisk}\n\tKillDisk debuted as a data wiper malware. It affected energy industry, finance sector, sea transport, and news agencies in 2015 and 2016. In early December 2016, the malware was updated to integrate a ransomware component. The KillDisk ransomware targets not only Windows operating system but also Linux workstations and servers, which magnifies its damage potential. It targets every drive (local and network) that the victim can access.\n\tBoth the Windows and Linux variant work differently. \n\t\\par\t\t\t\n\tThe Windows variant, detected by CyberX~\\cite{killdisk_cyberx}, encrypts each file with a separate AES-256 encryption key (generated using CryptGenRandom function from the Windows CryptoAPI library). After use, it encrypts the AES keys using a public RSA-1028 key. It obtains the RSA public key via the Telegram API from the C\\&C.  It uses the whitelisting technique to avoid sandbox analysis. \n\t\\par The Linux variant, detected by ESET~\\cite{killdisk}, performs encryption using Triple-DES applied to 4096-byte blocks where each file is encrypted using a different set of 64-bit encryption keys generated locally. However, the encryption keys are neither stored locally nor sent to the C\\&C, which means that the decryption is virtually impossible. Furthermore, it also makes the machine unbootable as it rewrites the boot sector and uses the GRUB bootloader to show the ransom note.\n\t\\par\n\tBoth the variants show the same ransom note, asking an enormous ransom of 222~BTC to be paid on the same Bitcoin address. The address is listed in Table~\\ref{killdisk_address}. The address cluster generated by our framework also had only one Bitcoin address, which did not receive any ransom.\n\t\n\t\n\n\t\n\t\n\t\\subsection{FindZip}\n\tFindZip ransomware, also known as Filecoder, was discovered and reported by ESET researchers~\\cite{findzip} on February~22,~2017. It is written in Swift programming language to infect systems with macOS operating system. It encrypts all mounted external and network storage. Upon execution, it locally generates a 25-characters long random string, which it uses to create a separate encrypted .zip file for each user file using the ``zip'' shell command. Next, it deletes all the original files by the ``rm'' command and sets the encrypted file's time to February~13,~2010 using the ``touch'' command. \n\tIt was distributed as a ``Patcher'' application from torrent distribution sites. The torrent file downloads a single ZIP archive file that contains fake patching applications for premium software such as Adobe Premiere Pro and Microsoft Office for Mac. However, the applications are not signed with an Apple-recognized key.\n\t\\par\n\tThe ransom message is hardcoded inside the ransomware. Hence, it uses the same Bitcoin address for each victim. It demands 0.25 BTC for decrypting the files and instructs the victims to wait for 24 hours after paying the ransom. But it promises to start the decryption in 10 minutes if the victim pays 0.45 BTC. The address cluster generated from the addresses listed in Table~\\ref{findzip_address} did not collect any payment.\n\t\n\n\t\n\t\n\t\\subsection{ThunderCrypt}\n\tThunderCrypt emerged in the first week of May~2017. It targeted primarily Taiwanese Windows users for ransom extortion. It carefully encrypts the user's data by a hybrid RSA-2048 public key encryption algorithm. It does not encrypt the essential files of the operating system so that the system keeps on working and has an active internet connection. To distribute the ransomware, the cybercriminals injected a malicious script into a Taiwanese forum ``ENVY.'' The script triggers a pop-up, which requests permission to run a fake Adobe Flash Player installer. The bogus installer was designed to drop the ThunderCrypt payload on the victim machine.\n\t\\par\n\tThunderCrypt demands exactly 0.345 Bitcoin (roughly USD~500) from the victims. And like most ransomware, it threatens the victim to erase the key from the server if the ransom is not paid. Additionally, it allows the victim to decrypt one file to prove that the decryption is possible. The addresses belonging to ThunderCrypt that we found are listed in Table~\\ref{thundercrypt_address}. However, the cluster generated from these addresses did not receive any payment. \\ankit{It is also worth mentioning that in communication\\footnote{wccftech.com\/thundercrypt-ransomware-taiwanese-man\/} with a victim, cybercriminals admitted that their campaign failed.\n\t\n\t\n\n\t\n\t\n\t\\subsection{DoubleLocker}\t\t\n\tOn October~13,~2017, ESET researchers~\\cite{doublelocker_addr_1} reported the first-ever ransomware that targets Android operating system. DoubleLocker is rewritten from an Android banking Trojan named ``Android.BankBot.211.origin.'' It abuses Android's accessibility services to elevate privileges on the victim system. Unlike its banking parent, it does not steal victims' banking credentials. It rather changes the device's PIN code and encrypts device's primary storage using the AES-256 encryption algorithm, which leaves the device inaccessible to the user. Similar to its banking parent, the attackers distributed it as a fake Adobe Flash Player app over compromised websites. To release the decryption key, DoubleLocker asks the victims to pay 0.0130~BTC (about USD~50) within 24~hours and waits for three~confirmations of the payment. The address cluster (generated from the addresses listed in Table~\\ref{doublelocker_address}) received only one payment, which is far too less than the asked ransom.\n\t\n\t\n\n\t\n\t\n\t\\subsection{Bad Rabbit}\n\n\tBad Rabbit started to spread from October~24,~2017. Similar to NotPetya, it encrypts files as well as the MFT on the Windows machine, and it also replaces the MBR with a custom bootloader. For file encryption, it uses the AES-128 encryption algorithm in CBC mode, and the RSA-2048 encryption secures the keys. It uses DiskCryptor driver in AES-XTS mode to encrypt disk partitions on the infected system. Nevertheless, it is not a wiper like NotPetya. The attackers distributed it via drive-by attack as a dropper-file named ``install\\_flash\\_player.exe,'' which prompts a standard UAC to elevate administrative privileges. Additionally, it exploits the EternalBlue exploit to infect machines in the local network. The ransom note asks the victims to use a Tor website to make a payment of 0.05~BTC within 42~hours, after which the price of decryption goes up~\\cite{securelist_badrabbit}. The payments collected by the address cluster (generated from the addresses listed in Table~\\ref{badrabbit_address}) are significantly lesser than the asked ransom.\t\t\t\t\n\t\n\n\t\n\t\n\t\\section{Limitations}\n\t\\label{limitation}\n\t\\ankit{\n\t\n\t\tOne of the most important and decisive elements for the quality of the outcomes of our framework is the address identification module, presented in Section~\\ref{identify}. It relies on the Bitcoin addresses collected from the public sources; the quality of data collected from the public sources could be a concern. One of the promising alternatives is to collect binaries of the ransomware and execute them several times in a virtual environment to witness\/obtain Bitcoin addresses. However, the question of integrity and authenticity of the binaries remains the same. Given the nature of the problem, we followed the approach used in the previous studies~\\cite{liao2016behind,spagnuolo2014bitiodine} and took extreme precaution while collecting addresses from the public sources.\n\t}\n\t\\par\n\tThe fundamental principles of the Bitcoin protocol implicitly impart two types of flaws in our address identification module: overestimation and underestimation. Our methodology would overestimate when multiple users pool their transactions into a single transaction; as in the case of mixers. On another side, it would underestimate when there exists no evidence (in the blockchain) of an address owned by a user being used in conjunction with any other address of the same user. However, in a given scenario, it would report more accurate results as compared to the existing approaches due to its attributes of ransom classifications.\n\t\n\t\n\t\\section{Conclusion and Future Work}\n\t\\label{futurework}\n\tPseudo-anonymity and irreversibility of Bitcoin transaction protocol have made Bitcoin a dexterous utility among cybercriminals. Unlike genuine users, who seek to transact securely and efficiently; cybercrooks exploit these characteristics to commit immutable and presumably untraceable monetary fraud. In this paper, we have presented our comprehensive and longitudinal study on twenty recent Bitcoin ransomware along with their renamed\/rebranded versions. We have also introduced our framework to identify, collect, and analyze Bitcoin addresses that belong to the cybercriminals behind the ransomware. Moreover, we elaborated the characteristics and the functionality of the ransomware as well as reported the economic impact of such ransomware from the Bitcoin payment perspective.\n\n\t\n\n\t\t\t\t\n\t\t\n\t\t\n\t\\par\n\tIn the future, we will extend our identification framework to other cryptography-based currencies. We will also investigate the ransoms extorted via other payment options; we~hope~to~present a comprehensive report that will include ransom payments from all payment option endorsed by the ransomware. Finally, we will attempt to trace how the received ransoms were used and by whom.\n\t\n\t\n\n\t\\section*{Acknowledgment}\n\tAnkit Gangwal is pursuing his Ph.D. with a fellowship for international students funded by Fondazione Cassa di Risparmio di Padova e Rovigo~(CARIPARO). This work is partially supported by the EU TagItSmart! Project (agreement H2020-ICT30-2015-688061), the EU-India REACH Project (agreement ICI+\/2014\/342-896), the grant n. 2017-166478 (3696) from Cisco University Research Program Fund and Silicon Valley Community Foundation, and by the grant ``Scalable IoT Management and Key security aspects in 5G systems'' from~Intel.\n\t\n\t\n\n\n\n\t%\n\n\n\n\n\t\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe development of the diffusion concept has always relied on the mutually-beneficial relationship between theory and experiments. Since Perrin's experiments proving Einstein's diffusion theory~\\cite{Perrin1910}, Brownian (usual) diffusion is well-known to display a Gaussian distribution and a linear time dependence of the mean square displacement (MSD). However, as deviations from these usual behaviors started to appear in experimental studies of disordered media and biological systems, the need to understand underlying microscopic mechanisms of these unusual dynamics has given rise to breakthrough theories in statistical physics. \n\nThe anomalous diffusion era started with the concept of waiting-time distribution in random walks, proposed independently by L\\'evy~\\cite{Levy1954}, Smith~\\cite{Smith1955}, and Montroll and Weiss~\\cite{Montroll1965}; but the continuous-time random walk (CTRW) emerged as the foundation of anomalous transport only after the works of Scher and Lax~\\cite{Scher1973a,Scher1973b} on unusual results for charge transport in amorphous semiconductors. These works use CTRW to describe heterogeneities of a medium in an annealed way, where the waiting-time distribution represents the environment randomness. The ``second youth'' of the CTRW~\\cite{Masoliver2017} is usually marked by its relationship with fractional diffusion equations~\\cite{Klafter1987,Hilfer1995,Compte1996,Barkai2002}, a formalism that becomes known as an efficient phenomenological description of anomalous diffusion in complex systems~\\cite{Metzler2000,West2016,Masoliver2017}.\n\nPercolation theory~\\cite{Broadbent1957} represents another significant breakthrough for the description of anomalous diffusion. As emphasized by Havlin and Ben-Avraham~\\cite{Havlin2002}, the percolation model is a simple and purely geometrical approach to describe disordered media. While random walk processes are generalized in the CTRW model, in percolation processes, usual random walks take place in a disordered environment. In contrast to the CTRW framework, percolation theory thus describes a quenched disorder, where the randomness associated with geometrical constraints are constant in time. Under this context, de Gennes~\\cite{deGennes1976} coined the term ``the ant in a labyrinth'' to describe random walks in percolation lattices and established a paradigm of anomalous diffusion caused by geometrical structures~\\cite{Bouchaud1990,Bok2009}. This paradigm becomes well established mainly due to fractal geometry~\\cite{Mandelbrot1989}, and it is essential in the study of porous media~\\cite{Havlin2002}.\n\nDiffusion also becomes an essential noninvasive tool to probe and characterize systems ranging from materials to living organisms~\\cite{Sen2004,Waigh2005,Kirstein2007,Wirtz2009,Novikov2014,Papaioannou2017,Assaf2019,Song2019}. These recent empirical results revealed a myriad of complex patterns that are usually not well described by analytical tools developed for amorphous solids and porous media. As argued by Metzler~\\cite{MetzlerBiophysJ2017}, these novel experimental findings require researchers to come up with novel models. In this context, hybrid or mixed models of anomalous diffusion emerged as a significant modeling possibility~\\cite{Meroz2010,Weigel2011,Jeon2011,Tabei2013,Miyaguchi2015,Golan2017,Furnival2017}. Examples include CTRW on fractals~\\cite{Meroz2010,Weigel2011,Golan2017}, CTRW combined with fractal Brownian motion~\\cite{Jeon2011,Tabei2013}, quenched-trap model and fractal lattices~\\cite{Miyaguchi2015}, and CTRW combined with percolation theory~\\cite{Furnival2017}. In general, these models share the idea of combining annealed (temporal) and quenched (spatial or static) disorder mechanisms. \n\nAnother model of anomalous diffusion of particular importance to the present work is the comb-model; a model emerged from studies of percolation threshold and anomalous diffusion on fractal structures~\\cite{Coniglio1981,Stanley1984,White1984,Havlin1987}. This model describes a diffusive process on a comb-like structure consisting of a ``backbone'' (a single infinite line in the $x$-direction) and ``branches'' (parallel lines in the $y$-direction that intersect the $x$-axis). The comb-model is a simplified description of the fractal geometry of percolation clusters, where the backbone represents the large bond and branches are the remaining bonds or ``dangling ends'' of percolation clusters (Figures~\\ref{fig:1}a and \\ref{fig:1}b). The comb-model retains essential properties of diffusion on fractals, with the advantage of providing exact results on such complicated systems. Moreover, random walks on comb-like structures established the sojourn times of walkers in the teeth as the underlying mechanism of anomalous diffusion in the backbone.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{fig1}\n\\caption{The comb-model as simplified description of diffusion on percolation cluster. (a) Illustration of a percolation cluster where the continuous lines indicate the large bond and dashed lines are the dangling ends. The comb-model shown in (b) and modeled by Eq.~(\\ref{Comb0}) is a simplified description of the geometry in percolation clusters. Panels (c) and (d) compare the usual two-dimensional diffusive process with the diffusion on comb structure.  The two-dimensional diffusion imposes no geometrical restriction on random walkers, while horizontal shifts occur only when $y=0$ in the comb-model; to accesses different branches, walkers must return to the backbone. Subdiffusive behavior in the backbone is a direct consequence of branches acting like traps.}\n\\label{fig:1}\n\\end{figure}\n\nGiven the previously-mentioned experimental findings and because the quenched disorder is intrinsic to the comb structure, it is essential to account for annealed disorder in the comb-model. Here we propose such hybrid comb-models by generalizing the usual comb diffusion equation~\\cite{ArBa} via different fractional time-derivative operators. In our hybrid comb-models, different comb-like structures describe quenched disorder, and fractional operators account for annealed disorder. By exploring different configurations where fractional operators act on the branches, backbone, or simultaneously on both, and also by replacing the usual comb structure by a generalized fractal structure, we find a series of nontrivial results that are useful for describing some recent empirical results reported for anomalous diffusion. Among other findings, we observe that these generalized comb-models describe restricted diffusion, Brownian diffusion, and crossovers from subdiffusion to restricted or Brownian diffusions.\n\nThe rest of this manuscript is organized as follows. In Section~\\ref{sec:2}, we define our generalized version of the comb-model and investigate its solutions under different situations. In Section~\\ref{sec:3}, we consider a fractal grid in place the single backbone structure and explore the effects of this modification on the diffusive behavior. Finally, we conclude this work in Section~\\ref{sec:4} with a discussion and summary of our findings.\n\n\\section{Generalized Comb-Models with Fractional Operators}\\label{sec:2}\n\nThe diffusion equation for a comb structure was proposed by Arkhincheev and Baskin~\\cite{ArBa} and represents a two-dimensional Einstein's diffusion equation where the diffusive term in the $x$-direction is multiplied by a Dirac delta function $\\delta(y)$, that is,\n\\begin{eqnarray}\n\\label{Comb0}\n\\frac{\\partial}{\\partial t}\\rho(x,y;t)= \n{\\cal{D}}_{y} \\frac{\\partial^2}{\\partial y^2} \\rho(x,y;t)+\n\\delta(y){\\cal{D}}_{x} \\frac{\\partial^2}{\\partial x^2} \\rho(x,y;t).\n\\end{eqnarray}\nBecause of the delta function, diffusion in the $x$-direction only occurs over the backbone structure (when $y=0$). The diffusion in the $y$-direction creates the branch structures; a walker can only leave a branch or access other branches by returning to the backbone structure (Figures~\\ref{fig:1}c and \\ref{fig:1}d). The geometrical restrictions in Eq.~(\\ref{Comb0}) mimic all features of early comb-models, including subdiffusive behavior in the backbone. The trapping times over the branches are also equivalent to a power-law behavior in the waiting-time distributions of a CTRW. The solutions of Eq.~(\\ref{Comb0}) are related to a time-fractional diffusion equation (with an anomalous exponent $\\alpha_{x}=1\/2$) describing the spreading behavior in the backbone~\\cite{ArBa,Arkhincheev1999,Wakil2002,IominPRE2005}. The diffusion over the backbone is also described by a time-fractional diffusion equation with exponent $\\alpha_{x}=1\/4$ in a three-dimensional comb structure and $\\alpha_{x}=1\/2N$ for an $N$-dimensional case~\\cite{Arkhincheev1999}. Extensions of Eq.~(\\ref{Comb0}) have been used to obtain a fractional diffusion equation with an absorbent term and a linear external force~\\cite{Zahran2009} as well as to deal with generalized fractal structures in the backbone and branches (namely the fractal comb-model)~\\cite{Iomin2011,pre2015,jpa2016,sandev2017anomalous}.\n\nIn this context, we propose to generalize the comb-model by including different fractional time-derivative operators on the diffusion terms, that is,\n\\begin{eqnarray}\n\\label{Comb1}\n\\frac{\\partial}{\\partial t}\\rho(x,y;t)= \n{\\cal{F}}_{t,y}\\left({\\cal{D}}_{y} \\frac{\\partial^2}{\\partial y^2} \\rho(x,y;t)\\right) +\n\\delta(y){\\cal{F}}_{t,x}\\left({\\cal{D}}_{x} \\frac{\\partial^2}{\\partial x^2} \\rho(x,y;t)\\right)\\,,\n\\end{eqnarray}\nwhere ${\\cal{F}}_{t,i}\\{ \\cdots \\}$ is an operator defined by the time derivative of a convolution integral between a function $f(x,y,t)$ and a memory kernel $\\mathcal{K}_{i}(t)$ ($i\\in \\{x,y\\}$), that is,\n\\begin{eqnarray}\n\\label{KK}\n{\\cal{F}}_{t,i} \\{ f(x,y;t) \\}=\n\\frac{\\partial}{\\partial t} \\int_{0}^{t}f(x,y;t')\\mathcal{K}_{i}(t-t')\\,dt' \\;.\n\\end{eqnarray}\nThe use of fractional derivatives in front of spatial operators is motivated by a possible connection with the linear-response theory~\\cite{Sokolov2001}. The memory kernel can also be connected with the waiting-time distribution of CTRW and represents a coarse-grained description of the environment's randomness. Specifically, the kernel of the time-convoluted operator represents a density memory (a property of a collection of trajectories) and not a trajectory memory~\\cite{Cakir2007}. A derivation of this integro-differential operator and the physical meaning of the memory kernel are given by Sokolov and Klafter~\\cite{Sokolov2005}. {It is worth mentioning that different operators~\\cite{sokolov2002solutions,PhysRevE.81.011116,PhysRevE.66.046129} have been used to extend diffusion equations. For instance, the operator $\\int_{0}^{t}f(x,y;t')\\mathcal{K}_{i}(t-t')\\,dt'$ was considered by Sokolov~\\cite{sokolov2002solutions} for identifying memory kernels that lead non-negative solutions (safe ones) and those that this condition is not guaranteed (dangerous ones).}\n\nThe memory kernels $\\mathcal{K}_{i}(t)$ define the integro-differential operators in Eq.~(\\ref{Comb1}) and establish a connection with fractional time-derivative operators. Thus, Eq.~(\\ref{KK}) represents a unified description for a broad class of situations where either singular or non-singular kernels describe different relaxation processes. Moreover, distinct kernels for the $x$ and $y$ directions yield anisotropic diffusion. Equations~(\\ref{Comb1}) and (\\ref{KK}) recover the usual comb-model [Eq.~(\\ref{Comb0})] when\n{{$\\mathcal{K}_{x}(t) = \\mathcal{K}_{y}(t)=1$}}. In the usual case, there are no memory effects, and geometrical restrictions of the comb-like structure are the only mechanism tied to the anomalous diffusion~\\cite{pre2015,IominBook}.\n\nDifferent choices for $\\mathcal{K}_{x}(t)$ and $\\mathcal{K}_{y}(t)$ imply in extending the comb-model to different contexts that combine quenched and annealed disorders. One possibility is to consider power-law functions such as \n\\begin{eqnarray}\n\\label{KK11}\n\\mathcal{K}_{i}(t)=  \n \\frac{t^{\\alpha_{i}-1}}{\\Gamma\\left(\\alpha_{i}\\right)}, \n\\end{eqnarray}\nwhich are directly related to the Riemann-Liouville fractional operator~\\cite{Livro1} for $0<\\alpha_{i}<1$. This fractional operator has been used to investigate several physical contexts, in particular the ones related to anomalous diffusion~\\cite{Metzler2000,Barkai2,Book2018}. \n\nAnother possibility is to assume an exponential behavior for the kernel\n\\begin{eqnarray}\\label{KK22e}\n\\mathcal{K}_{i}(t)&=& {\\cal{R}}(\\alpha'_{i})\\exp\\left(-\\alpha'_{i} \\, t\\right)\\,,\n\\end{eqnarray}\nwhere ${\\cal{R}}(\\alpha'_{i})$ is a normalization constant. {This choice corresponds to the Caputo-Fabrizio operator with $\\alpha'_{i}=\\alpha_{i}\/(1-\\alpha_{i})$ \\cite{nonsingular1,hristov2017derivatives,tateishifp}}. A remarkable feature of this exponential kernel is its connection with resetting processes~\\cite{tateishifp}. In particular, by combining Eqs.~(\\ref{Comb1}),~(\\ref{KK}), and~(\\ref{KK22e}), we find\n\\begin{eqnarray}\n\\label{Comb2}\n\\frac{\\partial}{\\partial t}\\rho(x,y;t)= \n{\\cal{D}}_{y} \\frac{\\partial^2}{\\partial y^2} \\rho(x,y;t) +\n\\delta(y){\\cal{D}}_{x} \\frac{\\partial^2}{\\partial x^2} \\rho(x,y;t)-\\tilde{\\alpha}\\left(\\rho(x,y,t)-\\varphi(x,y)\\right)\\,,\n\\end{eqnarray}\nwhere $\\alpha_{x}=\\alpha_{y}=\\tilde{\\alpha}$ and $\\varphi(x,y)$ is the initial condition. Equation~(\\ref{Comb2}) extends the standard expressions used to analyze resetting processes by including a geometric constraint between the $x$ and $y$ directions. {It is worth noticing that an exponential kernel leads to the Cattaneo equation in the approach of Sokolov~\\cite{sokolov2002solutions}, that is, a diffusion-wave equation different from Eq.~(\\ref{Comb2}).} The kernel ${\\cal K}_{\\alpha_{i}}(t)\\propto \\,E_{\\alpha}\\left(-\\overline{\\alpha}t^{\\alpha}\\right)$, where \n\\begin{equation}\nE_{\\alpha}(z)=\\sum_{k=0}^{\\infty}\\frac{z^k}{\\Gamma(\\alpha k+1)}  \n\\end{equation}\nis the Mittag-Leffler function~\\cite{Livro1} with parameter $\\alpha$ and $\\overline{\\alpha}$ a constant, somehow interpolates between the power-law and exponential cases and has been recently associated with fractional-time derivatives of distributed order~\\cite{tateishifp}. It is worth mentioning that these non-singular kernels have been used to investigate different contexts such as diffusion~\\cite{tateishifp}, heat processes~\\cite{EBasa1}, groundwater flow~\\cite{AAtangana1}, and electrical circuits~\\cite{JFGomez-Aguilar1}. \n\nWe now focus on the solutions of Eq.~(\\ref{Comb1}) in the Fourier-Laplace domain by using the Green function approach. After, we analyze particular cases related to the previous kernels. We consider Eq.~(\\ref{Comb1}) subjected to the initial condition $\\rho(x,y,0)= \\varphi(x,y)$, where $\\varphi(x,y)$ is a normalized function, that is, $ \\int_{-\\infty}^{\\infty}dx\\int_{-\\infty}^{\\infty}dy\\,\\varphi(x,y)=1$. We further assume $\\rho(\\pm \\infty,y;t)=0$ and $\\rho(x,\\pm \\infty;t) = 0$ as boundary conditions. These unlimited boundary conditions avoid possible effects of confinement in a limited domain, making more explicit the impact of geometrical restrictions and fractional operators on the spreading behavior. \n\nTo obtain the solutions of Eq.~(\\ref{Comb1}), we first apply the Laplace transform $\\left({\\mathcal{L}}\\left\\{\\rho(x,y;t)\\right\\}\\right.=$ $\\left.\\int_{0}^{\\infty}e^{-st}\\rho(x,y;t)\\,dt =\\bar{\\rho}(x,y;s) \\right)$, yielding \n\\begin{eqnarray}\\label{CombLaplace}\ns\\bar{\\rho}(x,y;s)-\\varphi(x,y) = s\\bar{\\mathcal{D}}_{y}(s)\\frac{\\partial^2}{\\partial y^2} \\bar{\\rho}(x,y;s)+\\delta(y)s\\bar{{\\cal{D}}}_{x}(s)\\frac{\\partial^2}{\\partial x^2} \\bar{\\rho}(x,y;s),\n\\end{eqnarray}\nwhere $\\bar{\\mathcal{D}}_{y}(s) = {\\cal{D}}_{y}\\bar{\\mathcal{K}}_{y}(s)$ and $\\bar{\\mathcal{D}}_{x}(s) = {\\cal{D}}_{x}\\bar{\\mathcal{K}}_{x}(s)$. We next apply the Fourier transform $\\left({\\mathcal{F}}\\left\\{\\rho(x,y;t)\\right\\}=\\tilde{\\rho}(k_x,y;s)=\\right.$ $\\left.\\int_{-\\infty}^{\\infty}e^{-ik_{x}x}\\rho(x,y;t)\\,dx  \\right.)$ on the $x$ variable of  Eq.~(\\ref{CombLaplace}), yielding\n\\begin{eqnarray}\n\\label{Equation1}\ns\\bar{{\\cal{D}}}_{y}(s)\\frac{d^{2}}{dy^{2}}\\bar{\\rho}(k_{x},y;s) - \ns\\left(1-\\delta(y) \\bar{{\\cal{D}}}_{x}(s) k_{x}^{2}\\right)\\bar{\\rho}(k_{x},y;s)  = -\\varphi(k_{x},y)\\;.\n\\end{eqnarray}\nBy using the Green functions approach, the solution for Eq.~(\\ref{Equation1}) is written as\n\\begin{eqnarray}\n\\label{SolutionLplacerho}\n\\bar{\\rho}(k_{x},y;s) = - \\int_{-\\infty}^{\\infty}dy'\\,\\varphi(k_{x},y')\\bar{{\\cal{G}}}(k_{x},y,y';s)\\,,\n\\end{eqnarray}\nwhere the Green function ${\\cal{G}}(k_{x},y,y';s)$ is the solution of\n\\begin{eqnarray}\n\\label{Green0}\ns\\bar{{\\cal{D}}}_{y}(s)\\frac{d^{2}}{dy^{2}}\\bar{{\\cal{G}}}(k_{x},y,y';s) -\\left(s +\n\\delta(y) s\\bar{{\\cal{D}}}_{x}(s) k_{x}^{2}\\right)\\bar{{\\cal{G}}}(k_{x},y,y';s)= \\delta(y-y')\n\\end{eqnarray}\nsubjected to the condition $\\bar{{\\cal{G}}}(k_{x}, \\pm\\infty,\\bar{y};s)=0$. \n\nAfter some calculations, we can show that the solution for Eq.~(\\ref{Green0}) is\n\\begin{eqnarray}\n\\tilde{\\bar{{\\cal{G}}}}(k_{x},y,y';s) =\n- \\frac{1}{2s\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}e^{-\\frac{1}{\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}|y-y'|} -\n\\frac{\\bar{{\\cal{D}}}_{x}(s)k_{x}^{2}}{2\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}e^{-\\frac{1}{\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}|y|} \\bar{{\\cal{G}}}(k_{x},0,y';s),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{BackboneGreen}\n\\!\\!\\!\\!\\!\\!\\tilde{\\bar{{\\cal{G}}}}(k_{x},0,y';s) = \\frac{e^{-\\frac{1}{\\sqrt{{\\cal{D}}_{y}(s)}}|y'|} }{ s\\left({\\cal{D}}_{x}(s)k_{x}^{2} +2\\sqrt{{\\cal{D}}_{y}(s)}\\right)}\\;\n\\end{eqnarray}\nrepresents the propagator for the backbone structure (at $y=0$). The term ${\\cal{D}}_{y}(s)$ in Eq.~(\\ref{BackboneGreen}) indicates that the backbone diffusion explicitly depends on the diffusion occurring along the branches; in other words, memory effects on branches directly affect the diffusion on the backbone.  \n\nThe Green function related to Eq.~(\\ref{Green0}) subjected to previous boundary condition is thus given by\n\\begin{eqnarray}\n\\!\\!\\!\\!\\!\\!\\bar{{\\cal{G}}}(k_{x},y,y';s) &=& \n- \\frac{1}{2s\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}e^{-\\frac{1}{\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}|y-y'|} +\n\\frac{\\bar{{\\cal{D}}}_{x}(s)k_{x}^{2}}{ \\bar{{\\cal{D}}}_{x}(s)k_{x}^{2}+ 2\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}} \\frac{1}{2s\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}e^{-\\frac{1}{\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}\\left(|y|+|y'|\\right)} \\nonumber\\\\\n&=& -\\frac{1}{2s\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}\\left(\ne^{-\\frac{1}{\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}|y-y'|} -  e^{-\\frac{1}{\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}\\left(|y|+|y'|\\right)} \\right) \\nonumber \\\\  &-&\\frac{1}{s\\left(2\\sqrt{\\bar{{\\cal{D}}}_{y}(s)} + \\bar{{\\cal{D}}}_{x}(s)k_{x}^{2} \\right)}e^{-\\frac{1}{\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}\\left(|y|+|y'|\\right)} \\;.\n\\end{eqnarray}\nAfter performing the inverse Fourier transform on $x$-direction $( {\\mathcal{F}}^{-1}\\left\\{\\tilde{\\rho}(k_x,y;t)\\right\\}$ $=$ $\\rho(x,y;t)$ $=$ $\\int_{-\\infty}^{\\infty}e^{ik_{x}x}\\tilde{\\rho}(k_x,y;t)\\,dk_x)$, we obtain\n\\begin{eqnarray}\n\\label{Green1}\n{\\cal{G}}(x,y,y';s) &=& \n-\\frac{\\delta(x)}{2s\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}} \\left( e^{-\\frac{|y-y'|}{\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}} -  e^{-\\frac{1}{\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}\\left(|y|+|y'|\\right)} \\right)\n\\nonumber \\\\ &-&\n\\frac{ 1}{2s\\sqrt{2\\bar{{\\cal{D}}}_{x}(s)\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}}}\ne^{-\\sqrt{\\frac{2\\sqrt{\\bar{{\\cal{D}}}_{y}(s) }}{\\bar{{\\cal{D}}}_{x}(s)}} |x|}e^{-\\frac{1}{\\sqrt{{\\cal{D}}_{y}(s)}}\\left(|y|+y'|\\right)}\\;.\n\\end{eqnarray}\n\nThe result in Eq.~(\\ref{Green1}) is completely general and can be used to describe different diffusive processes depending on the kernel of the integro-differential operator. For example, for $\\bar{{\\cal{K}}}_{x}(s)=1\/s^{\\alpha_{x}}$ and $\\bar{{\\cal{K}}}_{y}(s)=1\/s^{\\alpha_{y}}$, the inverse Laplace transform of Eq.~(\\ref{Green1}) is\n\\begin{eqnarray}\n\\label{GreenRL}\n&&{\\cal{G}}(x,y,y';t)=- \\frac{\\delta(x)}{2\\sqrt{{\\cal{D}}_{y}t^{\\alpha_{y}}}}\\left\\{\n{{\\Large{H}}}_{1,1}^{1,0} \\left[ \\frac{|y-y'|}{\\sqrt{{\\cal{D}}_{y}t^{\\alpha_{y}}}}\n\\left|_{\\left(0, 1 \\right) }^{\\left(1-\\frac{\\alpha_{y}}{2},  \\frac{\\alpha_{y}}{2} \\right)} \\right. \\right]-\n{{\\Large{H}}}_{1,1}^{1,0} \\left[ \\frac{|y|+|y'|}{\\sqrt{{\\cal{D}}_{y}t^{\\alpha_{y}}}}\n\\left|_{\\left(0, 1 \\right) }^{\\left(1-\\frac{\\alpha_{y}}{2},  \\frac{\\alpha_{y}}{2} \\right)} \\right. \\right]\\right\\}\\nonumber \\\\ &-& \\frac{ 1}{2\\sqrt{2{\\cal{D}}_{x}\\sqrt{{{\\cal{D}}}_{y}}}}\n\\int_{0}^{t}\\frac{dt'}{(t-t')t'^{\\alpha_{+\n}}}{{\\Large{H}}}_{1,1}^{1,0} \\left[ \\sqrt{\\frac{2\\sqrt{{\\cal{D}}_{y}t^{\\alpha_{y}}}}{{\\cal{D}}_{x}t^{\\alpha_{x}}}}|x|\n\\left|_{\\left(0, 1 \\right) }^{\\left(1-\\alpha_{+}, \\alpha_{-}\\right)} \\right. \\right]{{\\Large{H}}}_{1,1}^{1,0} \\left[ \\frac{|y|+|y'|}{\\sqrt{{\\cal{D}}_{y}t^{\\gamma_{y}}}}\n\\left|_{\\left(0, 1 \\right) }^{\\left(0, \\frac{\\alpha_{y}}{2} \\right)} \\right. \\right]\\;,\\nonumber\\\\\n\\end{eqnarray}\nwhere $\\alpha_{+}=\\alpha_{x}\/2+\\alpha_{y}\/4$, $\\alpha_{-}=\\alpha_{x}\/2-\\alpha_{y}\/4$, and $H_{p,q}^{m,n}\\left[z\\left|^{(a_p,A_p)}_{(b_q,B_q)} \\right.\\right]$ is the Fox H function~\\cite{Fox}. The case ${\\cal{K}}_{x}(s)=1\/(s+ \\alpha)$ and ${\\cal{K}}_{y}(s)=1\/(s+\\alpha)$ (where $\\alpha'_{x}=\\alpha'_{y}=\\alpha$) lead us to\n\\begin{eqnarray}\n\\label{CombModelResetting}\n&&{\\cal{G}}(x,y,y';t)=-\\delta(x)\\int_{0}^{t}dt'\\,k(t,t')\\frac{e^{-\\alpha t'}}{\\sqrt{4\\pi D_{y}t}}\\!\\left\\{\\!e^{-\\frac{|y-y'|^2}{4 D_{y}t'}}-e^{-\\frac{(|y|+|y'|)^2}{4 D_{y}t'}}\\right\\}\\nonumber \\\\ &-& \\int_{0}^{t}\\!\\!dt'\\,k(t,t')\\!\\!\\int_{0}^{t'}\\!\\!dt''\\,\\!\\frac{|y|+|y'|}{\\sqrt{8D_{x}t''\\sqrt{D_{y}t''}}}{{\\Large{H}}}_{1,1}^{1,0} \\left[ \\left(\\frac{4D_{y}}{D_{x}^{2}t''}\\right)^{\\frac{1}{4}}|x|\n\\left|_{\\left(\\frac{3}{4},  \\frac{1}{4} \\right) }^{\\left(\\frac{1}{4},  \\frac{1}{4} \\right)} \\right. \\right]\\frac{e^{-\\frac{(|y|+|y'|)^{2}}{4D_{y}(t'-t'')}}}{\\sqrt{4\\pi\\left[ D_{y} \\left(t'-t''\\right)\\right]^{3}}}\\;,\\nonumber\\\\\n\\end{eqnarray}\nwith $k(t,t')=\\alpha+\\delta(t-t')$ and $D_{x(y)}={\\cal{D}}_{x(y)}{\\cal{R}}(\\alpha_{x(y)})$. Differently from Eq.~(\\ref{GreenRL}), Eq.~(\\ref{CombModelResetting}) has a stationary solution (a time-independent solution ${\\cal{G}}_{st}(x,y,y')=\\lim_{t\\rightarrow\\infty}{\\cal{G}}(x,y,y';t)$) given by\n\\begin{eqnarray}\n\\label{CombModelResetting_Stationary}\n{\\cal{G}}_{st}(x,y,y') &=&    -\n\\sqrt{\\frac{\\alpha}{4{D}_{y}}} \\delta(x)\\left( e^{-\\sqrt{\\frac{\\alpha}{D_{y}}}|y-y'|} -  e^{-\\sqrt{\\frac{\\alpha}{D_{y}}}\\left(|y|+|y'|\\right)} \\right)\n\\nonumber \\\\ &-&\n\\sqrt{\\frac{ \\alpha\\sqrt{\\alpha}}{8D_{x}\\sqrt{D_{y}}}}\ne^{-\\sqrt{\\frac{2\\alpha}{D_{x}}\\sqrt{\\frac{D_{y} }{\\alpha}}} |x|}e^{-\\sqrt{\\frac{\\alpha}{{D}_{y}}}\\left(|y|+|y'|\\right)}\\;.\n\\end{eqnarray}\nThis result is obtained from an arbitrary condition where\n\\begin{eqnarray}\n\\lim_{s\\rightarrow 0}\\bar{{\\cal{K}}}_{x}(s)=const \\;\\;\\;{\\text{and}}\\;\\;\\;\n\\lim_{s\\rightarrow 0}\\bar{{\\cal{K}}}_{y}(s)=const, \\nonumber\n\\end{eqnarray}\nwhich implies in $\\lim_{s\\rightarrow 0}\\left(s\\bar{{\\cal{G}}}(x,y,y';s)\\right)={\\cal{G}}_{st}(x,y,y')$. Figure~\\ref{fig:2} illustrates the behavior of Eq.~(\\ref{SolutionLplacerho}) when considering the Green function of Eq.~(\\ref{CombModelResetting_Stationary}). In this stationary limit, the solution in Eq.~(\\ref{SolutionLplacerho}) is\n\\begin{eqnarray}\n\\label{SolutionIndependentTrho}\n\\rho_{st}(x,y) = \\int_{-\\infty}^{\\infty}dx'\\int_{-\\infty}^{\\infty}dy'\\,\\varphi(x',y'){\\cal{G}}_{st}(x-x',y,y')\\;.\n\\end{eqnarray}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{fig2.pdf}\n\\caption{Behavior of the stationary distribution obtained from Eq.~(\\ref{SolutionIndependentTrho}) with Green function given by Eq.~(\\ref{CombModelResetting_Stationary}). For simplicity, we consider the initial condition $\\varphi(x,y)\\propto e^{-x^{2}\/\\sigma_{x}^2-y^{2}\/\\sigma_{y}^2}\/\\left(\\sigma_{x}\\sigma_{y}\\right)$, where $\\sigma_{x}=\\sigma_{y}=1\/5$), $\\sqrt{\\alpha\/D_{y}}=1$, and $\\sqrt{\\alpha\/D_{x}}=1$ (in arbitrary units). }\n\\label{fig:2}\n\\end{figure}\n\nThe MSD also carries information about the medium structure. We thus use the previous results to investigate how the MSD in each direction changes under those different scenarios. To do so and avoid transient behaviors related to the initial position of the walkers, we consider the initial condition $\\varphi(x,y)=\\delta(x)\\delta(y)$. Under these assumptions, the MSD in the Laplace domain for each direction is\n\\begin{eqnarray}\n\\label{msdy}\n\\bar{\\sigma}_{y}^{2}(s)=\\left\\langle \\left(y- \\langle y \\rangle\\right)^{2}\\right\\rangle &=&  2\\bar{{\\cal{D}}}_{y}(s)\/s \\;\\;\\; {\\text{and}}\\\\ \n\\label{msdx}\n\\bar{\\sigma}_{x}^{2}(s)=\\left\\langle \\left(x- \\langle x \\rangle \\right)^{2} \\right \\rangle &=&  \\bar{{\\cal{D}}}_{x}(s)\/\\left(s\\sqrt{\\bar{{\\cal{D}}}_{y}(s)}\\right).\n\\end{eqnarray}\nEquations~(\\ref{msdy}) and (\\ref{msdx}) show that the MSD in the $y$-direction depends only on its memory kernel, while the MSD in the $x$-direction depends on both memory kernels. These features naturally emerge in the time-domain; indeed, by performing the inverse Laplace transform, we find\n\\begin{eqnarray}\n\\sigma_{y}^{2}(t)=2\\int_{0}^{t}dt'\\,{\\cal{D}}_{y}(t') \\;\\;\\; {\\text{and}}\\;\\;\\; \n\\sigma_{x}^{2}(t) =  \n\\frac{{\\cal{D}}_{x}}{\\sqrt{{\\cal{D}}_{y}}}\\int_{0}^{t}dt'\\,\\zeta_{x,y}(t') \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\zeta_{x,y}(t)=\\mathcal{L}^{-1}\\left\\{\\frac{\\bar{\\mathcal{K}}_{x}(s)}{\\sqrt{\\bar{\\mathcal{K}}_{y}(s)}}\\right\\}.    \n\\end{equation}\nThese dependencies are a direct consequence of the comb structure and are somehow related to the results of Ref.~\\cite{Ribeiro2014}. The authors of that work have simulated fractional Brownian walks on a comb-like structure and reported that memory effects (associated with Hurst exponents) in $x$-direction do not affect the diffusive behavior in the $y$-direction; but, in the backbone, they found a nontrivial interplay between long-range memories in $x$ and $y$-directions.\n\nWe now consider the behavior of the system in each direction. To do so, we note that the Green function for the probability distribution function along the backbone $\\mathcal{G}_{1}(x,t)=\\int_{-\\infty}^{\\infty}dy\\,\\rho(x,y,t)$ satisfies the following generalized diffusion equation\n\\begin{eqnarray}\\label{pdf x}\n\\frac{\\partial}{\\partial t}\\mathcal{G}_{1}(x,t)=\\frac{\\mathcal{D}_{x}}{2\\sqrt{\\mathcal{D}_{y}}}\\frac{\\partial}{\\partial t}\\int_{0}^{t}dt'\\,\\zeta_{x,y}(t-t')\\frac{\\partial^{2}}{\\partial x^{2}}\\mathcal{G}_{1}(x,t')\\;.\n\\end{eqnarray}\nSimilarly, the corresponding generalized diffusion equation for the Green function along the branches $\\mathcal{G}_{2}(y,t)=\\int_{-\\infty}^{\\infty}dx\\,\\rho(x,y,t)$ is\n\\begin{eqnarray}\\label{pdf y}\n\\frac{\\partial}{\\partial t}\\mathcal{G}_{2}(y,t)=\\mathcal{D}_{y}\\frac{\\partial}{\\partial t}\\int_{0}^{t}dt'\\,\\mathcal{K}_{y}(t-t')\\frac{\\partial^{2}}{\\partial y^{2}}\\mathcal{G}_{2}(y,t').\n\\end{eqnarray}\nThese two forms suggest that both Eqs.~(\\ref{pdf x}) and (\\ref{pdf y}) have similar mathematical properties. By following Refs.~\\cite{csf2017,fcaa2018}, we can verify that the probability distribution functions $\\mathcal{G}_{1}(x,t)$ and $\\mathcal{G}_{2}(y,t)$ are non-negative if: $1\/\\left[s\\bar{\\zeta}_{x,y}(s)\\right]$  and $1\/\\left[s\\bar{\\mathcal{K}}_{y}(s)\\right]$ are completely monotone functions; and $1\/\\bar{\\zeta}_{x,y}(s)$ and $1\/\\bar{\\mathcal{K}}_{y}(s)$ are Bernstein functions. Moreover, by following the results of Ref.~\\cite{Sokolov2001}, we can further verify that Eqs.~(\\ref{pdf x}) and (\\ref{pdf y}) fulfill the Nyquist theorem, and consequently, their solutions are thermodynamically sound.\n\nTo investigate the interplay between mechanisms of annealed (memory kernels) and quenched (comb-like structure) disorders, we start considering several definitions for the fractional time operators. For the sake of comparison, it is worth remembering that ordinary derivatives (usual comb-model) imply in $\\mathcal{K}_{x}(t) = \\mathcal{K}_{y}(t)=\\delta(t)$, that in turn lead to $\\sigma_{y}^{2}(t)\\propto t$ and $\\sigma_{x}^{2}(t)\\propto t^{1\/2}$~\\cite{ArBa}.\n\nWe first consider the Riemann-Liouville operator, yielding the kernels $\\bar{{\\cal{K}}}_{\\alpha_{x}}(s)\\propto 1\/s^{\\alpha_{x}}$ ($0<\\alpha_{x}<1$) and $\\bar{{\\cal{K}}}_{\\alpha_{y}}(s)\\propto 1\/s^{\\alpha_{y}}$ ($0<\\alpha_{y}<1$). These memory kernels are related to the Green function given by Eq.~(\\ref{GreenRL}), and their corresponding MSDs are\n\\begin{eqnarray}\n\\label{y2RL}\n\\sigma_{y}^{2}(t)&=&  2\\mathcal{D}_{y}\\,\\mathcal{L}^{-1}\\left\\{s^{-\\alpha_{y}-1}\\right\\}=2\\mathcal{D}_{y}\\frac{t^{\\alpha_{y}}}{\\Gamma(1+\\alpha_{y})} \\;\\;\\;\\;\\ \\text{and} \\\\ \n\\label{x2RL}\n\\sigma_{x}^{2}(t) &=&  \\frac{\\mathcal{D}_{x}}{\\sqrt{\\mathcal{D}_{y}}}\\mathcal{L}^{-1}\\left\\{s^{-\\alpha_{x}+\\alpha_{y}\/2-1}\\right\\}=\\frac{\\mathcal{D}_{x}}{\\sqrt{\\mathcal{D}_{y}}}\\frac{t^{\\alpha_{x}-\\alpha_{y}\/2}}{\\Gamma(1+\\alpha_{x}-\\alpha_{y}\/2)}.\n\\end{eqnarray}\nEquation~(\\ref{y2RL}) shows that the diffusion in branches is independent of the backbone dynamics; it only depends on its memory effects. However, the MSD in the backbone depends on memory effects in both directions, as shown in Eq.~(\\ref{x2RL}).\n\nBy imposing the conditions of non-negativity to the corresponding solution, we find that $\\alpha_{x}>\\alpha_{y}\/2$. To better understand this condition, let us examine some limiting cases. When $\\alpha_{x}=1$ (ordinary derivative in the backbone) and $0<\\alpha_{y}<1$ (memory effects in the branches), the backbone diffusion is enhanced if $1\/2< 1-\\alpha_{y}\/2<1$. This result is intriguing and counter-intuitive because, as the branches act like traps, we could initially presume that the slower the diffusion in the branches, the more subdiffusive is the diffusion on the backbone; but quite the opposite happens. When the spread over the branches is subdiffusive, the walkers stay closer to the backbone and their probability of returning to the backbone increases, enhancing the diffusion in the $x$-direction. This phenomenon is related to the so-called ``subdiffusion paradox'' reported in cell environments~\\cite{Sereshki2012,BarkaiPhysTod2012,Barr2015}. Although subdiffusion reduces the exploration area, it increases the likelihood of walkers to stay close to specific targets~\\cite{Golding2006,Guigas2008}. In contrast, for $1\/2<\\alpha_{x}<1$ (memory effects in the backbone) and $\\alpha_{y}=1$ (ordinary derivative in the branches), the spread in the backbone is even more subdiffusive if $0< \\alpha_{x}-1\/2<1\/2$. Thus, the backbone subdiffusion is governed by the interplay of two mechanisms: the trapping in the branches and the memory effects in the backbone. Furthermore, an essential feature of Eqs.~(\\ref{y2RL}) and (\\ref{x2RL}) is scale invariance, that is, the effects of geometrical restrictions and memory effects are the same in all time-scales. This last behavior is illustrated in Figure~\\ref{fig:3} for $\\alpha_{x}=1$ and $\\alpha_{y}=1$ (solid black line).\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{fig3.pdf}\n\\caption{Behavior of the MSD $\\sigma_{x}^{2}$ versus $t$ for different kernels. The solid black line refers to subdiffusion when $\\bar{{\\cal{K}}}_{x}(s)\\propto 1\/s^{\\alpha_{x}}$ and $\\bar{{\\cal{K}}}_{y}(s)\\propto 1\/s^{\\alpha_{y}}$, where we have chosen ${\\cal{D}}_{x}\/\\sqrt{{\\cal{D}}_{y}}=1$, $\\alpha_{x}=1$, and $\\alpha_{y}=1$. The dot-dashed blue line corresponds to usual diffusion obtained as the asymptotic behavior for $\\bar{{\\cal{K}}}_{x}(s)\\propto 1\/s^{\\alpha_{x}}$ and $\\bar{{\\cal{K}}}_{y}(s)\\propto 1\/\\left(s+\\alpha'_{y}\\right)$, where, for simplicity, we have chosen ${\\cal{D}}_{x}\/\\sqrt{{\\cal{D}}_{y}{\\cal{R}}(\\alpha_{y})}=1$, $\\alpha_{x}=1$, and $\\alpha_{y}=1$.  The dashed red line corresponds to $\\bar{{\\cal{K}}}_{x}(s)\\propto 1\/(s+\\alpha_{x})$ with $\\bar{{\\cal{K}}}_{y}(s)\\propto 1\/(s+ \\alpha_{y})$ (the case with stationary state), where we have chosen ${\\cal{R}}(\\alpha_{x}){\\cal{D}}_{x}\/\\sqrt{{\\cal{D}}_{y}{\\cal{R}}(\\alpha_{y}})=1$, $\\alpha_{x}=1$, and $\\alpha_{y}=1$.}\n\\label{fig:3}\n\\end{figure}\n\nAs a second example, we investigate an anisotropic case characterized by different memory kernels for $x$ and $y$ directions. We maintain the same power-law kernel for the backbone, that is, ${\\cal{K}}_{\\alpha_{x}}(s)\\propto 1\/s^{\\alpha_{x}}$ {(with $1\/2\\leq\\alpha_{x}<1$ to ensure non-negative solutions)}, and consider an exponential memory kernel ${\\cal{K}}_{\\alpha_{y}}(s)\\propto 1\/\\left(s+\\alpha'_{y}\\right)$ for the branches, with $\\alpha'_{y}=\\alpha_{y}\/(1-\\alpha_{y})$ and $0<\\alpha_{y}<1$. These choices correspond to the Riemann-Liouville fractional operator in the $x$-direction and the Caputo-Fabrizio operator in the $y$-direction. Under these conditions, we find the MSDs\n\\begin{eqnarray}\n\\label{y2RLCF}\n\\sigma_{y}^{2}(t) &=&  2D_{y}\\,\\mathcal{L}^{-1}\\left\\{\\frac{s^{-1}}{s+\\alpha'_{y}}\\right\\}=2\\frac{D_{y}}{\\alpha'_{y}}\\left(1-e^{-\\alpha'_{y}t}\\right) \\;\\;\\; \\text{and} \\;\\;\\; \\\\ \n\\label{x2RLCF}\n\\sigma_{x}^{2}(t) &=&  \\frac{\\mathcal{D}_{x}}{\\sqrt{D_{y}}}\\mathcal{L}^{-1}\\left\\{\\frac{s^{-\\alpha_{x}-1}}{\\left(s+\\alpha'_{y}\\right)^{-1\/2}}\\right\\}=\\frac{\\mathcal{D}_{x}}{\\sqrt{D_{y}}}t^{\\alpha_{x}-1\/2}E_{1,\\alpha_{x}+1\/2}^{-1\/2}(-\\alpha_{y}t),\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{ML three}\nE_{\\alpha,\\beta}^{\\delta}(z)=\\sum_{k=0}^{\\infty}\\frac{(\\delta)_k}{\\Gamma(\\alpha k+\\beta)}\\frac{z^k}{k!}\n\\end{equation}\nis the three-parameter Mittag-Leffler function~\\cite{Prabhakar}, and $(\\delta)_k=\\Gamma(\\gamma+k)\/\\Gamma(\\gamma)$ represents the Pochhammer symbol~\\cite{Prabhakar}. For the calculations of Eqs.~(\\ref{y2RLCF}) and (\\ref{x2RLCF}) we have used that~\\cite{Prabhakar}\n\\begin{equation}\\label{laplaceML}\n\\mathcal{L}\\left\\{t^{\\beta-1}E_{\\alpha,\\beta}^{\\delta}(-\\nu t^{\\alpha})\\right\\}(s)=\\frac{s^{\\alpha\\delta-\\beta}}{\\left(s^{\\alpha}+\\nu\\right)^{\\delta}},\n\\end{equation}\nwhere $\\Re(s)>|\\nu|^{1\/\\alpha}$. \n\nEquation~(\\ref{y2RLCF}) shows the isolated effects of the exponential memory kernel. We notice that the exponential term approaches zero for long times, and the MSD thus reaches a plateau of saturation. This behavior describes a confined (localized, restricted, or corralled) diffusion, where $\\alpha'_{y}$ can be associated with a saturation rate, and the asymptotic value of the MSD represents the magnitude of the confinement region. MSDs having the general form of Eq.~(\\ref{y2RLCF}), that is, \n\\begin{equation}\n\\label{ConfinedDiffusion}\n    \\sigma^{2}(t) = A_{\\infty}(1-e^{-\\xi t}),\n\\end{equation}\nare well-known to emerge in Ornstein-Uhlenbeck processes~\\cite{Uhlenbeck1945} and restricted diffusion confined within reflecting boundaries~\\cite{kusumi1993}. However, the restricted diffusion observed here occurs without external forces or finite boundary conditions, a remarkable feature of the Caputo-Fabrizio operator that has also been reported in Ref.~\\cite{tateishifp}.\n\nThe confined diffusion observed in Eq.~(\\ref{ConfinedDiffusion}) suggests a relationship between the exponential kernels and stationary states (stochastic localization phenomena). Indeed, the same equation emerges in the work M\\'endez and Campos~\\cite{Campos2016}, where a CTRW model for diffusion with resetting (walkers return to the origin with a resetting probability $r$) was proposed. A connection between fractional diffusion equations with the Caputo-Fabrizio operator and diffusion with stochastic resetting was also established in Ref.~\\cite{tateishifp}. M\\'endez and colleagues \\cite{Mendez2015} also studied a CTRW on a comb structure subjected to a bias parameter on the branches, where Eq.~(\\ref{ConfinedDiffusion}) appears as an asymptotic behavior for the backbone diffusion when the walker is biased to stay along the branches. The authors of Ref.~\\cite{Ribeiro2014} verified that normal diffusion emerges on the backbone when a fractional Brownian motion with long-range anti-persistent correlations occurs on the branches. By studying a minimal random walk model with infinite memory (walkers preferentially return to previously visited sites), Boyer and Solis-Salas~\\cite{Boyer2014} established a connection between long-range memory and stationary states of MSD and demonstrated how to infer memory strength use in animals (monkeys).\n\nIn this context, we can verify whether the relation between the Caputo-Fabrizio operator and confined diffusion is valid for the comb-model from the evolution of Eq.~(\\ref{x2RLCF}). To do so, we calculate the asymptotic limits of Eqs.~(\\ref{y2RLCF}) and (\\ref{x2RLCF}) for short- and long-times, that is,\n\\begin{eqnarray}\n\\label{y2limits}\n\\sigma_{y}^{2}(t) &\\sim&  2\\mathcal{D}_{y}\\,\\left\\lbrace \\begin{array}{c l} \n\\alpha'_{y}t, \\quad & t\\rightarrow0,\\\\\n1, \\quad & t\\rightarrow\\infty,  \n\\end{array}\\right.~~\\text{and}\\\\ \n\\label{x2limits}\n\\sigma_{x}^{2}(t) &\\sim&  \\frac{\\mathcal{D}_{x}}{\\sqrt{D_{y}}}\\left\\lbrace \\begin{array}{c l} \n\\frac{t^{\\alpha_{x}-1\/2}}{\\Gamma(\\alpha_{x}+1\/2)}, \\quad & t\\rightarrow0,\\\\\n\\frac{t^{\\alpha_{x}}}{\\Gamma(\\alpha_{x}+1)}, \\quad & t\\rightarrow\\infty.\n\\end{array}\\right.\n\\end{eqnarray}\nIn previous calculations, we use the formula~\\cite{fcaa2015,GG}\n\\begin{equation}\nE_{\\alpha,\\beta}^{\\delta}(-z)=\\frac{z^{-\\delta}}{\\Gamma(\\delta)}\\sum_{k=0}^{\\infty}\\frac{\\Gamma(\\delta+k)}{\\Gamma(\\beta-\\alpha(\\delta+n))}\\frac{(-z)^{-n}}{n!},\n\\end{equation}\nfor $0<\\alpha<2$ and $z\\rightarrow\\infty$, from which we find the asymptotic behavior \n\\begin{equation}\nE_{\\alpha,\\beta}^{\\delta}(-t^{\\alpha})\\simeq\\frac{t^{-\\alpha\\delta}}{\\Gamma(\\beta-\\alpha\\delta)}, \\quad t\\rightarrow\\infty.\n\\end{equation} \nEquations~(\\ref{y2limits}) and (\\ref{x2limits}) show that Brownian motion governs the branches dynamics at short-times, promoting enhanced subdiffusion in the backbone with $0<\\alpha'_{x}-1\/2<1\/2$ and $\\alpha'_{y}=1$ (as discussed earlier). In the long-time limit, a stationary state emerges in the branches, and the backbone dynamics only depends on its power-law memory kernel. In particular, there is a crossover from subdiffusion ($\\sigma_{x}^{2}(t) \\propto t^{1\/2}$) to Brownian diffusion ($\\sigma_{x}^{2}(t) \\propto t$) when $\\alpha_{x}=1$ (dot-dashed blue line in Figure~\\ref{fig:3}). We thus find an intriguing result where the interplay between geometrical restriction and memory effects (mechanisms associated with subdiffusion) produces usual Brownian motion. Similar behavior also emerges for power-law memory kernels when $\\alpha_{x}\\rightarrow 1$ and $\\alpha_{y}\\rightarrow 0$, for suitable combinations of Hurst exponents in fractional Brownian motions over a comb structure (Figure~5 of Ref.~\\cite{Ribeiro2014}), and without memory effects when the branches of the comb are finite~\\cite{Havlin2002, Berez2015,Arkhincheev2007}. However, the results of Eqs.~(\\ref{y2limits}) and (\\ref{x2limits}) are obtained with no memory effects in the $x$-direction and the stationary state is a consequence of the exponential memory kernel valid for $0<\\alpha'_{y}<1$.\n\nThe crossover is an essential feature of our model and provides insights into the time scale that each mechanism of subdiffusion is most relevant. As we already discussed, the dynamics in short-times is the same as the usual comb-model; therefore, subdiffusion is caused by geometrical restrictions. On the other hand, the exponential memory kernel produces a dynamics similar to a random walk with a high probability of returning to the origin. Since the backbone is at the origin, walkers along the branches tend to stay confined near the backbone because of memory effects, which in turn produces Brownian diffusion on the backbone. The memory effects along the branches thus dominate longer time scales. This interpretation is somehow in agreement with the results on anisotropic diffusion of entangled biofilaments reported in Ref.~\\cite{Tsang2017}, where the authors have written: {\\it ``The physical reason is that linear macromolecules become transiently localized in directions transverse to their backbone but diffuse with relative ease parallel to it.''} In particular, they obtained an empirical MSD $\\sim t^{0.2}$ for the transversal direction and a MSD $\\sim t^{0.9}$ for the parallel direction of such macromolecules. Liang Hong and co-workers also reported a gradual crossover from subdiffusion to Brownian diffusion on the mobility of water molecules on protein surfaces~\\cite{HongPRL2018}. They further argued that a broad distribution of trapping times causes the subdiffusion; however, water molecules start jumping to the empty sites as the trappings become occupied, resulting in the Brownian diffusion.\n\nWe can further investigate the effects of exponential memory kernels simultaneously acting on the backbone and branches, that is, $\\bar{{\\cal{K}}}_{\\alpha_{x}}(s)\\propto 1\/(s+\\alpha'_{x})$ and $\\bar{{\\cal{K}}}_{\\alpha_{y}}(s)\\propto 1\/(s+ \\alpha'_{y})$. These memory kernels are related to the Green functions given by Eqs.~(\\ref{CombModelResetting}) and (\\ref{CombModelResetting_Stationary}), where $\\alpha'_{x}=\\alpha'_{y}=\\alpha$ {is also a condition ensuring the non-negativity of the corresponding solution}. If our interpretation of the exponential memory kernel is valid, we expected stationary states to emerge even in the backbone dynamics. This hypothesis is corroborated by Figure~\\ref{fig:2} and the results for MSD \n\\begin{eqnarray}\n\\sigma_{y}^{2}(t) &=&  2\\mathcal{D}_{y}\\,\\mathcal{L}^{-1}\\left\\{\\frac{s^{-1}}{s+\\alpha}\\right\\}=2\\frac{D_{y}}{\\alpha}\\left(1-e^{-\\alpha t}\\right) \\;\\;\\; \\text{and} \\\\ \n\\sigma_{x}^{2}(t) &=&  \\frac{D_{x}}{\\sqrt{D_{y}}}\\mathcal{L}^{-1}\\left\\{\\frac{\\sqrt{s+\\alpha}}{s(s+\\alpha)}\\right\\}=\\frac{D_{x}}{\\sqrt{D_{y}}}\\frac{1}{\\sqrt{\\alpha}}\\mathrm{erf}\\left(\\sqrt{\\alpha t}\\right),\n\\end{eqnarray}\nwhere $\\mathrm{erf}(x)$ is the error function. These results show that the behavior on both directions reaches a stationary state for long times, that is, $\\sigma_{y}^{2}(t)$ and $\\sigma_{x}^{2}(t)$ approach a constant plateau when $t \\rightarrow \\infty$. Figure~\\ref{fig:3} shows the behavior of $\\sigma_{x}^{2}(t)$ for different time scales (dashed red line). Once again, a crossover characterizes the backbone dynamics and the system evolves from subdiffusion to confined diffusion (stationary state). The quenched mechanism dominates at short-time scales and the annealed mechanism predominates in the long-run. It is noteworthy that confined diffusion on both $x$ and $y$ directions has been experimentally observed in the crowded environment of living cells such as in lateral diffusion of membrane receptors~\\cite{kusumi1993} and diffusion of protein aggregates in live {\\it E. coli} cells~\\cite{Coquel2013}. \n\n\\section{Generalized Fractal Structure of Backbones}\\label{sec:3}\n\nWe now focus on generalizing the quenched disorder mechanism of the comb-model (Eq.~\\ref{Comb1}) by changing its geometrical restrictions. Instead of multiplying the diffusion term in the $x$-direction by a single delta function $\\delta(y)$, we consider a multiplication by $\\sum_{l_{j}\\in\\mathcal{S}_{\\nu}}\\delta(y-l_{j})$ to obtain an infinite number of backbones, where the position of the backbones $l_{j}$ $(j=1,2,\\dots)$ belong to a fractal set $\\mathcal{S}_{\\nu}$ with fractal dimension $0<\\nu<1$. These geometrical restrictions characterize a fractal grid~\\cite{pre2015,jpa2016,sandev2017anomalous}, as illustrated in Figure~\\ref{fig4} for the one-third Cantor set ($\\nu \\simeq 0.631$) at the third step of the iteration. Under these conditions, the diffusion equation for the fractal comb-model is\n\\begin{eqnarray}\n\\label{Comb1 grid}\n\\frac{\\partial}{\\partial t}\\rho(x,y;t)= \n{\\cal{F}}_{t,y}\\left({\\cal{D}}_{y} \\frac{\\partial^2}{\\partial y^2} \\rho(x,y;t)\\right) +\n\\sum_{l_{j}\\in\\mathcal{S}_{\\nu}}\\delta(y-l_{j}){\\cal{F}}_{t,x}\\left({\\cal{D}}_{x} \\frac{\\partial^2}{\\partial x^2} \\rho(x,y;t)\\right).\n\\end{eqnarray}\n\nWe proceed by defining the generalized diffusion equation governing the dynamics on the backbones. By applying the Laplace transform in Eq.~(\\ref{Comb1 grid}), we have\n\\begin{eqnarray}\\label{comb laplace2}\n\\!\\!\\!\\! s\\bar{\\rho}(x,y;s)-\\varphi(x,y) = s{\\cal{D}}_{y}\\bar{\\mathcal{K}}_{y}(s)\\frac{\\partial^2}{\\partial y^2} \\bar{\\rho}(x,y;s)+\\sum_{l_{j}\\in\\mathcal{S}_{\\nu}}\\delta(y-l_{j})s{\\cal{D}}_{x}\\bar{\\mathcal{K}}_{x}(s)\\frac{\\partial^2}{\\partial x^2} \\bar{\\rho}(x,y;s).\n\\end{eqnarray}\nBy following the procedures of Ref.~\\cite{jpa2016} and considering a suitable initial condition, we can represent the probability distribution function as\n\\begin{eqnarray}\\label{pdf grid}\n\\bar{\\rho}(x,y;s)=\\bar{f}(x,s)\\exp\\left(-\\frac{|y|}{\\sqrt{\\mathcal{D}_{y}\\bar{\\mathcal{K}}_{y}(s)}}\\right).\n\\end{eqnarray}\nNote that the exponential term carries information about the diffusion in the branches. From Eq.~(\\ref{pdf grid}), we find a relation for the probability distribution of the $x$-direction in the Laplace domain, that is,\n\\begin{eqnarray}\n\\label{Green grid}\n\\bar{\\mathcal{G}}_{1}(x;s)=2\\sqrt{\\mathcal{D}_{y}\\bar{\\mathcal{K}}_{y}(s)}\\bar{f}(x,s).\n\\end{eqnarray}\nOn the other hand, we also have that $\\bar{\\mathcal{G}}_{1}(x,t)=\\int_{-\\infty}^{\\infty}dy\\,\\bar{\\rho}(x,y,t)$ and by integrating Eq.~(\\ref{comb laplace2}), we find\n\\begin{eqnarray}\\label{comb laplace2 backbone}\ns\\bar{\\mathcal{G}}_{1}(x;s)-\\Phi(x) = s{\\cal{D}}_{x}\\bar{\\mathcal{K}}_{x}(s)\\frac{\\partial^2}{\\partial x^2} \\sum_{l_{j}\\in\\mathcal{S}_{\\nu}}\\bar{\\rho}(x,y=l_{j};s) \\;,\n\\end{eqnarray}\nwhere $\\Phi(x)=\\int_{-\\infty}^{\\infty}dy\\,\\varphi(x,y)$, and $\\bar{\\rho}(x,y=l_{j};s)$ represents the probability distribution function for the backbone in the position $l_{j}$. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{fig4}\n\\caption{Example of a fractal comb structure. The one-third Cantor set (at the third step of construction) provides the rule to locate the backbones perpendicularly to the $y$-axis. The location of the branches is distributed continuously along the backbones, that is, walkers access the branches through any position on the backbones. This spatial configuration characterizes a fractal grid.}\n\\label{fig4}\n\\end{figure} \n\nThe summation $\\sum_{l_{j}\\in\\mathcal{S}_{\\nu}}\\bar{\\rho}(x,y=l_{j};s)$ can be formally replaced by integration to fractal measure $\\mu_{\\nu}\\sim l^{\\nu}$, where $\\sum_{l_j\\in\\mathcal{S}_{\\nu}}\\delta(l-l_{j})\\rightarrow\\frac{1}{\\Gamma(\\nu)}l^{\\nu-1}$ is the fractal density, that is, $d\\mu_{\\nu}=\\frac{1}{\\Gamma(\\nu)}l^{\\nu-1}\\,dl$ \\cite{tarasov}. Therefore, from Eqs.~(\\ref{pdf grid}) and (\\ref{Green grid}), we obtain  \n\\begin{eqnarray}\\label{fractal summation}\n\\sum_{l_{j}\\in\\mathcal{S}_{\\nu}}\\bar{\\rho}(x,y=l_{j};s)&=&\\sum_{l_{j}\\in\\mathcal{S}_{\\nu}}\\bar{f}(x,s)\\exp\\left(-\\frac{|l_{j}|}{\\sqrt{\\mathcal{D}_{y}\\bar{\\mathcal{K}}_{y}(s)}}\\right)\\nonumber\\\\\n&=&\\bar{f}(x,s)\\frac{1}{\\Gamma(\\nu)}\\int_{0}^{\\infty}dl\\,l^{\\nu-1}e^{-\\frac{l}{\\sqrt{\\mathcal{D}_{y}\\bar{\\mathcal{K}}_{y}(s)}}}\\nonumber\\\\\n&=&\\bar{f}(x,s)\\left[\\mathcal{D}_{y}\\bar{\\mathcal{K}}_{y}(s)\\right]^{\\nu\/2}=\\frac{1}{2\\mathcal{D}_{y}^{\\frac{1-\\nu}{2}}}\\left[\\bar{\\mathcal{K}}_{y}(s)\\right]^{-\\frac{1-\\nu}{2}}\\bar{\\mathcal{G}}_{1}(x;s).\n\\end{eqnarray}\nWe observe in Eq.~(\\ref{fractal summation}) that the power-law exponents of the diffusion coefficient and memory kernel of $y$-direction contains all information about the fractal structure of backbones. Hence, by using the result of Eq.~(\\ref{fractal summation}) in Eq.~(\\ref{comb laplace2 backbone}), we find\n\\begin{eqnarray}\\label{comb laplace2 backbone2}\ns\\bar{\\mathcal{G}}_{1}(x;s)-\\Phi(x) = \\frac{\\mathcal{D}_{x}}{2\\mathcal{D}_{y}^{\\frac{1-\\nu}{2}}}s\\frac{\\bar{\\mathcal{K}}_{x}(s)}{\\left[\\bar{\\mathcal{K}}_{y}(s)\\right]^{\\frac{1-\\nu}{2}}}\\frac{\\partial^2}{\\partial x^2}\\bar{\\mathcal{G}}_{1}(x;s).\n\\end{eqnarray}\n\nThe inverse Laplace transform of Eq.~(\\ref{comb laplace2 backbone2}) yields the generalized diffusion equation\n\\begin{eqnarray}\\label{comb2 backbone2}\n\\frac{\\partial}{\\partial t}\\mathcal{G}_{1}(x;t) = \\left(\\frac{\\mathcal{D}_{x}}{2\\mathcal{D}_{y}^{\\frac{1-\\nu}{2}}}\\right)\\frac{\\partial}{\\partial t}\\int_{0}^{t}dt'\\,\\eta(t-t')\\frac{\\partial^2}{\\partial x^2}\\mathcal{G}_{1}(x;t'),\n\\end{eqnarray}\nwhere the memory kernel $\\eta(t)$ depends on both annealed and quenched disorder mechanisms, as given by\n\\begin{eqnarray}\n\\eta(t)=\\mathcal{L}^{-1}\\left\\{\\frac{\\bar{\\mathcal{K}}_{x}(s)}{\\left[\\bar{\\mathcal{K}}_{y}(s)\\right]^{\\frac{1-\\nu}{2}}}\\right\\}.\n\\end{eqnarray}\nBy using Eq.~(\\ref{comb2 backbone2}), we obtain the general expression for the MSD of a comb structure related to the fractal dimension $0<\\nu<1$ in the Laplace domain, that is, \n\\begin{eqnarray}\\label{msd general fractal mesh}\n\\sigma_{x}^{2}(t)=2\\left(\\frac{\\mathcal{D}_{x}}{2\\mathcal{D}_{y}^{\\frac{1-\\nu}{2}}}\\right)\\mathcal{L}^{-1}\\left\\{\\frac{\\bar{\\mathcal{K}}_{x}(s)}{s[\\bar{\\mathcal{K}}_{y}(s)]^{\\frac{1-\\nu}{2}}}\\right\\}.\n\\end{eqnarray}\nWe note again that probability distributions along the backbones structure should be non-negative; therefore: $1\/\\left[s\\bar{\\eta}(s)\\right]=\\left[\\bar{\\mathcal{K}}_{y}(s)\\right]^{\\frac{1-\\nu}{2}}\/\\left[s\\bar{\\mathcal{K}}_{x}(s)\\right]$ should be completely monotone function, and $1\/\\bar{\\eta}(s)=\\left[\\bar{\\mathcal{K}}_{y}(s)\\right]^{\\frac{1-\\nu}{2}}\/\\bar{\\mathcal{K}}_{x}(s)$ should be a Bernstein function.\n\nTo investigate the isolated effects of geometrical restrictions of the fractal grid, we consider the case $\\bar{\\mathcal{K}}_{x}(t)=\\bar{\\mathcal{K}}_{y}(t)=1$ (that is,  $\\bar{\\mathcal{K}}_{x}(s)=\\bar{\\mathcal{K}}_{y}(s)=1\/s$) where the MSD recovers the result of Ref.~\\cite{pre2015}, that is,\n\\begin{eqnarray}\n\\label{MSD2015}\n\\sigma_{x}^{2}(t)=2\\left(\\frac{\\mathcal{D}_{x}}{2\\mathcal{D}_{y}^{\\frac{1-\\nu}{2}}}\\right)\\mathcal{L}^{-1}\\left\\{\\frac{s^{-1}}{s^{1-\\frac{1-\\nu}{2}}}\\right\\}=2\\frac{\\mathcal{D}_{x}}{2\\mathcal{D}_{y}^{\\frac{1-\\nu}{2}}}\\frac{t^{\\frac{1+\\nu}{2}}}{\\Gamma\\left(1+\\frac{1+\\nu}{2}\\right)}.\n\\end{eqnarray}\nThe power-law behavior ($\\sigma_{x}^{2}(t)\\simeq t^{\\frac{1+\\nu}{2}}$) indicates a scale invariance, and because the exponent depends on $\\nu$, we infer that the fractal grid affects the spreading dynamics in all time scales. The overall effect is a subdiffusive dynamics with $\\frac{1}{2}<\\frac{1+\\nu}{2}<1$; however, the subdiffusion in a fractal grid is faster than the usual comb-model ($\\sigma_{x}^{2}(t)\\simeq t^{1\/2}$~\\cite{ArBa}, corresponding to $\\nu=0$ in our case). This behavior occurs because the set of backbones increases the possibilities of diffusion in the $x$-direction. For example, a fractal grid with $\\nu \\simeq 0.631$ (the one-third Cantor set) implies in $\\sigma_{x}^{2}(t)\\simeq t^{0.816}$. Moreover, Eq.~(\\ref{MSD2015}) connects the anomalous diffusion exponent with the fractal dimension of the backbone structure, a result that was experimentally observed in diffusion in porous and structurally inhomogeneous media~\\cite{zhokh}. \n\nOne example of the interplay between this generalized geometrical restriction and the effect of memory kernels is given by $\\bar{\\mathcal{K}}_{x}(s)=\\bar{\\mathcal{K}}_{y}(s)=1\/s^{\\alpha}$. This choice yields a fractional diffusion equation for a fractal grid given by\n\\begin{eqnarray}\n\\label{Comb_fract alpha}\n\\frac{\\partial}{\\partial t}\\rho(x,y;t) = {_{RL}}D_{t}^{1-\\alpha}\\left[\n{\\cal{D}}_{y}\\frac{\\partial^2}{\\partial y^2} \\rho(x,y;t) +\n{\\cal{D}}_{x}\\sum_{l_{j}\\in\\mathcal{S}_{\\nu}}\\delta(y-l_{j})\\frac{\\partial^2}{\\partial x^2} \\rho(x,y;t)\\right]\\,,\n\\end{eqnarray}\nand whose MSD is\n\\begin{eqnarray}\n\\sigma_{x}^{2}(t)=2\\left(\\frac{\\mathcal{D}_{x}}{2\\mathcal{D}_{y}^{\\frac{1-\\nu}{2}}}\\right)\\mathcal{L}^{-1}\\left\\{s^{-\\frac{\\alpha+\\nu}{2}-1}\\right\\}=2\\left(\\frac{\\mathcal{D}_{x}}{2\\mathcal{D}_{y}^{\\frac{1-\\nu}{2}}}\\right)\\frac{t^{\\frac{\\alpha+\\nu}{2}}}{\\Gamma\\left(\\frac{\\alpha+\\nu}{2}\\right)}.\n\\end{eqnarray}\nThe power-law exponent $\\frac{\\alpha+\\nu}{2}$ is associated with the two anomalous diffusion mechanisms: memory effects (related to $\\alpha$) and the fractal structure restriction (given by $\\nu$). The interplay between these mechanisms produces subdiffusive regimes between the limit cases of restricted and Brownian diffusion, that is, $0<\\frac{\\alpha+\\nu}{2}<1$. The case of a single backbone ($\\nu=0$) yields $\\sigma_{x}^{2}(t)\\simeq t^{\\alpha\/2}$, as reported in Ref.~\\cite{mmnp}. From the general formula of Eq.~(\\ref{msd general fractal mesh}), we further conclude that the MSD along the $x$-direction is stationary (case of localization) if  $\\bar{\\mathcal{K}}_{x}(s)\\propto\\left[\\bar{\\mathcal{K}}_{y}(s)\\right]^{\\frac{1-\\nu}{2}}$, and that normal diffusion along the $x$-direction occurs for $\\bar{\\mathcal{K}}_{x}(s)\\propto s^{-1}\\left[\\bar{\\mathcal{K}}_{y}(s)\\right]^{\\frac{1-\\nu}{2}}$. A remarkable feature of these conditions is that both memory kernels must have the same effects of geometrical restrictions.\n\n\\section{Discussion and Conclusions}~\\label{sec:4}\nOver this work, we showed that our generalized comb-models account for annealed and quenched disorder mechanisms. We believe these comb-models with fractional time-derivative operators are a reasonable abstraction for systems where the interplay between temporal and spatial disorders is present. For these hybrid models, we considered different memory kernels for the backbone and branch structures, and a fractal generalization of the geometrical restrictions. We obtained general solutions for the diffusion propagator and the MSD in terms of these memory kernels. With these solutions, we discussed particular cases based on the temporal evolution of the MSD and inferred time scales associated with each disorder mechanism. These results thus provide theoretical knowledge about the importance of interactions between geometrical restrictions and memory effects on anomalous diffusion.  \n\nWe argued that the behaviors obtained from our models are consistent with other theoretical and experimental results. In its usual form, the comb-model is a subdiffusive model with $\\sigma_{x}^{2}\\propto t^{1\/2}$. However, depending on the memory kernels and number of backbones (single or fractal set), our generalized comb-models also describe restricted diffusion, Brownian diffusion, and display a crossover from subdiffusion to these situations. \n\nFor power-law memory kernels, the MSD is scale-invariant and the diffusive regime depends on the values of memory exponents $0<\\alpha_{x}<1$ and $0<\\alpha_{y}<1$ with $\\alpha_{x}>\\alpha_{y}\/2$. Scale invariance is also a feature of the fractal grid structure of backbones. In this case, there is an enhancement of the diffusion (when compared with the usual comb-model) and the anomalous exponent depends on the fractal dimension ($\\frac{1}{2}<\\frac{1+\\nu}{2}<1$). By including power-law memory effects on the fractal grid, we obtained an anomalous exponent $0<\\frac{\\alpha+\\nu}{2}<1$ with $\\alpha= \\alpha_{x}=\\alpha_{y}$. Overall, these results from our generalized comb-models are consistent with simulations of fractal Brownian motion~\\cite{Ribeiro2014} and a biased CTRW~\\cite{Mendez2015} on comb-like structures. \n\nWhen the memory kernel acting on the branches is exponential, we showed that the behavior of the MSD in the $y$-direction is similar to those reported in diffusion with explicit confinement (external forces or limited boundaries) or having a bias to return to the origin. The spread in the branches thus follows Brownian diffusion for short-time scales and becomes confined near to the backbone for long times. Because of this high probability of returning to the backbone, the long-time behavior in $x$-direction only depends on its memory kernel. The overall effect of an exponential kernel is a crossover between diffusive regimes. This crossover occurs from subdiffusion to Brownian diffusion when walkers have no memory in the backbone. Thus, we observed that the interplay between two subdiffusion mechanisms (geometrical restrictions and memory effects) may lead to the usual Brownian motion. This crossover and its physical explanation also appear consistent with experimental results of anisotropic diffusion of entangled biofilaments~\\cite{Tsang2017} and the mobility of water molecules on protein surfaces~\\cite{HongPRL2018}. On the other hand, when an exponential memory kernel acts on the backbone, we obtained a crossover from subdiffusion to confined diffusion. This result also appears consistent with the MSD observed in lateral diffusion of membrane receptors~\\cite{kusumi1993} and diffusion of protein aggregates in live {\\it E. coli} cells \\cite{Coquel2013}. Moreover, we found that the confinement effects of exponential memory kernels are independent of the parameters $\\alpha'_{y}$ and $\\alpha'_{x}$. \n\nIn spite of its simplicity, our generalized comb-model may have an important role in the statistical mechanics of disordered media. This model can be used as a simple explanation for unusual transport properties caused by quenched disorder or to annealed disorder. It also has the advantage of providing exact solutions related to sub- and superdiffusive behaviors. Our comb-models are also relevant for investigating the interplay between temporal and spatial disorder mechanisms and for describing crossovers from subdiffusion to confined or Brownian diffusion. \n\n\n\\section*{Acknowledgments}\n\nE.K.L. thanks partial financial support of the CNPq under Grant No. 302983\/2018-0. EKL also thanks the National Institutes of Science and Technology of Complex Systems -- INCT-SC for partially supporting this work. HVR thanks the financial support of the CNPq under Grants 407690\/2018-2 and 303121\/2018-1. TS and IP acknowledge the support by the bilateral research project MK 07\/2018, WTZ Mazedonien S$\\&$T Macedonia 2018--20 funded under the inter-governmental Macedonian-Austrian agreement. TS was supported by the Alexander von Humboldt Foundation.\n\n\\linespread{1.01}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA recurrent theme in classical projective geometry is the study of\nspecial subvarieties of some given family of varieties, e.g. how\nmany isolated singular points can a surface of degree $d$ in\n$\\mathbb{P}^3$ have? when is it true that the members of a certain family\nof varieties contain rational curves? contain a linear space of\nsome positive dimension?  Other examples of similar questions can be\neasily provided by the reader.\n\nThe study of the special case of complete intersection\nsubvarieties of hypersurfaces in $\\mathbb{P}^n$ has been the subject of a great deal of research. It\nwas known to Severi \\cite{Severi} that for $n \\geq 4$ the only\ncomplete intersections, of codimension one, on a general hypersurface are obtained by\nintersecting that hypersurface with another.\n\nThis observation was extended to $\\mathbb{P}^3$ by Noether (and\nLefschetz)  \\cite{Lefschetz, GH85} for general hypersurfaces of\ndegree $\\geq 4$.  These ideas were further generalized by\nGrothendieck \\cite{SGA2}.\n\nIn \\cite{CaChGe}, we proposed a new approach to the problem of\nstudying complete intersection subvarieties of hypersurfaces. This\napproach used a mix of projective geometry and commutative algebra\nand is more elementary and direct than, for example, the\napproach of Grothendieck.  With our approach we were able to give a complete\ndescription of the situation for complete intersections of\ncodimension $r$ in $\\mathbb{P}^n$ which lie on a general hypersurface of\ndegree $d$ whenever $2r \\leq n+2$. The main result of\n\\cite{CaChGe} is the following:\n\n\\begin{thm}\\label{nostro}\nLet $X\\subset\\mathbb{P}^n$ be a generic degree $d$ hypersurface, with\n$n,d>1$. Then $X$ contains a complete intersection of type\n$(a_1,\\ldots,a_r)$, with $2r\\leq n+2$, and the $a_i$ all less than\n$d$, in the following (and only in the following) instances:\n\\medskip\n\n\\begin{itemize}\n\\item $n=2$: then $r=2$, $d$ arbitrary and $a_1$ and $a_2$ can\nassume any value less than $d$;\n\n\\item $n=3$, $r=2 $:  for $d \\leq 3$ we have that $a_1$ and $a_2$\ncan assume any value less than $d$;\n\n\\item $n=4$, $r = 3$: for $d \\leq 5$ we have that $a_1, a_2$ and\n$a_3$ can assume any value less than $d$;\n\n\\item $n=6,r = 4$ or $n=8,r = 5$: for $d \\leq 3$ we have that\n$a_1, \\ldots ,a_r$ can assume any value less than $d$;\n\n\\item $n=5,7$ or $n>8$, $2r=n+1$ or $2r = n+2$:  we have only\nlinear spaces on quadrics, i.e. $d = 2$ and $a_1 = \\ldots =a_r =\n1$.\n\n\n\\end{itemize}\n\n\\end{thm}\n\nIn this paper, we are interested in the first case not covered by\nTheorem \\ref{nostro}. Namely, the case  $n=3,r=3$, i.e.\ncomplete intersection points on surfaces of $\\mathbb{P}^3$. Although this\na very natural question, we are not aware of any reference to the\nsubject in the literature. Using the methods of \\cite{CaChGe} we prove the\nfollowing:\n\n\\begin{thm}\\label{mainTHM} For non-negative integers $a,b,c,d$, such that $a\\leq b\\leq c<\nd$ we have the following:\n\n\\begin{itemize}\n\n\\item if $a\\leq 4$, then the generic degree $d$ surface of $\\mathbb{P}^3$ contains a\n$CI(a,b,c)$;\n\n\\item if $a=5,b\\leq 11$, then the generic degree $d$\nsurface of $\\mathbb{P}^3$ contains a $CI(5,b,c)$; if $a=5,b=12$ and\n$c=12$ then the generic degree $d$ surface of $\\mathbb{P}^3$ contains a\n$CI(5,12,12)$; if $a=5,b=12$ and $c\\geq 13$ then the generic\ndegree $d\\geq 2c+15$ surface does not contain a $CI(5,12,c)$; if\n$a=5$ and $b\\geq 13$, then the generic degree $d\\geq b+c+2$\nsurface does not contain a $CI(5,b,c)$;\n\n\\item if $a=6,b\\leq 7$, then the generic degree $d$\nsurface of $\\mathbb{P}^3$ contains a $CI(6,b,c)$; if $a=6,b=8$ and\n$c=8,9$ then the generic degree $d$ surface of $\\mathbb{P}^3$ contains a\n$CI(6,8,c)$; if $a=6,\\ b=8$ and $c\\geq 10$ then the generic degree\n$d\\geq 2c+12$ surface does not contain a $CI(6,8,c)$; if $a=6$ and\n$b\\geq 9$, then the generic degree $d\\geq b+c+3$ surface does not\ncontain a $CI(6,b,c)$;\n\n\\item if $a\\geq 7$, then the generic degree $d\\geq a+b+c-3$ surface of $\\mathbb{P}^3$\ndoes not contain a $CI(a,b,c)$.\n\\end{itemize}\n\\end{thm}\n\nNotice that Theorem \\ref{mainTHM} gives a complete asymptotic solution to the existence problem for $CI(a,b,c)$ on a general surface of degree $d$ in $\\mathbb{P}^3$.  More precisely,\n\n\\begin{cor} Let $a \\leq b \\leq c <d$ be integers.  Then for $d\\gg 0$ the generic degree $d$ surface contains a $CI(a,b,c)$ if:\n\\begin{itemize}\n  \\item $a \\leq 4$;\n  \\item $a = 5,\\ b\\leq 11$;\n  \\item $a = 5,\\ b=12,\\ c=12$;\n  \\item $a=6,\\ b\\leq 7$;\n  \\item $a=6,\\ b=8,\\ c=8,9$.\n \\end{itemize}\nand does not contain a $CI(a,b,c)$ in all other cases.\n\\end{cor}\n\n\\begin{rem}\nWe also notice that the kind of asymptotic problem we solved above can only be considered for points.  More precisely, if we choose a family $\\mathcal{F}$ of subschemes of $\\mathbb{P}^n$ we can ask the following: is it true that for $d\\gg 0$ the generic degree $d$ hypersurface of $\\mathbb{P}^n$ contains an object of the family $\\mathcal{F}$?\n\nUsing a standard incidence correspondence argument, it is easy to see that a positive answer can be given only if\n$$\n\\dim {\\mathcal{F}} + 1 - h_{\\mathcal{F}}(d) \\geq 0,\n$$\nwhere $h_{\\mathcal{F}}(d)$ is the Hilbert polynomial of the objects in $\\mathcal{F}$.  Clearly this can be the case only if $h_{\\mathcal{F}}(d)$ is bounded and hence constant.  This implies that $\\mathcal{F}$ is a family parameterizing $0$-dimensional schemes.\n\\end{rem}\n\nThe paper is structured as follows: in Section \\ref{questionSEC},\nwe formalize the question we want to study and we treat the first\nsimple instances; in Section \\ref{technicalFACTSsec}, we recall\nthe results we need from \\cite{CaChGe}; in Sections\n\\ref{aleq4SEC}, \\ref{ageq5nonexistSEC} and \\ref{ageq5existSEC} we\napply our method to produce the intermediate results necessary to\nprove Theorem \\ref{mainTHM}. Finally, in Section \\ref{finalSEC},\nwe prove Theorem \\ref{mainTHM} and we state a conjecture for the\nexpected behavior in the cases which still remain open.\n\nIn the proof of Theorem \\ref{ageq5THM} we used the\ncomputer algebra system CoCoA \\cite{cocoa} for which we thank the\ndevelopers of the software.\n\nThe first author wishes to thank Queen's University for its kind\nhospitality during the writing of this paper, the research group\nGNSAGA of INDAM and the special fund ``Fondo giovani ricercatori\"\nof the Politecnico di Torino for financial support.  The first and third author also wish to thank NSERC (Canada) for its financial support during the writing of this paper.\n\n\n\n\\section{The question}\\label{questionSEC}\nIn this paper we study complete intersection points in projective\nthree space. We say that $\\mathbb{X}\\subset\\mathbb{P}^3$ is a complete\nintersection 0-dimensional scheme if its ideal\n$I_\\mathbb{X}=(F,G,H)$ where the forms $F,G$ and $H$ are a regular\nsequence in the ring $R=\\mathbb{C}[x_0,\\ldots,x_3]$. Moreover, if\n$\\deg F=a,\\deg G=b$ and $\\deg H=c$ we say that $\\mathbb{X}$ is a\n{\\it complete intersection of type} $(a,b,c)$. We will always\nassume $a\\leq b\\leq  c$ and we will write $CI(a,b,c)$\nto describe a complete intersection of type $(a,b,c)$.\n\nOur basic question is: {\\em for which integers $a,b,c$ and $d$\ndoes the general degree $d$ surface of $\\mathbb{P}^3$ contain a $CI(a,b,c)$}?\n\nThere are cases where the answer is straightforward. If $d=c$, the\nanswer is clearly affirmative as we are cutting a complete\nintersection curve of type $(a,b)$ with a surface of degree $d$\n(similarly for $d=a$ or $d=b$). If $d<a$, the answer is negative\nas no form of degree less than $a$ belongs to the ideal of a\n$CI(a,b,c)$, and similarly for $a<d<b$ as a generic form is\nirreducible. If $b<d<c$, then we are really looking for a complete\nintersection of type $(a,b)$ on the generic degree $d$ surface,\nand this is dealt with in Theorem \\ref{nostro}. Hence, it is\nenough to focus on the following refinement of our question: {\\em\nfor which integers $a,b,c$ and $d$, $a\\leq b\\leq c<d$, does the\ngeneral degree $d$ surface of $\\mathbb{P}^3$ contain a $CI(a,b,c)$}?\n\n\n\n\n\n\\section{Technical facts}\\label{technicalFACTSsec}\n\nWe will treat this question using the method introduced in\n\\cite{CaChGe}. Our method proceeds as follows: translate the problem of finding a $CI(a,b,c)$ on a general surface of degree $d$, say $M = 0$, as the problem of writing $M$ as\n$$\nM = F F^\\prime + G G^\\prime + H H^\\prime\n$$\nwhere $F,\\ G,\\ H$ and $F^\\prime,\\ G^\\prime, \\ H^\\prime$ are forms of degree $a,\\ b,\\ c$ and $d-a,\\ d-b,\\ d-c$ respectively.  As $M$ is generic, this decomposition problem is actually a problem about joins of varieties of splitting forms.  Then we use Terracini's lemma to translate the computation of the dimension of the join, into a Hilbert function computation.  Namely, as first observed in \\cite{Mamma}, the tangent space to the variety of splitting forms at the point $[FF^\\prime]$ corresponds to the degree $d$ homogeneous piece of the ideal $ (F, F^\\prime )$.  Thus, the tangent space at $M$ to the join corresponds to the degree $d$ homogeneous piece of the ideal spanned by $F,F^\\prime, G, G^\\prime, H, H^\\prime$.  For more details we refer the reader to \\cite{CaChGe}.\n\nIn particular we will need the following (see \\cite[Lemma 4.3]{CaChGe}):\n\n\\begin{lem}\\label{equivalence}\nFor given integers $a,b,c$ and $d$, such that $a\\leq b\\leq c<\nd$, the following are equivalent facts:\n\\begin{enumerate}\n\\item  The general degree $d$ surface of $\\mathbb{P}^3$ contains a\n$CI(a,b,c)$; \\item\\label{algebraicQUESTION} For a generic choice\nof forms $F,G,H,H',G',F'\\in R$ of degrees $a,b,c,d-c,d-b,d-a$ one\nhas that\n    \\[H\\left({R\\over (F,G,H,H',G',F')},d\\right)=0\\]\nwhere $H(\\cdot,d)$ denotes the Hilbert function in degree $d$.\n\\end{enumerate}\n\\end{lem}\n\nUsing Lemma \\ref{equivalence} we translate our geometric question\ninto a purely algebraic one. In particular, we can take advantage of\nresults about the Lefschetz property \\cite{Stanley,Anick} to deal\nwith our question.\n\nAs $F,G,H$ and $H'$ are a\nregular sequence in $R$ we have a good understanding of the ring\n\\[W={R\\over (F,G,H,H')}\\]\nand we will use this to study the Hilbert function of the ring\n\\[{R\\over(F,G,H,H',G',F')}\\simeq {W\\over ([F'],[G'])},\\]\nwhere $[\\cdot]$ denotes the class in $W$.\n\nVia the Koszul complex we compute the minimal free resolution of $W$:\n\\begin{equation}\\label{Wres}\n0\\leftarrow W\\leftarrow M_0 \\leftarrow M_1\\leftarrow M_2\\leftarrow\nM_3\\leftarrow M_4\\leftarrow 0\n\\end{equation}\nwhere\n\\[M_0=R,\\]\n\\[M_1=R(-a)\\oplus R(-b)\\oplus R(-c)\\oplus R(-d+c),\\]\n\\[M_2=R(-a-b)\\oplus R(-a-c)\\oplus R(-a-d+c)\\oplus R(-b-c)\\oplus R(-b-d+c)\\oplus R(-d),\\]\n\\[M_3=R(-a-b-c)\\oplus R(-a-b-d+c)\\oplus R(-a-d)\\oplus R(-b-d)\\]\n\\[M_4=R(-a-b-d).\\]\n\nWe also notice that (see  \\cite[Lemma 4.1 and Remark 4.2]{CaChGe}):\n\n\\begin{lem}\\label{switchLEM}\nThe following are equivalent:\n\\begin{itemize}\n\\item for integers $a\\leq b\\leq c<d$ a $CI(a,b,c)$ exists on the generic degree $d$ surface of $\\mathbb{P}^3$;\n\\item for integers $a'\\leq b'\\leq c'\\leq {d}$ a $CI(a',b',c')$ exists on the generic degree $d$ surface of $\\mathbb{P}^3$, where $a=a'$ or $a+a'=d$, and $b=b'$ or $b+b'=d$, and $c=c'$ or $c+c'=d$.\n\\end{itemize}\n\\end{lem}\n\n\\begin{rem}\\label{dover2LEM}\nUsing Lemma \\ref{switchLEM} we can study our question for integers $a\\leq b\\leq c<d\/2$ and produce a complete answer for the general case. In fact, either $a\\leq d\/2$ or $a'\\leq d\/2$.\n\\end{rem}\n\n\n\\section{The $a\\leq 4$ case}\\label{aleq4SEC}\nHere we use Stanley's result \\cite{Stanley} showing that the\nquotient of $R=\\mathbb{C}[x_0,\\ldots,x_3]$ by four generic forms\nhas the Strong Lefschetz Property. More precisely, given generic\nforms $F,G,H,F',G'\\in R$, of degrees $a,b,c,d-c,d-b$, we consider\n$W={R\/ (F,G,H,H')}$. Then the multiplication by the class of $G'$\nhas maximal rank. Hence the sequence\n\\[W(-d+b){\\rightarrow} W\\rightarrow {W\\over ( [G'])}\\rightarrow 0\\]\nproduces $H(W\/([G'])),d)=\\max\\{H(W,d)-H(W,b),0\\}$.\n\n\\begin{prop}\\label{asymptoticamenodi5}\nFor any choice of $a,b,c$ and $d$ positive integers such that\n$a\\leq 4\\leq b\\leq c$ and $d\\geq a+b+c-3$, the general degree $d$ surface in $\\mathbb{P}^3$\ncontains a $CI(a,b,c)$.\n\n\\end{prop}\n\\begin{proof} Using Lemma \\ref{dover2LEM} it is enough to consider the case when $a\\leq b\\leq c\\leq {d\\over 2}$. Using Proposition \\ref{equivalence} part \\eqref{algebraicQUESTION} we only have to show that\n\\[H(W\/([G']),d)=\\max\\{H(W,d)-H(W,b),0\\}=0.\\]\nBy the resolution of $W$ given in (\\ref{Wres}) we immediately get:\n\\begin{itemize}\n\\item if $b<c$, then\n\\begin{eqnarray*}\nH(W,b) & = & {b+3\\choose 3}-{b-a+3\\choose 3}-1 \\\\\n       & = & 1\/6 a^{3}-1\/2a^{2}b + 1\/2ab^{2} - a^{2} + 2ab + 11\/6a-1;\n\\end{eqnarray*}\n\\item if $b=c$, then\n\\begin{eqnarray*}\nH(W,b) & = & {b+3\\choose 3}-{b-a+3\\choose 3}-2 \\\\\n       & = & 1\/6 a^{3}-1\/2a^{2}b + 1\/2ab^{2} - a^{2} + 2ab + 11\/6a-2.\n\\end{eqnarray*}\n\\end{itemize}\n\nLed by the resolution of $W$, we also consider the following polynomial\n\n\\[h(W,d)=\\]\n\\[={d+3\\choose 3}-\\left[{d-a+3\\choose 3}+{d-b+3\\choose\n3}+{d-c+3\\choose 3}+\\right. \\left. {c+3\\choose 3}\\right]+\\]\n\\[+{d-a-b+3\\choose\n3}+{d-a-c+3\\choose 3}+{c-a+3\\choose 3}+{d-b-c+3\\choose 3}+\\]\n\n\\[+{c-b+3\\choose3}+1-\\left[{d-a-b-c+3\\choose 3}\\right. \\left. +{c-a-b+3\\choose 3}\\right],\\]\nwhere ${x \\choose 3}$ is the polynomial ${1\\over 6}x(x-1)(x-2)$. Making the computation we get\n\\[h(W,d)=1\/2 a^{2}b + 1\/2ab^{2}-2ab + 1.\\]\n\nNotice that, for given $a,b$ and $c$ such that $c-a-b\\geq -3$ and\n$d\\geq a+b+c-3$, the evaluation of $h(W,d)$ coincides with the\nHilbert function of $W$ in degree $d$, i.e. $h(W,d)=H(W,d)$.\nMoreover, the inequalities\n\\[a\\leq 4 \\mbox{ and } c-a-b\\leq -4\\]\nonly hold when $a=4$ and $b=c$ (recall that $a\\leq b\\leq c$) and in this case\n\\[H(W,d)=h(W,d)+{c-a-b+3\\choose 3}=h(W,d)-1.\\]\n\nFinally we compute $H(W,d)-H(W,b)$ distinguishing two cases.\n\nThe $a< 4$ or $b<c$ case. If $b<c$ we use the value of $H(W,b)$\nand the equality $H(W,d)=h(W,d)$  previously determined to  get\n\\[H(W,d)-H(W,b)=-1\/6 a^{3} + a^{2}b + a^{2}-4ab-11\/6a + 2.\\]\nThis polynomial is linear in $b$ and it does not involve $d$ and\nit is easy to see that for $a\\leq 4$\n\\[H(W,d)-H(W,b)\\leq 0.\\]\nWhen $a < 4$ and $b=c$ a completely analogous argument can be applied.\n\nThe $a=4$ and  $b=c$ case. Mutatis mutandis, we compute again and we get\n\\[H(W,d)-H(W,b)=-1\/6 a^{3} + a^{2}b + a^{2}-4ab-11\/6a + 2,\\]\nhence the same polynomial of the previous case and this finishes the proof.\n\\end{proof}\n\nProposition \\ref{asymptoticamenodi5} gives an asymptotic result\nyielding that, when one of the degree of the $CI$ is at most 4, then\nfor $d$ big enough a complete intersection of the given type\nexists on a generic surface of degree $d$. With a slightly more\ncareful analysis this can be improved and the condition on $d$ can\nbe dropped.\n\n\\begin{thm}\\label{aleq4THM}\nLet $a,b,c$ and $d$ be integers such that $a\\leq  b\\leq c<d$.\nIf $a\\leq 4$, then a $CI(a,b,c)$ exists on the generic degree $d$\nsurface in $\\mathbb{P}^3$.\n\\end{thm}\n\\begin{proof}\nUsing Proposition \\ref{asymptoticamenodi5} we have only to check\nvalues of $d$ in the range $c<d\\leq a+b+c-4$. The idea is to\nuse Lemma \\ref{switchLEM} to reduce the degree of the complete\nintersection not changing $d$ so that Proposition\n\\ref{asymptoticamenodi5} can be applied.\n\nFor $a=2$, we consider the existence of a $CI(2,b,c)$ on the\ngeneric degree $d$ surface for $c<d\\leq b+c-2$. Such a\ncomplete intersection exists if the same happen for a\n$CI(2,d-c,d-b)$. But, by Proposition \\ref{asymptoticamenodi5},\nthis is the case as soon as\n\\[d\\geq 2+(d-c)+(d-b)-3\\]\nand this equivalent to $d\\leq b+c-1$ which is actually the case. Similarly for $a=3$.\n\nThe case $a=4$ is treated in analogy with the previous ones,\nexcept for $d=b+c$. In this situation, applying Lemma\n\\ref{switchLEM}, we have to study $CI(4,b,b)$'s on a generic\nsurface of degree $d\\geq b$. Repeating the same argument above we\nhave to treat values of $d$ in the range $b\\leq d\\leq 2b$.\nProposition \\ref{existdbigPROP} gives the existence for all $d$,\nbut for $d=2b$. By Proposition $\\ref{equivalence}$ we have to\nconsider the coordinate ring $W$ of a complete intersection of\ntype $(b,b,b,b)$ and its Hilbert function $H(\\cdot)$. Using the\nfact that multiplication by one form  has maximal rank in $W$, we\nhave only to compare $H(2b)$ and $H(2b-4)$, but these values are\nthe same being $W$ a Gorenstein ring with socle degree $4b-4$, and\nthis finishes the proof.\n\\end{proof}\n\n\n\\section{The case $a> 4$: non-existence results}\\label{ageq5nonexistSEC}\n\nIn this section we will prove asymptotic non-existence results\nwhen $a> 4$. For non-negative integers $a,b,c$ and $d$, $a\\leq\nb\\leq c<d$, we consider generic forms $F,G,H,H',G',F'\\in R$ of\ndegrees $a,b,c,d-c,d-b$ and $d-a$. Consider the ring $W={R\/\n(F,G,H,H')}$ and notice that, by a straightforward dimensional\nargument, if\n\\[H(W,a)+H(W,b)-H(W,d)<0\\]\nthen\n\\[H\\left({W\\over ([G'],[F'])},d\\right)\\neq 0.\\]\nHence, by Lemma \\ref{equivalence} \\eqref{algebraicQUESTION},  if\n$H(W,d)-H(W,a)-H(W,b)<0$, then the generic degree $d$ surface of\n$\\mathbb{P}^3$ does not contain a\n$CI(a,b,c)$. Using this idea we prove the following:\n\n\\begin{thm}\\label{nonEXISTa5}\nLet $a\\leq b\\leq c$ and $d$  be non-negative integers such that\n\\[a=5 \\mbox{ and } b\\geq 13\\]\nor\n\\[a=6 \\mbox{ and } b\\geq 9\\]\nor\n\\[a \\geq 7.\\]\n\nThen, for $d\\geq a+b+c-3$ the generic degree $d$ surface of\n$\\mathbb{P}^3$ does not contain a\n$CI(a,b,c)$.\n\\end{thm}\n\nIn order to prove this theorem we need the following technical\nfact:\n\n\\begin{lem}\\label{inequalityLEMMA}\nLet $a,b,c$ be non-negative integers, such that $4<a$ .\nAssume that, for integers $c_0$ and $d$ such that\n\\[c_0\\geq b \\mbox{ and } d>a+b+c_0-4,\\]\none has the Hilbert function inequality\n\\[H(W,a)+H(W,b)-H(W,d)<0,\\]\nwhere $W$ is the ring\n\\[\nW={R\\over (F,G,H,H')}\n\\]\nand the forms $F,G,H$ and $H'$ are generic and have degrees\n$a,b,c_0$ and $d-c_0$.\n\nThen, if $A,\\ B,\\ C$ and $D$ are forms of degrees $a,\\ b,\\ c\\geq c_0$ and $d-c$ and $$W^\\prime = {R\\over (A,B,C,D)}$$ then the following inequality holds:\n\\[H(W^\\prime,a)+H(W^\\prime,b)-H(W^\\prime,d)<0,\\]\nfor $d>a+b+c-4$.\n\\end{lem}\n\\begin{proof}\nThe key observation is that\n\\[H(W,d)=H(W,a+b-4).\\]\nIn fact, being $W$ a Gorenstein ring,  its Hilbert function is symmetric and $H(W,x)=H(W,y)$ if $x+y=d+a+b-4$.\nThen, we compute\n\\[H(W,a)+H(W,b)-H(W,a+b-4)\\]\nusing the formulae in the proof of Proposition\n\\ref{asymptoticamenodi5}, for which we need the assumption on $d$.\nOne sees that the final expression does not involve neither $c$ or\n$d$ and the proof follows. For example, in the case $a<b$ one gets\n$H(W,a)+H(W,b)-H(W,a+b-4)<0$ if and only if\n\\[b\\geq \\frac{{1\\over 2}a^3-a^2+{11\\over 2}a -4}{(a^2-4a)}.\\]\n\\end{proof}\n\nWe can now prove Theorem \\ref{nonEXISTa5}.\n\n\\begin{proof}[Proof of Theorem \\ref{nonEXISTa5}]\nWe let $b=c$ and we show that for the required values of $a$ and\n$b$ we have the inequality $H(W,a)+H(W,b)-H(W,d)<0$. Then we apply\nLemma \\ref{inequalityLEMMA} to get the result when $c\\geq b$.\n\nAgain, we notice that $H(W,d)=H(W,a+b-4)$.\n\nWe divide the proof in two cases depending on whether $a=b$ or\n$a<b$.\n\n\\noindent{\\bf Case $a<b$.}\n\nUsing the resolution of the ring $W$, the\ninequality\n\\[H(W,a)+H(W,b)-H(W,a+b-4)\\geq 0\\]\nis readily seen to be equivalent to\n\\[b\\leq \\frac{{1\\over 2}a^3-a^2+{11\\over 2}a -4}{(a^2-4a)}.\\]\nRecalling that $a<b$ we get\n\\[H(W,a)+H(W,b)-H(W,d)\\geq 0\\]\nonly if\n\\[-{1\\over 2}a^3+3a^2+{11\\over 2}a-4\\geq 0\\]\nand this inequality holds if and only if\n\\[a=5 \\mbox{ or } a=6.\\]\nHence\n\\[H(W,a)+H(W,b)-H(W,d)\\geq 0\\]\nimplies\n\\[a=5,b\\leq 12\\]\nor\n\\[a=6,b\\leq 8.\\]\n\n{\\bf Case $a=b$.}\n\n Computing we get\n\\[H(W,a)+H(W,b)-H(W,d)=-{2\\over 3}a^3+4a^2+{11\\over 3}a-3\\geq 0\\]\nonly if $a< 7$ and this finishes the proof.\n\\end{proof}\n\nTo prove some more non-existence results, we need the following:\n\n\\begin{prop}\\label{nonexistEVENTUALLY}\n\nLet $a\\leq b\\leq c< {d}$ and $d> 2c+b+a-3$. If no $CI(a,b,c)$\nexists on the generic degree $d$ hypersurface, then it does not\nexist on the generic hypersurface of degree $d'>d$ either.\n\\end{prop}\n\\begin{proof}\nIt is enough to treat the case $d'=d+1$.  For generic forms $F,\\ G,\\ H$ of degrees $a,\\ b$ and $c$ let\n$$\nA = {\\mathbb{C}[x_0,\\cdots,x_3]\\over(F,G,H)}.\n$$\nBy hypothesis, for the\ngeneric choice of $F',G'$ and $H'$ of degrees $d-a,d-b$ and $d-c$\nin $A$, we know that the degree $d$ part of\n\\[{A\\over (F',G',H')}\\]\nis not zero. Now, consider elements $F'',G''$ and $H''$ of degrees\n$d+1-a,d+1-b$ and $d+1-c$. Notice that\n\n\\[d-a\\geq d-b\\geq d-c >c+b+a-3\\]\nand recall that $A_i\\simeq A_j$ as $\\mathbb{C}$ vector spaces if $i$ and $j$ are $>2c + b + a - 3$. Thus for a general linear form\n$L$ we have\n\\[F''=LF^*, G''=LG^*\\mbox{ and } H''=LH^*\\]\nand the forms $F^*,G^*$ and $H^*$ have degrees $d-a,d-b$ and\n$d-c$. Hence we have a isomorphism\n\\[(F'',G'',H'')_{d+1}\\simeq (F^*,G^*,H^*)_d\\]\nand this is enough to conclude that the degree $d+1$ part of\n\\[{A\\over (F'',G'',H'')}\\]\nis not zero and the result follows.\n\\end{proof}\n\n\n\\begin{lem}\\label{nonexistb8and12}\nIf $c\\geq 13$, then the generic degree $d\\geq 2c+15$ surface of\n$\\mathbb{P}^3$ does not contain a $CI(5,12,c)$.\n\nIf $c\\geq 10$, then the generic degree $d\\geq 2c+12$ surface of\n$\\mathbb{P}^3$ does not contain a $CI(6,8,c)$.\n\\end{lem}\n\n\\begin{proof}\nWe begin with the study of $CI(5,12,c)$. Let $c=13+x$,\n$d=2c+a+b-2=41+2x$ and consider the ring\n\\[\nW={R\\over (F,G,H,H')}\n\\]\nwhere the forms $F,G,H$ and $H'$ have degrees $5,12,13+x$ and\n$28+x$. The generic degree $d$ surface does not contain a\n$CI(5,12,c)$ if $H(W,5)+H(W,12)-H(W,d)<0$, where\n$H(W,d) = H(W,41+2x)=H(W,14)$. Now we compute\n\\[H(5)={8\\choose 3}-1=55,\\]\n\\[H(12)={15\\choose 3}-{10\\choose 3}-1=334,\\]\n\\[H(14)=\\left\\lbrace\n\\begin{array}{lr}\n{17\\choose 3}-{12\\choose 3}-{5\\choose 3}=450 & \\mbox{ if } x>1 \\\\ \\\\\n449 & \\mbox{ if } x=1 \\\\ \\\\\n446 & \\mbox{ if } x=0\n\\end{array}\n\\right. .\\] Hence, $H(W,5)+H(W,12)-H(W,d)<0$, and by Proposition\n\\ref{nonexistEVENTUALLY} we conclude that the generic degree $d'$\nsurface does not contain a $CI(5,12,c)$ for $d'\\geq\nd=41+2x=15+2c$.\n\nThe case of $CI(6,8,c)$ is solved by completely analogous computations.\n\\end{proof}\n\n\\section{The case $a> 4$: existence results}\\label{ageq5existSEC}\n\nTheorem \\ref{nonEXISTa5} does not cover small values of $a$ and $b$.\nIn this Section we derive a result analogous to Theorem\n\\ref{aleq4THM} in these cases.\n\nWe begin with proving two technical facts.\n\n\\begin{prop}\\label{existdbigPROP}\nLet $a\\leq b\\leq c\\leq {d}$ and $d\\geq a+b+c-3$. If a $CI(a,b,c)$\nexists on the generic degree $d$ surface, then it also exists on\nthe generic surface of degree $d'>d$.\n\\end{prop}\n\\begin{proof}\nLet $d'=d+1$ and notice that it is enough to treat this case. The\nhypothesis reads as follows: the degree $d$ part of the ring\n\\[{A\\over (F',G',H')}\\]\nis zero for generic forms $F',G'$ and $H'$ of degrees $d-a,d-b$\nand $d-c$ where $A=R\/(F,G,H)$. If $L\\in A$ is a generic linear\nform, by \\cite{Stanley}, we know that multiplication by $L$ is\nan isomorphism in degree bigger than or equal to $a+b+c-4$. Hence,\nthe degree $d+1$ piece of\n\\[{A\\over (LF',LG',LH')}\\]\nis zero and this is enough to complete the proof since, if three special\nforms, namely $LF',LG'$ and $LH'$, have maximal span then the same property holds\nfor a generic choice.\n\\end{proof}\n\n\n\\begin{lem}\\label{existdbigLEM}\nLet  $a,b$ and $d$ be non-negative integers such that $4<a\\leq b$\nand $d=2a+2b-6$. If the generic degree $d$ surface in $\\mathbb{P}^3$\ncontains a $CI(a,b,a+b-3)$, then the generic degree $d'$ surface\ncontains a $CI(a,b,c)$ for any $d'\\geq a+b+c-3$ and any $c\\geq\na+b-3$.\n\\end{lem}\n\\begin{proof}\nNotice that, by Proposition \\ref{existdbigPROP}, the generic\ndegree $d+\\epsilon$ surface in $\\mathbb{P}^3$ contains a $CI(a,b,a+b-3)$\nfor all $\\epsilon\\geq 0$. Hence, by Lemma \\ref{switchLEM}, the\nsame holds for $CI(a,b,a+b-3+\\epsilon)$ and surfaces of degree\n$d+\\epsilon$. Making $\\epsilon$ vary and again applying\nProposition \\ref{existdbigPROP} the result follows.\n\\end{proof}\n\n\\begin{thm}\\label{ageq5THM}\nLet  $a,b,c$ and $d$ be non-negative integers such that $a\\leq\nb\\leq c<d$. If $a=5$ and $b\\leq 11$, or $a=6$ and $b\\leq 7$,\nthen a $CI(a,b,c)$ exists on the generic degree $d$\nsurface of $\\mathbb{P}^3$. If $a=5,b=12$ and $c=12$, or $a=6,b=8$ and $c=8,9$,\nthen a $CI(a,b,c)$ exists on the generic degree $d$\nsurface of $\\mathbb{P}^3$.\n\\end{thm}\n\n\\begin{proof}\nTo prove the thesis we combine all the previous results and\ntechnical facts. Crucial ingredients are also some explicit\ncomputations that we performed using the computer algebra system\nCoCoA \\cite{cocoa}.\n\nTo determine whether a $CI(a,b,c)$ exists on the generic surface of degree $d$ in $\\mathbb{P}^3$, we proceed as follows:\n\\begin{itemize}\n\\item if $c\\leq a+b-3$, we make explicit computations for all\n$d\\leq a+b+c-3$; a positive answer for $d=a+b+c-3$ solves the\ncases for bigger $d$'s by Proposition \\ref{existdbigPROP};\n\\item if $c=a+b-3$ and $d\\geq 2a+2b-6$, we verify each statement\nwith an explicit computation for $d=2a+2b-6$; if the answer is\npositive we conclude the same for $c\\geq a+b-3$ and $d\\geq\na+b+c-3$ by Lemma \\ref{existdbigLEM}.\n\\end{itemize}\n\nWe sketch this procedure for $a=6$, the case $a=5$ is completely\nanalogous but lengthier. We need to perform explicit computations in the following cases:\n\\begin{itemize}\n\\item $CI(6,6,c_1)$, for $c_1\\leq 9$ and $d_1\\leq 9+c_1$;\n\\item $CI(6,7,c_2)$, for $c_2\\leq 10$ and $d_3\\leq 10+c_2$;\n\\end{itemize}\nThe computations (see Example \\ref{computationEX}) show that the  complete intersections exist on\nthe generic surfaces of the required degrees. Hence we conclude\nthat the generic surface of degree $d$ contains a $CI(6,b,c)$ for\nall $b\\leq 7$ and any $c,d$ such that $d> c$. We conclude the proof for $a=6$ by verifying existence in the cases: $CI(6,8,8)$ for $d=19$, and $CI(6,8,9)$ for $d=20$.\n\n\\end{proof}\n\n\\begin{ex}\\label{computationEX}\nWe begin with verifying that the generic surface of degree $7\\leq d\\leq 15$ contains a\n$CI(6,6,6)$. Using Proposition \\ref{equivalence} it is enough to\nshow that the ring\n\\[S={\\mathbb{C}[x_0,\\ldots,x_3]\\over(F,G,H,H',G',F')}\\]\nis zero in degree $d$, where the forms $F,G,H,H',G'$ and $F'$ are\ngeneric and have degrees $6,6,6,d-6,d-6$ and $d-6$. Hence, for each $d$, we choose\nrandom forms with rational coefficients of the  required degrees.\nThen we ask CoCoA \\cite{cocoa} to compute the Hilbert function of\n$S$ in degree $d$. Since for all $d$'s we get $H(S,d)=0$, we conclude (by semicontinuity)\nthat this is the case for a generic choice of forms of the\nappropriate degrees. In particular, as $15=6+6+6-3$ and $H(S,15)=0$, Proposition \\ref{existdbigPROP} yields that a $CI(6,6,6)$ exists on the generic degree $d\\geq 15$ surface of $\\mathbb{P}^3$.\n\nThe same argument works in complete analogy for $c\\leq 8$. For $c=9$ we make an explicit computation for $d=18$ and using Lemma \\ref{existdbigLEM} we show existence of a $CI(6,6,c)$ on the generic degree $d$ surface for $c\\geq 9$ and $d\\geq c+9$. The cases for $c<d<c+9$ are solved using Lemma \\ref{equivalence} and the results for $c\\leq 8$ and $a\\leq 4$.\n\\end{ex}\n\n\\section{Main theorem and final remarks}\\label{finalSEC}\n\nWe can now prove our main theorem:\n\\begin{proof}[Proof of Theorem \\ref{mainTHM}]\nThe existence part for the case $a\\leq 4$ is Theorem\n\\ref{aleq4THM} while existence for the remaining cases is Theorem\n\\ref{ageq5THM}. The asymptotic non-existences are given by Lemma\n\\ref{nonexistb8and12} and Theorem \\ref{nonEXISTa5}.\n\\end{proof}\n\nTheorem \\ref{mainTHM} produces a complete asymptotic answer to our\noriginal question. We also get many existence and non existence\nresults for small value of $d$. However, there are still\ninfinitely many cases which we have not solved, e.g. $a=7$ any\n$b,c$ and $d$ such that $7\\leq b\\leq c$ and $c+5\\leq d\\leq\na+b+c-4$.\n\nWe state a conjecture completing Theorem \\ref{mainTHM}:\n\\begin{quote}{\\it{ \\textbf{Conjecture}:}\ngiven non-negative integers $a,b,c$ and $d$ such that $ a\\leq\nb\\leq c<d$, there exists a function $d(a,b,c)$, possibly assuming\nthe value $+\\infty$, such that the generic degree $d$ surface in\n$\\mathbb{P}^3$ contains a $CI(a,b,c)$ if and only if $d<d(a,b,c)$.\n}\\end{quote}\n\nAs support for this conjecture, notice that it fits with the\nasymptotic statement and with the other results of Theorem\n\\ref{mainTHM}. For example, $d(a,b,c)=+\\infty$ for $a\\leq 4$ and\n$d(a,b,c)<a+b+c-3$ for $7\\leq a$.\n\n\n\n\n\n\n\\bibliographystyle{alpha}\n\n\\def$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn arithmetically Cohen-Macaulay (for short, aCM) sheaf on a projective scheme $X$ is a coherent sheaf supporting $X$, which has trivial intermediate cohomology and the stalk at each point whose depth equals the dimension of $X$. ACM vector bundles correspond to maximal Cohen-Macaulay modules over the associated graded ring and they reflect the properties of the graded ring. It is believed that the category generated by aCM sheaves on $X$ measures the complexity of $X$. Indeed, a classification of aCM varieties was proposed as {\\it finite, tame or wild} representation type according to the complexity of this category in \\cite{DG} and there are several contributions to this trichotomy such as \\cite{EH,BGS,CMP,FM}. It is only recent when such a representation type is determined for each aCM variety that is not a cone; see \\cite{FP}. \n\nIn this article, we pay our attention to the representation type of surfaces in three-dimensional projective space. Since the aCM vector bundles on smooth surfaces of degree at most two are completely classified due to the work by Horrocks and \\cite{Kapranov, Knorrer}, we may focus on surfaces of degree at least three. The case of cubic surfaces is dealt in \\cite{CH, faenzi} and the case of quartic surfaces is from \\cite{madonna}. Our main result is the following, which implies that the surfaces in Theorem \\ref{thth} are of wild representation type. \n\n\\begin{theorem}\\label{thth}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be a surface of degree at least four with $X_{\\mathrm{reg}} \\ne \\emptyset$ and assume either $\\mathrm{Pic}(X) =\\enm{\\mathbb{Z}}\\langle \\enm{\\cal{O}}_X(1)\\rangle$ or that $X$ is integral. For every even and positive integer $r$, there exists a family $\\{\\enm{\\cal{E}} _\\lambda\\}_{\\lambda \\in \\Lambda}$ of indecomposable aCM vector bundles of rank $r$ such that $\\Lambda$ is an integral quasi-projective variety with $\\dim \\Lambda=r$ and $\\enm{\\cal{E}} _{\\lambda}\\ncong \\enm{\\cal{E}} _{\\lambda'}$ for all $\\lambda \\ne \\lambda'$ in $\\Lambda$.\n\\end{theorem}\n\nIt has to be noticed that although the result in \\cite{FP} is more general than the implication of Theorem \\ref{thth} regarding the wildness of the representation type, Theorem \\ref{thth} provides a concrete way of constructing families of indecomposable aCM `vector bundles' with prescribed rank, even on singular surfaces. \n\nOn the other hand, every non-integral aCM projective schemes of arbitrary dimension at least two is of `very wild' representation type, in a sense that there exist arbitrarily large dimensional families of pairwise non-isomorphic aCM non-locally free sheaves of rank one; see Proposition \\ref{zz3}. \n\nHere we summarize the structure of this article. In Section \\ref{prem} we collect several definitions and basic results that are used throughout the article. In Section \\ref{sec3} we state the main result in Theorem \\ref{i1}, which would automatically imply Theorem \\ref{thth}. We also give a proof of Theorem \\ref{i1} in special case and suggest a number of its variation to construct aCM vector bundles. Then we spend the whole Section \\ref{sec4} for the proof of Theorem \\ref{i1}; basically we use induction on rank and the main ingredient for the proof is Lemma \\ref{x1} and the use of monodromy argument. Then we show in Section \\ref{sec5} the wildness of any aCM projective scheme of dimension at least two by investigating non-locally free ideal sheaves. \n\n\n\\section{Preliminary}\\label{prem}\nThroughout the article our base field $\\mathbf{k}$ is algebraically closed of characteristic $0$. We always assume that our projective schemes $X\\subset \\enm{\\mathbb{P}}^N$ are arithmetically Cohen-Macaulay, namely, $h^1(\\enm{\\cal{I}} _{X, \\enm{\\mathbb{P}}^N}(t)) =0$ for all $t\\in \\enm{\\mathbb{Z}}$ and $h^i(\\enm{\\cal{O}} _X(t)) =0$ for all $t\\in \\enm{\\mathbb{Z}}$ and all $i=1,\\dots ,\\dim X-1$, of pure dimension at least two. Then by \\cite[Th\\'eor\\`eme 1 in page 268]{serre} all local rings $\\enm{\\cal{O}} _{X,x}$ are Cohen-Macaulay of dimension $\\dim X$. From $h^1(\\enm{\\cal{I}} _{X, \\enm{\\mathbb{P}}^N})=0$ we see that $X_{\\mathrm{red}}$ is connected. Since in all our results we have $N =\\dim X+1=3$, the reader may just assume that $X$ is a surface in $\\enm{\\mathbb{P}}^3$. For a vector bundle $\\enm{\\cal{E}}$ of rank $r\\in \\enm{\\mathbb{Z}}$ on $X$, we say that $\\enm{\\cal{E}}$ {\\it splits} if all its indecomposable factors are $\\enm{\\cal{O}}_X(t)$ for some $t\\in \\enm{\\mathbb{Z}}$; $\\enm{\\cal{E}} \\cong \\oplus_{i=1}^r \\enm{\\cal{O}}_X(t_i)$ for some $t_i\\in \\enm{\\mathbb{Z}}$ with $i=1,\\ldots, r$.\n\nWe always fix the embedding $X\\subset \\enm{\\mathbb{P}}^N$ and the associated polarization $\\enm{\\cal{O}}_X(1)$. For a coherent sheaf $\\enm{\\cal{E}}$ on a closed subscheme $X$ of a fixed projective space, we denote $\\enm{\\cal{E}} \\otimes \\enm{\\cal{O}}_X(t)$ by $\\enm{\\cal{E}}(t)$ for $t\\in \\enm{\\mathbb{Z}}$. For another coherent sheaf $\\enm{\\cal{G}}$, we denote by $\\mathrm{hom}_X(\\enm{\\cal{F}}, \\enm{\\cal{G}})$ the dimension of $\\op{Hom}_X(\\enm{\\cal{F}}, \\enm{\\cal{G}})$, and by $\\mathrm{ext}_X^i(\\enm{\\cal{F}}, \\enm{\\cal{G}})$ the dimension of $\\mathrm{Ext}_X^i (\\enm{\\cal{F}}, \\enm{\\cal{G}})$. Finally we denote the canonical sheaf of $X$ by $\\omega_X$. \n\n\\begin{definition}\\label{deff}\nA coherent sheaf $\\enm{\\cal{E}}$ on $X$ is called {\\it arithmetically Cohen-Macaulay} (for short, aCM) if the following hold:\n\\begin{itemize}\n\\item [(i)] $\\enm{\\cal{E}}$ is locally Cohen-Macaulay, that is, the stalk $\\enm{\\cal{E}}_x$ has depth equal to $\\dim \\enm{\\cal{O}}_{X,x}$ for any point $x$ on $X$, and\n\\item [(ii)] $H^i(\\enm{\\cal{E}}(t))=0$ for all $t\\in \\enm{\\mathbb{Z}}$ and $i=1, \\ldots, \\dim (X)-1$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{remark}\\label{rremm}\nIn the condition (i) of Definition \\ref{deff}, we may only require that the stalk $\\enm{\\cal{E}}_x$ has positive depth for any point $x\\in X$; see \\cite[Remark 2.2]{BHMP} and \\cite[Th\\'eor\\`eme 1 in page 268]{serre}. \n\\end{remark}\n\nIf $\\enm{\\cal{E}}$ is a coherent sheaf on a closed subscheme $X$ of a fixed projective space, then we may consider its Hilbert polynomial $\\mathrm{P}_{\\enm{\\cal{E}}}(t)\\in \\enm{\\mathbb{Q}} [t]$ with the leading coefficient $\\mu(\\enm{\\cal{E}})\/d!$, where $d$ is the dimension of $\\mathrm{Supp}(\\enm{\\cal{E}})$ and $\\mu=\\mu(\\enm{\\cal{E}})$ is called the {\\it multiplicity} of $\\enm{\\cal{E}}$. The {\\it normalized} Hilbert polynomial $p_{\\enm{\\cal{E}}}(t)$ of $\\enm{\\cal{E}}$ is defined to be the Hilbert polynomial of $\\enm{\\cal{E}}$ divided by $\\mu (\\enm{\\cal{E}})$.\n\n\\begin{definition}\\label{def}\nIf $\\dim \\mathrm{Supp}(\\enm{\\cal{E}})=\\dim (X)$, then the {\\it rank} of $\\enm{\\cal{E}}$ is defined to be\n$$\\mathrm{rank}(\\enm{\\cal{E}})=\\frac{\\mu(\\enm{\\cal{E}})}{\\mu(\\enm{\\cal{O}}_X)}.$$\nOtherwise it is defined to be zero.\n\\end{definition}\nFor an integral scheme $X$, the rank of $\\enm{\\cal{E}}$ is the dimension of the stalk $\\enm{\\cal{E}}_x$ at the generic point $x\\in X$. But in general $\\mathrm{rank}(\\enm{\\cal{E}})$ needs not be integer.\n\n\\begin{lemma}\\label{z1}\nLet $(X, \\enm{\\cal{O}}_X(1))$ be an aCM projective scheme of dimension $n\\ge 2$. For a fixed coherent sheaf $\\enm{\\cal{G}}$ with pure depth $n$ on $X$, assume the existence of $t_0\\in \\enm{\\mathbb{Z}}$ such that $s:= h^1(\\enm{\\cal{G}} (t_0))> 0$. Then the vector space $W:= H^1(\\enm{\\cal{G}} (t_0))$ induces the following unique extension up to isomorphisms\n\\begin{equation}\\label{eqz1}\n0 \\to \\enm{\\cal{G}} \\to \\enm{\\cal{E}} \\to \\enm{\\cal{O}} _X(-t_0)\\otimes W^\\vee \\to 0\n\\end{equation}\nand the sheaf $\\enm{\\cal{E}}$ in the middle satisfies the following:\n\\begin{itemize}\n\\item [(i)] $h^1(\\enm{\\cal{E}} (t)) = h^1(\\enm{\\cal{G}} (t))$ for all $t\\ne t_0$, and $h^1(\\enm{\\cal{E}} (t_0)) =0$;\n\\item [(ii)] $h^i(\\enm{\\cal{E}} (t)) =h^i(\\enm{\\cal{G}} (t))$ for all $t\\in \\enm{\\mathbb{Z}}$ and all $i$ with $2\\le i \\le n-1$.\n\\end{itemize}\nIf $\\enm{\\cal{G}}$ is locally free, then $\\enm{\\cal{E}}$ is locally free.\n\\end{lemma}\n\n\\begin{proof}\nAll statements, except the one concerning $h^1(\\enm{\\cal{E}} (t_0))$, are true for any sheaf $\\enm{\\cal{E}}$ fitting into (\\ref{eqz1}). The vanishing\nof $H^1(\\enm{\\cal{E}} (t_0))$ is equivalent to the bijectivity of the coboundary map $\\delta : H^0(\\enm{\\cal{O}} _X)\\otimes W^\\vee \\rightarrow H^1(\\enm{\\cal{G}} (t_0))$ associated to the twist by $\\enm{\\cal{O}} _X(t_0)$ of (\\ref{eqz1}). The bijectivity of $\\delta$ is a standard result on the extension functor.\n\\end{proof}\n\n\\begin{theorem}\\label{e2}\nLet $X\\subset \\enm{\\mathbb{P}}^N$ be a projective Gorenstein scheme with pure dimension two and pure depth two, satisfying that \n\\begin{itemize}\n\\item $h^1(\\enm{\\cal{O}}_X(t))=0$ for all $t\\in \\enm{\\mathbb{Z}}$ and $h^1(\\enm{\\cal{I}}_{X,\\enm{\\mathbb{P}}^N})=0$; \n\\item $X_{\\mathrm{reg}} \\ne \\emptyset$ and $\\deg (\\omega _X)+\\deg (X) \\ge 0$.\n\\end{itemize}\nThen there exists a two-dimensional family of pairwise non-isomorphic aCM vector bundles of rank two on $X$ whose very general member is indecomposable; here ``very general\" means outside countably many proper subvarieties.\n\\end{theorem}\n\n\\begin{proposition}\\label{e1}\nLet $X\\subset \\enm{\\mathbb{P}}^N$ be as in Theorem \\ref{e2}. Assume $X_{\\mathrm{reg}} \\ne \\emptyset$ and fix $p\\in X_{\\mathrm{reg}}$. Then there exists an aCM vector bundle $\\enm{\\cal{E}} _p$ of rank two on $X$ fitting into the exact sequence\n\\begin{equation}\\label{eqe1}\n0 \\to \\omega _X(1)\\to \\enm{\\cal{E}} _p\\to \\enm{\\cal{I}} _{p,X}\\to 0.\n\\end{equation}\nMoreover, if $\\deg (\\omega _X)+\\deg (X) \\ge 0$ and $p,q\\in X_{\\mathrm{reg}}$ with $p\\ne q$, then we have $\\enm{\\cal{E}} _p\\ncong \\enm{\\cal{E}}  _q$.\n\\end{proposition}\n\n\\begin{proof}\nSince $X$ is Gorenstein, $\\omega _X(1)$ is a line bundle and we get\n$$\\op{Ext}_X^1(\\enm{\\cal{I}} _{p,X},\\omega _X(1)) \\cong H^1(\\enm{\\cal{I}} _{p,X}(-1))^\\vee \\cong \\mathbf{k}.$$ \nSo up to isomorphism there exists a unique sheaf $\\enm{\\cal{E}}_p$ fitting into an extension (\\ref{eqe1}) with a nonzero extension class. Since $h^0(\\enm{\\cal{O}} _X(-1)) =0$ and $p\\in X_{\\mathrm{reg}}$, the Cayley-Bacharach condition is satisfied for (\\ref{eqe1}) and so $\\enm{\\cal{E}}_p$ is locally free; see \\cite{cat}. Note that the restriction map \n$$H^0(\\enm{\\cal{O}} _X(t)) \\to H^0(\\enm{\\cal{O}} _X(t)_{|\\{p\\}})$$\nis surjective for any $t\\ge 0$. This implies that $h^1(\\enm{\\cal{I}}_{p,X}(t))=0$ for any $t\\ge 0$, because we have $h^1(\\enm{\\cal{O}}_X(t))=0$. Then we see from (\\ref{eqe1}) that $h^1(\\enm{\\cal{E}}_p(t))=0$ for any $t\\ge 0$. On the other hand, from $\\det (\\enm{\\cal{E}}_p) \\cong \\omega_X(1)$, we get that $h^1(\\enm{\\cal{E}}_p(t))=h^1(\\enm{\\cal{E}}_p^\\vee\\otimes \\omega_X(-t))=h^1(\\enm{\\cal{E}}_p(-t-1))=0$ for $t<0$ by Serre's duality. Thus $\\enm{\\cal{E}}_p$ is aCM. \n\nFor the second assertion, assume $\\enm{\\cal{E}} _p\\cong \\enm{\\cal{E}} _q$. From the assumption $\\deg (\\omega _X(1)) \\ge 0$, we get $h^0(\\omega _X^\\vee (-1)) \\le 1$ with equality if and only if $\\omega _X \\cong \\enm{\\cal{O}} _X(-1)$. In particular, we have $h^0(\\enm{\\cal{I}} _{p,X}\\otimes \\omega _X^\\vee (-1)) =0$. Then from the assumption $h^1(\\enm{\\cal{O}} _X)=0$ and (\\ref{eqe1}), we get $h^0(\\enm{\\cal{E}} _p\\otimes \\omega _X^\\vee (-1)) =1$ and that $p$ is the only zero of a nonzero section of $H^0(\\enm{\\cal{E}} _p\\otimes \\omega _X^\\vee (-1))$. Thus we get $p=q$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{e2}:]\nBy assumption $X_{\\mathrm{reg}}$ is a two-dimensional quasi-projective smooth variety. By Proposition \\ref{e1} there is a flat family of aCM vector bundles $\\{\\enm{\\cal{E}} _p\\}_{p\\in X_{\\mathrm{reg}}}$ of rank two such that if $p, q\\in X_{\\mathrm{reg}}$ and $p\\ne q$, then $\\enm{\\cal{E}} _p\\ncong \\enm{\\cal{E}} _q$. Now assume that $\\enm{\\cal{E}}_p$ is decomposable for some $p\\in X_{\\mathrm{reg}}$, say $\\enm{\\cal{E}} _p\\cong \\enm{\\cal{A}}_1 \\oplus \\enm{\\cal{A}}_2$ with each $\\enm{\\cal{A}}_i$ a line bundle on $X$. Since $\\det (\\enm{\\cal{E}} _p)\\cong \\omega _X(1)$, we have $\\enm{\\cal{A}}_2 \\cong \\enm{\\cal{A}}_1^\\vee\\otimes \\omega _X(1)$. Now from the assumption that $h^1(\\enm{\\cal{O}} _X)=0$, we see that $\\mathrm{Pic}(X)$ is discrete and countable. This implies that there can exist only countably many decomposable vector bundles in the family. Since the base field $\\mathbf{k}$ is algebraically closed and so uncountable, there exists some indecomposable vector bundle in the family $\\{\\enm{\\cal{E}} _p\\}_{p\\in X_{\\mathrm{reg}}}$ and for a very general point $o$ on any connected component of $X_{\\mathrm{reg}}$ the vector bundle $\\enm{\\cal{E}} _o$ is indecomposable.\n\\end{proof}\n\nThroughout the article, as in Proposition \\ref{e1}, our construction of aCM sheaf of rank two on $X$ is in terms of the following extension\n\\begin{equation}\\label{popo}\n0\\to \\omega_X \\to \\enm{\\cal{E}} \\to \\enm{\\cal{I}}_{Z,X}(a) \\to 0\n\\end{equation}\nwith $Z$ a locally complete intersection of codimension two in $X$ and $a\\in \\enm{\\mathbb{Z}}$. Such extensions are parametrized by $\\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{Z,X}(a), \\omega_X)$. In case when $X$ is a surface, the coboundary map associated to (\\ref{popo}) is\n$$\\delta_1 : H^1(\\enm{\\cal{I}}_{Z,X}(a)) \\to H^2(\\omega_X)\\cong \\mathbf{k}$$ \nand by Serre's duality in \\cite[Theorem 3.12]{H} its dual is\n$$\\mathbf{k}\\cong \\mathrm{Hom}_X(\\omega_X, \\omega_X) \\to \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{Z,X}(a), \\omega_X),$$\nwhich is obtained by applying the functor $\\mathrm{Hom}_X(-, \\omega_X)$ to (\\ref{popo}). Thus the coboundary map $\\delta_1$ is surjective if and only if (\\ref{popo}) is a non-trivial extension. Since we assume $h^1(\\enm{\\cal{O}}_X)=h^1(\\omega_X)=0$, this implies that $h^1(\\enm{\\cal{E}})=h^1(\\enm{\\cal{I}}_{Z,X}(a))-1$. \n\n\n\n\n\\section{aCM vector bundle on surfaces in $\\enm{\\mathbb{P}}^3$}\\label{sec3}\nWe always assume that $X\\subset \\enm{\\mathbb{P}}^3$ is a surface of degree $m$, not necessarily smooth. In particular, its dualizing sheaf is $\\omega_X \\cong \\enm{\\cal{O}}_X(m-4)$ and we get $h^2(\\enm{\\cal{O}}_X)=\\binom{m-1}{3}$. We also have $h^0(\\enm{\\cal{O}} _X)=1$ and $h^1(\\enm{\\cal{O}} _X)=0$. \n\n\\begin{lemma}\\label{c2}\nEach line bundle $\\enm{\\cal{O}} _X(t)$ with $t \\in \\enm{\\mathbb{Z}}$, is stable as an $\\enm{\\cal{O}} _{\\enm{\\mathbb{P}}^3}$-sheaf with pure depth $2$.\n\\end{lemma}\n\n\\begin{proof}\nIt is enough to deal with the case $t=0$. Assume the contrary and take a subsheaf $\\enm{\\cal{A}} \\subsetneq \\enm{\\cal{O}} _X$ such that $\\enm{\\cal{B}}:=\\enm{\\cal{O}} _X\/\\enm{\\cal{A}}$ has depth $2$ and normalized Hilbert polynomial at least the one of $\\enm{\\cal{O}} _X$. Since $\\enm{\\cal{B}}$ is a quotient of $\\enm{\\cal{O}} _X$ with depth $2$ and $X$ has no embedded component, we get $\\enm{\\cal{B}} \\cong \\enm{\\cal{O}} _T$ for $T$ a union of some of the irreducible components of $X_{\\mathrm{red}}$ with at most the multiplicities appearing in $X$. This implies that $T\\in |\\enm{\\cal{O}} _{\\enm{\\mathbb{P}}^3}(d)|$ for some integer $d$ with $1\\le d < m$. Now the Hilbert polynomial of $\\enm{\\cal{O}} _X$ is \n\\begin{align*}\n\\mathrm{P}_{\\enm{\\cal{O}}_X}(t)&=\\binom{t+3}{3} -\\binom{t-m+3}{3} \\\\\n&= \\left (\\frac{m}{2}\\right) t^2 + \\left ( 2m-\\frac{m^2}{2}\\right) t + \\left(\\frac{m^3}{6} -m^2 + \\frac{11m}{6} \\right). \n\\end{align*}\nSimilarly, we get the Hilbert polynomial $\\mathrm{P}_{\\enm{\\cal{O}}_T}(t)$ of $\\enm{\\cal{O}}_T$ by replacing $m$ in $\\mathrm{P}_{\\enm{\\cal{O}}_X}(t)$ by $d$. Then we see that $p_{\\enm{\\cal{O}}_X}(t)<p_{\\enm{\\cal{O}} _T}(t)$ for $t\\gg 0$, a contradiction.\n\\end{proof}\n\n\\begin{remark}\\label{c3}\nIf either $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}} \\langle \\enm{\\cal{O}}_X(1)\\rangle$ or $X$ is integral, then every line bundle is stable. Note also that the proof of Lemma \\ref{c2} shows that the ideal sheaf $\\enm{\\cal{I}} _{Z,X}$ for any zero-dimensional subscheme $Z\\subset X$, is also stable. If $X$ is integral, then any sheaf of rank $1$ with positive depth is stable. Thus these sheaves are indecomposable.\n\\end{remark}\n\n\n\n\n\\begin{proposition}\\label{a1.1}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be a surface of degree $m\\ge 2$ with $X_{\\mathrm{reg}} \\ne \\emptyset$. Fix $p \\in X_{\\mathrm{reg}}$, and let $\\enm{\\cal{E}}_p$ be the unique non-trivial extension  \n\\begin{equation}\\label{eqb1}\n0 \\to \\enm{\\cal{O}} _X(m-3) \\to \\enm{\\cal{E}}_p \\to \\enm{\\cal{I}} _{p,X}\\to 0.\n\\end{equation}\nThen $\\enm{\\cal{E}}_p$ is an aCM vector bundle of rank two on $X$ and $\\enm{\\cal{E}} \\ncong \\enm{\\cal{O}} _X(a)\\oplus \\enm{\\cal{O}} _X(b)$ for any $a, b\\in \\enm{\\mathbb{Z}}$. If one of the following holds, then $\\enm{\\cal{E}}$ is indecomposable. \n\\begin{itemize}\n\\item [(i)] $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}}\\langle \\enm{\\cal{O}} _X(1)\\rangle $,\n\\item [(ii)] $\\enm{\\cal{O}} _X(t)$ for $t\\in \\enm{\\mathbb{Z}}$ are the only aCM line bundles on $X$, or \n\\item [(iii)] $m\\ge 4$ and $X$ is integral.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nBy Proposition \\ref{e1} it remains to deal with indecomposability of $\\enm{\\cal{E}}_p$. First show that there are no integers $a, b$ such that $\\enm{\\cal{E}}_p \\cong \\enm{\\cal{O}} _X(a)\\oplus \\enm{\\cal{O}} _X(b)$. Assume that such $a,b$ exist, say $a\\ge b$. Since $h^0(\\enm{\\cal{E}}_p (3-m)) =1$ and $h^0(\\enm{\\cal{E}}_p (2-m)) =0$, we get $(a,b)=(m-3,0)$ and $m\\ge 3$. Then we get $h^0(\\enm{\\cal{E}}_p) = \\binom{m}{3} +1$, while (\\ref{eqb1}) gives $h^0(\\enm{\\cal{E}}_p )=\\binom{m}{3}$. \n\nNow assume that $\\enm{\\cal{E}}_p$ is decomposable. Since $\\enm{\\cal{E}}_p$ is locally free and it has rank $2$, we have $\\enm{\\cal{E}}_p \\cong \\enm{\\cal{A}}_1 \\oplus \\enm{\\cal{A}}_2$ with each $\\enm{\\cal{A}}_i\\in \\mathrm{Pic}(X)$. Since $\\enm{\\cal{E}}_p$ is aCM, each $\\enm{\\cal{A}}_i$ is aCM. In cases (i) and (ii) the assertion holds by above. Thus we assume the case (iii). By Lemma \\ref{c2} and Remark \\ref{c3}, (\\ref{eqb1}) is the HN filtration of $\\enm{\\cal{E}}_p$. Applying the functor $\\mathrm{Hom}_X(\\enm{\\cal{E}}_p, -)$ to (\\ref{eqb1}), we get \n$$0\\to \\mathrm{Hom}_X(\\enm{\\cal{E}}_p, \\enm{\\cal{O}}_X(m-3)) \\to \\mathrm{Hom}_X(\\enm{\\cal{E}}_p, \\enm{\\cal{E}}_p) \\to \\mathrm{Hom}_X(\\enm{\\cal{E}}_p, \\enm{\\cal{I}}_{p,X})\\to \\mathrm{Ext}_X^1(\\enm{\\cal{E}}_p, \\enm{\\cal{O}}_X(m-3)).$$\nNote that $\\hom_X(\\enm{\\cal{E}}_p, \\enm{\\cal{O}}_X(m-3))=h^2(\\enm{\\cal{E}}_p(-1))=h^0(\\enm{\\cal{E}}_p)=\\binom{m}{3}$ by Serre's duality. By applying the functor $\\mathrm{Hom}_X(-, \\enm{\\cal{I}}_{p,X})$ to (\\ref{eqb1}), we get \n$$\\hom_X(\\enm{\\cal{E}}_p, \\enm{\\cal{I}}_{p,X})=\\hom_X(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{I}}_{p,X})=1.$$ \nThus we have \n$$\\binom{m}{3}\\le \\hom_X (\\enm{\\cal{E}}_p ,\\enm{\\cal{E}}_p )\\le 1 + \\binom{m}{3}.$$\nSince $h^0(\\enm{\\cal{O}} _X) =1$, we have $\\hom_X (\\enm{\\cal{A}}_i ,\\enm{\\cal{A}}_i ) = 1$ for each $i$. So we get\n$$\\hom_X (\\enm{\\cal{E}}_p ,\\enm{\\cal{E}}_p ) =2+\\hom_X (\\enm{\\cal{A}}_1 ,\\enm{\\cal{A}}_2 )+\\hom_X (\\enm{\\cal{A}}_2,\\enm{\\cal{A}}_1 ).$$ \nSince $X$ is integral, each $\\enm{\\cal{A}}_i$ is stable and we get either $\\enm{\\cal{A}}_1\\cong \\enm{\\cal{A}}_2$ or $\\hom_X (\\enm{\\cal{A}}_i ,\\enm{\\cal{A}}_{3-i}) =0$ for each $i$. In the latter case we have $\\hom_X(\\enm{\\cal{E}}_p, \\enm{\\cal{E}}_p)=2<\\binom{m}{3}$, a contradiction. In the former case, we have $\\mathrm{hom}_X(\\enm{\\cal{E}}_p, \\enm{\\cal{E}}_p)=4$ and the only possibility is $m=4$. But this is also impossible, since we would get $\\enm{\\cal{A}}_1^{\\otimes 2} \\cong \\det (\\enm{\\cal{E}}_p) \\cong \\enm{\\cal{O}}_X(1)$.  \n\\end{proof}\n\n\n\n\\begin{proposition}\\label{z2}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be a surface of degree $m\\ge 2$ and let $Z\\subset X$ be a zero-dimensional subscheme of degree $3$, which is not collinear. Assume that $Z$ is a locally complete intersection inside $X$, i.e. for each $p\\in Z_{\\mathrm{red}}$ the ideal sheaf of $Z$ at $\\enm{\\cal{O}} _{X,p}$ is generated by two elements of $\\enm{\\cal{O}} _{X,p}$. Then there is a vector bundle $\\enm{\\cal{G}}$ of rank two fitting into an exact sequence\n\\begin{equation}\\label{eqz2}\n0 \\to \\enm{\\cal{O}} _X(m-4)\\to \\enm{\\cal{G}} \\to \\enm{\\cal{I}} _{Z,X}\\to 0\n\\end{equation}\nwith $h^1(\\enm{\\cal{G}} (t)) =0$ for all $t\\ne 0$ and $h^1(\\enm{\\cal{G}})=1$. There is also an exact sequence\n\\begin{equation}\\label{eqz3}\n0 \\to \\enm{\\cal{G}} \\to \\enm{\\cal{E}} \\stackrel{u}{\\to} \\enm{\\cal{O}} _X\\to 0,\n\\end{equation}\nwhere $\\enm{\\cal{E}}$ is an aCM vector bundle of rank three such that $\\enm{\\cal{E}} \\ncong \\enm{\\cal{O}} _X(a_1)\\oplus\n\\enm{\\cal{O}} _X(a_2)\\oplus \\enm{\\cal{O}} _X(a_3)$ for any $(a_1, a_2, a_3)\\in \\enm{\\mathbb{Z}}^{\\oplus 3}$. Moreover, if $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}} \\langle \\enm{\\cal{O}}_X(1) \\rangle$, then $\\enm{\\cal{E}}$ is indecomposable. \n\\end{proposition}\n\n\\begin{proof}\nSince $\\omega _X\\cong \\enm{\\cal{O}} _X(m-4)$, we have $h^0(\\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X)=1$ and $\\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X$ is globally generated. Since $\\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X$ is globally generated, we have $h^0(\\enm{\\cal{I}} _{p,X}\\otimes \\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X)=0$ for all $p\\in Z_{\\mathrm{red}}$. Since $Z$ is a locally complete intersection, the Cayley-Bacharach condition is satisfied and so there is a locally free $\\enm{\\cal{G}}$ fitting into (\\ref{eqz2}); see \\cite{cat}. From (\\ref{eqz2}) we immediately get $h^1(\\enm{\\cal{G}} (t)) =0$ for all $t>0$, because $Z$ is not collinear. Note that $\\det (\\enm{\\cal{G}} ) \\cong \\enm{\\cal{O}} _X(m-4)$ and $\\enm{\\cal{G}}$ is a vector bundle of rank two. This implies $\\enm{\\cal{G}} ^\\vee \\cong \\enm{\\cal{G}} (4-m)$. For $t<0$, we have $h^1(\\enm{\\cal{G}} (t)) = h^1(\\enm{\\cal{G}} ^\\vee (-t)\\otimes \\omega _X) = h^1(\\enm{\\cal{G}} (-t)) =0$ by Serre's duality. Now consider the coboudnary map $\\delta _1: H^1(\\enm{\\cal{I}} _{Z,X}) \\rightarrow H^2(\\enm{\\cal{O}} _X(m-4))\\cong \\mathbf{k}$ with $\\mathrm{ker}(\\delta _1) =H^1(\\enm{\\cal{G}})$. The dual of $\\delta_1$ is the map \n$$\\mathrm{Hom}_X(\\enm{\\cal{O}}_X(m-4), \\enm{\\cal{O}}_X(m-4)) \\to \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{Z,X}, \\enm{\\cal{O}}_X(m-4)) $$\nsending the identity map to the element corresponding to $\\enm{\\cal{G}}$. This implies that $\\delta_1$ is surjective and $h^1(\\enm{\\cal{G}})=1$. \n\nNow we apply Lemma \\ref{z1} to $\\enm{\\cal{G}}$ to obtain an aCM vector bundle $\\enm{\\cal{E}}$ of rank three fitting into (\\ref{eqz3}). Since $h^1(\\enm{\\cal{G}} )=1$ and $h^1(\\enm{\\cal{E}} )=0$, (\\ref{eqz2}) and (\\ref{eqz3}) give $h^0(\\enm{\\cal{E}} )=h^0(\\enm{\\cal{G}} )=\\binom{m-1}{3}$. Assume the existence of integers $a_1\\ge a_2\\ge a_3$ such that $\\enm{\\cal{E}} \\cong \\oplus_{i=1}^3 \\enm{\\cal{O}} _X(a_i)$. Since $\\det (\\enm{\\cal{E}} )\\cong \\enm{\\cal{O}} _X(m-4)$, we have $a_1+a_2+a_3 =m-4$. If $2 \\le m\\le 3$, then we have $a_1\\ge 0$ from $a_1+a_2+a_3 =m-4$. This implies that $h^0(\\enm{\\cal{O}} _X(a_1)) > 0 =\\binom{m-1}{3}=h^0(\\enm{\\cal{E}})$, a contradiction. If $m=4$, then we have $h^0(\\enm{\\cal{E}} )=1$. Since $a_1+a_2+a_3 =0$, we have $\\sum _{i=1}^{3} h^0(\\enm{\\cal{O}} _X(a_i)) >1$, a contradiction. Finally assume $m>4$. From (\\ref{eqz2}) and (\\ref{eqz3}) we see that $\\enm{\\cal{O}} _X(m-2)$ is the first non-trivial sheaf in the HN filtration of $\\enm{\\cal{E}}$. Thus $a_1 =m-4$ and $h^0(\\enm{\\cal{O}} _X(a_1)) =\\binom{m-1}{3}$. Since $a_2+a_3=0$, we have $h^0(\\enm{\\cal{O}} _X(a_2)) >0$ and so $h^0(\\enm{\\cal{E}} )> \\binom{m-1}{3}$, a contradiction. Hence we get $\\enm{\\cal{E}} \\ncong \\oplus_{i=1}^3\\enm{\\cal{O}}_X(a_i)$ for any triple of integers $(a_1, a_2, a_3)$. \n\nIt remains to show the last assertion. Assume $\\mathrm{Pic}(X) \\cong\\enm{\\mathbb{Z}}\\langle \\enm{\\cal{O}} _X(1)\\rangle$ and that $\\enm{\\cal{E}}$ is decomposable; by the previous assertion we have $\\enm{\\cal{E}} \\cong \\enm{\\cal{A}}_1 \\oplus \\enm{\\cal{A}}_2$ with $\\mathrm{rank}(\\enm{\\cal{A}}_i ) =i$ for each $i$ and $\\enm{\\cal{A}}_2$ indecomposable. Set $\\enm{\\cal{A}}_1 \\cong \\enm{\\cal{O}} _X(a)$ for $a\\in \\enm{\\mathbb{Z}}$. Since $h^0(\\enm{\\cal{E}} )=\\binom{m-1}{3}$, we have $a\\le m-4$. From (\\ref{eqz2}) and (\\ref{eqz3}) we get the existence of a subsheaf $\\enm{\\cal{F}} \\subset \\enm{\\cal{E}}$ such that $\\enm{\\cal{F}} \\cong \\enm{\\cal{O}} _X(m-4)$ and $\\enm{\\cal{E}} \/\\enm{\\cal{F}}$ is an extension $\\enm{\\cal{H}}$ of $\\enm{\\cal{O}} _X$ by $\\enm{\\cal{I}} _{Z,X}$. Note that $\\enm{\\cal{H}}$ is not locally free, because $\\enm{\\cal{I}} _{Z,X}$ has not depth $2$. In particular, $\\enm{\\cal{H}}$ is not isomorphic to $\\enm{\\cal{A}}_2$ and we get $\\enm{\\cal{A}}_1 \\ncong \\enm{\\cal{F}}$. So we have $a<m-4$. Now consider a restriction map\n$$u_{|\\{0\\}\\oplus \\enm{\\cal{A}}_2} : \\{0\\}\\oplus \\enm{\\cal{A}}_2 \\to \\enm{\\cal{O}}_X.$$\nIf this restriction map is surjective, then its kernel is a line bundle, say $\\enm{\\cal{O}}_X(b)$. Since $X$ is aCM, we get $\\enm{\\cal{A}}_2 \\cong \\enm{\\cal{O}}_X\\oplus \\enm{\\cal{O}}_X(b)$, a contradiction. Thus the restriction map is not surjecitve and so the other restriction map $u_{|\\enm{\\cal{A}}_1 \\oplus \\{0\\}} $ is not zero. In particular, we get $a\\le 0$. If $a=0$, then we have $\\enm{\\cal{A}}_1 \\cong \\enm{\\cal{O}}_X$ and the map $u_{|\\enm{\\cal{A}}_1 \\oplus \\{0\\}}$ is an isomorphism. Thus (\\ref{eqz3}) splits and we get $h^1(\\enm{\\cal{E}} )\\ge h^1(\\enm{\\cal{G}} )>0$, a contradiction. Hence we get $a<0$. Since there is no nonzero map $\\enm{\\cal{F}} \\rightarrow \\enm{\\cal{A}}_1$ from $a<m-4$, $\\enm{\\cal{F}}$ is isomorphic to a subsheaf $\\enm{\\cal{F}}_1$ of $\\enm{\\cal{A}}_2$ and we get $\\enm{\\cal{H}} \\cong \\enm{\\cal{O}}_X(a)\\oplus \\enm{\\cal{A}}_2\/\\enm{\\cal{F}}_1$. From $a<0$ we see that there is no nonzero map $\\enm{\\cal{I}}_{Z,X} \\rightarrow \\enm{\\cal{O}}_X(a)$. Since $\\enm{\\cal{H}}$ is an extension of $\\enm{\\cal{O}}_X$ by $\\enm{\\cal{I}}_{Z,X}$, we get that $\\enm{\\cal{I}}_{Z,X} \\cong \\enm{\\cal{A}}_2\/\\enm{\\cal{F}}_1$ and so $\\enm{\\cal{O}}_X(a)\\cong \\enm{\\cal{O}}_X$, a contradiction. \n\\end{proof}\n\n\\begin{remark}\nIn case $m=1$, i.e. $X=\\enm{\\mathbb{P}}^2$, we fail in obtaining an indecomposable aCM vector bundle of rank three, using the method in Proposition \\ref{z2}. Indeed, we get $\\enm{\\cal{G}} \\cong \\Omega_{\\enm{\\mathbb{P}}^2}^1$ and the corresponding  vector bundle of rank three is $\\enm{\\cal{E}} \\cong \\enm{\\cal{O}} _{\\enm{\\mathbb{P}}^2}(-1)^{\\oplus 3}$.\n\\end{remark}\n\n\\begin{corollary}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be union of multiple planes in which at least one plane occurs with multiplicity $1$. Then there is an indecomposable aCM vector bundle of rank three on $X$. If $m>4$, we have a family of such aCM vector bundles of dimension $6$.\n\\end{corollary}\n\n\\begin{proof}\nAssume that $X$ has one component $H$ with multiplicity $1$. In this case we take as $Z$ a set of $3$ general points in $H$. Then the first assertion follows from Proposition \\ref{z2}. Note that the set of all such $Z$ has dimension $6$. Now assume that $X$ has a component $H$ with multiplicity $3$. Fix a general point $p\\in H$ and take a general line $L\\subset \\enm{\\mathbb{P}}^3$ with $p\\in L$. Then set $Z$ to be the connected component of the scheme $X\\cap L$ with $p$ as its reduction. Then we may get the assertion from Proposition \\ref{z2} and that $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}} \\langle \\enm{\\cal{O}} _X(1)\\rangle$ by \\cite[Lemma 2.5]{BHMP}. \n\\end{proof}\n\n\\begin{proposition}\\label{z2+0}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be a surface of degree $m\\ge 4$ with an irreducible component $Y$ appearing with multiplicity $2$ in $X$. Fix $p\\in Y_{\\mathrm{reg}}$ so that $T$ is the only irreducible component of $X$ containing $p$. For a general line $L\\subset \\enm{\\mathbb{P}}^3$ containing $p$, let $Z\\subset X$ be the connected component of $L\\cap X$ with $p$ as its reduction. We have $\\deg (Z)=2$ and there is an aCM  vector bundle $\\enm{\\cal{E}}_Z$ of rank two fitting into an exact sequence\n\\begin{equation}\\label{eqz2+0}\n0 \\to \\enm{\\cal{O}} _X(m-4)\\to \\enm{\\cal{E}} _Z\\to \\enm{\\cal{I}} _{Z,X}\\to 0.\n\\end{equation}\nThe set of all isomorphism classes of $\\enm{\\cal{E}} _Z$ is uniquely parametrized by a $4$-dimensional irreducible quasi-projective variety $\\Delta$ satisfying the following. \n\\begin{itemize}\n\\item [(i)] For any $\\enm{\\cal{E}} _Z\\in \\Delta$, there are no integers $a, b$ with $\\enm{\\cal{E}} _Z \\cong \\enm{\\cal{O}} _X(a)\\oplus \\enm{\\cal{O}} _X(b)$.\n\\item [(ii)] A very general $\\enm{\\cal{E}} _Z\\in \\Delta$ is indecomposable.\n\\item [(iii)] If $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}} \\langle \\enm{\\cal{O}} _X(1)\\rangle $, then each $\\enm{\\cal{E}} _Z\\in \\Delta$ is indecomposable.\n\\item [(iv)] If $\\enm{\\mathbb{Z}} \\langle \\enm{\\cal{O}} _X(1)\\rangle $ are the only aCM line bundles on $X$, then each $\\enm{\\cal{E}} _Z\\in \\Delta$ is indecomposable.\n\\end{itemize} \n\\end{proposition}\n\n\\begin{proof}\nSince no other component of $X$ than $Y$ contains $p$ and $p$ is a smooth point of $X$, we have $\\deg (Z)=2$; it is sufficient to take as $L$ any line through $p$ not contained in the tangent plane $T_pY$ of $Y$ at $p$. \n\nSince $\\omega _X\\cong \\enm{\\cal{O}} _X(m-4)$, we have $h^0(\\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X)=1$ and $\\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X$ is globally generated. Thus we have $h^0(\\enm{\\cal{I}} _{p,X}\\otimes \\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X)=0$. Since $Z$ is a locally complete intersection, the Cayley-Bacharach condition is satisfied for (\\ref{eqz2+0}) and so there is a locally free $\\enm{\\cal{E}}_Z$ fitting into (\\ref{eqz2+0}); see \\cite{cat}. \n\nSince $\\enm{\\cal{O}} _X(1)$ is very ample and $\\deg (Z) =2$, we get $h^1(\\enm{\\cal{E}}_Z (t)) =0$ for all $t>0$ by (\\ref{eqz2}). Note that $\\det (\\enm{\\cal{E}}_Z ) \\cong \\enm{\\cal{O}} _X(m-4)$ and $\\enm{\\cal{E}}_Z$ is a vector bundle of rank two. This implies $\\enm{\\cal{E}}_Z ^\\vee \\cong \\enm{\\cal{E}}_Z (4-m)$. For $t<0$, we have $h^1(\\enm{\\cal{E}}_Z (t)) = h^1(\\enm{\\cal{E}}_Z ^\\vee (m-t-4)) = h^1(\\enm{\\cal{E}}_Z (-t)) =0$ by Serre's duality. Now consider the coboudary map $\\delta _1: H^1(\\enm{\\cal{I}} _{Z,X}) \\rightarrow H^2(\\enm{\\cal{O}} _X(m-4))\\cong \\mathbf{k}$ with $\\mathrm{ker}(\\delta _1) =H^1(\\enm{\\cal{E}}_Z)$. The dual of $\\delta_1$ is the map \n$$\\mathrm{Hom}_X(\\enm{\\cal{O}}_X(m-4), \\enm{\\cal{O}}_X(m-4)) \\to \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{Z,X}, \\enm{\\cal{O}}_X(m-4)) $$\nsending the identity map to the element corresponding to $\\enm{\\cal{E}}_Z$. This implies that $\\delta_1$ is non-zero and hence and $h^1(\\enm{\\cal{E}}_Z)=0$. Thus $\\enm{\\cal{E}}_Z$ is aCM.\n\nThe set of all $p\\in Y_{\\mathrm{reg}}$ such that $Y$ is the only irreducible component of $X$ containing $p$ is an irreducible $2$-dimensional variety $\\Delta '$. For each $p\\in \\enm{\\mathbb{P}}^3$ the set of all lines through $p$ is a $\\enm{\\mathbb{P}}^2$. Define a variety $\\Delta$ as follows:\n$$\\Delta:=\\{(p,L)~|~p\\in \\Delta' \\text{ and }L \\text{ a line in }\\enm{\\mathbb{P}}^3 \\text{ with }p\\in L \\text{ and } L\\nsubseteq T_pY\\}.$$\nSince $m\\ge 4$, we have $h^0(\\enm{\\cal{I}} _{Z,X}(4-m)) =0$. Thus (\\ref{eqz2+0}) gives $h^0(\\enm{\\cal{E}} _Z(4-m)) =1$. Thus the isomorphism classes of $\\enm{\\cal{E}} _Z$ uniquely determines $Z$, i.e. if $\\enm{\\cal{E}} _Z\\ncong \\enm{\\cal{E}} _{Z'}$, then we get $Z\\ne Z'$. For two elements $(p_1, L_1), (p_2, L_2)\\in \\Delta$, let $Z_i$ be the subscheme of degree $2$ determined by $(p_i, L_i)$ for each $i=1,2$. Since each $p_i$ is the reduction of $Z_i$ and $L_i$ is the line spanned by $Z_i$, the variety $\\Delta$ uniquely parametrizes the isomorphism classes of the aCM vector bundles $\\enm{\\cal{E}}_Z$.\n\nAssume $\\enm{\\cal{E}}_Z \\cong \\enm{\\cal{O}} _X(a)\\oplus \\enm{\\cal{O}} _X(b)$ for some integers $a, b$ with $a\\ge b$. Since $\\det (\\enm{\\cal{E}}_Z ) \\cong \\enm{\\cal{O}} _X(m-4)$, we have $b= m-4-a$. But since $h^0(\\enm{\\cal{E}}_Z (4-m)) =1$, the only possibility is that $a=4-m$ and $b<0$, a contradiction. Thus we get (i). We may get (ii) as in the proof of Theorem \\ref{e2}. Now assume that $\\enm{\\cal{E}}_Z$ is decomposable, say $\\enm{\\cal{E}}_Z \\cong \\enm{\\cal{A}}_1 \\oplus \\enm{\\cal{A}}_2$ with each $\\enm{\\cal{A}}_i$ a line bundle. Since $\\enm{\\cal{E}}_Z$ is aCM, each $\\enm{\\cal{A}}_i$ is also aCM. Thus (iii) and (iv) follow from (i).\n\\end{proof}\n\n\\begin{remark}\nIn case $m=2$, i.e. $X=2H$ the double plane with a hyperplane $H\\subset \\enm{\\mathbb{P}}^3$, the vector bundle $\\enm{\\cal{E}} _Z$ described in Proposition \\ref{z2+0} is the vector bundle $\\enm{\\cal{O}} _X(-1)^{\\oplus 2}$.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{theorem}\\label{i1}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be a surface of degree $m\\ge 4$ with $X_{\\mathrm{reg}} \\ne \\emptyset$, i.e. $X$ has an irreducible component $Y$ appearing with multiplicity $1$.  We further assume that either $\\mathrm{Pic}(X) =\\enm{\\mathbb{Z}}\\langle \\enm{\\cal{O}} _X(1)\\rangle $ or $X$ is integral. For a fixed integer $s>0$ and a set $S\\subset X_{\\mathrm{reg}}\\cap Y$ with $\\sharp (S)=s$, a general sheaf $\\enm{\\cal{E}} _S$ fitting into an exact sequence\n\\begin{equation}\\label{eqi1}\n0 \\to \\enm{\\cal{O}} _X(m-3)^{\\oplus s} \\stackrel{v}{\\to} \\enm{\\cal{E}} _S \\to \\oplus _{p\\in S}\\enm{\\cal{I}}_{p,X}\\to 0, \n\\end{equation}\nis a locally free, indecomposable and aCM sheaf of rank $2s$. Moreover, if $S'\\subset X_{\\mathrm{reg}}\\cap Y$ is another set with $\\sharp (S')=s$ and $S'\\ne S$, then we have $\\enm{\\cal{E}} _{S'}\\ncong \\enm{\\cal{E}} _S$.\n\\end{theorem}\n\n\nWe have $\\mathrm{ext}_X^1(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X(m-3))=h^1(\\enm{\\cal{I}}_{p,X}(-1))=1$ for each $p\\in X_{\\mathrm{reg}}$ by Serre's duality. So the extension $\\enm{\\cal{E}}_S$ corresponds to an element in a finite dimensional vector space\n$$\\mathbb{E}(S):=\\mathrm{Ext}_X^1(\\oplus_{p\\in S}\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X(m-3)^{\\oplus s})\\cong \\mathbf{k}^{s^2}.$$\nIf $s=1$, say $S=\\{p\\}$, the dimension of $\\mathbb{E}$ is one. Thus there exists a unique non-trivial extension. Denote this non-trivial extension simply by $\\enm{\\cal{E}}_p$. \n\nIn Theorem \\ref{i1}, a ``general'' choice of $\\enm{\\cal{E}}_S$ means that there exists a non-empty Zariski open subset $\\mathbb{U}\\subset \\mathbb{E}(S)$ such that the middle term of any extension in $\\mathbb{U}$ is aCM, locally free and indecomposable.\n\n\\begin{proof}[Proof of Theorem \\ref{thth}:]\nThe family $\\Sigma$ of all $S\\subset X_{\\mathrm{reg}}$ with $\\sharp (S) =s$ clearly has dimension $2s$. By Theorem \\ref{i1}, if $S$ and $S'$ are two distinct sets in $\\Sigma$, then we get $\\enm{\\cal{E}}_{S} \\ncong \\enm{\\cal{E}}_{S'}$. Now there is a universal family on any $\\mathrm{Ext}^1$-group of families of sheaves with $\\Sigma \\times X$ as its base. Thus, we get a family of aCM locally free and indecomposable vector bundles with as a parameter space a rank $s^2$ vector bundle over $\\Sigma$; the fibre of this vector bundle over $S\\in \\Sigma$ is $\\mathbb{E}(S)$, corresponding to $S$. Taking a non-empty open subset $V$ of $\\Sigma$ on which this vector bundle is trivial we get a family of pairwise non-isomorphic sheaves, at least if we restrict $V$, so that all sheaves in the family are locally free, aCM and indecomposable.\n\\end{proof}\n\n\\begin{remark}\nFor a surface $X$ as in Theorem \\ref{i1} and Theorem \\ref{thth}, the algebraic group $\\mathrm{Aut}(X)$ has finite dimension; it is often zero-dimensional. Hence there exists an integer $t_0$ such that for every even integer $r$, $X$ has a family of dimension at least $r-t_0$, consisting of indecomposable aCM vector bundles of rank $r$ on $X$, such that for any two distinct elements $\\enm{\\cal{E}}$, $\\enm{\\cal{E}}'$ in the family there is no $f\\in \\mathrm{Aut}(X)$ with $f^\\ast (\\enm{\\cal{E}} )\\cong \\enm{\\cal{E}}'$.\n\\end{remark}\n\n\n\\section{Proof of Theorem \\ref{i1}}\\label{sec4}\n\nSet $\\mathbb{E}'(S)$ to be the set of all elements in $\\mathbb{E}(S)$ whose corresponding middle term is locally free and aCM.\n\n\\begin{lemma}\\label{x01}\n$\\mathbb{E}'(S)$ is a non-empty open subset of $\\mathbb{E}(S)$.\n\\end{lemma}\n\n\\begin{proof}\nSince being locally free and aCM are both open properties in a flat family, $\\mathbb{E}'(S)$ is an open subset of $\\mathbb{E}(S)$. Thus it is sufficient to prove that $\\mathbb{E}'(S)  \\ne \\emptyset$. Proposition \\ref{a1.1} gives the case $s=1$. For $s>1$, we may find a direct sum of aCM vector bundles of rank two fitting into (\\ref{eqi1}), i.e. take $\\oplus _{p\\in S} \\enm{\\cal{E}}_p$. This implies $\\mathbb{E}'(S) \\ne \\emptyset$.\n\\end{proof}\n\n\\begin{remark}\\label{x00} \nIn the set-up of (\\ref{eqi1}) set $\\enm{\\cal{A}} := v(\\enm{\\cal{O}} _X(m-3)^{\\oplus s})$. By Lemma \\ref{c2} and Remark \\ref{c3} together with the assumption $m\\ge 3$, we see that $\\enm{\\cal{A}}$ is the first term of the HN filtration of $\\enm{\\cal{E}}_S$. Thus we get $f(\\enm{\\cal{A}})\\subseteq \\enm{\\cal{A}}$ for any $f\\in \\op{End} (\\enm{\\cal{E}}_S )$.\n\\end{remark}\n\n\\begin{lemma}\\label{x0}\nIf $\\enm{\\cal{E}}$ is the middle term of an extension $\\epsilon \\in \\mathbb{E}'(S)$, then $\\enm{\\cal{E}}$ has no line bundle as a factor.\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\enm{\\cal{L}}$ is a line bundle that is a factor of $\\enm{\\cal{E}}$, i.e. $\\enm{\\cal{E}} = \\enm{\\cal{L}} \\oplus \\enm{\\cal{G}}$ for some aCM vector bundle $\\enm{\\cal{G}}$ of rank $2s-1$. Since $m\\ge 3$, we have \n$$h^0(\\enm{\\cal{L}} (3-m)) +h^0(\\enm{\\cal{G}} (3-m)) =h^0(\\enm{\\cal{E}}(3-m))=s.$$ \nFirst assume $h^0(\\enm{\\cal{L}} (3-m)) =0$ and $h^0(\\enm{\\cal{G}} (3-m)) =s$. Then we have $v(\\enm{\\cal{O}}_X(m-3)^{\\oplus s}) \\subset \\{0\\}\\oplus \\enm{\\cal{G}}$ in (\\ref{eqi1}) and so $\\enm{\\cal{L}} \\cong \\enm{\\cal{I}}_{p,X}$ for some $p\\in S$, a contradiction. Thus we have $h^0(\\enm{\\cal{L}} (3-m)) >0$ and so $h^0(\\enm{\\cal{G}} (3-m)) <s$. In particular, there is a nonzero map $u: \\enm{\\cal{O}} _X(m-3) \\rightarrow \\enm{\\cal{L}}$. Assume for the moment that $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}}\\langle \\enm{\\cal{O}} _X(1)\\rangle$ and write $\\enm{\\cal{L}} \\cong \\enm{\\cal{O}} _X(a)$ for some $a\\in \\enm{\\mathbb{Z}}$. The map $u$ gives $a\\ge m-3$. Since $m\\ge 3$, (\\ref{eqi1}) is the HN-filtration of $\\enm{\\cal{E}}$ and we get $a=m-3$. Thus $\\enm{\\cal{G}}$ fits into an exact sequence\n$$0\\to \\enm{\\cal{O}}_X(m-3)^{\\oplus (s-1)} \\to \\enm{\\cal{G}} \\to \\oplus _{p\\in S}\\enm{\\cal{I}} _{p,X}\\to 0. $$ \nThen we get $h^1(\\enm{\\cal{G}}(-1))\\ge 1$ from $h^1(\\enm{\\cal{I}}_{p,X}(-1))=1$ and $h^2(\\enm{\\cal{O}}_X(m-4))=1$. Thus $\\enm{\\cal{G}}$ is not aCM, a contradiction. If $X$ is integral, then every line bundle is stable and so (\\ref{eqi1}) is the HN-filtration of $\\enm{\\cal{E}}$, we get either $\\enm{\\cal{L}} \\cong \\enm{\\cal{O}} _X(m-3)$; we get a contradiction as above, or $\\enm{\\cal{L}} $ is a factor of $\\oplus _{p\\in S} \\enm{\\cal{I}} _{p,X}$, which is not locally free, a contradiction.\n\\end{proof}\n\n\n\n\nLet $\\enm{\\mathbb{F}} (S)$ (resp. $\\enm{\\mathbb{F}} '(S)$) be the set of isomorphism classes of middle terms of extensions in $\\enm{\\mathbb{E}} (S)$ (resp. $\\enm{\\mathbb{E}} '(S)$). Let us denote by $\\enm{\\cal{E}}=\\enm{\\cal{E}}(\\epsilon)$ the middle term of the extension corresponding to $\\epsilon \\in \\enm{\\mathbb{E}}'(S)$.\n\n\\begin{lemma}\\label{x2}\nFor two non-empty finite sets $S_1, S_2\\subset X_{\\mathrm{reg}}$ with $\\sharp (S_i)=s_i$, take $\\enm{\\cal{E}}_i\\in \\enm{\\mathbb{F}}'(S_i)$ and call $\\enm{\\cal{A}}_i$ the subsheaf of $\\enm{\\cal{E}}_i$ isomorphic to $\\enm{\\cal{O}}_X(m-3)^{\\oplus s_i}$ for each $i=1,2$. If there exists a map $f: \\enm{\\cal{E}}_1 \\rightarrow \\enm{\\cal{E}}_2$ with $f(\\enm{\\cal{E}}_1) \\subset \\enm{\\cal{A}}_2$, then we have $S_1 \\cap S_2\\ne \\emptyset$. \n\\end{lemma}\n\n\\begin{proof}\nSince $\\op{Hom}_X (\\enm{\\cal{O}} _X(m-3),\\enm{\\cal{I}} _{p,X}) =0$ for all $p\\in X$, we have $f(\\enm{\\cal{A}} _1)\\subseteq \\enm{\\cal{A}} _2$. In particular, $f$ induces a nonzero map $\\tilde{f}: \\oplus _{p\\in S_1}\\enm{\\cal{I}} _{p,X} \\rightarrow \\oplus _{q\\in S_2} \\enm{\\cal{I}} _{q,X}$. This implies that $S_1\\cap S_2 \\ne \\emptyset$.\n\\end{proof}\n\n\\begin{lemma}\\label{x1}\nAssume that $\\enm{\\cal{E}}\\in \\enm{\\mathbb{F}}'(S)$ is decomposable; $\\enm{\\cal{E}} \\cong  \\enm{\\cal{E}} _1\\oplus \\cdots \\oplus \\enm{\\cal{E}} _h$ with each $\\enm{\\cal{E}} _i$ indecomposable. Then there is a partition $S = \\sqcup _{i=1}^{h} S_i$ with $\\enm{\\cal{E}} _i\\in \\enm{\\mathbb{F}} '(S_i)$ for each $i$. If there is another decomposition $\\enm{\\cal{E}} \\cong \\enm{\\cal{E}} _1'\\oplus \\cdots \\oplus \\enm{\\cal{E}} _k'$ with each $\\enm{\\cal{E}}_j'$ indecomposable, then we get $k=h$ and there is a permutation $\\sigma : \\{1,\\dots ,h\\}\\rightarrow\\{1,\\dots ,h\\}$ such that $\\enm{\\cal{E}} _{\\sigma (i)}' \\cong \\enm{\\cal{E}} _i$ for all $i$ and $\\enm{\\cal{E}} _{\\sigma (i)}'\\in \\enm{\\mathbb{F}} (S_{\\sigma\n(i)})$.\n\\end{lemma}\n\n\\begin{proof}\nWe use induction on $s$. The case $s=1$ is true, because each $\\enm{\\cal{E}} _p$ for $p\\in X_{\\mathrm{reg}}$ is indecomposable by Proposition \\ref{a1.1}. Since $\\enm{\\cal{E}}$ is aCM by the definition of $\\enm{\\mathbb{F}} (S)$, each $\\enm{\\cal{E}} _i$ is also aCM. We consider the subsheaf $\\enm{\\cal{A}} \\cong \\enm{\\cal{O}}_X(m-3)^{\\oplus s}\\subset \\enm{\\cal{E}}$ as in Remark \\ref{x00} and set $\\enm{\\cal{G}} _i:= \\enm{\\cal{A}} \\cap \\enm{\\cal{E}}_i$. Since the HN filtration of $\\enm{\\cal{E}}$ is obtained from the ones of each factors, we have \n$$\\enm{\\cal{A}} \\cong \\oplus _{i=1}^{h} \\enm{\\cal{G}} _i \\phantom{AA}\\text{and}\\phantom{AA}\\oplus _{p\\in S}\\enm{\\cal{I}}_{p,X}\\cong \\oplus_{i=1}^h \\enm{\\cal{E}} _i\/\\enm{\\cal{G}} _i.$$\nBy Lemma \\ref{x0} we have $\\enm{\\cal{G}} _i\\subsetneq \\enm{\\cal{E}} _i$ for all $i$. By Remark \\ref{c3} we may write $S = \\sqcup _{i=1}^{h} S_i $ with $\\enm{\\cal{E}}_i\/\\enm{\\cal{G}} _i\\cong \\oplus _{p\\in S_i}\\enm{\\cal{I}}_{p, X}$. Since $\\enm{\\cal{E}} _i\/\\enm{\\cal{G}} _i \\ne 0$, we have $S_i\\ne \\emptyset$ for all $i$. Thus the set $\\{S_1,\\dots ,S_h\\}$ gives a partition of $S$. To prove the first part of the lemma it is sufficient to prove that $\\sharp (S_i) =\\mathrm{rank}(\\enm{\\cal{G}} _i)\/2$ for all $i$. If this is not true, then there is $i\\in \\{1,\\dots ,h\\}$ with $\\sharp (S_i) > \\mathrm{rank}(\\enm{\\cal{G}} _i)\/2$, i.e. $\\mathrm{rank}(\\enm{\\cal{G}} _i\\cap \\enm{\\cal{A}})> \\sharp (S_i)$. The exact sequence\n$$0 \\to \\enm{\\cal{A}}\\cap \\enm{\\cal{G}} _i\\to \\enm{\\cal{G}} _i\\to \\oplus _{p\\in S_j}\\enm{\\cal{I}} _{p,X}\\to 0$$ \ngives $h^1(\\enm{\\cal{G}} _i)\\ge \\sharp (S_i)-\\mathrm{rank}(\\enm{\\cal{G}} _i\\cap\\enm{\\cal{A}}) >0$. In particular, $\\enm{\\cal{G}} _i$ is not aCM, a contradiction.\n\nNow we check the last assertion of the lemma. Take two partitions \n$$S = S_1\\sqcup \\cdots \\sqcup S_h= S_1'\\sqcup \\cdots \\sqcup S_k'$$ \nsuch that there is a decomposition \n$$\\enm{\\cal{E}} \\cong  \\enm{\\cal{E}} _1\\oplus \\cdots \\oplus \\enm{\\cal{E}} _h \\cong  \\enm{\\cal{E}} _1'\\oplus \\cdots \\oplus \\enm{\\cal{E}} _k'$$ \nwith $\\enm{\\cal{E}}_i\\in \\enm{\\mathbb{F}}'(S_i) $ and $\\enm{\\cal{E}} _j'\\in \\enm{\\mathbb{F}}'(S_j')$ indecomposable. By the Krull-Schmidt theorem in \\cite{Atiyah}, we get $h=k$ and there is a permutation $\\sigma : \\{1,\\dots ,h\\}\\rightarrow \\{1,\\dots ,h\\}$ such that $\\enm{\\cal{B}} _{\\sigma (i)} \\cong \\enm{\\cal{E}} _i$ for all $i$. By renaming $\\{\\enm{\\cal{E}} _1',\\dots ,\\enm{\\cal{E}}_h'\\}$, we may assume that $\\enm{\\cal{E}} _i' \\cong \\enm{\\cal{E}} _i$ for all $i$. This implies\n$$\\sharp (S_i)=\\mathrm{rank}(\\enm{\\cal{E}}_i)\/2=\\mathrm{rank}(\\enm{\\cal{E}}_i')\/2=\\sharp(S_i').$$\nNow fix an isomorphism $f_i: \\enm{\\cal{E}} _i\\rightarrow \\enm{\\cal{E}} _i'$ for each $i$. Since (\\ref{eqi1}) gives the HN filtrations of $\\enm{\\cal{E}} _i$ and $\\enm{\\cal{E}}_i'$, the map $f$ induces an isomorphism $\\tilde{f_i}: \\oplus _{p\\in S_i}\\enm{\\cal{I}} _{p,X}\\rightarrow  \\oplus _{p\\in S_i'} \\enm{\\cal{I}} _{p,X}$. Since $p$ is the unique point of $X$ at which $\\enm{\\cal{I}} _{p,X}$ is not locally free, we get $S_i=S_i'$. For each $i$, let $\\enm{\\cal{A}}_i$ be the unique subsheaf of $\\enm{\\cal{E}} _i$ isomorphic to $\\enm{\\cal{O}} _X(m-3)^{\\sharp (S_i)}$. Then for any embedding $u: \\enm{\\cal{E}} _i \\rightarrow  \\enm{\\cal{E}} _1\\oplus \\cdots \\oplus \\enm{\\cal{E}} _h$, the composition $v_j\\circ \\pi _j \\circ u$\n$$\\enm{\\cal{E}}_i \\stackrel{u}{\\to} \\enm{\\cal{E}}_1\\oplus \\cdots \\oplus \\enm{\\cal{E}}_h \\stackrel{\\pi_j}{\\to} \\enm{\\cal{E}}_j \\stackrel{v_j}{\\to} \\oplus_{p\\in S_j}\\enm{\\cal{I}}_{p,X}$$\nis zero for any $j\\ne i$ by Lemma \\ref{x2}, where $\\pi _j: \\enm{\\cal{E}} \\rightarrow \\enm{\\cal{E}} _j$ is the projection and $v_j: \\enm{\\cal{E}} _j\\rightarrow \\oplus _{p\\in S_j}\\enm{\\cal{I}} _{p,X}$ is the surjection in (\\ref{eqi1}) for $S_j$. Since $u$ is an embedding, we see that $v_i\\circ \\pi _i \\circ u$ is surjective. Thus $\\enm{\\cal{G}}:= \\pi _i(u(\\enm{\\cal{E}} _i))$ is a subsheaf with $v_i(\\enm{\\cal{G}} ) = \\oplus _{p\\in S_i}\\enm{\\cal{I}} _{p,X}$. \n\\end{proof}\n\n\\begin{lemma}\\label{ppo}\nWith the setting as in Theorem \\ref{i1}, we have $\\mathrm{ext}_X^1(\\enm{\\cal{E}}_p, \\enm{\\cal{E}}_q)\\ge 2$ for two points $p,q\\in X_{\\mathrm{reg}}$, possibly $p=q$. \n\\end{lemma}\n\n\\begin{proof}\nSet $\\enm{\\cal{F}} _o:= \\enm{\\cal{E}} _o(3-m)$ for $o\\in \\{p,q\\}$. Since $\\op{Ext} ^i_X(\\enm{\\cal{E}} _p,\\enm{\\cal{E}} _q) \\cong \\op{Ext}_X ^i(\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _q)$, we have $\\chi (\\enm{\\cal{E}} _p\\otimes \\enm{\\cal{E}} _q^\\vee ) = \\chi (\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _q^\\vee)$. Since Euler's characteristic is constant in a flat family of vector bundles and $p,q\\in X_{\\mathrm{reg}}$, it is sufficient to compute $\\chi (\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _q^\\vee)$ when $X$ is smooth. Since a smooth surface in $\\enm{\\mathbb{P}}^3$ is connected, the same observation applied to a family of vector bundles on $X$ shows $\\chi (\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _q^\\vee) =\\chi (\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee)$.\n\nWe have an exact sequence\n\\begin{equation}\\label{eqii1}\n0 \\to \\enm{\\cal{O}} _X \\stackrel{v}{\\to} \\enm{\\cal{F}} _p \\stackrel{w}{\\to} \\enm{\\cal{I}} _{p,X}(3-m)\\to 0\n\\end{equation}\nwith $\\det (\\enm{\\cal{F}} _p)\\cong \\enm{\\cal{O}} _X(3-m)$ and $c_2(\\enm{\\cal{F}} _p)=1$. Since $X\\subset \\enm{\\mathbb{P}}^3$ is a surface of degree $m$, we have $c_1(\\enm{\\cal{F}} _p)^2 = m(m-3)^2$. By Riemann-Roch for $\\mathcal{E}nd (\\enm{\\cal{F}} _p)$, we have \n\\begin{align*}\n\\chi (\\mathcal{E}nd (\\enm{\\cal{F}} _p ))& = c_1(\\enm{\\cal{F}} _p)^2 -4c_2(\\enm{\\cal{F}} _p) + 4\\chi (\\enm{\\cal{O}} _X) = m(m-3)^2 -4 +4\\binom{m-1}{3} +4\\\\\n&=\\frac{1}{6}\\left(10m^3-60m^2+98m-24\\right).\n\\end{align*}\nIn particular, we have $\\chi \\sim \\frac{5}{3}m^3$ for $m\\gg 0$. Note that by Serre's duality we have $h^2(\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee )=h^0(\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee (m-4))$.\n\n\\quad \\emph{Claim 1:} We have $\\mathrm{hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p) =1+\\binom{m}{3}$.\n\n\\quad \\emph{Proof of Claim 1:} We have $\\mathrm{hom}_X (\\enm{\\cal{I}} _{p,X}(3-m),\\enm{\\cal{O}} _X) =h^0(\\enm{\\cal{O}} _X(m-3)) = \\binom{m}{3}$ and any nonzero map $\\enm{\\cal{I}} _{p,X}(3-m) \\rightarrow \\enm{\\cal{O}} _X$ induces an element in $\\op{Hom}_X(\\enm{\\cal{F}}_p, \\enm{\\cal{F}}_p)$ with rank one as the following composition:\n$$\\enm{\\cal{F}}_p \\stackrel{w}{\\to} \\enm{\\cal{I}}_{p,X}(3-m) \\to \\enm{\\cal{O}}_X \\stackrel{v}{\\to} \\enm{\\cal{F}}_p.$$\nThe vector space $\\op{Hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p)$ also contains the nonzero multiples of the identity map $\\enm{\\cal{F}} _p\\rightarrow \\enm{\\cal{F}} _p$ and these maps have rank two. Thus we get $h^0(\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee )\\ge 1+\\binom{m}{3}$. On the other hand, for any $f\\in \\op{Hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p)$ we get $w\\circ f \\circ (v(\\enm{\\cal{O}} _X)) \\subseteq v(\\enm{\\cal{O}} _X)$ from $h^0(\\enm{\\cal{I}} _{p,X}(3-m)) =0$. Thus $w\\circ f\\circ v$ induces a map $f_1: \\enm{\\cal{O}} _X\\rightarrow \\enm{\\cal{O}}_X$, which is induced by the multiplication by $c\\in \\mathbf{k}$. Hence $f-c{\\cdot}\\mathrm{Id} _{\\enm{\\cal{F}} _p}$ is induced by a unique $g\\in \\op{Hom}_X (\\enm{\\cal{I}} _{p,X}(3-m),\\enm{\\cal{F}} _p)$. Since $\\enm{\\cal{F}} _p$ is locally free and $X$ is smooth at $p$, we have $\\op{Hom}_X (\\enm{\\cal{I}} _{p,X}(3-m),\\enm{\\cal{F}} _p) = H^0(\\enm{\\cal{F}} _p(m-3))$. By\n(\\ref{eqii1}) we have $h^0(\\enm{\\cal{F}}_p (m-3)) =\\binom{m}{3}$ and so $\\mathrm{hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p) \\le 1+\\binom{m}{3}$. \\qed\n\n \\quad \\emph{Claim 2:} We have $\\mathrm{hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p(m-4)) \\ge \\binom{2m-4}{3} +2\\binom{m-1}{3} -\\binom{m-4}{3}-1$.\n\n\\quad \\emph{Proof of Claim 2:} For any $f \\in \\op{Hom}_X (\\enm{\\cal{F}} _p, \\enm{\\cal{F}} _p(4-m))$, set $f_1:= f_{|v(\\enm{\\cal{O}} _X)}$. Since $h^0(\\enm{\\cal{O}} _X(-1)) =0$, we have $w\\circ f_1 =0$ and so $f_1(v(\\enm{\\cal{O}} _X)) \\subset v(\\enm{\\cal{O}} _X(m-4)))$. Take $f$ with $f_1\\equiv 0$. Such a map $f$ is uniquely determined by an element in $\\op{Hom}_X (\\enm{\\cal{I}} _{p,X}(3-m), \\enm{\\cal{F}} _p(m-4))$ and the converse also holds. Since $\\enm{\\cal{F}} _p(m-4)$ is locally free and $X$ is smooth at $p$, we have $\\op{Hom}_X (\\enm{\\cal{I}} _{p,X}(3-m), \\enm{\\cal{F}} _p(m-4)) = \\op{Hom}_X (\\enm{\\cal{O}} _X(3-m),\\enm{\\cal{F}} _p(m-4)) = H^0(\\enm{\\cal{F}} _p(2m-7))$. Since $h^1(\\enm{\\cal{O}} _X(t)) =0$ for any $t\\in \\enm{\\mathbb{Z}}$, (\\ref{eqii1}) gives \n$$h^0(\\enm{\\cal{F}} _p(2m-7)) = h^0(\\enm{\\cal{O}} _X(2m-7)) +h^0(\\enm{\\cal{O}} _X(m-4))-1 = \\binom{2m-4}{3} -\\binom{m-4}{3}+\\binom{m-1}{3}-1.$$ \nNote that a map $f$ obtained by a composition\n$$\\enm{\\cal{F}}_p \\stackrel{w}{\\to} \\enm{\\cal{I}}_{p,X}(3-m) \\to \\enm{\\cal{O}}_X(m-4) \\stackrel{v}{\\to} \\enm{\\cal{F}}_p(m-4)$$\nhas $f_1 \\equiv 0$. Now for any linear subspace $W\\subset \\op{Hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p(m-4))$ such that $f_1\\not\\equiv 0$ for any $f\\in W\\setminus \\{0\\}$, we would get \n$$\\mathrm{hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p(m-4)) \\ge \\binom{2m-4}{3} -\\binom{m-4}{3}+\\binom{m-1}{3}-1+\\dim W.$$\nWe may choose $W$ to consist of the compositions of the identity map $\\enm{\\cal{F}} _p\\rightarrow \\enm{\\cal{F}} _p$ with the multiplication by an element of $H^0(\\enm{\\cal{O}} _X(m-4))$. Then we have $\\dim W=\\binom{m-1}{3}$.   \\qed\n\nCombining Claims 1 and 2, we get\n\\begin{align*}\nh^0(\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee) +h^2(\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee )&\\ge \\binom{2m-4}{3} +\\binom{m}{3}+2\\binom{m-1}{3}-\\binom{m-4}{3}\\\\\n&=\\frac{1}{6}\\left( 10m^3-60m^2+98m-12\\right).\n\\end{align*}\nThus we have \n$$\nh^1(\\enm{\\cal{F}}_p\\otimes \\enm{\\cal{F}}_p^\\vee)=h^0(\\enm{\\cal{F}}_p\\otimes \\enm{\\cal{F}}_p^\\vee)+h^2(\\enm{\\cal{F}}_p\\otimes \\enm{\\cal{F}}_p^\\vee)-\\chi(\\mathcal{E}nd(\\enm{\\cal{F}}_p))\\ge 2\n$$\nand so we get the assertion. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{i1}:]\nBy Remark \\ref{x00} (\\ref{eqi1}) is the HN filtration of $\\enm{\\cal{E}}_S$. Proposition \\ref{a1.1} gives the case $s=1$. For $s>1$, we may find a direct sum of $s$ vector bundles of rank $2$ from the case $s=1$, fitting into (\\ref{eqi1}): just take $\\oplus _{p\\in S} \\enm{\\cal{E}}_p$. So a general extension in $\\mathbb{E}(S)$ has a locally free and aCM middle term, because being local free and aCM are both open conditions. \n\nNote that $h^0(\\enm{\\cal{E}}_S(3-m))=s$ from (\\ref{eqi1}). In particular there is a unique subsheaf $\\enm{\\cal{A}}\\subset \\enm{\\cal{E}}_S$ isomorphic to $\\enm{\\cal{O}} _X(m-3)^{\\oplus s}$ and for each $f\\in \\mathrm{Hom}(\\enm{\\cal{O}} _X(m-3),\\enm{\\cal{E}}_S)$ we have $f(\\enm{\\cal{O}} _X(m-3)) \\subseteq \\enm{\\cal{A}}$. Now by Lemma \\ref{c2} and Remark \\ref{c3}, the extension (\\ref{eqi1}) is the HN filtration of $\\enm{\\cal{E}}_S$. By uniqueness of the HN filtration, we get $\\enm{\\cal{E}}_S \\ncong \\enm{\\cal{E}}_{S'}$ for $S\\ne S'$. \n\nNow it remains to show the indecomposability of $\\enm{\\cal{E}}_S$. By Lemma \\ref{x0}, there is no rank one factor of $\\enm{\\cal{E}}_S$.\n\n\\quad \\emph{Claim 1:} For two distinct points $p, q$ in $X_{\\mathrm{reg}}$, we have \n$$\\mathrm{Hom}_X(\\enm{\\cal{I}} _{p,X},\\enm{\\cal{I}} _{q,X}) =0, \\mathrm{Hom}_X(\\enm{\\cal{E}}_p,\\enm{\\cal{I}} _{q,X}) =0 \\text{ and } \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{p,X} , \\enm{\\cal{I}}_{q,X})=0.$$  \n\n\\quad \\emph{Proof of Claim 1:} By an extension theorem for locally free sheaves in \\cite[Exercise I.3.20]{Hartshorne}, we have $\\mathrm{Hom}_X(\\enm{\\cal{I}} _{p,X},\\enm{\\cal{I}} _{q,X}) = \\mathrm{Hom}_X(\\enm{\\cal{O}} _X,\\enm{\\cal{I}} _{q,X}) =0$. The second vanishing is obtained from the first vanishing and $\\mathrm{Hom}_X(\\enm{\\cal{O}} _X(m-3),\\enm{\\cal{I}} _{q,X}) =0$. For the last vanishing, we apply the functor $\\mathrm{Hom}_X(\\enm{\\cal{I}}_{p,X}, -)$ to the standard exact sequence for $\\enm{\\cal{I}}_{q,X}\\subset \\enm{\\cal{O}}_X$ and obtain an exact sequence\n$$0 \\to \\mathrm{Hom}_X(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X) \\to \\mathrm{Hom}_X(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_q) \\to \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{I}}_{q,X}) \\to \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X)$$\nby the first vanishing in the Claim. Here we have\n$$\\mathrm{Hom}_X(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X) \\cong \\mathrm{Hom}_X(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_q) \\cong \\mathbf{k}$$\nand $\\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X) \\cong H^1(\\enm{\\cal{I}}_{p,X}(m-4))^\\vee$ by Serre's duality. Then we get the assertion from the assumption that $m\\ge 4$.  \\qed\n\n\\quad {(a)} First assume $s=2$ and take  two distinct points $p, q$ in $X_{\\mathrm{reg}}$.\n\n\\quad \\emph{Claim 2:} If there exists a sheaf $\\enm{\\cal{G}}\\ncong \\enm{\\cal{E}} _p\\oplus \\enm{\\cal{E}} _q$ fitting into the exact sequence\n\\begin{equation}\\label{eqa3.1}\n0 \\to \\enm{\\cal{E}} _p\\stackrel{u}{\\to} \\enm{\\cal{G}} \\stackrel{v}{\\to} \\enm{\\cal{E}} _q\\to 0,\n\\end{equation}\nthen the case $s=2$ is true.\n\n\\quad \\emph{Proof of Claim 2:} Such a sheaf $\\enm{\\cal{G}}$ would be locally free and aCM with rank $4$. Since $h^1(\\enm{\\cal{O}} _X)=0$ and (\\ref{eqi1}) gives the HN filtrations of $\\enm{\\cal{E}} _p$ and $\\enm{\\cal{E}} _q$ by Lemmas \\ref{c2}\nand Remark \\ref{c3}, $\\enm{\\cal{G}}$ has a subsheaf $\\enm{\\cal{F}} \\cong \\enm{\\cal{O}} _{X}(m-3)^{\\oplus 2}$ such that $\\enm{\\cal{G}} \/\\enm{\\cal{F}}$ is an extension of $\\enm{\\cal{I}} _{q,X}(1)$ by $\\enm{\\cal{I}} _{p,X}(1)$. Claim 1 gives $\\enm{\\cal{G}}\/\\enm{\\cal{F}} \\cong \\enm{\\cal{I}}_{p,X}\\oplus \\enm{\\cal{I}}_{q,X}$ and so we get $\\enm{\\cal{G}} \\cong \\enm{\\cal{E}} _S$ with $S =\\{p,q\\}$.\\qed\n\n\\quad \\emph{Claim 3:} If $\\enm{\\cal{G}} \\cong \\enm{\\cal{E}} _p\\oplus \\enm{\\cal{E}} _q$ for all $\\enm{\\cal{G}}$ in (\\ref{eqa3.1}), then we have $\\op{Ext}_X ^1(\\enm{\\cal{E}} _q,\\enm{\\cal{E}} _p)=0$.\n\n\\quad \\emph{Proof of Claim 3:} Let $\\enm{\\cal{G}} \\cong \\enm{\\cal{E}} _p\\oplus \\enm{\\cal{E}} _q$ fitting into (\\ref{eqa3.1}) correspond to $\\epsilon \\in \\op{Ext}_X ^1(\\enm{\\cal{E}} _q,\\enm{\\cal{E}} _p)$. Then it is sufficient to prove that $\\epsilon =0$, or $\\mathrm{ker}(v) \\cong \\enm{\\cal{E}}_p\\oplus \\{0\\}$. But since $\\mathrm{ker}(v) \\cong \\enm{\\cal{E}} _p$, it is sufficient to prove that either $\\enm{\\cal{E}} _p\\oplus \\{0\\} \\supseteq \\mathrm{ker}(v)$ or $\\enm{\\cal{E}} _p\\oplus \\{0\\} \\subseteq \\mathrm{ker}(v)$. Assume $v(\\enm{\\cal{E}} _p\\oplus \\{0\\}) \\ne 0$. Since $\\mathrm{Hom}_X(\\enm{\\cal{E}}_p,\\enm{\\cal{I}} _{q,X}) =0$ by Claim 1, we have $v(\\enm{\\cal{E}} _p\\oplus \\{0\\})\\subseteq \\enm{\\cal{O}} _X(m-3)$. This implies that the restriction of the surjection $\\enm{\\cal{E}} _q\\rightarrow \\enm{\\cal{I}} _{q,X}$ to $v(\\{0\\}\\oplus \\enm{\\cal{E}} _q)$ is surjective. Since $h^0(\\enm{\\cal{O}} _X)=1$ and $\\op{Hom}_X (\\enm{\\cal{O}} _X(m-3),\\enm{\\cal{I}} _{q,X}) =0$, we get either $v(\\{0\\}\\oplus \\enm{\\cal{O}} _X(m-3)) =0$ or $v$ induces an isomorphism $\\{0\\}\\oplus \\enm{\\cal{O}} _X(m-3)\\rightarrow \\enm{\\cal{O}} _X(m-3)$. Assume for the moment $v(\\{0\\}\\oplus \\enm{\\cal{O}} _X(m-3)) =0$. Since $v(\\enm{\\cal{E}} _p\\oplus \\{0\\})$ maps to $0$ in $\\enm{\\cal{I}} _{q,X}$, we get that $v(\\{0\\}\\oplus \\enm{\\cal{E}} _q)$ is a subsheaf of $\\enm{\\cal{E}} _q$ which maps isomorphically onto $\\enm{\\cal{I}} _{q,X}$. So we get $\\enm{\\cal{E}} _q\\cong \\enm{\\cal{O}} _X(m-3)\\oplus \\enm{\\cal{I}} _{q,X}$, a contradiction. Now assume $v(\\{0\\}\\oplus \\enm{\\cal{O}} _X(m-3)) =\\enm{\\cal{O}} _X(m-3)$. Since $v(\\{0\\}\\oplus \\enm{\\cal{E}} _q)$ maps surjectively onto $\\enm{\\cal{I}} _{q,X}$, the surjection $v$ induces an isomorphism $\\{0\\}\\oplus \\enm{\\cal{E}} _q\\rightarrow \\enm{\\cal{E}} _q$. Hence we get $\\enm{\\cal{E}} _p\\oplus \\{0\\} \\subseteq \\mathrm{ker}(v)$. \\qed\n\n\\noindent Since $\\op{Ext}_X ^1(\\enm{\\cal{E}}_q,\\enm{\\cal{E}} _p) \\ne 0$ by Lemma \\ref{ppo}, Claim 3 concludes the proof of the case $s=2$.\n\n\\quad {(b)} Assume $s>2$ and that Theorem \\ref{i1} holds for smaller numbers. On $\\mathbb{E}(S)$ there is a universal family of extensions, i.e. a coherent sheaf $\\enm{\\cal{V}}$ over $\\mathbb{E}(S)\\times X$ such that\nfor each $\\epsilon \\in \\mathbb{E}(S)$ the sheaf $\\enm{\\cal{V}} _{|\\{\\epsilon \\}\\times X}$ is the middle term $\\enm{\\cal{E}} (\\epsilon )$ of the extension corresponding to $\\epsilon$; in general, if we take $\\enm{\\mathbb{P}}(\\mathbb{E}(S))$ as a parameter space, then no such a universal sheaf exists. We call $\\enm{\\cal{V}} '$ the restriction of of $\\enm{\\cal{V}}$ to $\\enm{\\mathbb{E}} '(S)\\times X$; we thus consider the family of aCM vector bundles induced from the extensions in $\\enm{\\mathbb{E}}'(S)$. \n\nDefine a set $\\Gamma(S)$ as follows:\n$$\\Gamma(S) :=\\left\\{(\\epsilon ,\\phi)~|~\\epsilon \\in \\enm{\\mathbb{E}}' (S) \\text{ and } \\phi \\in \\op{End} (\\enm{\\cal{E}} (\\epsilon )) \\text{ with }\\phi ^2 = \\phi \\right \\}.$$ \nNote that $\\phi$ is a projection of $\\enm{\\cal{E}} (\\epsilon)$ onto a factor of $\\enm{\\cal{E}} (\\epsilon)$, with the exception when $\\phi = \\mathrm{Id}_{\\enm{\\cal{E}} (\\epsilon)}$ or $\\phi \\equiv 0$; if $\\enm{\\cal{E}} (\\epsilon )$ is indecomposable, only $(\\epsilon ,\\mathrm{Id}_{\\enm{\\cal{E}} (\\epsilon)})$ and $(\\epsilon ,0)$ are contained in $\\Gamma(S)$. Indeed, for any vector bundle $\\enm{\\cal{G}}$, there exists a one-to-one correspondence:\n$$\\{ \\phi \\in \\op{End}(\\enm{\\cal{G}}) ~|~ \\phi^2=\\phi\\} \\leftrightarrow \\{ \\text{factors of }\\enm{\\cal{G}}\\}$$ \nvia $\\phi \\mapsto \\mathrm{Im}(\\phi) = \\mathrm{ker}(\\mathrm{Id}_{\\enm{\\cal{G}}} -\\phi)$, with $\\enm{\\cal{G}}$ being associated to $\\mathrm{Id}_{\\enm{\\cal{G}}}$ and $0$ associated to the zero map. Thus $\\enm{\\cal{G}}$ is decomposable if and only if $\\op{End} (\\enm{\\cal{G}} )$ has a non-trivial idempotent. Note that $\\Gamma(S)$ is a closed in the total space of the vector bundle $\\mathcal{H}om(\\enm{\\cal{V}}' , \\enm{\\cal{V}}')$ over $\\enm{\\mathbb{E}} '(S)\\times X$. By Lemma \\ref{x1}, for each $\\enm{\\cal{E}} (\\epsilon )$ there is a unique partition of $S$ associated to any decomposition of $\\enm{\\cal{E}} (\\epsilon )$ with only finitely many indecomposable factors by the Krull-Schmidt theorem in \\cite{Atiyah}. By Lemma \\ref{x1} for each $\\enm{\\cal{E}} \\in \\enm{\\mathbb{F}}' (S)$ each isomorphism class of factors of $\\enm{\\cal{E}}$ corresponds to a unique subset of $S$; $\\enm{\\cal{E}}$ and $0$ correspond to $S$ and $\\emptyset$, respectively. For each $(\\epsilon,\\phi)\\in \\Gamma(S)$, let $S(\\phi)$ be the subset of $S$ associated to $\\mathrm{Im}(\\phi)$ by Lemma \\ref{x1}. Set \n$$\\Gamma _0(S) :=\\left\\{(\\epsilon, \\phi)\\in \\Gamma(S) ~\\big|~ \\phi\\ne 0 \\text{ and } \\phi\\ne \\mathrm{Id}_{|\\enm{\\cal{E}}(\\epsilon)}\\right\\}. $$\nThe goal is to show that $\\Gamma_0(S)$ is not dominant over $\\enm{\\mathbb{F}}(S)$ for a general $S$. \n\nNote that up to now we did not use that $S$ is contained in the same connected component $Y\\cap X_{\\mathrm{reg}}$ of $X_{\\mathrm{reg}}$. In particular the case $s=2$ holds even if $X$ has more than one irreducible components with multiplicity one and the two points of $S$ belong to different connected components of $X_{\\mathrm{reg}}$. \n\nNow we use a monodromy argument, which requires that $S$ is contained in a connected component of $T:=X_{\\mathrm{reg}}\\cap Y$ and that $S$ is general in $Y$. Set $S=\\{p_1, \\ldots, p_s\\}$ and fix an ordering of the points in $S$, along which we get an ordering of the indecomposable factors of the sheaf $\\oplus _{p\\in S} \\enm{\\cal{I}}_{p,X}$. Together with the usual ordering on the factors of $\\mathcal {O}_X(m-3)^{\\oplus s}$, we may see any $\\epsilon \\in \\mathbb{E}(S)$ as an $(s\\times s)$-square matrix, say $\\epsilon =(\\epsilon_{ij})$ with $1\\le i, j\\le s$, where $\\epsilon _{ij}$ is an element of the $1$-dimensional vector space $\\op{Ext}_X ^1(\\enm{\\cal{I}} _{p_j,X},\\enm{\\cal{O}}_X(m-3))$. Note that for a fixed integer $j$, each $\\epsilon_{ij}$ with $i=1,\\dots ,s$, is an element of the same $1$-dimensional vector space. We write $\\enm{\\cal{O}} _X(m-3)^{\\oplus s} = \\enm{\\mathbb{C}} ^s\\otimes \\enm{\\cal{O}} _X(m-3)$. \n\n\\quad \\emph{Claim 4:} $\\enm{\\cal{E}}=\\enm{\\cal{E}}(\\epsilon)$ has two indecomposable factors, one of them being $\\mathrm{Im}(\\phi)$ and the other one being $\\mathrm{ker}(\\phi)$.\n\n\\quad \\emph{Proof of Claim 4:} Since $\\phi^2=\\phi$, we have $\\enm{\\cal{E}} \\cong \\enm{\\cal{F}}_1 \\oplus \\enm{\\cal{F}}_2$ with $\\enm{\\cal{F}} _1:= \\mathrm{Im}(\\phi)$ and $\\enm{\\cal{F}}_2= \\mathrm{ker}(\\phi)$. By the definition of $A$, we get an exact sequence\n\\begin{equation}\\label{eert1}\n0\\to \\enm{\\cal{O}}_X(m-3)^{\\oplus k} \\to \\enm{\\cal{F}}_1 \\to \\oplus_{p\\in A}\\enm{\\cal{I}}_{p,X} \\to 0,\n\\end{equation}\nwith $k:= \\sharp (A)$. Since neither $\\phi \\equiv 0$ nor $\\phi =\\mathrm{Id} _{\\enm{\\cal{E}}}$, we have $0<k<s$. Then by Lemma \\ref{x1} we get an exact sequence\n\\begin{equation}\\label{eert2}\n0\\to \\enm{\\cal{O}}_X(m-3)^{\\oplus (s-k)} \\to \\enm{\\cal{F}}_2 \\to \\oplus_{p\\in S\\setminus A}\\enm{\\cal{I}}_{p,X} \\to 0.\n\\end{equation}\nNow we need to prove that each $\\enm{\\cal{F}}_i$ is indecomposable. By the inductive assumption it is sufficient to prove that $\\enm{\\cal{F}}_1$ and $\\enm{\\cal{F}}_2$ are the middle terms of general extensions (\\ref{eert1}) and (\\ref{eert2}), respectively. Since (\\ref{eqi1}) gives the HN filtration of each $\\enm{\\cal{F}}_i$, there are linear subspaces $V_1,V_2\\subset \\enm{\\mathbb{C}}^s$ such that $\\dim V_1 =k$, $\\dim V_2 =s-k$ and \n$$v(\\enm{\\mathbb{C}} ^s\\otimes \\enm{\\cal{O}} _X(m-3))\\cap \\enm{\\cal{F}}_i = V_i\\otimes \\enm{\\cal{O}} _X(m-3)$$\nfor each $i$. From $\\enm{\\cal{E}} \\cong \\enm{\\cal{F}}_1\\oplus \\enm{\\cal{F}}_2$ we see that $\\enm{\\mathbb{C}}^{s} =V_1\\oplus V_2$. Now we reorder  the points in $S$ so that all points of $A$ are smaller than any points of $S\\setminus A$. Then $\\epsilon$ can be understood as an $(s\\times s)$-square matrix in a block form:\n\\[\n\\epsilon=\n\\left[\n\\begin{array}{c|c}\nB_{11} & B_{12}\\\\\n\\hline\nB_{21} & B_{22}\n\\end{array}\n\\right]\n\\]\nHere the $(k\\times k)$-matrix $B_{11}$ in the upper left corner, is associated to the extension (\\ref{eert1}) and similarly the $((s-k)\\times (s-k))$-matrix $B_{22}$ in the lower right corner, is associated to the extension (\\ref{eert2}). The matrix of $\\epsilon$ also has a $(k\\times (s-k))$-submatrix $B_{12}$ and an $((s-k)\\times k)$-submatrix $B_{21}$. Since $\\epsilon$ is general, all the entries in each $B_{ij}$ are also general. In particular, $B_{11}$ and $B_{22}$ are general and this implies that each  $\\enm{\\cal{F}}_i$ is general. The inductive assumption gives that each $\\enm{\\cal{F}}_i$ is indecomposable.\\qed\n\nAssume that a general $\\enm{\\cal{E}}=\\enm{\\cal{E}}(\\epsilon)$ has two indecomposable factor, i.e. the set $\\Gamma_0(S)$ is dominant over $\\enm{\\mathbb{F}}(S)$. Let $\\Gamma'(S)$ be an irreducible component of $\\Gamma_0(S)$ dominant over $\\enm{\\mathbb{F}} (S)$ and set $A:= S(\\phi)$, where $(\\epsilon ,\\phi )$ is any element of $\\Gamma'(S)$. Now assume that $(\\epsilon ,\\phi)$ is general in $\\Gamma'(S)$ and set $\\enm{\\cal{E}} := \\enm{\\cal{E}} (\\epsilon)$. Note that the subset $A\\subset S$ is invariant as $(\\epsilon, \\phi)$ varies in $\\Gamma_0(S)$, due to the irreducibility of $\\Gamma_0(S)$. Below we find a contradiction under the assumptions that $\\enm{\\cal{E}}$ is decomposable and that $S$ is general in $\\mathrm{Sym}^s(T)$. \n\nLet $\\widetilde{\\Gamma}$ be the set of all triples $(S,\\enm{\\cal{E}} ,\\phi)$ with $S\\in \\mathrm{Sym} ^s(T)$ and $(\\enm{\\cal{E}} ,\\phi )\\in \\Gamma_0 (S)$. Then $\\widetilde{\\Gamma}$ is an algebraic subset whose fibre over $S\\in \\mathrm{Sym}^s(T)$ is $\\Gamma_0(S)$, with a projection map $u: \\widetilde{\\Gamma} \\rightarrow  \\mathrm{Sym} ^s(T)$. If $u$ is not dominant, then it would imply that there exists a $2s$-dimensional family of pairwise not isomorphic indecomposable aCM vector bundles of rank $2s$ on $X$. Thus we may assume that $u$ is dominant. We fix a general $S\\in \\mathrm{Sym} ^s(T)$ and fix an irreducible component $\\Gamma '(S)$ of $\\Gamma (S)$ to which we apply the previous construction with the partition $A\\sqcup (S\\setminus A)$ of $S$ attached to $\\Gamma'(S)$. Let $\\widetilde{\\Gamma}'$ be any irreducible component of $\\widetilde{\\Gamma}$ containing $\\Gamma '(S)$ such that $u_{|\\widetilde{\\Gamma} '}$ is dominant.\n\nLet $\\enm{\\cal{V}}$ denote a non-empty Zariski open subset of $\\mathrm{Sym}^s(T)$ containing $S$ such that for every $T\\in \\enm{\\cal{V}}$ a general $\\enm{\\cal{E}}_T \\in \\enm{\\mathbb{E}} (T)$ has exactly two indecomposable factors, one associated to a subset $F$ of $T$ with $|F| = |A|=k$ and the other one associated to $T\\setminus E$. Now we fix $p\\in A$ and $q\\in S\\setminus A$. Since $Y_{\\mathrm{reg}}$ is a connected manifold and $p, q\\in Y_{\\mathrm{reg}}$, there exists a connected smooth affine curve $U\\subset \\AA^1(\\mathbf{k})$ with a map $\\phi :  U\\rightarrow  Y_{\\mathrm{reg}}$ such that $\\phi (t_0) = p$ and $\\phi (t_1) =q$ for some $t_0, t_1 \\in U$, and $\\phi(U)$ passes no other points of $S$. Similarly we may consider a map $\\phi' : U \\rightarrow Y_{\\mathrm{reg}}$ with $\\phi'(t_1)=p$ and $\\phi'(t_0)=q$ such that $\\phi(t) \\ne \\phi'(t)$ for any $t\\in U$. For each $t\\in U$, set \n$$A_t:= (A\\setminus \\{p\\})\\cup \\{\\phi(t)\\} \\phantom{A}, \\phantom{A}S_t:= (S\\setminus \\{p,q\\}) \\cup \\{\\phi(t) , \\phi'(t)\\},$$\ne.g. $(A_{t_0}, S_{t_0})=(A_{t_1}, S_{t_1})=(A,S)$. Restricting $U$ to an open neighborhood of $\\{t_0, t_1\\}$, we may assume that $S_t\\in \\enm{\\cal{V}}$ for all $t\\in U$. Then for each $t\\in U$ we have a partition $S_t = A_t\\sqcup (S_t\\setminus A_t)$ such that a general $\\enm{\\cal{E}} _{S_t}\\in \\Gamma' (S_t)$ has exactly two indecomposable factors, one associated to $A_t$ and the other associated to $S_t\\setminus A_t$, due to the choice of $\\widetilde{\\Gamma}'$. \n\nWe start from $t=t_0$ and vary $t$ in $U$ to arrive at $t=t_1$, where we have $S_{t_1} =S=A_q\\sqcup (S\\setminus A_q)$ with $A_q=(A\\setminus \\{p\\})\\cup \\{q\\}$. Since $s>2$, we have $\\{A,S\\setminus A\\} \\ne \\{A_q,S\\setminus A_q\\}$, contradicting the assumption that $\\enm{\\cal{E}} _S$ has exactly two indecomposable factors.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Non-locally free aCM sheaf}\\label{sec5}\nIn this section, we let $X\\subset \\enm{\\mathbb{P}}^N$ be a closed subscheme with pure dimension $n$ at least two.  Assume that each local ring $\\enm{\\cal{O}} _{X,x}$ with $x\\in X$, has depth $n$ and that $X$ is aCM with respect\nto $\\enm{\\cal{O}} _X(1)$, i.e. $h^i(\\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N}(t)) =0$ for all $t\\in \\enm{\\mathbb{Z}}$ and all $1\\le i\\le n-1$. The exact sequence\n$$0\\to \\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N}(t) \\to \\enm{\\cal{O}} _{\\enm{\\mathbb{P}}^N}(t) \\to \\enm{\\cal{O}} _X(t)\\to 0$$\nshows that $h^i(\\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N}(t)) = h^{i-1}(\\enm{\\cal{O}} _X(t))$ for all $i\\ge 2$. Hence we may restate our assumption as $h^1(\\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N}(t)) =0$ and $h^i(\\enm{\\cal{O}} _X(t)) =0$ for all $t\\in \\enm{\\mathbb{Z}}$ and $i=1,\\dots ,n-2$. By a theorem of Serre, the condition that $h^i(\\enm{\\cal{O}} _X(-x)) =0$ for $x\\gg 0$ and $i=1,\\dots ,n-2$, plus having positive depth at each $x\\in X$, is equivalent to all $\\enm{\\cal{O}} _{X,x}$ having depth $n$. Since $h^1(\\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N})=0$, we have $h^0(\\enm{\\cal{O}} _X)=1$ and in particular $X$ is connected. Since $h^1(\\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N}(1)) =0$, $X$ is linearly normal in the linear subspace of $\\enm{\\mathbb{P}}^N$ spanned by $X$. Since $n\\ge 2$ we have $h^1(\\enm{\\cal{O}} _X)=0$ an so $\\mathrm{Pic}(X)$ is a finitely generated abelian group. \n\nFix an irreducible component $Y$ of $X_{\\mathrm{red}}$. If $X$ is a hypersurface in $\\enm{\\mathbb{P}}^N$, then the multiplicity $\\mu \\ge 1$ is well-defined. In the general case we do not need the notion of the multiplicity $\\mu$ of $Y$ in $X$ at a general point of $Y$. In this section we need knowledge only on whether $\\mu =1$ or $\\mu >1$. We say that $Y$ has multiplicity $\\mu=1$ if $X$ is reduced at a general $x\\in Y$, i.e. there is a non-empty open subset $U\\subseteq Y$ such that $\\enm{\\cal{O}}_{X,x}=\\enm{\\cal{O}}_{Y,x}$ for all $x\\in U$. Otherwise we say that $Y$ has multiplicity $\\mu>1$. We are interested only in the case $X$ not integral; if $Y$ has multiplicity $1$, then we have other irreducible components of $X_{\\mathrm{red}}$.\n\n\n\\begin{lemma}\\label{zz1}\nLet $C\\subset X$ be a reduced aCM subvariety of pure dimension $n-1$. Then its ideal sheaf $\\enm{\\cal{I}} _{C,X}$ is an aCM $\\enm{\\cal{O}} _X$-sheaf such that\n\\begin{itemize}\n\\item it is locally free outside $C$ and \n\\item for any closed subscheme $Y\\subsetneq X$, it is not an $\\enm{\\cal{O}} _Y$-sheaf. \n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nSince $C$ is aCM as a closed subscheme of $\\enm{\\mathbb{P}}^N$ and $C$ has pure dimension $n-1$, we have $h^1(\\enm{\\cal{I}} _{C,\\enm{\\mathbb{P}}^N}(t)) =0$ for all $t\\in \\enm{\\mathbb{Z}}$. Thus the restriction map $\\rho _t: H^0(\\enm{\\cal{O}} _{\\enm{\\mathbb{P}}^N}(t)) \\rightarrow H^0(\\enm{\\cal{O}} _C(t))$ is surjective for any $t\\in \\enm{\\mathbb{Z}}$. Since $\\rho _t$ factors through the restriction map $\\eta _t: H^0(\\enm{\\cal{O}} _X(t)) \\rightarrow H^0(\\enm{\\cal{O}} _C(t))$, $\\eta _t$ is surjective. Since $\\eta _t$ is surjective and $h^1(\\enm{\\cal{O}} _X(t))=0$, we have $h^1(\\enm{\\cal{I}} _{C,X}(t)) =0$. This implies that $\\enm{\\cal{I}} _{C,X}$ is aCM. From $\\enm{\\cal{I}} _{C,X\\setminus C} \\cong \\enm{\\cal{O}} _{X\\setminus C}$, we see that $\\enm{\\cal{I}} _{C,X}$ is locally free and of rank $1$ outside $C$. Since $C$ is not an irreducible component of $X_{\\mathrm{red}}$ and $\\enm{\\cal{I}} _{C,X}$ is locally free of positive rank outside $C$, there is no closed subscheme $Y\\subsetneq X$ with $\\enm{\\cal{I}} _{C,X}$ an $\\enm{\\cal{O}} _Y$-sheaf.\n\\end{proof}\n\n\\begin{proposition}\\label{zz2}\nFix an irreducible component $Y$ of $X_{\\mathrm{red}}$. For a fixed integer $e>0$ and any integral divisor $C\\in |\\enm{\\cal{O}}_Y(e)|$, define\n$$\\Sigma _C:=\\left\\{ p\\in Y~|~ \\enm{\\cal{I}}_{C,X} \\text{ is not locally free at }p\\right\\}.$$\n\\begin{itemize}\n\\item [(i)] If $Y$ has multiplicity $\\mu>1$ in $X$, then we have $\\Sigma _C = C$, i.e. for all $p\\in C$ the sheaf $\\enm{\\cal{I}} _{C,X}$ is not locally free at $p$. For any two integral curves $C_1,C_2\\in |\\enm{\\cal{O}}_Y(e)|$, we have $\\enm{\\cal{I}} _{C_1,X}\\cong \\enm{\\cal{I}} _{C_2,X}$ if and only if $C_1=C_2$.\n\\item [(ii)] Assume that $Y$ has multiplicity $\\mu=1$ and that $X$ is not integral. Let $F\\in |\\enm{\\cal{O}} _Y(m-1)|$ be the complete intersection of $Y$ with the other components of $X$, counting multiplicities. If $F\\ne \\emptyset$, then $F$ has pure dimension $n-1$ and $F\\cap C \\ne \\emptyset$ with $\\Sigma _C =(F\\cap C)_{\\mathrm{red}}$.\n\\item [(iii)] For any two integral divisors $C_1,C_2\\in |\\enm{\\cal{O}}_Y(e)|$ such that $\\enm{\\cal{I}} _{C_1,X}\\cong \\enm{\\cal{I}} _{C_2,X}$, we have $\\Sigma_{C_1}=\\Sigma_{C_2}$; in case (i) we have the converse.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nBy Lemma \\ref{zz1} the sheaf $\\enm{\\cal{I}} _{C,X}$ is aCM and locally free with rank $1$ at all $p\\in X\\setminus C$.  Fix $p\\in C$ and assume that $\\enm{\\cal{I}} _{C,X}$ is locally free at $p$. Then there is $w\\in (\\enm{\\cal{I}} _{C,X})_p$ such that $w$ is not a zero-divisor of $\\enm{\\cal{O}} _{X,p}$ and $(\\enm{\\cal{I}} _{C,X})_p \\cong w\\enm{\\cal{O}} _{X,p}$ as a module over the local ring $\\enm{\\cal{O}} _{X,p}$. We get that in a neighborhood of $p$ the divisor $C$ is a Cartier divisor of $X$. Let $I\\subset \\enm{\\cal{O}} _{X,p}$ be the ideal of $Y$ and $J\\subset \\enm{\\cal{O}} _{X,p}$ the ideal of $C$. We have $I \\subset J$. First assume that $X$ is not reduced at a general point of $X$. Since the support of the nilradical $\\eta \\subset \\enm{\\cal{O}} _X$  of the structural sheaf $\\enm{\\cal{O}}_Y$ is a closed subset of $X_{\\mathrm{red}}$, $X$ is not reduced at any point of $Y$ and in particular it is not reduced at $p$. Thus there is a nonzero $h\\in I$ such that $h^m=0$ for some $m>0$. Since $I\\subset J$, we have $h\\in J$ and so $h$ is divided by $w$. Thus we get $w^m=0$ and so $w$ is a zero-divisor, a contradiction.\n\nNow assume that $X$ is reduced at a general point of $Y$. Since $X$ is not integral and it has pure depth $n$, $X_{\\mathrm{red}}$ has at least one another irreducible component. Since $h^0(\\enm{\\cal{O}} _X)=1$, $X$ is connected and so $F\\ne \\emptyset$. Fix any $x\\in F$. Since $\\enm{\\cal{O}} _{X,x}$ has depth $n\\ge 2$, it is connected in dimension $\\le n-1$, i.e. for any open neighborhood $W$ of $x$ in $X$ and any closed subscheme $V$ of $W$, there is a neighborhhod $U$ of $x$ in $W$ such that $U\\setminus (U\\cap V)$ is connected. Thus $F$ has pure dimension $n-1$. Since $C\\in| \\enm{\\cal{O}} _Y(e)|$, $C$ is a Cartier divisor of $Y$. Thus $C$ is a Cartier divisor of $X$ at all points of $C\\setminus (C\\cap F)$. Since $e>0$, $C$ is an ample divisor of $Y$. In particular, we get $F\\cap C \\ne \\emptyset$. Fix $p\\in F\\cap C$. Any local equation $w$ of $C$ at $p$ vanishes on each irreducible component of $X_{\\mathrm{red}}$ containing $p$, because $w$ is assumed to be a non-zero divisor of $\\enm{\\cal{O}} _{X,p}$. There is at least one another irreducible component of $X_{\\mathrm{red}}$ containing $p$, because $p\\in F$.\n\nPart (iii) is obvious.\n\\end{proof}\n\nAs a corollary of Proposition \\ref{zz2} we get the following result, which shows that $X$ is of wild representation type in a very strong form.\n\n\\begin{proposition}\\label{zz3}\nTake $X$ as above. For a fixed integer $w>0$, there is an integral quasi-projective variety $\\Delta$ and a flat family $\\{\\enm{\\cal{F}} _a\\}_{a\\in \\Delta}$ of aCM sheaf on $X$ with each $\\enm{\\cal{F}} _a$ locally free outside a one-codimensional subscheme $C_a$ and for each $a\\in \\Delta$ the set of all $b\\in \\Delta$ such that\n$\\enm{\\cal{F}} _b \\cong \\enm{\\cal{F}} _a$ is contained in an algebraic subscheme $\\Delta _a\\subset \\Delta$ with $\\dim \\Delta -\\dim \\Delta _a\\ge w$.\n\\end{proposition}\n\n\\begin{proof}\nFirst assume that $X$ has at least one irreducible component $Y$ with multiplicity at least $2$. Fix a positive integer $e$ such that $\\dim |\\enm{\\cal{O}} _Y(e)| \\ge w$ and take as $\\Delta$ the family of all integral $C\\in |\\enm{\\cal{O}} _Y(e)|$. Then we may apply (i) of Proposition \\ref{zz2}. In this case we may find $\\Delta$ with the additional condition that for all $a,b\\in \\Delta$ we have $\\enm{\\cal{F}} _a\\cong \\enm{\\cal{F}} _b$ if and only if $a=b$.\n\nNow assume that each irreducible component of $X$ has multiplicity $1$ and fix one of them, say $Y$. Write $F\\subset Y$ as in (ii) of Proposition \\ref{zz2}. Fix an integer $e >0$ such that $h^0(\\enm{\\cal{O}} _X(e)) -h^0(\\enm{\\cal{O}} _X(e)(-F) )> w$ and let $\\Delta$ be the set of all integral divisors $C\\in |\\enm{\\cal{O}} _X(e)|$ not contained in $F$\nand such that the scheme $F\\cap C$ is reduced. Since $F$ has pure dimension $n-1$\nand $C$ is an ample divisor,\nthe set $(F\\cap C)_{\\mathrm{red}}$ has pure dimension $2$. Note that if $C, D\\in \\Delta$ and $(C\\cap F)_{\\mathrm{red}} = (D\\cap F)_{\\mathrm{red}} $, then any equation of $C$ in $H^0(\\enm{\\cal{O}} _X(e))$ differs from an equation of $D$ by an element of $H^0(\\enm{\\cal{O}} _X(e)(-F))$. Then we may apply (ii) of Proposition \\ref{zz2}.\n\\end{proof}\n\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\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\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe fourfold spin-valley degenerate degrees of freedom in bulk graphene can support rich physics and novel applications\nassociated with multicomponent quantum Hall effects \\cite{young2012spin,yang2010hierarchy,islam2016scheme}\nand linear conductance filtering \\cite{grujic2014spin,wu2016full,soodchomshom2011strain,wang2016valley}.\nFor the spin-valley quantum Hall effects,\nit is experimentally demonstrated that, the approximate SU(4) isospin\nsymmetry of Landau levels (LL) in graphene can be broken\nby interactions such as strong Coulomb interaction, Zeeman effect, and lattice scale interactions,\nmanifesting as quantum Hall isospin ferromagnetic states \\cite{young2012spin}.\nSeveral other schemes have also been proposed theoretically, such as\nbreaking the fourfold degeneracy of the central LL by valley-scattering random potential, Zeeman interaction,\nand electron-phonon coupling \\cite{yang2010hierarchy};\nor generating quantum spin-valley Hall effect\nvia quantum pumping by adiabatically modulating a magnetic impurity and an electrostatic potential\nin strained graphene \\cite{islam2016scheme}.\n\n\nFor the spin-valley filtering effect,\nit is demonstrated that, strain in a graphene barrier with carrier mass and spin-orbit coupling\ncan enforce opposite cyclotron trajectories\nfor the filtered valleys in a spin-valley dependent gap, demonstrating simultaneous filtering\nof both valley and spin \\cite{grujic2014spin}.\nIn addition, in strained graphene with Rashba spin-orbit coupling and magnetic barrier,\nfull valley- and spin-polarization currents can be accessed simultaneously,\ndue to the coexistence of valley and single spin band gaps \\cite{wu2016full}.\nAlso, the interplay of modulation fields in a graphene ferromagnetic\/strained (spin splitting barrier)\/ferromagnetic\nstructure \\cite{soodchomshom2011strain,wang2016valley}\nare shown as possible schemes. \n\n\nWhile almost all these works concern about\nconductances (quantum Hall or linear response),\nalmost nothing is known about how to break the spin-valley degeneracy of electron \\emph{beams} in graphene.\n\n\n\n\n\n\n\n\n\nIn this work, we\npropose a spin-valley beam splitter\nbased on Goos-H\\\"{a}nchen (GH) shifts \\cite{note1} and their difference between spin-valley flavours.\nThe device we considered is based on a proximity-induced ferromagnetic interaction\nfrom a EuO film covered by a top gate \\cite{haugen2008spin,yang2013proximity,song2015spin},\nwhich lifts the spin degeneracy,\ncombined with a uniaxial strain, which lifts the valley degeneracy (see Fig. \\ref{fig2}(a)).\nWe first consider the full quasi-ballistic transport regime.\nIt is shown that, the spin band-gaps and valley wave-vectors in the ferromagnetic and strained  regions\nrespectively mimic spin- and valley-dependent refractive indices,\nacting as a spin or valley beam splitter near total reflections.\nWe subsequently consider the full quantum tunneling regime.\nIt is found that, resonant tunnelings can lead $\\sim-\\pi$ sudden phase jumps of the transmission components,\nresulting four giant GH shifts as large as the transverse beam width \\cite{note2}.\nMore importantly, the interplay of the spin- and valley-dependent imaginary wave vectors\nin the ferromagnetic and strained regions lead to four different resonant angles (energies) for\neach spin-valley flavours, thus demonstrating a spin-valley beam splitter\nthat lies in the intersection of spintronics and valleytronics.\nThe beam splitting behavior is found to be controllable by the gating as well as the strain.\nSpin \\emph{or} valley beam splitter in graphene devices was studied before \\cite{zhai2011valley,zhang2015valley,wang2014spin,zhang2014spin};\nhowever, the mechanism proposed in this work is novel, and previously unexplored.\n\n\n\n\\begin{figure}[b]\n\\centering  \\includegraphics[width=\\linewidth]{model-snell-new}\n\\caption{Schematic diagrams of (a) the proposed spin-valley beam splitter,\n(b) modulation profiles felt by electrons from different spin-valley flavours,\nand (c) the motion paths in the device for electrons from different spin-valley flavours.\nIn (b) the modulations along the $z$ and $y$ axis stand for energy and wave vector, respectively.\n}\\label{fig2}\n\\end{figure}\n\n\n\n\n\n\nFigure \\ref{fig2}(a) shows the proposed device.\nA graphene of length $L$ and width $W$ is grown on a substrate placed\nin the $x$-$y$ plane.\nAn EuO film is deposited upon the graphene \\cite{swartz2012integration} in the region of $(0,l_f)$,\nwith a top gate further grown on top of it.\nA uniaxial strain is applied on the substrate in the region of $(l-l_{st},l)$.\n$W$ is several times of $l$ to ensure that the edge effect is negligible \\cite{tworzydlo2006sub}.\nThe spin- and valley-resolved trajectories of the reflection\nand transmission beam components are shown in Fig. \\ref{fig2}(a), with the line width approximately\nstanding for the beam strength.\nFor clearness, the sublattices position differences of $1\/(k_F\\cos\\alpha)$ \\cite{beenakker2009quantum}\nare not shown ($k_F$ stands for the Fermi wave vector).\nA detector placed at a proper position in the transmission region can be used to receive the splitting beam\ncomponent.\n\n\n\\begin{figure}[t]\n\\centering  \\includegraphics[width=\\linewidth]{Snell-shift}\n\\caption{(a) GH and (b) Snell shifts (in units of Fermi wave length) of the four beam components\nas a function of incident angle at beam energy of $E$=20.\nThe parameters of the spin-valley beam splitter are $(l_f,l_w,l_{st})=(1,2,1), V_g=135.6$, and $A_{Sy}=4$.\n}\\label{fig7}\n\\end{figure}\n\nAn electron beam can be represented\nas a wave packet of a weighted superposition of plane-wave\nspinors \\cite{li2003negative,beenakker2009quantum,song2013ballistic},\n$\\boldsymbol{\\Psi}_{i}(x,y)=\\frac{1}{\\sqrt{2}}\\int dq f(q-\\bar{q})\\Phi_p^+(x,y),$\n$\\boldsymbol{\\Psi}_{r}=\\frac{1}{\\sqrt{2}}\\int dq r_{\\xi s}(q)f(q-\\bar{q})\\Phi_p^-(x,y),$\n$\\boldsymbol{\\Psi}_{t}=\\frac{1}{\\sqrt{2}}\\int dq t_{\\xi s}(q)f(q-\\bar{q})\\Phi_p^+(x,y).$\nHere the subscript $i,r,t$ stands for incident, reflection, and transmission, respectively,\n$\\bar{q}$ is the central transverse wave vector,\nand $f()$ is the spectral distribution of the transverse wave vector,\nwhich can be assumed to be of Gaussian profile.\nFor reflection and transmission, the weights\nare further modulated by reflection\/transmission coefficients that\nare spin (with index $s=\\pm1$ for spin up\/down ($\\alpha$\/$\\beta$))\nand valley (with index $\\xi =\\pm1 $ for valley $K$ and $K^\\prime$) dependent.\nIn the steady state approximation \\cite{li2003negative,beenakker2009quantum,song2013ballistic},\nthe locus of the beam components can be given by\n\\begin{equation}\n\\sigma_{s\\xi}^{r,t}=-\\left(\\frac{\\partial \\phi_{s\\xi}^{r,t}}{\\partial q_p}\\right)_{E_p},\\label{eq4}\n\\end{equation}\nwhere $\\phi_{s\\xi}^{r}=\\bar\\phi_{s\\xi}^{r}$ and $\\phi_{s\\xi}^{t}=\\bar\\phi_{s\\xi}^{t}+k_p l$.\nThe phases relate with the coefficients through\n$r_{\\xi s}= |r_{\\xi s}|e^{i\\bar\\phi^r_{\\xi s}}$ and\n$t_{\\xi s}=|t_{\\xi s}|e^{i\\bar\\phi^t_{\\xi s}}$.\n\n\nTo obtain the phases, we calculate the transmission\ncoefficients by solving the right- and left-going\nspinor eigenstates in the pristine ($\\Phi_p^\\pm$ shown in the wave packet),\nferromagnetic, and strained regions,\nand subsequently using the well-know transfer matrix method \\cite{born2000principles}\nhandling the continuity of them.\nThe eigenstates in each uniform region can be exactly resolved by decoupling\nthe two-order differential equation $H_j\\Phi_j=E_j\\Phi_j$.\nThe Hamiltonian in the pristine and strained ($j=p,s$) regions are well known \\cite{katsnelson2006chiral,pereira2009strain}.\nFor the ferromagnetic ($j=f$) region, we adopt a half-metal model \\cite{yang2013proximity,song2015spin,ang2016nonlocal}.\nRather than a simple Zeeman effect, this Hamiltonian roundly describes \\cite{yang2013proximity,song2015spin}\nthe induced charge mass or opening energy gaps ($\\Delta_s=(58+9s)$ meV),\nre-normalized Fermi velocities ($v_s=(1.4825-0.1455s)v_F$),\nand shifted Dirac points ($D_s=(-1.356+0.031 s)$ eV)\ndue to the proximity interaction,\nthat are all spin resolved.\nThe eigenstates in the pristine, ferromagnetic, and strained regions can be written in an uniform form\n\\begin{equation}\n\\Phi_j^\\pm=e^{\\pm ik_jx+iq_jy}\n\\left(\n\\begin{array}{cc}\n\\sqrt{E_j\/2(\\pm k_j+iq_j)}\\\\\n\\sqrt{(\\pm k_j+iq_j)\/2E_j}\n\\end{array}\n\\right),\\label{eq2}\n\\end{equation}\nwhere $+ (-)$ stands for the right- (left-) going propagation,\n$E_p=E_{st}=E$ and $E_f=(E-\\widetilde{D}_s+\\Delta_s)\/v_s$ with\n$\\widetilde{D}_s=D_s+V$ and $V$ the gate voltage.\nIn Eq. (2), $q_p=q_f=E\\sin\\alpha$ is the conserved transverse wave vector,\n$q_{st}=q+\\xi A_{Sy}$ with $A_{Sy}$ the $y$-component of the pseudo magnetic vector potentials \\cite{pereira2009strain};\n$k_p=\\textmd{sign}(E_p)\\sqrt{E_p^{2}-q_p^{2}}$,\n$k_f=\\textmd{sign}(E_f)\\sqrt{E_f E_f^\\prime-q_f^{2}}$ with $E_f^\\prime=(E-\\widetilde{D}_s-\\Delta_s)\/v_s$,\nand $k_{st}=\\textmd{sign}(E_{st})\\sqrt{E_{st}^{2}-q_{st}^{2}}$.\nFor brevity, we express all quantities in dimensionless form by means of\na characteristic length $l_0 = 56.5$nm and energy unit $E_0 = 10$meV.\n\n\n\n\n\n\n\n\n\nIn Fig. \\ref{fig7}, we look at the behavior of GH shifts in detail for the quasi-ballistic transport regime\n(the beam energy is so high that all components are transported quasi-ballistically).\nAlso shown are the `electron paths' (see Fig. \\ref{fig2}(c)) calculated in the geometric optics using Snell's Law.\nThe sine of refractive angles in the three regions read $\\sin\\alpha_p=q_p\/E_p$,\n$\\sin\\alpha_f=q_f\/\\sqrt{E_fE^\\prime_f}$,\nand $\\sin\\alpha_{st}=q_{st}\/E_{st}$.\nSpin- or valley-dependent refractive indices can be defined as \\cite{cheianov2007focusing,moghaddam2010graphene}\n$n^f_s=\\sqrt{(E_p-D_s-V)^2-\\Delta_s^2}\/(v_s|E_p|)$ and $n^{st}_\\xi=q_p\/(q_p+\\xi A_{Sy})$.\nThen the four Snell shifts are\n$\\sigma^{Snell}_{\\xi s}=l_f\\tan(\\sin^{-1}(\\sin\\alpha\/n_s))+l_w\\tan\\alpha+l_s\\tan(\\sin^{-1}(\\sin\\alpha\/n_\\xi))$.\nIt is clear that, the GH shifts simply oscillate around corresponding Snell shifts.\nIt is seen in Fig. \\ref{fig7}(b) that, there are four critical angles corresponding to the four refractive indices,\ni.e., $\\sin\\alpha_{c1}=n^f_\\alpha$, $\\sin\\alpha_{c2}=n^f_\\beta$, $\\sin\\alpha_{c3}=n^{st}_K$, and $\\sin\\alpha_{c4}=n^{st}_{K^\\prime}$\n($\\alpha_{c4}$ is negative and not shown).\nNear each critical angle, GH shifts\nof two beam components with a same spin or valley increases dramatically.\nThis is an enhancement effect due to total reflection.\nAs a result, spin beam splitter of $\\alpha$\/$\\beta$ can be achieved near $\\alpha_{c1}$\/$\\alpha_{c2}$;\nwhile valley beam splitter of $K$\/$K'$ near $\\alpha_{c3}$\/$\\alpha_{c4}$.\nIn Fig. \\ref{fig7}(a) it is also observed that,\nthe four Snell shifts are spin- and valley-dependent,\nespecially for incident angle smaller than $\\alpha_{c2}$.\nHowever, it is hard to achieve enough GH shift difference\nbetween any component and the other three ones.\nThus, spin-valley beam splitting is rather challenging in this transport regime.\n\n\n\n\n\n\\begin{figure}[t]\n\\centering  \\includegraphics[width=\\linewidth]{shift-angle}\n\\caption{The GH shifts (in units of Fermi wave length) of the four\nbeam components as a function of the incident angle at $E=2$.\nThe device parameters are the same as in Fig. \\ref{fig7}.}\\label{fig5}\n\\end{figure}\n\n\n\n\n\nIn Fig. \\ref{fig5} we consider the behavior of GH shifts in the full resonant tunneling regime.\nOne can see that, as the incident angle increases,\nthe two spin $\\alpha$ (spin $\\beta$) components undergo increasing\n(decreasing) GH shifts approximatively proportional to\n$\\tan\\alpha$ ($-\\tan\\alpha$).\nThis is still a quasi-ballistic transport behavior.\nIt is distinct that, around some specific angles, the four GH shifts display sharp peaks.\nThe corresponding phases in the insert show sudden jumps of about $-\\pi$.\nThis is a clear signal for a formation of a standing wave in resonant tunneling.\nRecalling the condition for ideal standing waves in an infinite potential well, i.e.,\n$\\sin k_p l_w=0$,\nthe resonant angles can be solved as $\\alpha=38.2^\\circ$ with an only allowed $n=1$.\nThe resonant angles for the four spin-valley flavours\n($K\\alpha$, $K\\beta$, $K'\\alpha$, and $K'\\beta$) are 49.1$^\\circ$, 53.5$^\\circ$, 32.7$^\\circ$, and 43.4$^\\circ$, respectively.\nThey all deviate from the ideal resonant angle, because the standing waves are formed in\nnon-ideal potential wells formed by walls of spin band-gaps and valley wave-vectors, see Fig. \\ref{fig2}(b).\nIn this case, electron waves can leak out through evanescent waves\nin the walls.\nThis generally lowers the quantization energies and wave vectors, resulting larger resonant angles\n(this is true except for the $K'\\alpha$ case).\nMore importantly,\ninterplay of the spin- and valley-dependent imaginary wave vectors\nin the modulation regions (see Fig. \\ref{fig2}(b)) lead different standing waves,\nimplying different wave vectors and hence different resonant angles\nfor different spin-valley flavours.\nAt each resonant angle,\nthe resonant GH shift exceeds the transverse beam width\nand the other three ones are small.\nWhen we tune the incident angle, the four resonant spin-valley beam components can be splitting in sequence,\nmanifesting a novel spin-valley beam splitting effect.\n\n\n\\begin{figure}[t]\n  \\centering\n \n  \\includegraphics[width=\\linewidth]{k2}\\\\\n  \\caption{Square of the four wave vectors in the two modulated regions\n  as a function of the beam energy.\n  $\\alpha$=30$^\\circ$, $V_g=135.6$, and $A_{Sy}=4$.\n}\\label{fig4}\n\\end{figure}\n\nIt is surprising that, the spin band-gaps and valley wave-vectors play such different roles\nfor beam splitter in different transport regimes.\nIn Fig. \\ref{fig4}, we plot the square of the four wave vectors as a function of beam energy for a fixed incident angle\n(note that a beam component is related to one spin wave vector plus one valley wave vector).\nThree energy ranges can be defined.\ni) The energy rang that exceeds 15 or -21.\nAll wave vectors are real,\nmeaning that all the four beam components transport through the sandwich structure quasi-ballistically.\nThis is the case we considered in Fig. \\ref{fig7}, giving behaviors similar to geometric optics\nand can be applied as spin or valley beam splitter near total reflection.\nii) The range between -2.7 and 2.1.\nIn this range, the four wave vectors all become imaginary,\nimplying that the four spin-valley beams all undergo evanescent transport or resonant tunneling.\nThis is the case we considered in Fig. \\ref{fig5}, showing spin-valley beam splitter\ndue to flavours dependent standing waves.\nThis is the central result of the present work.\niii) The energy range between.\nAt least one wave vector (say $k_\\beta$) is real.\nThe two related components ($K\\beta$ and $K'\\beta$) cannot undergo resonant tunneling.\nAs a result, they cannot be splitting from each other (although can be\nsplitting from the $K\\alpha$ and $K'\\alpha$ components).\nIn this case, only components $K\\alpha$ and $K'\\alpha$ can be splitting,\ndemonstrating a partial spin-valley beam splitting effect.\n\n\n\n\\begin{figure}[t]\n\\centering  \\includegraphics[width=\\linewidth]{shift-energy-30}\n\\caption{The GH shifts (in units of Fermi wave length) of the four beam components as a function of the incident energy at $\\alpha=30^\\circ$.\nThe device parameters are the same as in Figs. \\ref{fig7} and \\ref{fig5}.\n}\\label{fig3}\n\\end{figure}\n\n\n\n\n\nThe full spin-valley beam splitting is further shown in the energy space in Figure \\ref{fig3} .\nOne can see clear huge resonant peaks on background of GH shifts of the order of the Fermi wavelength.\nThe resonant energies are around the one of an ideal standing wave with $n=\\pm1$,\ni.e., $E_{resonant}=\\pm1.81$.\nThe sequence of components is found to be in opposite orders\ncompared with that in the incident angle space (Fig. \\ref{fig5}).\nThese behaviors mean that,\nin addition to by tuning incident angle at fixed beam energy,\nspin-valley beam splitting can also be achieved by tuning the beam energy at fixed incident angle.\nThis can have potential applications in splitting beam of wide energy distribution.\n\n\n\n\nWe at last consider the tunability of the full spin-valley beam splitter.\nFigure \\ref{fig6} shows the GH shifts of $K\\beta$ and $K^\\prime\\alpha$ flavours\nfor different gate voltages or strains.\nAs can be seen, the resonant GH shift moves to a smaller incident angle\nwhen a higher gate voltage or a bigger strain is applied.\nThis is because, the wave leaks become weaken and the energy of the quasi-standing\nwave increases.\nOn the other hand, the resonant angle becomes bigger when a lower gate voltage or a smaller\nstress is applied.\nAll these trends are the same for the other three components\n(components $K\\alpha$ and $K^\\prime\\beta$ are not shown for clearness).\nThe GH shifts of $\\beta$-related flavours decrease at a lower gating\nbecause the wave leaks increase.\nThe above results imply that,\nthe resonant angles of the splitting components can be chosen by tuning the gating or strain,\nprovided the transport remains in the full resonant tunneling regime.\nOn the other hand, the bigger the strain, the wider the intersection set\n(also limited by the two $K^f_s$ sets),\nbenefiting to the full resonant tunneling.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{parameters}\\\\\n\\caption{GH shifts of the $K\\beta$ and $K^\\prime\\alpha$ components\nas a function of the incident angle at $E=2$ for\ndifferent gate voltages or strains. $l_f=1$, $l_w=2$, and $l_s=1$.\n}\\label{fig6}\n\\end{figure}\n\n\nIn summary, we have demonstrated that the spin-valley fourfold degeneracy of electron beam in\ngraphene can be totally eliminated by flavour-dependent giant GH shifts\nin a full resonant tunneling regime.\nThis manifests a novel application of spin-valley beam splitter, lying in\nthe intersection of spintronics and valleytronics.\nThe formation of standing waves in the potential wells, \nwhich leads sudden jump of phase of about $-\\pi$,\nand the interplay of the spin- and valley-dependent wave leaks, \nwhich leads difference in resonant angles,\nare at the heart of the mechanism of the beam splitting effect.\nThe spin-valley beam splitting can be achieved by tuning incident angle\n at fixed beam energy or vice versa,\nand can be controlled by gating or strain.\n\nWe have also shown that, the spin band-gaps and valley wave-vectors play\ntotally different roles in the full ballistic transport regime.\nThey act as medias with spin or valley dependent refractive index\nand can be applied as spin or valley beam splitter near total reflections.\n\nWe encourage experimental works on the proposed device and mechanism.\n\n\n\n\n\n\n\nThis work was supported by the National Natural Science Foundation of\nChina (NSFC) under Grant No. 11404300\nand the Science Challenge Project (SCP) under Grant No. JCKY2016212A503.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLinear optical architectures  offer\nthe potential for reliable realizations\nof small-scale quantum computing~\\cite{KLM01}\nIn the recent past, numerous \nproof-of-principle demonstrations have relied on the precise state\nmanipulation that is available using linear optical elements. The \ndevelopment of better and brighter sources with good mode \nquality as well as new types of detectors have opened up new \nperspectives~\\cite{Further} in state preparation\nand manipulation, for six or more photons. \nSpecifically, integrated optical circuits allow for\nstate manipulation with little mode matching problems in interferometers~\\cite{Integrated}. \n\nNaturally, significant efforts has been devoted towards realizing instances\nof quantum gates. As primitives for such small-scale computing, two-partite quantum gates delivering a \ncontrolled phase-shift of $\\varphi$ have already been experimentally\ndemonstrated (see Refs.~\\cite{HT02,RLBW02,PPBS} for $\\varphi=\\pi$ and Ref.~\\cite{LBA+09} \nfor general phases). In this note, we focus on linear\noptical implementations of phase gates with arbitrary phases. In particular,\nwe will ask when arbitrary phases can \nbe realised in the first place, and---one of the main figures of merit in linear optical\napplications---what the optimum probabilities of success are, as any non-linear map is necessarily\nprobabilistic.\nA realisation of such gates seems interesting from the perspective of \n\\begin{itemize}\n\\item[(i)] gaining an understanding of the probabilistic character of \nquantum gates as well as \n\\item[(ii)] serving as a proof of principle\nrealisation of a kind of quantum gate that has several \napplications in linear optical quantum information processing.\n\\end{itemize}\n\nAs far as the first aspect is concerned, \none may well expect that there\nis a trade-off between the notorious problem of having a small\nprobability of success and the phase that is being realised in\nthe gate. In fact, the study in Ref.~\\cite{Eisert04}\nsuggests exactly such a behavior: \nthe presented upper bounds to the probability of success\nincrease from the \nminimum at $\\ensuremath{p_{\\mathrm{s}}}(\\varphi=\\pi)=1\/4$ to $\\ensuremath{p_{\\mathrm{s}}}(0)=1$. To\ninvestigate such a trade-off is interesting in its own right\nand helps in building intuition concerning the probabilistic behavior\nof linear optics. One may well develop the intuition that ``large phases are costly'' as far\nas the probability of success and hence the overhead or repetition are concerned.\n\n\nFurther, concerning the second aspect, there are several applications \nfor which such a trade-off is relevant. In linear optical architectures,\nit may be a good idea to have a smaller phase, if one only\nhas higher success probabilities. The new measurement-based\nquantum computational models~\\cite{GE07a} for example offer this \nperspective: One does not have to have controlled $\\pi$ phase gates\nto prepare cluster states, but one would in principle also get\naway with smaller phases. This may well (but does not have to be)\na significant advantage when preparing resources for\nmeasurement-based quantum computing different from cluster states~\\cite{RB01,GE07a}.\n\nOf course, in standard gate-based quantum computing, \none will typically encounter all kinds of controlled phase gates. For example,\nin the quantum Fourier transform~\\cite{nielsen}, \none has to implement several controlled phase gates.\nThey can again be decomposed into other sets of universal \ngates (like CNOT or CZ and local unitaries). But,\nin terms of resource requirements, it is obviously an advantage to directly\nimplement the relevant quantum gates with phases in the range \n$0<\\varphi<\\pi$.\nThere are also interesting trade-offs\nbetween resource requirements and success probabilities in a number of\nrelated contexts, like non-local gates in distributed quantum computation~\\cite{EJPP00,CDKL01Berry07}. Refs.~\\cite{CDKL01Berry07}, for example,\nstudy distributed controlled phase-gates which would need less entanglement\nand succeed with a higher probability.\nIn the field of linear optics gates, numerical results on direct implementation of arbitrary two-qubit gates\nare known (see Ref.~\\cite{dmitry} and references therein).\n\nInstead of resorting to decompositions in the circuit model, one could gain\nfrom implementing unitaries in a fashion ``natural'' to the respective architecture at hand.\nIn the case of linear optics it means to leave the computational sub-space given by the\nencoding of the qubits for the sake of taking a ``shortcut'' through higher dimensions~\\cite{LBA+09}.\nGiven that the fundamental information carriers are implemented using bosonic modes, this will\noccur when mixing those modes in beam splitter networks and is an inherent feature of\ngenuine linear optics implementations, in contrast to decompositions into standard gate sets.\n\nHere we will study post-selected gates, so not genuine ``event-ready'' quantum\ngates, but---as is common in linear optical architectures at least to date---those \nwhere one measures the output modes and whether the gate\nactually succeeded is determined only {\\it a posteriori} by accepting only those\noutcomes which lie in the computational dual-rail subspace of the Hilbert space of $n$ photons on $2n$ modes.\nIncorporating less constraints, these gates concern only a smaller number of modes\nand are still within reach of current experiments.\nIn principle non-demolition measurements of the output would be required\nfor an event-ready gate.\n\n\\section{Controlled phase gates}\n\\subsection{Single beam splitter}\nIn a post-selected phase gate on four modes in \nthe standard dual-rail encoding, two of the modes\nare merely involved as ``by-standers'', in that their amplitude is \ncompensated in exactly the same fashion as in Refs.~\\cite{HT02,RLBW02,PPBS}.\nIn this section, we will hence \nconcentrate on two modes forming the ``core'' of the\nscheme, giving rise to a two-qubit dual-rail phase gate on four physical modes. The core\nitself may be regarded as a single-rail phase gate in its own right. Later we will see that\nnot breaking the network into a core and by-stander modes will not give any advantage.\n\nSimilar to Ref.~\\cite{HT02} we will briefly investigate the consequences of simply having a single beam\nsplitter forming the core of the quantum gate.\nThe action on the photonic creation operators of the two involved\nmodes it is mixing is described by the matrix\n\\begin{equation}\n  U = \\mathrm{diag} (\\mathrm{e}^{\\imath\\phi_1},\\mathrm{e}^{-\\imath\\phi_1}) \\cdot B \\cdot \\mathrm{diag} (\\mathrm{e}^{\\imath\\phi_2},\\mathrm{e}^{-\\imath\\phi_2} )\n\\end{equation}\nwith\n\\begin{equation} \\label{eqn:orthogonal}\n  B = \\left[\\begin{array}{cc} \\sin(\\vartheta) & \\cos(\\vartheta) \\\\ -\\cos(\\vartheta) & \\sin(\\vartheta) \\end{array}\\right] \n\\end{equation}\nand appropriate phases $\\phi_1,\\phi_2\\in[0,2\\pi)$\nand mixing angle $\\vartheta\\in [0,2\\pi)$.\nThe phases can also be realised deterministically by local operations on the dual-rail qubits, which leaves\nthe relevant part of the gate $U'=B$. The matrix elements of the unitary $\\mathcal U'$ belonging to $U'$ for vacuum, single photon operation, and the two-photon component\nread\n\\begin{eqnarray}\n  \\braket{0,0|\\mathcal U'|0,0} &=& 1 ,\\label{eqn:cphase:def1} \\\\\n  \\braket{1,0|\\mathcal U'|1,0} &=& A_{1, 1} = \\sin(\\vartheta), \\\\\n  \\braket{0,1|\\mathcal U'|0,1} &=& A_{2\\ic2} = \\sin(\\vartheta), \\\\\n  \\braket{1,1|\\mathcal U'|1,1} &=& \\mathrm{per}( A) = \\cos(2\\vartheta) = 1-2\\sin^2\\vartheta, \\label{eqn:cphase:def2}\n\\end{eqnarray}\nrespectively.\nSince we are restricted to $n\\le2$ modes, these four quantities determine the action of the core completely.\nWith the constraint\n\\begin{equation}\n  1-2\\sin^2(\\vartheta) = \\sin^2(\\vartheta)\n\\end{equation}\nthat ensures equal single- and two-photon amplitudes (equal probabilities for all dual-rail states),\nonly the two solutions $\\vartheta=\\pm\\arcsin(3^{-1\/2})$ are possible, giving rise to\n$\\varphi=0,\\pi$, respectively.\n\nHence, one finds that in this way, one \\emph{can} implement quantum phase gates, but only\ntwo different ones: One is not doing anything, and the other ones effect is a controlled\nphase of $\\pi$. This is exactly the gate of Refs.~\\cite{HT02,RLBW02}. In other words, \nwithout invoking at least a single additional mode, one can not go beyond\nthe known $\\pi$-phase in this fashion.\n\n\\subsection{Arbitrary phases}\nHowever, we can extend this scheme: \nThe restriction to unitary two-mode beam splitters can be relaxed.\nInstead of starting with $U\\in SU(2)$, we use an arbitrary matrix $A\\in\\mathbbm{C}^{2\\times2}$.\nThen we will embed the two-mode matrix into a higher dimensional unitary, such that an \nappropriately rescaled $A$ forms a principle submatrix of a larger unitary matrix $A'$. The optimal rescaling is simply\ndictated by the largest singular value of $A$. We will see that in the two-mode case only a single additional mode is already the most general extension, so the\nfull set-up would consist of a transformation on three modes involving \nat most three beam splitters. In this class of gates, for each $\\varphi$, \nthe one with the optimal probability of success $\\ensuremath{p_{\\mathrm{s}}}(\\varphi)$\ncan be identified.\n\n\\begin{figure}\n  \\includegraphics{phases_04.eps}\n  \\caption{Optimal success probability $\\ensuremath{p_{\\mathrm{s}}}(\\varphi)$ of phase gates with vacuum ancillas (one vacuum ancilla is already optimal)\n    {\\it vs.}\\ the phase $\\varphi$ (solid line). At $\\varphi=\\pi$ the result of Refs.~\\cite{HT02,RLBW02},\n    $\\ensuremath{p_{\\mathrm{s}}}(\\pi) = 1\/9$, is reproduced.\n    The intuitive assumption of a monotonous $\\ensuremath{p_{\\mathrm{s}}}(\\varphi)$ is not fulfilled: indeed, the success probability is worse than $1\/9$ in the interval $\\pi\/3 < \\varphi < \\pi$.\n    Due to phases $\\varphi<\\pi$ not being implementable with a single beam splitter, the additional unitary extension requires further measurements and therefore decreases the probability\n    of success near $\\varphi=\\pi$.\n    \\label{fig:phases} }\n\\end{figure}\n\n\\begin{theorem}[Optimal post-selected dual-rail controlled phase gate]\n  Consider linear optics, an arbitrary number of auxiliary vacuum modes and photon number resolving detectors. When post-selecting the state of the signal modes onto the computational sub-space\n  and the auxiliary modes onto the vacuum,\n  the optimal network on four modes implementing the gate represented in the computational basis of two dual-rail qubits by\n  $U = \\mathrm{diag} ( 1,1,1,\\mathrm{e}^{\\imath\\varphi} )$, $\\varphi\\in[0,\\pi]$,\n  has a success probability (shown in Fig.~\\ref{fig:phases}) of\n  \\begin{equation}\n    \\ensuremath{p_{\\mathrm{s}}}(\\varphi) = \\left(1+2\\left|\\sin\\frac{\\varphi}{2}\\right|+2^{3\/2}\\sin\\frac{\\pi-\\varphi}{4}\\sqrt{\\left|\\sin\\frac{\\varphi}{2}\\right|}\\right)^{-2} .\n  \\end{equation}\n\\end{theorem}\n\n{\\it Proof:}\nIn order to find $\\ensuremath{p_{\\mathrm{s}}}$---the same for all possible input states---we will first construct the linear transformation of the relevant creation \noperators and identify the optimal unitary extension afterwards.\n\\begin{itemize}\n\\item \n{\\it Two-mode transformation:}\nThe two-mode transformation resulting from solving the equations (\\ref{eqn:cphase:def1})--(\\ref{eqn:cphase:def2}) imposed by the gate we want to build is\n\\begin{equation}\\label{eqn:cphase:solution}\n  A = \\ensuremath{p_{\\mathrm{s}}}^{1\/4}\\left[ \\begin{array}{cc} x & \\left(\\mathrm{e}^{\\imath\\varphi} - 1\\right)x\/y \\\\ y\/x & 1\/x  \\end{array} \\right] .\n\\end{equation}\n$x$ and $y$ are free non-zero complex parameters. By writing\n\\begin{equation}\n  A = \\ensuremath{p_{\\mathrm{s}}}^{1\/4}\\,\\mathrm{diag} (a,a^{-1})\n  \\cdot\\left[ \\begin{array}{cc} 1&\\mathrm{e}^{\\imath\\varphi}-1 \\\\ 1&1 \\end{array}\\right]\\cdot\\mathrm{diag}(b,b^{-1}) ,\n\\end{equation}\nwith $a=xy^{-1\/2}$ and $b=y^{1\/2}$ we see that the singular values of $A$ only depend on $|a|^2$ and $|b|^2$, so not on the phases of $a$,$b$, and $x$ and $y$.\n\nThe general solution to the dual-rail problem is actually composed of the transformation $A$ together with appropriate damping of the by-standers:\nthe probability of success can not be enhanced by considering a full transformation on all four modes.\nThis can be seen by writing the polynomial system given by the dual-rail problem similar\nto (\\ref{eqn:cphase:def1})--(\\ref{eqn:cphase:def2}), consisting of $16$ quadratic equations\nin the matrix elements of $B\\in\\mathbbm{C}^{4\\times 4}$. It turns out that by permuting modes and appropriate variable substitutions all solutions can be brought into the form\n$B \\propto \\mathbbm{1}_2 \\otimes A$,\nconsisting of the two-mode core given in Eqn.~(\\ref{eqn:cphase:solution}) and two by-passed modes.\n\n\\item {\\it Optimal extension:}\nGiven the $2\\times2$ matrix $A$ that realises the transformation we are looking for,\nthe optimal unitary \nextensions can be identified. Let us extend the first and second row vectors (denoted by $A_1$ and $A_2$) to dimension $3$ by appending $A_{1\\ic3}$ and $A_{2\\ic3}$,\nrespectively, in such a way as to allow for unitarity of the extended matrix, $A'\\in SU(3)$.\n\n\nTo see why a dimension of three is already sufficient consider a linear transformation of the creation operators of $n$ modes, described by its a (not necessarily unitary) matrix $A$.\nBy using the singular value decomposition (SVD)\nit can be decomposed as $A=V\\cdot D\\cdot W^{-1}$ where $V$ and $W$ are unitary (and therefore have immediate interpretations as physical beam splitter matrices themselves),\nand $D=\\{d_1,\\ldots,d_n\\}$ is a diagonal matrix with real non-negative entries $d_1\\ge d_2\\ge\\ldots\\ge d_n$, the singular values.\nIn terms of linear optics $D$ can be interpreted as mixing each mode $k=1,\\ldots,n$ with an additional\nmode $n+k$ in the vacuum state which will be post-selected in the vacuum afterwards~\\cite{MLU+06,Kieling08}.\nThen, $d_k$ describes the transmittivity of the beam splitter used to couple modes $k$ and $n_k$.\nWithout loss of generality one can assume $d_1=1$, vacuum mixing for the first mode.\nThis can be achieved by rescaling with the inverse of the largest singular value, so $A \\mapsto d_1^{-1}A$, which implies $d_k\\mapsto d_1^{-1}d_k$.\nNote that such a ``global'' rescaling of $A$ does not change the post-selected action on the computational sub-space but only the success probability of it according to $\\ensuremath{p_{\\mathrm{s}}}\\mapsto d_1^{-2n}\\ensuremath{p_{\\mathrm{s}}}$.\nTherefore, in general, there are $n-1$ additional\nvacuum modes required to extend an $n$-mode linear transformation to a unitary, and thus physical, network. Also please note, that the constraint $d_1=1$\nhas to be taken into account for any optimisation of success probabilities of $A$. Here we will not explicitly use this decomposition further, but the constraint will be implemented\nimplicitly by requiring the $n-1$-mode extension to be unitary.\n\n\nIn our specific $3$-mode extension we choose\n$A_{1\\ic3}$ and $A_{2\\ic3}$ such that the new row vectors are orthogonal. By multiplying them by the root of the inverse of their respective norms, $|A'_1|$ and $|A'_2|$,\nthey will be normalised. Finding a third orthogonal vector to fill the unitary matrix can be done with the complex cross product $(A'_1\\times A'_2)^*$, or\nin general by choosing a vector at random and orthogonalising it with respect to the given ones.\n\nThe dependence of the success probability on the extension is \n\\begin{equation}\n\t\\ensuremath{p_{\\mathrm{s}}}=\\left(|A'_1||A'_2|\\right)^{-2}.\n\\end{equation}\t\nTherefore, the objective is to\n\\begin{eqnarray}\n  \\mbox{minimise}   && f = |A'_1|^2 |A'_2|^2 \\\\\n  \\mbox{subject to} && A'_1 (A'_2)^{\\dagger} = A_1 A_2^{\\dagger} + A_{1\\ic3}A^{*}_{2\\ic3} = 0 \\label{eqn:subject} .\n\\end{eqnarray}\nThe first observation is that the row-scaling by $x$ is already included in the norm of the row vectors, leaving us with one parameter less.\nBy using the phase of $y$, we can assure that $A_1 A_2^{\\dagger}$ is real and \npositive and also $\\arg(A_{1\\ic3}) - \\arg(A_{2\\ic3}) = \\pm\\pi$. This \nconstrained minimisation problem in $A_{1\\ic2}$ and $A_{1\\ic3}$ can indeed be solved\n(by using Lagrange's multiplier rule and showing constraint qualification)\nand  we find $|y|=\\left(2(1-\\cos\\varphi)\\right)^{1\/4}$.\nThen an optimal solution (phases chosen conveniently) is\n\\begin{equation}\n  A_{1\\ic3} = A^*_{2\\ic3} = \\mathrm{e}^{{\\imath\\pi}\/{2}}\\left(\\sqrt{2\\left|\\sin\\frac{\\varphi}{2}\\right|}\\sin\n  \\frac{\\pi-\\varphi}{4}\\right)^{1\/2}\n\\end{equation}\nwith the probability of success given by\n\\begin{eqnarray}\\label{eqn:cphase:ps}\n  \\ensuremath{p_{\\mathrm{s}}}(\\varphi) &=& \\left(1+|y|^2+|y|\\sqrt{2-|y|^2}\\right)^{-2} \\\\\n      &=& \\left(1+2\\left|\\sin\\frac{\\varphi}{2}\\right|+2^{3\/2}\\sin\\frac{\\pi-\\varphi}{4}\\sqrt{\\left|\\sin\\frac{\\varphi}{2}\\right|}\\right)^{-2} .\n      \\nonumber\n\\end{eqnarray}\nThe reflectivities of the compensating beam splitters in the by-passed modes have to be chosen such that the success probability is constant for all dual-rail states, i.e., \n\\begin{equation}\n\tr=\\ensuremath{p_{\\mathrm{s}}}^{1\/4}.\n\\end{equation}\t\n\\end{itemize}\n\n\n\\begin{figure}\n  \\includegraphics{cphase_01.eps}\n  \\caption{Basic spatial modes based set-up obtained from translating an arbitrary $2\\times 2$ core into the language of linear optics. \n   The core extension is provided by mixing\n    with a vacuum mode on the central beam splitter. This mode, in turn, has to be post-selected in the vacuum state afterwards. The upper and lower beam splitters\n    implement the appropriate compensation by damping the by-passed modes (which is the same for both modes for the optimal solution which we consider).\n\\\\\n    The labels at the beam splitters will be used to identify them with the respective optical elements in Figs.~\\ref{fig:pgate03}--\\ref{fig:pgate:ppbs}.\n    In general, the parameters ({i.e.}, reflectivity and phases) of these elements depend on the gate's phase $\\varphi$.\n    Further, the notation $\\ket{\\mathfrak{0}}_i$ and $\\ket{\\mathfrak{1}}_i$ is used for the logical $0\/1$-modes of the $i$-th qubit to avoid confusion with Fock states. \\label{fig:pgate01} }\n\\end{figure}\n\n\nThe success probability $\\ensuremath{p_{\\mathrm{s}}}(\\varphi)$ of this gate\nis shown in Fig.~\\ref{fig:phases}. \nInterestingly---and quite surprisingly---the \nworst success probability is not achieved for the sign-flip ($\\varphi=\\pi$), but for\n$\\varphi\\approx2.05$. This means, gates delivering a phase shift slightly\nsmaller than $\\pi$ and thereby generating less entanglement will not give rise to a larger, but to a smaller success probability.\nAs expected, the success probability for very small \nphases increases and reaches unity for $\\varphi=0$: one can always do nothing \nat all with unit success probability.\n\n\n\n\n\\subsection{Integrated quantum photonics realizations}\nSophisticated circuits such as the one shown in Fig.~\\ref{fig:pgate:ppbs} can be built from \nbulk optical elements (mirrors, beamsplitters, etc.) Such circuits often\nrequire implementation of Sagnac interferometers (e.g., Ref.~\\cite{NOO+07}), partially polarizing beam splitters (PPBSs)~\\cite{PPBS}, or\nbeam displacers~\\cite{OPW+03,Process1} to achieve interferometric stability. For the most complicated circuits a combination of these elements is\nrequired. Indeed a (non-optimal) implementation of a two-qubit controlled unitary gate used a combination of beam displacers and\nPPBSs~\\cite{LBA+09}. However, such circuits are extremely challenging to align, are limited in performance by the quality of that alignment, and are ultimately not scalable.\n\nAn alternative approach based on lithographically fabricated integrated waveguides on a chip has recently been developed~\\cite{Integrated}.\nThis approach has demonstrated better performance, in terms of alignment and stability, as well as miniaturization and scalability.\nThe monolithic nature of these devices enables interferometers to be fabricated with precise phase and stability, making the Mach-Zehnder\ninterferometer shown in Fig.~\\ref{fig:pgate01} directly implementable without the need to stabilize the optical phase (either actively,\nor using the Sagnac-type architecture of Fig.~\\ref{fig:pgate:ppbs}), greatly simplifying the task of making complicated circuits:\nEssentially the circuit one draws on the blackboard can be directly `written' into the circuit.\nIndeed, integrated photonics circuits have been used to implement a circuit of several logic gates on four photons, to implement a compiled\nversion of Shor's quantum factoring algorithm in this way~\\cite{PMO09}. In fact laser direct write techniques have been\nused to `write' circuits in an even more direct fashion than the lithographic approach.\n\nAnother key advantage of integrated quantum photonics for the circuits described here is that on-chip phase control can be directly\nintegrated with the circuit~\\cite{IntegratedReconfigurable}, which could allow measurment of the success probability\ncurve (Fig.~\\ref{fig:phases} or Fig.~\\ref{fig:toffoli:ps} for example)\nto be directly mapped by sweeping the applied voltage.\n\n\\subsection{Experimental issues in free space}\n\\begin{figure}\n  \\figspace{\\vspace{.25cm}}\n  \\includegraphics{cphase_03a.eps}\\figspace{\\hspace{.25cm}\\vspace{.5em}}\n  \\caption{Setup for a controlled phase gate on two polarisation encoded dual-rail qubits. The logical modes of the two qubits are separated and re-united by means of polarising beam splitters (PBS).\n    Replacing the left and right beam splitters in Fig.~\\ref{fig:pgate01} by wave plates $\\lambda_1$ and $\\lambda_2$,\n    they become easier to tune to different $\\varphi$, and provide better stability. The lower beam splitter, $\\gamma$,\n    implements compensation of both, $\\ket{\\mathfrak{0}}_1$ and $\\ket{\\mathfrak{0}}_2$, modes. \n    $\\lambda_V$ is taken care of by the partially polarising beam splitter (PPBS, allowing for \n    different reflectivities for the two polarisations, further explained in Fig.~\\ref{fig:ppbs}) in the centre.\n    Additionally, one of the qubits has to be flipped prior to and after the circuit, which here is done by acting with a wave plate on the second qubit. \\label{fig:pgate03} }\n\\end{figure}\n\nIn order to render the proposed gates experimentally more feasible in free space, some simplifications\nhave to be done -- tailored to the specific physical implementation at hand.\nWaveguide based setups would not need further simplifications because stablity of the interferometers would be ensured by the rigid substrate.\nImplementations more suitable for beam displacer based setups are known~\\cite{LBA+09}.\nInfluenced by the gate model, those gates include a controlled $\\pi$-phase gate at the core. Operating at lower success probabilities,\nespecially at phases $\\varphi$ approaching $0$, the probability of success does not converge to $1$.\n\nIn the following we will discuss simplifications which shall allow for easier free-space implementation of the networks introduced above\nwhile still preserving optimality with respect to $\\ensuremath{p_{\\mathrm{s}}}$.\n\\begin{figure}\n  \\includegraphics{ppbs1.eps}\n  \\caption{From left to right: (i) A PPBS implementing a beam splitter with polarisation dependent reflectivity. (ii) It is equivalent to an interferometer between two PBS\n    where the PPBS' reflectivities are incorporated by means of wave plates, $\\lambda_1$ and $\\lambda_2$. (iii) By identifying the two PBS, the interferometer collapses into a closed\n    loop (which is more compact and more robust in experimental implementations), leaving only one PBS.\n    In the second and third circuit, a polarisation flip of the second qubit before and after the circuit is added by using a wave plate.\\label{fig:ppbs} }\n\\end{figure}\nA straightforward set-up on dual-rail encoding that realises a three-mode unitary and \ncompensates the amplitudes in the remaining modes is shown in Fig.~\\ref{fig:pgate01}.\nIt includes one interferometer, but the whole gate would sit inside a double interferometer,\nbecause local unitaries on the input and output qubits would require classical interference.\nThus, the complexity of this gate is best described as a nested three-fold interferometer.\nIn this first stage, the parameters (reflectivities and phases) of all five beam splitters\ndepend on $\\varphi$.\n\nTo get rid of some of the interferometers, polarisation encoding is convenient. Two modes can be united in one spatial mode, resulting in inherent stability (neglecting\nbirefringence of the optical medium) of some interferometers. Rotations on these modes can be carried out easily using wave plates. Because the core acts on modes coming from different\ndual-rail qubits, they have to be combined into a single spatial mode before. This is achieved with a PBS, thus permuting $H$ and $V$ modes. Damping of both by-passed modes can be done simultaneously\nby a single polarisation insensitive beam splitter coupled to the vacuum. A straightforward translation of Fig.~\\ref{fig:pgate01}\ninto polarisation encoding, thereby collapsing the network into a single interferometer, is shown in Fig.~\\ref{fig:pgate03}.\n\nDue to the asymmetry of the core, still one PPBS is used, the reflectivity of one polarisation component of which actually depends on $\\varphi$, the other one being $1$. Fig.~\\ref{fig:ppbs} shows\nhow a tunable PPBS can be constructed, introducing another interferometer.\n\nIterating the ideas that led to the compact PPBS implementation once more yields a collapsed form of the phase gate based on only two PBS and a couple of wave plates. Due\nto the paths of light being very similar, this set-up should also be more robust.\nThis is illustrated by Fig.~\\ref{fig:pgate:ppbs} since it has an overall Mach-Zehnder interferometer structure to it, by feeding one displaced sagnac loop into another.\n\n\n\\subsection{Simple decomposition}\nHere we want to obtain a simple decomposition and explicit $\\varphi$-dependence of the involved elements of the full $3\\times3$ unitary transformation $U\\in SU(3)$.\nTo do so, we interpret the effective core, acting on the $\\ket{\\mathfrak{1}}_1$ and $\\ket{\\mathfrak{1}}_2$ modes in Fig.~\\ref{fig:pgate01}, in a different way:\nin between $U_{\\lambda_1}$ (the unitary beam splitter matrix of $\\lambda_1$)\nand $U_{\\lambda_2}$, there acts a diagonal mode transformation such that the first mode is unaffected and the amplitude of the second one is damped\n(due to $U_{\\lambda_V}$ coupling it with reflectivity $r_V$ to a vacuum mode which will be projected onto the vacuum).\n\nNow considering the singular value decomposition (SVD) of the matrix representing the core transformation, $A=V\\cdot\\Sigma\\cdot W^{\\dagger}$ we can identify $V=U_{\\lambda_2}$,\n$W^{\\dagger}=U_{\\lambda_1}$ and $\\Sigma=\\mathrm{diag}\\{1,r_V\\}$. As we have seen earlier, optimal extensions of $2\\times2$ cores only require global rescaling, which commutes with the unitaries involved.\nTherefore we can use the much simpler original form of $A$ in Eqn.~(\\ref{eqn:cphase:solution}), and we choose $x=1$ and $y=-\\sqrt{\\mathrm{e}^{\\imath\\varphi}-1}$.\n\nWe find the singular values of $A$\n\\begin{equation}\n  \\sigma_{\\pm} = \\sqrt{1+2\\sin\\frac{\\varphi}{2}\\pm2(2-2\\cos\\varphi)^{1\/4}\\cos\\frac{\\varphi+\\pi}{4}} .\n\\end{equation}\nGlobal rescaling amounts to fixing the largest singular value to unity, so the new singular values are $1$ and $r_V=\\sigma_-\/\\sigma_+$.\n\nDue to $\\det U_{\\lambda_1}=\\det U_{\\lambda_2}=1$\nwe need to attach a phase to one singular value as well in order to apply the identification of the matrices introduced above. \nThen the SVD of $A$ yields\n\\begin{equation}\n  U_{\\lambda_1} = \\frac{1}{\\sqrt{2}}\\left[ \\begin{array}{cc} -1 & 1 \\\\ -1 & -1  \\end{array} \\right]\n\\end{equation}\nand $U_{\\lambda_2} = U_{\\lambda_1}^{-1}\\cdot\\mathrm{e}^{\\imath\\phi_+\\sigma_z}$ with\n\\begin{equation}\n  \\phi_\\pm=\\mathrm{arccot}\\left[ \\cot\\frac{\\varphi+\\pi}{4}\\pm\\left((2-2\\cos\\varphi)^{1\/4}\\sin\\frac{\\varphi+\\pi}{4}\\right)^{-1}\\right]\n\\end{equation}\nwhere the order of rows and columns in the matrices is as in Fig.~\\ref{fig:pgate01} from top to bottom.\n\nThe ``complex singular values'' are $1$ (the first mode is not affected) and $\\sigma_-\/\\sigma_+\\exp{\\imath \\phi_++\\phi_-}$. The latter can be achieved by using the aforementioned\ncoupling to the vacuum with a reflectivity of $r_V$ and a phase upon reflection of $\\phi_++\\phi_-$. The further ingredients are the beam splitters required for the ``damping'' of the\nby-standers as discussed earlier.\nBy confirming $1\/\\sigma_+^4 = \\ensuremath{p_{\\mathrm{s}}}$ in the range $0\\le\\varphi\\le\\pi$, the optimality of this construction is assured.\n\n\n\\section{Event-ready gates}\nComing from post-selected gates, the next step towards scalable quantum computation would be to build\ngates not requiring measurements on the output modes. Intuitively it is clear that the construction of\na controlled phase gate in this class will be more demanding with respect to the resources (such as\nthe number of auxiliary modes and photons, size and complexity of the network) involved.\n\nEspecially the number of additional photons will change drastically: having had none in the post-selected case of a controlled $\\pi$-phase gate,\ntwo are required in the class of event-ready gates. We will use this example as a motivation for a detour to discussing a number\nof different methods that could be useful for handling linear optics state preparation.\n\nTo do so, we notice that a controlled $\\pi$-phase gate is more constrained than\na device that creates EPR pairs from single photons.\nThis is meant in the sense that it not only amounts to\na state transformation from two single photons to an EPR pair, but a full unitary transformation\non the entire computational state space in dual-rail encoding (and creating an EPR pair when applied to a\ncertain product input corresponding to the product state of two photons).\nIn the following two sections we will be concerned with different methods to describe linear optics state preparation\nand will apply them to the specific example at hand (i.e., heralded dual-rail EPR pair generations from single photons).\n\nIt will turn out, that the construction of an EPR pair out of single photons by means of linear optics, vacuum modes,\none additional photon, and detectors is not possible.\nOf course, directly solving the polynomial equations in the matrix elements of $A$ (generalisation of Eqns.~(\\ref{eqn:cphase:def1}) to~(\\ref{eqn:cphase:def2})) will yield the same result -- no solutions unless two\nadditional photons are involved.\nHaving excluded the cases of zero and one auxiliary photons, a set-up with two of them \nis possible, proven by the existence of such a scheme (EPR construction~\\cite{ZBL+06} as well\nas controlled-$Z$ gate~\\cite{KLM01}).\n\n\n\\subsection{State transformations}\nAn obvious way of looking at states of exactly $2$ photons in $m$ bosonic field modes is the following. \nSuch a state vector can be written as\n\\begin{eqnarray}\n  |\\psi_M\\rangle &=& P({\\bf a}^\\dagger)|\\mathrm{vac}\\rangle \\nonumber\\\\ &=& \\sum\\limits_{i,j=1}^{m} M_{i,j} \n  a_i^\\dagger a_j^\\dagger |\\mathrm{vac}\\rangle = ({\\bf a}^\\dagger )^T M {\\bf a}^\\dagger |\\mathrm{vac}\\rangle ,\n\\end{eqnarray}\nwhere $M$ is a symmetric $m\\times m$ matrix~\\footnote{The $m=4$ case was used already in \nRefs.~\\cite{LCS99,Calsamiglia02}. Thanks to D.~Uskov for pointing out the nice structure of this.}.\nThe application of a unitary mode-transformation $U$---representing a linear optical network---is reflected by\n\\begin{eqnarray}\n  |\\psi_M\\rangle \\mapsto |\\psi_{M'}\\rangle &=& (U{\\bf a}^\\dagger)^T M (U{\\bf a}^\\dagger) |\\mathrm{vac}\\rangle \\\\\n                  &=& ({\\bf a}^\\dagger)^T M' {\\bf a}^\\dagger |\\mathrm{vac}\\rangle\n\\end{eqnarray}\nwith $M'=U^T MU$ clearly again being symmetric.\nAs a special case of the singular value decomposition~\\cite{HornJohnson}, a diagonal $M'$ can be achieved,\ngiven an arbitrary input state vector $|\\psi_M\\rangle$.\n\nNow let us choose $U$ such that $M'$ is diagonal. Then, labelled by\n\\begin{equation}\n  \\nu' = \\mathrm{rank} (M'), \n\\end{equation}\t\nthere are $m$ different classes of states~\\cite{Kieling08,Kieling09}, in each of which is the states are composed by superpositions of $2$ photons in either of $\\nu'$ modes.\nThese classes are separated by linear optical mode transformations requiring additional modes.\nDecreasing the rank is possible by allowing for auxiliary vacuum modes. However, to increase the rank by $1$ one additional photon is required.\n\nFurther, the state matrix $M$ of two single photons on four modes has rank $\\nu=2$ while an EPR pair corresponds to a matrix with rank $\\nu=4$.\nTherefore, the desired state transformation requires at least two additional single photons.\n\n\\subsection{Polynomial factorisation}\nAn alternative approach is the following~\\cite{Kieling08}: The polynomial describing \nthe objective state vector $|\\psi\\rangle = P({\\bf a}^\\dagger)|\\mathrm{vac}\\rangle$\nwith\n\\begin{equation}\n  P({\\bf a}^\\dagger) = 2^{-1\/2}\\left(a^{\\dagger}_1 a^{\\dagger}_3 + a^{\\dagger}_2 a^{\\dagger}_4\\right)\n\\end{equation}\t\ndoes not factorise over $\\mathbbm{C}$.\nUsing Lemmata~1 and~2 from Ref.~\\cite{Ruppert99}, the property of factorisation of a bivariate polynomial \n\\begin{equation}\n  p(x,y)=\\sum_{i,j=0}^mp_{i,j}x^iy^j \n\\end{equation}\nover $\\mathbbm{C}$ can be tested by assessing the rank of a \ncomplex $2m(2m-1)\\times(m+1)(2m-1)$ matrix. Further, \napplying Lemma~7 of Ref.~\\cite{Kaltofen95}, this technique can be extended to multi-variate polynomials.\nNow, a state can be constructed from a product state using linear optical gate arrays \niff the corresponding polynomial is factorisable.\nIn the case mentioned before (dual-rail EPR pair, so four variables), one can use the resulting $12\\times 9$ matrix to confirm in the language of polynomials of creation operators that\nadditional resources are in fact required.\n\n\\begin{figure}\n  \\figspace{\\vspace{.25cm}}\\includegraphics{cphase_08.eps}\n  \\caption{Compact implementation of a controlled phase gate by using a single loop to implement the central PPBS and the compensation beam splitters simultaneously. Additionally, the two PBS are identified,\n     resulting in a second loop. All omitted modes are initialised in the vacuum and post-selected in the vacuum state (which will be achieved in practice by counting the photons in the other output).\n     \\label{fig:pgate:ppbs} }\n\\end{figure}\n\n\n\\begin{figure}\n \\includegraphics{toffoli.eps}\n  \\caption{Optimal success probabilities of generalised Toffoli gates.\n    The features exhibited by $\\ensuremath{p_{\\mathrm{s}}}(\\varphi)$ are similar to the ones observed at the controlled phase gate (Fig.~\\ref{fig:phases}):\n    There is a shallow dip between $\\varphi=\\pi\/2$ and $\\varphi=\\pi$ below $\\ensuremath{p_{\\mathrm{s}}}(\\pi)=\\ensuremath{p_{\\mathrm{s}}}(\\pi\/2)$ and a steep incline (more pronounced than for the controlled phase gate) for small phases towards $\\ensuremath{p_{\\mathrm{s}}}(0)=1$.\n    \\label{fig:toffoli:ps}}\n\\end{figure}\n\n\n\\section{Toffoli gates}\nIn the same way as above, we can consider a generalised Toffoli gate, the effect of which on the \ncomputational basis realized as dual-rail encodings can be described by the unitary \n\\begin{equation}\n\tU=\\mathrm{diag}(1,1,1,1,1,1,1,\\mathrm{e}^{\\imath \\varphi}).\n\\end{equation}\t\nThe solutions to the polynomial equations describing the action on the three-mode core---up to mode permutations---can be parametrised by $x,y\\in\\mathbbm{R}$ and are given by the matrix\n\\begin{equation}\n  A=\\ensuremath{p_{\\mathrm{s}}}^{1\/6} \\left[\\begin{array}{ccc} 1 & \\frac{\\mathrm{e}^{\\imath\\varphi}-1}{xy} & 0 \\\\ 0 & 1 & y \\\\ x & 0 & 1 \\end{array}\\right] .\n\\end{equation}\nThis has to be understood similar to the $2$-mode core used by the controlled phase gate. However, here we do not solve the full optimisation problem, but only consider global rescaling.\n\nFor a unitary extension all singular values of $A$ have to be at most $1$. In order to avoid to formulate \ncubic singular values explicitly, we use the following constraints. Let\n$p_{AA^{\\dagger}}(\\lambda)=\\det(A A^{\\dagger}-\\lambda\\mathbbm{1}_3)$ be the characteristic polynomial of $A A^{\\dagger}$, the roots $\\lambda_{1,2,3}$ of which are the squared singular values of $A$.\nBy requiring $p_{AA^{\\dagger}}(1)=0$, one of the singular values has to be $1$. For the roots of $p_{AA^{\\dagger}}$ are real-valued and non-negative,\nthe condition that all other singular values are not larger \nthan $1$ is equivalent with the condition\nthat all derivatives of $p_{AA^{\\dagger}}$ have the same sign at $\\lambda=1$. \nMore formally, this results in further constraints of the form\n\\begin{equation}\n  (-1)^{\\alpha} p^{(k)}_{AA^{\\dagger}}(1) = (-1)^{\\alpha} \\left.\\frac{\\mathrm{d}p_{AA^{\\dagger}}(\\lambda)}{\\mathrm{d}\\lambda}\\right|_{\\lambda=1} \\ge0\n\\end{equation}\nfor $1\\le k<n$ where $\\alpha=0,1$ is fixed by the condition of $k=n$.\nFor $\\varphi=\\pi$ the optimal $\\ensuremath{p_{\\mathrm{s}}}$ compliant with these conditions is\n\\begin{equation}\n  \\ensuremath{p_{\\mathrm{s}}}(\\pi) = 1 + 3\\left( 2^{1\/3}-2^{2\/3} \\right) \\approx 1\/57 .\n\\end{equation}\nSee Fig.~\\ref{fig:toffoli:ps} for the maximum success probability in the range $0\\le\\varphi\\le\\pi$.\nThe corresponding networks could be constructed in the same way as above. However, they would consist of $3$-mode cores\n(an interferometer composed of three partially polarising beam splitters)\ninside separate interferometers\nfor each of the $3$ qubits.\nFor free space experiments more appealing approaches for the specific choice of \n$\\varphi=\\pi$ are presented in Refs.~\\cite{RRG07,Fiurasek06}, but \nleading only to success probabilities of at most $1\/72$.\n\n\\section{Remarks on process tomography}\n\nProcess tomography amounts to characterizing (or sometimes certifying) \nan unknown physical process. In practice, the task is to identify that completely\npositive map that is closest to the date with respect to some meaningful figure of merit.\nTo accomplish this task, \none has to consider a tomographically complete set of inputs and look at Hilbert-Schmidt\nscalar products of the output with observables, to faithfully reconstruct the matrix form of the channel \n\\cite{Process1,Process3}. \nPractically, the closest physical process can then be found by solving\na convex optimization problem. \nFormally equivalently, and in instances in an experimentally simpler fashion, one can, instead of\nsending in a full set of input states, submit half of a single fixed maximally entangled state, and hence \nreconstruct the channel from the Choi matrix, then referrred to as entanglement-assisted \nprocess tomography.\n\nThe latter technique can clearly also be applied in case of a postselected quantum gate\nlike a phase gate. Yet, even without entangled inputs and including the actual measurement,\none can  reconstruct the resulting POVM elements, to which essentially postselected gates\namount to when one faithfully includes also the actual measurement in the black-box description\nof the process. If one has a well-characterized source at hand, then the statistics of\n\\begin{equation}\n\tp_{j,k}= \\text{tr}[\\rho_j A_k],\n\\end{equation}\nuniquely characterize the process,\nwhere $\\{\\rho_j\\}$ form a tomographically complete well-characterized\ninput set, and $A_1,\\dots, A_K\\geq $ constitute a POVM, i.e., \n\\begin{equation}\n\t\\sum_{k=1}^K A_k=\\mathbbm{1}\n\\end{equation}\n(for an experimental realization of such an approach, see Ref.\\ \\cite{Process2}). In practice,\none uses methods of convex optimization to identify\nthe closest physical process to the given data.\n\nFor the purposes of the present work, \none of the outcomes $k=1$, that is, a specific pattern $A_1$\nof detection, then gives rise to the actual postselected linear\noptical quantum gate. In this way, one can reconstruct\npostselected quantum gates, without using an ancilla-based approach,\neven faithfully including the final measurement as part of the process.\n\n\\section{Conclusion}\n\nWe have shown how to obtain the maximum probability of success of controlled phase- and Toffoli-like gates in \nthe class of post-selected linear optics dual-rail gates without additional photons.\nFurther, constructions of networks for the smaller gates suitable for experimental implementation have been \ngiven, and techniques elaborated upon that allow for the assessment of the possibility\nof certain linear optical schemes.\nFor further progress concerning the eventual full optimization of linear optical processes \nit would be interesting\nto investigate (i) optimal constructions with respect to a class of gates inherently incorporating such \nexperimental constraints or (ii) identify further\ndecomposition techniques from a given linear optics mode transformation\ninto suitable physical networks, respecting these constraints.\n\n\\section*{Acknowledgements}\n\nFor insightful discussions concerning current \nexperimental \nabilities we would like to thank\nM.~Du{\\v{s}}ek,\nW.~Wieczorek,\nC.~Schmid, and\nJ.~Matthews. \nKK was supported by Microsoft Research through the \nEuropean PhD Programme and the EU (MINOS), \nJE by the EU (QAP, COMPAS, MINOS) \nand the EURYI award.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and preliminaries}\n\nIn  \\cite{P}, H. Poincar\\'e has constructed a flow on a torus \nwith an exceptional minimal set. \nA. Denjoy has constructed such a $C^1$-flow \\cite{D}. \nOn the other hand,  \nthe author has shown that \nthere are no exceptional minimal sets of $C^2$-flows on tori. \nThis result is generalized to the compact surface cases by Schwartz, A. J. \\cite{S}.\nOn the other hand, \nin \\cite{NZ}, \nthey have shown the characterization of \nthe non-wandering flows on compact surfaces with finitely many singularities. \nIn \\cite{Mar}, \none has given a description near orbits of the non-wandering flow with \nthe set of singularities which is totally disconnected. \nMoreover, \nin  \\cite{M}, \nit has shown that \nif a nontrivial recurrent point $x$ of a surface flow \nbelongs to the limit\nset of another nontrivial recurrent point $y$, \nthen \n$y$ \nbelongs to the limit set of $x$. \nWe sharpen these results to analysis surface flows. \nIn particular, \nit has shown that \nthe orbit class of each \nnontrivial weakly recurrent point is \nthe orbit closure in \nthe set of regular weakly recurrent points. \nIn this paper, \nwe show the non-existence of exceptional minimal sets of \ncontinuous non-wandering flows \nwith arbitrary singularities \non compact surfaces \nand obtain  \na topological characterization of \nnon-wandering flows \nand \na topological characterization of \nnon-periodic proper orbits \nof non-wandering flows.  \nMoreover, we give \na smoothability of non-wandering flows \nand \na characterization of topological transitivity \nfor continuous flows. \nIn addition, \nwe construct a smooth flow on $\\mathbb{T}^2$ \nand a foliation on an open manifold contained in $\\mathbb{R}^2$ \nsuch that \n$\\mathrm{LD}$ and $\\mathrm{P}$ are dense, \nwhere \n$\\mathrm{LD}$ is the union of locally dense orbits (resp. leaves) \nand \n$\\mathrm{P}$ is the union of non-closed proper orbits (resp. leaves). \nOn the other hand, \nif $\\mathop{\\mathrm{Sing}}(v)$ \nconsists of finitely many contractible connected components, \nthen $\\mathrm{LD}$ is open \nand $\\mathrm{P}$ \nconsists of finitely many orbits. \n\n\n\nBy flows, \nwe mean continuous $\\mathbb{R}$-actions on surfaces.\nLet \n$v: \\mathbb{R} \\times M \\to M$ be a flow on a compact surface $M$. \nPut $v_t( \\cdot ) := v(t, \\cdot )$ \nand $O_v( \\cdot ) := v(\\mathbb{R}, \\cdot )$.  \nA subset of $M$ is said to be saturated \nif \nit is a union of orbits. \nRecall that \na point $x$ of $M$ is \nsingular if \n$x = v_t(x)$ for any $t \\in \\R$, \nis regular if \n$x$ is not singular, \nand \nis periodic if \nthere is positive number $T > 0$\nsuch that \n$x = v_T(x)$ and  \n$x \\neq v_t(x)$ for any $t \\in (0, T)$. \nDenote by \n$\\mathop{\\mathrm{Sing}}(v)$ \n(resp. $\\mathop{\\mathrm{Per}}(v)$) \nthe set of singular (resp. periodic) points. \nA point $x$ is wandering if \nthere are a neighbourhood $U$ of $x$ and \na positive number $N$ \nsuch that \n$\\bigcup_{t > N} v_t(U) \\cap U = \\emptyset$, \nand is  \nnon-wandering if  \n$x$ is not wandering \n(i.e. for each neighbourhood $U$ of $x$ and \neach positive number $N$, \nthere is $t \\in \\mathbb{R}$ with $|t| > N$ such that \n$v_t(U) \\cap U \\neq \\emptyset$). \nAn orbit is non-wandering if \nit consists of non-wandering points \nand \nthe flow $v$ is non-wandering if \nevery point is non-wandering. \nFor a point $x \\in M$, \ndefine \nthe omega limit set $\\omega(x)$\nand \nthe alpha limit set $\\alpha(x)$ \nof $x$\nas follows: \n$\\omega(x) \n:= \\bigcap_{n\\in \\mathbb{R}}\\overline{\\{v_t(x) \\mid t > n\\}} \n$, \n$\\alpha(x) \n:= \\bigcap_{n\\in \\mathbb{R}}\\overline{\\{v_t(x) \\mid t < n\\}} \n$. \nA point $x$ of $M$ is \npositive recurrent (resp. negative recurrent) if \n$x \\in \\omega(x)$ \n(resp.  $x \\in \\alpha(x)$),  \nand that \n$x$ is recurrent (resp. weakly recurrent) \nif $x$ is positive and (resp. or) negative recurrent. \nA (weakly) recurrent orbit is an orbit of such a point. \nAn orbit is proper if \nit is embedded, \nlocally dense if \nthe closure of it has nonempty interior, \nand \nexceptional if \nit is neither proper nor locally dense.  \nA point is proper (resp. locally dense, exceptional) if \nso is its orbit. \nDenote by \n$\\mathrm{LD}$ \n(resp. $\\mathrm{E}$, $\\mathrm{P}$)\nthe union of locally dense orbits \n(resp. exceptional orbits, \nnon-closed proper orbits).  \nNote \n$\\mathrm{P}$ is the complement of \nthe set of weakly recurrent points. \nBy the definitions,  \nwe have a decomposition \n$\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{P}\n\\sqcup \\mathrm{LD} \n\\sqcup \\mathrm{E} = M$. \nA (weakly) recurrent orbit is nontrivial if \nit is not closed.  \nNote that \nthe union of nontrivial weakly recurrent orbits \ncorresponds with $\\mathrm{LD} \\sqcup \\mathrm{E}$.   \nA quasi-minimal set of $v$ is the closure of a nontrivial weakly recurrent orbit. \nIt's known that \nthe total number of quasi-minimal sets for $v$ \ncannot exceed $g$ if $M$ is an orientable surface of genus $g$ \\cite{M}, \nand $\\frac{p-1}{2}$ if $M$ is a non-orientable surface of genus $p$ \\cite{Ma}. \nTherefore \nthe closure \n$\\overline{\\mathrm{LD \\sqcup E}}$ consists of finitely many quasi-minimal sets. \nBy a limit cycle, we mean \na periodic orbit of $v$ which is the $\\alpha$-limit set \nor the $\\omega$-limit set of some point \nnot on the periodic orbit. \n\n\n\n\n\n\\section{A Topological characterization of \nnon-wandering surface flows with arbitrary singularities}\n\nLet $v$ be a continuous flow on a compact surface $M$. \nWe call \nthat a collar $A$ of an periodic orbit $O \\subset A$  is \nan annulus $A$ one of whose connected component of \n$\\partial A$ is $O$, \nwhere $\\partial A := \\overline{A} - \\mathrm{int} A$ \nis the topological boundary of $A$. \nFirst, we state a following easy observation. \n\n\\begin{lemma}\\label{lem00}\n$O \\subset \\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{P}$ \nfor an orbit $O$ with $\\overline{O} \\cap \\mathop{\\mathrm{Per}}(v) \\neq \\emptyset$.  \nMoreover \neach limit cycle is contained in \n$\\partial (\\mathrm{int} \\mathrm{P})$. \n\\end{lemma}\n\n\\begin{proof}\nLet $O$ be an non-periodic orbit with $\\overline{O} \\cap \\mathop{\\mathrm{Per}}(v) \\neq \\emptyset$. \nThen $\\overline{O} - O$ contains a limit cycle $\\gamma$. \nThe  flow box theorem (cf. Theorem 1.1, p.45\\cite{ABZ}) implies that \na limit cycle $\\gamma$ is covered by finitely many flow boxes $\\{ U_i' \\}$. \nSince $\\overline{O} \\cap \\gamma \\neq \\emptyset$,  \nthe one-sided holonomy of $\\gamma$ is contracting or expanding. \nFix a point $z$ of $O$ in a flow box $U_i'$. \nThen the point $z$ has an open neighbourhood $U \\subset U_i'$ \nwhich is a flow box such that \n$O \\cap U$ is one arc. \nThen $\\cup_{t \\in \\mathbb{R}} v_t(U)$ \nis an open neighbourhood of $O$ in \nwhich $O$ is closed.    \nHence $O \\subset \\mathrm{P}$.  \nLet $U' \\subset \\bigcup_i U_i'$ be \na sufficiently small collar of $\\gamma$ \nwhere the holonomy along $\\gamma$ is contracting or expanding.  \nThen the orbit closure of each point $y$ in $U' - \\gamma$ contains $\\gamma$ \nbut the orbit of $y$ is not closed. \nTherefore $y \\in \\mathrm{P}$\n and so $V := \\cup_{t \\in \\mathbb{R}} v_t(U') - \\gamma \\subseteq \\mathrm{P}$ \nis a saturated open subset with $\\gamma \\subseteq \\partial V \n\\subseteq \\partial (\\mathrm{int} \\mathrm{P})$. \nThis implies the second assertion. \n\\end{proof}\n\n\nRecall that \nthe orbit class $\\hat{O}$ of an orbit $O$ is \nthe union of points each of whose orbit closure \ncorresponds with $\\overline{O}$ \n(i.e.  $\\hat{O} := \\{ y \\in M \\mid \\overline{O} = \\overline{O_v(y)} \\}$). \nNow we show that \nthe orbit class of each \nnontrivial weakly recurrent point is \nthe orbit closure in \nthe set of regular weakly recurrent points, \nwhich \nrefine a Ma\\v \\i er-type result \\cite{M}.  \n\n\\begin{proposition}\\label{lem001}\nFor an \norbit $O \\subset \\mathrm{LD} \\sqcup \\mathrm{E}$,  \nwe have \n$\\hat{O} = \\overline{O} \n\\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P}) \n\\subset \\mathrm{LD} \\sqcup \\mathrm{E}$. \n\\end{proposition}\n\n\\begin{proof}\nLet $Q$ be a quasi-minimal set which is the closure of $O$. \nBy Lemma \\ref{lem00}, \nwe have $Q \\cap \\mathop{\\mathrm{Per}}(v)  = \\emptyset$. \nNote that \nthe inverse image of \nproper (resp. locally dense, exceptional) orbits \nby any finite covering \nare also proper (resp. locally dense, exceptional). \nBy taking a double covering of $M$ and the doubling of $M$ \nif necessary, \nwe may assume that $v$ is transversally orientable \nand $M$ is closed and orientable. \nFor any point $y \\in \\hat{O} - O$, \nwe have \n$\\overline{O_v(y)} = \\overline{O}$ \nand so \n$O \\subseteq \\omega(y) \\cup \\alpha(y)$. \nSince $y \\in \\overline{O}  - O$, \nwe obtain $y \\in \\omega(y) \\cup \\alpha(y)$. \nThen $y$ is not proper. \nThe regularity of $y$ implies \n$y \\notin \\mathop{\\mathrm{Sing}}(v)$ \nand so \n$y \\in Q \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P}) \n\\subseteq \\mathrm{LD} \\sqcup \\mathrm{E}$. \nThus \n$\\hat{O} \\subseteq Q \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P})$. \nOn the other hand, \nwe show that \n$\\hat{O} \\supseteq Q \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P})$. \nIndeed, \nif there is exactly one quasi-minimal set $Q$, \nthen \n$\\overline{O_v(x)} = Q$ for any \n$x \\in Q  \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P}) \n\\subseteq \\mathrm{LD} \\sqcup \\mathrm{E}$. \nThus we may assume that \nthere are at least two quasi-minimal sets. \nThe above Ma\\v \\i er work \\cite{M} \n(cf. Remark 2 \\cite{AZ}) impies that \nthe genus of  $M$ is at least two.  \nNote \nCherry has proved that a quasi-minimal set contains \na continuum of nontrivially recurrent orbits each of  which is dense\nin the quasi-minimal set (Theorem VI\\cite{C}).  \nTherefore $Q$ contains \nnontrivially recurrent orbits. \nFix a recurrent point $x \\in (\\mathrm{LD} \\sqcup \\mathrm{E}) \\cap Q$ \nwhose orbit closure is $Q$. \nFor \nany point $y \\in Q \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P})$, \nwe have \n$y \\in \\mathrm{LD} \\sqcup \\mathrm{E}$  \nand so \n$y$ is weakly recurrent. \nBy the above Cherry result, \nthere is a recurrent point $z \\in \\overline{O_v(y)}$ \nwhose orbit closure is $\\overline{O_v(y)}$. \nBy another Ma\\v \\i er theorem (cf. Theorem 4.2 \\cite{AZ}) \nand its dual, \nwe obtain \n$\\omega(x) = \\omega(z)$ \nand \n$\\alpha(x) = \\alpha(z)$.  \nThus \n$\\overline{O} = \nQ = \n\\overline{O_v(x)} = \n\\overline{O_v(z)} = \n\\overline{O_v(y)}$. \nThis means  \n$\\hat{O} = Q \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P})$.  \n\\end{proof}\n\nWe state a key lemma which is a relation between exceptional \nand proper orbits.\nRecall that a subset $S$ of a surface $M$ is essential if \nsome connected component of $S$ \nis neither null homotopic \nnor homotopic to a subset of the boundary $\\partial M$. \n\n\n\\begin{lemma}\\label{lem0bb}\n$\\overline{\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{LD}} \n\\cap \\mathrm{E}= \\emptyset$ \nand \n$\\overline{\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{E}} \n\\cap \\mathrm{LD}= \\emptyset$.  \nMoreover \n$\\mathrm{E}\\subset  \\mathrm{int} \\overline{\\mathrm{P}}$. \n\\end{lemma}\n\n\n\\begin{proof}\nBy taking a double covering of $M$ and the doubling of $M$ \nif necessary, \nwe may assume that $v$ is transversally orientable \nand $M$ is closed and orientable. \nBy the Ma\\v \\i er theorem, \n$\\overline{\\mathrm{E}}$\n(resp. $\\overline{\\mathrm{LD}}$) \nconsists of finitely many closures of \nexceptional (resp. locally dense) orbits.  \nBy Proposition \\ref{lem001}, \nwe have \n$\\overline{\\mathrm{LD}} \\cap \\mathrm{E} = \\emptyset$ \nand \n$\\mathrm{LD} \\cap \\overline{\\mathrm{E}} = \\emptyset$. \nRecall that \nthe flow box theorem \nimplies that \nthe orbits of $v$ on a surface $M - \\mathop{\\mathrm{Sing}}(v)$ \nform a foliation $\\mathcal{F}$. \nMoreover there is \na transverse foliation $\\mathcal{L}$ for this foliation $\\mathcal{F}$ by \nProposition 2.3.8 (p.18 \\cite{HH}). \nWe show \n$ \\overline{\\mathop{\\mathrm{Per}}(v)} \\cap (\\mathrm{E} \\sqcup \\mathrm{LD}) =  \\emptyset$.\nOtherwise\nthere is a sequence of periodic orbits $O_i$ \nsuch that \nthe closure $\\overline{\\cup_i O_i}$ \ncontains \na point $x \\in \\mathrm{E} \\sqcup \\mathrm{LD}$.  \nBy Lemma \\ref{lem00}, \nwe have  $\\overline{O_v(x)} \\cap \\mathop{\\mathrm{Per}}(v) = \\emptyset$. \nWe show that \nthere is $K \\in \\mathbb{Z}_{>0}$ such that \n$O_k$ is contractible in $M_K$ \nfor any $k > K$ \nwhere $M_K$ is the resulting closed surface \nof adding \n$2K$ disks to $M - (O_1 \\sqcup \\cdots \\sqcup O_K)$. \nIndeed,  \nwe may assume that \n$O_1$ is \nessential \nby renumbering. \nLet $M_1$ be \nthe resulting closed surface \nof adding two center disks to $M - O_1$. \nThen $g(M_1)< g(M)$, \nwhere $g(N)$ is the genus of a surface $N$. \nSince $M$ and $M_1$ are closed surfaces, \nby induction \nfor essential closed curves \nat most $g(M)$ times, \nthe assertion is followed. \nSince $M$ is normal, \nthere are open disjoint neighbourhoods $U_x$ and $V$ of \n$\\overline{O_v(x)}$ and $\\cup_{i \\leq K} O_i$ respectively.\nThen \nthere is \na transverse arc $\\gamma \\subset U_x \\cap l_x$ through $x$ \nwhich does not intersect $\\cup_{i \\leq K} O_i$, \nwhere $l_x \\in \\mathcal{L}$. \nSince $O_v(x)$ is exceptional or locally dense, \nwe have \n$x$ is a  weakly recurrent point and so \nthere is an arc $\\gamma'$ in $O_v(x)$ \nwhose boundaries are contained in $\\gamma$. \nSince $M$ is normal, \nthere is a neighbourhood of $\\gamma \\cup \\gamma'$ \nwhich does not intersect $\\cup_{i \\leq K} O_i$. \nBy the waterfall constraction (cf. Lemma 1.2, p.46\\cite{ABZ}), \nwe can construct \na closed transversal $T$ for $v$ through $x$ \nwhich does not intersect $\\cup_{i \\leq K} O_i$. \nLet $v_K$ be a resulting flow on $M_K$ by adding center disks. \nThen $T$ is also a closed transversal for $v_K$. \nHowever there is $k > K$ such that \n$T$ intersects $O_k$ which is contractible in $M_K$. \nThis is impossible. \nThe closedness of $\\mathop{\\mathrm{Sing}}(v)$ \nimplies the first assertion. \nSince $M = \n\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \n\\mathrm{LD}\\sqcup \\mathrm{P} \\sqcup \\mathrm{E}$, \nwe obtain that \n$\\mathrm{E}$ is contained in an open subset \n$M - \\overline{\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{LD}} \n\\subseteq \\mathrm{P} \\sqcup \\mathrm{E}$.  \nTherefore \n$\\mathrm{E} \\subseteq \\mathrm{int}(\\mathrm{P} \\sqcup \\mathrm{E})$. \nSince $\\overline{\\mathrm{E}}$ consists of finitely many closures of exceptional orbits, \nwe have that \n${\\mathrm{E}}$ is nowhere dense. \nThis implies \n$\\mathrm{E} \\subset \\overline{\\mathrm{P}}$ \nand so  \n$\\mathrm{E} \\subset \\mathrm{int}(\\mathrm{P} \\sqcup \\mathrm{E}) \n \\subset  \\mathrm{int} \\overline{\\mathrm{P}}$. \n\\end{proof}\n\n\n\n\n\n\n\nFrom now on, \nwe consider only non-wandering cases \nin this section. \n\n\n\\begin{lemma}\\label{lem0aa}\nLet $v$ a non-wandering flow on a compact surface $M$. \nThen \n$\\mathop{\\mathrm{Per}}(v)$ is open, \n$M = \n\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \n\\mathrm{LD}\\sqcup \\mathrm{P}$, \nand \n$\\overline{\\mathrm{LD}\\sqcup \\mathop{\\mathrm{Per}}(v)}  \n\\supseteq M - \\mathop{\\mathrm{Sing}}(v)$. \n\\end{lemma}\n\n\n\n\n\n\n\\begin{proof}\nBy taking a double covering of $M$ if necessary, \nwe may assume that $v$ is transversally orientable. \nBy Theorem III.2.12, III.2.15 \\cite{BS}, \nthe set of recurrence points is dense in $M$. \nBy Lemma \\ref{lem00}, \nthere are no limit cycles. \nWrite $U := M - \\overline{\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{LD}}$.  \nBy Lemma \\ref{lem0bb}, \nthis $U \\subseteq \\mathrm{P} \\sqcup \\mathrm{E}$ is an open \\nbd of \n$\\mathrm{E}$. \nSince each point of $\\mathrm{P}$ is not weakly recurrent, \nwe have $\\overline{\\mathrm{E}} \\supseteq U$. \nSince $\\mathrm{E}$ is nowhere dense, \nwe have $U$ is empty and so is $\\mathrm{E}$. \nHence \n$M = \n\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \n\\sqcup \n\\mathrm{LD}\\sqcup \\mathrm{P}$. \nSince there are no limit cycles, \nwe obtain \n$\\overline{\\mathrm{LD} \n} \\cap \\mathop{\\mathrm{Per}}(v) = \\emptyset$.  \nFix a periodic orbit $O$. \nThen there is an annular  \n\\nbd $V$ of $O$ \nwhich is a finite union of flow boxes \nsuch that \n$V \\cap \n(\\mathop{\\mathrm{Sing}}(v) \\sqcup \\overline{\\mathrm{LD}} )\n = \\emptyset$. \nHence \n$V \\subseteq \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{P}$.  \nSince the set of recurrence points is dense in $M$, \nwe have that \n$V \\cap \\mathop{\\mathrm{Per}}(v)$ \nis dense in $V$. \nTherefore \nthe holonomy of $O$ is \nidentical \nand so \n$V \\subseteq \\mathop{\\mathrm{Per}}(v)$. \nThis means that \n$\\mathop{\\mathrm{Per}}(v)$ is open. \nFor any $x \\in \\mathrm{P}$, \nsince the regular weakly recurrent points form \n$\\mathrm{Per}(v) \\sqcup \\mathrm{LD}$, \nby non-wandering property, \neach neighbourhood of $x$ meets \n$\\mathrm{Per}(v) \\sqcup \\mathrm{LD}$ \nand so \n$\\overline{\\mathrm{LD}\\sqcup \\mathop{\\mathrm{Per}}(v)} \\supseteq \\mathrm{P}$.  \n\\end{proof}\n\n\n\n\n\n\n\n\nNow we state the characterization of non-wandering flows. \n\n\\begin{theorem}\\label{prop0c}\nLet $v$ be a continuous flow on a compact surface $M$. \nThen \n$v$ is non-wandering \nif and only if \n$\\overline{\\mathrm{LD} \\sqcup \\mathop{\\mathrm{Per}}(v)} \\cup \\mathop{\\mathrm{Sing}}(v) \n= M$.  \nIn particular, \nif $v$ is non-wandering, \nthen \n$\\mathop{\\mathrm{Per}}(v)$ \nis open \nand \nthere are no exceptional orbits. \n\\end{theorem}\n\n\n\n\n\\begin{proof}\nSuppose that $v$ is non-wandering. \nBy Lemma \\ref{lem0aa}, \nwe have \n$\\overline{\\mathrm{LD}\\sqcup \\mathop{\\mathrm{Per}}(v)} \\cup \\mathop{\\mathrm{Sing}}(v) \n= M$.  \nConversely, \nsuppose that \n$\\overline{\\mathrm{LD}\\sqcup \\mathop{\\mathrm{Per}}(v)} \\cup \\mathop{\\mathrm{Sing}}(v) = M$.  \nFor any regular point $x$ of $M$, \nwe have \n$x \\in \\overline{\\mathrm{LD}\\sqcup \\mathop{\\mathrm{Per}}(v)}$. \nThis shows that $v$ is non-wandering. \n\\end{proof}\n\n\n\nWe state the following charcterization of \nthe union \n$\\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v)$ of non-periodic proper orbits. \n\n\n\\begin{proposition}\\label{lem32}\nLet $v$ be a continuous non-wandering flow \non a compact surface $M$. \nThen $\\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v) = \n\\{ x \\in M \\mid \\omega(x) \\cup \\alpha(x) \\subseteq \\mathop{\\mathrm{Sing}}(v) \\}$. \n\\end{proposition}\n\n\\begin{proof}\nWe may assume \n$M$ is connected. \nBy taking a double covering of $M$ and the doubling of $M$ \nif necessary, \nwe may assume that $v$ is transversally orientable \nand $M$ is closed and orientable. \nObviously \n$\\mathrm{P}  \\sqcup \\mathop{\\mathrm{Sing}}(v) \n\\supseteq \\{ x \\in M \\mid \\omega(x) \\cup \\alpha(x) \\subseteq \\mathop{\\mathrm{Sing}}(v) \\}$. \nTherefore \nit suffices to show this converse relation.\nFix a point $y \\in \\mathrm{P}$. \nBy Lemma \\ref{lem00}, \nthe non-wandering property implies  \n$\\overline{O_v(y)} \\cap \\mathop{\\mathrm{Per}}(v) = \\emptyset$. \nWe show \n$\\overline{O_v(y)} \\cap \\mathrm{LD} = \\emptyset$. \nOtherwise \n$\\overline{O_v(y)}$ contains a locally dense orbit $O$. \nSince $\\mathrm{int}\\overline{O}$ is an open \\nbd of $O$, \nwe have  \n$O_v(y) \\cap \\mathrm{int}\\overline{O} \\neq \\emptyset$. \nThe relation \n$y \\in \\overline{O} \\subseteq \\overline{O_v(y)}$ \nimplies that \n$y$ is not proper, which is a contradiction. \nSince $\\mathrm{E} = \\emptyset$, \nwe have \n$\\overline{O_v(y)} \\subset \\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v)$. \nSuppose that \n$y \\in \\overline{\\mathrm{LD}}$. \nThus there is a recurrent point in $\\mathrm{LD}$ \nwhose orbit closure contains $O_v(y)$. \nRemoving the singular points, \nTheorem 3.1\\cite{Marz} implies \n$\\omega(y) \\cup \\alpha(y) \\subseteq \\mathop{\\mathrm{Sing}}(v)$. \nThis means \n$y \\in \n\\{ x \\in M \\mid \\omega(x) \\cup \\alpha(x) \\subseteq \\mathop{\\mathrm{Sing}}(v) \\}$. \nOtherwise \n$y \\notin \\overline{\\mathrm{LD}}$. \nSuppose that \neach periodic orbit is null homotopic (i.e. non-essential). \nFix a saturated \\nbd $W$ of $y$. \nThen the intersection   \n$W \\cap \\mathop{\\mathrm{Per}}(v)$ \nis open dense in $W$.  \nTo apply Theorem 3.1\\cite{Marz}, \nwe need replace a small flow box near $O_v(y)$. \nTake a flow box $B \\subset W$ \nwhich can be identified with $[-1, 1] \\times [-1, 1]$ \nsuch that \n$O_v(y) \\cap B = [-1, 1] \\times \\{ 0 \\}$ \nand \neach orbit in $B$ is $[-1, 1] \\times \\{ a \\}$ for some $a \\in [-1, 1]$. \nFix a small transverse arc $\\gamma$ through $y$ in $B$. \nLet $\\gamma_+$ be a connected component of $\\gamma \\setminus O$. \nSince each periodic orbit is null homotopic, \neach periodic orbit intersects $\\gamma_+$ at most one point. \nThis means that \neach point $ p:= (1, \\varepsilon)$ in $\\mathop{\\mathrm{Per}}(v) \\cap (\\{ 1 \\} \\times [-1, 1])$ \nintersecting $\\gamma_+$ \ngoes back to a point $(-1, \\varepsilon)$ in $B$ \n(i.e. $O_v(p) \\cap B = \\{ 1 \\} \\times \\{ \\varepsilon \\}$). \nTherefore \nthe flow $v|_B$ induces \na homeomorphism \nfrom $\\{-1 \\} \\times [-1, 1]$ \nto $\\{ 1 \\} \\times [-1, 1]$ \nwhich can be identified with the identity mapping $1_{[-1, 1]}$ on $[-1, 1]$. \nReplacing $1_{[-1, 1]}$ \nwith a homeomorphism $f$ which is contracting near $0$ \n(e.g. $f(x) = x^3$), \nwe obtain the resulting continuous flow $w$ \nsuch that \n$O_v(y) = O_w(y)$ and \n$\\mathop{\\mathrm{Sing}}(w) = \n\\mathop{\\mathrm{Sing}}(v)$, \nmodifying $v$ in $B$. \nWe identify $O_v(y) \\cap B = [-1, 1] \\times \\{ 0 \\}$ \nwith $\\mathrm{dom} (f) = [-1, 1]$. \nBy the Baire category theorem, \nthe countable intersection \n$\\bigcap_{n \\in \\Z} f^{n}([-1, 1] \\cap \\mathop{\\mathrm{Per}}(v))$ \nis dense in $[-1, 1]$. \nThus \nthere is a point $z \\in \\mathop{\\mathrm{Per}}(v)$ \nsuch that \n$O_w(z)$ is proper \nand \n$y \\in \\overline{O_w(z)} - O_w(z)$.  \nRemoving the singular points, \nTheorem 3.1\\cite{Marz} for $w$ implies \n$\\overline{O_w(y)} - O_w(y) = \n\\alpha_w(y) \\cup \\omega_w(y) \\subseteq \\mathop{\\mathrm{Sing}}(w)$. \nSince \n$O_v(y) = O_w(y)$ and \n$\\mathop{\\mathrm{Sing}}(w) = \n\\mathop{\\mathrm{Sing}}(v)$, \nwe have \n$\\omega(y) \\cup \\alpha(y) \\subseteq \\mathop{\\mathrm{Sing}}(v)$. \nSuppose that \nthere are essential periodic orbits. \nLet $O$ be an essential periodic orbit. \nCutting this periodic orbit $O$, \nthe remainder $M - O$ has \nnew two boundaries. \nAdding two center disks to the two boundaries of $M - O$,  \nwe obtain \na new closed surface $M'$ whose genus is less than one of $M$ \nand the resulting non-wandering flow $v'$ on $M'$ \nwith \n$\\mathop{\\mathrm{Sing}}(v') = \\mathop{\\mathrm{Sing}}(v)$\nsuch that \n$\\alpha(y) = \\alpha_{v'}(y)$ \nand \n$\\omega(y) = \\omega_{v'}(y)$, \nwhere \n$\\alpha_{v'}(y)$ (resp. $\\omega_{v'}(y)$) is \nthe alpha (resp. omega) limit set of $y$ with respect to $v'$. \nBy finite iterations, \nwe obtain a new closed surface $M^{\\dagger}$ \nand the resulting non-wandering flow $v^{\\dagger}$ on $M^{\\dagger}$ \nwith \n$\\mathop{\\mathrm{Sing}}(v^{\\dagger}) = \\mathop{\\mathrm{Sing}}(v)$\nsuch that \n$\\alpha(y) = \\alpha_{v^{\\dagger}}(y)$ \nand \n$\\omega(y) = \\omega_{v^{\\dagger}}(y)$ \nsuch that \neach periodic orbit of $v^{\\dagger}$ is null homotopic. \nThis implies \n$\\omega(y) \\cup \\alpha(y) \\subseteq \\mathop{\\mathrm{Sing}}(v)$. \n\\end{proof}\n\n\n\n\n\n\n\n\nTaking a suspension of \na non-wandering circle homeomorphism, \nwe have the following statement. \n\n\n\n\n\\begin{corollary}\\label{cor26}\nEach non-wandering continuous homeomorphism \non $\\S^1$  \nis topologically conjugate to \na rotation. \n\\end{corollary}\n\n\n\n\\begin{proof}\nLet $v$ be the suspension of \na non-wandering circle homeomorphism $f$. \nThen $v$ is a non-wandering continuous flow on $\\T^2$.\nSince $v$ is a suspension, \n there are no singular points. \n By Proposition \\ref{lem32}, \n we have \n $\\mathrm{P} = \\emptyset$.  \nTheorem \\ref{prop0c} implies \n $\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{LD} = \\T^2$. \n Since $\\mathop{\\mathrm{Per}}(v)$ \nis open, \nthe complement \n$\\mathrm{LD} = \\T^2 - \\mathop{\\mathrm{Per}}(v)$ \nis closed. \nBy Lemma \\ref{lem0bb}, \nwe obtain \n$\\overline{\\mathop{\\mathrm{Per}}(v)} \n\\cap \\overline{\\mathrm{LD}} \n= \\overline{\\mathop{\\mathrm{Per}}(v)} \n\\cap {\\mathrm{LD}}  \n= \\emptyset$. \nThis implies that \nthe both of \n$\\mathop{\\mathrm{Per}}(v)$ \nand $\\mathrm{LD}$ \nare open and closed. \nSince $\\T^2$ is connected, \nwe have that \n$v$ is pointwise periodic or \nminimal. \nThen $f$ is topologically conjugate to periodic or \nminimal. \nThis means that \n$f$ is topologically conjugate to \na rotation. \n\\end{proof}\n\n\n\n\nBy the smoothing result \\cite{G}, \nwe have the following result.\n\n\\begin{corollary}\\label{cor21}\nEach non-wandering continuous flow on a compact surface $M$ \nis topologically equivalent to a $C^{\\infty}$ flow. \n\\end{corollary}\n\n\n\\begin{proof}\nWe may assume that \n$M$ is connected. \nBy Theorem \\ref{prop0c}, \nthere are no exceptional orbits. \nBy Proposition \\ref{lem32}, \nthe closure of each orbit in $\\mathrm{P}$ contains \nsingular points.  \nWe show that \nif the closure of locally dense orbit is minimal \nthen it is the whole surface which is $\\mathbb{T}^2$. \nIndeed, \nlet $O$ be a locally dense orbit whose closure is minimal. \nThen $\\overline{O}$ contains \nneither singular points\nnor periodic points. \nSince the closure of an orbit in $\\mathrm{P}$ contains \nsingular points, \nthe minimal set $\\overline{O}$ consists of \nlocally dense orbits. \nSince each point of $\\overline{O}$ is contained \nin the interior of $\\overline{O}$, \nthis minimal set $\\overline{O}$ is closed and open, and so  \nis the whole surface  $M$ which is homeomorphic to $\\mathbb{T}^2$.\nTherefore each minimal set is \neither a closed orbit or $\\mathbb{T}^2$. \nBy the smoothing theorem \\cite{G}, \nthis continuous flow is topologically equivalent to a $C^{\\infty}$ flow. \n\\end{proof}\n\n\n\n\nWe state uniformity of $\\mathop{\\mathrm{Per}}(v)$ \nof a non-wandering continuous flow $v$. \n\n\\begin{corollary}\\label{cor29}\nLet $v$ be a non-wandering continuous flow \non a compact surface $M$.  \nThen \neach connected component of $\\mathop{\\mathrm{Per}}(v)$ \nis either \na connected component of $M$,   \nan open annulus, or \nan open M\\\"obius band. \nIf \n$\\mathop{\\mathrm{Sing}}(v)$ \nconsists of finitely many contractible connected components, \nthen $\\mathrm{P}$ consists of finitely many separatrices,  \n$\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P}$ is closed, \nand \n$\\mathrm{LD}$ is open. \n\\end{corollary}\n\n\n\\begin{proof} \nSince $\\mathop{\\mathrm{Per}}(v)$ is open, \nthe flow box theorem \nimplies that \neach periodic orbit has a saturated \\nbd \nwhich is either an annulus or a M\\\"obius band. \nNotice that  \na union of one saturated M\\\"obius band and one saturated annulus with one intersection \nis a M\\\"obius band \nand that \na union of two saturated annuli (resp. M\\\"obius bands) with one intersection \nis an annulus (resp. a Klein bottle).\nFix a connected component $U$ of $\\mathop{\\mathrm{Per}}(v)$.  \nIf $\\partial U = \\emptyset$, \nthen $U$ is a connected component of $M$. \nIf $\\partial U \\neq \\emptyset$, \nthen $U$ is \neither an annulus \nor a M\\\"obius band. \nSuppose that $\\mathop{\\mathrm{Sing}}(v)$ \nconsists of finitely many contractible connected components. \nCollapsing each connected component of \n$\\mathop{\\mathrm{Sing}}(v)$ into a point, \nby Theorem 1\\cite{RS}, \nthe resulting space is homeomorphic to \nthe original space $M$ \nand the resulting flow is non-wandering. \nThus we may assume that \n$\\mathop{\\mathrm{Sing}}(v)$ is finite.  \nBy Theorem 3\\cite{CGL}, \neach singular point is \neither a center or a multi-saddle. \nBy Proposition \\ref{lem32}, \neach orbit in $\\mathrm{P}$ is \na separatrix. \nTherefore \n$\\mathrm{P}$ consists of finitely many separatrices and so   \n$\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P}$ is closed.   \nSince $\\overline{\\mathop{\\mathrm{Per}}(v)} \\cap \\mathrm{LD} = \\emptyset$, \nthe complement \n$\\mathrm{LD} = M - (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P} \\sqcup \\mathop{\\mathrm{Per}}(v))$ is open. \n\\end{proof} \n\n\n\nWe show that \n$\\mathrm{LD}$ is not open in general.  \nIn fact, \nwe construct \na non-wandering flow on $\\T^2$ \nsuch that \n$\\mathrm{P}$ and \n$\\mathrm{LD}$ are dense as follows. \n\n\n\n\n\\begin{example}\\label{ex1}\nConsider an irrational rotation $v$ on $\\T^2$. \nFix any points $p \\in \\T^2$. \nLet  $( t_i )_{i \\in \\Z}$ of $\\R$ be \na sequence such that \n$\\lim_{i \\to \\infty} t_i = \\infty$, \n$\\lim_{i \\to - \\infty} t_i = - \\infty$, \nand \n$\\lim_{i \\to \\infty} p_i \n= \\lim_{i \\to - \\infty} p_i \n= p_0$, \nwhere $p_i := v(t_i, p_0) \\in O_v(p)$.  \nUsing dump functions, \nreplace $O_v(p)$ \nwith a union of countably many singular points $p_i$ ($i \\in \\Z$) \nand countably many proper orbits. \nLet $v'$ be the resulting vector field. \nFor any point $x \\in \\T^2 - O(p)$, \nwe have $O_v(x) = O_{v'}(x)$. \nMoreover $O_v(p) - \\mathop{\\mathrm{Sing}}(v') = \\mathrm{P}(v')$.  \nBy construction, \n$\\mathrm{LD}$ is not open but \n$\\overline{\\mathrm{P}} = \\overline{\\mathrm{LD}} = \\T^2$. \nThis example also shows that \nthe finiteness condition in \nCorollary \\ref{cor29} \nis necessary. \n\\end{example}\n\nIn \\cite{H2}, \na foliation $\\F$  on a manifold $M$ is \nsaid to be ``rare species'' \nif \neither \nall leaves are exceptional \nor \n$\\F$ has at least two of three type (i.e. proper, locally dense, exceptional) \nand the union of leaves of each type is dense. \nThe author \nhave constructed \nsome kind of   \ncodimension one ``rare species''  foliations on compact $3$-manifolds.\nAnalogically we call that \na flow $v$ on $M$ is \n``rare species''\nif \neither \n$\\mathrm{E} = M$ \nor \n$v$ has at least two of three type (i.e. proper, locally dense, exceptional) \nand the union of orbits of each type is dense. \nNote \nthat ``rare species'' surface flows \nare non-wandering and so \nhave no exceptional orbits. \nIn contrast to \nfoliations, \nthis implies \nthere is only one possible kind of \n``rare species'' flows on compact surfaces.  \nNotice that \nthe above example is \na ``rare species'' flows on $\\mathbb{T}^2$. \nTherefore, \nwe obtain the following statement. \n\n\n\n\n\\begin{proposition}\\label{prop}\nThere are smooth ``rare species'' flows on $\\mathbb{T}^2$.   \nOn the other hand, \nfor each ``rare species'' flow $v$ on a compact surface $M$, we have \n$M = \\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{LD} \\sqcup \\mathrm{P}$ \nand \n$\\overline{\\mathrm{LD}} = \n\\overline{\\mathrm{P}} = M$.   \n\\end{proposition}\n\n\nNote that \nall known codimension one foliations on \ncompact manifolds are not $C^1$ but continuous. \nIn contrast to codimension one foliations on \ncompact manifolds,  \neach ``rare species'' flow $v$ on a compact surface \nis topologically equivalent \nto a $C^{\\infty}$ flow. \nMoreover, \nwe obtain following examples. \n\n\\begin{corollary}\\label{cor}\nThere are codimension one ``rare species'' foliations \non open surfaces contained in $\\R^2$.  \nIn particular, \nthe union of locally dense leaves \nis not open. \n\\end{corollary}\n\n\\begin{proof}\nFix a minimal codimension one foliation $\\F$ \non open surfaces contained in $\\R^2$  \n(e.g. a foliation in Theorem \\cite{F}). \nReplace a locally dense leaf $L$ \ninto countably many singular points and \nproper leaves connecting two singular points,  \nand  \nremove the singular points as above construction for flows. \nThen we can obtain a desired foliation. \n\\end{proof}\n\n\n\n\\section{Applications of this characterization}\n\n\n\n\n\n\n\n\n\n\n\nRecall that \n$v$ is topologically transitive if \nit has a dense orbit.  \nClosed orbits are singular points or periodic orbits.\nA subset is said to be co-connected if \nthe complement of it is connected. \nWe have a following characterization of \ntransitivity for surface flows. \n\n\n\n\\begin{theorem}\\label{th0}\nLet $v$ be a continuous flow on a compact surface $M$. \nThen \nthe following are equivalent: \n\\\\\n1. \n$v$ is topologically transitive. \n\\\\\n2.  \n$v$ is non-wandering \nsuch that \n$\\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v)$ is co-connected \nand \n$ \\mathop{\\mathrm{int}}(\\mathop{\\mathrm{Per}}(v) \n\\sqcup \\mathop{\\mathrm{Sing}}(v)) = \\emptyset$.  \n\\\\\n3. \n$v$ is non-wandering \nsuch that \nthe set of regular weakly recurrent points is connected \nand \nthe interior of \nthe union of closed orbits is empty.  \n\\\\\n4. \n$\\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v)$ is co-connected \nand \n$\\mathop{\\mathrm{int}} \\mathop{\\mathrm{Sing}}(v) = \n\\mathop{\\mathrm{int}} \\mathop{\\mathrm{Per}}(v) = \n\\mathop{\\mathrm{int}} \\mathrm{P} = \n\\emptyset$.  \n\nIn each case, \n$M \n= \\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathrm{P} \\sqcup \\mathrm{LD} \n= \\overline{\\mathrm{LD}}$ \nand \neach locally dense orbit is dense. \n\\end{theorem}\n\n\n\\begin{proof}\nObviously, \nthe conditions 2 and 3 are equivalent. \nSince $\\mathrm{P}$ is the complement of \nthe set of the weakly recurrent points, \n$v$ is non-wandering if \nand only if $\\mathop{\\mathrm{int}} \\mathrm{P} = \\emptyset$.  \nLemma \\ref{lem0aa} implies that \nthe conditions 2 and 4 are equivalent. \nSuppose that \n$v$ is topologically transitive. \nThen $v$ is non-wandering. \nThe openness of  $\\mathop{\\mathrm{Per}}(v)$ \nimplies $\\mathop{\\mathrm{Per}}(v) = \\emptyset$. \nBy transitivity, \nthere is a dense orbit $O$ \nand so $\\overline{\\mathrm{LD}} = M$. \nSince $O$ is connected \nand $O \\subseteq \\mathrm{LD} \n\\subseteq \\overline{O} = M$, \nwe have that \n$M - (\\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v)) = \\mathrm{LD}$ is connected. \nConversely, \nsuppose \n$v$ is non-wandering \nsuch that \n$ \\mathop{\\mathrm{int}}(\\mathop{\\mathrm{Per}}(v) \n\\sqcup \\mathop{\\mathrm{Sing}}(v)) = \\emptyset$ \nand \n$M - (\\mathrm{P} \n\\sqcup \\mathop{\\mathrm{Sing}}(v))$ is connected. \nPut $U = M - (\\mathrm{P} \n\\sqcup \\mathop{\\mathrm{Sing}}(v))$. \nSince $\\mathop{\\mathrm{Per}}(v)$ is open, \nwe have \n$\\mathop{\\mathrm{Per}}(v)  = \\emptyset$ \nand so \n$\\mathop{\\mathrm{int}}\\mathop{\\mathrm{Sing}}(v) = \\emptyset$.  \nSince $v$ is non-wandering, \nwe have \n$\\overline{\\mathrm{LD}} = M = \n\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathrm{LD} \\sqcup \\mathrm{P}$. \nThus $\\mathrm{LD} = U$ is connected. \nFix an locally dense orbit $O$. \nFor any $z \\in \\hat{O}$, \nProposition \\ref{lem32} implies \n$z \\in \\mathrm{LD}$ \nand \nso \n$z \\in \\mathrm{int}\\overline{O_v(z)} = \\mathrm{int}\\overline{O}$. \nHence \n$\\hat{O} \\subseteq \\mathrm{int} \\overline{O}$ \nand so \n$\\hat{O} = \n\\overline{O} \\cap \\mathrm{LD} = \n\\mathrm{int} \\overline{O} \\cap \\mathrm{LD}$. \nThis means that \n$\\hat{O}$ is closed and open in $\\mathrm{LD}$ \nand so \n$\\overline{O} \\cap \\mathrm{LD} = \\mathrm{LD}$. \nTherefore  \n$M = \n\\overline{\\mathrm{LD}} = \n\\overline{O}$. \n\\end{proof}\n\n\nSince an area-preserving flow is non-wandering, \nProposition \\ref{lem32} implies \nthe following statement which is a generalization of Theorem A \\cite{MS}. \n\n\n\\begin{corollary}\\label{cor}\nLet $v$ be a continuous flow on a compact surface $M$. \nThen \n$v$ is topologically transitive \nif and only if \n$v$ is non-wandering \nsuch that \n$ \\mathop{\\mathrm{int}}(\\mathop{\\mathrm{Per}}(v) \n\\sqcup \\mathop{\\mathrm{Sing}}(v)) = \\emptyset$ \nand \n$\\{ x \\in M \\mid \\omega(x) \\cup \\alpha(x) \\subseteq \\mathop{\\mathrm{Sing}}(v) \\}$ is co-connected. \n\\end{corollary}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and main results}\nLet $W_{t}$ be a Brownian motion in $\\mathbb{R}^{d}$ $(d\\geq1)$ with generator $\\Delta$\nand $T_{t}$ be an independent $\\alpha\/2$-stable subordinator\nwith $\\alpha\\in(0,2)$. Then the subordinate process\n$X_{t}:=W_{T_{t}}$ is an isotropic $\\alpha$-stable\nprocess and its infinitesimal generator is the fractional Laplacian operator $-(-\\Delta^{\\alpha\/2})$ which is given by\n$$\n-(-\\Delta^{\\alpha\/2})f(x):= \\int_{{\\mathbb R}^d}\\big[f(x+z)-f(x)-1_{|z|\\leqslant\n1}z\\cdot\\nabla f(x)\\big]\\frac{c_{d,\\alpha}}{|z|^{d+\\alpha}}{\\mathord{{\\rm d}}} z,\\quad\\forall f\\in C^2_c({\\mathbb R}^d),\n$$\nwhere $c_{d,\\alpha}$ is a positive constant.\nIt is well known that the heat kernel $p(t,x,y)$ of $-(-\\Delta^{\\alpha\/2})$ (which is\nalso the transition density of $X:=(X_t)_{t\\geqslant 0}$)\nhas the following estimates: for every $t>0$ and $x,y\\in{\\mathbb R}^d$,\n\\begin{align}\np(t,x,y)\\asymp\\left(t^{-d\/\\alpha}\\wedge\\frac{t}{|x-y|^{d+\\alpha}}\\right).  \\label{Heat}\n\\end{align}\nHere and below, for two non-negative functions $f$ and $g$, the notation $f \\asymp g$ means that there are positive constants $c_1$ and $c_2$ such that $c_1g(x)\\leqslant f(x)\\leqslant c_2g(x)$ in the common domain of $f$ and $g$.\n\n\nIn \\cite{Bo-Ja0}, by using Duhamel's formula,\nBogdan and Jakubowski studied the following perturbation of\n$-(-\\Delta^{\\alpha\/2})$ by a gradient operator:\n$$\n{\\mathscr L}^b:=-(-\\Delta^{\\alpha\/2})+b(x)\\cdot\\nabla,\\quad\\alpha\\in(1,2),\n$$\nwhere $b=(b^1, \\cdots, b^d): {\\mathbb R}^d\\to{\\mathbb R}^d$ with $b^j$, $j=1, \\dots, d$, belonging\nto the Kato class ${\\bf K}^{\\alpha-1}_d$ defined\nas follows: for $\\gamma>0$,\n$$\n{\\bf K}^{\\gamma}_d:=\\left\\{f\\in L^1_{loc}({\\mathbb R}^d): \\lim_{r\\downarrow 0}\\sup_{x\\in{\\mathbb R}^d}\n\\int_{B(x,r)}\\frac{|f(y)|}{|x-y|^{d-\\gamma}}{\\mathord{{\\rm d}}} y=0\\right\\},\n$$\nand $B(x,r)$ denotes the open ball centered at $x\\in\\mathbb{R}^{d}$ with radius $r$.\nLet $p^b(t, x, y)$ be the heat kernel of ${\\mathscr L}^b$.\nSmall time sharp two-sided estimates for $p^b$,  of the form \\eqref{Heat},\nwere established in \\cite[Theorems 1 and 2]{Bo-Ja0}.\nThe key points of the perturbation method used in \\cite{Bo-Ja0} are that, on one hand, a nice bound on $\\nabla_x p(t,x,y)$ is known, and on the other hand, the following 3-P inequality concerning $p(t,x,y)$ holds: there exists $C_0>0$ such that for any $0<s<t$ and $x,y,z\\in {\\mathbb R}^d$,\n\\begin{align}\n\\frac{p(t-s,x,z)p(s,z,y)}{p(t,x,y)}\\leqslant C_0\\Big(p(t-s,x,z)+p(s,z,y)\\Big).  \\label{ppp}\n\\end{align}\nSee also \\cite{C-K-K,Ch-Ku2,C-K-W,J-S,K-S,W-Z,Xi-Zh} and the references therein for two-sided heat kernel estimates of more general non-local operators in the whole space ${\\mathbb R}^d$.\n\n\n\\vspace{2mm}\nLet $D$ be an open subset of ${\\mathbb R}^d$, we can kill the process $X$ upon exiting D and obtain a\nsubprocess $X^D$ known as the killed isotropic $\\alpha$-stable process.\nThe infinitesimal generator of $X^D$ is the Dirichlet fractional Laplacian $-(-\\Delta)^{\\alpha\/2}|_D$,\nthat is, the fractional Laplacian with zero exterior condition.\nDue to the complication near the boundary, two-sided\nestimates for the Dirichlet heat kernel of $-(-\\Delta)^{\\alpha\/2}|_D$ (or equivalently, the transition density of $X^D$) are much more difficult to obtain. To state the related results, we first recall that an open set $D$ in ${\\mathbb R}^{d}$ is said to be $C^{1,1}$ if there exist $r_{0}>0$ and\n$\\Lambda>0$ such that for every $Q\\in\\partial D$, there exist a $C^{1,1}$-function $\\phi=\\phi_{Q}: \\mathbb{R}^{d-1}\\rightarrow\\mathbb{R}$ satisfying\n$\\phi(0)=\\nabla\\phi(0)=0$, $\\|\\nabla\\phi\\|_{\\infty}\\leqslant\\Lambda$, $|\\nabla\\phi(x)-\\nabla\\phi(z)|\\leqslant\\Lambda|x-z|$ and an orthonormal coordinate system $y= (y_{1}, \\cdots, y_{d-1},\ny_{d}):=(\\tilde{y}, y_{d})$ such that $B(Q, r_{0})\\cap D=B(Q,r_{0})\\cap \\{y: y_{d}>\\phi(\\tilde{y})\\}$. The pair $(r_{0},\\Lambda)$ is called the characteristics of the $C^{1,1}$ open set $D$. In \\cite{C-K-S-2}, Chen, Kim and Song proved that when $D$ is a $C^{1,1}$ open set in ${\\mathbb R}^d$, the heat kernel $p^D(t,x,y)$ of $-(-\\Delta)^{\\alpha\/2}|_D$ has the following two-sided\nestimates: for every $T>0$ and $(t,x,y)\\in(0,T]\\times D\\times D$,\n\\begin{align}\np^D(t,x,y)\\asymp\n\\left(1\\wedge\\frac{\\rho(x)^{\\alpha\/2}}{\\sqrt{t}}\\right)\\left(1\\wedge\\frac{\\rho(y)^{\\alpha\/2}}{\\sqrt{t}}\\right)p(t,x,y),\\label{Heat2}\n\\end{align}\nwhere $\\rho(x)$ denotes the distance between $x$ and $D^{c}$.\n\nBy reversing the order of subordination and killing, one can obtain a process $Y^D$ which is different from\n$X^D$. More precisely, we first kill the Brownian motion $W$ at $\\tau_{D}$, the first exit time of $W$ from $D$, and then subordinate the killed Brownian motion $W^D$ using the independent $\\alpha\/2$-stable subordinator $T_{t}$. That is, $Y^D:=(W^{D})_{T_{t}}$ is defined as\n\\begin{align*}\nY^D_{t}:=\n\\begin{cases}\nW_{T_{t}}, & T_{t}<\\tau_{D}\\\\\n\\partial, & T_{t}\\geqslant\\tau_{D}\n\\end{cases}\n=\n\\begin{cases}\nW_{T_{t}}, & t<A_{\\tau_{D}}\\\\\n\\partial, & t\\geqslant A_{\\tau_{D}},\n\\end{cases}\n\\end{align*}\nwhere $\\partial$ is a cemetery state, $A_{t}:=\\inf\\{s>0: T_{s}\\geqslant t\\}$ is the inverse of $T$ and the last equality follows from the fact $\\{T_{t}<\\tau_{D}\\}=\\{t<A_{\\tau_{D}}\\}$.\nThe process $Y^D$ is called a subordinate killed Brownian motion. For the differences and relationship between\nthe processes $X^{D}$ and $Y^D$, see \\cite{S-V2}. The infinitesimal generator of $Y^D$ is the spectral fractional Laplacian $-(-\\Delta|_{D})^{\\alpha\/2}$, which is defined as a fractional power of the negative Dirichlet Laplacian. It is a very useful object in analysis and partial differential equations (see \\cite{B-S-V,P-A,S-Z}) and has been intensively studied (see \\cite{A-D,D-M-Z,G-P-R-S-S-V,S-V} and the references therein).\nWhen $D$ is a bounded $C^{1,1}$ domain,\nthe following sharp estimates for the heat kernel $r^D(t,x,y)$ of $-(-\\Delta|_{D})^{\\alpha\/2}$ (which is also the transition density of $Y^D$) were obtained in \\cite[Theorem 4.7]{Song}: for every $T>0$ and $(t,x,y)\\in(0,T]\\times D\\times D$,\n\\begin{align*}\nr^D(t,x,y)\\asymp\\left(1\\wedge\\frac{\\rho(x)\\rho(y)}{\\left(|x-y|+t^{1\/\\alpha}\\right)^{2}}\\right)p(t,x,y).\n\\end{align*}\nIn Lemma \\ref{sharp} below, we will give the following\nalternative form of the estimates above: for $(t,x,y)\\in(0,T]\\times D\\times D$,\n\\begin{align*}\nr^D(t,x,y)\\asymp\\left(1\\wedge\\frac{\\rho(x)}{|x-y|+t^{1\/\\alpha}}\\right)\\left(1\\wedge\\frac{\\rho(y)}{|x-y|+t^{1\/\\alpha}}\\right)p(t,x,y),\n\\end{align*}\nwhich is more convenient to use.\n\n\n\\vspace{2mm}\nGradient perturbations of Dirichlet operators have also been widely studied in recent years.\nIn \\cite{C-K-S-1}, Chen, Kim and Song studied the following gradient perturbation of the Dirichlet fractional Laplacian:\n$$\n{\\mathscr L}^{b,D}:=\\left(-(-\\Delta)^{\\alpha\/2}+b(x)\\cdot\\nabla\\right)|_{D},\\quad\\alpha\\in(1,2).\n$$\nUnder the condition that $b\\in{\\bf K}^{\\alpha-1}_d$ and $D$ is a\nbounded $C^{1,1}$ open set in $\\mathbb{R}^{d}$ with $d\\geq2$, Chen,\nKim and Song \\cite[Theorem 1.3]{C-K-S-1} showed that the heat kernel $p^{b,D}(t,x,y)$ of ${\\mathscr L}^{b,D}$ has the same\nestimates as in (\\ref{Heat2}). This result was generalized to unbounded $C^{1,1}$ open sets by \\cite{P-R}. Unlike the whole space case, there was no good estimate on\n$\\nabla_x p^D(t, x, y)$, thus \\cite{C-K-S-1, P-R} used Duhamel's formula\nfor the Green function and the probabilistic road-map designed in \\cite{C-K-S-2} for establishing\nthe estimates \\eqref{Heat2}.\n\nIn the recent paper \\cite{K-R},  Kulczycki and Ryznar  proved the following gradient estimate\nfor $p^D(t, x, y)$: for any $T>0$, there exists a constant $C=C(d,T)>0$ such that for any $(t, x, y)\\in (0, T]\\times D\\times D$,\n\\begin{align*}\n|\\nabla_x p^{D}(t,x,y)|\\leqslant  \\frac{C}{\\rho(x)\\wedge\nt^{1\/\\alpha}}p^{D}(t,x,y).\n\\end{align*}\nUsing this result, we gave, in the recent preprint \\cite{C-S-X-X}, a direct proof of the main results in \\cite{C-K-S-1, P-R} by using Duhamel's formula, with drift $b=(b^1, \\cdots, b^d): D\\to {\\mathbb R}^d$, where each $b^j$, $j=1, \\dots, d$, belongs to the following  Kato class:\n$$\n{\\bf K}_{D}^{\\alpha-1}:=\\left\\{f\\in L^1_{loc}(D): \\lim_{r\\downarrow0}\\sup_{x\\in D}\\int_{D\\cap\nB(x,r)}\\frac{|f(y)|}{|x-y|^{d+1-\\alpha}} {\\mathord{{\\rm d}}} y=0\\right\\}.\n$$\nMoreover, we also obtain a gradient estimate for $p^{b,D}(t,x,y)$. Notice that by H\\\"older's inequality, $L^p(D)\\subseteq {\\bf K}_{D}^{\\alpha-1}$ provided $d\/(\\alpha-1)<p\\leqslant \\infty$.\n\n\\vspace{2mm}\nThe aim of this paper is to study the following perturbation of spectral fractional Laplacian by a time-dependent gradient operator:\n$$\n{\\mathscr L}^{D,b}:=-(-\\Delta|_D)^\\frac{\\alpha}{2}+b(t,x)\\cdot\\nabla,\\quad\\alpha\\in(1,2),\n$$\nwith $b(t, x)=(b^1(t, x), \\cdots, b^d(t, x)): (0, \\infty)\\times D\\to{\\mathbb R}^d$ satisfying certain conditions.\nWe shall derive sharp two-sided estimates for the heat kernel $r^{D, b}(t, x, y)$\nof ${\\mathscr L}^{D,b}$ in bounded $C^{1,1}$ domains.\nMoreover, we also obtain a\ngradient estimate as well as the H\\\"older continuity of the gradient of $r^{D, b}(t, x, y)$, which are of independent interest.\n\nTo state our main result, let us first introduce our local Kato\nclass of space-time functions used in this paper.\n\n\\begin{definition}\\label{Definition}\nLet $D$ be a domain in ${\\mathbb R}^{d}$\nand $\\gamma\\geqslant 0$. For a real-valued function $f$ on $(0,\n\\infty)\\times D$, we define for every $\\delta>0$,\n\\begin{align*}\nK^{\\gamma}_{f}(\\delta):=\\sup_{t>0,x\\in\nD}\\delta^{\\gamma\/\\alpha}\\!\\int_{0}^{\\delta}\\!\\!\\!\\int_{D}&\\big[s^{-\\gamma\/\\alpha}+(\\delta-s)^{-\\gamma\/\\alpha}\\big]\\left(1\\wedge\\frac{\\rho(y)}{|x-y|+s^{1\/\\alpha}}\\right)\\\\\n&\\quad\\times\\frac{s}{(|x-y|+s^{1\/\\alpha})^{d+\\alpha+1}}\\cdot|f(t\\pm s,y)|{\\mathord{{\\rm d}}} y{\\mathord{{\\rm d}}} s.\n\\end{align*}\nWe say that the function $f$ belongs to thel Kato class ${\\mathbb K}_{D}^{\\gamma}$ if\n$\\lim_{\\delta\\downarrow0}K_{f}^{\\gamma}(\\delta)=0$.\n\\end{definition}\n\n\\begin{remark}\nWe note that our Kato class is time-dependent, which is needed when consider parabolic problems, see \\cite{J-S,ZhQ}. One can easily check that if $0\\leqslant\\gamma_1<\\gamma_2$, then ${\\mathbb K}_D^{\\gamma_2}\\subseteq{\\mathbb K}_D^{\\gamma_1}$.\nBy Lemma \\ref{include} below, we have that ${\\bf K}_D^{\\alpha-1}\\subset {\\mathbb K}_D^{0}$\nand that, for  $1<p,q\\leqslant \\infty$, $L^q({\\mathbb R};L^p(D))\\subseteq {\\mathbb K}_D^{\\gamma}$ provided $\\tfrac{d}{\\alpha p}+\\tfrac{1}{q}<1-\\tfrac{1+\\gamma}{\\alpha}$.\n\\end{remark}\n\nIn the remainder of this paper, we always assume that $b=(b^1, \\cdots, b^d): (0, \\infty)\\times D\\to{\\mathbb R}^d$ and each $b^j$, $j=1, \\dots, d$, belongs to ${\\mathbb K}_{D}^0$ .\n\nBy Duhamel's formula, the heat\nkernel $r^{D,b}(s,x;t,y)$ of ${\\mathscr L}^{D,b}$ should satisfy the following integral equation: for $0\\leqslant s<t$ and $x,y\\in D$,\n\\begin{align}\\label{Duhamel}\nr^{D,b}(s,x;t,y)=r^D(t-s,x,y)+\\int_s^t\\!\\!\\!\\int_{D}r^{D,b}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr^D(t-r,z,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r,\n\\end{align}\nor\n\\begin{align}\\label{duhamel}\nr^{D,b}(s,x;t,y)=r^{D}(t-s,x,y)+\\int_s^t\\!\\!\\!\\int_D\nr^{D}(r-s,x,z)b(r,z)\\cdot\\nabla_z r^{D,b}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.\n\\end{align}\nNotice that in (\\ref{Duhamel}) the derivative of the unknown heat kernel is not involved, and hence easier to solve.\nWhile (\\ref{duhamel}) is connected directly to the mild solutions of the corresponding parabolic equations, and from which one can easily derive the H\\\"older continuity of the gradient of the unknown heat kernel.\nFor convenience, we define for $t>0$ and $x,y\\in D$,\n\\begin{equation}\\label{qq}\nq^D(t, x, y):=\\left(1\\wedge\\frac{\\rho(x)}{|x-y|+t^{1\/\\alpha}}\\right)\\left(1\\wedge\\frac{\\rho(y)}{|x-y|+t^{1\/\\alpha}}\\right)p(t,x,y).\n\\end{equation}\nThe following is the main result of this paper.\n\n\\begin{theorem}\\label{main}\nLet $D$ be a bounded $C^{1,1}$ domain in $\\mathbb{R}^{d}$\nand $b\\in{\\mathbb K}^{0}_{D}$.\nThen there exists a unique function $r^{D,b}(s,x;t,y)$ on $(0,\n\\infty)\\times D\\times D$ satisfying (\\ref{Duhamel}) such that:\n\n\\begin{enumerate}[(i)]\n\n\\item (Two-sided estimates)\nfor any $\\delta>0$,\nthere exists a constant $C_1>1$ such that for all $0\\leqslant s<t\\leqslant s+\\delta$\nand $x, y\\in D$, we have\n\\begin{align}\\label{estimate}\nC_1^{-1}q^D(t-s,x,y)\\leqslant r^{D,b}(s,x;t,y)\\leqslant C_1 q^D(t-s,x,y);\n\\end{align}\n\n\\item (Gradient estimate)\nfor any $\\delta>0$, there exists a constant $C_2>0$ such that for all $0\\leqslant s<t\\leqslant s+\\delta$\nand $x, y\\in D$,\n\\begin{align}\\label{vi}\n|\\nabla_x r^{D,b}(s,x;t,y)|\\leqslant C_2\n\\frac{1}{\\rho(x)\\wedge\\left(|x-y|+(t-s)^{1\/\\alpha}\\right)}q^D(t-s,x,y),\n\\end{align}\nand $r^{D,b}(s,x;t,y)$ also satisfies (\\ref{duhamel});\n\n\\item (C-K equation) for all $0\\leqslant s<r<t$ and $x,y\\in D$, the following Chapman-Kolmogorov's equation holds:\n\\begin{align}\\label{eq21}\n\\int_{D}r^{D,b}(s,x;r,z)r^{D,b}(r,z;t,y){\\mathord{{\\rm d}}} z=r^{D,b}(s,x;t,y);\n\\end{align}\n\n\\item (Generator) for any $f\\in C^2_c(D)$, we have\n\\begin{align}\\label{eq23}\nR_{s,t}^{D,b}f(x)=f(x)+\\int_s^tR_{s,r}^{D,b}\\mathcal{L}^{D,b}f(x){\\mathord{{\\rm d}}} r,\n\\end{align}\nwhere $R^{D,b}_{s,t}f(x):=\\int_{D}r^{D,b}(s,x;t,y)f(y){\\mathord{{\\rm d}}} y$;\n\n\\item (Continuity) for any uniformly continuous function $f(x)$ with compact supports, we have\n\\begin{align}\n\\lim_{t\\downarrow s} \\|R^{D,b}_{s,t}f-f\\|_\\infty=0;   \\label{con}\n\\end{align}\n\n\\item (H\\\"older continuity)\nif we further assume that $b\\in {\\mathbb K}_D^{\\gamma}$ for some $\\gamma\\in (0, \\alpha-1)$,\nthen for any $\\delta>0$,\nthere exists a constant $C_3>0$ such that for any $0\\leqslant s<t\\leqslant s+\\delta$\nand $x,x',y\\in D$, we have\n\\begin{align}\n|\\nabla_x r^{D,b}& (s,x;t,y)-\\nabla_x r^{D,b}(s,x';t,y)| \\leqslant C_3|x-x'|^\\gamma (t-s)^{-\\gamma\/\\alpha}\\nonumber\\\\\n&\\times\\frac{1}{\\rho(\\widetilde x)\\wedge\\left(|\\widetilde x-y|+(t-s)^{1\/\\alpha}\\right)}q^D(t-s,\\widetilde x,y),  \\label{ho}\n\\end{align}\nwhere $\\widetilde x$ stands the point among $x$ and $x'$ which is closer to $y$.\n\\end{enumerate}\n\\end{theorem}\n\nWe remark that the gradient estimates (\\ref{vi}) and (\\ref{ho}) are new even in the case $b\\equiv0$.\nWe now briefly describe the main idea of our argument.\nDue to the difference between the processes $Y^D$ and $X^D$,\nthe method used in \\cite{C-K-S-1,P-R}  does not work for ${\\mathscr L}^{D,b}$.\nInstead, we will use Duhamel's formula (\\ref{Duhamel}) to obtain the sharp two-sided estimates of the heat kernel. As mentioned before, two main ingredients are needed: the gradient estimate for $r^D(t,x,y)$ and the corresponding 3-P inequality, both of which are unknown. In fact, by Remark \\ref{pp} below, we shall see that the 3-P inequality of the form (\\ref{ppp}) does not hold for the heat kernel $r^D(t,x,y)$. Because of these, we will first derive an estimate on $\\nabla_x r^D(t,x,y)$, and then establish a generalized 3-P type inequality for $r^D(t,x,y)$.\nThe gradient estimate and the H\\\"older estimate for $r^{D,b}(s,x;t,y)$ follow as easy by-products of our perturbation argument.\n\n\\vspace{2mm}\nThe remainder of this paper is organized as follows. In Section 2, we\nprepare some important inequalities for $r^D(t,x,y)$ and derive its first and second order gradient estimates. The proof of the\nmain result, Theorem \\ref{main}, will be given in Section 3.\n\n\nWe conclude this introduction by spelling out some conventions that\nwill be used throughout this paper. The letter C with or without\nsubscripts will denote an unimportant constant and $f\\preceq g$\nmeans that $f\\leqslant Cg$ for some $C\\geq1$. The letter $\\mathbb{N}$\nwill denote the collection of positive integers, and\n$\\mathbb{N}_{0}=\\mathbb{N}\\cup\\{0\\}$. We will use $:=$ to denote a\ndefinition, and we assume that all the functions\nconsidered in this paper are Borel measurable.\n\n\n\\section{Estimates for $r^{D}(t,x,y)$}\n\n\nIn the remainder of this paper, $D$ denotes a bounded $C^{1,1}$ domain in ${\\mathbb R}^d$.\nFor simplicity, we first introduce some functions for latter\nuse. Given $d\\geqslant 1$, $\\vartheta\\in{\\mathbb R}$ and $\\alpha\\in(0,2]$, we define for $t>0$ and $x,y\\in D$,\n$$\n\\varrho_{d}^\\vartheta(t,x):=\\frac{t^\\vartheta}{(|x|+t^{1\/\\alpha})^{d+\\alpha}},\n$$\nand\n\\begin{align}\n\\hat q_\\alpha(t,x,y):=1\\wedge\\frac{\\rho(x)}{|x-y|+t^{1\/\\alpha}},\\quad q_\\alpha(t,x,y):=\\hat q_\\alpha(t,x,y)\\hat q_\\alpha(t,y,x).\\label{q}\n\\end{align}\nThen we have $p(t, x, y)\\asymp \\varrho_{d}^1(t,x-y)$ and $q^D(t, x, y)=q_\\alpha(t,x,y)\np(t, x, y)$.\n\nWe will first establish a generalized 3-P type inequality for $r^D(t,x,y)$, and then derive its first and second order\ngradient estimates, which will be essential in\nconstructing the solution to the integral equation (\\ref{Duhamel}).\n\n\\subsection{Generalized 3-P inequality}\nLet $T>0$ be fixed. Recall that $r^D(t,x,y)$ is the heat kernel of\n$-(-\\Delta|_D)^{\\frac{\\alpha}{2}}$, and for any $t\\in(0, T]$ and $x, y\\in\nD$, we have\n\\begin{align*}\nr^D(t,x,y)\\asymp\\left(1\\wedge\\frac{\\rho(x)\\rho(y)}{(|x-y|+t^{1\/\\alpha})^{2}}\\right)\\varrho^1_{d}(t,x-y).\n\\end{align*}\nThe estimates above are not very convenient for our application since $\\rho(x)$ and $\\rho(y)$\nare intertwined together.\nWe prove the following result.\n\n\\begin{lemma}\\label{sharp}\nFor any $t\\in(0,\nT]$ and $x, y\\in D$, we have\n\\begin{align}\nr^D(t,x,y)\\asymp q_\\alpha(t,x,y)\\varrho^1_{d}(t,x-y)\\asymp q^D(t, x, y).\n\\label{er}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nThe second comparison follows immediately the sentence after \\eqref{q}. So we will only prove\nthe first comparison.\nIt is obvious that\n$$\nq_\\alpha(t,x,y)\\preceq 1\\wedge\\frac{\\rho(x)\\rho(y)}{(|x-y|+t^{1\/\\alpha})^{2}}.\n$$\nThus we only need to show that\n\\begin{align}\nq_\\alpha(t,x,y)\\succeq 1\\wedge\\frac{\\rho(x)\\rho(y)}{(|x-y|+t^{1\/\\alpha})^{2}}.    \\label{00}\n\\end{align}\nOne can easily see that the above inequality holds when\n$$\n\\rho(x)\\vee\\rho(y)\\leqslant |x-y|+t^{1\/\\alpha}\\quad\\text{or}\\quad\\rho(x)\\wedge\\rho(y)\\geqslant|x-y|+t^{1\/\\alpha}.\n$$\nBy symmetry, it suffices to prove (\\ref{00}) in the case when\n$$\n\\rho(x)\\leqslant |x-y|+t^{1\/\\alpha}\\leqslant\\rho(y).\n$$\nUsing the fact that $\\rho(y)\\leqslant\\rho(x)+|x-y|$, we can deduce\n\\begin{align*}\n1\\wedge\\frac{\\rho(x)\\rho(y)}{(|x-y|+t^{1\/\\alpha})^{2}}&\\leqslant 1\\wedge\\frac{\\rho(x)(\\rho(x)+|x-y|)}{(|x-y|+t^{1\/\\alpha})^{2}}\\\\\n&\\leqslant 1\\wedge\\frac{\\rho(x)^2}{(|x-y|+t^{1\/\\alpha})^{2}}+1\\wedge\\frac{\\rho(x)\\cdot|x-y|}{(|x-y|+t^{1\/\\alpha})^{2}}\\\\\n&\\preceq 1\\wedge\\frac{\\rho(x)}{|x-y|+t^{1\/\\alpha}},\n\\end{align*}\nwhich implies the desired result.\n\\end{proof}\n\n\\begin{remark}\\label{pp}\nBy (\\ref{er}) and the same argument as in \\cite[Remark 2.3]{C-K-S-3}, one can see that for all $t\/4<s<3t\/4$ and $x,y,z\\in D$ with $2|x-y|\\geqslant |x-z|+|z-y|$, it holds that\n\\begin{align*}\n\\frac{r^D(t+s,x,y)[r^D(t,x,z)+r^D(s,z,y)]}{r^D(t,x,z)r^D(s,z,y)}&\\preceq \\left(\\frac{\\rho(x)[\\rho(z)+|x-y|+(t+s)^{1\/\\alpha}]}{\\rho(z)[\\rho(x)+|x-y|+(t+s)^{1\/\\alpha}]}\\right)\\\\\n&\\quad+\\left(\\frac{\\rho(y)[\\rho(z)+|x-y|+(t+s)^{1\/\\alpha}]}{\\rho(z)[\\rho(y)+|x-y|+(t+s)^{1\/\\alpha}]}\\right),\n\\end{align*}\nwhich goes to zero as $\\rho(x)=\\rho(y)\\rightarrow0$. This means that, unlike (\\ref{ppp}), the inequality\n$$\n\\frac{r^D(t,x,z)r^D(s,z,y)}{r^D(t+s,x,y)}\\preceq r^D(t,x,z)+r^D(s,z,y)\n$$\ncan not be true for all $t,s>0$ and $x,y,z\\in D$, even for balls.\n\\end{remark}\n\nWe now proceed to prove a generalized 3-P type inequality for $r^D(t,x,y)$. Let us start with the following result.\n\\begin{lemma} For any $t,s\\geqslant 0$ and\n$x,y,z\\in D$, we have\n\\begin{align}\n\\frac{q_\\alpha(t,x,z)q_\\alpha(s,z,y)}{q_\\alpha(t+s,x,y)}\\preceq [\\hat q_\\alpha(t,z,x)]^2+[\\hat q_\\alpha(s,z,y)]^2. \\label{3q}\n\\end{align}\n\n\\end{lemma}\n\\begin{proof}\nNote that, for any $a, b>0$, it holds that\n\\begin{equation}\\label{elmentary}\n1\\wedge \\frac{a}{b}\\asymp \\frac{a}{a+b}.\n\\end{equation}\nThus,\n\\begin{align*}\n\\frac{\\hat q_\\alpha(t,x,z)\\hat q_\\alpha(s,y,z)}{q_\\alpha(t+s,x,y)}&\\asymp \\frac{\\left((t+s)^{1\/\\alpha}+|x-y|+\\rho(x)\\right)\n\\left((t+s)^{1\/\\alpha}+|x-y|+\\rho(y)\\right)}{\\left(t^{1\/\\alpha}+|x-z|+\\rho(x)\\right) \\left(s^{1\/\\alpha}+|z-y|+\\rho(y)\\right)}\\\\\n&\\preceq 1+\\frac{t^{1\/\\alpha}+|x-z|}{s^{1\/\\alpha}+|z-y|+\\rho(y)}+\\frac{s^{1\/\\alpha}+|z-y|}{t^{1\/\\alpha}+|x-z|+\\rho(x)}.\n\\end{align*}\nBy  (\\ref{q}), we have\n\\begin{align*}\n{\\mathcal I}:=\\frac{q_\\alpha(t,x,z)q_\\alpha(s,z,y)}{q_\\alpha(t+s,x,y)}&=\\frac{\\hat q_\\alpha(t,x,z)\\hat q_\\alpha(s,y,z)}{q_\\alpha(t+s,x,y)}\\hat q_\\alpha(t,z,x)\\hat q_\\alpha(s,z,y)\\\\\n&\\preceq \\hat q_\\alpha(t,z,x)\\hat q_\\alpha(s,z,y)+\\frac{\\rho(z)}{s^{1\/\\alpha}+|z-y|+\\rho(y)}\\hat q_\\alpha(s,z,y)\\\\\n&\\qquad+\\frac{\\rho(z)}{t^{1\/\\alpha}+|x-z|+\\rho(x)}\\hat q_\\alpha(t,z,x).\n\\end{align*}\nUsing the fact\n$$\n\\rho(x)+|x-z|\\asymp\\rho(z)+|x-z|,\n$$\nwe further get that\n$$\n\\frac{\\rho(z)}{t^{1\/\\alpha}+|x-z|+\\rho(x)}\\asymp\\frac{\\rho(z)}{t^{1\/\\alpha}+|x-z|+\\rho(z)}\\asymp \\hat q_\\alpha(t,z,x),\n$$\nand similarly\n$$\n\\frac{\\rho(z)}{s^{1\/\\alpha}+|z-y|+\\rho(y)}\\asymp\\frac{\\rho(z)}{s^{1\/\\alpha}+|z-y|+\\rho(z)}\\asymp \\hat q_\\alpha(s,z,y).\n$$\nThus, we have\n\\begin{align*}\n{\\mathcal I}\\preceq \\hat q_\\alpha(t,z,x)\\hat q_\\alpha(s,z,y)+[\\hat q_\\alpha(t,z,x)]^2+[\\hat q_\\alpha(s,z,y)]^2\\preceq [\\hat q_\\alpha(t,z,x)]^2+[\\hat q_\\alpha(s,z,y)]^2.\n\\end{align*}\nThe proof is finished.\n\\end{proof}\n\nAs a direct consequence, we can obtain the following generalized 3-P type inequality for $r^D(t,x,y)$.\n\n\\begin{lemma}\nLet $T>0$. For any $0\\leqslant s,t\\leqslant T $ and $x, y, z\\in D$, it holds that\n\\begin{align}\n\\frac{r^D(t,x,z)r^D(s,z,y)}{r^D(t+s,x,y)}\\preceq (t\\wedge s)\\Big(&[\\hat q_\\alpha(t,z,x)]^2\\varrho_{d}^0(t,x-z)+[\\hat q_\\alpha(s,z,y)]^2\\varrho_{d}^0(s,z-y)\\Big).   \\label{3r}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nCombining (\\ref{er}) and (\\ref{3q}), we get that\n\\begin{align*}\n{\\mathcal J}:=\\frac{r^D(t,x,z)r^D(s,z,y)}{r^D(t+s,x,y)}\\preceq \\Big([\\hat q_\\alpha(t,z,x)]^2+[\\hat q_\\alpha(s,z,y)]^2\\Big)\n\\frac{\\varrho_{d}^1(t,x-z)\\varrho_{d}^1(s,z-y)}{\\varrho_{d}^1(t+s,x-y)}.\n\\end{align*}\nNote that\n\\begin{align*}\n\\big(|x-y|+(t+s)^{1\/\\alpha}\\big)^{d+\\alpha}\\preceq \\big(|x-z|+t^{1\/\\alpha}\\big)^{d+\\alpha}+\\big(|z-y|+s^{1\/\\alpha}\\big)^{d+\\alpha}.\n\\end{align*}\nThus\n\\begin{align}\n\\frac{\\varrho_{d}^1(t,x-z)\\varrho_{d}^1(s,z-y)}{\\varrho_{d}^1(t+s,x-y)}&=\\frac{t\\cdot s}{t+s}\\cdot\\frac{\\varrho_{d}^0(t,x-z)\\varrho_{d}^0(s,z-y)}{\\varrho_{d}^0(t+s,x-y)}\\nonumber\\\\\n&\\preceq (t\\wedge\ns)\\Big(\\varrho_{d}^0(t,x-z)+\\varrho_{d}^0(s,z-y)\\Big).  \\label{333}\n\\end{align}\nHence\n\\begin{align*}\n{\\mathcal J}&\\preceq (t\\wedge s)\\Big([\\hat q_\\alpha(t,z,x)]^2+[\\hat q_\\alpha(s,z,y)]^2\\Big)\\Big(\\varrho_{d}^0(t,x-z)+\\varrho_{d}^0(s,z-y)\\Big)\\\\\n&\\preceq (t\\wedge s)\\Big([\\hat q_\\alpha(t,z,x)]^2\\varrho_{d}^0(t,x-z)+[\\hat q_\\alpha(s,z,y)]^2\\varrho_{d}^0(s,z-y)\\Big),\n\\end{align*}\nwhere in the last inequality we have used the fact\n\\begin{align}\n\\hat q_\\alpha(t,z,x)\\preceq \\hat q_\\alpha(s,z,y)\\,\\,\\Leftrightarrow\\,\\,\\varrho_{d}^0(t,x-z)\\preceq \\varrho_{d}^0(s,z-y) \\label{semi}\n\\end{align}\nand the symmetry in $x$ and $y$. The proof is finished.\n\\end{proof}\n\n\\subsection{Gradient estimates}\nIn this subsection, we derive gradient estimates for $r^D(t,x,y)$.\nRecall that $r^D(t,x,y)$ is the transition density of $Y^D$. By the construction of $Y^D$,\nit holds (see \\cite[(2.2)]{Song}) that\n\\begin{align}\nr^D(t,x,y)=\\int_0^\\infty p^D_{2}(s,x,y)\\mu(t,s){\\mathord{{\\rm d}}} s, \\label{rtxy}\n\\end{align}\nwhere $p^{D}_{2}(t,x,y)$ is the Dirichlet heat kernel of $\\Delta|_{D}$, and $\\mu(t,s)$ is the density of the subordinator $T_t$. To derive gradient estimates for $r^D(t,x,y)$, we need to recall some estimates for $p^D_2(t,x,y)$.\n\nFor any $\\gamma, \\lambda\\in {\\mathbb R}$ and $(t, x)\\in (0, \\infty)\\times{\\mathbb R}^d$, we define\n$$\n\\xi^\\gamma_{\\lambda}(t,x):=t^{-(d+\\gamma)\/2}{\\mathrm{e}}^{-\\lambda |x|^2\/t}.\n$$\nIt is known (see \\cite[Theorems 3.1 and 3.2]{Song} for instance) there exist constants $\\lambda_1,\\lambda_2>0$, $C_1>1$ and $C_2<1$ such that\n\\begin{align}\np^{D}_{2}(t,x,y)&\\leqslant C_1\n\\left(1\\wedge\\frac{\\rho(x)\\rho(y)}{t}\\right)\\xi^0_{\\lambda_1}(t,x-y), \\quad (t, x, y)\\in (0, \\infty)\\times D\\times D,\\label{pd1ub}\\\\\np^{D}_{2}(t,x,y)&\\geqslant C_2\n\\left(1\\wedge\\frac{\\rho(x)\\rho(y)}{t}\\right)\\xi^0_{\\lambda_2}(t,x-y), \\quad  (t, x, y)\\in (0, 1]\\times D\\times D.\\label{pd1lb}\n\\end{align}\nMoreover, it follows from \\cite[Theorem 2.1]{ZQS} that, for any $T>0$,\nthere exists a constant $C_T>0$ such that for all $t\\in(0,T]$ and $x,y\\in D$,\n\\begin{equation}\n|\\nabla_x p^D_2(t,x,y)|\\leqslant \\left\\{\\begin{aligned}\n&\\frac{C_T}{\\rho(x)}p^D_2(t,x,y),\\qquad\\qquad\\qquad\\quad\\quad \\textrm{if}\\,\\,\\,\\rho(x)\\leqslant \\sqrt{t};\\\\\n&\\frac{C_T}{\\sqrt{t}}\\left(1+\\frac{|x-y|}{\\sqrt{t}}\\right)p^D_2(t,x,y),\\qquad \\textrm{if}\\,\\,\\,\\rho(x)>\\sqrt{t}.\\label{pd3}\n\\end{aligned}\n\\right.\n\\end{equation}\nIt turns out that  (\\ref{pd1ub}), \\eqref{pd1lb} and (\\ref{pd3}) are not very convenient to use.\nTo get easy-to-use forms of the estimates above, we first do some manipulations on $p^D_2(t,x,y)$.\nWe want to separate the terms $\\rho(x)$ and $\\rho(y)$.\nThe following elementary observation will be important.\n\n\\begin{lemma}\nFor any $\\lambda_2>\\lambda_1>0$ and $\\gamma\\in{\\mathbb R}$, it holds for all $t>0$ and $x,y\\in D$ that\n\\begin{align}\n\\left(1\\wedge\\frac{\\rho(x)\\rho(y)}{t}\\right)\\xi^\\gamma_{\\lambda_2}(t,x-y)\\preceq \\left(1\\wedge\\frac{\\rho(x)}{\\sqrt{t}}\\right)\n\\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\xi^\\gamma_{\\lambda_1}(t,x-y).  \\label{pdd}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nIn light of \\eqref{elmentary}, it suffices to show that  for any $\\lambda_0>0$\n\\begin{align*}\n\\Big(\\rho(x)+\\sqrt{t}\\Big)\\left(\\rho(y)+\\sqrt{t}\\right)\\preceq\\big(\\rho(x)\\rho(y)+t\\big){\\mathrm{e}}^{\\lambda_0\\frac{|x-y|^{2}}{t}}.\n\\end{align*}\nIn fact, using symmetry and the elementary inequality\n$$\n\\rho(x)\\leqslant \\rho(y)+|x-y|,\n$$\nwe have\n$$\n\\rho(x)^{2}+\\rho(y)^{2}\\preceq \\rho(x)\\rho(y)+|x-y|^{2}.\n$$\nThus, we can deduce that\n\\begin{align*}\n\\left(\\rho(x)+\\sqrt{t}\\right)\\left(\\rho(y)+\\sqrt{t}\\right)&\\preceq \\rho(x)\\rho(y)+t+\\rho(x)^{2}+\\rho(y)^{2}\\\\\n&\\preceq \\rho(x)\\rho(y)+t+|x-y|^{2}.\n\\end{align*}\nNote that for any  $\\lambda_0>0$, we have\n$$\n|x-y|^{2}\\preceq t\\cdot{\\mathrm{e}}^{\\lambda_0\\frac{|x-y|^{2}}{t}}.\n$$\nThe desired result follows immediately.\n\\end{proof}\n\nRecall the definition of $q_\\alpha(t,x,y)$ in (\\ref{q}). We give a better form of (\\ref{pd1ub})\nand \\eqref{pd1lb} as follows.\n\\begin{lemma}\nThere exist constants $\\lambda_1,\\lambda_2>0$, $C_1>1$ and $C_2<1$ such that\n\\begin{align}\np^{D}_{2}(t,x,y)&\\leqslant C_1q_2(t,x,y)\\xi^0_{\\lambda_1}(t,x-y), \\quad  (t, x, y)\\in (0, \\infty)\\times D\\times D,\\label{pd2ub}\\\\\np^{D}_{2}(t,x,y)&\\geqslant C_2q_2(t,x,y)\\xi^0_{\\lambda_2}(t,x-y), \\quad (t,x,y)\\in (0, 1]\\times D\\times D.\n\\label{pd2lb}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nThe lower bound \\eqref{pd2lb} is obvious, we only need to prove the upper bound \\eqref{pd2ub}. Combining (\\ref{pd1ub})\nand \\eqref{pd1lb} with (\\ref{pdd}), we have that for any $\\lambda_0>0$,\n\\begin{align*}\np^{D}_{2}(t,x,y)&\\preceq\n\\left(1\\wedge\\frac{\\rho(x)}{\\sqrt{t}}\\right)\\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\xi^0_{\\lambda_0}(t,x-y).\n\\end{align*}\nThus, (\\ref{pd2ub}) is true when $|x-y|\\leqslant \\sqrt{t}$. On the other hand, notice that for $0<\\tilde{\\lambda}_0<\\lambda_0$ we have\n\\begin{align}\\label{new}\n\\frac{\\rho(x)}{\\sqrt{t}}{\\mathrm{e}}^{-\\lambda_0\\frac{|x-y|^{2}}{t}}=\\frac{\\rho(x)}{|x-y|}\\cdot\\frac{|x-y|}{\\sqrt{t}}{\\mathrm{e}}^{-\\lambda_0\\frac{|x-y|^{2}}{t}}\n\\preceq \\frac{\\rho(x)}{|x-y|}{\\mathrm{e}}^{-\\tilde{\\lambda}_0\\frac{|x-y|^{2}}{t}}.\n\\end{align}\nCombining \\eqref{new} with \\eqref{pd1ub} gives the desired result for $|x-y|>\\sqrt{t}$.\n\\end{proof}\n\n\n\nNow we prove the first and second order gradient estimates for\n$p^{D}_{2}(t,x,y)$.\n\n\\begin{lemma}\nLet $T>0$. There exist constants $C_T, \\lambda_3>0$ such that for $j=1,2$, \\\\\ni) for all $t\\in(0,T]$ and $x,y\\in D$,\n\\begin{align}\\label{p1}\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|\\leqslant\nC_T\\hat q_2(t,y,x)\\xi^j_{\\lambda_3}(t,x-y);\n\\end{align}\nii) for all $t\\in(T,\\infty)$ and $x,y\\in D$,\n\\begin{align}\\label{p2}\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|\\leqslant\\frac{C_T}{T^{j\/2}}\\hat q_2(t,y,x)\\xi^0_{\\lambda_3}(t,x-y),\n\\end{align}\nwhere $\\nabla^j_x$ denotes the $j$-order derivative with respect to the $x$ variable.\n\\end{lemma}\n\\begin{proof}\nFor (\\ref{p1}), we only need to show that there exist $\\lambda_3>0$ and $C_T>0$ such that for every $t\\in(0,T]$ and $x, y\\in D$,\n$$\n|\\nabla^j_xp^{D}_{2}(t,x,y)|\\leqslant C_T\\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\xi^j_{\\lambda_3}(t,x-y).\n$$\nThen applying \\eqref{new}, we can get (\\ref{p1}).\nBy \\cite[VI.2, Theorem 2.1]{G-M}, we have that for every $t\\in(0,T]$ and $x, y\\in D$,\n$$\n|\\nabla^j_xp^{D}_{2}(t,x,y)|\\preceq \\xi^j_{\\lambda_3}(t,x-y).\n$$\nUsing the Chapman-Kolmogorov equation, we have\n\\begin{align*}\n|\\nabla^j_xp^{D}_{2}(t,x,y)|&\\leqslant \\int_D\\big|\\nabla^j_xp^{D}_{2}(t\/2,x,z)\\big|\\cdot p^{D}_{2}(t\/2,z,y){\\mathord{{\\rm d}}} z\\\\\n&\\preceq \\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\int_D\\xi^j_{\\lambda_3}(t\/2,x-z)\\xi^0_{\\lambda_1}(t\/2,z-y){\\mathord{{\\rm d}}} z\\\\\n&\\preceq \\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\xi^j_{\\lambda_3}(t,x-y).\n\\end{align*}\nThus (\\ref{p1}) is valid. We now prove (\\ref{p2}). Similarly, it suffices to show that for every $t>T$ and $x, y\\in D$,\n$$\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|\\leqslant\n\\frac{C_T}{T^{j\/2}}\\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\xi^0_{\\lambda_2}(t,x-y).\n$$\nBy (\\ref{pd2ub}), (\\ref{p1}) and the Chapman-Kolmogorov equation, we have for $t>T$,\n\\begin{align}\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|&\\leqslant\\int_{D}\\left|\\nabla^j_x p^{D}_{2}(T,x,z)\\right|\\cdot p^{D}_{2}(t-T,z,y){\\mathord{{\\rm d}}} z \\nonumber\\\\\n&\\leqslant C_T\\int_{{\\mathbb R}^{d}}\\xi^j_{\\lambda_2}(T,x-z)\\xi^0_{\\lambda_1}(t-T,z-y){\\mathord{{\\rm d}}} z\\nonumber\\\\\n&\\leqslant \\frac{C_T}{T^{j\/2}}\\xi^0_{\\lambda_2}(t,x-y).  \\label{gr0}\n\\end{align}\nFurthermore, for $t\\in(T,2T]$ the same argument yields that\n\\begin{align*}\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|&\\leqslant\\int_{D}\\left|\\nabla^j_x p^{D}_{2}(t\/2,x,z)\\right|\\cdot p^{D}_{2}(t\/2,z,y){\\mathord{{\\rm d}}} z\\\\\n&\\leqslant C_T\\int_{{\\mathbb R}^d}\\xi^j_{\\lambda_2}(t\/2,x-z)\\frac{\\rho(y)}{\\sqrt{t\/2}}\\xi^0_{\\lambda_1}(t\/2,z-y){\\mathord{{\\rm d}}} z\\\\\n&\\leqslant \\frac{C_T}{T^{j\/2}}\\rho(y)\\xi^1_{\\lambda_2}(t,x-y)=\\frac{C_T}{T^{j\/2}}\\frac{\\rho(y)}{\\sqrt{t}}\\xi^0_{\\lambda_2}(t, x-y).\n\\end{align*}\nUsing (\\ref{gr0}) we get that for any $t\\in(2T,\\infty)$,\n\\begin{align*}\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|&\\leqslant\\int_{D}\\left|\\nabla^j_x p^{D}_{2}(t\/2,x,z)\\right|\\cdot p^{D}_{2}(t\/2,z,y){\\mathord{{\\rm d}}} z\\\\\n&\\leqslant \\frac{C_T}{T^{j\/2}}\\int_{{\\mathbb R}^d}\\xi^0_{\\lambda_2}(t\/2,x-z)\\frac{\\rho(y)}{\\sqrt{t\/2}}\\xi^0_{\\lambda_1}(t\/2,z-y){\\mathord{{\\rm d}}} z\\\\\n&\\leqslant \\frac{C_T}{T^{j\/2}}\\rho(y)\\xi^1_{\\lambda_2}(t,x-y)=\\frac{C_T}{T^{j\/2}}\\frac{\\rho(y)}{\\sqrt{t}}\\xi^0_{\\lambda_2}(t, x-y).\n\\end{align*}\nCombining the above computations, we get the\ndesired result.\n\\end{proof}\n\n\\begin{remark}\nIn fact, in the form of (\\ref{pd3}), our result means that for every $t\\in(0,T]$,\n$$\n|\\nabla^j_x p^D_{2}(t,x,y)|\\leqslant\nC_T\\frac{(|x-y|+\\sqrt{t})^{1-j}}{\\rho(x)\\wedge(|x-y|+\\sqrt{t})}q_2(t,x,y)\\xi^0_{\\lambda_3}(t,x-y).\n$$\nCompared with (\\ref{pd1ub}), (\\ref{pd1lb}) and (\\ref{pd3}),\nthe additional term $|x-y|$ in (\\ref{pd2ub})--(\\ref{pd2lb}) and (\\ref{p1})--(\\ref{p2}) is of critical importance\nin our derivation of the gradient estimates of $r^D(t,x,y)$ below.\n\\end{remark}\n\nRecall the definition of $q^D(t,x,y)$ in (\\ref{qq}). Now, we are ready to derive the following gradient estimates for the Dirichlet heat kernel\n$r^D(t,x,y)$.\n\n\\begin{lemma}\nLet $T>0$. There exists a constant $C_T>0$ such that for $j=1,2$,\nall $t\\in(0,T]$ and $x,y\\in D$,\n\\begin{align}\\label{grad}\n|\\nabla^j_x\nr^D(t,x,y)|\\leqslant C_T\\frac{(|x-y|+t^{1\/\\alpha})^{1-j}}{\\rho(x)\\wedge(|x-y|+t^{1\/\\alpha})}q^D(t,x,y).\n\\end{align}\nMoreover, for any $\\vartheta\\in(0,1)$ and $t\\in(0,T]$, $x,x',y\\in D$, we have\n\\begin{align}\n|\\nabla_x r^D(t,x,y)-\\nabla_x r^D(t,x',y)| \\leqslant C_T|x-x'|^\\vartheta \\hat q_{\\alpha}(t,y,\\widetilde x)\\varrho_{d+1+\\vartheta}^1(t,\\widetilde x-y), \\label{hold}\n\\end{align}\nwhere $\\widetilde x$ is the point among $x$ and $x'$ which is closer to $y$.\n\\end{lemma}\n\\begin{proof}\nWe claim that for $j=1,2$,\n\\begin{align}\n|\\nabla^j_x r^D(t,x,y)|\\preceq\n\\hat q_\\alpha(t,y,x)\\varrho^1_{d+j}(t,x-y).\\label{gr22}\n\\end{align}\nAs a consequence of this claim, we get\n\\begin{align*}\n|\\nabla^j_x r^D(t,x,y)|&\\preceq\\frac{1}{(|x-y|+t^{1\/\\alpha})^j\\hat q_\\alpha(t,x,y)} \\hat q_\\alpha(t,x,y)\\hat q_\\alpha(t,y,x)\\varrho_{d}^1(t,x-y)\\\\\n&\\asymp\\frac{(|x-y|+t^{1\/\\alpha})^{1-j}}{\\rho(x)\\wedge(|x-y|+t^{1\/\\alpha})}q^D(t,x,y).\n\\end{align*}\nNow we prove the claim (\\ref{gr22}).\nFrom \\cite[(4.1)]{Song}, we know that for all\n$\\xi\\in\\mathbb{R}^{d},$\n\\begin{align*}\n\\int_{0}^{\\infty}\\!s^{-d\/2}{\\mathrm{e}}^{-\\frac{|\\xi|^{2}}{s}}\\mu(t,s){\\mathord{{\\rm d}}}\ns\\asymp\\varrho^1_{d}(t,\\xi).\n\\end{align*}\nCombining this with (\\ref{rtxy}), (\\ref{p1}) and (\\ref{p2}), we can\nget\n\\begin{align}\n|\\nabla^j_x r^D(t,x,y)|&\\leqslant \\int_{0}^{1}\\left|\\nabla^j_x p^{D}_{2}(s,x,y)\\right|\\mu(t,s){\\mathord{{\\rm d}}} s+\\int_{1}^{\\infty}\\left|\\nabla^j_x p^{D}_{2}(s,x,y)\\right|\\mu(t,s){\\mathord{{\\rm d}}} s\\nonumber\\\\\n&\\preceq\\left(1\\wedge\\frac{\\rho(y)}{|x-y|}\\right)\\left[\\int_{0}^{\\infty}\\!\\xi^j_{\\lambda_3}(s,x-y)\\mu(t,s){\\mathord{{\\rm d}}} s\n+\\int_{0}^{\\infty}\\!\\xi^0_{\\lambda_3}(s,x-y)\\mu(t,s){\\mathord{{\\rm d}}} s\\right]\\nonumber\\\\\n&\\asymp\\left(1\\wedge\\frac{\\rho(y)}{|x-y|}\\right)\\Big[\\varrho^1_{d+j}(t,x-y)+\\varrho_{d}^1(t,x-y)\\Big]\\nonumber\\\\\n&\\preceq\\left(1\\wedge\\frac{\\rho(y)}{|x-y|}\\right)\\varrho^1_{d+j}(t,x-y),\n\\label{e1}\n\\end{align}\nwhere in the last inequality we have used the fact that $D$ is bounded and $t\\in(0,T]$. Thus, (\\ref{gr22}) is true when $|x-y|\\geqslant t^{1\/\\alpha}$. For the case that $|x-y|<t^{1\/\\alpha}$,\nwe may argue similarly to get that\n\\begin{align*}\n|\\nabla^j_x r^D(t,x,y)|\n&\\preceq \\rho(y)\\left[\\int_{0}^{\\infty}\\!\\xi^{j+1}_{\\lambda_3}(s,x-y)\\mu(t,s){\\mathord{{\\rm d}}} s\n+\\int_{0}^{\\infty}\\!\\xi^1_{\\lambda_3}(s,x-y)\\mu(t,s){\\mathord{{\\rm d}}} s\\right]\\\\\n&\\asymp\\rho(y)\\Big[\\varrho^1_{d+j+1}(t,x-y)+\\varrho_{d+1}^1(t,x-y)\\Big]\\\\\n&\\preceq\\frac{\\rho(y)}{t^{1\/\\alpha}}\\varrho^1_{d+j}(t,x-y).\n\\end{align*}\nThis together with estimate (\\ref{e1}) implies (\\ref{gr22}).\n\nFor (\\ref{hold}), without loss of generality, we may assume that $|x-y|\\leqslant |x'-y|$. Using (\\ref{gr22}) with $j=1$, we can get that when $|x-x'|\\geqslant (|x-y|+t^{1\/\\alpha})\/2$,\n\\begin{align*}\n{\\mathscr Q}&:=|\\nabla_x r^D(t,x,y)-\\nabla_x r^D(t,x',y)|\\\\\n&\\leqslant C_T|x-x'|^\\vartheta(|x-y|+t^{1\/\\alpha})^{-\\vartheta}\\Big(\\hat q_\\alpha(t,y,x)\\varrho^{1}_{d+1}(t,x-y)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\\qquad\\quad+\\hat q_\\alpha(t,y,x')\\varrho^{1}_{d+1}(t,x'-y)\\Big)\\\\\n&\\leqslant C_T|x-x'|^\\vartheta\\hat q_\\alpha(t,y,x)\\varrho_{d+1+\\vartheta}^{1}(t,x-y).\n\\end{align*}\nWhen $|x-x'|<(|x-y|+t^{1\/\\alpha})\/2$, we have by the mean value theorem and (\\ref{gr22}) with $j=2$ that for some $\\varepsilon\\in[0,1]$,\n\\begin{align*}\n{\\mathscr Q}&\\leqslant C_T|x-x'|\\hat q_\\alpha\\big(t,y,x+\\varepsilon(x'-x)\\big)\\varrho^{1}_{d+2}\\big(t,x+\\varepsilon(x'-x)-y\\big)\\\\\n&\\leqslant C_T|x-x'|\\hat q_\\alpha(t,y,x)\\varrho_{d+2}^{1}(t, x-y)\\\\\n&\\leqslant C_T|x-x'|^\\vartheta\\hat q_\\alpha(t,y,x)\\varrho_{d+1+\\vartheta}^{1}(t,x-y).\n\\end{align*}\nThe proof is finished.\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{main}}\n\n Let\n\\begin{align*}\n\\widehat{\\bf K}_{D}^{\\alpha-1}:=\\bigg\\{f\\in L^1_{loc}(D): \\lim_{t\\downarrow0}\\sup_{x\\in D}\\int_{D}&\\left(1\\wedge \\frac{\\rho(y)}{|x-y|}\\right)\\\\\n&\\times\\bigg(\\frac{1}{|x-y|^{d+1-\\alpha}}\\wedge\\frac{t^2}{|x-y|^{d+\\alpha+1}}\\bigg)|f(y)|{\\mathord{{\\rm d}}} y=0\\bigg\\}.\n\\end{align*}\nWe first give the following result about our Kato class.\n\n\\begin{lemma}\\label{include}\nWe have ${\\bf K}_D^{\\alpha-1}\\subset \\widehat{\\bf K}_D^{\\alpha-1}\\subset{\\mathbb K}_D^{0}$.\nMoreover, for any $\\gamma\\geqslant 0$, if $1<p,q\\leqslant \\infty$ satisfy\n\\begin{align}\n\\frac{d}{\\alpha p}+\\frac{1}{q}<1-\\frac{1+\\gamma}{\\alpha},    \\label{pq}\n\\end{align}\nthen $L^q({\\mathbb R};L^p(D))\\subseteq{\\mathbb K}_D^\\gamma$.\n\\end{lemma}\n\\begin{proof}\nIt follows from \\cite[Lemma 2.1]{C-S-X-X}, which follows from \\cite[Corollary 12]{Bo-Ja0},\nthat a real-valued function $f$ belongs to ${\\bf K}^{\\alpha-1}_D$ if and only if\n\\begin{equation*}\n\\lim_{t\\rightarrow0}\\sup_{x\\in D}\\int_{D}\\bigg(\\frac{1}{|x-y|^{d+1-\\alpha}}\\wedge\\frac{t^2}{|x-y|^{d+\\alpha+1}}\\bigg)|f(y)|{\\mathord{{\\rm d}}} y=0.\n\\end{equation*}\nThus the first inclusion is obvious.\nTo show that a real-valued time-independent function $f$ on $D$\nbelongs to ${\\mathbb K}_D^0$,\nit suffices to show that\n$$\n\\int_0^t\\varrho_{d+1}^1(s,x-y){\\mathord{{\\rm d}}} s\\preceq \\frac{1}{|x-y|^{d+1-\\alpha}}\\wedge\\frac{t^2}{|x-y|^{d+\\alpha+1}}.\n$$\nThis follows directly from \\cite[Lemma 2.3]{C-S-X-X} with $\\gamma=1$.\nThus the second inclusion is valid. Now we prove the third inclusion. By H\\\"older's inequality, we get\n$$\nK_f^\\gamma(\\delta)\\leqslant\\Bigg(\\int_{{\\mathbb R}}\\left(\\int_{D}|f(s,y)|^p\n{\\mathord{{\\rm d}}} y\\right)^{\\frac{q}{p}}{\\mathord{{\\rm d}}} s\\Bigg)^{\\frac{1}{q}}I_{\\alpha,\\gamma}(\\delta),\n$$\nwhere\n$$\nI_{\\alpha,\\gamma}(\\delta):=\\delta^{\\frac{\\gamma}{\\alpha}}\\Bigg(\\int^\\delta_0\\big[s^{-\\gamma\/\\alpha}+(\\delta-s)^{-\\gamma\/\\alpha}\\big]^{q^*}\\Bigg(\\!\\int_{{\\mathbb R}^d}\\frac{s^{p^*}}\n{\\big(|y|+s^{1\/\\alpha}\\big)^{(d+\\alpha+1)p^{*}}}{\\mathord{{\\rm d}}} y\\Bigg)^{\\frac{q^*}{p^*}}{\\mathord{{\\rm d}}} s\\Bigg)^{\\frac{1}{q^{*}}},\n$$\nwith $q^*:=\\frac{q}{q-1}$ and $p^*:=\\frac{p}{p-1}$. Noticing that\n\\begin{align*}\n\\int_{{\\mathbb R}^d}\\frac{s^{p^*}}{\\big(|y|+s^{1\/\\alpha}\\big)^{(d+\\alpha+1)p^{*}}}{\\mathord{{\\rm d}}} y\n&\\leqslant s^{p^*}\\left(\\int_{|y|\\leqslant s^{1\/\\alpha}}s^{-\\frac{(d+\\alpha+1)p^{*}}{\\alpha}}{\\mathord{{\\rm d}}} y\n+ \\int_{|y|> s^{1\/\\alpha}}\\frac{{\\mathord{{\\rm d}}} y}{|y|^{(d+\\alpha+1)p^{*}}}\\right)\\\\\n&\\preceq s^{\\frac{d-(d+1)p^{*}}{\\alpha}},\n\\end{align*}\nwe have\n$$\nI_{\\alpha,\\gamma}(\\delta)\n\\preceq\\delta^{\\frac{\\gamma}{\\alpha}}\\Bigg(\\int^\\delta_0\\big[s^{-\\gamma\/\\alpha}+(\\delta-s)^{-\\gamma\/\\alpha}\\big]^{q^*}s^{\\frac{dq^*}{\\alpha p^*}-\\frac{(d+1)q^*}{\\alpha}}{\\mathord{{\\rm d}}} s\\Bigg)^{\\frac{1}{q^{*}}}.\n$$\nThus $I_{\\alpha,\\gamma}(\\delta)$ converges to zero as $\\delta\\to 0$ provided that\n$$\n-\\frac{\\gamma q^*}{\\alpha}+\\frac{dq^*}{\\alpha p^*}-\\frac{d+1}{\\alpha}q^*+1>0\\Leftrightarrow (\\ref{pq}).\n$$\nThe desired result follows.\n\\end{proof}\n\n\nThe following lemma is related to the smallness of $b\\cdot\\nabla$ as a perturbation of $-(-\\Delta|_{D})^{\\alpha\/2}$, which plays an important role in proving our main result.\n\n\\begin{lemma}\\label{integral1}\nLet $\\delta>0$ and $b\\in {\\mathbb K}_{D}^{0}$. Then for all $0\\leqslant s<t\\leqslant s+\\delta$ and $x, y\\in\nD$, we have\n\\begin{align*}\n\\int_s^t\\!\\!\\!\\int_{D}r^D(r-s,x,z)|b(r,z)|\\cdot|\\nabla_z\nr^D(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r \\leqslant C(\\delta)r^D(t-s,x,y),\n\\end{align*}\nwhere $C(\\delta)$ is a positive constant with $C(\\delta)\\rightarrow0$ as $\\delta\\downarrow0$.\n\\end{lemma}\n\\begin{proof}\nIn this proof we always assume that $0\\leqslant s<t\\leqslant s+\\delta$ and $x, y\\in D$.\nFor brevity, we write\n$$\n{\\mathcal W}:=\\frac{r^D(r-s,x,z)|\\nabla_z r^D(t-r,z,y)|}{r^D(t-s,x,y)}.\n$$\nIt follows from (\\ref{grad}) that\n\\begin{align*}\n{\\mathcal W}&\\preceq \\frac{r^D(r-s,x,z)r^D(t-r,z,y)}{r^D(t-s,x,y)}\\cdot\\frac{1}{\\rho(z)\\wedge\\left(|z-y|+(t-r)^{1\/\\alpha}\\right)}\\\\\n&\\preceq \\frac{r^D(r-s,x,z)r^D(t-r,z,y)}{r^D(t-s,x,y)}\\left(\\frac{1}{\\rho(z)}+\\frac{1}{|z-y|+(t-r)^{1\/\\alpha}}\\right)=:{\\mathcal W}_1+{\\mathcal W}_2.\n\\end{align*}\nBy (\\ref{3r}), we have that\n\\begin{align}\\label{i}\n{\\mathcal W}_1&\\preceq \\big((r-s)\\wedge (t-r)\\big)\\Big([\\hat q_\\alpha(r-s,z,x)]^2\\varrho_{d}^0(r-s,x-z)\\nonumber\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad+[\\hat q_\\alpha(t-r,z,y)]^2\\varrho_{d}^0(t-r,z-y)\\Big)\\frac{1}{\\rho(z)}\\nonumber\\\\\n&\\preceq \\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z)\n+\\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y).\n\\end{align}\nAgain by (\\ref{3r}), we have\n\\begin{align*}\n{\\mathcal W}_2&\\preceq \\big((r-s)\\wedge (t-r)\\big)\\Big([\\hat q_\\alpha(r-s,z,x)]^2\\varrho_{d}^0(r-s,x-z)\\\\\n&\\qquad\\qquad\\qquad\\qquad+[\\hat q_\\alpha(t-r,z,y)]^2\\varrho_{d}^0(t-r,z-y)\\Big)\\frac{1}{|z-y|+(t-r)^{1\/\\alpha}}.\n\\end{align*}\nBy the same argument as in (\\ref{semi}), in the case $|x-z|+(r-s)^{1\/\\alpha}\\leqslant |z-y|+(t-r)^{1\/\\alpha}$, we have\n\\begin{align*}\n{\\mathcal W}_2&\\preceq [\\hat q_\\alpha(r-s,z,x)]^2\\varrho_{d}^1(r-s,x-z)\\frac{1}{|x-z|+(r-s)^{1\/\\alpha}}\\\\\n&\\preceq \\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z).\n\\end{align*}\nIn the case $|x-z|+(r-s)^{1\/\\alpha}>|z-y|+(t-r)^{1\/\\alpha}$, we have\n\\begin{align*}\n{\\mathcal W}_2&\\preceq  [\\hat q_\\alpha(t-r,z,y)]^2\\varrho_{d}^1(t-r,z-y)\\frac{1}{|z-y|+(t-r)^{1\/\\alpha}}\\\\\n&\\preceq \\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y).\n\\end{align*}\nHence,\n$$\n{\\mathcal W}_2\\preceq\n\\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z)+\\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y),\n$$\nwhich together with (\\ref{i}) yields that\n$$\n{\\mathcal W}\\preceq \\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z)\n+\\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y).\n$$\nConsequently, by the definition of Kato class\n${\\mathbb K}_{D}^{0},$ it holds that\n\\begin{align*}\n&\\quad\\int_s^t\\!\\!\\!\\int_{D}r^D(r-s,x,z)|b(r,z)|\\cdot|\\nabla_{z}\nr^D(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\preceq\\int_s^t\\!\\!\\!\\int_{D}\\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z)|b(r,z)|{\\mathord{{\\rm d}}}\nz{\\mathord{{\\rm d}}} r\\cdot r^D(t-s,x,y)\\\\\n&\\quad+\\int_s^t\\!\\!\\!\\int_{D}\\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y)|b(r,z)|{\\mathord{{\\rm d}}}\nz{\\mathord{{\\rm d}}} r\\cdot r^D(t-s,x,y)\\\\\n&\\leqslant 2K_{b}^{0}(\\delta)r^D(t-s,x,y),\n\\end{align*}\nwhere $K_{b}^{0}(\\delta)$\nis defined in Definition \\ref{Definition}. The proof is thus finished.\n\\end{proof}\n\nTo derive the gradient estimate of the Dirichlet heat kernel, we shall also need the following\nresult.\n\n\\begin{lemma}\\label{integral2}\nLet $\\delta>0$ and $b\\in {\\mathbb K}_{D}^{0}$. Then for all $0\\leqslant s<t\\leqslant s+\\delta$ and $x, y\\in D$, we have\n\\begin{align*}\n&\\int_s^t\\!\\!\\!\\int_{D}|\\nabla_x r^D(r-s,x,z)||b(r,z)|\\cdot|\\nabla_zr^D(t-r,z,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant\n\\hat C(\\delta)\\frac{1}{\\rho(x)\\wedge(|x-y|+(t-s)^{1\/\\alpha})}r^D(t-s,x,y),\n\\end{align*}\nwhere $\\hat C(\\delta)$ is a positive constant with $\\hat C(\\delta)\\rightarrow0$ as $\\delta\\downarrow 0$.\n\\end{lemma}\n\\begin{proof}\nIn this proof we always assume that $0\\leqslant s<t\\leqslant s+\\delta$ and $x, y\\in D$.\nDefine\n\\begin{align*}\n{\\mathcal V}&:=\\frac{|\\nabla_{x}r^D(r-s,x,z)|\\cdot|\\nabla_{z}\nr^D(t-r,z,y)|}{\\hat q_\\alpha(t-s,y,x)\\varrho_{d+1}^1(t-s,x-y)}.\n\\end{align*}\nIt follows from (\\ref{gr22}) that\n$$\n{\\mathcal V}\\preceq {\\mathcal Q}\\cdot\\frac{\\varrho_{d+1}^1(r-s,x-z)\\varrho_{d+1}^1(t-r,z-y)}{\\varrho_{d+1}^1(t-s,x-y)},\n$$\nwhere\n\\begin{align*}\n{\\mathcal Q}&:=\\frac{\\hat q_\\alpha(r-s,z,x)\\hat q_\\alpha(t-r,y,z)}{\\hat q_\\alpha(t-s,y,x)}.\n\\end{align*}\nUsing \\eqref{new}, we get\n\\begin{align*}\n{\\mathcal Q}&\\asymp\\rho(z)\\cdot\\frac{\\rho(y)+|x-y|+(t-s)^{1\/\\alpha}}{(\\rho(z)+|x-z|+(r-s)^{1\/\\alpha})(\\rho(y)+|z-y|+(t-r)^{1\/\\alpha})}\\\\\n&\\preceq\\rho(z)\\cdot\\frac{\\rho(z)+|x-z|+(r-s)^{1\/\\alpha}+\\rho(z)+|z-y|+(t-r)^{1\/\\alpha}}{(\\rho(z)+|x-z|+(r-s)^{1\/\\alpha})(\\rho(z)+|z-y|+(t-r)^{1\/\\alpha})}\\\\\n&=\\frac{\\rho(z)}{\\rho(z)+|x-z|+(r-s)^{1\/\\alpha}}+\\frac{\\rho(z)}{\\rho(z)+|z-y|+(t-r)^{1\/\\alpha}}\\\\\n&\\asymp\n\\Big(\\hat q_{\\alpha}(r-s,z,x)+\\hat q_{\\alpha}(t-r,z,y)\\Big).\n\\end{align*}\nCombining this with (\\ref{333}), and by the same argument as in (\\ref{semi}), we further have that\n\\begin{align}\n{\\mathcal V}&\\preceq[(r-s)\\wedge (t-r)]\\Big(\\hat q_{\\alpha}(r-s,z,x)+\\hat q_{\\alpha}(t-r,z,y)\\Big)\\nonumber\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\times \\big(\\varrho^0_{d+1}(r-s,x-z)+\\varrho^0_{d+1}(t-r,z-y)\\big)\\nonumber\\\\\n&\\preceq \\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z)\n+\\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y).   \\label{3g}\n\\end{align}\nHence,\n\\begin{align*}\n&\\int_s^t\\!\\!\\!\\int_{D}|\\nabla_x r^D(r-s,x,z)||b(r,z)|\\cdot|\\nabla_zr^D(t-r,z,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\preceq\nK_b^0(\\delta)\\hat q_\\alpha(t-s,y,x)\\varrho_{d+1}^1(t-s,x-y)\\\\\n&\\leqslant K_b^0(\\delta)\\frac{1}{\\rho(x)\\wedge(|x-y|+(t-s)^{1\/\\alpha})}r^D(t-s,x,y),\n\\end{align*}\nwhich yields the desired result. The proof is finished.\n\\end{proof}\n\nWe now proceed to solve the integral equation (\\ref{Duhamel}). For\nall $0\\leqslant s<t$ and $x,y\\in D$, set $r_0(s,x;t,y):=r^D(t-s,x,y)$,\nand define inductively that for $k\\geq1$,\n\\begin{align}\\label{induction}\nr_k(s,x;t,y):=\\int_s^t\\!\\!\\!\\int_Dr_{k-1}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr_{0}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.\n\\end{align}\nThe following result is an easy consequence of Lemma \\ref{integral1}\nand Lemma \\ref{integral2}.\n\n\\begin{lemma}\nLet $\\delta>0$ and $b\\in{\\mathbb K}^{0}_D$.\nThen there exits a constant $c_1>1$ such that for all $k\\geqslant 1$,\n$0\\leqslant s<t\\leqslant s+\\delta$ and $x,y\\in D$, we have\n\\begin{align}\\label{3.5}\n|r_k(s,x;t,y)|\\leqslant [c_1C(\\delta)]^{k}q^D(t-s,x,y)\n\\end{align}\nand\n\\begin{align}\\label{GK}\n|\\nabla_x r_k(s,x;t,y)|\\leqslant [c_1\\hat C(\\delta)]^{k}\\frac{1}{\\rho(x)\\wedge(|x-y|+(t-s)^{1\/\\alpha})}q^D(t-s,x,y),\n\\end{align}\nwhere $C(\\delta)$ and $\\hat C(\\delta)$ are the constants in Lemma\n\\ref{integral1} and Lemma \\ref{integral2}, respectively. Moreover,\nit holds that\n\\begin{align}\nr_k(s,x;t,y)=\\int_s^t\\!\\!\\!\\int_Dr_{0}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr_{k-1}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.    \\label{pkpk}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nWe first prove (\\ref{3.5}) by induction. By Lemma\n\\ref{integral1} and the definition of $q^D(t,x,y)$, we know that (\\ref{3.5}) holds for $k=1$. Now\nsuppose that it holds for  $k>1$. Then by definition and\nusing Lemmas \\ref{integral1} and \\ref{sharp}, we have\n\\begin{align*}\n|r_{k+1}(s,x;t,y)|\n&\\leqslant\\int_s^t\\!\\!\\!\\int_D|r_{k}(s,x;r,z)|\\cdot|b(r,z)|\\cdot|\\nabla_z\nr_{0}(r,z;t,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant\n[c_1C(\\delta)]^{k}\\int_s^t\\!\\!\\!\\int_Dr^{D}(s,x;r,z)|b(r,z)|\\cdot|\\nabla_z\nr_{0}(r,z;s,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant [c_1C(\\delta)]^{k+1}q^D(t-s,x,y).\n\\end{align*}\nFollowing the same argument with Lemma \\ref{integral1} replaced by\nLemma \\ref{integral2}, we can show that (\\ref{GK}) is true. We\nproceed to prove (\\ref{pkpk}). It is obvious that (\\ref{pkpk}) holds\nfor $k=1$. Suppose that it is true for $k>1$. Then, we have\nby (\\ref{induction}) and Fubini's theorem that\n\\begin{align*}\nr_{k+1}(s,x;t,y)&=\\int_s^t\\!\\!\\!\\int_Dr_{k}(s,x;r,z)b(r,z)\\cdot\\nabla_z r_{0}(r,z;s,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&=\\int_s^t\\!\\!\\!\\int_D\\int_s^r\\!\\!\\!\\int_Dr_{0}(s,x;r',z')b(r',z')\\cdot\\nabla_{z'} r_{k-1}(r',z';r,z){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\times b(r,z)\\cdot\\nabla_z r_{0}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&=\\int_s^t\\!\\!\\!\\int_Dr_{0}(s,x;r',z')b(r',z')\\cdot\\!\\int_{r'}^{t}\\!\\!\\int_D\\nabla_{z'} r_{k-1}(r',z';r,z)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\times b(r,z)\\cdot\\nabla_z r_{0}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r{\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'\\\\\n&=\\int_s^t\\!\\!\\!\\int_Dr_{0}(s,x;r',z')b(r',z')\\cdot\\nabla_{z'}r_{k}(r',z';t,y){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'.\n\\end{align*}\nThe proof is complete.\n\\end{proof}\n\nNow, we are ready to give:\n\n\\begin{proof}[Proof of Theorem \\ref{main}]\nLet $r_k$ be defined by \\eqref{induction}. For $\\delta>0$, define\n${\\mathbb D}_\\delta:=\\{(s,x;t,y): x,y\\in D\\,\\, \\text{and}\\,\\,0\\leqslant s<t\\leqslant s+\\delta\\}$.\nIt follows from Lemma \\ref{integral1} that there exists a\n$\\delta_0\\in (0, 1]$ such that for all $0\\leqslant s<t\\leqslant s+\\delta_0$, we have\n$c_1C(\\delta_0)<1\/4$,\nwhere $c_1$ and $C(\\delta_0)$ are the constants from Lemma \\ref{induction}. Hence,\n\\begin{align}\n\\sum_{k=0}^{\\infty}|r_{k}(s,x;t,y)|\\leqslant\n\\frac43 q^{D}(t-s,x,y)\\,\\,\\,\\text{on}\\,\\,\\,{\\mathbb D}_{\\delta_0},   \\label{es}\n\\end{align}\nwhich means that the series $\\sum_{k=0}^{\\infty}r_{k}(s,x;t,y)$\nconverges on ${\\mathbb D}_{\\delta_0}$. Define\n$r^{D,b}(s,x;t,y):=\\sum_{k=0}^{\\infty}r_{k}(s,x;t,y)$ on ${\\mathbb D}_{\\delta_0}$.\nBy \\eqref{induction}, we have\n\\begin{equation*}\n\\sum_{k=0}^{n+1}r_k(s,x;t,y)=r_0(s,x;t,y)+\\int_s^t\\!\\!\\!\\int_D\\sum_{k=0}^nr_k(s,x;r,z)b(r,z)\\cdot\\nabla_z r_0(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.\n\\end{equation*}\nLetting $n\\rightarrow\\infty$ on both sides, we get (\\ref{Duhamel}).\n\n\\vspace{2mm} \\noindent(i) The upper bound on ${\\mathbb D}_{\\delta_0}$ follows by (\\ref{es}). As\nfor the lower bound on ${\\mathbb D}_{\\delta_0}$, we have\n\\begin{align*}\nr^{D,b}(s,x;t,y)\\geqslant r^D(t-s,x,y)-\\sum_{k=1}^\\infty |r_k(s,x;t,y)|\\geqslant\n\\frac{2}{3}r^D(t-s,x,y).\n\\end{align*}\nThus, (\\ref{estimate}) is valid on ${\\mathbb D}_{\\delta_0}$.\n\nNow let $\\tilde{r}^{D,b}(s,x;t,y)$ be another solution to\nthe integral equation (\\ref{Duhamel}) satisfying (\\ref{estimate}) on ${\\mathbb D}_{\\delta_0}$.\nWe claim that for every $k\\in \\mathbb{N}$, there exists a constant $C_0$ such that\non ${\\mathbb D}_{\\delta_0}$,\n\\begin{align}\\label{unique}\n|r^{D,b}(s,x;t,y)-\\tilde{r}^{D,b}(s,x;t,y)|\\leqslant C_0[c_1C(\\delta_0)]^{k}q^{D}(t-s,x,y).\n\\end{align}\nIndeed, for $k=1$, using (\\ref{Duhamel}), (\\ref{estimate}) and Lemma\n\\ref{integral1} we have\n\\begin{align*}\n&\\quad|r^{D,b}(s,x;t,y)-\\tilde{r}^{D,b}(s,x;t,y)|\\\\\n&\\leqslant\\int_{s}^{t}\\!\\!\\!\\int_{D}\\big(|r^{D,b}(s,x;r,z)|+|\\tilde{r}^{D,b}(s,x;r,z)|\\big)\\cdot|b(r,z)|\\cdot|\\nabla_z\nr^{D}(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant C_0\\int_{s}^{t}\\!\\!\\!\\int_{D}r^{D}(r-s,x,z)\\cdot|b(r,z)|\\cdot|\\nabla_z\nr^{D}(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\leqslant C_0c_1C(\\delta_0)q^{D}(t-s,x,y).\n\\end{align*}\nSuppose that \\eqref{unique} holds for some $k\\in \\mathbb{N}$. By\n(\\ref{Duhamel}), Lemma \\ref{integral1} and the induction hypothesis,\nwe have\n\\begin{align*}\n&\\quad|r^{D,b}(s,x;t,y)-\\tilde{r}^{D,b}(s,x;t,y)|\\\\\n&\\leqslant\\int_{s}^{t}\\!\\!\\!\\int_{D}|r^{D,b}(s,x;r,z)-\\tilde{r}^{D,b}(s,x;r,z)|\\cdot|b(r,z)|\\cdot|\\nabla_z\nr^D(r,z;t,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant\nC_0[c_1C(\\delta_0)]^{k}\\int_{s}^{t}\\!\\!\\!\\int_{D}r^D(r-s,x,z)\\cdot|b(r,z)|\\cdot|\\nabla_z\nr^D(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant C_0[c_1C(\\delta_0)]^{k+1}q^D(t-s,x,y).\n\\end{align*}\nSince $c_1C(\\delta_0)<1$, letting $k\\rightarrow\\infty$ we obtain the\nuniqueness.\n\n\\vspace{2mm} \\noindent(ii)\nBy choosing $\\delta_0$ smaller if necessary, we can assume that $c_1\\hat C(\\delta_0)<1$\nfor $0\\leqslant s<t\\leqslant s+\\delta_0$,\nwhere $c_1$ and $\\widehat{C}(\\delta_0)$ are the constants from Lemma \\ref{induction}.\nIt then follows from \\eqref{GK} that on ${\\mathbb D}_{\\delta_0}$,\n\\begin{align*}\n\\left|\\sum_{k=0}^{\\infty}\\nabla_x\nr_{k}(s,x;t,y)\\right|\\preceq\\frac{1}{\\rho(x)\\wedge(|x-y|+(t-s)^{1\/\\alpha})}q^D(t-s,x,y),\n\\end{align*}\nwhich means that (\\ref{vi}) is true. Moreover, by (\\ref{pkpk}) and\nFubini's theorem, we have\n\\begin{align*}\n&\\quad r^{D,b}(s,x;t,y)=\\sum_{k=0}^{\\infty}r_{k}(s,x;t,y)\\\\\n&=r^D(s,x;t,y)+\\sum_{k=0}^{\\infty}\\int_s^t\\!\\!\\!\\int_Dr_{0}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr_{k}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&=r^D(s,x;t,y)+\\int_s^t\\!\\!\\!\\int_Dr_{0}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr^{D,b}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.\n\\end{align*}\nThis yields (\\ref{duhamel}).\n\n\\vspace{2mm} \\noindent(iii) By Fubini's theorem, we have\n$$\n\\int_Dr^{D,b}(s,x;r,z)r^{D,b}(r,z;t,y){\\mathord{{\\rm d}}}\nz=\\sum_{n=0}^{\\infty}\\sum_{m=0}^n\\int_Dr_m(s,x;r,z)r_{n-m}(r,z;t,y){\\mathord{{\\rm d}}}\nz.\n$$\nThus, for proving (\\ref{eq21}), it suffices to show that for each\n$n\\in{\\mathbb N}_0$,\n\\begin{align}\n\\sum_{m=0}^n\\int_Dr_{m}(s,x;r,z)r_{n-m}(r,z;t,y){\\mathord{{\\rm d}}} z=r_n(s,x;t,y).\n\\label{ck}\n\\end{align}\nIt is clear that the above equality holds for $n=0$. Suppose now\nthat it holds for some $n\\in{\\mathbb N}$. Write\n$$\n\\sum_{m=0}^{n+1}\\int_Dr_{m}(s,x;r,z)r_{n+1-m}(r,z;t,y){\\mathord{{\\rm d}}}\nz={\\mathcal J}_1+{\\mathcal J}_2,\n$$\nwhere\n$$\n{\\mathcal J}_1:=\\int_Dr_{n+1}(s,x;r,z)p_{0}(r,z;t,y){\\mathord{{\\rm d}}} z\n$$\nand\n$$\n{\\mathcal J}_2:=\\sum_{m=0}^{n}\\int_Dr_{m}(s,x;r,z)p_{n+1-m}(r,z;t,y){\\mathord{{\\rm d}}} z.\n$$\nBy (\\ref{induction}) and Fubini's theorem, we have\n\\begin{align*}\n{\\mathcal J}_{1}&=\\int_{D}\\left(\\int_{s}^{r}\\!\\!\\!\\int_{D}r_{n}(s,x;r',z')b(r',z')\\cdot\n\\nabla_{z'} r_{0}(r',z';r,z){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'\\right)r_{0}(r,z;t,y){\\mathord{{\\rm d}}} z\\\\\n&=\\int_{s}^{r}\\!\\!\\!\\int_{D}r_{n}(s,x;r',z')b(r',z')\\cdot\\left(\\int_{D}\\nabla_{z'}\nr_{0}(r',z';r,z)r_{0}(r,z;t,y){\\mathord{{\\rm d}}} z\\right){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'\\\\\n&=\\int_{s}^{r}\\!\\!\\!\\int_{D}r_{n}(s,x;r',z')b(r',z')\\cdot\\nabla_{z'}\nr_{0}(r',z';t,y){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'.\n\\end{align*}\nSimilarly, by (\\ref{induction}) and the induction hypothesis, we\nhave\n\\begin{align*}\n{\\mathcal J}_{2}=\\int_{r}^{t}\\!\\!\\!\\int_{D}r_{n}(s,x;r',z')b(r',z')\\cdot\\nabla_{z'}\nr_{0}(r',z';t,y){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'.\n\\end{align*}\nHence,\n\\begin{align*}\n{\\mathcal J}_{1}+{\\mathcal J}_{2}=\\int_{s}^{t}\\!\\!\\!\\int_{D}r_{n}(s,x;r',z')b(r',z')\\cdot\\nabla_{z'}\nr_{0}(r',z';t,y){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'=r_{n+1}(s,x;t,y),\n\\end{align*}\nwhich gives (\\ref{ck}).\n\n\\vspace{2mm}\nNow, we can extend  $r^{D,b}(s,x;t,y)$ from ${\\mathbb D}_{\\delta_0}$ to\n$\\{(s,x;t,y): x,y\\in D\\,\\, \\text{and}\\,\\,0\\leqslant s<t<\\infty\\}$.\nThen it is routine to\nextend the above assertions (i)-(iii) on ${\\mathbb D}_{\\delta_0}$ to\n$D_{\\delta}$ for any$\\delta>0$.\n\n\\vspace{2mm} \\noindent(iv) Let $R_{s,t}f(x):=\\int_Dr^D(t-s,x,y)f(y){\\mathord{{\\rm d}}} y$.\nBy (\\ref{Duhamel}), we have for any $f\\in C_c^2(D)$,\n\\begin{align}\\label{RT5}\nR_{s,t}^{D,b}f(x)=R_{s,t}f(x)+\\int^t_s R_{s,r}^{D,b}(b\\cdot\\nabla R_{r,t}f)(x){\\mathord{{\\rm d}}} r.\n\\end{align}\nIt then follows that\n\\begin{align}\nR_{s,t}^{D,b}f(x)-f(x)&=R_{s,t}f(x)-f(x)+\\int^t_s R_{s,r}^{D,b}(b\\cdot\\nabla R_{r,t}f)(x){\\mathord{{\\rm d}}} r\\nonumber\\\\\n&=\\int_s^t R_{s,r}(-(-\\Delta|_D)^{\\alpha\/2})f(x){\\mathord{{\\rm d}}} r+\\int^t_s R_{s,r}^{D,b}(b\\cdot\\nabla R_{r,t}f)(x){\\mathord{{\\rm d}}} r,\\label{RT4}\n\\end{align}\nand, by \\eqref{RT5} and Fubini's theorem,\n\\begin{align*}\n&\\quad\\int_{s}^{t}R^{D,b}_{s,r}(-(-\\Delta|_D)^{\\alpha\/2})f(x){\\mathord{{\\rm d}}}\ns-\\int_s^t R_{s,r}(-(-\\Delta|_D)^{\\alpha\/2})f(x){\\mathord{{\\rm d}}} r\\\\\n&=\\int_{s}^{t}\\!\\!\\int_{s}^{r}R^{D,b}_{s,u}\\big(b\\cdot\\nabla R_{u,r}(-(-\\Delta|_D)^{\\alpha\/2})f\\big)(x){\\mathord{{\\rm d}}} u{\\mathord{{\\rm d}}} r\\\\\n&=\\int_{s}^{t}R^{D,b}_{s,u}b\\cdot\\nabla\\left(\\int_{u}^{t}R_{u,r}(-(-\\Delta|_D)^{\\alpha\/2})f(x){\\mathord{{\\rm d}}} r\\right){\\mathord{{\\rm d}}} u\\\\\n&=\\int_{s}^{t}R^{D,b}_{s,u}b\\cdot\\nabla\\Big(R_{u,t}f(x)-f(x)\\Big){\\mathord{{\\rm d}}} u.\n\\end{align*}\nCombining this with \\eqref{RT4}, we obtain\n\\begin{align*}\nR_{s,t}^{D,b}f(x)-f(x)=\\int_s^tR_{s,r}^{D,b}{\\mathscr L}^{D,b}f(x){\\mathord{{\\rm d}}}\nr,\n\\end{align*}\nwhich gives (\\ref{eq23}).\n\n\\vspace{2mm} \\noindent(v) Since $r^D(t,x,y)$ is the transition density\nof the process $Y^D$, so we have for any uniformly continuous function\n$f(x)$ with compact supports,\n\\begin{align*}\n\\lim_{t\\downarrow s}\\|R_{s,t}f-f\\|_\\infty=0.\n\\end{align*}\nMeanwhile, by (\\ref{estimate}) and Lemma \\ref{integral1} we have\n\\begin{align*}\n&\\quad\\left|\\!\\int_{D}\\!\\!\\!\\left(\\int_{s}^{t}\\!\\!\\!\\int_{D}r^{D,b}_{\\alpha}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr^D(t-r,z,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\!\\right)\\!f(y){\\mathord{{\\rm d}}} y\\right|\\\\\n&\\preceq\n\\|f\\|_{\\infty}\\int_{D}\\!\\!\\!\\left(\\int_{s}^{t}\\!\\!\\!\\int_{D}r^D(r-s,x,z)|b(r,z)|\\cdot|\\nabla_z\nr^D(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\!\\right){\\mathord{{\\rm d}}} y\\\\\n&\\leqslant C(\\delta)\\|f\\|_{\\infty}\\int_{D}r^D(t-s,x,y){\\mathord{{\\rm d}}} y\\leqslant\nC(\\delta)\\|f\\|_{\\infty},\n\\end{align*}\nwhich yields (\\ref{con}) by (\\ref{Duhamel}).\n\n\\vspace{2mm} \\noindent(vi) Set\n$$\n\\Phi(s,x;t,y):=\\int_s^t\\!\\!\\!\\int_D r^{D}(r-s,x,z)b(r,z)\\cdot\\nabla_z r^{D,b}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.\n$$\nIf we further assume that for $\\gamma\\in(0,\\alpha-1)$, $b\\in{\\mathbb K}_D^{\\gamma}$, then using (\\ref{hold}) we have for any $x,x',y\\in D$,\n\\begin{align*}\n&|\\Phi(s,x;t,y)-\\Phi(s,x';t,y)|\\preceq |x-x'|^{\\gamma}\\int_s^t\\!\\!\\!\\int_D\\hat q_{\\alpha}(r-s,z,\\tilde x)\\varrho_{d+1+\\gamma}^1(r-s,\\tilde x-z)\\\\\n&\\qquad\\qquad\\times|b(r,z)|\\hat q_{\\alpha}(t-r,y,z)\\varrho_{d+1}^1(t-r,z-y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\preceq |x-x'|^{\\gamma}\\int_s^t\\!\\!\\!\\int_D(r-s)^{-\\gamma\/\\alpha}\\hat q_{\\alpha}(r-s,z,\\tilde x)\\varrho_{d+1}^1(r-s,\\tilde x-z)\\\\\n&\\qquad\\qquad\\times|b(r,z)|\\hat q_{\\alpha}(t-r,y,z)\\varrho_{d+1}^1(t-r,z-y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\preceq |x-x'|^{\\gamma}\\hat q_{\\alpha}(t-s,y,\\tilde x)\\varrho_{d+1}^1(t-s,\\tilde x-y)\\int_s^t\\!\\!\\!\\int_D(r-s)^{-\\gamma\/\\alpha}|b(r,z)|\\\\\n&\\qquad\\qquad\\times\\Big(\\hat q_{\\alpha}(r-s,z,\\tilde x)\\varrho_{d+1}^1(r-s,\\tilde x-z)+\\hat q_{\\alpha}(t-r,z,y)\\varrho_{d+1}^1(t-r,z-y)\\Big){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\preceq |x-x'|^{\\gamma}(t-s)^{-\\gamma\/\\alpha}\\hat q_{\\alpha}(t-s,y,\\tilde x)\\varrho_{d+1}^1(t-s,\\tilde x-y),\n\\end{align*}\nwhere the third inequality is due to (\\ref{3g}), and the last inequality follows from the definition of ${\\mathbb K}_D^\\gamma$. Combining this with (\\ref{duhamel}) and (\\ref{hold}), we get the desired result. The proof is complete.\n\\end{proof}\n\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:introduction}\nAs a result of hierarchically-structured, accreting molecular clouds, stars preferentially form in clustered environments. More than 70 \\% of massive stars are located in clusters and OB associations \\citep{ladalada03}, making young star clusters (YSCs) preferential sites from which radiative and mechanical feedback originates. Typically, only a minority of these stellar aggregates are gravitationally bound and will, therefore, survive as star clusters \\citep{kruijssen12a, grudic20}. These are the clusters that sit in the densest regions of the hierarchy and are therefore able to survive the removal of gas long enough to reach a high star formation efficiency and become dynamically relaxed and well-mixed \\citep{kruijssen19}. The remaining loosely-bound clusters will disperse in short timescales and blend into the field stellar population.\n\n\\par A detailed understanding of the process of stellar feedback is important as this mechanism leads to the suppression of global star formation in galaxies and to a multiphase and metal-enriched interstellar medium (ISM).\nIn particular, it will shed light on the origin of the diffuse ionised gas (DIG) that has been observed to constitute up to 50\\% of the total H$\\alpha$ luminosity in local spiral galaxies \\citep{ferguson96, hoopes96, zurita00, thilker02, oey07}. The origin of this gas is unclear \\citep[see e.g.][for a review]{mathis00, haffner09}, and it has been linked to various processes including leaking \\ion{H}{ii} regions~\\citep{zurita02, weilbacher18}, evolved field stars \\citep{hoopes00,zhang17}, shocks~\\citep{collins01}, and cosmic rays~\\citep{vandenbroucke18}.\nBoth theory and observation have shown that stellar feedback can facilitate the escape of ionising radiation from the star forming regions \\citep[see e.g.][]{bik15,menacho19} if the youngest star clusters have had time\nto clear out channels through the dense leftover gas from the parent giant molecular clouds (GMCs) from which they were born~\\citep{dale15,howard18}.\nIn order to study how the radiation from massive stars affects the ISM, it is therefore necessary to investigate the timescale for photons to find a channel through which they can leave the  region.\nVarious studies have found that small scale physics plays an important role in the escape of ionising radiation. In particular, the effect of temporal and spatial clustering of supernovae (SN) can have important consequences \\citep{gentry17,kim17,fielding18}, resulting in a transferred momentum up to an order of magnitude larger than for isolated SN.\nSimulations of individual galaxies do indeed indicate that the escape of ionising radiation is largely driven by the distribution of dense gas \\citep[see e.g.][]{paardekooper11}, and high-resolution GMC simulations have proven that photons are more likely to escape from less massive GMCs \\citep{howard18}. Consequently, late-emitted ionising photons are more likely to escape the star forming region as the gas surrounding the region is dispersed by ionising radiation and supernova feedback~\\citep{kim13a}. \nThe same trend has been recently observed in high-resolution cosmological simulations, showing that photons are preferentially leaking from star forming regions containing feedback-driven superbubbles~\\citep{ma20}.\nFinally, stars are also responsible of yielding metals that will enrich the ISM, and a detailed understanding of abundance variations in star forming regions is needed in order to constrain chemical evolution models. However, until recently, it has been unclear whether star forming galaxies feature, in general, a homogeneous metal enrichment or whether the enrichment varies in different star forming regions and within the regions themselves.\nRecent high-resolution studies \\citep[e.g.][]{james16, mcleod19} seem to indicate the latter, implying that feedback mechanisms might play an important role in local metallicity variations.\n\n\\par High resolution studies of star forming regions are therefore an essential piece of information in order to bridge the spatial scales that critically connect the sources of feedback (stars and clusters, i.e. a few parsec scales) and their immediate surroundings (10s parsec scale) to galactic scale dynamics (kpc scale).\nSuch studies, initially limited to narrow-band photometry \\citep[e.g.][]{pellegrini12}, were revolutionised with the advent of instruments with high spatial resolution integral field spectroscopy capabilities, such as the Multi Unit Spectroscopic Explorer~\\citep[MUSE,][]{bacon10} instrument at ESO Very Large Telescope (VLT). Recently, \\citet{mcleod19,mcleod20} exploited the MUSE capability for a detailed study of \\ion{H}{ii} regions in the Large Magellanic Cloud and in the dwarf spiral NGC 300, mapping the ionisation structure of the regions and constraining their escape of ionising radiation.\nIn this work, we perform a similar study in the local spiral galaxy NGC 7793 at a spatial resolution of $\\sim$ 10 pc with MUSE. In \\citet[][henceforth Paper~\\textsc{i}]{paperI} we presented the data, constructed a  sample of \\ion{H}{ii} regions, determined the fraction of DIG in our field of view (FoV) and studied the properties of the ionised gas in emission line diagrams.\nHere, we combine the MUSE results on the ionised gas with the stellar and cluster population studied with the Hubble Space Telescope (HST) and the molecular gas mapped with the Atacama Large Millimeter\/submillimeter Array (ALMA),\nconstruct a photoionisation budget for each \\ion{H}{ii} region and study how it relates to its stellar and GMC content and to its ionisation structure.\n\n\\par This work is structured as follows.\nIn Sect.~\\ref{section:data} we give a brief overview of the data.\nIn Sect.~\\ref{section:stellar_pop} we present the stellar census in our FoV and analyse the distribution of YSC properties.\nIn Sect.~\\ref{section:extinction} we compare extinction maps derived from individual stellar extinction and from the ionised gas and in Sect.~\\ref{section:metallicity} we derive the oxygen abundance of the \\ion{H}{ii} regions.\nIn Sect.~\\ref{section:optical_depth} we analyse the ionisation structure of the gas, and in Sect.~\\ref{section:budget} we compile a photoionisation budget for the entire FoV and in more detail for a subset of the \\ion{H}{ii} regions.\nFinally, in Sect.~\\ref{section:discussion} we discuss our findings and in Sect.~\\ref{section:conclusions} we give a brief summary and present our conclusions.\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=16cm]{stellar_census.pdf}\n\\caption{Stellar census in the MUSE FoV. The location of candidate O stars (white circles), YSCs (green triangles) and WR stars (blue crosses) is indicated, overlaid on a map of the H$\\alpha$ emission. Black contours indicate the \\ion{H}{ii} regions identified in Paper~\\textsc{i}, light blue filled contours denote the position of GMCs.}\n\\label{fig:stellar_census}\n\\end{figure*}\n\n\n\\section{Data description}\n\\label{section:data}\nThe data and data reduction steps are described in detail in Paper~\\textsc{i}: we give here a brief summary. We observed the nearby flocculent spiral galaxy NGC 7793 with MUSE\\footnote{ESO programme 60.A-9188(A), PI Adamo.}.\nThe dataset consists of two pointings in the Wide Field Mode (WFM) adaptive optics assisted (AO) configuration, with the extended wavelength setting. On the final datacube, we measure a PSF of $0.62 \\arcsec$ FWHM (at 7000~\\AA{}), corresponding to a spatial resolution of $\\sim$ 10 pc at the distance of NGC 7793.\n\\par NGC 7793 has also been observed in the framework of the HST Treasury programme LEGUS\\footnote{HST GO\u201313364.}~\\citep{calzetti15} and in ALMA CO(J = $2-1$).\nThe catalogues of YSCs derived from the LEGUS data are publicly available\\footnote{\\url{https:\/\/archive.stsci.edu\/prepds\/legus\/dataproducts-public.html}}: in this work, we make use of the catalogue generated with the Geneva stellar evolution models and Milky Way extinction. We are moreover selecting only clusters labelled as `Class 1' (compact and symmetric), `Class 2' (concentrated but with some degree of asymmetry), and `Class 3' (multiple peak systems). For more details on the cluster catalogues (e.g. cluster identification, classification and photometry) see~\\citet{adamo17}. \nLee et al. (in preparation) have identified O stars based on HST photometry. The catalogue is described in \\citet{wofford20}, and consists of\nmain sequence O stars of $M > 20~M_\\odot$, selected in a colour-magnitude and colour-$Q$ diagram\\footnote{Here, $Q$ is the reddening-free parameter introduced by \\citet{johnson53}.}.\nThe resulting completeness of this stellar catalogue is estimated to be about 75\\% at the lower mass limit of $20~M_\\odot$ and rapidly growing to 90\\% at stellar masses above $25~M_\\odot$\\footnote{We note that the mass of the candidates is not constrained by the selection method. However, mass estimates have been obtained from completeness simulations, in which the candidates are compared to synthetic stars.}. The authors report also an average contamination rate $\\sim$ 30\\%. We use the latter rate to account for uncertainties in the contribution of O stars to the ionisation budget.\n\nThe catalogue of GMCs built on the ALMA dataset is presented in \\citet{grasha18}. The ALMA data have an angular resolution of $0.85''$ and a velocity resolution of 1.2 km s$^{-1}$. The data are sensitive to emission up to $11''$ and are mapping GMC down to masses $\\sim 10^4 M_{\\sun}$.\n\n\\section{Properties of the stellar population}\n\\label{section:stellar_pop}\n\n\\begin{figure*\n\\center\n\\includegraphics[width=18cm]{ysc_age_mass_FOV.pdf}\n\\caption{Spatial distribution of YSC ages (\\textit{left panel}) and masses (\\textit{right panel}) in the MUSE FoV, overlaid on the H$\\alpha$ map shown in Fig.~\\ref{fig:stellar_census}. Circles and diamonds indicate, respectively, clusters younger and older than 10 Myr.\nAges and masses are best estimates computed from the median of the corresponding PDF (see Sect.~\\ref{section:ysc_properties}).\nBlack contours indicate the \\ion{H}{ii} regions identified in Paper~\\textsc{i}.}\n\\label{fig:ysc_distr_FOV}\n\\end{figure*}\n\n\\begin{figure*\n\\center\n\\includegraphics[width=17cm]{ysc_agemassq0_hist.png}\n\\caption{Posterior probability distributions of YSC age (\\textit{left}) mass (\\textit{centre}) and ionising flux (\\textit{right}) for clusters located in \\ion{H}{ii} regions (in blue) and for field clusters (in orange). Thick lines indicate the combined PDFs, thin lines correspond to the PDFs of single clusters. The grey dashed lines indicate the location of the main modes of the combined distributions.}\n\\label{fig:ysc_agemass_hist}\n\\end{figure*}\n\n\\begin{figure\n\\center\n\\includegraphics[width=7cm]{mass_distr_gmc_ysc.pdf}\n\\caption{Distribution of YSC mass (clusters with $t < 10$ Myr, in blue), GMC mass (in green) and \\ion{H}{ii} regions luminosity (in black, reddening corrected) in our FoV. YSC masses are best estimates computed from the median of the corresponding PDF (see Sect.~\\ref{section:ysc_properties}). The histograms are normalised to a peak value of 1.}\n\\label{fig:mass_distr_gmc_ysc}\n\\end{figure}\n\n\\subsection{Stellar census}\nWe present here the full census of the stellar population in our FoV. Figure~\\ref{fig:stellar_census} shows a map of the H$\\alpha$ emission line obtained from the integration of the MUSE stellar continuum subtracted datacube in the rest-frame wavelength range $6559 - 6568$~\\AA{}. Overlaid on the H$\\alpha$ emission, we show the position of candidate O stars from Lee et al. (in prep., white circles), YSCs from LEGUS (green triangles) and of Wolf Rayet (WR) stars that we spectroscopically identified in Paper~\\textsc{i} (listed in Table~\\ref{table:wr_stars}; blue crosses).\nWe also show the contours of \\ion{H}{ii} regions identified in Paper~\\textsc{i} (in black, corresponding to a flux brightness cut in H$\\alpha$ = $\\SI{6.7e-18}{\\erg \\per \\second \\per \\centi \\metre \\squared}$spaxel$^{-1}$) and of GMCs (light blue filled contours). \nThe total stellar census in our FoV consists of 651 O stars, 93 YSC, and 9 WR stars.\n\nFrom Fig.~\\ref{fig:stellar_census}, we observe that both YSCs and candidate O stars are abundant over the entire FoV. However, we note that several of the smaller and medium-bright \\ion{H}{ii} regions do not seem to host any cluster. Overall, we find\n$\\sim$ 49\\% field\\footnote{Throughout the paper, we refer to stars and clusters located outside \\ion{H}{ii} regions as `field' objects.} clusters (35\\% if considering only clusters with median age $<$ 10 Myr, as motivated in Sect.~\\ref{section:fesc_individual}) and\n$\\sim$ 38\\% field candidate O stars. \nWe note that the numbers of O-stars quoted here are not corrected for completeness, so that the latter fraction could in part be due to the high contamination rate of catalogue (see Sect.~\\ref{section:data}).\nWe observe that WR stars are populating the \\ion{H}{ii} regions, with the exception of one object (located in the upper left corner and catalogued as WR 3 in Table~\\ref{table:wr_stars}). \\citet{rate20} recently found that isolated WR stars are not an exception, and amount to at least 64\\% in the Milky Way \\citep[as observed in][]{gaia_DR2}. The authors explore the possible origin of field WR stars in simulations, finding that the most frequent mechanisms are the formation in a low density association that then expands during the star lifetime and the ejection from a cluster through dynamical interaction or binary disruption. This has been observed in galactic star clusters \\citep[e.g. Westerlund 2 and NGC 3603][]{drew18, drew19}.\nThe fact that WR 3 is located $\\sim$ 100 pc away from a nearby HII-region cluster (see Fig.~\\ref{fig:stellar_census} and~\\ref{fig:sii_oiii}) could support the latter formation scenario.\nIn order to investigate this hypothesis, we have computed the minimum velocity needed for the star to be ejected from the nearby cluster. We assume an age range of (4 $\\pm$ 1) Myr for the WR star\\footnote{We note that, despite the spectral information available, the age also depends on the mass of the progenitor O stars and on rotation effects.}.\nAssuming that the star has been ejected immediately after birth, we find $v_{proj, min} = 25_{-8}^{+5}$ km\/s, which meets the standard runaway criterium of $v > 25$ km\/s \\citep[e.g.][]{drew18, portegieszwart00}.\n\n\\subsection{YSCs age, mass, and ionising flux distribution}\n\\label{section:ysc_properties}\nIn this subsection, we investigate the physical properties of the YSCs, namely their age, mass, and ionising photon flux. The latter, $Q(H^0)$ [s$^{-1}$], is a measure of the number of Lyman continuum (LyC) hydrogen-ionising photons ($h\\nu > 13.6$ eV) emitted per unit time.\n\nFor this purpose, we use the stochastic stellar population synthesis code \\textsc{slug} \\citep[][v2]{dasilva12,krumholz15a}. \n\\textsc{Slug} implements a Monte Carlo technique by randomly drawing stars from an input initial mass function (IMF). This approach allows one to account for the effect of stochastic sampling of the IMF, which is especially important when populating low mass clusters \\citep[see e.g.][]{hannon19}.\nThis is of a particular relevance in our target: NGC 7793 has a very low SFR $\\sim$ 0.1 $M_\\sun$ yr$^{-1}$ \\citep{calzetti15}, resulting in \\ion{H}{ii} regions hosting low mass ($\\sim$ a few 1000s $M_\\sun$) clusters.\nThe \\textsc{SLUG} software package includes the \\texttt{cluster\\_slug} tool~\\citep{krumholz15a}, that implements Bayesian inference techniques to calculate posterior probability distribution functions (PDFs) of physical parameters of star clusters based on  observed cluster photometry, as described in \\citet{krumholz15b}.\nThis is achieved by comparing the data to a large library of simulated clusters. We modified \\texttt{cluster\\_slug} to also return the ionising photon luminosity $Q(H^0)$ as a further output parameter and ran the code on HST broadband photometry of the YSCs. We consider age $t$, mass $M$ and visual extinction $A_V$ as free parameters, and assume a flat prior in $A_V$ and in $\\log t$, and a $\\log(M) \\sim 1\/M$ prior on the mass. We use the library of mock star clusters described in~\\citet{ashworth18}, a Milky Way extinction law by \\citet{fitzpatrick99} and the non-rotating solar metallicity stellar population models from \\citet{genevamodels12}. The abundance map derived in Sect.~\\ref{section:metallicity} confirms that NGC 7793 has a metallicity close to solar \\citep[see also][]{pilyugin14}; the same metallicity was also assumed in the \\textsc{slug} models of \\citet{krumholz15b}. We discuss the impact of including binary stars and the effect of stellar rotation on the resulting physical properties in Sect.~\\ref{section:discussion}.\n\n\\par Figure~\\ref{fig:ysc_distr_FOV} shows the spatial distribution across the FoV of YSC ages (left panel) and masses (right panel). We are showing here as best estimate the median of each cluster PDF.\nAs remarked in \\citet{krumholz15b}, reducing a full PDF to a single point inevitably leads to some imprecision, especially if the PDF is multi-peaked. We follow the method tested in \\citet{krumholz15b} and we refer the interested reader to this work for evaluations of the methodology and effects on the recovered cluster physical parameters.\nIn Fig.~\\ref{fig:ysc_agemass_hist}, we show the age, mass, and ionising flux PDF for YSCs located inside \\ion{H}{ii} regions (in blue) and for field clusters (in orange). Thin and thick lines in the plot indicate, respectively, the PDFs of single clusters and the combined PDFs obtained by summing the fractional probability of each cluster in every age, mass, and flux bin, and re-normalising the resulting distribution.\nIn the case of the age, both the single and combined PDFs are (most often) multi-peaked; the main peak of the combined PDF\nis located at $t \\sim 5$~Myr, both for field and \\ion{H}{ii} region clusters. However, we see that the PDF of field clusters flattens towards older ages, while the probability distribution of \\ion{H}{ii} region clusters is more peaked at ages below 10 Myr, that is clusters located in \\ion{H}{ii} regions have a higher probability to be young and potentially ionising sources.\nThe mass distribution is single-peaked for both categories of clusters, with a tail towards higher masses. The peak of the distribution is located at $\\log(M) = 3.0$ and 3.4~$M_\\odot$ for \\ion{H}{ii} region clusters and field clusters respectively.\nThe $Q(H^0)$ distribution is, as the age, multi-peaked (\\text{both the single and combined PDFs}) with two main modes (both for \\ion{H}{ii} region and field clusters) at $\\log(Q(H^0)) = 44.1$ and 48.8 s$^{-1}$.\nNevertheless, we notice that clusters in \\ion{H}{ii} regions have a higher probability to be associated with the high $Q(H^0)$ value, while the opposite is true for field clusters. In general however, the inferred masses and $Q(H^0)$ are on the low range for YSCs: for a reference, the peak at $\\log(Q(H^0)) = 48.8$ s$^{-1}$ corresponds to the emission of a single O7.5 star \\citep{martins05}. We comment more on the mass distribution below (Fig.~\\ref{fig:mass_distr_gmc_ysc} and associated text).\n\nIn synthesis, we observe that YSCs in \\ion{H}{ii} regions have a higher probability to be younger, slightly lower mass, but a more significant source of ionising photons than clusters in the field. \nThese trend can easily be visualised in the plots of Fig.~\\ref{fig:ysc_distr_FOV}, where we colour code the positions of the clusters in the FoV according to their median ages and masses.\n\n\\par The age range observed for the clusters within the \\ion{H}{ii} regions is in agreement with studies of YSCs in local galaxies \\citep{whitmore11, hollyhead15, hannon19, grasha19}, that find that clusters emerge from their natal gas in $< 4 - 5$ Myr, but that the process can start as early as $2 - 3$ Myr. Also simulations of cluster formation including solely the effect of photoionisation and stellar winds \\citep{dale14, bending20}, have shown that within 3 Myr, and before the onset of SN, several clusters have vacated some channels with low gas density. These channels are easily ionised; therefore ionising radiation can escape from the \\ion{H}{ii} regions even if star formation is still taking place. In Fig.~\\ref{fig:stellar_census} we observe that a significant amount of molecular gas is still detectable in the \\ion{H}{ii} regions hosting YSCs, in agreement with the findings of \\citet{dale14}.\n\n\\par Finally, in Fig.~\\ref{fig:mass_distr_gmc_ysc} we show the total demographics of GMCs (in green), YSCs with age $< 10$ Myr (in blue) and \\ion{H}{ii} regions (in black) within our FoV, highlighting the various phases of the star formation cycle. We observe that GMCs, hosting the molecular gas that will give rise to the next generation of stars, span a rather low range of masses compared to what observed in other local galaxies \\citep[$\\log M_{GMC} \\sim 4.5 - 8 M_\\odot$,][]{hughes16}.\n\nThe collapse of the clouds results in turn in low-mass YSCs. While there is not a one-to-one correlation between the mass spectrum of GMCs and YSCs, we observe that overall the two distributions would suggest an integrated SFE of a few percent of what is typically observed in local galaxies \\citep[see e.g. the review by][]{krumholz19}.\nThe bulk of the YSCs have masses around 1000 $M_\\odot$, and the distribution reaches up to $\\sim 10^{3.5}~M_\\odot$. These mass distributions are typical of galaxies with low SFR, such as M31 \\citep{larsen09, johnson17, adamo20}.\nIf we consider that the ionising feedback from a cluster of 1000 $M_\\odot$ gives rise to an H$\\alpha$ luminosity\nof $\\log(L) \\sim 37.3$ erg s$^{-1}$,\\footnote{\\textsc{Starburst99} \\citep{leitherer99} estimate assuming a \\citet{kroupa01} IMF in the stellar mass range $0.1-120~M_\\odot$.}\nthen the observed luminosity distribution is consistent with the mass distributions of the YSCs in the FoV. If compared to studies of \\ion{H}{ii} regions luminosity functions in the local universe, our sample occupies the low-L end of the luminosity function of local spirals \\citep[$\\log(L ) \\sim 37 - 40$ erg s$^{-1}$,][]{kennicutt89a}. Overall, the low masses and luminosity observed are in agreement with the flocculent morphology and low SFR of our target.\n\n\\begin{figure*\n\\includegraphics[width=8.3cm]{E_BV_map_leguscontour.pdf}\n\\includegraphics[width=9.6cm]{E_BV_map_24microncontour.pdf}\n\\caption{Extinction map derived from the ionised gas observed with MUSE assuming an intrinsic H$\\alpha$\/H$\\beta$ ratio (from Paper~\\textsc{i}) compared to: \\textit{Left panel}: contours of the extinction map derived by~\\citet{kahre18} from the individual stellar extinctions obtained from HST data. The contours correspond to values of $E(B-V) = 0.2 - 0.5$, in steps of 0.033. \\textit{Right panel}: 24~$\\mu$m emission from Spitzer\/MIPS. The contours correspond to a flux of [1.5, 2, 3, 4, 6, 9, 11] MJy\/sr. The thick orange contours indicate the position of GMCs.}\n\\label{fig:extinction_map}\n\\end{figure*}\n\n\\section{Comparison between stellar and ionised gas extinction with maps of cold dense gas and hot dust emission}\n\\label{section:extinction}\nThe HST, MUSE, ALMA, and Spitzer\/MIPS\\footnote{Publicly available in the NED archive \\url{http:\/\/ned.ipac.caltech.edu}.} coverage gives us the rare opportunity of comparing extinctions as traced by stellar reddening and the ionised gas phase to the distribution of the cold dense gas and the hot dust emission.\nIn Paper~\\textsc{i}, we have derived an extinction map from the MUSE H$\\alpha$\/H$\\beta$ ratio using \\textsc{pyneb} \\citep{luridiana15}. Hereby, we assumed a theoretical H$\\alpha$\/H$\\beta_{int}$ = 2.863 and we obtained the observed ratio by binning the stellar continuum subtracted cube with the Voronoi technique of \\citet{cappellari03} to a S\/N = 20 in H$\\beta$ and fitting both lines with a single Gaussian profile.\nHere, we compare the gas extinction map derived in Paper~\\textsc{i} with the map derived by~\\citet{kahre18} from the individual stellar extinctions obtained from HST, and correlate the two maps with the position of GMCs traced by ALMA.\nFigure~\\ref{fig:extinction_map} (left panel) shows overlaid on the gas extinction map the contours of the weighted average map with adaptive resolution from~\\citet{kahre18}. The latter was derived by spatially binning the HST data in the smallest possible regions (of varying size between 1 and 10 arcsec$^2$) hosting at least 10 O stars. In order to compare the two maps, we have converted the colour excess E(V$-$I) to E(B$-$V) assuming a Milky Way extinction law with $R_V = 3.1$, as in \\citet{kahre18}.\nWe observe a clear correspondence between the two maps, despite the difference in binning and in the method used to derive the extinction. In general, regions in which we measure a high extinction also feature a comparable stellar extinction, although the position of the extinction peaks is slightly shifted in the two maps. Conversely, the ionised gas extinction seems to miss some of the regions of enhanced stellar extinction. We notice that these regions lie outside the bright ionised gas, where H$\\beta$ is poorly detected and the ionised gas might not be a sensitive enough tracer of extinction.\nHowever, we observe that the gas extinction seems to better correlate with the position of the GMCs (orange contours in Fig.~\\ref{fig:extinction_map}). We also note that there seems to be a better agreement between gas extinction peaks and GMC position within \\ion{H}{ii} regions.\n\\par\nIn the right panel of Fig.~\\ref{fig:extinction_map}, we furthermore show overlaid on the gas extinction map contours of 24~$\\mu$m emission from Spitzer\/MIPS. The 24~$\\mu$m emission is tracing hot dust in the galaxy, and inside the \\ion{H}{ii} regions it is indicative of where star formation is taking place. We observe a good correlation between gas extinction, GMC position and 24~$\\mu$m emission, indicative of the fact that, in HII regions, the dust is well mixed with gas. We comment more on this fact in relation with the effect of removal of LyC photons by dust prior to their absorption by neutral hydrogen in Sect.~\\ref{section:discussion}.\n\n\\begin{figure*\n\\includegraphics[width=9.2cm]{logO_H_map_1.pdf}\n\\includegraphics[width=9.0cm]{logO_H_map_2.pdf}\n\\caption{Oxygen abundance for the subset of \\ion{H}{ii} regions labelled in Fig.~\\ref{fig:sii_oiii}, determined with the $S$-strong line calibration of \\citet{pilyugin16}. The location of YSCs is indicated with filled circles, colour-coded based on their age (best estimates derived from the median of each PDF). Red crosses and purple stars correspond to the position of WR stars and PNe, respectively. Dark red contours indicate a ratio of [\\ion{S}{ii}]\/[\\ion{O}{iii}] = 0.5 (see Fig.~\\ref{fig:sii_oiii}).}\n\\label{fig:logO_H_map}\n\\end{figure*}\n\n\\section{Mapping the Oxygen abundance within the \\ion{H}{ii} regions}\n\\label{section:metallicity}\nWe estimate an oxygen abundance in the \\ion{H}{ii} regions using the strong-line method of \\citet{pilyugin16}.\nIn Paper~\\textsc{i}, we derived electron temperature and density from the [\\ion{S}{iii}]6312\/9069 and [\\ion{S}{ii}]6716\/6731 line ratios\\footnote{As remarked in Paper~\\textsc{i}, the [NII]5755 `auroral' line used for temperature diagnostics falls  in the wavelength range blocked out in the MUSE\/AO mode.}; however, due to the weakness of the [\\ion{S}{iii}]6312 line and to the fact that the [\\ion{S}{ii}] ratio is largely insensitive to $n_e < 30$ cm$^{-3}$, we were only able to obtain a coarse $T_e$ map and marginally constrained $n_e$, preventing us from measuring direct abundances.\nThe \\citet{pilyugin16} method bases on ratios of strong lines calibrated on a sample of $\\sim$ 300 \\ion{H}{ii} regions. Here we make use of the $S$ calibration, based on the following three line ratios:\n$$N_2 = (I_{[\\ion{N}{ii}]}\\lambda6548 + I_{[\\ion{N}{ii}]}\\lambda6584) \/ I_{H\\beta},$$\n$$S_2 = (I_{[\\ion{S}{ii}]}\\lambda6716 + I_{[\\ion{S}{ii}]}\\lambda6731) \/ I_{H\\beta},$$\n$$R_3 = (I_{[\\ion{O}{iii}]}\\lambda4959 + I_{[\\ion{O}{iii}]}\\lambda5007) \/ I_{H\\beta}.$$\nWe use the upper branch of the calibration, valid for $\\log N_2 \\gtrsim -0.6$, as our data lie well above this limit throughout the FoV. The line ratios are determined from the reddening corrected fluxes, which have been obtained by binning the data with the Voronoi technique to a S\/N = 20 in H$\\beta$ as described in Sect.~\\ref{section:extinction}.\nFigure~\\ref{fig:logO_H_map} shows the resulting metallicity variation in a subset of the \\ion{H}{ii} region sample, for which we construct a detailed photoionisation budget in Sect.~\\ref{section:budget}. The exact location of these regions in the FoV is shown in Fig.~\\ref{fig:sii_oiii}. We observe that the abundance ranges from $12 + \\log(O\/H) \\sim 8.25$ to 8.50, with a median value of 8.37.\nThis range of values is in agreement with the strong-line estimate of \\citet{pilyugin14} for NGC 7793 and with the metallicity range inferred in Paper~\\textsc{i} when comparing our data with the models of star forming galaxies from \\citet{levesque10} in `BPT' diagrams \\citep{baldwin81}.\n\n\\par In small and isolated \\ion{H}{ii} regions, such as regions 4, 5, 6, and 7, we notice that the oxygen abundance varies of about 0.1 dex at most. The largest \\ion{H}{ii} region complexes, such as regions 1, 2, 3, and 8 show larger gradients and pockets of enriched gas coincident or arranged in shell-like structures around clusters and massive stars\\footnote{We omit to plot the position of O stars, which are also a source of enrichment, to avoid overcrowding and facilitate the visualisation of the abundance variation. We refer the reader to Fig.~\\ref{fig:stellar_census}.}.\nThis is particularly visible in region 4, where the YSC is surrounded by an enriched shell, and in regions 6 and 8 (centre left and bottom right) where we estimate a local enhancement in the oxygen abundance at the location of YSCs.\nWe furthermore observe that some of the WR stars are associated with enriched shells or pocket of ionised gas: this is the case in region 1 (WR2) and 8 (WR4), likely due to nitrogen-enriched gas expelled by their stellar winds. In other cases, we do not observe any enrichment surrounding the stars: this is especially visible in region 3 and in the north-west, and south of region 1.\nFinally, we observe a large enhancement of the abundance at the location of one of the planetary nebulae identified in Paper~\\textsc{i} (PN2).\n\n\\par We would like to point out that, although such indirect methods are generally calibrated for unresolved \\ion{H}{ii} regions, they have recently been applied also to study abundance variations within resolved \\ion{H}{ii} regions \\citep[e.g.][]{james16, mcleod19}.\nUsing narrow-band HST maps, \\citet{james16} show chemical inhomogeneity ($\\sim$ 0.1 dex) on scales smaller than 50 pc within the star forming region Mrk 71. \\citet{mcleod19} also report abundance variations and structures similar to what we observe here in two star-forming regions of the Large Magellanic Cloud, resolved at scales of a few parsec. \nHowever, as remarked, for example, by \\citet{mcleod19}, abundances derived with a strong line method can have a secondary non-trivial dependence on temperature and density, resulting for instance in falsely low abundances in regions with a high ionisation state~\\citep{ercolano12,mcleod19,mcleod16a}. In Fig. \\ref{fig:logO_H_map}, we show contours of [\\ion{S}{ii}]\/[\\ion{O}{iii}] = 0.5 (dark red), delimiting strongly ionised regions in the nebulae (see Sect.~\\ref{section:optical_depth}). We observe indeed a close correspondence between the contours and areas of low metallicity in all the regions, pointing to the fact that a degeneracy between oxygen abundance and ionisation conditions is likely playing a role closest to the young massive stars. \nIn the future, IFU having a spectral and spatial capability comparable to MUSE, but covering a bluer wavelength range \\citep[e.g. BlueMUSE,][]{bluemuse}\nwill help us to better understand abundances in resolved \\ion{H}{ii} regions by addressing all such degeneracies.\n\n\\begin{figure\n\\center\n\\includegraphics[width=8cm]{cloudy_model_2Myr.pdf}\n\\caption{\\textsc{cloudy} model illustrating the ionisation structure of an ideal ionisation-bound nebula surrounding a 2 Myr YSC of $M = 10^4~M_\\odot$ at the metallicity of NGC 7793. The radial variation in ionic fraction of \\ion{H}{i} (dashed black), \\ion{O}{iii} (dashed purple), \\ion{S}{ii} (solid green) and the \\ion{S}{ii}\/\\ion{O}{iii} ratio (solid grey) are shown. The nebula consists of two main zones: an inner zone dominated by \\ion{S}{iii} and \\ion{O}{iii}, and an outer zone dominated by \\ion{S}{ii} and \\ion{O}{ii}.}\n\\label{fig:cloudy_model}\n\\end{figure}\n\n\\begin{figure*\n\\center\n\\includegraphics[width=14cm]{s2o3_selected_regions.pdf}\n\\caption{[\\ion{S}{ii}]6716,31\/[\\ion{O}{iii}]4959,5007 ratio map (reddening corrected), tracing the ionisation structure of the gas.\nThe cyan contours indicate a ratio of 0.5, which we use as fiducial limit for the transition to optically thick gas.\nNumbered black contours correspond to the \\ion{H}{ii} regions analysed in Sect~\\ref{section:budget}. The position of WR stars (yellow crosses) and planetary nebulae (white plus symbols) is also indicated.}\n\\label{fig:sii_oiii}\n\\end{figure*}\n\n\\begin{figure\n\\center\n\\includegraphics[width=7.8cm]{hii_class.pdf}\n\\caption{Values of [\\ion{S}{ii}]6716,31\/[\\ion{O}{iii}]4959,5007 along the contour of the \\ion{H}{ii} regions labelled in Fig.~\\ref{fig:sii_oiii}. The solid points and error bars show, respectively, the median and the range in values spanned by the ratio. The cyan line indicates a ratio of 0.5, which we use as fiducial limit for the transition to optically thick gas. Regions for which the range in [\\ion{S}{ii}]\/[\\ion{O}{iii}] extends below this limit are classified as featuring optically thin channels (CH); all other regions are classified as ionisation bounded (IB).}\n\\label{fig:hii_class}\n\\end{figure}\n\n\\begin{figure\n\\center\n\\includegraphics[width=9cm]{fesc_vs_appearance.pdf}\n\\caption{Ratio of expected to observed ionising luminosity $Q(H^0)$ for the regions labelled in Fig.~\\ref{fig:sii_oiii} versus their visual appearance: ionisation bounded regions (IB, green stars) vs regions featuring optically thin channels (CH,  blue stars). The dashed horizontal line indicates an $f_{esc} = 0$.}\n\\label{fig:fesc_vs_appearance}\n\\end{figure}\n\n\\section{Ionisation structure of the \\ion{H}{ii} regions}\n\\label{section:optical_depth}\n\nThe ionisation structure of a nebula can be studied through the relative emission of two ions with different ionisation potential. The structure of an ideal ionisation bounded \\ion{H}{ii} region is shown in Fig.~\\ref{fig:cloudy_model}.\nThe figure illustrates a model generated with \\textsc{cloudy} for a nebula at the metallicity of our target ($Z \\sim 0.008$, see Sect.~\\ref{section:metallicity}), surrounding a 2 Myr YSC with a mass of $10^4~M_\\odot$. The simulation is run until the gas interface reaches an optical depth $\\tau = 2$.\nTwo zones can be distinguished in the nebula: a central zone with a higher ionisation state, dominated by S++ and O++ and an external zone with a lower ionisation state, where these ions are singly ionised. The ratio of S+\/O++ (thick grey) can then be used as a proxy for the optical depth of the nebula, that progresses from optically thin in the inner zone to optically thick in the outer zone, with a sharp cutoff at the Str\u00f6mgren radius.\nIn reality, the inner zone is often not, or only partially embedded in the optically thick outer envelope, so that a fraction of hydrogen-ionising photons are not absorbed and can escape into the ISM.\nStudying the spatial extent of the two zones through emission from their most abundant ions can provide us with information about the ionisation structure of an \\ion{H}{ii} region. \\citet{pellegrini12} exploited this dependency to develop the ionisation parameter mapping (IPM) method and applied it to study the optical depth of \\ion{H}{ii} regions in the Large and Small Magellanic Clouds (LMC and SMC) with narrow-band photometry in [\\ion{S}{ii}], [\\ion{O}{iii}] and H$\\alpha$. Regions were classified based on the fraction of the central high-ionisation region being surrounded by a low-ionisation envelope in an [\\ion{S}{ii}]\/[\\ion{O}{iii}] emission map. Regions completely embedded in the envelope are labelled as optically thick, regions with a low [\\ion{S}{ii}]\/[\\ion{O}{iii}] throughout as optically thin and those with an in between morphology as `blister nebulae'. \nEach morphological class was then assigned an escape fraction; we discuss the results of this approach in the light of our work in Sect.~\\ref{section:discussion}.\n\n\\par Figure~\\ref{fig:sii_oiii} shows a map of the [\\ion{S}{ii}]6716,31\/[\\ion{O}{iii}]4959,5007 ratio in our FoV. The map has been reddening corrected assuming an intrinsic H$\\alpha$\/H$\\beta$ ratio of 2.863, as described in Paper~\\textsc{i}. The data have been Voronoi binned to a S\/N = 20 in H$\\beta$. The numbered black contours in Fig.~\\ref{fig:sii_oiii} indicate the subset of the \\ion{H}{ii} regions - out of the full sample identified in Paper~\\textsc{i} and shown in Fig.~\\ref{fig:stellar_census} - that we study in detail in Sect.~\\ref{section:budget}.\nThe bar shows increasing ratios of [\\ion{S}{ii}]\/[\\ion{O}{iii}], from black to yellow, corresponding to an increasing optical depth of the medium (or, a decrease in its transparency to ionising photons). A ratio of 1 (purple) indicates where the abundance of S+ prevails over O++, as shown in the model in Fig.~\\ref{fig:cloudy_model}\\footnote{We caution however that Fig.~\\ref{fig:cloudy_model} shows the ratio of the ionic fractions, which does not compare directly to a line emissivity ratio.}.\n\n\\par We observe that the DIG has a lower ionisation state than the \\ion{H}{ii} regions (higher ratios of [\\ion{S}{ii}]\/[\\ion{O}{iii}]),\nas we also observed in Paper~\\textsc{i} by analysing the ratios of [\\ion{S}{ii}] and [\\ion{N}{ii}] to H$\\alpha$ and in agreement with measurements in nearby galaxies \\citep{haffner09}.\nIn about half of the regions we observe zones of optically thin gas extending until the regions edge, which could potentially indicate escape channels cleared by stellar feedback. In Sect.~\\ref{section:budget} we explore the link between the fraction of photons escaping the regions and the ionisation structure traced by [\\ion{S}{ii}]\/[\\ion{O}{iii}].\n\nIn Table~\\ref{table:obs_properties_regions} we summarise the visual appearance of the regions in [\\ion{S}{ii}]\/[\\ion{O}{iii}], with a similar approach as \\citet{pellegrini12}. Regions are classified\ninto a `classical' ionisation bounded nebula (`IB'), with an ionisation structure as the model in Fig.~\\ref{fig:cloudy_model}, or a nebula featuring one (or more) optically thin channels (`CH') cutting through the low ionisation zone and reaching the regions edge.\nWe base our classification on the value of [\\ion{S}{ii}]\/[\\ion{O}{iii}] along the region contour, as shown in Fig.~\\ref{fig:hii_class}. In the Figure, the solid points indicate the median value of the ratio along the contour, and the grey bars span the entire range of observed values. In order to determine which regions provide a potential escape path, we consider [\\ion{S}{ii}]\/[\\ion{O}{iii}] = 0.5 as fiducial limit for the transition to optically thick gas: regions having a ratio below this threshold along their contours are classified as CH; all other regions are classified as IB.\nWe test the effect of spatial binning on this classification by using both a finer and a coarser binning pattern, corresponding to S\/N = 15 and 25 in H$\\beta$. The classification remains unchanged, suggesting that spatial tessellation does not play a decisive role.\nWe stress that this classification does not take into account the geometry of the nebula and can only be conclusive when having a statistically significant sample of regions that allows to eliminate selection effects.\n\n\\begin{table*}\n\\centering\n\\caption{Spectral classification of WR stars.}\n\\begin{tabular}{ccccc}\n\\hline \\hline\nObj. ID & Ra (J2000) & Dec (J2000) & Features\/Reference & Classification \\\\ \\hline\nWR1 & 23:57:40.81 & -32:35:36.5 & (1) & WN2 \\\\\nWR2 & 23:57:41.43 & -32:35:35.9 & (1) & WC4 \\\\  \nWR3 & 23:57:45.74 &-32:34:37.4 & Nebular lines + \\ion{He}{ii} 4686 are weak, H$\\beta$ absorption & WN + O star \\\\\nWR4 & 23:57:44.75 &-32:34:25.1 & \\ion{N}{iii} 4640, \\ion{He}{ii} 4886 & WN (late-type) \\\\\nWR5 & 23:57:40.90 &-32:35:35.6 & \\ion{He}{ii} 6560 (strong and broad) & WC4\\\\\nWR6 & 23:57:40.53 & -32:35:33.1 & \\ion{N}{iii} 4640, \\ion{He}{ii} 4686 & WN (late-type) \\\\\nWR7 & 23:57:41.37 &-32:35:52.5 & \\ion{N}{v} 4603 + \\ion{N}{iii} 4640 (in equal strength) & WN (mid-type) \\\\\nWR8 & 23:57:43.27 & -32:35:49.6 & \\ion{He}{ii} 4686 (narrow), \\ion{N}{iii} 4640 & WN (late-type)\\\\\nWR9 & 23:57:40.94 &-32:34:47.3  & Analogue to WR8, slightly broader \\ion{He}{ii} & WN (late-type)\\\\\n\\hline\n\\end{tabular}\n\\tablebib{(1) \\citet{bibby10}}\n\\label{table:wr_stars}\n\\end{table*}\n\n\\section{Ionisation budget}\n\\label{section:budget}\n\nWe construct a ionisation budget for the entire FoV and inspect the \\ion{H}{ii} regions labelled in Fig.~\\ref{fig:sii_oiii} in detail.\nThe latter regions were selected to lie for their most part in the FoV, and not at the edge of the field.\nWe note that we consider here the outermost contours of the \\ion{H}{ii} regions sample, based on a flux brightness cut in H$\\alpha$ = $\\SI{6.7e-18}{\\erg \\per \\second \\per \\centi \\metre \\squared}$spaxel$^{-1}$ as defined in Paper~\\textsc{i}. We therefore consider the largest \\ion{H}{ii} region complexes (such as regions 1 and 8 in Fig.~\\ref{fig:sii_oiii}) in their entirety. This choice is made because dividing the regions within smaller subregions would imply arbitrary choices of their outermost border and division of their fluxes.\n\n\\par For each region, we model the expected $Q(H^0)$ from the stellar content as described below, and compare it to the observed ionising flux computed from the (reddening corrected) H$\\alpha$ luminosity, following the calibration of \\citet{kennicutt98}:\n$$ Q(H^0)_{obs} = 7.31 \\times 10^{11} L(H\\alpha) \\quad [\\mbox{erg s}^{-1}],$$\nbased on the assumption of case B recombination and an electron temperature $T_e \\sim \\num{10 000}$ K.\n\n\n\\subsection{O stars}\nDue to the selection method for the O stars catalogue (see Sect.~\\ref{section:data}), we lack information such as mass, age or spectral class of the O stars. In order to compute what $Q(H^0)$ is expected from the O stars population, we therefore have to assume a mass distribution. We consider a Salpeter mass function \\citep{salpeter55}:\n$$ p(m) = A m^{-\\alpha},$$\nwhere $p(m)$ denotes the probability of finding a star of mass $m$, and $\\alpha = 2.35$. We sample the distribution from $m_{min} = 20 M_\\odot$, the lowest stellar mass probed by the catalogue, to $m_{max} = 60 M_\\odot$ with 1000 Montecarlo realisations.\nThis upper limit corresponds to bright HII regions in the Milky Way \\citep{feigelson13} and is the highest stellar mass tabulated in the work of \\citet{martins05} (see below). We also obtain a lower limit for $Q(H^0)$ of field stars by sampling the Salpeter distribution up to a maximum mass  $m_{max} = 30~M_\\odot$, corresponding to the most massive stars observed in typical galactic star forming regions \\citep{bik10, gennaro12, bik12, ellerbroek13}. Moreover, stars of mass $M > 20~M_\\odot$ are generally expected to emit enough ionising radiation to be embedded in an \\ion{H}{ii} region, and if observed in the field are most likely runaway objects.\nFor each Montecarlo realisation, we compute the total number of O stars by taking into account the 30\\% contamination rate and $\\sim$ 70\\% completeness of the stellar catalogue (see Sect.~\\ref{section:data}):\n$$n_{Ostars,tot} = \\mathcal{N}(n_{Ostars}, 0.3),$$\nwhere $\\mathcal{N}$ denotes a normal distribution. Hereby, we also excluded O stars coinciding with the position of WR stars and YSCs (r $<$ 0.4'').\nWe then draw $n_{Ostars,tot}$ times a mass\n$$m_i = p(m < m') = \\int_{m_{min}}^{m'} p(m) dm = m_{min} \\left[C \\cdot \\mbox{unif}(0,1) + 1 \\right]^{1\/(1 - \\alpha)}$$\n$$ C = \\left( \\frac{m_{max}}{m_{min}} \\right)^{1 - \\alpha} - 1,$$\nwhere \\texttt{unif} denotes a uniform distribution. The resulting masses are then converted into $Q(H^0)$ values by interpolation of Table 1 from \\citet{martins05}, and the total flux of the region $Q(H^0)_{tot}$ is stored for each iteration.\nWe finally compute the best $Q(H^0)$ value and its uncertainty from, respectively, the median $m$ and $m \\pm 1 \\sigma$ of the resulting $Q(H^0)_{tot}$ distribution.\n\n\\subsection{YSC}\nWe determine $Q(H^0)$ for the YSC using the \\textsc{slug} code, as described in Sect.~\\ref{section:ysc_properties}.\nWe obtain a distribution of $Q(H^0)_{tot}$ from 1000 Montecarlo realisations: for each realisation, we sample one value from the PDF of each cluster in the region, and sum the resulting quantities. As for the O stars, we then compute the best $Q(H^0)$ value and its uncertainty from, respectively, the median $m$ and $m \\pm 1 \\sigma$ of the resulting $Q(H^0)_{tot}$ distribution.\nHereby, we also check that no YSCs are coincident with the position of WR stars, within a radius of 0.4''.\n\n\\subsection{WR stars}\nLastly, we compute the contribution of WR stars. We classify each star as Carbon-dominated (WC type) or \\ion{He}{ii}-dominated (WN-type), based on its spectral features as listed in Table~\\ref{table:wr_stars}. We then compute $Q(H^0)_{exp}$ from Tables 3 and 4 in \\citet{smith02}, assuming a temperature $T = \\num{60000}$ K for WN-type stars and $T = \\num{120000}$ K for WC-type stars. We compute the uncertainty on $Q(H^0)$ with 1000 Montecarlo realisations of $T$ in the range $[40 - 80] \\times 10^3$ K for WN-type stars and $[100 - 140] \\times 10^3$ K for WC-type. Also in this case, we consider the median $m$ and $m \\pm 1 \\sigma$ of the resulting $Q(H^0)_{tot}$ distribution.\n\n\\begin{table*}\n\\centering\n\\caption{Observed properties of the sub-sample of regions shown in Fig.~\\ref{fig:sii_oiii} and of the overall \\ion{H}{ii} regions and DIG population. We note that the number of O stars indicated here is not completeness-corrected. $t_{max}$ indicates the age of the oldest cluster in each region (fiducial value of 50 $\\pm$ 25 percentile of the age PDF; masking cluster with ages $> 10$ Myr). In the case of region 5, which is hosting exclusively O stars, we have assumed 3 Myr as an upper limit for the age. The indicated E(B-V) is the median extinction in the region (estimated from the Balmer decrement).\nThe total $L(H\\alpha)$ in each region is reddening corrected. Regions are classified as described in Sect.~\\ref{section:optical_depth} into `classical' ionisation bounded regions (IB) and regions featuring optically thin channels (CH).}\n\\begin{tabular}{lcccccccc}\n\\hline \\hline\nRegion ID &\n\\multicolumn{3}{c}{Stellar content} &\n$t_{max}$ &\n$\\log M_{GMC}$ & E(B-V) &\n$\\log L(H\\alpha$) & Classification \\\\\n & \\small{YSC} & \\small{O$\\star$} & \\small{WR} & [Myr] & $[M_\\odot]$ & & $[\\mbox{erg s}^{-1}]$\\\\\n\\hline\n\\textbf{1} & 24 & 111 & 6 & $5.7_{-1.8}^{+1.8}$ & 6.1 & 0.14 & 39.11 & CH\\\\[0.1cm]\n\\textbf{2} & 3  & 30  & 1 & $4.3_{-2.9}^{+2.6}$ & 6.0 & 0.13 & 38.68 & CH\\\\[0.1cm]\n\\textbf{3} & 4  & 26  & 1 & $3.9_{-2.5}^{+2.9}$ & 5.3 & 0.19 & 38.73 & CH\\\\[0.1cm]\n\\textbf{4} & 1  & 19  & 0 & $5.7_{-1.4}^{+1.2}$ & 4.5 & 0.12 & 38.25 & IB\\\\[0.1cm]\n\\textbf{5} & 0  & 5   & 0 & $\\leq 3.0$ & 3.4 & 0.14 & 37.35 & IB\\\\[0.1cm]\n\\textbf{6} & 1  & 2   & 0 & $1.7_{-1.2}^{+2.2}$ & --- & 0.09 & 37.41 & CH\\\\[0.1cm]\n\\textbf{7} & 1  & 2   & 0 & $3.2_{-2.1}^{+3.6}$ & --- & 0.11 & 37.01 & IB\\\\[0.1cm]\n\\textbf{8} & 8  & 110 & 1 & $4.3_{-2.6}^{+2.6}$ & 6.3 & 0.17 & 38.89 & CH\\\\[0.1cm]\n\\textbf{Tot \\ion{H}{ii}} & 47 & 404 & 8 & & & & \\\\[0.1cm]\n\\textbf{Tot DIG} & 49 & 271 & 1 & & & &\\\\\n\\hline\n\\end{tabular}\n\\label{table:obs_properties_regions}\n\\end{table*}\n\n\n\\begin{table*}\n\\centering\n\\caption{Ionisation budget of the regions shown in Fig.~\\ref{fig:sii_oiii} and of the overall \\ion{H}{ii} regions and DIG population. The last row indicates a lower limit for the modelled ionising flux in the DIG, obtained by considering an upper limit $m_{max} = 30 M_\\odot$ for the field candidate O stars (as opposed to $m_{max} = 60 M_\\odot$ assumed elsewhere). All the uncertainties indicated are $\\pm 1 \\sigma$ errors on the $Q(H^0)$ distribution resulting from the Montecarlo sampling. The uncertainties on $\\log Q^0_{obs}$ are all of order $\\leq 0.001$.}\n\\begin{tabular}{lccccccc}\n\\hline \\hline\nRegion ID &\n$\\log Q^0_{exp, Ostars}$ &\n$\\log Q^0_{exp, YSC}$ &\n$\\log Q^0_{exp, WR}$ &\n$\\log Q^0_{exp, tot}$ &\n$\\log Q^0_{obs}$&\n$Q^0_{exp, tot}\/Q^0_{obs}$ &\n$f_{esc}$\n\\\\\n& [s$^{-1}$] & [s$^{-1}$]& [s$^{-1}$]& [s$^{-1}$] & [s$^{-1}$] & & \\\\\n\\hline\n\n\\textbf{1}\n& $51.11_{-0.14}^{+0.14}$\n& $49.60_{-0.34}^{+1.50}$\n& $50.06_{-0.02}^{+0.02}$\n& $51.16_{-0.13}^{+0.17}$\n& $50.74$\n& $2.71_{-0.78}^{+0.84}$\n& $0.63_{-0.15}^{+0.09}$  \\\\[0.2cm]\n\n\\textbf{2}\n& $50.54_{-0.13}^{+0.16}$\n& $48.54_{-0.42}^{+1.59}$\n& $49.40_{-0.09}^{+0.00}$\n& $50.58_{-0.13}^{+0.16}$\n& $50.31$\n& $1.83_{-0.49}^{+0.65}$\n& $0.45_{-0.20}^{+0.14}$  \\\\[0.2cm]\n\n\\textbf{3}\n& $50.48_{-0.15}^{+0.17}$\n& $48.54_{-0.43}^{+1.59}$\n& $49.40_{-0.09}^{+0.00}$\n& $50.52_{-0.15}^{+0.17}$\n& $50.35$\n& $1.50_{-0.47}^{+0.52}$\n& $0.33_{-0.30}^{+0.17}$  \\\\[0.2cm]\n\n\\textbf{4}\n& $50.34_{-0.15}^{+0.18}$\n& $48.53_{-0.43}^{+0.59}$\n& ---\n& $50.34_{-0.15}^{+0.19}$\n& $49.98$\n& $2.31_{-0.78}^{+0.94}$\n& $0.57_{-0.22}^{+0.13}$  \\\\[0.2cm]\n\n\\textbf{5}\n& $49.70_{-0.25}^{+0.30}$\n& ---\n& ---\n& $49.70_{-0.25}^{+0.30}$\n& $49.10$\n& $3.94_{-2.30}^{+2.74}$\n& $0.75_{-0.36}^{+0.10}$  \\\\[0.2cm]\n\n\\textbf{6}\n& $49.07_{-0.35}^{+0.94}$\n& $48.53_{-0.43}^{+0.59}$\n& ---\n& $49.18_{-0.37}^{+0.86}$\n& $49.14$\n& $1.24_{-0.86}^{+1.81}$\n& $0.20_{-1.83}^{+0.48}$  \\\\[0.2cm]\n\n\\textbf{7}\n& $49.09_{-0.34}^{+0.86}$\n& $48.53_{-0.43}^{+0.59}$\n& ---\n& $49.19_{-0.36}^{+0.80}$\n& $48.75$\n& $3.17_{-2.15}^{+4.09}$\n& $0.68_{-0.67}^{+0.18}$  \\\\[0.2cm]\n\n\\textbf{8}\n& $51.10_{-0.13}^{+0.14}$\n& $48.85_{-0.42}^{+2.04}$\n& $49.39_{-0.08}^{+0.01}$\n& $51.11_{-0.13}^{+0.15}$\n& $50.53$\n& $3.85_{-1.15}^{+1.19}$\n& $0.74_{-0.11}^{+0.06}$  \\\\[0.2cm]\n\n\\textbf{Tot \\ion{H}{ii}}\n& $51.68_{-0.13}^{+0.13}$\n& $49.90_{-0.33}^{+1.04}$\n& $50.23_{-0.02}^{+0.02}$\n& $51.70_{-0.13}^{+0.14}$\n& $51.22$\n& $3.06_{-0.84}^{+0.93}$\n& $0.67_{-0.12}^{+0.08}$  \\\\[0.2cm]\n\n\\textbf{Tot DIG}\n& $51.50_{-0.12}^{+0.13}$\n& $49.95_{-0.35}^{+0.91}$\n& $49.40_{-0.09}^{+0.00}$\n& $51.52_{-0.13}^{+0.15}$\n& $50.65$\n& $7.48_{-2.02}^{+2.39}$\n& $0.87_{-0.05}^{+0.03}$\\\\[0.2cm]\n\n\\textbf{Tot DIG}$_{lower \\thinspace limit}$\n& $50.94_{-0.13}^{+0.13}$\n&\n&\n& $50.99_{-0.15}^{+0.20}$\n&\n& $2.30_{-0.64}^{+0.75}$\n& $0.57_{-0.17}^{+0.11}$\\\\\n\\hline\n\\end{tabular}\n\\label{table:photo_budget}\n\\end{table*}\n\n\n\n\\subsection{Resulting budget}\n\\label{section:budget_results}\nTable~\\ref{table:obs_properties_regions} and \\ref{table:photo_budget} summarise, respectively, the observed properties and the photoionisation budget for the entire FoV\nand in detail for the eight regions labelled in Fig.~\\ref{fig:sii_oiii}.\nIn Table~\\ref{table:obs_properties_regions} we list the stellar content, the age of the oldest star or cluster in each region, the total GMC mass, median $E(B - V)$ (estimated from the Balmer decrement) and H$\\alpha$ luminosity (reddening corrected), as well as the morphological classification derived in Sect.~\\ref{section:optical_depth}.\nIn Table~\\ref{table:photo_budget} we indicate the total flux $Q(H^0)_{exp}$ modelled from the stellar content, as well as the relative contribution from O stars, YSC, and WR stars, and the observed ionising photon flux $Q(H^0)_{obs}$ derived from the reddening corrected H$\\alpha$ luminosity. We also list the ratio of expected to observed flux and the corresponding escape fraction $f_{esc} = 1 - Q(H^0)_{obs}\/Q(H^0)_{exp}$.\nWe note that the extremely large uncertainties on regions 5 and 7 are driven by the fact that these regions host (almost) exclusively O stars, for which $Q(H^0)$ is more loosely constrained. We also note that in region 6 the uncertainty ranges to unphysical values $f_{esc} < 0$, indicating that either the models are underestimating the photon flux, or the observed luminosity is being overestimated, possibly due to the reddening correction or to the exact location of the region boundaries.\n \n\\par From Table~\\ref{table:photo_budget}, we see that WR and O stars dominate the contribution to the $Q^0_{exp, tot}$ value of the regions. Overall, we find an $f_{esc, \\ion{H}{ii}} = 0.67_{-0.12}^{+0.08}$ for the entire population of \\ion{H}{ii} regions (black contours in Fig.~\\ref{fig:stellar_census}). We also observe that the stellar population in the DIG produces a more than sufficient amount of ionising photons ($Q(H^0)_{exp} > Q(H^0)_{obs}$), and that the DIG is therefore consistent with being self-ionised, with $f_{esc, DIG} = 0.87_{-0.05}^{+0.03}$. This holds also if considering a maximum mass of $30~M_\\odot$ for field O stars (last row in Table~\\ref{table:photo_budget}), in which case we find $f_{esc, DIG} = 0.57_{-0.17}^{+0.11}$.\nIn our FoV, we observe hence that the sources of ionising photons produce a photon flux that is more than sufficient to explain the emission of the ionised ISM, both within and outside the \\ion{H}{ii} regions.\n\n\\subsection{Escape fraction from individual \\ion{H}{ii} regions}\n\\label{section:fesc_individual}\n\nIn this section, we focus in more detail on individual \\ion{H}{ii} regions. In order to better visualise the link between $f_{esc}$ and the visual appearance of the regions, in Fig.~\\ref{fig:fesc_vs_appearance} we plot the ratio of $Q(H^0)_{exp}\/Q(H^0)_{obs}$, and label each region according to the morphological criteria described in Sect.~\\ref{section:optical_depth} (IB: ionisation bounded, CH: optically thin channels). We do not see any clear trend with visual appearance; we discuss this result in the light of other studies in Sect.~\\ref{section:discussion}.\n\n\\par In Fig.~\\ref{fig:fesc_vs_agerange}, we furthermore investigate the dependence of the $Q(H^0)_{exp}\/Q(H^0)_{obs}$ ratio on the age of the oldest cluster in each region.\nWe consider the latter as a proxy for the age range spanned by the stellar population inside each \\ion{H}{ii} region, assuming that all regions are currently forming stars.\nWe remark that this approximation might not be accurate for regions 6 and 7, in which no GMCs are detected with ALMA and the extinction traced by the ionised gas is very low (see Table~\\ref{table:obs_properties_regions}); however, both regions are still hosting young candidate O stars.\nWhen determining the age of the oldest cluster, we use as best estimate the median of each cluster PDF, and we exclude clusters with $t > 10$ Myr, as such objects are not anymore associated with \\ion{H}{ii} regions, and are most likely line-of-sight objects. For region 5, which is hosting exclusively O stars, we assume an upper limit of 3 Myr for the age. \nWe also colour-code the regions according to their median $E(B - V)$ (left panel) and total GMC mass (right panel), to indicate which regions are more affected by reddening and are still actively forming stars.\nDust can indeed absorb part of the LyC photons before they have the possibility to ionise hydrogen, and re-emit them at longer wavelengths, mimicking a larger $f_{esc}$.\nWe see that for regions 3 and 8, this could in part explain the $f_{esc} > 0$ observed, whereas - given their low dust and GMC content - regions 4 and 5 could be a true leaking region. We comment more on the effect of photons removal by dust in Sect.~\\ref{section:discussion}. In our small sample of regions and due to the large uncertainties, we do not find any trend between $f_{esc}$ and the age of the stellar population in the region. This is discussed\nin more detail in Sect.~\\ref{section:discussion}.\n\n\\begin{figure*\n\\includegraphics[width=8cm]{fesc_tmax_ebv.pdf}\n\\includegraphics[width=8cm]{fesc_tmax_gmc.pdf}\n\\caption{Ratio of expected to observed ionising photon flux $Q(H^0)$ for the regions labelled in Fig.~\\ref{fig:sii_oiii}, versus the age of the oldest cluster in each region (best-value estimate obtained from the median of the age PDF; masking cluster with ages $> 10$ Myr). For region 5, which is hosting exclusively O stars, we assume an upper limit of 3 Myr for the age. The dashed horizontal line indicates an $f_{esc} = 0$.\n\\textit{Left panel}: the points are colour-coded according to the median $E(B - V)$ in each region (Balmer decrement estimate). \\textit{Right panel}: the regions are colour-coded according to their total GMC mass. Empty black circles indicate regions hosting no GMCs.}\n\\label{fig:fesc_vs_agerange}\n\\end{figure*}\n\n\\section{Discussion}\n\\label{section:discussion}\nIn Paper~\\textsc{i}, we found a DIG fraction $f_{DIG, obs} = 0.15$ from the H$\\alpha$ luminosity in our FoV\\footnote{Revised estimate for the H$\\alpha$-selected \\ion{H}{ii} regions sample and corrected for extinction.}.\nThis is in good agreement with the fraction of diffuse gas expected by modelling the stellar and cluster population:\n$$f_{DIG, exp} = \\frac{Q^0_{exp,DIG} }{ Q^0_{exp,DIG}  + Q^0_{exp,HII}} \\gtrsim 0.17.$$\nMoreover, we assessed from our BPT diagram analysis in paper~\\textsc{i} that the bulk of the DIG emission in our FoV was consistent with being photoionised. Here we confirm that the DIG is consistent with being self-ionised by field stars and clusters, with an overabundance of ionising photons $f_{esc} \\sim 0.87_{-0.05}^{+0.03}$; we confirm this trend also when considering our lower-limit estimate.\nThis result is perhaps surprising considering the fact that we observe in Fig.~\\ref{fig:sii_oiii} that the DIG is not optically thin to Lyman continuum emission (high [\\ion{S}{ii}]\/[\\ion{O}{iii}] ratios). However, if the DIG has a clumpy structure, the emission line ratio could be tracing the denser, optically thick clumps; the apparent lack of H$\\alpha$ emission can originate from photons escaping through the dilute, optically thin interclump medium, which through its low density will not contribute significantly to the observed emission line intensities.\nWe also observe that overall the \\ion{H}{ii} regions are leaking ionising photons at a rate $f_{esc} \\sim 0.67_{-0.12}^{+0.08}$, and that seven out of eight have an $f_{esc} \\gtrsim 0.3$.\n\n\\par We note that the uncertainties on $f_{esc}$ are rather large, as we try to account for a variety of effects, including: the contamination rate $\\sim 30\\%$ in the O stars catalogue, the large range in $Q(H^0)$ obtained by Montecarlo sampling O stars of different mass,\nthe range in values spanned by the PDFs of the clusters and the assumption on the temperature of the WR stars.\nThe large uncertainties recovered are consistent with the work of \\citet{niederhofer16}: by varying parameters of synthetic YSCs, the authors conclude that escape fractions derived from broadband photometry data are typically dominated by uncertainties in the spectral types of the stars.\nHowever, even when considering the uncertainties, we find $f_{esc} > 0$ for the DIG overall and for six out of the eight \\ion{H}{ii} regions inspected  (see Table~\\ref{table:photo_budget} and Fig.~\\ref{fig:fesc_vs_appearance}). \n\n\\par An additional factor that is not taken into account in the computation of $f_{esc}$ is the effect of absorption of LyC photons by dust within the \\ion{H}{ii} regions before these photons have the chance of ionising hydrogen atoms. In Sect.~\\ref{section:extinction}, we have compared the extinction map derived from the H$\\alpha$\/H$\\beta$ ratio with 24~$\\mu$m emission from Spitzer\/MIPS (Fig.~\\ref{fig:extinction_map}, right panel), and found a good correspondence, indicative that dust and gas are well mixed \\citep[see e.g.][]{choi20}. The resolution of mid-IR data is currently insufficient to confirm potential dust absorption in proximity of the sources of Lyman continuum photons at the distance of our target, and will only become possible with new generation IR telescopes such as JWST. However, in local \\ion{H}{ii} regions, it is typically observed that a large fraction of LyC photons contributes to hydrogen ionisation. This fraction anti-correlates with metallicity \\citep[e.g.][]{inoue01}, and at solar metallicity, an increasing fraction of Lyman continuum photons are absorbed by dust, instead. In the metallicity range spanned by our target (12 + log(O\/H) $\\sim  8.25 - 8.5$, see Sect.~\\ref{section:metallicity}), \\citet{inoue01} estimate that $\\gtrsim 80\\%$ Lyman continuum photons are absorbed by neutral hydrogen. Therefore, we expect that the effect of dust absorption in the \\ion{H}{ii} regions in our FoV can reduce $Q(H^0)_{obs}$ of up to $\\sim$ 20\\%, which lies well within the given uncertainties.\n\n\\par Other studies targeting \\ion{H}{ii} regions in nearby galaxies at high resolution found similar escape rates.\n\\citet{doran13} have estimated an $f_{esc} > 0.5$ in the 30 Dor star forming region in the LMC, based on a complete spectroscopic census of hot, luminous stars.\nIn a sample of \\ion{H}{ii} regions in the LMC and SMC studied with narrowband photometry, \\citet{pellegrini12} have found $f_{esc} > 0.4$ for all the regions. \\citet{mcleod19} have studied in more detail two of the largest \\ion{H}{ii} region complexes in the LMC with MUSE, finding $f_{esc} > 0.2$ for the complexes and the respective sub-regions.\nFinally, in two regions recently observed with MUSE and HST in the nearby dwarf galaxy NGC 300, \\citet{mcleod20} have estimated an $f_{esc} \\gtrsim 0.3$ .\n\nFollowing the non-negligible escape of Lyman continuum photons from \\ion{H}{ii} regions, several studies of local and nearby galaxies have proven the DIG consistent with being photoionised. For example, \\citet{pellegrini12} have estimated that both in the LMC and SMC the total galactic escape fraction would be sufficient to account for the observed $f_{DIG}$, and that, when considering the additional contribution of field stars, the galaxies could leak a substantial amount of ionising radiation into the circumgalactic medium. Similarly, in a recent MUSE+HST study of the Antennae merger system, \\citet{weilbacher18} have estimated a sufficient fraction of Lyman-continuum leakage in order to explain the amount of DIG observed.\n\n\\par We do not find any correlation between ionising photon leakage in the HII regions and their visual appearance (`classic' ionisation bound regions vs regions featuring optically thin channels).\nThis is somewhat in contrast with the results of \\citet{pellegrini12} that, by comparing the fiducial $f_{esc}$ assigned to \\ion{H}{ii} regions in the LMC and SMC based on their appearance in [\\ion{S}{ii}]\/[\\ion{O}{iii}] (see Sect.~\\ref{section:optical_depth}) with previous $f_{esc}$ estimated based on the stellar population, found a good agreement overall. On the other hand, a systematic trend is not apparent from the MUSE results of \\citet{mcleod19} on the LMC, that used the [\\ion{O}{ii}]7320,7330\/[\\ion{O}{iii}]4959,5007\nratio as optical depth tracer. We interpret this apparent tension as likely being a simple effect of 3D geometry in number limited samples.\n\\par In our small sample of regions, we do not find evidence for a correlation between age range of the stellar population and $f_{esc}$. Increasing the sample size will be essential to investigate any such correlation, that could be expected in light of the results of galactic and cosmological simulations.\n\\citet{kim13a} simulated ionising radiation and supernova feedback in a low-redshift galactic disk, at a high spatial resolution $\\sim 4$ pc. The study indicated that photons emitted at later ages are more likely to escape the star forming region, as the gas surrounding the region is dispersed by ionising radiation and supernova feedback. \nA similar result was obtained in the recent high-resolution cosmological simulations of \\citet{ma20}, that studied $\\sim$ 30 zoom-in simulations of galaxies at $z \\gtrsim 5$.\nThe simulations suggest that ionising photons are preferably leaking from star forming regions containing a kpc-size superbubble, likely created by clustered SNe set off by stars of age $>$ 3 Myr. As the bubble expands, new stars are formed at its edge, inside a dense shell of compressed gas. The shell keeps expanding while forming stars and, as a consequence, the young stars formed in it end up inside the superbubble and are able to fully ionise channels of low-column density  pre-cleared by feedback from the previous population of stars. Therefore, regions hosting stars spanning a large range in age seem to be advantaged in leaking photons. Also observations have already pointed to the importance of the local star formation history in shaping the ionisation conditions in \\ion{H}{ii} regions. By analysing a sample of $\\sim$ 5000 \\ion{H}{ii} regions from CALIFA \\citep[Calar Alto Large Integral Field Area survey,][]{sanchez12}, \\citet{sanchez15} found for example a correlation between the position occupied by the regions in BPT diagrams and the age and metallicity of their stellar population.\n\n\\par Finally, we comment on the impact of using different stellar population model in our \\texttt{cluster\\_slug} analysis on the cluster physical properties. The use of non rotating and single star models produces a lower limit to the total estimated $Q(H^0)$ and therefore to $f_{esc}$, in particular at ages larger than 3-4 Myr. Binaries and rotating stars have the same effect, that is they increase the production of ionising photons at ages older than 3 Myr with respect to predictions from non-rotating single star models \\citep[see][]{leitherer14, gotberg19}.\nAny existing trend between $f_{esc}$ and age range of the stellar population would therefore be reinforced by different model assumptions, as also reported in the simulations of \\citet{ma20}.\n\n\\section{Conclusions}\n\\label{section:conclusions}\nWe have studied the ionised gas in the nearby galaxy NGC 7793 with MUSE, and complemented our observations with HST and ALMA data tracing the stellar content and the molecular gas.\n\n\\par We have constructed a census of YSCs, O stars, and WR stars in the MUSE FoV, and studied the properties of the stellar population. We modelled the age, mass, and ionising flux of YSCs with the stochastic stellar population synthesis code \\textsc{slug}, finding that clusters located in \\ion{H}{ii} regions have a higher probability to be younger, less massive, and to emit a higher number of ionising photons than clusters in the field.\n\n\\par We have investigated the link between the stellar population and the dense and ionised gas. We have contrasted the reddening map derived from individual stellar extinctions with the one constructed from the H$\\alpha$\/H$\\beta$ ratio and found that they compare well, but that the latter correlates better with the position of GMCs traced by ALMA. We have estimated an oxygen abundance for the \\ion{H}{ii} regions from the MUSE data, using the $S$-strong line method from \\citet{pilyugin16}. We found a median abundance of $12 + \\log(O\/H) \\sim 8.37$ with a scatter of 0.25 dex, in agreement with the previous estimate by \\citet{pilyugin14}. The abundance map appears to be rich in substructures, especially surrounding YSCs and WR stars. We caution however against possible degeneracies with for example the temperature and density, and we do indeed observe a correlation with the ionisation state of the regions.\n\n\\par We have studied the ionisation structure of the \\ion{H}{ii} regions using the [\\ion{S}{ii}]6716,31\/[\\ion{O}{iii}]4959,5007 ratio as a proxy for the optical depth. We have focused on a subset of regions and classified them based on the value of [\\ion{S}{ii}]\/[\\ion{O}{iii}] along their border into ionisation bounded or as featuring channels of optically thin gas. Finally, we have compiled a photoionisation budget for the entire FoV and for the subset of \\ion{H}{ii} regions.\n\n\\par Overall, we find an escape fraction $f_{esc} = 0.67_{-0.12}^{+0.08}$ for the population of \\ion{H}{ii} regions, and that the DIG in our FoV is more than consistent with being self-ionised, with an $f_{esc} = 0.87_{-0.05}^{+0.03}$. \nThis holds even when considering a lower-limit estimate for the DIG flux, derived by assuming a maximum mass of $30~M_\\odot$ for the field O stars. We furthermore find that the $f_{DIG, exp} \\gtrsim 0.17$ obtained by modelling the DIG stellar population is in good agreement with the DIG fraction derived from the observed H$\\alpha$ luminosity in Paper~\\textsc{i}, $f_{DIG, obs} = 0.15$. \n\n\\par We observe an $f_{esc} \\gtrsim 0.3$ in seven out of the eight studied regions, and investigate how the $f_{esc}$ is linked to the regions properties. Among others, we do not find any trend between $f_{esc}$ and the visual appearance of the regions; we read this as the effect of 3D geometry in our number limited sample.\n\n\\begin{acknowledgements}\nWe would like to thank the anonymous referee for the constructive feedback on the second draft of the manuscript.\nThis work is based on observations collected at the European Southern Observatory under ESO programme 60.A-9188(A).\nA.A. acknowledges the support of the Swedish Research Council, Vetenskapsr\\aa{}det, and the Swedish National Space Agency (SNSA).\nThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 757535).\nThis research made use of Astropy\\footnote{http:\/\/www.astropy.org}, a community-developed core Python package for Astronomy \\citep{astropy:2013, astropy:2018}.\n\\end{acknowledgements}\n\\bibpunct{(}{)}{;}{a}{}{,}\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n Currently,\n direct Dark Matter detection experiments\n searching for Weakly Interacting Massive Particles (WIMPs)\n are one of the promising methods\n for understanding the nature of Dark Matter (DM)\n and identifying them among new particles produced at colliders\n as well as\n studying the (sub)structure of our Galactic halo\n \\cite{SUSYDM96,\n       Drees12, Saab12, Baudis12c}.\n\n In our earlier work\n \\cite{DMDDf1v},\n we developed methods\n for reconstructing the (moments of the)\n time--averaged one--dimensional velocity distribution of halo WIMPs\n by using\n the measured recoil energies directly.\n This analysis requires\n no prior knowledge about\n the WIMP density near the Earth\n nor about their scattering cross section on nucleus,\n the unique required information\n is the mass of incident WIMPs.\n We therefore turned to develop the method for\n determining the WIMP mass model--independently\n by combining two experimental data sets\n with two different target nuclei\n \\cite{DMDDmchi}.\n By combining these methods\n and using two or three experimental data sets\n with different detector materials,\n one could reconstruct the one--dimensional velocity distribution\n of Galactic WIMPs directly.\n However,\n as presented in Ref.~\\cite{DMDDf1v},\n with a few hundreds or even thousands recorded WIMP events,\n only estimates of the reconstructed velocity distribution\n with pretty large statistical uncertainties\n at a few ($<$ 10) points\n could be obtained.\n\n Therefore,\n in order to offer more detailed information\n about the Galactic WIMP velocity distribution,\n we introduce in this paper the Bayesian analysis\n into our model--independent reconstruction procedure\n developed in Ref.~\\cite{DMDDf1v}\n to be able to determine,\n e.g.~the position of the peak of\n the one--dimensional velocity distribution function\n and the concrete values of\n the characteristic Solar and Earth's Galactic velocities.\n\n The remainder of this paper is organized as follows.\n In Sec.~2,\n we first review the model--independent method\n for reconstructing the time--averaged\n one--dimensional velocity distribution of halo WIMPs\n by using data from direct DM detection experiments directly.\n Then\n we introduce the Bayesian analysis\n and give the basic formulae\n needed in the extended reconstruction process.\n In Sec.~3,\n we present numerical results of\n the reconstructed WIMP velocity distribution functions\n based on Monte--Carlo simulations\n for different generating\n and fitting velocity distributions.\n Different input WIMP masses\n as well as\n impure (pseudo--)data sets\n mixed with (artificially added) unrejected background events\n will also be considered.\n We conclude in Sec.~4.\n Some technical details for our analysis\n will be given in Appendix.\n\\section{Formalism}\n In this section,\n we develop the formulae\n needed for our Bayesian reconstruction of\n the one--dimensional velocity distribution function of halo WIMPs $f_1(v)$\n by using direct Dark Matter detection data directly.\n\n We first review the model--independent method\n for reconstructing the time--averaged\n WIMP velocity distribution\n by using experimental data,\n i.e.~measured recoil energies,\n directly from direct detection experiments.\n These ``reconstructed data''\n (with estimated statistical uncertainties)\n will be used as input information\n for the further Bayesian analysis.\n Then,\n in the second part of this section,\n we review the basic concept of the Bayesian analysis\n and give the formulae\n needed in our extended reconstruction procedure.\n\n\\subsection{Model--independent reconstruction of\n            one--dimensional WIMP velocity distribution}\n In this subsection,\n we review briefly the method for reconstructing\n the one--dimensional WIMP velocity distribution\n from experimental data directly.\n Detailed derivations and discussions\n can be found in Ref.~\\cite{DMDDf1v}.\n\n\\subsubsection{From the recoil spectrum}\n The basic expression for the differential event rate\n for elastic WIMP--nucleus scattering is given by \\cite{SUSYDM96}:\n\\beq\n   \\dRdQ\n = \\calA \\FQ \\int_{\\vmin}^{\\vmax} \\bfrac{f_1(v)}{v} dv\n\\~.\n\\label{eqn:dRdQ}\n\\eeq\n Here $R$ is the direct detection event rate,\n i.e.~the number of events\n per unit time and unit mass of detector material,\n $Q$ is the energy deposited in the detector,\n $F(Q)$ is the elastic nuclear form factor,\n $f_1(v)$ is the one--dimensional velocity distribution function\n of the WIMPs impinging on the detector,\n $v$ is the absolute value of the WIMP velocity\n in the laboratory frame.\n The constant coefficient $\\calA$ is defined as\n\\beq\n        \\calA\n \\equiv \\frac{\\rho_0 \\sigma_0}{2 \\mchi \\mrN^2}\n\\~,\n\\label{eqn:calA}\n\\eeq\n where $\\rho_0$ is the WIMP density near the Earth\n and $\\sigma_0$ is the total cross section\n ignoring the form factor suppression.\n The reduced mass $\\mrN$ is defined by\n\\beq\n        \\mrN\n \\equiv \\frac{\\mchi \\mN}{\\mchi + \\mN}\n\\~,\n\\label{eqn:mrN}\n\\eeq\n where $\\mchi$ is the WIMP mass and\n $\\mN$ that of the target nucleus.\n Finally,\n $\\vmin$ is the minimal incoming velocity of incident WIMPs\n that can deposit the energy $Q$ in the detector:\n\\beq\n   \\vmin\n = \\alpha \\sqrt{Q}\n\\label{eqn:vmin}\n\\eeq\n with the transformation constant\n\\beq\n        \\alpha\n \\equiv \\sfrac{\\mN}{2 \\mrN^2}\n\\~,\n\\label{eqn:alpha}\n\\eeq\n and $\\vmax$ is the maximal WIMP velocity\n in the Earth's reference frame,\n which is related to\n the escape velocity from our Galaxy\n at the position of the Solar system,\n $\\vesc~\\gsim~600$ km\/s.\n\n In our earlier work \\cite{DMDDf1v},\n it was found that,\n by using a time--averaged recoil spectrum $dR \/ dQ$\n and assuming that no directional information exists,\n the normalized one--dimensional velocity distribution function\n of incident WIMPs, $f_1(v)$, can be solved\n from Eq.~(\\ref{eqn:dRdQ}) directly as\n\\beq\n   f_1(v)\n = \\calN\n   \\cbrac{ -2 Q \\cdot \\dd{Q} \\bbrac{ \\frac{1}{\\FQ} \\aDd{R}{Q} } }\\Qva\n\\~,\n\\label{eqn:f1v_dRdQ}\n\\eeq \n where the normalization constant $\\calN$ is given by\n\\beq\n   \\calN\n = \\frac{2}{\\alpha}\n   \\cbrac{\\intz \\frac{1}{\\sqrt{Q}}\n                \\bbrac{ \\frac{1}{\\FQ} \\aDd{R}{Q} } dQ}^{-1}\n\\~.\n\\label{eqn:calN_int}\n\\eeq\n Here the integral\n goes over the entire physically allowed range of recoil energies:\n starting at $Q = 0$,\n and the upper limit of the integral has been written as $\\infty$.\n Note that,\n because $f_1(v)$ in Eq.~(\\ref{eqn:f1v_dRdQ})\n is the normalized velocity distribution,\n the normalization constant $\\cal N$ here is independent of\n the constant coefficient $\\cal A$\n defined in Eq.~(\\ref{eqn:calA}).\n Hence,\n as the most important consequence,\n the velocity distribution function of halo WIMPs\n reconstructed by Eq.~(\\ref{eqn:f1v_dRdQ})\n is independent of the local WIMP density $\\rho_0$\n as well as\n of the WIMP--nucleus cross section $\\sigma_0$.\n However,\n not only the overall normalization constant $\\calN$\n given in Eq.~(\\ref{eqn:calN_int}),\n but also the shape of the velocity distribution,\n through the transformation $Q = v^2 \/ \\alpha^2$\n in Eq.~(\\ref{eqn:f1v_dRdQ}),\n depends on the WIMP mass $\\mchi$\n (involved in the coefficient $\\alpha$\n  defined in Eq.~(\\ref{eqn:alpha})).\n\n\\subsubsection{From experimental data directly}\n In order to use the expressions\n (\\ref{eqn:f1v_dRdQ}) and (\\ref{eqn:calN_int})\n for reconstructing $f_1(v)$,\n one needs a functional form for the recoil spectrum $dR \/ dQ$.\n In practice\n this requires usually a fit to experimental data.\n However,\n data fitting will re--introduce some model dependence\n and make the error analysis more complicated.\n Hence,\n expressions that allow to reconstruct $f_1(v)$\n directly from data\n (i.e.~measured recoil energies)\n have also been developed \\cite{DMDDf1v}.\n We started by considering experimental data described by\n\\beq\n     {\\T Q_n - \\frac{b_n}{2}}\n \\le \\Qni\n \\le {\\T Q_n + \\frac{b_n}{2}}\n\\~,\n     ~~~~~~~~~~~~\n     i\n =   1,~2,~\\cdots,~N_n,~\n     n\n =   1,~2,~\\cdots,~B.\n\\label{eqn:Qni}\n\\eeq\n Here the entire experimental possible\n energy range between the minimal and maximal cut--offs\n $\\Qmin$ and $\\Qmax$\n has been divided into $B$ bins\n with central points $Q_n$ and widths $b_n$.\n In each bin,\n $N_n$ events will be recorded.\n\n As argued in Ref.~\\cite{DMDDf1v},\n the statistical uncertainty on\n the ``slope of the recoil spectrum'',\n $\\bbrac{d\/dQ \\~ (dR \/ dQ)}_{Q = Q_n}$,\n appearing in the expression (\\ref{eqn:f1v_dRdQ}),\n scales like the bin width to the power $-1.5$.\n In addition,\n the wider the bin width,\n the more the recorded events in this bin,\n and thus the smaller the statistical uncertainty\n on the estimator of $\\bbrac{d\/dQ \\~ (dR \/ dQ)}_{Q = Q_n}$.\n Hence,\n since the recoil spectrum $dR \/ dQ$ is expected\n to be approximately exponential \\cite{DMDDf1v},\n in order to approximate the spectrum\n in a rather wider range,\n instead of the conventional standard linear approximation,\n the following exponential ansatz\n for the measured recoil spectrum\n (before normalized by the exposure $\\calE$)\n in the $n$th bin has been introduced \\cite{DMDDf1v}:\n\\beq\n        \\adRdQ_{{\\rm expt}, \\~ n}\n \\equiv \\adRdQ_{{\\rm expt}, \\~ Q \\simeq Q_n}\n \\equiv \\rn  \\~ e^{k_n (Q - Q_{s, n})}\n\\~.\n\\label{eqn:dRdQn}\n\\eeq\n Here $r_n$ is the standard estimator\n for $(dR \/ dQ)_{\\rm expt}$ at $Q = Q_n$:\n\\beq\n   r_n\n = \\frac{N_n}{b_n}\n\\~,\n\\label{eqn:rn}\n\\eeq\n $k_n$ is the logarithmic slope of\n the recoil spectrum in the $n$th $Q-$bin,\n which can be computed numerically\n from the average value of the measured recoil energies\n in this bin:\n\\beq\n   \\bQn\n = \\afrac{b_n}{2} \\coth\\afrac{k_n b_n}{2}-\\frac{1}{k_n}\n\\~,\n\\label{eqn:bQn}\n\\eeq\n where\n\\beq\n        \\bQxn{\\lambda}\n \\equiv \\frac{1}{N_n} \\sumiNn \\abrac{\\Qni - Q_n}^{\\lambda}\n\\~.\n\\label{eqn:bQn_lambda}\n\\eeq\n Then the shifted point $Q_{s, n}$\n in the ansatz (\\ref{eqn:dRdQn}),\n at which the leading systematic error\n due to the ansatz\n is minimal \\cite{DMDDf1v},\n can be estimated by\n\\beq\n   Q_{s, n}\n = Q_n + \\frac{1}{k_n} \\ln\\bfrac{\\sinh(k_n b_n\/2)}{k_n b_n\/2}\n\\~.\n\\label{eqn:Qsn}\n\\eeq\n Note that $Q_{s, n}$ differs from the central point of the $n$th bin, $Q_n$.\n\n Now,\n substituting the ansatz (\\ref{eqn:dRdQn})\n into Eq.~(\\ref{eqn:f1v_dRdQ})\n and then letting $Q = Q_{s, n}$,\n we can obtain that\n\\beq\n   f_{1, {\\rm rec}}(v_{s, n})\n = \\calN\n   \\bBigg{\\frac{2 Q_{s, n} r_n}{F^2(Q_{s, n})}}\n   \\bbrac{\\dd{Q} \\ln \\FQ \\bigg|_{Q = Q_{s, n}} - k_n}\n\\~.\n\\label{eqn:f1v_Qsn}\n\\eeq\n Here\n\\beq\n   v_{s, n}\n = \\alpha \\sqrt{Q_{s, n}}\n\\~,\n\\label{eqn:vsn}\n\\eeq\n and the normalization constant $\\calN$\n given in Eq.~(\\ref{eqn:calN_int})\n can be estimated directly from the data by\n\\beq\n   \\calN\n = \\frac{2}{\\alpha}\n   \\bbrac{\\sum_{a} \\frac{1}{\\sqrt{Q_a} \\~ F^2(Q_a)}}^{-1}\n\\~,\n\\label{eqn:calN_sum}\n\\eeq\n where the sum runs over all events in the sample.\n\n\\subsubsection{Windowing the data set}\n As mentioned above,\n the statistical uncertainty on\n the slope of the recoil spectrum\n around the central point $Q_n$,\n $\\bbrac{d\/dQ \\~ (dR \/ dQ)}_{Q \\simeq Q_n}$,\n is approximately proportional to $b_n^{-1.5}$.\n Thus,\n in order to reduce the statistical uncertainty\n on the velocity distribution\n reconstructed by Eq.~(\\ref{eqn:f1v_Qsn}),\n it seems to be better\n to use large bin width.\n However,\n neither the conventional linear approximation:\n\\beq\n   \\adRdQ_{{\\rm expt}, \\~ Q = Q_n}\n = \\frac{N_n}{b_n}\n\\eeq\n nor the exponential ansatz\n given in Eq.~(\\ref{eqn:dRdQn})\n can describe the real (but as yet unknown)\n recoil spectrum exactly.\n The neglected terms of higher powers of $Q - Q_n$\n could therefore induce some uncontrolled systematic errors\n which increase with increasing bin width.\n Moreover,\n since the number of bins scales inversely with their size,\n by using larger bins we would be able to estimate $f_1(v)$\n only at a smaller number of velocities.\n Additionally,\n once a quite large bin width is used,\n it would correspondingly lead to a quite large value\n of the first reconstructible point of $f_1(v)$,\n i.e.~$f_{1, {\\rm rec}}(v_{s, 1})$,\n since the central point $Q_1$\n as well as\n the shifted point $Q_{s, 1}$\n of the first bin would be quite large.\n Finally,\n choosing a fixed bin size,\n as one conventionally does,\n would let errors on the estimated logarithmic slopes,\n and hence also on the estimates of $f_1(v)$,\n increase quickly with increasing $Q$ or $v$.\n This is due to the essentially exponential form\n of the expected recoil spectrum,\n which would lead to a quickly falling number of events\n in equal--sized bins.\n By some trial--and--error analyses\n it was found that\n the errors are roughly equal in all bins\n if the bin widths increase linearly \\cite{DMDDf1v}.\n\n Therefore,\n it has been introduced in Ref.~\\cite{DMDDf1v} that\n one can first collect experimental data\n in relatively small bins\n and then combining varying numbers of bins\n into overlapping ``windows''.\n In particular,\n the first window would be identical with the first bin.\n One starts by binning the data,\n as in Eq.~(\\ref{eqn:Qni}),\n where the bin widths satisfy\n\\beq\n   b_n\n = b_1 + (n - 1) \\delta\n\\~,\n\\label{eqn:bn_delta}\n\\eeq\n i.e.\n\\beq\n   Q_n\n = \\Qmin + \\abrac{n - \\frac{1}{2}} b_1 + \\bfrac{(n - 1)^2}{2} \\delta\n\\~.\n\\label{eqn:Qn_delta}\n\\eeq\n Here the increment $\\delta$ satisfies\n\\beq\n   \\delta\n = \\frac{2}{B (B - 1)} \\aBig{\\Qmax - \\Qmin - B b_1}\n\\~,\n\\label{eqn:delta_B}\n\\eeq\n $B$ being the total number of bins,\n and $Q_{\\rm (min, max)}$ are\n the experimental minimal and maximal cut--off energies.\n Assume up to $n_W$ bins are collected into a window,\n with smaller windows at the borders of the range of $Q$.\n\n In order to distinguish the numbers of bins and windows,\n hereafter Latin indices $n,~m,~\\cdots$ are used to label bins,\n and Greek indices $\\mu,~\\nu,~\\cdots$ to label windows.\n For $1 \\leq \\mu \\leq n_W$,\n the $\\mu$th window simply consists of the first $\\mu$ bins;\n for $n_W \\leq \\mu \\leq B$,\n the $\\mu$th window consists of bins\n $\\mu-n_W + 1,~\\mu-n_W + 2,~\\cdots,~\\mu$;\n and for $B \\leq \\mu \\leq B+n_W-1$,\n the $\\mu$th window consists of the last $n_W - (\\mu - B)$ bins.\n This can also be described by introducing\n the indices $n_{\\mu-}$ and $n_{\\mu+}$\n which label the first and last bins\n contributing to the $\\mu$th window,\n with\n\\cheqna\n\\beq\n\\renewcommand{\\arraystretch}{1.3}\n   n_{\\mu-}\n = \\cleft{\\begin{array}{l c l}\n           1,         & ~~~~~~ & {\\rm for}~\\mu \\leq n_W, \\\\\n           \\mu-n_W+1, &        & {\\rm for}~\\mu \\geq n_W,\n          \\end{array}}\n\\label{eqn:n_mu_minus}\n\\eeq\n and\n\\cheqnb\n\\beq\n\\renewcommand{\\arraystretch}{1.3}\n   n_{\\mu+}\n = \\cleft{\\begin{array}{l c l}\n           \\mu, & ~~~~~~ & {\\rm for}~\\mu \\leq B, \\\\\n           B,   &        & {\\rm for}~\\mu \\geq B.\n          \\end{array}}\n\\label{eqn:n_mu_plus}\n\\eeq\n\\cheqn\n The total number of windows\n defined through Eqs.~(\\ref{eqn:n_mu_minus}) and (\\ref{eqn:n_mu_plus})\n is evidently $W = B + n_W - 1$,\n i.e.~$1 \\leq \\mu \\leq B + n_W - 1$.\n\n As shown above,\n the basic observables needed\n for the reconstruction of $f_1(v)$ by Eq.~(\\ref{eqn:f1v_Qsn})\n are the number of events in the $n$th $Q-$bin, $N_n$,\n as well as\n the average value of the measured recoil energies\n in this bin, $\\bQn$.\n For a ``windowed'' data set,\n one can easily calculate\n the number of events per window as\n\\beq\n   N_{\\mu}\n = \\sum_{n = n_{\\mu-}}^{n_{\\mu+}} N_n\n\\~,\n\\label{eqn:N_mu}\n\\eeq\n as well as\n the average value of the measured recoil energies\n\\beq\n   \\Bar{Q - Q_{\\mu}}|_{\\mu}\n = \\frac{1}{N_{\\mu}}\n   \\abrac{\\sum_{n = n_{\\mu-}}^{n_{\\mu+}} N_n \\Bar{Q}|_{n}} - Q_{\\mu}\n\\~,\n\\label{eqn:wQ_mu}\n\\eeq\n where $Q_{\\mu}$ is the central point of the $\\mu$th window.\n The exponential ansatz in Eq.~(\\ref{eqn:dRdQn})\n is now assumed to hold over an entire window.\n We can then estimate the prefactor as\n\\beq\n   r_{\\mu}\n = \\frac{N_{\\mu}}{w_{\\mu}}\n\\~,\n\\label{eqn:r_mu}\n\\eeq\n $w_{\\mu}$ being the width of the $\\mu$th window.\n The logarithmic slope of the recoil spectrum\n in the $\\mu$th window, $k_{\\mu}$,\n as well as\n the shifted point $Q_{s, \\mu}$\n (from the central point of each ``window'', $Q_{\\mu}$)\n can be calculated as in Eqs.~(\\ref{eqn:bQn}) and (\\ref{eqn:Qsn})\n with ``bin'' quantities replaced by ``window'' quantities.\n Finally,\n note that,\n due to the combination of bins\n into overlapping windows,\n these quantities are all correlated (for $n_W \\neq 1$).\n The covariance matrix of\n the estimates of $f_1(v)$\n at adjacent values of $v_{s, \\mu} = \\alpha \\sqrt{Q_{s, \\mu}}$\n is given b\n\\footnote{\n Note that\n Eq.~(\\ref{eqn:cov_f1v_Qs_mu}) should in principle\n also include contributions involving\n the statistical error on the estimator for $\\calN$\n in Eq.~(\\ref{eqn:calN_sum}).\n However,\n this error and its correlations\n with the errors on the $r_{\\mu}$ and $k_{\\mu}$\n have been found to be negligible\n compared to the errors included in Eq.~(\\ref{eqn:cov_f1v_Qs_mu})\n \\cite{DMDDf1v}.\n}\n\\beqn\n \\conti {\\rm cov}\\aBig{f_{1, {\\rm rec}}(v_{s, \\mu}), f_{1, {\\rm rec}}(v_{s, \\nu})}\n        \\non\\\\\n \\=     \\bfrac{f_{1, {\\rm rec}}(v_{s, \\mu}) f_{1, {\\rm rec}}(v_{s, \\nu})}\n              {r_{\\mu} r_{\\nu}}\n        {\\rm cov}\\abrac{r_{\\mu}, r_{\\nu}}\n       +\\abrac{2 \\calN}^2\n        \\bfrac{Q_{s, \\mu} Q_{s, \\nu} r_{\\mu} r_{\\nu}}{F^2(Q_{s, \\mu}) F^2(Q_{s, \\nu})}\n        {\\rm cov}\\abrac{k_{\\mu}, k_{\\nu}}\n        \\non\\\\\n\\conti ~~~~~~~~~~~~\n       -\\calN\n        \\cbrac{ \\bfrac{f_{1, {\\rm rec}}(v_{s, \\mu})}{r_{\\mu}}\n                \\bfrac{2 Q_{s, \\nu} r_{\\nu} }{F^2(Q_{s, \\nu})}\n                {\\rm cov}\\abrac{r_{\\mu}, k_{\\nu}}\n               +\\aBig{\\mu \\lgetsto \\nu}}\n\\~.\n\\label{eqn:cov_f1v_Qs_mu}\n\\eeqn\n\n\\subsection{Bayesian analysis}\n In this subsection,\n we review the basic concept of the Bayesian analysis\n \\cite{Barlow-Statistics}\n (for its applications in physics,\n  see e.g.~Ref.~\\cite{DAgostini95},\n  and for its recent applications\n  in direct DM detection phenomenology,\n  see e.g.~Refs.~\\cite{Akrami,\n                       Pato,\n                       Arina,\n                       Strege12,\n                       Cerdeno13,\n                       Kavanagh13})\n and extend this method to the use of\n the ``reconstructed data''\n offered by our model--independent reconstruction process\n described in the previous subsection.\n\n\\subsubsection{Baye's theorem}\n We start with the {\\em Baye's theorem}:\n the probability of $A$ given that $B$ is true\n multiplies the probability of that $B$ is true\n is equal to\n the probability of that\n $A$ and $B$ happen simultaneously,\n which\n is also equal to\n the probability of $B$ given that $A$ is true\n multiplies the probability of that $A$ is true.\n This can simply be expressed as\n\\beq\n    {\\rm p}(A | B) \\~ {\\rm p}(B)\n =  {\\rm p}(A \\cap B)\n =  {\\rm p}(B | A) \\~ {\\rm p}(A)\n\\~,\n\\eeq\n where ${\\rm p}(A | B)$ is called\n the ``conditional probability'' of $A$\n given that $B$ is true.\n As long as ${\\rm p}(B) \\neq 0$,\n the above equation can be rewritten as\n\\beqn\n     {\\rm p}(A | B)\n \\=  \\frac{{\\rm p}(B | A) \\~ {\\rm p}(A)}{{\\rm p}(B)}\n     \\non\\\\\n \\=  \\frac{{\\rm p}(B | A) \\~ {\\rm p}(A)}\n          {{\\rm p}(B | A) \\~ {\\rm p}(A) + {\\rm p}(B | \\Bar{A}) \\~ {\\rm p}(\\Bar{A})}\n\\~,\n\\label{eqn:Bayes_theorem}\n\\eeqn\n where\n\\(\n     {\\rm p}(\\Bar{A})\n =   1 - {\\rm p}(A)\n\\)\n is the probability of the complement of $A$,\n i.e.~the probability of that\n $A$ is not happen.\n\n\\subsubsection{Bayesian statistics}\n Applying the Baye's theorem\n described\n above,\n one can directly have that\n\\beq\n     {\\rm p}({\\rm theory} | {\\rm result})\n  =  \\frac{{\\rm p}({\\rm result} | {\\rm theory})}\n          {{\\rm p}({\\rm result})} \\cdot {\\rm p}({\\rm theory})\n\\~.\n\\label{eqn:Bayesian_statistics}\n\\eeq\n This means that,\n given the observed result,\n the probability of that\n a specified theory is true\n is proportional to\n the probability of the observed result\n in the specified theory\n multiplies\n the probability of the specified theory.\n\n This statement can be understood as follows.\n If the observed result is predicted by a specified theory\n to be highly unlikely or even impossible\/forbidden,\n then this observation\n makes the ``degree of belief'' of this specified theory small\n or even disproves this theory.\n In contrast,\n the observation of a prediction\n by a specified theory with a high probability\n will strengthen one's belief in this theory.\n\n\\subsubsection{Bayesian analysis}\n By extending Eq.~(\\ref{eqn:Bayesian_statistics}),\n we can obtain that\n\\beq\n     {\\rm p}(\\Theta | {\\rm data})\n  =  \\frac{{\\rm p}({\\rm data} | \\Theta)}\n          {{\\rm p}({\\rm data})} \\cdot {\\rm p}(\\Theta)\n\\~.\n\\label{eqn:Bayesian_analysis}\n\\eeq\n Here\n $\\Theta = \\cbig{a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}}}$\n denotes a specified (combination of the) value(s)\n of the fitting parameter(s);\n ${\\rm p}(\\Theta)$,\n called ``prior probability'',\n represents our degree of belief about\n $\\Theta$ being the true value(s) of fitting parameter(s),\n which is often given\n in form of the (multiplication of the)\n probability distribution(s) of the fitting parameter(s).\n ${\\rm p}({\\rm data})$,\n called ``evidence'',\n is the total probability of obtaining the particular set of data,\n which is in practice\n irrespective of the actual value(s) of the\n parameter(s) and\n can be treated as a normalization constant;\n it will not be of interest in our further discussion.\n ${\\rm p}({\\rm data} | \\Theta)$\n denotes the probability of the observed result,\n once the specified (combination of the) value(s)\n of the fitting parameter(s)\n happens,\n which can usually be described\n by the {\\em likelihood} function of $\\Theta$,\n ${\\cal L}(\\Theta)$.\n Finally,\n ${\\rm p}(\\Theta | {\\rm data})$,\n called the ``posterior probability density function''\n for $\\Theta$,\n representes\n the probability of that\n the specified (combination of the) value(s)\n of the fitting parameter(s)\n happens,\n given the observed result.\n\n\\subsubsection{Bayesian reconstruction of \\boldmath$f_1(v)$}\n Now,\n we can describe\n the procedure of our Bayesian reconstruction of\n the one--dimensional WIMP velocity distribution function\n in detail.\n\n First,\n by using Eqs.~(\\ref{eqn:f1v_Qsn})\n to (\\ref{eqn:calN_sum}),\n one can obtain $W = B + n_W - 1$ reconstructed data points:\n $\\aBig{v_{s, \\mu}, f_{1, {\\rm rec}}(v_{s, \\mu}) \\pm \\sigma_{f_1, s, \\mu}}$,\n for $\\mu = 1,~2,~\\cdots,~W$,\n where\n\\beq\n        \\sigma_{f_1, s, \\mu}\n \\equiv \\sqrt{{\\rm cov}\\aBig{f_{1, {\\rm rec}}(v_{s, \\mu}), f_{1, {\\rm rec}}(v_{s, \\mu})}}\n\\label{eqn:sigma_f1v_Qs_mu}\n\\eeq\n denote the square roots of\n the diagonal entries of the covariance matrix\n given in Eq.~(\\ref{eqn:cov_f1v_Qs_mu}).\n Choosing a theoretical prediction of\n the one--dimensional velocity distribution\n of halo WIMPs:\n $f_{1, \\rm th}(v; a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}})$,\n where $\\abig{a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}}}$\n are the $N_{\\rm Bayesian}$ fitting parameters,\n and\n assuming that\n the reconstructed data points\n is {\\em Gaussian}--distributed\n around the theoretical predictions\n $f_{1, \\rm th}(v_{s, \\mu}; a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}})$,\n the likelihood function for ${\\rm p}({\\rm data} | \\Theta)$\n can then be defined by\n\\beqn\n \\conti\n     {\\cal L}\\aBig{f_{1, {\\rm rec}}(v_{s, \\mu}),~\\mu = 1,~2,~\\cdots,~W;~\n                   a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}}\n     \\non\\\\\n \\eqnequiv\n     \\prod_{\\mu = 1}^{W}\n     {\\rm Gau}\\aBig{v_{s, \\mu},\n                    f_{1, {\\rm rec}}(v_{s, \\mu}),\n                    \\sigma_{f_1, s, \\mu};\n                    a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}}}\n\\~,\n\\label{eqn:calL}\n\\eeqn\n where\n\\beqn\n \\conti\n     {\\rm Gau}\\aBig{v_{s, \\mu},\n                    f_{1, {\\rm rec}}(v_{s, \\mu}),\n                    \\sigma_{f_1, s, \\mu};\n                    a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}}}\n     \\non\\\\\n \\eqnequiv\n     \\frac{1}{\\sqrt{2 \\pi} \\~ \\sigma_{f_1, s, \\mu}} \\~\n     e^{-\\bbig{  f_{1, {\\rm rec}}(v_{s, \\mu})\n               - f_{1, \\rm th}(v_{s, \\mu}; a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}})}^2 \\big\/\n         2 \\sigma_{f_1, s, \\mu}^2}\n\\~.\n\\label{eqn:Bayesian_DF_Gau}\n\\eeqn\n Or,\n equivalently,\n we can use the {\\em logarithmic} likelihood function\n given by\n\\beqn\n \\conti\n    \\ln{\\cal L}\\aBig{f_{1, {\\rm rec}}(v_{s, \\mu}),~\\mu = 1,~2,~\\cdots,~W;~\n                     a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}}\n     \\non\\\\\n \\=- \\frac{1}{2}\n     \\sum_{\\mu = 1}^{W}\n     \\frac{\\bBig{  f_{1, {\\rm rec}}(v_{s, \\mu})\n                 - f_{1, \\rm th}(v_{s, \\mu}; a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}})}^2}\n          {\\sigma_{f_1, s, \\mu}^2}\n   - \\sum_{\\mu = 1}^{W}\n     \\ln \\abrac{\\sqrt{2 \\pi} \\~ \\sigma_{f_1, s, \\mu}}\n\\~.\n\\label{eqn:ln_calL}\n\\eeqn\n Note that\n in practical use\n the second term in Eq.~(\\ref{eqn:ln_calL}) can be neglected,\n since it is just a constant\n for all scanned (combinations of) values of\n the fitting parameter(s)\n $\\abig{a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}}}$.\n\n Finally,\n choosing the probability distribution function\n for each fitting parameter $a_i$,\n ${\\rm p}_i(a_i)$,\n the posterior probability density\n on the left--hand side of Eq.~(\\ref{eqn:Bayesian_analysis})\n can then be given by\n\\beqn\n \\conti\n     {\\rm p}\\aBig{a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}~\\Big|~\n                  f_{1, {\\rm rec}}(v_{s, \\mu}),~\\mu = 1,~2,~\\cdots,~W}\n     \\non\\\\\n \\eqnpropto\n     {\\cal L}\\aBig{f_{1, {\\rm rec}}(v_{s, \\mu}),~\\mu = 1,~2,~\\cdots,~W;~\n                   a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}}\n     \\prod_{i = 1}^{N_{\\rm Bayesian}} {\\rm p}_i(a_i)\n\\~.\n\\label{eqn:P_Bayesian}\n\\eeqn\n\n\\section{Numerical results}\n In this section,\n we present numerical results\n of our Bayesian reconstruction of\n the one--dimensional velocity distribution function\n of halo WIMPs\n based on Monte--Carlo simulations.\n\n In order to test\n whether we can reconstruct $f_1(v)$ correctly\n with even an improper adopted model\n and\/or incorrect expectation value(s) of the fitting parameter(s),\n different choices of the WIMP velocity distribution function\n for generating signal events\n as well as\n different assumptions of $f_1(v)$ and\/or\n (slightly) different expectation value(s) of the fitting parameter(s)\n from the commonly adopted values\n for the Bayesian fitting process\n will be condisered in our simulations.\n In Tables \\ref{tab:setup_Gau},\n \\ref{tab:setup_sh} and \\ref{tab:setup_Gau_k},\n we list the {\\em input} setup\n for the chosen {\\em input} velocity distribution functions\n used for {\\em generating} WIMP signals\n as well as\n the scanning ranges,\n the expectation values of\n and 1$\\sigma$ uncertainties on\n the fitting parameters\n used for different {\\em fitting} velocity distributions.\n Additionally,\n a common maximal cut--off\n on the one--dimensional WIMP velocity distribution\n has been set as \\mbox{$\\vmax = 700$ km\/s}.\n\n The WIMP mass $\\mchi$\n involved in the coefficient $\\alpha$\n in Eqs.~(\\ref{eqn:vsn}) and (\\ref{eqn:calN_sum})\n for estimating the reconstructed points $v_{s, \\mu}$\n as well as\n the normalization constant $\\calN$\n has been assumed\n either to be known precisely with a negligible uncertainty\n from other (e.g.~collider) experiments\n or determined from\n direct detection experiments\n with {\\em different} data sets.\n As in Ref.~\\cite{DMDDf1v},\n a $\\rmXA{Ge}{76}$ nucleus has been chosen\n as our detector material for reconstructing $f_1(v)$,\n whereas a $\\rmXA{Si}{28}$ target\n and a {\\em second} $\\rmXA{Ge}{76}$ target\n have been used for determining $\\mchi$\n \\cite{DMDDmchi}.\n\n As in Refs.~\\cite{DMDDf1v, DMDDbg-f1v},\n the WIMP--nucleus cross section in Eq.~(\\ref{eqn:calA})\n has been assumed\n to be only spin--independent (SI),\n \\mbox{$\\sigmapSI = 10^{-9}$ pb},\n and\n the commonly used analytic form\n for the elastic nuclear form factor:\n\\beq\n   F_{\\rm SI}^2(Q)\n = \\bfrac{3 j_1(q R_1)}{q R_1}^2 e^{-(q s)^2}\n\\label{eqn:FQ_WS}\n\\eeq\n has been adopted.\n Here $Q$ is the recoil energy\n transferred from the incident WIMP to the target nucleus,\n $j_1(x)$ is a spherical Bessel function,\n\\(\n   q\n = \\sqrt{2 m_{\\rm N} Q}\n\\)\n is the transferred 3-momentum,\n for the effective nuclear radius we use\n\\(\n   R_1\n = \\sqrt{R_A^2 - 5 s^2}\n\\)\n with\n\\(\n        R_A\n \\simeq 1.2 \\~ A^{1\/3}~{\\rm fm}\n\\)\n and a nuclear skin thickness\n\\(\n        s\n \\simeq 1~{\\rm fm}\n\\).\n\n In our simulations,\n the experimental threshold energies\n have been assumed to be negligible ($\\Qmin = 0$\n\\footnote{\n Note that,\n once the experimental threshold energy\n of the data set\n used for reconstructing $f_1(v)$\n is non--negligible,\n the estimate (\\ref{eqn:calN_sum})\n of the normalization constant $\\calN$\n would need to be modified properly,\n especially for light WIMPs.\n}\n and the maximal cut--off energies\n are set as \\mbox{$\\Qmax = 100$ keV}\n for all target nucle\n\\footnote{\n Note that,\n due to the maximal cut--off\n on the one--dimensional WIMP velocity distribution,\n $\\vmax$,\n a kinematic maximal cut--off energy\n\\beq\n   Q_{\\rm max, kin}\n = \\frac{\\vmax^2}{\\alpha^2}\n\\label{eqn:Qmax_kin}\n\\eeq\n has also been taken into account.\n};\n the widths of the first energy bin in Eq.~(\\ref{eqn:bn_delta})\n have also been set commonly as \\mbox{$b_1 = 10$ keV}.\n Additionally,\n we assumed that\n all experimental systematic uncertainties\n as well as\n the uncertainty on the measurement of the recoil energy\n could be ignored.\n The energy resolution of most existing and next--generation detectors\n should be good enough so that\n the very small measurement uncertainties can be neglected\n compared to the statistical uncertainties\n on our reconstructed results\n with only few events.\n Energy range between $\\Qmin$ and $\\Qmax$\n have been divided into five bins\n and up to three bins\n have been combined to a window.\n (\\mbox{3 $\\times$}) 5,000 experiments with 500 total events on average\n in one experiment have been simulated.\n In Secs.~3.1 to 3.3,\n the input WIMP mass has been fixed as \\mbox{$\\mchi = 100$ GeV}.\n In Sec.~3.4,\n we consider the cases with\n a light WIMP mass of \\mbox{$\\mchi = 25$ GeV}\n and a heavy one of \\mbox{$\\mchi = 250$ GeV}.\n In addition,\n in Sec.~3.5,\n we consider also briefly\n the effects of unrejected background events\n for different input WIMP masses\n \\cite{DMDDbg-mchi, DMDDbg-f1v}.\n\n Note that,\n for our numerical simulations\n presented in this section,\n the actual numbers of generated signal and background events\n in each simulated experiment\n are Poisson--distributed around their expectation values\n {\\em independently}.\n This means that,\n for example,\n for simulations shown in Figs.~\\ref{fig:dRdQ-bg-ex-const-000-100-20-100}\n we generate 400 (100) events {\\em on average} for WIMP--signals (backgrounds)\n and the total event number recorded in one experiment\n is then the sum of these two numbers.\n\n Regarding our degree of belief about\n each fitting parameter $a_i$,\n i.e.~${\\rm p}_i(a_i)$ in Eq.~(\\ref{eqn:P_Bayesian}),\n two probability distribution functions\n have been considered.\n The simplest one is the flat--distribution:\n\\beq\n    {\\rm p}_i(a_i)\n =  1\n\\~,\n    ~~~~~~~~~~~~~~~~\n    {\\rm for~} a_{i, \\rm min} \\le a_i \\le a_{i, \\rm max},\n\\label{eqn:Bayesian_DF_a_flat}\n\\eeq\n where $a_{i, \\rm (min, max)}$ denote\n the minimal and maximal bounds of the scanning interval of\n the fitting parameter $a_i$.\n On the other hand,\n for the case that\n we have already prior knowledge about\n one fitting parameter,\n a Gaussian--distribution:\n\\beq\n    {\\rm p}_i(a_i; \\mu_{a, i}, \\sigma_{a, i})\n =  \\frac{1}{\\sqrt{2 \\pi} \\~ \\sigma_{a, i}} \\~\n    e^{-(a_i - \\mu_{a, i})^2 \/ 2 \\sigma_{a, i}^2}\n\\label{eqn:Bayesian_DF_a_Gau}\n\\eeq\n with the expectation value $\\mu_{a, i}$ of and\n the 1$\\sigma$ uncertainty $\\sigma_{a, i}$ on\n the fitting parameter $a_i$\n is used.\n\n Note that,\n in one simulated experiment,\n we scan the parameter space\n $(a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}})$\n in the volume\n $a_i \\in [a_{i, {\\rm min}}, a_{i, {\\rm max}}]$,\n $i = 1, 2, \\cdots, N_{\\rm Bayesian}$\n to find a particular point\n $(a_1^{\\ast}, a_2^{\\ast}, \\cdots, a_{N_{\\rm Bayesian}}^{\\ast})$,\n which maximizes the (numerator of the)\n posterior probability density: \\\\\n ${\\rm p}\\aBig{a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}~\\Big|~\n               f_{1, {\\rm rec}}(v_{s, \\mu}),~\\mu = 1,~2,~\\cdots,~W}$.\n After that\n all simulations have been done,\n we determine the median value of\n the (1$\\sigma$ lower and upper bounds of the) velocity distribution\n reconstructed by Eq.~(\\ref{eqn:f1v_Qsn}) from all experiments,\n denoted as\n $f_{1, {\\rm median}}\\abrac{\\alpha_{\\rm (median)} \\sqrt{Q_{s, \\mu, {\\rm Bayesian}}}}$,\n for $\\mu = 1,~2,~\\cdots,~W$,\n and shown as solid black crosses in the (top--)left frame(s) of,\n e.g.~Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat}.\n And\n we define further\n\\beqn\n \\conti    {\\rm p}_{\\rm median}\\aBig{a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}}\n           \\non\\\\\n \\eqnequiv {\\rm p}\\aBig{a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}~\\Big|~\n                        f_{1, {\\rm median}}\\abrac{\\alpha_{\\rm (median)} \\sqrt{Q_{s, \\mu, {\\rm Bayesian}}}},\n                        \\mu = 1,~2,~\\cdots,~W}\n\\~,\n           \\non\\\\\n\\label{eqn:P_Bayesian_median}\n\\eeqn\n and check the points\n $(a_1^{\\ast}, a_2^{\\ast}, \\cdots, a_{N_{\\rm Bayesian}}^{\\ast})$\n obtained from all simulated experiments\n to find the special (``best--fit'') point\n $(a_{1, {\\rm max}}, a_{2, {\\rm max}}, \\cdots, a_{N_{\\rm Bayesian}, {\\rm max}})$,\n which maximizes ${\\rm p}_{\\rm median}(a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian})$.\n\n\\subsection{Simple Maxwellian velocity distribution}\n\n\\begin{table}[t!]\n\\InsertSetupTable\n{simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{Simple\n & $v_0$      [km\/s] &        220 & [160,    300] &   230 & 20   \\\\\n}\n{The input setup for\n the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n used for generating WIMP events\n as well as\n the scanning range,\n the expectation value of\n and the 1$\\sigma$ uncertainty on\n the unique fitting parameter $v_0$.\n}\n{tab:setup_Gau}\n\n We consider first\n the simplest isothermal spherical Galactic halo model\n for generating WIMP events.\n The normalized one--dimensional\n simple Maxwellian velocity distribution function\n can be expressed as\n \\cite{SUSYDM96, DMDDf1v}:\n\\beq\n   f_{1, \\Gau}(v)\n = \\frac{4}{\\sqrt{\\pi}}\n   \\afrac{v^2}{v_0^3} e^{-v^2 \/ v_0^2}\n\\~,\n\\label{eqn:f1v_Gau}\n\\eeq\n where $v_0 \\approx 220$ km\/s\n is the Solar orbital velocity in the Galactic frame.\n\n In Table \\ref{tab:setup_Gau},\n we list the input setup for\n the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n used for generating WIMP events\n as well as\n the scanning range,\n the expectation value of\n and the 1$\\sigma$ uncertainty on\n the unique fitting parameter $v_0$.\n Note that,\n for generating WIMP signals,\n \\mbox{$v_0 = 220$ km\/s} has been used,\n whereas for Bayesian fitting of this unique parameter\n we used a slightly different value of \\mbox{$v_0 = 230$ km\/s}\n and set a 1$\\sigma$ uncertainty\n of \\mbox{$\\sigma(v_0) = 20$ km\/s}.\n\n\\subsubsection{Simple Maxwellian velocity distribution}\n\n As the simplest Galactic halo model,\n we consider first the use of\n the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n given in Eq.~(\\ref{eqn:f1v_Gau})\n with an unique fitting parameter $v_0$\n to fit the reconstructed--input data\n given by Eqs.~(\\ref{eqn:f1v_Qsn}) to (\\ref{eqn:calN_sum})\n and (\\ref{eqn:cov_f1v_Qs_mu}).\n\n In Fig.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat}(a),\n we show the reconstructed\n simple Maxwellian velocity distribution function\n for an input WIMP mass of \\mbox{$\\mchi = 100$ GeV}\n with a $\\rmXA{Ge}{76}$ target.\n Here the flat distribution\n given by Eq.~(\\ref{eqn:Bayesian_DF_a_flat})\n for the fitting parameter $v_0$\n has been used.\n The black crosses are the velocity distribution\n reconstructed by Eqs.~(\\ref{eqn:vsn}) and (\\ref{eqn:f1v_Qsn}):\n the vertical error bars show\n the square roots of the diagonal entries of the covariance matrix\n given in Eq.~(\\ref{eqn:cov_f1v_Qs_mu})\n (i.e.~$\\sigma_{f_1, s, \\mu}$\n  given in Eq.~(\\ref{eqn:sigma_f1v_Qs_mu}))\n and\n the horizontal bars indicate\n the sizes of the windows used\n for estimating $f_{1, {\\rm rec}}(v_{s, \\mu})$,\n respectively.\n The solid red curve\n is the {\\em generating} simple Maxwellian velocity distribution\n with an input value of \\mbox{$v_0 = 220$ km\/s}.\n While\n the dashed green curve indicates\n the {\\em reconstructed} simple Maxwellian velocity distribution\n with the fitting parameter $v_0$\n given by the {\\em median} value of all simulated experiments,\n the dash--dotted blue curve indicates\n the {\\em reconstructed} simple Maxwellian velocity distribution\n with\n $v_0$\n which maximizes ${\\rm p}_{\\rm median}\\abrac{a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}}$\n defined in Eq.~(\\ref{eqn:P_Bayesian_median}).\n\n\\plotGeGauGauflat\n\\plotGeGauGauGau\n\n Meanwhile,\n the light--green (light--blue) area shown here\n indicate the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n of the Bayesian reconstructed velocity distribution function,\n which has been determined as follows.\n After scanning\n the reconstructed fitting parameter $v_0$\n obtained from all simulated experiments\n and ordering\n according to their ${\\rm p}_{\\rm median}$ values\n defined in Eq.~(\\ref{eqn:P_Bayesian_median})\n {\\em descendingly},\n we can not only\n determine the point\n which maximizes ${\\rm p}_{\\rm median}$\n (labeled with the subscript ``max'' in our plot\n\\footnote{\n Note that\n the subscript ``input'' in Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat}\n and \\ref{fig:f1v-Ge-100-0500-Gau-Gau-Gau}\n indicate that\n the WIMP mass needed\n in Eqs.~(\\ref{eqn:vsn}) and (\\ref{eqn:f1v_Qsn})\n has been given as the input WIMP mass;\n whereas,\n the subscript ``algo'' in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-flat}\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-Gau}\n indicate that\n the WIMP mass\n is reconstructed by the algorithmic procedure\n developed in Ref.~\\cite{DMDDmchi}.\n}),\n but also\n the smallest and largest values of\n the first 68.27\\% (95.45\\%) of all these reconstructed $v_0$'s.\n We then use the smallest (largest) value of\n the first 68.27\\% (95.45\\%) reconstructed $v_0$'s\n to give the $1\\~(2)\\~\\sigma$ lower (upper) boundaries\n of the Bayesian reconstructed velocity distribution function.\n This means that\n all of the velocity distributions with $v_0$'s\n which give the largest 68.27\\% (95.45\\%) ${\\rm p}_{\\rm median}$ values\n should be in the $1\\~(2)\\~\\sigma$ light--green (light--blue) areas.\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 218.8 & $218.8\\~^{+12.6}_{-11.2}\\~(^{+33.6}_{-22.4})$ & [207.6, 231.4] & [196.4, 252.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 221.6 & $221.6 \\pm  9.8\\~(^{+23.8}_{-18.2})$ & [211.8, 231.4] & [203.4, 245.4] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 220.2 & $218.8\\~^{+21.0}_{-16.8}\\~(^{+49.0}_{-33.6})$ & [202.0, 239.8] & [185.2, 267.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 220.2 & $221.6\\~^{+15.4}_{-12.6}\\~(^{+33.6}_{-26.6})$ & [209.0, 237.0] & [195.0, 255.2] \\\\\n}\n{The reconstructed results of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_Gau_Gau}\n\n On the other hand,\n Fig.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat}(b) shows\n the distribution of the Bayesian reconstructed fitting parameter $v_0$\n in all simulated experiments.\n The red vertical line indicates the true (input) value of $v_0$,\n which has been labeled with the subscript ``gen''.\n The green vertical line indicates\n the median value of the simulated results,\n whereas\n the blue one\n indicates the value which maximizes\n ${\\rm p}_{\\rm median}$.\n In addition,\n the horizontal thick (thin) green bars show\n the $1\\~(2)\\~\\sigma$ ranges of the reconstructed result\n\\footnote{\n Note that\n the $1\\~(2)\\~\\sigma$ ranges given here mean that,\n according to the order of the reconstructed values of $v_0$ {\\em along}\n and centered at their {\\em median} value $v_{0, {\\rm median}}$,\n 68.27\\% (95.45\\%) of the reconstructed values in the simulated experiments\n are in this range.\n}.\n Note that\n the bins at \\mbox{$v_0 = 160$ km\/s} and \\mbox{$v_0 = 300$ km\/s}\n are ``overflow'' bins,\n which contain also the experiments\n with the best--fit $v_0$ value of\n either \\mbox{$v_0 < 160$ km\/s} or \\mbox{$v_0 > 300$ km\/s}.\n\n In Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat},\n it can be seen clearly that,\n {\\em without} a prior knowledge about\n the Solar Galactic velocity,\n one could in principle pin down the parameter $v_0$ very precisely\n with $1\\~(2)\\~\\sigma$ statistical uncertainties of only\n \\mbox{$^{+12.6}_{-11.2}\\~(^{+33.6}_{-22.4})$ km\/s}\n (see Table \\ref{tab:results_Gau_Gau}).\n Moreover,\n by using the Bayesian reconstruction of\n the one--dimensional velocity distribution function,\n the large (1$\\sigma$) statistical uncertainty\n given by Eq.~(\\ref{eqn:sigma_f1v_Qs_mu})\n can be reduced significantly:\n the band of the 2$\\sigma$ statistical uncertainty\n would be approximately equal to or even smaller than\n the (solid black) vertical 1$\\sigma$ uncertainty bars!\n\n Furthermore,\n in Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-Gau}\n we consider the case\n with a rough prior knowledge about the Solar Galactic velocity $v_0$.\n It has been found that,\n firstly,\n by using a Gaussian probability distribution\n for $v_0$ with a 1$\\sigma$ uncertainty of \\mbox{20 km\/s},\n one could reduce the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the Bayesian reconstructed parameter $v_0$\n to \\mbox{$\\pm  9.8\\~(^{+23.8}_{-18.2})$ km\/s}\n (see Table \\ref{tab:results_Gau_Gau}).\n Secondly and more importantly,\n although an expectation value of \\mbox{$v_0 = 230$ km\/s},\n which differs (slightly) from the true (input) one,\n is used,\n the value of the fitting parameter $v_0$\n could still be pinned down precisely\n with a tiny systematic deviation\n ($<$ 2 km\/s).\n\n\\plotGeSiGeGauGauflat\n\\plotGeSiGeGauGauGau\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-flat}\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-Gau},\n we consider the case that\n the WIMP mass $\\mchi$\n needed in Eqs.~(\\ref{eqn:vsn}) and (\\ref{eqn:calN_sum})\n is reconstructed by the algorithmic procedure\n developed in Ref.~\\cite{DMDDmchi}\n with a $\\rmXA{Si}{28}$ target\n and a second $\\rmXA{Ge}{76}$ target.\n Note that,\n while the vertical bars show\n the 1$\\sigma$ statistical uncertainties\n estimated by Eq.~(\\ref{eqn:sigma_f1v_Qs_mu})\n taking into account an extra contribution from\n the 1$\\sigma$ statistical uncertainty\n on the reconstructed WIMP mass,\n the horizontal bars shown here\n indicate the 1$\\sigma$ statistical uncertainties\n on the estimates of $v_{s, \\mu}$\n given in Eq.~(\\ref{eqn:vsn})\n due to the uncertainty on the reconstructed WIMP mass;\n the statistical and systematic uncertainties\n due to estimating of $Q_{s, \\mu}$\n have been neglected here.\n\n It can be seen that,\n due to the extra {\\em statistical fluctuation}\n on the reconstructed WIMP mass\n \\cite{DMDDmchi}\n and in turn\n the uncertainty on the estimated $v_{s, \\mu}$\n (the horizontal bars in the left frames),\n the statistical uncertainties on the Bayesian reconstructed $v_0$\n with both of the flat and the Gaussian probability distributions\n become \\mbox{$\\sim 30$\\%} to \\mbox{$\\sim 60$\\%} larger.\n However,\n same as the case with an input WIMP mass,\n the use of the Gaussian probability distribution\n can not only reduce\n the statistical uncertainty on $v_0$ significantly\n and therefore improve\n the Bayesian reconstructed velocity distribution,\n but also alleviate\n the ``imprecisely'' expected value of $v_0$.\n\n In Table \\ref{tab:results_Gau_Gau},\n we list the reconstructed results of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the {\\em median} values of $v_0$.\n It would be worth to emphasize that,\n the statistical uncertainties shown\n in Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-Gau}\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-Gau}\n are (much) smaller than the {\\em input} uncertainty\n on the expectation value of $v_0$ of \\mbox{20 km\/s}.\n\n\\subsection{Shifted Maxwellian velocity distribution}\n\n\\begin{table}[t!]\n\\InsertSetupTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{Simple\n & $v_0$      [km\/s] & $\\sim 295$ & [160,    400] &   280 & 40   \\\\\n \\hline\n 1--para. shifted\n & $v_0$      [km\/s] &        220 & [160,    300] &   230 & 20   \\\\\n \\hline\n \\multirow{2}{*}{Shifted}\n & $v_0$      [km\/s] &        220 & [160,    300] &   230 & 20   \\\\\n & $\\ve$      [km\/s] &        231 & [160,    300] &   245 & 20   \\\\\n \\hline\n \\multirow{2}{*}{Variated shifted}\n & $v_0$      [km\/s] &        220 & [160,    300] &   230 & 20   \\\\\n & $\\Delta v$ [km\/s] &         11 & [$-50$,   80] &    15 & 20   \\\\\n}\n{The input setup for\n the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n used for generating WIMP signals\n as well as\n the theoretically estimated values,\n the scanning ranges,\n the expectation values of\n and the 1$\\sigma$ uncertainties on\n the fitting parameters\n used for different fitting velocity distribution functions.\n}\n{tab:setup_sh}\n\n By taking into account\n the orbital motion of the Solar system around our Galaxy\n as well as\n that of the Earth around the Sun,\n a more realistic\n shifted Maxwellian velocity distribution of halo WIMPs\n has been given by\n \\cite{SUSYDM96, DMDDf1v}:\n\\beq\n   f_{1, \\sh}(v)\n = \\frac{1}{\\sqrt{\\pi}} \\afrac{v}{v_0 \\ve}\n   \\bBig{  e^{-(v - \\ve)^2 \/ v_0^2}\n         - e^{-(v + \\ve)^2 \/ v_0^2}  }\n\\~.\n\\label{eqn:f1v_sh}\n\\eeq\n Here\n $\\ve$ is the time-\u2013dependent\n Earth's velocity in the Galactic frame\n \\cite{Freese,\n       SUSYDM96}:\n\\beq\n   \\ve(t)\n = v_0 \\bbrac{1.05 + 0.07 \\cos\\afrac{2 \\pi (t - t_{\\rm p})}{1~{\\rm yr}}}\n\\~,\n\\label{eqn:ve}\n\\eeq\n with $t_{\\rm p} \\simeq$ June 2nd is the date\n on which the velocity of the Earth relative to the WIMP halo is maxima\n\\footnote{\n In our simulations,\n the time dependence of the Earth's velocity in the Galactic frame,\n the second term of $\\ve(t)$,\n will be ignored,\n i.e.~$\\ve = 1.05 \\~ v_0$\n is used.\n}.\n\n In Table \\ref{tab:setup_sh},\n we list the input setup for\n the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n used for generating WIMP signals\n as well as\n the theoretically estimated values,\n the scanning ranges,\n the expectation values of\n and the 1$\\sigma$ uncertainties on\n the fitting parameters \n used for different fitting velocity distribution functions.\n\n\\subsubsection{Simple Maxwellian velocity distribution}\n\n\\plotGeSiGeshGauflat\n\n\\plotGeSiGeshGauGau\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 296.8 & $296.8\\~^{+21.6}_{-19.2}\\~(^{+45.6}_{-36.0})$ & [277.6, 318.4] & [260.8, 342.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 294.4 & $292.0\\~^{+16.8}_{-12.0}\\~(^{+31.8}_{-26.4})$ & [280.0, 308.8] & [265.6, 323.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 299.2 & $296.8\\~^{+40.8}_{-31.2}\\~(^{+86.4}_{-60.0})$ & [265.6, 337.6] & [236.8, 383.2] \\\\\n \\cline{3-7}\n & & Gaussian &\n 294.4 & $294.4 \\pm 26.4\\~(\\pm 52.8)$ & [268.0, 320.8] & [241.6, 347.2] \\\\\n}\n{The reconstructed results of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_sh_Gau}\n\n We consider first the simplest case of\n the simple Maxwellian velocity distribution function $f_{1, \\Gau}(v)$\n with the unique fitting parameter $v_0$\n to fit the reconstructed--input data points\n given by Eqs.~(\\ref{eqn:f1v_Qsn}) and (\\ref{eqn:cov_f1v_Qs_mu}).\n\n As in Sec.~3.1.1,\n in Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}\n we use first the flat probability distribution\n for the fitting parameter $v_0$\n with either the precisely known (input) (upper)\n or the reconstructed (lower) WIMP mass,\n respectively.\n Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}(a)\n and\n (c)\n show clearly that,\n although an ``improper'' choice for\n the fitting velocity distribution function\n and a simple flat probability distribution for $v_0$\n (i.e.~without any prior knowledge about $v_0$)\n have been used,\n the 1$\\sigma$ statistical uncertainty bands\n of the reconstructed WIMP velocity distribution function\n could in principle still cover the true (input) distribution.\n More precisely and quantitatively,\n the deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n are only \\mbox{$\\sim 10$ km\/s}.\n Note that\n the 1$\\sigma$ statistical uncertainty\n on the reconstructed fitting parameter $v_0$\n is \\mbox{$\\sim 20$ km\/s} or even \\mbox{$\\sim 40$ km\/s}\n (see Table \\ref{tab:results_sh_Gau}).\n\n However,\n our simulations show also that,\n with an ``improper'' assumption about\n the fitting velocity distribution function,\n one would obtain an ``unexpected'' result\n for the fitting parameter $v_0$:\n 2.5$\\sigma$\n (with the reconstructed WIMP mass,\n  Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}(d))\n to 4$\\sigma$\n (with the input WIMP mass,\n  Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}(b))\n deviations of the\n reconstructed Solar Galactic velocity\n from the theoretical estimate of\n \\mbox{$v_0 \\approx 220$ km\/s}.\n Such observation would indicate clearly that\n our initial assumption about\n the fitting velocity distribution function\n would be incorrect \n or at least need to be modified.\n\n Moreover,\n in Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-Gau}\n we assume that\n a rough prior knowledge about\n the Solar Galactic velocity $v_0$ exists\n and use the Gaussian probability distribution\n for $v_0$\n with an expectation value of \\mbox{$v_0 = {\\it 280}$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{{\\em 40} km\/s}.\n As observed in Sec.~3.1.1,\n with a prior knowledge about the fitting parameter $v_0$,\n one could reconstruct the velocity distribution function better:\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $v_0$\n could be reduced to \\mbox{$\\sim 60$\\%}.\n Additionally,\n the reconstructed 1$\\sigma$ statistical uncertainties on $v_0$\n with both of the input and the reconstructed WIMP masses\n are much smaller than\n the input 1$\\sigma$ value of \\mbox{40 km\/s}.\n\n In Table \\ref{tab:results_sh_Gau},\n we list the reconstructed values of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values of $v_0$.\n\n\\subsubsection{One--parameter shifted Maxwellian velocity distribution}\n\n\\plotGeSiGeshshvflat\n\n\\plotGeSiGeshshvGau\n\n In the previous Sec.~3.2.1,\n we have found that,\n by assuming (improperly)\n the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$,\n in both cases with and without an expectation value of\n the fitting parameter $v_0$,\n one would obtain a {\\em much higher} reconstruction result:\n \\mbox{$v_{\\rm 0, rec} \\simeq 295$ km\/s},\n which is 2$\\sigma$ to 4$\\sigma$ apart from\n the theoretical estimate of\n \\mbox{$v_0 \\approx 220$ km\/s}.\n This observation implies the need of\n a more suitable fitting velocity distribution function.\n Hence,\n as the second trial,\n we consider now the use of\n the {\\em shifted} Maxwellian velocity distribution $f_{1, \\sh}(v)$\n given in Eq.~(\\ref{eqn:f1v_sh})\n with {\\em only one} fitting parameter,\n i.e.~the Solar Galactic velocity $v_0$.\n In this case,\n we fix simply that\n\\beq\n   \\ve\n = 1.05\\~v_0\n\\~,\n\\label{eqn:ve_ave}\n\\eeq\n and neglect the time--dependence of $\\ve(t)$\n\\footnote{\n Note that\n hereafter we use $f_{1, \\sh, v_0}(v)$\n to denote the ``one--parameter''\n shifted Maxwellian velocity distribution function,\n in order to distinguish this\n from the ``original'' one\n given in Eq.~(\\ref{eqn:f1v_sh})\n with $v_0$ and $\\ve$\n as two independent fitting parameters.\n}\n\n In Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-flat},\n we consider first the flat probability distribution\n for the fitting parameter $v_0$\n with either the precisely known (input) (upper)\n or the reconstructed (lower) WIMP mass,\n respectively.\n It can be seen clearly that,\n with a more suitable assumption about\n the fitting function,\n one could indeed reconstruct the WIMP velocity distribution\n much closer to the true (input) one.\n Although\n no prior knowledge about $v_0$\n is used,\n this most important characteristic parameter\n could in principle be pinned down very precisely:\n the difference between the {\\em median} values of\n the reconstructed $v_0$\n and the true (input) one\n would be \\mbox{$\\lsim\\~{\\it 5}$ km\/s}\n (see also Table \\ref{tab:results_sh_sh_v0}).\n\n In addition,\n with the input WIMP mass,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the reconstructed $v_0$\n are only \\mbox{$^{+14.0}_{-11.2}\\~(^{+29.4}_{-23.8})$ km\/s}.\n Even with the reconstructed WIMP mass,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n could still be limited as small as only\n \\mbox{$^{+25.2}_{-22.4}\\~(^{+56.0}_{-44.8})$ km\/s}.\n It would be worth to emphasize that,\n compared to the uncertainty\n on the astronomical measurement of $v_0$\n of \\mbox{$\\sim 20$ km\/s} (or probably larger),\n the result offered by our Bayesian reconstruction method\n would be a pretty precise estimate\n and could help us to confirm\n the astronomical measurement of $v_0$.\n\n Moreover,\n in Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau}\n we give the reconstruction results\n with the Gaussian probability distribution\n for $v_0$\n with an expectation value of \\mbox{$v_0 = 230$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{20 km\/s}.\n As summarized in Table \\ref{tab:results_sh_sh_v0},\n with a prior knowledge of the parameter $v_0$,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n could be reduced significantly\n to be \\mbox{$\\lsim$ 70\\%}.\n Remind here that\n the expectation value of the Gaussian probability distribution\n of $v_0$\n has been set as \\mbox{$v_0 = 230$ km\/s},\n a bit different from the input value.\n Nevertheless,\n our simulations show that\n this ``artificial'' (systematic) error\n could be corrected in our reconstruction process for $v_0$.\n\n In Table \\ref{tab:results_sh_sh_v0},\n we list the reconstructed results of $v_0$\n for all four considered cases\n with the one--parameter shifted Maxwellian velocity distribution\n $f_{1, \\sh, v_0}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values of $v_0$.\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 217.4 & $217.4\\~^{+14.0}_{-11.2}\\~(^{+29.4}_{-23.8})$ & [206.2, 231.4] & [193.6, 246.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 221.6 & $221.6\\~^{+ 9.8}_{- 8.4}\\~(\\pm 18.2)$ & [213.2, 231.4] & [203.4, 239.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 218.8 & $218.8\\~^{+25.2}_{-22.4}\\~(^{+56.0}_{-44.8})$ & [196.4, 244.0] & [174.0, 274.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 221.6 & $221.6\\~^{+16.8}_{-15.4}\\~(\\pm 33.6)$ & [206.2, 238.4] & [188.0, 255.2] \\\\\n}\n{The reconstructed results of $v_0$\n for all four considered cases\n with the one--parameter shifted Maxwellian velocity distribution\n $f_{1, \\sh, v_0}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_sh_sh_v0}\n\n\\subsubsection{Shifted Maxwellian velocity distribution}\n\n\\plotGeshshGau\n\\plotGeSiGeshshGau\n\n Now we release the fixed relation between $v_0$ and $\\ve$\n given in Eq.~(\\ref{eqn:ve_ave})\n and consider the reconstruction of these two parameters\n {\\em simultaneously} and {\\em independently}.\n In addition,\n we assume here that,\n from the (naive) trials with the simple and one--parameter shifted\n Maxwellian velocity distributions\n done previously,\n one could already obtain a rough idea about\n the shape of the velocity distribution of halo WIMPs.\n This information would in turn give us\n the prior knowledge about\n the expectation values of the Solar and Earth's Galactic velocities\n $v_0$ and $\\ve$.\n Hence,\n we consider here only the Gaussian probability distribution\n for both fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\ve = 245$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n (see Table \\ref{tab:setup_sh}\n\\footnote{\n Remind that\n both of the expectation values of the fitting parameters $v_0$ and $\\ve$\n differ slightly from the true (input) values.\n Note also that\n the time--dependence of the Earth's Galactic velocity\n is ignored here\n and $\\ve$ is thus treated as\n a {\\em time--independent} fitting parameter.\n}.\n\n In Fig.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}(a),\n we show the reconstructed\n shifted Maxwellian velocity distribution function\n as well as\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands.\n Here the true (input) WIMP mass has been used.\n Comparing to Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau}(a),\n it can be seen clearly that,\n with a prior knowledge about\n the Solar and Earth's Galactic velocities $v_0$ and $\\ve$,\n one could in principle reconstruct the velocity distribution function\n with two fitting parameters\n more precisely:\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n are much thinner and\n the deviations of $v_0$ and $\\ve$ are {\\em only a few} km\/s\n (see also Table \\ref{tab:results_sh_sh}\n\\footnote{\n Note however that,\n for using velocity distributions\n with two or more fitting parameters\n {\\em without} constraints on these parameters\n (i.e.~the use of the ``flat'' probability distribution),\n the distribution of the reconstructed results\n by our Bayesian analysis\n would be pretty wide,\n a part of them would even be\n on the boundary of the scanning ranges of these parameters.\n}.\n\n Fig.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}(b) shows\n the distribution of\n the Bayesian reconstructed fitting parameter $v_0$ and $\\ve$\n in all simulated experiments\n on the $v_0 - \\ve$ plane.\n The light--green (light--blue, gold) points\n indicate the $1\\~(2)\\~(> 2)\\~\\sigma$ areas\n of the reconstructed combination of $v_0$ and $\\ve$.\n Note here that\n these $1\\~(2)\\~(> 2)\\~\\sigma$ areas are determined\n according to the {\\em descending} order of the ${\\rm p}_{\\rm median}$ values\n of the reconstructed combination of $v_0$ and $\\ve$.\n This means that\n the light--green (light--blue, gold) areas\n are the reconstructed combinations of $v_0$ and $\\ve$\n which give the largest 68.27\\% (95.45\\%) ${\\rm p}_{\\rm median}$ values\n in all of the simulated experiments.\n\n Moreover,\n the red upward--triangle indicates\n the input values of $v_0$ and $\\ve$,\n which has been labeled with the subscript ``gen''.\n The green disk shows\n the median values of the simulated results,\n whereas\n and blue downward--triangle\n the point which maximizes\n ${\\rm p}_{\\rm median}$.\n In addition,\n the thick (thin) green crosses show\n the $1\\~(2)\\~\\sigma$ (68.27\\% (95.45\\%)) ranges of the reconstructed results\n according to the order of\n the reconstructed values of $v_0$ or $\\ve$ {\\em along}\n (centered at their median values\n  $v_{0, {\\rm median}}$ or $v_{{\\rm e, median}}$).\n\n Additionally,\n in Figs.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}(c) and (d),\n we give\n the distributions of\n the Bayesian reconstructed fitting parameters $v_0$ and $\\ve$\n as well as\n the $1\\~(2)\\~\\sigma$ statistical uncertainty ranges\n on the $v_0-$ and $\\ve-$ axes\n {\\em separately}.\n Note here that\n we project all reconstructed combinations of $\\abig{v_0, \\ve}$\n on the $v_0-$ or $\\ve-$axis.\n Comparing Fig.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}(c)\n to Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau}(b),\n it can be found that,\n while\n the deviations of $v_0$ are a little bit larger\n then the results\n with the ``one--parameter'' fitting velocity distribution,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n are reduced to \\mbox{$\\sim$ 70\\%}\n (see Table \\ref{tab:results_sh_sh} and\n  Table \\ref{tab:results_sh_sh_v0}).\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 223.0 & $223.0 \\pm  7.0\\~(\\pm 14.0)$ & [216.0, 230.0] & [209.0, 237.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 223.0 & $223.0 \\pm 12.6\\~(^{+26.6}_{-23.8})$ & [210.4, 235.6] & [199.2, 249.6] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 238.4 & $238.4\\~^{+ 7.0}_{- 8.4}\\~(^{+15.4}_{-16.9})$ & [230.0, 245.4] & [221.6, 253.8] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 238.4 & $238.4 \\pm 12.6\\~(^{+25.2}_{-26.6})$ & [225.8, 251.0] & [211.8, 263.6] \\\\\n}\n{The reconstructed results of $v_0$ and $\\ve$\n with the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_sh_sh}\n\n As a comparison,\n in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau}\n we use the reconstructed WIMP mass.\n Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau}(b)\n shows that,\n as expected,\n the distribution of $v_0$ and $\\ve$ on the $v_0 - \\ve$ plane\n becomes wider and\n extends toward the directions of\n higher (lower)--$v_0$ and higher (lower)--$\\ve$.\n Nevertheless,\n the ``best--fit'' combination of them\n which maximizes ${\\rm p}_{\\rm median}$\n as well as\n the median values of $v_0$ and $\\ve$\n consistant very well\n with the true (input) values.\n\n In Table \\ref{tab:results_sh_sh_v0},\n we list the reconstructed results of $v_0$ and $\\ve$\n with the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n\n\\subsubsection{Variated shifted Maxwellian velocity distribution}\n\n\\plotGeshshDvGau\n\\plotGeSiGeshshDvGau\n\n In the previous Sec.~3.2.3,\n it has been found that,\n by using the shifted Maxwellian velocity distribution\n given in Eq.~(\\ref{eqn:f1v_sh})\n with two fitting parameters:\n $v_0$ and $\\ve$,\n one could reconstruct\n the ($1\\~(2)\\~\\sigma$ statistical uncertainty bands of the)\n velocity distribution function\n as well as\n pin down the Solar Galactic velocity $v_0$\n pretty precisely.\n However,\n as shown in Figs.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}(d)\n and \\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau}(d),\n the deviations of the reconstructed Earth's Galactic velocity $\\ve$\n from the true (input) value\n seem to be (much) larger than\n the deviations of the reconstructed $v_0$.\n This might be caused by\n the strong (anti--)correlation\n between $v_0$ and $\\ve$.\n Hence,\n we consider now a variation of\n the shifted Maxwellian distribution function,\n in the hope that\n this pretty large systematic deviation of $\\ve$\n could be reduced.\n\n We rewrite the shifted Maxwellian velocity distribution\n given in Eq.~(\\ref{eqn:f1v_sh})\n to the following ``variated'' form:\n\\beq\n   f_{1, \\sh, \\Delta v}(v)\n = \\frac{1}{\\sqrt{\\pi}} \\bfrac{v}{v_0 \\abrac{v_0 + \\Delta v}}\n   \\cbigg{  e^{-\\bbrac{v - \\abrac{v_0 + \\Delta v}}^2 \/ v_0^2}\n          - e^{-\\bbrac{v + \\abrac{v_0 + \\Delta v}}^2 \/ v_0^2}  }\n\\~,\n\\label{eqn:f1v_sh_Dv}\n\\eeq\n where\n\\beq\n        \\Delta v\n \\equiv \\ve - v_0\n\\label{eqn:Delta_v}\n\\eeq\n is the difference between $v_0$ and $\\ve(t)$\n\\footnote{\n As in the previous Sec.~3.2.3,\n the time--dependence of the Earth's Galactic velocity\n is ignored here\n and $\\Delta v$ is thus treated as\n a {\\em time--independent} fitting parameter.\n}\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$}\n{\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 221.6 & $221.6 \\pm  8.4\\~(\\pm 15.4)$ & [213.2, 230.0] & [206.2, 237.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 221.6 & $221.6 \\pm 14.0\\~(\\pm 29.4)$ & [207.6, 235.6] & [192.2, 251.0] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n  11.1 & $ 11.1 \\pm  5.2\\~(^{+10.4}_{-11.7})$ & [5.9,  16.3] & [$- 0.6$,  21.5] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n  11.1 & $ 11.1 \\pm  7.8\\~(^{+15.6}_{-14.3})$ & [3.3,  18.9] & [$- 3.2$,  26.7] \\\\\n}\n{The reconstructed results of $v_0$ and $\\Delta v$\n with the variated shifted Maxwellian velocity distribution\n $f_{1, \\sh, \\Delta v}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_sh_sh_Dv}\n\n As in the previous Sec.~3.2.3,\n we assume here that,\n we have already a rough idea about\n the shape of the velocity distribution of halo WIMPs,\n and thus\n the prior knowledge about\n the expectation values of the (difference between the)\n Solar and Earth's Galactic velocities\n $v_0$ and $\\Delta v$.\n Hence,\n we consider here only the Gaussian probability distribution\n for both of the fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\Delta v = 15$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n (see Table \\ref{tab:setup_sh}).\n Two cases with both of\n the true (input) and the reconstructed WIMP masses\n have been considered.\n\n By comparing Figs.~\\ref{fig:f1v-Ge-100-0500-sh-sh_Dv-Gau}\n to Figs.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}\n and Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_Dv-Gau}\n to Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau},\n it can be seen obviously that\n the reconstructed velocity distribution function\n could indeed match the true (input) one much precisely,\n with however\n (slightly) wider $1\\~(2)\\~\\sigma$ statistical uncertainty bands.\n The systematic deviations of\n both fitting parameters\n from the true (input) values\n would also be much smaller than\n those given in the $v_0 - \\ve$ case.\n This implies importantly that,\n the use of our variated shifted velocity distribution function\n given in Eq.~(\\ref{eqn:f1v_sh_Dv})\n could indeed offer an estimate of\n the Earth's Galactic velocity $\\ve = v_0 + \\Delta v$\n with a much higher precision.\n Such a trick would be helpful to improve\n the estimation(s) of the Earth's Galactic velocity $\\ve$\n (and\/or other fitting parameter(s)).\n\n Note however that,\n as shown in Tables \\ref{tab:results_sh_sh}\n and \\ref{tab:results_sh_sh_Dv},\n by using the variated shifted Maxwellian velocity distribution,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the reconstructed $v_0$\n would be \\mbox{$\\sim$ 10\\%} larger.\n Meanwhile,\n from Eq.~(\\ref{eqn:Delta_v})\n the statistical uncertainty on $\\ve$\n can be estimated b\n\\footnote{\n Since,\n according to Eq.~(\\ref{eqn:Delta_v}),\n for a fixed value of $\\ve$\n two fitting parameters $v_0$ and $\\Delta v$\n should be ``anti--correlated''.\n}\n\\beqn\n      \\sigma\\abrac{\\ve}\n \\=   \\sqrt{  \\sigma^2\\abrac{v_0}\n            + \\sigma^2\\abrac{\\Delta v}\n            + 2 \\~ {\\rm cov}\\abrac{v_0, \\Delta v} \\sigma\\abrac{v_0} \\sigma\\abrac{\\Delta v}}\n      \\non\\\\\n \\eqnle  \\sqrt{  \\sigma^2\\abrac{v_0}\n            + \\sigma^2\\abrac{\\Delta v}}\n\\~.\n\\label{eqn:sigma_ve}\n\\eeqn\n Then,\n according to the results\n given in Table \\ref{tab:results_sh_sh_Dv},\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $\\ve$\n would be maximal \\mbox{$\\lsim~15\\%$} enlarged,\n whereas\n the systematic deviation of $\\ve$\n is much smaller.\n\n In Table \\ref{tab:results_sh_sh_Dv},\n we list the reconstructed results of $v_0$ and $\\Delta v$\n with the variated shifted Maxwellian velocity distribution\n $f_{1, \\sh, \\Delta v}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n\n\\subsection{Modified Maxwellian velocity distribution}\n\n In this subsection,\n we consider another recently often used theoretical\n one--dimensional WIMP velocity distribution function.\n In Refs.~\\cite{Lisanti10, Pato},\n a modification of the simple Maxwellian velocity distribution\n in Eq.~(\\ref{eqn:f1v_Gau})\n has been suggested:\n\\beq\n   f_{1, \\Gau, k}(v)\n = \\left\\{\\renewcommand{\\arraystretch}{1.5}\n          \\begin{array}{l c c}\n          \\frac{v^2}{N_{f, k}}\n          \\abrac{  e^{-v^2     \/ k v_0^2}\n                 - e^{-\\vmax^2 \/ k v_0^2}  }^k,\n          & ~~~~~~~~~~~~ &\n          ({\\rm for}~v \\le \\vmax), \\\\\n          0,\n          & ~~~~~~~~~~~~ &\n          ({\\rm for}~v >   \\vmax). \\\\\n          \\end{array}\n   \\right.\n\\label{eqn:f1v_Gau_k}\n\\eeq\n Here\n $\\vmax$ is the maximal cut-\u2013off\n velocity of $f_{1, \\Gau, k}(v)$\n and $N_{f, k}$ is the normalization constant\n depending on the value of the power index $k$.\n\n\\plotfvGauk\n\n\\begin{table}[b!]\n\\InsertSetupTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k}(v)$}\n{Simple\n & $v_0$      [km\/s] & $\\sim 220$ & [160,    300] &   230 & 20   \\\\\n\\hline\n 1--para. shifted\n & $v_0$      [km\/s] & $\\sim 175$ & [100,    300] &   200 & 40   \\\\\n\\hline\n \\multirow{2}{*}{Shifted}\n & $v_0$      [km\/s] & $\\sim 175$ & [100,    300] &   200 & 40   \\\\\n & $\\ve$      [km\/s] & $\\sim 185$ & [50,     300] &   200 & 40   \\\\\n\\hline\n \\multirow{2}{*}{Variated shifted}\n & $v_0$      [km\/s] & $\\sim 175$ & [100,    300] &   200 & 40   \\\\\n & $\\Delta v$ [km\/s] & $\\sim  10$ & [$-120$,  80] & $-20$ & 40   \\\\\n\\hline\n \\multirow{2}{*}{Modified}\n & $v_0$      [km\/s] &        220 & [160,    300] &   230 & 20   \\\\\n & $k$               &          2 & [0.5,    3.5] &     1 &  0.5 \\\\\n}\n{The input setup for\n the modified Maxwellian velocity distribution $f_{1, \\Gau, k}(v)$\n used for generating WIMP signals\n as well as\n the theoretically estimated values,\n the scanning ranges,\n the expectation values of\n and the 1$\\sigma$ uncertainties on\n the fitting parameters\n used for different fitting velocity distribution functions.\n}\n{tab:setup_Gau_k}\n\n In Fig.~\\ref{fig:f1v_Gau_k},\n we give\n the {\\em normalized}\n modified Maxwellian velocity distribution function $f_{1, \\Gau, k}(v)$\n with a common value of the Solar Galactic velocity\n \\mbox{$v_0 = 220$ km\/s}\n and different power indices $k$:\n $k = 1$ (dashed light--green),\n $k = 2$ (solid red),\n $k = 3$ (dash--dotted blue) and\n $k = 4$ (dotted magenta).\n As a comparison,\n the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n with \\mbox{$v_0 = 220$ km\/s}\n is also given\n as the short--dashed black curve.\n\n In Table \\ref{tab:setup_Gau_k},\n we list the input setup for\n the modified Maxwellian velocity distribution $f_{1, \\Gau, k}(v)$\n used for generating WIMP signals\n as well as\n the theoretically estimated values,\n the scanning ranges,\n the expectation values of\n and the 1$\\sigma$ uncertainties on\n the fitting parameters\n used for different fitting velocity distribution functions.\n\n\\subsubsection{Simple Maxwellian velocity distribution}\n\n\\plotGeSiGeGaukGauflat\n\n\\plotGeSiGeGaukGauGau\n\n As in Sec.~3.2,\n we start to fit to the reconstructed--input data points\n given by Eqs.~(\\ref{eqn:f1v_Qsn}) and (\\ref{eqn:cov_f1v_Qs_mu})\n with the simple Maxwellian velocity distribution function $f_{1, \\Gau}(v)$.\n\n In Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-flat},\n we use first the flat probability distribution\n for the fitting parameter $v_0$\n with either the precisely known (input) (upper)\n or the reconstructed (lower) WIMP mass,\n respectively.\n As shown in Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat}\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-flat},\n although\n prior knowledge about the Solar Galactic velocity is {\\em absent},\n one could in principle pin down the fitting parameter $v_0$ very precisely\n with systematic deviations of {\\em only a few} km\/s and\n $1\\~(2)\\~\\sigma$ statistical uncertainties of only\n \\mbox{$^{+11.2}_{-12.6}\\~(^{+36.4}_{-22.4})$ km\/s} and\n \\mbox{$^{+21.0}_{-18.2}\\~(^{+53.2}_{-32.2})$ km\/s}.\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k = 2}(v)$}\n{simple   Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 217.4 & $217.4\\~^{+11.2}_{-12.6}\\~(^{+36.4}_{-22.4})$ & [204.8, 228.6] & [195.0, 253.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 218.8 & $220.2 \\pm  9.8\\~(^{+26.6}_{-18.2})$ & [210.4, 230.0] & [202.0, 246.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 218.8 & $217.4\\~^{+21.0}_{-18.2}\\~(^{+53.2}_{-32.2})$ & [199.2, 238.4] & [185.2, 270.6] \\\\\n \\cline{3-7}\n & & Gaussian &\n 220.2 & $220.2\\~^{+15.4}_{-14.0}\\~(^{+36.4}_{-26.6})$ & [206.2, 235.6] & [193.6, 256.6] \\\\\n}\n{The reconstructed results of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_Gau_k_Gau}\n\n In Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-Gau},\n we use the Gaussian probability distribution\n for $v_0$\n with an expectation value of \\mbox{$v_0 = 230$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{20 km\/s}.\n As summarized in Table \\ref{tab:results_Gau_k_Gau},\n the statistical uncertainties on the reconstructed $v_0$\n could now in principle be reduced by\n \\mbox{$\\sim 10$\\%} to \\mbox{$\\sim 30$\\%}.\n In addition,\n even with the reconstructed WIMP mass\n and thus a larger statistical fluctuation,\n the 1$\\sigma$ statistical uncertainties shown\n in Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-Gau}\n are (much) smaller than the given uncertainty\n on the expectation value of $v_0$ (\\mbox{20 km\/s}).\n Moreover,\n although an expectation value of \\mbox{$v_0 = 230$ km\/s},\n which differs (slightly) from the true (input) one,\n is used,\n the value of the fitting parameter $v_0$\n could still be pinned down precisely\n with a negligible systematic deviation.\n\n Furthermore,\n remind here that,\n either,\n as shown in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-flat},\n two reconstructed\n (dashed green and dash-dotted blue)\n velocity distribution functions\n match the true (input) (solid red) one\n very well,\n but the reconstructed Solar Galactic velocity $v_0$\n would shift slightly away from the true (input) value,\n or,\n as shown in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-Gau},\n $v_0$ could be determined very precisely,\n but the reconstructed velocity distribution functions\n would differ slightly from the true (input) one.\n This is because of the fact that\n we have {\\em artificially} used an input value of $k = 2$\n and,\n as shown in Fig.~\\ref{fig:f1v_Gau_k},\n the modified Maxwellian velocity distribution function\n with $k = 2$\n is slightly different from the simple Maxwellian distribution.\n\n In Table \\ref{tab:results_Gau_k_Gau},\n we list the reconstructed results of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values of $v_0$.\n\n\\subsubsection{One--parameter shifted Maxwellian velocity distribution}\n\n\\plotGeSiGeGaukshvflat\n\n\\plotGeSiGeGaukshvGau\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k = 2}(v)$}\n{one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 162.0 & $160.0\\~^{+10.0}_{- 8.0}\\~(^{+28.0}_{-14.0})$ & [152.0, 170.0] & [146.0, 188.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 162.0 & $162.0\\~^{+10.0}_{- 8.0}\\~(^{+26.0}_{-14.0})$ & [154.0, 172.0] & [148.0, 188.0] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 162.0 & $162.0 \\pm 14.0\\~(^{+36.0}_{-24.0})$ & [148.0, 176.0] & [138.0, 198.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 164.0 & $162.0\\~^{+16.0}_{-12.0}\\~(^{+38.0}_{-24.0})$ & [150.0, 178.0] & [138.0, 200.0] \\\\\n}\n{The reconstructed results of $v_0$\n for all four considered cases\n with the one--parameter shifted Maxwellian velocity distribution\n $f_{1, \\sh, v_0}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_Gau_k_sh_v0}\n\n Now,\n as a comparison of Sec.~3.3.1,\n we consider as the next trail the reconstruction with\n the one--parameter shifted Maxwellian velocity distribution function\n to fit the modified simple Maxwellian velocity distribution.\n\n As usual,\n in Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_v0-flat}\n we use first the flat probability distribution\n for the fitting parameter $v_0$\n with either the precisely known (input) (upper)\n or the reconstructed (lower) WIMP mass,\n respectively.\n It can be seen that,\n firstly,\n although the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n could still cover the true (input) velocity distribution\n (somehow well),\n the systematic deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n would be \\mbox{$\\sim 15$ km\/s}.\n The comparisons between reconstructed results shown in\n Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_v0-flat}(a) and (c)\n to Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-flat}(a) and (c)\n could indicate a high probability of the improper assumption of\n the fitting\n (one--parameter) shifted Maxwellian velocity distribution.\n\n Moreover,\n Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_v0-flat}(b) and (d)\n (see also Table \\ref{tab:results_Gau_k_sh_v0})\n show clearly 4$\\sigma$ to even 6$\\sigma$ deviations of\n the reconstructed $v_0$\n from the true (input) value of \\mbox{$v_0 = 220$ km\/s}.\n As discussed in Sec.~3.2.1,\n this observation implies also\n an improper use of\n the (one--parameter) shifted Maxwellian velocity distribution\n as our fitting function.\n\n In Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_v0-Gau},\n we use the Gaussian probability distribution\n for $v_0$\n with a {\\em slightly smaller} expectation value of \\mbox{$v_0 = {\\it 200}$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{{\\em 40} km\/s}.\n In contrast to our observations presented before,\n for this case\n the use of the Gaussian probability distribution\n would {\\em not} reduce the $1\\~(2)\\~\\sigma$ statistical uncertainties\n for both cases with the true (input) and\n the reconstructed WIMP masses\n (see Table \\ref{tab:results_Gau_k_sh_v0}).\n This would be caused by\n the large difference between\n the given expectation value\n and the (theoretically estimated) most suitable one of the parameter $v_0$\n (\\mbox{175 km\/s},\n  see Table \\ref{tab:setup_Gau_k}).\n This would indicate that,\n in practical use of our Bayesian reconstruction method,\n some trial--and--error tests\n for determining a more suitable expectation value of $v_0$\n (and the other fitting parameters)\n would be necessary and\n could then improve the fitting results.\n\n In Table \\ref{tab:results_Gau_k_sh_v0},\n we list the reconstructed results of $v_0$\n for all four considered cases\n with the one--parameter shifted Maxwellian velocity distribution\n $f_{1, \\sh, v_0}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values of $v_0$.\n\n\\subsubsection{Shifted Maxwellian velocity distribution}\n\n\\plotGeGaukshGau\n\\plotGeSiGeGaukshGau\n\n As in Sec.~3.2.3,\n we release now the fixed relation between $v_0$ and $\\ve$\n given in Eq.~(\\ref{eqn:ve_ave})\n and consider the reconstruction of these two parameters\n simultaneously and independently.\n In addition,\n we also assume here that,\n from the (naive) trials with the simple and one--parameter shifted\n Maxwellian velocity distributions\n done previously,\n one could already obtain a rough idea about\n the shape of the velocity distribution of halo WIMPs\n as well as\n expectation values of the Solar and Earth's Galactic velocities\n $v_0$ and $\\ve$.\n Hence,\n we consider here only the Gaussian probability distribution\n for both fitting parameters\n with a common expectation value of \\mbox{$v_0 = \\ve = {\\it 200}$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{{\\em 40} km\/s}\n (see Table \\ref{tab:setup_Gau_k}).\n Note also that,\n after some trial--and--error tests,\n we set the scanning ranges of $v_0$ and $\\ve$\n as \\mbox{$v_0 \\in [100, 300]$ km\/s} and\n \\mbox{$\\ve \\in [50, 300]$ km\/s}.\n\n In Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh-Gau},\n we consider first the case with the true (input) WIMP mass.\n As observed in Sec.~3.3.2,\n although the (low--velocity parts of the)\n $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n could cover the true (input) velocity distribution,\n the systematic deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n would be \\mbox{$\\sim 10$ km\/s}\n (a little bit better then results shown\n  in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_v0-flat}(a) and (c)).\n Moreover,\n Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh-Gau}(c) and (d)\n show \\mbox{$\\gsim 4\\sigma$} deviations of\n the reconstructed values of $v_0$ and $\\ve$\n from the true (input, estimated) values of\n \\mbox{$v_0 = 220$ km\/s} and\n \\mbox{$\\ve = 231$ km\/s}.\n Additionally,\n the best--fit value of $\\ve$ is now even\n (\\mbox{$\\sim 10$ km\/s}) {\\em smaller} than\n the best--fit value of $v_0$.\n Hence,\n as discussed in Sec.~3.3.2,\n this observation\n (combined with the results given in Sec.~3.3.2)\n would indicate a high probability of\n the improper assumption of the fitting\n shifted Maxwellian velocity distribution.\n Moreover,\n in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh-Gau}\n we show our simulations with the reconstructed WIMP mass.\n The $1\\~(2)\\~\\sigma$ statistical uncertainties on $v_0$ and $\\ve$\n would be \\mbox{$\\sim$ 10\\%} to \\mbox{$\\sim$ 50\\%} enlarged.\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k = 2}(v)$}\n{shifted  Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 172.0 & $170.0 \\pm 12.0\\~(^{+38.0}_{-24.0})$ & [158.0, 182.0] & [146.0, 208.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 172.0 & $170.0\\~^{+18.0}_{-14.0}\\~(^{+44.0}_{-30.0})$ & [156.0, 188.0] & [140.0, 214.0] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 162.5 & $162.5\\~^{+17.5}_{-15.0}\\~(^{+42.5}_{-27.5})$ & [147.5, 180.0] & [135.0, 205.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 162.5 & $162.5\\~^{+20.0}_{-15.0}\\~(^{+48.1}_{-30.0})$ & [147.5, 182.5] & [132.5, 210.6] \\\\\n}\n{The reconstructed results of $v_0$ and $\\ve$\n with the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_Gau_k_sh}\n\n Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh-Gau} and\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh-Gau}\n as well as\n Table \\ref{tab:results_Gau_k_sh}\n show that\n one could fit an improper model\n to experimental data (somehow) well\n with combinations of special, probably unusual values of\n the fitting parameters.\n On one hand,\n the reconstructed results could offer us (rough) information\n about the (shape of the) velocity distribution of Galactic WIMPs.\n On the other hand,\n however,\n the observation that\n the reconstructed values of $v_0$ and $\\ve$\n are $2\\sigma$ to even $> 4\\sigma$ different\n from our (theoretically) expected values\n would indicate evidently the improper assumption about\n the fitting velocity distribution function.\n\n In Table \\ref{tab:results_Gau_k_sh},\n we list the reconstructed results of $v_0$ and $\\ve$\n with the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n\n\\subsubsection{Variated shifted Maxwellian velocity distribution}\n\n\\plotGeGaukshDvGau\n\\plotGeSiGeGaukshDvGau\n\n As in Sec.~3.2.4,\n in order to correct\n the systematic deviations of the results given\n with the shifted Maxwellian velocity distribution function\n in Eq.~(\\ref{eqn:f1v_sh}),\n we consider here the use of\n its variation\n given in Eq.~(\\ref{eqn:f1v_sh_Dv}).\n As previously,\n we consider here only the Gaussian probability distribution\n for both fitting parameters $v_0$ and $\\Delta v$,\n with a common 1$\\sigma$ uncertainty of \\mbox{{\\em 40} km\/s}.\n Moreover,\n according to the fitting results given in Sec.~3.3.3\n and some trial--and--error tests,\n we set \\mbox{$v_0 = {\\it 200}$ km\/s} and \\mbox{$\\Delta v = {\\it -20}$ km\/s}\n as the expectation values\n as well as\n \\mbox{$v_0 \\in [100, 300]$ km\/s} and\n \\mbox{$\\Delta v \\in [-120, 80]$ km\/s}\n as the scanning ranges.\n\n Comparing Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh_Dv-Gau}\n to Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh-Gau},\n it can be seen clearly that\n the use of the variated shifted Maxwellian velocity distribution\n could indeed offer a more reasonable and preciser reconstruction:\n not only that\n the ($1\\~(2)\\~\\sigma$ statistical uncertainty bands of the)\n reconstructed velocity distribution functions\n can match the true (input) one more closer,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $v_0$\n is also only \\mbox{$\\lsim$ 70\\%} of the uncertainties\n shown in Fig.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh-Gau}(c).\n Additionally,\n by using Eq.~(\\ref{eqn:sigma_ve}),\n the upper bound of the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on $\\ve$\n can be (approximately) given as:\n \\mbox{$^{+16.12}_{-12.80}\\~(^{+41.23}_{-24.41})$ km\/s},\n which is also maximal (approximately) equal to\n or even smaller than the uncertainties\n given in Table \\ref{tab:results_Gau_k_sh}.\n Hence,\n we would like to conclude that\n our introduction of the variated\n shifted Maxwellian velocity distribution\n (and\/or probably some other variations)\n could indeed be a useful trick\n for the practical use of our Bayesian reconstruction procedure.\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k = 2}(v)$}\n{variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$}\n{\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 180.0 & $178.0 \\pm  8.0\\~(^{+26.0}_{-14.0})$ & [170.0, 186.0] & [164.0, 204.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 180.0 & $178.0\\~^{+14.0}_{-12.0}\\~(^{+34.5}_{-22.0})$ & [166.0, 192.0] & [156.0, 212.5] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n $-34.0$ & $-32.0\\~^{+14.0}_{-10.0}\\~(^{+32.0}_{-20.0})$ & [$-42.0$, $-18.0$] & [$-52.0$,   0.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n $-32.0$ & $-32.0\\~^{+14.0}_{-10.0}\\~(^{+30.5}_{-20.0})$ & [$-42.0$, $-18.0$] & [$-52.0$,  $- 1.5$] \\\\\n}\n{The reconstructed results of $v_0$ and $\\Delta v$\n with the variated shifted Maxwellian velocity distribution\n $f_{1, \\sh, \\Delta v}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_Gau_k_sh_Dv}\n\n On the other hand,\n in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_Dv-Gau}\n we use the reconstructed WIMP mass.\n Comparing Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_Dv-Gau}\n to Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh_Dv-Gau}\n (see also Table \\ref{tab:results_Gau_k_sh_Dv}),\n although\n the width of the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n of the reconstructed velocity distribution functions\n as well as\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $v_0$\n would be clearly larger,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $\\Delta v$\n are approximately equal to results\n with the true (input) WIMP mass.\n Meanwhile,\n even for this case with the reconstructed WIMP mass,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $v_0$ and $\\Delta v$\n are (much) smaller than the given 1$\\sigma$ uncertainty\n of their Gaussian probability distributions\n (\\mbox{40 km\/s}).\n\n Remind that,\n our simulations of the use of\n the variated shifted Maxwellian velocity distribution function\n for fitting data generated by (modified) simple Maxwellian one\n shown here\n indicates again that,\n by using an ``improper'' assumption about\n the fitting velocity distribution function\n with prior knowledge about\n the fitting parameters ($v_0$, $\\ve$ or $\\Delta v$),\n one would still reconstruct\n an approximate shape of the WIMP velocity distribution;\n the deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n could even be \\mbox{$\\lsim {\\it 10}$ km\/s},\n much smaller than the commonly used 1$\\sigma$ uncertainty\n on the Solar Galactic velocity of \\mbox{$\\sim 20$ km\/s}.\n\n However,\n our simulations show also that,\n with an ``improper'' assumption about\n the fitting velocity distribution,\n one would obtain an ``unexpected'' result\n about each single fitting parameter.\n E.g.~here we get 3$\\sigma$ to 5$\\sigma$ deviations\n of the reconstructed Solar Galactic velocity\n from the theoretical estimate\n (see\n  Table\n  \\ref{tab:results_Gau_k_sh_Dv})\n and large ``negative'' values for\n the difference between the Solar and Earth's Galactic velocities.\n This observation indicates clearly that\n our initial assumption about\n the fitting velocity distribution function\n would be incorrect \n or at least need to be modified.\n\n In Table \\ref{tab:results_Gau_k_sh_Dv},\n we list the reconstructed results of $v_0$ and $\\Delta v$\n with the variated shifted Maxwellian velocity distribution\n $f_{1, \\sh, \\Delta v}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n\n\\subsubsection{Modified Maxwellian velocity distribution}\n\n\\plotGeGaukGaukGauflat\n\\plotGeSiGeGaukGaukGauflat\n\n As the last tested fitting velocity distribution function\n with data generated by the modified Maxwellian velocity distribution\n given by Eq.~(\\ref{eqn:f1v_Gau_k}),\n we consider now the reconstruction of\n the modified Maxwellian velocity distribution itself\n with two fitting parameters:\n the Solar Galactic velocity $v_0$ and\n the power index $k$.\n\n In Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-Gau_k-Gau-flat}\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau_k-Gau-flat},\n we use the Gaussian probability distribution\n for the fitting parameter $v_0$\n with an expectation value of \\mbox{$v_0 = 230$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n but\n the flat distribution\n for the parameter $k$,\n with either the precisely known (input)\n or the reconstructed WIMP mass,\n respectively.\n Remind that,\n in Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-Gau_k-Gau-flat}(d)\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau_k-Gau-flat}(d)\n the bins at $k = 0.5$ and $k = 3.5$ are ``overflow'' bins.\n This means that\n they contain also the experiments\n whose best--fit values of $k$ would be\n either $k < 0.5$ or $k > 3.5$.\n\n First,\n as shown in Sec.~3.3.1,\n the Solar Galactic velocity $v_0$\n could be pinned down very precisely:\n a small systematic deviation of \\mbox{$<$ 10 km\/s}\n and 1$\\sigma$ statistical uncertainty of\n \\mbox{$\\lsim\\~10$ km\/s}\n could be achieved.\n However,\n Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-Gau_k-Gau-flat}(b) and (d)\n as well as\n Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau_k-Gau-flat}(b) and (d)\n show that,\n due to the small difference between\n the modified Maxwellian velocity distribution\n with different power indices $k$\n (see Fig.~\\ref{fig:f1v_Gau_k})\n and the large statistical fluctuation and\n 1$\\sigma$ reconstruction uncertainty\n with only 500 WIMP events (on average),\n our Bayesian reconstruction of the velocity distribution\n would be {\\em totally non--sensitive}\n on the second fitting parameter (power index) $k$.\n\n Nevertheless,\n the wide spread of the reconstructed power index $k$\n (in particular,\n  the high cumulative numbers in the (overflow) bin $k = 3.5$)\n and,\n in contrast,\n the narrow widths of\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands of\n the reconstructed velocity distribution functions\n imply that,\n for reconstructing the rough information about\n the one--dimensional WIMP velocity distribution,\n a precise value of the power index $k$\n would {\\em not be crucial},\n and\n the {\\em simple} Maxwellian velocity distribution\n $f_{1, \\Gau}(v)$ given in Eq.~(\\ref{eqn:f1v_Gau})\n would already be a good approximatio\n\\footnote{\n As described in Ref.~\\cite{Lisanti10},\n the modification of the simple Maxwellian velocity distribution function\n $f_{1, \\Gau, k}(v)$ given in Eq.~(\\ref{eqn:f1v_Gau_k})\n has significant difference from\n the simple Maxwellian one given in Eq.~(\\ref{eqn:f1v_Gau})\n in the high--velocity tail.\n Moreover,\n in Refs.~\\cite{YYMao, Kuhlen13},\n another empirical modification of\n the simple Maxwellian velocity distribution\n with also a significant difference\n in the high--velocity tail\n has been introduced.\n Unfortunately,\n our simulations presented here\n show that\n it would be {\\em impossible}\n to distinguish these subtle modifications\n by our Bayesian reconstruction method\n (with only a few hundreds of recorded WIMP events).\n\n On the other hand,\n simulations with other (well--motivated) halo models\n and different fitting velocity distribution functions\n can be tested on the {\\tt AMIDAS} website\n \\cite{AMIDAS-web,\n       AMIDAS-II}.\n}.\n\n In Table \\ref{tab:results_Gau_k_Gau_k},\n we give\n the reconstructed results of $v_0$ and $k$\n with the modified Maxwellian velocity distribution\n $f_{1, \\Gau, k}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n Note that,\n since the 1$\\sigma$ lower and upper bounds of the median values of $k$\n are already beyond our scanning range,\n the 2$\\sigma$ bounds are meanless to give here.\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k = 2}(v)$}\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k}(v)$}\n{\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 228.6 & $227.2\\~^{+ 7.0}_{- 8.4}\\~(^{+19.6}_{-18.2})$ & [218.8, 234.2] & [209.0, 246.8] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 227.2 & $227.2\\~^{+ 9.8}_{-12.6}\\~(^{+29.4}_{-28.0})$ & [214.6, 237.0] & [199.2, 256.6] \\\\\n\\hline\n \\multirow{2}{*}{$k$} &\n Input &\n Flat &\n 3.47 & $3.38                  $ & [0.5, 3.5] & $\\times$ \\\\\n \\cline{2-7}\n & Reconst. &\n Flat &\n 2.72 & $3.29                  $ & [0.5, 3.5] & $\\times$ \\\\\n}\n{The reconstructed results of $v_0$ and $k$\n with the modified Maxwellian velocity distribution\n $f_{1, \\Gau, k}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n Note that,\n firstly,\n we use here the Gaussian probability distribution for $v_0$\n but the flat one for $k$.\n Secondly,\n since the 1$\\sigma$ lower and upper bounds of the median values of $k$\n are already beyond our scanning range,\n the 2$\\sigma$ bounds are meanless to give here.\n}\n{tab:results_Gau_k_Gau_k}\n\n\\subsection{For different WIMP masses}\n\n\\plotGeSiGeshGauflatL\n\n\\plotGeSiGeshshGauL\n\n\\plotGeSiGeshshDvGauL\n\n In the previous Secs.~3.1 to 3.3,\n we fixed the input WIMP mass as \\mbox{$\\mchi = 100$ GeV}.\n As a further test of our Bayesian reconstruction method\n for the one--dimensional WIMP velocity distribution,\n in this subsection,\n we consider the cases for\n either a light WIMP mass of \\mbox{$\\mchi = 25$ GeV}\n or a heavy WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n\n Here we consider\n only the {\\em shifted} Maxwellian velocity distribution\n given in Eq.~(\\ref{eqn:f1v_sh})\n for generating WIMP events;\n three fitting functions:\n simple,\n shifted\n and variated shifted\n Maxwellian velocity distributions\n will be tested.\n All input setup and fitting parameters\n are the same as in Sec.~3.2\n (see Table \\ref{tab:setup_sh}).\n Additionally,\n only the {\\em reconstructed} WIMP mass is used.\n\n\\subsubsection{For a light WIMP mass}\n\n We consider first\n a rather light WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n Note that,\n firstly,\n since for our tested targets $\\rmXA{Si}{28}$ and $\\rmXA{Ge}{76}$,\n the kinematic maximal cut--off energies\n given in Eq.~(\\ref{eqn:Qmax_kin})\n are only \\mbox{$Q_{\\rm max, kin, Si} = 68.12$ keV}\n and \\mbox{$Q_{\\rm max, kin, Ge} = 52.65$ keV},\n respectively,\n the maximal experimental cut--off energy for both targets\n in our simulations demonstrated here\n has been reduced to only \\mbox{$\\Qmax = 50$ keV}.\n Secondly,\n since the lighter the WIMP mass,\n the shaper the expected recoil energy spectrum,\n the width of the first energy bin in Eq.~(\\ref{eqn:bn_delta})\n has been set as \\mbox{$b_1 = 5$ keV}\n and the total number of bins\n has been reduced to only {\\em four} bins\n between $\\Qmin$ and $\\Qmax$ ($B = 4$);\n up to {\\em two} bins have been combined to a window\n and thus {\\em four} windows ($W = 4$)\n will be reconstructe\n\\footnote{\n The last window is neglected automatically\n in the \\amidas\\ code,\n due to a very few expected event number\n in the last bin (window).\n}$^{,~}\n\\footnote{\n It has been found that,\n by reducing the total number of the energy bins\n (and in turn that of the windows)\n and thus collecting more events in one bin (window),\n the Bayesian reconstructed velocity distribution\n could be improved (significantly).\n}.\n\n\\paragraph{Simple Maxwellian velocity distribution \\\\}\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 296.8 & $294.4\\~^{+14.4}_{-12.0}\\~(^{+31.2}_{-21.6})$ & [282.4, 308.8] & [272.8, 325.6] \\\\\n \\cline{3-7}\n & & Gaussian &\n 294.4 & $294.4\\~^{+ 9.6}_{-12.0}\\~(^{+24.0}_{-21.6})$ & [282.4, 304.0] & [272.8, 318.4] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 292.0 & $289.6\\~^{+28.8}_{-24.0}\\~(^{+62.4}_{-43.2})$ & [265.6, 318.4] & [246.4, 352.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 292.0 & $289.6 \\pm 21.6\\~(^{+48.0}_{-40.8})$ & [268.0, 311.2] & [248.8, 337.6] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$} \\\\\n\\hline\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 220.2 & $218.8\\~^{+ 9.8}_{- 7.0}\\~(^{+19.6}_{-14.0})$ & [211.8, 228.6] & [204.8, 238.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 221.6 & $221.6 \\pm  7.0\\~(^{+15.4}_{-14.0})$ & [214.6, 228.6] & [207.6, 237.0] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 217.4 & $214.6\\~^{+21.0}_{-16.8}\\~(^{+44.8}_{-30.8})$ & [197.8, 235.6] & [183.8, 259.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 218.8 & $218.8 \\pm 15.4\\~(^{+32.2}_{-29.4})$ & [203.4, 234.2] & [189.4, 251.0] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 221.6 & $221.6 \\pm  7.0\\~(^{+15.4}_{-12.9})$ & [214.6, 228.6] & [208.7, 237.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 218.8 & $220.2\\~^{+12.6}_{-14.0}\\~(^{+28.0}_{-26.6})$ & [206.2, 232.8] & [193.6, 248.2] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 237.0 & $237.0\\~^{+ 7.0}_{- 8.4}\\~(^{+14.0}_{-16.8})$ & [228.6, 244.0] & [220.2, 251.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 234.2 & $234.2 \\pm 12.6(\\pm 25.2)$ & [221.6, 246.8] & [209.0, 259.4] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 221.6 & $221.6 \\pm  5.6\\~(^{+12.6}_{-11.2})$ & [216.0, 227.2] & [210.4, 234.2] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 218.8 & $218.8\\~^{+14.0}_{-12.6}\\~(^{+29.4}_{-23.8})$ & [206.2, 232.8] & [195.0, 248.2] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n   9.8 & $ 11.1\\~^{+ 3.9}_{- 5.2}\\~(^{+ 9.1}_{-11.7})$ & [5.9,  15.0] & [$- 0.6$,  20.2] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n   9.8 & $  9.8 \\pm  6.5\\~(^{+14.3}_{-13.0})$ & [3.3,  16.3] & [$- 3.2$,  24.1] \\\\\n}\n{The reconstructed results\n with four fitting velocity distribution functions\n for the input WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n}\n{tab:results_sh_L}\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-025-0500-sh-Gau-flat}\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}(c) and (d)),\n we use first the (improper)\n simple Maxwellian velocity distribution function\n with one parameter $v_0$\n to fit the reconstructed--input data.\n As the first trial of the reconstruction of\n the one--dimensional WIMP velocity distribution\n {\\em without} prior knowledge about the Solar Galactic velocity,\n the flat probability distribution\n has been used here\n (results with the Gaussian probability distribution\n  are given in Table \\ref{tab:results_sh_L}).\n\n Although we use the improper assumption about\n the fitting velocity distribution\n and have only {\\em four} available data points\n (solid black vertical bars),\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n could still cover the true (input) velocity distribution\n with a systematic deviation of the peak of\n the reconstructed velocity distribution\n of \\mbox{$\\lsim$ 15 km\/s}\n from that of the true (input) one.\n However,\n the best--fit values of the Solar Galactic velocity\n are now \\mbox{$\\simeq 294$ km\/s}\n and \\mbox{$\\sim 3\\sigma$} apart from the theoretically expected value.\n\n Moreover,\n our further simulations\n with the Gaussian probability distribution\n for the fitting parameter $v_0$\n with an expectation value of \\mbox{$v_0 = {\\it 280}$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{{\\em 40} km\/s}\n show that,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n for such a light WIMP mass\n could be reduced to \\mbox{$\\sim 60$\\%} to \\mbox{$\\sim 80$\\%}\n (compare Table \\ref{tab:results_sh_L} to Table \\ref{tab:results_sh_Gau}).\n\n\\paragraph{Shifted Maxwellian velocity distribution \\\\}\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh-Gau}\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau}),\n the shifted Maxwellian velocity distribution\n with two fitting parameters $v_0$ and $\\ve$\n has been tested to fit to the reconstructed--input data.\n Only the Gaussian probability distribution\n for both fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\ve = 245$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n is considered here.\n\n Astonishingly,\n with {\\em only four} available data points\n the reconstructed velocity distribution functions\n could match the true (input) one very precisely:\n the systematic deviation of $v_0$ is negligible\n and that of $\\ve$ is {\\em only a few} km\/s,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on two fitting parameters are also only\n \\mbox{$^{+12.6}_{-14.0}\\~(^{+28.0}_{-26.6})$} and\n \\mbox{$\\pm 12.6(\\pm 25.2)$},\n respectively.\n\n\\paragraph{Variated shifted Maxwellian velocity distribution \\\\}\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh_Dv-Gau}\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_Dv-Gau}),\n the variated shifted Maxwellian velocity distribution\n with two parameters $v_0$ and $\\Delta v$\n has been tested to fit to the reconstructed--input data.\n Only the Gaussian probability distribution\n for both fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\Delta v = 15$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n is considered here.\n\n It can be seen that,\n although\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n are a bit wider than\n those given with the shifted Maxwellian distribution,\n with only four available data points\n the reconstructed velocity distribution function\n could also match the true (input) one very well.\n Moreover,\n as shown in Secs.~3.2.4 and 3.3.4,\n the second fitting parameter $\\Delta v$\n could also be pinned down very precisely\n with a negligible systematic deviation\n (see Table \\ref{tab:results_sh_L}).\n\n ~\n\n In Table \\ref{tab:results_sh_L},\n we give the reconstructed results\n with all four fitting velocity distribution functions\n for the input WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n Both cases with\n the true (input) and the reconstructed WIMP masses\n have been simulated and summarized.\n\n\\subsubsection{For a heavy WIMP mass}\n\n\\plotGeSiGeshGauflatH\n\n\\plotGeSiGeshshGauH\n\n\\plotGeSiGeshshDvGauH\n\n We consider now\n a rather heavy WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n The maximal experimental cut--off energy for both targets\n in our simulations demonstrated here\n has been set again as \\mbox{$\\Qmax = 100$ keV}.\n And,\n as usual,\n the width of the first energy bin in Eq.~(\\ref{eqn:bn_delta})\n has been set as \\mbox{$b_1 = 10$ keV},\n {\\em five} bins between $\\Qmin$ and $\\Qmax$ are used ($B = 5$)\n and up to {\\em three} bins have been combined to a window ($W = 6$).\n\n\\paragraph{Simple Maxwellian velocity distribution \\\\}\n\n For a heavy WIMP mass,\n due to the (large) statistical fluctuation\n discussed in Ref.~\\cite{DMDDmchi},\n our reconstructed velocity distribution functions\n given in Fig.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-Gau-flat}(a)\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}(c)\n  and \\ref{fig:f1v-Ge-SiGe-025-0500-sh-Gau-flat}(a))\n have clearly\n a (much) wider $1\\~(2)\\~\\sigma$ statistical uncertainty bands;\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the reconstructed Solar Galactic velocity\n are also (much) larger as\n \\mbox{$^{+33.6}_{-31.2}\\~(^{+64.8}_{-60.0})$ km\/s}.\n And,\n as shown in Fig.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-Gau-flat}(b),\n a considerable fraction of the reconstructed $v_0$\n would excess our scanning upper bound of \\mbox{400 km\/s}.\n Remind that\n the bin at \\mbox{$v_0 = 400$ km\/s}\n is an ``overflow'' bin,\n which contains also the experiments\n with the best--fit $v_0$ value of \\mbox{$> 400$ km\/s}.\n\n Nevertheless,\n comparing to the much larger 1$\\sigma$ statistical uncertainty\n on the reconstructed--input data (solid black vertical bars),\n our Bayesian reconstruction with\n an in fact improper fitting velocity distribution\n could still offer an approximation\n with only \\mbox{$\\lsim\\~15$ km\/s}\n systematic deviation of the peak of\n the reconstructed velocity distribution\n from that of the true (input) one.\n\n\\paragraph{Shifted Maxwellian velocity distribution \\\\}\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-sh-Gau}\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau}\n  and \\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh-Gau}),\n the shifted Maxwellian velocity distribution\n with two fitting parameters $v_0$ and $\\ve$\n has been tested to fit to the reconstructed--input data.\n Only the Gaussian probability distribution\n for both fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\ve = 245$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n is considered here.\n\n With a proper fitting velocity distribution\n and {\\em one more} fitting parameter,\n the reconstructed velocity distribution\n could now match the true (input) one very precisely\n with much narrower $1\\~(2)\\~\\sigma$ statistical uncertainty bands.\n Additionally,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the fitting parameters $v_0$ and $\\ve$\n can be significantly reduced to only\n \\mbox{$\\pm 9.8\\~(^{+22.4}_{-18.2})$} and\n \\mbox{$^{+14.0}_{-11.2}\\~(^{+26.6}_{-21.0})$},\n respectively.\n Note that,\n as shown in Table \\ref{tab:results_sh_H},\n the use of the {\\em one--parameter}\n shifted Maxwellian velocity distribution\n with {\\em only one} fitting parameter $v_0$\n and the fixed relation between $v_0$ and $\\ve$\n would give\n a (much) wider $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n of the reconstructed velocity distribution\n as well as\n a (much) larger $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the reconstructed Solar Galactic velocity:\n \\mbox{$\\sim 50$\\%} to a factor of \\mbox{$\\sim 2$} larger.\n\n\\paragraph{Variated shifted Maxwellian velocity distribution \\\\}\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 287.2 & $287.2\\~^{+28.8}_{-21.6}\\~(^{+69.6}_{-40.8})$ & [265.6, 316.0] & [246.4, 356.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 284.8 & $284.8\\~^{+19.2}_{-16.8}\\~(^{+36.0}_{-31.2})$ & [268.0, 304.0] & [253.6, 320.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 289.6 & $289.6\\~^{+55.2}_{-43.2}\\~(^{+110.4}_{-74.4})$ & [246.4, 344.8] & [215.2, 400.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 289.6 & $287.2\\~^{+33.6}_{-31.2}\\~(^{+64.8}_{-60.0})$ & [256.0, 320.8] & [227.2, 352.0] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$} \\\\\n\\hline\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 209.0 & $209.0\\~^{+18.2}_{-15.4}\\~(^{+43.4}_{-26.6})$ & [193.6, 227.2] & [182.4, 252.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 218.8 & $218.8 \\pm  9.8\\~(\\pm 19.6)$ & [209.0, 228.6] & [199.2, 238.4] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 211.8 & $210.4\\~^{+37.8}_{-29.4}\\~(^{+89.6}_{-50.4})$ & [181.0, 248.2] & [160.0, 300.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 218.8 & $220.2 \\pm 18.2\\~(^{+36.4}_{-35.0})$ & [202.0, 238.4] & [185.2, 256.6] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 223.0 & $223.0\\~^{+ 5.6}_{- 7.0}\\~(^{+12.6}_{-14.0})$ & [216.0, 228.6] & [209.0, 235.6] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 223.0 & $224.4 \\pm  9.8\\~(^{+22.4}_{-18.2})$ & [214.6, 234.2] & [206.2, 246.8] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 237.0 & $237.0 \\pm  7.0\\~(^{+14.0}_{-15.4})$ & [230.0, 244.0] & [221.6, 251.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 237.0 & $237.0\\~^{+14.0}_{-11.2}\\~(^{+26.6}_{-21.0})$ & [225.8, 251.0] & [216.0, 263.6] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 218.8 & $218.8\\~^{+ 9.8}_{- 8.4}\\~(\\pm 18.2)$ & [210.4, 228.6] & [200.6, 237.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 220.2 & $220.2\\~^{+15.4}_{-16.8}\\~(^{+32.2}_{-30.8})$ & [203.4, 235.6] & [189.4, 252.4] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n   9.8 & $  9.8 \\pm  5.2\\~(^{+ 9.1}_{-10.4})$ & [4.6,  15.0] & [$- 0.6$,  18.9] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n   9.8 & $  9.8\\~^{+ 9.1}_{- 7.8}\\~(^{+18.2}_{-15.6})$ & [2.0,  18.9] & [$- 5.8$,  28.0] \\\\\n}\n{The reconstructed results\n with four fitting velocity distribution functions\n for the input WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n}\n{tab:results_sh_H}\n\n Finally,\n in Figs.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-sh_Dv-Gau}\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_Dv-Gau}\n  and \\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh_Dv-Gau}),\n we test the possibility of\n improving the reconstruction precision\n by the use of\n the variated shifted Maxwellian velocity distribution\n with two fitting parameters $v_0$ and $\\Delta v$.\n Only the Gaussian probability distribution\n for both fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\Delta v = 15$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n is considered here.\n\n While\n the reconstructed velocity distribution function\n could still match the true (input) one very precisely\n with however\n wider $1\\~(2)\\~\\sigma$ statistical uncertainty bands,\n the ``best--fit'' values of\n both parameters $v_0$ and $\\Delta v$\n (and in turn $\\ve$)\n could again be very precisely determined\n with negligible systematic deviations\n (see Table \\ref{tab:results_sh_H}).\n\n ~\n\n Note that,\n since the heavier the WIMP mass,\n the smaller the transformation constant $\\alpha$\n defined in Eq.~(\\ref{eqn:alpha}),\n for an experimental maximal cut--off energy\n \\mbox{$\\Qmax \\approx 100$ GeV},\n the reconstructible velocity range\n of our model--independent data analysis method\n would be much smaller than our maximal cut--off velocity $\\vmax$\n (e.g.~\\mbox{$\\sim 285$ km\/s}\n  for \\mbox{$\\mchi = 250$ GeV}\n  and the $\\rmXA{Ge}{76}$ target).\n Therefore,\n our simulations shown in this subsection\n demonstrate meaningfully that,\n our Bayesian reconstruction of\n the one--dimensional WIMP velocity distribution\n would be an important improvement\n for offering more and preciser information\n about the Galactic halo,\n e.g.~the position of the peak of the WIMP velocity distribution,\n for the WIMP mass between \\mbox{$\\cal O$(20) GeV}\n and even \\mbox{$\\cal O$(500) GeV}.\n\n In Table \\ref{tab:results_sh_H},\n we give the reconstructed results\n with all four fitting velocity distribution functions\n for the input WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n Both cases with the true (input) and the reconstructed WIMP masses\n have been simulated and summarized.\n\n\\subsection{Background effects}\n\n\\plotdRdQbg\n\n In this last part of\n our presentation of the numerical simulations\n of the Bayesian reconstruction of\n the WIMP velocity distribution function,\n we consider the effects of {\\em unrejected} background events.\n Similar to our earlier works\n in Refs.~\\cite{DMDDbg-mchi, DMDDbg-f1v},\n we take into account a small fraction of\n {\\em artificially} generated background events\n in the fake experimental data sets\n and want to study\n how well the WIMP velocity distribution\n as well as\n the fitting parameters\n could be reconstructed.\n\n In all simulations demonstrated in this subsection,\n a combination of\n the {\\em target--dependent exponential} form\n of the residue background spectrum\n introduced in Ref.~\\cite{DMDDbg-mchi}\n with a small {\\em constant} component\n has been used:\n\\beq\n    \\aDd{R}{Q}_{\\rm bg}\n =  \\aDd{R}{Q}_{\\rm bg, ex}\n  + r_{\\rm const} \\aDd{R}{Q}_{\\rm bg, const}\n\\~,\n\\label{eqn:dRdQ_bg}\n\\eeq\n where\n\\beq\n   \\aDd{R}{Q}_{\\rm bg, ex}\n = \\exp\\abrac{-\\frac{Q \/{\\rm keV}}{A^{0.6}}}\n\\~,\n\\label{eqn:dRdQ_bg_ex}\n\\eeq\n and\n\\beq\n   \\aDd{R}{Q}_{\\rm bg, const}\n = 1\n\\~.\n\\label{eqn:dRdQ_bg_const}\n\\eeq\n Here $Q$ is the recoil energy,\n $A$ is the atomic mass number of the target nucleus.\n The power index of $A$, 0.6, is an empirical constant,\n which has been chosen so that\n the exponential background spectrum is\n somehow similar to,\n but still different from\n the expected recoil spectrum of the target nuclei;\n otherwise,\n there is in practice no difference between\n the WIMP scattering and background spectr\n\\footnote{\n Note that,\n among different possible choices,\n we use in our simulations the atomic mass number $A$\n as the simplest, unique characteristic parameter\n in the general analytic form (\\ref{eqn:dRdQ_bg_ex})\n for defining the artificial residue background spectra\n for different target nuclei.\n However,\n it does not mean that\n the (superposition of the real) background spectra\n would depend simply\/primarily on $A$ or\n on the mass of the target nucleus, $\\mN$.\n In other words,\n it is practically equivalent to\n use the expression (\\ref{eqn:dRdQ_bg_ex})\n or $(dR \/ dQ)_{\\rm bg, ex} = e^{-Q \/ 13.5~{\\rm keV}}$ directly\n for a $\\rmXA{Ge}{76}$ target\n (cf.~\\cite{Green-mchi08}).\n}.\n Additionally,\n $r_{\\rm const}$ is the ratio\n between the exponential and constant components\n in the total {\\em background} spectrum,\n which has been fixed as $r_{\\rm const} = 0.05$\n for all simulations.\n\n Note that,\n firstly,\n as argued in Ref.~\\cite{DMDDbg-mchi},\n the exponential form of background spectrum\n is rather naive;\n but,\n since we consider here\n only {\\em a few tens residue} background events\n induced by {\\em several different} sources,\n pass all discrimination criteria,\n and then mix with other WIMP--induced events\n in our data sets of a few hundreds {\\em total} events,\n an exact form of background spectrum\n for each target nucleus\n would not be crucial and\n the exponential + constant form of background spectrum\n in Eq.~(\\ref{eqn:dRdQ_bg})\n should practically not be unrealistic.\n Secondly,\n as demonstrated in Refs.~\\cite{DMDDf1v, DMDDmchi}\n and in the previous subsections,\n our Bayesian reconstruction of\n the one--dimensional WIMP velocity distribution\n requires only measured recoil energies\n and occasionally prior knowledge about\n the Solar and Earth's Galactic velocities.\n Hence,\n for applying this method to future real experimental data,\n prior knowledge about (different) background source(s)\n is {\\em not required at all}.\n\n In Figs.~\\ref{fig:dRdQ-bg-ex-const-000-100-20-100},\n we show\n the measured recoil energy spectrum (solid red histogram)\n for a $\\rmXA{Ge}{76}$ (a) and a $\\rmXA{Si}{28}$ (b) targets\n with an input WIMP mass of \\mbox{$\\mchi = 100$ GeV}.\n The dotted blue curve is\n the elastic WIMP--nucleus scattering spectrum\n for {\\em generating} signal events,\n whereas\n the dashed green curve shows\n the {\\em artificial} background spectrum:\n the exponential background spectrum\n given in Eq.~(\\ref{eqn:dRdQ_bg_ex})\n accompanied with an extra constant component,\n normalized to fit to the background ratio of 20\\%.\n\n\\plotGeSiGeshshvGaubg\n\n\\plotGeSiGeshshGaubg\n\n\\subsubsection{For a moderate WIMP mass}\n\n We consider first a moderate input WIMP mass of\n \\mbox{$\\mchi = 100$ GeV}.\n The {\\em shifted} Maxwellian velocity distribution\n given in Eq.~(\\ref{eqn:f1v_sh})\n is used for generating WIMP signals.\n All input setup and fitting parameters\n are the same as in Sec.~3.2\n (see Table \\ref{tab:setup_sh})\n and\n a fraction of 20\\% background events\n has been taken into account.\n Additionally,\n as in Sec.~3.4,\n we consider only the use of\n the Gaussian probability distribution\n for the fitting parameters:\n $v_0$ and $\\ve$ or $\\Delta v$\n as well as\n the use of the reconstructed WIMP mass.\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 284.8 & $284.8\\~^{+24.0}_{-19.2}\\~(^{+52.8}_{-40.8})$ & [265.6, 308.8] & [244.0, 337.6] \\\\\n \\cline{3-7}\n & & Gaussian &\n 284.8 & $284.8 \\pm 16.8\\~(^{+36.0}_{-33.6})$ & [268.0, 301.6] & [251.2, 320.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 253.6 & $253.6\\~^{+36.0}_{-31.2}\\~(^{+79.2}_{-60.0})$ & [222.4, 289.6] & [193.6, 332.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 258.4 & $258.4\\~^{+28.8}_{-26.4}\\~(^{+57.6}_{-55.2})$ & [232.0, 287.2] & [203.2, 316.0] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$} \\\\\n\\hline\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 211.8 & $211.8\\~^{+16.8}_{-15.4}\\~(^{+36.4}_{-30.8})$ & [196.4, 228.6] & [181.0, 248.2] \\\\\n \\cline{3-7}\n & & Gaussian &\n 217.4 & $218.8\\~^{+ 9.8}_{-11.2}\\~(^{+21.0}_{-23.8})$ & [207.6, 228.6] & [195.0, 239.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 188.0 & $188.0\\~^{+26.6}_{-22.4}\\~(^{+56.0}_{-28.0})$ & [165.6, 214.6] & [160.0, 244.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 202.0 & $202.0\\~^{+18.2}_{-16.8}\\~(^{+36.4}_{-35.0})$ & [185.2, 220.2] & [167.0, 238.4] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 225.8 & $225.8\\~^{+ 7.0}_{- 8.4}\\~(\\pm 16.8)$ & [217.4, 232.8] & [189.4, 239.8] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 214.6 & $214.6\\~^{+12.2}_{-12.6}\\~(\\pm 25.2)$ & [202.0, 226.8] & [189.4, 239.8] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 231.4 & $231.4 \\pm 8.4\\~(^{+16.8}_{-18.2})$ & [223.0, 239.8] & [213.2, 248.2] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 220.2 & $220.2\\~^{+12.6}_{-11.2}\\~(^{+25.2}_{-23.8})$ & [209.0, 232.8] & [196.4, 245.4] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 220.2 & $220.2\\~^{+ 8.4}_{- 9.8}\\~(\\pm 19.6)$ & [210.4, 228.6] & [200.6, 239.8] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 204.8 & $204.8\\~^{+16.8}_{-15.4}\\~(^{+32.2}_{-30.8})$ & [189.4, 221.6] & [174.0, 237.0] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n   5.9 & $  5.9\\~^{+ 5.2}_{- 6.5}\\~(^{+10.4}_{-13.0})$ & [$- 0.6$,  11.1] & [$- 7.1$,  16.3] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n $- 0.6$ & $- 0.6 \\pm  7.8\\~(\\pm 15.6)$ & [$- 8.4$,   7.2] & [$-16.2$,  15.0] \\\\\n}\n{The reconstructed results\n with four fitting velocity distribution functions\n for data sets mixed with 20\\% background events\n and the input WIMP mass of \\mbox{$\\mchi = 100$ GeV}.\n}\n{tab:results_sh_bg}\n\n\\paragraph{One--parameter shifted Maxwellian velocity distribution \\\\}\n\n In Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(a),\n it can be seen first that,\n due to the extra background events\n in both of the low and high energy ranges\n (see Figs.~\\ref{fig:dRdQ-bg-ex-const-000-100-20-100}),\n the reconstructed--input data (solid black vertical bars)\n would be shifted (strongly) to the low--velocity rang\n\\footnote{\n Note that,\n as shown in Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(a)\n and \\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau-bg}(a),\n the reconstructed WIMP mass\n is now {\\em overestimated}:\n \\mbox{$m_{\\chi, {\\rm rec}} \\approx 136$ GeV}.\n}:\n the peak of the solid black crosses is now at \\mbox{$\\sim 220$ km\/s},\n i.e.~\\mbox{$\\sim 90$ km\/s} smaller then\n the position of the true (input) velocity distribution.\n However,\n our simulation shown in\n Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(a)\n indicates clearly and importantly that,\n by assuming the shifted Maxwellian WIMP velocity distribution\n and the {\\em time--averaged} relation\n between the Solar and Earth's Galactic velocities,\n the reconstructed WIMP velocity distributions\n could alleviate this systematic shift:\n the deviations of the peaks of\n the ($1\\~(2)\\~\\sigma$ statistical uncertainty bands of the)\n reconstructed velocity distributions\n would only be \\mbox{$\\sim\\~30\\~^{+30}_{-20}\\~(^{+60}_{-40})$ km\/s}.\n\n In fact,\n it has also been found that,\n once an (approximately) precisely determined (true) WIMP mass\n could be used,\n the reconstructed WIMP velocity distribution\n could match the true (input) one very precisely:\n the deviation of the reconstructed $v_0$\n would be \\mbox{$\\lsim\\~10$ km\/s} (flat)\n or even negligible (Gaussian)\n (see Table \\ref{tab:results_sh_bg}).\n\n Note that,\n although a fraction of 20\\% unrejected background events\n has been mixed (artificially) into the analyzed (pseudo--)data sets,\n the (1$\\sigma$ statistical uncertainty on the)\n median value of the reconstructed $v_0$'s\n (\\mbox{$202.0\\~^{+18.2}_{-16.8}$ km\/s})\n would still cover the true (input) Solar Galactic velocity of\n \\mbox{$v_0 = 220$ km\/s}.\n Moreover,\n once we take into account the statistical fluctuation\n of the reconstructed--input data (the solid black vertical bars),\n the effect of 20\\% residue background events\n on reconstructing information about\n the (shape of the) WIMP velocity distribution function\n would {\\em not be very significant}.\n\n\\paragraph{Shifted Maxwellian velocity distribution \\\\}\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau-bg},\n we release the fixed relation between $v_0$ and $\\ve$\n and determine these two parameters\n simultaneously and independently.\n\n It can be seen that,\n firstly,\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n are obviously narrower than those shown\n in Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(a);\n the deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n would be reduced to only \\mbox{$\\lsim\\~15$ km\/s}.\n In addition,\n the systematic deviations and\n the (1$\\sigma$ statistical uncertainties on the)\n median values of the reconstructed fitting paramaters $v_0$ and $\\ve$\n shown in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau-bg}(c) and (d)\n are also (much) smaller than that shown in\n Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(b)\n (see Table \\ref{tab:results_sh_bg}).\n Note here that,\n as given in Table \\ref{tab:results_sh_bg},\n once an (approximately) precisely determined (true) WIMP mass\n could be used,\n one could reconstruct\n the WIMP velocity distribution function\n very precisely\n with very small or even negligible systematic deviations\n of both two fitting parameters.\n This means that,\n our Bayesian reconstruction method\n for the WIMP velocity distribution function\n would {\\em not be affected (significantly)}\n by a fraction of \\mbox{$\\sim 20$\\%}\n unrejected background events\n mixed in the analyzed data sets\n (for a WIMP mass of ${\\cal O}(100)$ GeV). \n\n ~\n\n In Table \\ref{tab:results_sh_bg},\n we give the reconstructed results\n with all four fitting velocity distribution functions\n for data sets mixed with 20\\% background events\n and the input WIMP mass of \\mbox{$\\mchi = 100$ GeV}.\n Both cases with the true (input) and the reconstructed WIMP masses\n have been simulated and summarized.\n\n\\subsubsection{For a light WIMP mass}\n\n Now,\n we consider the case\n with a light input WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n Simulation setup is the same as in Sec.~3.4.1\n and\n a fraction of 20\\% background events\n has been taken into account.\n\n\\plotGeSiGeshshvGaubgL\n\n\\plotGeSiGeshshGaubgL\n\n\\paragraph{One--parameter shifted Maxwellian velocity distribution \\\\}\n\n Since the reconstructed WIMP mass\n would be \\mbox{$\\sim 30$\\%} {\\em overestimated}\n (\\mbox{$m_{\\chi, {\\rm rec}} \\approx 31.5$ GeV})\n due to the extra background events\n and {\\em only four} reconstructed--input data points are available,\n Fig.~\\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh_v0-Gau-bg}(a)\n shows that\n the ``best--fit'' {\\em one--parameter}\n shifted Maxwellian velocity distribution functions\n would match\n not the true (input) velocity distribution (solid red curve),\n but the analyzed data points (solid black crosses)\n well.\n Nevertheless,\n at least,\n the 2$\\sigma$ statistical uncertainty band\n of the reconstructed velocity distributions\n could cover the true (input) one;\n the systematic deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n would also only be \\mbox{$\\sim 30$ km\/s}.\n Meanwhile,\n the 2$\\sigma$ statistical uncertainty on the\n median value of the reconstructed $v_0$'s\n (\\mbox{$200.6^{+29.4}_{-26.6}$ km\/s})\n would still cover\n the true (input) Solar Galactic velocity of\n \\mbox{$v_0 = 220$ km\/s}.\n This could be further improved\n by using an (approximately) precisely determined (true) WIMP mass\n to be\n \\mbox{$227.2\\~^{+ 9.8}_{- 8.4}$ km\/s} (flat) and\n \\mbox{$228.6 \\pm  7.0$ km\/s} (Gaussian)\n (see Table \\ref{tab:results_sh_bg_L}).\n\n\\paragraph{Shifted Maxwellian velocity distribution \\\\}\n\n As the case of the \\mbox{100 GeV} WIMP mass\n shown in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(a)\n and \\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau-bg}(a),\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n reconstructed with the shifted Maxwellian velocity distribution\n with two independent fitting parameters $v_0$ and $\\ve$\n shown in Fig.~\\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh-Gau-bg}(a)\n would clearly be much narrower than those\n reconstructed with only one parameter $v_0$.\n Meanwhile,\n in contrast to other (presented) cases,\n our simulations with the {\\em true} ({\\em input}) WIMP mass\n show that\n the reconstructed--input data\n as well as\n the ($1\\~(2)\\~\\sigma$ statistical uncertainty bands of the)\n reconstructed velocity distribution function\n would slightly shift to the {\\em high}--velocity range.\n\n ~\n\n Furthermore,\n comparing results given in Table \\ref{tab:results_sh_bg_L}\n to those in Table \\ref{tab:results_sh_bg},\n it has been found interesting and probably importantly that,\n for an (input) WIMP mass of \\mbox{$\\cal O$(20) GeV},\n the use of our {\\em variated} shifted Maxwellian velocity distribution\n given in Eq.~(\\ref{eqn:f1v_sh_Dv})\n could offer preciser reconstruction results\n with (relatively) smaller statistical uncertainties,\n although\n fewer (four in our simulations) data points are available.\n\n In Table \\ref{tab:results_sh_bg_L},\n we give the reconstructed results\n with all four fitting velocity distribution functions\n for data sets mixed with 20\\% background events\n and the input WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n Both cases with the true (input) and the reconstructed WIMP masses\n have been simulated and summarized.\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 308.8 & $306.4\\~^{+16.8}_{-12.0}\\~(^{+36.0}_{-24.0})$ & [294.4, 323.2] & [282.4, 342.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 304.0 & $304.0 \\pm 12.0\\~(\\pm 24.0)$ & [292.0, 316.0] & [280.0, 328.0] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 260.8 & $260.8\\~^{+24.0}_{-21.6}\\~(^{+51.0}_{-40.8})$ & [239.2, 284.8] & [220.0, 311.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 265.6 & $263.2\\~^{+21.6}_{-19.2}\\~(^{+43.2}_{-38.4})$ & [244.0, 284.8] & [224.8, 306.4] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$} \\\\\n\\hline\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 228.6 & $227.2\\~^{+ 9.8}_{- 8.4}\\~(^{+22.4}_{-15.4})$ & [218.8, 237.0] & [211.8, 249.6] \\\\\n \\cline{3-7}\n & & Gaussian &\n 228.6 & $228.6 \\pm  7.0\\~(^{+15.4}_{-14.0})$ & [221.6, 235.6] & [214.6, 244.0] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 193.6 & $193.6\\~^{+16.8}_{-15.4}\\~(^{+35.4}_{-29.4})$ & [178.2, 210.4] & [164.2, 229.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 200.6 & $200.6\\~^{+15.4}_{-14.0}\\~(^{+29.4}_{-26.6})$ & [186.6, 216.0] & [174.0, 230.0] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 230.0 & $230.0 \\pm  7.0\\~(\\pm 14.0)$ & [223.0, 237.0] & [216.0, 244.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 207.6 & $207.6 \\pm 12.6\\~(^{+23.8}_{-25.2})$ & [195.0, 220.2] & [182.4, 231.4] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 239.8 & $239.8\\~^{+ 8.4}_{- 7.0}\\~(\\pm 15.4)$ & [232.8, 248.2] & [224.4, 255.2] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 218.8 & $220.2\\~^{+11.2}_{-12.6}\\~(^{+23.8}_{-25.2})$ & [207.6, 231.4] & [195.0, 244.0] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 228.6 & $228.6 \\pm  7.0\\~(^{+14.0}_{-12.6})$ & [221.6, 235.6] & [216.0, 242.6] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 203.4 & $204.8 \\pm 12.6\\~(\\pm 23.8)$ & [192.2, 217.4] & [181.0, 228.6] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n  12.4 & $ 11.1 \\pm  5.2\\~(\\pm 10.4)$ & [5.9,  16.3] & [0.7,  21.5] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n   2.0 & $  2.0 \\pm  6.5\\~(\\pm 13.0)$ & [$- 4.5$,   8.5] & [$-11.0$,  15.0] \\\\\n}\n{The reconstructed results\n with four fitting velocity distribution functions\n for data sets mixed with 20\\% background events\n and the input WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n}\n{tab:results_sh_bg_L}\n\n\\subsubsection{For a heavy WIMP mass}\n\n As the last case,\n we consider here\n a heavy input WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n Simulation setup is the same as in Sec.~3.4.2.\n Note however that,\n since the constant component of the background spectrum\n used in our simulations\n would cause a strongly {\\em overestimated} WIMP mass,\n in particular,\n once WIMPs are heavy\n (e.g.~the \\mbox{250 GeV} input WIMP mass\n  would now be reconstructed as \\mbox{$\\simeq 338$ GeV})\n \\cite{DMDDbg-mchi},\n the ratio of the background events\n in the analyzed data sets\n has been set as {\\em only 10\\%}\n \\cite{DMDDbg-f1v}.\n\n\\paragraph{One--parameter shifted Maxwellian velocity distribution \\\\}\n\n As usual,\n we consider first the one--parameter\n shifted Maxwellian velocity distribution function\n to fit the reconstructed--input data points.\n\n\\plotGeSiGeshshvGaubgH\n\n\\plotGeSiGeshshGaubgH\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-sh_v0-Gau-bg},\n we can see unexpectedly that,\n although the input WIMP mass is pretty heavy,\n the systematic deviations of the peak positions of\n the reconstructed WIMP velocity distribution functions\n from that of the true (input) one\n would be only \\mbox{$\\sim 10$ km\/s}\n (for 10\\% background ratio!).\n This might be due to that,\n as shown in Sec.~3.4.2,\n for our used experimental maximal cut--off energy\n \\mbox{$\\Qmax = 100$ GeV}\n and the $\\rmXA{Ge}{76}$ target,\n the reconstructible velocity range\n would only be\n \\mbox{$\\sim 270$ km\/s}\n (shifted slightly to the {\\em low}--velocity range\n  due to the {\\em overestimated} WIMP mass)\n and thus\n this maximal reconstructible velocity\n is theoretically smaller than\n the position of the peak of the velocity distribution function\n (see e.g.~Fig.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-sh_v0-Gau-bg}(a)).\n This means that\n the approximately {\\em monotonically increased} shape of\n the reconstructed--input data points (solid black vertical bars)\n with pretty large 1$\\sigma$ statistical uncertainties\n would alleviate the effects of the overestimations of\n the analyzed (reconstructed--input) data points\n and the reconstructed WIMP mass\n caused by the extra background events.\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 277.6 & $277.6\\~^{+26.4}_{-21.6}\\~(^{+64.8}_{-38.4})$ & [256.0, 304.0] & [239.2, 342.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 277.6 & $278.8 \\pm 18.0\\~(^{+37.2}_{-32.4})$ & [260.8, 296.8] & [246.4, 316.0] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 263.2 & $263.2\\~^{+48.0}_{-38.4}\\~(^{+115.2}_{-62.4})$ & [224.8, 311.2] & [200.8, 378.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 268.0 & $268.0\\~^{+33.6}_{-31.2}\\~(^{+67.2}_{-55.2})$ & [236.8, 301.6] & [212.8, 335.2] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$} \\\\\n\\hline\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 203.4 & $203.4\\~^{+19.2}_{-14.0}\\~(^{+44.8}_{-28.0})$ & [189.4, 222.6] & [175.4, 248.2] \\\\\n \\cline{3-7}\n & & Gaussian &\n 216.0 & $216.0\\~^{+ 9.8}_{-11.2}\\~(^{+19.6}_{-21.0})$ & [204.8, 225.8] & [195.0, 235.6] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 195.0 & $193.6\\~^{+33.6}_{-28.0}\\~(^{+78.4}_{-33.6})$ & [165.6, 227.2] & [160.0, 272.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 211.8 & $210.4 \\pm 18.2\\~(^{+36.4}_{-35.0})$ & [192.2, 228.6] & [175.4, 246.8] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 221.6 & $223.0\\~^{+ 5.6}_{- 8.4}\\~(^{+12.6}_{-15.4})$ & [214.6, 228.6] & [207.6, 235.6] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 218.8 & $220.2\\~^{+ 9.8}_{- 8.4}\\~(^{+22.4}_{-18.2})$ & [211.8, 230.0] & [202.0, 242.6] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 232.8 & $234.2\\~^{+ 7.0}_{- 8.4}\\~(^{+14.0}_{-16.8})$ & [225.8, 241.2] & [217.4, 248.2] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 230.0 & $230.0\\~^{+12.6}_{- 9.8}\\~(^{+25.2}_{-21.0})$ & [220.2, 242.6] & [209.0, 255.2] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 216.0 & $217.4\\~^{+ 8.4}_{- 9.8}\\~(^{+18.2}_{-19.6})$ & [207.6, 225.8] & [197.8, 235.6] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 210.4 & $211.8\\~^{+16.8}_{-15.4}\\~(^{+33.6}_{-30.8})$ & [196.4, 228.6] & [181.0, 245.4] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n   7.2 & $  7.2\\~^{+ 5.2}_{- 5.2}\\~(^{+10.4}_{-11.7})$ & [2.0,  12.4] & [$- 4.5$,  17.6] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n   4.6 & $  4.6\\~^{+ 7.8}_{- 7.8}\\~(^{+16.9}_{-14.3})$ & [$- 3.2$,  12.4] & [$- 9.7$,  21.5] \\\\\n}\n{The reconstructed results\n with four fitting velocity distribution functions\n for data sets mixed with {\\em 10\\%} background events\n and the input WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n}\n{tab:results_sh_bg_H}\n\n\\paragraph{Shifted Maxwellian velocity distribution \\\\}\n\n Now\n we use the shifted Maxwellian velocity distribution function\n with two independent fitting parameters $v_0$ and $\\ve$\n to fit the reconstructed--input data points.\n Astonishingly and unexpectedly\n (probably accidentally),\n Figs.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-sh-Gau-bg}\n show that\n both of the ``best--fit'' results of the parameters $v_0$ and $\\ve$\n are {\\em almost exact} as the true (input) values\n and\n the 1$\\sigma$ statistical uncertainties on $v_0$ and $\\ve$\n are only \\mbox{$\\sim 10$ km\/s}.\n\n Meanwhile,\n in contrast to our simulation results\n with the variated shifted Maxwellian velocity distribution function\n shown previously,\n for the case of the 250 GeV WIMP mass\n with 10\\% background ratio,\n the Bayesian reconstructed parameter $\\Delta v$\n could have a (much) larger deviations\n from the true (estimated) value\n (see Table \\ref{tab:results_sh_bg_H})!\n\n ~\n\n In Table \\ref{tab:results_sh_bg_H},\n we give the reconstructed results\n with all four fitting velocity distribution functions\n for data sets mixed with {\\em 10\\%} background events\n and the input WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n Both cases with the true (input) and the reconstructed WIMP masses\n have been simulated and summarized.\n\n\\section{Summary and conclusions}\n\n In this paper,\n we extended our earlier work on\n the development of the model--independent data analysis method\n for the reconstruction of the (time--averaged) one--dimensional\n velocity distribution of Galactic WIMPs\n and introduced the Bayesian fitting procedure\n of the theoretical velocity distribution functions.\n\n In this fitting procedure,\n the (rough) velocity distribution\n reconstructed by using raw experimental data,\n i.e.~measured recoil energies,\n with one or more different target nuclei\n has been used as reconstructed--input data (points).\n By assuming a fitting WIMP velocity distribution function\n and scanning the parameter space\n based on the Bayesian analysis,\n the (fitting) astronomical characteristic parameters,\n e.g.~the Solar and Earth's Galactic velocities $v_0$ and $\\ve$,\n would be pinned down as the output results\n and thus\n the functional form of the one--dimensional velocity distribution\n can be reconstructed\n (instead of only a few discrete points).\n\n As the first test of our Bayesian reconstruction method\n for the one--dimensional WIMP velocity distribution function,\n we used the simplest isothermal spherical Galactic halo model\n for both generating WIMP--signal events\n and as the assumed velocity distribution\n with the unique fitting parameter:\n the Solar Galactic velocity $v_0$.\n Our simulations show that,\n with (only) 500 recorded events (on average)\n and without prior knowledge about the Solar Galactic velocity,\n $v_0$ could in principle be pinned down\n with a negligible deviation and\n a 1$\\sigma$ statistical uncertainty of only \\mbox{$\\sim 12$ km\/s}\n (with a precisely known WIMP mass)\n or \\mbox{$\\sim 20$ km\/s}\n (with a reconstructed WIMP mass),\n respectively.\n Moreover,\n once (rough) information about the Solar Galactic velocity\n can be given,\n the statistical uncertainties on the reconstructed $v_0$\n could even be reduced to \\mbox{$\\sim 70$\\%}.\n \n For more realistic consideration,\n we then took into account\n the orbital motion of the Solar system around our Galaxy\n as well as\n that of the Earth around the Sun\n and\n turned to use the shifted Maxwellian velocity distribution function\n for generating WIMP signals.\n As comparisons,\n four different fitting\n functions\n have been considered:\n the simple and\n the (one--parameter and variated) shifted Maxwellian velocity distributions.\n It has been found\n that,\n firstly,\n with an improper assumed fitting function\n (e.g.~the simple Maxwellian velocity distribution here),\n the WIMP velocity distribution\n could still be reconstructed and\n offer some important information about Galactic WIMPs,\n e.g.~the rough position of the peak of\n the one--dimensional velocity distribution function.\n The deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n would be only \\mbox{$\\sim 10$ km\/s}.\n However,\n the best--fit value(s) of the fitting parameter(s)\n would be unexpected\/unreasonable.\n For instance,\n the reconstructed Solar Galactic velocity $v_0$\n would be\n 2$\\sigma$ (with the reconstructed WIMP mass)\n or even\n 4$\\sigma$ (with the input WIMP mass)\n apart from its theoretical estimate.\n Such an observation\n could in turn be an important criterion\n on the assumption of fitting velocity distribution function.\n\n Moreover,\n our simulations with\n the (one--parameter and variated)\n shifted Maxwellian velocity distributions\n show that,\n although in all of these three cases\n the reconstructed velocity distributions\n could match the true (input) one pretty precisely,\n with two fitting parameters\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n of the reconstructed velocity distributions\n would be narrower then those\n with only one fitting parameter.\n In addition,\n the use of the variation of the shifted Maxwellian velocity distribution\n could (strongly) reduce the systematic deviations of\n the determinations of the characteristic\n Solar and Earth's Galactic velocities $v_0$ and $\\ve$,\n with however a bit larger statistical uncertainties.\n\n Furthermore,\n we considered also a modification of\n the simple Maxwellian velocity distribution\n with an extra power index\n as the generating WIMP velocity distribution.\n First,\n we used the simple Maxwellian velocity distribution\n without the power index\n as the test fitting function.\n Since the difference between the modification and the original\n simple Maxwellian velocity distributions\n are very tiny,\n the reconstructed velocity distribution function\n could match the true (input) one very precisely\n and the characteristic Solar Galactic velocity\n could also be reconstructed\n with negligible systematic deviation.\n\n Meanwhile,\n our simulations with the\n (one--parameter and variated)\n shifted Maxwellian velocity distribution functions\n show that,\n although the positions of the peak of\n the reconstructed velocity distribution\n would be only \\mbox{$\\lsim\\~10$ km\/s} deviated\n from the true (input) one,\n a clear 2$\\sigma$ to even 6$\\sigma$ difference\n between the best--fit values of\n the Solar and\/or the Earth's Galactic velocities\n and the true (input) ones\n could be observed.\n Such results would in turn\n indicate evidently the improper assumption of\n the shifted Maxwellian velocity distribution function.\n\n Moreover,\n from the simulations with\n the modified simple Maxwellian velocity distribution\n with the power index\n as the second fitting parameter,\n it has been found that\n our Bayesian reconstruction of the WIMP velocity distribution\n would be (totally) non--sensitive\n on the power index.\n This means that,\n unfortunately,\n with only a few hundreds of recorded WIMP events\n it would still be impossible\n to distinguish (evidently)\n different subtle variations of\n the (simple and shifted) WIMP velocity distribution functions.\n\n As comparisons,\n we considered also a light and a heavy input WIMP masses.\n For the case of light WIMPs,\n due to the sharp shapes of the recoil energy spectra\n and the small kinetic maximal cut--off energies,\n the recorded WIMP events\n would need to be separated into fewer bins\/windows.\n However,\n our simulations show that,\n with only four available reconstructed--input data points,\n the true (input) velocity distribution function\n could astonishingly be reconstructed very precisely.\n On the other hand,\n once WIMPs are heavy,\n the statistical fluctuation on the reconstructed WIMP mass\n becomes pretty large\n and hence\n the Bayesian reconstructions of the velocity distribution\n as well as\n of the Solar and Earth's Galactic velocities\n would have large statistical uncertainties.\n Nevertheless,\n the reconstructed velocity distribution function\n with the best--fit characteristic Solar and Earth's Galactic velocities\n could still match the true (input) one very precisely.\n\n Finally,\n the effects of residue (unrejected) background events\n mixed in data sets to analyze\n have also been considered.\n Three different WIMP masses\n with background ratios of 10\\% or 20\\%\n have been tested.\n Although,\n due to the choice of our artificial residue background spectrum,\n the reconstructed WIMP masses\n would be overestimated\n and the (rough shape of) the reconstructed--input data points\n would thus be shifted (significantly) to lower velocities,\n the functional forms of the chosen fitting velocity distributions\n could somehow alleviate these systematic shifts\n and the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n could still cover the true (input) velocity distribution.\n In particular,\n for heavy WIMPs,\n since the reconstructed--input data points\n should be in the velocity range smaller than\n the position of the peak of the velocity distribution function,\n its approximately monotonically increased shape\n with pretty large 1$\\sigma$ statistical uncertainties\n would alleviate the effects of the overestimations of\n the analyzed (reconstructed--input) data points\n and the reconstructed WIMP mass\n caused by the extra background events.\n The reconstructed velocity distribution function\n could then match the true (input) one pretty well.\n\n It would be worth to emphasize that,\n first,\n comparing to the pretty large\n (1$\\sigma$) statistical uncertainties\n on the reconstructed--input data points\n (offered by our model--independent method\n  developed in Ref.~\\cite{DMDDf1v}\n  with raw experimental measured recoil energies),\n our Bayesian reconstruction of\n the WIMP velocity distribution function\n introduced here\n with only a few km\/s deviation and\n \\mbox{$\\cal O$(10) km\/s} 1$\\sigma$ statistical uncertainties\n on the reconstructed Solar and Earth's Galactic velocities\n would be a remarkable improvement.\n\n Second,\n all our simulations show importantly that,\n even initial values different slightly from the true (input) setup\n have been used as the expectation values\n for the Gaussian probability distribution of the fitting parameters,\n these fitting parameters\n could still be pinned down (pretty) precisely.\n As long as\n a proper assumed fitting velocity distribution function\n is used,\n the best--fit values of the reconstructed parameters\n could always be less than 1$\\sigma$\n apart from the true (input\/theoretical) values.\n This observation indicates that\n rough, slightly incorrect prior knowledge\n about our fitting parameters\n would not affect (significantly)\n the reconstructed results\n in our Bayesian reconstruction procedure.\n\n Moreover,\n by rewriting the functional form of\n the (basic) fitting velocity distribution function,\n one could not only pin down the fitting parameters more precisely,\n but also occasionally reduce the statistical uncertainties\n on the reconstructed parameters.\n\n In summary,\n we developed in this paper\n the Bayesian reconstruction procedure\n for fitting theoretically predicted models of\n the one--dimensional WIMP velocity distribution function\n to data (points),\n which can be reconstructed directly\n from experimental measured recoil energies.\n Hopefully,\n this extension of our earlier work\n could offer more useful information about the Dark Matter halo,\n which could further be used\n in e.g.~indirect DM detection experiments.\n\n\\subsubsection*{Acknowledgments}\n The author appreciates Mei-Yu Wang\n for useful discussions\n about models of the velocity distribution of Galactic WIMPs.\n The author would also like to thank\n the Physikalisches Institut der Universit\\\"at T\\\"ubingen\n for the technical support of the computational work\n presented in this paper\n as well as\n the friendly hospitality of\n the Graduate School of Science and Engineering for Research,\n University of Toyama,\n the Institute of Modern Physics,\n Chinese Academy of Sciences,\n the Center for High Energy Physics,\n Peking University,\n and\n the Xinjiang Astronomical Observatory,\n Chinese Academy of Sciences,\n where part of this work was completed.\n This work\n was partially supported\n by the National Science Council of R.O.C.\n under the contracts no.~NSC-98-2811-M-006-044 and\n no.~NSC-99-2811-M-006-031\n as well as\n the CAS Fellowship for Taiwan Youth Visiting Scholars\n under the grant no.~2013TW2JA0002.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe giant dipole resonance (GDR) is the best-known fundamental mode of nuclear excitations at high frequencies. The GDR built on the ground state of heavy nuclei has a small width ($\\sim$ 4 - 5 MeV) and the integrated cross section up to around 30 MeV that exhausts the Thomas-Reich-Kuhn (TRK) sum rule. The GDR built on highly excited compound (CN) nuclei was observed for the first time in 1981~\\cite{Newton}, and at present a wealth of experimental data has been accumulated for the GDR widths at finite temperature $T$ and angular momentum $J$ in various medium and heavy nuclei formed in heavy ion fusions~\\cite{Schiller}, deep inelastic scattering of light particles on heavy targets~\\cite{Baumann,Heckman}, and $\\alpha$ induced fusions~\\cite{Kolkata}. The common features of the hot GDR are: (1) Its energy is nearly independent of $T$ and $J$, (2) Its full width at half maximum (FWHM) remains mostly unchanged in the region of $T\\leq$ 1 MeV, but increases sharply with $T$ within 1$\\leq T\\leq$ 2.5 - 3 MeV, and seems to saturate at $T\\geq$ 4 MeV. As a function of $J$, a significant increase in the GDR width is seen only at $J\\geq$ 25 - 27$\\hbar$. In Ref. \\cite{Kusnezov} some GDR data were reanalyzed by adding the pre-equlibrium $\\gamma$ emission  and it was claimed that the GDR width does not saturate. However, it was realized later that the pre-equilibrium emission is proportional to the asymmetry between projectiles and targets and lowers the CN excitation energy, which alters the conclusion on the role of pre-equlibrium emission. The recent measurements in $^{88}$Mo at $T\\geq 3$ MeV and $J>$ 40$\\hbar$ did not show any significant effect of pre-equilibrium emission on the GDR width~\\cite{Maj}.   The evaporation width owing to the quantal mechanical uncertainty in the energies of the CN states was also proposed to be added into the total GDR width~\\cite{Chomaz}. However, the high-energy $\\gamma$-ray spectra resulting from the complete {\\scriptsize CASCADE} calculations~\\cite{Gervais} including the evaporation width turned out to be essentially identical to those obtained by neglecting this width even up to excitation energy higher than 120 MeV for $^{120}$Sn (i.e., at $T >$  3.3 MeV). This indicates that the effect of evaporation width, if any, may become noticeable only at much higher values of $T$ ($\\gg$ 3.3 MeV) and $J$ ($\\gg$ 30$\\hbar$). \nIn a classical representation of the GDR as a damped spring mass system, the damping width of the oscillator (the GDR width) should be smaller than its frequency (the GDR energy) otherwise the spring mass system cannot make any oscillation. This means that the GDR width is upper-bounded by its energy. This implies the saturation of the GDR width.\n\nThe present lecture summarizes the achievements of the Phonon Damping Model (PDM)~\\cite{PDM1a,PDM2,PDMJ} in the description of the the GDR width and shape at finite $T$ and $J$ (Sec. \\ref{TJ}). As two applications, the GDR parameters predicted by the PDM and experimentally extracted are used to calculate the shear viscosity of finite hot nuclei, which is also employed to test the recent preliminary data of the GDR width at high $T$ and $J$ in $^{88}$Mo (Sec. \\ref{visco}). Conclusions are drawn in the last section.\n\n\\section{Damping of GDR in highly excited nuclei}\n\\label{TJ}\n\\subsection{GDR width and shape in hot nuclei}\n\\label{T}\nThe width of the GDR built of the ground state ($T=$ 0) (the quantal width $\\Gamma_{Q}$), consists of the three components: (i) the Landau width $\\Gamma^{LD}$, which is essentially the variance $\\sqrt{\\langle E^2\\rangle - \\langle E\\rangle^2}$ of the $ph$-state distribution, (ii) the spreading width $\\Gamma^{\\downarrow}$ caused by coupling of $1p1h$ states to more complicated configurations such as $2p2h$ ones, and (iii) the escape width $\\Gamma^{\\uparrow}$ owing to the direct particle decay into hole states of the residual nucleus because of coupling to continuum. In medium and heavy nuclei, the major contribution to $\\Gamma_Q$ is given by $\\Gamma^{\\downarrow}$, whereas $\\Gamma^{LD}$ and $\\Gamma^{\\uparrow}$ account for a small fraction. The calculations within the microscopic models such as the particle+vibration model~\\cite{Bortignon} and the quasiparticle-phonon model~\\cite{QPM} have shown that $\\Gamma^{\\downarrow}$ does not increase width $T$. Therefore the mechanism of the width's increase width $T$ should be sought beyond the one that causes $\\Gamma^{\\downarrow}$.\n\nThe PDM's Hamiltonian consists of the \nindependent single-particle (quasiparticle) field, GDR phonon field, and the coupling between \nthem [Eq. (1) in Ref. \\cite{PDM1a}].   The Woods-Saxon potentials at $T =$ 0 are often \nused to obtain the single-particle energies $\\epsilon_k$. The GDR width $\\Gamma(T)$ is a sum: $\\Gamma(T)=\\Gamma_{\\rm Q}+\\Gamma_{\\rm T}$ of\nthe quantal width, $\\Gamma_{\\rm Q}$, and thermal width, $\\Gamma_{\\rm T}$.\nIn the presence of superfluid pairing, the quantal and thermal widths \nare given as~\\cite{PDM2}\n\\begin{equation}\n\\Gamma_{\\rm Q}=2\\gamma_Q(E_{GDR})=2\\pi F_{1}^{2}\\sum_{ph}[u_{ph}^{(+)}]^{2}(1-n_{p}-n_{h})\n\\delta[E_{\\rm GDR}-E_{p}-E_{h}]~,\n\\label{GammaQ}\n\\end{equation}\n\\begin{equation}\n\\Gamma_{\\rm T}=2\\gamma_T(E_{GDR})=2\\pi F_{2}^{2}\\sum_{s>s'}[v_{ss'}^{(-)}]^{2}(n_{s'}-n_{s})\n\\delta[E_{\\rm GDR}-E_{s}+E_{s'}]~, \n\\label{GammaT}\n\\end{equation}\nwhere $u_{ph}^{(+)} = u_pv_h+u_hv_p$, $v_{ss'}^{(-)}=u_su_{s'}-v_sv_{s'}$ ($ss' = pp', hh'$) with\n$u_k$ and $v_k$ being the coefficients of Bogolyubov's transformation from particle operators to \nquasiparticle ones, $E_k\\equiv\\sqrt{(\\epsilon_k-\\lambda)^2+\\Delta^2}$, with superfluid pairing gap $\\Delta$,\nare quasiparticle energies, $n_k$ are quasiparticle occupations numbers, which, for medium and heavy nuclei, can be well approximated with the Fermi-Dirac distribution for\nindependent quasiparticles, $n_k = [\\exp(E_k\/T)+1]^{-1}$. The parameter $F_1$ is chosen so that $\\Gamma_Q$ at $T=$ 0 is equal to GDR's width at $T=$ 0, whereas the parameter $F_2$ is chosen so that, with varying $T$,  the GDR energy $E_{GDR}$ does not change significantly. The latter is found as the solution of the equation $E_{GDR} - \\omega_{q}-P_q(E_{GDR})=0$, where $\\omega_q$ is the energy of the GDR phonon before the coupling between the phonon and single-particle mean fields is switched on, and $P_q(\\omega)$ is the polarization operator owing to this coupling, whose explicit expression in given in Refs. \\cite{PDM2}. The GDR strength function is calculated as \n\\begin{equation}\nS_{q}(\\omega) = \\frac{1}{\\pi}\\frac{\\gamma_Q(\\omega) + \\gamma_T(\\omega)}{(\\omega-E_{GDR})^2+[\\gamma_Q(\\omega) + \\gamma_T(\\omega)]^{2}}~.\\label{S}\n\\end{equation}\nIn numerical calculations the representation $\\delta(x) =\\lim_{\\varepsilon\\rightarrow 0}\\varepsilon\/[\\pi(x^{2}+\\varepsilon^2)]$ is used for the $\\delta$-functions in Eqs. (\\ref{GammaQ}) and (\\ref{GammaT}) with $\\epsilon=$ 0.5 MeV.\n\n\\begin{figure}\n\\center{\n\\includegraphics[width=10.3cm]{widthSnPb.eps}}\n\\caption{GDR widths for $^{120}$Sn (a) and $^{208}$Pb (b) predicted by the PDM (thick solid), pTSFM (dot-dashed), AM (double dot-dashed), and FLDM (thin solid) as functions of $T$ in comparison with experimental data in tin and lead regions. (c): Exact canonical neutron (N) and proton (Z) pairing gaps for $^{201}$Tl as functions of $T$. (d): GDR width for $^{201}$Tl obtained within the PDM as a function of $T$ (thick solid) including the exact canonical gaps in (c) in comparison with the experimental data for $^{201}$Tl (black circles) and $^{208}$Pb (open boxes). The thin solid line is the PDM result without the effect of thermal pairing. The dotted line is the PDM result for $^{208}$Pb [the same as the thick solid line in (b)]. \n\\label{widthSnPb}}\n\\end{figure}\nThe GDR widths predicted by the PDM, the two versions of thermal shape fluctuation model (TSFM), namely the phenomenological TSFM (pTSFM)~\\cite{pTSFM} and the adiabatic model (AM)~\\cite{AM}, and the Fermi liquid drop model (FLDM)~\\cite{FLDM} for $^{120}$Sn and $^{208}$Pb are shown in Figs. \\ref{widthSnPb} (a) and \\ref{widthSnPb} (b) in comparison with the experimental systematics. The PDM results for $^{120}$Sn include the effect of non-vanishing thermal pairing gap because of thermal fluctuations owing to finiteness of nuclei. The figure clear shows that among the models under consideration, the PDM is the only one that is able to describe well the experimental data in the entire temperature region including $T\\leq$ 1 MeV, where the other model fail. It is also able to reproduce the very recent data for the GDR width in $^{201}$Tl at 0.8$\\leq T<$ 1.2 MeV [Fig. \\ref{widthSnPb} (d)] after including the exact canonical \ngaps for neutrons and protons shown in Fig. \\ref{widthSnPb} (c)~\\cite{Tl201}.\n\n\\begin{figure}\n\\center{\n\\includegraphics[width=9.3cm]{SE.eps}}\n\\caption{Experimental (shaded areas) and theoretical shapes for GDR in $^{120}$Sn generated by the {\\scriptsize CASCADE} code at various excitation energies. (a) -- (c): predictions generated by using the PDM strength functions. (d) and (e)~\\cite{Gervais}: dotted and dashed lines are generated by using the TSFM strength functions, exhausting 80$\\%$ and 100$\\%$ TRK sum rule, respectively.\\label{SE}}\n\\end{figure}\nFor an adequate description of not only the width but also the entire GDR shape, the PDM strength functions were incorporated into all the decay steps of the full statistical calculations and the generated results are compared with those obtained from the measured $\\gamma$-ray spectra in Fig. \\ref{SE}, which shows that the PDM describes fairly well the GDR shape [Fig. \\ref{SE} (a)], whereas the TSFM fails in doing so [Fig. \\ref{SE} (b)].\n\\subsection{GDR width and shape in hot and rotating nuclei}\n\\label{J}\nTo describe the non-collective rotation of a spherical nucleus, the $z$-projection $M$ of the total angular momentum $J$ is added into the PDM Hamiltonian as  $- \\gamma\\hat{M}$, where $\\gamma$ is the rotation frequency~\\cite{PDMJ}. The latter and the chemical potential are defined, in the absence of pairing,  from the equation $M = \\sum_k m_k(f_{k}^{+}-f_{k}^{-})~,$ and  $N =\\sum_k(f_{k}^{+}+f_{k}^{-})~$, where $N$ is the particle number and $f_k^{\\pm}$ are the single-particle occupation numbers, $f_{k}^{\\pm} =1\/[\\exp(\\beta E_k^{\\mp})+1]$, and $E_k^{\\mp} = \\epsilon_k-\\lambda\\mp\\gamma m_k~$. With the smoothing of $\\delta$-functions by using the Breit-Wigner distribution mentioned in Sec. \\ref{T}, the final form of phonon damping $\\gamma_q(\\omega)$ becomes\n\\begin{equation}\n\\gamma_q(\\omega) = \\varepsilon\\sum_{kk'}[{\\cal F}_{kk'}^{(q)}]^{2}\n\\bigg[\\frac{f_{k'}^{+}-f_{k}^{+}}\n{(\\omega-E_k^{-} + E_{k'}^{-})^2+\\varepsilon^2}+\\frac{f_{k'}^{-}-f_{k}^{-}}{(\\omega-E_k^{+} + E_{k'}^{+})^{2}+\\varepsilon^{2}}\\bigg]~,\n\\label{gamma1}\n\\end{equation}\nwhere $(k,k') = ph, pp', hh'$. The GDR strength function is calculated by using the same Eq. (\\ref{S}) where \n$\\gamma_Q(\\omega) +\\gamma_T(\\omega)$ is replaced with $\\gamma_q(\\omega)$. The explicit expression for the polarization operator $P_q(\\omega)$ is given in Eq. (13) of Ref. \\cite{PDMJ}.\n\n\\begin{figure}\n     \\center{\\includegraphics[width=9cm]{streJSn1.eps}\n     \\caption{(a) -- (f): GDR strength functions  for\n     $^{106}$Sn at $T=$ 0.5, 1, 2, 3, 4, and 5 MeV as shown at the curves and $M=$ 0, 10, 20, 40, 60, and 80$\\hbar$ as shown in the panels.\n     (g): FWHM of GDR for $^{106}$Sn as a function of $T$\n     at several values of $M$ (in $\\hbar$) shown at the curves, (h): FWHM of GDR for $^{106}$Sn as a function of $M$ at several values of $T$ (in MeV) shown at the curves. The experimental data for GDR in $^{106}$Sn (solid circles), $^{109,110}$Sn (solid and open boxes) are adapted from Refs. \\cite{Mattiuzzi,Bra}.}    \n      \\label{streJSn}}\n\\end{figure}\nShown in  Fig. \\ref{streJSn}  are the GDR strength functions $S(\\omega)$ and the widths in\n$^{106}$Sn at various $T$ and $M$.  The GDR shape becomes smoother  as $T$ and $M$ increase, and the smoothing caused by the angular momentum is stronger than that caused by thermal effects (Figs. \\ref{streJSn} (a) -- \\ref{streJSn} (f)). The GDR width increases with both $T$ and $M$ and stronger at low $T$ and $M$. This increase in the width approaches a saturation at moderate and high $T$ and\/or $M$. As a function of $T$, the saturation begins at $T>$ 4 MeV in $^{106}$Sn [Fig. \\ref{streJSn} (g)], whereas as a function of $M$ it takes place in $^{106}$Sn already at $T\\geq$ 3 MeV [Fig. \\ref{streJSn} (h)].  Experimental data for $^{106}$Sn~\\cite{Mattiuzzi} and $^{109,110}$Sn~\\cite{Bra} are also shown in Figs. \\ref{streJSn} (g) and \\ref{streJSn} (h), which are in fair agreement with theory.\n\\section{Shear viscosity of hot nuclei}\n\\label{visco}\nIn the verification of the condition for applying hydrodynamics to \nnuclear system, the quantum mechanical \nuncertainty principle requires a finite viscosity for any thermal \nfluid. Kovtun, Son and Starinets (KSS)~\\cite{KSS} conjectured\nthat the ratio $\\eta\/s$ of shear viscosity $\\eta$ to the entropy \nvolume density $s$ is bounded below for all fluids, namely the value\n${\\eta}\/{s}= {\\hbar}\/(4\\pi k_{B})$  \nis the universal lower bound (KSS bound or unit).\nFrom the viewpoint of collective theories, one of the fundamental \nexplanations for the giant resonance damping is the friction term (or viscosity) \nof the neutron and proton fluids.  By using the Green-Kubo's relation, it has been shown in Ref. \\cite{visco} that the shear viscosity $\\eta(T)$ \nat finite $T$ is expressed in terms of the GDR's parameters at zero and finite $T$ as\n\\begin{equation}\n\\eta(T)=\\eta(0)\\frac{\\Gamma(T)}{\\Gamma(0)}\n\\frac{E_{GDR}(0)^{2}+[\\Gamma(0)\/2]^{2}}{E_{GDR}(T)^{2}+[\\Gamma(T)\/2]^{2}}~.\n\\label{eta1}\n\\end{equation}\n\\begin{figure}\n     \\center{\\includegraphics[width=9.0cm]{etaratio.eps}\n     \\caption{Shear viscosity $\\eta(T)$ [(a) and (b)] and ratio \n     $\\eta\/s$ [(c) and (d)] as functions of $T$ for nuclei in tin [(a) and (c)], and lead [(b) and (d)] regions. The  \n     gray areas are the PDM predictions by using $0.6u\\leq\\eta(0)\\leq \n     1.2u$ with $u=$ 10$^{-23}$ Mev s fm$^{-3}$.}\n     \\label{eta&ratio}}\n\\end{figure}\nThe predictions for the shear viscosity $\\eta$ and the ratio $\\eta\/s$ by the PDM,  pTSFM, AM, and \n FLDM for $^{120}$Sn and $^{208}$Pb are plotted as functions of $T$ \nin Fig. \\ref{eta&ratio} in comparison with the empirical results. \nThe latter are extracted from the \nexperimental systematics for GDR in tin and lead regions~\n\\cite{Schiller} making use of Eq. (\\ref{eta1}).  It is seen in Fig. \\ref{eta&ratio} that the predictions by the PDM have the best overall \nagreement with the empirical results. It produces an increase \nof $\\eta(T)$ with $T$ up to 3 - 3.5 MeV and a saturation of $\\eta(T)$ within (2 - 3)$u$ \nat higher $T$ [with \n$\\eta(0)=$ 1$u$, $u=$ 10$^{-23}$ Mev s fm$^{-3}$]. The ratio $\\eta\/s$ decreases sharply with \nincreasing $T$ up to $T\\sim$ 1.5 MeV, starting from which the decrease \ngradually slows down to reach (2 - 3) KSS units \nat $T=$ 5 MeV. The FLDM has a similar trend as that of the PDM up to \n$T\\sim$ 2 - 3 MeV, but at higher $T$ ($T>$ 3 MeV for $^{120}$Sn or 2 MeV for \n$^{208}$Pb) it produces an increase of both $\\eta$ and $\\eta\/s$ with $T$. \nAt $T=$ 5 MeV the FLDM model predicts the ratio $\\eta\/s$ within (3.7 - 6.5) KSS units, which are \nroughly 1.5 -- 2 times larger than the PDM predictions. The AM and pTSFM show a similar trend for $\\eta$ and $\\eta\/s$. \nHowever, in order to obtain such similarity, $\\eta(0)$ in the pTSFM \ncalculations has to be reduced to 0.72$u$ instead of 1$u$. They all \noverestimate $\\eta$ at $T<$ 1.5 MeV. \n\nA model-independent estimation for the high-$T$ limit of the ratio $\\eta\/s$ can also be inferred \ndirectly from Eqs. (\\ref{eta1}). Assuming that,\nat the highest $T_{max}\\simeq$ 5 - 6 MeV where the GDR can still exist, the \nGDR width $\\Gamma(T)$ cannot exceed $\\Gamma_{max}\\simeq \n3\\Gamma(0)\\simeq 0.9E_{GDR}(0)$~\\cite{Auerbach1}, and $E_{GDR}(T)\\simeq E_{GDR}(0)$, one obtains from \nEq. (\\ref{eta1}) $\\eta_{max} \\simeq 2.551\\times\\eta(0)$. By noticing \nthat, $S_{F}\\to 2\\Omega\\ln{2}$ at $T\\to\\infty$ \nbecause $n_{j}\\to$ 1\/2, where $\\Omega=\\sum_{j}(j+1\/2)$ for \nthe spherical single-particle basis or sum of all doubly-degenerate levels for the \ndeformed basis and that the particle-number conservation requires $A = \n\\Omega$ since all single-particle occupation numbers are equal to \n1\/2, one obtains the high-$T$ limit of entropy density $s_{max} = 2\\rho\\ln{2}\\simeq 0.222~(k_{B})$.\nDividing $\\eta_{max}$ by $s_{max}$ yields the high-$T$ limit (or lowest bound) for $\\eta\/s$ \nin finite nuclei, that is $({\\eta}\/{s})_{min}\\simeq 2.2^{+0.4}_{-0.9}$ KSS units, where the empirical values for $\\eta(0) = \n1.0^{+0.2}_{-0.4}~u$ are used~\\cite{Auerbach1,fission}. Based on these results, one can conclude that\nthe value of $\\eta\/s$ for medium and heavy nuclei at $T=$ 5 MeV is in \nbetween (1.3 - 4.0) KSS units, which is about (3 - 5) times smaller \n(and of much less uncertainty) that the value between (4 - 19) KSS units predicted by \nthe FLDM for heavy nuclei~\\cite{Auerbach}, where the same lower value $\\eta(0)=$0.6$u$ was used.\n\nFinally, by using the temperature dependence of $\\eta\/s$ and the KSS lower bound conjecture, it is possible to examine the recent preliminary data for the GDR width in $^{88}$Mo in Ref. \\cite{Maj}. Shown in Fig. \\ref{Stest} is the strength function \n$S_L(\\omega) = {\\omega}[S(\\omega, E_{GDR})-S(\\omega, -E_{GDR})]\/E_{GDR}$, where $S(\\omega, \\pm E_{GDR})$ are the PDM strength functions (\\ref{S}) at finite $T$ and $J$ for the GDR located at $\\pm E_{GDR}$. The PDM predictions are shown at the initial temperature $T_{max}$ of the compound nucleus ($T_{max} =$ 3 and 4 MeV in Figs. \\ref{Stest} (a) and \\ref{Stest} (b), respectively), and also at $T=$ 2.5  MeV (Figs. \\ref{Stest} (a)) and 3.2 MeV (Figs. \\ref{Stest} (b)), i.e. within the error bars of the average temperature $\\langle T\\rangle$ obtained by averaging over all the GDR decay steps ($\\langle T\\rangle =$ 2$\\pm$0.6 and 2.6$\\pm$0.8 MeV for $E^{*}=$ 300 and 450 MeV, respectively~\\cite{Maj1}). While the PDM strength functions and experimental line shapes of the GDR agree fairly well at $M=$ 41 $\\hbar$ with the FWHM $\\Gamma$ predicted by the PDM between 9.6 MeV (T = 2.5 MeV) and 11 MeV (T = 3 MeV), they strongly mismatch at $M=$ 44 $\\hbar$, where the experimental GDR peak becomes noticeably narrower with a width $\\Gamma_{ex}\\simeq$ 7.5 MeV. By using this value $\\Gamma_{ex}$ and $\\eta(0)=$ 0.6 $u$, one ends up with the value of $\\eta\/s=$ 0.85 KSS units. Including the error bars in $\\Gamma_{ex}$ leads to $\\Gamma_{ex}^{<}\\simeq$ 6 MeV and $\\Gamma_{ex}^{>}\\simeq$ 8.5 MeV, which give the values of $\\eta\/s$ equal to 0.69 and 0.94 KSS, respectively. All these values are smaller than the KSS lower bound conjecture. This may indicate that either (i) the data analysis in extracting the experimental GDR strength function for $^{88}$Mo at $E^{*} = 450$ MeV (Fig. \\ref{Stest} (b)) is inaccurate, or (ii) a violation of the KSS conjecture has been experimentally confirmed for the first time ever. The reanalysis of the data is now underway to clarify which one from these two conclusions holds~\\cite{Maj1}. \\begin{figure}\n\\center{\n     \\includegraphics[width=11 cm]{STE1_rev.eps}\n     \\caption{GDR strength function $S_L(\\omega)$ for $^{88}$Mo at $M=$ 41 $\\hbar$ (a) and \n    $M=$ 44 $\\hbar$ (b) predicted by the PDM in comparison with the preliminary data from Ref. \\cite{Maj}.}     \n      \\label{Stest}}\n\\end{figure}\n \n\n\\section{Conclusions}\nThe PDM generates the damping of GDR through its couplings to $ph$ configurations, causing the quantal width, as well as to $pp$ and\/or $hh$ configurations, causing the thermal width. This leads to an overall increase in the GDR width at low and moderate $T$, and its saturation at high $T$. At very low T $<$ 1 MeV the GDR width remains nearly constant because of thermal pairing. The GDR width also increases with angular momentum $M$ and saturates at high $M$, but this saturation goes beyond the value of maximal angular momentum that the nucleus can sustain without violating the KSS conjecture, that is 46 and 55$\\hbar$ for $^{88}$Mo and $^{106}$Sn, respectively, if the value $\\eta(0) =$ 0.6$\\times 10^{-23}$ Mev s fm$^{-3}$ for the shear viscosity at $T=$ 0 is used.   The PDM predictions agree well with the experimental systematics for the GDR width and shape in various medium and heavy nuclei. The PDM also predicts the shear viscosity to the entropy-density ratio $\\eta\/s$ between (1.3 - 4.0) KSS units for medium and heavy nuclei at $T=$ 5 MeV, almost the same at that of the quark-gluon-plasma like matter at $T>$ 170 MeV discovered at RHIC and LHC. The PDM and the KSS conjecture are also used to show that the recent preliminary experimental data for GDR in $^{88}$Mo~\\cite{Maj} need to be reanalyzed. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{secIntro}\n\nThe Peak Over Threshold (POT) method, described in details by \\citet[][Chapter 4]{coles2001introduction},  \nhas been used in many fields to identify extremal events such as loads,\nwave heights, floods, wind velocities, etc. This method provides a\nmodel for independent exceedances over a high threshold.\nAccording the  \nPickands-Balkema-de Haan theorem \\citep{balkema:dehaan:1974,pickands:1975}\nfor a threshold $u$  sufficiently high, the unnormalized conditional tail distribution of a random variable $Y$ can be approximated as \n$$\n\\Pr(Y-u\\le y|Y>u)\\approx \tH_{\\xi}(y\/\\psi_u),\\qquad \\text{for}\\, y\\ge 0\\quad \\text{and}\\quad \\psi_u >0\n$$\nwhere\n\\begin{equation}\\label{eq:GPD}\n\tH_{\\xi}(z) =\n\t\\begin{cases}\n\t\t1 - {(1+\\xi y)}_+^{-1\/\\xi} & \\text{for $\\xi \\neq 0$}\\\\\n\t\t1-\\exp\\{-y \\}& \\text{for $\\xi = 0$}\n\t\\end{cases}  \n\\end{equation}\nand  $a_+ = \\max (a, 0)$. \n\n The limiting distribution \\eqref{eq:GPD} is referred to as the Generalized Pareto  distribution (GPD) and the parameters $\\xi$ and $\\psi_u$ are respectively the shape and the scale parameters. \n The scale parameter depend on the threshold $u$, but \nin  remainder of the paper the subscript $u$ is  dropped in the  notation.\n\nModeling using GPD has become very popular since the paper of \\citet{davison:smith:1990} where estimation and model-checking procedures for univariate and regression data are developed.\n Such procedure have then been extended  by allowing the GPD parameters to be represented as smooth functions of covariates \\citep{chavez-demoulin:davison:2005}.\n \n Since then several R packages have been developed to fit GPDs  and contributed to the Comprehensive R Archive Network (CRAN) in the field of extreme value analysis \\citep{team2013r}. Among them, \\texttt{ismev} \\citep{heffernan2016ismev}, providing tools for the  extreme value analyses presented in \\citet{coles2001introduction}, \\texttt{evd} \\citep{stephenson2002evd}, \\texttt{evir} \\citep{pfaff2018package}, \\texttt{extRemes} \\citep{gilleland2016extremes} and \\texttt{mev} \\citep{belzile1mev}, have offered various approaches for fitting univariate and multivariate GPDs. \\citet{gilleland2013software} and, more recently, \\citet{Belzile-et-al:2022} review them while the \\citet{dutang2022cran}'s CRAN Task View provides an up-to-date list of the R packages in this field. In particular, some of them offer regression-based models for extreme distributions, where GPD parameters are allowed to vary with covariates. \nBeyond the linear forms introduced in the packages \\texttt{ismev} and \\texttt{evd},  \na few R packages allow to fit GPDs with parameters following additive forms i.e. when the parameters or their transformations can be modelled as additive smooth functions of the covariates. \nIn the sequel we term this form a Generalized Additive Model (GAM) \\citep{Hastie:Tibshirani:1990,wood2017generalized}. Thus \\texttt{VGAM} \\citep{yee2007vector}, \\texttt{gamlss} \\citep{rigby2005generalized} and \\texttt{evgam} \\citep{youngman2020evgam} provide this functionality for GPDs, as reviewed among other alternatives in \\citet{youngman2020evgam}. \n\n\nIt is clear that GPD  needs\nthe determination of a threshold which is neither too high.\nThe choice of an appropriate threshold is  challenging because the stability plots and the mean residual life\n\\citep[][Chapter 4]{coles2001introduction} plot do not give an obvious value of a suitable threshold. \nThe problem is tackled  in a regression setting by\n\\citet{eastoe:tawn:2009}. \nHowever the question of the threshold choice and then simulation of the whole spectrum of the random variable considered remains problematic. \n\nForming mixtures of Exponential distributions \\citep{wilks1998multisite,keller2015implementation} or conditional Gamma distributions \\citep{kenabatho2012stochastic,chen2014multi} remains a competitive alternative e.g. for heavy-tailed precipitation modeling and simulation. Approaches of this type are implemented in the package \\texttt{GSM}, that uses a Bayesian approach for estimating a mixture of Gamma distributions in which the mixing occurs over the shape parameter and the procedure only requires to specify a prior distribution for a single parameter. The package \\texttt{evmix} fits an extreme value mixture model with normal behavior for the bulk distribution up to the threshold and conditional GPD above the threshold. Options for profile likelihood estimation for threshold and fixed threshold approach are provided. Finally, \\texttt{distributionsrd} provides the double Pareto-lognormal distribution. \n\n\nAlthough interesting, in practice these packages do not allow to fit nonlinear models as flexible as the GAM options allowed for the GPD alone.\nBesides, the mixture approaches tend to quickly inflate the number of parameters and the inference of the latter may be problematical operationally \\citep{naveau2016modeling}.\n\n\\citet{papastathopoulos2013extended} opened another route in the field of heavy-tail distribution modeling by replacing the uniform draws in the inverse cumulative distribution function (CDF) transform to generate simulations of the GPD, by draws from a richer distribution e.g. type Beta distribution. Using this approach, \\citet{naveau2016modeling} presented a concise parametric formulation (later extended to a semi-parametric formulation \\citep{tencaliec2020flexible}) for simulating the whole range of a positive random variable (namely precipitations), that makes use of GPD to model not only the high quantiles but the smallest ones as well and bypass the issue of threshold selection. \nThe main idea rooted from composing a cumulative distribution function with the GPD (see section 2 for more details). Very recently, a similar idea was taken up by \\citet{stein2021env,stein2021ext}.\n\n\nIn the sequel, we refer to the extension of GPD in the sense of \\citet{naveau2016modeling} as Extended Generalized Pareto distribution (EGPD).\nAn implementation of EGPD with four parametric families, together with the fitting procedure, can be found  in the R package \\texttt{mev}.  \n\n\nThe aim of this work is to extend the applicability of the EGPD in the presence of covariates that can modify the parameters of the distribution.\nThe extension takes place in the same spirit as  GAM. \nWe also want to present an add-on to the \\texttt{gamlss} package that allows to make inference on the parameter.\n\nWe have chosen the \\texttt{gamlss} package because it provides univariate distributional regression models, where parameters of the assumed distribution for the response can be modeled as additive functions of the covariates. It is flexible enough to allow the encoding of novel families of distribution, as long as their density, cumulative, quantile and random generative function and, optionally, first and second (cross-) derivatives of the likelihood can be computed.  We seize this potential to implement the parametric EGPD family in a generic way, allowing to introduce new parametric forms, up to 4 parameters so far (due to limitations of \\texttt{gamlss}), satisfying the requirement of the EGPD family.\n\nThe remaining of this article will first introduce the EGPD and the  distribution  regression model (Section \\ref{secTheory}), followed by the key commands to use the EGPD within the \\texttt{gamlss} package and our add-on in Section \\ref{secTuto}. \nSection \\ref{secApplRainfalls} implements them to perform an analysis of hourly positive rainfalls on the North-west of France and discuss specific advantages and limitations of distributional regression modeling and its implementation in R, while Section \\ref{secConclu} concludes the paper. \n\n\\section{A distributional regression model}\n\\label{secTheory}\n\n\\subsection{Extended-GPD}\n\\label{secEGPDmod}\n\nThe EGPD \\citep{naveau2016modeling}  accounts simultaneously and in a parsimonious way for both extreme and non extreme values of a random variable $Y$. Its cumulative distribution function (CDF) reads:\n\\begin{equation}\n\\label{CDFegpd}\n    F(y) = G\\{ H_{\\xi}(y\/\\psi) \\} \\:, \\: \\: \\text{for all}\\quad y > 0.\n\\end{equation}\nThe function $G(\\cdot)$ is a continuous CDF on the unit interval. It is constrained by several assumptions: 1) we need to ensure that the upper-tail behavior of $F$ follows a GPD; 2) the low values of $Y$, seen as the upper tail of $-Y$ when $y \\rightarrow 0$, follows a GPD as well. \n\nThis leads to the following consequences for any candidate CDF $G(\\cdot)$: \n\\begin{enumerate}\n\t\\item[a.]  the upper-tail behavior\n\tof $1-F(y)$ is equivalent to the original GPD tail used to build $F(y)$ ; \n\t\\item[b.]  when $y \\rightarrow 0$, $\\displaystyle F(y) \\sim \\frac{c}{\\psi^s}y^s $ where $\\displaystyle c = \\lim_{y \\rightarrow 0} \\frac{G(y)}{y^s}$ is positive and finite for some real $s$.\n\\end{enumerate}\n\nFour parametric families, $G(\\cdot,\\kappa)$, $\\kappa=(\\kappa_1,\\ldots,\\kappa_k)^T$, that satisfies the above-mentioned constraints are proposed in \\citet{naveau2016modeling}.\n\n\\begin{enumerate}\n    \\item \\label{c1} \n    $G(u;\\kappa)=u^{\\kappa_1}$, $\\kappa_1 >0$\n    \\item \\label{c2}\n    $G(u;\\kappa)=\\kappa_3 u^{\\kappa_1}+(1-\\kappa_3)u^{\\kappa_2}$, $\\kappa_2 \\geq \\kappa_1 >0$ and $\\kappa_3 \\in [0,1]$\n    \\item \\label{c3}\n    $G(u;\\kappa)=1-Q_{\\kappa_1}\\{(1-u)^{\\kappa_1}\\}$, $\\kappa_1>0$ where $Q_{\\kappa_1}$ is the CDF of a Beta random variable with parameters $1\/\\kappa_1$ and $2$, that is:\n    $$Q_{\\kappa_1}(u)=\\frac{1+\\kappa_1}{\\kappa_1} u^{1\/\\kappa_1} \\Big( 1-\\frac{u}{1+\\kappa_1} \\Big)$$\n    \\item \\label{c4}\n    $G(u;\\kappa)=[1-Q_{\\kappa_1}\\{(1-u)^{\\kappa_1}\\}]^{\\kappa_2\/2}$, $\\kappa_1, \\kappa_2>0$\n\\end{enumerate}\n\nIn model \\ref{c1}, $\\xi$ controls the rate of upper tail decay, $\\kappa_1$ the shape of the lower tail and $\\psi$ is a scale parameter. $\\kappa_1=1$ allows us to recover the GPD model, while varying it gives flexibility in the description of the lower tail. \n\nModel \\ref{c2}, as a mixture of power laws, allows to increase the flexibility provided by model \\ref{c1}. Therein, $\\kappa_1$ controls the lower tail behavior while $\\kappa_2$ modifies the shape of the density in its central part. \n\nModel \\ref{c3} is connected to the work of \\citet{falk2010laws}. $\\xi$ keeps controlling the upper extreme tail, $\\kappa_1$ the central part of the distribution in a threshold tuning way. However, this model imposes a behavior type $\\chi^2$ for the lower tail (null density at $0$) instead of allowing the data to constrain it. \n\nHence Model \\ref{c4}, which adds an extra parameter $\\kappa_2$ to circumvent this limitation. In this last model, $\\kappa_2, \\kappa_1, \\xi$ control respectively the lower, moderate and upper parts of the distribution. By construction, the lower and upper tails are GPD of shape parameters $\\kappa_2$ and $\\xi$ respectively. \n\n\nIn the sequel we denote the vector of parameters $(\\xi,\\psi,\\kappa^T)^T$ with $\\theta = (\\theta_1, ..., \\theta_d)^T$ and the density of a parametrized EGPD with $EGPD(y;\\theta)$.\nTo simulate from Equation (\\ref{CDFegpd}), the user randomly draws a uniform variable $U$ in $[0,1]$ and then applies the quantile function $Y=F^{-1}(U)$, given for $p \\in [0,1]$ by:\n\\begin{equation}\n\\label{quantileegpd}\n     y_p=F^{-1}(p) =\n     \\begin{cases}\n       \\frac{\\psi}{\\xi}[\\{ 1-G^{-1}(p) \\}^{-\\xi} - 1] & \\text{for $\\xi \\neq 0$}\\\\\n       -\\psi \\log{ \\{ 1-G^{-1}(p) \\}} & \\text{for $\\xi = 0$}\n    \\end{cases}  \n\\end{equation}\n\n\n\n\\subsection{Additive model specification}\n\\label{GAM}\n\nIn practice, the parameters of the distribution of $Y$ may depend on  some covariates $\\boldsymbol{x}=(x_1,\\ldots,x_m)^T$, i.e.\n\\begin{equation}\\label{eq:EGPD}\nY|\\boldsymbol{x}\\sim EGPD(\\cdot,\\theta(\\boldsymbol{x}))\n\\end{equation}\nwhere $\\theta(\\boldsymbol{x})=(\\theta_1(\\boldsymbol{x}),\\ldots,\\theta_d(\\boldsymbol{x}))^T$.\nThe previous specification is an instance  of a  distributional regression model \\citep{Stasinopoulos-et-al-2018}. \n\nFor relating the different distributional parameters  $(\\theta_1(\\boldsymbol{x}),\\ldots,\\theta_d(\\boldsymbol{x}))$ to the covariates, we rely on additive predictors of the form\n\\begin{equation}\\label{eq:predictors}\n\\eta_i(\\boldsymbol{x})=f_{i1}(\\boldsymbol{x})+\\cdots+f_{iJ_i}(\\boldsymbol{x})\t\n\\end{equation}\nwhere $f_{i1}(\\cdot),\\ldots, f_{iJ_i}(\\cdot)$ are smooth functions of the covariates $\\boldsymbol{x}$.\n\nThe representation \\eqref{eq:predictors} is particularly flexible and allows to capture complex dependence patterns between distribution parameters and covariates\n$\\boldsymbol{x}$. For instance a semiparametric model with  two covariates $(x_1,x_2)$ can be written as\n$$\n\\eta_i(\\boldsymbol{x})=\\beta_{i1}+ \\beta_{i2}x_1+f_{i2}(x_2)\n$$\nAnother example is when we want to model  space-time data observed at site $(s_1,s_2)$ and at time $t$. In this case $\\boldsymbol{x}=(s_1,s_2,t)^T$\n$$\n\\eta_i(\\boldsymbol{x})=f_{i1}(t)+ f_{i2}(s_1,s_2)\n$$\n\n\n\nIn analogy to Generalized Linear Model the predictors are  linked to the distributional parameters via known monotonic and twice differentiable link functions $h_i(\\cdot)$.\n\nIn the case of model 1, i.e. $G(u,\\kappa)=u^\\kappa_1$, common link functions are\n$$\n\\xi(\\boldsymbol{x})=\\eta_\\xi(\\boldsymbol{x}) \\mbox{ (i.e. the identity function)},\\quad \\psi(\\boldsymbol{x})=\\exp(\\eta_\\psi(\\boldsymbol{x})),\\quad \\kappa_1(\\boldsymbol{x})=\\exp(\\eta_{\\kappa_1}(\\boldsymbol{x})).\n$$\n\n\\begin{equation}\\label{eq:link}\n\\theta_i(\\boldsymbol{x})=h_i(\\eta_i(\\boldsymbol{x})), \\qquad i=1,\\ldots, d\n\\end{equation}\nThe functions $f_{ij}$in  \\eqref{eq:predictors} are approximated in terms of  basis function expansions\n\n\\begin{equation}\\label{eq:basis}\n\tf_{ij}(\\boldsymbol{x})=\\sum_{k=1}^{K_{ij}} \\beta_{ij,k}B_k(\\boldsymbol{x}),\n\\end{equation}\nwhere $B_k(\\boldsymbol{x})$ are the basis functions and $\\beta_{ij,k}$ denote the corresponding basis coefficients. These basis can be of different types, thoroughly presented e.g. in \\citet{stasinopoulos2017flexible} or \\citet{wood2017generalized}.\n\nThe basis function expansions can be written as \n$$\nf_{ij}(\\boldsymbol{x})=  \\boldsymbol{z}_{ij}(\\boldsymbol{x})^T\\boldsymbol{\\beta}_{ij}\n$$\nwhere $\\boldsymbol{z}_{ij}(\\boldsymbol{x})$ is still a vector of transformed covariates that depends on the basis functions.\nand $\\boldsymbol{\\beta}_{ij}=(\\beta_{ij,1},\\ldots, \\beta_{ij,K_{ij}})^T$ is a parameter vector to be estimated.\n\nTo ensure regularization of the functions $f_{ij}(\\boldsymbol{x})$  so-called penalty terms are added to the objective log-likelihood function. \nUsually, the penalty for each function $f_{ij}(\\boldsymbol{x})$ are quadratic penalty $\\lambda \\boldsymbol{\\beta}_{ij}^T \\boldsymbol{G}_{ij}(\\boldsymbol{\\lambda}_{ij}) \\boldsymbol{\\beta}_{ij} $ where $\\boldsymbol{G}_{ij}(\\boldsymbol{\\lambda}_{ij})$ is a known semi-definite matrix and the vector  $\\boldsymbol{\\lambda}_{ij}$ regulates the amount of smoothing\nneeded for the fit. A special case is when $\\boldsymbol{G}_{ij}(\\boldsymbol{\\lambda}_{ij})=\\lambda_{ij}\\boldsymbol{G}_{ij}$. Therefore the type and properties of the smoothing functions are controlled by the vectors  $\\boldsymbol{z}_{ij}(\\boldsymbol{x})$ and the matrices $\\boldsymbol{G}_{ij}(\\boldsymbol{\\lambda}_{ij})$.\n\nThe penalized log-likelihood function for the latter models reads:\n\\begin{equation}\\label{eq:penlik}\n\tl_p = l - \\frac{1}{2}\\sum_{i=1}^d \\sum_{j=1}^{J_i} \\boldsymbol{\\beta}_{ij}^T \\boldsymbol{G}_{ij}(\\boldsymbol{\\lambda}_{ij}) \\boldsymbol{\\beta}_{ij}\n\\end{equation}\nwhere $l$ represents the log-likelihood function for the model \\eqref{eq:EGPD}.\n\n\\section{Distributional regression  model for EGPD within the R package \\texttt{gamlss}}\n\\label{secTuto}\n\nIn this section we introduce an add-on script in R \nthat implements the EGPD regression model described in the previous section. \nThe script, available in\n\\texttt{github.com\/noemielc\/egpd4gamlss}, is flexible enough as it only requires the specification of a parametric CDF $G(\\cdot)$.\n\n\nThe Generalized Additive Model for Location, Scale and Shape (GAMLSS) is an example of distributional regression model and the companion package in R \\texttt{gamlss} \\citep{stasinopoulos2017flexible}  allows a straightforward fitting of multiple built-in and well-known family distributions with parameters that vary flexibly with $\\boldsymbol{x}$ as in \\eqref{eq:predictors}-\\eqref{eq:link}.\n\nAn interesting feature is the possibility to extend the family of distributions described in \\citet{rigby2019distributions}.\nThe only limitation is the dimension of the parameter vector $\\theta$, $d=4$,\nwhere the four parameters are denoted with $\\mu$, $\\sigma$, $\\nu$ and $\\tau$ in \\texttt{gamlss}.\n\nDue to the restriction on the number of parameters, our code allows the implementation of parametric distributions for $G$ with only 2 parameters, namely $\\kappa=(\\kappa_1,\\kappa_2)$.\nIn the following, the parameters $(\\xi,\\psi,\\kappa_1,\\kappa_2)$ of the EGPD distribution are identified with $(\\mu,\\sigma,\\nu,\\tau)$.\n\nThe generic implementation of a EGPD is provided in the file \\texttt{GenericEGPDg.R}. The \\texttt{MakeEGPD} function assembles a new EGPD family   distribution exploiting an user-built $G(\\cdot)$ as a function of $u$ and (at most) two parameters $\\nu$ and $\\tau$. \nThe required  first,  second and  cross derivatives of the density in \\eqref{eq:EGPD}, necessary for the fitting procedure in \\texttt{gamlss}, are obtained by   the symbolic derivation  using the R package \\texttt{Deriv}. Moreover the inverse function $G^{-1}(u)$ is  computed numerically. \n\n\nIn the following, we exemplify the use of \\texttt{MakeEGPD} by assembling   Model 1, $G(u;\\kappa)=u^{\\kappa_1}, \\kappa_1 > 0$.\n\n\\begin{lstlisting}[language=R]\nlibrary(gamlss)\nsource (\"GenericEGPDg.R\")\nEGPD1Family <- MakeEGPD (function (z,nu) z^nu, Gname = \"Model1\")\nEGPDModel1 <- EGPD1Family()\n\\end{lstlisting}\nThe last command is necessary and allows  to create the object EGPDModel1 (name concatenating \\texttt{EGPD} and \\texttt{Gname}) from the \\texttt{EGPD1Family} function.\n\nFor given values $\\mu_0, \\sigma_0$ and $\\nu_0$, the R functions of\ndensity, cumulative distribution function, quantile function and random generation are ready to use.\n\\begin{lstlisting}[language=R]\ndEGPDModel1(x,mu=mu0,sigma=sigma0,nu=nu0)\npEGPDModel1(x,mu=mu0,sigma=sigma0,nu=nu0)\nqEGPDModel1(u,mu=mu0,sigma=sigma0,nu=nu0)\nrEGPDModel1(n,mu=mu0,sigma=sigma0,nu=nu0)\n\\end{lstlisting}\n\nThe reserved names  for the distribution parameters in \\texttt{gamlss} are $\\mu$ \\texttt{mu} ($\\mu$), \\texttt{sigma} ($\\sigma$), \\texttt{nu} ($\\nu$), and \\texttt{tau} ($\\tau$).\nTo change the link functions $\\eta_i=g_i(\\theta_i)$ by means of which each EGPD parameter $\\theta_i$ (here $\\mu,\\sigma$ or $\\nu$) is indirectly optimised, or the default starting values for the parameters to fit, or to print these links, we use the commands:\n\n\\begin{lstlisting}[language=R]\nEGPD1Family(mu.link=\"identity\",mu.init=1, # log , inverse, own , ...\n            sigma.link=\"log\",sigma.init=3, nu.link=\"log\",nu.init=0.5)\nshow.link(EGPD1Family)\n\\end{lstlisting}\n\n\nIt is possible to create an original link function \"own\", by gathering the expressions of the link function $\\eta_i$, its inverse, the derivative of the latter w.r.t. parameter $\\theta_i$ and its area of validity. For instance to create a log-link function shifted to the right for $\\mu$:\n\n\\begin{lstlisting}[language=R]\n# the link function defining the predictor eta  as a function of the current distribution parameter (referred to as mu), i.e. eta = g(mu)\nown.linkfun <- function (mu) {\n    .shift = 0.0001\n    log (mu - .shift) \n}\n#  the inverse of the link function as a function of the predictor eta, i.e.mu = g-1 (eta)\nown.linkinv <- function (eta) {\n    shift = 0.0001 \n    thresh <- - log (.Machine$double.eps)\n    eta <- pmin (thresh, pmax(eta, -thresh))\n    exp (eta) + shift \n} \n# the first derivative of the inverse link with respect to eta\nown.mu.eta <- function (eta) {\n    shift = 0.0001  \n    pmax (exp (eta), .Machine$double.eps) \n}\n# the range in which values of eta are defined.\nown.valideta <- function (eta) TRUE\nEGPD1Family(mu.link=\"own\")\n\\end{lstlisting}\n\nFor more details on link function definition, we refer to   \\citet[][pag. 179]{stasinopoulos2017flexible}.\n\n\n\\section{Application to a hourly rainfall dataset in France}\n\\label{secApplRainfalls}\n\n\\subsection{Data}\n\nWe illustrate our add-on to package \\texttt{gamlss} with the analysis of the hourly precipitations recorded over the north-west area of France from 2016 to 2018 both included. This database, containing precipitations as well as other meteorological variables whose values are registered every 6 minutes over the 3 years of interest for each of the 274 stations of the network (see Figure \\ref{figSets}), is freely accessible  \\citep[see][fro more details]{MeteoNet}.\n\nLet us note $Y(\\boldsymbol{x})$ the random variable for strictly positive hourly precipitations (dry and wet events are traditionally treated separately when it comes to rainfall simulation \\citep{ailliot2015stochastic}), with covariates $\\boldsymbol{x}=(x_l, x_L, x_t)$, where $x_l$ and $x_L$ represents the position in space (longitude, latitude) and $x_t$ a cyclic time index, namely the day or the month of the year. The work of \\citet{naveau2016modeling} and our tests show that EGPD fitting is improved by censoring the data, so in practice we remove values $Y(\\boldsymbol{x}) < 0.5$ (mm). Other thresholds ($\\leq 0.2$ and $\\leq 0.1$mm) have been tested but $Y(\\boldsymbol{x}) < 0.5$ (mm) gives the  best results, in particular for the upper tails. Besides, as noted in \\citet{evin2018stochastic,naveau2016modeling}, negative shape parameters $\\xi$ are not expected for rainfall distributions at these time scales, so we use a link function constraining the parameter $\\xi$ to positive values. Finally, as performed in \\citet{naveau2016modeling}, we only keep one every three hourly records of precipitation so as to ensure independency between samples.\n\n\\subsection{Model fitting}\n\nWe fit different specifications of EGPD-based models, where EGPD takes form 1, 3 or 4, and compare them with equivalently specified Gamma-based models. The Gamma (GA) distribution is a classical choice that works well for modeling the bulk of precipitations at a given site \\citep{katz1977precipitation,vrac2007stochastic,vlvcek2009daily}. However non-negligible deviations are observed when one is interested in capturing extreme rainfalls \\citep{katz2002statistics}, typically underestimated due to the lightness of the GA tail. For a marginal model  \\texttt{M=EGPD1,EGPD3,EGPD4,GA}, the different specifications read:\n\\begin{description}\n\t\\item \\texttt{M.0} where\n$$\t\\eta_a(\\boldsymbol{x})=\\beta_0 , \\qquad a=\\mu,\\sigma,\\nu,\\tau,$$\ni.e.\t the parameters $\\theta(\\boldsymbol{x})$ do not depend on the covariates ;\n\t\\item  \\texttt{M.t} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ s_t (x_t) , \\qquad a=\\mu,\\sigma,\\nu,\\tau,\n\t$$\n\t\tuses cyclic cubic splines $s_t$ to model the nonlinear dependency in time while considering no variation in space ;\n\t\\item \\texttt{M.tnomu} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ s_t (x_t) , \\qquad a=\\sigma,\\nu,\\tau,\n\t$$ does the same as before but considers as constant the parameter $\\mu$ ;\n\t\\item \\texttt{M.st} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ TP(x_l,x_L)+ s_t (x_t), \\qquad a=\\mu,\\sigma,\\nu,\\tau,\n\t$$\n\tuses smooth surface fitting (thin-plate splines) to model the dependency in space as well as cyclic cubic splines for the dependency in time ;\n\t\\item \\texttt{M.st2mu} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ TP(x_l,x_L)+ s_t (x_t), \\qquad a=\\mu,\\sigma,\n\t$$\n\tdoes the same as before but considers constant in space the parameters that are not $\\mu$ or $\\sigma$ ;\n\t\\item \\texttt{M.st2nomu} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ TP(x_l,x_L)+ s_t (x_t), \\qquad a=\\sigma,\\nu,\n\t$$\n\tdoes the same as before but considers constant in space the parameters $\\mu$ and $\\tau$ ;\n\t\\item \\texttt{M.st3nomu} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ TP(x_l,x_L)+ s_t (x_t), \\qquad a=\\sigma,\\nu,\\tau,\n\t$$\n\tdoes the same as before but applies to \\texttt{M=EGPD4} only and considers parameter $\\mu$ as a constant over space.\n\\end{description}\n\nThese variations within each model class will allow us to assess whether the smooth regression of distribution parameters over covariates is justified or not. \nBelow we reproduce the code to fit \\texttt{megpd1.0}, \\texttt{megpd1.tnomu} and \\texttt{mga.st} on the training set (\\texttt{training}), that contains 60 \\% of the stations, as reported on Figure \\ref{figSets}. The dataset consisting of the remaining stations form the validation set (\\texttt{validation}). The \\texttt{gamlss()} function reads:\n\n\\begin{lstlisting}[language=R]\n# Notations: lat, lon and cyc represent, respectively, covariates xl, xL and xt, while y is given by precip\n# Set algorithm control parameters:\ncon <- gamlss.control (n.cyc = 200, mu.step = 0.01, sigma.step = 0.01, nu.step = 0.01,tau.step = 0.01, autostep=TRUE)\nmegpd1.0 <- gamlss(precip ~ 1,\n       data = training, \n                  family = EGPD1Family(mu.link = \"own\"),\n                  control = con,\n                  method=CG())         \nmegpd1.tnomu <- gamlss(precip ~ 1,\n                  sigma.fo =~ pbc(cyc),\n                  nu.fo =~ pbc(cyc),\n                  data = training, \n                  family = EGPD1Family(mu.link = \"own\"),\n                  control = con,\n                  method=CG()) \n                      \n# Load library to interface with the mgcv package and get ga() for multidimensional tensor spline smoothers:\nlibrary(gamlss.add)\nfmla_ga <- list(~s(lon,lat,bs='tp',k=30)+s(cyc,bs=\"cc\",k = 50),\n                ~s(lon,lat,bs='tp',k=30)+s(cyc,bs=\"cc\",k = 50))\nGA.st <- gamlss(precip ~ga(~s(lon,lat,bs='tp',k=30)+s(cyc,bs=\"cc\",k = 50),                                          method=\"REML\"), \n                      ~ga(~s(lon,lat,bs='tp',k=30)+s(cyc,bs=\"cc\",k = 50), \n                      method=\"REML\"),\n                      ~ga(~s(lon,lat,bs='tp',k=30)+s(cyc,bs=\"cc\",k = 50), \n                      method=\"REML\"),\n                      data = training, \n                      family = GA,\n                      control = con,\n                      method=CG())\n\\end{lstlisting}\n\n\\subsection{Model assessment}\n\nIn order to evaluate the goodness of fit on the training set, we use summary statistics: Global Deviance (GD), Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC).\nThe \\texttt{GAIC(modelName,k=a)} function allows to compute GD (minus twice the fitted log-likelihood, $a=0$), the AIC ($a=2$) and the BIC ($a=\\log(\\mbox{number of observations})$) for each fitted model \\texttt{modelName} and to compare them as performed on Figure \\ref{figInfCriteria}. Besides, Table \\ref{table1} indicates the effective degree of freedom (df) computed for each model. The function call is reproduced below for the models of class EGPD1.\n\n\\begin{lstlisting}[language=R]\nGAIC_BIC_m1 <-GAIC(megpd1.0,megpd1.t,megpd1.tnomu,megpd1.st,megpd1.st2mu,megpd1.st2nomu,k=log(length(training$precip))\n\\end{lstlisting}\n\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.5\\columnwidth]{P3_SituationMapSets.png}\n\t\t\\caption{French stations from the MeteoNet public dataset used for our experiments, split into training and validation sets.\n\t\t\t{\\label{figSets}}\n\t\t}\n\t\\end{center}\n\\end{figure}\n\n\\begin{table}[h]\n\t\\begin{center}\n\t\t\\begin{tabular}{||l c||} \n\t\t\t\\hline\n\t\t\tModel & \\multicolumn{1}{c||}{df} \\\\ [0.5ex] \n\t\t\t\\hline\\hline\n\t\t\t\\texttt{megpd4.st}  & 242.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.st3nomu}  & 199.3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.st2nomu}  & 138.3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.st2mu}  & 122.5 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.t}  & 72.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.tnomu} & 38.0 \\\\ \n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.0} & 4 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.st}  & 183.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.st2nomu}  & 139.3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.st2mu}  & 121.3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.t}  & 54.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.tnomu}  & 37.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.0}  & 3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.st}  & 181.6 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.st2nomu}  & 137.3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.st2mu}  & 122.9 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.t}  & 54.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.tnomu}  & 37.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.0}  & 3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{mga.st}  & 120.2 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{mga.stnomu}  & 78.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{mga.t}  & 36.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{mga.0}  & 2 \\\\ [1ex] \n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\label{table1} Effective degree of freedom (df) for the fitted models.}\n\\end{table}\n\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.85\\columnwidth]{P4_InfCrit_m1_f.png}\n\t\t\\includegraphics[width=0.85\\columnwidth]{P4_InfCrit_m3f.png}\n\t\t\\includegraphics[width=0.85\\columnwidth]{P4_InfCrit_m4f.png}\n\t\t\\includegraphics[width=0.85\\columnwidth]{P4_InfCrit_gaf.png}\n\t\t\\caption{Values above the global minimum of each information criterion associated with fitted models: global deviance (GD), AIC, BIC on fitting set, and GD on validation set (GD-v). From top to bottom, we compare the variations of EGPD1, EGPD3, EGPD4 and Gamma-based models. The model variations are indicated in full letters on the right side of the points, in an order following their relative ranking.\n\t\t\t{\\label{figInfCriteria}}%\n\t\t}\n\t\\end{center}\n\\end{figure}\nFrom Figure \\ref{figInfCriteria}, we can see that, as could be expected, for each classes of models (where EGPD1-based models represent one class, EGPD3-based ones another, etc) the GD ranks first models with the full space-time description \\texttt{M.st}. All other variations are rather close in fitting performance, at the exception of the constant-parameters one \\texttt{M.0}, that is largely behind. This shows the clear interest of allowing a smooth variation of distribution parameters as a function of the chosen covariates. However, when it comes to modeling efficiency, where the trade-off between number of parameters and performances is taken into account (through the AIC and BIC), the full space-time model \\texttt{M.st} becomes much less desirable, falling just  above and occasionally behind the constant parameter models \\texttt{M.0}. This is all the more through that we consider the BIC indice, that penalizes more the number of parameters than the AIC, and a model with a large number of parameters (EGPD4 versus GA). The best trade-off, according to the BIC, are the time-only full model \\texttt{M.t} for the GA class and the time-only without the shape parameter $\\mu$ otherwise. This may also be an indication that the seasonal variation of distribution parameters is more important than their spatial variation over the area of interest. \n\nThe ranking of model variations within a same model class allows us to draw conclusions w.r.t. the advantages or challenges of each class. Thus, for EGPD4 the GD ranks model variations following decreasing complexity: \\texttt{megpd4.st} followed by \\texttt{megpd4.st3nomu}, etc. We can deduce that the estimation of $\\mu$ is not that sensitive when it comes to the fitting set (at least with enough data). On the contrary, \\texttt{megpd1.st2nomu} is preferred to \\texttt{megpd1.st} and \\texttt{megpd1.st2mu} which can be interpreted as $\\mu$ being a rather sensitive parameter to estimate for EGPD1. Considering it constant in space consequently improves the global fitting. \n\nPerformances on the validation set are obtained by means of the \\texttt{getTGD()} and \\texttt{TGD()} functions, providing the validation global deviance (GD-v), that is the GD evaluated using predictive values for the parameters at the validation sample.\n\\begin{lstlisting}[language=R]\n\tgg.megpd4.st <- getTGD(megpd4.st,newdata=validation)\n\tgg.megpd4.0 <- getTGD(megpd4.0,newdata=validation)\n\t...\n\tTGD(gg.megpd4.0,gg.megpd4.t,gg.megpd4.tnomu,gg.megpd4.st,gg.megpd4.st2mu,gg.megpd4.st2nomu,gg.megpd4.st3nomu)\n\\end{lstlisting}\nThus, when it comes to extrapolating parameters for new covariate values, the GD-v ranking displayed in Figure \\ref{figInfCriteria} reveals that models considering only the dependence in time (\\texttt{M.t}, \\texttt{M.tnomu}) are systematically preferred, followed by space-time models not considering $\\mu$ (\\texttt{M.st2nomu}), before the full space-time models (\\texttt{M.st}). This supports the fact that the rainfall distributions are relatively close over the studied geographical area, compared to their time evolution that shows stronger variations. Consequently, given the limited size of the dataset, keeping low the number of parameters to regress on covariates (by considering the seasonality only) and in particular avoiding in this category the shape parameter $\\mu$, makes estimations more robust. This agrees with observations from \\citet{evin2018stochastic}, who noted that fitting the shape parameter is the most challenging part of (extended)-GPD modeling. They concluded that mono-site fitting is not robust, leading e.g. to negative shape parameters that do not match observations on rainfall intensities. To circumvent this issue, the authors use constant parameters within regions of influence defined by homogeneity tests.\n\n\\subsubsection{Model verification and comparison}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.48\\columnwidth]{P4_Uplot_w.png}\n\t\t\\includegraphics[width=0.48\\columnwidth]{P4_Uplot_sp.png}\n\t\t\\includegraphics[width=0.48\\columnwidth]{P4_Uplot_su.png}\n\t\t\\includegraphics[width=0.48\\columnwidth]{P4_Uplot_au.png}\n\t\t\\caption{Assessment of the quality of each model (EGPD1, EGPD3, EGPD4 and GA) for all stations from the verification set in (from left to right and top to bottom) winter, spring, summer and autumn. The closer to the diagonal, the better the underlying model distribution. See text for theoretical development. We compare the \\texttt{st2nomu} version of each model class (that is where parameters $\\mu,\\tau$ are constant in space). Theoretical probabilities $\\leq 0.4$ is overall equivalent to $y \\leq 1$ mm where $y$ designates the hourly rainfall amount.\n\t\t\t{\\label{figUplots}}%\n\t\t}\n\t\\end{center}\n\\end{figure}\n\nNow that it is admitted that regressing distributional parameters over time and possibly space is justified for the case study at hand, we want to assess how good is the modeling and compare model classes at hand.\nA way to verify whether the fitted models actually represents well the empirical distribution is to draw so-called P-P plots. \nA simple result in probability states if $Y(\\mathbf{x}) \\sim F(\\cdot;\\theta(\\mathbf{x})$ then $U(\\mathbf{x}):=F(Y(\\mathbf{x});\\theta(\\mathbf{x}))$\nis uniformly distributed on the unit interval. \n\nTherefore we can consider the  residuals $u(\\mathbf{x}) = F(y(\\mathbf{x}),\\widehat{\\theta}(\\mathbf{x}))$ where $\\widehat{\\theta}(\\mathbf{x})$ are the parameters\n of the fitted model and, after having ordered them,  compare them with  $\\Big( \\frac{1}{n+1}, ..., \\frac{n}{n+1} \\Big)$. Here $n$ is the number of observations $y(\\mathbf{x})$ at hand. The closer points are to the diagonal, the better the model fit. Results are broken down at each season for model variations \\texttt{M.st2nomu} on the verification set in Figure \\ref{figUplots}. \nThey show that at all seasons, EGPD4 outperforms the other models. This is all the more true that we consider not too small rainfalls, namely hourly amounts $x \\geq 1$ mm. Below, we keep wiggly artefacts of the discrete sampling scale of observations. Indeed, rainfall amounts are sampled with a precision of $0.2$ mm, which makes the distribution of small values more discrete than continuous. Behind, come models GA, EGPD3 and finally EGPD1. EGPD3 is often better than GA for small hourly rain amounts (below around $2$ mm) however it may be outperformed for larger ones, especially in winter. EGPD1 performs as EGPD3 for intermediate and large amounts of rain, but is significantly behind all other models for small amounts.\n\n\\subsubsection{Visualisation of the fitted functions}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.75\\columnwidth]{P3_parEvol_statNN.png}\n\t\t\\includegraphics[width=0.75\\columnwidth]{P3_ParEvol_stationS.png}\n\t\t\\caption{EGPD4 coefficients fitted for the models \\texttt{megpd4.0} (constant parameters), \\texttt{megpd4.t} (smooth time dependence only), and \\texttt{megpd4.st} (full smooth space time dependence) in two stations: (top two rows) on the north-west coast of Brittany ($x_L=48.7$ $x_l=-4.33$) and (bottom two rows) in the south-east countryside of the studied area ($x_L=46.7$ $x_l=1.25$). Coefficients are fitted against the covariate $cyc$, namely time of the year. The shadowed areas correspond to the associated standard errors.\n\t\t\t{\\label{figParEvol}}%\n\t\t}\n\t\\end{center}\n\\end{figure}\n\nFigure \\ref{figParEvol} represents the fitted functions for the variations \\texttt{megpd4.0} (constant parameters), \\texttt{megpd4.t} (smooth time dependence only), and \\texttt{megpd4.st} (full smooth space time dependence) of the EGPD4 model in two \\textit{a priori} rather different stations when it comes to the rainfall regime: the north-west coast of Brittany ($x_L=48.7$ $x_l=-4.33$) and the south-east countryside of our area of interest ($x_L=46.7$ $x_l=1.25$).\n \nThe functions \\texttt{predict()} or \\texttt{predictAll()} are used to provide fitted values of the functions of the covariates. Standard errors can be extracted for fitted values by means of the argument \\texttt{se.fit=TRUE}. However this option is not supported by \\texttt{gamlss} for parameter estimates on new covariate values at the moment of writing this paper. \n\\begin{lstlisting}[language=R]\n# Fitted parameter values:\nmuFit <- predict(megpd4.st,what=\"mu\",type=\"response\")\n\n# Estimates for new covariate values, gathered in the database \"validation\":\nall <- predictAll(megpd4.st,data=training,newdata=validation)\nmuVal <- all$mu\nsigmaVal <- all$sigma\nnuVal <- all$nu\n\nmuFit <- predict(megpd4.st,what=\"mu\",type=\"response\",data=training,se.fit=TRUE)\nStandardErrorMu <- muFit$se.fit\n\\end{lstlisting}\nWe observe rather similar temporal behaviors of the parameters estimated in both stations, highlighting the relative spatial homogeneity of distributions compared to their seasonal dependence. The standard errors decrease with the simplicity of the model ; they are generally negligible for for parameters $\\sigma$ and $\\tau$ but can become significant for parameters $\\nu$ and $\\mu$ when we are dealing with space-time models. Beyond that, the estimations for the three different versions of the EGPD4-based model are consistent. \nWe note (second station) that a shift between \\texttt{megpd4.t} and \\texttt{megpd4.st}'s estimates of parameter $\\sigma$ is compensated by an opposite shift between the associated estimates of parameter $\\tau$.\nFrom our observations and experience on EGPD fitting, a given EGPD-based distribution is matched by several local optima (i.e. parameter combinations). Along this \"Pareto front\", $\\nu$ is increasing for EGPD1 when $\\sigma$ decreases. Similarly  $\\tau$ is increasing for EGPD4 when $\\sigma$ decreases. \n\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\columnwidth]{P3_egp4st_12.png}\n\t\t\\includegraphics[width=\\columnwidth]{P3_egp4st_4.png}\n\t\t\\includegraphics[width=\\columnwidth]{P3_egp4st_8.png}\n\t\t\\caption{Median parameter maps for the full EGPD4 model \\texttt{megpd4.st} for the months of December (top row), April (middle row) and August (last row).\n\t\t\t{\\label{figParMaps}}%\n\t\t}\n\t\\end{center}\n\\end{figure}\n\nFinally, Figure \\ref{figParMaps} represents the spatial distribution of the median of the EGPD4 parameters estimated from \\texttt{megpd4.st} at three different times of the year: december, april and august. We indeed observe spatial variations of the parameter values, in addition to the monthly changes already discussed. $\\mu$ the most spatially homogeneous parameter, which goes in line with previous observations that favored spatially constant $\\mu$-models over non-constant ones.\nWe also note that the $\\tau$ map follows an opposite trend w.r.t. $\\sigma$'s map: the larger $\\tau$, the smaller $\\sigma$, which follows the general idea of parameter Pareto front developed earlier.\n\n\n\\subsubsection{Goodness-of-fit w.r.t. the upper tail}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext099_w.png}\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext0995_w.png} \\\\\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext099_sp.png}\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext0995_sp.png} \\\\\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext099_su.png}\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext0995_su.png} \\\\\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext099_au.png}\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext0995_au.png}\n\t\t\\caption{Assessment of the quality of model EGPD4's variations and GA for all stations from the validation set in (from top to bottom) winter, spring, summer and autumn. The closer to the diagonal, the better the underlying model distribution, see Figure \\ref{figUplots}. We focus on extremes, with theoretical probabilities $\\geq 0.99$ (left column) equivalent to hourly rainfall amounts above $11$ mm for EGPD4 models and above $7$ mm for GA. Theoretical probabilities $\\geq 0.995$ (right column) are equivalent to rain amounts above $15$ mm for EGPD4 models and above $7.6$ mm for GA.\n\t\t{\\label{figUplotext}}%\n\t\t}\n\t\\end{center}\n\\end{figure}\nWe finish the presentation of the results by comparing in Figure \\ref{figUplotext} the EGPD4 model to a GA fit, for all stations of the validation set and for each season. We  want to check the added value of the EGPD4 model over the GA when it comes to extremes, as well as to assess the best modeling option for the EGPD4 shape parameter in order to ensure a correct representation of extremes.\nThe series of P-P plots presented on Figure \\ref{figUplotext} focus on the upper tail of the distribution (above the quantiles 0.99 and 0.995 respectively) and allow us to do that. We first note that whatever the season, EGPD4 performs significantly better than GA when it comes to extremes. Most often, the constant parameter model \\texttt{megpd4.0} is enough to outperform the GA, and sometimes is clearly the best option as in spring and autumn. In winter and summer, it is not enough to capture correctly the upper tail of the empirical distribution. Making EGPD4 parameters dependent on time only is sufficient to obtain correct upper tail representations, whether $\\mu$ is considered constant (\\texttt{megpd4.tnomu}) or not (\\texttt{megpd4.t}). The difference between the last two models is very small however using the constant shape parameter option can slightly improve results, especially for larger quantiles (0.995 and above versus 0.99). Adding spatial dependence to the temporal one improves even more the accuracy of the EGPD4 upper tail modeling. Making spatially dependent only two parameters usually performs better. However $\\mu$ needs to be included in these two parameters (\\texttt{megpd4.st2} versus \\texttt{megpd4.st2nomu}). \nThis highlights the fact that a smooth space-time estimation of the shape parameter $\\mu$ is important to capture correctly the non-stationary behavior of the upper tail  of the rainfall distributions. However, due to limited data and numerical sensitivity of that estimation, it may be preferable to only limit the smooth spatial dependence to two parameters instead of four, including $\\mu$, in order to make it more robust for extrapolation.\n\n\\section{Conclusions}\n\\label{secConclu}\n\nThis article introduced an add-on to the R-package \\texttt{gamlss}, to model non-stationary fields whose margins are EGPD distributed. A number of R packages target the modeling of heavy-tailed distributions. Yet, although some allow distributional regression models, for which distribution parameters can be modeled as additive functions of covariates, none of these packages address the extended-GPD distribution nor another form of unique and closed-form formulation for heavy-tailed distributions that models small, intermediate and large values in a synthetic manner. This add-on implements the EGPD family in a generic way in the \\texttt{gamlss} package, allowing to test any new parametric form, up to 4 parameters so far, satisfying the requirement of the EGPD family. \nApplications include space-time modeling of rainfall marginal distributions. The example developed in this paper shows 1) that modeling hourly rainfall amounts is significantly improved when distribution parameters are non-stationary in space and time; and that 2) the EGPD class clearly improves the modeling of intermediate values and upper tail compared to a classical Gamma option. This illustrates the benefits of our add-on with respect to existing GAM forms for modeling non-stationary margins (typically the Gamma model for our application). We compare three parametric EGPD models and discuss the modeling strategies to best capture non stationary upper tails with limited datasets. This case study also reveals the added-value of EGPD4 (and to a less extent EGPD3) over EGPD1 to correctly model hourly rainfall distributions in a space-time context, while previous works \\cite{naveau2016modeling,evin2018stochastic,bertolacci2018comparison} restricted themselves to EGPD1 as the best option.\n\n\n\\section*{Software and data availability}\n\nThe add-on to the R-package \\texttt{gamlss} \\citep{gamlssP} presented here for the generic implementation of EGPD consists in the script \\texttt{GenericEGPDg.R}, available at \\url{https:\/\/github.com\/noemielc\/egpd4gamlss}. On this same webpage, supplementary material consisting in a reproducible tutorial with code is presented.\n\nThe data used in this paper is freely accessible on the MeteoNet French database \\citep{MeteoNet}: \\url{https:\/\/meteofrance.github.io\/meteonet\/english\/data\/summary\/}\n\n\\section*{Acknowledgement}\n\nScientific activity was performed as part of the research program Venezia2021, coordinated by CORILA, with the contribution of the Provveditorato for the Public Works of Veneto, Trentino Alto Adige and Friuli Venezia Giulia (Italy).\nThe authors thank Nicolas Berthier for helpful R suggestions.\n\n\n\\bibliographystyle{elsarticle-harv.bst}\n\n\t\\setlength\\baselineskip{.96\\baselineskip}%\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Acknowledgments}\n\nThis work received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7\/20072013\/ERC grant agreement no. [319456] dHCP project). The research was supported by the Wellcome\/EPSRC Centre for Medical Engineering at King's College London [WT 203148\/Z\/16\/Z]; the Medical Research Council [MR\/K006355\/1]; and the National Institute for Health Research (NIHR) Biomedical Research Centre based at Guy's and St Thomas' NHS Foundation Trust and King's College London. The views expressed are those of the author and not necessarily those of the NHS, the NIHR or the Department of Health. The author also acknowledges the Department of Perinatal Imaging \\& Health at King's College London and advice and support from Edgar Dobriban, Jo V. Hajnal and Daan Christiaens.\n\n\\clearpage\n\n\\bibliographystyle{apalike}\n\n\\section{Introduction}\n\n\\label{sec:INTR}\n\nRandom matrix theory is at the core of modern high dimensional statistical inference~\\citep{Yao15} with applications in physics, biology, economics, communications, computer science or imaging~\\citep{Couillet13,Paul14,Bun17}. In the large dimensional scenario, classical asymptotics where the available number of samples of a given population $N$ is much larger than the dimension $M$ are no longer valid. A key contribution in this setting is the Mar\\v{c}enko-Pastur theorem~\\citep{Marcenko67,Silverstein95a}, which characterizes the limiting behaviour of the empirical spectral distribution (ESD) for matrices with random entries when $N\\to\\infty$. Namely, there exists a fixed point equation relating the eigenvalues of the empirical and population distributions, which can be used for inference under the mediation of appropriate numerical techniques. Using this characterization, subsequent asymptotics-based inference can be performed, for instance, for dimensionality reduction, hypothesis testing, signal retrieval, classification or covariance estimation~\\citep{Yao15}. In addition,~\\cite{Silverstein95b} develops on ideas outlined in~\\cite{Marcenko67} to analyze the support of the ESD based on the monotonicity of the inverse of the function involved in the fixed point equation. Emerging from statistical models in array processing, the extension in~\\cite{Wagner12} focuses on generalizing previous results to the case where the matrix rows are independent but drawn from a collection of population distributions. Here, the relation between the population covariances and the limiting behaviour of the sample eigenvalues is governed by a system of non-linear equations and, although asymptotic sample eigenvalue confinement has also been proved~\\citep{Kammoun16}, a simple description of the support is no longer at hand.\n\nDespite the variety of potential applications, not many works have practically confronted the numerical issues involved in computing the ESD from a given population distribution. The most flexible package that we have identified has been described in~\\cite{Dobriban15} (SPECTRODE), where the fixed point equation in~\\cite{Silverstein95a} is transformed into an ordinary differential equation (ODE) with starting point obtained by the solution of the fixed point equation on a single point within the support. In this work the author shows that the proposed method compares favourably with straightforward fixed point solvers, both in terms of accuracy and computational efficiency. In addition, accuracy limitations of Monte Carlo simulations~\\citep{Jing10} are showcased and, while recognizing their independent interest, arguments are provided about the practical limitations~\\citep{Rao08} or limited applicability~\\citep{Olver13} of other approaches. In our experiments, this package has revealed an exquisite accuracy and efficiency in computing the ESD and determining its support. However, the applicability or efficiency of an ODE approach may be compromised for more general models such as~\\cite{Wagner12}. This is mainly due to the increased computational complexity of evaluating the Jacobian of the system of fixed point equations, which no longer depends on the eigenvalues of the populations but on a combination of their covariances. Another interesting tool, conceived to solve the more general problem of recovering the population distribution from the observed eigenvalues by means of the so called QuEST function, has been described in~\\cite{Ledoit17}. Solving this problem is required, for instance, for covariance estimation. The literature review in this work points to a systematic limitation of most precedent methods, as they are only capable to obtain estimates for very particular forms of the population distribution. In addition, it introduces an interesting feature not present in~\\cite{Dobriban15}, the use of a non-uniform grid with increased resolution near the support edges, which allows more efficient approximations. However, once again, this technique is designed for cases where a single nonlinear equation is to be solved while generalizations to systems of equations do not seem straightforward.\n\nThis technical note describes a series of tools that have been developed to compute the ESD for the mixture of populations case in~\\citep{Wagner12}. This model or certain analogous forms, has attracted interest in areas such as telecommunications~\\citep{Moustakas07,Couillet11,Dupuy11}, machine learning~\\citep{Benaych-Georges16,Couillet16}, medical imaging~\\citep{Cordero-Grande19} or genetics~\\citep{Fan19}. Our method is based on directly solving the system of nonlinear equations. However, the results in~\\cite{Dobriban15} and our own analysis, have identified certain limitations in commonly reported algorithms based on fixed point iteration solvers, so a set of technical refinements are proposed in this note. These include the use of Anderson mixing to accelerate the fixed point iterations, an homotopy continuation method to prevent non-admissible solutions, a set of heuristics to detect the support of the distribution and to adapt the approximation grid to the ESD shape, and a formulation that allows for efficient computations in graphical processing units (GPU). We validate our method by comparison with~\\cite{Dobriban15} and~\\cite{Ledoit17}, both in terms of efficiency and accuracy. As our methods are envisaged to operate on more general models than those contemplated in~\\cite{Dobriban15,Ledoit17}, they do not make use of any precomputed information about the distribution support. Nevertheless, we show that they are reliable enough and their efficiency and accuracy is comparable or superior to that in~\\cite{Dobriban15,Ledoit17}. In addition, comparisons with Monte Carlo simulations show that they are also capable of providing accurate results for more general models. A \\textsmaller{\\textsc{MATLAB}} implementation of our approach, that we will refer to as the MIXANDMIX (Mixtures by Anderson Mixing) method, including the scripts required to replicate the experiments in this note, has been made publicly available at \\url{https:\/\/github.com\/mriphysics\/MixAndMix\/releases\/tag\/1.1.0}. This note is organized as follows: in~\\S~\\ref{sec:THEO} we review different random matrix models, in~\\S~\\ref{sec:METH} we describe the main features of our method, in~\\S~\\ref{sec:RESU} we validate the proposed technique, in~\\S~\\ref{sec:DISC} we discuss the implications of the obtained results and in~\\S~\\ref{sec:CONC} we end up with some conclusions.\n\\section{Theory}\n\n\\label{sec:THEO}\n\nConsider a complex random matrix $\\mathbf{X}$ of size $N\\times M$. We are interested in the eigenvalue distribution of the sample covariance $\\mathbf{Y}=\\displaystyle N^{-1}\\mathbf{X}^H\\mathbf{X}$ in the asymptotic regime where both $M\\to\\infty$ and $N\\to\\infty$ but they maintain a fixed ratio $\\gamma=M\/N$ with $\\gamma>0$. For simplicity we assume the entries of the matrix are zero mean Gaussian distributed, but keeping in mind that the literature contemplates different generalizations. Three main scenarios are contemplated:\n\n\\subsection{IID standard entries}\n\n\\label{sec:IIDS}\n\nWithin this model, that we call \\emph{standard} model, we can write $\\mathbf{X}^H\\sim\\boldsymbol{\\mathcal{CN}}\\displaystyle\\left(\\mathbf{0}_{MN},\\mathbf{I}_{MN}\\right)$, with $\\boldsymbol{\\mathcal{CN}}$ denoting a circularly symmetric complex Gaussian distribution, $\\mathbf{0}_{MN}$ the $M\\times N$ matrix with zero entries, and $\\mathbf{I}_{MN}$ the $MN\\times MN$ identity matrix.~\\cite{Marcenko67} showed that in this case the distribution of eigenvalues of the sample covariance asymptotically converges to\n\\begin{equation}\n\\label{ec:MPLA}\nf_{\\gamma}^{\\mathbf{I}}(x)=\\begin{cases}\\displaystyle\\frac{\\sqrt{(\\gamma_{+}-x)(x-\\gamma_{-})}}{2\\pi\\gamma x} & \\mbox{if }\\gamma_{-}\\leq x\\leq \\gamma_{+}\\\\ 0 & \\mbox{otherwise,}\\end{cases}\n\\end{equation}\nwith $\\gamma_{-}=(1-\\sqrt\\gamma)^2$ and $\\gamma_{+}=(1+\\sqrt\\gamma)^2$ defining the support of the distribution. Note that if $\\gamma\\geq 1$, the distribution has a $1-\\gamma^{-1}$ point mass ($\\gamma>1$) or is locally unbounded ($\\gamma=1$) at $x=0$. Thus, in this case,~\\eqref{ec:MPLA} gives an explicit characterization of the ESD.\n\n\\subsection{IID rows}\n\n\\label{sec:IIDC}\n\nIn this scenario, we can write $\\mathbf{X}^H\\sim\\boldsymbol{\\mathcal{CN}}\\displaystyle\\left(\\mathbf{0}_{MN},\\boldsymbol{\\Lambda}_M\\otimes\\mathbf{I}_N\\right)$, with $\\boldsymbol{\\Lambda}_M$ a given population covariance matrix.~\\cite{Marcenko67} and~\\cite{Silverstein95a} respectively derived expressions for probabilistic and almost sure limits of the sample covariance spectrum using its Stieltjes transform\n\\begin{equation}\nm(z)=\\int_{\\mathbb{R}}\\frac{f_{\\gamma}^{\\boldsymbol{\\Lambda}_M}(x)}{x-z}dx,\\quad z\\in\\mathbb{C}\\setminus\\mathbb{R}\n\\end{equation}\nfor which the inversion formula would give back the ESD by\n\\begin{equation}\n\\label{ec:INST}\nf_{\\gamma}^{\\boldsymbol{\\Lambda}_M}(x)=\\frac{1}{\\pi}\\lim_{\\epsilon\\to 0^{+}}\\Im\\left\\{m(x+i\\epsilon)\\right\\}.\n\\end{equation}\nIf, in a discrete setting, we denote the increasingly sorted eigenvalues of $\\boldsymbol{\\Lambda}_M$ by $\\{\\lambda_1,\\ldots,\\lambda_M\\}$, when $N\\to\\infty$ the eigenvalue distribution of the matrix $\\tilde{\\mathbf{Y}}=\\displaystyle N^{-1}\\mathbf{X}\\mathbf{X}^H$ converges to a function whose Stieltjes transform $\\tilde{m}(z)$ is related to the population distribution by the fixed point equation\\newline $\\tilde{m}(z)=\\displaystyle\\left(-z+\\frac{\\gamma}{M}\\sum_{m=1}^M\\frac{\\lambda_m}{1+\\lambda_m \\tilde{m}(z)}\\right)^{-1}$ and to the Stieltjes transform of the corresponding limiting function for the spectral distribution of the sample covariance matrix $\\mathbf{Y}$, $m(z)$, by $\\tilde{m}(z)=\\gamma m(z)+(\\gamma-1)\/z$. In addition,~\\cite{Silverstein95b}, following the guidelines in~\\cite{Marcenko67}, showed that the support of the ESD can be characterized by studying the zeros of the derivative of the inverse map, $z'(\\tilde{m})$, in appropriate domains. This property, together with the analyticity of the empirical density within its support are the keys to the SPECTRODE approach in~\\cite{Dobriban15}.\n\nTo clarify the connections between the different models, it is more convenient to express the previous relations in terms of auxiliary functions $e(z)=-\\displaystyle\\frac{1}{\\gamma z\\tilde{m}(z)}-\\frac{1}{\\gamma}$ for an equivalent fixed point equation,\n\\begin{equation}\n\\label{ec:CFST}\ne(z)=\\frac{1}{M}\\sum_{m=1}^M\\lambda_m\\left(\\frac{\\lambda_m}{1+\\gamma e(z)}-z\\right)^{-1},\n\\end{equation}\nand an expression for the Stieltjes transform of the ESD,\n\\begin{equation}\n\\label{ec:STSD}\nm(z)=\\frac{1}{M}\\sum_{m=1}^M\\left(\\frac{\\lambda_m}{1+\\gamma e(z)}-z\\right)^{-1}.\n\\end{equation}\nWe refer to this model as the \\emph{single population} model.\n\n\\subsection{Independent rows}\n\n\\label{sec:INCO}\n\nIn this \\emph{mixture of populations} model, the matrix is drawn from $\\mathbf{X}^H\\mathbin{\\sim}\\boldsymbol{\\mathcal{CN}}\\displaystyle\\left(\\mathbf{0}_{MN},\\sum_{k=1}^{K}\\boldsymbol{\\Lambda}^k_M\\otimes\\mathbf{D}^k_N\\right)$, where $\\mathbf{D}^k_N$ is a diagonal indicator matrix with ones in the diagonal elements corresponding to those rows sampled according to the population covariance $\\boldsymbol{\\Lambda}^k_M$ and zero otherwise. The equations relating the empirical and population distributions for $N\\to\\infty$ have been derived in~\\cite{Wagner12} also making use of the Stieltjes transform of the limiting spectral distribution $f_\\gamma^{\\prescript{}{K}{\\boldsymbol{\\Lambda}}^{}_{M}}$, $m(z)$, and the auxiliary functions $e_j(z)$, $1\\leq j\\leq K$. These functions are related to the population covariances by a system of nonlinear equations\n\\begin{equation}\n\\label{ec:CSST}\ne_j(z)=\\frac{1}{M}\\tr\\left(\\boldsymbol{\\Lambda}_j\\left(\\sum_{k=1}^{K}\\frac{\\alpha_k\\boldsymbol{\\Lambda}_k}{1+\\gamma e_k(z)}-z\\mathbf{I}_M\\right)^{-1}\\right),\n\\end{equation}\nfor which there is a unique solution in $\\mathbb{C}\\setminus\\mathbb{R}^{+}$. $m(z)$ is expressed in terms of these functions as\n\\begin{equation}\n\\label{ec:SSSD}\nm(z)=\\frac{1}{M}\\tr\\left(\\left(\\sum_{k=1}^{K}\\frac{\\alpha_k\\boldsymbol{\\Lambda}_k}{1+\\gamma e_k(z)}-z\\mathbf{I}_M\\right)^{-1}\\right),\n\\end{equation}\nwith $\\alpha_k=\\tr(\\mathbf{D}^k)\/N$. Note that~\\eqref{ec:CSST} and~\\eqref{ec:SSSD} reduce to~\\eqref{ec:CFST} and~\\eqref{ec:STSD} when $K=1$. There are two main limitations to extend the SPECTRODE method to this setting. First, we are unaware of studies characterizing the support of the measures inducing $e_j(z)$ by means of some analogue to~\\cite{Silverstein95b}. Second, both potential extensions of support characterizations or usage of ODE solvers would require the Jacobian of~\\eqref{ec:CSST}, which involves additional matrix multiplications, with a penalty in computational efficiency. Thus, we have focused on developing a reliable method to solve the system~\\eqref{ec:CSST} not requiring the Jacobian.\n\\section{Methods}\n\n\\label{sec:METH}\n\nIn this Section we describe our numerical solver for the system of equations in~\\eqref{ec:CSST}.\n\n\\subsection{Support detection}\n\n\\label{sec:SUDE}\n\nWhen $\\gamma\\to 0$ the ESD tends to the population distribution for the standard and single population models, while for the mixture of populations it is governed by an effective single population distribution $f_\\gamma^{\\prescript{}{K}{\\boldsymbol{\\Lambda}}^{}_{M}}=f_0^{\\overline{\\boldsymbol{\\Lambda}}_M}$ with $\\overline{\\boldsymbol{\\Lambda}}_M=\\displaystyle\\sum_{k=1}^K\\alpha_k\\boldsymbol{\\Lambda}_M^k$, so that we have $f_0^{\\overline{\\boldsymbol{\\Lambda}}_M}(x)=\\displaystyle\\frac{1}{M}\\sum_{m=1}^M\\delta(x-\\overline{\\lambda}_m)$, with $\\overline{\\lambda}_m$ the $m$-th eigenvalue of $\\overline{\\boldsymbol{\\Lambda}}_M$. Thus, in this limiting case the ESD support only includes the eigenvalues of an equivalent single population distribution. On the other side, for $\\gamma>0$, $[a,b]=\\left [t^{-1}(1-\\sqrt{\\gamma})^2\\displaystyle\\min_k\\lambda_1^k,t(1+\\sqrt{\\gamma})^2\\max_k\\lambda_M^k\\right ]$, with $\\lambda_m^k$ denoting the $m$-th eigenvalue of $\\boldsymbol{\\Lambda}_M^k$ and $t>1$, provide lower and upper bounds on the limiting support.\n\nTo consider these two extreme cases, we group and sort the set of $P=M(K+1)$ eigenvalues $\\lambda_p=\\{\\lambda_m^k,\\overline{\\lambda_m}\\}$, $1<k<K$, $1<m<M$. Our method attends to the overlap of the set of intervals\n\\begin{equation}\n\\label{ec:OVER}\n[a_p,b_p]=[t^{-1}(1-\\sqrt{\\gamma})^2\\lambda_p,t(1+\\sqrt{\\gamma})^2\\lambda_p],\n\\end{equation}\ni.e., the intervals generated by the maximum possible spectral dispersion of each grouped eigenvalue, including a numerical safety margin $t$, which we have set to $t=1.001$ (and constraining $|\\gamma-1|\\geq 10^{-9}$ to avoid the potential singularity at $x=0$ when $\\gamma=1$).\n\nThe support of the limiting distribution will be contained in $[a,b]$, as this interval considers the maximum possible eigenvalue dispersion as produced by minorizing ($a$) and majorizing ($b$) point mass population distributions governed by~\\eqref{ec:MPLA}. On the contrary, there is no guarantee for $\\displaystyle\\bigcup_p [a_p,b_p]$ to contain the whole support of the limiting distribution. However, the intervals in~\\eqref{ec:OVER} serve to cover the $\\gamma\\to 0$ regime, which originates challenging highly localized support intervals, as in this case our construction guarantees that the support will be contained in the defined intervals. Although we have empirically observed cases where part of the support lies on $[a,b]\\setminus\\displaystyle\\bigcup_p [a_p,b_p]$, we have never observed that these intervals contain any of the support edges. In this situation, the adaptive regridding procedure in~\\S~\\ref{sec:ADRE} should generally be enough to interpolate the distribution estimates on $[a,b]\\setminus\\displaystyle\\bigcup_p [a_p,b_p]$.\n\nLack of overlap among adjacent intervals is detected by checking the condition $b_p<a_{p+1}$. Assuming this is observed $I\\leq P-1$ times, we can get a partition into $I+1$ segments, each one induced by $P_i$ eigenvalues. Each of these support segments is gridded using a total of $\\max(M^{\\text{(o)}}P_i,M^{\\text{(i)}})$ points, with $M^{\\text{(i)}}$ the minimum number of points per segment and $M^{\\text{(o)}}\\leq M^{\\text{(i)}}$ the minimum ratio of points per number of grouped eigenvalues. Importantly, to improve detectability, by noting the multiplicative dependence of the spectral dispersion width with the spectral location in~\\eqref{ec:OVER}, gridding is performed uniformly in logarithmic units. Then, we can call the solver of~\\eqref{ec:CSST} (to be described in~\\S~\\ref{sec:HOMO}), and compute~\\eqref{ec:SSSD} and~\\eqref{ec:INST} over the defined grid locations. \n\nThe output of this first step is a set of estimates for the distribution in a non-uniform grid $\\mathbf{x}=x_p$ with $1\\leq p\\leq P_0$ and $M^{\\text{(o)}}P\\leq P_0\\leq M^{\\text{(i)}}P$. The lower and upper bounds on the number of grid points, $M^{\\text{(o)}}P$ and $M^{\\text{(i)}}P$, are respectively attained for large and small enough $\\gamma$. Fixing $M^{\\text{(o)}}=3$ and $M^{\\text{(i)}}=15$ has detected at least a single point within all the segments conforming the support of the distribution for the range of problems studied in~\\S~\\ref{sec:RESU}. For problems where the total geometric multiplicity of the discretized population covariances is given by $M^{\\text{(u)}}$, equivalent expressions of the problem can be obtained for any $M=SM^{\\text{(u)}}$ with $S\\in\\mathbb{N}_{>0}$. In our experiments $S$ has been selected by defining a minimum number of grid points to approximate the densities by $M^{\\text{(m)}}$, and making $M=\\max(M^{\\text{(u)}},M^{\\text{(m)}})$ with $M^{\\text{(m)}}=100$.\n\n\\subsection{Adaptive regridding}\n\n\\label{sec:ADRE}\n\nAdditional points are added to the grid by pursuing $g'(x)\\sqrt{xf''(x)}=c$, with $g(x)$ the grid mapping function and $c$ a constant. This non-uniform grid construction criterion is based on both the second order derivative of the density $f''(x)$ and the grid value $x$. The first feature, $f''(x)$, favours the allocation of grid points near the support edges, in accordance to the $\\sqrt{x-x_0}$ behaviour of the distribution at the boundaries~\\citep{Silverstein95b}, as well as in those areas where linear interpolation results in larger approximation errors. This is similar in spirit to the arcsine criterion in~\\cite{Ledoit17} but does not use any prior information about the support edges as it is not available in the mixture of populations model. The second feature, $x$, favours the allocation of grid points near the upper edge of the support, which could be important for applications related to signal detection~\\citep{Nadakuditi14,Dobriban19}. After $P_l=R_lP_0$ points are added to the grid, the solver for $f(\\mathbf{x})$ is called on the new set of points to allow for an update of the $f''(x)$ values to be used at the next iterative regridding step. This whole process is repeated $L$ times, so $R_l$, $1\\leq l\\leq L$ control the final resolution of the distribution computations. The parameters by default in our implementation are $R_l=1$ $\\forall l$ and $L=1$.\n\n\\subsection{Homotopy continuation}\n\n\\label{sec:HOMO}\n\nThe calculation of the ESD involves a pass to the limit in~\\eqref{ec:INST} as the Stieltjes transform does not converge in the real line. In addition, the solution of~\\eqref{ec:CSST} is not unique on the real line. Numerically, this may provoke spurious fixed point convergence when the current solution is far away from the optimum and the computations are being performed in locations that are close to the real line. To prevent these situations, we have emulated~\\eqref{ec:INST} by homotopy continuation. We start by obtaining an approximate solution for~\\eqref{ec:CSST} in a grid given by $\\mathbf{z}=\\mathbf{x}+\\boldsymbol{\\xi}^2i$, with $\\boldsymbol{\\xi}=\\xi^0\\mathbf{1}_{P_l}$ for big enough $\\xi^0$ common for all $1\\leq p\\leq P_l$. At each iteration $i$ we compute $\\varepsilon^i_p=\\displaystyle\\max_k|e^i_k(x_p)-e^{i-1}_k(x_p)|$, where the updates on $e$ are to be described in~\\S~\\ref{sec:ANDE}. Considering that, as discussed in~\\cite{Dobriban15}, to obtain an accuracy of at least $\\epsilon$ for the distribution $f(x)$, we need to solve the system of equations in a complex grid given by $x+i\\epsilon^2$, we can perform the update $\\xi^{j+1}_p=\\max(\\xi^j_p\/\\beta,\\epsilon)$ whenever $\\varepsilon^i_p\\leq\\varepsilon^{i-1}_p$. The iterations at the grid location indexed by $p$ are terminated when the prescribed accuracy is reached, namely when $\\xi^j_p=\\epsilon$ and $\\varepsilon^i_p<\\epsilon$. In our experiments we have used $\\xi^0=1$ and $\\beta=10$, for which we have observed robust and efficient performance. \n\n\\subsection{Anderson acceleration}\n\n\\label{sec:ANDE}\n\nThe experiments in~\\cite{Dobriban15} showed that when solving the IID rows problem in~\\S~\\ref{sec:IIDC} by a straightforward fixed point algorithm, in our context when performing the updates on $e(z)$ directly using~\\eqref{ec:CFST}, convergence is often very slow, so they reported situations where their SPECTRODE code could be $1000\\times$ quicker while simultaneously obtaining $1000\\times$ higher accuracy. In this note we show that this apparent limitation of the fixed point iterates can be overcome by using techniques to accelerate their convergence. As discussed in~\\S~\\ref{sec:INCO}, the accelerated convergence achievable by methods requiring the Jacobian of the fixed point identities may not compensate for the increased cost per iteration involved in computing the Jacobian. Thus, we have resorted to Anderson mixing~\\citep{Anderson65}, a technique not requiring explicit Jacobian calculations that has demonstrated good practical performance, in occasions providing competitive results when compared to gradient-based approaches~\\citep{Ramiere15}.\n\nConsidering a given multidimensional fixed point mapping $\\mathbf{g}(\\mathbf{e})$ such as~\\eqref{ec:CSST}, Anderson iterations are computed as\n\\begin{equation}\n\\mathbf{e}^{i+1}=\\mathbf{g}(\\mathbf{e}^{i})-\\sum_{q=1}^{Q_i}(\\mathbf{g}(\\mathbf{e}^{i-Q_i+q})-\\mathbf{g}(\\mathbf{e}^{i-Q_i+q-1}))\\nu^i_q,\n\\end{equation}\nwith $Q_i$ denoting the number of iterations whose history is used for the update at iteration $i$ and $\\boldsymbol{\\nu}^i=(\\nu^i_1,\\ldots,\\nu^i_{Q_i})^T$ obtained by solving a linear least squares problem involving the fixed point updates $\\mathbf{h}^i(\\mathbf{e}^i)=\\mathbf{g}(\\mathbf{e}^i)-\\mathbf{e}^i$ and their differences $\\Delta\\boldsymbol{h}^i=\\boldsymbol{h}^{i}-\\boldsymbol{h}^{i-1}$ arranged in a $K\\times Q_i$ matrix $\\Delta\\boldsymbol{H}^i=[\\Delta\\boldsymbol{h}^{i-Q_i+1},\\ldots,\\Delta\\boldsymbol{h}^i]$. Due to the potential ill-posedness of this system, we have actually solved a damped version~\\citep{Scieur19,Henderson19}\n\\begin{equation}\n\\boldsymbol{\\nu}^i=\\argmin_{\\boldsymbol{\\nu}}\\|\\boldsymbol{h}^i-\\Delta\\boldsymbol{H}^i\\boldsymbol{\\nu}\\|_2^2+\\lambda^i\\|\\boldsymbol{\\nu}\\|_2^2,\n\\end{equation}\nwith damping parameter given by $\\lambda^i=\\displaystyle 0.1\\max_{k,q}|\\Delta H^i_{k,q}|$. We have set $Q_i=\\min(2,i-1)$ on the basis of our empirical testing and in agreement with the experimental results in~\\cite{Ramiere15}.\n\n\\subsection{GPU acceleration}\n\n\\label{sec:GPUA}\n\nGPU-based implementations of the SPECTRODE method~\\citep{Dobriban15} appear involved due to the sequential nature of ODE solvers. In contrast, acceleration of the ESD computation in the QuEST method~\\citep{Ledoit17} seems more plausible as it solves for a zero of a function independently for the different grid locations, but the authors have not discussed this aspect. Our code has been architectured so that most demanding routines support both CPU and GPU based parallel computations. This includes the parallel computation of the solutions of the system of equations in~\\eqref{ec:CSST} for the different grid locations but also the parallel computation of different ESDs, required, for instance, in patch-based image denoising applications~\\citep{Cordero-Grande19}. \n\n\\section{Results}\n\n\\label{sec:RESU}\n\nIn this Section we first validate the beneficial effects of Anderson acceleration and homotopy continuation in ESD computations (\\S~\\ref{sec:BEFU}), then compare our method to the SPECTRODE and QuEST proposals in those regimes in which they can operate (\\S~\\ref{sec:COLI}) and finally provide some results on the application of our technique to the mixture of populations model (\\S~\\ref{sec:MIPO}). Unless otherwise stated experiments are performed using CPU computations.\n\n\\subsection{Validation of introduced refinements}\n\n\\label{sec:BEFU}\n\nIn~\\cite{Dobriban15} an experiment was performed illustrating the limitations of fixed point iterations to obtain accurate results in ESD calculations. Their method is compared with a fixed point iteration solver for the standard model in~\\S~\\ref{sec:IIDS}, using the closed form density in~\\eqref{ec:MPLA} with $\\gamma=0.5$ to assess the accuracy. Here we repeat that experiment adding our Anderson acceleration technique to the fixed point solver. The results are presented in Fig.~\\ref{fig:FIG1}. First, we have been able to replicate the results in~\\cite{Dobriban15}; the fixed point algorithm, grossly equivalent to the MIXANDMIX implementation with $Q=0$, i.e., without Anderson mixing, is only able to provide very moderate accuracies, despite being run for $1\/\\epsilon$ iterations. However, when introducing the Anderson acceleration scheme, the results are dramatically better, with MIXANDMIX and SPECTRODE demonstrating comparable performance. In this experiment the curves show a slightly better accuracy (Fig.~\\ref{fig:FIG1}a) and worse computational efficiency (Fig.~\\ref{fig:FIG1}b) for MIXANDMIX, but this should be taken with caution as these tests have been conducted without considering the influence of grid sizes on the approximation, which will be taken into account in the experiments in~\\S~\\ref{sec:COLI}. In addition, we show (Fig.~\\ref{fig:FIG1}c) that despite the accuracy obtained by a straightforward fixed point algorithm is poor everywhere within the support, the accuracy curve after Anderson mixing remains below the SPECTRODE curve almost everywhere.\n\\begin{figure}[!htb]\n\\begin{minipage}{.32\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig01a}\n\\end{minipage}\n\\begin{minipage}{.32\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig01b}\n\\end{minipage}\n\\begin{minipage}{.32\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig01c}\n\\end{minipage}\n\\caption{\\textbf{a)} Averaged accuracy of different methods $\\overline{\\Delta f}=\\displaystyle\\sum_{p=1}^{P}|\\hat{f}_{0.5}^{\\mathbf{I}}(x_p)-f_{0.5}^{\\mathbf{I}}(x_p)|\/P$ with $\\hat{f}$ denoting the computed density. \\textbf{b)} Computation times $t$. \\textbf{c)} Accuracy $\\Delta f(x)=|\\hat{f}_{0.5}^{\\mathbf{I}}(x)-f_{0.5}^{\\mathbf{I}}(x)|$ throughout the support (case $\\epsilon=10^{-5}$).}\n\\label{fig:FIG1}\n\\end{figure}\n\nIn Fig~\\ref{fig:FIG2}a we compare the results of our method without and with homotopy continuation to those of SPECTRODE. The SPECTRODE method and ours with homotopy continuation are observed to overlap at the scale of the plot. However, when running MIXANDMIX without homotopy continuation, we can see that there exist some grid points for which the computations spuriously converge to the zero solution. We know this solution is infeasible because it provokes discontinuities in the distribution, which contradicts its expected analyticity properties~\\citep{Silverstein95b}. To illustrate the reasons for these numerical issues, we have taken a grid point corresponding to one of these infeasible results, $x=2.2$. For this point, Figs.~\\ref{fig:FIG2}b-e show the squared magnitude of the residuals of the fixed point maps, $|h(e)|^2$, at different complex plane values of the auxiliary function $e(z)=e(x+\\delta i)$ as we approach the real line with $\\delta=\\{1,0.1,0.01,0.001\\}$. First, for $z=x+1i$ there is a unique minimum in $\\mathbb{C}^{+}$ whose basin of attraction covers the whole of $\\mathbb{C}^{+}$. As we decrease $\\delta$ (see for instance $z=x+0.1i$) we can track this minimum in a neighborhood of its previous location and check that it is still the only one in the upper half of the complex plane. However, a new local minimum has emerged in the lower half of the plane but so close to the real line that its basin of attraction extends to the upper half. As we keep decreasing $\\delta$, the basin of attraction of this minimum in $\\mathbb{C}^{+}$ gets bigger; however, by analyticity there has to be an area around the global optimum for which the method should still converge to the global optimum, which can be ensured by homotopy continuation. This explains the problem we are observing in the left hand side: the fixed point algorithm has entered the basin of attraction of the minimum in the lower half and it has not been able to escape from this area. In addition, the location of the attractor explains why the spurious distribution value obtained by the fixed point algorithm is generally pushed to $0$ when failing to converge to the right optimum.\n\\begin{figure}[!htb]\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig02a}\n\\end{minipage}\n\\begin{minipage}{.66\\textwidth}\n\\includegraphics[width=0.49\\textwidth]{figs\/Fig02b.png}\n\\includegraphics[width=0.49\\textwidth]{figs\/Fig02c.png}\\\\\n\\includegraphics[width=0.49\\textwidth]{figs\/Fig02d.png}\n\\includegraphics[width=0.49\\textwidth]{figs\/Fig02e.png}\n\\end{minipage}\n\\caption{\\textbf{a)} Calculated ESDs when using the SPECTRODE method, MIXANDMIX including homotopy continuation and MIXANDMIX without homotopy continuation. \\textbf{b-e)} Base $10$ logarithm of the squared magnitude of the fixed point update, i.e., $\\log_{10}(|h(e)|^2)$, together with corresponding isolines at \\textbf{b)} $z=2.2+1i$, \\textbf{c)} $z=2.2+0.1i$, \\textbf{d)} $z=2.2+0.01i$ and \\textbf{e)} $z=2.2+0.001i$.}\n\\label{fig:FIG2}\n\\end{figure}\n\n\\subsection{Comparison with the literature}\n\n\\label{sec:COLI}\n\nIn Fig.~\\ref{fig:FIG3} we compare the accuracy and computational efficiency of MIXANDMIX with the SPECTRODE and QuEST approaches for two population distributions that admit a closed form expression for the ESD. In Figs.~\\ref{fig:FIG3}a,b, we show respectively the averaged accuracy and computation times for a set of aspect ratios ranging from $\\gamma=0.05$ to $\\gamma=0.95$ in steps of $0.1$, $\\gamma=1$, and reciprocal $1\/\\gamma$ ranging from $\\gamma=0.05$ to $\\gamma=0.95$ in steps of $0.1$ for the standard distribution (MP) in~\\eqref{ec:MPLA}. Corresponding accuracies throughout the support together with the gold standard density (in a logarithmic scale) are shown in Fig.~\\ref{fig:FIG3}c for the $\\gamma=0.5$ case. Analogous plots are provided in Figs.~\\ref{fig:FIG3}d-f for a two-delta ($\\delta\\delta$) distribution with equiprobable eigenvalues at $\\lambda_1=1$ and $\\lambda_2=8$, for which the ESD can be obtained by solving a third order polynomial equation~\\citep{Dobriban15,Rao08}. The SPECTRODE and MIXANDMIX approaches have been run with an accuracy parameter providing similar computation times than those of the QuEST method using $100$ grid points, which corresponds to $\\epsilon=10^{-6}\/\\epsilon=10^{-5}$ (MP, $\\gamma<1\/\\gamma\\geq 1$) and $\\epsilon=10^{-5}\/\\epsilon=10^{-4}$ ($\\delta\\delta$, $\\gamma<1\/\\gamma\\geq 1$) for SPECTRODE and $\\epsilon=10^{-5}$, $L=3$ (both) for MIXANDMIX. To account for the relative grid complexities of different methods, linearly interpolated densities are compared with close-form solutions in a uniform grid comprised of $10000$ evenly distributed points along the support. MIXANDMIX is roughly $2$ and $1$ orders of magnitude more accurate than QuEST and SPECTRODE respectively. We observe that the computation times of SPECTRODE largely depend on the aspect ratio, with increments of several orders of magnitude as $\\gamma\\to 1$, while they are much more uniform for MIXANDMIX and QuEST. As for the accuracy distributions, they are generally satisfactory for all methods, but MIXANDMIX seems to provide improved results near the lower edge for the MP case and throughout the support for the $\\delta\\delta$ case. The grid sizes used by each method for $\\gamma=0.5$ have been $102\/104$ by QuEST ($100$ plus some additional points to localize the support limits), $2930\/5734$ by SPECTRODE and $1200\/1200$ by MIXANDMIX, for the MP \/ $\\delta\\delta$ problems.\n\\begin{figure}[!htb]\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig03a}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig03b}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig03c}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig03d}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig03e}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig03f}\n\\end{minipage}\n\\caption{\\textbf{a,d)} Averaged accuracy, \\textbf{b,e)} computation times, and \\textbf{c,f)} accuracy throughout the domain ($\\gamma=0.5$) for \\textbf{a-c)} the MP problem and \\textbf{d-f)} the $\\delta\\delta$ problem with equiprobable $\\boldsymbol{\\lambda}=\\{1,8\\}$.}\n\\label{fig:FIG3}\n\\end{figure}\n\nMIXANDMIX is potentially limited by risks of failures at detecting all the segments comprising the distribution support. Thereby, we have conducted some experiments to test the support detection reliability in challenging scenarios. In Fig.~\\ref{fig:FIG4} we cover analogous experiments to those in Fig.~\\ref{fig:FIG3} for skewed $\\delta\\delta$ problems with $\\boldsymbol{\\lambda}=\\{1,100\\}$ with respective multiplicity ratios $\\mathbf{w}=\\{0.99,0.01\\}$ (Figs.~\\ref{fig:FIG4}a-c), left skewed (LS) problem, and $\\mathbf{w}=\\{0.01,0.99\\}$ (Figs.~\\ref{fig:FIG4}d-f), right skewed (RS) problem. Note that due to the scale differences in the population eigenvalue locations, results on the accuracy throughout the domain are more conveniently represented in a logarithmic scale. SPECTRODE parameters for approximately matched average computation times have now been modified to $\\epsilon=10^{-4}\/\\epsilon=10^{-3}$ (LS, $\\gamma<1\/\\gamma\\geq 1$) and $\\epsilon=10^{-3}\/\\epsilon=10^{-2}$ (RS, $\\gamma<1\/\\gamma\\geq 1$) while MIXANDMIX parameters could be kept the same as in previous experiments while still matching QuEST computation times. Once again, results in Figs.~\\ref{fig:FIG4}a,d show an improvement of averaged accuracy by several orders of magnitude when using the MIXANDMIX method. Results on Figs.~\\ref{fig:FIG4}c,f show that MIXANDMIX has been capable to approximate the support for both problems and to provide more accurate results than the other two methods almost everywhere within the support. In this case the grid sizes have been $104\/104$ for QuEST, $6963\/9130$ for SPECTRODE and $1248\/1248$ for MIXANDMIX for the LS \/ RS problems.\n\\begin{figure}[!htb]\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig04a}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig04b}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig04c}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig04d}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig04e}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig04f}\n\\end{minipage}\n\\caption{\\textbf{a,d)} Averaged accuracy, \\textbf{b,e)} computation times, and \\textbf{c,f)} accuracy throughout the domain ($\\gamma=0.5$) for the $\\delta\\delta$ problem with $\\boldsymbol{\\lambda}=\\{1,100\\}$ skewed towards \\textbf{a-c)} the smallest ($\\mathbf{w}=\\{0.99,0.01\\}$) and \\textbf{d-f)} largest ($\\mathbf{w}=\\{0.01,0.99\\}$) eigenvalue.}\n\\label{fig:FIG4}\n\\end{figure}\n\nIn Fig.~\\ref{fig:FIG5} we apply the three methods to a challenging comb-like~\\citep{Dobriban15} problem. In this case the population eigenvalues come from $100$ equiprobable point masses uniformly distributed in the interval $[0.1,10]$, thus with a large number of components that differ by as much as two orders of magnitude. We show the results at selected areas of the support for $\\gamma=0.025$, $\\gamma=0.5$ and $\\gamma=0.975$ respectively in Figs.~\\ref{fig:FIG5}a-c, again using logarithmic scaling for improved visualization. When tuning the SPECTRODE parameter for similar computation times to those of QuEST and MIXANDMIX, which turned out to happen at $\\epsilon=10^{-3}$, the visual impression is of limited performance. In contrast, MIXANDMIX appears to be powerful, showing strengths not only in detecting all the support intervals, but also in picking-up the density oscillations observed for $\\gamma=0.025$ and the steeped lower edge for $\\gamma=0.975$. We have quantitatively assessed this perception by running the final experiment including also the SPECTRODE results at the accuracy level where we could not visually detect any differences with MIXANDMIX results, $\\epsilon=10^{-6}$ for both $\\gamma=0.025$ and $\\gamma=0.5$, or the maximum code accuracy, $\\epsilon=10^{-8}$, for $\\gamma=0.975$. Grid sizes of the different methods were $1950\/1113\/1216$ for SPECTRODE with matched computation times, $140\/106\/102$ for QuEST, $1248\/1200\/1200$ for MIXANDMIX, and $11678\/19156\/244296$ for SPECTRODE with improved accuracy. Results show no visual differences between MIXANDMIX and SPECTRODE with improved accuracy for $\\gamma=0.025$ and $\\gamma=0.5$, and more plausible functional shape of the former around the left edge for $\\gamma=0.975$, even though approximately $200\\times$ less computational resources were used. Finally, QuEST results show a remarkable ability to correctly determine the support intervals, but limitations to accurately approximate the spiked density areas or capture the density oscillations.\n\\begin{figure}[!htb]\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig05a}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig05b}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig05c}\n\\end{minipage}\n\\caption{ESD for the Comb problem with $100$ equiprobable point masses evenly distributed in the interval $[0.1,10]$. \\textbf{a)} $\\gamma=0.025$, \\textbf{b)} $\\gamma=0.5$ and \\textbf{c)} $\\gamma=0.975$.}\n\\label{fig:FIG5}\n\\end{figure}\n\n\\subsection{ESD of population mixtures}\n\n\\label{sec:MIPO}\n\nIn this Section we provide an illustration of the MIXANDMIX results for the mixture of populations model and the benefits of the GPU architecture. Fig.~\\ref{fig:FIG6}a shows the calculated ESD for a mixture of populations with $K=6$ equiprobable populations, so $\\alpha_k=1\/K$, drawn from population covariances $\\boldsymbol{\\Lambda}_{M\\text{(DIAG)}}^{k}=\\Lambda_{mn\\text{(DIAG)}}^k=((m+k)\\bmod K)+1)\\delta[m,n]$, i.e., a diagonal matrix whose elements in the diagonal grow from $1$ to $K$ with period $K$ with this pattern being shifted across the populations. Fig.~\\ref{fig:FIG6}b extends this DIAG problem to non-diagonal population covariances using $\\boldsymbol{\\Lambda}_{M\\text{(CORR)}}^{k}=\\Lambda_{mn\\text{(CORR)}}^k=\\rho^{|m-n|^l}\\sqrt{\\Lambda_{mm\\text{(DIAG)}}^k\\Lambda_{nn\\text{(DIAG)}}^k}$, $\\rho<1$, $l>0$, so this CORR problem reduces to the DIAG problem when $l\\to\\infty$. Namely, Fig.~\\ref{fig:FIG6}b  shows the results for $\\rho=0.2$, $l=0.25$ and $\\gamma=0.5$, here with $M=120$ for convenience. We can see that calculations are in agreement with simulations (with these being obtained using the biggest matrix sizes that we would fit in our GPU memory) for both the DIAG and CORR problems, with the CORR results showing a larger spectral dispersion due to the larger condition number of the non-diagonal matrix. Finally, Fig.~\\ref{fig:FIG6}c shows the GPU computation times for the CORR problem for a number of populations ranging from $K=1$ to $K=6$. First, we can appreciate a significant penalty when moving from $K=1$ to $K=2$, as that switches the problem from solving a single equation based only on the eigenvalues to a system of equations based on the whole structure of the covariance matrices. Second, we observe that for $K\\geq 2$ the computation times grow with $K$ following a lower than $1$ ratio. This is to be attributed to an increased degree of parallelization of the GPU implementation for bigger problems and a stable fixed point Anderson acceleration in the multidimensional case.\n\\begin{figure}[!htb]\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig06a}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig06b}\n\\end{minipage}\n\\begin{minipage}{.33\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/Fig06c}\n\\end{minipage}\n\\caption{Calculated ESDs and simulations for the \\textbf{a)} DIAG and \\textbf{b)} CORR problems. \\textbf{c)} computation times for solving the CORR problem for different values of $\\gamma$ and $K$.}\n\\label{fig:FIG6}\n\\end{figure}\n\n\\section{Discussion}\n\n\\label{sec:DISC}\n\nWe have presented a set of numerical techniques to aid in the computation of the ESD in a mixture of populations model. These include the use of Anderson mixing to accelerate the fixed point iterations, homotopy continuation for robust convergence to the right optimum, adaptive grid construction to efficiently detect the support and approximate the distribution, and a parallel architecture to tackle the increased computational demands in this setting. Results have shown that our method offers favorable practical efficiency-accuracy tradeoffs when compared with related approaches while being able to address more general models than available tools. \n\nOur tool has focused on the model described in~\\cite{Wagner12} but it could straightforwardly be adapted and optimized to address several analogous models in the literature, such as those mentioned in~\\S~\\ref{sec:INTR}. Nevertheless, we have not contemplated even more general models such as the Kronecker model~\\citep{Zhang13}, with a recent contribution for the separable case in~\\cite{Leeb19}, or more general couplings between the matrix elements~\\citep{Wen11,Lu16}. However, these models generally involve the solution of more intricate systems of nonlinear equations with more auxiliary functions, but with strong functional resemblances to the system we have studied here, so there is potential to reuse or adapt our tools to tackle them. The population model discussed in this paper admits a free probability~\\citep{Mingo17} based formulation given by\n\\begin{equation}\n\\label{ec:FPFP}\nf_\\gamma^{\\prescript{}{K}{\\boldsymbol{\\Lambda}}^{}_{M}}=\\displaystyle\\op_{k=1}^K\\alpha_k(f_0^{\\boldsymbol{\\Lambda}_M^k}\\boxtimes f^{\\mathbf{I}}_{\\gamma \/\\alpha_k}),\n\\end{equation}\nwhere $\\boxtimes$ represents the free multiplicative convolution and $\\op$ the free additive convolution over the summands. This implies that the machinery described in~\\cite{Belinschi17} could be used to obtain the system of equations in~\\S~\\ref{sec:INCO}. However, the models in~\\cite{Belinschi17} are more general, including self-adjoint polynomials not necessarily built from a combination of Mar\\v{c}enko-Pastur and atomic distributions, but simply from asymptotically freely independent matrices. Thus, investigations are required to discern under what conditions our numerical tools can be extended to cope with these models.\n\nThe proposed technique does not fully exploit the existing descriptions used for support detection in the single population model because we are not aware of any such descriptions for mixtures of populations (although recent results such as those in~\\cite{Bao19,Ji19} may pave the way for them). Anyhow, the gridding procedure has shown a robust behaviour in challenging practical scenarios, even when compared with methods that exploit the properties of the support. The main reason is the introduction of a logarithmic grid, that enables efficient support searches at different spectral scales. This has been synergistically combined with an adaptive grid refinement making use of the second order derivative of the density (with a bias towards the upper edge information) generally with more efficient approximations than provided by previous methods. Although this grid refinement criterion has proven effective for all the tested cases, generally providing finer spectral resolvability than previously proposed methods, other criteria may be more appropriate for different applications. In this regard, we should mention that our software is built on top of a generic grid refinement method that allows to test other possibilities by simply defining different criteria for grid cell subdivision, with some exemplary alternatives already included in the code. In addition, support detection correctness can generally be inferred from the numerical integration of the estimated measures, which is provided as an output.\n\nAnother difference with previous approaches is the dependence of the method on more parameters. Although this may add an extra degree of complexity for users, we have observed good behaviour for all test cases without resorting to parameter tuning, so the tool should be readily usable for many applications. From a different perspective, the combined inspection of simulations and manipulation of these parameters may allow to fine tune the method in most challenging scenarios, some of them, as shown in some of the experiments in~\\S~\\ref{sec:COLI}, not being adequately covered by the reduced flexibility of related approaches. In summary, grid detection robustness can be improved by increasing $M^{\\text{(i)}}$, robustness of approximation by increasing $\\xi^0$ and\/or decreasing $\\beta$ and accuracy by decreasing $\\epsilon$ and\/or increasing $L$. However, a line for future research could involve the incorporation of refined techniques for parameter selection. A possibility could be to investigate more efficient and robust interplays between homotopy continuation and nonlinear acceleration, for instance based on adaptive regularization schemes for nonlinear acceleration~\\citep{Scieur19}.\n\\section{Conclusions}\n\n\\label{sec:CONC}\n\nThis work has introduced a set of techniques to compute the ESD in a mixture of populations model. A generic procedure using only the functional form of the fixed point equations relating the population and limiting empirical distributions has been proposed. Efficient convergence is achieved by Anderson acceleration and homotopy continuation and novel strategies for grid construction have been provided. The method has compared well with related proposals in the literature which, to our knowledge, are only capable to address more restricted models. By providing this detailed description of our solution, we expect our distributed tool to be of practical interest for statisticians working in the field.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn \\cite{GO}, Spatzier communicated the conjecture that any irreducible higher-rank ${\\mathbb Z}^d$ Anosov action is $C^\\infty$-conjugate to an algebraic action. Later, in \\cite{KS}, Kalinin and Spatzier stated a refinement of this conjecture that contends that any irreducible higher-rank ${\\mathbb Z}^d$ Anosov action on any compact manifold has a finite cover $C^\\infty$-conjugate to an algebraic action. The asserted global rigidity was motivated in part by earlier results of Katok and Lewis in \\cite{KL1} and \\cite{KL2},  and a more recent result by Rodriguez Hertz in \\cite{RH}. In the latter, global rigidity has been shown for any higher-rank ${\\mathbb Z}^d$ Anosov action on ${\\mathbb T}^n$ whose action on homology has simple eigenvalues and whose course Lyapunov spaces are one or two dimensional, plus additional conditions. A partial confirmation of the refined assertion of global rigidity is provided in \\cite{KS} for  higher-rank ${\\mathbb Z}^d$ Anosov $C^\\infty$ actions each of whose course Lyapunov spaces are one-dimensional, plus additional conditions. If a higher-rank ${\\mathbb Z}^d$ Anosov action on ${\\mathbb T}^n$ is $C^\\infty$-conjugate to an algebraic one and that algebraic action has a  common real eigenvector, then that higher-rank ${\\mathbb Z}^d$ Anosov action preserves a  one-dimensional $C^\\infty$ foliation of ${\\mathbb T}^n$ determined by that common real eigenvector, i.e., generated by a equilibrium-free $C^\\infty$ flow.\n\nThis paper intertwines the global rigidity of  higher-rank ${\\mathbb Z}^d$ Anosov $C^\\infty$ actions on ${\\mathbb T}^n$ with the classification of equilibrium-free $C^\\infty$ flows on ${\\mathbb T}^n$ that possess nontrivial generalized symmetries. The intertwining centers on presence of a single  one-dimensional $C^\\infty$ distribution determined by an equilibrium-free $C^\\infty$ flow that is invariant under a ${\\mathbb Z}^d$ Anosov $C^\\infty$ action, without a priori conditions on all the course Lyapunov spaces. As shown in Section 3, any generalized symmetry of an equilibrium-free flow is nontrivial if it is Anosov (see Corollary \\ref{multiplierAnosov}). Furthermore, any equilibrium-free flow that possesses a nontrivial generalized symmetry does not have any uniformly hyperbolic compact invariant sets (see Corollary \\ref{nonhyper}). The intertwining juxtaposes an equilibrium-free $C^\\infty$ flow that is not Anosov with a smooth ${\\mathbb Z}^d$ action that is Anosov. In the $C^\\infty$ topology, this is a counterpoint to the result of Palis and Yoccoz in \\cite{PY} on the triviality of centralizers for an open and dense subset of Anosov diffeomorphisms on ${\\mathbb T}^n$, and also to the result of Sad in \\cite{Sa} on the local triviality of centralizers for an open and dense subset of Axiom A vector fields that satisfy the strong transversality condition (as applied to vector fields on ${\\mathbb T}^n$). In particular, it is quite rare for an equilibrium-free $C^\\infty$ flow on ${\\mathbb T}^n$ (or more generally, on a closed Riemannian manifold) to possess a nontrivial generalized symmetry (see Corollary \\ref{rare}).\n\nThe first aspect of the intertwining on ${\\mathbb  T}^n$ relates the global rigidity of a ${\\mathbb Z}^d$ Anosov $C^\\infty$ action with an equilibrium-free $C^\\infty$ flow that is quasiperiodic. As detailed in Section 2, the generalized symmetry group of a $C^\\infty$ flow $\\Phi$ is the subgroup $S_\\Phi$ of ${\\rm Diff}^\\infty({\\mathbb T}^n)$ each of whose elements $R$ sends (via the pushforward) the generating vector field $X_\\Phi$ of $\\Phi$ to a uniform scalar multiple $\\rho_\\Phi(R)$ of itself. The multiplier group $M_\\Phi$ of $\\Phi$ is the abelian group of these scalars. As shown in Section 4, the elements of $M_\\Phi\\setminus\\{1,-1\\}$ when $\\Phi$ is quasiperiodic (or more generally, minimal) are algebraic integers of degree between $2$ and $n$ inclusively (see Corollary \\ref{algebraicnature}).\n\n\\begin{theorem}\\label{Anosovalgebraic} On ${\\mathbb T}^n$, suppose $\\alpha$ is a\\, ${\\mathbb Z}^d$ Anosov $C^\\infty$ action, and $\\Phi$ is equilibrium-free $C^\\infty$ flow. If $\\alpha({\\mathbb Z}^d)\\subset S_\\Phi$ and $\\Phi$ is quasiperiodic $($i.e., $C^\\infty$-conjugate to an irrational flow$)$, then $\\alpha({\\mathbb Z}^d)$ is $C^\\infty$-conjugate to an affine action, a finite index subgroup of $\\alpha({\\mathbb Z}^d)$ is\\, $C^\\infty$-conjugate to an algebraic action, and $M_\\Phi$ contains a ${\\mathbb Z}^d$ subgroup.\n\\end{theorem}\n\n\\noindent Relevant definitions and the proof are given in Section 6. The proof holds not only for $d\\geq 2$ but also for $d=1$. It uses a semidirect product characterization of the structure of the generalized symmetry group for an irrational flow (as shown in Section 5). It also uses the existence of a common fixed point for a finite index subgroup of the ${\\mathbb Z}^d$ Anosov action, a device used in other global rigidity results (for example, see \\cite{KL2}).\n\nThe second aspect of the intertwining on ${\\mathbb T}^n$ relates the classification of an equilibrium-free $C^\\infty$ flow with a ${\\mathbb Z}^d$ Anosov $C^\\infty$ action that is topologically irreducible. As detailed in Section 7, an irrational flow $\\phi$ on ${\\mathbb T}^n$ is of Koch type if a uniform scalar multiple of its frequencies form a ${\\mathbb Q}$-basis for a real algebraic number field of degree $n$ (also see \\cite{KO} and \\cite{LD}). For an $R\\in S_\\Phi$ of an equilibrium-free $C^\\infty$ flow $\\Phi$, the quantity $\\log\\vert \\rho_\\Phi(R)\\vert$ is the value of the Lyapunov exponent $\\chi_R$ of $R$ in the direction of $X_\\Phi$ (see Theorem \\ref{Lyapunov}).\n\n\\begin{theorem}\\label{flowKoch} On ${\\mathbb T}^n$, suppose $\\alpha$ is a higher-rank\\, ${\\mathbb Z}^d$ Anosov $C^\\infty$ action, and $\\Phi$ is equilibrium-free $C^\\infty$ flow. If $\\alpha({\\mathbb Z}^d)\\subset S_\\Phi$, and $\\alpha$ is topologically irreducible and $C^\\infty$-conjugate to an algebraic ${\\mathbb Z}^d$ action, and for an Anosov element $R\\in{\\alpha({\\mathbb Z}^d})$, the multiplicity of the value $\\log\\vert \\rho_\\Phi(R)\\vert$ of $\\chi_R$ is one at some point of\\, ${\\mathbb T}^n$, then $\\Phi$ is  projectively $C^\\infty$-conjugate to an irrational flow of Koch type.\n\\end{theorem}\n\n\\noindent Relevant definitions and the proof are given in Section 7. The proof uses the Oseledets decomposition for an Anosov diffeomorphism (see \\cite{BP} and \\cite{KH}) to show that the flow is $C^\\infty$-conjugate to one generated by a constant vector field. Then by the topological irreducibility and results of Wallace in \\cite{WA}, the components of a scalar multiple of the constant vector field are shown to form a ${\\mathbb Q}$-basis for a real algebraic number field.\n\n\\section{Flows with Nontrivial Generalized Symmetries}\n\nGeneralized symmetries extend the classical notions of time-preserving and time-reversing symmetries  of flows. To simplify notations for these and for proofs of results, it is assumed throughout the remainder of the paper that all manifolds, flows, vector fields, diffeomorphisms, distributions, etc., are smooth, i.e., of class $C^\\infty$. Let $P$ be a closed (i.e., compact without bounday) manifold. Let ${\\rm Flow}(P)$ denote the set of flows on $P$. Following \\cite{BC}, a {\\it generalized symmetry}\\, of $\\psi\\in{\\rm Flow}(P)$ is an $R\\in{\\rm Diff}(P)$ such that there is $\\mu\\in{\\mathbb R}^\\times = {\\mathbb R}\\setminus\\{0\\}$ (the multiplicative real group) for which\n\\[ R\\psi(t,p) = \\psi(\\mu t, R(p)){\\rm\\  for\\ all\\ } t\\in{\\mathbb R}{\\rm\\ and\\ all\\ } p\\in P.\\]\nIt is easy to show that $R$ being a generalized symmetry of $\\psi$ is equivalent to $R$ satisfying \n\\[ R_*X_\\psi = \\mu X_\\psi{\\rm\\ for\\ some\\ }\\mu\\in{\\mathbb R}^\\times.\\]\nHere $X_\\psi(p)=(d\/dt)\\psi(t,p)\\vert_{t=0}$ is the vector field that generates $\\psi$, and $R_*X_\\psi = {\\bf T}R X_\\psi R^{-1}$ is the push-forward of $X_\\psi$ by $R$ where ${\\bf T}R$ is the derivative map. The {\\it generalized symmetry group}\\, of $\\psi$ is the set $S_\\psi$ of all the generalized symmetries that $\\psi$ possesses. There is a homomorphism $\\rho_\\psi:S_\\psi \\to {\\mathbb R}^\\times$ taking $R\\in S_\\psi$ to its unique multiplier $\\rho_\\psi(R) = \\mu$. The multiplier group of $\\psi$ is $M_\\psi = \\rho_\\psi(S_\\psi)$.\n\nThe generalized symmetry group and the multiplier group of a flow are invariants for the equivalence relation of projective conjugacy. Two $\\psi,\\phi\\in{\\rm Flow}(P)$ are {\\it projectively conjugate}\\, if there are $h\\in{\\rm Diff}(P)$ and $\\vartheta\\in{\\mathbb R}^\\times$ such that $h_*X_\\psi = \\vartheta X_\\phi$. Projective conjugacy is an equivalence relation on ${\\rm Flow}(P)$. Projective conjugacy reduces to smooth conjugacy when $\\vartheta = 1$. For $h\\in{\\rm Diff}(P)$, let $\\Delta_h$ be the inner automorphism of ${\\rm Diff}(P)$ given by $\\Delta_h(R) = h^{-1}Rh$ for $R\\in{\\rm Diff}(P)$. If $h_*X_\\psi = \\vartheta X_\\phi$, then $\\Delta_h(S_\\phi) = S_\\psi$ (see Theorem 4.1 in \\cite{BC} which states that $S_\\psi$ is conjugate to the generalized symmetry group for the flow determined by $\\vartheta X_\\phi$, which is exactly the same as $S_\\phi$.) Furthermore, if $\\psi$ and $\\phi$ are projectively conjugate, then $M_\\psi=M_\\phi$ (see Theorem 2.2 in \\cite{BA5}), i.e., the multiplier group is an absolute invariant of projective conjugacy.\n\nAny $R\\in S_\\psi$ is a trivial generalized symmetry of $\\psi$ if  $\\rho_\\psi(R)=1$ (i.e., $R$ is time-preserving), or if $\\rho_\\psi(R)=-1$ (i.e., $R$ is time-reversing). Any $R\\in S_\\psi$ with $\\vert\\rho_\\psi(R)\\vert\\ne 1$ is a {\\it nontrivial generalized symmetry}\\, of $\\psi$. A flow $\\psi$ (or equivalently its generating vector field $X_\\psi$) is said to possess a nontrivial generalized symmetry when $M_\\psi\\setminus\\{1,-1\\}\\ne\\emptyset$. \n\n\\begin{theorem}\\label{noperiodic} Let $\\psi$ be a flow on a closed Riemannian manifold $P$. If $\\psi$ has a periodic orbit and $M_\\psi\\setminus\\{1,-1\\}\\ne\\emptyset$, then $\\psi$ has a nonhyperbolic equilibrium.\n\\end{theorem}\n\n\\begin{proof} Suppose for $p_0\\in P$ that ${\\mathcal O}_\\psi(p_0)=\\{\\psi_t(p_0):t\\in{\\mathbb R}\\}$ is a periodic orbit whose fundamental period is $T_0>0$. (Here $\\psi_t(p) = \\psi(t,p)$.) Assuming that $M_\\phi\\setminus\\{1,-1\\}\\ne\\emptyset$ implies there is $Q\\in S_\\psi$ such that $\\rho_\\psi(Q)\\ne \\pm 1$. If $\\vert \\rho_\\psi(Q)\\vert >1$, then $\\vert\\rho_\\psi(Q^{-1})\\vert<1$, since $\\rho_\\psi$ is a homomorphism. Hence there is $R\\in S_\\psi$ such that $\\vert\\rho_\\psi(R)\\vert<1$. Then \n\\[ R(p_0) = R\\psi(T,p_0) = \\psi\\big(\\rho_\\psi(R)T_0,R(p_0)\\big).\\]\nFor $p_1= R(p_0)$, this implies that ${\\mathcal O}_\\psi(p_1)$ is a periodic orbit with period $T_1 = \\vert \\rho_\\psi(R)\\vert T_0$. Suppose that $T_1$ is not a fundamental period for ${\\mathcal O}_\\psi(p_1)$, i.e., there is $0<T_1^\\prime<T_1$ such that $p_1 = \\psi(T_1^\\prime,p_1)$. This implies that\n\\[ R(p_0) = p_1 = \\psi(T_1^\\prime,p_1) = \\psi\\big(T_1^\\prime,R(p_0)\\big) = R\\psi(T_1^\\prime\/\\rho_\\psi(R),p_0).\\]\nSince $R$ is invertible, this gives $p_0 = \\psi(T_1^\\prime\/\\rho_\\psi(R), p_0)$. With $T_0$ as the fundamental period for ${\\mathcal O}_\\psi(p_0)$, there is $m\\in{\\mathbb Z}^+$ such that $\\vert T_1^\\prime\/\\rho_\\psi(R)\\vert = mT_0$. Hence\n\\[ T_1^\\prime = m T_0 \\vert \\rho_\\psi(R)\\vert \\geq T_0\\vert \\rho_\\psi(R)\\vert = T_1,\\]\na contradiction to $T_1^\\prime < T_1$. Thus, $T_1$ is the fundamental period for ${\\mathcal O}_\\psi(p_1)$. Since $\\vert \\rho_\\psi(R)\\vert<1$, it follows that ${\\mathcal O}_\\psi(p_0)$ and ${\\mathcal O}_\\psi(p_1)$ have different fundamental periods, and hence are distinct periodic orbits. Iteration gives a sequence of distinct periodic orbits ${\\mathcal O}_\\psi(p_i)$, where $p_i = R^i(p_0)$, whose fundamental periods $T_i = \\vert \\rho_\\psi(R)\\vert^i T_0$ decrease to $0$ since $\\vert \\rho_\\psi(R)\\vert <1$.\n\nLet $l(p_i)$ be the arc length of ${\\mathcal O}_\\psi(p_i)$. In terms of the Riemannian norm $\\Vert\\cdot\\Vert$ on ${\\bf T}P$, the tangent bundle of $P$,\n\\[ l(p_i) = \\int_0^{T_i} \\Vert X_\\psi\\big(\\psi(t,p_i)\\big)\\Vert \\ dt.\\]\nContinuity of $X_\\psi$ on the compact manifold $P$ implies that $M=\\sup_{p\\in P} \\Vert X_\\psi(p)\\Vert$ is finite. Thus \n\\[ l(p_i)\\leq MT_i \\to 0.\\]\nBy the compactness of $P$, there is a convergence subsequence $p_{i_m}$ with limit $p_\\infty$. If $\\Vert X_\\psi(p_\\infty)\\Vert\\ne 0$, then the smoothness of $X_\\psi$ implies by the Flow Box Theorem that there are no periodic orbits of $\\psi$ completely contained in a sufficiently small neighborhood $U$ of $p_\\infty$. But since $l(p_{i_m})\\to0$ and $p_{i_m}\\to p_\\infty$, there are periodic orbits of $\\psi$ completely contained in $U$, a contradiction. Thus $p_\\infty$ is an equilibrium for $\\psi$. If $p_\\infty$ were hyperbolic, then the Hartman-Grobman Theorem would imply that there are no periodic orbits completely contained in a sufficiently small neighborhood of $p_\\infty$, again a contradiction. Therefore, $\\phi$ has a nonhyperbolic equilibrium.\n\\end{proof}\n\nThe flows on a closed Riemannian manifold which possess nontrivial generalized symmetries are rare or non-generic in the sense of Baire category theory. Let ${\\mathcal X}(P)$ denote the set of vector fields on $P$. With $P$ compact, there is a one-to-one correspondence between ${\\mathcal X}(P)$ and ${\\rm Flow}(P)$. Equipped with the usual $C^\\infty$ topology, ${\\mathcal X}(P)$ is a Baire space. A residual subset of ${\\mathcal X}(P)$ is one that contains a countable intersection of open, dense subsets of ${\\mathcal X}(P)$. Because ${\\mathcal X}(P)$ is a Baire space, any residual subset is dense. \n\n\\begin{corollary}\\label{rare} For a closed Riemannian manifold $P$, the set of vector fields on $P$ which do not possess nontrivial generalized symmetries is a residual subset of ${\\mathcal X}(P)$.\n\\end{corollary}\n\n\\begin{proof} Let ${\\mathcal V}$ be the set of $X$ in ${\\mathcal X}(P)$ such that any equilibrium, if any, of the flow $\\psi$ induced by $X$ is hyperbolic, and there is at least one periodic orbit for $\\psi$. By Theorem \\ref{noperiodic}, none of the vector fields in ${\\mathcal V}$ possess nontrivial generalized symmetries. Let ${\\mathcal H}$ be the subset of $X$ in ${\\mathcal X}(P)$ such that the flow $\\psi$ induced by $X$ has periodic orbits all of which are hyperbolic, and any equilibrium it has is hyperbolic. By definition, ${\\mathcal H}\\subset {\\mathcal V}$, and by the Kupka-Smale Theorem, ${\\mathcal H}$ is a residual subset of ${\\mathcal X}(P)$.\n\\end{proof}\n\nAccording to Theorem \\ref{noperiodic}, flows which might possess nontrivial generalized symmetries are those that are equilibrium-free and without periodic orbits. The possession of a nontrivial generalized symmetry for an equilibrium-free flow places dynamical restrictions on the compact invariant sets of the flow. An invariant set $\\Lambda$ for an equilibrium-free $\\Phi\\in{\\rm Flow}(P)$ is {\\it uniformly hyperbolic}\\, if there is a continuous ${\\bf T}\\Phi$-invariant splitting ${\\bf T}_p\\Lambda = E^s(p) \\oplus{\\rm Span}(X_\\Phi(p))\\oplus E^u(p)$ for $p\\in\\Lambda$ and constants $C\\geq 1$ and $\\lambda\\in(0,1)$ such that\n\\[ \\Vert {\\bf T}_p \\Phi_t(v)\\Vert \\leq C\\lambda^t \\Vert v\\Vert {\\rm\\ for\\ }v\\in E^s(p) {\\rm\\ and\\ }t\\geq 0\\]\nand\n\\[ \\Vert {\\bf T}_p \\Phi_{-t}(v)\\Vert \\leq C\\lambda^t \\Vert v\\Vert {\\rm\\ for\\ }v\\in E^u(p) {\\rm\\ and\\ }t\\geq 0.\\]\nRecall that if $\\Lambda = P$ is uniformly hyperbolic for $\\Phi$, then $\\Phi$ is called Anosov.\n\n\\begin{corollary}\\label{nonhyper} If $\\Phi$ is an equilibrium-free flow on a closed Riemannian manifold $P$ which possesses a nontrivial generalized symmetry, then any compact invariant set for $\\Phi$ is not uniformly hyperbolic $($and, in particular, $\\Phi$ is not Anosov$)$.\n\\end{corollary}\n\n\\begin{proof} Suppose $\\Lambda$ is a uniformly hyperbolic compact invariant set of $\\Phi$. By the Anosov Closing Lemma, there exists a periodic orbit for $\\Phi$. But with $\\Phi$ being equilibrium-free and possessing a nontrivial generalized symmetry, Theorem \\ref{noperiodic} implies that $\\Phi$ does not have periodic orbits. Thus $\\Lambda$ can not be uniformly hyperbolic.\n\\end{proof}\n\n\n\\section{Multipliers and Lyapunov Exponents}\n\nThe multiplier of a generalized symmetry for an equilibrium-free flow is related to the Lyapunov exponents of that generalized symmetry. For a closed Riemannian manifold $P$, the {\\it Lyapunov exponent}\\, of $R\\in{\\rm Diff}(P)$ at $p\\in P$ is\n\\[ \\chi_R (p,v) = \\limsup_{m\\to\\infty} \\frac{\\log\\Vert {\\bf T}_p R^m(v)\\Vert}{m}, \\ v\\in {\\bf T}_p P\\setminus\\{0\\}.\\]\nIt is independent of the Riemannian norm on $P$ because $P$ is compact. As the following result shows, any trivial generalized symmetry for an equilibrium-free flow has zero Lyapunov exponents everywhere.\n\n\\begin{theorem}\\label{Lyapunov} If $\\psi$ is an equilibrium-free flow on a closed Riemannian manifold $P$, then for each $R\\in S_\\psi$, the one-dimensional distribution $E\\subset {\\bf T}P$ determined by $E(p) = {\\rm Span}\\big(X_\\psi(p)\\big)$ is $R$-invariant and\n\\[ \\chi_R\\big(p,X_\\psi(p)\\big) = \\log\\vert \\rho_\\psi(R)\\vert {\\rm \\ for\\ all\\ }p\\in P.\\]\n\\end{theorem}\n\n\\begin{proof} Each $R\\in S_\\psi$ satisfies ${\\bf T}_pR\\big(X_\\psi(R^{-1}(p)\\big) = \\rho_\\psi(R) X_\\psi(p)$ for all $p\\in P$. Then ${\\bf T}_pR(X_\\psi(p)) = \\rho_\\psi(R)X_\\psi(R(p))$. The distribution $E\\subset {\\bf T}P$ determined by $E(p) = {\\rm Span}\\big(X_\\psi(p)\\big)$ is $R$-invariant and one-dimensional because $\\rho_\\psi(R)\\ne 0$ and $X_\\psi(p)\\ne 0$ for all $p\\in P$. Furthermore, it follows for all $m\\geq 1$ that \n\\[ {\\bf T}_pR^m(X_\\psi(p)) = [\\rho_\\psi(R)]^m X_\\psi\\big(R^m(p)\\big),\\]\nand so\n\\[ \\Vert {\\bf T}_pR^m(X_\\psi(p))\\Vert = \\vert \\rho_\\psi(R)\\vert^m \\Vert X_\\psi\\big(R^m(p)\\big)\\Vert.\\]\nThe flow $\\psi$ being smooth and equilibrium-free on the compact manifold $P$ implies that $\\Vert X_\\psi\\big(R^m(p)\\big)\\Vert$ is both bounded away from $0$ and bounded above uniformly in $m$ for each $p\\in P$. Therefore\n\\[\\chi_R\\big(p,X_\\psi(p)\\big) = \\limsup_{m\\to\\infty} \\frac{ m \\log\\vert \\rho_\\psi(R)\\vert + \\log \\Vert X_\\psi\\big(R^m(p)\\big)\\Vert}{m} = \\log\\vert \\rho_\\psi(R)\\vert\\]\nfor all $p\\in P$.\n\\end{proof}\n\nGlobal uniform hyperbolicity is a dynamical condition on a generalized symmetry of an equilibrium-free flow that guarantees that it is nontrivial. An $R\\in{\\rm Diff}(P)$ is {\\it Anosov} if there is a continuous ${\\bf T}R$-invariant splitting $T_p P = E^s(p)\\oplus E^u(p)$ for $p\\in P$, and constants $c>0$ and $\\lambda\\in(0,1)$ independent of $p\\in P$, such that\n\\[ \\Vert {\\bf T}_p R^m(v)\\Vert \\leq c \\lambda^m \\Vert v\\Vert {\\rm \\ for\\ }v\\in E^s(p){\\rm\\ and \\ }m\\geq 0,\\]\nand\n\\[ \\Vert {\\bf T}_p R^{-m}(v)\\Vert \\leq c \\lambda^m \\Vert v\\Vert {\\rm \\ for\\ }v\\in E^u(p){\\rm\\ and \\ }m\\geq 0\\]\nwith the angle between $E^s(p)$ and $E^u(p)$ bounded away from $0$ (see \\cite{BP}).\n\n\\begin{theorem}\\label{multiplierAnosov} Let $\\psi$ be an equilibrium-free flow on a closed Riemannian manifold $P$. If $R\\in S_\\psi$ is Anosov, then $\\vert\\rho_\\psi(R)\\vert \\ne 1$.\n\\end{theorem}\n\n\\begin{proof} Suppose $R\\in S_\\psi$ is Anosov and $\\vert\\rho_\\psi(R)\\vert=1$. If $\\rho_\\psi(R)=-1$, then replacing $R$ with $R^2$ gives $\\rho_\\psi(R) =1$ with $R$ Anosov. Let $E^s(p)\\oplus E^u(p)$ be the continuous ${\\bf T}R$-invariant splitting with its associated contraction estimates. The generating vector field for $\\psi$ has a continuous decomposition $X_\\psi(p) = v^s(p) + v^u(p)$ for $v^s(p)\\in E^s(p)$ and $v^u(p)\\in E^u(p)$. From the contraction estimates of ${\\bf T}_pR$ on $E^s(p)$ and $E^u(p)$ it follows for all $p\\in P$ that\n\\[ \\Vert {\\bf T}_pR^m(v^s(p))\\Vert \\to 0, \\ \\ \\Vert {\\bf T}_p R^{-m}(v^u(p))\\Vert\\to 0{\\rm\\ as\\ } m\\to\\infty.\\]\nWith $\\rho_\\psi(R) = 1$ and $X_\\psi(p) = v^s(p) + v^u(p)$, the equation $R_*X_\\psi = \\rho_\\psi(R)X_\\psi$ becomes\n\\[ {\\bf T}_p R^m (v^s(p) + v^u(p)) = X_\\psi(R^m(p)) = v^s(R^m(p))+ v^u(R^m(p))\\]\nfor  all $p\\in P$ and $m\\in {\\mathbb Z}$. The ${\\bf T}R$-invariance of $E^u(p)$ implies that\n\\[  {\\bf T}_p R^m(v^u(p)) = v^u(R^m(p)).\\]\nFor a fixed $p\\in P$, there is by the compactness of $P$, a subsequence ${\\rm R}^{m_i}(p)$ converging to a point, say $p_\\infty$, as $m_i\\to \\infty$. Hence\n\\begin{align*}\nX_\\psi(p_\\infty)\n& = \\lim_{i\\to\\infty} X_\\psi(R^{m_i}(p)) \\\\\n& = \\lim_{i\\to\\infty} {\\bf T}_pR^{m_i}(v^s(p)+v^u(p)) \\\\\n& = \\lim_{i\\to\\infty} \\big[{\\bf T}_p R^{m_i}(v^s(p)) + v^u(R^{m_i}(p))\\big] \\\\\n& = v^u(p_\\infty).\n\\end{align*}\nIt now follows that\n\\[ \\lim_{m\\to\\infty}X_\\psi(R^{-m}(p_\\infty)) = \\lim_{m\\to\\infty}{\\bf T}_{p_\\infty} R^{-m}(X_\\psi(p_\\infty)) = \\lim_{m\\to\\infty}{\\bf T}_{p_\\infty} R^{-m}(v^u(p_\\infty))=0.\\]\nCompactness of $P$ implies that $X_\\psi$ has a zero. But $\\psi$ is an equilibrium-free flow, and therefore $\\vert\\rho_\\psi(R)\\vert\\ne 1$.\n\\end{proof}\n\nAny equilibrium-free flow with a generalized symmetry that is Anosov does not have any periodic orbits according to Lemma \\ref{noperiodic} and Theorem \\ref{multiplierAnosov}. On the other hand, for an equilibrium-free flow that is without nontrivial generalized symmetries, Theorem \\ref{multiplierAnosov} implies that none of its generalized symmetries can be Anosov. However, the converse of Theorem \\ref{multiplierAnosov} is false. As illustrated next, a partially hyperbolic diffeomorphism can be a nontrivial generalized symmetry of an equilibrium-free flow without periodic orbits.\n\n\\begin{example}\\label{nonquasi}{\\rm Let $P={\\mathbb T}^n={\\mathbb R}^n\/{\\mathbb Z}^n$, the $n$-torus, equipped with global coordinates $(\\theta_1,\\theta_2,\\dots,\\theta_n)$. The flow $\\psi$ generated by\n\\[ X_\\psi = \\frac{\\partial}{\\partial\\theta_1} + \\frac{\\partial}{\\partial\\theta_2} + \\cdot\\cdot\\cdot + \\frac{\\partial}{\\partial\\theta_{n-2}} + \\frac{\\partial}{\\partial\\theta_{n-1}}  + \\sqrt 2 \\frac{\\partial}{\\partial\\theta_n}\\]\nis equilibrium-free and without periodic orbits. Let $R\\in{\\rm Diff}({\\mathbb T}^n)$ be induced by the ${\\rm GL}(n,{\\mathbb Z})$ matrix\n\\[ B = \\begin{bmatrix} 1 & 0 & \\hdots & 0 & 0 & 1 \\\\ 0 & 1 & \\hdots & 0 & 0 & 1 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & \\hdots & 1 & 0 & 1 \\\\ 0 & 0 & \\hdots & 0 &  1 & 1 \\\\ 0 & 0  & \\hdots & 0 &  2 & 1\\end{bmatrix},\\]\nthat is, ${\\bf T}R = B$. This $R$ is a nontrivial generalized symmetry of $\\psi$ because $R_*X_\\psi = (1+\\sqrt 2)X_\\psi$, i.e., $\\rho_\\psi(R) = 1+\\sqrt 2$. However $B$ has an eigenvalue of $1$ of multiplicity $n-2$, and the eigenspace corresponding to this eigenvalue is an $(n-2)$-dimensional center distribution for $R$. The other two eigenvalues of $B$ are $1\\pm\\sqrt 2$ and the corresponding eigenspaces are the $1$-dimensional unstable and stable distributions for $R$ respectively. Thus $R$ is partially hyperbolic.\n}\\end{example}\n\n\\section{Restrictions on Multipliers}\n\nAlgebraic restrictions may occur on $\\rho_\\Phi(R)$ for a nontrivial generalized symmetry $R$ of an equilibrium-free flow $\\Phi$ on a compact manifold $P$ without boundary. This happens when there are interactions beyond $R_*X_\\Phi = \\rho_\\Phi(R)X_\\Phi$ between the dynamics of $R$ and $\\Phi$ on submanifolds diffeomorphic to ${\\mathbb T}^k$ for some $2\\leq k\\leq {\\rm dim}(P)$. A {\\it real algebraic integer}\\, is the root of a monic polynomial with integer coefficients, and its degree is the degree of its minimal polynomial. As shown by Wilson in \\cite{WI}, there is for any integer $k$ with $2\\leq k\\leq {\\rm dim}(P)-2$, an equilibrium-free flow $\\Phi$ on $P$ which has an invariant submanifold $N$ diffeomorphic to ${\\mathbb T}^k$ on which $\\Phi$ is {\\it minimal}, i.e., every orbit of $\\Phi$ in $N$ is dense in $N$. The well-known prototype of a minimal flow on ${\\mathbb T}^k$ is an {\\it irrational flow}, i.e., a $\\psi$ on ${\\mathbb T}^k$ for which $X_\\psi$ is a constant vector field whose components (or frequencies) are rationally independent (see \\cite{FA}), i.e., linearly independent over ${\\mathbb Q}$.  According to Basener in \\cite{BA}, any minimal flow on ${\\mathbb T}^n$ is topologically conjugate to an irrational flow.\n\n\\begin{lemma}\\label{subset} If a flow $\\phi$ on a compact manifold $N$ without boundary is topologically conjugate to a flow $\\psi$ on ${\\mathbb T}^k$ $(k={\\rm dim}(N))$ and $\\psi$ is minimal, then $M_\\phi\\subset M_\\psi$ and each $\\mu\\in M_\\phi\\setminus\\{1,-1\\}$ is an algebraic integer of degree between $2$ and $k$ inclusively.\n\\end{lemma}\n\n\\begin{proof} Suppose there is a homeomorphism $h:N\\to{\\mathbb T}^k$ such that $h\\phi_t = \\psi_t h$ for all $t\\in{\\mathbb R}$. Without loss of generality, it is assumed that $\\psi$ is an irrational flow, since any minimal flow on ${\\mathbb T}^k$ is topologically conjugate to an irrational flow.\n\nLet $V\\in S_\\phi$ and set $\\mu=\\rho_\\phi(V)$. In terms of the homeomorphism $Q=hVh^{-1}$ on ${\\mathbb T}^k$, the multiplier $\\mu$ passes through $h$ to $\\psi$:\n\\[ Q\\psi(t,\\theta) = hR\\phi(t,h^{-1}(\\theta)) = h\\phi(\\mu t,Rh^{-1}(\\theta)) = \\psi(\\mu t, Q(\\theta)), {\\rm\\ for\\ all\\ }t\\in{\\mathbb R}, \\theta \\in{\\mathbb T}^k,\\] \ni.e., $Q\\psi_t = \\psi_{\\mu t}Q$. This does not say yet that $Q$ is a generalized symmetry of $\\psi$ with multiplier $\\mu$ because $Q$ is only a homeomorphism at the moment.\n\nFollowing \\cite{AP}, the homeomorphism $Q$ of ${\\mathbb T}^k$ lifts uniquely to $\\hat Q(x)=\\hat L(x) + \\hat U(x) +\\hat c$ on ${\\mathbb R}^k$, i.e., $\\pi \\hat Q = Q\\pi$ where $\\pi:{\\mathbb R}^k\\to{\\mathbb T}^k$ is the covering map. Here the linear part of this lift is $\\hat L(x) = Bx$ for $B\\in {\\rm GL}(k,{\\mathbb Z})$; the periodic part is $\\hat U(x)$, i.e., $\\hat U(x+\\nu) = \\hat U(x)$ for all $x\\in{\\mathbb R}^k$ and all $\\nu\\in {\\mathbb Z}^k$, is continuous and satisfies $\\hat U(0)=0$; and the constant part is $\\hat c\\in[0,1)^k$. A lift of the irrational flow $\\psi$ to ${\\mathbb R}^n$ is $\\hat \\psi(t,x) = x+td$ where $d=X_\\psi$ and $x\\in{\\mathbb R}^k$. For all $t\\in{\\mathbb R}$, a lift of $Q\\psi_t$ to ${\\mathbb R}^k$ is $\\hat Q\\hat \\psi_t$ and a lift of $\\psi_{\\mu t}Q$ is $\\hat \\psi_{\\mu t}\\hat Q$. These two lifts differ by a constant $m\\in{\\mathbb Z}^k$ since $Q\\psi_t = \\psi_{\\mu t}Q$ for all $t\\in{\\mathbb R}$:\n\\[ \\hat Q\\hat \\psi(t,x)  = \\hat \\psi(\\mu t,\\hat Q(x)) + m {\\rm\\ for\\ all\\ }t\\in{\\mathbb R}, x\\in{\\mathbb R}^k.\\]\nSince\n\\[ \\hat \\psi(t,x) = Bx + tBd + \\hat U(x+td) + \\hat c\\]\nand \n\\[ \\hat\\psi(\\mu t,\\hat Q(x)) = Bx + \\hat U(x) + \\hat c + \\mu t d,\\]\nit follows that\n\\[ \\hat U(x+td) - \\hat U(x) = - t(B-\\mu I)d + m {\\rm\\ for\\ all\\ }t\\in{\\mathbb R}, x\\in{\\mathbb R}^k,\\]\nwhere $I$ is the identity matrix. Evaluation of this at $x=0$ gives\n\\[ \\hat U(td) = -t(B-\\mu I)d + m {\\rm\\ for\\ all\\ }t\\in{\\mathbb R}.\\]\nHowever, $\\hat U$ is bounded since it is continuous and periodic. This boundedness implies that $(B-\\mu I)d = 0$, and so $\\hat U(td) = m$ for all $t\\in{\\mathbb R}$. Evaluation of this at $t=0$ shows that $m=0$ because $\\hat U(0)=0$. Thus\n\\[ 0=\\hat U(td) = \\hat U (\\hat \\psi_t(0)){\\rm \\ for\\  all\\ }t\\in{\\mathbb R}.\\] Since $\\hat U$ is periodic and continuous on ${\\mathbb R}^n$, it is a lift of a homeomorphism $U$ on ${\\mathbb T}^n$. A lift of $U\\psi_t$ is $\\hat U\\hat\\psi_t$, and so\n\\[ 0=\\pi \\hat U(\\hat \\psi_t(0)) = U(\\psi_t(0)) {\\rm\\ for\\ all\\ }t\\in{\\mathbb R}.\\]\nThe minimality of $\\psi$ implies that $U$ is $0$ on a dense subset of ${\\mathbb T}^n$. By continuity, $U = 0$ which implies that $\\hat U=0$. Thus $\\hat Q$ is $C^\\infty$, and so $Q$ is a diffeomorphism, whence $Q\\in S_\\psi$ with $\\rho_\\psi(Q) = \\mu$. Since $\\mu\\in M_\\phi$, then $M_\\phi\\subset M_\\psi$.\n\nThe multiplier $\\mu \\in M_\\psi$ is a real algebraic integer of degree at most $k$ because it satisfies $(B-\\mu I)d = 0$ for nonzero $d$, i.e., the characteristic polynomial of $B$ is a monic polynomial of degree $k$ with integer coefficients. The only rational roots this characteristic polynomial can have are $\\pm 1$ since $\\psi$ is an irrational flow, i.e., $M_\\psi\\cap{\\mathbb Q}=\\{1,-1\\}$ (see Corollary 4.4 in \\cite{BA2}). However, if $\\mu \\ne \\pm 1$, then the minimal polynomial for $\\mu$ has degree between $2$ and $k$ inclusively.\n\\end{proof}\n\n\\begin{theorem}\\label{algebraicmultiplier} Suppose for an equilibrium-free flow $\\Phi$ on $P$ there is a $\\Phi$-invariant compact submanifold $N$ without boundary and $R\\in S_\\Phi$ with $\\vert\\rho_\\Phi(R)\\vert \\ne 1$ such that $R(N)\\cap N\\ne \\emptyset$. If ${\\rm dim}(N)=2$ and $N$ is orientable, then $\\rho_\\Phi(R)$ is a real algebraic integer of degree $2$. If ${\\rm dim}(N)\\geq 3$ with $N$ diffeomorphic to ${\\mathbb T}^{{\\rm dim}(N)}$ and $\\Phi\\vert N$ is a minimal flow, then $\\rho_\\Phi(R)$ is a real algebraic integer of degree between $2$ and ${\\rm dim}(N)$ inclusively.\n\\end{theorem}\n\n\\begin{proof} Let $\\mu=\\rho_\\Phi(R)$ with $\\vert\\mu\\vert\\ne 1$ and $k={\\rm dim}(N)$. By Theorem \\ref{noperiodic} there are no periodic orbits for the equilibrium-free flow $\\Phi\\vert N$. If $k=2$ and $N$ is orientable, the Poincar\\'e-Bendixson Theorem implies that $N$ is diffeomorphic to ${\\mathbb T}^2$, and that $\\Phi\\vert N$ is minimal. If $k\\geq 3$, it is assumed that $N$ is diffeomorphic to ${\\mathbb T}^k$ and that $\\Phi\\vert N$ is minimal. \n\nThe submanifold $R(N)$ is $\\Phi$-invariant because $N$ is $\\Phi$-invariant and $R\\in S_\\Phi$, i.e., for $p\\in R(N)$ and $q= R^{-1}(p)\\in N$,\n\\[ \\Phi(t,p) = \\Phi(t,R(q)) = R\\Phi(t\/\\mu,q) \\subset R(N) {\\rm \\ for\\ all\\ }t\\in{\\mathbb R}.\\]\nBy hypothesis, there is $\\tilde p\\in R(N)\\cap N$. By the $\\Phi$-invariance of $N$ and $R(N)$ and the minimality of $\\Phi\\vert N$ it follows that $\\overline{{\\mathcal O}_\\Phi(\\tilde p)} = N$ and $\\overline{{\\mathcal O}_\\Phi(\\tilde p)} = R(N)$. This gives $R(N) = N$, i.e., that $N$ is $R$-invariant.\n\nThe nontrivial generalized symmetry $R$ restricts to a nontrivial generalized symmetry of $\\Phi\\vert N$ because $N$ is $\\Phi$-invariant and $R$-invariant. If $V=R\\vert N$ and $\\phi$ is the flow on $N$ determined by $X_\\phi = X_\\Phi\\vert N$, then $R_*X_\\Phi = \\mu X_\\Phi$ becomes \n\\[ {\\bf T} V X_\\phi (p) = \\mu X_\\phi (V(p)){\\rm\\ for\\ }p\\in N.\\]\nSince $V\\in{\\rm Diff}(N)$, then $V\\in S_\\phi$ with $\\rho_\\phi(V) = \\mu$.\n\nBy \\cite{BA}, minimality of $\\phi=\\Phi\\vert N$ with $N$ diffeomorphic to ${\\mathbb T}^n$ implies that $\\phi$ is topologically conjugate to an irrational flow $\\psi$. Applying Lemma \\ref{subset} shows that $\\mu$ is an algebraic integer of degree between $2$ and $k$ inclusively.\n\\end{proof}\n\n\\begin{corollary}\\label{algebraicnature} Suppose $\\Phi$ is a minimal flow on $\\mathbb T^n$, and $R\\in S_\\Phi$. If $\\vert \\rho_\\Phi(R)\\vert \\ne 1$, then $\\rho_\\Phi(R)$ is an algebraic integer of degree between $2$ and $n$ inclusively.\n\\end{corollary}\n\n\\begin{proof} Any nontrivial generalized symmetry $R$ of $\\Phi$ together with $N=\\mathbb T^n$ satisfy the conditions of Theorem \\ref{algebraicmultiplier}.\n\\end{proof}\n\n\\section{A Group-Theoretical Characterization of Irrational Flows}\n\nMinimality of an equilibrium-free flow on ${\\mathbb T}^n$ places a semidirect product structure the generalized symmetry group of that flow. A group ${\\mathcal S}$ is the {\\it semidirect product}\\, of two subgroups ${\\mathcal N}$ and ${\\mathcal H}$ if ${\\mathcal N}$ is a normal subgroup of ${\\mathcal S}$, if ${\\mathcal S}={\\mathcal N}{\\mathcal H}$, and if ${\\mathcal N}\\cap {\\mathcal H}$ is the identity element of ${\\mathcal S}$. Notational this is written\n\\[ {\\mathcal S} = {\\mathcal N}\\rtimes_\\Gamma {\\mathcal H},\\]\nwhere $\\Gamma: {\\mathcal H}\\to {\\rm Aut}({\\mathcal N})$ is the conjugating homomorphism of the semidirect product, i.e., $\\Gamma({\\mathfrak h})({\\mathfrak n}) = {\\mathfrak h}{\\mathfrak n}{\\mathfrak h}^{-1}$ for ${\\mathfrak h}\\in {\\mathcal H}$ and ${\\mathfrak n}\\in {\\mathcal N}$. A normal subgroup of $S_\\psi$ is $\\ker \\rho_\\psi$ for any flow $\\psi$. A normal subgroup of ${\\rm Diff}(\\mathbb T^n)$ is the abelian group ${\\rm Trans}(\\mathbb T^n)$ of translations. Each {\\it translation}\\, on ${\\mathbb T}^n$ is of the form ${\\mathcal T}_c(\\theta) = \\theta + c$ for $c\\in{\\mathbb T}^n$. If $\\psi$ is a flow on ${\\mathbb T}^n$ with $X_\\psi$ a constant, then ${\\rm Trans}({\\mathbb T}^n)\\subset {\\rm ker}\\rho_\\psi$ because $({\\mathcal T}_c)_* X_\\psi = X_\\psi$ for all $c\\in{\\mathbb T}^n$.\n\n\\begin{lemma}\\label{kernel} Let $\\psi$ be a flow on ${\\mathbb T}^n$ for which $X_\\psi$ is a constant. If ${\\rm ker}\\rho_\\psi = {\\rm Trans}({\\mathbb T}^n)$, then $\\psi$ is irrational.\n\\end{lemma}\n\n\\begin{proof} Suppose that $\\psi$ is not irrational. Then the components of $X_\\psi$ are not rationally independent. Up to a permutation of the coordinates $\\theta_1,\\dots,\\theta_n$ on ${\\mathbb T}^n$, i.e., a smooth conjugacy, it can be assumed the first $l$ components of $X_\\psi$ are the smallest subset of the components that are linearly dependent  over ${\\mathbb Q}$. Specifically, writing $X_\\psi = [a_1,a_2,\\dots,a_n]^T$, there is a least integer $l$ with $1\\leq l\\leq n$ such that\n\\[ k_1a_1+\\cdot\\cdot\\cdot + k_la_l =0\\]\nwith $k_i\\in{\\mathbb Z}\\setminus\\{0\\}$ for all $i=1,\\dots,l$. The existence of an $R\\in{\\rm ker}\\rho_\\psi\\setminus{\\rm Trans}({\\mathbb T}^n)$ will be exhibited separately in the cases of $l<n$ and $l=n$.\n\nCase $1\\leq l\\leq n-1$. The $R\\in{\\rm Diff}({\\mathbb T}^n)$ induced by the ${\\rm GL}(n,{\\mathbb Z})$ matrix\n\\[ B  = \\begin{bmatrix} 1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & 0 \\\\ 0 & 1 & \\dots & 0 & 0 & 0 & \\dots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & 1 & 0 & 0 & \\dots & 0 & 0 \\\\ 0 & 0 & \\dots & 0 & 1 & 0 & \\dots & 0 & 0 \\\\ 0 & 0 & \\dots & 0 & 0 & 1 & \\dots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & 0 & 0 & 0 & \\dots & 1 & 0 \\\\  k_1 & k_2 & \\dots & k_{l-1} & k_l & 0 & \\dots & 0 & 1\\end{bmatrix}\\]\nsatisfies $R_*X_\\psi = X_\\psi$. So $R\\in {\\rm ker}\\rho_\\psi$, but $R\\not\\in{\\rm Trans}({\\mathbb T}^n)$.\n\nCase $l=n$. Here none of the $k_1,k_2,\\dots,k_n$ are zero, and the integer-entry matrix\n\\[ B = \\begin{bmatrix} 1 & 0 & \\dots & 0 & 0 \\\\ 0 & 1 & \\dots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & 1 & 0 \\\\ k_1 & k_2 & \\dots & k_{n-1} & k_n\\end{bmatrix}\\]\ninduces a $k_n$-to-$1$ smooth surjection $g$ of ${\\mathbb T}^n$ to itself. The flow $\\phi$ on ${\\mathbb T}^n$ generated by $X_\\phi = [a_1,\\dots,a_{n-1},0]^T$ is a factor of $\\psi$ i.e., $g\\psi_t = \\phi_t g$ for all $t\\in\\mathbb R$, because $BX_\\psi = X_\\phi$.\n\nThe components $a_1,\\dots,a_{n-1}$ of $X_\\phi$ are rationally independent. For, if they were not, there would then be integers $k_1^\\prime,\\dots,k_{n-1}^\\prime$, not all zero, such that $k_1^\\prime a_1 + \\cdot\\cdot\\cdot + k_{n-1}^\\prime a_{n-1}=0$, which would contradict the minimality of $l=n$. With $e_1,\\dots,e_n$ as the standard basis for ${\\mathbb R}^n$ and $E={\\rm Span}(e_1,\\dots,e_{n-1})$ an $(n-1)$-dimensional subspace of ${\\mathbb R}^n$, the closure of ${\\mathcal O}_\\phi(0)$ is $\\pi(E)$. Thus $\\overline{{\\mathcal O}_\\phi(0)}$ is an embedded submanifold of ${\\mathbb T}^n$ diffeomorphic to ${\\mathbb T}^{n-1}$.\n\nThe semiconjugacy $g$ between $\\psi$ and $\\phi$ along with $g(0)=0$ imply that $\\overline{{\\mathcal O}_\\psi(0)}$ is an embedded submanifold of ${\\mathbb T}^n$ that is diffeomorphic to ${\\mathbb T}^{n-1}$. In particular, the $(n-1)$-dimensional subspace $U$ of ${\\mathbb R}^n$ determined by $B(U) = E$ satisfies $\\pi(U) = \\overline{{\\mathcal O}_\\psi(0)}$, where $U={\\rm Span}(u_1,u_2,\\dots,u_{n-1})$ for $B(u_i) = e_i$, $i=1,\\dots,n-1$. Furthermore, \n\\[ X_\\psi = a_1u_1 + \\cdot\\cdot\\cdot + a_{n-1}u_{n-1} \\in U\\]\nbecause $k_1a_1+\\cdot\\cdot\\cdot + k_na_n = 0$.\n\nFor $\\epsilon>0$, $V$ a nonempty subset of ${\\mathbb R}^n$, and $m\\in{\\mathbb Z}^n$, define $N_\\epsilon(V)$ to be the set of points in ${\\mathbb R}^n$ less than a distance of $\\epsilon$ from $V$, and define $V+m$ to be the translation of $V$ by $m$. By the definition of $E$, if $E+m\\ne E$, then $N_{1\/2}(E+m)\\cap N_{1\/2}(E)=\\emptyset$. Since $B(U)=E$ and $B({\\mathbb Z}^n)\\subset {\\mathbb Z}^n$, there exists $\\epsilon>0$ such that if $U+m\\ne U$, then $N_\\epsilon(U+m)\\cap N_\\epsilon(U)=\\emptyset$.\n\nThe vector $e_n\\not \\in U$. Let $x_1,\\dots, x_n$ be the coordinates on ${\\mathbb R}^n$ that correspond to the basis $u_1,\\dots,u_{n-1},e_n$. Let $f:{\\mathbb R}\\to{\\mathbb R}$ be a smooth bump function with $f(0)=1$ and whose support has length smaller than $\\epsilon$. In terms of the coordinates $x_1,\\dots, x_n$, define a smooth vector field on $N_\\epsilon(U)$ by\n\\[ Y = f(x_n)\\frac{\\partial}{\\partial x_n}.\\]\nExtend this vector field to all of ${\\mathbb R}^n$ by translation to $N_\\epsilon(U+m)$ for those $m\\in{\\mathbb Z}^m$ for which $U+m\\ne U$, and to the remainder of ${\\mathbb R}^n$ as $0$. The extended vector field is globally Lipschitz, and so determines a flow $\\xi$ on ${\\mathbb R}^n$.\n\nSince the vector field generating $\\xi$ is invariant under translations by $m\\in{\\mathbb Z}$, the time-one map $\\xi_1$ is also invariant under these translations. Thus $\\xi_1$ is a lift of an $R\\in{\\rm Diff}({\\mathbb T}^n)$. In terms of the coordinates $x_1,\\dots, x_n$, the derivative of $\\xi_1$ at any point $x\\in{\\mathbb R}^n$ is of the form\n\\[ {\\bf T}_x\\xi_1 = \\begin{bmatrix} 1 & 0 & \\dots & 0 & 0 \\\\ 0 & 1 & \\dots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & 1 & 0 \\\\ 0 & 0 & \\dots & 0  & \\ast \\end{bmatrix},\\]\nand so ${\\bf T}_x\\xi _1(u) = u$ for $u\\in U$. Since $X_\\psi\\in U$, this means that ${\\bf T}_x \\xi_1(X_\\psi) =  X_\\psi$. Since $X_\\psi$ is a constant vector field, then $(\\xi_1)_* X_\\psi = X_\\psi$. Since $\\xi_1$ is a lift of $R$, it now follows that $R_*X_\\psi = X_\\psi$. Therefore $R\\in {\\rm ker}\\rho_\\psi$ but $R\\not\\in{\\rm Trans}({\\mathbb T}^n)$.\n\\end{proof}\n\nThe group ${\\rm Aut}({\\mathbb T}^n)$ of automorphisms of $\\mathbb T^n$ is naturally identified with ${\\rm GL}(n,{\\mathbb Z})$. For $\\mathcal T_c\\in {\\rm Trans}(\\mathbb T^n)$ and $B\\in {\\rm GL}(n,\\mathbb Z)$, the composition of $\\mathcal T_c$ with $B$ is written $\\mathcal T_c  B= B+c$. \n\n\\begin{theorem}\\label{characterization} Let $\\psi$ be a flow on ${\\mathbb T}^n$ with $X_\\psi$ a nonzero constant vector. Then $\\psi$ is irrational if and only if there exists a subgroup $H$ of ${\\rm GL}(n,{\\mathbb Z})$ isomorphic to $M_\\psi$ such that $S_\\psi = {\\rm Trans}({\\mathbb T}^n) \\rtimes_\\Gamma H$.\n\\end{theorem}\n\n\\begin{proof} Suppose that $\\psi$ is irrational. This implies (by Theorem 5.5 in \\cite{BA2}) that\n\\[ S_\\psi = {\\rm ker}\\rho_\\psi \\rtimes_\\Gamma H_\\psi,\\]\nwhere $H$ is a subgroup of ${\\rm GL}(n,{\\mathbb Z})$ isomorphic to $M_\\psi$. Furthermore, the irrationality of $\\psi$ implies (by Corollary 4.7 in \\cite{BA2}) that ${\\rm ker}\\rho_\\psi={\\rm Trans}({\\mathbb T}^n)$.\n\nNow suppose that $\\psi$ is not irrational. By Lemma \\ref{kernel}, there is $R\\in {\\rm ker}\\rho_\\psi\\setminus{\\rm Trans}({\\mathbb T}^n)$. If there were a subgroup $H$ of ${\\rm GL}(n,{\\mathbb Z})$ isomorphic to $M_\\psi$ such that $S_\\psi = {\\rm Trans}(\\mathbb T^n)\\rtimes_\\Gamma H$, then $R=\\mathcal T_c B$ for some $c\\in{\\mathbb T}^n$ and $B\\in H$. Hence $1=\\rho_\\psi(R) = \\rho_\\psi(\\mathcal T_c)\\rho_\\psi(B)= \\rho_\\psi(B)$. However, since $H$ is isomorphic to $M_\\psi$, there is only one element of $H$ which corresponds to the multiplicative identity $1$ of $\\mathbb R^\\times$, and that is $I$, the identity matrix. This means that $B=I$, and so $R=\\mathcal T_c$, a contradiction.\n\\end{proof}\n\n\n\\section{Global Rigidity for Certain ${\\mathbb Z}^d$ Anosov Actions}\n\nGlobal rigidity is about when a ${\\mathbb Z}^d$ Anosov action, which as is well-known is topologically conjugate to an algebraic ${\\mathbb Z}^d$ action (see \\cite{KK}), is smoothly conjugate to an algebraic ${\\mathbb Z}^d$ action. A ${\\mathbb Z}^d$ action on ${\\mathbb T}^n$ is a monomorphism $\\alpha:{\\mathbb Z}^d\\to{\\rm Diff}({\\mathbb T}^n)$. It is Anosov if there is $m\\in{\\mathbb Z}^d\\setminus\\{0\\}$ with $\\alpha(m)$ Anosov, is {\\it algebraic}\\, if $\\alpha({\\mathbb Z}^d)\\subset {\\rm GL}(n,{\\mathbb Z})$, and more generally is {\\it affine}\\, if $\\alpha({\\mathbb Z}^d)\\subset{\\rm Trans}({\\mathbb T}^n)\\rtimes_\\Gamma{\\rm GL}(n,{\\mathbb Z})$. Algebraic ${\\mathbb Z}^d$-actions are found in algebraic number theory (see \\cite{KKS} and \\cite{Sc}). \n\n\\vskip0.2cm\n\\noindent{\\it Proof of Theorem \\ref{Anosovalgebraic}.} Although well-known, the argument for the existence of a common fixed point of a finite index subgroup of a ${\\mathbb Z}^d$ Anosov action is included for completeness. Let $m_0\\in{\\mathbb Z}^d$ be such that $\\alpha(m_0)$ is Anosov. Then $\\alpha(m_0)$ is topologically conjugate to a hyperbolic automorphism of ${\\mathbb T}^n$ (see \\cite{Ma}), and so $\\alpha(m_0)$ has a finite number of fixed points, $f_1,\\dots,f_l$. Let $\\alpha(m_1),\\dots,\\alpha(m_d)$ be a generating set for $\\alpha({\\mathbb Z}^d)$. Since $\\alpha({\\mathbb Z}^d)$ is abelian, then for all $i=1,\\dots,d$ and $j=1,\\dots,l$,\n\\[ \\alpha(m_0)\\alpha(m_i)(f_j) = \\alpha(m_i)\\alpha(m_0)(f_j) = \\alpha(m_i)(f_j).\\]\nThis means that $\\alpha(m_i)(f_j)$ is one of the finitely many fixed points of $\\alpha(m_0)$. Since each $\\alpha(m_i)$ is invertible, there is a positive finite integer $r_i$ such that $\\alpha(m_i)^{r_i}(f_1) = f_1$. Thus the finite index subgroup of $\\alpha({\\mathbb Z}^d)$ generated by $\\alpha(m_1)^{r_1}, \\dots,\\alpha(m_d)^{r_d}$ has $f_1$ as a common fixed point.\n\nBy hypothesis, there is a $g\\in{\\rm Diff}({\\mathbb T}^n)$ and an irrational flow $\\phi$ on ${\\mathbb T}^n$ for which $X_\\phi = g_* X_\\Phi$. By Theorem \\ref{characterization}, there is a subgroup $H_\\phi$ of ${\\rm GL}(n,\\mathbb Z)$ isomorphic to $M_\\phi$ such that $S_\\phi={\\rm Trans}({\\mathbb T}^n)\\rtimes_\\Gamma H_\\phi$. For $h\\in{\\rm Diff}({\\mathbb T}^n)$ given by $h={\\mathcal T}_{-g(f_1)}\\circ g$ it follows that $h_*X_\\Phi = X_\\phi$ because $({\\mathcal T}_c)_*X_\\phi = X_\\phi$ for any $c\\in{\\mathbb T}^n$. Then $\\Phi$ and $\\phi$ are projectively conjugate, and so, as mentioned in Section 2, $\\Delta_h(S_\\phi) = S_\\Phi$. The inclusion $\\alpha({\\mathbb Z}^d)\\subset S_\\Phi$ implies that\n\\[ \\Delta_{h^{-1}}(\\alpha({\\mathbb Z}^d))\\subset {\\rm Trans}({\\mathbb T}^n)\\rtimes_\\Gamma H_\\phi.\\]\nThis means that $\\alpha$ is $C^\\infty$-conjugate to an affine ${\\mathbb Z}^d$-action.\n\nFor each $\\alpha(m)$ in the finite index subgroup generated by $\\alpha(m_1)^{r_1},\\dots,\\alpha(m_d)^{r_d}$ there are $B_m\\in H_\\phi$ and $c_m\\in {\\mathbb T}^n$ such that $\\Delta_{h^{-1}}(\\alpha(m)) = B_m + c_m$. Since $h(f_1) = 0$, then\n\\[ c_m=(B_m+c_m)(0) = h\\circ\\alpha(m)\\circ h^{-1}(0) = h \\circ \\alpha(m)(f_1) = h(f_1) = 0.\\]\nThis means that $\\Delta_{h^{-1}}(\\alpha(m))\\in H_\\phi$. Hence the finite index subgroup of $\\alpha({\\mathbb Z}^d)$ generated by $\\alpha(m_1)^{r_1},\\dots,\\alpha(m_d)^{r_d}$, which is isomorphic to ${\\mathbb Z}^d$, is $C^\\infty$-conjugate to an algebraic ${\\mathbb Z}^d$-action. Since $H_\\phi$ contains a ${\\mathbb Z}^d$ subgroup, and $M_\\phi$ is isomorphic to $H_\\phi$, then $M_\\phi$ contains a ${\\mathbb Z}^d$ subgroup. By absolute invariance of the multiplier group under projective conjugacy, $M_\\Phi$ contains a ${\\mathbb Z}^d$ subgroup.  \\hfill $\\Box$\n\n\\vskip0.2cm\nAny quasiperiodic flow on ${\\mathbb T}^n$ whose generalized symmetry group contains a ${\\mathbb Z}^d$ Anosov action must possess nontrivial generalized symmetries since its multiplier group contains a ${\\mathbb Z}^d$ subgroup by Theorem \\ref{Anosovalgebraic} (cf.\\,Theorem \\ref{multiplierAnosov}). This puts a necessary condition on the quasiperiodic flows to which Theorem \\ref{Anosovalgebraic} does apply. The quasiperiodic flows of Koch type mentioned in the next section satisfy this necessary condition. However, there are quasiperiodic flows that do not, as is illustrated next.\n\n\\begin{example}\\label{trivialquasi}{\\rm On ${\\mathbb T}^n$, let $\\psi$ be the flow generated by\n\\[ X_\\psi = \\frac{\\partial}{\\partial\\theta_1} + \\pi\\frac{\\partial}{\\partial\\theta_2} + \\cdot\\cdot\\cdot + \\pi^{n-1} \\frac{\\partial}{\\partial\\theta_n}.\\]\nThis flow is quasiperiodic, since if its frequencies $1,\\pi,\\dots,\\pi^{n-1}$ were linearly dependent over ${\\mathbb Q}$ then $\\pi$ would be algebraic. Quasiperiodicity of $\\psi$ implies that each $\\mu\\in M_\\psi$ is a real algebraic integer of degree at most $n$, and moreover, $M_\\psi\\cap{\\mathbb Q}=\\{1,-1\\}$ (see Corollary 4.4 in \\cite{BA2}). Suppose $\\mu\\in M_\\psi\\setminus\\{1,-1\\}$. Then there is $R\\in S_\\psi$ such that $R_*X_\\psi = \\mu X_\\psi$. Quasiperiodicity of $\\psi$ implies that ${\\bf T}R=B\\in{\\rm GL}(n,{\\mathbb Z})$ (see Theorem 4.3 in \\cite{BA2}). Then $R_*X_\\psi = \\mu X_\\psi$ becomes $BX_\\psi = \\mu X_\\psi$, and for $B=(b_{ij})$, it follows that\n\\[ \\mu = b_{11} + b_{12}\\pi + \\cdot\\cdot\\cdot + b_{1n}\\pi^{n-1}.\\]\nIf $b_{12}=\\cdot\\cdot\\cdot = b_{1n} = 0$, then $\\mu = b_{11}\\in{\\mathbb Z}$. But $M_\\psi\\cap{\\mathbb Q}=\\{1,-1\\}$, and so $\\mu = \\pm 1$. This contradiction means that one of $b_{12},\\dots,b_{1n}$ is nonzero. Because each multiplier of $\\psi$ is a real algebraic integer of degree at most $n$, there is a monic polynomial $l(z)$ in the polynomial ring ${\\mathbb Z}[z]$ such that $l(\\mu)=0$. But this implies that $\\pi$ is a root of a polynomial in ${\\mathbb Z}[z]$, making $\\pi$ algebraic. This shows that $M_\\psi=\\{1,-1\\}$, and so $\\psi$ does not possess nontrivial generalized symmetries.\n}\\end{example}\n\n\n\\section{Classification of Certain Equilibrium-Free Flows}\n\nQuasiperiodic flows of Koch type are algebraic in nature and provide foliations which are often preserved by a topologically irreducible ${\\mathbb Z}^d$ Anosov action (see \\cite{BA5}, \\cite{BA6}, and \\cite{KKS} for such examples). A flow on ${\\mathbb T}^n$ is {\\it quasiperiodic of Koch type}\\, if it is projectively conjugate to an irrational flow whose frequencies form a ${\\mathbb Q}$-basis for a real algebraic number field ${\\mathbb F}$ of degree $n$ over ${\\mathbb Q}$. For a quasiperiodic flow $\\Phi$ of Koch type, the real algebraic number field ${\\mathbb F}$ of degree $n$ associated to it is unique, and its multiplier group is a finite index subgroup of the group of units ${\\mathfrak o}_{\\mathbb F}^\\times$ in the ring of integers of ${\\mathbb F}$ (see Theorem 3.3 in \\cite{BA5}).  By Dirichlet's Unit Theorem (see \\cite{SD}), there is $d\\geq 1$ such that ${\\mathfrak o}_{\\mathbb F}^\\times$ is isomorphic to ${\\mathbb Z}_2\\oplus{\\mathbb Z}^d$, and so every quasiperiodic flow of Koch type always possesses nontrivial generalized symmetries.\n\nTopological irreducibility of a ${\\mathbb Z}^d$ action $\\alpha$ is a condition on the topological factors that $\\alpha$ has. A ${\\mathbb Z}^d$ action $\\alpha^\\prime$ on ${\\mathbb T}^{n^\\prime}$ is a {\\it topological factor}\\, of $\\alpha$ if there is a continuous surjection $h:{\\mathbb T}^n\\to{\\mathbb T}^{n^\\prime}$ such that $h\\circ \\alpha = \\alpha^\\prime\\circ h$. A topological factor $\\alpha^\\prime$ of $\\alpha$ is {\\it finite}\\, if the continuous surjection $h$ is finite-to-one everywhere. A ${\\mathbb Z}^d$ action $\\alpha$ is {\\it topologically irreducible}\\, if every topological factor $\\alpha^\\prime$ of $\\alpha$ is finite.\n\nFor an algebraic ${\\mathbb Z}^d$ action $\\alpha$, there is a stronger sense of irreducibility, one that uses the group structure of ${\\mathbb T}^n$. An algebraic ${\\mathbb Z}^d$ action $\\alpha^\\prime$ on ${\\mathbb T}^{n^\\prime}$ is an {\\it algebraic factor}\\, of $\\alpha$ if there is a continuous homomorphism $h:{\\mathbb T}^n\\to{\\mathbb T}^{n^\\prime}$ such that $h\\circ \\alpha = \\alpha^\\prime\\circ h$. An algebraic factor $\\alpha^\\prime$ of $\\alpha$ is {\\it finite} if the continuous homomorphism $h$ is finite-to-one everywhere. An algebraic ${\\mathbb Z}^d$ action $\\alpha$ is {\\it algebraically irreducible}\\, if every algebraic factor $\\alpha^\\prime$ of $\\alpha$ is finite. Algebraic irreducibility of a higher rank algebraic $\\mathbb Z^d$ action $\\alpha$ is equivalent to there being an $m\\in \\mathbb Z^d$ such that $\\alpha(m)$ has an irreducible characteristic polynomial (see Proposition 3.1 on p.\\,726 in \\cite{KKS}; cf.\\,\\cite{BE}).\n\n\n\\vskip0.2cm\n\\noindent{\\it Proof of Theorem \\ref{flowKoch}.} Identify ${\\bf T}{\\mathbb T}^n$ with ${\\mathbb T}^n\\times{\\mathbb R}^n$, and place on the fiber the standard Euclidean norm $\\Vert\\cdot\\Vert$. By the hypotheses, there is $h\\in{\\rm Diff}({\\mathbb T}^n)$ and a hyperbolic $B\\in{\\rm GL}(n,{\\mathbb Z})$ such that $\\Delta_{h^{-1}}(\\alpha(m_0)) = B$. Every point of ${\\mathbb T}^n$ is Lyapunov regular for $B$. The Oseledets decomposition associated with $\\chi_B$ is\n\\[ {\\bf T}_\\theta {\\mathbb T}^n = \\bigoplus_{i=1}^k E^i_B,\\]\nwhere $E_B^i$, $i=1,\\dots,k$, are the invariant subspaces of $B$ which are independent of $\\theta$. Since $\\Delta_{h^{-1}}(\\alpha(m_0))=B$, every point of ${\\mathbb T}^n$ is Lyapunov regular for $\\alpha(m_0)$. Set\n\\[ E^i_{\\alpha(m_0)}(\\theta) = {\\bf T}_{h(\\theta)}h^{-1}\\big( E^i_B), \\ \\ \\theta\\in{\\mathbb T}^n.\\]\nThe Oseledets decomposition associated with $\\chi_{\\alpha(m_0)}$ is then\n\\[ {\\bf T}_\\theta {\\mathbb T}^n = \\bigoplus_{i=1}^k E^i_{\\alpha(m_0)}(\\theta).\\]\n\nFor $\\mu = \\rho_\\Phi(\\alpha(m_0))$, the hypothesis that the multiplicity of the value $\\log\\vert \\mu \\vert$ of $\\chi_{\\alpha(m_0)}$ is one at a point $\\hat \\theta\\in{\\mathbb T}^n$ implies that there is $1\\leq l\\leq k$ such that ${\\rm dim}\\big(E^l_{\\alpha(m_0)}(\\hat\\theta)\\big) = 1$ and\n\\[ \\chi_{\\alpha(m_0)}(\\hat\\theta,v) = \\log\\vert \\mu \\vert {\\rm\\ for\\ } v\\in E^l_{\\alpha(m_0)}(\\hat\\theta)\\setminus\\{0\\}.\\]\nBy the definition $E^l_{\\alpha(m_0)}(\\hat\\theta) = {\\bf T}_{h(\\hat\\theta)}h^{-1}\\big( E^l_B)$, it follows that ${\\rm dim}\\big( E^l_B\\big) = 1$. Furthermore, since $\\Delta_{h^{-1}}(\\alpha(m_0))=B$ and $\\chi_B$ is independent of $\\theta$, it follows that $\\chi_B(\\theta,v) = \\log\\vert\\mu\\vert$ for all $v\\in E^l_B\\setminus\\{0\\}$ and for all $\\theta\\in{\\mathbb T}^n$. The definition $E^l_{\\alpha(m_0)}(\\theta) = {\\bf T}_{h(\\theta)}h^{-1}\\big( E^l_B)$ and the independence of $E^l_B$ from $\\theta$ imply for all $\\theta\\in{\\mathbb T}^n$ that ${\\rm dim} \\big(E^l_{\\alpha(m_0)}(\\theta)\\big) = 1$ and $\\chi_{\\alpha(m_0)}(\\theta,v) = \\log\\vert\\mu\\vert$ for all $v\\in E^l_{\\alpha(m_0)}(\\theta)\\setminus\\{0\\}$. Hence, the multiplicity of $\\log\\vert\\mu\\vert$ for $\\chi_{\\alpha(m_0)}$ is one for all $\\theta\\in{\\mathbb T}^n$.\n\nBy Theorem \\ref{Lyapunov}, the $\\alpha(m_0)$-invariant one-dimensional distribution $E$ given by $E(\\theta) = {\\rm Span}(X_\\Phi(\\theta))$ satisfies $\\chi_{\\alpha(m_0)}(\\theta,X_\\Phi(\\theta))=\\log\\vert\\mu\\vert$ for all $\\theta\\in{\\mathbb T}^n$. If $E(\\theta)\\ne E^l_{\\alpha(m_0)}(\\theta)$ at some $\\theta\\in{\\mathbb T}^n$, then $E(\\theta)+E^l_{\\alpha(m_0)}(\\theta)$ is a two-dimensional subspace of  ${\\bf T}_\\theta {\\mathbb T}^n$ for which $\\chi_{\\alpha(m_0)}(\\theta, v) = \\log\\vert\\mu\\vert$ for all $v\\in \\big(E(\\theta)+ E^l_{\\alpha(m_0)}(\\theta)\\big)\\setminus\\{0\\}$. This contradicts the multiplicity of $\\log\\vert\\mu\\vert$ for $\\chi_{\\alpha(m_0)}$ being one at every $\\theta$. Thus $E^l_{\\alpha(m_0)}(\\theta) = E(\\theta) = {\\rm Span}(X_\\Phi(\\theta))$ for all $\\theta\\in{\\mathbb T}^n$.\n\nThe vector field $h_*X_\\Phi$ satisfies $h_*X_\\Phi(\\theta)\\in E^l_B$ for all $\\theta\\in{\\mathbb T}^n$ because ${\\rm Span}(X_\\Phi(\\theta)) = {\\bf T}_{h(\\theta)} h^{-1}(E^l_B)$ for all $\\theta\\in{\\mathbb T}^n$. Let $\\psi$ be the flow for which $X_\\psi = h_*X_\\Phi$.  Since $E^l_B$ is a one-dimensional invariant subspace of $B$ and $\\chi_B(\\theta,v) = \\log\\vert\\mu\\vert$ for all $v\\in E^l_B\\setminus\\{0\\}$, it follows for all $\\theta\\in{\\mathbb T}^n$ that\n\\[ \\Vert B^kX_\\psi(\\theta)\\Vert = \\vert \\mu\\vert^k  \\Vert X_\\psi(\\theta)\\Vert {\\rm \\ for\\ all\\ }k\\in{\\mathbb Z}.\\]\n\nHyperbolicity of $B$ implies that there is $\\bar\\theta\\in{\\mathbb T}^n$ such that ${\\mathcal O}_B(\\bar\\theta)=\\{ B^k(\\bar\\theta):k\\in{\\mathbb Z}\\}$ is dense in ${\\mathbb T}^n$. Since $\\alpha(m_0)_*X_\\Phi = \\mu X_\\Phi$ and $X_\\psi = h_*X_\\Phi$, the matrix $B$ satisfies $BX_\\psi =\\mu X_\\psi B$. Then $B^k X_\\psi = \\mu^k X_\\psi B^k$, and so $X_\\psi B^k = \\mu^{-k} B^k X_\\psi$. Thus,\n\\[ \\Vert X_\\psi(B^k\\bar\\theta)\\Vert = \\vert \\mu\\vert^{-k} \\Vert B^k X_\\psi(\\bar\\theta)\\Vert = \\vert\\mu\\vert^{-k}\\vert\\mu\\vert^k \\Vert X_\\psi(\\bar\\theta)\\Vert = \\Vert X_\\psi(\\bar\\theta)\\Vert {\\rm \\ for\\ all\\ }k\\in{\\mathbb Z}.\\]\nDenseness of ${\\mathcal O}_B(\\bar\\theta)$ and continuity of $X_\\psi$ imply that $\\Vert X_\\psi(\\theta)\\Vert = \\Vert X_\\psi(\\bar\\theta)\\Vert$ for all $\\theta\\in{\\mathbb T}^n$. The one-dimensionality of $E^l_B$ to which $X_\\psi$ belongs implies that $X_\\psi$ is a constant vector. Thus $X_\\psi$ is an eigenvector of $B$.\n\nThe assumed topological irreducibility of $\\alpha$ and the inclusion $\\Delta_{h^{-1}}(\\alpha({\\mathbb Z}^d))\\subset {\\rm GL}(n,{\\mathbb Z})$ imply that $\\Delta_{h^{-1}}(\\alpha(\\mathbb Z^d))$ is algebraically irreducible. Thus there is $B^\\prime\\in \\Delta_{h^{-1}}(\\alpha({\\mathbb Z}^d))$ with an irreducible characteristic polynomial. Since $BB^\\prime = B^\\prime B$ and $B^\\prime$ has an irreducible characteristic polynomial, the eigenvector $X_\\psi$ of $B$ is an eigenvector of $B^\\prime$ too. Then there is $\\vartheta\\in{\\mathbb R}^\\times$ such that the components of $\\vartheta^{-1}X_\\psi$ form a ${\\mathbb Q}$-basis for an algebraic number field of degree $n$ over ${\\mathbb Q}$ (see Propositions 1 and  8 in \\cite{WA}). Thus the flow $\\phi$ determined by $X_\\phi = \\vartheta^{-1}X_\\psi$ is irrational of Koch type for which $h_*X_\\Phi = X_\\psi = \\vartheta X_\\phi$. \\hfill $\\Box$\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusions}\n\\vskip -0.39em\nWe introduce a general method to train a CNN which can switch the resolution during inference, and thus its running speed can be selected to fit different computational constraints. Specifically, we propose the parallel training to handle multi-resolution images, using shared parameters and private BNs. We analyze the interaction effects of resolutions from the perspective of train-test discrepancy. And we propose to learn an ensemble distillation based on the same image instances with different resolutions, which improves accuracies to a large extent.\n\n\n\\section*{Acknowledgement}\n\\vskip -0.35em\nThis work is  jointly supported by the National Science Foundation of China (NSFC) and the German Research Foundation (DFG) in project Cross Modal Learning, NSFC 61621136008\/DFG TRR-169. We thank Aojun Zhou for the insightful discussions. \n\n\n\\clearpage\n\n\n\n\\section*{\\LARGE Appendix}\n\n\\section{Related Work}\n\\label{rw}\n\\subsubsection{Image Recognition.}  Image recognition acts as a benchmark to evaluate models and is a core task for computer vision. Advances on the large-scale image recognition datasets, like ImageNet \\cite{DBLP:conf\/cvpr\/DengDSLL009}, can translate to improved results on a number of other applications \\cite{DBLP:journals\/ijcv\/GordoARL17,DBLP:conf\/cvpr\/KornblithSL19,DBLP:conf\/cvpr\/OquabBLS14}. \nIn order to enhance the model generalization for image recognition, data augmentation strategies, e.g., random-size crop, are adopted during training \\cite{DBLP:conf\/cvpr\/HeZRS16,DBLP:journals\/corr\/abs-1812-01187,DBLP:conf\/cvpr\/SzegedyLJSRAEVR15}. Besides, common models are usually trained and tested with fixed-resolution inputs. \\cite{DBLP:journals\/corr\/abs-1906-06423} shows that for a model trained with the default $224\\times224$ resolution and tested at lower resolutions, the accuracy quickly deteriorates (e.g., drops 11.6\\% at test resolution $128\\times128$ on ResNet50). \n\n\\subsubsection{Accuracy-Efficiency Trade-Off.} \n\\vskip-1em There have been many attempts to balance accuracy and efficiency by model scaling. Some of them adjust the structural configurations of networks. For example, \nResNets \\cite{DBLP:conf\/cvpr\/HeZRS16} provide several choices of network depths from shallower to deep. MobileNets \\cite{DBLP:journals\/corr\/HowardZCKWWAA17,DBLP:conf\/cvpr\/SandlerHZZC18} and ShuffleNets \\cite{DBLP:conf\/cvpr\/ZhangZLS18} can reduce network widths by using smaller width multipliers. While some other works \\cite{DBLP:journals\/corr\/abs-1905-02244,DBLP:journals\/corr\/HowardZCKWWAA17,DBLP:journals\/corr\/abs-1908-03888,DBLP:conf\/cvpr\/SandlerHZZC18} reduce the computational complexity by decreasing image resolutions at input, which is also our focus. Modifying the resolution usually does not make changes to the number of network parameters, but significantly affects the computational complexity \\cite{DBLP:journals\/corr\/HowardZCKWWAA17}. \n\n\n\\subsubsection{Knowledge Distillation.} \n\\vskip-1em \nA student network can be improved by imitating feature representations or soft targets of a larger teacher network \\cite{DBLP:journals\/corr\/HintonVD15,DBLP:conf\/eccv\/LiH16,DBLP:conf\/cvpr\/ZhangXHL18}. The teacher is usually pre-trained beforehand and fixed, and the knowledge is transferred in one direction \\cite{DBLP:journals\/corr\/RomeroBKCGB14}. Yet \\cite{DBLP:conf\/cvpr\/ZhangXHL18} introduces a two-way transfer between two peer models. \\cite{DBLP:conf\/cvpr\/SunYZZ19} performs mutual learning within one single network assisted by intermediate classifiers. \\cite{DBLP:conf\/nips\/LanZG18} learns a native ensemble design based on multiple models for distillation. \\cite{DBLP:journals\/corr\/abs-1903-05134} conducts the knowledge distillation between the whole model and each split smaller model. Regarding the supervised image recognition, existing distillation works rely on different models, usually needing another teacher network with higher-capacity than low-capacity students. While our design is applied in a shared model, which is data-driven, collecting complementary knowledge from the same image instances with different resolutions.\n\n\\section{Introduction}\n\\label{intro}\nConvolutional Neural Networks (CNNs) have achieved great success on image recognition tasks \\cite{DBLP:conf\/cvpr\/HeZRS16,DBLP:conf\/nips\/KrizhevskySH12}, and well-trained recognition models usually need to be deployed on mobile phones, robots or autonomous vehicles \\cite{DBLP:conf\/iclr\/CaiZH19,DBLP:journals\/corr\/HowardZCKWWAA17}. To fit the resource constraints of devices, extensive research efforts have been devoted to balancing between accuracy and efficiency, by reducing computational complexities of models. Some of these methods adjust the structural configurations of networks, e.g., by adjusting the network depths \\cite{DBLP:conf\/cvpr\/HeZRS16}, widths \\cite{DBLP:journals\/corr\/HowardZCKWWAA17,DBLP:conf\/iclr\/YuYXYH19} or the convolutional blocks \\cite{DBLP:conf\/eccv\/MaZZS18,DBLP:conf\/cvpr\/ZhangZLS18}. Besides that, adjusting the image resolution is another widely-used method for the accuracy-efficiency trade-off \\cite{DBLP:journals\/corr\/abs-1905-02244,DBLP:journals\/corr\/HowardZCKWWAA17,DBLP:journals\/corr\/abs-1908-03888,DBLP:conf\/cvpr\/SandlerHZZC18}. If input images are downsized, all feature resolutions at different convolutional layers are reduced subsequently with the same ratio, and the computational cost of a model is nearly proportional to the image resolution ($H\\times W$) \\cite{DBLP:journals\/corr\/HowardZCKWWAA17}. However, for a common image recognition model, when the test image resolution differs from the resolution used for training, the accuracy quickly deteriorates \\cite{DBLP:journals\/corr\/abs-1906-06423}. To address this issue, existing works \\cite{DBLP:journals\/corr\/abs-1905-02244,DBLP:conf\/cvpr\/SandlerHZZC18,DBLP:conf\/icml\/TanL19} train an individual model for each resolution. As a result, the total number of models to be trained and saved is proportional to the amount of resolutions considered at runtime. Besides the high storage costs, each time adjusting the resolution is accompanied with the additional latency to load another model which is trained with the target resolution.\n\nThe ability to switch the image resolution at inference meets a common need for real-life model deployments. By switching resolutions, the running speeds and costs are adjustable to flexibly handle the real-time latency and power requirements for different application scenarios or workloads. Besides, the flexible latency compatibility allows such model to be deployed on a wide range of resource-constrained platforms, which is friendly for application developers. In this paper, we focus on switching input resolutions for an image recognition model, and propose a general and economic method to improve overall accuracies. Models trained with our method are called \\textbf{Resolution Switchable Networks (RS-Nets)}. Our contribution is composed of three parts.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.5]{pics\/mflops.pdf}\n\\vskip-0.8em\n\\label{mflops_all_nets}\n\\caption{ImageNet accuracy vs. FLOPs (Multiply-Adds) of our \\textbf{single} models and the corresponding \\textbf{sets} of individual models. A single RS-Net model is executable at each of the resolutions, and even achieves significantly higher accuracies than individual models. The results of two state-of-the-art switchable networks (switch by varying network widths) S-MobileNetV2 \\cite{DBLP:conf\/iclr\/YuYXYH19} and US-MobileNetV2 \\cite{DBLP:journals\/corr\/abs-1903-05134} are provided for comparison.\nDetails are in Table \\ref{tabs:main_results} and Table \\ref{tabs:mflops}.} \n\\vskip-1.4em\n\\end{figure}\n\n\nFirst, we propose a parallel training framework where images with different resolutions are trained within a single model. As the resolution difference usually leads to the difference of activation statistics in a network \\cite{DBLP:journals\/corr\/abs-1906-06423}, we adopt shared network parameters but privatized Batch Normalization layers (BNs) \\cite{DBLP:conf\/icml\/IoffeS15} for each resolution. Switching BNs enables the model to flexibly switch image resolutions, without needing to adjust other network parameters.\n\nSecond, we associate the multi-resolution interaction effects with a kind of train-test discrepancy (details in Section \\ref{sec:discrepancy}). Both our analysis and empirical results reach an interesting conclusion that the parallel training framework tends to enlarge the accuracy gaps over different resolutions. On the one hand, accuracy promotions at high resolutions make a stronger teacher potentially available. On the other hand, the accuracy drop at the lower resolution indicates that the benefits of parallel training itself are limited. Both reasons encourage us to further propose a design of ensemble distillation to improve overall performance.\n\nThird, to the best of our knowledge, we are the first to propose that a data-driven ensemble distillation can be learnt on the fly for image recognition, based on the same image instances with different resolutions. Regarding the supervised image recognition, the structure of our design is also different from existing ensemble or knowledge distillation works, as they focus on the knowledge transfer among different models, e.g., by stacking multiple models \\cite{DBLP:conf\/cvpr\/ZhangXHL18}, multiple branches \\cite{DBLP:conf\/nips\/LanZG18}, pre-training a teacher model, or splitting the model into sub-models \\cite{DBLP:journals\/corr\/abs-1903-05134}, while our model is single and shared, with little extra parameters.\n\nExtensive experiments on the ImageNet dataset validate that RS-Nets are executable given different image resolutions at runtime, and achieve significant accuracy improvements at a wide range of resolutions compared with individually trained models. Illustrative results are provided in Fig. \\ref{mflops_all_nets}, which also verify that our proposed method can be generally applied to modern recognition backbones.\n\n\\section{Proposed Method}\n\\vskip-0.1em\nA schematic overview of our proposed method is shown in Fig. \\ref{framework}. In this section, we detail the insights and formulations of parallel training, interaction effects and ensemble distillation based on the multi-resolution setting.\n\n\\subsection{Multi-Resolution Parallel Training}\n\\label{sec:parallel_training}\n\nTo make the description self-contained, we begin with the basic training of a CNN model. Given training samples, we crop and resize each sample to a fixed-resolution image $\\bm{x}^i$. We denote network inputs as $\\{(\\bm{x}^i,y^i)|i\\in\\{1,2,\\cdots,N\\}\\}$, where $y^i$ is the ground truth which belongs to one of the $C$ classes, and $N$ is the amount of samples. Given network configurations with parameters $\\bm{\\theta}$, the predicted probability of the class $c$ is denoted as $p(c|\\bm{x}^i, \\bm{\\theta})$. The model is optimized with a cross-entropy loss defined as:\n\\vskip-0.3em\n\\begin{equation}\n\\label{ce}\n\\mathcal{H}(\\bm{x},\\bm{y})=-\\frac{1}{N}\\sum_{i=1}^N\\sum_{c=1}^C\\delta(c,y^i)\\log\\big(p(c|\\bm{x}^i, \\bm{\\theta})\\big),\n\\end{equation}\nwhere $\\delta(c,y^i)$ equals to $1$ when $c=y^i$, otherwise $0$.\n\n\t\n\\begin{figure}[t]\n\\centering\n\\hskip-0.3em\n\\resizebox{\\linewidth}{!}{\n\\includegraphics[scale=0.42]{pics\/framework_frame.pdf}}\n\\caption{Overall framework of training a RS-Net. Images with different resolutions are trained in parallel with shared Conv\/FC layers and private BNs. The ensemble logit ($\\bm{z}_0$) is learnt on the fly as a weighted mean of logits ($\\bm{z}_1,\\bm{z}_2,\\cdots,\\bm{z}_S$), shown as green arrows. Knowledge distillations are shown as red arrows. For inference, one of the $S$ forward paths is selected (according to the image resolution), with its corresponding BNs, for obtaining its corresponding prediction $\\bm{p}_s, s\\in\\{1,2,\\cdots,S\\}$. \\emph{The ensemble and knowledge distillation are not needed during inference.}}\n\\label{framework}\n\\vskip-0.7em\n\\end{figure}\n\nIn this part, we propose \\textbf{multi-resolution parallel training}, or called \\textbf{parallel training} for brevity, to train a single model which can switch image resolutions at runtime. During training, each image sample is randomly cropped and resized to several duplicate images with different resolutions. Suppose that there are $S$ resolutions in total, the inputs can be written as $\\{(\\bm{x}_1^i, \\bm{x}_2^i, \\cdots, \\bm{x}_S^i, y^i)|i\\in\\{1,2,\\cdots,N\\}\\}$. Recent CNNs for image recognition follow similar structures that all stack Convolutional (Conv) layers, a Global Average Pooling (GAP) layer and a Fully-Connected (FC) layer. In CNNs, if input images have different resolutions, the corresponding feature maps in all Conv layers will also vary in resolution. Thanks to GAP, features are transformed to a unified spatial dimension ($1\\times1$) with equal amount of channels, making it possible to be followed by a same FC layer. During our parallel training, we share parameters of Conv layers and the FC layer, and therefore the training for multiple resolutions can be realized in a single network. The loss function for parallel training is calculated as a summation of the cross-entropy losses:\n\\vskip-2mm\n\\begin{equation}\\label{loss_cls}\n\\mathcal{L}_{cls}=\\sum_{s=1}^S \\mathcal{H}(\\bm{x}_s,\\bm{y}).\n\\end{equation}\n\nSpecializing Batch Normalization layers (BNs) \\cite{DBLP:conf\/icml\/IoffeS15} is proved to be effective for efficient model adaption \\cite{DBLP:conf\/cvpr\/ChangYSKH19,DBLP:conf\/iclr\/MudrakartaSZH19,DBLP:conf\/iclr\/YuYXYH19,DBLP:journals\/corr\/abs-1909-00182}. In image recognition tasks, resizing image results in different activation statistics in a network \\cite{DBLP:journals\/corr\/abs-1906-06423}, including means and variances used in BNs. Thus during parallel training, we privatize BNs for each resolution. Results in the left panel of Fig. \\ref{share} verify the necessity of privatizing BNs. For the $s^{\\text{th}}$ resolution, each corresponding BN layer normalizes the channel-wise feature as follows:\n\\begin{equation}\\label{sbn}\n\\bm{y}'_s=\\bm{\\gamma}_s\\frac{\\bm{y}_s-\\bm{\\mu}_s}{\\sqrt{\\bm{\\sigma}_s^2+\\epsilon}}+\\bm{\\beta}_s, s\\in\\{1,2,\\cdots,S\\},\n\\end{equation}\n\nwhere $\\bm{\\mu}_s$ and $\\bm{\\sigma}_s^2$ are running mean and variance; $\\bm{\\gamma}_s$ and $\\bm{\\beta}_s$ are learnable scale and bias. Switching these parameters enables the model to switch resolutions.\n\n\n\\subsection{Multi-Resolution Interaction Effects}\n\\label{sec:discrepancy}\n\n\n\\begin{figure}[t]\n\\centering\n\\resizebox{115mm}{!}{\n\\includegraphics[scale=0.13]{pics\/crop_224_96.pdf}}\n\\vskip-0.5em\n\\caption{\\textbf{Left:} An illustration of interaction effects for the parallel training with two resolutions. Each red box indicates the region to be cropped, and the size of each blue dotted box is the apparent object (in this sample is a cup) size. For this example, in either one of the models, the apparent object size at testing is smaller than at training. \\cite{DBLP:journals\/corr\/abs-1906-06423} reveals that this relation still holds when averaging all data, which is called the train-test discrepancy. The data pre-processing for training or testing follows the standard image recognition method, which will be described in Section \\ref{details}. \\textbf{Right:} CDF curves for comparing the value distributions of feature activations. All curves are plotted on the validation dataset of ImageNet, but are based on different data pre-processing methods as annotated by (train) or (test).}\n\\label{pic_crop}\n\\vskip-1.4em\n\\end{figure}\n\n\nIn this section, restricted to large-scale image datasets with fine-resolution images, we analyze the interaction effects of different resolutions under the parallel training framework. We start by posing a question: compared with individually trained models, how does parallel training affect test accuracies at different resolutions? As multi-resolution can be seen as a kind of data augmentation, we analyze from two aspects as follows.\n \n\nThe first aspect is straightforward. The model meets a wide range of image resolutions, which improves the generalization and reduces over-fitting. Thus if the setting of resolutions is suitable (e.g., not too diverse), the parallel training tends to bring overall accuracy gains at testing, especially for a high-capacity network such as ResNet50.\n\n\nThe second aspect is based on the specialty of large-scale image recognition tasks, where objects of interest randomly occupy different portions of image areas and thus the random-size crop augmentation is used during training. \\cite{DBLP:journals\/corr\/abs-1906-06423} reveals that for common recognition works, as the random-size crop is used for training but not for testing, there exists a train-test \\textbf{discrepancy} that the average ``apparent object size\" at testing is smaller than that at training. Besides, \\cite{DBLP:journals\/corr\/abs-1906-06423} achieves accuracy improvements by alleviating such discrepancy, but is accompanied with the costs of test-time resolution augmentations and finetuning. Note that we do not aim to modify the data pre-processing method to alleviate the discrepancy. Instead, we are inspired to use the concept of the discrepancy to analyze multi-resolution interaction effects. We take the parallel training with two resolutions $224\\times224$ and $96\\times96$ for example. According to the left panel of Fig. \\ref{pic_crop} (the analysis in colors), compared with the model using only $224\\times224$ images, the parallel training can be seen as augmenting $96\\times96$ images, which reduces the average apparent object size at training and thus alleviates the discrepancy. On the contrary, compared with the individual model using $96\\times96$ images, augmenting $224\\times224$ images increases the discrepancy. Thus in this aspect, this parallel training tends to increase the test accuracy at $224\\times224$ (actually +1.6\\% for ResNet18, as shown in Fig. \\ref{parallel}), while tends to reduce the test accuracy at $96\\times96$ (-0.8\\%). The right panel of Fig. \\ref{pic_crop} plots the Cumulative Distribution Function (CDF)\\footnote{The CDF of a random variable $X$ is defined as $F_X(x)=P(X\\le x)$, for all $x\\in \\mathbb{R}$.} of output components of the Global Average Pooling (GAP) layer for a well-trained ResNet18 (as a ReLU layer is before the GAP, all components are nonnegative). We plot CDF to compare the value distributions of feature activations when using training or testing data pre-processing method. The parallel training seems to narrow the train-test gap for the $224\\times224$ model, but widen the gap for the $96\\times96$ model.\n\n\nWe take ResNet18 as an example and summarize the two aforementioned aspects. For the parallel training with two resolutions, the test accuracy at the high resolution increases compared with its individual model, as the two aspects reach an agreement to a large degree. As for the lower resolution, we find that the accuracy slightly increases if the two resolutions are close, otherwise decreases. Similarly, when multiple resolutions are used for parallel training, test accuracies increase at high resolutions but may decrease at lower resolutions, compared with individual models. Results in Table \\ref{tabs:main_results} show that for the parallel training with five resolutions, accuracies only decrease at $96\\times96$ but increase at the other four resolutions. Detailed results in Fig. \\ref{parallel} also verify our analysis.\n\nFor image recognition, although testing at a high resolution already tends to achieve a good accuracy, using the parallel training makes it even better. This finding opens up a possibility that a stronger teacher may be available in this framework, and seeking a design based on such teacher could be highly effective.\n\n\n\n\\subsection{Multi-Resolution Ensemble Distillation}\n\\label{mred}\n\nIn this section, we propose a new design of ensemble distillation. Regarding the supervised image recognition, unlike conventional  distillation works that rely on transferring knowledge among different models, ours is data-driven and can be applied in a shared model. Specifically, our design is learnt on the fly and the distillation is based on the same image instances with different resolutions.\n\n\n\\input{tabs\/samples.tex}\n\nAs is commonly known, for image recognition tasks, models given a high resolution image are easy to capture fine-grained patterns, and thus achieve good performance \\cite{DBLP:conf\/cvpr\/SzegedyVISW16,DBLP:conf\/icml\/TanL19}. However, according to the sample statistics in the middle column of Table \\ref{tab_proportion}, we find that there always exists a proportion of samples which are correctly classified at a low resolution but wrongly classified at another higher resolution. Such results indicate that model predictions at different image resolutions are complementary, and not always the higher resolution is better for each image sample. Therefore, we propose to learn a teacher on the fly as an ensemble of the predictions w.r.t. all resolutions, and conduct knowledge distillation to improve the overall performance. Our design is called \\textbf{Multi-Resolution Ensemble Distillation (MRED)}. \n\n\nDuring the training process of image recognition, for each input image $\\bm{x}^i$, the probability of the class $c$ is calculated using a softmax function:\\vskip-0.2em\n\\begin{equation}\\label{p}\np(c|\\bm{x}^i, \\bm{\\theta})=p(c|\\bm{z}^i)=\\frac{\\exp(z_c^i)}{\\sum_{j=1}^{C}\\exp(z_j^i)}, c\\in\\{1,2,\\cdots,C\\},\n\\end{equation}\nwhere $\\bm{z}^i$ is the logit, the unnormalized log probability outputted by the network, and probabilities over all classes can be denoted as the model prediction $\\bm{p}$.\n\nIn the parallel training framework, each image is randomly cropped and resized to $S$ images with different resolutions. To better benefit from MRED, these $S$ images need to be resized from a same random crop, as illustrated in the left-most part of Fig. \\ref{framework}. The necessity will be verified in Section \\ref{veri_kd}.\n\nAs each image sample is resized to $S$ resolutions, there are $S$ corresponding logits $\\bm{z}_1,\\bm{z}_2,\\cdots,\\bm{z}_S$. We learn a group of importance scores $\\bm{\\alpha}\\!=[\\alpha_1\\,\\alpha_2\\,\\cdots\\,\\alpha_S]$, satisfying $\\bm{\\alpha}\\!\\ge\\!0$, $\\sum_{s=1}^S\\alpha_s\\!=\\!1$, which can be easily implemented with a softmax function. We then calculate an ensemble logit $\\bm{z}_0$ as the weighted summation of the $S$ logits:\n\\begin{equation}\\label{ens_logits}\n\\hskip-0.8em\n\\bm{z}_0=\\small{\\sum_{s=1}^S}\\alpha_s \\bm{z}_s.\n\\end{equation}\n\nTo optimize $\\bm{\\alpha}$, we temporally froze the gradients of the logits $\\bm{z}_1,\\bm{z}_2,\\cdots,\\bm{z}_S$. Based on the ensemble logit $\\bm{z}_0$, the corresponding prediction $\\bm{p}_0$, called ensemble prediction,  can be calculated via Eq. \\ref{p}. Then $\\bm{\\alpha}$ is optimized using a cross-entropy loss between $\\bm{p}_0$ and the ground truth, which we call the ensemble loss $\\mathcal{L}_{ens}$:\n\\begin{equation}\\label{loss_ens}\n\\mathcal{L}_{ens}=-\\frac{1}{N}\\sum_{i=1}^N\\sum_{c=1}^C\\delta(c,y^i)\\log\\big(p(c|\\bm{z}_0^i)\\big).\n\\end{equation}\n\n\nIn knowledge distillation works, to quantify the alignment between a teacher prediction $\\bm{p}_t$ and a student prediction $\\bm{p}_s$, Kullback Leibler (KL) divergence is usually used:\n\\begin{equation}\\label{kl}\n\\mathcal{D}_{kl}\\big(\\bm{p}_t\\|\\bm{p}_s\\big)=\\frac{1}{N}\\sum_{i=1}^N\\sum_{c=1}^C p(c|\\bm{z}_t^i)\\log\\frac{p(c|\\bm{z}_t^i)}{p(c|\\bm{z}_s^i)}.\n\\end{equation}\n\n\nWe force predications at different resolutions to mimic the learnt ensemble prediction $\\bm{p}_0$, and thus the distillation loss $\\mathcal{L}_{dis}$ could be obtained as:\n\\begin{equation}\\label{loss_kd_v1}\n\\mathcal{L}_{dis}=\\sum_{s=1}^S\\mathcal{D}_{kl}\\big(\\bm{p}_0\\|\\bm{p}_s\\big).\n\\end{equation}\n\nFinally, the overall loss function is a summation of the classification loss, the ensemble loss and the distillation loss, without needing to tune any extra weighted parameters:\n\\begin{equation}\\label{loss}\n\\mathcal{L}=\\mathcal{L}_{cls}+\\mathcal{L}_{ens}+\\mathcal{L}_{dis},\n\\end{equation}\nwhere in practical, optimizing $\\mathcal{L}_{ens}$ only updates $\\bm{\\alpha}$, with all network weights temporally frozen; optimizing $\\mathcal{L}_{cls}$ and $\\mathcal{L}_{dis}$ updates network weights. \n\nWe denote the method with Eq. \\ref{loss_kd_v1} as our vanilla-version MRED. Under the parallel training framework, as the accuracy at a high resolution is usually better than at a lower resolution, accuracies can be further improved by offering dense guidance from predications at high resolutions toward predictions at lower resolutions. Thus the distillation loss can be extended to be a generalized one:\\vskip-0.2em\n\\begin{equation}\\label{loss_kd_v2}\n\\mathcal{L}_{dis}=\\frac{2}{S+1}\\sum_{t=0}^{S-1}\\sum_{s=t+1}^S\\mathcal{D}_{kl}\\big(\\bm{p}_t\\|\\bm{p}_s\\big),\n\\end{equation}\nwhere the index $t$ starts from $0$ referring to the ensemble term; as the summation results in $S(S+1)\/2$ components in total, we multiply $\\mathcal{L}_{dis}$ by a constant ratio $2\/(S+1)$ to keep its range the same as $\\mathcal{L}_{cls}$. We denote the method with Eq. \\ref{loss_kd_v2} as our full-version MRED, which is used in our experiments by default. \n\n\nThe proposed design involves negligible extra parameters (only $S$ scalars), without needing extra models. Models trained with the parallel training framework and MRED are named Resolution Switchable Networks (RS-Nets). An overall framework of training a RS-Net is illustrated in Fig. \\ref{framework}. \\textbf{During inference}, the network only performs one forward calculation at a given resolution, without ensemble or distillation, and thus both the computational complexity and the amount of parameters equal to a conventional image recognition model. \n\n\n\\section{Experiments}\nWe perform experiments on ImageNet (ILSVRC12) \\cite{DBLP:conf\/cvpr\/DengDSLL009,DBLP:journals\/ijcv\/RussakovskyDSKS15}, a widely-used image recognition dataset containing about 1.2 million training images and 50 thousand validation images, where each image is annotated as one of 1000 categories. Experiments are conducted with prevailing CNN architectures  including a lightweight model MobileNetV2 \\cite{DBLP:conf\/cvpr\/SandlerHZZC18} and ResNets \\cite{DBLP:conf\/cvpr\/HeZRS16}, where a basic-block model ResNet18 and a bottleneck-block model ResNet50 are both considered. Besides, we also evaluate our method in handling network quantization problems, where we consider different kinds of bit-widths.\n\n\n\\subsection{Implementation Details}\n\\label{details}\nOur basic experiments are implemented with PyTorch$\\,$\\cite{DBLP:conf\/nips\/PaszkeGMLBCKLGA19}.$\\,$For quantization tasks, we apply our method to LQ-Nets \\cite{DBLP:conf\/eccv\/ZhangYYH18} which show state-of-the-art performance in training CNNs with low-precision weights or both weights and activations. \n\nWe set $\\mathbb{S}=\\{224\\times224,192\\times192,160\\times160,128\\times128,96\\times96\\}$, as commonly adopted in a number of existing works \\cite{DBLP:journals\/corr\/HowardZCKWWAA17,DBLP:journals\/corr\/abs-1908-03888,DBLP:conf\/cvpr\/SandlerHZZC18}. During training, we pre-process the data for augmentation with an area ratio ($\\text{cropped area}\/\\text{original area}$) uniformly sampled in $[0.08, 1.0]$, an aspect ratio $[3\/4, 4\/3]$ and a horizontal flipping. We resize images with the bilinear interpolation. Note that both $[0.08, 1.0]$ and $[3\/4, 4\/3]$ follow the standard data augmentation strategies for ImageNet \\cite{DBLP:conf\/cvpr\/HuangLMW17,DBLP:conf\/cvpr\/SzegedyLJSRAEVR15,DBLP:journals\/corr\/abs-1906-06423}, e.g., \\emph{RandomResizedCrop} in PyTorch uses such setting as default. During validation, we first resize images with the bilinear interpolation to every resolution in $\\mathbb{S}$ divided by 0.875 \\cite{DBLP:journals\/corr\/abs-1908-08986,DBLP:journals\/corr\/abs-1908-03888}, and then feed central regions to models. \n\n\nNetworks are trained from scratch with random initializations. We set the batch size to 256, and use a SGD optimizer with a momentum 0.9. For standard ResNets, we train 120 epochs and the learning rate is annealed from 0.1 to 0 with a cosine scheduling \\cite{DBLP:journals\/corr\/abs-1812-01187}. For MobileNetV2, we train 150 epochs and the learning rate is annealed from 0.05 to 0 with a cosine scheduling. For quantized ResNets, we follow the settings in LQ-Nets \\cite{DBLP:conf\/eccv\/ZhangYYH18}, which train 120 epochs and the learning rate is initialized to 0.1 and divided by 10 at 30, 60, 85, 95, 105 epochs. The weight decay rate is set to 1e-4 for all ResNets and 4e-5 for MobileNetV2.\n\n\n\\subsection{Results}\nAs mentioned in Section \\ref{intro}, common works with multi-resolution settings train and deploy multiple individual models separately for different resolutions. We denote these individual models as \\textbf{I-Nets}, which are set as baselines. We use I-\\{resolution\\} to represent each individual model, e.g., I-$224$.\n\n\n\\input{tabs\/main_results.tex}\n\n\\subsubsection{Basic Results.} \n\\vskip-0.3em\nIn Table \\ref{tabs:main_results}, we report results on ResNet18, ResNet50 and MobileNetV2 (M-NetV2 for short). Besides I-Nets, we also report accuracies at five resolutions using the individual model which is trained with the largest resolution (I-$224$). For our proposed method, we provide separate results of the parallel training (parallel) and the overall design (RS-Net). As mentioned in Section \\ref{intro}, I-Nets need several times of parameter amount and high latencies for switching across models. We also cannot rely on an individual model to switch the image resolutions, as accuracies of I-$224$ are much lower than I-Nets at other resolutions (e.g., 15\\%$\\sim$20\\% accuracy drop at the resolution $96\\times96$). Similarly, each of the other individual models also suffers from serious accuracy drops, as can be seen in the right panel of Fig. \\ref{share}. Our parallel training brings accuracy improvements at the four larger resolutions, while accuracies at $96\\times96$ decrease, and the reason is previously analyzed in Section \\ref{sec:discrepancy}. Compared with I-Nets, our RS-Net achieves large improvements at all resolutions with only 1$\/$5 parameters. For example, the RS-Net with ResNet50 obtains about 2.4\\% absolute top-1 accuracy gains on average across five resolutions. Note that the number of FLOPs (Multiply-Adds) is nearly proportional to the image resolution \\cite{DBLP:journals\/corr\/HowardZCKWWAA17}. Regarding ResNet18 and ResNet50, accuracies at $160\\times160$ of our RS-Nets even surpass the accuracies of I-Nets at $224\\times224$, significantly reducing about 49\\% FLOPs at runtime. Similarly, for MobileNetV2, the accuracy at $192\\times192$ of RS-Net surpasses the accuracy of I-Nets at $224\\times224$, reducing about 26\\% FLOPs.\n\n\n\\input{tabs\/quanti_results.tex}\n\\input{tabs\/mflops.tex}\n\n\\subsubsection{Quantization.}\n\\vskip -1em\nWe further explore the generalization of our method to more challenging quantization problems, and we apply our method to LQ-Nets \\cite{DBLP:conf\/eccv\/ZhangYYH18}. Experiments are performed under two typical kinds of quantization settings, including the quantization on weights (2$\/$32) and the more extremely compressed quantization on both weights and activations (2$\/$2). Results of I-Nets and each RS-Net based on quantized ResNets are reported in Table \\ref{tabs:quan_results}. Again, each RS-Net outperforms the corresponding I-Nets at all resolutions. For quantization problems, as I-Nets cannot force the quantized parameter values of each individual model to be the same, 2-bit weights in I-Nets are practically stored with more digits than a 4-bit model\\footnote{The total bit number of five models with individual 2-bit weights is $\\log_2(2^2\\times5)\\approx4.3$.}, while the RS-Net avoids such issue. For quantization, accuracy gains of RS-Net to ResNet50 are more obvious than those to ResNet18, we conjecture that under compressed conditions, ResNet50 better bridges the network capacity and the augmented data resolutions. \n\n\n\n\n\\subsubsection{Switchable Models Comparison.} \n\\vskip -1em\nBased on MobileNetV2, results in Table \\ref{tabs:mflops} indicate that under comparable FLOPs (Multiply-Adds), adjusting image resolutions (see I-Nets-r) achieves higher accuracies than adjusting network widths (see I-Nets-w). For example, adjusting resolution to $96\\times96$ brings 2.5\\% higher absolute accuracy than adjusting width to $0.35\\times$, with even lower FLOPs. Results of S-MobileNetV2 (S) \\cite{DBLP:conf\/iclr\/YuYXYH19} and US-MobileNetV2 (US) \\cite{DBLP:journals\/corr\/abs-1903-05134} are also provided for comparison. As we can see, our RS-Net significantly outperforms S and US at all given FLOPs, achieving 1.5\\%$\\sim$4.4\\% absolute gains. Although both S and US can also adjust the accuracy-efficiency trade-off, they have marginal gains or even accuracy drops (e.g., at width $1.0\\times$) compared with their baseline I-Nets-w. Our model shows large gains compared with our baseline I-Nets-r (even they are mostly stronger than I-Nets-w). Note that adjusting resolutions does not conflict with adjusting widths, and both methods can be potentially combined together.\n\n\n\n\\begin{figure}[t]\n\\hskip-0.1em\n\\includegraphics[scale=0.39]{pics\/basic_ablation.pdf}\n\\vskip -1em\n\\caption{\\textbf{Left:} Comparison of parallel trainings with shared BNs and unshared (private) BNs, based on MobileNetV2. Individual models and Parallel (unshared BNs) + MRED (i.e., RS-Net) are provided for reference. \\textbf{Right:} Comparison of our RS-Net and each individual model (from I-$224$ to I-$96$) tested at denser resolutions (the interval is 16), based on ResNet18. Each individual model suffers from serious accuracy drops at other resolutions, but our RS-Net avoids this issue.}\n\\label{share}\n\\vskip -0.5em\n\\end{figure}\n\n\\subsection{Ablation Study}\n\n\\subsubsection{Importance of Using Private BNs.}\nA quick question is that, why not share BNs as well? Based on MobileNetV2, the left panel of Fig. \\ref{share} shows that during the parallel training, privatizing BNs achieves higher accuracies than sharing BNs, especially at both the highest resolution (+1.7\\%) and the lowest (+2.3\\%). When BNs are shared, activation statistics of different image resolutions are averaged, which differ from the real statistics especially at two ends of resolutions.\n\n\n\\subsubsection{Tested at New Resolutions.} \n\\vskip -1em\nOne may concern that if a RS-Net can be tested at a new resolution. In the right panel of Fig. \\ref{share}, we test models at different resolutions with a smaller interval, using ResNet18. We follow a simple method to handle a resolution which is not involved in training. Suppose the resolution is sandwiched between two of the five training resolutions which correspond to two groups of BN parameters, we apply a linear interpolation on these two groups and obtain a new group of parameters, which are used for the given resolution. We observe that the RS-Net maintains high accuracies. RS-Net suppresses the serious accuracy drops which exist in every individual model. \n\n\n\\begin{figure}[t]\n\\centering\n\\hskip-0.3em\\includegraphics[scale=0.36]{pics\/joint_all.pdf}\n\\vskip-0.6em\n\\caption{Absolute top-1 accuracy variations (\\%) (compared with individual models) of parallel trainings, based on ResNet18. The result of each individual model (from I-$96$ to I-$224$) is used as the baseline. We use single numbers to represent the image resolutions.}\n\\label{parallel}\n\\vskip-1em\n\\end{figure}\n\n\n\\subsubsection{Multi-Resolution Interaction Effects.} \n\\vskip-0.7em\nThis part is for verifying the analysis in Section \\ref{sec:discrepancy}, and we do not apply MRED here. Parallel training results of ResNet18 are provided in Fig. \\ref{parallel}, which show top-1 accuracy variations compared with each individual model. The left panel of Fig. \\ref{parallel} illustrates the parallel training with two resolutions, where all accuracies increase at $224\\times224$. As the gap of the two resolutions increases, the accuracy variation at the lower resolution decreases. For example, compared with I-$224$ and I-$96$ respectively, the parallel training with $224\\times224$ and $96\\times96$ images increases the accuracy at $224\\times224$ from 71.0\\% to 72.6\\%, but decreases the accuracy at $96\\times96$ from 62.6\\% to 61.8\\%. The right panel of Fig. \\ref{parallel} illustrates the parallel training with multiple resolutions, where we observe that most accuracies are improved and accuracies at the lowest resolutions may decrease. These results verify our analysis. \n\n\n\n\\subsubsection{Verification of Ensemble Distillation.} \n\\vskip-0.7em\n\\label{veri_kd}\nIn Section \\ref{mred}, we define two versions of MRED. The vanilla-version has only the distillation paths starting from the ensemble prediction $\\bm{p}_0$ toward all the other predictions, while the full-version has additional paths from predications at high resolutions toward predictions at lower resolutions. In Table \\ref{tabs:kd_results}, we compare the performance of the two versions as well as two other variants that omit the distillations from $\\bm{p}_0$, based on ResNet18. Results indicate that all kinds of the proposed distillations are indispensable. Besides, we also emphasize in Section \\ref{mred} that for training a RS-Net, each image should be randomly cropped only once (called single-crop) and then resized to multiple resolutions (called multi-resolution). Results in Table \\ref{tabs:kd_crop_results} indicate that using multi-crop (applying a random crop individually for each resolution) will weaken the benefits of MRED, as accuracies at low resolutions are lower compared with using single-crop. We also verify the importance of multi-resolution by replacing the multi-resolution setting by five identical resolutions, called single-resolution. As resolutions are identical, an individual experiment is needed for each resolution. Results indicate that applying ensemble distillation to predictions of different crops has very limited benefits compared with applying it to predictions of different resolutions.\n\n\n\\input{tabs\/kd.tex}\n\\input{tabs\/kd_crop.tex}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\t\nThroughout, let $n$ be a positive integer, let $[n]$ denote the set $\\{1,2,\\dots,n\\}$, and let\n$B_n$ be the power set of $[n]$, ordered by inclusion. An \\emph{antichain} is a family of subsets\nof $[n]$ none of which contains any other. An antichain $\\mathcal{A}$ such that no\n$B \\subseteq [n]$ with $B \\not\\in \\mathcal{A}$ can be added to $\\mathcal{A}$ without destroying the\nantichain property is called a \\emph{maximal antichain}.\nBy Sperner's Theorem \\cite{Sperner1928}, any antichain in $B_n$ has at most $\\binom{n}{\\lceil n\/2\\rceil}$ elements.\nThe question studied here is: For which\nintegers $m$ with $1\\leq m\\leq\\binom{n}{\\lceil n\/2\\rceil}$ does there exist a maximal antichain of\nsize $m$?\nIn~\\cite{Griggs2021a}, we answered this question for $m$ close to $\\binom{n}{\\lceil\n  n\/2\\rceil}$, the largest possible size of an antichain.\n  \n\\begin{theorem}[Theorem 1 in~\\cite{Griggs2021a}]\\label{thm:large_sizes}\n  Let $k=\\lceil n\/2\\rceil$, and let $m$ be a positive integer with\n  $\\binom{n}{k}-k^2\\leq m\\leq\\binom{n}{k}$. There exists a maximal antichain of size $m$ in $B_n$ if\n  and only if\n  \\[m=\\binom{n}{k}-tl+\\binom{a}{2}+\\binom{b}{2}+c\\] for some integers $l\\in\\{k,n-k\\}$,\n  $t\\in\\{0,\\dots,k\\}$, $a\\geq b\\geq c\\geq 0$, $1\\leq a+b\\leq t$.\n\\end{theorem}\n\nIn this paper, we complete the solution by constructing maximal antichains of all sizes $m$ less than those \naddressed in Theorem \\ref{thm:large_sizes}. Following \\cite{Griggs2021a} let us introduce the notation\n\\[w(n)=\\binom{n}{\\lceil n\/2 \\rceil} - \\left\\lceil\\frac{n}{2}\\right\\rceil  \\left\\lceil\\frac{n+2}{4}\n\\right\\rceil.\\]\nHere is the answer to our question:\n\n\\begin{theorem}\\label{cor:main_result}\n  Let $m$ be an integer with\n  $1 \\le m \\le \\binom{n}{\\lceil n\/2\\rceil}$. If $m \\le w(n)$, then there is a maximal antichain of size\n  $m$ in $B_n$. For $m>w(n)$, there is a maximal antichain of size $m$ if and only if $m$ is of the\n  form that is stated in Theorem~\\ref{thm:large_sizes}.\n\\end{theorem}\n\nFor given $n$, denote by $\\phi(n)$ the smallest value of $m$ for which there is no maximal antichain of size $m$ in $B_n$.\nIn view of Theorem \\ref{cor:main_result} here and Theorem 3 in \\cite{Griggs2021a} we immediately obtain:\n\n\\begin{theorem}\\label{thm:asymptotics}\nFor $n\\to\\infty$, $\\displaystyle\\phi(n)=\\binom{n}{\\lceil n\/2\\rceil}-\\left(\\frac12+o(1)\\right)n^{3\/2}$.\n\\end{theorem}\n\nLet us mention a related problem of interest. A family of subsets of $[n]$ is \\emph{flat}\\\/ if all sets in the \nfamily have the same size within one. So a flat antichain in $B_n$ is completely contained in some two \nconsecutive levels of $B_n$. For what values $m$ does there exist a maximal antichain in $B_n$ that is flat?\nIn \\cite{Griggs2021a} it is shown that values $m$ covered in Theorem \\ref{thm:large_sizes} (which are large) \nhave this property. Our third paper \\cite{Griggs2021c} will take us down all the way below $n^2$, by constructing \nflat maximal antichains for all possible sizes at least $\\binom n2 - \\lfloor (n+1)^2\/8 \\rfloor$.\nIt turns out that not for all very small sizes flat maximal antichains exist.\n\nThe rest of the paper consists of the proof of our main result, Theorem \\ref{cor:main_result}. \nIn the next section we set up the actual result we need to prove, by induction on $n$.\nIt is helpful to define $S(n)=\\{\\abs{\\mathcal A}\\,:\\,\\mathcal A\\text{ is a maximal antichain in }B_n\\}$.\nIn \\Cref{sec:construction} we present our general construction of maximal antichains on three consecutive\nlevels, and derive that $S(n)$ contains certain intervals. In \\Cref{sec:large_n}, we verify that the\nintervals of sizes obtained from this construction are sufficient for $n\\geq 20$. In\n\\Cref{sec:small_n} we provide an additional construction to fill the gaps for $7\\leq n\\leq 19$.\n\n\n\\section{Inductive reduction}\\label{sec:reduction}\n\nIn view of Theorem \\ref{thm:large_sizes}, to prove Theorem \\ref{cor:main_result} it is enough to construct \nmaximal antichains of all sizes below $\\binom nk-k^2$. Note that this gets below the maximum antichain size\n$\\binom nk$ by $k^2\\sim n^2\/4$. However, for our induction to work nicely, we only go below the maximum\nantichain size by $\\sim n^2\/8$. Specifically, Theorem \\ref{cor:main_result} is implied by this statement:\n\n\\begin{theorem}\\label{thm:main_theorem}\n  Let $m$ be an integer with $1\\leq m\\leq w(n)$.\n  Then there exists a maximal antichain in $B_n$ with size $m$.\n\\end{theorem}\n\nTo prove Theorem \\ref{thm:main_theorem}, it will suffice by induction to construct antichains of larger sizes $m$,\nspecifically consider:\n\n\\begin{claim}\\label{lem:induction_step}\n  If $n\\geq 7$, then $[w(n-1)+2,w(n)]\\subseteq S(n)$.\n\\end{claim}\n\nAssuming this claim, the proof of \\Cref{thm:main_theorem} is easy. In the proof, we shall use\nsome more notation. For a set $M$ and a positive integer $k$ we denote the\nset of all $k$-subsets of $M$ by $\\binom{M}{k}$, and we call $\\binom{[n]}{k}$ the $k$-th level of\n$B_n$. The \\emph{shadow} of a family $\\mathcal F$ of $k$-sets, denoted by $\\Delta\\mathcal F$, is the\nfamily of $(k-1)$-sets which are a subset of some member of $\\mathcal F$. Similarly, the\n\\emph{shade} (or \\emph{upper shadow}) of $\\mathcal F$ (with respect to the fixed ground set $[n]$), denoted by\n$\\nabla\\mathcal F$, is the family of all\n$(k+1)$-subsets of $[n]$ which are a superset of some member of $\\mathcal F$:\n\\begin{align*}\n  \\Delta\\mathcal F &= \\left\\{A\\,:\\,\\abs{A}=k-1,\\,A\\subseteq X\\text{ for some }X\\in\\mathcal F\\right\\},\\\\\n  \\nabla\\mathcal F &= \\left\\{A\\,:\\,A\\subseteq[n],\\,\\abs{A}=k+1,\\,X\\subseteq A\\text{ for some }X\\in\\mathcal F\\right\\}.\n\\end{align*}\n\n\\begin{proof}[Proof of \\Cref{thm:main_theorem} (assuming Claim \\ref{lem:induction_step})]\n  We proceed by induction on $n$. As we want to use Claim \\ref{lem:induction_step} for the induction\n  step, we have to establish the result for $n\\leq 6$ as the base case.\n  For $1\\leq m\\leq n$, a maximal antichain of size $m$ is given by\n  $\\{\\{1\\},\\{2\\},\\dots,\\{m-1\\},\\{m,\\dots,n\\}\\}$. For $n\\leq 5$, this is already sufficient, because\n  then $w(n)\\leq n$. As $w(6)=14$, we need maximal antichains of sizes\n  $m=7,8,\\dots,14$ for $n=6$. Such are given by:\n  \\begin{description}\n  \\item[$m=7$] $\\mathcal A=\\{\\{1,2,3\\},\\{1,2,4\\},\\{1,2,5\\},\\{3,4\\},\\{3,5\\},\\{4,5\\},\\{6\\}\\}$,\n  \\item[$m=8$] $\\mathcal A=\\binom{[4]}{2}\\cup\\{\\{5\\},\\{6\\}\\}$,\n  \\item[$m=9$]\n    $\\mathcal\n    A=\\{\\{1,2,5\\},\\{1,2,6\\},\\{3,4,5\\},\\{3,4,6\\},\\{5,6\\},\\{1,3\\},\\{1,4\\},\\{2,3\\},\\{2,4\\}\\}$,\n  \\item[$m=10$]\n    $\\mathcal\n    A=\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{2,5\\},\\{3,6\\}\\}\\cup\\binom{[6]}{3}\\setminus\\nabla\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{2,5\\},\\{3,6\\}\\}$,\n  \\item[$m=11$]\n    $\\mathcal\n    A=\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{2,5\\},\\{3,5\\}\\}\\cup\\binom{[6]}{3}\\setminus\\nabla\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{2,5\\},\\{3,5\\}\\}$,\n  \\item[$m=12$]\n    $\\mathcal\n    A=\\{\\{1,2\\},\\{2,3\\},\\{4,5\\}\\}\\cup\\binom{[6]}{3}\\setminus\\nabla\\{\\{1,2\\},\\{2,3\\},\\{4,5\\}\\}$,\n  \\item[$m=13$]\n    $\\mathcal\n    A=\\{\\{1,2\\},\\{2,3\\},\\{3,4\\}\\}\\cup\\binom{[6]}{3}\\setminus\\nabla\\{\\{1,2\\},\\{2,3\\},\\{3,4\\}\\}$,\n  \\item[$m=14$]\n    $\\mathcal A=\\{\\{1,2\\},\\{3,4\\}\\}\\cup\\binom{[6]}{3}\\setminus\\nabla\\{\\{1,2\\},\\{3,4\\}\\}$.\n  \\end{description}\n  For the induction step, we assume $n\\geq 7$ and $[1,w(n-1)]\\subseteq S(n-1)$. Noting that\n  $1\\in S(n)$ for every $n$, and adding the singleton $\\{n\\}$ to each of the maximal antichains with\n  sizes $m\\in[1,w(n-1)]$ in $B_{n-1}$, we obtain $[1,w(n-1)+1]\\subseteq S(n)$, and we use \n  Claim \\ref{lem:induction_step} to conclude $[1,w(n)]\\subseteq S(n)$ as required.\n\\end{proof}\n\nThe rest of the paper is devoted to the proof of Claim \\ref{lem:induction_step}.\n\n\n\\section{The main construction}\\label{sec:construction}\nBefore going into the details of the construction we outline the general idea. A crucial role is\nplayed by \\emph{maximal squashed flat antichains}. These are maximal antichains of the form\n$\\mathcal{A} = \\mathcal{I} \\cup\\binom{[n]}{k-1} \\setminus \\Delta \\mathcal{I}$ with\n$\\mathcal{I} \\subseteq \\binom{[n]}{k}$ for some $k$ such that $\\mathcal{I}$ is an initial segment\nof $\\binom{[n]}k$ in \\emph{squashed order} (also known as \\emph{colexicographic order}), i.e.,\n$\\mathcal{I}$ consists of the first $\\abs{\\mathcal{I}}$ sets of $\\binom{[n]}{k}$, where $A \\subseteq [n]$\nprecedes $B \\subseteq [n]$ if $\\max \\left[ (A\\setminus B) \\cup (B \\setminus A)\\right] \\in B$.\nBy the Kruskal-Katona Theorem \\cite{Kruskal1963, Katona1968}, the squashed order is known to solve the\nShadow Minimization Problem for families of $k$-sets.\n\nFrom a maximal squashed flat antichain $\\mathcal A'\\subseteq\\binom{[n]}{k}\\cup\\binom{[n]}{k-1}$ with\n$\\binom{[k+3]}{k}\\subseteq\\mathcal A'$, we can obtain a new maximal antichain $\\mathcal A$ in the\nfollowing way. We replace $\\binom{[k+3]}{k}$ in $\\mathcal{A}'$ by a maximal antichain\n$\\mathcal{F}'\\subseteq \\binom{[k+3]}{k+1}\\cup\\binom{[k+3]}{k}$, where\n$\\mathcal{F}:=\\mathcal{F}'\\cap \\binom{[k+3]}{k+1}$ contains only few sets (about $k\/2$).\nClearly, $\\mathcal{F}'=\\mathcal{F}\\cup\\binom{[k+3]}{k}\\setminus\\Delta\\mathcal{F}$.\nBy results from \\cite{Griggs2021a} about the shadow spectrum, we can vary $\\mathcal F$ without changing\nits cardinality to get $k$ consecutive values for the size of $\\Delta \\mathcal{F}$.\n(For a few small values of $n$ we only get $k-1$ or $k-2$ consecutive shadow sizes,\nand we close the resulting gaps by extra constructions.)\nHence, from every maximal squashed $\\mathcal A'\\subseteq\\binom{[n]}{k}\\cup\\binom{[n]}{k-1}$,\nwe obtain an interval of $k$ consecutive sizes of maximal antichains in $B_n$. We then\nvary $\\mathcal A'$.\nWe say that two distinct maximal squashed antichains\n$\\mathcal{A}_1,\\mathcal{A}_2\\subseteq\\binom{[n]}{k}\\cup\\binom{[n]}{k-1}$ are \\emph{consecutive}\\\/\nif there is no maximal squashed antichain $\\mathcal{A}_3\\subseteq\\binom{[n]}{k}\\cup\\binom{[n]}{k-1}$\nwith $|\\mathcal{A}_1\\cap\\binom{[n]}{k}|< |\\mathcal{A}_3\\cap\\binom{[n]}{k}| < |\\mathcal{A}_2\\cap\\binom{[n]}{k}|$.\n(Note that if $\\mathcal{A}_1,\\mathcal{A}_2$ are consecutive, their sizes might not, i.e., there can be an\n$\\mathcal{A}_3$ with $|\\mathcal{A}_1|<|\\mathcal{A}_3|<|\\mathcal{A}_2|$.)\nObserving that the sizes of consecutive maximal squashed flat antichains differ\nby at most $k+1$, we deduce that the maximal antichains on levels $k+1$, $k$ and $k-1$ obtained from\nall these $\\mathcal A'$ yield an interval $I(n,k)$ of sizes of maximal antichains. We check\nthat these intervals for consecutive values of $k$ overlap, so that their union is an interval containing the\nrequired $[w(n-1)+2,w(n)]$.\n\nFor an integer $k$ with $\\lfloor n\/2 \\rfloor \\le k \\le n-3$ we denote by $\\mathcal{M}(n,k)$ the set\nof all maximal squashed flat antichains $\\mathcal{A}$ with\n$\\binom{[k+3]}{k} \\subseteq \\mathcal{A} \\subseteq \\binom{[n]}{k}\\cup \\binom{[n]}{k-1}$. Moreover, we\nlet $Y(n,k)=\\{\\abs{\\mathcal{A}}\\,:\\,\\mathcal{A} \\in \\mathcal{M}(n,k)\\}$ be the set of their\nsizes. The \\emph{shadow spectrum} $\\sigma(t,k)$ has been introduced in~\\cite{Griggs2021a} as the set\nof shadow sizes for a $t$-family of $k$-sets:\n$\\sigma(t,k) = \\left\\{\\abs{\\Delta \\mathcal{F}}\\,:\\, \\mathcal{F} \\text{ is a family of }k\\text{-sets with }\n    \\abs{\\mathcal{F}} = t \\right\\}$.\n\nThe following lemma was used in \\cite{Griggs2021a} already.\n\\begin{lemma}\\label{lem:standard_ac_is_max}\n  Let $1\\le k \\le n$. For every $\\mathcal{F} \\subseteq \\binom{[n]}{k}$ with $\\abs{\\mathcal{F}}<k$,\n  the antichain $\\mathcal{A} = \\mathcal{F} \\cup \\binom{[n]}{k-1} \\setminus \\Delta \\mathcal{F}$ is\n  maximal.\n\\end{lemma}\n\\begin{proof}\n  For $X\\in\\binom{[n]}{k}\\setminus\\mathcal F$, $\\abs{\\Delta X\\cap\\Delta\\mathcal F}\\leq\\abs{\\mathcal\n    F}<k=\\abs{\\Delta X}$. Therefore $\\Delta X\\not\\subseteq\\Delta\\mathcal F$. Therefore $\\mathcal\n  A\\cup\\{X\\}$ is not an antichain, hence $\\mathcal A$ is a maximal antichain.\n\\end{proof}\n\n\nThe following result is proved using the construction outlined above.\n\\begin{lemma}\\label{lem:construction}\n  Let $n$, $k$ and $t$ be integers with $n\\geq 7$, $\\frac{n-1}{2}\\le k \\le n-3$ and $1\\leq\n  t\\leq k$. Then for every $m\\in Y(n,k)$ and $s\\in \\sigma(t,k)$ there is a maximal antichain\n  $\\mathcal A \\subseteq \\binom{[k+3]}{k+1} \\cup \\binom{[n]}{k} \\cup \\binom{[n]}{k-1}$ with\n  $\\abs{\\mathcal A}=m-s$.\n\\end{lemma}\n\\begin{proof}\n  Fix $m\\in Y(n,k)$, $s\\in\\sigma(t,k)$, and $\\mathcal A'\\in\\mathcal{M}(n,k)$ with\n  $\\abs{\\mathcal A'}=m$. By~\\cite[Theorem~2 and Remark~2]{Griggs2021a}, $s+t\\in\\sigma(t,k+1)$, and\n  there is a family $\\mathcal F\\subseteq\\binom{[k+3]}{k+1}$ with $\\abs{\\mathcal F}=t$ and\n  $\\abs{\\Delta\\mathcal F}=s+t$. By \\Cref{lem:standard_ac_is_max}\n  $\\mathcal F'=\\mathcal F\\cup\\binom{[k+3]}{k}\\setminus\\Delta\\mathcal F$ is a maximal antichain in\n  $B_{k+3}$. As a consequence,\n  $\\mathcal A=\\left[\\mathcal A'\\setminus\\binom{[k+3]}{k}\\right]\\cup\\mathcal F'$ is a maximal\n  antichain in $B_n$ with\n  \\[\\abs{\\mathcal A}=\\abs{A'}-\\binom{k+3}{k}+\\abs{\\mathcal F'}=m-\\binom{k+3}{k}+\\abs{\\mathcal\n      F}-\\binom{k+3}{3}+\\abs{\\Delta\\mathcal F}=m+t-(s+t)=m-s.\\qedhere\\]\n\\end{proof}\nWe know from \\cite[Theorem 2 and Proposition 1]{Griggs2021a} that for $t\\leq k+1$,\n\\[\\left[tk - \\binom{t-j^*(t)}{2} - \\binom{j^*(t)+1}{2},\\,tk\\right] \\subseteq \\sigma(t,k),\\]\nwhere $j^*(t)=\\lceil \\sqrt{2t}-5\/2\\rceil$ is the smallest non-negative integer $j$ with\n$\\binom{j+3}{2}\\geq t$. Thus we have the following consequence of Lemma~\\ref{lem:construction}.\n\\begin{corollary}\\label{cor:mini_intervals}\n  Let $n\\geq 7$ and $k$ be integers with $\\frac{n-1}{2}\\le k \\le n-3$. Let $m \\in Y(n,k)$ and set\n  $t=\\lfloor \\frac{k+3}{2} \\rfloor$, $j=j^*(t)$. Then\n  \\[\\left[m-tk,\\,m-tk+\\binom{t-j}{2}+\\binom{j+1}{2}\\right]\\subseteq S(n).\\]\n\\end{corollary}\nIn the next lemma we bound the lengths of the intervals in \\Cref{cor:mini_intervals}.\n\\begin{lemma}\\label{lem:last_shadow_interval}\n  Let $k\\geq 3$, $t=\\left\\lfloor\\frac{k+3}{2}\\right\\rfloor$, $j=j^*(t)$, and set\n  $L=\\binom{t-j}{2}+\\binom{j+1}{2}$.\n  \\begin{enumerate}[(i)]\n  \\item If $k\\geq 9$ or $k\\in\\{3,7\\}$, then $L\\geq k$.\n  \\item If $k\\in\\{4,5,8\\}$, then $L=k-1$.\n  \\item If $k=6$, then $L=4$.\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n   For $3\\leq k\\leq 32$, the statement can be verified by calculating $L$. We now assume $k\\geq\n   33$. Using $\\frac{k+2}{2}\\leq t\\leq\\frac{k+3}{2}$, we bound $j$ by\n   \\[j=\\left\\lceil\\sqrt{2t}-\\frac52\\right\\rceil\\leq\\sqrt{2t}\\leq\\sqrt{k+3}.\\]\n   Then\n   \\[\nL\\geq\\binom{t-j}{2}\\geq\\frac12\\left(\\frac{k+2}{2}-\\sqrt{k+3}\\right)\\left(\\frac{k}{2}-\\sqrt{k+3}\\right)\\geq\\frac{1}{2}\\left(\\frac{k}{2}-\\sqrt{k+3}\\right)^2=\\frac{k^2}{8}-\\frac12k\\sqrt{k+3}+\\frac{k+3}{2},\\]\n   and\n   \\begin{multline*}\n     L-k\\geq\\frac{k^2}{8}-\\frac12k\\sqrt{k+3}-\\frac{k}{2}=\\frac{k}{8}\\left(k-4\\sqrt{k+3}-4\\right)\\geq\\frac{k}{8}\\left(k-4\\sqrt{k+3}-9\\right)\\\\\n    =\\frac{k}{8}\\left((k+3)-4\\sqrt{k+3}-12\\right) =\\frac{k}{8}\\left(\\sqrt{k+3}+2\\right)\\left(\\sqrt{k+3}-6\\right)\\geq 0,\n   \\end{multline*}\n   where the last inequality follows from $k+3\\geq 36$.\n\\end{proof}\nTo show that the union of the intervals from \\Cref{cor:mini_intervals} is an interval, we need an\nupper bound for the gaps between the elements of $Y(n,k)$. This is provided in the next lemma.\n\\begin{lemma}\\label{lem:squashed_gaps}\n  The sizes of any two consecutive maximal squashed flat antichains on levels $k$ and $k-1$ in $B_n$\n  differ by at most $\\max\\{k-1,n-k\\}$. In particular, for $k \\ge \\frac{n+1}{2}$ any two consecutive\n  elements of $Y(n,k)$ differ by at most $k-1$, for $k=\\frac{n}{2}$ they differ by at most $k$, and\n  for $k=\\frac{n-1}{2}$ they differ by at most $k+1$.\n\\end{lemma}\n\\begin{proof}\n  Let $A_m$ be the $m$-th $k$-set in squashed order, and let $\\mathcal F(k,m)=\\{A_1,\\dots,A_m\\}$ be\n  the collection of the first $m$ $k$-sets in squashed order. Let $\\Delta' A_m$ denote the marginal\n  shadow of $A_m$, that is, $\\Delta' A_m =\\Delta A_m\\setminus\\Delta\\mathcal F(k,m-1)$. For\n  $m<\\binom{n}{k}$, the collection\n  $\\mathcal A(k,m,n)=\\mathcal F(k,m)\\cup\\binom{[n]}{k-1}\\setminus\\Delta\\mathcal F(k,m)$ is a maximal\n  antichain in $B_n$ if and only if $\\Delta' A_{m+1}\\neq\\emptyset$.\n  Note that $\\Delta' A_{m+1}\\neq\\emptyset$ if and only if $1\\in A_{m+1}$ because if $1\\in A_{m+1}$ then\n  $A_{m+1}\\setminus\\{1\\}\\in\\Delta'A_{m+1}$ and if $1\\notin A_{m+1}$ and $x\\in A_{m+1}$, then\n  $A_{m+1}\\setminus\\{x\\}\\in \\Delta(A_{m+1}\\setminus\\{x\\}\\cup\\{1\\})$.\n\n  Suppose that $\\mathcal A=\\mathcal A(k,m,n)$ is a maximal squashed antichain and let\n  $\\mathcal A'=\\mathcal A(k,m',n)$ be the next maximal squashed antichain. We will conclude the\n  proof by verifying that $\\abs{\\mathcal A}-(k-1)\\leq\\abs{\\mathcal A'}\\leq\\abs{\\mathcal A}+(n-k)$.\n  If $\\Delta' A_{m+2}\\neq\\emptyset$ then $m'=m+1$, and the claim follows from $\\abs{\\mathcal A'}=\\abs{\\mathcal\n  A}+1-\\abs{\\Delta' A_{m+1}}$. \n  If $\\Delta'A_{m+2}=\\emptyset$, then $1\\in A_{m+1}$, $2\\notin A_{m+1}$ and $A_{m+2}=A_{m+1}\\setminus\\{1\\}\\cup\\{2\\}$.\n  It follows that $\\Delta'A_{m+1}=\\{A_{m+1}\\setminus\\{1\\}\\}$. Moreover, with $y:=\\min A_{m'}$ we have \n  $A_{m+i}=A_{m'}\\setminus\\{y\\}\\cup\\{i\\}$ for $i=1,2,\\dots,m'-m$. Hence, $2\\le m-m'=y\\le n-k-1$, \n  $\\abs{\\Delta' A_{m+1}}=1$ and $\\Delta'A_{m+2}=\\dots=\\Delta' A_{m'}=\\emptyset$.\n  The claim follows with $\\abs{\\mathcal A'}=\\abs{\\mathcal A}+(m'-m)-\\abs{\\Delta'A_{m+1}\\cup\\dots\\cup\\Delta'A_{m'}}$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:interval_n_k}\n  Let $n\\geq 7$, $\\lfloor n\/2\\rfloor\\leq k\\leq n-3$, $t=\\lfloor\\frac{k+3}{2}\\rfloor$ and\n  $j=j^*(t)$. Then $I(n,k)\\subseteq S(n)$, where\n  \\[I(n,k)=\\left[\\min Y(n,k)-tk,\\,\\max Y(n,k)-tk+\\binom{t-j}{2}+\\binom{j+1}{2}\\right].\\]\n\\end{lemma}\n\\begin{proof}\n  If $k\\geq 9$ or $k\\in\\{3,7\\}$, then by \\Cref{lem:squashed_gaps,lem:last_shadow_interval}\n  \\[I(n,k) = \\bigcup_{m \\in Y(n,k)} \\left[m-tk,\\,m-tk+\\binom{t-j}{2}+\\binom{j+1}{2}\\right],\\]\n   and the claim follows by \\Cref{cor:mini_intervals}.\n  For $k\\in\\{4,5,8\\}$, the claim follows if\n  $n\\leq 2k$, and for $n=2k+1$ there is exactly one gap of length $k+1$ in $Y(n,k)$: the difference between\n  the sizes of the last two maximal squashed flat antichains which have sizes $\\binom{n}{k}-(k+1)$\n  and $\\binom{n}{k}$, respectively. As a consequence, the only element of $I(n,k)$ which is not\n  covered by \\Cref{cor:mini_intervals} is $\\binom{n}{k}-tk-1$. For $k=6$, we still need to do\n  something for $n=12$ and $n=13$. For $n=12$, we only need a maximal antichain of size\n  $\\binom{12}{6}-25$ by the same argument as above. For $n=13$, we need to look for the gaps in\n  $Y(13,6)$ which have length $6$ or $7$. As before, there is only one gap of size 7. Writing\n  down the elements of $Y(n,k)$ in the order in which they appear when we run through the antichains\n  in $\\mathcal M(n,k)$ ordered by the number of $k$-sets, then there are a few more differences of size\n  6. Fortunately, in most cases these gaps are filled by maximal squashed antichains that occur\n  earlier or later. Using a computer, we verify that when we write down the elements of $Y(n,k)$ in\n  increasing order, the only gaps of size at least six appear between the three largest values, that is, between\n  $\\binom{13}{6}-13$, $\\binom{13}{6}-7$ and $\\binom{13}{6}$. As a consequence, we only need the sizes\n  $\\binom{13}{6}-25$, $\\binom{13}{6}-26$ and $\\binom{13}{6}-32$. Applying \\Cref{lem:construction}\n  with $\\binom{n}{k}\\in Y(n,k)$, it is sufficient to verify that the missing sizes can be written in\n  the form $\\binom{n}{k}-s$ for some $s\\in\\sigma(t',k)$ with $t'\\leq k$. The proof is completed by observing that\n  \\begin{itemize}\n   \\item $\\binom{9}{4}-13\\in S(9)$ because $13\\in\\sigma(4,4)$,\n   \\item $\\binom{11}{5}-21\\in S(11)$ because $21\\in\\sigma(5,5)$,\n   \\item $\\left\\{\\binom{13}{6}-25,\\,\\binom{13}{6}-26,\\,\\binom{13}{6}-32\\right\\}\\subseteq\n    S(13)$ because $\\{25,26\\}\\subseteq\\sigma(5,6)$ and $32\\in\\sigma(6,6)$,\n   \\item $\\binom{12}{6}-25\\in S(12)$ because $25\\in\\sigma(5,6)$,\n   \\item $\\binom{17}{8}-41\\in S(17)$ because $41\\in\\sigma(6,8)$.\\qedhere\n\\end{itemize}\n\\end{proof}\nWe use results from~\\cite{Griggs2021} to bound $\\min Y(n,k)$. To state these results, let $C_l$ be\nthe sum of the first $l$ Catalan numbers: $C_l=\\sum_{i=1}^l\\frac{1}{i+1}\\binom{2i}{i}$.\n\\begin{lemma}[Section 4.1 in \\cite{Griggs2021}]\\label{lem:results_order_paper} For $1<k<n$, the\n  minimum size of a maximal squashed flat antichain is\n  \\[\\min\\left\\{\\binom{n}{k-1},\\binom{n}{k}\\right\\}-C_{\\min\\{k-1,n-k\\}}.\\]\n  The largest number of $k$-sets in a maximal squashed flat antichain of minimum size is\n  $\\binom{a_k}{k}+\\dots+\\binom{a_2}{2}$ where\n  \\begin{equation}\\label{eq:last_min}\n    a_i=\n    \\begin{cases}\n      2i-2 & \\text{for }i\\leq n-k+1,\\\\\n      n-k-1+i & \\text{for }i\\geq n-k+2.\n    \\end{cases}\n  \\end{equation}\n\\end{lemma}\nFrom \\Cref{lem:results_order_paper} we obtain the following values for $\\min Y(n,k)$.\n\\begin{lemma}\\label{lem:min_Y} If $5 \\le k \\le n-4$ then $\\displaystyle\\min Y(n,k)=\\min\\left\\{\\binom{n}{k-1},\\binom{n}{k}\\right\\}-C_{\\min\\{k-1,n-k\\}}$.\n\\end{lemma}\n\\begin{proof}\n  To deduce this from \\Cref{lem:results_order_paper}, we verify\n  $\\binom{a_k}{k}+\\dots+\\binom{a_2}{2}\\geq\\binom{k+3}{k}$, where the $a_i$ are given\n  by~(\\ref{eq:last_min}). If $k\\leq n-k+1$ then\n  $\\binom{a_k}{k}+\\dots+\\binom{a_2}{2}\\geq\\binom{a_k}{k}=\\binom{2k-2}{k}\\geq\\binom{k+3}{k}$, where\n  the inequality follows from $k\\geq 5$. If $k\\geq n-k+2$, then\n  $\\binom{a_k}{k}+\\dots+\\binom{a_2}{2}\\geq\\binom{a_k}{k}=\\binom{n-1}{k}\\geq\\binom{k+3}{k}$, where the\n  inequality follows from $n-1\\geq k+3$.\n\\end{proof}\n\nWe are now ready to express the intervals $I(n,k)$ in terms of $n$ and $k$.\n\\begin{lemma}\\label{lem:size_of_I(nk)}\n  Let $k \\ge 5$, assume $\\frac{n-1}{2} \\le k \\le n-4$, and set\n  $t=\\left\\lfloor\\frac{k+3}{2}\\right\\rfloor$, $j=j^*(t)$.\n  \\begin{enumerate}[(i)]\n  \\item If $k > \\frac{n+1}{2}$ then\n \\[I(n,k) \\supseteq \\left[ \\binom{n}{k} - C_{n-k} -tk,\\,\\binom{n}{k-1} + \\binom{k+3}{k} -\n     \\binom{k+3}{k-1} -tk + \\binom{t-j}{2} + \\binom{j+1}{2} \\right].\\]\n\\item If $k \\le \\frac{n+1}{2}$ then $\\displaystyle I(n,k) = \\left[ \\binom{n}{k-1} - C_{k-1} -tk,\\,\\binom{n}{k} -tk + \\binom{t-j}{2} + \\binom{j+1}{2}\\right]$.\n\\end{enumerate}\nMoreover, $I(7,3)=[16,29]$, $I(8,4)=[41,61]$, and $I(9,4)=[69,117]$.\n\\end{lemma}\n\\begin{proof}\n  For $k\\geq 5$, by \\Cref{lem:min_Y},\n  \\[\\min I(n,k)=\\min\\left\\{\\binom{n}{k-1},\\,\\binom{n}{k}\\right\\}-C_{\\min\\{k-1,n-k\\}}-tk,\\]\n  and this gives the left ends of the intervals in (i) and (ii). For $k>\\frac{n+1}{2}$, the right\n  end of the interval comes from\n  $\\binom{[k+3]}{k}\\cup\\binom{[n]}{k-1}\\setminus\\binom{[k+3]}{k-1}\\in\\mathcal M(n,k)$. For\n  $k\\leq\\frac{n+1}{2}$, $\\binom{[n]}{k}\\in\\mathcal M(n,k)$ implies\n  \\[\\max I(n,k)= \\binom{n}{k} -tk + \\binom{t-j}{2} + \\binom{j+1}{2}.\\]\n  This is also valid for $k\\in\\{3,4\\}$, and gives the right ends of the intervals for\n  $(n,k)\\in\\{(7,3),(8,4),(9,4)\\}$. For the left ends of these intervals we do an exhaustive search\n  over $\\mathcal M(n,k)$ and find the following antichains. For $(n,k)=(7,3)$, the maximal squashed\n  flat antichain with $21$ $3$-sets has size $25$. For $(n,k)=(8,4)$, the maximal squashed flat\n  antichain with $37$ $4$-sets has size $53$. For $(n,k)=(9,4)$, the maximal squashed flat antichain\n  with $37$ $4$-sets has size $81$.\n\\end{proof}\n\n\\section{Proof of Claim \\ref{lem:induction_step} for large $n$}\\label{sec:large_n}\nIn this section, we prove Claim \\ref{lem:induction_step} for $n\\geq 20$.\n\\begin{lemma}\\label{lem:induction_large_n}\n  If $n\\geq 20$ then $[w(n-1)+2,w(n)]\\subseteq S(n)$.\n\\end{lemma}\n\\begin{proof}\n  For $20\\leq n\\leq 199$, we use \\Cref{lem:size_of_I(nk),lem:interval_n_k} (and a computer) to\n  verify the statement. For $n\\geq 200$, we conclude using the following inequalities which are\n  proved below:\n  \\begin{itemize}\n  \\item For $k=\\left\\lceil\\frac{9n}{10}\\right\\rceil$, $\\min I(n,k)<w(n-1)$ (\\Cref{lem:lower_bound}).\n  \\item For $k=\\lceil n\/2\\rceil$, $\\max I(n,k)\\geq w(n)$ (\\Cref{lem:upper_bound}).\n  \\item For $\\left\\lceil\\frac{n+2}{2}\\right\\rceil\\leq\n  k\\leq\\left\\lceil\\frac{9n}{10}\\right\\rceil$, $\\max I(n,k)\\geq\\min I(n,k-1)$ (\\Cref{lem:bridges}).\\qedhere\n  \\end{itemize}\n\\end{proof}\n\\begin{lemma}\\label{lem:lower_bound}\n  For $n\\geq 200$ and $k=\\left\\lceil\\frac{9n}{10}\\right\\rceil$, $\\min I(n,k)<w(n-1)$.\n\\end{lemma}\n\\begin{proof}\n  By \\Cref{lem:size_of_I(nk)}, we have to show\n  \\[\\binom{n}{k}-C_{n-k}-\\left\\lfloor\\frac{k+3}{2}\\right\\rfloor k\\leq w(n-1).\\]\n  We prove the stronger inequality\n  $\\binom{n}{k}\\leq\\binom{n-1}{\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor}-n^2$. We bound the left-hand side by\n  \\[\\binom{n}{k}=\\binom{n}{n-k}\\leq\\left(\\frac{en}{n-k}\\right)^{n-k}\\leq\\left(\\frac{200e}{19}\\right)^{n\/10}<1.4^n.\\]\n  For the right-hand side, we start with\n  \\[\\binom{n-1}{\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor}\\geq 2^{\\left\\lfloor(n-1)\/2\\right\\rfloor}\\geq\n    2^{\\frac{99n}{200}}.\\]\n  Combining this with $\\displaystyle n^2\\leq 2^{\\frac{98n}{200}}\\leq 2^{\\frac{98n}{200}}\\left(2^{\\frac{n}{200}}-1\\right)$,\n  we obtain $\\displaystyle\\binom{n-1}{\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor}-n^2\\geq 2^{\\frac{49n}{100}}>1.4^n$.\n\\end{proof}\n\\begin{lemma}\\label{lem:upper_bound}\n  For $n\\geq 200$ and $k=\\lceil n\/2\\rceil$, $\\max I(n,k)\\geq w(n)$.\n\\end{lemma}\n\\begin{proof}\n  Set $t=\\left\\lfloor\\frac{k+3}{2}\\right\\rfloor$ and $j=j^*(t)$. By \\Cref{lem:size_of_I(nk)}, we\n  have to show\n  \\[\\binom{n}{k}-tk+\\binom{t-j}{2}+\\binom{j+1}{2}\\geq \\binom{n}{k}-k\\left\\lceil\\frac{k+1}{2}\\right\\rceil.\\]\n  We prove the stronger inequality $\\displaystyle\\binom{n}{k}-tk+\\binom{t-j}{2}\\geq\\binom{n}{k}-k\\left\\lceil\\frac{k+1}{2}\\right\\rceil$,\n  or, equivalently,\n  \\[\\binom{t-j}{2}\\geq\n    k\\left(\\left\\lfloor\\frac{k+3}{2}\\right\\rfloor-\\left\\lceil\\frac{k+1}{2}\\right\\rceil\\right).\\]\n  If $k$ is even then the right-hand side is $0$, and there is nothing to do. For odd $k$, the\n  right-hand side is equal to $k$, and we can show $\\binom{t-j}{2}\\geq k$ as in the proof of\n  \\Cref{lem:last_shadow_interval}. The assumption $k\\geq 33$ made in this proof follows from $n\\geq 200$.\n\\end{proof}\n\\begin{lemma}\\label{lem:bridges}\n  For $n\\geq 200$ and $\\left\\lceil\\frac{n+2}{2}\\right\\rceil\\leq k\\leq\\left\\lceil\\frac{9n}{10}\\right\\rceil$, $\\max I(n,k)\\geq\\min I(n,k-1)$.\n\\end{lemma}\n\\begin{proof}\n Set $t=\\left\\lfloor\\frac{k+3}{2}\\right\\rfloor$, $t'=\\left\\lfloor\\frac{k+2}{2}\\right\\rfloor$ and\n $j=j^*(t)$. By \\Cref{lem:size_of_I(nk)}, we have to show that\n \\[\\binom{n}{k-1} + \\binom{k+3}{k} -\n     \\binom{k+3}{k} -tk + \\binom{t-j}{2} + \\binom{j+1}{2}\\geq\\min I(n,k-1).\\]\n  For $k=\\frac{n+2}{2}$, this is\n  \\[\\binom{n}{k-1}+\\binom{k+3}{k}-\\binom{k+3}{k-1}-tk+\\binom{t-j}{2}+\\binom{j+1}{2}\\geq\\binom{n}{k-2}-C_{k-2}-t'(k-1).\\]\n  From $n^5\\leq\\binom{n}{k-1}$, we obtain\n  \\[n^4\\leq\\frac1k\\binom{n}{k-1}=\\binom{n}{k-1}-\\binom{n}{k-2},\\]\n  and this can be used to bound the left-hand side:\n  \\[\\binom{n}{k-1}+\\binom{k+3}{k}-\\binom{k+3}{k-1}-tk+\\binom{t-j}{2}+\\binom{j+1}{2}\\geq\\binom{n}{k-1}-n^4\\geq\\binom{n}{k-2},\\]\n  which is obviously larger than the right-hand side. Now we assume $k\\geq\\left\\lfloor\\frac{n+4}{2}\\right\\rfloor$, and the claim will follow from\n  \\[\\binom{n}{k-1}+\\binom{k+3}{k}-\\binom{k+3}{k-1}-tk+\\binom{t-j}{2}+\\binom{j+1}{2}\n    \\geq\\binom{n}{k-1}-C(n-k+1)-t'(k-1).\\]\n  The left-hand side is larger than $\\binom{n}{k-1}-n^4$ and the right-hand side is smaller than\n  \\[\\binom{n}{k-1}-C_{n-k+1}\\leq\\binom{n}{k-1}-C_{\\lfloor n\/10\\rfloor}\\leq\\binom{n}{k-1}-\\frac{1}{\\lfloor n\/10\\rfloor+1}\\binom{2\\lfloor n\/10\\rfloor}{\\lfloor n\/10\\rfloor}.\\]\n    Therefore, it is sufficient to\n  verify $\\frac{1}{l+1}\\binom{2l}{l}\\geq n^4$ for $l=\\lfloor n\/10\\rfloor$. We use the bound\n  \\[\\frac{1}{l+1}\\binom{2l}{l}\\geq \\frac{4^l}{(l+1)\\left(\\pi l\\frac{4l}{4l-1}\\right)^{1\/2}}\\]\n  from~\\cite{Dutton1986}. From $n\\geq 200$, it follows that $l\\geq 20$, and this implies $4^l\\geq\n  25000l^{11\/2}$. Bounding the denominator by\n  \\[(l+1)\\left(\\pi l\\frac{4l}{4l-1}\\right)^{1\/2}\\leq\\frac{23l}{22}\\left(\\pi\n      l\\frac{88}{87}\\right)^{1\/2}\\leq 2l^{3\/2},\\]\n  we obtain\n  \\[\\frac{4^l}{(l+1)\\left(\\pi l\\frac{4l}{4l-1}\\right)^{1\/2}}\\geq 12500 l^4\\geq (11l)^4\\geq n^4,\\]\n  which concludes the proof.\n\\end{proof}\n\n\n\\section{Proof of Claim \\ref{lem:induction_step} for small $n$}\\label{sec:small_n}\nIn this Section, we prove Claim \\ref{lem:induction_step} for $7\\leq n\\leq 19$. We need a few auxiliary results. Let\n\\[S(n,k)=\\left\\{\\abs{\\mathcal A}\\,:\\,\\mathcal A\\subseteq\\binom{[n]}{k-1}\\cup\\binom{[n]}{k}\\text{ is\n      a maximal antichain in }B_n\\right\\}.\\]\n\n\\begin{lemma}\\label{lem:recursion}\n  For $k\\geq 2$ and $n\\geq k+1$, $\\displaystyle S(n,k)\\supseteq S(n-1,k)+\\binom{n-1}{k-2}$ and\n  $\\displaystyle S(n,k)\\supseteq S(n-1,k-1)+\\binom{n-1}{k}$.\n\\end{lemma}\n\\begin{proof}\n  let $\\mathcal A'\\subseteq\\binom{[n-1]}{[k-1]}\\cup\\binom{[n-1]}{[k]}$ be a maximal antichain in\n  $B_{n-1}$. Then\n  \\[\\mathcal A=\\mathcal\n    A'\\cup\\left\\{A\\cup\\{n\\}\\,:\\,A\\in\\binom{[n-1]}{k-2}\\right\\}\\subseteq\\binom{[n]}{k-1}\\cup\\binom{[n]}{k}\\]\n  is a maximal antichain in $B_n$, and this implies the first inclusion. For the second one, let\n  $\\mathcal A'\\subseteq\\binom{[n-1]}{[k-2]}\\cup\\binom{[n-1]}{[k-1]}$ be a maximal antichain in\n  $B_{n-1}$. Then\n  \\[\\mathcal A=\\left\\{A\\cup\\{n\\}\\,:\\,A\\in\\mathcal\n      A'\\right\\}\\cup\\binom{[n-1]}{k}\\subseteq\\binom{[n]}{k-1}\\cup\\binom{[n]}{k}\\]\n  is a maximal antichain in $B_n$.\n\\end{proof}\n\n\\begin{definition}\n  A flat antichain $\\mathcal A\\subseteq\\binom{[n]}{k-1}\\cup\\binom{[n]}{k}$ is called\n  \\emph{$\\{1,2\\}$-separated} if $\\{1,2\\}\\subseteq A$ for every $k$-set $A\\in\\mathcal A$\n  $\\lvert A\\cap\\{1,2\\}\\rvert\\leq 1$ for every $(k-1)$-set $A\\in\\mathcal A$.\n\\end{definition}\n\n\\begin{observation}\\label{obs:12_separated_recursion}\n  Let $\\mathcal A'\\subseteq\\binom{[n-1]}{k-1}\\cup\\binom{[n-1]}{k}$ and\n  $\\mathcal A''\\subseteq\\binom{[n-1]}{k-2}\\cup\\binom{[n-1]}{k-1}$ be $\\{1,2\\}$-separated\n  maximal antichains in $B_{n-1}$.  Then\n  $\\mathcal A=\\mathcal A'\\cup\\{A\\cup\\{n\\}\\,:\\,A\\in\\mathcal\n  A''\\}\\subseteq\\binom{[n-1]}{k-1}\\cup\\binom{[n-1]}{k}$ is a $\\{1,2\\}$-separated maximal antichain\n  in~$B_n$.\n\\end{observation}\nThis can be used in an induction to establish the following result.\n\\begin{lemma}\\label{lem:separated_intervals}\n  Let $n$, $k$ and $m$ be integers with $k\\geq 2$, $n\\geq k+1$, and\n  \\[\\binom{n-1}{k-1}\\leq m\\leq \\binom{n}{k-1}-2\\binom{n-3}{k-3}-\\binom{n-4}{k-5}.\\]\n  Then there exists a $\\{1,2\\}$-separated antichain\n  $\\mathcal A\\subseteq\\binom{[n]}{k-1}\\cup\\binom{[n]}{k}$ with $\\abs{\\mathcal A}=m$.\n\\end{lemma}\n\\begin{proof}\n  For $k=2$, we have $m\\in\\{n-1,n\\}$. Then $\\binom{[n]}{1}$ is a $\\{1,2\\}$-separated maximal\n  antichain of size $n$, and $\\{\\{1,2\\}\\}\\cup\\left(\\binom{[n]}{1}\\setminus\\{\\{1\\},\\,\\{2\\}\\}\\right)$\n  is a $\\{1,2\\}$-separated maximal antichain of size $n-1$. For $k\\geq 3$, $n=k+1$, we have\n  $k\\leq m\\leq 2k-2$. For $l\\in\\{3,4,\\dots,n\\}$,\n  \\[\\mathcal A= \\left\\{[n]\\setminus\\{j\\}\\,:\\,3\\leq j\\leq l\\right\\}\\ \\cup\\ \\{\\{3,4,\\dots,n\\}\\}\\ \\cup\\\n    \\{\\{i\\}\\cup([3,n]\\setminus\\{j\\})\\,:\\,i\\in\\{1,2\\},\\,j\\in\\{l+1,\\dots,n\\}\\}.\\] is a maximal\n  $\\{1,2\\}$-separated antichain with $\\abs{\\mathcal A}=l-2+2(n-l)+1=2k-l+1$. For $n\\geq k+2$, we\n  proceed by induction, and note that \\Cref{obs:12_separated_recursion} implies that we\n  get all sizes $m$ with\n  \\[\\binom{n-2}{k-1}+\\binom{n-2}{k-2}\\leq\n    m\\leq\\binom{n-1}{k-1}-2\\binom{n-4}{k-3}-\\binom{n-5}{k-5}+\\binom{n-1}{k-2}-2\\binom{n-4}{k-4}-\\binom{n-5}{k-6},\\]\n  and this simplifies to the claimed range.\n\\end{proof}\n\\begin{lemma}\\label{lem:bridges_2}\n  For $\\displaystyle 3\\leq k\\le 10$ and $2k\\leq n\\leq 20$, $\\displaystyle\\left[\\binom{n-1}{k-1},\\binom{n}{k-1}\\right]\\subseteq S(n,k)$.\n\\end{lemma}\n\\begin{proof}\n  We proceed by induction on $k$, and for fixed $k$ by induction on $n$. For the base case\n  $(k,n)=(3,6)$, we have to verify $[10,15]\\subseteq S(6,3)$. For $m\\in[10,14]$ we can use the\n  maximal antichains on levels $2$ and $3$ in $B_6$ that are given in the proof of\n  \\Cref{thm:main_theorem} at the end of \\Cref{sec:introduction}, and for $m=15$ we use the maximal\n  antichain $\\binom{[6]}{2}$. Next, we look at the induction step from $(k,n-1)$ to $(k,n)$ for\n  $n\\geq 2k+1$. From the induction hypothesis and \\Cref{lem:recursion},\n  \\[S(n,k)\\supseteq\\left[\\binom{n-2}{k-1},\\,\\binom{n-1}{k-1}\\right]+\\binom{n-1}{k-2}=\\left[\\binom{n-2}{k-1}+\\binom{n-1}{k-2},\\,\\binom{n}{k-1}\\right],\\]\n  and the claim follows with \\Cref{lem:separated_intervals} and\n  \\[\\binom{n}{k-1}-2\\binom{n-3}{k-3}-\\binom{n-4}{k-5}\\geq\\binom{n-2}{k-1}+\\binom{n-1}{k-2}\\]\n  for all $(k,n)$ with $4\\leq k\\leq 10$ and $2k+1\\leq n\\leq 20$. To complete the induction argument, we check that the claim for $k\\geq 4$ and $n=2k$ is implied by\n  the statement for $k-1$. We need four ingredients:\n  \\begin{itemize}\n  \\item By \\Cref{lem:separated_intervals},\n    \\[\\left[\\binom{2k-1}{k-1},\\,\n        \\binom{2k}{k-1}-2\\binom{2k-3}{k-3}-\\binom{2k-4}{k-5}\\right]\\subseteq S(2k,k).\\]\n  \\item By \\Cref{lem:recursion,lem:separated_intervals},\n    \\[S(2k,k)\\supseteq\\left[\\binom{2k-2}{k-1},\\,\\binom{2k-1}{k-1}-2\\binom{2k-4}{k-3}-\\binom{2k-5}{k-5}\\right]+\\binom{2k-1}{k-2}.\\]\n  \\item By \\Cref{lem:recursion}, $S(2k,k) \\supseteq S(2k-1,k) + \\binom{2k-1}{k-2} \\supseteq S(2k-2,k-1)+\\binom{2k-2}{k}+\\binom{2k-1}{k-2}$,\n    and with induction,\n    \\[S(2k,k)\\supseteq\\left[\\binom{2k-3}{k-2},\\,\\binom{2k-2}{k-2}\\right]+\\binom{2k-2}{k}+\\binom{2k-1}{k-2}.\\]\n  \\item By \\Cref{lem:recursion} and induction,\n    \\[S(2k,k)\\supseteq\\left[\\binom{2k-2}{k-2},\\,\\binom{2k-1}{k-2}\\right]+\\binom{2k-1}{k}=\\left[\\binom{2k-2}{k-2}+\\binom{2k-1}{k},\\,\\binom{2k}{k-1}\\right].\\]\n  \\end{itemize}\n  We check that for $4\\leq k\\leq 10$, the union of these four intervals is\n  $[\\binom{2k-1}{k-1},\\,\\binom{2k}{k-1}]$, as required.\n\\end{proof}\nWe need one more simple construction before we can complete the argument for $n\\leq 19$.\n\\begin{lemma}\\label{lem:final_case}\n  Let $n \\ge 2$ be an integer. If $m \\in S(n-1)$ and $m > \\binom{n-2}{\\lfloor \\frac{n-2}{2} \\rfloor} +1$, then $m+n-1 \\in S(n)$.\n\\end{lemma}\n\\begin{proof}\n  Let $m \\in S(n-1)$ with $m > \\binom{n-2}{\\lfloor \\frac{n-2}{2} \\rfloor} +1$, and $\\mathcal{A}$ be\n  a maximal antichain in $B_{n-1}$ with size $m$. Then, by Sperner's Theorem, $\\mathcal{A}$ cannot\n  contain a singleton. Thus, $\\mathcal{A} \\cup \\{\\{i,n\\}\\,:\\, i \\in [n-1]\\}$ is a maximal antichain in\n  $B_n$ of size $m+n-1$.\n\\end{proof}\n\nFinally, we combine \\Cref{lem:interval_n_k,lem:size_of_I(nk),lem:bridges_2,lem:final_case} to prove\nClaim \\ref{lem:induction_step} for $7\\leq n\\leq 19$.\n\\begin{lemma}\\label{lem:induction_step_small_n}\n  For all $n\\in\\{7,\\dots,19\\}$, $\\left[w(n-1)+2,w(n)\\right] \\subseteq S(n)$.\n\\end{lemma}\n\\begin{proof} For each value of $n$ we list the sizes coming from the various lemmas.\n  \\begin{description}\n  \\item[$n=7$] We need the interval $[w(6)+2,w(7)]=[16,23]$, and \\Cref{lem:size_of_I(nk)} yields\n    $I(7,3)=[16,29]$.\n  \\item[$n=8$] We need $[w(7)+2,w(8)]=[25,58]$. From \\Cref{lem:size_of_I(nk)} we have $\\mathcal\n    I(8,4)=[41,61]$, and from \\Cref{lem:bridges_2},\n    $\\left[\\binom{7}{2},\\binom{8}{2}\\right]\\cup\\left[\\binom{7}{3},\\binom{8}{3}\\right]=[21,28]\\cup[35,56]$. Finally,\n    Lemma~\\ref{lem:final_case} with $[22,27]\\subseteq S(7)$ yields $[29,34]\\subseteq S(8)$.\n  \\item[$n=9$] We need $[w(8)+2,w(9)]=[60,111]$. From \\Cref{lem:size_of_I(nk)} we have $\\mathcal\n    I(9,4)=[69,117]$, and from \\Cref{lem:bridges_2},\n    $\\left[\\binom{8}{3},\\binom{9}{3}\\right]=[56,84]$.\n  \\item[$n=10$] We need $[w(9)+2,w(10)]=[113,237]$. From \\Cref{lem:size_of_I(nk)}\n    we have\n    \\[I(10,5)\\cup I(10,6)=[164,236],\\]\n    and from \\Cref{lem:bridges_2},\n    $\\left[\\binom{9}{3},\\binom{10}{3}\\right]\\cup\\left[\\binom{9}{4},\\binom{10}{4}\\right]=[84,120]\\cup[126,210]$. From\n    \\Cref{lem:final_case} and $[112,116]\\subseteq S(9)$, we deduce $[121,125]\\subseteq S(10)$, and\n    finally, a maximal antichain of size $237$ is given by $\\mathcal A=\\mathcal\n    F\\cup\\binom{[10]}{5}\\setminus\\Delta\\mathcal F$ where $\\mathcal F=\\{\\{1, 2, 3, 4, 5, 6\\},\\,\\{1, 2, 3, 4, 7, 8\\},\\,\\{1, 2, 3, 4, 9, 10\\}\\}$.\n  \\item[$n=11$] We need $[w(10)+2,w(11)]=[239,438]$. From \\Cref{lem:size_of_I(nk)}\n    we have\n    \\[I(11,5)\\cup I(11,7)=[273,446],\\]\n    and from \\Cref{lem:bridges_2}, $\\left[\\binom{10}{4},\\binom{11}{4}\\right]=[210,330]$.\n  \\item[$n=12$] We need $[w(11)+2,w(12)]=[440,900]$. From \\Cref{lem:size_of_I(nk)}\n    we have\n    \\[I(12,6)\\cup I(12,7)=[693,904],\\]\n    and from \\Cref{lem:bridges_2}, $\\left[\\binom{11}{4},\\binom{12}{4}\\right]\\cup\\left[\\binom{11}{5},\\binom{12}{5}\\right]=[330,792]$.\n  \\item[$n=13$] We need $[w(12)+2,w(13)]=[902,1688]$. From \\Cref{lem:size_of_I(nk)}\n    we have\n    \\[I(13,7)\\cup I(13,8)=[1138,1688],\\]\n    and from \\Cref{lem:bridges_2}, $\\left[\\binom{12}{5},\\binom{13}{5}\\right]=[792,1287]$.\n  \\item[$n=14$] We need $[w(13)+2,w(14)]=[1690,3404]$. From \\Cref{lem:size_of_I(nk)}\n    we have\n    \\[I(14,7)\\cup I(14,8)=[2767,3404],\\]\n    and from \\Cref{lem:bridges_2}, $\\left[\\binom{13}{5},\\binom{14}{5}\\right]\\cup\\left[\\binom{13}{6},\\binom{14}{6}\\right]=[1287,3003]$.\n  \\item[$n=15$] We need $[w(14)+2,w(15)]=[3406,6395]$. From \\Cref{lem:size_of_I(nk)}\n    we have\n    \\[I(15,8)\\cup I(15,9)=[4755,6402],\\]\n    and from \\Cref{lem:bridges_2}, $\\left[\\binom{14}{6},\\binom{15}{6}\\right]=[3003,5005]$.\n  \\item[$n=16$] We need $[w(15)+2,w(16)]=[\\num{6397},\\num{12830}]$. From \\Cref{lem:size_of_I(nk)}\n    we have\n    \\[I(16,8)\\cup I(16,10)=[\\num{7752},\\num{12837}],\\]\n    and from \\Cref{lem:bridges_2}, $\\left[\\binom{15}{6},\\binom{16}{6}\\right]=[\\num{5005}, \\num{8008}]$.\n  \\item[$n=17$] We need $[w(16)+2,w(17)]=[\\num{12832},\\num{24265}]$. From \\Cref{lem:size_of_I(nk)}\n    we have\n    \\[ I(17,9)\\cup I(17,10)=[\\num{18763},\\num{24267}],\\]\n    and from \\Cref{lem:bridges_2}, $\\left[\\binom{16}{7},\\binom{17}{7}\\right]=[\\num{11440}, \\num{19448}]$.\n  \\item[$n=18$] We need $[w(17)+2,w(18)]=[\\num{24267},\\num{48575}]$. From \\Cref{lem:size_of_I(nk)}\n    we have\n    \\[ I(18,9)\\cup I(18,10)\\cup I(18,11)=[\\num{31122},\\num{48577}],\\]\n    and from \\Cref{lem:bridges_2}, $\\left[\\binom{17}{7},\\binom{18}{7}\\right]=[\\num{19448}, \\num{31824}]$.\n  \\item[$n=19$] We need $[w(18)+2,w(19)]=[\\num{48577},\\num{92318}]$. From \\Cref{lem:size_of_I(nk)}\n    we have\n    \\[ I(19,10)\\cup I(19,11)\\cup I(19,12)=[\\num{49679},\\num{92329}],\\]\n    and from \\Cref{lem:bridges_2}, $\\left[\\binom{18}{7},\\binom{19}{7}\\right]=[\\num{31824}, \\num{50388}]$.\\qedhere\n  \\end{description}\n\\end{proof}\n\n\n\\printbibliography\n\n\n\n\\end{document}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:introduction}\nQuasars are among the most powerful objects in the Universe, found from low redshift to redshifts beyond 7 \\citep{Mortlock2011}. Tracing the properties of quasars can help understand supermassive black holes in massive galaxies and the coevolution of black holes and their host galaxies \\citep{Kormendy2013}. Large quasar surveys are important for finding the clustering of quasars and lensed quasars, and for probing the galaxy merger scenario and measuring the mass distribution of halos \\citep[e.g.,][]{Oguri2006, Hennawi2010}. So far, more than 346,000 quasars have been spectroscopically identified in the SDSS \\citep{Schneider2010, Paris2017}.\n\nMassive spectroscopic surveys require a large amount of telescope time, so it is usually very expensive to obtain spectroscopic redshifts for large quasar samples. Photometric redshifts (photo-$z$s), derived from photometric data, provide an alternative technique to measure redshifts. Photometric quasar samples have been used to do many important studies, such as the clustering of quasars \\citep{Myers2006, Myers2007a, Myers2007b}, quasar number count statistics \\citep[e.g.,][]{Richards2009b, Richards2015}, cosmic magnification \\citep{Scranton2005}, and the Integrated Sachs-Wolfe effect \\citep{Giannantonio2006}. Besides, photo-$z$ estimation is very useful for quasar candidate selection in spectroscopic redshift surveys \\citep{Richards2004, Richards2009a, Richards2009b, Richards2015}.\n\nNowadays, more and more photometric data are being acquired. For example, the Pan-STARRS1 Telescope \\citep[PS1;][]{Kaiser2002} carried out a distinct set of imaging synoptic sky surveys that are useful for quasar searches in the southern sky. In the near future, the Large Synoptic Survey Telescope \\citep[LSST;][]{Tyson2002} will bring more opportunities for photo-$z$ estimates and cosmology research based on photo-$z$ quasar samples. The Dark Energy Spectroscopic Instrument \\citep[DESI;][]{DESI1,DESI2} is the successor to the Stage-III BOSS redshift survey, and will study baryon acoustic oscillations (BAO) and the growth of structure through redshift-space distortions (RSD) with a wide-area galaxy and quasar redshift survey. High efficiency quasar candidate selection would save a lot of follow-up observation time. We aim to improve the photo-$z$ accuracy of quasars and develop an efficient quasar candidate selection algorithm for a wide range of redshift and magnitude. With carefully defined selection completeness and efficiency correction, a photometrically selected quasar sample has the potential to be used to derive the quasar luminosity function (QLF), and reach a fainter magnitude limit than a spectroscopically identified sample. Moreover, photometrically selected quasars combined with multi-epoch and multi-band LSST data will be powerful for studies such as measuring black hole mass through photometric reverberation mapping \\citep[e.g.,][]{Hernitschek2015, Zu2016}; detecting changing-look quasars \\citep[e.g.,][]{Gezari2017}; and characterizing the variability of quasars \\citep[e.g.,][]{MacLeod2010, Zuo2012}.\n\nDifferent methods have been put forward to estimate the photo-$z$s of quasars, including quasar template fitting \\citep[e.g.,][]{Budavari2001, Babbedge2004, Salvato2009}, the empirical color-redshift relation (CZR) \\citep[e.g.,][]{Richards2001a, Wu2004, Weinstein2004, Wu2010, Wu2012}, Machine Learning \\citep[e.g.,][]{Ball2007, Yeche2010, Laurino2011, Brescia2013, Zhang2013} and the XDQSOz method \\citep{Bovy2012}. In the COSMOS field, the template fitting method is efficient with the photometry from 30 bands. But there are few fields with such rich photometry available. Apart from the template fitting method, the photo-$z$ regression method needs a training sample, usually a spectroscopically identified quasar sample. The redshift and magnitude distributions of spectroscopically identified quasars are affected by their target selection methods and the incompleteness of spectroscopic observations. So, dividing the spectroscopically identified quasar training sample into a grid of redshift and magnitude is helpful, considering the dependence of quasar colors on redshift and luminosity. Quasars are usually bluer when brighter, and the equivalent width (EW) of their emission lines are anti-correlated with the continuum flux \\citep[Baldwin effect;][]{Baldwin1997}. The slope of the power law continuum, the EW and FWHM of emission lines span wide ranges \\citep[e.g.,][]{Vanden2001, Telfer2002}. In addition, the redward flux of the Lyman-$\\alpha$ emission profile in a quasar spectrum is affected by the absorption lines of the Lyman-$\\alpha$ forest from neutral hydrogen along the line-of-sight to the quasar. The color distribution of quasars, even in a narrow redshift and magnitude bin, differs from a Gaussian distribution. It is obviously skewed and shows tails even when excluding broad absorption line (BAL) quasars. A significant population of red quasars exists \\citep[e.g.,][]{Webster1995, Richards2001b, Richards2003, Hopkins2004}. \\citet{Richards2003} defined a quasar to be dust-reddened with relative color $\\Delta (g^*-i^*)$ redder than 0.2, corresponding to $E(B-V)=0.04$, and find 6\\% quasars fall into the redder quasar category. Dust reddening at the redshift of the quasar is the primary explanation for the red tail in quasar color distribution. \\citet{Hopkins2004} modeled the color distribution as a Gaussian convolved with an exponential function to represent the dust. The Skew-t function can be used to describe data with skewed and tail features. The Skew-t distribution is widely used for multivariate skew distributions in statistics, quantitative risk management, and insurance. We choose skew functions instead of Gaussian functions to model the posterior distributions of quasars. Details about the Skew-t function will be provided in \\ref{subsec:skewt}.\n\nIn addition to the systematics of photo-$z$, quasar candidate selection is also a key issue. There are diverse methods used to select quasars. For example, the ultraviolet excess (UVX) method \\citep{Sandage1965, Green1986} for $z<2.2$ quasars; X-ray sources \\citep[e.g.,][]{Trump2009}; radio sources such as from the VLA FIRST survey \\citep[e.g.,][]{Becker2000}; quasar variability \\citep[e.g.,][]{NPD2011}; optical color box selection for the 2dF-SDSS LRG and QSO Survey \\citep[2SLAQ,][]{Croom2009}, and for the SDSS target selection \\citep[e.g.,][]{Richards2002}; more complex methods with optical (and infrared photometry), including non-parametric Bayesian classification and Kernel Density Estimator \\citep[KDE,][]{Richards2004, Richards2009b}, XDQSO \\citep{Bovy2011}, the neural network approach \\citep{Yeche2010}, the Bayesian likelihood approach \\citep{Kirkpatrick2011}; and selection combining different methods \\citep[e.g.,][]{Ross2012}. When a survey goes fainter, the contamination of point-like galaxies becomes significant, with the contamination rate as a function of magnitude. Fitting a training sample with all point-like objects is not efficient with regard to quasar selection at different magnitudes. A training sample consisting of all point-like objects will include stars, quasars, and point-like galaxies, thus it is hard to fit their posterior distribution all together. To separate quasars from stars, we estimate the number counts and colors (relative fluxes) of stars from a Milky Way synthetic simulation with the Besan\\c{c}on model. We also do galaxy template fitting to help distinguish galaxies from quasars.\n\nThe paper is organized as follows. In Section \\ref{sec:data}, we introduce the spectroscopically identified quasar sample and photometric data used in this work. In Section \\ref{sec:method}, we describe the photo-$z$ regression algorithm. We compare the photo-$z$ results obtained by different photo-$z$ methods using the same optical photometric data in Section \\ref{sec:results}. We also present photo-$z$ results using SDSS, SDSS-WISE, PS1-WISE and DECaLS-WISE photometry in Section \\ref{sec:results}. We present the classification method in Section \\ref{sec:classification}, including the stellar simulation, the galaxy template fitting, and the Bayesian classification method. Quasar candidate selection using the DECaLS and WISE photometry is presented in Section \\ref{sec:discussion}. We test the results in some deep fields and present the quasar number count statistics in the SDSS Stripe 82 (S82) region. We summarize the paper in Section \\ref{sec:summary}. We will make the photo-$z$ and classification code publicly available\\footnote{\\url{https:\/\/github.com\/qian-yang\/Skewt-QSO}} with the current version archived in Zenodo \\citep{SkewtQSOv1}. In the paper, all magnitudes are expressed in the AB system. The galactic extinction of extragalactic objects is corrected using the dust reddening map of \\citet{Schlegel1998}. We discuss only type 1 quasars (or AGNs) in this work. We use a $\\Lambda$CDM cosmology with $\\Omega_{\\Lambda}=0.728$, $\\Omega_m=0.272$, $\\Omega_b=0.0456$, and $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ \\citep{Komatsu2009}.\n\n\\section{The Data} \\label{sec:data}\n\\subsection{Spectroscopically Identified Quasar Sample} \\label{subsec:sample}\nWe use a sample of spectroscopically identified quasars consisting of quasars from the SDSS Data Release 7 Quasar catalog (DR7Q) \\citep{Schneider2010} and the Data Release 12 Quasar catalog (DR12Q) \\citep{Paris2017}. There are 105,783 quasars in the DR7Q, and 297,301 quasars in the DR12Q, including 25,275 quasars in both catalogs. BAL quasars are anomalously redder than most quasars and are excluded from our analysis. There are 29,580 quasars identified as BAL quasars in the DR12Q, and 6,214 quasars in the DR7Q identified as BAL quasars by \\citet{Shen2011}. After removal of the BAL quasars, there are 346,464 quasars in our quasar sample (DR7\\&12). Since, in comparison with the SDSS photometric bands, there are more high redshift quasars detected by the redder PS1 $y$ band and deeper DECaLS $z$ band, it is now possible to construct color models for high redshift quasars. We also include some quasars, which are not in the SDSS DR7 or DR12 catalog. A high redshift quasar catalog with 437 $z>4.5$ (called the BONUS high redshift sample) was constructed from the literature (Table 1 and Table 3 in \\citet{Wang2015} and references therein; Table 7 in \\citet{Banados2016} and references therein; \\citealt{Jiang2016}; \\citealt{Yang2017}; \\citealt{Wang2017}).\n\n\\subsection{SDSS Photometry} \\label{subsec:sdss}\nWe use the point spread function \\citep[PSF;][]{Lupton1999} photometry in the five SDSS bands $ugriz$ \\citep{Fukugita1996}. The magnitude limits\n(95\\% completeness for point sources) in the five bands are 22.0, 22.2, 22.2, 21.3, and 20.5 mag, respectively. We queried the photoObjAll table in the SDSS CASJOB, and got the SDSS photometry for 304,241 quasars with restrictions on mode and flags \\citep{Stoughton2002, Bovy2011, Richards2015}. The Galactic extinction coefficients for E(B-V) used are $Au, Ag, Ar, Ai, Az = 5.155, 3.793, 2.751, 2.086, 1.479$. The $u$ band and $z$ band are converted to the AB system using $u_{AB} = u_{SDSS} - 0.04$ mag and $z_{AB} = z_{SDSS} + 0.02$ mag \\citep{Fukugita1996}.\n\n\\subsection{PS1 Photometry} \\label{subsec:ps1}\nWe use the PSF photometry in the PS1 survey. The 5$\\sigma$ median limiting AB magnitudes in the five PS1 bands $grizy$ are 23.2, 23.0, 22.7, 22.1, and 21.1 mag, respectively. We queried the StackObjectThin table in the PS1 CASJOB with restrictions on primaryDetection and infoFlag, and got PS1 photometry for 344,318 quasars. Due to the difference between the absorbing column of the atmosphere at the two survey sites, the extinction coefficients for the SDSS and PS1 filters are different. The Galactic extinction coefficients for E(B-V) are $Ag, Ar, Ai, Az, Ay = 3.172, 2.271, 1.682, 1.322, 1.087$ \\citep{Schlafly2011}.\n\n\\subsection{WISE Photometry} \\label{subsec:wise}\nWISE \\citep{Wright2010} mapped the sky at 3.4, 4.6, 12, and 22 $\\mu$m (W1, W2, W3, W4). The 5$\\sigma$ limiting magnitudes of the ALLWISE catalog in W1, W2, W3 and W4 bands are 19.6, 19.3, 16.7 and 14.6 mag. We use only WISE W1 and W2 photometric data, because the other two bands are much shallower. Out of 346,464 quasars in the DR7\\&12 spectroscopic quasar catalog, 261,614 (76\\%) and 256,606 (74\\%) quasars are detected within 2 arcseconds in the WISE ALLWISE W1 and W2 bands, respectively. The WISE magnitudes are converted from Vega magnitude to AB magnitude with $\\Delta m = $ 2.699 and 3.339 for the W1 and W2 bands, respectively.\n\n\\subsection{DECaLS Photometry} \\label{subsec:decals}\nThe DESI Legacy imaging survey (DELS; Dey et al. 2017, in preparation) will provide images for target selection, including the DECam Legacy Survey (DECaLS) in the $g$, $r$ and $z$ bands, the Beijing-Arizona Sky Survey \\citep[BASS;][]{Zou2017} in the $g$ and $r$ bands, and the Mayall $z$-band Legacy Survey (MzLS). The 5$\\sigma$ point-source magnitude limits in $g$, $r$, and $z$ will be roughly 24.7, 23.9, and 23.0 mag. With depths of $1.5-2.5$ mag fainter than in the SDSS, the DELS will be useful in searching for fainter quasars than the SDSS spectroscopic quasars, and also high redshift quasars \\citep{Wang2017}. In this work, we use the three band ($grz$) photometry from the DECaLS DR3. There are 194,529 known quasars detected in the DECaLS DR3 catalogs, and 98,481 quasars observed in all three bands in DR3. There are 235 quasars in the BONUS high redshift sample detected in the DECaLS DR3, and 149 of them were observed in the $g$, $r$, and $z$ bands. The unWISE coadds the WISE imaging and has better resolution \\citep{Lang2014}. The unWISE $5\\sigma$ detection rates for our spectroscopic quasar sample are higher than those for WISE, 87.6\\% and 77.1\\% for the W1 and W2 bands, respectively. The unWISE photometry is available in the DECaLS catalogs. For objects with detections lower than 5$\\sigma$, the unWISE data are still included in the DECaLS catalogs with corresponding larger photometric errors. We use the unWISE W1 and W2 band photometry, instead of ALLWISE, when using the DECaLS optical photometry.\n\n\\begin{figure}[htbp]\n\\hspace*{-0.3cm}\n\\epsscale{1.2}\n\\plotone{f1_flux_ratio.pdf}\n\\caption{\\label{fig:skewt}Examples of one dimensional relative flux $f_u\/f_r$, $f_g\/f_r$, $f_i\/f_r$ and $f_z\/f_r$ distributions for quasars with $1.5<z_s<1.6$ and $18.5<r<19.0$. The relative flux distributions are skewed and show tail features even in a small redshift and magnitude bin. Obviously, the Skew-t model (red solid line) fits the relative flux distributions better than the Gaussian model (black dotted line) and the Skew-Normal model (blue dashed line).}\n\\end{figure}\n\\section{The photo-$z$ regression algorithm} \\label{sec:method}\n\\subsection{Posterior probability with Multivariate Skew-t model} \\label{subsec:skewt}\nIn this work, we model the posterior probability distribution for the relative fluxes of quasars using multivariate Skew-t distributions, with (1) ``skew\" considering the asymmetric characteristic; (2) ``t\" (student distribution) considering the incompleteness of the spectroscopically identified quasar sample.\n\nMultivariate skew-normal densities extend the multivariate normal model by allowing a shape parameter to account for skewness \\citep{Azzalini1985, Azzalini1996}. The probability density function (PDF) of an n-dimensional multivariate skew-normal distribution is $SN_n(\\bm{\\mu}, \\bm{\\Sigma}, \\bm{\\lambda)}$, where $\\bm{\\mu}$ is the mean vector, $\\bm{\\Sigma}$ is the covariance matrix, and $\\bm{\\lambda}$ is the shape parameter vector. The distribution can be written as\n\\begin{equation} \\label{eq:1}\n 2\\phi_n(\\mathbf{x}|\\bm{\\mu}, \\bm{\\Sigma})\\Phi(\\bm{\\lambda}^{\\rm T}\\bm{\\Sigma}^{-1\/2}(\\mathbf{x}-\\bm{\\mu})),\n\\end{equation}\nwhere $\\phi_n(\\mathbf{x}|\\bm{\\mu}, \\bm{\\Sigma})$ is the PDF of the n-variate normal distribution, ${\\bm{\\lambda}}^{\\rm T}$ is the transform vector of $\\bm{\\lambda}$, and $\\Phi(\\bm{\\lambda}^{\\rm T}\\bm{\\Sigma}^{-1\/2}(\\mathbf{x}-\\bm{\\mu}))$ is the cumulative distribution function (CDF) of the standard normal distribution. When $\\bm{\\lambda} = 0$, the skew normal distribution becomes the normal distribution $ N_n(\\bm{\\mu}, \\bm{\\Sigma}$).\n\nThe Student-t distribution is used to estimate the mean of a normally distributed population when a sample size is small and its standard deviation is unknown. Adding a parameter for the number of degrees of freedom $\\nu$, the PDF of the multivariate student-t distribution can be expressed as \\citep[e.g.,][]{Johnson1994, Lachos2014},\n\\begin{equation} \\label{eq:2}\n  \\frac{\\Gamma(\\frac{n+\\nu}{2})}{\\Gamma(\\frac{\\nu}{2})(\\nu\\pi)^{n\/2}}|\\bm{\\Sigma}|^{-1\/2}(1+\\frac{d}{\\nu})^{-(\\frac{n+\\nu}{2})},\n\\end{equation}\nwhere $\\Gamma$ is the gamma function, and d is the Mahalanobis distance $d = (\\mathbf{x} - \\bm{\\mu})^{\\rm T}\\bm{\\Sigma}^{-1}(\\mathbf{x} - \\bm{\\mu})$.\nWhen $\\nu = \\infty$, the student-t distribution becomes the the normal distribution.\n\nWith a shape parameter vector $\\bm{\\lambda}$ and a degree of freedom parameter $\\nu$, the PDF of the multivariate Skew-t distribution $ST_n(\\bm{\\mu}, \\bm{\\Sigma}, \\bm{\\lambda}, \\nu)$ can be described as \\citep[e.g.,][]{Johnson1994, Lachos2014},\n\\begin{equation} \\label{eq:3}\n  2t_n(\\mathbf{x}|\\bm{\\mu}, \\bm{\\Sigma}, \\nu)T_n(\\mathbf{x}|\\bm{\\mu}, \\bm{\\Sigma}, \\nu),\n\\end{equation}\nwhere $t_n(\\mathbf{x}|\\bm{\\mu}, \\bm{\\Sigma}, \\nu)$ and $T_n(\\mathbf{x}|\\bm{\\mu}, \\bm{\\Sigma}, \\nu)$ are the PDF and CDF of the student-t distribution. When $\\nu = \\infty$ the Skew-t distribution becomes the Skew-normal distribution. Figure \\ref{fig:skewt} shows the distributions of quasar relative fluxes $f_u\/f_r$, $f_g\/f_r$, $f_i\/f_r$, $f_z\/f_r$ in a narrow bin with $1.5<z_s<1.6$ and $18.5<r<19.0$. These relative flux distributions are obviously skewed and show tails even when excluding broad absorption line (BAL) quasars. We present the Anderson-Darling goodness of fit tests \\citep{Marsaglia2004} using the R $ADGofTest$ package\\footnote{https:\/\/cran.r-project.org\/web\/packages\/ADGofTest} for distributions shown in Figure \\ref{fig:skewt}. The probability values fit with the Gaussian function, the Skew-normal function, and the Skew-t function for $f_u\/f_r$ distribution are $4.46\\mathrm{e}{-7}$, 0.0168, and 0.768; for $f_g\/f_r$ distribution are $4.46\\mathrm{e}{-7}$, 0.111, and 0.986; for $f_i\/f_r$ distribution are $4.87\\mathrm{e}{-6}$, $5.48\\mathrm{e}{-4}$, and 0.539; and for $f_z\/f_r$ distribution are $4.47\\mathrm{e}{-7}$, $7.72\\mathrm{e}{-5}$, and 0.913. This indicates clearly that the advantage of Skew-t model is statistically significant. The Skew-t functions fit the quasar relatve fluxes better than the Gaussian and the Skew-Normal functions do.\n\nThe quasar sample is divided into a grid of redshifts and magnitudes with bin sizes of $\\Delta z = 0.05$ and $\\Delta m = 0.1$. A redshift bin 0.05 is acceptable for photo-$z$ regression. A smaller redshift bin will lead to poor statistics in any single bin. We use the R $sn$ package\\footnote{https:\/\/cran.r-project.org\/web\/packages\/sn\/index.html}$^{,}$\\footnote{http:\/\/azzalini.stat.unipd.it\/SN\/} to do a maximum penalized likelihood estimation to model the multivariate relative flux distribution in each redshift and magnitude bin, and get the $\\bm{\\mu}(z, m)$, $\\bm{\\Sigma}(z, m)$, $\\bm{\\lambda}(z, m)$ and $\\nu(z, m)$ parameters. The $g$-band magnitude of the SDSS reaches a fainter depth than other bands, and the second faint band is the $r$ band \\citep{McGreer2013}. Due to the Lyman-$\\alpha$ emission shifting out of the $g$ band and the Lyman forest absorptions, g-band magnitudes of $z>4.6$ quasars become faint. Quasars are therefore divided into magnitude bins based on the $r$-band magnitude, and we use the relative fluxes between other band fluxes and the $r$-band flux, for example $f_u\/f_r$, $f_g\/f_r$, $f_i\/f_r$ and $f_z\/f_r$ when using the SDSS five-band photometry. Each relative flux is a dimension in the Skew-t multi-dimensional model. The covariance between relative fluxes is accounted for by the covariance matrix $\\bm{\\Sigma}$.\n\nTo calculate the PDF, we weigh relative fluxes using photometric uncertainties as follows. For example, for the four relative fluxes of the SDSS photometry $f_u\/f_r$, $f_g\/f_r$, $f_i\/f_r$ and $f_z\/f_r$, with flux uncertainties $e_{u}$, $e_{g}$, $e_{r}$, $e_{i}$ and $e_{z}$ in the five SDSS bands, the relative flux covariance matrix can be derived from the error propagation equations as,\n\\begin{equation} \\label{eq:4}\n\\bm{\\Sigma}_0 =\n\\left(\n\\begin{array}{ccccc}\n\\frac{e_u^2 f_r^2 + e_r^2 f_u^2}{f_r^4} & \\frac{f_u f_g e_r^2}{f_r^4} & \\frac{f_u f_i e_r^2}{f_r^4} & \\frac{f_u f_z e_r^2}{f_r^4} \\\\\n\\frac{f_u f_g e_r^2}{f_r^4} & \\frac{e_g^2 f_r^2 + e_r^2 f_g^2}{f_r^4} & \\frac{f_g f_i e_r^2}{f_r^4} & \\frac{f_g f_z e_r^2}{f_r^4} \\\\\n\\frac{f_u f_i e_r^2}{f_r^4} & \\frac{f_g f_i e_r^2}{f_r^4} & \\frac{e_i^2 f_r^2 + e_r^2 f_i^2}{f_r^4} & \\frac{f_i f_z e_r^2}{f_r^4} \\\\\n\\frac{f_u f_z e_r^2}{f_r^4} & \\frac{f_g f_z e_r^2}{f_r^4} & \\frac{f_i f_z e_r^2}{f_r^4} & \\frac{e_z^2 f_r^2 + e_r^2 f_z^2}{f_r^4}\\\\\n\\end{array}\n\\right)\n.\n\\end{equation}\n\\vspace{0cm}\n\nWhen combining optical photometry and mid-infrared photometry that are taken separately for years, the quasar variability introduces extra uncertainties into the relative fluxes, such as $f_{W1}\/f_r$ and $f_{W2}\/f_r$. To reduce the uncertainties from quasar variability, we use $f_{W1}\/f_r$ and $f_{W2}\/f_{W1}$ for quasar photo-$z$ estimation. In the case of using the DECaLS $g$, $r$, $z$ and WISE W1, W2 photometry, the relative fluxes used are $f_g\/f_r$, $f_z\/f_r$, $f_{W1}\/f_r$ and $f_{W2}\/f_{W1}$, and the covariance matrix $\\bm{\\Sigma}_0$ is written as\n\\begin{equation} \\label{eq:5}\n\\left(\n\\begin{array}{ccccc}\n\\frac{e_g^2 f_r^2 + e_r^2 f_g^2}{f_r^4} & \\frac{f_g f_z e_r^2}{f_r^4} & \\frac{f_g f_{W1} e_r^2}{f_r^4} & 0\\\\\n\\frac{f_g f_z e_r^2}{f_r^4} & \\frac{e_z^2 f_r^2 + e_r^2 f_z^2}{f_r^4} & \\frac{f_z f_{W1} e_r^2}{f_r^4} & 0\\\\\n\\frac{f_g f_{W1} e_r^2}{f_r^4} & \\frac{f_z f_{W1} e_r^2}{f_r^4} & \\frac{e_{W1}^2 f_r^2 + e_r^2 f_{W1}^2}{f_r^4} & -\\frac{e_{W1}^2 f_{W2}}{f_{W1}^2}\\\\\n0 & 0 & -\\frac{e_{W1}^2 f_{W2}}{f_{W1}^2} & \\frac{e_{W2}^2 f_{W1}^2 + e_{W1}^2 f_{W2}^2}{f_{W1}^4} \\\\\n\\end{array}\n\\right).\n\\end{equation}\n\nThen the covariance matrix is $\\bm{\\Sigma}^*(z, m) = \\bm{\\Sigma}(z, m) + \\bm{\\Sigma}_0$. The posterior probability is expressed as\n\\begin{equation} \\label{eq:6}\nP_{\\rm QSO}(\\bm{f}|z, m) = ST_n(\\bm{\\mu}(z, m), \\bm{\\Sigma^*}(z, m), \\bm{\\lambda}(z, m), \\nu(z, m)),\n\\end{equation}\nwhere $\\bm{f}$ represents the relative fluxes.\n\nPS1, DECaLS, and WISE photometry are based on multiple epochs of imaging data, and the effect of variability is mitigated. It happens\nthat PS1, DECaLS, and WISE are all averages over a roughly similar timeframe (although DECaLS is mostly a couple of years after PS1),\nwhereas SDSS is about a decade earlier than the others. Therefore, combinations of SDSS and the other surveys will be the most impacted\nby long-term variability.\n\n\\begin{figure}[htbp]\n\\hspace*{-0.5cm}\n\\epsscale{1.3}\n\\plotone{f2_QLF_part.pdf}\n\\vspace{-0.5cm}\n\\caption{\\label{fig:qlf}The quasar number prior $N_{\\rm QSO}(z, m)$ per deg$^2$ as a function of redshift and magnitude with $\\Delta z = 0.05$ and $\\Delta g = 0.1$, derived from the QLF in \\citet{NPD2016}. The curves, from top to bottom, are for $g = $ 22.0, 21.5, 21.0, 19.5, 19.0, 18.5, 18.0 mag, respectively.}\n\\end{figure}\n\n\\subsection{Prior Probability from the QLF} \\label{subsec:qlf}\nThe number density of quasars depends on the redshift and luminosity \\citep[e.g.,][]{Ross2013, NPD2016}. The QLF characterizes quasars through the evolution of their number density with luminosity and redshift. \\citet{NPD2016} present the QLF using quasars from the extended Baryon Oscillation Spectroscopic Survey of the Sloan Digital Sky Survey (SDSS-IV\/eBOSS). Their quasar sample is 80\\% complete to $g = 20$ mag and 50\\% complete to $g = 22.5$ mag, and the QLF has been corrected for incompleteness. We derive the quasar number prior $N_{\\rm QSO}(z, m)$ per deg$^2$ as a function of redshift and magnitude with $\\Delta z = 0.05$ and $\\Delta g = 0.1$ from the QLF in \\citet{NPD2016} derived in the SDSS $g$ band, and with the k-corrections as a function of both redshift and luminosity \\citep{McGreer2013}. Figure \\ref{fig:qlf} shows the number distribution as a function of redshift for quasars with $g = $ 22.0, 21.5, 21.0, 19.5, 19.0, 18.5, 18.0 mag from top to bottom.\n\nThus the PDF is obtained with the posterior probability and the number prior as\n\\begin{equation} \\label{eq:7}\n  P_{\\rm QSO}(z) = P_{\\rm QSO}(\\bm{f}|z, m) N_{\\rm QSO}(z, m).\n\\end{equation}\nUsing the PDF as a function of redshift, $P_{\\rm QSO}(z)$, the photo-$z$ can be estimated by a maximum probability method or a peak recognition with maximum integrated probability. We identify peaks in a PDF curve using the $findpeaks$ function in R $pracma$ package\\footnote{https:\/\/cran.r-project.org\/web\/packages\/pracma\/index.html}, and calculate the photo-$z$ as the peak with the largest integrated PDF within a redshift range $(z_1, z_2)$. A parameter $P_{\\rm prob}$ describes the probability that the redshift locates within $(z_1, z_2)$ is\n\\begin{equation} \\label{eq:8}\nP_{\\rm prob} = \\frac{\\int_{z_1}^{z_2} P_{\\rm QSO}(z) dz}{\\int P_{\\rm QSO}(z) dz}\n.\n\\end{equation}\n\nThe logarithmic likelihood ($L$) of an object to be a quasar over the whole redshift range is written as\n\\begin{equation} \\label{eq:9}\nL_{\\rm QSO} = {\\rm log}(P_{\\rm QSO}) = {\\rm log}\\int P_{\\rm QSO}(z) dz .\n\\end{equation}\n\nTo assess the impact of the prior distribution on the photo-$z$ regression results, we also present the results with photo-$z$ derived only from the posterior distribution. The PDF from the posterior probability is\n\\begin{equation} \\label{eq:10}\n  P_{\\rm QSO}'(z) = P_{\\rm QSO}(\\bm{f}|z, m).\n\\end{equation}\nThe logarithmic likelihood of an object to be a quasar from the posterior distribution over the whole redshift range is written as\n\\begin{equation} \\label{eq:11}\nL'_{\\rm QSO} = {\\rm log}(P_{\\rm QSO}') = {\\rm log}\\int P_{\\rm QSO}'(z) dz .\n\\end{equation}\nThe influence of prior distribution on the photo-$z$ regression and quasar candidate selection is discussed in Sections \\ref{subsec:photoz_infrared} and \\ref{subsec:selection}.\n\n\\begin{figure}[htbp]\n\\hspace*{-0.5cm}\n\\epsscale{1.3}\n\\plotone{f3_compare.pdf}\n\\vspace{-0.5cm}\n\\caption{\\label{fig:compare}The photo-$z$ distributions from different photo-$z$ methods, including Skew-t (red solid), XDQSOz (blue dashed) and CZR (blue dotted), compared with the spectroscopic redshift $z_s$ distribution (gray shade), using the SDSS five-band photometry for the same quasar test sample. The photo-$z$ distribution from the Skew-t model is more similar to the $z_s$ distribution, while the CZR method identifies more $z\\sim 0.8$ quasars and the XDQSOz method identifies more $z \\sim 2.2$ quasars.}\n\\end{figure}\n\n\\begin{figure*}\n  \\centering\n  \\hspace{0cm}\n    \\subfigure{\n    \\includegraphics[width=3.6in]{f4a_sdss.pdf}}\n  \\hspace{-1.4cm}\n  \\subfigure{\n    \\includegraphics[width=3.6in]{f4b_sdss_wise.pdf}}\\\\\n  \\vspace{-0.3cm}\n  \\hspace{0cm}\n  \\subfigure{\n    \\includegraphics[width=3.6in]{f4c_ps1_wise.pdf}}\n  \\hspace{-1.4cm}\n  \\subfigure{\n    \\includegraphics[width=3.6in]{f4d_decals_wise.pdf}}\\\\\n  \\caption{\\label{fig:photoz}Photo-$z$ ($z_p$) compared with spectroscopic redshifts ($z_s$) for SDSS, SDSS-WISE, PS1-WISE and DECaLS-WISE photometry, respectively. The degeneracy between $z\\sim 0.8$ and $z\\sim2.2$ is obvious when using only the SDSS photometry, and is alleviated by combining optical data with mid-infrared photometry.}\n  \\vspace{0.5cm}\n\\end{figure*}\n\n\\begin{figure}[htbp]\n\\hspace*{-1cm}\n\\epsscale{1.4}\n\\plotone{f5_offset.pdf}\n\\vspace{-0.5cm}\n\\caption{\\label{fig:offset}The normalized distributions of $\\Delta z$ for the different sources of photometry listed in Table \\ref{tab:photoz_infrared}. The histograms are SDSS (S, gray shade), SDSS-WISE (SW, blue solid), PS1-WISE (PW, black dashed), and DECaLS-WISE (SW, red dotted), respectively.}\n\\end{figure}\n\n\\begin{figure*}\n  \\centering\n  \\hspace{-0.7cm}\n    \\subfigure{\n    \\includegraphics[width=3.7in]{f6a_ratio_redshift.pdf}}\n  \\hspace{-1.2cm}\n  \\subfigure{\n    \\includegraphics[width=3.7in]{f6b_ratio_mag.pdf}}\n  \\caption{\\label{fig:raito}Photo-$z$ accuracy $R_{0.1}$ as a function of redshift (left panel) and magnitude (right panel) for different combinations of photometry, including SDSS (gray open dot-line), SDSS-WISE (black pentagon-line), PS1-WISE (black diamond-dashed line) and DECaLS-WISE (red triangle-dotted line). The photo-$z$ accuracies of PS1-WISE and DECaLS-WISE are lower than that of SDSS-WISE, mainly due to lack of $u$ band photometry in PS1 and DECaLS.}\n\\end{figure*}\n\n\\begin{table}\n\\small\n\\begin{center}\n\\tablewidth{1pt}\n\\caption{Photo-$z$ results for different methods with the same test sample of SDSS photometry} \\label{tab:photoz}\n\\begin{tabular}{lcrcccccc}\n\\tableline\\tableline\n${\\rm Method}$ & $\\sigma(\\Delta z)$ & $\\overline{\\Delta z}$ & $R_{0.1}$ & $R_{0.2}$ & ${\\rm Time}$\\\\% & $Photometry$ & $Number$ \\\\\n\\tableline\nSkew-t & 0.27 & -0.02  & 74.2\\% & 81.5\\% & 1 \\\\% & SDSS & 151,385\\\\\nXDQSOz & 0.31 & -0.04 & 72.8\\% & 79.3\\% & 17 \\\\%& SDSS & 151,385 \\\\\nKDE & 0.35 & -0.06 & 70.6\\% & 77.9\\% & 0.002 \\\\%& SDSS & 151,385 \\\\\nCZR & 0.29 & 0.05 & 68.0\\% & 73.9\\% & 0.005  \\\\%& SDSS & 151,385\\\\\n\\tablelin\n\\end{tabular}\n\\tablecomments{$R_{0.1}$ ($R_{0.2}$) is the fraction of quasars with $|\\Delta z|$ smaller than 0.1 (0.2). Time is calculated by using the same machine, and the time used by the Skew-t method to obtain the photo-$z$ results for the test sample is normalized to 1. A test calculation of 100,000 objects with SDSS five bands data using Skew-t method took 23 minutes (by one processor computer with 3.1 GHz Intel Core i7 CPU).}\n\\end{center}\n\\end{table}\n\n\\begin{table*}\n\\small\n\\begin{center}\n\\tablewidth{1pt}\n\\caption{Photo-$z$ results for different sources of photometry} \\label{tab:photoz_infrared}\n\\begin{tabular}{ccccrccc}\n\\tableline\\tableline\n${\\rm Photometry}$ & $N_{\\rm bands}$ & ${\\rm PDF}$ & $\\sigma(\\Delta z)$ & $\\overline{\\Delta z}$ & $R_{0.1}$ & $R_{0.2}$ & ${\\rm Numbe}r$ \\\\\n\\tableline\nSDSS & 5 & posterior \\& prior  & 0.27 & -0.024  & 74.9\\% & 82.0\\% &304,241 \\\\\n- & -  & posterior & 0.29 & -0.025  & 73.5\\% & 81.1\\% & - \\\\\n\\tableline\nSDSS-WISE & 7 & posterior \\& prior & 0.15 & -0.005 & 87.0\\% & 93.3\\% & 229,653 \\\\\n- & - & posterior & 0.16 & -0.007 & 85.8\\% & 92.5\\% & - \\\\\n\\tableline\nPS1-WISE & 7 & posterior \\& prior & 0.18 & -0.006 & 79.1\\% & 89.5\\% & 254,349 \\\\\n- & - & posterior & 0.22 & -0.03 & 77.0\\% & 87.6\\% & - \\\\\n\\tableline\nDECaLS-WISE & 5 & posterior \\& prior & 0.17 & 0.020 & 72.4\\% & 88.0\\% & 98,450 \\\\\n- & - & posterior & 0.23 & -0.002 & 72.3\\% & 87.5\\% & - \\\\\n\\tablelin\n\\end{tabular}\n\\tablecomments{The photo-$z$ results are calculated with the PDF derived from the posterior and prior distributions in Equation (\\ref{eq:7}) or from the posterior distribution in Equation (\\ref{eq:10}).}\n\\end{center}\n\\end{table*}\n\n\\section{photo-$z$ Results} \\label{sec:results}\n\\subsection{Comparing Photo-$z$ Results using the SDSS Photometric Data} \\label{subsec:photoz_sdss}\nWe compare the performance of our photo-$z$ regression algorithm with other methods by testing with the same sample of photometric data. We randomly divide the quasar sample with the SDSS photometric data into two subsamples, one as the training sample, and the other one as the test sample. We also try a KDE method mapping the two dimensional color-color distributions of $u-g$ versus $g-r$, $g-r$ versus $r-i$, and $r-i$ versus $i-z$ in redshift bins. The KDE photo-$z$ code is based on the KDE method in \\citet{Silverman1986}. The CZR photo-$z$ is calculated based on the CZR method in \\citet{Weinstein2004}. The XDQSOz photo-$z$ is calculated with the XDQSOz code \\citep{Bovy2012}. The photo-$z$ results of the Skew-t, XDQSOz, KDE and CZR methods are listed in Table \\ref{tab:photoz}.\n\nThe Skew-t photo-$z$ algorithm performs well compared with other photo-$z$ methods. The difference between the photo-$z$ ($z_p$) and the spectroscopic redshift ($z_s$) is expressed as $\\Delta z = (z_s-z_p)\/(1+z_s)$. $R_{0.1}$ ($R_{0.2}$) is the fraction of quasars with $|\\Delta z|$ smaller than 0.1 (0.2). The standard deviation of $\\Delta z$, $\\sigma(\\Delta z)$, from the Skew-t photo-$z$ is 0.27, slightly better than 0.31 and 0.29 from the XDQSOz and CZR methods. The KDE method is memory consuming when the number of dimensions is high. The KDE method also strongly depends on its training sample. When the test sample is the same as the training sample, the $R_{0.1}$ is as high as 85\\% if 4 SDSS colors are used. It decreases to 70\\% when the test sample is different from the training sample. So, the KDE photo-$z$ method is easily over-trained. For the Skew-t method, when the test sample is the same as the training sample, the accuracy $R_{0.1}$ changes by less than 1\\% (74.2\\% in Table \\ref{tab:photoz} and 74.9\\% in Table \\ref{tab:photoz_infrared}).\n\n\\begin{figure}[htbp]\n\\hspace*{-1cm}\n\\epsscale{1.4}\n\\plotone{f7_hist_simulation.pdf}\n\\vspace{-0.5cm}\n\\caption{\\label{fig:hist}The number distribution as a function of the $r$-band magnitude for the DECaLS ``PSF\" type objects (black), ``PSF\" type objects with $\\Delta \\chi^2>40$ (gray dashed), and stellar simulation (blue) within a 20 deg$^2$ test region in S82 ($340^{\\circ}<{\\rm R.A.}<350^{\\circ}$, $-1^{\\circ}<{\\rm Decl.}<1^{\\circ}$). The contaminations of point-like galaxies become prominent at the faint end.}\n\\end{figure}\n\n\\begin{figure*\n  \\centering\n  \\hspace{-0.7cm}\n    \\subfigure{\n    \\includegraphics[width=3.5in]{f8a_gr_rz.pdf}}\n  \\hspace{-0.2cm}\n  \\subfigure{\n    \\includegraphics[width=3.5in]{f8b_zW1_W12.pdf}}\n  \\caption{\\label{fig:colorcolor}Color-color diagram of $g-r$ versus $r-z$ (left panel) and $z-W1$ versus $W1-W2$ (right panel) for $19<r<20$ objects in a 20 deg$^2$ test region ($340^{\\circ}<{\\rm R.A.}<350^{\\circ}$, $-1.0^{\\circ}<{\\rm Decl.}<1.0^{\\circ}$). The gray dots are the DECaLS ``PSF\" type objects, and the blue contours are stellar simulation colors using DECAM and WISE filters.}\n\\end{figure*}\nFigure \\ref{fig:compare} shows the spectroscopic redshift distribution compared with the Skew-t, XDQSOz, and CZR photo-$z$ distributions. The SDSS colors of $z \\sim 2.2$ quasars look similar to those of $z\\sim 0.8$ quasars, as the C III] and MgII lines shift into the $g$ band at $z \\sim 2.2$ and $z \\sim 0.8$, respectively. This is a degeneracy, if we use only the SDSS photometry to calculate the photo-$z$ of $z\\sim 2.2$ and $z\\sim 0.8$ quasars. We present Anderson-Darling goodness of fit tests with the R $kSamples$ package\\footnote{https:\/\/cran.r-project.org\/web\/packages\/kSamples}. Randomly choosing 1000 objects from the sample and comparing with spectroscopic redshift, the probability values of photo-$z$ from method Skew-t, XDQSOz, and CZR are 0.207, 0.102, and approximately 0, respectively. The Skew-t method performs better than other methods on this problem. A better result requires more photometric data, such as infrared photometry.\n\n\\subsection{Photo-$z$ Results Using Optical and Mid-Infrared Photometry} \\label{subsec:photoz_infrared}\nAdding the infrared photometric data will make the photo-$z$ results more robust \\citep{Wu2010, Wu2012}. Using SDSS\/PS1\/DECaLS optical photometry with WISE W1 and W2 mid-infrared photometry, the $\\sigma(\\Delta z)$ values are reduced to 0.15, 0.18, and 0.17, respectively (Table \\ref{tab:photoz_infrared}). The photo-$z$ accuracy $R_{0.1}$ is enhanced to 87.0\\%, 89.1\\%, and 72.4\\%, respectively. As WISE data are shallower than optical data, using WISE photometry will reduce the number of quasars detected. Figure \\ref{fig:photoz} shows the photo-$z$ versus spectroscopic redshift for SDSS, SDSS-WISE, PS1-WISE, and DECalS-WISE, respectively. Figure \\ref{fig:offset} shows their $\\Delta z$ distributions. Figure \\ref{fig:raito} shows the photo-$z$ accuracy $R_{0.1}$ as a function of redshift (left panel) and $r$-band magnitude (right panel), respectively. The degeneracy problem between $z \\sim 2.2$ and $z \\sim 0.8$ is alleviated with the inclusion of mid-infrared data. PS1 and DECaLS do not have $u$-band data, so the photo-$z$ results are less accurate than those for SDSS-WISE. At $z>3.4$, the Lyman limit moves out of the $u$ band, and then the $u$ band photometry is not important for the photo-$z$ regression any more. Optical and mid-infrared data are sufficient for photo-$z$ regression at $3.4<z<5$. The photo-$z$ results derived without prior probability in Equation (\\ref{eq:10}) are also listed in Table \\ref{tab:photoz_infrared}. The $\\sigma(\\Delta z)$ values increase 0.1-0.5, and the photo-$z$ accuracy decreases 0.1\\%-2.1\\% without prior probability.\n\n\\section{The Classification method} \\label{sec:classification}\n\\subsection{Stellar Simulation} \\label{subsec:simulation}\nStars are the main contaminants for quasar candidate selection. \\citet{Fan1999} simulated the SDSS colors of Galactic stars and showed that the simulated colors of stars are in good agreement with observations in the SDSS. He proved that stellar simulation can be used as a tool to separate stars and quasars. \\citet{Robin2003} built a synthesis model of stellar populations (the Besan\\c{c}on Galaxy Model\\footnote{http:\/\/model.obs-besancon.fr}) consisting of stars from 4 populations including thin disk, thick disk, spheroid, and bulge. Each population is described by a star formation rate history, an Initial Mass Function and an age (or age-range) (see Table 1 in \\citet{Robin2003}). Density laws for the thin disk are constrained self-consistently by the potential in the Boltzmann equation (see Table 3 in \\citet{Robin2003}). \\citet{Sharma2011} presented a code called GALAXIA creating a synthetic survey of the Milky Way based on parameters in the Besan\\c{c}on model. We built up the stellar color distribution with a simulation of the Milky Way using the GALAXIA code within 30 kpc, and updated the Padova isochrones to the PARSEC isochrones \\citep[PARSEC v1.2S\\footnote{http:\/\/stev.oapd.inaf.it\/cmd};][]{Bressan2012, Chen2014, Chen2015, Tang2014}.\n\nFigure \\ref{fig:hist} shows an example of a stellar simulation with DECAM filters in a 20 deg$^2$ test region ($340^{\\circ}<{\\rm R.A.}<350^{\\circ}$, $-1.0^{\\circ}<{\\rm Decl.}<1.0^{\\circ}$). Figure \\ref{fig:hist} shows the $r$-band magnitude distribution of the DECaLS point objects (${\\rm type} =``{\\rm PSF}\"$, black histogram) compared with the simulated stars (blue histogram). The gray dashed histogram shows ``PSF\" type objects with $\\Delta \\chi^2>40$, where $\\Delta \\chi^2$ is the $\\chi^2$ difference between fitting to a PSF model and a simple galaxy model. Smaller $\\Delta \\chi^2$ means that the object is more likely to be similar to a galaxy morphology. Point-like galaxy is possible to be identified as ``PSF\" morphology but remains smaller $\\Delta \\chi^2$ comparing to those objects with true ``PSF\" morphologies. The difference between the simulation and observation at $r \\sim 24$ are mainly caused by the contaminations of point-like galaxies. At fainter magnitudes, the contamination by point-like galaxies becomes more and more prominent. At the faint end, the observation is not consistent with the simulation due to the magnitude limit of the imaging survey, which is roughly 23.9 mag in the r band for DELS. Figure \\ref{fig:colorcolor} shows the $g-r$ versus $r-z$ (left panel) and $z-W1$ versus $W1-W2$ (right panel) color-color diagrams of stars with $19<r<20$ in this region with DECAM and WISE detections. The simulated colors (blue contours) trace the observed stellar locus (gray dots) well, so objects deviating from the stellar locus can be easily found.\n\nThe probability of one object to be a star is expressed as\n\\vspace{-0.1cm}\n\\begin{equation} \\label{eq:12}\n\\vspace{-0.2cm}\n  P_{\\rm Star} = \\sum_{i}^{N_{\\rm Star}(m)}{P_{\\rm Star}(\\bm{f}|m)},\n\\end{equation}\nwhere $N_{\\rm Star}(m)$ is the number of stars with magnitude bin $\\Delta m=0.1$ in 1 deg$^2$, and $P_{\\rm Star}(\\bm{f}|m)$ is expressed as a multivariate Gaussian distribution\n\\begin{equation} \\label{eq:13}\n  P_{\\rm Star}(\\bm{f}|m) = N(\\bm{\\mu}, \\bm{\\Sigma^*}),\n\\end{equation}\nwhere $\\mu$ comes from the relative fluxes of simulated stars. For example, in the case of using the DECaLS $grz$ and WISE W1 and W2 magnitudes, we use $f_g\/f_r$, $f_z\/f_r$, $f_{W1}\/f_r$ and $f_{W2}\/f_{r}$, as most stars do not vary significantly and the photometric error in the $r$-band is smaller than that in the W1 band. There are no errors assigned to the simulated relative fluxes, and $\\bm{\\Sigma^*}$ is the covariance matrix for a target with flux uncertainties $e_g, e_r, e_z, e_{W1}, e_{W2}$, namely\n\\begin{equation} \\label{eq:14}\n{\\bm{\\Sigma}^*} =\n\\left(\n\\begin{array}{ccccc}\n\\frac{e_g^2 f_r^2 + e_r^2 f_g^2}{f_r^4} & 0 & 0 \\\\\n0 & \\frac{e_z^2 f_r^2 + e_r^2 f_z^2}{f_r^4} & 0 & 0 \\\\\n0 & 0 & \\frac{e_{W1}^2 f_r^2 + e_r^2 f_{W1}^2}{f_r^4} & 0 \\\\\n0 & 0 & 0 & \\frac{e_{W2}^2 f_r^2 + e_r^2 f_{W2}^2}{f_r^4}\\\\\n\\end{array}\n\\right)\n\\end{equation}\nWe do not apply Galactic extinction to the observed data or the simulated relative fluxes when calculating the probability to be a star. The logarithmic likelihood of an object to be a star is defined as\n\\begin{equation} \\label{eq:15}\nL_{\\rm Star} = {\\rm log}(P_{\\rm Star}).\n\\end{equation}\n\n\\begin{figure}[htbp]\n\\hspace*{-0.5cm}\n\\epsscale{1.2}\n\\plotone{f10_flowchart_skew.pdf}\n\\caption{\\label{fig:flowchart}Quasar candidate selection flowchart. The classification process includes morphological selection and fitting the relative fluxes with quasar, star and galaxy fluxes. Comparing the three probabilities with Bayesian theory, we can tell whether an object is a quasar or not. The $x_1$ criterion is used to insure that the colors are not far away from the quasar colors, $x_2$ is used to establish that it is more probable to be a quasar than a star, and $x3$ to tell that it is more probable to be a quasar than a galaxy.}\n\\end{figure}\n\n\\begin{figure*}\n  \\centering\n  \\hspace{-0.3cm}\n    \\subfigure{\n    \\includegraphics[width=3.7in]{f9a_noPrior.pdf}}\n  \\hspace{-1.0cm}\n  \\subfigure{\n    \\includegraphics[width=3.7in]{f9b_withPrior.pdf}}\n  \\caption{\\label{fig:prior}The logarithmic likelihoods of objects to be quasars from the posterior probability $L'_{\\rm QSO}$ in Equation (\\ref{eq:11}) (left panel), and those from a convolution of the posterior probability and the prior probability $L_{\\rm QSO}$ in Equation (\\ref{eq:9}) (right panel). The x-axis is $r$-band magnitude. The photometry used in these two panels are the DECaLS $g, r,$ and $z$-band photometry, and WISE W1 and W2-band photometry. Density contours are in blue. For fainter objects, the posterior probabilities decrease due to the larger photometric uncertainties. According to the QLF, a fainter object is more likely to be a quasar compared to a brighter object. $L_{\\rm QSO}$ is more uniform than $L'_{\\rm QSO}$ over the range of magnitudes.}\n\\end{figure*}\n\n\\subsection{Galaxy template fitting} \\label{subsec:galaxy}\nIt becomes more difficult to classify a fainter object as star or galaxy by morphology, thus the contamination from point-like galaxies becomes more significant. We reduce the contamination of point-like galaxies by a galaxy template fitting procedure. For the dataset that includes WISE photometry, we use a subsample of 18 galaxy spectral energy distributions (SEDs) from \\citet{Brown2014} with wavelength coverage from ultraviolet to mid-infrared. \\citet{Benitez2000} presented a BPZ algorithm to estimate the photo-$z$ of galaxies. They derived the galaxy prior probability as $p(z, {\\rm type}, m)=p({\\rm type}|m)p(z|{\\rm type}, m)$, where $p({\\rm type}|m)$ is the galaxy type fraction as a function of magnitude, and $p(z|{\\rm type}, m)$ is the redshift distribution of a given galaxy type and magnitude. As many imaging observations were taken under 1\\arcsec seeing, galaxies with half-light radius $R_e<0.5\\arcsec$ \\citep{Fan1999} are treated as point-like galaxies. \\citet{Shen2003} obtained the size distributions of early and late type galaxies and their dependence on luminosity from 140,000 SDSS galaxies. We derive the prior distribution $p(z, {\\rm type}, m)$ of point-like galaxy for spectral types corresponding to E\/S0, Sbc\/Scd and Irr (parameters from Table 1 in \\citet{Benitez2000}) with $R_e<0.5\\arcsec$. The probability of one object being a galaxy over the whole redshift range is expressed as\n\\begin{equation} \\label{eq:16}\n  P_{\\rm Galaxy} = \\sum\\sum_{i}\\int{P_{\\rm Galaxy}(z)N_{\\rm Galaxy}(z, {\\rm type},m)dz},\n\\end{equation}\nwhere $\\Sigma$ is the sum of all galaxy types, and $\\Sigma_{i}$ is the sum of all galaxy SEDs for each type, and $N_{\\rm Galaxy}(z, {\\rm type},m)$ is the number of a certain type of galaxy at redshift z and magnitude m within $\\Delta z = 0.05$ and $\\Delta m = 0.1$ per $deg^2$. $P_{\\rm Galaxy}(z)$ is expressed as a multivariate Gaussian distribution\n\\begin{equation} \\label{eq:17}\n  P_{\\rm Galaxy}(z) = N(\\bm{\\mu}(z), \\bm{\\Sigma}^*) ,\n\\end{equation}\nwhere $\\mu(z)$ comes from the relative fluxes of galaxy templates as a function of redshift, and $\\Sigma^*$ is described in Equation (\\ref{eq:14}).\n$N_{\\rm Galaxy}(z, {\\rm type},m)$ is derived from the prior distribution $p(z, {\\rm type}, m)$ using a scale factor c, and defining $P'_{\\rm Galaxy}$ as\n\\begin{equation} \\label{eq:18}\n  P'_{\\rm Galaxy} = \\sum\\sum_{i}\\int{P_{\\rm Galaxy}(z)p(z, {\\rm type},m)dz},\n\\end{equation}\nThe logarithmic likelihood of an object being a galaxy is defined as\n\\begin{equation} \\label{eq:19}\nL_{\\rm galaxy} = {\\rm log}(P_{\\rm galaxy}) = {\\rm log(c)} + L'_{\\rm galaxy},\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:20}\nL'_{\\rm galaxy} = {\\rm log}(P'_{\\rm galaxy}).\n\\end{equation}\n\n\\subsection{Quasar Candidate Selection Flowchart} \\label{subsec:flowchart}\nQuasars can be selected based on Bayesian probabilities \\citep[e.g.,][]{Richards2009b, Richards2015, Kirkpatrick2011}. Considering a point-like object that is likely to be a quasar, star or galaxy, the Bayesian probability of being a quasar is expressed as\n\\begin{equation} \\label{eq:21}\nP({\\rm QSO}) = \\frac{P_{\\rm QSO}}{P_{\\rm QSO} + P_{\\rm Star} + P_{\\rm Galaxy}}\n\\end{equation}\nwhere $P_{\\rm QSO}$, $P_{\\rm Star}$ and $P_{\\rm Galaxy}$ are expressed in Equations (\\ref{eq:7}), (\\ref{eq:12}) and (\\ref{eq:17}), respectively. A Bayesian probability criterion is usually defined as $P({\\rm QSO})>x$, namely\n\\begin{equation} \\label{eq:22}\n\\frac{P_{\\rm Star}}{P_{\\rm QSO}} + \\frac{P_{\\rm Galaxy}}{P_{\\rm QSO}} < \\frac{1}{x} - 1.\n\\end{equation}\nHere we suggest three Bayesian probability criteria as\n\\begin{equation} \\label{eq:23}\nL_{\\rm QSO} > x_1,\n\\end{equation}\n\\begin{equation} \\label{eq:24}\nL_{\\rm QSO} - L_{\\rm Star} > x_2,\n\\end{equation}\n\\begin{equation} \\label{eq:25}\nL_{\\rm QSO} - L'_{\\rm Galaxy} > x_3,\n\\end{equation}\nwhich correspond to\n\\begin{equation} \\label{eq:26}\nP_{\\rm QSO} > 10^{x_1}\n\\end{equation}\n\\begin{equation} \\label{eq:27}\n\\frac{P_{\\rm Star}}{P_{\\rm QSO}} < 10^{x_2}\n\\end{equation}\n\\begin{equation} \\label{eq:28}\n\\frac{P_{\\rm Galaxy}}{P_{\\rm QSO}} < c10^{x_3}\n\\end{equation}\nThese criteria mean that (1) the object has relative fluxes similar to quasars, (2) the object is more likely to be a quasar than a star, (3) the object is more likely to be a quasar than a galaxy. The quasar candidate selection flowchart is shown in Figure \\ref{fig:flowchart}. For a given object, we measure its relative fluxes and magnitudes, and then apply a morphology criterion that most quasars are point-like objects. Then we calculate the probability of the object being (1) a quasar, with a prior probability derived from the QLF, and a posterior probability modeled with a multivariate Skew-t distribution as a function of magnitude and redshift; (2) a star, with a prior probability from number counts and distribution of stellar parameters from a Milky Way synthetic simulation, and a posterior distribution modeled by a multivariate Gaussian distribution with relative fluxes from the Padova isochrones; (3) a galaxy, with a prior probability from the BPZ prior distribution for point-like galaxies, and a posterior probability modeled by a multivariate Gaussian distribution with relative fluxes from galaxy templates. We obtain quasar candidates, as well as photo-$z$s, with the three Bayesian probability criteria in Equations (\\ref{eq:23}), (\\ref{eq:24}), (\\ref{eq:25}).\n\nFor fainter objects, the posterior probabilities decrease due to larger photometric uncertainties. The left panel in Figure \\ref{fig:prior} shows the logarithmic likelihoods of the objects to be quasars integrated from the posterior probability ($L'_{\\rm QSO}$ in Equation (\\ref{eq:11})). According to the QLF, a fainter object is more likely to be a quasar compared to a brighter object. The right panel in Figure \\ref{fig:prior} shows the logarithmic likelihoods integrated from posterior probabilities and prior probabilities ($L_{\\rm QSO}$ in Equation (\\ref{eq:9})). $L_{\\rm QSO}$ is more uniform than $L'_{\\rm QSO}$ over the range of magnitudes. A criterion of a $L_{\\rm QSO}$ cut is more reasonable than a simple $\\chi^2$ cut or a probability cut without considering photometric uncertainties. As a consequence, the selection completeness will be affected by the prior distribution from the QLF. For example, if the bright end of the QLF is underestimated, some bright quasars with colors deviating from bright normal quasars may be missed.\n\n\\begin{figure*\n\\centering\n\\hspace{0cm}\n\\epsscale{1.1}\n\\plotone{f12_x123.pdf}\n\\caption{\\label{fig:roc} Trade-off between completeness and efficiency in the classfication procedure. Left panel: the dots are results with $x_1 \\in (-3.5, -0.5)$, $x_2 \\in (0, 15)$ and $x_3 \\in (0, 10)$, and the color map shows the value of $x_1$. Black pentagons mark $x_1 = -2$, which are obviously located at the edge of the PR diagram. Right panel: results with $x_1 = -2.0$, $x_2 \\in (0, 15)$ and $x_3 \\in (0, 10)$. The blue dot-line denotes $x_2 = 4.4$, and is located at the edge of the curves for a wide range of $x_3$ values. The magenta open dot-line denotes $x_3 = 5.4$, and is located at the edge of the curves when the completeness is in the range of $\\sim 79\\%$ to $\\sim 82\\%$. The black unfilled star marks the point $x_1 = -2$, $x_2 = 4.4$, and $x_3 = 5.4$.}\n\\end{figure*}\n\n\\begin{figure*\n\\centering\n\\hspace{0cm}\n\\epsscale{1.1}\n\\plotone{f11_L1_L2.pdf}\n\\caption{\\label{fig:ll} The $L_{\\rm QSO}-L'_{\\rm Galaxy}$ versus $L_{\\rm QSO}-L_{\\rm Star}$ diagram. The gray scale hexagon show the density of point sources (left panel) and quasars (right panel). The blue vertical line is the cut of $L_{\\rm QSO}-L_{\\rm Star} = 4.4$, and the blue horizontal line is the cut of $L_{\\rm QSO}-L'_{\\rm Galaxy} = 5.4$. Quasars span a much larger space in the $L_{\\rm QSO}-L'_{\\rm Galaxy}$ versus $L_{\\rm QSO}-L_{\\rm Star}$ diagram, and 11\\% of known quasars with ``PSF\" type and $L_{\\rm QSO} >-2$ are located below these two cuts. Meanwhile, 85\\% of point objects with $L_{\\rm QSO}>-2$ are excluded by these two cuts.}\n\\end{figure*}\n\n\\section{Discussion} \\label{sec:discussion}\n\\subsection{Quasar Candidate Selection using DECaLS and WISE photometry} \\label{subsec:selection}\nWe test the quasar candidate selection algorithm described in Section \\ref{subsec:flowchart} with the DECaLS $g$, $r$, $z$ and WISE W1 and W2 photometry. There is a 15 deg$^2$ region ($36^{\\circ}<{\\rm R.A.}<42^{\\circ}$ and $-1.25^{\\circ}<{\\rm Decl.}<1.25^{\\circ}$) in S82 with spectroscopically identified quasars as dense as 167 per deg$^2$. We exclude quasars in this region from the quasar training sample, and model the quasar relative fluxes posterior distribution using the method in Section \\ref{subsec:skewt}. For selection criteria, there is a trade-off between completeness and efficiency. We define the completeness as the completeness of selecting the spectroscopically identified quasars at $r<23$ mag and redshift $z<5.4$. The efficiency is defined as\n\\begin{equation} \\label{eq:29}\n{\\rm efficiency} = \\frac{N_{\\rm QLF}(r) * {\\rm completeness}(r)}{N_{{\\rm photo}-z}(r)},\n\\end{equation}\nwhere $N_{\\rm QLF}(r)$ is calculated from the QLF \\citep{NPD2016} (PLE+LEDE model). The completeness also includes $\\sim 5\\%$ incompleteness from the ``PSF\" morphological criterion. It is worth noting that the completeness is probably overestimated, because the spectroscopic sample is not complete \\citep{Ross2012, Ross2013} at $r<23$ mag even in this dense S82 region. Therefore, the efficiency might be also overestimated. Figure \\ref{fig:roc} shows the efficiency vesus completeness, the Precision-Recall (PR) diagram \\citep{Davis2006}, with parameters $x_1 \\in (-3.5, -0.5)$, $x_2 \\in (0, 15)$ and $x_3 \\in (0, 10)$. These parameter ranges are large enough to cover a wide range of the completeness and efficiency space. We determine $x_1$, $x_2$, and $x_3$ sequentially. First, for the above given ranges of $x_2$ and $x_3$, the best value $x_1$ is --2. The black pentagons mark $x_1 = -2$, which is located at the edge of the PR diagram, with relatively larger efficency with the same completeness. With the criterion $L_{\\rm QSO} >-2$, 97\\% of the known quasars (``PSF\" type) are selected, and 87\\% of the point sources are excluded. For the fixed $x_1=-2$ and the $x_3$ range given above, the best $x_2$ value is 4.4. The blue dot-line denotes $x_2 = 4.4$, which is located at the edge of the PR with a wide range of $x_3$ values. Finally, with the best values of $x_1$ and $x_2$ determined above, we find the best $x_3$ to be 5.4. The magenta open dot-line shows where $x_3 = 5.4$, which is located at the edge when the completeness is in the range $\\sim 79\\%$ to $\\sim 82\\%$. The black star marks $x_2 = 4.4$ and $x_3 = 5.4$. Figure \\ref{fig:ll} shows the $L_{\\rm QSO}-L'_{\\rm Galaxy}$ versus $L_{\\rm QSO}-L_{\\rm Star}$ diagram. Quasars span a much larger space in the $L_{\\rm QSO}-L'_{\\rm Galaxy}$ versus $L_{\\rm QSO}-L_{\\rm Star}$ diagram, and 11\\% of the known quasars with ``PSF\" type and $L_{\\rm QSO} >-2$ are located below these two cuts. Meanwhile, 85\\% of the point objects with $L_{\\rm QSO}>-2$ are excluded by these two cuts. Larger cuts will result in lower selection completeness and higher efficiency. For example, specifically a $L_{\\rm QSO}-L_{\\rm Star} > 10.0$ cut excludes 86\\% of the point objects with $L_{\\rm QSO}>-2$, but meanwhile causes 12\\% more selection incompleteness at $z\\sim 2.8$. Therefore, we suggest the criteria for quasar selection when using the DECaLS $g$, $r$, $z$, and WISE W1 and W2 to be\n\\begin{equation} \\label{eq:30}\n{\\rm type} = ``{\\rm PSF}\",\n\\end{equation}\n\\begin{equation} \\label{eq:31}\n{\\rm flux}(r)>0,\n\\end{equation}\n\\begin{equation} \\label{eq:32}\nL_{\\rm QSO}>-2,\n\\end{equation}\n\\begin{equation} \\label{eq:33}\nL_{\\rm QSO} - L_{\\rm Star}>4.4,\n\\end{equation}\n\\begin{equation} \\label{eq:34}\nL_{\\rm QSO} - L'_{\\rm Galaxy}>5.4.\n\\end{equation}\n\n\\begin{figure*}[htbp]\n\\hspace*{-0.5cm}\n\\epsscale{1.2}\n\\plotone{f13_completeness_all.pdf}\n\\caption{\\label{fig:completeness}The completeness of the classification method used for the spectroscopic quasar sample as a function of redshift (left panels) and $r$-band magnitude (right panels). The top panels show the completeness results applying the logarithmic likelihood criteria, including criterion $L_{\\rm QSO}>-2$ in Equation (\\ref{eq:32}) (blue diamonds), criterion $L_{\\rm QSO} - L_{\\rm Star}>4.4$ in Equation (\\ref{eq:33}) (orange dots), criterion $L_{\\rm QSO} - L'_{\\rm Galaxy}>5.4$ in Equation (\\ref{eq:34}) (magenta open pentagons), and criteria all above in Equations (\\ref{eq:32})-(\\ref{eq:34}) (black open stars). The bottom panels show the completeness results applying the ``PSF\" morphology criterion in Equation (\\ref{eq:30}) (blue dot-line), and criteria combined all logarithmic likelihood criteria and ``PSF\" morphology criterion in Equations (\\ref{eq:30})-(\\ref{eq:34}) (red star-line). The incompleteness for $z<1$ is probably caused by quasar variability, non-PSF morphology, and host galaxy contamination. $z\\sim 2.8$ quasars are close to the stellar locus, and the completeness of $z\\sim2.8$ decreases to $\\sim 70\\%$. The completeness decreases to lower than 50\\% at $r>22.3$ as the WISE data are shallower than the DECaLS data.}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\hspace{-0.5cm}\n    \\subfigure{\n    \\includegraphics[width=3.7in]{f14a_vvds.pdf}}\n  \\hspace{-1.2cm}\n  \\subfigure{\n    \\includegraphics[width=3.7in]{f14b_cosmos.pdf}}\n  \\caption{\\label{fig:deep}Objects in fields with deep spectroscopic surveys, in the VVDS deep field (left panel) and the COSMOS field (right panel). The y-axis is $r$-band magnitude. The black stars are spectroscopically identified AGNs that are selected by our classification method, and the blue open circles are AGNs missed by our method. Most missed AGNs are fainter than 23 mag. The gray dots are photo-$z$ selected quasars without spectra. The red crosses are AGNs missed by our method because their morphology types are not ``PSF\". The orange diamonds show non-AGN objects selected by our method, and they mainly show up at $z<1$ or at the faint end.}\n\\vspace{0.5cm}\n\\end{figure*}\n\\begin{table*}\n\\small\n\\begin{center}\n\\tablewidth{1pt}\n\\caption{Quasar Candidate Selection Test in Spectroscopic Surveys} \\label{tab:deep}\n\\begin{threeparttable}\n\\begin{tabular}{lcrcccccc}\n\\tableline\\tableline\n${\\rm Field}$ & ${\\rm Area}$ & Spec\\tnote{a} & photo\\tnote{b} & QSO\\tnote{c} & Star\\tnote{d} & Galaxy\\tnote{e} & completeness\\tnote{f} & efficiency\\tnote{g} \\\\\n& $\/deg^2$ & QSO & QSO & selected & selected & selected & spec & spec\\\\\n\\tableline\nVVDS (deep)  & 0.7 & 70 & 273 & 53 & 0 &  17 & 71\\% & 76\\% \\\\\nCOSMOS & 2.1 & 156 & 255 & 119 & 1 & 28 & 76\\% & 77\\% \\\\\nS82 (2.5h) & 15 & 153 & 271 & 129 & 8 & 3  & 84\\% & 91\\%\\\\\nS82 (22.7h-3h) & 162.5 & 71 & 251 & 64 & 2 & 1 & 90\\% & 95\\%\\\\\n\\tablelin\n\\end{tabular}\n\\tablecomments{Collumns in the table are as follows ($r<22.5$),}\n\\begin{tablenotes}\n        \\item[a] spectroscopically identified quasar number per $deg^2$.\n        \\item[b] photometrically selected quasar number per $deg^2$.\n        \\item[c\/d\/e] photometric method selected objects that are spectroscopically identified as quasars\/stars\/galaxies per $deg^2$.\n        \\item[f] The completeness is calculated from the spectroscopically identified quasars at $r<22.5$.\n        \\item[g] The efficiency is calculated as the ratio of photometric method selected, spectroscopically identified quasars from all spectroscopically identified objects (quasars, stars, and galaxies) at $r<22.5$.\n    \\end{tablenotes}\n   \\end{threeparttable}\n   \\end{center}\n\\end{table*}\n\nWith the criteria in Equations (\\ref{eq:30})-(\\ref{eq:34}), the selection completeness of spectroscopically identified quasars in the dense region is 81\\%. For 98,450 quasars with DECaLS photometry, we recover 84,639 quasars (86\\%). Figure \\ref{fig:completeness} shows the completeness for the spectroscopically identified quasar sample as a function of redshift (left panel) and $r$-band magnitude (right panel). In the top panels, the blue diamonds show the completeness after applying the criteria in Equation (\\ref{eq:32}). The completeness decreases when the redshift is less than 1, and one possible reason is the uncertainties from variability, because the DECaLS images and WISE images were not taken simultaneously. The incompleteness at $z>4.5$ is mainly caused by the limited number of high redshift quasars and larger photometric uncertainty in the $r$ band. Better Bayesian probability selection for $z>4.5$ quasars is potentially possible if we use simulated quasar fluxes \\citep{McGreer2013}, relative fluxes divided by the $z$, $y$ or $J$-band flux, and the QLF at high redshift. The orange dots show the completeness using the criterion in Equation (\\ref{eq:33}), the completeness at $z \\sim 2.8$ decreases as quasars move close to the stellar locus \\citep[e.g. ][]{Fan1999}. This criterion also causes an incompleteness at $r>21$ mag. The magenta open pentagons show the completeness with the criterion in Equation (\\ref{eq:34}). The completeness decreases rapidly at $r>21.5$ mag, as the WISE photometric uncertainties increase dramatically. The black open stars show the completeness when applying all the criteria in Equations (\\ref{eq:32})-(\\ref{eq:34}). The blue dot-line in Figure \\ref{fig:completeness} (bottom panels) shows the completeness applying the ``PSF\" morphology criterion as a function of redshift (bottom left panel) and magnitude (bottom right panel). The completeness with the morphology criterion decreases rapidly as redshift decrease at $z<1$. The fraction of known quasars satisfying the morphology criterion decreases from 92\\% with redshift at $0.5<z<1$ to 53\\% at $z<0.5$. This criterion also causes an incompleteness at the bright magnitude end. As the $r$-band magnitude goes fainter than 22.7, the completeness with the morphology criterion begins to decrease. The fraction decrease from 96\\% at $22.5<r<23$ to 93\\% at $23<r<23.5$. The solid red stars show the completeness when applying the three criteria above in Equations (\\ref{eq:32})-(\\ref{eq:34}) and the ``PSF\" morphology criterion in Equation (\\ref{eq:30}). Because the resolutions of the WISE images and the DECaLS images are different, the extended morphology introduces high photometric uncertainties at $z<1$. Furthermore, light from host galaxies also contaminate the colors of quasars at $z<1$.\n\nThe number counts of stars vary in different locations in the sky, but such variations have little effect on quasar selection completeness and efficiency if we consider only relatively high Galactic latitude $|b|\\geq 30^{\\circ}$. We ran a simulation for an area of 20 deg$^2$ at $b=80^{\\circ}$, and the star counts are reduced by 30\\% compared to the number at $b = -50^{\\circ}$. We performed tests with the relative fluxes of the simulated stars and the number counts at $b=80^{\\circ}$, and applied the criteria in Equations (\\ref{eq:30})-(\\ref{eq:34}). The selection completeness increases by 0.05\\%. A test in the 15 deg$^2$ S82 region shows that the efficiency differs by 0.22\\% at $r<22.5$mag. The star number counts do not strongly affect the selection results. In this work, we use only the stellar simulation from a 20 deg$^2$ region in S82 described in Section \\ref{subsec:simulation} (Galactic latitude $b\\sim -50^{\\circ}$).\n\n\\begin{figure}[htbp]\n\\hspace*{-0.5cm}\n\\epsscale{1.2}\n\\plotone{f15_dist.pdf}\n\\caption{\\label{fig:dist}The $r$-band magnitude versus redshift distribution of photometric quasars in the S82 catalog. The colorbar shows the number density.}\n\\end{figure}\n\n\\subsection{Test Quasar Candidate Selection in Deep Fields} \\label{subsec:deep}\n\nFor the dense quasar region in S82 (R.A.$\\sim$2.5h), we recover 84\\% of the quasars at $r<22.5$ mag (129 of 153 per $deg^2$). Only 8 stars per $deg^2$ at $r<22.5$ with spectra in the SDSS DR13 catalog\\footnote{https:\/\/data.sdss.org\/sas\/dr13\/sdss\/spectro\/redux\/specObj-dr13.fits} are selected, and 3 galaxies per $deg^2$ at $r<22.5$ with spectra in the SDSS DR13 catalog are selected. We test the quasar selection method using the DECaLS $g$, $r$, $z$ and WISE W1, W2 photometry in some fields with deeper spectroscopic surveys. There are 104 AGNs, and 9789 galaxies in one of the VVDS deep fields \\citep[$``{\\rm vvds\\_spF02}\"$,][]{Gavignaud2006}. In the COSMOS field, there are 409 spectroscopically identified AGNs \\citep{Prescott2006, Trump2009, Lilly2009}, and in the SDSS DR7\\&DR12. We recover 71\\% and 76\\% of $r<22.5$ AGNs in the VVDS deep field and COSMOS field, respectively. More detailed results of the quasar candidate selection in some deep spectroscopic surveys at $r<22.5$ are listed in Table \\ref{tab:deep}. Figure \\ref{fig:deep} shows quasar selection in the VVDS deep field (left panel) and the COSMOS field (right panel). The AGNs missed are mostly because of the morphology criterion or magnitude fainter than 23 mag. The deep survey results confirm that our method performs well in quasar candidate selection.\n\n\\begin{figure*}\n  \\centering\n  \\hspace{0cm}\n    \\subfigure{\n    \\includegraphics[width=3.5in]{f16a_counts_z01.pdf}}\n  \\hspace{-0.7cm}\n  \\subfigure{\n    \\includegraphics[width=3.5in]{f16b_counts_z12.pdf}}\n  \\vspace{-0.5cm}\n  \\hspace{-0.5cm}\n  \\subfigure{\n    \\includegraphics[width=3.5in]{f16c_counts_z23.pdf}}\n  \\hspace{-0.7cm}\n  \\subfigure{\n    \\includegraphics[width=3.5in]{f16d_counts_z34.pdf}}\n  \\vspace{1cm}\n  \\caption{\\label{fig:counts}The quasar number counts per deg$^2$ per 0.5mag in S82 ($340^{\\circ}<{\\rm R.A.}<45^{\\circ}$, $-1.25^{\\circ}<{\\rm Decl.}<1.25^{\\circ}$) as a function of $g$-band magnitude. The black open boxes and gray open pentagons are results calculated from the QLF \\citep{NPD2016} PLE+LEDE model and PLE model, respectively. The blue diamonds represent the statistical results from spectroscopically identified quasars in this region, and the red dots show the results from our photo-$z$ QSO sample using DECaLS and WISE photometry. The photometric results of \\citet{Richards2015} from the SDSS and WISE\/Spitzer photometric data are also presented (cyan pentagons). The photo-$z$ QSO sample with DECaLS reaches a fainter magnitude limit than the spectroscopically identified samples. The photo-$z$ QSO sample is complete and efficient except for at $2.5<z<3.7$.}\n\\vspace{0.5cm}\n\\end{figure*}\n\n\\begin{figure}[htbp]\n\\hspace*{-1cm}\n\\epsscale{1.4}\n\\plotone{f17_completeness.pdf}\n\\vspace{-0.5cm}\n\\caption{\\label{fig:eff}The completeness (black dots) and efficiency (blue diamonds) as a function of $g$-band magnitude at $2.5<z<3.7$. The efficiency decreases to $\\sim$ 40\\% at $18<r<22$.}\n\\end{figure}\n\n\\subsection{Quasar Number Count Statistics in S82} \\label{subsec:counts}\nIn a larger S82 region within $340^{\\circ}<{\\rm R.A.}<45^{\\circ}$, $-1.25^{\\circ}<{\\rm Decl.}<1.25^{\\circ}$ (roughly $b<-50^{\\circ}$), there are 45,505 $r<23$ mag objects that satisfy the criteria of Equations (\\ref{eq:30})-(\\ref{eq:33}), listed in Table \\ref{tab:catalog}. There are 12,332 spectroscopically identified quasars in this region, and the photometric quasar catalog covers 10,457 of them, with a completeness of 86\\% at $r<22.5$ mag. \\citet{Jiang2006} presented a spectroscopic sample of faint quasars, including 414 quasars down to $g = 22.5$ mag over $\\sim 3.9$ deg$^2$. There are 282 quasars in their sample located within our catalog region, and 241 (85\\%) of them are included in the photometric quasar catalog, and 32 are missed because of the morphology criterion.\n\nFigure \\ref{fig:dist} shows the $r$-band magnitude versus photo-$z$ distribution of the photometric quasars. There is a desert region at redshift $2.5<z<3.7$ and magnitude $r>21.5$. Because the colors of $2.5<z<3.7$ quasars are very close to those of stars, these quasars are excluded by criteria Equation \\ref{eq:33} and \\ref{eq:34}, shown in Figure \\ref{fig:completeness}. The main contaminants show up at photo-$z\\sim 2.7$ or photo-$z\\sim 3.2$ and at $18.5<r<21.5$ (stars or galaxies). The selection completeness and efficiency at this redshift range is relatively low. There are some contaminant galaxies showing up at photo-$z<0.7$ and $20.5<r<22.5$. Slight contaminants show up at photo-$z\\sim0.15$ and $r\\sim 17.7$ (stars), photo-$z\\sim 1$ (mainly stars), and photo-$z\\sim 4.3$ (stars or galaxies).\n\nFigure \\ref{fig:counts} shows the number counts of the photometric quasar catalog as a function of $g$-band magnitude in redshift bins $0.3<z<1$ (top left panel), $1<z<2.5$ (top right panel), $2.5<z<3.7$ (bottom left panel) and $3.7<z<4.1$ (bottom right panel). The black open boxes and gray open pentagons are results calculated from the QLF \\citep{NPD2016} PLE+LEDE model and PLE model, respectively. The results from our photometric quasar sample are shown as red dots, and the spectroscopic quasar number counts are shown as blue diamonds. \\citet{Richards2015} presented a Bayesian quasar classification based on the SDSS optical data, and WISE\/Spitzer mid-infrared photometry, and near-infrared data when available. We also plot the photometric quasar results (cyan pentagon) from \\citet{Richards2015}. Because of the inclusion of deep DECaLS photometry, our photometric quasar sample reaches a fainter magnitude than the spectroscopic quasar sample in S82. In addition, we achieve a higher completeness even in the bright end compared to the photometric quasar sample in \\citet{Richards2015} (R15). However, there are more $2.5<z<3.7$ photometric quasars showing up than the QLF. Figure \\ref{fig:eff} shows the completeness and efficiency of quasar candidates in the range $2.5<z<3.7$. The efficiency decreases to $\\sim$ 40\\% at $18<r<22$. Only $g$, $r$, $z$, W1, and W2 are used in this case. More robust photometric quasar selection in this redshift range would need more photometric data in other bands. The selection completeness and efficiency at $3.7<z<4.1$ is high, so it is a redshift interval useful for spectroscopic surveying to study the QLF at redshift $z\\sim4$.\n\n\\begin{table*}\n\\small\n\\begin{center}\n\\tablewidth{1pt}\n\\caption{Photometric quasar sample in S82} \\label{tab:catalog}\n\\begin{tabular}{ccccrcccccc}\n\\tableline\n\\tableline\n${\\rm R.A.(J2000)}$ & ${\\rm Decl.(J2000)}$ & ${\\rm g}$ & ${\\rm r}$ & ${\\rm photo-}z$ & $z_1$ & $z_2$ & $P_{\\rm prob}$ & $L_{\\rm QSO}$ & $L_{\\rm star}$ & $L'_{\\rm Galaxy}$ \\\\\n(deg) & (deg) & (mag) & (mag) & & & & & & & \\\\\n\\tableline\n0.00193188&\t-0.22927938&\t20.23&\t19.67&\t3.45&\t3.35&\t3.60&\t0.328&\t-1.053&\t-18.464&\t-321.392\\\\\n0.00370984&\t-0.23459069&\t23.28&\t22.53&\t1.45&\t1.40&\t1.55&\t0.448&\t-1.783&\t-32.538&\t-13.792\\\\\n0.00677980& 0.58319355&\t21.87&\t21.81&\t2.10&\t1.85&\t2.20&\t0.548&\t0.294&\t-44.691&\t-39.144\\\\\n0.00741207&\t0.30379374&\t22.44&\t22.33&\t1.75&\t1.70&\t1.90&\t0.546&\t-1.892&\t-50.650&\t-19.015\\\\\n0.00758513&\t1.07482973&\t23.47&\t22.48&\t1.45&\t1.40&\t1.50&\t0.465&\t-1.682&\t-56.578&\t-23.730\\\\\n0.00783493&\t-0.34638540&\t21.93&\t20.61&\t0.30&\t0.25&\t0.35&\t0.420&\t-1.795&\t-37.288&\t-13.842\\\\\n\\tableline\n\\end{tabular}\n\\tablecomments{$r<23$ mag photometric quasars selected from DECaLS and WISE photometry with the criteria in Equations (\\ref{eq:30})-(\\ref{eq:34}) in the S82 region within $340^{\\circ}<{\\rm R.A.}<45^{\\circ}$ and $-1.25^{\\circ}<{\\rm Decl.}<1.25^{\\circ}$ (roughly $b<-50^{\\circ}$). $z_1$ and $z_2$ are the lower and upper limits of ${\\rm photo-}z$ in Equation \\ref{eq:8}. (This table is available in its entirety in machine-readable form.)}\n\\end{center}\n\\end{table*}\n\n\\subsection{Classification using Random Forests}\nAn alternative method to decide the quasar criteria is machine learning classification procedure. We present a test using the Breiman and Cutler's Random Forests\\footnote{https:\/\/www.stat.berkeley.edu\/~breiman\/RandomForests\/} \\citep{Breiman2001} classification with R $randomForest$ package\\footnote{https:\/\/cran.r-project.org\/web\/packages\/randomForest}. In this test, we use the same training data with that used in Section \\ref{subsec:selection}, namely quasars and other point sources in a 15 $deg^2$ area in S82, and the same parameters, which are $L_{\\rm QSO}$, $L_{\\rm QSO} - L_{\\rm Star}$, and $L_{\\rm QSO}-L'_{\\rm Galaxy}$. We use 100 trees, and it is sufficient for this classification case. Running a test on 98,450 quasars with DECaLS photometry, random forests method recovers 71\\% of them. Only 19,853 objects are selected in the whole S82 region, and there are 0.3 stars per $deg^2$ and 0.4 galaxies per $deg^2$ at $r < 22.5$ with spectra in the SDSS DR13 catalog. Therefore, to select quasars, random forests classification method achieves higher efficiency but lower completeness than the selection criteria described in Section \\ref{subsec:selection}.\n\n\\section{Summary} \\label{sec:summary}\nWe present a new photo-$z$ regression algorithm for quasars considering the skew features of quasar color distributions, and use multivariate Skew-t funcitons to model the posterior relative flux distribution of quasars as a function of redshift and magnitude. The photo-$z$s are calculated by combining the posterior probability with the prior probability from the QLF. Photometric uncertainties are considered both in the photo-$z$ regression and classification procedures. The Skew-t photo-$z$ algorithm achieve a higher photo-$z$ accuracy than the XDQSOz and CZR method, and a higher calculation speed than the XDQSOz method. In the case that only the five SDSS bands are used, we achieve a photo-$z$ accuracy $R_{0.1}$ of 74\\%. When combining SDSS\/PS1\/DECaLS optical photometry with WISE mid-infrared photometry, the photo-$z$ accuracy $R_{0.1}$ is enhanced to 87\\%, 79\\%, and 72\\%, respectively. With WISE photometry, the degeneracy between $z\\sim0.8$ and $z\\sim 2.2$ is alleviated. The photo-$z$ accuracy decreases at $z<3.5$ due to the lack of $u$-band photometry when using PS1 or DECaLS photometric data.\n\nTo separate quasars from stars and galaxies, we perform a Milky Way synthetic simulation with the Besan\\c{c}on model and galaxy template fitting. Quasars are selected with Bayesian probability criteria. We test the classification method based on the DECaLS optical and WISE mid-infrared data. The quasar selection completeness is higher than 70\\% for a wide redshift range $0.5<z<4.5$, and a wide magnitude range $18<r<21.5$ mag. The photo-$z$ QSO sample with DECaLS reaches roughly mag fainter than the SDSS photo-$z$ QSO in R15. We find that the completeness at $z\\sim2.8$ drops to $70\\%$ if using only the $g$, $r$, $z$, W1 and W2 bands, because $z\\sim 2.8$ quasars are close to the stellar locus. The completeness decreases at $z<1$, likely caused by quasar variability, morphology, and host galaxy. In a S82 test region with a high surface density of spectroscopically identified quasars, we recover 84\\% of the quasars using our classification method. Meanwhile, only a small fraction of stars and galaxies with spectra in the SDSS DR13 are selected in this region. We also test the classification method in the VVDS deep field and COSMOS field. We recover 71\\% and 76\\% of the spectroscopically identified AGNs at $r<22.5$ mag. We present a catalog of 45,505 photometric quasars with $r<23$ mag in S82 using the DECaLS $grz$ and WISE W1 and W2 photometry. The sample is highly complete at $r < 22$ mag. The selection efficiency is high except for those with redshift at $2.5<z<3.7$. More photometric data in other bands are needed to improve quasar selection at $2.5<z<3.7$.\n\nOur photo-$z$ algorithm has a potential for the future LSST survey. To derive the QLF from the photometric quasar sample, careful correction for the selection completeness and efficiency is needed. A simulated quasar sample can be used to check the selection completeness \\citep[e.g.,][]{Fan2001, Fan2003, Jiang2008, McGreer2013, Yang2016}. A simulated star sample and a galaxy sample can be used to check the effects of stars and galaxies on the quasar selection efficiency, respectively. Underestimation of the simulated star number counts, underestimation of the galaxy luminosity function, and overestimation of the galaxy size distribution will lead to overestimation of the selection efficiency, and overestimation of the QLF. Our quasar candidate selection method can be extended for multi-band photometric data, such as the optical photometric data from PS1; some future dataset, such as LSST data, Euclid \\citep{Laureijs2011} data, and the Wide-Field Infrared Survey Telescope \\citep[WFIRST,][]{Spergel2013} data; near-infrared data, such as the UKIRT Infrared Deep Sky Survey \\citep[UKIDSS,][]{Lawrence2007}, the UKIRT Hemisphere Survey \\citep[UHS,][]{Lawrence2013}, the VISTA Hemisphere Survey \\citep[VHS,][]{McMahon2013} and the VISTA Kilo-degree INfrared Galaxy survey (Viking); and mid-infrared data WISE, NEOWISE \\citep{Mainzer2011} and unWISE. More robust quasar candidate selection can be achieved by combining the probability classification with other methods, such as variability \\citep[e.g.,][]{Geha2003, Schmidt2010, NPD2011} and proper motion \\citep[e.g.,][]{Koo1986, Brunzendorf2001, Richards2009a}, radio surveys \\citep[e.g.,][]{Becker2000, Lu2007}, and X-ray surveys \\citep[e.g.,][]{Boyle1993, Trump2009}.\\\\\n\nWe gratefully acknowledge the support from the Ministry of Science and Technology of China under grant 2016YFA0400703, NSFC grants No.11373008 and 11533001, and the National Key Basic Research Program of China 2014CB845700.\n\nWe thank Yang Huang, Jo Bovy, Leo Girardi, Arjun Dey, and N. Palanque-Delabrouille for very helpful discussions. We thank David G. Grier for providing their KDE codes, Eduardo Ba{\\~n}ados for help providing the PS1 photometric data, Chengpeng Zhang for providing their density contour plots code, and Michael Brown for providing the galaxy SEDs.\n\nWe acknowledge the use of SDSS photometric data. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III website is \\url{http:\/\/www.sdss3.org\/}. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State\/Notre Dame\/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.\n\nWe acknowledge the use of PS1 photometric data. The PS1 has been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation under Grant No. AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE).\n\nWe acknowledge the use of DECaLS photometric data, and the website is \\url{http:\/\/legacysurvey.org}. This publication makes use of data products from the \\emph{Wide-field Infrared Survey Explorer}, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory\/California Institute of Technology, funded by the National Aeronautics and Space Administration. This work has made use of the TOPCAT \\citep{Taylor2005}. We thank the Chinese Virtual Observatory, and the website is \\url{http:\/\/www.china-vo.org}.\n\n\\software{%\n$ADGofTest$    (\\url{https:\/\/cran.r-project.org\/package=ADGofTest}),\nGALAXIA \\citep{Sharma2011},\n$kSamples$     (\\url{https:\/\/cran.r-project.org\/package=kSamples}),\n$pracma$       (\\url{https:\/\/cran.r-project.org\/package=pracma}),\n$randomForest$ (\\url{https:\/\/cran.r-project.org\/package=randomForest}),\n$sn$           (\\url{https:\/\/cran.r-project.org\/package=sn}),\nSkewt-QSO \\citep{SkewtQSOv1},\nXDQSOz \\citep{Bovy2012}\n}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe purpose of this paper is to analyze some optimization problems related to the first eigenvalue of the fractional $p-$laplacian with mixed boundary conditions of Neumann and Dirichlet type, where the optimization variable is the region in which the Dirichlet condition is imposed.\n\nLet us be more specific.  Let $\\Omega \\subset \\mathbb{R}^n$ be an open, bounded and Lipschitz domain. \nThe complement of $\\Omega$ is then divided into two sets\n$$\n\\mathbb{R}^n\\setminus \\Omega = N\\cup D, \\quad N\\cap D=\\emptyset,\n$$\nand , for $0<s<1<p<\\infty$,\nwe consider the fractional $p-$laplacian operator of order $s$ with homogeneous Neumann condition on $N$ and homogeneous Dirichlet condition on $D$. That is\n$$\n(-\\Delta_p)^s u(x) = 2K\\ \\text{p.v.} \\int_{\\Omega\\cup D} \\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\\, dy,\n$$\nwhere $K=K(n,s,p)$ is a normalization constant given by\n\\begin{equation}\\label{defK}\nK:= (1-s) \\left(\\operatorname {--\\!\\!\\!\\!\\!\\int\\!\\!\\!\\!\\!--}_{\\partial B_1} |e_1\\cdot y|^p\\, dS_y\\right)^{-1} = (1-s) \\frac{\\sqrt{\\pi}\\Gamma(\\frac{n+p}{2})}{\\Gamma(\\frac{n}{2})\\Gamma(\\frac{p+1}{2})},\n\\end{equation}\nhere $\\Gamma(z) = \\int_0^\\infty t^{z-1}e^{-t}\\, dt$ is the Gamma function.\n\nThis operator is thought as acting in the fractional Sobolev space\n$$\nW^{s,p}_D(\\Omega) := \\{u\\in W^{s,p}(\\Omega\\cup D)\\colon u|_D = 0 \\text{ a.e.}\\}\n$$\nwhere, as usual, the fractional Sobolev space $W^{s,p}(U)$ is defined as \n$$\nW^{s,p}(U)=\\left \\{u\\in L^{p}(U)\\colon \\frac{|u(x)-u(y)|}{|x-y|^{\\frac{n}{p}+s}}\\in L^{p}(U\\times U)\\right \\}.\n$$\nHence, $(-\\Delta_p)^s\\colon W^{s,p}_D(\\Omega)\\to [W^{s,p}_D(\\Omega)]'$ and\n$$\n\\langle (-\\Delta_p)^s u, v\\rangle = K \\iint_{(\\Omega\\cup D)\\times (\\Omega\\cup D)} \\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+sp}} dxdy.\n$$\n\nWith these definitions, the associated eigenvalue problem is\n\\begin{equation}\\label{eigenvalue}\n\\begin{cases}\n(-\\Delta_p)^s u = \\lambda |u|^{p-2} u & \\text{in }\\Omega\\\\\nu = 0 & \\text{in } D.\n\\end{cases}\n\\end{equation}\n\nThe set $N$ does not enter in our formulation and is interpreted as a Neumann condition over it, since the system does not interact with $N$.\n\nThis problem is analogous in the nonlocal fractional setting to the eigenvalue problem for the\nlocal $p-$Laplacian\n$$\n\\begin{cases}\n-\\Delta_p u = - \\mbox{div} (|\\nabla u|^{p-2} \\nabla u) = \\lambda |u|^{p-2} u & \\text{in }\\Omega\\\\\nu = 0 & \\text{on } \\partial \\Omega \\setminus \\Gamma \\\\\n|\\nabla u|^{p-2} \\partial_\\nu u = 0 & \\text{on } \\Gamma .\n\\end{cases}\n$$\nHere $\\Gamma\\subset \\partial \\Omega$. When $\\Gamma= \\emptyset$ we have the usual Dirichlet problem\n(with a positive first eigenvalue, $\\lambda_1>0$); \nwhile for $\\Gamma=\\partial \\Omega$ we have the Neumann problem (and here the first eigenvalue is zero, $\\lambda_1^N=0$).\n\nRecall that the fractional Sobolev space $W^{s,p}(U)$ is a Banach space when one considers the norm\n\\begin{equation}\\label{normaSobolevclasico}\n\\|u\\|_{W^{s,p}(U)}=\\left(\\int_U |u|^p\\,dx+\\iint_{U\\times U} \\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\\,dx \\,dy \\right)^{1\/p}.\n\\end{equation}\nMoreover, for $0<s<1<p<\\infty$ it is a reflexive, uniformly convex and separable Banach space.\nThe term\n\\begin{equation}\\label{seminormaSobolev}\n[u]_{W^{s,p}(U)}^p = [u]_{s,p; U}^p= \\iint_{U\\times U} \\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\\,dx \\,dy,\n\\end{equation}\nis called the {\\em Gagliardo seminorm} of $u$. We refer the interested reader to \\cite{DiNezza-Palatucci-Valdinoci} for a throughout introduction to these spaces and operators.\n\n\nIt is straightforward to see that $\\lambda\\in\\mathbb{R}$ is an eigenvalue of \\eqref{eigenvalue} if and only if is a critical value of the functional\n$$\n\\mathcal{J}(v) := K \\iint_{(\\Omega\\cup D)\\times (\\Omega\\cup D)} \\frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}}\\, dxdy = K [v]_{s,p; \\Omega\\cup D}^p,\n$$\nrestricted to the unit ball of $L^p(\\Omega)$. Moreover, eigenfunctions of \\eqref{eigenvalue} associated to $\\lambda$ are critical points of $\\mathcal{J}$ on the unit ball of $L^p(\\Omega)$ with critical value $\\lambda$.\n\nOf particular importance is the {\\em first eigenvalue} of \\eqref{eigenvalue}, that is given by\n$$\n\\lambda_s(D) = \\inf_{v\\in W^{s,p}_D(\\Omega)} \\frac{\\mathcal{J}(v)}{ \\|v\\|_{p;\\Omega}^p}.\n$$\n\nIt is an easy consequence of the direct method of the Calculus of Variations (c.f. Section 2) that\nthe following assertions hold:\n\\begin{itemize}\n\\item the above infimum is achieved, and we can assume that a function $u\\in W^{s,p}_D(\\Omega)$ that realize the infimum (we will called such a function an {\\em extremal}) is normalized in $L^p(\\Omega)$, that is, $\\|u\\|_{p;\\Omega}=1$;\n\n\\item the number $\\lambda_s(D)$ is in fact an eigenvalue of \\eqref{eigenvalue};\n\n\\item $\\lambda_s(D)$ is the first (smallest) eigenvalue, i.e. if $\\lambda\\in\\mathbb{R}$ is an eigenvalue of \\eqref{eigenvalue}, then $\\lambda\\ge \\lambda_s(D)$;\n\n\\item any eigenfunction associated to $\\lambda_s(D)$ has constant sign;\n\n\\item $\\lambda_s(D)>0$ if $|D|>0$.\n\\end{itemize} \n\nThe main problem that we address here is the optimization of the {\\em principal eigenvalue} $\\lambda_s(D)$ with respect to the region $D$ where the Dirichlet data is imposed. With that in mind, we fix a constant $\\alpha>0$, define the class\n$$\n\\mathcal{D}_\\alpha := \\{D\\subset \\mathbb{R}^n\\setminus \\Omega\\colon \\text{measurable and } |D|=\\alpha\\}\n$$\nand consider the optimization problems\n\\begin{equation}\\label{lambda.s.+-}\n\\Lambda_s^+(\\alpha) := \\sup_{D\\in \\mathcal{D}_\\alpha} \\lambda_s(D)\\quad \\text{and} \\quad \\Lambda_s^-(\\alpha) := \\inf_{D\\in \\mathcal{D}_\\alpha} \\lambda_s(D).\n\\end{equation}\n\nAs we will see, $\\Lambda_s^-(\\alpha) = 0$ since pushing $D$ to infinity force that in the limit, so in order to recover a nontrivial constant for the  minimization problem one has to restrict the sets $D$ to be uniformly bounded. Therefore, given $R>0$ large, we define\n$$\n\\mathcal{D}_\\alpha^R := \\{D\\in \\mathcal{D}_\\alpha\\colon D\\subset B_R(0)\\},\n$$\nand\n\\begin{equation}\\label{lambda.s.-R}\n\\Lambda_s^{-,R}(\\alpha) := \\inf_{D\\in \\mathcal{D}_\\alpha^R} \\lambda_s(D).\n\\end{equation}\nThis value $\\Lambda_s^{-,R}(\\alpha)$ is strictly positive.\n\nThe main results in this paper are contained in the following theorem.\n\\begin{thm}\\label{teo.main}\nLet $\\Omega\\subset \\mathbb{R}^n$ be bounded and open. \nTake $0<s<1<p<\\infty$ and fix $\\alpha>0$. Let $\\Lambda_s^+(\\alpha)$, $\\Lambda_s^-(\\alpha)$, $\\Lambda_s^{-,R}(\\alpha)$ be given by \\eqref{lambda.s.+-} and \\eqref{lambda.s.-R}. Then the following hold:\n\\begin{itemize}\n\\item $\\Lambda_s^-(\\alpha)=0$ while $\\Lambda_s^{-,R}(\\alpha)>0$;\n\n\\item $\\lim_{s\\uparrow 1}\\Lambda_s^+(\\alpha) = \\lambda_1$, where $\\lambda_1$ is the first Dirichlet eigenvalue of the local $p-$laplacian in $\\Omega$;\n\n\\item $\\lim_{s\\uparrow 1} \\Lambda_s^{-,R}(\\alpha)= 0$.\n\\end{itemize}\n\nMoreover, for every $0<s<1$ there exists an optimal set $D_s^R$ for the constant $\\Lambda_s^{-,R}(\\alpha)$.\n\nFinally, given a sequence of quasi-optimal sets for $\\Lambda_s^+(\\alpha)$, i.e. $D_s\\subset \\mathcal{D}_\\alpha$ such that\n$$\n\\Lambda_s^+(\\alpha) = \\lambda_s(D_s) + o(1),\n$$\nwhere $o(1)\\to 0$ as $s\\uparrow 1$, we have that $D_s$ ``surrounds'' the boundary of $\\Omega$ in the sense that for any $x\\in \\partial\\Omega$ and any $\\varepsilon>0$, there exists $s_0$ such that if $s_0<s<1$, $|D_s\\cap B_\\varepsilon(x)|>0$.\n\\end{thm}\n\nNotice that the first Neumann eigenvalue for the $p-$Laplacian in $\\Omega$ is $\\lambda_1^N =0$ (with\na constant function as eigenfunction). Therefore, as we have that for every set $D$ of measure $\\alpha$\nin $B_R \\setminus \\Omega$\n$$\n\\Lambda_s^+(\\alpha) \\geq \\lambda_s (D) \\geq \\Lambda_s^{-,R}(\\alpha),\n$$\nwe conclude that for $s$ close to $1$ we have that $\\lambda_s (D)$ \nasymptotically lies between \nthe first Dirichlet eigenvalue and the first Neumann eigenvalue for the local $p-$Laplacian in $\\Omega$.\nIn fact, for every $\\varepsilon $ there exists $s_0$ such that for $s_0 <s<1$ it holds\n$$\n \\lambda_1 + \\varepsilon \\geq \\lambda_s (D) \\geq  \\lambda_1^N =0.\n$$\nThe surprising fact of these asymptotic bounds as $s\\uparrow 1$ is that they are independent of the size of the Dirichlet part in our nonlocal problem and they are also independent of the radius of the ball that bounds everything.\n\nTo emphasize that in the limit as $s\\uparrow 1$ for the lower bound for the eigenvalues we obtain a local problem with Neumann boundary conditions in $\\Omega$ we remark that with minor modifications of our arguments we can deal with the eigenvalue problem with a potential $V$. In fact, let us consider $V\\in L^\\infty(\\mathbb{R}^n)$ a potential such that \n\\begin{equation}\\label{cota.V}\n0<v_1\\leq V (x) \\leq v_2 <+\\infty\n\\end{equation} \nfor some numbers $v_1,v_2$ and then we take\n$$\n\\lambda_{s,V}(D) = \\inf_{v\\in W^{s,p}_D(\\Omega)} \\frac{\\mathcal{J}_V(v)}{ \\|v\\|_{p;\\Omega}^p},\n$$\nwith\n$$\n\\mathcal{J}_V(v) := K \\iint_{(\\Omega\\cup D)\\times (\\Omega\\cup D)} \\frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}}\\, dxdy \n+ \\int_\\Omega |u(x)|^p V(x)\\, dx .\n$$\nAssociated with this functional we have the optimal constants\n\\begin{equation}\\label{LambdaV}\n\\Lambda_{s,V}^+(\\alpha) := \\sup_{D\\in \\mathcal{D}_\\alpha} \\lambda_{s,V}(D),\\qquad \\Lambda_{s,V}^-(\\alpha) := \\inf_{D\\in \\mathcal{D}_\\alpha} \\lambda_{s,V}(D)\n\\end{equation}\nand\n\\begin{equation}\\label{LambdaVR}\n\\Lambda_{s,V}^{-,R}(\\alpha) := \\inf_{D\\in \\mathcal{D}_\\alpha^R} \\lambda_{s,V}(D).\n\\end{equation}\nIn this case, we obtain that\n$$\n\\Lambda_{s,V}^+(\\alpha) \\to \\lambda_1(V)\\qquad \\mbox{and} \\qquad\n\\Lambda_{s,V}^{-,R}(\\alpha) \\to \\lambda_1^N(V),\n$$\nas $s\\uparrow 1$. Here $\\lambda_1 (V)$ and $\\lambda_1^N (V)$ are the first eigenvalues for the local $p-$Laplacian with the potential $V$ with Dirichlet and Neumann boundary condition respectively, that are given by\n\\begin{equation}\\label{localVD}\n\\lambda_1 (V) = \\min_{v\\in W^{1,p}_0 (\\Omega)}  \\frac{ \\displaystyle\n  \\int_\\Omega |\\nabla v|^p\\, dx + \\int_\\Omega |v|^p V(x)\\, dx }{\\|v\\|_{p;\\Omega}^p}\n\\end{equation}\nand \n\\begin{equation}\\label{localVN}\n\\lambda_1^N (V) = \\min_{v\\in W^{1,p} (\\Omega)}  \\frac{ \\displaystyle\n\\int_\\Omega |\\nabla v|^p\\, dx + \\int_\\Omega |v|^p V(x)\\, dx}{\\|v\\|_{p;\\Omega}^p}.\n\\end{equation}\nNotice that in this case $\\lambda_1^N (V) \\neq 0$.\n\nWhen we consider a potential $V$ one can also check that  if we don't constraint $D$ into a large ball we still have\n$$\n\\lim_{s\\uparrow 1}\\Lambda_{s,V}^{-}(\\alpha) = \\lambda_1^N(V).\n$$\nIn fact, this limit can be deduced taking the limit as $s\\uparrow 1$ in the following two\ninequalities\n$$\n\\Lambda_{s,V}^{-}(\\alpha) \\leq \\Lambda_{s,V}^{-,R}(\\alpha) \n$$\nand\n$$\n\\Lambda_{s,V}^{-}(\\alpha) \\geq \\min_{v\\in W^{s,p} (\\Omega)}\n\\frac{ \\displaystyle  K \\iint_{\\Omega\\times \\Omega} \\frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}}\\, dxdy  + \\int_\\Omega |v(x)|^p V(x)\\, dx}{ \\|v\\|_{p;\\Omega}^p}.\n$$\nWe include some details in Section 5.\n\nWith these preliminaries, our second result is the following:\n\\begin{thm}\\label{teo.main2}\nLet $\\Omega\\subset \\mathbb{R}^n$ be bounded and open. Let $0<s<1<p<\\infty$ be fixed and take $\\alpha>0$. Let $V\\in L^\\infty(\\mathbb{R}^n)$ a potential such that \\eqref{cota.V} is satisfied.\n\nLet $\\Lambda_{s,V}^+(\\alpha)$, $\\Lambda_{s,V}^-(\\alpha)$, $\\Lambda_{s,V}^{-,R}(\\alpha)$ be the constants defined in \\eqref{LambdaV} and \\eqref{LambdaVR}. Then the following hold\n\\begin{itemize}\n\\item $\\lim_{s\\uparrow 1}\\Lambda_{s,V}^+(\\alpha) = \\lambda_1(V)$, where $\\lambda_1(V)$ is given by \\eqref{localVD};\n\n\\item $\\lim_{s\\uparrow 1}\\Lambda_{s,V}^{-,R}(\\alpha) = \\lambda_1^N(V)$, where $\\lambda_1^N(V)$ is given by \\eqref{localVN}.\n\n\\item $\\lim_{s\\uparrow 1}\\Lambda_{s,V}^{-}(\\alpha) = \\lambda_1^N(V)$.\n\\end{itemize}\n\\end{thm}\n\nA very brief comment on related bibliography is in order. Optimal configurations related to eigenvalue problems\nis by now a classical subject, \njust to mention a few references we quote \\cite{Bucur2, Bucur1, Ch, henrot1, S1, S2}.\nOn the other hand, nonlocal problems are quite popular nowadays, we just refer to \\cite{DiNezza-Palatucci-Valdinoci}\nand for references concerning eigenvalues for the nonlocal $p-$Laplacian to \\cite{Brasco-Parini-Squassina, Bonder-Spedaletti} and references therein.\n\n\\subsection*{Organization of the paper} After this introduction, the rest of the paper is organized as follows: \nIn Section 2 we revise some preliminary notions on fractional Sobolev spaces that are needed in the paper. In Section 3 we study the maximization problem and in Section 4 the minimization problem. Finally, in Section 5, we prove Theorem \\ref{teo.main2}. Since the proof is similar to the one of Theorem \\ref{teo.main} we only sketch it and stress the differences.\n\n\n\n\n\n\\section{Preliminaries}\nIn this section, we review some definitions on fractional Sobolev spaces and on the $p-$fractional Laplace operator. We believe that most of these results are known to experts and constitute part of the ``folklore'' on the subject, \nbut we have chosen to include some proofs of the facts that are needed just for the reader's convenience.\n\n\n\\subsection{A probabilistic interpretation for the mixed boundary conditions}\n\nLet $\\Omega\\subset \\mathbb{R}^n$ and take $N, D\\subset \\mathbb{R}^n\\setminus\\Omega$ such that $N\\cap D=\\emptyset$ and $N\\cup D=\\mathbb{R}^n\\setminus \\Omega$.\n\nSuppose that $u=u(x,t)$ measures the density of some substance in space \n$x\\in\\mathbb{R}^n$ and time $t>0$. Assume that the probability that a particle jumps from a position $x$ to a different position $y$ by time unit is given by a kernel $k = k(x-y)$. Assume moreover that the kernel is symmetric, i.e. $k(z)=k(-z)$.\n\nNow, we further assume that the particles inside $\\Omega$ do not interact with the particles in $N$ and inside $D$ the density is zero (every particle that jumps into $D$ is automatically killed). Then, the conservation law for the mass gives rise to the equation\n\\begin{equation}\\label{conservation}\n\\begin{split}\nu_t(x,t) &= \\int_\\Omega k(x-y) u(y,t)\\, dy - \\int_{\\Omega\\cup D} k(x-y) u(x,t)\\, dy\\\\\n&= -\\int_{\\Omega\\cup D} k(x-y) (u(x,t)-u(y,t))\\, dy.\n\\end{split}\n\\end{equation}\n\nIn this paper we consider the case where the kernel $k$ behaves like a power of the distance, that is\n$$\nk(z) \\sim |z|^{-(n+2s)},\n$$\nwhich means that the particle tends with high probability to stay close to the original position but can jump with positive probability to positions far away.\nThis type of kernel give rise to the so-called fractional Laplace operator defined as in the introduction,\n$$\n(-\\Delta)^s v(x) = 2 K(n,s,2)\\, \\text{p.v.}\\int_U \\frac{v(x)-v(y)}{|x-y|^{n+2s}}\\, dy,\n$$\nwhere p.v. stands for {\\em principal value}.\n\nIn some models, the kernel $k$ depends not only on the distance but also on the difference in concentration, making more likely to jump when the difference in concentration is large. i.e.\n$$\nk = k(x-y, v(x)-v(y)).\n$$\nHere we consider the case where\n$$\nk \\sim \\frac{|v(x)-v(y)|^{p-2}}{|x-y|^{n+sp}},\n$$\nand these type of kernels give rise to the so-called fractional $p-$laplace operator defined as\n\\begin{equation}\\label{eq.plap}\n(-\\Delta_p)^s v(x) = 2K(n,s,p) \\, \\text{p.v.} \\int_U \\frac{|v(x)-v(y)|^{p-2}(v(x)-v(y))}{|x-y|^{n+sp}}\\, dy.\n\\end{equation}\n\nOne key factor to analyze the behavior of the solutions of \\eqref{conservation} is the first eigenvalue of the associated operator, i.e. the smallest value of $\\lambda\\in \\mathbb{R}$ such that there exists a nontrivial solution to\n$$\n\\begin{cases}\n(-\\Delta_p)^s v = \\lambda |v|^{p-2} v & \\text{in }\\Omega\\\\\nv = 0 & \\text{in } D\n\\end{cases}\n$$\nso the purpose of this paper is to analyze this problem and in particular the optimization of this first eigenvalue with respect to the region $D$ and the asymptotic behavior of this optimal eigenvalue when the fractional parameter $s$ tends to 1.\n\n\\subsection{The fractional $p-$laplacian}\nIn this subsection, we recall some basic facts about the fractional $p-$laplacian given in \\eqref{eq.plap}.\n\n\\begin{lemp}[\\cite{Bonder-Spedaletti}, Lemma 2.2]\\label{plap.distribucional}\nLet $0<s<1<p<\\infty$ be fixed and let $U\\subset\\mathbb{R}^n$ be open. For every $u\\in W^{s,p}(U)$, the fractional $p-$laplacian given by \\eqref{eq.plap} defines a distribution $\\mathcal D'(U)$. Moreover,\n$$\n\\langle (-\\Delta_p)^s u, \\phi\\rangle = K \\iint_{U\\times U} \\frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))(\\phi(x)-\\phi(y))}{|x-y|^{n+sp}}\\, dx\\, dy,\n$$\nfor every $\\phi\\in C^\\infty_c(\\Omega)$, where $K$ is defined in \\eqref{defK}.\n\\end{lemp}\n\n\n\\begin{rk}\\label{def.lap}\nFrom Lemma \\ref{plap.distribucional} one observe that the fractional $p-$laplacian is a bounded operator between $W^{s,p}(U)$ and its dual $[W^{s,p}(U)]'$.\n\\end{rk}\n\n\n\\begin{rk}\\label{rem.ponce}\nThe choice of the constant $K$ in the definition of the fractional $p-$laplacian is made in order for the operators $(-\\Delta_p)^s$ to converge to the local $p-$laplacian $-\\Delta_p$ as $s\\uparrow 1$. In fact, it is easy to see, from the Gamma convergence results in \\cite{Ponce} that for any $f\\in L^{p'}(\\Omega)$, if $u_s\\in W^{s,p}_0(\\Omega)$ is the weak solution of \n$$\n\\begin{cases}\n(-\\Delta_p)^s u_s = f & \\text{in }\\Omega\\\\\nu_s = 0 & \\text{in }\\mathbb{R}^n\\setminus \\Omega,\n\\end{cases}\n$$\nthen $u_s\\to u$ as $s\\uparrow 1$ strongly in $L^p(\\Omega)$ where $u\\in W^{1,p}_0(\\Omega)$ is the weak solution to\n$$\n\\begin{cases}\n-\\Delta_p u = f & \\text{in }\\Omega\\\\\nu = 0 & \\text{on }\\partial \\Omega.\n\\end{cases}\n$$\n\\end{rk}\n\nWith all of these preliminaries, we establish the definition of {\\em weak solution} for mixed boundary value problem for the fractional $p-$laplacian.\n\\begin{de}\\label{de.sol}\nLet $0<s<1<p<\\infty$ be fixed and let $\\Omega\\subset\\mathbb{R}^n$ be open and let $D\\subset \\mathbb{R}^n\\setminus \\Omega$. Given $f\\in L^{p'}(\\Omega)$ (or more generally, $f\\in [W^{s,p}_D(\\Omega)]'$), we say that $u\\in W^{s,p}_D(\\Omega)$ is a weak solution of \n\\begin{equation}\\label{eq.sp}\n\\begin{cases}\n(-\\Delta_p)^s u = f & \\text{in }\\Omega,\\\\\nu= 0 & \\text{in } D,\n\\end{cases}\n\\end{equation}\nif the equality holds in the distributional sense. That is, if\n$$\nK \\iint_{(\\Omega\\cup D)\\times (\\Omega\\cup D)} \\frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+sp}}\\, dx\\, dy = \\int_\\Omega f v\\, dx,\n$$\nfor every $v\\in W^{s,p}_D(\\Omega)$, where $K$ is given in \\eqref{defK}.\n\\end{de}\n\n\nThe next Poincar\\'e-type inequality, although its proof is elementary, is crucial in the remaining of the paper.\nThis inequality is stablished in \\cite[Proposition 2.10]{Bonder-Spedaletti}. We include here a proof for the reader's convenience.\n\\begin{prop}\\label{poincare}\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bounded, open set and let $D\\subset \\mathbb{R}^n\\setminus\\Omega$ be a measurable bounded set. Then \n$$\n[u]_{s,p; \\Omega\\cup D}^p \\ge d^{-(n+sp)}|D| \\|u\\|_{p; \\Omega}^p,\n$$\nfor every $u\\in W^{s,p}_D(\\Omega)$, where\n$$\nd=d(\\Omega, D) = \\sup\\{|x-y|\\colon x\\in\\Omega,\\ y\\in D\\}.\n$$\n\\end{prop}\n\n\\begin{proof}\nThe proof is rather simple. In fact, given $u\\in W^{s,p}_D(\\Omega)$ we have\n\\begin{align*}\n[u]_{s,p;\\Omega\\cup D}^p &= \\iint_{(\\Omega\\cup D)\\times (\\Omega\\cup D)} \\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\\, dxdy\\\\\n&\\ge \\int_{\\Omega} |u(x)|^p\\left(\\int_D \\frac{1}{|x-y|^{n+sp}}\\, dy\\right)\\, dx.\n\\end{align*}\nNow, just observe that for every $x\\in\\Omega$, one has\n$$\n \\int_D \\frac{1}{|x-y|^{n+sp}}\\, dy \\ge d^{-(n+sp)} |D|, \n$$\nand the proof is complete.\n\\end{proof}\n\nWe now want to remove the hypothesis that $D$ is bounded in Proposition \\ref{poincare}.\n\\begin{corol}\\label{poincare2} Assume that $D$ is only measurable, then there is a positive constant $\\theta >0$ such that\n$$\n[u]_{s,p; \\Omega\\cup D}^p \\ge \\theta \\|u\\|_{p; \\Omega}^p.\n$$\n\\end{corol}\n\n\\begin{proof}\nTake $R>0$ large so that $D_R := D\\cap B_R(0)$ has positive measure. Hence\n$$\n[u]_{s,p; \\Omega\\cup D}^p\\ge [u]_{s,p; \\Omega\\cup D_R}^p\\ge \\theta \\|u\\|_{p;\\Omega}^p; \n$$\nwhere $\\theta=\\theta_R$ is obtained from Proposition \\ref{poincare}. The proof is complete.\n\\end{proof}\n\n\n\\begin{rk}\\label{poincare3}\nif $|D|=\\alpha<\\infty$, one can take $R>0$ in the above proof such that $|D_R|=\\frac12 \\alpha$. Hence one arrives at\n$$\n[u]_{s,p; \\Omega\\cup D}^p\\ge d_R^{-(n+sp)}\\frac{|D|}{2} \\|u\\|_{p;\\Omega}^p,\n$$\nwhere $d_R=\\sup\\{|x-y|\\colon x\\in\\Omega,\\ |y|=R\\}$. Therefore one can take\n$$\n\\theta = d_R^{-(n+sp)}\\frac{|D|}{2}\n$$\nfor $R$ large.\n\\end{rk}\n\nFrom Corollary \\ref{poincare2} it is not difficult to show the existence of an extremal for the constant $\\lambda_s(D)$. The main difficulty is that, even if $\\Omega$ is smooth, since we do not want to make any regularity assumptions on $D$ we cannot assume that the injection $W^{s,p}(\\Omega\\cup D)\\subset L^p(\\Omega\\cup D)$ is compact.\n\\begin{thm}\\label{existe.extremal}\nLet $\\Omega\\subset\\mathbb{R}^n$ be a bounded domain with Lipschitz boundary. Then, given $D\\subset \\mathbb{R}^n\\setminus \\Omega$ measurable with positive measure, there exists $u\\in W^{s,p}_D(\\Omega)$ such that\n$$\n\\lambda_s(D) = \\frac{K[u]_{s,p; \\Omega\\cup D}^p}{\\|u\\|_{p; \\Omega}^p},\n$$\nwhere $K$ is given in \\eqref{defK}. Moreover, the extremal can be taken to be normalized in $L^p(\\Omega)$, i.e. $\\|u\\|_{p;\\Omega} = 1$.\n\\end{thm}\n\n\\begin{proof}\nThe scheme of the proof is standard. Let $\\{v_k\\}_{k\\in\\mathbb{N}}\\subset W^{s,p}_D(\\Omega)$ be a normalized minimizing sequence for $\\lambda_s(D)$, i.e.\n$$\n\\mathcal{J}(v_k) = K [v_k]_{s,p;\\Omega\\cup D}^p\\to \\lambda_s(D)\\quad \\text{and}\\quad \\|v_k\\|_{p;\\Omega} = 1.\n$$\n\nThen, $\\{v_k\\}_{k\\in \\mathbb{N}}\\subset W^{s,p}(\\Omega\\cup D)$ is bounded and hence, up to some subsequence, we can assume that $v_k\\rightharpoonup u$ weakly in $W^{s,p}(\\Omega\\cup D)$. \n\nSince $v_k=0$ in $D$ for every $k\\in\\mathbb{N}$ implies that $u=0$ in $D$, so $u\\in W^{s,p}_D(\\Omega)$.\n\nMoreover, $v_k\\rightharpoonup u$ weakly in $W^{s,p}(\\Omega)$ and since $\\Omega$ is Lipschitz, this implies that the injection $W^{s,p}(\\Omega)\\subset L^p(\\Omega)$ is compact (see \\cite[Theorem 7.1]{DiNezza-Palatucci-Valdinoci}) and so $\\|u\\|_{p;\\Omega} = 1$.\n\nTherefore\n$$\n\\lambda_s(D)\\le \\mathcal{J}(u)\\le \\liminf_{k\\to\\infty} \\mathcal{J}(v_k) = \\lambda_s(D).\n$$\n\nThe proof is complete.\n\\end{proof}\n\n\n\\begin{rk}\nThe Lipschitz regularity on $\\partial\\Omega$ is needed in order for the compactness of the embedding  $W^{s,p}(\\Omega)\\subset L^p(\\Omega)$ to hold. See  \\cite[Theorem 7.1]{DiNezza-Palatucci-Valdinoci}. In fact, what is needed is that $\\Omega$ be a {\\em bounded extension domain}. That is the existence of a bounded extension operator $E\\colon W^{s,p}(\\Omega)\\to W^{s,p}(\\mathbb{R}^n)$. Lipschitz boundary imply that $\\Omega$ is a bounded extension domain. See  \\cite[Theorem 5.4]{DiNezza-Palatucci-Valdinoci}.\n\\end{rk}\n\n\n\\section{The maximization problem}\n\nIn this section we study the problem of maximization of $\\lambda_s(D)$. That is,  given $\\alpha>0$ we define\n$$\n\\Lambda_s^+(\\alpha) = \\sup\\{\\lambda_s(D)\\colon D\\subset \\mathbb{R}^n\\setminus\\Omega,\\ |D|=\\alpha\\}.\n$$\nIn particular, we are interested in the asymptotic behavior of the constant $\\Lambda_s^+(\\alpha)$ when $s\\uparrow 1$.\nAs we will see, the behavior of $\\Lambda_s^+(\\alpha)$ when $s\\uparrow 1$ is independent of the value of the constant $\\alpha>0$. \n\nWe begin with a simple lemma.\n\\begin{lemp}\\label{orden}\nLet $D_1\\subset D_2\\subset \\mathbb{R}^n\\setminus\\Omega$ be measurable sets with positive measure. Then $\\lambda_s(D_1)\\le \\lambda_s(D_2)$.\n\\end{lemp}\n\n\\begin{proof}\nLet $u\\in W^{s,p}_{D_2}(\\Omega)$ be such that $\\|u\\|_{p;\\Omega}=1$. Then, $u\\in W^{s,p}_{D_1}(\\Omega)$ and so\n$$\n\\lambda_s(D_1)\\le K [u]_{s,p; \\Omega\\cup D_1}^p \\le K [u]_{s,p; \\Omega\\cup D_2}^p.\n$$\ntaking infimum on $u\\in W^{s,p}_{D_2}(\\Omega)$ the conclusion follows.\n\\end{proof}\n\nAs a consequence of this Lemma we have the following upper bound for $\\Lambda_s^+(\\alpha)$\n$$\n\\Lambda_s^+(\\alpha) \\le \\lambda_s(\\mathbb{R}^n\\setminus \\Omega).\n$$\nObserve that $\\lambda_s(\\mathbb{R}^n\\setminus\\Omega)$ is the first eigenvalue of the fractional $p-$laplacian with Dirichlet Boundary conditions. i.e. is the first eigenvalue of the problem\n\\begin{equation}\\label{frac.dir.eig}\n\\begin{cases}\n(-\\Delta_p)^s u = \\lambda |u|^{p-2} u & \\text{in }\\Omega\\\\\nu = 0 & \\text{in } \\mathbb{R}^n\\setminus\\Omega.\n\\end{cases}\n\\end{equation}\n\nProblem \\eqref{frac.dir.eig} was studied, among others, in \\cite{Brasco-Parini-Squassina} where, based on the famous results of Bourgain-Brezis-Mironescu \\cite{Bourgain-Brezis-Mironescu}, the asymptotic behavior of the eigenvalues when $s\\uparrow 1$ is obtained. Namely,\n$$\n\\lim_{s\\uparrow 1} \\lambda_s(\\mathbb{R}^n\\setminus\\Omega) = \\lambda_1,\n$$\nwhere $\\lambda_1$ is the first eigenvalue of the local $p-$laplacian in $\\Omega$ with homogeneous Dirichlet boundary conditions, i.e.\n$$\n\\lambda_1 = \\inf_{v\\in W^{1,p}_0(\\Omega)}\\frac{\\displaystyle\\int_\\Omega |\\nabla v|^p\\, dx}{\\displaystyle\\int_\\Omega |v|^p\\, dx}.\n$$\n\nOn the other hand, take $D= \\Omega_r\\setminus \\overline{\\Omega}$ where $\\Omega_r = \\bigcup_{x\\in\\Omega} B_r(x)$ is the usual fattening of $\\Omega$ and $r$ is chosen in such a way as $|D|=\\alpha$.\n\nThen\n$$\n\\lambda_s(D)\\le \\Lambda^+_s(\\alpha), \n$$\nand so\n$$\n\\liminf_{s\\uparrow 1} \\lambda_s(D)\\le \\liminf_{s\\uparrow 1} \\Lambda^+_s(\\alpha)\\le \\limsup_{s\\uparrow 1} \\Lambda^+_s(\\alpha) \\le \\lambda_1(\\Omega).\n$$\n\nNow, Let $u_s\\in W^{s,p}_D(\\Omega)$ be the eigenfunction associated to $\\lambda_s(D)$ normalized in $L^p(\\Omega)$. Then, by \\cite[Theorem 4]{Bourgain-Brezis-Mironescu} it follows that there exist a sequence $s_k\\uparrow 1$ and a function $u\\in W^{1,p}_0(\\Omega)$, such that $u_{s_k}\\to u$ in $L^p(\\Omega)$ and\n$$\n\\|\\nabla u\\|_{p;\\Omega}^p\\le \\liminf_{k\\to\\infty} K(n,s_k,p) [u_{s_k}]^p_{s_k,p; \\Omega\\cup D},\n$$\nwhere $K(n,s,p)$ is given in \\eqref{defK}. Hence\n$$\n\\|\\nabla u\\|_{p;\\Omega}^p\\le \\lambda_1(\\Omega)\n$$\nwith $\\|u\\|_{p,\\Omega}=1$.\nTherefore, $u$ is the normalized eigenfunction of the $p-$laplacian in $\\Omega$ and one can conclude that\n\\begin{equation}\\label{limLambda+}\n\\lim_{s\\uparrow 1} \\Lambda^+_s(\\alpha) = \\lambda_1(\\Omega). \n\\end{equation}\n\n\nIn order to finish the study of the maximization problem, we need to study the quasi-optimal Dirichlet sets $D$. \nNotice that, since we are considering a maximization problem, it is not clear that there is\nan optimal set for $\\Lambda_s^+(\\alpha)$. Hence, we deal with quasi-optimal sequences as $s\\uparrow 1$, that is, we take\n$D_s\\subset \\mathcal{D}_\\alpha$ such that\n$$\n\\Lambda_s^+(\\alpha) = \\lambda_s(D_s) + o(1),\n$$\nwhere $o(1)\\to 0$ as $s\\uparrow 1$. Our next result says that the quasi-optimal sets $D_s$ ``covers'' the boundary of $\\Omega$ \nas $s\\uparrow 1$.\n\n\\begin{thm}\nGiven any quasi-optimal configuration $D_s$ for $\\Lambda_s^+(\\alpha)$, any $x\\in \\partial\\Omega$ and $\\varepsilon>0$, there exists $s_0\\in (0,1)$ such that for every $s_0<s<1$ it holds\n$$\n|D_s\\cap B_\\varepsilon(x)|>0.\n$$\n\\end{thm}\n\n\\begin{proof}\nAssume that the conclusion is false. Then, there exists a sequence $s_k\\uparrow 1$, sets $D_k\\subset\\mathbb{R}^n\\setminus \\Omega$ and $\\varepsilon_0>0$ such that\n$$\n\\lambda_{s_k}(D_k) \\ge \\Lambda_{s_k}^+(\\alpha) - \\frac{1}{k}\\quad \\text{and} \\quad |D_k\\cap B_{\\varepsilon_0}(x)|=0.\n$$\n\nNow, by \\eqref{limLambda+}, we have\n\\begin{equation}\\label{liminf}\n\\lambda_1 \\le \\liminf_{k\\to \\infty} \\lambda_{s_k}(D_k).\n\\end{equation}\n\nOn the other hand, let $\\Gamma = \\partial\\Omega\\cap B_{\\varepsilon_0}(x)$ \nand consider the following eigenvalue problem (that was already mentioned in the introduction)\n\\begin{equation}\\label{mixed}\n\\begin{cases}\n-\\Delta_p u = \\lambda |u|^{p-2}u & \\text{in }\\Omega\\\\\nu= 0 & \\text{on } \\partial\\Omega\\setminus \\Gamma\\\\\n|\\nabla u|^{p-2}\\partial_\\nu u = 0 & \\text{on } \\Gamma.\n\\end{cases}\n\\end{equation}\n\nLet us denote by $\\lambda_1(\\Gamma)$ the first eigenvalue of \\eqref{mixed}, i.e.\n$$\n\\lambda_1(\\Gamma) = \\inf \\frac{\\displaystyle\\int_\\Omega |\\nabla v|^p\\, dx}{\\displaystyle\\int_{\\Omega} |v|^p\\, dx}\n$$\nwhere the infimum is taken over all functions $v\\in W^{1,p}(\\Omega)$ such that $v=0$ on $\\partial\\Omega\\setminus \\Gamma$.\n\nIt is straightforward to see that $$0= \\lambda_1^N \\leq \\lambda_1(\\Gamma) \\leq \\lambda_1$$\nand that the above inequalities are strict when $\\Gamma \\neq \\emptyset, \\partial \\Omega$.\n\nNow, denote by $u_1\\in W^{1,p}(\\Omega)$ the eigenfunction of \\eqref{mixed} associated to $\\lambda_1(\\Gamma)$ extended by 0 on $D_k$ normalized in $L^p(\\Omega)$. Then $u_1\\in W^{s,p}_{D_k}(\\Omega)$ and so\n\\begin{align*}\n\\limsup_{k\\to\\infty} \\lambda_{s_k}(D_k) &\\le \\lim_{k\\to\\infty} K(n,s_k,p) [u_1]_{s_k, p}^p\\\\\n& = \\|\\nabla u_1\\|_p^p = \\lambda_1(\\Gamma)\\\\\n& <  \\lambda_1.\n\\end{align*}\nThis contradicts \\eqref{liminf} and the proof is complete.\n\\end{proof}\n\n\\section{The minimization problem}\n\nNow we analize the minimization problem\n\\begin{equation}\\label{inf.l}\n\\Lambda^-_s(\\alpha) := \\inf_{D\\in \\mathcal{D}_\\alpha} \\lambda_s(D).\n\\end{equation}\nAs we mentioned in the introduction, this problem is of little interest since taking a sequence of domains to infinity makes the eigenvalues go to zero.\n\n\\begin{prop}\nLet $\\Omega\\subset\\mathbb{R}^n$ be bounded and open and let $0<s<1<p<\\infty$ be fixed. Then\n$$\n\\Lambda_s^-(\\alpha)=0,\n$$\nfor every $\\alpha>0$, where $\\Lambda^-_s(\\alpha)$ is the constant defined in \\eqref{inf.l}.\n\\end{prop}\n\n\\begin{proof}\nTake $r>0$ such that $|B_r(0)|=\\alpha$ and define $D_k= B_r(ke_1)$ where $e_1=(1,0,\\dots,0)\\in \\mathbb{R}^n$ so $D_k\\in \\mathcal{D}_\\alpha$.\n\nNow, let $u_k$ be defined as\n$$\nu_k(x) = \\begin{cases}\n|\\Omega|^{-\\frac{1}{p}} & x\\in\\Omega\\\\\n0 & x\\in D_k.\n\\end{cases}\n$$\nObserve that $u_k\\in W^{s,p}_{D_k}(\\Omega)$ for $k$ large and $\\|u_k\\|_{p;\\Omega} = 1$.\n\nSo,\n$$\n\\Lambda_s^-(\\alpha)\\le \\lambda_s(D_k) \\le K [u_k]_{s,p;\\Omega\\cup D_k}^p,\n$$\nwith\n$$\n[u_k]_{s,p;\\Omega\\cup D_k}^p = \\frac{2}{|\\Omega|} \\int_{D_k}\\int_{\\Omega} \\frac{1}{|x-y|^{n+sp}}\\, dxdy.\n$$\nNow, just observe that $|x-y|\\sim k$ for $x\\in\\Omega$ and $y\\in D_k$ and so there exists a constant $C>0$ independent of $k$ such that\n$$\n[u_k]_{s,p;\\Omega\\cup D_k}^p \\le C k^{-(n+sp)}.\n$$\nThis completes the proof.\n\\end{proof}\nWe consider now the problem $\\Lambda_s^{-,R}(\\alpha)$ defined by\n\\begin{equation}\\label{inf.ll}\n\\Lambda_s^{-,R}(\\alpha)=\\inf_{\\mathcal{D}_\\alpha^R}\\lambda_s(D).\n\\end{equation}\nIn this case the admissible sets are forced to be inside  of the ball $B_R(0)$. Note that in this case we have\n$\\Lambda_s^{-,R}(\\alpha)>0$.\nTaking the limit as $s\\uparrow 1$ in $\\Lambda_s^{-,R}(\\alpha)$ we obtain that this quantity tends to zero. This is contained in the following proposition.\n\n\\begin{prop}\\label{-R}\nLet $\\Omega\\subset\\mathbb{R}^n$ be bounded and open, let $0<s<1<p<\\infty$ be fixed. Then \n$$\n\\lim_{s\\uparrow 1} \\Lambda_s^{-,R}(\\alpha)=0,\n$$\nfor every $\\alpha>0$, where $\\Lambda_s^{-,R}(\\alpha)$ is the constant defined in \\eqref{inf.ll}. \n\\end{prop}\n\\begin{proof}\nLet $D\\in \\mathcal{D}_\\alpha^R$ be such that $\\mathop{\\text{\\normalfont dist}}(D, \\Omega)>0$. Now we define the function\n$$\nu(x) = \\begin{cases}\n|\\Omega|^{-\\frac{1}{p}} & x\\in\\Omega\\\\\n0 & x\\in D,\n\\end{cases}\n$$\n so $u\\in W_D^{s,p}(\\Omega)$ and $\\|u\\|_{p;\\Omega}=1$.\n\nWe can then estimate $\\Lambda_s^{-,R}(\\alpha)$ as follows,\n$$\n\\Lambda_s^{-,R}(\\alpha)\\le \\lambda_s(D)\\le K [u]_{s,p,\\Omega\\cup D}^p.\n$$\nOn the other side\n$$\n[u]_{s,p,\\Omega\\cup D}^p = \\frac{2}{|\\Omega|}\\int_{D} \\int_{\\Omega}\\frac{1}{|x-y|^{n+sp}}\\,dxdy\\le \\frac{2\\alpha}{\\mathop{\\text{\\normalfont dist}}(D,\\Omega)^{n+sp}},\n$$\nthen \n$$\nK[u]_{s,p,\\Omega\\cap D_R}\\le C (1-s),\n$$\nwhere the constant $C$ does not depend on $s$. This implies the desired result.\n\\end{proof}\n\nTo end this section, we show that for any fixed $0<s<1$ there exists an optimal configuration for the constant $\\Lambda_s^{-,R}(\\alpha)$. Let us observe that this does not follows by a direct application of the {\\em Direct Method of the Calculus of Variations} since for a minimizing sequence of domains, we do not have enough compactness in a topology of domains and, what is more important, the associated eigenfunctions do not lie in the same functional space.\n\nWe begin with the following Lemma.\n\\begin{lemp}\\label{lema1}\nGiven $u\\in W^{s,p}(\\Omega)$, there exists $D_u\\subset B_R\\setminus\\Omega$, $|D_u|=\\alpha$ such that \n$\\phi=\\chi_{D_u}$ is a solution to the following minimization problem\n$$\n\\min_\\phi \\left\\{\\int_\\Omega \\int_{B_R\\setminus \\Omega} \\frac{\\phi(y) |u(x)|^p}{|x-y|^{n+sp}}\\, dy\\, dx\n\\ \\colon \\ 0\\le \\phi\\le 1, \\int_{B_R\\setminus \\Omega} \\phi(y)\\, dy=\\alpha\\right\\}.\n$$\n\\end{lemp}\n\n\\begin{proof}\nThe proof is a direct consequence of the {\\em Bathtube Principle} \\cite[Theorem 1.14]{Lieb-Loss}.\nIn fact, let \n$$\nf(y) := \\int_\\Omega \\frac{|u(x)|^p}{|x-y|^{n+sp}}\\, dx,\n$$\nand observe that the level sets of $f$, $\\{f\\le t\\}$, has finite measure.\nSo, the Bathtube principle says that the problem\n$$\n\\inf_{\\phi}\\left\\{\\int_{B_R\\setminus \\Omega} \\phi(y) f(y)\\, dy\\colon 0\\le \\phi\\le 1,\\ \\int_{B_R\\setminus \\Omega} \\phi(y)\\, dy=\\alpha\\right\\}\n$$\nhas a solution of the form $\\phi = \\chi_D$, where $\\{f<t\\}\\subset D\\subset \\{f\\le t\\}$, and $|D|=\\alpha$ for some $t>0$.\n\nThis completes the proof of the Lemma.\n\\end{proof}\n\nWe can now prove the existence of the optimal configuration for $\\Lambda^{-,R}_s(\\alpha)$.\n\n\n\\begin{thm}\\label{existe.config}\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bounded domain, let $\\alpha>0$ and let $R>0$ be such that $|B_R\\setminus \\Omega|>\\alpha$. Then, there exists an optimal set $D\\subset B_R\\setminus \\Omega$, that is, $D$ such that $|D|=\\alpha$ and\n$$\n\\Lambda^{-,R}_s(\\alpha) = \\lambda_s(D).\n$$\n\\end{thm}\n\n\\begin{proof}\nLet $D_k\\subset B_R\\setminus \\Omega$ be a minimizing sequence for $\\Lambda^{-,R}_s(\\alpha)$, that is\n$$\n|D_k|=\\alpha \\qquad \\text{and}\\qquad \\lambda_s(D_k)\\to \\Lambda^{-,R}_s(\\alpha).\n$$\nLet $u_k\\in W^{s,p}_{D_k}(\\Omega)$ be the normalized, nonnegative eigenfunction associated to $\\lambda_s(D_k)$. Then, it is easy to see that the sequence $\\{u_k\\}_{k\\in\\mathbb{N}}$ is bounded in $W^{s,p}(\\Omega)$ and so, there exists $u\\in W^{s,p}(\\Omega)$ such that, up to a subsequence,\n\\begin{align*}\nu_k\\rightharpoonup u & \\text{ weakly in } W^{s,p}(\\Omega)\\\\\nu_k\\to u & \\text{ strongly in } L^p(\\Omega) \\text{ and a.e. in }\\Omega.\n\\end{align*}\n\nNow, let $D=D_u$ be the optimal set given in Lemma \\ref{lema1}. Let us check that this set is optimal for $\\Lambda^{-,R}_s(\\alpha)$.\n\nIn fact, passing to a further subsequence, if necessary, we can assume that there exists $\\phi\\in L^\\infty(B_R\\setminus\\Omega)$, such that\n$$\n\\chi_{D_k} \\stackrel{*}{\\rightharpoonup} \\phi \\quad *-\\text{weakly in } L^\\infty(B_R\\setminus \\Omega).\n$$\nMoreover, this function $\\phi$ satisfies\n$$\n0\\le \\phi\\le 1\\quad \\text{and}\\quad \\int_{B_R\\setminus\\Omega}\\phi(y)\\, dy=\\alpha.\n$$\n\nObserve now, that given $x\\in \\Omega$, the function $y\\mapsto |x-y|^{-(n+sp)}$ belongs to $L^1(B_R\\setminus\\Omega)$. So, for any $x\\in \\Omega$, we have that\n$$\n\\int_{D_k} \\frac{1}{|x-y|^{n+sp}}\\, dy\\to \\int_{B_R\\setminus\\Omega} \\frac{\\phi(y)}{|x-y|^{n+sp}}\\, dy \\quad \\text{as }k\\to\\infty.\n$$\nWe can then apply Fatou's Lemma to conclude that\n$$\n\\int_\\Omega\\int_{B_R\\setminus\\Omega} \\frac{\\phi(y)|u(x)|^p}{|x-y|^{n+sp}}\\, dy\\, dx \\le \\liminf_{k\\to\\infty} \\int_\\Omega\\int_{D_k} \\frac{|u_k(x)|^p}{|x-y|^{n+sp}}\\, dy\\, dx\n$$\nand so, from our choice of $D$ (c.f. Lemma \\ref{lema1}), we obtain\n\\begin{equation}\\label{complemento}\n\\int_\\Omega\\int_{D} \\frac{|u(x)|^p}{|x-y|^{n+sp}}\\, dy\\, dx \\le \\liminf_{k\\to\\infty} \\int_\\Omega\\int_{D_k} \\frac{|u_k(x)|^p}{|x-y|^{n+sp}}\\, dy\\, dx\n\\end{equation}\nAlso, by the weak semicontinuity of the Gagliardo seminorm, we have\n\\begin{equation}\\label{gagliardo}\n\\iint_{\\Omega\\times\\Omega} \\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\\, dxdy \\le \\liminf_{k\\to\\infty} \\iint_{\\Omega\\times\\Omega} \\frac{|u_k(x)-u_k(y)|^p}{|x-y|^{n+sp}}\\, dxdy\n\\end{equation}\n\nObserve now that if we extend $u$ to $D$ by zero, we have that\n$$\n[u]_{s,p;\\Omega\\cup D}^p = \\iint_{\\Omega\\times\\Omega} \\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\\, dxdy + 2\\int_\\Omega\\int_{D} \\frac{|u(x)|^p}{|x-y|^{n+sp}}\\, dy\\, dx \n$$\nand so \\eqref{complemento} and \\eqref{gagliardo} imply that $u\\in W^{s,p}_D(\\Omega)$ and that\n$$\n\\lambda_s(D)\\le K [u]_{s,p; \\Omega\\cup D}^p\\le \\liminf_{k\\to\\infty} K [u_k]_{s,p; \\Omega\\cup D_k}^p = \\lim_{k\\to\\infty}\\lambda_s(D_k) = \\Lambda^{-,R}_s(\\alpha).\n$$\n\nThe proof is complete.\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{teo.main2}}\n\n\nThe results for the maximization problem follows exactly as those in Section 3 with the obvious modifications. \n\nIt remains to check the results about the minimization problem.\n\nFor this, we first observe that, by Lemma \\ref{orden} (with the obvious modifications to include the potential function $V$), it holds that\n$$\n\\lambda_{s,V}(\\emptyset) \\le \\lambda_{s,V}(D),\n$$\nfor every $D\\in \\mathcal{D}_\\alpha$, from where it follows that\n\\begin{equation}\\label{primera.cota}\n\\lambda_{s,V}(\\emptyset)\\le \\Lambda_{s,V}^-(\\alpha)\\le  \\Lambda_{s,V}^{-,R}(\\alpha)\n\\end{equation}\nfor any $R>0$.\n\nRecall that $\\lambda_{s,V}(\\emptyset)$ is the first eigenvalue of the fractional $p-$laplacian plus a potential with homogeneous Neumann boundary conditions and is given by\n$$\n\\lambda_{s,V}(\\emptyset) = \\inf_{v\\in W^{s,p}(\\Omega)} \\frac{\n\\displaystyle K[v]_{s,p; \\Omega}^p + \\int_\\Omega V(x) |v|^p\\, dx}{\\|v\\|_{p;\\Omega}^p}.\n$$\n\nIt is not difficult to check (c.f. with Remark \\ref{rem.ponce}) that $\\lambda_{s,V}(\\emptyset)$ converges to the first eigenvalue of the local $p-$laplacian plus a potential with homogeneous Neumann boundary conditions. i.e.\n\\begin{equation}\\label{segunda.cota}\n\\lim_{s\\uparrow 1} \\lambda_{s,V}(\\emptyset) = \\lambda_1^N(V),\n\\end{equation}\nwhere\n$$\n\\lambda_1^N(V) = \\inf_{v\\in W^{1,p}(\\Omega)} \\frac{\n\\displaystyle \\int_\\Omega |\\nabla v|^p\\, dx + \\int_{\\Omega} V(x) |v|^p\\, dx}{\\displaystyle \\int_\\Omega |v|^p\\, dx}.\n$$\n\nLet us now give an estimate for $\\Lambda_{s,V}^{-,R}(\\alpha)$. This estimate uses the ideas from Proposition \\ref{-R}.\n\nLet $D\\in \\mathcal{D}_\\alpha^R$ be such that $\\mathop{\\text{\\normalfont dist}}(D,\\Omega)=d>0$. So $\\Lambda_{s,V}^{-,R}(\\alpha)\\le \\lambda_{s,V}(D)$.\n\nDenote by $u_1\\in W^{1,p}(\\Omega)$ be the eigenfunction associated to $\\lambda_1^N(V)$ normalized such that $\\|u_1\\|_{p;\\Omega}=1$ and extend $u_1$ to $D$ by zero. Since $\\mathop{\\text{\\normalfont dist}}(D,\\Omega)>0$ we have that $u_1\\in W^{s,p}_D(\\Omega)$ and hence\n$$\n\\lambda_{s,V}(D) \\le K[u_1]_{s,p;\\Omega\\cup D}^p + \\int_\\Omega V(x) |u_1|^p\\, dx.\n$$\n\nNow\n\\begin{align*}\nK[u_1]_{s,p;\\Omega\\cup D}^p &= K[u_1]_{s,p;\\Omega}^p + 2K \\int_D\\int_\\Omega \\frac{|u_1(x)|^p}{|x-y|^{n+sp}}\\, dx\\, dy\\\\\n&= I + II. \n\\end{align*}\nUsing the results of \\cite{Bourgain-Brezis-Mironescu} it follows that\n$$\nI = K[u_1]_{s,p;\\Omega}^p \\to \\|\\nabla u_1\\|_{p;\\Omega}^p \\quad \\text{as } s\\uparrow 1.\n$$\nFinally\n$$\nII\\le 2K \\frac{\\|u_1\\|_{p;\\Omega}^p |D|}{d^{n+sp}} = \\frac{2\\alpha K}{d^{n+sp}}\\to 0 \\quad \\text{as } s\\uparrow 1\n$$\n(recall that $K\\sim (1-s)$, from \\eqref{defK}). Combining all this we arrive at\n\\begin{equation}\\label{tercera.cota}\n\\lim_{s\\uparrow 1} \\lambda_{s,V}(D) \\le \\|\\nabla u_1\\|_{p;\\Omega}^p + \\int_\\Omega V(x) |u_1|^p\\, dx = \\lambda_1^N(V).\n\\end{equation}\n\nPutting together \\eqref{primera.cota}, \\eqref{segunda.cota} and \\eqref{tercera.cota} we arrive at\n$$\n\\lim_{s\\uparrow 1} \\Lambda_{s,V}^-(\\alpha) = \\lim_{s\\uparrow 1} \\Lambda_{s,V}^{-,R}(\\alpha) = \\lambda_1^N(V).\n$$\n\nTo finish this section just observe that the existence of an optimal configuration for the constant $\\Lambda_{s,V}^{-,R}(\\alpha)$ follows without change as in the proof of Theorem \\ref{existe.config}.\n\n\n\\section*{Acknowledgements}\n\nThis paper was partially supported by Universidad de Buenos Aires under grant UBACyT 20020130100283BA, by ANPCyT under grant PICT 2012-0153 and by CONICET under grant PIP2015 11220150100032CO. J. Fern\\'andez Bonder and J. D. Rossi are members of CONICET.\n\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn recent years, there has been great interest in characterizing the fixed-error  \nasymptotics (e.g. dispersion, the Gaussian approximation) \nof source coding and channel coding problems, and the behavior is well-understood\nfor a variety of single-user settings \\cite{Strassen,Hayashi09,PPV10}.  \nOn the other hand, analogous studies of multi-user\nproblems have generally had significantly less success, with the main exceptions being\nSlepian-Wolf coding \\cite{TK12,Nom14}, the Gaussian interference channel with strictly very strong\ninterference \\cite{LeTan14}, and the Gaussian multiple-access channel (MAC) with degraded \nmessage sets~\\cite{ScaTan13}.\n\nIn this paper, we complement our work on the latter problem by considering its  discrete\ncounterpart. By obtaining matching achievability and converse results, we provide\nthe first complete characterization of the second-order \nasymptotics for a discrete channel-type network information theory problem. \n\n\\subsection{System Setup}\n\nWe consider the two-user discrete memoryless MAC (DM-MAC) with degraded message \nsets \\cite[Ex.~5.18]{elgamal}, with input alphabets $\\mathcal{X}_1$ and $\\mathcal{X}_2$ and \noutput alphabet $\\mathcal{Y}$.  As usual, there are two messages $m_1$ and $m_2$,\nequiprobable on the sets $\\{1,\\dotsc,M_1\\}$ and $\\{1,\\dotsc,M_2\\}$ respectively. \nThe first user knows both messages, whereas the second user only knows $m_2$.  Given\nthese messages, the users transmit the codewords $\\mathbf{x}_1(m_1,m_2)$ and $\\mathbf{x}_2(m_2)$ from their \nrespective codebooks, and the decoder receives a noisy output sequence which is generated according\nto the memoryless transition law $W^n(\\mathbf{y}|\\mathbf{x}_1,\\mathbf{x}_2) = \\prod_{i=1}^{n}W(y_i|x_{1,i},x_{2,i})$.  \nAn estimate $(\\hat{m}_1,\\hat{m}_2)$ is formed, and an error is said to have occurred if\n$(\\hat{m}_1,\\hat{m}_2) \\ne (m_1,m_2)$.\n\nThe capacity region $\\mathcal{C}$ is given by the set of rate pairs $(R_1,R_2)$ satisfying \\cite[Ex. 5.18]{elgamal}\n\\begin{align}\n    R_1       \\le I(X_1;Y|X_2) \\label{eqn:cap_region1} \\\\\n    R_1 + R_2 \\le I(X_1,X_2;Y) \\label{eqn:cap_region2}\n\\end{align} \nfor some  input joint distribution $P_{X_1X_2}$, where the mutual information quantities are \nwith respect to $P_{X_1X_2}(x_1,x_2)W(y|x_1,x_2)$.  The achievability part is proved\nusing superposition coding.\n\nWe formulate the second-order asymptotics according to the following definition \\cite{Nom14}.\n\n\\begin{definition}[Second-Order Coding Rates] \\label{def;second}\n    Fix $\\varepsilon\\in (0,1)$, and let $(R_1^*, R_2^*)$ be a pair of rates on the boundary of $\\mathcal{C}$. A pair $(L_1, L_2)$ is {\\em $(\\varepsilon,R_1^*,R_2^*)$-achievable} \n    if there exists a sequence of codes with length $n$, number of codewords\n    for message $j=1,2$ equal to $M_{j,n}$, and average error probability $\\varepsilon_n$, such that  \n    \\begin{gather}\n    \\liminf_{n\\to\\infty}\\frac{1}{\\sqrt{n}}(\\log M_{j,n} - nR_j^*) \\ge L_j,\\quad j = 1,2, \\label{eqn:second_def0}\\\\\n    \\limsup_{n\\to\\infty}\\varepsilon_n \\le \\varepsilon. \\label{eqn:second_def}\n    \\end{gather}\n    The {\\em $(\\varepsilon,R_1^*,R_2^*)$-optimal second-order coding rate region} \n    $\\mathcal{L}(\\varepsilon;R_1^*,R_2^*) \\subset\\mathbb{R}^2$ is defined to be the closure of the \n    set of all $(\\varepsilon,R_1^*,R_2^*)$-achievable rate pairs  $(L_1, L_2)$. \n\\end{definition}\n\nThroughout the paper, we write non-asymptotic rates as \n$R_{1,n} := \\frac{1}{n}\\log{M_{1,n}}$ and $R_{2,n} := \\frac{1}{n}\\log{M_{2,n}}$.\nRoughly speaking, the preceding definition is concerned with $\\varepsilon$-reliable codes \nsuch that $R_{j,n} \\ge R_{j}^* + \\frac{1}{\\sqrt n}L_{j} + o\\big(\\frac{1}{\\sqrt n}\\big)$ for $j=1,2$.\n\nWe will also use the following standard definition: A rate pair $(R_1,R_2)$ is \n\\emph{$(n,\\varepsilon)$-achievable} if there exists a length-$n$ code having an average \nerror probability no higher than $\\varepsilon$, and whose rate is at least $R_{j}$\nfor message $j=1,2$.\n\n\\subsection{Notation}\n\nExcept where stated otherwise,\\footnote{For example, the vectors in \\eqref{eqn:rate_vec}--\\eqref{eqn:L_vec},  do not adhere to this convention.} \nthe $i$-th entry of a vector \n(e.g. $\\mathbf{y}$) is denoted using a subscript (e.g. $y_i$).\nFor two vectors of the same length $\\mathbf{a},\\mathbf{b}\\in\\mathbb{R}^d$, \nthe notation $\\mathbf{a}\\le\\mathbf{b}$ means that $a_j\\le b_j$ for \nall $j$. The notation $\\mathcal{N}(\\mathbf{u};\\bm{\\mu}, \\bm{\\Lambda})$ \ndenotes the  multivariate Gaussian probability density function \n(pdf) with mean $\\bm{\\mu}$ and covariance $\\bm{\\Lambda}$. \nWe use the standard asymptotic notations $O(\\cdot)$, $o(\\cdot)$,\n$\\Theta(\\cdot)$, and $\\omega(\\cdot)$.\nAll logarithms have base $e$, and all rates have units of nats.\nThe closure operation is denoted by $\\mathrm{cl}(\\cdot)$.\n\nThe set of all probability distributions on an alphabet $\\mathcal{X}$ \nis denoted by $\\mathcal{P}(\\mathcal{X})$, and the set of all types \\cite[Ch. 2]{Csi97}\nis denoted by $\\mathcal{P}_n(\\mathcal{X})$.  For a given type $Q_X \\in \\mathcal{P}_n(\\mathcal{X})$,\nwe define the type class $T^n(Q_X)$ to be the set of sequences having type $Q_X$.\nSimilarly, given a conditional type $Q_{Y|X}$ and a sequence $\\mathbf{x} \\in T^n(Q_X)$,  \nwe define $T_{\\mathbf{x}}^n(Q_{Y|X})$ to be the set of sequences $\\mathbf{y}$ such that\n$(\\mathbf{x},\\mathbf{y}) \\in T^n(Q_X \\times Q_{Y|X})$.\n\n\\section{Main Result}\n\n\\subsection{Preliminary Definitions}\n\nGiven the rate pairs $(R_{1,n}, R_{2,n})$ and $(R_1^*,R_2^*)$, we define\n\\begin{align}\n    \\mathbf{R}_n := \\begin{bmatrix}\n    R_{1,n} \\\\ R_{1,n} + R_{2,n}\n    \\end{bmatrix}, \\quad \n    \\mathbf{R}^* := \\begin{bmatrix}\n    R_1^* \\\\ R_1^* + R_2^*\n    \\end{bmatrix} \\label{eqn:rate_vec}\n\\end{align}\nSimilarly, given the second-order rate pair $(L_1,L_2)$, we write\n\\begin{align}\n    \\mathbf{L} := \\begin{bmatrix}\n    L_1 \\\\ L_1 + L_2 \\label{eqn:L_vec}\n    \\end{bmatrix}\n\\end{align}\nGiven a joint input distribution $P_{X_1X_2}\\in\\mathcal{P}(\\mathcal{X}_1\\times\\mathcal{X}_2)$, we define \n$P_{X_1X_2Y} := P_{X_1X_2} \\times W$, and denote the induced\nmarginals by $P_{Y|X_1}$, $P_Y$, etc.  We define the following {\\em information density vector}, \nwhich implicitly depends on $P_{X_1X_2}$:\n\\begin{align} \n    \\mathbf{j}(x_1, x_2  , y) &:= \\begin{bmatrix}\n    j_1(x_1, x_2  , y)   &  j_{12}(x_1, x_2,y)\n    \\end{bmatrix}^T  \\\\\n    &= \\begin{bmatrix}\n    \\log\\dfrac{W(y|x_1,x_2)}{P_{Y|X_2}(y|x_2)}  & \\log\\dfrac{W(y|x_1,x_2)}{P_{Y}(y)}  \n    \\end{bmatrix}^T.\n    \\label{eqn:info_dens}\n\\end{align} \nThe mean and conditional covariance matrix are given by\n\\begin{align}\n    \\mathbf{I}(P_{X_1X_2}) &=\\mathbb{E}\\big[\\, \\mathbf{j}(X_1,X_2,Y)\\big],  \\label{eqn:mean_v}\\\\\n    \\mathbf{V}(P_{X_1X_2}) &=\\mathbb{E}\\big[\\cov \\big(\\mathbf{j}(X_1,X_2,Y) \\, \\big|\\, X_1,X_2 \\big)\\big]. \\label{eqn:cov_v}\n\\end{align}\nObserve that the entries of $\\mathbf{I}(P_{X_1X_2})$ are the mutual informations\nappearing in \\eqref{eqn:cap_region1}--\\eqref{eqn:cap_region2}.  We write \nthe entries of $\\mathbf{I}$ and $\\mathbf{V}$ using subscripts as follows:\n\\begin{align}\n    \\mathbf{I}(P_{X_1X_2}) &= \\begin{bmatrix}\n    I_1(P_{X_1X_2}) \\\\ I_{12}(P_{X_1X_2})\n    \\end{bmatrix} \\label{eqn:mi_vec}, \\\\\n    \\mathbf{V}(P_{X_1X_2}) &= \\begin{bmatrix}\n    V_1(P_{X_1X_2})  & V_{1,12}(P_{X_1X_2}) \\\\ V_{1,12}(P_{X_1X_2})  & V_{12}(P_{X_1X_2}) \\end{bmatrix}, \\label{eqn:inf_disp_matr}\n\\end{align}\nFor a given point $(z_1, z_2) \\in \\mathbb{R}^2$ and a positive semi-definite matrix $\\mathbf{V}$, \nwe define the multivariate Gaussian cumulative distribution function (CDF)\n\\begin{align}\n    \\Psi(z_1, z_2;\\mathbf{V}) :=\\int_{-\\infty}^{z_2}\\int_{-\\infty}^{z_1}\\mathcal{N}(\\mathbf{u};\\mathbf{0},\\mathbf{V})\\,\\mathrm{d} \\mathbf{u},\n\\end{align}\nand for a given $\\varepsilon\\in (0,1)$, we define the corresponding ``inverse\" set\n\\begin{align}\n    \\Psi^{-1}(\\mathbf{V},\\varepsilon):=\\left\\{ (z_1, z_2)\\in\\mathbb{R}^2:\\Psi(-z_1,- z_2;\\mathbf{V}) \\ge  1-\\varepsilon\\right\\}. \\label{eqn:psiinv}\n\\end{align}\nSimilarly, we let $\\Phi(\\cdot)$ denote the standard Gaussian CDF, and we denote\nits functional inverse by $\\Phi^{-1}(\\cdot)$.  Moreover, we let \n\\begin{align}\n    \\Pi(R_1^*,R_2^*) := \\big\\{ P_{X_1X_2} \\,:\\,   \\mathbf{I}(P_{X_1X_2}) \\ge \\mathbf{R}^* \\big\\} \\label{eqn:def_Pi}\n\\end{align}\nbe the set of input distributions achieving a given point $(R_1^*,R_2^*)$ of \nthe boundary of $\\mathcal{C}$. Note that in contrast with the single-user setting \n\\cite{Strassen, PPV10, Hayashi09}, this definition uses an inequality rather\nthan an equality, as one of the mutual information quantities may be \nstrictly larger than the corresponding entry of $\\mathbf{R}^*$ and yet be \nfirst-order optimal. For example, assuming that the capacity region on the\nleft of Figure \\ref{fig:tangents} is achieved by a single input distribution,\nall points $(R_1^*,R_2^*)$ on the vertical boundary satisfy $I_{12}(P_{X_1 X_2}) > R_1^*+R_2^*$.\n\nThe preceding definitions are analogous to those appearing in previous works such\nas \\cite{TK12}, while the remaining definitions are somewhat less standard.  Given\nthe boundary point $(R_1^*,R_2^*)$, we let\n$\\hat{\\mathbf{T}}_{-} := \\hat{\\mathbf{T}}_{-}(R_1^*,R_2^*)$ and $\\hat{\\mathbf{T}}_{+} := \\hat{\\mathbf{T}}_{+}(R_1^*,R_2^*)$ denote the\nleft and right unit tangent vectors along the boundary of $\\mathcal{C}$ in $(R_1,R_2)$\nspace; see Figure \\ref{fig:tangents} for an illustration. Furthermore, we define\n\\begin{align}\n    {\\mathbf{T}}_{-} := \\begin{bmatrix} \\hat{T}_{-,1} \\\\ \\hat{T}_{-,1} + \\hat{T}_{-,2} \\end{bmatrix}, \\quad\n    {\\mathbf{T}}_{+} := \\begin{bmatrix} \\hat{T}_{+,1} \\\\ \\hat{T}_{+,1} + \\hat{T}_{+,2} \\end{bmatrix}.\n\\end{align}\nIt is understood that $\\hat{\\mathbf{T}}_{-}$ and ${\\mathbf{T}}_{-}$ (respectively, $\\hat{\\mathbf{T}}_{+}$ and ${\\mathbf{T}}_{+}$)\nare undefined when $R_1^* = 0$ (respectively, $R_2^* = 0$).  As is observed\nin Figure \\ref{fig:tangents}, we have $\\hat{\\mathbf{T}}_{-}=-\\hat{\\mathbf{T}}_{+}$ on the curved\nand straight-line parts of $\\mathcal{C}$, and $\\hat{\\mathbf{T}}_{-}\\ne-\\hat{\\mathbf{T}}_{+}$ when\nthere is a sudden change in slope (e.g. at a corner point).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.95\\columnwidth]{DegMAC_Example}\n\n\\vspace{-3mm}\n\\caption{Unit tangent vectors $\\hat{\\mathbf{T}}_{-}$ and $\\hat{\\mathbf{T}}_{+}$ for two boundary points $(R_1^*,R_2^*)$ of two hypothetical capacity regions.} \n\\vspace{-3mm}\n\n\\label{fig:tangents}\n\\end{figure}\n\nThe following set of vectors can be thought of as those that point strictly inside  \n$\\mathcal{C}$ when placed at $(R_1^*,R_2^*)$:\n\\begin{multline}\n    \\hat{\\mathcal{V}}(R_1^*,R_2^*) := \\\\ \\big\\{ \\mathbf{v}\\in\\mathbb{R}^2 \\,:\\, (R_1^*,R_2^*) + \\alpha \\mathbf{v} \\in \\mathcal{C} \\text{ for some } \\alpha>0 \\big\\}.\n\\end{multline}\nUsing this definition, we set\n\\begin{align}\n    \\mathcal{V}(R_1^*,R_2^*) := \\mathrm{cl}\\bigg( \\bigcup_{(v_1,v_2) \\in \\hat{\\mathcal{V}}(R_1^*,R_2^*)}\\big\\{ (v_1,v_1+v_2) \\big\\}\\bigg). \\label{eqn:setV}\n\\end{align}\nDue to the closure operation, it is readily verified that ${\\mathbf{T}}_{-} \\in \\mathcal{V}$ and\n${\\mathbf{T}}_{+} \\in \\mathcal{V}$.\n\n\\subsection{Statement of Main Result}\n\nFor a given boundary point $(R_1^*,R_2^*)$ and input distribution $P_{X_1X_2} \\in \\Pi(R_1^*,R_2^*)$, \nwe define the set $\\mathcal{L}_0(\\varepsilon;R_1^*,R_2^*,P_{X_1X_2})$ separately for the following three cases:\n\n\\emph{(i)} If $R_1^* = I_1(P_{X_1X_2})$ and  $R_1^* + R_2^* < I_{12}(P_{X_1X_2})$, then\n    \\begin{align}\n        \\mathcal{L}_0 = \\Big\\{ (L_1,L_2) \\,:\\, L_1 \\le \\sqrt{V_1(P_{X_1X_2})}\\Phi^{-1}(\\varepsilon) \\Big\\} \\label{eqn:L0_1}\n    \\end{align}\n        \n\\emph{(ii)} If $R_1^* < I_1(P_{X_1X_2})$ and  $R_1^* + R_2^* = I_{12}(P_{X_1X_2})$, then\n    \\begin{align}\n        \\hspace*{-0.1ex}\\mathcal{L}_0 = \\Big\\{ (L_1,L_2) \\,:\\, L_1+L_2 \\le \\sqrt{V_{12}(P_{X_1X_2})}\\Phi^{-1}(\\varepsilon) \\Big\\} \\label{eqn:L0_12}\n    \\end{align}\n        \n\\emph{(iii)} If $R_1^* = I_1(P_{X_1X_2})$ and  $R_1^* + R_2^* = I_{12}(P_{X_1X_2})$, then\n    \\begin{align}\n        \\mathcal{L}_0 = \\Big\\{ (L_1,L_2) \\,:\\, \\mathbf{L} \\in \\bigcup_{\\beta \\ge 0} \\big\\{ \\beta{\\mathbf{T}}_{-} + \\Psi^{-1}(\\mathbf{V}(P_{X_1X_2}),\\varepsilon) \\big\\} \\Big\\} & \\nn \\\\\n            \\cup~\\Big\\{ (L_1,L_2) \\,:\\, \\mathbf{L} \\in \\bigcup_{\\beta \\ge 0} \\big\\{ \\beta{\\mathbf{T}}_{+} + \\Psi^{-1}(\\mathbf{V}(P_{X_1X_2}),\\varepsilon) \\big\\} \\Big\\}  &, \\label{eqn:L0_both}\n    \\end{align}\nwhere the first (respectively, second) set in the union is understood to be empty\nwhen $R_1^*=0$ (respectively, $R_2^*=0$).\n    \nWe are now in a position to state our main result.\n\n\\begin{theorem} \\label{thm:local}\n    For any point $(R_1^*,R_2^*)$ on the boundary of the capacity region, and any $\\varepsilon\\in(0,1)$, we have\n    \\begin{equation}\n        \\mathcal{L}(\\varepsilon;R_1^*,R_2^*) \\!=\\! \\bigcup_{P_{X_1X_2} \\in \\Pi(R_1^*,R_2^*)} \\mathcal{L}_0(\\varepsilon;R_1^*,R_2^*,P_{X_1X_2}). \\label{eqn:local}\n    \\end{equation}\n\\end{theorem}\n\\begin{proof}\n    See Section \\ref{sec:prf_local}.\n\\end{proof}\n\nSuppose that $\\mathcal{X}_1=\\emptyset$ and $(R_1^*, R_2^*)=(0,C)$, where $C := \\max_{P_{X_2}}I(P_{X_2} , W)$ and $W:\\mathcal{X}_2\\to\\mathcal{Y}$.  Clearly $L_1$ plays no role, and Theorem \\ref{thm:local} states that the achievable values of \n$L_2$ are precisely those in  the set \n\\begin{equation}\n \\mathcal{L}_2(\\varepsilon  ):=    \\bigcup_{P_{X_2} \\in\\Pi} \\Big\\{L_2: L_2 \\le  \\sqrt{V(P_{X_2}) }\\Phi^{-1}(\\varepsilon) \\Big\\},\n\\end{equation}\nwhere $\\Pi := \\{P_{X_2}  :  I(P_{X_2}, W) = C \\}$, and $V(\\cdot) := V_{12}(\\cdot)$ is the conditional information variance \\cite{PPV10}.\nLetting $L^*  := \\sup\\mathcal{L}_2(\\varepsilon  )$ be the {\\em second-order coding \nrate} \\cite{Hayashi09} of the discrete memoryless channel (DMC) $W:\\mathcal{X}_2\\to\\mathcal{Y}$, we readily obtain\n\\begin{equation}\n    L^*  = \\left\\{  \\begin{array}{cc}\n    \\sqrt{\\min_{P_{X_2} \\in\\Pi} V(P_{X_2}) } \\, \\Phi^{-1}(\\varepsilon) &\\varepsilon < \\frac12 \\\\\n    \\sqrt{ \\max_{P_{X_2} \\in\\Pi} V(P_{X_2})  }\\,\\Phi^{-1}(\\varepsilon) &\\varepsilon \\ge \\frac12.\n    \\end{array}\n    \\right.\n\\end{equation}\nThus, our main result reduces to the classical result of Strassen~\\cite[Thm.~3.1]{Strassen}  \nfor the single-user setting\n(see also \\cite{Hayashi09,PPV10}). This illustrates the necessity of the \nset $\\Pi(R_1^*, R_2^*)$ in the characterization of $\\mathcal{L}(\\varepsilon;R_1^*,R_2^*)$ \nin Theorem~\\ref{thm:local}. Such a set is not needed in the Gaussian setting~\\cite{ScaTan13}, \nas every boundary point is achieved uniquely by a single multivariate Gaussian distribution. \nAnother notable difference in Theorem \\ref{thm:local} compared to \\cite{ScaTan13} is the \nuse of left and right tangent vectors instead of a single derivative vector.\n\nBoth of the preceding differences were also recently observed in an achievability \nresult for the standard MAC \\cite{Scarlett13b_arxiv}.  However, no converse results\nwere given in \\cite{Scarlett13b_arxiv}, and the main novelty of the present paper is in the converse proof.\n\nIt is not difficult to show that $\\mathcal{L}$ equals a half-space whenever \n$\\hat{\\mathbf{T}}_{-}=-\\hat{\\mathbf{T}}_{+}$, as was observed in \\cite{ScaTan13,Scarlett13b_arxiv}.\nA less obvious fact is that the unions over $\\beta$ in \\eqref{eqn:L0_both}\ncan be replaced by $\\beta=0$ whenever the corresponding input distribution\n$P_{X_1X_2}$ achieves all of the boundary points in a neighborhood\nof $(R_1^*,R_2^*)$. We refer the reader to \\cite{ScaTan13,Scarlett13b_arxiv} \nfor further discussions and illustrative numerical examples.\n\n\\section{Proof of Theorem \\ref{thm:local}} \\label{sec:prf_local}\n\nDue to space constraints, we do not attempt to make the proof self-contained.\nWe avoid repeating the parts in common with \\cite{Scarlett13b_arxiv,ScaTan13}, and\nwe focus on the most novel aspects.\n\n\\subsection{Achievability}\n\nThe achievability part of Theorem \\ref{thm:local} is proved using a similar\n(yet simpler) argument to that of the standard MAC given in \\cite{Scarlett13b_arxiv}, \nso we only provide a brief outline.\n\nWe use constant-composition superposition coding with coded \ntime sharing \\cite[Sec. 4.5.3]{elgamal}.  We set $\\mathcal{U} := \\{1,2\\}$, fix a joint\ndistribution $Q_{UX_1X_2}$ (to be specified shortly), and let $Q_{UX_1X_2,n}$ be \nthe closest corresponding joint type. We write the marginal distributions\nin the usual way (e.g. $Q_{X_1|U,n}$).\nWe let $\\mathbf{u}$ be a deterministic time-sharing sequence with $nQ_{U,n}(1 )$ ones and\n$nQ_{U,n}(2 )$ twos.  We first generate the $M_{2,n}$ codewords of user 2 independently according to the\nuniform distribution on  $T_{\\mathbf{u}}^n(Q_{X_1|U,n})$.  For\neach $m_2$, we generate $M_{1,n}$ codewords for user 1 conditionally independently according  \nto the uniform distribution on $T_{\\mathbf{u}\\mathbf{x}_2}^n(Q_{X_1|X_2U,n})$, where $\\mathbf{x}_2$ is  \nthe codeword for user 2 corresponding to $m_2$.  \n\nWe fix $\\beta\\ge0$ and choose $Q_{UX_1X_2}$ such that $Q_U(1) = 1 - \\frac{\\beta}{\\sqrt n}$\nand $Q_U(2) = \\frac{\\beta}{\\sqrt n}$, let $Q_{X_1X_2|U=1}$ be an input distribution $P_{X_1X_2}$ achieving\nthe boundary point of interest, and let $Q_{X_1X_2|U=2}$ be an input distribution $P'_{X_1X_2}$ achieving \na different boundary point.\nWe define the shorthands $\\mathbf{I} := \\mathbf{I}(P_{X_1X_2})$, $\\mathbf{V} := \\mathbf{V}(P_{X_1X_2})$ and $\\mathbf{I}' := \\mathbf{I}(P'_{X_1X_2})$.\nUsing the generalized Feinstein bound given in \\cite{ScaTan13} along with the multivariate \nBerry-Esseen theorem, we can use the arguments of \\cite{Scarlett13b_arxiv} to conclude that\nall rate pairs $(R_{1,n},R_{2,n})$ satisfying\n\\begin{equation}\n    \\mathbf{R}_n \\in \\mathbf{I} + \\frac{1}{\\sqrt n}\\Big(\\beta(\\mathbf{I}'-\\mathbf{I}) + \\Psi^{-1}(\\mathbf{V},\\varepsilon) \\Big) + g(n)\\mathbf{1} \\label{eq:global_ach}\n\\end{equation}\nare $(n,\\varepsilon)$-achievable for some $g(n)=O\\big(n^{1\/4}\\big)$ depending on $\\varepsilon$, \n$\\beta$, $P_{X_1X_2}$ and $P'_{X_1X_2}$.  Note that this argument may require \na reduction to a lower dimension for singular dispersion matrices; an analogous\nreduction will be given in the converse proof below.\n\nThe achievability part of Theorem \\ref{thm:local} now follows as in \\cite{Scarlett13b_arxiv}.\nIn the cases corresponding to \\eqref{eqn:L0_1}--\\eqref{eqn:L0_12}, we eliminate one of\nthe two element-wise inequalities from \\eqref{eq:global_ach} to obtain the desired result.\nFor the remaining case corresponding to \\eqref{eqn:L0_both}, we obtain the first\n(respectively, second) term in the union by letting $P_{X_1X_2}'$ achieve a boundary  \npoint approaching $(R_1^*,R_2^*)$ from the left (respectively, right).\n\n\\subsection{Converse}\n\nThe converse proof builds on that for the Gaussian case \\cite{ScaTan13}, but contains\nmore new ideas compared to the achievability part.  We thus provide a more detailed\ntreatment. \n\n\\subsubsection{A Reduction from Average Error to Maximal Error}\n\nUsing an identical argument to the Gaussian case \\cite{ScaTan13} (which itself\nbuilds on~\\cite[Cor.~16.2]{Csi97}), we can show\nthat $\\mathcal{L}(\\varepsilon;R_1^*,R_2^*)$ is identical when the average error probability\nis replaced by the maximal error probability in Def.~\\ref{def;second}.\nWe may thus proceed by considering the maximal error probability.  Note that\nneither this step nor the following step are possible for the standard MAC;\nthe assumption of degraded message sets is crucial.\n\n\\subsubsection{A Reduction to Constant-Composition Codes}\n\nUsing the previous step and the fact that the number of joint types \non $\\mathcal{X}_1 \\times \\mathcal{X}_2$ is polynomial in $n$, we can again follow an\nidentical argument to the Gaussian case \\cite{ScaTan13} to show that \n$\\mathcal{L}(\\varepsilon;R_1^*,R_2^*)$ is unchanged when the codebook\nis restricted to contain codeword pairs $(\\mathbf{x}_1,\\mathbf{x}_2)$ sharing a common joint\ntype.  We thus limit our attention to such codebooks; we denote the corresponding\nsequence of joint types by $\\{P_{X_1X_2,n}\\}_{n\\ge1}$.\n\n\\subsubsection{Passage to a Convergent Subsequence} \\label{step:passage}\n\nSince  $\\mathcal{P}(\\mathcal{X}_1 \\times \\mathcal{X}_2)$ is compact, the sequence\n$\\{P_{X_1X_2,n}\\}_{n\\ge1}$ must have a convergent subsequence, say indexed by a\nsequence $\\{n_k\\}_{k\\ge1}$ of block lengths.\nWe henceforth limit our attention to values of $n$ on this subsequence.  To avoid cumbersome\nnotation, we continue writing $n$ instead of $n_k$. \nHowever, it should be understood that asymptotic\nnotations such as $O(\\cdot)$ and $(\\cdot)_n \\to (\\cdot)$ \nare taken with respect to this subsequence.\nThe idea is that it suffices to prove the converse result \nonly for values of $n$ on an arbitrary subsequence of $(1,2,3,\\dotsc)$, since \nwe used the $\\liminf$ in \\eqref{eqn:second_def0} and the \n$\\limsup$ in \\eqref{eqn:second_def}.  \n\n\\subsubsection{A Verd\\'u-Han-Type Converse Bound}\n\nWe make use of the following non-asymptotic converse bound from \\cite{ScaTan13}:\n\\begin{align}\n    \\varepsilon_n \\ge 1 - \\Pr\\bigg(\\frac{1}{n}\\sum_{i=1}^{n}\\mathbf{j}(X_{1,i},X_{2,i},Y_i) \\ge \\mathbf{R}_n - \\gamma\\mathbf{1} \\bigg) - 2e^{-n\\gamma}, \\label{eqn:vh}\n\\end{align}\nwhere $\\gamma$ is an arbitrary constant, $(\\mathbf{X}_1,\\mathbf{X}_2)$ is the random pair induced by\nthe codebook, and $\\mathbf{Y}$ is the resulting output.  The output distributions defining $\\mathbf{j}$ \nare those induced by the fixed input joint type $P_{X_1 X_2, n}$.  By the above \nconstant-composition reduction and a simple symmetry argument, we may replace \n$(\\mathbf{X}_1,\\mathbf{X}_2)$ by a fixed pair $(\\mathbf{x}_1,\\mathbf{x}_2) \\in T^n(P_{X_1X_2,n})$.\n\n\\subsubsection{Handling Singular Dispersion Matrices}\n\nDirectly applying the multivariate Berry-Esseen theorem (e.g. see \\cite[Sec. VI]{TK12}) \nto \\eqref{eqn:vh} is problematic, since the dispersion matrix $\\mathbf{V}(P_{X_1X_2,n})$ may\nbe singular or asymptotically singular.  We therefore proceed by handling such matrices,\nand reducing the problem to a lower dimension as necessary.\n\nWe henceforth use the shorthands $\\mathbf{I}_n := \\mathbf{I}(P_{X_1X_2,n})$ and $\\mathbf{V}_n := \\mathbf{V}(P_{X_1X_2,n})$.\nAn eigenvalue decomposition yields\n\\begin{align}\n    \\mathbf{V}_n = \\mathbf{U}_n\\mathbf{D}_n\\mathbf{U}_n^T,\n\\end{align} \nwhere $\\mathbf{U}_n$ is unitary  (i.e. $\\mathbf{U}_n\\mathbf{U}_n^T$ is the identity matrix) and $\\mathbf{D}_n$ is diagonal.  \nSince we passed to a convergent subsequence\nin Step 3 and the eigenvalue decomposition map $\\mathbf{V}_n \\to (\\mathbf{U}_n,\\mathbf{D}_n)$ is continuous, \nwe conclude that both $\\mathbf{U}_n$ and $\\mathbf{D}_n$ converge, say to $\\mathbf{U}_{\\infty}$ and $\\mathbf{D}_{\\infty}$.\nWhen $\\rank(\\mathbf{D}_{\\infty})=2$ (i.e. $\\mathbf{D}_{\\infty}$ has full rank), we directly use the multivariate\nBerry-Esseen theorem.  We proceed by discussing lower rank matrices.\n\nSince $\\mathbf{V}_n$ is the covariance matrix of $\\mathbf{A}_n := \\frac{1}{\\sqrt n}\\big( \\sum_{i=1}^{n}\\mathbf{j}(x_{1,i},x_{2,i},Y_i) - n\\mathbf{I}_n \\big)$\n(with $Y_i\\sim W(\\cdot|x_{1,i},x_{2,i})$),\nwe see that $\\mathbf{D}_n$ is the covariance matrix of $\\tilde{\\mathbf{A}}_n := \\mathbf{U}_n^T\\mathbf{A}_n$.  In the case that\n$\\rank(\\mathbf{D}_{\\infty})=1$, we may write\n\\begin{align}\n    \\tilde{\\mathbf{A}}_n := \\big[ \\tilde{A}_{n,1} ~ \\tilde{A}_{n,2}]^T,\n\\end{align}\nwhere $\\var[\\tilde{A}_{n,1}]$ is bounded away from zero, and  $\\var[\\tilde{A}_{n,2}] \\to 0$. \nSince $\\mathbf{U}_n$ is unitary, we have\n\\begin{align}\n    \\mathbf{A}_n = \\mathbf{U}_n\\tilde{\\mathbf{A}}_n  \n  = \\mathbf{U}_{n,1}\\tilde{A}_{n,1} + \\bm{\\Delta}_n, \\label{eqn:An2}\n\\end{align}\nwhere $\\mathbf{U}_{n,i}$ denotes the $i$-th column of $\\mathbf{U}_n$, and $\\bm{\\Delta}_n := \\mathbf{U}_{n,2}\\tilde{A}_{n,2}$.\nSince $\\mathbf{A}_n$ has mean zero by construction, the same is true of $\\tilde{\\mathbf{A}}_n$ and hence $\\bm{\\Delta}_n$.\nMoreover, since $\\tilde{A}_{n,1}$ has vanishing variance, the same is true of each entry of $\\bm{\\Delta}_n$.\nThus, Chebyshev's inequality implies that, for any $\\delta_n>0$,\n\\begin{align}\n    \\Pr\\big(\\|\\bm{\\Delta}_n\\|_{\\infty} \\ge \\delta_n \\big) \\le \\frac{\\psi_n}{\\delta_n^2}, \\label{eqn:Cheby}\n\\end{align} \nwhere $\\psi_n := \\max_{i=1,2}\\var[\\Delta_{n,i}] \\to 0$.\n\nWe can now bound the probability in \\eqref{eqn:vh} as follows:\n\\begin{align}\n    & \\Pr\\bigg(\\frac{1}{n}\\sum_{i=1}^{n}\\mathbf{j}(X_{1,i},X_{2,i},Y_i) \\ge \\mathbf{R}_n - \\gamma\\mathbf{1} \\bigg) \\nn \\\\\n        & \\,\\, = \\Pr\\Big( \\mathbf{A}_n \\ge \\sqrt{n}\\big(\\mathbf{R}_n - \\mathbf{I}_n - \\gamma\\mathbf{1} \\big) \\Big) \\\\\n        & \\,\\, = \\Pr\\Big( \\mathbf{U}_{n,1}\\tilde{A}_{n,1} + \\bm{\\Delta}_n \\ge \\sqrt{n}\\big(\\mathbf{R}_n - \\mathbf{I}_n - \\gamma\\mathbf{1} \\big) \\Big) \\\\\n        & \\,\\, \\le \\Pr\\Big( \\mathbf{U}_{n,1}\\tilde{A}_{n,1} \\ge \\sqrt{n}\\big(\\mathbf{R}_n - \\mathbf{I}_n - \\gamma\\mathbf{1}\\big) - \\delta_n\\mathbf{1} \\Big) \\nn \\\\\n        & \\,\\,\\qquad\\qquad\\qquad\\qquad\\qquad   + \\Pr\\big(\\|\\bm{\\Delta}_n\\|_{\\infty} \\ge \\delta_n \\big) \\\\\n        & \\,\\, \\le \\Pr\\Big( \\mathbf{U}_{n,1}\\tilde{A}_{n,1} \\!\\ge\\! \\sqrt{n}\\big(\\mathbf{R}_n\\! -\\! \\mathbf{I}_n\\! -\\! \\gamma\\mathbf{1}\\big) - \\delta_n\\mathbf{1} \\Big) \\!+\\! \\frac{\\psi_n}{\\delta_n^2}, \\label{eqn:rank_one}\n\\end{align}\nwhere the last three steps respectively follow from \\eqref{eqn:An2}, \\cite[Lemma 9]{TK12}, \nand \\eqref{eqn:Cheby}.  We now choose $\\delta_n=\\psi_n^{1\/3}$, so that both $\\delta_n$ and \n$\\frac{\\psi_n}{\\delta_n^2}$ are vanishing.  Equation \\eqref{eqn:rank_one} permits an application\nof the {\\em univariate} Berry-Esseen theorem, since the variance of $\\tilde{A}_{n,1}$ is bounded \naway from zero.\n\nThe case $\\rank(\\mathbf{D}_{\\infty})=0$ is handled similarly using Chebyshev's inequality, and we thus \nomit the details and merely state that \\eqref{eqn:rank_one} is replaced by\n\\begin{align}\n    \\mathds{1}\\Big( \\sqrt{n}\\big(\\mathbf{R}_n - \\mathbf{I}_n - \\gamma\\mathbf{1}\\big) \\le \\delta_n\\mathbf{1} \\Big) + \\delta_n' \\label{eqn:rank_zero}\n\\end{align}  \nwhere $\\delta_n \\to 0$ and $\\delta_n' \\to 0$.\n\n\\subsubsection{Application of the Berry-Esseen Theorem}\n\nLet $\\mathbf{I}_{\\infty}$ and $\\mathbf{V}_{\\infty}$ denote the limiting values (on the convergent\nsubsequence of block lengths) of $\\mathbf{I}_n$ and $\\mathbf{V}_n$. In this step, we will use the fact \nthat $\\Psi^{-1}(\\cdot,\\varepsilon)$ is continuous in the following sense:\n\\begin{align}\n    \\Psi^{-1}(\\mathbf{V}_n,\\varepsilon) - \\delta\\mathbf{1} \\subset \\Psi^{-1}(\\mathbf{V}_{\\infty},\\varepsilon) \\subset \\Psi^{-1}(\\mathbf{V}_n,\\varepsilon) + \\delta\\mathbf{1} \\label{eqn:continuity} \n\\end{align}\nfor any $\\delta>0$ and sufficiently large $n$.  This is proved using a \nTaylor expansion when $\\mathbf{V}_{\\infty}$ has full rank, and is proved similarly to \n\\cite[Lemma 6]{ScaTan13} when $\\mathbf{V}_{\\infty}$ is singular.\n\nWe claim that the preceding two steps, along with the choice $\\gamma := \\frac{\\log n}{n}$,\nimply that the rate pair $(R_{1,n},R_{2,n})$ satisfies\n\\begin{equation}\n    \\mathbf{R}_n \\in \\mathbf{I}_n + \\frac{1}{\\sqrt n}\\Psi^{-1}(\\mathbf{V}_{\\infty},\\varepsilon) + g(n)\\mathbf{1} \\label{eq:global_conv}\n\\end{equation}\nfor some $g(n)=o\\big(\\frac{1}{\\sqrt n}\\big)$ depending on $P_{X_1,X_2,n}$ and $\\varepsilon$.  \nIn the case $\\rank(\\mathbf{D}_{\\infty})=2$ (see the preceding step), this follows by applying the multivariate\nBerry-Esseen theorem with a positive definite covariance matrix, re-arranging to obtain\n\\eqref{eq:global_conv} with $\\mathbf{V}_n$ in place of $\\mathbf{V}_{\\infty}$, and then using \\eqref{eqn:continuity}.\n\nIn the case $\\rank(\\mathbf{V}_{\\infty})=1$, we obtain \\eqref{eq:global_conv} by applying the univariate \nBerry-Esseen theorem to \\eqref{eqn:rank_one} and similarly applying rearrangements and \n\\eqref{eqn:continuity}.  The resulting expression can be written in the multivariate form\nin \\eqref{eq:global_conv} by a similar argument to \\cite[p. 894]{TK12}.\n\nWhen $\\rank(\\mathbf{V}_{\\infty})=0$, we have $\\mathbf{V}_{\\infty}=\\mathbf{0}$, and $\\Psi^{-1}(\\mathbf{V}_{\\infty},\\varepsilon)$ is simply the \nquadrant $\\{\\mathbf{z} \\,:\\, \\mathbf{z} \\le \\mathbf{0} \\}$.  We thus obtain \\eqref{eq:global_conv} by\nnoting that the indicator function in \\eqref{eqn:rank_zero} is zero for sufficiently\nlarge $n$ whenever either entry of $\\mathbf{R}_n$ exceeds the corresponding entry of $\\mathbf{I}_n$\nby $\\Theta\\big(\\frac{1}{\\sqrt n}\\big)$. \n\n\\subsubsection{Establishing the Convergence to $\\Pi(R_1^*,R_2^*)$}\n\nWe use a proof by contradiction to show that the limiting value $P_{X_1X_2,\\infty}$\nof $P_{X_1X_2,n}$ (on the convergent subsequence of block lengths) must lie\nwithin $\\Pi(R_1^*,R_2^*)$.  Assuming the contrary, we observe from\n\\eqref{eqn:def_Pi} that at least one of the strict inequalities\n$I_1(P_{X_1X_2,\\infty}) < R_1^*$ and $I_{12}(P_{X_1X_2,\\infty}) < R_1^* + R_2^*$ must hold.\nIt thus follows from \\eqref{eq:global_conv} and the continuity of $\\mathbf{I}(P_{X_1X_2})$ that there exists $\\delta>0$\nsuch that either $R_{1,n} \\le R_1^* - \\delta$ or $R_{1,n} + R_{2,n} \\le R_1^* + R_2^* - \\delta$\nfor sufficiently large $n$, in contradiction with the convergence of $(R_{1,n},R_{2,n})$\nto $(R_1^*,R_2^*)$ implied by \\eqref{eqn:second_def0}.\n\n\\subsubsection{Completion of the Proof for Cases (i) and (ii)}\n\nHere we handle distributions $P_{X_1X_2,\\infty}$ corresponding to the\ncases in \\eqref{eqn:L0_1}--\\eqref{eqn:L0_12}.  We focus on case (ii), \nsince case (i) is handled similarly.  \n\nIt is easily verified from \\eqref{eqn:psiinv} that each point $\\mathbf{z}$ in $\\Psi^{-1}(\\mathbf{V},\\varepsilon)$ \nsatisfies $z_1 + z_2 \\le \\sqrt{V_{12}}\\Phi^{-1}(\\varepsilon)$.  We can thus weaken \\eqref{eq:global_conv} to\n\\begin{align}\n    R_{1,n}+R_{2,n} \\le I_{12}(P_{X_1X_2,n}) + \\sqrt{\\frac{V_{\\infty,12}}{n}}\\Phi^{-1}(\\varepsilon) + g(n).\n\\end{align}\nWe will complete the proof by showing that \n$I_{12}(P_{X_1X_2,n}) \\le R_1^* + R_2^*$ for all $n$. \nSince $\\bigcup_{P_{X_1X_2}}\\big\\{I_{12}(P_{X_1X_2})\\big\\}$ is the set of all achievable (first-order) sum rates,\nit suffices to show that any boundary point corresponding to \\eqref{eqn:L0_12} is\none maximizing the sum rate.  We proceed by establishing that this is true.\n\nThe conditions stated before \\eqref{eqn:L0_12} state that $(R_1^*,R_2^*)$ lies on\nthe diagonal part of the achievable trapezium corresponding to $P_{X_1X_2}$, and away\nfrom the corner point.  It follows that $\\mathbf{p}_1 := (R_1^* - \\delta,R_2^* + \\delta)$ and\n$\\mathbf{p}_2 := (R_1^* + \\delta,R_2^* - \\delta)$ are achievable for sufficiently small $\\delta$.\nIf another point $\\mathbf{p}_0$ with a strictly higher sum rate were achievable, then all points\nwithin the triangle with corners defined by $\\mathbf{p}_0$, $\\mathbf{p}_1$ and $\\mathbf{p}_2$ would also \nbe achievable.  This would imply the achievability of $(R_1^* + \\delta',R_2^* + \\delta')$\nfor sufficiently small $\\delta'>0$, which contradicts the assumption that $(R_1^*,R_2^*)$ is a boundary point.\n\n\n\\subsubsection{Completion of the Proof for Case (iii))}\n\nWe now turn to the remaining case in \\eqref{eqn:L0_both}, corresponding to \n$\\mathbf{I}_{\\infty} = \\mathbf{R}^*$.  Again using the fact that $\\mathbf{I}(P_{X_1X_2})$ \nis continuous in $P_{X_1X_2}$, we have \n\\begin{align}\n    \\mathbf{I}_n = \\mathbf{R}^* + \\bm{\\Delta}(P_{X_1X_2,n}), \\label{eqn:Idiff}\n\\end{align}\nwhere $\\|\\bm{\\Delta}(P_{X_1X_2,n})\\|_{\\infty} \\to 0$.  We claim \nthat $\\bm{\\Delta}(P_{X_1X_2,n}) \\in \\mathcal{V}(R_1^*,R_2^*)$ \n(see \\eqref{eqn:setV}). Indeed, if this were not the case, then \n\\eqref{eqn:Idiff} would imply that the pair $(I_{n,1},I_{n,12}-I_{n,1})$ \nlies outside the capacity region, in contradiction with the fact\nthat rates satisfying \\eqref{eqn:cap_region1}--\\eqref{eqn:cap_region2}\nare (first-order) achievable for {\\em all} $P_{X_1X_2}$.\n\nAssuming for the time being that $\\|\\bm{\\Delta}(P_{X_1X_2,n})\\|_{\\infty} = O\\big(\\frac{1}{\\sqrt n}\\big)$,\nwe immediately obtain the outer bound\n\\begin{align}\n    & \\mathcal{L}(\\varepsilon,R_{1}^{*},R_{2}^{*})\\subseteq\\bigg\\{(L_{1},L_{2})\\,:\\, \\nn \\\\\n    & \\mathbf{L}\\in\\bigcup_{P_{X_{1}X_{2}}\\in\\Pi,\\mathbf{T}\\in\\mathcal{V}}\\Big\\{\\Psi^{-1}(\\mathbf{V}(P_{X_{1}X_{2}}),\\varepsilon)+\\mathbf{T}\\Big\\}\\bigg\\}. \\label{eqn:alt_form}\n\\end{align}\nThe set in \\eqref{eqn:alt_form} clearly includes $\\mathcal{L}_0$\nin \\eqref{eqn:L0_both}.  We proceed by showing that the reverse inclusion\nholds, and hence the two sets are identical.  Since $\\hat{\\mathbf{T}}_{-}$\nand $\\hat{\\mathbf{T}}_{+}$ are tangent vectors, any vector $\\mathbf{T}\\in\\mathcal{V}$\ncan have one or more of its components increased to yield a vector $\\mathbf{T}'$\nwhose direction coincides with either $\\mathbf{T}_{-}$ or $\\mathbf{T}_{+}$.  The fact\nthat the magnitude of $\\mathbf{T}'$ may be arbitrary is captured by the unions\nover $\\beta\\ge0$ in \\eqref{eqn:L0_both}.\n\nIt remains to handle the case that \n$\\|\\bm{\\Delta}(P_{X_1X_2,n})\\|_{\\infty}$ is not $O\\big(\\frac{1}{\\sqrt n}\\big)$. \nBy performing another pass to a subsequence of block lengths if necessary, we can assume that\n$\\|\\bm{\\Delta}(P_{X_1X_2,n})\\|_{\\infty} = \\omega\\big(\\frac{1}{\\sqrt n}\\big)$.\nSuch scalings can be shown to play no role in characterizing $\\mathcal{L}$, similarly to\n\\cite{ScaTan13}; we provide only an outline here.  \nLet $\\Delta_{n,1}$ and $\\Delta_{n,12}$ denote the entries of $\\bm{\\Delta}(P_{X_1X_2,n})$,\nand let $\\Delta_{n,2} := \\Delta_{n,12} - \\Delta_{n,1}$.  If either $\\Delta_{n,1}$\nor $\\Delta_{n,2}$ is negative and decays with a rate $\\omega\\big(\\frac{1}{\\sqrt n}\\big)$, then\nno value of the corresponding $L_j$ ($j\\in\\{1,2\\}$) can satisfy \nthe condition in \\eqref{eqn:second_def0}, so the converse is trivial.  \nOn the other hand, if either $\\Delta_{n,1}$ or $\\Delta_{n,2}$ is \npositive and $\\omega\\big(\\frac{1}{\\sqrt n}\\big)$, we simply recover the right-hand side\nof \\eqref{eqn:alt_form} in the limiting case that either $T_1$ or\n$T_{12} - T_1$ (where $\\mathbf{T} := [T_1~~T_{12}]^T$) grows unbounded.  Thus,\nthe required converse statement for this case is already captured by \\eqref{eqn:alt_form}. \n\n \n\\subsubsection*{Acknowledgments}\nThe second author  is supported by NUS startup grants WBS R-263-000-A98-750\/133.\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Motivation}\n\\label{sec:introduction}\n\nThe Standard Model (SM) of particle physics has been the most successful theory of elementary particles that provides excellent explanations of many physical phenomena occurring in Nature~\\cite{Workman:2022ynf}. However, there are a plethora of motivations for physicists to go beyond the SM, for example, to explain the observed matter-antimatter asymmetry in the Universe, the existence of dark matter and dark energy, and the non-zero neutrino mass. The observed mass-induced flavor transition~\\cite{Super-Kamiokande:2004orf, Mohapatra:2005wg, Strumia:2006db, Gonzalez-Garcia:2007dlo, Fantini:2018itu} that requires the neutrinos to be massive provides the first experimental signature of physics beyond the SM. The standard three-flavor neutrino oscillation framework involves three mixing angles ($\\theta_{12}$, $\\theta_{13}$, and $\\theta_{23}$), one Dirac CP phase ($\\delta_{\\rm CP}$), and two independent mass-squared differences, $\\Delta{m}^2_{21}~ (\\equiv{m}^2_2-{m}^2_1~\\text{in the solar sector})$ and $\\Delta{m}^2_{31}~(\\equiv{m}^3_3-{m}^2_1~\\text{in the atmosperic sector})$. Now that the phenomenon of neutrino oscillation has been well established, the focus has been shifted to measure the oscillation parameters with utmost precision. Marvelous data from various neutrino oscillation experiments such as Super-K-Solar~\\cite{Super-Kamiokande:2016yck}, SNO~\\cite{SNO:2011hxd}, BOREXINO~\\cite{BOREXINO:2014pcl}, Super-K-Atmospheric~\\cite{Super-Kamiokande:2004orf, Super-Kamiokande:2010tar, Super-Kamiokande:2017yvm, Super-Kamiokande:2019gzr}, IceCube-DeepCore~\\cite{IceCube:2017lak}, ANTARES~\\cite{ANTARES:2018rtf}, Daya Bay~\\cite{DayaBay:2018yms}, RENO~\\cite{RENO:2018dro}, Double Chooz~\\cite{DoubleChooz:2019qbj}, MINOS~\\cite{MINOS:2013utc}, Tokai-to-Kamioka (T2K)~\\cite{T2K:2019bcf}, and NuMI Off-axis $\\nu_{e}$ Appearance (NO$\\nu$A)~\\cite{NOvA:2019cyt, NOvA:2021nfi} have already provided a first order picture of the lepton mixing pattern in three-flavor scenario.\n\nThere are three major issues in the three-flavor neutrino oscillation paradigm that are yet to be resolved, namely, the value of the CP phase ($\\delta_{\\rm CP}$), the octant of the atmospheric mixing angle ($\\theta_{23}$), and the neutrino mass ordering. The Deep Underground Neutrino Experiment (DUNE)~\\cite{DUNE:2015lol, DUNE:2020lwj, DUNE:2020ypp, DUNE:2020jqi, DUNE:2021cuw, DUNE:2021mtg} with its wide band neutrino beam, will play a crucial role in establishing the deviation of the atmospheric mixing angle ($\\theta_{23}$) from its maximal value and settling down its correct octant with outstanding precision~\\cite{Agarwalla:2021bzs}. DUNE can measure the value of atmospheric mass splitting at several $L\/E$ values and settle the issue of neutrino mass ordering at high confidence level exploiting the Earth's matter effect that it possess due to its long baseline~\\cite{DUNE:2020ypp}. DUNE is also capable to establish leptonic CP violation by measuring the value of $\\delta_{\\rm CP}$ precisely~\\cite{Agarwalla:2022xdo}. Another proposed long-baseline experiment which spans from Tokai to Hyper-Kamiokande (T2HK)~\\cite{Hyper-KamiokandeWorkingGroup:2014czz, Hyper-KamiokandeProto-:2015xww, Hyper-Kamiokande:2018ofw} will also shed light on these pressing issues. In the T2HK setup, one detector is placed in Japan, which is 295 km away from the J-PARC facility and will receive a highly intense narrow-band neutrino beam from the J-PARC source at an off-axis angle of $2.5^\\circ$. T2HK will have a four times smaller baseline than DUNE, which in turn will face negligible Earth's matter effect and hence can measure the value of $\\delta_{\\rm CP}$ and establish leptonic CP violation without having the interference of the fake CP-asymmetry induced by Earth's matter~\\cite{Agarwalla:2022xdo}.\n\nApart from measuring the standard oscillation parameters with high precision, the long-baseline experiments will also search for new physics beyond the Standard Model (BSM)\\footnote{For an extensive review on this topic, see Refs.~\\cite{Arguelles:2019xgp, Arguelles:2022tki}.} through neutrino oscillation, namely, eV-scale sterile neutrino~\\cite{Berryman:2015nua, Agarwalla:2016mrc, Agarwalla:2016xxa, Agarwalla:2016xlg, Agarwalla:2018nlx, KumarAgarwalla:2019blx}, neutrino non-standard interactions~\\cite{Coloma:2015kiu, Agarwalla:2016fkh}, non-unitary neutrino mixing~\\cite{Escrihuela:2016ube, Agarwalla:2021owd}, long-range interactions~\\cite{Chatterjee:2015gta}, neutrino decay~\\cite{Choubey:2017dyu, Coloma:2017zpg}, and Lorentz Invariance Violation (LIV)~\\cite{Barenboim:2018ctx, KumarAgarwalla:2019gdj, Fiza:2022xfw}. In this work, we mainly focus on LIV. It is a well-established fact that the Lorentz symmetry is an exact symmetry of Nature. As a consequence, the Standard Model of particle physics conserves Lorentz symmetry. However, there exist several models, unifying SM and general relativity, that violate the Lorentz and CPT symmetry at the Planck scale ($\\sim10^{19}$ GeV). This can be realized at a low energy scale accessible to the current experiments under Standard Model Extension (SME) framework. Various neutrino experiments are at the forefront to test the Lorentz Invariance Violation at a low energy scale. For example, in an attempt to understand the excess of $\\nu_e$ signal events in the $\\nu_{\\mu}$ beam, LSND collaboration~\\cite{LSND:2005oop}, searched for the possible signature of LIV in the context of neutrino oscillation. They did not find any signature of LIV and put strong constraints on relevant LIV parameters.\n\nSeveral other experiments have made efforts to search for LIV, which include MINOS~\\cite{MINOS:2008fnv, MINOS:2010kat, MINOS:2012ozn}, MiniBooNE~\\cite{MiniBooNE:2011pix}, Double Chooz~\\cite{DoubleChooz:2012eiq}, Super-K~\\cite{Super-Kamiokande:2014exs}, IceCube~\\cite{IceCube:2010fyu} and T2K~\\cite{Abe:2017eot}. None of these experiments found any positive signal of LIV and set competitive bounds on various LIV parameters. In addition to the aforementioned studies by the experimental collaborations, there are various independent works towards the exploration of LIV with accelerator neutrinos in long-baseline experiments~\\cite{Dighe:2008bu, Barenboim:2009ts, Rebel:2013vc, Diaz:2015dxa, deGouvea:2017yvn, Barenboim:2017ewj, Barenboim:2018lpo, Barenboim:2018ctx, Majhi:2019tfi, Fiza:2022xfw, Majhi:2022fed}, reactor antineutrinos in short-baseline experiments~\\cite{Giunti:2010zs}, atmospheric neutrinos~\\cite{Datta:2003dg, Chatterjee:2014oda, SinghKoranga:2014mxh, Sahoo:2021dit, Sahoo:2022rns}, solar neutrinos~\\cite{Diaz:2016fqd}, and high-energy neutrinos from astrophysical sources~\\cite{Hooper:2005jp, Tomar:2015fha, Liao:2017yuy}. Recently, the KATRIN experiment, using the data from the first scientific run, placed limits on some of the oscillation-free LIV parameters that can not be probed by the time-of-flight or neutrino oscillation experiments~\\cite{KATRIN:2022qou}. An exhaustive list of the constraints on all the CPT-violating and CPT-conserving LIV parameters can be found in Ref.~\\cite{Kostelecky:2008ts}.\n\nIn the present work, we mainly focus on the capability of the upcoming long-baseline experiments, DUNE and T2HK in isolation and combination, to constrain the LIV parameters. We derive the sensitivities of these experiments to place competitive limits on the off-diagonal CPT-violating LIV parameters ($a_{\\alpha\\beta}$ where $\\alpha, \\beta = e, \\mu, \\tau$ and $\\alpha\\neq\\beta$) and for the first time, the off-diagonal CPT-conserving LIV parameters ($c_{\\alpha\\beta}$ where $\\alpha, \\beta = e, \\mu, \\tau$ and $\\alpha\\neq\\beta$). We study the impact of these LIV parameters and their associated phases at the probability level. Then we shift our attention to explore the possible degeneracies between the standard oscillation parameters ($\\theta_{23}$ and $\\delta_{\\rm CP}$) and the above-mentioned LIV parameters. Finally, we derive the expected constraints on these LIV parameters using the upcoming long-baseline experiments, DUNE and T2HK in standalone mode and also in combination, considering their state-of-the-art simulation details. To understand various interesting features of our numerical results, we derive simple and compact analytical expressions of the oscillation probabilities for both appearance and disappearance channels.\n\nThis paper is organized as follows. In section~\\ref{sec:LIV}, we discuss the formalism of neutrino oscillation in the presence of Lorentz Invariance Violation and the effects of various LIV parameters on the appearance and disappearance probabilities. In section~\\ref{sec:LBL}, we give the details of the long-baseline experimental setups considered for our work and discuss the expected synergies between DUNE and T2HK in various aspects. Also, this section discusses the effect of LIV parameters at the event level. We dedicate section~\\ref{sec:RnA} to describe the numerical technique used for our analyses. We present our results in section~\\ref{sec:results}, where we show the correlations among different LIV parameters, the atmospheric mixing angle ($\\theta_{23}$), and the Dirac CP-phase ($\\delta_{\\rm CP}$) and finally the expected bounds on the CPT-conserving and CPT-violating LIV parameters. We summarize our results and give our concluding remarks in section~\\ref{sec:SnC}. In appendix~\\ref{appndx}, we compare the numerical (exact) and analytical (approximate) probabilities.\n\\section{Neutrino Oscillation in Presence of Lorentz Invariance Violation}\n\\label{sec:LIV}\n\\subsection{Theoretical Formalism of LIV}\nThe Lorentz invariance has been considered to be an inalienable part in both the Standard Model (SM) of particle physics and General Relativity (GR) for the global as well as the local variables. However, a few proposed models in String Theory~\\cite{Polyakov:1987ez, Kostelecky:1988zi, Kostelecky:1989jp, Kostelecky:1990pe, Kostelecky:1991ak, Kostelecky:1995qk, Kostelecky:1999mu, Kostelecky:2000hz} and Loop Quantum Gravity~\\cite{Gambini:1998it, Alfaro:2002xz, Sudarsky:2002ue, Amelino-Camelia:2002aqz, Ng:2003jk} allow the Lorentz invariance violation (LIV) while attempting a unification of gravity with the SM gauge fields at the Planck scale ($M_P \\sim 10^{19}$ GeV). Here, we consider the mechanism proposed in the string theory, which spontaneously breaks the CPT and Lorentz symmetry at a higher dimension $(>4)$ of space-time. The plausible extension of such a violation in Lorentz and CPT symmetries in the realistic four-dimensional space-time, can be attempted by introducing new interaction coefficients to the minimal SM of particle physics as a tiny perturbation. In an observer-scalar effective field theory~\\cite{Weinberg:1979sa}, the strength of such interaction is expected to be suppressed by order of ($1\/M_P$)~\\cite{Colladay:1998fq, Kostelecky:2003fs, Colladay:1996iz, Kostelecky:2000mm, Kostelecky:2003cr, Bluhm:2005uj} manifesting the effect of Planck scale physics at low energy. The impacts of such LIV interactions can be experienced by the fundamental particles in a broad category of experiments via coherent, interference, or extreme effects.\n\nBy virtue of mass-induced neutrino flavor oscillations, the neutrinos are sensitive to the LIV effects while propagating through space-time. Under the minimal SM extension (SME), the Lagrangian density of the induced renormalizable and gauge-invariant LIV interaction terms for the left-handed neutrinos can be expressed as ~\\cite{Kostelecky:2011gq, KumarAgarwalla:2019gdj, Antonelli:2020nhn, Sahoo:2021dit, Sahoo:2022rns} :\n\\begin{align}\n\t\\mathcal{L}_{\\rm LIV} & = -\\frac{1}{2}\\left[a^{\\mu}_{\\alpha\\beta}\\,\\overline{\\psi}_\\alpha\\,\\gamma_{\\mu}\\,P_L\\,\\psi_\\beta - i c^{\\mu\\nu}_{\\alpha\\beta}\\,\\overline{\\psi}_\\alpha\\,\\gamma_{\\mu}\\,\\partial_\\nu P_L\\,\\psi_\\beta \\right] + h.c.\\,, \n\t\\label{Eq:LIV-1}\n\\end{align}\nwhere $P_L$ is the projection operator, $a^{\\mu}$ and $c^{\\mu\\nu}$ are the CPT-violating and CPT-conserving parameters, respectively. Here, $(\\mu,\\,\\nu)$ are space-time indices, and $\\alpha,\\,\\beta$ are the neutrino-flavor indices. The CPT-violating coefficient changes its sign under CPT transformation while the CPT-conserving one does not (see Refs.~\\cite{Sahoo:2021dit, Kostelecky:2003cr}). Now, considering a realistic scenario where the neutrinos can propagate through the Earth matter, the effective Hamiltonian of an ultra-relativistic left-handed neutrino, in a three neutrino mixing scenario, can be expressed in the flavor basis as ~\\cite{Kostelecky:2011gq, Sahoo:2021dit, Sahoo:2022rns}:\n\\begin{align}\n\t\\mathcal{H}_{\\rm eff} & = \\frac{1}{2E}\\,U\\,\\Delta m^2\\,U^\\dagger\n\t+ \\frac{1}{E}\\big(a^\\mu_{L} p_\\mu\n\t- c^{\\mu\\nu}_{L} p_\\mu p_\\nu \\big) + \\sqrt{2}G_FN_e\\tilde{I}.\n\t\\label{Eq:LIV-2}\n\\end{align} \nThe first term in the above equation, $U$ represents the three-neutrino unitary mixing matrix, also known as PMNS matrix, and the $\\Delta m^2$ part contains two independent mass-squared splittings in the form of a diagonal-matrix: ${\\rm diag}(0,\\,\\Delta{m}^2_{21},\\,\\Delta{m}^2_{31})$. The second term shows the strength of induced potential with left-handed neutrino due to LIV, and $p$ is the four-momenta. The third term defines the effective matter-potential induced due to the elastic charged-current scattering between $\\nu_e$ and electron. Here, $G_F$ is an electro-weak coupling constant, also known as the Fermi constant, $N_e$ is the number density of the ambient electrons present in matter, and $\\tilde{I}$ is a diagonal matrix with components $(1,\\,0,\\,0)$. The scalar part of the last term can be parameterized with matter density as $\\sqrt{2}G_FN_e$ $\\approx$ $7.6\\,\\times 10^{-23}\\cdot Y_e \\cdot \\rho\\left(\\rm g\/cm^3\\right)$ GeV, where $Y_e$ is the relative electron number density in the ambient matter having an average Earth-matter density $\\rho$.\\\\\nIn this study, we only focus on the time-like component ($\\mu,\\,\\nu\\,=\\,0$) of LIV coefficients in an isotropic space-time coordinate. From here onwards, we will consider $(a^0_L)_{\\alpha\\beta} \\equiv a_{\\alpha\\beta}$ and $(c^{00}_L)_{\\alpha\\beta} \\equiv c_{\\alpha\\beta}$.  Using the Sun-centered celestial-equatorial coordinate (see Ref.~\\cite{Kostelecky:2008ts}) as an approximated inertial frame of reference, the eq. (\\ref{Eq:LIV-2}) can be re-written as follows :\n\n\\begin{align}\n\t\\mathcal{H}_{\\rm eff} = \\; \\frac{1}{2E} \n\tU\\left(\\begin{array}{ccc}\n\t\t0 & 0 & 0                 \\\\\n\t\t0 & \\Delta m^{2}_{21} & 0 \\\\\n\t\t0 & 0 & \\Delta m^{2}_{31} \\\\\n\t\\end{array}\\right)U^{\\dagger}\n\t+&\\left(\\begin{array}{ccc}\n\t\ta_{ee} & a_{e\\mu} & a_{e\\tau} \\\\\n\t\ta^*_{e\\mu} & a_{\\mu\\mu} & a_{\\mu\\tau} \\\\\n\t\ta^*_{e\\tau} & a^*_{\\mu\\tau} & a_{\\tau\\tau}\n\t\\end{array} \\right) \\nonumber \\\\\n\t-&\\frac{4}{3} E\n\t\\left(\n\t\\begin{array}{ccc}\n\t\tc_{ee} & c_{e\\mu} & c_{e\\tau} \\\\\n\t\tc^*_{e\\mu} & c_{\\mu\\mu} & c_{\\mu\\tau} \\\\\n\t\tc^*_{e\\tau} & c^*_{\\mu\\tau} & c_{\\tau\\tau}\n\t\\end{array}\n\t\\right) \n\t+ \\sqrt{2}G_{F}N_{e}\n\t\\left(\\begin{array}{ccc}\n\t\t1 & 0 & 0 \\\\ \n\t\t0 & 0 & 0 \\\\ \n\t\t0 & 0 & 0\n\t\\end{array}\\right).\n\t\\label{Eq:LIV-3}\n\\end{align}\nFor the case of right-handed antineutrino $U \\to U^*$, $a_{\\alpha\\beta} \\to -a_{\\alpha\\beta}^*$, $c_{\\alpha\\beta} \\to c_{\\alpha\\beta}^*$ and $\\sqrt{2}G_FN_e \\to -\\sqrt{2}G_FN_e$. Note that an extra fraction $4\/3$ appears due to the choice of isotropic coordinate. It is essential to note that at the origin of the LIV coefficients $a_{\\alpha\\beta}$ and $c_{\\alpha\\beta}$ are real-valued quantities. However, due to the hermiticity, the off-diagonal elements of these LIV interaction matrices can have imaginary components while implanting them in an effective Hamiltonian.\n\nTo have an estimate of the strength of LIV parameters which may affect the outcome of the long-baseline experiments under consideration, we compare the first three terms in the neutrino propagation Hamiltonian as shown in Eq.~\\ref{Eq:LIV-3}. The first term governs the neutrino oscillation in vacuum, whereas the second and third terms are the contribution from CPT-violating and CPT-conserving LIV, respectively. For typical long-baseline experiments with neutrino energy in the GeV range, the relevant scale for standard atmospheric neutrino oscillation is $\\Delta m^2_{31}\/{2E}\\sim 10^{-22}$ GeV. So, in order to have noticeable effects from LIV, the strength of the parameters in the second and third terms should be around the same order as the standard neutrino oscillation scale. So, we observe that both CPT-violating ($a_{\\alpha\\beta}$) and CPT-conserving ($E\\times c_{\\alpha\\beta}$) parameters should have the strength of the order $10^{-22}$ GeV to have visible effects over standard neutrino oscillation in the matter.\n\nFollowing the above-discussed formalism, various neutrino experiments have given bounds on the CPT-violating and CPT-conserving LIV parameters. In particular, a recent publication by the IceCube collaboration~\\cite{IceCube:2017qyp}, where the analysis has been performed in an effective two-flavor oscillation scenario, presented the most stringent bounds on the CPT-violating and the CPT-conserving LIV parameters in $\\mu-\\tau$ sector. Apart from this, there are also limits on both CPT-violating and CPT-conserving LIV parameters using the atmospheric neutrino data from Super-K~\\cite{Super-Kamiokande:2014exs}. In Table~\\ref{tab:existing_bounds}, we tabulate the existing limits on various off-diagonal LIV parameters from these two experiments.\nNote that in our work, we represent the off-diagonal LIV coefficients as $|R|\\cdot e^{i\\phi}$, where $|R|$ represents the magnitude of the LIV coefficient, and $\\phi$ is the phase of that corresponding quantity. \n\n\\begin{table}[h!]\n\t\\centering\n\t\\begin{center}\n\t\t\\begin{adjustbox}{width=1\\textwidth}\n\t\t\t\\begin{tabular}{|c| c| c| c|}\n\t\t\t\t\\hline \\hline\n\t\t\t\t\\multicolumn{4}{|c|}{Existing constraints on CPT-violating LIV parameters} \\\\ \\hline\n\t\t\t\tExperiments & $a_{e\\mu} ~[ \\rm GeV ]$ &  $a_{e\\tau} ~[ \\rm GeV ]$ & $a_{\\mu\\tau} ~[ \\rm GeV ]$ \\\\ \\hline\n\t\t\t\t\\multirow{2}{*}{Super-K (95\\% C.L.)} & $\\mathrm{Re}(a_{e\\mu}) < 1.8\\times10^{-23}$  & $\\mathrm{Re}(a_{e\\tau}) < 4.1\\times10^{-23}$ & $\\mathrm{Re}(a_{\\mu\\tau}) < 0.65\\times10^{-23}$ \\\\\n\t\t\t\t\n\t\t\t\t& $\\mathrm{Im}(a_{e\\mu}) < 1.8\\times10^{-23}$  & $\\mathrm{Im}(a_{e\\tau}) < 2.8\\times10^{-23}$ & $\\mathrm{Im}(a_{\\mu\\tau}) < 0.51\\times10^{-23}$ \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multirow{2}{*}{IceCube (99\\% C.L.)} &\\multirow{2}{*}{--} & \\multirow{2}{*}{--} & $|\\mathrm{Re}(a_{\\mu\\tau})| < 0.29\\times10^{-23}$ \\\\ \n\t\t\t\t&\\multirow{2}{*}{--} & & $|\\mathrm{Im}(a_{\\mu\\tau})| < 0.29\\times10^{-23}$ \\\\ \n\t\t\t\t\\hline \n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{4}{|c|}{Existing constraints on CPT-conserving LIV parameters} \\\\ \\hline \n\t\t\t\t\n\t\t\t\tExperiments & $c_{e\\mu}$ &  $c_{e\\tau}$ & $c_{\\mu\\tau}$ \\\\ \\hline\n\t\t\t\t\n\t\t\t\t\\multirow{2}{*}{Super-K (95\\% C.L.)} & $\\mathrm{Re}(c_{e\\mu}) < 8.0\\times10^{-27}$  & $\\mathrm{Re}(c_{e\\tau}) < 9.3\\times10^{-25}$ & $\\mathrm{Re}(c_{\\mu\\tau}) < 4.4\\times10^{-27}$ \\\\\n\t\t\t\t\n\t\t\t\t& $\\mathrm{Im}(c_{e\\mu}) < 8.0\\times10^{-27}$  & $\\mathrm{Im}(c_{e\\tau}) < 1.0\\times10^{-24}$ & $\\mathrm{Im}(c_{\\mu\\tau}) < 4.2\\times10^{-27}$ \\\\\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\t\\multirow{2}{*}{IceCube (99\\% C.L.)} &\\multirow{2}{*}{--} & \\multirow{2}{*}{--} & $|\\mathrm{Re}(c_{\\mu\\tau})| < 0.39\\times10^{-27}$ \\\\ \n\t\t\t\t&\\multirow{2}{*}{--} & & $|\\mathrm{Im}(c_{\\mu\\tau})| < 0.39\\times10^{-27}$\n\t\t\t\t\\\\\\hline\\hline\n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\\end{center}\n\t\\mycaption{Existing constraints on the off-diagonal CPT-violating and CPT-conserving LIV parameters from Super-K~\\cite{Super-Kamiokande:2014exs} and IceCube~\\cite{IceCube:2017qyp}.}\n\t\\label{tab:existing_bounds}\n\\end{table}\n\\FloatBarrier\n\\subsection{Analytical Expressions of the Oscillation Probabilities with LIV}\n\nThe presence of LIV terms in the neutrino Hamiltonian would affect the neutrino propagation through a medium, consequently modifying the neutrino flavor transition probability. So, it is possible to probe LIV from various neutrino oscillation experiments. In order to have an analytical understanding of the neutrino flavor transition probabilities in the presence of LIV, we follow the approach given in Refs.~\\cite{Kikuchi:2008vq, Agarwalla:2016fkh, KumarAgarwalla:2019gdj}, where authors use perturbation theory to calculate the neutrino evolution matrix in various BSM scenarios like the presence of neutral-current NSI and CPT violating LIV parameters. We use $\\alpha~(\\equiv \\Delta m^2_{21}\/\\Delta m^2_{31})$, $\\sin^2\\theta_{13}$, and LIV parameters $a'_{\\alpha\\beta}~(\\equiv a_{\\alpha\\beta}\/\\sqrt{2}G_F N_e)$ and $c'_{\\alpha\\beta}~(\\equiv c_{\\alpha\\beta}E\/\\sqrt{2}G_F N_e)$ ($\\alpha,\\beta = e,\\mu,\\tau$) as the expansion parameters. In this work, we mainly probe the three off-diagonal CPT-violating and CPT-conserving LIV parameters $a_{\\alpha\\beta}$ and $c_{\\alpha\\beta}$ ($\\alpha,\\beta=e,\\mu,\\tau;\\,\\alpha\\neq\\beta$), respectively.\n\\newline\n\\newline\n\\noindent\n$\\bullet$  \\textbf{$\\nu_\\mu\\to\\nu_e$ Appearance Channel:}\\\\\n\\\\\nThe expression for the $\\nu_\\mu\\to\\nu_e$ transition probability, considering terms up to first-order in the above mentioned expansion parameters, can be written as~\\cite{KumarAgarwalla:2019gdj},\n\\begin{align}\\label{eq:pme_liv}\n\t\\centering\n\tP_{\\mu e} \\simeq P_{\\mu e}(\\text{SI}) + P_{\\mu e}(a_{e\\beta}\/c_{e\\beta}) + \\mathcal{O}(\\alpha^2, \\alpha \\sin^2\\theta_{13},a'^2_{e\\beta},c'^2_{e\\beta},a'^2_{\\mu\\tau},c'^2_{\\mu\\tau}),\\,\\,\\,\\,\\,\\,\\,\\beta=\\mu,\\tau.\n\\end{align}\nThe first term in right-hand-side (RHS) is the standard $\\nu_\\mu\\to\\nu_e$ appearance probability in the absence of any new physics parameters,\n\\begin{align}\\label{eq:p_si}\n\tP_{\\mu e}(\\text{SI}) \n\t&\\simeq \\mathbb{X} + \\mathbb{Y}\\cos(\\delta_{\\text{CP}} + \\Delta),\n\\end{align}\nwhere,\n\\begin{align}\\label{eq:si_coeff}\n\t&\\mathbb{X} = \\sin^22\\theta_{13}\\sin^2\\theta_{23} \\frac{\\sin^{2}\\big[(1-\\hat{A})\\Delta \\big]}{(1-\\hat{A})^{2}};\\nonumber \\\\\n\t&\\mathbb{Y} = \\alpha \\sin2\\theta_{12} \\sin2\\theta_{13} \\sin2\\theta_{23} \\frac{\\sin \\hat{A}\\Delta}{\\hat{A}}  \\frac{\\sin \\big[(1-\\hat{A})\\Delta\\big]}{1-\\hat{A}}; \\nonumber \\\\\n\t&\\hat{A} = \\frac{2\\sqrt{2}G_{F}N_{e}E}{\\ensuremath{\\Delta m_{31}^2}}, \\qquad  \\Delta = \\frac{\\ensuremath{\\Delta m_{31}^2} L}{4E}.\n\\end{align}\nThe second term in RHS of Eq.~\\ref{eq:pme_liv} are the contributions from the LIV parameters $a_{e\\beta}\/c_{e\\beta}$ ($\\beta = \\mu,\\tau$). For CPT-violating LIV cases, the expression of this term is;\n\\begin{align}\\label{eq:p_cptv_liv}\n\t&P_{\\mu e}(a_{e\\beta}) \n\t\\simeq \n\t\t2|a_{e\\beta}|L\\sin\\ensuremath{\\theta_{13}}\\sin2\\ensuremath{\\theta_{23}} \\sin\\Delta\\big[\\mathbb{Z}_{e\\beta}\\sin(\\delta_{\\rm CP} + \\varphi_{e\\beta}) +\n\t\t\\mathbb{W}_{e\\beta}\\cos(\\delta_{\\rm CP} + \\varphi_{e\\beta})\\big],\\nonumber\\\\\n\\end{align}\nand for CPT-conserving case,\n\\begin{align}\\label{eq:p_cptc_liv}\n\t&P_{\\mu e}(c_{e\\beta}) \n\t\\simeq\\frac{-8}{3}|c_{e\\beta}|EL\\sin\\ensuremath{\\theta_{13}}\\sin2\\theta_{23}\\sin\\Delta \\big[\\mathbb{Z}_{e\\beta}\\sin(\\delta_{\\rm CP} + \\varphi_{e\\beta}) +\n\t\t\\mathbb{W}_{e\\beta}\\cos(\\delta_{\\rm CP} + \\varphi_{e\\beta})\\big],\\nonumber\\\\\n\\end{align}\nwhere,\n\\begin{align}\\label{eq:liv_coeff}\n\t&\\mathbb{Z}_{e\\beta} = \n\t\\begin{cases}\n\t\t- c_{23}  \\sin \\Delta, & \\text{if}\\ \\beta=\\mu. \\\\\n\t\ts_{23}  \\sin \\Delta, & \\text{if}\\ \\beta=\\tau.\n\t\\end{cases} \\nonumber \\\\\n\t&\\mathbb{W}_{e\\beta} = \n\t\\begin{cases}\n\t\tc_{23}  \\big(\\frac{s_{23}^{2}\\sin \\Delta}{c_{23}^{2}\\Delta} + \\cos \\Delta \\big), & \\text{if}\\ \\beta=\\mu. \\\\\n\t\ts_{23}  \\big(\\frac{\\sin \\Delta}{\\Delta} - \\cos \\Delta \\big), & \\text{if}\\ \\beta=\\tau.\n\t\\end{cases} \n\\end{align}\nNote that the other off-diagonal LIV parameter $\\ensuremath{a_{\\mu\\tau}}\/\\ensuremath{c_{\\mu\\tau}}$ does not appear in the first-order terms. However, it may be present in higher-order terms and has a relatively smaller impact on the appearance probabilities. Note that for the appearance probability in the antineutrino case, one needs to apply ($a_{\\alpha\\beta}\\rightarrow-a^{*}_{\\alpha\\beta}$) and $c_{\\alpha\\beta} \\to c_{\\alpha\\beta}^*$ and $\\hat{A}\\to -\\hat{A}$ in Eqs.~[\\ref{eq:p_si} -\\ref{eq:liv_coeff}]. In Appendix~\\ref{appndx}, we show the validity of the approximate analytical expression of the appearance probability derived in this section by comparing it with the exact probability calculated numerically.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{.\/plots\/prob_CPTV_app_f1.pdf}\n\t\\vspace*{-10mm}\n\t\\mycaption{$\\nu_\\mu\\rightarrow\\nu_e$ appearance probability as a function of energy in the presence of off-diagonal CPT-violating LIV parameters $a_{e\\mu}$ (left column), $a_{e\\tau}$ (middle column), and $a_{\\mu\\tau}$ (right column). The top and bottom rows correspond to the baselines of DUNE ($L=1285$ km) and T2HK ($L=295$ km), respectively. The black line in each panel shows the SI case and four colored lines correspond to four benchmark values of the phases associated with the LIV parameters: $0^\\circ$, $90^\\circ$, $180^\\circ$, and $270^\\circ$ with LIV strength $|a_{\\alpha\\beta}|=2.0\\times10^{-23}$ GeV ($\\alpha,\\beta=e,\\mu,\\tau;\\alpha\\neq\\beta$). The vertical grey-dashed lines in each panel show the energies at the first and second oscillation maxima. The values of the standard oscillation parameters used in this plot are mentioned in Table~\\ref{tab:params_value}.}\n\t\\label{fig:app_prob}\n\\end{figure}\n\n\\begin{table}[htb!]\n\t\\centering\n\t\\begin{tabular}{|c|c|c|c|c|c|}\n\t\t\\hline\n\t\t$\\theta_{12}$ & $\\theta_{13}$ & $\\theta_{23}$ & $\\delta_{\\text{CP}}$ & $\\Delta{m^2_{21}}~[\\rm{eV}^2]$ & $\\Delta{m^2_{31}}~[\\rm{eV}^2]$\\\\\n\t\t\\hline\n\t\t$33.45^\\circ$ & $8.62^\\circ$ & $42.1^\\circ$ & $230^\\circ$ & $7.42\\times10^{-5}$ & $2.51\\times10^{-3}$\\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{The benchmark values of the oscillation parameters used in our analysis~\\cite{Esteban:2020cvm}. We consider normal mass ordering (NMO) throughout this work.}\n\t\\label{tab:params_value}\n\\end{table}\n\nIn Fig.~\\ref{fig:app_prob}, we plot $\\nu_\\mu\\rightarrow\\nu_e$ oscillation probability as a function of energy for the baseline $L=1285$ km (top row) and $L=295$ km (bottom row) in SI case and in the presence of the CPT-violating LIV parameters. To plot exact oscillation probability, we use GLoBES software~\\cite{Huber:2004ka, Huber:2007ji} and modified the probability calculator accordingly to introduce LIV. The values of the standard oscillation parameters used to calculate the probability are given in Table~\\ref{tab:params_value}. The left, middle, and right columns correspond to the appearance probability in the presence of  $a_{e\\mu}$, $a_{e\\tau}$, and $a_{\\mu\\tau}$ one-at-a-time, respectively with strength $|a_{\\alpha\\beta}|=2\\times10^{-23}$ GeV. In each panel, the solid black curve shows the SI case. Four colored curves correspond to the probabilities in the presence of CPT-violating LIV for the four chosen values of the associated phase, namely, 0, $90^\\circ$, $180^\\circ$, and $270^\\circ$. It is clear from the figure that the impact of $\\ensuremath{a_{\\mu\\tau}}$ is marginal compared to $\\ensuremath{a_{e\\mu}}$ and $\\ensuremath{a_{e\\tau}}$. This behavior is obvious from our analytical expression in Eq.~\\ref{eq:pme_liv} where $\\ensuremath{a_{\\mu\\tau}}$ does not appear at the leading order, contrary to the other two off-diagonal CPT-violating LIV parameters. In the presence of $\\ensuremath{a_{e\\mu}}$, the appearance probability shows a significant deviation from the SI case depending on the value of the associated phase. Near first oscillation maxima, probability is maximum when $\\phi_{e\\mu}=90^\\circ$ and minimum when $\\phi_{e\\mu}=270^\\circ$. It can be explained using terms in Eq.~\\ref{eq:p_cptv_liv}, which shows the contribution from LIV in oscillation probability. The sign of $\\mathbb{Z}$ ($\\mathbb{W}$) in Eq.~\\ref{eq:p_cptv_liv} is negative (positive) near the first oscillation maxima. So, the appearance probability will be maximum (minimum) when the terms associated with $\\mathbb{Z}$ and $\\mathbb{W}$ is negative (positive) and positive (negative), respectively. Since $\\delta_{\\rm CP}=230^\\circ$, this happens at $\\phi_{e\\mu} = 90^\\circ$ ($270^\\circ$). However, in the middle panels, we observe that in the presence of $\\ensuremath{a_{e\\tau}}$, the appearance probability is maximum (minimum) at $\\phi_{e\\tau}=180^\\circ$ ($0^\\circ$). This happens because now both $\\mathbb{Z}$ and $\\mathbb{W}$ are positive. So, the maximum (minimum) probability corresponds to the value of $\\phi_{e\\tau}$ for which both $\\sin(\\delta_{\\rm CP} + \\varphi_{e\\beta})$ and $\\cos(\\delta_{\\rm CP} + \\varphi_{e\\beta})$ in Eq.~\\ref{eq:p_cptv_liv} is positive (negative). This occurs at $\\phi_{e\\tau} = 180^\\circ$ ($0^\\circ$) considering our benchmark value of $\\delta_{\\text{CP}}$.\nEven though all these features can be observed both in the top and the bottom rows, we notice that the spread of the oscillation probability due to phase is significantly lesser in the lower row, where we consider a comparatively smaller baseline ($L=295$ km). It is because of $L$ dependency in Eq.~\\ref{eq:p_cptv_liv}, which adds the contribution from LIV parameters.\n\n\\FloatBarrier\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{.\/plots\/prob_CPTC_app_f1.pdf}\n\t\\vspace*{-10mm}\n\t\\mycaption{$\\nu_\\mu\\rightarrow\\nu_\\mu$ disappearance probability as a function of energy in the presence of off-diagonal CPT-conserving LIV parameters $c_{e\\mu}$ (left column), $c_{e\\tau}$ (middle column), and $c_{\\mu\\tau}$ (right column). The top and bottom rows correspond to the baselines of DUNE ($L=1285$ km) and T2HK ($L=295$ km), respectively. The black line in each panel shows the SI case and four colored lines correspond to four benchmark values of the phases associated with the LIV parameters: $0^\\circ$, $90^\\circ$, $180^\\circ$, and $270^\\circ$ with LIV strength $|c_{\\alpha\\beta}|=1.0\\times10^{-24}$ ($\\alpha,\\beta=e,\\mu,\\tau;\\alpha\\neq\\beta$). The vertical grey-dashed lines in each panel show the energies at first and second oscillation maxima. The values of the standard oscillation parameters used in this plot are mentioned in Table~\\ref{tab:params_value}.}\n\t\\label{fig:cptc_app_prob}\n\\end{figure}\n\nIn Fig.~\\ref{fig:cptc_app_prob}, we plot the appearance probability in the presence of three off-diagonal CPT-conserving LIV parameters, $\\ensuremath{c_{e\\mu}}$ (left column), $\\ensuremath{c_{e\\tau}}$ (middle column), and $\\ensuremath{c_{\\mu\\tau}}$ (right column) with strength $1.0\\times10^{-24}$.\nHere also, we use $L=1285$ km (top row) and $L=295$ km (bottom row). For DUNE ($L=1285$ km), in the presence of $c_{e\\beta}$ ($\\beta=\\mu,\\tau$), we observe that the impact of phase is in the opposite order compared to the corresponding CPT-violating LIV case in Fig.~\\ref{fig:app_prob}. It is because of the opposite sign of the LIV contributing term in the case of CPT-conserving LIV, as shown in Eq.~\\ref{eq:p_cptc_liv}. As expected, $c_{\\mu\\tau}$ has an almost negligible effect on the probability in the case of DUNE. For T2HK ($L=295$ km), all three parameters have almost no impact on the oscillation probabilities because of the smaller baseline and comparatively lower neutrino energy.\n\\begin{figure}[t]\n\t\\centering\t\n\t\\includegraphics[width=\\textwidth]{.\/plots\/disapp_prob_f1.pdf}\n\t\\vspace*{-10mm}\n\t\\mycaption{$\\nu_\\mu\\rightarrow\\nu_\\mu$ disappearance probability as a function of energy in the presence of off-diagonal LIV parameters $a_{\\mu\\tau}$ (left column), $c_{\\mu\\tau}$ (right column). The top and bottom rows correspond to the baselines of DUNE ($L=1285$ km) and T2HK ($L=295$ km), respectively. The black line in each panel shows the SI case and four colored lines correspond to four benchmark values of the phases associated with the LIV parameters: $0^\\circ$, $90^\\circ$, $180^\\circ$, and $270^\\circ$ with LIV strength $|a_{\\alpha\\beta}|=2.0\\times10^{-23}$ GeV, $|c_{\\alpha\\beta}|=1.0\\times10^{-24}$ ($\\alpha,\\beta=e,\\mu,\\tau;\\alpha\\neq\\beta$). The vertical grey-dashed lines in each panel show the energies at first and second oscillation maximums. The values of the standard oscillation parameters used in this plot are mentioned in Table~\\ref{tab:params_value}.}\n\t\\label{fig:disapp_prob}\n\\end{figure}\n\\newline\n\\newline\n\\noindent\n$\\bullet$  \\textbf{$\\nu_\\mu\\to\\nu_\\mu$ Disappearance Channel:}\\\\\n\\\\\nNow we discuss $\\nu_\\mu\\to\\nu_\\mu$ disappearance probability, another relevant oscillation channel for the long-baseline experiments.\nFollowing the same strategy as the appearance channel, we derive the compact analytical expression for $\\nu_\\mu\\to\\nu_\\mu$ disappearance probability,\n\\begin{align}\\label{eq:pmm}\n\tP_{\\mu\\mu} \\simeq P_{\\mu\\mu}(\\text{SI}) + P_{\\mu\\mu}(\\ensuremath{a_{\\mu\\tau}}\/\\ensuremath{c_{\\mu\\tau}}) +\\mathcal{O}(\\alpha^2, \\alpha \\sin^2\\theta_{13},a'^2_{e\\beta},c'^2_{e\\beta},a'^2_{\\mu\\tau},c'^2_{\\mu\\tau}),\\,\\,\\,\\,\\,\\,\\,\\beta=\\mu,\\tau.\n\\end{align}\nThe first term in the RHS is the standard disappearance probability without any new physics contribution,\n\\begin{align}\n\tP_{\\mu\\mu}(\\text{SI}) = P_{\\mu\\mu}(\\text{vacuum},\\text{two flavor})+\\alpha\\mathbb{P}+\\sin^2\\theta_{13}\\mathbb{Q}+\\alpha\\sin\\theta_{13}\\mathbb{R},\n\\end{align}\nwhere,\n\\begin{align}\n\tP_{\\mu\\mu}(\\text{vacuum},\\text{two flavor}) = 1-\\sin^22\\theta_{23}\\sin^2\\Delta.\n\\end{align}\n$\\mathbb{P}$, $\\mathbb{Q}$, and $\\mathbb{R}$ are defined as follows,\n\\begin{align}\n\t&\\mathbb{P} = \\cos^2\\theta_{12}\\sin^22\\theta_{23}\\sin2\\Delta,\\\\\n\t&\\mathbb{Q} = -4\\sin^2\\theta_{23}\\frac{\\sin^2(\\hat{A}-1)\\Delta}{(\\hat{A}-1)^2}-\\frac{2}{\\hat{A}-1}\\sin^22\\ensuremath{\\theta_{23}}\\left(\\sin\\Delta\\cos \\hat{A}\\Delta\\frac{\\sin(\\hat{A}-1)\\Delta}{\\hat{A}-1}-\\frac{\\hat{A}}{2}\\Delta\\sin2\\Delta\\right),\\\\\n\t&\\mathbb{R}= 2\\sin2\\theta_{12}\\sin2\\theta_{23}\\cos\\delta_{\\rm CP}\\cos\\Delta\\frac{\\sin \\hat{A}\\Delta}{\\hat{A}}\\frac{\\sin(\\hat{A}-1)\\Delta}{\\hat{A}-1}.\n\\end{align}\nNote that in the above expression, we have also neglected the term with $\\alpha\\sin\\ensuremath{\\theta_{13}}\\cos2\\ensuremath{\\theta_{23}}$, since it is of the same order as $\\alpha \\sin^2\\theta_{13}$. The contribution from the LIV parameters\\footnote{Only off-diagonal LIV parameters that appear at the first order in the disappearance probability is $\\ensuremath{a_{\\mu\\tau}}\/\\ensuremath{c_{\\mu\\tau}}$. This has already been discussed in Refs.~\\cite{Kopp:2007ne,Kikuchi:2008vq} in case of NSI.} is given by,\n\\begin{align}\\label{eq:Pmm_liv}\n\tP_{\\mu\\mu}(\\ensuremath{a_{\\mu\\tau}}\/\\ensuremath{c_{\\mu\\tau}}) = \\frac{\\sin^22\\ensuremath{\\theta_{23}}}{2}\\left[2\\sin^2\\ensuremath{\\theta_{13}}\\Delta-\\mathbb{S}\\right] \\sin2\\Delta,\n\\end{align}\nwhere,\n\\begin{align}\\label{eq:S}\n\t\\mathbb{S} = \\begin{cases}\n\t\t2L\\sin2\\ensuremath{\\theta_{23}}|a_{\\mu\\tau}|\\cos\\phi_{\\mu\\tau}, & \\text{CPT-violating LIV}. \\\\\n\t\t-\\frac{8}{3}E L \\sin2\\ensuremath{\\theta_{23}}|c_{\\mu\\tau}|\\cos\\phi_{\\mu\\tau}, & \\text{CPT-conserving LIV}.\n\t\\end{cases} \n\\end{align}\nIn Appendix~\\ref{appndx}, we compare the disappearance probability calculated using the analytical expressions derived above with the same calculated numerically. \n\nIn Fig.~\\ref{fig:disapp_prob}, we show the $\\nu_\\mu\\to\\nu_\\mu$ disappearance probability as a function of energy for the baseline $L=1285$ (top row) km and $L=295$ km (bottom row) in SI case as well as in the presence of LIV parameters. In the left column, we show the impact of CPT-violating LIV parameter $\\ensuremath{a_{\\mu\\tau}}$, whereas the right column shows the CPT-conserving LIV parameter $c_{\\mu\\tau}$. As discussed earlier, the impact of the other two off-diagonal parameters does not appear at the leading order and is expected to have a negligible impact on disappearance probability. From the left panels, we observe that the phase associated with $\\ensuremath{a_{\\mu\\tau}}$, shows a significant impact with positive (negative) deviation for $\\phi_{\\mu\\tau}=180^\\circ$ ($0^\\circ$) from the SI case. We can explain this feature using the LIV contributing terms in the analytical expression (ref. to Eq.~\\ref{eq:pme_liv}). It is clear that when $\\mathbb{S}$ (shown in Eq.~\\ref{eq:S}) is negative (positive), the disappearance probability is larger (smaller) than its corresponding SI value. Similar to the appearance probability, here also, we observe the impact of the phases become smaller for a smaller baseline, as shown in lower panels, because the LIV contributing term is proportional to the baseline length $L$. In the right column, we show the impact of the CPT-conserving LIV parameters. Here we observe that for $L=1285$ km, disappearance is maximum (minimum) for $\\phi_{\\mu\\tau}=0$ ($180^\\circ$). This can be understood from the sign of the term $\\mathbb{S}$ in Eq.~\\ref{eq:S}; when it is negative ($\\phi_{\\mu\\tau}=0$), $P_{\\mu\\mu}(c_{\\mu\\tau})$ is larger compared to the case when it is positive ($\\phi_{\\mu\\tau}=180^\\circ$). \n\n\\section{Long-baseline Experiments: DUNE and T2HK}\n\\label{sec:LBL}\n\\subsection{Essential Features of the Experimental Setups}\nAccelelarator-based neutrino oscillation experiments are playing a very important role in resolving issues in the standard $3\\nu$ paradigm and exploring various BSM physics in the neutrino sector. Precise information about the neutrino flux, cross-section, and baseline make these experiments unique. In this work, we are going to probe LIV in the context of next-generation long-baseline experiments DUNE and T2HK. DUNE is a Fermilab-based future long-baseline experiment with an on-axis, high-intensity wide-band neutrino beam produced at Fermilab~\\cite{DUNE:2020lwj, DUNE:2020jqi, DUNE:2021cuw, DUNE:2021mtg}. The detector would be a 40 kt Liquid argon time projection chamber (LArTPC) placed underground at Homestake mine, 1285 km from the source. On the other hand, T2HK~\\cite{Hyper-KamiokandeProto-:2015xww, Hyper-Kamiokande:2018ofw} is another next-generation long-baseline experiment with off-axis, narrow band beam produced at the J-PARC proton synchrotron facility. The beam would be detected at Hyperkamiokande, a 187 kt water Cherenkov detector placed at a distance of 295 km from the source with an off-axis angle $2.5^\\circ$. In Table~\\ref{tab:exp_details}, we tabulate the other relevant information on these two experiments.\n\n\\begin{table}\n\t\\centering\n\t\\begin{tabular}{|c|c|c|}\n\t\t\\hline \\hline\n\t\t& DUNE & T2HK\\\\ \n\t\t\\hline\n\t\tDetector Mass& 40 kt LArTPC & 187 kt WC   \\\\\n\t\t\\hline\n\t\tBaseline & 1285 km & 295 km \\\\\n\t\t\\hline\n\t\tProton Energy & 120 GeV & 80 GeV \\\\\n\t\t\\hline\n\t\tBeam type & Wide-band, on-axis & Narrow-band, off-axis ($2.5^{\\circ}$)\\\\\n\t\t\\hline\n\t\tBeam power & 1.2 MW & 1.3 MW \\\\\n\t\t\\hline\n\t\tP.O.T.\/year& $1.1\\times10^{21}$ & $2.7\\times10^{21}$\\\\ \n\t\t\\hline\n\t\tRun time ($\\nu+\\bar{\\nu}$) & 5 yrs + 5 yrs & 2.5 yrs + 7.5 yrs  \\\\\n\t\t\\hline\n\t\tNormalization error & 2\\% (app.) 5\\% (disapp.) & 5\\% (app.), 3.5\\% (disapp.)\\\\\n\t\t\\hline\\hline\n\t\\end{tabular}\n\t\\mycaption{Major features of LBL experiments, DUNE~\\cite{DUNE:2020lwj} and T2HK~\\cite{Hyper-KamiokandeProto-:2015xww} used in our simulation.}\n\t\\label{tab:exp_details}\n\\end{table}\n\nAs mentioned earlier, the DUNE setup uses an on-axis, wide-band neutrino beam with an energy range between 0.1 to 12 GeV. This allows DUNE to explore both the oscillation maximum and minima for the baseline 1285 km, which are around 2.5 GeV and 0.6 GeV, respectively. On the other hand, T2HK has an off-axis narrow band beam with an energy peak around 0.6 GeV, which is the first oscillation maxima for T2HK. The monochromatic beam will give the advantage of high statistics at the first oscillation maxima, where the impact of various physics can be significant. DUNE will have equal runtime for neutrino and antineutrino mode, which will give a larger number of expected neutrino events than the antineutrino, since the neutrino has almost three times the cross-section of the antineutrino. For T2HK, antineutrino runtime is three times neutrino in order to compensate for suppression in the cross-section. So, depending on the physics under probe, different ratios of the neutrino and antineutrino event can be useful. Also, the baseline of DUNE is 1285 km allowing it to have a large matter effect compared to T2HK, which has a baseline of 295 km. Apart from the complementarity at the neutrino flux, baseline, and runtime, the detector properties of the two setups are different. As proposed by the collaborations, DUNE has systematics uncertainties of 2.5\\% in the appearance channel and 5\\% in the disappearance channel and for T2HK, it is 5\\% and 3.5\\%, respectively. Lower systematics in the appearance channel in DUNE can allow it to have comparatively larger sensitivity to some physics that have a larger impact on the appearance channel. A similar argument can be given for T2HK in the disappearance channel.\n\n\\subsection{Expected Event Rates in the Presence of LIV}\n\\label{subsec:LBL-Evt-Sim}\n\nAs mentioned earlier, in this work, we consider two experimental configurations, namely, DUNE and T2HK, for our analysis. We calculate the expected event rate of these two configurations using the GLoBES software~\\cite{Huber:2004ka, Huber:2007ji}. For the oscillation analysis with LIV parameters, we use GLoBES-extension $snu.c$~\\cite{Kopp:2007ne}.\n\nIn Table.~\\ref{tab:total_events}, we give the expected $\\nu_e\/\\bar{\\nu}_e$ and $\\nu_\\mu\/\\bar{\\nu}_\\mu$ event rate from DUNE and T2HK in SI case and in the presence of various LIV parameters. Assumed configurations of the experiments are discussed in detail in Sec.~\\ref{sec:LBL} (see Table~\\ref{tab:exp_details}). While generating the events, the strength of the CPT-violating (CPT-conserving) LIV parameters is considered to be $2.0\\times10^{-23}$ GeV ($1.0\\times10^{-24}$), one at-a-time. We consider the phase associated with the LIV parameters to be zero. \n\n\t\\begin{table}[h!]\t\t\n\t\t\\centering\n\t\t\\begin{adjustbox}{width=1\\textwidth}\n\t\t\t\\begin{tabular}{|ccc|*{8}{c|}}\t\t\n\t\t\t\t\\hline\\hline\n\t\t\t\t\n\t\t\t\t\\multicolumn{3}{|c|}{\\multirow{2}{*}{}} & \\multicolumn{2}{|c}{$\\nu_e$ appearance} & \\multicolumn{2}{|c|}{$\\bar\\nu_e$ appearance}& \\multicolumn{2}{|c}{$\\nu_\\mu$ disappearance} & \\multicolumn{2}{|c|}{$\\bar\\nu_\\mu$ disappearance}\\\\ \n\t\t\t\t\n\t\t\t\t\\cline{4-11}\n\t\t\t\t\n\t\t\t\t\\multicolumn{3}{|c|}{} & DUNE & T2HK & DUNE & T2HK & DUNE & T2HK & DUNE & T2HK \\\\\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\t\\multicolumn{3}{|c|}{SI} & 1614 & 1613  & 292 & 727 & 15624 & 9577  & 9052 & 9074 \\\\\n\t\t\t\t\n\t\t\t\t\\cline{1-11}\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\t\\multicolumn{3}{|c|}{$|a_{e\\mu}|$ (= $2\\times10^{-23}$ GeV)} & 2276 & 1731 & 666 & 901  & 15446 & 9581  & 8891 & 9042\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{3}{|c|}{$|a_{e\\tau}|$ (= $2\\times10^{-23}$ GeV)} & 817 & 1387  & 226 & 697 & 15613 & 9565   & 9063 & 9084\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{3}{|c|}{$|a_{\\mu\\tau}|$ (= $2\\times10^{-23}$ GeV)} & 1567 & 1598  & 303 & 735 & 14404 & 9366  & 9237 & 9373\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{3}{|c|}{$|c_{e\\mu}|$ (= $1.0\\times10^{-24}$)} & 1757 & 1610 & 480 & 735  & 15315 & 9575  & 8800 & 9071\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{3}{|c|}{$|c_{e\\tau}|$ (= $1.0\\times10^{-24}$)} & 1792 & 1623  & 296 & 724 & 15634 & 9578   & 9056 & 9074\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{3}{|c|}{$|c_{\\mu\\tau}|$ (= $1.0\\times10^{-24}$)} & 1620 & 1614  & 295 & 727 & 16263 & 9600  & 9411 & 9099\\\\\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\mycaption{Total signal rate for the $\\nu_{e}$ appearance channel and  $\\nu_{\\mu}$ disappearance channel both in neutrino and anti-neutrino mode for DUNE and T2HK setups in SI case as well as in the presence off-diagonal LIV parameters. The relevant features of these facilities are given in Table~\\ref{tab:exp_details}. The strength of the CPT-violating (CPT-conserving) LIV parameters is taken to be $2\\times10^{-23}$ GeV ($1.0\\times10^{-24}$). The phases associated with the off-diagonal LIV parameters are considered to be zero. The values of the other standard oscillation parameters used to calculate event rate are quoted in Table~\\ref{tab:params_value}.}\n\t\n\t\\label{tab:total_events}\n\\end{table}\n\nWe make following observations from Table~\\ref{tab:total_events}:\n\\begin{itemize}\n\n\\item In the presence of $a_{e\\mu}$ ($a_{e\\tau}$), $\\nu_e$ event deviates from the SI case by 41\\% (49\\%) for DUNE and by 7.3\\% (14\\%) for T2HK. \n\n\\item Similarly, we observe that the presence of LIV parameter $a_{\\mu\\tau}$ changes the expected $\\nu_{\\mu}$ disappearance event rates by 7.8\\% for DUNE and 2.2\\% for T2HK from the SI case. In the presence of other LIV parameters, changes in the event rates are very small ($\\leq 1\\%$).\n\n\\item In presence of $c_{e\\mu}$ ($c_{e\\tau}$), the $\\nu_{e}$ appearance event rates changes by $\\sim$9\\% (11\\%) for DUNE, but for T2HK, the changes are very minute ($<$ 1\\%). However $c_{\\mu\\tau}$ changes the $\\nu_{\\mu}$ event rates by 4\\% for DUNE and 0.2\\% for T2HK.\n\n\\end{itemize}\nAll these observations are consistent with the results seen at the probability level in the previous section.\n\n\\section{Numerical Analysis}\n\\label{sec:RnA}\nOne of the major goals of this work is to study the ability of DUNE and T2HK to constrain various LIV parameters.  To do this, we examine the sensitivity of these two setups in realizing LIV.\nWe define our sensitivity in terms of the Poissonian $\\chi^2$ defined as,\n\\begin{equation}\n\\chi^2 (\\vec{\\lambda}, \\xi_{s}, \\xi_{b}) = \\min_{\\vec\\rho, \\xi_{s}, \\xi_{b}}~\\Big[{{2\\sum_{i=1}^{n}}\\big(y_{i}-x_{i}-x_{i} \\text{ln}\\frac{y_{i}}{x_{i}}\\big) + \\xi_{s}^2 + \\xi_{b}^2}~\\Big],\n\\end{equation} \nwhich gives the median sensitivity of the experiment where $n$ is the total number of reconstructed energy bins.\n\\begin{equation}\ny_{i} = N^{th}_{i}\n(\\vec\\rho)~[1 + \\pi^{s}\\xi_{s}] + N^{b}_{i}(\\vec\\rho)~[1+\\pi^b\\xi_{b}],\n\\end{equation}\n\nwhere $N^{th}_{i}$ is the expected number of signal events in the $i$-th bin with the set of oscillation parameters $\\vec\\lambda$ = \\{$\\theta_{12}, \\theta_{13}, \\theta_{23}, \\Delta{m}^2_{21}, \\Delta{m}^2_{31}, \\delta_{\\rm CP}, a_{\\alpha\\beta}, \\phi^{a}_{\\alpha\\beta}, c_{\\alpha\\beta}, \\phi^{c}_{\\alpha\\beta}$\\}. $\\phi^{a}_{\\alpha\\beta}$ and $\\phi^{c}_{\\alpha\\beta}$ are the phases associated with the CPT-violating and CPT-conserving LIV parameters, respectively. $N^{b}_{i}$ is the number of background events in the $i$-th energy bin. The systematic pulls on the signal and background are denoted by the variables $\\xi_{s}$ and $\\xi_{b}$, respectively. We marginalize the $\\chi^2$ over the set of parameters $\\vec\\rho$ and also over the systematic pulls ($\\xi_{s}$ and $\\xi_{b}$) in the fit. The variables $\\pi^s$ and $\\pi^b$ stand for the normalization error on the signal and background. $x_{i} = N^{obs}_{i} + N^{b}_{i}$ embodies the prospective data from the experiment, where $N^{obs}_{i}$ is the number of charged current signal events and $N^b_{i}$, as mentioned before, is the number of background events. \n\nWe quantify our results in terms of the statistical significance given by Poissonian $\\Delta{\\chi}^2$ defined as,\n\\begin{equation}\n\t\\Delta{\\chi}^2 = \\min_{\\vec\\rho, \\xi_{s}, \\xi_{b}}\\Big[\\chi^2 (a_{\\alpha\\beta}\/c_{\\alpha\\beta} \\neq 0) - \\chi^2 (a_{\\alpha\\beta}\/c_{\\alpha\\beta} = 0)\\Big],\\,\\,\\,\\,\\,\\alpha,\\beta=e,\\mu,\\tau;\\alpha\\neq\\beta.\n\\end{equation}\nThe first term in the right-hand side of the above equation is obtained when we fit the prospective data from the experiment with the theory in the presence of Lorentz Invariance Violation, and the second term is calculated by fitting with the standard case with no LIV present in theory. Due to suppression in the statistical fluctuations, we can take $\\chi^2 (a_{\\alpha\\beta}\/c_{\\alpha\\beta} = 0)$ $\\sim$ 0 while obtaining the median sensitivity of the experiment using frequentist approach~\\cite{Blennow:2013oma}. Here $\\vec\\rho$ is the set of parameters over which the $\\chi^2$ is marginalized, and $\\xi_{s}$ and $\\xi_{b}$ are the systematic pulls on the signal and background, respectively.\nFor our analysis, we use the true values of the standard oscillation parameters as given in Table~\\ref{tab:params_value}. In theory, we keep two mixing angles $\\theta_{12}$, $\\theta_{13}$ and two mass-splittings $\\Delta m^2_{21}$, $\\Delta m^2_{31}$ fixed at the same value as these parameters have been well measured. Also, we do not marginalize over the neutrino mass ordering since there is a hint towards the normal mass ordering from the global oscillation data. Also, for the correlation between LIV parameters and $\\delta_{\\mathrm{CP}}$, we marginalize over the $\\theta_{23}$ in its allowed $3\\sigma$ range~\\cite{Esteban:2020cvm} and the phase $\\phi$ associated with the off-diagonal LIV parameters in the range [0, $360^\\circ$]. Similarly, for the LIV parameter and $\\theta_{23}$ correlation analysis, we marginalize over $\\delta_{\\mathrm{CP}}$ in its allowed $3\\sigma$ range and $\\phi$ in its entire permitted range.\nWhile deriving the limits on the off-diagonal LIV parameters, we marginalize over $\\theta_{23}$, $\\delta_{\\rm CP}$, and $\\phi$.\n\n\\section{Our Results}\n\\label{sec:results}\nIn this section, we present our results in two parts. First, we discuss the correlation between the LIV parameters and most unsettled standard oscillation parameters $\\delta_{\\mathrm{CP}}$ and $\\theta_{23}$. This helps us to find if there is any degeneracy between the LIV parameters and the standard oscillation parameters. In the second part, we present the expected constraints on the LIV parameters from DUNE, T2HK individually, and their combination DUNE+T2HK.\n\n\\subsection{Correlations in test ($\\delta_{\\rm CP} - |a_{\\alpha\\beta}|\/|c_{\\alpha\\beta}|$) and test ($\\theta_{23} - |a_{\\alpha\\beta}|\/|c_{\\alpha\\beta}|$) Planes}\n\\label{subsec:correlation}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{.\/plots\/cor-dcp-CPTV-f4.pdf}\n\t\\vspace*{-10mm}\n\t\\mycaption{95\\% C.L. (2 d.o.f.) contours in the $\\delta_{\\rm CP}-|a_{e\\mu}|$ (top row), $\\delta_{\\rm CP}-|a_{e\\tau}|$ (middle row), and $\\delta_{\\rm CP}-|a_{\\mu\\tau}|$ (bottom row) planes for DUNE, T2HK, and DUNE+T2HK. Three benchmark values of $\\delta_{\\mathrm{CP}}$ considered in the data are $180^\\circ$ (left column), $230^\\circ$ (middle column), and $300^\\circ$ (right column), as shown by black dots in each panel. True values of other oscillation parameters are given in Table~\\ref{tab:params_value}. In the fit, we marginalize over $\\theta_{23}$ in its allowed $3\\sigma$ range and new phase $\\phi$ in its entire allowed range.}\n\t\\label{fig:correlation_dcp_cptv}\n\\end{figure}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{.\/plots\/cor-dcp-CPTC-f4.pdf}\n\t\\vspace*{-10mm}\n\t\\mycaption{95\\% C.L. (2 d.o.f.) contours in the $\\delta_{\\rm CP}-|c_{e\\mu}|$ (top row), $\\delta_{\\rm CP}-|c_{e\\tau}|$ (middle row), and $\\delta_{\\rm CP}-|c_{\\mu\\tau}|$ (bottom row) planes for DUNE, T2HK, and DUNE+T2HK. Three benchmark values of $\\delta_{\\mathrm{CP}}$ considered in the data are $180^\\circ$ (left column), $230^\\circ$ (middle column), and $300^\\circ$ (right column), as shown by black dots in each panel. True values of other oscillation parameters are given in Table~\\ref{tab:params_value}. In the fit, we marginalize over $\\theta_{23}$ in its allowed $3\\sigma$ range and new phase $\\phi$ in its entire allowed range.}\n\t\\label{fig:correlation_dcp_cptc}\n\\end{figure}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{.\/plots\/cor-th23-CPTV-f4.pdf}\n\t\\vspace*{-10mm}\n\t\\mycaption{ 95\\% C.L. (2 d.o.f.) contours in the $\\ensuremath{\\theta_{23}}-|a_{e\\mu}|$ (top row), $\\ensuremath{\\theta_{23}}-|a_{e\\tau}|$ (middle row), and $\\ensuremath{\\theta_{23}}-|a_{\\mu\\tau}|$ (bottom row) planes for DUNE, T2HK, and DUNE+T2HK. Three benchmark values of $\\ensuremath{\\theta_{23}}$ considered in the data are $42.1^\\circ$ (left column), $45^\\circ$ (middle column), and $47.9^\\circ$ (right column), as shown by black dots in each panel. True values of other oscillation parameters are given in Table~\\ref{tab:params_value}. In the fit, we marginalize over $\\delta_{\\mathrm{CP}}$ in its allowed $3\\sigma$ range and new phase $\\phi$ in its entire allowed range.}\n\t\\label{fig:correlation_th23_cptv}\n\\end{figure}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{.\/plots\/cor-th23-CPTC-f4.pdf}\n\t\\vspace*{-10mm}\n\t\\mycaption{95\\% C.L. (2 d.o.f.) contours in the $\\ensuremath{\\theta_{23}}-|c_{e\\mu}|$ (top row), $\\ensuremath{\\theta_{23}}-|c_{e\\tau}|$ (middle row), and $\\ensuremath{\\theta_{23}}-|c_{\\mu\\tau}|$ (bottom row) planes for DUNE, T2HK, and DUNE+T2HK. Three benchmark values of $\\ensuremath{\\theta_{23}}$ considered in the data are $42.1^\\circ$ (left column), $45^\\circ$ (middle column), and $47.9^\\circ$ (right column), as shown by black dots in each panel. True values of other oscillation parameters are given in Table~\\ref{tab:params_value}. In the fit, we marginalize over $\\delta_{\\mathrm{CP}}$ in its allowed $3\\sigma$ range and new phase $\\phi$ in its entire allowed range.}\n\t\\label{fig:correlation_th23_cptc}\n\\end{figure}\n\nIn Fig.~\\ref{fig:correlation_dcp_cptv}, we show the correlations between the CPT-violating LIV parameters and the standard CP phase $\\delta_{\\rm CP}$. In the fit, we marginalize over the $\\theta_{23}$ in its allowed $3\\sigma$ range~\\cite{Esteban:2020cvm} and the phase $\\phi$ associated with the off-diagonal LIV parameters in the range [0, $360^\\circ$]. As mentioned earlier, all the other standard oscillation parameters are fixed at their best-fit values given in Table~\\ref{tab:params_value}, both in data and theory. \nTop, middle, and bottom rows correspond to non-zero $a_{e\\mu}$, $a_{e\\tau}$, and $a_{\\mu\\tau}$, respectively, where consider these LIV parameters \none at-a-time in the data and fit. We take three different choices for the value of $\\delta_{\\rm CP}$ in the data that are allowed in the current $3\\sigma$ limits, namely, $180^\\circ$ (left panels), $270^\\circ$ (middle panels), $300^\\circ$ (right panels) as shown by the black dot in each panel. The red, blue, and black curves in each plot correspond to DUNE, T2HK, and the combination DUNE+T2HK, respectively. Each contour represents the allowed regions at 95\\% C.L. (2 d.o.f.). We observe from the figure that for all the LIV parameters and all choices of the $\\delta_{\\mathrm{CP}}$ in the data, the allowed regions in $\\delta_{\\rm CP}-|a_{\\alpha\\beta}|$ planes are significantly small for DUNE compared to T2HK. One can understand it from the analytical expression of the oscillation probabilities discussed in  Sec.~\\ref{sec:LIV}. From Eq.~\\ref{eq:p_cptv_liv} and Eq.~\\ref{eq:Pmm_liv}, we see that contribution from the CPT-violating LIV parameters is directly proportional to $L$. So, DUNE being the experiment with a longer baseline, shows better sensitivity to the CPT-violating LIV parameters. Hence, it has smaller allowed regions in  $\\delta_{\\rm CP}-|a_{\\alpha\\beta}|$ plane as compared to T2HK.\n\nIn case of $a_{e\\mu}$, we notice another degenerate solution in $\\delta_{\\rm CP}-|a_{\\alpha\\beta}|$ plane which is centered around some non-zero value of  $|a_{e\\mu}|$. This happens for both DUNE and T2HK around $|a_{e\\mu}|\\approx1\\times 10^{-23}$ GeV and  $|a_{e\\mu}|\\approx4\\times 10^{-23}$ GeV, respectively. This mainly occurs due to the degeneracy between $\\theta_{23}$ and the complex phases ($\\delta_{\\rm CP}~\\rm and~ \\phi$) in the LIV contributing term in the appearance channel (see Eq.~\\ref{eq:p_cptv_liv}), which plays a major role in constraining this parameter. For some combination of $\\theta_{23}$ and $\\delta_{\\mathrm{CP}}+\\phi$, this term minimizes at some non-zero value of $|a_{e\\mu}|$ resulting in degeneracy with the standard oscillation case. As a result, we observe an allowed region at around that value of $|\\ensuremath{a_{e\\mu}}|$. However, when we take the combined setup DUNE+T2HK, this degeneracy disappears as the values of $L$ and $E$ in the LIV contributing terms are now different for these two experiments, and we do not observe any allowed region centred around some non-zero value of $|\\ensuremath{a_{e\\mu}}|$. We also observe that upon combining these two setups, the allowed regions shrink further. In the case of $|\\ensuremath{a_{e\\tau}}|$ (middle row) and $|\\ensuremath{a_{\\mu\\tau}}|$ (bottom row), we do not observe such degenerate solutions at 95\\% C.L. (2 d.o.f.). In both cases, DUNE+T2HK shows a small improvement in the sensitivity as compared to DUNE. \n\nIn Fig.~\\ref{fig:correlation_dcp_cptc}, we repeat the above analysis for three off-diagonal CPT-conserving LIV parameters, $c_{e\\mu}$ (top row), $c_{e\\tau}$ (middle row), and $c_{\\mu\\tau}$ (bottom row). It is clear from the figure that DUNE shows a noticeable correlation between $c_{\\alpha\\beta}$ ($\\alpha,\\beta=e,\\mu,\\tau;\\alpha\\neq\\beta$) and $\\delta_{\\mathrm{CP}}$. However, for T2HK, there is almost no correlation between those two parameters in all three cases. One can explain it from the fact that CPT-conserving parameters have negligible impact on both appearance and disappearance probabilities for T2HK as shown in the bottom row of Fig.~\\ref{fig:app_prob} and bottom right panel of Fig.~\\ref{fig:disapp_prob}. As discussed earlier,\nthis happens because of $L\\times E$ dependencies in LIV contributing terms in the CPT-conserving case (see Eq.~\\ref{eq:pme_liv} and Eq.~\\ref{eq:Pmm_liv}). Since T2HK has a short baseline and low energy neutrino beam compared to DUNE, it shows almost no sensitivity to CPT-conserving LIV parameters. \n\nIn Fig.~\\ref{fig:correlation_th23_cptv}, we show the correlation between the CPT-violating LIV parameters $a_{\\alpha\\beta}$ ($\\alpha,\\beta=e,\\mu,\\tau$; $\\alpha\\neq\\beta$) and $\\theta_{23}$. We consider three values of $\\theta_{23}$ to in data, namely $42.1^\\circ$ (left column) in the lower octant, 45$^\\circ$ (middle column) maximal mixing case, and $47.9^\\circ$ (right column) in the upper octant\\footnote{We choose $\\theta_{23}=42.1^\\circ$ in lower octant as it is the current best-fit value from the global-fit of the oscillation parameters~\\cite{Esteban:2020cvm}. For simulation, we marginalize over $\\delta_{\\mathrm{CP}}$ in its allowed $3\\sigma$ range and the corresponding LIV phases in their entire allowed range. We consider the corresponding value ($47.9^\\circ$) in the upper octant.}. We observe that for both DUNE and T2HK, the best result is obtained when the true value of $\\theta_{23}$ is in the lower octant, where the allowed region is relatively small compared to the other two cases. \nSimilar to $\\delta_{\\mathrm{CP}}-a_{\\alpha\\beta}$ correlation, DUNE perform significantly better compared to T2HK for all three choices of true $\\theta_{23}$. As we discussed earlier, it is because of $L$ dependencies in CPT-violating LIV contributing terms in the oscillation probabilities.  \nHere also, we observe degenerate allowed region at a non-zero value of $a_{e\\mu}$ and $a_{e\\tau}$ that appear at the opposite octant of $\\theta_{23}$ for both DUNE and T2HK. This happens because of the degeneracy between $\\theta_{23}$ and CP phases $\\delta_{\\mathrm{CP}}$, $\\phi$ in the LIV contributing term of the oscillation probabilities.\nWhen data from the DUNE and T2HK are added, the allowed region becomes smaller, and interestingly, degenerate region appearing for the individual setup vanishes, as shown by the black contour in each panel. In Fig.~\\ref{fig:correlation_th23_cptc}, we show the same for CPT-conserving LIV parameters. Here also, the best results is obtained when $\\theta_{23}$ is in lower octant for the individual setups. Here we observe that, although DUNE shows a notable correlation with $\\theta_{23}$, whereas T2HK shows almost no correlation with the CPT-conserving LIV parameters. As mentioned before, T2HK has almost no sensitivity to the CPT-conserving LIV parameters because of its smaller baseline and lower neutrino energy.\nHowever, we observe a slight improvement in the allowed region when we combine the data from DUNE and T2HK.\n\n\\subsection{Constraints on CPT-violating and CPT-conserving LIV parameters}\n\\label{subsec:constraints}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{.\/plots\/chi-square-profile-f.pdf}\n\t\\vspace*{-10mm}\n\t\\mycaption{ The expected limits on the CPT-violating and CPT-conserving LIV parameters\n\t\tfrom DUNE (red curves), T2HK (blue curves), and DUNE+T2HK (black curves).\n\t\tThe upper (lower) panels show the $\\Delta\\chi^2$ for the off-diagonal CPT-violating (CPT-conserving) LIV parameters considering one-at-a-time. The true values of $\\theta_{23}$ and $\\delta_{\\mathrm{CP}}$ are kept at their best fit values given in table~\\ref{tab:params_value}. We marginalize over $\\theta_{23}$ and $\\delta_{\\mathrm{CP}}$ in their 3$\\sigma$ allowed range in the fit. Apart from $\\theta_{23}$ and $\\delta_{\\mathrm{CP}}$, we also marginalize over the associated LIV phases in their total allowed range [$0^{\\circ}$, $360^{\\circ}$].}\n\t\\label{fig:cptv_liv_bounds}\n\\end{figure}\n\nIn the previous section, we have discussed the correlation of the LIV parameters with the most uncertain standard oscillation parameters, $\\delta_{\\mathrm{CP}}$ and $\\theta_{23}$ in the context of DUNE, T2HK, and their combination. In this section, we present the limits on the off-diagonal LIV parameters that would be obtained by these three setups. As discussed earlier, in our simulation, we marginalized over $\\theta_{23}$, $\\delta_{\\text{CP}}$, and the phase associated with the off-diagonal LIV parameters in the fit (see Sec.~\\ref{sec:RnA} for details). \n\nIn Fig.~\\ref{fig:cptv_liv_bounds}, we show $\\Delta \\chi^2$ as a function of the off-diagonal CPT-conserving (top row) and CPT-violating (bottom row) LIV parameters. The red, blue, and black lines in each panel correspond to the sensitivity of DUNE, T2HK, and DUNE+T2HK setups, respectively. The top left panel correspond to $a_{e\\mu}$ case where we see that DUNE shows better sensitivity compared to T2HK at 95\\% C.L.. Interestingly, for both DUNE and T2HK, there is a local minima in the $\\Delta\\chi^2$ around $1\\times10^{-23}$ GeV and $4\\times10^{-23}$ GeV, respectively. This feature can be explained using the correlation of the LIV parameters with standard oscillation parameters $\\theta_{23}$ and $\\delta_{\\mathrm{CP}}$ discussed in Sec.~\\ref{subsec:correlation}. We observe that there is allowed region in the $\\delta_{\\mathrm{CP}}-|a_{e\\mu}|$ and $\\theta_{23}-|a_{e\\mu}|$ plane at around the same value of $|\\ensuremath{a_{e\\mu}}|$ where the local minima occurs. Since this parameter is mainly constrained by the appearance channel, it hints towards a degeneracy between the appearance probability in absence of any new physics and the same in presence of the LIV for some combination of $\\theta_{23}, \\delta_{\\mathrm{CP}}$ and new phase $\\phi$. However, this degeneracy vanishes as we combine the data from DUNE and T2HK, giving a more stringent limit on $|\\ensuremath{a_{e\\mu}}|$. In the top middle panel, we show the sensitivity for $|\\ensuremath{a_{e\\tau}}|$. Here also, a local minima of $\\Delta\\chi^2$ is observed for the individual setup DUNE and T2HK, which again occurs due to degeneracy between the $\\theta_{23}$, $\\delta_{\\mathrm{CP}}$, and $\\phi$ (see top middle panel of Fig.~\\ref{fig:correlation_dcp_cptv} and \\ref{fig:correlation_th23_cptv}). Adding the data from the two experiments solves the issue of local minima.\nTop right panel shows the constraints on $\\ensuremath{a_{\\mu\\tau}}$ for the three setups. We observe that DUNE gives significantly better limits for $\\ensuremath{a_{\\mu\\tau}}$ as compared to T2HK. Unlike $\\ensuremath{a_{e\\mu}}$ and $\\ensuremath{a_{e\\tau}}$, we do not observe local minima of $\\Delta\\chi^2$ here. It happens because $\\ensuremath{a_{\\mu\\tau}}$ is mainly constrained by the disappearance channel, where such degeneracy among the $\\theta_{23}$ and the phases does not occur.\nIn the lower panels of Fig.~\\ref{fig:cptv_liv_bounds}, we show the constraints on CPT-conserving LIV parameters. As it is clear from the oscillation probability plots in Fig.~\\ref{fig:cptc_app_prob} (see bottom row) and Fig.~\\ref{fig:disapp_prob} (see bottom right panel), T2HK has almost no sensitivity on the CPT-conserving LIV parameters. However, when the data from T2HK and DUNE are added, sensitivity is slightly improved for all the three off-diagonal parameters.\n\nWe tabulate our result by showing the limits at 95\\% C.L. in Table~\\ref{tab:constraints_a}. The Second and third columns show the limits from DUNE and T2HK independently. The last column is the ultimate limit on LIV parameters from the combination of DUNE and T2HK.\nNote that for the bound on $|\\ensuremath{a_{e\\mu}}|$ and $|\\ensuremath{a_{e\\tau}}|$, we consider the most conservative scenarios, $\\textit{i.e.}$ the largest value of $a_{e\\beta}$, which reaches 95\\% C.L. value. For $a_{e\\beta}$ ($\\beta = e,\\mu$), the obtained constraints from DUNE are almost five times better than that of T2HK. Also, combining the data from DUNE and T2HK, the limits improved further by a factor of $\\approx 2$ for $\\ensuremath{a_{e\\mu}}$  and $\\approx 3$ for $\\ensuremath{a_{e\\tau}}$. For $\\ensuremath{a_{\\mu\\tau}}$ also, constraints from DUNE outperform T2HK approximately by a factor of five. However, for DUNE+T2HK, improvement is small ($\\approx 12\\%$) compared to the other two off-diagonal CPT-violating LIV parameters. In the case of CPT-conserving LIV parameters, the constraints from DUNE are incomparable to that of T2HK, as the latter shows almost no sensitivity to CPT-conserving LIV parameters. However, adding the data from the two experiments shows marginal improvement in the limits. \n\n\\begin{table}[h!]\n\t\\begin{center}\n\t\t\\begin{adjustbox}{width=0.7\\textwidth}\n\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\\hline\\hline\n\t\t\t\t&DUNE& T2HK & DUNE+T2HK & T2K+NO$\\nu$A\\\\ \n\t\t\t\t\\hline\n\t\t\t\t$|a_{e\\mu}|~[10^{-23}~\\rm {GeV}]$ &  $<$ 1.0 & $<$ 5.15 & $<$ 0.32 & $<$ 6.1\\\\ \n\t\t\t\t\\hline\n\t\t\t\t$|a_{e\\tau}|~[10^{-23}~\\rm {GeV}]$& $<$ 1.05 & $<$ 5.3 & $<$ 0.55& $<$ 7.0\\\\\n\t\t\t\t\\hline\n\t\t\t\t$|a_{\\mu\\tau}|~[10^{-23}~\\rm {GeV}]$& $<$ 1.26 & $<$ 5.5 & $<$ 1.1& $<$ 8.3\\\\\n\t\t\t\t\\hline\n\t\t\t\t$|c_{e\\mu}|~[10^{-24}]$ &  $<$ 0.66 &  $<$ 17.1  &   $<$ 0.64 & $<$ 11.0 \\\\ \n\t\t\t\t\\hline\n\t\t\t\t$|c_{e\\tau}|~[10^{-24}]$& $<$ 1.65 &  $<$ 71.1 &   $<$ 1.49& $<$ 37.5\\\\\n\t\t\t\t\\hline\n\t\t\t\t$|c_{\\mu\\tau}|~[10^{-24}]$& $<$ 0.97 &  $<$ 42.4 &   $<$ 0.95 & $<$ 29.0\\\\\n\t\t\t\t\\hline\\hline\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\mycaption{Expected bounds on the off-diagonal CPT-violating and CPT-conserving LIV parameters at 95\\% C.L. (1 d.o.f.) using DUNE, T2HK, and the combination of DUNE and T2HK. Last column shows the results using the combination of T2K and NO$\\nu$A with their full exposures.  }\n\t\t\\label{tab:constraints_a}\n\t\\end{center}\n\\end{table}\n\nFor a comparison with currently running long-baseline experiments T2K and NO$\\nu$A, in the last column, we provide expected bounds from the combination of T2K and NO$\\nu$A considering their full exposure. We assume a total exposure of 84.4 kt$\\cdot$MW$\\cdot$yrs for T2K~\\cite{T2K:2001wmr, T2K:2011qtm, T2K:2014xyt} with five years of total runtime divided equally in neutrino and anti-neutrino modes. For NO$\\nu$A~\\cite{Ayres:2002ws, NOvA:2004blv, NOvA:2007rmc, Patterson:2012zs}, we consider total exposure of 58.8 kt$\\cdot$MW$\\cdot$yrs with six years of runtime with three years each in neutrino and anti-neutrino mode. We observe that DUNE puts significantly better constraints on both CPT-violating and CPT-conserving LIV parameters than T2K+NO$\\nu$A setup because the former has a larger baseline and better systematic uncertainties. However, the constraints on CPT-violating LIV parameters from T2K+NO$\\nu$A setup are close to that of T2HK. Since NO$\\nu$A has a comparatively larger baseline ($L=810$ km), it has the upper hand in putting stringent bound on the CPT-violating parameters. However, T2HK has less systematics uncertainties to compensate for its small baseline. Similarly, for the CPT-conserving LIV parameters, limits from the T2K+NO$\\nu$A setup are of the same order as T2HK, with the former giving slightly better constraints. It is because, apart from the larger baseline, the energy of the neutrino beam is also prominent for NO$\\nu$A, which plays a vital role in constraining CPT-conserving LIV parameters.\n\nThe fourth column of Table~\\ref{tab:constraints_a} shows the ultimate constraints of off-diagonal CPT-conserving and CPT-violating LIV parameters that would be set from the combination of two next-generation long-baseline experiments DUNE and T2HK at 95\\% C.L. In Table~\\ref{tab:existing_bounds}, we show the existing bounds on some CPT-violating and CPT-conserving LIV parameters from the atmospheric neutrino experiment Super-K and astrophysical neutrino experiment IceCube. Comparing it with our result in Table~\\ref{tab:constraints_a}, we observe that DUNE alone may be able to give better constraints of $\\ensuremath{a_{e\\mu}}$ and $\\ensuremath{a_{e\\tau}}$ as compared to the existing bounds from Super-K. Combining DUNE and T2HK can improve the constraints for $a_{e\\beta}$ ($\\beta=e,\\mu$) by almost one order of magnitude. One reason for this is that $a_{e\\beta}$ ($\\beta=e,\\mu$) are mainly constrained by the appearance channel, which is the most important oscillation channel for a next-generation long-baseline experiment like DUNE, which has a considerably larger baseline. For the atmospheric neutrino experiment like Super-K, the major channel is $\\nu_\\mu\\to\\nu_\\mu$ disappearance channel, in which these two LIV parameters do not appear in the leading order. For $\\ensuremath{a_{\\mu\\tau}}$, projected results from the DUNE+T2HK setup are of the same order as Super-K, with the later having a slightly better limit. In the case of CPT-conserving LIV parameters, we observe that the existing limits from Super-K are at least one order better for $\\ensuremath{c_{e\\mu}}$ and $\\ensuremath{c_{\\mu\\tau}}$ compared to our results for DUNE+T2HK. For $\\ensuremath{c_{e\\tau}}$, limits are comparable, with Super-K having a slightly better limit. This mainly happens because the contribution from CPT-conserving LIV parameters to oscillation probabilities in both the oscillation channels is proportional to both neutrino energy and the baseline. Atmospheric neutrino experiment like Super-K probes a significantly larger range of neutrino energies and baselines as compared to the LBL experiment like DUNE. So we expect Super-K to have better limits on the CPT-conserving LIV parameters.\n\n\\section{Summary and Conclusions}\n\\label{sec:SnC}\n\nIn the past few decades, data from outstanding neutrino oscillation experiments that are either completed or currently operational has almost settled the issue of measuring standard three-flavor neutrino oscillation parameters with excellent precision. Apart from resolving a few remaining issues in the three-neutrino paradigm, another major goal of the next-generation neutrino oscillation experiments will be to search for various physics beyond the standard model, which will open up a new era in particle physics. \nWith that motivation, in this work, we probe the Lorentz invariance violation and its impact on neutrino flavor transition in the context of the two most anticipated upcoming long-baseline experiments, DUNE and T2HK. Lorentz invariance violation can be realized in low energy effective field theories where the LIV interaction terms in the lagrangian come as multiplication of Lorentz violating coefficients and Lorentz violating operators of arbitrary mass dimensions. The coefficients of the dimension-three and dimension-four operators are, respectively, CPT-violating and CPT-conserving. In this work, for the first time, we have explored the CPT-conserving LIV parameters in the context of long-baseline experiments. Here, we focus on the isotropic components of the CPT-violating and CPT-conserving LIV parameters. The presence of non-zero CPT-violating and CPT-conserving parameters modify the neutrino propagation Hamiltonian and hence the oscillation probabilities, making them worth studying in neutrino oscillation experiments.\n\nTo have an analytical understanding about the impact of various LIV parameters on the neutrino oscillation probability, we derive simple approximate analytical expression of the $\\nu_\\mu\\to\\nu_e$ appearance probability and $\\nu_\\mu\\to\\nu_\\mu$ disappearance probability using a perturbative approach up to first order in $\\alpha$, $\\sin^2\\theta_{13}$, and LIV parameters $a_{\\alpha\\beta}\/c_{\\alpha\\beta}$ ($\\alpha,\\beta = e,\\mu,\\tau;\\alpha\\neq\\beta$). We found that for the appearance channel, $a_{e\\mu}\/c_{e\\mu}$ and $a_{e\\tau}\/c_{e\\tau}$ appear at the leading order, whereas for the disappearance channel, only $a_{\\mu\\tau}\/c_{\\mu\\tau}$ presents. Our analytical expressions explain various features of oscillation probabilities shown in Figs.~[\\ref{fig:app_prob}-\\ref{fig:disapp_prob}], where we plot the exact oscillation probabilities numerically. We explain how the impact of LIV on oscillation probabilities depends on the values of phase associated with the off-diagonal LIV parameters with the help of LIV contributing term in the oscillation probability given in Eqs.~\\ref{eq:p_cptv_liv},~\\ref{eq:p_cptc_liv}, and \\ref{eq:Pmm_liv}. Also, we find that the LIV contributing terms in our analytical expression in CPT-violating LIV and CPT-conserving LIV are proportional to $L$ and $L\\times E$, respectively, both in the appearance and disappearance channel. As a result, DUNE, the experiment with a larger baseline and higher energy of the neutrino beam, shows significantly larger sensitivity to LIV than T2HK. As shown in lower panels of  Fig.~\\ref{fig:cptc_app_prob}, T2HK shows negligible sensitivity to the CPT-conserving LIV parameters.\n\nUsing the configuration of the DUNE and T2HK as tabulated in Table~\\ref{tab:exp_details}, we calculate the expected total event rate of the two setups in the standard case and in the presence of off-diagonal CPT-conserving and CPT-violating LIV parameters considered one-at-a-time (see Table~\\ref{tab:total_events}). As expected from the probability analysis, we observe a major change in the event rate from the SI case in the presence of $a_{e\\mu}$ ($a_{e\\tau}$), which shows 41\\% (49\\%) increase in the event rate for DUNE and 7.3\\%(14\\%) for T2HK when we consider $|a_{e\\beta}|=2\\times10^{-23}$ GeV ($\\beta=\\mu,\\tau$). In the disappearance channel, the presence of $a_{\\mu\\tau}$ leads to modifications in the event rate by 7.8\\% for DUNE and 2.2\\% for T2HK. For the CPT-conserving for which we consider the strength $|c_{\\alpha\\beta}|=1\\times 10^{-24}$ shows comparatively small changes in the event rate with maximum 11\\% changes for DUNE and $<1\\%$ for T2HK.\n\nWe discuss the correlation between various LIV parameters and most uncertain oscillation parameters $\\theta_{23}$ and $\\delta_{\\mathrm{CP}}$ (see Sec.~\\ref{subsec:correlation}). To demonstrate this, we show allowed regions at 95\\% C.L. (2 d.o.f.) in  $\\delta_{\\mathrm{CP}}-|a_{\\alpha\\beta}|\/|c_{\\alpha\\beta}|$ (see Fig.~\\ref{fig:correlation_dcp_cptv} and Fig.~\\ref{fig:correlation_dcp_cptc})  and $\\theta_{23}-|a_{\\alpha\\beta}|\/|c_{\\alpha\\beta}|$ (see Fig.~\\ref{fig:correlation_th23_cptv} and Fig.~\\ref{fig:correlation_th23_cptc}) planes. In all cases, DUNE shows a more constrained allowed region in the above-mentioned planes, as expected from the probability plots. We notice almost no correlation between the CPT-conserving LIV parameters and standard oscillation parameters for T2HK, as discussed earlier. For $\\delta_{\\mathrm{CP}}-|a_{e\\mu}|$ case, we observe some allowed regions centered around a non-zero value of $|a_{e\\mu}|$, both for DUNE and T2HK. It happens due to degeneracy between the $\\theta_{23}$ and the phases $\\delta_{\\mathrm{CP}}$ and $\\phi$, which minimizes the contribution from LIV at some non-zero value of the $|a_{e\\mu}|$. Also, in the case of $\\theta_{23}-|a_{e\\mu}|$ correlation, we observe such degenerate regions in opposite octant to the value of $\\theta_{23}$ considered in the data. This also happens due to degeneracy between these three parameters. In both cases, the degenerate allowed regions vanish upon combining the data from DUNE and T2HK. Also, there is improvement in allowed region in both $\\delta_{\\mathrm{CP}}-|a_{\\alpha\\beta}|$ and $\\theta_{23}-|a_{\\alpha\\beta}|$ planes for the DUNE+T2HK setup. For the CPT-conserving case, this improvement is marginal. \n \nFrom the discussion on the correlation between the LIV parameters and standard oscillation parameters $\\delta_{\\mathrm{CP}}$ and $\\theta_{23}$ in DUNE and T2HK, we get an idea about the sensitivity of these two setups to the off-diagonal CPT-conserving and CPT-violating LIV parameters. To quantify the sensitivity, we present the limits (see Fig.~\\ref{fig:cptv_liv_bounds} and Table~\\ref{tab:constraints_a}) on the LIV parameters that DUNE, T2HK, and their combination is expected to set out with their full exposure. For CPT-violating LIV parameters, DUNE shows almost five times better constraints than T2HK.\nAlso, for $|a_{e\\beta}|$ ($\\beta = \\mu,\\tau$), we observe a local minimum in $\\Delta\\chi^2$ which results in deterioration in constraints from DUNE and T2HK. It happens due to the degeneracy between $\\theta_{23}$ and the phases $\\delta_{\\mathrm{CP}}$ and $\\phi$ which is clear from the correlation plots. \nFor the DUNE+T2HK combination, limits improve significantly for $|a_{e\\beta}|$ ($\\beta = \\mu, \\tau$) as the above discussed degeneracies vanish. For $a_{\\mu\\tau}$ also, DUNE outperforms T2HK, and their combination results in a small improvement in the limits. T2HK shows almost no sensitivity for the CPT-conserving LIV parameters. As a result, T2HK produces incomparably worse constraints relative to DUNE. To compare the limits from the future LBL experiments with the currently operating ones, we show the expected constraints by combining T2K and NO$\\nu$A setups with their full exposures (see the last column of Table~\\ref{tab:constraints_a}). We observe for CPT-violating LIV parameters, though bounds from T2K+NO$\\nu$A are worse than DUNE, it is comparable to that of T2HK. It is mainly because of the larger baseline of NO$\\nu$A than T2HK that has better systematic uncertainties than the former. For the CPT-conserving parameters, T2K+NO$\\nu$A gives slightly better constraints than T2HK, again because the former has a longer baseline and higher energy of the neutrino beam. We also compare our results with the existing limits on CPT-violating and CPT-conserving parameters listed in Table~\\ref{tab:existing_bounds} from Super-K. We find that for CPT-violating parameters, especially $a_{e\\beta}$ ($\\beta=\\mu,\\tau$), projected limits at 95\\% C.L. for the DUNE and T2HK combination is better compared to Super-K. It happens because the limits on $a_{e\\beta}$ are mainly driven by the appearance channel, which is a major oscillation channel for LBL experiments, whereas the atmospheric neutrino experiment like Super-K mainly probe disappearance channel. For CPT-conserving parameters, Super-K shows almost two-order better constraints compared to the DUNE+T2HK setup. It is because of the large energy of the neutrinos in atmospheric neutrino experiments.\n\nIn this work, we discuss the impact of LIV on neutrino flavor transition probabilities in the context of next-generation long-baseline experiments DUNE and T2HK and explore the ability of these two setups to constrain the CPT-violating and CPT-conserving LIV parameters. We conclude that although these two setups have good complementarity in their configurations, a smaller baseline and lower energy of the neutrino beam for T2HK make it less favorable to probe LIV as compared to DUNE. However, combining DUNE and T2HK can improve the limits on LIV parameters up to a certain extent. We hope that our present study can be an important addition to the several interesting beyond the Standard Model scenarios which can be probed in the next-generation long-baseline neutrino oscillation experiments.\n\n\\newpage \n\n\\subsubsection*{Acknowledgments}\n\nWe acknowledge the support from the Department of Atomic Energy (DAE), Govt. of India, under the Project Identification Number RIO 4001. S.K.A. is supported by the Young Scientist Research Grant [INSA\/SP\/YSP\/144\/2017\/1578] from the Indian National Science Academy (INSA). S.K.A. acknowledges the financial support from the Swarnajayanti Fellowship (sanction order No. DST\/SJF\/PSA- 05\/2019-20) provided by the Department of Science and Technology (DST), Govt. of India, and the Research Grant (sanction order No. SB\/SJF\/2020-21\/21) provided by the Science and Engineering Research Board (SERB), Govt. of India, under the Swarnajayanti Fellowship project. We thank M. Singh and A. Kumar for useful communications. S.K.A would like to thank the United States-India Educational Foundation for providing the financial support through the Fulbright-Nehru Academic and Professional Excellence Fellowship (Award No. 2710\/F-N APE\/2021). The numerical simulations are carried out using SAMKHYA: High-Performance Computing Facility at Institute of Physics, Bhubaneswar.\n\n\\begin{appendix}\n\t\\renewcommand\\thefigure{A\\arabic{figure}}\n\t\\renewcommand\\theequation{A\\arabic{equation}}\n\t\\setcounter{figure}{0} \n\t\\setcounter{equation}{0}\n\\section{Comparison Between Numerical and Analytical Probabilities}\n\\label{appndx}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.86\\textwidth]{plots\/Probability-Pme.pdf}\n\t\\mycaption{Comparison between exact $\\nu_{\\mu}\\to\\nu_e$ appearance probability calculated numerically (dashed lines) with the same calculated analytically (solid lines) using Eq.~\\ref{eq:pme_liv} for a baseline $L=1285$ km. The top (bottom) row corresponds to the probability in the presence of CPT-violating (CPT-conserving) LIV parameters $a_{e\\beta}$ ($c_{e\\beta}$) with $\\beta=\\mu $ in left column, and $\\beta=\\tau$ in right column. Four colored curves correspond to four values of the associated phase, as mentioned in the legend. The values of the standard oscillation parameters used in the plot are given in Table~\\ref{tab:params_value} with NMO.\\label{fig:Pme-comparison}}\n\\end{figure}\n\nIn this section, we check the accuracy of the approximate analytical expressions of the oscillation probabilities derived in Sec.~\\ref{sec:LIV}. In Fig.~\\ref{fig:Pme-comparison}, we compare the $\\nu_\\mu\\rightarrow\\nu_e$ appearance probability in the presence of $a_{e\\mu}\/c_{e\\mu}$ and $a_{e\\tau}\/c_{e\\tau}$, calculated using the analytical expression with the exact oscillation probability calculated numerically using GLoBES software, for DUNE ($L = 1285$ km). We consider only these LIV parameters since they appear in the first order in the expansion parameters. We show the results for four values of the phase associated with the LIV parameters, namely, $0^\\circ$, $90^\\circ$, $180^\\circ$, and $270^\\circ$. We observe that for the CPT-violating LIV parameters, matching with the exact appearance probability is slightly worse compared to the CPT-conserving case, as the assumed strength of LIV parameters in the CPT-violating case is one order larger than the corresponding CPT-conserving case.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.86\\textwidth]{plots\/Probabilty-Pmm.pdf}\n\t\\mycaption{Comparison between exact $\\nu_\\mu\\to\\nu_{\\mu}$ disappearance probability calculated numerically (dashed lines) with the same calculated analytically (solid lines) using Eq.~\\ref{eq:pme_liv} for a baseline $L=1285$ km. The Left (right) column corresponds to the probability in the presence of CPT-violating (CPT-conserving) LIV parameter $a_{\\mu\\tau}$ ($c_{\\mu\\tau}$). Four colored curves correspond to four values of the associated phase, as mentioned in the legend. The values of the standard oscillation parameters used in the plot are given in Table~\\ref{tab:params_value} with NMO.\\label{fig:Pmm-comparison}}\n\\end{figure}\n\nHowever, the features of the appearance probabilities for different values of LIV-phases are preserved by the analytical expression. In Fig.~\\ref{fig:Pmm-comparison}, we show the same for the $\\nu_\\mu\\to\\nu_\\mu$ disappearance probability. Here, we show the effect from the LIV parameters, $a_{\\mu\\tau}\/c_{\\mu\\tau}$, since only these terms appear till the first order. Again, we consider four values of associated phases as in Fig.~\\ref{fig:Pme-comparison}. We find the oscillation probabilities calculated analytically match quite well with the numerical ones. It is valid for all assumed values of the associated phases, both in CPT-violating and CPT-conserving scenarios. The same is true for T2HK.\n\n\\end{appendix}\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Vanishing residuals and weak convergence}\\label{section:weak}\n\nAn operator $T:\\mathcal{H}\\to \\mathcal{H}$ is {\\it quasi-nonexpansive} if $\\Fix(T)\\neq\\emptyset$ and $\\|Ty-p\\|\\le\\|y-p\\|$ for all $y\\in \\mathcal{H}$ and $p\\in\\Fix(T)$. This implies, in particular, that\n\\begin{equation} \\label{E:quasi-cocoercive}\n\t2\\pint{y-p}{Ty-y}\\le-\\|Ty-y\\|^2\n\\end{equation} \nfor all $y\\in \\mathcal{H}$ and $p\\in\\Fix(T)$. \\\\\n\nIn this section, we consider a family $(T_k)$ of quasi-nonexpansive operators on $\\mathcal{H}$, with $F:=\\bigcap_{k\\ge 1}\\Fix(T_k)\\neq\\emptyset$, along with a sequence $(x_k,y_k)$ satisfying \\eqref{E:Algorithm}, where $(\\alpha_k)$ is a nondecreasing sequence\\footnote{This is just to simplify the proof and is sufficiently general for practical purposes.} in $[0,1)$, and $(\\lambda_k)$ is a sequence in $(0,1)$ such that $\\inf_{k\\ge 1}\\lambda_k>0$. \\\\\n\nTo simplify the notation, given $p\\in F$, we set\n\\begin{equation} \\label{E:def_C_Delta_delta}\n\t\\left\\{\\begin{array}{ccl} \n\t\t\\nu_k & = & \\left(\\lambda_{k}^{-1}-1\\right)\n\t\t\\smallskip  \\\\\n\t\t\\delta_k & = & \\nu_{k-1}(1-\\alpha_{k-1})\\|x_k-x_{k-1}\\|^2, \\smallskip  \\\\\n\t\t\\Delta_k(p) & = & \\|x_k-p\\|^2-\\|x_{k-1}-p\\|^2,\\quad\\Delta_1(p)=0 \\smallskip \\\\\n\t\tC_k(p) & = & \\|x_k-p\\|^2-\\alpha_{k-1}\\|x_{k-1}-p\\|^2+\\delta_k,\\quad C_1(p)=\\|x_1-p\\|^2.\n\t\\end{array} \\right.\n\\end{equation} \n\nThe following auxiliary result will be useful in the sequel:\n\n\\begin{lemma}\\label{L:1}\n\tLet $(T_k)$ be a family of quasi-nonexpansive operators on $\\mathcal{H}$, with $F:=\\bigcap_{k\\ge 1}\\Fix(T_k)\\neq\\emptyset$, and let $(x_k,y_k)$ satisfy \\eqref{E:Algorithm}. For each $k\\ge 1$ and $p\\in F$, we have\n\t\\begin{equation} \\label{E:proof5}\n\t\t\\Delta_{k+1}(p)+\\delta_{k+1}+\\nu_k\\alpha_k\\|x_{k+1}-2x_k+x_{k-1}\\|^2 \\le \\alpha_k\\Delta_k(p) + \\big[\\alpha_k(1+\\alpha_k)+\\nu_k\\alpha_k(1-\\alpha_k)\\big]\\|x_k-x_{k-1}\\|^2.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\tTake $p\\in F$. From \\eqref{E:Algorithm}, it follows that\n\t\\begin{equation} \\label{E:proof1}\n\t\t\\|x_{k+1}-p\\|^2 = \\|y_k-p\\|^2 +\\lambda_k^2\\|y_k-T_ky_k\\|^2+2\\lambda_k\\pint{y_k-p}{T_ky_k-y_k} \\le \\|y_k-p\\|^2 -\\lambda_k(1-\\lambda_k)\\|y_k-T_ky_k\\|^2,\n\t\\end{equation} \n\twhere the inequality is given by \\eqref{E:quasi-cocoercive}. From\n\t$$\\|y_k-p\\|^2 = \\|x_k-p+\\alpha_k(x_k-x_{k-1})\\|^2 = \\|x_k-p\\|^2+\\alpha_k^2\\|x_k-x_{k-1}\\|^2+2\\alpha_k\\pint{x_k-p}{x_k-x_{k-1}}$$\n\tand\n\t$$2\\alpha_k\\pint{x_k-p}{x_k-x_{k-1}} = \\alpha_k\\|x_k-p\\|^2+\\alpha_k\\|x_k-x_{k-1}\\|^2-\\alpha_k\\|x_{k-1}-p\\|^2,$$\n\twe deduce that\n\n\t\n\t\n\t\n\t\\begin{equation} \\label{E:proof2}\n\t\t\\|y_k-p\\|^2 = (1+\\alpha_k)\\|x_k-p\\|^2+\\alpha_k(1+\\alpha_k)\\|x_k-x_{k-1}\\|^2-\\alpha_k\\|x_{k-1}-p\\|^2.\n\t\\end{equation} \n\tBy combining expressions \\eqref{E:proof1} and \\eqref{E:proof2}, we obtain\n\n\n\n\t$$\n\t\\|x_{k+1}-p\\|^2 \\le (1+\\alpha_k)\\|x_k-p\\|^2+\\alpha_k(1+\\alpha_k)\\|x_k-x_{k-1}\\|^2 -\\alpha_{k}\\|x_{k-1}-p\\|^2-\\lambda_k(1-\\lambda_k)\\|y_k-T_ky_k\\|^2.\n\t$$\n\n\tRecalling from \\eqref{E:def_C_Delta_delta} that $\\Delta_k(p)=\\|x_k-p\\|^2-\\|x_{k-1}-p\\|^2$, we rewrite the latter as\n\n\t\n\t\n\t\\begin{equation} \\label{E:proof3}\n\t\t\\Delta_{k+1}(p) \\le \\alpha_k\\Delta_k(p) + \\alpha_k(1+\\alpha_k)\\|x_k-x_{k-1}\\|^2 -\\lambda_k(1-\\lambda_k)\\|y_k-T_ky_k\\|^2.\n\t\\end{equation}\n\tIn turn,\n\t\\begin{equation} \\label{E:proof3.5}\n\t\t\\lambda_k^2\\|y_k-T_ky_k\\|^2 = \\|x_{k+1}-x_k\\|^2+\\alpha_k^2\\|x_k-x_{k-1}\\|^2-2\\alpha_k\\pint{x_{k+1}-x_k}{x_k-x_{k-1}},\n\t\\end{equation}\n\tand\n\t$$-2\\alpha_k\\pint{x_{k+1}-x_k}{x_k-x_{k-1}} = \\alpha_k\\|x_{k+1}-2x_k+x_{k-1}\\|^2-\\alpha_k\\|x_{k+1}-x_k\\|^2-\\alpha_k\\|x_k-x_{k-1}\\|^2,$$\n\ttogether give\n\t$$\\lambda_k^2\\|y_k-T_ky_k\\|^2 = (1-\\alpha_k)\\|x_{k+1}-x_k\\|^2-\\alpha_k(1-\\alpha_k)\\|x_k-x_{k-1}\\|^2+\\alpha_k\\|x_{k+1}-2x_k+x_{k-1}\\|^2.$$\n\tBy multiplying the latter by $\\nu_k=(1-\\lambda_k)\/\\lambda_k$, and using the definition of $\\delta_k$ in \\eqref{E:def_C_Delta_delta}, we rewrite this as\n\t\\begin{equation} \\label{E:proof34}\n\t\t\\delta_{k+1}+\\nu_k\\alpha_k\\|x_{k+1}-2x_k+x_{k-1}\\|^2=\\nu_k\\alpha_k(1-\\alpha_k)\\|x_k-x_{k-1}\\|^2 +\\lambda_k(1-\\lambda_k)\\|y_k-T_ky_k\\|^2.\n\t\\end{equation}\n\tSumming \\eqref{E:proof3} and \\eqref{E:proof34}, we obtain \\eqref{E:proof5}.\n\\end{proof}\n\nWe are now in a position to show that the sequence $(x_n)$ remains anchored to the set $F$, while both the residuals $\\|y_k-T_ky_k\\|$ and the speed $\\|x_k-x_{k-1}\\|$ tend to 0 faster than $1\/\\sqrt{k}$, as occurs with standard KM iterations.\n\n\\begin{proposition} \\label{P:1}\n\tLet $(T_k)$ be a family of quasi-nonexpansive operators on $\\mathcal{H}$, and let $(x_k,y_k)$ satisfy \\eqref{E:Algorithm}. Take $p\\in F=\\bigcap_{k\\ge 1}\\Fix(T_k)$.\n\t\\begin{itemize}\n\t\t\\item [i)] Assume that there is $k_0\\ge 1$ such that\n\t\t\\begin{equation} \\label{E:H:1_0}\n\t\t\t\\alpha_k(1+\\alpha_k)+(\\lambda_k^{-1}-1)\\alpha_k(1-\\alpha_k)-(\\lambda_{k-1}^{-1}-1)(1-\\alpha_{k-1}) \\le 0\n\t\t\\end{equation}\n\t\tfor all $k\\ge k_0$. Then, the sequence $\\big(C_k(p)\\big)_{k\\ge k_0}$ is nonincreasing and nonnegative, thus $\\lim\\limits_{k\\to\\infty}C_k(p)$ exists.\n\t\t\\item [ii)] Assume further that\n\t\t\\begin{equation} \\label{E:H:1}\n\t\t\t\\limsup_{k\\to\\infty}\\big[\\alpha_k(1+\\alpha_k)+(\\lambda_k^{-1}-1)\\alpha_k(1-\\alpha_k)-(\\lambda_{k-1}^{-1}-1)(1-\\alpha_{k-1})\\big]<0.\n\t\t\\end{equation}\n\t\tThen, the series \n\t\t$\\sum\\limits_{k\\ge 1} \\alpha_k\\|x_{k+1}-2x_k+x_{k-1}\\|^2$, $\\sum\\limits_{k\\ge 1}\\|x_k-x_{k-1}\\|^2$, $\\sum\\limits_{k\\ge 1}\\delta_k$ and $\\sum\\limits_{k\\ge 1}\\|y_k-T_ky_k\\|^2$ are convergent. Moreover, $\\lim\\limits_{k\\to\\infty}\\|x_k-p\\|$ exists, and $$\\lim\\limits_{k\\to\\infty}k\\|y_k-T_ky_k\\|^2=\\lim\\limits_{k\\to\\infty}k\\|x_k-x_{k-1}\\|^2=0.$$\n\t\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\n\tWith either \\eqref{E:H:1_0} or \\eqref{E:H:1}, there exist $\\varepsilon\\ge 0$ and $k_0\\ge 1$ such that\t\n\t\\begin{equation} \\label{E:H:1_bis}\n\t\\alpha_k(1+\\alpha_k)+(\\lambda_k^{-1}-1)\\alpha_k(1-\\alpha_k) \\le (\\lambda_{k-1}^{-1}-1)(1-\\alpha_{k-1})-\\varepsilon\n\t\\end{equation}\t\n\tfor all $k\\ge k_0$ (if \\eqref{E:H:1} holds, then $\\varepsilon>0$; otherwise, $\\varepsilon=0$). Without any loss of generality, we may assume that $k_0=1$. Combining this with \\eqref{E:proof5}, we obtain\n\t\\begin{eqnarray} \n\t\t\\Delta_{k+1}+\\delta_{k+1}+\\nu_k\\alpha_k\\|x_{k+1}-2x_k+x_{k-1}\\|^2 & \\le & \\alpha_k\\Delta_k + \\big[\\nu_{k-1}(1-\\alpha_{k-1})-\\varepsilon\\big]\\|x_k-x_{k-1}\\|^2 \\nonumber \\\\\n\t\t& = & \\alpha_k\\Delta_k +\\delta_k-\\varepsilon\\|x_k-x_{k-1}\\|^2. \\label{E:proof6}\n\t\\end{eqnarray}\n\tOn the one hand, \\eqref{E:proof6} immediately gives \n\t\\begin{equation} \\label{E:L:1_1}\n\t\t\\Delta_{k+1}  \\le  \\alpha_k\\Delta_k+\\delta_k. \n\t\\end{equation}\n\tOn the other, since $(\\alpha_k)$ is nondecreasing, we have \n\t\\begin{eqnarray*}\n\t\tC_{k+1}(p)-C_k(p) = \\Delta_{k+1}-\\big(\\alpha_k\\|x_k-p\\|^2-\\alpha_{k-1}\\|x_{k-1}-p\\|^2\\big) +\\delta_{k+1}-\\delta_k \\le \\Delta_{k+1}+\\delta_{k+1}-\\alpha_k\\Delta_k -\\delta_k.\n\t\\end{eqnarray*}\n\tTherefore, \\eqref{E:proof6} implies\n\t\\begin{equation} \\label{E:L:1_2}\n\t\tC_{k+1}(p)+\\nu_k\\alpha_k\\|x_{k+1}-2x_k+x_{k-1}\\|^2+\\varepsilon\\|x_k-x_{k-1}\\|^2 \\le C_k(p).\n\t\\end{equation}\n\tIt ensues that $\\big(C_k(p)\\big)$ is nonincreasing. To show that it is nonnegative, suppose that $C_{k_1}(p)<0$ for some $k_1\\ge 1$. Since $\\big(C_k(p)\\big)$ is nonincreasing, \n\t$$\\|x_k-p\\|^2-\\alpha_{k-1}\\|x_{k-1}-p\\|^2\\le C_k(p)\\le C_{k_1}(p)<0$$ \n\tfor all $k\\ge k_1$. If follows that $\\|x_k-p\\|^2 \\le \\|x_{k-1}-p\\|^2+ C_{k_1}(p)$, and so\n\t$$0\\le \\|x_k-p\\|^2\\le\\|x_{k-1}-p\\|^2+C_{k_1}(p) \\le \\dots \\le \\|x_{k_1}-p\\|^2+(k-k_1)C_{k_1}(p)$$\n\tfor all $k\\ge k_1$, which is impossible. As a consequence $\\big(C_k(p)\\big)$ is nonnegative, and $\\lim\\limits_{k\\to\\infty}C_k(p)$ exists. \\\\\n\tFor ii), Inequality \\eqref{E:H:1_bis} holds with $\\varepsilon>0$. The summability of the first two series follows at once from \\eqref{E:L:1_2}. The third one is a consequence of the second one, since $\\inf_{k\\ge 1}\\lambda_k>0$. For the last one, we use \\eqref{E:proof3.5}. \\\\\n\tNow, denoting the positive part of $d\\in\\mathbb{R}$ by $[d]_+$, we obtain from \\eqref{E:L:1_1} that\n\t$$[\\Delta_{k+1}]_+  \\le  \\alpha[\\Delta_k]_++\\delta_k,$$\n\twhere $\\alpha:=\\sup_{k\\ge 1}\\alpha_k<1$ by virtue of \\eqref{E:H:1}. Iterating this inequality, we deduce that\n\t$$\\big[\\Delta_{j+1}\\big]_+\\le\\alpha^j\\big[\\Delta_1\\big]_++\\sum_{k=1}^j\\alpha^{j-k}\\delta_k,$$\n\tand so\n\t$$\\sum_{j=1}^\\infty\\big[\\Delta_{j+1}\\big]_+\\le\\frac{1}{1-\\alpha}\\left[\\big[\\Delta_1\\big]_++\\sum_{j=1}^\\infty\\delta_j\\right]<\\infty.$$\n\tNow, write $h_k=\\|x_k-p\\|^2-\\sum_{j=1}^k\\big[\\Delta_{j}\\big]_+$. We have $$h_{k+1}-h_k=\\Delta_{k+1}-\\big[\\Delta_{k+1}\\big]_+\\le 0,$$\n\tfrom which we conclude that $\\lim\\limits_{k\\to\\infty}\\|x_k-p\\|$ exists. Finally, from \\eqref{E:proof6}, we deduce that\n\t$$[\\Delta_{k+1}]_+ +\\delta_{k+1}  \\le  [\\Delta_k]_+ +\\delta_k.$$\n\tWe have already shown that \n\t$$\\sum_{k=1}^\\infty\\big([\\Delta_k]_+ +\\delta_k\\big)<\\infty.$$\n\tAn elementary property of real sequences (see Lemma \\ref{L:small_o} below) implies that \n\t$\\lim\\limits_{k\\to\\infty}k\\,[\\Delta_k]_+=\\lim\\limits_{k\\to\\infty}k\\,\\delta_k=0$, which allows us to conclude.\n\\end{proof}\n\n\\begin{remark} \\label{R:1}\n\tInequalities \\eqref{E:H:1_0} and \\eqref{E:H:1} are closely related, but different, from the hypotheses used in \\cite{AttouchCabot2019} for forward-backward iterations. In the non-inertial case $\\alpha=0$, \\eqref{E:H:1} is just $\\limsup_{k\\to\\infty}\\lambda_k<1$. On the other hand, since $(\\alpha_k)$ is nondecreasing and bounded, we have $\\alpha_k\\to\\alpha\\in[0,1]$. If $\\lambda_k\\to \\lambda$, then \\eqref{E:H:1} is reduced to\n\t\\begin{equation} \\label{E:H2_constant}\n\t\t\\lambda(1-\\alpha+2\\alpha^2)<(1-\\alpha)^2.\n\t\\end{equation} \n\tFor each $\\alpha\\in[0,1)$, there is $\\lambda_\\alpha>0$ such that \\eqref{E:H2_constant} holds for all $\\lambda<\\lambda_\\alpha$. \n\\end{remark}\n\nIn order to prove the weak convergence of the sequences generated by Algorithm \\eqref{E:Algorithm}, we shall use the following nonautonomous extension of the concept of demiclosedness. \\\\\n\nThe family of operators $(I-T_k)$ is {\\it asymptotically demiclosed at $0$} if for every sequence $(z_k)$ such that $z_k\\rightharpoonup z$ and $z_k-T_kz_k\\to 0$, we must have $z\\in F=\\bigcap_{k\\ge 1}\\Fix(T_k)$. \\\\\n\nOf course, if $T:\\mathcal{H}\\to \\mathcal{H}$ is nonexpansive and $T_k\\equiv T$, then $I-T_k$ is asymptotically demiclosed. We shall discuss other examples in the next section.\n\n\\begin{theorem} \\label{T:1}\n\tLet $(T_k)$ be a family of quasi-nonexpansive operators on $\\mathcal{H}$, with $F=\\bigcap_{k\\ge 1}\\Fix(T_k)\\neq\\emptyset$. Let $(x_k,y_k)$ satisfy \\eqref{E:Algorithm}, and assume \\eqref{E:H:1} holds. If $(I-T_k)$ is asymptotically demiclosed at $0$, then both $x_k$ and $y_k$ converge weakly, as $k\\to\\infty$, to a point in $F$.\n\\end{theorem}\n\n\\begin{proof}\n\tRecall that $\\lim\\limits_{k\\to\\infty}\\|y_k-T_ky_k\\|=\\lim\\limits_{k\\to\\infty}\\|x_k-x_{k-1}\\|=0$, by part ii) of Proposition \\ref{P:1}. From \\eqref{E:Algorithm}, we deduce that $(y_k)$ and $(x_k)$ have the same (weak and strong) limit points. Suppose $x_{n_k}\\rightharpoonup x$. Then, $y_{n_k}\\rightharpoonup x$ as well. Since $y_{n_k}-T_ky_{n_k}\\to 0$, the asymptotic demiclosedness implies $x\\in F$. Opial's Lemma \\cite{opial1967weak} (see, for instance, \\cite[Lemma 5.2]{peypouquet2015convex}) yields the conclusion. \n\\end{proof}\n\t\nIn the proof of Proposition \\ref{P:1}, we used the following result concerning real sequences:\n\n\\begin{lemma} \\label{L:small_o}\nLet $(\\zeta_k)$ be a nonincreasing sequence of positive numbers such that $\\sum\\limits_{k\\ge 1}\\zeta_k<\\infty$. Then $\\lim\\limits_{k\\to\\infty}\\big[k\\,\\zeta_k\\big]=0$.\n\\end{lemma}\n\n\\begin{proof}\nOn the one hand, since $(\\zeta_k)$ is nonincreasing, for each $k\\ge 1$, we have\n$$k\\,\\zeta_k-(k-1)\\,\\zeta_{k-1}=\\zeta_k+(k-1)(\\zeta_k-\\zeta_{k-1})\\le\\zeta_k.$$\nBut $k\\,\\zeta_k\\ge 0$ and the right-hand side is summable, so $\\lim\\limits_{k\\to\\infty}\\big[k\\,\\zeta_k\\big]$ exists. On the other hand, since\n$$\\sum_{k\\ge 1}\\frac{1}{k}\\big[k\\,\\zeta_k\\big]=\\sum_{k\\ge 1}\\zeta_k<\\infty\\qquad\\hbox{and}\\qquad \\sum_{k\\ge 1}\\frac{1}{k}=\\infty,$$\nwe must have $\\lim\\limits_{k\\to\\infty}\\big[k\\,\\zeta_k\\big]=0$.  \n\\end{proof}\n\t\n\n\n\\section{Strong and linear convergence}\\label{section:strong}\n\n\nWe now focus on the strong convergence of the sequences generated by \\eqref{E:Algorithm}, and their convergence rate. As before, we assume that $(\\alpha_k)$ is nondecreasing but we do not assume, in principle, that $\\inf_{k\\ge 1}\\lambda_k>0$. \\\\\n\nGiven $q\\in(0,1)$, an operator $T:\\mathcal{H}\\to \\mathcal{H}$ is $q$-{\\it quasi-contractive} if $\\Fix(T)\\neq\\emptyset$ and $\\|Ty-p\\|\\le q\\|y-p\\|$ for all $y\\in \\mathcal{H}$ and $p\\in\\Fix(T)$. If $T$ is $q$-quasi-contractive, then $\\Fix(T)=\\{p^*\\}$.\n\nGiven $\\lambda,q\\in(0,1)$ and $\\xi\\in[0,1]$, we define    \\begin{equation} \\label{E:Q}\n    Q(\\lambda,q,\\xi):=\\xi\\big(1-\\lambda +\\lambda q^2\\big)+(1-\\xi)(1-\\lambda+\\lambda q)^2=(1-\\lambda+\\lambda q)^2+\\xi\\lambda(1-\\lambda)(1-q)^2.\n\\end{equation}\n\nNotice that $Q(\\lambda,q,\\xi)$ decreases as $\\lambda$ increases, or as either $q$ or $\\xi$ decreases. The quantity $Q(\\lambda,q,\\xi)$ will play a crucial role in the linear convergence rate of the sequences satisfying \\eqref{E:Algorithm}. The inclusion of the auxiliary parameter $\\xi$ will also allow us to establish convergence rates, with and without inertia, in a unified manner (see the discussion in Subsection \\ref{SS:remarks}).\t\\\\\n\nThe following result establishes a bound on the distance to a solution after performing a standard KM step:\n\n\\begin{lemma}\n    Let $T:\\mathcal{H}\\to \\mathcal{H}$ be $q$-quasi-contractive with fixed point $p^*$, and let $x,y\\in \\mathcal{H}$ and $\\lambda>0$ be such that $x=(1-\\lambda)y+\\lambda Ty$. Then, for each $\\xi\\in[0,1]$, we have\n    \\begin{equation} \\label{E:Lemma2_1}\n    \\|x-p^*\\|^2 \\le Q(\\lambda,q,\\xi)\\|y-p^*\\|^2-\\xi\\lambda(1-\\lambda)\\|Ty-y\\|^2.\n    \\end{equation}\n\\end{lemma}\t\n\t\n\\begin{proof}\nOn the one hand, the equalities\n$$\\begin{array}{rcccl}\n\\|x-p^*\\|^2 & = & \\|y-p^*+\\lambda(Ty-y)\\|^2 & = & \\|y-p^*\\|^2+\\lambda^2\\|Ty-y\\|^2+2\\lambda\\pint{y-p^*}{Ty-y} \\smallskip\\\\\n-\\lambda\\|Ty-p^*\\|^2 & = & -\\lambda\\|Ty-y+y-p^*\\|^2 & = & -\\lambda\\|Ty-y\\|^2-\\lambda\\|y-p^*\\|^2-2\\lambda\\pint{y-p^*}{Ty-y},\n\\end{array}$$\ntogether give\n\\begin{eqnarray} \\label{E:Lemma2_P1}\n\\|x-p^*\\|^2 & = & (1-\\lambda)\\|y-p^*\\|^2+\\lambda\\|Ty-p^*\\|^2-\\lambda(1-\\lambda)\\|Ty-y\\|^2 \\nonumber\\\\\n& \\le &  \\big(1-\\lambda +\\lambda q^2\\big)\\|y-p^*\\|^2-\\lambda(1-\\lambda)\\|Ty-y\\|^2.\n\\end{eqnarray}\nOn the other hand, we have\n\\begin{equation} \\label{E:Lemma2_P2}\n\\|x-p^*\\|=\\|(1-\\lambda)(y-p^*)+\\lambda(Ty-p^*)\\|\\le(1-\\lambda)\\|y-p^*\\|+\\lambda\\|Ty-p^*\\|\\le (1-\\lambda+\\lambda q)\\|y-p^*\\|.\n\\end{equation}\nThen, inequality \\eqref{E:Lemma2_1} is just a convex combination of \\eqref{E:Lemma2_P1} and the square of \\eqref{E:Lemma2_P2}.\n\\end{proof}\t\n\n\\subsection{Convergence analysis}\n\nWe now turn to the convergence of the sequences verifying \\eqref{E:Algorithm}.\tTo simplify the notation, for each $k\\in\\mathbb{N}$, we set\n$$\\tilde C_k(p)  =  \\|x_k-p^*\\|^2-\\alpha_{k-1}\\|x_{k-1}-p^*\\|^2+\\xi\\delta_k\\quad\\hbox{with}\\quad \\tilde C_1(p^*)=\\|x_1-p^*\\|^2.$$\n\nWe have the following:\n\t\n\\begin{proposition}\n    Let $(T_k)$ be a sequence of operators on $\\mathcal{H}$, such that $\\Fix(T_k)\\equiv \\{p^*\\}$ and $T_k$ is $q_k$-quasi-contractive for each $k\\in\\mathbb{N}$. Let $(x_k,y_k)$ satisfy \\eqref{E:Algorithm}, and let $\\xi\\in [0,1]$. Write $Q_k=Q(\\lambda_k,q_k,\\xi)$, where $Q$ is defined in \\eqref{E:Q}. For each $k\\in\\mathbb{N}$, we have\n    \\begin{eqnarray} \\label{E:Lemma3}\n\t\\|x_{k+1}-p^*\\|^2 +\\xi\\delta_{k+1} & \\le & Q_k\\left[ (1+\\alpha_k)\\|x_k-p^*\\|^2-\\alpha_k\\|x_{k-1}-p^*\\|^2\\right] \\nonumber \\medskip \\\\\n\t&&\\, +\\big[Q_k\\alpha_k(1+\\alpha_k)+\\xi\\nu_k\\alpha_k(1-\\alpha_k)\\big]\\|x_k-x_{k-1}\\|^2.\n    \\end{eqnarray}  \n\tIf, moreover,\n\t\\begin{equation} \\label{E:H:2}\n\t\tQ_k\\alpha_k(1+\\alpha_k)+\\xi\\nu_k\\alpha_k(1-\\alpha_k)-\\xi Q_k\\nu_{k-1}(1-\\alpha_{k-1})\\le 0\n\t\\end{equation}\n\tfor all $k\\in\\mathbb{N}$, then\n\t\\begin{equation} \\label{E:Lemma3_1}\n\t\\tilde C_{k+1}(p^*)\\le \\left[\\prod_{j=1}^{k}Q_j\\right]\\|x_1-p^*\\|^2\n\t\\end{equation}\n\tand\n\t\\begin{equation} \\label{E:Lemma3_2}\n\t\\|x_{k+1}-p^*\\|^2 \\le \\left[\\alpha^k\n\t+ \\sum_{j=1}^k\\alpha^{k-j}\\left[\\prod_{i=1}^{j}Q_i\\right]\n\t\\right]\\|x_1-p^*\\|^2.\n\t\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n\tWe use \\eqref{E:Algorithm} and \\eqref{E:Lemma2_1} to obtain\n\t$$\\|x_{k+1}-p^*\\|^2=Q_k\\|y_k-p^*\\|^2 -\\xi\\lambda_k(1-\\lambda_k)\\|y_k-T_ky_k\\|^2.$$\n\n\tNow, by \\eqref{E:proof2}, we deduce that\n\t$$\\|x_{k+1}-p^*\\|^2 \\le Q_k\\left[ (1+\\alpha_k)\\|x_k-p^*\\|^2+\\alpha_k(1+\\alpha_k)\\|x_k-x_{k-1}\\|^2-\\alpha_k\\|x_{k-1}-p^*\\|^2\\right]-\\xi\\lambda_k(1-\\lambda_k)\\|y_k-T_ky_k\\|^2$$\n\tOn the other hand, from \\eqref{E:proof34}, we get\n\t$$\\xi\\delta_{k+1} \\le \\xi \\nu_k\\alpha_k(1-\\alpha_k)\\|x_k-x_{k-1}\\|^2 +\\xi \\lambda_k(1-\\lambda_k)\\|y_k-T_ky_k\\|^2,$$\n\tand the last two inequalities together imply \\eqref{E:Lemma3}. For the second part, inequalities \\eqref{E:Lemma3} and \\eqref{E:H:2} together give\n\t$$\\|x_{k+1}-p^*\\|^2 +\\xi\\delta_{k+1} \\le Q_k\\left[ (1+\\alpha_k)\\|x_k-p^*\\|^2-\\alpha_k\\|x_{k-1}-p^*\\|^2\\right]+\\xi Q_k\\delta_k.$$\t\n\tSubtracting $\\alpha_k\\|x_k-p^*\\|^2$, we are left with\n\t\\begin{eqnarray*} \n\t\t\\tilde C_{k+1}(p^*) & \\le & \\big(Q_k(1+\\alpha_k)-\\alpha_k\\big)\\|x_k-p^*\\|^2-\\alpha_k Q_k\\|x_{k-1}-p^*\\|^2+\\xi Q_k \\delta_k \\\\\n\t\t& \\le & Q_k\\|x_k-p^*\\|^2-Q_k\\alpha_{k-1}\\|x_{k-1}-p^*\\|^2+\\xi Q_k\\delta_k \\\\\n\t\t& = & Q_k\\tilde C_k(p^*).\n\t\\end{eqnarray*} \n\tThis gives \\eqref{E:Lemma3_1}, recalling that $\\tilde C_1(p^*)=\\|x_1-p^*\\|^2$.\n\tNow, since $\\|x_{k+1}-p^*\\|^2-\\alpha_{k}\\|x_{k}-p^*\\|^2\\le \\tilde C_{k+1}(p^*)$, we have \n\n\t$$\\|x_{k+1}-p^*\\|^2 \\le \\alpha_{k}\\|x_{k}-p^*\\|^2+\\left[\\prod_{j=1}^{k}Q_j\\right]\\|x_1-p^*\\|^2 \\le \\alpha\\|x_{k}-p^*\\|^2+\\left[\\prod_{j=1}^{k}Q_j\\right]\\|x_1-p^*\\|^2,$$\n\twhich we then iterate to obtain \\eqref{E:Lemma3_2}.\n\\end{proof}\n\nThe preceding estimations allow us to establish the main result of this section, namely:\n\n\\begin{theorem} \\label{T:2}\n\t Let $(T_k)$ be a sequence of operators on $H$, such that $\\Fix(T_k)\\equiv \\{p^*\\}$ and $T_k$ is $q_k$-quasi-contractive for each $k\\in\\mathbb{N}$. Let $(x_k,y_k)$ satisfy \\eqref{E:Algorithm}, and let $\\xi\\in [0,1]$. Write $Q_k=Q(\\lambda_k,q_k,\\xi)$, and assume that \\eqref{E:H:2} holds for all $k\\in\\mathbb{N}$. We have the following:\n\t\\begin{itemize}\n\t    \\item [i)] If $\\sum_{k=1}^\\infty\\lambda_k(1-q_k^2)=\\infty$, then $x_k$ converges strongly to $p^*$, as $k\\to\\infty$.\n\t    \\item [ii)] If $\\lambda_k\\ge\\lambda>0$ and $q_k\\le q<1$ for all $k\\in\\mathbb{N}$, then $x_k$ converges linearly to $p^*$, as $k\\to\\infty$. More precisely, \n\t\\begin{equation} \\label{E:rate}\n\t\\|x_k-p^*\\|^2 \\le\n\t\\left[ \\frac{\\alpha^{k+1}-Q(\\lambda,q,\\xi)^{k+1}}{\\alpha-Q(\\lambda,q,\\xi)}\\right]\\|x_1-p^*\\|^2=\\mathcal O\\left(Q(\\lambda,q,\\xi)^{k}\\right). \n\t\\end{equation} \n\t\\end{itemize}\n\\if{ \t$$\\|x_{k+1}-p^*\\|^2 \\le \\alpha^k\\left[1+ \\sum_{j=1}^k\\left[\\frac{1-\\lambda(1-q^2)}{\\alpha}\\right]^j\\right]\\|x_1-p^*\\|^2,$$\n\twhere $\\lambda=\\inf_{k\\ge 1}\\lambda_k$. If we set $\\beta_+=\\max\\{\\alpha,1-\\lambda(1-q^2)\\}$ and $\\beta_-=\\min\\{\\alpha,1-\\lambda(1-q^2)\\}$, then\n\t\\begin{equation} \\label{E:rate}\n\t\\|x_k-p^*\\|^2\\le \\frac{2}{\\beta_+-\\beta_-}\\left(\\beta_+^k-\\beta_-^k\\right)\\|x_1-p^*\\|^2=\\mathcal O\\left(\\beta_+^k\\right).\n\t\\end{equation} \n}\\fi \n\\end{theorem}\n\t\n\\begin{proof}\n    For part i), observe that if $\\sum_{k=1}^\\infty\\lambda_k(1-q_k^2)=\\infty$, then $\\prod_{k=1}^{\\infty}Q_k=0$. By \\eqref{E:Lemma3_1}, $\\lim_{k\\to\\infty}\\tilde C_k(p^*)=0$. As in the proof of Proposition \\ref{P:1}, we can show that the sum of the first two terms in $\\tilde C_k(p^*)$, namely $\\|x_k-p^*\\|^2-\\alpha_{k-1}\\|x_{k-1}-p^*\\|^2$, is nonnegative. Therefore, $\\lim_{k\\to\\infty}\\left[\\|x_k-p^*\\|^2-\\alpha_{k-1}\\|x_{k-1}-p^*\\|^2\\right]=0$. If $\\alpha_k\\equiv 0$, the conclusion is straightforward. Otherwise, given any $\\varepsilon>0$, there is $K\\in\\mathbb{N}$ such that \n    $$\\|x_k-p^*\\|^2\\le\\alpha_{1}\\|x_{k-1}-p^*\\|^2+\\varepsilon$$\n    for all $k\\ge K$, since $\\alpha_k$ is nondecreasing. This implies\n    $$\\|x_k-p^*\\|^2\\le\\alpha_{1}^{k-K}\\|x_{K}-p^*\\|^2+\\varepsilon(1-\\alpha_1)^{-1},$$\n    so that $\\limsup_{k\\to\\infty}\\|x_k-p^*\\|\\le\\varepsilon(1-\\alpha_1)^{-1}$, and the conclusion follows. \\\\\n    For ii), we know that $Q(\\lambda_k,q_k,\\xi)\\le Q(\\lambda,q,\\xi)$, because $Q$ increases either if $\\lambda$ decreases, and also if $q$ increases. Gathering the common factors in the second and third terms on the left-hand side of inequality \\eqref{E:H:2}, we deduce that $Q\\ge \\alpha$ (strictly if $\\alpha>0$). Using \\eqref{E:Lemma3_2}, and observing that the case $Q(\\lambda,q,\\xi)=\\alpha$ is incompatible with inequality \\eqref{E:H:2}, we deduce that\n    $$\\|x_{k+1}-p^*\\|^2 \\le \\alpha^k\\left[ \\sum_{j=0}^k\\left(\\frac{Q(\\lambda,q,\\xi)}{\\alpha}\\right)^j\\right]\\|x_1-p^*\\|^2=\\left[ \\frac{\\alpha^{k+1}-Q(\\lambda,q,\\xi)^{k+1}}{\\alpha-Q(\\lambda,q,\\xi)}\\right]\\|x_1-p^*\\|^2,$$\n    as claimed.\n\\end{proof}\n\n\n\n\\subsection{Some insights into inequality \\eqref{E:H:2} and the convergence rate} \\label{SS:remarks}\n\nTo fix the ideas, we comment on some special cases of inequality \\eqref{E:H:2}, especially with constant parameters:\n\n\\begin{enumerate}\n    \\item In the non-inertial case $\\alpha_k\\equiv 0$, \\eqref{E:H:2} holds if either $\\xi=0$ or $\\lambda_k\\le 1$ for all $k$, as in \\eqref{E:H:1_0}. This is less restrictive than \\eqref{E:H:1} (see Remark \\ref{R:1}). If we take $\\xi=0$, the best convergence rate is\n    $$\\|x_k-p^*\\|=\\mathcal O\\big(q^k\\big),$$\n    which is obtained from Theorem \\ref{T:2} with $\\lambda_k\\equiv 1$.\n    If $\\alpha_k>0$ for at least one $k$, the case $\\xi=0$ is ruled out. From a theoretical perspective, this seems to be an argument {\\it against} the use of inertia, since\n    $$q^2\\le(1-\\lambda+\\lambda q)^2= Q(\\lambda,q,0)\\le Q(\\lambda,q,\\xi)\\le Q(\\lambda,q,1)= 1-\\lambda+\\lambda q^2.$$\n    However, the results above correspond to worst-case scenarios, and need not be representative of concrete instances found in practice, in which inertia does improve the theoretical convergence rate guarantees (see Subsection \\ref{S:Gradient} below, and the commented references). Moreover, the numerical tests reported below show noticeable improvements in the performance of the selected algorithms. \n    \n    \\item In the limiting case $q_k\\equiv 1$, we have $Q_k\\equiv 1$. With constant parameters $\\lambda_k\\equiv \\lambda$, $\\alpha_k\\equiv\\alpha$, \\eqref{E:H:2} becomes\n    $$\\lambda\\alpha(1+\\alpha)-\\xi(1-\\lambda)(1-\\alpha)^2\\le 0.$$\n    If \n    \\begin{equation} \\label{E:feasibility} \\frac{\\alpha\\lambda(1+\\alpha)}{(1-\\lambda)(1-\\alpha)^2}\\le  1,\n    \\end{equation}\n    then, there is $\\xi_{\\alpha,\\lambda,1}\\in(0,1)$ such that \\eqref{E:H:2} holds for all $\\xi\\in[\\xi_{\\alpha,\\lambda,1},1]$. If $\\xi=1$, it is precisely the constant case in \\eqref{E:H:1_0} (see \\eqref{E:H2_constant} for a more direct comparison).\n    \n    \\item Keeping $\\lambda_k\\equiv \\lambda\\in(0,1)$, $\\alpha_k\\equiv\\alpha\\in(0,1)$, and fixing $\\xi=1$, let us take $q_k\\equiv q\\in(0,1)$. \n    In this case, condition \\eqref{E:H:2} is equivalent to\n        \\begin{equation} \\label{E:H_q_constant}\n        \\Psi(\\lambda):=(1+\\alpha^2)(1-q^2)\\lambda^2 -\\big(2\\alpha^2+(1-\\alpha)(2-q^2)\\big)\\lambda +(1-\\alpha)^2\\ge 0.\n    \t\\end{equation} \n        Observe that $\\Psi(0)=(1-\\alpha)^2>0$, while $\\Psi(1) = -\\alpha q^2(1+\\alpha)<0$. Since $\\Psi$ is quadratic, the equation $\\Psi(\\lambda)=0$ has exactly one root in $(0,1)$, which we denote by $\\lambda_{\\alpha,q}$. It follows that, for each $(\\alpha,q)\\in[0,1)\\times(0,1)$, inequality \\eqref{E:H_q_constant} holds for all $\\lambda\\le\\lambda_{\\alpha,q}$. The values of $\\lambda_{\\alpha,q}$ on $[0,1)\\times(0,1)$ are depicted in Figure \\ref{fig:heatmap_lambdas}. \n        \\begin{figure}[htbp]\n        \\centering\n        \\includegraphics[scale=0.3]{images\/heatmap_lambdas.eps}\n        \\caption{Values of $\\lambda_{\\alpha,q}$.  }\n        \\label{fig:heatmap_lambdas}\n        \\end{figure}\n        Once a value for the inertial parameter $\\alpha$ has been selected, the best theoretical convergence rate is\n        $$Q(\\lambda_{\\alpha,q},q,1)=1-\\lambda_{\\alpha,q}(1-q^2).$$\n        On the other hand, using the formula for the roots of a quadratic equation and some algebraic manipulations, we deduce that\n        $$\\left[\\frac{2\\alpha^2+(1-\\alpha)}{2\\alpha^2+(1-\\alpha)(2-q^2)}\\right]\\lambda_{\\alpha,1} \\le \\lambda_{\\alpha,q} \\le \\lambda_{\\alpha,1}$$\n        for every $(\\alpha,q)\\in[0,1)\\times(0,1)$. Therefore, $\\lambda_{\\alpha,q}\\to \\lambda_{\\alpha,1}$ as $q\\to 1$, and there is no discontinuity as the contractive character is lost. \\\\\n        The case $\\xi\\in(0,1)$ is more involved. Lower values of $\\xi$ make the constant $Q$ smaller, but may also restrict the possible values for $\\alpha$ and $\\lambda$, in view of inequality \\eqref{E:H:2}. In the fully general case, if $\\alpha$, $\\lambda$ and $q$ satisfy\n        $$\\left[\\frac{\\alpha\\lambda(1+\\alpha)}{(1-\\alpha)(1-\\lambda)}\\right]\\left[\\frac{1-\\lambda+\\lambda q^2}{1-\\lambda+\\lambda q^2-\\alpha}\\right] <1,$$\n        then, there is $\\xi_{\\alpha,\\lambda,q}\\in(0,1)$ such that \\eqref{E:H:2} holds for all $\\xi\\in[\\xi_{\\alpha,\\lambda,q},1]$. As $q\\to 1$, we recover \\eqref{E:feasibility} as a limit case.\n \\end{enumerate}\n\\section{Examples}\\label{section:examples}\n\t\n\\subsection{Averaged Operators}\\label{sec:averaged}\n\nAn operator $T:\\mathcal{H}\\to \\mathcal{H}$ is $\\gamma$-averaged if there is a nonexpansive operator $R:\\mathcal{H}\\to \\mathcal{H}$ such that $T=(1-\\gamma)I+\\gamma R$. In this case, $\\Fix(T)=\\Fix(R)$.\\\\\n\nLet $R:\\mathcal{H}\\to \\mathcal{H}$ be nonexpansive and let $(\\gamma_k)$ be a sequence in $(0,1)$. Setting $T_k=(1-\\gamma_k)I+\\gamma_kR$, \\eqref{E:Algorithm} can be rewritten as\n\\begin{equation} \\label{E:Algorithm_gamma}\n\t\\left\\{\n\t\\begin{array}{rcl}\n\t\ty_k & = & x_k+\\alpha_k(x_k-x_{k-1})\\\\\n\t\tx_{k+1} & = & (1-\\gamma_k\\lambda_k)y_k+\\gamma_k\\lambda_kR(y_k),\n\t\\end{array}\n\t\\right.\n\\end{equation} \nand \\eqref{E:H:1} becomes\n$$\\limsup_{k\\to\\infty}\\big[\\alpha_k(1+\\alpha_k)+\\big((\\gamma_k\\lambda_k)^{-1}-1\\big)\\alpha_k(1-\\alpha_k)-\\big((\\gamma_{k-1}\\lambda_{k-1})^{-1}-1\\big)(1-\\alpha_{k-1})\\big]<0.$$\nIf $\\gamma_k\\lambda_k\\to\\eta>0$, this is\n\\begin{equation} \\label{E:H2_constant_gamma}\n\t\\eta(1-\\alpha+2\\alpha^2)<(1-\\alpha)^2.\n\\end{equation}\nIt is not necessary to implement the algorithm using the operator $R$ explicitly. However, the interval for the relaxation parameters is enlarged, and it may be convenient to over-relax. We shall come back to this point in the numerical illustrations.\\\\\n\n\\subsection{Euler Iterations and Gradient Descent}  \\label{S:Gradient}\n\nAn operator $B$ is {\\it $\\beta$-cocoercive} with $\\beta>0$ if $\\inner{Bx-By}{x-y}\\geq \\beta \\norm{Bx - By}^2$ for all $x,y \\in \\mathcal{H}$. \\\\\n\nLet $B:\\mathcal{H}\\to \\mathcal{H}$ be cocoercive with constant $\\beta$, and let $(\\rho_k)$ be a sequence in $(0,2\\beta)$. For each $k\\ge 1$, set\n$$T_k=I-\\rho_kB.$$\nThen, $T_k$ is nonexpansive (thus quasi-nonexpansive) and $(\\rho_k\/2\\beta)$-averaged. If $\\rho_-:=\\inf_{k\\ge 1}\\rho_k>0$, the family $(I-T_k)$ is asymptotically demiclosed. \nIf $\\lambda_k\\rho_k\\to\\sigma$, \\eqref{E:H:1} becomes\n$$\\sigma(1-\\alpha+2\\alpha^2)<2\\beta(1-\\alpha)^2.$$\nNow, let $f:\\mathcal{H}\\to \\mathcal{H}$ be convex and differentiable, and assume $\\nabla f$ is Lipschitz-continuous with constant $L$. Then, $B=\\nabla f$ is cocoercive with constant $\\beta=1\/L$. If, moreover, $f$ is strongly convex with parameter $\\mu$ and $\\rho_k\\le 2\/(L+\\mu)$, then $T_k$ is $q_k$-quasi-contractive with \n$$q_k=1-\\frac{2\\mu L\\rho_k}{L+\\mu}\\le 1-\\frac{2\\mu L\\rho_-}{L+\\mu}=:q.$$\nTherefore, $(T_k)$ is $q$-quasi-contractive. Considering the non-inertial case ($\\alpha_k \\equiv 0$), $\\lambda_k \\equiv 1$ and the fixed-sted choice $\\rho_k=2\/(\\mu + L)$, the algorithm exhibits a rate of convergence\n$$f(x_k) - f^* \\leq \\dfrac{L}{2}\\left( \\dfrac{Q-1}{Q+1}\\right)^{2k}\\norm{x_0 - x^*}^2,$$\nwhere $Q=L\/\\mu$ is the {\\it condition number}  (\\cite[Theorem 2.1.15]{nesterov2018lectures}. Introducing the inertial term, and using  \n$$\\rho_k=1\/L\\qquad\\hbox{and}\\qquad \\alpha_k\\equiv\\left(\\frac{\\sqrt{L}-\\sqrt{\\mu}}{\\sqrt{L}+\\sqrt{\\mu}}\\right),$$ \nit turns into  {\\it constant step scheme, III} \\cite{nesterov2018lectures}, which has a rate of convergence of \n$$f(x_k) - f^* \\leq \\min \\left\\lbrace \\left( 1 - \\sqrt{\\dfrac{\\mu}{L}} \\right)^k, \\dfrac{4L}{(2\\sqrt{L}+k\\sqrt{\\mu})^2} \\right\\rbrace(f(x_0) - f^* + \\frac{\\mu}{2}\\norm{x_0-x^*}^2) $$\nHere, \\eqref{E:H:1} can be written as\n$$\\lambda<\\dfrac{2Q}{1-\\sqrt{Q}+2Q},$$\nwhich gives the condition for the convergence of Nesterov's constant step scheme with constant relaxation $\\lambda$.  \n\n\\subsection{Proximal and Forward-Backward Methods}\\label{s:FB}\n\nLet $M:\\mathcal{H}\\to 2^{\\mathcal{H}}$ be maximally monotone and let $(\\rho_k)$ be a positive sequence. The {\\it proximal method} consists in iterating\n\\begin{equation} \\label{E:T_PROX}\nz_{k+1}=(I+\\rho_k M)^{-1}z_k,\n\\end{equation} \nfor $k\\ge 1$. The operator $T_k=J_{\\rho_k M}:=(I+\\rho_k M)^{-1}$ is nonexpansive, $\\frac{1}{2}$-averaged, and $Z=\\bigcap_{k\\ge 1}\\Fix(T_k)=M^{-1}0$. If $\\lambda_k\\to\\lambda$, inequality \\eqref{E:H:1} is reduced to\n$$\t\\lambda(1-\\alpha+2\\alpha^2)<2(1-\\alpha)^2.$$\nAs before, the family $(I-T_k)$ is asymptotically demiclosed at $0$ if $\\inf_{k\\geq 1}\\rho_k>0$. To see this, let $(z_k)$ be a sequence in $\\mathcal{H}$ such that $z_k\\rightharpoonup z$ and $z_k-T_kz_k\\to 0$. We must show that $0\\in Mz$. By the definition of $T_k$, we have\n$$\\frac{1}{\\rho_k}(z_k-T_kz_k)\\in M(T_kz_k).$$\nThe left-hand side converges strongly to zero, while $T_kz_k\\rightharpoonup z$. We conclude by the weak-strong closedness of the graph of $M$. \\\\\n\nLet $A:\\mathcal{H}\\to 2^{\\mathcal{H}}$ be maximally monotone, let $B:\\mathcal{H}\\to \\mathcal{H}$ be cocoercive with parameter $\\beta$, and let $(\\rho_k)$ be a sequence in $(0,2\\beta)$. For each $k\\ge 1$, set\n$$T_k=(I+\\rho_k A)^{-1}(I-\\rho_kB).$$\nThen, $T_k$ is $\\gamma_k$-averaged with $\\gamma_k=2\\beta(4\\beta-\\rho_k)^{-1}$. If $\\rho_k\\to\\rho$ and $\\lambda_k\\to\\lambda$, then \\eqref{E:H:1} is equivalent to\n$$\t\\lambda(1-\\alpha+2\\alpha^2)<\\left(2-\\frac{\\rho}{2\\beta}\\right)(1-\\alpha)^2.$$\nAs in the proximal case, the family $(I-T_k)$ is asymptotically demiclosed at $0$ if $\\inf_{k\\ge 1}\\rho_k>0$. \n\n\\subsection{Douglas-Rachford and primal-dual splitting}\\label{S:DR}\n\nLet $A, B:\\mathcal{H}\\to 2^{\\mathcal{H}}$ be maximally monotone, and let $(r_k)$ be a positive sequence. The {\\it Douglas-Rachford} splitting method consists in iterating $z_{k+1}=T_{r_k}z_k$, for $k\\geq 1$, where\n\\begin{equation} \\label{E:DRS}\nT_{r}=J_{r A}\\circ\\big(2J_{r B}-I\\big)+\\big(I-J_{rB}\\big)=\\frac{1}{2}\\big(I+(2J_{r A}-I)\\circ (2J_{r B}-I)\\big).\n\\end{equation} \nThe second expression shows that $T_r$ is averaged. Using the weak-strong closedness of the graphs of $A$ and $B$, and a little algebra, one proves that the family $\\big(I-T_{r_k}\\big)$ is asymptotically demiclosed if $\\inf_{k\\ge 0} r_k>0$. Finally, observe that $\\Zer(A+B)=J_{rB}\\Fix(T_r)$. \\\\\n\nMore generally, let $X$ and $Y$ be Hilbert spaces, and consider the {\\it primal problem}, which is to find $\\hat x\\in X$ such that\n$$0\\in A\\hat x+L^*BL\\hat x,$$\nwhere $A:X\\to 2^X$ and $B:Y\\to 2^Y$ are maximally monotone operators, and $L:X\\to Y$ is linear and bounded. The {\\it dual problem} is to find $\\hat y\\in Y$ such that\n$$0\\in B^{-1}\\hat y-LA^{-1}(-L^*\\hat y).$$\nThe primal and dual solutions, namely $\\hat x$ and $\\hat y$, are linked by the inclusions\n$$-L^*\\hat y\\in A\\hat x\\qquad\\hbox{and}\\qquad L\\hat x\\in B^{-1}\\hat y.$$\n\n\\begin{remark}\nLet $f:X\\to \\mathbb{R}\\cup\\lbrace+\\infty\\rbrace$ and $g:Y\\to\\mathbb{R}\\cup\\lbrace+\\infty\\rbrace$ be closed and convex, and set $A=\\partial f$ and $B=\\partial g$. The inclusions above are the optimality conditions for the primal and dual (in the sense of Fenchel-Rockafellar) optimization problems\n\\begin{equation}\\label{eq:min_problem}\n \\min_{x\\in X}\\{f(x)+g(Lx)\\}\\qquad\\hbox{and}\\qquad \\min_{y\\in Y}\\{g^*(y)+f^*(-L^*y)\\},   \n\\end{equation}\nrespectively. Douglas-Rachford splitting applied to $ A=\\partial g^*$ and $ B=\\partial\\big(f^*\\circ(-L^*)\\big)$ yields the {\\it alternating direction method of multipliers} (see \\cite{gabay1976dual}).\n\\end{remark}\n\nIn order to find a primal-dual pair, the {\\it primal-dual} splitting algorithm (see \\cite{chambolle2011first}) iterates:\n\\begin{equation}\\label{eq:splitting_iterations}\n\\left\\{\\begin{array}{rcl}\nx_{k+1} & = & J_{\\tau A}\\big(x_k-\\tau L^*y_k\\big) \\\\\ny_{k+1} & = & J_{\\sigma B^{-1}}\\big(y_k+\\sigma L(2x_{k+1}-x_k)\\big), \n\\end{array}\\right.    \n\\end{equation}\nwith $\\tau\\sigma\\|L\\|^2\\le 1$. The algorithm can be expressed as \n$(x_{k+1},y_{k+1})=T(x_k,y_k)$, where $T: X\\times Y \\to X\\times Y$ is a $1\/2$-averaged operator (see \\cite[Remark 4.24]{bauschke2011convex}). \\\\% given by\n\nAn inertial version of the primal-dual iterations is given by\n\\begin{equation}\n    \\left\\{\\begin{array}{l}\n       (y_k,v_k)=(x_k,u_k)+\\alpha_k\\parenc{(x_k,u_k)-(x_{k-1},u_{k-1})}   \\\\\n        p_{k+1}=J_{\\tau A}(y_k-\\tau L^* v_k)\\\\\n       q_{k+1}=J_{\\sigma B^{-1}}(v_k+\\sigma L(2p_{k+1}-y_k))\\\\\n        (x_{k+1},u_{k+1})= (1-\\lambda_k)(y_k,v_k)+\\lambda_k(p_{k+1},q_{k+1}),\n    \\end{array}\\right.\n     \\label{seq:inertial_splitting}\n\\end{equation}\nwith appropriate sequences $\\alpha_k$ and $\\lambda_k$.\\\\\n\nIn \\cite{briceno2019primal}, the authors propose the {\\it Split Douglas-Rachford} algorithm \n\\begin{equation} \\label{E:SDRA}\n\\left\\{\\begin{array}{rcl}\nv_k & = & \\Sigma\\big(I-J_{\\Sigma^{-1}B}\\big)\\big(Lx_k+\\Sigma^{-1}y_k\\big) \\\\\nx_{k+1} & = & J_{\\Upsilon A}\\big(x_k-\\Upsilon L^*v_k\\big) \\\\\ny_{k+1} & = & \\Sigma L(x_{k+1}-x_k)+v_k, \n\\end{array}\\right.\n\\end{equation} \nwhere $\\Upsilon$ and $\\Sigma$ are elliptic linear operators that induce an {\\it ad-hoc} metric and account for preconditioning.\n\n\n\n\\subsection{Three Operator Splitting}\\label{section:3op}\n\nGiven three maximally monotone operators $A, B, C$ defined on the Hilbert space $H$, we wish to find $\\hat x \\in H$ such that \n\\begin{equation}\\label{prob:3op}\n    0 \\in A\\hat x + B\\hat x + C\\hat x.\n\\end{equation}\nIf $C$ is $\\beta$-cocoercive, the {\\it three-operator} splitting method \\cite{davis2017three} generates a sequence $(z_k)$ by \n\\begin{equation}\n    \\left\\{\\begin{array}{l}\n         x^B_k=J_{\\rho B}(z_k)  \\\\\n         x^A_k=J_{\\rho A}(2x^B_k-z_k-\\rho Cx_k^B)\\\\\n         z_{k+1}= z_k+\\lambda_k(x^A_k-x^B_k)\n    \\end{array}\\right.\n    \\label{seq:threeOperator}\n\\end{equation}\nstarting from a point $z_0\\in H$. Here $\\rho \\in(0,2\\beta)$,  $\\lambda_k\\in(0,1\/\\gamma)$ and \n\\begin{equation}\n    \\gamma=\\frac{2\\beta}{4\\beta-\\rho}.    \n    \\label{eq:gamma}\n\\end{equation}\nThis recurrence is generated by iterating the $\\gamma$-averaged operator\n$$T= I-J_{\\rho B}+J_{\\rho A}\\circ(2J_{\\rho B}-I-\\rho C\\circ J_{\\rho B}),$$\nand we have $\\Zer(A+B+C) = J_{\\rho B} (\\Fix T)$. Also, it gives the forward-backward method if $B=0$ and the Douglas-Rachford method if $C=0$. An inertial version is given by\n\\begin{equation}\n    \\left\\{\\begin{array}{l}\n        u_k=z_k+\\alpha_k(z_k-z_{k-1})\\\\\n         x^B_k=J_{\\rho B}(u_k)  \\\\\n         x^A_k=J_{\\rho A}(2x^B_k-u_k-\\rho Cx_k^B)\\\\\n         z_{k+1}= u_k+\\lambda_k(x^A_k-x^B_k),\n    \\end{array}\\right.\n    \\label{seq:inertialThreeOperator}\n\\end{equation}\nfor appropriate choices of $\\alpha_k$, $\\lambda_k$. One particular instance is given by the optimization problem\n\\begin{equation}\\label{prob:3opconvex}\n    \\min f(x) + g(x) + h(Lx),\n\\end{equation}\nwhere $f,g,h$ are closed and convex, $h$ has a $(1\/\\beta)$-Lipschitz-continuous gradient, and $L$ is a bounded linear mapping.   \n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\n\n\n\n\n\nKrasnoselskii-Mann (KM) iterations \\cite{Krasnosel1955two,mann1953mean} are at the core of numerical methods used in optimization, fixed point theory and variational analysis, since they include many fundamental splitting algorithms whose convergence can be analyzed in a unified manner. These include the {\\it forward-backward}  \\cite{LionsMercier,Passty} to approximate a zero of the sum of two maximally monotone operators, and its various particular instances: on the one hand, we have the {\\it gradient projection} algorithm \\cite{goldstein1964convex,levitin1966constrained}, the gradient method \\cite{cauchy1847methode} and the proximal point algorithm\n\\cite{martinet1970regularisation,rockafellar1976monotone,brezis1978produits,guler1991convergence}, to cite some abstract methods, as well as the {\\it Iterative Shrinkage-Thresholding Algorithm} (ISTA) \\cite{daubechies2004iterative,combettes2005signal}, to speak more concretely. KM iterations also encompass other splitting methods like  {\\it Douglas-Rachford} \\cite{DouglasRachford1956}, primal-dual methods \\cite{chambolle2011first,attouch2010parallel,combettes2004solving,vu2013splitting,condat2013primal} and the three-operator splitting \\cite{davis2017three}.  \\\\\n\nConvex optimization methods can be enhanced, by adding an inertial substep, motivated by physical considerations \\cite{polyakInertial,Nesterov1983AMF,alvarez_minimizing_2000}. Extensions beyond the optimization setting have been developed in \\cite{alvarez2001inertial} and, more recently, in \\cite{Lorenz2014,boct2015inertial,AttouchCabot2019,shehu2020inertial,iyiola2021new}, to mention a few. Interest in this type of methods increased remarkably in the past decade in view of theoretical advances in the convergence theory for the {\\it Fast Iterative Shrinkage-Thresholding Algorithm} (FISTA) \\cite{beck2009fast}, obtained in \\cite{ChambolleDossal2015,AttPeyRedMAPR2018,AttPey1_k2}. \\\\\n\nThe purpose of this work is to develop further insight into the convergence properties of {\\it inertial Krasnoselskii-Mann iterations} in their general form\n\\begin{equation}\\label{E:Algorithm}\n \t\\left\\{\n \t\\begin{array}{rcl}\n \t\ty_k & = & x_k+\\alpha_k(x_k-x_{k-1})\\\\\n \t\tx_{k+1} & = & (1-\\lambda_k)y_k+\\lambda_kT_k(y_k),\n \t\\end{array}\n \t\\right.\n\\end{equation} \nwhere $(T_k)$ is a family of operators defined on a real Hilbert space $\\mathcal{H}$, and the positive sequences $(\\alpha_k)$ and $(\\lambda_k)$ are the {\\it inertial} and {\\it relaxation} (or {\\it averaging}) parameters, respectively. \\\\\n\n\n\n\nThe paper is organized as follows: in Section \\ref{section:weak} we establish  the weak convergence of the iterations towards a common fixed point of the family of operators in the quasi-nonexpansive case, along with worst case estimates for the rate at which the residuals vanish. Section \\ref{section:strong} is devoted to the strong and linear convergence in the quasi-contractive setting. In both cases, we highlight the relationship with the non-inertial case, and show that passing from one regime to the other is a continuous process in terms of parameter hypotheses and convergence rates. In Section \\ref{section:examples}, we discuss several instances of KM iterations, which are relevant to the numerical illustrations provided in Section \\ref{section:numerical}, concerning an inertial primal-dual method and an inertial three-operator splitting algorithm.  \n\n\\section{Numerical Illustrations}\\label{section:numerical}\n\nFor this section, please refer to the full version following \\href{https:\/\/drive.google.com\/file\/d\/1HhtIIvlfHmlvOKdDmvsebvr3qkU9eDJu\/view?usp=sharing}{this link}.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn a recent work,~\\citet{bulbul:14,boyarsky:14} reported the detection of an unidentified 3.55~keV line in the analysis of stacked data on galaxy clusters observed by XMM-Newton\\xspace, in the  observations of individual galaxy clusters (Perseus) and  in the Andromeda galaxy. In the absence of obvious options for this line to be of instrumental\/astrophysical origin (see, however, \\citet{Jeltema:14}), the authors invoked the possibility for the line to be produced in the decay of Dark Matter (DM) particles populating the halos of the considered structures. Assuming a sterile-neutrino nature of the decaying DM (see e.g. \\citet{Dodelson:93,Asaka:05a,Lattanzi:07} and references therein), \\citet{bulbul:14} obtained a value of $\\sin^2(2\\theta)\\sim6.8\\times 10^{-11}$ for the the mixing angle of the sterile neutrino and a particle mass $m_{DM}$ of 7.1~keV. These parameters are consistent with the previously-derived upper bounds on $\\sin^2(2\\theta)$ from observations of the extragalactic diffuse X-ray background~\\cite{Boyarsky:05,Abazajian:06b};\ngalaxy clusters~\\cite{Boyarsky:06b,Riemer-Sorensen:06b,Boyarsky:06e}; the Milky Way, Andromeda (M31) and Triangulum (M33) galaxies~\\cite{Abazajian:06b,Watson:06,Boyarsky:06b,Boyarsky:06c,Boyarsky:06d,Boyarsky:07b,Yuksel:07} and individual dwarf spheroidal (dSph) satellites of the Milky Way\n\\cite{Boyarsky:06c,Boyarsky:06f,Boyarsky:10b,Riemer-Sorensen:09,Loewenstein:09,Loewenstein:12,Kusenko:12}.\n\nThe signal from decaying DM is expected to be strongest from the most nearby source, our own Milky Way galaxy. However, the signal is distributed over the entire sky, so that it is not straightforward to look for this signal using narrow-field X-ray telescopes. Besides, the radial density profile of DM in the Milky Way is somewhat uncertain, especially in its central part, where a significant contribution to the overall matter content of the Galaxy comes from the baryons. Recently \\citet{riemer:14} analyzed Chandra data on the Galactic Center (GC) region, finding no clear evidence for the 3.55~keV line (see however \\citet{Boyarsky:14b}). The non-detection of the signal from the GC was found to be consistent with the existence of the DM sterile neutrino with parameters suggested by \\citet{bulbul:14} for the most conservative assumptions on the DM density profile in the innermost part of the Galaxy.  \n\nThe signal from the Milky Way halo is superimposed on approximately equally strong signal from the DM halos of nearby dSph galaxies in the direction of these sources. Combining the two signals and additionally stacking the signal from all the observed dSph systems, an improved sensitivity to the DM decay line can be obtained compared to the previously-reported constraints from the Milky Way or individual dSph galaxies. \n\nIn the following, we perform a stacked analysis of dSph galaxies using XMM-Newton\\xspace\\ data. In spite of a shorter overall exposure compared to the stacked galaxy cluster dataset analyzed by \\citet{bulbul:14}, the signal from the dSph is cleaner than in galaxy clusters, because the interstellar medium of dSph galaxies is not a source of thermal X-ray emission, contrary to the intracluster medium of the galaxy clusters.  The work presented here is organized as follows. In the next section, we discuss the expected flux level from the dSph galaxies (Sect.~\\ref{sec:dsphs_signal}) and the contribution of the Milky Way (MW) halo to the observed signal in Sect.\\ref{sec:mw_contribution}. The details of the XMM-Newton\\xspace observations and of the analysis procedure are given in Sect.~\\ref{sec:observations}. Finally, the results are summarized and discussed in Sect.~\\ref{sec:results_discussion}.\n\n\\section{Expected signal from DM decay}\n\\label{sec:expected_signal}\n\\subsection{dSph galaxies}\n\\label{sec:dsphs_signal}\n\nThe decay of DM particles of mass $m_{DM}$ results in a line at the energy $\\epsilon=m_{DM}\/2$ and flux\n\\begin{equation}\nF=\\frac{\\Gamma M_{DM,FoV}}{4\\pi d^2 m_{DM}}\n\\label{eq:flux}\n\\end{equation}\nwhere $M_{DM,FoV}$ is the total DM mass within the field of view (FoV) of the telescope. In the case of sterile neutrinos, the radiative decay width $\\Gamma$ can be written as \\cite{Pal:82}\n\\begin{eqnarray}\n\\Gamma&=&\\frac{9\\alpha G_F^2}{256 \\cdot 4\\pi^4}\\sin^2 2\\theta_{DM} m_{DM}^5\\nonumber\\\\\n&\\simeq& 1.7\\times 10^{-28}\\left[\\frac{\\theta_{DM}^2}{1.75\\times 10^{-11}}\\right]\\left[\\frac{m_{DM}}{7.1\\mbox{ keV}}\\right]^5\\mbox{ s}^{-1}\n\\end{eqnarray}\nwhere we have normalized the mixing angle $\\theta_{DM}$ and $m_{DM}$ to the values reported in~\\citet{bulbul:14}.\nSubstituting this expression into Eq.~\\ref{eq:flux} one finds the flux from dwarf spheroidal in the field of view to be \n\\begin{eqnarray}\n\\label{eq:flux_num}\nF_{dSph}\\simeq 2.4\\times 10^{-7}\\left[\\frac{\\theta_{DM}^2}{1.75\\times 10^{-11}}\\right]\\left[\\frac{m_{DM}}{7.1\\mbox{ keV}}\\right]^4\\nonumber\\\\\\left[\\frac{d}{100\\mbox{ kpc}}\\right]^{-2}\n\\left[\\frac{M_{DM,FoV}}{10^{7}M_\\odot}\\right]\\frac{\\mbox{ph}}{\\mbox{cm}^2\\mbox{s}}\n\\end{eqnarray}\n\n\n\nNote, that the flux~(\\ref{eq:flux_num}) does not depend on the exact DM distribution, but only on the total mass inside the telescope's FoV. The latter can be measured with greater accuracy than the measurement of a profile. The uncertainty in the expected DM decay flux~(\\ref{eq:flux_num}) can be directly propagated from the uncertainties in $M_{DM}$ measurements.\n\nIn our work we consider the sample of dwarf spheroidal galaxies presented by~\\citet{wolf:10}, where the authors obtained accurate estimates of the DM mass inside the half-light radius, i.e. the radius within which half of the total light is observed. The choice of this integration radius is motivated by the fact that this radius minimizes the uncertainty in the enclosed mass. In Table~\\ref{tab:dsphs_params} we summarize the available data on dSphs for which archival XMM-Newton\\xspace data are available. These data (from~\\citet{wolf:10}) quote the mass estimates $M_{1\/2}$ within the half-light radius sphere, together with the uncertainties and the values of the half-light radius both in physical ($r_{1\/2}$)  and angular ($\\theta_{1\/2}$) size. For comparison we also show the masses within the columns out to $r_{1\/2}$ from the recent work \\citet{geringer:14}, where available.\n\nFor a number of dSphs (e.g. Ursa Minor), the half-light radius exceeds the size of the XMM-Newton\\xspace FoV, $\\theta_{FoV}\\simeq 15'$. This implies that only a fraction of the DM decay signal is visible in a single XMM-Newton\\xspace pointing.\nWe take this into account for the estimation of the DM flux from these particular galaxies. Namely, we assume that the mass within the FoV scales as $M_{DM,FoV}\\sim \\theta_{FoV}$ for the sources with $\\theta_{1\/2}>\\theta_{FoV}$, taking into account the fact that the velocity dispersion profiles of dSphs are flat, see e.g.~\\citet{walker:07,walker:13} and references therein. The centres of the field-of-view for most observations are slightly displaced from the dSphs centre positions, so that the maximal angular distance from the centre is typically smaller than the full FoV size. For sources spanning the full FoV we restrict the analysis to the region $\\theta_{max}=12'$ around the source centre.\n\n\\begin{table}\n\\input{dsphs_params.tbl.tex}\n\\caption{{Parameters of dwarf spheroidal galaxies considered in the analysis.  The distances to the dwarves($d$, kpc), half-light radii ($r_{1\/2}$,kpc and $\\theta_{1\/2}$, arcmin) and the masses within half-light radii sphere ( $M_{1\/2}, 10^{7}M_\\bigodot$ ) are taken from \\citet{wolf:10}. The column masses within half-light radii are from \\citet{geringer:14}, see text for the details.}}\n\\label{tab:dsphs_params}\n\\end{table}\n\n\\subsection{Milky Way}\n\\label{sec:mw_contribution}\n\nThe DM decay flux from the Milky Way halo is typically comparable to the flux from isolated distant sources, like dSph galaxies or galaxy clusters \\cite{boyarsky:06}. The flux from DM decay in the Milky Way within the telescope field-of-view $\\Omega_{FoV}$,\n\\begin{equation}\nF=\\frac{\\Gamma\\Omega_{FoV}{\\cal S}}{4\\pi m_{DM}}\n\\end{equation}\nis determined by the column density of the DM\n\\begin{equation}\n{\\cal S}=\\int\\limits_{0}^{\\infty}\\rho_{DM}\\left(\\sqrt{r^2_{\\bigodot}-2zr_\\bigodot\\cos\\phi+z^2}\\right)dz\n\\label{eq:S_MW}\n\\end{equation}\nwhere $r_\\bigodot=8.5$~kpc is the distance from the Sun to the centre of our Galaxy and the angle $\\phi$ relates to the galactic coordinates $(l,b)$ as $\\cos\\phi=\\cos b \\cos l$.\n\nFor a source of angular size $\\theta$, the contribution of the MW to the DM decay flux is \n\\begin{eqnarray}\nF_{MW}=1.1\\times 10^{-6}\\left[\\frac{\\theta}{\\theta_{FoV}}\\right]^2\\left[\\frac{S}{10^{22}\\mbox{GeV\/cm}^2}\\right]\\nonumber\\\\\n\\left[\\frac{\\theta_{DM}^2}{1.75\\times 10^{-11}}\\right]\\left[\\frac{m_{DM}}{7.1\\mbox{ keV}}\\right]^4\\mbox{ph\/cm}^2\\mbox{s}\n\\label{eq:flux_MW}\n\\end{eqnarray}\n\nTo estimate the column density $S$ of Galactic DM in different directions, we adopt the models of the DM halo of the Milky Way discussed by~\\citet{klypin:02}. \\citet{klypin:02} have adopted the Navarro-Frenk-White (NFW, \\citet{nfw}) profile\n\\begin{equation}\n\\rho_{NFW}(r)=\\frac{\\rho_s r_s^{3}}{r(r+r_s)^2}\n\\label{eq:nfw}\n\\end{equation}\nin which the characteristic density $\\rho_s$ and radius $r_s$ are free parameters estimated from the data. \n\nThe uncertainty in the column density as well as in the radial DM density profile arises from the difficulty of disentangling contributions from the visible and DM components to the Galaxy rotation curve. In what follows we consider the ``favoured NFW'' model of \\citet{klypin:02} for the estimate of the \\emph{mean} Milky Way DM column density in the direction of individual dSphs (with $\\rho_s=4.9\\times 10^6$~M$_\\bigodot$kpc$^{-3}$, $r_s=21.5$~kpc). To estimate how the uncertainty propagates to the limits on \nthe mixing angle of sterile neutrino DM, we also consider the column densities of Galactic DM deduced from the  ``maximal disk model'' ($\\rho_s=0.6\\times 10^6$~M$_\\bigodot$kpc$^{-3}$, $r_s=46$~kpc) of \\citet{klypin:02}. We refer to this estimate as the \\emph{minimal} Galactic DM contribution to the signal.  For each position in the sky the \\emph{minimal} DM column density is  2-3~times lower than the estimated \\emph{mean} column density.\n\nThe expected Galactic contribution to the DM decay flux from individual dSphs is given in Table \\ref{tab:predicted_flux}. The total expected DM decay flux from  the direction of each dSph galaxy is the sum of the fluxes given in Eqs.  (\\ref{eq:flux_num}) and (\\ref{eq:flux_MW}). The expected fluxes $F_{dSph}$ are given in two column format for the mass estimations by \\citet{wolf:10} and \\citet{geringer:14} correspondingly.\n\n\n\n\n\\begin{table}\n\\input{predicted.tbl.tex}\n\\caption{Expected fluxes from DM decay line with parameters corresponding to \\citet{bulbul:14}. For each dwarf spheroidal galaxy the expected flux is split into two components: the signal from the dwarf itself ($F_{dsph}$, see Eq.~\\ref{eq:flux_num}) and from the Milky Way ($F_{MW}$, see Eq.~\\ref{eq:flux_MW}). For the MW contribution flux estimates are given for both mean and minimal DM profiles.The two columns for F$_{dSph}$ correspond to the flux values from $M_{1\/2}$ mass estimations by \\citet{wolf:10} and \\citet{geringer:14} correspondingly.}\n\\label{tab:predicted_flux}\n\\end{table}\n\n\\section{Observations and data analysis}\n\\label{sec:observations}\n\nThe details of the XMM-Newton\\xspace observations of the dSph galaxies selected for this analysis are summarized in Table~\\ref{tab:observations}. We collected all publicly-available observations of eight dSphs with exposures exceeding 10~ks. The total exposure time of the observations is $\\sim 0.6$~Msec. In most cases, the observations were already used for the search of the DM decay signal and upper bounds on the sterile-neutrino DM mixing angle were already derived on a source-by-source basis~\\citep{Loewenstein:09,Riemer-Sorensen:09,Loewenstein:12}. The goal of our re-analysis of these data is to stack the signal from all the dSphs to increase our sensitivity to the DM decay line.\n\nWe processed the raw data with the ESAS (v.0.9.28 as part of XMM SAS v.13.5) reduction scheme \\citep{Snowden:08}\\footnote{See e.g. http:\/\/heasarc.gsfc.nasa.gov\/docs\/xmm\/esas \/esasimage\/esasimage\\_thread.html} using the calibration files from May 2014. Within this scheme, we produced cleaned event files by removing the periods of soft proton flares. After the flare removal, we performed a self-consistency check on the level of contribution of residual soft protons to the background flux, following \\citet{leccardi:08}. This contribution is found to be less than $\\sim~20\\%$ in all observations. In any case, the spectrum of the soft protons is featureless, so our line search is unaffected by soft-proton contamination. \n\nWe masked all the detected point sources in the FoV using the ESAS task \\texttt{cheese}, such that only the signal from the extended DM halo of the dSphs is taken into account. We then extracted spectra and images of the extended emission using the ESAS tasks \\texttt{mos-spectra} and \\texttt{pn-spectra}. These tools use a collection of closed-filter data to estimate the local non X-ray background (NXB) by scaling the normalization of the closed-filter spectra to match the count rates measured in the unexposed corners of each of the three EPIC detectors. \n\n\\begin{table}[t!]\n\\input{xmm_obs.tbl.tex}\n\\caption{XMM-Newton\\xspace observations of dwarf spheroidals considered in this analysis. The total exposure time is given as the sum of effective exposures for the MOS1, MOS2 and pn cameras individually. The total clean exposure is $\\sim 0.6$~Msec}\n\\label{tab:observations}\n\\end{table}\n\nWe stacked the spectra of all the dSph observations using the {\\it addspec} routine, to obtain mean MOS1+MOS2 and pn spectra. We then fitted the resulting spectra in the 0.7-10~keV energy band with the sum of models representing the astrophysical and NXB contributions. We ignored the energy interval 1.2-1.8~keV from the fit, as this range is affected by the presence of strong and time-variable Si K$\\alpha$ and Al K$\\alpha$ fluorescence lines.  \n\nInstead of subtracting the NXB spectra directly from the data, we modelled the NXB spectrum using a phenomenological model including all known fluorescence lines (see Appendix B of \\citet{leccardi:08}) and added this model as an additive component to the fit. This method has the advantage of retaining the original statistics of the spectrum and allowing for possible variations of the NXB level, e.g. caused by soft protons. During the fitting procedure, we fixed the spectral shape of the particle-induced continuum to the values obtained from the closed-filter data, since this component is known to be stable with time \\citep{leccardi:08}. On the other hand, we leave the normalizations of the instrumental lines free while fitting. In the case of the pn camera we also need to model the Ca line at $\\sim 4.6$~keV. We note that no instrumental line is observed between 3 and 4 keV, thus this analysis is suitable for the detection of an additional emission line in this energy range. For the details of the analysis procedure, we refer the reader to \\citet{Eckert:14}.\n\nThe astrophysical background and foreground contribution is modelled as the sum of a power law representing the cosmic X-ray background (CXB) and two {\\it apec} models for the Galactic emission. The combined fit is reasonably good, with $\\chi^2\/DOF=595.5\/576$. In Fig. \\ref{fig:spectra} we show the stacked spectra and best-fit model for pn (red) and MOS (black). The residuals from the fit are displayed in the bottom panel of the figures. The main parameters of the fit are given in Table \\ref{tab:fit_params}. The fit requires the presence of a rather hot component at the temperature of $\\sim0.9$ keV, which is significantly higher than the typical temperature \\citep{Mccammon:02}; however the temperature of the Galactic halo is known to vary significantly from one direction to another, and thus such a result is not unusual \\citep{Masui:09}. In any case, we note that the foreground emission from the MW halo is very soft compared to the energy range of interest for this study (see Fig. \\ref{fig:spectra}), such that the exact temperature of the foreground component has little influence on our analysis. Moreover, the spectral slope of the CXB, which is the dominant sky component beyond $\\sim1$ keV, is in excellent agreement with the canonical value of 1.4 \\citep[e.g.][]{DeLuca:04}. The normalization of the CXB in each individual observation was also found to agree with the measurement of \\citet{DeLuca:04}. A closer look at the residuals in the range of interest for this analysis, i.e. the range between 2 and 4 keV, is shown in Fig. \\ref{fig:residuals}. \n\n\\begin{table}[t!]\n\\input{parameters.tbl.tex}\n\\caption{Parameters of the fit of the stacked spectra of dSphs with the reference background model.}\n\\label{tab:fit_params}\n\\end{table}\n\n\n\\begin{figure*}[t!]\n\\resizebox{0.9\\hsize}{!}{\\includegraphics[angle=270]{spectrum.eps}}\n\\vspace{0.6cm}\n\\caption{MOS1+2 (black) and pn (red) stacked spectra of dSph galaxies. The top panel shows the data and the best-fit model, broken into the astrophysical contribution (green) and the NXB (blue), while the bottom panel shows the residuals from the model.}\n\\label{fig:spectra}\n\\end{figure*}\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.5\\textwidth]{zoom}\n\\caption{Residual flux from the combined MOS~1,2  (green) and pn (red) cameras in the 2-4~keV energy band including the expected 3.55~keV line.}\n\\label{fig:residuals}\n\\end{figure}\n\nTo search for the DM decay line, we added to the model a narrow gaussian line at a fixed energy of $3.55$~keV. The addition of such a line does not provide a significant improvement to the fit. To compute upper limits to the line flux marginalizing over all uncertainties, we sampled the likelihood using a Markov chain Monte Carlo (MCMC) method as implemented in XSPEC v12.8. After an initial burning phase of 5,000 steps, we performed 50,000 MCMC steps and drew the posterior distributions from the resulting chain. The output distribution for the line flux is shown in Fig. \\ref{fig:mcmc}. From this distribution we obtained upper limits to the line flux of $2.97\\times10^{-7}$ phot cm$^{-2}$ s$^{-1}$ (90\\% confidence level) and $4.74\\times10^{-7}$ phot cm$^{-2}$ s$^{-1}$ (3$\\sigma$). In addition, we also performed a search for the DM decay line in the entire energy range 2-10~keV, which did not give any positive results. From this analysis we derived an energy-dependent upper bound on the line flux.\n\n\\begin{figure}\n\\resizebox{0.9\\hsize}{!}{\\includegraphics[angle=270]{mcmc.eps}}\n\\vspace{0.2cm}\n\\caption{Posterior probability distribution for the 3.55 keV line flux in the stacked dSph dataset, obtained through 50,000 MCMC sampling of the likelihood function.}\n\\label{fig:mcmc}\n\\end{figure}\n\nThe contribution of the individual dSph galaxies to the XMM-Newton\\xspace\\ signal could be calculated using the information on the instrument effective area $A_{i}$ (found using the {\\it plot efficiency} command in XSPEC) and the clean exposure $t_i$ in each observation. If the expected flux of the DM decay line in the $i$th observation is $F_i$, the expected number of DM decay photons is $N_{i}=F_{i} A_i t_i$. The mean DM decay flux in the entire stacked dataset is then \n\\begin{equation}\n\\left<F\\right>=\\frac{\\sum\\limits_i F_{i} A_i t_i} {\\sum\\limits_i A_i t_i },\n\\label{eq:Fmean}\n\\end{equation}\nwith the sum being performed over all dSphs observations. \n\nTable \\ref{tab:predicted_flux} gives the information on $A_i$ and $t_i$ in each observation, together with the estimate of the expected DM decay line flux calculated assuming the sterile neutrino DM parameters suggested by the observations of \\citet{bulbul:14}.  \n\n\n\\section{Results and discussion}\n\\label{sec:results_discussion}\n\nOur analysis of the stacked sample of dSph galaxies provides a moderate improvement of constraints on the parameters for sterile neutrino DM, compared to the previously-derived bounds based on the previous observations of the diffuse X-ray background, of individual dSph galaxies, of the Milky Way and of galaxy clusters. \n\nAs it follows from Table~\\ref{tab:predicted_flux}, the flux from DM decay expected to be seen in the combination of observations of all dwarves is $F_{mean}\\sim 6.2\\pm 0.7\\times 10^{-7}$~cts\/s\/cm$^2$ for mean dark matter column density in the MW and $F_{min}\\sim 4.6\\pm 0.6\\times 10^{-7}$~cts\/s\/cm$^2$ for minimal Galactic DM column density for DM mass estimates from \\citet{wolf:10}. In the case of \\citet{geringer:14} mass estimates the reference values are $F_{mean}\\sim 7.3\\pm 0.8\\times 10^{-7}$~cts\/s\/cm$^2$ and $F_{min}\\sim 5.7\\pm 0.8\\times 10^{-7}$~cts\/s\/cm$^2$. Such a flux was expected to be detected at  $3.5\\sigma (4.1\\sigma)$  or $2.8\\sigma (3.2\\sigma)$ levels for the mean \/ minimal Galactic DM column density models in the stacked dSph dataset for \\citet{wolf:10} (\\citet{geringer:14}) mass estimates. The non-detection of the DM line in the analyzed data sets is, therefore, inconsistent with the assumption that the unidentified line at 3.55~keV is produced by decaying sterile-neutrino DM with parameters suggested by the analysis of \\citet{bulbul:14}. {\n\n\nThis is illustrated by Fig. \\ref{fig:exclusion_plot}, where we plot the $2\\sigma$ upper limits on the mixing angle of the DM sterile neutrino as a function of the DM particle mass, assuming quoted by \\citet{geringer:14} dSphs masses. The value of $\\sin^2(2\\theta)$ derived by \\citet{bulbul:14} is above the $2\\sigma$ upper bound for both the \\emph{mean} and \\emph{minimal} Galactic DM column density models. \n\n\\begin{figure}[t!]\n\\includegraphics[width=\\linewidth]{exclusion_3.5_revised.eps}\n\\caption{Exclusion plot on sterile neutrino mass -- mixing angle plane. All parameter values above the curves are excluded. The solid dark green and light green lines show the $2\\sigma$ constraints for minimal and mean dark matter column density in the Milky Way, respectively. The red point with error bars indicates the parameter values reported by \\citet{bulbul:14}. The dashed line indicates the M31 constraints from \\citet{horiuchi:14}.}\n\\label{fig:exclusion_plot}\n\\end{figure}\n\nThe calculation of the DM decay line flux from the direction of galaxy clusters by \\citet{bulbul:14} does not include the flux from the foreground DM halo of the Milky Way. This is justified if the foreground flux is subtracted in the analysis of the spectra of the galaxy clusters. This is not the case in the analysis of \\citet{bulbul:14}, who modelled the cluster spectra together with the instrumental and sky background \/ foreground in a way similar to the approach adopter here. In this case, the flux from DM decay in the MW halo should, in principle, be included in the calculation of the line flux \\cite{boyarsky:06}. This should result in a somewhat lower value of the sterile neutrino mixing angle $\\sin^2(2\\theta)$ from the observed line flux. This effect might potentially relax the inconsistency of the \\citet{bulbul:14} result with the dSph data reported here. \n\nThe estimates of  $\\sin^2(2\\theta)$ derived by \\citet{boyarsky:14} from the analysis of M31 and of the Perseus galaxy cluster are much more uncertain than those derived from the stacked galaxy cluster sample, because of the much larger uncertainty in the DM column density in the two particular individual sources. Taking into account a roughly order-of-magnitude uncertainty in the estimate of  $\\sin^2(2\\theta)$  derived from the analysis of \\citet{boyarsky:14}, one could see that our constraints on the DM parameters derived from the dSph data are still consistent with the results of \\citet{boyarsky:14}.\n\nHowever, the central value of the estimate of $\\sin^2(2\\theta)\\sim 5\\cdot 10^{-11}$ from \\citet{boyarsky:14} is inconsistent with the limits from dSphs at the level of $3.4\\sigma (2.5\\sigma)$ assuming the mean (minimal) Galactic DM column density and taking the column (rather than within-a-ball) mass within half-light radius estimates of \\citet{geringer:14}.\n\n\n\nOur analysis is only marginally ruling out the possibility of the DM decay origin of the unidentified line at 3.55~keV. \nAn increase of the sensitivity by a factor of $\\sim 2$ is necessary to firmly rule out the DM decay line hypothesis for the line origin. This is possible already with XMM-Newton\\xspace (rather than with the next-generation telescopes like ASTRO-H), via a moderate increase of exposure towards selected dSph galaxies (e.g. Ursa Minor, Ursa Major II), which are characterized by strong DM decay line flux, but are currently not dominating the stacked dSph signal because of the relatively short exposures. Deeper XMM-Newton\\xspace observations of these dSphs galaxies would thus be sufficient to test conclusively the DM origin of the 3.55 keV line.\n\\newline\n\\textit{Acknowledgements.} The authors would like to thank Alex Geringer-Sameth for providing the raw data from the recent analysis \\citet{geringer:14}.\n\n\n\n\n\n\n\n\n\n\\input{journals.tex}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe study of birationality of algebraic varieties is a classical \nand well studied subject, with many open problems.\nIn some cases, it is interesting to study birational maps\npreserving additional structure, for example\ngroup actions, symplectic forms, or volume forms. \nSuch a study is already implicit in many questions of birational geometry,\n\\eg,  in the notion of crepant resolution of singularities.\n\nIn this paper, we consider the case of varieties\nendowed with volume forms with logarithmic poles\nand develop  a formalism of Burnside rings \nalong the lines of their counterpart introduced\nby~\\citet{KontsevichTschinkel-2019} to establish\nthe specialization of rationality, and \nits equivariant version by~\\citet{KreschTschinkel-2022}.\n\nLet $k$ be a field of characteristic zero.\nFor each integer~$n$,\nwe define \n\\[\u00a0\\Burn_n(k) \\]\nas the free abelian group\non birational equivalence classses of pairs $(X,\\omega)$\nconsisting of an integral smooth  proper $k$-variety~$X$ of dimension~$n$\nequipped with an $n$-form~$\\omega$ with at most logarithmic poles.\n\nThe graded abelian group\n\\[\u00a0\\Burn(k) = \\bigoplus_{n\\in\\N} \\Burn_n(k) \\]\ncarries a ring structure, induced by taking products of varieties,\ndecomposed into irreducible components,\nand equipped with the external product of the volume forms. \n\nIn section~\\ref{sec.residues},\nwe define morphisms of abelian groups\n\\[\u00a0\\partial \\colon \\Burn_n(k) \\to \\Burn_{n-1}(k) .\\]\nWhen $X$ is smooth and the polar divisor of~$\\omega$ has strict\nnormal crossings, the image of the class $[X,\\omega]$ is given\nby the following formula. Let $(D_\\alpha)_{\\alpha\\in\\mathscr A}$\nbe the family of irreducible components of the polar divisor of~$\\omega$.\nFor each subset~$A$ of~$\\mathscr A$, the intersection $D_A=\\bigcap_{\\alpha\\in A} D_\\alpha$ is a union of integral smooth varieties of codimension~$\\Card(A)$;\ntaking iterated residues, we may equip it \nwith a volume form with logarithmic poles~$\\omega_A$.\nThen\n\\[\u00a0\\partial ([X,\\omega]) = \\sum_{\\emptyset\\neq A \\subset\\mathscr A}\n (-1)^{\\Card(A)-1} [D_A, \\omega_A] \\cdot \\mathbf T^{\\Card(A)-1}, \\]\nwhere \n\\[\u00a0\\mathbf T = [\\P^1, \\mathrm dt\/t]. \\]\nIn particular, the existence of the map~$\\partial$ relies on the \nbirational invariance of this expression,\nsee theorem~\\ref{theo.residue-bir}.\n\nThis construction is reminiscent of the boundary map in polar homology\n\\citep{KhesinRosly-2003,KhesinRoslyThomas-2004,GorchinskiyRosly-2015}.\nHowever, apart from the obvious difference that we only record birational types\nof strata, rather than the strata themselves,\nour formula takes into account strata of all codimensions, rather\nthan those of codimension one.\n\nThe map~$\\partial$  is additive.\nFurthermore, we prove in theorem~\\ref{theo.partial-deriv} that\n\\[\u00a0\\partial(a \\cdot b) = \\eps^n \\cdot \\partial (a)\\cdot  b\n+ a \\cdot \\partial(b) - \\mathbf T \\cdot  \\partial(a) \\cdot \\partial(b), \\]\nwhen $a\\in\\Burn_m(k)$ and $b\\in\\Burn_n(k)$.\nHere $\\eps$ is the class of the point~$\\Spec(k)$\nequipped with the volume form equal to~$-1$.\n\nIn theorem~\\ref{theo.dd=0}, we show that \n\\[\u00a0\\partial \\circ \\partial  = 0 . \\]\n\nThese formulas may look complicated.\nHowever, as  we explain in \\S\\ref{sec.alg},\nthey simplify significantly after inverting~$2$.\n\nInspired by the constructions of~\\cite{LinShinderZimmermann-2020,LinShinder-2022,KreschTschinkel-2022a}, we define in~\\S\\ref{sec.biraut}\na homomorphism \n\\[\u00a0\\mathbf c \\colon \\Bir(X,\\omega) \\to \\Burn_{n-1}(k), \\]\nfrom the group of birational automorphisms of the pair\n$(X,\\omega)$, where $X$ is an $n$-dimensional integral proper smooth variety\nequipped with a logarithmic volume form~$\\omega$.\nAs in the above references, our map~$\\mathbf c$ is defined\nat the groupoid level of birational maps preserving logarithmic volume forms.\n\nWhen the birational isomorphism~$\\phi\u00a0\\colon (X,\\omega) \\dashrightarrow (Y,\\eta)$ is described\nby a diagram \n\\[\u00a0\\begin{tikzcd}[column sep = small]\n & W \\ar{dl}[swap]{p} \\ar{dr}{q} \\\\\nX \\ar[dashrightarrow]{rr}{\\phi} && Y \\end{tikzcd}\u00a0\\]\nof smooth proper integral $k$-varieties,\nwith birational morphisms~$p$ and~$q$, the two logarithmic\nvolume forms~$p^*\\omega$ and~$q^*\\eta$ on~$W$ are equal,\nand the element~$\\mathbf c(\\phi)\\in\\Burn_{n-1}(k)$ is given by\n\\[\u00a0\\mathbf c(\\phi) = \n\\sum_{E \\in\\operatorname{Exc}(q)} [E, p^*\\omega_E] - \n\\sum_{D \\in\\operatorname{Exc}(p)} [D, q^*\\eta_D] \\]\nwhere $\\operatorname{Exc}(p)$ is the set of irreducible components\nof the exceptional divisor of~$p$, and where, for each such component~$D$,\nthe logarithmic volume form~$p^*\\omega_D$ on~$D$\nis obtained by taking the residue of~$p^*\\omega$ along~$D$\n(and similarly for~$q$).\n\nFinally, consider a discrete valuation ring\nwith residue field~$k$ and field of fractions~$K$\nand let $t$ be a uniformizing element.\nIn this context, we define a specialization map\n\\[\u00a0\\rho_t \\colon \\Burn_n(K)\\to \\Burn_n(k). \\]\nThe image of the class $[X,\\omega]$  involves the combinatorics of a \ngood model $(\\mathscr X,\\omega)$ over the valuation ring, \nand a certain subcomplex of the Clemens complex of the special fiber.\nIn the particular case where $\\mathscr X$ is smooth,\nthe polar divisor of~$\\omega$ is a relative divisor with normal crossings,\nand denoting by~$\\omega_o$ the restriction of~$\\omega$ \nto the special fiber~$\\mathscr X_o$,\none has\n\\[\u00a0\\rho_t ([X,\\omega])  = [\\mathscr X_o,\\omega_o]. \\]\nNote that \nthe existence of such a specialization map implies,\nas in theorem~1 of~\\cite{KontsevichTschinkel-2019},\nor as in \\citep{NicaiseShinder-2019},\nthat birational equivalence of varieties with\nlogarithmic volume forms  is preserved under \"good specializations\".\n\nIn the geometric case, where the valuation is the local ring of a curve~$C$\nat point~$o$, the construction of the specialization map\ncan be viewed as a restriction to the special fiber\nof a normalization of a global residue map~$\\partial$\nthat takes place on a proper model whose special fiber is\n a divisor with normal crossings.\nThe normalization procedure extracts a subcomplex\nof the Clemens complex of the special fiber.\nA similar situation appeared in the study of Tamagawa\nmeasures on analytic manifolds \\citep{Chambert-LoirTschinkel-2010}.\n\nRelated constructions\nemerged from the seminal work of~\\citet{KontsevichSoibelman-2006}\ninspired by mirror symmetry,\nand its subsequent developments,\n\\eg, by \\citet{MustataNicaise-2015a, NicaiseXu-2016a, BoucksomJonsson-2017, JonssonNicaise-2020}.\n\nOur constructions use essentially only formal properties of the residue maps.\nConsequently, one can envision analogous theories \nfor logarithmic forms of smaller degree, Milnor $K$-theory,\nor even for the cycle modules of \\citet{Rost-1996}.\n\n\\subsection*{Acknowledgments}\nWe are grateful to Hsueh-Yung Lin and Evgeny Shinder for having pointed out a lapsus \nin~\\S\\ref{sec.biraut}.\nThe third author was partially supported by NSF grant 2000099.\n\n\\section{Logarithmic differential forms}\n\n\\subsection{K\u00e4hler differentials}\nLet $k$ be a field of characteristic zero\nand let $K$ be a finitely generated extension of~$k$;\nlet $n$ be its transcendence degree.\nThe space of K\u00e4hler differentials $\\Omega_{K\/k}$\nis the $K$-vector space generated\nby symbols $\\mathrm da$, for $a\\in K$, subject to the relations:\n\\begin{enumerate}\n\\item For $a\\in k$, one has $\\mathrm da = 0$;\n\\item For $a,b\\in K$, one has $\\mathrm d(a+b)=\\mathrm da+\\mathrm db$\nand $\\mathrm d(ab)=a\\mathrm db + b(\\mathrm da)$.\n\\end{enumerate}\n\nFor any integer~$m\\geq0$, we may consider its $m$th exterior\npower $\\Omega^{m}_{K\/k}$, which is a $K$-vector space\nof dimension~$\\binom nm$; in particular, it vanishes if $m>n$,\n$\\Omega^1_{K\/k}$ has dimension~$n$,\nand $\\Omega^n_{K\/k}$ has dimension~one.\nOne has $\\Omega^0_{K\/k}=k$, canonically.\n\nElements of $\\Omega^n_{K\/k}$, for $n=\\trdeg_k(K)$,\nare also called \\emph{volume forms}.\n\nFor $a\\in K^\\times$, we also write $\\dlog a=\\mathrm da\/a\\in\\Omega_{K\/k}$.\n\n\n\\subsection{Models}\nLet $m$ be an integer and \nlet $\\omega\\in\\Omega^m_{K\/k}$.\nA \\emph{model} of~$K$ is an integral $k$-scheme~$X$\ntogether with a $k$-isomorphism $K\\simeq k(X)$; we say\nthat this model is proper, resp.\\ smooth if $X$ is proper,\nresp.\\ smooth over~$k$. \nGiven such a model,\n $\\omega$ induces a meromorphic global section~$\\omega_X$\nof~$\\Omega^m_{X\/k}$.\nThe polar ideal of~$\\omega_X$ \nis the subsheaf of~$\\mathscr O_X$ whose local sections \nare the $a\\in\\mathscr O_X$\nsuch that $a\\omega_X$ is induced by a regular $m$-form. \nLet $D$ be the zero-locus of this ideal. Its complement~$U$\nis the largest open subscheme of~$X$ \nsuch that $\\omega_X$ is induced by a regular $m$-form on~$U$.\nIf $X$ is smooth, then $\\omega_X$ is locally free,\nhence the scheme~$D$ is an effective divisor (Hartogs's principle);\nwe call it the \\emph{polar divisor} of~$\\omega$ on~$X$.\n\n\\subsection{Logarithmic forms}\nBy Hironaka's theorem on embedded resolution of singularities,\nthere exist smooth projective  models $(X,\\omega_X)$ of $(K,\\omega)$\nsuch that the polar divisor~$D$ of~$\\omega_X$ has normal crossings.\n\nFollowing \\cite[chap.~2, \\S3]{Deligne-1970}, we then say that  $\\omega_X$ \nhas at most logarithmic poles, or that $\\omega$\nhas at most logarithmic poles on~$X$, \nif both $\\omega_X$ and $\\mathrm d\\omega_X$ have at most simple poles along~$D$. \n\nThe following lemma implies that this condition \nis essentially independent of the choice of~$X$\nsuch that the polar divisor of~$\\omega_X$ has normal crossings.\n\n\\begin{lemm}\\label{lemm.logarithmic-functoriality}\nLet $g\\colon X'\\to X$ be a morphism of smooth $k$-varieties,\nlet $D$ be a divisor with normal crossings in~$X$ and\nlet $D'$ be a divisor with normal crossings in~$X'$\nsuch that $D'=g^{-1}(D)$.\nLet $\\omega$ be a regular $m$-form on $X\\setminus D$\nand let $\\omega'=g^*\\omega$.\n\n\\begin{enumerate}\n\\item If $\\omega$ has at most logarithmic poles along~$D$, \nthen $\\omega'$ has at most logarithmic poles along~$D'$.\n\n\\item The converse holds if $g$ is proper and surjective.\n\\end{enumerate}\n\\end{lemm}\n\\begin{proof}\nThe first assertion is \\citep[chap.~II, prop.~3.2, (iv)]{Deligne-1970}.\nLet us prove the second one. \n\nConsider the\ngeneric point~$\\eta$ of~$X$ and a point $\\eta'\\in X'\\setminus D'$\nwhich is algebraic over~$k(\\eta)$.\nThe Zariski closure~$X'_1$ of~$\\eta'$ \nis proper and generically finite over~$X$,\nand $D'_1=D'\\cap X'_1$ is a divisor.\nThere is a proper modification $h\\colon X'_2\\to X'_1$ \nsuch that $D'_2=h^{-1}(D'_1)$ has normal crossings.\nBy the first part, the form $h^*\\omega'|_{X'_1}$ has at most\nlogarithmic poles along~$D'_2$. \nReplacing~$g$ by $g\\circ h$, we may assume \nthat $g$ is generically finite. \n\nSince the sheaf of forms with at most logarithmic poles along~$D$\nis locally free and $X$ is smooth, we can \ndelete from~$X$ a subset of codimension at least~2.\nThus, we may assume that $g$ is flat,\n$D$ is smooth and irreducible,\nand $g$ is \u00e9tale outside of $D$. \nIt suffices to argue \u00e9tale locally at the generic point of~$D$.\nBy the local description of ramified morphisms,\nthere are \u00e9tale local coordinates $(z_1,\\dots,z_n)$\non~$X$ such that $D_\\red=V(z_1)$, \nlocal coordinates $(z'_1,\\dots,z'_n)$ on~$X'$\nsuch that $g^*z_1=(z'_1)^e$, $g^*z_2=z'_2$, etc.,\nwhere $e$ is the ramification index of~$g$ along~$D$.\nLet $d$ be the order of the pole of~$\\omega$ along~$D$;\nwrite $\\omega = \\alpha\/z_1^d + \\beta \\wedge \\mathrm dz_1\/z_1^d$,\nwhere $\\alpha,\\beta$ are regular forms which do not involve~$\\mathrm dz_1$.\nThen \n\\[\u00a0\\omega'=g^*\\omega=g^*\\alpha\/(z'_1)^{de} \n+ e\\, g^*\\beta \\wedge \\mathrm dz'_1\/(z'_1)^{1+(d-1)e}. \\]\nAssume, by contradiction, that $d\\geq 2$, so that $de\\geq 2$\nand $1+(d-1)e\\geq 2$.\nSince $\\omega'$ has at most logarithmic poles along~$D$,\nwe get $g^*\\alpha=0$ and $g^*\\beta=0$.\nThis implies that both $\\alpha$ and $\\beta$\nare multiples of~$z_1$, contradicting the hypothesis\nthat $d$ was the order of the pole of~$\\omega$ along~$D$.\nTherefore, $d\\leq 1$. This concludes the proof.\n\\end{proof}\n\n\\subsection{}\nWe say that an $m$-form $\\omega\\in\\Omega^m_{K\/k}$ is  \\emph{logarithmic}\nif for all proper smooth models~$X$ of~$K$\nsuch that the polar divisor of~$\\omega_X$ has normal crossings,\nthe meromorphic differential form $\\omega_X$ \nhas at most logarithmic poles.\n\nBy resolution of singularities \\citep{Hironaka-1964},\ntwo models are dominated by a third one, \nhence lemma~\\ref{lemm.logarithmic-functoriality},\nimplies that it suffices that this condition \nis satisfied on some proper smooth model for which\nthe polar divisor of~$\\omega_X$ has normal crossings.\n\nAnalogously, if $X$ is a reduced $k$-variety, then we \nsay that a meromorphic $m$-form~$\\omega$ on~$X$ is logarithmic \"everywhere\"\nif for all proper birational models $(X',\\omega')$ of~$(X,\\omega)$,\nthe meromorphic $m$-form~$\\omega'$ on~$X'$ has at most logarithmic poles.\nIt suffices that this holds on one such model.\n\n\\section{Burnside rings for logarithmic forms}\n\n\\subsection{Burnside rings}\\label{ssec.burn}\nLet $k$ be a field of characteristic zero and $n$\nan integer such that $n\\geq0$.\n\\citet{KontsevichTschinkel-2019}\ndefined the Burnside group $\\burn_n(k)$ as the free abelian group\non isomorphism classes of \nfinitely generated extensions of~$k$\nof transcendence degree~$n$.\n\nAny integral $k$-variety~$X$ of dimension~$n$\nhas a class $[X]$ in $\\burn_n(k)$.\nThis gives rise to alternative useful presentations of~$\\burn_n(k)$,\nfor example involving only classes of integral projective smooth varieties.\n\nThe group\n\\[\n\\burn(k)=\\bigoplus_{n\\ge 0}  \\burn_n(k)\n\\]\ncarries a natural commutative ring structure, \nwith multiplication defined by taking products of (smooth projective) $k$-varieties:\n\\[\n[X]\\cdot [X'] = [X\\times X'].\n\\]\n\n\n\\subsection{Definition of a Burnside group for volume forms}\nLet $k$ be a field of characteristic zero and let $n$ be an integer\n$\\geq0$.\nWe define $\\Burn_n(k)$ to be the free abelian group\non isomorphisms classes of pairs $(K,\\omega)$,\nwhere \n\\begin{itemize}\n\\item\n$K$ is a finitely generated extension of $k$ of transcendence degree~$n$ and \n\\item\n$\\omega\\in\\Omega^n_{K\/k}$ is a logarithmic volume form.\n\\end{itemize}\nWe write \n\\[\n[K,\\omega] \\in \\Burn_n(k)\n\\]\nfor the class of a pair $(K,\\omega)$.\n\n\\begin{rema}\nThis definition has obvious more geometric formulations.\nFor example, we can take for generators equivalence classes\nof pairs $(X,\\omega)$, where \n\\begin{itemize}\n\\item\n$X$ is a smooth integral $k$-scheme of dimension~$n$,\nand \n\\item\n$\\omega$ a regular volume form on~$X$ which is logarithmic \n\"everywhere\",\n\\end{itemize}\nmodulo the smallest equivalence relation\nthat identifies $(X,\\omega)$ and $(X',\\omega')$\nif there exists an open immersion $f\\colon X'\\to X$\nsuch that $\\omega'=f^*\\omega$. \n\nAlternatively, we can assume that $X$ is proper, smooth and integral, \nthe form $\\omega$ is a logarithmic volume form on~$X$,\nand consider the smallest equivalence relation\nthat identifies $(X,\\omega)$ and $(X',\\omega')$\nif there exists a proper birational morphism $f\\colon X'\\to X$\nsuch that $\\omega'=f^*\\omega$.\nBy the weak factorization theorem of~\\citep{AbramovichKaruMatsukiEtAl-2002},\nthis equivalence relation is generated by such morphisms~$f$\nwhich are blowing-ups along smooth centers in good\nposition with respect to the polar divisor of~$X$.\n\nIn both contexts, if $X$ is an $n$-dimensional $k$-variety\nand $\\omega$ is a meromorphic $n$-form on~$X$ \nwhich is logarithmic \"everywhere\",\nthen we\ndefine $[X,\\omega]$ to be the sum, over all irreducible components~$Y$\nof~$X$ which have dimension~$n$,\nof the classes $[Y,\\omega|_Y]$.\n\\end{rema}\n\n\\begin{exem}\\label{exem.eps}\nFinitely generated extensions of~$k$ of transcendence degree~$0$\nare finite extensions of~$k$. Let $K$ be such an extension.\nSince $k$ has characteristic zero, \none has $\\Omega^1_{K\/k}=0$. However, $\\Omega^0_{K\/k}$,\nwhich is its $0$th exterior power, is canonically isomorphic to~$K$.\nConsequently, \n$\u00a0\\Burn_0(k) $ is the free  abelian group on isomorphism\nclasses of pairs $(K,\\lambda)$, where $K$ is a finite extension of~$k$\nand $\\lambda\\in K$.\n\nWe will let $\\mathbf 1=[\\Spec(k),1]$\nand $\\eps = [\\Spec(k),-1]$.\n\\end{exem}\n\n\\begin{exem}\\label{exem.T}\nLet $K=k(t)$. The differential form $\\mathrm dt\/t$ \nis a logarithmic volume form; indeed $X=\\P^1_k$ is a model\nof~$K$ and this form has poles of order~$1$ at~$0$ and~$\\infty$,\nand no other poles. We write $\\mathbf T$ for the class of\n$(k(t),\\mathrm dt\/t)$. \n\nNote that the $k$-isomorphism of $K$ that maps~$t$ to~$1\/t$\nmaps $\\mathrm dt\/t$ to its opposite;  consequently,\nwe also have $\\mathbf T=[k(t),-\\mathrm dt\/t]=\\eps\\cdot\\mathbf T$.\n\nIn the context of birational\ngeometry in presence of logarithmic volume forms,\n\"rational varieties\" would have class in~$\\mathbf T^n$,\nand similarly for stable birationality.\n\\end{exem}\n\n \n\\subsection{Multiplicative structure}\\label{ss.commut}\nWe view the direct sum \n\\[\u00a0\\Burn(k)=\\bigoplus_{n\\in\\N}\\Burn_n(k) \\]\nas a graded abelian group.\nIt is endowed with a multiplication such that\n\\[\u00a0[X,\\omega] \\cdot [X',\\omega'] =  [X\\times X', \\omega\\wedge \\omega'] \\]\nwhen $X,X'$ are proper, smooth and integral\nand $\\omega$, resp.~$\\omega'$ are logarithmic volume forms on~$X$, resp.~$X'$,\nand $Y$ ranges over the set of irreducible components of $X\\times X'$. \n\nLet $s\\colon X'\\times X\\to X\\times X'$ be the isomorphism\nexchanging the two factors. One has\n\\[\u00a0s^*(\\omega \\wedge \\omega') = (-1)^{nn'} \\omega'\\wedge\\omega,  \\]\nif $n=\\dim(X)$, $n'=\\dim(X')$, $\\omega$ is a volume form on~$X$ \nand $\\omega'$ is a  volume form on~$X'$.\nConsequently,\n\\[ a \\cdot b = \\eps^{nn'} \\cdot b \\cdot a \\]\nfor $a\\in\\Burn_n(k)$ and $b\\in \\Burn_{n'}(k)$.\nIn particular, classes in $\\Burn_n(k)$, for even~$n$, are central in~$\\Burn(k)$.\n\nWe remark that \nthe element~$\\mathbf T\\in\\Burn_1(k)$ is central as well.\nLet indeed $a\\in\\Burn_n(k)$. If $n$ is even, then $a\\cdot\\mathbf T=\\mathbf T\\cdot a$.\nOtherwise, we have $a\\cdot \\mathbf T=\\eps \\cdot \\mathbf T\\cdot a$,\nbut we have seen in example~\\ref{exem.T}\nthat \n$\\mathbf T=\\eps\\cdot\\mathbf T$. \nAs a consequence, $a\\cdot \\mathbf T=\\mathbf T\\cdot a$.\n\nHowever, the ring~$\\Burn(k)$ is \\emph{not} commutative.\nIndeed, consider curves $X$, $X'$ \nwithout automorphisms\nand no nonconstant morphism between them.\nThen the switch is the only isomorphism\nfrom $X'\\times X$ to~$X\\times X'$. \nTake nonzero logarithmic $1$-forms $\\omega,\\omega'$ on~$X,X'$ respectively. \nThe classes $[X\\times X',\\omega\\wedge\\omega']$\nand $[X'\\times X,\\omega'\\wedge\\omega]$ are then distinct.\n\n\n\\subsection{Functoriality}\nLet $k'$ be an extension of~$k$. Then there is a natural\nring homomorphism \n\\[\u00a0\\Burn(k)\\to\\Burn(k') \\]\ndescribed as follows.\nLet $(X,\\omega)$ be an integral $k$-variety\nof dimension~$n$ equiped with a logarithmic $q$-form. \nLet $X'=X\\otimes_k k'$\nbe its base change to~$k'$, and let $\\omega'$\nbe the volume form on~$X'$ deduced from~$\\omega$\nby base change. \nThen the class of $(X,\\omega)$ maps to \nthe sum of classes $(Y,\\omega'|_Y)$, where $Y$ runs\nthe (finite) set of irreducible components of~$X'$.\n\nIf $k'$ is a finite extension of~$k$,\nwe also have a trace map\n\\[\u00a0\\Tr_{k'\/k} \\colon \\Burn(k')\\to \\Burn(k) \\]\nobtained by averaging over a set of representatives\nof automorphisms of the Galois closure of~$k'$ over~$k$ modulo \nthose preserving~$k'$.\n\n\\subsection{Relation with the classical Burnside group}\nForgetting the form~$\\omega$ gives a ring morphism\n\\[\u00a0\\pi\\colon \\Burn(k) \\to \\burn(k). \\]\nOn the other hand, if $K$ is a finitely generated\nextension of~$k$ of transcendence degree~$n$, \nwe can endow it with the zero $n$-form.\nThe resulting map\n\\[\u00a0\\varpi\\colon \\burn(k) \\to \\Burn(k) \\]\nidentifies $\\burn(k)$ with an ideal of $\\Burn(k)$.\nOne has $\\pi\\circ\\varpi=\\id$.\n\n\\subsection{Variations on the theme}\nThe construction of the Burnside ring~$\\Burn(k)$ admits\nseveral natural variants that are relevant in more specific contexts.\nSome of them will be used in later sections.\n\n\\subsubsection{A relative ring}\nLet $n$ be an integer.\nFor any $k$-scheme~$S$, we define $\\Burn_n(S\/k)$\nas the free abelian group on triples $(X,\\omega,u)$\nwhere $X$ is an integral smooth $n$-dimensional $k$-scheme,\n$\\omega\\in\\Omega^n_{X\/k}$ is a regular volume form\nwhich is logarithmic \"everywhere\",\nand $u\\colon X\\to S$ is a morphism, \nmodulo the smallest equivalence relation \nthat identifies $(X,\\omega,u)$ and $(X',\\omega',u')$\nif there exists an open immersion $f\\colon X'\\to X$\nsuch that $\\omega'=f^*\\omega$ and $u'=u\\circ f$.\n\nLet $h\\colon S\\to T$ be a morphism of $k$-schemes.\nIt induces a morphism of abelian groups \n\\[\nh_*\\colon \\Burn_n(S\/k)\\to \\Burn_n(T\/k)\n\\]\nsuch that $h_*([X,\\omega,u])=[X,\\omega,h\\circ u]$\nfor any triple $(X,\\omega,u)$ as above.\n\n\\subsubsection{Pluriforms}\nOne can replace volume forms with volume $r$-pluriforms,\nthat is, elements of~$(\\Omega^n_{K\/k})^{\\otimes r}$,\nfor some given integer~$r$. \nThe corresponding logarithmic condition requires\nthat the pluriform has poles of order at most~$r$ on \nan adequate model.\nNote that when $r$ is even, the obtained ring is commmutative.\n\n\\subsubsection{Forms up to scalars}\nIn the construction, we may wish to identify\n$(K,\\omega)$ and $(K',\\omega')$\nif there exists $\\lambda\\in k^\\times$, resp. $\\lambda \\in\\{\\pm1\\}$,\nand a $k$-isomorphism $f\\colon K\\to K'$ such that $f^*\\omega'=\\lambda\\omega$.\nThese variants also give rise to a commutative ring.\n\n\\subsubsection{Group actions}\\label{sss.group}\nLet $G$ be a profinite group scheme over~$k$.\nOne can also consider pairs $(K,\\omega)$,\nwhere the field~$K$ is endowed with an action of~$G$\nleaving the form~$\\omega$ invariant.\nThe obtained ring will be denoted by~\n$\u00a0\\Burn^G(k)$.\n\n\n\\section{Residues}\\label{sec.residues}\n\n\\subsection{Residue of a volume form}\nLet $X$ be an equidimensional smooth $k$-variety of dimension~$n$.\n\nLet $D$ be a smooth divisor on~$X$.\nWe denote by $\\Omega^m_{X\/k}(\\log D)$\nthe sheaf of $m$-forms on~$X$ with logarithmic poles along~$D$, \nlocally of the form $\\eta \\wedge \\mathrm d\\log f + \\eta' $,\nwhere $\\eta$ and $\\eta'$ are regular and $f$ is a local equation of~$D$.\nThe residue map is the homomorphism  of $\\mathscr O_X$-modules\n\\[\n\\rho_D\\colon \\Omega^{m}_{X\/k}(\\log D)\\to \\Omega^{m-1}_{D\/k},\n\\]\ncharacterized by the relation\n\\[\n\\rho_D(\\eta \\wedge \\mathrm d \\log f + \\eta' )\n=\\eta|_D\n\\] \nfor every local sections $\\eta\\in\\Omega^{m-1}_{X\/k}$\nand $\\eta'\\in\\Omega^m_{X\/k}$, \nand any local generator~$f$ of the ideal of~$D$.\n\nIf $\\omega$ is a logarithmic $m$-form on~$X$,\nthere is an open subset~$U$ of~$X$ such that $U\\cap D\\neq\\emptyset$\nand such that $\\omega|_U$ belongs to $\\Omega^m_{X\/k}(\\log D)$.\nIts residue $\\rho_D(\\omega|_U)$ is then\na meromorphic section of~$\\Omega^{m-1}_{D\/k}$.\n\n\\begin{lemm}\\label{lemm.residue-log}\nLet $\\omega$ be a logarithmic differential form of degree~$m$ on~$X$.\nThen $\\rho_D(\\omega)$ is a logarithmic $(m-1)$-form on~$D$.\n\\end{lemm}\n\\begin{proof}\nWe may assume that the sum of~$D$ and of the polar divisor of~$\\omega$\nhas strict normal crossings. The assertion \nis then evident in  local  coordinates.\n\\end{proof}\n\n\n\n\n\\subsection{Blowing-ups and normal bundles}\n \nLet $Y$ be a smooth closed subscheme of~$X$.\nThe blow-up $\\Bl_Y(X)$ of~$X$ along~$Y$ is a smooth $k$-variety.\nThe blowing-up morphism $b_Y\\colon \\Bl_Y(X)\\to X$ is an isomorphism\nover the complement of~$Y$.\nIf $Y$ is nowhere dense and nonempty,\nthen $E_Y=b_Y^{-1}(Y)$ is a smooth divisor in~$\\Bl_Y(X)$. \n\nIn general, $E_Y=b_Y^{-1}(Y)$ identifies, as an $Y$-scheme, with\nthe projectivization of the normal bundle $\\mathscr N_Y(X)$\nof~$Y$ in~$X$. \n\nLet $W$ be a closed smooth subscheme of~$X$. Assume that $W$ and~$Y$ \nare transversal. Then the Zariski closure of $b_Y^{-1}(W\\setminus (Y\\cap W))$\nis called the strict transform of~$W$ in~$\\Bl_Y(X)$.\nIt identifies with $\\Bl_{Y\\cap W}(W)$.\n\nLet now $\\omega$ be a logarithmic $m$-form on~$X$.\nThen the form $b_Y^*\\omega$ on~$\\Bl_Y(X)$ is logarithmic;\nassuming that $Y$ is nonempty and nowhere dense, we can consider\nthe residue $\\rho_Y(\\omega)$\nof~$b_Y^*\\omega$ along~$E_Y$. It is a logarithmic $(n-1)$-form\non $\\P(\\mathscr N_Y(\\mathscr X))$.\n\n\\begin{defi}\\label{defi.rho}\nLet $X$ be an irreducible proper smooth $k$-variety,\nlet $n$ be its dimension and\nlet $\\omega\\in\\Omega^n_{X\/k}$ be  logarithmic volume form\nwhose polar divisor~$D$ has strict normal crossings.\nLet $(D_\\alpha)_{\\alpha\\in\\mathscr A}$ be the family of its\nirreducible components; for $A\\subset\\mathscr A$,\nwe let $D_A=\\bigcap_{\\alpha\\in A}D_\\alpha$.\nWe then define an element $\\rho(X,\\omega)$ in $\\Burn_{n-1}(X\/k)$\nby the formula:\n\\[\u00a0\\rho(X,\\omega) = \\sum_{\\emptyset\\neq A\\subset \\mathscr A}\n (-1)^{\\Card(A)-1}  \\rho_{D_A}(X,\\omega) .\\]\n\\end{defi}\n(In this formula and all similar ones below, it is always\nimplicit that the terms where $D_A=\\emptyset$ are omitted.)\n\n\\subsection{Iterated residues}\\label{ss.iter}\nWe retain the notation of definition~\\ref{defi.rho}\n\nFix \na logarithmic volume form~$\\omega$ on~$X$\nand \na nonempty subset~$A$ of~$\\mathscr A$\nsuch that $D_A\\neq\\emptyset$.\nIt will be useful to compute  inductively\nthe logarithmic volume form $\\rho_{D_A}(\\omega)$\nthat appears in definition~\\ref{defi.rho}.\n\nLet $b_A\\colon \\tilde X\\to X$ be the blowing-up of~$X$\nalong~$D_A$ and let $E$ be its exceptional divisor.\n\nWhen $A=\\{\\alpha\\}$ has a single element, \n$D_A$ is the divisor~$D_\\alpha$,\nthe blowing-up morphism~$b_A$ is an isomorphism\nand the exceptional divisor identifies with~$D_A$.\nThen \n\\[\u00a0\\rho_{D_A}(X,\\omega)=[D_\\alpha,\\rho_{D_\\alpha}(\\omega),j_\\alpha], \\]\nwhere  $j_\\alpha$ is the immersion of~$D_\\alpha$ into~$X$.\n\nThis construction can be pursued in higher codimension,\nusing iterated residues.\nFix a total order on~$\\mathscr A$.\nThere is a unique, strictly increasing sequence\n$(\\alpha_1,\\dots,\\alpha_m)$ in~$\\mathscr A$ such that \n$A=\\{\\alpha_1,\\dots,\\alpha_m\\}$.\nGiven the chosen order on~$\\mathscr A$,\nwe may apply the iterated residues construction \nand obtain a logarithmic form of degree~$n-m$\n\\[\u00a0\\rho_{D_A} (\\omega) \n= \\rho_{D_{\\alpha_1}} \\circ \\dots \\circ \\rho_{D_{\\alpha_m}}(\\omega). \\]\nOn a nonempty open subset~$U$ of~$X$ that meets~$D_A$, we may write\n\\[\u00a0\\omega=\\eta \\wedge \\dlog(f_{\\alpha_1})\\wedge \\dots\n\\dlog (f_{\\alpha_m}), \\]\nfor a regular form~$\\eta$, \nand then one has $\\rho_{D_A}(\\omega)=\\eta|_{U\\cap D_A}$.\n\nDenote by~$b_A$ the blowing-up of~$X$ along~$D_A$\nand by~$E_A$ its exceptional divisor;\nrecall that $E_A$ identifies with the projectivized normal\nbundle~$\\mathscr N_{D_A}(X)$ of~$D_A$ in~$X$.\nUsing local equations for the divisors~$D_\\alpha$, for $\\alpha\\in A$,\nwe trivialize $\\mathscr N_{D_A}(X)$ on a dense open subscheme of~$D_A$;\nthis gives a birational isomorphism of~$E_A$ with~$D_A \\times \\P^{m-1}$\n(with $m=\\Card(A)$), and a local computation gives the formula\n\\[\u00a0\\rho_{D_A}(X,\\omega) = \n[D_A,\\rho_{D_A}(\\omega)]\\cdot \\mathbf T^{m-1} \\]\n in $\\Burn_{n-1}(D_A\/k)$.\n\nWhen $m\\geq 2$, \nthe definition of~$\\rho_{D_A}$ actually depends \non the chosen order of~$\\mathscr A$, but only \nup to a sign, so that  the class $[D_A,\\rho_{D_A}(\\omega)]$\nis well defined up to multiplication\nby the class~$\\eps\\in\\Burn_0(k)$. \nOn the other hand,\nit is multiplied by $\\mathbf T^{m-1}$\nand we have observed that $\\eps \\cdot \\mathbf T=\\mathbf T$.\n\n\n\\begin{prop}\\label{prop.good-blow-up}\nLet $(X,\\omega)$, $D$, and $(D_\\alpha)_{\\alpha\\in \\mathscr A}$\nbe as in definition~\\ref{defi.rho}.\nLet $Y$ be a strict irreducible subvariety of~$X$;\nlet $\\mathscr A_Y$ be the set of all $\\alpha\\in \\mathscr A$\nsuch that $Y\\not\\subset D_\\alpha$;\nwe assume that $\\sum_{\\alpha\\in \\mathscr A_Y} D_\\alpha$\nmeets~$Y$ transversally.\n\nLet $g\\colon X'\\to X$ be the blowing-up of~$X$ along~$Y$\nand let $\\omega'=g^*\\omega$; it is a logarithmic\nform, its polar divisor\nhas strict normal crossings, and we have \n\\[\u00a0g_* \\rho (X',\\omega') = \\rho(X,\\omega) \\]\nin $\\Burn(X\/k)$.\n\\end{prop}\n\\begin{proof}\nLet $E=g^{-1}(Y)$ be the exceptional divisor;\nfor each $\\alpha\\in\\mathscr A$, let $D'_\\alpha$\nbe the strict transform of~$D_\\alpha$.\nThe blow-up~$X'$\nis smooth;\nthe  divisor $E+\\sum_{\\alpha\\in\\mathscr A}D'_\\alpha$\nhas strict normal crossings and\ncontains the polar divisor of~$\\omega'$.\n\nLet $B$ be the set of all~$\\beta\\in\\mathscr A$ such that $Y\\subset D_\\beta$,\nso that $D_B$ is the minimal stratum containing~$Y$.\n\nWe now split the discussion into two cases.\n\n\\begin{enumerate}\n\\item\n\\emph{Assume that $\\dim(Y)< \\dim(D_B)$.} \nSince $g$ is ramified along~$E$, its Jacobian vanishes along~$E$.\nSince $\\omega$ has poles of order at most~one, \nthe form $\\omega'=g^*\\omega$ is regular at the generic point of~$E$.\nConsequently, the polar divisor of~$\\omega'$ does not contain~$E$\nand we have to compare\n\\[\u00a0\\sum_{\\emptyset\\neq A\\subset \\mathscr A} (-1)^{\\Card(A)-1}\n  \\rho_{D'_A} (\\omega') \n\\]\nwith\n\\[\u00a0\\sum_{\\emptyset\\neq A\\subset \\mathscr A} (-1)^{\\Card(A)-1}\n  \\rho_{D_A} (\\omega) .\\]\n\nSince $g$ is a local isomorphism around\nthe generic points of~$D_\\alpha$, for $\\alpha\\in\\mathscr A$,\nwe see that the polar divisor of~$\\omega'$ \nis equal to $\\sum_{\\alpha\\in{\\mathscr A}}D'_\\alpha$.\nFor every nonempty subset~$A$ of~$\\mathscr A$,\none has\n\\[\u00a0g_* \\rho_{D_A}(X',\\omega')  = \\rho_{D_A}(X,\\omega) \\]\nfor every nonempty subset~$A$ of~$\\mathscr A$,\nwhich implies the desired formula in this case.\n\n\\item\n\\emph{Assume that $\\dim(Y)=\\dim(D_B)$.}\nIn this case, $Y$ is an irreducible component of~$D_B$. \nSince $D_{\\emptyset}=X$ and $Y\\neq X$, we have $B\\neq\\emptyset$.\nWe have to compare the expression\n\\[\u00a0\\sum_{\\emptyset\\neq A\\subset \\mathscr A} (-1)^{\\Card(A)-1}\n  \\rho_{D'_A} (\\omega') \n+ \\sum_{A\\subset\\mathscr A} (-1)^{\\Card(A)}  \\rho_{E\\cap D'_A}(\\omega') \\]\nwith\n\\[\u00a0\\sum_{\\emptyset\\neq A\\subset \\mathscr A} (-1)^{\\Card(A)-1}\n  \\rho_{D_A} (\\omega) .\\]\nThe argument takes place in a neighborhood of~$Y$, \nwhich allows us to assume that $Y=D_B$.\n\nLet $A$ be a nonempty subset of~$\\mathscr A$.\nOne has $D'_A=\\emptyset$ whenever $B\\subset A$, and the corresponding\nterms are absent from the second expression.\nOn the other hand, if $B\\not\\subset A$, \nthe morphism~$g$  identifies~$D'_A$ with the blow-up\nof~$D_A$ along~$D_{A}\\cap Y=D_{A\\cup B}$. \nIn particular, $g$ induces a birational isomorphism from~$D'_A$ to~$D_A$,\nso that $g_* \\rho_{D'_A}(X',\\omega')=\\rho_{D_A}(X,\\omega)$.\nMoreover, $E\\cap D'_A$ is the projectivized\nnormal bundle $\\mathbf P\\mathscr N_{D_{A\\cup B}}(D_A)$,\nand \n\\[\u00a0g_* \\rho_{E\\cap D'_A} (X',\\omega')= \\rho_{D_{A\\cup B}}(X,\\omega). \\]\nSimilarly, one has \n\\[\u00a0g_*\u00a0\\rho_E(X',\\omega') = \\rho_{D_B}(X,\\omega). \\]\nThis gives a formula of the form \n\\begin{align*}\n\u00a0g_*\\rho(X',\\omega') & = \\sum_{\\substack{\\emptyset \\neq A \\subset \\mathscr A \\\\ B \\not\\subset A}}  (-1)^{\\Card(A)-1} \\rho_{D_A}(X,\\omega)\n+ \\sum_{\\substack{ A \\subset \\mathscr A \\\\ B \\not\\subset A}} (-1)^{\\Card(A)} \\rho_{D_{A\\cup B}}(X,\\omega)  \\\\\n& = \\sum_{\\emptyset \\neq A \\subset\\mathscr A} n'_A \\rho_{D_A}(X,\\omega), \n\\end{align*}\nwhere\n\\[\n\u00a0n'_A = \\begin{cases} (-1)^{\\Card(A)-1} & \\text{if $B\\not\\subset A$,} \\\\\n \\sum_{\\substack{ C \\subset \\mathscr A \\\\ B \\not\\subset C \n \\\\ C \\cup B = A }} (-1)^{\\Card(C)} & \\text{if $B\\subset A$.} \\end{cases} \\]\nIt suffices to prove that $n'_A=n_A$ \nfor any nonempty subset~$A$ of~$\\mathscr A$.\nThis is obvious when $B\\not\\subset A$, so let us assume that $B\\subset A$.\nIn the sum that defines~$n'_A$,  we write $C=(C\\setminus B)\\cup C'$,\nwhere $C'=C\\cap B$ is a subset of~$B$;\nthe condition $C\\cup B=A$ means $C\\setminus B=A\\setminus B$;\nthe condition $B\\not\\subset C$ means $C'\\neq B$.\nConsequently, we have\n\\begin{align*}\n\u00a0n'_A & = (-1)^{\\Card(A\\setminus B)}\n\\sum_{\\substack{C' \\subset B \\\\\u00a0C' \\neq B}} (-1)^{\\Card(C')} \\\\\n& = (-1)^{\\Card(A\\setminus B)}\n\\left( \\sum_{C' \\subset B } (-1)^{\\Card(C')} \n- (-1)^{\\Card(B)}\\right) \\\\\n& = (-1)^{\\Card(A\\setminus B)}\n\\left( (1-1)^{\\Card(B)} - (-1)^{\\Card(B)}\\right) \\\\\n& = (-1)^{\\Card(A)-1}, \n\\end{align*}\nsince $\\Card(B)\\geq1$.\nThis concludes the proof of the proposition.\n\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{theo}\\label{theo.residue-bir}\nLet $(X,\\omega)$ be as in definition~\\ref{defi.rho}.\nIf $X$ is proper, then the image of $\\rho(X,\\omega)$\nin~$\\Burn_{n-1}(k)$ only depends on the class $[X,\\omega]\\in\n\\Burn_n(k)$.\nIt gives rise to a morphism of abelian groups\n\\[\u00a0\\partial_n\\colon \\Burn_n(k)\\to \\Burn_{n-1}(k). \\]\n\\end{theo}\n\\begin{proof}\nBy the definition of $\\Burn_n(k)$ involving\npairs $(X,\\omega)$ where $X$ is proper,\nit suffices to consider two pairs $(X,\\omega)$\nand $(X',\\omega')$ as in definition~\\ref{defi.rho}\nwhich are related by a proper birational morphism\n$g\\colon X'\\to X$ such that $g^*\\omega=\\omega'$.\nBy the weak factorization theorem of~\\cite{AbramovichKaruMatsukiEtAl-2002},\nin order to prove the theorem,\nwe may assume that $g$ is a blowing-up\nof~$X$ along a smooth subvariety which is transversal\nto the polar divisor of~$\\omega$. In\nthis case, proposition~\\ref{prop.good-blow-up} \nasserts that $g_*\\rho(X',\\omega')=\\rho(X,\\omega)$\nin $\\Burn(X\/k)$. In particular,\nthe images in~$\\Burn(k)$\nof $\\rho(X',\\omega')$ and $\\rho(X,\\omega)$\nare equal.\n\\end{proof}\n\n\\begin{exem}\\label{exem.residue-torus}\nThe meromorphic differential form $\\mathrm dt\/t$ on~$\\P^1_k$\nhas residues~$1$ and~$-1$ at~$0$ and~$\\infty$ respectively.\nBy construction, we thus have \n\\[\u00a0\\partial_1(\\mathbf T) = [\\Spec(k), 1] + [\\Spec(k),-1]\n = \\mathbf 1 + \\eps . \\]\n\nLet $n$ be an integer such that $n\\geq 2$ \nand let us compute $\\partial_n(\\mathbf T^n)$.\nWe view $\\mathbf T^n$ as the class of~$\\P^n$, with homogeneous\ncoordinates $[1:x_1:\\dots:x_n]$, and with the toric differential\nform\n\\[\n\\omega_n=(\\mathrm dx_1\/x_1)\\wedge \\dots (\\mathrm dx_n\/x_n).\n\\]\nIts divisor is the sum of the toric hyperplanes $D_0,\\dots,D_n$.\nEach of these hyperplanes identifies with $\\P^{n-1}$, \nand $\\rho_{D_j}(\\omega_n)$ is $(-1)^{n-j}\\omega_{n-1}$.\nLet $\\mathscr A=\\{0,\\dots,n\\}$.\nIf $A=\\mathscr A$, then $D_A=\\emptyset$.\nOtherwise, we see by induction\nthat  $D_A$ is isomorphic to~$\\P^{n-\\Card(A)}$\nand $\\rho_{D_A}(\\omega_n)$ identifies with $\\pm \\omega_{n-\\Card(A)}$,\nso that \n\\[\u00a0[D_A,\\rho_{D_A}(\\omega_n)]\\cdot \\mathbf T^{\\Card(A)-1} = \n[\\gm^{n-1},\\pm \\omega_{n-1}] = \\mathbf T^{n-1}, \\]\nsince $n-1\\geq 1$.\nThen,\n\\begin{align*}\n\\partial_n(\\mathbf T^n) \n& = \\sum_{\\emptyset \\neq A\\subset \\mathscr A}\n (-1)^{\\Card(A)-1} [D_A, \\rho_{D_A}(\\omega_n)]\u00a0\\cdot \\mathbf T^{\\Card(A)-1} \\\\\n& = \\sum_{\\emptyset \\neq A\\subsetneq \\mathscr A} (-1)^{\\Card(A)-1} \\mathbf T^{n-1}.\n\\end{align*}\nNow,\n\\[\u00a0\\sum_{\\emptyset \\neq A\\subsetneq\\mathscr A} (-1)^{\\Card(A)-1}\n= 1 - (1-1)^{n+1}+ (-1)^{n+1} = \n\\begin{cases} \n 2 & \\text{if $n$ is odd;} \\\\\n0 & \\text{if $n$ is even}. \n\\end{cases} \\]\nWe get \n$\u00a0\\partial_n(\\mathbf T^n) = 2 \\mathbf T^{n-1}$ \nif $n$ is odd and $\\partial_n(\\mathbf T^n)=0$ if $n$ is even.\n(Remind that $n\\geq 2$.)\nSince $\\mathbf T=\\eps\\cdot\\mathbf T$, the following formula unifies \nthe various cases: for $n\\geq 1$, we have\n\\[\u00a0\\partial_n(\\mathbf T^n) = (1+(-1)^{n-1}\\eps)\\cdot \\mathbf T^{n-1}. \\]\n\\end{exem}\n\n\\begin{prop}\\label{prop.residue-times-T}\nFor every class $b\\in\\Burn_n(k)$, we have\n\\[\u00a0\\partial_{n+1} (b\\cdot\\mathbf T)\n = - \\partial_n(b) \\cdot\\mathbf T+ b\\cdot \\partial_1\\mathbf T. \\]\n\\end{prop}\n\\begin{proof}\nWe may assume that $b=[X,\\omega]$, where \n$X$ is a proper integral smooth variety of dimension~$n$,\nand $\\omega$ is a logarithmic volume form on~$X$\nwhose polar divisor has strict normal crossings.\nLet $(D_\\alpha)_{\\alpha\\in\\mathscr A}$ be the family of its irreducible\ncomponents.\nWe view $b\\cdot\\mathbf T$\n as the class of $[X\\times \\mathbf P^1,\\omega \\wedge \\mathrm dt\/t]$.\nThe polar divisor of~$\\omega \\wedge\\mathrm dt\/t$ is equal to\n\\[\u00a0\\sum_{\\alpha\\in\\mathscr A} D_\\alpha \\times \\P^1 +\n X \\times \\{0\\} + X \\times \\{\\infty\\}. \\]\nIt has strict normal crossings, and its strata are \nof the form $D_A \\times \\P^1$, for nonempty $A\\subset\\mathscr A$,\nor $D_A \\times\\{0\\}$, or $D_A \\times\\{\\infty\\}$, for $A\\subset\\mathscr A$.\nThis decomposes $\\partial_{n+1}(b\\times\\mathbf T)$ as \nthe sum of three terms.\n\nThe first one is\n\\[\u00a0\\sum_{\\emptyset\\neq  A\\subset\\mathscr A} [D_A\\times\\P^1,\\rho_{D_A\\times\\P^1}(\\omega\\wedge\\mathrm dt\/t)]\\cdot \\mathbf T^{\\Card(A)-1}. \\]\nFor any nonempty subset~$A$ of~$\\mathscr A$, one has \n\\[\u00a0\\rho_{D_A\\times\\P^1}(\\omega\\wedge \\mathrm dt\/t) = \\pm \\rho_{D_A}(\\omega)\\wedge \\mathrm dt\/t, \\]\nso that\n\\[\u00a0 [D_A\\times\\P^1,\\rho_{D_A\\times\\P^1}(\\omega\\wedge\\mathrm dt\/t)]\\cdot \\mathbf T^{\\Card(A)-1}\n= [D_A,\\rho_{D_A}(\\omega)]\u00a0\\cdot \\mathbf T\\cdot \\mathbf T^{\\Card(A)-1}.  \\]\nConsequently, the first term equals\n$\u00a0\\partial_n(b)\\times\\mathbf T$.\n\nWrite $D_0=X\\times\\{0\\}$ and $D_\\infty=X\\times\\{\\infty\\}$, and\nidentify both divisors to~$X$.\nFor a subset~$A$ of~$\\mathscr A$, \nwe have\n\\[\u00a0\\rho_{D_{A\\cup\\{0\\}}} (\\omega\\wedge\\mathrm dt\/t)\n= \\rho_{D_A} \\circ \\rho_{D_0} (\\omega\\wedge\\mathrm dt\/t)\n = \\rho_{D_A} (\\omega). \\]\nConsequently, the second term is equal to\n\\[\u00a0\\sum_{A\\subset \\mathscr A}\n (-1)^{\\Card(A)} [D_A,\\rho_{D_A}(\\omega)] \\cdot \\mathbf T^{\\Card(A)}\n= [X,\\omega]\u00a0 - \\partial_n(b) \\cdot \\mathbf T. \\]\n\nSimilarly, the third term is equal to\n\\[\u00a0[X,-\\omega] - \\partial_n(b)\\cdot \\mathbf T. \\]\n\nSumming up these three terms, we get\n\\[\u00a0\\partial_{n+1}(b\\times\\mathbf T)\n= - \\partial_n(b) \\cdot \\mathbf T + [X,\\omega] + [X,-\\omega]. \\]\nWe now recall that $\\partial_1(\\mathbf T)=[\\Spec(k),1]+[\\Spec(k),-1]$,\nso that\n\\[\u00a0[X,\\omega] + [X,-\\omega] = [X,\\omega]\u00a0\\cdot \\partial_1(\\mathbf T)\n= b \\cdot \\partial_1(\\mathbf T). \\]\nThis concludes the proof.\n\\end{proof}\n\n\n\\begin{theo}\\label{theo.partial-deriv}\nLet $a\\in\\Burn_m(k)$ and $b\\in\\Burn_n(k)$;\nwe have\n\\[\u00a0\\partial_{m+n}(a\\cdot b)\n= \\eps^n\\cdot   \\partial_m(a) \\cdot b + a \\cdot \\partial_n(b) - \\mathbf T \\cdot \\partial_m(a)\\cdot \\partial_n(b) \\]\nin $\\Burn_{m+n-1}(k)$.\n\\end{theo}\n\\begin{proof}\nIt suffices to treat the case where $a$ and $b$ are classes \nof proper integral smooth varieties $(X,\\omega)$,\n$(Y,\\eta)$, endowed with meromorphic volume forms\nwhose polar divisors have strict normal crossings and no multiplicities.\nLet $(D_\\alpha)_{\\alpha\\in\\mathscr A}$ be the irreducible\ncomponents of the polar divisor of~$\\omega$,\nlet $(E_\\beta)_{\\beta\\in\\mathscr B}$ be the irreducible\ncomponents of the polar divisor of~$\\eta$.\nThen $[X,\\omega]\\cdot [Y,\\eta]$\nis the class of $[X\\times Y, \\omega\\wedge\\eta]$;\nthe polar divisor of~$\\omega\\wedge\\eta$ is equal to\n\\[\u00a0\\sum_{\\alpha\\in\\mathscr A} D_\\alpha \\times Y + \\sum_{\\beta\\in \\mathscr B} X \\times E_\\beta. \\]\nWe fix a total order on the disjoint union of~$\\mathscr A$\nand~$\\mathscr B$ such that the elements of~$\\mathscr A$ \nare smaller than those of~$\\mathscr B$.\nFor any subsets $A,B$ of~$\\mathscr A$ and~$\\mathscr B$,\nobserve that we have\n\\[\u00a0\\rho_{D_{A\\cup B}} (\\omega\\wedge\\eta) \n = \\pm \\rho_{D_A}(\\omega) \\wedge \\rho_{E_B}(\\eta), \\]\nwhere $\\rho_{D_A}$ has to be understood as\nthe identity when $A$ is empty, and similarly for~$\\rho_{E_B}$.\nThe sign is~$1$ when $A=\\emptyset$;\nwhen $B=\\emptyset$, it is equal to~$(-1)^{\\Card(A)n}$;\nwe won't need to use its explicit value in the other cases.\nThen we can write\n$\u00a0\\partial ([X,\\omega]\\cdot [Y,\\eta]) $ as\n\\[\n \\sum_{\\substack{A\\subset \\mathscr A \\\\ B\\subset \\mathscr B \\\\\n A\\cup B\\neq\\emptyset}}\n(-1)^{\\Card(A)+\\Card(B)-1}\n[\u00a0D_A\\times E_B, \\pm \\rho_{A}(\\omega)\\wedge\\rho_{E_B}(\\eta)]\n \\cdot \\mathbf T^{\\Card(A\\cup B)-1} \\]\nand we split it into the sum of three terms,\naccording to which $B=\\emptyset$, or $A=\\emptyset$, \nor none of them is empty.\nThe first two terms are respectively equal to \n\\[\u00a0\\sum_{\\emptyset\\neq A\\subset\\mathscr A}\n(-1)^{\\Card(A)-1} [D_A\\times Y,(-1)^{n\\Card(A)}\\rho_{D_A}(\\omega)\\wedge\\eta]\n\\cdot \\mathbf T^{\\Card(A)-1}\n= \\partial([X,(-1)^n \\omega])\\cdot [Y,\\eta] \\]\nand\n\\[\u00a0\\sum_{\\emptyset\\neq B\\subset\\mathscr B}\n(-1)^{\\Card(B)-1} [X\\times E_B, \\omega\\wedge \\rho_{E_B}(\\eta)]\n\\cdot \\mathbf T^{\\Card(B)-1}\n= [X,\\omega]\\cdot \\partial([Y,\\eta]), \\]\nsince $\\mathbf T$ belongs to the center of~$\\Burn(k)$.\nAs for the third one, \nwe obtain\n\\[ - \\sum_{\\emptyset \\neq B\\subset \\mathscr B} \n (-1)^{\\Card(B)-1} \\sum_{\\emptyset \\neq A\\subset \\mathscr A}\n (-1)^{\\Card(A)-1}\u00a0[D_A, \\rho_{D_A}(\\omega)]\u00a0\\cdot [E_B,\\rho_{E_B}(\\eta)]\n \\cdot \\mathbf T^{\\Card(A)+\\Card(B)-2}  \\]\nwhich equals\n\\[\u00a0- \\partial ([X,\\omega]) \\cdot \\partial([Y,\\eta])\\cdot \\mathbf T. \\]\nFinally, we get\n\\begin{align*}\n\u00a0\\partial_{m+n}(a\\cdot b)  & = \n\u00a0\\partial_{m+n}([X,\\omega]\\cdot [Y,\\eta])  \\\\\n& = \\partial_m([X,(-1)^n\\omega) \\cdot [Y,\\eta]\n+ [X,\\omega] \\cdot \\partial_n([Y,\\eta]) \\\\\n& \\qquad{}  - \\mathbf T \\cdot \\partial_m([X,\\omega]) \\cdot \\partial_n([Y,\\eta]) \\\\\n& = \\eps^n \\cdot \\partial_m(a) \\cdot b + a \\cdot \\partial_n(b) \n  - \\mathbf T \\cdot \\partial_m(a) \\cdot \\partial_n(b) \n\\end{align*}\nas was to be shown. \n\\end{proof}\nIn particular, using the computation of example~\\ref{exem.residue-torus},\nwe obtain the following generalization of proposition~\\ref{prop.residue-times-T}.\n\\begin{coro} \nFor any $a\\in\\Burn_m(k)$ and any integer~$n$, we have\n\\[\u00a0\\partial_{m+n}(a\\cdot \\mathbf T^n) \n= \\begin{cases}\n\\partial_m(a) \\cdot \\mathbf T^n  & \\text{if $n$ is even;} \\\\ \n-\\partial_m(a) \\cdot \\mathbf T^n +  a \\cdot\\partial_n(\\mathbf T^{n}) \n& \\text{if $n$ is odd.} \\end{cases}\n\\]\n\\end{coro}\n\n\\begin{rema}\nFor the variant of $\\Burn(k)$ where we consider forms up to sign,\nthe formula of theorem~\\ref{theo.partial-deriv}\nsimplifies to \n\\[\u00a0\\partial_{m+n}(a\\cdot b)\n=  \\partial_m(a) \\cdot b + a \\cdot \\partial_n(b) - \\mathbf T \\cdot \\partial_m(a)\\cdot \\partial_n(b). \\]\n\\end{rema}\n\n\\section{A complex of Burnside rings}\n\n\\begin{theo}\\label{theo.dd=0}\nFor any integer $n\\geq 2$, we have\n\\[\u00a0\\partial_{n-1} \\circ \\partial_n = 0. \\]\n\\end{theo}\nIn other words,\nthe residue morphisms of Burnside groups give rise to a complex\n\\[\u00a0\\dots \\to \\Burn_n(k) \\to \\Burn_{n-1}(k)\u00a0\\to \\dots \\to \\Burn_1(k) \\to\\Burn_0(k) \\]\n\n\\begin{proof}\nIt suffices to prove the following result:\nLet $(X,\\omega)$ be an integral proper smooth variety \nof dimension~$n$\nequipped with a meromorphic volume form~$\\omega$\nwhose polar divisor has strict normal crossings and \nno multiplicities;\nthen $\\partial_{n-1}(\\partial_n ([X,\\omega]))=0$.\n\nLet $(D_\\alpha)_{\\alpha\\in\\mathscr A}$ be the family\nof irreducible components of the polar divisor of~$\\omega$ in~$X$.\nBy definition, one has\n\\[\u00a0\\partial_n ([X,\\omega]) = \n\\sum_{\\emptyset\\neq A\\subset \\mathscr A}\n(-1)^{\\Card(A)-1}  \\rho_{D_A}(X,\\omega). \\]\nFix a total order on~$\\mathscr A$.\nLet $(\\alpha_1,\\dots,\\alpha_m)$ be a strictly increasing sequence\nin~$\\mathscr A$ and let $A=\\{\\alpha_1,\\dots,\\alpha_m\\}$.\nWe have seen in~\\S\\ref{ss.iter}\nthat $\\rho_{D_A}(X,\\omega)$ can be defined \nvia iterated residue maps: \n\\[\u00a0\\rho_{D_A}([X,\\omega])=\n[D_A, \\rho_{D_{\\alpha_1}}\\circ \\dots \\circ \\rho_{D_{\\alpha_m}} (\\omega)]\n\\cdot \\mathbf T^{\\Card(A)-1}\n= [D_A,\\omega_A]\\cdot \\mathbf T^{\\Card(A)-1} \n \\]\nwhere we wrote $\\omega_A$ for\nthe composition \n$\\rho_{D_{\\alpha_1}}\\circ \\dots \\circ \\rho_{D_{\\alpha_m}}(\\omega)$.\nWhen $\\Card(A)$ is odd, we have \n\\[\u00a0\\partial (\\rho_{D_A}([X,\\omega])) = \\partial([D_A,\\omega_A])\n\\cdot \\mathbf T^{\\Card(A)-1}, \\]\nwhile when $\\Card(A)$ is even, we have \n\\[\u00a0\\partial (\\rho_{D_A}([X,\\omega])) \n= - \\partial([D_A,\\omega_A])\\cdot \\mathbf T^{a-1}\n+ [D_A,\\omega_A] \\cdot \\partial(\\mathbf T^{\\Card(A)-1}) . \\]\nConsequently, we have\n\\[\u00a0\\partial \\circ\\partial ([X,\\omega])\n= \\sum_{\\emptyset\\neq A\\subset \\mathscr A}\n \\partial ([D_A,\\omega_A])\\cdot \\mathbf T^{\\Card(A)-1}\n- \\sum_{\\substack{\\emptyset\\neq A\\subset \\mathscr A \\\\\u00a0\\text{$\\Card(A) $ even}}}\n[D_A,\\omega_A]\\cdot \\partial(\\mathbf T^{\\Card(A)-1}). \n\\]\n\nThe polar divisor of the form\n$\\omega_A $ on~$D_A$\nis equal to $\\sum_{\\beta\\not\\in A} D_\\beta \\cap D_A$, \nso that, by definition (and computation of~$\\partial$ via iterated residues), \n\\[\u00a0\\partial([D_A,\\omega_A])\n= \\sum_{\\emptyset \\neq B \\subset \\complement A}\n(-1)^{\\Card(B)-1} [D_{AB},\\omega_{A\\cup B}]\u00a0\\cdot \\mathbf T^{\\Card(B)-1}.\\]\nAlso, when $A$ is nonempty and of even cardinality, \n$\\partial(\\mathbf T^{\\Card(A)-1})=2\\mathbf T^{\\Card(A)-2}$.\nWhen we put these two formulas into the antepenultimate one\nand collect the various terms, we obtain\n\\[\u00a0\\partial \\circ\\partial ([X,\\omega])\n= \\sum_{\\substack{ C\\subset \\mathscr A \\\\ 2\\leq \\Card(C)}} n_C [D_C,\\omega_C]\u00a0 \\cdot \\mathbf T^{\\Card(C)-2},\n\\]\nwhere\n\\[\u00a0n_C = - \\sum_{\\substack{\\emptyset \\neq A, B  \\\\\n A \\cup B = C, A \\cap B = \\emptyset}} (-1)^{\\Card(B)}\n - \u00a02 \\delta_{\\text{$\\Card(C)$ is even}}. \\]\nIn the first sum, the terms $A=\\emptyset $ or $B=\\emptyset$ are omitted,\nwhile  if we put them in, we obtain\n\\[ \\sum_{\\substack{A \\cup B=C \\\\ A \\cap B=\\emptyset}}\n(-1)^{\\Card(B)} = \\sum_{b=0}^{\\Card(C)} \\binom {\\Card(C)}b (-1)^b\n= (1-1)^{\\Card(C)}=0 \\]\nsince $\\Card(C)\\geq 1$.\nConsequently, \n\\[\u00a0n_C = 1 + (-1)^{\\Card(C)} - 2 \\delta_{\\text{$\\Card(C)$  is even}}\n= 0. \\]\nThis concludes the proof.\n\\end{proof}\n\n\\section{Algebraic structure of \\(\\Burn(k)\\) after localization at~\\(2\\)}\n\\label{sec.alg}\n\nIn this section, we study the algebraic structure of\nthe Burnside ring $\\Burn(k)$, endowed with its elements~$\\eps$, $\\mathbf T$\nand the operator~$\\partial$. \n\n\\subsection{}\nBy construction, $\\Burn(k)=\\bigoplus_{n\\geq 0}\\Burn_n(k)$\nis an associative unital $\\Z_{\\geq0}$-graded ring, $\\eps\\in\\Burn_0(k)$,\n$\\mathbf T\\in\\Burn_1(k)$ and $\\partial$ is a homogeneous\nadditive map of degree~$-1$. They satisfy the following relations,\nfor homogeneous elements $a,b\\in\\Burn(k)$:\n\\begin{align*}\n\\tag{1}\n & b\\cdot a = \\eps^{\\abs a\\abs b} \\cdot a\\cdot b \n\t&& \\text{(\\S\\ref{ss.commut});} \\\\\n\\tag{2}\n & \\eps^2 = 1 && \\text{(example~\\ref{exem.eps});}  \\\\\n\\tag{3}\n & \\mathbf T = \\eps\\cdot \\mathbf T && \\text{(example~\\ref{exem.T});} \\\\\n\\tag{4}\n & \\partial(\\mathbf T) = 1+\\eps && \\text{(example~\\ref{exem.residue-torus});} \\\\\n\\tag{5}\n & \\partial(a\\cdot b) = \\eps^{\\abs b}\\cdot\\partial(a)\\cdot b+ a\\cdot \\partial(b) -\\mathbf T\\cdot\\partial(a)\\cdot\\partial(b) && \\text{(theorem~\\ref{theo.partial-deriv});} \\\\\n\\tag{6}\n & \\partial(\\partial(a)) = 0 &&  \\text{(theorem~\\ref{theo.dd=0}).} \n\\end{align*}\nBy~(1), the element~$\\eps$ is central, and by~(2), we may view~$\\Burn(k)$\nas an algebra over $\\Z[\\eps]\/(\\eps^2-1)$. After inverting~2, \nthe algebra $\\Burn(k)$ splits into two components\n$\\Burn^{\\eps=1}(k)$ and $\\Burn^{\\eps=-1}(k)$, one over which $\\eps=1$,\nand the other over which $\\eps=-1$.\n\n\\emph{In the rest of this section, we implicitly assume that $2$ is inverted,\nwithout changing the notation.}\n\n\n\\subsection{Sector $\\eps=-1$}\\label{sect.sectorminus}\nHere, we have $\\mathbf T=-\\mathbf T$, \nhence $\\mathbf T=0$ since $2$ is invertible.\nAs a consequence, after replacing~$\\partial$ with\n$\\partial'\\colon a\\mapsto (-1)^{1+\\abs a}\\partial(a)$,\none gets from~(5) the usual graded Leibniz rule \n\\[\u00a0\\partial'(a\\cdot b)=\\partial'(a)\\cdot b+(-1)^{\\abs a}a\\cdot\\partial'(b) \\]\nand therefore\n$\\Burn^{\\eps=-1}(k)$ \nis a classical differential graded (super\\nobreak-)commutative algebra,\nsimilar to, \\eg, the de Rham complex.\n\n\\subsection{Sector $\\eps=1$}\nThe algebra $\\Burn^{\\eps=1}$ is now commutative (and not graded commutative).\nThis reflects the intuition in our constructions that they\nspeak about volume forms (as opposed to top-degree differential forms) \nfor which we have commutativity (as reflected by the change of order\nof integration in multiple integrals). \n\n\\begin{lemm}\nThe map $F\\colon a \\mapsto  a- \\mathbf T\\cdot \\partial(a)$\nis a ring endomorphism of $\\Burn^{\\eps=1}(k)$, and $F^2=\\id$.\nMoreover, one has $F \\circ \\partial =\\partial = - \\partial \\circ F$.\n\\end{lemm}\n\\begin{proof}\nThis map is additive. One has \n$\u00a0F(1) = 1 - \\mathbf T\\cdot \\partial(1)=1$. Let us show multiplicativity.\nIndeed, for $a,b\\in \\Burn^{\\eps=1}(k)$, one has\n\\begin{align*}\nF(a)\\cdot F(b) & = (a-\\mathbf T\\cdot \\partial(a))\\cdot (b-\\mathbf T\\cdot\\partial(b)) \\\\\n& = a\\cdot b - \\mathbf T\\cdot \\partial(a)\\cdot b-\\mathbf T\\cdot a\\cdot\\partial(b)\n+\\mathbf T^2 \\cdot\\partial(a)\\cdot\\partial(b)\\\\\n& = a\\cdot b - \\mathbf T\\cdot (\\partial(a)\\cdot b+ a\\cdot\\partial(b)\n-\\mathbf T \\cdot\\partial(a)\\cdot\\partial(b)) \\\\\n& = a\\cdot b - \\mathbf T\\cdot \\partial(a\\cdot b) & \\text{(using~(5))} \\\\\n& = F(a\\cdot b). \n\\end{align*}\n\nSince $\\partial^2=0$, one has \n\\[\u00a0F (\\partial(a))= \\partial(a)-\\mathbf T\\cdot \\partial(\\partial(a))=\\partial(a). \\]\nOn the other hand,\n\\begin{align*}\n\\partial(F(a))& = \\partial (a-\\mathbf T\\cdot \\partial(a)) \u00a0\\\\\n& = \\partial(a) - \\partial(\\mathbf T\\cdot \\partial(a)) \\\\\n& = \\partial(a) - \\partial(\\mathbf T) \\cdot \\partial(a)\n- \\mathbf T \\cdot \\partial(\\partial(a))+\\mathbf T\\cdot \\partial(\\mathbf T)\n\\cdot \\partial(\\partial(a)) \\\\\n& = -\\partial(a)\n\\end{align*}\nusing that $\\partial(\\mathbf T)=2$ and $\\partial^2=0$.\n\nConsequently, for $a\\in\\Burn^{\\eps=1}(k)$, we have\n\\[\nF^2(a)  = F(a) - \\mathbf T\\cdot \\partial (F(a)) \n = a - \\mathbf T\\cdot \\partial(a) + \\mathbf T \\cdot \\partial (a) \n = a \\]\nsince $\\partial\\circ F=-\\partial$.\n\\end{proof}\n\n\\subsection{}\nTo simplify the notation, write $\\mathscr B=\\Burn^{\\eps=1}(k)$.\nSince $F^2=\\id$ and $2$ is invertible, the algebra $\\mathscr B$\nsplits as a direct sum \n\\[\u00a0\\mathscr B= \\mathscr B_+ \\oplus \\mathscr B_-, \\]\nsuch that $F$ acts as~$\\id$ on $\\mathscr B_+$\nand as $-\\id$ on~$\\mathscr B_-$. Moreover, $\\mathscr B_+$ is a subalgebra.\n\nSince the operator~$\\partial$ anticommutes with~$F$,\nit induces maps\n\\[\u00a0\\partial_{\\pm}\\colon \\mathscr B_+\u00a0\\to \\mathscr B_-,\n\\qquad \\partial_{\\mp}\\colon \\mathscr B_-\u00a0\\to \\mathscr B_+. \\]\n\nNote that \n\\[\u00a0F(\\mathbf T) = \\mathbf T- \\mathbf T \\cdot \\partial(\\mathbf T)\n= - \\mathbf T , \\]\nso that $\\mathbf T\\in\\mathscr B_-$.\nConsequently, the multiplication by~$\\mathbf T$ map induces\ntwo maps\n\\[\u00a0t_{\\pm}\\colon \\mathscr B_+\\to\\mathscr B_-, \\qquad \nt_{\\mp}\\colon \\mathscr B_-\\to\\mathscr B_+. \\]\n\n\\begin{lemm}\nThe map~$\\partial$ vanishes on~$\\mathscr B_+$.\nEquivalently, $\\partial_{\\pm}=0$. \n\nThe maps $\\frac12 \\partial_{\\mp}$ and~$t_{\\pm}$ are inverses\nthe one of the other.\n\\end{lemm}\n\\begin{proof}\nFor $a\\in\\mathscr B_+$, one has $\\partial(a)=-\\partial(F(a))=-\\partial(a)$,\nsince $\\partial\\circ F=-\\partial$, hence $\\partial(a)=0$.\n\nOn the other hand, for $a\\in\\mathscr B_+$, one has\n\\[\u00a0\\partial (\\mathbf T\\cdot a)\n= 2 \\cdot a+\\mathbf T\\cdot \\partial(a)- 2\\mathbf T\\cdot \\partial(a)\n= 2a - \\mathbf T\\cdot\\partial(a)=a + F(a) = 2a \\]\nwhile for $a\\in\\mathscr B_-$, we have \n\\[\u00a0\\mathbf T\\cdot \\partial(a) = a-F(a) = 2a. \\]\nThis concludes the proof of the lemma.\n\\end{proof}\nIn particular, we see that the cohomology of the differential $\\partial$ vanishes in the sector $ \\Burn^{\\eps=1}(k) =\\mathscr B$.\n\n\\subsection{}\nIt follows from the lemma that we have a ring isomorphism\n\\[\u00a0\\mathscr B = \\mathscr B_+[t](t^2-\\mathbf T^2), \\]\nfrom which we see that all the algebraic structure of~$\\mathscr B_+$\n(namely $\\delta$, $\\mathbf T$, $F$) can be canonically reconstructed from a unital commutative associative $\\Z_{\\geq0}$-graded ring~$\\mathscr B_+$ endowed with an element in degree $+2$ (namely, the element $\\mathbf T^2$).\n\\begin{rema}\nThe situation clarifies even more if we invert the class~$\\mathbf T$.\nThen we can write $\\partial(a)=(a-F(a))\/\\mathbf T$,\nand all relations happen to follow from the fact that $F$ is an involution\nsuch that $F(\\mathbf T)=-1$.\nIndeed,\n\\[\n\\partial^2(a) = \\frac{\\partial(a)-F(\\partial(a))}{\\mathbf T}\n= \\frac1{\\mathbf T} \\left( \\frac{a-\\partial(a)}{\\mathbf T}\n- F( \\frac{a-\\partial(a)}{\\mathbf T} \\right) = 0 \\]\nexplains that $\\partial^2=0$.\nMoreover, for $a,b\\in\\mathscr B$, we  have\n\\begin{align*}\n\\partial(a\\cdot b) & = \\frac{a\\cdot b - F(a\\cdot b)}{\\mathbf T}\n = \\frac{a\\cdot b - F(a)\\cdot F(b)}{\\mathbf T} \\\\\n& = \\frac{a-F(a)}{\\mathbf T} \\cdot b + a \\cdot \\frac{b-F(b)}{\\mathbf T}\u00a0\n - \\mathbf T \\cdot \\frac{a- F(a)}{\\mathbf T} \\cdot \\frac{b-F(b)}{\\mathbf T}\u00a0\\\\\n & =\\partial(a)\\cdot b+a\\cdot\\partial(b)-{\\mathbf T}\\cdot\\partial(a)\\cdot\\partial(b).\n\\end{align*}\n\\end{rema}\n\n\n\n\\section{Birational morphisms preserving volume forms}\n\\label{sec.biraut}\n\n\\subsection{}\nLet $(X,\\omega_X)$ be a smooth integral $k$-variety of dimension~$n$\nequipped with a logarithmic volume form, \nand let $f\\colon Y\\to X$ be a proper birational morphism.\n\nLet $E$ be an exceptional divisor in~$Y$,\nthat is, such that $\\dim(f(E))<\\dim(E)$. \nBy lemma~\\ref{lemm.logarithmic-functoriality},\nand lemma~\\ref{lemm.residue-log}, \nthe residue $\\rho_E(f^*\\omega_X)$ along~$E$ \nof the meromorphic form~$f^*\\omega$ is a logarithmic volume\nform on~$E$.\n\nWe define $\\mathbf c(f;X,\\omega)$ to be the sum\nof all such classes $[E,\\rho_E(f^*\\omega)]$\nin the free abelian group $\\Burn_{n-1}(k)$.\n\n\\begin{lemm}\\label{lemm.c-additivity}\nLet $g\\colon Z\\to Y$ be a proper birational morphism\nof smooth integral varieties  of dimension~$n$.\nThen $g\\circ f$ is a proper birational morphism\nand one has\n\\[\u00a0\\mathbf c(g\\circ f;X,\\omega) = \\mathbf c(g;Y,f^*\\omega) + \\mathbf c(f;X,\\omega) \\]\nin $\\Burn_{n-1}(k)$.\n\\end{lemm}\n\\begin{proof}\nAn integral divisor~$F$ in~$Z$ is exceptional for\n$g\\circ f$ if and only if one of the two \nmutually excluding situations happen:\n\\begin{itemize}\n\\item The divisor~$F$ is exceptional for~$g$;\n\\item Or $g(F)$ is a divisor in~$Y$ which is exceptional for~$f$.\n\\end{itemize}\nMoreover, any divisor~$E$ in~$Y$ which is exceptional\nfor~$f$ appears once and only as  a divisor of the form~$g(F)$.\nThe contribution of~$F$ to~$\\mathbf c(g\\circ f;X,\\omega)$\nis given by the volume form~$\\rho_F((g\\circ f)^*\\omega)$.\nIn the first case, we write\n$\\rho_F((g\\circ f)^*\\omega)=\\rho_F(g^*(f^*\\omega))$,\nso that the contribution of~$F$ coincides with its contribution\nto the term $\\mathbf c(g;Y,f^*\\omega)$.\nIn the second case, $g$ induces a birational isomorphism\nfrom~$F$ to~$E=g(F)$;\nwriting $\\rho_F((g\\circ f)^*\\omega)=g^*(\\rho_F(f^*\\omega))$,\nwe see that the contribution of~$F$ coincides with the\ncontribution of~$E$ to $\\mathbf c(f;X,\\omega)$.\nThis concludes the proof.\n\\end{proof}\n\n\\subsection{}\nLet $(X,\\omega_X)$ and $(Y,\\omega_Y)$ be proper smooth $k$-varieties\nequipped with logarithmic volume forms and \nlet \n\\[\n\\phi\\colon (X,\\omega_X) \\dashrightarrow (Y,\\omega_Y)\n\\]\nbe a birational map\npreserving the volume forms.\nBy definition, this means that there exists a diagram\n\\[\u00a0\\begin{tikzcd}[column sep = small]\n & W \\ar{dl}[swap]{p} \\ar{dr}{q} \\\\\nX \\ar[dashrightarrow]{rr}{\\phi} && Y \\end{tikzcd}\u00a0\\]\nof integral $k$-varieties\nsuch that that $p$ and~$q$ are proper and birational,\nand such that $p^*\\omega=q^*\\omega'$ on~$W$.\nIn this situation, we may assume that $W$ is smooth.\n\n\\begin{lemm}\\label{lemm.c-independence}\nWith this notation, the element  \n\\[\n\\mathbf c(\\phi)=\\mathbf c(q)-\\mathbf c(p) \\in \\Burn_{n-1}(k)\n\\]\nonly depends on the birational map~$\\phi$,\nand not on the choice of the triple $(W,p,q)$.\n\\end{lemm}\n\n\\begin{proof}\nConsider two possible diagrams \n$X\\xleftarrow p V\\xrightarrow q Y$ \nand $X\\xleftarrow r W\\xrightarrow s Y$\ndescribing~$\\phi$. \nConsidering for example a resolution\nof singularities~$U$ of $V\\times_X W$,\nwe can fit these two diagrams in a common commutative diagram\nof the following form:\n\\[\u00a0\\begin{tikzcd}[column sep = small]\n && U \\ar{dl}[swap]{u} \\ar{dr}{v} \\\\\n & V \\ar{dl}[swap]{p} \\arrow{drrr}[very near start]{q}\n \t&& W \\ar[crossing over]{dlll}[swap,very near start]{r\\vphantom q} \\ar{dr}{s} \\\\\nX \\ar[dashrightarrow]{rrrr}{\\phi} &&&& \nY \n\\end{tikzcd}\u00a0\\]\nThe equalities $p^*\\omega_X=q^*\\omega_Y$\nand $r^*\\omega_X=s^*\\omega_Y$ imply\nthat \n\\[\u00a0(p\\circ u)^*\\omega_X = \nu^*p^*\\omega_X = u^* q^*\\omega_Y\n=(q\\circ u)^*\\omega_Y = (s\\circ v)^*\\omega_Y. \\]\nBy lemma~\\ref{lemm.c-additivity}, we then have\n\\[\n\\mathbf c(p)-\\mathbf c(q) \n = \\mathbf c(p\\circ u)-\\mathbf c(q\\circ u)  \n =  \\mathbf c(r\\circ v)-\\mathbf c(s\\circ v)  \n = \\mathbf c(r)-\\mathbf c(s) . \n\\]\nThis concludes the proof.\n\\end{proof}\n\n\\begin{theo}\\label{theo.additivity}\nIf $\\psi\\colon (Y,\\omega_Y)\\dashrightarrow (Z,\\omega_Z)$\nis another birational map preserving volume forms,\nthen one has\n\\[\u00a0\\mathbf c(\\psi\\circ\\phi) = \\mathbf c(\\psi)+\\mathbf c(\\psi). \\]\n\\end{theo}\n\\begin{proof}\nConsider two diagrams \n$X\\xleftarrow p V\\xrightarrow q Y$ \nand $Y\\xleftarrow r W\\xrightarrow s Y$\ndescribing~$\\phi$ and~$\\phi$. \nConsidering for example a resolution\nof singularities~$U$ of $V\\times_Y W$,\nwe can fit these two diagrams in a common commutative diagram\nof the following form:\n\\[\u00a0\\begin{tikzcd}[column sep = small]\n && U \\ar{dl}[swap]{u} \\ar{dr}{v} \\\\\n & V \\ar{dl}[swap]{p} \\arrow{dr}{q}\n \t&& W \\ar{dl}[swap]{r\\vphantom q} \\ar{dr}{s} \\\\\nX \\ar[dashrightarrow]{rr}{\\phi} && Y \\ar[dashrightarrow]{rr}{\\psi} && Z \n\\end{tikzcd}\u00a0\\]\nand the diagram $X\\xleftarrow{p\\circ u} U\\xrightarrow{s\\circ v}$\ndescribes the birational map $\\psi\\circ \\phi$.\nSince $q\\circ u=r\\circ v$, we then have\n\\begin{align*}\n\\mathbf c(\\psi\\circ \\phi) & = \n\\mathbf c(p\\circ u)-\\mathbf c(s\\circ v) \\\\\n& = \\mathbf c(p\\circ u)-\\mathbf c(q\\circ u)\n+ \\mathbf c(r\\circ v)- \\mathbf c(s\\circ v) \\\\\n& = \\mathbf c(p)-\\mathbf c(q) + \n+ \\mathbf c(r)- \\mathbf c(s) \\\\\n& = \\mathbf c(\\phi) + \\mathbf c(\\psi), \n\\end{align*}\nas was to be shown.\n\\end{proof}\n\n\\begin{coro}\nLet $\\Bir(X,\\omega)$ be the set of birational automorphisms\nof~$X$ preserving~$\\omega$. The map~$\\mathbf c$\ninduces a homomorphism of abelian groups\n\\[\n\\Bir(X,\\omega)\\to \\Burn_{n-1}(k).\n\\]\nIts kernel contains the group of automorphisms of~$X$\nthat preserve~$\\omega$.\n\\end{coro}\n\n\n\n\\section{Specialization}\n\nLet $K$ be the field of fractions of a discrete valuation ring~$R$\nwith residue field~$k$. \nFix a uniformizer~$t\\in R$.\n\nIn this context, \n\\citet{KontsevichTschinkel-2019} have defined\ntwo (distinct) \\emph{specialization morphisms}\n\\[\u00a0\\rho_t \\colon \\burn_n(K)\\to\\burn_n(k), \\]\nrelating the Burnside groups of~$K$ and~$k$ (see \\ref{ssec.burn}),\none of which is a ring homomorphism.\n(The latter homomorphism actually\ndepends on the choice of~$t$, \nsee example~6.2 of~\\citep{KreschTschinkel-2022}.)\n\nThe goal of this section is to define a similar homomorphism\n\\[\u00a0\\mathbf\\rho_t \\colon \\Burn(K) \\to \\Burn(k)  \\]\nfor varieties with logarithmic volume forms.\n\n\\subsection{}\\label{sec.kt}\nLet $\\mathscr X$ be an integral proper scheme over~$R$,\nof relative dimension~$n$,\nwhose special fiber~$\\Delta$ is a divisor with strict normal crossings.\n\nLet $(\\Delta_\\alpha)_{\\alpha\\in\\mathscr A}$  \nbe the family of irreducible components of the special fiber~$\\Delta$;\nfor $\\alpha\\in\\mathscr A$, let $e_\\alpha$ be the multiplicity\nof~$\\Delta_\\alpha$ in~$\\Delta$.\nFor every nonempty subset~$A$ of~$\\mathscr A$,\nlet $\\Delta_A$ be the intersection of all divisors~$\\Delta_\\alpha$,\nfor $\\alpha\\in\\mathscr A$\nand $e_A$ be the greatest common divisor of\nthe~$e_\\alpha$, for $\\alpha\\in A$; let also $\\Delta_A^\\circ$\nbe the complement $\\Delta_A\\setminus \\bigcup_{\\alpha\\not\\in A}\\Delta_\\alpha$.\n\nThe first specialization morphism of~\\citep{KontsevichTschinkel-2019}\nis defined by\n\\begin{equation}\n\u00a0\\rho_t( [\\mathscr X_K]) \n= \\sum_{\\emptyset\\neq A\\subset\\mathscr A} (-1)^{\\Card(A)-1} [\\Delta_A] \n\\Lef^{\\Card(A)-1},\n\\end{equation}\nwhere $\\Lef\\in\\burn(k)$ is the class of the affine line.\n\nAlthough this map is not multiplicative, it proved sufficient for\nmany applications to rationality problems.\n\nTo ensure multiplicativity, a more delicate construction\nwas necessary, valued in the Burnside ring \n\\[\u00a0\\burn^{\\widehat\\mu}(k) \\]\nof varieties endowed with an action of the profinite group~$\\widehat\\mu$,\nlimit of finite groups of roots of unity.\n\nFix a nonempty subset~$A$ of~$\\mathscr A$.\nWe identify the normal bundle of~$\\Delta_A$ in~$\\mathscr X$\nas a direct sum of line bundles:\n\\[\u00a0\\mathscr N_{\\Delta_A}(\\mathscr X) \\simeq \\bigoplus_{\\alpha\\in A}\n  \\mathscr N_{\\Delta_\\alpha}(\\mathscr X)|_{\\Delta_A} . \\]\nLet us consider its open subscheme~$\\mathscr N_{\\Delta_A}^\\circ(\\mathscr X)$\nobtained by restricting to~$\\Delta_A^\\circ$\nand taking out all \"coordinate\" hyperplanes.\nThis furnishes a morphism\n\\[\u00a0\\nu_A \\colon \\mathscr N_{\\Delta_A}^\\circ(\\mathscr X) \\to\n \\bigotimes_{\\alpha\\in A} \\mathscr N_{\\Delta_\\alpha}(\\mathscr X)^{\\otimes e_\\alpha}|_{\\Delta_A^\\circ} . \\]\nSince the uniformizer~$t$ has divisor \n$-\\sum_{\\alpha\\in\\mathscr A} e_\\alpha \\Delta_\\alpha$ on~$\\mathscr X$,\nit trivializes the line bundle on the target of~$\\nu_A$.\nWe set $\\Delta'_A=\\nu_A^{-1}(t)$.\nBy construction, the projection $\\Delta'_A\\to \\Delta_A$ \nis a torsor with group~$\\mu_{e_A}$.\n\nWith this notation, the correct, multiplicative, specialization\nmap of~\\citep{KontsevichTschinkel-2019} is given by the formula\n\\[\u00a0\\widehat\\rho_t(X)  = \\sum_{\\emptyset\\neq A\\subset \\mathscr A}\n (-1)^{\\Card(A)-1} [\\Delta'_A]\u00a0\\Lef ^{\\Card(A)-1} \\]\nin $\\burn^{\\widehat\\mu}(k)$.\n\n\\begin{rema}\nThe relation between the two specialization morphisms\nis as follows. Fix a nonempty subset~$A$ of~$\\mathscr A$.\nThe group~$\\gm$ acts diagonally on~$\\mathscr N_{\\Delta_A}^\\circ(\\mathscr X)$\n(the factors of index $\\alpha\\notin A$ don't act),\nand this induces an action \nof the finite group of roots of unity of order~$e_A$\non~$\\Delta'_A$,\nhence an action of~$\\widehat\\mu$,\nso that $\\widehat\\rho_t(X)$ naturally lives\nin the equivariant Burnside ring $\\burn^{\\widehat\\mu}(k)$.\nMoreover, taking the $\\widehat\\mu$-invariants of~$\\Delta'_A$,\nwe get $\\Delta_A^\\circ$, so that the specialization map~$\\rho_t$\nis the composition of~$\\widehat\\rho_t$ with the map\n\\[\u00a0\\burn^{\\widehat\\mu}(k) \\to \\burn(k) \\]\nobtained by taking~$\\widehat\\mu$-invariants.\n\nTaking invariants does not commute with taking products, in general,\nso that $\\rho_t$ is not multiplicative.\n\\end{rema}\n\n\n\\subsection{}\nLet us explain how to define analogous specialization\nhomomorphisms in our context\nof Burnside groups with volume forms.\n\nFor simplicity, we only consider\nthe case where $K$ has transcendence degree~$1$ over~$k$,\nin which case the idea can be explained geometrically as follows.\nWe assume that there exists an smooth integral curve~$C$\ntogether with a $k$-point $o\\in C(k)$\nsuch that $K=k(C)$ and $R=\\mathscr O_{C,o}$.\nWe fix a local parameter $t\\in R$ such that $V(t)=o$.\n\nLet us consider a pair $(X,\\omega)$ consisting of an integral\nproper $K$-variety~$X$ of dimension~$n$\nand a logarithmic $n$-form~$\\omega$ on~$X$.\n\n\\subsection{}\nConsider a regular flat proper model~$\\mathscr X$ is of~$X$ over~$C$,\nlet $\\Delta=(\\mathscr X_o)_\\red$ be its reduced special fiber, and \nconsider a divisor~$\\mathscr D$ with relative normal crossings on~$\\mathscr X$.\nWe assume that the divisor $\\mathscr D+\\Delta$ has normal crossings.\nIn this situation, \n\\citet[\\S3.3.2]{Deligne-1970} says that \na meromorphic relative differential $m$-form on~$\\mathscr X\/C$\nis logarithmic with respect to~$\\mathscr D+\\Delta$ \nif it is (locally) \nthe image of a logarithmic $m$-form~$\\tilde\\omega$\nin~$\\Omega^m_{\\mathscr X\/k}$ with poles $\\mathscr D+\\Delta$\nunder the  natural\nmorphism $\\Omega^m_{\\mathscr X\/k}\\to\\Omega^m_{\\mathscr X\/C}$.\n\nConsider a logarithmic relative $n$-form~$\\omega$ on~$\\mathscr X\/C$.\nWe consider an associated volume form~$\\omega'$ on~$\\mathscr X$, \ndefined locally by\n\\[\u00a0\\omega' = \\tilde \\omega \\wedge \\mathrm dt\/t, \\]\nwhere $\\tilde\\omega$ is any local lift of~$\\omega$.\nThis form $\\omega'$ is logarithmic\nand we can compute its \"residue along~$\\Delta$\" as\nin~\\S\\ref{sec.residues}, only taking into account\nthe strata of the polar divisor of~$\\omega'$\nwhich are contained in the special fiber~$\\Delta$.\n\nThere exists a subset $\\mathscr A_o$ of~$\\mathscr A$\nand a subset~$\\mathscr B_o$ of~$\\mathscr B$\nsuch that \nthe polar divisor of~$\\omega'$ is given by \n\\[\u00a0\\sum_{\\alpha\\in\\mathscr A_o}  \\Delta_\\alpha + \\sum_{\\beta\\in\\mathscr B_o} \\mathscr D_\\beta. \\]\nWe thus set\n\\[\u00a0\\rho_t(\\mathscr X,\\omega)\n = \\sum_{\\substack{\\emptyset \\neq A \\subset \\mathscr A_o \\\\\u00a0B \\subset\\mathscr B_o} }(-1)^{\\Card(A)+\\Card(B)-1}\n\\rho_{\\Delta_A\\cap \\mathscr D_B}(\\mathscr X,\\omega). \\]\nThis is an element of $\\Burn_n(\\mathscr X_o\/k)$.\n\\begin{prop}\nLet $\\mathscr Y$ be an irreducible closed subscheme of~$\\mathscr X$\nwhich is transverse to $\\mathscr D+\\Delta$\nand let $g\\colon \\mathscr X'\\to\\mathscr X$ \nbe the blowing-up of~$\\mathscr X$ along~$\\mathscr Y$.\nThe form $g^*\\omega$ on~$\\mathscr X'$ is logarithmic and \nwe have\n\\[\u00a0g_* \\rho_t (\\mathscr X',g^*\\omega) = \\rho_t (\\mathscr X,\\omega) \\]\nin $\\Burn_n(\\mathscr X_o\/k)$.\n\\end{prop}\n\\begin{proof}\nWith the notation of~\\S\\ref{sec.residues}, the difference\n\\[\u00a0\\rho (\\mathscr X,\\tilde\\omega)  - \\rho_t(\\mathscr X,\\omega) \\]\nis exactly the  part of $\\rho(\\mathscr X,\\tilde\\omega)$ \nwhich lies over the complement of the special fiber~$\\mathscr X_o$\nin~$\\mathscr X$.\nWe have seen in theorem~\\ref{theo.residue-bir} that\n\\[\u00a0g_* \\rho(\\mathscr X',\\tilde\\omega') = \\rho(\\mathscr X,\\omega), \\]\nand a similar formula holds over $\\mathscr X\\setminus\\mathscr X_o$.\nThis implies the proposition.\n\\end{proof}\n\n\\subsection{}\nStarting from a smooth proper $K$-variety~$X$\nand a logarithmic volume form~$\\omega$ on~$X$,\nwe can define a model~$\\mathscr X\/C$, with $\\mathscr D$ and~$\\Delta$\nas above, but the form~$\\omega$\nwill not necessarily extend to a logarithmic relative form with respect\nto $\\mathscr D+\\Delta$,\nnor does \nthe volume form~$\\tilde\\omega$ on~$\\mathscr X$.\nHowever, this can be achieved by multiplying~$\\omega$\nby a suitable power of the uniformizing element.\n\nLet us write the polar divisor of~$\\tilde\\omega$ on~$\\mathscr X$ as\n\\[\u00a0\\div_{\\mathscr X}(\\tilde\\omega) = D+\\Delta\n= \\sum_{\\alpha\\in\\mathscr A} d_\\alpha \\Delta_\\alpha\n + \\sum_{\\beta\\in\\mathscr B} d_\\beta \\mathscr D_\\beta. \\]\nWith this notation, the condition for~$\\tilde\\omega$ to be logarithmic\non~$\\mathscr X$ is just that \n\\[\u00a0d_\\alpha \\geq -1, \\qquad d_\\beta\\geq -1. \\]\nIn particular, while the conditions at the horizontal components follow \nfrom their counterparts on the generic fiber, \nthose for the vertical components are not automatic.\nOn the other hand, for any $\\kappa\\in\\Z$, the form $t^\\kappa \\tilde\\omega$\nis logarithmic if and only if \n\\[\u00a0\\kappa e_\\alpha + d_\\alpha \\geq -1 \\]\nfor all $\\alpha\\in\\mathscr A$, that is, if and only if\n$\u00a0\\kappa \\geq \\kappa(\\omega)$, \nwhere $\\kappa(\\omega)$ is defined by \n\\[\u00a0\\kappa(\\omega) = \\inf_{\\alpha\\in\\mathscr A} \\frac{1-d_\\alpha}{e_\\alpha}. \\]\n\nSince the rational number\n$\\kappa(\\omega)$ is defined in terms of logarithmic forms,\nit only depends on the class of $(X,\\omega)$ in~$\\Burn_n(K)$,\nand not on the actual model which is chosen to compute it.\n\n\\subsection{}\nWe assume for the moment that $\\kappa(\\omega) \\in\\Z$. \nThis holds in particular if the special fiber~$\\mathscr X_o$ is reduced.\nLet then $\\mathscr A_o$ be the subset of~$\\mathscr A$\nconsisting of all $\\alpha$ such that \n\\[\u00a0\\kappa (\\omega) e_\\alpha+d_\\alpha = -1,\\]\nand let $\\mathscr B_o$ be the subset of~$\\mathscr B$\nconsisting of all $\\beta$ such that $d_\\beta=-1$.\nThe polar divisor of $t^\\kappa\\tilde\\omega$ is equal to\n\\[\u00a0\\sum_{\\alpha\\in\\mathscr A_o} \\Delta_\\alpha\n + \\sum_{\\beta\\in\\mathscr B_o} \\mathscr D_\\beta, \\]\nand we set\n\\[\u00a0\\rho_t(\\mathscr X,\\omega) = \\rho_t(\\mathscr X,t^{\\kappa(\\omega)}\\omega) \\]\nin $\\Burn_n(\\mathscr X_o\/k)$.\n\nIn the particular case where $\\mathscr D$ is empty, \nthe strata of the Clemens complex of the special fiber \nthat actually appear in the definition of this class\nare those defined by~\\cite{KontsevichSoibelman-2006},\nmore precisely, by the adjustment provided by~\\cite{MustataNicaise-2015a}.\n\n\n\\subsection{}\nIn the general case, the rational number~$\\kappa(\\omega)$\nis not an integer.  \nLet us consider the finite ramified extension $K_d=K(t^{1\/d})$\nof~$K$, \nwhose ramification index~$d$\nis a multiple of the denominator of~$\\kappa(\\omega)$,\nbut which induces an isomorphism on the residue field.\nGeometrically, this furnishes a morphism $\\pi\\colon C_d\\to C$\nwhich is ramified at the point~$o$, together\nwith a lift of~$o$ in~$C_d(k)$ (still denoted by~$o$),\nand a distinguished uniformizing element~$t^{1\/d}$.\n\nWe consider the extension of $(X,\\omega)$ to~$K_d$ and\nintroduce a model $(\\mathscr X_d,\\omega_d)$ as above, over~$C_d$. \nNow, the corresponding $\\kappa$-parameter is integral, so that \nany choice of a uniformizing element~$t^{1\/d}$ in~$R_d$\ninduces a class \n$\\rho_{t^{1\/d}}(\\mathscr X_d,\\omega_d)$\nin~$\\Burn(k)$.\nIn fact, we can assume\nthat the scheme~$\\mathscr X_d$ carries an action\nof the group scheme~$\\mu_d$ of $d$th roots of unity\ninduced by its action on~$\\Spec(R_d)$, leaving the\nlogarithmic form~$\\omega_d$ invariant.\nIn other words, we obtain a class in the group $\\Burn^{\\widehat\\mu}(k)$.\n\nCombinining these classes, we obtain the desired\ngroup homomorphism\n\\[\u00a0\\widehat\\rho_t \\colon \\Burn(K)  \\to \\Burn^{\\widehat\\mu}(k). \\]\n\nIn fact, as explained in \\cite[\\S2.3]{Nicaise-2013},\nespecially proposition~2.3.2,\none can compute the normalisation of~$\\mathscr X\\otimes R_d$\nin terms of the given model~$\\mathscr X$.\nThis gives an explicit decomposition of~$\\widehat\\rho_t(X,\\omega)$\nas a sum\n\\[\u00a0\\sum_{\\emptyset \\neq A\\subset \\mathscr A_o}\n (-1)^{\\Card(A)-1} [D'_A,\\nu_A^*\\omega_A] \\cdot \\mathbf T^{\\Card(A)-1}, \\]\nwhere $\\nu_A\\colon D'_A\\to D_A$ is the $\\mu_{d_A}$-torsor\nintroduced in~\\S\\ref{sec.kt} for the definition of the classical\nspecialization map.\n\n\\begin{rema}\nIn the case  of specialization of rationality, it has proved\nfruitful to consider models with singularities on the special fiber,\nmild enough so that \nthe special fiber computes the specialization of\nthe birational type of the generic fiber.\nThis is in particular the case for rational double points.\n\nA parallel study can be developped in the context\nof varieties with logarithmic forms.\n\\end{rema}\n\nFollowing \\citep{KontsevichTschinkel-2019} and \nkeeping track of the various logarithmic volume forms on the strata,\nwe have:\n\\begin{theo}\nThe morphism~$\\widehat\\rho_t$ is a ring  homomorphism.\n\\end{theo}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the main efforts of modern cosmology is to determine what is responsible for the observed accelerated expansion of the universe \\cite{riess:1998a, perlmutter:1999a}. Alcock and Paczynski \\cite{alcock:1979a} proposed a cosmological test (hereafter denoted AP) based on the assumption of statistical isotropy around any comoving location. For regions of space-time that expand with the background, observed angles and redshifts can be translated into proper distances using the angular diameter distance $d_A(z)$ and the reciprocal of the Hubble parameter $H(z)$. Requiring isotropy in proper distance, after translating from angle and redshift measurements, leads to measurements of the product $d_A(z)H(z)$.\n\nBecause radial information comes from redshifts, AP measurements are traditionally limited by peculiar velocities, also known as comoving velocities \\cite{ballinger:1996a, simpson:2009a}. These add to expansion-driven redshifts, leading to apparent anisotropic clustering if redshifts are assumed to be completely cosmological in origin, even if the correct $d_A(z)$ and $H(z)$ are used to analyze redshifts. These redshift-space distortions (hereafter RSD) are degenerate with the AP effect, removing signal \\cite{ballinger:1996a, simpson:2009a}, unless assumptions are made such as the Universe following a FLRW metric \\cite{samushia:2011a}. In fact, it is simply standard convention that makes us split redshift into cosmological and peculiar velocity components: considering that pairs of galaxies move due to local space-time curvature shows that the expansion rate and the RSD component can be strongly correlated. In the extreme case of bound systems, for example, the combined pairwise velocity is not dependent on background evolution, i.e.\\ the expansion-driven redshift difference across a pair is exactly canceled by the RSD signal (see Appendix \\ref{app:bound_systems}).\n\nMarinoni and Buzzi \\cite{marinoni:2010a} recently proposed a method to derive cosmological constraints from pairs of galaxies for which peculiar velocities can be modeled. They provided a fitting formula for the observed distribution of velocities, which can then be used to help break the AP-RSD degeneracy. They assume the normalization of the galaxy velocity distribution to be redshift and cosmology independent, whereas a more recent work by \\citet{jennings:2012a} questioned this statement using N-body simulations. In the work presented here we investigate this further, considering how well pairs of galaxies, selected using different properties, trace the cosmological expansion. \n\nWe use the Millennium simulation \\cite{springel:2005a} to test how the pairwise velocity of galaxy pairs may contain information about the background expansion of the Universe. We argue that the local density in which the pairs are found may affect the amount of information these pairs carry on cosmology, because each patch of the Universe expands in a way that depends on the local density. Our analysis suggests that selecting isolated pairs, as considered by Marinoni and Buzzi, can result in average pairwise velocities more in line with the Hubble expansion, i.e.\\ they need smaller, less cosmology dependent peculiar velocity corrections. We also find a better match if low-mass tracers are used. \n\n\nThe layout of this paper is as follows: in \\sref{sec:AP} we briefly review the AP effect. In \\sref{sec:millennium} we describe the Millennium simulation and the two semi-analytic models used: \\citet{guo:2011a} and \\citet{font:2008a}. In \\sref{sec:all_pairs} we present and discuss the results we obtained by analyzing all galaxy pairs regardless of their local density, while in \\sref{sec:isolated_pairs} we consider only isolated pairs. The effects of varying galaxy properties are studied in \\sref{sec:galaxy_properties}. In \\sref{sec:comparison_catalogs} we compare the results from the two different semi-analytic galaxy formation models used. We then conclude in \\sref{sec:conclusions}. \n\n\n\n\\section{The Alcock Paczynski effect}  \\label{sec:AP}\n\nConsider a distribution of particles expanding with the\nHubble flow, in the redshift interval ($z-\\Delta z\/2,\nz+\\Delta z\/2$) and subtended by an angle $\\Delta\\theta$. Assuming a\nFLRW cosmology, the proper size of the object perpendicular to our line of\nsight is given by\n\\begin{equation}\n   d_1=d_A(z)\\Delta\\theta, \n\\end{equation} \nwhere $d_A(z)$ is the angular diameter distance to the object. The size of the object parallel to the line of sight is given by\n\\begin{equation} \\label{eq:d2}\n  d_2=\\frac{\\Delta z}{(1+z)H(z)},\n\\end{equation}\nwhere $\\Delta z$ is the difference in the redshift of objects closest\nand furthest away from the observer and $H(z)$ is the Hubble parameter\nat the central redshift of the distribution.\n\nAssuming that the collection of particles statistically does not have a preferred direction with respect to one line of sight, then\n$\\langle d_1\\rangle =\\langle d_2 \\rangle$, allowing a statistical cosmological measurement \\cite{alcock:1979a}, from a sufficient number of pairs, of\n\\begin{equation}\n  H(z)d_A(z) = \\frac{\\Delta z}{(1+z)\\Delta\\theta}.\n\\end{equation}\nNote that $\\Delta z$, $z$ and $\\Delta \\theta$ are all directly\nobservable quantities. The AP effect, as described above assumes that $\\Delta z$ as measured only depends on the cosmological expansion. In fact, the relative  velocity of pairs of particles depends on the local curvature of space, so this is not necessarily a good approximation.\n\n\n\n\\section{The Millennium simulation and semi-analytic galaxy models} \\label{sec:millennium}\n\nIn order to quantify how the dynamics of galaxy pairs may be affected by factors like redshift, isolation radius, mass of the halo etc., we have considered a population of galaxies from  the Millennium simulation \\cite{springel:2005a}. This traces the evolution of 2160$^3$ dark matter particles of mass $1.18\\times 10^9\\ \\text{M}_\\odot$ from redshift 127 to the present day inside a periodic box of side \\mpch{500}. The simulation assumes a \\lcdm cosmology with parameters based on a combined analysis of the 2dFGRS \\cite{colless:2001a} and the first-year WMAP data \\cite{spergel:2003a}. The parameters are $\\Omega_m=0.25,\\ \\Omega_b=0.045,\\ \\Omega_\\Lambda=0.75,\\ n=1,\\ \\sigma_8=0.9$ and $H_0=73$ km s$^{-1}$ Mpc$^{-1}$.\n\nData on dark matter particles were stored at 64 different times. At each output time, the post-processing pipeline produced a friends-of-friends (FOF) catalog by linking particles with a separation less than 0.2 of the mean value. The SUBFIND algorithm \\cite{springel:2001a} was then applied to each FOF group to identify all the substructures. The merger trees, vital for galaxy formation modeling, were then constructed by linking each subhalo found in a given ``snapshot\" to one and only one descendant in the subsequent output time-slice.\n\nWe used the data from two semi-analytic models of galaxy formation based on the Millennium simulation: \\citet{guo:2011a} and \\citet{font:2008a}. Most of our analysis uses the semi-analytic model developed by \\citet{guo:2011a} which is based on the growth of and merging of the population of subhaloes. Within this catalog, each FOF group hosts a central galaxy which sits in the minimum of the potential of the main subhalo. Other galaxies associated to the same FOF group may sit at the potential minima of smaller subhaloes or may no longer correspond to a resolved dark matter substructure, the latter being know as `orphans'. The last two collectives of galaxies are referred to as satellites, although in \\citet{guo:2011a}, the physical processes affecting satellite galaxies only begin to differ from those affecting central galaxies when a satellite first enters within the virial radius of the larger system. This is the radius of the largest sphere with its center at the center of the FOF group and a mean overdensity exceeding 200 times the critical value.\n\nFor our analysis, we have varied several parameters from the galaxy catalogs, namely redshift, mass of the subhalo that hosts the galaxy, stellar mass and \\ensuremath{r\\text{-band}}\\xspace rest-frame magnitude.  Unless differently stated, the redshift shown in our plots corresponds to $z=0.989$.\n\n\n\n\\section{All galaxy pairs} \\label{sec:all_pairs}\nIn this section we study the average pairwise velocity of galaxies regardless of local density and galaxy properties.  We shall compare our findings with predictions from linear theory to examine general trends, and test the possibility of using randomly selected galaxy pairs to trace cosmological expansion. \n\n\n\\subsection{Method}\\label{sec:method_allpairs}\n\nFor each galaxy pair, we compute the comoving separation $ d $, the pairwise velocity $ v_{12} $ and its square $ v_{12}^2 $.  We define the pairwise velocity as:\n\\begin{equation}\n\\label{eq:pairwise_velocity_definition}\n  v_{12} = \\frac{\\operatorname{d}d}{\\operatorname{d}t} = \\frac{(\\vec{v_1} - \\vec{v_2}) \\, \\cdot \\, (\\vec{x_1} - \\vec{x_2})}{d} \\;,\n\\end{equation}\nwhere $ t $ is cosmic time, and $\\vec{x}$ and $\\vec{v}$ are galaxy positions and velocities.  Note that, following the Millennium simulation, we work in coordinates that are comoving with the Hubble flow, hence, $ v_{12} $ represents the peculiar, non-Hubble component of the pairwise velocity.  In the plots that follow, we shall always show the $ - H(z)\\,d $ curve and denote it as ``static solution\".  We shall also highlight the zero line, which in these plots represents the Hubble flow, and denote it as ``comoving solution\".  Any data point above the comoving solution represents pairs where the two galaxies are receding from each other faster than the Hubble expansion, while below they are moving towards each other in comoving coordinates.\n\n\nWe group pairs in bins according to their separation $ d $ and, for each bin, compute the average $ \\avg{v_{12}} $ and the variance $ \\text{var}(v_{12}) = \\avg{v_{12}^2} - \\avg{v_{12}}^2 $ of the pairwise velocity.  Our definition of expectation value is:\n\\begin{equation}\n\\label{eq:average_definition}\n  \\avg{v^n} \\equiv \\frac{1}{N_\\text{pairs}} \\, \\sum\\limits_{i=1}^{N_\\text{pairs}} (v_i)^n \\;,\n\\end{equation}\nwhere $ N_\\text{pairs} $ is the number of pairs in the separation bin.  In all the plots in this paper, we shall always show $ \\avg{v_{12}} $ as a function of pair separation, with the error bars for each bin taken as $ \\sqrt{\\text{var}(v_{12}) \/ N_\\text{pairs}} $.\n\nBy default, we employ a logarithmic binning in galaxy separation.  In order to better visualize the data, however, in some plots we combine underpopulated bins together so that each bin represents at least a minimum number of galaxy pairs, $ N_\\text{min}$.  In this section we use $ N_\\text{min} \\geq 1000 $ while, due the poor statistics, we shall employ $ N_\\text{min} \\geq 2 $ for some of the ``isolated\" plots in Sections \\ref{sec:isolated_pairs} and \\ref{sec:galaxy_properties}.\n\n\n\\subsection{Results}\\label{sec:results_all_pairs}\nIn \\fref{fig:generic_all_pairs} we show the average pairwise velocity $ v_{12} $ of all the galaxies within the \\citet{guo:2011a} semi-analytic model at redshift $ z=0 $, as a function of separation $ d $.  The velocity curve is represented by the red dots with one-sigma error bars, while the blue and black lines are, respectively, the static and comoving solutions. %\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0_z0_noiso_theory.pdf}\n\\caption{Average relative velocity for all galaxy pairs at redshift $z=0$. The blue solid line represents the static solution, followed by pairs that have already virialized and do not feel the background Hubble flow. The green dashed line represents the linear theory model for pairwise velocities according to the prescription found in References \\cite{fisher:1995a, reid:2011a} with a bias of $b=1$. Error bars, which are too small to be clearly seen are plotted at the one sigma confidence limit, assuming Poisson statistics.}\n\\label{fig:generic_all_pairs}\n\\end{figure}\n\nWe also plot the prediction of linear perturbation theory for $ v_{12} $ as the green dashed line, obtained using the prescription from \\cite{fisher:1995a, reid:2011a}:\n\\begin{equation}\n  v_{12}(d) = -\\frac{fb}{\\pi^2}\\int dk\\ k\\ P_m(k)j_1(kd),\n\\end{equation}\nwhere we have chosen a bias of $b=1$, which is in agreement with that measured from clustering within the galaxy catalogue.  \n\nFor separations larger than  $d>\\mpch{10}$, we can see that the average peculiar velocity is correctly predicted by linear theory.  As we follow the velocity curve into the nonlinear regime, galaxy pairs approach the static line until they cross it at $d\\sim\\mpch{3}$.  This crossing marks the beginning of the infall regime, where the galaxies in the pairs get closer to each other, but with smaller velocities as their separation decreases.  On the smallest scales, the pairs asymptote to the static solution.\n\nTo use galaxy pairs as tracers of the cosmological expansion, we need their peculiar velocity to be small with respect to the Hubble flow or modellable.  A smaller correction is required if $ v_{12} $ is closer to zero comoving velocity than to the static solution. As we noted above, only for $d>\\mpch{10}$ are the velocities closer to the comoving solution than the static solution, which is the regime of linear perturbation theory.  On scales $d\\lesssim\\mpch{3}$, galaxy pairs follow closely the static solution on average.  Their peculiar velocity component is equal and opposite to the Hubble flow, therefore they do not carry any cosmological information (refer to Appendix \\ref{app:bound_systems}).\n\n\\section{Isolated pairs} \\label{sec:isolated_pairs}\nIn this section we investigate how the dynamics of galaxy pairs changes when an isolation criterion is imposed and how this depends on the isolation radius, the allowed number of galaxies within this radius, and redshift. We use the same galaxy sample as in \\sref{sec:all_pairs}.\n\n\n\\subsection{Method}\\label{sec:method_isolated_pairs}\nWe initially define a galaxy pair to be isolated within a radius \\ensuremath{r_\\text{iso}}\\xspace if each galaxy in the pair has exactly 1 neighbor within \\ensuremath{r_\\text{iso}}\\xspace, and that neighbor is the other galaxy in the pair.  This is equivalent to drawing two spheres of radius \\ensuremath{r_\\text{iso}}\\xspace centered on the galaxies, and imposing the absence of galaxies extraneous to the pair in each of the spheres.  We shall weaken this requirement, allowing for the maximum number of neighbors \\ensuremath{N_\\text{neigh}}\\xspace in each sphere to be larger than 1.  Thus we can interpolate between the dynamics of galaxy pairs in the fully isolated case ($ \\ensuremath{N_\\text{neigh}}\\xspace = 1 $) and in the unconstrained case ($ \\ensuremath{N_\\text{neigh}}\\xspace \\rightarrow \\infty $). We implement such isolation criterion with arbitrary number of neighbors in a two-step process.  We first determine the number of neighbors for each galaxy in the simulation, and later use this information to select only those pairs where each galaxy has less than \\ensuremath{N_\\text{neigh}}\\xspace neighbors.\n\nWe fix the isolation radius to $ \\ensuremath{r_\\text{iso}}\\xspace = \\mpch{4} $, which matches the definition of isolation adopted by \\citet{marinoni:2010a}.  Note that in the plots that follow, we only look at separations less than the isolation radius to ensure that the pair is truly isolated.\n\n\n\n\\subsection{Results}\nIn the upper panel of \\fref{fig:generic_isolated} we present the relative motion of galaxy pairs as a function of their separation for galaxies that are isolated within a \\mpch{4} radius. The galaxies are taken from the \\citet{guo:2011a} semi-analytic model applied to the Millennium simulation at redshift $ z = 0 $. The data points are plotted in red with one-sigma error bars. The blue line is the static solution and the black line the comoving solution -- see \\sref{sec:method_allpairs} for their definition.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0_z0_isolation4_ratio.pdf}\n\\caption{Upper panel: average pairwise velocity for isolated galaxy pairs at redshift $z=0$. The isolation radius is taken to be \\mpch{4} for each member of the pair. The blue solid line represents the static solution, showing the virialization of pairs. The three shaded regions represent three different regimes. The left blue area represents the virialization regime: galaxies within these separations have virialized and do not experience the background expansion. The middle red area shows the infall regime, where galaxies start to collapse to form bound systems. The right green region shows what we denote as the `void effect': on average, the isolated pair feels a stronger gravitational pull separating the pair rather than making it closer. The error bars shown are the one sigma confidence limit, assuming Poisson statistics. Lower panel: ratio of average pairwise velocities to the static solution $Hd$. Note that the y axis of this panel is in logarithmic scale. The error bars shown are the propagated one-sigma errors from the upper panel.}\n\\label{fig:generic_isolated}\n\\end{figure}\n\nOur first comment on \\fref{fig:generic_isolated} regards the error bars, which are much larger than those in the non-isolated case of \\fref{fig:generic_all_pairs}.  The reason is that the imposition of an isolation criterion results in a drastic reduction of the galaxy pairs found, which in the case of \\fref{fig:generic_isolated} are only $ 694 $\n\\footnote{Equivalent to one isolated pair every $10^6$ other pairs for $ d = \\mpch{1} $}.\nThis number is in line with \\citet{marinoni:2010a}, who find $ 721 $ pairs for their low-redshift SDSS sample, and $ 509 $ for their DEEP2 sample. \n\nThe most striking feature about the dynamics of isolated pairs is the roughly logarithmic growth of the peculiar velocity $ v_{12} $ for scales larger than $ \\sim \\mpch{0.2} $.\nThe behavior of $ v_{12} $ can be explained when we recognize that the dynamics is determined by the combined effect of two competing forces: the mutual attraction between the two galaxies, dominant for small separations, and the disrupting outflow from the void -- the \\emph{void effect}, dominant for separations close to the isolation radius.\nFor small separations, the mutual attraction of the members of the pair overcomes the void effect and we see an infall regime.  As we study objects with larger separations, the void effect becomes dominant and we see a logarithmically growing pairwise velocity $ v_{12} $.\n\nIn the lower panel of \\fref{fig:generic_isolated} we plot $ v_{12}\/H d $, that is the ratio of peculiar velocity to Hubble flow. For separations $ 0.4 < d < \\mpch{4} $ we find an almost comoving regime where the peculiar velocities are less than $ 20\\% $ of the Hubble flow.  In such a regime,  the RSD corrections are small, and one could hope that they are easier to model so that $ v_{12} $ becomes a proxy of the expansion rate.  In the following subsections we shall investigate how the comoving regime depends on the isolation radius, the local density and redshift.\n\n\n\\subsubsection{Varying the isolation radius}\\label{sec:shrinking}\nIt is interesting to investigate whether the void effect, giving the approximately logarithmic growth of $ v_{12}(d) $ for isolated pairs, is still present when we allow the size of the isolation radius to vary.  We demonstrate this in \\fref{fig:isolation_radii}, where we show the peculiar velocities of galaxy pairs with $ \\ensuremath{r_\\text{iso}}\\xspace $ varying from \\mpch{0.1} to \\mpch{4}, at redshift $ z = 0.989 $.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{guo2010_isolation_radii01to4_z0989_np0.pdf}\n\\caption{Average pairwise velocity for isolated galaxy pairs at redshift $z=0.989$ for different isolation radii ranging from \\mpch{0.1} to \\mpch{4}.}\n\\label{fig:isolation_radii}\n\\end{figure}\n\nFor $ \\ensuremath{r_\\text{iso}}\\xspace \\geqslant \\mpch{0.6} $ the presence of the void severely affects the dynamics of galaxy pairs.  The logarithmic growth of the peculiar velocity is visible, even though it is just a hint for the $ \\ensuremath{r_\\text{iso}}\\xspace = \\mpch{0.6} $ data points.  The cosmological scale $ d_0 $, defined as the separation where the peculiar velocity $ v_{12} $ vanishes, occurs at $ d_0 \\simeq \\mpch{0.8} $ and appears to be independent of the isolation radius. \n\nAt small separations, we cannot see any noticeable differences between the various $ \\ensuremath{r_\\text{iso}}\\xspace $ datasets.  They all follow the static solution line for $d<\\mpch{0.05}$, suggesting that isolated galaxy pairs tend to virialize on the smallest scales just as non-isolated ones do.\n\nThe number of isolated galaxy pairs at $ z=0.989 $ increases from $ 435 $ to $ 71,201 $ when reducing the isolation radius from $ 4 $ to \\mpch{2}, a factor of roughly $ 160 $.  As we have noted above, the $ \\ensuremath{r_\\text{iso}}\\xspace = \\mpch{2} $ pairs still experience a regime where peculiar velocities are negligible with respect to the Hubble flow.  Hence their velocity difference still traces the cosmological expansion, although the maximum separation for which this is the case is halved with respect to the $ \\ensuremath{r_\\text{iso}}\\xspace = \\mpch{4} $ case.  We conclude that using pairs isolated within a \\mpch{2} radius as cosmological tracers would drastically reduce the statistical error with respect to the \\mpch{4} case, while still needing minimal corrections for RSD, provided that the cosmological dependence of the RSD could be modeled. We shall discuss this in more detail later.\n\n\n\\subsubsection{Varying the isolation density criterion}\\label{sec:populating}\nHere, we investigate how the dynamics of galaxy pairs changes if we relax the isolation criterion by increasing the allowed number of galaxies $ \\ensuremath{N_\\text{neigh}}\\xspace $ within the isolation sphere of \\mpch{4}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{guo2010_neighbours1to5000_z0989_isolaton4_np0_lin.pdf}\n\\caption{Average pairwise velocity for isolated galaxy pairs at redshift $z=0.989$ for different number of neighbors within the isolation sphere of \\mpch{4}. The blue solid line represents the static solution, showing the virialization of pairs. The green dashed line represents the average pairwise velocity for non-isolated pairs. The error bars shown are the one sigma confidence limit, assuming Poisson statistics.}\n\\label{fig:neighbours}\n\\end{figure}\n\nIn the linear plot of \\fref{fig:neighbours}, we present the average peculiar velocity $ v_{12} $ at $ z=0.989 $ for 7 values of $ \\ensuremath{N_\\text{neigh}}\\xspace $ ranging from $ \\ensuremath{N_\\text{neigh}}\\xspace = 1 $ (equivalent to the pure isolated case of \\fref{fig:generic_isolated}) to $ \\ensuremath{N_\\text{neigh}}\\xspace = 5000 $.  We also plot $ v_{12} $ for the non-isolated galaxy pairs, as already shown in \\fref{fig:generic_all_pairs}, as a dashed green curve.  As $ \\ensuremath{N_\\text{neigh}}\\xspace $ increases, the different $ v_{12} $ curves monotonically fill the gap between the pure isolated case and the non-isolated case.  A good agreement between the dynamics of pairs with and without isolation criterion is reached once we allow each galaxy in the pair to have $ 5000 $ other neighbors.\n\nIn the $ \\ensuremath{N_\\text{neigh}}\\xspace = 10 $ case, we found $ 250,670 $ pairs, roughly a factor $ 600 $ more pairs than in the fully isolated case of $ \\ensuremath{N_\\text{neigh}}\\xspace = 1 $.  Nonetheless, the $ v_{12} $ curve for $ \\ensuremath{N_\\text{neigh}}\\xspace = 10 $ is strikingly similar to the $ \\ensuremath{N_\\text{neigh}}\\xspace = 1 $ one.  In particular, the void effect still seems to trigger the logarithmic growth of the peculiar velocity, with $ v_{12} $ crossing the zero line at $ d_0 \\simeq \\mpch{1} $ (in the fully isolated case, we have $ d_0 \\simeq \\mpch{0.8}) $. Hence, we suggest that pairs that are not completely isolated trace the cosmological expansion almost as well as the fully isolated ones, with the added benefit of a much better statistics.\n\n\n\\subsubsection{Varying redshift} \\label{sec:varying_redshift}\n\n\nWe illustrate the redshift dependence of the peculiar velocity for isolated pairs in the top panel of \\fref{fig:iso_redshift_variation}, where we plot $ v_{12}(d) $ for the four redshifts $ z = 0, 0.5085, 0.989, 1.504 $.  It is remarkable that for separations $ d \\gtrsim \\mpch{0.2} $ the peculiar velocity depends only slightly on redshift.  The scale $ d_0 $, defined as the separation where $ v_{12} $ vanishes, ranges from \\mpch{0.6} at $ z=0 $ to \\mpch{0.9} at $ z=1.504 $.  This is a small variation if we consider that at $ z=1.504 $ the Universe was at one third of its current age.\n\n\\begin{figure}[ht]\n\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0_z0to1504_isolation4_ratio_errorbars.pdf}\n\\caption{Upper panel: variation of the average pairwise velocity of isolated galaxy pairs with redshift as a function of separation. The isolation radius is taken to be \\mpch{4} for each member of the pair. Lower panel: range in pairwise velocity for each separation bin over range in the static solutions at each separation bin. Note that the y axis of this panel is in logarithmic scale. The error bars shown are the propagated one-sigma errors from the upper panel.}\n\\label{fig:iso_redshift_variation}\n\\end{figure}\n\nIn the bottom panel of \\fref{fig:iso_redshift_variation} we plot the ratio between the range in $ v_{12} $ and the range in $ H d $ at a given separation.  For separations $ 1 < d < \\mpch{4} $, the change of peculiar velocity with redshift is only $ 10 \\% $ of the change in Hubble flow.  This implies a weak dependence of $ v_{12} $ on cosmological expansion on those scales.\n\n\\subsection{Cosmological implications}\\label{sec:cosmological_implications}\n\\fref{fig:generic_isolated} shows that for $ 0.4 < d < \\mpch{4} $ isolated galaxy pairs at $ z=0 $ are nearly comoving with the Hubble flow.  Thus, they move with the cosmological expansion and, for this cosmology and epoch, only need a small RSD correction. Using different redshift slices as a way to test different cosmological expansion rates, in the lower panel of \\fref{fig:iso_redshift_variation}, we identify a second regime for $ 1 < d < \\mpch{4} $ where the peculiar velocity $ v_{12}(z) $ depends only slightly on the cosmological expansion. This suggests the intriguing possibility of isolated pairs behaving in the same way on those scales regardless of the assumed cosmology. In particular, the lower panel of \\fref{fig:iso_redshift_variation} shows that the variation in RSD model is less than the variation in expansion rates. Thus we conclude that there is cosmological signal to be extracted here.\n\nWe refer to the intersection of these regimes, where we have almost comoving pairs with small redshift evolution, as a cosmological regime, since we might be able to use these pairs as cosmological tracers. Measuring galaxy pairs in the cosmological regime would still induce a systematic error due to the fact that peculiar velocities are non-zero.  As the correction is of the $10\\%$ level -- see lower panel of \\fref{fig:generic_isolated} -- and we might suppose to be able to model this at the same level, we would have a $1\\%$ systematic correction to contend with.  This claim is little more than a speculation at this stage; in order to falsify or confirm it, one needs to model isolated pairs in detail and to analyze N-body simulations with different underlying cosmologies. \n\n\nPairs isolated within a radius of \\mpch{4} are rare objects -- see \\fref{fig:number_pairs} -- and this might result in significant statistical error when dealing with observations.  In Sections \\ref{sec:shrinking} and \\ref{sec:populating} we found that one can drastically increase the number of pairs while keeping the RSD correction small by either reducing the isolation radius to \\mpch{2} or allowing up to $ 10 $ galaxies to be neighbors of the pair.\n\n\n\n\\section{Varying galaxy properties} \\label{sec:galaxy_properties}\nThe main result of the previous section, illustrated in Fig.\\ \\ref{fig:generic_isolated} and \\ref{fig:iso_redshift_variation}, is that there is a regime where isolated galaxy pairs may be used as tracers of expansion with correction for RSD that is weaker than the signal to be measured.  Such a finding relies on the ability of measuring the redshift of galaxies regardless of their mass or luminosity.  This is clearly not the case when dealing with actual galaxy surveys, whose flux sensitivity is limited.   To model such a selection bias, we need to investigate ways of selecting galaxies from simulations which mimic the selection process of surveys.   In this section we address this issue by forming subsamples where galaxies are selected according to subhalo mass, stellar mass and magnitude.  We also study the redshift dependence of our results by analyzing 4 different redshifts: $ z = 0,\\,0.5085,\\,0.989,\\,1.504 $.\n\n\n\\subsection{Method} \\label{sec:method_varying}\n\\begin{table*}[t]\n    \\begin{tabular}{ r*{8}r }\n        $\\ensuremath{n_\\text{p}}\\xspace$  &\\:\\:\\:  $m(\\msun\/h)$  &\\:\\:\\: $\\ensuremath{r\\text{-mag}}\\xspace$  &\\:\\:\\:  $\\ensuremath{m_{\\star}}\\xspace(\\msun\/h)$  &\\:\\:\\:  n(z=0) (Mpc$^{-3}$)  &\\:\\:  \\%(z=0)  &\\:\\:\\:  \\%(z=0.5085)  &\\:\\:\\:  \\%(z=0.989)  &\\:\\:\\:  \\%(z=1.504)  \\\\\n        \\hline \\hline\n        20   & \\sci{1.72}{10} &        &                & \\sci{8.1}{-2} &   100  &   100  &   100  &   100 \\\\\n        100  & \\sci{8.6}{10}  & -19.27 & \\sci{1.40}{9}  & \\sci{1.4}{-2} & 17.10  & 17.81  & 17.77  & 17.49 \\\\\n        500  & \\sci{4.3}{11}  & -21.72 & \\sci{1.59}{10} & \\sci{3.1}{-3} &  3.85  &  3.73  &  3.45  &  3.09 \\\\\n        1000 & \\sci{8.6}{11}  & -22.17 & \\sci{2.60}{10} & \\sci{1.6}{-3} &  2.01  &  1.88  &  1.67  &  1.41 \\\\\n        5000 & \\sci{4.3}{12}  & -22.86 & \\sci{5.36}{10} & \\sci{3.4}{-4} &  0.42  &  0.35  &  0.27  &  0.19\n    \\end{tabular}\n    \\caption{Cuts imposed on our galaxy sample from the \\citet{guo:2011a} semianalytic model. The first line corresponds to no cut at all; in that case, for the particle number column, we report the \\citet{guo:2011a} resolution limit of $ \\ensuremath{n_\\text{p}}\\xspace = 20 $.  The columns with a percentage sign denote the percentage of galaxies surviving the cut at a given redshift.  The \\ensuremath{r\\text{-mag}}\\xspace and \\ensuremath{m_{\\star}}\\xspace columns refer to the cuts performed at $ z = 0.989 $, as it is the only redshift we plot for these quantities (see Figures \\ref{fig:rmag_variation} and \\ref{fig:stellar_mass_variation}).  Note that the \\ensuremath{r\\text{-mag}}\\xspace cuts are intended to be upper limits, while the mass cuts are lower limits.}\n\\label{tab:cuts}\n\\end{table*}\n\n\n\\begin{figure*}[t]\n\\subfloat[No isolation]{\\label{fig:npairs_noiso}\\includegraphics[width=0.5\\textwidth]{guo2010_npairs_np10to5000_z0989_noiso}}\n\\subfloat[Isolation of \\mpch{4}]{\\label{fig:npairs_isolation4}\\includegraphics[width=0.5\\textwidth]{guo2010_npairs_np10to5000_z0989_isolation4}}\n\\caption{Number of galaxy pairs for different mass cuts as a function of separation at redshift $z=0.989$.  In the right panel we omitted the $ m > \\msunh{8.6}{11} $ curve for the sake of readability.}\n\\label{fig:number_pairs}\n\\end{figure*}\n\nWe select galaxy subsamples from the semi-analytic models by applying cuts on galaxy properties.  We then study the dynamics of each subsample by applying the same analyses of Sections \\ref{sec:all_pairs} and \\ref{sec:isolated_pairs}.  Initially we look at the number \\ensuremath{n_\\text{p}}\\xspace of dark matter particles of the subhalo the galaxy is in%\n\\footnote{For reference, this is the \\emph{np} field of the \\emph{Guo2010a} database in the Millennium simulation servers.}.\nWe consider $ \\ensuremath{n_\\text{p}}\\xspace = 100,\\,500,\\,1000,\\,5000 $ corresponding to masses $ m = \\sci{8.6}{10},\\,\\sci{4.3}{11},\\,\\sci{8.6}{11},\\,\\msunh{4.3}{12} $.  Note that, where no cut is made, the number of particles in each subhalo is always $ \\ensuremath{n_\\text{p}}\\xspace > 20 $, corresponding to \\msunh{1.72}{10}, since this is the threshold that defines a bound subhalo according to \\citet{guo:2011a}. \n\nWe choose the dark matter mass as our main cut because it is directly related to the pairwise velocity dynamics, which is the main subject of this paper.  In order to make a more direct link with observations, we also study the pairwise velocity statistics when varying the rest-frame $ r $-band magnitude%\n\\footnote{More precisely, this is the rest-frame total absolute magnitude in the SDSS r-band, corresponding to the \\emph{r\\_mag} field in the \\emph{Guo2010a} database of the Garching mirror and to the \\emph{r\\_SDSS} field in the \\emph{Font2008a} database of the Durham mirror.}\nand stellar mass of galaxies.\n\nThe limits on $ r $-band magnitude (hereafter \\ensuremath{r\\text{-mag}}\\xspace) and stellar mass (hereafter \\ensuremath{m_{\\star}}\\xspace) are chosen such that, for a given \\ensuremath{n_\\text{p}}\\xspace cut, the corresponding \\ensuremath{r\\text{-mag}}\\xspace and \\ensuremath{m_{\\star}}\\xspace cuts yield the same number of surviving galaxies. \\tref{tab:cuts} reports the values of the limits used, together with the resulting fraction of surviving galaxies at each redshift. We apply these cuts to the data sets before running the pair-finder algorithms.  Thus, a pair that is isolated within its subsample may not be isolated when considering the full catalog, \\ie our isolation criterion is sample dependent.  This implementation is in line with an analysis of an actual galaxy survey, limited by these cuts.\n\n\nIn \\fref{fig:number_pairs} we show how many galaxy pairs we find after imposing the cuts given in \\tref{tab:cuts}.  The number of unconstrained pairs (left panel) decreases monotonically with increasing mass cut.  Note that the drop-off in the number of pairs at large separations is due to the size of the individual boxes we consider and has no physical meaning.  For isolated pairs (right panel), the number density increases as we increase the mass cut, as a higher mass cut results in a sparser distribution of galaxies where it is easier to find isolated pairs.  Only for our most stringent mass cut, that is for $ m>\\msunh{4.3}{12} $, do we see a slight decrease in the density of isolated pairs due to the small number of high mass galaxies.\n\n\\begin{figure*}[ht]\n\\subfloat[Redshift z=0]{\\label{fig:z_0}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z0_noiso.pdf}}\n\\subfloat[Redshift z=0.5085]{\\label{fig:z_0.5}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z05085_noiso.pdf}}\\\\\n\\subfloat[Redshift z=0.989]{\\label{fig:z_1}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z0989_noiso.pdf}}\n\\subfloat[Redshift z=1.504]{\\label{fig:z_1.5}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z1504_noiso.pdf}}\n\\caption{Variation of the average pairwise velocity of all galaxy pairs with redshift for different mass cuts (number of dark matter particles) as a function of separation.}\n\\label{fig:redshift_variation_noiso}\n\\end{figure*}\n\n\n\\subsection{Results}\n\n\\subsubsection{All galaxy pairs}\nIn \\fref{fig:redshift_variation_noiso} we present the average pairwise velocity $ v_{12} $ for non-isolated pairs above different subhalo mass thresholds.  The four panels show the same selection procedure at different redshifts. The mass cuts range from \\msunh{1.72}{10} up to \\msunh{4.3}{12} as tabulated in \\tref{tab:cuts}.  Note that the lowest mass cut corresponds to the smallest subhalo in the \\citet{guo:2011a} semianalytic model, consisting of $ 20 $ dark matter particles.\n\n\n\n\nThe imposition of a mass cut has a significant impact on non-linear scales. Independent of redshift, \\fref{fig:redshift_variation_noiso} shows that massive galaxy pairs experience an infall regime for separations smaller than $ d \\lesssim \\mpch{3} $.  While in such a regime, peculiar velocity increases with mass, with the most massive galaxies ranging from $ v_{12} \\simeq \\kms{330} $ at $ z=0 $ and $ v_{12} \\simeq \\kms{600} $ at $ z=1.504 $.  Lower mass galaxies, on the other hand, seem to follow the static solution up to higher separations, especially at low redshift. Our interpretation for such behavior is that galaxies in high mass pairs are more affected at small separations by their mutual attraction than the underlying density field.\n\n\\begin{figure*}[ht]\n\\subfloat[Redshift z=0]{\\label{fig:z_0_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z0_isolation4.pdf}}\n\\subfloat[Redshift z=0.5085]{\\label{fig:z_0.5_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z05085_isolation4.pdf}}\\\\\n\\subfloat[Redshift z=0.989]{\\label{fig:z_1_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z0989_isolation4.pdf}}\n\\subfloat[Redshift z=1.504]{\\label{fig:z_1.5_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z1504_isolation4.pdf}}\n\\caption{Variation of the average pairwise velocity of isolated galaxy pairs with redshift for different mass cuts (number of dark matter particles) as a function of separation. The isolation radius is taken to be \\mpch{4} for each member of the pair.}\n\\label{fig:redshift_variation_isolation4}\n\\end{figure*}\n\n\\begin{figure}[h]\n\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np10to5000_z0989_isolation4_scaling.pdf}\n\\caption{Same plot as in \\fref{fig:z_1_iso} where both the separation and the average pairwise velocity for each curve have been scaled by a factor $(M_{\\text{ref}}\/M)^{1\/3}$ where $M_{\\text{ref}}$ is the minimum mass of a subhalo: $M_{\\text{ref}}=\\sci{1.72}{10} \\msun\/h$. Note that the errors for each curve have been scaled accordingly.}\n\\label{fig:scaling_z1}\n\\end{figure}\n\nFor separations $ d > \\mpch{10} $, the velocity curves at each redshift seem to converge to a common asymptote.  In \\sref{sec:all_pairs}, we have shown that this limit is correctly predicted by linear theory -- see the agreement between the green curve and $ v_{12} $ in \\fref{fig:generic_all_pairs}.  This means that, even though the non-linear dynamics of the different mass limit pairs differs, their behavior at large separations seems to be predicted by the same linear theory.\n\nA remarkable feature of \\fref{fig:redshift_variation_noiso} is the different redshift dependence of the various $ v_{12} $ curves.  The pairwise velocity of the heaviest galaxies (cyan triangles) greatly increases with redshift, while in the uncut case (red circles) it decreases.  The intermediate curves seem to experience smaller variations. \n\n\n\\subsubsection{Isolated pairs}\n\n\\begin{figure*}[ht]\n\\subfloat[\\;No mass cut\\;\\;(\\:$ \\ensuremath{n_\\text{p}}\\xspace \\geqslant 20 \\:)$]\n {\\label{fig:np_0_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0_z0to1504_isolation4.pdf}}\n\\subfloat[Mass $ \\geqslant 4.3\\times10^{11}$ M$_\\odot\/h$\\, \\;\\;(\\:$ \\ensuremath{n_\\text{p}}\\xspace \\geqslant 500 \\:)$] {\\label{fig:np_100_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np500_z0to1504_isolation4.pdf}}\\\\\n\\subfloat[Mass $ \\geqslant 8.6\\times10^{11}$ M$_\\odot\/h$\\, \\;\\;(\\:$ \\ensuremath{n_\\text{p}}\\xspace \\geqslant 1000 \\:)$] {\\label{fig:np_1000_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np1000_z0to1504_isolation4.pdf}}\n\\subfloat[Mass $ \\geqslant 4.3\\times10^{12}$ M$_\\odot\/h$\\, \\;\\;(\\:$ \\ensuremath{n_\\text{p}}\\xspace \\geqslant 5000 \\:)$] {\\label{fig:np_5000_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np5000_z0to1504_isolation4.pdf}}\n\\caption{Variation of the average pairwise velocity of isolated galaxy pairs with redshift for different mass cuts (number of dark matter particles) as a function of separation. The isolation radius is taken to be \\mpch{4} for each member of the pair. Note that panel \\fref{fig:np_0_iso} is equivalent to the upper panel of \\fref{fig:iso_redshift_variation}.}\n\\label{fig:masscut_variation_isolation4}\n\\end{figure*}\n\n\nHaving analyzed the dynamics of non-isolated pairs with varying subhalo mass, we now do the same for pairs isolated within a \\mpch{4} radius. In \\fref{fig:redshift_variation_isolation4} we show the average pairwise velocity $ v_{12} $ for different mass cuts, with the redshift varying form panel to panel. This is the same setup as in \\fref{fig:redshift_variation_noiso}; note, however, that here we only plot separations up to \\mpch{4}.\n\n\nAll curves in \\fref{fig:redshift_variation_isolation4}, regardless of redshift, present the same features found in the uncut sample shown in \\fref{fig:generic_isolated} and explained in \\sref{sec:isolated_pairs}.  Namely, we see a virialized region on the smallest scales, an infalling regime on intermediate scales, and a roughly logarithmic growth due to the void effect on the largest separations analyzed.  The main difference from the uncut case is the separations at which these different regimes hold.  Most importantly, we can see that the logarithmic growth of $ v_{12} $ begins at larger separations for higher masses.  This is intuitive, since we expect the mutual attraction to be stronger in heavier pairs, thus overcoming the void effect even when the galaxies are closer to the edge of the void.\n\nAs a result of this stronger mutual attraction, the peculiar velocity contribution increases as we consider heavier pairs.  This means that when it comes to isolated galaxy pairs, low-mass pairs trace the cosmological expansion better than high-mass ones.  More quantitatively, the scale $ d_0 $ where the peculiar velocity vanishes is reached at larger separations for massive pairs.  At $ z=0.989 $, $ d_0 $ ranges from \\mpch{0.8} in the uncut case to almost \\mpch{4} for pairs with $ m>\\msunh{4.3}{11} $.   For higher masses, $ v_{12} $ does not even cross the zero line.\n\n\\begin{figure*}[ht]\n\\subfloat[No isolation]{\\label{fig:noiso_rmag}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_rmag-22to99_z0989_noiso}}\n\\subfloat[Isolation of \\mpch{4}]{\\label{fig:iso_rmag}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_rmag-22to99_z0989_isolation4}}\n\\caption{Average pairwise velocity of galaxy pairs at $z=0.989$ varying r-band magnitude.}\n\\label{fig:rmag_variation}\n\\end{figure*}\n\n\n\\begin{figure*}[ht]\n\\subfloat[No isolation]{\\label{fig:noiso_stellarm}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_stellarm0to536_z0989_noiso}}\n\\subfloat[Isolation of \\mpch{4}]{\\label{fig:iso_stellarm}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_stellarm0to536_z0989_isolation4}}\n\\caption{Average pairwise velocity of galaxy pairs  at $z=0.989$ varying stellar mass.}\n\\label{fig:stellar_mass_variation}\n\\end{figure*}\n\n\nTo illustrate the redshift dependence of the peculiar velocity in more detail, in \\fref{fig:masscut_variation_isolation4} we plot $ v_{12} $ for a given mass-cut at four different redshifts, with the mass-cuts varying across the panels.  Increasing the mass-cut makes the redshift evolution of $ v_{12} $ more evident.  As a result, for $ m > \\msunh{4.3}{11} $, we cannot identify a cosmological regime where the pairs are comoving and have a redshift independent peculiar velocity. Where ``independence\" here means that the evolution is significantly less than the change in expansion rate.\n\nA closer look at \\fref{fig:redshift_variation_isolation4} shows a pattern in the different mass cuts. We notice that, as we increase the mass cut, the absolute value of the pairwise velocity increases and the minimum shifts to the right. This suggests that applying a mass dependent scaling to both separation and velocity may stack the curves. This is shown in \\fref{fig:scaling_z1}, where we have have taken as an example the plot at redshift $z=0.989$ (\\fref{fig:z_1_iso}) and scaled both the x and y axis by a factor $(M_{\\text{ref}}\/M)^{1\/3}$ where $M_{\\text{ref}}$ is the minimum mass of a subhalo: $M_{\\text{ref}}=\\sci{1.72}{10} \\msun\/h$. The physical motivation for this scaling is to have the same orbital period for all curves in the Keplerian regime \\ie on small scales. Indeed we see on this plot that such scaling collapses the curves specially in the infalling and virialized regions.\n\n\\subsubsection{Magnitude \\& stellar mass}\n\n\\begin{figure*}[ht]\n\\subfloat[No isolation]{\\label{fig:noiso_font}\\includegraphics[width=0.5\\textwidth]{font2008_vinfall_rmag-22to99_z0989_noiso}}\n\\subfloat[Isolation of \\mpch{4}]{\\label{fig:iso_font}\\includegraphics[width=0.5\\textwidth]{font2008_vinfall_rmag-22to99_z0989_isolation4}}\n\\caption{Average pairwise velocity of galaxy pairs for the \\citet{font:2008a} semi-analytic model at $ z=0.989 $.  This figure should be compared with \\fref{fig:rmag_variation}, where we plotted the same curves obtained for the \\citet{guo:2011a} model.}\n\\label{fig:rmag_font}\n\\end{figure*}\n\n\nWe now make a more direct link with observations and study the dependence of $ v_{12} $ on $ r $-band absolute magnitude and stellar mass.\n\nIn the left panel of \\fref{fig:rmag_variation}, we show the average pairwise velocity $ v_{12} $ of all galaxy pairs at $ z = 0.989 $ for the \\ensuremath{r\\text{-mag}}\\xspace cuts given in \\tref{tab:cuts}.  Although the curves retain their qualitative shape, there are two major differences with respect to the mass-cut sample in \\fref{fig:z_1}.  Firstly, the \\ensuremath{r\\text{-mag}}\\xspace selected pairs have smaller average velocities.  Secondly, the velocity minima are all approximately aligned at the same scale of $r_\\text{min}\\sim$ \\mpch{2}, while for the subhalo mass cuts the different velocity curves have their minima at different separations. These two differences are also seen when we apply the cuts in stellar mass (\\fref{fig:noiso_stellarm}).\n\nThe velocity differences can be explained by the fact that, although the subsamples chosen based on limits in $ r $-band magnitude and stellar mass preserve the number density of galaxies selected, these are not the same galaxies as the ones selected by the subhalo mass cuts. In particular, most massive galaxies do not necessarily coincide with the most luminous ones. In general, dark matter haloes trace the velocity of galaxies more directly than stellar mass or \\ensuremath{r\\text{-band}}\\xspace magnitude. Cuts based on stellar mass or luminosity add an additional dispersion, affecting the position of the minima with respect to subhalo mass cuts.\n\nThe right panels of \\fref{fig:rmag_variation} and \\fref{fig:stellar_mass_variation} show the average pairwise velocity $ v_{12} $ for pairs with an isolation radius of \\mpch{4} for cuts in \\ensuremath{r\\text{-band}}\\xspace magnitude and stellar mass respectively. These plots should be compared with the corresponding cuts in subhalo mass at redshift $z=0.989$ (\\fref{fig:z_1_iso}). Even though we again appreciate that the pairwise velocities in the \\ensuremath{r\\text{-mag}}\\xspace and stellar mass cut plots are smaller, the general dynamics shown on the plots are the same. It is worth noting that the almost comoving regime mentioned in \\sref{sec:cosmological_implications} for each curve remains unchanged both for the \\ensuremath{r\\text{-band}}\\xspace magnitude and the stellar mass cuts. It is clear that the effects of galaxy selection (be it subhalo mass, stellar mass or \\ensuremath{r\\text{-band}}\\xspace magnitude) play an important role in the behavior of pairwise velocities for isolated galaxy pairs.\n\n\n\\section{Comparison of two catalogs} \\label{sec:comparison_catalogs}\nThe results presented in the previous sections were based on the semi-analytic model of \\citet{guo:2011a}. To check the robustness of these results, we also compute the average peculiar velocities $ v_{12} $ for the semi-analytic model in \\citet{font:2008a}. This catalog is an improvement over the one presented in \\citet{bower:2006a} to better match the colors of satellite galaxies observed in the SDSS sample. In order to do this, the main modification introduced in \\citet{font:2008a} is the stripping of hot gaseous haloes of satellite galaxies into the GALFORM semi-analytic model for galaxy formation, while \\citet{guo:2011a} concentrate more on the independence of satellite galaxies from the FOF group. Both semi-analytic models have similar galaxy luminosity functions that fit the data well.\n\nIn \\fref{fig:rmag_font} we show the average pairwise velocity $ v_{12} $ as a function of separation for all the galaxy pairs (left panel) and for isolated pairs with an isolation radius of \\mpch{4} (right panel).  Each curve corresponds to one of the \\ensuremath{r\\text{-mag}}\\xspace cuts in \\tref{tab:cuts}, and should be compared with the matching curve for the \\citet{guo:2011a} catalog in \\fref{fig:rmag_variation}.   Note that we omitted to plot $ v_{12} $ for our most stringent cut of $ \\ensuremath{r\\text{-mag}}\\xspace < -22.86 $ because of poor statistics.  In general, we found that \\citet{font:2008a} has significantly less bright galaxies with $ \\ensuremath{r\\text{-mag}}\\xspace < -22.17 $ than \\citet{guo:2011a}, as can be seen by the large error bars in \\fref{fig:rmag_font}. \n\nA direct comparison between Figures \\ref{fig:rmag_font} and \\ref{fig:rmag_variation} shows that galaxy pairs have very similar dynamics regardless of the semi-analytic model used.   Not only do we see almost the same $ v_{12} $ range, but also the almost comoving regime introduced in \\sref{sec:cosmological_implications} is found approximately in the same range.  Such findings suggest that the parameters of the semi-analytic model used do not significantly affect the average pairwise velocities of galaxies. \n\n\n\\section{Conclusions}  \\label{sec:conclusions}\nWe have investigated the pairwise velocities of galaxies with a view towards using them as cosmological tracers by means of an AP style test, as recently proposed by \\citet{marinoni:2010a}.  We have analyzed the dynamics of such objects within the semi-analytic models of \\citet{guo:2011a} and \\citet{font:2008a} applied to the Millennium simulation \\cite{springel:2005a}, and studied the dependence of their relative velocity on local density, redshift, mass of the hosting subhaloes, \\ensuremath{r\\text{-band}}\\xspace magnitude and stellar mass.\n\nWe have first analyzed the dynamics of all galaxy pairs at redshift $ z=0 $ (see \\fref{fig:generic_all_pairs}).  We have found that, on scales $ d>\\mpch{10} $, the peculiar velocity is correctly predicted by linear theory \\cite{fisher:1995a, reid:2011a}.  On the other hand, for separations $ d<\\mpch{3} $, the pairs are decoupled from the Hubble flow and close to the static solution.  We argue that pairs in this regime cannot be used as cosmological tracers (see Appendix \\ref{app:bound_systems}).  \n\nBeing interested in investigating the claims by \\citet{marinoni:2010a}, we have studied the dynamics of galaxy pairs that are isolated within a radius of $ \\mpch{4} $.  At $ z=0 $, isolated galaxy pairs are almost comoving already for separations of $ 0.4<d<\\mpch{4} $ and only need up to 20\\% RSD correction (see \\fref{fig:generic_isolated}).   By analyzing redshift slices up to $ z=1.504 $, we have found that the peculiar velocities are only weakly dependent on the cosmological expansion ($ < 10\\% $ variation) for separations of $ 1<d<\\mpch{4} $ (see \\fref{fig:iso_redshift_variation}).  Since expansion is the main property characterizing a cosmological model, we might assume that in this regime the dynamics of isolated pairs are independent of the underlying cosmology.  Hence, we argue that isolated pairs in this regime could possibly be used as cosmological tracers with minimal RSD corrections.  \n\nImposing an isolation criterion of \\mpch{4}, as done in Ref.\\ \\cite{marinoni:2010a}, greatly reduces the number of pairs (see \\fref{fig:number_pairs}).  We have found that one can drastically increase the statistics while keeping the RSD corrections small by either reducing the isolation radius to \\mpch{2} or allowing up to $ 10 $ galaxies to be neighbors of the pair.  When dealing with observations, these adjustments may be helpful to reduce possibly large statistical uncertainties.\n\nAs galaxy surveys are flux limited, we have studied the feasibility of a measurement by varying the following properties of galaxy pairs: mass of the subhaloes that host the galaxies, \\ensuremath{r\\text{-band}}\\xspace absolute magnitude in the rest-frame, and stellar mass.   Low-mass pairs appear to be the best cosmological tracers, as RSD corrections increase with mass.  More precisely, a nearly comoving regime is reached in our analysis only for subhalo masses of $ m \\lesssim \\msunh{4.3}{11} $, corresponding to \\ensuremath{r\\text{-band}}\\xspace magnitudes of $ \\ensuremath{r\\text{-mag}}\\xspace \\gtrsim -21.27 $ and stellar masses of $ \\ensuremath{m_{\\star}}\\xspace \\lesssim \\msunh{1.59}{10} $ (see, respectively, Figures \\ref{fig:redshift_variation_isolation4}, \\ref{fig:rmag_variation} and \\ref{fig:stellar_mass_variation}).  We have also found that the peculiar velocities of galaxy pairs becomes more redshift-dependent as we increase the subhalo mass (see \\fref{fig:iso_redshift_variation}).  Therefore, we suggest that isolated pairs may not be adequate as cosmological tracers if their mass or luminosity is above the given thresholds.\n\nMarinoni and Buzzi \\cite{marinoni:2010a} selected isolated galaxy pairs from the DEEP2 galaxy survey \\cite{davis:2003a, newman:2012a} with comoving transverse separation $ \\ensuremath{r_\\perp}\\xspace $ in the range $ \\unit[20]{kpc\/h} $ -- $ \\mpch{0.7} $.  DEEP2 galaxies are known to reside in dark matter haloes of approximately $ \\unit[10^{12}]{\\msun\/h} $ \\cite{newman:2012a, conroy:2005a}.  The results of our analysis imply that such galaxy pairs do require corrections for evolution and cosmology dependent RSD component, which is significant with respect to the evolution being measured. \n\nIndeed, the primary concern for observational studies is the extent to which RSD ``corrections\" need to be modeled. Cosmological measurements from Baryon Acoustic Oscillation and RSD measurements on large-scales are reaching a precision at the 2--5\\% level \\cite{anderson:2012a,reid:2012a,padmanabhan:2012a}, and it is therefore reasonable to suppose that this is also the level at which we need to understand RSD corrections in order to make a useful contribution to the field from small-scale measurements. We have investigated whether selection based on local density can reduce the modeling burden, and find that low-mass, isolated galaxy pairs are preferred. However, even for these galaxies, the corrections depend on sample properties, and would need to be recalculated for each cosmological model to be tested: the only currently available way to do this is via numerical simulations. We conclude that observations of close-pairs of galaxies do show promise for AP-style cosmological measurements, particularly for low mass, isolated galaxies. However, it is likely that modeling limitations will continue to be the limiting factor for the foreseeable future.\n\n\\section{Acknowledgments}\nABB acknowledges financial support from the Spanish Ministry \nof Education (ME) within the FPU grant program with ref.\\ number AP2008-02679. \nShe also thanks the Institute of Cosmology and Gravitation for hospitality \nduring her stay from September 2011 to January 2012. WJP thanks the European Research Council and the UK Science and Technology Facilities Research Council for financial support. The authors would like to thank David Bacon, Marco Bruni, Phil Bull, Diego Capozzi, Chris Clarkson, Timothy Clifton, Robert Crittenden, Chris D'Andrea, Sean February, Pedro G.\\ Ferreira, Juan Garc\\'ia-Bellido, Alan Heavens, Elise Jennings, Marc Manera, Francesco Pace, John A.\\ Peacock, Mat Pieri and Lado Samushia for useful discussions. We would also like to thank the referee for useful suggestions.\n\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Credits}\n\n\n\\section{Introduction}\n\nKnowledge graph(KG) has received a lot of traction in the recent past and led to much research in this area. Most of the research is focused on the generation of Knowledge graphs and consumption of the information enshrined in the Knowledge Graph. Some of the earlier works to create KG are YAGO\\citep{Suchanek2007}, Freebase\\citep{Bollacker2008}, DBpedia\\citep{Lehmann2012} and WikiData\\citep{Vrandecic2014}. The evolution of the Knowledge Graph starts with the seminal paper from Berners-Lee \\citep{Berners-Lee2001}. The knowledge graph has evolved in three phases. In the first phase, the Knowledge representation was brought to the level of Web standard. The core focus shifted to Data management, linked data, and its application in the second phase. In the third phase, the focus shifted on the real-world application \\citep{Bonatti2018KnowledgeGN}. The real-world application ranges from Semantic parsing\\citep{Berant2013, Heck2014}, recommender system\\citep{Sun2020, Wang}, question answering\\citep{ Saxena}, named entity disambiguation\\citep{Lin}, information extraction \\citep{Xiong, Xiong, Liu2018a, Dietz2019} etc. The knowledge graph is a representation of structured relational information in the form of Entities and relations between them. It is a multi-relational graph where nodes are entities and edges are relations. Entities are real-world objects or abstract information. The representation entities and relations between them are represented as triple. For e.g. \\textit{(New Delhi, IsCapitalOf, India)} is an example of a triple. \\textit{New Delhi} and \\textit{Delhi} are entities and \\textit{IsCapitalOf} is relation. Though the representation looks scientific but consuming these in the real-world application is not an easy task. The consumption of information enshrined in the knowledge graph will be very easy if it can be converted to numerical representation. \\textit{Knowledge graph embedding} is a solution to incorporate the knowledge from the knowledge graph in a real-world application. \\\\\nThe motivation behind Knowledge graph embedding \\citep{Bordes} is to preserve the structural information, i.e., the relation between entities, and represent it in some vector space. This makes it easier to manipulate the information. Most of the work in \\textit{Knowledge graph embedding \\textbf{(KGE)}} is focused on generating a continuous vector representation for entities and relations and apply a relationship reasoning on the embedding. The relationship reasoning is supposed to optimize some scoring function to learn the embedding. Researchers have used different approaches to learn the embedding Path-based learning\\citep{Toutanova2016}, Entity-based learning, textual-based learning, etc. A lot of work was focused were translation model \\citep{Bordes} and semantic-based model\\citep{Bordes2014}. The representation of the triple results in a lot of information lost because it fails to take \"textual information\" into account. With the proposal of Graph attention network \\citep{Velickovic2017a} the representation of the entities has become more contextualized. In recent years, the proposal of multi-modal graph has extended the spectrum to a new level. In multi-modal knowledge graph, Knowledge graph can have multi-modal information like image and text etc. \\citep{Sun2020, Wei2019}. \\\\\nPrevious work on survey has focused on the KG Embedding \\citep{Wang}, KG Embedding and application\\citep{Ji2020b} , KG embedding with textual data \\citep{Lu2020a}, KG embedding based on the deep-learning \\citep{Wang2020a}. This work shall focus on the KG embedding from translation-based model, semantic-based model, embedding with enriched representation from textual data and multi-modal data and their application. In section 2, we shall provide the details of KGE; in section 3, we shall present the application area. In the summary section, we shall try to put emerging areas of research in KGE. \n\n\\section{Knowledge Graph embedding}\n\\label{sec:length}\n\nKnowledge Graph embedding is an approach to transform Knowledge Graphs (nodes, edges, and their feature vectors) into a low dimensional continuous vector space that preserves various graph structure, information, etc. These approaches are broadly classified into two groups: $\\textit{translation models}$ and $\\textit{semantic matching models}$.\n\n\\subsection{Translation Models}\nThe translation-based model uses distance-based measures to generate the similarity score for a pair of entities and their relationships. The translation-based model aims to find a vector representation of entities with relation to the translation of the entities. It maps entities to a low-dimensional vector space. \n\\subsubsection{TransE \\citep{Bordes2014}}\nThe first model proposed was TransE. It is an energy based model. If a triple \\textit{h, r, t} holds then the vector representation \\textit{h} and \\textit{r} should be as close as possible. It can be graphically represented as Figure-\\ref{fig:my_label}. Mathematically, it can be stated like $\\textit{h} + \\textit{r} \\approx \\textit{t}$. The energy of a triple is  \\textit{d(h + r,t)} for some similarity measure d. To learn the embedding, minimization of ranking based loss function over the training set.\n\\begin{equation}\n    \\mathcal{L= \\sum_{\\textit{h,r,t} \\in S}  \\sum_{\\textit{\\^{h},r,\\^{t}} \\in \\^{S} } [\\gamma +\\textit{d(h + r,t)} - \\textit{d(\\^{h},r,\\^{t})}] }\n\\end{equation}\n\\textit{d(\\^{h},r,\\^{t})} represents the set of the corrupt triples. This loss function will be optimized so that the valid triples are ranked above the corrupt triples. \nThis model fails in case of the one to many relation and many to many relation. To overcome this deficit new model TransH is proposed.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\linewidth]{Trans E.png}\n    \\caption{The intuition of TransE model. Image credit \\cite{Wang}}\n    \\label{fig:my_label}\n\\end{figure}\n\n\\subsubsection{TransH \\cite{Wang2014KnowledgeGE}}\nIt was proposed to address the limitations of TransE. This model enables an entity to\nhave distributed representations based on their involvement in the relation. The representation of \\textit{$h_\\bot$}, \\textit{$t_\\bot$} are projected in a relation specific hyperplane. The relation between then is \\textit{$d_r$}. As in the Figure-\\ref{fig:my_label_2}, the vectors \\textit{h} and \\textit{t} are projected in the relation hyper-plane. The loss function and intuition remain similar to TransE. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\linewidth]{TransH.png}\n    \\caption{TransH model. Image taken from \\cite{Wang}}\n    \\label{fig:my_label_2}\n\\end{figure}\n\n\\subsubsection{TransR \\cite{10.5555\/2886521.2886624}}\nIt proposes that an entity may have multiple attributes and various relations. Each relation may focus on different attributes of entities. TransR models entities  and relation in different embedding space. It means that two different spaces: entity space and relation space are modelled. Each entity is mapped into relation space. The translation construct is applied on the projected representation in the relationship space. The Figure-\\ref{fig:my_label_3} presents the intuition behind TransR model.\\\\ \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\linewidth]{TransR.png}\n    \\caption{TransR model. Image taken from \\cite{10.5555\/2886521.2886624}}\n    \\label{fig:my_label_3}\n\\end{figure}\nTo further refine the representation, a new model TransD was proposed by \\cite{ji-etal-2015-knowledge}. TransR captures the possibility of relations and their embedding from the relation space. But the TransD, extends it to the entity space as well. Here Entity-relation pair is considered as the first-class object. \n\\subsubsection{RotatE \\cite{DBLP:journals\/corr\/abs-1902-10197}}\nIn knowledge graphs, often, the relation is symmetric\/anti-symmetric, inversion and composition. e.g., \"Marriage\" is a symmetric relation, \"My niece is my sister's daughter\" is a composition, etc. The models discussed above are not capable of predicting these relations. The model proposed here is based on the intuition that the relation from head to tail is modeled as rotation in a complex plane. It is motivated by Euler's identity. \n    \\begin{equation}\n        e^{\\iota\\theta} = \\sin \\theta + \\iota \\cos \\theta\n    \\end{equation}\n\n\\noindent\nFor a triplet \\textit{(h,r,t)}, the relation among them can be represented as \\textit{t = h $\\circ$ r}. Where \\textit{h, r, t} is the k-dimension embedding of the head, relation and tail, restricting the $|{r_i}|$ = 1. It means we are in the unit circle and $\\circ$ represents element wise product. For each dimension $t_i = r_i h_i$ subjected to constraint $|{r_i}|$ = 1.\nUnder these condition the relationship is symmetric $\\iff$  for all the values of i, $e^{0\/\\iota\\theta} = \\pm 1$, The relationship is inverse $\\iff$ $r_j = \\bar{r}_i$, i.e. both are the complex conjugate. Two relations are composite $\\iff$ $r_j = r_i \\circ r_k$. It means the relation $r_j$ can be obtained by a combined rotation of $r_i$ and $r_k$, $\\theta_j = \\theta_i + \\theta_k$. The scoring function measures the angular distance.\n\\begin{equation}\n    d_r(h,t) = \\left\\| h \\circ r - t \\right\\|\n\\end{equation}\nThe Figure-\\ref{fig:my_label_4} shows the comparison of RotaE and TransE. \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\linewidth]{rotaE.png}\n    \\caption{Representation of RotaE in comparison to TransE. In RotatE, the relationship between \\textit{h} and \\textit{t} is represented as angle of rotation. This image depicts the relationship in 1-dimension space. \\citep{DBLP:journals\/corr\/abs-1902-10197} }\n    \\label{fig:my_label_4}\n\\end{figure}\n\\subsubsection{HakE \\cite{DBLP:journals\/corr\/abs-1911-09419}}\nThe approaches we have discussed so far fails to capture the semantic hierarchies. In HakE, the authors have proposed to model the hierarchy in the entities as concentric circles in polar coordinate. The entity with smaller radius belongs to higher up in the hierarchy. The angle between them represents the variation in the meaning. \nTo represent a point on the circle, we should have $(r, \\theta)$. Similarly, this model has two components, one to map the modulus and the other one to map the angle. The modulus part considers the depth of the tree as moduli. \nLet $h_m, t_m$ and $r_m$ are the representation in modulus space then $h_m \\circ r_m = t_m$, where $h_m, t_m$ is a k-dim vector. \nThe distance function is similar to RotatE with modification to consider only modulo part. \n\\begin{equation}\n    d_{r,m}(h_m,t_m) = \\left\\| h_m \\circ r_m - t_m \\right\\|_2\n\\end{equation}\n    \nSimilarly, the phase part can be formulated as \n$(h_p + r_p)mod 2\\pi = t_p$ where $h_p, r_p, t_p \\in [0,2\\pi)^k$. The distance function is \n\\begin{equation}\n    d_{r,p}(h_p,t_p) = \\left\\|\\sin ( (h_p \\circ r_p - t_p)\/2) \\right\\|_1\n\\end{equation}\nBy combining both the part, an entity can be mapped in the polar coordinate space. \n\n\n\\subsection{Semantic Matching Models}\nSemantic Matching is one of the core tasks in Natural Language Processing. As we have seen that Translational Distance Models use distance-based scoring functions to calculate the similarity between the different entities and relations thus build the embedding accordingly. On the other hand, Semantic Matching Models use similarity-based scoring function. There are several Knowledge Graph Embedding algorithms comes under this model. Some of the algorithms are described below. \n\n\\subsubsection{RESCAL \\cite{nickel2011three}}\nRESCAL follows Statistical Relational Learning Approach which is based on a Tensor Factorization model that takes the inherent structure of relational data into account. A Tensor\\cite{kolda2009tensor} is a multidimensional array. More formally we can say that a first order Tensor is a vector, second order Tensor is a matrix and, Tensor with more than two order is called as higher order Tensor. Tensor Factorization is expressing a Tensor as a sequence of elementary operations acting on other, often a simpler Tensors.  Statistical Relational Learning inherits from Probability Theory and Statistics to address uncertainty and complexity of relational structures. \\\\\n\\cite{nickel2011three} models the Knowledge Graph triplet of the form (head, relation, tail) into three-way tensor, $\\mathcal{X}$ as shown in  Figure 5.\n\n\\begin{figure}[htb]\n    \\includegraphics[width=.9\\linewidth]{Fig_1.png}\n    \\caption{Tensor model for relational data. $E_{1} ... E_{n}$ denote the entities and $R_{1} ... R_{m}$ denote the relation in the domain\\cite{nickel2011three} }\n    \\label{fig: Menu}\n\\end{figure}\n\n\\noindent\nIn  $\\mathcal{X}$, two modes holds the concatenated entities (head and tail), and the third mode holds relation (relation). A Tensor entity $\\mathcal{X}_{ijk}$ = 1  denotes that there exist a relation and if $\\mathcal{X}_{ijk}$ = 0 denotes that there is unknown relation. It is assumed that data is given as n * n * m Tensor. Where n is the number of entities and m is the number of relations. RESCAL \"explains triples via pairwise interaction of latent features\". It performs the rank-r factorization on each slice of $\\mathcal{X}$ (relational data) and the score of a fact (head, relation, tail ) is given by the following bi-linear function.\n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} = \\sum_{i=0}^{d-1}\\sum_{j=0}^{d-1} [\\textbf{M}_r]_{ij} .[\\textbf{h}]_i . [\\textbf{t}]_j\\]\nwhere h,t $\\in$ $\\mathbb{R}^d$ are vector representation of entities, and $M_{r}$ $\\in$ $\\mathbb{R}^{d*d}$ is a matrix representation of $r^{th}$ relation. Thus from this equation we are able to calculating the score of the triple using the weighted sum of all the pairwise interactions between the latent features of the entities $h$ and $t$ as shown in the Figure 6.\n\n\\begin{figure}[htb]\n    \\includegraphics[width=.9\\linewidth]{Fig_2.png}\n    \\caption{ RESCAL Representation. Here number of latent features for entities are 3 and number of latent features for relations are 3. \\cite{Nickel_2016}  }\n    \\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nThis method require $O(d^{2})$ parameters per relation and the space complexity of $O(nd + md^{2})$ where n is the number of entities and m is the number of relations.\n\n\\subsubsection{TATEC \\cite{10.1007\/978-3-662-44848-9_28}}\nTATEC stands for \\textit{Two And Three-way Embeddings Combination}. The main disadvantage of RESCAL is that it is a Three-way model which performs fairly good for relationships which occur frequently but it performs poor for the rare relationships and leads to major over-fitting. The issue of major over-fitting for rare relationships can be controlled by regularizing or reducing the expressivity of the model and former method is not feasible. The second method of reducing the expressivity is Two-way interaction which is implemented in TransE and SME. Two-way interaction approaches overperform the Three-way approaches on many datasets from which we can conclude that Two-way interactions are more efficient for the datasets and specially for those datasets which have more rare relationships. But the problem with the two-way interaction is that they are limited and are not able to represent all kind of relations with entities.\\\\\nTATEC is a latent factor model which is capable of incorporating the high capacity Three-way model with well-controlled two-way interactions and take  the advantage of both of them. Since two-way and three-way models do not use the the same kind of data pattern and do not encode the same kind of information in the embedding. So, in TATEC during first stage they used two different embeddings and then combined and fine-tuned them in the later stage. The scoring function of TATEC is given by $f_{r}(h,t) = f_{r}^1(h,t) + f_{r}^2(h,t) $ which is a linear combination of bi-gram and tri-gram terms, where $f_{r}^1(h,t)$ is a two-way interaction score and $f_{r}^2(h,t)$ is a three-way interaction score. These can be calculates as follows:\\\\\n1) Two-way interactions terms  can be given by:\n\\[  f_{r}^1(h,t) = \\textbf{h}^T\\textbf{r} + \\textbf{t}^T\\textbf{r} + \\textbf{h}^T\\textbf{D}\\textbf{t} \\]\n\\noindent\nwhere D is the diagonal matrix shared across all the different relations and does not depend on input triple and r $\\in$ $\\mathbb{R}^d$ is a vector  that depends on relationships.\\\\\n2) Three-way interactions terms  can be given by:\n\\[f_{r}^2(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} \\]\n\n\\noindent\nThe final scoring function of TATEC is given by \n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} + \\textbf{h}^T\\textbf{r} + \\textbf{t}^T\\textbf{r} + \\textbf{h}^T\\textbf{D}\\textbf{t}\\]\n\n\\begin{figure}[htb]\n    \\includegraphics[width=.9\\linewidth]{Fig_3.png}\n    \\caption{ Link prediction results \\cite{10.1007\/978-3-662-44848-9_28} }\n    \\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nAuthors of \\cite{10.1007\/978-3-662-44848-9_28} compared this model to the other existing models such as RESCAL \\cite{nickel2011three}, TransE \\citep{Bordes}, LFM, SE and, SME for link prediction on FB15k dataset and as a result TATEC performs better than all other available models as shown in  Figure 7.\n\n\\noindent\nTime complexity and the space complexity of TATEC is same as RESCAL as TATEC extends RESCAL. The time complexity of TATEC is $O(d^{2})$ parameters per relation and the space complexity of $O(nd + md^{2})$ where n is the number of entities and m is the number of relations.\n\n\\subsubsection{DistMult \\cite{yang2014embedding}}\nThis model compares with the NTN neural model, TransE and  bi-linear models like RESCAL. The problem with NTN is that it is the most expensive model it incorporate both linear and bilinear relation operations. Similarly TransE parameterizes  the linear operations with one dimensional vectors. DistMult is a simplified RESCAL, which uses the basic bilinear scoring function.\n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} \\]\n\n\\noindent\nThese bilinear formulations are combined with different forms of regularization to make different models. In DistMult authors considered a simpler approach where they reduced the number of parameters by imposing restrictions on $M_{r}$ to be a diagonal matrix. This results in a simpler model and this model enjoys the same scalable properties of TransE as well as it achieves better performance over TransE. Thus the final scoring function is given as \n\\[ f_{r}(h,t) = \\textbf{h}^T diag(\\textbf{r}) \\textbf{t} = \\sum_{i=0}^{d-1} [\\textbf{r}]_{i} .[\\textbf{h}]_i . [\\textbf{t}]_i \\]\n\n\\noindent\nwhere for each relation r, r $\\in \\mathbb{R}^d$ is a vector that depends on relationships. \n\n\\begin{figure}[htb]\n    \\includegraphics[width=.9\\linewidth]{DistMult.png}\n    \\caption{ DistMult simple Illustration }\n    \\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nTime complexity and the space complexity of DistMult is more efficient as compared to RESCAL or TATEC. The time complexity of DistMult is $O(d)$ parameters per relation and the space complexity of $O(nd + md)$ where n is the number of entities and m is the number of relations. Due to its over-simplified mature of the model, this model is not enough powerful for the use in case of general Knowledge Graphs because it is only able to work efficiently  with symmetric relations.\n\n\\subsubsection{HolE \\cite{nickel2016holographic}}\nHolE stands for Holographic Embedding. HolE tried to overcome the problem of Tensor Product used in RESCAL by using circular correlation. Tensor product uses pairwise multiplicative interactions between feature vectors which results in increase in dimensionality of the representation i.e., $\\mathbb{R}^{d^2}$ thus increase the computational demand.\n\\[ \\textbf{h} \\circ \\textbf{t} = \\textbf{h} \\otimes \\textbf{t} \\in \\mathbb{R}^{d^2} \\]\nWhere \\textbf{a},\\textbf{b} $\\in \\mathbb{R}^d$ are entity embeddings. Tensor products are very rich in capturing the interactions but are computational intensive. On the other hand HolE  use Circular Correlation which can be seen as compression of the Tensor Product. The main advantage of Circular Correlation over Tensor Product is that it won't increase the dimensionality of the representation.\n\\[ \\textbf{h} \\circ \\textbf{t} = \\textbf{h} * \\textbf{t} \\in \\mathbb{R}^d \\]\nWhere * : $ \\mathbb{R}^d \\times \\mathbb{R}^d \\rightarrow \\mathbb{R}^d$ denotes the circular correlation. \n\\[ [\\textbf{h} * \\textbf{t}]_{i} = \\sum_{k=0}^{d-1} [\\textbf{h}_{k}] . \\textbf{t}_{(k+i)mod(d)}\\]\nThe final score of the fact in  HolE is given by matching the compositional vector ($\\textbf{h} * \\textbf{t}$) with the relational representation,i.e.,\n\n\\[ f_{r}(h,t) = \\textbf{r}^T  (\\textbf{h} * \\textbf{t} ) = \\sum_{i=0}^{d-1} [\\textbf{r}]_{i} \\sum_{k=0}^{d-1}[\\textbf{h}]_k . [\\textbf{t}]_{(k+i)mod(d)} \\]\n\n\\begin{figure}[htb]\n    \\includegraphics[width=.9\\linewidth]{HolE.png}\n    \\caption{ HolE simple Illustration (It requires only d components) \\cite{nickel2016holographic} }\n    \\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nSo HolE is more efficient as compared to RESCAL or TransE. HolE take $O(d log d)$ parameters per relation and the space complexity of $O(nd + md)$ where n is the number of entities and m is the number of relations. Another advantage of HolE is that Circular Correlation is not commutative (  $\\textbf{h} * \\textbf{t} \\neq \\textbf{t} * \\textbf{h}$ ) thus HolE is able to model asymmetric relations (directed graphs) with compositional representations which is not possible in RESCAL.\n\n\\subsubsection{ComplEx\\cite{Trouillon2016}}\nKnowledge graphs represent the relation between entities. The entities may be termed as subjects and objects respectively of a given relation. However not all relations may be present in a given KG. One of the application of KG is the ability to predict missing relations or entities. \\\\\nDot product of vector embedding of KG triplets is being successfully used for symmetric, reflexive, anti-reflexive and even transitive relations \\citep{bouchard2015} however it can't be used for anti-symmetric relations. For example the relation capitalOf(New Delhi, India) is not symmetric since we cannot interchange subject and object entity in this relation therefore we need to have different embedding for an entity as subject and as object which increases the number of parameters.\\\\\nComplex embedding facilitates joint learning of subject and object entities while preserving the asymmetry of the relation. It uses Hermitian dot product of embedding of subject entities and object entities. The Eigen Vector decomposition is used to identify a low rank diagonal matrix W such that there exists X = Re($EWE^{T}$) such that X has same sign pattern as Y. The low rank diagonal matrix W is then used to predict missing relations by applying Re($e_{s}^{T}W\\bar{e}_{o}$).\n\\subsubsection{ANALOGY\\cite{Liu2017}}\nANALOGY is based on a multiplicative model where the relation (s,r,o) is scored by multiplying the vector representations of of subject (s), relation (r) and object (o). The relation present in the knowledge graph are expected to have higher score i.e. $\\phi(s,r,0)$ = $e_{s}^{T}W_{r}e_{o}$ will be high if the triplet (s,r,o) exists. \\\\\nAn example of analogy may be branch:tree::petal:leaf, in such an analogy, the relation $W_{r}$ \"is part of\" may be used to predict the missing entity from the analogy. The foundation of ANALOGY model is the linear maps of matrix representation of relations from the triplets present in the KG. Basically, the model uses the fact that there can be multiple paths to arrive at the entity from a given entity through a linear map of relations $W_{r1}, W_{r2}, W_{r3}$ and so on, and application of such relations in any sequence will give the same result. Hence such linear maps may be used to predict entities missing from the knowledge graph.\\\\\nFor example Let's consider the entities teacher (t), school (s), professor (p) and college (c). We have two relations in this set up: teachesAt (t,s) \\& teachesAt(p,c) and juniorOf(t,p) \\& juniorOf(s,c) therefore the relation between teacher and college is teachesAt*juniorOf = juniorOf*teachesAt. However such linear maps are feasible only if the commutative property holds for such relations. ANALOGY uses such linear maps to predict entities.\n\n\\subsection{Enrichment based embedding}\nIn the recent times, new emerging research areas are focusing on contextualized embedding. Under this, the entity under the consideration is enriched information from the neighbourhood information. A few notable approaches are Graph attention network (GAT) \\citep{Velickovic2017a} based information enrichment. Two methods based on the KGAT \\citep{Wei2019}, MMGAT \\citep{Sun2020} has proposed models to embed contextual information for an entity. Under MMGAT, they proposed model to combine the embedding from multi-modal data with an attention framework, as adopted from GAT's attention framework. Both the frameworks, were using translation model to learn the representation after the enrichment. The new emerging research areas are try to learn the structural information as well as path based information, multi-modal data. There are other research areas in the embedding are: \\textit{Text-enhanced embedding}, \\textit{Logic-enhanced embedding}, \\textit{Image-enhanced embedding} \\citep{Bianchi2020KnowledgeGE} etc. \n\n\\section{Applications of Knowledge graph embedding}\n\nThere are many applications of KG embedding learning methods. This section explores three of them namely- link prediction, triplet classification and recommender systems. \n\n\\noindent\nThe first two are In-KG applications, which are conducted within the scope of the KG. The last is an example of Out-of-KG applications that scale to broader domains \\citep{Wang2017}.\n\n\\subsection{Link Prediction}\n\nThe set of edges in a knowledge graph is a subset of \\textit{Entities$\\times$Relations$\\times$Entities}. The link prediction task focuses on finding an entity that can be represented as a fact (edge) together with a given relation and entity i.e., \\textit{(entity, relation, ?)} or \\textit{(?, relation, entity)} where ? refers to the missing entity. For e.g. \\textit{(New Delhi, isCapitalOf, ?)} or \\textit{(?, isCapitalOf, India)}. Link prediction is a way of Knowledge graph augmentation \\citep{paulheim2017knowledge}. It deduces missing information from the knowledge graph itself. \n\n\\noindent\nThe datasets for LP are constructed by sampling from the original knowledge graph. Then, the links removed can be used in validation set or the test set \\citep{bordes2013translating, dettmers2018convolutional}. The structure of such graphs play a vital role \nfor improving the results, multiple source entities making learning effective and multiple destination entities making learning difficult \\citep{Rossi_2021}.\n\n\\noindent\nThe LP models assigns a score to the triplet corresponding to each possible entity to fill the question mark (?). The triplets are then ranked by a function and entity corresponding to the lowest rank is predicted. If the predicted facts in the ranked predictions are already present in the Knowledge graph, they may or may not be excluded while calculating the ranks called raw and filtered rankings respectively \\citep{bordes2013translating}. For e.g. if the training knowledge graph contains the fact that \\textit{(Arjuna, isSonOf, Kunti)}, and the test query is \\textit{(?, isSonOf, Kunti)}. The target answer is \\textit{(Yudhishtra, isSonOf, Kunti)} and the system ranks  \\textit{(Arjuna, isSonOf, Kunti)} and then \\textit{(Yudhishtra, isSonOf, Kunti)}. The raw ranking of the triplet \\textit{(Yudhishtra, isSonOf, Kunti)} will be two and filtered ranking will be one. \n\n\\noindent\nThere are several tie breaking policies that are used by the ranking system. Assigning the \\textit{minimum} or the \\textit{\nmaximum} rank, or a \\textit{random} or the \\textit{average} rank to the targeted entity \\citep{Rossi_2021}.\n\n\\noindent\nThe ranks obtained are used to compute metrics such as \\textit{Mean Rank} (average of all the ranks), \\textit{Mean Reciprocal Rank} (average of the inverse of ranks), or \\textit{Hits@M} (proportion of ranks \\textit{$\\leq$ M}).\n\n\n\\subsection{Triple Classification}\nTriple Classification is the problem of identifying whether a given triple is correct. It aims to give a yes or no answer to questions such as \\textit{is New Delhi capital of India?} which can be written in the form of a triple \\textit{(New Delhi, isCapitalOf, India)} \\citep{socher2013reasoning}.\n\n\\noindent\nA scoring function is used to calculate score of a triple similar to the link prediction. If the score is greater than a certain threshold, then it is considered a fact else a wrong triple \\citep{Wang2017}. \n\n\\noindent\nBoth the classical methods, such as \\textit{micro} and \\textit{macro averaging}, and ranking methods such as \\textit{Mean Rank} are used as evaluation metrics \\citep{guo2016jointly}. \n\n\\subsection{Recommender Systems}\nRecommender system (RS) assists the user in an environment where multiple options are available by providing a certain ordering of choices that the recommendation algorithm infers. This inference can be based on the similarity of the choices and behaviour pattern of different users. This type of recommendation methods falls into the domain of collaborative filtering methods \\citep{adomavicius2005toward}. \n\n\\noindent\nThe CF methods suffer from problems of Data sparsity and cold start. Data sparsity arises from the fact that only a small proportion of items are rated by the users and most options have only limited feedback from the users. Cold start problem is the problem of having no historical data about the new users and options. To deal with these problems different types of side information about a user and item are utilized by the RS \\citep{Sun_2019}.\n\n\\noindent\nKG is utilised for side information in CF. It acts as a heterogeneous graph that represent entities as nodes and relation as edges. The KG connects various entities via latent relationships and also provide explainability in recommendations \\citep{Wang_2018}.\n\n\\noindent\nThe KG embedding based methods for RS use two modules - Graph embedding and Recommendation Module. The way that these modules are coupled lead to categorization of embedding based methods in a). two stage learning methods, b). joint learning method and c). multi task learning method\\citep{Guo_2020}. \n\n\\noindent\nTwo stage learning methods first uses graph embedding module to obtain the embeddings using various KG algorithms and then use recommendation module to infer. The advantages of this method lies in its simplicity and scalability but since the two modules are loosely coupled the embeddings might not be suitable for recommendation tasks.\n\n\\noindent\nJoint learning methods train both the modules in an end to end fashion. Thus, recommendation module guides the training in graph embedding layer. \n\n\\noindent\nMulti task learning method train the recommendation module with the guidance of KG related task such as KG completion. The primary intuition behind this method is that the bipartite graph of user and item in recommendation task share structures with the corresponding KG entities.\n\n\\section{Summary}\nKnowledge graphs provide an effective way of presenting real-world relationships. As a result Knowledge graphs have an inherent advantage w.r.t serving the information need. KG in itself is a growing area of research. KG embedding is a technique to represent all the components of the KG in vector form. These vectors represent the latent properties of the components of the graph. Various models for embedding methods are based on different combinations of vector algebra which present an interesting area of research. In this work, we have surveyed the embedding methods that started this active area of research, state-of-art models and the new frontiers which are being explored in the KG embedding. \\\\\nThe KG embedding methods progressed from translation-based models which are based on vector addition. In this work, we have presented how translation-based models improved over time to overcome shortcomings of the earlier models. While translation-based models used vector addition, semantic models can be clubbed together as multiplicative models. We have included the transition from basic semantic models to the more advanced semantic models which may be used to explain different types of real world relationships such as symmetric, anti-symmetric,inverse or composition.\\\\\nNew research areas have broaden the scope from structural embedding to more contextual embedding by encoding additional information in the learned representation. The latest area of research in this field is enrichment based embedding models. In this work, we have introduced those briefly. \\\\\nVector space representation has paved a way to use the information from Knowledge graph directly into the real world application. In this work, we have described a few real world applications of KG embedding such as link prediction, triple classification and recommender systems.\n\n\n\n\\chapter*{}\n\\setcounter{page}{1}%\n\\vspace{-4.65cm}\n\\noindent\n\\begin{tabular}[H]{lr}\\toprule [1pt] \\vspace{-0.33cm}\\\\\n\\hspace{-2mm}\\Journal{Journal Name} & \\hspace{49mm}\\multirow{5}*{\\includegraphics[height=22mm,width=58mm]{SciencePGLOGO.pdf}}\\\\\n\\hspace{-2mm}\\Address{2021; X(X): XX-XX} &\\\\\n\\hspace{-2mm}\\Address{http:\/\/www.sciencepublishinggroup.com\/x\/xxx} &\\\\\n\\hspace{-2mm}\\Address{doi: 10.11648\/j.XXXX.2021XXXX.XX} &\\\\\n\\hspace{-2mm}\\Address{ISSN: xxxx-xxxx (Print); ISSN: xxxx-xxxx (Online)} \\\\\\bottomrule [2.5pt]\n\\end{tabular}\n\\parskip=12pt\n\\begin{flushleft}\n\\noindent \\Head{A Survey of Knowledge Graph Embedding and Their Applications}\n\\end{flushleft}\n\\parskip=8pt\n\\noindent \\Name{Shivani Choudhary^{1}$, Tarun Luthra$^{2}$, Ashima Mittal$^{2}$, Rajat Singh$^{2} $}\n\\parskip=8pt\n\n\n\\noindent \\Address{$^1$School of Interdisciplinary Research, Indian Institute of Technology Delhi, Delhi, India\\\\$^2$Department of Computer Science and Engineering, Indian Institute of Technology Delhi, Delhi, India}\n\\vspace{5pt}\n\n\\parskip=4pt\n\\noindent \\Emailaddress{Email address:}\\\\\n\\noindent \\Emailaddresscontent{shivani@sire.iitd.ac.in (Shivani Choudhary), mcs202475@cse.iitd.ac.in (Tarun Luthra), anz208486@cse.iitd.ac.in (Ashima Mittal), csz208507@cse.iitd.ac.in (Rajat Singh)}\n\n\\parskip=8pt\n\\noindent \\Citation{To cite this article:}\n\\begin{flushleft}\n\\vspace{-0.56cm}\n\\noindent \\Citationcontent{Authors Name. Paper Title. {\\it International Journal of XXXXXX}. Vol. x, No. x, 2021, pp. x-x. doi: 10.11648\/j.xxx.xxxxxxxx.xx.}\n\\vspace{-0.3cm}\n\\end{flushleft}\n\\parskip=10pt\n\n\\noindent \\Date{{\\bf Received:} MM DD, 2021; {\\bf Accepted:} MM DD, 2021; {\\bf Published:} MM DD, 2021}\n\n\\parskip=5pt\n\\noindent \\rule{\\textwidth}{1pt}\n\n\\parskip=6pt\n\\noindent \\Abstract{Abstract: }\\Abstractcontent{\\baselineskip=12pt Knowledge Graph represents a network of real-world entities. The information embedded in the Knowledge graph, though structured, is challenging to consume in a real-world application. Consuming this information for real-world task necessitate a numerical representation of entities. Knowledge graph embedding focuses on approaches to generate a dense embedding. These techniques find their application in various tasks such as completion of knowledge graph to predict missing information, recommender systems, question answering, query expansion, etc. Knowledge graph embedding enables the real-world application to consume information to improve performance. Knowledge graph embedding is an active research area. Most of the embedding methods focus on structure-based information. Recent research has extended the boundary to include multimodal text-based information and image-based information to generate improved entity embedding. Efforts have been made to enhance the representation with context information. This paper introduces the growth in the field of KG embedding from simple translation-based models to enrichment-based models. Further, This paper covers the areas where Knowledge Graph finds its application. }\n\n\\parskip=8pt\n\\hangafter=1\n\\setlength {\\hangindent}{6.2em}\n\\noindent \\Abstract{Keywords: }\\Abstractcontent{knowledge Graph, Recommender Systems, Question Answering, Query Expansion}\n\\parskip=0pt\n\n\\noindent \\rule{\\textwidth}{1pt}\n\\parskip=0pt\n\\columnseprule=0pt\n\\setlength{\\columnsep}{1.5em}\n\\vspace{-12pt}\n\\begin{multicols}{2}\n\\section{Introduction}\n\\vspace{-6pt}\n\nKnowledge graph(KG) has received a lot of traction in the recent past and led to much research in this area. Most of the research is focused on the generation of Knowledge graphs and consumption of the information enshrined in the Knowledge Graph. Some of the earlier works to create KG are YAGO\\citep{Suchanek2007}, Freebase\\citep{Bollacker2008}, DBpedia\\citep{Lehmann2012} and WikiData\\citep{Vrandecic2014}. The evolution of the Knowledge Graph starts with the seminal paper from Berners-Lee \\citep{Berners-Lee2001}. The knowledge graph has evolved in three phases. In the first phase, the Knowledge representation was brought to the level of Web standard. The core focus shifted to Data management, linked data, and its application in the second phase.\n In the third phase, the focus shifted on the real-world application \\citep{Bonatti2018KnowledgeGN}. The real-world application ranges from Semantic parsing\\citep{Berant2013, Heck2014}, recommender system\\citep{Sun2020, Wang}, question answering\\citep{ Saxena}, named entity disambiguation\\citep{Lin}, information extraction \\citep{Xiong, Xiong, Liu2018a, Dietz2019} etc. The knowledge graph is a representation of structured relational information in the form of Entities and relations between them. It is a multi-relational graph where nodes are entities and edges are relations. Entities are real-world objects or abstract information. The representation entities and relations between them are represented as triple. For e.g. \\textit{(New Delhi, IsCapitalOf, India)} is an example of a triple. \\textit{New Delhi} and \\textit{Delhi} are entities and \\textit{IsCapitalOf} is relation. Though the representation looks scientific but consuming these in the real-world application is not an easy task. The consumption of information enshrined in the knowledge graph will be very easy if it can be converted to numerical representation. \\textit{Knowledge graph embedding} is a solution to incorporate the knowledge from the knowledge graph in a real-world application. \\\\\nThe motivation behind Knowledge graph embedding \\citep{Bordes} is to preserve the structural information, i.e., the relation between entities, and represent it in some vector space. This makes it easier to manipulate the information. Most of the work in \\textit{Knowledge graph embedding \\textbf{(KGE)}} is focused on generating a continuous vector representation for entities and relations and applying a relationship reasoning on the embedding. The relationship reasoning is supposed to optimize some scoring functions to learn the embedding. Researchers have used different approaches to learn the embedding of Path-based learning\\citep{Toutanova2016}, Entity-based learning, textual-based learning, etc. MuchA lot of work was focused onwere translation model \\citep{Bordes} and semantic-based model\\citep{Bordes2014}. The representation of the triple results in a lot of information lost because it fails to take \"textual information\" into account. With the proposal of Graph attention network \\citep{Velickovic2017a} the representation of the entities haves become more contextualized. In recent years, the proposal of a multi-modal graph has extended the spectrum to a new level. In the multi-modal knowledge graph, Knowledge graph can have multi-modal information like image and text, etc. \\citep{Sun2020, Wei2019}. \\\\\nPrevious work on survey has focused on the KG Embedding \\citep{Wang}, KG Embedding and application\\citep{Ji2020b} , KG embedding with textual data \\citep{Lu2020a}, KG embedding based on the deep-learning \\citep{Wang2020a}. This work shall focus on the KG embedding from translation-based model, semantic-based model, embedding with enriched representation from textual data, and multi-modal data and their application. In section 2, we shall provide the details of KGE; in section 3, we shall present the application area. In the summary section, we shall try to put emerging areas of research areas in KGE. \n\n\\section{Knowledge Graph embedding}\n\\label{sec:length}\n\nKnowledge Graph embedding is an approach to transform Knowledge Graphs (nodes, edges, and their feature vectors) into a low dimensional continuous vector space that preserves various graph structures, information, etc. These approaches are broadly classified into two groups: $\\textit{translation models}$ and $\\textit{semantic matching models}$.\n\n\\subsection{Translation Models}\nThe translation-based model uses distance-based measures to generate the similarity score for a pair of entities and their relationships. The translation-based model aims to find a vector representation of entities with relation to the translation of the entities. It maps entities to a low-dimensional vector space. \n\\subsubsection{TransE \\citep{Bordes2014}}\nThe first model proposed was TransE. It is an energy based model. If a triple \\textit{h, r, t} holds then the vector representation \\textit{h} and \\textit{r} should be as close as possible. It can be graphically represented as Figure-\\ref{fig:my_label}. Mathematically, it can be stated like $\\textit{h} + \\textit{r} \\approx \\textit{t}$. The energy of a triple is  \\textit{d(h + r,t)} for some similarity measure d. To learn the embedding, minimization of ranking based loss function over the training set.\n\\begin{equation}\n    \\mathcal{L= \\sum_{\\textit{h,r,t} \\in S}  \\sum_{\\textit{$\\hat{h}$,r,$\\hat{t}$} \\in \\hat{S} } [\\gamma+\\textit{d(h+r),t}-\\textit{d($\\hat{h}$,r,$\\hat{t}$)}]}\n\\end{equation}\n\\textit{d(\\^{h},r,\\^{t})} represents the set of the corrupt triples. This loss function will be optimized so that the valid triples are ranked above the corrupt triples. \nThis model fails in the case of the one- to- many relation and many- to- many relation. To overcome this deficit new model, TransH is proposed.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.9\\linewidth]{TransE.png}\n\\caption{The intuition of TransE model. Image credit \\cite{Wang}}\n\\label{fig:my_label}\n\\end{figure}\n\n\\subsubsection{TransH \\cite{Wang2014KnowledgeGE}}\nIt was proposed to address the limitations of TransE. This model enables an entity to\nhave distributed representations based on their involvement in the relation. The representation of \\textit{$h_\\bot$}, \\textit{$t_\\bot$} are projected in a relation specific hyperplane. The relation between then is \\textit{$d_r$}. As in the Figure-\\ref{fig:my_label_2}, the vectors \\textit{h} and \\textit{t} are projected in the relation hyper-plane. The loss function and intuition remain similar to TransE. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.9\\linewidth]{TransH.png}\n\\caption{TransH model. Image taken from \\cite{Wang}}\n\\label{fig:my_label_2}\n\\end{figure}\n\n\\subsubsection{TransR \\cite{10.5555\/2886521.2886624}}\nIt proposes that an entity may have multiple attributes and various relations. Each relation may focus on different attributes of entities. TransR models entities  and relation in different embedding space. It means that two different spaces: entity space and relation space are modelled. Each entity is mapped into relation space. The translation construct is applied on the projected representation in the relationship space. The Figure-\\ref{fig:my_label_3} presents the intuition behind TransR model.\\\\ \n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.9\\linewidth]{TransR.png}\n\\caption{TransR model. Image taken from \\cite{10.5555\/2886521.2886624}}\n\\label{fig:my_label_3}\n\\end{figure}\nTo further refine the representation, a new model TransD was proposed by \\cite{ji-etal-2015-knowledge}. TransR captures the possibility of relations and their embedding from the relation space. But the TransD, extends it to the entity space as well. Here Entity-relation pair is considered as the first-class object. \n\\subsubsection{RotatE \\cite{DBLP:journals\/corr\/abs-1902-10197}}\nIn knowledge graphs, often, the relation is symmetric\/anti-symmetric, inversion and composition. e.g., \"Marriage\" is a symmetric relation, \"My niece is my sister's daughter\" is a composition, etc. The models discussed above are not capable of predicting these relations. The model proposed here is based on the intuition that the relation from head to tail is modeled as rotation in a complex plane. It is motivated by Euler's identity. \n    \\begin{equation}\n        e^{\\iota\\theta} = \\sin \\theta + \\iota \\cos \\theta\n    \\end{equation}\n\n\\noindent\nFor a triplet \\textit{(h,r,t)}, the relation among them can be represented as \\textit{t = h $\\circ$ r}. Where \\textit{h, r, t} is the k-dimension embedding of the head, relation and tail, restricting the $|{r_i}|$ = 1. It means we are in the unit circle and $\\circ$ represents element wise product. For each dimension $t_i = r_i h_i$ subjected to constraint $|{r_i}|$ = 1.\nUnder these condition the relationship is symmetric $\\iff$  for all the values of i, $e^{0\/\\iota\\theta} = \\pm 1$, The relationship is inverse $\\iff$ $r_j = \\bar{r}_i$, i.e. both are the complex conjugate. Two relations are composite $\\iff$ $r_j = r_i \\circ r_k$. It means the relation $r_j$ can be obtained by a combined rotation of $r_i$ and $r_k$, $\\theta_j = \\theta_i + \\theta_k$. The scoring function measures the angular distance.\n\\begin{equation}\n    d_r(h,t) = \\left\\| h \\circ r - t \\right\\|\n\\end{equation}\nThe Figure-\\ref{fig:my_label_4} shows the comparison of RotaE and TransE. \n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=.9\\linewidth]{rotaE.png}\n    \\caption{Representation of RotaE in comparison to TransE. In RotatE, the relationship between \\textit{h} and \\textit{t} is represented as angle of rotation. This image depicts the relationship in 1-dimension space. \\citep{DBLP:journals\/corr\/abs-1902-10197} }\n    \\label{fig:my_label_4}\n\\end{figure}\n\\subsubsection{HakE \\cite{DBLP:journals\/corr\/abs-1911-09419}}\nThe approaches we have discussed so far fails to capture the semantic hierarchies. In HakE, the authors have proposed to model the hierarchy in the entities as concentric circles in polar coordinate. The entity with smaller radius belongs to higher up in the hierarchy. The angle between them represents the variation in the meaning. \nTo represent a point on the circle, we should have $(r, \\theta)$. Similarly, this model has two components, one to map the modulus and the other one to map the angle. The modulus part considers the depth of the tree as moduli. \nLet $h_m, t_m$ and $r_m$ are the representation in modulus space then $h_m \\circ r_m = t_m$, where $h_m, t_m$ is a k-dim vector. \nThe distance function is similar to RotatE with modification to consider only modulo part. \n\\begin{equation}\n    d_{r,m}(h_m,t_m) = \\left\\| h_m \\circ r_m - t_m \\right\\|_2\n\\end{equation}\n    \nSimilarly, the phase part can be formulated as \n$(h_p + r_p)mod 2\\pi = t_p$ where $h_p, r_p, t_p \\in [0,2\\pi)^k$. The distance function is \n\\begin{equation}\n    d_{r,p}(h_p,t_p) = \\left\\|\\sin ( (h_p \\circ r_p - t_p)\/2) \\right\\|_1\n\\end{equation}\nBy combining both the part, an entity can be mapped in the polar coordinate space. \n\n\n\\subsection{Semantic Matching Models}\nSemantic Matching is one of the core tasks in Natural Language Processing. As we have seen that Translational Distance Models use distance-based scoring functions to calculate the similarity between the different entities and relations thus build the embedding accordingly. On the other hand, Semantic Matching Models use similarity-based scoring function. There are several Knowledge Graph Embedding algorithms comes under this model. Some of the algorithms are described below. \n\n\\subsubsection{RESCAL \\cite{nickel2011three}}\nRESCAL follows Statistical Relational Learning Approach which is based on a Tensor Factorization model that takes the inherent structure of relational data into account. A Tensor\\cite{kolda2009tensor} is a multidimensional array. More formally we can say that a first order Tensor is a vector, second order Tensor is a matrix and, Tensor with more than two order is called as higher order Tensor. Tensor Factorization is expressing a Tensor as a sequence of elementary operations acting on other, often a simpler Tensors.  Statistical Relational Learning inherits from Probability Theory and Statistics to address uncertainty and complexity of relational structures. \\\\\n\\cite{nickel2011three} models the Knowledge Graph triplet of the form (head, relation, tail) into three-way tensor, $\\mathcal{X}$ as shown in  Figure 5.\n\n\\begin{figure}[H]\n    \\includegraphics[width=.9\\linewidth]{Fig_1.png}\n    \\caption{Tensor model for relational data. $E_{1} ... E_{n}$ denote the entities and $R_{1} ... R_{m}$ denote the relation in the domain\\cite{nickel2011three} }\n    \\label{fig: Menu}\n\\end{figure}\n\n\\noindent\nIn  $\\mathcal{X}$, two modes holds the concatenated entities (head and tail), and the third mode holds relation (relation). A Tensor entity $\\mathcal{X}_{ijk}$ = 1  denotes that there exist a relation and if $\\mathcal{X}_{ijk}$ = 0 denotes that there is unknown relation. It is assumed that data is given as n * n * m Tensor. Where n is the number of entities and m is the number of relations. RESCAL \"explains triples via pairwise interaction of latent features\". It performs the rank-r factorization on each slice of $\\mathcal{X}$ (relational data) and the score of a fact (head, relation, tail ) is given by the following bi-linear function.\n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} = \\sum_{i=0}^{d-1}\\sum_{j=0}^{d-1} [\\textbf{M}_r]_{ij} .[\\textbf{h}]_i . [\\textbf{t}]_j\\]\nwhere h,t $\\in$ $\\mathbb{R}^d$ are vector representation of entities, and $M_{r}$ $\\in$ $\\mathbb{R}^{d*d}$ is a matrix representation of $r^{th}$ relation. Thus from this equation we are able to calculating the score of the triple using the weighted sum of all the pairwise interactions between the latent features of the entities $h$ and $t$ as shown in the Figure 6.\n\n\\begin{figure}[H]\n    \\includegraphics[width=.9\\linewidth]{Fig_2.png}\n    \\caption{ RESCAL Representation. Here number of latent features for entities are 3 and number of latent features for relations are 3. \\cite{Nickel_2016}  }\n    \\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nThis method require $O(d^{2})$ parameters per relation and the space complexity of $O(nd + md^{2})$ where n is the number of entities and m is the number of relations.\n\n\\subsubsection{TATEC \\cite{10.1007\/978-3-662-44848-9_28}}\nTATEC stands for \\textit{Two And Three-way Embeddings Combination}. The main disadvantage of RESCAL is that it is a Three-way model which performs fairly good for relationships which occur frequently but it performs poor for the rare relationships and leads to major over-fitting. The issue of major over-fitting for rare relationships can be controlled by regularizing or reducing the expressivity of the model and former method is not feasible. The second method of reducing the expressivity is Two-way interaction which is implemented in TransE and SME. Two-way interaction approaches overperform the Three-way approaches on many datasets from which we can conclude that Two-way interactions are more efficient for the datasets and specially for those datasets which have more rare relationships. But the problem with the two-way interaction is that they are limited and are not able to represent all kind of relations with entities.\\\\\nTATEC is a latent factor model which is capable of incorporating the high capacity Three-way model with well-controlled two-way interactions and take  the advantage of both of them. Since two-way and three-way models do not use the the same kind of data pattern and do not encode the same kind of information in the embedding. So, in TATEC during first stage they used two different embeddings and then combined and fine-tuned them in the later stage. The scoring function of TATEC is given by $f_{r}(h,t) = f_{r}^1(h,t) + f_{r}^2(h,t) $ which is a linear combination of bi-gram and tri-gram terms, where $f_{r}^1(h,t)$ is a two-way interaction score and $f_{r}^2(h,t)$ is a three-way interaction score. These can be calculates as follows:\\\\\n1) Two-way interactions terms  can be given by:\n\\[  f_{r}^1(h,t) = \\textbf{h}^T\\textbf{r} + \\textbf{t}^T\\textbf{r} + \\textbf{h}^T\\textbf{D}\\textbf{t} \\]\n\\noindent\nwhere D is the diagonal matrix shared across all the different relations and does not depend on input triple and r $\\in$ $\\mathbb{R}^d$ is a vector  that depends on relationships.\\\\\n2) Three-way interactions terms  can be given by:\n\\[f_{r}^2(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} \\]\n\n\\noindent\nThe final scoring function of TATEC is given by \n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} + \\textbf{h}^T\\textbf{r} + \\textbf{t}^T\\textbf{r} + \\textbf{h}^T\\textbf{D}\\textbf{t}\\]\n\n\\begin{figure}[H]\n\\includegraphics[width=.9\\linewidth]{Fig_3.png}\n\\caption{ Link prediction results \\cite{10.1007\/978-3-662-44848-9_28} }\n\\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nAuthors of \\cite{10.1007\/978-3-662-44848-9_28} compared this model to the other existing models such as RESCAL \\cite{nickel2011three}, TransE \\citep{Bordes}, LFM, SE and, SME for link prediction on FB15k dataset and as a result TATEC performs better than all other available models as shown in  Figure 7.\n\n\\noindent\nTime complexity and the space complexity of TATEC is same as RESCAL as TATEC extends RESCAL. The time complexity of TATEC is $O(d^{2})$ parameters per relation and the space complexity of $O(nd + md^{2})$ where n is the number of entities and m is the number of relations.\n\n\\subsubsection{DistMult \\cite{yang2014embedding}}\nThis model compares with the NTN neural model, TransE and  bi-linear models like RESCAL. The problem with NTN is that it is the most expensive model it incorporate both linear and bilinear relation operations. Similarly TransE parameterizes  the linear operations with one dimensional vectors. DistMult is a simplified RESCAL, which uses the basic bilinear scoring function.\n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} \\]\n\n\\noindent\nThese bilinear formulations are combined with different forms of regularization to make different models. In DistMult authors considered a simpler approach where they reduced the number of parameters by imposing restrictions on $M_{r}$ to be a diagonal matrix. This results in a simpler model and this model enjoys the same scalable properties of TransE as well as it achieves better performance over TransE. Thus the final scoring function is given as \n\\[ f_{r}(h,t) = \\textbf{h}^T diag(\\textbf{r}) \\textbf{t} = \\sum_{i=0}^{d-1} [\\textbf{r}]_{i} .[\\textbf{h}]_i . [\\textbf{t}]_i \\]\n\n\\noindent\nwhere for each relation r, r $\\in \\mathbb{R}^d$ is a vector that depends on relationships. \n\n\\begin{figure}[H]\n\\includegraphics[width=.9\\linewidth]{DistMult.png}\n\\caption{ DistMult simple Illustration }\n\\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nTime complexity and the space complexity of DistMult is more efficient as compared to RESCAL or TATEC. The time complexity of DistMult is $O(d)$ parameters per relation and the space complexity of $O(nd + md)$ where n is the number of entities and m is the number of relations. Due to its over-simplified mature of the model, this model is not enough powerful for the use in case of general Knowledge Graphs because it is only able to work efficiently  with symmetric relations.\n\n\\subsubsection{HolE \\cite{nickel2016holographic}}\nHolE stands for Holographic Embedding. HolE tried to overcome the problem of Tensor Product used in RESCAL by using circular correlation. Tensor product uses pairwise multiplicative interactions between feature vectors which results in increase in dimensionality of the representation i.e., $\\mathbb{R}^{d^2}$ thus increase the computational demand.\n\\[ \\textbf{h} \\circ \\textbf{t} = \\textbf{h} \\otimes \\textbf{t} \\in \\mathbb{R}^{d^2} \\]\nWhere \\textbf{a},\\textbf{b} $\\in \\mathbb{R}^d$ are entity embeddings. Tensor products are very rich in capturing the interactions but are computational intensive. On the other hand HolE  use Circular Correlation which can be seen as compression of the Tensor Product. The main advantage of Circular Correlation over Tensor Product is that it won't increase the dimensionality of the representation.\n\\[ \\textbf{h} \\circ \\textbf{t} = \\textbf{h} * \\textbf{t} \\in \\mathbb{R}^d \\]\nWhere * : $ \\mathbb{R}^d \\times \\mathbb{R}^d \\rightarrow \\mathbb{R}^d$ denotes the circular correlation. \n\\[ [\\textbf{h} * \\textbf{t}]_{i} = \\sum_{k=0}^{d-1} [\\textbf{h}_{k}] . \\textbf{t}_{(k+i)mod(d)}\\]\nThe final score of the fact in  HolE is given by matching the compositional vector ($\\textbf{h} * \\textbf{t}$) with the relational representation,i.e.,\n\n\\[ f_{r}(h,t) = \\textbf{r}^T  (\\textbf{h} * \\textbf{t} ) = \\sum_{i=0}^{d-1} [\\textbf{r}]_{i} \\sum_{k=0}^{d-1}[\\textbf{h}]_k . [\\textbf{t}]_{(k+i)mod(d)} \\]\n\n\\begin{figure}[H]\n\\includegraphics[width=.9\\linewidth]{HolE.png}\n\\caption{ HolE simple Illustration (It requires only d components) \\cite{nickel2016holographic} }\n\\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nSo HolE is more efficient as compared to RESCAL or TransE. HolE take $O(d log d)$ parameters per relation and the space complexity of $O(nd + md)$ where n is the number of entities and m is the number of relations. Another advantage of HolE is that Circular Correlation is not commutative (  $\\textbf{h} * \\textbf{t} \\neq \\textbf{t} * \\textbf{h}$ ) thus HolE is able to model asymmetric relations (directed graphs) with compositional representations which is not possible in RESCAL.\n\n\\subsubsection{ComplEx\\cite{Trouillon2016}}\nKnowledge graphs represent the relation between entities. The entities may be termed as subjects and objects respectively of a given relation. However not all relations may be present in a given KG. One of the application of KG is the ability to predict missing relations or entities. \\\\\nDot product of vector embedding of KG triplets is being successfully used for symmetric, reflexive, anti-reflexive and even transitive relations \\citep{bouchard2015} however it can't be used for anti-symmetric relations. For example the relation capitalOf(New Delhi, India) is not symmetric since we cannot interchange subject and object entity in this relation therefore we need to have different embedding for an entity as subject and as object which increases the number of parameters.\\\\\nComplex embedding facilitates joint learning of subject and object entities while preserving the asymmetry of the relation. It uses Hermitian dot product of embedding of subject entities and object entities. The Eigen Vector decomposition is used to identify a low rank diagonal matrix W such that there exists X = Re($EWE^{T}$) such that X has same sign pattern as Y. The low rank diagonal matrix W is then used to predict missing relations by applying Re($e_{s}^{T}W\\bar{e}_{o}$).\n\\subsubsection{ANALOGY\\cite{Liu2017}}\nANALOGY is based on a multiplicative model where the relation (s,r,o) is scored by multiplying the vector representations of of subject (s), relation (r) and object (o). The relation present in the knowledge graph are expected to have higher score i.e. $\\phi(s,r,0)$ = $e_{s}^{T}W_{r}e_{o}$ will be high if the triplet (s,r,o) exists. \\\\\nAn example of analogy may be branch:tree::petal:leaf, in such an analogy, the relation $W_{r}$ \"is part of\" may be used to predict the missing entity from the analogy. The foundation of ANALOGY model is the linear maps of matrix representation of relations from the triplets present in the KG. Basically, the model uses the fact that there can be multiple paths to arrive at the entity from a given entity through a linear map of relations $W_{r1}, W_{r2}, W_{r3}$ and so on, and application of such relations in any sequence will give the same result. Hence such linear maps may be used to predict entities missing from the knowledge graph.\\\\\nFor example Let's consider the entities teacher (t), school (s), professor (p) and college (c). We have two relations in this set up: teachesAt (t,s) \\& teachesAt(p,c) and juniorOf(t,p) \\& juniorOf(s,c) therefore the relation between teacher and college is teachesAt*juniorOf = juniorOf*teachesAt. However such linear maps are feasible only if the commutative property holds for such relations. ANALOGY uses such linear maps to predict entities.\n\n\\subsection{Enrichment based embedding}\nIn the recent times, new emerging research areas are focusing on contextualized embedding. Under this, the entity under the consideration is enriched information from the neighbourhood information. A few notable approaches are Graph attention network (GAT) \\citep{Velickovic2017a} based information enrichment. Two methods based on the KGAT \\citep{Wei2019}, MMGAT \\citep{Sun2020} has proposed models to embed contextual information for an entity. Under MMGAT, they proposed model to combine the embedding from multi-modal data with an attention framework, as adopted from GAT's attention framework. Both the frameworks, were using translation model to learn the representation after the enrichment. The new emerging research areas are try to learn the structural information as well as path based information, multi-modal data. There are other research areas in the embedding are: \\textit{Text-enhanced embedding}, \\textit{Logic-enhanced embedding}, \\textit{Image-enhanced embedding} \\citep{Bianchi2020KnowledgeGE} etc. \n\n\\section{Applications of Knowledge graph embedding}\n\nThere are many applications of KG embedding learning methods. This section explores three of them namely- link prediction, triplet classification and recommender systems. \n\n\\noindent\nThe first two are In-KG applications, which are conducted within the scope of the KG. The last is an example of Out-of-KG applications that scale to broader domains \\citep{Wang2017}.\n\n\\subsection{Link Prediction}\n\nThe set of edges in a knowledge graph is a subset of \\textit{Entities$\\times$Relations$\\times$Entities}. The link prediction task focuses on finding an entity that can be represented as a fact (edge) together with a given relation and entity i.e., \\textit{(entity, relation, ?)} or \\textit{(?, relation, entity)} where ? refers to the missing entity. For e.g. \\textit{(New Delhi, isCapitalOf, ?)} or \\textit{(?, isCapitalOf, India)}. Link prediction is a way of Knowledge graph augmentation \\citep{paulheim2017knowledge}. It deduces missing information from the knowledge graph itself. \n\n\\noindent\nThe datasets for LP are constructed by sampling from the original knowledge graph. Then, the links removed can be used in validation set or the test set \\citep{bordes2013translating, dettmers2018convolutional}. The structure of such graphs play a vital role \nfor improving the results, multiple source entities making learning effective and multiple destination entities making learning difficult \\citep{Rossi_2021}.\n\n\\noindent\nThe LP models assigns a score to the triplet corresponding to each possible entity to fill the question mark (?). The triplets are then ranked by a function and entity corresponding to the lowest rank is predicted. If the predicted facts in the ranked predictions are already present in the Knowledge graph, they may or may not be excluded while calculating the ranks called raw and filtered rankings respectively \\citep{bordes2013translating}. For e.g. if the training knowledge graph contains the fact that \\textit{(Arjuna, isSonOf, Kunti)}, and the test query is \\textit{(?, isSonOf, Kunti)}. The target answer is \\textit{(Yudhishtra, isSonOf, Kunti)} and the system ranks  \\textit{(Arjuna, isSonOf, Kunti)} and then \\textit{(Yudhishtra, isSonOf, Kunti)}. The raw ranking of the triplet \\textit{(Yudhishtra, isSonOf, Kunti)} will be two and filtered ranking will be one. \n\n\\noindent\nThere are several tie breaking policies that are used by the ranking system. Assigning the \\textit{minimum} or the \\textit{\nmaximum} rank, or a \\textit{random} or the \\textit{average} rank to the targeted entity \\citep{Rossi_2021}.\n\n\\noindent\nThe ranks obtained are used to compute metrics such as \\textit{Mean Rank} (average of all the ranks), \\textit{Mean Reciprocal Rank} (average of the inverse of ranks), or \\textit{Hits@M} (proportion of ranks \\textit{$\\leq$ M}).\n\n\n\\subsection{Triple Classification}\nTriple Classification is the problem of identifying whether a given triple is correct. It aims to give a yes or no answer to questions such as \\textit{is New Delhi capital of India?} which can be written in the form of a triple \\textit{(New Delhi, isCapitalOf, India)} \\citep{socher2013reasoning}.\n\n\\noindent\nA scoring function is used to calculate score of a triple similar to the link prediction. If the score is greater than a certain threshold, then it is considered a fact else a wrong triple \\citep{Wang2017}. \n\n\\noindent\nBoth the classical methods, such as \\textit{micro} and \\textit{macro averaging}, and ranking methods such as \\textit{Mean Rank} are used as evaluation metrics \\citep{guo2016jointly}. \n\n\\subsection{Recommender Systems}\nRecommender system (RS) assists the user in an environment where multiple options are available by providing a certain ordering of choices that the recommendation algorithm infers. This inference can be based on the similarity of the choices and behaviour pattern of different users. This type of recommendation methods falls into the domain of collaborative filtering methods \\citep{adomavicius2005toward}. \n\n\\noindent\nThe CF methods suffer from problems of Data sparsity and cold start. Data sparsity arises from the fact that only a small proportion of items are rated by the users and most options have only limited feedback from the users. Cold start problem is the problem of having no historical data about the new users and options. To deal with these problems different types of side information about a user and item are utilized by the RS \\citep{Sun_2019}.\n\n\\noindent\nKG is utilised for side information in CF. It acts as a heterogeneous graph that represent entities as nodes and relation as edges. The KG connects various entities via latent relationships and also provide explainability in recommendations \\citep{Wang_2018}.\n\n\\noindent\nThe KG embedding based methods for RS use two modules - Graph embedding and Recommendation Module. The way that these modules are coupled lead to categorization of embedding based methods in a). two stage learning methods, b). joint learning method and c). multi task learning method\\citep{Guo_2020}. \n\n\\noindent\nTwo stage learning methods first uses graph embedding module to obtain the embeddings using various KG algorithms and then use recommendation module to infer. The advantages of this method lies in its simplicity and scalability but since the two modules are loosely coupled the embeddings might not be suitable for recommendation tasks.\n\n\\noindent\nJoint learning methods train both the modules in an end to end fashion. Thus, recommendation module guides the training in graph embedding layer. \n\n\\noindent\nMulti task learning method train the recommendation module with the guidance of KG related task such as KG completion. The primary intuition behind this method is that the bipartite graph of user and item in recommendation task share structures with the corresponding KG entities.\n\n\\section{Summary}\nKnowledge graphs provide an effective way of presenting real-world relationships. As a result Knowledge graphs have an inherent advantage w.r.t serving the information need. KG in itself is a growing area of research. KG embedding is a technique to represent all the components of the KG in vector form. These vectors represent the latent properties of the components of the graph. Various models for embedding methods are based on different combinations of vector algebra which present an interesting area of research. In this work, we have surveyed the embedding methods that started this active area of research, state-of-art models and the new frontiers which are being explored in the KG embedding. \\\\\nThe KG embedding methods progressed from translation-based models which are based on vector addition. In this work, we have presented how translation-based models improved over time to overcome shortcomings of the earlier models. While translation-based models used vector addition, semantic models can be clubbed together as multiplicative models. We have included the transition from basic semantic models to the more advanced semantic models which may be used to explain different types of real world relationships such as symmetric, anti-symmetric,inverse or composition.\\\\\nNew research areas have broaden the scope from structural embedding to more contextual embedding by encoding additional information in the learned representation. The latest area of research in this field is enrichment based embedding models. In this work, we have introduced those briefly. \\\\\nVector space representation has paved a way to use the information from Knowledge graph directly into the real world application. In this work, we have described a few real world applications of KG embedding such as link prediction, triple classification and recommender systems.\n\n\n\n\\section*{Acknowledgements}\nThe preferred spelling of the word `acknowledgment\" in America is without an ``e\" after the ``g\". Avoid the stilted expression, ``One of us (R. B. G.) thanks . . .\" Instead, try ``R. B. G. thanks\". Put sponsor acknowledgments in the unnum-bered footnote on the first page.\n\n\n\n\\section{Introduction}\nConversational systems have acquired the center stage in NLP research. Compared to the conventional information retrieval task where we have to extract the passage or document from a vast collection of documents, the Conversational system requires extracting related information to respond to a series of questions. The turns in the conversation may follow the previous question. Complexity in this task arises due to the way we form the queries, which often have a reference to previous information using pronouns, co-reference. The presence of pronouns and unresolved co-references induces ambiguity in the query. Resolving the contextual dependency is one of the most challenging tasks in the Conversational system. \n\nThe Conversational assistance track (CAsT) has started in 2019. The long-term vision of the CAsT is ``to support natural conversations between a person and a search engine to satisfy information needs and support complex information tasks''. The task in both years of CAsT remained the same, to retrieve relevant passages depending upon the context evolution from the subsequent queries. \nraw\\_utterance\nCAsT-2019 had two tacks viz: automatic and manual track. Under Automatic track, response extraction was based on the raw utterances while the manual track has queries rewritten by humans. Co-reference and pronouns were resolved in the manually rewritten queries based on the historical context. Manually rewritten queries contain all of the information required to represent the single turn of the underlying information need. In CAsT-2019, additional information like description and title for the session was also provided \\citep{Dalton2020}. Three corpora, namely MSMARCO, TREC CAR (Wikipedia) paragraph Corpus V2.0\\footnote{http:\/\/trec-car.cs.unh.edu\/datareleases\/} were used. Initially, it had also considered WaPo, but it was later dropped. CAsT-2020 had some changes, title and description was \\textbf{removed from the query} info, and document id as a canonical response to the query is added in the task. CAsT-2020 had three tracks - automatic, automatic-canonical, and manual. Under automatic, only raw utterance is to be used, while automatic-canonical can use the provided canonical response. The manual track was similar to last year based on the manually rewritten queries \\citep{Dalton2021}. The dataset was MSMARCO and CAR \\citep{Dietz2018TRECCA}. CAsT-2021 has three tracks similar to CAsT-2020, but with a modification that we need to extract a specific passage out of the extracted document. The idea of specific passage to be extracted from the document centered around the idea that responses should be crisp and could be used by automatic voice assistants like Alexa, Google, etc. CAsT-2021 used MSMARCO (2019\/20 dump), WAPO-2020 and KILT \\citep{petroni-etal-2021-kilt}.  \n\n\\begin{table}[]\n    \\centering\n    \\begin{tabular}{|l|p{0.8\\linewidth}|}\n       \n       \n       \n       \n         \\hline\n         Turn & Queries  \\\\\n         \\hline\n         1 & I just had a breast biopsy for cancer. What are the most common types? \\\\\n         \\hline\n         2 & Once it breaks out, how likely is it to spread?\\\\\n         \\hline\n        3 &\tHow deadly is it?\\\\\n        \\hline\n        4 &\tWhat? No, I want to know about the deadliness of lobular carcinoma in situ. \\\\\n        \\hline\n        5 & Wow, that's better than I thought.  What are common treatments? \\\\\n        \\hline\n        6 & ... \\\\\n        \\hline\n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n    \\end{tabular}\n    \\caption{A sample query from CAsT-2021 Automatic Evaluation Topics }\n    \\label{tab:my_label}\n\\end{table}\n\n\\section{Problem Description}\nIn CAsT track, A conversation session $S$ has a series of utterances $\\{u_1, u_2, u_3, u_4\\dots\\}$ called as turns. Our task is to predict a set of top-K passages from the collection for each turn. \n\n\\textbf{Tracks:} There are three tracks of submission for CAsT-2021 - Automatic, Automatic-canonical, and Manual. Manual track has rewritten queries as input.\n\n\\textbf{Dataset}: CAsT-2021 uses two dataset corpus for the task - MSMARCO, KILT \\citep{petroni-etal-2021-kilt}  and WAPO. \n\nWe have attempted the Automatic track. Each turn (Queries) in this session is raw in nature. The onus to resolve references and pronouns lies on the model itself. \n\n\\section{Motivation and Method}\nWe have gone through the CAsT-2020 submissions, and most of the models were multistage models. In the initial stage, a set of documents were retrieved from the collection, followed by re-ranking using neural model. Most of the models used query rewriting using generative model based on Transformer architecture \\citep{DBLP:journals\/corr\/VaswaniSPUJGKP17}. For query rewriting, T5, BART, GPT-2 was used, and the re-ranker engine was based on BERT, ALBERT, and T5.  \n\nOur retrieval framework has three components: Query rewrite, document retrieval with pseudo-relevance feedback, and neural engine-based re-ranker. The Query rewrite framework is based on chatty-goose \\footnote{https:\/\/github.com\/castorini\/chatty-goose} and re-ranker is based on pygaggle \\footnote{https:\/\/github.com\/castorini\/pygaggle}. Core retrieval engine is based on the HQE \\citep{Yang2019QueryAA} and PQE  \\citep{Al-Thani2021}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth] {trec.jpg}\n    \\caption{Multi-stage retrieval pipeline}\n    \\label{fig:my_label}\n\\end{figure}\n\n\\subsection{Historical Query expansion (\\textbf{HQE})}\nThis step is based on the submission from CAsT-2019 \\citep{Yang2019QueryAA}. It is a three-stage algorithm. In the first step, it extracts the keyword for the session and query level followed by measurement of ambiguity in query in second step, and in the last stage, query expansion with the session keyword and query level keyword is done. \n\nA keyword is deemed important if it strongly relates with the documents in the index. HQE uses BM25 score between a keyword and its highest-ranked document in the index as its measure of importance. HQE computes this measure for each token in every utterance when presented with a topic. The most informative ones are then selected for query expansion.\nFor a given topic, keywords can be locally important (i.e., it strongly relates to the topic being discussed in the current utterance) or globally important (i.e., it relates to the overall conversation theme). HQE thus creates two sets of expansion keywords, session and query, during the extraction phase.\nWhether the keyword is considered a session keyword or a query keyword is decided by two separate cutoffs, $Q_s$, and $Q_t$. If the importance score of a keyword is greater than either of the cutoff scores, it is added to corresponding expansion sets. Note that session keywords are always used for expansion, while query keywords are only used when current utterance is identified as ambiguous. An utterance is deemed ambiguous when BM25 score between it and its highest ranked document in index is low (i.e. by itself utterance is not important). If the ambiguity score is less than a certain threshold $\\theta$, then the query expansion will take place with query keywords.\n\nThe cutoff values for $Q_s$, $Q_t$, and $\\theta$ determine the performance of HQE. We follow a greedy approach to find the optimal values of these cutoffs. Note that a query is always expanded with current session keywords and occasionally with preceding query keywords. Thus, we start by tuning $Q_s$ session cutoff. We set $Q_t$ and $\\theta$ to arbitrarily high values allowing query expansion only via session keywords. Then we do a line search for $Q_s$ over the training set. Second, we set $Q_t$ to some fixed value ($> Q_s$) and in similar fashion tune $\\theta$. Finally, we tune $Q_s$. \n\n\\begin{table*}[]\n    \\centering\n    \\begin{tabular}{p{0.4\\linewidth}|c|c|c|c|c|c}\n        \\hline\n        Run Name & $K1$ & $b$ & NDCG@3 & NDCG@5 & P@5 & AP@500\\\\\n        \\hline\n        IITD-RAW\\_U\\_T5\\_1 & $0.9$ & $0.4$ & 0.3712 & 0.3631 & 0.5025 & 0.1759\\\\\n        \\hline\n        IITD-RAW\\_U\\_T5\\_2 & $1.2$ & $0.75$ & 0.3801 & 0.3731 & 0.5203  & 0.1874 \\\\\n         \\hline\n    \\end{tabular}\n    \\caption{Evaluation score of Submitted manual runs}\n    \\label{tab:my_label_2}\n\\end{table*}\n\n\n\\subsection{Passage Query Expansion (PQE)}\nPseudo-relevance feedback enriches the query by incorporating features from top-k relevant documents. PQE uses a pseudo-relevance feedback mechanism as follows:\n\\begin{itemize}\n\n\\item The expanded query is from HQE is used to fetch an initial set of top-k documents. These documents combined together form the corpus of responses for the query.\n\\item Individual tokens in the corpus are scored using TF-IDF. An IDF vector is pre-computed on complete MSMARCO documents.\n\\item The topmost unique tokens are then considered for query expansion.\n\\end{itemize}\nNote that PQE can be computationally prohibitive since it involves complete document retrieval. In order to avoid this, \\cite{Al-Thani2021} proposes to use a simple rule to decide whether a query is to be expanded using PQE or not. \\cite{Al-Thani2021} only perform PQE expansion when the query has at least one pronoun.\n\nThis process has two HyperParameters, top-k documents, and top-k tokens from the document based on the TF-IDF score. The TF-IDF score helps the system to select the important terms. While selecting the term, we have also placed a criterion that it should have a DF between 0.001 and 0.2. In the PQE step, we noticed that some digits also appeared in top-k terms. Later, we filter out those instances. \n\n\n\n\n\n\\subsection{T5 query rewriter and reranker}\nIn automatic track, query are presented in the bare format. In order to preserve the context of the query, we have used query rewriter. It uses T5 based model \\footnote{https:\/\/huggingface.co\/castorini\/t5-base-canard}. The model is trained on CANARD dataset \\citep{Elgohary:Peskov:Boyd-Graber-2019} that contains a set of rewritten queries based upon the history. To produce a $n^{th}$ rewitten utterance, we have supplied history of turns $\\{u_1, u_2, u_3, u_4\\dots, u_{n-1}\\}$ concatenated with $u_n$. Reformulated queries were used for the final stage ranking of passages. \n\nOn the other hand T5 based ranking model \\citep{DBLP:journals\/corr\/NguyenRSGTMD16} is trained on the MSMARCO, Robust04, Core17 and Core18. This model takes query $Q$ and a list of passages $\\{p_1, p_2, p_3, .. p_n\\}$. It returns submitted passages according to the descending order of their relevance.  \n\n\\subsection{Document Indexing}\nAll three datasets were pre-processed and converted into jsonl format. We have use Pyserini\\footnote{https:\/\/github.com\/castorini\/pyserini} to generate an index for faster retrieval of documents. Index was generated with an option to keep a copy of raw documents. We choose this option because the raw content was required at HQE and PQE stage. \n\n\n\\section{Evaluation Matrix}\nCAsT-2021 used following matrix to present the results of the participants NDCG@3, NDCG@5, NDCG@500 and AP@500.\n\n\\section{Results and Discussion}\n\nIndex for the cleaned document was generated using Pyserini\\footnote{https:\/\/github.com\/castorini\/pyserini} with the default setting. Default setting did not restrict us to keeping $K1$ and $b$ fixed. It can be changed during the retrieval of the documents. First, we performed our extraction and tuning on the 2019 training set. The best performance for HQE was obtained with $Q_s$ = 4, $Q_t$ = 4, $\\theta$ = 10 and PQE with top-k = 5, top-k token = 3.  \n\nWe have participated in automatic track and submitted two runs in the CAsT-2021. Two different set of parameters for BM25, $K1 = 0.9$, $b = 0.4$ and $K1 = 1.2$, $b = 0.75$ was selected for the retrieval of the document. The MAP is very low in both the results, and the main issue lies with the recall. Our retrieval engine extracted the top 100 documents from the corpus. Extracted documents were later chunked and re-ranked. In this process, lower retrieval numbers left the ranker with fewer relevant chunks, resulting in a lower than expected performance. \n\nIITD-RAW\\_U\\_T5\\_2 has produced a mean NDCG@3 performance better than the median model. Model's performance on evaluation query has a noticeable variation in scores compared with median scores. Results for queries like -- 115 and 119 were better than the median; however, results on the evaluation queries like -- 111 and 117 were worse than the median benchmark for NDCG@3. Analysis of query expansion term for the first stage retrieval suggests that query expansion term has carried the intent of the conversation to higher depths. While the poor performing queries have expansion terms with less relevant keywords, generic keywords. Query expansion terms for query 119 have terms like \\textit{swelling, shaking, infection, ear} that carried the conversation context to higher depth led to better performance. \n\nWe have also analyzed the hqe and pqe keywords for query expansion. We can take evaluation number 106; the first few queries are listed in Table-\\ref{tab:my_label}. For turn 3, query expansion terms were \\textit{biopsy, breast, deadly, cancer, comment and cell}. This turn has only one term, ``deadly'', that can explain the intent of the query. But, it is too generic in nature which has led to poor recall. On the flip side, turn five is expanded with terms \\textit{carcinoma, wow, deadly, situ, lobular, deadliness, biopsy, breast} which has specific terms related to cancer like ``lobular, situ, carcinoma'' has a good recall. Here the word ``specific'' means relevance to the topic. \n\nOur submission to CAsT-2021 aimed to preserve the key terms and the context in all subsequent turns and use classical Information retrieval methods. It was aimed to pull as relevant documents as possible from the corpus. It appears that it can retain some of the keywords in subsequent turns. But, it fails when the key term itself is generic. The performance of this model can be improved further by including the context word vectors in the HQE and PQE stages. Context vector-based selection may help to keep top-k terms that are relevant to the conversation theme. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider a polynomial \nof complex variables $\\zz = (z_1, \\dots, z_n)$\nwhich is given by \n\\[\nP(\\zz, \\bar{\\zz}) := \\sum_{i = 1}^{m} c_{i}\\zz^{\\nu_{i}}\\bar{\\zz}^{\\mu_{i}}, \n\\]\nwhere $c_{i} \\in \\Bbb{C}^*$, \n$\\zz^{\\nu_{i}} = z^{\\nu_{i,1}}_1 \\cdots z^{\\nu_{i,n}}_n$ and \n$\\bar{\\zz}^{\\mu_{i}} = \\bar{z}_{1}^{\\mu_{i,1}} \\cdots \\bar{z}_{n}^{\\mu_{i,n}}$ \nfor $\\nu_{i} = (\\nu_{i,1}, \\dots, \\nu_{i,n})$ and \n$\\mu_{i} = (\\mu_{i,1}, \\dots, \\mu_{i,n})$ respectively.  \nHere $\\bar{\\zz}^{\\mu_{i}}$ \nrepresents the complex conjugate of $\\zz^{\\mu_{i}} = z_{1}^{\\mu_{i,1}} \\cdots z_{n}^{\\mu_{i,n}}$. \nWe call $P(\\zz, \\bar{\\zz})$ \\textit{a mixed polynomial} of complex variables $\\zz = (z_1, \\dots, z_n)$. \nA point $\\ww \\in \\Bbb{C}^{n}$ is called \\textit{a mixed singular point of $P(\\zz, \\bar{\\zz})$} \nif~the gradient vectors of $\\Re P$ and $\\Im P$ are linearly dependent at~$\\ww$ \\cite{O1, O2}. \nSuppose that $P(0,\\dots, 0) = 0$ and $P$ has an isolated singularity at the origin. \nThere exist positive real numbers $\\varepsilon$ and $\\delta$ with $\\delta \\ll \\varepsilon \\ll 1$ such that \nthe map \n\\[\nP : D^{2n}_{\\varepsilon} \\cap P^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta} \n\\]\nis a locally trivial fibration over $\\partial D^{2}_{\\delta}$, where \n$D^{2n}_{\\varepsilon} = \\{ \\zz \\in \\Bbb{C}^{n} \\mid \\| \\zz \\| \\leq \\varepsilon \\}$ and \n$D^{2}_{\\delta} = \\{ \\eta \\in \\Bbb{C} \\mid \\lvert \\eta \\rvert \\leq \\delta \\}$.  \nThis map is called \\textit{the Milnor fibration of $P$ at the origin}. \n\nRuas, Seade and Verjovsky \\cite{RSV} and Cisneros-Molina \\cite{C} introduced \nthe following classes of mixed polynomials. \nLet $p_{1}, \\dots, p_{n}$ and $q_{1}, \\dots, q_{n}$ be integers such that \n$\\gcd(p_1, \\dots, p_n) = \\gcd(q_1,\\dots,q_n) = 1$. \nWe define the~$S^1$-action and the~$\\Bbb{R}^{*}$-action on~$\\Bbb{C}^{n}$ as follows: \n\\begin{equation*}\n\\begin{split}\ns \\circ \\zz = (s^{p_{1}}z_{1}, \\dots, s^{p_{n}}z_{n}), \\ \\ \\ \nr \\circ \\zz = (r^{q_1}z_{1}, \\dots, r^{q_n}z_{n}), \n\\end{split}\n\\end{equation*}\nwhere $s \\in S^{1}$ and $r \\in \\Bbb{R}^{*}$. \nA mixed polynomial $P(\\zz, \\bar{\\zz})$ is called \\textit{a polar weighted homogeneous polynomial} \nif there exists a positive integer $d_p$ such that $P(\\zz, \\bar{\\zz})$ satisfies \n\\[\nP(s^{p_{1}}z_{1}, \\dots, s^{p_{n}}z_{n}, \\bar{s}^{p_1}\\bar{z}_{1}, \\dots, \\bar{s}^{p_1}\\bar{z}_{n})\n = s^{d_p}P(\\zz, \\bar{\\zz}), \\ \\ s \\in S^{1}.  \\notag \\\\\n\\]\nThe weight vector $(p_{1}, \\dots, p_{n})$ is called \\textit{the polar weight} and $d_p$ is called \n\\textit{the polar degree}. \nSimilarly $P(\\zz, \\bar{\\zz})$ is called \\textit{a~radial weighted homogeneous polynomial} \nif there exists a positive integer $d_{r}$ such that \n\\[\nP(r^{q_1}z_{1}, \\dots, r^{q_n}z_{n}, r^{q_1}\\bar{z}_{1}, \\dots, r^{q_n}\\bar{z}_{n}) \n= r^{d_{r}}P(\\zz, \\bar{\\zz}), \\ \\ r \\in \\Bbb{R}^{*}. \n\\]\nThe integer $d_r$ is called \\textit{the radial degree}. \nIf $P$ is  polar and radial weighted homogeneous,  \n$P$ admits the global Milnor fibration $P : \\Bbb{C}^{n} \\setminus P^{-1}(0) \\rightarrow \\Bbb{C}^{*}$ \nand the monodromy of the Milnor fibration is given by the $S^1$-action, see for instance \\cite{RSV, C, O1, O2}. \n\n\nWe study the topology of the Milnor fibration of a mixed polynomial $P$ by using a deformation of $P$. \nHere \\textit{a deformation of $P$} is a~polynomial map \n$F : \\Bbb{C}^{n} \\times \\Bbb{R} \\rightarrow \\Bbb{C}, (\\zz, t) \\mapsto F_{t}(\\zz)$, with $F_{0}(\\zz) = P(\\zz, \\bar{\\zz})$. \nA deformation of $P$ is useful to study the Milnor fibration of $P$. \nFor complex isolated singularities, \nit is known that there exist a~neighborhood $U$ of the origin of $\\Bbb{C}^{n}$ and \na~deformation $F_{t}$ of a complex polynomial $P(\\zz)$ \nsuch that $F_{t}(\\zz)$ is also a~complex polynomial and any singularity of $F_{t}(\\zz)$ in $U$ \nis a Morse singularity for any $0 < t \\ll 1$, see for instance \\cite[Chapter 4]{Lg}. \nA sufficiently small compact neighborhood of each Morse singularity can be regarded as a $2n$-dimensional $n$-handle and \nthus we have a decomposition \n\\[\nD^{2n}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta}) \\cong \n\\bigr(D^{2n}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})\\bigl) \\cup_{\\varphi} \\bigr(\\sqcup_{i=1}^{\\ell} (\\text{$n$-handle})_{i}\\bigl), \n\\]\nwhere $\\ell$ is the Milnor number of the singularity of $P$ at $(0, \\dots, 0)$, \n$\\varphi = (\\varphi_{1}, \\dots \\varphi_{\\ell})$ is the $\\ell$-tuple of the attaching map $\\varphi_{i}$ of \n$(n\\text{-handle})_i$ \nand $D^{2}_{\\delta_{t}}$ is a $2$-disk centered at $0$ with radius $\\delta_{t}$ \nsuch that $\\delta_{t} < \\delta$ and $F_{t} \\mid_{F_{t}^{-1}(D^{2}_{\\delta_{t}})}$ has no singularities. \nNote that the framing of each handle attaching is determined by the vanishing cycle of the corresponding Morse singularity \\cite{K}. \nSuch a decomposition induces a decomposition of the monodromy of the Milnor fibration into those of \nthe Morse singularities. \n\nIn this paper, we observe analogous deformations for mixed singularities. \nLet $P$ be a $2$-variable polar and radial weighted homogeneous \npolynomial which has an isolated singularity at the origin of $\\Bbb{C}^{2}$ \nand let $F_t$ be a deformation of $P$. \nSet $S_{k}(F_{t}) = \\{ \\zz \\in U \\mid \\text{rank}\\>dF_{t}(\\zz) = 2-k \\}$ for $k = 1, 2$. \nWe assume that $F_t$ satisfies the following properties: \n\\begin{enumerate}\n\\item\n$F_t$ is polar weighted homogeneous for any $0 \\leq t \\ll 1$, which implies that, for each $0 < t \\ll 1$, \nthe singular set $S_{1}(F_{t}) \\cup S_{2}(F_{t})$ \nconsists of the union of a finite number of orbits of the $S^1$-action \\cite[Proposition 2]{In3} and \n$F_{t}(S_{1}(F_{t}))$ consists of circles centered at $0$ except the origin; \n\\item\nFor each point $\\ww \\in S_{1}(F_{t})$, \nthere exist local coordinates $(x_{1}, x_{2}, x_{3}, x_{4})$ centered at $\\ww$  \nsuch that $F_{t}$ is given by \n\\[\n(F_{t}\/\\lvert F_{t} \\rvert, \\lvert F_{t} \\rvert) = \\bigr(x_{1}, -x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+c_{t, \\ww}\\bigl), \n\\]\nwhere $c_{t,\\ww} = \\lvert F_{t}(\\ww)\\rvert$ for $\\ww \\in S_{1}(F_{t})$ and $0 < t \\ll 1$. \nIn particular, $S_{1}(F_{t})$ consists of indefinite fold singularities; \n\\item\n$S_{2}(F_{t}) = \\{ \\mathbf{o}\\}$ or $\\emptyset$. \n\\end{enumerate}\nIn \\cite{In3}, we focused on the mixed singularity of $f\\bar{g}$, \nwhere $f$ and $g$ are $2$-variable weighted homogeneous complex polynomials which have no common branches, \nand showed that there exists a deformation $F_t$ of $f\\bar{g}$ \nsuch that $F_t$ satisfies the above conditions. \nIt is important to notice that there exist polar weighted homogeneous polynomials which do not admit \ndeformations into smooth maps with only Morse singularities, see \\cite[Theorem 1]{In1}, \\cite[Corollary 1 and Corollary 2]{In2}. \n\n\nBy the condition $(2)$, the absolute value $\\lvert F_{t}\\rvert$ of $F_t$ defines a Morse function on \nthe orbit space $(D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta}))\/S^{1}$ of the $S^1$-action for $t>0$ and \noutside the image of the origin, and the indices of the Morse singularities are always $1$. \nThen the handle decomposition of the orbit space according to this Morse function induces a decomposition \nof $D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta})$ into a tubular neighborhood of a singular fiber over \nthe origin and a finite number of round $1$-handles, that is, we have \n\\begin{equation}\n\\label{0.1}\n\\tag{0.1}\nD^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta}) \\cong \n\\bigr(D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})\\bigl) \\cup_{\\varphi} \n\\bigr(\\sqcup_{i=1}^{\\ell} (\\text{round $1$-handle})_{i}\\bigl) \n\\end{equation}\nwhere $\\ell$ is the number of the singularities of the Morse function on the orbit space outside the origin \nand $\\varphi = (\\varphi_{1}, \\dots \\varphi_{\\ell})$ is the attaching map of $\\ell$ copies of a \nround $1$-handle. \nHere we may assume that the images of $\\varphi_{1}, \\dots, \\varphi_{\\ell}$ in \n$\\partial(D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}}))$ are disjoint. \n\nIn this paper, we describe the structure of this decomposition more precisely. To state our theorem, \nwe introduce the notion of negative link components. \nLet $P$ be a polar weighted homogeneous polynomial. Then the link of $P$ at the origin is a Seifert link, \ndenoted by $L(P, \\mathbf{o})$. \nA fiber surface of the Seifert link induces an orientation to each link component canonically. \nA connected component of $L(P, \\mathbf{o})$ is called a \\textit{positive component} if the orientation of the link \ncomponent coincides with that of the $S^1$-action, and otherwise it is called a \\textit{negative component}. \nLet $\\lvert L^{+}(P, \\mathbf{o}) \\rvert$ and $\\lvert L^{-}(P, \\mathbf{o}) \\rvert$ \ndenote the number of positive components of $L(P, \\mathbf{o})$ and \nthe number of negative components of $L(P, \\mathbf{o})$, respectively. \nThen the decomposition is given as follows: \n\n\n\n\n\\begin{theorem}\\label{rthm}\nLet $P$ be a $2$-variable polar and radial weighted homogeneous \npolynomial which has an isolated singularity at the origin and \nlet $F_t$ be a deformation of $P$ satisfying the conditions $(1), (2)$ and $(3)$. \nThen \n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\begin{enumerate}\n\\item\n$D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})$ is diffeomorphic to \nthe disjoint union of a $4$-ball and $\\ell$ copies of $S^{1} \\times B^{3}$, \nwhere $B^3$ is a $3$-ball, and each $\\varphi_{i}$ of the attaching map \n$\\varphi = (\\varphi_{1}, \\dots, \\varphi_{\\ell})$ \nmaps the two attaching regions of the $i$-th round $1$-handle to \nboth of the boundary of the $4$-ball and that of the $i$-th $S^1 \\times B^3$, \nand these $\\ell+1$ connected components are connected by $\\ell$ round $1$-handles attached by the map $\\varphi$; and\n\\item\nthe number $\\ell$ of round $1$-handles in the decomposition $\\eqref{0.1}$ is given as \n\\[\n\\ell = \\lvert L^{+}(P, \\mathbf{o})\\rvert - \\lvert L^{+}(F_{t}, \\mathbf{o})\\rvert = \\lvert L^{-}(P, \\mathbf{o})\\rvert - \\lvert L^{-}(F_{t}, \\mathbf{o})\\rvert \n\\]\nfor $0 < t \\ll 1$. \n\\end{enumerate}\n\\end{theorem}\nAs we mentioned, in \\cite{In3}, a deformation of a mixed singularity of type $(f\\bar{g}, \\mathbf{o})$ is given explicitly. \nIn that case, the number $\\ell$ is determined by the polar degree and the radial degree of $P$ as \n$\\ell = \\frac{d_{r} - d_{p}}{2pq}$. \nFrom the decomposition in $\\eqref{0.1}$, we can observe that the Milnor fiber of $(P, \\mathbf{o})$ is obtained from \nthe Milnor fiber of $(F_{t}, \\mathbf{o})$ by removing $2d_{p}$ disks from \ntwo connected components of $D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})$ \nand gluing these boundary circles by $d_p$ annuli for each $i = 1, \\dots \\ell$. \nMoreover, we see that monodromy exchanges these $\\ell$ copies of the union of $d_p$ annuli \nand $2d_{p}$ disks cyclically. \n\nThis paper is organized as follows. In Section $2$ \nwe give the definitions of fold singularities and round handles and \nintroduce deformations of polar weighted homogeneous polynomials. \nIn Section $3$ we prove Theorem $1$. In Section $4$ we make a few comments on the monodromy of the Milnor fibration of $F_t$ \nand a specific deformation of $f\\bar{g}$. \n\nThe author would like to thank Professor Masaharu Ishikawa for precious comments. \n\n\n\\section{Preliminaries}\n\\subsection{Fold singularities}\nLet $X$ be a $n$-dimensional manifold and $W$ be a $2$-dimensional manifold. \nWe denote $C^{\\infty}(X, W)$ the set of smooth maps from $X$ to $W$. \nIt is known that the subset of smooth maps from $X$ to $W$ whose singularities are \nonly definite fold singularities, indefinite fold singularities or cusps \nis open and dense in $C^{\\infty}(X, W)$ topologized with \nthe $C^{\\infty}$-topology \\cite{L1}. \nHere \\textit{a fold singularity} is the singularity of the following map \n\\[\n(x_{1}, \\dots, x_{n}) \\mapsto (x_{1}, \\sum_{j=2}^{n}\\pm x_{j}^{2}), \n\\] \nwhere $(x_{1}, \\dots, x_{n})$ are coordinates of a small neighborhood of the singularity. \nIf the coefficients of $x_j$ for $j = 2, \\dots, n$ is \neither all positive or all negative, we say that \n$x$ is a \\textit{definite fold} singularity, \notherwise it is an \\textit{indefinite fold} singularity. \n\n\n\n\n\\subsection{Deformations of polar weighted homogeneous polynomials}\nLet $P : \\Bbb{C}^{2} \\rightarrow \\Bbb{C}$ be a polar weighted homogeneous polynomial map which has an isolated singularity at the origin of $\\Bbb{C}^2$. \nThen $P$ admits a Milnor fibration, i.e., there exist positive real numbers $\\varepsilon$ and $\\delta$ such that \nthe map \n\\[\nP : D^{4}_{\\varepsilon} \\cap P^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta} \n\\]\nis a locally trivial fibration over $\\partial D^{2}_{\\delta}$, where \n$D^{4}_{\\varepsilon} = \\{ \\zz \\in \\Bbb{C}^{2} \\mid \\| \\zz \\| \\leq \\varepsilon \\}$ and \n$D^{2}_{\\delta} = \\{ \\eta \\in \\Bbb{C} \\mid \\lvert \\eta \\rvert \\leq \\delta \\}$. \nWe fix such  positive real numbers $\\varepsilon$ and $\\delta$. \nLet $F$ be the fiber surface of a polar weighted homogeneous polynomial $P$. \nIn \\cite{O1, O2}, the monodromy $h : F \\rightarrow F$ of the Milnor fibration of $P$ is given by \n\\[\n(z_{1}, z_{2}) \\mapsto \\Bigr(\\exp\\Bigr(\\frac{2p\\pi i}{d_p}\\Bigl)z_{1}, \\exp\\Bigr(\\frac{2q\\pi i}{d_p}\\Bigl)z_{2}\\Bigl), \n\\] \nwhere $(p,q)$ is the polar weight of $P$. \nNote that the link $K_{P} = \\partial D^{4}_{\\varepsilon} \\cap P^{-1}(0)$ is an invariant set of the $S^1$-action. \nSo the link $K_{P}$ is a Seifert link in the $3$-sphere \\cite{EN}. \n\nLet $F_t$ be a deformation of $P$ which satisfies the conditions $(1), (2)$ and $(3)$. \nSince the fiber surface $F_{0}^{-1}(c)$ intersects $\\partial D^{4}_{\\varepsilon}$ transversely, \n$F_{t}^{-1}(c)$ intersects $\\partial D^{4}_{\\varepsilon}$ transversely for each $c \\in D^{2}_{\\delta}$ \nand $0 \\leq t \\ll 1$. \nBy the Ehresmann's fibration theorem \\cite{W}, the map \n\\[\nF_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}\n\\] \nis a locally trivial fibration for $0 \\leq t \\ll 1$. \nThe polar weight of $F_t$ coincides with that of $F_{0}$ for $0 < t \\ll 1$. \nThus the monodromy of $F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nis given by the same $S^1$-action on $\\Bbb{C}^{2}$ for each $0 \\leq t \\ll 1$. \n\n\\begin{lemma}\nThe Milnor fibration \n$F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nis isomorphic to the fibration \n$F_{0}\/\\lvert F_{0} \\rvert : \\partial D^{4}_{\\varepsilon} \\setminus \\emph{Int}N(K_{F_{0}}) \\rightarrow S^{1}$ \nfor $0 \\leq t \\ll 1$, \nwhere $N(K_{F_{0}}) = \\{\\zz \\in \\partial D^{4}_{\\varepsilon} \\mid \\lvert F_{0}(\\zz)\\rvert \\leq \\delta\\}$. \n\\end{lemma}\n\\begin{proof}\nSince the fiber surface $F_{0}^{-1}(c)$ is transversal to $\\partial D^{4}_{\\varepsilon}$, \n$F_{t}^{-1}(c)$ is transversal to $\\partial D^{4}_{\\varepsilon}$ for any $c \\in D^{2}_{\\delta}$ \nand $0 \\leq t \\ll 1$. \nFix such a positive real number $t$. \nWe set \n\\[\\begin{split}\n\\partial \\mathcal{E}(\\delta, \\varepsilon)& := \\{(\\zz, t') \\in \\Bbb{C}^{2} \\times [0, t] \\mid \\lvert F_{t'}(\\zz)\\rvert = \\delta, \\| \\zz \\| \\leq \\varepsilon \\}\\\\\n\\partial ^2\\mathcal{E}(\\delta, \\varepsilon)& := \\{(\\zz, t') \\in \\Bbb{C}^{2}\\times [0, t] \\mid \\lvert F_{t'}(\\zz)\\rvert = \\delta, \\| \\zz \\| = \\varepsilon \\}.\n\\end{split}\n\\]\nThen the projection \n\\[\n\\pi' :( \\partial \\mathcal{E}(\\delta, \\varepsilon),\\partial^2\\mathcal{E}(\\delta, \\varepsilon)) \\rightarrow [0, t], \\ \\ \n(\\zz, t') \\mapsto t'\n\\]\nis a~proper submersion. \nBy the Ehresmann's fibration theorem \\cite{W}, \n$\\pi'$ is a~locally trivial fibration over $[0, t]$. \nThus the projection \n$\\pi'$ induces \na~family of isomorphisms $\\psi_{t'} : \\partial E_{0}(\\delta, \\varepsilon) \\rightarrow \\partial E_{t'}(\\delta, \\varepsilon)$ \nof fibrations \nsuch that the following diagram is commutative for $0 \\leq t' \\leq t$: \n\\def\\mapright#1{\\smash{\\mathop{\\longrightarrow}\\limits^{{#1}}}}\n\\def\\mapdown#1{\\Big\\downarrow\\rlap{$\\vcenter{\\hbox{$#1$}}$}}\n\\[\\begin{matrix}\n\\partial E_{0}(\\delta, \\varepsilon)&\\mapright{\\psi_{t'}}& \\partial E_{t'}(\\delta, \\varepsilon) \\\\\n\\mapdown{F_{0}}&&\\mapdown{F_{t'}}\\\\\nS^{1}_{\\delta}&=& S^{1}_{\\delta} \n\\end{matrix},\\]\nwhere $\\partial E_{t'}(\\delta, \\varepsilon) = \n\\{\\zz \\in \\Bbb{C}^{2} \\mid \\lvert F_{t'}(\\zz)\\rvert = \\delta, \\| \\zz \\| \\leq \\varepsilon \\}$. \nThus the two fibrations \n$F_{t'} : D^{4}_{\\varepsilon} \\cap F_{t'}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ and \n$F_{0} : D^{4}_{\\varepsilon} \\cap F_{0}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nare isomorphic for $0 \\leq t' \\leq t$. \nBy \\cite[Theorem 37]{O2}, the two fibrations \n\\[\nF_{0} : \\partial E_{0}(\\delta, \\varepsilon) \\rightarrow S^{1}_{\\delta}, \\ \\ \\\nF_{0}\/\\lvert F_{0}\\rvert : S^{3}_{\\varepsilon} \\setminus \\text{Int}N(K_{F_{0}}) \\rightarrow S^{1} \n\\]\nare isomorphic for $\\varepsilon > 0, \\delta > 0$ and $\\delta \\ll \\varepsilon$. \nThus $F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nis isomorphic to \n$F_{0}\/\\lvert F_{0} \\rvert : \\partial D^{4}_{\\varepsilon} \\setminus \\text{Int}N(K_{F_{0}}) \\rightarrow S^{1}$ \nfor $0 \\leq t \\ll 1$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{c1}\nThe orbit space of  \n$D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta})$ \nof the $S^1$-action is homeomorphic to \na holed $2$-sphere for $0 \\leq t \\ll 1$. \n\\end{lemma}\n\\begin{proof}\nThe monodromy of $F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nis given by the same $S^1$-action on $\\Bbb{C}^{2}$ for each $0 \\leq t \\ll 1$. \nBy Lemma $1$, $(D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}))\/S^{1}$ is homeomorphic to \n$(\\partial D^{4}_{\\varepsilon} \\setminus \\text{Int}N(K_{F_{0}}))\/S^{1}$. \n\n\nSince the orbit space $\\partial D^{4}_{\\varepsilon}\/S^{1}$ is homeomorphic to a $2$-sphere and \n$K_{F_{0}}$ is an invariant set of the $S^1$-action, \nthe orbit space $(D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}))\/S^{1}$ is a holed $2$-sphere.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Round handles}\nLet $X$ and $Y$ be $n$-dimensional smooth manifolds. \nAccording to \\cite{As, M}, we say that \\textit{$X$ is obtained from $Y$ by attaching a round $k$-handle} if \n\\begin{enumerate}\n\\item there are disk bundles over $S^{1}$, $E_{s}^{k}$ and $E_{u}^{n-k-1}$, \\\\\n\\item there exists an embedding $\\varphi : \\partial E_{s}^{k} \\times_{S^{1}} E_{u}^{n-k-1} \\rightarrow \\partial Y$ \nsuch that $X \\cong Y \\cup_{\\phi} E_{s}^{k} \\oplus E_{u}^{n-k-1}$, \n\\end{enumerate}\nwhere $E_{s}^{k} \\oplus E_{u}^{n-k-1}$ is the Whitney sum of $E_{s}^{k}$ and $E_{u}^{n-k-1}$ over $S^1$. \nThe bundle $E_{s}^{k} \\oplus E_{u}^{n-k-1}$ over $S^1$ is called \n\\textit{an $n$-dimensional round $k$-handle} and $\\varphi$ is called \n\\textit{the attaching map of $E_{s}^{k} \\oplus E_{u}^{n-k-1}$}. \nNote that a sufficiently small compact neighborhood of a connected component of the set of fold singularities \ncan be regarded as an $n$-dimensional round handle. \nIn our case, a sufficiently small compact neighborhood of each connected component of $S_{1}(F_{t})$ \nis regarded as a $4$-dimensional round $1$-handle. \n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{rthm}}\n\\subsection{Round $1$-handles determined by $S_{1}(F_{t})$}\nBy the condition $(3)$, the origin $\\mathbf{o}$ is an isolated singularity of $F_t$. \nThere exist positive real numbers \n$\\varepsilon_{t}$ and $\\delta_{t}$ such that $\\delta_{t} \\ll \\varepsilon_{t}$ and the map \n\\[\nF_{t} : D^{4}_{\\varepsilon'_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta'_{t}}) \\rightarrow \\partial D^{2}_{\\delta'_{t}} \n\\]\nis a locally trivial fibration over $\\partial D^{2}_{\\delta'_{t}}$, where \n$D^{4}_{\\varepsilon'_{t}} = \\{ \\zz \\in \\Bbb{C}^{2} \\mid \\| \\zz \\| \\leq \\varepsilon'_{t} \\}$ and \n$D^{2}_{\\delta'_{t}} = \\{ \\eta \\in \\Bbb{C} \\mid \\lvert \\eta \\rvert \\leq \\delta'_{t} \\}$ \nfor $0 < \\varepsilon'_{t} \\leq \\varepsilon_{t}, 0 < \\delta'_{t} \\leq \\delta_{t}$ and \n$\\delta'_{t} \\ll \\varepsilon'_{t}$, see \\cite{W} . \nThus $F_{t}^{-1}(c)$ intersects $\\partial D^{4}_{\\varepsilon_{t}}$ transversely for any $c \\in D^{2}_{\\delta_{t}}$ \nand $0 \\leq t \\ll 1$. \nWe assume that $\\varepsilon_{t}$ and $\\delta_{t}$ satisfy the following properties: \n\\[\nD^{4}_{\\varepsilon_{t}} \\cap S_{1}(F_{t}) = \\emptyset, \\ \\ \\ \nD^{2}_{\\delta_{t}} \\cap F_{t}(S_{1}(F_{t})) = \\emptyset. \n\\]\nSee Figure~\\ref{fig1}.\n\\begin{figure}[h]\n\\centering\n  \\includegraphics[scale=1.1]{F_t.eps}\n  \\caption{$D^{2}_{\\delta_{t}}$ and $F_{t}(S_{1}(F_{t}))$}\n  \\label{fig1}\n\\end{figure\nPut $M_{0} = D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})$. \n\n\nFix $t$ with $0 < t \\ll 1$ and \nlet $\\ell$ be the number of singularities of $\\lvert F_{t} \\rvert$. \nNote that $\\lvert F_{t} \\rvert$ is a Morse function by the conditions $(1)$ and $(2)$ except the origin. \nLet $C_{1}, \\dots, C_{\\ell}$ be the connected components of $S_{1}(F_{t})$, where \nthe number of the connected components of $S_{1}(F_{t})$ is $\\ell$ because of the conditions $(1), (2)$ and $(3)$. \nWe may assume that $\\lvert F_{t} \\rvert$ satisfies \n\\[\n\\lvert F_{t}(c_{1}) \\rvert \\leq \\lvert F_{t}(c_{2}) \\rvert \\dots \\leq \\lvert F_{t}(c_{\\ell}) \\rvert\n\\]\nfor $c_{i} \\in C_{i}$ and $i = 1, \\dots, \\ell$. \nLet $c'_{i}$ be the singularity of $\\lvert F_{t} \\rvert$ corresponding to $C_{i}$ and \n$N'_{i}$ be a sufficiently small compact neighborhood of $c'_{i}$ for $i = 1, \\dots, \\ell$. \nSince $\\lvert F_{t} \\rvert$ is a Morse function, each $N'_{i}, i = 1,\\dots, \\ell$, \ncan be regarded as a $3$-dimensional $1$-handle $[-1, 1] \\times D^{2}_{i}$, \nwhere $D^{2}_{i}$ is a $2$-disk. \nWe set $M'_{0} = M_{0}\/S^{1}$ and $M'_{i} := M'_{i-1} \\cup_{\\varphi'_{i}} N'_{i}$, \nwhere $\\varphi'_{i} : \\{\\pm 1\\} \\times D^{2}_{i} \\rightarrow \\partial M'_{i-1}$ is the attaching map of $N'_{i}$ \nfor $i = 1,\\dots, \\ell$. \nWe may assume that $\\varphi'_{i}(\\{\\pm 1\\} \\times D^{2}_{i}) \\subset \\partial M'_{0}$ for $i = 1,\\dots, \\ell$. \nThen the orbit space $D^{4}_{\\varepsilon}\/S^{1}$ is a topological manifold obtained from \n$M_{0}'$ by attaching $3$-dimensional $1$-handles $N'_{1}, \\dots, N'_{\\ell}$. \nNote that $D^{4}_{\\varepsilon}\/S^{1}$ is homeomorphic to a $3$-ball. \n\n\n\\begin{lemma}\\label{l2}\nLet $M_{0}^{*}$ be a connected component of $M'_{0}$. \nThen $\\varphi'_{i}(\\{\\pm 1\\} \\times D^{2}_{i}) \\not\\subset \\partial M_{0}^{*}$ for $i = 1,\\dots, \\ell$. \n\\end{lemma}\n\\begin{proof}\nAssume that there exist \n$i \\in \\{ 1, \\dots, \\ell \\}$ and \na connected component $M_{0}^{*}$ of $M'_{0}$ such that \n$\\varphi'_{i}(\\{\\pm 1\\} \\times D^{2}_{i})$ is contained in $\\partial M_{0}^{*}$. \nThen the genus of $\\partial M'_{i}$ is greater than $0$. \nAfter attaching $1$-handles, the genus of the boundary of the orbit space does not decrease. \nThus the genus of $\\partial M'_{\\ell}$ is greater than $0$. \nAs Lemma \\ref{c1}, the genus of $\\partial M'_{\\ell}$\nis equal to $0$. This is a~contradiction. \n\\end{proof}\nLet $M_{i}$ and $N_i$ be $4$-dimensional manifolds such that \n$M_{i}\/S^{1} = M'_{i}$ and $N_{i}\/S^{1} = N'_{i}$ respectively for $i = 1, \\dots, \\ell$. \nThen $N_i$ can be regarded as a $4$-dimensional round $1$-handle and \n$M_{i}$ is a manifold obtained from $M_{i-1}$ by attaching $N_{i}$ for $i = 1, \\dots, \\ell$. \nBy Lemma \\ref{l2}, $N_{i}$ connects two connected components of $M_{0}$. \nNote that $M_{\\ell}$ is diffeomorphic to \n$D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta})$. \n\n\\subsection{The fiber surface of $F_{t} : D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}}) \\rightarrow \\partial D^{2}_{\\delta_{t}}$} \nWe consider the restricted Milnor fibration  \n$F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}}) \\rightarrow \\partial D^{2}_{\\delta_{t}}$ and \nconnected components of $M_0$. \n\n\\begin{lemma}\\label{l3} \nLet $S_{0}$ be the fiber surface of \n$F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}}) \\rightarrow \\partial D^{2}_{\\delta_{t}}$. \nThen $S_{0}$ \nis diffeomorphic to the disjoint union of the fiber surface of \n$F_{t} \\mid_{D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}})}$ and  \n$\\ell$ copies of an annulus, where $\\ell$ is the number of connected components of $S_{1}(F_{t})$. \n\\end{lemma}\n\\begin{proof}\nLet $M_{0}^{0}, M_{0}^{1}, \\dots, M_{0}^{k}$ denote the connected components of $M_{0}$ such that \n$\\mathbf{o} \\in M_{0}^{0}$. \nThen $M_{0}^{0} \\cap D^{4}_{\\varepsilon_{t}} \\neq \\emptyset$. \nThe restricted map $F_{t} : D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}}) \\rightarrow D^{2}_{\\delta_{t}}$ \nhas a~unique singularity at the origin $\\mathbf{o}$ of $\\Bbb{C}^{2}$.  \nBy \\cite[Lemma 11.3]{Mi}, \n$D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}})$ \nis homeomorphic to $\\partial D^{4}_{\\varepsilon_{t}} \\setminus \\text{Int}N(K_{F_{t}})$, where \n$N(K_{F_{t}}) = \\{\\zz \\in \\partial D^{4}_{\\varepsilon_{t}} \\mid \\lvert F_{t}(\\zz)\\rvert \\leq \\delta_{t} \\}$. \nSo any fiber surface of \n$F_{t}: D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}}) \\rightarrow \\partial D^{2}_{\\delta_{t}}$ \nis connected. \nThe boundary of the orbit space \n$(D^{4}_{\\varepsilon_{t}} \\cap M_{0}^{0})\/S^{1}$ is homeomorphic to a~$2$-sphere and  \n\\[\nM_{0}^{j} \\cap D^{4}_{\\varepsilon_{t}} = \\emptyset \n\\]\nfor $j = 1, \\dots, k$. \n\nLet $S_{0}^{0}$ be a fiber surface of $F_{t}\\mid_{M_{0}^{0} \\cap (D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}}))}$. \nWe divide the surface $S_{0}^{0}$ as follows: \n\\[\nS_{0}^{0} = (S_{0}^{0} \\cap D^{4}_{\\varepsilon_{t}}) \\cup (S_{0}^{0} \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})). \n\\]\nSince $F_{t} : (D^{4}_{\\varepsilon} \\setminus \n\\text{Int$D^{4}_{\\varepsilon_{t}}$}) \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}}) \\rightarrow D^{2}_{\\delta_{t}}$ \nhas no singularities and  \n$F_{t}^{-1}(c)$ intersects $\\partial D^{4}_{\\varepsilon} \\sqcup \\partial D^{4}_{\\varepsilon_{t}}$ \ntransversely for any $c \\in D^{2}_{\\delta_{t}}$, \n$F_{t}^{-1}(c) \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})$ \nis diffeomorphic to $F_{t}^{-1}(0) \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})$. \nNote that $F_{t}^{-1}(0)$ is an invariant set of the $S^{1}$-action and \n$F_{t}^{-1}(0)\/S^{1}$ is a $1$-dimensional algebraic set. \nThe orbit space $F_{t}^{-1}(0)\/S^{1}$ is diffeomorphic to $[0, 1]$. \nThus the connected component of \n$F_{t}^{-1}(c) \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})$ is diffeomorphic to an annulus. \nSo any connected component of \n$S_{0}^{0} \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})$ is an annulus. \nSince any fiber of $F_{t}$ intersects $\\partial D^{4}_{\\varepsilon_{t}}$ transversely, \n$S^{0}_{0} \\cap \\partial D^{4}_{\\varepsilon_{t}}$ consists of circles and \n$S_{0}^{0} \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})$ is diffeomorphic to \n$(S^{0}_{0} \\cap \\partial D^{4}_{\\varepsilon_{t}}) \\times [0, 1]$. \nSo we have \n\\begin{equation*}\n\\begin{split}\nS_{0}^{0} &= (S_{0}^{0} \\cap D^{4}_{\\varepsilon_{t}}) \\cup (S_{0}^{0} \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})) \\\\ \n&\\cong (S_{0}^{0} \\cap D^{4}_{\\varepsilon_{t}}) \\cup ((S^{0}_{0} \\cap \\partial D^{4}_{\\varepsilon_{t}}) \\times [0, 1]) \\\\\n&\\cong S_{0}^{0} \\cap D^{4}_{\\varepsilon_{t}}.\n\\end{split} \n\\end{equation*}\n\n\nWe consider $M_{0}^{j}$ for $j = 1, \\dots, k$. \nThe restricted map $F_{t} : M_{0}^{j} \\rightarrow D^{2}_{\\delta_{t}}$ has no singularities. \nFor any $c \\in D^{2}_{\\delta_{t}} \\setminus \\{0\\}$ \nand $j = 1, \\dots, k$, \n$F_{t}^{-1}(c) \\cap M_{0}^{j}$ is diffeomorphic to $F_{t}^{-1}(0) \\cap M_{0}^{j}$. \nSince $F_{t}^{-1}(0)$ is an invariant set of the~$S^{1}$-action, \nthe orbit space $F_{t}^{-1}(0)\/S^{1}$ is a $1$-dimensional algebraic set. \nSo $F_{t}^{-1}(0)\/S^{1}$ is diffeomorphic to $[0, 1]$ or $S^{1}$. \nAssume that $F_{t}^{-1}(0)\/S^{1} = S^{1}$. \nThen $F_{t}^{-1}(c)$ is a torus and \nthe orbit space $F_{t}^{-1}(c)\/S^{1}$ is also a torus. \nSince the boundary of $M_{\\ell}'$ is a~$2$-sphere, \nthis is a contradiction. \nLet $S_{0}^{j}$ denote the fiber surface of $F_{t}\\mid_{M_{0}^{j}}$. \nThen $S_{0}^{j}\/S^{1}$ is diffeomorphic to $[0, 1]$ and \nthe fiber surface $S_{0}^{j}$ is diffeomorphic to an~annulus for $j = 1, \\dots, k$. \n\nBy Lemma \\ref{l2}, each $N_{i}$ connects two connected components of $M_0$. \nSince $M_{\\ell}$ is connected, we have $k+1-\\ell = 1$. \nThus the number of connected components of $S_0$ other than that of \n$F_{t} : D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}})$ \nis equal to $\\ell$. \n\\end{proof}\n\n\\begin{lemma}\\label{c2}\nThe connected component $M_{0}^{0}$ of $M_{0}$ is diffeomorphic to a $4$-ball and \n$M_{0}^{j}$ is diffeomorphic to $S^1 \\times B^{3}$, where $B^3$ is a $3$-ball, for $j =1, \\dots, \\ell$. \n\\end{lemma}\n\\begin{proof}\nThe two fibrations $F_{t}: D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta'}) \\rightarrow \\partial D^{2}_{\\delta'}$ \nand $F_{t}\/\\lvert F_{t}\\rvert : \\partial D^{4}_{\\varepsilon_{t}} \\setminus (\\partial D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(0)) \\rightarrow S^{1}$  \nare isomorphic for any $0 < \\delta' \\leq \\delta_{t}$. \nThus $M_{0}^{0}$ is diffeomorphic to a $4$-ball. \n\nThe map $F_{t}\\mid_{M_{0}^{j}}$ has no singularities for $j \\neq 0$. \nThen the fiber surface of \n$F_{t} : M_{0}^{j} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta'}) \\rightarrow \\partial D^{2}_{\\delta'}$\nis diffeomorphic to $S_{0}^{j}$ for $0 < \\delta' \\leq \\delta_{t}$. \nSince the monodromy of $F_{t} : M_{0}^{j} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta'}) \\rightarrow \\partial D^{2}_{\\delta'}$ \nis given by the $S^1$-action, \nthe orbit space $(M_{0}^{j} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}} \\setminus \\{0\\}))\/S^{1}$ \nis homeomorphic to $(S_{0}^{j}\/S^{1}) \\times (0, 1]$. \nBy Lemma \\ref{l3}, $S_{0}^{j}\/S^{1}$ is also an annulus. We identify $S_{0}^{j}\/S^{1}$ with $S^{1} \\times [0,1]$. \nSince $S^{1} \\times (0, 1]$ is diffeomorphic to $D^{2}\\setminus \\{0\\}$, \n$(M_{0}^{j} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}} \\setminus \\{0\\}))\/S^{1}$ is homeomorphic to \n$D^{2} \\times [0,1] \\setminus (\\{0\\}\\times [0,1])$, where $D^2$ is a $2$-disk centered at $0$. \nSince $F_{t}^{-1}(0)$ is an invariant set of the $S^1$-action, \nthe orbit space of $F_{t}^{-1}(0)$ is homeomorphic to $\\{0\\} \\times [0, 1]$.\nThus the orbit space of $M_{0}^{j}$ is homeomorphic to $D^{2} \\times [0,1]$. \nThe manifold $M_{0}^{j}$ is diffeomorphic to $S^{1} \\times B^{3}$. \n\\end{proof}\n\n\n\n\n\\subsection{The number of connected components of $S_{1}(F_{t})$}\nTo complete the proof of Theorem~\\ref{rthm}, \nit is enough to show the equality in Theorem \\ref{rthm} $(\\text{ii})$. \nWe set $\\tilde{M}_{0} = D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}})$ and \n$\\tilde{M}_{i} = \\tilde{M}_{i-1} \\cup_{\\varphi_{i}} \\partial N_{i}$ for $i = 1, \\dots, \\ell$. \nSince $F_t$ is polar weighted homogeneous, \nthe monodromy of $F_{t}\\mid_{\\tilde{M}_{i}}$ is given by the $S^{1}$-action on~$\\Bbb{C}^{2}$. \nBy the condition $(2)$, a fiber of $\\lvert F_{t}\\rvert : N'_{i} \\rightarrow \\Bbb{R}$ is as follows: \n\\begin{equation*}\n\\lvert F_{t}\\rvert^{-1}(\\lvert u\\rvert) \\cap N'_{i} \\cong \\begin{cases}                                             \n          \\text{two $2$-disks} & 0 < c_{t,\\ww} - \\lvert u\\rvert \\ll 1 \\\\                                                         \n          \\text{an annulus} &  0 < \\lvert u\\rvert - c_{t,\\ww} \\ll 1           \n          \\end{cases}. \n\\end{equation*}\nSince the polar degree of $F_t$ is equal to $d_p$ and $N_i$ is a neighborhood of an orbit of the $S^1$-action, \na fiber of $F_{t}: N_i \\rightarrow D^{2}_{\\delta}$ is a $d_p$-fold cover over a fiber of $\\lvert F_{t}\\rvert$. \nThus we have \n\\begin{equation*}\nF_{t}^{-1}(u) \\cap N_{i} \\cong \\begin{cases}                                             \n          (\\sqcup_{j=1}^{d_{p}} D^{2}_{1,j}) \\sqcup (\\sqcup_{j=1}^{d_{p}} D^{2}_{2,j}) & 0 < c_{t,\\ww} - \\lvert u\\rvert \\ll 1 \\\\                                                         \n          \\sqcup_{j=1}^{d_{p}} A_{j} &  0 < \\lvert u\\rvert - c_{t,\\ww} \\ll 1           \n          \\end{cases}, \n\\end{equation*}\nwhere $D^{2}_{k, j}$ is a $2$-disk and $A_j$ is an annulus for $k=1, 2$ and $j = 1, \\dots, d_{p}$. \nBy Lemma \\ref{l2}, we may assume that $\\sqcup_{j=1}^{d_{p}} D^{2}_{1,j}$ is contained in a connected component of $M_0$ \nwhich does not contain $\\sqcup_{j=1}^{d_{p}} D^{2}_{2,j}$. \nLet $S_{i}$ be the fiber surface of $F_{t}\\mid_{\\tilde{M}_{i}}$ for $i = 0, 1, \\dots, \\ell$. \nSet $A' = \\sqcup_{j=1}^{d_{p}}A_{j}, \\partial A' = \\sqcup_{j=1}^{d_{p}}\\partial A_{j}$ and \n$D'_{k} = \\sqcup_{j=1}^{d_{p}}D^{2}_{k,j}$ for $k = 1, 2$. \nThen $S_i$ is the surface obtained from $S_{i-1}$ by replacing $D'_{1} \\sqcup D'_{2}$ by $A'$. \nThe Euler characteristic $\\chi(S_{i})$ of $S_i$ is equal to $\\chi(S_{i-1}) - 2d_{p}$, where \n$d_p$ is the polar degree of $F_t$ for $i = 1, \\dots, \\ell$. \nThus we have \n\\[\n\\chi(S_{\\ell}) - \\chi(S_{0}) = - 2\\ell d_{p}. \n\\]\n\n\\begin{lemma}\\label{c3} \nLet $\\ell$ be the number of connected components of $S_{1}(F_{t})$. \nThen $\\ell$ is equal to $\\lvert L^{+}(P, \\mathbf{o})\\rvert - \\lvert L^{+}(F_{t}, \\mathbf{o})\\rvert$ and also to \n$\\lvert L^{-}(P, \\mathbf{o})\\rvert - \\lvert L^{-}(F_{t}, \\mathbf{o})\\rvert$. \n\\end{lemma}\n\\begin{proof}\nSince the fibration $F_{t}\\mid_{\\tilde{M}_{\\ell}}$ is isomorphic to \n$P : D^{4}_{\\varepsilon} \\cap P^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ and \n$L(P, \\mathbf{o})$ is the Seifert link in $\\partial D^{4}_{\\varepsilon}$, \nwe have \n\\[\n\\chi(S_{\\ell}) = 1 - \\{pq(m+n) - p - q\\}(m-n), \n\\]\nwhere $m = \\lvert L^{+}(P, \\mathbf{o})\\rvert$ and $n = \\lvert L^{-}(P, \\mathbf{o})\\rvert$, see \\cite[Theorem 11.1]{EN}. \nBy Lemma \\ref{l3}, the fiber surface $S_{0}$ of $F_{t}\\mid_{\\tilde{M}_{0}}$ \nis diffeomorphic to $S^{0}_{0} \\sqcup S^{1}_{0} \\sqcup \\dots \\sqcup S^{k}_{0}$ \nand $S^{j}_{0}$ is an annulus for $j \\neq 0$. \nThe Euler characteristic $\\chi(S_{0})$ of $S_0$ is equal to $\\chi(S^{0}_{0})$. \nSince the link $L(F_{t}, \\mathbf{o})$ is also the Seifert link in a $3$-sphere, \n$\\chi(S^{0}_{0})$ is given by \n\\[\n\\chi(S^{0}_{0}) = 1 - \\{pq(m'+n') - p -q\\}(m'-n'), \n\\] \nwhere $m' = \\lvert L^{+}(F_{t}, \\mathbf{o})\\rvert$ and $n' = \\lvert L^{-}(F_{t}, \\mathbf{o})\\rvert$.\nOn the other hand, $\\chi(S_{\\ell})$ is equal to $\\chi(S_{0}) - 2\\ell d_{p}$. \nThe polar degree $d_p$ is equal to $pq(m-n)$ and also to $pq(m'-n')$ \\cite{EN}. \nThen we have \n\\begin{equation*}\n\\begin{split}\n\\chi(S_{\\ell}) - \\chi(S_{0}) &= - \\{pq(m+n) - p - q\\}(m-n) + \\{pq(m'+n') - p -q\\}(m'-n') \\\\\n                             &= -pq(m-n)(m+n) + pq(m'-n')(m'+n') \\\\\n                             &= -d_{p}\\{(m+n)-(m'+n')\\} \\\\\n                             &= - 2\\ell d_{p}. \n\\end{split}\n\\end{equation*}\nSo $2\\ell$ is equal to $m+n - (m'+n')$. Since $m-m'$ is equal to $n-n'$, \n$\\ell$ is equal to $m - m'$ and also to $n-n'$. \n\\end{proof}\n\nWe give an example of Lemma $\\ref{c3}$ which is considered in \\cite{In3}. \n\\begin{Example}\nSet $f(\\zz) = z_{1}^{m} + z_{2}^{m}$ and $g(\\zz) = z_{1} + 2z_{2}$ where $m \\geq 3$. \nWe consider a deformation $F_{t} = f(\\zz)\\overline{g(\\zz)} + t(z_{1}^{m}\\bar{z}_{1} + z_{1}^{m-1} + \\gamma z_{2}^{m-1})$ \nof $f(\\zz)\\overline{g(\\zz)}$ where $\\gamma \\in \\Bbb{C}$. \nIn \\cite{In3}, we take a~coefficient $\\gamma$ of $h(\\zz)$ such that \n\\[\n\\overline{\\gamma} \\neq \n\\frac{-(2\\alpha'f(z, 1) - mg(z, 1))(mz\\bar{z}^{m-1}r^{2} + (m-1)\\bar{z}^{m-2} - \\alpha' z^{m}r^{2})}\n{(m-1)(m\\bar{z}^{m-1}g(z, 1) - \\alpha' f(z, 1))} \n\\]\nwhere $(z^{m}+1)\\overline{(z+2)} = \\alpha'\\overline{(z^{m}+1)}(z+2), \\alpha' \\in S^1$. \nThen $S_{1}(F_{t})$ is the set of indefinite fold singularities and \nthe link $S^{3}_{\\varepsilon} \\cap F_{t}^{-1}(0)$ is a $(m-1, m-1)$-torus link, \nwhere \n$S^{3}_{\\varepsilon} = \\{ (z_{1}, z_{2}) \\in \\Bbb{C}^{2} \\mid  \\lvert z_{1} \\rvert^{2} + \\lvert z_{2} \\rvert^{2} = \\varepsilon \\}$, \n$\\varepsilon \\ll 1$. \nBy Lemma $\\ref{c3}$, the number of connected components of $S_{1}(F_{t})$ is equal to~$1$. \n\\end{Example}\n\n\n\\begin{proof}[Proof of Theorem 1]\nBy Lemma $\\ref{l2}$ and Lemma $\\ref{c2}$, \n$D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})$ consists of a $4$-ball and $\\ell$-copies of \n$S^{1} \\times B^{3}$ \nand the image of the attaching map $\\varphi_{i}$ of $i$-th round $1$-handle is \ncontained in both of the boundary of a $4$-ball and that of $S^{1} \\times B^{3}$ for $i=1, \\dots, \\ell$. \nBy Lemma $\\ref{c3}$, the number of connected components of $S_{1}(F_{t})$ is equal to \n$\\lvert L^{-}(P, \\mathbf{o})\\rvert - \\lvert L^{-}(F_{t}, \\mathbf{o})\\rvert$. \n\\end{proof}\n\n\n\n\n\\section{Remarks}\n\\subsection{Monodromy and characteristic polynomials}\nLet $h : S \\rightarrow S$ be a homeomorphism of a surface $S$. We define \n\\[\n\\Delta_{*}(h) = \\frac{\\Delta_{1}(h)}{\\Delta_{0}(h)}, \n\\]\nwhere $\\Delta_{i}(h)$ is the characteristic polynomial of the homological map from \n$H_{i}(S, \\Bbb{Z})$ to itself induced by $h$ for $i = 0, 1$. \n\nLet $h_{i} : S_{i} \\rightarrow S_{i}$ be the monodromy of $F_{t}\\mid_{\\tilde{M}_{i}}$ for $i=1, \\dots, \\ell$. \nSince $h_{i}$ is given by the $S^1$-action on $\\Bbb{C}^{2}$, $h_{i} : S_{i} \\rightarrow S_{i}$ satisfies the following conditions:  \n\\renewcommand{\\theenumi}{\\Roman{enumi}}\n\\begin{enumerate}\n\\item\n$h_{i}(S_{i-1} \\setminus (D'_{1} \\sqcup D'_{2})) = S_{i-1} \\setminus (D'_{1} \\sqcup D'_{2})$ and \n$h_{i}\\mid_{S_{i-1}\\setminus (D'_{1} \\sqcup D'_{2})} = h_{i-1}\\mid_{S_{i-1}\\setminus (D'_{1} \\sqcup D'_{2})}$, \n\\item\n$h_{i}\\mid_{D'_{k}}$ and $h_{i}\\mid_{A'}$ are periodic maps which satisfy \n$D^{2}_{k,j} \\rightarrow D^{2}_{k,j+1}$ and \n$A_{j} \\rightarrow A_{j+1}$ \n\\end{enumerate}\nfor $i = 1, \\dots, \\ell, j = 1,\\dots, d_{p}$ and $k = 1, 2$. Here $D^{2}_{k,d_{p}+1} = D^{2}_{k,1}$ and $A_{d_{p}+1} = A_{1}$. \nWe calculate $\\Delta_{*}(h_{i})$ from $\\Delta_{*}(h_{i-1})$ by using a round $1$-handle $N_i$. \n\n\n\n\\begin{lemma}\\label{l4} \nLet $S_{i}$ be the fiber surface of $F_{t}\\mid_{\\tilde{M}_{i}}$ and \n$h_{i} : S_{i} \\rightarrow S_{i}$ be the monodromy of $F_{t}\\mid_{\\tilde{M}_{i}}$ for $i = 1, \\dots, \\ell$. \nThen the characteristic polynomial of $h_i$ satisfies \n\\[\n\\Delta_{*}(h_{i}) = \\Delta_{*}(h_{i-1})(t^{d_{p}}-1)^{2}. \n\\]\n\\end{lemma}\n\\begin{proof}\nSince $S_i$ is the surface obtained from $S_{i-1}$ by replacing $D'_{1} \\sqcup D'_{2}$ by $A'$ \nand $h_i$ satisfies the above properties, \nwe have \n\\[\n\\Delta_{*}(h_{i}) = \\frac{\\Delta_{*}\\bigr(h_{i}\\mid_{S_{i-1} \\setminus (D'_{1} \\sqcup D'_{2})}\\bigl)\n\\Delta_{*}\\bigr(h_{i}\\mid_{A'}\\bigl)}\n{\\Delta_{*}\\bigr(h_{i}\\mid_{\\partial A'}\\bigl)}. \n\\]\nBy the condition $(\\text{II})$, the monodromy matrices of $H_{0}(D'_{k}, \\Bbb{Z}), H_{i}(A', \\Bbb{Z})$ and \n$H_{i}(\\partial A', \\Bbb{Z})$ are equal to \n\\begin{eqnarray*}\n \\left(\n        \\begin{array}{@{\\,}cccccccc@{\\,}}\n0 & 1 & 0 & \\ldots & 0 \\\\\n0 & 0 & 1 & \\ddots & \\vdots \\\\ \n\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\n0 & \\ldots & 0 & 0 & 1 \\\\\n1 & 0 & \\ldots & 0 & 0\n\\end{array}\n       \\right)\n\\end{eqnarray*}\nfor $k = 1, 2$ and $i = 0, 1$. \nThe characteristic polynomial of the above matrix is equal to $t^{d_{p}} - 1$. \nSo $\\Delta_{*}\\bigr(h_{i}\\mid_{A'}\\bigl)$ and $\\Delta_{*}\\bigr(h_{i}\\mid_{\\partial A'}\\bigl)$\nare equal to $1$. \nAs the condition $(\\text{I})$, we have\n\\[\n\\Delta_{*}\\bigr(h_{i}\\mid_{S_{i-1} \\setminus (D'_{1} \\sqcup D'_{2})}\\bigl)\n\\Delta_{*}\\bigr(h_{i-1}\\mid_{D'_{1}}\\bigl)\\Delta_{*}\\bigr(h_{i-1}\\mid_{D'_{2}}\\bigl) = \n\\Delta_{*}\\bigr(h_{i-1}\\bigl).\n\\] \nThus the characteristic polynomial satisfies  \n\\begin{equation*}\n\\begin{split}\n\\Delta_{*}(h_{i}) \n&= \\Delta_{*}\\bigr(h_{i}\\mid_{S_{i-1} \\setminus (D'_{1} \\sqcup D'_{2})}\\bigl) \\\\\n&= \\Delta_{*}(h_{i-1})(t^{d_{p}} - 1)^{2}. \n\\end{split}\n\\end{equation*}\n\\end{proof}\n\nSince the two fibrations \n$P : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ and \n$F_{t} : D^{4}_{\\varepsilon} \\cap F_{0}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nare isomorphic, we have the following theorem. \n\n\\begin{theorem}\nLet $h$ be the monodromy of $P : D^{4}_{\\varepsilon} \\cap P^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$. \nThen $\\Delta_{*}(h)$ is equal to $\\Delta_{*}(h_{0})(t^{d_{p}}-1)^{2\\ell}$, \nwhere $\\ell$ is the number of connected components of $S_{1}(F_{t})$. \n\\end{theorem}\n\\begin{remark}\nThe algebraic set $P^{-1}(0) \\cap \\partial D^{4}_{\\varepsilon}$ is a fibered Seifert link in the $3$-sphere. \nThus the characteristic polynomial of the monodromy of the Milnor fibration of $P$ at the origin \ncan also be calculated from the splice diagram \\cite{EN}. \n\\end{remark}\n\n\\subsection{A specific deformation of $f\\bar{g}$}\nWe introduce a deformation of $f\\bar{g}$ given in \\cite{In3}, where $f$ and $g$ are $2$-variables convenient \ncomplex polynomials and $f\\overline{g}$ has an isolated singularity at the origin $\\mathbf{o}$. \nWe define the $\\Bbb{C}^{*}$-action on $\\Bbb{C}^{2}$ as follows: \n\\[\nc \\circ (z_{1}, z_{2}) := (c^{q}z_{1}, c^{p}z_{2}), \\ \\ c \\in \\Bbb{C}^{*}. \n\\]\nAssume that $f(\\zz)$ and $g(\\zz)$ satisfy\n\\[ \nf(c \\circ \\zz) = c^{pqm}f(\\zz), \\ \\ g(c \\circ \\zz) = c^{pqn}g(\\zz), \\ \\ m > n. \n\\]\nThen $f(\\zz)$ and $g(\\zz)$ are weighted homogeneous polynomials. \nTwo complex polynomials $f(\\zz)$ and $g(\\zz)$ can be written as \n\\[\n f(\\zz) =  \\textstyle\\prod\\limits_{j=1}^{m} (z^{p}_{1} + \\alpha_{j}z^{q}_{2}), \\ \\  \n g(\\zz) = \\textstyle\\prod\\limits_{j=1}^{n}(z^{p}_{1} + \\beta_{j}z^{q}_{2}), \\ \\  \\gcd(p, q) = 1, \n\\]\nwhere $\\alpha_{j} \\neq \\alpha_{j'}, \\beta_{j} \\neq \\beta_{j'} \\ (j \\neq j')$ and \n$\\alpha_{k} \\neq \\beta_{k'}$ for $1 \\leq k \\leq m$ and $1 \\leq k' \\leq n$. \nThe mixed polynomial $f(\\zz)\\overline{g(\\zz)}$ is a polar and radial weighted homogeneous polynomial, \ni.e., $f(\\zz)\\overline{g(\\zz)}$ satisfies that \n$f(s \\circ \\zz)\\overline{g(s \\circ \\zz)} = s^{pq(m - n)}f(\\zz)\\overline{g(\\zz)}$ and \n$f(r \\circ \\zz)\\overline{g(r \\circ \\zz)} = r^{pq(m + n)}f(\\zz)\\overline{g(\\zz)}$, \nwhere $s \\in S^{1}$ and $r \\in \\Bbb{R}^{*}$ \\cite{O1}. \nWe define a deformation of $f(\\zz)\\overline{g(\\zz)}$ as follows: \n\\[\nF_{t}(\\zz) := f(\\zz)\\overline{g(\\zz)} + th(\\zz),\n\\]\nwhere $0 < t \\ll 1$ and  \n\\begin{equation*}\nh(\\zz) = \\begin{cases}\n            \\gamma_{1}z_{1}^{p(m-n)} + \\gamma_{2}z_{2}^{q(m-n)}  & \\text{$(g(\\zz) \\neq z_{1} + \\beta z_{2})$}, \\\\\n            z_{1}^{m}\\overline{z}_{1} + z_{1}^{m-1} + \\gamma z_{2}^{m-1} & \\text{$(g(\\zz) = z_{1} + \\beta z_{2})$}. \n         \\end{cases}\n\\end{equation*} \nThen $F_{t}(\\zz)$ is also a polar weighted homogeneous polynomial with the polar degree $pq(m-n)$. \nBy \\cite[Theorem 1]{In3}, there exists $h(\\zz)$ such that $F_{t}(\\zz)$ satisfies the conditions $(1), (2)$ and $(3)$ \nfor $0 < t \\ll 1$. \nThe above deformation $F_t$ of $f\\bar{g}$ satisfies that $\\lvert L^{-}(F_{t}, \\mathbf{o})\\rvert = 0$. \nBy Lemma $\\ref{c3}$, the number $\\ell$ of connected components of $S_{1}(F_{t})$ is equal to $n$. \nSince the radial degree $d_r$ and the polar degree $d_p$ are equal to $pq(m+n)$ and $pq(m-n)$ respectively, \nwe have the following proposition. \n\\begin{proposition}\nLet $F_t$ be the above deformation of $f\\bar{g}$. \nThen the number $\\ell$ of connected components of $S_{1}(F_{t})$ is equal to $\\frac{d_{r} - d_{p}}{2pq}$, \nwhere $d_r$ is the radial degree of $f\\bar{g}$ and $d_p$ is the polar degree of $f\\bar{g}$. \n\\end{proposition}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\n\nHadron therapy is a novel treatment against cancer. The main advantage\nof this therapy causes less side effect in comparison to X-ray\nirradiation methods. Hadron therapy is just ahead of a significant\nbreakthrough since this technique can be more precise, applying proton\ncomputer tomograph (pCT) to map the stopping power in the tissues.\n\nThe research and development of a pCT require a fast detector to\nmeasure the energy of hadrons behind the patient. The best detector\noption is called hadron-tracking calorimeter, which consists of sandwich\nlayers of silicon tracking detectors and absorber layers. The\ncombination of measuring the trajectory (tracking process), and, in parallel, the\nenergy of relativistic particles, can provide high-resolution hadron\nimaging. This semiconductor-based technology requires stable\ntemperature and homogeneous cooling.\n\nI have worked in the development of this detector in the Bergen pCT Collaboration for two years. Last year my work was to investigate the temperature distribution in the\ncalorimeter and examine two cooling concepts in detail. I performed\nboth analytical and numerical calculations to analyze the temperature\ndistribution of the calorimeter. The final decision about the design takes into account\nmany engineering aspects, such as reliability, flexibility, and\nperformance. \n\\end{titlepage}\n\n\\tableofcontents\n\n\\newpage\n\\section{Introduction}\n\\subsection{Cancer}\n\nCancer has become one of the leading reasons for death in the developed world. It is responsible for 25\\% of all death in Hungary \\cite{Cancerstatistics}. It affects all age groups, but the risk of cancer is increasing with age, as one can see in Figure \\ref{fig:procure}.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{16cm}\n\\centering\n\\includegraphics[width=16cm]{Pictures\/Rak_statisztika2.png}\n\\caption{Cancer affects all age groups, but the risk of cancer is increasing with age. This statistic was made in Scotland in 2016 \\cite{cancerage}.}\n\\label{fig:cancerstatistics}\n\\end{minipage}\n\\end{figure}\n\nThis section is about the nature of cancer, briefly summarized based on the literature \\cite{oncology1, oncology2, oncology3}.\n\nThe cell cycle is an important period of life of the cells because they divide and replace dead cells. In healthy tissues, there are stem cells, which can divide at any time. However, for most of the cells, the number of divisions and their times are limited. This is not true for the cancer cells because a DNA mutation turns off this limitation.\n\nUsually, a cell dies after a lifetime, or in case of serious injury in its DNA. This function is called apoptosis, which, in general, does not work in cancer cells.\n\nThere are two types of cancer. The first is called bengilus. In the case of bengilus cancer, the tumor does not reduce the life expectancy and quality of life of the patient. The second one is called malignus, which means risk in life or the life quality of the patient. It is hard to determinate whether a tumor is bengilus or malignus. Usually, bengilus tumors expand slower, and they have a tegument, but the lack of these effects does not evidence that the tumor is malignus.\n\nUnfortunately, tumors can expand in the surrounding tissues. This phenomenon is called an invasion. Cells of the tumor can spread far from the original one, this is called metastasis. After the treatment of cancer, it is still possible to find other cancer cells in the surrounding tissues that can cause a new, so-called residual tumor.\n\nThere are status groups, called stages, to categorize the tumors. The traditional staging has five stages:\n\\\\0. stage: there is no invasion,\n\\\\I. stage: small, local tumor, with small invasion (without metastasis),\n\\\\II. stage: local tumor with more significant invasion, probably with metastasis in the local  lymph nodes,\n\\\\III. stage: local tumor with significant invasion with high probability to has big metastasis in the local  lymph nodes,\n\\\\IV. stage: local tumor with huge invasion and with far-reaching metastasis.\n\n\\subsection{Mapping methodes in the detection and treatment of cancer}\n\nThere are lots of methods to measure or find a tumor in the human body. \nThis subsection presents them briefly based on \\cite{oncology4}.\n\nUltrasound method uses sound waves to take a picture from the inside of the body. Applying this method, the sound wave penetrates into the body of the patient and is scattered on the boundaries of the organs with different speed of sound. A microphone records that reflection and a computer reconstructs the map.  A usual ultrasound machine takes a 2D picture from a segment of the human body, but nowadays it is also possible to obtain a 3D picture. The ultrasound technology does not have any side effect.  \n\nAn X-ray computer tomograph uses X-ray beams to fluoroscope through the human body. It creates a 3D picture from the nucleon density of the body. This type of imaging causes radiation in the body, which can damage healthy tissues.\n\nThe MR technology is based on the measurement of the magnetic field caused by the water molecules in the human body. The water molecules are excited by a strong magnetic field generated by superconductor magnets. The detector measures the changes of the magnetic field after the magnets are turned off. For instance, it makes visible the blood flow in the veins. MR technology does not cause any side effect, based on the current knowledge of science. The mapping process takes a long time, and the patient have to lie in a narrow, and closed space that causes feeling of being locked up in $4-6\\%$ of the patients.\n\nMapping with radiative isotopes can give information about the life cycle of a tumor. This technology is based on the density of the chemical material which is absorbed by the tissues and the tumor. Usually, the tumor absorbes different amount of chemical element than the surrounding healty tissues. Thus, the concentration difference of the isotopes is measured here. The directional dependence of the radiation makes possible to construct the 3D map. Moreover, the amount of the absorbed radiation offers information about the type, size and the current state of the tumor. It is also useful to find metastasis; however, the irradiation of healthy tissues is unavoidable.\n\n\\subsection{Treatment of cancer}\n\nThe treatment aims to ensure high life expectancy (or as high as possible) and life quality (or as high as possible). In order to reach this goal, modern health care uses surgery, radiotherapy, chemotherapy, and hormone therapy. These treatments can be applied either standalone or combined as a complex therapy to improve the effectiveness.\nThat subsection is summarized based on \\cite{oncology5, oncology6, oncology7}.\n\nEvery treatment starts with canceling the tumor.  It is crucial not to leave any cancer cell in the surrounding tissues because it can cause a residual tumor later. If it is not possible due to some reason, then it becomes necessary to kill these cells in the second part of the treatment. That part is about to kill all the remaining cancer cells in order to avoid metastasis. These cells are probably to be a metastasis, or cells in the vascular and lymphatic system, which try to find a new living space and create a new metastasis.\n\nSurgery is the leading treatment of cancer. It is the oldest treatment of tumor diseases. In general, it is used to remove the original tumor, but it is useful in removing the metastasis, too. Its one of the most significant advantage is that the cancer cells cannot become resistive to the surgery. During this process, doctors cut out the tumor and usually remove the local lymph nodes also from the body to reduce the risk of metastasis. The surgery burdens the body of the patient. If the patient is not in adequate conditions, the operation becomes impossible. There are other situations in which the outcome is the same: either the tumor is in a hardly available location, or too big to cut out. In these cases, doctors have to find another type of treatment or use complex methods.\n\nRadiotherapy uses the effects of the ionization of X-ray or ion beams. The beam exerts its effect in two ways, directly and indirectly. The direct way is when the particles of the beam damage the DNA and the critical parts of the cell. The indirect way is when the particles of the beam ionize the water molecules, and these molecules damage the DNA of the cancer cells. In both cases, the damage induces a self-destruct function of the cancer cell, thus the cell destroys itself. This self-destruction process is called apoptosis. Unfortunately, there are such cancer cells which have resistance to the radiation, the apoptosis process is prevented. \n\nUsually, chemotherapy uses apoptosis to destroy cancer cells. In the case of chemotherapy, doctors give medicine to the patient, which consists of chemical elements, inducing the apoptosis process. The most common chemicals are called cytostatics. \nThe cytostatics affect the cancer cells more seriously than healthy cells. Besides, there are other chemicals to reduce the effect of cytostatics in the healthy regions or increase their exterminating effect in the cancer cells.\n\nSurgery as a standalone treatment is usual for a tumor in the I.~or II.~stages. In the III.~stage, radiotherapy is recommended. Chemotherapy is typical for IV.~stage. \n\nBeyond these possibilities, sometimes the surgeon must remove the entire organ, this is called `organ removal surgery'. However, this is not preferred and tried to be avoided. For instance, using radiotherapy first reduces the size of the tumor, hence it is possible to preserve the organ as much as possible.\nMoreover, chemotherapy helps to reduce the risk of metastasis with canceling the remaining cells in the vascular and lymphatic system.\n\n\\subsection{Radiotherapy}\n\nRadiotherapy uses the ionizing effect of radiation to destroy cancer cells. There are different ways to transfer the radiation to the tumor \\cite{OrvosiBiofizika}. As a first option, one can use beams, which penetrates the tissues and the tumor, and in the meantime, it damages them. This treatment is called external beam radiation therapy. Its advantage is that it is usable in many segments of the body, without using surgery. Secondly, one can situate a radioactive source in the vicinity of the tumor, for example, into a body cavity. This technology is called brachytherapy. The advantage is the same, no operation is needed. As a third option, it is possible to ingest certain radioactive isotopes into the body. In fact, a tumor absorbs more isotopes than any of the other healthy organs. This treatment is called systemic radiation therapy. The disadvantage of this technology is that the patient becomes radioactive for a short time, so he or she has to stay in the hospital for some days.\n\nOne of the primary development in radiotherapy is to reduce the side effects caused by the radiation in healthy tissues. If one improves the ratio of the ionization in the tumor, it reduces the amount of ionization in the healthy tissues, thus it reduces the side effects.\n\nX-ray photon beams are the most common in external beam radiation therapy. The usage of protons or heavier ions instead of X-ray photons results in less ionization in the healthy tissues \\cite{OrvosiBiofizika, protonTherapy0}. Its advantage is based on the radiation concentration distribution. The X-ray photons generate the highest radiation after the entry point in the body, then the radiation is monotonously decreasing. Using ions instead of X-ray photons, it is possible to concentrate the radiation onto the tumor, since the ions stop in the tumor mostly. Ions create the highest radiation right before they stop.\n One can see the resulted radiation by X-ray photons and protons in Figure \\ref{fig:EnergiaLeadas}. One can also see the radiation of each treatment in a cross-section of a head in Figure \\ref{fig:procure}.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/EnergialeadasFotonproton.jpg}\n\\caption{The energy-loss of X-ray photons and protons in function of the amount of material in their previous path \\cite{ProCure}.}\n\\label{fig:EnergiaLeadas}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/treatement_more_angles.png}\n\\caption{Radiation in a head in case of proton and X-ray therapy \\cite{ProCure}.}\n\\label{fig:procure}\n\\end{minipage}\n\\end{figure}\n\nAs it is apparent, the application of ions in the external beam radiation therapy is beneficial. Nowadays, the number of hadron therapy centers is increasing. One can see the treatment room of a hadron therapy facility in Figure \\ref{fig:ProtonKezeles}. However, there is a design barrier in the spread of hadron therapy centers. This barrier is the lack of an accurate imaging tool for  dose planning. \nDoctors use X-ray computer tomographs (CTs) to take a three-dimensional map from the nucleon density of the body. However, the energy loss of ions depends on the electron density of the body. It is possible to calculate the electron density using the measured nucleon density, but there is no simple relation between them, because of the different ratio between proton and nucleon number of atoms of the human body. As a result, one can calculate the electron density with $1.7\\%$ statistical error from nucleon density \\cite{CTcomparison}. If one could measure the electron density map of the human body directly with the usage of ions, the statistical error could be reduced to $0.5\\%$ without significant error \\cite{CTcomparison}. Overall, the development of a hadron computer tomography (usually called proton computer tomography or pCT) is beneficial.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Proton_therapy2.jpg}\n\\caption{The patient is fixed into a bed during the radiotherapy. The patient is moved with the bed by a robot arm that can be moved in all directions. First of all they take a 3D map of the tissues nearby the tumor of the patient, after they treat the tumor by a proton beam \\cite{pKezeles}.}\n\\label{fig:ProtonKezeles}\n\\end{minipage}\n\\end{figure}\n\n\\subsection{Proton computer tomography}\n\nThe original idea of computer tomography (CT) comes from Allan M. Cormack (1963) \\cite{Cormack}. He won the Nobel Prize with this idea in 1979.\n\nA CT equipment moves around the body and takes pictures from the inside in several directions, and a computer calculates the 3D map using these pictures. In case of proton CT, one can use a beam to fluoroscope through the body. One scans the body with this beam in a particular direction, so one obtains a two-dimensional picture about the body. After, the detector is turned into a different direction, and the scanning process starts again. One can see the scanning and rotating process in Figure \\ref{fig:workingofapCT}. Finally, a computer algorithm calculates the three-dimensional map of the body from the two-dimensional pictures.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Working_of_proton_CT.pdf}\n\\caption{Left: it is visible how the proton beam scans the body. Right: it is visible how the detector and the beam rotate around the patient. }\n\\label{fig:workingofapCT}\n\\end{minipage}\n\\end{figure}\n\nDuring the scanning process, one has to use another detector at the other side of the patient in order to measure the energy and the path of particles coming from the beam. Such a detector is called a tracking calorimeter since it measures both the energy and the tracks of hadrons. $10^7-10^9~\\frac{particle}{second}$ particle rate is necessary to obtain a three-dimensional picture within a reasonable time for clinical use \\cite{Dieter}. To measure the path of the particles with such a rate was impossible five years ago. However, thanks to the development of tracking silicon pixel detectors in the last years in CERN LHC, there are available tracking detectors that can meet with this particle rate now \\cite{ALICETEC}. These tracking detectors can measure the path of the particles, but there is not any available detector that can measure the energy of each particle with this particle rate.\n\nThe Bergen pCT Collaboration aims to develop a calorimeter (calorimeter is a detector, which measures the energy of hadrons) build of alternating tracking detector and aluminum absorber layers, as a sandwich structure \\cite{Dieter}. One can see the concept of this detector in Figure \\ref{fig:Concept1}. As a tracking calorimeter, we are going to use a tracking detector, which was developed in ALICE, CERN LHC, and which is called ALPIDE.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Patient_and_detector.pdf}\n\\caption{Concept of the proton CT and its tracking calorimeter.}\n\\label{fig:Concept1}\n\\end{minipage}\n\\end{figure}\n\nMy role in the Bergen pCT Collaboration is the investigation of the temperature distribution in the calorimeter and compare two cooling system concept. That work is crucial to ensure the accuracy of the detector. That part of the detector development is clearly separated from the other tasks related to the design. Thus the present thesis reflects my work.\n\n\n\n\\subsection{ALPIDE tracking detector}\n\nALPIDE was developed to upgrade the Inner Tracking System (ITS) of `A Large Ion Collider Experiment' (ALICE) in 2019-2020, which is one of the four large experiment programs of the Large Hadron Collider of the European Organization for Nuclear Research (CERN) \t\\cite{ALICE_ALL}. One can see the ALPIDE detector in Figure \\ref{fig:ALPIDE}.\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{10cm}\n\\centering\n\\includegraphics[width=10cm]{Pictures\/pALPIDE.png}\n\\caption{ALPIDE detector \\cite{MoniKonyve}.}\n\\label{fig:ALPIDE}\n\\end{minipage}\n\\end{figure}\n\nALPIDE is a monolithic active pixel sensor (MAPS) type silicon detector \\cite{ALICETEC}, which contains the sensitive sensors together with the readout electronics on the same silicon layer, see Figure \\ref{fig:MAPS} for details. \n When a charged particle goes through the sensitive (light blue area, called Epitaxial Layer P-) area of the detector, it generates electron-hole pairs. The electrons and the holes are transferred in the sensitive area by diffusion. The holes are absorbed by the Substrate P++ layer (dark blue). There is an area (white), which is evacuated by electric voltage. When an electron reaches this zone, the electric field of the evacuated zone transfers it into the NWELL diode (green). This diode collects the electrons, then a microelectronic amplifier amplifies the signal. Another microelectronics decides whether the signal is higher than a threshold or not. If so, the microerectronic device sends a signal to the readout electronics with the coordinates of the pixel. This is a state of the art solution because the sensitive silicon layer contains the complete amplifier electronics. It is possible, thanks to the DEEP PWELL (red), which separates the NWELL parts of the microelectronics, thus the NWELL parts cannot operate as an undesirable diode.\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{10cm}\n\\centering\n\\includegraphics[width=10cm]{Pictures\/technology.pdf}\n\\caption{Main parts of a monolithic active pixel sensor in a section view of one pixel \\cite{ALICETEC}.}\n\\label{fig:MAPS}\n\\end{minipage}\n\\end{figure}\n\nThe electrons, generated by the same particle, can be transfered to more than one pixels. This effect causes that more than one pixel sends signals to the readout electronics. We call as cluster the pixels send signals after a particle go through the detector. The size of the cluster depends on the absorbed energy in the detector, so it is possible to use this information to increase the accuracy of the calorimeter. However the cluster size also depends on the temperature of the detector, so it is very important to know the accurate temperature of the detector, because a small difference can cause error in the energy measurement.\n\n\\newpage\n\\section{Detector}\n\\subsection{Concept of the detector}\n\nLet us recall the concept of the detector in Figure \\ref{fig:Concept}. The detector is built of two parts. The first part is a tracking detector, made of two layers of ALPIDE sensors. This part measures the direction of incoming particles. The second part of the detector is a calorimeter. It has a sandwich structure, and it is made of 35 ALPIDE sensor layers, separated by 4 mm thick aluminum absorber layers. This calorimeter measures the energy of the particles.\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Patient_and_detector.pdf}\n\\caption{Concept of the proton CT and its tracking calorimeter.}\n\\label{fig:Concept}\n\\end{minipage}\n\\end{figure}\n\nOne can see the structure of a detector layer in Figure \\ref{fig:Layers}. The ALPIDEs are glued on both sides of the aluminum absorber layer, alternately with a small overlap, hence the absorber functions as a support structure also. Here, two definitions are introduced for the layers, depicted in Fig.~\\ref{fig:Layers}. Its first meaning is about the engineering aspects, and consists of the aluminium absorber with APLIDE sensors on its both sides. Its second meaning is about the data analysis aspects, which considers the ALPIDE sensors between two aluminium absorbers. \n\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/What_is_a_LAYER.pdf}\n\\caption{This figure shows the structure of a layer. Gray: absorber, red: ALPIDE, blue: chip cable and orange: glue.}\n\\label{fig:Layers}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Stack_on_a_Layer.pdf}\n\\caption{A stack is a group of ALPIDEs, which is connected to the readout electronics with the same chip cable.}\n\\label{fig:Stacks}\n\\end{minipage}\n\\end{figure}\n\nThe ALPIDEs are grouped into stacks that form a readout unit. One can see a stack on one side of the absorber in Figure \\ref{fig:Stacks}. The stacks are placed in this way, but they are mirrored.\nThe thickness of the glue, ALPIDE, and chip cable are small (Figure \\ref{fig:Layers}) but not necessarily negligible. \nThe resulted temperature difference $\\Delta T$ in the perpendicular direction of the layer can be calculated with the following equation in steady-state \\cite{hokozles}:\n\n\\begin{equation}\n\\Delta T=\\frac{q \\cdot h}{\\lambda},\n\\end{equation}\nwhere $q$ is the heat flux, $\\lambda$ stands for the thermal conductivity, and $h$ being the thickness. One can find the constants and the results in Table \\ref{tab:smalllayers}. Indeed, these $\\Delta T$ values are small, thus in later calculations their thermal resistances are neglected. \nA layer model is depicted in Figure \\ref{fig:Layer_Mosell}. The volumetric heat generation occuring in the ALPIDE detectors is modelled  as an equivalent volumetric heat generation in the absorber layer.\n\n\\begin{table}[h!]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|l|r|r|r|r|r|}\n      \\hline\n      \\textbf{Layer} & \\textbf{Material} & \\textbf{Thermal} & \\textbf{Thickness} &  \\textbf{Typical} & \\textbf{Temperature}\\\\\n      \\textbf{} & \\textbf{} & \\textbf{conductivity} & \\textbf{}  & \\textbf{heat flux} & \\textbf{difference}\\\\\n      \\hline\n       &  & $\\lambda~\\big [\\frac{W}{mK}\\big]$ & $h~[\\mu m]$ & $q~\\big[\\frac{W}{m^2}\\big]$ & $\\Delta T~[K]$\\\\\n      \\hline\n      Glue & Glue & 0.22 & 10 & 410 & 0.0186\\\\\n      ALPIDE & Silicone & 149 & 50 & 410 & $1.38\\times10^{-4}$\\\\\n      Chip cable & Polyamide & 0.24 & 50 & 410& 0.0854\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Thermal resistance parameters of ALPIDE, chip cable and glue.}\n  \\label{tab:smalllayers}\n\\end{table}\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Detector_and_absolber_connection.pdf}\n\\caption{Comparing the real detector to the model. $Q_{v_1}$ is the real volumetric heat generation in ALPIDEs, and $Q_{v_2}$ is the equivalent heat generation in absorber layer.}\n\\label{fig:Layer_Mosell}\n\\end{minipage}\n\\end{figure}\n\nOne can see the essential geometrical parameters and the thermal load of a real layer in Table \\ref{tab:layerinreality}. It is not worth to model it in its original arrangement. Consequently, a simplified model is prepared using the parameters in Table \\ref{tab:layermodell}. There are several layers side by side in which almost equal heat generation occurs. Thus, applying a periodic boundary condition, one can define a `unit' in which half of the air gap appears on both sides together with one layer. Between the layer and the air heat convection occurs. \n\n\\begin{table}[h!]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|l|r|r|r|r|r|}\n      \\hline\n      \\textbf{Part} & \\textbf{Material} & \\textbf{Height} & \\textbf{Width} &  \\textbf{Thickness} & \\textbf{Heat generation}\\\\\n      \\hline\n       &  & [mm] & $[mm]$ & $[mm]$ & $\\big [\\frac{W}{m^3}\\big ]$\\\\\n      \\hline\n      Absorber & Pure aluminum & 200 & 300 & 4 & -\\\\\n      ALPIDE & Silicone & 15 & 30 & 0.05 & $8.2\\cdot 10^6$\\\\\n      Chip cable & Polyamide & 27 & 285 & 0.05& -\\\\\n      Glue & Glue & 27 & 270 & 0.01 & -\\\\\n      Air gap & Air & 200 & 300 & 1 & -\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The essential parameters of the real layer.}\n  \\label{tab:layerinreality}\n\\end{table}\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|l|r|r|r|r|r|}\n      \\hline\n      \\textbf{Part} & \\textbf{Material} & \\textbf{Height} & \\textbf{Width} &  \\textbf{Thickness} & \\textbf{Heat generation}\\\\\n      \\hline\n       &  & $[mm]$ & $ [mm]$ & $[mm]$ & $\\big [\\frac{W}{m^3}\\big]$\\\\\n      \\hline\n      Absorber & Pure aluminum & 200 & 300 & 4 & $8.3\\cdot 10^4$\\\\\n      Air gap & Air & 200 & 300 & 1 & -\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Overall dimensions and the heat load in the layer model.}\n  \\label{tab:layermodell}\n\\end{table}\n\n\\subsection{Without any active cooling system}\n\nHere, I consider the detector that is covered by a casing, it protects the detector from external effects. As a first approximation, I neglect the inner temperature difference, and the entire detector is modeled as a lumped capacitance. Despite this simplification, it offers quantitative information about the characteristic heating of the device, without any active cooling system.\nOne can see the material data in Table \\ref{tab:materialmodell}, which are used to characterize the detector. In Table \\ref{tab:freeairmodell}, further parameters are summarized for modeling. Table \\ref{tab:generatedheat} contains the number of ALPIDEs and the heat generation is specified for each part. The amount of heat generation is determined based on the description of ALPIDE elements, provided by the research group. One can see the heat generated by the particles of the beam in Table \\ref{tab:particleheat}. The heat generated by the beam is several orders of magnitude smaller than the heat generated by the electronics, so I neglected it. Regarding the heat transfer coefficient, I assumed air at rest around the detector casing.\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|r|}\n      \\hline\n      \\textbf{Part} & \\textbf{Density} & \\textbf{Mass} & \\textbf{Specific heat } &  \\textbf{Heat capacity}\\\\\n      \\hline\n       & $\\big [\\frac{kg}{m^3}\\big ]$ & $[kg]$ & $\\big [\\frac{J}{kg\\cdot K}\\big ]$ & $\\big [\\frac{J}{K}\\big ]$\\\\\n      \\hline\n      One layer & 2850 & 0.684 & 434 & 297\\\\\n      35 layers & 2850 & 23.94 & 434 & 10395\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Thermal parameters for the lumped capacitance model.}\n  \\label{tab:materialmodell}\n\\end{table}\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|r|r|}\n      \\hline\n      \\textbf{Height} & \\textbf{Width} &  \\textbf{Thickness} & \\textbf{Surface area} & \\textbf{Heat transfer coefficient}\\\\\n      \\hline\n      $[mm]$ & $[mm]$ & $[ mm]$ & $[m^2]$ & $\\big[\\frac{W}{m^2\\cdot K}\\big ]$\\\\\n      \\hline\n      200 & 300 & 175 & $0.295$ & $ 5 $\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The dimensions of the detectors, and the applied heat transfer coefficient. }\n  \\label{tab:freeairmodell}\n\\end{table}\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|r|}\n      \\hline\n      & \\textbf{ALPIDE} & \\textbf{Stack} &  \\textbf{Layer} & \\textbf{Detector}\\\\\n      \\hline\n      Number of ALPIDEs & 1 & 9 & 108 & 3780\\\\\n      \\hline\n      Generated power & 185 mW & 1.66 W & 19.9 W & 697 W\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The generated heat and the number of ALPIDEs.}\n  \\label{tab:generatedheat}\n\\end{table}\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|}\n      \\hline\n      \\textbf{Energy of a particles} & \\textbf{Particle rate} &  \\textbf{Generated power}\\\\\n      \\hline\n      $[MeV]$ & $[\\frac{particle}{s}]$ & $[mW]$\\\\\n      \\hline\n      200 & $10^9$ &  35.2\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The generated heat by the particles of the beam.}\n  \\label{tab:particleheat}\n\\end{table}\n\nThe lumped capacitance method is the mathematical formulation of the I.~law of thermodynamics, i.e., one has to formulate the balance of internal energy. On the left hand side, there is the time evolution of the internal energy, considering constant specific heat $c$ and mass density $\\rho$. On the right hand side, I consider the thermal load and the heat transfer by convection \\cite{hokozles}:\n\\begin{equation}\n\\label{eqn:powerballance}\n\\frac{dT}{dt}\\cdot m\\cdot c=Q-(T-T_{\\infty})\\cdot \\alpha \\cdot A~.\n\\end{equation}\nThe solution of the differential equation (\\ref{eqn:powerballance}) reads:\n\\begin{equation}\n\\label{eqn:powerballancesolution}\nT(t)=C_{const}\\cdot e^{-\\frac{ \\alpha \\cdot A}{m \\cdot c}t}+T_{\\infty}+\\frac{Q}{\\alpha \\cdot A},\n\\end{equation}\nin which I can define the time constant:\n\\begin{equation}\n\\label{eqn:timeconstant}\n\\tau=\\frac{m \\cdot c}{\\alpha \\cdot A}~,\n\\end{equation}\nand the $C_{const}$ is about to consider the initial condition\n\\begin{equation}\nT(0)=T_0.\n\\end{equation}\nOne obtains the following form of the $T(t)$ function  \\eqref{eqn:powerballancesolution}:\n\\begin{equation}\nT(t)=\\frac{Q}{\\alpha \\cdot A} + T_{\\infty}+(T_0 - T_{\\infty} - \\frac{Q}{\\alpha \\cdot A}) \\cdot e^{\\frac{-t}{\\tau}}~.\n\\end{equation}\nThe maximum allowable temperature for the detector is $40~^\\circ C$, which can be exceeded easily in continous operation, even for a one hour interval (see  Figure \\ref{fig:T_t_freeair}), and Table \\ref{tab:calsulationdata} summarizes the constants used in the calculation. The allowable $40~^\\circ C$ exceeded under $230$ s, which time interval is too short for the detection. Moreover, one should wait too much time to use the detector again. Consequently, an active cooling system is desired.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Python_code_analitic\/Angular_gap\/T_without_active_cooling_t18000.png}\n\\caption{The temperature history of the detector without active cooling.}\n\\label{fig:T_t_freeair}\n\\end{minipage}\n\\end{figure}\n\n\\begin{table}[h!]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|r|r|r|}\n      \\hline\n      \\textbf{$T_0$} & \\textbf{$T_{\\infty}$} &  \\textbf{$Q$} & \\textbf{$\\alpha \\cdot A$} & \\textbf{$m \\cdot c$} & \\textbf{$\\tau$}\\\\\n      \\hline\n      [$^\\circ C$] & [$^\\circ C$] & $W$ & $\\big [\\frac{W}{K}\\big ]$ & $\\big[\\frac{kJ}{K}\\big]$ & $[s]$\\\\\n      \\hline\n      25 & 25 & 697 & 1.48 & 10.4 & 7030\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The constants used to determine the temperature history.}\n  \\label{tab:calsulationdata}\n\\end{table}\n\n\\subsection{Cooling concepts}\nHere, I compare three significantly different cooling concepts:\n\\begin{itemize}\n\\item water cooling at the edge of aluminum layers,\n\\item air cooling between the layers,\n\\item water cooling inside the aluminum layers.\n\\end{itemize}\nTwo of them use water as coolant and one concept uses air. Let me review them in the following.\n\n\\textbf{A) Water cooling at the edge of aluminum layers.}\nIn concept A, the detector layers are cooled at their top and bottom edges. The scheme of this concept is depicted in Figure \\ref{fig:Water_Geom}. The benefit, compared with concept C, is that it does not need any extra material between the layers, thus it does not cause any extra scattering, consequently, it does not have any negative effect in the data reconstruction.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/WatercoolingGeometry.pdf}\n\\caption{Schematic arrangement of concept A.}\n\\label{fig:Water_Geom}\n\\end{minipage}\n\\end{figure}\n\nOn one hand, this is a compact and quiet arrangement, so it is perfect to use in treatment as well. It does not need a high coolant volume flow because of the high density and heat capacity of water. One needs a heat pump to cool the coolant that can be set far from the detector, and it makes possible to change the mean temperature of the detector. On the other hand, we cannot influence the temperature distribution. That could be a problem when a significant temperature difference occurs in a layer. In this case, one cannot use this concept of detector cooling. It reveals that the knowledge about the temperature distribution could be essential.\n\n\\textbf{B) Air cooling between layers.} In concept B, there is forced airflow between the detector layers, which cools the detectors directly. One can see this cooling concept in Figure \\ref{fig:Air_Geom}. Using this cooling method, almost the entire surface of the detector is exposed to forced convection. Here, I expect a more advantageous temperature distribution with less significant temperature gradient within a layer. \n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/AircoolingGeometry.pdf}\n\\caption{Schematic arrangement of concept B.}\n\\label{fig:Air_Geom}\n\\end{minipage}\n\\end{figure}\n\nIt could be easier to realize this concept since one can use a simple fan instead of a water circuit system, thus the engineering design is easier than the others. Moreover, it becomes completely irrelevant about how to protect the electronics from the coolant. However, it uses the air of the room as a coolant which restricts the minimum temperature and the cooling rate. It may result in high airspeed and disadvantageous vibrations in the structure. It is also an interesting question to investigate the temperature distribution in this case. \n\n\\textbf{C) Water cooling inside aluminum layers.}\nThis concept can produce the most homogenous temperature distribution. However, it is disadvantageous for the data analysis due to the flowing medium among the layer. Moreover, it would be significantly harder to protect the electronics in case of any failure. Later, this concept is excluded from further investigations.\n\n\n\n\\newpage\n\\section{Comparing the concepts}\nNow, my aim is to present the basic differences between concepts A and B by comparing the resulting temperature distributions to each other. In the following, I consider only one-dimensional heat conduction along the layer (direction of x axis). In case of concept A I also used symmetry boundary condition at the middle ($x=0$) of the layer.\n\n\\subsection{Concept A}\n\nIn general, the heat transfer coefficient can be high enough to replace the convection boundary condition with a so-called first-type boundary in which the temperature is prescribed directly. This is what we apply here. \nIntegrating the Fourier heat equation in Cartesian coordinate system, one obtains \\cite{hokozles}\n\\begin{equation}\nT(x)= - \\frac{q_v}{2 \\lambda} x^2 + \\frac{q_v}{2 \\lambda} \\left(\\frac{L_1}{2}\\right)^2 + T_0,\n\\end{equation}\nin which $T_0$ is the coolant temperature that used as a boundary condition, and $L_1$ stands for the height of the detector.\nDue to the symmetrical arrangement, the maximum temperature occurs at the middle, i.e., considering $x=0$,\n\\begin{equation}\nT_{max}=\\frac{q_v}{2 \\lambda} \\left(\\frac{L_1}{2}\\right)^2 + T_0.\n\\end{equation}\nThe desired operation condition requires as small as possible temperature difference inside a layer, thus the difference between $T_{max}$ and $T_0$ has the importance:\n\\begin{equation}\nT_{diff}=T_{max}-T_0=\\frac{q_v}{2 \\lambda} \\left(\\frac{L_1}{2}\\right)^2.\n\\end{equation}\nOne can see the temperature distribution in Figure \\ref{fig:Tdist_water}, and the parameters can be found in Table \\ref{tab:constantsofwatercooling}.\n\n\\begin{figure}[!h]\n\\centering\n\\centering\n\\includegraphics[width=10.5cm]{Python_code_analitic\/Angular_gap\/Tdistribution_waterCooling.png}\n\\caption{Temperature distribution in the layer, where the direction of x axis is visible in Figure \\ref{fig:Water_Geom}.}\n\\label{fig:Tdist_water}\n\\end{figure}\n\n\\begin{table}[!h]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|}\n      \\hline\n      \\textbf{Constant} & \\textbf{Notation} &  \\textbf{Value}\\\\\n      \\hline\n      Coolant temperature & $T_0$ & $25~^\\circ C$\\\\\n      Thermal conductivity of the absorber & $\\lambda$ & $237~\\frac{W}{m \\cdot K}$\\\\\n      Height of the detector & $L_1$ & $0.2~m$\\\\\n      Equivalent volumetric heat generation in the absorber & $q_v$ & $8.3 \\cdot 10^4~\\frac{W}{m^3}$\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Constants used to the calculation of the temperature distribution in concept A.}\n  \\label{tab:constantsofwatercooling}\n\\end{table}\n\nIn the current state of the detector design, the research group did not decide the material of the absorber. Most probably, it will be made from an aluminium alloy. Previously, the material properties of pure aluminium is used. However, I have found important to investigate the possible temperature difference $T_{diff}$ when the thermal conductivity is changed, accounting the fact that the material might be different in the real design. Thus thermal conductivity from $100 \\frac{W}{mK}$ to $237 \\frac{W}{mK}$ is considered, which are the realistic values for an aluminum alloy. The maximum temperature has a strong dependence on this material property as one can see in Figure \\ref{fig:Tmax_TC_Absorber_Water}. Most likely, the material will be Al1050, which has a thermal conductivity of $222 \\frac{W}{mK}$. The difference in maximum temperature between pure aluminum and Al1050 is $0.12~^\\circ C$, which is small, and the calculations with pure aluminum are also a good approximation for Al1050 in steady-state situations.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10.5cm]{Python_code_analitic\/Angular_gap\/Tmax_TCofAbsolber_WaterCooling.png}\n\\caption{Maximum layer temperature as a function of thermal conductivity of the absorber layer. The temperature of the edge of the detector is set to 25 $^\\circ C$.}\n\\label{fig:Tmax_TC_Absorber_Water}\n\\end{figure}\n\n\\subsection{Concept B}\n\n\\subsubsection{Heat transfer coefficient}\n\nThe calculation of the heat transfer coefficient is not evident in this case. There is a wide and long, but very narrow gap between the layers. Considering it as a tube with an equivalent tube diameter does not work evidently. The heat transfer coefficient in narrow annular gaps was measured previously by Tachibana and Fukui \\cite{HTC_article1}. \nNow, I am considering the equivalent dimension between a narrow annular gap and a gap with rectangular cross-section by wrapping the rectangular one around a cylinder (see Figure \\ref{fig:Equ_Geom}). One can find the geometrical results in Table \\ref{tab:equvalentgeom}. The equivalent tube diameter is necessary for the calculation of Reynolds (Re) number. It is calculated based on the dimensions of the annular gap, but one would obtain almost the same results using the rectangular cross-section directly.\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Equivalent_geometry.pdf}\n\\caption{Transformation between a rectangular cross-section gap and a narrow annular gap.}\n\\label{fig:Equ_Geom}\n\\end{minipage}\n\\end{figure}\n\n\\begin{table}[h!]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|}\n      \\hline\n      \\textbf{Dimension} & \\textbf{Notation} &  \\textbf{Calculation} & \\textbf{Value [mm]}\\\\\n      \\hline\n      Hight of the air gap & $L_1$ & - & 200\\\\\n      Width of the air gap & $L_2$ & - & 300\\\\\n      Thickness of the air gap & $L_3$ & - & 1\\\\\n      Equivalent outer diameter & $D_1$ & $\\frac{L_2}{\\pi} + L_3$ & 96,5\\\\\n      Equivalent inner diameter & $D_2$ & $\\frac{L_2}{\\pi} - L_3$ & 94,5\\\\\n      Equivalent tube diameter & $D_e$ & $\\frac{(D_1^2 - D_2^2)\\pi}{(D_1 + D_2)\\pi}=D_1 - D_2=2L_3$ & 2\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Geometry of the air gap between two layers and the equivalent annular air gap.}\n  \\label{tab:equvalentgeom}\n\\end{table}\n\nThe general strategy to determines the heat transfer coefficient requires an empirical (or semi-empirical) relation between the Nusselt (Nu) number (the dimensionless heat transfer coefficient) and some other dimensionless numbers such as the Reynolds (Re), Prandtl (Pr), Grashof (Gr), depending on the particular situation.\n\nHere, I use the following relation to determine the Nusselt number \\cite{HTC_article1}:\n\\begin{equation}\nNu=0.017\\left(1+2.3\\frac{D_e}{L}\\right)\\left(\\frac{D_2}{D_1}\\right)^{0.45} Re^{0.8} Pr^{1\/3},\n\\end{equation}\nwhere the Reynolds number and the Prandtl number are calculated using the equations \\cite{hokozles}:\n\n\\begin{figure}[H]\n        \\centering\n        \\begin{minipage}{6cm}\n        \\begin{equation}\n        Re=\\frac{wD_e}{\\nu}\n        \\end{equation}\n        \\end{minipage}\n        \\qquad\n        \\begin{minipage}{2cm}\n        ~~~~~~~~~~~~~~and\n        \\end{minipage}\n        \\qquad\n        \\begin{minipage}{6cm}\n        \\begin{equation}\n        Pr=\\frac{\\nu}{a}~.\n        \\end{equation}\n        \\end{minipage}\n\\end{figure}\n\nUsing the Nu number, I can determine the heat transfer coefficient as\n\\begin{equation}\n{\\alpha}=\\frac{Nu \\lambda}{D_e}.\n\\end{equation}\nFor this particular situation, one can see the heat transfer coefficient as a function of airspeed in Figure \\ref{fig:HTC_w}, and the constants are summarized in Table \\ref{tab:constantsofair}.\n\n\\begin{figure}[h!]\n\\centering\n\\centering\n\\includegraphics[width=12cm]{Python_code_analitic\/Angular_gap\/HTCoefficient_Airspeed.png}\n\\caption{The heat transfer coefficient as a function of airspeed.}\n\\label{fig:HTC_w}\n\\end{figure}\n\n\\begin{table}[!h]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|}\n      \\hline\n      \\textbf{Constant} & \\textbf{Notation} &  \\textbf{Value} & \\textbf{Unit}\\\\\n      \\hline\n      Air temperature & $T$ & $25$ & [$^\\circ C$]\\\\\n      Density & $\\rho$ & $1.19$ & $\\big[\\frac{kg}{m^3}\\big]$\\\\\n      Kinematic viscosity & $\\nu$ & $1.58 \\cdot 10^{-5}$ & \\big[$\\frac{m^2}{s}\\big]$\\\\\n      Specific heat  & $c$ & $1013$ & $\\big[\\frac{J}{kg \\cdot K}\\big]$\\\\\n      Thermal conductivity & $\\lambda$ & $0.0261$ & $\\big[\\frac{W}{m \\cdot K}\\big]$\\\\\n      Thermal diffusivity & $a$ & $2.17 \\cdot 10^{-5}$ & $\\big[\\frac{m^2}{s}\\big]$\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Parameters of the air.}\n  \\label{tab:constantsofair}\n\\end{table}\n\n\\subsubsection{Temperature distribution}\n\nIn order to obtain the temperature distribution, I have applied the following simple numerical approximation and discretization. \nThe solution is based on the internal energy balance, accounting the heat transfer among each cell. The aluminium plate and the air gap are splitted into $n$ parts, see Figure \\ref{fig:Al_and_Air} for details. Every cell of the aluminum (`w-type' cell) are connected by heat conduction and connected by heat convection to the air (`a-type' cell). Heat conduction among the a-type cells is not considered. It is possible to determine the steady-state temperature distribution by iteration among the cells.\n\n\\begin{figure}[h!]\n\\centering\n\\centering\n\\includegraphics[width=6cm]{Python_code_analitic\/Drawing_from_the_aircooling_numerical_solution.png}\n\\caption{The red cells denote the aluminum layer, the blue cells denote the air.}\n\\label{fig:Al_and_Air}\n\\end{figure}\n\nThe results are presented in Figure \\ref{fig:T_air_w5}. It makes apparent how the temperature of air increases between the entry and the exit points. More importantly, it influences the temperature distribution of a layer as well. It makes the heat transfer less intensive at the end, thus higher temperature difference can occur together with less homogeneous distribution. Consequently, the highest temperature can be found at the uttermost point from the entry of the air.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/AirandWalltemperature_w10.png}\n\\caption{The air and the layer temperature distribution in a streamline. The direction of x axis is visible in Figure \\ref{fig:Air_Geom}. Average airspeed is set to 10 $\\frac{m}{s}$, and the heat transfer coefficient is 62 $\\frac{W}{m^2 \\cdot K}$.}\n\\label{fig:T_air_w5}\n\\end{figure}\n\n\\subsubsection{Maximum temperature}\n\nOne can see the maximum temperature as a function of airspeed in the Figure \\ref{fig:Tmax_airspeed}. The maximum temperature is lower than the allowed $40~^\\circ C$ if the airspeed is higher than $5 \\frac{m}{s}$. However, it is safer to ensure $15 \\frac{m}{s}$ airspeed, but it is not worth to increase it. The optimum interval could be between $10$ to $15$ m\/s. In the further calculations, $10$ m\/s is used. Unfortunately, that speed could be too noisy and too high for clinical applications and may cause other problems as well.\n\n\\begin{figure}[!h]\n\\centering\n\\centering\n\\includegraphics[width=11cm]{Python_code_analitic\/Angular_gap\/MaxWalltemperature_Airspeed2.png}\n\\caption{Maximum temperature as a function of airspeed. The room temperature is 25 $^\\circ C$.}\n\\label{fig:Tmax_airspeed}\n\\end{figure}\n\nOne significant uncertainty is the determination of the heat transfer coefficient. The heat convection between the wall and airflow is a very complicated process because of the irregularity and uncertainty in the shape of the surface. Hence it is important to investigate the effect of changing the heat transfer coefficient. One can see the maximum temperature as a function of heat transfer coefficient in the Figure \\ref{fig:Tmax_HTC}. The domain of interest starts at $40~\\frac{W}{m^2 \\cdot K}$ to $80~\\frac{W}{m^2 \\cdot K}$. Within this domain, the maximum temperature is safely lower than the allowed $40~^\\circ C$.\n\n\\begin{figure}[!h]\n\\centering\n\\centering\n\\includegraphics[width=11cm]{Python_code_analitic\/Angular_gap\/MaxT_HTCoefficient_w10.png}\n\\caption{The maximum temperature as a function of the heat transfer coefficient. The room temperature is set to 25 $^\\circ C$.}\n\\label{fig:Tmax_HTC}\n\\end{figure}\n\nPreviously, I mentioned the uncertainty due to the material change that should be investigated. Now, I handle the thermal conductivity of the absorber layer as a parameter, see Fig.~\\ref{fig:Tmax_TC} for details. The domain of interest is $100 \\frac{W}{mK}$ to $237 \\frac{W}{mK}$ which covers all the possibilities for aluminium alloys. \n The maximum temperature variation is found to be $0.3~^\\circ C$ in this interval, hence the thermal conductivity of the absorber does not have a big impact on the maximum. One of the most likely aluminum alloy for the material of the detector is Al1050, which has the thermal conductivity of $222 \\frac{W}{mK}$. The difference between the maximum temperature in case of pure aluminum and Al1050 is $0.02~^\\circ C$, which is negligible.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/MaxTemperature_tC_of_Al_w10.png}\n\\caption{The maximum temperature as a function of thermal conductivity.}\n\\label{fig:Tmax_TC}\n\\end{figure}\n\n\\subsection{Comparison}\n\nThe maximum temperature difference in a layer is very important, since the cluster size and the noise rate of the ALPIDE are temperature dependent. I compared the maximum temperature differences for concepts A and B as a function of the airspeed, thermal conductivity and heat transfer coefficient. \n\nOne can see the maximum temperature difference as a function of airspeed in case of air and water cooling in the Figure \\ref{fig:Tdiff_a_w}. As one can see, there is no significant difference in this value. If the airspeed is higher than $8 \\frac{m}{s}$ the temperature difference is slightly lower for concept B. Otherwise, the temperature difference is higher than for water cooling. The temperature difference in case of concept B, considering $10 \\frac{m}{s}$ airseed, is $1.6~^\\circ C$. The maximum temperature difference in case of concept A is given as a benchmark, and it is constantly $1.8~^\\circ C$. In both cases, the temperature difference is lower than the allowed 5 $^\\circ C$.\n\n\\begin{figure}[h!]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/temperatureDifference_Air_vs_Water.png}\n\\caption{The temperature difference in a  layer in the case of air and water cooling. The temperature difference in case of concept A is given as benchmark as it is independent of the airspeed.}\n\\label{fig:Tdiff_a_w}\n\\end{figure}\n\nThe maximum temperature difference depends on the thermal conductivity of the layer, as one can see in Figure \\ref{fig:Tdiff_a_w_TC}, using $10 \\frac{m}{s}$ airspeed. One can see that the temperature difference is increasing in both cases with decreasing the thermal conductivity. The concept A, using water cooling, seems to be more sensitive for this parameter. Considering again the Al1050 alloy, the temperature difference in the layer is $1.9~^\\circ C$ for concept A and $1.7~^\\circ C$ in case of concept B.\n\n\\begin{figure}[!h]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/temperatureDifference_thermalConductivityofAir_W10.png}\n\\caption{Temperature difference in layer in case of air- and watercooling in function of the thermal conductivity of the layer. The average airspeed is set to 10 $\\frac{m}{s}$ in case of concept B.}\n\\label{fig:Tdiff_a_w_TC}\n\\end{figure}\n\nChanges in the heat transfer coefficient between the layer and the air also has an effect in the maximum temperature difference. One can see the outcome in the Figure \\ref{fig:Tdiff_a_w_HTC}. If the heat transfer coefficient is increasing the temperature difference is also increasing. The reason behind this effect is the changes in the temperature distribution in the streamline. In Figure \\ref{fig:T_line_lowHTC}, one can see the temperature distribution using unrealistically low heat transfer coefficient, and in Figure \\ref{fig:T_line_highHTC}, with unrealistically high  heat transfer coefficient. The realistic zone of the heat transfer coefficient is between $40~\\frac{W}{m^2 \\cdot K}$ and $80~\\frac{W}{m^2 \\cdot K}$.\nIn this domain, the maximum temperature difference starts from $1.2~^\\circ C$ to $1.8~^\\circ C$ in case of concept B. In case of concept A, the maximum temperature difference is not affected by the heat transfer coefficient, so it is only given as a benchmark, it is constantly $1.8~^\\circ C$.\n\n\\begin{figure}[!h]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/DeltaT_HTCoefficient_w10.png}\n\\caption{The temperature difference in a layer in case of air and water cooling as a function of the heat transfer coefficient. The temperature difference in case of concept A is given as a benchmark.}\n\\label{fig:Tdiff_a_w_HTC}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/AirandWalltemperature_w10_alpha15.png}\n\\caption{The layer and the air temperature with unrealistically low (15 $\\frac{W}{m^2 \\cdot K}$) heat transfer coefficient, in stream direction. The direction of x axis is visible in Figure \\ref{fig:Air_Geom}. The average air speed is set to 10 $\\frac{m}{s}$.}\n\\label{fig:T_line_lowHTC}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/AirandWalltemperature_w10_alpha200.png}\n\\caption{The layer and the air temperature with unrealistically high (200 $\\frac{W}{m^2 \\cdot K}$) heat transfer coefficient, in stream direction. The direction of x axis is visible in Figure \\ref{fig:Air_Geom}. The average air speed is set to 10 $\\frac{m}{s}$.}\n\\label{fig:T_line_highHTC}\n\\end{figure}\n\n\\newpage\n\\section{Cost estimation and environmental effects}\n\nAs a result of the previous calculations, I considered that both concept A and concept B meets with the requirements of the hadron-tracking calorimeter, so the decision between these concepts can be made based on other aspects. One of these aspects is the cost of the cooling system, so I compare the possible cost of concept A and concept B in this chapter. This comparison is a approximation of the costs, so it contains only the cost of the main elements of the cooling system. The prices in Table \\ref{tab:costofA} and Table \\ref{tab:costofB} are given as an example for the price of the parts.\n\n\\subsection{Concept A}\n\nIn case of concept \"A\" I use a water cooling circle. To protect the hadron-tracking calorimeter from a possible leak, I need to use water cooling circle, which operates under room pressure. If the cooling system operates under room pressure, in case of a leak the air of the room move into the tube, not the water move out. To operate under room pressure I need to use a vacuum pump. In case of a leak the vacuum pump has to be able to remove all water from the cooling circle, so I need the insert a vacuum tank between the cooling circle and the vacuum pump. In normal working this tank mainly filled with air. If there is a leak in the cooling circle this tank will be filled with the water of the cooling circle. The cooling circle contains a cooler, to cool the water under the temperature of the room. The cooler contains a pump, which circulates the water in the cooling circle, so I did not calculate the price of the pump separately.  The cooling circle also contains cooling plates to cool the detector. I only find smaller cooling plates then the top or the bottom of the hadron-tracking calorimeter, so I calculated with two cooling plate in the top and two cooling plate in the bottom of the detector. The cooling circle also contains tubes and fittings. I just estimated price of them, because without an exact plan of the cooling system I could not find price for them. The elements of concept A is visible in Figure \\ref{fig:A_concept_gazd}. One can see the estimated cost of this solution in Table \\ref{tab:costofA}.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Pictures\/Aconcept_gazdasagielemzes.pdf}\n\\caption{The water cooling circuit of concept A.}\n\\label{fig:A_concept_gazd}\n\\end{figure}\n\n\\begin{table}[!h]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|}\n      \\hline\n      \\textbf{Component} & \\textbf{Main operation data} &  \\textbf{Number of components} & \\textbf{Price}\\\\\n      \\hline\n\tWater cooler \\cite{cooler} & $2400~W,~16.3~\\frac{l}{min},~2.1~bar$ & 1 & 4090 \\texteuro \\\\\n\tVacuum pump \\cite{vacuumpump} & 100 mbar & 1 & 640 \\texteuro \\\\\n\tCooling plate \\cite{coolingplate} & 304.8 mm $\\cdot$ 88.9 mm& 4 & 110 \\texteuro \\\\\n\tVacuum tank \\cite{vacuumtank}  & 25 liter & 1 & 50 \\texteuro \\\\\n\tTubes and fittings & - & - & 300 \\texteuro \\\\\n\tOverall & - & - & 5520 \\texteuro \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Cost of concept \"A\".}\n  \\label{tab:costofA}\n\\end{table}\n\n\\subsection{Concept B}\n\nI need to calculate the volume flow through the detector and the pressure drop in the detector to find a corresponding fan to concept B.\n\nPressure drop in case of concept B, if the temperature of the room is $T_{room}=25~^\\circ C$. The equivalent diameter is:\n\\begin{equation}\nd_{eq}=4 \\cdot \\frac{L_2 \\cdot L_3}{2 \\cdot L_2 + 2 \\cdot L_3}=1.99~mm,\n\\end{equation}\n where $L_2=300~mm$ and $L_3=1~mm$. The airspeed is $w_{air}=10~\\frac{m}{s}$, and the kinematic viscosity is $\\nu=1.58 \\cdot 10^{-5}$ $\\frac{m^2}{s}$. The Reynolds-number of the airflow is:\n\\begin{equation}\nRe=\\frac{w_{air} \\cdot d_{eq}}{\\nu}=1250,\n\\end{equation}\nwhich means the airflow is laminar between the detector layers. The pressure drop can be calculated with this equation\n\\begin{equation}\n\\Delta p=\\frac{3 \\cdot L_1 \\nu \\cdot \\rho w_{air}}{\\left(\\frac{L_3}{2}\\right)^2}=438~Pa\n\\end{equation}\nbased on Newton's law of viscosity, where $\\rho=1.19~\\frac{kg}{m^3}$ and $L_1=200~mm$. The volume flow between the layers is\n\\begin{equation}\nQ=n_{layers} \\cdot L_3 \\cdot L_2 \\cdot w_{air}=0.105~\\frac{m^3}{s}=378~\\frac{m^3}{h},\n\\end{equation}\nwhere $n_{layers}=35$.\n\nI need to have a fun in the cooling system to generate airflow in the detector. The fan is as big as the hadron-tracking calorimeter and it is possible the fan generate vibrations, so it has to operate separately from the detector connected with a tube. I also need an air filter before the detector, to protect the hadron-tracking calorimeter from the dust in the air. The elements of the cooling system is visible in Figure \\ref{fig:B_concept_gazd}.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=3.5cm]{Pictures\/Bconcept_gazdasagielemzes.pdf}\n\\caption{The elements of B cooling system concept.}\n\\label{fig:B_concept_gazd}\n\\end{figure}\n\n\\begin{table}[!h]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|}\n      \\hline\n      \\textbf{Component} & \\textbf{Main operation data} &  \\textbf{Number of components} & \\textbf{Price}\\\\\n      \\hline\n\tFan \\cite{fan} & 950 Pa, $400~\\frac{m^3}{h}, 485 W$ & 1 & 890 \\texteuro \\\\\n\tTube \\cite{tube} & l=10 m, d=250 mm & 1 & 50 \\texteuro \\\\\n\tFittings and filter & - & - & 300 \\texteuro \\\\\n\tOverall & - & - & 1240 \\texteuro \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Cost of concept \"B\".}\n  \\label{tab:costofB}\n\\end{table}\n\n\\subsection{Comparison}\n\nAs a result of the cost estimation one can conclude that the concept B is cheaper than concept A.  As environmental effect one can consider that concept B use less energy to operate than concept A, because the water cooler needs more energy to cool the water under room temperature, than the fan needs to transfer air through the detector.\n\nIt is possible to use a concept like A, but use thermoelectric cooler and heat sinks with heat pipe and funs to cool the top and the bottom of the detector. This solution results same temperature distribution than concept A, but it can be cheaper, smaller than concept A. In case of this concept it is also possible to cool the detector under room temperature and the maximum temperature also adjustable. This concept can be more advantageous than concept A in case of beam test, but concept A is more suitable for clinical use, because in this case the generated heat is transported out of the treatment room.\n\n\\newpage\n\\section{Finite element simulation of concept A}\n\nMy next work in the research team was the more accurate calculation of the temperature distribution in case of concept A. I decided to check the effect of contact thermal resistances and inhomogeneous load on the temperature distribution. I took into account the heat generation of the ALPIDEs as a heat flux in the contact surfaces of them. I calculated with a two dimensional model (Figure \\ref{fig:Tresistance_layer_structure}) when I calculate the effect of the contact resistances, and I calculate with a three dimensional model (Figure \\ref{fig:Stacks}) when I calculated the effect of the inhomogeneous load. I used $\\lambda_{Al1050}=222~\\frac{W}{m \\cdot K}$ as the thermal conductivity of the absorber layers in both calculation.\n\n\\subsection{The effect of contact thermal resistances}\n\nFirstly I calculated with 4 mm thick absorber layer. In reality the layer will be build from 2 mm thick absorber layer in the middle, and two 1 mm thick layer in both side. The ALPIDEs will be glued into the 1 mm thick plates, as one can see in \\ref{fig:Tresistance_layer_structure}. There will be contact resistances between this plates and between the cooling plate and this plates. I calculate the contact resistance between the plates as the resistance of the average air gap between the layers, based on \\cite{contactRes}. I calculate the average air gap between two plate with $\\delta_1=\\frac{2 \\cdot R_z}{2}=R_z$ equation. The aluminum plate is cold rolled aluminum, so the mean roughness depth is $R_z=12.5~\\mu m$. The average distance is $\\delta_1=12.5~\\mu m$. Between the cooling plate and the edge of the absolber plates I calculated with $\\delta_2=200~\\mu m$, because of the uncertainty of the positioning of the plates. It is necessary to use thermal conductive grease between the cooling plate and the edge of the absolber plates. I considered $\\lambda_{air}=0.0261~\\frac{W}{m \\cdot K}$ as the thermal conductivity of air and $\\lambda_{conductive}=8.5~\\frac{W}{m \\cdot K}$ \\cite{thermalpaste} as the thermal conductivity of a typical thermal conductive paste. The thermal resistance between two layer is $R_1=\\frac{\\delta_1}{\\lambda_{air}}=4.79 \\cdot 10^{-4}~\\frac{m^2 \\cdot K}{W}$ and between the cooling plate and the edge of the absorber layers is $R_2=\\frac{\\delta_2}{\\lambda_{conductive}}=2.35 \\cdot 10^{-5}~\\frac{m^2 \\cdot K}{W}$.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Pictures\/ContactResistances.pdf}\n\\caption{The model of the layer with thermal resistances.}\n\\label{fig:Tresistance_layer_structure}\n\\end{figure}\n\nOne can see the model of the layer and the contact resistances in Figure \\ref{fig:Tresistance_layer_structure} and the result of the two dimensional simulation in Figure \\ref{fig:Tresistance_results1} and Figure \\ref{fig:Tresistance_results2}. I calculated the maximum temperature difference in that part of the surface of the absorber layer, where ALPIDEs are glued.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{SzakdogaAnsys\/Thermal_Resistance_Water_Cooling\/Results\/Delta_T_Thermal_Insulance.png}\n\\caption{The effect of the thermal resistance between layers in the maximum temperature difference.}\n\\label{fig:Tresistance_results1}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{SzakdogaAnsys\/Thermal_Resistance_Water_Cooling\/Results\/Delta_T_Thermal_Insulance_Cooling_Plate.png}\n\\caption{The effect of the thermal resistance between the cooling plate and absorber the layers in the maximum temperature difference.}\n\\label{fig:Tresistance_results2}\n\\end{figure}\n\nI have checked the mesh dependence of the solution in case when  $R_1$ is $4.79 \\cdot 10^{-4}~\\frac{m^2 \\cdot K}{W}$ and $R_2$ is $2.35 \\cdot 10^{-5}~\\frac{m^2 \\cdot K}{W}$. I checked the maximum temperature difference ($\\Delta T_{max}$) in y=2 mm line, which is visible in Table \\ref{tab:Tresistance_mesh}. I found that the mesh with 0.5 mm edge length seems to be the most accurate. The maximum temperature difference between 1 mm edge length and 0.5 mm edge length is small compared with the measured temperature differences, so the solution is mesh independent.\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|r|}\n      \\hline\n      $L_{max}$ in x direction (mm) & $L_{max}$ in y direction (mm) & Number of elements & $\\Delta T_{max}~(^\\circ C)$\\\\\n      \\hline\n\t1 & 1 & 800 & - \\\\\n\t0.5 & 0.5 & 3200 & 0.017\\\\\n\t2 & 0.5 & 824 & - \\\\\n\t1 & 0.25 & 3216 & 0.104\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Outcome of the mesh dependence test, where $\\Delta T_{max}$ is the maximum difference of temperature compared with the previous mesh. $L_{max}$ is the maximum edge length in the given direction.}\n  \\label{tab:Tresistance_mesh}\n\\end{table}\n\nI considered based on Figure \\ref{fig:Tresistance_results1} and Figure \\ref{fig:Tresistance_results2} that the contact thermal resistance between the layers has minimal effect on the temperature difference, and the contact thermal resistance between the cooling plate and the layers has only marginal effect on the temperature distribution.\n\n\\subsection{The effect of inhomogeneous load}\n\nI obtained the temperature distribution in the case, if only the middle of the detector is used. I expect the maximum temperature difference will be smaller than in case of full load, but it is interesting how much it will be, because the partial load will be a typical usage, if we have to take image from a smaller part of the human body then the size of the full detector area.\n\nI used $P_{standby}=60~mW$ heat generation in every ALPIDE detector, which was inactive in the measurement, which is the power consumption of waiting for hits. I used $P_{active}=185~mW$ heat generation in every active ALPIDE, which is the heat generation working with maximum readout capacity.\n\nI obtained the temperature distribution in an x direction path, defined by z=0 mm and y=0.002 mm. The temperature distribution in this path is visible compared with the temperature distribution of full load in Figure \\ref{fig:T_x_partial_load1}.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{SzakdogaAnsys\/Inhomogenity_Water_Cooling\/Results\/Full_and_partial_load_x.png}\n\\caption{The temperature distribution in case of full and partial load in path z=0 mm and y=0.002 mm.}\n\\label{fig:T_x_partial_load1}\n\\end{figure}\n\nOne can see the temperature in a path, defined by x=0 mm and y=0.002 mm, in Figure \\ref{fig:T_x_partial_load2}.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{SzakdogaAnsys\/Inhomogenity_Water_Cooling\/Results\/Full_and_partial_load_z.png}\n\\caption{The temperature distribution in case of full and partial load in path x=0 mm and y=0.002 mm.}\n\\label{fig:T_x_partial_load2}\n\\end{figure}\n\nThe result of the mesh dependence test is visible in Table \\ref{tab:T_inhomogen_mesh}.\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|r|}\n      \\hline\n      Maximum edge size (mm) & Number of elements & $\\Delta T_{max}~(^\\circ C)$ full load & $\\Delta T_{max}~(^\\circ C)$ full load \\\\\n      \\hline\n\t7.5 & 1234 & - & - \\\\\n\t5 & 2400 & 0.171 & 0.077 \\\\\n\t2.5 & 19200 & 0.086 & 0.038 \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Outcome of the mesh dependence test in case of full and partial load, where the $\\Delta T_{max}$ is the maximum temperature difference compared with the previous mesh.}\n  \\label{tab:T_inhomogen_mesh}\n\\end{table}\n\nThe temperature difference was $2.1~^\\circ C$ in case of full load and $1.1~^\\circ C$ in case of partial load.\n\n\\newpage\n\\section{Calculation of transient behavior}\n\nIn this subsection I focus on how much time does the heat up takes, because it effects both the clinical use, both the measurements during the development of the detector. I used the same geometry than I used in Chapter 3.1, so I used a one dimensional model. I calculated with the thermal conductivity of Al1050, which is $222~\\frac{W}{m \\cdot K}$. The heat transfer is described by the following equation:\n\\begin{equation}\n\\label{mainEq}\n\\frac{\\partial T}{\\partial t}=\\alpha \\cdot \\frac{\\partial^2 T}{\\partial x^2} + \\frac{\\dot q_v}{\\rho \\cdot c},\n\\end{equation}\nwhere $\\alpha=\\frac{\\lambda}{\\rho \\cdot c}$.\n\nInitial condition:\n\\begin{equation}\n\\label{eqn:init1}\nT(x,0)=T_0=25~^\\circ C.\n\\end{equation}\nBoundary conditions:\n\\begin{figure}[H]\n        \\centering\n        \\begin{minipage}{6cm}\n        \\begin{equation}\n\tT(0,t)=T_1=25~^\\circ C\n        \\end{equation}\n        \\end{minipage}\n        \\qquad\n        \\begin{minipage}{2cm}\n        ~~~~~~~~~~~~~~and\n        \\end{minipage}\n        \\qquad\n        \\begin{minipage}{6cm}\n        \\begin{equation}\n\tT(0,t)=T_2=25~^\\circ C.\n        \\end{equation}\n        \\end{minipage}\n\\end{figure}\n\nThe temperature can be calculated as the sum of steady state and the transient temperature:\n\\begin{equation}\nT(x,t)=T_s(x) + T_h(x,t).\n\\end{equation}\nThe steady state temperature is \\cite{hokozles}:\n\\begin{equation}\nT_s(x)=-\\frac{\\dot q_v}{\\lambda} \\cdot \\frac{x^2}{2} + C_1 \\cdot x + C_2,\n\\end{equation}\nwhere $C_2=T_1$ and $C_1=\\frac{T_2 - T_1}{L} + \\frac{\\dot q_v}{\\lambda} \\cdot \\frac{L}{2}$. The transient temperature is \\cite{hokozles}:\n\\begin{equation}\nT_h(x,t)=\\sum_{n=1}^{n_{max}} D_n \\cdot e^{-\\Lambda_n \\cdot t} \\cdot sin \\left(\\frac{n \\cdot \\pi}{L} \\cdot x \\right),\n\\end{equation}\nwhere $\\Lambda_n=\\alpha \\cdot \\left( \\frac{n \\cdot \\pi}{L} \\right)^2$ and $D_n=\\frac{\\int_{0}^{L} (T(x,0)-T_s(x)) \\cdot sin \\left( \\frac{n \\cdot \\pi}{L} \\cdot x \\right) dx}{\\int_{0}^{L} sin^2 \\left( \\frac{n \\cdot \\pi}{L} \\cdot x \\right) dx}$.\n\nOne can see the temperature of the middle of the detector in function of time in Figure \\ref{fig:Tmiddle_t}, and the parameters, which was used to the calculation, in Table \\ref{tab:consHeatup}.\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{SzakdogaAnsys\/Heatup_T_t.png}\n\\caption{The temperature of the middle point of the detector in function of time.}\n\\label{fig:Tmiddle_t}\n\\end{figure}\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|}\n      \\hline\n      Parameter & Value\\\\\n      \\hline\n\tInitial temperature of the layer ($T_1$) & $25~^\\circ C$\\\\\n\tTemperature of the edge of the layer ($T_1$ and $T_2$) & $25~^\\circ C$\\\\\n\tThermal conductivity of the layer ($\\lambda$) & $222~\\frac{W}{m \\cdot K}$ \\\\\n\tMass density of the layer ($\\rho$) & $2710~\\frac{kg}{m^3}$ \\\\\n\tSpecific heat of the layer (c) & $434~\\frac{J}{kg \\cdot K}$ \\\\\n\tEquivalent volumetric heat generation ($q_v$) & $8.3 \\cdot 10^4~\\frac{W}{m^3}$ \\\\\n\tMaximal item of Fourier series ($n_{max}$) & 50 \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Parameters used in the heat up calculation.}\n  \\label{tab:consHeatup}\n\\end{table}\n\nIt takes 134 s, while the temperature of the middle point of the layer reach $\\pm 0.1~^\\circ C$ zone around the steady state temperature of the middle point. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\section{Looking Ahead}\n\nIn this work I analysed the temperature distribution of a layer of the hadron-tracking calorimeter. The detector is going to contain two layer before the calorimeter, which measure the path of the particles. This layers are going to contains a carbon composite layer in the middle instead of an aluminum absorber layer. This carbon composite layer is going to work as a support structure, and it is thinner than aluminum absorber layer. The thermal conductivity of carbon composite is anisotropic, it is reasonable in longitudinal direction, but low in perpendicular direction. It is essential to analyze the temperature distribution in this layers, and design new cooling concepts for these layers, if it is necessary.\n\nAnother important engineering task in the development of  proton computer tomograph is the comparison of the temperature distribution between layers. This task requires more information about the power consumption of the tracking detectors and the number of particles, which goes through them. It is possible to reach this information with test beam measurements, so one of the next engineering task should be the measurement of the electric power consumption in case of different particle fluxes.\n\n\\newpage\n\\section{Summary}\n\nCancer started to become the leading cause of death in the developed world, as it is responsible for 25\\% of all death in Hungary. Mainly, there are three ways of treatment: surgery, radiotherapy, and chemotherapy. Hadron therapy is a novel treatment in radiotherapy. It is beneficial because it generates less radiation in healthy tissues than X-ray radiotherapy, thus it leads to fewer side effects, and it allows a higher daily dose, which increases the effectiveness of treatment.\n\nHadron therapy requires an accurate three-dimensional map about the electron density of the body. The most accurate solution to obtain the map uses hadrons, however, it requires the development of a proton computer tomograph (pCT). Our group, the Bergen pCT Collaboration, aims to develop a pCT that can meet the requirements of the clinical use. The detector of the pCT is built of silicon pixel tracking detectors and aluminum absorber layers.\n\nIn the collaboration, my role was to investigate the temperature distribution of the detector, which is essential because of the accuracy of the energy measurement based on the temperature of the detector. It is indispensable in the development of the pCT detector.\n\nI applied the lumped capacitance method to estimate the thermal behaviour of the detector, containing 35 absorber layers. This simple analysis shows the necessity for cooling. Hence, two cooling concepts are elaborated in this work. In concept A, water cools the edges of the aluminum absorber layers. In concept B, the air is circulated between the detector layers. My goal was to quantify the differences in the steady-state temperature distribution and compare the concepts to each other.\n\nFirstly I calculated with analytic methods the steady state temperature distribution in case of both concepts. Both of the them satisfies the first requirement, which is to keep the maximum temperature lower than 40  $^\\circ C$. The allowed temperature difference is 5 $^\\circ C$. Both of the concepts meets this requirement also, but concept B has a bit better performance as the temperature difference is 1.6 $^\\circ C$ in this case, compared with 1.8 $^\\circ C$ in case of concept A.\n\nSecondly I estimated the cost of both concept. I considered concept A is more expensive than concept B. It is possible to use a third solution, which cools the top and the bottom of the hadron-tracking calorimeter, like concept A, but contains thermoelectric coolers, heat pipes, heat sinks and funs instead of a water cooling circuit. This concept can be smaller and cheaper than concept A, but it has the advantageous properties of concept A.\n\nThirdly I used finite element simulations to determinate the effect of the contact thermal resistances and partial load in the temperature distribution, in case of concept A. I obtained that a reasonable contact thermal resistance has small effect on the temperature distribution, as it cause maximum $0.2~^\\circ C$ increase in the temperature difference. In case of partial load the temperature difference is decrease, as it was expected. I calculated the time, which is required to reach steady state temperature distribution. It takes 134 s, which is suitable for clinical use and test beam measurements.\n\n\\newpage\n\\section{\u00d6sszefoglal\u00f3}\n\nNapjainkban a r\u00e1k az egyik vezet\u0151 hal\u00e1lokk\u00e1 kezd v\u00e1llni a fejledt vil\u00e1gban, Magyarorsz\u00e1gon a teljes elhal\u00e1loz\u00e1s 25\\%-\u00e1\u00e9rt felel\u0151s. A r\u00e1knak h\u00e1rom f\u00e9le f\u0151 gy\u00f3gy\\-m\u00f3d\\-ja van: az els\u0151 a seb\u00e9szet, a m\u00e1sodik a sug\u00e1rter\u00e1pia \u00e9s a harmadik a kemoter\u00e1pia. A hadronter\u00e1pia egy \u00faj sug\u00e1rter\u00e1pi\u00e1s kezel\u00e9s. El\u0151ny\u00f6s, mert kevesebb ioniz\u00e1ci\u00f3t okoz az eg\u00e9szs\u00e9ges sz\u00f6vetekben mint a hagyom\u00e1nyos, r\u00f6ntgen sug\u00e1rz\u00e1ssal v\u00e9gzet sug\u00e1rter\u00e1pia. Ennek k\u00f6sz\u00f6nhet\u0151en kevesebb mell\u00e9khat\u00e1ssal j\u00e1r, \u00e9s nagyobb napi d\u00f3zist tesz lehet\u0151v\u00e9, ezzel n\u00f6velve a kezel\u00e9s hat\u00e1soss\u00e1g\u00e1t.\n\nA hadronter\u00e1pi\u00e1hoz sz\u00fcks\u00e9ges egy pontos t\u00e9rbeli k\u00e9pet k\u00e9sz\u00edteni a p\u00e1ciens test\u00e9nek elektrons\u0171r\u0171s\u00e9g\u00e9r\u0151l. A legpontosabb megold\u00e1s, ha ezt a k\u00e9pet hadronok haszn\u00e1lat\u00e1val k\u00e9sz\u00edtj\u00fck el, de ez egy proton computer tomogr\u00e1f (pCT) kifejleszt\u00e9s\u00e9t ig\u00e9nyli. A Bergen pCT egy\u00fcttm\u0171k\u00f6d\u00e9s c\u00e9lja, hogy kifejlesszen egy ilyen eszk\u00f6zt, ami a klinikai felhaszn\u00e1l\u00e1si szempontoknak eleget tesz. A pCT detektora szil\u00edciumpixel-nyomk\u00f6vet\u0151 detektorokb\u00f3l \u00e9s alum\u00ednum abszorber r\u00e9tegekb\u0151l \u00e9p\u00fcl fel.\n\nAz egy\u00fcttm\u0171k\u00f6d\u00e9sben bet\u00f6lt\u00f6tt szerepem a detektor h\u0151m\u00e9rs\u00e9klet-eloszl\u00e1s\u00e1nak meg\\-ha\\-t\u00e1\\-ro\\-z\u00e1\\-s\u00e1\\-ra ir\u00e1nyult. Ennek vizsg\u00e1lata elengedhetetlen a pontos m\u00e9r\u00e9sek elv\u00e9gz\u00e9s\u00e9hez. El\u0151sz\u00f6r, egy egy\\-sze\\-r\u0171, koncentr\u00e1lt param\u00e9ter\u0171 modellel becs\u00fcltem a 35 r\u00e9tegb\u0151l \u00e1ll\u00f3 detektor termikus viselked\u00e9s\u00e9t. Ebb\u0151l egy\u00e9rtelm\u0171en l\u00e1tni, hogy a h\u0171t\u00e9s elengedhetetlen.\nK\u00e9t f\u00e9le h\u0171t\u00e9si koncepci\u00f3t dolgoztam ki. Az ``A'' koncepci\u00f3 eset\u00e9n v\u00edzzel h\u0171tj\u00f6m az abszorber r\u00e9tegek sz\u00e9leit, m\u00edg a ``B'' koncepci\u00f3 eset\u00e9n leveg\u0151t keringtetek a r\u00e9tegek k\u00f6z\u00f6tt. C\u00e9lom a kialakul\u00f3, \u00e1lland\u00f3sult \u00e1llapot\u00fa h\u0151m\u00e9rs\u00e9klet-eloszl\u00e1sok meghat\u00e1roz\u00e1sa \u00e9s a k\u00e9t koncepci\u00f3 \u00f6sszehasonl\u00edt\u00e1sa.\n\nEl\u0151sz\u00f6r analitikusan meghat\u00e1roztam az \u00e1lland\u00f3sult h\u0151m\u00e9rs\u00e9kleteloszl\u00e1st mindk\u00e9t koncepci\u00f3 eset\u00e9ben. A megengedett maximum h\u0151m\u00e9rs\u00e9klet a detekteorban $40~^\\circ C$, amit mindk\u00e9t koncepci\u00f3 teljes\u00edt. A megengedett h\u0151m\u00e9rs\u00e9klet-k\u00fcl\u00f6nbs\u00e9g $5~^\\circ C$, amit \u00fagyszint\u00e9n mindk\u00e9t koncepci\u00f3 teljes\u00edt, de a B koncepci\u00f3 ilyen szempontb\u00f3l el\u0151ny\u00f6sebb, mivel ebben az esetben a h\u0151m\u00e9rs\u00e9klet-k\u00fcl\u00f6nbs\u00e9g csak $1.6~^\\circ C$, am\u00edg az A koncepci\u00f3 eset\u00e9ben $1.8~^\\circ C$.\n\nM\u00e1sodszor megbecs\u00fcltem a k\u00e9t koncepci\u00f3 lehets\u00e9ges k\u00f6lts\u00e9geit. Eredm\u00e9ny\u00fcl azt kaptam, hogy az A koncepci\u00f3 l\u00e9nyegesen dr\u00e1g\u00e1bb. Lehets\u00e9ges egy harmadik k\u00f6ncepci\u00f3 alkalmaz\u00e1s is, amely eset\u00e9ben ugyan\u00fagy a detektor als\u00f3 \u00e9s fels\u0151 fel\u00fclet\u00e9t h\u0171tj\u00fak, mint az A koncepci\u00f3 eset\u00e9ben, azonban v\u00edzh\u0171t\u00e9s helyett Peltier-elemek, h\u0151cs\u00f6vek, bord\u00e1k \u00e9s ventill\u00e1torok seg\u00edts\u00e9g\u00e9vel val\u00f3s\u00edtjuk meg a h\u0171t\u00e9st. Ez a koncepci\u00f3 kisebb \u00e9s olcs\u00f3bb megold\u00e1st jelenthet az A koncepci\u00f3n\u00e1l, azonban rendelkezik annak az el\u0151ny\u00f6s tulajdons\u00e1gaival.\n\nHarmadszor v\u00e9geselemes-szimul\u00e1ci\u00f3k seg\u00edts\u00e9g\u00e9vel vizsg\u00e1ltam a kontakt-h\u0151ellen\u00e1ll\u00e1sok \u00e9s az r\u00e9szleges terhel\u00e9s hat\u00e1s\u00e1t a h\u0151m\u00e9rs\u00e9klet eloszl\u00e1sra az A koncepci\u00f3 eset\u00e9ben. Eredm\u00e9ny\u00fcl azt kaptam, hogy a re\u00e1lis kontakt-h\u0151ellen\u00e1ll\u00e1sok maximum $0.2~^\\circ C$ h\u0151m\u00e9rs\u00e9kletk\u00fcl\u00f6nbs\u00e9g-n\u00f6veked\u00e9st okoz. R\u00e9szleges terhel\u00e9s eset\u00e9n a maxim\u00e1lis h\u0151m\u00e9rs\u00e9klet-k\u00fcl\u00f6nbs\u00e9g  kisebb, mint teljes terhel\u00e9s eset\u00e9n, ahogyan ezt el\u0151re v\u00e1rtuk. Analitikus sz\u00e1m\u00edt\u00e1ssal meghat\u00e1roztam az \u00e1lland\u00f3sult h\u0151m\u00e9rs\u00e9kleteloszl\u00e1s kialakul\u00e1s\u00e1hoz sz\u00fcks\u00e9ges id\u0151t. Ez 134 m\u00e1sodpercnek ad\u00f3dott, ami megfelel\u0151, mind az orvosi felhaszn\u00e1l\u00e1s, mind a tesztm\u00e9r\u00e9sek sz\u00e1m\u00e1ra.\n\n\\newpage\n\\addcontentsline{toc}{section}{Acknowledgements}\n\\section*{Acknowledgements}\n\nI would like to thank for the work of my supervisor, R\u00f3bert Kov\u00e1cs, and my consultant M\u00f3nika Varga-K\u0151farag\u00f3. Both of them helped me a lot. First, I would like to thank to M\u00f3nika, who found this topic for me, and helped a lot in the initial difficulties. I studied a lot from her about data analysis, programming, and how to behave as a scientist while we worked together. I would like to thank R\u00f3bert for his help that I got from him during the creation of this thesis. I studied a lot from him about thermodynamics, and about how to work as an engineer. I hope we will continue the work together, and I can learn a lot more from him.\n\nI would like to thank Gergely G\u00e1bor Barnaf\u00f6ldi, who helped me a lot with his advice and encouragement. I started to learn from him how to write not just grammatically correct, but readable sentences, that will be useful in my entire career to become a scientist.\nI would like to thank Dieter R\u00f6rich, who gave me the possibility to spend three months in Bergen and study a lot from him and his colleagues. I also would like to thank Shruti Vineet Mehendale, who helped me a lot in my work during the last half-year. I owe a lot of gratitude for the Wigner \\-ALICE Group and the Bergen pCT Group for the possibility of joining to their project. Finally, this work would not be possible without the support of NKFIH\/OTKA K 120660, Hungarian Scientific Research Fund \u2013 OTKA.\n\n\\newpage\n\\addcontentsline{toc}{section}{Bibliography}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{I. Introduction}\n\nThe Nobel Laureate Richard Feynman first suggested that a new type of\ncomputer, running on the principles of quantum rather than classical\nmechanics, might have an exponential speed-up in computation time over a\nclassical device for some tasks[1]. This observation engendered the\ndevelopment of a theory of a digital quantum computer by David Deutsch of\nCambridge University[2], in which he demonstrated the intrinsic massive\nparallelism of quantum computation by operating on a coherent superposition of\na large number of states. In particular, a single computation acting on N\nquantum bits can achieve the same effect as 2N simultaneous computations\nacting on classical bits. For N=300, this is a number greater than the number\nof atoms in the visible universe. Obviously, no classical computer could ever\nbe built to compete with this kind of processing power. Exact methods and\napplications that harness this enormous potential are currently the subject of\nintense research. In 1994 Peter Shor of Bell Labs announced the discovery of\nan algorithm which, if running on a quantum computer, could find prime factors\nof large numbers in polynomial time[3]. There is currently no known classical\ncounterpart that can accomplish the same task in less than exponential time.\nShor's factoring algorithm was the impetus for an explosion of research aimed\ntoward the theory and experimental implementation of a working digital quantum\ncomputer[4]. In a subsequent development, Lov Grover discovered an algorithm\nthat shows a square-root speed-up over a classical machine for the task of\nsearching a large unordered database[5]. Even though Grover's algorithm is\nonly polynomially faster than a classical algorithm, the wide range of its\napplicability to data search makes it very enticing. Another extremely\nimportant class of problems is represented by the so-called ''intractable''\nNP-complete set. These problems are at the core of a wide range of practical\nrouting, layout, and other logistics issues. A prototypical example here is\nthe traveling salesman problem (TSP). It is widely believed that no classical\nalgorithm that can solve this problem, or any other member of this class,\nexactly in less than exponential time. Such problems are now a challenge for\nquantum computers and algorithm developers.\n\nVarious approaches are currently in process for laboratory realization of\nessential elements to perform quantum computing, namely, the construction of\nqubits and demonstration of quantum logic operations. The work of Jeff Kimble\nand his colleagues at the California Institute of Technology (Cal Tech)\nemphasizes the use of entangled photon states using microcavities to construct\nquantum logic gates [6]. Correspondingly, David Wineland and his group at\nNIST, following the theoretical predictions of Peter Zoller and coworkers[7],\nachieved what amounts to a two bit quantum register using laser excitation\ncontrol of atoms in cold ion traps[8]. These methods require large scale,\nstate-of-the-art laboratory facilities and defy scalability beyond a few\nqubits. Aside from the requirements of extremely sophisticated laboratory\nfacilities and procedures, quantum decoherence occurs in these schemes at a\ntime scale of nanoseconds at best, and is a major obstacle. Recently, Neil\nGershenfeld and Isaac Chuang introduced a revolutionary scheme for quantum\ncomputing using nuclear spins and the methods of nuclear magnetic resonance\n(NMR) to construct and manipulate quantum logic[9]. Free temporal evolution\ntakes place virtually without decoherence due to the robust isolation of\nnuclear spins caused by screening by the atoms electrons. This version of\nquantum computation can be controlled and processed with current\nstate-of-the-art technology. More recently, Chuang, Gershenfeld, and Kubinec\n(CGK) reported, using this scheme, the first ever laboratory demonstration of\na genuine quantum computation [10] by execution of Grover's quantum search\nalgorithm [5]. This included loading of the registers, unitary execution of\nthe quantum calculation, and read-out of the results. Their results, though it\nutilized only four input registers, constitute a significant step forward\ntoward the practical realization of a useful quantum computer. One of the\nproblems in scaling, in this regard, to a practical device is that the nuclear\nspins within a molecule that form the computational basis must rely on\nchemical shifts to be distinguished from one another. These shifts are usually\nvery small and consequently the rf pulses that must control the computation\nare required to be sufficiently long so as not to couple distinct spins by\nresonance overlap. This is usually of the order of microseconds. Thus,\ncomputation is required to be exceedingly slow. A second, and more severe\nproblem in scaling is that the read-out signal necessarily diminishes\nexponentially with the size of the quantum bit register. The current\nstate-of-the-art limits this size to no more than 15 bits. We recognized early\non [11] that these difficulties could be mitigated by selectively coupling the\nnuclear and electronic intrinsic spin angular momenta via hyperfine or\ntransferred hyperfine interactions [12,13]. Further emphasis has been recently\nenunciated in the use of electron-nuclear spin coupling and control for\nquantum computation by David DiVincenzo[14]. These comments were stimulated by\nthe specific seminal suggestions by Kane[15] for nanostructures in Si. The\nscheme proposed by Kane surmounts the problems of scalability and individual\nnuclear spin addressability that is inherent in the liquid state NMR methods\n[9,10]. However, it suffers in principle from the requirement of electron\ncharge transfer, and the measurement of single electronic charge by\nnanostructured gates for signal read-in and read-out. Otherwise, electronic\ncharge migration is required for mediation of nuclear spin coupling as well as\nthe shift of individual nuclear spins in and out of resonance with a constant\nrf field. Electronic charge and spin migration is controlled by nanofabricated\ngates. The requirements of the procedure are beyond current fabrication\ntechnology, but this is likely to change in the reasonably near future.\nHowever, there remain problems connected with charge transfer control for\nnuclear spin identification, and single charge measurements for read-in and read-out.\n\nOur purpose here is to present a scheme for quantum computation that is\nuniversal and that does not suffer from the limiting conditions of liquid\nstate NMR and that may offer a more viable alternative to the proposed scheme\nof Kane. We draw upon the favorable decoherence properties of nuclear spin\nsystems utilized in NMR quantum computing, and the solid state distributed\nimplanted nuclear spin elements system coupled via electron hyperfine and\ntransferred hyperfine interactions proposed by Kane. Our scheme involves a\ndistributed individually addressable system of electron spin\/nuclear spin\nqubits and state-of-the-art microwave single electron spin resonance for\nsignal read-in and sensitive optical fluorescence for read-out. Our treatment\nhere is generic. Specific materials and optimizations are relegated to a\nfuture publication.\n\nIn the next section we present and discuss the irreducible requirements for a\nuniversal quantum computer. Our generic scheme is presented in Section III.\nQuantum logic, entanglement, nonlocality and quantum information memory are\ndiscussed in Section IV where it is shown that all the conditions of Section\nII are met. The last section is used for discussion and comparison with other\nproposed methods and schemes, and for concluding remarks.\n\n\\bigskip\n\n\\section{II. Requirements for Universal Quantum Computer}\n\n\\bigskip\n\nQuantum computational (QC) algorithms are of only esoteric significance apart\nfrom the hardware capability to execute them. The resource and technological\ninvestiture necessary to develop such a device would necessitate design\nrequirements that fulfill the capability to perform any given quantum\ncomputation and accommodate the execution of any algorithm, including initial\nstate preparation, read-in and read-out. A quantum computer design meeting\nthese requirements will be termed universal. The design must meet, therefore,\ncertain irreducible requirements that are both necessary and sufficient for\nthe execution of any given QC algorithm. Establishing these requirements\nnecessitates the identification of the set of elemental operations common to\nthe execution of all quantum computational algorithms that, in a particular\noperational sequence, are sufficient for the execution of any given algorithm.\nA universal quantum computer must be capable of execution of the elemental operations.\n\nIt is conjectured that the identification of universal elemental operations\ncommon to al QC algorithms and sufficient in specific combination are simple\nHadamard transforms, together with a generalized quantum phase shift operation\n[16]. The work of Cleve, et.al. [16] is the basis for this conjecture. A\nsimple Hadamard transform, $H$ , on a qubit $\\left|  q\\right\\rangle $, $q=0,1$\nis given by [16]%\n\n\\begin{align}\n& \\left|  0\\right\\rangle \\overset{H}{\\rightarrow}\\frac{1}{\\sqrt{2}}(\\left|\n0\\right\\rangle +\\left|  1\\right\\rangle )\\label{eq1}\\\\\n& \\left|  1\\right\\rangle \\overset{H}{\\rightarrow}\\frac{1}{\\sqrt{2}}(\\left|\n0\\right\\rangle -\\left|  1\\right\\rangle )\n\\end{align}\nThis operation is exactly equivalent to a uniform single particle beam\nsplitter[17]., and is a special case of the more general Quantum Fourier\nTransform (QFT),\n\\begin{equation}\n\\left|  a\\right\\rangle \\overset{F_{2^{m}}}{\\rightarrow}\\sum_{y=0}^{2^{m}%\n-1}e^{2\\pi iay\/2^{m}}\\left|  y\\right\\rangle\n\\end{equation}\nIn terms of a two-bit operation a subsequent phase shift operation can impart\na conditional phase shift to one component of the transformed wave function,\n\\begin{equation}\n\\left|  0\\right\\rangle \\left|  1\\right\\rangle \\overset{H}{\\rightarrow}\\frac\n{1}{\\sqrt{2}}(\\left|  0\\right\\rangle +\\left|  1\\right\\rangle )\\left|\nu\\right\\rangle \\overset{c-u}{\\rightarrow}\\frac{1}{\\sqrt{2}}(\\left|\n0\\right\\rangle +e^{i\\varphi}\\left|  1\\right\\rangle )\\left|  u\\right\\rangle\n\\end{equation}\nOr a conditional transition of the auxilliary qubit as well,\n\\begin{equation}\n\\frac{1}{\\sqrt{2}}(\\left|  0\\right\\rangle +\\left|  1\\right\\rangle )\\left|\nu\\right\\rangle \\overset{C-NOT}{\\rightarrow}\\frac{1}{\\sqrt{2}}(\\left|\n0\\right\\rangle \\left|  u\\right\\rangle +e^{i\\varphi}\\left|  1\\right\\rangle\n\\left|  v\\right\\rangle )\n\\end{equation}\nthus forming an entangled state and constituting a controlled NOT-gate\n(C-NOT). Ultimately, quantum trajectories are brought together as in a\nMach-Zehnder interferometer to produce quantum interference[17]. The\nintermediate conditional operation, phase shift and\/or entanglement, requires\nauxiliary qubits and a unitary operation[16]. The corresponding network\nrepresentation is given in fig. 1.%\n\n\n\\par\\noindent Fig. 1. Network representation of quantum Mach-Zender interferometer with\ngeneralized phase shifter.\n\n\\bigskip\n\nThus, the elemental operations are conjectured to be constituted by a simple\nQFT, a conditional phase shift and\/or entanglement, and ultimately a QFT that\nbrings together quantum trajectories to produce quantum interference. The\nlater constitutes the results of a computation, i.e. collapse of the wave\nfunction. These operations are entirely equivalent to the unique\ncharacterization of quantum computation: a) quantum superposition, b) quantum\nentanglement, and c) quantum interference.\n\nIn addition to the ability to facilitate the operational requirements\ndiscussed above, a universal digital quantum computer must satisfy the\nfollowing criteria:\n\n\\bigskip\n\ni)\\qquad A distinct set of distributed qubits must be defined that are\nindividually addressable.\n\nii) \\ \\ \\ \\ \\ The qubits must be linkable via a binary interaction.\n\niii) \\ \\ \\ \\ Must be capable of performing two-bit logic operations, i.e.\nC-NOT gate.\n\niv) \\ \\ \\ The speed of operation must satisfy the Preskill criterion, i.e. the\ncapability must exist for the execution of at \\qquad\\qquad\\ \\ \\ least 104\ndistinct operations within the quantum decoherence time.\n\nv) \\ \\ \\ \\ \\ Arbitrary initial state preparation must be clearly executable.\n\nWe present our generic scheme for realization of a quantum computer in the\nnext section. In section IV we discuss the fulfillment of the above criteria\nin our proposed scheme.\n\n\\section{III. Realization of a Universal Quantum Computer}\n\nOur proposed scheme is similar to the seminal proposal of Kane[15], but does\nnot depend upon electronic charge migration for read-in and read-out, nor wave\nfunction displacement to distinguish nuclear spin qubits. Our scheme, that we\nfeel offers a viable alternative, is represented in fig.2.%\n\n\\par\\noindent Fig. 2. Schematic representation of distributed qubit array as discussed in\nthe text. Atoms C and A are embedded in the Si wave guide as shown. The Stark\ngates are depicted above atoms C external to the guide, and the bias gates,\nlocated on the top and edge of the guide are also shown.\n\n\\bigskip\n\nAs depicted in fig. 2 the two-qubit system is constituted by the unpaired\nsingle electron spin of atom C, and the nuclear spin of atom A. The\nspecification of the atomic constituents is that atom C be composed of a\nsingle loosely bound unpaired electron and isotope of nuclear spin $I_{C}=0$,\nwhereas, atom A is composed of isotope with $I_{A}=1\/2$ and no unpaired\nelectrons. The configuration of atoms A and C, shown in fig. 2, is embedded in\nan optical waveguide composed of pure Si of isotope $I_{SI}=0$. The substrate\nalso consists of Si with $I_{SI}=0$ to avoid background interference. The\nelectron spin of atom C is coupled to the nuclear spin of atom A by electronic\nwavefunction overlap as shown in fig 2. The coupling is via Fermi contact\ninteraction[18] given generically for isotropic transferred hyperfine\ninteraction by\n\\begin{equation}\nA_{s}=\\frac{8\\pi\\beta\\hbar\\beta_{n}}{3S}\\left[  a_{1s}^{2}\\left|  \\psi\n_{1s}\\left(  0\\right)  \\right|  ^{2}+a_{2s}^{2}\\left|  \\psi_{2s}\\left(\n0\\right)  \\right|  ^{2}+2a_{1s}a_{2s}\\left|  \\psi_{1s}\\left(  0\\right)\n\\right|  \\left|  \\psi_{2s}\\left(  0\\right)  \\right|  \\right]\n\\end{equation}\n\nwhere $\\beta$ is the Bohr magneton, $\\beta_{n}$ the nuclear magneton, and the\nwave functions $\\psi_{ks}(0)$ , k = 1, 2, are the 1s and 2s orbitals of atom A\nevaluated at its nucleus, and the a2s=are overlap integrals coupling the\nelectron wave function of atom C with the 1s and 2s inner shell wave functions\nof atom A. Thus, the unpaired electron spin of atom C is coupled with the\nnuclear spin of atom A by contact interaction mediated by the inner shell s -\nstate orbitals of atom A.\n\nThe interaction is controlled by laser field induced electronic excitation of\natom C from its ground s - state (no overlap interaction with atom A) to an\nelectronic excited state with a p -, or d- orbital, with strong overlap and\nsuperhyperfine interaction with one or both of its nearest neighbor atoms A.\nEnhanced, as well as suppressed, interaction is controlled by gates mounted\nexternal to the waveguide, that produce a positive or negative bias. Outer\nshell, loosely bound, electronic wave functions may be as extensive as 20nm in\na solid of sufficiently high dielectric constant, so nearest neighbor C- A\natom configurations must be within this range. Selective excitation is\nprovided by a gate that sandwiches atom C used to Stark shift atom C into and\nout of resonance with the cw laser field in the waveguide, thus enabling $\\pi$\n- excitation \/ deexcitation. The nuclear spin flips are controlled by\nmicrowave induced simultaneous electron - nuclear spin flips mediated by the\ntransferred hyperfine interaction.\n\nThe elementary, time independent, spin Hamiltonian for this system is,\n\\begin{equation}\nH=g\\beta H_{0}S_{z}+S.A.I.-g_{n}\\beta_{n}H_{0}I_{z}%\n\\end{equation}\n\nwhere $g$ and $g_{n}$ are the electron and nuclear gyromagnetic ratios. Here,\nwe have omitted the weak transverse time dependent part of the Hamiltonian for\nsimplicity. The first and third terms are the Zeeman terms for the electronic\nand nuclear spins, respectively, coupling to the constant magnetic field,\n$H_{0}$. The second term expresses the transferred hyperfine coupling between\nthe electron spin S of atom C and the nuclear spin I of atom A. These are\ncoupled via the superhyperfine tensor A, that for isotropic homogeneous\ninteraction is diagonal in the representation where $S_{z}$ and $I_{z}$ are\naligned with $H_{0}$, and is given in magnitude by (6). For simplicity we\nassume isotropy in what follows unless stated otherwise. To first order, and\nwith the condition\n\\begin{equation}\ng\\beta>A_{s}>>g_{n}\\beta_{n}%\n\\end{equation}\nthe eigen energies of (7) with associated quantum numbers, $m_{s}$ and $m_{I}\n$ for electron spin and nuclear spin in the representation in which the Zeeman\nterms are diagonal are given by,\n\\begin{equation}\nE(m_{s},m_{I})=g\\beta H_{0}m_{s}+A_{s}m_{s}m_{I}%\n\\end{equation}%\n\n\\par\\noindent Fig. 3. Zeeman and superhyperfine energy level splitting and microwave\nfrequency transition for simultaneous electron-nuclear spin flips.\n\n\\bigskip\n\nThe corresponding Zeeman energy level diagram is shown in fig. 3. It is seen\nthat simultaneous electron-nuclear spin flips can be induced by an applied\nmicrowave field of frequency $\\omega$, according to the selection rule $\\Delta\nm_{F}=0$, where F = S + I is the total spin angular momentum. Thus the\n$E_{4}\\rightarrow E_{2}$ transition, corresponding to simultaneous electron -\nnuclear spin flips, is allowed under the proper selection rules, $\\Delta\nm_{F}=\\pm1,0$, but the $E_{3}\\rightarrow E_{1}$ transition is forbidden.\nExplicitly, for an isotropic tensor A, i.e. diagonal in this representation,\ntransitions are induced according to the second term in (7) by the operators\n$S^{\\pm}I^{\\mp}$, consistent with the selection rule for simultaneous electron\n- nuclear spin transitions indicated in fig. 3. However, provided the symmetry\nis such that A is not diagonal in this representation, or if there is an\nelement of symmetry breaking, then the superhyperfine interaction described by\nthe second term in (7) can contain contributions from mixed terms $S_{x}S_{y}%\n$, $S_{y}S_{x}$ that can cause the transition $E_{3}\\rightarrow E_{4}$. Such\nconditions can, in principle, be regulated by local impurities or the natural\nlocal symmetry environment.\n\nWith these conditions, simultaneous electron - nuclear spin flips can be\ncontrolled by externally applied microwave pulses mediated by transferred\nhyperfine interactions. The interaction can be regulated, i.e. turned on -\noff, by laser pulse electronic excitation \/ deexcitation between the ground s\n- state of atom C (no electronic wave function overlap with the nuclear spin\nof atom A) and an electronic excited state of nonzero orbital angular momentum\n(strong overlap with the nucleus of atom A) as indicated in fig. 4. This\nintroduces also the option of storing information in the nuclear spin system.\nMost important, however, is the single qubit selectivity facilitated by\nselective laser pulse excitation using the Stark shift gates discussed\nearlier. In the next Section we discuss some useful specific performance\naspects of the model.%\n\n\n\\par\\noindent Fig. 4. Zeeman and superhyperfine (shf) energy level splittings and laser and microwave\n\npulse sequence for shf interaction control and simultaneous electron- nuclear\nspin flips.\n\n\\bigskip\n\n\\section{IV. Quantum Calculational Features}\n\nWith reference to figs. 3-4 we identify the calculational basis,\n\\begin{equation}\n\\left|  \\downarrow\\right\\rangle _{e}\\left|  \\uparrow\\right\\rangle _{n},\\left|\n\\uparrow\\right\\rangle _{e}\\left|  \\downarrow\\right\\rangle _{n},\\left|\n\\downarrow\\right\\rangle _{e}\\left|  \\downarrow\\right\\rangle _{n},\\left|\n\\uparrow\\right\\rangle _{e}\\left|  \\uparrow\\right\\rangle _{n}%\n\\end{equation}\n\nwhere $\\left|  {}\\right\\rangle _{e}$ and $\\left|  {}\\right\\rangle _{n}$\nrepresent electron, and nuclear spin states, respectively. The Zeeman split\nelectron spin manifold depicted in fig. 3 can be driven by a microwave pulse\nof frequency $\\omega$ into a coherent superposition described by a single spin\nrotation unitary transformation, U(t),\n\\begin{equation}\nU(t)\\left|  \\downarrow\\right\\rangle _{e}\\left|  \\uparrow\\right\\rangle\n_{n}=\\alpha(t)\\left|  \\downarrow\\right\\rangle _{e}\\left|  \\uparrow\n\\right\\rangle _{n}+\\beta\\left|  \\uparrow\\right\\rangle _{e}\\left|\n\\downarrow\\right\\rangle _{n}%\n\\end{equation}\nwhere $|\\alpha|^{2}+|\\beta\\}^{2}=1.$ The operation (10a), together with\n\\begin{equation}\nU(t)\\left|  \\downarrow\\right\\rangle _{e}\\left|  \\downarrow\\right\\rangle\n_{n}=\\alpha(t)\\left|  \\downarrow\\right\\rangle _{e}\\left|  \\downarrow\n\\right\\rangle _{n}+\\beta\\left|  \\uparrow\\right\\rangle _{e}\\left|\n\\downarrow\\right\\rangle _{n}%\n\\end{equation}\nform a quantum controlled NOT-gate (C-NOT-gate). The target bit $\\left|\n{}\\right\\rangle _{n}$ is flipped contingent upon the state of the control bit\n$\\left|  {}\\right\\rangle _{e}$ .The externally applied coherent microwave\npulse induced unitary transformation drives the electron-nuclear spin system\nground and excited states into an entangled pair via the transferred hyperfine\ninteraction, and the entanglement is manifestly nonlocal. Since two-bit gates\nare universal in quantum computing [19], the identification of the set of\nqubits and the calculational basis, together with the demonstrated C-NOT gate,\n(12), fulfills the requirement for universal quantum computation, i.e., any\nalgorithm is executable in the system provided the system is scalable.\n\nWorth noting here is that if the target state is initially $\\left|\n\\uparrow\\right\\rangle _{n}$ , then the unitary transformation is to the\ntwo-bit sub-manifold, $\\{\\left|  \\downarrow\\right\\rangle _{e}\\left|\n\\uparrow\\right\\rangle _{n},\\left|  \\uparrow\\right\\rangle _{e}\\left|\n\\downarrow\\right\\rangle _{n}\\}$, (11); whereas, given the target state\n$\\left|  \\downarrow\\right\\rangle _{n}$, the same transformation results in the\ntransformation to the submanifold, $\\{\\left|  \\downarrow\\right\\rangle\n_{e}\\left|  \\downarrow\\right\\rangle _{n},\\left|  \\uparrow\\right\\rangle\n_{e}\\left|  \\uparrow\\right\\rangle _{n}\\}$. These alternatives are isomorphic\nto the oracle of the Deutsch-Jozsa Promise Algorithm [20], and the initial\nnuclear spin target state can be determined by application of that algorithm\nin a single unitary operation.\n\nA different calculational basis can be identified, independently, with respect\nto the nuclear spins. Selective rf field induced nuclear spin flips can be\ninduced by laser pulse controlled transferred hyperfine interaction. The\nselective transferred hyperfine interaction can be used to induce significant\nnuclear spin level shifts. This interesting alternative and option will not be\npursued further here, but will be treated elsewhere. We feel that we have\npresented here the simplest approach to quantum computation with respect to\nthis particular scheme.\n\nSo far, we have discussed results for selective coupling between an atom C,\nwith respect to its electronic component, and a single atom A, with its\nnuclear component. Now we focus attention on electronic wave function mediated\ncoupling of two adjacent nuclear spins, fig. 2. For this case, instead of (7)\nwe have the Hamiltonian\n\\begin{equation}\nH=g\\beta H_{0}S_{z}+A_{k}S.I_{k}+A_{k+1}S.I_{k+1}-g_{nk}\\beta H_{0}I_{z}%\n^{(k)}-g_{nk+1}\\beta_{n}H_{0}I_{z}^{(k+1)}%\n\\end{equation}\nand to first order and within the approximation (8), the associated energy\nlevels are given by\n\\begin{equation}\nE(m_{s},m_{Ik},m_{Ik+1})=g\\beta H_{0}m_{s}+A_{s}m_{s}(m_{I}+m_{Ik+1})\n\\end{equation}\nHere, k labels the location of atom $A_{k}$, and $m_{Ik}$, and $m_{Ik+1}$ are\nthe spin quantum numbers for nuclei of atoms $A_{k}$ and $A_{k+1}%\n$respectively. The corresponding energy level diagram and transitions are\ndisplayed in fig. 5.%\n\n\n\\par\\noindent Fig.5 Zeeman and superhyperfine energy level splittings and microwave field\ninduced transitions for electron spin mediated nuclear spin flips (see fig.\n2). Here, the left-hand arrow corresponds to the nuclear spin of the kth atom\nof type A whereas the right-hand arrow corresponds to the nuclear spin of the\nk+1 atom of type A.\n\n\\bigskip\n\n\\bigskip\n\nAs an example, consider the $\\Delta m_{I}=-1$ microwave excitation from the\nground initial state\n\\begin{equation}\n\\left|  \\psi_{l}\\right\\rangle =\\left|  \\downarrow\\right\\rangle _{e}\\left|\n\\uparrow\\uparrow\\right\\rangle _{n}%\n\\end{equation}\nto the state\n\\begin{equation}\nU_{1}(t)\\left|  \\downarrow\\right\\rangle _{e}\\left|  \\uparrow\\uparrow\n\\right\\rangle _{n}=\\left|  \\downarrow\\right\\rangle _{e}\\left|  \\uparrow\n\\downarrow+\\downarrow\\uparrow\\right\\rangle _{n}%\n\\end{equation}\nCorresponding to the transition $E_{6}\\longrightarrow E_{2}$ , fig.5. Here,\nthe left arrow in $\\left|  {}\\right\\rangle _{n}$ always refers to atom $A_{k}$\n, and the right arrow refers to atom $A_{k+1}$ nuclear spin. This is followed\nby the $\\Delta m_{I}=0$ microwave pulse deexcitation,\n\\begin{equation}\nU_{1}(t)\\left|  \\uparrow\\right\\rangle _{e}\\left|  \\uparrow\\downarrow\n+\\downarrow\\uparrow\\right\\rangle _{n}=\\left|  \\downarrow\\right\\rangle\n_{e}\\left|  \\uparrow\\downarrow+\\downarrow\\uparrow\\right\\rangle _{n}%\n\\end{equation}\nCorresponding to the transformation $E_{2}\\longrightarrow E_{5}$ .\nSubsequently, this is followed by laser pulse induced adiabatic electron\nspin-nuclear spin decoupling\n\\begin{equation}\n\\left|  \\downarrow\\right\\rangle _{e}\\left|  \\uparrow\\downarrow+\\downarrow\n\\uparrow\\right\\rangle _{n}\\longrightarrow\\left|  \\uparrow\\downarrow\n+\\downarrow\\uparrow\\right\\rangle _{n}%\n\\end{equation}%\n\\begin{equation}\nA_{s}\\longrightarrow0\n\\end{equation}\nthus storing entangled state information in the nuclear spin system,\n\\begin{equation}\n\\left|  \\psi_{f}\\right\\rangle _{n}=\\left|  \\uparrow\\downarrow+\\downarrow\n\\uparrow\\right\\rangle _{n}%\n\\end{equation}\n\nNow, suppose the superhyperfine coupling is again induced adiabatically via\nlaserpulse electronic excitation, except that the overlap is enhanced with\nrespect to the left- hand atom $A_{n}$, and at the same time suppressed with\nregard to the right hand atom, $A_{n+1}$ by means of the outer gate elements,\nfig. 3. The electron spin of atom C is now coupled to the nuclear spin of atom\n$A_{n}$ only. This is represented by\n\\begin{align}\n\\left|  \\uparrow\\downarrow+\\downarrow\\uparrow\\right\\rangle _{n}  &\n\\longrightarrow\\left|  \\downarrow\\right\\rangle _{e}\\left|  \\uparrow\n\\downarrow+\\downarrow\\uparrow\\right\\rangle _{n}\\\\\nA_{s}  & \\neq0\n\\end{align}\nThen, microwave excitation from the initial state (18) gives\n\\begin{equation}\nU_{1}(t)\\left|  \\downarrow\\right\\rangle _{e}\\left|  \\uparrow\\downarrow\n+\\downarrow\\uparrow\\right\\rangle _{n}=\\left|  \\uparrow\\right\\rangle\n_{e}\\left|  \\uparrow\\downarrow+\\downarrow\\uparrow\\right\\rangle _{n}%\n\\end{equation}\nThen, adiabatic laser pulse induced electron-spin, nuclear-spin decoupling\nyields the entangled nuclear spin state,\n\\begin{equation}\n\\left|  \\psi_{f}\\right\\rangle _{n}=\\left|  \\uparrow\\uparrow+\\downarrow\n\\downarrow\\right\\rangle _{n}%\n\\end{equation}\nThus, we have manipulated the system to induce two distinct entangled pairs of\nspin states, (20) and (24), and this information can be stored in the nuclear\nspin system. In a similar manner we can produce a uniform superposition of\n(20) and (24).\n\n\\section{V. Universality}\n\n\\bigskip\n\nWe are now in position to discuss the universality of our scheme in relation\nto the criteria of Section II.: i) We have identified a distinct set of qubits\nand a computational basis, (10), fig.2; ii)The qubits are distinct,\ndistributed, and individually addressable, fig. 3; iii) The coupling of\nelectron and nuclear spins is via transferred hyperfine interaction, (5-6),\nfig. 3; iv) We have demonstrated a C- NOT gate, (11,12); v) The speed of\noperation is governed by the rate at which an externally applied microwave\nfield can cause simultaneous electron-nuclear spin flips. This rate is limited\nby the strength of the transferred hyperfine interaction, As 10KHz - 100MHz.\nThe nuclear spin flip relaxation times in Si:31P are measured to be within the\n1-10 hour range at low temperatures [21], whereas the direct electron spin\nrelaxation is found to be on the order of one hour. The electron spin\nresonance (ESR) line width for 106 Phosphorus ions \/ cm3 in Si was observed to\nbe 1kHz using spin-echo techniques [22], much narrower than the ESR frequency\non the order 10 GHz. Thus, Preskill's criterion [23] is well satisfied for the\nconfiguration of our QC scheme; vi) We demonstrated selective single qubit\naddressability in Section IV. Initial state preparation can be established\neither globally by the usual techniques of electron-nuclear spin resonance,\nNMR, and ESR, or by selective single bit preparation using laser pulse\nselective electronic excitation and subsequent microwave pulse initial state\npreparation. There are a variety of combinations of techniques that can be\nused; however, the selective single qubit preparation is sufficient to\nguarantee arbitrary initial state preparation capability; vii) Input\ninformation can be imparted selectively to either the nuclear or electronic\nspin systems, or both, by microwave pulse excitation following selective laser\npulse electronic excitation. Output information can be rendered efficiently\nvia optical fluorescence from electronic excited states. Output information is\ntherefore imparted to the excited state electronic spin system. The\nfluorescence yield conveys all information involving the electron-spin\nnuclear-spin system. The specifics of optical fluorescence and detection for\nthis system will be the subject of another publication [24]; vii) scalability\nwas clearly identified in the previous Section, (15-24).\n\n\\section{\\bigskip VI. Summary and Conclusions}\n\n\\bigskip\n\nWe have demonstrated a scheme for quantum computing and shown that it\nsatisfies the irreducible set of criteria for universality. Our scheme has\ndefinitive advantages over other schemes involving solid state systems,\nnotably that of Kane [15], and Yablonovitch [25]. These schemes, each, involve\nelectron charge transfer and readin \/ readout requiring single electron charge\nmeasurements. Our method is more robust, depending upon microwave pulse\nexcitation control for reading and sensitive optical fluorescence and single\nphoton detection for readout.\n\nFabrication technology requirements for the scheme proposed here is currently\nbeyond the state-of-the-art, but this current impasse is expected to be\nalleviated within a few years due to the rapid progress in Nanoscience of\nmaterials and fabrication, especially in regard to silicone fabrication\ntechnology. Another concern involves the specific choice of atomic species to\nsatisfy the conditions required for the qubit design. This requires\nsimultaneous materials study and optimization analysis. The spectroscopy\ntechniques required in this scheme provide a particular challenge. The NMR,\nESR, and double resonance requirements correspond to well developed\ntechniques; however, this combined with integrated laser pulse excitation adds\na third component to constitute a triple resonance spectroscopy. This may lead\nto the cultivation of an interesting and useful spin-off as a novel method in spectroscopy.\n\nIt is felt that the proposed scheme combines several novel ideas to provide a\nrealistic multi-purpose method of exploiting quantum parallelism,\nentanglement, and interference. We anticipate that the scheme and analysis\nexpressed here can serve as a template for advancement toward realization of a\npractical universal quantum computer.\n\n\\bigskip\n\n\\section{References}\n\n\\bigskip\n\n1. Feynman, R. P., 1982, Int. J. Theor. Phys.21, 467.\n\n2. Deutsch, D., 1985, Proc. Roy. Soc. London, Ser. A 400, 96.\n\n3. Shor, P., 1994, Proc. 35th Annual Symp. On Foundations of Computer Science,\nLos Alamitas, CA, IEEE \\qquad\\qquad\\qquad\\qquad\\ \\ \\ \\ Press, 124.\n\n4. Ekert, A., and Jozsa, R., 1996, Rev. Mod. Phys. 68, 733.\n\n5. Grover, L. K., 1997, Phys. Rev. Lett. 78, 325.\n\n6. Turchette, Q. A., Hood, C. J., Lange, W., Mabuchi, H., and Kimble, H. J.,\n1995, Phys. Rev. Lett. 75, 4710.\n\n7. Cirac, J. I., and Zoller, P., 1995, Phys. Rev. Lett. 74, 4091.\n\n8. Monroe, C., Meekhof, D. M., King, B. E., Itono, W. M., ans Wineland, D. J.,\n1995, Phys. Rev. Lett. 75, 4714.\n\n9. Gershenfeld, N. A., and Chuang, I. L., 1997, Science 275, 350.\n\n10. Chuang, I. L., Gershenfeld, N. A., and Kubinec, M., 1998, Phys Rev. Lett.\n80, 3408.\n\n11. Bowden, C. M., Dowling, J. P., and Hotaling, S. P., 1997, Proc. SPIE 1111\nAnnual International Symposium on Aerospace \/ Defence, Sensing, Simulations,\nand Control, Orland, FL, April.\n\n12. Bowden, C. M., and Miller, J. E., 1967, Phys. Rev. Lett. 19, 4.\n\n13. Marshall, W., and Stuart, R., 1961, Phys. Rev. 123, 2048.\n\n14. DiVincenzo, D. P., 1998, Nature 393, 113.\n\n15. Kane, B. E., 1998, Nature 393, 133.\n\n16. Cleve, R., Ekert, A., Macchiavello, C., and Mosca, M., 1997, quant-ph\/9708016.\n\n17. Scully, M. O., and Zubairy, M. S., 1997, Quantum Optics (Cambridge), Ch.4, 17.\n\n18. Slichter, C. P., 1963, (Harper \\& Row), Ch. 7.\n\n19. DiVincenzo, D. P., 1995, Phys. Rev. A 51, 1015.\n\n20. Deutsch, D., and Jozsa, R., 1992, Proc. Roy. Soc. London A 439, 553.\n\n21. Feher, G., and Gere, E. A., 1959, Phys. Rev. 114, 1245.\n\n22. Chiba, M. and Hirai, A., 1972, J. Phys. Soc. Japan 33, 730.\n\n23. Preskill, J., 1998, Royal Soc. London, A 454, 385.\n\n24. Bowden, C. M., Pethel, S. D., and Bloemer, M. J., in preparation.\n\n25. Yablonovitch, E., private communication.\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Preliminaries}\nWe begin with the definition of the process that we are going to study. It is a two component Markov process $(X_t^\\varepsilon,I_t^\\varepsilon)$ to which we refer in all the work with \\textit{\"molecular motors model\"} or \\textit{\"motor proteins model\"}.\n\\begin{definition}[Molecular motors]\\label{contmodel}\nGiven an integer $J$, we consider the setting $E = {{\\mathbb R}}^d \\times \\{1,\\dots,J\\}$. For all $i,j$ in $\\{1,\\dots,J\\}$, let $r_{ij} \\in C^\\infty(\\mathbb{R}^d\\times {{\\mathbb R}}^d; [0,\\infty))$ denote nonnegative smooth maps, $\\psi^i \\in C^\\infty({{\\mathbb R}}^d\\times{{\\mathbb R}}^d)$ a smooth and $\\nabla \\psi^i$ its gradient with respect to $x$. We suppose that $\\psi$ grows at most linearly in the first component and is periodic in the second one. Finally, given the following operator\n\\begin{equation}\n\\widetilde{A}_\\varepsilon f(x,i)\n:=-\\nabla \\psi^i(\\varepsilon x,x) \\cdot \\nabla_x f(\\cdot,i)(x)\n+\n \\frac{1}{2} \\Delta_x f(\\cdot,i)(x)\n+\n\\sum_{j = 1}^J r_{ij}(\\varepsilon x,x) \n\\left[\nf(x,j) - f(x,i)\n\\right],\n\\label{eq:intro:MP_with_switching_L_varepsilon}\n\\end{equation} \nwe define the $E$-valued Markov process $(X^\\varepsilon_t, I^\\varepsilon_t)|_{t \\geq 0}$ as the solution to the martingale problem  corresponding to $\\widetilde{A}_\\varepsilon$. More precisely, $(X^\\varepsilon_t, I^\\varepsilon_t)$ is such that for all $f\\in D(\\widetilde{A}_\\varepsilon)$,\n\\begin{equation}\nf(X^\\varepsilon(t), I^\\varepsilon(t))- f(X^\\varepsilon(0),I^\\varepsilon(0)) - \\int_0^t \\widetilde{A}_\\varepsilon f(X^\\varepsilon(s),I^\\varepsilon(s))\\, ds\n\\end{equation}\nis a martingale.\n\n\n\n\\end{definition}\n\n\n\\begin{remark}\nIn our case $r_{ij}$ is regular enough that the martingale problem associated to $A_\\varepsilon$ is well posed (see \\cite{EithKurtz} and \\cite{StroVar}).\n\\end{remark}\n\\begin{remark}\nIt is straightforward to see that the above defined process solves the stochastic differential equation \\eqref{sde} given in the introduction.\n\\end{remark}\nWe firstly study the above particular model for which we prove the large deviations property. Then, using this model, we lead to a theorem for a general class of processes called \\textit{Switching Markov process} (see Section \\ref{general}).\n\nAs mentioned in the introduction, we will work with the rescaled process $(Y_{t}^\\varepsilon, \\bar{I}_t^\\varepsilon)=$\\\\ $=\\left(\\varepsilon X_{t\/\\varepsilon}^\\varepsilon, I_{t\/\\varepsilon}^{\\varepsilon}\\right)$. \nThen, by the chain rule, the generator becomes\n\\begin{equation} \\label{gen}\nA_\\varepsilon f(x,i) = - \\nabla \\psi^i \\left(x, \\frac{x}{\\varepsilon}\\right) \\nabla_x f(\\cdot,i)(x) + \\frac{\\varepsilon}{2} \\Delta_x f(\\cdot,i)(x) + \\frac{1}{\\varepsilon}\\sum_{j=1}^J r_{ij}\\left(x, \\frac{x}{\\varepsilon}\\right)[f(x,j)-f(x,i)]. \n\\end{equation}\nWe will assume in the main theorem that the matrix $(R_{ij}(x))_{ij} := (\\sup_{y \\in \\mathbb{R}^d}r_{ij}(x,y))_{ij}$ is irreducible. Here we give the rigorous definition.\n\n\\begin{definition}\\label{irreducible:def}\n We say that a matrix $A=(A_{ij}(x))_{ij\\in \\{1,\\dots,J\\},x\\in {{\\mathbb R}}^d}$ is irreducible if there is no decomposition of $\\{1,\\dots,J\\}$ into two disjoint sets $\\mathcal{J}_1$ and $\\mathcal{J}_2$ such that $A_{ij} = 0$ on $\\mathbb{R}^d$ whenever $i \\in \\mathcal{J}_1$ and $j \\in \\mathcal{J}_2$.\n\\end{definition}\n\nThe main goal of this work is to prove that the spatial component $Y^\\varepsilon$ of the above Markov process verifies the large deviation principle. Here we give the main definitions which are written down in terms of a general Polish space $\\mathcal{X}$ but will later on be applied for e.g. $\\mathcal{X} = \\mathbb{R}^d$ or $\\mathcal{X} = C_{\\mathbb{R}^d}([0,\\infty))$.\n\n\\begin{definition}\\label{def:LDP}Let $\\{X_\\varepsilon\\}_{\\varepsilon>0}$ be a sequence of random variables on a Polish space $\\rchi$. Given a function $I : \\rchi \\to [0,\\infty]$, we say that\n\\begin{enumerate}[(i)]\n\\item the function $I$ is a \\textit{good rate function} if the set $\\{x \\,|\\, I(x) \\leq c\\}$ is compact for every $c\\ge0$.\n\\item the sequence $\\{X_\\varepsilon\\}_{\\varepsilon>0}$ satisfies the \\textit{large deviation principle} with good rate function $I$ if for every closed set $A\\subseteq\\rchi$, we have\n\\begin{equation}\n\\limsup_{\\varepsilon\\to0} \\varepsilon\\log \\mathbb{P}[X_\\varepsilon \\in A] \\leq -\\inf_{x\\in A} I(x)\n\\end{equation}\nand, for every open set $U \\subseteq \\rchi$,\n\\begin{equation}\n\\liminf_{\\varepsilon\\to0} \\varepsilon\\log\\mathbb{P}[X_\\varepsilon \\in U] \\geq -\\inf_{x\\in U} I(x).\n\\end{equation}\n\\end{enumerate}\n\\end{definition}\nWe recall the definition of exponentially tightness and the compact containment condition, typical properties that come out in a large deviations context.\n\\begin{definition}[Exponential tightness]\nA sequence of probability measures $\\{P_\\varepsilon\\}$ on a Polish space $\\rchi$ is said to be \\textit{exponentially tight} if for each $a>0$, there exists a compact set $K_a \\subset \\rchi$ such that\n\\begin{equation}\n\\limsup_{\\varepsilon \\to 0} \\varepsilon \\log P_\\varepsilon(K_a^c) \\leq a.\n\\end{equation}\nIn the next we will consider for the above definitions the space $\\rchi=C_{{{\\mathbb R}}^d}[0,\\infty)$.\n\nA sequence $\\{X_\\varepsilon\\}$ of $E$-valued random variables is exponentially tight if the corresponding\nsequence of distributions is exponentially tight.\n\\end{definition}\n\\begin{definition}\nWe say that the processes $(Z_\\varepsilon(t))$ satisfy the \\textit{exponential compact containment condition} if for all $T >0$ and $a > 0$ there is a compact set $K= K(T,a) \\subseteq E$ such that \n\\begin{equation*}\n\\limsup_{\\varepsilon\\rightarrow 0} \\varepsilon\\log \\mathbb{P}\\left[Z_\\varepsilon(t) \\notin K \\text{ for some } t \\in [0,T] \\right] \\leq - a.\n\\end{equation*}\n\\end{definition}\n\n\\subsection{Statement of the main theorem}\nNow we state the main theorem in which we prove sufficient conditions for large deviation property for the spatial component of the switching process defined in Definition \\ref{contmodel}. \n\\begin{theorem}[Large deviation for the \"molecular motors model\"]\\label{maintheorem}\nLet $(X^\\varepsilon_t, I^\\varepsilon_t)$ be the Markov process of Definition \\ref{contmodel}. Suppose that the matrix $R_{ij} = (\\sup_{y \\in \\mathbb{R}^d}r_{ij}(y))_{ij}$ is irreducible. Denote $Y^\\varepsilon_t=\\varepsilon X^\\varepsilon_{t\/\\varepsilon}$ the rescaled process.\nSuppose further that at time zero, the family of random variables $\\{Y^\\varepsilon(0)\\}_{\\varepsilon > 0}$ satisfies a large deviation principle in $\\mathbb{R}^d$ with good rate function $\\mathcal{I}_0 :C_{{{\\mathbb R}}^d}[0,\\infty)  \\rightarrow [0,\\infty]$. Then, the spatial component $\\{Y_t^\\varepsilon\\}$ satisfies a large deviation principle in $C_{\\mathbb{R}^d}[0,\\infty)$ with good rate function $\\mathcal{I} : C_{{{\\mathbb R}}^d}[0,\\infty) \\to [0,\\infty]$ given in the integral form \n\\begin{equation}\n\\mathcal{I}(x)=\\begin{cases}\n\\mathcal{I}_0(x(0)) +\\int_0^\\infty\\mathcal{L} \\left(x(t),\\dot{x}(t)\\right) \\, dt &\\quad \\text{if } x\\in AC([0,\\infty); \\mathbb{R}^d),\\\\\n\\infty & \\quad \\text{else},\n\\end{cases}\n\\end{equation}\nwith $\\mathcal{L}(x,v)=\\sup_p\\{pv-{{\\mathcal H}}(x,p)\\}$ the Legendre transform of a Hamiltonian ${{\\mathcal H}}(x,p)$ given in variational form by\n\\begin{equation} \\label{eigen:repr}\n\\mathcal{H}(x,p)= \\sup_{\\mu\\in\\mathcal{P}(E^\\prime)}\\left[\\int_{E^\\prime} V_{x,p}(z) \\,d\\mu(z) - I_{x,p}(\\mu)\\right],\n\\end{equation}\nwhere $E^\\prime = \\mathbb{T}^d \\times \\{1, \\dots, J\\}$, \\begin{equation}\n\tV_{x,p} (y,i) = \\frac{1}{2} p^2 - p\\cdot \\nabla_x\\psi^i\\left(x,y\\right)\n\\end{equation}\nand the map $I_{x,p} : \\mathcal{P}(E^\\prime) \\rightarrow [0,\\infty]$ is the Donsker--Varadhan function, i.e.\n\\begin{equation}\nI_{x,p}(\\mu)= -\\inf_{\\varphi}\\int_{E^\\prime} e^{-\\varphi} L_{x,p} (e^{\\varphi}) \\, d\\mu,\n\\end{equation}\nwhere the infimum is taken over vectors of functions $\\varphi(\\cdot,i) \\in C^2(\\mathbb{T}^d)$, and $L_{x,p}$ is the operator defined by\n\\begin{equation}\\label{generator}\nL_{x,p} u(z,i) =\\frac{1}{2} \\Delta_z u(z,i) + (p-\\nabla_x \\psi^i(x,z))\\cdot \\nabla_z u(z,i)+\\sum_{j = 1}^J r_{ij}(x,z) \\left[u(z,j) - u(z,i)\\right].\n\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\n$E^\\prime$ captures the periodic behaviour and the internal state. In the homogenisation context described in the introduction, $E^\\prime$ is exactly what is being homogenised over while $L_{x,p}$ describes the dynamics on it.\n\\end{remark}\n\n\\subsection{Law of large numbers and speed of the limit process}\nThe following corollary characterises the limit process.\n\\begin{corollary}\\label{velocity:cor}\nConsider the same assumptions of Theorem \\ref{maintheorem} for the Markov process $(Y_t^\\varepsilon, I_t^\\varepsilon)$. Then, the spatial component converges almost surely to the path with velocity given by\n\\begin{equation}\n\\partial_t x = \\partial_p \\mathcal{H}(x,0) = - \\int_{E'} \\nabla_x \\psi^i (x,y) \\, d\\mu_x^*(y),\n\\end{equation}\nwith $\\mu_x^*$ the unique stationary measure of the operator $L_{x,0}$ given in \\eqref{generator}.\n\\end{corollary}\n\\begin{proof}\nBy Theorem A.2 of \\cite{PelSch}, the spatial component $Y_t^\\varepsilon$ converges almost surely to the set of minimizers of the rate function. More precisely, \n\\begin{equation}\nd(Y_t^\\varepsilon, \\{\\mathcal{I}=0\\})\\to 0 \\qquad \\text{a.s. as $\\varepsilon \\to 0$}\n\\end{equation}\nwhere $\\{\\mathcal{I} = 0\\}=\\{x\\in C_{{{\\mathbb R}}^d}[0,\\infty) \\, :\\, \\mathcal{I}(x)=0\\}$. We now prove that this set is actually a singleton and then characterise the unique element.\n\nWith this aim, note that by \\autocite[Theorem 23.5]{Ro70}, $v$ is a minimizer of $\\mathcal{L}$ if and only if $v \\in \\partial_p{{\\mathcal H}}(x,0)$. Moreover, by \\autocite[Thoerem 4.4.2]{HULe01},\n\\begin{equation}\n\\partial_p {{\\mathcal H}} (x,0)= co \\left\\{ \\bigcup_{\\mu\\in \\mathcal{P}(E^\\prime)} \\partial \\left[ \\int_{E^\\prime}V_{x,0}\\, d\\mu - I_{x,0}(\\mu)\\right]\\, \\text{for all $\\mu$ s.t.} \\, {{\\mathcal H}}(x,0)=  \\int_{E^\\prime}V_{x,0}\\, d\\mu - I_{x,0}(\\mu) \\right\\},\n\\end{equation}\nwhere with $co$ we refer to the convex hull of a set and $\\partial \\left[ \\int_{E^\\prime}V_{x,0}\\, d\\mu - I_{x,0}(\\mu)\\right]$ is the differential of the convex functions $\\int_{E^\\prime}V_{x,p}\\, d\\mu- I_{x,p}(\\mu)$ for $p=0$.\n\nWe know that ${{\\mathcal H}}(x,0)=0$ and $V_{x,0}(z)=0$ for all $z\\in E^\\prime$. Then, if $\\mu^*_x$ is the optimal measure for ${{\\mathcal H}}(x,0)$, we have that $$0={{\\mathcal H}}(x,0)= \\int_{E^\\prime} V_{x,0}(z) \\,d\\mu^*_x - I_{x,0}(\\mu^*_x) = I_{x,0}(\\mu^*_x).$$ We can conclude that the optimal $\\mu^*_x$ is the unique stationary measure of $L_{x,0}$ (see Proposition \\ref{prop:stationarymeasure} in the appendix for existence and uniqueness of $\\mu^*_x$). We thus find that $\\partial_p {{\\mathcal H}}(x,0)= \\left\\{\\frac{\\partial}{\\partial p}{{\\mathcal H}}(x,p)\\vert_{p=0}\\right\\}$ and hence $\\mathcal{I}(x)=0 \\iff \\partial_t x=\\frac{\\partial}{\\partial p}\\mathcal{H}(x,0)$ for almost all $t$\nand \n\\begin{align}\n\\partial_t x &=\\frac{\\partial}{\\partial p}\\mathcal{H}(x,0)= \\int_{E'} \\frac{\\partial V_{x,p}(z)}{\\partial p} \\biggl\\vert_{p=0} \\, d\\mu_x^*(z)\\\\\n&=-\\int_{E'} \\nabla_x \\psi(x,z) \\, d\\mu_x^*(z).\n\\end{align}\n\n\\end{proof}\n\n\n\\begin{remark}\nThe above corollary confirms the suggestion of Figure \\ref{limit} that, when there is no dependence on $x$ in the drift, the spatial component is converging to a path with constant speed. Indeed, for small $\\varepsilon$, $X_t^\\varepsilon$ tends to concentrate around a path with a constant velocity $v= \\partial_p{{\\mathcal H}}(0)$.\n\\end{remark}\n\n\\end{section}\n\\section{Connection with Hamilton-Jacobi equations and strategy of proof}\\label{method:sub}\nWe will now present a brief overview of the technical aspects of the Hamilton-Jacobi approach introduced by Feng and Kurtz \\cite{FK} to the path-space large deviations theory for Markov processes.\nIn the next we give an outline of the main steps of this argument.\n\nFeng and Kurtz in \\cite[Theorem 5.15]{FK} used a variation of the projective limit method (\\cite{DawGar}, \\cite{deAcosta}) to prove that the large deviations property can be obtained as a consequence of the large deviations of the finite dimensional distributions and the exponential tightness of the process. By Bryc's theorem and the Markov property, large deviations for the finite dimensional distributions follows by the convergence of the conditional \"moment generating functions\" that forms the semigroup \n\\begin{equation*} \\label{semigroup}\nV_\\varepsilon(t)f(x)= \\varepsilon \\log \\mathbb{E}\\left[e^{f(X_\\varepsilon(t))\/\\varepsilon}\\vert X(0)=x\\right]=\\varepsilon \\log \\int_E e^{f(y)\/\\varepsilon}\\mathbb{P}_\\varepsilon(t,x,dy),\n\\end{equation*}\nwith $\\mathbb{P}_\\varepsilon(t,x,dy)$ the transition probabilities of $X^\\varepsilon_t$. Note that $V_\\varepsilon(t)f = \\varepsilon \\log S_\\varepsilon(t) e^{f \/ \\varepsilon}$, where $S_\\varepsilon$ is the linear semigroup associated to the generator $A_\\varepsilon$.\nComputing $V_\\varepsilon$ and verifying its convergence is usually hard. In analogy to results for linear semigroups and their generators, the convergence of $V_\\varepsilon$ follows from the convergence of the \\textit{nonlinear generators} $H_\\varepsilon$. Formally applying the chain rule to $V_\\varepsilon(t)$ in terms of the linear semigroup $S_\\varepsilon(t)$ yields the following definition, that can be put on more rigorous grounds as exhibited in \\cite{FK,Kr21}.\n\n\\begin{definition}[Nonlinear generator]\\label{nonlineargenerators:def}\nLet $A_\\varepsilon$ the generator of a process $X^\\varepsilon_t$. The \\textit{nonlinear generator} of $X^\\varepsilon_t$ is the map defined in the domain $D(H_\\varepsilon)=\\left\\{f\\in C(E)\\, : \\, e^{f(\\cdot)\/\\varepsilon}\\in D(A_\\varepsilon)\\right\\}$ by\n\\begin{equation}\\label{eq:nonlinear generator}\nH_\\varepsilon f(x) = \\varepsilon e^{-f(x)\/\\varepsilon} A_\\varepsilon e^{f(x)\/\\varepsilon}.\n\\end{equation}\n\\end{definition}\nMore precisely, the problem comes down to two steps. First one needs to prove the convergence of the generators $H_\\varepsilon \\to H$ in a suitable sense. \nThen, one has to show that the limiting semigroup generates a semigroup. The sufficient conditions are, using Crandall--Liggett Theorem \\cite{CraLigg}, the \\textit{range condition} and the \\textit{dissipativity} property.\n\\begin{definition}[Range condition]\nLet $E$ be an arbitrary metric space and $H: D(H) \\subseteq C_b(E) \\to C_b(E)$ a nonlinear operator. We say that $H$ satisfies the range condition if\n\\begin{equation}\n\\exists \\lambda_0>0 \\, : \\, D(H) \\subset \\overline{\\mathcal{R}(I-\\lambda H)} \\qquad \\text{for all $0<\\lambda <\\lambda _0$.}\n\\end{equation}\n\\end{definition}\n\n\\begin{definition}[Dissipative operator]\nWe say that an operator $(H,D(H))$ is dissipative if for all $\\lambda>0$,\n\\begin{equation*}\n\t\\vn{(f_1 - \\lambda Hf_1) - (f_2 - \\lambda H f_2)} \\geq \\vn{f_1 - f_2}\n\\end{equation*}\nfor all $f_1,f_2 \\in \\mathcal{D}(H)$.\n\\end{definition}\n\nThe range condition corresponds to existence of classical solutions for the equation $(1- \\lambda H)u=h$. Hence, we can conclude that, in order to prove large deviations, we need the convergence of the nonlinear generators to a dissipative operator $H$ such that the existence of classical solutions for $(1- \\lambda H)u=h$ holds. However, it is well known that the existence of classical solutions is a too strong condition to make this method work in most cases. As observed by Crandall -- Lions in \\cite{CraLio}, the use of \\textit{viscosity solutions} allows to overcome this problem. These weak solutions are defined in order to create an extension $\\tilde{H}$ of $H$ that automatically satisfies the range condition and that is still dissipative.\nBelow the definitions for both single and multi-valued operators.\n\\begin{definition}[Sub- and supersolutions for single valued operators]\nLet $H:\\mathcal{D}(H)\\subseteq C(E)\\rightarrow C(E)$ be a nonlinear operator. Then for $\\lambda>0$ and $h\\in C(E)$, define viscosity sub- and supersolutions of $(1 - \\lambda H) u = h$ as follows:\n\\begin{itemize}\n\\item[(i)] \nWe say that $u:E \\to {{\\mathbb R}}$ is a viscosity subsolution if it is bounded and upper semicontinuous and, for every $f \\in \\mathcal{D}(H)$, there exists a sequence $x_n \\in E$ such that\n\\begin{gather*}\n\\lim_{n \\uparrow \\infty} u(x_n) - f(x_n)  = \\sup_x u(x) - f(x), \\\\\n\\lim_{n \\uparrow \\infty} u(x_n) - \\lambda H f(x_n) - h(x_n) \\leq 0.\n\\end{gather*}\n\\item[(ii)]\nWe say that $v:E\\to{{\\mathbb R}}$ is a viscosity supersolution if it is bounded and lower semicontinuous and, for every $f \\in \\mathcal{D}(H)$, there exists a sequence $x_n \\in E$ such that\n\\begin{gather*}\n\\lim_{n \\uparrow \\infty} v(x_n) - f(x_n)  = \\inf_x v(x) - f(x), \\\\\n\\lim_{n \\uparrow \\infty} v(x_n) - \\lambda Hf(x_n) - h(x_n) \\geq 0.\n\\end{gather*}\n\n\\end{itemize} \nA function $u\\in C(E)$ is called a viscosity solution of $(1 - \\lambda H) u = h$ if it is both a viscosity sub- and supersolution.\n\n\\end{definition}\n\n\\begin{definition}[Sub - and supersolutions for multivalued operators]\n\\label{def:viscosity_solution}\nLet $H \\subseteq C(E) \\times C(E\\times E^\\prime)$ be a multivalued operator with domain $\\mathcal{D}(H) \\subseteq C(E)$. Then for $h\\in C(E)$ and $\\lambda>0$, define viscosity solutions of $(1 - \\lambda H) u = h$ as follows:\n\\begin{itemize}\n\\item[(i)] $u:E\\to {{\\mathbb R}}$ is a viscosity subsolution of $(1 - \\lambda H) u = h$ if it is bounded and upper semicontinuous and, for all $(f,g) \\in H$, \nthere exists a sequence $(x_n,z_n) \\in E\\times E'$ such that\n\\begin{gather*}\n\\lim_{n \\uparrow \\infty} u(x_n) - f(x_n)  = \\sup_x u(x) - f(x), \\\\\n\\limsup_{n \\uparrow \\infty} u(x_n) - \\lambda g(x_n,z_n) - h_1(x_n) \\leq 0.\n\\end{gather*}\n\\item[(ii)] $v:E\\to {{\\mathbb R}}$ is a viscosity supersolution of $(1 - \\lambda H) u = h$ if it is bounded and lower semicontinuous and, for all $(f,g) \\in H$, \nthere exists a sequence $(x_n,z_n) \\in E\\times E'$ such that\n\\begin{gather*}\n\\lim_{n \\uparrow \\infty} v(x_n) - f(x_n)  = \\inf_x v(x) - f(x), \\\\\n\\liminf_{n \\uparrow \\infty} v(x_n) - \\lambda g(x_n,z_n) - h_2(x_n) \\geq 0.\n\\end{gather*}\n\\end{itemize}\n\nA function $u\\in C(E)$ is called a viscosity solution of $(1 - \\lambda H) u = h$ if it is both a viscosity sub- and supersolution.\n\n\\end{definition}\n\\begin{remark} \\label{remark:existence of optimizers}\nConsider the definition of subsolutions. Suppose that the test function $f \\in \\mathcal{D}(H)$ has compact sublevel sets, then instead of working with a sequence $x_n$, there exists $x_0  \\in E$ such that\n\\begin{gather*}\nu(x_0) - f(x_0)  = \\sup_x u(x) - f(x), \\\\\n(x_0) - \\lambda H f(x_0) - h(x_0) \\leq 0.\n\\end{gather*}\nA similar simplification holds in the case of supersolutions.\n\\end{remark}\n\nIn the classical context, the range condition, combined with the dissipativity of the operator can be shown to imply unique solvability of the equation $u - \\lambda H u = h$. However, for viscosity solutions this argument does not work anymore. The main reason is that viscosity solutions are in general not in the domain of $H$. In order to address this issue, an option can be to suppose that the following \\textit{comparison principle} (implying uniqueness) holds.\n \\begin{definition}[Comparison Principle]\nWe say that a Hamilton-Jacobi equation $(1 - \\lambda H) u = h$  satisfies the comparison principle if for any viscosity subsolution $u$ and viscosity supersolution $v$, $u\\leq v$ holds on $E$.\n\nFor two operators $H_\\dagger, H_\\ddagger \\subseteq C(E) \\times C(E \\times E^\\prime)$, we say that the comparision principle holds if for any viscosity subsolution $u$ of $(1 - \\lambda H_\\dagger) u = h$ and viscosity supersolution $v$ of $(1 - \\lambda H_\\ddagger) u = h$, $u\\leq v$ holds on $E$.\n\\end{definition}\n\nThe theory above was made rigorous in \\cite{FK}, \\cite{Kr21}. We present the key result in our context and notations.\n\\begin{theorem}[Adaptation of 7.18 of \\cite{FK} to our context]\\label{FengKurtz:th}\nLet $(X_\\varepsilon,I_\\varepsilon)$ be a Markov process with generator $A_\\varepsilon$ and nonlinear generator $H_\\varepsilon$ as in Definition \\ref{nonlineargenerators:def}. Consider the semigroup $V_\\varepsilon$ defined in \\eqref{semigroup} and suppose the following \n\\begin{enumerate}[(i)]\n\\item $\\exists \\, H$ \\quad \\text{s.t.} \\quad $H_\\varepsilon$ converges to $H$ in the sense of Definition \\ref{def:convergence} ,\n\\item $\\forall \\lambda>0, h \\in C(E)$, the comparison principle holds for $(1-\\lambda H)u=h$,\n\\item $X_\\varepsilon$ is exponentially tight,\n\\item $X_\\varepsilon(0)$ satisfies large deviation principle with rate function $I_0$.\n\\end{enumerate}\nThen, there exists a unique viscosity solution $R(\\lambda)h$ to $(1 - \\lambda H ) f = h$ and a unique semigroup $V(t):C_b(E)\\to C_b(E)$ such that\n\\begin{enumerate}\n\\item $\\lim_{m\\to \\infty} R(\\frac{t}{m})^m f(x) = V(t)f(x)$ for every $f\\in C_b(E), t\\geq 0$ and every $x\\in E$,\n\\item $V_\\varepsilon$ converges to $V$ in the sense that for any sequence of functions $f_\\varepsilon \\in C(E)$ and $f\\in C(E)$, \n\\begin{equation}\n\\text{if \\quad $\\Vert f_\\varepsilon - f \\Vert_E \\xrightarrow[]{\\text{$\\varepsilon \\to 0$}} 0$ \\quad then \\quad $\\Vert V_\\varepsilon(t) f_\\varepsilon \\to V(t) f \\Vert_E \\xrightarrow[]{\\text{$\\varepsilon \\to 0$}}0$.}\n\\end{equation}\n\\end{enumerate}\nMoreover, $X_\\varepsilon$ satisfies the large deviation principle with rate function $I: \\mathbb{C}_E [0,\\infty) \\to [0,\\infty]$ given by\n\\begin{equation} \\label{rate:eq}\nI(x)= I_0(x(0)) + \\sup_{k\\in \\mathbb{N}} \\sup_{(t_1,\\dots,t_k)} \\sum_{i=1}^k I_{t_i-t_{i-1}}(x(t_i)\\vert x(t_{i-1}))\n\\end{equation}\nwith $I_t(z\\vert y)=\\sup_{f\\in C(E)}[f(z)-V(t)f(y)]$ and the supremum above is taken over all finite tuples $t_0=0 < t_1 < t_2 <\\dots< t_k$.\n\\end{theorem}\n\n\n\n\\section{Proof of the main theorem}\\label{proof}\n\n\nUsing the discussion of the previous section and Theorem \\ref{FengKurtz:th}, we can prove Theorem \\ref{maintheorem}.\n\\begin{proof}[Proof of Theorem \\ref{maintheorem}]\nWe claim the following five facts:\n\\begin{enumerate}\n\\item The nonlinear generators $H_\\varepsilon f=\\varepsilon e^{-f\/\\varepsilon} A_\\varepsilon e^{f\/\\varepsilon}$ of $X^\\varepsilon_t$ converge to a multivalued operator\n$H:=\\left\\{(f, H_{f,\\varphi}) \\, : \\,f \\in C^2(\\mathbb{R}^d), \\; H_{f, \\varphi} \\in C( \\mathbb{R}^d\\times E^\\prime) \\text{ and }\\varphi \\in C^2(E^\\prime)\\right\\}$,\n\\item there exists $\\tilde{\\varphi}$ such that $H_{f, \\tilde{\\varphi}}(x,z)=H_{\\tilde{\\varphi}}(x,p,z)=\\mathcal{H}(x,p)$ for all $z \\in E'$ and $p=\\nabla f$,\n\\item the comparison principle for $ (1-  \\lambda H)u= h$ holds,\n\\item $X_\\varepsilon$ verifies the exponential tightness property,\n\\item the rate function \\eqref{rate:eq} can be represented in the following integral form\n\\begin{equation}\n\\mathcal{I}(x)=\\mathcal{I}_0(x(0)) +\\int_0^\\infty\\mathcal{L} \\left(x(t),\\dot{x}(t)\\right) \\, dt\n\\end{equation} \nwith $\\mathcal{L}(x,v)=\\sup_{p \\in \\mathbb{R}^d}\\left[p \\cdot v  - \\mathcal{H}(x,p)\\right]$ is the Legendre transform of $\\mathcal{H}(x,p)$ in \\eqref{eigen:repr}.\n\\end{enumerate}\nWe prove the above claims respectively in Propositions \\ref{convergence:th}, \\ref{hamiltonian}, \\ref{theorem:comparison}, \\ref{proposition:exponential_compact_containment}, \\ref{ratefunction:th} in the following subsections.\nThen, once the above facts are proved, we can apply Theorem \\ref{FengKurtz:th} and the required large deviation property follows.\n\\end{proof}\n\\subsection{The convergence of generators and an eigenvalue problem}\\label{eigenvalueProb}\nThe first step of the proof of large deviations is based on operator convergence. Since the process and its limit do not live in the same space, we can not work with the usual definition. In the following, we introduce a new definition of limit for functions and multivalued operator on different spaces.\n\\begin{definition}\nLet $f_\\varepsilon \\in C({{\\mathbb R}}^d \\times \\{1, \\dots, J\\})$ and $f\\in C^2({{\\mathbb R}}^d)$. We say that $LIM f_n=f$ if \n\\begin{equation}\n\\Vert f_\\varepsilon - f \\circ \\eta_\\varepsilon \\Vert_{{{\\mathbb R}}^d\\times \\{1,\\dots,J\\}} = \\sup_{{{\\mathbb R}}^d\\times \\{1,\\dots,J\\}} \\vert f_\\varepsilon - f \\circ \\eta_\\varepsilon \\vert  \\to 0 \\quad \\text{as $\\varepsilon \\to 0$},\n\\end{equation}\nwhere $\\eta_\\varepsilon: {{\\mathbb R}}^d \\times \\{1,\\dots,J\\} \\to {{\\mathbb R}}^d$ is the projection \n$$ \\eta_\\varepsilon (x,i)=  x.$$\n\\end{definition}\n\\begin{definition}[extended limit of multivalued operators]\\label{def:convergence}\nLet $H_\\varepsilon\\subseteq C({{\\mathbb R}}^d\\times\\{1,\\dots,J\\})$. Define $ex - LIM H_\\varepsilon$ as the set\n\\begin{multline}\nex - LIM H_\\varepsilon=\\\\\n=\\bigg \\{ (f,H)\\in C^2({{\\mathbb R}}^d)\\times C({{\\mathbb R}}^d \\times \\mathbb T^d\\times \\{1,\\dots,J\\})\\lvert \\, \\exists f_\\varepsilon\\in D(H_\\varepsilon) \\,:\\, f=LIM f_\\varepsilon \\\\\n\\text{and} \\,  \\Vert H\\circ \\eta'_\\varepsilon - H_\\varepsilon f_\\varepsilon \\Vert_{{{\\mathbb R}}^d\\times\\{1,\\dots,J\\}}\\to 0 \\bigg \\},\n\\end{multline}\nwhere $\\eta'_\\varepsilon : {{\\mathbb R}}^d \\times \\{1,\\dots,J\\} \\to {{\\mathbb R}}^d \\times \\mathbb T^d \\times \\{1, \\dots, J\\}$ is the function $\\eta'_\\varepsilon(x,i)=\\left(x, \\left[\\frac{x}{\\varepsilon}\\right]_{{{\\mathbb Z}}^n}, i \\right)$.\n\\end{definition}\nThe following basic example gives the idea of the intuition behind the definitions above. \n\\textbf{Example:}\nGiven $H_\\varepsilon f (x,i) = f'(x) + \\varepsilon \\Delta f (x)$, for every $(f,H) \\in ex - LIM H_\\varepsilon$ we define \n$$f_\\varepsilon (x,i) = f(x) + \\varepsilon \\varphi \\left(\\frac{x}{\\varepsilon}, i \\right) \\qquad \\text{and} \\qquad H(x,y,i) = \\Delta \\varphi ^i (y)$$\nthen, the two definitions of limit hold.\n\\begin{proposition}[Convergence of nonlinear generator]\\label{convergence:th}\nLet $E={{\\mathbb R}}^d \\times \\{1,\\dots,J\\}$ and let $(X_t^\\varepsilon, I_t^\\varepsilon)$ be a Markov process from Definition \\ref{contmodel} with generator $A_\\varepsilon$ from \\eqref{gen} and let $H_\\varepsilon$ be the nonlinear generators defined in Definition \\ref{nonlineargenerators:def}. Then,  the multivalued operator $H\\subseteq C({{\\mathbb R}}^d) \\times C({{\\mathbb R}}^d \\times \\mathbb T^d\\times \\{1,\\dots, J\\})$ given by\n$$ H:= \\left \\{ (f, H_{f,\\varphi}) \\, : \\, f\\in C^2({{\\mathbb R}}^d), H_{f,\\varphi} \\in C({{\\mathbb R}}^d\\times E') \\, \\text{and}\\,  \\varphi \\in C^2(E')\\right\\},$$\nwhere the images $H_{f,\\varphi} : \\mathbb{R}^d \\times \\mathbb T^d \\times \\{1, \\dots, J\\} \\to \\mathbb{R}$ are\n\\begin{multline}\nH_{f,\\varphi}(x,y,i):= \\frac{1}{2}\\Delta_y\\varphi^i(y) + \\frac{1}{2} \\big|\\nabla f(x)+\\nabla_y\\varphi^i(y) \\big|^2 - \\nabla_x\\psi^i(x,y)(\\nabla f(x) + \\nabla_y\\varphi^i(y))\\\\ + \\sum_{j = 1}^J r_{ij}(x,y)\\left[e^{\\varphi(y, j)-\\varphi(y, i)}-1\\right],\n\\end{multline}\nis such that $H\\subseteq ex - LIM H_\\varepsilon$. Moreover, for all $\\varphi$ parametrising the images we have a map\n$\nH_\\varphi : \\mathbb{R}^d \\times \\mathbb{R}^d \\times \\mathbb{T}^d\\times \\{1,\\dots,J\\} \\to \\mathbb{R} $\nsuch that for all\n $f \\in \\mathcal{D}(H)$ and any $x \\in \\mathbb{R}^d$, the images $H_{f,\\varphi}$ of $H$ are given by\n$$\nH_{f,\\varphi}(x,z^\\prime) = H_\\varphi(x ,\\nabla f(x),z^\\prime), \\quad \\text{ for all } z^\\prime \\in \\mathbb{T}^d\\times \\{1,\\dots,J\\}. $$\n\\end{proposition}\n\\begin{proof}\nWe want to prove that $H_\\varepsilon$ converges to $H$ in terms of Definition \\ref{def:convergence}.\nWith this aim, note that, by the definitions of $A_\\varepsilon$ and $H_\\varepsilon$, we have \n\\begin{align}\nH_\\varepsilon f(x,i)= \\frac{\\varepsilon}{2} \\Delta_x f^i(x) +\\frac{1}{2} \\left| \\nabla_x f^i(x)\\right|^2 - &\\nabla\\psi^i\\left(x,\\frac{x}{\\varepsilon}\\right) \\nabla_x f^i(x)\\\\& + \\sum r_{ij} \\left(x,\\frac{x}{\\varepsilon}\\right)\\left(e^{(f(x,j)-f(x,i))\/\\varepsilon} -1\\right).\n\\end{align}\nChoosing functions $f_\\varepsilon(x,i)$ of the form\n$$\nf_\\varepsilon(x,i)= f(x)+\\varepsilon\\,\\varphi\\left(\\left[\\frac{x}{\\varepsilon}\\right]_{{{\\mathbb Z}}^n}, i\\right)=f\\circ \\eta_\\varepsilon (x,i)  +\\varepsilon\\,\\varphi\\left(\\left[\\frac{x}{\\varepsilon}\\right]_{{{\\mathbb Z}}^n}, i\\right),$$\n\t%\nwe obtain, \n\\begin{multline*}\nH_\\varepsilon( f_\\varepsilon)(x,i)= \\frac{\\varepsilon}{2}\\Delta f(x) + \\frac{1}{2}\\Delta_y\\varphi^i\\left(\\left[\\frac{x}{\\varepsilon}\\right]_{{{\\mathbb Z}}^n}\\right) + \\frac{1}{2}\\big|\\nabla f(x)+\\nabla_y\\varphi^i\\left(\\left[\\frac{x}{\\varepsilon}\\right]_{{{\\mathbb Z}}^n}\\right)\\big|^2 \\\\ -\n\\nabla\\psi^i\\left(x,x\/\\varepsilon \\right) \\left(\\nabla f(x)+\\nabla_y\\varphi^i\\left(\\left[\\frac{x}{\\varepsilon}\\right]_{{{\\mathbb Z}}^n}\\right)\\right)\n+\n\\sum_{j = 1}^J r_{ij}\\left(x, x\/\\varepsilon \\right)\n\\left[\ne^{\\varphi\\left(\\left[\\frac{x}{\\varepsilon}\\right]_{{{\\mathbb Z}}^n}, j\\right)-\\varphi\\left(\\left[\\frac{x}{\\varepsilon}\\right]_{{{\\mathbb Z}}^n}, i\\right)}-1\\right],\n\\end{multline*}\nwhere $\\nabla_y$ and $\\Delta_y$ denote the gradient and Laplacian with respect to the variable $y=x\/\\varepsilon$.\nWe can conclude that\n\\begin{equation}\n\\|f\\circ\\eta_\\varepsilon - f_\\varepsilon\\|_{E}=\\|f(x) -f_\\varepsilon (x, i) \\|_{E}=\\varepsilon\\|\\varphi\\|_{E} \\to 0 \\, \\text{as $\\varepsilon \\to 0$},\n\\end{equation}\nand\n\n\\begin{align*}\n\\|H_{f,\\varphi}\\circ\\eta_\\varepsilon^\\prime-H_\\varepsilon f_\\varepsilon\\|_{E} &=\n\\sup_{(x,i)\\in E}\\bigg{|} H_{f,\\varphi}\\left(x,\\left[\\frac{x}{\\varepsilon}\\right]_{{{\\mathbb Z}}^n},i\\right)-H_\\varepsilon f_\\varepsilon(x,i)\\bigg{|}\\\\\n\t&= \\frac{\\varepsilon}{2}\\;{\\sup_{(x,i)\\in E}|\\;\\Delta f(x)|}\n\t\t\\xrightarrow{\\varepsilon\\rightarrow 0}0.\n\\end{align*}\n\\end{proof}\n\\begin{remark}\\label{limit_repr}\t\nNote that for all $f \\in D(H)$ the image $H_\\varphi$ has the representation\n$$H_\\varphi(x,p,z)=e^{- \\varphi(z)}\\left[B_{x,p} +V_{x,p}+R_x\\right]e^\\varphi (z)$$\nwith $p=\\nabla f(x)$ and \n\\begin{align*}\n(B_{x,p} h)(y,i) &:= \\frac{1}{2} \\Delta_y h(y,i) + \\left( p - \\nabla_x \\psi^i(x,y)\\right)\\cdot \\nabla_y h(y,i) \\\\\n(V_{x,p} h)(y,i)&:= \\left(\\frac{1}{2} p^2 - p\\cdot \\nabla_x\\psi^i(x,y)\\right) h(y,i), \\\\\n(R_x\\,h)(y,i)&:=\\sum_{j = 1}^J r_{ij}(x,y) \\left[h(y,j) - h(y,i)\\right].\n\\end{align*}\n\\end{remark}\n\\begin{proposition}[Existence of an eigenvalue]\\label{eigenvalue:th}\nLet $E'=\\mathbb T^d \\times \\{1, \\dots, J\\}$ and let $H_\\varphi :{{\\mathbb R}}^d\\times{{\\mathbb R}}^d \\times E' \\to R$ the images of $H$ given in Proposition \\ref{convergence:th}. Then, for all $p\\in {{\\mathbb R}}^d$ there exists an eigenfunction $g_{x,p} \\in C^2(E^\\prime\\times \\{1,\\dots,J\\})$ with $g_{x,p}^i>0$ and an eigenvalue $\\lambda_{x,p}$ such that\n\\begin{equation}\n\\left[B_{x,p} +V_{x,p}+R_x\\right]g_{x,p}=\\lambda_{x,p}g_{x,p}.\n\\end{equation}\n\\end{proposition}\t\n\n\\begin{proof}\nWe want to solve the following eigenvalue problem\n\\begin{equation}\\label{problem}\n\\left[L_{x,p} +R_x\\right]g_{x,p} = \\lambda_{x,p} g_{x,p}\n\\end{equation}\nwhere $L_{x,p}$ is a diagonal matrix with $(L_{x,p})_{ii}=(B_{x,p})_{i}+(V_{x,p})_i$ and $(R_x)_{ij}=r_{ij}$ for $i\\neq j$ and $(R_x)_{ii}=\\sum_{j = 1}^J r_{ij}$.\\\\\nGuido Sweers showed (see \\cite{Sweers1992}) that there exists $\\gamma_{x,p}$ and $g_{x,p} >0$ such that \n\\begin{equation}\n\\left[-L_{x,p} -R_x\\right]g_{x,p} = \\gamma_{x,p} g_{x,p}\n\\end{equation}\nwhen $L_{x,p}$ is a diagonal matrix with $(L_{x,p})_{ii}$ of the type $-\\Delta + p\\cdot\\nabla +c$. Hence, in our case, the equality \\eqref{problem} is verified by taking $\\lambda_{x,p}=-\\gamma_{x,p}$.\n\\end{proof}\nIn the next proposition we prove that the images $H_{\\varphi}$ depend only on $x$ and $p$. \n\\begin{proposition}\\label{hamiltonian:prop}\nConsider the same setting of Proposition \\ref{eigenvalue:th} and let ${{\\mathcal H}}(x,p)$ be the constant depending on $p$ and $x$ given in \\eqref{eigen:repr}.\nThen, for all $x,p \\in {{\\mathbb R}}^d$ there exist a function $\\varphi_{x,p} \\in C^2(E')$ such that \n\\begin{equation}\\label{hamiltonian}\nH_{\\varphi_{x,p}}(x,p,z)= {{\\mathcal H}}(x,p) \\qquad \\text{for all $z \\in E'$}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{eigenvalue:th}, there exists a function $g_{x,p}$ and a constant $\\lambda_{x,p}$ that satisfy the eigenvalue problem for the operator $L_{x,p} +R_{x,p}$ defined in \\eqref{problem}. By the variational representation established by Densker and Varadhan in \\cite{DV2}, the eigenvalue is equal to the constant ${{\\mathcal H}}(x,p)$ defined in \\eqref{eigen:repr}. Then, equality \\eqref{hamiltonian} follows from Remark \\ref{limit_repr} and Proposition \\ref{eigenvalue:th} by choosing $\\varphi_{x,p}= \\log g_{x,p}$.\n\\end{proof}\n\\subsection{Regularity of the Hamiltonian}\\label{sub:regularity}\nBefore proving the comparison principle, we first show that the map $p \\mapsto \\mathcal{H}(x,p)$, constructed out of the eigenvalue problem in Propositions \\ref{eigenvalue:th} and \\ref{hamiltonian:prop}, is convex, coercive and continuous uniformely with respect to $x$. \n\\begin{proposition}[Convexity and Coercivity of $\\mathcal{H}$]\\label{conv:th}\nThe map ${{\\mathcal H}}: (x,p)\\to\\mathcal{H}(x,p)$ in \\eqref{eigen:repr} is convex in $p$ and coercive in $p$ uniformly with respect to $x$. Precisely, \n\\begin{equation}\\label{coercivity:eq}\n\\lim _{|p| \\to \\infty} \\inf_{x\\in K} \\mathcal{H}(x,p) = \\infty\n\\end{equation}\nfor every $K$ compact set. Moreover, ${{\\mathcal H}}(x,0)=0$ for all $x \\in {{\\mathbb R}}^d$.\n\\end{proposition}\n\n\\begin{proof}\nBy Proposition \\ref{hamiltonian:prop} the eigenvalue $\\mathcal{H}(x,p)$ admits the representation\n\\begin{align*}\n\\mathcal{H}(x,p)\n&=\n- \\sup_{g > 0} \\inf_{z^\\prime \\in E^\\prime} \n\\left\\{ \n\\frac{1}{g(z^\\prime)}\\left[\n(-B_{x,p} - V_{x,p} - R_x)g\n\\right](z^\\prime) \n\\right\\} \\\\\n&=\n\\inf_{g > 0} \\sup_{z^\\prime \\in E^\\prime}\n\\left\\{\n\\frac{1}{g(z^\\prime)}\n\\left[\n(B_{x,p} + V_{x,p} + R_x)g\n\\right](z^\\prime)\n\\right\\} \\\\\n&=\n\\inf_{\\varphi} \\sup_{z^\\prime \\in E^\\prime}\n\\left\\{\ne^{-\\varphi(z^\\prime)}\n\\left[(B_{x,p} + V_{x,p} + R_x)e^{\\varphi}\n\\right] (z^\\prime)\n\\right\\}\n=:\n\\inf_{\\varphi} \\sup_{z^\\prime \\in E^\\prime} F(x,p,\\varphi)(z^\\prime),\n\\end{align*}\nwhere the map $F$ is given by\n\\begin{align*}\nF(x,p,\\varphi)(y,i)\n=\n\\frac{1}{2}\\Delta\\varphi^i(y)\n+\n\\frac{1}{2} |\\nabla\\varphi^i(y)+p|^2\n&-\n\\nabla_x\\psi^i(x,y)(\\nabla\\varphi^i(y)+p)\n\\\\&+\n\\sum_{j = 1}^J r_{ij}(x,y)\n\\left[ e^{\\varphi^j(y)-\\varphi^i(y)}-1 \\right].\n\\end{align*}\nNote that $F$ is jointly convex in $p$ and $\\varphi$. \nRegarding coercivity of $\\mathcal{H}(x,p)$, we isolate the $p^2$ term in $V_{x,p}$, to obtain\n\\begin{align*}\n\\mathcal{H}(x,p)\n&=\n\\frac{p^2}{2}\n+\n\\inf_{\\varphi}\\sup_{E^\\prime}\n\\left\\{\ne^{-\\varphi}\n\\left[\nB_{x,p} - p\\cdot\\nabla_x \\psi + R_x\n\\right]e^{\\varphi}\n\\right\\}.\n\\end{align*}\nAny $\\varphi \\in C^2(E^\\prime)$ admits a minimum $(y_m,i_m)$ on the compact set $E^\\prime$, and with the thereby obtained uniform lower bound\n\\begin{align*}\n\\Gamma(x,p,\\varphi) \n&=\\sup_{E'} \\left\\{e^{-\\varphi(z_m)}\\left[ B_{x,p} -p\\cdot \\nabla_x\\psi +R_x\\right]e^{\\varphi(z_m)}\\right\\} \\\\\n&\\geq\ne^{-\\varphi(z_m)}\\left[ B_{x,p} -p\\cdot \\nabla_x\\psi +R_x\\right]e^{\\varphi(z_m)} \\\\\n&=\n\\underbrace{\\frac{1}{2} \\Delta_y \\varphi(y_m,i_m)}_{ \\displaystyle \\geq 0} \n+\n\\frac{1}{2}|\\underbrace{\\nabla_y \\varphi(y_m,i_m)}_{\\displaystyle = 0}|^2\n+ (p-\n\\nabla_x \\psi^{i_m}(x,y_m))\\cdot \\underbrace{\\nabla_y \\varphi(y_m,i_m)}_{\\displaystyle = 0}\\\\\n&+\n\\sum_{j\\neq i} r_{ij}(x,y_m) \n\\underbrace{\\left[ e^{\\varphi(y_m,j) - \\varphi(y_m,i_m)} - 1 \\right]}_{\\displaystyle \\geq 0} - p \\cdot \\nabla_x\\psi^{i_m}(x,y_m)\n\\geq\n- p \\cdot \\nabla_x\\psi^{i_m}(x,y_m).\n\\end{align*}\nUsing the lower bound $\\Gamma(x,p,\\varphi)\\geq -p\\cdot \\nabla_x\\psi^{i_m} (x,y_m)\\geq \\inf_{E'} (-p \\cdot \\nabla_x\\psi)$, it follows that, if $K$ is a compact set\n\\begin{align*}\n\\inf_{x\\in K} \\mathcal{H}(x,p)\n&\\geq\n\\frac{p^2}{2} - \\sup_{x\\in K}\\sup_{E^\\prime} (p\\cdot \\nabla _x\\psi^i(x,y))\n\\geq\n\\frac{1}{4}p^2 - \\sup_{x\\in K}\\sup_{E^\\prime}|\\nabla_x\\psi^i(x,y)|^2 \\xrightarrow{|p|\\rightarrow \\infty} \\infty.\n\\end{align*}\nRegarding ${{\\mathcal H}}(x,0)=0$, note that $\\Gamma(x,0,\\varphi)\\leq0$ for all $x$ and $\\varphi$. Then we have the first inequality ${{\\mathcal H}}(x,0)\\geq \\inf_\\varphi \\Gamma(x,0,\\varphi)\\geq 0$. For the opposite inequality we choose the function $\\varphi = (1,\\dots,1)$ in the representation of ${{\\mathcal H}}$.\n\\end{proof}\n\n\\begin{proposition}[Continuity of $\\mathcal{H}$]\\label{continuity-th}\nThe map ${{\\mathcal H}}:(x,p)\\in {{\\mathbb R}}^d\\times{{\\mathbb R}}^d\\to {{\\mathcal H}}(x,p)\\in {{\\mathbb R}}$ is continuous.\n\\end{proposition}\n\nWe will prove the continuity of ${{\\mathcal H}}$ by showing that it is lower and upper semicontinuous. For that, we need the following auxiliary results. In particular, for the lower semicontinuity we will make use of the $\\Gamma$--convergence in the sense expressed in the following lemma in which we prove that property in a general context. Later, we will use it for $\\mathcal{J}(x,p,\\theta)=I_{x,p}(\\theta)$.\n\n\\begin{lemma}[$\\Gamma$-convergence]\\label{gammaConv}\nGiven two sets $U,V\\subseteq {{\\mathbb R}}^d$ and a constant $M\\geq0$ we define $\\Theta_{U,V,M}$ as\n$$\\Theta_{U,V,M} = \\bigcup_{x\\in U, p\\in V} \\{\\theta \\in \\Theta \\vert \\mathcal{J}(x,p,\\theta)\\leq M \\}.$$\nLet $\\mathcal{J}:{{\\mathbb R}}^d \\times {{\\mathbb R}}^d \\times\\Theta\\to[0,\\infty]$ satisfy the following assumptions:\n\\begin{enumerate}[(i)]\t\n\\item \\label{item:assumption:I1} \n The map $(x,p,\\theta) \\mapsto \\mathcal{J}(x,p,\\theta)$ is lower semi-continuous on ${{\\mathbb R}}^d \\times {{\\mathbb R}}^d\\times\\Theta$.\n \n\n\\item \\label{item:assumption:I4}\nFor every $x$ and $p$ fixed and $M\\geq 0$, there exist $U_x$ and $U_p$ open and bounded neighbourhoods and a constant $M'$ such that \n$$ \\mathcal{J}(y,q,\\theta) \\leq M' \\qquad \\text{for all $y \\in U_x$, $q \\in U_p$ and $\\theta \\in \\Theta_{\\{x\\},\\{p\\},M}$}$$.\n\\item \\label{item:assumption:I5}For all compact sets $K_1 \\subseteq {{\\mathbb R}}^d$ and $K_2 \\subseteq {{\\mathbb R}}^d$ and each $M \\geq 0$ the collection of functions $\\{\\mathcal{J}(\\cdot,\\cdot, \\theta)\\}_{\\theta \\in \\Theta_{K_1,K_2,M}}$ is equi-continuous.\n\t\n\\end{enumerate}\n\n\n\nThen if $x_n\\to x$ and $p_n \\to p$, the functionals $\\mathcal{J}_n$ defined by\n\t\n\\begin{equation*}\n\\mathcal{J}_n(\\theta) := \\mathcal{J}(x_n,p_n,\\theta)\n\\end{equation*}\n\t\nconverge in the $\\Gamma$-sense to $\\mathcal{J}_\\infty(\\theta) := \\mathcal{J}(x,p,\\theta)$. That is:\n\\begin{enumerate}\n\\item If $x_n \\rightarrow x$, $p_n \\to p$ and $\\theta_n \\rightarrow \\theta$, then $\\liminf_{n\\to\\infty} \\mathcal{J}(x_n,p_n,\\theta_n) \\geq \\mathcal{J}(x,p,\\theta)$,\n\\item For $x_n \\rightarrow x$ and $p_n \\to p$ and all $\\theta \\in \\Theta$ there are controls $\\theta_n \\in \\Theta$ such that $\\theta_n \\rightarrow \\theta$ and $\\limsup_{n\\to\\infty} \\mathcal{J}(x_n,p_n,\\theta_n) \\leq \\mathcal{J}(x,p,\\theta)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $x_n\\to x$ and $p_n \\to p$ in ${{\\mathbb R}}^d$. If $\\theta_n\\to \\theta$, then by lower semicontinuity~\\ref{item:assumption:I1},\n\\begin{equation*}\n\\liminf_{n\\to\\infty}\\mathcal{J}(x_n,p_n,\\theta_n) \\geq \\mathcal{J}(x,p,\\theta).\n\\end{equation*}\nFor the $\\text{lim-sup}$ bound, let $\\theta\\in\\Theta$. If $\\mathcal{J}(x,p,\\theta)=\\infty$, there is nothing to prove. Thus suppose that $\\mathcal{J}(x,p,\\theta)$ is finite, i.e., $\\theta \\in \\Theta_{\\{x\\},\\{p\\},M}$ for some $M$. Then, by~\\ref{item:assumption:I4}, there exist a bounded neighborhood $U_x$ of $x$, a bounded neighborhood $U_p$ of $p$ and a constant $M' $ such that for any $y\\in U_x$ and $q\\in U_p$,\n\\begin{equation*}\n\\mathcal{J}(y,q,\\theta) \\leq M'.\n\\end{equation*}\nSince $x_n\\to x$ and $p_n \\to p$, the sequences $x_n$ and $p_n$ are, for $n$ large, contained in $U_x$ and $U_p$, respectively. Taking the constant sequence $\\theta_n:=\\theta$, we thus get that $\\mathcal{J}(x_n,p_n,\\theta_n) \\leq M'$ for all $n$ large enough. By~\\ref{item:assumption:I5}, the family of functions $\\{\\mathcal{J}(\\cdot,\\cdot, \\theta)\\}_{\\theta \\in \\Theta_{\\bar{U}_{x},\\bar{U}_p,M'}}$ is equi-continuous, and hence\n\\begin{equation*}\n\\lim_{n\\to\\infty}|\\mathcal{J}(x_n,p_n,\\theta_n)-\\mathcal{J}(x,p,\\theta)| \\leq 0,\n\\end{equation*\nand the $\\text{lim-sup}$ bound follows.\n\\end{proof}\n\nWe can now prove that the function $I_{x,p}$ in \\eqref{eigen:repr} is $\\Gamma$-convergent.\n\\begin{proposition}[$\\Gamma$-convergence of $I_{x,p}$]\\label{prop:gamma-conv}\nLet $I_{x,p}: \\Theta \\to [0,\\infty]$ the function defined in \\eqref{eigen:repr}. If $x_n \\to x$ and $p_n \\to p$, the functionals $I_n(\\theta) := I_{x_n,p_n}(\\theta)$ converge in the $\\Gamma$-sense to $I_\\infty (\\theta):= I_{x,p}(\\theta)$.\n\\end{proposition}\n \n\\begin{proof}\nUsing Lemma \\ref{gammaConv}, we only need to prove that $I_{x,p}$ verifies the assumptions.\n\n\\textbf{Assumption \\ref{item:assumption:I1}.}\nFor any fixed function $u\\in\\mathcal{D}(L_{x,p})$ such that $u > 0$, the function $(L_{x,p}u\/u)$ is continuous. Thus, for any such fixed $u > 0$ it follows that\n\\begin{equation*}\n(x,p,\\theta)\\mapsto \\int_{E' }\\frac{L_{x,p}u}{u}\\,d\\theta\n\\end{equation*}\nis continuous on ${{\\mathbb R}}^d\\times{{\\mathbb R}}^d\\times\\Theta$. As a consequence $I(x,p,\\theta)$ is lower semicontinuous as the supremum over continuous functions.\n\n\\textbf{Assumption \\ref{item:assumption:I4}.}\nFix $x$, $p$ and $M\\geq0$. Let $\\theta \\in \\Theta_{{x},{p},M}$. Then, $I_{x,p}(\\theta)=I(x,p,\\theta)\\leq M$. It follows from\n\\autocite[Theorem 3]{Pin07} that the density $\\frac{d\\theta}{dz}$ exists. Moreover, by the same theorem, for all $y$ and $q$ there exist constants $c_1(y,q),c_2(y,q)$ positive, depending continuously on $y$ and $q$, but not on $\\theta$, such that\n\\begin{equation} \n I_{y,q}(\\theta) \\leq c_1(y,q) \\int_{E^\\prime}|\\nabla g_\\theta |^2\\,dz + c_2(y,q),\n\\end{equation}\nwhere $g_\\theta = (d\\theta\/dz)^{1\/2}$ is the square root of the Radon--Nykodym derivative.\nAs the dependence is continuous in $y$ and $q$, we can find two open neighbourhoods, $U \\subseteq {{\\mathbb R}}^d$ of $x$ and $V\\subseteq {{\\mathbb R}}^d$ of $p$, such that there exist constants $c_1,c_2$ positive, that do not depend on $\\theta$, such that for any $y \\in U$ and $q \\in V$:\n\\begin{equation}\nI_{y,q}(\\theta) \\leq c_1 \\int_{E'}|\\nabla g_\\theta |^2\\,dz + c_2:= M',\n\\end{equation}\t\nobtaining then \\ref{item:assumption:I4}.\n\n\\textbf{Assumption \\ref{item:assumption:I5}.} \n By the continuity of $r_{ij}$ and $\\psi$, assumption \\ref{item:assumption:I5} follows from Theorem 4 of \\cite{Pin07}.\n \n\n\\end{proof}\n\nThe following technical lemma will give us the upper semi-continuity of $\\mathcal{H}$.\n\n\\begin{lemma}[Lemma 17.30 in~\\cite{AlBo} ] \\label{lemma:upper_semi}\nLet $\\mathcal{X}$ and $\\mathcal{Y}$ be two Polish spaces. Let $\\phi : \\mathcal{X} \\rightarrow \\mathcal{K}(\\mathcal{Y})$, where $\\mathcal{K}(\\mathcal{Y})$ is the space of non-empty compact subsets of $\\mathcal{Y}$. Suppose that $\\phi$ is upper hemi-continuous, that is if $x_n \\rightarrow x$ and $y_n \\rightarrow y$ and $y_n \\in \\phi(x_n)$, then $y \\in \\phi(x)$. \nLet $f : \\text{Graph} (\\phi) \\rightarrow {{\\mathbb R}}$ be upper semi-continuous. Then the map $m(x) = \\sup_{y \\in \\phi(x)} f(x,y)$ is upper semi-continuous.\n\\end{lemma} \n\n\\medspace\n\nWe can finally prove the continuity of $\\mathcal{H}(x,p)$.\n\\begin{proof}[Proof of Proposition \\ref{continuity-th}]\nWe have already showed that $I_{x,p}(\\mu)$ is lower semicontinuous and, since $V_{p}(x,i)$ is continuous and bounded, $\\int_{E^\\prime} V_{x,p} \\,d\\mu$ is continuous. Then, $f(x,p,\\mu):=\\int_{E^\\prime} V_{x,p} \\,d\\mu - I_{x,p}(\\mu)$ is upper semi-continuous. \\\\\nLet $x,p\\in {{\\mathbb R}}^d$. We know, by Proposition \\ref{prop:stationarymeasure} in the appendix, that there exists a unique stationary measure $\\theta_{x,p}^0$ such that for all $g \\in D(L_{x,p})$,\n\\begin{equation}\\label{stationary}\n\\int_{E'} L_{x,p}g(z,i) d\\theta_{x,p}^0=0.\n\\end{equation}\n\nLet $L_{x,p}^{\\lambda}= \\lambda (\\lambda - L_{x,p})^{-1} L_{x,p}$ the Hille-Yosida approximation of $L_{x,p}$. Then we have\n\\begin{align}\n&- \\int_{E^\\prime} \\frac{L_{x,p} u}{u} d\\theta_{x,p}^0 = - \\int_{E^\\prime} \\frac{L_{x,p}^{\\lambda} u}{u} d\\theta_{x,p}^0 +\\int_{E^\\prime} \\frac{\\left(L_{x,p}^{\\lambda}-L_{x,p}\\right)u}{u} d\\theta_{x,p}^0\\\\\n&\\leq -\\int_{E^\\prime} \\frac{L_{x,p}^{\\lambda}u}{u} d\\theta_{x,p}^0 + \\frac{1}{\\inf_{E^\\prime} u} \\Vert{(L_{x,p}^{\\lambda} - L_{x,p})u}\\Vert_{E^\\prime} \\\\\n&\\leq  -\\int_{E^\\prime} L_{x,p}^\\lambda \\log u \\, d\\theta_{x,p}^0 +o(1).\n\\end{align}\nSending $\\lambda \\to 0$ and using \\eqref{stationary} we have that $I_{x,p}(\\theta^0_{x,p})=0$.\nThen, $\\mathcal{H}(x,p) \\geq \\int_{E'}V_{x,p} d\\theta^0_{x,p}$.\nThus, it suffices to restrict the supremum over $\\theta \\in \\phi(x,p)$ where \n\\begin{equation*}\n\\phi(x,p) := \\left\\{\\theta \\in \\mathcal{P}(E') \\, \\middle| \\, I_{x,p}(\\theta) \\leq 2 \\Vert{\\Pi(x,p,\\cdot)}\\Vert_{\\mathcal{P}(E')}\\right\\},\n\\end{equation*}\nwhere $\\Vert{\\cdot}\\Vert_{\\mathcal{P}(E')}$ denotes the supremum norm on $\\mathcal{P}(E')$ and we called for simplicity $\\Pi(x,p,\\theta)=\\int_{E'} V_{x,p} d\\theta.$\\\\\nNote that $\\Vert{\\Pi(x,p,\\theta)}\\Vert_{\\mathcal{P}(E')} < \\infty$ by definition of $V_{x,p}$.\n It follows that\n\\begin{equation*}\n\\mathcal{H}(x,p) = \\sup_{\\theta\\in \\phi(x,p)}\\left[\\int_{E^\\prime} V_{x,p} \\,d\\mu-I_{x,p}(\\mu)\\right].\n\\end{equation*}\n$\\phi(x,p)$ is non-empty as $\\theta_{x,p}^0 \\in \\phi(x,p)$ and it is compact because any closed subset of $\\mathcal{P}(E')$ is compact. We are left to show that $\\phi$ is upper hemi-continuous.\nLet $(x_n,p_n,\\theta_n) \\rightarrow (x,p,\\theta)$ with $\\theta_n \\in \\phi(x_n,p_n)$. We establish that $\\theta \\in \\phi(x,p)$. By the lower semi-continuity of $I$ and the definition of $\\phi$ we find\n\\begin{equation*}\nI_{x,p}(\\theta) \\leq \\liminf_n I_{x_n,p_n}(\\theta_n) \\leq \\liminf_n 2\\Vert{\\Pi(x_n,p_n,\\cdot})\\Vert_{\\mathcal{P}(E')} = 2 \\Vert{\\Pi(x,p,\\cdot)}\\Vert_{\\mathcal{P}(E')}\n\\end{equation*}\nwhich implies indeed that $\\theta  \\in \\phi(x,p)$. Thus, upper semi-continuity follows by an application of Lemma \\ref{lemma:upper_semi}.\n\nWe proceed with proving lower semi-continuity of $\\mathcal{H}$. Suppose that $(x_n,p_n) \\rightarrow (x,p)$, we prove that $\\liminf_n \\mathcal{H}(x_n,p_n) \\geq \\mathcal{H}(x,p)$. \nLet $\\theta$ be the measure such that $\\mathcal{H}(x,p) = \\Pi(x,p,\\theta) - I_{x,p}(\\theta)$. By Proposition \\ref{prop:gamma-conv}, there are $\\theta_n$ such that $\\theta_n \\rightarrow \\theta$ and $\\limsup_n I_{x_n,p_n}(\\theta_n) \\leq I_{x,p}(\\theta)$. \nMoreover, $\\Pi(x_n,p_n,\\theta_n)$ converges to $\\Pi(x,p,\\theta)$ by continuity.\nTherefore,\n\\begin{align*}\n\\liminf_{n\\to\\infty}\\mathcal{H}(x_n,p_n)&\\geq \\liminf_{n\\to\\infty} \\left[\\Pi(x_n,p_n,\\theta_n)-I_{x_n,p_n}(\\theta_n)\\right]\\\\\n&\\geq \\liminf_{n\\to\\infty}\\Pi(x_n,p_n,\\theta_n)-\\limsup_{n\\to\\infty}I_{x_n,p_n}(\\theta_n)\\\\\n&\\geq \\Pi(x,p,\\theta)-I_{x,p}(\\theta) = \\mathcal{H}(x,p),\n\\end{align*}\nestablishing that $\\mathcal{H}$ is lower semi-continuous.\n\\end{proof}\n\n\n\\subsection{Comparison principle}\\label{comparison-sub}\nIn this section we prove the comparison principle for the Hamilton--Jacobi equation in terms of $H$ by relating it to a set of Hamilton-Jacobi equations constructed from $\\mathcal{H}$ (Figure  \\ref{SF:fig:CP-diagram-in-proof-of-CP}). We introduce the operators $H_\\dagger,H_\\ddagger$ and $H_1,H_2$. In both cases, the new Hamiltonians will serve as natural upper and lower bounds for $\\mathbf{H}f(x)={{\\mathcal H}}(x,\\nabla f(x))$ and $H$ respectively, where $\\mathcal{H}$ and $H$ are the operators introduced in Propositions \\ref{hamiltonian:prop} and \\ref{convergence:th}. These new Hamiltonians are defined in terms of a containment function $\\Upsilon$, which allows us to restrict our analysis to compact sets.\nHere we give the rigorous definition.\n\\begin{definition}[Containment function]\\label{contfun_def}\n    A function $\\Upsilon: {{\\mathbb R}}^d \\to [0,\\infty)$ is a containment function for $V_{x,p}$ in \\eqref{eigen:repr}, if $\\Upsilon \\in C^1({{\\mathbb R}}^d)$ and it is such that\n    \\begin{itemize}\n        \\item $\\Upsilon$ has compact sub-level sets,  i.e. for every $c\\geq0$ the set $\\{x\\vert \\Upsilon(x)\\leq c\\}$ is compact ;\n        \\item $\\sup_{x\\in{{\\mathbb R}}^d, z\\in E^\\prime} V_{x,\\nabla \\Upsilon(x)}(z) < \\infty$.\n    \\end{itemize}\n\\end{definition}\n\n\\begin{lemma}\\label{contfun_log_lemma}\n    The function $\\Upsilon(x)= \\frac{1}{2}\\log\\left(1+\\abs{x}^2\\right)$ is a containment function for $V_{x,p}$.\n\\end{lemma}\n\\begin{proof}\nFirstly note that $\\Upsilon$ has compact sub-level sets. \nRegarding the second property, by the definition of $V_{x,p}$, we have for every $x\\in {{\\mathbb R}}^d$ and $z=(y,i)\\in \\mathbb{T}^d\\times \\{1,\\dots,J\\}$,\n\\begin{align}\n    V_{x,\\nabla \\Upsilon(x)} (y,i) =\\frac{x^2}{2(1+\\abs{x}^2)^2} - \\nabla_x \\psi^i(x,y) \\frac{x}{1+\\abs{x}^2}.\n\\end{align}\nRecalling that $\\psi$ grows at most linearly in $x$, we can conclude that $\\sup_{x,z}V_{x,\\nabla \\Upsilon}(z)< \\infty$.\n\\end{proof}\nUsing the above lemma we are now able to define the auxiliary operators in terms of $\\Upsilon$.\nIn the following we will denote by $C_l^\\infty(E)$ the set of smooth functions on $E$ that have a lower bound and by $C_u^\\infty(E)$ the set of smooth functions on $E$ that have an upper bound.\n\\begin{definition} \\label{aux}\nFix $\\eta \\in (0,1)$ and  given $\\Upsilon(x)= \\frac{1}{2}\\log\\left(1+|x|^2\\right)$, $C_\\Upsilon := \\sup_{x,z} V_{x,\\nabla\\Upsilon(x)}(z)$ and $\\mathbf{H}f(x)=\\mathcal{H}(x,\\nabla f(x))$, we define\n\\begin{itemize}\n\\item For $f \\in C_l^\\infty(E)$,\n\\begin{gather*}\nf^\\eta_\\dagger := (1-\\eta) f + \\eta \\Upsilon, \\\\\nH_{\\dagger,f}^\\eta(x) := (1-\\eta) \\mathbf{H} f(x) + \\eta C_\\Upsilon,\n\\end{gather*}\nand set\n\\begin{equation*}\nH_\\dagger := \\left\\{(f^\\eta_\\dagger,H_{\\dagger,f}^\\eta) \\, \\middle| \\, f \\in C_l^\\infty(E), \\eta \\in (0,1) \\right\\}.\n\\end{equation*} \n\\item For $f \\in C_u^\\infty(E)$,\n\n\\begin{gather*}\nf^\\eta_\\ddagger := (1+\\eta) f - \\eta \\Upsilon, \\\\\nH_{\\ddagger,f}^\\eta(x) := (1+\\eta) \\mathbf{H} f(x) - \\eta C_\\Upsilon,\n\\end{gather*}\nand set\n\\begin{equation*}\nH_\\ddagger := \\left\\{(f^\\eta_\\ddagger,H_{\\ddagger,f}^\\eta) \\, \\middle| \\, f \\in C_u^\\infty(E), \\eta \\in (0,1) \\right\\}.\n\\end{equation*} \n\\end{itemize}\n\\end{definition}\n\n\\begin{definition}\nFix $\\eta \\in (0,1)$ and  given $\\Upsilon(x)= \\frac{1}{2}\\log\\left(1+|x|^2\\right)$, $C_\\Upsilon := \\sup_{x,z} V_{x,\\nabla\\Upsilon(x)}(z)$ and $\\mathbf{H}f(x)=\\mathcal{H}(x,\\nabla f(x))$, we define\n\\begin{itemize}\n\\item For $f \\in C_l^\\infty(E)$ , $\\varphi\\in C^2(E^\\prime)$, $\\eta \\in (0,1)$ set \n\\begin{gather*}\nf^\\eta_1 := (1-\\eta) f + \\eta \\Upsilon, \\\\\nH^\\eta_{1,f,\\varphi}(x,z) :=\n(1-\\eta) H_{f,\\varphi}(x,z) + \\eta C_\\Upsilon ,\n\\end{gather*}\nand set\n\\begin{equation*}\nH_1 := \\left\\{(f^\\eta_1,H^\\eta_{1,f,\\varphi}) \\, \\middle| \\, f \\in C_l^\\infty(E), \\varphi \\in C^2(E^\\prime), \\eta \\in (0,1) \\right\\}.\n\\end{equation*} \n\\item For $f \\in C_u^\\infty(E)$, $\\varphi\\in C^2(E^\\prime)$, $\\eta \\in (0,1)$ set \n\\begin{gather*}\nf^\\eta_2 := (1+\\eta) f - \\eta \\Upsilon, \\\\\nH^\\eta_{2,f,\\varphi}(x,z) :=(1+\\eta) H_{f,\\varphi}(x,z) - \\eta C_\\Upsilon,\n\\end{gather*}\nand set\n\\begin{equation*}\nH_2 := \\left\\{(f^\\eta_2,H^\\eta_{2,f,\\varphi}) \\, \\middle| \\, f \\in C_u^\\infty(E), \\varphi\\in C^2(E^\\prime), \\eta \\in (0,1)\\right\\}.\n\\end{equation*}\n\\end{itemize}\n\\label{definition:H1H2}\n\\end{definition}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}\n\\matrix (m) [matrix of math nodes,row sep=1em,column sep=4em,minimum width=2em]\n{\n{ } & H_1 &[7mm] H_\\dagger &[5mm] { } \\\\\nH & { } & { } & \\mathbf{H} \\\\\n{ }  & H_2 & H_\\ddagger & { } \\\\};\n\\path[-stealth]\n(m-2-1) edge node [above] {sub} (m-1-2)\n(m-2-1) edge node [below] {super \\qquad { }} (m-3-2)\n(m-1-2) edge node [above] {sub \\qquad { }} (m-1-3)\n(m-3-2) edge node [below] {super \\qquad { }} (m-3-3)\n(m-2-4) edge node [above] {\\qquad sub} (m-1-3)\n(m-2-4) edge node [below] {\\qquad super} (m-3-3);\n\\begin{pgfonlayer}{background}\n\\node at (m-2-3) [rectangle,draw=blue!50,fill=blue!20,rounded corners, minimum width=1cm, minimum height=2.5cm]  {comparison};\n\\end{pgfonlayer}\n\\end{tikzpicture}\n\\end{center}\n\\captionsetup{width=.9\\textwidth}\n\\caption{An arrow connecting an operator $A$ with operator $B$ with subscript 'sub' means that viscosity subsolutions of $(1 - \\lambda A)f= h$ are also viscosity subsolutions of $(1 - \\lambda B)f = h$. Similarly for arrows with a subscript 'super'. The box around the operators $H_\\dagger$ and $H_\\ddagger$ indicates that the comparison principle holds for subsolutions of $(1 - \\lambda H_\\dagger) f = h$ and supersolutions of $(1 - \\lambda H_\\ddagger) f = h$.}\n\\label{SF:fig:CP-diagram-in-proof-of-CP}\n\\end{figure}\n\nWe now prove the comparison principle for $f - \\lambda Hf = h$ based on the results summarized in Figure~\\ref{SF:fig:CP-diagram-in-proof-of-CP}. \n\\begin{theorem}[Comparison principle] \\label{theorem:comparison}\nLet $h \\in C_b(E)$ and $\\lambda >0$. Let $u$ and $v$ be, respectively,  any subsolution and any supersolution to $(1 - \\lambda H)f = h$. Then we have that \n\\begin{equation*}\n\\sup_x u(x) - v(x) \\leq 0.\n\\end{equation*}\n\\end{theorem}\n\\begin{proof}\n\nFix~$h\\in C_b(E)$ and~$\\lambda>0$. Let~$u$ be a viscosity subsolution and~$v$ be a viscosity supersolution to~$(1-\\lambda H)f=h$. By Figure~\\ref{SF:fig:CP-diagram-in-proof-of-CP}, the function~$u$ is a viscosity subsolution to~$(1-\\lambda H_\\dagger)f=h$ and~$v$ is a viscosity supersolution to~$(1-\\lambda H_\\ddagger)f=h$. Hence by the comparison principle for $H_\\dagger,H_\\dagger$ established in Theorem \\ref{theorem:comparison_from_otherpaper} below,~$\\sup_x u(x) - v(x) \\leq 0 $, which finishes the proof.\n\\end{proof}\n\nThe rest of this subsection is devoted to establishing Figure~\\ref{SF:fig:CP-diagram-in-proof-of-CP}. More precisely, we establish Figure \\ref{SF:fig:CP-diagram-in-proof-of-CP} in results \\ref{theorem:comparison_from_otherpaper}, \\ref{H12}, \\ref{lemma:viscosity_solutions_arrows_based_on_eigenvalue} and \\ref{lemmaRepre}.\n \\\\\n\nThe next theorem contains the comparison principle for $H_\\dagger$ and $H_\\ddagger$. The proof follows standard ideas that can be found for instance in \\cite{BaCaDo} and \\cite{CrIsLi}. In order to be able to use both the subsolution and supersolution properties in the estimate of $\\sup_x u(x) - v(x)$, we use the following strategy based on the introduction of double variables.\n\\begin{enumerate}\n\\item First of all, note that the supremum over $x$ of $u(x)-v(x)$ can be replaced, sending $\\varepsilon \\to 0$, with the supremum over $x$ and $y$ of the double variables function $u(x)-v(y)- (2\\varepsilon)^{-1}(x-y)^2$\n\\item Once the supremum (x,y) is found, we are able to use the sub-super solution properties in the following way: \n\\begin{itemize}\n\\item fixing $y$ and optimising over $x$, it can be used in the application of the subsolution property of $u$\n\\item fixing $x$ and optimising over $y$, it can be used in the application of the supersolution property of $v$.\n\\end{itemize}\n\\end{enumerate}\n\\begin{theorem} \\label{theorem:comparison_from_otherpaper}\nLet $h \\in C_b(E)$ and $\\lambda >0$. Let $u$ be any subsolution to $(1 - \\lambda H_\\dagger) f = h$ and let $v$ be any supersolution to $(1- \\lambda H_\\ddagger) f = h$. Then we have that \n\\begin{equation*}\n\\sup_x u(x) - v(x) \\leq 0.\n\\end{equation*}\n\\end{theorem}\n\n\n\\begin{proof}\nFollowing the above steps we define the double variables function \n\\begin{equation}\n\\Phi_{\\varepsilon,\\beta}(x,y) = \\frac{u(x)}{1-\\beta} - \\frac{v(y)}{1+\\beta} - \\frac{|x-y|^2}{2\\varepsilon} - \\frac{\\beta}{1-\\beta}\\Upsilon(x) -\\frac{\\beta}{1+\\beta}\\Upsilon(y).\n\\end{equation}\nNote that the containment function $\\Upsilon$ is introduced in order to be able to work in a compact set, and the positive constant $\\beta$ will allow us to use the convexity of $\\mathcal{H}$.\nSince $\\Phi_{\\varepsilon,\\beta}$ is upper semicontinuous and $\\lim_{|x|+|y| \\to \\infty}\\Phi (x,y) =-\\infty$, for every $\\varepsilon\\in(0,1)$ there exists $(x_{\\varepsilon},y_{\\varepsilon})$ such that\n\\begin{equation}\\label{supremum}\n \\Phi_{\\varepsilon,\\beta}(x_{\\varepsilon},y_{\\varepsilon})=\\sup_{{{\\mathbb R}}^d\\times{{\\mathbb R}}^d} \\Phi_{\\varepsilon,\\beta}(x,y).\n \\end{equation}\nSuppose by contradiction that $\\delta= u(\\tilde{x})-v(\\tilde{x})> 0 $ for some $\\tilde{x}$. We choose $\\beta$ such that $\\frac{2\\beta}{(1-\\beta)(1+\\beta)} \\Upsilon(\\tilde{x})<\\delta\/2$ and $\\frac{2\\beta}{1-\\beta^2}\\left(\\vn{h} + C_\\Upsilon\\right)<\\delta\/2$. Then, \n\\begin{equation}\\label{positivity}\n\\Phi_{\\varepsilon,\\beta}(x_\\varepsilon,y_\\varepsilon) \\geq \\Phi_{\\varepsilon,\\beta}(\\tilde{x},\\tilde{x}) > \\delta - \\frac{2\\beta}{(1-\\beta)(1+\\beta)} \\Upsilon(\\tilde{x})>\\frac{\\delta}{2}> 0,\n\\end{equation}\nand \n$$\\frac{\\beta}{1-\\beta} \\Upsilon(x_\\varepsilon) + \\frac{\\beta}{1+\\beta} \\Upsilon(y_\\varepsilon) \\leq \\sup\\left(\\frac{u}{1-\\beta}\\right)+\\sup\\left(\\frac{-v}{1+\\beta}\\right) <\\infty.$$\nTherefore there exists $R_\\beta>0$ such that $x_\\varepsilon$ and $y_\\varepsilon$ belong to $B(0,R_\\beta)$.\n\nNext we observe that by Lemma 3.1 of \\cite{CrIsLi}, \n\\begin{equation}\\label{conv2}\n\\frac{|x_\\varepsilon -y_\\varepsilon|^2}{\\varepsilon} \\to 0 \\qquad \\text{as $\\varepsilon \\to 0^+$},\n\\end{equation}\nand, as a consequence, $\\vert x_\\varepsilon - y_{\\varepsilon} \\vert \\to 0$ as $\\varepsilon \\to 0^+$.\nDefine the functions $\\varphi_1^{\\varepsilon,\\beta} \\in D(H_\\dagger)$ and $\\varphi_2^{\\varepsilon,\\beta} \\in D(H_\\ddagger)$ by\n\\begin{align}\n&\\varphi_1^{\\varepsilon,\\beta} (x)= (1-\\beta) \\left [ \\frac{v(y_\\varepsilon)}{1+\\beta} +\\frac{|x-y_\\varepsilon|^2}{2\\varepsilon} +\\frac{\\beta}{1-\\beta} \\Upsilon (x) + \\frac{\\beta}{1+\\beta}\\Upsilon(y_\\varepsilon)+(1-\\beta)(x-x_{\\varepsilon})^2 \\right] \\\\\n& \\varphi_2^{\\varepsilon,\\beta}(y) = (1+ \\beta) \\left [ \\frac{u(x_\\varepsilon)}{1-\\beta} - \\frac{|x_\\varepsilon-y|^2}{2\\varepsilon} -\\frac{\\beta}{1-\\beta} \\Upsilon (x_\\varepsilon) - \\frac{\\beta}{1+\\beta}\\Upsilon(y) - (1+\\beta) (y-y_{\\varepsilon})^2 \\right].\n\\end{align}\nUsing \\eqref{supremum}, observe that $u - \\varphi_1^{\\varepsilon,\\beta}$ attains its supremum at $x = x_{\\varepsilon}$, and thus\n\\begin{equation*}\n\\sup_E (u-\\varphi^{\\varepsilon,\\beta}_1) = (u-\\varphi^{\\varepsilon,\\beta}_1)(x_{\\varepsilon}).\n\\end{equation*}\nMoreover, by addition of the $(1-\\beta)(x-x_\\varepsilon)^2$ term, this supremum is the unique optimizer of $u- \\varphi_1^{\\varepsilon,\\beta}$.\nThen, by the subsolution and supersolution properties, taking into account Remark \\ref{remark:existence of optimizers},\n\\begin{equation}\\label{subsolution}\nu(x_\\varepsilon) -\\lambda \\left[ (1-\\beta)\\mathcal{H}\\left(x_\\varepsilon, \\frac{x_\\varepsilon -y_\\varepsilon}{\\varepsilon}\\right) + \\beta \\,C_\\Upsilon \\right] \\leq h(x_\\varepsilon).\n\\end{equation}\nWith a similar argument for $u_2$ and $\\varphi^{\\varepsilon}_2$, we obtain by the supersolution inequality that\n\\begin{equation}\\label{eq:proof-CP:supersol-ineq}\nv(y_{\\varepsilon}) - \\lambda \\left[(1+\\beta)\\mathcal{H}\\left(y_{\\varepsilon}, \\frac{x_\\varepsilon -y_\\varepsilon}{\\varepsilon} \\right) - \\beta C_\\Upsilon\\right] \\geq h(y_{\\varepsilon}).\n\\end{equation}\nBy the coercivity property obtained in \\ref{conv:th} on page \\pageref{conv:th} and by the inequality \\eqref{eq:proof-CP:supersol-ineq}, $p_\\varepsilon := \\frac{x_\\varepsilon-y_\\varepsilon}{\\varepsilon}$ is bounded in $\\varepsilon$, allowing us to extract a converging subsequence $p_{\\varepsilon_k}$. \n\nWe conclude that for each $\\beta$\n\t\\begin{align}\n\t\t& \\liminf_{\\varepsilon\\to0}\\Phi(x_\\varepsilon,y_\\varepsilon) \\\\\n\t\t& \\leq \\liminf_{\\varepsilon \\to 0} \\frac{u(x_\\varepsilon)}{1-\\beta} - \\frac{v(y_\\varepsilon)}{1+\\beta} \\\\\n\t\t& \\leq  \\liminf_{k \\rightarrow \\infty} \\lambda  \\mathcal {H}\\left(x_{\\varepsilon_k},  p_{\\varepsilon_k}\\right)+\\frac{\\beta}{1-\\beta} C_\\Upsilon - \\lambda \\mathcal{H}\\left(y_{\\varepsilon_k},p_{\\varepsilon_k}\\right) + \\frac{\\beta}{1+\\beta} \\, C_\\Upsilon\\\\\n\t\t&\\qquad + \\frac{h(x_{\\varepsilon_k})}{1-\\beta} - \\frac{h(y_{\\varepsilon_k})}{1+\\beta} \\\\\n\t\t& \\leq \\liminf_{k \\rightarrow \\infty} \\lambda  \\left[ \\mathcal{H}\\left(x_{\\varepsilon_k},  p_{\\varepsilon_k}\\right) - \\mathcal{H}\\left(y_{\\varepsilon_k},p_{\\varepsilon_k}\\right)\\right] +\\frac{h(x_{\\varepsilon_k}) - h(y_{\\varepsilon_k})}{1-\\beta^2} + \\frac{2\\beta}{1-\\beta^2}\\left(\\vn{h} + C_\\Upsilon\\right) \\\\\n\t\t& \\leq  \\frac{2\\beta}{1-\\beta^2}\\left(\\vn{h} + C_\\Upsilon\\right).\n\t\\end{align}\nAs $\\beta$ is chosen such that $\\frac{2\\beta}{1-\\beta^2}\\left(\\vn{h} + C_\\Upsilon\\right)<\\delta\/2$, we obtain a contradiction with \\eqref{positivity}, establishing the comparison principle. \n\\end{proof}\nBelow, we complete the figure by proving the left-hand side of Figure \\ref{SF:fig:CP-diagram-in-proof-of-CP}.\n\\begin{lemma}\\label{H12}\nFor all $h \\in C(\\mathbb{R}^d)$ and $\\lambda > 0$, viscosity subsolutions of\n$(1 - \\lambda H) f =h$\nare viscosity subsolutions of $(1 - \\lambda H_1) f =h,$ and viscosity supersolutions of $(1 - \\lambda H) f = h $ are viscosity supersolutions of $(1 - \\lambda H_2) f =h.$\n\\end{lemma}\n\n\n\\begin{proof}\n\nFix $\\lambda > 0$ and $h \\in C_b(E)$. Let $u$ be a subsolution to $(1 - \\lambda H) f = h$. We prove it is also a subsolution to $(1 - \\lambda H_1) f = h$. Fix $\\eta \\in (0,1)$, $\\varphi\\in C^2(E')$ and $f \\in C_{l}^\\infty(E)$, so that $(f^\\eta_1,H^\\eta_{1,f,\\varphi}) \\in H_1$ with $f_1^\\eta$ and $H^\\eta_{1,f,\\varphi}$ as in Definition \\ref{definition:H1H2}. We will prove that there are $(x_n,z_n)$ such that\n\t\t\\begin{gather} \n\t\t\t\\lim_n u(x_n) - f^\\eta_1(x_n) = \\sup_x u(x) - f^\\eta_1(x),\\label{eqn:proof_HH1_conditions_for_subsolution_first} \\\\\n\t\t\t\\limsup_n u(x_n) - \\lambda H^\\eta_{1,f,\\varphi}(x_n,z_n) - h(x_n) \\leq 0. \\label{eqn:proof_HH1_conditions_for_subsolution_second}\n\t\t\\end{gather}\n\t\nGiven $M:= \\eta^{-1}\\sup_y u(y)-(1-\\eta)f(y)<\\infty$, as $u$ is bounded and $f\\in C_l^\\infty(E)$, we have that the sequence $x_n$ along which the limit in \\eqref{eqn:proof_HH1_conditions_for_subsolution_first} is\nattained, is contained in the compact set $K:=\\left\\{x\\vert \\Upsilon(x)\\leq M\\right\\}$. We define $\\gamma:{{\\mathbb R}} \\to {{\\mathbb R}}$ as a smooth increasing function such that \n\\begin{equation}\n\\gamma(r)=\n\\begin{cases}\nr \\qquad &\\text{if $r\\leq M$},\\\\\nM+1 \\qquad &\\text{if $r\\geq M+2$.}\n\\end{cases}\n\\end{equation}\nDenote by $f_\\eta$ the function on $E$ defined by\n$$ f_\\eta(x)=\\gamma((1-\\eta)f(x)+\\eta\\Upsilon(x))=\\gamma(f_1^\\eta(x)).$$\nBy construction, $f_\\eta$ is smooth and constant outside a compact set and thus lies in $\\mathcal{D}(H)$. We conclude that $(f_\\eta, H_{f_{\\eta},(1-\\eta)\\varphi}) \\in H$. \nAs $u$ is a viscosity subsolution for $(1 - \\lambda H)u = h$, there exist $x_n \\in E$ and $z_n \\in E'$ with\n\\begin{gather}\n\\lim_n u(x_n) - f_\\eta(x_n) = \\sup_x u(x) - f_\\eta(x), \\label{eqn:visc_subsol_sup} \\\\\n\\limsup_n u(x_n) - \\lambda H_{f_{\\eta},(1-\\eta)\\varphi}(x_n,z_n) - h(x_n) \\leq 0. \\label{eqn:visc_subsol_upperbound}\n\\end{gather}\nSince $f_1^\\eta$ equals $f_\\eta$ in $K=\\left\\{x\\vert \\Upsilon(x)\\leq M\\right\\}$, we also have that \n\\begin{equation*}\n\\lim_n u(x_n) - f_1^\\eta(x_n) = \\sup_x u(x) - f_1^\\eta(x),\n\\end{equation*}\nestablishing \\eqref{eqn:proof_HH1_conditions_for_subsolution_first}. Convexity of $H_{f,\\varphi}(x,z)=H_\\varphi(x,\\nabla f (x), z)$ in $p$ and $\\varphi$ yields for arbitrary $(x,z)$ the elementary estimate\n\\begin{align*}\nH_{f_{\\eta},(1-\\eta)\\varphi}(x,z) = & H_{(1-\\eta)\\varphi}(x, (1-\\eta)\\nabla f (x)+ \\eta \\nabla \\Upsilon(x), z) \\\\\n& \\leq (1-\\eta) H_\\varphi(x,\\nabla f(x), z) +\\eta H_0(x, \\nabla \\Upsilon (x), z) \\\\\n&= (1-\\eta)H_{\\varphi}(x,\\nabla f(x), z) + \\eta V_{x,\\nabla \\Upsilon(x)}(z)\\\\\n& \\leq H^\\eta_{1,f,\\varphi}(x,z).\n\\end{align*} \nCombining the above inequality with \\eqref{eqn:visc_subsol_upperbound}, we have\n\\begin{align*}\n\\limsup_n  u(x_n) - \\lambda H^\\eta_{1,f,\\varphi}(x,z) - h(x_n) \\leq \\limsup_n u(x_n) - \\lambda H_{f_{\\eta},(1-\\eta)\\varphi}(x_n,z_n) - h(x_n) \\leq 0,\n\\end{align*}\nestablishing \\eqref{eqn:proof_HH1_conditions_for_subsolution_second}.\nThe supersolution statement follows in the same way.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:viscosity_solutions_arrows_based_on_eigenvalue}\nFix $\\lambda > 0$ and $h \\in C_b(E)$. \n\\begin{enumerate}[(a)]\n\\item Every subsolution to $(1 - \\lambda H_1) f = h$ is also a subsolution to $(1 - \\lambda H_\\dagger) f = h$.\n\\item Every supersolution to $(1 - \\lambda H_1 )f = h$ is also a supersolution to $(1 - \\lambda H_\\ddagger) f = h$.\n\\end{enumerate}\n\\end{lemma}\n\nThe definition of viscosity solutions,  Definition \\ref{def:viscosity_solution}, is written down in terms of the existence of a sequence of points that maximizes $u-f$ or minimizes $v-f$. To prove the lemma above, we would like to have the subsolution and supersolution inequalities for any point that maximizes or minimizes the difference. This is achieved by the following auxiliary lemma.\n\t\n\\begin{lemma} \\label{lemma:strong_viscosity_solutions}\n\t\tFix $\\lambda > 0$ and $h \\in C_b(E)$. \n\t\t\\begin{enumerate}[(a)]\n\t\t\t\\item Let $u$ be a subsolution to $(1 - \\lambda H_1) f = h$, then for all $(f,g) \\in H_1$ and $x_0 \\in E$ such that\n\t\t\t\\begin{equation*}\n\t\t\t\tu(x_0) - f(x_0) = \\sup_x u(x) - f(x)\n\t\t\t\\end{equation*}\n\t\t\tthere exists a $z \\in E'$ such that \n\t\t\t\\begin{equation*}\n\t\t\t\tu(x_0) - \\lambda g(x_0,z)  \\leq h(x_0).\n\t\t\t\\end{equation*}\n\t\t\t\\item Let $v$ be a supersolution to $(1 - \\lambda H_2) f = h$, then for all $(f,g) \\in H_2$ and $x_0 \\in E$ such that\n\t\t\t\\begin{equation*}\n\t\t\t\tv(x_0) - f(x_0) = \\inf_x v(x) - f(x)\n\t\t\t\\end{equation*}\n\t\t\tthere exists a $z \\in E'$ such that \n\t\t\t\\begin{equation*}\n\t\t\t\tv(x_0) - \\lambda g(x_0,z)  \\geq h(x_0).\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{lemma}\nFor a proof of the above Lemma see Lemma 5.7 of \\cite{KraSch}.\n\n\\begin{proof}[Proof of Lemma \\ref{lemma:viscosity_solutions_arrows_based_on_eigenvalue}]\nWe only prove the subsolution statement. Fix $\\lambda > 0$ and $h \\in C_b(E)$. \nLet $u$ be a subsolution of $(1 - \\lambda H_1 ) f = h$. We prove that it is also a subsolution of $(1 - \\lambda H_\\dagger) f = h$. Let $f^\\eta_1 = (1-\\eta)f + \\eta \\Upsilon \\in \\mathcal{D}(H_1)$ and let $x_0$ be such that\n\\begin{equation*}\nu(x_0) - f^\\eta_1(x_0) = \\sup_x u(x) - f_1^\\eta(x).\n\\end{equation*}\nFor each $\\delta > 0$, since $\\mathcal{H}(x,p)$ is a principal eigenvalue for $L_{x,p}+R_x$ (as remarked in Proposition \\ref{hamiltonian:prop}, there exists a function $g$ such that \n\\begin{equation} \\label{eqn:inequality_based_on_ev}\n\\mathcal{H}(x_0,\\nabla f(x_0)) = g^{-1}\\left( L_{x_0,\\nabla f(x_0)} + R_{x_0}\\right)g.\n\\end{equation}\nAs\n\\begin{equation*}\n\\left(f^\\eta_1, (1-\\eta) g^{-1}\\left( L_{x_0,\\nabla f(x_0)} + R_{x_0}\\right)g + \\eta C_\\Upsilon \\right) \\in H_1,\n\\end{equation*}\nwe find by the subsolution property of $u$ and that there exists $z$ such that\n\\begin{align}\nh(x_0) & \\geq u(x_0) - \\lambda \\left((1-\\eta) g^{-1}\\left( L_{x_0,\\nabla f(x_0)} + R_{x_0}\\right)g + \\eta C_\\Upsilon\\right) \\\\\n& = u(x_0) - \\lambda \\left((1-\\eta)  \\mathcal{H}(x_0,\\nabla f(x_0)) + \\eta C_\\Upsilon \\right)\n\\end{align}\nwhere the second inequality follows by \\eqref{eqn:inequality_based_on_ev} and it establishes that $u$ is a subsolution for $(1 - \\lambda H_\\dagger )f = h$.\n\\end{proof}\n\nWe conclude this subsection proving the right part of Figure \\ref{SF:fig:CP-diagram-in-proof-of-CP}.\n\\begin{proposition}\\label{right-diag:prop}\nLet the map $\\mathcal{H}: \\mathbb{R}^d\\times {{\\mathbb R}}^d \\to \\mathbb{R}$ be the eigenvalue \\eqref{eigen:repr} and let $\\mathbf{H} : \\mathcal{D}(\\mathbf{H}) \\subseteq C^1(\\mathbb{R}^d) \\to C(\\mathbb{R}^d)$ be the operator $\\mathbf{H} f(x) := \\mathcal{H}(x,\\nabla f(x))$. Then, for all $\\lambda > 0$ and $h \\in C(\\mathbb{R}^d)$, every viscosity subsolution of $(1 - \\lambda \\mathbf{H}) f = h$ is also a viscosity subsolutions of $(1 - \\lambda H_\\dagger) f = h$ and every viscosity supersolution of $(1-\\lambda \\mathbf{H})f=h$ is also a viscosity supersolution of $(1-\\lambda H_\\ddagger)f=h$.\n\\end{proposition}\n\\begin{proof}\nFix $\\lambda > 0$ and $h \\in C_b(E)$. Let $u$ be a subsolution to $(1- \\lambda \\mathbf{H})f = h$. We prove it is also a subsolution to $(1 - \\lambda H_\\dagger) f = h$.\n\\smallskip\nFix $\\eta > 0 $ and $f\\in C_\\ell^\\infty(E)$ and let $(f^\\varepsilon_\\dagger,H^\\eta_{\\dagger,f}) \\in H_\\dagger$ as in Definition \\ref{aux}. We will prove that\n\\begin{equation}\n\\left(u-f_\\dagger^\\eta\\right)(x) = \\sup_{x\\in E}\\left(u(x)-f_\\dagger^\\eta(x) \\right),\\label{eqn:proof_lemma_conditions_for_subsolution_first}\n\\end{equation}\nimplies\n\\begin{equation}\nu(x)-\\lambda H_{\\dagger,f}^\\eta(x) - h(x)\\leq 0.\n\\label{eqn:proof_lemma_conditions_for_subsolution_second}\n\\end{equation}\nAs $u$ is a viscosity subsolution for $(1 - \\lambda \\mathbf{H})f = h$ and $f_\\dagger^\\eta \\in D(\\mathbf{H})$, if\n\\begin{equation}\n\\left(u-f^\\eta_\\dagger\\right)(x) = \\sup_x \\left(u(x)-f^\\eta_\\dagger(x)\\right), \\label{eqn:visc_subsol_sup_dagger}\n\\end{equation}\nthen,\n\\begin{equation}\nu(x) - \\lambda \\mathbf{H} f^\\eta_\\dagger(x) - h(x) \\leq 0. \\label{eqn:visc_subsol_upperbound_dagger}\n\\end{equation}\nConvexity of $p \\mapsto \\mathcal{H}(x,p)$ yields the estimate\n\\begin{align*}\n\\mathbf{H} f_\\eta(x) &= \\mathcal{H}(x,\\nabla f_\\eta(x)) \\\\\n& \\leq (1-\\eta) \\mathcal{H}(x,\\nabla f(x)) + \\eta \\mathcal{H}(x,\\nabla \\Upsilon(x)) \\\\\n&\\leq (1-\\eta) \\mathcal{H}(x,\\nabla f(x)) + \\eta C_\\Upsilon = H^\\eta_{\\dagger,f}(x).\n\\end{align*} \nCombining this inequality with \\eqref{eqn:visc_subsol_upperbound_dagger}, we have\n\\begin{equation}\nu(x) - \\lambda H^\\eta_{\\dagger,f}(x) - h(x)\n\\leq u(x) - \\lambda \\mathbf{H} f^\\eta_\\dagger(x) - h(x) \\leq 0,\n\\end{equation}\nestablishing \\eqref{eqn:proof_lemma_conditions_for_subsolution_second}. The supersolution statement follows in a similar way.\n\\end{proof}\n\n\\subsection{Exponential tightness}\\label{sub:expo}\n\t\nTo establish exponential tightness, we first note that by \\cite[Corollary 4.19]{FK} it suffices to establish the exponential compact containment condition. This is the content of the next proposition.\n\\begin{proposition} \\label{proposition:exponential_compact_containment}\nFor all $K\\subset E$ compact, $T >0$ and $a > 0$ there is a compact set $\\hat{K}_{K,T,a} \\subset E$ such that \n\\begin{equation}\\label{compactcontainment}\n\\limsup_{\\varepsilon \\rightarrow 0} \\varepsilon \\log \\mathbb{P}\\left[\\bigcup_{t\\in[0,T]}\\left\\{ X_\\varepsilon(t) \\notin \\hat{K}_{K,T,a}\\right\\}\\neq \\emptyset \\right] \\leq \\max\\{- a, \\limsup_{\\varepsilon \\to 0} \\varepsilon \\log \\mathbb{P}(X_\\varepsilon(0) \\notin K)\\}.\n\\end{equation}\t\n\\end{proposition}\t\n\\begin{remark}\nNote that, since $X_\\varepsilon(0)$ satisfies the large deviations principle by assumption, inequality \\eqref{compactcontainment} gives the searched compact containment condition.\n\\end{remark}\n\\begin{proof}[Proof of Proposition \\ref{proposition:exponential_compact_containment}]\nFirst of all let's consider $\\varphi \\equiv 0$. Note that, by Lemma \\ref{contfun_log_lemma}, we have $\\sup_{x,z} H_0(x,\\nabla\\Upsilon,z)= \\sup_{x,z}V_{x,\\nabla \\Upsilon(x)}(z)\\leq C_{\\Upsilon}$. Choose $\\beta > 0$ such that $T C_\\Upsilon - \\beta \\leq -a$. \nSince $\\Upsilon$ is continuous, there is some $c$ such that the set $G := \\left\\{x \\, \\middle| \\, \\Upsilon(x) < c + \\beta \\right\\}$ is non empty. Note that $G$ is open and let $\\overline{G}$ be the closure of $G$. Then, $\\overline{G}$ is compact.\nLet $f(x) := \\iota \\circ \\Upsilon $ where $\\iota$ is some smooth increasing function such that\n\\begin{equation*}\n\\iota(r) = \\begin{cases}\nr & \\text{if } r \\leq \\beta +c, \\\\\n2\\beta+ c & \\text{if } r \\geq \\beta + c + 2.\n\\end{cases}\n\\end{equation*}\nIt follows that $\\iota \\circ \\Upsilon$ equals $\\Upsilon$ on $\\overline{G}$ and is constant outside of a compact set. Set $f_\\varepsilon = f \\circ \\eta_\\varepsilon$, $g_\\varepsilon = H_\\varepsilon f_\\varepsilon$ and $g = H_{f,\\varphi}$. Note that $g(x,z) = H_{\\varphi}(x,\\nabla \\Upsilon(x),z)$ if $x \\in \\overline{G}$. Therefore, we have $\\sup_{x \\in \\overline{G}, z \\in E'} g(x,z) \\leq C_{\\Upsilon}$.\nLet $\\tau$ be the stopping time $\\tau := \\inf \\left\\{t \\geq 0 \\, \\middle| \\, X_\\varepsilon(t) \\notin \\overline{G} \\right\\}$ and let\n\\begin{equation*}\nM_\\varepsilon(t) := \\exp\\left\\{\\frac{1}{\\varepsilon}\\left( f(X_\\varepsilon(t)) - f(X_\\varepsilon(0)) - \\int_0^t g_\\varepsilon(X_\\varepsilon(s),I_\\varepsilon(t))  \\mathrm{d} s \\right)\\right\\}.\n\\end{equation*}\nBy construction $M_\\varepsilon$ is a martingale. Let $K\\subset E$ be compact. We have\n\\begin{align*}\n& \\mathbb{P}\\left[\\bigcup_{t\\in[0,T]}\\left\\{X_\\varepsilon(t) \\notin \\overline{G}\\right\\}\\neq \\emptyset \\right] \\\\\n& \\leq \\mathbb{P}\\left(X_\\varepsilon(0) \\in K , \\bigcup_{t \\in [0,T]}\\left\\{X_\\varepsilon(t) \\notin \\overline{G}\\right\\}\\right) + \\mathbb{P}\\left(X_\\varepsilon (0) \\notin K\\right)\\\\\n& = \\mathbb{E}\\left[\\mathbbm{1}_{\\{X_\\varepsilon(0)\\in K\\}}\\mathbbm{1}_{\\left\\{\\bigcup_{t \\in [0,T]}\\left\\{X_\\varepsilon(t) \\notin \\overline{G}\\right\\}\\right\\}}M_\\varepsilon(\\tau) M_\\varepsilon(\\tau)^{-1} \\right] +\\mathbb{P}\\left(X_\\varepsilon (0) \\notin K\\right) \\\\\n& \\leq \\exp\\left\\{ -\\frac{1}{\\varepsilon} \\left(\\inf_{y_1 \\notin \\overline{G}} f(y_1)- f(X_\\varepsilon(0)) \\right. \\right. \\\\\n& \\hspace{4cm} \\left. \\left. - T \\sup_{y_2 \\in \\overline{G}, i \\in \\{1,\\dots,J\\}} g_\\varepsilon(y_2,i) \\right) \\right\\} \\\\\n& \\hspace{2.5cm} \\times  \\mathbb{E}\\left[\\mathbbm{1}_{\\{X_\\varepsilon(0)\\in K\\}}\\mathbbm{1}_{\\left\\{\\bigcup_{t \\in [0,T]}\\left\\{X_\\varepsilon(t) \\notin \\overline{G}\\right\\}\\right\\}}M_\\varepsilon(\\tau) \\right] + \\mathbb{P}\\left(X_\\varepsilon (0) \\notin K\\right).\n\\end{align*}\nSince $\\sup_{x \\in \\overline{G}, z \\in E'} g(x,z) \\leq C_{\\Upsilon,\\varphi}$ and $g$ is the limit of $g_\\varepsilon$ for $\\varepsilon \\to 0$ in the sense of Definition \\ref{def:convergence}, we obtain that the term in the exponential is bounded by $ \\frac{1}{\\varepsilon}\\left(C_\\Upsilon T - \\beta \\right) \\leq -  \\frac{1}{\\varepsilon} a$ for sufficiently small $\\varepsilon$. The expectation is bounded by $1$ due to the martingale property of $M_\\varepsilon(\\tau)$.\nWe can conclude that\n\\begin{equation*}\n\\limsup_{\\varepsilon \\to 0} \\varepsilon \\log \\mathbb{P} \\left[\\bigcup_{t\\in[0,T]} \\left\\{X_\\varepsilon(t) \\notin K_{T,a}\\right\\}\\neq\\emptyset\\right] \\leq \\max\\{-a, \\limsup_{\\varepsilon \\to 0}\\varepsilon \\log\\mathbb{P}\\left(X_\\varepsilon (0) \\notin K\\right)\\}\n\\end{equation*}\nwhere $\\hat{K}_{K,T,a}=\\overline{G}$.\n\\end{proof}\n\n\n\\begin{subsection}{Action-integral representation of the rate function}\\label{rate}\nIn this section we establish a representation of the rate function as an integral of a Lagrangian function $\\mathcal{L}$. We refer to this representation as the \"action-integral representation\" of the rate function $\\mathcal{I}$. We argue on basis of Section 8 of \\cite{FK} for which we need to check the following two conditions.\n\\begin{lemma}\\label{lemmaRepre}\nLet $\\mathcal{H}: \\mathbb{R}^d\\times {{\\mathbb R}}^d \\to \\mathbb{R}$ be the map given in \\eqref{eigen:repr} and $\\mathbf{H} : \\mathcal{D}(\\mathbf{H}) \\subseteq C^1(\\mathbb{R}^d) \\to C(\\mathbb{R}^d)$ the operator $\\mathbf{H} f(x) := \\mathcal{H}(x,\\nabla f(x))$. Then:\n\\begin{enumerate}[(i)]\n\\item\nThe Legendre-Fenchel transform $\\mathcal{L}(x,v) := \\sup_{p \\in \\mathbb{R}^d} (p\\cdot v  - \\mathcal{H}(x,p))$ and the operator $\\mathbf{H}$ satisfy Conditions 8.9, 8.10 and 8.11 of \\cite{FK}.\n\\item For all $\\lambda > 0$ and $h \\in C(\\mathbb{R}^d)$, the comparison principle holds for\n$(1 - \\lambda \\mathbf{H}) u = h.$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nTo prove the first aim, we will show that following items (a), (b) and (c) imply Condition 8.9, 8.10 and 8.11 of \\cite{FK}. Then, the proof of (a), (b), (c) is shown in \\cite[Proposition 6.1]{PelSch}.\n\\begin{enumerate}[(a)]\n\\item The function $\\mathcal{L}:\\mathbb{R}^d\\times {{\\mathbb R}}^d \\rightarrow[0,\\infty]$ is lower semicontinuous and for every $C \\geq 0$, the level set\n$\n\\{(x,v)\\in \\mathbb{R}^d\\times {{\\mathbb R}}^d\\,:\\,\\mathcal{L}(x,v)\\leq C\\}\n$\nis relatively compact in $\\mathbb{R}^d \\times {{\\mathbb R}}^d$.\n\\item For all $f\\in\\mathcal{D}(H)$ there exists a right continuous, nondecreasing function $\\psi_f:[0,\\infty)\\rightarrow[0,\\infty)$ such that for all $(x,v)\\in \\mathbb{R}^d \\times \\mathbb{R}^d$,\n\\[\n|\\nabla f(x)\\cdot v|\\leq \\psi_f(\\mathcal{L}(x,v))\\qquad \\text{and} \\qquad\n\\lim_{r\\rightarrow\\infty}\\frac{\\psi_f(r)}{r}=0.\n\\]\n\\item For each $x_0\\in {{\\mathbb R}}^d$ and every $f\\in\\mathcal{D}(\\mathbf{H})$, there exists an absolutely continuous path $x : [0,\\infty) \\to \\mathbb{R}^d$ such that $x_0=x(0)$ and\n\\begin{equation}\n\\int_0^t \\mathcal{H}(x(s), \\nabla f (x(s))) \\, ds\n=\n\\int_0^t \\left[\n\\nabla f(x(s)) \\cdot \\dot{x}(s) - \\mathcal{L}(x(s),\\dot{x}(s))\n\\right] \\, ds.\n\\label{eq:action_integral:optimal_path_for_H}\n\\end{equation}\n\\end{enumerate}\nThen regarding Condition 8.9, the operator $A f(x,v) := \\nabla f(x) \\cdot v$ on the domain $\\mathcal{D}(A) = \\mathcal{D}(H)$ satisfies (1). For (2), we can choose $\\Gamma = \\mathbb{T}^d \\times \\mathbb{R}^d$, and for $x_0 \\in \\mathbb{T}^d$, take the pair $(x,\\lambda)$ with $x(t) = x_0$ and $\\lambda(dv \\times dt) = \\delta_{0} (dv) \\times dt$. Part (3) is a consequence of (a) from above. Part (4) can be verified as follows. Let $\\Upsilon$ the containment function used in Definition \\ref{aux} and note that the sub-level sets of $\\Upsilon$ are compact. Let $\\gamma \\in \\mathcal{AC}$ with $\\gamma(0)\\in K$ and such that the control\n\\begin{equation}\n\\int_0^T \\mathcal{L}(\\gamma(s),\\dot{\\gamma}(s))\\, ds \\leq M\n\\end{equation}\nimplies $\\gamma(t) \\in \\hat{K}$ for all $t\\leq T$, with $\\hat{K}$ compact. Then,\n\\begin{equation}\n\\begin{aligned}\n\\Upsilon(\\gamma(t))&=\\Upsilon(\\gamma(0))+\\int_0^t \\left\\langle\\nabla\\Upsilon(\\gamma(s)),\\dot{\\gamma}(s)\\right\\rangle \\,ds\\\\\n&\\leq \\Upsilon(\\gamma(0))+\\int_0^t \\mathcal{L}(\\gamma(s),\\dot{\\gamma}(s)) + \\mathcal{H}(\\gamma(s),\\nabla \\Upsilon(\\gamma(s))) \\,ds\\\\\n& \\leq \\sup_{y\\in K} \\Upsilon(y) + M +\\int_0^T \\sup_z {{\\mathcal H}}(z, \\nabla \\Upsilon(z)) \\,ds\\\\\n&:= C < \\infty.\n\\end{aligned}\n\\end{equation}\nHence, we can take $\\hat{K} = \\{z \\in {{\\mathbb R}}^d | \\Upsilon(z) \\leq C\\}$.\n\nPart (5) is implied by (b) from above. Condition 8.10 is implied by Condition 8.11 and the fact that $\\mathbf{H}1 = 0$, by Theorem \\ref{conv:th} (see Remark 8.12 (e) in \\cite{FK}). Finally, Condition 8.11 is implied by (c) above, with the control $\\lambda(dv \\times dt) = \\delta_{\\dot{x}(t)}(dv) \\times dt$.\n\nThe comparison principle for $\\mathbf{H}$ follows from Proposition \\ref{right-diag:prop} and Theorem \\ref{theorem:comparison_from_otherpaper}.\n\\end{proof}\nIn the following, we prove the integral representation of the rate function. Firstly, let's recall that Theorem \\ref{FengKurtz:th} gives the existence of a semigroup $V(t)$ and a family of functions $R(\\lambda)$ and let $\\mathbf{V}(t):C({{\\mathbb R}}^d)\\to C({{\\mathbb R}}^d)$ be the Nisio semigroup with cost function $\\mathcal{L}$, that is\n\\begin{equation}\n\\mathbf{V}(t) f(x)=\\sup_{ \\substack{ \\gamma \\in \\mathrm{AC}_{\\mathbb{R}^d}[0,\\infty) \\\\ \\gamma(0) = x}}\\left[f(\\gamma(t)) - \\int_0^t \\mathcal{L}(\\gamma(t),\\dot{\\gamma}(s)) \\, ds\\right].\n\\end{equation}\nLet $\\mathbf{R}(\\lambda)h$ be the operator given by\n\\begin{equation*}\n\t\t\t\\mathbf{R}(\\lambda) h(x) = \\sup_{\\substack{\\gamma \\in \\mathcal{A}\\mathcal{C}\\\\ \\gamma(0) = x}} \\int_0^\\infty \\lambda^{-1} e^{-\\lambda^{-1}t} \\left[h(\\gamma(t)) - \\int_0^t \\mathcal{L}(\\gamma(s),\\dot{\\gamma}(s))\\right] \\,  \\mathrm{d} t.\n\\end{equation*}\n\nThe proof of the result below is based on the following four main steps.\n\\begin{itemize}\n\\item Figure \\ref{SF:fig:CP-diagram-in-proof-of-CP} on page \\pageref{SF:fig:CP-diagram-in-proof-of-CP} shows that $R(\\lambda)h$ is the unique function that is a sub- and supersolution to the equations $(1 - \\lambda H_\\dagger )f = h$ and $(1 - \\lambda H_\\ddagger) f = h$ respectively.\n\\item $\\mathbf{R}(\\lambda) h$ has been proven to be the unique viscosity solution to $(1 - \\lambda \\mathbf{H}) f = h$. Then, again by  Figure \\ref{SF:fig:CP-diagram-in-proof-of-CP}, we must have $R(\\lambda) h = \\mathbf{R}(\\lambda)h$. \n\\item Starting from the equality of resolvents we work to an equality for the semigroups $V(t)$ and $\\bf{V}(t)$. \n\\item Recalling that the rate function in Theorem \\ref{maintheorem} is given by,\n\\begin{equation}\nI(x)= I_0(x(0)) + \\sup_{k\\in \\mathbb{N}} \\sup_{(t_1,\\dots,t_k)} \\sum_{i=1}^k I_{t_i-t_{i-1}}(x(t_i)\\vert x(t_{i-1}))\n\\end{equation}\nwith $I_t(z\\vert y)=\\sup_{f\\in C(E)}[f(z)-V(t)f(y)]$, it is not difficult to realise that, if $V(t)=\\mathbf{V}(t)$, it follows that $I_t(y\\vert z)= \\inf_{\\substack{\\gamma : \\gamma(0)=z, \\\\ \\gamma(t)=y}} \\int_0^t \\mathcal{L}(\\gamma(s),\\dot{\\gamma}(s))\\, ds$.\n\\end{itemize}\n\n\n\\begin{theorem}[Integral representation of the rate function]\\label{ratefunction:th}\nThe rate function of Theorem \\ref{maintheorem} has the following representation\n\\begin{equation}\n\\mathcal{I}(x)=\\begin{cases}\n\\mathcal{I}_0(x(0)) +\\int_0^\\infty\\mathcal{L} \\left(x(t),\\dot{x}(t)\\right) \\, dt &\\quad \\text{if } x\\in AC([0,\\infty); \\mathbb{R}^d),\\\\\n\\infty & \\quad \\text{else},\n\\end{cases}\n\\end{equation}\nwhere %\n$\\mathcal{L}(x,v) =\\sup_{p \\in \\mathbb{R}^d}\\left[p \\cdot v  - \\mathcal{H}(x,p)\\right]\n$\nis the Legendre transform of $\\mathcal{H}$.\n\\end{theorem}\n\n\\begin{proof}\nFollowing the above mentioned steps, we recall that, as stated by Theorem \\ref{FengKurtz:th}, there exists a family of operators $R(\\lambda) : C_b({{\\mathbb R}}^d) \\rightarrow C_b({{\\mathbb R}}^d)$, such that for $\\lambda > 0$ and $h \\in C_b({{\\mathbb R}}^d)$, the function $R(\\lambda)h$ is the unique function that is a viscosity solution to $(1- \\lambda H ) f = h$ and such that\n\\begin{equation} \\label{eqn:convergence_R_to_V}\n\\lim_{m \\rightarrow \\infty} \\vn{R\\left(\\frac{t}{m}\\right)^m f  - V(t)f  } = 0 \\qquad \\text{for all $f$ in a dense set $D\\subseteq C_b({{\\mathbb R}}^d)$.}\n\\end{equation}\nSee also \\autocite[Theorem 7.10]{Kraaij2020} or \\autocite[Theorem 7.17]{FK} for the construction of the operators $R(\\lambda)$.\nBy \\autocite[Proposition 6.1]{KraSch} (or \\cite[Chapter 8]{FK}), $\\mathbf{R}(\\lambda)$ is the unique viscosity solution to $(1 - \\lambda \\mathbf{H}) f = h$. Then, Figure \\ref{SF:fig:CP-diagram-in-proof-of-CP} on page \\pageref{SF:fig:CP-diagram-in-proof-of-CP} shows that it must equal $R(\\lambda) h $. \nMoreover, we find by \\cite[Lemma 8.18]{FK} (whose assumptions are implied by Lemma \\ref{lemmaRepre} above) that for all $f \\in C_b({{\\mathbb R}}^d)$ and $x \\in {{\\mathbb R}}^d$\n\\begin{equation} \\label{eqn:convergence_bfR_to_bfV}\n\t\t\t\\lim_{m \\rightarrow \\infty} \\mathbf{R}\\left(\\frac{t}{m}\\right)^m f(x) = \\mathbf{V}(t)f(x).\n\\end{equation}\nWe conclude from \\eqref{eqn:convergence_R_to_V} and \\eqref{eqn:convergence_bfR_to_bfV} that $V(t)f = \\mathbf{V}(t)f$ for all $t$ and $f \\in D$.\nNow recall that $D$ is sequentially strictly dense so that equality for all $f \\in C_b({{\\mathbb R}}^d)$ follows if $V(t)$ and $\\mathbf{V}(t)$ are sequentially continuous. The first statement follows by Theorems \\cite[Theorem 7.10]{Kraaij19} and \\cite[Theorem 6.1]{Kraaij2020}. The second statement follows by \\cite[Lemma 8.22]{FK}. We conclude that $V(t)f = \\mathbf{V}(t)f$ for all $f \\in C_b({{\\mathbb R}}^d)$ and $t \\geq 0$. Using Theorem 8.14 of \\cite{FK} and the convexity of $v\\mapsto \\mathcal{L}(x,v)$ we get the integral representation.\n\n\n\n\n\\end{proof}\n\n\\end{subsection}\n\n\n\n\n\n\\begin{section}{A more general theorem}\\label{general}\n\nAnalysing the proofs in the previous sections, we can state the following facts:\n\\begin{itemize}\n\\item In the proof of large deviations principle, the main steps are:\n\\begin{enumerate}\n\\item Convergence of the nonlinear operators $H_\\varepsilon$ to a multivalued operator $H$,\n\\item comparison principle for $(1-\\lambda H)f= h$.\n\\end{enumerate}\n\\item The existence of an eigenvalue ${{\\mathcal H}} (x,p)$ and its convexity, coercivity and continuity are crucial for the proof of comparison principle and\n\\begin{itemize}\n\\item the arguments for existence, convexity and coercivity (proofs of Propositions \\ref{eigenvalueProb} and \\ref{conv:th}) are based on the fact that ${{\\mathcal H}} (x,p)$ is the eigenvalue of an operator of the type $B_{x,p} + V_{x,p} + R_x$ with the three operators that verify particular properties such as coercivity and the maximum principle,\n\\item to show the continuity of ${{\\mathcal H}}$ the representation \\eqref{eigen:repr} is needed. In particular, some properties of $V$ and $I$, like $\\Gamma$- convergence, are necessary.\n\\end{itemize}\n\\end{itemize}\nThe above observations allow for a straightforward generalization in Theorem \\ref{maintheorem_gen} and justify the assumptions of the next subsection. In this section we indeed prove the large deviation principle for a general switching Markov process. In particular, we will study the Markov process $(X_t^\\varepsilon, I_t^\\varepsilon)$, that is the solution to the  Martingale problem corresponding to the following operator\n\n\\begin{equation}\\label{generator_gen}\nA_\\varepsilon f(x,i)\n:=\nA_\\varepsilon^{(i)} f(\\cdot,i) (x)\n+\n\\sum_{j = 1}^J r_{ij}(x,x\/\\varepsilon) \n\\left[\nf(x,j) - f(x,i)\n\\right]\n\\end{equation}\nwith $\nA_\\varepsilon^{(i)}  : \\mathcal{D}(A_\\varepsilon^{(i)}) \\subseteq C({{\\mathbb R}}^d) \\rightarrow C({{\\mathbb R}}^d)\n$\nbe the generator of a strong ${{\\mathbb R}}^d$-valued Markov process, with domain $\\mathcal{D}(A_\\varepsilon^{(i)})$.\n \n\\begin{subsection}{Assumptions}\\label{assumption}\n\nHere we give the assumptions needed.\n\n\\begin{assumption}\\label{assumption1}\nThe nonlinear generators $H_\\varepsilon f = \\varepsilon e^{-f\/\\varepsilon} A_\\varepsilon e^{f\/\\varepsilon}$ admits an extended limit $H \\subseteq ex - LIM H_\\varepsilon$ with $H$ of the type \n\\begin{align*}\nH:=\\left\\{(f, H_{f,\\varphi}) \\, : \\,f \\in C^2(\\mathbb{R}^d), \\; H_{f, \\varphi} \\in C( \\mathbb{R}^d\\times E^\\prime) \\text{ and }\\varphi \\in C^2(E^\\prime)\\right\\}.\n\\end{align*}\nFor all $\\varphi$ there exist a map $H_{\\varphi}:{{\\mathbb R}}^d \\times {{\\mathbb R}}^d \\times E^\\prime$ such that for all $f\\in D(H)$, $x\\in{{\\mathbb R}}^d$ and $z\\in E^\\prime$, $H_{\\varphi,f}(x,z)=H_{\\varphi}(x,\\nabla f, z)$.\nMoreover, the image $H_\\varphi$ has the representation\n$$H_\\varphi(x,p,z)=e^{- \\varphi(z)}\\left[B_{x,p} +V_{x,p}+R_x\\right]e^\\varphi (z)$$\nwith $p=\\nabla f(x)$ and $B_{x,p}, V_{x,p}, R_x$ such that\n\\begin{enumerate}[(i)]\n\\item For all $p\\in {{\\mathbb R}}^d $ there exists an eigenfunction $g_{x,p} \\in C^2({{\\mathbb R}}^d \\times J)$ with $g_{x,p}^i>0$ and an eigenvalue $\\mathcal{H}(x,p)$ such that\n$\n\\left[B_{x,p} +V_{x,p}+R_x\\right]g_{x,p}=\\mathcal{H}(x,p)g_{x,p}.\n$\n\\item $T_{x,p}=B_{x,p} + R_x$ verifies the maximum principle : \n\nif $(i_m,y_m)=\\argmin {\\varphi}$ then $e^{-\\varphi(i_m,y_m)}T_{x,p}e^{\\varphi(i_m,y_m)} \\geq 0$.\n\\item $p \\mapsto V_{x,p}$ is coercive uniformly with respect to $x$.\n\\item $p \\mapsto B_{x,p}$ and $p \\to V_{x,p}$ are convex uniformly on compact sets.\n\\end{enumerate}\n\n\\end{assumption}\n\nThe above assumption implies the convergence of the nonlinear operators and the existence of the principal eigenvalue $\\mathcal{H}$. Moreover, it will imply convexity and coercivity of $\\mathcal{H}(x,p)$.\n\n\\begin{assumption}\\label{assumption2}\nThe eigenvalue $\\mathcal{H}$ is of the type $\\mathcal{H}(x,p) = \\sup_{\\mu \\in \\mathcal{P}(E')} \\left[\\Lambda (x,p,\\mu) - I_{x,p}(\\mu)\\right]$ with \n\\begin{equation}\n\\Lambda(x,p,\\mu)=\\int_{E^\\prime}V_{x,p}\\,d\\mu, \\qquad\\text{and}\\qquad I_{x,p}(\\mu)= -\\inf_{u>0}\\int_{E^\\prime}\\frac{(B_{x,p}+R_x)u}{u}\\, d\\mu,\n\\end{equation}\nand the following properties hold\n\\begin{enumerate}[(i)]\n\\item $I_{x,p}$ satisfies the assumption of Lemma \\ref{gammaConv},\n\\item $\\Lambda (x,p,\\mu)$ is continuous and $\\norma{\\Lambda(x,p,\\mu)}_\\Theta < \\infty$,\n\\item there exists a containment function $\\Upsilon$ for $\\Lambda$ in the sense of Definition \\ref{contfun_def},\n\\item for all $x$, there exists a unique measure $\\mu_x^*$ such that $I_{x,0}(\\mu_x^*)=0$.\n\\end{enumerate}\n\\end{assumption}\nAssumption \\ref{assumption2} implies the continuity of $\\mathcal{H}$.\n\n\\end{subsection}\n\n\\begin{subsection}{Large Deviation for a Switching Markov process}\\label{mainTheorem-sub}\n\nWe are ready to state the general theorem. \n\\begin{theorem}[Large deviation for a Switching Markov process]\\label{maintheorem_gen}\nLet $(X_t^\\varepsilon, I_t^\\varepsilon)$ be the solution of the Martingale problem corresponding to the operator given in \\eqref{generator_gen}.\nIf Assumptions \\ref{assumption1} and \\ref{assumption2} hold and suppose further that at time zero, the family of random variables $\\{X^\\varepsilon(0)\\}_{\\varepsilon > 0}$ satisfies a large deviation principle in $\\mathbb{R}^d$ with good rate function $\\mathcal{I}_0 :\\mathbb{R}^d \\rightarrow [0,\\infty]$. Then, the spatial component $\\{X_t^\\varepsilon\\}$ satisfies a large deviation principle in $C_{\\mathbb{R}^d}[0,\\infty)$. \n\\end{theorem}\nThe proof of the above theorem follows the same lines of what is done in Subsection \\ref{comparison-sub}. \n\\end{subsection}\n\\end{section}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\review{Fluid-structure interaction (FSI) problems often occur in  Engineering (aircraft and automotive industries, wind turbines) as well as in medical applications (cardiovascular systems, artificial organs, artificial valves, medical devices, etc.). Today the design of such systems usually requires advanced studies and high-fidelity (HF) numerical simulations become an essential tool of computed-aided analysis. However, computational FSI is known to be very time-consuming even on high-performance computing facilities. Usually, engineering problems are parameterized and the search of suitable designs require numerous computer experiments leading to prohibitive computational times. \nFor particular applications such as the tracking of drug carrier capsules flowing in blood vessels, it would be ideal to have real-time simulations for a better understanding of the behaviour of the dynamics and for efficiency assessment. Unfortunately, today high-fidelity real-time FSI simulations are far from being reached with current High Performance Computing (HPC) facilities. \n\nA current trend is to use machine learning (ML) or artificial intelligence (AI) tools such as artificial neural networks (ANN). Such tools learn numerical simulations from HF solvers and try to map entry parameters with output criteria in an efficient way, with response times far less than HF ones, say 3 or 4 orders of magnitude smaller. In some sense, heavy HF computations and training stage are done in an offline stage, and learned ANNs can be used online for real time evaluations and analysis. However, ML and ANN today are not fully satisfactory for dynamical problems, and\/or the training stage itself may be time consuming, thus requiring more Central Processing Unit (CPU) time.\nAnother option is the use of model order reduction (MOR). Reduced-order modeling (ROM) can be seen as a 'grey-box' supervized ML methodology, taking advantage of the expected low-order dimensionality of the FSI mechanical problem. By 'grey-box' we mean that the low-dimensional encoding of the ML process is based on mechanical principles and a man-made preliminary dimensionality reduction study. This allows one for a better control of the ROM accuracy and behaviour. \nThere are two families of MOR: intrusive and non-intrusive approaches. The intrusive approaches use physical equations. The low-order model is derived by setting the physical problem on a suitable low-dimensional space. The accuracy can be very good, but the price to pay is the generation of a new code which can be a tedious and long task. The non-intrusive approach does not require heavy code development. It is based on HF simulation results used as entry data. Although it is not based on high-fidelity physical equations, a non-intrusive approach can include a priori physical informations, like e.g. meaningful physical features, prototype of system of equations, pre-computed principal components, consistency with physical principles, etc.\n\nIn the recent literature, efficient intrusive ROMs for FSI\nhave been proposed e.g. in \\citep{Quarteroni2016}. But to our knowledge\nthere are far less contributions in non-intrusive ROMs dedicated to FSI. \n\n\nIn this paper, we propose a data-driven model order reduction approach for FSI problem which is consistent with the equations of kinematics and is designed from a low-order meaningful system of equations.\nAs case of study, we focus on the motion of a microcapsule, a droplet surrounded by a membrane, subjected to a confined and unconfined Stokes flow.  \n\nArtificial microcapsules can be used in various industrial applications such as in cosmetics \\citep{Miyazawa2000, Casanova2016}, food industry \\citep{Yun2021} and biotechnology, where drug targeting is a high potential application \\citep{Ma2013,Abuhamdan2021, Ghiman2022}.\nOnce in suspension in an external fluid, capsules are subjected to hydrodynamics forces, which may lead to large membrane deformation, wrinkle formation or damage. \nThe numerical model must be able to capture the time-evolution of the nonlinear 3D large deformations of the capsule membrane. \nDifferent numerical strategies are possible to solve the resulting large systems of equations \\citep{Lefebvre2007, Hu2012, Ye2017, Tran2020}. However, they all have long computational times.\n\nDifferent approaches have been used over the past decade to accelerate the computations, such as HPC (e.g. \\cite{Zhao2010}) and Graphics Processing units (e.g. \\cite{Matsunaga2014}).\nMore recently, reduced order models have been proposed to predict the motion of capsules suspended in an external fluid flow.\nIn~\\citet{Quesada2021}, the authors used the large amount of data generated by numerical simulations to show how relevant it is to recycle these data to produce lower-dimensional problem using physics-based reduced order models. However, their method can only  predict the steady-state capsule deformed shape. \n\\citet{Boubehziz2021} show for the first time the efficiency of data-driven model-order reduction technique to predict the dynamics of the capsule in a microchannel. However, the method is cumbersome as it requires two POD bases, one to predict the velocity field, the other to capture the shape evolution over time. And then they reconstruct the solution in the parameter space thanks to diffuse approximation (DA) strategy.\n\n\nThe proposed method serves different objectives.\nWe have designed the  method to be non intrusive for practical uses of  existing high-fidelity FSI solver (also referred to as the Full-Order Model, or FOM). That means that the ROM methodology should be data-driven.\nWe also want the ROM to be consistent with the equations of kinematics. The model must thus return the displacement  $\\{u\\}$ and velocity $\\{v\\}$ fields from a few snapshots provided by the FOM. \nIt must otherwise be able to predict the solution for any parameter vector in predefined admissible domain.\nFinally, the kinematics-consistent data-driven reduced-order model of capsule dynamics must ideally open the way to real-time simulations. \nTo do so, we use a coupling between Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD), as well as a Tikhonov regularization for robustness purposes and interpolation to predict solutions for any parameter value. \n\n\nAs indicated above, we mainly consider the case of an initially spherical capsule flowing in a microfluidic channel with a square cross-section.\nThe corresponding FOM was developed by \\citet{Hu2012} and used to get a complete numerical database of the three-dimensional capsule dynamics as a function of the parameters of the problem: the capsule-to-tube confinement ratio, hereafter referred to as size ratio $a\/\\ell$ and the capillary number $Ca$, which measures the ratio between the viscous forces acting onto the capsule membrane and the membrane elastic forces. \nFor clarity reasons, different ROMs are introduced  with increasing levels of generality, as detailed in Table~\\ref{tab:1}. First, we consider a fixed parameter vector, and get a space-time ROM in the form of a low-order dynamical system.  Next, we generate such $N$ ROMs for the~$N$ parameter samples that fill the admissible parameter domain, and then assess the uniform accuracy (space-time accuracy over the whole sample set). Finally, we propose a strategy to derive a general space-time-parameter ROM for any value of the parameter vector $(Ca,a\/\\ell)$ in the admissible space.\n\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{|c|c|c|c|}\\hline\n        Nb of parameter & ROM output type & Verification & Related Section(s)  \\\\ \n        samples for data && (accuracy) & in the paper\\\\ \\hline\\hline\n       1  & 1 space-time ROM & Space-time accuracy & Sections \\ref{sec:ROM} and  \\ref{sec:1example} \\\\ \\hline\n       $N$ & $N$ space-time ROM & Uniform space-time  & Section~\\ref{sec:database} \\\\ \n       && accuracy on the & \\\\ \n       && sample set & \\\\ \\hline\n       $N$ & 1 space-time-parameter ROM & Uniform accuracy & Section~\\ref{sec:interpolation} \\\\\n       & (any parameter couple) && \\\\ \\hline\n    \\end{tabular}\n    \\caption{\\review{General methodology, stepwise procedure for ROM construction of increasing level of generality.}}\n    \\label{tab:1}\n\\end{table}\n\n\nThe paper is organized as follows. First, we present the physics of the problem and the FOM in Section \\ref{sec:FOM}. \nThe  strategy used to develop a non-intrusive space-time ROM is detailed in Section \\ref{sec:ROM}. The results are presented for a given configuration in Section \\ref{sec:1example}. The accuracy is then estimated in Section \\ref{sec:database} on the entire database, formed by all the cases that have reached a stationary state. Finally in Section \\ref{sec:interpolation} we present the methodology of space-time-parameter ROM. The ROM accuracy is confirmed by some numerical experiments.}\n\n\n\\section{Full-order microcapsule model, parameters and quantities of interest}\n\\label{sec:FOM}\n\n\\subsection{Problem description}\n\nAn initially spherical capsule of radius $a$ flows within a long microfluidic channel having a constant square section of side $2\\ell$ (Figure \\ref{IllustrationPb}). The suspending fluid and capsule liquid core are incompressible Newtonian fluids with the same kinematic viscosity $\\eta$.  \n\nThe capsule liquid core is enclosed by a hyperelastic isotropic membrane. Its thickness is assumed to be negligible compared to the capsule dimension. The membrane is thus modeled as a surface devoid of bending stiffness with surface shear modulus $G_S$. \nThe two non-dimensional governing parameters of the problem are the size ratio $a\/\\ell$ and the capillary number \n\\begin{equation}\nCa= \\eta V\/G_S\n\\end{equation}\nwhere $V$ is the mean axial velocity of the undisturbed external Poiseuille flow.\\\\\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{CapsuleInChannel.eps}\n\\caption{Sketch of the model geometry showing an initially spherical capsule of radius $a$ placed in a channel  with a constant square section of side $2\\ell$. }\n\\label{IllustrationPb} \n\\end{figure}\n\nThe flow Reynolds number is assumed to be very small. We solve the Stokes equations in the external ($\\beta = 1$) and internal fluids ($\\beta = 2$), together with the membrane equilibrium equation to determine the dynamics of the deformable capsule within the microchannel. \n\nFor the fluid problem, we denote $\\vbold^{(\\beta)}$, $\\boldsymbol{\\sigma}^{(\\beta)}$ and $p^{(\\beta)}$ the velocity, stress and pressure fields in the two fluids. These parameters are non-dimensionalized using $\\ell$ as characteristic length, $\\ell\/V$ as characteristic time and $G_S \\ell$ as characteristic force. \nThe non-dimensional Stokes equations \n \\begin{equation}\n\\nabla p^{(\\beta)} = Ca \\nabla^2\\vbold^{(\\beta)},\\quad\\nabla\\cdot\\vbold^{(\\beta)}=0,\\quad\\beta=1,2\n\\label{eq:Stokes}\n\\end{equation}\nare solved in the domain bounded by the cross sections $S_{in}$ at the tube entrance and $S_{out}$ at the exit. These cross sections are assumed to be both located far from the capsule. The reference frame $(O,\\bm{x}, \\bm{y}, \\bm{z})$ is centered at each time step on the capsule center of mass $O$ in the high-fidelity code, but the displacement of the capsule center of mass along the tube axis~$\\bm{Oz}$ is computed.\n\nThe boundary conditions of the problem are the following ones:\n\n\\begin{itemize}\n\\item~The velocity field is assumed to be the unperturbed flow field  on $S_{in}$ and $S_{out}$, i.e. the flow disturbance vanish far from the capsule.\n\\item~The pressure is uniform on $S_{in}$ and $S_{out}$.\n\\item~A no-slip boundary condition is assumed at the channel wall $W$ and on the capsule membrane $M$:\n\\begin{equation}\n\\forall \\bm{x}\\in W, \\vbold(\\bm{x})=\\bm{0};\\quad \n\\forall \\bm{x} \\in M,\\ \\vbold(\\bm{x})=\\frac{\\partial \\ubold}{\\partial t}.\n\\label{eq:noslipcondition}\n\\end{equation}\n\\item~The normal load $\\bm{n}$ on the capsule membrane $M$ is continuous, i.e. the non-dimensionalized external load per unit area $\\bm{q}$ exerted by both fluids  is due to the viscous traction jump:\n\\begin{equation} \\label{eq:traction_jump}\n(\\boldsymbol{\\sigma}^{(1)}-\\boldsymbol{\\sigma}^{(2)})\\bcdot\\boldsymbol{n} =  \\bm{q}\n\\end{equation} where $\\boldsymbol{n}$ is the unit normal vector pointing towards the suspending fluid. \n\\end{itemize}\n\nTo close the problem, the external load $\\bm{q}$ on the membrane is deduced from the local equilibrium equation, which, in absence of internia, can be written as\n\\begin{equation}\n  \\bnabla_s \\bcdot \\boldsymbol \\tau + \\bm{q} = \\boldsymbol 0\n  \\label{eq:equi-mb}\n\\end{equation}\nwhere $\\bm{\\tau}$ is the non-dimensionalized Cauchy tension tensor (forces per unit arclength in the deformed plane of the membrane) and $\\bnabla_s \\bcdot$ is the surface divergence operator. We assume that the membrane deformation is governed by the strain-softening neo-Hookean law. The principal Cauchy tensions can then be expressed as\n\\begin{equation}\n\\tau_1 = \\frac{G_S}{\\lambda_1 \\lambda_2}\\left[ \\lambda_1^2 - \\frac{\n1}{(\\lambda_1 \\lambda_2)^2}\\right] ~(\\text{likewise for } \\tau_2),\n\\end{equation}\nwhere $\\lambda_1$ and $\\lambda_2$ are the principal extension ratios measuring the in-plane deformation. \n\n\n\n\\subsection{Numerical procedure}\n\nThe FSI problem is solved by coupling a finite element method that determines the capsule membrane mechanics with a boundary integral method that solves for the fluid flows \\citep{Walter2010,Hu2012}. Thanks to the latter, only the boundaries of the flow domain, i.e the channel entrance $S_{in}$ and exit $S_{out}$, the channel wall and the capsule membrane have to be discretized to solve the problem.\nThe mesh of the initially spherical capsule is generated by subdividing the faces of the icosahedron (regular polyhedron with 20 triangular faces) inscribed in the sphere until reaching the desired number of triangular elements. At the last step, nodes are added at the middle of all the element edges to obtain a capsule mesh with 1280 $P_2$ triangular elements and 2562 nodes, which correspond to a characteristic mesh size $\\Delta h_C= 0.075\\,a$. The channel mesh of the entrance surface $S_{in}$ and exit surface $S_{out}$ and of the channel wall is generated using \n\\texttt{Modulef} (INRIA, France). The central portion of the channel, where the capsule is located, is refined. The channel mesh comprises 3768 $P_1$~triangular elements and 1905~nodes. \n\n\nAt time $t=0$, a spherical capsule is positioned with its center of mass $O$ on the channel axis. \nAt each time step, the in-plane stretch ratio $\\lambda_1$ and $\\lambda_2$ are computed from the nodes deformation.\n The elastic tension tensor $\\boldsymbol \\tau$ is then deduced from the values of $\\lambda_1$ and $\\lambda_2$. The finite element method is used to solve the weak form of the membrane equilibrium equation~\\eqref{eq:equi-mb} and determine the external load $\\bm{q}$.\n\nApplying the boundary integral method, the velocity of the nodes on the capsule membrane reads \\citep{Pozrikidis1992}:\n\\begin{equation}\n\\vbold(\\bm{x}) = \\vbold^\\infty(\\bm{x}) - \\frac{1}{8\\pi\\mu_F} \\left[ \\int_M \\bm{J}(\\bm{r})\\cdot\\bm{q} dS(\\bm{y}) + \\int_W \\bm{J}(\\bm{r})\\cdot \\bm{f} dS(\\bm{y}) -\\Delta P \\int_{S_{out}} \\bm{J}(\\bm{r})\\cdot \\bm{n}\\, dS(\\bm{y})\\right]\n\\label{eq:flo1}\n\\end{equation}\nfor any $\\bm{x}$ in the spatial domain when the suspending and internal fluids have the same viscosity. The vector $\\bm{f}$ is the disturbance wall friction due to the capsule, $\\Delta P$ is the additional pressure drop and $\\bm{r}=\\bm{y}-\\bm{x}$. \n\nTo update the position of the membrane nodes, the nodal displacement $\\ubold$ is computed by integrating equation \\eqref{eq:noslipcondition} in time.\nThe procedure is repeated until the desired non-dimensional time $VT\/\\ell$.\n\nFor later development, it is more convenient to work on the condensed abstract form of the system. The full order semi-discrete FSI system to solve consists of the kinematics and the membrane equilibrium algebraic equations:\n\\begin{align}\n& \\dot\\{u\\rbrace = \\{v\\rbrace, \\qquad t\\in [0,T],\\label{eq:flo2}\\\\\n& \\{v\\rbrace = \\varphi(\\{u\\rbrace) \\label{eq:flo3}\n\\end{align}\nwhere $\\varphi$ is a nonlinear mapping from $\\mathbb{R}^{3d}$ to~$\\mathbb{R}^{3d}$ and $d$ is the number of nodes on the membrane.\nRegarding time discretization, a Runge-Kutta Ralston scheme is used:\n\\begin{align*}\n    & \\{\\widehat u^{n+2\/3}\\} = \\{u^n\\} + \\frac{2}{3}\\Delta t \\, \\{v^{n}\\},\n    \\\\\n    & \\{\\widehat v^{n+2\/3}\\} = \\{\\varphi\\}(\\{\\widehat u^{n+2\/3}\\}), \\\\\n    & \\{u^{n+1}\\} = \\{u^n\\} + \\Delta t \\, \\left(\\frac{1}{4}\\{v^{n}\\}+\\frac{3}{4}\\{\\widehat v^{n+2\/3}\\}\\right),\\\\\n    & \\{v^{n+1}\\} = \\{\\varphi\\}(\\{u^{n+1}\\}), \\\\\n    & \\{u^0\\} = \\{0\\},\\ \\{v^{0}\\} = \\{\\varphi\\}(\\{0\\})\n\\end{align*}\n\\noindent where $\\Delta t>0$ is a constant time step and $\\{u\\rbrace^n$ and $\\{v\\rbrace^n$ respectively represent the discrete membrane displacement field and the discrete membrane velocity field at discrete time~$t^n=n\\Delta t$. The initial condition is simply $\\{u\\rbrace^0=\\lbrace 0\\rbrace $.\n\nThe whole numerical scheme is subject to some Courant-Friedrichs-Lewy (CFL) type stability condition on the time step \\citep{Walter2010} because of its explicit nature.\nThe numerical method is conditionally stable if the time step $\\Delta t$ satisfies\n\\begin{equation}\n\\frac{V}{\\ell}\\Delta t < O\\left( \\frac{\\Delta h_C}{\\ell}Ca\\right).\n\\label{StabilityCondition}\n\\end{equation}\nFrom the computational point of view, the resolution of~\\eqref{eq:flo3} at each time step requires i) the computation of  the disturbance wall friction $\\bm{f}$ at all the wall nodes, ii)  the additional pressure drop $\\Delta P$, iii) the traction jump $\\bm{q}$ at the membrane nodes and iv) the boundary integrals for each node.\nThe resulting numerical FOM may thus be time-consuming, depending on the membrane discretization and the number of time steps. Figure \\ref{SimuTime} shows that the evolution of the computational cost when $a\/\\ell=0.7$, considering the mesh discretization described above and a workstation equipped with 2  processors Intel\\textsuperscript{\\textregistered} Xeon\\textsuperscript{\\textregistered} Gold 6130 CPU (2.1 GHz). A week of computation is sometimes necessary to simulate the dynamics of an initially spherical capsule in a microchannel over the non-dimensional time $VT\/\\ell=10$.    \n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{SimuTime.eps}\n\\caption{Simulation time of the dynamics of the capsule over a non-dimensional time $Vt\/\\ell=10$ ($a\/\\ell=0.7$) according to the time step. Simulations were performed on a workstation equipped with 2  processors Intel\\textsuperscript{\\textregistered} Xeon\\textsuperscript{\\textregistered} Gold 6130 CPU (2.1 GHz).}\n\\label{SimuTime} \n\\end{figure}\n\n\n\n\nFor that reason, a model-order reduction (MOR) strategy is studied in this paper, in order to reduce the computational time by several orders of magnitude. ROMs try to approximate solutions of the initial problem by strongly lowering the dimensionality of the numerical model, generally using a reduced basis (RB) of suitable functions, then derive a low-order system of equations.  \n\n\nIn the case of differential algebraic equations (DAE) like~\\eqref{eq:flo2}-\\eqref{eq:flo3}, the reduced system of equations to find should also be of DAE nature. Remark that it is often possible to reformulate DAEs as a system of ordinary differential equations (ODEs) \\citep{Ascher1998}. \nIn the next section, we give details on the chosen ROM methodology for the particular case and context of FSI capsule problem.  \n\n\\section{\\review{Non-intrusive space-time model-order reduction strategy}}\n\\label{sec:ROM}\n\n\\review{In this section, the parameter couple $\\bm{\\theta}=(Ca,a\/\\ell)$ is fixed, thus we omit the dependency of the solutions with respect to $\\bm{\\theta}$ for the sake of simplicity}.\nFor the derivation of the ROM model, we consider the semi-discrete time-continuous version of the FOM, i.e.~\\eqref{eq:flo2}-\\eqref{eq:flo3}.\n\\subsection{Dimensionality reduction and reduced variables for displacements and velocities}\n    %\nAssume first that, for any $t\\in [0,T]$, the discrete velocity field can be accurately approximated according to the expansion\n\\begin{equation}\n    \\{v\\rbrace(t) \\approx \\sum_{k=1}^K \\beta_k(t)\\, \\lbrace \\phi_k\\rbrace \n\\label{eq:flo6}\n\\end{equation}\n\n\\noindent for some orthonormal modes $\\lbrace \\phi_k\\rbrace \\in\\mathbb{R}^d$ and real coefficients $\\beta_k(t)$.\nThe truncation rank $K\\leq d$ is of course expected to be far less than $d$ as expected in a general ROM methodology. From the kinematics equations we have\n\\begin{align*}\n\\{u\\rbrace(t) &= \\int_0^t \\{v\\rbrace(s)\\, ds\\\\\n       &\\approx \\int_0^t \\beta_k(s)\\, \\lbrace \\phi_k\\rbrace \\ ds\n\\end{align*}\nso that the displacement field can be accurately represented by \n\\begin{equation}\n\\{u\\rbrace(t) \\approx \\sum_{k=1}^K\\alpha_k(t)\\, \\lbrace \\phi_k\\rbrace \n\\label{eq:flo7}\n\\end{equation}\nwhere $\\displaystyle{\\alpha_k(t)=\\int_0^t \\beta_k(s)\\, ds}$. The coefficients\n$(\\alpha_k(t))_k$ and $(\\beta_k(t))_k$ are called the reduced variables. For the  sake of readability and mental correspondence between full-order unknowns and reduced ones, we will use the convenient notations\n\\[\n\\bm{\\alpha}(t)=(\\alpha_1(t),...,\\alpha_K(t))^T,\\quad\n\\bm{\\beta}(t)=(\\beta_1(t),...,\\beta_K(t))^T\n\\]\nwhere the exponent $^T$ denotes the transpose of the matrix.\nThe condensed matrix forms of~\\eqref{eq:flo7} and~\\eqref{eq:flo6}\nrespectively are\n\\begin{equation}\n\\{u\\rbrace(t) \\approx Q\\, \\bm{\\alpha}(t),\\quad\n\\{v\\rbrace(t) \\approx Q\\, \\bm{\\beta}(t),\n\\end{equation}\nwhere $Q=[\\lbrace \\phi_1\\rbrace ,...,\\lbrace \\phi_K\\rbrace ]\\in\\mathscr{M}_{dK}$. Since the modes $\\lbrace \\phi_k\\rbrace $ are assumed to be orthonormal (for the standard Euclidean inner product), the matrix $Q$ is a semi-orthogonal matrix, i.e. $Q^TQ=I_K$. In particular, we have\n$\\bm{\\alpha}(t)\\approx Q^T\\, \\{u\\rbrace(t)$ and $\\bm{\\beta}(t)=Q^T\\, \\{v\\rbrace(t)$.\n\n\\review{Note that the modes $\\lbrace \\phi_k\\rbrace $ and reduced variables $\\bm{\\alpha}$, $\\bm{\\beta}$  are determined for each parameter set ($Ca, a\/\\ell$), but a common value of the truncation rank $K$  is chosen for all the sets. Its practical computation will be detailed in a next subsection, as well as that of the modes $\\lbrace \\phi_k\\rbrace $}.\n\\subsection{ROM prototype}\nThe expressions $\\lbrace \\tilde u\\rbrace (t)=Q\\, \\bm{\\alpha}(t)$ and\n$\\lbrace \\tilde v\\rbrace (t)=Q\\, \\bm{\\beta}(t)$ provide low-order representations of displacement and velocity fields respectively. We can now write equations for the reduced vectors $\\bm{\\alpha}(t)$ and $\\bm{\\beta}(t)$ respectively. In this subsection, let us consider a projection Galerkin-type approach. Let us denote $\\langle.,.\\rangle$\nthe standard Euclidean scalar product in $\\mathbb{R}^d$. Considering \na test vector $\\lbrace w\\rbrace $ in $W=span(\\lbrace \\varphi_1\\rbrace ,...,\\lbrace \\varphi_K\\rbrace )$, we look for an approximate displacement field $\\tilde \\{u\\rbrace(t)$ solution of the projected kinematics equations\n\\[\n\\langle \\frac{d}{dt}\\lbrace \\tilde u\\rbrace (t),\\lbrace w\\rbrace \\rangle = \n\\langle \\lbrace \\tilde v\\rbrace (t),\\lbrace w\\rbrace  \\rangle\n\\quad \\forall\\, \\lbrace w\\rbrace \\in W.\n\\]\nBy considering each test vector $\\lbrace w\\rbrace =\\lbrace \\varphi_k\\rbrace $, we get the consistent reduced kinematics equation\n\\begin{equation}\n\\dot\\bm{\\alpha} = \\bm{\\beta}.\n\\label{eq:flo9}\n\\end{equation}\nConsider now the projected field $\\lbrace \\tilde v\\rbrace (t)$ which is solution of the\nsystem of algebraic equations (Galerkin approach):\n\\begin{equation}\n\\langle \\lbrace \\tilde v\\rbrace (t),\\lbrace w\\rbrace \\rangle = \\langle \\varphi(\\lbrace \\tilde u\\rbrace (t)),\\lbrace w\\rbrace  \\rangle \\quad \\forall\\, \\lbrace w\\rbrace \\in W.\n\\label{eq:flo10}\n\\end{equation}\n\\review{\nAgain by taking the test vector $\\{w\\}=\\{\\phi_k\\}$, we have\n\\[\n\\{\\phi_k\\}^T Q \\beta(t) = \\{\\phi_k\\}^T  \\varphi(Q \\bm{\\alpha}(t)).\n\\]\nConsidering all $k$ in $\\{1,...,K\\}$, since $Q=[\\lbrace \\phi_1\\rbrace ,...,\\lbrace \\phi_K\\rbrace ]$ and $Q^TQ=I_K$ we get}\n\\[\nQ^T Q \\bm{\\beta}(t) = \\bm{\\beta}(t) = Q^T \\varphi(Q \\bm{\\alpha}(t)).\n\\]\nIt is in the form\n\\begin{equation}\n\\bm{\\beta}(t) = \\varphi_r(\\bm{\\alpha}(t))\n\\label{eq:flo11}\n\\end{equation}\nwith the mapping $\\varphi_r:\\mathbb{R}^K\\rightarrow\\mathbb{R}^K$\ndefined by $\\varphi_r(\\bm{\\alpha})=Q^T\\varphi(Q\\bm{\\alpha})$. \nWe get a reduced-order algebraic equilibrium equation.\nUnfortunately,\nbecause of the nonlinearities in $\\varphi$, the computation of\n$\\varphi_r(\\bm{\\alpha})$ requires high-dimensional $O(d)$ operations, making this approach irrelevant from the performance point of view.\nA possible solution to deal with the nonlinear terms would be to use for example Empirical Interpolation Methods (EIM) \\citep{Barrault2004} but from the algorithm and implementation point of view, this would lead to an intrusive approach with specific code developments. We here rather adopt a linearization strategy in the following sense: by derivating~\\eqref{eq:flo11} with respect to time, we get\n\\[\n\\dot\\bm{\\beta}(t) = \\frac{\\partial\\varphi_r}{\\partial \\bm{\\alpha}}(\\bm{\\alpha}(t))\n\\, \\dot\\bm{\\alpha}(t).\n\\]\nThanks to the reduced kinematics equation~\\eqref{eq:flo9}, we get\n\\begin{equation}\n\\dot\\bm{\\beta}(t) = \\frac{\\partial\\varphi_r}{\\partial \\bm{\\alpha}}(\\bm{\\alpha}(t))\n\\ \\bm{\\beta}(t).\n\\label{eq:12}\n\\end{equation}\nSince $\\varphi_r$ is hard to evaluate, it is even  harder to evaluate its differential. But the differential $\\frac{\\partial\\varphi_r}{\\partial \\bm{\\alpha}}(\\bm{\\alpha}(t))$ can be approximated itself, or replaced by some matrix  $A(t)$.\nThen we get a ROM structure (ROM prototype) in the form\n\\begin{align}\n& \\dot\\bm{\\alpha} = \\bm{\\beta}(t), \\label{eq:flo13} \\\\\n& \\dot\\bm{\\beta}(t) = A(t) \\ \\bm{\\beta}(t) \\label{eq:flo14}.\n\\end{align}\nThe differential system~\\eqref{eq:flo13}-\\eqref{eq:flo14} is linear with variable coefficient matrix $A(t)\\in \\mathscr{M}_K(\\mathbb{R})$. It can be written in matrix form\n\\begin{equation}\n\\frac{d}{dt}\\begin{pmatrix} \\bm{\\alpha}(t)\\\\ \\bm{\\beta}(t) \\end{pmatrix}\n= \\underbrace{\\begin{pmatrix} [0] & I_K \\\\ [0] & A(t) \\end{pmatrix}}_{=\\mathbb{A}(t)}\n\\begin{pmatrix} \\bm{\\alpha}(t)\\\\ \\bm{\\beta}(t) \\end{pmatrix}.\n\\label{eq:flo15}\n\\end{equation}\nThe spectral properties of the differential system~\\eqref{eq:flo15} are related to the spectral properties of matrix $A(t)$. In particular, if all the (complex) eigenvalues $\\lambda_k(t)$ of $A(t)$ are such that\n$\\Re(\\lambda_k(t))<0$ for all~$k$ (uniformly distributed in time), then the system is dissipative.\n\\subsection{Nonintrusive approach, SVD decomposition and POD modes}\nOne of the requirements of this work is to achieve a non-intrusive reduced-order model, meaning that the effective implementation of the ROM does not involve tedious low-level code development into the FOM code. For that, a data-based approach is adopted: from the FOM code, it is possible to compute FOM solutions $(\\{u\\rbrace^n,\\{v\\rbrace^n)$ at discrete times $t^n$, $n=0,...,N$ ($t^N=N\\Delta t=T$), then store some snapshot solutions (called snapshots) into a database for data analysis and later design of a ROM. Proper Orthogonal Decomposition (POD) \\citep{Berkooz1993} is today a well-known dimensionality reduction approach to determine the principal components from solutions of partial differential equations. The Sirovich's snapshot variant approach \\citep{Sirovich1987} is based on snapshot solutions from a FOM to get \\textit{a posteriori} empirical POD modes $\\lbrace \\varphi_k\\rbrace $. For the sake of simplicity, assume that the snapshot solutions are all the discrete FOM solution at simulation instants.\nApplying a singular value decomposition (SVD) to the displacement  snapshot matrix\n\\[\n\\mathbb{S}^u = \\left[\\ubold^1,\\ubold^2,...,\\ubold^N\\right],\n\\]\nof size $d\\times N$, we get the SVD decomposition \n\\begin{equation}\n\\mathbb{S}^u = U \\Sigma V^T\n\\end{equation}\nwith orthogonal matrices $U\\in\\mathscr{M}_d(\\mathbb{R})$,\n$V\\in\\mathscr{M}_N(\\mathbb{R})$ and the singular value matrix\n$\\Sigma=diag(\\sigma_k)\\in\\mathscr{M}_{d\\times N}(\\mathbb{R})$, with $\\sigma_k\\geq 0$ for all $k$ organized in decreasing order:\n$\\sigma_1\\geq \\sigma_2\\geq ...\\geq\\sigma_{\\min(d,N)}\\geq 0$.\nFrom SVD theory, for a given accuracy threshold $\\varepsilon>0$,\nthe truncation rank $K=K(\\varepsilon)$ is computed as the smallest integer such that the inequality\n\\begin{equation}\n \\frac{\\displaystyle{\\sum_{k=K+1}^{\\min(d,N)} \\sigma_k^2}}{\\displaystyle{\\sum_{k=1}^{\\min(d,N)}\\sigma_k^2}}\\leq \\varepsilon\n\\label{eq:RIC}\n\\end{equation}\nholds \\citep{Shawe2004}. Proceeding like that, it is shown that the relative orthogonal projection error of the snapshots~$\\{v\\rbrace^n$ onto the linear subspace $W$ spanned by the $K$ first eigenvectors of $U$ is controlled by $\\varepsilon$. Denoting~$\\pi^W$ the linear orthogonal projection operator over $W$, we have:\n\\[\n\\sum_{n=1}^N \\|\\{v\\rbrace^n-\\pi^W\\{v\\rbrace^n\\|^2 \\leq \\varepsilon\n\\sum_{n=1}^N \\|\\{v\\rbrace^n\\|^2.\n\\]\nThe matrix $Q$ is obtained as the restriction of $U$ to its $K$ first columns. \n\n\\subsection{Data-driven identification of coefficient matrix}\\label{sec34}\nThe system~\\eqref{eq:flo13}-\\eqref{eq:flo14} is still not closed since the coefficient matrices $A(t)$ are unknowns.\nFrom FOM data, one can try to identify the matrices by minimizing some residual function that measures the model discrepancy.\nThe simplest linear model corresponds to the case where $A(t)$ is searched as a time-constant matrix~$A$. In this case, equation~\\eqref{eq:flo14} becomes\n$\\dot\\bm{\\beta}(t) = A \\, \\bm{\\beta}(t)$. This is the scope of this article. From the time continuous problem, one could determine the matrix $A$ by minimizing the least square functional\n\\[\n\\min_{A\\in\\mathscr{M}_K(\\mathbb{R})} \\frac{1}{2}\\int_0^T\n\\|\\dot\\bm{\\beta}(t)-A \\bm{\\beta}(t)\\|^2\\, dt.\n\\]\nBut practically, we only have velocity snapshot data at discrete times and\nwe do not have access to the time derivatives of the velocity fields. \nSo the following numerical procedure is adopted: from the velocity snapshot matrix \n$\\mathbb{S}^v=[\\{v\\rbrace^1,...,\\{v\\rbrace^N]$, we compute first the reduced snapshots variables:\n\\[\n\\bm{\\beta}^n = Q\\,  \\{v\\rbrace^n,\\quad n=1,...,N.\n\\]\nNext, we determine a matrix $A$ that minimizes the least square cost function:\n\\begin{equation}\n\\min_{A\\in\\mathscr{M}_K(\\mathbb{R})}\\ \\frac{1}{2} \\sum_{n=1}^{N-1}\n\\left\\|\\frac{\\bm{\\beta}^{n+1}-\\bm{\\beta}^n}{\\Delta t}-A\\bm{\\beta}^n \\right\\|^2\n\\label{eq:flo17}\n\\end{equation}\nIn~\\eqref{eq:flo17}, the finite difference $\\dfrac{\\bm{\\beta}^{n+1}-\\bm{\\beta}^n}{\\Delta t}$\nis a first-order approximation (in $\\Delta t$) of $\\dot\\bm{\\beta}$ at time $t^n$.\nIn appendix \\ref{app:a}, we provide a mathematical analysis of the effect of time discretization in~\\eqref{eq:flo17} about the impact on the stability of the resulting identified differential system compared to the initial one. \n\nThe minimization problem~\\eqref{eq:flo17} can be written in condensed matrix form\n\\begin{equation}\n\\min_{A\\in\\mathscr{M}_K(\\mathbb{R})}\\ \\frac{1}{2} \\|\\mathbb{Y}-A\\mathbb{X}\\|_F^2\n\\label{eq:flo18}\n\\end{equation}\nwith the two data matrices\n\\begin{equation}\n\\mathbb{X} = \\left[\\bm{\\beta}^1,\\bm{\\beta}^2,...,\\bm{\\beta}^{N-1}\\right],\\quad\n\\mathbb{Y} = \\left[\\frac{\\bm{\\beta}^2-\\bm{\\beta}^1}{\\Delta t},...,\\frac{\\bm{\\beta}^N-\\bm{\\beta}^{N-1}}{\\Delta t} \\right].\n\\label{eq:flo19}\n\\end{equation}\nBecause $\\mathbb{X}$ and $\\mathbb{Y}$ store reduced variables (of size $K$), for a sufficient number of discrete snapshot times, these two matrices are\nhorizontal ones. We will assume that the rank of $\\mathbb{X}$ is always maximal, i.e. equal to $K$.\nThe least-square solution $A$ of~\\eqref{eq:flo18} is then given by\n\\begin{equation}\nA = \\mathbb{Y}\\mathbb{X}^\\dagger\n\\label{eq:flo20}\n\\end{equation}\nwhere $\\mathbb{X}^\\dagger=\\mathbb{X}^T(\\mathbb{X}\\mathbb{X}^T)^{-1}$ is the Moore-Penrose pseudoinverse matrix of $\\mathbb{X}$.\nThis least square approach has close connections with SVD-based Dynamic Mode Decomposition (DMD) \\citep{Schmid2010,Kutz2016}.\n    \\subsection{Tikhonov least-square regularized formulation}\n    %\n    From standard spectral theory arguments, it is expected that the POD coefficients rapidly decay when $k$ increases as soon as both displacement and velocity fields are smooth enough. \n    A possible side effect is the bad condition number of the matrix $\\mathbb{X}$, since the last rows of $\\mathbb{X}$ have small coefficients (thus leading to row vectors close to zero 'at the scale' of the first row of $\\mathbb{X}$). Even if the solution $A$ in~\\eqref{eq:flo20} always exists, the solution may be sensitive to perturbations, noise or round-off errors.\n    In order to get a robust identification approach, one can regularize the least-square problem~\\eqref{eq:flo18} by adding a Tikhonov regularization term (see e.g. \\citep{Aster2005})\n    %\n    \\begin{equation}\n    \\min_{A\\in\\mathscr{M}_K(\\mathbb{R})}\\ \\frac{1}{2} \\|\\mathbb{Y}-A\\mathbb{X}\\|_F^2 + \\frac{\\mu}{2} \\|\\mathbb{X}\\|_F^2\\, \\|A\\|_F^2\n    \\label{eq:flo21}\n    \\end{equation}\n    %\n    where the scalar $\\mu>0$ is the regularization coefficient. The factor\n    $\\|\\mathbb{X}\\|_F^2$ in the regularization term has been added for scaling purposes.\n    The solution $A_\\mu$ of~\\eqref{eq:flo21} is given by\n    %\n    \\begin{equation}\n    A_\\mu = \\mathbb{Y} \\mathbb{X}^T\\left(\\mathbb{X}\\mathbb{X}^T+\\mu \\|\\mathbb{X}\\|_F^2\\, I_K\\right)^{-1}.\n    \\label{eq:flo22}\n    \\end{equation}\n    %\n    \\subsubsection*{Choice of optimal regularization coefficient}\n    Of course, the solution matrix $A_\\mu$ depends on the regularization coefficient $\\mu$ and one can ask what is the optimal choice for $\\mu$.\n    There is a trade-off between the approximation quality measured by the residual $\\|\\mathbb{Y}-A_\\mu\\mathbb{X}\\|_F$ and the norm solution $\\|A_\\mu\\|_F$. The minimization of $\\|A_\\mu\\|_F$ should ensure that unneeded features will not appear in the regularized solution.\n    When plotted on the log-log scale, the curve of optimal values \n    $\\mu\\mapsto \\|A_\\mu\\|_F$ versus the residual $\\mu\\mapsto\\|\\mathbb{Y}-A_\\mu\\mathbb{X}\\|_F$ often takes on a characteristic L shape \\citep{Aster2005}. A design of experiment with\n    the test of \n    different values of $\\mu$ (starting say from \\review{$10^{-12}$ to $10^{-5}$})\n    generally allow to find quasi-optimal values of $\\mu$ located at the corner of the L-curve, thus providing a good trade-off between the two criteria.\n    %\n    \\subsection{Reduced-order continuous dynamical system}\n    %\n    Once the matrix $A_\\mu$ has been determined, we get the reduced-order continuous dynamical system\n    %\n    \\begin{align}\n    & \\dot\\bm{\\alpha} = \\bm{\\beta}, \\label{eq:flo23}\\\\\n    & \\dot\\bm{\\beta} = A_\\mu\\, \\bm{\\beta} \\label{eq:flo24}\n    \\end{align}\n    %\n    with initial conditions $\\bm{\\alpha}(0)=\\bm{0}$, $\\vbold(0)=Q^T \\varphi(\\lbrace 0\\rbrace )$. At any time $t$, one can go back to the high-dimensional physical space using the POD modes: $\\{u\\rbrace(t)=Q\\bm{\\alpha}(t)$, $\\{x\\rbrace(t)=\\lbrace X\\rbrace +\\{u\\rbrace(t)$, \n    $\\{v\\rbrace(t)=Q \\bm{\\beta}(t)$. As already mentioned, the system can be written in condensed matrix form\n    %\n    \\begin{equation}\n    \\dot{\\bm{w}} = \\mathbb{A}_\\mu\\, \\bm{w}\n    \\label{eq:flo25}\n    \\end{equation}\n    %\n    where $\\bm{w}(t)=(\\bm{\\alpha}(t),\\bm{\\beta}(t))^T$ and\n    $\\mathbb{A}_\\mu = \\begin{pmatrix}[0]_K & I_K\\\\ [0]_K & A_\\mu. \\end{pmatrix}$.\n    \n    The exact analytical solution of~\\eqref{eq:flo25} is\n    \\begin{equation}\n    \\bm{w}(t) = \\exp(\\mathbb{A}_\\mu t)\\,\\bm{w}(0).\n    \\label{eq:flo26}\n    \\end{equation}\n    %\n    The stability of the differential system depends on the spectral structure of $\\mathbb{A}_\\mu$, or equivalently on the spectrum of $A_\\mu$. Because of the stability of the fluid-capsule coupled system and from accurate solutions of the FOM solver, one can hope that the solution $A_\\mu$ of the least-square identification problem has the expected spectral properties. This will be studied and discussed in the numerical experimentation section. From the kinetic energy point of view, it is shown in appendix \\ref{app:b} that the stability of the kinetic energy is linked to the property of the (real) spectrum of the symmetric matrix $(A_\\mu+A_\\mu^T)\/2$. \n    %\n    \\subsubsection*{Model consistency with steady states} \n    %\n    A steady state in our context is defined by a capsule that reaches a constant\n    velocity $\\{v\\rbrace_\\infty$, so that the motion becomes a translation flow in time\n    in the direction $\\{v\\rbrace_\\infty$. From~\\eqref{eq:flo6}, this shows that \n    $\\bm{\\beta}(t)$ also reaches a constant vector $\\bm{\\beta}_\\infty$, and\n    $\\dot\\bm{\\beta} = 0$ at steady state. As a consequence, from~\\eqref{eq:flo24},\n    we get $A_\\mu \\bm{\\beta}_\\infty=0$, meaning that $0$ is an eigenvalue of $A_\\mu$ with $\\bm{\\beta}_\\infty$ as eigenvector. As a conclusion, the matrix $A_\\mu$ must have zero\n    in its spectrum in order to be consistent with the existence of steady states.\n    %\n    \\subsection{Reduced-order discrete dynamical system}\n    %\n    Of course, it is also possible to derive a discrete dynamical system from the continuous one by using a standard time advance scheme. For example the explicit forward Euler scheme with a constant time step $\\Delta t$  gives\n    %\n    \\begin{align}\n    & \\bm{\\alpha}^{n+1} = \\bm{\\alpha}^n + \\Delta t\\, \\bm{\\beta}^n,  \\label{eq:flo27} \\\\\n    & \\bm{\\beta}^{n+1} = \\bm{\\beta}^n + \\Delta t\\, A_\\mu \\bm{\\beta}^n. \\label{eq:flo28}\n    \\end{align}\n    %\n    By multiplying~\\eqref{eq:flo27} by $Q$ we get the space-time approximate solution\n    \\[\n    \\{u\\rbrace^{n+1} = \\{u\\rbrace^n + \\Delta t\\, \\{v\\rbrace^n,\n    \\]\n    so the ROM model is completely consistent with the kinematics equation.\n    Stability properties of the discrete system are linked to the spectral properties of the matrix\n    \\[\n    A_\\mu^\\Delta = \\begin{pmatrix}\n    I_K & \\Delta t\\, I_K \\\\\n    [0]_K & (I_K+\\Delta t\\, A_\\mu) \n    \\end{pmatrix}\n    \\]\n    For unconditional stability in time, it is required for the eigenvalues\n    of $I_K+\\Delta t A_\\mu$ to stay in the unit disk of the complex plane.\n    \n    More generally, it is possible to use any other time advance scheme, according to the expected order of accuracy or stability domain.\n    %\n    \\subsection{Accuracy criteria and similarity distances between ROM and FOM solutions}\n    \nIn order to quantify the error induced by approximations, we introduce 3 accuracy criteria. \nThe first accuracy criterion  is the relative information content (RIC), defined by \n\\[\n\\text{RIC}(K) = \\frac{\\displaystyle{\\sum_{k=K+1}^{\\min(d,N)} \\sigma_k^2}}{\\displaystyle{\\sum_{k=1}^{\\min(d,N)}\\sigma_k^2}},\n\\]  \nquantifies the relative amount of neglected information when truncating the number of modes at rank $K$. \nThe truncation rank is determined such that the RIC is inferior to the accuracy threshold~$\\varepsilon$. The accuracy threshold $\\varepsilon$ is fixed to $10^{-6}$.\n\n\nThe second accuracy criterion  is the relative time residual $\\mathcal{R}$. It quantifies the relative error induced by the minimization of the least square cost function (\\ref{eq:flo17}) using $A_{\\mu}$. It is given by\n\\[\n\\mathcal{R}(j)=\\frac{\\Vert A_{\\mu} \\mathbb{X}_j-\\mathbb{Y}_j \\Vert_1}{\\Vert  \\mathbb{Y}_j \\Vert_1}\n\\]\nwhere $\\mathbb{X}_j$ represents the $j^{th}$ column of $\\mathbb{X}$ and $\\mathbb{Y}_j$ the $j^{th}$ column of $\\mathbb{Y}$. The index $j$ is thus linked to the snapshots ($j\\in \\{1, ..., N\\}$). To better draw a parallel between the evolution of this parameter and the capsule dynamics, this parameter will be represented as a function of the non-dimensional time $Vt\/\\ell$ hereafter.\n\nThe third accuracy criteria $\\varepsilon_{\\text{Shape}}(Vt\/\\ell)$ measures the difference between the 3D reference capsule shape given by the FOM ($\\mathcal{S}_{\\text{FOM}}$) and the 3D shape predicted by the ROM ($\\mathcal{S}_{\\text{ROM}}$). It is defined  at a given non-dimensional time $Vt\/\\ell$ as the ratio between the modified Hausdorff distance (MHD)  computed between $\\mathcal{S}_{FOM}$ and $\\mathcal{S}_{ROM}$ and non-dimensionalized by $\\ell$\n\\[\n\\varepsilon_{\\text{Shape}}(Vt\/\\ell) =\\frac{\\text{MHD}(\\mathcal{S}_{\\text{FOM}}(Vt\/\\ell), \\mathcal{S}_{\\text{ROM}}(Vt\/\\ell))}{\\ell}\n\\]\nThe modified Hausdorff distance is the maximum value of the mean distance between $\\mathcal{S}_{\\text{FOM}}$ and $\\mathcal{S}_{\\text{ROM}} $ and the mean distance between $\\mathcal{S}_{\\text{ROM}}$ and $\\mathcal{S}_{\\text{FOM}}$ \\citep{Dubuisson1994}.\n\n\n\n\\section{Numerical experimentation on a given configuration}\n\\label{sec:1example}\n\nThe method is first applied to a given configuration, in order to set the model parameters  and to study its stability and precision. We consider the dynamics of an initially spherical capsule flowing in a microchannel when $Ca=0.17$ and $a\/\\ell=0.8$.  The time step between each snapshot $\\Delta t$ equals to 0.04. The dynamics predicted by the FOM is illustrated  in Fig. \\ref{DynamicFOM} up to a nondimensional time $VT\/\\ell=10$. As the capsule flows, its membrane is gradually deformed by the hydrodynamic forces inside the channel during a temporary time until a steady state is reached. We assume that the capsule has reached its steady-state shape, when the surface area of the capsule varies by less than $5\\times10^{-4}\\times(4 \\pi a^2)$ over a non-dimensional time $Vt\/\\ell=1$. For $(Ca=0.17$, $a\/\\ell=0.8)$, the steady state is reached at $Vt\/\\ell=2.6$ and is characterized by a parachute capsule shape (Figure \\ref{DynamicFOM}). \n\n\\begin{figure}\n\\centering\n\n(a)\\includegraphics[width=0.18\\textwidth]{newfigures\/CuttedPlane.eps}\n(b)\\includegraphics[width=0.7\\textwidth]{newfigures\/Comp_capsule_al0.8_Ca0.17_FOM.eps}\n\\caption{Dynamics of a microcapsule flowing in a microchannel with a square cross-section predicted by FOM in the vertical cutting plane represented in grey in (a). The in-plane capsule profiles are shown for $Ca=0.17$ and $a\/\\ell=0.8$ at the non-dimensional times $Vt\/\\ell=$ 0, 0.4, 2, 4, 6 in (b). The horizontal lines on (b) represent the channel borders. The capsule will always be represented flowing from left to right.} \n\\label{DynamicFOM}\n\\end{figure}\n\n\\subsection{Proper orthogonal decomposition, truncation and modes}\n\nThe singular value decomposition is first applied to the displacement snapshot matrix. To determine the truncation rank, the evolution of 1- RIC is illustrated in Figure~\\ref{RIC} as a function of number of modes considered. The RIC is about 1\\% only with one mode. The more modes is kept, the less information is neglected.  In the following, we fix the number of modes to 20. The accuracy threshold $\\varepsilon$ is thus equal to $10^{-6}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{newfigures\/RIC_Ca0.17_al0.8.eps}\n\\caption{Evolution of the relative amount of neglected information 1-RIC, as a function of the number of modes considered ($Ca=0.17, a\/\\ell=0.8$).}\n\\label{RIC}\n\\end{figure}\n\nThe modes are determined from the displacement snapshot matrix. They are added to the sphere of radius 1 and amplified by a factor 2 to be  visualized. The first six modes are represented in Figure~\\ref{modesRepresentation} for ($Ca=0.17, a\/\\ell=0.8$).\nThe first six modes are mostly dedicated to change the shape of the rear of the capsule. The following modes appear to become noisy (not shown). However, these modes are not negligible, if one wants to get an accuracy of $10^{-6}$.\n\n\\begin{figure}\n\\centering\n(a)\\includegraphics[width=0.28\\textwidth]{newfigures\/Modes2_al0.80_Ca0.170.eps}\n(b)\\includegraphics[width=0.28\\textwidth]{newfigures\/Modes3_al0.80_Ca0.170.eps}\n(c)\\includegraphics[width=0.28\\textwidth]{newfigures\/Modes4_al0.80_Ca0.170.eps}\n(d)\\includegraphics[width=0.28\\textwidth]{newfigures\/Modes5_al0.80_Ca0.170.eps}\n(e)\\includegraphics[width=0.28\\textwidth]{newfigures\/Modes6_al0.80_Ca0.170.eps}\n(f)\\includegraphics[width=0.28\\textwidth]{newfigures\/Modes7_al0.80_Ca0.170.eps}\n\\caption{Representation of the first six modes of the capsule dynamics when $a\/\\ell =0.80$ and $Ca=0.17$. To be vizualized the modes of displacement were added to the sphere of radius 1 and amplified by a factor 2.} \n\\label{modesRepresentation}\n\\end{figure}\n    \n\\subsection{Dynamic Mode Decomposition: empirical regularization}\n\nBefore determining the matrix $A$, we check the condition number of the matrices $\\mathbb{X}$ and $\\mathbb{XX^T}$. They are respectively equal to $7.9\\times 10^4$ and $6.2 \\times 10^9$. The condition numbers of these matrices are very high and the matrix $A$, determined by solving (\\ref{eq:flo20}), may be sensitive to perturbations or noise. To improve the robustness, a Tikhonov regularization is applied to solve the least-square problem (\\ref{eq:flo17}) and the matrix $A_{\\mu}$ is computed using (\\ref{eq:flo22}), which depends on the regularization coefficient $\\mu$.\nTo determine the optimal value of $\\mu$, the relative least square error $\\Vert A_{\\mu}\\mathbb{X}-\\mathbb{Y}\\Vert_F\/\\Vert\\mathbb{Y}\\Vert_F$ is represented according to the norm solution $\\Vert A_{\\mu}\\Vert_F$ when 20 modes are considered and when $\\mu$ is varied between \\review{$10^{-12}$ and $10^{-5}$} (Figure \\ref{muDetermination}). \nThe least square error $\\Vert A_{\\mu}\\mathbb{X}-\\mathbb{Y}\\Vert_F$ and the norm solution $\\Vert A_{\\mu}\\Vert_F$ are minimal when $\\mu=10^{-9}$. In the following, $\\mu$ is thus fixed to $\\mu=10^{-9}$ and the number of modes to 20. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{newfigures\/DeterminationMu_Ca0.17_al0.8_Xrelatif.eps}\n\\caption{Evolution of the norm solution $\\Vert A_{\\mu}\\Vert_F$ as a function of\nthe least square error $\\Vert A_{\\mu}\\mathbb{X}-\\mathbb{Y}\\Vert_F \/ \\Vert \\mathbb{Y}\\Vert_F$\n when the number of modes is fixed to 20 and $(Ca=0.17, a\/\\ell=0.8)$.} \n\\label{muDetermination}\n\\end{figure}\n\n\\subsection{Validity check of the ROM: spectral study of the resulting matrix}\n\nIn order to detect anomalies, a spectral analysis of the reduced-order model learned by the DMD method is carried out. The spectrum of the matrix $A_{\\mu}$ is represented in Figure~\\ref{eigenvalues}. All the eigenvalues $\\lambda_k$ of the matrix $A_{\\mu}$ have non-positive real parts. The resulting linear ROM is thus stable. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{newfigures\/ValeurPropreCa0.17Al0.8_ModelKBfModelA_X.eps}\n\\caption{Eigenvalues $\\lambda_k$ of $A_{\\mu}$ when 20 modes are considered, $\\mu=10^{-9}$ and $(Ca=0.17, a\/\\ell=0.8)$. } \n\\label{eigenvalues}\n\\end{figure}\n\nThe temporal evolution of the residual $\\mathcal{R}$ (Figure~\\ref{ErrorModel}) shows that the error is less than 4\\%. The maximal value is reached\nat the beginning of the simulation ($Vt\/\\ell < 0.3$) and $\\mathcal{R}$ decreases afterwards. \nWhen $Vt\/\\ell \\lesssim 2$, i.e. before the capsule has reached its steady state, high  frequency oscillations are observed. This probably means that a high frequency mode is neglected, even if 20 modes are considered. For $Vt\/\\ell >6$, $\\varepsilon_{ROM}$ is of order $0.01\\%$. The stationary state is thus well predicted by the model and the error during the transient stage is more than acceptable. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{newfigures\/ErrorSnapCa0.17Al0.8_X.eps}\n\\caption{Temporal evolution of  the normalized time residual when 20 modes are considered, $\\mu=10^{-9}$ and $(Ca=0.17, a\/\\ell=0.8)$.} \n\\label{ErrorModel}\n\\end{figure}\n\n\n\n\\subsection{ROM online stage and accuracy assessment}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{newfigures\/resultROM3D.eps}\n\\caption{Dynamics of a microcapsule flowing in microchannel with a square cross-section predicted by ROM at the non-dimensional time $Vt\/\\ell=$ 0.4, 2.8, 5.2, 7.6, 10 when $Ca=0.17$ and $a\/\\ell=0.8$. The initial spherical capsule is shown on the left by transparency. The number of modes is fixed to 20 and $\\mu=10^{-9}$.} \n\\label{DynamicsROM3D}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{newfigures\/ROM_FOM_Profilcapsule.eps}\n\\caption{Comparison of the capsule contours given by the FOM (dotted line) and estimated by the ROM (orange line). The capsule is shown for $(Ca=0.17, a\/\\ell=0.8$ at the non-dimensional time $Vt\/\\ell=$ 0, 0.4, 2, 4, 6. The horizontal lines represent the channel borders. The number of modes is fixed to 20 and $\\mu=10^{-9}$.} \n\\label{CompProfils}\n\\end{figure}\n\n\n\n\n\n\nThe displacement of all the nodes of the capsule mesh estimated by the ROM is then added to the corresponding node of the sphere of radius 1 to visualize the temporal evolution of the capsule shape in three dimensions. Figure \\ref{DynamicsROM3D} shows the capsule dynamics for the reference case ($Ca=0.17, a\/\\ell=0.8$). The ROM allows us to reproduce the capsule deformation from the initial state up to the parachute-shaped steady state. \nFor the FOM and the ROM, the capsule profile is then determined in the\n cutting plane passing through the middle of the microchannel. Figure \\ref{CompProfils} shows that the two capsule profiles perfectly overlap  at $Vt\/\\ell=0, 0.4, 2, 4, 6$.\nThe temporal evolution of $\\varepsilon_{\\text{Shape}}$ is shown in Figure \\ref{TimeLearning}a. The maximum value of the error committed on the 3D shape $\\varepsilon_{\\text{Shape}}$ equals to 0.022\\%.\n The error on the capsule shape $\\varepsilon_{\\text{Shape}}$ is thus negligible. The deformation of the capsule from its initially spherical shape to its steady state over an non-dimensional time $Vt\/\\ell=10$ can thus be estimated very precisely with the developed reduced-order model.  \n\n\n\n\n\\review{\\subsection{Sensitivity of the ROM on the learning time  }\n\nThe DMD method predicts the capsule displacement at time $t^{n+1}$ from that at time $t^n$. The model has been constructed until now by considering the dynamics of the capsule over a non-dimensional time $Vt\/\\ell$ of 10.\n\nIn order to study its influence, we increase the learning  $T_L$, i.e. the non-dimensional time over which the model is trained, from 2 to 8 and estimate the capsule dynamics using the ROM up to a non-dimensional time $Vt\/\\ell$ of 10. The number of modes is always equal to 20 and $\\mu=10^{-9}$. The estimated shape is then compared to the one simulated with the FOM. The time evolution of $\\varepsilon_{\\text{Shape}}$ is shown in Figure \\ref{TimeLearning}a. The error between the 3D shape increases, as soon as we exceed the learning time. The longer the learning time, the smaller the error $\\varepsilon_{\\text{Shape}}$ at $Vt\/\\ell=10$ (Figure \\ref{TimeLearning}b). The comparison of the capsule profile estimated by the ROM at $Vt\/\\ell=10$ with the one simulated with the FOM (Figure \\ref{TimeLearning}c) confirms that considering only the dynamics of the capsule up to a non-dimensional time $Vt\/\\ell$ of 2 is not sufficient. In fact, the capsule is still in the transient phase at $Vt\/\\ell=2$. When $T_L=4$, a small difference persists in the parachute at $Vt\/\\ell=10$ and $\\varepsilon_{\\text{Shape}}=1.5\\%$. However, when the learning time is increased above 6, $\\varepsilon_{\\text{Shape}}$ falls below 0.19\\% at $Vt\/\\ell=10$.  This learning time is sufficient to estimate the overall shape of the capsule.}\n\n\n\\begin{figure}\n\\centering\na)\\includegraphics[width=0.6\\textwidth]{newfigures\/Influence_TimeLearning_avecInsert.eps}\\\\\nb)\\includegraphics[width=0.6\\textwidth]{newfigures\/Error_TimeLearning_insert.eps}\\\\\nc)\\includegraphics[width=0.95\\textwidth]{newfigures\/Influence_TimeLearning_snap.eps}\n\\caption{\\review{ a) Influence of the learning time $T_L$ on the temporal evolution of the error on the capsule shape $\\varepsilon_{\\text{Shape}}$. The error during the learning time is shown in solid line. b) Evolution of $\\varepsilon_{\\text{Shape}}$ measured at $Vt\/\\ell=10$ as a function of the learning time $T_L$. c) Comparison of the capsule contours given by the FOM (dotted line) and estimated by the ROM (orange line) for the different learning times $T_L$. For this case, the parameters are 20 modes, $\\mu=10^{-9}$, $Ca = 0.17$ and $a\/\\ell=0.8$. } } \n\\label{TimeLearning}\n\\end{figure}\n\n\\section{\\review{Space-time ROM accuracy assessment over the full parameter sample set}}\n\\label{sec:database}\nThe capillary number  $Ca$ and the aspect ratio $a\/\\ell$ are now considered as variable parameters.\nA database of 119 simulations of the deformation of an initially spherical capsule in a microchannel has been generated using the FOM with the same time step and mesh size as in section \\ref{sec:1example}. Figure~\\ref{DB} shows the different values of $Ca$ and $a\/\\ell$ for which the simulations have been computed to create the training database. \nWhen the capsule initial radius is close to or larger than the microchannel cross-dimension ($a\/\\ell \\geq 0.90$), the capsule is predeformed into a prolate spheroid to fit in the channel. For a given $a\/\\ell$, a limit value of $Ca$ exists beyond which a capsule does not reach a steady-state (Figure~\\ref{DB}). This is due to the softening behavior of the neo-Hookean law. \n\n\\begin{figure} \n\\centering\n\\includegraphics[width=0.55\\textwidth]{GraphDataBaseLearning.eps}\n\\caption{Values of $Ca$ and $a\/\\ell$ included in the training database. The dotted line delimits the domain where a steady-state capsule deformation exists for capsules following the neo-Hookean law.}\n\\label{DB}\n\\end{figure}\n\nFor all the couples ($Ca, a\/\\ell$) of the training database, the capsule shape is reconstructed from the ROM results at given non-dimensional times and compared to the shape predicted by the FOM at the same non-dimensional time. The evolution of the error committed on the capsule shape $\\varepsilon_{Shape}$ on the full database is illustrated in Figure~\\ref{ErreurBaseA20} at $Vt\/\\ell=0, 0.4, 1, 2, 5, 10$.  $\\varepsilon_{Shape}$ is null at $Vt\/\\ell=0$. The ROM is therefore able to  predict the initial capsule shape correctly, whether it is spherical or slightly ellipsoidal. Until $Vt\/\\ell\\leq2$, $\\varepsilon_{Shape}$  essentially remains zero on the majority of the database. Otherwise, it is equal to 0.15\\% at maximum. At $Vt\/\\ell = 5$ and 10, the error $\\varepsilon_{Shape}$ slightly increases for most of the couples ($Ca, a\/\\ell$) of the database. It remains fully acceptable since it is equal to 0.35\\% at maximum. When considering 20 modes and $\\mu = 10^{-9}$, the developed ROM allows us to estimate with great precision the dynamics of an initially spherical capsule in a  microchannel with a square cross-section. \n\n\\begin{figure}\n\\centering\n(a)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_DB0_20modes_mu10-8.eps}\n(b)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_DB0.4_20modes_mu10-8.eps}\n(c)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_DB1_20modes_mu10-8.eps}\n(d)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_DB2_20modes_mu10-8.eps}\n(e)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_DB5_20modes_mu10-8.eps}\n(f)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_DB10_20modes_mu10-8.eps}\n\\caption{Heat maps of $\\epsilon_{Shape}$ on the training database as a function of $Ca$ and $a\/\\ell$ at (a) $\\dot{\\gamma}t$=0, (b) 0.4, (c) 1, (d) 2, (e) 5, (f) 10. 20 modes and $\\mu=10^{-9}$ are considered. The dotted line delimits the domain where a steady-state capsule deformation exists.} \n\\label{ErreurBaseA20}\n\\end{figure}\n\nTo respect the stability condition (see Equation~\\ref{StabilityCondition}), the time step imposed to simulate the capsule dynamics with the FOM decreases, when the $Ca$ decreases. The lower the $Ca$, the longer the simulation lasts (Figure~\\ref{SimuTime}). The time needed to calculate the capsule shape and write the results was estimated on the same workstation used to simulate and generate the result files with the FOM (2-CPU Intel\\textsuperscript{\\textregistered} Xeon\\textsuperscript{\\textregistered} Gold 6130, 2.1 GHz). The speedup is the ratio between the FOM runtime and the ROM runtime. Its evolution according to the FOM time step is illustrated in Figure~\\ref{Speedup}. It was estimated from the ROM and FOM simulation time obtained when $a\/\\ell=0.7$. The speedup varies between 52106 for a FOM time step of $10^{-4}$ (i.e for the lowest value of $Ca$ tested) and 4200 for $5 \\times 10^{-4}$ (i.e $Ca\\leq 0.05$). It is thus possible to estimate the capsule dynamics very precisely with the developed ROM, while considerably reducing  the computational time.\n\n\\review{Another significant advantage is the gain in storage of the simulation results. By storing only the reduced variables  $\\bm{\\alpha}$, $\\bm{\\beta}$, the modes $\\lbrace \\phi_k\\rbrace $ and the initial position of the nodes of each couple $\\theta=(Ca,a\/\\ell)$, the training database is reduced from 1.9 GB, when computed with the FOM, to 0.15 GB with the ROM. It can therefore be more easily shared. }\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{Speedup.eps}\n\\caption{Evolution of the speedup as function to the time step imposed to simulate the capsule dynamics with the FOM ($a\/\\ell=0.7$).}\n\\label{Speedup} \n\\end{figure}\n\n\\section{\\review{Full space-time-parameter ROM (for any admissible  parameter value)}}\n\\label{sec:interpolation}\n\\review{\n\\subsection{General methodology}\n\nIt is here again assumed that a training database of $N$ precomputed FOM results is available. Now we would like to derive a ROM for any\nparameter couple $\\bm{\\theta}=(Ca,a\/\\ell)$ in the admissible parameter domain. The proposed space-time-parameter ROM is made of two steps. The first step consists in predicting the space-time solution $\\{\\ubold\\}(t;\\bm{\\theta})$ \nby means of a robust interpolation procedure. The second step consists in deriving a ROM in the form of a low-order dynamical system\nby using the predicted solutions of the first step as training data. Then we apply the former procedure detailed in Section \\ref{sec:ROM}. Below we give a detailed explanation of the two steps. \\medskip\n\n\\textbf{Step 1: predictor step.} Considering a parameter couple $\\bm{\\theta}$, we first search the three nearest neighbor parameters in the sample set that form a nondegenerate triangle in the plane $(Ca,a\/\\ell)$. Let us denote them by $\\bm{\\theta}_1$, $\\bm{\\theta}_2$ and $\\bm{\\theta}_3$. We will define a linear operator in the triangle $(\\bm{\\theta}_1,\\bm{\\theta}_2,\\bm{\\theta}_3)$. For that, let us introduce the barycentric coordinates $(\\lambda_1,\\lambda_2,\\lambda_3)$, $\\lambda\\in[0,1]$, $i=1,2,3$ such that\n\\begin{align}\n   \\lambda_1 + \\lambda_2 + \\lambda_3 & = 1, \\label{eq:baryc1}\\\\\n   \\bm{\\theta}_1 \\lambda_1 + \\bm{\\theta}_2 \\lambda_2 + \\bm{\\theta}_3\\lambda_3 & = \\bm{\\theta}. \\label{eq:baryc2}\n\\end{align}\nThe $3\\times 3$ linear system~\\eqref{eq:baryc1},\\eqref{eq:baryc2} is invertible as soon as the triangle $(\\bm{\\theta}_1,\\bm{\\theta}_2,\\bm{\\theta}_3)$ is nondegenerate. Notice that the $\\lambda_i$ ($i=1,2,3$) are actually functions of $\\bm{\\theta}$.\nLet us now denote by $\\{u_1\\}$, $\\{u_2\\}$ and\n$\\{u_3\\}$ the displacement fields for the parameter vectors\n$\\bm{\\theta}_1$, $\\bm{\\theta}_2$ and~$\\bm{\\theta}_3$ respectively. Then we can consider the predicted velocity field $\\hat\\ubold(t;\\bm{\\theta})$\ndefined by\n\\begin{equation}\n\\{\\hat u\\}(t,\\bm{\\theta}) =\n\\lambda_1 \\{u_1\\}(t) + \\lambda_2 \\{u_2\\}(t) + \\lambda_3 \\{u_3\\}(t).\n\\label{eq:hatv}\n\\end{equation}\n\n\\textbf{Step 2: low-order dynamical system ROM.}\nExpression~\\eqref{eq:hatv} can be evaluated at some discrete instants\nin order to generate new training data. Then the SVD-DMD ROM methodology presented in Section~\\ref{sec:ROM} can be applied to these data to get a reduced dynamical system in the form\n\\begin{align*}\n    & \\dot\\bm{\\alpha}(\\bm{\\theta}) = \\bm{\\beta}(\\bm{\\theta}), \\\\\n    & \\dot\\bm{\\beta}(\\bm{\\theta}) = A_\\mu(\\bm{\\theta})\\, \\bm{\\beta}(\\bm{\\theta}).\n\\end{align*}\n\nWe also have a matrix $Q(\\bm{\\theta})$ of orthogonal POD modes and we can go back to the high-dimensional physical space by the standard operations\n\\begin{equation}\n\\{\\hat u\\}(t,\\bm{\\theta}) \\approx Q(\\bm{\\theta})\\, \\bm{\\alpha}(t,\\bm{\\theta}),\\quad\n\\{\\hat v\\}(t,\\bm{\\theta}) \\approx Q(\\bm{\\theta})\\, \\bm{\\beta}(t,\\bm{\\theta}).\n\\end{equation}\nNotice that the capsule position field $\\{\\bm{x}\\}(t,\\bm{\\theta})$ is given by\n\\[\n\\{ x\\}(t;\\bm{\\theta}) = \\{X\\}(\\bm{\\theta}) + \\{\\hat u\\}(t,\\bm{\\theta})\n\\]\nwith an initial capsule position $\\{X\\}(\\bm{\\theta})$ that may depend on $\\bm{\\theta}$ because of the pre-deformation preprocessing\nif $a\/\\ell \\geq 0.95$.\n}\n\n    \n\\review{\\subsection{Numerical experiments, ROM accuracy assessment}\n\\begin{figure} \n\\centering\n\\includegraphics[width=0.55\\textwidth]{GraphDataBaseLearningTesting.eps}\n\\caption{Values of $Ca$ and $a\/\\ell$ included in the testing database (open circle). The filled squares represent the cases in the training database.\nThe dotted line delimits the domain where a steady-state capsule deformation exists for capsules following the neo-Hookean law.}\n\\label{DB2}\n\\end{figure}\n\nA testing database is created using the FOM as in Section \\ref{sec:database} and considering $(Ca,a\/\\ell)$-couples which are not in the training database. A set of 110 $(Ca,a\/\\ell)$-couples are included in this database (Figure \\ref{DB2}). \nFor all the $(Ca,a\/\\ell)$-couples of the testing database, the capsule dynamics is interpolated from the dynamics of the 3 closest neighbors at a given  non-dimensional time. \nCapsule shapes obtained by the ROM are compared to the ones predicted by the FOM at the same nondimensional time. Figure~\\ref{ErreurBaseTestingA20} represents the evolution of the error committed on the capsule shape $\\varepsilon_{Shape}$ on the training database at $Vt\/\\ell=0,\\ 0.4,\\ 1,\\ 2,\\ 5,\\ 10$. At initial time, $\\varepsilon_{\\text{Shape}}$ is zero. The interpolation method is therefore able to capture the initial capsule shape. When the time increases, $\\varepsilon_{\\text{Shape}}$ increases and greater than if we apply directly the POD-DMD method on the FOM results and reconstruct the dynamics. However, $\\varepsilon_{\\text{Shape}}$ remains less than 0.3\\% on the majority of the testing database.  It remains fully acceptable.  $\\varepsilon_{\\text{Shape}}$ is more important near the steady-state limit and when we approach the lowest values of $Ca$ because we are close to the limits of the training base.}\n\n\n\n\n\n\\begin{figure}\n\\centering\n(a)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_outDB0_20modes_mu10-8_ROMFOM.eps}\n(b)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_outDB0.4_20modes_mu10-8_ROMFOM.eps}\n(c)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_outDB1_20modes_mu10-8_ROMFOM.eps}\n(d)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_outDB2_20modes_mu10-8_ROMFOM.eps}\n(e)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_outDB5_20modes_mu10-8_ROMFOM.eps}\n(f)\\includegraphics[width=0.45\\textwidth]{newfigures\/GraphMHD_outDB10_20modes_mu10-8_ROMFOM.eps}\n\\caption{\\review{Heat maps of $\\varepsilon_{\\text{Shape}}$ on the testing database as a function of $Ca$ and $a\/\\ell$ at (a) $\\dot{\\gamma}t$=0, (b) 0.4, (c) 1, (d) 2, (e) 5, (f) 10. The dotted line delimits the domain for which a steady-state capsule deformation exists.} }\n\\label{ErreurBaseTestingA20}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n(a)\\includegraphics[height=0.17\\textwidth]{newfigures\/SSFCuttedPlane.eps}\n(b)\\includegraphics[height=0.17\\textwidth]{newfigures\/SSF_1.eps}\n(c)\\includegraphics[height=0.17\\textwidth]{newfigures\/SSF_2.eps}\n(d)\\includegraphics[height=0.17\\textwidth]{newfigures\/SSF_4.eps}\\\\\n\\caption{\\review{Dynamics estimated by the ROM of a capsule subjected to a simple shear flow. The capsule is shown for $Ca=0.3$ at the non-dimensional time (a) $\\dot{\\gamma}=$ 0,(b) 1.6,(c) 4.8,(d) 6.4. 15 modes and $\\mu=10^{-6}$ are considered. The red point is a membrane point. }} \n\\label{SSF3D}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{newfigures\/SSF2D_2plan.eps}\\\\\n\\caption{\\review{Capsule subjected to a simple shear flow: Comparison of the  contours given by the FOM (dotted line) and estimated by the ROM (orange line). The capsule is shown for $Ca=0.3$ in the shear plane in the top and in the cross plane in the bottom. 15 modes and $\\mu=10^{-6}$ are considered. } } \n\\label{SSFCompProfils}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{newfigures\/SSFErrorShape.eps}\\\\\n\\caption{\\review{Evolution of the maximum error committed on the shape of a capsule subjected to a simple shear flow as a function of the capillary number $Ca$. The capsule dynamics was simulated up to a non-dimensional time $\\dot{\\gamma}=10$.} } \n\\label{SSFError}\n\\end{figure}\n\n\\section{\\cdd{Discussion and conclusion}}\n\\review{\nAs a summary, in this paper we have considered a $\\bm{\\theta}$-parametrized reduced-order model of microcapsule dynamics in the form\n\\begin{align*}\n    & \\dot\\bm{\\alpha}(\\bm{\\theta}) = \\bm{\\beta}(\\bm{\\theta}), \\\\\n    & \\dot\\bm{\\beta}(\\bm{\\theta}) = A_\\mu(\\bm{\\theta})\\, \\bm{\\beta}(\\bm{\\theta}).\n\\end{align*}\n\n\\noindent The vector  $\\bm{\\theta}=(Ca,a\/\\ell)$ contains the governing parameters, the coefficients $\\alpha_k(t,\\bm{\\theta})$ and $\\beta_k(t,\\bm{\\theta})$ are spectral coefficients of \nPOD decomposition for the displacement and velocity fields respectively, and the \nmatrix $A_\\mu(\\bm{\\theta})$ is identified from data using a dynamic mode decomposition least-square procedure.\nWe have numerically proven for a broad range of capillary numbers $Ca$ and aspect ratios $a\/\\ell$ that it is able to capture the dynamics up to the steady state of a capsule flowing in a channel and its large deformations.\nAs a first approach, we have presently chosen to use a DMD method that is linear in time to build the ROM model. Still the ROM model captures spatial non-linearity by means of the POD modes. \nThe resulting reduced-order model is of great fidelity, weak discrepancies being only observed in the early transient stage.\nWe have also shown that the learning time need to be larger than the transient stage duration and that we can go beyond the FOM time window used for the training of the ROM model.\n\nFor generalisation, we have computed the capsule dynamics for any parameter set. The generalization algorithm is based on interpolation: we first pre-calculate the ROM dynamic model at a finite number of points in the parameter space domain and determine the $\\alpha$, $\\beta$ and $\\phi_k$ (and thus the capsule displacement) at these points. For any other value of the parameters, we first predict the time-evolution of the capsule node displacements using a linear interpolation procedure in the parameter space and then build a dynamical system based the DMD methodology. The error is mostly below 0.3\\% over the entire domain, which proves the precision and utility of the ROM approach.\n\nLike any other data-driven model, the model requires a certain number of high-fidelity simulations to provide accurate predictions. By discretizing the parameter space in a regular and homogeneous way (Figure \\ref{DB}), we have not presently tried to optimize the number of FOM simulations. But sampling strategies like the Latin Hypercube Sampling (LHS) exist and result in a net reduction in FOM simulation number. \nThe empirical law, conventional among the data-driven model community, is that one needs between $10\\times D$ and $50 \\times D$ points, where $D$ is the dimension of the problem ($D = 2$ in our case). This law shows that the number of high-fidelity simulations does not explode with the problem dimension, owing to the linear dependence of the law.\n\nTo prove the generality of the proposed approach, we additionally apply the ROM to a capsule in simple shear flow. This classical case was extensively studied  over the past years \\citep{ramanujan98,lac_dbb:05,Li2008,Walter2010,Foessel2011,dbb2010book,dupont2015}.\nWe build a ROM model to predict the evolution of an initially spherical capsule subjected to a shear rate $\\dot{\\gamma}$ until $\\dot{\\gamma}t=10$ with 15 modes, a learning time of $T_L=10$  and $\\mu=10^{-6}$. \nThe time step~$\\Delta t$ between each snapshot is equal to 0.04.\nWe retrieve that the initial capsule is elongated in the straining direction by the external flow and that the membrane rotates around the deformed shape due to the flow vorticity (Figure~\\ref{SSF3D}). The ROM is thus able to recover the tank-treading motion. The capsule contours in both shear and  perpendicular planes predicted by the ROM and simulated by the FOM are in very good agreement (Figure \\ref{SSFCompProfils}).\nFigure~\\ref{SSFError} shows the evolution of the maximum error on the  capsule shape for different values of $Ca$. At $Ca=0.1$, folds appear periodically on the capsule, which prevents the ROM from reproducing precisely the wrinkling phenomenon. \nBut from  $Ca\\geq 0.3$, the error is reduced by an order of magnitude and is below 0.2\\%.\n\n\nThe linear differential model is stable as soon as the eigenvalues of $A_\\mu$ have nonpositive real parts, and is consistent with steady states as soon as zero is an eigenvalue. Numerical experiments show that identified matrices $A_\\mu$\nfrom data have eigenvalues with negative real parts and one of the eigenvalues is very close to zero.\n\nAs it is often the case with spectral-like methods, there is a trade-off between accuracy and ill-conditioning effects: when a large number of POD modes are used ($K>20$), the data matrix~$\\mathbb{X}$ of snapshot POD coefficients is ill-conditioned. For the determination of $A_\\mu$, we have used a Tikhonov regularization in the least square cost function (see~\\eqref{eq:flo21}) in order to have a better conditioned problem and a $L$-curve procedure to determine the best regularization coefficient $\\mu$. Unfortunately we observe some limitations in the accuracy. A perspective would be to use a proximal approach: within an iterative procedure, at iteration~$(p+1)$, compute the matrix $A_\\mu^{(p+1)}$ solution of\n\\[\nA_\\mu^{(p+1)} = \\arg \n    \\min_{A\\in\\mathscr{M}_K(\\mathbb{R})}\\ \\frac{1}{2} \\|\\mathbb{Y}-A\\mathbb{X}\\|_F^2 + \\frac{\\mu}{2} \\|\\mathbb{X}\\|_F^2\\, \\|A-A_\\mu^{(p)}\\|_F^2\n\\]\nusing $A_\\mu^{(0)}=0$. At convergence, one can observe that the regularization term vanishes, so that one can expect better accuracy with this approach. This will be investigated in a future work.\n\nWe have proposed a  successful and very efficient ROM for FSI problems.\nIt is an alternative to the use of HPC. It must be seen as a complimentary (and non-competing) approach to full-order models, and has many advantages. \nAmong them, one can mention the easiness in implementation. \nIt leads to a very handy set of ODEs, that are easy to determine from an algorithmic point of view. Furthermore, the system can be run on any computer. The size of the matrices is, indeed, reduced from ($3 \\times 2562 \\text{nodes} \\times 250 $snapshots) to about ($3 \\times 2562 \\text{nodes} \\times (K+1)$), where the number of modes is $K$ = 20. The computation required time is a few milliseconds for one parameter set. The current  speedups are between 5 000 and 52 000, which out-performs any full-order model approach. We believe that this work is an encouraging milestone to move toward real time simulation of general coupled problems and to deal with high-level parametric studies, sensitivity analysis, optimization and uncertainty quantification.\n\n\nThe next milestone following this work would be to go toward nonlinear differential  dynamical systems as reduced-order models. There is three natural ways for that. The first one is to use Kernel Dynamic Model Decomposition (KDMD) rather than DMD. But we have recently shown in \\cite{DeVuyst2022} that a non-linear low-order dynamical model does not provide significant improvement.\nThe second one is to use Extended Dynamic Model Decomposition (EDMD) \\citep{Williams2015}. The EDMD method adds some suitable nonlinear observables (or features) in the data, so that a linear 'augmented' dynamical system is searched for. \nA third option is would be to directly use artificial neural networks (ANN), in particular recurrent neural networks (RNN) \\citep{Trischler2016}. The RNN training would replace the DMD procedure, and would be trained with the same POD coefficient matrices $\\mathbb{X}$ and~$\\mathbb{Y}$. As shown in the recent study by \\cite{Lin2021}, artificial intelligent may prove to be efficient and precise to predict capsule deformation.\n}\n\n\\backsection[Acknowledgements]{\nThe authors warmly thank Prof. Pierre Villon for fruitful discussions on model-order reduction and related topics.}\n\n\\backsection[Funding]{This project has received funding from the European Research Council (ERC) under the European Union's Horizon\n2020 research and innovation programme (Grant agreement No. ERC-2017-COG - MultiphysMicroCaps). }\n\n\\backsection[Declaration of interests]{The authors report no conflict of interest.}\n\n\\backsection[Author ORCID]{C. Dupont, https:\/\/orcid.org\/0000-0002-7727-3846; F. De Vuyst, https:\/\/orcid.org\/0000-0003-0854-4670;  A.-V. Salsac, https:\/\/orcid.org\/0000-0001-8652-5411}\n\n\\backsection[Author contributions]{A.V.S and F.D.V. created the research plan and formulated the numerical problem. C.D. implemented the numerical method and performed the tests.  All authors contributed to analysing data and reaching conclusions, and in writing the paper.}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuantum tunneling through a barrier is a fundamental physical effect\n\\cite{AG,Lee}, and the neutral atoms trapped in optical lattices\nhave given us an opportunity to study the aspects of  tunneling,\nincluding coherent dynamics on macroscopic scale. For the symmetric\ndouble-well system which has long served as a paradigm of quantum\nphysics, the eigenvalue structures are well-known for the  low lying\nexcited states \\cite{WKB1,WKB2,WKB3,WKB4,GDJH}, and it has been pointed out that, by\nadding a specific driving force, the tunneling dynamics can be\nbrought to a complete standstill known as coherent destruction of\ntunneling (CDT) \\cite{GDJH}. Recently this coherent destruction has\nbeen visualized in single particle tunneling \\cite{KSSTO} as well as\nin the tunneling of Bose-Einstein condensates (BECs) \\cite{Valle}.\nOn the other hand, in experiments, localized wave packets have been\nprepared in one well of an asymmetric double-well potential, and the\ntunneling dynamics has been observed by turning off the asymmetric\npart of the potential \\cite{HADGJ,DJ,KSSTO}. It is also known that\nthe density distributions of the BECs of interacting particles are\nasymmetric in the asymmetric double-well potentials\n\\cite{AG,HWAHS,SH,DHC}, to result in the non-vanishing relative\nphase evolution rate \\cite{JS1,JS2}.\n\nIn this article,\nwe will find the energy eigenvalues of low lying excited states of\nan asymmetric double-well potential $V(x)$ which has one local maximum\nat $x=x_c$ between the wells, by constructing WKB wave functions with\nthe quadratic connection formula.\nTo quantify the degree of tunneling,  we define the  {\\em tunneling\nvisibility} for a wave function $\\psi(x,t)$, as ${\\cal V}= (P_{max}\n-P_{min})\/(P_{max} +P_{min}),$ where $P_{max}$ $(P_{min})$ denotes\nthe maximum (minimum) value of $P_r(t)=\\int_{x_c}^\\infty \\psi^*(x,t)\n\\psi(x,t) dx$ during the time evolution. We find that the wave\nfunctions of (almost) arbitrary  ${\\cal V}$ ranging from 0 to 1 can\nbe realized from a Gaussian wave packet, by controlling the\npotential energy difference between the bottoms of the double well.\nThe case of ${\\cal V}\\approx 0$ with $P_{max}\\approx 1$ amounts to\nthe CDT, and this case can be realized when the two wells can be\nconsidered to be separate.\nA Gaussian wave packet at a stand still  can be realized also in the\nwell of higher bottom.\n\nWhile the results for a single particle can be applied only for\nthe systems of noninteracting particles, we note that interacting systems\nhave been extensively studied \\cite{MCWW,NK,JI,JMP}. Particularly,\nfor the systems of a few bosons, highly delayed pair tunneling\nanalogous to nonlinear self-trapping  has\nbeen found for the medium range of the interaction strength \\cite{JMS}.\nThough we only consider the double-well potentials bounded\nfrom below, the system of a particle in a periodic potential\nwith an additional constant force has been of great interest \\cite{Sias,BWK,Holthaus},\nand we note that this system has been analyzed through the instanton method\n\\cite{LFZLC}\nwhich is intimately related with the WKB analysis \\cite{WKB1}.\n\nIn the next section, before presenting the main {\\em analytic} results,\ntwo asymmetric systems will be numerically studied.\nIn Sec.~III, we will construct the WKB wave functions for a general potential\n$V(x)$.  It will be shown that the asymmetric systems\ncan be classified into two different regimes: In the one regime, an eigenfunction\nof low lying excited states has\nsignificant amplitude in both wells as in the symmetric systems,\nwhile, in the other, the eigenfunction describes the\nparticle mostly localized in just one of the wells.\nIn Sec.~IV, we will develop formulas for the estimation of the energy eigenvalues\nin the regime of the localized eigenfunctions, and for the estimation of\nthe tunneling visibility in the other.\nIn Sec.~V, the asymmetric double oscillator model\nwill be numerically solved to give an implication on the eigenvalue structure\nof a general double-well system, and to show that {\\em WKB description} could be\n{\\em remarkably accurate}.\nThe last section will be devoted to a summary and discussions.\n\n\n\n\n\n\n\\section{Coherent control of Tunneling: Numerical examples}\nIn this section, we will study two systems numerically to indicate that\nthe coherent control method could  also be useful in a general potential\nwhose wells are {\\em not} exactly quadratic,\nand to expose that gravity may be used to control the tunneling dynamics in\nthe settings of the recent experiments \\cite{KSSTO,HADGJ}.\n\n\nFirst,   we\nconsider the system of a particle of mass $m$ in the quartic\ndouble-well potential\n\\begin{equation}\nV_Q(\\alpha;x)=\\hbar\\omega\\left[\n-\\frac{x^2}{4l_{ho}^2}+\\frac{x^4}{96l_{ho}^4}+\\frac{\\alpha\nx}{8\\sqrt{3}l_{ho}}+C(\\alpha)\\right],\n\\end{equation}\nwith $l_{ho}=\\sqrt{\\frac{\\hbar}{m\\omega}}$, where $C(\\alpha)$ is\nintroduced  to ensure that the minimum of the potential is 0. While\nthe term proportional to $\\alpha$ is added to give the asymmetry,\n$V_Q(0;x)$ is a special case of the well-known potentials\n\\cite{GDJH,WKB1,WKB2,WKB3,WKB4}.\nFor $V_Q(0;x)$, the angular frequency for small oscillations\nat the bottoms of each well is $\\omega$, and the barrier height is\n$3\\hbar\\omega\/2$.\n\nFor the Gaussian wave packet\n$\\phi_G(\\alpha;x)=<x|\\phi_G(\\alpha)>=\\exp[-(x-a(\\alpha))^2\/(2l_{r}^2)]\/(l_r^2\\pi)^{1\/4}$\ncentered at the bottom of the right well, $a(\\alpha)$, with\n$l_r^{-4}=\\frac{m^2\\omega_r^2(\\alpha)}{\\hbar^2}=\\frac{m}{\\hbar^2}\\frac{d^2\nV_\\alpha(x)}{dx^2}|_{x=a(\\alpha)}$,  in Fig.~\\ref{quartic}, we\nevaluate the probabilities ${\nP}_i(\\alpha)=|<\\phi_G(\\alpha)|\\varphi_i(\\alpha)>|^2$, where\n$\\varphi_i(\\alpha;x)=<x|\\varphi_i(\\alpha)>$ are the eigenfunctions\narranged in the order of ascending energy eigenvalue $E_i(\\alpha)$\n$(i=0,1,2,\\cdots)$. When we define the (unnormalized) wave function $\\psi_\\alpha (x,t)$ as\n\\begin{eqnarray}\n&\\psi_\\alpha(x,t)&\\cr\n&=&\\sqrt{P_1(\\alpha)}\\varphi_1(\\alpha;x)\\exp\\left[\\frac{-iE_1(\\alpha)t}{\\hbar}\\right]\\cr\n&&+\\sqrt{P_2(\\alpha)}\\varphi_2(\\alpha;x)\\exp\\left[\\frac{-iE_2(\\alpha)t}{\\hbar}\\right],\n\\end{eqnarray}\nFig.~\\ref{quartic} indicates that, in this system of deep quantum regime,\n$\\psi_\\alpha(x,0)$ closely describes the Gaussian wave function $\\phi_G(\\alpha;x)$\nfor $0.01\\le \\alpha $ in the given range of $\\alpha$.\nFurther, plot (e) shows that, if $ 0.01<\\alpha<0.9$,\nthe Gaussian wave\npacket is closely described by an eigenfunction,\n$\\varphi_1(\\alpha;x),$ which is in turn mostly localized in the\nright well of the higher bottom (We note that localized eigenfunctions have also\nbeen known in the buried double-well systems \\cite{WM}).\n\n\\begin{figure}\n\\includegraphics[width=3.3in]{1a.eps}\n\\includegraphics[width=3.3in]{1b.eps}\n\\includegraphics[width=3.3in]{1cd.eps}\n\\includegraphics[width=3.3in]{1e.eps}\n\\caption{(Color online) Tunneling dynamics in $V_Q(\\alpha;x)$.  (a)\nA potential (thick solid line) and three lowest eigenvalues (thin\nblack, green, red solid lines), for $\\alpha=\\alpha_0=0.985$. The\ndotted and dashed, dashed, solid vertical lines indicate the values\nof  $a(\\alpha_0)$,  $-b(\\alpha_ 0)$, $x_c(\\alpha_0)$, respectively,\nwhere $V_Q(\\alpha_0;-b(\\alpha_0))=0$. (b) $|\\phi_G(\\alpha_0;x)|^2$\n(solid line), and $|\\psi_{\\alpha_0}(x,t)|^2$ at $t=0$ (dotted line)\nand at $t=\\hbar \\pi\/(E_2(\\alpha_0 )-E_1(\\alpha_0 ))$ (dashed line).\n (c) and (d)\nThe calculated tunneling visibility for $\\psi_\\alpha(x,t)$ (solid\nlines) and an estimation (dotted lines). The estimation is made from\nEq.~(\\ref{visibility}). For (c),\n$\\delta_\\epsilon\/\\delta_a=2V_\\alpha(a(\\alpha))\/(E_1(0)-E_0(0)).$ For\n(d), since $E_2(\\alpha)-E_1(\\alpha)$ has the minimum at\n$\\alpha=\\alpha_1=1.00$,\n$2[V_\\alpha(a(\\alpha))-V_{\\alpha_1}(a(\\alpha_1))]\/(E_2(\\alpha_1)-E_1(\\alpha_1))$\nis used as $\\delta_\\epsilon\/\\delta_a$.   The \"{\\bf +}\" mark  denotes\nthe value of $\\alpha_0$.  } \\label{quartic}\n\\end{figure}\n\nAs is well-known in the systems of symmetric potentials \\cite{WKB1,WKB3},\nthe wave function $\\frac{1}{\\sqrt{2}}\\left(\\varphi_0(0;x)e^{\\frac{-iE_0(0)t}{\\hbar}}\n+\\varphi_1(0;x)e^{\\frac{-iE_1(0)t}{\\hbar}}\\right)$ describes the system\nwhich tunnels back and forth between an almost Gaussian state localized\nin the left well and the state localized in the right well,\nwith ${\\cal V} \\approx 1$.\nAs, for $0.01\\le\\alpha\\le 0.9$, the ground state wave function $\\varphi_0(\\alpha;x)$\nand the first excited state wave function $\\varphi_1(\\alpha;x)$\nare mostly localized in the left and right wells with the shapes close to a Gaussian, respectively,\nif we could change $\\alpha$ from 0 to a value\nwhich is larger than 0.01 but smaller than 0.9, without distorting the wave function\nby the change,\nthen we will have a system where the probability density is almost stationary.\nFor this stationary system, the probability of finding the particle in the left\n(or right) well depends on the details of changing $\\alpha$.\nIf the change could be made in a much shorter period of time compared with\nthe period of tunneling $2\\hbar \\pi\/(E_1(0 )-E_0(0 ))$, the probability of finding the\nparticle in one of the wells\ncrucially depends on the timing of the change.\nIf we tune $\\alpha$ back to 0 or to 1, tunneling dynamics appears again,\nbut in this time the tunneling visibility ${\\cal V}$ could be less than 1\ndepending on the  process.\n\n\n\nSecond, for an atomic spinor trapped in a double-well of the potential\n\\begin{eqnarray}\nV_L(\\beta;x)&=&C E_R[\\cos^2 kx +\\xi\\cos^2\n(\\frac{kx}{2})-{\\xi}\/{2}+{\\xi^2}\/{16}]\\cr\n&&+m\\beta gx,\n\\end{eqnarray}\nwith $E_R=(\\hbar\nk)^2\/2m$  \\cite{KSSTO,HADGJ}, we explicitly consider the case of\n$C=10$ and $\\xi=\\frac{1}{2}$ which is similar to an experimental\nsituation \\cite{KSSTO}. For $\\beta=0$, with the vanishing boundary condition\nat $x=\\frac{2j\\pi}{k}$ ($j:$integer), there are three energy\neigenvalues $E_i^L~(i=0,1,2)$ under the barrier height, with\n$E_1^L-E_0^L=0.122E_R$, $E_1^L+E_0^L=5.70E_R$, and $E_2^L=7.27E_R$.\n\nIf $\\lambda=2\\pi\/k=811$ nm, $g=980$ ${\\rm cm\/s^2}$ and\ncesium atoms are in $V_L(1;x)$, the fact\n$mg\\lambda\/2=0.580E_R=4.75(E_1^L-E_0^L)$ then suggests that the\ntunneling visibility is very low for a cesium atom trapped in the\nvertical optical lattice aligned along the Earth's gravity. If the\nright minimum of a double-well is located at $x_r$, we construct a\nGaussian wave packet\n$\\phi(f;x)=<x|\\phi(f)>=\\exp[-(x-x_r)^2\/(2f^2l_{r}^2)]\/(f^2l_r^2\\pi)^{1\/4}$,\nhere, with  a fitting factor $f$. Indeed, for the first excited state\n$|\\varphi_1^L>$ of $\\beta=1$, we find that\n$|<\\phi(0.759)|\\varphi_1^L>|^2=0.980$ and\n$|<\\phi(1.00)|\\varphi_1^L>|^2=0.945$,\nwhich shows that the eigenfunction\n$\\varphi_1^L(x)$ is closely described by a Gaussian wave\npacket, to prove the low visibility of the wave packet.\nAs the visibility is high for the symmetric horizonal lattice, this shows that\ngravity may be used to control the tunneling dynamics in the optical\nlattices.\n\n\\section{WKB wave functions}\n\nIn this section, we will construct WKB wave functions for a general potential\n$V(x)$, assuming that  $V(x)$ is written as $\\frac{m\\omega_l^2}{2}(x+b)^2$ and as\n$\\frac{m\\omega_r^2}{2}(x-a)^2+\\epsilon\\hbar\\omega_r$ around the\nbottoms of the left and right wells, respectively. The eigenvalue structure\nwill then be found by requiring the WKB wave functions to be asymptotically matched,\nin the overlapping regions, onto the exact solutions of the quadratic wells.\n\n\nIn the quadratic\nregions of $V(x)$, the eigenfunction is described by the parabolic\ncylinder function $D_\\eta(z)$, and the eigenfunction of an energy\neigenvalue $\\hbar\\omega_r (\\nu+\n\\epsilon+\\frac{1}{2})[\\equiv\\hbar\\omega_l (\\mu +\\frac{1}{2})]$ is\nwritten as\n\\begin{equation}\nC_L D_{\\mu}\\left(- \\frac{\\sqrt{2}(x+b)}{l_{l}} \\right) ~{\\rm and}~\nC_R D_{\\nu}\\left( \\frac{\\sqrt{2}(x-a)}{l_{r}} \\right),\n\\end{equation}\nnear the bottoms of the left and right wells, respectively,  with\n$l_i=\\sqrt{\\frac{\\hbar}{m\\omega_i}}$ $(i=l,r)$. On the other hand,\nby taking $x_c=0$,\nin the region of the barrier we have an approximate solution for the\neigenfunction through the WKB method \\cite{WKB1,WKB2,WKB3}, as\n\\begin{eqnarray}\n\\psi_{WKB}(x)&=&  \\frac{N_R\\sqrt{\\hbar}}{\\sqrt{l_{ho}\np(x)}}\\exp\\left[\\int_0^x\\frac{p(y)}{\\hbar} dy\\right]\\ \\cr\n&&+\\frac{N_L\\sqrt{\\hbar}}{\\sqrt{l_{ho}\np(x)}}\\exp\\left[-\\int_0^x\\frac{p(y)}{\\hbar} dy\\right], \\label{WKB}\n\\end{eqnarray}\nwhere\n$p(y)=\\sqrt{2m[V(y)-(\\nu+\\epsilon+\\frac{1}{2})\\hbar\\omega_r]}$.  The\nrelations between the real coefficients $C_L,~C_R,~N_L,~N_R$ may be\ngiven by comparing the eigenfunctions in the regions where the\ndescriptions by the parabolic cylinder function and by the WKB\nfunction are both valid. On the negative real axis, the asymptotic\nexpansion of the parabolic cylinder function is \\cite{AS}\n\\begin{eqnarray}\nD_\\eta(z) &\\sim\n&\\frac{\\sqrt{2\\pi}}{\\Gamma(-\\eta)}\\exp\\left[\\frac{z^2}{4}\\right]\\frac{1}{|z|^{\\eta+1}}\n\\left[1 +O\\left(\\frac{\\eta^2}{z^2}\\right)  \\right]\\cr &&+\\cos(\\eta\n\\pi)\n\\exp[-\\frac{z^2}{4}]|z|^\\eta\\left[1+O\\left(\\frac{\\eta^2}{z^2}\\right)\\right].\n~~~~~ \\label{pcf asymp}\n\\end{eqnarray}\n\n\n\\begin{figure}\n\\includegraphics[width=3.3in]{vxf.eps}\n\\caption{(Color online) An illustration of turning points.} \\label{vxf}\n\\end{figure}\n\n\n\nThe WKB wave function can also be expanded, for instance, as\n\\begin{eqnarray}\n&&\\psi_{WKB}(x) \\cr && \\sim2^\\frac{1}{4}N_R\n\\sqrt{\\frac{l_r}{l_{ho}}}\\left(\\frac{e}{\\nu+\\frac{1}{2}}\\right)\n                   ^{\\frac{\\nu}{2}+\\frac{1}{4}}\\left(\\frac{\\sqrt{2}(a-x)}{l_{r}}\\right)^\\nu\n                   \\cr\n&&~~~~~~~~~~~~~\\times\n\\exp\\left({-\\frac{(a-x)^2}{2l_{r}^2}+\\int_0^{a_\\nu}\n\\frac{p(y)}{\\hbar} dy}\\right)\\cr &&~~~ +2^\\frac{1}{4}N_L\n\\sqrt{\\frac{l_r}{l_{ho}}} \\left(\\frac{\\nu+\\frac{1}{2}}{e}\\right)\n                   ^{\\frac{\\nu}{2}+\\frac{1}{4}}\\left(\\frac{l_{r}}{\\sqrt{2}(a-x)}\\right)^{\\nu+1}\n                   \\cr\n&&~~~~~~~~~~~~~\\times \\exp\\left(\\frac{(a-x)^2}{2l_{r}^2}-\\int_0^{a_\\nu} \\frac{p(y)}{\\hbar} dy\\right),\n\\label{WKB asymptotic}\n\\end{eqnarray}\nwhen $\\frac{a-x}{a-a_\\nu} \\gg 1$ and $x$ is in the region of\nquadratic potential of the right well,  where $a_\\nu, -b_\\mu$ are\nturning points satisfying\n$V(a_\\nu)=V(-b_\\mu)=(\\nu+\\epsilon+\\frac{1}{2})\\hbar\\omega_r$ ~~$(a>\na_\\nu>0,~b>b_\\mu >0)$ (see Fig.~\\ref{vxf}). By comparing the leading terms in the\nparabolic cylinder function description which originates from the\nfirst term in the right hand side (r.h.s.) of Eq.~(\\ref{pcf asymp})\nwith the relevant terms in the WKB description, we have\n\\begin{eqnarray}\nN_R&=&\\frac{\\sqrt{\\sqrt{2}\\pi}}{\\Gamma(-\\mu)}\\sqrt{\\frac{\\l_{ho}}{l_l}}\n             \\left(\\frac{e}{\\mu+\\frac{1}{2}}\\right)^{\\frac{\\mu}{2}+\\frac{1}{4}}\n             e^{\\int_{-b_\\mu}^0 \\frac{p(y)}{\\hbar} dy}C_L,\\cr\nN_L&=&\\frac{\\sqrt{\\sqrt{2}\\pi}}{\\Gamma(-\\nu)}\\sqrt{\\frac{\\l_{ho}}{l_r}}\n             \\left(\\frac{e}{\\nu+\\frac{1}{2}}\\right)^{\\frac{\\nu}{2}+\\frac{1}{4}}\n             e^{\\int_0^{a_\\nu} \\frac{p(y)}{\\hbar} dy}C_R.\n\\label{necessary condition}\n\\end{eqnarray}\nIf any of $\\mu$ or $\\nu$ is not close to a non-negative integer,  in\nthe large separation limit of $e^{\\int_{-b_\\mu}^0 \\frac{p(y)}{\\hbar}\ndy}\\gg 1$ and $e^{\\int_0^{a_\\nu} \\frac{p(y)}{\\hbar} dy}\\gg 1$,\nEq.~(\\ref{necessary condition}) yields\n\\begin{equation}\n|N_R| \\gg |C_L| ~~~{\\rm and}~~~ |N_L|\\gg|C_R|.\n\\label{noninteger}\n\\end{equation}\nIf the WKB condition $p^2(x) \\gg \\hbar|\\frac{dp}{dx}|$ is satisfied,\nin the region of the barrier, the first term  in r.h.s.~of\nEq.~(\\ref{WKB}) with positive (negative) $N_R$ is a monotonically\nincreasing (decreasing) function and the second term with positive\n(negative)$N_L$ is a monotonically decreasing (increasing) function.\nEq.~(\\ref{noninteger}) then implies that if an eigenfunction exist\nfor such $\\nu$, it gives the probability distribution in which the\nprobability of finding the particle in the barrier region is\nconsiderable. Since $E_\\varphi >\\int \\varphi^*(x) V(x) \\varphi(x)$,\nif an eigenfunction gives a considerable probability in the barrier\nregion, the eigenvalue can not be much smaller than $V(0)$.\n\nOn the other hand, if any of $\\nu$ or $\\mu$ is close to a\nnon-negative integer, the relations in Eq.~(\\ref{noninteger}) are\nnot valid. If $\\nu$ is close to a non-negative integer, due to the\nsingularity in the gamma function, the second term of the r.h.s.~of\nEq.~(\\ref{pcf asymp}) can also be a leading term. By comparing this\ntype of leading term in the parabolic cylinder function description\nwith the relevant term in the WKB wave function, we have a relation\nbetween $N_R$ and $C_R$. By combining this relation with the first\none in Eq.~(\\ref{necessary condition}), we have\n\\begin{eqnarray}\n\\frac{C_L}{C_R}&=&\\sqrt{\\frac{l_l}{l_r}}\\frac{\\cos(\\nu\\pi)\\Gamma(-\\mu)}{\\sqrt{2\\pi}}\n     \\left( \\frac{\\nu+\\frac{1}{2}}{e}\\right)^{\\frac{\\nu}{2}+\\frac{1}{4}}\n     \\cr\n&&\\times   \\left(\n\\frac{\\mu+\\frac{1}{2}}{e}\\right)^{\\frac{\\mu}{2}+\\frac{1}{4}}\n\\exp\\left[- \\int_{-b_\\mu}^{a_\\nu} \\frac{p(y)}{\\hbar}dy\\right].\n\\label{integer nu}\n\\end{eqnarray}\nIf $\\mu$ is not close to an integer, Eqs.~(\\ref{necessary\ncondition}) and  (\\ref{integer nu}) imply that the eigenfunction\ngives the probability distribution in which the particle is mostly\nfound in the right well.\n\nIf $\\mu$ is close to a non-negative integer, we have the relation\n\\begin{eqnarray}\n\\frac{C_L}{C_R}&=&\\sqrt{\\frac{l_l}{l_r}}\\frac{\\sqrt{2\\pi}}{\\cos(\\mu\\pi)\\Gamma(-\\nu)}\n     \\left( \\frac{e}{\\nu+\\frac{1}{2}}\\right)^{\\frac{\\nu}{2}+\\frac{1}{4}}\\cr\n&&\\times \\left( \\frac{e}{\\mu+\\frac{1}{2}}\\right)^{\\frac{\\mu}{2}+\\frac{1}{4}}\n     \\exp\\left[ \\int_{-b_\\mu}^{a_\\nu} \\frac{p(y)}{\\hbar}dy\\right].\n\\label{integer mu}\n\\end{eqnarray}\nIn this case, if $\\nu$ is not close to an integer,\nEqs.~(\\ref{necessary condition})  and (\\ref{integer mu})  imply that\nthe eigenfunction gives the probability distribution of the particle\nmostly localized in the left well.\n\nFor an eigenstate whose eigenvalue is much lower than the barrier\nheight,  the eigenvalue thus must be close to\n$\\hbar\\omega_l(m+\\frac{1}{2})$ or\n$\\hbar\\omega_r(n+\\frac{1}{2}+\\epsilon)$ $(n,m=0,1,2,\\cdots)$, the\neigenvalues of the quadratic potentials of the wells. Furthermore,\nthe fact $D_k(\\sqrt{2}y)=e^{-y^2\/2}H_k(y)\/(\\sqrt{2})^k$ for a\nnon-negative integer $k$, implies that, if the eigenvalue is close\nto $\\hbar\\omega_r(n+\\frac{1}{2}+\\epsilon)$\n($\\hbar\\omega_l(m+\\frac{1}{2})$), the eigenfunction of the\ndouble-well system must be closely described by\n$\\psi_n^{ho}(a;l_r;x)$ ($\\psi_m^{ho}(-b;l_l;x)$) around the bottom\nof the right (left) well, with  an eigenfunction of a simple\nharmonic oscillator $\\psi_k^{ho}(c;l;x)$ ($\\equiv H_k(\\frac{x-c}{l})\\exp[-\\frac{(x-c)^2}{2l^2}]\/\\sqrt{\\sqrt{\\pi}l 2^k\nk!}$).\n\n\n\n\\section{Two different regimes}\n\nThe analysis of the previous section shows that an eigenfunction of\nthe low lying excited states in the large separation limit has\nsignificant amplitude either in both wells or in just one of the wells.\nIn this section, we will show that the eigenfunction of significant\namplitude in both wells must be accompanied by another eigenfunction\nto form a doublet.  As in the symmetric case, a linear combination\nof the doublet is responsible for the tunneling, and\nwe will calculate the tunneling visibility for the combination.\nFor the eigenfunctions localized in one of the wells, we will develop a\nformula for the energy eigenvalue estimation.\n\n\n\\subsection{Tunneling visibility}\n\n\nFor the case that both $\\mu$ and $\\nu$ are close to integers $m$ and $n$,\nrespectively,\nwe define $\\delta_\\epsilon= \\epsilon+n- \\frac{\\omega_l}{\\omega_r}m$,\nso that $|\\delta_\\epsilon|$ is equal or less than the minimum of\n$\\frac{1}{2}$ and $ \\frac{\\omega_l}{2\\omega_r}$. In this case,\nthe corresponding eigenfunction gives\nconsiderable probabilities in both of the left and right wells,\nand $\\mu$ and $\\nu$ should be written as $\\mu=m+\\delta_\\mu$ and\n$\\nu=n+\\delta_\\nu$ with $|\\delta_\\mu|,$ $|\\delta_\\nu|,$\n$|\\delta_\\epsilon|\\ll 1.$ From the fact that\n$\\delta_\\mu=\\frac{\\omega_r}{\\omega_l}(\\delta_\\nu+\\delta_\\epsilon)$,\nEqs.~(\\ref{integer nu}) and (\\ref{integer mu}) then yield\n$\\delta_\\nu^2+\\delta_\\epsilon\\delta_\\nu-\\delta_a^2=0,$ with\n\\begin{eqnarray}\n\\delta_a&=&\\sqrt{\\frac{\\omega_l}{\\omega_r}}\\sqrt{\\frac{1}{2\\pi n!m!}}\n  \\left(\\frac{n+\\frac{1}{2}}{e}\\right)^{\\frac{2n+1}{4}}\\cr\n&&~~~~ \\times  \\left( \\frac{m+\\frac{1}{2}}{e}\\right)^{\\frac{2m+1}{4}}\n \\exp\\left[- \\int_{-b_m}^{a_n} \\frac{p(y)}{\\hbar}dy\\right].~~~\n\\end{eqnarray}\n\nWith\n\\begin{equation}\n\\delta_\\pm= \\frac{1}{2}(-\\delta_{\\epsilon}  \\pm\n\\sqrt{\\delta_\\epsilon^2+4\\delta_a^2}),\n\\end{equation}\nwhen $\\delta_\\nu=\\delta_-$,\nthe eigenfunction is written as\n\\begin{equation}\n\\psi^-(x)=\n\\frac{(-1)^n\\delta_a\n\\psi_{m}^{ho}\n(-b;l_l;x)+\\delta_+\\psi_{n}^{ho}(a;l_r;x)}{\\sqrt{\\delta_a^2+\\delta_+^2}},\n\\end{equation}\nwhile the eigenfunction of $\\delta_\\nu=\\delta_+$ is\n\\begin{equation}\n\\psi^+(x)=\n\\frac{-(-1)^{n}\\delta_+ \\psi_{m}^{ho}\n(-b;l_l;x)+\\delta_a\\psi_{n}^{ho}(a;l_r;x)}{\\sqrt{\\delta_a^2+\\delta_+^2}}.\n\\end{equation}\nThe formal expression of  $\\delta_{\\pm}$ in terms of\n$\\delta_{\\epsilon}$ and $\\delta_a$  can be understood from the fact\nthat, for $|\\delta_\\epsilon|\\ll 1$, the tunneling dynamics is\nessentially described by that of a two-level system(TLS)\n\\cite{CDL,RMP,GDJH}. If  $\\psi_n^{ho}(a;l_r;x)$ is written as a linear\ncombination of $\\psi^-(x)$ and $\\psi^+(x)$, the visibility of the\nlinear combination is\n\\begin{equation}\n{\\cal V}=1\/[1+\\frac{1}{2}(\\frac{\\delta_\\epsilon}{\\delta_a})^2].\n\\label{visibility}\n\\end{equation}\n\n\nFor the visibility estimation in Fig.~\\ref{quartic}(c) and (d), the parameter is\ndetermined from the consideration that, when $\\delta_\\epsilon=0$, the\nenergy splitting is given as $2\\hbar\\omega_r\\delta_a$.\nThe fact that the tunneling dynamics significantly takes place for\n$|\\delta_\\epsilon|\\ll 1$, even with $m\\neq n$, may be closely\nrelated to the resonant enhancement of tunneling in the\nmultiple-well structures \\cite{WM,Sias}. As in the BEC loaded into\nan asymmetric double-well potential \\cite{HWAHS}, if there are\nnoninteracting $N$ atoms in the ground state of $V(x)$ of small\n$\\delta_\\epsilon$ ($\\ll \\delta_a$), the number difference between\nthe left and right wells is proportional to $\\delta_\\epsilon$.\n\n\\subsection{Energy eigenvalue estimation}\n\nFor an eigenfunction $\\psi(x)$ of the system of $V(x)$ with the\nenergy eigenvalue $E$,   we have the identity\n\\begin{eqnarray}\n&&E-\\hbar\\omega(n+\\frac{1}{2})\\cr\n&&=\\frac{\\int_{-\\infty}^{\\infty}\\left[ V(x)- V_r^{ho}(x)\\right]\n\\psi(x)\\psi_{n}^{ho}(a;l_r;x)\ndx}{\\int_{-\\infty}^{\\infty}\\psi(x)\\psi_{n}^{ho}(a;l_r;x) dx},~~~~~~~\n\\label{harmonic identity}\n\\end{eqnarray}\nwhere $V_r^{ho}(x)= \\frac{\\hbar\\omega_r}{2}\\frac{(x-a)^2}{l_r^2}$.\nIn numerical calculations, this identity may be efficiently used\nin estimating the energy eigenvalue of an eigenfunction which is close to\n $\\psi_{n}^{ho}(a;l_r;x)$.\n\n\nAs the visibility also implies, when $\\delta_\\epsilon$ is much\nlarger than $\\delta_0$, $\\psi(x)$ of $E\\approx \\hbar\nw_r(n+\\epsilon+\\frac{1}{2})$ is mostly localized in the right well,\nand  around the bottom it will be closely described by\n$\\psi_{n}^{ho}(a;l_r;x)$, to give $\\psi_{n}^{app}\n(l_r;x)$ an approximation of $\\psi(x)$ in this well.\nIn the other regions, Eqs.~(5,7) and the\nWKB method can be used to find $\\psi_{n}^{app}\n(l_r;x)$. Eq.~(\\ref{harmonic identity}) then may be\nused to find a correction to $\\nu$, as\n\\begin{eqnarray}\n&&\\nu-n +\\epsilon \\cr &&\\approx\\frac{\\int_{-\\infty}^{\\infty}[ V(x)-\nV_r^{ho}(x)]  \\psi_n^{app}(l_r;x)\\psi_{n}^{ho}(a;l_r;x) dx}\n{\\hbar\\omega_r\\int_{-\\infty}^{\\infty}\n\\psi_n^{app}(x)\\psi_{n}^{ho}(a;x) dx}.~~~~~~\n\\end{eqnarray}\nThis approximation of a localized eigenfunction (ALE) can also be made similarly, for the eigenstate\nof $E\\approx \\hbar w_l(m+\\frac{1}{2})$ which describes  a probability distribution mostly localized\nin the left well.\n\n\n\n\n\\section{ precision test: Asymmetric Double Oscillator}\n\nIn application of the WKB method for a symmetric double-well potential, it is\nknown that the energy splitting could be found accurately, if the (ground state)\nenergy eigenvalue\nand thus the turning points are appropriately chosen \\cite{WKB2}. If a well is quadratic with\nangular frequency $\\omega$, then $(j+\\frac{1}{2})\\hbar\\omega$ ($j$: nonnegative integer) may be\na good estimation for an energy eigenvalue.\n\nIn order to check the accuracy of the formalism\nwe have provided, avoiding the turning-point problem as much as possible,\nwe consider the system of the asymmetric double oscillator potential \\cite{Song}\n\\begin{equation}\nV_D(\\epsilon;x)=\\left\\{\\begin{array}{ll}\n          \\hbar\\omega\\left(\\frac{x+\\sqrt{a^2+2\\epsilon l_{h0}^2}}\n                   { \\sqrt{2} l_{ho}} \\right)^2  &\n              ~{\\rm for}~ x< 0,\\\\\n          \\hbar\\omega\\left[\\left(\\frac{x-a}{ \\sqrt{2} l_{ho}} \\right)^2+\\epsilon\\right]\n          & ~ {\\rm for}~ x\\geq 0.\n         \\end{array}\\right.\n\\end{equation}\nFor this system, since both wells are exactly quadratic, the eigenfunctions\nare described by the parabolic\ncylinder functions  on both  sides of $x=0$ \\cite{Song,Merzbacher} and the\ncontinuities of the eigenfunction and its derivative at $x=0$ can be\nused to find the eigenvalues $E_0^D(a),E_1^D(a),\\cdots$. As in\nFig.~\\ref{Double Osc}, the calculations indeed show that, when $a$ is\na few times of $l_{ho}$, the eigenvalues of the low lying excited\nstates are close to $\\hbar\\omega(n+\\epsilon+\\frac{1}{2})$ or\n$\\hbar\\omega(m+\\frac{1}{2})$. To expose that the estimation through\nthe ALE is not valid when $\\hbar\\omega\\delta_\\epsilon$  is order of\nor smaller than the energy difference of the adjacent energy\neigenstates, we add the ratio\n$2\\epsilon\\hbar\\omega\/(E_1^D(a)-E_0^D(a))$ (dotted and dashed line)\nin Fig.~\\ref{Double Osc}(b).\n\n\\begin{figure}\n\\includegraphics{DO.eps}\n\\caption{(Color online) Energy eigenvalues for some lowest\neigenstates of the system of $V_D(\\epsilon;x)$.  $\\epsilon$ is\nchosen as 0.01  for  (a) and (b), 0.3 for (c), and 0.5 for (d).\nCalculated values (Solid lines); estimated values through the ALEs\n(dotted lines) and through the approximation to a TLS (dashed\nlines). For the ALEs of  the ground and second excited states,\nEq.~(\\ref{integer mu}) and the turning points $a_n$, $b_n$\nsatisfying $V_D(\\epsilon;a_n)=V_D(\\epsilon;-b_n)=\n(n+\\frac{1}{2})\\hbar\\omega$  are used with $n=0$ and $n=1$,\nrespectively; and for the first and third excited states,\nEq.~(\\ref{integer nu}) and $a_n$, $b_n$ of\n$V_D(\\epsilon;a_n)=V_D(\\epsilon;-b_n)=\n(n+\\epsilon+\\frac{1}{2})\\hbar\\omega$  are used with $n=0$ and $n=1$,\nrespectively. In the approximation to a TLS, $a_0$ and $b_0$ are\ndetermined from  $V_D(0;a_0)=V_D(0;-b_0)= \\frac{1}{2}\\hbar\\omega$.}\n\\label{Double Osc}\n\\end{figure}\n\nWhen $\\epsilon$ is as large as 0.3, the ALE gives\nbetter results than the approximation to a TLS practically in the\nwhole range where both methods are applicable [Fig.~\\ref{Double Osc}(c)].\nAs the approximation to a TLS is suggested by the WKB method,\nFig.~\\ref{Double Osc} indeed shows that {\\em WKB description} could be\n{\\em very accurate}. Fig.~\\ref{quartic}(c) and (d) also suggests that\nthis accuracy is not limited to the systems of the wells\nwhich are exactly quadratic.\nThis with the reasons the WKB method provides implies that,\nif the potential $V(x)$ is quadratic until it reaches several times\nof the zero-point energies $\\hbar\\omega_l\/2$ and $\\hbar\\omega_r\/2$\nfrom the bottoms of the left\nand right well, respectively, the energy eigenvalues of the low\nlying excited states of the system must be close to the eigenvalues\nof the quadratic potentials.\n\n\n\n\n\n\n\n\n\\section{conclusions and outlook}\n\nWe have shown, through the WKB method of quadratic connection formula,\nthat the systems of asymmetric double-well potentials\ncan be classified into two different regimes. In the regime of\neigenfunctions giving significant amplitude in both wells, the tunneling\ndynamics could take place, while there is no tunneling\nin the regime of localized eigenfunctions.  In this respect, the systems of\nthe eigenfunctions mostly localized in just one of the wells are very different\nfrom those of the symmetric potentials. As  Fig.~\\ref{quartic}(c), (d) and\nFig.~\\ref{Double Osc} clearly show, the WKB description could be very accurate,\nand the results given here may be valid for a system of the potential\nwhose wells are not exactly quadratic.\n\n\nFor the regime of localized eigenfunctions, even in the deep quantum limit,\nit may be possible to confine a large number of noninteracting bosons\nin just one of the wells.\nFor single-component fermions, in the light of the particle density\n$\\rho_R(\\epsilon;x,t)=\\sum_i\\psi_{Ri}^*(\\epsilon;x,t)\\psi_{Ri}(\\epsilon;x,t)$\n(see, e.g., Ref.~\\cite{BB}),\nthe number of fermions which can be confined in one of the wells\nis limited by that of the localized eigenfunctions $\\psi_{Ri}(\\epsilon;x,t)$.\nFor the system of particles confined in just one of the wells, the\ntunneling dynamics can be initiated and controlled by adjusting $\\epsilon$\nthe potential energy difference between the bottoms of the double well, since,\nif we change $\\epsilon$ so that $\\delta_\\epsilon \\ll 1,$\n$\\psi_{Ri}(\\epsilon;x,t)$ turns into a linear combination of the\neigenfunctions of the new system.\n\nIn the periodic arrangement of double-wells of the optical lattice\nwhere the tunneling is accompanied by a precession of the atom's\nangular momentum \\cite{HADGJ,KSSTO}, a  considerable time-periodic\n{\\em fluctuation of the population} of atoms in a spin state could\nimply that the atomic spinors are in the states of {\\em high\nvisibility}. Since $\\delta_a$ is very small in the large separation limit\nand the period of tunneling is inversely proportional to $\\delta_a$,\nif a tunneling phenomenon can be established over a long period of time,\nit can be used for precision measurements.\n\n\n\n\\acknowledgments\nThe author thanks Professors Kyungwon An and Yong-il Shin for discussions\non experimental aspects.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nKnowledge is distinguished by the ability to evolve over time. \nThis progression of knowledge is usually incremental and its formation is related to the cognitive areas being studied. \nThe process of Knowledge Tracing (KT) defined as the task of predicting students' performance has attracted the interest of many researchers in recent decades \\cite{corbett1994knowledge}.\nThe Knowledge State (KS) of a student is the degree of his or her mastering the Knowledge Components (KC) in a certain domain, for example ``Algebra'' or ``Physics''.\nA knowledge component generally refers to a learnable entity, such as a concept or a skill, that can be used alone or in combination with other KCs in order to solve an exercise or a problem \\cite{koedinger2012knowledge}.\nKnowledge Tracing is the process of modeling and assessing a student's KS in order to predict his or her ability to answer the next problem correctly.\nThe estimation of the student's knowledge state is useful for improving the educational process by identifying the level of his\/her understanding of the various knowledge components. By exploiting this information it is possible to suggest appropriate educational material to cover the student's weaknesses and thus maximize the learning outcome.\n\n\nThe main problem of  Knowledge Tracing is the efficient management of the responses over time. \nOne of the factors which add complexity to the problem of KT is the student-specific learning pace. \nThe knowledge acquisition may differ from person to person and may also be influenced by already existing knowledge. \nMore specifically, KT is predominantly considered as a supervised sequence learning problem where the goal is to predict the probability that a student will answer correctly the future exercises, given his or her history of interactions with previous tests.\nThus, the prediction of the correctness of the answer is based on the history of the student's answers in combination with the skill that is currently examined at this time instance.\n\nMathematically, the KT task is expressed as the probability $P(r_{t+1} = 1|q_{t+1}, X_t)$ that the student will offer the correct response in the next interaction $x_{t + 1}$, where the students learning activities are represented as a sequence of interactions  $X_t = \\{ x_1,x_2,x_3,...,x_t\\}$ over time $T$. \nThe $x_t$ interaction consists of a tuple $(q_t, r_t)$ which represents the question $q_t$ being answered at time $t$ and the student response $r_t$ to the question.\nWithout loss of generality, we shall assume that knowledge components are represented by skills from a set $S = \\{s_1, s_2,..., s_m\\}$.\nOne simplifying assumption, used by many authors \\cite{zhang2017dynamic}, is that\nevery question in the set $Q = \\{q_1, q_2,..., q_T\\}$ is related to a unique skill from $S$.\nThen the knowledge levels of the student for each one of the skills in $S$ compose his or her knowledge state.\n\nThe dynamic nature of Knowledge Tracing leads to approa-ches that have the ability to model time-series or sequential data. \nIn this work we propose two dynamic machine learning models that are implemented by time-dependent methods, specifically recurrent and time delay neural networks. \nOur models outperform the current state-of-the-art approaches in four out of five benchmark datasets that we have studied. \nThe proposed models differ from the existing ones in two main architectural aspects:\n\\begin{itemize}\n    \\item we find that attention does not help improve the performance and therefore we make no use of attention layers\n    \\item we experiment with and compare between two different skill embedding types: (a) initialized by pre-trained embeddings of the textual descriptions of the skill names using standard methods such as Word2Vec and FastText and (b) randomly initialized embeddings based  on skill ids\n\\end{itemize}\n\n\nThe rest of the paper is organized as follows. \nSection 2 reviews the related works on KT and the existing models for student performance prediction. \nIn Section 3 we present our proposed models and describe their architecture and characteristics. \nThe datasets we prepared and used are present in Section 4 while the experiments setup and the results are explained in Section 5. \nFinally, Section 6 concludes this work and discusses the future works and extensions of the research.\n\n\n\\section{Related Works}\nThe problem of knowledge tracing is dynamic as student knowledge is constantly changing over time.\nThus, a variety of methods, highly structured or dynamic, have been proposed to predict students' performance. \nOne of the earlier methods is Bayesian Knowledge Tracing (BKT) \\cite{corbett1994knowledge} which models the problem as a Hidden Markov chain in order to predict the sequence of outcomes for a given learner.\nThe Performance Factors Analysis Model (PFA)  \\cite{pavlik2009performance} proposed  to tackle the knowledge tracing task by modifying the Learning Factor Analysis model.\nIt estimates the probability that a student will answer a question correctly by maximizing the likelihood of a logistic regression model. \nThe features used in the PFA model, although interpretable, are relatively simple and designed by hand, and may not adequately represent the students' knowledge state \\cite{yeung2019deep}.\n\nDeep Knowledge Tracing (DKT) \\cite{piech2015deep} is the first dynamic model proposed in the literature utilizing recurrent neural networks (RNN) and specifically the Long Short-Term Memory (LSTM) model \\cite{hochreiter1997long} to track student knowledge. \nIt uses one-hot encoded skill tags and associated responses as inputs and it trains the neural network to predict the next student response. \nThe hidden state of the LSTM can be considered as the latent knowledge state of a student and can carry the information of the past interactions to the output layer. \nThe output layer of the model computes the probability of the student answering correctly a question relating to a specific Knowledge Component.\n\nAnother approach for predicting student performance is the Dynamic Key-Value Memory Network (DKVMN) \\cite{zhang2017dynamic} which relies on an extension of \\textit{memory networks} proposed in \\cite{miller2016key}.\nThe model tries to capture the relationship between different concepts.\nThe DKVMN model outperforms DKT using memory slots as key and value components to encode the knowledge state of students. \nLearning or forgetting of a particular skill are stored in those components and controlled by read and write operations through the Least Recently Used Access (LRUA) attention mechanism \\cite{santoro2016meta}.\nThe key component is responsible for storing the concepts and is fixed during testing while the value component is updated when a concept state changes. \nThe latter means that when a student acquires a concept in a test the value component is updated based on the correlation between exercises and the corresponding concept.\n\nThe Deep-IRT model \\cite{yeung2019deep} is the newest approach that extends the DKVMN model. \nThe author combined the capabilities of DKVMN with the Item Response Theory (IRT) \\cite{hambleton1991fundamentals} in order to measure both student ability and question difficulty. \nAt the same time, another model, named Sequential Key-Value Memory Networks (SKVMN) \\cite{abdelrahman2019knowledge}, tried to overcome the problem of DKVMN to capture long term dependencies in the sequences of exercises and generally in sequential data. \nThis model combines the DKVMN mechanism with the Hop-LSTM, a variation of LSTM architecture and has the ability to discover sequential dependencies among exercises, but it skips some LSTM cells to approach previous concepts that are considered relevant.\nFinally, another newly proposed model is Self Attentive Knowledge Tracing (SAKT) \\cite{pandey2019self}. \nSAKT utilizes a self-attention mechanism and mainly consists of three layers: an embedding layer for interactions and questions followed by a Multi-Head Attention layer \\cite{vaswani2017attention} and a feed-forward layer for student response prediction. \n\nThe above models either use simple features (e.g. PFA) or they use machine learning approaches such as key-value memory networks or attention mechanisms that may add significant complexity. \nHowever we will show that similar and often, in fact, better performance can be achieved by simpler dynamic models combining embeddings and recurrent and\/or time-delay feed-forward networks as proposed next.\n\n\\section{Proposed Approach}\n\n\\subsection{Dynamic Models}\nAs referenced in the relative literature, knowledge change over time is often modeled by dynamic neural networks.\nThe dynamic models produce output based on a time window, called ``context window'', that contains the recent history of inputs and\/or outputs.\n\nThere are two types of dynamic neural networks (Figure \\ref{fig:DNN_architecture}):\n(a) Time-Delay Neural Networks (TDNN), with only feed-forward connections and finite-memory of length $L$ equal to the length of the context window, and\n(b) Recurrent Neural Networks (RNN) with feed-back connections that can have potentially infinite-memory although, practically, their memory length is dictated by a forgetting factor parameter.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[scale=0.6]{dynamic_models.png}\n  \\caption{Dynamic model architectures: (a) Time-Delay Neural Network (b) Recurrent Neural Network.}\n  \\label{fig:DNN_architecture}\n\\end{figure}\n\n\\subsection{The Proposed Models}\n\nWe approach the task of predicting the student response (0=wrong, 1=correct) on a question involving a specific skill as a dynamic binary classification problem.\nIn general, we view the response $r_t$ as a function of the previous student interactions:\n\\begin{equation}\n    r_t = h( q_t,q_{t-1},q_{t-2},\\dots,r_{t-1},r_{r-2},\\dots ) + \\epsilon_t\n    \\label{eq:total_recurrent_model}\n\\end{equation}\nwhere $q_t$, is the skill tested on time $t$ and $\\epsilon_t$ is the prediction error. The response is therefore a function of the current and the previous tested skills $\\{q_t, q_{t-1}, q_{t-2}, \\dots\\}$, as well as the previous responses $\\{r_{t-1}, r_{t-2}, \\dots\\}$ given by the student.\n\nWe implement $h$ as a dynamic neural model.\nOur proposed general architecture is shown in Figure \\ref{fig:EDM_architecture}.\nThe inputs are the skill and response sequences $\\{q\\}$, $\\{r\\}$ collected during a time-window of length $L$ prior to time $t$.\nNote that the skill sequence includes the current skill $q_t$ but the response sequence does not contain the current response which is actually what we want to predict.\nThe architecture consists of two main parts:\n\\begin{itemize}\n    \\item The Encoding sub-network. It is used to represent the response and skill input data using different embeddings.\n    Clearly, embeddings are useful for encoding skills since skill ids are categorical variables. \n    We found that using embeddings to encode responses is also very beneficial.\n    The details of the embeddings initialization and usage are described in the next section.\n   \n    \\item The Tracing sub-network. This firstly estimates the knowledge state of the student and then uses it to predict his\/her response.\n    Our model function consists of two parts: (i) the Knowledge-Tracing part, represented by the dynamic model $f$, which predicts the student knowledge state $\\mathbf{v}_t$ and (ii) the classification part $g$, which predicts the student response based on the estimated knowledge state:\n    \\begin{eqnarray}\n        \\mathbf{v}_t &=& f(q_t,q_{t-1},q_{t-2},\\dots,r_{t-1},r_{r-2},\\dots)\n        \\label{eq:kt_estimation}\n        \\\\\n        \\hat{r}_t &=& g(\\mathbf{v}_t)\n        \\label{eq:classification}\n    \\end{eqnarray}\n    Depending on the memory length, we obtain two categories of models:\n    \\begin{itemize}\n        \\item[(a)] models based on RNN networks which can potentially have infinite memory.\n        In this case the KT model is recurrent:\n        \\[\n        \\mathbf{v}_t = f(\\mathbf{v}_{t-1}, q_t,q_{t-1},\\dots,q_{t-L},r_{t-1},\\dots,r_{r-L})\n        \\]\n        \\item[(b)] models based on TDNN networks which have finite memory of length $L$.\n        In this case the KT model has finite impulse response $L$:\n        \\[\n        \\mathbf{v}_t = f(q_t,q_{t-1},\\dots,q_{t-L},r_{t-1},\\dots,r_{r-L})\n        \\]\n    \\end{itemize}\n\\end{itemize} \n   \nAlthough RNNs have been used in the relevant literature, it is noteworthy that TDNN approaches have not been investigated in the context of knowledge tracing. \nThe classification part is modeled by a fully-connected feed-forward network with a single output unit.\n\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[scale=0.62]{edm_architecture.png}\n  \\caption{General proposed architecture. The dynamic model can be either a Recurrent Neural Network (with a feedback connection from the output of the dynamic part into the model input) or a Time Delay Neural Network (without feedback connection).}\n  \\label{fig:EDM_architecture}\n\\end{figure}\n\nWe investigated two different architectures: one based on recurrent neural networks and another based on time delay neural networks.\nThe details of each proposed model architecture are described below.\n\n\\subsection{Encoding Sub-network}\nThe first part in all our proposed models consists of two parallel embedding layers with dimensions $d_q$ and $d_r$, respectively, which encode the tested skills and the responses given by the student.\nDuring model training the weights of the Embedding layers are updated. \nThe response embedding vectors are initialized randomly.\nThe skill embedding vectors, on the other hand, are initialized either randomly or using pretrained data. In the latter case we use pretrained vectors corresponding to the skill names obtained from Word2Vec \\cite{mikolov2013efficient} or FastText \\cite{joulin2016fasttext} methods.\n\nA 1D spatial dropout layer \\cite{tompson2015efficient} is added after each Embedding layer. \nThe intuition behind the addition of spatial dropout was the overfitting phenomenon that was observed in the first epochs of each validation set. We postulated that the correlation among skill name embeddings, that might not actually exist, confused the model.\n\n\\subsection{Tracing Sub-network}\n\nWe experimented with two types of main dynamic sub-net- \nworks, namely Recurrent Neural Networks and Time Delay Neural Networks. These two approaches are described next.\n\n\\subsubsection{RNN Approach: Bi-GRU Model}\n\nThe model architecture based on the RNN method for the knowledge tracing task is shown in Figure \\ref{fig:Bi_GRU}.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[scale=0.525]{GRUModel.png}\n  \\caption{Bi-GRU model}\n  \\label{fig:Bi_GRU}\n\\end{figure}\n\nThe Spatial Dropout rate following the input embedding layers is $0.2$ for most of used datasets.\nNext, we feed the skills and the responses input branches into a Convolutional layer consisting of 100 filters, with kernel size 3, stride 1, and ReLU activation function.\nThe Convolutional layer acts as a projection mechanism that reduces the input dimensions from the previous Embedding layer.\nThis is found to help alleviate the overfitting problem.\nTo the best of our knowledge, Convolutional layers have not been used in previously proposed neural models for this task.\nThe two input branches are then concatenated to feed a Bidirectional Gated Recurrent Unit (GRU) layer with 64 units \\cite{cho2014learning}.\nBatch normalization and ReLU activation layers are applied between convolutional and concatenation layers.\nThis structure has resulted after extensive experiments with other popular recurrent models such as LSTM, plain GRU and also bi-directional versions of those models and we found this to be the proposed architecture is the most efficient one.\nOn top of the RNN layer we append a fully connected sub-network  consisting of three dense layers with 50 and 25 units and one output unit respectively.\nThe first two dense layers have a ReLU activation function while the last one has sigmoid activation which is used to make the final prediction ($0 < \\hat{r}_t < 1$).\n\n\n\n\n\n\\subsubsection{TDNN Approach}\n\nIn our TDNN model (Figure \\ref{fig:Tdnn_v1}) we add a Convolutional layer after each embedding layer with 50 filters and kernel size equal to 5. \n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[scale=0.525]{TDNN_Model.png}\n  \\caption{TDNN model}\n  \\label{fig:Tdnn_v1}\n\\end{figure}\n\nBatch normalization is used before the ReLU activation is applied.\nAs with the RNN model, the two input branches are concatenated to feed the classification sub-network.\nIt consists of four dense layers with 20, 15, 10, and 5 units respectively, using the ReLU activation function.\nThis funnel schema of hidden layers (starting with wider layers and continuing with narrower ones) has helped achieve better results for all datasets we have experimented with. \nIn the beginning of the classification sub-network we insert a Gaussian Dropout layer \\cite{srivastava2014dropout} which multiplies neuron activations with a Gaussian random variable of mean value 1. This has been shown to work as good as the classical Bernoulli noise dropout and in our case even better.\n\n\n\\section{Datasets}\n\n\\begin{table*} [htb]\n\\centering\n  \\caption{Datasets Overview.}\n  \\label{tab:datasets}\n    \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n    Dataset & Skills &  Students & Responses & Baseline Accuracy \\\\ \\hline\n    ASSISTment09 & 110 & 4,151 & 325,637 & 65.84\\%\\\\ \\hline\n    ASSISTment09 corrected & 101 & 4,151 & 274,590 & 66.31\\%\\\\ \\hline\n    ASSISTment12 & 196 & 28,834 & 2,036,080 & 69.65\\%\\\\ \\hline\n    ASSISTment17 & 101 & 1,709 & 864,713 & 62.67\\%\\\\ \\hline\n    FSAI-F1toF3 & 99 & 310 & 51,283 & 52.98\\%\\\\\n    \\hline\n  \\end{tabular}\n\\end{table*}\n\nWe tested our models using four popular datasets from the ASSISTments online tutoring platform.\nThree of them, ``\\textit{ASSISTment09}'',  ``\\textit{ASSISTment09 corrected}''\\footnote{\\rule{0pt}{1.2\\baselineskip}https:\/\/sites.google.com\/site\/assistmentsdata\/home\/assist\nment-2009-2010-data\/skill-builder-data-2009-2010},\nand ``\\textit{ASSISTment12}''\\footnote{https:\/\/sites.google.com\/site\/assistmentsdata\/home\/2012-13-school-data-with-affect} were provided by the above platform.\nThe fourth dataset, named ``\\textit{ASSISTment17}'' was obtained from 2017 Data Mining competition page\\footnote{https:\/\/sites.google.com\/view\/assistmentsdatamining\/dat a-mining-competition-2017}.\nFinally a fifth dataset, ``\\textit{FSAI-F1toF3}'' provided by ``Find Solution Ai Limited'' was also used in our experiments.\nIt is collected using data from the from the 4LittleTrees\\footnote{https:\/\/www.4littletrees.com} adaptive learning application. \n\n\n\\subsection{Datasets Descriptions}\n\nThe ASSISTments datasets contain data from student tests on mathematical problems \n\\cite{assistmentsdata} and the content is organized in columns style.\nThe student's interaction is recorded on each line.\nThere are one or more interactions recorded for each student. \nWe take into account the information concerning the responses of students to questions related with a skill.\nThus, we use the following columns:\n``\\textit{user\\_id}'', ``\\textit{skill\\_id}'',\n``\\textit{skill\\_name}'', \nand ``\\textit{correct}''. \nThe ``\\textit{skill\\_name}'' contains a verbal description of the skill tested.\nThe ``\\textit{correct}'' column contains the values of the students' responses which are either $1$ (for correct) or $0$ (for wrong).\n\nThe original ``\\textit{ASSISTment09}'' dataset contains 525,534 student responses.\nIt has been used extensively in the KT task from several researchers but according to \\cite{assistmentsdata} data quality issues have been detected concerning duplicate rows. \nIn our work we used the ``\\textit{preprocessed ASSISTment09}'' dataset found on DKVMN\\footnote{https:\/\/github.com\/jennyzhang0215\/DKVMN} and Deep-IRT\\footnote{https:\/\/github.com\/ckyeungac\/DeepIRT} models GitHubs. \nIn this dataset  the duplicate rows and the empty field values were cleaned, so that finally 1,451 unique students participate with 325,623 total responses and 110 unique skills.\n\nEven after this cleaning there are still some problems such as duplicate skill ids for the same skill name.\nThese problems have been corrected in the ''\\textit{Assistment09 corrected}'' dataset.\nThis dataset contains 346,860 students interactions and has been recently used in \\cite{xu2020dynamic}.\n\nThe ``\\textit{ASSISTment12}'' dataset contains students' data until the school year 2012-2013. The initial dataset contains 6,123,270 responses and 198 skills. \nSome of the skills have the same skill name but different skill id.\nThe total number of skill ids is 265. \nThe ``\\textit{Assistment17}'' dataset contains 942,816 students responses and 101 skills. \n\nFinally, the ``\\textit{FSAI-F1toF3}'' dataset is the smallest dataset we used. \nIt involves responses to mathematical problems from 7th grade to 9th grade Hong Kong students and consists of 51,283 students responses from 310 students on 99 skills and 2,266 questions. As it is commonly the case in most studies using this dataset, we have used the question tag as the model input $q_t$.\n\n\n\\subsection{Data Preprocessing}\n\nNo preprocessing was performed on the ``\\textit{ASSISTment09}'' and ``\\textit{FSAI-F1toF3}'' datasets.\nFor the remaining datasets we followed three preparation steps. \n\nFirst, the skill ids had been repaired by replacement. \nIn particular, the ``\\textit{ASSISTments09 corrected}'' dataset contained skills of the form of ``\\textit{skill1\\_skill2}'' and ``\\textit{skill1\\_skill2\\_skill3}'' which correspond to the same skill names, so we have merged them into the first skill id, found before the underscore. \nIn other words, the skill ``\\textit{10\\_13}'' was replaced with skill ``\\textit{10}'' and so on.\nMoreover, few misspellings were observed that were corrected and the punctuations found in three skill names were converted to the corresponding words. \nFor example, in the skill name ``\\textit{Parts of a Polnomial Terms Coefficient Monomial Exponent Variable}'' we corrected the ``\\textit{Polnomial}'' with ``\\textit{Polynomial}''. Also, in the skill name ``\\textit{Order of Operations +,-,\/,*() positive reals}'' we replaced the symbols ``\\textit{+,-,\/,* ()}'' with the words that express these symbols, ie. ``\\textit{addition subtraction division multiplication parentheses}''. \nThe latter preprocessing action was preferred over the removal of punctuations since the datasets referred to mathematical methods and operations and without them, we would lose the meaning of each skill.  \nSimilar procedure has been followed for the ``\\textit{ASSISTments12}'' dataset. \nFurthermore, spaces after some skill names were removed  i.e. the skill name ``\\textit{Pattern Finding  }'' became ``\\textit{Pattern Finding}''. \nIn the ``\\textit{ASSISTment17}'' dataset we came across skill names as ``\\textit{application: multi-column subtraction}'' and corrected them by replacing punctuation marks such as ``\\textit{application multi column subtraction}''.\nThat text preparation operations made to ease the generation of word embeddings of the skill names descriptions.\nIn addition, in the ``\\textit{ASSISTment17}'' dataset, the problem ids are used instead of the skill ids. \nWe had to match and replace the problem ids with the corresponding skill ids with the aim of uniformity of the datasets between them.\n\n\nSecondly, all rows containing missing values were discarded. \nThus, after the preprocessing, the statistics of the data sets were formulated as described in the Table \\ref{tab:datasets}.\n\nFinally, we split the datasets so that 70\\% was used for training and 30\\% for testing.\nThen, the training subset was further split into five train-validation subsets using 80\\% for training and 20\\% for validation. \n\n\\section{Experiments}\n\nIn this section we experimentally validate the effectiveness of the proposed methods by comparing them with each other and also with other state-of-the-art performance prediction models.\nThe Area Under the ROC Curve (AUC) \\cite{ling2003auc} metric is used for comparing the predicting probability correctness of student's response. \n\nThe state-of-the-art knowledge tracing models we are compared with the DKT, DKVMN and Deep-IRT. \nWe performed the experiments for our proposed models Bi-GRU, TDNN as well as for each of the previous model for all datasets, using the code provided by the authors on their GitHubs. \nIt is worth noting that the python GitHub code\\footnote{\\rule{0pt}{1.2\\baselineskip}https:\/\/github.com\/lccasagrande\/Deep-Knowledge-Tracing} used for the DKT model experiments requires the entire dataset file and the train\/test splitting is performed during the code execution. \n\nAll the experiments were performed on a workstation with Ubuntu operating system, Intel i5 CPU and 16GB Titan Xp GPU card.\n\n\\begin{table*}[htb]\n    \\caption{Models experiments settings} \n    \\label{tab:settings}\n    \\centering\n    \\begin{tabular}{|l | r|  r| }\n        \\hline  \n            \\multicolumn{1}{|c|}{Parameters} & \n            \\multicolumn{1}{c|}{Bi-GRU} & \n            \\multicolumn{1}{c|}{TDNN}\\\\  \\hline\n            Learning rate & 0.001 & 0.001 \\\\ \\hline\n            Learning rate schedule & yes & no\\\\ \\hline\n            Training epochs& 30 & 30 \\\\ \\hline\n            Batch size & 32 & 50 \\\\ \\hline\n            Optimizer & Adam & AdaMax\\\\ \\hline\n            History window length & 50 & 50 \\\\\\hline\n            Skill embeddings dim. & 100 \\& 300 & 100 \\& 300\\\\ \\hline\n            Skill embeddings type & Random, W2V, FastText & Random, W2V, FastText\\\\ \\hline\n            Responses embeddings dim. & Same to skill dim. & Same to skill dim.\\\\ \\hline\n            Responses embeddings type & Random & Random\\\\\n        \\hline   \n\\end{tabular}\n\\end{table*}\n\\subsection{Skill embeddings initialization}\n\nAs mentioned earlier, skill embeddings are initialized either randomly or using pretrained vectors.\nRegarding the initialization of the skill embeddings with pretrained vectors we used two methods described next. \nIn first method we used the text files from Wikipedia2Vec\\footnote{https:\/\/wikipedia2vec.github.io\/wikipedia2vec\/} \\cite{yamada2020wikipedia2vec} that is based on Word2Vec method and contains pretrainable embeddings for the  word representation vectors in English language in 100 and 300 dimensions.\nIn second method we used the ``\\textit{SISTER}'' (SImple SenTence EmbeddeR)\\footnote{https:\/\/pypi.org\/project\/sister\/} library to prepare the skill name embeddings  based on FastText in 300 dimensions pretrained word embeddings.\nEach skill name consists of one or more words. \nThus, for the Word2Vec method, the skill name embeddings vector is created by adding the word embeddings vectors, while in case of FastText, the skill name embeddings are created by taking the average of the word embeddings. \n\nEspecially for the FsaiF1toF3 dataset, the question embeddings are initialized either randomly or using the pretrained word representations of the corresponding skill descriptions by employing the Wikipedia2Vec and SISTER methods as described above. Since many questions belong to the same skill, in this case the corresponding rows in the embedding matrix are initialized by the same vector. \n\n\n\\subsection{Experimental Settings}\nWe performed the cross-validation method for the 5 training and validation set pairs. \nThis was to choose the best architecture and parameter settings for each of the proposed models.\nUsing the train and test sets we evaluated the chosen architectures for all the datasets. \n\nOne of the basic hyperparameters of our models that affect to the inputs is the $L$.  It represents the student's interaction history window length. \nThe inputs with $L$ sequence of questions and $L-1$ sequence of responses. \nThe best results we succeeded are when using $L=50$ for the both Bi-GRU and TDNN models.\nThe batch sizes used in the models during the training are: 32 in Bi-GRU and 50 in TDNN. \n\nSince specific dimensions of the pretrained word embeddings are provided, we used the same dimensions in case of random embedding in order to take the comparable results. \nSkill embeddings and responses embeddings set in the same dimensions.   \n\nThe scheduler learning rate is implementing in Bi-GRU starting from 0.001 \nand reducing over the training operation of the models that performs for 30 epochs.\nDuring training we applied the following learning rate schedule depending on the epoch number $n$:\n\\[\n    lr = \\left\\{\n        \\begin{array}{ll}\n        r_{init} &\n        \\text{if } n < 10\n        \\\\\n        r_{init} \\times e^{(0.1\\cdot(10-n))} &\n        \\text{otherwise}\n        \\end{array}\n    \\right.\n\\]\n\nIn case of the TDNN-based model, the learning rate equals 0.001 and is the same during the whole training process for 30 epochs.\nWe used cross-entropy optimization criterion and the Adam or AdaMax \\cite{kingma2014adam} learning algorithms.\n \nDropout with rate = $0.2$ or $0.9$ is also applied to the Bi-GRU model while the dropout rate of the TDNN equals to one of the  $(0.2, 0.4, 0.6, 0.9)$ values through to the Gaussian dropout layer.\nWe observed a reduction of overfitting during model training by changing the Gaussian dropout rate relative to the dataset's size.\nThus, the smaller dataset size is, the bigger dropout rate has been used.\n\nThe various combinations of parameters settings were applied during the experimental process for all proposed models presented in Table \\ref{tab:settings}.     \n\n\n\\subsection{Experimental Results}\n\n\\begin{table*}[htb]\n    \\centering\n    \\caption{Comparison between our proposed models - AUC (\\%). (R) = random skill embedding initialization, (W) = skill embedding initialization using W2V, (F) = skill embedding initialization using FastText. Datasets: (a) \\textit{ASSISTment09}, (b) \\textit{ASSISTment09 corrected}, (c) \\textit{ASSISTment12}, (d) \\textit{ASSISTment17}, (e) \\textit{FSAI-F1toF3}}\n    \\label{tab:compare_our_models} \n    \\begin{tabular}{|c|c|c|c|c|c|}\n    \\hline\n         & $d_q=100$(R) & $d_q=300$(R) & $d_q=100$(W) & $d_q=300$(W) & $d_q=300$(F)\n         \\\\\n        \\hline\n        Bi-GRU & 82.55 & 82.45 & 82.52 & 82.55 & 82.39\n        \\\\ \\hline\n        TDNN & 81.54 & 81.67 & 81.59 & 81.50 & 81.53\n        \\\\\n        \\hline\n    \\end{tabular}\n    \\\\\n    (a)\n    \\\\\n    ~\\\\\n    \\begin{tabular}{|c|c|c|c|c|c|}\n        \\hline\n         & $d_q=100$(R) & $d_q=300$(R) & $d_q=100$(W) & $d_q=300$(W) & $d_q=300$(F)\n         \\\\\n        \\hline\n        Bi-GRU & 75.27 & 75.13 & 75.14 & 75.09 & 75.12\n        \\\\ \\hline\n        TDNN & 74.38 & 74.39 & 74.40 & 74.33 & 74.37\n        \\\\\n        \\hline\n    \\end{tabular}\n    \\\\\n    (b)\n    \\\\\n    ~\\\\\n        \\begin{tabular}{|c|c|c|c|c|c|}\n        \\hline\n         & $d_q=100$(R) & $d_q=300$(R) & $d_q=100$(W) & $d_q=300$(W) & $d_q=300$(F)\n         \\\\\n        \\hline\n        Bi-GRU & 68.37 & 68.37 & 68.40 & 68.23 & 68.27\n        \\\\ \\hline\n        TDNN & 67.95 & 67.97 & 67.99 & 67.95 & 67.91\n        \\\\\n        \\hline\n    \\end{tabular}\n    \\\\\n    (c)\n    \\\\\n    ~\\\\\n        \\begin{tabular}{|c|c|c|c|c|c|}\n        \\hline\n         & $d_q=100$(R) & $d_q=300$(R) & $d_q=100$(W) & $d_q=300$(W) & $d_q=300$(F)\n         \\\\\n        \\hline\n        Bi-GRU & 73.62 & 73.58 & 73.76 & 73.54 & 73.58\n        \\\\ \\hline\n        TDNN & 71.68 & 71.75 & 71.52 & 71.81 & 71.83\n        \\\\\n        \\hline\n    \\end{tabular}\n    \\\\\n    (d)\n    \\\\\n    ~\\\\\n        \\begin{tabular}{|c|c|c|c|c|c|}\n        \\hline\n         & $d_q=100$(R) & $d_q=300$(R) & $d_q=100$(W) & $d_q=300$(W) & $d_q=300$(F)\n         \\\\\n        \\hline\n        Bi-GRU & 70.47 & 69.34 & 70.24 & 69.80 & 69.51\n        \\\\ \\hline\n        TDNN & 70.03 & 69.80 & 69.80 & 70.11 & 70.06 \n        \\\\\n        \\hline\n    \\end{tabular}\n    \\\\\n    (e)\n\\end{table*}\n\n\n\\begin{table*}[htb]\n    \\caption{Comparison test results of evaluation measures - the AUC metric (\\%) }\n    \\label{tab:results}\n    \\centering\n    \\begin{tabular}{|l|c|c|c|c|c|}\n        \\hline\n            Dataset & DKT & DKVMN & Deep-IRT & Bi-GRU & TDNN\\\\ \\hline\n    ASSISTment09 & 81.56\\% & 81.61\\% & 81.65\\% &\\textbf{82.55}\\%$^{(1,2)}$ & 81.67\\%$^{(3)}$ \\\\ \\hline\n    ASSISTment09 corrected & 74.27\\% & 74.06\\% & 73.41\\% & \\textbf{75.27}\\%$^{(1)}$ & 74.40\\%$^{(2)}$\\\\ \\hline\n    ASSISTment12 & 69.40\\% & 69.26\\% & \\textbf{69.73}\\% & 68.40\\%$^{(4)}$ & 67.99\\%$^{(4)}$ \\\\ \\hline\n    ASSISTment17 & 66.85\\% & 70.25\\% & 70.54\\% & \\textbf{73.76}\\%$^{(4)}$ & 71.83\\%$^{(5)}$ \\\\ \\hline\n    FSAI-F1toF3 & 69.42\\% & 68.40\\% & 68.69\\% & \\textbf{70.47}\\%$^{(1)}$ & 70.11\\%$^{(2)}$ \\\\\n    \\hline\n    \\multicolumn{6}{l}{\n    \\rule{0pt}{3ex}\n    $^{(1)}$ $d_q=d_r=100$, Random, ~~\n    $^{(2)}$ $d_q=d_r=300$, W2V, ~~\n    $^{(3)}$ $d_q=d_r=300$, Random,\n    }\n    \\\\\n    \\multicolumn{6}{l}{\n    $^{(4)}$ $d_q=d_r=100$, W2V, ~~\n    $^{(5)}$ $d_q=d_r=300$, FastText\n    }\n    \\end{tabular}\n\\end{table*}\n\nThe experiments results of our models are shown in Table \\ref{tab:compare_our_models}. Comparing our models with each other we can see that the RNN-based Bi-GRU model outperforms the TDNN-based model in all datasets. \nIt achieved best results when 100d embeddings were used either in pretrained or the random initialization type. \n\nWe observed that in both Bi-GRU or TDNN, the embedding type is not the significant parameter that affects the models performance. \nThe differences between the results of the experiments showed that the size of embeddings dimensions not particularly contributed to the final result and the difference in performance of the models was small.\n\nExcept for our models, we performed experiments for all datasets on the previous models we compared. \nFor three of the datasets, specifically  for ``\\textit{ASSISTment09 corrected}'', ``\\textit{ASSISTment12}'' and ``\\textit{ASSISTment17}'' there were not available results in the corresponding papers. \nIn this paper, we present the results of the experiments we run using that models codes.\n\nThe best experimental results of the ours models in comparison with the previous models for each dataset are presented in Table \\ref{tab:results}.\nThe model that has the best performance for the four of datasets is the Bi-GRU. \nExcept for that, the TDNN-based model has better performance in comparison to the previous models for four datasets.\nThe only dataset, for which the previous models overcomed our models is the ``\\textit{ASSISTment12}''.\n\n\\subsection{Discussion}\n\nOur model architecture is loosely based on the DKT model and offers improvements in the aspects discussed below.\nFirst, we employ embeddings for representing both skills and responses.\nIt is known that embeddings offer more useful representations compared to one-hot encoding because they can capture the similarity between the items they represent \\cite{wang2020survey}.\nSecond, we thoroughly examined dynamical neural models for estimating the student knowledge state by trying both infinite-memory RNNs and finite-memory TDNNs.\nTo our knowledge, TDNNs have not been well studied in the literature with respect to this problem.\nThird, we used convolutional layers in the inputs encoding sub-net. We found that this layer functioned as a reducing mechanism of the embedding dimensions and in conjunction with the dropout layer mitigated the overfitting problem.  \nThe use of Convolutional layers is a novelty in models tackling the knowledge tracing problem. \nFourth, unlike DKT, we used more hidden layers in the classification sub-net.\nOur experiments demonstrate that this gives more discriminating capability to the classifier and improves the results.\nFinally, our experiments with key-value modules and attention mechanism did not help further improve our results and so these experiments are not reported here.\nIn the majority of the datasets we examined our model outperforms the state-off the models employing key-value mechanisms such as DKVMN and Deep-IRT.\n\n\\begin{table*}[hbt]\n\\centering\n    \\caption{Statistical significance testing results of Bi-GRU and TDNN}\n    \\label{tab:p_value}\n    \\begin{tabular}{|l|c|c|}\n    \\hline\n    Dataset  & P-value \\\\ \\hline\n    ASSISTment09 & 7.34 e-59 \\\\ \\hline\n    ASSISTment09 corrected & 2.31 e-52 \\\\ \\hline\n    ASSISTment12 & 1.45 e-203 \\\\ \\hline\n    ASSISTment17 & 7.96 e-44   \\\\ \\hline\n    FSAI-F1toF3  & 1.38 e-84 \\\\ \\hline\n \\end{tabular}\n\\end{table*}\n\nIn addition to the AUC metric which is typically used for evaluating the performance of our machine learning models, we applied statistical significance testing to check the similarity between out Bi-GRU and TDNN models.\nSpecifically, we performed a T-Test between the outcomes of the two models in all training data using the best configuration settings as shown in Table \\ref{tab:results}.\nThe results reported in Table \\ref{tab:p_value} show that the P-value calculated in all cases is practically zero which proves the hypothesis that the two models are significantly different.\n\n\\section{Conclusion and Future Work}\n\nIn this paper we propose a novel two-part neural network architecture for predicting student performance in the next exam or exercise based on their performance in previous exercises.\nThe first part of the model is a dynamic network which tracks the student knowledge state and the second part is a multi-layer neural network classifier.\nFor the dynamic part we tested two different models: a potentially infinite memory recurrent Bidirectional GRU model and a finite memory Time-Delay neural network (TDNN). \nThe experimental process showed that the Bi-GRU model achieves better performance compared to the TDNN model.\nDespite the fact that TDNN models have not been used for this problem in the past, our results have shown that they can be just as efficient or even better compared to previous state-of-art RNN models and only slightly worse than our proposed RNN model.\nThe model inputs are the student's skills and responses history which are encoded using embedding vectors. Skill embeddings are initialized either randomly or by pretrained vectors representing the textual descriptions of the skills.\nA novel feature of our architecture is the addition of spatial dropout and convolutional layers immediately after the embeddings layers.\nThese additions have been shown to reduce the overfitting problem.\nWe found that the choice of initialization of the skill embeddings has little effect on the outcome of our experiments.\nMoreover, noting that there is a different use of the same datasets in different studies, we described in detail the process of the datasets pre-processing, and we provide the  train, validation and test splits of the data that were used in our experiments on our GitHub repository\\footnote{\\rule{0pt}{1.2\\baselineskip} https:\/\/github.com\/delmarin35\/Dynamic-Neural-Models-for-Knowledge-Tracing}.\nThe extensive experimentation with more benchmark datasets as well as the study of variants of the proposed models will be the subject of our future work with the aim of even further improving the prediction performance of the models.\n\n\n\n\\section{Acknowledgments}\n\nWe would like to thank NVIDIA Corporation for the kind donation of an Titan Xp GPU card that was used to run our experiments.\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\nThe Hill-Valley Evolutionary Algorithm (HillVallEA) \\cite{Maree18, Maree18b} is a real-valued multi-modal evolutionary algorithm, that automatically detects niches in the search space, based on the Hill-Valley test. This test states that two solutions belong to the same niche (valley) when there is no hill in between. To do so, a number of intermediate solutions are sampled and evaluated. Hill-Valley Clustering (HVC) is an iterative approach to efficiently cluster an entire population of solutions into niches. The resulting clusters are used to initialze a population-based core search algorithm, in this case, AMaLGaM-Univariate \\cite{bosman08} is used.\n\n\\section{Adaptations to the HillVallEA19}\nSome small adaptations have been made to HillVallEA18 \\cite{Maree18b} to further enhance its performance. Source code of HillVallEA is available at \\url{github.com\/scmaree\/HillVallEA}.\n\n\\subsection{Adaptive initial population sampling}\nIn HillVallEA, after all local optimizers have been terminated (and there is still budget remaining), a new initial population is sampled. When this population is sampled uniformly random, previously explored basins get re-explored every time a new population is initialized. To reduce this computational overhead, we store for each solution of the previous initial population to which cluster it belonged.  Then, new solutions are sampled based on rejection sampling. A sample is rejected with probability $P = 0.9$ if its nearest $d+1$ solutions of the previous initial population belonged to the same cluster. In that case, it is very likely that this solution will end up exploring the same basin as the cluster of the previous generation. \n\nAdditionally, better spreading the initially sampled population has been shown to improve performance of evolutionary algorithms \\cite{wessing15}. Minimax sampling method or Latin hypercube sampling \\cite{wessing15} grow slow very quickly as the problem dimensionality or the sample size increase. We therefore use a greedy scattered subset selection method \\cite{rodrigues14}. To construct a population of $N$ solutions, we sample $2N$ using the strategy above, and use greedy scattered subset selection to reduce this to $N$ solutions.\n\n\n\\subsection{Force accept of low-fitness solutions in Hill-Valley test}\nPreviously in hill-valley clustering, all nearest-better solutions were tested with the hill-valley test. Especially for problems with a large number of low-fitness local optima, such as the Shubert function (problems 6 and 8 in the benchmark), this results in many resources spent on obtaining accurate low-fitness clusters that are later discarded. Therefore, during hill-valley clustering, the hill-valley test is only performed on solutions that belong to the fittest half of the selection, or on solutions-pairs that are more than the expected edge length (EEL) \\cite{Maree18} apart (i.e., $N_t > 1$).\n\n\n\\subsection{Recalibration of the recursion scheme}\nThe population sizing scheme within HillVallEA is parameterized as $\\xi = (N,N^{\\mbox{inc}}, N_C,N_C^{\\mbox{inc}}, )$,  with initial population size $N$, population size increment $N^{\\mbox{inc}}$, cluster size $N_C$, and cluster size increment $N_C^{\\mbox{inc}}$. Previously, these parameters were set to $\\xi = (2^8d, 2, 1, 1.2)$, where $d$ is the problem dimensionality. As the initial population sampling has been adapted, setting $\\xi = (2^6, 2, 0.8, 1.1)$ was found to enhance performance of the HillVallEA. That is, using both a smaller population and smaller clusters initially, and increasing cluster size at a slower pace.\n\n\\section{Experiment Setup}\nWe evaluate the performance of HillVallEA on the test problems in the CEC2013 niching benchmark suite \\cite{CEC2013NichingCompetition}. The benchmark consists of 20 problems, as shown in Table~\\ref{tab:2dbenchmarks}, to be solved within a predefined budget in terms of function evaluations. For each of the benchmark problems, the location of the optima and the corresponding fitness values are known, however, these are only used for to measure performance, and not used during optimization. All benchmark functions are defined on a bounded domain. All experiments are repeated 50 times, and resulting performance measures are averaged over all repetitions. Note that no problem-specific parameter tuning has been performed.\n\n\\subsection{Performance Metrics}\n\\label{sec:measures}\nTwo performance measures are used, from which three scoring scenarios are computed, according to the competition guidelines. Let $\\mathcal{O}$ be the set of presumed optima obtained by an algorithm, and let $g$ be the number of distinct global optima within $\\mathcal{O}$. Finally, let $G_p$ be the number of global optima for problem $p$. Then, we define the peak ratio (PR) as $\\mbox{PR} = g \/ G_p$ and the success rate (SR) as $\\mbox{SR} = g \/ |\\mathcal{O}|$. Both measures should be maximized, with maximum $1$. From these two measures, three scoring scenarios are constructed. \n\n\\begin{enumerate}\n\\item[S1] The first scenario is simply the PR. \n\\item[S2] The second scenario is known as the static $F_1$ measure, defined as $F_1 = \\frac{2 \\cdot PR \\cdot SR}{PR + SR}$.\n\\item[S3] The third and final scenario is the dynamic $F_1$ (dyn$F_1$), which is the area under the curve of the $F_1$ over time (in number of function evaluations). For this, sort the solutions in $\\mathcal{O}$ based on the number of function evaluations $f_i$ before a solution $o_i\\in\\mathcal{O}$ was considered a global optimum, with the first-obtained solution first. Let $\\mathcal{O}_{[1:t]}$ with $t\\in[1,|\\mathcal{O}|]$ be the subset of $\\mathcal{O}$ containing the first $t$ solutions and let $B_p$ be the function evaluation budget for problem $p$. Then we can write the dyn$F_1$ as,\n$$\\mbox{dyn}F_1 =  \\left(\\frac{B_p - f_{|\\mathcal{O}|}}{B_p}\\right) F_1(\\mathcal{O})  + \\sum_{i = 2}^{|\\mathcal{O}|} \\left( \\frac{f_i - f_{i-1} }{B_p}\\right) F_1(\\mathcal{O}_{[1:i-1]}).$$\n\\end{enumerate}\n\nAccording to the competition guidelines, a solution is marked as a distinct global optima for five different accuracy levels $\\varepsilon = \\{10^{-1}, 10^{-2}, 10^{-3}, 10^{-4}, 10^{-5}\\}$. Then, for each of the accuracy levels, for each problem, the scenario score is the average of the scores over the five accuracy levels.\n\n\\subsection{Algorithms}\n\\label{sec:algorithms}\nWe compare performance of HillVallEA-2019 to all algorithms that previously participated in the niching competitions held at the GECCO and CEC conferences in 2016, 2017 and 2018. The raw solution sets are used. The obtained solutions are re-evaluated and the scores under the different scenarios are computed given the definitions stated above. Note that the algorithms are not re-run. The included algorithms are,\nNEA2+ \\cite{preuss12}, RLSIS \\cite{wessing15}, RS-CMSA \\cite{ahrari17} , HillVallEA18 \\cite{Maree18b}, SDE-Ga (No known reference, developed by Jun-ichi Kushida) and finally the method that we discus in this paper, HillVallEA19. \n\n\n\\section{Results and discussion}\nTables~\\ref{tab:s1}, \\ref{tab:s2} and \\ref{tab:s3} shows the score per problem and per algorithm under respectively scenario S1, S2 and S3. Table~\\ref{tab:ranks} shows the overall score and the corresponding ranks. \n\nThe SR of HillVallEA is 1 in all cases, and almost always for RS-CMSA, which shows that the similar post-processing step that both algorithms perform is successful at removing duplicates and local optima. Problems 1-5 and 10 are fully solved by all methods in all runs for all accuracy levels (except for two runs of NEA2+), which suggesting that these problems are too simple. \n\nNo method can obtain the final two Weierstrass peaks of Composition Function 4 fully in any dimension (problem 15, 17, 19 and 20), which may indicate that this is a needle in a haystack problem. Similarly to problem 18, for which the obtained peak ratio was 0.667 for all methods (except NEA2+). Especially, SDE-Ga obtains for problems 13, 14, 16, and 18 a maximum peak ratio of 0.667, being unable to solve the Weierstrass function. \n\nTo conclude, HillVallEA19 was shown to be an improvement over HillVallEA18 under all scenarios. RS-CMSA comes directly after in all three scenarios. SDE-Ga performs well under S1 and S2, especially for problems 8 and 9, but performance deteriorates for the higher-dimensional problems.  SDE-Ga obtains solutions very late in the convergence process, resulting in a very low S3 score. Overall, HillVallEA19 performs best under all scenarios.\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Niching benchmark suite from the CEC2013 special session on multi-modal optimization \\cite{CEC2013NichingCompetition}. For each problem the function name, problem dimensionality $d$, number of global optima $\\#gopt$, and local optima $\\#lopt$ and budget in terms of function evaluations are given.}\n\\label{tab:2dbenchmarks}\n\\small\n\\begin{tabular}{cccccc}\n\\toprule\n\\# & Function name & $d$ & \\#gopt & \\#lopt & budget \\\\\n\\toprule\n1 & Five-Uneven-Peak Trap & 1 & 2 & 3 & 50K  \\\\\n2 & Equal Maxima & 1 & 5 & 0 & 50K  \\\\\n3 & Uneven Decreasing Maxima & 1 & 1 & 4 & 50K \\\\\n4 & Himmelblau & 2 & 4 & 0 & 50K\\\\\n5 & Six-Hump Camel Back & 2 & 2 & 5 & 50K \\\\\n6 & Shubert & 2 & 18 & many & 200K   \\\\\n7 & Vincent & 2 & 36 & 0 & 200K \\\\\n8 & Shubert & 3 & 81 & many & 400K  \\\\\n9 & Vincent & 3 & 216 & 0 & 400K  \\\\\n10 & Modified Rastrigin & 2 & 12 & 0 & 200K  \\\\\n11 & Composition Function 1 & 2 & 6 & many &200K \\\\\n12 & Composition Function 2 & 2 & 8 &many & 200K \\\\\n13 & Composition Function 3 & 2 & 6 &many & 200K  \\\\\n14 & Composition Function 3 & 3 & 6 &many & 400K  \\\\\n15 & Composition Function 4 & 3 & 8 &many & 400K  \\\\\n16 & Composition Function 3 & 5 & 6 &many & 400K \\\\\n17 & Composition Function 4 & 5 & 8 &many & 400K \\\\\n18 & Composition Function 3 & 10 & 6 & many &400K  \\\\\n19 & Composition Function 4 & 10 & 8 &many & 400K  \\\\\n20 & Composition Function 4 & 20 & 8 &many & 400K \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Scores obtained under Scenario S1 (peak ratio) for each of the algorithms per problem $p$. Higher is better, 1 is the maximum score. Scores are averaged over 50 runs and five accuracy levels. Average (avg.) score computed over all 20 problems. }\n\\label{tab:s1}\n\\smaller\n\\begin{tabular}{c|cccc|cc}\n\\toprule\n& & &  & & \\multicolumn{2}{|c}{HillVallEA} \\\\\n  p & NEA2+ & RLSIS & RS-CMSA & SDE-Ga & HillVallEA18 & HillVallEA19 \\\\ \n\\toprule\n   1 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n   2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n   3 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n   4 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n   5 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n   6 & 0.997 & 0.872 & 0.999 & 1.000 & 1.000 & 1.000 \\\\\n   7 & 0.840 & 0.920 & 0.997 & 1.000 & 1.000 & 1.000 \\\\\n   8 & 0.568 & 0.189 & 0.871 & 1.000 & 0.920 & 0.975 \\\\\n   9 & 0.552 & 0.584 & 0.730 & 0.992 & 0.945 & 0.972 \\\\\n  10 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n  11 & 0.955 & 1.000 & 0.997 & 0.733 & 1.000 & 1.000 \\\\\n  12 & 0.796 & 0.950 & 0.948 & 0.800 & 1.000 & 1.000 \\\\\n  13 & 0.947 & 0.938 & 0.997 & 0.667 & 1.000 & 1.000 \\\\\n  14 & 0.813 & 0.799 & 0.810 & 0.667 & 0.917 & 0.923 \\\\\n  15 & 0.721 & 0.720 & 0.748 & 0.750 & 0.750 & 0.750 \\\\\n  16 & 0.683 & 0.675 & 0.667 & 0.667 & 0.687 & 0.723 \\\\\n  17 & 0.723 & 0.738 & 0.703 & 0.703 & 0.750 & 0.750 \\\\\n  18 & 0.650 & 0.667 & 0.667 & 0.667 & 0.667 & 0.667 \\\\\n  19 & 0.505 & 0.515 & 0.502 & 0.555 & 0.585 & 0.593 \\\\\n  20 & 0.398 & 0.422 & 0.482 & 0.460 & 0.482 & 0.480 \\\\\n\n\\bottomrule\n avg  & 0.807 & 0.800 & 0.856 & 0.833 & 0.885 & 0.892 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Scores obtained under Scenario S2 (static $F_1$) for each of the algorithms per problem $p$. Higher is better, 1 is the maximum score. Scores are averaged over 50 runs and five accuracy levels. Average (avg.) score computed over all 20 problems. }\n\\label{tab:s2}\n\\smaller\n\\begin{tabular}{c|cccc|cc}\n\\toprule\n& & &  & & \\multicolumn{2}{|c}{HillVallEA} \\\\\n  \\# & NEA2+ & RLSIS & RS-CMSA & SDE-Ga & HillVallEA18 & HillVallEA19 \\\\ \n\\toprule\n   1 & 1.000 & 0.993 & 0.996 & 1.000 & 1.000 & 1.000 \\\\\n   2 & 1.000 & 0.993 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n   3 & 1.000 & 0.993 & 0.987 & 1.000 & 1.000 & 1.000 \\\\\n   4 & 0.960 & 0.978 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n   5 & 0.947 & 0.949 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n   6 & 0.997 & 0.924 & 0.999 & 1.000 & 1.000 & 1.000 \\\\\n   7 & 0.614 & 0.947 & 0.999 & 1.000 & 1.000 & 1.000 \\\\\n   8 & 0.723 & 0.315 & 0.931 & 1.000 & 0.958 & 0.987 \\\\\n   9 & 0.646 & 0.733 & 0.844 & 0.996 & 0.972 & 0.986 \\\\\n  10 & 0.997 & 0.988 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n  11 & 0.971 & 0.992 & 0.998 & 0.733 & 1.000 & 1.000 \\\\\n  12 & 0.881 & 0.967 & 0.972 & 0.800 & 1.000 & 1.000 \\\\\n  13 & 0.966 & 0.941 & 0.998 & 0.723 & 1.000 & 1.000 \\\\\n  14 & 0.894 & 0.865 & 0.893 & 0.799 & 0.953 & 0.958 \\\\\n  15 & 0.835 & 0.831 & 0.855 & 0.857 & 0.857 & 0.857 \\\\\n  16 & 0.811 & 0.795 & 0.800 & 0.800 & 0.813 & 0.837 \\\\\n  17 & 0.838 & 0.843 & 0.823 & 0.824 & 0.857 & 0.857 \\\\\n  18 & 0.787 & 0.794 & 0.800 & 0.800 & 0.800 & 0.800 \\\\\n  19 & 0.668 & 0.676 & 0.668 & 0.712 & 0.735 & 0.741 \\\\\n  20 & 0.563 & 0.590 & 0.650 & 0.627 & 0.650 & 0.647 \\\\\n\\bottomrule\n avg  & 0.855 & 0.855 & 0.911 & 0.884 & 0.930 & 0.934 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\caption{Scores obtained under Scenario S3 (dynamic $F_1$) for each of the algorithms per problem $p$. Higher is better, 1 is the maximum score. Scores are averaged over 50 runs and five accuracy levels. Average (avg.) score computed over all 20 problems. }\n\\label{tab:s3}\n\\smaller\n\\begin{tabular}{c|cccc|cc}\n\\toprule\n& & &  & & \\multicolumn{2}{|c}{HillVallEA} \\\\\n  \\# & NEA2+ & RLSIS & RS-CMSA & SDE-Ga & HillVallEA18 & HillVallEA19 \\\\ \n\\toprule\n   1 & 0.982 & 0.987 & 0.911 & 0.572 & 0.992 & 0.995 \\\\\n   2 & 0.917 & 0.983 & 0.959 & 0.626 & 0.987 & 0.989 \\\\\n   3 & 0.895 & 0.982 & 0.949 & 0.456 & 0.992 & 0.994 \\\\\n   4 & 0.959 & 0.975 & 0.932 & 0.528 & 0.973 & 0.977 \\\\\n   5 & 0.971 & 0.979 & 0.944 & 0.416 & 0.982 & 0.983 \\\\\n   6 & 0.917 & 0.699 & 0.933 & 0.525 & 0.951 & 0.966 \\\\\n   7 & 0.659 & 0.855 & 0.928 & 0.413 & 0.960 & 0.966 \\\\\n   8 & 0.464 & 0.209 & 0.715 & 0.324 & 0.750 & 0.805 \\\\\n   9 & 0.550 & 0.573 & 0.654 & 0.157 & 0.791 & 0.818 \\\\\n  10 & 0.988 & 0.983 & 0.984 & 0.539 & 0.979 & 0.982 \\\\\n  11 & 0.961 & 0.980 & 0.967 & 0.254 & 0.983 & 0.983 \\\\\n  12 & 0.829 & 0.859 & 0.909 & 0.303 & 0.958 & 0.963 \\\\\n  13 & 0.932 & 0.897 & 0.923 & 0.372 & 0.957 & 0.964 \\\\\n  14 & 0.862 & 0.834 & 0.832 & 0.434 & 0.867 & 0.882 \\\\\n  15 & 0.806 & 0.782 & 0.785 & 0.544 & 0.824 & 0.836 \\\\\n  16 & 0.799 & 0.788 & 0.777 & 0.449 & 0.776 & 0.793 \\\\\n  17 & 0.803 & 0.779 & 0.688 & 0.419 & 0.787 & 0.816 \\\\\n  18 & 0.719 & 0.748 & 0.730 & 0.160 & 0.721 & 0.763 \\\\\n  19 & 0.624 & 0.627 & 0.560 & 0.035 & 0.634 & 0.656 \\\\\n  20 & 0.491 & 0.496 & 0.502 & 0.003 & 0.514 & 0.524 \\\\\n\\bottomrule\n avg  & 0.806 & 0.801 & 0.829 & 0.376 & 0.869 & 0.883 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\caption{Algorithm ranks based on the three scenarios.  }\n\\label{tab:ranks}\n\\smaller\n\\begin{tabular}{c|cc|cc|cc|c}\n\\toprule\nAlgorithm & S1 & rank & S2 & rank & S3 & rank & average rank \\\\\n\\toprule\n                 RLSIS & 0.800 & 6& 0.855 &5  & 0.801 & 5& 5.3\\\\\n                 NEA2+ & 0.807 & 5& 0.855 & 6  & 0.806 &4 & 5\\\\ \n                SDE-Ga & 0.833 & 4& 0.884 & 4 & 0.376 & 6& 4.7\\\\  \n           RS-CMSA & 0.856 &3 & 0.911 &  3& 0.829 & 3& 3\\\\ \n          HillVallEA18 & 0.885 &2 & 0.930 & 2 & 0.869 &2 & 2\\\\ \n          HillVallEA19 & 0.892& 1& 0.934 &  1 & 0.883 & 1& 1\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Pure outputs}\\label{s:uno}\n\nProving that proper quantum mover $\\mbox{PQM}(p)$\ncannot be implemented on a physical system S is equivalent to showing the following:\n\n{\\em  Given $p \\in [0,1[$ there is no deterministic physical process \n$\\cal E$ which transforms  an unknown pure vector $|\\psi\\ran$ of a generic \nHilbert space ${\\cal H}_S$ according to the mapping\n\\begin{eqnarray}\n\t|\\psi\\ran \\;  \\stackrel{\\cal E}{\\longrightarrow} \\; |\\psi_p\\ran \\label{THETRA} \\;,\n\\end{eqnarray}\nwhere $|\\psi_p \\ran$ is a pure state of ${\\cal H}_S$ depending on $|\\psi\\rangle$ \nwhich satisfies the identity\n\\begin{eqnarray}\n\t|\\bra{\\psi} \\psi_p\\ran |^2=  p \\;.\n\\label{unnot}\n\\end{eqnarray}}\nThe adjective ``deterministic'' in the hypothesis plays an important  role.\nOn one hand it allows us to exclude from the analysis those processes  which \nimplement~(\\ref{THETRA}) probabilistically, i.e. allowing some probability of failure.\nAs in the case of the cloning and the U-NOT machines~\\cite{DEMA,PASQ,BUZEK,BUZEK1}\nsuch transformations are not prohibited by quantum mechanics and the above no-go theorem \ndoes not apply to them. In addition the term ``deterministic'' indicates that here \nwe are considering scenarios where the output state $|\\psi_p\\ran$ associated with \n$|\\psi\\ran$ is uniquely determined: i.e. if we run the transformation many times \non various copies of a certain input $|\\psi\\ran$, the output state will be always \nthe same vector $|\\psi_p\\ran$. This point is important since the vectors $|\\psi_p\\ran$ \nwhich satisfy the property~(\\ref{unnot}) form a dense subset of ${\\cal H}_S$. A transformation \n(\\ref{THETRA}) which randomly picks up one of these $|\\psi_p\\ran$ is described by \na $\\mbox{GQM}(p)$ map whose output state is not pure: these operations will be discussed in the \nnext section where we will see that the possibility of implementing them depends \nexplicitly on the dimension of ${\\cal H}_S$.\n\nThe most general deterministic physical transformations~(\\ref{THETRA}) \nof the system S  are the linear, completely positive,\ntrace preserving (LCPT) maps $\\cal E$ which for all $|\\psi\\ran \\in {\\cal H}_S$ \ngive\n\\begin{eqnarray}\n\t{\\cal E}(|\\psi\\rangle \\langle \\psi|) \n\t&=&| \\psi_p \\rangle \\langle \\psi_p | \n\\label{cpt}\\;,\n\\end{eqnarray}\nwith $|\\psi_p\\ran$ unit vector of ${\\cal H}_S$ such that Eq.~(\\ref{unnot}) \napplies~\\cite{NOTA0}.\nIn the qubit case ($d=2$)  and for $p<1\/3$ \nthe impossibility of implementing the transformation~(\\ref{cpt}) \ncan be established  by considering  the maximal fidelity achievable in performing \nan approximated U-NOT~\\cite{BUZEK,PASQ}. In fact, by expressing the trace of \n${\\cal E}(|\\psi\\rangle \\langle \\psi|)$ in the orthonormal basis $|\\psi\\rangle$, \n$|\\psi_\\perp\\rangle$ of ${\\cal H}_S$ we get \n\\begin{eqnarray}\n\t\\langle \\psi_\\perp | {\\cal E}(|\\psi\\rangle\n\t\\langle \\psi|) |\\psi_\\perp\\rangle = 1 - p\\;.\n\\label{ULTIMA}\n\\end{eqnarray}\nThe thesis then follows by observing that the left-hand side term of this \nexpression is the fidelity of the output state of $\\cal E$ with respect to \nthe orthogonal vector $|\\psi_\\perp\\rangle$ which according to~\\cite{BUZEK,PASQ} is\nupper bounded by $2\/3$. \n\nThe above argument does not apply if $d > 2$ and for generic $p$. One can still \nprove that no  transformations~(\\ref{cpt})  exist by considering a unitary \nrepresentation~\\cite{CHUANG} of $\\cale$, i.e.\n \\begin{eqnarray}\n{\\cal E}(| \\psi\\rangle \\langle \\psi|) = \n\\mbox{Tr}_A \\big[ \\; U \\; \\left( \n\\ket{\\psi}\\bra{\\psi} \\otimes \\ket{0}_A\\bra{0}\\right) \\; U^\\dag \\;\\big]\\;, \\label{cpt1}\n\\end{eqnarray}\nwhere $U$ is a unitary operator acting on S and on an ancillary system A,  \nwhile $|0\\ran_A$ is a fiduciary state of A. According to Eq.~(\\ref{cpt}) the pure states \nof ${\\cal H}_S$ are mapped into pure states of the same space, this implies that\nthe state $U \\; \\ket{\\psi} \\otimes \\ket{0}_A$ cannot be entangled, i.e.\n\\begin{eqnarray}\nU \\; \\ket{\\psi} \\otimes \\ket{0}_A = \\ket{\\psi_p} \\otimes \\ket{\\phi(\\psi)}_A \\label{cpt2}\n\\end{eqnarray}\nwith $\\ket{\\phi(\\psi)}_A$ a pure state of A which, in principle, depends on the input \nstate $|\\psi\\ran$ of S. By applying this relation to the unit vector\n\\begin{eqnarray}\n|\\psi\\ran = \\alpha \\; |k\\ran + \\beta \\; |k^\\prime\\ran\\label{VETTORI}\\;,\n\\end{eqnarray}\n(with $|k\\ran$, $|k^\\prime\\ran$ orthonormal states of ${\\cal H}_S$ and $\\alpha$, $\\beta$ complex\nsuch that $|\\beta|^2=1-|\\alpha|^2$) and using the linearity of $U$ one gets\n\\begin{eqnarray}\n\\ket{\\psi_p} \\otimes \\ket{\\phi(\\psi)}_A = \n\\alpha\\; |k_p\\ran \\otimes \\ket{\\phi(k)}_A + \\beta \\; |k_p^\\prime\\ran \\otimes\n\\ket{\\phi(k^\\prime)}_A \\nonumber \\\\\n\\label{cpt3}\n\\end{eqnarray}\nwith $|k_p\\ran$, $\\ket{\\phi(k)}_A$ and  $|k_p^\\prime\\ran$, $\\ket{\\phi(k^\\prime)}_A$\ndefined as in Eq.~(\\ref{cpt2}). In particular $|k_p\\ran$ and $|k_p^\\prime\\ran$ are, \nrespectively, the {\\em unit vectors} of ${\\cal H}_S$ which describes the \noutputs of the map $\\cale$ associated with $\\ket{k}$, $\\ket{k^\\prime}$: by  hypothesis \nthey satisfy the relations $\\bra{k_p}k\\ran = \\sqrt{p} e^{i \\phi}$, \n$\\bra{k^\\prime_p}k^\\prime \\ran = \\sqrt{p} e^{i \\phi^\\prime}$ for some \nphases $\\phi$ and $\\phi^\\prime$.\nBy projecting the left hand side of Eq.~(\\ref{cpt3}) on $|\\psi\\ran$ we derive the following \nidentity\n\\begin{eqnarray}\np&=& \\| \\big( |\\alpha|^2  \\sqrt{p} \\; e^{i \\phi} +\n \\alpha \\beta^*\\; \\bra{k^\\prime} k_p\\ran  \\; \\big) |\\phi(k)\\ran_A \\nonumber \\\\\n&&+ \\big(|\\beta|^2   \\sqrt{p} \\; e^{i \\phi^\\prime}\n+ \\alpha^*\\beta\\;  \\bra{k} k^\\prime_p\\ran \\;  \\big) |\\phi(k^\\prime)\\ran_A \\|^2 \\;,\\label{cpt10}\n\\end{eqnarray}\nwhich must hold for all the complex amplitudes $\\alpha$ and $\\beta$.\nOne can verify that \nthis condition is satisfied if and only if\n\\begin{eqnarray}\n \\bra{k^\\prime} k_p\\ran  = \\bra{k} k^\\prime_p\\ran = 0 \n\\label{cpt11}\\;,\n\\end{eqnarray}\n(see Appendix~\\ref{appendixA} for details).\nThe impossibility of  the transformation $\\cal E$\nfinally follows by observing that Eq.~(\\ref{cpt11}) is in contradiction\nwith $|k_p\\ran$ being a \nunit vector of ${\\cal H}_S$. In fact since Eq.~(\\ref{cpt11})\napplies for all $|k\\ran$ and\n$|k^\\prime\\ran$ orthonormal we have\n\\begin{eqnarray}\n\\| \\ket{k_p} \\|^2 = |\\bra{k} k_p\\ran|^2 + \\sum_{k^\\prime} \n|\\bra{k^\\prime} k_p\\ran|^2 = p \\;, \\label{NUOVA}\n\\end{eqnarray}\nwhere the sum is performed over the set  of vectors~$\\{|k^\\prime\\ran\\}_{k^\\prime}$ which\ncomplete $|k\\ran$ to an orthonormal basis of ${\\cal H}_S$. \nIt should be stressed that the above derivation holds for all value of $p<1$ and  for \nHilbert space  ${\\cal H}_S$ (also infinite-dimensional Hilbert spaces).\n\n\\section{Mixed outputs}\\label{s:due}\n\nConsider now the case of   generalized \nquantum movers $\\mbox{GQM}(p)$ with not necessarily pure\noutput states $\\rho_b$.\nThese mappings include all those transformations which implement a proper quantum mover\ntransformation~(\\ref{THETRA}) \nprobabilistically, i.e. where the output state is randomly selected among the set\nof the vectors $|\\psi_p\\ran$ which satisfies the condition of Eq.~(\\ref{unnot}).  \nDifferently from the previous case, these generalized quantum \nmovers are \nallowed at least for some restricted values of $p$. \n What is more interesting \nis that the generalized quantum movers $\\mbox{GQM}(p)$ can be implemented by means of \nLCPT maps only if $p$ exceeds a threshold $p_{\\mbox{\\small t}}(d)$ which  explicitly \ndepends on $d$. \nHere we provide both upper and \nlower bounds for this threshold, i.e.\n\\begin{eqnarray}\n 1\/(2 d-1)\\; \\leqslant \\; p_{\\mbox{\\small t}}(d) \\;\\leqslant \\; 1\/(d +1)  \\label{UPLOW}\\;.\n\\end{eqnarray}\nSuch bounds coincide for $d=2$ so that   $p_{\\mbox{\\small t}}(2)=1\/3$\nand the threshold of the qubit case is completely determined by the above inequalities.\nAs in the case of Eq.~(\\ref{ULTIMA}) this is \na direct consequence of the $2\/3$ bound on  the optimal fidelity associated with \nan approximate realization of a U-NOT~\\cite{BUZEK,PASQ}.\n\nTo derive the upper bound of Eq.~(\\ref{UPLOW}) it is sufficient to provide a  LCPT \ntransformation that satisfies Eq.~(\\ref{movers1}) for $p\\geqslant 1\/(d+1)$. \nFor instance let us analyze the  linear mapping which takes any trace-class \noperator $\\Lambda$ of ${\\cal H}_S$ to\n\\begin{eqnarray}\n\t{\\cal N}(\\Lambda) \\equiv \\frac{dp -1}{d-1} \\Lambda + \n\t\\frac{1-p}{d-1} \\mbox{Tr}[ \\Lambda ] \\; \\openone \\;,\n\\label{esempio}\n\\end{eqnarray}\nwhere $\\openone$ is the identity operator of ${\\cal H}_S$. The transformation ${\\cal N}$ \nbelongs to the set of universal inverter super-operators that have been introduced in \nRefs.~\\cite{BUZEK1,HORO} in the context of entanglement criteria for bipartite systems.\nMoreover, for $p=1\/(d+1)$ Eq.~(\\ref{esempio})  gives the approximated U-NOT \nmapping~\\protect\\cite{BUZEK,PASQ,DEMA} which provides optimal output fidelity for $d=2$ \nand, in the case $d>2$, provides optimal fidelity in the \nclass of  LCPT maps  which are covariant   \nwith respect to arbitrary unitary transformations of ${\\cal H}_S$~\\cite{BUZEK1}.\nOne easily verifies that the transformation (\\ref{esempio}) \nsatisfies the condition~(\\ref{movers1}) \nfor all~$|\\psi\\ran$. However, it turns out~\\cite{BUZEK1} that the map ${\\cal N}$ \nis completely positive only for \n$p\\geqslant \\; 1\/(d +1)$ and this fixes the upper bound for the threshold \n$p_{\\mbox{\\small t}}(d)$.\nIt should be stressed that in agreement with the no-go theorem \non the deterministic quantum movers, the map ${\\cal N}$ \nof Eq.~(\\ref{esempio}) transforms pure states of ${\\cal H}_S$ in mixed outputs.\nIn fact, the purity of the output state is\n$$\\mbox{Tr} \\left[ \\; \n{\\cal N}(|\\psi\\ran\\lan\\psi|)^2 \\; \\right] = \\left(\\frac{1-p}{d-1}\\right)^2 +\np^2, $$\nwhich is strictly less than $1$ for  $p\\geqslant \\; 1\/(d +1)$  and $d\\geqslant 2$. \n\n\nTo derive the lower bound of Eq.~(\\ref{UPLOW})\n assume that a LCPT map $\\cale$ satisfying Eq.~(\\ref{movers1}) for all \ninput states $|\\psi\\ran$ exists.\nBy applying this relation to the vector of Eq.~(\\ref{VETTORI}) we get the\nfollowing equation\n\\begin{eqnarray}\np&=&|\\alpha|^2 \\bra{k}\\: \\cale (|\\psi\\ran\\lan\\psi|)\\; |k\\ran\n+ |\\beta|^2 \\bra{k^\\prime}\\: \\cale (|\\psi\\ran\\lan\\psi|)\\; |k^\\prime\\ran \\nonumber \\\\\n&&+\\; [\\; \\alpha \\beta^* \\bra{k^\\prime}\\: \\cale (|\\psi\\ran\\lan\\psi|)\\; |k\\ran + \\mbox{c.c.} \\; ]\\;,\n\\label{equazione}\n\\end{eqnarray} \nwith\n\\begin{eqnarray} \n\\cale (|\\psi\\ran\\lan\\psi|) &\\equiv& |\\alpha|^2 \\cale (|k\\ran\\lan k|) \n+  |\\beta|^2 \\cale (|k^\\prime \\ran\\lan k^\\prime |)  \\nonumber\\\\\n&&+\\; \n [ \\; \\alpha \\beta^*  \\cale (|k\\ran\\lan k^\\prime|) + \\mbox{h.c.} \\; ] \\;.\\label{equazione1}\n\\end{eqnarray}\nAgain by requiring Eq.~(\\ref{equazione}) to hold for all $\\alpha$ and $\\beta$ we get the following \nconstraints on the map $\\cale$ which apply for all $|k\\ran$ and $|k^\\prime\\ran$\northonormal \n\\begin{eqnarray}\n\\left\\{ \n\\begin{array}{l}\n2p = \\lan k k^\\prime k^\\prime k\\ran\n+ \\lan k^\\prime k k k^\\prime \\ran\n+ (\\; \\lan k k k^\\prime k^\\prime \\ran + c.c \\;)\\\\\n\\lan k k^\\prime k k^\\prime \\ran= 0 \\\\\n\\lan k k k k^\\prime \\ran +\\lan k k^\\prime k k\\ran \n =0\n \\;,\n\\end{array}\\right. \\label{equo}\n\\end{eqnarray}\nwhere we defined the quantities\n$\\lan j k \\ell m \\ran \\equiv \\bra{j} \\cale (|k \\ran\\lan \\ell|)  |m\\ran \n= \\lan m \\ell k j \\ran^*$ (see Appendix~\\ref{appendixB}).\nThe previous equations can be supplemented by another constraint which can be obtained \nby applying  $(\\cale_S\\otimes{\\cal I}_A)$ \nto the entangled state \n$$|\\Psi\\ran_{SA} \\equiv \\frac{1}{\\sqrt{d}}\\sum_{k=1}^d |k\\ran_S\\otimes |k\\ran_A,$$\nand requiring its trace \nto be properly normalized to 1\n[In the above expression $\\{|k\\ran_{S,A}\\}_k$ are orthonormal bases of \n${\\cal H}_S$ and ${\\cal H}_A$,\nrespectively, and ${\\cal I}_A$ is the identity mapping on the ancillary\nspace ${\\cal H}_A$].\n This yields\n\\begin{equation}\n1= \\frac{1}{d} \\sum_{k=1}^{d}\\sum_{k^\\prime=1}^d \\lan k^\\prime kk k^\\prime \\ran \n =  p + \\sum_{k \\neq k^\\prime} \n\\lan k^\\prime kk k^\\prime \\ran \/d \\;,\n\\label{traccia} \n\\end{equation} \nwhere we used the hypothesis~(\\ref{movers1}) to write \n$\\lan k kk k \\ran = p$ for all $|k\\ran$.\nEquation~(\\ref{traccia}) can be further simplified by replacing the first relation\nof the system~(\\ref{equo}). This gives \n$$ \\sum_{k \\neq k^\\prime} \n\\lan k k k^\\prime k^\\prime \\ran = d(d p -1).$$\nThe lower bound of Eq.~(\\ref{UPLOW}) finally follows by comparing this\nrelation with the inequality\n$\n|\\lan k k k^\\prime k^\\prime\\ran | \\leqslant p\n$, obtained  by introducing a Kraus decomposition\nof the LCPT map $\\cale$ \nand applying the Cauchy-Schwarz inequality to the left hand side term~\\cite{FOOTNOTE4}. \n\nWe finally comment on the relations of  $\\mbox{GQM}(p)$ with the U-NOT. One can easily verify \nthat the set of the linear transformations which implement $\\mbox{GQM}(p)$\nis closed under convex combination, e.g. given ${\\cal E}_{p_i}$ LCPT implementations \nof $\\mbox{GQM}(p_i)$ on S and given $\\{\\lambda_i\\}_i$ a set of probabilities,\nthe transformation  $\\sum_i \\lambda_i {\\cal E}_{p_i}$ \nis a LCPT map which\nimplements  the generalized mover  $\\mbox{GQM}(p)$ on S \nwith $p=\\sum_i \\lambda_i p_i$ being the\naverage of the $p_i$'s.\nThis observation allows us to establish that if the U-NOT could be implemented through \na LCPT transformation, then convex combinations of such a map and the identity map will \nimplement the $\\mbox{GQM}(p)$ for $p<1$.\n\n\n\\section{Quantum Movers and Entanglement}\\label{s:tre}\nIn this last section we discuss an application of quantum movers in the\ncontext of entanglement witness. \nA central issue in quantum information \nis to determine whether or not a given density matrix $R$ of a \ncomposite system S+A is entangled. The identification of classes of impossible \ntransformations \n can become a \npowerful tool in this respect~\\cite{HORO3}. In our case, \none can construct a one parameter set of positive \n{\\em but} not completely positive linear maps $\\{{\\cal M}^{(p)}\\}_{p}$ which \nimplement the generalized quantum mover transformation $\\mbox{GQM}(p)$ on the system S.\nThe presence of entanglement in $R$ will then be detected by \nlooking for the presence of negative eigenvalues in  ${\\cal M}_S^{(p)}\\otimes {\\cal I}_A(R)$ \nfor some values of $p$. \nMoreover, one can use $p$  as an effective ``measure'' of the entanglement in $R$ as one \nexpects intuitively only highly entangled states to feel the non complete positivity of \nthe transformation ${\\cal M}^{(p)}$ with $p\\sim 1$. A suitable choice for the operators \n$\\{{\\cal M}^{(p)}\\}_{p}$ is obtained by adding to the transformations~(\\ref{esempio}) \na linear term which gives \nnull contribution to Eq.~(\\ref{movers1}), e.g.\n\\begin{eqnarray}\n{\\cal M}^{(p)} (\\Lambda) \\equiv \\frac{dp -1}{d-1} \\Lambda + \n\\frac{1-p}{d-1} \\mbox{Tr}[ \\Lambda ]\\; \\openone + i [ \\Lambda, \\Theta ]\n\\label{NEWesempio1}\n\\end{eqnarray}\nwith $\\Theta$ being a Hermitian operator on ${\\cal H}_S$. \nFor the sake of simplicity let us consider the $d=2$ case where we can select \n$\\Theta = \\sqrt{p(1-p)} \\; \\sigma_{\\hat{n}}$ with $\\sigma_{\\hat{n}}$ being the Pauli \noperator along the $\\hat{n}$ direction. With this choice Eq.~(\\ref{NEWesempio1}) yields a  \nmapping which, for all $p\\in[0,1[$ is positive but not completely positive (this can be \nverified for instance by a direct inspection of the corresponding Choi \nmatrix~\\cite{CHOI}). As a test consider the set of Werner states~\\cite{WER}\n\\begin{eqnarray}\nR_q \\equiv  \\frac{1-q}{4} \\, \\openone_S \\otimes \\openone_A \n\\, + \\, q \\, |\\Psi_-\\rangle_{SA} \\langle \\Psi_-| \\; \n\\label{werner},\n\\end{eqnarray}\nwhich  have concurrence~\\cite{CONC} $C= \\max \\{ 0, (3q-1)\/2\\}$ and are hence \nseparable if and only if $q\\leqslant 1\/3$  \n[in the above expression $|\\Psi_-\\rangle = (|01\\rangle_{SA} - |10\\rangle_{SA})\/\\sqrt{2}$\nis the singlet state].\nThe eigenvalues of  ${\\cal M}_S^{(p)}\\otimes {\\cal I}_A(R_q)$ can\nbe easily computed: three of them are always positive and the fourth\nis equal to\n\\begin{eqnarray}\n\\lambda(q,p)\\equiv  q(p-(3q-1)\/2q)\/2\\;. \n\\end{eqnarray}\nFor $R_q$ separable (i.e. $q\\leqslant 1\/3$) this quantity is positive for all $p$. \nHowever for  $R_q$ entangled (i.e. $q> 1\/3$) $\\lambda(q,p)$ is negative for all \n$p$ smaller than the critical value $(3 q -1)\/2q=(3\/2) [ 1-1\/(2 C+1)]$ which\nmonotonically increases with $C$. As intuitively expected the more entanglement is  in $R_q$\nthe more transformations ${\\cal M}_S^{(p)}\\otimes {\\cal I}_A(R_q)$ \nare able to detect it. \n\n\\section{Conclusions}\\label{s:conc}\nWe have introduced a new class of impossible\nquantum operations, the quantum movers which include, as a   \nspecial case, the U-NOT transformation of Refs.~\\cite{BUZEK,PASQ}.\nWe proved that it is not possible to transform an unknown state $|\\psi\\ran$\nof a system S under the requirement that the final output configuration $|\\psi_p\\ran$\nhas a fixed (non unitary) component along the direction of the initial state.\nBy relaxing the requirement on the purity of the final configuration of the system,\nthere exists a threshold $p_{\\mbox{\\small t}}(d)$ for the value of the input-output fidelity \nabove which mixed quantum movers can be implemented. We related the movers to the U-NOT\nand we showed an application of these transformation in the context of separability\ncriteria.\nThe approach introduced in this work can be further extended. One can, for example, \nanalyze the case in which the output fidelity of a quantum mover is bounded \nin a given interval, e.g. $F\\in [p_1,p_2]$. Moreover, as in the case of a U-NOT\ntransformation, \nit may be interesting to analyze under which conditions the $\\mbox{GQM}(p)$ with $p$ \nbelow the threshold value $p_t(d)$ can be realized approximatively.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nUltra high energy (UHE) cosmic neutrinos play a key role in the\ndetermination of the composition of the ultra high energy cosmic\nrays (UHECRs) and their origin. UHE neutrinos are expected to be\nproduced in astrophysical sources in the decays of charged pions\ncreated in the interactions of UHECRs with matter or radiation. They\ncan be also produced via interaction of the UHECRs with the cosmic\nmicrowave background during propagation to the Earth (cosmogenic\nneutrinos). UHE cosmic neutrinos are not deviated by magnetic fields\nand could point back to their sources.\n\nRecently, three neutrinos of energy 1-2 PeV, as well as tens of\nneutrinos above 10 TeV were detected with the IceCube experiment\n\\cite{IceCube:2014}. The cosmic neutrinos with energies near 1 EeV\nare detectable with the Surface Detector (SD) of the Pierre Auger\nObservatory (PAO) \\cite{PAO}. In order to isolate neutrino-induced\nevents at the SD of the PAO, it is necessary to look for deeply\npenetrating quasi-horizontal (inclined) air showers\n\\cite{Berezinsky:1969}-\\cite{Zas:2005}. The PAO can efficiently\nsearch for two types of neutrino-induced inclined air showers (see\nfig.~\\ref{nu_events}):\n\\begin{enumerate}\n  \\item\nDownward-going (DG) neutrino-induced showers. They are initiated by\nneutrinos moving with large zenith angle $\\theta$ which interact in\nthe atmosphere close to the SD. At the PAO the search is restricted\nto showers with $\\theta > 60^\\circ$ \\cite{Auger:2015}. Note that the\nbackground from hadronic showers above $10^{17}$ eV is\n$\\mathrm{O}(1)$ in 20 years, and it is negligible above $10^{19}$ eV\n\\cite{Anchordoqui:2010}.\n  \\item\nEarth-skimming (ES) showers induced by upward tau neutrinos. They\ncan interact in the Earth's crust producing tau leptons. The latter\nare efficiently produced at zenith angles $90^\\circ < \\theta <\n95^\\circ$ \\cite{Auger:2015}. The tau leptons escape the Earth and\ndecay in the atmosphere close to the SD\n\\cite{Bertou:2002}-\\cite{Feng:2002}.\n\\end{enumerate}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=10cm,clip]{nu_events}\n\\caption{Different types of showers induced by DG and ES neutrinos\n(fig.~1 in \\cite{Auger:2011}).}\n\\label{nu_events}\n\\end{figure}\n\nRecently, the Auger Collaboration reported on searches for DG\nneutrinos in the zenith angle bins $60^\\circ - 75^\\circ$ and\n$75^\\circ - 90^\\circ$, as well as for ES neutrinos. The data were\ncollected by the SD of the PAO from 1 January 2004 until 20 June\n2013 \\cite{Auger:2015}\n\\footnote{This search period is equivalent of 6.4 years of a\ncomplete Auger SD working continuously \\cite{Auger:2015}.}\nNo neutrino candidates were found. Assuming the diffuse flux of UHE\nneutrinos to be $dN\/dE_\\nu = k E_\\nu^{-2}$ in the energy range\n$1.0\\times10^{17}$ eV - $2.5\\times10^{19}$ eV, the stringent limit\nwas obtained:\n\\begin{equation}\\label{Auger_bound}\nk < 6.4 \\times 10^{-9} \\mathrm{\\ GeV \\ cm^2 \\ s^{-1} \\ sr^{-1}} \\;.\n\\end{equation}\nThis Auger limit is a factor 3.64 below the Waxman-Bachall bound on\nneutrino production in optically thin astrophysical sources\n\\cite{WB:2001}:\n\\begin{equation}\\label{WM_bound}\nE_\\nu^2 \\frac{dN}{dE_\\nu} = 2.33 \\times 10^{-8} \\mathrm{\\ GeV \\ cm^2\n\\ s^{-1} \\ sr^{-1}} \\;.\n\\end{equation}\n\nIn the Standard Model (SM) neutrino-nucleon cross sections are\nexpected to be very small in comparison with hadronic cross sections\neven at UHEs \\cite{Sarkar:2008}. That is why, the UHE cosmic\nneutrinos can be regarded as unique probes of new interactions. In\nthe present paper a theory with an extra dimension (ED) is\nconsidered to be a ``new physics'' theory. We will see that effects\ncoming from the ED can be significant or even dominant in the $\\nu\nN$-scattering at UHEs.\n\n\n\\section{General solution for the Randall-Sundrum metric}\n\\label{sec:RS}\n\nIn ref.~\\cite{Randall:1999} the 5-dimensional space-time with\nnon-factorizable geometry was suggested as an alternative to the\nmodels with flat extra dimensions. The Randall-Sundrum (RS) model\n\\cite{Randall:1999} predicts an existence of heavy Kaluza-Klein (KK)\nexcitations (massive gravitons). These massive KK resonances are\nintensively searched for at the LHC.\n\nThe RS scenario is described by the following background warped\nmetric\n\\begin{equation}\\label{RS_background_metric}\n\\quad ds^2 = e^{-2 \\sigma (y)} \\, \\eta_{\\mu \\nu} \\, dx^{\\mu} \\,\ndx^{\\nu} - dy^2 \\;,\n\\end{equation}\nwhere $\\eta_{\\mu\\nu}$ is the Minkowski tensor with the signature\n$(+,-,-,-)$, and $y$ is an extra coordinate. The periodicity\ncondition $y=y + 2\\pi r_c$ is imposed, and the points $(x_\\mu,y)$\nand $(x_\\mu,-y)$ are identified. Thus, one has a model of gravity in\nthe AdS$_5$ space-time compactified to the orbifold $S^1\\!\/Z_2$. The\norbifold has two fixed points, $y=0$ and $y=\\pi r_c$. It is assumed\nthat there are two branes located at these points. All the SM fields\nlive on one of these branes.\n\nThe classical action of the RS scenario is given by\n\\cite{Randall:1999}\n\\begin{align}\\label{action}\nS &= \\int \\!\\! d^4x \\!\\! \\int_{-\\pi r_c}^{\\pi r_c} \\!\\! dy \\,\n\\sqrt{G} \\, (2 \\bar{M}_5^3 \\mathcal{R}\n- \\Lambda) \\nonumber \\\\\n&+ \\int \\!\\! d^4x \\sqrt{|g^{(1)}|} \\, (\\mathcal{L}_1 - \\Lambda_1) +\n\\int \\!\\! d^4x \\sqrt{|g^{(2)}|} \\, (\\mathcal{L}_2 - \\Lambda_2) \\;,\n\\end{align}\nwhere $G_{MN}(x,y)$ is the 5-dimensional metric, with $M,N =\n0,1,2,3,4$, $\\mu = 0,1,2,3$. The quantities\n\\begin{equation}\ng^{(1)}_{\\mu\\nu}(x) = G_{\\mu\\nu}(x, y=0) \\;, \\quad\ng^{(2)}_{\\mu\\nu}(x) = G_{\\mu\\nu}(x, y=\\pi r_c)\n\\end{equation}\nare induced metrics on the branes, $\\mathcal{L}_1$ and\n$\\mathcal{L}_2$ are brane Lagrangians, $G = \\det(G_{MN})$, $g^{(i)}\n= \\det(g^{(i)}_{\\mu\\nu})$. $\\bar{M}_5$ is the \\emph{reduced}\n5-dimensional Planck scale. The quantity $\\Lambda$ is a\n5-dimensional cosmological constant, while $\\Lambda_{1,2}$ are brane\ntensions.\n\nFor the first time, the function $\\sigma(y)$ was obtained in\n\\cite{Randall:1999} (see also the expression at the bottom of\nfig.~\\ref{two_solutions}):\n\\begin{equation}\\label{sigma_1}\n\\sigma_0(y) = \\kappa |y| - C_0 \\;,\n\\end{equation}\nwhere $\\kappa$ is a parameter with a dimension of mass, which\ndefines the curvature of the 5-dimensional space-time.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=8cm,clip]{two_solutions}\n\\caption{Two equivalent solutions of Einstein-Hilbert's equations\nfor the function $\\sigma(y)$ related to the different branes located\nat $y=0$ and $y=\\pi r_c$.}\n\\label{two_solutions}\n\\end{figure}\nThis solution is consistent with the orbifold symmetry $y\n\\rightarrow - y$. However, it is attributed to the brane $y=0$ (see\nfig.~\\ref{two_solutions}), and it is\nnot symmetric with respect to the branes\n\\footnote{Note that in contrast to \\eqref{sigma_1}, the RS1 solution\n\\cite{Randall:1999} has no constant term.}\n\nInstead of \\eqref{sigma_1}, one can use another \\emph{equivalent}\nsolution related to the brane $y=\\pi r_c$ (see the expression at the\ntop of fig.~\\ref{two_solutions}):\n\\begin{equation}\\label{sigma_2}\n\\sigma_\\pi (y) =  -\\kappa |y - \\pi r_c| + |\\kappa| \\pi r_c - C_\\pi\n\\;,\n\\end{equation}\nIn order to get an expression symmetric with respect to both branes,\nwe take half the sum of expressions \\eqref{sigma_1} and\n\\eqref{sigma_2}. As a result, we come to the solution:\n\\begin{equation}\\label{sigma_symmetric}\n\\sigma (y) = \\frac{\\kappa}{2} ( |y| - |y - \\pi r_c| ) +\n\\frac{|\\kappa| \\pi r_c}{2} - C \\;,\n\\end{equation}\nwhere $C = (C_0 + C_\\pi)\/2$. The constant terms in\n\\eqref{sigma_symmetric} are chosen in such a way that one has\n$\\sigma (y) = \\kappa y - C$ within the interval $0 < y < \\pi r_c$.\n\nNote that neither original RS1 solution nor generalized RS-like\nsolution \\eqref{sigma_symmetric} obeys periodicity in $y$\n\\emph{explicitly}. One has to keep the periodicity condition in\n``mind''. Fortunately, a new solution was recently proposed\n\\cite{Kisselev:2015} which is both symmetric with respect to the\nbranes and \\emph{periodic} function of $y$:\n\\begin{equation}\\label{sigma}\n\\sigma(y) = \\frac{\\kappa r_c }{2} \\left[ \\left| \\mathrm{Arccos}\n\\left(\\cos \\frac{y}{r_c} \\right) \\right| - \\left| \\pi -\n\\mathrm{Arccos} \\left(\\cos \\frac{y}{r_c} \\right)\\right| \\right] +\n\\frac{\\pi \\,|\\kappa| r_c }{2} - C \\;,\n\\end{equation}\nwith the fine tuning relations:\n\\begin{equation}\\label{fine_tuning}\n\\Lambda = -24 \\bar{M}_5^3\\kappa^2 \\;, \\quad \\Lambda_1 = -\n\\,\\Lambda_2 = 24 \\bar{M}_5^3 \\kappa \\;.\n\\end{equation}\nHere $\\mathrm{Arccos(z)}$ is a \\emph{principal value} of the\nmultivalued inverse trigonometric function $\\arccos(z)$. It is\ndefine by the inequality (see, for instance, \\cite{Arccos}):\n\\begin{equation}\\label{Arccos}\n0 \\leqslant \\mathrm{Arccos}(z) \\leqslant \\pi \\;, \\quad -1 \\leqslant\nz \\leqslant 1 \\;.\n\\end{equation}\nIt means that \\cite{Arccos}\n\\begin{equation}\\label{arccos}\n\\mathrm{Arccos} (\\cos x) = \\left\\{\n\\begin{array}{ll}\n  x - 2n \\pi , & 2n \\pi \\leqslant x \\leqslant (2n +1)\\pi \\;, \\\\\n  - x + 2(n +1)\\pi  , & (2n +1)\\pi \\leqslant x \\leqslant 2(n +1)\\pi\n  \\;,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $n=0,1, \\ldots$ It follows from \\eqref{sigma}, (\\ref{arccos})\nthat $\\sigma(y) + C = \\kappa y$ for $0 < y < \\pi r_c$, $\\sigma(y) +\nC = \\kappa (2\\pi r_c - y)$ for $\\pi r_c < y < 2\\pi r_c$. and so on\n(see fig.~\\ref{two_solutions}).\n\nOur solution \\eqref{sigma} \\\\\n- is symmetric with respect to the\nbranes: $\\sigma(y)$ remains unchanged if\n$y \\rightarrow \\pi r_c - y$, $k \\rightarrow -k$\n\\footnote{Under the replacement $y \\rightarrow \\pi r_c - y$ the\npositions of the branes are interchanged ($y=0 \\rightleftarrows\ny=\\pi r_c$), while under the replacement $k \\rightarrow -k$ their\ntensions are interchanged ($\\Lambda_1 \\rightleftarrows \\Lambda_2$).}\n\\\\\n- obeys the orbifold $Z_2$-symmetry: $\\sigma(y) = \\sigma(-y)$;\n\\\\\n- reproduces the jumps of $\\sigma'(y)$ on both branes: $\\sigma''(y)\n= 2\\kappa [\\delta (y) - \\delta (y - \\pi r_c)]$, $0 \\leqslant y\n\\leqslant \\pi r_c$; \\\\\n- is the periodic function of the extra coordinate: $\\sigma(y + 2\\pi\nr_c n) = \\sigma(y)$, $n=\\pm 1, \\pm 2, \\ldots$ Note that the warped\nfunction $\\sigma(y)$ depends on the constant $C$.\n\n\n\\section{RS-like scenario with the small curvature}\n\\label{sec:RSSC}\n\nBy taking different values of $C$ in eq.~\\eqref{sigma}, we come to\nquite \\emph{diverse physical scenarios} \\cite{Kisselev:2015},\n\\cite{Kisselev:2016}. One of them ($C = 0$) is in fact the RS1 model\n\\cite{Randall:1999}. Another scheme ($C = \\kappa \\pi r_c$) describes\na geometry with a small curvature of five-dimensional space-time\n(RSSC model) \\cite{Giudice:2005}-\\cite{Kisselev:2006}. In the RSSC\nmodel the hierarchy relation looks like (we assume that $\\kappa \\pi\nr_c \\gg 1$)\n\\begin{equation}\\label{hierarchy_relation_RSSC}\n\\bar{M}_{\\mathrm{Pl}}^2  = \\frac{\\bar{M}_5^3}{\\kappa} \\, e^{2\\kappa\n\\pi r_c} \\;.\n\\end{equation}\nSo, one can take, for instance, $\\kappa \\sim 1$ GeV, $M_5 \\sim 1$\nTeV. Let us underline that the original RS scenario does not admit\nthe parameters to lie in this region, since $\\kappa \\sim \\bar{M}_5\n\\sim M_{\\mathrm{Pl}}$ TeV in it \\cite{Randall:1999}.\n\nThe masses of the KK gravitons are proportional to the curvature\nparameter $\\kappa$:\n\\begin{equation}\\label{m_n_2}\nm_n = x_n \\kappa \\;,\n\\end{equation}\nwhere $x_n$ are zeros of the Bessel function $J_1(x)$. The coupling\nof the massive gravitons to the SM fields is given by the constant:\n\\begin{equation}\\label{Lambda_pi_2}\n\\Lambda_\\pi = \\bar{M}_5 \\left( \\frac{\\bar{M}_5}{\\kappa}\n\\right)^{1\/2} .\n\\end{equation}\nThus, the RSSC scenario predicts a spectrum of the KK gravitons\nsimilar to that of the ADD model\n\\cite{Arkani-Hamed:1998}-\\cite{Antoniadis:1998}. For the LHC\nphenomenology of the RSSC model, see \\cite{Kisselev:2008},\n\\cite{Kisselev:2013}.\n\n\n\\section{Neutrino-nucleon amplitude in the RSSC model}\n\\label{sec:nu_N}\n\nFor the UHE cosmic neutrinos, the scattering takes place in the\ntrans-Planckian kinematical region:\n\\begin{equation}\\label{kinem_region}\n\\sqrt{s} \\gtrsim \\bar{M}_5 \\gg -t \\;,\n\\end{equation}\nwith $\\sqrt{s}$ being the colliding energy and $t = - q_{\\bot}^2$\nfour-dimensional momentum transfer. Remember that the inequality\n$\\kappa \\ll \\bar{M}_5$ is also satisfied. In the eikonal\napproximation, which is valid in the kinematical region\n\\eqref{kinem_region}, the neutrino-proton amplitude is of the form:\n\\begin{equation}\\label{ampl}\nA_{\\nu \\rm p}(s, t) = 4 \\pi i \\, s \\!\\! \\int\\limits_0^{\\infty} \\! db\nb J_0(b \\, q_{\\bot}) \\left\\{ 1 - \\exp [i \\chi(s, b)] \\right\\} \\;.\n\\end{equation}\nIn its turn, the eikonal is given by the Fourier-Bessel\ntransformation of the eikonal:\n\\begin{equation}\\label{eikonal}\n\\chi(s, b) = \\frac{1}{4 \\pi s} \\! \\int\\limits_0^{\\infty} \\!\ndq_{\\bot} q_{\\bot}  J_0(q_{\\bot}  b) \\, A_{\\nu \\rm p}^{\\rm B} (s, t)\n\\;.\n\\end{equation}\n\nThe Born amplitude is given by the sum of gravi-Reggeons, i.e.\nreggeized gravitons in the $t$-channel. Because of a presence of the\nED, the Regge trajectory of the graviton is splitting into an\ninfinite sequence of trajectories enumerated by the KK number\n$n$~\\cite{Kisselev:2004}:\n\\begin{equation}\\label{trajectories}\n\\alpha_n(t) = 2 + \\alpha_g' t  - \\alpha_g' \\, m_n^2, \\quad n = 0, 1,\n\\ldots.\n\\end{equation}\nIn string theories, the slope of the gravi-Reggeons is universal,\nand $\\alpha_g' = M_s^{-2}$, where $M_s$ is the string scale. As a\nresult, in the RSSC model the gravity Born amplitude for the\nneutrino scattering off \\emph{a point-like} particle looks like\n\\cite{Kisselev:2005}, \\cite{Kisselev:2009}:\n\\begin{equation}\\label{Born_part_ampl}\nA_{\\rm grav}^{\\mathrm{B}}(s, t) = \\frac{\\pi \\alpha'_g s^2}{2\n\\Lambda_{\\pi}^2} \\sum_{n \\neq 0} \\left[ i - \\cot\\frac{\\pi\n\\alpha_n(t)}{2} \\right] \\left( \\frac{s}{\\bar{M}_5}\n\\right)^{\\alpha_n(t) -2} \\;.\n\\end{equation}\n\nThe \\emph{hadronic} Born amplitude in \\eqref{eikonal} is defined by\na convolution of the gravity amplitude~\\eqref{Born_part_ampl} and\nskewed ($t$-dependent) PDFs $f_i(x,\\mu^2; t)$:\n\\begin{equation}\\label{Born_had_ampl}\nA_{\\nu \\rm p}^{\\rm B}(s, t) = \\sum_{i=q,\\bar{q},g} \\int\\limits_0^1\ndx A_{\\rm grav}^{\\mathrm{B}}(xs, t) \\, f_i(x,\\mu^2; t) \\;.\n\\end{equation}\nWe take the $t$-dependent PDFs in a factorized form:\n\\begin{equation}\\label{skewed dist}\nf_i(x,\\mu^2; t) = f_i(x, \\mu^2) D(t) \\;.\n\\end{equation}\nFor the PDFs $f_i(x, \\mu^2)$ we use the CT14 parametrization\n\\cite{CT14} and put $\\mu^2 = |t|$. As for a suppression factor\n$D(t)$, at hadron scales it was evaluated to be $D(t) =\n\\exp(tr_0^2)$, with $r_0^2 = 0.62 \\mathrm{\\ GeV}^{-2}$\n\\cite{Petrov:2002}. For larger $|t|$, we assume $D(t)$ to have a\npower-like behavior. As a result, we come to the parametrization:\n\\begin{equation}\\label{damping_factor}\nD(t) =\n\\left\\{\n  \\begin{array}{ll}\n    \\exp(t r_0^2) \\;, & |t| \\leqslant q_0^2 \\;, \\\\\n    \\left(\\mathrm{e} |t|\/q_0^2 \\right)^{-q_0^2 r_0^2} \\;, & |t| > q_0^2 \\;,\n  \\end{array}\n\\right.\n\\end{equation}\nwhere $q_0 = m_\\rho$ is the $\\rho$-meson mass, and $\\ln \\mathrm{e} =\n1$.\n\nThe total cross sections with the account of the contribution from\nthe KK gravitons are shown in fig.~\\ref{cross_sections}. We adopted\nthe SM $\\nu N$ cross sections from \\cite{Sarkar:2008}. One can see,\nthe  total cross section noticeably exceeds the SM one at energies\nhigher than $10^{18}$ or $10^{19}$ eV, depending on the reduced\n5-dimensional Planck scale $\\bar{M}_5$.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=7cm,clip]{cs_mm}\n\\caption{Solid lines (from above): the neutrino total cross sections\nfor $\\bar{M}_5 = 1.5$, 2.0 and 3.0 TeV. Dashed line: neutrino CC\ntotal cross section.}\n\\label{cross_sections}\n\\end{figure}\n\n\n\\section{Expected number of neutrino events at the PAO}\n\\label{sec:ev_num}\n\nIn \\cite{Anchordoqui:2010} the following functional dependence of\nthe DG event rate on the new physics cross section\n$\\sigma_{\\mathrm{NP}}$ was proposed for a bin in neutrino energy\n($10^{18.5} \\mathrm{\\ eV} < E_\\nu < 10^{19.5} \\mathrm{\\ eV}$):\n\\begin{equation}\\label{DG:BSM_vs_SM_prel}\nN_{\\mathrm{BSM}}^{\\mathrm{DG}} = N_{\\mathrm{SM}}^{\\mathrm{DG}} \\,\n\\frac{\\sigma_{\\mathrm{CC}} +\n\\sigma_{\\mathrm{NP}}}{\\sigma_{\\mathrm{CC}}} \\;,\n\\end{equation}\nwhere $N_{\\mathrm{SM}}^{\\mathrm{DG}}$ is the number of DG events in\nthe absence of new physics, $\\sigma_{\\mathrm{CC}}$ is a charged\ncurrent (CC) total cross section. $N_{\\mathrm{BSM}}^{\\mathrm{DG}}$\nis the number of DG events with the account of interactions beyond\nthe SM. We propose a more general formula:\n\\begin{equation}\\label{DG:BSM_vs_SM}\n\\mathcal{E}_{\\mathrm{BSM}}^{\\mathrm{DG}} (E_\\nu) =\n\\mathcal{E}_{\\mathrm{SM}}^{\\mathrm{DG}} (E_\\nu)\\,\n\\frac{\\sigma_{\\mathrm{SM}}^{\\mathrm{eff}}(E_\\nu) +\n\\sigma_{\\mathrm{NP}}(E_\\nu)}{\\sigma_{\\mathrm{SM}}^{\\mathrm{eff}}(E_\\nu)}\n\\;,\n\\end{equation}\nwhere $\\mathcal{E}_{\\mathrm{BSM}}^{\\mathrm{DG}}$\n($\\mathcal{E}_{\\mathrm{SM}}^{\\mathrm{DG}}$) is the \\emph{exposure}\nof the SD of the PAO with (without) account of the new interaction.\nIn addition, instead of $\\sigma_{\\mathrm{CC}}$, an effective SM\ncross section $\\sigma_{\\mathrm{SM}}^{\\mathrm{eff}}$ is introduce in\n\\eqref{DG:BSM_vs_SM}:\n\\begin{equation}\\label{sigma_eff}\n\\sigma_{\\mathrm{SM}}^{\\mathrm{eff}} = \\sigma_{\\mathrm{CC}}\n\\!\\!\\sum_{i=e,\\mu,\\tau} \\! \\, m_{\\mathrm{CC}}^i +\n3\\sigma_{\\mathrm{NC}} \\, m_{\\mathrm{NC}} + \\sigma_{\\mathrm{CC}}\\,\nm_{\\mathrm{mount}} \\;.\n\\end{equation}\nHere $m_{\\mathrm{CC}}^i$ and $m_{\\mathrm{NC}}$ are relative mass\napertures for charged current and neutral current (NC) interactions\nof the DG neutrinos at the PAO. The mass aperture\n$m_{\\mathrm{mount}}$ corresponds to the CC interaction of a $\\tau$\nneutrino within the mountains around the PAO (see\nfig.~\\ref{nu_events}). The relative mass apertures as functions of\nthe neutrino energy where calculated using data from Table~I of\nref.~\\cite{Auger:2011}\n\\footnote{In particular, $\\sum_{i=e,\\mu,\\tau} m_{\\mathrm{CC}}^i$ is\nequal to 0.69 (0.65) at $10^{18}$ ($10^{19}$) eV.}\nNote that $\\sum_{i=e,\\mu,\\tau} m_{\\mathrm{CC}}^i + 3 m_{\\mathrm{NC}}\n+ m_{\\mathrm{mount}} = 1$.\n\nIn contrast to the DG neutrino exposure, the exposure of the ES\nneutrinos \\emph{decreases} with the rise of the neutrino total\ncross section\n\\footnote{Is is a generalization of the formula for the rate for ES\nshowers in the range $10^{18.5} \\mathrm{\\ eV} < E_\\nu < 10^{19.5}\n\\mathrm{\\ eV}$ \\cite{Anchordoqui:2010}.}\n\\begin{equation}\\label{ES:BSM_vs_SM}\n\\mathcal{E}_{\\mathrm{BSM}}^{\\mathrm{ES}}(E_\\nu) =\n\\mathcal{E}_{\\mathrm{SM}}^{\\mathrm{ES}}(E_\\nu) \\,\n\\frac{\\sigma_{\\mathrm{CC}}^2(E_\\nu)}{[\\sigma_{\\mathrm{CC}}(E_\\nu) +\n\\sigma_{\\mathrm{NP}}(E_\\nu)]^2} \\;.\n\\end{equation}\n\nThe formulas \\eqref{DG:BSM_vs_SM} and \\eqref{ES:BSM_vs_SM} allowed\nus to calculate expected exposures of the SD of the PAO for the\nperiod 1 January 2004 - 20 June 2013. The Auger data on exposures\nfor the SM neutrino interactions in the region from\n$\\log(E_\\nu\/\\mathrm{eV}) = 17$ to 20.5 in steps of 0.5 were used\n(see also fig.~3 from \\cite{Auger:2015}). The results of our\ncalculations are presented in fig.~\\ref{Auger_exposures}.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=7cm,clip]{exp_DG_BSM}\n\\includegraphics[width=7cm,clip]{exp_ES_BSM}\n\\caption{Left panel: expected Auger exposures for the DG neutrinos\nwith zenith angle $75^\\circ < \\theta < 90^\\circ$. Red, green and\nblue line corresponds to $\\bar{M}_5 = 1.5$, 2.0 and 3.0 TeV,\nrespectively. Dashed line: SM exposure taken from \\cite{Auger:2015}.\nRight panel: the same but for the ES neutrinos.}\n\\label{Auger_exposures}\n\\end{figure}\n\nThe number of neutrino events is given by the formula:\n\\begin{equation}\\label{numbers_events}\nN_{\\mathrm{ev}} = \\int \\frac{dN_\\nu}{dE_\\nu}(E_\\nu) \\,\n\\mathcal{E}(E_\\nu) \\, dE_\\nu \\;.\n\\end{equation}\nWe assume differential neutrino flux $dN\/dE_\\nu \\sim E_\\nu^{-2}$, as\nwell as a ratio $\\nu_e : \\nu_\\mu : \\nu_\\tau = 1:1:1$. In\nfig.~\\ref{BSM_vs_SM} we compare the number of neutrino events\npredicted in the RSSC model with the expected number of neutrino\nevents in the absence of a new physics. Both the DG events with\nzenith angles $60^\\circ < \\theta < 90^\\circ$ and ES events was taken\ninto account in the full energy region of the sensitivity of the SD\n($1.0\\times10^{17} - 2.5\\times10^{19}$ eV).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=7cm,clip]{ev_BSM-SM_2_2_m}\n\\caption{The ratio of the neutrino events with and without effects\nfrom EDs as a function of the gravity scale $\\bar{M}_5$, expected at\nthe SD of the PAO. The event numbers correspond to the energy range\n$1.0\\times10^{17} - 2.5\\times10^{19}$ eV.}\n\\label{BSM_vs_SM}\n\\end{figure}\nAs was already mentioned above, no neutrino events were seen at the\nSD of the PAO. Thus, from fig.~\\ref{BSM_vs_SM} we obtain the lower\nbound on 5-dimensional \\emph{reduced} Planck scale:\n\\begin{equation}\\label{M5_bound}\n\\bar{M}_5 > 2.75 \\mathrm{\\ TeV} \\;.\n\\end{equation}\nCorrespondingly, for the 5-dimensional  Planck scale we find $M_5 >\n2.75 \\times (2\\pi)^{1\/3} = 5.07$ TeV.\n\nWe have also calculated an expected ratio of the ES neutrino showers\nto the DG neutrino showers with the zenith angles $75^\\circ < \\theta\n< 90^\\circ$ (see fig.~\\ref{ES_vs_DG}).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=7cm,clip]{ev_DG-ES_2_2}\n\\caption{The expected ratio of the ES neutrinos to the DG neutrinos\n(with zenith angle $75^\\circ < \\theta < 90^\\circ$) at the SD of the\nPAO as a function of the gravity scale $M_5$.}\n\\label{ES_vs_DG}\n\\end{figure}\nWe predict $N_{\\mathrm{ES}}\/N_{\\mathrm{DG}}$ to be 2.8 (5.7) for\n$\\bar{M}_5 = 2.8$ (4.0) TeV. For the SM interactions, this ratio was\nestimated by the Auger Collaboration to be equal to 6.0\n\\cite{Auger:2015}.\n\n\n\\section{Conclusions}\n\nIn the present paper we have studied the neutrino-induced inclined\n(quasi-horizontal) events at the Surface Detector of the Pierre\nAuger Observatory in the Randall-Sundrum scenario with the extra\ndimension and warped metric. We have presented the general solution\nfor the metric \\eqref{sigma} which is symmetric with respect to both\nbranes and explicitly periodic in extra variable. In the framework\nof the RS-like model with the small curvature of the 5-dimensional\nspace-time, the exposures of the SD of the PAO for the downward\ngoing and Earth-skimming neutrinos are estimated\n(fig.~\\ref{Auger_exposures}). The lower bound on the fundamental\ngravity scale $\\bar{M}_5$ is obtained \\eqref{M5_bound}. The ratio of\nthe number of the ES air showers to the number of the DG showers,\n$N_{\\mathrm{ES}}\/N_{\\mathrm{DG}}$, is calculated as a function of\n$\\bar{M}_5$ (fig.~\\ref{ES_vs_DG}).\n\n\n\\section*{Acknowledgements}\n\nThe authors are indebted to J. Alvarez-Mu\\~{n}iz for sending\nnumerical data on the Auger exposures shown in fig.~3 from\nref.~\\cite{Auger:2015}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Section1}\nConsider the time-dependent Ginzburg-Landau system of equations\n\\begin{subequations}\n\\label{eq:1}\n\\begin{alignat}{2}\n   \\frac{\\partial\\psi}{\\partial t}-  \\nabla_{\\kappa A}^2\\psi +i\\kappa\\phi \\psi=\\kappa^2(1-|\\psi|^2)\\psi & \\quad \\text{ in }(0,+\\infty)\\times  \\Omega \\,,\\\\\n \\frac{1}{c}\\Big(\\frac{\\partial A}{\\partial t}+\\nabla\\phi\\Big)+ \\text{\\rm curl\\,}^2A\n  =\\frac{1}{\\kappa}\\Im(\\bar\\psi\\nabla_{\\kappa A}\\psi) &\n  \\quad \\text{ in } (0,+\\infty)\\times   \\Omega\\,, \\\\\n  \\psi=0 &\\quad \\text{ on } (0,+\\infty)\\times  \\partial\\Omega_c\\,, \\\\\n  \\nabla_{\\kappa A}\\psi\\cdot\\nu=0 & \\quad \\text{ on } (0,+\\infty)\\times  \\partial\\Omega_i\\,, \\\\\n  \\frac{\\partial\\phi}{\\partial\\nu} = - c\\kappa J(x) &\\quad \\text{ on } (0,+\\infty)\\times \\partial\\Omega_c\\,, \\\\\n  \\frac{\\partial\\phi}{\\partial\\nu}=0  &\\quad \\text{ on } (0,+\\infty)\\times  \\partial\\Omega_i \\,, \\\\[1.2ex]\n  \\Xint-_{\\partial\\Omega}\\text{\\rm curl\\,} A(t,x)  \\, ds = \\kappa h_{ex} \\,,&\\quad {\\text{ on } (0,+\\infty)}\\\\\n  \\psi(0,x)=\\psi_0(x)  & \\quad \\text{ in } \\Omega\\,, \\\\\nA(0,x)=A_0(x) & \\quad \\text{ in } \\Omega \\,.\n\\end{alignat}\n\\end{subequations}\nIn the above $\\psi$ denotes the order parameter, $A$  the\nmagnetic potential, $\\phi$  the electric potential, $\\kappa$\nthe Ginzburg-Landau parameter, which is a material property, and\n$$c=\\kappa^2\/\\sigma$$ where $\\sigma$ is the normal conductivity of the\nsample. Finally, $h_{ex}$, or the average magnetic field on $\\partial\\Omega$\ndivided by $\\kappa$, is constant in time. Length has been scaled with\nrespect to the penetration depth (see \\cite{alhe14}). \\\\\n Unless otherwise stated we\nshall assume in the sequel that $$\\kappa\\geq1\\,.$$ We further assume that\n$(\\psi_0,A_0)\\in H^1(\\Omega,{\\mathbb{C}})\\times H^1(\\Omega,{\\mathbb{R}}^2)$ and that\n\\begin{equation}\\label{condinit}\n  \\|\\psi_0\\|_\\infty\\leq 1 \\,.\n\\end{equation}\nWe use the notation $\\nabla_A=\\nabla-iA$, $\\Delta_A=\\nabla_A^2$ and $ds$ for the induced\nmeasure on $\\pa \\Omega$.  We have also used above the standard notation\n\\begin{displaymath}\n  \\Xint-_{\\partial \\Omega}  = \\frac{1}{\\ell(\\partial \\Omega)}\\int \\,.\n\\end{displaymath}\nThe domain $\\Omega\\subset\\subset{\\mathbb{R}}^2$ has the same characteristics as in \\cite{alhe14},\nin particular its boundary $\\pa \\Omega$ contains a smooth interface,\ndenoted by $\\partial\\Omega_c\\,$, with a conducting metal which is at the normal\nstate. Thus, we require that $\\psi$ vanishes on $\\partial\\Omega_c$ in (\\ref{eq:1}c).\n\nWe make the following assumptions on the current $J$\n\\begin{equation}\n\\label{eq:2}\n(J1)\\quad J\\in C^2(\\overline{\\partial\\Omega_c},{\\mathbb{R}})\\,,\n\\end{equation}\n\\begin{equation}\n  \\label{eq:3}\n(J2) \\quad \\int_{\\partial\\Omega_c}J \\,ds=0 \\,,\n\\end{equation}\nand\n\\begin{equation}\\label{eq:75aa}\n(J3)\\quad  \\mbox{the sign of } J \\mbox{ is constant on each\nconnected component of } \\partial\\Omega_c\\,.\n\\end{equation}\nWe allow for $J\\neq0$ at the corners despite the fact that no current is\nallowed to enter the sample through the insulator.  The rest of\n$\\partial\\Omega$, denoted by $\\partial\\Omega_i$ is adjacent to an insulator. By\nconvention, we extend $J$ as equal to $0$ on $\\partial \\Omega_i$.\n\nTo simplify some of our regularity arguments we introduce the\nfollowing geometrical assumption (for further discussion we refer the\nreader to Appendix A in \\cite{alhe14}) on $\\partial\\Omega$:\n\\begin{equation}\\label{propertyR}\n(R1)\\,\\left\\{\n\\begin{array}{l}\n(a) \\; \\pa \\Omega_i \\mbox{ and } \\pa \\Omega_c  \\mbox{ are of class } C^3\\,;\\\\\n(b) \\mbox{ Near each edge, } \\pa\n\\Omega_i \\mbox{  and  } \\pa \\Omega_c\n \\mbox{ are} \\\\ \\quad  \\mbox{ flat and meet with  an angle of } \\frac \\pi 2\\,.\n\\end{array}\\right.\n\\end{equation}\n  We also require that\n\\begin{equation}\\label{hyptopolo}\n(R2) \\quad\\quad \\mbox{Both }  \\partial\\Omega_c \\mbox{  and } \\partial\\Omega_i \\mbox{  have two components}.\n\\end{equation}\n\\begin{figure}\n  \\centering\n\\setlength{\\unitlength}{0.0005in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  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 \\caption{Typical superconducting sample. The arrows denote the direction of the current flow ($J_{in}$  for the inlet, and $J_{out}$ for the outlet).}\n  \\label{fig:1}\n\\end{figure}\n\\nopagebreak\nFigure 1 presents a typical sample with properties (R1) and\n(R2), where the current flows into the sample from one connected\ncomponent of $\\partial\\Omega_c$, and exits from another part, disconnected from\nthe first one. Most wires would fall into the above class of domains.\n\nThe system \\eqref{eq:1} is invariant to the gauge transformation\n\\begin{equation}\n\\label{eq:4}\n   A'= A + \\nabla\\omega\\,, \\; \\phi'= \\phi-\n  \\frac{\\partial\\omega}{\\partial t}\\,, \\; \\psi' = \\psi e^{i\\omega} \\,.\n\\end{equation}\nWe thus choose, as in \\cite{alhe14}, the Coulomb gauge, i.e., we assume\n\\begin{equation}\n  \\label{eq:5}\n  \\begin{cases}\n    \\text{\\rm div\\,} A =0  & \\text{ in } (0,+\\infty)\\times \\Omega\\,, \\\\\n    A\\cdot\\nu =0 & \\text{ on }(0,+\\infty)\\times  \\partial\\Omega\\,,\n  \\end{cases}\n\\end{equation}\nwhere the divergence is computed with respect to the spatial coordinates only.\n\n\nFrom (\\ref{eq:25}b), (\\ref{eq:25}d), and (\\ref{eq:25}f), we know that $\\text{\\rm curl\\,}\nA$ is constant on each connected component of $\\partial \\Omega_i$. Let then\n$\\{\\partial\\Omega_{i,j}\\}_{j=1}^{2}$ denote the set of connected components of\n$\\partial\\Omega_i$. We can write, for $j=1,2\\,$,\n\\begin{equation}\n\\label{eq:6}\n\\text{\\rm curl\\,} A|_{\\partial\\Omega_{i,j}} =\nh_j \\,  \\kappa\\,,\n\\end{equation}\nwhere  $h_1$ and $h_2$ are constants.\n\nNote that $h_1$ and $h_2$ can be determined from\n$J$ and $h_{ex}$ via the formula \\cite{alhe14}\n\\begin{equation}\n\\label{eq:7}\n   h_j= h_{ex}  -  \\Xint-_{\\pa \\Omega} \\, |\\Gamma(\\tilde x, x_j)| \\,\n   J(\\tilde x)  ds (\\tilde x)\\, \\mbox{ for any } x_j  \\in \\pa\n   \\Omega_{i,j}\\,, \\quad j=1,2\\,,\n\\end{equation}\nwhere $\\Gamma(x,x_0)$ is the portion of $\\partial\\Omega$ connecting $x_0$ and $x$ in\nthe positive trigonometric direction.  We assume that $h_{ex}$ and $J$  are such\nthat\n\\begin{equation}\n  \\label{eq:8}\nh_1h_2 <0 \\,,\n\\end{equation}\nand without any loss of generality we can assume $h_2>0$.  Let\n\\begin{equation}\n  \\label{eq:9}\nh=\\text{\\rm max}(|h_1|,|h_2|)\\,.\n\\end{equation}\nWe assume that\n\\begin{equation}\n\\label{hyph}\nh>1\n\\end{equation}\nand distinguish below between two\n  different cases\n\\begin{subequations}\n\\label{eq:10}\n  \\begin{equation}\n \\label{eq:5a}\n  1<h\\leq \\frac{1}{\\Theta_0} \\quad \\mbox{or}  \\quad \\frac{1}{\\Theta_0}<h\\,, \\tag{\\ref{eq:10}a,b}\n\\end{equation}\n\\end{subequations}\nwhere $\\Theta_0$ is given by \\eqref{eq:20} ($\\Theta_0\\sim 0.59$).\n\nIn \\cite{alhe14} we have established the  global existence of solutions for\n\\eqref{eq:1}, such that\n\\begin{align*} \\label{prop1coul}\n &\\psi_c\\in C([0,+\\infty);W^{1+\\alpha,2}(\\Omega,{\\mathbb{C}}))\\cap\n H^1_{\\text{\\rm loc}}([0,+\\infty);L^2(\\Omega,{\\mathbb{C}}))\\,, \\, \\forall \\alpha <1\\,, \\\\\n &A_c\\in C([0,+\\infty); W^{1,p}(\\Omega,{\\mathbb{R}}^2))\\cap\n H^1_{\\text{\\rm loc}}([0,+\\infty);L^2(\\Omega,{\\mathbb{R}}^2))\\,, \\forall p\\geq 1\\,, \\\\\n&\\phi_c\\in L^2_{\\text{\\rm loc}}([0,+\\infty); H^1(\\Omega))\\,.\n\\end{align*}\n\nWe next define, as in \\cite[Subsection 2.2]{alhe14}, (in a slightly\ndifferent manner as the definition here is $c$-independent)  the\nnormal fields. They are defined as the weak solution -- $(\\phi_n,A_n)\\in\nH^1(\\Omega)\\times H^1(\\Omega,{\\mathbb{R}}^2)$ -- of \\eqref{eq:5} and\n\\begin{subequations}\n\\label{eq:11}\n  \\begin{empheq}[left={\\empheqlbrace}]{alignat=2}\n    &\\, \\text{\\rm curl\\,}^2A_n +  \\nabla\\phi_n = 0 \\qquad &\\text{in } &\\Omega\\,, \\\\\n    & - \\frac{\\partial\\phi_n}{\\partial\\nu} = J   \\qquad &\\text{on } &\\partial\\Omega\\,, \\\\\n    & \\Xint-_{\\partial\\Omega}\\text{\\rm curl\\,} A_n\\,ds = h_{ex}\\,, && \\\\\n    &\\int_\\Omega \\phi_n \\,dx =0\\, . &&\n\\end{empheq}\n\\end{subequations}\nNote that $(0,\\kappa A_n,c\\kappa\\phi_n)$ is a steady-state solution of \\eqref{eq:1}.\nBy \\eqref{eq:11} we have (cf. \\cite{alhe14})\n\\begin{equation}\n\\label{eq:12}\n  \\begin{cases}\n    \\Delta B_n = 0 & \\text{in } \\Omega\\,, \\\\\n    \\frac{\\partial B_n}{\\partial\\tau} = J & \\text{on } \\partial\\Omega\\,, \\\\\n   \\Xint-_{\\partial\\Omega}B_n(x)\\,ds = h_{ex} \\,. &\n  \\end{cases}\n\\end{equation}\nBy taking the\ndivergence of (\\ref{eq:11}a) we also  obtain\n\\begin{subequations}\n\\label{eq:11b}\n \\begin{empheq}[left={\\empheqlbrace}]{alignat=2}\n    & \\; \\Delta\\phi_n=0 & \\qquad \\text{in } \\Omega\\,, \\\\\n     &- \\frac{\\partial\\phi_n}{\\partial\\nu} = J   & \\qquad\\text{on } \\partial\\Omega\\,, \\\\\n   &\\;\\int_\\Omega  \\phi_n \\,dx =0\\, . &\n  \\end{empheq}\n\\end{subequations}\nWe recall from \\cite[(2.16) and (2.17)] {alhe14} that\n\\begin{equation}\\label{reguphi}\n B_n\\in W^{2,p}(\\Omega)\\,,\\, \\phi_n \\in W^{2,p}(\\Omega) \\,,\\, \\forall p>1\\,.\n \\end{equation}\n\n We focus attention in this work on the exponential decay of $\\psi$ in\n regions where $|B_n|>1$. For steady-state solutions of \\eqref{eq:1}\n in the absence of electric current ($J=0$) we may set $\\phi\\equiv0$ and the\n magnetic field is then constant on the boundary. The exponential\n decay of $\\psi$ away from the boundary has been termed ``surface\n superconductivity'' and has extensively been studied (cf.\n \\cite{fohe09} and the references within). More recently, the case of\n a non-constant magnetic field has been studied as well\n \\cite{attar2014ground,heka15}. In these works $\\phi$ still identically\n vanishes but nevertheless $\\nabla B_n\\neq0$ in view of the presence of a\n current source term $\\text{\\rm curl\\,} h_{ex}$ in (\\ref{eq:1}b).  In particular\n in \\cite{heka15} it has been established, in the large $\\kappa$ limit for\n the case $1\\ll h\\ll\\kappa$ that $\\psi$ is exponentially small away from\n $B_n^{-1}(0)$.\n\n In the absence of electric current the time-dependent case is of\n lesser interest, since every solution of \\eqref{eq:1} converges to a\n steady-state solution  \\cite{lid97, feta01}. This result has been obtained in \\cite{feta01} by\n using the fact that the Ginzburg-Landau energy functional is a\n Lyapunov function in this case. In contrast, when $J\\neq0$ this\n property of the energy functional is lost, and convergence to a\n steady-state is no-longer guaranteed. In \\cite{alhe14} the global\n stability of the normal state has been established for $h=\\mathcal O(\\kappa)$\n in the large $\\kappa$ limit. In the present contribution, for the same\n limit, we explore the behavior of the solution for $1<h\\ll\\kappa$ , and\n establish exponential decay of $\\psi$ in every subdomain of $\\Omega$ where\n $|B_n|>1$. We do that for both steady-state solutions (whose\n existence we need to assume) and time-dependent ones. We also study\n the large-domain limit, where we obtain weaker results for\n steady-state solutions only.\n\n\nLet,  for $j=1,2$,\n    \\begin{equation}\n\\label{eq:13}\n    \\omega_{j}  = \\{x\\in \\Omega \\, : \\, (-1)^jB_n(x)>1 \\} \\,.\n  \\end{equation}\nOur first theorem concerns  steady state solutions and their\nexponential decay, in certain subdomains of $\\Omega$, in the large $\\kappa$ limit.\n\\begin{theorem}\n\\label{mainsteadystate}\n Let for $\\kappa \\geq 1$, $(\\psi_\\kappa,A_\\kappa,  \\phi_\\kappa)$ be a time-independent solution of \\eqref{eq:1}.\n  Suppose that for some $j\\in\\{1,2\\}$ we have\n  \\begin{equation}\n\\label{eq:14}\n    1<|h_j| \\,.\n  \\end{equation}\n  Then,  for any compact  set $K\\subset \\omega_j \\cup \\partial \\Omega_c $, there\n  exist $C>0$,  $\\alpha >0$, and\n  $\\kappa_0\\geq 1 $, such that for any  $\\kappa\\geq \\kappa_0$ we have\n\\begin{equation}\n\\label{eq:15}\n\\int_{K} |\\psi_\\kappa(x)|^2 \\,dx \\leq C  e^{- \\alpha \\kappa} \\,.\n\\end{equation}\nIf, in addition,\n\\begin{equation}\n    \\frac{1}{\\Theta_0}<|h_j| \\,,\n  \\end{equation}\n  then (\\ref{eq:15}) is satisfied for any compact subset $K\\subset\\overline{\\omega_j}$.\n\\end{theorem}\nIn addition to the above exponential decay, which is limited to the\nregion where the normal magnetic field is large, we establish a weaker\ndecay of $\\psi_\\kappa$ in the entire domain.\n\\begin{proposition}\n\\label{entire}\n  Under the assumptions \\eqref{eq:2}-\\eqref{eq:10} there exists\n  $C(J,\\Omega)>0$ such that, for $\\kappa \\geq 1$,\n  \\begin{equation}\n\\label{eq:16}\n\\| \\psi_\\kappa \\|_2 \\leq C(J,\\Omega) \\, (1+c^{-1\/2})^{1\/3}\\kappa^{-1\/6}  \\,.\n  \\end{equation}\n\\end{proposition}\nTheorem \\ref{mainsteadystate} is extended to the time dependent case\nin the following way.\n\\begin{theorem}\n  \\label{main}\n  Let $(\\psi_\\kappa,A_\\kappa, \\phi_\\kappa)$ denote a time-dependent solution of\n  \\eqref{eq:1}. Assuming $c=1$, under the conditions of Theorem\n  \\ref{mainsteadystate} and \\eqref{condinit}, for any compact set $K\\subset\n  \\omega_j\\cup \\partial \\Omega_c$ there exist $C>0$ , $\\alpha >0$, and $\\kappa_0\\geq 1$, such\n  that for any $\\kappa\\geq \\kappa_0$ we have\n\\begin{equation}\n\\label{eq:17}\n\\limsup_{t\\to\\infty}\\int_{K}  |\\psi_\\kappa(t,x)|^2 \\,dx \\leq C   e^{- \\alpha \\kappa}  \\,.\n\\end{equation}\n\\end{theorem}\n\n\nFinally, we consider steady-state solutions of (\\ref{eq:1}) in the\nlarge domain limit, i.e., we set $\\kappa=c=1$ and stretch $\\Omega$ by a factor\nof $R \\gg 1 $.  Let $\\Omega^R$ be the image of $\\Omega$ under the map $x\\to Rx$.\nWe consider again steady-state solutions of \\eqref{eq:1}.\n\\begin{subequations}\n\\label{eq:18}\n\\begin{alignat}{2}\n& \\Delta_A\\psi + \\psi   \\left( 1 - |\\psi|^{2} \\right)-i\\phi\\psi =0 & \\quad \\text{ in } \\Omega^R\\, ,\\\\\n & \\text{\\rm curl\\,}^2A +  \\nabla\\phi  =  \\Im(\\bar\\psi\\, \\nabla_A\\psi) & \\quad \\text{ in }  \\Omega^R\\,,\\\\\n  &\\psi=0 &\\quad \\text{ on }  \\partial\\Omega_c^R\\,, \\\\\n &\\nabla_A\\psi\\cdot\\nu=0 & \\quad \\text{ on }  \\partial\\Omega_i^R\\,,\\\\\n & \\frac{\\partial\\phi}{\\partial\\nu}= \\frac{F(R)}{R}J &\\quad \\text{ on } \\partial\\Omega_c^R\\,,\\\\\n& \\frac{\\partial\\phi}{\\partial\\nu}= 0 &\\quad \\text{ on } \\partial\\Omega_i^R\\,,\\\\[1.2ex]\n&\\Xint-_{\\partial\\Omega^R}\\text{\\rm curl\\,} A(x)\\,ds = F(R)h_{ex}\\,.\n\\end{alignat}\n\\end{subequations}\nIn the above $F(R)=R^{\\gamma}$ for some $0<\\gamma<1$. We study the above\nproblem in the limit $R\\to\\infty$. Assuming again (\\ref{eq:2})-(\\ref{eq:10})\nwe establish the following result.\n\\begin{proposition}\n\\label{largedomain}\n Let $(\\psi,A,\\phi)$ denote a solution of \\eqref{eq:18}.\n Then,  there exists a compact  set $K\\subset\\Omega$, $C>0$, $R_0>0$, and $\\alpha >0$,\nsuch that for any  $R>R_0\\,$ we have\n\\begin{equation}\n\\label{eq:126a}\n\\int_{K_R} |\\psi (x)|^2 \\,dx \\leq C  e^{-\\alpha R} \\,,\n\\end{equation}\nwhere $K_R$ is the image of $K$ under the map $x\\to Rx$.\n\\end{proposition}\nNote that $h_{ex}$ must be of $\\mathcal O(J)$, otherwise $(0,A_n,\\phi_n)$ would\nbe the unique solution.\n\n\nPhysically \\eqref{eq:126a} demonstrates that there is a\nsignificant portion of the superconducting sample which remains,\npractically, at the normal state, for current densities which may be\nvery small. This result stands in\ncontrast with what one finds in standard physics handbooks\n\\cite{poetal99} where the critical current density, for which the\nfully superconducting state looses its stability, is tabulated a\nmaterial property. However, our results suggest that the critical\ncurrent depends also on the geometry of the superconducting sample. In\nfact, according to Proposition \\ref{largedomain}, this current density\nmust decay in the large domain limit. In two-dimensions, our result\nsuggests that one should search for a critical current (and not\ncurrent density), whereas in three-dimensions a density with respect\nto cross-section circumference (instead of area) should be obtained.\n\nWe note that Proposition \\ref{largedomain} is certainly not\noptimal. In fact, we expect the following conjecture to be true.\n\\begin{conjecture}\n  \\label{conj}\nUnder the  conditions of Proposition \\ref{largedomain}, for any\ncompact  set $K\\subset\\Omega\\setminus B_n^{-1}(0)$, there exist $R_0>0$, $C>0$, and $\\alpha >0$,\nsuch that for any  $R>R_0$ \\eqref{eq:126a} is satisfied.\n\\end{conjecture}\n\n\nThe rest of this contribution is organized as follows.  In the next\nsection we establish some preliminary results related to the\neigenvalues of the magnetic Laplacian in the presence of\nDirichlet-Neumann corners. We use these results in Section 3 where we\nestablish Theorem \\ref{mainsteadystate} and Proposition~\\ref{entire}.\nIn Section 4 we consider the time-dependent problem and establish, in\nparticular, Theorem~\\ref{main}. Finally, in the last section, we\nobtain some weaker results for steady-state solutions of (\\ref{eq:1})\nin the large domain limit.\n\n\\section{Magnetic Laplacian Ground States}\n\\label{sec:2}\nIn this section, we analyze the spectral properties of the Schr\\\"odinger\noperator with constant magnetic field in a sector. The Neumann problem\nhas been addressed by V. Bonnaillie-No\\\"el in \\cite{Bon}. In the sequel\nwe shall need, however, a lower bound for the ground state energy of\nthe above operator on a Dirichlet-Neumann sector, i.e., a Dirichlet\ncondition is prescribed on one side of the sector and the magnetic\nNeumann condition on the other side. We begin by the following\nauxiliary lemma whose main idea has been introduced to us by M.~Dauge\n\\cite{da14}.  Hereafter the norms in the Lebesgue spaces $L^p(\\Omega)$,\n$L^p(\\Omega,\\mathbb  R^2)$ and $L^p(\\Omega,\\mathbb  C)$ will be denoted by\n$\\|\\cdot\\|_{L^p(\\Omega)}$ or $\\|\\cdot\\|_p$, and the norms in the Sobolev spaces\n$W^{k,p}(\\Omega,\\mathbb R)$, $W^{k,p}(\\Omega,\\mathbb  R^2)$ and $W^{k,p}(\\Omega,\\mathbb  C)$\nwill be denoted by $\\|\\cdot\\|_{W^{k,p}(\\Omega)}$ or $\\|\\cdot\\|_{k,p}$.\n \\begin{lemma}\n\\label{lem:Dirichlet-Neumann}\n Let $S_\\alpha$ denote an infinite sector of angle $\\alpha \\in (0,\\pi]$, i.e.,\n \\begin{displaymath}\n   S_\\alpha = \\{ (x,y)\\in{\\mathbb{R}}^2 \\,: \\, 0<\\arg(x+iy)<\\alpha \\}\\,.\n \\end{displaymath}\nLet further\n \\begin{displaymath}\n   {\\mathcal H}_\\alpha = \\{ u\\in H^1(S_\\alpha) \\,: \\, u(r\\cos\\alpha,r\\sin\\alpha)=0\\,, \\; \\forall r>0\\}\\,,\n \\end{displaymath}\nand\n\\begin{displaymath}\n  \\Theta^{DN}_\\alpha  = \\inf_{\\begin{subarray}{c}\n      u\\in {\\mathcal H}_\\alpha \\\\\n      \\|u\\|_2 =1\n    \\end{subarray}}\n\\int_{S_\\alpha} |(\\nabla-iF) u|^2 \\,dx \\,,\n\\end{displaymath}\nwhere $F$ is a magnetic potential satisfying $\\text{\\rm curl\\,} F = 1$ in\n$S_\\alpha$. Then,\n \\begin{equation}\n   \\label{eq:19}\n  \\Theta^{DN}_\\alpha =\\Theta_0\\,,\n \\end{equation}\nwhere\n  \\begin{equation}\n\\label{eq:20}\n  \\Theta_0  = \\inf_{\\begin{subarray}{c}\n u\\in H^1(S_\\pi)  \\\\\n      \\|u\\|_2 =1\n    \\end{subarray}}\n\\int_{S_\\pi} |(\\nabla-iF)u|^2 \\,dx \\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n  Let $0<\\alpha_1<\\alpha_2\\leq\\pi$ and $u\\in{\\mathcal H}_{\\alpha_1}$. Let further\n  $\\tilde{u}\\equiv u$ in $S_{\\alpha_1}$ and $\\tilde{u}\\equiv0$ in\n  $S_{\\alpha_2}\\setminus S_{\\alpha_1}$. Clearly $\\tilde{u}\\in{\\mathcal H}_{\\alpha_2}$, and hence it\n  follows that $\\Theta^{DN}_{\\alpha_1}\\geq\\Theta^{DN}_{\\alpha_2}$. Consequently,\n  \\begin{displaymath}\n      \\Theta^{DN}_\\alpha \\geq \\Theta^{DN}_\\pi\\,, \\quad \\forall\\alpha\\leq\\pi.\n  \\end{displaymath}\n  From the definition of $\\Theta_0$ it follows, however, that\n  $\\Theta^{DN}_\\pi\\geq\\Theta_0$, and hence, $ \\Theta^{DN}_\\alpha \\geq \\Theta_0$ for all\n  $\\alpha\\in(0,\\pi]$. The proof of \\eqref{eq:19} now easily follows \n    from the proof of Persson's Theorem \\cite[Appendix B]{fohe09}\n    providing the upper bound $\\Theta_0$ for the essential spectrum of the\n    magnetic Dirichlet-Neumann Laplacian in $S_\\alpha$.\n\\end{proof}\n\nLet ${\\mathcal D}\\subset\\Omega$ have a smooth boundary, except at the corners of $\\partial\\Omega$. As\nin \\cite{alhe14} we let\n\\begin{equation}\n  \\label{eq:21}\n    \\mu_\\epsilon(\\mathcal A,{\\mathcal D}) = \\inf_{\n    \\begin{subarray}{c}\n      u\\in {\\mathcal H}({\\mathcal D}) \\\\\n      \\|u\\|_2 =1\n    \\end{subarray}} \\int_{\\mathcal D} |\\epsilon\\nabla-i\\mathcal A u|^2 \\,dx \\,,\n\\end{equation}\nwherein\n\\begin{displaymath}\n   {\\mathcal H}({\\mathcal D}) = \\{ u\\in H^1({\\mathcal D})\\,:\\, u=0 \\text{ on } \\partial{\\mathcal D}\\setminus (\\partial{\\mathcal D}\\cap \\partial\\Omega_i)\\,\\} \\,.\n\\end{displaymath}\nLet\n\\begin{displaymath}\n  {\\mathcal S} = \\overline{\\partial\\Omega_c}\\cap\\overline{\\partial\\Omega_i}\\cap\\overline{\\partial{\\mathcal D}}\n\\end{displaymath}\ndenote the corners of $\\Omega$ belonging to  $\\partial{\\mathcal D}$.  Following \\cite{boda06}\nwe set for a given magnetic potential $\\mathcal A \\in C^1(\\overline{\\Omega},\\mathbb  R^2)$,\n\\begin{displaymath}\n  b=\\inf_{x\\in{\\mathcal D}}|\\text{\\rm curl\\,} \\mathcal A| \\quad ; \\quad b^\\prime = \\inf_{x\\in\\partial{\\mathcal D}\\cap\\partial\\Omega_i}|\\text{\\rm curl\\,} \\mathcal A|\\,.\n\\end{displaymath}\nThe following proposition is similar to a result in \\cite{boda06}\nobtained for a Neumann boundary condition. Here we treat a\nDirichlet-Neumann boundary condition and allow, in addition, some\ndependence of the magnetic potential on the semi-classical parameter.\n\\begin{proposition}\n  Let $a\\in W^{1,\\infty}({\\mathcal D},\\mathbb  R^2)$. There exist $C>0$ and $\\epsilon_0>0$ such that for all\n$0<\\epsilon<\\epsilon_0$ we have\n\\begin{equation}\n  \\label{eq:22}\n \\mu_\\epsilon(\\mathcal A+\\epsilon^{1\/2}a,{\\mathcal D}) \\geq \\epsilon \\min(b,\\Theta_0b^\\prime) [1-C(1+\\|\\nabla a\\|_\\infty^2)\\epsilon^{1\/3}]\\,.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\n  The case $b=0$ being trivial, we assume that $b>0$.  We begin by\n  introducing for any $\\epsilon >0$ a partition of unity (cf. also\n  \\cite{hemo96}), i.e., families $\\{\\eta_i\\}_{i=1}^K\\subset C^\\infty(\\Omega)$, and\n  $\\{x_i\\}_{i=1}^K\\subset{\\mathcal D}$ satisfying\n\\begin{displaymath}\n  \\sum_{i=1}^K\\eta_i^2=1 \\quad ; \\quad {\\rm supp}\\,\\eta_i \\subset B(x_i,\\epsilon^{1\/3})  \\quad\n  ; \\quad \\sum_{i=1}^K|\\nabla\\eta_i|^2 \\leq \\frac{C}{\\epsilon^{2\/3}} \\,,\n\\end{displaymath}\nwhere $C>0$ is independent of $\\epsilon$.\n\nIt can be easily verified that for any $u\\in H^1(\\Omega,{\\mathbb{C}})$\n\\begin{equation}\n\\label{eq:23}\n  \\|(\\epsilon\\nabla-i[\\mathcal A+\\epsilon^{1\/2}a])u\\|_2^2=\\sum_{i=1}^K\\big[\n  \\|(\\epsilon\\nabla-i[\\mathcal A+\\epsilon^{1\/2}a])(\\eta_iu)\\|_2^2 - \\epsilon^2\\|u\\nabla\\eta_i\\|_2^2 \\big]\\,.\n\\end{equation}\nWe now set\n\\begin{displaymath}\n  v_i=\\eta_iu\\exp(-i\\epsilon^{-1\/2}a(x_i)\\cdot x) \\,,\n\\end{displaymath}\nto obtain that\n$$\\aligned\n& \\|(\\epsilon\\nabla-i[\\mathcal A+\\epsilon^{1\/2}a])(\\eta_iu)\\|_2^2\\\\\n=&\\|(\\epsilon\\nabla-i[\\mathcal A+\\epsilon^{1\/2}(a-a(x_i))])v_i\\|_2^2\\\\\n \\geq&\n(1-\\epsilon^{1\/3})\\|(\\epsilon\\nabla-i\\mathcal A)v_i\\|_2^2 - \\epsilon^{2\/3}\\|a-a(x_i))v_i\\|_2^2\\\\\n \\geq&\n(1-\\epsilon^{1\/3})\\mu_\\epsilon(\\mathcal A,{\\mathcal D}) \\|v_i\\|_2^2  - C\\epsilon^{4\/3}\\|v_i\\|_2^2\\,.\n\\endaligned\n$$\nSubstituting the above into \\eqref{eq:23} yields\n\\begin{equation}\n\\label{eq:24}\n   \\|(\\epsilon\\nabla-i[\\mathcal A+\\epsilon^{1\/2}a])u\\|_2^2 \\geq (1-\\epsilon^{1\/3})\\mu_\\epsilon(\\mathcal A,{\\mathcal D}) \\|u\\|_2^2  - C\\epsilon^{4\/3}(1+\\|\\nabla a\\|_\\infty^2)\\|u\\|_2^2\\,.\n\\end{equation}\nBy following the same steps of the proof  of Theorem 7.1 in \\cite{boda06}\nwe can establish that\n\\begin{equation}\\label{eq:74a}\n \\mu_\\epsilon(\\mathcal A,{\\mathcal D}) \\geq   \\epsilon \\min(b,\\Theta_0b^\\prime,\\Theta^{DN}_\\alpha\\inf_{x\\in{\\mathcal S}}|\\text{\\rm curl\\,} \\mathcal A| )\n (1-C\\epsilon^{1\/2})\\,.\n\\end{equation}\nThe lemma now follows from \\eqref{eq:19}, \\eqref{eq:24}, and\n\\eqref{eq:74a}.\n\\end{proof}\n\n\n\n\n\\section{Steady-State Solutions}\\label{sec:3}\n\nWe begin by considering steady-state solutions of \\eqref{eq:1}\n$(\\psi_k,A_\\kappa,\\phi_\\kappa)\\in H^1(\\Omega,{\\mathbb{C}}\\times{\\mathbb{R}}^2\\times{\\mathbb{R}})$ in\nthe limit $\\kappa\\to + \\infty$. Hence, we look at the\nsystem \\cite[Section 5]{alhe14}\n\\begin{subequations}\n\\label{eq:25}\n\\begin{alignat}{2}\n  - & \\nabla_{\\kappa A_\\kappa}^2\\psi_\\kappa+i\\kappa\\phi_\\kappa \\psi_\\kappa =\\kappa^2(1-|\\psi_\\kappa|^2)\\psi_\\kappa & \\quad \\text{ in } \\Omega \\,,\\\\\n  - &\\text{\\rm curl\\,}^2A_\\kappa+ \\frac{1}{c}\\nabla\\phi_\\kappa  =\\frac{1}{\\kappa}\\Im(\\bar\\psi_\\kappa\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa) &\n  \\quad \\text{ in }  \\Omega\\,, \\\\\n  &\\psi_\\kappa =0 &\\quad \\text{ on }  \\partial\\Omega_c\\,, \\\\\n  &\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\cdot\\nu=0 & \\quad \\text{ on }  \\partial\\Omega_i\\,, \\\\\n  & \\frac{\\partial\\phi_\\kappa}{\\partial\\nu} = - c\\kappa J(x) &\\quad \\text{ on } \\partial\\Omega_c\\,, \\\\\n  &\\frac{\\partial\\phi_\\kappa}{\\partial\\nu}=0  &\\quad \\text{ on }  \\partial\\Omega_i \\,, \\\\[1.2ex]\n  &\\Xint-_{\\partial\\Omega}\\text{\\rm curl\\,} A_\\kappa \\, ds = \\kappa h_{ex} \\,,&\n\\end{alignat}\n\\end{subequations}\nwith the additional gauge restriction \\eqref{eq:5}.  In the above\n$(\\psi_k,A_\\kappa,\\phi_\\kappa)$ is the same as $(\\psi,A,\\phi)$ in \\eqref{eq:1}. The\nsubscript $\\kappa$ has been added to emphasize the limit we consider\nhere. We assume in addition (\\ref{eq:2})-(\\ref{eq:10}).  By the strong\nmaximum principle we easily obtain that\n\\begin{equation}\n\\label{eq:26}\n  \\|\\psi_\\kappa\\|_\\infty <1 \\,.\n\\end{equation}\n\n\n\nLet $h$ be given by\n\\eqref{eq:9}. It has been demonstrated in \\cite{alhe14} that for\nsome $h_c>0$, when $h<h_c\\kappa$, the normal state looses its stability.\nSince we consider cases for which $1<h\\ll\\kappa$ it is reasonable to expect\nthat other steady-state solutions would exist. We note, however, that\nin contrast with the case $J=0$, where the existence of steady-state\nsolutions can be proved using variational arguments (inapplicable in\nour case), existence of steady-state solutions to \\eqref{eq:25} is yet an open problem\nwhen an electric current is applied. We shall address time-dependent\nsolutions in the next section.\n\n\n\nNext we set\n\\begin{subequations}\n\\label{eq:27}\n  \\begin{equation}\n\\label{eq:28}\n A_{1,\\kappa}=A_\\kappa-\\kappa A_n\\,,\\quad\n\\phi_{1,\\kappa}=\\phi_\\kappa-c\\kappa\\phi_n\\,.\\tag{\\ref{eq:27}a,b}\n\\end{equation}\n\\end{subequations}\nSet further\n\\begin{subequations}\n\\label{eq:29}\n  \\begin{equation}\n\\label{eq:30}\nB_\\kappa =\\text{\\rm curl\\,} A_\\kappa\\,,\\quad\nB_{1,\\kappa}=\\text{\\rm curl\\,} A_{1,\\kappa} \\,. \\tag{\\ref{eq:29}a,b}\n\\end{equation}\n\\end{subequations}\nBy\n(\\ref{eq:25}b) we then have\n\\begin{subequations}\n\\label{eq:31}\n  \\begin{empheq}[left={\\empheqlbrace}]{alignat=2}\n  &\\text{\\rm curl\\,} B_{1,\\kappa} +  \\frac{1}{c}\\nabla\\phi_{1,\\kappa}  =  \\frac{1}{\\kappa}\\Im(\\bar\\psi_\\kappa\\, \\nabla_{\\kappa A_\\kappa}\\psi_\\kappa)  &  \\text{ in }\n  \\Omega\\,,\\\\\n   &\\frac{\\partial\\phi_{1,\\kappa}}{\\partial\\nu}= 0 & \\text{ on } \\partial\\Omega\\,, \\\\\n&\\Xint-_{\\partial\\Omega}B_{1,\\kappa}(x)\\,ds = 0 \\,. &\\label{eq:42c}\n\\end{empheq}\n\\end{subequations}\nNote that since $\\partial B_{1,\\kappa}\/\\partial\\tau=\\partial\\phi_{1,\\kappa}\/\\partial\\nu=0$ on $\\partial\\Omega$\nwe must have by (\\ref{eq:31}c) that\n\\begin{equation}\n\\label{eq:32}\n  B_{1,\\kappa}|_{\\partial\\Omega}\\equiv 0\\,.\n\\end{equation}\nTaking the divergence of (\\ref{eq:25}b) yields, with the aid of the\nimaginary part of (\\ref{eq:25}a),  that $\\phi_\\kappa $ is a weak\nsolution of\n\\begin{equation}\n  \\label{eq:33}\n  \\begin{cases}\n    -\\Delta\\phi_\\kappa + c \\,|\\psi_\\kappa|^2 \\phi_\\kappa =0 & \\text{ in } \\Omega\\,, \\\\\n  \\frac{\\partial\\phi_\\kappa}{\\partial\\nu}= c\\kappa J &  \\text{ on } \\partial\\Omega_c\\,,\\\\\n \\frac{\\partial\\phi_\\kappa}{\\partial\\nu}= 0 &  \\text{ on } \\partial\\Omega_i\\,.\n  \\end{cases}\n\\end{equation}\nBy assumption $\\phi_\\kappa\\in H^1(\\Omega)$ and hence, by \\cite[Proposition\nA.2]{alhe14} we obtain that $\\phi_\\kappa\\in W^{2,p}(\\Omega)$ for all $p\\geq2$, hence $\\phi_\\kappa \\in C^1(\\overline{\\Omega})$.  By\n(\\ref{eq:27}b) we then have\n\\begin{displaymath}\n  \\begin{cases}\n    -\\Delta\\phi_{1,\\kappa} + c\\,  |\\psi_\\kappa|^2 \\phi_{1,\\kappa} =- \\kappa c^2|\\psi_k|^2 \\phi_n & \\text{ in } \\Omega\\,, \\\\\n  \\frac{\\partial\\phi_{1,\\kappa}}{\\partial\\nu}= 0 &  \\text{ on } \\partial\\Omega \\,.\n  \\end{cases}\n\\end{displaymath}\nLet $K=\\|\\phi_n\\|_\\infty$ and $w=\\phi_{1,\\kappa} +K\\kappa c$.  Clearly,\n\\begin{displaymath}\n  \\begin{cases}\n    -\\Delta w+ c\\, |\\psi_\\kappa|^2  w =- \\kappa c^2|\\psi_k|^2(\\phi_n-K)\\geq 0 & \\text{ in } \\Omega\\,, \\\\\n  \\frac{\\partial w}{\\partial\\nu}= 0 &  \\text{ on } \\partial\\Omega \\,.\n  \\end{cases}\n\\end{displaymath}\nIt can be easily verified that $w$ is the minimizer in $H^1(\\Omega)$ of\n\\begin{displaymath}\n  {\\mathcal J}(v) = \\|\\nabla v\\|_2^2 + c\\|\\psi_\\kappa v\\|_2^2+\\kappa c^2\\langle|\\psi_k|^2(\\phi_n-K),v\\rangle \\,.\n\\end{displaymath}\nAs\n\\begin{displaymath}\n   {\\mathcal J}(v_+) \\leq    {\\mathcal J}(v)\\,,\n\\end{displaymath}\nit easily follows that $ w\\geq 0$, that is\n\\begin{displaymath}\n  \\phi_{1,\\kappa}+K\\kappa c \\geq 0 \\,.\n\\end{displaymath}\nIn a similar manner we obtain\n\\begin{displaymath}\n  \\phi_{1,\\kappa}-K\\kappa c \\leq 0 \\,,\n\\end{displaymath}\nwhich together with (\\ref{eq:27}b) yields\n\\begin{equation}\n\\label{eq:34}\n  \\|\\phi_\\kappa\\|_\\infty  \\leq C(\\Omega,J)\\, c\\, \\kappa \\,.\n\\end{equation}\nWe now apply again Proposition A.2 in \\cite{alhe14} to obtain that for\nany $p\\geq2$\n\\begin{equation}\\label{estlp}\n  \\|\\phi_\\kappa\\|_{2,p} \\leq C(\\Omega,J)\\, c\\, \\kappa \\,.\n\\end{equation}\nWe note that all elliptic estimates must be taken with special care\nsince $\\Omega$ possesses corners.  The necessary details (with references\ntherein) can be found in Appendices A and B of \\cite{alhe14} .\n\nNext we set for $\\delta >0$  and $\\kappa\\geq 1$,\n\\begin{displaymath}\n  D_\\delta(\\kappa) = \\{ x\\in\\Omega \\,: \\,|B_\\kappa(x)|>(1+\\delta)\\kappa \\} \\,,\n\\end{displaymath}\nand\n\\begin{equation}\n\\label{eq:35}\n   S_\\delta = \\{ x\\in\\Omega \\,: \\,|B_n(x)|>(1+\\delta)\\}\\,.\n\\end{equation}\nBy either (\\ref{eq:10}a) or (\\ref{eq:10}b), it follows that for $0<\\delta <\nh-1$, $S_\\delta\\neq\\emptyset\\,$. Below we show that the same is true for $ D_\\delta(\\kappa)$.\nNote that \\eqref{eq:8} implies that $S_\\delta$ consists of two disjoint\nsets:\n\\begin{equation}\\label{sdj}\nS_{\\delta} = S_{\\delta,1} \\cup S_{\\delta,2}\\,,\n\\end{equation}\none near $\\partial\\Omega_{i,1}$ denoted by $S_{\\delta,1}$, and one\nnear $\\partial\\Omega_{i,2}$ denoted by $S_{\\delta,2}\\,$. \\\\\nWe then let\n\\begin{equation}\n  \\label{eq:36}\n\\mathcal C_{\\delta,j}=\\partial S_{\\delta,j}\\setminus (\\partial\\Omega\\cap\\partial S_{\\delta,j})\\,,\\quad j=1,2\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{Cdi}\n \\mathcal C_\\delta=\\mathcal C_{\\delta,1}\\cup \\mathcal C_{\\delta,2} \\,.\n\\end{equation}\n\nWe can now state and prove\n\\begin{lemma}\\label{lemma3.1}\n  For any $0<\\alpha<1$ there exists $ \\kappa_0=\\kappa_0(\\Omega,J,\\alpha)>0$ such that for\n  all $\\kappa \\geq \\kappa_0$ and $0<\\delta<h-1$, we have\n  \\begin{equation}\n \\label{eq:37}\nS_{\\delta+\\kappa^{-\\alpha}} \\subset  D_\\delta(\\kappa) \\,.\n  \\end{equation}\n\\end{lemma}\n\\begin{proof} {\\it Step 1: Prove that\nfor some $C(\\Omega,J)>0$\n\\begin{equation}\n\\label{eq:43}\n\\|A_\\kappa\\|_\\infty \\leq C\\, \\kappa  \\,.\n\\end{equation}}\n\nTaking the divergence of (\\ref{eq:31}a)  yields, with the aid of  (\\ref{eq:31}b),\n\\begin{equation}\n\\label{eq:38}\n  \\begin{cases}\n    - \\Delta\\phi_{1,\\kappa}   = - \\frac{c}{\\kappa}\\, \\text{\\rm div\\,} \\Im(\\bar\\psi_\\kappa\\, \\nabla_{\\kappa A_\\kappa\n    }\\psi_\\kappa) & \\text{ in } \\Omega\\,, \\\\\n     \\frac{\\partial\\phi_{1,\\kappa}}{\\partial\\nu}= 0 & \\text{ on } \\partial\\Omega \\,.\n  \\end{cases}\n\\end{equation}\nMultiplying the above equation by $\\phi_{1,\\kappa }$ and integrating by parts\nthen yields, with the aid of \\eqref{eq:26} and (\\ref{eq:25}c,d),\n\\begin{equation}\n\\label{eq:39}\n     \\|\\nabla \\phi_{1,\\kappa} \\|_2 \\leq \\frac{c}{\\kappa}\\,  \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_2 \\,.\n \\end{equation}\nTaking the inner product of (\\ref{eq:25}a) with $\\psi_\\kappa$ yields, after\nintegration by parts\n\\begin{equation}\n\\label{eq:40}\n   \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_2^2 = \\kappa^2\\, \\|\\psi_\\kappa\\|_2^2 \\,.\n\\end{equation}\nBy \\eqref{eq:39} we then obtain that\n\\begin{displaymath}\n\\|\\nabla\\phi_{1,\\kappa}\\|_2 \\leq Cc\\,.\n\\end{displaymath}\nSince $\\text{\\rm curl\\,} B_{1,\\kappa}=\\nabla_\\perp B_{1,\\kappa}\\,$, the boundedness of $ \\|\\nabla\nB_{1,\\kappa}\\|_2$ then easily follows from the above and\n\\eqref{eq:31}. Consequently,\n  \\begin{equation}\n\\label{eq:41}\n    \\frac{1}{c}\\|\\nabla \\phi_{1,\\kappa} \\|_2+ \\|\\nabla B_{1,\\kappa}\\|_2 \\leq C\\,.\n\\end{equation}\nNote that $\\nabla\\phi_{1,\\kappa}$ and $\\nabla_\\perp B_{1,\\kappa}$ are respectively the  $L^2$ projections\nof $\\Im(\\bar\\psi_\\kappa\\, \\nabla_{\\kappa A_\\kappa}\\psi_\\kappa)$ on\n\\begin{displaymath}\n   H^0_0(\\text{\\rm curl\\,}, \\Omega)=\\{\\widehat V \\in L^2(\\Omega,\\mathbb R^2)\\,:\\, \\text{\\rm curl\\,} \\widehat V=0\\}\\,,\n\\end{displaymath}\nand\n\\begin{displaymath}\n  \\mathcal H^0_d:= \\{\\widehat W \\in L^2(\\Omega,\\mathbb R^2)\\,:\\, \\text{\\rm div\\,} \\widehat\n  W=0\\mbox{ and } \\widehat W \\cdot \\nu =0\\mbox{ on } \\pa \\Omega \\}\\,.\n\\end{displaymath}\n\nNext, we attempt to estimate $\\|\\nabla \\phi_{1,\\kappa} \\|_p$ and $\\|\\nabla B_{1,\\kappa}\\|_p$ for\nany \\linebreak $p>2$. Since $\\Omega$ is simply-connected, we may conclude\nfrom \\eqref{eq:5}, (\\ref{eq:29}a) and Remark B.2 in \\cite{alhe14} that\nthere exists for any $p>2$ a constant \\linebreak $C(p,\\Omega)>0$ such that\n\\begin{displaymath}\n  \\|A_\\kappa\\|_{1,p} \\leq C\\, \\|B_\\kappa\\|_p \\,,\n\\end{displaymath}\nfor all $\\kappa\\geq 1$.  Sobolev embeddings then imply\n\\begin{equation}\n  \\label{eq:42}\n\\|A_\\kappa\\|_\\infty \\leq C\\,  \\|B_\\kappa\\|_p\\,.\n\\end{equation}\nSince $ \\|\\nabla B_{1,\\kappa}\\|_2$ is uniformly bounded for all $\\kappa \\geq 1$, we\nobtain from \\eqref{eq:32}, the Poincar\\'e inequality, and Sobolev embeddings\nthat, for any $p>2$ there exists  a constant  $C(p,\\Omega)>0$ such that we have\n\\begin{equation}\\label{estB1p}\n \\|B_{1,\\kappa}\\|_p \\leq  C(p,\\Omega)\\,.\n\\end{equation}\n\nHence, recalling from \\eqref{reguphi} that $B_n\\in L^p$ and independent\nof $c$ and $\\kappa$, as $J$ is independent of $\\kappa$, there\nexists a constant $C>0$ such that\n\\begin{equation}\\label{estBp}\n  \\|B_\\kappa\\|_p=\\|B_{1,\\kappa}+\\kappa B_n\\|_p\\leq C\\, \\kappa\\,.\n\\end{equation}\nCombining the above computations with \\eqref{eq:42} then yields\n\\eqref{eq:43}.\n\\vspace{1ex}\n\n\n\n{\\it Step 2: Prove \\eqref{eq:37}.}\n\\vspace{1ex}\n\nWe first rewrite (\\ref{eq:25}a,c,d)  in the following form\n\\begin{displaymath}\n  \\begin{cases}\n\\Delta\\psi_\\kappa   = 2i\\kappa A_\\kappa\\cdot\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa  +  |\\kappa A_\\kappa|^2\\psi_\\kappa- \\kappa^2\\psi_\\kappa\\big( 1 - |\\psi_\\kappa|^{2}\n\\big)+i\\kappa\\phi_\\kappa\\psi_\\kappa  & \\text{ in } \\Omega\\, ,\\\\\n  \\psi_\\kappa =0  & \\text{on }  \\partial\\Omega_c \\,, \\\\\n\\frac{\\partial\\psi_\\kappa}{\\partial\\nu}=i\\kappa  (A_\\kappa \\cdot\\nu) \\psi_\\kappa=0\\, &  \\text{ on }  \\partial\\Omega_i \\,,\\\\\n  \\end{cases}\n\\end{displaymath}\nwhere the last equality follows from (\\ref{eq:5}).  By \\eqref{eq:43},\n\\eqref{eq:34}, the fact $\\|\\psi_\\kappa\\|_\\infty\\leq 1$, Proposition~A.3 and\nRemark~A.4 in \\cite{alhe14} (note that $\\psi_\\kappa$ vanishes at the\ncorners) we obtain that for some $C(\\Omega,p,J)$\n\\begin{displaymath}\n  \\|\\psi_\\kappa\\|_{2,p} \\leq C\\big[\\kappa^4 + \\kappa^2\\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\big]\\,,  \\quad \\forall p>2\\,.\n\\end{displaymath}\nSobolev embedding and \\eqref{eq:43} then yield\n\\begin{equation}\n\\label{eq:44}\n  \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_\\infty \\leq C\\big[\\kappa^4 +\n  \\kappa^2\\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\big] \\,.\n\\end{equation}\nWe now use a standard interpolation theorem to obtain that\n\\begin{displaymath}\n  \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\leq \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_2^{2\/p} \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_\\infty^{1-2\/p}\\,.\n\\end{displaymath}\n\n\n\n\nSubstituting \\eqref{eq:44} in conjunction with \\eqref{eq:40} into the\nabove inequality then yields\n\\begin{equation}\n\\label{eq:45}\n   \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\leq C\\Big[\\kappa^4 + \\kappa^2\\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\Big]^{1-2\/p}\\kappa^{2\/p}\\,.\n\\end{equation}\nSuppose first that\n\\begin{displaymath}\n \\kappa^2< \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p \\,.\n\\end{displaymath}\nThen, we have\n\\begin{displaymath}\n  \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p \\leq C\\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p^{1-2\/p}\\kappa^{2(1-1\/p)}\\,.\n\\end{displaymath}\nHence,\n\\begin{equation}\n\\label{eq:46}\n  \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p \\leq C\\kappa^{p-1} \\,.\n\\end{equation}\nNext, assume that\n\\begin{displaymath}\n  \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\leq \\kappa^2 \\,,\n\\end{displaymath}\nto obtain that\n\\begin{displaymath}\n   \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\leq C\\kappa^{4-6\/p} \\,.\n\\end{displaymath}\nFrom the above, together with \\eqref{eq:46} we easily conclude that,\nfor any $2<p\\leq3$, there exists a constant $C$ such that\n\\begin{equation}\n\\label{eq:47}\n    \\frac{1}{\\kappa}\\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\leq C\\kappa^{3(1-2\/p)}\\,.\n  \\end{equation}\n\n  To continue, we need a $W^{1,p}$ estimates for the solution of $\\Delta\n  u=f$ where $f\\in W^{-1,p}$.  We thus apply \\cite[Theorem 7.1]{gima12},\nwhich is valid for any domain which is bilipschitz equivalent to the\nunit cube, to \\eqref{eq:38}.  This yields that, for some $C(p,\\Omega)>0$,\nwe have\n\\begin{equation}\\label{reglp}\n  \\|\\nabla\\phi_{1,\\kappa}\\|_p \\leq  C \\frac{c}{\\kappa} \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\,,  \\quad  \\forall\\kappa \\geq 1\\,,\\; 2<p\\leq3\\,.\n\\end{equation}\nFrom \\eqref{eq:47} and \\eqref{eq:31} we then obtain that\n \\begin{equation}\n\\label{eq:48}\n   \\frac{1}{c}\\|\\nabla \\phi_{1,\\kappa} \\|_p + \\|\\nabla B_{1,\\kappa} \\|_p \\leq C\\,  \\kappa^{3(p-2)\/p}\\,.\n\\end{equation}\nUpon \\eqref{eq:48} and \\eqref{eq:32} we use the Poincar\\'e inequality\ntogether with Sobolev embeddings to conclude that\n\\begin{equation}\n\\label{B1k}\n  \\|B_{1,\\kappa}\\|_\\infty  \\leq C \\kappa^{3(p-2)\/p}\\,,\\quad  \\forall\\kappa>1\\,,\\; 2<p\\leq3\\,.\n\\end{equation}\n\n\nLet $x\\in S_{\\delta+\\kappa^{-\\alpha}}\\,$, namely\n\\begin{displaymath}\n|B_n(x)|>(1+\\delta+\\kappa^{-\\alpha})\\,.\n\\end{displaymath}\nFrom \\eqref{B1k}, for some $C(p,\\Omega,J)>0$ we have that\n\\begin{displaymath}\n  \\aligned\n  |B_\\kappa(x)|>&\\kappa|B_n(x)| - |B_{1,\\kappa}(x)| \\geq(1+\\delta+\\kappa^{-\\alpha})\\kappa  - C \\kappa^{3(p-2)\/p}\\\\\n  =&(1+\\delta)\\kappa+[\\kappa^{1-\\alpha}- C \\kappa^{3(p-2)\/p}].\n  \\endaligned\n\\end{displaymath}\nBy choosing\n\\begin{displaymath}\n  2<p<\\min\\Big(3,\\frac{6}{2+\\alpha}\\Big)\\,,\n\\end{displaymath}\nwe have $\\kappa^{1-\\alpha}- C \\kappa^{3(p-2)\/p}>0$ for sufficiently large $\\kappa\\,$. Thus\n$$|B_\\kappa(x)|>(1+\\delta)\\kappa\\,,\n$$\nand hence $x\\in D_\\delta(\\kappa)$. Consequently, $S_{\\delta+\\kappa^{-\\alpha}}\\subset  D_\\delta(\\kappa)\\,$.\n\\end{proof}\n\nAs a byproduct of the proof, we also obtain\n\\begin{proposition}\n  For any $2<p\\leq 3 $, there exists $\\kappa_0\\geq 1$ and $C>0$ such that\n  \\begin{equation}\n    \\label{eq:49}\n\\|A_{1,\\kappa}\\|_{2,p} \\leq C \\kappa^{3(p-2)\/p}\\,,\\quad \\forall \\kappa \\geq \\kappa_0\\,.\n  \\end{equation}\n\\end{proposition}\n\\begin{proof}\n  The proof follows immediately from \\eqref{eq:31}, \\eqref{eq:48}, and\n  Proposition B.3 in \\cite{alhe14}.\n\\end{proof}\n\nWe can now prove the following semi-classical Agmon estimate for\n$\\psi_\\kappa$, establishing  that it must be exponentially small in $S_\\delta$.\n\\begin{proposition}\nSuppose that $h$ satisfies (\\ref{eq:10}b). Let then $j\\in\\{1,2\\}$ be such that\n$h_j>1\/\\Theta_0$.  There exist $C>0$  and\n$\\delta_0>0$, such that, for any $0<\\delta\\leq \\delta_0\\,$, some $\\kappa_0(\\delta)$ can be found,\nfor which\n  \\begin{equation}\n \\label{eq:50}\n\\kappa \\geq \\kappa_0(\\delta) \\Rightarrow\\int_{S_{\\delta,j}} \\exp\\Big(\\delta^{1\/2}\\kappa \\,\n  d(x,\\mathcal C_{\\delta,j})\\Big) |\\psi_\\kappa|^2 \\,dx \\leq \\frac{C}{\\delta^{3\/2}} \\,,\n  \\end{equation}\nwhere $S_{\\delta,j}$ is introduced in \\eqref{sdj} and  $\\mathcal C_{\\delta,j}$ in  \\eqref{eq:36}.\n\\end{proposition}\n\\begin{proof}\nFor $\\delta >0$,  let  $\\eta\\in C^\\infty(\\Omega,[0,1])$ satisfy\n\\begin{equation}\n\\label{eq:51}\n    \\eta(x)=\n    \\begin{cases}\n      1 & x\\in S_{\\delta,j}\\,, \\\\\n      0 & x\\in\\Omega\\setminus S_{\\delta\/2,j} \\,.\n    \\end{cases}\n  \\end{equation}\n  By \\eqref{eq:12} and \\eqref{reguphi}, it follows that $\\nabla B_n$ is\n  bounded and independent of both $\\delta$ and $\\kappa$.  Consequently,\nthere exists a constant $C_1 >0$ such that\n  \\begin{displaymath}\n    d(\\mathcal C_{\\delta,j},\\mathcal C_{\\delta\/2,j}) \\geq \\frac{\\delta}{C_1}\\,,\n  \\end{displaymath}\nand hence,  for some $C(\\Omega,J)$ and all $0<\\delta<\\delta_0$ we can choose  $\\eta$ such  that\n  \\begin{displaymath}\n   |\\nabla\\eta| \\leq \\frac{C}{\\delta} \\,.\n  \\end{displaymath}\nLet further $$\\zeta=\\chi\\, \\eta$$ where\n\\begin{displaymath}\n  \\chi=\n  \\begin{cases}\n    \\exp(\\alpha_\\delta\\kappa d(x,\\mathcal C_{\\delta,j})) &\\text{if } x\\in S_{\\delta,j}\\,, \\\\\n    1 &\\text{if } x\\in\\Omega\\setminus S_{\\delta,j}\\,.\n  \\end{cases}\n\\end{displaymath}\nWe leave the determination of $\\alpha_\\delta$ to a later stage.  We further\ndefine, for any $r\\in (0,r_0)$,  $\\eta_r\\in C^\\infty(\\Omega,[0,1])$ and $\\tilde \\eta_r\\in C^\\infty(\\Omega,[0,1])$ such that\n\\begin{equation}\n\\label{eq:52}\n     \\eta_r(x)=\n    \\begin{cases}\n      1 & d(x,\\partial\\Omega_i)>r \\\\\n      0 & d(x,\\partial\\Omega_i)<r\/2 \\,,\n    \\end{cases}\n    \\qquad\\text{and}\\qquad\n|\\nabla\\eta_r|^2 + |\\nabla \\tilde \\eta_r|^2  \\leq \\frac{C}{r^2} \\,,\n\\end{equation}\nand\n$$\n \\eta_r^2 + \\tilde{\\eta}_r ^2 = 1\\,.\n$$\n\n\nFix $0<\\alpha<1$. Multiplying (\\ref{eq:25}a)  by $\\zeta^2\\bar{\\psi}$,\nintegrating by parts yields for the real part\n$$\\aligned\n& \\|\\nabla_{\\kappa A_\\kappa}(\\zeta\\tilde{\\eta}_{\\kappa^{-1\/2}}\\psi_\\kappa)\\|_2^2+\n \\|\\nabla_{\\kappa A_\\kappa}(\\zeta\\eta_{\\kappa^{-1\/2}}\\psi_\\kappa)\\|_2^2 \\\\\n \\leq& \\kappa^2\\|\\zeta\\psi_\\kappa\\|_2^2\n +\\|\\zeta\\psi_\\kappa\\nabla\\eta_{\\kappa^{-1\/2}}\\|_2^2+\\|\\zeta\\psi_\\kappa\\nabla\\tilde{\\eta}_{\\kappa^{-1\/2}}\\|_2^2 + \\|\\psi_\\kappa\\nabla\\zeta\\|_2^2 \\,.\n\\endaligned\n$$\nObserving that $\\langle\\psi_\\kappa\\nabla\\chi,\\psi_\\kappa\\nabla\\eta\\rangle=0$, we obtain\n\\begin{displaymath}\n   \\|\\psi_\\kappa\\nabla\\zeta\\|_2^2 \\leq  \\alpha_\\delta^2\\kappa^2\\|\\psi_\\kappa\\zeta\\|_2^2 +   \\|\\psi_\\kappa\\nabla\\eta\\|_2^2 \\,.\n\\end{displaymath}\nHence,\n\\begin{equation}\\label{eq:53}\n\\aligned\n&  \\|\\nabla_{\\kappa A_\\kappa}(\\zeta\\tilde{\\eta}_{\\kappa^{-1\/2}}\\psi_\\kappa)\\|_2^2+\n \\|\\nabla_{\\kappa A_\\kappa}(\\zeta\\eta_{\\kappa^{-1\/2}}\\psi_\\kappa)\\|_2^2 \\\\\n\\leq& \\kappa^2\\big(1+ \\alpha_\\delta^2+C\\kappa^{-1}\\big)\\|\\zeta\\psi_\\kappa\\|_2^2 + \\|\\psi_\\kappa\\nabla\\eta\\|_2^2\\,.\n\\endaligned\n\\end{equation}\n\n\nWe now use \\eqref{eq:22} and \\eqref{eq:49} to obtain, for sufficiently\nsmall $\\delta$,\n\\begin{equation}\n\\label{eq:54}\n\\aligned\n \\|\\nabla_{\\kappa A_\\kappa}(\\zeta\\tilde{\\eta}_{\\kappa^{-1\/2}}\\psi_\\kappa)\\|_2^2\n\\geq&\n\\kappa^4\\mu_{\\kappa^{-2}}(A_n+\\kappa^{-1}A_{1,\\kappa},S_{\\delta\/2})\\|\\zeta\\tilde{\\eta}_{\\kappa^{-1\/2}}\\psi_\\kappa\\|_2^2\\\\\n\\geq & \\kappa^2\\min(\\Theta_0h_j,1+\\delta\/2)\n [1-C\\kappa^{-2\/3}]\\|\\zeta\\tilde{\\eta}_{\\kappa^{-1\/2}}\\psi_\\kappa\\|_2^2 \\\\\n\\geq &  (1+\\frac \\delta 2 )\\, \\kappa^2[1-C\\kappa^{-2\/3}] \\|\\zeta\\tilde{\\eta}_{\\kappa^{-1\/2}}\\psi_\\kappa\\|_2^2  \\,.\n\\endaligned\n\\end{equation}\nBy \\cite[Theorem 2.9]{AHS} we have, since $\\zeta\\eta_{\\kappa^{-1\/2}}\\psi_\\kappa$ vanishes on $\\partial \\Omega$,\n\\begin{equation}\n\\label{eq:55}\n  \\|\\nabla_{\\kappa A_\\kappa}(\\zeta\\eta_{\\kappa^{-1\/2}}\\psi_\\kappa)\\|_2^2 \\geq\n  (1+\\delta\/2-\\kappa^{-1\/2})\\kappa^2\\|\\zeta\\eta_{\\kappa^{-1\/2}}\\psi_\\kappa\\|_2^2\\,.\n\\end{equation}\nConsequently, by  \\eqref{eq:53}, \\eqref{eq:54}, and\n\\eqref{eq:55}, and  by choosing\n$$\n\\alpha_\\delta^2 = \\frac{\\delta}{4}\\,$$ we obtain, that for $\\kappa\\geq \\kappa(\\delta)$, with\n$\\kappa(\\delta)$ sufficiently large:\n\\begin{displaymath}\n  \\kappa^2 \\frac{\\delta}{8}  \\|\\zeta\\psi_\\kappa\\|_2^2 \\leq  \\kappa^2 \\left( \\frac{\\delta}{4} - \\widehat C \\kappa^{-\\frac 12}\\right)\\, \\|\\zeta\\psi_\\kappa\\|_2^2 \\leq  \\|\\psi_\\kappa\\nabla\\eta\\|_2^2\\,,\n  \\end{displaymath}\nfrom which \\eqref{eq:50} easily follows.\n\\end{proof}\n\nNext we consider currents satisfying only  (\\ref{eq:10}a). Let, for $j=1,2$,\n \\begin{equation} \\label{eq:56}\n  \\omega_{\\delta,j}  = \\{x\\in \\Omega \\,: \\, (-1)^jB_n(x)>1+\\delta \\;;\\;d(x,\\partial\\Omega_i) > \\delta\\,\\} \\,,\\end{equation}\nand\n\\begin{equation} \\label{Gamma}\n \\Gamma_{\\delta,j}=  \\partial\\omega_{\\delta,j} \\setminus\\partial\\Omega_c\\cap\\partial\\omega_{\\delta,j} \\,.\n\\end{equation}\nWe can now state\n\\begin{proposition}\n  Suppose that for some $j\\in\\{1,2\\}$ we have\n  \\begin{equation}\n\\label{eq:57}\n    1<|h_j| \\,.\n  \\end{equation}\nThen, there exist $C>0$, $\\delta_0>0$, such that  for any $0<\\delta<\\delta_0$,\nsome $\\kappa_0 (\\delta)>0$ can be found, for which\n \\begin{equation}   \\label{eq:58}\n\\kappa\\geq \\kappa_0(\\delta)\\Rightarrow \\int_{\\omega_{\\delta,j}} \\exp\\Big(\\delta^{1\/2}\\kappa d(x,\n\\Gamma_{\\delta,j})\\Big) |\\psi_\\kappa|^2 \\,dx \\leq \\frac{C}{\\delta^{3\/2}}  \\,.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nWithout loss of generality we may assume $h_j>0$; otherwise we apply\n  to \\eqref{eq:25} the transformation\n  $(\\psi_\\kappa,A_\\kappa,\\phi_\\kappa)\\to(\\bar{\\psi_\\kappa},-A_\\kappa,-\\phi_\\kappa)$.  Let\n   \\begin{equation}\\label{check1}\n { \\check{\\chi}=}\n  \\begin{cases}\n    \\exp\\Big(\\frac 12 \\delta^{1\/2}\\kappa d(x,\\Gamma_{\\delta,j})\\Big) &\\text{if } x\\in \\omega_{\\delta,j}\\,,  \\\\\n    1 &\\text{if } x\\in\\Omega\\setminus \\omega_{\\delta,j}\\,.\n  \\end{cases}\n\\end{equation}\nLet further $\\eta$  and  $\\eta_r$  be given by \\eqref{eq:51} and\n\\eqref{eq:52} respectively.  Then set\n\\begin{equation}\\label{check2}\n\\check{\\zeta}= \\eta_\\delta\\, \\eta\\, \\check \\chi \\,.\n\\end{equation}\nThe proof proceeds in the same manner as in the previous proposition\nwith $\\zeta$ replaced by $\\check{\\zeta}$ with the difference that now\n$\\check{\\zeta} \\psi_\\kappa(x)$ vanishes for all $ x\\in\\partial \\Omega_i$. Consequently,\n(\\ref{eq:10}a) is no longer necessary (see \\eqref{eq:55}). We use (\\ref{eq:10}b)\nto establish that $\\omega_{\\delta,j}$ is not empty.\n\\end{proof}\n\nWe conclude this section by showing that for $\\mathcal O(\\kappa)$ currents\n(i.e. when $J$ is independent of $\\kappa$) $\\|\\psi_\\kappa\\|_2$ must be small.\nTo this end we define $\\Phi_n$ as the solution of (\\ref{eq:11b}a,b),  and\n\\begin{equation}\n\\label{eq:59}\n  \\int_\\Omega |\\psi_\\kappa|^2\\Phi_n \\,dx =0\\, .\n\\end{equation}\nThe above condition is a natural choice as by \\eqref{eq:33} we have\nthat\n\\begin{displaymath}\n   \\int_\\Omega |\\psi_\\kappa|^2\\phi_\\kappa  \\,dx =0\\, .\n\\end{displaymath}\nIt can be easily verified from \\eqref{eq:20} that\n\\begin{equation}\\label{calconst}\n  \\Phi_n=\\phi_n+ C(\\kappa,c)\\,,\n\\end{equation}\nwhere $\\phi_n$ denotes the solution of \\eqref{eq:11b}.\nThe constant can be extracted from \\eqref{eq:59}:\n$$\nC (\\kappa,c) = - \\frac{\\int_\\Omega \\phi_n |\\psi_\\kappa|^2\\, dx }{\\int_\\Omega  |\\psi_\\kappa|^2\\, dx}\\,,\n$$\nfrom which we get  the following  upper bound (independent of $\\kappa$ and $c$)\n\\begin{equation}\n\\label{eq:60}\n|C (\\kappa,c) | \\leq \\|\\phi_n\\|_\\infty < +\\infty\\,.\n\\end{equation}\n\n\\begin{proposition}\n  Under Assumptions \\eqref{eq:2}-\\eqref{eq:10} there exists \\linebreak\n  $C(J,\\Omega)>0$  and $\\kappa_0>0$ such that for any $\\kappa\\geq \\kappa_0$,\n  \\begin{equation}\n    \\label{eq:61}\n\\| \\psi_\\kappa \\|_2 \\leq C (J,\\Omega) (1+c^{-1\/2})^{1\/3}\\kappa^{-1\/6}  \\,.\n  \\end{equation}\n\\end{proposition}\n\\begin{proof}\nLet $\\Phi_{1,\\kappa}=\\phi_k-c\\kappa\\Phi_n$.\n An immediate consequence of \\eqref{eq:33} is that\n\\begin{equation}\n\\label{eq:62}\n  \\begin{cases}\n    -\\Delta\\Phi_{1,\\kappa}  + c|\\psi|^2_\\kappa \\Phi_{1,\\kappa} = - c^2|\\psi|^2_\\kappa \\kappa\\Phi_n & \\text{ in } \\Omega\\,, \\\\\n  \\frac{\\partial\\Phi_{1,\\kappa}}{\\partial\\nu}= 0 &  \\text{ on } \\partial\\Omega\\,.\n  \\end{cases}\n\\end{equation}\nTaking the inner product with $\\Phi_n$ yields, with the aid of (\\ref{eq:59}),\n\\begin{equation}\n\\label{eq:63}\n   \\aligned\n& \\kappa\\| \\psi_\\kappa \\Phi_n\\|_2^2 = - \\frac{1}{c^2}\\langle\\nabla\\Phi_n,\\nabla\\Phi_{1,\\kappa}\\rangle  +\\frac{1}{c}\\langle \\langle|\\psi_\\kappa|\n  \\Phi_n,|\\psi_\\kappa| (\\Phi_{1,\\kappa}-(\\Phi_{1,\\kappa})_\\Omega \\rangle\\\\\n\\leq& \\frac{1}{c^2}\\|\\nabla\\Phi_n\\|_2\\|\\nabla\\Phi_{1,\\kappa}\\|_2 + \\frac{1}{c}\\| \\psi_\\kappa\n  \\Phi_n\\|_2 \\| \\psi_\\kappa(\\Phi_{1,\\kappa}-(\\Phi_{1,\\kappa})_\\Omega )\\|_2\\,,\n\\endaligned\n\\end{equation}\nwhere $(\\Phi_{1,\\kappa})_\\Omega $ is the average of $\\Phi_{1,\\kappa}$ in $\\Omega$.\n\nWith the aid of (\\ref{eq:41}) (note that $\\nabla \\phi_{1,\\kappa}=\\nabla \\Phi_{1,\\kappa}$),\nthe fact that \\break $|\\psi_\\kappa|\\leq1$, and the Poincar\\'e inequality we then\nobtain\n\\begin{equation}\n  \\label{eq:64}\n \\| \\psi_\\kappa \\Phi_n\\|_2 \\leq  C\\kappa^{-1\/2}(1+ c^{-1\/2}) \\,.\n\\end{equation}\nWe now set\n\\begin{displaymath}\n  {\\mathcal U}_\\kappa = \\{ x\\in\\Omega \\,:\\, |\\Phi_n(x) |< (1+c^{-1\/2})^{2\/3}\\kappa^{-1\/3}\\,\\} \\,.\n\\end{displaymath}\nBy \\eqref{eq:59}  the level set $\\Phi_n^{-1}(0)$ lies inside\n$\\Omega$.\nLet $x_0\\in\\Phi_n^{-1}(\\tau)$ for some $\\tau\\neq0$, and set\n$$\\Gamma_\\perp=B_n^{-1}(B_n(x_0)).\n$$\nBy (\\ref{eq:11}a) $B_n$ is the conjugate harmonic function of $\\Phi_n$,\nand hence $\\Gamma_\\perp$ must be perpendicular to $\\Phi_n^{-1}(\\tau)$ at $x_0$. Note\nthat in \\cite[(2.3)]{alhe14} we showed that $B_n^{-1}(\\mu)$ is a simple\nsmooth curve connecting the two connected components of $\\partial\\Omega_c$ for any\n$h_1<\\mu<h_2$. We denote by $\\tilde{\\Gamma}_\\perp$ the subcurve of $\\Gamma_\\perp$\n  originating from $x_0$ in the direction where $\\Phi_n$ decreases if\n$\\tau>0$ or increases if $\\tau<0$, and terminating either on $\\Phi_n^{-1}(0)$ or\non the boundary.  Clearly,\n\\begin{displaymath}\n  |\\tilde{\\Gamma}_\\perp|\\inf_{x\\in\\Omega}|\\nabla\\Phi_n|\\leq  \\Big|\\int_{\\tilde{\\Gamma}_\\perp} \\nabla\\Phi_n  ds\\Big| \\leq |\\tau|\\,,\n\\end{displaymath}\nwhere $\\int_\\Gamma \\vec{V}$ denotes the circulation of $\\vec{V}$ along the path $\\Gamma$.\nIn \\cite[\\S2.3]{alhe14} we have\nestablished that $|\\nabla\\Phi_n|=|\\nabla B_n|>0$ in $\\bar{\\Omega}$. It follows that\n\\begin{equation}\n\\label{eq:65}\n  d(x_0,\\partial\\Omega\\cup\\Phi_n^{-1}(0)) \\leq   |\\tilde{\\Gamma}_\\perp| \\leq C|\\tau|\\,.\n\\end{equation}\nLet\n\\begin{displaymath}\n  \\tilde{{\\mathcal U}}_\\kappa(r)= \\{x\\in\\Omega \\,| \\, d(x,\\partial\\Omega\\cup\\Phi_n^{-1}(0)) \\leq\n  r(1+c^{-1\/2})^{2\/3}\\kappa^{-1\/3}\\}\\,.\n\\end{displaymath}\nBy \\eqref{eq:65} we obtain that for sufficiently large $r$ there\nexists $\\kappa_0(r)$ such that for all $\\kappa>\\kappa_0$ and\n$c\\in{\\mathbb{R}}$ we have ${\\mathcal U}_\\kappa \\subseteq \\tilde{{\\mathcal U}}_\\kappa(r)$. Consequently,\n\\begin{equation}\n\\label{eq:66}\n  | {\\mathcal U}_\\kappa| \\leq C(1+c^{-1\/2})^{2\/3}\\kappa^{-1\/3}(|\\Phi_n^{-1}(0)|+|\\partial\\Omega|)\\leq C(1+c^{-1\/2})^{2\/3}\\kappa^{-1\/3}\\,.\n\\end{equation}\n\nBy \\eqref{eq:64} we have that\n\\begin{displaymath}\n  \\| \\psi_\\kappa\\|_{L^2(\\Omega\\setminus{\\mathcal U}_\\kappa)} \\leq C(1+c^{-1\/2})^{1\/3}\\kappa^{-1\/6} \\,,\n\\end{displaymath}\nwhereas from \\eqref{eq:66} and \\eqref{eq:26} we learn that\n\\begin{displaymath}\n   \\| \\psi_\\kappa\\|_{L^2({\\mathcal U}_\\kappa)} \\leq C(1+c^{-1\/2})^{1\/3}\\kappa^{-1\/6}  \\,.\n\\end{displaymath}\nThe proposition can now be  readily verified.\n\\end{proof}\nAn immediate conclusion is that whenever $c\\kappa\\gg1$, $|\\psi_\\kappa|$ is small. If\n$c=\\mathcal O(\\kappa^{-1})$, $|\\psi_\\kappa|$ may not tend to $0$ as $\\kappa\\to\\infty$. Further research\nis necessary to establish this point.\n\\begin{remark}\n If, for some $0<\\alpha<1$, we assume that $J=J(\\cdot, \\kappa)$ satisfies\n$$\n\\|J\\|\\leq C\\kappa^{-\\alpha}\\,,\n$$\nthen (\\ref{eq:63}) and\n  (\\ref{eq:41}) remain valid. Assuming $c=1$, and using  this time (\\ref{eq:48}), we obtain\n  instead of (\\ref{eq:64}),   for any $0<\\beta$, that\n  \\begin{displaymath}\n     \\| \\psi_\\kappa \\Phi_n\\|_2 \\leq  C_\\beta \\kappa^{-1\/2}(\\kappa^{\\alpha\/2}+\\kappa^\\beta\\|\\psi_\\kappa\\|_2^{1\/2}) \\,.\n  \\end{displaymath}\nUsing the above, with sufficiently small $\\beta$, we obtain,  similarly to the derivation of  (\\ref{eq:61})\n\\begin{displaymath}\n   \\| \\psi_\\kappa\\|_2\\leq C\\kappa^{-(1-\\alpha)\/6}\\,,\n\\end{displaymath}\nwhich implies\n$$\\|\\psi_\\kappa\\|_2\\xrightarrow[\\kappa\\to + \\infty]{}0\\,.\n$$\nThis result stands in sharp contrast with the behavior obtained\nin the absence of electric potential \\cite{sase03,attar2014ground}.\n\\end{remark}\n\n\\section{Time-Dependent Analysis}\n\\label{sec:4}\nIn this section we return to the time-dependent problem as introduced\nin \\eqref{eq:1}.  For convenience we set here $$c=1\\,.$$\n\\begin{subequations}\n\\label{eq:67}\n\\begin{alignat}{2}\n & \\frac{\\partial\\psi_\\kappa}{\\partial t}-  \\nabla_{\\kappa A_\\kappa}^2\\psi_\\kappa+i\\kappa\\phi_\\kappa \\psi_\\kappa =\\kappa^2(1-|\\psi_\\kappa|^2)\\psi_\\kappa & \\text{ in } (0,+\\infty)\\times \\Omega& \\,,\\\\\n  &\\frac{\\partial A_\\kappa}{\\partial t}+\\nabla\\phi_\\kappa +\\text{\\rm curl\\,}^2A_\\kappa\n  =\\frac{1}{\\kappa}\\Im(\\bar\\psi_\\kappa\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa) &\n  \\text{ in } (0,+\\infty)\\times  \\Omega&\\,, \\\\\n  &\\psi =0 &\\text{ on } (0,+\\infty)\\times  \\partial\\Omega_c&\\,, \\\\\n&  \\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\cdot\\nu=0 &  \\text{ on } (0,+\\infty)\\times  \\partial\\Omega_i&\\,, \\\\\n&  \\frac{\\partial\\phi_\\kappa}{\\partial\\nu} = - \\kappa J(x) & \\text{ on }(0,+\\infty)\\times  \\partial\\Omega_c&\\,, \\\\\n & \\frac{\\partial\\phi_\\kappa}{\\partial\\nu}=0  & \\text{ on } (0,+\\infty)\\times  \\partial\\Omega_i &\\,, \\\\[1.2ex]\n&  \\Xint-_{\\partial\\Omega}\\text{\\rm curl\\,} A_\\kappa \\, ds = \\kappa h_{ex} &\\quad \\text{ on } (0,+\\infty)&\\,, \\\\\n&  \\psi(0,x)=\\psi_0(x)  & \\text{ in } \\Omega&\\,, \\\\\n&A(0,x)=A_0(x) & \\text{ in } \\Omega & \\,.\n\\end{alignat}\n\\end{subequations}\nWe assume again \\eqref{condinit}-(\\ref{eq:6}), \\eqref{eq:8}, and\n\\eqref{eq:10}.  Since in the time dependent case $\\phi_\\kappa$ is determined\nup to a constant in view of \\eqref{eq:4}\n and \\eqref{eq:5}, we can\nfurther  impose\n\\begin{equation}\\label{eq:68}\n  \\int_\\Omega\\phi_\\kappa (t,x)\\,dx =0\\,, \\quad \\forall t>0 \\,.\n\\end{equation}\nIt follows from \\eqref{condinit} by the maximum\nprinciple (see \\cite[Theorem 2.6] {alhe14}) that\n\\begin{equation}\n\\label{eq:69}\n  \\|\\psi_\\kappa(t, \\cdot)\\|_\\infty\\leq 1 \\,, \\quad \\forall t\\geq0\\,.\n\\end{equation}\nWe recall from \\cite  [Subsection 2.4] {alhe14} the following spectral entity\n\\begin{subequations}\n\\label{eq:70}\n  \\begin{equation}\n\\lambda= \\inf_{\n    \\begin{subarray}{2}\n      V\\in{\\mathcal H}_d \\\\\n      \\|V\\|_2 =1\n    \\end{subarray}} \\| \\text{\\rm curl\\,} V\\|_2^2 \\,,\n\\end{equation}\nwhere\n\\begin{equation}\n  {\\mathcal H}_d = \\big\\{ V\\in H^1(\\Omega,{\\mathbb{R}}^2)\\,:\\, \\text{\\rm div\\,} V=0 \\,, V\\big|_{\\partial\\Omega}\\cdot\\nu=0 \\big\\}  \\,.\n\\end{equation}\n\\end{subequations}\nWe further recall from \\cite [Proposition 2.5] {alhe14} that, under condition $(R_1)$ on $\\partial \\Omega$,\n\\begin{displaymath}\n  \\lambda=\\lambda^D:=\\inf_{\n    \\begin{subarray}{2}\n      u\\in H^1_0(\\Omega) \\\\\n      \\|u\\|_2 =1\n    \\end{subarray}} \\| \\nabla u\\|_2^2>0 \\,.\n\\end{displaymath}\n\nWe retain our definition of the normal fields $(A_n,\\phi_n)$ via \\eqref{eq:11}. For\nthe solution $(A_\\kappa,\\phi_\\kappa)$ of \\eqref{eq:67} we set\n\\begin{equation}\\label{time-dep-AB}  \\aligned\n&A_{1,\\kappa}(t,x)=A_\\kappa(t,x)-\\kappa A_n(x),\\\\\n&\\phi_{1,\\kappa}(t,x)=\\phi_\\kappa(t,x)-\\kappa\\phi_n(x),\\\\\n&B_\\kappa(t,x)=\\text{\\rm curl\\,} A_\\kappa(t,x),\\\\\n&B_{1,\\kappa}(t,x)=\\text{\\rm curl\\,} A_{1,\\kappa}(t,x).\n\\endaligned\n\\end{equation}\nClearly,\n\\begin{subequations}\n\\label{eq:71}\n  \\begin{alignat}{2}\n  \\frac{\\partial A_{1,\\kappa}}{\\partial t}+\\nabla\\phi_{1,\\kappa}  + \\text{\\rm curl\\,} B_{1,\\kappa}\n & =  \\frac{1}{\\kappa}\\Im(\\bar\\psi_\\kappa\\, \\nabla_{\\kappa A_\\kappa}\\psi_\\kappa)  &  \\text{ in }\n (0,+\\infty)\\times  \\Omega\\,,\\\\\n   \\frac{\\partial\\phi_{1,\\kappa}}{\\partial\\nu}&= 0 & \\text{ on } (0,+\\infty)\\times \\partial\\Omega\\,, \\\\\n\\Xint-_{\\partial\\Omega}B_{1,\\kappa}(t,x)\\,ds& = 0  & \\text{ in } (0,+\\infty) \\,.\n\\end{alignat}\n\\end{subequations}\n\nWe begin by the following auxiliary estimate. We recall that \\break $\\|A(t,\\cdot)\\|_{1,2}=\\|A(t,\\cdot)\\|_{H^1(\\Omega,{\\mathbb{R}}^2)}$.\n\\begin{lemma}\n\\label{lem:parabolic-A1}\nLet $A_{1,\\kappa}$ and $B_{1,\\kappa}$ be defined by \\eqref{time-dep-AB}.\nSuppose that \\linebreak $\\|A_{1,\\kappa}(\\cdot,0)\\|_2\\leq M$ (where $M$ may depend on\n$\\kappa$). Then, under the above assumptions, there exists $ t^*(M)$ and a\nconstant $C=C(\\Omega,t^*)>0$ such that for all $t>t^*$ and $\\kappa\\geq 1$ we\nhave\n\\begin{equation}\n  \\label{eq:72}\n\\|A_{1,\\kappa}(t,\\cdot)\\|_{1,2} +  \\|A_{1,\\kappa}\\|_{L^2(t,t+1,H^2(\\Omega))} \\leq C.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n  By \\cite[Lemma 5.3]{alhe14} there exists a constant $C=C(\\Omega)>0$ such that for\n  sufficiently large\n  $\\kappa$\n  \\begin{displaymath}\n    \\|A_{1,\\kappa}(t,\\cdot)\\|_2^2 \\leq \\Big(\\|A_{1,\\kappa}(0,\\cdot)\\|_2^2+\\frac{C}{\\kappa^2}\\Big)\ne^{-\\lambda t} + C   \\int_0^te^{-\\lambda(t-\\tau)}\\|\\psi_\\kappa(\\tau,\\cdot)\\|_2^2 \\,d\\tau \\,.\n  \\end{displaymath}\nFrom \\eqref{eq:69},  we then get\n   \\begin{displaymath}\n    \\|A_{1,\\kappa}(t,\\cdot)\\|_2^2 \\leq \\Big(\\|A_{1,\\kappa}(0,\\cdot)\\|_2^2+\\frac{C}{\\kappa^2}\\Big)\\,\ne^{-\\lambda t} + \\frac{C}{\\lambda}\n  \\end{displaymath}\nWe thus have\n     \\begin{equation}\\label{anytime}\n    \\|A_{1,\\kappa}(t,\\cdot)\\|_2^2 \\leq  (M+C) e^{-\\lambda t} + \\frac{C}{\\lambda}\n  \\end{equation}\nHence, there exists $t_0^*(M)$, such that for $t \\geq t_0^*(M)$,  we have\n\\begin{equation}\n\\label{eq:73}\n    \\|A_{1,\\kappa}(t,\\cdot)\\|_2 \\leq \\frac{2C}{\\lambda} \\,.\n\\end{equation}\nNext, we apply \\cite[Theorem C.1 (Formula C.4)]{alhe14} to the operator\n$\\mathcal L^{(1)}$ (as introduced there  in Example (4) above this theorem) to\nobtain that\n\\begin{equation}\\label{consc1}\n  \\aligned\n& \\|A_{1,\\kappa}\\|_{L^\\infty(t_0,t_0+1,H^1(\\Omega))}+\n  \\|A_{1,\\kappa}\\|_{L^2(t_0,t_0+1,H^2(\\Omega))}\\\\& \\qquad\n\\leq  \\frac{C}{\\kappa}\\|\\Im \\{\\bar{\\psi}_\\kappa\\nabla_{\\kappa\n    A_\\kappa}\\psi_\\kappa\\}\\|_{L^2(t_0,t_0+1,L^2(\\Omega))} +\n  C  \\|A_{1,\\kappa}(t_0,\\cdot)\\|_{1,2} \\,.\n\\endaligned\n\\end{equation}\nwith a constant $C$ independent of $t_0$.\nSince from (\\ref{eq:67}a) (cf. \\cite{alhe14}) we can easily get\nthat\n\\begin{equation}\n\\label{eq:74}\n      \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa (t,\\cdot)\\|_2^2 \\leq  \\kappa^2\\|\\psi_\\kappa(t,\\cdot)\\|_2^2-\\frac{1}{2}\n      \\frac{d\\|\\psi_\\kappa (t,\\cdot)\\|_2^2}{dt} \\,,\n\\end{equation}\nwe obtain by integrating over $(t_0,t_0+1)$\n\\begin{equation}\\label{eq:70a}\n  \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa \\|^2_{L^2(t_0,t_0+1,L^2(\\Omega))} \\leq  \\kappa^2\\|\\psi_\\kappa \\|_{L^2(t_0,t_0+1,L^2(\\Omega))}^2 + \\frac 12 \\|\\psi_\\kappa(t_0, \\cdot)\\|_2^2\\,,\n \\end{equation}\nand note for later reference that it implies\n \\begin{equation}\\label{eq:70aa}\n  \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa \\|_{L^2(t_0,t_0+1,L^2(\\Omega))} \\leq   C(\\Omega)\\, \\kappa\\,.\n \\end{equation}\nImplementing the upper bound \\eqref{eq:70a}  in \\eqref{consc1}, yields\n$$\\aligned\n& \\|A_{1,\\kappa}\\|_{L^\\infty(t_0,t_0+1,H^1(\\Omega))}+ \\|A_{1,\\kappa}\\|_{L^2(t_0,t_0+1,H^2(\\Omega))} \\\\\n\\leq& C\\Big[1+\n  \\|\\psi_\\kappa\\|_{L^2(t_0,t_0+1,L^2(\\Omega))}  +\n  \\frac{1}{\\kappa}\\|\\psi_\\kappa(t_0, \\cdot)\\|_2+\\|A_{1,\\kappa}(t_0,\\cdot)\\|_{1,2}\\Big] \\,.\n\\endaligned\n$$\nWe next apply \\cite[Theorem C.1 (Formula C.2)]{alhe14} to obtain in\nprecisely the same manner\n$$\n\\begin{array}{l}\n\\|A_{1,\\kappa}\\|_{L^\\infty(t_0,t_0+1,H^1(\\Omega))}\\\\\n\\qquad  \\leq C\\Big[1+\n  \\|\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}  +\n  \\frac{1}{\\kappa}\\|\\psi_\\kappa(t_0, \\cdot)\\|_2 + \\|A_{1,\\kappa}(t_0-1,\\cdot)\\|_2 \\Big] \\,.\n  \\end{array}\n$$\nThe above together with \\eqref{eq:69} and \\eqref{eq:73} yields, for\n $t_0\\geq t_0^* +1$,\n\\begin{equation}\n\\label{est-of-B}\n\\|A_{1,\\kappa}\\|_{L^\\infty(t_0,t_0+1,H^1(\\Omega))}+ \\|A_{1,\\kappa}\\|_{L^2(t_0,t_0+1,H^2(\\Omega))}\n\\leq C\\,,\n\\end{equation}\nwhich implies  \\eqref{eq:72}, with $t^* = t_0^* (M) +1$.\n\\end{proof}\n\\begin{remark}\\label{usefulrem}\n  Since our interest is in the limit as $t \\to +\\infty$, Lemma~\\ref{lem:parabolic-A1} allows us to assume in the sequel, without\n  any loss of generality, that (\\ref{eq:72}) is satisfied for all\n  $t\\geq0$. We have just to make a translation $t \\mapsto t -t^*$ and to\n  observe that $\\psi_\\kappa(t^*,\\cdot)$ has the same properties as $\\psi_0$.\n  \\end{remark}\n\\begin{proposition}\n  \\label{lem:asymp-exp}\n  Let $\\omega_{\\delta,j}$ ($j\\in\\{1,2\\}$) be defined in\n  \\eqref{eq:56}.\nSuppose that\n  for some $j\\in\\{1,2\\}$ we have that\n  \\begin{equation}\n\\label{eq:75}\n    1<|h_j| \\,.\n  \\end{equation}\n Then, there exist $C>0$ and $\\delta_0>0$, and, for any $0<\\delta<\\delta_0$,  $\\kappa_0(\\delta)\\geq 1$ such that, for  $\\kappa\\geq \\kappa_0(\\delta)$,\n\\begin{equation}\n\\label{eq:76}\n\\limsup_{t\\to\\infty}\\int_{\\omega_{\\delta,j} }|\\psi_\\kappa|^2(t,x) \\,dx \\leq\n  \\frac{C_\\delta }{\\kappa^2}  \\,.\n\\end{equation}\n\\end{proposition}\n\\begin{proof} \n  Without loss of generality we may assume $h_j>0\\,$; otherwise we apply\nto \\eqref{eq:67} the transformation\n$(\\psi_\\kappa,A_\\kappa,\\phi_\\kappa)\\to(\\bar{\\psi_\\kappa},-A_\\kappa,-\\phi_\\kappa)$.\n\\vspace{1ex}\n\n{\\it Step 1: Let, for $n\\geq1$,\n\\begin{displaymath}\n  a_n = \\|{\\widehat \\zeta}\\psi_\\kappa\\|_{L^\\infty(n-1, n,L^2(\\Omega))} \\,.\n\\end{displaymath}\nProve that for all $\\delta \\in (0,1)$  and $\\kappa \\geq \\kappa_0(\\delta)$,\n\\begin{equation}\\label{inega}\n  a_n^2 \\leq  C \\delta^{-3} \\Big(\\kappa^{-2} + \\kappa^{-1} (a_{n} + a_{n-1})\\Big)\\,.\n\\end{equation}}\n\\vspace{2ex}\n\nLet $\\eta$ and $\\eta_r$ be given by \\eqref{eq:51} and \\eqref{eq:52}\nrespectively. Then, set\n\\begin{equation}\n\\label{eq:77}\n  {\\widehat \\zeta}=\\eta \\, \\eta_\\delta\\, .\n\\end{equation}\nMultiplying (\\ref{eq:67}a) by ${\\widehat \\zeta}^{\\,2} \\bar{\\psi_\\kappa}$ and integrating by\n parts yields\n\\begin{displaymath}\n  \\frac{1}{2}\\frac{d}{dt} \\left(  \\|{\\widehat \\zeta}\\psi_\\kappa (t,\\cdot)\\|_2^2\\right)  \\, +  \\|\\nabla_{\\kappa A_\\kappa}({\\widehat \\zeta}\\psi_\\kappa (t,\\cdot))\\|_2^2 \\leq\n  \\kappa^2\\|{\\widehat \\zeta}\\psi_\\kappa (t,\\cdot)\\|_2^2 + \\|\\psi_\\kappa (t,\\cdot)\\, \\nabla{\\widehat \\zeta}\\|_2^2 \\,.\n\\end{displaymath}\nBy \\cite[Theorem 2.9] {AHS},\nwe have\n$$\\aligned\n  \\|\\nabla_{\\kappa A_\\kappa}({\\widehat \\zeta}\\psi_\\kappa (t,\\cdot))\\|_2^2 \\geq& \\langle\\kappa B_\\kappa (t,\\cdot) {\\widehat \\zeta}\\psi_\\kappa(t,\\cdot),{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\rangle \\\\ =&\n  \\kappa^2\\langle B_n{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot),{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot) \\rangle + \\langle\\kappa B_{1,\\kappa}(t,\\cdot)\\,{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot),{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\rangle  \\\\\n  \\geq&\n  \\kappa^2\\Big(1+\\frac{\\delta}{2}\\Big)\\|{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2+ \\langle\\kappa B_{1,\\kappa}{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\,,\\,{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\rangle \\,.\n\\endaligned\n$$\nWe can thus write\n\\begin{displaymath}\n\\begin{array}{l}\n   \\frac{1}{2}\\frac{d}{dt} \\left( \\|{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot) \\|_2^2\\right) +   \\frac{\\kappa^2\\delta}{2}\\|{\\widehat \\zeta}\\psi_\\kappa (t,\\cdot) \\|_2^2\\\\ \\quad  \\leq\n  \\|\\psi_\\kappa (t,\\cdot) \\nabla\\eta\\|_2^2 + \\|\\psi_\\kappa(t,\\cdot) \\nabla\\eta_\\delta\\|_2^2- \\langle\\kappa B_{1,\\kappa}(t,\\cdot) {\\widehat \\zeta}\\psi_\\kappa(t,\\cdot),{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\rangle \\,.\n  \\end{array}\n\\end{displaymath}\nSince\n\\begin{displaymath}\n  \\|\\psi_\\kappa (t,\\cdot)\\nabla\\eta\\|_2^2 + \\|\\psi_\\kappa(t,\\cdot) \\nabla\\eta_\\delta\\|_2^2 \\leq \\frac{C}{\\delta^2} \\,,\n\\end{displaymath}\nwe obtain that\n\\begin{equation}\n  \\label{eq:78}\n  \\begin{array}{l}\n   \\frac{1}{2}\\frac{d}{dt} \\left( \\|{\\widehat \\zeta}\\psi_\\kappa (t,\\cdot)\\|_2^2 \\right)\n    +   \\frac{\\kappa^2\\delta}{2}\\|{\\widehat \\zeta}\\psi_\\kappa (t,\\cdot)\\|_2^2\\\\ \\qquad\n   \\leq \\frac{C}{\\delta^2}  - \\langle \\kappa B_{1,\\kappa} (t,\\cdot) {\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\,,\\, {\\widehat \\zeta}\\psi_\\kappa (t,\\cdot) \\rangle \\,.\n\\end{array} \\end{equation}\nFrom \\eqref{eq:78} we can conclude that\n\\begin{equation}\\label{eq:79}\n\\aligned\n \\|{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2 \\leq& \\|{\\widehat \\zeta}\\psi_0\\|_2^2\\, e^{-\\delta\\kappa^2t} +\n  \\frac{C}{\\delta^3\\kappa^2}\\\\\n  &+2 \\int_0^t\n  e^{-\\delta\\kappa^2(t-\\tau)}\\big|\\langle\\kappa B_{1,\\kappa}(\\tau,\\cdot) {\\widehat \\zeta}\\psi_\\kappa(\\tau,\\cdot) ,{\\widehat \\zeta}\\psi_\\kappa(\\tau,\\cdot) \\rangle \\big|\\,d\\tau \\,.\n\\endaligned\n\\end{equation}\n \nTo estimate the last term on the right-hand-side of \\eqref{eq:79},  we start from\n $$ \\aligned\n&  \\int_0^t  e^{-\\delta\\kappa^2(t-\\tau)}\\big|\\langle\\kappa\n  B_{1,\\kappa}(\\tau,\\cdot) {\\widehat \\zeta}\\psi_\\kappa (\\tau,\\cdot) ,{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot) \\rangle\\big|\\,d\\tau  \\\\\n\\leq &   \\kappa\\, \\Big[\\int_0^t  e^{-\\delta\\kappa^2(t-\\tau)}\n  {  \\|B_{1,\\kappa}(\\tau, \\cdot)\\|_2^2 \\,d\\tau} \\cdot \\int_0^t  e^{-\\delta\\kappa^2(t-\\tau)}\n  \\|{\\widehat \\zeta}{ \\psi_\\kappa(\\tau, \\cdot )\\|_4^4}\\,d\\tau\\Big]^{1\/2}\\,. \\\\\n   \\endaligned\n  $$\nWith Remark \\ref{usefulrem} in mind,  we use \\eqref{eq:69} to obtain\n$$\n \\int_0^t  e^{-\\delta\\kappa^2(t-\\tau)}\n    \\|B_{1,\\kappa}(\\tau, \\cdot)\\|_2^2 \\,d\\tau  \\leq \\frac{C}{\\delta \\kappa^2} \\,.\n$$\nImplementing the above estimate, we obtain\n$$\n\\begin{array}{l}\n\\int_0^t  e^{-\\delta\\kappa^2(t-\\tau)}\\big|\\langle\\kappa\n  B_{1,\\kappa}(\\tau,\\cdot) {\\widehat \\zeta}\\psi_\\kappa (\\tau,\\cdot) ,{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot) \\rangle\\big|\\,d\\tau\\\\\n \\qquad\n   \\leq C \\delta^{-\\frac 12} \\Big[\\int_0^t  e^{-\\delta\\kappa^2(t-\\tau)}\n  \\|{\\widehat \\zeta}{\\psi_\\kappa(\\tau, \\cdot )\\|_4^4}\\,d\\tau\\Big]^{1\/2}\\,.\n  \\end{array}\n$$\nTo control of the right hand side we now write for $t\\geq 1$\n$$\\aligned\n& \\Big[\\int_0^t  e^{-\\delta\\kappa^2(t-\\tau)}\n  \\|{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_4^4\\,d\\tau\\Big]^{1\/2} \\\\\n& \\leq  \\Big[\\int_0^{t-1}  e^{-\\delta\\kappa^2(t-\\tau)}\n  \\|{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_4^4\\,d\\tau\\Big]^{1\/2} +  \\Big[\\int_{t-1} ^t  e^{-\\delta\\kappa^2(t-\\tau)}\n  \\|{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_4^4\\,d\\tau\\Big]^{1\/2}\\\\\n & \\leq C  \\Big[\\int_{t-1} ^t  e^{-\\delta\\kappa^2(t-\\tau)}\n  \\|{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_2^2\\,d\\tau\\Big]^{1\/2}  + C \\delta^{-\\frac 12}  \\kappa^{-1}  e^{- \\frac{\\delta }{2} \\kappa^2}\\\\\n& \\leq  C \\delta^{-\\frac 12} \\kappa^{-1} \\|\\widehat \\zeta\n  \\psi_\\kappa\\|_{L^\\infty(t-1,t,L^2(\\Omega)) } + C \\delta^{-\\frac 12}  \\kappa^{-1}  e^{-\n    \\frac{\\delta }{2} \\kappa^2} \\,.\n\\endaligned\n$$\n\nSubstituting the above into \\eqref{eq:79} yields, with a new constant\n$C$, for $\\kappa$ large enough, and for $t\\geq1\\,$,\n$$\n\\begin{array}{l}\n   \\|{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2 \\\\ \\quad   \\leq C \\delta^{-3}  \\kappa^{-2} +  C \\delta^{-1} \\kappa^{-1}  e^{ - \\frac{\\delta}{2} \\kappa^2}\n    + C \\delta^{-1} \\kappa^{-1} \\|\\widehat \\zeta \\psi_\\kappa\\|_{L^\\infty(t-1,t,L^2(\\Omega)) }  \\,.\n     \\end{array}\n$$\nFrom which we easily obtain \\eqref{inega}.\n\\vspace{1ex}\n\n\n{\\it Step 2: Prove \\eqref{eq:76}.}\n\\vspace{1ex}\n\nBy (\\ref{eq:69}) we have \n$$\n0 < a_n \\leq C \\,,\n$$\nwhich readily yields\n$$\na_n \\leq  C \\delta^{-\\frac 32}\\kappa^{-\\frac 12}\\,.\n$$\nWe improve the above estimate by reimplementing \\eqref{inega}.\nTo this end we set\n$$\\widehat C := C \\delta^{- \\frac 32 }\\,,$$ and then let\n\\begin{displaymath}\n  a_n= \\frac{\\widehat C}{\\kappa}\\alpha_n \\,.\n\\end{displaymath}\nSubstituting into (\\ref{inega}) yields\n\\begin{equation}\\label{inegaga}\n  \\alpha_n^2 \\leq 1 + \\alpha_{n-1} +\\alpha_n \\,.\n\\end{equation}\nSuppose that for some $N\\geq0$, we have $\\alpha_N\\leq 1 + \\sqrt{2} \\,$,  then\\break $\\alpha_{N+1}\\leq 1 +\\sqrt{2}$ and\nhence $\\alpha_n\\leq 1 + \\sqrt{2}$ for all $n\\geq N$.\n\nIf $\\alpha_{n-1} > 1+ \\sqrt{2}$ for any $n$,  we have, with $\\hat \\alpha_n = \\alpha_n - \\frac 12\\,$,\n\\begin{displaymath}\n \\hat  \\alpha_n^2\\leq \\frac 74 + \\hat \\alpha_{n-1} < \\hat \\alpha_{n-1}^2  \\,.\n\\end{displaymath}\nHence, $\\hat \\alpha_n<\\hat \\alpha_{n-1}$ which means that $\\hat \\alpha_n$\nconverges as a positive decreasing sequence, and necessarily to a\nlimit smaller than $1\/2 + \\sqrt{2}$. We thus conclude that\n\\begin{equation}\n  \\label{eq:80}\n\\limsup \\alpha_n \\leq 1 + \\sqrt{2} \\,,\n\\end{equation}\nand hence\n\\begin{equation}\n\\label{ineq-1} \\limsup a_n \\leq \\frac{C (1 +\\sqrt{2})}{\\delta^{3\/2}\\kappa},\n\\end{equation}\nfrom which \\eqref{eq:76} can easily be  deduced.\n\\end{proof}\n\nWe next obtain the following improvement over (\\ref{eq:70aa}) for\n$\\nabla_{\\kappa A_\\kappa}(\\widehat \\zeta \\psi_\\kappa)$.\n\n\\begin{proposition}\n  Let $p \\geq 2$.  For any $\\delta >0$, there exists $\\kappa_0 (\\delta)$ and $C(\\delta)$\n  such that for $\\kappa \\geq \\kappa (\\delta)$ we have, with $u={\\widehat \\zeta}\\psi_\\kappa\\,,\n  $ the following estimate:\n\\begin{equation}\\label{eq:89a}\n  \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^p(t_0,t_0+1; L^p(\\Omega))}\\leq C (\\delta) \\kappa^{6(1-2\/p)}\\,.\n  \\end{equation}\n\\end{proposition}\n\n\\begin{proof} {\\it Step 1: Prove that for some $C(\\delta)>0$ we have, for\n  sufficiently large $\\kappa$ that\n\\begin{equation}\n  \\label{eq:87}\n\\|u\\|_{L^2(t_0-1,t_0+1,H^2(\\Omega))} \\leq C\\kappa^3\\,.\n\\end{equation}}\n\\vspace{1ex}\n\nWe rewrite (\\ref{eq:67}a, c, d) in the form\n\\begin{displaymath}\n  \\begin{cases}\n\\frac{\\partial\\psi_\\kappa}{\\partial t}-\\Delta\\psi_\\kappa   = -2i\\kappa A_\\kappa\\cdot\\nabla\\psi_\\kappa  -  |\\kappa A_\\kappa|^2\\psi_\\kappa+ \\kappa^2\\psi_\\kappa\\big( 1 - |\\psi_\\kappa|^{2}\n\\big)\\\\\n\\qquad\\qq\\qquad \\qquad   - i\\kappa\\phi_\\kappa\\psi_\\kappa&\\text{ in } (0,+\\infty)\\times\\Omega,\\\\\n  \\psi_\\kappa =0   &\\text{ on } (0,+\\infty)\\times  \\partial\\Omega_c \\,, \\\\\n\\frac{\\partial\\psi_\\kappa}{\\partial\\nu}=i\\kappa A_\\kappa\\psi_\\kappa\\cdot\\nu=0  &\\text{ on }(0,+\\infty)\\times   \\partial\\Omega_i \\,.\\\\\n  \\end{cases}\n\\end{displaymath}\nClearly, by our choice of $\\widehat \\zeta$,\n\\begin{displaymath}\n    \\begin{cases}\n\\frac{\\partial u}{\\partial t}-\\Delta u\n= {\\widehat \\zeta}\\big[-2i\\kappa A_\\kappa\\cdot\\nabla\\psi_\\kappa -  |\\kappa\nA_\\kappa|^2\\psi_\\kappa + \\kappa^2\\psi_\\kappa\\big( 1 - |\\psi_\\kappa|^{2}\n\\big)&  \\\\\n\\qquad\\qquad\\quad - i\\kappa\\phi_\\kappa\\psi_\\kappa\\big]  + 2\\nabla{\\widehat \\zeta}\\cdot\\nabla\\psi_\\kappa\n  + \\psi_\\kappa \\Delta{\\widehat \\zeta}   & \\text{ in } (0,+\\infty)\\times  \\Omega\\, ,\\\\\n  u =0  & \\text{on }  (0,+\\infty)\\times \\partial\\Omega \\,.\n\\end{cases}\n\\end{displaymath}\nBy  \\cite[Theorem C.1]{alhe14} (this time applied to the Dirichlet\nLaplacian in $\\Omega$)  in the interval $(t_0-1,t_0+1)$\n\\begin{equation}\\label{eq:81}\n\\aligned\n&\\|u\\|_{L^2(t_0,t_0+1,H^2(\\Omega))} \\\\\n\\leq &\\big\\|{\\widehat \\zeta}\\big[-2i\\kappa A_\\kappa\\cdot\\nabla\\psi_\\kappa -  |\\kappa\nA_\\kappa|^2\\psi_\\kappa + \\kappa^2\\psi_\\kappa\\big( 1 - |\\psi_\\kappa|^{2}\n\\big)- i\\kappa\\phi_\\kappa\\psi_\\kappa\\big]\\big\\| _{L^2(t_0-1,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))} \\\\\n&+ \\|2\\nabla{\\widehat \\zeta}\\cdot\\nabla\\psi_\\kappa\n  + \\psi_\\kappa \\Delta{\\widehat \\zeta}\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))} + C\\|u(t_0-1,\\cdot)\\|_{2} \\,.\n\\endaligned\n\\end{equation}\n\nBy \\eqref{eq:69} we have that\n\\begin{equation}\n\\label{eq:82}\n  \\|2\\nabla{\\widehat \\zeta}\\cdot\\nabla\\psi_\\kappa  + \\psi_\\kappa \\Delta{\\widehat \\zeta}\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}\n\\leq C(1+\\|\\nabla\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))})  \\,.\n\\end{equation}\nAs\n\\begin{displaymath}\\\n  \\|\\nabla\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))} \\leq\n  \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}+\\|\\kappa A_\\kappa\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}\\,,\n\\end{displaymath}\nwe obtain in view of (\\ref{eq:70a})  and (\\ref{eq:72}) that\n\\begin{equation}\n\\label{eq:83}\n  \\|\\nabla\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))} \\leq C\\kappa^2 \\,.\n\\end{equation}\nSubstituting the above into (\\ref{eq:82}) yields\n\\begin{equation}\n  \\label{eq:84}\n\\|2\\nabla{\\widehat \\zeta}\\cdot\\nabla\\psi_\\kappa  + \\psi_\\kappa \\Delta{\\widehat \\zeta}\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))} \\leq C\\kappa^2\\,.\n\\end{equation}\n\nWe next observe that\n\\begin{displaymath}\n \\|{\\widehat \\zeta} |\\kappa A_\\kappa|^2\\psi_\\kappa \\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))} \\leq\n \\kappa^2\\|A_\\kappa\\|_{L^4(t_0-1,t_0+1,L^\\infty(\\Omega,{\\mathbb{R}}^2))}^2 \\|{\\widehat \\zeta}\\psi_\\kappa \\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}\\,.\n\\end{displaymath}\nSince $\\widehat \\zeta$ is supported in the set $\\omega_{\\delta,j}$, we may use\n\\eqref{eq:76}, which together with Agmon's inequality \\cite[Lemma\n13.2]{ag65}, (\\ref{time-dep-AB}), and \\eqref{eq:72}, yield, for $t_0$ large enough,\n\\begin{equation}\\label{eq:85}  \\aligned\n& \\|{\\widehat \\zeta} |\\kappa A_\\kappa|^2\\psi_\\kappa \\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}\\\\\n\\leq& C\\kappa\\|A_\\kappa\\|_{L^\\infty(t_0-1,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))}\n\\|A_\\kappa\\|_{L^2(t_0-1,t_0+1,H^2(\\Omega,{\\mathbb{R}}^2))}\\\\\n\\leq& C\\kappa^3\\,.\n\\endaligned\n\\end{equation}\nSimilarly,\n\\begin{equation}\\label{eq:86}\n\\aligned\n& \\|{\\widehat \\zeta}\\kappa A_\\kappa\\cdot\\nabla\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}\\\\\n\\leq& \\kappa\\|A_\\kappa\\|_{L^4(t_0,t_0+1,L^\\infty(\\Omega,{\\mathbb{R}}^2))}\n  \\big[\\|u\\|_{L^4(t_0-1,t_0+1,H^1(\\Omega))}\n  +\\|\\psi_\\kappa\\nabla{\\widehat \\zeta}\\|_{L^4(t_0-1,t_0+1,L^2(\\Omega))}\\big]\\\\\n\\leq&   C\\kappa^2 +\n  C\\kappa^2\\|u\\|^{1\/2}_{L^2(t_0-1,t_0+1,H^2(\\Omega))}\\|u\\|^\\frac 12 _{L^\\infty(t_0-1,t_0+1,L^2(\\Omega))} \\\\\n\\leq & C\\Big[\\kappa^2 + \\kappa^{3\/2}\\|u\\|^\\frac 12_{L^2(t_0-1,t_0+1,H^2(\\Omega))}\\Big]\\,.\n\\endaligned\n\\end{equation}\nSubstituting \\eqref{eq:86} together with \\eqref{eq:84},\nand\n\\eqref{eq:85} into \\eqref{eq:81} yields with the aid of \\eqref{eq:69}\n\\begin{equation}\\label{eq:80a}\n\\|u\\|_{L^2(t_0,t_0+1,H^2(\\Omega))} \\leq C\\Big[\\kappa^3 + \\kappa^{3\/2}\\|u\\|^{\\frac 12}_{L^2(t_0-1,t_0+1,H^2(\\Omega))}\\Big]\\,\n\\end{equation}\n\n\n\nProceeding as in the proof of Proposition \\ref{lem:asymp-exp}, we can\nassume $C\\geq 1$ in \\eqref{eq:80a} and set\n$$\n\\alpha_n = C^{-1} \\kappa^{-\\frac 32}  \\|u\\|_{L^2(n,n+1,H^2(\\Omega))} ^\\frac 12\\,.\n$$\nWe now can  rewrite \\eqref{eq:80a} in the form\n$$\n\\alpha_n^2 \\leq ( 1 +\\alpha_{n-1} + \\alpha_n)\\,,\n$$\nwhich is precisely \\eqref{inegaga}.\nWe can thus conclude (\\ref{eq:80}), and hence, for a new value of $C$,\n\\eqref{eq:87} easily follows.\n\\vspace{1ex}\n\n{\\it Step 2: Prove that\n\\begin{equation}\n  \\label{eq:89}\n  \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^2(t_0,t_0+1,H^1(\\Omega,{\\mathbb{R}}^2))}  \\leq C \\kappa^3 \\,.\n\\end{equation}}\n\\vspace{2ex}\n\nIt can be easily verified that\n\\begin{equation}\n\\label{eq:88}\n  \\|\\nabla_{\\kappa A_\\kappa}u(t,\\cdot)\\|_{1,2}\\leq \\|u(t,\\cdot)\\|_{2,2} + \\kappa\\|\\,|A_\\kappa|\\nabla u(t,\\cdot)\\|_2 +\n  \\kappa\\|u\\nabla A_{\\kappa}(t,\\cdot)\\|_2 \\,.\n\\end{equation}\nFurthermore, in the same manner we have obtained \\eqref{eq:86} we\nobtain, with the aid of \\eqref{eq:87}\n\\begin{displaymath}\n \\kappa  \\|\\,|A_\\kappa|\\nabla u\\|_{L^2(t_0,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))} \\leq C\n  \\kappa^{3\/2}\\|u\\|_{L^2(t_0,t_0+1,H^2(\\Omega,{\\mathbb{R}}^2))}^{1\/2}\\leq C\\kappa^3 \\,.\n\\end{displaymath}\nBy \\eqref{eq:69} and \\eqref{eq:72} we have that\n\\begin{displaymath}\n  \\kappa\\|u\\nabla A_{\\kappa}\\|_{L^2(t_0,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))} \\leq C\\kappa^2 \\,.\n\\end{displaymath}\nWe can now conclude \\eqref{eq:89} from \\eqref{eq:88}. \n\\vspace{1ex}\n\n\n{\\it Step 3: Prove \\eqref{eq:89a}.}\n\\vspace{1ex}\n\n In \\cite[(5.35)]{alhe14} it was shown that\n \\begin{equation}\n\\label{eq:90}\n    \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_{L^\\infty(t_0,t_0+1; L^2(\\Omega))}\\leq C\\kappa^3\\,.\n \\end{equation}\n (Note that while the setting in \\cite{alhe14} is different then - in\n particular, we assume there $J\\sim\\mathcal O(\\kappa)$ - the estimate  is\n still valid in the present case because $c=1$.)\nHence, we get\n\\begin{equation}\n\\label{eq:91}\n  \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^\\infty(t_0,t_0+1; L^2(\\Omega))}\\leq C\\kappa^3\\,.\n\\end{equation}\nWe now use Gagliardo-Nirenberg interpolation inequality (see  \\cite{ni59}) to\nobtain\n$$\n\\aligned\n &  \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^p(t_0,t_0+1; L^p(\\Omega))}^p\\qquad  \\\\\n& \\leq C \\int_{t_0}^{t_0+1}\n   \\|\\nabla_{\\kappa A_\\kappa}u(t,\\cdot)\\|_{1,2}^{p-2}\\|\\nabla_{\\kappa A_\\kappa}u(t,\\cdot)\\|_2^2 \\,dt\\\\\n& \\leq C   \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^2(t_0,t_0+1; H^1(\\Omega))}^{p-2} \\|\\nabla_{\\kappa\n    A_\\kappa}u\\|_{L^{\\frac{4}{4-p}}(t_0,t_0+1; L^2(\\Omega))}^2 \\,.\n\\endaligned\n$$\nConsequently,\n\\begin{equation}\\label{eq:92}\n\\aligned\n&\\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^p(t_0,t_0+1; L^p(\\Omega))}^p \\\\\n\\leq &C \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^2(t_0,t_0+1; H^1(\\Omega))}^{p-2} \\|\\nabla_{\\kappa\n    A_\\kappa}u\\|_{L^2(t_0,t_0+1; L^2(\\Omega))}^{4-p} \\|\\nabla_{\\kappa\n    A_\\kappa}u\\|_{L^\\infty(t_0,t_0+1; L^2(\\Omega))}^{p-2} \\,.\n\\endaligned\n\\end{equation}\n Multiplying (\\ref{eq:67}a) by ${\\widehat \\zeta}^2\\bar{\\psi}_\\kappa$ and integrating over\n$\\Omega$ we obtain for the real part  that\n\\begin{displaymath}\n  \\|\\nabla_{\\kappa A_\\kappa(t,\\cdot)}u(t,\\cdot)\\|_2^2 \\leq \\kappa^2\\|u (t,\\cdot)\\|_2^2-\\frac{1}{2}\n      \\frac{d\\|u (t,\\cdot)\\|_2^2}{dt}+ \\|\\psi_\\kappa (t,\\cdot) \\nabla{\\widehat \\zeta}\\|_2^2 \\,,\n\\end{displaymath}\nIntegrating over $(t_0,t_0+1)$ and using \\eqref{eq:76} we then obtain, for sufficiently large $t_0$,\n\\begin{displaymath}\n   \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^2(t_0,t_0+1; L^2(\\Omega))}^2  \\leq C \\,.\n\\end{displaymath}\nSubstituting the above, \\eqref{eq:89}, and \\eqref{eq:91}, into\n\\eqref{eq:92} we then obtain \\eqref{eq:89a}.\n\\end{proof}\n\n\nWe can now obtain the following improved regularity for $B_{1,\\kappa}$\n\\begin{proposition}\nLet $2<p\\leq12\/5$. For $0<\\delta<\\delta_0$,  there exists a constant $C=C(\\Omega,\\delta)>0$ such that for all $t_0>1$\n  and $\\kappa>\\kappa_0(\\delta)$ we have\n\\begin{equation}\n \\label{eq:93}\n\\|B_{1,\\kappa}\\|_{L^p(t_0,t_0+1,W^{1,p}(\\omega_{\\delta,j}))}  \\leq C \\,.\n\\end{equation}\n\\end{proposition}\n\\begin{proof} \nTaking the curl of \\eqref{eq:71} yields that $B_{1,\\kappa}$ is a weak\nsolution of\n\\begin{equation}\n\\label{eq:94}\n  \\begin{cases}\n       \\frac{\\partial  B_{1,\\kappa}}{\\partial t} - \\Delta B_{1,\\kappa}  = \\frac{1}{\\kappa}\\text{\\rm curl\\,}\n       \\Im(\\bar{\\psi}_\\kappa\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa) & \\text{ in } (0,+\\infty)\\times  \\Omega \\\\\n   B_{1,\\kappa} =0 & \\text{ on } (0,+\\infty)\\times  \\partial\\Omega \\,.\n  \\end{cases}\n\\end{equation}\n Let\n\\begin{displaymath}\n  \\mathcal B_{\\widehat \\zeta} = {\\widehat \\zeta}B_{1,\\kappa} \\,,\n\\end{displaymath}\nwhere the cutoff function $\\widehat{\\zeta}$ is defined by (\\ref{eq:77}).\nLet further $\\hat{\\Omega}(\\delta)\\subset\\Omega$ be smooth  and satisfy\n$$\n{\\rm supp}\\,\\widehat{\\zeta}\\subset \\hat{\\Omega}(\\delta)\\,.$$\nAs for any $V\\in H^1(\\Omega,{\\mathbb{R}}^2)$ we have that\n\\begin{displaymath}\n  \\text{\\rm curl\\,} V = \\text{\\rm div\\,} V_\\perp\\,,\n\\end{displaymath}\nit can be  easily verified\nfrom (\\ref{eq:94}) that\n\\begin{displaymath}\n  \\frac{\\partial  \\mathcal B_{\\widehat \\zeta}}{\\partial t} - \\, \\Delta \\mathcal B_{\\widehat \\zeta}  =\n    \\frac{1}{\\kappa}{\\widehat \\zeta}\\text{\\rm div\\,}\\big( \\Im(\\bar{\\psi}_\\kappa\\nabla_{\\kappa\n      A_\\kappa}\\psi_\\kappa)_\\perp\\big) -2\\text{\\rm div\\,}(B_{1,\\kappa}\\nabla{\\widehat \\zeta)}) +\n    B_{1,\\kappa}\\Delta\\widehat \\zeta\\,.\n\\end{displaymath}\nConsequently,\n\\begin{equation}\n\\label{eq:95}\n\\left\\{\n  \\begin{array}{rll}\n    \\frac{\\partial  \\mathcal B_{\\widehat \\zeta}}{\\partial t} - \\, \\Delta \\mathcal B_{\\widehat \\zeta}  &=\n    \\frac{1}{\\kappa}{\\widehat \\zeta}\\, \\text{\\rm div\\,}\\,\\big( \\Im(\\bar{\\psi}_\\kappa\\nabla_{\\kappa\n      A_\\kappa}\\psi_\\kappa)_\\perp\\big)& \\\\\n      &\\quad  -\\text{\\rm div\\,}(2  B_{1,\\kappa}\\nabla{\\widehat \\zeta} +\n  \\nabla \\Delta_D^{-1}(B_{1,\\kappa}\\Delta{\\widehat \\zeta}) ) & \\text{in } (t_0-1,t_0+1)\\times\\hat{\\Omega} \\\\\n   \\mathcal B_{\\widehat \\zeta} &=0 & \\text{on }  (t_0-1,t_0+1)\\times\\partial\\hat{\\Omega} \\\\\n\\mathcal B_{\\widehat \\zeta} (t_0-1,\\cdot) &  =  \\widehat \\zeta\\, \\mathcal B_{1,\\kappa} (t_0-1,\\cdot)  &\\text{in } \\hat{\\Omega}  \\,.\n  \\end{array}\n  \\right.\n\\end{equation}\nIn the above $\\Delta_D^{-1}$ denotes the inverse Dirichlet Laplacian in $\n\\hat{\\Omega}$.\n\nIn order to apply\n  \\cite[Theorem 1.6]{by07}  which is devoted to the case of parabolic\n  operators written in divergence form and with zero initial condition\n  we first decompose the solution of \\eqref{eq:95} into two\n  Cauchy-Dirichlet problems.\n  The first of them is:\n  \\begin{equation} \\label{eq:117a}\n\\left\\{\n  \\begin{array}{rll}\n    \\frac{\\partial U_1}{\\partial t} - \\, \\Delta U_1 &= \\text{\\rm div\\,} f_1\n    & \\text{ in } (t_0-1,t_0+1)\\times\\hat{\\Omega}\\,, \\\\\n   U_1 &=0 & \\text{ on }  (t_0-1,t_0+1)\\times\\partial\\hat{\\Omega} \\,,\\\\\nU_1 (t_0-1,\\cdot)& = 0 &\\text{ in } \\hat{\\Omega}  \\,,\n  \\end{array}\n  \\right.\n\\end{equation}\nin which\n\\begin{equation}\\label{eq:117aa}\nf_1 =   \\frac{1}{\\kappa}{\\widehat \\zeta}\\, \\,\\big( \\Im(\\bar{\\psi}_\\kappa\\nabla_{\\kappa\n      A_\\kappa}\\psi_\\kappa)_\\perp\\big)  - 2  B_{1,\\kappa}\\nabla{\\widehat \\zeta} -\n  \\nabla \\Delta_D^{-1}(B_{1,\\kappa}\\Delta{\\widehat \\zeta}) \\,.\n \\end{equation}\nThe second one is:\n\\begin{equation}\n\\label{eq:117b}\n\\left\\{\n  \\begin{array}{rll}\n    \\frac{\\partial U_2}{\\partial t} - \\, \\Delta U_2  &= F_2\n & \\text{in } (t_0-1,t_0+1)\\times\\hat{\\Omega} \\,,\\\\\n   U_2 &=0 & \\text{on }  (t_0-1,t_0+1)\\times\\partial\\hat{\\Omega} \\,,\\\\\nU_2(t_0-1,\\cdot )&  =  \\widehat \\zeta\\, \\mathcal B_{1,\\kappa} (t_0-1,\\cdot)  &\\text{in } \\hat{\\Omega}  \\,,\n  \\end{array}\n  \\right.\n\\end{equation}\nwhere\n \\begin{equation} \\label{eq:117ba}\n F_2:= -  \\frac{1}{\\kappa}  \\,\\big( \\Im(\\bar{\\psi}_\\kappa \\nabla {\\widehat \\zeta}\\cdot (\\nabla_{\\kappa\n      A_\\kappa}\\psi_\\kappa)_\\perp) \\big)\\,.\n \\end{equation}\nBy uniqueness of the weak solution, we have\n\\begin{equation} \\label{eq:117c}\n \\mathcal B_{\\widehat \\zeta}  = U_1 + U_2  \\text{ in } (t_0-1,t_0+1)\\times\\hat{\\Omega}\\,.\n \\end{equation}\nWe now separately estimate $U_1$ and $U_2$ in  $L^p (t_0,t_0+1, W^{1,p}( \\hat \\Omega))\\,$.\n\\vskip0.05in\n\n{\\it Estimate of $U_1$}\n\\vspace{1ex}\n\nWe apply  \\cite[Theorem 1.6]{by07} to obtain\n\\begin{equation} \\label{eq:96}\n\\| U_1\\|_{L^p(t_0,t_0+1,W^{1,p}(\\Omega))}\\\\\n\\leq C \\,\\|f_1\\|_{L^p(t_0,t_0+1,L^p(\\Omega,{\\mathbb{R}}^2)}\\,.\n\\end{equation}\nIt can be easily verified, by the Gagliardo-Nirenberg interpolation\ninequality \\cite{ni59} that, for\nall $2<p<4$, there exists a constant $C$ such that,\nfor any $\\phi \\in L^2(t_0,t_0+1,H^1(\\Omega))\\cap L^\\infty (t_0,t_0+1,L^2(\\Omega))$, we have \\begin{multline}\\label{eq:76a}\n  \\|\\phi \\|_{L^p(t_0,t_0+1,L^p(\\Omega,{\\mathbb{R}}^2))}^p\\leq C \\int_{t_0}^{t_0+1}\\|\\phi (\\tau,\\cdot)\n  \\|_2^2 \\|\\phi (\\tau,\\cdot) \\|_{1,2}^{p-2}\\,d\\tau \\\\ \\leq C  \\| \\phi \\|_{L^{\\frac{4}{4-p}}(t_0,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))}^2\\|\\phi\\|_{L^2(t_0,t_0+1,H^1(\\Omega,{\\mathbb{R}}^2))}^{p-2}\\,.\n\\end{multline}\nBy \\eqref{eq:76a},  \\eqref{eq:72}, Remark~\\ref{usefulrem}, and Sobolev embeddings we\nhave:\n\\begin{equation}\\label{eq:97}\n\\| \\mathcal B_{\\widehat \\zeta} (t_0,\\cdot)\\|_p +\n\\|\\mathcal B_{1,\\kappa} \\nabla{\\widehat \\zeta}\\|_{L^p(t_0,t_0+1,L^p(\\Omega))} +\n\\|\\mathcal B_{1,\\kappa}\n\\Delta{\\widehat \\zeta}\\|_{W^{-1,p}(t_0,t_0+1,L^p(\\Omega,{\\mathbb{R}}^2))}\n\\leq C\\,.\n\\end{equation}\nFurthermore, by \\eqref{eq:69} we have that\n\\begin{displaymath}\n  \\begin{array}{l}\n\\|{\\widehat \\zeta}\\Im(\\bar{\\psi}_\\kappa\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa)\\|_{L^p(t_0,t_0+1,L^p(\\Omega,{\\mathbb{R}}^2))}\\\\\n\\qquad   \\leq\n\\|\\psi_\\kappa\\nabla{\\widehat \\zeta}\\|_{L^p(t_0,t_0+1,L^p(\\Omega,{\\mathbb{R}}^2))}+  \\|\\nabla_{\\kappa\n  A_\\kappa}({\\widehat \\zeta}\\psi_\\kappa)\\|_{L^p(t_0,t_0+1,L^p(\\Omega,{\\mathbb{R}}^2))} \\\\\n\\qquad \\leq \\|\\nabla_{\\kappa A_\\kappa}({\\widehat \\zeta}\\psi_\\kappa)\\|_{L^p(t_0,t_0+1,L^p(\\Omega,{\\mathbb{R}}^2))} +\nC\\,.\n\\end{array}\n\\end{displaymath}\nSubstituting the above together with \\eqref{eq:97}\ninto \\eqref{eq:96} yields for $t_0$ large enough\n\\begin{equation}\\label{eq:98}\n\\| U_1\\|_{L^p(t_0,t_0+1,W^{1,p}(\\Omega))} \\leq C\\Big(1 +\\frac{1}{\\kappa}\n \\|\\nabla_{\\kappa A_\\kappa}({\\widehat \\zeta}\\psi_\\kappa)\\|_{L^p(t_0,t_0+1,L^p(\\Omega,{\\mathbb{R}}^2))}\\Big)  \\,.\n\\end{equation}\n Substituting \\eqref{eq:89a}\n into \\eqref{eq:98} yields\n\\begin{equation}\\label{eq:79a}\n\\| U_1\\|_{L^p(t_0,t_0+1,W^{1,p}(\\Omega))}\n\\leq C(1+\\kappa^{5-12\/p})\n \\leq  \\widehat C\\,,\n\\end{equation}\nsince $2< p\\leq 12\/5$.\n\\vspace*{2ex}\n\n{\\it Estimate of $U_2$}\n\\vspace{1ex}\n\n\nHere we apply first $L^2$ estimates and then combine them with Sobolev's\nestimates. It is in this part that we need the information on $F_2$ and $U_2$\nin $[t_0-1,t_0+1)\\times \\hat \\Omega$ in order to bound the various norms on\n$(t_0,t_0+1)\\times \\hat \\Omega$. We begin by applying once again \\cite[Theorem\nC.1]{alhe14} (combining $(C.1)$ and $(C.2)$ there) to obtain\n $$\n \\begin{array}{l}\n \\| U_2\\|_{ L^2(t_0,t_0+1, H^2)} + \\| U_2 \\|_{L^\\infty (t_0,t_0+1,H^1)}\\\\ \\qquad\n  \\leq C \\left(  \\| F_2\\|_{L^2 ((t_0-1,t_0 +1)\\times \\hat \\Omega)} + \\| U_2(t_0-1,\\cdot)\\|_{L^2(\\hat \\Omega)} \\right)\\,,\n  \\end{array}\n  $$\n  where $F_2$ and $U_2(t_0-1,\\cdot)$ given in \\eqref{eq:117ba} and\n  \\eqref{eq:117b}.  Applying Gagliardo-Nirenberg's inequality yields,\n  for $ 2 <p <4$,\n  \\begin{equation}\\label{eq:89b}\n   \\| U_2\\|_{ L^p(t_0,t_0+1,W^{1,p})} \\leq C \\left(  \\| F_2\\|_{L^2\n       ((t_0-1,t_0 +1)\\times \\hat \\Omega)} + \\| U_2(t_0-1,\\cdot)\\|_{L^2(\\hat \\Omega)}\n   \\right)\\,,\n  \\end{equation}\nBy \\eqref{eq:70aa} we have that\n\\begin{displaymath}\n  \\|  F_2\\|_{L^2((t_0-1,t_0 +1)\\times \\hat \\Omega)} \\leq C \\,.\n\\end{displaymath}\nFurthermore, using \\eqref{eq:72}, with Remark \\ref{usefulrem} in mind\nyields\n\\begin{displaymath}\n  \\| U_2(t_0-1,\\cdot)\\|_2 \\leq C \\,.\n\\end{displaymath}\nConsequently, by \\eqref{eq:89b}, there exist, for any $2 <p <4$ and\nany $\\delta >0\\,,$ constants $C(\\delta)$ and $\\kappa(\\delta)$ such that for any $\\kappa\n\\geq \\kappa_0(\\delta)$ and any $t_0 >1$ we have\n   \\begin{equation}\n\\label{eq:89c}\n   \\| U_2\\|_{ L^p(t_0,t_0+1,W^{1,p})} \\leq C (\\delta)\\,.\n  \\end{equation}\n The combination of \\eqref{eq:79a} and \\eqref{eq:89c}  together with\n \\eqref{eq:117c} completes the proof of the proposition.\n\\end{proof}\n\nWe can now establish the exponential decay of $\\psi_\\kappa$.\n\\begin{proposition}\n  \\label{asymp-exp-2}\n  Let $\\omega_{\\delta,j}$ ($j\\in\\{1,2\\}$) be given by \\eqref{eq:56}.  Suppose\n  that for some $k\\in\\{1,2\\}$ \\eqref{eq:75} is satisfied.  Then, there\n  exist $C>0$ and $\\delta_0>0$, and, for any  $0<\\delta<\\delta_0$,\n  $\\kappa_0(\\delta)$, such that for any  $\\kappa\\geq \\kappa_0(\\delta)$ we have\n\\begin{equation}\n\\label{eq:99}\n\\limsup_{t\\to\\infty}\\int_{\\omega_{\\delta,j}} \\exp \\Big(\\delta^{1\/2}\n \\kappa d(x,\n\\Gamma_{\\delta,j})\\Big) |\\psi_\\kappa|^2(t,x) \\,dx \\leq C  \\,.\n\\end{equation}\n\\end{proposition}\n\\begin{proof} \n  Without loss of generality we may, as before,  assume $h_j>0$.\nLet $\\check{\\chi}$ and $\\check{\\zeta}$ be defined by \\eqref{check1} and\n\\eqref{check2}.\n\\vspace{1ex}\n \n{\\it Step 1: Prove that\n\\begin{equation}\n\\label{eq:101}\n  \\|\\check{\\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2 \\leq \\|\\check{\\zeta}\\psi_0\\|_2^2e^{-2\\gamma\\kappa^2t} +\n  \\frac{C(\\delta)}{\\kappa^4}+ \\int_0^t\n  e^{-2\\gamma\\kappa^2(t-\\tau)}\\big|\\langle\\kappa B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle(\\tau)\\big|\\,d\\tau \\,.\n\\end{equation}}\n\\vspace{1ex}\n      \nMultiplying\n(\\ref{eq:67}a) by $\\check{\\zeta}^2 \\bar{\\psi}$ and integrating by parts yields\n\\begin{displaymath}\n  \\frac{1}{2}\\frac{d}{dt}\\left( \\|\\check{\\zeta}\\psi_\\kappa(t,\\cdot) \\|_2^2 \\right)  +  \\|\\nabla_{\\kappa A_\\kappa(t,\\cdot)}(\\check{\\zeta}\\psi_\\kappa(t,\\cdot))\\|_2^2 \\leq\n  \\kappa^2\\|\\check{\\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2 + \\|\\psi_\\kappa (t,\\cdot) \\nabla\\check{\\zeta}\\|_2^2 \\,.\n\\end{displaymath}\nBy Theorem 2.9 in \\cite{AHS} we have\n$$\\aligned\n  \\|\\nabla_{\\kappa A_\\kappa}(\\check{\\zeta}\\psi_\\kappa)\\|_2^2\n&  \\geq \\langle\\kappa B_\\kappa(t,\\cdot) \\check{\\zeta}\\psi_\\kappa(t,\\cdot),\\check{\\zeta}\\psi_\\kappa (t,\\cdot)\\rangle\\\\  & =\n  \\kappa^2\\langle B_n\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle + \\langle\\kappa B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle  \\\\\n&\\geq\n  \\kappa^2\\Big(1+\\frac{\\delta}{2}\\Big)\\|\\check{\\zeta}\\psi_\\kappa\\|_2^2+ \\langle\\kappa B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle \\,.\n\\endaligned\n$$\nWe can thus write\n\\begin{equation}\\label{eq3.32}\n\\aligned\n&\\frac{1}{2}\\frac{d}{dt}\\left( \\|\\check{\\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2\\right)  +\n\\kappa^2\\Big( \\frac{\\delta}{2}- \\frac{\\delta}{4}\n\\Big)\\|\\check{\\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2\\\\\n\\leq&\n  \\|\\psi_\\kappa (t,\\cdot) \\nabla\\eta\\|_2^2 + \\|\\tilde \\chi\\,\\psi_\\kappa (t,\\cdot)\\nabla\\eta_\\delta\\|_2^2- \\langle\\kappa B_{1,\\kappa}(t,\\cdot)\\check{\\zeta}\\psi_\\kappa(t,\\cdot),\\check{\\zeta}\\psi_\\kappa(t,\\cdot)\\rangle \\,.\n\\endaligned\n\\end{equation}\nBy \\eqref{eq:76}, for every $0<\\delta\\leq\\delta_0$, we have\n\\begin{displaymath}\n  \\|\\psi_\\kappa (t,\\cdot) \\nabla\\eta\\|_2^2 + \\|\\tilde \\chi\\, \\psi_\\kappa (t,\\cdot) \\,\\nabla\\eta_\\delta\\|_2^2 \\leq \\frac{C}{\\kappa^2} \\,,\n\\end{displaymath}\nwhich when substituted into \\eqref{eq3.32} yields\n\\begin{equation}\n \\label{eq:100}\n   \\frac{1}{2}\\frac{d}{dt}\\left( \\|\\check{\\zeta}\\psi_\\kappa (t,\\cdot)\\|_2^2\\right)  +   \\kappa^2\\gamma\\|\\check{\\zeta}\\psi_\\kappa (t,\\cdot) \\|_2^2\n   \\leq  \\frac{C}{\\kappa^2} - \\langle\\kappa B_{1,\\kappa}(t,\\cdot) \\check{\\zeta}\\psi_\\kappa(t,\\cdot) ,\\check{\\zeta}\\psi_\\kappa (t,\\cdot)\\rangle \\,,\n\\end{equation}\nwhere $\\gamma=\\delta\/4$. We now get \\eqref{eq:101} from \\eqref{eq:100}. \n\\vspace{1ex}\n\n\n{\\it Step 2: Prove that,  for all $n\\geq2$,\n\\begin{equation}\n\\label{eq3.33}\n\\aligned\n&\\|\\check{\\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_{L^\\infty(t^*+n-1,t^*+n,L^2(\\Omega))}^2 \\\\\n\\leq& C \\kappa^{-(p-2)\/p}\\|\\check{\\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_{L^\\infty(t^*+n-2,t^*+n,L^2(\\Omega))}^2  +{C\\over\\kappa^4} \\,,\n\\endaligned\n\\end{equation}\nwhere $C$ is independent of $\\kappa\\,$.}\n\\vspace{1ex}\n\nTo prove \\eqref{eq3.33} we need to estimate the last term on the\nright-hand-side of \\eqref{eq:101}. To this end we first write\n\\begin{multline*}\n  \\int_0^t  e^{-2\\gamma\\kappa^2(t-\\tau)}\\big|\\langle\\kappa\n  B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle (\\tau) \\big|\\,d\\tau  \\\\\n\\leq \\kappa \\left(\\int_{t-1}^t  e^{-2\\gamma\\kappa^2(t-\\tau)}\n{  \\|B_{1,\\kappa}(\\tau, \\cdot\n  )\\|_{L^\\infty(\\omega_{\\delta\/2,j})}}\\,d\\tau\\right)\\,\\cdot\\|\\check{\\zeta}\\psi_\\kappa\\|_{L^\\infty(t-1,t;L^2(\\Omega))}\\\\\n+ \\kappa\\int_0^{t-1}\ne^{-2\\gamma\\kappa^2(t-\\tau)}\\|B_{1,\\kappa}(\\tau,\\cdot)\\|_1\\,d\\tau\n\\|\\check{\\zeta}\\psi_\\kappa\\|_{L^\\infty(0,t-1;L^2(\\Omega))}\\,.\n\\end{multline*}\nFor the last term on the right-hand-side we have in view of Remark~\\ref{usefulrem},  \\eqref{check1}, \\eqref{check2}, and \\eqref{eq:69} that\n\\begin{displaymath}\n  \\kappa\\int_0^{t-1}\ne^{-2\\gamma\\kappa^2(t-\\tau)}\\|B_{1,\\kappa}(\\tau,\\cdot)\\|_1\\,d\\tau\n\\|\\check{\\zeta}\\psi_\\kappa\\|_{L^\\infty(0,t-1;L^2(\\Omega))}\\leq Ce^{-2\\gamma\\kappa^2}e^{C\\kappa}\\,.\n\\end{displaymath}\nHence for sufficiently large $\\kappa$ we have\n\\begin{multline}\n\\label{eq:102}\n  \\int_0^t  e^{-2\\gamma\\kappa^2(t-\\tau)}\\big|\\langle\\kappa\n  B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle (\\tau) \\big|\\,d\\tau \\leq \\\\\n\\kappa \\left(\\int_{t-1}^t  e^{-2\\gamma\\kappa^2(t-\\tau)}  \\|B_{1,\\kappa}(\\tau, \\cdot\n  )\\|_{L^\\infty(\\omega_{\\delta\/2,j})}\\,d\\tau\\right)\\,\\cdot\\|\\check{\\zeta}\\psi_\\kappa\\|_{L^\\infty(t-1,t;L^2(\\Omega))}+ Ce^{-\\gamma\\kappa^2}\\,.\n\\end{multline}\nSince by Sobolev embeddings\n\\begin{displaymath}\n  \\|B_{1,\\kappa}\\|_{L^p(t-1,t,L^\\infty(\\omega_{\\delta\/2,j}))} \\leq C\\|B_{1,\\kappa}\\|_{L^p(t-1,t,W^{1,p}(\\omega_{\\delta\/2,j}))} \\,,\n\\end{displaymath}\nwe can use \\eqref{eq:93}  to obtain, for\nsufficiently large $t$ and $\\kappa$, and for any $2<p\\leq12\/5\\,$,\n$$ \\aligned\n& \\int_{t-1}^t  e^{-2\\gamma\\kappa^2(t-\\tau)}\\|B_{1,\\kappa}(\\tau, \\cdot)\\|_{L^\\infty(\\omega_{\\delta\/2,j})}\\,d\\tau \\\\\n\\leq&\n \\|B_{1,\\kappa}(\\tau, \\cdot)\\|_{L^p(t-1,t;L^\\infty(\\omega_{\\delta\/2,j}))}\\,\\Big[\\int_{t-1}^t\n e^{-\\frac{2p}{p-1}\\gamma\\kappa^2(t-\\tau)}\\,d\\tau\\Big]^{\\frac{p-1}{p}} \\\\\n\\leq&\n C\\kappa^{-2(p-1)\/p}\\,.\n\\endaligned\n$$\nSubstituting the above into \\eqref{eq:102} yields\n$$\\aligned\n&\\int_0^t  e^{-2\\gamma\\kappa^2(t-\\tau)}\\big|\\langle\\kappa B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle\\big|\\,d\\tau \\\\ \n \\leq& C\\kappa^{-(p-2)\/p}\\|\\check{\\zeta}\\psi_\\kappa\\|_{L^\\infty(t-1,t;L^2(\\Omega))} + Ce^{- \\frac 14 \\delta \\kappa^2 } \\,,\n\\endaligned\n$$\nwhich, when substituted into \\eqref{eq:101},  yields \\eqref{eq3.33}. \n\\\\\n\n\n{\\it Step 3: Prove \\eqref{eq:99}.}\n\\\\\n\nLet now\n\\begin{displaymath}\n  b_n = \\|\\check{\\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_{L^\\infty(t^*+n-1,t^*+n,L^2(\\Omega))}^2 \\,.\n\\end{displaymath}\nFrom \\eqref{eq3.33} we get that if $p=12\/5$ then for sufficiently large\n$\\kappa$  it holds that\n\\begin{displaymath}\nb_n \\leq C \\left( \\kappa^{-1\/6}(b_{n-1} + b_n)+  1\\right)\\,,\n\\end{displaymath}\nwhere $C$ is independent of $\\kappa$.\nFor another  constant $\\widehat C$, we get for sufficiently large $\\kappa$,\n\\begin{displaymath}\nb_n \\leq \\widehat C (\\kappa^{-1\/6} b_{n-1} +  1)\\,.\n\\end{displaymath}\nThis immediately implies, for $\\kappa$ large enough so that $\\widehat C\n\\kappa^{-\\frac 16} \\leq \\frac 12$, the upperbound\n\\begin{displaymath}\n  \\limsup_{n\\to\\infty}b_n \\leq C_0\\,,\n\\end{displaymath}\nwhere $C_0$ is independent of $\\kappa$. Consequently,\n$$\n\\limsup_{t\\to\\infty}\\int_{\\omega_{\\delta,j}} \\exp\\Big(\\delta^\\frac 12 \\kappa d(x,\n\\partial\\omega_{\\delta,j})\\Big) |\\psi_\\kappa(t,x)|^2 \\,dx \\leq C_0\\,,\n$$\nwhich readily yields \\eqref{eq:99}.\n\\end{proof}\n\\newpage\n\n\\section{Large domains}\n\\label{sec:5}\nThe main goal of this section is to prove Proposition\n\\ref{largedomain}. To this end it is more convenient to consider \\eqref{eq:18}\nin a fixed domain. Assuming that $0\\in\\Omega$ we set $\\epsilon=1\/R$ (hence we have $\\epsilon \\ll 1$)\nand apply the transformation,\n\\begin{equation}\n\\label{eq:103}\n\\psi_\\epsilon(\\epsilon x) = \\psi(x)   \\, , \\qquad A(x) = \\epsilon^{-1}A_\\epsilon(\\epsilon x)\\,, \\qquad\n  \\phi(x) = \\phi_\\epsilon(\\epsilon x) \\,,\n\\end{equation}\nIf we write $y=\\epsilon x$, we have:\n  \\begin{displaymath}\n     \\text{\\rm curl\\,}_x^2A  =  \\epsilon\\,\\text{\\rm curl\\,}_y^2A_\\epsilon \\quad ; \\quad\n     \\nabla_x\\phi = \\epsilon\\,\\nabla_y\\phi_\\epsilon  \\quad ;\n     \\quad  \\nabla_A\\psi  =  \\epsilon\\,\\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon \\,,\n  \\end{displaymath}\n  leading to the following system for $(\\psi_\\epsilon, A_\\epsilon, \\phi_\\epsilon)$\n\\begin{subequations}\n\\label{eq:104}\n\\begin{alignat}{2}\n& \\Delta_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon + \\frac{\\psi_\\epsilon}{\\epsilon^2}   \\big( 1 - |\\psi_\\epsilon|^{2}\n\\big)-\\frac{i}{\\epsilon^2}\\phi_\\epsilon\\psi_\\epsilon =0 & \\quad \\text{ in } \\Omega\\, ,\\\\\n & \\text{\\rm curl\\,}^2A_\\epsilon +  \\nabla\\phi_\\epsilon  =  \\Im\\big(\\bar\\psi_\\epsilon\n \\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon\\big)  & \\quad \\text{ in }  \\Omega\\,,\\\\\n  &\\psi_\\epsilon=0 &\\quad \\text{ on }  \\partial\\Omega_c \\,, \\\\\n & \\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon\\cdot\\nu=0 & \\quad \\text{ on }  \\partial\\Omega_i \\,,\\\\\n & \\frac{\\partial\\phi_\\epsilon}{\\partial\\nu}= f(\\epsilon)J  &\\quad \\text{ on } \\partial\\Omega \\,,\\\\\n& \\frac{\\partial\\phi_\\epsilon}{\\partial\\nu}= 0 &\\quad \\text{ on } \\partial\\Omega_i\\,,\\\\[1.2ex]\n&\\Xint-_{\\partial\\Omega}\\text{\\rm curl\\,} A_\\epsilon(x)\\,ds = f(\\epsilon)h_{ex}\\,.\n\\end{alignat}\n\\end{subequations}\nIn the above\n$$\nf(\\epsilon)=F(1\/\\epsilon)= \\epsilon^{-\\alpha}\\,.\n$$\nIt follows from \\eqref{eq:6} that\n\\begin{displaymath}\n  h_j = b_j\\epsilon^{-\\alpha}\\,, \\quad j=1,2\\,,\n\\end{displaymath}\nwhere $b_j$ is independent of $\\epsilon$ for $j=1,2\\,$.\\\\\n\nWe assume that $A_\\epsilon$ is in the Coulomb gauge space \\eqref{eq:5}, and\nsuppose that a weak solution $(\\psi_\\epsilon,A_\\epsilon,\\phi_\\epsilon)\\in H^1(\\Omega,{\\mathbb{C}})\\times\nH^1(\\Omega,{\\mathbb{R}}^2)\\times L^2(\\Omega)$ exists.  Proposition \\ref{largedomain} can now be\nreformulated in the following way:\n\n\\begin{proposition}\n\\label{semiclassical}\n Let $(\\psi_\\epsilon,A_\\epsilon,\\phi_\\epsilon)$ denote a solution of \\eqref{eq:104}, and let\n$h$ be given by \\eqref{eq:9}.\n  Suppose that for some $0<\\gamma<1$ and $\\epsilon_0>0$  we have\n  \\begin{displaymath}\n    \\epsilon^{-\\gamma}<h\\,, \\quad \\forall\\, 0<\\epsilon<\\epsilon_0\\,.\n  \\end{displaymath}\nThen,  there exists a compact  set $K\\subset\\Omega$, $C>0$, and $\\alpha >0$,\nsuch that for any  $0\\,<\\,\\epsilon\\,<\\epsilon_0\\,$ we have\n\\begin{equation}\n\\label{eq:105}\n\\int_{K} |\\psi_\\epsilon(x)|^2 \\,dx \\leq C  e^{-\\alpha\/\\epsilon} \\,.\n\\end{equation}\n\\end{proposition}\n\nWe split the proof of Proposition \\ref{semiclassical} into several\nsteps, to each of them we dedicate a separate lemma. We begin by observing,\nas in Section \\ref{sec:3}, that\n\\begin{equation}\n  \\label{eq:106}\n\\|\\psi_\\epsilon\\|_\\infty\\leq 1\\,.\n\\end{equation}\nLet\n\\begin{equation}\n\\label{eq:107}\n  A_{1,\\epsilon}=A_\\epsilon-\\epsilon^{-\\alpha}A_n,\\quad\n\\phi_{1,\\epsilon}=\\phi_\\epsilon-\\epsilon^{-\\alpha}\\phi_n,\n\\end{equation}\nSet further\n\\begin{equation}\n\\label{eq:108}\n  B_{1,\\epsilon}= \\text{\\rm curl\\,} A_{1,\\epsilon}\\,;  \\quad B_\\epsilon= \\text{\\rm curl\\,} A_\\epsilon  \\,.\n\\end{equation}\nBy (\\ref{eq:104}b) and (\\ref{eq:11}a),   we then have\n\\begin{subequations}\n\\label{eq:109}\n\\begin{empheq}[left={\\empheqlbrace}]{alignat=2}\n  &\\nabla_\\perp B_{1,\\epsilon} +  \\nabla\\phi_{1,\\epsilon}  =  \\Im(\\bar\\psi_\\epsilon\\, \\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon)  &  \\text{ in }\n  \\Omega\\,,\\\\\n   &\\frac{\\partial\\phi_{1,\\epsilon}}{\\partial\\nu}= 0 & \\text{ on } \\partial\\Omega \\,, \\\\\n&\\Xint-_{\\partial\\Omega}B_{1,\\epsilon}(x)\\,ds = 0 \\,. &\n\\end{empheq}\n\\end{subequations}\nNote that since $\\partial B_{1,\\epsilon}\/\\partial\\tau=\\partial\\phi_{1,\\epsilon}\/\\partial\\nu=0$ on $\\partial\\Omega$\nwe must have by (\\ref{eq:109}c) that\n\\begin{equation}\n\\label{eq:110}\n  B_{1,\\epsilon}|_{\\partial\\Omega}\\equiv 0\\,.\n\\end{equation}\n\n\n We begin with the following auxiliary estimate.\n\\begin{lemma}\n  \\label{lem:magnetic-bound}\nLet $w_\\epsilon$ denote the solution of\n\\begin{equation}\\label{eq:111}\n \\left\\{\\aligned\n&    \\Delta w_\\epsilon - \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2w_\\epsilon = 0\\quad & \\text{\\rm in }\\Omega\\;, \\\\\n&    w_\\epsilon = B_\\epsilon - 1 \\quad& \\text{\\rm on } \\partial\\Omega.\n\\endaligned\n\\right.\n\\end{equation}\nUnder the assumptions on $J$ and $\\Omega$ in (\\ref{eq:3})-(\\ref{hyptopolo}) we have\n\\begin{equation}\n \\label{eq:112}\n\\|B_\\epsilon-1-w_\\epsilon\\|_\\infty \\leq  \\frac{1}{2} \\,.\n\\end{equation}\nFurthermore, we have\n\\begin{equation}\n\\label{eq:113}\n\\|\\nabla\\phi_\\epsilon\\|_2+ \\|\\phi_\\epsilon\\|_\\infty \\leq  C\\epsilon^{-\\alpha}   \\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n  As can be easily verified from (\\ref{eq:104}a) we have (see in\n  \\cite{al02} (formula (2.4) in the case when $\\phi_\\epsilon=0$),\n  \\begin{equation}\n    \\label{eq:114}\n \\frac 12 \\Delta|\\psi_\\epsilon|^2 = -\\frac{|\\psi_\\epsilon|^2}{\\epsilon^2}(1-|\\psi_\\epsilon|^2) +\n|\\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon|^2 \\,.\n  \\end{equation}\nFurthermore, taking the curl of (\\ref{eq:104}b) yields\n\\begin{displaymath}\n  \\Delta B_\\epsilon - \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2B_\\epsilon  =\n   -\\Im (\\nabla_{\\epsilon^{-2}A_\\epsilon}\\bar\\psi_\\epsilon\\times\\nabla_{\\epsilon^{-2}A_\\epsilon} \\psi_\\epsilon) \\,.\n\\end{displaymath}\nNote that\n$$\\aligned\n&\\text{\\rm curl\\,} \\Im(\\bar\\psi_\\epsilon\\, \\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon) \\\\\n=&\\Im(\\nabla\\bar\\psi_\\epsilon\\times \\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon\n  -i\\epsilon^{-2}\\bar\\psi_\\epsilon A_\\epsilon\\times\\nabla\\psi_\\epsilon) -  \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2B_\\epsilon\\\\\n=&\\Im (\\nabla_{\\epsilon^{-2}A_\\epsilon}\\bar \\psi_\\epsilon\\times\\nabla_{\\epsilon^{-2}A_\\epsilon} \\psi_\\epsilon) -\n  \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2B_\\epsilon \\,.\n\\endaligned\n$$\nLet \n$$u_\\epsilon=B_\\epsilon-1 + {|\\psi_\\epsilon|^2\\over 2}-w_\\epsilon\\,.\n$$\nCombining the above and \\eqref{eq:114}\nyields that (cf. also \\cite{al02})\n\\begin{displaymath}\n  \\begin{cases}\n    \\Delta u_\\epsilon - \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2u_\\epsilon = |\\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon|^2 -\\Im (\\nabla_{\\epsilon^{-2}A_\\epsilon}\\bar \\psi_\\epsilon\\times\\nabla_{\\epsilon^{-2}A_\\epsilon} \\psi_\\epsilon)\n + \\frac{1}{2\\epsilon^2} |\\psi_\\epsilon|^4\\geq 0 &\n\\text{in } \\Omega, \\\\\nu_\\epsilon = \\frac{|\\psi_\\epsilon|^2}{2} & \\text{on } \\partial\\Omega.\n  \\end{cases}\n\\end{displaymath}\nBy the weak maximum principle (cf. for instance \\cite[Theorem\n8.1]{GT}) and \\eqref{eq:106} we obtain that for sufficiently small $\\epsilon$\n\\begin{displaymath}\n  u_\\epsilon (x)\\leq \\frac{1}{2}\\,.\n\\end{displaymath}\nThe lower bound in \\eqref{eq:112} follows easily by setting\n$$\\tilde{u}_\\epsilon = -B_\\epsilon-1 + {|\\psi_\\epsilon|^2\\over 2}+w_\\epsilon\n$$ \nto obtain\n\\begin{displaymath}\n  \\begin{cases}\n    \\Delta \\tilde{u}_\\epsilon - \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2\\tilde{u}_\\epsilon = |\\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon|^2  +\n\\Im (\\nabla_{\\epsilon^{-2}A_\\epsilon}\\bar \\psi_\\epsilon\\times\\nabla_{\\epsilon^{-2}A_\\epsilon} \\psi_\\epsilon)  + \\frac{1}{2\\epsilon^2} |\\psi_\\epsilon|^4 \\geq 0 &\n\\text{in } \\Omega, \\\\\n\\tilde{u}_\\epsilon = \\frac{|\\psi_\\epsilon|^2}{2} & \\text{on } \\partial\\Omega,\n  \\end{cases}\n\\end{displaymath}\nupon which we use again the weak maximum principle.\n\nTo prove \\eqref{eq:113} we first obtain for $\\phi_\\epsilon$, in the same manner\nused to derive \\eqref{eq:33}, the following problem:\n\\begin{equation}\\label{eq:115}\n\\left\\{\n\\aligned\n-&\\Delta\\phi_\\epsilon + \\frac{\\rho^2_\\epsilon}{\\epsilon^2}\\phi_\\epsilon =0 \\quad& \\text{\\rm in } \\Omega\\,, \\\\\n&  \\frac{\\partial\\phi_\\epsilon}{\\partial\\nu}= \\epsilon^{-\\alpha}J \\quad&  \\text{\\rm on } \\partial\\Omega_c\\,,\\\\\n& \\frac{\\partial\\phi_\\epsilon}{\\partial\\nu}= 0 \\quad&  \\text{\\rm on } \\partial\\Omega_i\\,.\n\\endaligned\\right.\n\\end{equation}\nThen, we follow the same steps as in the proof of \\eqref{eq:34} to\nobtain the bound $\\|\\phi_\\epsilon\\|_\\infty\\leq C \\epsilon^{-\\alpha}$. Multiplying \\eqref{eq:115} by $\\phi_\\epsilon$ and\nintegrating by parts yields, using the preceding $L^\\infty$ bound \\eqref{eq:112}, we find that\n\\begin{displaymath}\n  \\|\\nabla\\phi_\\epsilon\\|_2^2 \\leq \\int_{\\partial\\Omega} \\phi_\\epsilon  \\frac{\\partial\\phi_\\epsilon}{\\partial\\nu}\\,ds \\leq C\\epsilon^{-2\\alpha}\\,.\n\\end{displaymath}\n\\end{proof}\nAs a corollary we get:\n\\begin{corollary}\n  \\begin{equation}\n\\label{eq:116}\n\\|B_\\epsilon\\|_\\infty \\leq  \\text{\\rm max}(|b_1|,b_2)\\epsilon^{-\\alpha} + \\frac{1}{2} \\,.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\n  By \\eqref{eq:112} and the maximum principle we have that\n  \\begin{displaymath}\n    \\|B_\\epsilon-1\\|_\\infty\\leq \\|w_\\epsilon\\|_\\infty +\\frac{1}{2} \\leq  \\text{\\rm max}(|b_1|,b_2)\\epsilon^{-\\alpha} - \\frac{1}{2}\\,,\n  \\end{displaymath}\nwhich readily yields \\eqref{eq:116}.\n\\end{proof}\n\nWe continue with the following auxiliary estimate:\n\\begin{lemma}\\label{Lem3}\n  Let $\\Omega$ and $J$ satisfy (\\ref{eq:3})-(\\ref{hyptopolo}), and $w_\\epsilon$ be a solution of \\eqref{eq:111}.\nThere exist positive constants $C$ and $\\epsilon_0$, such that, for all $0<\\epsilon<\\epsilon_0$,\n  \\begin{equation}\n\\label{eq:117}\n\\|\\nabla w_\\epsilon \\|_\\infty \\leq\n\\frac{C}{\\epsilon^{1+\\alpha}} \\,.\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\n  For convenience of notation we drop the subscript $\\epsilon$ in the proof\n  and bring only its main steps, as it rather standard. We first apply\n  the inverse transformation of \\eqref{eq:103} to \\eqref{eq:111} to\n  obtain\n\\begin{displaymath}\n   \\begin{cases}\n    \\Delta w - |\\psi|^2w = 0 & \\text{in }\\Omega_R\\,, \\\\\n    w = B - 1 & \\text{on } \\partial\\Omega_R \\,,\n  \\end{cases}\n\\end{displaymath}\nwhere $B=\\text{\\rm curl\\,} A$.\\\\\nWe distinguish in the following between interior estimates and\nboundary estimates.  Let $x_0\\in \\partial \\Omega_R$ and\n$D_r=D_r(x_0)=B(x_0,r)\\cap\\Omega_R\\,.\n$\\\\\nBy the standard elliptic estimates we then have, in view of\n\\eqref{eq:106},\n\\begin{equation}\n\\label{eq:124}\n  \\|w\\|_{H^2(D_r)} \\leq C(\\|w\\|_{L^2(D_{2r})}+ \\| \\epsilon^{-\\alpha}B_n-1\\|_{H^2(D_{2r})}) \\,.\n\\end{equation}\nTo obtain the above we first observe that $B = \\epsilon^{-\\alpha} B_n$ on the\nboundary, and then use the fact that the trace of $B_n$ in $H^\\frac\n32 (\\partial\\Omega)$ and is therefore bounded from above by a proper $H^2$ norm.\n\nSimilarly,\n\\begin{equation}\n\\label{eq:125}\n  \\|w\\|_{H^3(D_r)} \\leq C\\Big(\\|w\\|_{H^1(D_{2r})} + \\|\\epsilon^{-\\alpha}B_n-1\\|_{H^3(D_{2r})}\n +\\|w\\|_{L^\\infty(D_{2r})} \\||\\psi|^2\\|_{H^1(D_{2r})}\\Big) \\,.\n\\end{equation}\nUsing Kato's inequality and  (\\ref{eq:18}a) yields\n\\begin{displaymath}\n  \\|\\nabla|\\psi|\\,\\|_{L^2(D_r)} \\leq \\|\\nabla_A\\psi\\|_{L^2(D_r)} \\leq C\\|\\psi\\|_{L^2(D_{2r})}\\,,\n\\end{displaymath}\nthen we obtain by \\eqref{eq:112} that\n\\begin{equation}\\label{112aa}\n  \\|w\\|_{H^3(D_r)} \\leq C\\epsilon^{-\\alpha}\\,.\n\\end{equation}\n\nAn interior estimate is even easier.  Consider $x_0\\in \\Omega_R$ such that\n$D(x_0,2r) \\subset \\Omega_R$. There is no need in this case to include a\nboundary term in \\eqref{eq:124} and \\eqref{eq:125}. We then obtain\n\\eqref{112aa} in this case in a similar manner.  \n\n\nThe proof of \\eqref{eq:117} now follows from Sobolev embeddings and\n\\eqref{eq:103}.\n\\end{proof}\n\n\nWe next define the following subdomain of $\\Omega$:\n \\begin{displaymath}\n{\\mathcal D}_\\delta(\\epsilon) = \\{ x\\in\\Omega \\,:\\,  |B_\\epsilon(x)|<\\delta\\epsilon^{-\\alpha} \\} \\,.\n \\end{displaymath}\n Let further\n \\begin{displaymath}\n   d_{\\delta,j}(\\epsilon) = d( {\\mathcal D}_\\delta(\\epsilon) ,\\partial\\Omega_{i,j}),\\quad j=1,2\\,,\n \\end{displaymath}\nwhere, as in the introduction,  $\\{\\partial\\Omega_{i,j}\\}_{j=1}^{2}$ denotes the set of connected components of\n$\\partial\\Omega_i$. We now obtain a lower bound of\n\\begin{equation}\\label{d-delta}\nd_\\delta(\\epsilon)=\\text{\\rm max}_{j\\in\\{1,2\\}}d_{\\delta,j}(\\epsilon) .\n\\end{equation}\n\\begin{lemma}\n  Let\n  \\begin{equation}\n    \\label{eq:118}\n\\delta_0=\\min(|b_1|,|b_2|)\\,.\n  \\end{equation}\n Under the conditions of Lemma \\ref{Lem3}, there exists, for any   $0<\\delta<\\delta_0\\,$, a\n  positive  $C_\\delta$ such that for sufficiently small $\\epsilon$ we have\n  \\begin{equation}\n\\label{eq:119}\nd_\\delta(\\epsilon) \\geq  C_\\delta\\, \\epsilon  \\,.\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\n  Let $x\\in\\partial\\Omega_i$ and $y\\in{\\mathcal D}_\\delta(\\epsilon)$.  By \\eqref{eq:117} we have\n\\begin{equation}\n\\label{eq:120}\n |w_\\epsilon(x)-w_\\epsilon(y)| \\leq   \\frac{C}{\\epsilon^{1+\\alpha}} |x-y|\\,.\n\\end{equation}\nBy \\eqref{eq:112} we have\n\\begin{displaymath}\n  |w_\\epsilon(x)-w_\\epsilon(y)| \\geq |B_\\epsilon(x)-B_\\epsilon(y)| -\\frac{1}{2}\\geq  (\\delta_0-\\delta)\\epsilon^{-\\alpha}-\\frac{1}{2}\\,.\n\\end{displaymath}\nCombining the above with \\eqref{eq:120} yields\n\\begin{displaymath}\n   |x-y| \\geq (\\delta_0-\\delta)\\epsilon-\\frac{1}{2}\\epsilon^{1+\\alpha} \\,,\n\\end{displaymath}\nwhich readily yields \\eqref{eq:119}.\n\\end{proof}\n\n\nNext we show\n\\begin{lemma} Under the conditions of Lemma \\ref{Lem3}, there exist $C>0$, $\\epsilon_0>0$ and $\\delta_0>0$ such that,  for\n   $0<\\epsilon\\leq \\epsilon_0 $ and $0< \\delta \\leq \\delta_0\\,$,\n  \\begin{equation}\n\\label{eq:121}\n\\int_{\\Omega\\setminus{\\mathcal D}_\\delta(\\epsilon)} \\exp \\left( [2\\delta \\Theta_0\\epsilon^{-\\alpha}]^{1\/2}(4\\epsilon)^{-1}d(x,{\\mathcal D}_\\delta(\\epsilon))\\right)  |\\psi_\\epsilon|^2 \\,dx \\leq \\frac{C}{\\delta^{3\/2}} \\,.\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\n  Let $\\eta\\in C^\\infty(\\Omega,[0,1])$ satisfy\n  \\begin{displaymath}\n    \\eta(x)=\n    \\begin{cases}\n      1 & x\\in \\Omega\\setminus {\\mathcal D}_\\delta(\\epsilon)\\,, \\\\\n      0 & x\\in {\\mathcal D}_{\\delta\/2}(\\epsilon) \\,.\n    \\end{cases}\n  \\end{displaymath}\n  With the aid of \\eqref{eq:117} and \\eqref{eq:112}, it can be easily\n  verified, for some $C=C(p,J,\\Omega)>0$ which is independent of both $\\delta$\n  and $\\epsilon$, that for all $0<\\delta<\\delta_0$ we can construct $\\eta$ with the\n  additional property\n  \\begin{displaymath}\n   |\\nabla\\eta| \\leq \\frac{C}{\\delta\\epsilon} \\,.\n  \\end{displaymath}\nLet further\n$$\\zeta=\\chi\\eta\\,,\n$$\nwhere\n\\begin{displaymath}\n  \\chi=\n  \\begin{cases}\n    \\exp\\big(\\alpha_\\delta\\epsilon^{-1-\\alpha\/2}d(x,{\\mathcal D}_\\delta(\\epsilon))\\big) &\\text{if } x\\in\\Omega\\setminus {\\mathcal D}_\\delta(\\epsilon)\\,, \\\\\n    1 &\\text{if } x\\in {\\mathcal D}_\\delta(\\epsilon)\\,.\n  \\end{cases}\n\\end{displaymath}\nWe leave the determination of\n$\\alpha_\\delta$ to a later stage.\n\nMultiplying (\\ref{eq:25}a) by $\\zeta^2\\bar{\\psi_\\epsilon}$ and integrating by parts yields\n\\begin{displaymath}\n  \\|\\nabla_{\\epsilon^{-2}A_\\epsilon}(\\zeta\\psi_\\epsilon)\\|_2^2 =\\frac{1}{\\epsilon^2}\\big[ \\|\\zeta\\psi_\\epsilon\\|_2^2 -\n  \\|\\zeta^{1\/2}\\psi_\\epsilon\\|_4^4\\big] + \\|\\psi_\\epsilon\\nabla\\zeta\\|_2^2 \\,.\n\\end{displaymath}\nBy \\eqref{eq:22} we have that for sufficiently small $\\epsilon$\n\\begin{displaymath}\n \\|\\nabla_{\\epsilon^{-2}A_\\epsilon}(\\zeta\\psi_\\epsilon)\\|_2^2  \\geq\\frac{\\Theta_0\\delta}{2\\epsilon^2}\\epsilon^{-\\alpha}\n \\|\\zeta\\psi_\\epsilon\\|_2^2 \\,,\n\\end{displaymath}\nwhere $\\Theta_0$ is defined in \\eqref{eq:20}.  Consequently,\n\\begin{equation}\n\\label{eq:122}\n   \\frac{\\Theta_0\\delta \\epsilon^{-\\alpha}-2}{2\\epsilon^2}\\|\\zeta\\psi_\\epsilon\\|_2^2 \\leq \\|\\psi_\\epsilon\\nabla\\zeta\\|_2^2 \\,.\n\\end{equation}\nFrom \\eqref{eq:122} we learn that\n\\begin{displaymath}\n  \\|\\zeta\\psi_\\epsilon\\|_2 \\leq \\Big[\\frac{2\\epsilon^2}{\\Theta_0\\delta \\epsilon^{-\\alpha}-2}\\Big]^{1\/2} \\Big(\\alpha_\\delta\n  \\epsilon^{-(1+\\alpha\/2)}\\|\\zeta\\psi_\\epsilon\\|_2 +\n  \\|\\psi_\\epsilon\\nabla\\eta\\|_2\\Big) \\,,\n\\end{displaymath}\nwhere we have used the fact that $|\\nabla d(x,{\\mathcal D}_\\delta(\\epsilon))|\\leq1$  a.e.\n\nChoosing\n\\begin{displaymath}\n  \\alpha_\\delta=\\frac{1}{2}\\Big[\\frac{\\Theta_0\\delta}{2}\\Big]^{1\/2}\n\\end{displaymath}\nwe obtain that for sufficiently small $\\epsilon$\n\\begin{displaymath}\n  \\|\\zeta\\psi_\\epsilon\\|_2 \\leq 4\\Big[\\frac{2}{\\delta}\\Big]^{1\/2} \\epsilon^{1+\\alpha\/2}\\|\\psi_\\epsilon\\nabla\\eta\\|_2 \\leq\n  \\frac{C\\epsilon^{\\alpha\/2}}{\\delta^{3\/2}}\\|\\psi_\\epsilon\\|_{L^2({\\mathcal D}_\\delta(\\epsilon)\\setminus {\\mathcal D}_{\\delta\/2}(\\epsilon))}\\,.\n\\end{displaymath}\nThe proof of \\eqref{eq:121} now follows from \\eqref{eq:106}.\n\\end{proof}\n\nWe now obtain an improved lower bound.\n\\begin{proposition}\n  Let $\\delta_0$ be given by \\eqref{eq:118}.  Under the conditions of Lemma\n  \\ref{Lem3}, for any $0<\\delta<\\delta_0$,  there exist positive constants  $C_\\delta$ and $\\epsilon_\\delta$ such that for\n   $0< \\epsilon \\leq \\epsilon_\\delta$ we have\n  \\begin{equation}\n\\label{eq:123}\nd_\\delta(\\epsilon) \\geq  C_\\delta \\,.\n  \\end{equation}\n\\end{proposition}\n\\begin{proof}\n  We begin by noticing that by (\\ref{eq:104}b), \\eqref{eq:113}, and\n  \\eqref{eq:121} we have\n\\begin{displaymath}\n  \\|\\nabla B_\\epsilon\\|_{L^2(\\Omega\\setminus{\\mathcal D}_\\delta(\\epsilon))} \\leq\n  \\|\\nabla\\phi_\\epsilon\\|_{L^2(\\Omega\\setminus{\\mathcal D}_\\delta(\\epsilon))}+Ce^{-1\/\\epsilon} \\leq \\widehat C \\epsilon^{-\\alpha} \\,.\n\\end{displaymath}\nThen, we write\n\\begin{displaymath}\n  (\\delta_0-\\delta)\\epsilon^{-\\alpha} \\leq \\|\\nabla B_\\epsilon\\|_{L^1(\\Omega\\setminus{\\mathcal D}_\\delta(\\epsilon))} \\leq Cd_\\delta^{1\/2}\n  \\|\\nabla B_\\epsilon\\|_{L^2(\\Omega\\setminus{\\mathcal D}_\\delta(\\epsilon))} \\leq Cd_\\delta^{1\/2}\\epsilon^{-\\alpha}\\,,\n\\end{displaymath}\nfrom which \\eqref{eq:123} readily follows.\n\\end{proof}\n\\begin{remark}\n\\label{rem:size}\n  Note that by \\eqref{eq:121} and  the above arguments, \\eqref{eq:123}\n  holds even if we use $d_\\delta$ to measure the distance of ${\\mathcal D}_\\delta(\\epsilon)$ from\n  any point on $\\partial\\Omega$ where $B_\\epsilon>\\delta_1\\epsilon^{-\\alpha}$ with $\\delta_1>\\delta$.\n\\end{remark}\n\nProposition \\ref{semiclassical} now follows from \\eqref{eq:121} and \\eqref{eq:123}.\n\n\n\n\n\\paragraph{\\bf Acknowledgements}\nThe authors gratefully acknowledge Phuc Nguyen for introducing to them\nByun's result \\cite{by07} and Monique Dauge for the idea behind the\nproof of Lemma \\ref{lem:Dirichlet-Neumann}. Y. Almog was partially\nsupported by NSF grant DMS-1109030 and by US-Israel BSF grant\nno.~2010194.  B. Helffer was partially supported by the ANR programme\nNOSEVOL and Simons foundation visiting fellow at the Newton Institute\nin Cambridge during the completion of this article. X.B. Pan was\npartially supported by the National Natural Science Foundation of\nChina grant no. 11171111 and no. 11431005. \n\\vspace{2ex}\n\nThe authors declare that\nthey have no conflict of interest. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusions}\n\\label{sec:conclusions}\nWe have studied the utilization of BERT for the task of passage re-ranking in light of its strict limitation on passage length. Using three BERT-based LTR methods, we conducted experiments on three datasets, with different passage lengths and segmentation configurations.\n\nOur findings are that using BERT representations for mid-sized (SeqLen=256 tokens) of (query, passage) pairs give the best results. We have further shown that, by breaking passages to smaller chunks and aggregating their (query, chunk) representations, we can re-rank larger passages, with only a moderate degradation in quality. Finally, we found out that point-wise and pair-wise LTR achieved similar results on the three datasets. \n\n\n\n\\section{Experimentation Analysis}\nThe primary goal of our analysis is to evaluate the behavior of the three proposed BERT-based LTR methods under various passage-length and segmentation settings.\nOur implementation of the various LTR models is based on the TensorFlow\\footnote{https:\/\/www.tensorflow.org\/} version of BERT\\footnote{ https:\/\/github.com\/google-research\/bert}. \nIn all the experiments we used the pre-trained BERT-Base, Uncased model (12-layer, 768-hidden, 12-heads, 110M parameters).\n \nWe start with a short description of our datasets and setup, followed by the empirical results of the analysis.\n\\label{sec:experiments}\n\n\\subsection{Datasets}\n\\label{sec:datasets}\nWe used three different datasets that were previously used for the non-factoid QA task~\\cite{hybrid:2018,wikiPassageQA:2018}. These datasets are next shortly described while their statistics are summarized in Table~\\ref{tab:datasets}.\n\n\\noindent \\textbf{nfL6}\\footnote{https:\/\/ciir.cs.umass.edu\/downloads\/nfL6}~\\cite{hybrid:2018} consists of 87,361 Yahoo's non-factoid questions. Each query has a single correct passage.\n\n\\noindent \\textbf{WebAP}\\footnote{https:\/\/ciir.cs.umass.edu\/downloads\/WebAP\/}~\\cite{hybrid:2018} consists of 82\\footnote{Two queries have no relevant answers so are omitted from our experiments.} queries with labeled passages from the TREC GOV2 web collection. Queries in this dataset are more open ended than the former dataset, and can have a variety of relevant passages.\nLabeling was done by first retrieving the top-50 web documents for each query and then marking contiguous sequences by a five graded-relevance scores (4 - Perfect, 3 - Excel, 2- Good, 1 - fair, 0 - None). Similar to~\\cite{hybrid:2018}, we took the relevant sequences (i.e., relevance $>0$) as positive passages and divided the irrelevant sequences to negative ones. Since there was not enough details in~\\cite{hybrid:2018} on how the latter was done, we assumed passage lengths are normally distributed with mean and standard deviation estimated using the relevant passages.\n\n\\noindent \\textbf{WikiPassageQA}\\footnote{https:\/\/ciir.cs.umass.edu\/downloads\/wikipassageqa\/}~\\cite{wikiPassageQA:2018} contains 4,165 non-factoid queries with passages extracted from Wikipedia articles. Here, it is assumed that for each query, there is a single Wikipedia article that contains relevant passages, which are manually marked as contiguous sequences. Unlike WebAP, each article is divided to fixed passages of $6$ sentences each, where a passage is labeled relevant if it has at least $15\\%$ overlap with a marked relevant sequence.\n\n\\begin{table}[tbh]\n\t\\center\n    \\small\n\t\t\\caption{Datasets}\n        \\setlength\\tabcolsep{2.0pt}\n\t\t\\begin{tabular}{l|c|c|c|c|c|}\n\t\t    \\cline{2-6}\n\t\t\t & \\textbf{queries} &\\textbf{psgs} & \\textbf{min} &\n\t\t\t    \\textbf{max} & \\textbf{avg}   \\\\\n\t\t\t\\hline\n\t\t\t\\multicolumn{1}{|l|}{nfL6} &\n\t\t\t87,362 &  87,362 & 2 & 809  & 42.4  \\\\\n\t\t\n\t\t\t\\hline\n\t\t\t\\multicolumn{1}{|l|}{WebAP} &\n\t\t\t80 & 489,042 & 0 & 10,851 & 74.5 \\\\\n\t\t\n             \\hline\n\t\t\t\\multicolumn{1}{|l|}{WikiPassageQA} &\n\t\t\t4,165 & 50,612 & 10 & 1,332 & 134.2\\\\\n\t\t\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t  \\label{tab:datasets}\n\\end{table}\n\n\n\\subsection{Setup}\n\\label{sec:train}\nSimilar to~\\cite{hybrid:2018}, for each query, we\nretrieved the top-$k$ BM25 results in the nfL6 and WebAP datasets ($k=10$ for nfL6 and $k=100$ for WebAP). To this end, we indexed each passage as a document in an ElasticSearch\\footnote{https:\/\/www.elastic.co} index. For WikiPassageQA, we simply took all passages from the query's Wikipedia article.\n\nFor fine-tuning the three\nBERT models, we used the retrieved top-$k$ passages of each query $q$, and created their corresponding triplets set as follows. \n\nGiven a query $q$ with $m$ positive examples and $n$ negative examples (usually $m << n$), each positive passage participates in different $n \/ m$ triplets.\nTo cope with a reasonable number of triplets, we down-sampled nfL6 (which has a large number of queries) to have at most two negative examples for each positive one.  For WikiPassageQA we down-sampled to five negative examples per each positive one. Since WebAP has a small number of queries, we used all the negative examples.\n\nWikiPassageQA is already divided into $3,332$ train, $417$ development and $416$ test queries. For the other two datasets, we used a 5-fold cross validation. Similar to~\\cite{BERT:2018}, we used a fixed number of three train epochs, hence we omitted the development set.\n\nTo evaluate the performance of the various ranking methods, we re-ranked the top-$k$ passages returned by the BM25 baseline ranker. Similar to~\\cite{hybrid:2018}, for the nfL6 and WebAP  datasets, we added a correct passage at position $k$ ($k=10$ for nfL6 and $k=100$ for WebAP) if it was not retrieved within the BM25 top-$k$ passages.\n\nWe used standard Information Retrieval (IR) metrics: Precision@1 (P@1), Mean Average Precision (MAP) and Mean Reciprocal Rank (MRR).\n\nTraining was done using the default BERT setup~\\cite{BERT:2018}, with Adam optimizer, learning rate $2e$-$5$ over three epochs.\nAll experiments were run on a Tesla K40m GPU, with a memory clock-rate of 0.745Ghz and 12Gb memory.\n\n\\subsection{Results}\n\\label{sec:results}\n\nWe compared the three studied BERT-based LTR methods (Section~\\ref{sec:triplets}) to two baselines, namely BM25 and the previously best known deep-learning (DL) method. Similar to the reported DL methods, we use P@1 for the first two datasets (nfL6 and webAP)~\\cite{hybrid:2018} and MAP for the third one (WikiPassageQA)\n~\\cite{,wikiPassageQA:2018}. Figure~\\ref{fig:all-map} depicts the results obtained by each ranker.  The three BERT methods use 256 tokens for each pair (SeqLen in the BERT jargon), which was found to be the optimal size, as we discuss in Section~\\ref{sec:block_size} below.\n\nWe can observe that the three BERT methods significantly outperform the two baselines. The improvements range from $31\\%$ improvement on MAP in WikiPassageQA to $120\\%$ on P@1 in nfL6 over the best DL baseline.\nAmong the three BERT methods, the point-wise\n(\\texttt{BERT[PW]}) performed better on webAP (with P@1=$0.55$, compared to $0.53$ by the pair-wise (\\texttt{BERTlets})), while \\texttt{BERTlet} performed better on WikiPassageQA (with MAP=$0.74$ compared to $0.71$ by \\texttt{BERT[PW]}).  \nAmong the two pair-wise BERT variants, the \\texttt{BERTlets} method performs better.\nThe experiments in the subsequent sections are based on \\texttt{BERTlets}.\n\n\\input{figs\/all_map}\n\n\\subsubsection{Effect of Passage Length}\n\\label{sec:block_size}\n\nAs was mentioned above, BERT imposes a limit of maximum $512$ token\n\\footnote{BERT ignores input tokens after the first SeqLen tokens} (SeqLen) in each pair when using a GPU. Figure~\\ref{fig:passage_size_map}\ndepicts the \n\\texttt{BERTlets} results for varying SeqLen. We observe that, the best P@1 and MAP are achieved for $256$ tokens.  Lower values ($64$ and $128$) actually hurt the performance while increasing to $384$ does not improve and even hurts it mainly on WikiPassageQA.\n\nThe effect of the small SeqLen is mainly manifested on the WikiPassageQA which has the largest average passage-length as shown in Table~\\ref{tab:datasets}. This is actually expected, since short passages may lack enough content, while too long ones may include additional superfluous content (``noise'') to allow to fully capture such passages semantics through the BERT representation.\n\n\n\\input{figs\/passage_size_map}\n\n\\subsubsection{Effect of passage segmentation}\n\\label{sec:sentences}\n\nTo overcome the limitations of BERT on passage-length, we further segmented each passage into equal number of chunks. Given a query $q$ and passage $p$, we derived $\\texttt{BERT}(q,c)$, for chunks $c \\in p$,  and combined their representation through an attention mechanism~\\cite{Attention:2016}.  We tried to break to two and three chunks and pass them with SeqLen$=128$, using an attention size of $192$. The results are summarized in Table~\\ref{tab:sentences}.\n\n\\begin{table}[h]\n\t\\center\n    \\small\n\t\t\\setlength\\tabcolsep{1.8pt}\n\t\t\\caption{Effect of breaking paragraphs to chunks of SeqLen$=128$ (with attention)}\n\t\t\\begin{tabular}{l|c|c||c|c||c|c|}\n\t\t\t\\cline{2-7}\n\t\t\t\\multirow{2}{*}{} &\n\t\t\t\t\\multicolumn{2}{c||}{nfL6} & \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\multicolumn{2}{c||}{WebAP} &\n\t\t\t\t\\multicolumn{2}{c|}{WikiPassageQA} \\\\\n\t\t\t\\cline{2-7}\n \t\t\t\t\t& P@1 & MRR & P@1 & MRR & MAP & MRR \\\\ \\hline\n\t\t\t\t\t\\multicolumn{1}{|l|}{BERTlets (2*128)} & .744 & .835 & .513  & .662 & .715 & .778  \\\\ \\hline\n\t\t\t\t\t\\multicolumn{1}{|l|}{BERTlets (3*128)} & .748 & .839 & .488 & .634 & .715  & .791 \\\\ \\hline\n\t\t\t\t\t\\multicolumn{1}{|l|}{BERTlets (256)} & .776 & .857 & .525 & .683 & .736 & .810 \\\\ \\hline\n\t\t\\end{tabular}\n    \\label{tab:sentences}\n\\end{table}\n\nAs we can observe, using passage segmentation results in a slightly lower performance.\nHowever, given the BERT constraints on passage length, this segmentation can be used to handle large passages above the BERT limitation of $512$ tokens. We further tried to combine the chunks representations using \\textit{Max-Pooling} but it gave similar results to the attention method.\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\n\nWithin a question-answering (QA) setting, the passage-retrieval task, retrieves a (relatively) short text fragment (passage)\nthat provides a focused answer to a given question.\nAs opposed to the factoid-QA task, which requires a succeeding step of answer extraction, the non-factoid QA task aims at retrieving passages as the answers.\n\nIn this work, we focus on the passage retrieval task and more specifically, on the passage re-ranking task, i.e., given an initial ranked-list of passages, retrieved by some basic passage-retrieval method (e.g., BM25~\\cite{psgbm25}), our goal is to re-rank the passages in the list so as to position the most relevant ones higher. Similar to the latest works on this task, our approach is based on Deep-Learning (DL).\nExisting DL methods commonly train their networks directly on the labeled datasets~\\cite{hybrid:2018, wikiPassageQA:2018,mitra2019duet,tan2015lstmbased}. While achieving better results than more ``traditional'' methods, such methods still suffer from the relatively small labeled data available for training.\n\nTrying to overcome such data limitations, more recent works have utilized the pre-trained Bidirectional Encoder Representations from Transformers (BERT) language representation models~\\cite{BERT:2018}.\nEach BERT model was trained on large corpora of Wikipedia and news data,\nconsisting of a large number of layers (12\/24 in the base\/large model) based on the Transformer~\\cite{Transformer:2017} architecture.\nWhen trained on pairs of sentences, BERT allows to capture high-order interactions between a given pair of sentences. Such pre-trained models can be then fine-tuned for a specific downstream task with a relatively small amount of labeled data. Previously, such end-to-end networks have been shown to be highly effective in several NLP tasks, and very recently also for passage-retrieval~\\cite{multiTaskBert:2019, passageRerankingBert:2019}.\n\nWhile utilizing BERT for passage re-ranking is promising, there are still several technical limitations that are imposed by BERT's basic architecture. First, due to the large number of layers of BERT, it should be run on a GPU, enforcing a maximum of 512 input tokens \nper pair.\nAmong these tokens are the special \\textsf{[CLS]} token and two \\textsf{[SEP]} tokens that BERT adds to each pair. Moreover, BERT's tokenizer breaks tokens not in its dictionary into sub-tokens, which further limits the total number of tokens that can be passed for each pair.\n\nThe primary goal of our work is to study how BERT can be best utilized for the passage re-ranking task, within the non-factoid QA setting, in spite of its input length limitations. To this end, we report on an extensive empirical analysis with different passage lengths. We show that, too short passages may suffer from a lack of content, while too large passages may generate noisy representations. We further report on experiments that try to breakdown passages into smaller chunks and combine their representations into a passage-level representation.\n\n\\section{Related work}\n\\label{sec:related}\n\nFollowing the success in applying BERT~\\cite{BERT:2018} for multiple NLP tasks, some very recent works have applied it to\nthe factoid QA~\\cite{BERTserini} task, as well as to the non-factoid QA task~\\cite{multiTaskBert:2019,passageRerankingBert:2019,qiao2019understanding}.\nIn this work we focus on the non-factoid QA task.\n\nSimilar to the MRPC paraphrase task in the original BERT work, the authors of~\\cite{passageRerankingBert:2019} have fine-tuned BERT for passage re-ranking by casting it as a binary classification problem. To this end, correct passages were used as positive examples and the rest as negative examples. At run time, the classification score of passages was used for ranking. In~\\cite{multiTaskBert:2019}, the authors further fine-tuned BERT with a variation of the triplet-loss for ranking passages within a factoid QA task.\n\nIn this work we study the behaviour of BERT with respect to passage length properties.\nA couple of recent works have also investigated BERT's behaviour for retrieval tasks. In~\\cite{qiao2019understanding} an analysis of BERT's attention allocation between query-document tokens in its Transformer layers was performed, further examining its difference from soft match patterns learned by a traditional learning-to-rank (LTR) neural model. The work by~\\cite{padigela2019investigating} has compared BERT's performance to that obtained by a BM25 model with respect to various query properties (e.g., query-length and term frequencies).\n\n\n\n\n\\section{Passage Re-ranking with BERT\n\\label{sec:triplets}\n\nHere, we shortly discuss several options for fine-tuning BERT for the passage re-ranking task.\nTo this end, we exploit a given pre-trained BERT model\\footnote{ https:\/\/github.com\/google-research\/bert} for assigning an effective representation $\\texttt{BERT}(s_1,s_2)$ for any given pair of sentences $(s_1,s_2)$. As the representation itself, we consider it to be that of the \\textsf{[CLS]} token~\\cite{BERT:2018} which serves as the aggregated pair representation.\n\nWe now assume the availability of some labeled examples of queries, where each query ($q$) is potentially associated with both positive ($p^{+}$) and negative ($p^{-}$) passages.\nAs a first step, we create training examples, consisting of (positive and negative) query-passage pairs for fine-tuning BERT.\nIn our study we explore two main learning-to-rank~\\cite{Li2011ASI} (LTR) strategies with BERT, namely \\textbf{point-wise} and \\textbf{pair-wise}.\n\n\\paragraph{Point-wise}\nWithin a point-wise LTR approach, each passage is scored independently of other passages~\\cite{Li2011ASI}. To this end, we use the MRPC (Microsoft Research Paraphrase Corpus) classifier as described in~\\cite{BERT:2018}. The classifier has two labels, $\\textsf{0}$ - for negative examples (i.e., ($q,p^{-}$) pairs) and $\\textsf{1}$ - for positive examples (i.e., ($q,p^{+}$) pairs). Training the LTR model is simply implemented by minimizing the cross-entropy loss. Once trained, the classifier's confidence is used as the score. We refer to this ranking method as \\texttt{BERT[PW]}.\n\n\\paragraph{Pair-wise}\nCompared to the point-wise LTR approach, a pair-wise LTR approach is trained to learn a preference (ordering) among pairs of passages~\\cite{Li2011ASI}.\nIn this work we implement two such methods.  The first method, termed \\texttt{BERTlets}, uses a \\textit{Triplet-Network}~\\cite{Triplet:2018} as follows.\nLet $v$ be a vector, having the same dimension as the obtained BERT representation\\footnote{$768$\/$1024$ for the base\/large BERT model.}. We define the score of a passage $p$ for query $q$ as:\n\\vspace{-0.1in}\n\\begin{equation}\\label{eq:score}\n  score(q;p) = \\texttt{BERT}(q,p) \\cdot v,\n\\end{equation}\n\nwhere \\texttt{BERT}(q,p) is the BERT pooled representation of the CLS token of the last layer.\n\nGiven a triplet $t = (q,p^{+},p^{-})$,\nlet $s^{+} = score(q;p^{+})$ and $s^{-} = score(q;p^{-}))$ be the scores assigned to the positive and the negative passage samples for a given query $q$, respectively.\nOur goal is to tune $v$, such that \n$(q,p^{+}) \\prec (q,p^{-})$ (or $s^{+} > s^{-}$). To achieve that, we train the network with a Hinge loss~\\cite{hinge}:\n\\begin{equation}\\label{eq:loss}\n   loss_t = \\max(m - (\\hat{s}^{+} -\\hat{s}^{-}), 0),\n\\end{equation}\n\nwhere $\\hat{s}^{+} = \\frac {\\exp{s^{+}}}{\\exp{s^{+}}  + \\exp{s^{-}}}$, $\\hat{s}^{-} = \\frac {\\exp{s^{-}}}{\\exp{s^{+}}  + \\exp{s^{-}}}$ and $m$ is the margin hyperparameter (e.g., 0.2).\n\nThe second pair-wise method, termed \\texttt{BERT[CE]}, is inspired by~\\cite{multiTaskBert:2019} and serves as an alternative to \\texttt{BERTlets}. This method uses a similar technique as \\texttt{BERTlets}, but instead of triplets, it is trained by comparing a given positive pair $(q,p^{+})$ with any given negative pair $(q,p^{-})$. It also learns a vector $v$ and assigns a similar $score(q,p)$ as in Eq.~\\ref{eq:score}. Yet, instead of the Hinge loss, its loss is the negative log-likelihood of the positive example (i.e., $-log (\\hat{s}^{+})$).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\label{}}\n\n\\vspace{-0.5in}\n\\section{INTRODUCTION}\nThe ATLAS combined reconstruction software uses several approaches to calculating the missing transverse energy (Missing\nET \\cite{csc-ref}). The simplest approach is to use as input all calorimeter cells, which are calibrated based on the cell energy density and position,\n and muons which leave little energy in the calorimeter. This approach is expected to be robust in early data and is constructed from three terms:\nThe \\emph{calo term} is calculated from all calorimeter cells, the \\emph{muon term} from muons measured in the muon system\nwith $|{\\eta}|$ $<$ 2.7 and the \\emph{cryostat term} corrects for energy deposited in the cryostat between the electromagnetic\nand hadronic calorimeters. A more sophisticated approach, called the Refined Missing ET, is to modify these calorimeter cell \nweights according to the object (electron, jet etc) that they are attached to. \n\n\\vspace{-0.2in}\n\\section{PERFORMANCE}\n\nThe linearity is defined to be:\n\\begin{equation}\n\\frac{|E_{TMiss}^{True}|-|E_{TMiss}|}{|E_{TMiss}^{True}|}\n\\end{equation}\nwhere $E_{TMiss}^{True}$ is the true Missing ET calculated from non-interacting truth particles. The performance is shown for a number of samples in Figure 1. In the left plot points with true Missing ET of 20 GeV are from $Z$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$, at 35 GeV from $W$ $\\rightarrow$ e\/$\\mu\\nu$, at 68 GeV from semileptonic $t$$\\bar{t}$ events, at 124 GeV from $A^{0}$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ with $m_{A}$ = 800 GeV and at 280 GeV from events with supersymmetric particles with a mass scale of 1 TeV. The right plot shows a more detailed version for $A^{0}$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ with $m_{A}$ = 800 GeV - the non-linearity at low true Missing ET is due to the finite detector resolution and is not a bias in the algorithm.\n\\begin{figure}[h]\n\\begin{center}\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\multicolumn{1}{l}{\\mbox{\\bf (a)}} &\n        \\multicolumn{1}{l}{\\mbox{\\bf (b)}} \\\\ [-0.53cm]\n\\hspace{-0.4in}\n\\includegraphics[width=7.cm,height=3.7cm]{.\/Fig3_1.eps} &\n        \\includegraphics[width=7.cm,height=3.7cm]{.\/fig3_2.eps} \\\\ [-0.4cm]\n\\end{array}$\n\\end{center}\n\\caption{Linearity as function of true Missing ET for different stages in the calibration. The global calibration refers to the hadronic calibration cell weights applied to the cells in the \\emph{calo term}. }\n\\label{ETMiss_Lin}\n\\end{figure}\n\nThe resolution is shown as a function of scalar missing transverse energy sum ($\\sum{E_{T}}$) for a number of processes in Figure 2. The resolution follows a stochastic behaviour, $a\\sqrt{\\sum{E_T}}$, with $a$ between 0.53 and 0.57.\n\\begin{figure}[h]\n\\begin{center}\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\multicolumn{1}{l}{\\mbox{\\bf (a)}} &\n        \\multicolumn{1}{l}{\\mbox{\\bf (b)}} \\\\ [-0.53cm]\n\\hspace{-0.4in}\n\\includegraphics[width=7.cm,height=3.7cm]{.\/Fig4_1.eps} &\n        \\includegraphics[width=7.cm,height=3.7cm]{.\/Fig4_2.eps} \\\\ [-0.4cm]\n\\end{array}$\n\\end{center}\n\\caption{Resolution as function  scalar missing transverse energy ($\\sum{E_{T}}$) of for a number of physics processes. }\n\\label{ETMiss_Lin}\n\\end{figure}\n\\vspace{-0.3in}\n\\section{FAKE MISSING ET}\nThere are a number of sources of fake Missing ET: instrumental effects which can be understood using real data - beam gas, beam halo and noisy or dead cells, inefficiencies in reconstructing high $p_{T}$ muons, fake muons from e.g. jets punching through calorimeters and jet leakage from calorimeters. The effects of fake sources of Missing ET is shown for QCD di-jet events in the left plot of Figure 3 where there is a significant shape difference between the true Missing ET distribution and the fake Missing ET Distribution.  \n\n\\begin{figure}[h]\n\\begin{center}\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\multicolumn{1}{l}{\\mbox{\\bf (a)}} &\n        \\multicolumn{1}{l}{\\mbox{\\bf (b)}} \\\\ [-0.53cm]\n\\hspace{-0.2in}\n\\includegraphics[width=7.cm,height=3.7cm]{.\/Fig10_1.eps} &\n        \\includegraphics[width=7.cm,height=3.7cm]{.\/Fig18.eps} \\\\ [-0.4cm]\n\\end{array}$\n\\end{center}\n\\caption{True and fake Missing ET distributions for di-jets sample with $p_{T}$ of the hard scatter between 560 and 1120 GeV (left). Missing ET resolution is shown as a function of $\\sum{E_{T}}$. $W$ $\\rightarrow$ $\\mu\\nu$ corresponds to $\\sum{E_T}$ of 150 GeV, $W$ $\\rightarrow$ $e\\nu$ to 165 GeV, $Z$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ to 210 GeV, $t$$\\bar{t}$ to 470 GeV, $A^{0}\/H^{0}$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ to 843 GeV, di-jets with $p_{T}$ of the hard scatter in the range 280 to 560 GeV corresponds to 800 GeV and SUSY to 906 GeV (right). }\n\\label{ETMiss_Lin}\n\\end{figure}\n\\vspace{-0.3in}\n\\section{VALIDATION WITH FIRST DATA}\nThe first data will be used to calibrate the calorimeter and understand instrumentation failures. Subsequently the $Z$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ process can be used to determine the Missing ET scale in situ with a systematic uncertainty of 8$\\%$. The $W$ $\\rightarrow$ $l\\nu$ decays can be used to test the Missing ET reconstruction over the energy range 20 to 150 GeV. Top pair production will allow the reconstruction of genuine Missing ET to be tested in an enviroment relevant to Supersummetry (SUSY). The Missing ET resolution is shown as a function of $\\sum{E_{T}}$ for various processes at four different stages of the Refined Missing ET calibration in the right plot of Figure 3.\n\nThe minimum bias sample will have large statistics in the first data and thus it will be very useful for commissioning the Missing ET reconstruction. Events are required to pass one of three minimum bias triggers, to have at least 20 semiconductor tracker space points to reject empty events and to have at least one good reconstructed track to reject beam gas and halo events. The mean true Missing ET is 0.06 GeV and mean expected Missing ET is 4.3 GeV because of the calorimeter energy resolution and acceptance. At low $\\sum{E_{T}}$ the stochastic term dominates the resolution, illustrated in the left plot of Figure 4. The Missing ET is observed to scale with the $\\sum{E_{T}}$ as expected when compared to QCD di-jet events (right plot in Figure 4).\n\\begin{figure}[h]\n\\begin{center}\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\multicolumn{1}{l}{\\mbox{\\bf (a)}} &\n        \\multicolumn{1}{l}{\\mbox{\\bf (b)}} \\\\ [-0.53cm]\n\\hspace{-0.4in}\n\\includegraphics[width=7.cm,height=3.4cm]{.\/Fig19_1.eps} &\n        \\includegraphics[width=7.cm,height=3.4cm]{.\/Fig19_2.eps} \\\\ [-0.4cm]\n\\end{array}$\n\\end{center}\n\\caption{Resolution for different $\\sum{E_{T}}$ regimes in minimum bias events (left) and Missing ET resolution as function of $\\sum{E_{T}}$ for both minimum bias and di-jet events. }\n\\label{ETMiss_MinBias}\n\\end{figure}\n\nTo select the $Z$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ sample the Missing ET is required to be larger than 20 GeV to reject QCD events and the transverse mass, calculated from the Missing ET and lepton (defined to be the leading and isolated lepton, electron or muon, with $P_{T}^{l}$ $>$ 15 GeV and $|{\\eta}|$ $<$ 2.5), is required to be less than 50 GeV, to suppress semileptonic $W$ decays. A cut on $\\sum{E_{T}}$ $<$ 400 GeV further suppresses QCD events. No $b$-tagged jets are allowed in order to suppress bottom and top quark pair production. At least one tau-jet with $p_{T}$ $>$ 15 GeV, $|{\\eta}|$ $<$ 2.5 and a track multiplicity of one or three is required. The azimuthal distance between the isolated lepton and tau-jet is required to be in the range 1 - 2.8 radians to reject badly reconstructed events and further suppress backgrounds. The invariant mass of the $\\tau\\tau$ system shows good sensitivity to the Missing ET Scale (left plot in Figure 5) and allows the determination of this scale with 8$\\%$ accuracy.\nIn the semileptonic top pair sample 7000 events are selected in 200 $pb^{-1}$ when 3 jets with $p_{T}$ $>$ 40 GeV, one more jet with $p_{T}$ $>$ 20 GeV, Missing ET $>$ 20 GeV and an isolated lepton (electron or muon) with $p_{T}$ are required. The effect of scaling the true Missing ET by 0.8 and 1.2 is shown in the right plot of Figure 5 - Gaussian fits indicate the peak shifts by $\\pm$ 7 GeV with a statistical error of 0.5 $\\%$.\n\n\\begin{figure}[!h]\n\\begin{center}\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\multicolumn{1}{l}{\\mbox{\\bf (a)}} &\n        \\multicolumn{1}{l}{\\mbox{\\bf (b)}} \\\\ [-0.53cm]\n\\hspace{-0.4in}\n\\includegraphics[width=7.cm,height=3.4cm]{.\/Fig20_2.eps} &\n        \\includegraphics[width=7.cm,height=3.4cm]{.\/Fig23_1.eps} \\\\ [-0.4cm]\n\\end{array}$\n\\end{center}\n\\caption{The invariant mass of the $\\tau\\tau$ system as function of the Missing ET scale (left). The transverse mass distribution for three semileptonic top quark pair samples, each with different Missing ET scales (right).}\n\\label{ETMiss_MinBias}\n\\end{figure}\n\n\\vspace{-0.4in}\n\\section{CONCLUSIONS}\nThe linearity of the Missing ET is expected to be within 5$\\%$ over a wide range of Missing ET at the beginning of data taking in ATLAS. The resolution follows a stochastic behaviour, $a\\sqrt{\\sum{E_T}}$, with $a$ between 0.53 and 0.57 except at very low and very high $\\sum{E_T}$. Minimum bias events will allow initial validation of the Missing ET reconstruction to be applied. The Missing ET scale can be measured with an 8$\\%$ accuracy using $Z$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ events. The semileptonic top quark pair sample will allow to check for shifts in the Missing ET scale in a kinematic regime relevant for SUSY.\n\n\\vspace{-0.2in}\n\n\\begin{acknowledgments}\nI would like to thank the organisers of the ICHEP08 for the invitation to present this poster and the ATLAS collaboration.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Background and Related Work}\nNetwork densification through the deployment of more access points(APs)\/base stations (BSs) per unit area  and connecting them in a multi-tier network architecture is a natural way to improve cellular network capacity~\\cite{HetNet_1}. However, network densification gives rise to critical issues such as increased  signal interference (both inter-cell interference [ICI] and inter-user interference [IUI]), as well as increased network complexity and capital and operating expenditures.\nMany interference-tolerant and low-cost technologies including massive multiple-input multiple-output antenna (mMIMO), transmit\/receive diversity (TRD), and distributed antenna system (DAS)~\\cite{massiveMIMO_1,R_Diversity_1,R_Diversity_2,Distributed_Ant_1} have therefore been developed. \nThe performance gains in these technologies are mainly achieved by increasing the received power of the desired signal without any coordination among  interference sources in other cells~\\cite{UltraDense_1}. The interference can however be minimized by coordinating wireless transmitters and\/or receivers (either APs or user equipment [UE]).  For instance, coordinated multi-point (CoMP) technique allows adjacent APs to coordinate their transmission and reception to serve cell-edge UEs. This will turn the strongest interference signals into a desired ones at the cost of decreased spectral efficiency \\cite{CoMP6146494}.\n\nRecently, the concept of cooperation among distributed APs has been generalized in which all APs serve all users  (i.e. both cell-edge and cell-center users) within the network coverage area. \nThis concept can be implemented through several techniques such as cloud radio access network (C-RAN)  \\cite{CRAN,CRAN_1}, cell-free networks (or cell-less networks) \\cite{Cell_Less_1,Cell_Less_2,Cell_Less_3,Cell_Less_4} and Generalized CoMP (GCoMP) networks~\\cite{DOMA,GCoMP}. \nIn a C-RAN, a group of distributed remote radio heads (RRHs) that are geographically distributed in the network coverage area are used to perform all radio functionalities such as high frequency amplification, frequency up\/down conversion and A\/D and D\/A conversion. \nThe users' baseband signals coming from all RRHs are then sent to a virtual baseband unit (BBU) pool that is located in the network cloud and is responsible to perform all baseband functionalities such as signal detection, etc. \nIn a cell-free network architecture, a group of distributed APs cooperate to serve all active users within the network coverage area simultaneously using the same frequency-time resources. This makes the entire network act as a massive distributed antenna system (DAS) that covers the entire network coverage area \\cite{DAS_1}. \nThis network architecture however will require a huge processing capability of the BBU pool (or the central processing unit [CPU]), especially with the massive increase on the number of users per unit area in beyond 5G\/6G networks \\cite{6G8412482,DBLP:journals\/corr\/abs-1901-07106}. \n\nSeveral studies on cell-free networks have investigated performance metrics such as per-user transmission rate \\cite{Cell_Less_3},  per-user packet delay \\cite{Cell_Less_Delay_1}, implementation issues such as pilot contamination \\cite{Cell_Less_1}, network backhauling\/fronthauling \\cite{Fronthaul_1,Fronthaul_2}, beamforming techniques \\cite{Cell_Less_Beamforming_1}. \nIn \\cite{Cell_Less_1}, it was found that channel estimation error resulting from non-orthogonal pilot sequences tends to significantly decrease the network performance.  \nThe authors in \\cite{Fronthaul_2} evaluated the per user performance under limited fronthaul link capacity and showed that the network performance degrades as the number of active users increases under limited fronthaul link capacity. Many uplink\/downlink beamforming techniques were developed for cell-free networks. In \\cite{Cell_Less_2}, the authors proposed the use of  conjugate beamforming on the downlink and matched filtering on the uplink to multiplex\/de-multiplex signals to\/from different users. \nIt was also shown that using these beamforming techniques, the inter-user interference (IUI) goes to zero as the number of APs goes to infinity. \nDifferent uplink\/downlink optimization techniques for cell-free decoding\/precoding vectors that are based on maximizing the weighted sum-rate (WSR) or users' minimum rate were proposed in the literature. \nFor instance, the authors in \\cite{Cell_Free_Sum_Rate_1} proposed a low-complexity algorithm that solve the problem of maximizing the WSR of downlink cell-free system (mixed non-convex and combinatorial optimization problem) while maximizing the minimum per-user rate was modeled as a quasi-convex problem and solved by different iterative algorithms (such as bisection search and gradient descent) \\cite{Cell_Less_1,Cell_Less_2,Cell_Less_3}. \n\n\nTo further enhance the performance of cell-free networks, non-orthogonal multiple access (NOMA) method can be used to suppress some of the in-band interfering signals\n\\cite{Cell_Free_NOMA_3,Cell_Free_NOMA_1,Cell_Free_NOMA_5}. \nIn \\cite{Cell_Free_NOMA_3}, downlink users were assumed to be divided into a number of NOMA clusters with successive interference cancellation (SIC) operation applied at every cluster members and the achievable per user transmission rate was calculated. The authors in \\cite{Cell_Free_NOMA_1} proposed an adaptive NOMA\/OMA selection scheme such that the downlink per-user transmission rate is maximized. It was also found that the performance of NOMA scheme outperforms that of cell-free OMA when the number of users is relatively high. Additionally, in \\cite{Cell_Free_NOMA_5}, a spectral efficiency maximization algorithm for uplink NOMA-enabled cell-free network was proposed where the authors showed that a better performance can be achieved by controlling the per-user transmission power. \n\n\\subsection{Motivation and Contributions}\n\nIn the existing literature, performance analysis of cell-free networks has focused only on the derivation of an approximate mathematical limits of the per-user transmission rate.\nThis is due to the fact that other important  performance metrics such as the probability of outage [or alternatively coverage probability] are significantly difficult to be derived in a closed-form. Furthermore, for the cell-free networks, beamforming techniques (uplink and downlink) were basically developed and tested for relatively small number of users and APs. The reason is that computing the beamforming vectors for massive numbers of users and APs using the conventional algorithms (such as bisection search and gradient descent) gives rise to real-time implementation issues such as convergence time and computational complexity. In  \\cite{GCoMP}, we proposed a downlink variable-order clustering scheme for APs that divides the APs into subgroups where every group is dedicated to serve a subset of the active users. This decreases the lengths of the beamforming vectors for every subgroup of APs by a factor that is linearly proportional to the clustering order. The performance degradation caused by clustering of the APs was compensated by adopting NOMA detection  in every cluster. Additionally,  clustering of APs was based on the channel gains of different users and APs.\n\nThis paper extends the work in \\cite{GCoMP} for massive {\\em uplink} multiple access in cell-free networks. For a static cell-free network, we first derive a closed-form expression of the per-user probability of outage. To reduce the complexity of joint processing of signals from all users in a static cell-free network, we propose a dynamic clustering scheme for APs. For real-time implementation of both dynamic AP clustering and uplink beamforming, we develop a deep reinforcement learning (DRL) scheme, namely, the hybrid Deep Deterministic Policy Gradient (DDPG)-deep double Q-network (DDQN)scheme. The major contributions on this paper can be summarized as follows:\n\n\\begin{itemize}\n    \\item For uplink static cell-free networks, we derive an accurate closed-form expression for the per-user probability of outage by exploiting the \\textit{Welch-Satterthwaite} approximation \\cite{Satterthwaite1946}.\n    \\item To  significantly decrease the signal processing complexity at the CPU for the static cell-free network, we propose a clustering scheme that dynamically partitions the APs into subsets with each subset acting as a virtual AP within a DAS system. We also propose an SIC-based signal detection scheme for non-orthogonal multiple access in a dynamic cell-free network and a modified DAS combining scheme that considers inter-user interference (IUI).\n    \\item We formulate a general problem to jointly optimize the clustering of APs and the beamforming vectors such that the uplink users' sum-rate is maximized or the minimum user rate is maximized.\n    \\item To solve the general optimization problem, we propose and design a novel hybrid DRL scheme based on DDPG-DDQN model.\n    \\item We study and compare different performance metrics of conventional static and those of dynamic cell-free networks under different number of users and APs.  \n\\end{itemize}\nThe rest of this paper is organized as follows.  The static cell-free network model and the analysis of outage performance for this network model are presented in Section II. In Section III, we present the dynamic cell-free network architecture and the corresponding outage performance analysis. For the dynamic cell-free network model, in Section IV, we present a successive interference cancellation (SIC)-aided signal detection scheme and a diversity combining scheme. Also, the joint optimization problem of AP clustering and  beamforming design for the dynamic cell-free model is formulated in this section. In Section V, we propose a novel hybrid DRL method that jointly performs AP clustering and optimization of the beamforming vectors. Numerical and simulation results are presented in Section VI\nbefore the paper is concluded in Section VII. \n\n\\textbf{Notations:} For a random variable (rv) $X$,  $F_X(x)$ and $f_X(x)$  represent cumulative distribution function (CDF) and probability density function (PDF), respectively.  $\\mathbb P ( \\cdot )$ and $\\mathbb E[\\cdot ]$ denote probability and  expectation. For a given matrix $\\bm{A}\\in \\mathbb{C}^{M\\times N}$, $A^H$ represents the Hermitian transpose of $\\bm{A}$. A PDF expression of Nakagami-$\\mathcal{M}$ rv is given by $f_X(x)=\\frac{2\\mathcal{M}^{\\mathcal{M}}}{\\Gamma(\\mathcal{M}) {\\Omega}^{\\mathcal{M}}}x^{2\\mathcal{M}-1}e^{\\frac{\\mathcal{M}}{\\Omega}x^2}$ while a Gamma  rv is  denoted by $X \\thicksim  \\mathcal G (\\alpha, \\beta)$, with PDF as $f_{X}(x) = \\frac{\\beta^\\alpha }{\\Gamma(\\alpha)} x^{\\alpha -1} e^{- \\beta x },  \\quad x >0$, where $ \\beta >0$, $ \\alpha \\geq  1 $, and $ \\Gamma(z)$ is the Euler's Gamma function. The base of $\\log(x) $ is  $2$. \n\\section{Static Cell-Free Network Architecture}\n\n\\subsection{Network Model}\nWe consider an uplink  network with $M$ single-antenna APs and $K$ single-antenna UEs that are located at fixed locations within a certain coverage area (Fig.~\\ref{System_Model}).\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=6cm, width=6cm]{System_Model_Centr.jpg}\n\t\n\t\t\\caption{Static cell-free network model.}\\label{System_Model}\n\t\\end{figure}\nAll APs are connected to each other through backhaul links to form a static cell-free network architecture \\cite{Cell_Less_1}. \nThis network setup enables the distributed APs to collaborate simultaneously to serve all users within the network coverage area.\nThe network is equipped with a CPU pool that performs the last-stage processing tasks to detect\/decode signals from every user.\t\n\nThe channel state information (CSI), which will be required at the CPU for signal detection for individual UEs, is assumed to be achieved through training the pilot sequences that are not completely orthogonal.\nWe assume that the channel gain between the $k\\text{-th}$ user and the $m\\text{-th}$ AP follows the following probabilistic model:\n\\begin{equation}\\label{Channel_Model}\n    g_{mk}={L}_{mk}^{-\\kappa}\\,h_{mk},\n\\end{equation}\nwhere $L_{mk}=||d_{mk}||$ is the Euclidean distance between the $k\\text{-th}$ user and the $m\\text{-th}$ AP, $\\kappa$ is the path-loss exponent ($\\kappa\\geq 2$) and $h_{mk}$ is the small-scale channel fading gain between the $k\\text{-th}$ UE and the $m\\text{-th}$ AP. \nWe assume that $h_{mk}$ follows Nakagami-$\\mathcal{M}_{mk}$ distribution with spreading and shape parameters $\\mathcal{M}_{mk}$ and $\\Omega_{mk}$, respectively, i.e. $|h_{mk}|^2\\thicksim \\mathcal{G}(\\alpha_{mk},\\beta_{mk})$, where $\\alpha_{mk}=\\mathcal{M}_{mk}$ and $\\beta_{mk}=\\frac{\\mathcal{M}_{mk}}{\\Omega_{mk}}$. \nFor simplicity of analysis, we assume that $L_{mk}^{-\\kappa} (\\forall~m$ and $k$) is known. We have $|g_{mk}|^2\\thicksim \\mathcal{G}(\\alpha_{mk},\\beta_{mk}\/L_{mk}^{-2\\kappa})$.\nWe assume that $g_{mk}(\\forall m=1, \\dots, M$ and $k=1, \\dots, K$) belong to a set of independent but not identically distributed (i.n.d) rvs. \n\\subsection{Uplink Network Training (CSI Acquisition)}\nUnder the assumption that large-scale fading gain ($L_{ {m}k}^{-\\kappa}$) for all users is known, the aim of network training is to estimate the small-scale fading component of the overall channel gain ($h_{ {m}k}$). \nThis can be achieved by assigning different pilot sequences for every user denoted by ${\\bm{\\varphi}}_{ {m} k}=\\left[\\varphi_{mk, 1} \\dots \\varphi_{mk, \\tau_p} \\right]^H$ such that $||{\\bm{\\varphi}}_{ {m} k}||^2=1$, where $\\tau_p\\leq \\tau_c$ is the length of the length of pilot training sequence (in samples), which is less than or equal the channel coherence time ($\\tau_c$). Accordingly, the received pilot vector at $m$-th AP is given by \n\\begin{equation}\n\\label{Y_Pilot_Vector}\n\\bm{y}_{\\textit{p}, {m} }=\\sum_{k=1}^{K}\\sqrt{\\tau_p \\rho_k}g_{ {m} k}\\bm{\\varphi}_{mk}+\\bm{\\eta}_{ {m} },\n\\end{equation}\nwhere $\\rho_k$ is the normalized transmitted power for each symbol of the $k$-th user's pilot vector and $\\bm{\\eta}_m\\in \\mathbb{C}^{\\tau_p\\times 1}$ is the additive white Gaussian noise (AWGN) vector related to pilot symbols with independent and identically distributed (i.i.d) rvs, i.e. ${\\eta}_{i,m}\\thicksim \\mathcal{CG}\\left(\\mu_{i,m}, \\sigma_{i,m}^2\/2\\right), \\forall i=1, \\dots, \\tau_p$.\nThe goal is to find the best estimate of $g_{ {m} k}$ (denoted by $ \\hat{g}_{ {m} k}$) given the vector of observations ($\\bm{y}_{\\textit{p}, {m} k}$). \nThis can be optimally achieved by using the maximum {\\em a posteriori} decision rule (MAP).\nThe Bayesian estimator of $g_{ {m} k}$ is found to be identical to that of the minimum mean square method (MMSE) \\cite{Statistical_Signal_Processing, MMSE_1}.\nAccordingly, the best estimate of $g_{ {m} k}$ can be expressed as \\cite{Cell_Less_2}\n\\begin{dmath}\n    \\label{g_Estimate_1}\n     \\hat{g}_{ {m} k}=\\frac{\\sqrt{\\tau_p \\rho_k}L_{ {m} k}^{-\\kappa}}{\\rho_c\\sum_{l=1, l\\neq k}^{K}\\rho_kL^{-\\kappa}_{ {m} l}|\\bm{\\varphi}_{mk}^H\\bm{\\varphi}_{ml}|^2+1}\\bm{\\varphi}^H_{mk}\\bm{y}_{\\text{p}, {m} }\n     \\\\\n     = \\mathcal{E}_{mk}\\sqrt{\\tau_p\\rho_k}g_{mk}+\\sum_{l=1, l\\neq k}^K \\mathcal{E}_{mk}\\sqrt{\\tau_p\\rho_k}|\\bm{\\varphi}^H_{mk} \\bm{\\varphi}_{ml}|g_{ml}+\\mathcal{E}_{mk}|\\bm{\\varphi}^H_{mk} \\bm{\\eta}_m|,\n\\end{dmath}\nwhere $\\mathcal{E}_{mk}=\\frac{\\sqrt{\\tau_p \\rho_k}L_{ {m} k}^{-\\kappa}}{\\rho_c\\sum_{l=1, l\\neq k}^{K}\\rho_kL^{-\\kappa}_{ {m} l}|\\bm{\\varphi}_{mk}^H\\bm{\\varphi}_{ml}|^2+1}$. \nNote that, if all users assigned a set of mutually orthogonal pilot sequences (i.e. $|\\bm{\\varphi}^H_{mk} \\bm{\\varphi}_{ml}|=0, k\\neq l$), the estimated small-scale channel fading in (\\ref{g_Estimate_1}) reduces to a scaled version of the exact fading gain plus a relatively small AWGN noise portion.\nHowever, for certain applications (e.g. mMTC applications) and due to length limitations of $\\tau_p$, non-orthogonal pilot signals has to be used among some active users.\n\n\\subsection{Outage Performance}\n\\subsubsection{Distribution of Users' SINR}\nIn the static cell-free uplink network with $M$ APs and $K$ users, the distributed APs receive signals from all users within their coverage area and then forward the baseband signal components through the fronthaul links to the CPU pool. The CPU performs the detection of each user's signal separately. This process can be implemented by optimizing a user's transmission power and\/or the beamforming vectors at every user's detector.\nAt a certain CPU detector module, the received signal used in the detection of the $k\\text{-th}$ user's component is given by\n\\begin{dmath}\n y_k=\\sum_{m=1}^Mw_{mk}\\left[\\sum_{l=1}^K\\hat{g}_{ml}\\sqrt{p_l}x_l+\\tilde{\\eta}_m\\right]\n    =\n    \\underbrace{\n    \\sqrt{\\tau_p\\rho_kp_k}x_k\\sum_{m=1}^Mw_{mk}\\mathcal{E}_{mk}g_{mk}}\n    _{\\text{Desired Signal}}\n    +\n    \\sum_{m=1}^M\n    w_{mk}\n     \\left[\n     \\underbrace{    \\sum_{l=1, l\\neq k}^K\\sqrt{\\tau_p\\rho_lp_l}x_l\\mathcal{E}_{ml}g_{ml}\n    }\n    _{\\text{Inter-User Interference}}\n    \\\\\n    +\n    \\underbrace{\n    \\sum_{l=1, l\\neq k}^K\\sqrt{\\tau_p\\rho_kp_k}x_k\\mathcal{E}_{mk}|\\bm{\\varphi}_{mk}^H\\bm{\\varphi}_{ml}|g_{ml}\n    +\n    \\sum_{l=1, l\\neq k}^K \\sum_{\\tilde{l}=1, \\tilde{l}\\neq l}^K\\sqrt{\\tau_p\\rho_lp_l}x_l\\mathcal{E}_{ml}|\\bm{\\varphi}_{ml}^H\\bm{\\varphi}_{m\\tilde{l}}|g_{m\\tilde{l}}}\n    _{\\text{Non-Orthogonal Pilots' Related Estimation error}}\n        \\right]\n    \\\\\n    +\n    \\sum_{m=1}^M\n    w_{mk}\n    \\left[\n          \\underbrace{ \\sqrt{p_k}x_k|\\bm{\\varphi}_{mk}^H\\bm{\\eta}_m|\n    +\n    \\sum_{l=1, l\\neq k}^K\\sqrt{p_l}x_l|\\bm{\\varphi}_{ml}^H\\bm{\\eta}_m|\n    }\n    _{\\text{AWGN's Related Estimation Error}}\n    +\n              \\underbrace{  \\tilde{\\eta}_m}\n    _{\\text{AWGN Component}}\n    \\right]\n    ,\\label{Received_Signal_1} \n\\end{dmath}\nwhere $w_{mk}$ is the $m$-th element of the $k$-th user's beamforming vector such that $0\\leq w_{mk}\\leq 1$, $p_k$ is the uplink transmission power of the $k$-th user such that $0\\leq p_k\\leq P_k$, where $P_k$ is the maximum allowable transmission power for the $k\\text{-th}$ user, $x_k$ is the transmitted symbol from the $k\\text{-th}$ user such that $\\mathbb{E}[|x_k|^2]=1$ and ${\\tilde{\\eta}}_m$ is the AWGN at the input of the $m$-th AP such that $\\tilde{\\eta}_m\\thicksim \\mathcal{CG}\\left(\\tilde{\\mu}_m, \\tilde{\\sigma}_m^2\/2\\right)$. \nWe assume that $\\tilde{{\\eta}}_m, \\forall m=1, \\dots, M$ are from a set of i.i.d rvs.\n\nAccordingly, the SINR of the $k\\text{-th}$ user related to $y_k$ at (\\ref{Received_Signal_1}) can be expressed as\n\\begin{dmath}\n    \\gamma_k=\n    \\frac\n    {\n    \\sum_{m=1}^M|\\tilde{g}_{mk}|^2\n    }\n    {\n    \\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K|\\tilde{g}_{ml}|^2\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    |\\tilde{g}_{m{\\ddot{l}}}|^2\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K|\\tilde{g}_{m{\\Breve{l}}}|^2\n    \\right]+1\n    }\n\n\n,\\label{SINR_1}\n\\end{dmath}\nwhere $|\\tilde{g}_{mk}|^2\\thicksim \\mathcal{G}\\left(\\tilde{\\alpha}_{mk},\\tilde{\\beta}_{mk} \\right)$, $|\\tilde{g}_{ml}|^2\\thicksim \\mathcal{G}\\left(\\tilde{\\alpha}_{ml},\\tilde{\\beta}_{ml} \\right)$, $|\\tilde{g}_{m{\\dot{l}}}|^2\\thicksim \\mathcal{G}\\left(\\tilde{\\alpha}_{m\\dot{l}},\\tilde{\\beta}_{m\\dot{l}} \\right)$,  $|\\tilde{g}_{m\\ddot{l}}|^2\\thicksim \\mathcal{G}\\left(\\tilde{\\alpha}_{m\\ddot{l}},\\tilde{\\beta}_{m\\ddot{l}} \\right)$ and \n$|\\tilde{g}_{m\\Breve{l}}|^2\\thicksim \\mathcal{G}\\left(\\tilde{\\alpha}_{m\\Breve{l}},\\tilde{\\beta}_{m\\Breve{l}} \\right)$\n with\n $\\tilde{\\alpha}_{mk}=\\mathcal{M}_{mk}$, \n $\\tilde{\\beta}_{mk}=\\frac{\\mathcal{M}_{mk}\\dot{\\sigma}_{mk}L_{mk}^{2\\kappa}}{\\Omega_{mk}    w_{mk}^2\\tau_p\\rho_kp_k\\mathcal{E}_{mk}^2}$, \n $\\tilde{\\alpha}_{ml}=\\mathcal{M}_{ml}$,  \n $\\tilde{\\beta}_{ml}=\\frac{\\mathcal{M}_{ml}\\dot{\\sigma}_{mk}L_{ml}^{2\\kappa} }{\\Omega_{ml}  w_{mk}^2\\tau_p\\rho_kp_k\\mathcal{E}_{mk}^2|\\bm{\\varphi}_{mk}^H\\bm{\\varphi}_{ml}|^2}$, \n $\\tilde{\\alpha}_{m\\ddot{l}}=\\mathcal{M}_{m\\ddot{l}}$,  \n $\\tilde{\\beta}_{m\\ddot{l}}=\\frac{\\mathcal{M}_{m\\ddot{l}}\\dot{\\sigma}_{mk}L_{m\\ddot{l}}^{2\\kappa}}{\\Omega_{m\\ddot{l}}   w_{mk}^2\\tau_p\\rho_{\\dot{l}}p_{\\dot{l}}\\mathcal{E}_{m{\\dot{l}}}^2|\\bm{\\varphi}_{m\\dot{l}}^H\\bm{\\varphi}_{m\\ddot{l}}|^2}$,\n $\\tilde{\\alpha}_{m\\Breve{l}}=\\mathcal{M}_{m\\Breve{l}}$,  \n $\\tilde{\\beta}_{m\\Breve{l}}=\\frac{\\mathcal{M}_{m\\Breve{l}}\\dot{\\sigma}_{mk}L_{m\\Breve{l}}^{2\\kappa}}{\\Omega_{m\\Breve{l}}   w_{mk}^2\\tau_p\\rho_{\\Breve{l}}p_{\\dot{l}}\\mathcal{E}_{m{\\dot{l}}}^2}$.\\\\ \n $\\dot{\\sigma}_{mk}=\\sum_{m=1}^M\\left[\\sum_{t=1}^{\\tau_p}\\frac{w_{mk}^2\\sigma_m^2}{2}\\left(p_k\\varphi_{mk,t}+\\sum_{l=1, l\\neq k}^Kp_l\\varphi_{ml,t}\\right)+ w_{mk}^2\\tilde{\\sigma}^2_m\/2\\right]$.\n \n \nNote that the cross-product terms between channel gains are averaged to zero due to the i.i.d assumption. \nFurthermore, the AWGN term in the denominator (denoted by $\\dot{\\sigma}_m$) is replaced by the PSD (variance) of the sum of $M$ i.i.d rvs. \nOur goal is to first find a closed-form expression for the PDF and CDF of $\\gamma_k$ and then utilize the derived expressions in deriving some fundamental performance limits for the static cell-free network. \\textbf{Lemma~\\ref{l_pdfs}} gives an accurate approximation for the PDF of $\\gamma_k$ in (\\ref{SINR_1}).\n\t\\begin{lemma} \\label{l_pdfs}\n    Let \n    $ |\\tilde{g}_{mk}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{mk}} {\\tilde{\\beta}_{mk}} $\n    , \n    $ |\\tilde{g}_{ml}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{ml}} {\\tilde{\\beta}_{ml}} $\n    , \n    $ |\\tilde{g}_{m\\ddot{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{m\\ddot{l}}} {\\tilde{\\beta}_{m\\ddot{l}}} $ \n    and \n    $ |\\tilde{g}_{m\\Breve{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{m\\Breve{l}}} {\\tilde{\\beta}_{m\\Breve{l}}} $\n    where \n    $|\\tilde{g}_{mk}|^2$\n    ,\n    $|\\tilde{g}_{ml}|^2$\n    ,\n    $|\\tilde{g}_{m\\ddot{l}}|^2$\n    and\n    $|\\tilde{g}_{m\\Breve{l}}|^2$ are independent rvs with $m=1, \\dots, M$ and $k=1, \\dots, K$.  The PDF of  $ \\gamma_k$ in (\\ref{SINR_1}) with relatively large $M$ and $K$ is \n    \\begin{dmath}\tf_{\\gamma_k}\\left(\\gamma\\right)=\n    \\frac{\\Dot{\\beta}_{mk}^{\\Dot{\\alpha}_{mk}}\\Dot{\\beta}_{mk'}^{\\Dot{\\alpha}_{mk'}}e^{-\\frac{\\Dot{\\beta}_{mk}\\gamma-\\Dot{\\beta}_{mk'}}{2}}}\n    {\\Gamma\\left(\\Dot{\\alpha}_{mk}\\right)\\left(\\Dot{\\beta}_{mk}\\gamma+\\Dot{\\beta}_{mk'} \\right)^{\\frac{\\Dot{\\alpha}_{mk}+\\Dot{\\alpha}_{ml}+1}{2}}}\n    W_{\\frac{\\Dot{\\alpha}_{mk}-\\Dot{\\alpha}_{mk'}+1}{2},\\frac{-\\Dot{\\alpha}_{mk}-\\Dot{\\alpha}_{mk'}}{2}}\\left(\\Dot{\\beta}_{mk}\\gamma+\\Dot{\\beta}_{mk'} \\right) \n    ,\\label{PDF_Centralized}\t\t\\end{dmath}\n\t\twhere  \n        $\\Dot{\\alpha}_{mk}$, $\\Dot{\\alpha}_{mk'}$, $\\Dot{\\beta}_{mk}$ and $\\Dot{\\beta}_{mk'}$ are defined in \\textbf{Appendix A} and  $W_{\\lambda,\\mu}(x)$ is the  Whittaker function with parameters $\\lambda$ and $\\mu$ \\cite[Eq. 9.222.1]{gradshteyn2000}.\n\t\\end{lemma}\n    \\begin{proof}\nSee \\textbf{Appendix A}.\n\\end{proof}\n\n\\subsubsection{SINR Outage}\nA receiver is in an outage if the SINR of the received signal falls bellow a certain predefined threshold value (denoted by $\\gamma_{\\text{th}}$). \n\\textbf{Theorem \\ref{Theorem_Outage}} below gives an accurate approximation for the average probability of outage of the $k\n$-th user.\n\t\\begin{theorem}\\label{Theorem_Outage} \\label{l_cdfs}\n    If \n    $ |\\tilde{g}_{mk}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{mk}} {\\tilde{\\beta}_{mk}} $\n    , \n    $ |\\tilde{g}_{ml}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{ml}} {\\tilde{\\beta}_{ml}} $\n    , \n    $ |\\tilde{g}_{m\\ddot{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{m\\ddot{l}}} {\\tilde{\\beta}_{m\\ddot{l}}} $ \n    and \n    $ |\\tilde{g}_{m\\Breve{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{m\\Breve{l}}} {\\tilde{\\beta}_{m\\Breve{l}}} $\n    where \n    $|\\tilde{g}_{mk}|^2$\n    ,\n    $|\\tilde{g}_{ml}|^2$\n    ,\n    $|\\tilde{g}_{m\\ddot{l}}|^2$\n    and\n    $|\\tilde{g}_{m\\Breve{l}}|^2$ are independent rvs with $m=1, \\dots, M$ and $k=1, \\dots, K$.  The probability of outage of the $k$-th user in a static cell-free network is \n    \\begin{dmath}\tP_{\\text{out}}^{(k)}=1-\\frac{\\left(\\frac{\\Dot{\\beta}_{mk'}}{\\gamma_{\\text{th}}\\Dot{\\beta}_{mk}} \\right)^{\\Dot{\\alpha}_{mk'}}\\Gamma\\left(\\Dot{\\alpha}_{mk}+\\Dot{\\alpha}_{mk'} \\right)}{\\Dot{\\alpha}_{mk'}\\Gamma\\left(\\Dot{\\alpha}_{mk} \\right)\\Gamma\\left(\\Dot{\\alpha}_{mk'} \\right)}\n    {}_{2}F_{1}\\left(\\Dot{\\alpha}_{mk'},\\Dot{\\alpha}_{mk}+\\Dot{\\alpha}_{mk'};1+\\Dot{\\alpha}_{mk'};-\\frac{\\Dot{\\beta}_{mk'}}{\\Dot{\\beta}_{mk}} \\frac{1}{\\gamma_{\\text{th}}}\\right)\n    ,\\label{e_outage_1}\t\t\\end{dmath}\n\t\twhere  \n        $\\gamma_{\\text{th}}$ is the SINR threshold value and  ${}_2F_{1}(.)$ is the  Gauss hypergeometric function function \\cite[Eq. 9.11.1]{gradshteyn2000}.\n\t\\end{theorem}\n    \\begin{proof}\nSee \\textbf{Appendix B}.   \n\\end{proof}\n\nOne major drawback of the static cell-free network architecture is the requirement of joint processing of signals related to all active users which will require the CSI for the links from all users to all APs. This becomes very challenging for massive numbers of active users and APs in the network. In the following section, we propose a dynamic cell-free network model for uplink access that reduces the complexity of joint processing, compared to that of static cell-free network model, in order to serve a massive number of users. \n\n\\section{Dynamic Cell-Free Architecture}\n\n\n\n\\subsection{Network Model and Assumptions}\\label{s_sys}\nIn the proposed dynamic cell-free network model (Fig.~\\ref{System_Model_2}), the  APs in the network are partitioned among a set of $\\tilde{M}$ subgroups (known as clusters) in which every cluster consists of $\\mathcal{N}_{\\tilde{m}}$ APs such that $1\\leq \\mathcal{N}_{\\tilde{m}}\\leq M-(\\tilde{M}-1)$ \\footnote{Note that when $\\tilde{M}=M$, every cluster will contain a single AP and this falls back to the conventional static cell-free network model.}.\nAdditionally, the $\\tilde{m}$-th cluster, where $\\tilde{m}=1, \\dots, \\tilde{M}$,  acts as an single virtual AP with a $\\mathcal{N}_{\\tilde{m}}$-antenna DAS that combine the overall received signals using a predefined combining technique (such as maximal ratio combining [MRC], equal gain combining [EGC], selection combining [SC], etc.)\\cite{simonmk05}.\nThe clustering of APs is assumed to be performed in each time slot (or transmission interval) based on the current CSI of the network.\nSimilar to the static cell-free network model, we assume that channel gain between the $k$-th user and the ${n}_{\\tilde{m}}$-th antenna at the $\\tilde{m}$-th cluster (the virtual AP) follows the following distribution:\n\\begin{equation}\n    g_{\\tilde{m}n_{\\tilde{m}}k}={L}_{\\tilde{m}n_{\\tilde{m}}k}^{-\\kappa}h_{\\tilde{m}n_{\\tilde{m}}k},\n\\end{equation}\nwhere $L_{\\tilde{m}n_{\\tilde{m}}k}=||d_{\\tilde{m}n_{\\tilde{m}}k}||$ is the Euclidean distance between the $k\\text{-th}$ user and the $n_{\\tilde{m}}$-th antenna at the $\\tilde{m}$-th cluster, $\\kappa$ is the path-loss exponent with a value depends on the propagation environment with $\\kappa\\geq 2$ and $h_{\\tilde{m}n_{\\tilde{m}}k}$ is the small-scale channel fading between the $k\\text{-th}$ user and the $n_{\\tilde{m}}$-th antenna at the $\\tilde{m}$-th cluster. \nSimilarly,  $h_{\\tilde{m}n_{\\tilde{m}}k}$ is assumed to follow a Nakagami-$\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}$ distribution, i.e. $|h_{\\tilde{m}n_{\\tilde{m}}k}|^2\\thicksim \\mathcal{G}(\\alpha_{\\tilde{m}n_{\\tilde{m}}k},\\beta_{\\tilde{m}n_{\\tilde{m}}k})$, where $\\alpha_{\\tilde{m}n_{\\tilde{m}}k}=\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}$ represents the shape parameter of Nakagami-$\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}$ fading and $\\beta_{\\tilde{m}n_{\\tilde{m}}k}=\\frac{\\mathcal{M}_{mnk}}{\\Omega_{\\tilde{m}n_{\\tilde{m}}k}}$ is defined as the shape parameter normalized by spreading parameter of Nakagami-$\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}$ fading. \nAgain, for simplicity of analysis, we assume that $L_{\\tilde{m}n_{\\tilde{m}}k}^{-\\kappa}, \\forall~\\tilde{m},n_{\\tilde{m}},k$ is known. Accordingly, we have $|g_{\\tilde{m}n_{\\tilde{m}}k}|^2\\thicksim \\mathcal{G}(\\alpha_{\\tilde{m}n_{\\tilde{m}}k},\\beta_{\\tilde{m}n_{\\tilde{m}}k}')$, where $\\beta_{\\tilde{m}n_{\\tilde{m}}k}'=\\beta_{\\tilde{m}n_{\\tilde{m}}k}\/L_{\\tilde{m}n_{\\tilde{m}}k}^{-2\\kappa}$. \nLet us denote $\\bm{\\mathcal{C}}=\\{\\mathcal{C}_1 \\dots \\mathcal{C}_N \\}$ as the set of all possible clustering configurations of APs such that every cluster contains at least one  AP. \nAs an example, with $M=8$ and $\\tilde{M}=3$, one possible set is\n\\[\n\\mathcal{C}_n=\\{ \\underbrace{ \\{ AP_3, AP_6, AP_8 \\}}_{\\mathcal{N}_{\\tilde{m}=1}=3}, \\dots, \\underbrace{\\{AP_2\\}}_{\\mathcal{N}_{\\tilde{m}=2}=1},\\underbrace{\\{ \\overbrace{AP_1}^{n_{\\tilde{m}=3}=1}, AP_4, AP_5, AP_7 \\}}_{\\mathcal{N}_{\\tilde{m}=\\tilde{M}=3}=4}  \\}.\n\\]\n\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=6cm, width=6.5cm]{System_Model_Dist.jpg}\n\t\n\t\t\\caption{Dynamic cell-free network model.}\\label{System_Model_2}\n\t\\end{figure}\nNote that in this network setup, the CSI and the instantaneous clustering information will be only required at the CPU for decoding the signals from the UEs. \n\n\\subsection{Outage Performance}\nFor the proposed dynamic cell-free model, the received signal used in the detection of the $k$-th user's component is given by  \n\\begin{dmath}\n\\scriptsize\ny_k=\n\\sum_{\\tilde{m}=1}^{\\tilde{M}}w_{\\tilde{m}k}\n\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}\n\\left[\\sum_{l=1}^K\\hat{g}_{\\tilde{m}n_{\\tilde{m}}l}\\sqrt{p_l}x_l+{\\tilde{\\eta}}_{\\tilde{m}n_{\\tilde{m}}}\\right]\n  =\n    \\underbrace{\n    \\sqrt{\\tau_p\\rho_kp_k}x_k\\sum_{\\tilde{m}=1}^{\\tilde{M}}w_{\\tilde{m}k}\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}\n    \\sum_{n_{n_{\\tilde{m}}}}^{\\mathcal{N}_{n_{\\tilde{m}}}}G_{{\\tilde{m}}n_{\\tilde{m}}k}\n    g_{\\tilde{m}n_{\\tilde{m}}k}}\n    _{\\text{Desired Signal}}+\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    w_{\\tilde{m}k}\n    \\sum_{n_{{\\tilde{m}}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{{\\tilde{m}}n_{\\tilde{m}}k}\n     \\left[\n     \\underbrace{    \\sum_{l=1, l\\neq k}^K\\sqrt{\\tau_p\\rho_lp_l}x_l\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}l}g_{\\tilde{m}n_{\\tilde{m}}l}\n    }\n    _{\\text{Inter-User Interference}}\n    \\\\\n    +\n    \\underbrace{\n    \\sum_{l=1, l\\neq k}^K\\sqrt{\\tau_p\\rho_kp_k}x_k\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}k}^H\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}|g_{\\tilde{m}n_{\\tilde{m}}l}\n    +\n    \\sum_{l=1, l\\neq k}^K \\sum_{\\tilde{l}=1, \\tilde{l}\\neq l}^K\\sqrt{\\tau_p\\rho_lp_l}x_l\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}l}|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}^H\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}{\\tilde{l}}}|g_{\\tilde{m}n_{\\tilde{m}}\\tilde{l}}}\n    _{\\text{Non-Orthogonal Pilots' Related Estimation error}}\n        \\right]\n    \\\\\n    +\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    w_{\\tilde{m}k}\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{{\\tilde{m}}n_{\\tilde{m}}k}\n    \\left[\n          \\underbrace{ \\sqrt{p_k}x_k|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}k}^H\\bm{\\eta}_{\\tilde{m}n_{\\tilde{m}}}|\n    +\n    \\sum_{l=1, l\\neq k}^K\\sqrt{p_l}x_l|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}^H\\bm{\\eta}_{\\tilde{m}n_{\\tilde{m}}}|\n    }\n    _{\\text{AWGN's Related Estimation Error}}\n    +\n              \\underbrace{  \\tilde{\\eta}_{\\tilde{m}n_{\\tilde{m}}}}\n    _{\\text{AWGN}}\n    \\right], \\label{Received_Signal_2} \n\\end{dmath}\nwhere $w_{\\tilde{m}k}$ is the $\\tilde{m}$-th element of the $k$-th user's beamforming vector such that $0\\leq w_{{\\tilde{m}}k}\\leq 1$, \n$G_{\\tilde{m}n_{\\tilde{m}}k}$ is a design gain parameter for the $\\mathcal{N}_{\\tilde{m}}$-antenna DAS at the $\\tilde{m}$-th virtual AP,\n$p_{k}$ is the uplink transmission power of the $k$-th user such that $0\\leq p_k\\leq P_k$, where $P_k$ is the power budget of the $k\\text{-th}$ user, $x_k$ is the transmitted symbol from the $k\\text{-th}$ user such that $\\mathbb{E}[|x_k|^2]=1$, and $\\tilde{\\eta}_{\\tilde{m}n_{\\tilde{m}}}$ is the AWGN at the $m$-th AP with $\\tilde{\\eta}_{\\tilde{m}n_{\\tilde{m}}}\\thicksim \\mathcal{CG}\\left(\\tilde{\\mu}_{\\tilde{m}n_{\\tilde{m}}}, \\tilde{\\sigma}_{\\tilde{m}n_{\\tilde{m}}}^2\/2\\right)$.\nWe assume that $\\tilde{\\eta}_{\\tilde{m}n_{\\tilde{m}}}, \\forall \\{\\tilde{m}~ \\&~ n_{\\tilde{m}}\\}$ are from a set of i.n.d rvs. \n\nFor uplink network training, we follow a procedure similar to that in Section II.B. \nTherefore, we have ${\\bm{\\varphi}}_{\\tilde{m}n_{\\tilde{m}}k}=\\left[\\varphi_{{\\tilde{m}n_{\\tilde{m}}k}, 1} \\dots \\varphi_{{\\tilde{m}n_{\\tilde{m}}k}, \\tau_p} \\right]^H$  as the set of $\\tau_p$-dimensional training symbols such that $||{\\bm{\\varphi}}_{\\tilde{m}n_{\\tilde{m}}k}||^2=1$ and $\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}=\\frac{\\sqrt{\\tau_p \\rho_k}L_{\\tilde{m}n_{\\tilde{m}}k}^{-\\kappa}}{\\rho_c\\sum_{l=1, l\\neq k}^{K}\\rho_kL^{-\\kappa}_{\\tilde{m}n_{\\tilde{m}}l}|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}k}^H\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}|^2+1}$ is the MMSE channel estimation constant. \nNote that for every cluster set ($\\mathcal{C}_n, n=1, \\dots, N$), different SINR values ($\\gamma_k$) will result  for different users. \n\n\n\nThe SINR $\\gamma_k$ of the $k$-th user related to $y_k$ in (\\ref{Received_Signal_2}) and under the $n$-th cluster can be expressed as   \n\\begin{dmath}\n   \\text{\\hspace{-3mm}} \\gamma_k^{\\{\\mathcal{C}_n\\}}=\n    \\frac\n    {\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_{\\bar{m}}=1}^{\\mathcal{N}_{\\bar{m}}}\n    |\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2\n    }\n    {\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K|\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    |\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\ddot{l}}}|^2\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K|\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\Breve{l}}}|^2\n    \\right]+1\n    } \n,\\label{SINR_2}\n\\end{dmath}\n where\n    $ |\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}} {\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}} $\n    , \n    $ |\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}} {\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}} $\n    , \n    $ |\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}} {\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}} $ \n    and\\\\ \n    $ |\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}} {\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}} $ \n with\n $\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}=\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}$, \n \n $\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}=\\frac{\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}\\dot{\\sigma}_{\\tilde{m}n_{\\tilde{m}}k}L_{\\tilde{m}n_{\\tilde{m}}k}^{2\\kappa}}{\\Omega_{\\tilde{m}n_{\\tilde{m}}k}    w_{mk}^2G_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_kp_k\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}^2}$, \n $\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}=\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}l}$,  \n \n $\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}=\\frac{\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}l}\\dot{\\sigma}_{\\tilde{m}n_{\\tilde{m}}l}L_{\\tilde{m}n_{\\tilde{m}}l}^{2\\kappa} }{\\Omega_{\\tilde{m}n_{\\tilde{m}}l}  w_{mk}^2G_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_kp_k\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}^2|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}k}^H\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}|^2}$, \n $\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}=\\mathcal{M}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}$, \n \n $\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}=\\frac{\\mathcal{M}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}\\dot{\\sigma}_{{\\tilde{m}n_{\\tilde{m}}}k}L_{{{\\tilde{m}n_{\\tilde{m}}}}\\ddot{l}}^{2\\kappa}}{\\Omega_{{{\\tilde{m}n_{\\tilde{m}}}}\\ddot{l}}   w_{mk}^2G_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_{\\dot{l}}p_{\\dot{l}}\\mathcal{E}_{{{\\tilde{m}n_{\\tilde{m}}}}{\\dot{l}}}^2|\\bm{\\varphi}_{{{\\tilde{m}n_{\\tilde{m}}}}\\dot{l}}^H\\bm{\\varphi}_{{{\\tilde{m}n_{\\tilde{m}}}}\\ddot{l}}|^2}$,\n \n $\\tilde{\\alpha}_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}=\\mathcal{M}_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}$,  \n \n $\\tilde{\\beta}_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}=\\frac{\\mathcal{M}_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}\\dot{\\sigma}_{{\\tilde{m}n_{\\tilde{m}}}k}L_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}^{2\\kappa}}{\\Omega_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}   w_{mk}^2\\tau_p\\rho_{\\Breve{l}}p_{\\dot{l}}\\mathcal{E}_{{{\\tilde{m}n_{\\tilde{m}}}}{\\dot{l}}}^2}$, and\n \\\\\n $\\dot{\\sigma}_{\\tilde{m}n_{\\tilde{m}}k}=\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n w_{\\tilde{m}k}^2\n \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n \\left[\\sum_{t=1}^{\\tau_p}\\frac{\\sigma_{\\tilde{m}n_{\\tilde{m}}}^2}{2}\\left(p_k\\varphi_{{\\tilde{m}n_{\\tilde{m}}k},t}+\\sum_{l=1, l\\neq k}^Kp_l\\varphi_{{\\tilde{m}n_{\\tilde{m}}l},t}\\right)+ \\tilde{\\sigma}^2_{\\tilde{m}n_{\\tilde{m}}}\/2\\right]$.\\\\\nNote that $G_{\\tilde{m}nk}$ is a one-time design parameter while $w_{\\tilde{m}k}$ is an optimization parameter.\nGenerally, different performance metrics of the dynamic cell-free network will depend on the clustering algorithm that assigns different APs to different clusters.\n\\textbf{Corollary \\ref{Dynamic_Performance}} below states the outage performance of a dynamic cell-free network under a random clustering scheme.\n\\begin{corollary} \\label{Dynamic_Performance}\n     If \n    $ |\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}} {\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}} $\n    , ~\n    $ |\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}} {\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}} $\n    , ~\n    $ |\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}} {\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}} $ \n    and \n    $ |\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}} {\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}} $\n    where \n    $|\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2$\n    ,\n    $|\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2$\n    ,\n    $|\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}|^2$\n    and\n    $|\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}|^2$ are independent rvs with ${\\tilde{m}}=1, \\dots, \\tilde{M}$, $n_{\\tilde{m}}=1, \\dots, \\mathcal{N}_{\\tilde{m}}$ such that $1\\leq \\mathcal{N}_{\\tilde{m}}\\leq M-(\\tilde{M}-1)$ and $\\{k, l, \\ddot{l}, \\Breve{l}\\}=1, \\dots, K$ such that $k\\neq l\\neq \\ddot{l}\\neq \\Breve{l}$. The  probability of outage $P_{\\text{out}}^{(k)}$ of the $k$-th user in the proposed dynamic cell-fee network under random AP clustering is identical to that given in \\textbf{Theorem 1} with \n    \\[\n    \\Dot{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}=\n    \\frac{\n    \\left(\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_m}^{\\mathcal{N}_{\\tilde{m}}}\n    \\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}\/ \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}\n    \\right)^2\n    }\n    {\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}\/\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}^2\n    },~~~~~~\n    \\Dot{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}=\\frac{\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}\/\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}^2}{\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}\/ \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}},\n\\]\n\\[\n\\Dot{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k'}=\\frac{\n\\left(\n\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{    \\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}}{ \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}}{     \\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}}{        \\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}}\n    \\right]\n\\right)^2\n}\n{\n\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}}{    \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}^2}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac\n    {\n    \\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}\n    }\n    {\n    \\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}^2\n    }\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\n    \\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}\n    }\n    {\n    \\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}^2\n    }\n    \\right]\n},\n\\]\n\\[\n\\Dot{\\beta}_{\\tilde{m}n_{\\tilde{m}}k'}=\n\\frac{\n\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}}{    \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}^2}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}}{\n    \\tilde{\\beta}_{m\\ddot{l}}^2}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{    \\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}}{\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}^2}\n    \\right]\n}\n{\n \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}}{     \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}\n}}{ \\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}\n}}{\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}}\n    \\right]\n}\n.\n\\]\n\t\\end{corollary}\n    \\begin{proof}\nThis can be easily concluded due to the fact that equations (\\ref{SINR_1}) and (\\ref{SINR_2}) are different from each other only by the scaling parameters of the rvs.\n\\end{proof}\nNote that using the AP clustering scheme with DAS at every virtual AP, there will be no requirements for CSI from other virtual APs. Additionally, a significant reduction of the lengths of the beamforming vectors will be achieved with a direct positive effect on processing speed and complexity  for the beamforming optimization.\nIt is important to mention that the dynamic cell-free network model is a generalization of the static cell-free network model. \nThis can be inferred by setting $\\tilde{M}=M$ which will result in $\\mathcal{N}_{\\tilde{m}}=1,~\\forall \\tilde{m}=1, \\dots, \\tilde{M}(\\equiv M)$. Here,  $G_{\\tilde{m}n_{\\tilde{m}k}}$ will be merged with $w_{\\tilde{m}n_{\\tilde{m}}k}(\\equiv w_{mk})$.  \n\\section{Clustering Technique and Beamforming Design}\nThis section formulates the  general problem of jointly performing the clustering of APs and designing the beamforming vectors.\nHowever, before presenting the problem formulation, we describe the SIC technique for  detection of the signals received by the CPU and also the diversity technique to combine the signals from multiple antennas. SIC can be achieved by utilizing the fact that the signals from all users are superimposed in the power domain, and with the proper beamforming design, the signal related to every user can have a distinct power level. \n\\subsection{SIC-Enabled Signal Detection}\nNote that in a cell-free network, signals from different users use the same time-frequency resource. The main idea of power-domain SIC technique is that a proper power control is performed such that signals from different users have a distinct power levels\\footnote{This is similar to traditional power-domain non-orthogonal multiple access (NOMA), where signals from different users are multiplexed in the same time-frequency resource and SIC techniques are used at the receivers.}. \nAdditionally, to detect signal from a certain user, the SIC unit first detects signals of users with higher power levels, subtracts their contributions from the overall received signal, and then detects the intended signal.\nSpecifically, for the proposed dynamic cell-free network, when detecting the signal from the $k$-th user, the uplink beamforming vectors are designed such that $p_1\\sum_{\\tilde{m}=1}^{\\tilde{M}}~\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}1}|^2 \n    \\leq \\dots \\leq\n    p_k\\sum_{\\tilde{m}=1}^{\\tilde{M}} \n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2~~\n    \\leq\n    \\dots\n    \\leq\np_K\\sum_{\\tilde{m}=1}^{\\tilde{M}} \n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}K}|^2$.\n    \nAccordingly, the SINR $\\gamma_k$ of the $k$-th user can be expressed as\n\\begin{dmath}\n    \\gamma_k^{\\{\\mathcal{C}_n\\}}=\n    \\frac\n    {\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    w_{\\tilde{m}k}^2\n    \\sum_{n_{\\bar{m}}=1}^{\\mathcal{N}_{\\bar{m}}}\n    | \\check{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2\n    }\n    {\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    w_{\\tilde{m}k}^2\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K|\\check{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    |\\check{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\ddot{l}}}|^2\n    +\n    \\sum_{\\Breve{l}=1}^{k-1}|\\check{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\Breve{l}}}|^2\n    \\right]+1\n    } \n,\\label{SINR_3}\n\\end{dmath}\nwhere $|\\Breve{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2=\\frac{\n    G_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_kp_k\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}^2}\n    {   \n \\dot{\\sigma}_{\\tilde{m}n_{\\tilde{m}}k} }|{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2$, \n$|\\Breve{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2\n=\\frac{\n   G_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_kp_k\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}^2|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}k}^H\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}|^2}\n    {\n     \\dot{\\sigma}_{\\tilde{m}n_{\\tilde{m}}l} \n    }|{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2\n$,\\\\\n$\n|\\Breve{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\ddot{l}}}|^2\n=\n\\frac{\nG_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_{\\dot{l}}p_{\\dot{l}}\\mathcal{E}_{{{\\tilde{m}n_{\\tilde{m}}}}{\\dot{l}}}^2|\\bm{\\varphi}_{{{\\tilde{m}n_{\\tilde{m}}}}\\dot{l}}^H\\bm{\\varphi}_{{{\\tilde{m}n_{\\tilde{m}}}}\\ddot{l}}|^2\n}\n{\n \\dot{\\sigma}_{{\\tilde{m}n_{\\tilde{m}}}k}\n}|{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\ddot{l}}}|^2\n$, \nand \n$\n|\\Breve{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\Breve{l}}}|^2\n=\n\\frac{\n\\tau_p\\rho_{\\Breve{l}}p_{\\dot{l}}\\mathcal{E}_{{{\\tilde{m}n_{\\tilde{m}}}}{\\dot{l}}}^2\n}\n{\n \\dot{\\sigma}_{{\\tilde{m}n_{\\tilde{m}}}k}\n}|{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\Breve{l}}}|^2\n$.\nNote that     \n    $ |{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2$\n    , \n    $ |{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2 $\n    , \n    $ |{g}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}|^2$, \n    and \n    $ |{g}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}|^2$\nrepresent different order statistical rvs of gamma rvs which prevents the direct utilization of \\textit{Welch-Satterthwaite} approximation as in \\textbf{Appendix A}. \nThis is due to the introduction of random variable ordering required for SIC-based detection which requires the pre-ordering of different users' signals based on their different power levels. However, a closed-form expression for the outage probability can be derived using order statistics theory as in \\cite{GCoMP}.\nWe can also notice from (\\ref{SINR_3}) that the CSI estimation error contains portions from all users including those with higher power levels than that of the $k$-th user. This is because the SIC operation is also affected by channel estimation errors \\cite{SIC_IMPERFECT_CSI}.\n\\subsection{Diversity Combining Scheme}\nSeveral diversity techniques can be used at multi-antenna transmitters\/receivers to combat multipath fading. However, maximal ratio combining (MRC) and selection combining (SC) are the most common diversity combining techniques \\cite{simonmk05}.\nIn this work, we propose the use of a modified version of \\textit{Wiener-Hopf} multiple antennas combining scheme under the existence of co-channel interference \\cite{iet:\/content\/books\/ew\/sbew046e}.\nAccordingly, the gain factor $\\bm{G}_{\\tilde{m}k}=\\left[G_{\\tilde{m}1k} \\dots G_{\\tilde{m}{\\mathcal{N}}_{\\tilde{m}}k}\\right]^H$ that is used in combining signals at the $\\tilde{m}$-th AP for the detection of $k$-th user signal is defined as \\cite{iet:\/content\/books\/ew\/sbew046e}\n\\begin{equation}\n    \\bm{G}_{\\tilde{m}k}=\\bm{R}_{\\tilde{m}}^{-1}\\hat{\\bm g}_{\\tilde{m}k},\n\\end{equation}\nwhere $\\bm{R}_{\\tilde{m}}=\\text{Cov}\\left(\\sum_{l=1, l\\neq k}^K\\sqrt{{p}_l}\\bm{\\hat g}_{\\tilde{m}l}+\\tilde{\\bm \\eta}_{\\tilde{m}}\\right)$,   $\\bm{\\hat g}_{\\tilde{m}l}=\\left[{\\hat g}_{\\tilde{m}1l} \\dots {\\hat g}_{\\tilde{m}\\mathcal{N}_{\\tilde{m}} l} \\right]^H$, \nand\n$\\tilde{\\bm \\eta}_{\\tilde{m}}=\\left[\\tilde{ \\eta}_{\\tilde{m}1} \\dots \\tilde{ \\eta}_{\\tilde{m}\\mathcal{N}_{\\tilde{m}}} \\right]^H$.  \nFurthermore, the covariance matrix $\\bm{R}_{\\tilde{m}}$ conditioned on instantaneous CSI can be written as\n\\begin{equation}\n\\bm{R}_{\\tilde{m}}=\\sum_{l=1, l\\neq k}^K p_l\\hat{\\bm g}_{\\tilde{m}l}\\hat{\\bm g}_{\\tilde{m}l}^H+ \\bm{\\tilde{\\Sigma}}_{\\tilde{m}},\n\\end{equation}\nwhere $\\bm{\\tilde{\\Sigma}}_{\\tilde{m}}\\in \\mathbb{C}^{\\mathcal{N}_{\\tilde{m}}\\times \\mathcal{N}_{\\tilde{m}}}$ is a diagonal matrix with $n_{\\tilde{m}}$-th diagonal element given by ${\\tilde{\\Sigma}}_{\\tilde{m}n_{\\Tilde{m}}}=\\tilde{\\sigma}_{\\tilde{m}n_{\\tilde{m}}}^2\/2$.\nNote that, when $K=1$, we have $G_{\\tilde{m}k}=2\\hat{g}_{\\tilde{m}k}\/\\tilde{\\sigma}_{\\tilde{m}n_{\\tilde{m}}}$ which is identical to the MRC scheme that represents the optimal combining scheme under no IUI. \nAdditionally, note that we use the estimated values of CSI ($\\hat{g}_{\\tilde{m}n_{\\tilde{m}}k}$) without considering the CSI estimation error. However, several works in the literature discussed the optimal diversity combining scheme under imperfect CSI estimation \\cite{DAS_IMPERFECT_CSI_1,DAS_IMPERFECT_CSI_2}, which is however not investigated in this paper for brevity.  \n\\subsection{Formulation of Optimization Problem }\nFor the proposed dynamic cell-free network, the process of AP clustering needs to be performed jointly with beamforming design.\nThis can be achieved by forming a general optimization problem that jointly finds the optimal cluster set-beamforming vector pairs such that a certain objective is satisfied.\nTo achieve the best gain from SIC detection, we propose to order users based on their overall weighted power gain of the communication channel such that\n$p_1\\sum_{\\tilde{m}=1}^{\\tilde{M}}~\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}1}|^2 \n    \\leq \\dots \\leq\n    p_k\\sum_{\\tilde{m}=1}^{\\tilde{M}} \n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2~~\n    \\leq\n    \\dots\n    \\leq\np_K\\sum_{\\tilde{m}=1}^{\\tilde{M}} \n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}K}|^2$.\nAccordingly, the general problem of AP clustering and beamforming optimization can be formulated as\n\\begin{equation}\n\\begin{aligned}\n& ~\\textbf{P}_1: \\underset{n, \\bm{W}}{\\text{max}}~\n& \\text{\\hspace{-40mm}} \\sum_{k=1}^K\\log_2\\left(1+\\gamma_{k}^{\\{\\mathcal{C}_n\\}} \\right)\\\\\n& ~\\text{Subject to:} \\\\\n& ~\\textbf{C}_1: \\sum_{m=1}^{M} \\left(w_{m \\delta_l}^2-\\sum_{i=\\delta_l+1}^{l}w_{mi}^2 \\right)\\bar{\\gamma}_{\\tilde{m}n_{\\tilde{m}}l}\\geq P_{\\text{s}}, \\\\\n& ~  \\textbf{C}_2: 0\\leq w_{\\tilde{m}k}\\leq 1 ,\n\\\\\n&  ~  \\textbf{C}_3:||\\bm{w}_{k}||^2\\leq 1,\n\\\\\n&~~\\forall k,  \\forall~\\delta_l= 1, \\dots, l-1\\;\\mbox{and}\\; l=2, \\dots, K,\\\\\n\\end{aligned}\\label{OptimizationProb}\n\\end{equation}\nwhere $\\bar{\\gamma}_{\\tilde{m}n_{\\tilde{m}}l}=\\frac{2p_lG_{\\tilde{m}n_{\\tilde{m}l}}}{\\tilde{\\sigma}_{\\tilde{m}n_{\\tilde{m}}}^2}|\\hat{g}_{\\tilde{m}n_{\\tilde{m}l}}|^2$. \nThe constraints $\\textbf{C}_1$ refer to the set of $\\sum_{l=2}^K(l-1)=\\frac{K(K-1)}{2}$ conditions required for successful SIC operation with receiver sensitivity of $P_{\\text{s}}$. For maximizing the minimum rate, the objective function of the optimization problem will be replaced by $\\underset{k}{\\text{min}} \\log_2\\left(1+\\gamma_{k}^{\\{\\mathcal{C}_n\\}} \\right)$.\nNote that the receiver deals only with measured channel values (estimated) which include the estimation error as well as the AWGN part. However, the achieved SINR value after the SIC procedure will be decreased by pilot contamination components (as can be noticed from the second part of Eq. (\\ref{SINR_3})).  \n\nTo optimally solve the problem $\\textbf{P}_1$ in (\\ref{OptimizationProb}), \nwe need to simultaneously solve for $\\mathcal{C}_n$ and $\\bm{W}$.\nSpecifically, for every possible clustering configuration, there is a related optimal beamforming vector that maximizes the objective function.\n{\\em The globally optimal solution is then the one that gives the best performance among all clustering configurations and their corresponding optimized beamforming vectors ($\\bm{W}$).\nSuch a problem belongs to the set of complete non-deterministic polynomial-time hard (NP-hard) problems.\nFurthermore, given that a certain clustering configuration is selected, solving $\\textbf{P}_1$ w.r.t $\\bm{W}$ grows exponentially with $\\tilde{M}$ and\/or $K$ which makes the cell-free network model  impractical for a network with a massive number of devices communicate through a massive number of APs within a  geographic area.\nTherefore, for  practical implementation of a dynamic cell-free network, we design a novel hybrid DDPG-DDQN-based DRL system that produces the best clustering configuration and the beamforming vectors such that the objective function in $\\textbf{P}_1$ is maximized.}\n\n\n\\section{DRL Implementation of AP Clustering and Beamforming Design}\n\nIn this section, we introduce a novel DRL model that solves the general optimization problem (i.e. problem $\\textbf{P}_1$) of jointly optimizes the clustering of APs and the beamforming vectors. \n\n\\subsection{Theoretical Preliminaries}\nThe concept of reinforcement learning (RL) refers to the learning process of an agent interacting with its environment after receiving certain observations. The environment provides a reward to the agent for every interaction and\nthe RL agent aims to select the right action for the next interaction in order to maximize the discounted reward over a time horizon. \nThis problem can be formulated as a \\textit{Markov decision\nprocess} (MDP). An MDP is a tuple ($\\mathcal{\\bm S}$, $\\mathcal{A}$, $\\mathcal{P}$, $\\mathcal{R}$, $\\zeta$), where $\\mathcal{S}$ represents the states space which contains the set of $K$-dimensional states (with each state at time $t$ denoted by $\\bm{s}_t$), $\\mathcal{A}$ is the action space that contains a finite set of actions from which the agent can choose, $\\mathcal{P}$ : $\\mathcal{S}$ $\\times$ $\\mathcal{A}$ $\\times$ $\\mathcal{S}$ $\\rightarrow$ [0, 1] is a transition probability in which $\\mathcal{P}(\\bm{s}, a, \\bm{s}')$ defines the probability of observing state $\\bm{s}'$ after executing action $a$ in the state $\\bm{s}$, $\\mathcal{R}: \\mathcal{S} \\times \\mathcal{A} \\rightarrow \\mathbb{R}$ is the expected reward after being in state $\\bm{s}$ and taking action $a$, and $\\zeta$ $\\in [0, 1)$ is the discount factor.\nTo solve the MDP,  RL algorithms have been developed to learn and find a discrete value function or a ``policy\". Such a discretization can lead to lack of generalization and significantly increase the problem's dimensionality. \nTherefore, deep RL (DRL) algorithms based on function approximation by deep neural networks (DNNs) have been proposed.\n\nDRL algorithms can be classified into three types: (i) \\textit{value-based} methods such as  deep Q-learning (DQL) and SARSA which only learn the so-called  value function to find a policy, (ii) \\textit{policy-based} methods which learn the policy directly by following the gradient with respect to the policy, and (iii) \\textit{actor-critic} methods which are a hybrid of the value-based for the critic and policy-based methods for the actor.\n\\iffalse\n\\cite{DRL_1}:\n\\begin{itemize}\n    \\item \\textit{Value-based} methods such as  deep Q-learning (DQL) and SARSA which only learn the so-called  value function to find a policy.\n    \\item \\textit{Policy-based} methods which learn the policy directly by following the gradient with respect to the policy itself in  order to keep improving the policy constantly. These methods suffer from noisy gradients and high variance.\n    \\item \\textit{Actor-critic} methods which are a hybrid of the value-based methods for the critic and the policy-based methods for the actor. The critic reduces the variance of policy gradient methods by estimating the action-value function $Q(\\bm{s},a)$.\n\\end{itemize}\n\\fi\n\nThe standard DQL method represents the most popular update algorithm in the literature due to the availability of a mature theory. Basically, the DQL update equation at time $t$ for a network agent with parameters $\\theta^Q$ after taking action $a_t$ in state $\\bm{s}_t$ and observing the\nimmediate reward $r_{t+1}$ and resulting state $\\bm{s}_{t+1}$ is:\n\\begin{equation}\n\\label{eq:dqn-update equation}\n\\begin{aligned}\nQ(\\bm{s}, a \\,|\\, \\theta^Q_{t+1}) &= Q(\\bm{s}, a\\,|\\, \\theta^Q_t) + \\nu \\bigg[r_{t+1} + \\zeta  \\max_{a'} Q(\\bm{s}_{t+1}, a'\\,|\\, \\theta^Q_t) - Q(\\bm{s}_t, a_t\\,|\\, \\theta^Q_t)\\bigg]\\\\\n&\\text{\\hspace{-30mm}}= Q(\\bm{s}, a\\,|\\, \\theta^Q_t) + \\nu \\bigg[r_{t+1} + \\zeta \\max_{a'} Q(\\bm{s}_{t+1}, \\argmax_{a'} Q(\\bm{s}_{t+1},a'\\,|\\, \\theta^Q_t)\\,|\\, \\theta^Q_t) - Q(\\bm{s}_t, a_t\\,|\\, \\theta^Q_t)\\bigg],\n\\end{aligned}\n\\end{equation}\nwhere $\\nu$ is the learning rate.\nComputing the term $ \\underset{a'}{\\text{max}}~ Q(\\bm{s}_{t+1}, \\underset{a'}{\\text{argmax}}  Q(\\bm{s}_{t+1},a'\\,|\\, {\\theta}^Q_t)\\,|\\, \\theta^Q_t)$ introduces a systematic overestimation of the Q-values during the learning that is accentuated by the use of bootstrapping, i.e. learning estimates from estimates. The Q-learning update in (\\ref{eq:dqn-update equation}) uses the same Q-network $Q(\\bm{s}, a\\,|\\, {\\theta}^Q_t)$ both to select and to evaluate an action. After highlighting the overestimation bias in experiments across different Atari game environments,  Hasselt et al. \\cite{van2016deep} decoupled the action selection and evaluation by introducing two deep Q-networks,  a  $Q$ network and a target network $Q'$ with different parameters $\\theta^Q$ and ${\\theta}^{Q'}$, respectively, to avoid the maximization bias.\nThe  $Q'$ network is used for action selection while the $Q$ network is used for action evaluation.\nThis is known as {\\em deep double Q-learning algorithm (DDQL)}.\nThe DDQL update equation of the network can be expressed as:\n\\begin{dmath}\n\\text{\\hspace{-2mm}}Q(\\bm{s}, a \\,|\\, {\\theta}^Q_{t+1}) = Q(\\bm{s}, a\\,|\\, {\\theta}^Q_t) \n\\\\\n+ \\nu \\bigg[r_{t+1} + \\zeta \\max_{a'} Q(\\bm{s}_{t+1}, \\argmax_{a'} Q'(\\bm{s}_{t+1},a'\\,|\\, {\\theta}^{Q'}_t)\\,|\\, {\\theta}^Q_t) - Q(\\bm{s}_t, a_t\\,|\\, {\\theta}^Q_t)\\bigg].\n\\label{eq:ddqn-update-q1-equation}\n\\end{dmath}\nThe parameters ${\\theta}^{Q'}$ of the  $Q'$ network periodically hard-copy the parameters $\\theta^Q$ of $Q$ network after $t_0$ time steps using the Polyak averaging method with parameter $\\tau \\in [0,1]$:\n\\begin{equation}\n\\label{eq:ddqn-update-q2-equation}\n{\\theta}^{Q'}_{t+t_0} = (1-\\tau) \\,{\\theta}^{Q'}_t + \\tau \\,\\theta^Q_t.\n\\end{equation}\n\nThe DDQL networks show a better performance that standard DQL \\cite{van2016deep}; however, due to the discretization requirements of the DNN outputs (the action space $\\mathcal{A}$), it results in a huge expansion of the action space dimensionality when used in the optimization of an objective function of continuous dependent variables.\n{\\em This dimensionality issue makes it an unattractive solution for solving the beamforming problem under massive number of users and APs.}\nHowever, it is a relevant candidate for the clustering problem of the APs since it avoids the need for an extremely inefficient exhaustive search method. \nThis  motivates us to utilize the ``DDPG\" policy for the beamforming design problem.\n\nDDPG belongs to the class of actor-critic algorithms. It concurrently learns a Q-function network approximation $Q(\\bm{s}, a|\\theta^{Q})$ called the critic, and a policy network approximation $\\mu(\\bm{s}|\\theta^{\\mu})$ called the actor. \nThe Q-function network is trained using the Bellman equation, while the policy network is learnt using the Q-function. \nUnlike the DQL policies which output the probability distribution $\\pi(a|\\bm{s})$ across a discrete action space $\\mathcal{A}$,\nthe policy network of DDPG directly maps states to actions. \nSpecifically, at every time step $t$, it maximizes its loss function defined as:\n\\begin{equation}\n\\label{eq:loss-function-actor}\nJ(\\theta) = \\mathbb{E}\\bigg[Q(\\bm{s}, a) \\;|\\; \\mathcal{S}=\\bm{s}_t, a=\\pi(a|\\bm{s}_t)\\bigg]\n\\end{equation}\nand updates its weights $\\theta$ by following the gradient of (\\ref{eq:loss-function-actor}):\n\\begin{equation}\n\\label{eq:gradient-loss-function-actor}\n\\nabla J_{\\theta^{\\mu}}(\\theta) \\approx \\nabla_{a} Q(\\bm{s},a) \\,\\nabla \\mu(s|\\theta^{\\mu}).\n\\end{equation}\nThis update rule represents the deterministic policy gradient (DPG) theorem, rigorously proved by Silver et al. in the supplementary material of \\cite{silver2014deterministic}.\nThe term $\\nabla_{a} Q(\\bm{s},a)$ is obtained from a Q-network  $Q(\\bm{s}, a|\\theta^{Q})$ called the critic by backpropagating its output w.r.t. the action input $\\mu(\\bm{s}|\\theta^{\\mu})$. When the number of actions is very large, this actor-critic training procedure solves the intractability problem of DQN \\cite{mnih2015human}  by using the following approximation:\n\\begin{equation}\n\\label{eq:max-approx-ddpg}\n\\max_{a} Q(\\bm{s}, a) \\approx Q(\\bm{s}, a |\\theta^{Q})|_{a=\\mu(\\bm{s}|\\theta^{\\mu})}.\n\\end{equation}\nSimilar to DQN, two tricks are employed to stabilize the training of the DDPG actor-critic architecture, namely, 1) the experience replay buffer $R$ to train the critic, and 2) target networks for both the actor and the critic which are updated using the polyak averaging in the same way it was done in (\\ref{eq:ddqn-update-q2-equation}).\nNow that we have provided a brief DRL background on the two methods used by our proposed algorithm (DDQL and DDPG), \na detailed description of the proposed DRL-based AP clustering and beamforming design for a dynamic cell-free network can be presented in the following subsection.\n\\subsection{DRL Agent Design for AP Clustering and Beamforming Optimization}\nOur goal is to design a DRL system that jointly optimizes the clustering of APs and the beamforming vectors given a certain CSI matrix\n$\\bm{H}=\n\\left[\n{\\tilde{\\bm{g}}}_{1}\n\\dots \n{\\tilde{\\bm{g}}}_{K}\n\\right]$.\nIn this context, we develop a hybrid DDPG-DDQL DRL scheme that simultaneously learns the best clustering subsets-beamforming vector pairs given a certain CSI matrix $\\bm{H}$ (or any other metric related to $\\bm{H}$).\nFig. \\ref{AI_Model} shows a schematic block diagram of the developed hybrid DDPG-DDQL DRL scheme.\n   \t\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[scale=0.35]{DQL_MODEL.jpg}\n\t\n\t\t\\caption{Hybrid DDPG-DDQL model for AP clustering and uplink beamforming design. }\\label{AI_Model}\n\t\\end{figure} \nTo find the beamforming decoding matrix of the $k$-th user $\\bm{W}_k=\\text{Reshap}(\\bm{w}_{k})$, where $\\bm{W}_k\\in \\mathbb{R}^{\\nint*{\\sqrt{\\sum_{\\tilde{m}=1}^{\\tilde{m}}\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}} \\bm{w}_{k}}}\n\\times\n\\nint*{\\sqrt{\\sum_{\\tilde{m}=1}^{\\tilde{m}}\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}} \\bm{w}_{k}}}}$\\footnote{Note that the empty elements on $\\bm{W}_k$ are zero padded.} and  $\\bm{w}_{k}=\\left[w_{1k} \\dots w_{\\tilde{M}k}\\right]^H$, we opt for the actor-critic DDPG algorithm \\cite{lillicrap2015continuous} since all the  elements $w_{i,j}$ of $\\bm{W}_k$ are in the continuous range $[0,1]$.\nFor the AP clustering problem, we use the well-known DDQL algorithm \\cite{van2016deep} to find the best access point clustering configuration since the possible number of clustering configurations is finite and the index of each possible configuration is an integer number. \n\nAs shown in Fig.~\\ref{AI_Model}, the two algorithms interact with a simulated cell-free network environment to solve the optimization problem $\\textbf{P}_1$ in (\\ref{OptimizationProb}).\nThe design of the cell-free network environment involves the specification of the environment state $\\bm{s}$ and the definition of immediate reward function $r$ required by the DRL algorithms to approximate the policies and the Q-values. \nThe state of the environment is a vector of the SINR values of users, where the state $\\bm{s}_k$ of the $k^{th}$-user is chosen to be the SINR value $\\gamma_{k}^{\\mathcal{C}_n}$ which is a varying function of the instantaneous CSI matrix.\nUnder this setup, the action space $\\mathcal{A}$ is the pair of actions $a=(a^{b}, a^{\\mathcal{C}}) = (\\bm{W}_k,\\mathcal{C}_n)$ that the DDPG and the DDQN output separately. \nThe superscripts $b$ and $\\mathcal{C}$ refer to  ``\\textit{beamforming}\" and ``\\textit{clustering}\", respectively. \nAfter receiving the environment's state $\\bm{s}$, the DDPG algorithm outputs the action $a^{b} = \\bm{W}_k$ and the DDQL outputs  the action $a^{\\mathcal{C}}$ representing the cluster partition $\\mathcal{C}_n$.\n\nTable \\ref{Table3} summarizes the environment design by specifying the additional problem parameters.\n\\begin{table}[h!]\n    \\centering\n      \\caption{DRL agent design}\n\\begin{tabular}{|c|c|}\n\\hline\nEnvironment Variables & System Equivalence  \\\\\n\\hline\n\\hline\nState ${\\bm{s}}=\\{{s}_1, \\dots, {s}_K \\}$ & $\\{\\gamma_{1}^{\\mathcal{C}_n}, \\dots, \\gamma_{K}^{\\mathcal{C}_n}\\}$\\\\\n\\hline\nReward $r$ & $ \\sum_{k=1}^K  \\;\\log\\left(1+\\gamma_{k}^{\\mathcal{C}_n}\\right)$\\\\\n\\hline\nAction $\\mathcal{A}$ & $(a^{b}, a^{\\mathcal{C}}) = (\\bm{W}_k, \\mathcal{C}_n)$ \\\\\n\\hline\n$n=1, \\dots, N$ & Index of possible Clustering \\\\\n\\hline\n$\\mathcal{C}_n$ & $n$-th clustering configuration \\\\\n\\hline\n$K$ & Number of active users \\\\\n\\hline\n\\end{tabular}\n    \\label{Table3}\n\\end{table}\n{\\em Note that when the number of clusters equals to the total number of APs ($\\tilde{M}=M$), the network will reduce to the static cell-free network with $\\mathcal{C}_n=\\mathcal{C}, ~  n=1$}. \nAccordingly, only DDPG model will be required to optimize the beamforming vectors ($\\bm{W}_k$).\\\\\nOur TensorFlow\/Keras implementation of the actor and critic networks (including their corresponding target networks) have two hidden layers of 256 and 128 neurons, respectively. The DDQN networks have two \nfully-connected layers of 64 neurons followed with an activation function \\textit{relu} each, and a final linear fully-connected layer. We use a discount factor $\\zeta = 0.99$, a learning rate $\\nu = 5 \\,\\cdot\\,10^{-5}$, a Polyak averaging parameter $\\tau=10^{-3}$, and an experience replay buffer of size $R=20000$. The critic optimizer is Adam with its default hyperparameters $\\beta_1=0.9$ and $\\beta_2=0.999$. We train all the networks on 2500 episodes, with 500 time step each.\n\\subsection{Description of the Hybrid DDPG-DDQL Algoritm}\n\nFor the static cell-free network, only the DDPG algorithm is trained (one clustering configuration). However, for the dynamic cell-free network, both the DDPG and DDQL networks are trained.\n\\begin{itemize}\n    \\item \\underline{DDPG network}:\n    \\begin{itemize}\n        \\item \\textit{The actor network $\\mu (\\bm{s} | \\theta^{\\mu})$} maps the SINR values of users to the beamforming matrix $\\bm{W}_k$. The output of the network is $a^{b}$, i.e. a flatten list of all the elements $w_{ij} \\in [0,1]$.\n        \\item \\textit{The target actor network $\\mu' (\\bm{s} | \\theta^{\\mu'})$}: time-delayed copy of the actor network $\\mu (\\bm{s} | \\theta^{\\mu})$.\n        \\item \\textit{The critic network $Q(\\bm{s}, a^b | \\theta^{Q})$}: maps the SINR values and the output action of $\\mu (\\bm{s} | \\theta^{\\mu})$ to their corresponding Q-value.\n        \\item \\textit{The target critic network $Q'(\\bm{s}, a^b | \\theta^{Q'})$}: time-delayed copy of the critic network $Q(\\bm{s}, a^b | \\theta^{Q})$.\n    \\end{itemize}\n    \\item \\underline{DDQL network}:\n    \\begin{itemize}\n        \\item \\textit{The $Q_c$-network $Q_c(\\bm{s}, a^{\\mathcal{C}} | \\theta^{Q_c})$} maps the SINR values of users to the the Q-values of the state and all the clustering partitions.\n        \\item \\textit{The target $Q_c$-network $Q_c'(\\bm{s}, a^{\\mathcal{C}} | \\theta^{Q_c'})$}: time-delayed copy of the $Q_c$-network $Q(\\bm{s}, a^{\\mathcal{C}} | \\theta^{Q_c})$.\n    \\end{itemize}\n\\end{itemize}\n\nWe describe all of the training steps of our algorithm in \\textbf{Algorithm \\ref{algo:DRL-for-free-cell-network}}. We start by initializing all neural networks and their targets for the beamforming and clustering problems as well as a replay buffer $R$ (lines \\ref{algo:init1}--\\ref{algo:init5}). For every episode, we initialize the environment with $N$ access points and $K$ users by setting the initial state $\\bm{s}_0$ to a random vector of SINR values of size $K$ (line \\ref{algo:init-env}). At every time step $t$ of the episode, the DDQL and DDPG agents pick an action $a^{\\mathcal{C}}_t$ and $a^b_t$ respectively (lines \\ref{algo:pick-ab}--\\ref{algo:pick-ac}). The combined action $a_t = (a^b_t, a^{\\mathcal{C}}_t)$ is sent to the free-cell network environment which will transit to a new state $\\bm{s}_{t+1}$, and this new state will be returned together with the immediate reward $r_t$ (lines \\ref{algo:combine-ab-ac}--\\ref{algo:get-state-reward}). After storing the transition tuple $(\\bm{s}_{t}, a_{t}, r_{t}, s_{t+1})$ in the replay buffer $R$ (line \\ref{algo:store-transition}), we randomly sample from the experience replay buffer $N$ transitions to train the DDPG and DDQL networks (line \\ref{algo:sample-replay-buffer}). We start the DDPG training in line (\\ref{algo:ddpg-td-target}) by computing the TD target for the Q-network $Q(\\bm{s}, a | \\theta^{Q})$ using the target Q-network $Q'(\\bm{s}, a | \\theta^{Q'})$. We update the critic $Q(\\bm{s}, a | \\theta^{Q})$'s parameters $\\theta^{Q}$ in line \\ref{algo:update-critic} using the gradient of the MSE of the loss function of the TD target and the output of the critic. The update of the actor's parameters $\\theta^{\\mu}$ uses the Monte Carlo approximation of gradient in line \\ref{eq:gradient-loss-function-actor}. \nThe target critic and target policy networks get updated slowly every $P$ iterations (lines \\ref{algo:update-critic-target}--\\ref{algo:update-actor-target}). Finally, we update the parameters $\\theta^{Q_c}$ of the DDQL Q-network using the Bellman equation in line (\\ref{algo:update-ddql-q-network}) after selecting the action using the target Q-network $Q'_c(\\bm{s}, a | \\theta^{Q'_c})$ in line \\ref{algo:ddql-action-selection}. Similar to the DDPG target network, we update in line \\ref{algo:update-ddql-target} the DDQL target Q-network every $P$ iterations.\n\\begin{algorithm}\n\\small\n\\caption{Hybrid DDPG-DDQL algorithm for uplink beamforming and clustering}\\label{algo:DRL-for-free-cell-network}\n\\begin{algorithmic}[1]\n\\State Randomly initialize the critic $Q(\\bm{s}, a | \\theta^{Q})$ and the actor $\\mu(\\bm{s} | \\theta^{\\mu})$ with weights $\\theta^{Q}$ and $\\theta^{\\mu}$ \\label{algo:init1}\n\\State Initialize target network $Q'$ and $\\mu'$ with weights $\\theta^{Q^{\\prime}} \\leftarrow \\theta^{Q}$, $\\theta^{\\mu^{\\prime}} \\leftarrow \\theta^{\\mu}$ \\label{algo:init2}\n\\State Randomly initialize the $Q_c$-network $Q_c(\\bm{s}, a | \\theta^{Q_c})$ \\label{algo:init3}\n\\State Initialize the target network $Q'_c(\\bm{s}, a | \\theta^{Q'_c})$ with weights $\\theta^{Q_c^{\\prime}} \\leftarrow \\theta^{Q_c}$ \\label{algo:init4}\n\\State Initialize replay buffer R \\label{algo:init5}\n\\For {$episode=1,\\dots, E$}\n\\State Receive initial observation state $\\bm{s}_1$ after initializing the environment \\label{algo:init-env}\n\\For {$t=1,\\dots, T$}\n\\State  Select the beamforming action $a^{b}_{t}=\\mu\\left(\\bm{s}_{t} | \\theta^{\\mu}\\right)$ \\label{algo:pick-ab}\n\\State  Select the clustering action $a^{\\mathcal{C}}_t=\\arg \\max _{a^{\\mathcal{C}}} Q_c\\left(\\bm{s}_t, a^{\\mathcal{C}}\\right)$ \\label{algo:pick-ac}\n\\State Define $a_t = (a^b_t, a^{\\mathcal{C}}_t)$ \\label{algo:combine-ab-ac}\n\\State Execute action $a_{t}$ and observe reward $r_{t}$ and observe new state $\\bm{s}_{t+1}$ \\label{algo:get-state-reward}\n\\State Store transition $(\\bm{s}_{t}, a_{t}, r_{t}, \\bm{s}_{t+1})$ in $R$ \\label{algo:store-transition}\n\\State Sample a random minibatch of $\\mathcal{L}$ transitions $(\\bm{s}_{i}, a_{i}, r_{i}, \\bm{s}_{i+1})$ from $R$ \\label{algo:sample-replay-buffer}\n\\State Get $a^b_i$ and $a^{\\mathcal{C}}_i$ from $a_i$\n\\Statex\\LeftComment{2}{\\underline{Training the DDPG networks}}\n\\State Compute the TD target $y_{i}=r_{i}+\\zeta\\, Q^{\\prime}\\left(\\bm{s}_{i+1}, \\mu^{\\prime}\\left(\\bm{s}_{i+1} | \\theta^{\\mu^{\\prime}}\\right) | \\theta^{Q^{\\prime}}\\right)$  \\label{algo:ddpg-td-target}\n\\State Update the critic $Q(\\bm{s}, a | \\theta^{Q})$ by minimizing the loss: $L=\\frac{1}{\\mathcal{L}} \\sum_{i}\\left(y_{i}-Q\\left(\\bm{s}_{i}, a^b_{i} | \\theta^{Q}\\right)\\right)^{2}$ \\label{algo:update-critic}\n\\Statex \\qquad \\quad  Update the actor policy $\\mu(\\bm{s} | \\theta^{\\mu})$ using a monte-carlo approximation of (\\ref{eq:gradient-loss-function-actor}):\n\\State \\qquad \\qquad$\\left.\\left.\\nabla_{\\theta^{\\mu}} J \\approx \\frac{1}{\\mathcal{L}} \\sum_{i} \\nabla_{a} Q\\left(\\bm{s}, a | \\theta^{Q}\\right)\\right|_{\\mathcal{S}=\\bm{s}_{i}, a=\\mu\\left(\\bm{s}_{i}\\right)} \\nabla_{\\theta^{\\mu}} \\mu\\left(\\bm{s} | \\theta^{\\mu}\\right)\\right|_{\\mathcal{S}=\\bm{s}_{i}}$\n\\Statex \\qquad \\quad Update the DDPG target networks $Q'$ and $\\mu'$ \\textbf{if} $mod(t, P)=0$:\n\\State \\qquad \\qquad ${\\theta^{Q^{\\prime}} \\leftarrow \\tau \\theta^{Q}+(1-\\tau) \\theta^{Q^{\\prime}}}$ \\label{algo:update-critic-target}\n\\State \\qquad \\qquad ${\\theta^{\\mu^{\\prime}} \\leftarrow \\tau \\theta^{\\mu}+(1-\\tau) \\theta^{\\mu^{\\prime}}}$ \\label{algo:update-actor-target}\n\\Statex\\LeftComment{2}{\\underline{Training the DDQL networks}}\n\\State select $a^{*}=\\arg \\max _{a} Q'_c\\left(\\bm{s}_{i+1}, a| \\theta^{Q'_c}\\right)$   \\label{algo:ddql-action-selection}\n\\Statex \\qquad \\quad Update the $Q_c$ usin{}g:\n\\State \\qquad \\qquad $Q_c(\\bm{s}_i, a^c_i | \\theta^{Q_c}) \\leftarrow Q_c(\\bm{s}_i, a^c_i| \\theta^{Q_c})+\\nu\\,\\left(r_i+\\zeta\\, Q_c\\left(s_{i+1}, a^{*}| \\theta^{Q_c}\\right)-Q_c(\\bm{s}_i, a^{\\mathcal{C}}_i| \\theta^{Q_c})\\right)$ \\label{algo:update-ddql-q-network}\n\\Statex \\qquad \\quad Update the DDQL target networks $Q_c'$ \\textbf{if} $mod(t, P)=0$:\n\\State \\qquad \\qquad ${\\theta^{Q_1^{\\prime}} \\leftarrow \\tau \\theta^{Q_1}+(1-\\tau) \\theta^{Q_1^{\\prime}}}$ \\label{algo:update-ddql-target}\n\\EndFor\n\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\nNote that the DRL model can be easily extended to  optimize also the  uplink transmission power for users $P_k, \\forall k=1, \\dots, K$. However, the power control information will then need to be transmitted to the users, which will increase the signaling traffic significantly.\n\\section{Numerical Results and Discussions}\n\n\\subsection{Parameters and Assumptions}\n\n\n\n\n\n\\begin{table}[h!]\n    \\centering\n      \\caption{Simulation parameters}\n\\begin{tabular}{|c|c|}\n\\hline \nParameter & Value  \\\\\n\\hline\n\\hline\nAWGN PSD per UE &\n$-169$ dBm\/Hz  \\\\\n\\hline\nPath-loss exponent, $\\kappa$ & $2$\\\\\n\\hline\nNakagami parameters, $(\\mathcal{M},\\Omega)$ & $(1,1)$\\\\\n\\hline\nTraining sequence length, $\\tau_p$ & $K$ Samples\\\\\n\\hline\nPilot transmission power, $\\rho_k$ & $100$ mW, ${\\forall k}$\\\\\n\\hline\nSIC sensitivity, $P_s$  & $1$ dBm\n\\\\\n\\hline\n\\end{tabular}\n    \\label{Table2}\n\\end{table}\n\nTable \\ref{Table2} presents  the  main  system  parameters  used\nto obtain simulation and analytical results. To simplify simulation and analysis of this work and to concentrate on the most insightful conclusions, we establish some operating assumptions related to channel training and CSI fading models.\nWe assume that users' pilot signals are mutually orthonormal such that $\\tau_p=K\\leq \\tau_c$ with zero AWGN component in CSI estimation, i.e. $|\\bm{\\varphi}_{lmn}^H\\bm{\\varphi}_{xyz}|=1$ if $(l,m,n)=(x,y,z)$ or zero otherwise.\n$\\forall~(l,x)=1, \\dots ,\\tilde{M}$, $(m,y)=1, \\dots, \\mathcal{N}_{(l,x)}$ and $(n,z)=1, \\dots, K$.\nThis will result in removal of the CSI estimation error parts (due to both pilot contamination and AWGN component, $\\bm{\\eta}_{\\tilde{m}n_{\\tilde{m}}}$) and considering $\\hat{g}_{xyz}={g}_{xyz}$.\nAdditionally, we assume that all channel fading gains $h_{\\tilde{m}n_{\\tilde{m}}k}$ are independent and identically distributed (i.i.d) with $\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}=\\mathcal{M}=1$ and $\\Omega_{\\tilde{m}n_{\\tilde{m}}k}=\\Omega=1$.\nSimilarly, we assume that all AWGN values belong to a set of i.i.d rvs with PSD $\\sigma_{\\tilde{m}n_{\\tilde{m}}}\/2=-169$ dBm\/Hz. \nAs for large-scale fading, we assume that all APs and users are uniformly distributed over a disc of radius 18m (corresponding to a network total coverage area of $1 \\text{km}^2$.\n\nFirst, we  investigate the accuracy of \\textit{Welch-Satterthwaite} sum of gamma rvs approximation method \n(Fig. \\ref{Welch_Figure_1}).\n\\begin{figure}[htb]\n\t\t\\centering\n\t\n\t\t\\includegraphics[scale=0.6]{Sum_Gamma.pdf}\n\t\t\\caption{An illustrative example of \\textit{Welch-Satterthwaite} approximation with different $\\#$ of rvs ($K$).}\\label{Welch_Figure_1}\n\t\\end{figure}\n\tIt can be noticed from Fig. \\ref{Welch_Figure_1} that a satisfactory accuracy for the sum of non-identically distributed (different $\\beta$) gamma rvs can be achieved by \\textit{Welch-Satterthwaite} approximation for small, medium, and high numbers of random variables.\n\tThis justifies the use of such an approximation instead of the use of the central limit theorem which requires a relatively large number of random variables.\n\\subsection{Outage Performance}\nThis section discusses the outage performance of cell-free network in a massive communication regime. \nEach simulation value is obtained via  $2\\times10^6$ Monte-Carlo simulation runs. In Fig. \\ref{Figure_Outage_DRL_1} (a),\nwe evaluate the  outage probability for a user in a static cell-free  network under different values of $K$.\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=9cm, width=14cm]{Figure_Outage_DRL_1.pdf}\n\t\t\\caption{User performance: (a) Probability of outage,  (b) Transmission rate per user. }\\label{Figure_Outage_DRL_1}\n\t\\end{figure}\nFig.~\\ref{Figure_Outage_DRL_1} (a) shows that an excellent outage performance can be achieved when the number of APs ($M$) is much grater than the number of users $K$, i.e. $M>>K$. \nHowever, when $K$ becomes comparable to $M$, the outage performance deteriorates significantly with almost total blockage at $K\\approx 0.5 M$.\nNote that this figure is produced assuming no precoding scheme, i.e. $w_{mk}=c, \\forall m, k$. A better performance can be achieved by first solving problem $\\textbf{P}_1$ to obtain $\\bm{W}_k$ and then substituting $\\bm{W}_k$ into (\\ref{e_outage_1}).\n\n\n\\subsection{DRL Model}\nIn this section, we study the performance of using the proposed DRL model in solving problem $\\textbf{P}_1$.\nFig. \\ref{Figure_Outage_DRL_1} (b) shows the per-user transmission rate in the cell-free network with the proposed DRL method.\nWe can notice that the performance reaches a certain level after around $500$ training episodes and then stays around that level.\nCompared with the max-min objective, we notice a significant performance gain with the max-sum objective.\nThis is due to the fact that, with the max-sum objective, we optimize $\\bm{W}_k, \\forall k=1, \\dots, K$ while with max-min objective, we maximize only $\\bm{W}_{k'}$ and use $W_k, k\\neq k'~\\&~k=1, \\dots, K$, where $k'$, denoting the index of the user with minimum achievable rate, is found from previous iterations.\n\nTo evaluate the performance of the proposed dynamic cell-free clustering scheme with SIC detection, Fig. \\ref{Figure_Two_DRL_1} a shows the per-user transmission rate for both static and dynamic cell-free networks.  \nIt can be noticed that utilizing SIC at the receiver side significantly compensates for the performance loss caused by clustering of the APs. Note that the per-user transmission rate achieved without SIC in presence of AP clustering is significantly lower than that in a static cell-free network.\n \nAdditionally, in Fig. \\ref{Figure_Two_DRL_1}(b), we compare the performance of the proposed DRL-based solution approach compared to traditional methods.\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=9cm, width=14cm]{Figure_Two_DRL_1.pdf}\n\t\t\\caption{Per-user transmission rate: (a) with SIC, (b) optimal vs. DRL.}\\label{Figure_Two_DRL_1}\n\\end{figure}\nWe can notice that DRL can achieve per user transmission rate of around $78\\%$ of that with conventional bisection method (subgradient method). \nThis performance degradation comes with a remarkable decrease in the computational complexity. With the proposed DRL-based design, the beamforming vectors can be obtained in an online manner for practical implementation of non-orthogonal multiple access in cell-free networks.\n\nFigs. \\ref{RMSE} (a) and (b) evaluate the convergence rate of the proposed DRL algorithm for two different network setups.\nSpecifically, we plot the absolute value of the difference between the normalized rate values generated by the DRL model and that by the bisection method (using the Matlab optimization toolkit).\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=9cm, width=14cm]{DRL_Conv_Figure.pdf}\n\t\t\\caption{DRL convergence rate: (a) Medium-scale scenario, (b) large-scale scenario.}\\label{RMSE}\n\\end{figure}\nIt can be noticed that the proposed DRL model shows a better convergence rate than the bisection method under medium-scale scenario with $M = 150$ and $K = 50$ (Fig. \\ref{RMSE} [a]) compared to that of large-scale scenario with $M = 200$ and $K = 60$ (Fig. \\ref{RMSE} [b]).\nHowever, the proposed DRL model under large-scale scenario is found to achieve a better performance by achieving an approximate of $92\\%$ of the normalized rate achieved by bisection method compared to only $78\\%$ achieved under medium-scale scenario.   \n\nFinally, to evaluate the performance of the cell-free network under different values of per-user transmission power at the large-scale regime, Fig. \\ref{DRL_Vs_Power} illustrates the variation in per-user transmission rate in a large-scale static cell-free network for different transmit power values.\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=10cm, width=11cm]{DRL_Vs_Power.pdf}\n\t\t\\caption{Per-user transmission rate vs. $P$.}\\label{DRL_Vs_Power}\n\\end{figure}\nIt can be noticed that rate enhancement becomes smaller when the transmission power is increased in a high $P$ regime (e.g. from $P = 37$ dBm to $P = 50$ dBm). \nHowever, a good rate enhancement still can be achieved when $P$ is changed from a relatively small value to a significantly larger value (e.g. from $P = 20$ dBm to $P = 37$ or $P=50$ dBm). Furthermore, the DRL model is found to show a faster convergence with larger values of $P$.  \n\\section{Conclusion}\n We have first derived closed-form expressions for the  probability of outage for an uplink user in a static  cell-free network.\nNext, we have proposed a novel dynamic cell-free network that partitions the distributed APs among a set of subgroups with each subgroup acting as a virtual AP equipped with a distributed antenna system (DAS).\nSuch a clustering is performed based on the current channel state information (CSI) among the APs and all users within the network coverage area. The dynamic cell-free network model can reduce the complexity for joint signal processing. For the dynamic cell-free model, we have  formulated the general optimization problem of  clustering the APs and designing the beamforming vectors. To solve this optimization problem, we have proposed a hybrid deep deterministic policy gradients (DDPG)-double deep Q-network(DDQN) scheme that jointly selects the optimal network clustering scheme with its optimal beamforming vector values. Possible extensions of this work would include the design and evaluation of more comprehensive DRL models that jointly estimate CSI, select the best clustering configuration for APs, and optimize different beamforming vectors. Also, benchmarking different DRL-based algorithms for optimizing resource allocation in different cell-free network architectures will be valuable.\n\n\n\t\\appendices\n\t\\section{}\n\n\\setcounter{equation}{0}\nFirst, let us define $X=\\sum_{m=1}^M|\\tilde{g}_{mk}|^2$ and\n\\[\nY =    \\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K|\\tilde{g}_{ml}|^2\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    |\\tilde{g}_{m{\\ddot{l}}}|^2\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K|\\tilde{g}_{m{\\Breve{l}}}|^2\n    \\right].\n    \\]\n    Let $W=Y+1\\approx Y$. (This assumption is valid for large values of $K$ and $M$.) \nThe derivation of closed-form expression for $X$ and $Y$ is very complicated due to the non-equal rate parameters ($\\tilde{\\beta}_{m,k}$ and $\\tilde{\\beta}_{m,l}$) which makes the MGF method unusable to derive $f_X(x)$ and $f_Y(y)$. Here, we use an accurate approximation of the sum of independent Gamma rvs with different shape and rate parameters, which is referred to as the \\textit{Welch-Satterthwaite} approximation \\cite{Satterthwaite1946}. Accordingly, we have $X\\thicksim \\mathcal{G}\\left(\\Dot{\\alpha}_{mk}, \\Dot{\\beta}_{mk} \\right)$ and \n$Y\\thicksim \\mathcal{G}\\left(\\Dot{\\alpha}_{mk'}, \\Dot{\\beta}_{mk'} \\right)$, \nwhere \n\\[\\Dot{\\alpha}_{mk}=\\frac{\\left(\\sum_{m=1}^M\\tilde{\\alpha}_{mk}\/ \\tilde{\\beta}_{mk}\\right)^2}{\\sum_{m=1}^M\\tilde{\\alpha}_{mk}\/\\tilde{\\beta}_{mk}^2},~~~~\n\\Dot{\\beta}_{mk}=\\frac{\\sum_{m=1}^M\\tilde{\\alpha}_{mk}\/\\tilde{\\beta}_{mk}^2}{\\sum_{m=1}^M\\tilde{\\alpha}_{mk}\/ \\tilde{\\beta}_{mk}},\n\\]\n\\[\\Dot{\\alpha}_{mk'}=\\frac{\n\\left(\n\\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{    \\tilde{\\alpha}_{ml}}{ \\tilde{\\beta}_{ml}}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{    \\tilde{\\alpha}_{m\\ddot{l}}}{ \\tilde{\\beta}_{m\\ddot{l}}}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{m\\Breve{l}}}{        \\tilde{\\beta}_{m\\Breve{l}}}\n    \\right]\n\\right)^2\n}\n{\n\\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{ml}}{    \\tilde{\\beta}_{ml}^2}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{    \\tilde{\\alpha}_{m\\ddot{l}}}{\\tilde{\\beta}_{m\\ddot{l}}^2}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{m\\Breve{l}}}{    \\tilde{\\beta}_{m\\Breve{l}}^2}\n    \\right]\n},\n\\]\n\\[\\Dot{\\beta}_{mk'}=\n\\frac{\n\\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{ml}}{\\tilde{\\beta}_{ml}^2}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{    \\tilde{\\alpha}_{m\\ddot{l}}}{\\tilde{\\beta}_{m\\ddot{l}}^2}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{m\\Breve{l}}}{    \\tilde{\\beta}_{m\\Breve{l}}^2}\n    \\right]\n}\n{\n \\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{    \\tilde{\\alpha}_{ml\n}}{\\tilde{\\beta}_{ml}}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{\\tilde{\\alpha}_{m\\ddot{l}\n}}{    \\tilde{\\beta}_{m\\ddot{l}}}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{    \\tilde{\\alpha}_{m\\Breve{l}\n}}{\\tilde{\\beta}_{m\\Breve{l}}}\n    \\right]\n}.\n\\]\nNote that this approximation becomes identical to the exact expression even for small to moderate number of APs and\/or UEs. This makes it useful for modeling large-scale cell-free architectures. \nThe PDF of the ratio $Z=\\frac{x}{w}$ can be defined as\n\\begin{equation}\n    f_Z(z)=\\frac{d}{dz}\\int_{0}^{\\infty}\\text{P}\\left(x\\leq z  w\\right)f_Y(w)dw=\\int_{0}^{\\infty}wf_X(zw)f_Y(w)dw.\\label{PDF_Integral_1}\n\\end{equation}\n Substituting $f_X(zw)$ and $f_Y(w-1)$ into \\textbf{A}.1, utilizing \\cite[Eq. 3.383.4]{gradshteyn2000}, and simplifying, we obtain  Eq. (\\ref{PDF_Centralized}).\n\\section{}\n\\setcounter{equation}{0}\nThe CDF of the ratio distribution ($Z=\\frac{X}{W}$) can be expressed as\n\\begin{equation}\n    F_Z(z)=\\text{P}\\left(x\\leq z W\\right) =\\int_{0}^{\\infty}\\text{P}\\left(x\\leq z w\\right)f_Y(w)dw.\\label{CDF_Integral_1}\n\\end{equation}\nBy substituting $f_Y(w)$ and $P\\left(x\\leq zw \\right)=1-\\gamma\\left(\\Dot{\\alpha}_{mk},\\Dot{\\beta}_{mk}zw\\right)$, where $\\gamma(.)$ is the lower incomplete gamma function \\cite[Eq. 6.5.2]{1965}, and utilizing \\cite[Eq. 6.455.2]{gradshteyn2000} and \\cite[Eq. 15.3.7]{1965}, and simplifying, we obtain Eq. (\\ref{e_outage_1}).\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAn intersection representation $\\calR$ of a graph $G$ is a collection\nof sets $\\{R(v) : v \\in V(G)\\}$ such that\n$R(u) \\cap R(v) \\neq \\emptyset$ if and only if $uv \\in\nE(G)$. Important classes of graphs are obtained by restricting the\nsets $R(v)$ to some specific geometric objects.  In an \\emph{interval\n  representation} of a graph, each set $R(v)$ is a closed interval of\nthe real line; \nand in a\n\\emph{circular-arc representation}, the sets $R(v)$ are closed arcs of\na circle;\nsee Fig.~\\ref{fig:cK222}.\n A graph is an \\emph{interval graph} if it admits an interval representation\n and it is a \\emph{circular-arc graph} if it admits a circular-arc\n representation.  We also denote the corresponding classes of graphs by \\hbox{\\rm \\sffamily INT}\\xspace\n and \\hbox{\\rm \\sffamily CA}\\xspace, respectively.\n \n\nIn many cases, the availability of a geometric representation makes\ncomputational problems tractable that are otherwise \\hbox{\\rm \\sffamily NP}-complete.  For\nexample, maximum clique can be solved in polynomial time for both\ninterval graphs and circular-arc graphs.  Another example is the\ncoloring problem, which can be solved in polynomial time for interval\ngraphs but remains \\hbox{\\rm \\sffamily NP}-complete for circular-arc graphs~\\cite{GareyJMP80}.\n\nA key problem in the study of geometric intersection graphs is the\n\\emph{recognition problem}, which asks whether a given graph has a\nspecific type of intersection representation.  It is a classic result\nthat interval graphs can be recognized in linear time.  For\ncircular-arc graphs the first polynomial-time recognition algorithm\nwas given by Tucker~\\cite{tucker1980efficient}.  McConnell gave a linear-time\nrecognition algorithm~\\cite{mcconnell2003linear}.\n\nIn this paper, we are interested in a generalization of the\nrecognition problem.  For a class $\\calX$ of intersection representations, the\n\\emph{partial representation extension\n  problem for $\\calX$} ({\\textsc{RepExt}}{$(\\calX)$} for short) is defined as follows.\n  In addition to a graph $G$, the input consists of a\n\\emph{partial representation}~$\\calR'$ that is a representation of an\ninduced subgraph $G'$ of $G$.  The question is whether there exists a\nrepresentation $\\calR \\in \\calX$ of $G$ that \\emph{extends} $\\calR'$ in the\nsense that $R(u) = R'(u)$ for all $u \\in V(G')$.  The recognition\nproblem is the special case where the partial representation is empty.\nThe partial representation extension problem has been recently studied\nfor many different classes of intersection graphs, e.g., interval\ngraphs~\\cite{KlavikKOSV17}, proper\/unit interval\ngraphs~\\cite{klavik2017extending}, function and permutation\ngraphs~\\cite{klavik2012extending}, circle graphs~\\cite{CFK13}, chordal\ngraphs~\\cite{KKOS15}, and trapezoid graphs~\\cite{KrawczykW17}.\nRelated extension problems have also been considered, e.g., for planar\ntopological~\\cite{adfjk-tppeg-15,jkr-kttpp-13} and\nstraight-line~\\cite{p-epsld-06} drawings, for contact\nrepresentations~\\cite{chaplick2014contact}, and rectangular\nduals~\\cite{abs-2102-02013}.\n\nIn many cases, the key to solving the partial representation extension\nproblem is to understand the structure of all possible\nrepresentations.  For interval representations, the basis for this is\nthe characterization of Fulkerson and Gross~\\cite{maximal_cliques},\nwhich establishes a bijection between the combinatorially distinct\ninterval representations of a graph $G$ on the one hand and the linear\norderings~$\\preceq$ of the maximal cliques of $G$ where for each\nvertex $v$ the cliques containing $v$ appear consecutively\nin~$\\preceq$ on the other hand.  This not only forms the basis for the\nlinear-time algorithm using PQ-trees by Booth and\nLueker~\\cite{PQ_trees}, but also shows that a PQ-tree can compactly\nstore the set of all possible interval representations of a graph.\nThe partial representation problem for interval graphs can be solved\nefficiently by searching this set for one that is compatible with the\ngiven partial representation.\n\nDespite the fact that circular-arc graphs straightforwardly generalize\ninterval graphs, the structure of their representations is much less\nunderstood.  It is not clear whether there exists a way to compactly\nrepresent the structure of all representations of a circular-arc\ngraph.  There are two structural obstructions to this aim.  First, in\ncontrast to interval graphs, it may happen that two arcs have\ndisconnected intersection, namely in the case when their union covers\nthe entire circle.  Secondly, intervals of the real line satisfy the\n\\emph{Helly property}: if any pair of sets in a set system intersects,\nthen the intersection of the entire set system is non-empty.\nConsequently, the maximal cliques of interval graphs can be associated\nto distinct points of the line and also the number of maximal cliques\nin an interval graph is linear in the number of its vertices.  In\ncontrary, arcs of a circle do not necessarily satisfy the Helly property\nand indeed the number of maximal cliques can be exponential.  The\ncomplement of a perfect matching $nK_{2}$ is an example of this\nphenomenon, see Fig.~\\ref{fig:cK222}b.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1.0,page=1]{cK222}\n\\caption{(a) The graph $\\overline{3K_{2}}$ and (b) its circular-arc representation.\n(c) A non-Helly representation of $K_4$.}\n\\label{fig:cK222}\n\\end{figure}\n\nTo capture the above properties, that may have substantial impact on\nexplorations of  circular-arc graphs, the following specific subclasses of\ncircular-arc graphs have been defined and intensively\nstudied~\\cite{Tucker1971MatrixCO,gavril1974algorithms,LinS2009survey,lin2013normal}:\n\n\\begin{compactitem}\n\\item \\emph{Normal circular-arc graphs} ($\\hbox{\\rm \\sffamily NCA}\\xspace$) are circular-arc graphs that\n  have an intersection representation in which the intersection of any two arcs\n  is either empty or connected. \n\\item \\emph{Helly circular-arc graphs} ($\\hbox{\\rm \\sffamily HCA}\\xspace$) have an intersection\n  representation that satisfies the Helly property, i.e there are no $k\\ge 3$\n  pairwise intersecting arcs without a point in common.\n\\item \\emph{Proper circular-arc graphs} ($\\hbox{\\rm \\sffamily PCA}\\xspace$) are circular-arc graphs that\n  have an intersection representation in which no arc properly contains\n  another.\n\\item \\emph{Unit circular-arc graphs} ($\\hbox{\\rm \\sffamily UCA}\\xspace$) are circular-arc graphs with an\n  intersection representation in which every arc has a unit length. \n\\end{compactitem}\n\nThe above properties can be combined together in the sense that a\nsingle representation shall satisfy more properties simultaneously,\ne.g. \\emph{Proper Helly circular-arc graphs} ($\\hbox{\\rm \\sffamily PHCA}\\xspace$) are\ncircular-arc graphs with an intersection representation that is both\nproper and Helly~\\cite{lin2007proper}. This is stronger than requiring\nthat a graph is a proper circular-arc graph as well as a Helly\ncircular-arc graph (with each property guaranteed by a different\nrepresentation), i.e., $\\hbox{\\rm \\sffamily PHCA}\\xspace \\subsetneq \\hbox{\\rm \\sffamily PCA}\\xspace \\cap \\hbox{\\rm \\sffamily HCA}\\xspace$.\n\nAnalogously, since $C_4$ has a unique representation, the wheel $W_4$\nis a graph with a Helly representation (the universal vertex covers\nall four clique points) or a normal representation (it covers three\ncliquepoints) but not normal Helly representation. Thus also\n$\\hbox{\\rm \\sffamily NHCA}\\xspace \\subsetneq \\hbox{\\rm \\sffamily NCA}\\xspace \\cap \\hbox{\\rm \\sffamily HCA}\\xspace$.\n\nMoreover, Tucker~\\cite{Tucker1974} proved that every representation of a proper\n(Helly) circular-arc graph that is not normal can be transformed into a normal\nrepresentation.\nHence, the following graph classes coincide $\\hbox{\\rm \\sffamily PCA}\\xspace=\\hbox{\\rm \\sffamily NPCA}\\xspace$ and $\\hbox{\\rm \\sffamily PHCA}\\xspace=\\hbox{\\rm \\sffamily NPHCA}\\xspace$.\nFig.~\\ref{fig:ca_inclusions}a shows inclusions between the defined graph classes.\n\nWe use an analogous notation for the classes of possible representations, \ni.e., \nfor $X\\subseteq\\{\\hbox{\\rm \\sffamily N,P,H}\\}$ the symbol $X$\\hbox{\\rm \\sffamily CAR}{} for the\nclass of all $X$\\hbox{\\rm \\sffamily CA}\\xspace{} representations, see Fig.~\\ref{fig:ca_inclusions}b.  We\nnote that whether a graph $G$ with a partial representation $\\cal R'$ admits an\nextension depends crucially on the class of allowed representations, as\nillustrated by the example of $W_4$ above.\n \n\\myparagraph{Our results.}\nWhile for many classes efficient algorithms for the representation extension\nproblem have been found, the problem has been open for circular arc graphs for\nnine years~\\cite{DBLP:journals\/corr\/abs-1207-6960}.\nWe prove that ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ is \\hbox{\\rm \\sffamily NP}-hard. To the best of our knowledge,\nit is the first known representation class for which the extension problem is\n\\hbox{\\rm \\sffamily NP}-hard while the recognition problem is in \\hbox{\\rm \\sffamily P}.\nOur reduction also works for ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$.\n\n\\begin{restatable}{theorem}{hcanpc}\\label{thm:hca_npc}\n  The problems ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$ and ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ are \\hbox{\\rm \\sffamily NP}-hard.\n  ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ is also \\hbox{\\rm \\sffamily NP}-hard if the predrawn arcs have pairwise distinct endpoints.\n\\end{restatable}\n\nWe complement this result by showing tractability for several subclasses,\nincluding all Helly variants; see Figure~\\ref{fig:ca_inclusions}b. \nLinear-time algorithms for recognizing\nHelly circular-arc graphs~\\cite{lin2006characterizations,joeris2011linear} use\nMcConnell's~\\cite{mcconnell2003linear} algorithm to construct a circular-arc\nrepresentation and transform it to a Helly circular-arc representation. This\ncannot be exploited in the case of partial representation extension.\n\n \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1,page=2]{ca_inclusions}\n\\caption{ (a) Relationships between classes of circular-arc graphs.  (b)\n  Relationships between classes of circular-arc representations. Classes\n  studied in this paper are underlined. {\\textsc{RepExt}}{} is polynomial for blue, while\n  \\hbox{\\rm \\sffamily NP}-complete for red.  \n}\n\\label{fig:ca_inclusions}\n\\end{figure}\n\nDeng et al.~\\cite{deng1996linear} and Lin et\nal.~\\cite{lin2006characterizations} characterize proper and proper Helly\ncircular-arc representations in terms of vertex orderings of the graph.  They\nshow that these orderings are unique under certain conditions.  Building on\nthese results, we prove the following two theorems.\n\n\\begin{restatable}{theorem}{nphcapoly}\n  \\label{thm:nphca_poly}\n  The problem ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$ can be solved in linear time.\n\\end{restatable}\n\n\\begin{restatable}{theorem}{phcared}\n  \\label{thm:phca_red}\n  The problem ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily PHCAR})$ can be solved in linear time.\n\\end{restatable}\n\nRecall that in the case of interval graphs, PQ-trees can be used to capture all\nplausible linear orderings of the maximal cliques.   Klav\\'{\\i}k et\nal.~\\cite{klavik2017extending} use this to solve ${\\textsc{RepExt}}$ for interval\nrepresentation by determining an order that is represented by the PQ-tree and\nthat extends a partial order that is derived from the partial representation.\n\nThe fact that Gavril~\\cite{gavril1974intersection} shows that a graph $G$ is a Helly\ncircular-arc graph if and only if there exists a \\emph{cyclic\n  ordering} $\\lhd$ of its maximal cliques such that for every vertex\n$v$, the maximal cliques containing $v$ appear consecutively in\n$\\lhd$ and that Hsu and McConnell~\\cite{hsu2003pc} use PC-trees to capture\nall plausible cyclic orderings of the maximal cliques of a Helly\ncircular-arc graph makes it tempting to simply apply the same techniques to\ngeneralize the algorithm of Klav\\'ik et al.  However, this cannot be\nstraightforwardly applied for two reasons.  First, the clique ordering carries\nlittle information about whether a representation is normal or not, and, even\nmore severely, extending a partial cyclic ordering is \\hbox{\\rm \\sffamily NP}-complete, even\nwithout requiring that the order be additionally represented by some given\nPC-tree~\\cite{gm-con-77}. We overcome this by working with suitably linearized\npartial orders to show the following results.\n\n\\begin{restatable}{theorem}{nhcapoly}\\label{thm:nhca_poly}\nThe problem ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NHCAR})$ can be solved in $\\calO(n^3)$ time.\n\\end{restatable}\n\n\\begin{restatable}{theorem}{hcapoly}\\label{thm:hca_poly}\nThe problem ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$ can be solved in $\\calO(n^3)$ time if the partial\nrepresentation consists of arcs with pairwise distinct endpoints.\n\\end{restatable}\n\nIt follows that Helly representations used in our reduction essentially\ninvolve arcs that share endpoints. \nThis is surprising since non-degeneracy assumptions like this are often made\nwithout much consideration of the impact on the problem when working with graph\nrepresentations.\n\n\nThe bottleneck of our NHCA-algorithms is the testing of the consecutivity\nconstraints for all universal pairs of vertices.  A closer exploration of the\nstructure of the set of universal pairs may yield improvements of the running\ntime upper bound.\n\nFinally, we show that involving the most tight constraints on arc lengths, the\nproblem becomes again computationally difficult.\n\n\n\\begin{restatable}{theorem}{ucanpc}\\label{thm:uca_npc}\nThe problem ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily UCAR})$ is \\hbox{\\rm \\sffamily NP}-complete.\n\\end{restatable}\n\nThe \\hbox{\\rm \\sffamily NP}-hardness of Theorems~\\ref{thm:hca_npc} and\n\\ref{thm:uca_npc} follows by a reduction from the $3$-${\\textsc{Partition}}$\nproblem~\\cite{garey1975complexity}.  For the unit case, the membership\nin \\hbox{\\rm \\sffamily NP} can be seen by a linear programming argument.\n\n\\section{Preliminaries}\n\n\\myparagraph{Cyclic order.}\nLet $<\\ = v_0,\\dots,v_{n-1}$ and $<'\\ = u_0,\\dots,u_{n-1}$ be two linear orders\non a finite set $S$.\nWe say that $<$ and $<'$ are \\emph{cyclically equivalent} if there is\n$k\\in\\{0,\\dots,n-1\\}$ such that $v_i = u_{i+k}$, where the addition is modulo\n$n$.\nClearly, this is an equivalence relation on the set of all linear orders on $S$.\nA \\emph{cyclic order} $\\lhd$ on $S$ is an equivalence class of this relation.\nFor a linear order $<$, we denote the corresponding cyclic ordering by $[<]$.\n\n\nEvery linear order $<$ on $S$ induces a linear order $<'$ on a subset\n$S'\\subseteq S$ by omitting all ordered pairs in which the elements of\n$S\\setminus S'$ occur.\nIn this case we say that $<$ \\emph{extends} $<'$ and similarly that the cyclic\norder $[<]$ \\emph{extends} $[<']$.\n\n\\iffalse\nFor a cyclic order $\\lhd$ a subset $S$ of elements is \\emph{consecutive} if it\nis consecutive in some linear order $<$ with $[<]=\\lhd$.  We say $b$ is\n\\emph{between} $a$ and $c$ if $a<b<c$ for a linear order $<$ with $[<]=\\lhd$.\nIf no element is between $a$ and $c$, then $a$ and $c$ are \\emph{consecutive}\nand we also say that $a$ is \\emph{immediately before} $c$ and that $c$ is\n\\emph{immediately after} $a$. \n\\fi\n\n\\myparagraph{Circular-arc representations.}\nFor any circular-arc representation $\\calR$ and each connected component $C$ \nof a graph $G$ the set $\\bigcup_{v \\in V(C)}R(v)$ is a connected subset of the circle.\nTherefore, if $G$ is a disconnected circular-arc graph, then each connected component of\n$G$ has to be an interval graph. These cases can be treated with the\ncorrepsonding algorithms for interval graphs\nof~\\cite{klavik2017extending,KlavikKOSV17}. Hence without loss of generality we\nrestrict ourselves to connected graphs in this paper.\n\nLet $\\calR$ be a representation of a circular-arc graph $G$.\nFor a vertex $v$ of $G$, we call the \\emph{tail} $R(v)_t$ and the \\emph{head}\n$R(v)_h$ the two endpoints of $R(v)$. We use the convention of traversing the\narc from the tail to the head in the clockwise direction along the circle. \nWe denote such an arc as $R(v)=[R(v)_t,R(v)_h]$, and its complement\n$(R(v)_h,R(v)_t)$ as $R(v)^c.$\n\n\nLet $\\calR$ be a Helly representation of a circular-arc graph $G$.\nDenote by $\\calC$ the set of maximal cliques of $G$.\nWe assign every maximal clique $C\\in \\calC$ a unique point\n$\\cp(C)\\in\\bigcap_{v\\in C} R(v)$ and call it the \\emph{clique-point} of $C$. \n\n\\begin{lemma}[Gavril~\\cite{gavril1974intersection}]\n\\label{lem:gavril_hca}\nA graph $G$ is a Helly circular-arc graph if and only if there exists a cyclic\nordering $\\lhd$ of its maximal cliques such that for every vertex $v$,\nthe maximal cliques containing $v$ appear consecutively in $\\lhd$.\n\\end{lemma}\n\nNote that if we distribute clique points on the circle according to a cyclic\nordering $\\lhd$ of Lemma~\\ref{lem:gavril_hca}, then a representation $\\calR$ of\n$G$ can be obtained by choosing for each vertex $v$ an arc $R(v)$ that covers\nexactly the clique-points $v$ belongs to.\n\n\\myparagraph{PC-Trees and the Reordering Problem.}\nA \\emph{PC-tree} $T$ on a set $L$ of leaves is a tree whose inner\nnodes have one of two types: \\emph{P-nodes} and \\emph{C-nodes}.  The\nneighbors around a P-node can be permuted arbitrarily, whereas the order of the\nneighbors of a C-node is fixed up to reversal.  In this way a PC-tree\nrepresents a set of cyclic orderings of its leaf set $L$.  The\nusefulness of PC-trees derives from the fact that they can represent\ncyclic orderings subject to consecutivity constraints.  Namely, given\na set $L$ and sets $X_1,\\dots,X_r \\subseteq L$, a PC-tree that\nrepresents precisely those cyclic orderings of $L$ where each of the subsets\n$X_1,\\dots,X_r$ is consecutive can be computed in\n$\\calO(|L| + \\sum_{i=1}^r |X_i|)$ time~\\cite{hsu2003pc}.\nIn our setting, in the spirit of Lemma~\\ref{lem:gavril_hca}, the leaf set $L$\nwill always be the set of all maximal cliques of a Helly circular-arc graph.\n\nWe make use of the following reordering problem, which can be solved\nanalogously to the topological sorting of PQ-trees~\\cite{klavik2017extending}.\n The input consists of a PC-tree $T$, a leaf $u$ of $T$, and a partial ordering\n (not cyclic) $<$ of the remaining leaves $L' = L\\setminus\\{u\\}$ of $T$. The\n question is whether $T$ represents a cyclic order $<'$ which induces a linear\n extension of $<$ on $L'$. \n If this is the case, $T$ is called \\emph{compatible with $<$ with respect to $u$}.  \nWe denote an instance of this problem by ${\\textsc{Reorder}}(T,u,<)$.\n\n\\begin{restatable}{lemma}{reorderingLinear} \n  \\label{lem:reordering_linear}\nAn instance ${\\textsc{Reorder}}(T, u, <)$ can be solved in time $\\calO(\\ell + c)$, where\n$\\ell$ is the number of leaves of $T$ and $c$ is the number of comparable pairs\nin the partial ordering $<$.\n\\end{restatable}\nFor a detailed proof see Section~\\ref{sec:PQtree}.\n\\newcommand{\\reorderingLinearProof}{\n\\begin{proof}\nWe root $T$ by $u$. We represent the ordering $<$ by a digraph $D$ having $c$\nedges. The algorithm reorders the nodes from the bottom to the root and modifies\n$D$ by contractions. Once we finish reordering a subtree, it is never modified\nlater.  After reordering a subtree, the corresponding vertices in $D$ are\ncontracted. We process a node of $T$ when all its subtrees are finished and the\ncorresponding digraphs are contracted to single vertices. Note that since the\ntree $T$ is now rooted, the equivalent transformations (i) and (ii) correspond\nto permuting the children of a P-node and reversing the children of\na C-node.\n\nFor a P-node, we then permute its children according to any topological sort of\nthe subdigraph $D'$ induced by the vertices corresponding to the children of the\nP-node. Note that $D'$ is acyclic since $<$ is a partial ordering. For a C-node,\nthere are two possible orderings of its children and we check whether one of\nthem is feasible. The resulting PC-tree $T'$ is compatible with $<$.\n\\relax\n\\end{proof}}\n\n\n\n\\section{Complexity}\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[page=2]{HCA_Reduction}\n  \\caption{Illustration of the proof of Theorem~\\ref{thm:hca_npc}. (a) predrawn\n    universal vertices (b) the instance of ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ obtained from\n    $\\{1,1,2,2,2,3,3,4\\}$ of  {\\sc 3-Partition}. For clarity we have omitted 24\n    predrawn universal vertices and instead indicated their enpoints.   \n    (c) the corresponding solution (d) variant without shared endpoints.\n  }\n  \\label{fig:hca_npc}\n\\end{figure}\n\n\\begin{proof}[Sketch of proof for Theorem~\\ref{thm:hca_npc}]\nWe first reduce the strongly \\hbox{\\rm \\sffamily NP}-complete\nproblem {\\sc 3-Partition}~\\cite{garey1975complexity} to ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$.\nFigure~\\ref{fig:hca_npc} illustrates the proof.  Figure~\\ref{fig:hca_npc}a\nshows a representation with four universal vertices (blue).  The green dots\nindicate the endpoints that are shared by these arcs.  The red \\emph{blocker\nvertex} covers all other points of the circle shared by all universal vertices.\nThe key insight is that each vertex that is not adjacent to the blocker vertex\nmust not be contained in the complement of a universal vertex and thus contain\nat least one of the green points.\nFigure~\\ref{fig:hca_npc}b shows an instance resulting from our reduction, and\nFigure~\\ref{fig:hca_npc}c shows a corresponding solution from the instance\n$\\{1,1,2,2,2,3,3,4\\}$.\n\nFor ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$, the same construction works. However, we can avoid \nshared endpoints in the partial representation with a simple modification. \nNamely, we slightly shorten the arc for each universal vertex; see\nFigure~\\ref{fig:hca_npc}d. \nThen we have for each former green dot a green area between the corresponding\narc ends. Each vertex that is not adjacent to the blocker vertex must now contain \na green area.\nNote that in every solution  each leaf of a star in $G$ must contain exactly\none green area. It thus violates the Helly property with the two universal arcs\nending there.\n\\end{proof}\n\nFor details see Section~\\ref{ssec:CARnp}.\n\n\n\\section{Normal Proper Helly Circular-Arc Graphs}\n\nWe show how to extend partial representations of normal proper Helly\ncircular-arc graphs in linear time.  To do this, we give a characterization of\nall partial representations that are extendable.  This generalizes the\ncharacterization of the extendable partial representations of proper interval\ngraphs~\\cite[Lemma 2.4]{klavik2017extending}.\nWe first simplify the possible instances by reducing the number of vertices\nwith the same neighborhood as follows.\n\n\\myparagraph{Twin vertices.}\nRecall that vertices $u,v\\in V(G)$ are called twins if $N[u] = N[v]$.  It is\npossible to find the equivalence classes of twin vertices in linear\ntime~\\cite{rose1976algorithmic}.  If vertices $u$ and $v$ are twins and if\neither $u$ is not predrawn or if both $u$ and $v$ are predrawn with the same\narc, then we may remove $u$ and in the final representation we can set $R(u) =\nR(v)$.  This allows us to assume that each twin class either consists of a\nsingle vertex, which is not predrawn, or it consists only of predrawn\nvertices that are represented by distinct arcs.\n\n\\myparagraph{Consecutive orderings of vertices.}\nLet $\\lhd = [v_0, \\dots, v_{n-1}]$ be a cyclic ordering of the vertices of some\ngraph $G$. \nWe say that $\\lhd$ is \\emph{consecutive} if the closed neighborhood $N[v]$ of\neach $v\\in V(G)$ is consecutive in $\\lhd$.\nNote that $N[v_i]$ is consecutive in $\\lhd$ if there exist positive integers\n$a$ and $b$ such that $a + b \\leq n$ and \\[N[v_i] = \\{v_{i-a}, \\dots, v_{i-1},\nv_i, v_{i+1}, \\dots, v_{i+b}\\},\\] where the addition is performed modulo $n$.\n\nRoberts characterized connected proper interval graphs in terms of consecutive linear\norderings~\\cite{roberts1968representations} that are unique up to permuting\ntwin vertices and a complete reversal.  The following lemma, proved in slightly\ndifferent terminology by Deng et al. as Corollaries 2.7 and 2.9.\nin~\\cite{deng1996linear} give a characterization of proper circular-arc\ngraphs by cyclic ordering:\n\n\\begin{lemma}\\label{lem:pca_charac}\n  A graph $G$ belongs to $\\hbox{\\rm \\sffamily PCA}\\xspace$ if and only if $V(G)$ has a consecutive cyclic\n  ordering.\n\\end{lemma}\n\nThey also showed that this ordering is unique up to permuting twin vertices and\na complete reversal only in the case when the complement is connected or\nnonbipartite. For example, the complement of a perfect matching\n$K_{2,2,\\dots,2}$ on at least 6 vertices allows several distinct consecutive\norderings, like $[1,2,3,4,5,6]$ and $[1,3,2,4,6,5]$ for the graph depicted in\nFig.~\\ref{fig:cK222}a. \nFor normal proper Helly circular-arc graphs, the following strengthening of\nLemma~\\ref{lem:pca_charac} follows from the results summarized in Lin et\nal.~\\cite{lin2013normal}:\n\n\\begin{lemma}\\label{lem:nphca_unique}\nEvery $\\hbox{\\rm \\sffamily NPHCA}\\xspace$ graph $G$ has a \\emph{unique} consecutive cyclic ordering of\n$V(G)$, up to permuting twins and a reversal. Such an ordering can be obtained\nin  $O(m+n)$ time, where $n$ and $m$ stand for the number of vertices and\nedges, respectively.\n\\end{lemma}\n\nNote that each such consecutive cyclic ordering of vertices corresponds to the\ncyclic order on tails of the arcs in some normal proper Helly representation.\n\n\\myparagraph{Characterization of extendable instances.}\nLet $G \\in \\hbox{\\rm \\sffamily NPHCA}\\xspace$ and let $\\calR'$ be a partial representation of $G$.\nBy the discussion above, we may assume that if $u$ and $v$ are twins, then both\nare predrawn with $R(u)\\neq R(v)$. \n\nA constraint specific to the representation extension of proper circular-arc\ngraphs is imposed by any pair of touching arcs, for which  the tail of one\ncoincides with the head of the other, i.e., $R'(v_i)_h = R'(v_j)_t$. In this\nsituation, it is impossible to place another tail or head between $R'(v_i)_h$\nand $R'(v_j)_t$ -- in fact on this point of the circle -- as the resulting\nrepresentation would not be proper.\n\nWe use this fact as well as the uniqueness of the ordering of\nLemma~\\ref{lem:nphca_unique} to characterize all instances that allow a\nrepresentation extension; see Section~\\ref{sec:DetailsNPHCA}.\n\n\\begin{restatable}{lemma}{nphcaChar}\\label{lem:nphca_char}\nA partial normal proper Helly circular-arc representation $\\calR'$ of a\nconnected graph $G$, where all twins are predrawn by distinct arcs, is\nextendable if and only if $V(G)$ has a consecutive cyclic ordering $\\lhd =\n[v_0,\\dots,v_{n-1}]$ satisfying:\n\\begin{compactenum}[(1)]\n  \\item \\label{itm:nphca_prop1}\nThe ordering $\\lhd$ extends the cyclic ordering $\\lhd'$ of the predrawn\nvertices corresponding to the clockwise cyclic ordering of the tails of their\npredrawn arcs.\n  \\item \\label{itm:nphca_prop2}\nIf $R'(v_i)$ and $R'(v_j)$, are distinct touching predrawn arcs such that\n$R'(v_i)_h = R'(v_j)_t$, then\n$N[v_i] = \\{v_a, \\dots,v_i,\\dots,v_j\\} \\quad \\text{and} \\quad N[v_j] =\n\\{v_i,\\dots,v_j,\\dots,v_{b}\\}$, for some non-negative integers $a, b$.\n\\end{compactenum}\n\\end{restatable}\n\n\n\\newcommand{\\nphcaCharProof}{\n\\begin{proof}\nSuppose that there is a normal proper Helly representation $\\calR$ extending\n$\\calR'$.  Let $\\lhd$ be the cyclic ordering $[v_0,\\dots,v_{n-1}]$ of $V(G)$\ninduced by the clockwise cyclic ordering $[t_0,\\dots,t_{n-1}]$ of tails\n$t_i=R(v_i)_t$.  For convenience, we also write $h_i$ for $R(v_i)_h$.\nThe ordering $\\lhd$ extends $\\lhd'$ since $\\calR'$ is contained in $\\calR$, so\nthe condition~(\\ref{itm:nphca_prop1}) is satisfied.\n\nTo verify condition~(\\ref{itm:nphca_prop2}), suppose that there are two\npredrawn arcs $R(v_i)$ and $R(v_j)$ such that $h_i = t_j$.\nIf $v_i$ is a universal vertex (adjacent to all other vertices), then the\ncondition on $N[v_i]$ holds trivially.\nOtherwise, since $N[v_i]\\neq V(G)$ and $N[v_i]$ is consecutive, $\\lhd$ induces\na linear order on $N[v_i]$, where the minimum and maximum elements $v_a$ and\n$v_b$ correspond to the first and the last neighbor of $v_i$ in clockwise\ndirection.\nSince $h_i = t_j$, it must be that $b = j$. An analogous argument holds for $v_j$.\n\n\nFor the opposite implication, let $\\lhd = [v_0,\\dots,v_{n-1}]$ be a consecutive\ncyclic ordering of $V(G)$ satisfying all three conditions. \nWithout loss of generality we may assume that $G$ is not a complete graph, as\notherwise the problem is trivial -- we pick any predrawn arc and replicate it\nfor every non-predrawn vertex.\nTo construct the representation $\\calR = \\{R(v_0),\\dots,R(v_{n-1})\\}$ of $G$,\nwe need to determine the position of the endpoints of $R(v_i)$, for every $v_i$\nthat is not predrawn. \n\nFirst, we construct a cyclic order $\\prec$ on the auxiliary set\n\\[A = \\{t_0,\\dots,t_{n-1}, h_0,\\dots,h_{n-1}\\}.\\]\nWe construct $\\prec$ in several steps.\nInitially, we set $\\prec\\ = [t_0,\\dots,t_{n-1}]$.\nNext, we insert $h_i$ into $\\prec$, for $i = 0,\\dots,n-1$, in the following way.\nFor each $i=0,\\dots,n-1$ we distinguish two cases, depending on whether $v_i$\nis universal or not, see Fig.~\\ref{fig:nphca_order}:\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1,page=1]{nphca_order}\n\\caption{(a) Position of the head $h_i$ in $\\prec$ for a non-universal $v_i$,\n and (b) for a universal $v_i$.}\n\\label{fig:nphca_order}\n\\end{figure}\n\n\\begin{compactenum}[(a)]\n\\item If $v_i$ is not universal, then we determine the last neighbor $v_b$ of\n  $v_i$, i.e., the vertex satisfying $N[v_i] = \\{v_a, \\dots,v_i,\\dots,v_b\\}$\n  and insert $h_i$ into $\\prec$ immediately before $t_{b+1}$.\n\\item Otherwise, i.e., when $v_i$ is universal, we determine the first\n  non-universal vertex $v_a$ for which $N[v_a] = \\{\\dots, v_a,\n    \\dots,v_i,\\dots\\}$, i.e., the arc $R(v_a)$ shall contain the tail\n    $R(v_i)_t$, and insert $h_i$ into $\\prec$ immediately before $t_a$. The\n    phrase \\emph{the first} means that all other non-universal vertices $a'$\n    which have $v_{a'}$ before $v_i$ in their neighborhood $N(v_{a'})$ are\n    between $v_a$ and $v_i$ in $\\lhd$.\n\\end{compactenum} \n\nWe compute the order $\\prec$ on the set $A$ from the cyclic vertex order $\\lhd$\nand from adjacencies between vertices of the graph $G$. As desribed, the order\n$\\prec$ depends uniquely on $\\lhd$ and $G$.\n\nThe representation $\\calR'$ imposes a cyclic ordering on the points\nrepresenting the tails and heads of the predrawn arcs. We extend this cyclic\nordering to the set of tails and heads of the predrawn arcs, i.e. on\n$\\{t_i,h_i: R'(v_i)\\in \\calR'\\}$. In the case when a single point of the circle\nis the tail and the head of touching arcs, i.e., when $R'(v_i)_h=R'(v_j)_t$, we\ninsert $h_i$ immediately after $t_j$. Note that as $G$ is a proper circular-ar\ngraph, only a single pair of arcs may touch on a single point.  Denote the\nresulting cyclic ordering by $\\prec'$.\n\nSince $\\calR'$ represents an induced subgraph of $G$ and as the cyclic ordering\n$\\lhd'$ on tails is a sub-ordering of $\\lhd$ due to (1), it follows that\n$\\prec'$ is a sub-ordering of $\\prec$. In particular, it could be obtained from\n$\\lhd'$ and $G'$ by the same process as $\\prec$ was constructed from $\\lhd$ and\n$G$.\n\nNow, we use the cyclic order $\\prec$ to construct the representation $\\calR$.\nWe identify every $t_i \\in A$ with $R(v_i)_t$ and $h_i \\in A$ with $R(v_i)_h$\nand with a slight abuse of notation we view the elements of $A$ as the points\nof the circle.\n\nConsider now two predrawn elements $e$ and $e'$ that are consecutive in\n$\\prec'$ and that corresponds to distinct points of the circle --- in other\nwords, the interior of the arc $[e,e']$ contains no other predrawn head or\ntail. \nEspecially we exclude here the case when $e$ and $e'$ represent the head and\nthe tail of touching arcs.  For any such pair $e$ and $e'$, let\n$e_0,\\dots,e_{\\ell-1}$ be the linear ordering of the elements in $\\prec$ that\nare between $e$ and $e'$ (if any). We place $e_0,\\dots,e_{\\ell-1}$\nequidistantly between $e$ and $e'$ on the circle. This way we obtain the\nrepresentation $\\calR$. By the same argument we have used for $\\calR'$, the\ncyclic ordering on the points representing the tails and heads can be extended\nto $\\lhd$ by putting heads of touching arcs immediately after the matching\ntails.\n\nSince we do not modify the endpoints of predrawn arcs, $\\calR$ extends\n$\\calR'$.\n\nConsider now any pair of vertices $v_i, v_j\\in V(G)$.\nWhen $v_iv_j \\in E(G)$, we distinguish the following cases: \n\\begin{compactitem}\n  \\item If both $v_i$ and $v_j$ are not universal, then we may without loss of\n    generality assume that $v_j$ appears after $v_i$ in $N[v_i]= \\{v_a,\n    \\dots,v_i,\\dots, v_j, \\dots, v_b\\}$.  In this case the head $h_i$ has been\n    inserted in $\\prec$ immediately before the tail $t_{b+1}$. As the tail $t_b$\n    is between $t_i$ and $t_{b+1}$ and $t_j=t_b$ or $t_j$ is between $t_i$ and\n    $t_b$, it follows that the tail $t_j$ is inside the arc $R(v_i)$, see\n    Fig.~\\ref{fig:nphca_order}a.\n  \\item If $v_i$ in universal but $v_j$ not, then either $v_i$ appears before\n    $v_j$ or after it in $N[v_j]$.  When $v_i$ is after $v_j$ in $N[v_j]$ we\n    deduce that $t_i$ is inside the arc $R(v_j)$ as in the previous case.\n    Otherwise the head $h_i$ has been inserted in $\\prec$ immediately before\n    $t_a$, the first non-universal vertex which has in $N[v_a]$ the vertex\n    $v_j$ positioned after $v_a$. Therefore $h_j$ is between $h_i$ and $t_i$\n    and the tail $t_j$ is inside the arc $R(v_i)$, see\n    Fig.~\\ref{fig:nphca_order}b.\n  \\item Finally, when both $v_i$ and $v_j$ are universal, then they are twins\n    and hence are predrawn and intersecting already in $\\calR'$. \n\\end{compactitem}\nIn all three cases, the arcs $R(v_i)$ and $R(v_j)$ intersect.\n\nOn the other hand, when $v_iv_j \\notin E(G)$, then none of these vertices is universal. \nIn the moment of insertion of $t_i$ in $\\prec$, i.e., immediately before\n$t_{b+1}$ for $N[v_i]= \\{v_a, \\dots,v_i,\\dots,v_b\\}$, the tail $t_i$ appeared\nbetween $h_i$ and $h_j$ as otherwise $N[v_i]$ would not be consecutive.\nAnalogously, the tail $t_j$ was inserted between $h_j$ and $h_i$.  Therefore,\nthe arcs $R(v_i)$ and $R(v_j)$ are disjoint.\n\\relax\n\\end{proof}}\n\nWe are ready to prove that ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$ can be solved in time $\\mathcal{O}{}(n+m)$,\nwhere~$n$ is the number of vertices and~$m$ is the number of edges of the given\ngraph $G$.\n\n\\nphcapoly*\n\\begin{proof}\nThe procedure is given as Algorithm~\\ref{alg:nphca}.  Its correctness follows\ndirectly from ~Lemma~\\ref{lem:nphca_char}. \n\n\\begin{algorithm}[tb]\n\\KwData{A graph $G$ and a partial representation $\\calR'$.}\n\\KwResult{A \\hbox{\\rm \\sffamily NPHCA}\\xspace{} representation $\\calR$ of $G$ extending $\\calR'$ or a message that it does not exist.}\n\\Begin{\nDetermine the twin classes and prune not predrawn twins\\;\nFind the ordering $<$  by Lemma~\\ref{lem:nphca_unique}\\;\nTransform $<$ to $\\lhd$ by reversal and permutation of predrawn twins\\;\n\\ForAll{arcs $R'(v_i)$ and $R'(v_j)$ such that \n$R'(v_i)_h = R'(v_j)_t$}{\n\\lIf{$N[v_i] \\cap N[v_j] \\ne \\{v_i,\\dots,v_j\\}$}{\\Return{$\\calR'$ has no extension.}}\n}\nConstruct the cyclic order $(A, \\prec)$ from the proof of Lemma~\\ref{lem:nphca_char}\\;\nConvert $(A, \\prec)$ into the representation $\\calR$ as descibed in  Lemma~\\ref{lem:nphca_char}\\;\nReplicate the pruned twins into $\\calR$ from the non-pruned ones\\;\n\\Return{$\\calR$}\n}\n\n\\caption{The algorithm for the ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$ problem.}\\label{alg:nphca}\n\\end{algorithm}\n\nFor the computational complexity note that:\n\\begin{compactitem}\n\\item Step 3 could be achieved in $O(n+m)$ time by Lemma~\\ref{lem:nphca_unique}. \n\\item Step 4. To check whether reversal is necessary it suffices to check two consecutive vetices belonging to discinct twin classes.\nThe correct ordering of the predrawn twins can be achieved by a single traversal of $<$.\n\\item\nThe for-loop at lines 5--7 has at most $n$ iterations since we check the head of every predrawn arc exactly once.\n\\item\nThe test at line $6$ can be done in constant time since it suffices to check whether $v_{i-1}v_j \\in E(G)$ or $v_{i}v_{j+1} \\in E(G)$.\n\\item\nStep 8.\nFollowing the proof of Lemma~\\ref{lem:nphca_char}, we perform a single\ntraversal of the initial order $[t_0,\\dots, t_{n-1}]$ and insert the elements\n$h_0,\\dots,h_{n-1}$ appropriately according to $\\lhd$.\n\\item\nFor step 9, the positions of heads and tails of non-predrawn arcs can be determined in $\\calO(n)$ by a single traversal of $(A, \\prec)$.\n\\item\nAt line 10, each pruned twin requires only constant time to be replicated.\n\\end{compactitem}\n\\relax\n\\end{proof}\n\n\nFor Theorem~\\ref{thm:phca_red}, which deals with ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily PHCA}\\xspace)$, note that\neven though all proper Helly circular-arc graphs allow also a normal proper\nrepresentation, we cannot reduce ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily PHCAR})$ to ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$\ndirectly, as the given partial representation need not to be normal.\n\nHowever, the existence of a proper Helly representation extending a pair of arcs in a\nnot-normal position imposes strong conditions on the structure of $G$: after\npruning all universal vertices only two disjoint cliques remain.\nSuch instances can be solved in linear time. For details see\nSection~\\ref{sec:PHCA}.\n\n\n\n\\section{Normal Helly Circular-Arc Graphs}\n\\label{sec:NHCA}\n\nWith the following lemma, we can avoid universal vertices since they allow \nconsidering instances of ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NHCAR})$ as instances for the interval case. \n\\begin{lemma}\n  Let $G$ be a graph with a universal vertex $u$. Then for every normal Helly\n  circular-arc representation of $G$ there exists a point on the circle that is\n  contained in no arc.\n  \\label{lem:nhcaGuniversal}\n\\end{lemma}\n\\begin{proof}\n  Let $\\calR$ be a normal Helly circular-arc representation of $G$.\n  Assume that every point of the circle is contained in some arc. \n  Without loss of generality, we may assume that $R(u)$ is not strictly contained in\n  any other arc of $\\calR$. We consider the complement $R^c(u)$ of $R(u)$.\n  Let $v_t$, $v_h$ be vertices whose arcs contain $R^c(u)_t$, $R^c(u)_h$\n  respectively, and whose arcs maximize the intersection with $R^c(u)$.\n\n  Note that, since $\\calR$ is normal, neither $R(v_t)$ nor $R(v_h)$ contain $R^c(u)$.\n  Assume that one of $R(v_t)$, $R(v_h)$, say $R(v_t)$ contains both endpoints of $R(u)$.\n  We then have $R^c(u)\\subseteq R(v_t)$ which by the maximality of $R(u)$\n  implies $R(v_t)=R(u)$. \n  By the choice of $v_t$, we find in this case a point close to $R(u)_t$ that\n  is not contained in any arc.\n\n  In the other case, each of $R(v_t)$ and $R(v_h)$ contains exactly one endpoint of $R(u)$.\n  In particular, we have $v_t\\ne v_h$.\n  Since $u$ is universal and since every point of $R^c(u)$ is contained in some arc,\n  it follows that $R^c(u)\\subseteq R(v_t)\\cup R(v_h)$. We obtain $R(v_t)\\cap\n  R(v_h) \\cap R^c(u)\\ne \\emptyset$ (recall that our arcs are closed sets).\n  Due to the normal property, it follows that $R(v_t)\\cap R(v_h) \\cap R(u)=\n  \\emptyset$, which contradicts the Helly property.\n  We conclude that there exists a point $p$ that is not contained in any arc.\n\\end{proof}\n\nWe assume for the rest of this section that our graph contains no universal vertices.\n\nLet $G=(V,E)$ be a graph. Then two adjacent vertices $u, v\\in V$ form a\n\\emph{universal pair} if each vertex $w\\in V$ is adjacent either to $u$ or to $v$.\n\n\\begin{lemma}\n  A graph $G$ without a universal vertex is a normal Helly circular-arc graph\n  if and only if there exists a cyclic ordering $\\lhd$ of its maximal cliques\n  such that \n  \\begin{compactenum}[(i)]\n    \\item for every vertex $v$, the maximal cliques containing $v$ appear\n  consecutively in $\\lhd$.\n    \\item for every universal pair $u$, $w$, the maximal cliques\n      containing $u$ and $w$ appear consecutively in $\\lhd$.\n  \\end{compactenum}\n\n  \\label{lem:nhcaG}\n\\end{lemma}\n\\begin{proof}\n If $G$ has a normal Helly circular-arc representation, then, due to the Helly\n property, we have for each maximal clique $C$ a clique point $\\cp(C)$ where\n the arcs of all vertices in $C$ intersect.\n For each vertex $v$ and for each universal pair $u$, $w$ the corresponding\n cliques are consecutive in $\\lhd$ since $R(v)$ and $R(u)\\cap R(w)$ each are\n connected (the latter due to the normal property).\n\n Next assume that we have a cyclic ordering $\\lhd$ of the maximal cliques\n of $G$ with properties (i) and (ii).\n We obtain a normal Helly circular-arc representation as follows;\n see Fig.~\\ref{fig:lem16}.\n We first arrange the maximal clique points on the circle according to $\\lhd$.\n Then, for each vertex $v$, we define the arc $R(v)$ as the smallest arc that\n contains exactly the clique points of the maximal cliques that contain $v$. Note\n that $R(v)$ is well-defined since $v$ is not universal.\n This defines a circular-arc representation $\\calR$ of $G$ since any two\n intersecting arcs share a clique point. Moreover, the existence of the clique\n points shows the Helly property. It remains to show that $\\calR$ is\n normal.\n\n Assume that there are two nodes $u$, $w$ such that $R(u)\\cap R(w)$ is not connected.\n Then $u$, $w$ is a universal pair. Thus the cliques containing $u$ and\n $w$ are consecutive in $\\lhd$ which contradicts $R(u)\\cap R(w)$ being not connected.\n Hence, the representation is normal.\n\\end{proof}\n\n\n\\myparagraph{Extendable Partial Representations.}\nWe characterize all partial representations of a normal Helly circular-arc\ngraph $G$ that are extendable. Let $\\calR'$ be a partial representation of $G$\nand let $C$ be a maximal clique of $G$. We define $\\Pre(C) = \\{R'(v) : v \\in\nC \\cap V(G')\\}$ to be the \\emph{predrawn arcs} corresponding to the vertices in $C$.\n\n\\begin{definition}\nFor a maximal clique $C\\in\\calC$, we define sets\n$$\\Reg^+(C) = \\bigcap\\limits_{w\\in\\Pre(C)} R'(w) \\qquad\\text{ and }\\qquad \\Reg^-(C) =\n\\bigcup\\limits_{w\\in \\calR'\\setminus \\Pre(C)} R'(w).$$\nThe \\emph{region} of $C$ is the set $\\Reg(C) =\n\\Reg^+(C)\\setminus \\Reg^-(C)$.\n\\end{definition}\n\nThe set $\\Reg^+(C)$ means the set of possible locations of $\\cp(C)$ with respect to predrawn arcs, \nwhile $\\Reg^-(C)$ means the forbidden locations.\nThus if the region $\\Reg(C)$ is empty for some clique $C\\in\\calC$, then the\ngiven partial representation is not extendable. We thus assume in the following\nthat no region is empty. The following lemmas give some useful properties that\nhold for the regions of maximal cliques.\n\n\\begin{lemma}\\label{lem:region_equivalence}\nFor maximal cliques $C$ and $D$, we have either $\\Reg(C) \\cap \\Reg(D) =\n\\emptyset$, or $\\Reg(C) = \\Reg(D)$.\n\\end{lemma}\n\n\\begin{proof}\nIf $\\Pre(C) = \\Pre(D)$, then clearly $\\Reg(C) = \\Reg(D)$. So, we assume that $\\Pre(C) \\neq\n\\Pre(D)$. We can further assume that there exists an arc $R'\\in\n\\Pre(C)\\setminus \\Pre(D)$.\nSince the clique point $\\cp(C)$ must be placed on the point of $R'$ and the clique point $\\cp(D)$ cannot be placed on any point\nof $R'$, we have that $\\Reg(C)$ and $\\Reg(D)$ are disjoint.\n\\relax\n\\end{proof}\n\nFor every maximal clique $C$, we call the connected components of $\\Reg(C)$\n\\emph{islands} and the connected components of its complement $\\Reg^c(C)$ we call \\emph{gaps of $C$}.\nWe say an island and a gap are \\emph{neighboring}, if they share an endpoint\n(where one end is open and the other is closed.) Note that every island has two\nneighboring gaps and every gap has two neighboring islands.\n\nObserve that if two maximal cliques $C,D\\in \\calC$ satisfy $\\Pre(C)=\\Pre(D)$\nthen $\\Reg(C)=\\Reg(D)$ by definition. In the other\ncase we obtain the following relationship:\n\n\n\\begin{lemma}\n\\label{lem:singleGap}\nLet $C$ and $D$ be two maximal cliques with $\\Pre(C)\\ne\\Pre(D)$.\nThen $D$ has a gap $J$ with $\\Reg(C)\\subseteq J$.\n\\end{lemma}\n\n\\begin{proof}\nFrom $\\Pre(C)\\ne\\Pre(D)$ we obtain $\\Reg(C)\\subseteq \\Reg^c(D)$.\nAssume that there are two gaps $J_1$, $J_2$ of $D$ that contain points $j_1,\nj_2\\in\\Reg(C)$. Let $I_1,I_2$ be the islands of $D$ neighboring $J_1$.\nLet $i_1\\in I_1$, $i_2\\in I_2$. \nFor $v\\in\\Pre(C)$ it holds that $j_1,j_2\\in R'(v)$ and thus $i_1\\in R'(v)$ or\n$i_2\\in R'(v)$ since $i_1$, $i_2$ separate $j_1$, $j_2$ on the circle; see\nFig.~\\ref{fig:lem15}.\nThis implies \n$\\Pre(C)\\subseteq\n\\{v\\in V(G'): i_1 \\in R'(v) \\lor i_2 \\in R'(v)\\} =\n\\{v\\in V(G'): i_1 \\in R'(v)\\} \\cup \\{v\\in V(G'): i_2 \\in R'(v)\\} \n= \\Pre(D)$ as both sets in the union are equal to $\\Pre(D)$.\nLikewise, we obtain $\\Pre(D)\\subseteq\\Pre(C)$. This contradicts $\\Pre(C)\\ne\\Pre(D)$.\nHence, $\\Reg(C)$ is contained in a single gap of $D$.\n\\end{proof}\n\nThis means that for any two maximal cliques $C$ and $D$, the clique point $\\cp(C)$\nmust be placed in a given gap of $D$. We obtain additional consecutivity constraints.\nFor every gap $J$, we define the set $S_J = \\{C\\in\\calC \\mid \\Reg(C)\\cap J\\ne \\emptyset\\} \n= \\{C\\in\\calC \\mid \\Reg(C)\\subseteq J\\}$. Recall that we assumed that no region is empty.\n\n\n\n\n\\begin{lemma}\n  \\label{lem:gapConsecutivity}\n  Let $\\calR$ be a NHCA extension of $\\calR'$. Let $J$ be a gap of some maximal clique $D$.\n  Let $\\le$ be the clique ordering derived from $\\calR$. Then in $\\le$ the set $S_J$\n  is consecutive.\n\\end{lemma}\n\\begin{proof}\n  Direct consequence of Lemma~\\ref{lem:singleGap} since all clique points of\n  that set must be placed on $J$ and all other clique points must be placed on\n  its complement $J^c$.\n\\end{proof}\n\n\n\n\\begin{lemma}\n  \\label{lem:singleIsland}\n  There exists a maximal clique $D$ with a single island.\n\\end{lemma}\n\\begin{proof}\nLet $D$ be a maximal clique with $|\\Pre(D)|$ maximal.\nThen $\\Reg^+(D)$ and $\\Reg^-(D)$ are disjoint as otherwise there would\nexist a point $p$ in $\\Reg^+(D)\\cap\\Reg^-(D)$ with \n$\\Pre(D)\\subsetneq \\{v\\in V(G'): p \\in R'(v)\\}$.\nThis implies the existence of a (maximal) clique $C$ in $G$ with\n$\\Pre(D)\\subsetneq \\Pre(C)$ in contradiction to the choice of $D$. \n\nSince $\\calR'$ is NHCA, this\nyields that $\\Reg(D) = \\Reg^+(D)$ is connected. In other words, $D$ has a single island.\n\\end{proof}\n\nFor the rest of the section, let $D$ be a maximal clique with a single island. \nLet $p_D$ be a point in the middle of $\\Reg(D)$\nand let $\\le$ be the linear order of points on the circle obtained by starting\nat $p_D$. We consider a point $p$ to be \\emph{on the left} of a point $q$ if we\nhave $p\\le q$.\n\nWe define a partial order $\\prec$ on the maximal cliques $\\calC$ such that \nfor each $C \\in \\calC\\setminus\\{D\\}$ we have $D\\prec C$ and \nfor each $C, C' \\in \\calC\\setminus \\{D\\}$ satisfying $\\forall p\\in\n\\Reg(C),q\\in \\Reg(C')\\colon p < q$ we set $C\\prec C'$.\nNote that every linearization of a clique ordering of an extension of $\\calR'$\nthat starts with $D$ extends $\\prec$.\nFor any vertex $v$ let $M_v$ denote the set of maximal cliques containing $v$.  \n\n\n\\begin{theorem}\n  \\label{the:NHCArepextChar}\n  Let $G$ be a graph without universal vertices and let $\\calR'$ be a partial\n  normal Helly circular-arc representation of $G$.  \n  %\n  There exists a normal Helly circular-arc\n  representation of $G$ that extends $\\calR'$ if and only if there exists a\n  linear extension $<$ of $\\prec$\n such that\n  \\begin{compactenum}\n    \\item  \\label{itm:linCh0} For any pair of distinct maximal cliques $C\\ne C'$ with $\\Reg(C)=\\Reg(C')$, the region $\\Reg(C)$ is not a single point.\n     \n    \\item \\label{itm:linCh1} For every vertex $v$, the set $M_v$ is consecutive in $[<]$. \n    \\item \\label{itm:linCh2} For every universal pair $u$, $w$, the set $M_{u}\\cap M_{w}$ is consecutive in $[<]$. \n    \\item \\label{itm:linCh4} For every gap $J$ of some $C \\in \\calC$, the set $S_J$ is consecutive in $[<]$. \n  \\end{compactenum}\n\\end{theorem}\n\n\\begin{proof}\nWe first show that, if there is an NHCA-extension of $\\calR'$, then these properties are satisfied.\nWe obtain $<$ as the linearization of a clique ordering of an extension of $\\calR'$\nstarting with $D$ where the clique point of $D$ is $p_D$. \nBy the construction of $\\prec$, the order $<$ is a linear extension of $\\prec$.\nBy Lemmas~\\ref{lem:nhcaG} and \\ref{lem:gapConsecutivity}, we obtain\nproperties~\\ref{itm:linCh1},~\\ref{itm:linCh2} and~\\ref{itm:linCh4}.\nProperty~\\ref{itm:linCh0} is necessary, since no two clique points can be\nplaced at the same point.\n    \nFor the opposite implication, let $< = C_1,C_2,\\dots,C_k$ be a linear extension of $\\prec$ such that\nproperties~\\ref{itm:linCh0},~\\ref{itm:linCh1},~\\ref{itm:linCh2} and~\\ref{itm:linCh4} are\nsatisfied. We show that each $C_i\\in \\calC$ can be assigned its clique point $\\cp(C_i)\\in \\Reg(C_i)$ such that\n$\\cp(C_j)<\\cp(C_i)$ whenever $j<i$.\n\nLet $\\varepsilon>0$ be the $\\frac{1}{2n+1}$-fraction of the length of the\nshortest nontrivial island. This allows to draw all new endpoints at\ndistance at least $\\varepsilon$ but still within any chosen island or side of\nisland $\\Reg(D)$.\nFor $C_1=D$ we place $\\cp(C_1)$ on $p_D$. In a greedy way, when the location of the clique points $\\cp(C_1),\\dots,\\cp(C_{i-1})$ is settled, we determine the set $P$ of feasible points for $\\cp(C_i)$ that is \n$P=\\Reg(C_i) \\cap \\{p: p> \\cp(C_{i-1})+\\varepsilon\\}$. If $P$ has minimum, we place $\\cp(C_i)$ there, otherwise\nwe put $\\cp(C_i)$ at $\\inf(P)+\\varepsilon$.\nWe argue that such choice always exists.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1,page=1]{nhca_char.pdf}\n\\caption{Argument for the existence of a placement of $\\cp(C_i)$.}\n\\label{fig:nhca_char}\n\\end{figure}\n\nAssume for a contradiction that $C_i$ is the first maximal clique in the order $<$ whose \nclique point $\\cp(C_i)$ cannot be properly placed. \nNote that a clique $C\\ne C_i$ can only have an island $I_C$ consisting of a\nsingle point if we have $I_C=\\Reg(C)$, since $\\Reg^+$ is a closed arc and all islands\nseparated by gaps within $\\Reg^+$ have an open end.  Hence, by the choice of\n$\\varepsilon$, we only place a clique $C$ at the very last point of $\\Reg(C)$, if\n$\\Reg(C)$ consists of a single point. With property~\\ref{itm:linCh0}, this can\nnot happen if $\\Reg(C) = \\Reg(C_i)$.\nTherefore, one clique point must be placed to the right of $\\Reg(C_i)$ before placing $\\cp(C_i)$.\nWe identify the first maximal clique $C_j, j<i$ that is\nplaced to the right of all points in $\\Reg(C_i)$. Since $\\cp(C_j)\\notin \\Reg(C_i)$, \nwe have that $\\Pre(C_i)\\ne\\Pre(C_j)$.\nBy Lemma~\\ref{lem:singleGap}, the maximal clique\n$C_j$ has a gap $J$ with $\\Reg(C_i)\\subseteq J$, see Fig.~\\ref{fig:nhca_char}.\n    \nConsider the neighboring island $I$ of $C_j$ to the left of $J$. Since\n$\\cp(C_j)$ was not placed on $I$, the clique point $\\cp(C_{j-1})$ has\nbeen placed to the right of $I$. By the choice of $C_j$, we have that\n$\\cp(C_{j-1})$ is not placed to the right of $J$ and thus $\\cp(C_{j-1})\\in J$. \nSince $\\cp(C_j)$ has been placed to the right of $\\Reg(C_i)$ and thus to\nthe right of $J$, we have $p_D\\not\\in J$ and thus $D\\not\\in S_J$.\nWith $D<C_{j-1}< C_j< C_i$, where $C_{j-1}, C_i\\in S_J$ and $D,C_j\\not\\in S_J$\nwe get a contradiction with the property~\\ref{itm:linCh4}, since $S_J$ is not consecutive in $[<]$.\n\nFrom the placement of clique points, \nwe obtain a NHCA-extension of $\\calR'$ so that for every not yet represented \nvertex $u \\notin V(G')$ we choose $R(u)$ to be the minimal arc containing \nexactly the clique points of the maximal cliques from $M_u$.\n\nBy the proof of Lemma~\\ref{lem:nhcaG}, we obtain from\nproperties~\\ref{itm:linCh2},~\\ref{itm:linCh4} that this results in an NHCA representation of $G$\nextending $\\calR'$, since the maximal clique points are placed\ncorrectly with regards to both predrawn arcs and new arcs, and moreover, two arcs \nintersect if and only if they are predrawn or share a maximal clique.\n\\end{proof}\n\n\\begin{algorithm}[t!]\n\\KwData{A graph $G$ and a partial representation $\\calR'$.}\n\\KwResult{A \\hbox{\\rm \\sffamily NHCA}\\xspace{} representation $\\calR$ of $G$ extending $\\calR'$ or a message that none exists.}\n\\Begin{\n\\lIf{$G$ has a universal vertex}{resolve ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily INTR})$ instead}\n\\Else{\nDetermine the set of maximal cliques $\\calC$ of $G$\\;\n\\ForEach{$C\\in \\calC$}{\nDetermine $\\Reg(C)$\\;\n\\lIf{$\\Reg(C)=\\emptyset$}{\\Return{$\\calR'$ has no extension}}\n}\n$\\varepsilon:=\\frac{1}{2n+1}\\min\\{|I|,I \\text{ is a non-trivial island}\\}$\\;\nFind $C_1\\in\\calC$ with a single island\\;\n\\lIf{such $C_1$ does not exist}{\\Return{$\\calR'$ has no extension}}\n Set $\\cp(C_1)$ to the middle of $\\Reg(C_1)$\\;\nDetermine the partial order $\\prec$\\;\nBuild a PC tree $T$ on $\\calC$ capturing the constraints stated in  Theorem~\\ref{the:NHCArepextChar}\\;\n\\lIf{such $T$ does not exist}{\\Return{$\\calR'$ has no extension}}\nSolve ${\\textsc{Reorder}}(T, C_1, \\prec)$ to get the order $C_1<C_2<\\dots<C_k$\\;\n\\lIf{such order $<$ does not exist}{\\Return{$\\calR'$ has no extension}}\n\\For{$i=2$ \\KwTo $k$}{ \n $P:=\\Reg(C_i) \\cap \\{p: p> \\cp(C_{i-1})+\\varepsilon\\}$\\;\n \\lIf {$\\min P$ exists}{$\\cp(C_i):=\\min P$}\n \\lElse{$\\cp(C_i):=\\inf(P)+\\varepsilon$}\n}\n\\lForEach{$u \\notin V(G')$}{draw $R(u)$ to cover exactly $\\{\\cp(C): C\\in M_u\\}$} \n\\Return{$\\calR$}\n}\n}\n\\caption{The algorithm for the ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NHCAR})$ problem.}\\label{alg:nhca}\n\\end{algorithm}\n\n\\nhcapoly*\n\\begin{proof}\nThe correctness of Algorithm~\\ref{alg:nhca} follows from the already mentioned arguments.\nFor the computational complexity of the more complex steps note that:\n\\begin{compactitem}\n\\item\nLine 2: ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily INTR})$ can be solved in linear time~\\cite{klavik2013linear}.\n\\item\nLine 4: We run the linear time recognition algorithm for Helly circular-arc graphs~\\cite{lin2006characterizations} on $G$ and read its at most $n$ maximal cliques from any of its representation.\n\n\\item\nLines 5--8: The regions can be obtained in time $\\calO(n)$ by traversing the circle once.\nDuring the traversal two consecutive predrawn endpoints specify possible islands which can be assigned to appropriate maximal cliques.\n\\item\nLine 12: The comparable pairs of the partial order $\\prec$ can also be determined during the traversal in Steps 5--8.\n\\item\n  Line 13: The $\\calO(n^2)$ constraints can be computed in $O(n^3)$ time. The\n  construction of the PC-tree from~\\cite{hsu2003pc} also runs in $\\calO(n^3)$\n  time.\n\\item\nLine 16: The ${\\textsc{Reorder}}(T, C_1, \\prec)$ problem can be solved in $\\calO(n^2)$ time by Lemma~\\ref{lem:reordering_linear}.\n\\end{compactitem}\n\\end{proof}\n\nIn light of the hardness result from Theorem~\\ref{thm:hca_npc}, it is unlikely that this\nresult can be generalized to ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$. However, the hardness result for\nHelly representations crucially relies on predrawn arcs sharing endpoints.\nIndeed, if all predrawn arcs have distinct endpoints, the problem can be solved\nin a similar fashion.\n\n\\hcapoly*\n\\begin{proof}[Proof sketch]\n  We characterize extendable $\\hbox{\\rm \\sffamily HCA}\\xspace$ instances as in  \n  Theorem~\\ref{the:NHCArepextChar} (see Theorem~\\ref{the:HCArepextChar})\n  with Lemma~\\ref{lem:gavril_hca} instead of Lemma~\\ref{lem:nhcaG}.  Since the\n  predrawn arcs have pairwise distinct endpoints, we have no islands consisting\n  of a single point.\n  %\n  Note that Lemma~\\ref{lem:singleIsland} no longer applies and thus the\n  placement of $p_D$ cannot be chosen freely.  Instead, observe that\n  every maximal clique $D$ in a Helly circular-arc representation of $\\calR'$\n  has a clique point $\\cp(D)$ that can be chosen as~$p_D$.\n  %\n  Thus, we choose an arbitrary clique as $C_1$ and apply the remaining\n  procedure for every island $I$ of $C_1$, choosing $\\cp(C_1)\\in I$.\n  In contrast to our method for ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NHCAR})$\n  we have no special procedure for universal vertices.\n\\end{proof}\n\nFor details see Section~\\ref{sec:HCA}.\n\n\n\n\n\n\\section{Details fo PC-trees and The Reordering Problem}\n\\label{sec:PQtree}\n\n \\reorderingLinear*\n \\reorderingLinearProof\n\n \\section{Detailed Hardness Proofs}\n \\subsection{${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ and ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$}\n \\label{ssec:CARnp}\n\n\\hcanpc*\n\\begin{proof}\nWe reduce the $3$-{\\textsc{Partition}}\\ problem~\\cite{garey1975complexity}. \nLet $S=\\{s_1,\\dots, s_{3n}\\}$ be its instance, i.e., a collection of $3n$\nintegers summing up to $nt$ for some positive integer $t$\nsuch that each satisfies $\\frac{t}4<s_i<\\frac{t}2$, where the goal is to\npartition $S$ into $n$ disjoint subsets whose sum is always $t$. Note that the\nsize constraints on $s_i$'s ensure that every subset, which sums exactly to\n$t$, is a suitable triple.\n\nFrom $S$ and $t$ we construct a graph $G$ as follows: For the set of vertices\nwe choose\n$V(G)=\\{u_1,\\dots,u_{(t+1)n},v_1,\\dots,v_n,w_1,\\dots,w_{tn},z_1,\\dots,z_{3n}\\}$,\nwhere $u_1,\\dots,u_{(t+1)n}$ are universal vertices; vertices\n$v_1,\\dots,v_n,w_1,\\dots,w_{tn}$ form an independent set; and each $z_i$ is\nconnected to all of $u_1,\\dots,u_{(t+1)n}$ and    $s_i$ private vertices among\n$w_1,\\dots,w_{tn}$.\n\n\n\nObserve that this graph has exactly $(t+1)n$ maximal cliques, each identified\nby a vertex from $v_1,\\dots,v_n,w_1,\\dots,w_{tn}$. Moreover, each $z_i$ belongs\nto exactly $s_i$ maximal cliques.\n\nLet $p_1,\\dots, p_{(t+1)n}$ be distinct points of a circle ordered in a\nclockwise direction.\nFor each $u_i$ let $R'(u_i)=[p_i,p_{i-1}]$. For each $v_i$, let\n$R'(v_i)=[p_{(t+1)i},p_{(t+1)i}]$, (alternatively these degenerate intervals\ncould be extended to nondegenerate by expanding them by $\\frac{1}{3}$ distance\nof two nearest points $p_i,p_{i+1}$). The partial representation $\\calR'$ is\nthe just described representation of the subgraph induced by\n$\\{u_1,\\dots,u_{(t+1)n},v_1,\\dots,v_n\\}$, see Fig.~\\ref{fig:hca_npc} for an\nexample.\n\nWhen $G$ allows a representation $\\calR$ extending $\\calR'$, then each maximal\nclique with some $w_i$ will be placed on some point $p_j$ where $j\\not\\equiv 0\n\\pmod{t+1}$. Moreover clique points corresponding to some $z_i$ must be\nconsecutive and do not correspond to any other $z_{i'}$ or any $v_j$.\n\nTherefore the set system $\\{\\{s_i:\nR(z_i)\\cap\\{p_{(t+1)j+1},\\dots,p_{(t+1)j+t}\\}\\ne \\emptyset \\}:\nj\\in\\{1,\\dots,n\\}\\}$ yields the desired partition of $S$. A construction of the\nrepresentation from the set system is straightforward.\n\nTo argue that this reduction works also for \\hbox{\\rm \\sffamily NP}-hardness of ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$\nobserve that we did not put any condition on the representation to enforce the\nHelly property, it is guaranteed by the partial representation and the\nconstruction itself. \n\nThis \\hbox{\\rm \\sffamily NP}-hardness reduction for ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ can be altered to the situation \nwhen the predrawn arcs are required to have pairwise distinct endpoints just by \nshrinking each arc $R'(t_i)$ by moving its tail by $\\frac{1}{5}$ distance of\nthe two nearest points $p_i,p_{i+1}$. The argument is the same, only to\nguarantee that each $w_i$ is adjacent to all universal vertices $u_1,\\dots,\nu_{(t+1)n}$, the corresponding arc $R(w_i)$ would have to cover not only a\nsingle point $p_j$ for some $j$, but instead the whole arc to the nearest tail\nin the clockwise direction, formally the arc $[p_j,R'(u_j)_t]$. Observe that by\nthis modofication we loose the Helly property as the universal vertices share\nno common point.\n\\end{proof}\n\n\n\n\n\n\\subsection{${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily UCAR})$}\n\nWe first argue the \\hbox{\\rm \\sffamily NP}-hardness of Theorem~\\ref{thm:uca_npc} by showing a\nreduction from the $3$-${\\textsc{Partition}}$ problem defined already in the proof of\nTheorem~\\ref{thm:hca_npc}.\n\nThen we provide a linear programming argument for the membership in the class\n\\hbox{\\rm \\sffamily NP}.\n\n\\ucanpc*\n\\begin{proof}\nFor a given instance of $3$-${\\textsc{Partition}}$, we construct a unit circular-arc\ngraph $G$ and its partial representation $\\calR'$. For technical reasons, we\nassume that $t \\geq 8$.\n\nLet $P_{2\\ell}$ be a path of length $2\\ell$. There exists a unit circular-arc\nrepresentation $P_{2\\ell}$ such that it spans $\\ell + \\varepsilon$ units, for some\n$\\varepsilon > 0$. To see this, note that $P_{2\\ell}$ has two independent sets of size\n$\\ell$ and each of this independent sets needs at least $\\ell + \\varepsilon$. Let $a,\nb, c$ be positive integers such that $a + b + c = t$. It follows that the\ndisjoint union of $P_{2a}$, $P_{2b}$, and $P_{2c}$ has a representation such\nthat it spans $t + \\varepsilon$ units, for some $\\varepsilon > 0$, and therefore, it can be\nfit into $t+1$ units.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.85]{uint_proof}\n\\caption{Example of packing of three paths corresponding to the triple\n$5+3+2=10$. The predrawn arcs are depicted in bold.}\n\\label{fig:unit_proof}\n\\end{figure}\n\nLet $x_0, \\dots, x_{n(t+2)-1}$ be points of the circle that divide it into\n$n(t+2)$ equal parts, i.e., vertices of a regular $n(t+2)$-gon.  The graph $G$\nis a disconnected graph consisting of $4n$ connected components.  For each\n$s_i$, we take the path $P_{2s_i}$. We further add an isolated vertex $v_j$,\nfor $j = 0, \\dots, n-1$. The partial representation $\\calR'$ is the collection\n$\\{R(v_j) : j = 0,\\dots, n-1\\}$, where $R(v_j)$ is the arc of the circle from\n$x_{j(t+2)}$ to $x_{j(t+2)+1}$ in the clockwise direction.\n\nThe predrawn arcs $R(v_0), \\dots, R(v_{n-1})$ split the circle into $n$ gaps,\nwhere each gap has exactly $t+1$ units. By the discussion above, if the $s_i$'s\ncan be partitioned into $n$ triples such that each triple sums to $t$, then a\nrepresentation of the disjoint union of the paths corresponding to a triple can\nbe placed in one of the $n$ gaps, see Fig.~\\ref{fig:unit_proof}. If the partial\nrepresentation $\\calR'$ can be extended, then we a have partition of the\n$s_i$'s into $n$ triples such that each triple sums to $t$.\n\nThe certificate for the membership in \\hbox{\\rm \\sffamily NP} of an instance $G$, $\\calR'$ is a\nlinear order of $\\prec$ on $V_G$ corresponding to the cyclic ordering of the\nintervals together with a subset of edges $E'\\subseteq E_G$.\nWithout loss of generality we assume that the predrawn arcs are\n$R(v_1),\\dots,R(v_k)$.\n\nThe following linear program, where for each $i\\in\\{1,\\dots,n\\}$ the variable\n$x_i$ corresponds to the tail coordinate $R(v_i)_t$ has a feasible solution if\nand only if $\\calR'$ can be extended to a unit interval representation $\\calR$\nwhere its cyclic ordering of intervals (as the equivalence class) contains\n$\\prec$, and where the set $E'$ specifies pairs of adjacent vertices with tail\ncoordinates at least $\\ell-1$ apart. \n\n$$\n\\begin{array}{rcll}\nx_i &=& R(v_i)_t & \\text{ for all } i\\in\\{1,\\dots,k\\} \\\\\nx_i &\\ge& 0 &  \\text{ for all } i\\in\\{k+1,\\dots,n\\} \\\\\nx_i &<& \\ell & \\text{ for all } i\\in\\{k+1,\\dots,n\\} \\\\\nx_j - x_i &\\ge& 0 & \\text{ if } i\\prec j \\text{ and } (v_i,v_j)\\in E_G \\setminus E' \\\\\nx_j - x_i &\\le& 1 &\\text{ if } i\\prec j \\text{ and } (v_i,v_j)\\in E_G \\setminus E' \\\\\nx_j - x_i &\\ge& \\ell-1 &\\text{ if } i\\prec j \\text{ and } (v_i,v_j)\\in E' \\\\\nx_j - x_i &>& 1 &\\text{ if } i\\prec j \\text{ and } (v_i,v_j)\\notin E_G \\\\\nx_j - x_i &<& \\ell-1 &\\text{ if } i\\prec j \\text{ and } (v_i,v_j)\\notin E_G \\\\\n\\end{array}\n$$\n\n\\relax\n\\end{proof}\n\n\n \\section{Additional Details for Normal Proper Helly Circular Arc Graphs}\n  \\label{sec:DetailsNPHCA}\n\n  \\nphcaChar*\n  \\nphcaCharProof\n\n  \\section{Proper Helly Circular-Arc Graphs}\n  \\label{sec:PHCA}\nNote that even though all proper Helly circular-arc graphs allow also a normal\nproper Helly circular-arc representation~\\cite{lin2013normal}, we cannot use\nAlgorithm~\\ref{alg:nphca} directly, as the given partial representation may not\nadmit a normal extension.  We can, however, still use the same machinery.\n\n\\begin{lemma}[{\\cite[Lemma~1]{lin2013normal}}]\n  \\label{lem:non-normal-universal}\n  Let $\\calR$ be a proper circular-arc representation of a graph $G$. Any two\n  vertices whose arcs are in non-normal position are both universal, i.e.\n  adjacent to all vertices of $G$.\n\\end{lemma}\n\\begin{proof}\n  Let $u$, $v$ be two vertices with arcs in non-normal position.  Then a\n  non-neighbor of $u$ would have to be represented as a proper sub-arc of\n  $R(v)$ (and vice-versa) in contradiction to $\\calR$ being proper.\n\\end{proof}\n\n\\begin{lemma}\n  \\label{lem:phca-nphca}\n  A yes-instance of ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily PHCAR})$ is a yes-instance of ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$\n  if and only if no two predrawn arcs are in non-normal position.\n\\end{lemma}\n\n\\begin{proof}\n  Clearly, if the partial representation $\\calR'$ contains two\n  predrawn arcs in non-normal position, then there is no normal\n  extension of $\\calR'$.\n\n  Conversely, assume that there is no such pair and consider a\n  \\hbox{\\rm \\sffamily PHCAR}-extension $\\calR$ of $\\calR'$.\n  By Lemma~\\ref{lem:non-normal-universal}, only universal vertices can have arcs in\n  non-normal position.\n  If $\\calR'$ prescribes no universal vertex, we modify~$\\calR$ so\n  that all universal vertices are represented by the same arc\n  $R(v)$ for some arbitrary universal vertex $v$.  If $\\cal R'$\n  prescribes a universal vertex $v$, we modify~$\\calR$ by representing\n  all unprescribed universal vertices by the same arc $R(v)$.\n  \n  After this modification, if there still exists a non-normal pair of\n  arcs, there also exists a non-normal pair of prescribed arcs.  Since\n  such a pair does not exist by assumption, it follows that the\n  modified representation is an $\\hbox{\\rm \\sffamily NPHCAR}$-extension of $\\calR'$.\n\\end{proof}\n\n\n\\phcared*\n\\begin{proof}\n  Without loss of generality we assume that $G$ is not complete as otherwise \n  an extension can always be obtained by duplication of any predrawn arc. \n\n  Let $\\calR'$ be a partial representation of a graph $G\\in \\hbox{\\rm \\sffamily PHCA}\\xspace$.\n  Without loss of generality we also assume that predrawn arcs are distinct\n  as otherwise the identical arcs must correspond to twins and \n  it suffices to keep only one.\n\n  If there is no pair of prescribed arcs in non-normal position,\n  Lemma~\\ref{lem:phca-nphca} implies that the problem can be solved\n  with Theorem~\\ref{thm:nphca_poly}.\n  Hence assume that there exists a pair $R'(u),R'(v)$ of prescribed arcs in\n  non-normal position that by Lemma~\\ref{lem:non-normal-universal} must\n  correspond to universal vertices.\n  \nDenote by $A$ and $B$ the two disjoint circular arcs whose union is the\nintersection of $R'(u)$ and $R'(v)$ and by $C$ and $D$, resp., the arcs formed\nby the points of the circle that belong only to $R'(u)$ but not $R'(v)$ and\nvice-versa. The two closed arcs $A$, $B$ together with the two open $C$ and $D$\ncover the whole circle, see Fig.~\\ref{fig:phca}a.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1,page=1]{phca.pdf}\n\\caption{(a) Partition of the circle into sections $A, B, C$ and $D$. (b) Violating\nthe Helly property by vertices for $v, w_A$ and $w_B$. }\n\\label{fig:phca}\n\\end{figure}\n\nIn order to characterize all graphs that allow a representation extension,\nconsider now any such graph $G$ together with its proper Helly representation\n$\\calR$. \n\nSince $\\calR$ is proper, any arc $R(w)\\in\\calR$ distinct from $R(v)$ \nmust intersect $C$ to guarantee $R(w)\\not\\subset R(v)$.\nWith an analogous condition on $D$ we get that either \n$A\\subseteq R(w)$ or $B\\subseteq R(w)$.\n\nThen let $V_A=\\{w_A\\in V(G): A\\subseteq R(w_A)\\}$ and $V_B=\\{w_B\\in V(G): B\\subseteq\nR(w_B)\\}$ be the sets of vertices whose arcs contain $A$ and $B$, respectively.\nBy the definition $V_A$ and $V_B$ are two cliques covering $V(G)$.\nSince $G$ is not complete, both $V_A$ and $V_B$ contain a non-universal vertex.\nAny arc intersecting $A$ and $B$ contains either $C$ or $D$, thus due to proper\nand distinct positions of arcs we have $V_A \\cap V_B = \\{u,v\\}$.\n\nObserve that when the arc representing some $w_A\\in V_A$ intersects $B$ (or vice\nversa), then $w_A$ is universal --- it would intersect all vertices from $V_B$ on\n$R(w_A)\\cap B$. On the other hand, we claim that every arc of a universal\nvertex  intersects both $A$ and $B$. Assume by a contradiction that such arc\n$R(w_A)$ would intersect only $A$.\nBy Lemma~\\ref{lem:non-normal-universal}, it is in normal position with respect\nto the arc $R(w_B)$ of some non-universal vertex $w_B\\in V_B$. Without loss of\ngenerality the intersection \nof these two arcs is in $C$. Then as depicted in Fig.~\\ref{fig:phca}b the arcs\nof $w_A,w_B$ and $v$ violate the Helly property, a contradiction. In summary,\neach universal arc  includes one of $A$ or $B$, one of $C$ or $D$ and\nintersects the remaining two.\n\nBy identical arguments in slightly altered context we show that after pruning\nall universal vertices from $G$ the resulting graph $G'$ is disjoint union of\ntwo cliques $V_A'=V_A\\setminus V_B$ and $V_B'=V_B\\setminus V_A$.\nAssume for contradiction that $G'$ contains two adjacent vertices \n$w_A\\in V_A'$ and $w_B\\in V_B'$.\nSince $w_A$, $w_B$ are not universal, $R(w_A)$, $R(w_B)$ must be in normal position by\nLemma~\\ref{lem:non-normal-universal}.\nTherefore, the intersection $R(w_A)\\cap R(w_B)$ is fully contained either in $C$ or in\n$D$. In the first case we get a contradiction as the arcs $R(v)$, $R(w_A)$,\n$R(w_B)$ violate the Helly property, see again Fig.~\\ref{fig:phca}b. In the\nother case, the arcs $R(u)$, $R(w_A)$ and $R(w_B)$ also violate the Helly\nproperty. \n\nBy Lemma~\\ref{lem:non-normal-universal}, any \narc $R(w_A)$ for a non universal vertex $w_A\\in V_A'$ contains exactly one\nendpoint of the each arc $R(x)$ representing a universal vertex.\nThe other endpoint of $R(x)$ must then belong to $R(w_B)$ of each $w_B\\in V_B'$. \nThis way we get a partition of the endpoints of universal arcs into two consecutive sets.\nWe indeed describe this partition more precisely: The arc $R(w_A)$ contains \n$\\{h_i: C\\subset R(u_i)\\} \\cup \\{t_i: D\\subset R(u_i)\\}  $, \nwhile  \n$\\{t_i: C\\subset R(u_i)\\} \\cup \\{h_i: D\\subset R(u_i)\\} \\subset R(w_B)$.\nWhen $C\\subset R(u_i)$ then $h_i$ belongs either to $A$ or $D$. The first case\nis enforced immediately and in the latter $t_i\\in B$. Hence we also have to put\n$t_i$ into $R(w_A)$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1,page=2]{phca.pdf}\n\\caption{Position of $R(w_A)_t$ and $R(w_B)_h$ with respect to \nuniversal arcs containig $D$.}\n\\label{fig:phca-2}\n\\end{figure}\n\nThis necessary condition is also sufficient, namely a partial representation\n$\\calR'$ of a non-complete graph proper-Helly circular graph $G$ with arcs\n$R(u),R(v)$ in not-normal position allows an extension if and only if the sets\n$\\{h_i: C\\subset R'(u_i)\\} \\cup \\{t_i: D\\subset R'(u_i)\\}$, \nand   \n$\\{t_i: C\\subset R'(u_i)\\} \\cup \\{h_i: D\\subset R'(u_i)\\}$ are consecutive ---\nif no $w_A\\in V_A'$ is pre-drawn, we choose $R(w_A)$ to be the shortest arc\ncontaining the first of these two sets within $A\\cup C\\cup D$ and analogously\nfor any $w_B\\in V_B'$, see Fig.~\\ref{fig:phca-2}.\nThe remaining arcs could be obtained by replication.\n\nOur arguments convert directly to the following algorithm:\nAccept if $G$ is complete. If the partial proper Helly\nrepresentation has no pair of arcs in non-normal position, use\nAlgorithm~\\ref{alg:nphca}. \nOtherwise check whether $G'$ is isomorphic to the\nunion of two disjoint cliques; if not, reject. \nFinally, check whether the endpoints of the predrawn arcs of universal vertices\ncould be split into two consecutive sets described above and if the predrawn\narcs non-universal ones cover these sets appropriately. If not, reject.\n\nIn the affirmative case, a representation could be constructed by replication\nof arcs of three representatives: a universal vertex, one vertex from $V_A'$\nand one from $V_B'$.\nA minor simplification could applied when a representative from $V_A'$ or\n$V_B'$ is already pre-drawn and recognized. \n\\end{proof}\n\n\n\\section{Illustrations for the Normal Helly Case}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics{lem16.pdf}\n\\caption{An example of a construction of a normal Helly circular-arc representation.}\n\\label{fig:lem16}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics{lem15.pdf}\n\\caption{The region of a maximal clique cannot intersect two gaps of another.}\n\\label{fig:lem15}\n\\end{figure}\n\n\n\n\\section{Helly Circular Arc Graphs}\n\\label{sec:HCA}\n\nSurprisingly, when the Helly property is required, we can solve\n${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$ similarly to ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NHCAR})$\nwhen the predrawn arcs have pairwise distinct endpoints. The most notable\ndifference is that Lemma~\\ref{lem:singleIsland} no longer applies and we have\nto test every island of clique $D$ for placing $p_D$.\n\n\\begin{theorem}\n  \\label{the:HCArepextChar}\n  Let $\\calR'$ be a partial $\\hbox{\\rm \\sffamily HCA}\\xspace$ representation of $G$ where the arcs have\n  pairwise distinct endpoints and let $D$ be a maximal clique of $G$.\n  %\n  There exists an $\\hbox{\\rm \\sffamily HCA}\\xspace$ representation of $G$ that extends\n  $\\calR'$ if and only if for some point $p_D\\in \\Reg(D)$ with the corresponding\n  partial order $\\prec$ as defined in Section~\\ref{sec:NHCA}\n  there exists a linear extension $<$ of $\\prec$ such\n  that:\n  \\begin{compactenum}\n    \\item \\label{itm:HlinCh1} For every vertex $v$, $M_v$ is consecutive in $[<]$. %\n    \\item \\label{itm:HlinCh4} For every gap $J$ of a $C\\in \\calC$, the set\n      $S_J$ is consecutive in $[<]$. \n  \\end{compactenum}\n\\end{theorem}\n\n\n\n\\begin{proof}\nWe first show that, if there is such an $\\hbox{\\rm \\sffamily HCA}\\xspace$-extension of $\\calR'$, then\nthese properties are satisfied.\n\nWe obtain $<$ as the linearization of a clique ordering of an extension of $\\calR'$\nstarting with $D$ where the clique point of $D$ is $p_D$. \nBy the construction of $\\prec$, the order $<$ is a linear extension of $\\prec$.\nBy Lemmas~\\ref{lem:gavril_hca} and \\ref{lem:gapConsecutivity}, we obtain\nproperties~\\ref{itm:HlinCh1} and~\\ref{itm:linCh4}.\n    \nFor the opposite implication, let $p_D\\in \\Reg(D)$ be on an island $I_1$ of $D$\nand let $< = C_1,C_2,\\dots,C_k$ be a linear extension of $\\prec$ such that\nproperties~\\ref{itm:HlinCh1} and~\\ref{itm:HlinCh4} are\nsatisfied. We show that each $C_i\\in \\calC$ can be assigned its clique point\n$\\cp(C_i)\\in \\Reg(C_i)$ such that $\\cp(C_j)<\\cp(C_i)$ whenever $j<i$.  We\nrelocate $p_D$ to the middle of island $I_1$.  Note that the\nproperties~\\ref{itm:HlinCh1} and~\\ref{itm:HlinCh4} remain\nsatisfied for the new ordering $\\prec$ since the only possible change is that\nrelations to cliques with island $I_1$ may get lost. \n\nLet $\\varepsilon>0$ be the $\\frac{1}{2n+1}$-fraction of the length of the\nshortest nontrivial island. This choice allows us to draw all new endpoints at\ndistance at least $\\varepsilon$ but still within any chosen island or side of\nisland $I_1$.\n\nFor $C_1 = D$ we place $\\cp(C_1)$ on $p_D$. In a greedy way, when the location\nof the clique points $\\cp(C_1),\\dots,\\cp(C_{i-1})$ is settled, we determine the\nset $P$ of feasible points for $\\cp(C_i)$ that is $P = \\Reg(C_i) \\cap \\{p: p>\n\\cp(C_{i-1})+\\varepsilon\\}$. If $P$ has minimum, we place $\\cp(C_i)$ there,\notherwise we put $\\cp(C_i)$ at $\\inf(P)+\\varepsilon$.\nWe argue that such choice always exists.\n\nAssume for a contradiction that $C_i$ is the first maximal clique in the order $<$ whose \nclique point $\\cp(C_i)$ cannot be properly placed. \nNote that a maximal clique $C\\ne C_i$ can never have an island $I_C$ consisting of a\nsingle point since this can only occur as intersection of two predrawn arcs\nending at that point. Hence, by the choice of\n$\\varepsilon$, we never place a clique $C$ at the very last point of $\\Reg(C)$.\nTherefore, one clique point must be placed to the right of $\\Reg(C_i)$ before placing $\\cp(C_i)$.\nWe identify the first maximal clique $C_j, j < i$ that is\nplaced to the right of all points in $\\Reg(C_i)$. Since $\\cp(C_j)\\notin \\Reg(C_i)$, \nwe have that $\\Pre(C_i)\\ne\\Pre(C_j)$.\nBy Lemma~\\ref{lem:singleGap}, the maximal clique\n$C_j$ has a gap $J$ with $\\Reg(C_i)\\subseteq J$, see Fig.~\\ref{fig:nhca_char}.\n    \nConsider the neighboring island $I$ of $C_j$ to the left of $J$. Since\n$\\cp(C_j)$ was not placed on $I$, the clique point $\\cp(C_{j-1})$ has\nbeen placed to the right of $I$. By the choice of $C_j$, we have that\n$\\cp(C_{j-1})$ is not placed to the right of $J$ and thus $\\cp(C_{j-1})\\in J$. \nSince $\\cp(C_j)$ has been placed to the right of $\\Reg(C_i)$ and thus to\nthe right of $J$, we have $p_D\\not\\in J$ and thus $D\\not\\in S_J$.\nWith $D<C_{j-1}< C_j< C_i$, where $C_{j-1}, C_i\\in S_J$ and $D,C_j\\not\\in S_J$\nwe get a contradiction with the property~\\ref{itm:HlinCh4}, since $S_J$ is not consecutive in $[<]$.\n\nFrom the placement of clique points, \nwe obtain an $\\hbox{\\rm \\sffamily HCA}\\xspace$-extension of $\\calR'$ so that for every not yet represented \nvertex $u \\notin V(G')$ we choose $R(u)$ to be a minimal arc containing \nexactly the clique points of the maximal cliques from $M_u$.\n\nBy the Lemma~\\ref{lem:gavril_hca}, we obtain from\nproperties~\\ref{itm:HlinCh1},~\\ref{itm:HlinCh4} that this results in an $\\hbox{\\rm \\sffamily HCA}\\xspace$ representation of $G$\nextending $\\calR'$, since the maximal clique points are placed\ncorrectly with regards to both predrawn arcs and new arcs, and moreover, two arcs \nintersect if and only if they are predrawn or share a maximal clique.\nNote that we can perturbate the endpoints of the non-predrawn arcs to avoid\nshared endpoints.\n\\end{proof}\n\n\n\n\n\\begin{algorithm}[t!]\n  \\SetKw{Continue}{continue} \n  \\KwData{A graph $G$ and a partial representation $\\calR'$ consisting of arcs with pairwise distinct ends.}\n  \\KwResult{An \\hbox{\\rm \\sffamily HCA}\\xspace{} representation $\\calR$ of $G$ extending $\\calR'$ or a message that it does not exist.}\n  \\Begin{\n    Determine the set of maximal cliques $\\calC$ of $G$\\;\n    \\ForEach{$C\\in \\calC$}{\n      Determine $\\Reg(C)$\\;\n      \\lIf{$\\Reg(C)=\\emptyset$}{\\Return{$\\calR'$ has no extension}}\n    }\n    $\\varepsilon:=\\frac{1}{2n+1}\\min\\{|I|,I \\text{ is an island}\\}$\\;\n    Choose any $C_1\\in\\calC$\\;\n    \\ForEach{island $I$ of $C_1$}{\n      Set $\\cp(C_1)$ to the middle of $I$\\;\n      Determine the partial order $\\prec$\\;\n      Build a PC tree $T$ on $\\calC$ capturing the constraints stated in  Theorem~\\ref{the:HCArepextChar}\\;\n      \\lIf{such $T$ does not exist}{\\Continue}\n      Solve ${\\textsc{Reorder}}(T, C_1, \\prec)$ to get the order $C_1<C_2<\\dots<C_k$\\;\n      \\lIf{such order $<$ exists}{\n        \\For{$i=2$ \\KwTo $k$}{ \n          $P:=\\Reg(C_i) \\cap \\{p: p> \\cp(C_{i-1})+\\varepsilon\\}$\\;\n          \\lIf {$\\min P$ exists}{$\\cp(C_i):=\\min P$}\n          \\lElse{$\\cp(C_i):=\\inf(P)+\\varepsilon$}\n        }\n        \\lForEach{$u \\notin V(G')$}{draw $R(u)$ to cover exactly $\\{\\cp(C): C\\in M_u\\}$} \n        \\Return{$\\calR$}\n      }\n    }\n  }\n  \\Return{$\\calR'$ has no extension}\\;\n  \\caption{The algorithm for the ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$ problem.}\\label{alg:hca}\n\\end{algorithm}\n\n\\hcapoly*\n\\begin{proof}\n  The correctness of Algorithm~\\ref{alg:hca} follows from the already mentioned arguments.\n  For the computational complexity of the more complex steps note that:\n  \\begin{compactitem}\n    \\item\n      Line 2:We run the linear time recognition algorithm for Helly\n      circular-arc graphs~\\cite{lin2006characterizations} on $G$ and read its\n      at most $n$ maximal cliques from any of its representation.\n    \\item\n      Lines 3--5: The regions can be obtained in time $\\calO(n)$ by traversing\n      the circle once.  During the traversal two consecutive predrawn\n      endpoints specify possible islands which can be assigned to appropriate\n      maximal cliques.\n    \\item\n      Line 8: There are at most $O(n)$ islands for $C_1$.\n    \\item \n      Line 10: The comparable pairs of the partial order $\\prec$ can also be\n      determined by a traversal as in Steps 3--5.\n    \\item\n      Line 11: The construction of the PC-tree follows the standard approach\n      from the recognition of Helly circular-arc graphs~\\cite{hsu2003pc}. \n      Each arc is at the left start of at most one gap and each of the $O(n)$\n      maximal cliques has at most one gap without an arc inside. Hence, there\n      are $O(n)$ gaps and thus $O(n)$ constraints. $T$ can thus be constructed\n      in $O(n^2)$ time~\\cite{hsu2003pc}.\n    \n    \n    \n    \\item\n      Line 13: The ${\\textsc{Reorder}}(T, C_1, \\prec)$ problem can be solved in\n      $\\calO(n^2)$ time by Lemma~\\ref{lem:reordering_linear}.\n  \\end{compactitem}\n\\end{proof}\n\n\n \n\n\n\n\\section{Conclusions and Open Problems}\n\\label{sec:conclusions}\n\nOur study of the ${\\textsc{RepExt}}$ problem has been restricted in two ways:\n\nFirst, we have considered mostly representations satisfying Helly property as\nthis allows us to consider the clique points of maximal cliques. For\nrepresentations that do not have this property one would involve a completely\ndifferent approach.\n\nSecondly, for the recognition problem it is irrelevant whether arcs are closed\nor open, but this is not the case for the representation extension. Observe\nthat touching intervals in ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$ in Lemma~\\ref{lem:nphca_char}\nimply constraints on the ordering. For the sake of completeness it might be\nworth to check whether use of open or semi-open intervals would yield\nsignificant impact on the computational complexity.\n\n\n\\paragraph*{Acknowledgement.}\n\\label{sec:acknowledgement}\nWe thank Bartosz Walczak for inspiring comments, in particular for his hint to\nextend Theorem~\\ref{thm:hca_npc} to the case of $\\hbox{\\rm \\sffamily CAR}$ with distinct endpoints.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nIn this paper we prove the following identity for a particular sum\nover two $9j$ and one $6j$ symbol which, to the best of our knowledge,\nseems not to appear previously as a distinct entity in the\nliterature:\n\\begin{eqnarray}\n\\lefteqn{\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}\n\\sixj{j_3}{j_4}{j_5}{\\Delta}{j^\\prime_5}{j^\\prime_4}\n\\sixj{j^\\prime_1}{j^\\prime_2}{j_3}{j^\\prime_4}{j^\\prime_5}{j^\\prime_6}\n}\n&& \\nonumber \\\\\n&=& \\sum_{k,\\ell} [k][\\ell] (-1)^\\Phi \n\\ninej{j_1}{j^\\prime_1}{k}{j_6}{j^\\prime_6}{\\ell}{j_5}{j^\\prime_5}\n{\\Delta}\n\\ninej{j_2}{j^\\prime_2}{k}{j_6}{j^\\prime_6}{\\ell}{j_4}{j^\\prime_4}\n{\\Delta}\n\\sixj{j_1}{j_2}{j_3}{j^\\prime_2}{j^\\prime_1}{k}\n\\, , \\label{newident}\n\\end{eqnarray}\nwhere $[j] \\equiv 2j + 1$ is the multiplicity of the spin-$j$\nirreducible representation, and the argument of the phase $(-1)^\\Phi$\nis given by\n\\begin{equation} \\label{phasearg}\n\\Phi = j_1 + j_2 + j_6 - j^\\prime_4 - j^\\prime_5 - j^\\prime_6 + k -\n\\Delta \\ .\n\\end{equation}\n\nThis identity is a generalization of the well-known Biedenharn-Elliott\n(BE) sum rule [Eq.~(\\ref{BE}) below], to which (as we shall show) it\nreduces when $\\Delta$ is set to zero.\n\nThis identity arises when considering the angular momentum coupling\nscheme\n\\begin{equation} \\label{coupling}\n\\begin{array}{ccc}\n{\\bf j_3} = {\\bf j_1} + {\\bf j_2} , \\ &\n{\\bf j_3} = {\\bf j^\\prime_1} + {\\bf j^\\prime_2} , \\ &\n{\\bf j^\\prime_1} = {\\bf j_1} + {\\bf k} , \\\\\n{\\bf j_6} = {\\bf j_2} + {\\bf j_4} , \\ &\n{\\bf j^\\prime_6} = {\\bf j^\\prime_2} + {\\bf j^\\prime_4} , \\ &\n{\\bf j_2} = {\\bf j^\\prime_2} + {\\bf k} , \\\\\n{\\bf j_5} = {\\bf j_3} + {\\bf j_4} , \\ &\n{\\bf j^\\prime_5} = {\\bf j_3} + {\\bf j^\\prime_4} , \\ & \n{\\bf j^\\prime_4} = {\\bf j_4} + {\\bf \\Delta} \\\\\n{\\bf j_5} = {\\bf j_1} + {\\bf j_6} , \\ &\n{\\bf j^\\prime_5} = {\\bf j^\\prime_1} + {\\bf j^\\prime_6} , \\ &\n{\\bf j^\\prime_5} = {\\bf j_5} + {\\bf \\Delta} , \\\\\n{\\bf j^\\prime_6} = {\\bf j_6} + {\\xbf \\ell} , \\ &\n{\\bf \\Delta} = {\\bf k} + {\\xbf \\ell} .\n\\end{array}\n\\end{equation}\nThe unprimes and primes suggest (for example) initial- and final-state\nquantum numbers whose inequality is enforced by a ``spurion'' ${\\bf\n\\Delta}$.  Other physically useful coupling schemes may be obtained by\nglobally replacing any of these angular momenta {\\bf j} by their\ntime-reversed forms $\\tilde{\\bf j} \\equiv -{\\bf j}$, where $\\tilde{\\bf\nj}$ is the angular momentum operator whose eigenstates are related to\nthose $( \\left| j m \\right>)$ of ${\\bf j}$ by $(-1)^{j+m} \\left| j -m\n\\right>$.  This manipulation gives a well-defined meaning to the\nconcept of subtracting angular momentum operators~\\cite{Edmonds}.\n\nAs an explicit physical example using this coupling scheme, consider\nmeson-baryon scattering ($\\phi B \\to \\phi^\\prime B^\\prime$) in which\nan additional isoscalar meson $f$ is produced with total angular\nmomentum $J_f$ with respect to the other final-state particles: $\\phi\nB \\to \\phi^\\prime B^\\prime f$.  The specified observables are isospins\nand angular momenta: $I_{\\phi (\\phi^\\prime)}$, $J_{\\phi\n(\\phi^\\prime)}$ for the mesons (as usual, $J$ denotes spin and orbital\nangular momenta combined), and $I_{B (B^\\prime)}$, $S_{B (B^\\prime)}$\nfor the baryons.  The total $s$-channel quantum numbers are ${\\bf I_s}\n\\! \\equiv \\! {\\bf I}_\\phi \\! + {\\bf I}_B \\! = \\! {\\bf I}_{\\phi^\\prime}\n\\! + {\\bf I}_{B^\\prime}$, ${\\bf J_s} \\! = \\! {\\bf J}_\\phi \\! + {\\bf\nS}_B$, ${\\bf J^\\prime_s} \\!  = \\! {\\bf J}_{\\phi^\\prime} \\! + {\\bf\nS}_{B^\\prime}$, and ${\\bf J^\\prime_s} \\!  = \\! {\\bf J_s} \\! - \\! {\\bf\nJ}_f$.  In a chiral soliton model or the $1\/N_c$ expansion of QCD, the\nvector sum of isospin and angular momentum for each particle assumes\nextra significance: The stable baryons (such as nucleons) are\nzero-eigenvalue states of the operators ${\\bf I}_B \\! + \\!  {\\bf S}_B$\nand ${\\bf I}_{B^\\prime} \\! + \\!  {\\bf S}_{B^\\prime}$, and the\nscattering is characterized by the ``grand spins'' ${\\bf K} \\! \\equiv\n\\! {\\bf I_s} \\! + \\! {\\bf J_s}$, ${\\bf K}^\\prime \\! \\equiv \\!  {\\bf\nI_s} \\! + \\! {\\bf J^\\prime_s}$.  The application of\nEq.~(\\ref{newident}) arises when one considers processes such as this\nnot in the $s$-channel but the $t$-channel~\\cite{cross}: Then ${\\bf\nI}_{\\phi^\\prime} \\! = \\! {\\bf I}_\\phi \\! + {\\bf I_t}$ and ${\\bf\nJ}_{\\phi^\\prime} \\! = \\! {\\bf J}_\\phi \\! + {\\bf J_t}$.  The full\nidentification using the notation of Eq.~(\\ref{coupling}) is ${\\bf\nj_1} \\! \\to \\!  {\\bf I}_\\phi$, ${\\bf j_2} \\! \\to \\! {\\bf I}_B$, ${\\bf\nj_3} \\! \\to \\!  {\\bf I_s}$, ${\\bf j_4} \\! \\to \\! {\\bf J_s}$, ${\\bf\nj_5} \\! \\to \\! {\\bf K}$, ${\\bf j_6} \\! \\to \\! {\\bf J}_\\phi$ (and\nanalogously for the primed ${\\bf j}$'s), and ${\\bf k} \\! \\to \\! {\\bf\nI_t}$, ${\\xbf \\ell} \\! \\to \\!  {\\bf J_t}$, and ${\\bf \\Delta} \\! \\to \\!\n-{\\bf J}_f$.  The transcription of Eq.~(\\ref{newident}) then reads\n\\begin{eqnarray}\n\\lefteqn{\n\\sixj{I_\\phi}{I_B}{I_s}{J_s}{K}{J_\\phi}\n\\sixj{I_s}{J_s}{K}{J_f}{K^\\prime}{J_s^\\prime}\n\\sixj{I_{\\phi^\\prime}}{I_{B^\\prime}}{I_s}{J_s^\\prime}{K^\\prime}\n{J_{\\phi^\\prime}}\n}\n&& \\nonumber \\\\\n&=& \\sum_{I_t,J_t} [I_t][J_t] (-1)^\\Phi \n\\ninej{I_\\phi}{I_{\\phi^\\prime}}{I_t}{J_\\phi}{J_{\\phi^\\prime}}{J_t}{K}\n{K^\\prime}{J_f}\n\\ninej{I_B}{I_{B^\\prime}}{I_t}{J_\\phi}{J_{\\phi^\\prime}}{J_t}{J_s}\n{J_s^\\prime}{J_f}\n\\sixj{I_\\phi}{I_B}{I_s}{I_{B^\\prime}}{I_{\\phi^\\prime}}{I_t}\n\\, , \\label{example}\n\\end{eqnarray}\nwhere now $\\Phi \\! = \\! I_\\phi \\! + \\! I_B \\! + \\! J_\\phi \\! - \\!\nJ_s^\\prime \\! - \\! K^\\prime \\! - \\! J_{\\phi^\\prime} \\! + \\! I_t \\! -\n\\! J_f$.  The three $6j$ symbols on the left-hand side (expressed\nsolely in terms of $s$-channel quantities) appear as coefficients in\nexpressions for partial-wave scattering amplitudes for the process\n$\\phi B \\to \\phi^\\prime B^\\prime f$ written in terms of underlying\n``reduced'' amplitudes labeled by $K$ values; for example, the case in\nwhich $f$ is absent (effectively, $J_f \\! = \\! 0$) has been studied\nfor quite some time~\\cite{MP}.  Expressing amplitudes in terms of\n$t$-channel quantities [as is manifest on the right-hand side of\nEq.~(\\ref{example})] is of particular interest because such amplitudes\nscale as $1\/N_c^{|I_t \\!  - \\!  J_t|}$~\\cite{cross}, thereby creating\na hierarchy of dominant and subdominant amplitudes in the $1\/N_c$\nexpansion of QCD.\n\nThe most illuminating proof of Eq.~(\\ref{newident}) uses diagrammatic\ntechniques, an approach we summarize in Sec.~\\ref{diagrammatic}.  We\npresent the diagrammatic proof in Sec.~\\ref{proof}, and finally an\nalgebraic proof, using standard SU(2) identities, in\nSec.~\\ref{algebraic}.\n\n\\section{Diagrammatic Method for Coupling Angular Momenta}\n\n\\label{diagrammatic}\n\n\\subsection{Notation}\n\nAlgebraic techniques for manipulating Clebsch-Gordan coefficients\n(CGC) to obtain invariants (quantities independent of magnetic quantum\nnumbers $m$, such as $6j$ and $9j$ symbols) are certainly\nstraightforward and appear in all standard treatments of the\ntopic~\\cite{Edmonds}.  However, at a certain point of complexity these\ntechniques become particularly cumbersome, and the bookkeeping\nnecessary to impose the required identities for simplifying such\nexpressions [particularly with regard to the numerous phases $(-1)^n$\nthat arise] becomes increasingly onerous.  A much cleaner strategy is\nto use diagrammatic techniques introduced originally by Jucys\n(alternate spellings {\\it Yutsis}, {\\it Iutsis}), Levinson, and\nVanagas (JLV)~\\cite{JLV}.  In this method, each angular momentum $j$\nis represented as a line, and each vertex represents a $3j$ symbol (or\nCGC).  The quantum number $m$ corresponding to $j$ is summed if and\nonly if line $j$ is internal to the diagram.\n\nThe diagrammatic technique is particularly valuable because of two\nfeatures: First, the identities involved in combining large complexes\nof angular momenta become topological in nature, and the ability to\nidentify them reduces to one's cunning in picturing how to connect the\nlines.  Second, the phases endemic to CGC are incorporated in the\ndiagrams very neatly (as too are factors of $[j]$, but we do not need\nthem here), appearing as either signs at the vertices or arrows on the\nlines, in the manner described below.  The JLV technique is laid out\npedagogically in the text by Lindgren \\& Morrison (LM)~\\cite{LM} or\nthe review by Wormer and Paldus~\\cite{WP}.  Here we list only the\nfeatures essential for this paper.\n\nFor starters, the vertex in Fig.~\\ref{fig:3j} represents a $3j$\nsymbol:\n\\begin{eqnarray}\n\\left(\\begin{array}{ccc}\nj_1 & j_2 & j_3 \\\\\nm_1 & m_2 & -m_3\n\\end{array}\n\\right)(-1)^{j_3-m_3},\n\\end{eqnarray}\n\\begin{figure}[ht]\n\\epsfxsize 1.8 in \\epsfbox{3j.eps}\n\\caption{Graphical representation of the $3j$ symbol.}\n\\label{fig:3j}\n\\end{figure}\nwith the specific ordering of the $j$'s and $m$'s as\n(counter)clockwise defining the vertex orientation as (positive)\nnegative, as indicated by a sign at the vertex in the diagram.  The\narrow on $j_3$ introduces the phase $(-1)^{j_3-m_3}$; were it pointing\ntoward the vertex, the phase introduced would be $(-1)^{j_3+m_3}$.\nNote that we follow the LM arrow convention, which is opposite that of\nJLV convention, since as argued in Ref.~\\cite{LM} it is more closely\nanalogous to the flow of momentum in scattering diagrams.\n\nSeveral manipulations help simplify such calculations:\n\\begin{enumerate}\n\\item Two arrows pointing in opposite direction on the same line can be\nremoved.\n\\item Reversing the direction of an arrow introduces an additional\n$(-1)^{2j}$.\n\\item Reversing the orientation of the vertex (changing the sign symbol)\nintroduces an additional $(-1)^{j_1+j_2+j_3}$.\n\\item Introducing three arrows all pointing inward or outward at a\nvertex does not change the value of the diagram.\n\\end{enumerate}\n\n\\subsection{Invariants}\n\nThe combination of several vertices (with no external lines) forms a\ndiagram representing a higher-order $3nj$ symbol; for example, the\nirreducible combination of 4 vertices, as depicted in\nFig.~\\ref{fig:6j}, forms the $6j$ symbol,\n\\begin{equation}\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6} ,\n\\end{equation}\n\\begin{figure}[ht]\n\\epsfxsize 1.8 in \\epsfbox{6j.eps}\n\\caption{Graphical representation of the $6j$ symbol.}\n\\label{fig:6j}\n\\end{figure}\nand the irreducible combination of 6 vertices, as depicted in\nFig.~\\ref{fig:9j}, forms the $9j$ symbol,\n\\begin{equation}\n\\ninej{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}{j_7}{j_8}{j_9} ,\n\\end{equation}\n\\begin{figure}[ht]\n\\epsfxsize 1.8 in \\epsfbox{9j.eps}\n\\caption{Graphical representation of the $9j$ symbol.}\n\\label{fig:9j}\n\\end{figure}\n\n\\subsection{Theorems}\n\nThe true power of the JLV approach arises through a serious of\ntheorems~\\cite{JLV,LM}, reminiscent of the factorization theorems of\nquantum field theory, that allow one to cut diagrams internally\nconnected only by a small number $n$ of lines.  Consider a diagram\nconsisting of two such blocks, $\\overline{\\alpha}$ and $\\beta$, such\nthat $\\overline{\\alpha}$ is {\\it closed\\\/} (no external lines) and in\n{\\it normal form\\\/} (every internal line on $\\overline{\\alpha}$\ncarries an arrow, and any arrows on the lines connecting with $\\beta$\nare pushed into the block $\\beta$).  Then one obtains a series of\ntheorems JLV$n$, $n \\! = \\! 1, 2, \\ldots$.  Of greatest interest to us\nhere are JLV3 (depicted in Fig.~\\ref{fig:JLV3}) and JLV4\n(Fig.~\\ref{fig:JLV4}).  JLV3, for example, applied to a system in\nwhich block $\\beta$ is empty, turns out to be none other than the\nWigner-Eckart theorem.\n\n\\begin{figure}[ht]\n\\epsfxsize 5.8 in \\epsfbox{JLV3.eps}\n\\caption{The JLV3 theorem.}\n\\label{fig:JLV3}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\epsfxsize 4.8 in \\epsfbox{JLV4.eps}\n\\caption{The JLV4 theorem.}\n\\label{fig:JLV4}\n\\end{figure}\n\n\\section{Proof of the Identity} \\label{proof}\n\nThe Biedenharn-Elliott sum rule~\\cite{Edmonds} reads\n\\begin{eqnarray} \\label{BE}\n\\lefteqn{\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}\n\\sixj{j^\\prime_1}{j^\\prime_2}{j_3}{j_4}{j_5}{j^\\prime_6}\n} \\nonumber \\\\ & = &\n\\sum_{\\cal J} (-1)^\\sigma [{\\cal J}]\n\\sixj{j_1}{j_6}{j_5}{j^\\prime_6}{j^\\prime_1}{\\cal J}\n\\sixj{j_2}{j_6}{j_4}{j^\\prime_6}{j^\\prime_2}{\\cal J}\n\\sixj{j_1}{j_2}{j_3}{j^\\prime_2}{j^\\prime_1}{\\cal J} ,\n\\end{eqnarray}\nwhere $\\sigma$ is the sum of the 9 distinct arguments on the left-hand\nside, plus ${\\cal J}$.  Here we wish to find the analog of the BE sum\nrule for the following three $6j$ symbols, the left-hand side of\nEq.~(\\ref{newident}), represented graphically in\nFig.~\\ref{fig:three_6j}:\n\\begin{equation} \\label{three6js}\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}\n\\sixj{j_3}{j_4}{j_5}{\\Delta}{j^\\prime_5}{j^\\prime_4}\n\\sixj{j^\\prime_1}{j^\\prime_2}{j_3}{j^\\prime_4}{j^\\prime_5}{j^\\prime_6}\n\\end{equation}\n\n\\begin{figure}[ht]\n\\epsfxsize 5.8 in \\epsfbox{three_6j.eps}\n\\caption{The three $6j$ symbols of Eq.~(\\ref{three6js}).}\n\\label{fig:three_6j}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\epsfxsize 4.0 in \\epsfbox{12j.eps}\n\\caption{Result of combining the three $6j$ symbols of\nEq.~(\\ref{three6js}) or Fig.~\\ref{fig:three_6j}.}\n\\label{fig:12j}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\epsfxsize 4.8 in \\epsfbox{6j_12j.eps}\n\\caption{The result of applying JLV4 to Fig.~\\ref{fig:12j}.}\n\\label{fig:6j_12j}\n\\end{figure}\n\nBy means of the JLV3 theorem, the three $6j$ symbols of\nEq.~(\\ref{three6js}) combine into a diagram with 12 quantum numbers\nshown in Fig.~\\ref{fig:12j}.  In order to introduce the quantum number\n$k$ of Eq.~(\\ref{newident}), we cut the diagram on the four lines,\n$j_1$, $j^\\prime_1$, $j_2$, and $j^\\prime_2$ using the JLV4 theorem.\nThe summed quantum number in this diagram is indeed the desired $k$,\nand the result is a $6j$ symbol and a $12j$ {\\it symbol of the second\nkind\\\/}, as shown in Fig.~\\ref{fig:6j_12j}.  Since the latter object\nis surely obscure to most readers, we pause to point out that, while\nhigher $3nj$ symbols are not difficult to generate and manipulate\nusing the diagrammatic approach, and while they possess remarkable and\nintricate symmetry properties, they may always be reduced to products\nover convenient sums of $6j$ and $9j$ symbols using the JLV\ntheorems~\\cite{BC,MB}.  One other fine point is that the attractive\nsquare diagram in Fig.~\\ref{fig:6j_12j} actually differs from the true\n$12j$ symbol by a phase $(-1)^{j_1 - j_2 + j^\\prime_4 - j^\\prime_5}$,\nbut this distinction is purely formal; like Eq.~(\\ref{newident}), the\ndiagram as depicted is symmetric upon exchange of primed and unprimed\nquantum numbers.\n\nTo introduce the quantum number $\\ell$, we make another four-line cut,\nhere on $j_6$, $j^\\prime_6$, $k$, and $\\Delta$ in the $12j$\nsymbol of Fig.~\\ref{fig:6j_12j}.  The result of this action is shown\nin Fig.~\\ref{fig:6j_two_9j}.  The hexagonal figures are none other\nthan standard $9j$ symbols in canonical JLV form.\n\n\\begin{figure}[ht]\n\\epsfxsize 6.45 in \\epsfbox{6j_two_9j.eps}\n\\caption{The result of applying JLV4 to Fig.~\\ref{fig:6j_12j},\nwhich is expressed algebraically in Eq.~(\\ref{newident}).}\n\\label{fig:6j_two_9j}\n\\end{figure}\nIn fact, we have expressed a particular product of three $6j$ symbols\n[Eq.~(\\ref{three6js})] as a $6j$ and two $9j$ symbols summed over two\nnew quantum numbers, $k$ and $\\ell$.  To repeat Eq.~(\\ref{newident}),\n\\begin{eqnarray}\n\\lefteqn{\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}\n\\sixj{j_3}{j_4}{j_5}{\\Delta}{j^\\prime_5}{j^\\prime_4}\n\\sixj{j^\\prime_1}{j^\\prime_2}{j_3}{j^\\prime_4}{j^\\prime_5}{j^\\prime_6}\n}\n&& \\nonumber \\\\\n&=& \\sum_{k,\\ell} [k][\\ell] (-1)^\\Phi \n\\ninej{j_1}{j^\\prime_1}{k}{j_6}{j^\\prime_6}{\\ell}{j_5}{j^\\prime_5}\n{\\Delta}\n\\ninej{j_2}{j^\\prime_2}{k}{j_6}{j^\\prime_6}{\\ell}{j_4}{j^\\prime_4}\n{\\Delta}\n\\sixj{j_1}{j_2}{j_3}{j^\\prime_2}{j^\\prime_1}{k}\n\\, ,\n\\end{eqnarray}\nwith the phase given by\n\\begin{equation}\n\\Phi = j_1 + j_2 + j_6 - j^\\prime_4 - j^\\prime_5 - j^\\prime_6 + k -\n\\Delta \\ .\n\\end{equation}\nWhile $\\Phi$ is not primed-unprimed symmetric, neither are the two new\n$9j$ symbols, because switching their initial and final quantum\nnumbers requires column exchanges; when the necessary permutations are\ntaken into account, it is quite straightforward to show this explicit\nsymmetry.  To the best of our knowledge, Eq.~(\\ref{newident}) actually\nrepresents a new SU(2) identity~\\cite{why_new}, one that reduces for\n$\\Delta \\! = \\! 0$ to the BE sum rule.  This reduction becomes\napparent when one simplifies using special cases\n\\begin{equation}\n\\sixj{j_3}{j_4}{j_5}{0}{j^\\prime_5}{j^\\prime_4} =\n\\frac{(-1)^{j_3 + j_4 + j_5}}{\\sqrt{[j_4][j_5]}}\n\\delta_{j^{\\vphantom\\dagger}_4 j^\\prime_4}\n\\delta_{j^{\\vphantom\\dagger}_5 j^\\prime_5} ,\n\\end{equation}\n\\begin{equation}\n\\ninej{j_1}{j^\\prime_1}{k}{j_6}{j^\\prime_6}{\\ell}{j_5}{j^\\prime_5}{0}\n= \\frac{(-1)^{j^\\prime_1 + j_6 + j_5 + k}}{\\sqrt{[k][j_5]}}\n\\delta_{j^{\\vphantom\\dagger}_5 j^\\prime_5}\n\\delta_{k^{\\vphantom\\dagger}_{\\vphantom\\bullet}\n\\ell^{\\vphantom\\dagger}_{\\vphantom\\bullet}}\n\\sixj{j_1}{j_6}{j_5}{j^\\prime_6}{j^\\prime_1}{k} ,\n\\end{equation}\nand\n\\begin{equation}\n\\ninej{j_2}{j^\\prime_2}{k}{j_6}{j^\\prime_6}{\\ell}{j_4}{j^\\prime_4}{0}\n= \\frac{(-1)^{j^\\prime_2 + j_6 + j_4 + k}}{\\sqrt{[k][j_4]}}\n\\delta_{j^{\\vphantom\\dagger}_4 j^\\prime_4}\n\\delta_{k^{\\vphantom\\dagger}_{\\vphantom\\bullet}\n\\ell^{\\vphantom\\dagger}_{\\vphantom\\bullet}}\n\\sixj{j_2}{j_6}{j_4}{j^\\prime_6}{j^\\prime_2}{k} ,\n\\end{equation}\nin which case both $k$ and $\\ell$ reduce to ${\\cal J}$ of\nEq.~(\\ref{BE}).\n\n\\section{Algebraic Proof} \\label{algebraic}\n\nEquation~(\\ref{newident}) is also fairly straightforward to verify\nalgebraically, once the right-hand side is known.  Here we reduce\nthis side of the equation.  We use the symmetry properties of $6j$\nsymbols and $9j$ symbols: $6j$ symbols are invariant under the\npermutation of any two columns or under the exchange of upper and\nlower entries for any two columns, while $9j$ symbols are invariant\nunder even permutations of any two rows or columns.  We also use the\nBE sum rule, Eq.~(\\ref{BE}).  Furthermore, $9j$ symbols may be\nexpanded in terms of $6j$ symbols using the standard\nidentity~\\cite{Edmonds}\n\\begin{equation} \\label{9jexpand}\n\\ninej{j_6}{j^\\prime_6}{\\ell}{j_4}{j^\\prime_4}{\\Delta}{j_2}\n{j^\\prime_2}{k}\n= \\sum_x (-1)^{2x} [x] \\sixj{j_6}{j_4}{j_2}{j^\\prime_2}{k}{x}\n\\sixj{j^\\prime_6}{j^\\prime_4}{j^\\prime_2}{j_4}{x}{\\Delta}\n\\sixj{\\ell}{\\Delta}{k}{x}{j_6}{j^\\prime_6} ,\n\\end{equation}\nwhere the $9j$ symbol is equivalent to the second one in\nEq.~(\\ref{newident}).  The first $9j$ symbol of Eq.~(\\ref{newident})\nand the last $6j$ symbol of Eq.~(\\ref{9jexpand}) (arguments\nrearranged), along with their sum over $\\ell$, may be re-expressed\nusing another standard identity~\\cite{Edmonds}:\n\\begin{equation} \\label{Edmonds6.4.8}\n\\sum_{\\ell} [\\ell]\n\\ninej{j_6}{j^\\prime_6}{\\ell}{j_5}{j^\\prime_5}{\\Delta}{j_1}\n{j^\\prime_1}{k} \\sixj{j_6}{j^\\prime_6}{\\ell}{\\Delta}{k}{x} = (-1)^{2x}\n\\sixj{j_5}{j^\\prime_5}{\\Delta}{j^\\prime_6}{x}{j^\\prime_1}\n\\sixj{j_1}{j^\\prime_1}{k}{x}{j_6}{j_5} .\n\\end{equation}\nThe phase $(-1)^{2x}$ cancels between Eqs.~(\\ref{9jexpand}) and\n(\\ref{Edmonds6.4.8}), and the remaining expression reads\n\\begin{equation} \\label{preBE}\n\\sum_x [x] \\sum_k [k] (-1)^\\Phi \n\\sixj{j^\\prime_6}{j^\\prime_4}{j^\\prime_2}{j_4}{x}{\\Delta}\n\\sixj{j_5}{j^\\prime_5}{\\Delta}{j^\\prime_6}{x}{j^\\prime_1}\n\\sixj{j_6}{j_4}{j_2}{j^\\prime_2}{k}{x}\n\\sixj{j_1}{j^\\prime_1}{k}{x}{j_6}{j_5}\n\\sixj{j_1}{j_2}{j_3}{j^\\prime_2}{j^\\prime_1}{k} .\n\\end{equation}\n\nThe summed angular momentum $k$ appears only in the last three of\nthese $6j$ symbols, and the phase argument $\\Phi$\n[Eq.~(\\ref{phasearg})] conveniently contains a factor of $k$,\nsuggesting that they can be simplified using the BE sum rule.  Indeed,\nwriting the sum of the ten angular momenta in the last three $6j$\nsymbols as $\\sigma$, one has\n\\begin{equation} \\label{firstBE}\n\\sum_k (-1)^\\sigma [k]\n\\sixj{j_2}{j_6}{j_4}{x}{j^\\prime_2}{k}\n\\sixj{j_6}{j_5}{j_1}{j^\\prime_1}{k}{x}\n\\sixj{j_2}{j_1}{j_3}{j^\\prime_1}{j^\\prime_2}{k} =\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}\n\\sixj{j_4}{j_5}{j_3}{j^\\prime_1}{j^\\prime_2}{x} .\n\\end{equation}\nThe first $6j$ symbol appears on the left-hand side of\nEq.~(\\ref{newident}).  The remaining factors to be simplified (those\ncontaining $x$) are the first two $6j$ symbols in Eq.~(\\ref{preBE}),\nthe second in Eq.~(\\ref{firstBE}), and the phase $(-1)^{\\Phi-\\sigma}\n\\! = \\!  (-1)^{\\sigma - \\Phi}$ (since $\\Phi$ and $\\sigma$ are integers)\n$\\equiv \\! (-1)^{\\tilde\\sigma}$.  The BE sum rule again simplifies the\nexpression:\n\\begin{equation} \\label{secondBE}\n\\sum_x (-1)^{\\tilde\\sigma} [x]\n\\sixj{j_4}{\\Delta}{j^\\prime_4}{j^\\prime_6}{j^\\prime_2}{x}\n\\sixj{\\Delta}{j^\\prime_5}{j_5}{j^\\prime_1}{x}{j^\\prime_6}\n\\sixj{j_4}{j_5}{j_3}{j^\\prime_1}{j^\\prime_2}{x} =\n\\sixj{j_3}{j_4}{j_5}{\\Delta}{j^\\prime_5}{j^\\prime_4}\n\\sixj{j^\\prime_1}{j^\\prime_2}{j_3}{j^\\prime_4}{j^\\prime_5}{j^\\prime_6}\n,\n\\end{equation} \nsince $\\tilde\\sigma$ is easily shown to be the sum of the ten angular\nmomenta on the left-hand side.  These two $6j$ symbols, along with the\nfirst from Eq.~(\\ref{firstBE}), complete the left-hand side of\nEq.~(\\ref{newident}), and hence complete the proof.\n\n\\section*{Acknowledgments}\nThis work was supported by the NSF under Grant No.\\ PHY-0456520.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\uppercase{Introduction}}\\label{sec:introduction}\nIn last decade, Government of India has taken several e-Governance initiatives such as a unique digital identity (referred to as \\textit{Aadhaar} \\cite{aadhaar}) for every resident, online Aadhaar based authentication and several online citizen centric services such as \\textit{eKYC}, \\textit{eSign}, and \\textit{DigiLocker}. At present, most of these services are built using traditional Public Key Infrastructure (PKI) with limited data privacy in which specifying authorized entities beforehand which are permitted to access data may not be possible and even if possible, the solution may not scale.\n\nIn \\textit{DigiLocker} \\cite{digilocker}, documents of subscribers are hosted on public cloud which is assumed to be a trusted entity. However, cloud storage may not be trustworthy and may be susceptible to insider attacks. Moreover, instead of providing a \\textit{reactive access authorization to a single requester} (using OAuth2 \\cite{oauth2}), a\n subscriber may want to provide a \\textit{proactive access authorization to multiple requester} meeting certain criteria of attributes.\n\nCiphertext-Policy Attribute-Based Encryption (CP-ABE) \\cite{bethencourt2007ciphertext} is a recent cryptographic mechanism which can improve data privacy, but the right implementation and efficiency of it are still some of the major concerns for its wide deployment. This paper presents a scheme to implement privacy enhanced DigiLocker based on CP-ABE.\n\n\\section{\\uppercase{Related Work}}\n\\noindent Recent developments in cryptography have introduced Attribute-Based Encryption (ABE) \\cite{goyal2006attribute} in which encryption is done under a set of attributes. ABE is classified in Key-Policy ABE (KP-ABE) \\cite{goyal2006attribute} and CP-ABE. In KP-ABE, access policy is encoded in subscriber's private key and a set of attributes are encoded in ciphertext. In CP-ABE, access policy is encoded in ciphertext and a set of attributes are encoded in subscriber's private key. In CP-ABE, only if the set of required attributes encoded in receiver's private key satisfies the access policy encoded in received ciphertext, will the receiver be able to decrypt the ciphertext. Since the introduction of CP-ABE, researchers have proposed innovative mechanisms to use it to improve data privacy \\cite{zhou2012efficient}, \\cite{ji2014privacy}.\n\n\\section {\\uppercase{Digital Lockers in India}}\n\\noindent \\textit{DigiLocker} is an Aadhaar based online service which facilitates \\textit{subscribers} to store e-documents, \\textit{issuer} agencies to provide e-documents and \\textit{requester} applications to get access to e-documents. An \\textit{e-document} is a digitally signed electronic document. \\textit{Repositories} are provided by issuers to host collection of e-documents. \\textit{Digital Locker} is a storage space provided to each subscriber to store e-documents. \\textit{Requester} is an application which seeks access to some e-document. All participating entities must adhere to \\textit{Digital Locker Technology Specification (DLTS)} \\cite{dlts}. \n\nAn e-document is uniquely identified by a \\textit{Unique Resource Identifier (URI)} which is a triplet of the form $\\mathrm{\\langle{IssuerID::DocType::DocID}\\rangle}$, where $IssuerID$ is a unique identifier ot the issuer, e.g., $\\mathtt{CBSE}$, for Central Board of Secondary Education. $DocType$ is a classification of e-documents as defined by the issuer. For example, CBSE may classify certificates into $\\mathtt{MSTN}$ for 10th mark sheet and $\\mathtt{KVYP}$ for certificates issued to KVPY scholarship fellow. $DocType$ also helps issuers to use different repositories for different types of e-documents. $DocID$ is an issuer defined unique identifier (an alphanumeric string) of the e-document within a document type. Some hypothetical examples of e-document URI are $\\langle{\\mathtt{CBSE::MSTN::22636726}}\\rangle$, $\\langle{\\mathtt{DLSSB::HSMS::GJSGEJXS}\\rangle}$. DigiLocker ensures data integrity of e-documents by mandating that all e-documents are digitally signed by issuers.\n\nWhen an issuer is registered, it provides two APIs, namely, $\\mathtt{PullDoc}$ to pull an e-document based on a given URI and $\\mathtt{PullUri}$ to pull all URIs meeting a given search criteria. When a requester application is registered, it is given a unique requester identifier, a secret key which is shared between DigiLocker and requester application and a $\\mathtt{FetchDoc}$ API is given to access e-documents based on URI. Based on the URI, $\\mathtt{FetchDoc}$ forwards the request to appropriate issuers to retrieve the e-document. DigiLocker ensures secure data access of e-documents by API license keys, secure transport, an explicit authentication (if required by \\textit{DocType}) and all requests and responses to be digitally signed.\n\n\\section{\\uppercase{Preliminaries}}\n\\noindent This section briefly describes some of the necessary background.\n\n\\subsection{Bilinear pairings \\cite{zhang2004efficient}}\nLet $\\mathbb{G}_1$ and $\\mathbb{G}_2$ are elliptic groups of order $\\mathrm{p}$, $\\mathbb{G}_{\\mathrm{T}}$ is a multiplicative group of order $\\mathrm{p}$, $\\mathrm{g_1}$ is a generator of $\\mathbb{G}_{1}$, $\\mathrm{g_2}$ is a generator of $\\mathbb{G}_2$, $\\mathrm{P} \\in \\mathbb{G}_1$, $\\mathrm{Q}\\in \\mathbb{G}_2$ and $\\mathrm{a}$, $\\mathrm{b}$ $\\in \\mathbb{Z}_p$, then a bilinear pairing is a map $\\mathrm{e}: \\mathbb{G}_1 \\times \\mathbb{G}_2 \\rightarrow \\mathbb{G}_{\\mathrm{T}}$ that satisfies the following three properties.\n\\begin{enumerate}\n\t\\item [1] Bilinearity: $\\mathrm{e(P^a, Q^b) = e(P, Q)^{ab}}$\n\t\\item [2] Non-Degeneracy: $\\mathrm{e(g_1, g_2) \\neq 1}$\n\t\\item [3] Computability: $\\mathrm{e(P, Q)}$ can be computed efficiently.\n\\end{enumerate}\n\n\\subsection {Decision Bilinear Diffie-Hellman (DBDH) assumption \\cite{yacobi2002note}}\nLet $\\mathbb{G}$, $\\mathbb{G}_T$ are cyclic groups of prime order $\\mathrm{p} > 2^{\\lambda}$ where $\\lambda \\in \\mathbb{N}$, $\\mathrm{g}$ is the generator of $\\mathbb{G}$, $\\mathrm{e}: \\mathbb{G} \\times \\mathbb{G} \\rightarrow \\mathbb{G}_{\\mathrm{T}}$ is an efficiently computable symmetric bilinear pairing map and $\\mathrm{a, b, c, z} \\in \\mathbb{Z}_{\\mathrm{p}}$ are random numbers. The DBDH assumption states that no probabilistic polynomial time algorithm can distinguish between $\\mathrm{\\langle{{g},{g^a},{g^b},{g^c},{e(g,g)^{abc}}}\\rangle}$ and $\\mathrm{\\langle{{g},{g^a},{g^b},{g^c},{e(g,g)^{z}}}\\rangle}$ with more than a negligible advantage.\n\n\\subsection {Security Model}\nThe security model of proposed scheme is based on the following IND-sAtt-CPA game \\cite{ibraimi2009efficient} between a challenger and an adversary $\\mathcal{A}$. \n\\begin{enumerate}[leftmargin=0pt]\n\t\\item []\\textit{Init Phase}: Adversary $\\mathcal{A}$ chooses a challenge access tree $\\mathcal{T}^*$ and gives it to challenger.\n\t\n\t\\item []\\textit{Setup Phase}: Challenger runs a \\textit{setup} procedure to generate $\\mathrm{\\langle{ASK, APK}\\rangle}$ and gives the public key $\\mathrm{APK}$ to adversary $\\mathcal{A}$.\n\t\n\t\\item []\\textit{Phase I}: Adversary $\\mathcal{A}$ makes an attribute- based private key request to the key generation oracle for any attribute set with the restriction that the attribute set should not include any attribute which is part of $\\mathcal{T}^*$. Challenger generates the key as described in section \\ref{key_generation} and returns the same to adversary $\\mathcal{A}$.\n\t\n\t\\item []\\textit{Challenge Phase}: Adversary $\\mathcal{A}$ sends two equal length messages $\\mathrm{m_0}$ and $\\mathrm{m_1}$ to challenger. Challenger chooses a random number $\\mathrm{b} \\in_{\\mathrm{R}} \\{0, 1\\}$, encrypts $\\mathrm{m_b}$ using $\\mathcal{T}^*$ and $\\mathrm{APK}$ as is described in section \\ref{encryption}.\n\t\n\t\\item []\\textit{Phase II}: Adversary $\\mathcal{A}$ can send multiple requests to generate attribute-based private key with the same restriction as in Phase I.\n\t\n\t\\item []\\textit{Guess Phase}: Adversary $\\mathcal{A}$ outputs a guess $\\mathrm{b\\prime} \\in \\{0, 1\\}$.\n\\end{enumerate}\nThe advantage of adversary $\\mathcal{A}$ in this game is defined to be $\\epsilon = \\vert {\\mathrm{Pr[b\\prime = b]} - \\frac{1}{2}} \\vert$. Only if any polynomial time adversary $\\mathcal{A}$ has a negligible advantage, the scheme is considered secure against an adaptive chosen plaintext attack (CPA).\n\n\\subsection {Access tree} \\label{access_tree}\nAccess tree structure is a means to specify an access policy during encryption that must be satisfied by attribute-based private keys in order to decrypt. Let $\\mathcal{T}$ be a tree representing an access structure. Each non-leaf node of the tree represents a threshold gate, described by its children and a threshold value. If $\\mathrm{num_x}$ is the number of children and $\\mathrm{k_x}$ is the threshold value of a node $\\mathrm{x}$, then, $\\mathrm{k_x=1}$ represents an OR gate and $\\mathrm{k_x=num_x}$ represents an AND gate. Each leaf node $\\mathrm{x}$ of the tree is described by an attribute and a threshold value $\\mathrm{k_x = 1}$. \n\nLet $\\mathcal{T}_x$ denotes the subtree rooted at node $\\mathrm{x}$. If a set of attributes $\\lambda$ satisfies the subtree $\\mathcal{T}_x$, it is represented as $\\mathcal{T}_x(\\lambda) = 1$. $\\mathcal{T}_x(\\lambda)$ is computed recursively as follows. If $\\mathrm{x}$ is a non-leaf node, evaulate $\\mathcal{T}(y)$ for all children nodes $\\mathrm{y}$ of node $\\mathrm{x}$. $\\mathcal{T}_x(\\lambda)$ returns $1$ if and only if at least $\\mathrm{k_x}$ children return 1. If $\\mathrm{x}$ is a leaf node, then $\\mathcal{T}_x(\\lambda)$ returns $1$ if and only if $\\mathtt{attr(x)} \\in \\lambda$.\n\n\\section {\\uppercase{Our Construction}}\n\n\\noindent The proposed scheme introduces two new roles, namely, \\textit{Attribute Authority Manager (AAM)} and \\textit{Attribute Authority (AA)}. AAM is an entity which manages the universe of attributes and AA is an entity which manages a set of attributes (as assigned by AAM). DigiLocker is proposed to assume the role of AAM and individual issuers are proposed to assume the role of AA. A subscriber is assigned a set of attributes from each issuer which holds at least one e-document of the subscriber. Each requester application is assigned a set of attributes from DigiLocker based on certain criteria such as purpose of access, for how long the data is going to be used, etc. To create a privacy enhanced e-document for a subscriber, issuer and subscriber mutually creates an attribute-based token (which will be used later in encryption) for an access policy, generates a symmetric key, encrypts the document with symmetric key, encrypts the symmetric key with attribute-based token, creates an e-document enclosing both the encrypted symmetric key and the encrypted document, creates a URI for this e-document and pushes it to subscriber's digital locker using $\\mathtt{PushURI}$ API. When this e-document is shared with a requester application, the requester will be able to decrypt the encrypted symmetric key only if the requester is associated with a set of attributes which satisfies the access policy used to encrypt the symmetric key. Only when the requester obtains the symmetric key, will he be able to decrypt and retrieve the document.\n\nIn \\textit{Setup($\\kappa$)} procedure, AAM chose a cyclic group ${\\mathbb{G}_0}$ of large prime order $\\mathrm{p}$ ($\\kappa$ defines the size of group) on which discrete logarithm problem is assumed to be hard, generator $\\mathrm{g}$, a bilinear map $\\mathrm{e}:{\\mathbb{G}_0}\\times {\\mathbb{G}_0}\\rightarrow {\\mathbb{G}_1}$ for which bilinear diffie hellman problem is assumed to be hard, a hash function $\\mathrm{H}: \\{0, 1\\}^{*} \\rightarrow \\mathbb{G}_0$ which maps a binary string encoded attribute to a group element, chose random numbers $\\alpha, \\beta \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$ and set its private key $\\mathrm{ASK}$ and public key $\\mathrm{APK}$ as below.\n\\begin{center}\n\t\\begin{tabular}{ l c l }\n\t\t$\\mathrm{ASK}$ & = & $\\{\\beta, \\mathrm{g}^{\\alpha} \\}$ \\\\ \n\t\t$\\mathrm{APK}$ & = & $\\{\\mathrm{g}^{\\beta}, \\mathrm{e(g, g)}^{\\alpha}, \\mathbb{G}_0, \\mathrm{g}\\}$ \\\\  \n\t\\end{tabular}\n\\end{center}\n\\subsection {Attribute Assignment}{\\label{attribute_assignment}}\nAn attribute can be any characteristic of a subscriber or requester and is represented by a binary string $\\{0,1\\}^*$. Attribute assignment to both subscribers and requesters is proposed to be done lazily in the background with the aim to keep the list of associated attributes in DigiLocker up to date.\n\nFor subscriber's attribute assignment and modification, two APIs are proposed to be introduced. First is  $\\mathtt{PullAttrs(ID_i)}$ which is provided by issuers and is consumed by DigiLocker to pull updated list of attributes of subscriber with Aadhaar number $\\mathrm{ID_i}$. Second is $\\mathtt{PushAttrs(ID_i, NewAttrs)}$ which is provided by DigiLocker and is consumed by issuer to push any change in attributes of subscriber with Aadhaar number $\\mathrm{ID_i}$. For requester applications, attributes are assigned and updated by DigiLocker.\n\nIt is important to take appropriate measures to handle load of a voluminous country like India. One such measure could be to prepone part of the encryption process. This preponed encryption process generates a token with mutual cooperation between subscriber and issuer. This token can be reused every time for a given subscriber and for a given access policy. \n\nA helper procedure $\\mathtt{encPartial(\\mathcal{T}, r)}$ is assumed to be present which works as follows. It chooses a polynomial $\\mathrm{q_x}$ for each node $\\mathrm{x}$ (including the leaves) in the tree $\\mathcal{T}$. These polynomials are chosen in the following way in a top-down manner, starting from the root node $\\mathrm{R}$. For each node $\\mathrm{x}$ in the tree, set the degree $\\mathrm{d_x}$ of the polynomial $\\mathrm{q_x}$ to be one less than the threshold value $\\mathrm{k_x}$ of that node, that is, $\\mathrm{d_x = k_x - 1}$. Starting with the root node $\\mathrm{R}$ the procedure chooses a random $\\mathrm{r}\\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$ and sets $\\mathrm{q_r(0)=r}$. Then, it chooses $\\mathrm{d_R}$ other points of the polynomial $\\mathrm{q_R}$ randomly to define it completely. For any other node $\\mathrm{x}$, it sets $\\mathrm{q_x(0)} = \\mathrm{q_{parent(x)}(index(x))}$ and chooses $\\mathrm{d_x}$ other points randomly to completely define $\\mathrm{q_x}$.\n\n\\subsection {Token Generation}{\\label{token_generation}}\n\\begin{center}\n\\begin{figure}\n\t\\caption{Example of an access policy tree}\n\t\\label{fig:AP}\n\t\\resizebox {0.9\\columnwidth}{!} {\n\t\t\\begin{tikzpicture} [level\tdistance = 50pt,\n\t\tsibling distance = 20pt, \n\t\tedge from parent\/.style = {\n\t\t\tdraw, \n\t\t\tedge from parent path = {(\\tikzparentnode) -- (\\tikzchildnode.north)}\n\t\t}\n\t\t]\n\t\t\\tikzset{every internal node\/.style\t= {draw, circle, black, font = \\Large}}\n\t\t\\tikzset{every leaf node\/.style\t\t= {draw, black, regular polygon, regular polygon sides = 3, inner sep = 1pt}}\n\t\t\n\t\t\\Tree [.\\node(1)[label={[label distance=0.25cm]10:\\LARGE $R$}]{$\\lor$};\n\t\t[.\\node(2)[label={[label distance=0.25cm]10:\\LARGE ${R_S}_{iv}$}]{$\\wedge$};\n\t\t\\node(3){${A_R}_1$};\n\t\t\\node(4){${A_R}_n$};\n\t\t] \n\t\t[.\\node(5)[label={[label distance=0.25cm]10:\\LARGE ${R_I}_{iv}$}]{$\\wedge$};\n\t\t\\node(6){${A_R}_1$};\n\t\t\\node(7){${A_R}_n$};\n\t\t] \n\t\t]\n\t\t\n\t\t\\node[draw, dashed, inner xsep=3mm, inner ysep=6mm, fit=(2)(3)(4)](r1){};\n\t\t\\node[draw, dashed, inner xsep=3mm, inner ysep=6mm, fit=(5)(6)(7)](r2){};\n\t\t\\node[draw, dashed, inner xsep=7mm, inner ysep=9mm, fit=(3)(1)(7)](r3){};\n\t\t\n\t\t\\node [above, inner sep=5pt] at (r1.south) {\\large ${\\mathcal{T}_S}_{iv}$};\n\t\t\\node [above, inner sep=5pt] at (r2.south) {\\large ${\\mathcal{T}_I}_{iv}$};\n\t\t\\node [above, inner sep=5pt] at (r3.south) {\\large ${\\mathcal{T}_{iv}}$};\n\t\n\t\t\n\t\t\n\t\t\\end{tikzpicture}\n\t}\n\\end{figure}\n\\end{center}\n\nAn access tree $\\mathcal{T}_{iv}$ is comprised of access subtree $\\mathcal{T}_{S_{iv}}$ from subscriber $\\mathrm{S_i}$ and access subtree $\\mathcal{T}_{I_{iv}}$ from issuer $\\mathrm{I_v}$ (refer figure \\ref{fig:AP}). If issuer $\\mathrm{I_v}$ needs to generate its part of token for subscriber $\\mathrm{S_i}$, for access tree $\\mathcal{T}_{iv}$, it generates a random number $\\mathrm{r_i} \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$, and generates following partial-token using $\\mathrm{APK}$ and $\\mathtt{encPartial(\\mathcal{T}_{I_{iv}}, r_i)}$. Let $\\mathrm{Y_I}$ is the set of leaf nodes in $\\mathcal{T}_{I_{iv}}$.\n\n$\\mathrm{CTtok_{I_{iv}}} =\t\\begin{cases}\n\t\\quad \\mathcal{T}_{I_{iv}}\\\\\n\t\\quad \\mathrm{C1_I} = \\mathrm{e(g, g)}^{\\alpha \\mathrm{r_i}}\\\\\n\t\\quad \\mathrm{C2_I} = \\mathrm{g}^{\\beta \\mathrm{r_i}}\\\\\n\t\\begin{rcases}\n\t\t\\quad \\mathrm{C3_{I_y} = g^{q_y(0)}}\\\\\n\t\t\\quad \\mathrm{C4_{I_y} = H(\\mathtt{attr(y)})^{q_y(0)}}\\\\\n\t\\end{rcases} {\\!\\scriptscriptstyle\\mathrm{\\forall\\ y \\in Y_I}}\\\\\n\\end{cases}$\nIssuer notifies subscriber to provide its part of the token. Subscriber $\\mathrm{S_i}$ generates a random number $\\mathrm{r_s} \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$ and generates following partial-token using $\\mathrm{APK}$ and $\\mathtt{encPartial(\\mathcal{T}_{S_{iv}})}$. Let $\\mathrm{Y_S}$ is the set of leaf nodes in $\\mathcal{T}_{S_{iv}}$.\n\n$\\mathrm{CTtok_{S_{iv}}} = \\begin{cases}\n\t\\quad \\mathcal{T}_{S_{iv}}\\\\\n\t\\quad \\mathrm{C1_S} = \\mathrm{e(g, g)}^{\\alpha \\mathrm{r_s}}\\\\\n\t\\quad \\mathrm{C2_S} = \\mathrm{g}^{\\beta \\mathrm{r_s}}\\\\\n\t\\begin{rcases}\n\t\t\\quad \\mathrm{C3_{S_y} = g^{q_y(0)}}\\\\\n\t\t\\quad \\mathrm{C4_{S_y} = H(\\mathtt{attr(y)})^{q_y(0)}}\\\\\n\t\\end{rcases} {\\!\\scriptscriptstyle\\mathrm{\\forall\\ y \\in Y_S}}\\\\\n\\end{cases}$\n\nSubscriber provides its part of partial-token to issuer. Issuer creates the final token by combining the two partial-tokens and keeps it securely with it.\n\n$\\mathrm{CTtok_{iv}} = \\begin{cases}\n\t\\quad \\mathcal{T}_{iv} = \\mathcal{T}_{S_{iv}} \\cup \\mathcal{T}_{I_{iv}}\\\\\n\t\\quad \\mathrm{C1} = \\mathrm{C1_S}.\\mathrm{C1_I}\\\\\n\t\\qquad\\ \\ = \\mathrm{e(g, g)}^{\\alpha \\mathrm{r_s}}\\mathrm{e(g, g)}^{\\alpha \\mathrm{r_i}}\\\\\n\t\\quad \\mathrm{C2} = \\mathrm{C2_S}.\\mathrm{C2_I} = \\mathrm{g}^{\\beta \\mathrm{r_s}}\\mathrm{g}^{\\beta \\mathrm{r_i}}\\\\\n\t\\begin{rcases}\n\t\t\\quad \\mathrm{C3 = C3_{S_y} \\cup C3_{I_y}}\\\\\n\t\t\\qquad\\ \\  = \\mathrm{g^{q_y(0)}}\\\\\n\t\t\\quad \\mathrm{C4 = C4_{S_y} \\cup C4_{I_y}}\\\\\n\t\t\\qquad\\ \\ = \\mathrm{H(\\mathtt{attr(y)})^{q_y(0)}}\\\\\n\t\\end{rcases} {\\!\\scriptscriptstyle\\forall\\ y \\in Y_S \\cup Y_I}\\\\\n\\end{cases}$\n\n\\subsection{Encryption} {\\label {encryption}}\nA new $DocType$ $\\mathtt{PRIV}$ is proposed to be introduced for privacy enhanced e-documents. To create a privacy enhanced e-document, issuer creates a URI $\\mathrm{\\langle{I_v::PRIV::D_w}\\rangle}$ where $\\mathrm{I_v}$ is the issuer identifier and $\\mathrm{D_w}$ is the document identifier within the document type $\\mathtt{PRIV}$. Now, issuer generates a random number $\\mathrm{r_{ie}} \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$, generates a symmetric key $\\mathrm{SK_{ivw}}$, encrypts e-document $\\mathrm{m}$ with $\\mathrm{SK_{ivw}}$, encrypts $\\mathrm{SK_{ivw}}$ with $\\mathrm{CTtok_{I_{iv}}}$ and produces the following ciphertext.\n\n$\\mathrm{CT_{ivw}} = \\begin{cases}\n\t\\quad \\mathcal{T}_{iv} = \\mathcal{T}_{S_{iv}} \\cup \\mathcal{T}_{I_{iv}}\\\\\n\t\\quad \\mathrm{C1} = \\mathrm{e(g, g)}^{\\alpha \\mathrm{r_s r_{ie}}}\\mathrm{e(g, g)}^{\\alpha \\mathrm{r_i r_{ie}}}\\mathrm{SK_{ivw}}\\\\\n\t\\quad \\mathrm{C2} = \\mathrm{g}^{\\beta \\mathrm{r_s r_{ie}}}\\mathrm{g}^{\\beta \\mathrm{r_i r_{ie}}}\\\\\n\t\\begin{rcases}\n\t\t\\quad \\mathrm{C3_{y} = g^{r_{ie}(q_y(0))}}\\\\\n\t\t\\quad \\mathrm{C4_{y} = H(\\mathtt{attr(y)})^{q_y(0)}}\\\\\n\t\\end{rcases} {\\!\\scriptscriptstyle\\mathrm{\\forall\\ y \\in Y_{iv}}}\\\\\n\t\\quad \\mathrm{C5 = \\{m\\}_{SK_{ivw}}}\\\\\n\\end{cases}$\n\n\\subsection {Key Generation} {\\label{key_generation}\nA new API $\\mathtt{GenABPvtKey(ID_i, IS_j)}$ is proposed to be provided by DigiLocker to generate an attribute-based private key for a subscriber with Aadhaar identifier $\\mathrm{ID_i}$ and with attributes from issuers in the set $\\mathrm{IS_j}$. Let $\\mathrm{S_{ij}}$ is the set of all attributes assigned to $\\mathrm{S_i}$ by all issuers in set $\\mathrm{IS_j}$. DigiLocker generates random numbers $\\mathrm{r} \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$, $\\mathrm{r_j} \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$ for each attribute $\\mathrm{j}\\in_{\\mathrm{R}} \\mathrm{S_{ij}}$, computes attribute-based private key $\\mathrm{ASK_{ID_{i}IS_{j}}}$ as below and keeps this key securely with it.\n\n\n\t$\\mathrm{ASK_{{ID_i}{IS_j}}} = \\begin{cases}\n\t\t\\quad \\mathrm{D} = \\mathrm{g}^{(\\alpha+\\mathrm{r})\/\\beta}\\\\\n\t\t\\begin{rcases}\n\t\t\t\\quad \\mathrm{D_j} = \\mathrm{g^{r}.H(j)^{r_j}}\\\\\n\t\t\t\\quad \\mathrm{D_j\\prime} = \\mathrm{g^{r_j}}\\\\\n\t\t\\end{rcases} {\\!\\scriptscriptstyle\\mathrm{\\forall\\ j \\in S_{ij}}}\\\\\n\t\\end{cases}$\n\nNote that multiple attribute based private keys can exist for a subscriber for different set of attributes. If anyone issuer set $\\mathrm{IS_i}$ is a proper subset of another issuer set $\\mathrm{IS_j}$, the key corresponding to $\\mathrm{IS_i}$ is redundant and can be removed.\n\n\t\n\\subsection {Decryption} {\\label{decryption}}\nA new API $\\mathtt{FetchPrivDoc{URI}}$ is proposed to be provided by DigiLocker for decryption purpose. This API facilitates a requester with identifier $\\mathrm{ID_R}$ to retrieve ciphertext $\\mathrm{CT_{ivw}}$ of e-document from URI $\\mathrm{\\langle{I_v::PRIV::D_W}\\rangle}$ of subscriber $\\mathrm{S_i}$. DigiLocker extracts the set of attribute issuers $\\mathrm{IS_k}$ from $\\mathrm{CT_{ivw}}\\rightarrow\\mathcal{T}_{iv}$, retrieves $\\mathrm{ASK_{ID_RIS_k}}$ and calls $\\mathtt{Decrypt(CT_{ivw}, ASK_{ID_RIS_k})}$. A helper procedure $\\mathtt{DecryptNode(CT_{ivw}, ASK_{ID_RIS_K})}$ is defined as below. Let $\\mathrm{S_k}$ is the set of all attributes from issuers in set $\\mathrm{IS_k}$. If $\\mathrm{x}$ is a leaf node and if $\\mathtt{attr(x)} \\notin \\mathrm{S_k}$, then $\\mathtt{DecryptNode(CT_{ivw}, ASK_{{ID_R}{IS_k}}, x)} = \\perp$ else if $\\mathtt{attr(x)}\\in \\mathrm{S_k}$, then the procedure is defined as below.\n\\begin{tabular}{ m{1cm} l }\n\t\\multicolumn{2}{l}{ $\\mathtt{DecryptNode(CT_{ivw}, ASK_{{ID_R}{IS_k}}, x)}$ }\\\\\n\t& = $ \\mathrm{\\frac{e(D_x, C4_y)}{e(D_x\\prime, C5_y)}}$ \\\\ \n\t& = $ \\mathrm{\\frac{e(g^r.H(\\mathtt{attr(x)})^{r_j}, g^{r_{ie}q_y(0)})}{e(g^{r_j}, H(\\mathtt{attr(x)})^{q_x(0)})}}$ \\\\  \n\t& = $ \\mathrm{e(g, g)^{rr_{ie}q_x(0)}}$\n\\end{tabular}\n\nIf $\\mathrm{x}$ is a non-leaf node, the recursive procedure is defined as follows. For all children nodes $\\mathrm{z}$ of $\\mathrm{x}$, $\\mathtt{DecryptNode(CT_{ivw}, ASK_{i}, x)}$ is called and their output is stored in $\\mathrm{F_z}$. Let $\\mathrm{S_x}$ be an arbitrary $\\mathrm{k_x}$ sized set of child nodes $\\mathrm{z}$ such that $\\mathrm{F_z} \\neq \\perp$. If no such set exists then the node was not satisfied and the function returns $\\perp$. Otherwise, $\\mathrm{F_x}$ is computed as below.\n\n\\begin{tabular}{ m{0cm} l }\n\t\\multicolumn{2}{l}{$\\mathrm{F_x}$ = $  \\mathrm{\\displaystyle\\prod_{z\\in S_x} F_{z}^{\\Delta_{i, S_x\\prime}(0)}\\quad {\\tiny where \\{ {i=index(z)} \\atop  {S_x\\prime = \\{index(z): z \\in S_x} \\} }} $ }\\\\\n\t& = $ \\mathrm{\\displaystyle\\prod_{z\\in S_x} F_{z}^{\\Delta_{i, S_x\\prime}(0)}}$ \\\\ \n\t& = $ \\mathrm{\\displaystyle\\prod_{z\\in S_x} (e(g, g)^{r.r_{ie}.q_z(0)})^{\\Delta_{i, S_x\\prime}(0)}}$\\\\  \n\t& = $ \\mathrm{\\displaystyle\\prod_{z\\in S_x} (e(g, g)^{r.r_{ie}.q_{parent(z)}(index(z))})^{\\Delta_{i, S_x\\prime}(0)}}$\\\\\n\t& = $ \\mathrm{\\displaystyle\\prod_{z\\in S_x} e(g, g)^{r.r_{ie}.q_x(i)\\Delta_{i, S_x\\prime}(0)}}$\\\\\n\t& = $ \\mathrm{e(g, g)^{rr_{ie}q_x(0)}\\ (\\tiny{using\\ polynomial\\ interpolation})}$\n\\end{tabular}\n$\\mathtt{Decrypt}(\\mathtt{CT_{ivw}}, \\mathtt{ASK_{{ID_R}{IS_k}}})$ calls $\\mathtt{DecryptNode(}$ $\\mathtt{CT_{ivw}}, \\mathtt{ASK_{{ID_R}{IS_k}}}, \\mathtt{R})$ where $\\mathrm{R}$ is root node of $\\mathrm{{T}_{iv}}$. If the access tree is satisfied by attributes in $\\mathrm{S_k}$, set $\\mathrm{A =}$ $\\mathtt{DecryptNode(CT_{ivw}, ASK_{{ID_R}{IS_k}}, R)} =$ $\\mathrm{e(g, g)}^{\\mathrm{r}.\\mathrm{r_{ie}}.(\\mathrm{r_s}+\\mathrm{r_i})}$. Now the procedure obtains symmetric key $\\mathrm{SK_{ivw}}$ by computing \n\\begin{tabular}{ m{1cm} l }\n\t\\multicolumn{2}{l}{$\\mathrm{\\dfrac{C1}{\\dfrac{e(C2, D)}{A}}}$ = $\\dfrac{\\mathrm{e(g,g)}^{\\alpha \\mathrm{r_s r_{ie}}}\\mathrm{e(g,g)}^{\\alpha \\mathrm{r_i r_{ie}}}\\mathrm{SK_{ivw}}}{\\dfrac{\\mathrm{e}(\\mathrm{g}^{\\beta \\mathrm{r_s r_{ie}}}\\mathrm{g}^{\\beta \\mathrm{r_i r_{ie}}}, \\mathrm{g}^{(\\alpha+\\mathrm{r})\/\\beta})}{\\mathrm{e(g, g)}^{\\mathrm{r r_{ie} (r_s+r_i) )} }}}$}\\\\\n\t& = $\\dfrac{\\mathrm{e(g,g)}^{\\alpha \\mathrm{r_s r_{ie}}}\\mathrm{e(g,g)}^{\\alpha \\mathrm{r_i r_{ie}}}\\mathrm{SK_{ivw}}} {\\dfrac {\\mathrm{e}(\\mathrm{g}^{\\beta \\mathrm{r_{ie} (r_s+r_i)}}, \\mathrm{g}^{(\\alpha+\\mathrm{r})\/\\beta})} {\\mathrm{e(g, g)}^{\\mathrm{r r_{ie} (r_s+r_i))}}}}$ \\\\ \n\t& = $\\dfrac{\\mathrm{e(g,g)}^{\\alpha \\mathrm{r_{ie} (r_s + r_i)}}\\mathrm{SK_{ivw}}} {\\dfrac {\\mathrm{e(g, g)}^{\\mathrm{r_{ie} (r_s+r_i)} (\\alpha+\\mathrm{r})}} {\\mathrm{e(g, g)}^{\\mathrm{r r_{ie} (r_s+r_i))}}}}$\\\\  \n\t& = $\\mathrm{SK_{ivw}}$\\\\\n\\end{tabular}\n\nSymmetric key $\\mathrm{SK_{ivw}}$ is now used to decrypt the encrypted e-document.\n\n\\begin{tabular}{m{0cm} l}\n\t\\multicolumn{2}{l}{$\\mathrm{m = \\{CT_{ivw}\\rightarrow{C5}\\}_{SK_{ivw}}}$}\n\\end{tabular}\n\nDigiLocker returns the decrypted document $\\mathrm{m}$ to requester.\n\n\\section{\\uppercase{Security Analysis}}\n\\noindent If the proposed scheme is not secure than an adversary $\\mathcal{A}$ can win IND-sAtt-CPA game and solve the DBDH assumption with advantage $\\epsilon\/2$. If the DBDG assumption is solved by adversary $\\mathcal{A}$, a simulator $\\beta$ can be built which can solve DBDH assumption with advantage $\\epsilon\/2$. Challenger chose a group $\\mathbb{G}_0$, a generator $\\mathrm{g}$, a bilinear map $\\mathrm{e}$ and chose random numbers $\\mathrm{a, b, c,} \\theta \\in_{\\mathrm{R}} \\mathbb{Z}_p^*$. The challenger selects at random $\\mu \\in_{\\mathrm{R}} {0, 1}$ and sets $\\mathrm{Z}_\\mu$ as below.\n\\begin{align*}\n\t\\mathrm{Z}_\\mu = \\begin{cases}\n\t\\mathrm{(g, g)^{abc}}, &\\mathrm{if}\\ \\mu = 0\\\\\n\t\\mathrm{e(g, g)}^{\\theta}, &\\mathrm{if}\\ \\mu = 1\\\\\n\t\\end{cases}\t\n\\end{align*}\n\nChallenger provides DBDB challenge to the simulator: $\\langle{\\mathrm{g, A, B, C, Z}_{\\mu}}\\rangle$  $\\langle{\\mathrm{g, g^a, g^b, g^c, Z}_{\\mu}}\\rangle$. In IND-sAtt-CPA game, simulator $\\beta$ plays the role of challenger for adversary $\\mathcal{A}$.\n\t\n\\begin{itemize}[leftmargin=*]\n\t\t\\item [] \\textit{Init Phase}: The adversary chose the challenge access tree $\\mathcal{T}^{*}$ and gives it to simulator.\n\t\t\n\t\t\\item [] \\textit{Setup Phase}: The challenger chose a random number $\\mathrm{x\\prime} \\in \\mathbb{Z}_p$, sets $\\alpha = \\mathrm{ab + x\\prime}$ and computes $\\mathrm{y}$ as below.\n\t\t\\[\n\t\t\\mathrm{y = e(g, g)}^{\\alpha} = \\mathrm{e(g, g)^{ab}e(g, g)^{x\\prime}}\n\t\t\\]\n\t\tNow, challenger chose a random numbers $\\mathrm{r \\in_R \\mathbb{Z}_p}$ and $\\mathrm{r_i \\in_R \\mathbb{Z}_p}$ for ($\\mathrm{1 \\le i \\le \\vert{\\mathtt{U}}\\vert}$) and for all $\\mathrm{a_j \\in \\mathbb{U}}$, computes $\\mathrm{d_j}$ and $\\mathrm{d_j\\prime}$ as below.\n\t\t\\begin{align*}\n\t\t\t\\begin{rcases}\n\t\t\t\t\\mathrm{d_j} &= \\begin{cases}\n\t\t\t\t\t\\mathrm{g^{r\/b} H(j)^{r_j}} & \\mathrm{...if\\ a_j} \\notin \\mathcal{T}^*\\\\\n\t\t\t\t\t\\mathrm{g^{r} H(j)^{r_j}} & \\mathrm{...if\\ a_j} \\in \\mathcal{T}^*\\\\\n\t\t\t\t\t\\end{cases}\\\\\n\t\t\t\t\t\\mathrm{d_j\\prime} &= \\mathrm{g^{r_j}}\n\t\t\t\t\\end{rcases} \n\t\t\t\\mathrm{(1 \\le j \\le \\vert{U}\\vert)}\n\t\t\\end{align*}\n\t\tNow, challenger sends public parameters $\\mathrm{APK} = \\{\\mathrm{g}^\\beta, \\mathrm{e(g, g)}^\\alpha, \\mathbb{G}, \\mathrm{g}\\}$ to adversary $\\mathcal{A}$.\n\t\t\n\t\t\\item [] \\textit{Phase 1}: In this phase, adversary $\\mathcal{A}$ sends requests for private key for any set of attributes $w_j$ which does not contain any attribute in $\\mathcal{T}^*$.\n\t\t\\begin{align*}\n\t\t\t\\mathrm{w_j} = \\{\\mathrm{a_j} \\mid (\\mathrm{a_j} \\in \\mathbb{U} \\wedge \\mathrm{a_j} \\notin \\mathcal{T}^*)\\}\n\t\t\\end{align*}\n\t\tFor each query from adversary $\\mathcal{A}$, challenger chose a random number $\\mathrm{r\\prime \\in_R\\mathbb{Z}_p}$, sets $\\mathrm{r= - b(r\\prime + a)}$ and computes $\\mathrm{D}$ as below.\n\t\t\\begin{align*}\n\t\t\t\\mathrm{D} &= \\mathrm{g}^{(\\alpha+\\mathrm{r})\/\\beta} = (\\mathrm{g}^{(\\alpha+\\mathrm{r})})^{1\/\\beta} = (\\mathrm{g}^{((\\mathrm{ab+x\\prime})-\\mathrm{b(r\\prime+a)})})^{1\/\\beta}\\\\\n\t\t\t&= (\\mathrm{g^{x\\prime-br\\prime})}^{1\/\\beta} = (\\mathrm{g^{x\\prime}.(g^{b})^{-r\\prime})}^{1\/\\beta}\n\t\t\\end{align*}\n\t\tBecause of restriction $\\mathrm{a_j} \\notin \\mathcal{T}^*$ in this phase, $\\mathrm{D_j}$ can be computed as below.\n\t\t\\begin{align*}\n\t\t\t\\mathrm{D_j} &= \\mathrm{g^{r\/b}H(j)^{r_j} = g^{r\/b}H(j)^{r_j} = g^{-(r_\\prime+a)}H(j)^{r_j}}\\\\\n\t\t\t    &= \\mathrm{(g^{a})^{-1}g^{-r\\prime}H(j)^{r_j}}\n\t\t\\end{align*}\n\t\tNow, challenger sends private key $\\mathrm{ASK_{w_j}} = \\mathrm{D}, \\mathrm{(D_j, D_j\\prime)}\\mid\\forall \\mathrm{a_j\\in w_j}$ to adversary $\\mathcal{A}$\n\t\t\n\t\t\\item [] \\textit{Challenge Phase}: In this phase, adversary $\\mathcal{A}$ submits two plaintext messages $\\mathrm{m_0}$ and $\\mathrm{m_1}$ to the challenger. Challenger selects a random plaintext message $\\mathrm{m_b}$ from the two messages where $\\mathrm{b \\in_R \\{0, 1\\}}$, sets $\\mathrm{r_{ie}=1}$, chose random variables $\\mathrm{r_i}$ and $\\mathrm{r_s}$ such that $\\mathrm{c=r_i+r_c}$. Now, set value of root node $\\mathcal{T}^*$ to $\\mathrm{c}$ and assign values to leaf nodes of $\\mathcal{T}^*$ as described in section \\ref{access_tree} to arrive at $\\mathrm{C3_y}$ and $\\mathrm{C4_y}$. The final ciphertext $\\mathrm{CT}_{\\mathcal{T}^*}$ is computed as below. The ciphertext is returned to adversary $\\mathcal{A}$.\n\t\t\n\t\t$CT_{\\mathcal{T}^*} = \\begin{cases}\n\t\t\t\\quad \\mathcal{T}_{iv} = \\mathcal{T}^*\\\\\n\t\t\t\\quad \\mathrm{C1} = \\mathrm{e(g, g)}^{\\alpha \\mathrm{r_s}}\\mathrm{e(g, g)}^{\\alpha \\mathrm{r_i}}\\mathrm{m_b}\\\\\n\t\t\t\\qquad\\ \\  = \\mathrm{e(g, g)}^{\\alpha(\\mathrm{r_s+r_i})}\\mathrm{m_b}\\\\\n\t\t\t\\qquad\\ \\  = \\mathrm{e(g, g)^{c}m_b}\\\\\n\t\t\t\\quad \\mathrm{C2} = \\mathrm{g}^{\\beta \\mathrm{r_s}}\\mathrm{g}^{\\beta \\mathrm{r_i}} = \\mathrm{g}^{\\beta(\\mathrm{r_s+r_i})}\\\\\n\t\t\t\\qquad\\ \\  = \\mathrm{g}^{\\beta} \\mathrm{g^{c}}\\\\\n\t\t\t\\begin{rcases}\n\t\t\t\t\\quad \\mathrm{C3_{y} = g^{(q_y(0))}}\\\\\n\t\t\t\t\\quad \\mathrm{C4_{y} = H(\\mathtt{attr(y)})^{q_y(0)}}\\\\\n\t\t\t\\end{rcases} {\\mathrm{\\forall\\ y \\in Y_{iv}}}\\\\\n\t\t\\end{cases}$\n\t\t\n\t\t\\item [] \\textit{Phase 2}: In this phase, adversary $\\mathcal{A}$ can continue to send secret key generation requests with the same restriction as in $Phase 1$, i.e., $\\mathrm{a_j} \\notin \\mathcal{T}^*$.\n\t\t\n\t\t\\item [] \\textit{Guess Phase}: In this phase, adversary $\\mathcal{A}$ outputs a guess $\\mathrm{b\\prime} \\in \\{0, 1\\}$. If $\\mathrm{b\\prime} = \\mathrm{b}$, the simulator $\\beta$ will guess that $\\mu = 0$ and $\\mathrm{Z}_\\mu = \\mathrm{e(g, g)^{abc}}$, otherwise will guess that $\\mu = 1$ and $\\mathrm{Z}_\\mu = \\mathrm{e(g, g)}^{\\theta}$. When $\\mathrm{Z}_u = \\mathrm{e(g, g)^{abc}}$ the simulator $\\beta$ gives the perfect simulation and $\\mathrm{c}_{\\mathcal{T}^*}$ is a valid ciphertext. Therefore, the advantage of the adversary is \n\t\t\\begin{align*}\n\t\t\t\\mathrm{Pr} [ \\mathrm{b\\prime} = \\mathrm{b} \\mid \\mathrm{Z}_\\mu = \\mathrm{e(g, g)^{abc}} ] = \\frac{1}{2} + \\epsilon\n\t\t\\end{align*}\n\t\tIf $\\mu = 1$ then $\\mathrm{Z}_\\mu = \\mathrm{e(g, g)}^\\theta$ and $\\mathrm{c}_{\\mathcal{T}^*}$ is random ciphertext for the adversary, and the adversary does not gain information about $\\mathrm{m_b}$. Hence, we have\n\t\t\\begin{align*}\n\t\t\t\\mathrm{Pr} [ \\mathrm{b\\prime} \\neq \\mathrm{b} \\mid \\mathrm{Z}_\\mu = \\mathrm{e(g, g)}^{\\theta} ] = \\frac{1}{2}\n\t\t\\end{align*}\n\t\tSince the simulator $\\beta$ guesses $\\mu\\mathrm{\\prime} = 0$ when $\\mathrm{b\\prime} = \\mathrm{b}$ and $\\mu\\mathrm{\\prime} = 1$ when $\\mathrm{b\\prime} \\neq \\mathrm{b}$, the overall advantage of $\\beta$ to solve DBDH assumption is \n\t\t\\begin{align*}\n\t\t\\dfrac{1}{2} \\mathrm{Pr}[\\mu\\mathrm{\\prime} = \\mu \\mid \\mu = 0] + \\dfrac{1}{2} \\mathrm{Pr}[\\mu\\mathrm{\\prime} = \\mu \\mid \\mu = 1] - \\dfrac{1}{2} = \\dfrac{\\epsilon}{2}\n\t\t\\end{align*}\n\t\tIf the adversary $\\mathcal{A}$ has the above advantage $\\epsilon$ to win the IND-sAtt-CPA game, the challenger can solve the DBDH assumption problem with $\\epsilon\/2$ advantage with the help of adversary $\\mathcal{A}$. However, there are no effective polynomial algorithms which can solve the DBDH assumption problem with non-negligible advantage according to the DBDH assumption. Hence, the adversary cannot win the IND-sAtt-CPA game with the above advantage $\\epsilon$, namely the adversary having no advantage to break through the proposed scheme.\n\t\\end{itemize}\n\t\n\\section {\\uppercase{Conclusion}}\n\\noindent\nThis paper presented a scheme to improve data privacy in DigiLocker by using CP-ABE. The scheme also proposed to prepone part of the encryption process to increase performance. This preponed process creates a token which can be reused later. The proposed scheme is proved to be secure against IND-sAtt-CPA game. The proposed scheme can further be enhanced by using homomorphic encryption which allows processing on encrypted data and using post-quantum ABE schemes, for both of which, though schemes exist but are still non-trivial and not practical. \n\n\\bibliographystyle{apalike}\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Appendix A:  Three-qubit bit-flip code}\nIn this section, we explain how the deviation of the physical GKP qubits is projected onto the deviation of the ancillae. Fig.\\ref{sfiga} shows a quantum circuit for the QEC with the three-qubit bit-flip code. This circuit looks almost the same as the circuit for DV apart from the third ancilla qubit. However, the error identification for the GKP qubits is substantially different from that for DV-QEC. In this circuit, the sum of deviations of the physical GKP qubits $i$ and $i+1$ ($i = 1, 2$) are projected onto the ancilla $i$. The deviation of the physical GKP qubit 3 is projected onto ancilla 3. First, a single logical qubit $\\ket{\\widetilde{\\psi}}_{{\\rm L}}$ is prepared by two controlled-not (CNOT) gates acting on the data qubit $\\ket{\\widetilde{\\psi}}_{\\rm 1}$ = $\\alpha \\ket{\\widetilde{0}}_{\\rm 1}$ +$\\beta \\ket{\\widetilde{1}}_{\\rm 1}$ and two ancillae $\\ket{\\widetilde{0}}_{i}$ ( $i$ = 2, 3). The CNOT gate, which corresponds to the operator exp(-$i\\hat{q}_{\\rm L}\\hat{p}_{\\rm A}$), transforms\n\\begin{align*}\n\\hat{q}_{\\rm L} \\to   \\hat{q}_{\\rm L}, \\ \\ \n\\hat{p}_{\\rm L} \\to  \\hat{p}_{\\rm L} - \\hat{p}_{\\rm A} ,  \\nonumber \\\\\n\\hat{q}_{\\rm A} \\to   \\hat{q}_{\\rm A} +  \\hat{q}_{\\rm L}, \\ \\\n\\hat{p}_{\\rm A} \\to  \\hat{p}_{\\rm A},\n\\tag{A1}\n\\end{align*}\nwhere $\\hat{q_{\\rm L}}$ ($\\hat{q_{\\rm A}}$) and $\\hat{p_{\\rm L}}$ ($\\hat{p_{\\rm A}}$) are the $q$ and $p$ quadrature operators of the logical (ancilla) qubit, respectively. Then, the GQC displaces the $q$ and $p$ quadratures randomly and independently, and increases the variance of the three physical GKP qubits. After the GQC, the bit-flip error correction is implemented using the three ancillae $\\ket{\\widetilde{0}}_{{\\rm A}j}$ ($j$=1, 2) and $\\ket{\\widetilde{+}}_{{\\rm A}3}$. Before the CNOT gates in the error correction circuit, the true deviation values of the physical GKP qubits and ancillae in $q$ quadrature, which obey Gaussian distribution with mean zero, are denoted by $\\overline{\\Delta}_{j}$ and $\\overline{\\Delta}_{{\\rm A}j}$ ($j$=1, 2, 3), respectively. For simplicity, because the ancilla qubits are fresh, we assume that the initial variance is much smaller than that of the physical qubits of the logical qubit. Then, the CNOT gates change the true deviation values of three ancillae $\\overline{\\Delta}_{{\\rm A}j}$ in $q$ quadrature as follows:\n\\begin{figure}[t]\n    \\includegraphics[angle=270, width=1.0\\columnwidth]{sfiga.ps}\n    \\caption{\\label{sfiga}A quantum circuit of the QEC for the three-qubit bit-flip code with GKP qubits using the proposed method. The data qubit $\\ket{\\widetilde{\\psi}}_{\\rm 1}$ and two GKP qubits $\\ket{\\widetilde{0}}_{\\rm 2}$ and $\\ket{\\widetilde{0}}_{\\rm 3}$ encode a single logical qubit. $\\ket{\\widetilde{0}}_{\\rm A1}, \\ket{\\widetilde{0}}_{\\rm A2}, {\\rm and} \\ket{\\widetilde{+}}_{\\rm A3}$ denote ancilla qubits for the QEC. The GQC and $M_{q}$ denote the GQC and measurements of ancillae in $q$ quadrature, respectively.}   \n\\end{figure}\n\\begin{align*}\n\\overline{\\Delta}_{{\\rm A1}}  \\to   \\overline{\\Delta}_{{\\rm A1}} + \\overline{\\Delta}_{{\\rm 1}} + \\overline{\\Delta}_{{\\rm 2}}= \\overline{\\Delta}_{{\\rm 1}} + \\overline{\\Delta}_{{\\rm 2}}\\ \\nonumber ,\\\\\n \\overline{\\Delta}_{{\\rm A2}}  \\to   \\overline{\\Delta}_{{\\rm A2}} + \\overline{\\Delta}_{{\\rm 2}} + \\overline{\\Delta}_{{\\rm 3}}=\\overline{\\Delta}_{{\\rm 2}} + \\overline{\\Delta}_{{\\rm 3}}\\ \\nonumber , \\\\\n \\overline{\\Delta}_{{\\rm A3}} \\to  \\overline{\\Delta}_{{\\rm A3}} + \\overline{\\Delta}_{{\\rm 3}} = \\overline{\\Delta}_{{\\rm 3}} .\n \\tag{A2}\n\\end{align*}\nTherefore, the sum of deviations of the physical GKP qubits $i$ and $i+1$ ($i = 1, 2)$ are projected onto the ancilla $i$. The deviation of physical GKP qubit 3 is projected onto ancilla 3.\n\n\\section*{Appendix B: $C_{4}\/C_{6}$ code}\nThe error correction in the $C_{4}\/C_{6}$ code is based on quantum teleportation, where the logical qubit $\\ket{\\widetilde{\\psi}}_{{\\rm L}}$ encoded by the $C_{4}\/C_{6}$ code is teleported to the fresh encoded Bell state, as shown in Fig.\\ref{sfigb}. The quantum teleportation process refers to the outcomes $M_{p}$ and $M_{q}$ of the Bell measurement on the encoded qubits, and determines the amount of displacement. We obtain the Bell measurement outcomes of bit values $m_{pi}$ and $m_{qi}$ for the $i$-th physical GKP qubit of the encoded data qubit and encoded qubit of the encoded Bell state, respectively. In addition to bit values, we also obtain deviation values $\\Delta_{p{\\rm m}i}$ and $\\Delta_{q{\\rm m}i}$ for the $i$-th physical GKP qubit. Therefore, the proposed likelihood method can improve the error tolerance of the Bell measurement.\n\nAs a simple example to explain our method for the Bell measurement, we describe the level-1 $C_{4}\/C_{6}$ code, that is, the $C_{4}$ code. The $C_{4}$ code is the $[[\\rm{4,2,2}]]$ code and consists of four physical GKP qubits to encode a level-1 qubit pair; thus, it is not the error-correcting code but the error-detecting code in the conventional method. The logical bit value of the $C_{4}$ code is $k$ (=0,1) when the bit value of the level-1 qubit pair is ($k$,0) or ($k$,1), that is, the bit value of the first qubit $k$ defines a logical bit value of a qubit pair. As the parity check of the $Z$ operator for the first and second qubits $ZIZI$ and $IIZZ$ indicates, the bit value of the level-1 qubit pair (0,0) corresponds to the bit value of the physical GKP qubits $(m_{q{\\rm1}}, m_{q{\\rm 2}}, m_{q{\\rm 3}}, m_{q{\\rm 4}})$ = (0,0,0,0) or (1,1,1,1)~\\cite{Knill}. The bit values of the pairs (0,1), (1,0), and (1,1) correspond to the bit values of the physical GKP qubits (0,1,0,1) or (1,0,1,0), (0,0,1,1) or (1,1,0,0), and (0,1,1,0) or (1,0,0,1), respectively. Therefore, if the measurement outcome of the physical GKP qubits is (0,0,1,0) for the $Z$ basis, then we consider two error patterns, assuming the level-1 qubit pair (0,0). The first pattern is a single error on the physical qubit 3 and the second pattern is the triple errors on the physical qubits 1, 2, and 4. We then calculate the likelihood for the level-1 qubit pair (0,0) $F_{0,0}$ as\n\\begin{widetext}\n\\begin{equation*}\nF_{0 ,0} = f(\\sqrt{\\pi}-|\\Delta_{q{\\rm m1}}|)f(\\sqrt{\\pi}-|\\Delta_{q{\\rm m2}}|)f(\\Delta_{q{\\rm m3}})f(\\sqrt{\\pi}-|\\Delta_{q{\\rm m4}}|)\n                 + f(\\Delta_{q{\\rm m1}})f(\\Delta_{q{\\rm m2}})f(\\sqrt{\\pi}-|\\Delta_{q{\\rm m3}}|)f(\\Delta_{q{\\rm m4}}).\\tag{B1}\n\\end{equation*}\n\\end{widetext} \nWe similarly calculate the $ F_{0,1}, F_{1,0},$ and $F_{1,1}$ likelihood for the bit value of qubit pairs (0,1), (1,0), and (1,1). Finally, we determine the level-1 logical bit value for the $Z$ basis by comparing $F_{0,0}+F_{0,1}$ with $F_{1,0}+F_{1,1}$, which refer to the likelihood functions for the logical bit values zero and one, respectively. If $F_{0,0}+F_{0,1} > F_{1,0}+F_{1,1}$, then we determine that the level-1 logical bit value for the $Z$ basis is zero, and vice versa. The level-1 logical bit value for the $X$ basis can be determined by the parity check of the $X$ operator for the first and second qubits $XXII$ and $IXIX$ in a similar manner. In the conventional likelihood method~\\cite{Pou,Goto} $F_{0,0}$, $F_{0,1}$, $F_{1,0}$, and $F_{1,1}$ are given by the same joint probability\n\\begin{equation*}\np_{\\rm corr}^{3} (1-p_{\\rm corr}) + p_{\\rm corr}(1-p_{\\rm corr})^{3},\\tag{B2}\n\\end{equation*}\n\\begin{figure}[b]\n    \\includegraphics[angle=270, width=1.0\\columnwidth]{sfigb.ps}  \n    \\caption{\\label{sfigb}Error correction by quantum teleportation. The encoded data qubit $\\ket{\\widetilde{\\psi}}_{\\rm L}$, two encoded qubits $\\ket{\\widetilde{+}}_{\\rm L}$, and $\\ket{\\widetilde{0}}_{\\rm L}$ are encoded by $C_{4}\/C_{6}$ code. GQC and  MLD denote the GQC and a maximum-likelihood decision, respectively.}\n\\end{figure}\nwhere the probability $p_{corr}$ is defined by Eq. (1) in the main text.\nBecause $F_{0,0}+F_{0,1} = F_{1,0}+F_{1,1}$, the $C_{4}$ code is not error-correcting code but error-detecting code in the conventional method, whereas it is the error-correcting code in our method. For higher levels of concatenation, the likelihood for the level-$l$ ($l\\geqq2$) bit value can be calculated by the likelihood for the level-$(l-1)$ bit value in a similar manner.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Plane curves approximating varieties near the tangent cone.}\\label{s:pc}\n\n In this section, we generalize the notion of multiplicity of intersection\n used in \\cite[Chapter~3,~\\S4, Definition~1]{clo}.  Proposition~3 in\n \\cite[Chapter~9,~\\S6]{clo},\n has a criterion which says that a line lies in the tangent space\n of the variety iff the line meets the variety with multiplicity at least 2.\n Here, we show that through a singular point of an affine variety\n we can always draw a smooth plane curve inside the ambient affine space,\n which has the multiplicity of intersection with the variety at least 3.\n This basic result does not seem to appear anywhere in\n the literature.\n\n\n Let $K$ be an arbitrary field.\n An algebraic curve $C\\subset{\\Bbb A}^{n}$ can be\n parameterized at its smooth point $p$ by Taylor's series\n $$G(t)=p+t\\cdot v_1+t^2\\cdot v_2+\\cdots,$$\n where $t$ is a local parameter.\n Using this parameterization, we can define multiplicity of intersection\n of an affine variety with the curve $C$ at the point $p$.\n\n \\begin{defn}\n Let $m$ be a nonnegative integer and $X\\subset{\\Bbb A}^{n}$\n an affine variety, given by\n $\\text{\\bf I}(X)\\subset K[x_1,\\ldots,x_n]$. Suppose that we have\n a curve $C$, smooth at a point $p$ and with the parameterization as above.\n Then $C$ {\\bf meets} $X$ {\\bf with multiplicity} $m$  at $p$\n if $t=0$ is a zero of multiplicity at least $m$ of the series\n $f\\circ G(t)$  for all $f\\in\\text{\\bf I}(X)$, and for some $f$, of multiplicity\n exactly $m$. We denote this multiplicity of intersection by $I(p,C,X)$.\n \\end{defn}\n\n \\begin{rem}\n This definition does not depend on the choice of the parameterization.\n To check this one can use the following criterion:\n $t=0$ is a zero of multiplicity $m$ of $h(t)\\in K[[t]]$ if and only if\n $h(0)=h'(0)=\\cdots=h^{(m-1)}(0)=0$ but $h^{(m)}(0)\\ne 0$.\n \\end{rem}\n\n\n\n\n Also, we denote by $T_{p}X$ ($TC_{p}X$)\n the tangent space (cone) of a variety $X$ at a point $p$.\n The next result contains a necessary condition for a line to be in the tangent\n cone.\n Unfortunately, it is not a sufficient one.\n\n \\begin{thm}\\label{t:m}\n Let $X\\subset{\\Bbb A}^{n}$ be an affine variety and $p\\in X$ a singular\n point. Then for each $v\\in TC_{p}X$ there exists a smooth\n curve $C\\subset{\\Bbb A}^{n}$ through $p$ such that\n $v\\in T_{p}C$ and $I(p,C,X)\\geq3$.\n \\end{thm}\n\n \\begin{pf} Let  $v\\in TC_{p}X$.\n We can choose affine coordinates $x_{1},\\ldots,x_{n}$, so that\n $p=(0,\\ldots,0)$. Let $C$ be a curve through\n $p$ parameterized\n by $G(t)=t\\cdot v+t^{2}\\cdot\\gamma$, where\n $v\\in TC_{p}X$,\n $\\gamma=(\\gamma_1,\\ldots,\\gamma_{n})$.\n It suffices  to show that there is $\\gamma$, which is  zero or not a scalar\n multiple\n of $v$, such that $t=0$ is a zero\n of multiplicity $\\geq3$ of $f\\circ G(t)$  for all $f\\in\\text{\\bf I}(X)$.\n In this case, $C$ will be a smooth curve with the local parameter $t$\n at $p$.\n\n For each $f\\in\\text{\\bf I}(X)$, denote by\n $l_{f}(x)=l_{1}x_{1}+\\cdots+l_{n}x_{n}$\n  and\n $q_{f}(x)$ the  linear and quadratic parts of $f$.\n According to this notation we define\n $$W=\\{(l_{1},\\ldots,l_{n},q_{f}(v)):\\,f\\in\\text{\\bf I}(X)\\}\\subset\n k^{n+1},$$ which is a  subspace  since $\\text{\\bf I}(X)$ is. For all\n $f\\in\\text{\\bf I}(X)$,    we have\n $$f(t\\cdot v+t^{2}\\cdot\\gamma)=l_{f}(t\\cdot v)+l_{f}(t^{2}\\cdot\\gamma)+\n q_{f}(t\\cdot v)+\\text{ terms of degree }\\geq3\\text{ in }t.$$\n Since $v\\in T_{p}X$, we get $l_{f}(v)=0$ for any $f\\in\\text{\\bf I}(X)$.\n Hence,\n $$f(t\\cdot v+t^{2}\\cdot\\gamma)\\equiv\n t^{2}(l_{1}\\gamma_{1}+\\cdots+l_{n}\\gamma_{n}+q_{f}(v))$$\n modulo terms of degree $\\geq3$ in $t$. So we need to resolve\n equations $a_{1}\\gamma_{1}+\\cdots+a_{n}\\gamma_{n}+a_{n+1}=0$ in variables\n $\\gamma_{1},\\ldots,\\gamma_{n}$ for all possible\n $(a_{1},\\ldots,a_{n+1})\\in W$.\n\n Consider the scalar product $k^{n+1}\\times k^{n+1}\\rightarrow k$,\n which sends $a\\times b$ to $a\\cdot b:=\n \\sum^{n+1}_{i=1}a_{i}b_{i}$.\n Using this scalar product, we will find the vector\n $(\\gamma_{1},\\ldots,\\gamma_{n},1)\\in W^{\\perp}:=\n \\{b\\in k^{n+1}:\\,b\\cdot a=0\\text{ for all } a\\in W\\}$.\n If $W\\subseteq k^{n}\\times\\{0\\}$, then we can put $\\gamma=(0,\\ldots,0)$\n to obtain the required curve.\n Otherwise, consider the  projection\n $\\pi:\\,k^{n+1}\\rightarrow k^{n}\\times\\{0\\}$, which assigns\n $(a_{1},\\ldots,a_{n},0)$\n  to\n $(a_{1},\\ldots,a_{n+1})$.\n And, let $\\tilde{\\pi}:\\,W\\rightarrow k^{n}\\times\\{0\\}$ be induced by $\\pi$.\n The projection  $\\tilde{\\pi}$ is injective, because $v\\in TC_{p}X$ and\n $l_{f}=0$ imply $q_{f}(v)=0$.\n If we denote $\\widetilde{W}=\\tilde{\\pi}(W)$, then\n $\\dim W^{\\perp}=\\dim\\widetilde{W}^{\\perp}$, by injectivity of $\\tilde{\\pi}$.\n Now, if $W^{\\perp}$ is of the form $W_{1}\\times\\{0\\}$,\n then by construction $\\widetilde{W}^{\\perp}=W_{1}\\times k$,\n contradicting with the equality of the dimensions.\n Therefore, there exists a vector\n $(\\gamma_{1},\\ldots,\\gamma_{n},1)\\in W^{\\perp}$,\n i.e., $l_{f}(\\gamma)+q_{f}(v)=0$ for all $f\\in I(X)$.\n We claim that this $\\gamma$ is linearly independent of $v$.\n Indeed, since $W$ is not included into $k^{n}\\times\\{0\\}$, we get\n $q_{f}(v)\\ne0$\n  for\n some $f\\in\\text{\\bf I}(X)$, which implies $l_{f}(\\gamma)\\ne0$.\n On the other hand, since $v\\in T_{p}X$, we get $l_{f}(v)=0$. Thus, we have found\n the desired $\\gamma$.\n \\end{pf}\n\n \\begin{rem}\n This result cannot be improved in the following sense. For the curve $X$,\n given by the equation $x^{2}=y^{3}$,\n in the affine plane, and $p=(0,0)$\n there is no smooth curve $C$ such that $I(p,C,X)\\geq4$.\n \\end{rem}\n\n \\begin{rem}\n In the case of analytic varieties the theorem is also valid.\n \\end{rem}\n\n\n\n \\section{An application to the affine schemes of algebras.}\\label{s:ap}\n\nThis section  shows that at least theoretically one may still be\nable to answer the\n problem of Shafarevich discussed in the introduction by solving quadratic equations arising from\n the second order obstructions to deformations of algebras.\n In particular, we find that the vectors, contributing to the discrepancy\n between the tangent spaces to the scheme of associative multiplications and the\n scheme\n of the degree 3 nilpotent multiplications, can be easily ``killed'' by the\n obstructions in many cases.  Then Theorem~\\ref{t:m} implies that  the tangent cones\n of the reduced schemes are the same, which means that the irreducible\n component of one variety is the component of the other one.\n\n Let us recall the notation from \\cite{sh,am}.\n The affine scheme $C_n$ of all multiplications on a fixed $n$-dimensional vector\n space $V$\n over a field $K$ (${\\rm char} K\\ne2$),\n which represent associative commutative algebras,\n is given by the equations of commutativity and associativity:\n $$c_{ij}^k=c_{ji}^k,\\qquad \\sum_{s=1}^n c_{ij}^s c_{sk}^l=\\sum_{s=1}^n c_{is}^l\n c_{jk}^s$$\n in the structure constants of multiplication\n $e_ie_j=\\sum_{k=1}^n c_{ij}^k e_k$\n of the basis $\\{e_1,\\ldots,e_n\\}$.\n Similar equations determine the affine scheme $A_n$ of commutative  nilpotent\n degree 3\n multiplications.  The reduced schemes associated to the above schemes are denoted\n $C_n^{\\rm red}$ and $A_n^{\\rm red}$, respectively.\n From \\cite{sh}, the irreducible components of $A_n^{\\rm red}$ have a very simple\n description\n $$A_{n,r}=\\{N\\in A_n^{\\rm red} |\\, \\dim N^2\\le r\\le \\dim {\\rm Ann}_N N\\},$$\n $1\\le r\\le(n-1)(n-r+1)\/2$, where $N$ denotes the algebra represented by the\n corresponding\n multiplication, $N^2$ is the square of the algebra and $\\rm Ann$ is the\n annihilator.\n\n Let $d:=n-r$, then, for $r=1,2$ and $r>(d^2-1)\/3$, Shafarevich in \\cite{sh} showed that\n $A_{n,r}$ is not a component of\n $C_n^{\\rm red}$, by constructing\n a line, contained in  $C_n^{\\rm red}$ but not in $A_n^{\\rm red}$, through a point of\n $A_{n,r}$.\n For other $r$, one has to compare\n the tangent spaces to the non-reduced schemes $A_n$ and $C_n$.\n The smooth set of $A_n^{\\rm red}$ is the union of\n $$U_{n,r}=\\{N\\in A_n^{\\rm red} |\\, \\dim N^2= r= \\dim {\\rm Ann}_N N\\}.$$\n If $W\\subset V$ is a subspace of dimension $r$, then\n the space $S_{n,r}=L(S^2(V\/W),W)$ of all linear maps from the symmetric product of\n $V\/W$ to $W$\n is naturally included as an affine subspace into  $A_{n,r}$. The group $G={\\rm\n GL}(V)$ acts\n on $C_n$ and $G S_{n,r}=A_{n,r}$.\n Then, the tangent space to $A_{n,r}$ at a point $N\\in S_{n,r}$ is\n $$T_N A_{n,r}=L(S^2(N\/N^2),N^2)+T_N G N,$$\n where the tangent space $T_N G N$ to the orbit is the space of\n maps from $L(S^2N,N)$ given by the coboundaries (in a Hochschild complex,\n see \\cite{am}) $x\\circ y= x\\varphi(y)-\\varphi(xy)+y\\varphi(x)$\n for some linear map $\\varphi:N@>>>N$.\n\n The tangent space $T_N C_n$\n always includes  $T_N A_{n,r}+F$, where\n the space $F$ consists of $x\\circ y=xf(y)+yf(x)$ for some $f\\in L(N\/N^2,K)\\subset\n L(N,K)$.\n To explain this difference\n Shafarevich embedded the scheme $C_n$ into the scheme $\\widetilde{C}_n$\n of commutative associative multiplications on a $(n+1)$-dimensional space which\n represent algebras with a unit $e$:\n $$C_n\\hookrightarrow\\widetilde{C}_n, \\qquad N\\mapsto N\\oplus Ke.$$\n In this situation, the subgroup $\\widetilde G$ of $GL(V\\oplus Ke)$ which fixes the\n unit\n acts on $\\widetilde{C}_n$. It turns out that\n $T_N GN+F$ is the tangent space to\n $\\widetilde{G} S_{n,r}$, whence  the equality\n \\begin{equation}\\label{e:tan}\n T_N \\widetilde{C}_n=T_N A_{n,r}+F\n \\end{equation}\n was enough to conclude that $A_{n,r}$ is the component of $C_n^{\\rm red}$.\n The equality was shown in \\cite{sh} for $3\\le r\\le (d+1)(d+2)\/6$,\n and this also holds $r=(5d^2-8d)\/16$ and $d$ divisible by 4\n by \\cite[Section~2.1]{am}.\n\n The original Shafarevich's method is very special to this situation and\n is  impractical\n to use it in general to explain the nilpotents in the structure sheaf of a scheme.\n We expect that the equality (\\ref{e:tan}) holds for all $3\\le r<(d^2-1)\/3$,\n which would leave only one case $r=(d^2-1)\/3$ unsettled. However, in this last case\n the tangent space to $C_n$ at a generic point in $A_{n,r}$ is too big:\n \\begin{equation}\\label{e:for}\n T_N C_n= L(S^2(N\/N^2),N^2)+T_N G N+L(S^2N_1,N_1),\n \\end{equation}\n where $N=N_1\\oplus N^2$ is a fixed decomposition.\n To show this one have to use Lemma~1 in \\cite{am}:\n a  generic\n algebra $N\\in A_{n,r}$, for $r\\le(d^2-1)\/3$, can be given by $d$ generators and a\n $(d(d+1)\/2-r)$-dimensional space  of  homogeneous degree 2 relations among the\n generators,\n so that\n the nilpotence of the third degree follows from this relations.\n But, for $r=(d^2-1)\/3$, this condition implies that there are no nontrivial\n relations among the\n homogeneous degree 2 relations, i.e., all of the relations $\\sum_i n_iz_i=0$ in\n $S^3N_1$,\n $n_i\\in N_1$, $z_i\\in Z:=ker(\\mu:S^2N_1@>>>N^2)$ ($\\mu$ is the multiplication\n on $N$),  are induced by a trivial in $N_1\\otimes Z$ element\n $\\sum_i n_i\\otimes z_i$. Indeed, $\\dim S^3N_1=d(d+1)(d+2)\/6$, while the number of\n equations\n that we get from the homogeneous degree 2 relations is equal to\n $$\\dim N_1\\otimes Z=d(d(d+1)\/2-(d^2-1)\/3)=d(d+1)(d+2)\/6.$$\n So, if there is a nontrivial relation among the\n homogeneous degree 2 relations, there would be not enough equations to deduce\n $N^3=0$.\n Since the only restriction in \\cite[Lemma~1]{am} for a vector from\n $L(S^2N_1,N_1)\\subset L(S^2N,N)$ to be tangent to the scheme $C_n$\n was arising from the nontrivial relations,\n we conclude (\\ref{e:for}).\n\n Anan'in suggested in \\cite{am} to use the second order obstructions to deformations\n of algebras\n to eliminate the excessive space   $L(S^2N_1,N_1)$. To be precise, suppose that\n $N\\in A_{n,r}$\n is a smooth point of $C_n^{\\rm red}$ and $\\circ\\in L(S^2N_1,N_1)$. Then, if $\\circ$ is\n tangent to $C_n^{\\rm red}$, the sum $\\circ+L(S^2(N\/N^2),N^2)$ also lies in the tangent\n space\n $T_N C_n^{\\rm red}$. A deformation of a commutative algebra on a vector space $V$\n with multiplication $x\\cdot y$ can be thought as\n a smooth curve $$x\\cdot y+(x\\circ y)t+(x\\star y)t^2+\\cdots$$ in\n the affine space $L(S^2V,V)$, where $t$ is a local parameter.\n It is a well know fact in algebraic geometry that\n through a smooth point $p$ of a variety $X$ we can  find\n a smooth curve inside $X$, whose tangent vector at $p$ is a given one in $T_p X$.\n Therefore, the smooth curves in $C_n^{\\rm red}$ with tangent vectors\n $\\circ+L(S^2(N\/N^2),N^2)$\n   give a lot of equations (the second order obstructions) arising from the\n associativity of\n multiplication:\n $$(x\\tilde\\circ y)\\tilde\\circ z-x\\tilde\\circ(y\\tilde\\circ z)=\n x(y\\star z)-(xy)\\star z+x\\star(yz)-(x\\star y) z$$\n for some $\\star\\in L(S^2N,N)$, where $\\tilde\\circ\\in\\circ+L(S^2(N\/N^2),N^2)$.\n A nice thing about these quadratic equations on $\\circ$ is that one can linearize\n them:\n $$(x\\circ y)* z+(x*y)\\circ z-x\\circ(y* z)-x*(y\\circ z)=\n x(y\\star z)-(xy)\\star z+x\\star(yz)-(x\\star y) z$$\n for all $*\\in L(S^2(N\/N^2),N^2)$.\n These equations have been used to prove Theorem~1 in \\cite{am} which implies that\n $A_{n,r}$ is the component of $C_n^{\\rm red}$ for $d(d+1)\/9\\le r\\le[d\/3](d-3)$\n (almost all cases that are not covered by \\cite{sh})\n if a certain algebra $N\\in A_{n,r}$ is a smooth point\n of $C_n^{\\rm red}$.\n Unfortunately, it seems impossible to prove that $N$ is the smooth point unless we\n know the tangent space at $N$.\n\n Now, we can apply Section~\\ref{s:pc}.\n Let $N\\in A_{n,r}$ with multiplication $x\\cdot y$ and $\\circ\\in TC_N C_n^{\\rm red}$.\n By Theorem~\\ref{t:m}, there exists a smooth plane curve $C$\n $$x\\cdot y+(x\\circ y)t+(x\\star y)t^2$$\n through $N$\n in the affine space $L(S^2V,V)$ of commutative\n multiplications on the vector space $V$, such that\n $\\circ$ is the tangent vector to the curve at $N$ and the multiplicity of\n intersection $I(p,C,C_n^{\\rm red})\\ge3$. Hence, the associativity equations in the\n structure\n constants imply\n \\begin{equation}\\label{e:ob}\n (x\\circ y)\\circ z-x\\circ(y\\circ z)=\n x(y\\star z)-(xy)\\star z+x\\star(yz)-(x\\star y) z\n \\end{equation}\n for some $\\star\\in L(S^2N,N)$. Note that we do not assume the smoothness condition\n at $N$.\n\n We want to deduce  the effective part of the obstructions to $\\circ$.\n As in \\cite[\\S1]{am}, decompose $\\circ$ and $\\star$ into the sum of linear maps\n $f_{ij}^k$ and $g_{ij}^k$ in $L(N_i\\otimes N_j,N_k)$, respectively,\n where $1\\le i,j,k\\le2$ and $N=N_1\\oplus N_2$,\n $N_2:=N^2$.\n For a sufficiently generic $N\\in A_{n,r}$, besides commutativity,\n $f_{ij}^k$\n  satisfy\n $$f_{12}^1=f_{22}^1=f_{22}^2=0,$$\n \\begin{equation}\\label{e:co}\n xf_{11}^1(y,z)+f_{12}^2(x,yz)=f_{12}^2(z,xy)+f_{11}^1(x,y)z,\n \\end{equation}\n by \\cite[\\S1]{am}. Moreover,\n $f_{12}^2$ satisfying (\\ref{e:co}) is unique for\n a given $f_{11}^1$, while its existence condition is described in\n \\cite[Lemma~1]{am}.\n The missing $f_{11}^2\\in L(S^2(N\/N^2),N^2)$ is always tangent\n to $C_n$.\n\n From (\\ref{e:ob}) we get\n \\begin{equation}\\label{e:ob1}\n f_{11}^1(f_{11}^1(x,y),z)-f_{11}^1(x,f_{11}^1(y,z))=\n -g_{12}^1(z,xy)+g_{12}^1(x,yz),\n \\end{equation}\n \\begin{equation}\\label{e:ob2}\n f_{12}^2(x,f_{12}^2(y,zt))-f_{12}^2(y,f_{12}^2(x,zt))=\n xg_{12}^1(y,zt)-yg_{12}^1(x,zt)\n \\end{equation}\n and\n $$f_{12}^2(f_{11}^1(x,y),zt)-f_{12}^2(x,f_{12}^2(y,zt))=xg_{12}^1(y,zt)-\n g_{22}^2(xy,zt),$$\n where $x,y,z,t\\in N_1$.\n The last\n equation determines $g_{22}^2$, and, one can check that\n commutativity of $g_{22}^2$ (surprisingly)\n follows  from  (\\ref{e:co}),\n (\\ref{e:ob1}) and (\\ref{e:ob2}).\n So, the only restrictions for $\\circ$ to lie in the tangent cone to $C_n$\n are (\\ref{e:co}),\n (\\ref{e:ob1}) and (\\ref{e:ob2}). The same argument as in \\cite[\\S1]{am}\n shows that $g_{12}^1$, satisfying (\\ref{e:ob1}), is unique  for a given\n $f_{11}^1$, and the existence condition is similar to \\cite[Lemma~1]{am}.\nWe have the following result:\n\n\\begin{thm}\\label{t:m1}\nThe variety $A_{n,r}$ is an irreducible component\n of $C_n^{\\rm red}$ when $r\\le(d^2-1)\/3$ if and only if,\n for some sufficiently generic $N\\in A_{n,r}$,\na solution $\\circ\\in L(S^2N,N)$  to (\\ref{e:co}),\n (\\ref{e:ob1}) and (\\ref{e:ob2}) has its part\n$f^1_{11}=0$ on the kernel of the multiplication\n$\\mu:S^2N_1@>>>N^2$ of the algebra $N$.\n\\end{thm}\n\n\\begin{pf}\nIf $A_{n,r}$ is an irreducible component\n of $C_n^{\\rm red}$, then, for some sufficiently generic $N\\in A_{n,r}$,\n the tangent cone to $C_n^{\\rm red}$ at $N$ coincides with\n$$T_N A_{n,r}=L(S^2(N\/N^2),N^2)+T_N G N.$$ But we know that the\nsolution to (\\ref{e:co}),\n (\\ref{e:ob1}) and (\\ref{e:ob2}) gives rise to a vector $\\circ$ from the\n tangent cone. Hence, $f^1_{11}=0$ on the kernel of the multiplication\n$\\mu:S^2N_1@>>>N^2$, because the vectors of $T_N G N$ are of the\nform $x\\circ y= x\\varphi(y)-\\varphi(xy)+y\\varphi(x)$\n for some linear map $\\varphi:N@>>>N$, while the vectors from\n $L(S^2(N\/N^2),N^2)$ clearly satisfy the property.\n\n Conversely, suppose that a vector $\\circ\\in L(S^2N,N)$ belongs to the tangent\n cone to the scheme $C_n$ at $N$. So, by the discussion above, it\n satisfies\n  (\\ref{e:co}),\n (\\ref{e:ob1}) and (\\ref{e:ob2}). Hence,  $\\circ$ has\n$f^1_{11}=0$ on the kernel of the multiplication of some\nsufficiently generic $N\\in A_{n,r}$. Then $f_{11}^1$ with\n$f_{12}^2(x,yz)=-xf_{11}^1(y,z)$ and\n$g_{12}^1(x,yz)=-f_{11}^1(x,f_{11}^1(y,z))$ are unique solutions\nto (\\ref{e:co}), (\\ref{e:ob1}) and (\\ref{e:ob2}). This shows that\n$\\circ$ is actually from $T_N A_{n,r}$.\n\\end{pf}\n\n\n We will finish this section showing the result of\n \\cite[Theorem~2]{am} with application of the above theorem and\n without the use of  Shafarevich's embedding $C_n\\hookrightarrow\\widetilde{C}_n$.\n\n\\begin{cor}\n$A_{n,r}$ is a  component\n of $C_n^{\\rm red}$ for the values $r=(5d^2-8d)\/16$, $d$ is divisible by 4.\n\\end{cor}\n\n\\begin{pf}\n In \\cite[Section~2.1]{am}, it was already shown that\n $T_N{C}_n=T_N A_{n,r}+F$ for a sufficiently generic algebra $N$.\n This implies that the part $f_{11}^1$ of a vector from $T_N{C}_n$ is determined by\n $f(x)y+f(y)x$, for some $f\\in L(N\/N^2,K)$, on the kernel of the multiplication\n $\\mu$ on $N$.\n The algebra $N$ had the property that the square of all $d$ generators is zero and\n for each generator $u$ there was a distinct generator $v$\n such that their product $uv$ also vanishes in $N$.\n Taking $x=y=u$ and $z=v$ in (\\ref{e:ob1}), we have\n $$f_{11}^1(2f(u)u,v)-f_{11}^1(u,f(u)v+f(v)u)=0.$$\n Since $uv$ and $uu$ are in the kernel of multiplication $\\mu$, we further get\n $$2f(u)(f(u)v+f(v)u)-f(u)(f(u)v+f(v)u)-f(v)(2f(u)u)=0,$$\n whence $f(u)^2v-f(u)f(v)u=0$. Therefore, $f(u)=0$ for all generators, and\n $f_{11}^1=0$\n on the kernel of $\\mu$ as it was required in Theorem~\\ref{t:m1}.\n\\end{pf}\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Appendix A}\\label{app:non_canonical}\n\nWe show, through an example,\nthat on the level of data languages we cannot hope for unique minimal\nQDA. Consider the QDA in Figure~\\ref{fig:even-sorted-QDA} over $PV =\n\\emptyset$ and $Y = \\{y_1,y_2\\}$.  It accepts all valuation words in\nwhich $d(y_1) \\le d(y_2)$ if $y_1$ is before $y_2$ and $y_1,y_2$ are\n  both on even positions, and all valuation words in which $y_2 < y_1$\n  or at least one of $y_1,y_2$ is not on an even position.\nHence, the data language define by this QDA consist of all data words\nsuch that the data on the even list positions is sorted. Since the QDA\nhas to check that each variable occurs exactly once, the number of states\nis minimal for defining this data language.\n\nHowever, the same data language would also be defined if the\n$\\blank$-transition from $q_3$ would be redirected to $q_5$. Then the\nsortedness property would only be checked for all $y_1,y_2$ with $y_2\n= y_1+2$, which is sufficient. This shows that the transition\nstructure of a state minimal QDA for a given data language is not\nunique.\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[node distance=40]\n\\node[state](q0){$q_0$};\n\\node[state,right of=q0](q5){$q_5$};\n\\node[state,accepting,right of=q5](q6){$q_6$};\n\\node[state,left of=q0](q7){$q_7$};\n\\node[state,below of=q0](q1){$q_1$};\n\\node[state, right of=q1](q2){$q_2$};\n\\node[state,right of=q2](q3){$q_3$};\n\\node[state,accepting, right of=q3](q4){$q_4$};\n\\node[state,right of=q2](q3){$q_3$};\n\\path (q4) node[right,xshift=10]{$\\scriptstyle d(y_1) \\le d(y_2)$}\n -- (q6) node[right,xshift=10]{\\scriptsize \\emph{true}}\n;\n\\draw[<-, shorten <=1pt] (q0.north) -- +(0, .3);\n\\draw[->]\n(q0) edge[bend left]  node{$\\scriptstyle \\blank$} (q1)\n     edge             node[above]{$\\scriptstyle (b,y_1)$} (q5)\n     edge node[swap]{$\\scriptstyle (b,y_2)$} (q7)\n(q1) edge  node[left]{$\\scriptstyle \\blank$} (q0)\n     edge             node[above]{$\\scriptstyle (b,y_1)$} (q2)\n\t   edge             node[left, very near start]{$\\scriptstyle (b,y_2)$} (q7)\n(q2) edge  node[above]{$\\scriptstyle \\blank$} (q3)\n     edge  node[very near end, yshift=-7] {$\\scriptstyle (b, y_2)$} (q6)\n(q3) edge[bend left]  node[below]{$\\scriptstyle \\blank$} (q2)\n     edge             node[above]{$\\scriptstyle (b,y_2)$} (q4)\n(q4) edge[loop above] node{$\\scriptstyle \\blank$} ()\n(q5) edge[loop below] node{$\\scriptstyle \\blank$} ()\n     edge             node[above]{$\\scriptstyle (b,y_2)$} (q6)\n(q6) edge[loop below] node{$\\scriptstyle \\blank$} ()\n(q7) edge[loop below] node{$\\scriptstyle \\blank$} ()\n     edge[bend left=40]             node[very near start, above, yshift=5]{$\\scriptstyle (b,y_1)$} (q6)\n;\n\\end{tikzpicture}\n\\caption{A QDA expressing the property that the data on the even\n  positions in the list is sorted.} \\label{fig:even-sorted-QDA}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Appendix B}\n\nAngluin \\cite{DBLP:journals\/iandc\/Angluin87} introduced a popular learning framework, which is originally designed to learn regular languages. In this framework, a \\emph{learner} (or learning algorithm) learns a regular language $L$, the so-called \\emph{target language}, over an a priory fixed alphabet $\\Sigma$ by actively querying a \\emph{teacher}. The teacher is capable of answering two different kinds of queries: \\emph{membership} and \\emph{equivalence queries}. On a membership query, the learner presents a word $w \\in \\Sigma^\\ast$, and the teacher replies ``yes'' or ``no'' depending on whether $w$ belongs to $L$ or not. On an equivalence query, the learner conjectures a regular language $H \\subseteq \\Sigma^\\ast$, typically given as a finite automaton, and the teacher checks whether $H$ is an equivalent description of $L$. If this is the case, he replies ``yes''. Otherwise, he returns a \\emph{counter-example} $w \\in L \\Leftrightarrow w \\not\\in H$.\n\nIn \\cite{DBLP:journals\/iandc\/Angluin87}, Angluin presented a learning algorithm that learns a regular language in time polynomial in the size of the (unique) minimal deterministic finite automaton accepting the target language and the length of the longest counter-example returned by the teacher. Angluin's algorithm maintains a prefix-closed set $S \\subseteq \\Sigma^\\ast$, a suffix-closed set $E \\subseteq \\Sigma^\\ast$, and stores the learned data in a table (realized as a mapping $T\\colon (S \\cup S\\Sigma) E \\to \\{0,1\\}$), whose rows are labeled with strings from $S$ and whose columns are labeled with string from $E$. The key idea of the algorithm is to approximate the Nerode congruence of the target language using strings from $S$ as representatives for the equivalence classes and strings from $E$ as samples to distinguish these classes. New strings are added to $S$ and $E$ whenever necessary until an equivalence query reveals that the conjectured automaton is equivalent to the target language. \n\nAlthough originally introduced to learn regular languages, this algorithm can be easily lifted to the learning of Moore machines. In this setting, the ``target language'' is a finite-state computable mapping $\\lambda \\colon \\Sigma^\\ast \\to \\Gamma$ (i.e., a mapping computable by a Moore machine) that maps each word $w \\in \\Sigma^\\ast$ to some output $\\lambda(w)$ taken from a finite set $\\Gamma$ of output symbols. (We obtain Angluin's setting for $\\Gamma = \\{0,1\\}$.) Moreover, membership queries ask now for the output---or classification---of a word rather then whether it belongs to a language or not. Finally, on an equivalence query, the learner proposes a Moore machine $\\mathcal M$. If $\\mathcal M$ is not equivalent to the target language, the teacher returns a counter-example $w$ such that the output of $\\mathcal M$ on $w$ is different from $\\lambda(w)$.\n\n\nAdapting Angluin's algorithm to work with Moore machines is straightforward. Since the Nerode congruence can easily be lifted to the Moore machine setting, it is indeed enough to change the table to a mapping $T\\colon (S \\cup S\\Sigma) E \\to \\Gamma$; everything else can be left unchanged.  \nThis adapted algorithm also learns the minimal Moore machine for the target language in time polynomial in this minimal Moore machine and the length of the longest counter-example returned by the teacher.\n\n\n\\section*{Appendix C}\n\n\nIn this appendix we describe, in a greater detail, the translation from an EQDA $\\mathcal{A}$ to a formula $\\mathcal{T}(\\mathcal{A})$ expressed in $\\Strand$ or the APF such that the set of data words accepted by $\\mathcal{A}$ corresponds to the program configurations $\\mathcal{C}$ which model $\\mathcal{T}(\\mathcal{A})$.\n\nRecall the formal definition of an EQDA from Section~\\ref{sec:elastic}. In an EQDA $\\mathcal{A} = (Q, q_0, \\Pi, \\delta, f)$ over program variables $PV$ and universal variables $Y$, each transition on $\\blank$ is a self loop. Without restricting the class of languages accepted by $\\mathcal{A}$ we assume, for the purpose of translation, that our EQDAs have three additional properties.\n\nFirstly, we assume that any path in the EQDA, along which a universal variable occurs together with auxiliary variables like $nil$ which are introduced by the encoding from Section~\\ref{sec:modeling}, goes to the formula $true$. This does not change the language accepted by the automaton as it still accepts all data words respecting the formula constraints for other valuations of the universal variables and where the data at these auxiliary variables can be any data value.\n\n\\begin{figure}\n\\vspace{-0.4cm}\n\t\\centering\n\t\\subfigure[]{\n\t\t\\begin{tikzpicture}\n\t\t\t\\node[state]             (1) {$q_1$};\n\t\t\t\\node[state, right of=1] (2) {$q_2$};\n\t\t\t\\draw[<-] (1.west) -- +(-.5, 0);\n\t\t\t\\draw[->] (2.east) -- +(.5, 0);\n\t\t\t\\draw[->] (1) edge             node {\\scriptsize $(b, y)$} (2)\n\t\t                   (2) edge[loop above] node {\\scriptsize $\\blank$} ();\n\t\t\\end{tikzpicture}\n\t\t\\label{fig:relevant-loops-a}\n\t}\n\t%\n\t\\hskip 3em\n\t%\n\t\\subfigure[]{\n\t\t\\begin{tikzpicture}\n\t\t\t\\node[state]             (1) {$q_1$};\n\t\t\t\\node[state, right of=1] (2) {$q_2$};\n\t\t\t\\draw[<-] (1.west) -- +(-.5, 0);\n\t\t\t\\draw[->] (2.east) -- +(.5, 0);\n\t\t\t\\draw[->] (1) edge             node {\\scriptsize $(b, y)$} (2)\n\t\t                       edge[loop above] node {\\scriptsize $\\blank$} ();\n\t\t\\end{tikzpicture}\n\t\t\\label{fig:relevant-loops-b}\n\t}\n\\caption {Base cases for detecting irrelevant self-loops.}\n\\vspace{-0.3cm}\n\\end{figure}\n\nSecondly, we assume that the EQDA has no \\emph{irrelevant} loops which are defined inductively as follows:\nfix a simple path $p$ of an EQDA $\\A$ that leads from the initial to an\naccepting state, and on $p$ consider a state $q_1$ (see\nFig~\\ref{fig:relevant-loops-a}) which reads a universal variable $(b,\ny)$ on a transition to state $q_2$. If $q_2$ has a self loop on the\nblank symbol, i.e. $\\delta(q_2, \\underline{b}) = q_2$, then this loop\nis inductively defined to be \\emph{irrelevant on $p$} if $q_1$ has no\nself-loop, or if the self-loop at $q_1$ is also irrelevant on\n$p$. Symmetrically, a self-loop at $q_1$ is irrelevant on $p$ if $q_2$\nhas no self-loop or has one which is irrelevant on $p$ (see\nFig~\\ref{fig:relevant-loops-b}).  If a self-loop is irrelevant on $p$,\nthen it can be omitted for words accepted along the path $p$.  To see\nwhy, consider a valuation word $v = \\dots\n(b, y)~ \\blank\\dots$ that is accepted along $p$ using the self-loop in\nFigure~\\ref{fig:relevant-loops-a}.  A different valuation $v' = \\dots \\blank~(b, y)\\dots$ of the\nsame data word is rejected by $\\mathcal{A}$\nsince $q_1$ has no transition on $\\underline{b}$. Hence, the data word\ncorresponding to $v$ is not accepted by $\\A$.\nWe can remove irrelevant loops from $\\A$ without changing the accepted\ndata language by simply removing those loops that are irrelevant on\neach path they occur on, or by splitting states if they have a\nself-loop that is irrelevant only on some paths.\n\n\nThirdly, we assume that the universal variables are read by the EQDA in a particular order and all paths in the EQDA that do not respect this order lead to the formula $\\true$.\nThe translation that we give below considers each path of the automaton separately. Thus, if the automaton does not satisfy the above property, then for any path that does not read the variables in the correct order we rename the variables on the transitions and in the data formulas along that path accordingly before the translation.\n\nLet us now turn to the translation of the paths. We observe that all variables appear exactly once in any valuation word accepted by $\\mathcal{A}$. Since we disallow universal variables to appear together, this is ensured by adding some dummy symbols where these variables can appear in case the valuation word is too short. A consequence of this property is that there can be no cycle in our EQDA model which shares an edge labeled with a (universal or pointer) variable. Consider a simple path $p$ of the automaton from the initial state $q_0$ to the output state $q_p$,\n$q_0 \\xrightarrow{\\pi_0} \\ldots \\xrightarrow{\\pi_{n-1}} q_p$ ($\\pi_i \\in \\Pi \\neq \\underline{b}$). Below we informally describe the translation $\\mathcal{T}$  from path $p$ to a formula $\\phi_p$ which captures the relative positions of the pointer and universal variables along $p$ and forms the guard of a universally quantified implication in a conjunct of the translated formula.\nAt a higher level, whenever a state $q$  in path $p$ has a self-loop on the blank symbol $\\underline{b}$, pointers and universal variables $v_1, v_2 \\in PV \\cup Y$ labeled along the incoming and outgoing transitions of this state are constrained by the relation $v_1 < v_2$ or $v_1 \\rightarrow^+ v_2$. The presence of a self loop ensures that the variables are related by an elastic relation which is required for decidability in \\Strand and APF. On the other hand, if  $q$ has no transition on $\\blank$, then the pointers labeled along the incoming and outgoing transitions are constrained by the successor relation. Note that successor is an inelastic relation and is not allowed to relate two universal variables. In this case we identify a state $q'$ on path $p$, closest to $q$, which has a transition on some pointer (non-universal) variable $pv$. Since we have already stripped our EQDA of all \\emph{irrelevant } loops, the subpath from $q'$ to $q$ has no self-loops. Thus, the universal variables at $q$ can be constrained to be a fixed distance away from the pointer $pv$. This is allowed in APF using arithmetic on the pointer variables. For \\Strand, the same effect can be obtained by introducing a monadic predicate which tracks the distance of the universal variable from the pointer variable $pv$.\n\nWe skip a formal description of the translation. A subtle case to note, however, is when a state $q$ in path $p$ has a self loop on the blank symbol $\\underline{b}$ and the incoming and outgoing transitions on $q$ are both labeled by letters of the form $(b, y)$ where $y \\in Y$. Unlike \\Strand, APF forbids two adjacent universal variables $y_1, y_2$ to be related by $<$. And so for the case of arrays, translation $\\mathcal{T}(p)$ constrains these universal variables as $y_1 \\leq y_2$.  Moreover, we modify the output of the final state\nalong this path $f_{\\mathcal{A}}(q_p)$ to include the data constraint $d(y_1) = d(y_2)$ if it was not already implied by the output formula. Note that at this point the constraint does not capture the exact semantics of the automaton.\n\n\nThe universally quantified formula that is captured by this particular path $p$ is\n$\\forall Y. ~\\phi_p \\Rightarrow f_{\\mathcal{A}}(q_p)$. We construct these\nimplications for all simple paths in the EQDA and conjunct them to get the\nfinal formula. All other paths in $\\mathcal{A}$ semantically go to\n$\\false$. Hence, we also add a conjunct $\\forall Y.~ \\neg (\\bigvee_p\n\\phi_p) \\Rightarrow \\mathit{false}$. So for an EQDA $\\mathcal{A}$,\n$\\mathcal{T}(\\mathcal{A}) = \\bigwedge_p \\forall Y.~ \\phi_p \\Rightarrow f_{\\mathcal{A}}(q_p) ~\\bigwedge~ \\forall Y.~\\neg (\\bigvee_p \\phi_p) \\Rightarrow \\textit{false}$.  \nSince negation is arbitrarily allowed over atomic formulas\nin \\Strand, $\\mathcal{T}(\\mathcal{A})$ is clearly in the decidable\nfragment of \\Strand. APF also allows negation over atomic formulas\nwhich relate two pointer variables or a universal variable with a\npointer variable.  However, negation of an atomic formula $y_1 \\leq\ny_2$ is not allowed in APF. But since we assume for the translation\nthat the automaton considers a fixed variable ordering on $Y$ and all\nother paths with a different ordering lead to $\\true$, we can simply\nremove negations of formulas $y_1 \\leq y_2$ from $\\neg \\bigvee_p\n\\phi_p$.\n\n\n\n\\section*{Appendix D}\n\nIn this appendix, we sketch how an active learning algorithm can be used to learn program invariants expressible in the array property fragment and the \\Strand decidable fragment over lists.\nInvariant synthesis \ncan be achieved using\n\\emph{two} distinct procedures: (a) building the learner according to the learning algorithm described in Section~\\ref{sec:learning}, and (b) building\na teacher which can answer questions about invariant for a particular program.\nAn acceptable invariant for a program, in general, has to satisfy \\emph{three} properties:\n it must include the pre-condition,\n it must be contained in the post-condition, and\n it must be inductive.\nMoreover, in order to certify the above indeed hold, the invariant should be expressible in a logic that permits\na decidable satisfiability problem for the above conditions.\n\nBuilding an adequate teacher is not easy as the\ninvariant is \\emph{unknown}, and the whole point of learning is to find the invariant.\nStill the teacher certainly has \\emph{some knowledge} about the set of structures\nin the invariant and can answer certain questions with certainty. For example, when asked whether a data word $w$ belongs\nto the invariant $I$, if $w$ belongs to the pre-condition (or the strongest post-condition of the pre-condition), \n she can definitely say that $w$ belongs to $I$.\nAlso, when $w$ belongs to the negation of the post-condition (or to the weakest pre-condition of the negated post-condition), the teacher can definitely answer that $w$ does not belong to $I$.\nFor other queries, in general, the teacher\ngives arbitrary answers\nand these answers determine the kind of invariant that is finally learned.\n Turning to equivalence queries, if the learned invariant falls within a decidable fragment (as is ensured by the above learning algorithm) and the pre\/post-condition and the program body is such that the verification conditions \nare expressed in appropriate decidable logics\n(\\Strand\/APF),\na teacher is able to check if the conjectured\ninvariant is adequate and satisfies the above three conditions.\nIf the invariant is inadequate and does \nnot include the pre-condition or intersects the negation of the postcondition, then the teacher can find an appropriate counterexample\nto report to the learner. If the inadequacy is due to the conjecture\nnot being inductive, then the teacher would find a pair of configurations\n$(c,c')$ such that $c$ is allowed by the conjecture\nwhile $c'$ is reachable from $c$ and is excluded from it,\nand decide\nto either report $c$ or $c'$ as a counterexample. This choice \nagain determines the final invariant being learned, similar to membership queries that the teacher is unsure about.\n\nThe idea is to pit such a teacher against a learner in order to learn the invariant, \\emph{despite} the fact that the teacher\ndoes not know the invariant herself. The learner's objective is to learn the \\emph{simplest} data automaton that captures the knowledge the teacher has.\nThe key property that the learner relies on is Occam's razor --- that the simplest set (i.e., the automaton with the least number of states) consistent with the queries answered by the teacher is a likely invariant. Note that the learner will not, in general, simply learn an automaton that captures precisely the knowledge of the inadequate teacher;\nrepresentation of this knowledge is often far more complex than a true invariant. In other words,\nthe learner will learn the simplest automaton that \\emph{generalizes} the partial knowledge the teacher has.\n\n\n\n\\newpage\n\\section*{Appendix E}\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[align=center,node distance=40]\n\\node[state](q0){$q_0$};\n\\node[state,below right=1.7cm and 1.0cm of q0](q1){$q_1$};\n\\node[state,below left=1.7cm and 1.0cm of q0](q2){$q_2$};\n\\node[state,below of=q2](q8){$q_8$};\n\\node[state,below left=1.7cm and 1.5cm of q8](q7){$q_7$};\n\\node[state,below right=2.2cm and 0.3cm of q1](q6){$q_6$};\n\\node[state,below left=1.3cm and 1.8cm of q6](q4){$q_4$};\n\\node[state,below left=1.7cm and 0.5cm of q4](q10){$q_{10}$};\n\\node[state,below right=1.7cm and 1.3cm of q4](q12){$q_{12}$};\n\\node[state,accepting,below left=1.0cm and 0.5cm of q10](q13){$q_{13}$};\n\\node[state,accepting,below right=1.0cm and 0.5cm of q12](q11){$q_{11}$};\n\\node[state,below right=1.8cm and 1.6cm of q6](q5){$q_{5}$};\n\\node[state,below right=1cm and 1.9cm of q5](q3){$q_3$};\n\\node[state,accepting,below of=q3](q9){$q_9$};\n\\node[state,above=4cm of q3](q14){$q_{14}$};\n\n\\path (q13) node[below,yshift=-10]{$\\scriptstyle d(y_1) \\le d(y_2) \\wedge$}\n -- (q13) node[below,yshift=-20]{$\\scriptstyle d(y_1) < k \\wedge d(y_2) < k$}\n -- (q11) node[below,yshift=-10]{$\\scriptstyle d(y_1) \\le d(y_2) \\wedge d(y_1) < k$}\n -- (q9) node[below,yshift=-15]{$\\scriptstyle d(y_1) \\le d(y_2)$}\n\n;\n\\draw[<-, shorten <=1pt] (q0.north) -- +(0, .3);\n\\draw[->]\n(q0) edge node{$\\scriptstyle (\\{nil\\}, \\yblank)$} (q1)\n(q0) edge node[left]{$\\scriptstyle (\\{cur,nil\\}, \\yblank)$} (q2)\n(q2) edge node[near start]{$\\scriptstyle (\\{head\\}, \\yblank)$} (q8)\n(q8) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q8) edge node[near end, yshift=4]{$\\scriptstyle (b, y_1)$} (q7)\n(q2) edge[bend right] node[left]{$\\scriptstyle (\\{head\\}, y_1)$} (q7)\n(q7) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q7) edge node[left]{$\\scriptstyle (b, y_2)$} (q13)\n(q13) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q1) edge[bend right] node[left, yshift=-17, xshift=-2]{$\\scriptstyle (\\{head\\}, y_1)$} (q4)\n(q4) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q4) edge node[left]{$\\scriptstyle (b, y_2)$} (q10)\n(q10) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q10) edge node[xshift=-3]{$\\scriptstyle (\\{cur\\}, \\yblank)$} (q13)\n(q12) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q11) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q12) edge node[left]{$\\scriptstyle (b, y_2)$} (q11)\n(q4) edge node[left, very near end, yshift=6, xshift=-3]{$\\scriptstyle (\\{cur\\}, \\yblank)$} (q12)\n(q4) edge[bend left] node[right]{$\\scriptstyle (\\{cur\\}, y_2)$} (q11)\n\n(q1) edge node[left, very near end, yshift=7]{$\\scriptstyle (\\{head\\}, \\yblank)$} (q6)\n(q6) edge node[left, near start]{$\\scriptstyle (b, y_1)$} (q4)\n(q6) edge[loop left] node{$\\scriptstyle \\blank$} ()\n(q6) edge[bend right] node[left, yshift=14, xshift=-12]{$\\scriptstyle (\\{cur\\}, y_1)$} (q3)\n(q3) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q5) edge node[xshift=-6]{$\\scriptstyle (b, y_1)$} (q3)\n(q5) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q6) edge node[very near start, yshift=-8]{$\\scriptstyle (\\{cur\\}, \\yblank)$} (q5)\n(q1) edge[bend left] node[left, yshift=3]{$\\scriptstyle (\\{head, cur\\}, y_1)$} (q3)\n\n(q3) edge node{$\\scriptstyle (b, y_2)$} (q9)\n(q9) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q14) edge[loop right] node{$\\scriptstyle \\blank$} ()\n(q1) edge node[above, near end, yshift=5, xshift=-2]{$\\scriptstyle (\\{head, cur\\}, \\yblank)$} (q14)\n(q14) edge node{$\\scriptstyle (b, y_1)$} (q3)\n;\n\\end{tikzpicture}\n\\caption{The EQDA expressing the invariant of the program which finds a key $k$ in a sorted list. Here \\emph{head} and \\emph{cur} are pointer variables and {\\emph k} is an integer variable in the program.}\n\\end{figure}\n\n\n\n\n\\section*{Appendix B}\n\n\n\n\\vspace{-0.1cm}\n\\begin{theorem}\n\\label{thm-canonical}\nFor each QDA $\\A$ there is a unique minimal QDA $\\A'$ that accepts the\nsame set of valuation words.\n\\end{theorem}\n\\vspace{-0.1cm}\n\\emph{Proof}\nConsider a language $\\Lval$ of valuation words that can be accepted by\na QDA, and let $w \\in \\Pi^*$ be a symbolic word. Then there must be a\nformula in the lattice that characterizes precisely the data\nextensions $v$ of $w$ such that $v$ in $\\Lval$. Since we assume that all\nthe formulas in the lattice are pairwise non-equivalent, this formula is uniquely determined. \nIn fact, take any QDA $\\A$ that accepts $\\Lval$. Then $w$ leads to some state $q$ in\n$\\A$ that outputs some formula $f(q)$. If $w$ leads to any other\nformula in another QDA $\\A'$, then $\\A'$ accepts a different language of\nvaluation words.\n\nThus, a language of valuation words can be seen as a function that\nassigns to each symbolic word a uniquely determined formula, and a QDA can be viewed as a\nMoore machine that\ncomputes this function. For each such Moore machine there exists a\nunique minimal one that computes the same function, hence the theorem.\n\\qed\n\n\nAs the proof above shows, we can view a language of valuation words as a function that\nmaps to each symbolic word a uniquely determined formula, and a QDA can be viewed as a\nMoore machine (an automaton with output function on states) that\ncomputes this function.\n\n\n\n\n\n\\section*{Appendix C}\n\n\n\n\nThe learning algorithm that we use to synthesize QDAs does not\nconstruct EQDAs in general. However, we can show the following surprising \nresult that every QDA $\\A$ can be \\emph{uniquely over-approximated}\nby a language of valuation words that can be\naccepted by an EQDA $\\AEL$. \nWe will refer to this construction, which we outline below, as \\emph{elastification}.\nThis construction crucially\nrelies on the particular structure that elastic automata have, which forces\na unique set of words to be added to the language in order to make it elastic.\n\nLet $\\A = (Q, q_0, \\Pi, \\delta, f)$ be a QDA and for a state $q$ let\n$R_{\\blank}(q):= \\{q' \\mid q \\xrightarrow{\\blank}^* q' \\}$\nbe the set of states reachable from $q$ by a (possibly empty) sequence\nof $\\blank$-transitions.  For a set $S \\subseteq Q$ we let\n$R_{\\blank}(S) := \\bigcup_{q \\in S} R_{\\blank}(q)$.\n\nThe set of states of $\\AEL$ consists of sets of states of $\\A$ that\nare reachable from the initial state $R_{\\blank}(q_0)$ of $\\AEL$ by\nthe following transition function (where $\\delta(S,a)$ denotes the\nstandard extension of the transition function of $\\A$ to sets of\nstates):\\\\\n$\\delta_{\\text{el}}(S,a) =\n\\begin{cases}\nR_{\\blank}(\\delta(S,a)) & \\mbox{if } a \\not= \\blank \\\\\nS  & \\mbox{if } a = \\blank \\mbox{ and $\\delta(q,\\blank)$ is defined  for some $q \\in S$} \\\\\n\\mbox{undefined}  & \\mbox{otherwise.} \\\\\n\\end{cases}\n$\\\\\nNote that this construction is similar to the usual powerset construction\nexcept that in each step we apply the transition function of $\\A$ to the\ncurrent set of states and take the $\\blank$-closure. However, if the\ninput letter is $\\blank$, $\\AEL$ loops on the current\nset if a $\\blank$-transition is defined for some state in the set.\n\nThe final evaluation formula for a set is the least upper bound of the\nformulas for the states in the set:\n$f_{\\text{el}}(S) = \\bigsqcup_{q \\in S}f(q)$. \nWe can now show that $\\AEL$  is the \n\\emph{most precise over-approximation} of the language of valuation words accepted by QDA $\\A$.\n\n\\vspace{-0.15cm}\n\\begin{theorem}\\label{thm:elastification}\nFor every QDA $\\A$, the EQDA $\\AEL$ satisfies \n$\\Lval(\\A) \\subseteq \\Lval(\\AEL)$, and \nfor every EQDA $\\B$ such that $\\Lval(\\A) \\subseteq \\Lval(\\B)$,\n $\\Lval(\\AEL) \\subseteq \\Lval(\\B)$ holds.\n\\end{theorem}\n\\emph{Proof:~}\nNote that $\\AEL$ is elastic by definition of $\\delta_{\\text{el}}$. It\nis also clear that $\\Lval(\\A) \\subseteq \\Lval(\\AEL)$ because for each\nrun of $\\A$ using states $q_0 \\cdots q_n$ the run of $\\AEL$ on the\nsame input uses sets $S_0 \\cdots S_n$ such that $q_i \\in S_i$, and by\ndefinition $f(q_n)$ implies $f_{\\text{el}}(S_n)$.\n\nNow let $\\B$ be an EQDA with $\\Lval(\\A) \\subseteq \\Lval(\\B)$. Let $w =\n(a_1,d_1) \\cdots (a_{n},d_{n})\\in \\Lval(\\AEL)$ and let $S$ be the\nstate of $\\AEL$ reached on $w$.  We want to show that $w \\in\n\\Lval(\\B)$.  Let $p$ be the state reached in $\\B$ on $w$. We show that\n$f(q)$ implies $f_\\B(p)$ for each $q \\in S$. From this we obtain\n$f_{\\text{el}}(S) \\Rightarrow f_\\B(p)$ because $f_{\\text{el}}(S)$ is\nthe least formula that is implied by all the $f(q)$ for $q \\in S$.\n\n\n\nPick some state $q \\in S$. By definition of $\\delta_{\\text{el}}$ we\ncan construct a valuation word $w'$ that leads to the state $q$ in\n$\\A$ and has the following property: if all letters of the form\n$(\\blank,d)$ are removed from $w$ and from $w'$, then the two\nremaining words are the same. In other words, $w$ and $w'$ can be\nobtained from each other by inserting and\/or removing\n$\\blank$-letters.\n\n\nSince $\\B$ is elastic, $w'$ also leads to $p$ in $\\B$. From this we\ncan conclude that $f(q) \\Rightarrow f(p)$ because otherwise there\nwould be a model of $f(q)$ that is not a model of $f(p)$ and by\nchanging the data values in $w'$ accordingly we could produce an input\nthat is accepted by $\\A$ and not by $\\B$.\n\\qed\n\n\\subsection*{A Teacher for Invariants over Linear Data Structures}\n\n\n\\noindent {\\bf Implementing the learner.}\nWe used the \\libalf library \\cite{DBLP:conf\/cav\/BolligKKLNP10} as an implementation of the active learning algorithm~\\cite{DBLP:journals\/iandc\/Angluin87}.\nWe adapted its implementation to our setting by modeling QDAs as Moore machines.\nIf the learned QDA is not elastic, we elastify it as described in Section~\\ref{sec:elastic}.\nThe result is then converted to a quantified formula over \\Strand or the APF\nand we check if the learned invariant was adequate using a constraint solver.\n\\begin{table*}[th]\n\t\\centering \\footnotesize\n\t\\vspace{-0.55cm}\n\n\n\n\n\n\n    \n     \n\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\t\n\\begin{tabular}{||l|| r| r||r|r|r|c|r||}\n\t\\hline\n\t~~~Example & \\#Test & $T_{teacher}$ & \\#Eq. & \\#Mem. & Size~~ & Elastification & $T_{learn}$ \\\\\n\t             &    inputs &   (s) &    &       & \\#states & required ?   & (s)\\\\\\hline\t\n\narray-find\t    &310\t& 0.05 &  2 & 121   & 8  & no  &  0.00 \\\\\\hline\narray-copy\t&7380 & 1.75   & 2  & 146   & 10 & no   & 0.00 \\\\\\hline\narray-comp\t&7380 & 0.51    & 2  & 146   & 10 & no  & 0.00 \\\\\\hline\nins-sort-outer  &363 & 0.19    & 3  & 305   & 11 & no   & 0.00 \\\\\\hline\nins-sort-innner &363& 0.30  & 7  & 2893  & 23 & yes  & 0.01 \\\\\\hline\nsel-sort-outer  &363& 0.18  & 3  & 306   & 11 & no   & 0.01 \\\\\\hline\nsel-sort-inner  &363& 0.55   & 9 &  6638  & 40 & yes  & 0.05 \\\\\\hline\\hline\n\n\nlist-find\t    &111& 0.04   & 6 & 1683   & 15 & yes & 0.01 \\\\\\hline\nlist-insert\t    &111& 0.04   & 3 & 1096   & 20 & no   & 0.01 \\\\\\hline\nlist-init\t    &310& 0.07   & 5 & 879    & 10 & yes  & 0.01 \\\\\\hline\nlist-max\t    &363& 0.08   & 7 & 1608   & 14 & yes  & 0.00 \\\\\\hline\nlist-merge\t    &5004& 10.50  & 7 & 5775   & 42 & no   & 0.06 \\\\\\hline\nlist-partition  &16395 & 11.40    & 10 & 11807 & 38 & yes  & 0.11 \\\\\\hline\nlist-reverse\t&27& 0.02    & 2 & 439    & 18 & no   & 0.00 \\\\\\hline\n\n\nlist-bubble-sort   &363& 0.19 & 3 & 447    & 12 & no   & 0.01 \\\\\\hline\nlist-fold-split\t  &1815& 0.21   & 2 & 287    & 14 & no  & 0.00 \\\\\\hline\nlist-quick-sort\t   &363 & 0.03 & 1 & 37     & 5  & no  & 0.00 \\\\\\hline\n\nlist-init-cmplx    &363& 0.05 & 1 & 57     & 6  & no   & 0.01 \\\\\\hline\\hline\n\nlookup\\_prev   &111& 0.04  & 3  & 1096  & 20  & no   & 0.01 \\\\\\hline\nadd\\_cachepage &716& 0.19  & 2  & 500   & 14  & no   & 0.01 \\\\\\hline\n\nsort (merge)\t\t&363& 0.04  & 1 & 37     & 5  & no   & 0.00 \\\\\\hline\ninsert\\_sorted\t\t&111& 0.04  & 2 & 530     & 15 & no   & 0.01 \\\\\\hline\n\ndevres\t\t   &372& 0.06 & 2  & 121   & 8   & no   & 0.00 \\\\\\hline\nrm\\_pkey\t\t   &372& 0.06 & 2  & 121   & 8   & no   & 0.00 \\\\\\hline\\hline\n\n\\multicolumn{8}{|c|}{Learning Function Pre-conditions}\\\\\\hline\nlist-find\t\t    &111& 0.01   & 1 & 37    & 5  & no   & 0.00 \\\\\\hline\nlist-init\t\t    &310& 0.02   & 1 & 26\t& 4  & no   & 0.00 \\\\\\hline\nlist-merge\t\t    &329& 0.06   & 3 & 683   & 19 & no   & 0.01 \\\\\\hline\n\n\t\\end{tabular}\n\t\\caption{Results of our experiments.\\label{tbl:experimental_results}}\n\\vspace{-1cm}\n\\end{table*}\n\n{\\bf Experimental Results.}\\footnote{Our prototype implementation along with the results for all our programs can be found at \\url{http:\/\/automata.rwth-aachen.de\/~neider\/learning_qda\/}}.\nWe evaluate our approach on a suite of programs  (see Table~\\ref{tbl:experimental_results}) for learning invariants and preconditions.\nFor every program, we report the\nthe number of test inputs and the time ($T_{teacher}$) taken to build the teacher from the samples collected along these test runs. We also report the number of equivalence and membership queries answered by the teacher in the active learning algorithm, the size of the final elastic automata, whether the learned QDA required any elastification and finally, the time ($T_{learn}$) taken to learn the QDA.\n\nThe array programs are programs for finding a key in an array, copying and comparing two arrays, and inner and outer loops of insertion and selection sort over an array.\nThe list programs find and insert a key in a sorted list, initialize a list, return the maximum data value in a list, merge two disjoint lists, partition a list into two lists depending on a predicate and reverse in-place a sorted list. The programs bubble-sort, fold-split and quick-sort are taken from~\\cite{celia}.\nThe program \\emph{list-init-cmplx}\nsorts an input array using heap-sort and then initializes a list with the contents of this sorted array.\nSince heap-sort is a complex algorithm that views an array as a binary tree,\nnone of the current automatic white-box techniques for invariant synthesis can handle such complex programs.\nHowever, our learning approach being black-box, we are able to learn the correct invariant, which is that the list is sorted. Similarly synthesizing post-condition annotations for recursive procedures like merge-sort and quick-sort is in general difficult for white-box techniques, like \\emph{interpolation}, which require a post-condition.\n\nThe methods \\emph{lookup\\_prev} and \\emph{add\\_cachepage} are from the module cachePage in a verified-for-security platform for mobile applications~\\cite{asplos13}. The module cachePage maintains a cache of the recently used disc pages as a priority queue based on a sorted list.\nThe method \\emph{sort} is a merge sort implementation and \\emph{insert\\_sorted} is a method for insertion into a sorted list. Both these methods are from Glib which is a low-level C library that forms basis of the GTK+ toolkit and the GNOME environment.\nThe methods \\emph{devres}\\footnote{method \\texttt{pcim\\_iounmap} in Linux kernel at \\texttt{linux\/lib\/devres.c}} and \\emph{rm\\_pkey}\\footnote{from InfiniBand device driver at \\texttt{drivers\/infiniband\/hw\/ipath\/ipath\\_mad.c}} are methods from the Linux kernel and an Infiniband device driver, both mentioned in~\\cite{wang-aplas10}.\n\n\n\n\nAll experiments were completed on an Intel Core i5 CPU at 2.4GHz with 6GB of RAM.\nFor all examples, our prototype implementation learns an adequate invariant really fast. Though the learned QDA might not be the smallest automaton representing the samples $S$ (because of the inaccuracies of the teacher),\nin practice we find that they are reasonably small (less than 50 states).\nMoreover, we verified that the learned invariants were adequate for proving\nthe programs correct by generating verification conditions and validating them using an SMT solver\n(these verified in less than 1s).\nLearnt invariants are complex in some programs; for example  the invariant QDA for the program \\emph{list-find} is presented in Appendix~E and corresponds to:\\\\\n{\\small\n$head \\neq nil \\wedge (\\forall y_1 y_2. head \\rightarrow^* y_1 \\rightarrow^* y_2 \\Rightarrow d(y_1) \\leq d(y_2)) \\wedge ((cur = nil \\wedge \\forall y_1. head \\rightarrow^* y_1 \\Rightarrow d(y_1) < k) \\vee (head \\rightarrow^* cur \\wedge \\forall y_1. head \\rightarrow^* y_1 \\rightarrow^+ cur \\Rightarrow d(y_1) < k))$.\n}\n\n\n\n\\subsection*{Formula Words}\n\nA \\emph{formula word} over $PV$, ${\\cal F}$, and $Y$ is a word over\n$(\\Pi^* \\times \\mathcal{F})$ where, as before, $\\Pi = \\Sigma \\times\n(Y \\cup \\{\\yblank\\})$ and each $p \\in PV$ and\n$y \\in Y$ occurs exactly once in the word. Note that a formula word\ndoes not contain elements of the data domain -- it simply consists of\nthe symbolic word that depicts the pointers into the list (modeled\nusing $\\Sigma$) and a valuation for the quantified variables in $Y$\n(modeled using the second component) as well as a formula over the\nlattice $\\mathcal{F}$. For example,\n$\\bigl( (\\{h\\},y_1) (b, \\yblank) (b, y_2) (\\{t\\}, \\yblank), d(y_1)\\leq d(y_2) \\bigr)$\nis a formula word, where $h$ points to the first element, $t$ to the last element, $y_1$ points to the first element, and $y_2$ to the third element; and the data formula is $d(y_1) \\leq d(y_2)$.\n\nBy using formula words we explicitly take the view of a QDA as a Moore machine that reads symbolic words and outputs data formulas. A formula word $(u, \\alpha)$ is accepted by a QDA $\\mathcal A$ if $\\mathcal A$ reaches the state $q$ after reading $u$ and $f(q) = \\alpha$. Hence, a QDA defines a unique language of formula words in the usual way.\nBy means of formula words, we can reduce the problem of learning QDAs to the problem of learning Moore machines.\nNext, we briefly sketch the learning framework we use for learning QDAs.\\\\\n\\noindent {\\bf Actively learning QDAs}:\nAngluin \\cite{DBLP:journals\/iandc\/Angluin87} introduced a popular learning framework in which a \\emph{learner}\nlearns a regular language $L$, the so-called \\emph{target language}, over an a priory fixed alphabet $\\Sigma$ by actively querying a \\emph{teacher} which is capable of answering  \\emph{membership} and \\emph{equivalence queries}.\nAngluin's algorithm\nlearns a regular language in time polynomial in the size of the (unique) minimal deterministic finite automaton accepting the target language and the length of the longest counterexample returned by the teacher.\nThis algorithm can however be easily lifted to the learning of Moore machines (see Appendix~B for details).\nMembership queries now ask for the output or classification of a word.\nOn an equivalence query, the teacher says ``yes'' or returns a counter-example $w$ such that the output of\nthe conjecture on $w$ is different from the output on $w$ in the target language.\nAs  QDAs can viewed as Moore languages (since it's just a set of words with output being data-formulas), \nwe can apply Angluin's algorithm directly in order to learn a QDA, and obtain the following theorem.\n\n\\vspace{-0.07cm}\n\\begin{theorem}\nGiven a teacher for a QDA-acceptable language of formula words that can answer membership and equivalence queries, the unique minimal QDA for this language can be learned in time polynomial in this minimal QDA and the length of the longest counterexample returned by the teacher.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection*{List and Array Invariants:}\n\nConsider a typical invariant in a sorting program over lists where the loop invariant is expressed as:\\\\\n$\\textit{head} \\rightarrow^* i ~~\\wedge ~~\\forall y_1, y_2. ((\\textit{head} \\rightarrow^* y_1 \\wedge \\textit{succ}(y_1, y_2) \\wedge y_2  \\rightarrow^* i ) \\Rightarrow d(y_1) \\leq d(y_2))~~~(1)$\\\\\nThis says that for all cells $y_1$ that occur somewhere in the list pointed to by $\\textit{head}$ and where $y_2$ is the successor of $y_1$, and where $y_1$ and $y_2$ are before the cell pointed to by a scalar pointer variable $i$, the data value stored at $y_1$ is no larger than the data value stored at $y_2$. \nThis formula is \\emph{not} in the decidable fragment of \\Strand since the universally quantified variables are involved in a non-elastic relation $\\textit{succ}$ (in the subformula $\\textit{succ}(y_1, y_2)$). Such an invariant for a program manipulating arrays can be expressed as:\\\\\n$~~~~~~~~~~~\\forall y_1, y_2. ((0 \\leq y_1 \\wedge y_2=y_1+1 \\wedge y_2 \\leq i) \\Rightarrow A[y_1] \\leq A[y_2])~~~~~~~~~~~~~~~~(2)$\\\\\nNote that the above formula is also not in the decidable array property fragment.\\\\\n\n{\\bf Quantified Data Automata:}\nThe key idea in this paper is an automaton model for expressing such constraints called \\emph{quantified data automata} (QDA). The above two invariants are expressed by the following QDA:\n\n\n\\begin{center}\n\t\\vspace{-3mm}\n\t\\begin{tikzpicture}\n\t\n\t\t\\node[state]                        (0) {$q_0$};\n\t\t\\node[state, right=2cm of 0]            (1) {$q_1$};\n\t\t\\node[state, right=2.5cm of 1]            (2) {$q_2$};\n\t\t\\node[state, right of=2]            (3) {$q_3$};\n\t\t\\node[state, right of=3, accepting] (4) {$q_4$};\n\t\t\\node[state, above right= 1cm and 0.5cm of 1, accepting] (5) {$q_5$};\n\n\t\n\t\t\\path (4) node[right, xshift=10] {$\\scriptstyle d(y_1) \\leq d(y_2)$}\n\t\t      -- (5) node[right,xshift=10,yshift=5]{$\\scriptstyle true$}\n\t\t;\n\t\n\t\t\\draw[<-, shorten <=1pt] (0.west) -- +(-.3, 0);\n\t\t\\draw[->] (0) edge                node       {$\\scriptstyle (\\mathrm{head}, \\yblank)$} (1)\n\t\t\t (0) edge[bend left] node[yshift=-3] {$\\scriptstyle (\\mathrm{\\{head, i\\}, *}), (\\mathrm{head}, y_2)$} (5)\n\t\t              edge[bend right=17] node[swap] {$\\scriptstyle (\\mathrm{head}, y_1)$}     (2)\n\t\t\t\t\t\t\t(1) edge                node       {$\\scriptstyle (b, y_1)$}                 (2)\n\t\t\t\t\t\t\t(1) edge node[right] {$\\scriptstyle (i, *), (b, y_2)$} \t\t    (5)\n\t\t\t\t\t\t\t\t\tedge[loop above]    node       {$\\scriptstyle \\blank$}                   ()\n\t\t\t\t\t\t\t(2) edge                node       {$\\scriptstyle (b, y_2)$}                 (3)\n\t\t\t\t\t\t\t(2) edge[bend right] node[right,xshift=2] {$\\scriptstyle \\blank,~ (i, \\yblank)$}  (5)\n\t\t\t\t\t\t\t    edge[bend right=25] node[swap] {$\\scriptstyle (i, y_2)$}                 (4)\n\t\t\t\t\t\t\t(3) edge                node       {$\\scriptstyle (i, \\yblank)$}             (4)\n\t\t\t\t\t\t\t\t\tedge[loop above]    node       {$\\scriptstyle \\blank$}                   ()\n\t\t\t\t\t\t\t(4) edge[loop above]    node       {$\\scriptstyle \\blank$}                   ()\n\t\t\t\t\t\t\t(5) edge[loop above]    node       {$\\scriptstyle *$}                   ()\n\t\t\t\t\t\t\t;\n\t\\end{tikzpicture}\n\\vspace{-3mm}\n\n\\end{center}\n\nThe above automaton reads (deterministically) data-words whose labels denote the positions\npointed to by the scalar pointer variables $\\textit{head}$ and\n$\\textit{i}$, as well as valuations of the quantified variables $y_1$\nand $y_2$. We use two \\emph{blank} symbols that indicate that no\nscalar variable (``$b$'') or no variable from $Y$ (``$\\yblank$'') is read\nin the corresponding component; moreover, $\\blank = (b, -)$.\nMissing transitions go to a sink state labeled \\textit{false}.\nThe above automaton accepts a data-word $w$ with a valuation $v$ for \nthe universally quantified variables $y_1$ and $y_2$  as follows: it stores the value\nof the data at $y_1$ and $y_2$ in two registers, and then checks\nwhether the formula annotating the final state it reaches holds for these data\nvalues. The automaton accepts the data word $w$ if for \\emph{all}\npossible valuations of $y_1$ and $y_2$, the automaton accepts the corresponding word with valuation.\nThe above automaton hence accepts precisely those set of\ndata words that satisfy the invariant formula.\\\\\n\n\\noindent{\\bf Decidable Fragments and Elastic Quantified Data Automata:}\nThe emptiness problem for QDAs is undecidable; in other words, the logical formulas that QDAs express fall into \nundecidable theories of lists and arrays. A common restriction in the array property fragment\nas well as the syntactic decidable fragments of \\Strand is that quantification is not permitted to be over \nelements that are only a \\emph{bounded} distance away. \nThe restriction allows quantified variables to only be related through \\emph{elastic}\nrelations (following the terminology of \\Strand~\\cite{strand,strandsas})\n\nFor instance, a formula equivalent to the formula in Eq.~1\nbut expressed in the decidable fragment\nof \\Strand over lists is:\\\\\n$\\textit{head} \\rightarrow^* i ~~\\wedge ~~\\forall y_1, y_2. ((\\textit{head} \\rightarrow^* y_1 \\wedge y_1 \\rightarrow^* y_2 \\wedge y_2  \\rightarrow^* i ) \\Rightarrow d(y_1) \\leq d(y_2))~~~(3)$\\\\\nThis formula compares data at $y_1$ and $y_2$ whenever $y_2$ occurs sometime after $y_1$, and this makes the formula fall in a decidable class.\nSimilarly, a formula equivalent to the formula Eq.~2 in the decidable array property fragment is:\\\\\n$~~~~~~~~~~~~~~~~~\\forall y_1, y_2. ((0 \\leq y_1 \\wedge y_1 \\leq y_2 \\wedge y_2 \\leq i) \\Rightarrow A[y_1] \\leq A[y_2])~~~~~~~~~~~~~~~~(4)$\\\\\nThe above two formulas are captured by a QDA that is the same as in the figure above, except that the $\\blank$-transition from $q_2$ to $q_5$ is replaced by a $\\blank$-loop  on $q_2$. \n\n\nWe identify a restricted form of quantified data automata, called \\emph{elastic quantified data automata} (EQDA) in Section~\\ref{sec:elastic}, which\nstructurally captures the constraint that quantified variables can be related only using elastic relations\n(like $\\rightarrow^*$ and $\\leq$).\nFurthermore, we show\nin Section~\\ref{sec:application} that EQDAs can be converted to formulas in the decidable fragment of \\Strand and the array property fragment, and hence expresses invariants that are amenable to decidable analysis across loop bodies.\n\nIt is important to note that QDAs are not necessarily a blown-up version of the\nformulas they correspond to. For a formula, the corresponding QDA can\nbe exponential, but for a QDA the corresponding formula can be\nexponential as well (QDAs are like BDDs, where there is sharing of common suffixes of constraints, which\nis absent in a formula).\n\n\n\n\n\\section{Introduction} \\label{sec:introduction}\n\\vspace{-0.2cm}\n\nSynthesizing invariants for programs is one of the most challenging problems in verification today.\nIn this paper, we are interested in using \\emph{learning} techniques to synthesize quantified\ndata-structure invariants.\n\nIn an \\emph{active} black-box learning framework, we look upon the invariant as a set of configurations of the program,\nand allow the learner to query the teacher for membership and equivalence queries on this set.\nFurthermore, we fix a particular representation class for these sets, and demand that the learner\nlearn the smallest (simplest) representation that describes the set. A learning algorithm that learns\nin time polynomial in the size of the simplest representation of the set is desirable.\nIn \\emph{passive} black-box learning, the learner is given a sample of examples and counter-examples\nof configurations, and is asked to synthesize the simplest representation that includes the examples\nand excludes the counter-examples. In general, several active learning algorithms that work in polynomial\ntime are known (e.g., learning regular languages represented as DFAs \\cite{DBLP:journals\/iandc\/Angluin87}) while passive polynomial-time\nlearning is rare (e.g., conjunctive Boolean formulas can be learned but general Boolean formulas cannot\nbe learned efficiently, automata cannot be learned passively efficiently)~\\cite{KearnsVazirani}.\n\nIn this paper, we build active learning algorithms for \\emph{quantified logical formulas describing\nsets of linear data-structures}. Our aim is to build algorithms that can learn formulas of the kind\n ``$\\forall y_1, \\ldots y_k ~\\varphi$'', where $\\varphi$ is quantifier-free, and that captures properties\n of arrays and lists (the variables range over indices for arrays, and locations for lists, and the\n formula can refer to the data stored at these positions and compare them using arithmetic, etc.).\nFurthermore, we show that we can build learning algorithms that learn  properties that are\nexpressible in known decidable logics. We then\nemploy the active learning algorithm in a \\emph{passive learning} setting where\nwe show that by building an imprecise teacher that answers the questions of the active learner,\nwe can build effective invariant generation algorithms that learn simply from a finite set of\nexamples.\n\n\\noindent{\\bf Active Learning of Quantified Properties using QDAs:}\n Our first technical contribution is a novel representation (normal form) for quantified properties of linear\n data-structures, called \\emph{quantified data automata} (QDA), and a polynomial-time active learning algorithm\n for QDAs.\n\n We model linear data-structures as \\emph{data words}, where each position is decorated with a finite alphabet modeling the program's pointer variables that point to that cell in the list or index variables that index into the cell of the array, and with data modeling the data value stored in the cell, e.g. integers.\n\nQuantified data automata (QDA) are a new model of automata over data words that are powerful enough to express \\emph{universally} quantified properties of data words. A QDA accepts a data word provided it accepts \\emph{all possible} annotations of the data word with valuations of a (fixed) set of variables $Y=\\{ y_1, \\ldots, y_k\\}$; for each such annotation, the QDA reads the data word, records the data stored at the positions pointed to by $Y$, and finally checks these data values against a data formula determined by the final state reached. QDAs are very powerful in expressing typical invariants of programs manipulating lists and arrays, including invariants of a wide variety of searching and sorting algorithms, maintenance of lists and arrays using insertions\/deletions, in-place manipulations that destructively update lists, etc.\n\nWe develop an efficient learning algorithm for QDAs. By using a combination of \\emph{abstraction} over a set of data formulas and Angluin's learning algorithm for DFAs~\\cite{DBLP:journals\/iandc\/Angluin87}, we build a learning algorithm for QDAs. We first show that for any set of valuation words (data words with valuations for the variables $Y$), there is a \\emph{canonical} QDA. Using this result, we show that learning valuation words can be reduced to learning \\emph{formula words} (words with no data but paired with data formulas), which in turn can be achieved using Angluin-style learning of Moore machines. The number of queries the learner poses and the time it takes is bound polynomial in the size of the canonical QDA that is learned. Intuitively, given a set of pointers into linear data structures, there are an exponential number of ways to permute the pointers into these and the universally quantified variables; the learning algorithm allows us to search this space using only polynomial time in terms of the\nactual permutations that figure in the set of data words learned.\n\n\\noindent {\\bf Elastic QDAs and a Unique Minimal Over-Approximation Theorem:}\nThe quantified properties that we learn in this paper (we can synthesize them from QDAs)\nare very powerful, and are, in general \\emph{undecidable}.\nConsequently, even if they are learnt in an invariant-learning application, we will be unable to \\emph{verify}\nautomatically whether the learnt properties are adequate invariants for the program at hand.\nThe goal of this paper is to also offer mechanisms to \\emph{learn invariants that are amenable to decision procedures}.\n\nThe second technical contribution of this paper is to identify a subclass of QDAs (called elastic QDAs) and show\ntwo main results for them: (a) elastic QDAs can be converted to \\emph{decidable} logical formulas, to the\narray property fragment when modeling arrays and the decidable \\Strand fragment when modeling lists;\n(b) a surprising \\emph{unique minimal over-approximation theorem} that says that for every QDA, accepting say a language $L$ of valuation-words,\nthere is a \\emph{minimal} (with respect to inclusion) language of valuation-words $L'$ that is accepted by an elastic QDA.\n\nThe latter result allows us to learn QDAs and then apply the unique minimal over-approximation (which is effective) to compute\nthe best over-approximation of it that can be expressed by elastic QDAs (which then is decidable). The result is proved by showing\nthat there is a unique way to minimally morph a QDA to one that satisfies the elasticity restrictions.\nFor the former, we identify a common property of the array property fragment and the syntactic decidable fragment of \\Strand, called \\emph{elasticity} (following the general terminology on the literature on \\Strand~\\cite{strand}). Intuitively, both the array property fragment and \\Strand prohibit quantified cells to be tested to be bounded distance away (the array property fragment does this by disallowing arithmetic expressions over the quantified index variables~\\cite{apf} and the decidable fragment of \\Strand disallows this by permitting only the use of $\\rightarrow^*$ or $\\rightarrow^+$ in order to compare quantified variables~\\cite{strand,strandsas}). We finally identify a \\emph{structural restriction} of QDAs that permits only elastic properties to be stated\nthat there is a unique way to minimally morph a QDA to one that satisfies the elasticity restrictions.\n\n\\noindent{\\bf Passive Learning of Quantified Properties:}\nThe active learning algorithm can itself be used in a verification framework, where the membership and equivalence queries are answered\nusing under-approximate and deductive techniques (for instance, for iteratively increasing values of $k$, a teacher can answer membership\nquestions based on bounded and reverse-bounded model-checking, and answer equivalence queries by checking if the invariant is adequate using\na constraint solver; see Appendix~D for details). In this paper, we do not pursue an implementation of active learning as above, but instead build a passive learning algorithm that uses the active learning algorithm.\n\nOur motivation for doing passive learning is that we believe (and we validate this belief using experiments)\nthat in many problems, a lighter-weight passive-learning algorithm which learns from a few randomly-chosen small data-structures is sufficient\nto divine the invariant. Note that passive learning algorithms, in general, often boil down to a guess-and-check algorithm of some kind, and often pay an exponential price in the property learned. Designing a passive learning algorithm using an active learning core allows us\nto build more interesting algorithms; in our algorithm, the inacurracies\/guessing is confined to the way the teacher answers\nthe learner's questions.\n\nThe passive learning algorithm works as follows. Assume that we have a finite set of configurations $S$, obtained from sampling\nthe program (by perhaps just running the program on various random small inputs). We are required to learn the simplest representation\nthat captures the set $S$ (in the form of a QDA). We now use an active learning algorithm for QDAs; membership questions are answered\nwith respect to the set $S$ (note that this is imprecise, as an invariant $I$ must include $S$ but need not be precisely $S$).\nWhen asked an equivalence query with a set $I$, we check whether $S \\subseteq I$; if yes, we can check if the invariant is adequate using a constraint-solver and the program.\n\nIt turns out that this is a good way to build a passive learning algorithm. First, enumerating random small data-structures\nthat get manifest at the header of a loop fixes for the most part the structure of the invariant, since the invariant is forced to be expressed as\na QDA. Second, our active learning algorithm for QDAs promises never to ask long membership queries (queried words are guaranteed to be less than the diameter of the  automaton), and often the teacher has the correct answers.\nFinally, note that the passive learning algorithm answers membership queries with respect to $S$; this is because we do not\nknow the true invariant, and hence err on the side of keeping the invariant semantically small.\nThis inaccuracy is common in most learning algorithms employed for verification (e.g, Boolean learning~\\cite{wang-aplas10}, compositional verification~\\cite{Giano,CAV05}, etc). This inaccuracy could lead to a non-optimal QDA being learnt, and is precisely\nwhy our algorithm need not work in time polynomial in the simplest representation of the concept (though it is polynomial in\nthe invariant  it finally learns).\n\nThe proof of the efficacy of the passive learning algorithm rests in the experimental evaluation.\nWe implement the passive learning algorithm (which in turn uses the active learning algorithm).  By building a teacher using dynamic test runs of the program and by pitting this teacher against the learner, we learn invariant QDAs, and then over-approximate them using EQDAs. These EQDAs are then transformed into formulas over decidable theories of arrays and lists. Using a wide variety  of programs manipulating arrays and lists, ranging from several examples\nin the literature involving sorting algorithms, partitioning, merging lists, reversing lists, and programs from the Glib list library,\nprograms from the Linux kernel, a device driver, and programs from a verified-for-security mobile application platform,\nwe show that we can effectively learn adequate quantified invariants in these settings.\nIn fact, since our technique is a black-box technique, we show that it can be used to infer pre-conditions\/post-conditions for methods as well.\n\n\\input{relatedwork}\n\n\n\\vspace*{-3.5mm}\n\\section{Overview} \\label{sec:overview}\n\\vspace{-0.2cm}\n\\input{overview}\n\\vspace{-3.5mm}\n\n\\section{Quantified Data Automata} \\label{sec:preliminaries}\n\n\n\\input{preliminaries}\n\n\n\n\n\\vspace{-0.2cm}\n\\section{Learning Quantified Data Automata} \\label{sec:learning}\n\\vspace{-0.2cm}\n{\\input{learning}}\n\n\\vspace{-0.55cm}\n\\section{Unique Over-approximation Using Elastic QDAs} \\label{sec:elastic}\n\\vspace{-0.2cm}\n\n\n\\input{elastification.tex}\n\n\\vspace{-0.4cm}\n\\section{Linear Data-structures to Words and EQDAs to Logics} \\label{sec:application}\n\\vspace{-0.15cm}\n\nIn this section, we sketch briefly how to model arrays and lists as data-words, and how to convert EQDAs to quantified\nlogical formulas in decidable logics.\n\n\n\\input{translation}\n\n\n\n\n\\vspace{-0.4cm}\n\\section{Implementation and Evaluation on Learning Invariants}\n\\label{sec:implementation}\n\\vspace{-0.15cm}\n\n\\input{implementation}\n{\\bf Future Work:}\n\\input{conclusion}\n\n\n\\vspace{-0.3cm}\n\\bibliographystyle{splncs}\n{\\footnotesize\n\n\\subsection*{Valuation words}\n\n\n\n\n\n\n\n\n\n\n\n\nWe are now ready to introduce the automaton model.  A quantified data\nautomaton (QDA) over a set of program variables $PV$, a data domain\n$D$, a set of universally quantified variables $Y$, and a formula\nlattice ${\\cal F}$ is of the form $\\A = (Q, q_0,\n\\Pi, \\delta, f)$ where \n$Q$ is a finite set of states, \n$q_0 \\in Q$ is the initial state, \n$\\Pi = \\Sigma \\times (Y \\cup \\{\\yblank\\})$, \n$\\delta: Q \\times \\Pi \\rightarrow Q$ is the transition function, and\n$f: Q \\rightarrow F$ is a \\emph{final-evaluation function} that\n   maps each state to a data formula.\nThe alphabet $\\Pi$ used in a QDA does not contain data. Words over\n$\\Pi$ are referred to as \\emph{symbolic words} because they do not contain\nconcrete data values. The symbol $(b,\\yblank)$ indicating that a\nposition does not contain any variable is denoted by $\\blank$.\n\nIntuitively, a QDA is a \\emph{register} automaton that reads the data\nword extended by a valuation that has a register for each $y \\in Y$,\nwhich stores the data stored at the positions evaluated for $Y$, and\nchecks whether the formula decorating the final state reached holds\nfor these registers. It accepts a data word $w \\in (\\Sigma \\times\nD)^*$ if it accepts \\emph{all possible} valuation words $v$ extending\n$w$ with a valuation over $Y$.\n\n We formalize this below.\nA configuration of a QDA is a pair of the form $(q,r)$ where $q \\in Q$\nand $r: Y \\rightarrow D$ is a partial variable assignment. The initial\nconfiguration is $(q_0, r_0)$ where the domain of $r_0$ is empty. For\nany configuration $(q,r)$, any letter $a \\in \\Sigma$, data value $d\n\\in D$, and variable $y \\in Y$ we define $ \\delta'((q,r), (a,d,y)) =\n(q',r')$ provided $\\delta(q,(a,y)) = q'$ and $r'(y') = r(y')$ for each\n$y' \\not= y$ and $r'(y) = d$, and\nwe let $ \\delta'((q,r),\n(a,d,\\yblank)) = (q',r)$ if $\\delta(q,(a,\\yblank)) = q'$.  We extend\nthis function $\\delta'$ to valuation words in the natural way.\n\nA valuation word $v$ is accepted by the QDA if $\\delta'((q_0,r_0), v)\n= (q,r)$ where $(q_0,r_0)$ is the initial configuration and $r \\models\nf(q)$, i.e., the data stored in the registers in the final\nconfiguration satisfy the formula annotating the final state reached.\nWe denote the set of valuation words accepted by $\\A$ as $\\Lval(\\A)$. We\nassume that a QDA verifies whether its input satisfies the constraints\non the number of occurrences of variables from $PV$ and $Y$, and that\nall inputs violating these constraints either do not admit a run\n(because of missing transitions) or are mapped to a state with final\nformula $\\false$.\n\nA data word $w$ is accepted by the QDA if for every valuation word\n$v$ such that the data word corresponding to $v$ is $w$, $v$ is\naccepted by the QDA. The language of the QDA, $L(\\A)$, is the set of\ndata words accepted by it.\n\n\n\n\n\n\\subsubsection*{Related Work:}\n{\\bf Related Work:}\nFor invariants expressing properties on the dynamic heap, \\emph{shape analysis} techniques are the most well known~\\cite{shapeanalysis}, where locations are classified\/merged using \\emph{unary} predicates (some dictated by the program and some given as instrumentation predicates by the user), and abstractions summarize all nodes with the same predicates into a single node. The data automata that we build also express an infinite set of linear data structures, but do so using automata, and further allow $n$-ary quantified relations between data elements.\nIn recent work,~\\cite{celia} describes an abstract domain, for analyzing list manipulating programs, that can capture quantified properties about the structure and the data stored in lists. This domain can be instantiated with any numerical domain for the data constraints and a set of user-provided patterns for capturing the structural constraints.\nHowever, providing these patterns for quantified invariants is in general a difficult task.\n\nIn recent years, techniques based on \\emph{Craig's interpolation}~\\cite{mcmillan03} have emerged as\na new method for invariant synthesis.\nInterpolation techniques,\nwhich are inherently white-box as well, are known for several theories, including\nlinear arithmetic, uninterpreted function theories, and even quantified properties\nover arrays and lists~\\cite{mcmillan06,mcmillan08,natasha,podelski}. These methods use different heuristics like\nterm abstraction~\\cite{natasha}, preferring smaller constants~\\cite{mcmillan06,mcmillan08} and use of existential ghost variables~\\cite{podelski}\nto ensure that the interpolant converges on an invariant from a \\emph{finite} set of spurious counter-examples.\nIC3~\\cite{ic3} is another white-box technique for generalizing inductive invariants from a set of counter-examples.\n\n\nA primary difference in our work, compared to all the work above, is that ours is a \\emph{black-box technique} that does not look at the code of the program, but synthesizes an invariant from a snapshot of examples and counter-examples\nthat characterize the invariant. The black-box approach to constructing invariants has both advantages and disadvantages. The main disadvantage is that information regarding what the program actually does is lost in invariant synthesis. However, this is the basis for its advantage as well---\nby \\emph{not} looking at the code, the learning algorithm promises to learn the sets with\nthe simplest representations in polynomial time, and can also be much more flexible.\nFor instance, even when the code of the program\nis complex, for example having non-linear arithmetic or complex heap manipulations that preclude\nlogical reasoning, black-box learning gives ways to learn simple invariants for them.\n\n\nThere are several black-box learning algorithms that have been explored in verification. Boolean formula learning has been investigated for finding quantifier-free program invariants~\\cite{bowyawwang}, and also extended to quantified invariants~\\cite{wang-aplas10}. However unlike us,~\\cite{wang-aplas10} learns a quantified formula given a set of data predicates as also the predicates which can appear in the guards of the quantified formula. \nRecently, machine learning techniques have also been explored~\\cite{NoriCAV}. \nVariants of the Houdini algorithm~\\cite{houdini} essentially use conjunctive Boolean\nlearning (which can be achieved in polynomial time) to learn conjunctive invariants over templates of atomic formulas (see also~\\cite{gulwani}).\nThe most mature work in this area is Daikon~\\cite{daikon}, which learns formulas over a template, by enumerating all formulas and checking \nwhich ones satisfy the samples, and where scalability is achieved in practice using several heuristics that reduce the enumeration space which is doubly-exponential. For quantified\ninvariants over data-structures, however, such heuristics aren't very effective, and Daikon often restricts learning only to formulas of very restricted\nsyntax, like formulas with a single atomic guard, etc. In our experiments Daikon was, for instance, not able to learn the loop invariant for the selection sort algorithm.\n\n\n\n\n\\subsection*{Modeling Lists and Arrays as Data Words}\n\\label{sec:modeling}\n\\vspace{-0.2cm}\n\nWe model a linear data structure as a word over ($\\Sigma \\times D$) with $\\Sigma = 2^{PV}$, where $PV$ is the set of pointer variables and $D$ is the data domain; scalar variables in the program are modeled as single element lists.\nThe encoding introduces a special pointer variable $nil$ which is always read in the beginning of the data word together with all other null-pointers in the configuration.\nFor arrays, the encoding also introduces  $nil\\_le\\_zero$ and $nil\\_geq\\_size$ which are read together with  all those index variables which are less than zero or which exceed the size of the respective array.\nThe data value at these variables is not important; they can be populated with any data value in $D$. Given a configuration, the corresponding data words read the scalar variables and the linear data structures one after the other, in some pre-determined order. In programs like copying one array to another, where both the arrays are read synchronously, the encoding models multiple data structures as a single structure over an extended data domain.\n\n\n\n\n\n\n\n\\vspace{-0.3cm}\n\\subsection*{From EQDAs to \\Strand and APF}\\label{sec:EQDA_to_logic}\n\\vspace{-0.2cm}\n\nNow we briefly sketch the translation from an EQDA $\\mathcal{A}$ to an equivalent formula $\\mathcal{T}(\\mathcal{A})$ in $\\Strand$ or the APF such that the set of data words accepted by $\\mathcal{A}$ corresponds to the program configurations $\\mathcal{C}$ which model $\\mathcal{T}(\\mathcal{A})$.\n\nGiven an EQDA $\\mathcal{A}$, the translation enumerates all simple paths in the automaton to an output state. For each such path $p$ from the initial state to an output state $q_p$, the translation records the relative positions of the pointer and universal variables\nas a structural constraint $\\phi_p$ and the formula $f_{\\mathcal{A}}(q_p)$ relating the data value at these positions. Each path thus leads to a universally quantified implication of the form $\\forall Y. ~\\phi_p \\Rightarrow f_{\\mathcal{A}}(q_p)$. All valuation words not accepted by the EQDA semantically go to the formula \\emph{false}, hence an additional conjunct $\\forall Y.~\\neg (\\bigvee_p \\phi_p) \\Rightarrow \\textit{false}$ is added to the formula. So the final formula ${ \\mathcal{T}(\\mathcal{A}) = \\bigwedge_p \\forall Y.~ \\phi_p \\Rightarrow f_{\\mathcal{A}}(q_p) ~\\bigwedge~ \\forall Y.~\\neg (\\bigvee_p \\phi_p) \\Rightarrow \\textit{false}}$.\nSee Appendix C for more details.\n\n\\begin{figure}\n\\vspace{-0.55cm}\n\t\\centering\n\t\\begin{tikzpicture}\n\t\n\t\t\\node[state]                        (0) {$q_0$};\n\t\t\\node[state, right of=0, xshift=23] (1) {$q_2$};\n\t\t\\node[state, right of=1]            (2) {$q_8$};\n\t\t\\node[state, right of=2]            (3) {$q_{18}$};\n\t\t\\node[state, right of=3, accepting] (4) {$q_{26}$};\n\t\n\t\t\\draw[<-] (0.west) -- +(-.3, 0);\n\t\t\\draw[->] (0) edge             node {\\scriptsize $(\\{ \\textit{cur}, \\textit{nil} \\}, \\yblank )$} (1)\n\t\t           (1) edge             node {\\scriptsize $(\\textit{h}, \\yblank)$}                          (2)\n\t\t           (2) edge             node {\\scriptsize $(\\textit{b}, y_1)$}                               (3)\n\t\t               edge[loop above] node {\\scriptsize $\\blank$}                                          ()\n\t\t           (3) edge             node {\\scriptsize $(\\textit{b}, y_2)$}                               (4)\n\t\t               edge[loop above] node {\\scriptsize $\\blank$}                                          ()\n\t\t           (4) edge[loop above] node {\\scriptsize $\\blank$}                                          ();\n\t\n\t\t\\node[right of=4, xshift=-22, inner sep=0] {\\scriptsize $\\varphi$};\n\t\t\\node[anchor=west, inner sep=0] at (-.1, -.8) {\\scriptsize $\\varphi \\coloneqq  \\begin{array}[t]{l} d(y_1) \\leq d(y_2) \\wedge d(y_1) < \\textit{k} \\wedge d(y_2) < \\textit{k} \\end{array}$};\n\t\\end{tikzpicture}\n\t\\caption{A path in the automata expressing the invariant of the program which finds a key $k$ in a sorted list.The full automaton is presented in Appendix ~E.\\label{fig:translation}}\n\\vspace{-0.3cm}\n\\end{figure}\n\n\\vspace{-0.3cm}\nWe next explain, through an example, the construction of the structural constraints $\\phi_p$. Consider program \\emph{list-find} which searches for a key in a sorted list. The EQDA corresponding to the loop invariant learned for this program is presented in Appendix E. One of the simple paths in the automaton (along with the associated self-loops on $\\blank$) is shown in Fig~\\ref{fig:translation}. The structural constraint $\\phi_p$ intuitively captures all valuation words which are accepted by the automaton along $p$; for the path in the figure $\\phi_p$ is $(cur =\nnil \\wedge h \\rightarrow^+ y_1 \\wedge y_1 \\rightarrow^+ y_2)$ and the formula $\\forall y_1 y_2. ~(cur =\nnil \\wedge h \\rightarrow^+ y_1 \\wedge y_1 \\rightarrow^+ y_2) \\Rightarrow (d(y_1) \\leq d(y_2) \\wedge d(y_1) < \\textit{k} \\wedge d(y_2) < \\textit{k})$ is the corresponding conjunct in the learned invariant. \n\nApplying this construction yields the following theorem.\n\n\\vspace{-0.05cm}\n\\begin{theorem} \\label{thm:correctness-STRAND-translation}\nLet $\\mathcal{A}$ be an EQDA, $w$ a data word, and $c$ the program configuration corresponding to $w$. If $w \\in \\mathcal{L}(\\mathcal{A})$, then $c \\models \\mathcal{T}(\\mathcal{A})$. Additionally, if $\\mathcal{T}(\\mathcal{A})$ is a $\\Strand$ formula, then the implication also holds in the opposite direction.\n\\end{theorem}\n\n\nAPF allows the universal variables to be related by $\\leq$ or $=$ and not $<$. Hence, along paths where $y_1 < y_2$, we over-approximate the structural constraint $\\phi_p$ to $y_1 \\leq y_2$ and, subsequently, the data formula $f_{\\mathcal{A}}(q_p)$ is abstracted to include $d(y_1) = d(y_2)$. This leads to an abstraction of the actual semantics of the QDA and is the reason Theorem~\\ref{thm:correctness-STRAND-translation} only holds in one direction for the APF. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStreaming algorithms for processing massive graphs have been studied for two decades \\cite{hrrav98}. In the \nmost traditional setting, the {\\em insertion-only model}, an algorithm receives a sequence of the edges of the \ninput graph in arbitrary order, and the objective is to solve a graph problem using as little space as possible. \nThe insertion-only model has received significant attention, and many problems, such as matchings (e.g. \\cite{kmm12,gkk12,kks14,kr16,kt17,k18,ps19,fhmrr20}), \nindependent sets (e.g. \\cite{hssw12,hhls16,cdk18,cdk19}), and subgraph counting (e.g. \\cite{kmss12,cj17,bc17}), \nhave since been studied in this model. See \\cite{m14} for an excellent survey.\n\nIn 2012, Ahn et al. \\cite{agm12} introduced the first techniques for addressing {\\em insertion-deletion} graph streams,\nwhere the input stream consists of a sequence of edge insertions and deletions. They showed that\nmany problems, such as \\textsf{Connectivity} and \\textsf{Bipartiteness}, can be solved using the same amount of space \nas in insertion-only streams up to poly-logarithmic factors. Various other works subsequently gave results of a similar \nflavor and presented insertion-deletion streaming algorithms with similar space complexity as their insertion-only\ncounterparts for problems including \\textsf{Spectral Sparsification} \\cite{kmmmnst20} and \\textsf{$\\Delta+1$-coloring} \n\\cite{ack19}. Konrad \\cite{k15} and \nAssadi et al. \\cite{akly16} were the first to give a separation result between the insertion-only graph stream model and the \ninsertion-deletion graph stream model: While it is known that a $2$-approximation to \\textsf{Maximum Matching} can be computed using space \n$\\mathrm{O}(n \\log n)$ in insertion-only streams, Konrad showed that space $\\Omega(n^{\\frac{3}{2} - 4 \\epsilon})$ is required \nfor an $n^{\\epsilon}$-approximation in insertion-deletion streams, and Assadi et al. gave a lower\nbound of $n^{2-3\\epsilon-o(1)}$ for such an approximation. Assadi et al. also presented an $\\tilde{\\mathrm{O}}(n^{2-3\\epsilon})$ space \nalgorithm that matches their lower bound up to lower order terms, which establishes that their lower bound is optimal \n(a different algorithm that matches this lower bound is given by Chitnis et al. \\cite{ccehmmv16}). \n\n\nBoth Konrad and Assadi et al. exploit an elegant connection between insertion-deletion streaming algorithms\nand linear sketches.\nAi et al. \\cite{ahlw16}, building on the work of Yi et al. \\cite{lnw14}, showed that insertion-deletion graph \nstreaming algorithms can be characterized as algorithms that essentially solely rely on the computation of linear sketches of the input \nstream. A consequence of this result is that space lower bounds for insertion-deletion streaming algorithms\ncan also be proved in the {\\em simultaneous model of communication}, since linear sketches can be implemented in this \nmodel. This provides an alternative to the more common approach of proving streaming lower bounds in the one-way model of \ncommunication. In particular, the lower bounds by Konrad and Assadi et al. are proved in the simultaneous model of communication.\n\nFrom a technical perspective, this model has various attractive features, however, it comes with a major disadvantage:\nThe characterization of Ai et al. only holds for insertion-deletion streaming algorithms that (1) are able to process \n``very long'' input streams, i.e., input streams of triple exponential length in $n$, the number of vertices of the \ninput graph, and (2) are able to process multi-graphs. In particular, this characterization does not hold for \ninsertion-deletion streaming algorithms that rely on the assumption that input streams are short and the graph \ndescribed by the input stream is always simple. Consequently, the lower bounds of Konrad and Assadi et al. \ndo not hold for such algorithms.\n\n\\vspace{0.1cm}\n\\textbf{Our Results.}\nIn this work, we prove an optimal space lower bound for \\textsf{Maximum Matching} in insertion-deletion streams \nvia the one-way two-party model of communication. Our lower bound construction yields insertion-deletion streams of \nlength $\\mathrm{O}(n^2)$ and does not involve multi-edges. Our lower bound therefore also holds for streaming algorithms that are \ndesigned for short input streams and simple graphs for which the characterization by Ai et al. does not hold.\nFurthermore, the optimal lower bound by Assadi et al. \\cite{akly16} only holds for streaming algorithms that\nnever output non-existing edges when the (randomized) algorithms fail. We do not require this restriction.\n\nOur lower bound method is simple and more widely applicable. Using the same method, we also give an optimal \nlower bound for \\textsf{Minimum Vertex Cover}, showing that computing a $n^\\epsilon$-approximation requires $\\Omega(n^{2-2\\epsilon})$\nspace. Assadi and Khanna mention in \\cite{ak17} that the $n^{2 - 3\\epsilon - o(1)}$ space lower bound \nfor \\textsf{Maximum Matching} given in \\cite{akly16} also applies to \\textsf{Minimum Vertex Cover}. Our lower bound\ntherefore improves on this result by a factor of $n^{{\\epsilon} + o(1)}$. Furthermore, we show that our lower bound is \noptimal up to a factor of $\\log n$: We give a very simple deterministic \ninsertion-deletion streaming algorithm for \\textsf{Minimum Vertex Cover} that uses space $\\mathrm{O}(n^{2-2\\epsilon} \\log n)$.\n\n\n\nWhile the main application of our lower bounds in the one-way two-party communication model are lower bounds for insertion-deletion \ngraph streaming algorithms, we believe that our lower bounds are of independent interest. Indeed, the one-way two-party communication \ncomplexity of \\textsf{Maximum Matching} without deletions has been addressed in \\cite{gkk12}, and our result can therefore also be understood\nas a generalization of their model to incorporate deletions.\n\n\n\\vspace{0.1cm}\n\\textbf{The Simultaneous Model of Communication.}\nThe lower bounds by Konrad \\cite{k15} and Assadi et al. \\cite{akly16} are proved in the simultaneous model of communication. \nIn this model,\na typically large number of parties $k$ hold not necessarily disjoint subsets of the edges of the input graph.\nEach party $P_i$ sends a message $M_i$ to a referee, who then outputs the result of the protocol. \nThe connection between insertion-deletion streaming algorithms and linear sketches by Ai et al. \\cite{ahlw16} then implies that \na lower bound on the size of any message $M_i$ yields a lower bound on the space requirements of any \ninsertion-deletion streaming algorithm. \n\nIn the lower bound of Assadi et al. \\cite{akly16} for \\textsf{Maximum Matching}, each party $P_i$ holds the edges $E_i$\nof a dense subgraph, which itself constitutes a {\\em Ruzsa-Szemer\\'{e}di graph}, i.e., a graph whose edge set can be partitioned \ninto large disjoint induced matchings. All previous streaming lower bounds for approximate \\textsf{Maximum Matching} rely on \nrealizations of Ruzsa-Szemer\\'{e}di graphs \\cite{gkk12,k15,akly16}. \nTheir construction is so that \nonly a single induced matching of every party $P_i$ is useful for the construction of a global large matching. \nDue to symmetry of the construction, the parties are unable to identify the important induced matching and therefore \nneed to send large messages that contain information about most of the induced matchings to the referee for them to \nbe able to compute a large global matching. Interestingly, none of the parties hold edge deletions in their construction. \n\n\\vspace{0.1cm}\n\\textbf{The One-way Model of Communication.}\nIn this paper, we give a lower bound in the one-way two-party model of communication. In this model, Alice holds\na set of edges $E$ of the input graph and sends a message $M$ to Bob. Bob holds a set of edge deletions $D \\subseteq E$\nand outputs a large matching in the graph spanned by the edges $E \\setminus D$. A standard reduction shows that \na lower bound on the size of message $M$ also constitutes a lower bound on the space requirements of an \ninsertion-deletion streaming algorithm.\nThe two models are illustrated in Figure~\\ref{fig:models}.\n\n\n\\begin{figure}[h!]\n\\begin{center}\n \\begin{tikzpicture}\n  \\node (ref) at (0,0) {\\textbf{Referee}};\n  \\node (out1) at (2,0) {result};\n  \\node (p1) at (-1.5, -1.5) {$\\mathbf{P_1}$};\n  \\node (p2) at (-0.5, -1.5) {$\\mathbf{P_2}$};\n  \\node (pdots) at (0.5, -1.5) {\\dots};\n  \\node (p3) at (1.5, -1.5) {$\\mathbf{P_k}$};\n  \n  \\node(e1) at (-1.9, -2) {$E_1 \\subseteq E$};\n  \\node(e2) at (-0.3, -2) {$E_2 \\subseteq E$};\n  \\node(e3) at (1.5, -2) {$E_k \\subseteq E$};\n  \n  \\draw[->] (ref) -- (out1) {};\n  \\draw[->] (p1) -- (ref) node [midway, left,xshift=-2pt,yshift=-1pt] {$M_1$};\n  \\draw[->] (p2) -- (ref) node [midway, right,yshift=-1pt,xshift=-1pt] {$M_2$};\n  \\draw[->] (p3) -- (ref) node [midway, right,yshift=-1pt, xshift=4pt] {$M_k$};\n  \n  \n  \\node (alice) at (5, -0.5) {\\textbf{Alice}};\n  \\node (bob) at (8, -0.5) {\\textbf{Bob}};\n  \\node (out2) at (9.7,-0.5) {result};\n  \n \n  \\draw[->] (alice) -- (bob) node [midway, above] {$M$};\n  \\draw[->] (bob) -- (out2) {};\n  \n  \\node (ae1) at (5, -1) {$E$};\n  \\node (be2) at (8, -1) {$D \\subseteq E$};\n \\end{tikzpicture}\n \\caption{The simultaneous (left) and the one-way two-party (right) models of communication. \\label{fig:models}}\n \\end{center}\n\\end{figure}\n\n\\vspace{0.1cm}\n\\textbf{Our Techniques.}\nTo prove our lower bound, we identify that an insertion-deletion streaming algorithm for \\textsf{Maximum Matching}\nor \\textsf{Minimum Vertex Cover} can be used to obtain a one-way two-party communication protocol for a\ntwo-dimensional variant of the well-known \\textsf{Augmented Index} problem that we denote by \\textsf{Augmented Bi-Index}, \nor \\textsf{BInd} in short.\nIn an instance of \\textsf{BInd}, Alice holds an $n$-by-$n$ binary matrix $A \\in \\{0, 1\\}^{n \\times n}$. \nBob is given a position $(x,y) \\in [n-k]^2$\nand needs to output the bit $A_{x,y}$. Besides $(x, y)$, he also knows the $k$-by-$k$ submatrix of $A$ with upper left\ncorner at position $(x, y)$, however with the bit at position $(x,y)$ missing - we will denote this $k$-by-$k$ submatrix \nwith $(x, y)$ missing by $A_{S(x, y)}$. We show that this problem has a one-way communication complexity of $\\Omega((n-k)^2)$\nby giving a reduction from the \\textsf{Augmented Index} problem.\n\nTo obtain a lower bound for \\textsf{Maximum Matching}, we show that Alice and Bob can construct a protocol for \\textsf{BInd}\ngiven an insertion-deletion streaming algorithm for \\textsf{Maximum Matching}. In our reduction, we will consider instances\nwith $k = n - \\Theta(n^{1-\\epsilon})$, for some $\\epsilon > 0$. Consider the following attempt: Suppose that the input matrix \n$A$ is a uniform random binary matrix and that $A_{x,y} = 1$ (we will get rid of these assumptions later). \nAlice and Bob interpret the matrix $A$ as the incidence matrix of a bipartite graph $G$. Bob interprets \nthe ``1'' entries in the submatrix $A_{S(x,y)}$ outside the diagonal, i.e., all ``1'' entries except those in positions\n$\\{ (x+j, y+j) \\ : \\ 0 \\le j < k \\}$, as edge deletions $F$. \nThe graph $G - F$ has a large matching: Since the diagonal of $A_{S(x,y)}$ is not deleted, and each entry in the diagonal is $1$\nwith probability $1\/2$, we expect that half of all potential edges in the diagonal of $S(x,y)$ are contained in $G - F$ and thus form a matching of size \n$\\Theta(k) = \\Theta(n - n^{1-\\epsilon})$. An $n^\\epsilon$-approximation algorithm for \n\\textsf{Maximum Matching} would therefore report $\\Omega(n^{1-\\epsilon})$ of these edges. Suppose that the\nalgorithm reported $\\Omega(n^{1-\\epsilon})$ uniform random edges from the diagonal in $A_{S(x, y)}$ (we will also get rid of this\nassumption). Then, by repeating this scheme $\\Theta(n^\\epsilon)$ times in parallel, with large constant probability\nthe edge corresponding to $A_{x,y}$ is reported at least once, which allows us to solve \\textsf{BInd}. \nThis reduction yields an optimal $\\Omega(n^{2-3\\epsilon})$ space lower bound for insertion-deletion streaming algorithm for \n\\textsf{Maximum Matching}, since $\\Theta(n^{\\epsilon})$ parallel executions are used to solve a problem that has a lower bound \nof $\\Omega((n-k)^2) = \\Omega(n^{2-2\\epsilon})$.\n\nIn the description above, we assumed that (1) $A$ is a uniform random binary matrix; (2) $A_{x,y} = 1$; and (3) the algorithm outputs\nuniform random positions from the diagonal of $A_{S(x, y)}$. To eliminate (1) and (2), Alice and Bob first sample\na uniform random binary matrix $X \\in \\{0, 1\\}^{n \\times n}$ from public randomness and consider the matrix obtained \nby computing the entry-wise XOR between $A$ and $X$, i.e., matrix $A \\oplus X$, instead. Observe that $A \\oplus X$ \nis a uniform random binary matrix (independently of $A$), and with probability $\\frac{1}{2}$, property (2), i.e., $(A \\oplus X)_{x,y} = 1$, holds. \nRegarding assumption (3), besides computing the XOR $A \\oplus X$, Alice and Bob also sample two random \npermutations $\\sigma_1, \\sigma_2: [n] \\rightarrow [n]$ from public randomness. Alice and Bob permute the rows \nand columns of $A \\oplus X$ using $\\sigma_1$ and $\\sigma_2$, respectively.\nThen, no matter which elements from the permuted relevant diagonal of $A \\oplus X$ are reported by the algorithm, \ndue to the random permutations, these elements could have originated from any other position in this diagonal. \nThis in turn makes every element along the diagonal equally likely to be reported, including the position $(x,y)$ \n(in the unpermuted) matrix that we are interested in.\n\nOur reduction for \\textsf{Minimum Vertex Cover} is similar but simpler. We show that only a constant number of parallel executions\nof an insertion-deletion streaming are required. \n\n\n\\vspace{0.1cm}\n\\textbf{Further Related Work.}\nHosseini et al. \\cite{hly19} were able to improve on the ``triple exponential length'' requirement of the input streams\nfor a characterization of insertion-deletion streaming algorithms in terms of linear\nsketches by Li et al. \\cite{lnw14} and Ai et al. \\cite{ahlw16}. They showed that in the case of XOR-streams \nand $0\/1$-output functions, input streams of length $\\mathrm{O}(n^2)$ are enough.\n\nVery recently, Kallaugher and Price \\cite{kp20} showed that if either the stream length or the maximum value \nof the stream (e.g. the maximum multiplicity of an edge in a graph stream) are substantially restricted, \nthen the characterization of turnstile streams as linear sketches cannot hold. For these situations they \ndiscuss problems where linear sketching is exponentially harder than turnstile streaming.\n\n\nBesides the \\textsf{Maximum Matching} problem, the only other separation result between the insertion-only and\nthe insertion-deletion graph stream models that we are aware of is a recent result by Konrad \\cite{k19}, who showed that \napproximating large stars is significantly harder in insertion-deletion streams.\n\n\n\\vspace{0.1cm}\n\\textbf{Outline.}\nWe give a lower bound on the communication complexity of \\textsf{Augmented Bi-Index} in Section~\\ref{sec:aug-bi-index}.\nThen, in Section~\\ref{sec:matching}, we show that a one-way two-party communication protocol for \\textsf{Maximum Matching} \ncan be used to solve \\textsf{Augmented Bi-Index}, which yield an optimal space lower bound for \\textsf{Maximum Matching} \nin insertion-deletion streams. We conclude with a similar reduction for \\textsf{Minimum Vertex Cover} in \nSection~\\ref{sec:vertex-cover}, which also implies an optimal space lower bound for \\textsf{Minimum Vertex Cover} in \ninsertion-deletion streams.\n\n\\section{Augmented Bi-Index} \\label{sec:aug-bi-index}\nIn this section, we define the one-way two-party communication problem \\textsf{Augmented Bi-Index} and prove a\nlower bound on its communication complexity. \n\n\\begin{problem}[\\textsf{Augmented Bi-Index}]\n    In an instance of {\\em \\textsf{Augmented Bi-Index}} {\\em $\\textsf{BInd}^{n,k}_\\delta$} we have two players denoted Alice and Bob.\n    Alice holds a binary matrix $A \\in \\lbrace 0,1 \\rbrace^{n \\times n}$. Bob holds indices $x, y \\in [n-k]$ and the incomplete\\footnote{We use $A_S$ to refer to the collection of entries indexed by the set $S$, so $A_S = (A_{i,j})_{(i,j)\\in S}$.} binary matrix $A_{S(x,y)}$ where\n    \\[\n        S(x,y) = \\{ (i,j) \\in [n]^2 \\,|\\, (x \\leq i < x + k) \\text{ and } (y \\leq j < y + k) \\} \\setminus \\{ (x,y) \\} \\ .\n    \\]    \n    Alice sends a single message $M$ to Bob who must output $A_{x, y}$ with probability at least $1-\\delta$.\n\\end{problem}\n\nOur lower bound proof consists of a reduction from the well-known \\textsf{Augmented Index} problem, which is known to \nhave large communication complexity.\n\n\n\n\n\\begin{problem}[\\textsf{Augmented Index}]\n    In an instance of {\\em \\textsf{Augmented Index}} {\\em $\\textsf{Ind}^{n}_\\delta$} we have two players denoted Alice and Bob.\n    Alice holds a binary vector $V \\in \\lbrace 0,1 \\rbrace^n$. Bob holds an index $\\ell \\in [n]$ and the vector suffix $V_{>\\ell} = (V_{\\ell+1}, V_{\\ell+2}, \\cdots, V_n)$. \n    Alice sends a single message $M$ to Bob who must output $V_\\ell$ with probability at least $1-\\delta$.\n\\end{problem}\n\nAs a consequence of Lemma~13 in \\cite{mnsw98}, we can see that this problem has linear communication complexity (see also Lemma~2 in \\cite{bjkk04} for a more direct proof technique).\n\n\\begin{theorem}[e.g. \\cite{mnsw98}]\\label{thm:augind-lower}\n    For $\\delta < 1\/3$, any randomised one-way communication protocol which solves $\\textsf{Ind}^{n}_\\delta$ must communicate $\\Omega(n)$ bits.\n\\end{theorem}\n\n\n\n\n\nWe are now ready to prove our lower bound for \\textsf{Augmented Bi-Index}.\n\\begin{theorem}\\label{thm:bind-lower}\n    For $\\delta < 1\/3$, any randomised one-way communication protocol which solves $\\textsf{BInd}^{n,k}_\\delta$ must communicate $\\Omega((n-k)^2)$ bits.\n\\end{theorem}\n\\begin{proof}\n    Let $\\mathcal{P}$ be a communication protocol for $\\textsf{BInd}^{n,k}_\\delta$ that uses messages of length at most \n    $S(n, k)$ bits. We will show how $\\mathcal{P}$ can be used to solve $\\textsf{Ind}^{(n-k)^2}_\\delta$ with the same message size.\n    \n    Let $V, \\ell$ be any instance of $\\textsf{Ind}^{(n-k)^2}_\\delta$. Alice builds the matrix $A \\in \\{ 0, 1 \\}^{n \\times n}$ by placing the bits of $V$ in lexicographical order in the top-left $(n-k)$-by-$(n-k)$ region:\n    \\[\n        A_{i,j} =   \\begin{cases}\n                        V_{j+(n-k)(i-1)} &\\text{for } i,j \\in [n-k]\\\\\n                        0 &\\text{otherwise}\n                    \\end{cases} \\ .\n    \\]    \n    This packing is illustrated in Figure~\\ref{fig:aug-index}(a).\n    \n    Alice runs protocol $\\mathcal{P}$ on $A$ and sends the resulting message $M$ to Bob.    \n    Now, Bob has the message $M$, the index $\\ell \\in [(n-k)^2]$ and the suffix $V_{>\\ell}$. Let $x, y \\in [n-k]$ be the unique pair of integers such that $\\ell = y + (n-k)(x-1)$. Observe that $A_{x,y} = V_\\ell$.\n    \n    For Bob to be able to complete protocol $\\mathcal{P}$ he needs to provide $A_{S(x,y)}$. Because of the way we packed the entries of $V$ onto $A$, the overlap between $V$ and $A_{S(x,y)}$ is a subset of the entries of $V_{>\\ell}$ (see Figure~\\ref{fig:aug-index}(b) for an illustration). Therefore Bob can complete the protocol and determine $A_{x,y} = V_\\ell$ with probability at least $1-\\delta$.    \n    By Theorem~\\ref{thm:augind-lower}, it must be that $S(n, k) = \\Omega((n-k)^2)$.\n\\end{proof}\n\n\\begin{figure}\n    \\centering\n    \\hspace{1cm}\n    \\subcaptionbox{Example packing of the bits of $V$ into matrix $A$ with $n=9$ and $k=4$.}[0.4\\textwidth]{\n        \\begin{tikzpicture}[\n            cell\/.style={rectangle, draw=black!35!white, minimum size =6mm, inner sep=0mm}\n        ]\n            \\fill[blue!20!white] (3mm, -3mm) rectangle (33mm, -33mm);\n            \\foreach \\y in {1,...,5}\n            {\n                \\foreach \\x in {1,...,5}\n                {                \n                    \\pgfmathtruncatemacro{\\i}{int(int((\\y-1)*5 + \\x))}\n                    \\node[cell] at (6*\\x mm, -6*\\y mm) {\\small $V_{\\i}$};\n                }\n                \\foreach \\x in {6,...,9} \\node[cell] at (6*\\x mm, -6*\\y mm) {$0$};\n            }\n            \\foreach \\y in {6,...,9}\n            {\n                \\foreach \\x in {1,...,9} \\node[cell] at (6*\\x mm, -6*\\y mm) {$0$};\n            }\n            \\draw[black, very thick] (3mm, -3mm) rectangle (57mm, -57mm);\n            \\draw[black, very thick] (3mm, -3mm) rectangle (33mm, -33mm);\n        \\end{tikzpicture}\n    }\n    \\hspace{1cm}\n    \\subcaptionbox{Bob can construct the area $A_{S(x,y)}$ given $V_{>\\ell}$, which is part of his input.}[0.4\\textwidth]{\n        \\begin{tikzpicture}[\n            cell\/.style={rectangle, minimum size =6mm, inner sep=0mm}\n        ]\n            \\fill[blue!20!white] (3mm, -3mm) rectangle (33mm, -33mm);\n            \\node[cell, fill=red!30!white] at (18mm, -18mm) {\\small $V_\\ell$};\n            \\fill[green!20!white] (3mm, -21mm) -- (3mm, -33mm) -- (33mm, -33mm) -- (33mm, -15mm) -- (21mm, -15mm) -- (21mm, -21mm) -- (3mm, -21mm);\n            \\draw[very thick, dashed] (15mm, -21mm) -- (15mm, -39mm) -- (39mm, -39mm) -- node[right] {$A_{S(x,y)}$} (39mm, -15mm) -- (21mm, -15mm) -- (21mm, -21mm) -- (15mm, -21mm);\n            \\draw[black, very thick] (3mm, -3mm) rectangle (57mm, -57mm);\n        \\end{tikzpicture}\n    }\n    \\caption{The construction of $A$ and $A_{S(x,y)}$ in Theorem~\\ref{thm:bind-lower}.}\n    \\label{fig:aug-index}\n\\end{figure}\n\n\\section{Maximum Matching} \\label{sec:matching}\nLet $\\mathbf{A}$ be a $C$-approximation insertion-deletion streaming algorithm for \\textsf{Maximum Matching} that errs \nwith probability at most $1\/10$. We will now show that $\\mathbf{A}$ can be used to solve $\\textsf{BInd}^{n,k}_{\\delta}$.\n\n\\subsection{Reduction}\nLet $A \\in \\{0, 1\\}^{n \\times n}, x \\in [n-k]$ and  $y \\in [n-k]$ be an instance of $\\textsf{BInd}^{n,k}_{\\delta}$. \nAlice and Bob first sample a uniform random binary matrix $X \\in \\{0,1\\}^{n \\times n}$ \nand random permutations $\\sigma_1, \\sigma_2: [n] \\rightarrow [n]$ from public randomness. Alice then computes matrix \n$A'$ which is obtained by first computing the entry-wise XOR of $A$ and $X$, denoted by $A \\oplus X$, \nand then by permuting the rows and columns\nof the resulting matrix by $\\sigma_1$ and $\\sigma_2$, respectively. Next, Alice interprets $A'$ as the incidence matrix\nof a bipartite graph $G(A')$. Alice runs algorithm $\\mathbf{A}$ on a random ordering of the edges of $G(A')$ and sends\nthe resulting memory state to Bob.\n\nNext, Bob also computes the entry-wise XOR between the part of the matrix $A$ that he knows about, $A_{S(x,y)}$, and $X$, followed by applying\nthe permutations $\\sigma_1$ and $\\sigma_2$. In doing so, Bob knows the matrix entries of $A'$ at positions \n$(\\sigma_1(i), \\sigma_2(j))$\nfor every $(i,j) \\in S(x,y)$. He can therefore compute the subset $E_S$ of the edges of $G(A')$ with\n\n$$E_S = \\{(\\sigma_1(i),\\sigma_2(j)) \\in [n]^2 \\ | \\ (i,j) \\in S(x,y) \\mbox{ and } A'(\\sigma_1(i),\\sigma_2(j)) = 1  \\} \\ .$$\n\nFurthermore, let $E_{diag} \\subseteq E_S$ be the set of edges $(\\sigma_1(i), \\sigma_2(j))$ so that $(i,j)$ lies on the same\ndiagonal in $A$ as $(x,y)$, or, in other words, there exists an integer $1 \\le q \\le k-1$ such that $(x+q, y+q) = (i, j)$.\nThen, let $E_{del} = E_S \\setminus E_{diag}$.\nBob continues the execution of algorithm $\\mathbf{A}$, as follows: for every edge $e \\in E_{del}$, \nBob introduces an edge deletion of $e$, in random order.\n\nLet $M'$ be the matching returned by $\\mathbf{A}$.\nFrom $M'$ Bob computes the matching $M$ as follows: If $|M' \\le 0.99 \\frac{k}{2C}|$ then Bob sets $M = \\varnothing$.\nOtherwise, Bob sets $M$ to be a uniform random subset of $M'$ of size exactly $0.99 \\frac{k}{2C}$.\n\n\\textbf{Parallel Executions.} Alice and Bob execute the previous process $\\ell = 100 \\cdot C$ times in parallel. Let $M^i$, $X^i$, \n$\\sigma_1^i$ and $\\sigma_2^i$ be $M$, $X$, $\\sigma_1$ and $\\sigma_2$ that are used in run $i$, respectively.\nLet $Q_i$ be the indicator random variable that is $1$ iff $M^i$ contains the edge $(\\sigma^i_1(x), \\sigma^i_2(y))$. \nWe also define $p = \\sum_i Q_i$ to be the total number of times the edges $(\\sigma^i_1(x), \\sigma^i_2(y))$ are reported. \nWhenever the edge $(\\sigma^i_1(x), \\sigma^i_2(y))$ is reported, we interpret this to be a claim that $A_{x,y} = \\neg X^i_{x,y}$. So depending on the value of $X^i_{x,y}$, this acts as a claim that $A_{x,y} = 0$ or $A_{x,y} = 1$. We define $p_0 = \\sum_{i: Q_i = 1} X^i_{x, y}$ (which counts how often\n$A_{x,y} = 0$ was claimed) and let $p_1 = p - p_0$ (the number of times $A_{x,y} = 1$ was claimed). Bob outputs $1$ as his estimator for $A_{x,y}$ if $p_1 \\ge p_0$ \nand $0$ otherwise.\n\n\n\\subsection{Analysis}\nLet $G$ be the bipartite graph with incidence matrix $A \\oplus X$, and let \n$$F = \\{ (i,j) \\in S(x,y) \\ | \\ (A \\oplus X)_{i,j} = 1 \\mbox{ and } \\nexists \\ q \\mbox{ s.t. } (i,j) = (x+q, y+q) \\} \\ .$$\nThen the graph $G - F$ is isomorphic to the graph $G(A') - E_{del}$. In particular, \n$G(A') - E_{del}$ is obtained from $G - F$ by relabeling the vertex sets of the two bipartitions\nusing the permutations $\\sigma_1$ and $\\sigma_2$. \n\nWe will first bound the maximum matching size in $G(A') - E_{del}$. To this end, we will bound the \nmaximum matching size in $G - F$, which is easier to do:\n\n\\begin{lemma}\\label{lem:matching-size}\n With probability $1-\\frac{1}{k^{10}}$, the graph $G(A') - E_{del}$ is such that:\n $$0.99 \\frac{k}{2} \\le \\mu(G(A') - E_{del}) \\le 1.01 \\frac{k}{2} + 2(n-k) \\ ,$$\n where $\\mu(G)$ denotes the \\emph{matching number} of $G$, i.e., the size of a maximum matching.\n\\end{lemma}\n\\begin{proof}\n We will consider the graph $G - F$ instead, since it is isomorphic to $G(A') - E_{del}$ and has the same\n maximum matching size.\n \n First, observe that $G$ is a random bipartite graph where every edge is included with probability $\\frac{1}{2}$.\n Let $U$ and $V$ denote the bipartitions in $G$, and consider the subsets $U' = [x, x+k)$ and $V' = [y, y+k)$.\n Observe that in the vertex induced subgraph $G[U' \\cup V']$\n all edges are deleted in $F$ except those that connect \n the vertices $x+i$ and $y+i$, for every $0 \\le i \\le k-1$. By a Chernoff bound, the number of edges and thus\n the maximum matching size in $G[U' \\cup V']$ is bounded by:\n $$0.99 \\cdot \\frac{k}{2} \\le \\mu(G[U' \\cup V']) \\le 1.01 \\cdot \\frac{k}{2} \\ , $$\n with probability $1 - \\frac{1}{k^{10}}$.\n \n Observe that, with probability $1 - \\frac{1}{k^{10}}$, the neighborhood $\\Gamma(U')$ is such that \n $$0.99 \\cdot \\frac{k}{2} \\le |\\Gamma(U')| \\le 1.01 \\cdot \\frac{k}{2} + (n-k) \\ .$$ \n The set $U'$ can therefore be matched to at most $1.01 \\cdot \\frac{k}{2} + (n-k)$ vertices in $V$. \n We thus obtain\n $$\\mu(G - F) \\le 1.01 \\cdot \\frac{k}{2} + 2(n-k) \\ ,$$\n since we may also be able to match all $n-k$ vertices of $U \\setminus U'$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:learn}\n Suppose that $M_i \\neq \\varnothing$. Then:\n \\begin{eqnarray*}\n\\frac{0.99}{2C} - \\frac{2(n - k)}{k} & \\le & \\Pr \\left[ Q_i = 1 \\right] \\le \\frac{0.99}{2C} \\ .\n \\end{eqnarray*}\n \n\n\\end{lemma}\n\\begin{proof}\n First, by construction of our reduction, since $M_i \\neq \\varnothing$ we have \n $|M_i| = 0.99 \\frac{k}{2C}$. \n Let \n $$U'_i = \\sigma_1^i([x, x+k)) \\mbox{ and } V'_i = \\sigma_2^i([y, y+k)) \\ .$$ \n Let $\\tilde{M}_i$ be the set of edges of $M_i$ connecting vertices in $U'_i$ to $V'_i$. Observe that there are\n $2(n-k)$ vertices in the graph outside the set $U'_i \\cup V'_i$. We thus \n have  \n $$|M_i| - 2(n-k) \\le |\\tilde{M}_i| \\le |M_i|  \\ .$$ \n Next, since the permutations $\\sigma^i_1, \\sigma^i_2$ are chosen uniformly at random, any edge of $\\tilde{M}_i$\n may have originated from any of the diagonal entries in $A_{S(x, y)}$. Hence, $\\tilde{M}_i$ claims the bits\n of at least $|M_i| - 2(n-k)$ and at most $|M_i|$ uniform random positions in the diagonal of $A_{S(x, y)}$.\n Every entry in the diagonal of $A_{S(x,y)}$ is thus claimed with the same probability.\n Since the diagonal of $A_{S(x,y)}$ is of length $k$, this probability is at least\n $$ \n \\frac{|M_i| - 2(n-k)}{k} = \\frac{0.99 \\frac{k}{2C} - 2(n-k)}{k} = \\frac{0.99}{2C} - \\frac{2(n - k)}{k} \\ ,\n $$\n and at most\n $$\\frac{|M_i|}{k} = \\frac{0.99 \\frac{k}{2C}}{k} = \\frac{0.99}{2C} \\ .$$ \n\\end{proof}\n\n\n\n\n\n\n\n\\begin{theorem}\\label{thm:reduction} \n Let $\\mathbf{A}$ be a $n^{\\epsilon}$-approximation insertion-deletion streaming algorithm \n for \\textsf{Maximum Matching} that errs with probability at most $1\/10$ and uses space $s$. Then there exists a communication \n protocol for $\\textsf{BInd}^{n,n - \\frac{1}{40} n^{1-\\epsilon}}_{0.05}$ that communication $\\mathrm{O}(n^\\epsilon \\cdot s)$ bits.\n\\end{theorem}\n\\begin{proof}\nLet $C = n^{\\epsilon}$ and let $k = n - \\frac{1}{40} n^{1-\\epsilon}$.\n First, by Lemma~\\ref{lem:matching-size}, with probability $1-\\frac{1}{k^{10}}$, the graph \n $G(A') - E_{del}$ contains a matching of size at least $0.99 k\/ 2$. By a union bound, \n the probability that this graph is of at least this size in each of the $\\ell$ iterations  \n is at least $1 - \\frac{\\ell}{k^{10}}$. Suppose from now on that this event happens.\n\n Let $\\ell_1$ be the number of times the algorithm $\\mathbf{A}$ succeeds, and let $\\ell_0$ be the number of times $\\mathbf{A}$\n errs. Then, $\\ell = \\ell_0 + \\ell_1$. Whenever $\\mathbf{A}$ succeeds, since $\\mathbf{A}$ is a $C$-approximation\n algorithm, the matching $M'_i$ is of size $0.99 \\frac{k}{2C}$, which further implies that $M_i$ is of size exactly\n $0.99 \\frac{k}{2C}$. Since the algorithm must return a correct matching, every time we have a claim (i.e. $Q_i = 1$), the claimed bit value must be correct. Thus, by Lemma~\\ref{lem:learn}, we get a correct claim on $A_{x,y}$ with probability at least \n \\begin{eqnarray*}\n  \\frac{0.99}{2C} - \\frac{2(n - k)}{k} & = & \\frac{0.99}{2n^{\\epsilon}} - \\frac{2(\\frac{1}{40} n^{1-\\epsilon})}{n - \\frac{1}{40} n^{1-\\epsilon}} \\ge  \\frac{0.99}{2n^{\\epsilon}} - \\frac{\\frac{1}{40} n^{1-\\epsilon}}{n} = \\frac{0.99}{2n^{\\epsilon}} - \\frac{1}{40n^{\\epsilon}} \\ge \\frac{2}{5 n^\\epsilon}   \\ , \\label{eqn:018}\n \\end{eqnarray*}\nwhere we used the inequality $\\frac{2x}{y-x} \\ge \\frac{x}{y}$, which holds for every $y > x$. \nWe thus expect to see the correct bit claimed at least $\\ell_1 \\cdot \\frac{2}{5 n^\\epsilon}$ times in total.\nOn the other hand, incorrect claims of the bit value can only occur when the algorithm errs. In the worst case, $\\mathbf{A}$ \nwill make as many false claims as possible - so we assume the algorithm never results in $M_i = \\emptyset$ when it errs. \nLemma~\\ref{lem:learn} also allows us to bound the probability of an incorrect claim for this bad algorithm by $\\frac{0.99}{2n^\\epsilon}$. We thus expect to see the wrong bit value claimed at most $\\ell_0 \\cdot \\frac{0.99}{2C} \\le \\frac{\\ell_0}{2n^\\epsilon}$ times.\n \n Recall that $\\ell = 100 n^\\epsilon$. Then, by standard concentration bounds, the probability that \n $\\ell_0 \\ge 2 \\cdot \\frac{\\ell}{10}$ is at most $\\frac{1}{100}$ (recall that the error probability of $\\mathbf{A}$ \n is at most $\\frac{1}{10}$). Suppose now that $\\ell_0 \\le \\frac{1}{5} \\ell$ holds, which also implies that $\\ell_1 \\ge \\frac{4}{5} \\ell$.\n We thus expect to learn the correct bit at least \n $$\\frac{4}{5} 100 n^{\\epsilon} \\cdot \\frac{2}{5 n^\\epsilon} = 32$$\n times, and using a Chernoff bound, it can be seen that the probability that we learn the correct bit less than \n $21$ times is at most $0.02$. \n Similarly, we expect to learn the incorrect bit at most \n $$\\frac{1}{5} 100 n^{\\epsilon} \\cdot \\frac{1}{2n^\\epsilon} = 10$$\n times, and by a Chernoff bound, it can be seen that the probability that we learn the incorrect bit at least $20$ times \n is at most $ 0.01$. Our algorithm therefore succeeds if all these events happen. Taking a union bound\n over all failure probabilities that occurred in this proof, we see that our algorithm succeeds with probability\n $$1 - \\frac{100n^{\\epsilon}}{k^{10}} - 0.01 - 0.02 - 0.01 \\ge 0.95 \\ .$$\n\\end{proof}\n\nSince by Theorem~\\ref{thm:bind-lower}, $\\textsf{BInd}^{n,n - \\frac{1}{40} n^{1-\\epsilon}}_{0.05}$ has randomized one-way \ncommunication complexity $\\Omega(n^{2-2\\epsilon})$, by Theorem~\\ref{thm:reduction} we obtain our main result of this section\n\\begin{corollary}\n Every insertion-deletion $n^\\epsilon$-approximation streaming algorithm  \n for \\textsf{Maximum Matching} that errs with probability at most $\\frac{1}{10}$ requires space $\\Omega(n^{2-3\\epsilon})$.\n\\end{corollary}\n\n\n\n\n\\section{Minimum Vertex Cover} \\label{sec:vertex-cover}\nLet $\\mathbf{B}$ be a $C$-approximation insertion-deletion streaming algorithm for \\textsf{Minimum Vertex Cover} that succeeds with\nprobability $1 - 1\/400$. Similar to the previous section, we will now show how $\\mathbf{B}$ can be used to solve $\\textsf{BInd}^{n,k}_{\\delta}$.\n\n\\subsection{Reduction}\nLet $A \\in \\{0, 1\\}^{n \\times n}, x \\in [n-k]$ and  $y \\in [n-k]$ be an instance of $\\textsf{BInd}^{n,k}_{\\delta}$. \nThe reduction for \\textsf{Minimum Vertex Cover} is very similar to the reduction for \\textsf{Maximum Matching} presented \nin the previous section. Alice's behaviour is in fact identical:\n\nFirst, Alice and Bob sample a uniform random binary matrix $X \\in \\{0,1\\}^{n \\times n}$ and random permutations \n$\\sigma_1, \\sigma_2: [n] \\rightarrow [n]$ from public randomness. \nAlice then computes matrix $A'$ which is obtained by first computing $A \\oplus X$\nand then  permuting the rows and then the columns of the resulting matrix by $\\sigma_1$ and $\\sigma_2$, respectively. \nAlice interprets $A'$ as the incidence matrix of a bipartite graph $G(A')$. Alice then runs algorithm $\\mathbf{B}$ \non a random ordering of the edges of $G(A')$ and sends the resulting memory state to Bob.\n\nNext, Bob also computes the entry-wise XOR between the part of the matrix $A$ that he knows about and $X$, followed by applying\nthe permutations $\\sigma_1$ and $\\sigma_2$. In doing so, Bob knows the matrix entries of $A'$ at positions \n$(\\sigma_1(i), \\sigma_2(j))$\nfor every $(i,j) \\in S(x,y)$. He can therefore compute the subset $E_S$ of the edges of $G(A')$ with\n\n$$E_S = \\{(\\sigma_1(i),\\sigma_2(j)) \\in [n]^2 \\ | \\ (i,j) \\in S(x,y) \\mbox{ and } A'(\\sigma_1(i),\\sigma_2(j)) = 1  \\} \\ .$$\n\nNext, Bob continues the execution of $\\mathbf{B}$ and introduces deletions {\\em for all edges in $E_S$} in random order. \nObserve that this \nstep is different to the reduction for \\textsf{Maximum Matching}. Let $I$ be the vertex cover produced by $\\mathbf{B}$.\n\n\\textbf{Parallel Executions.} Alice and Bob run the procedure above $40$ times in parallel. Denote by $I^i$, $X^i$, $E^i_S$,\n$A'^i$, $\\sigma_1^i$, and $\\sigma_2^i$ the variables $I, X, E_S, A', \\sigma_1$ and $\\sigma_2$ used in iteration $i$.  \nFurthermore, let $Q_i$ be \nthe indicator variable that is $1$ iff $\\{ \\sigma^i_1(x), \\sigma^i_2(y) \\} \\cap I_i \\neq \\varnothing$,\ni.e., the potential edge $(\\sigma^i_1(x), \\sigma^i_2(y))$ is covered by the vertex cover. \n\nIf there exists a run $j$ with $Q_j = 0$, then Bob predicts $A_{x,y} = X_{x,y}$ (if there are multiple such runs\nthen Bob breaks ties arbitrarily). Otherwise, Bob returns \\texttt{fail} and the algorithm errs.\n\n\\subsection{Analysis}\n\nThe first lemma applies to every parallel run $j$. For simplicity of notation, we will omit\nthe superscripts that indicate the parallel run in our random variables.\n\nWe first show an upper bound on the size of a minimum vertex cover in $G(A') - E_S$.\n\n\\begin{lemma}\\label{lem:vc-size}\n The size of a minimum vertex cover in $G(A') - E_S$ is at most $2(n-k) + 1$.\n\\end{lemma}\n\\begin{proof}\nLet $U, V$ be the bipartitions of the graph $G(A') - E_S$, let $U' = \\{\\sigma_1(a) \\ : \\ a \\in [x, x+k) \\}$ and let\n$V' = \\{\\sigma_2(b) \\ : \\ b \\in [y, y+k) \\}$. Observe that $(G(A') - E_S)[U' \\cup V']$ \ncontains at most one edge: The potential edge between $\\sigma_1(x)$ and $\\sigma_2(y)$. A valid vertex cover \nof $G(A') - E_S$ is therefore $(U \\setminus U') \\cup (V \\setminus V') + \\sigma_1(x)$,\nwhich is of size $2(n-k) + 1$.\n\\end{proof}\n\nNext, we prove the key property of our reduction: We show that if $A'_{\\sigma_1(x), \\sigma_2(y)} = 0$ (or equivalently, \n$A_{x,y} \\oplus X_{x,y} = 0$)\nthen neither $\\sigma_1(x)$ nor $\\sigma_2(y)$ is in the output vertex cover with large probability.\n\n\\begin{lemma} \\label{lem:covered}\n Assume that algorithm $\\mathbf{B}$ does not err in run $j$. Suppose that \n $A'^j_{\\sigma^j_1(x), \\sigma^j_2(y)} = 0$. Then the probability that $Q_j = 1$ is at most \n $$\\frac{3 C \\cdot (2(n-k) + 1)}{k} \\ .$$\n\\end{lemma}\n\\begin{proof}\n Consider the set $D  = \\{ (\\sigma^j_1(x + i), \\sigma^j_2(y+i)) \\ | \\ 0 \\le i \\le k-1 \\}$, i.e., the positions of the diagonal\n of $S(x,y) \\cup \\{x, y\\}$ permuted by $\\sigma^j_1$ and $\\sigma^j_2$. Then, since $A'^j$ is a uniform random matrix, with probability\n at least $1-\\frac{1}{k^{10}}$, the ``permuted diagonal'' $A'^j_{D}$ contains at least $0.99 k\/ 2$ entries with value $0$,\n or, in other words, graph $G(A'^j) - E^j_S$ contains at least $0.99 k\/ 2$ non-edges in the positions of the permuted diagonal $D$.\n By Lemma~\\ref{lem:vc-size}, the size of a minimum vertex cover in $G(A'^{j}) - E^{j}_S$ is at most $2(n-k) + 1$, and since\n $\\mathbf{B}$ has an approximation factor of $C$, the  vertex cover $I_j$ is of size at most $C \\cdot (2(n-k) + 1)$. Hence, \n at most $C \\cdot (2(n-k) + 1)$ non-edges in $D$ can be covered in $I_j$. However, since the permutations are random, the \n probability that the non-edge $(\\sigma^j_1(x), \\sigma^j_2(y))$\n is covered, which is identical to the event $Q_j = 1$, is therefore at most \n $$\\frac{C \\cdot (2(n-k) + 1)}{0.99 k\/ 2} \\le \\frac{3 C \\cdot (2(n-k) + 1)}{k} \\ .$$\n\\end{proof}\n\n\n\\begin{theorem} \\label{thm:vc-reduction}\n Let $\\mathbf{B}$ be a $n^{\\epsilon}$-approximation \n insertion-deletion streaming algorithm for \\textsf{Minimum Vertex Cover} that uses space $s$ and errs with probability \n at most $1\/400$. Then, there exists a communication protocol for $\\textsf{BInd}^{n,n - \\frac{1}{20}n^{1-\\epsilon}}_{\\frac{1}{3}}$ \n that communicates $\\mathrm{O}(s)$ bits.\n\\end{theorem}\n\\begin{proof}\nLet $k = n - \\frac{1}{40} n^{1-\\epsilon}$ and let $C= n^{\\epsilon}$.\n Consider the reduction given in the previous subsection. First, observe that since $\\mathbf{B}$ errs with probability\n at most $1\/400$, by the union bound the probability that $\\mathbf{B}$ errs at least once in the $40$ parallel executions of our\n reduction is at most $\\frac{1}{10}$. We assume from now on that the algorithm never errs.\n \n Observe that the matrices $A'^j$ are random matrices. Hence,\n\n\n the probability that there exists at least one run $i$ with $A'^i_{\\sigma^i_1(x), \\sigma^i_2(y)} = 0$ is at least \n $1 - (\\frac{1}{2})^{40}$. Suppose that this event happens. Let run $i$ be so that $A'^i_{\\sigma^i_1(x), \\sigma^i_2(y)} = 0$.\n Then, by Lemma~\\ref{lem:covered}, the probability that the non-edge $(\\sigma^i_1(x), \\sigma^i_2(y))$ is covered by $I_i$,\n or in other words, the probability that $Q_i = 1$, is at most \n \n $$\\frac{3 C \\cdot (2(n-k) + 1)}{k} = \\frac{3n^{\\epsilon} \\cdot (\\frac{1}{20} n^{1-\\epsilon} + 1) }{n - \\frac{1}{40}n^{1-\\epsilon}} = \n \\frac{\\frac{3}{20}n + 3n^{\\epsilon}}{n - \\frac{1}{40}n^{1-\\epsilon}} = \\frac{3}{20} + o(1) \\ . $$  \n \n Observe that whenever $Q_i = 0$, the algorithm outputs $X^i_{x,y}$ as a predictor for $A_{x,y}$. Since the algorithm $\\mathbf{B}$ \n does not err, we have $A_{x,y} \\oplus X^i_{x,y} = 0$. This implies that $A_{x,y} = X^i_{x,y}$, which establishes correctness.\n \n Last, we need to bound the error probability of our algorithm. First, the probability that at least one of the $40$ runs \n fails is at most $\\frac{1}{10}$. Next, the probability that none of the runs are such that $A'^j_{\\sigma^j_1(x), \\sigma^j_2(y)} = 0$\n is at most $(\\frac{1}{2})^{40}$. Furthermore, the probability that $Q_i = 1$ when $A'^i_{\\sigma^i_1(x), \\sigma^i_2(y)} = 0$ is at \n most $\\frac{3}{20} + o(1)$. Applying the union bound, we see that the overall error probability of our algorithm is at most\n $$\\frac{1}{10}+ (\\frac{1}{2})^{40} + \\frac{3}{20} + o(1) \\le \\frac{1}{3} \\ , $$\n for large enough $n$. \n\\end{proof}\nSince by Theorem~\\ref{thm:bind-lower}, $\\textsf{BInd}^{n,n - \\frac{1}{40}n^{1-\\epsilon}}_{\\frac{1}{3}}$ has a \ncommunication complexity of $\\Omega(n^{2-2\\epsilon})$, we\nobtain the following result:\n\\begin{corollary}\n Every insertion-deletion $n^\\epsilon$-approximation streaming algorithm for \\textsf{Minimum Vertex Cover} with error probability \n at most $\\frac{1}{400}$ requires space $\\Omega(n^{2-2\\epsilon})$.\n\\end{corollary}\n\n\\subsection{Insertion-deletion Streaming Algorithm for \\textsf{Minimum Vertex Cover}}\nWe now sketch a simple deterministic $n^\\epsilon$-approximation insertion-deletion streaming algorithm for \n\\textsf{Minimum Vertex Cover} on general graphs that uses space $\\mathrm{O}(n^{2-2\\epsilon} \\log n)$. \nLet $G=(V, E)$ be the graph described by the input stream. The algorithm proceeds as follows: \n\\begin{enumerate}\n \\item Arbitrarily partition $V$ into subsets $V_1, V_2, \\dots, V_{n^{1-\\epsilon}}$,\n each of size $n^{\\epsilon}$.\n \\item Consider the multi-graph $G'$ obtained from $G$ by contracting the sets $V_i$ into vertices.\n \\item \\textbf{While processing the stream:} For each pair of vertices $V_i, V_j$ in $G'$ deterministically maintain the number \n of edges connecting $V_i$ to $V_j$. \n \\item \\textbf{Post-processing:} Compute a minimum vertex cover $I'$ in the multi-graph $G'$. \n \\item \\textbf{Return} $I = \\cup_{V_j \\in I'} V_j$ as the vertex cover in $G$. \n\\end{enumerate}\n\n\\noindent \\textbf{Analysis:} Regarding space, the dominating space requirement is the maintenance of the number\nof edges between every pair $V_i, V_j$. Since there are $n^{2-2\\epsilon}$ such pairs, this requires space $\\mathrm{O}(n^{2-2\\epsilon} \\cdot \\log n)$.\n\nConcerning the approximation factor, let $I^*$ be a minimum vertex cover in $G$. \nRecall that $I'$ is an optimal cover in $G'$ and hence $|I'| \\le |I^*|$ (edge contractions cannot increase the size of \na minimum vertex cover). \nSince every set $V_j$ is of size $n^\\epsilon$, \nthe computed vertex cover $I$ is of size at most $|I'| \\cdot n^{\\epsilon} \\le |I^*|n^{\\epsilon}$, which proves the approximation factor.\nBy construction of the algorithm, every edge is covered.\n\n\\begin{theorem} \\label{thm:vc-algorithm}\n There is a deterministic $n^\\epsilon$-approximation insertion-deletion streaming algorithm for \\textsf{Minimum Vertex Cover}\n that uses space $\\mathrm{O}(n^{2-2\\epsilon} \\log n)$.\n\\end{theorem}\n\n \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\vrule}}\n\\def\\color[rgb]{0.2,0.6,0.1}{\\color[rgb]{0.2,0.6,0.1}}\n\\newcommand{\\color{blue}}{\\color{blue}}\n\\newcommand{\\color{red}}{\\color{red}}\n\\newcommand{\\color[rgb]{0.2,0.6,0.1}}{\\color[rgb]{0.2,0.6,0.1}}\n\\newcommand{\\eq}[1]{{\\begin{equation}#1\\end{equation}}}\n\\newcommand{\\spl}[1]{{\\begin{aligned}#1\\end{aligned}}}\n\\newcommand{\\eqn}[1]{\\begin{equation*}#1\\end{equation*}}\n\\newcommand{\\eqsp}[1]{{\\begin{equation}\\begin{aligned}#1\\end{aligned}\\end{\nequation}}}\n\\newcommand{\\quot}[1]{\\begin{itemize}\\item[]``#1\"\\end{itemize}}\n\n\\mathsurround 1pt\n\n\\newcommand{\\red}[1]{{\\color{red}#1}}\n\n\n\\title[Well-posedness of Hardy-H\\'enon equation]{Optimal well-posedness and forward self-similar solution for the Hardy-H\\'enon parabolic equation in critical weighted Lebesgue spaces}\n\n\\date\\today\n\n\\author[N. Chikami, M. Ikeda and K. Taniguchi]{Noboru Chikami, Masahiro Ikeda and Koichi Taniguchi}\n\\address[N. Chikami]\n\nGraduate School of Engineering, \nNagoya Institute of Technology, \nGokiso-cho, Showa-ku, Nagoya \n466-8555, Japan.}\n\\email{chikami.noboru@nitech.ac.jp}\n\\address[M. Ikeda]\n{Faculty of Science and Technology,\nKeio University, \n3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan\/ Center for Advanced Intelligence Project\nRIKEN, Japan.}\n\\email{masahiro.ikeda@keio.jp\/masahiro.ikeda@riken.jp}\n\\address[K. Taniguchi]\n{Advanced Institute for Materials Research,\nTohoku University,\n2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan.}\n\\email{koichi.taniguchi.b7@tohoku.ac.jp}\n\n\\keywords{Hardy-H\\'enon parabolic equation, well-posedness, global existence, nonexistence, self-similar solution}\n\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\n\\begin{document}\n\n\\footnote[0]\n{2010 {\\it Mathematics Subject Classification.}\nPrimary 35K05; Secondary 35B40;}\n\\maketitle\n\n\\begin{abstract}\nThe Cauchy problem for the Hardy-H\\'enon parabolic equation is studied in the critical and subcritical regime in weighted Lebesgue spaces on the Euclidean space $\\mathbb{R}^d$. \nWell-posedness for singular initial data and existence of non-radial forward self-similar solution of the problem are previously shown only for the Hardy and Fujita cases ($\\gamma\\le 0$) in earlier works. \nThe weighted spaces enable us to treat the potential $|x|^{\\gamma}$ as an increase or decrease of the weight, thereby we can prove well-posedness to the problem for all $\\gamma$ with $-\\min\\{2,d\\}<\\gamma$ including the H\\'enon case ($\\gamma>0$). \nAs a byproduct of the well-posedness, the self-similar solutions to the problem are also constructed for all $\\gamma$ without restrictions. \nA non-existence result of local solution for supercritical data is also shown. \nTherefore our critical exponent $s_c$ turns out to be optimal in regards to the solvability. \n\\end{abstract} \n\n\n\\section{Introduction}\\label{sec:1}\n\n\\subsection{Background and setting of the problem}\nWe consider the Cauchy problem of the Hardy-H\\'enon parabolic equation\n\\begin{equation}\\label{HH}\n\t\\begin{cases}\n\t\t\\partial_t u - \\Delta u = |\\cdot|^{\\gamma} |u|^{\\alpha-1} u,\n\t\t\t&(t,x)\\in (0,T)\\times D, \\\\\n\t\tu(0) = u_0 \\in L^q_{s}(\\mathbb{R}^d),\n\t\\end{cases}\n\\end{equation}\nwhere  $T>0,$ $d\\in \\mathbb{N}$, $\\gamma\\in \\mathbb{R},$ $\\alpha\\in \\mathbb{R},$ \n$D:=\\mathbb{R}^d$ if $\\gamma\\ge0$ and $D:=\\mathbb{R}^d\\setminus\\{0\\}$ if $\\gamma<0.$ \nHere, $\\partial_t:=\\partial\/\\partial t$ is the time derivative, \n$\\Delta:=\\sum_{j=1}^d\\partial^2\/\\partial x_j^2$ is the Laplace operator on $\\mathbb{R}^d$, \n$u=u(t,x)$ is the unknown real- or complex-valued function on \n$(0,T)\\times \\mathbb R^d$, and $u_0=u_0(x)$ is a prescribed real- or \ncomplex-valued function on $\\mathbb R^d$. \nIn this paper, we assume that the initial data $u_0$ belongs to weighted Lebesgue spaces $L^q_s(\\mathbb{R}^d)$ given by\n\\[L^q_s(\\mathbb{R}^d):=\\left\\{ f \\in \\mathcal{M} (\\mathbb{R}^d) \\,;\\, \n\t\\|f\\|_{L^q_s}\n\t< \\infty \\right\\}\n\\] endowed with the norm\n\\[\n\t\\|f\\|_{L^q_s} := \\left(\\int_{\\mathbb{R}^d} ( |x|^s |f(x)|)^q \\, dx \\right)^\\frac1{q},\n\\]\nwhere $s\\in \\mathbb{R}$ and $q\\in [1,\\infty]$ and $\\mathcal{M} (\\mathbb{R}^d)$ denotes the set of all\nLebesgue measurable functions on $\\mathbb{R}^d$. \nWe express the time-space-dependent function \n$u$ as $u(t)$ or $u(t,x)$ depending on circumstances. \nWe introduce a exponent $\\alpha_F(d,\\gamma)$ given by\n\\[\n    \\alpha_F(d,\\gamma):=1+\\frac{2+\\gamma}{d},\n\\]\nwhich is often referred as the {\\it Fujita exponent} and is known to divide the existence and \nnonexistence of positive global solutions (See \\cite[Theorem 1.6]{Qi1998}). \n\nThe equation \\eqref{HH} with $\\gamma<0$ is known as a \n{\\it Hardy parabolic equation} while that with $\\gamma>0$ \nis known as a {\\it H\\'enon parabolic equation}. \nThe elliptic part of \\eqref{HH}, that is, \n\\begin{equation}\\nonumbe\n    -\\Delta \\phi=|x|^{\\gamma}|\\phi|^{\\alpha-1}\\phi,\\ \\ \\ x\\in \\mathbb{R}^d,\n\\end{equation}\nwas proposed by H\\'enon as a model to study the rotating \nstellar systems (see \\cite{H-1973}), \nand has been extensively studied in the mathematical context, \nespecially in the field of nonlinear analysis and variational methods \n(see \\cite{GhoMor2013} for example). \nThe case $\\gamma=0$ corresponds to a heat equation with \na standard power-type nonlinearity, often called the {\\it Fujita equation}, \nwhich has been extensively studied in various directions. \nRegarding well-posedness of the Fujita equation ($\\gamma=0$) in Lebesgue \nspaces, we refer to \\cites{Wei1979, Wei1980, Gig86}, among many. \nConcerning the global dynamics and asymptotic behaviors, \nwe refer to \\cites{Ish2008,IT-arxiv,CIT-arxiv} for the Fujita \nand Hardy cases of \\eqref{HH} with Sobolev-critical exponents. \nArticles \\cites{HisIsh2018, HisTak-arxiv} give definitive results on \nthe optimal singularity of initial data to assure the solvability for $\\gamma\\le 0.$ \nIn \\cite{Tay2020}, unconditional uniqueness has been established for the Hardy case $\\gamma<0.$ Concerning earlier conditional uniqueness when $\\gamma<0$, \nwe refer to \\cites{BenTayWei2017, Ben2019}. \nLastly, we refer to \\cite{Maj-arxiv} for the analysis of the problem \n\\eqref{HH} with an external forcing term in addition to the nonlinear term. \n\nLet us recall that the equation \\eqref{HH} is invariant under the scale transformation \n\\begin{equation}\\label{scale}\nu_{\\lambda}(t,x) \n:= \\lambda^{\\frac{2+\\gamma}{\\alpha-1}} u(\\lambda^2 t, \\lambda x), \n\t\\quad \\lambda>0.\n\\end{equation}\nMore precisely, if $u$ is the classical solution to \\eqref{HH}, \nthen $u_{\\lambda}$ defined as above also solves the equation \nwith the rescaled initial data $ \\lambda^{\\frac{2+\\gamma}{\\alpha-1}} u_0(\\lambda x).$\nUnder \\eqref{scale}, the $L^q_{s}(\\mathbb{R}^d)$-norm scales as follows:\n$\\|u_\\lambda (0)\\|_{L^q_{s}} \n\t= \\lambda^{-s+\\frac{2+\\gamma}{\\alpha-1}-\\frac{d}{q}} \\|u(0)\\|_{L^q_{s}}.$ \nWe say that the space $L^q_{s}(\\mathbb{R}^d)$ is (scale-){\\sl critical} if $s=s_c$ with \n\\begin{equation}\\label{d:sc}\n\ts_c=s_c(q)= s_c(d,\\gamma,\\alpha,q) := \\frac{2+\\gamma}{\\alpha-1} - \\frac{d}{q}, \n\\end{equation}\n{\\sl subcritical} if $s<s_c,$ and {\\sl supercritical} if $s>s_c.$ \nIn particular, when $s=s_c = 0,$ $L^{\\frac{d(\\alpha-1)}{2+\\gamma}}(\\mathbb{R}^d)$ \nis a critical Lebesgue space. \n\nOne of our purposes in this article is to establish well-posedness results in the critical and subcritical cases ($s\\le s_c$) for all the range of the parameter $\\gamma$ such that \n$-\\min\\{2,d\\} < \\gamma,$ including the H\\'enon case ($\\gamma>0$). \nIn terms of well-posedness in function spaces containing sign-changing singular data, \nthe equation \\eqref{HH} has been studied mainly for $\\gamma<0$ (Hardy case). \nAs far as we know, there has been no result concerning well-posedness in the sense of Hadamard (Existence, uniqueness and continuous dependency) of the H\\'enon parabolic equation $\\gamma>0$ for sign-changing singular data. \nFor the Hardy and Fujita cases that are well-studied, our results provide \nwell-posedness in new function spaces (See Remark \\ref{r:HH.LWP}). \nWe stress that the use of weighted spaces enables us to treat the equations \nfor all $\\gamma$ in a unified manner. \n\nOur second purpose of this article is to \nprove the existence of forward self-similar solutions for all of \nHardy, Fujita and H\\'enon cases, without restrictions on the exponent $\\alpha.$ \nA forward self-similar solution is a solution such that $u_{\\lambda} = u$ for all $\\lambda>0,$ \nwhere $u_{\\lambda}$ is as in \\eqref{scale}. \nIn \\cite[Lemma 4.4]{Wan1993}, \nthe existence of radially symmetric self-similar solutions for \n$d\\ge3$, $\\gamma>-2$ and $\\alpha\\ge1+\\frac{2(2+\\gamma)}{d-2}$ \nis established. \nLater, the case $\\alpha_F(d,\\gamma)<\\alpha<1+\\frac{2(2+\\gamma)}{d-2}$ \nis treated in \\cite{Hir2008} under some additional restriction on $\\gamma,$ \nnamely $\\gamma\\le 0$ for $d\\ge4$ and $\\gamma\\le \\sqrt{3}-1$ for $d=3.$   \nIn \\cite[Theorem 1.4]{BenTayWei2017}, the existence of self-similar solutions that are not \nnecessarily radially symmetric has been proved for all $\\alpha>\\alpha_F(d,\\gamma),$ \nbut only for the Hardy case $\\gamma<0$ (See also \\cite{Chi2019}). \nOur result (Theorem \\ref{t:HH.self.sim}) covers all the previous results and asserts the existence of non-radial forward self-similar solutions for $\\gamma$ and $\\alpha$ \nsuch that $-\\min(2,d)<\\gamma$ and $\\alpha > \\alpha_F(d,\\gamma)$. \n\nIn earlier works, the crux of the matter has been \nthe handling of the singular potential $|x|^{\\gamma}.$ \nIf $\\gamma<0$, the conventional methods are to regard the potential \n$|x|^{\\gamma}$ as a function belonging either to the Lorentz space $L^{\\frac{d}{-\\gamma},\\infty}(\\mathbb{R}^d)$ \n(\\cites{BenTayWei2017, Tay2020}) \nor the homogeneous Besov space $\\dot B^{\\frac{d}{q}+\\gamma}_{q,\\infty}(\\mathbb{R}^d),$ \n$1\\le q \\le \\infty$ (\\cite{Chi2019}), \nand apply appropriate versions of H\\\"older's inequality to establish suitable \nheat kernel estimates. In contrast to their previous works, in this article, \nwe treat the potential $|x|^{\\gamma}$ as the increase or decrease of the order of \nthe weight in $L^q_s(\\mathbb{R}^d)$-norms, thereby covering the H\\'enon case ($\\gamma>0$) as well. \nIn this regard, the introduction of the weighted spaces is crucial to our results. \nIndeed, if the data only belongs to the critical Lebesgue space, then we \nmay only treat the Hardy case ($\\gamma<0$) in our main theorem \n(See Remark \\ref{r:HH.LWP} below). \nThe proofs of the well-posedness results rely on Banach's fixed point theorem. \nThe essential ingredient in the proof of various nonlinear estimates is the following linear estimate for the heat semigroup $\\{e^{t\\Delta}\\}_{t>0}$ on weighted Lebesgue spaces: \n\\begin{equation}\\nonumber\n\t\\| e^{t\\Delta} f\\|_{L^q_{s'}} \n\t\\le C t^{-\\frac{d}2 (\\frac1{p}-\\frac1{q}) - \\frac{s-s'}{2} } \t\n\t\t\\| f\\|_{L^p_{s}},\n\\end{equation}\n(see Lemma \\ref{l:wLpLq} for precise statement), which is known in the literatures such as \\cite{Tsu2011} except for the end-point cases. \nIn this article, we first extend the above estimate to the end-point cases $(i)$ $1<p<q=\\infty,$ $(ii)$ $p=q=1,$ $(iii)$ $1=p<q<\\infty,$ \n$(iv)$ $p=q=\\infty$ and $(v)$ $(p,q)=(1,\\infty)$. \n\nTo complete the picture of the admissible range of our well-posedness results, we also discuss the non-existence of positive distributional local solutions to \\eqref{HH} \nfor suitable supercritical data $u_0 \\in L^q_{s}(\\mathbb{R}^d)$ with $s>s_c.$ \n\n\\subsection{Main results}\nIn order to state our results, we introduce the following auxiliary function spaces. Let $\\mathscr{D}'([0,T)\\times\\mathbb{R}^d)$ be the space of distributions on $[0,T)\\times\\mathbb{R}^d$. \n\\begin{definition}[Kato class]\n\\label{def:Kato}\nLet $T \\in (0,\\infty],$ $s\\in\\mathbb{R}$ and $q\\in [1,\\infty].$ \n\\begin{enumerate}[(1)]\n\\item In the critical regime, i.e. $\\tilde s=s_c$, where $s_c$ is defined by \\eqref{d:sc}, for $s<\\tilde s$, the space $\\mathcal{K}^{s}(T)$ is defined by \n\\begin{equation}\\nonumber\n   \\mathcal{K}^{s}(T)\n   :=\\left\\{u\\in \\mathscr{D}'([0,T)\\times\\mathbb{R}^d) \\,;\\, \n   \t\\|u\\|_{\\mathcal{K}^{s}(T')}\n   <\\infty\\ \\text{for any } T' \\in (0,T)\\right\\}\n\\end{equation}\nendowed with a norm\n\\[\n\\|u\\|_{\\mathcal K^{s}(T)}\n\t:=\\sup_{0\\le t\\le T}t^{\\frac{s_c -s}{2}} \\|u(t)\\|_{L^q_s}.\n\\]\nWe simply write $\\mathcal{K}^{s}=\\mathcal{K}^{s}(\\infty)$ \nwhen $T=\\infty,$ if it does not cause confusion. \n\\item In the subcritical regime, i.e. $\\tilde s<s_c$, for $s<\\tilde s$, the space $\\tilde{\\mathcal{K}}^{s}(T)$ is defined by \n\\begin{equation}\\nonumber\n   \\tilde{\\mathcal{K}}^{s}(T)\n   :=\\left\\{u\\in \\mathscr{D}'([0,T)\\times\\mathbb{R}^d) \\,;\\, \n   \t\\|u\\|_{\\tilde{\\mathcal{K}}^{s}(T')}\n   <\\infty\\ \\text{for any } T' \\in (0,T)\\right\\}\n\\end{equation}\nendowed with a norm\n\\[\n\\|u\\|_{\\tilde{\\mathcal{K}}^{s}(T)}\n\t:=\\sup_{0\\le t\\le T}t^{\\frac{\\tilde s -s}{2}} \\|u(t)\\|_{L^q_s}.\n\\]\n\\end{enumerate}\n\\end{definition}\nFor $t\\in \\mathbb{R}_+$, we introduce the heat kernel $g_t:\\mathbb{R}^d\\rightarrow \\mathbb{R}_+$ given by\n\\begin{equation}\\label{d:h.krnl}\n\tg_t(x) := (4\\pi t)^{-\\frac{d}{2}} e^{-\\frac{|x|^2}{4t}}, \\ x \\in\\mathbb{R}^d. \n\\end{equation}\nWe denote by $\\{e^{t\\Delta}\\}_{t\n\\ge 0}$ the free heat semigroup defined by \n\\[\n(e^{t\\Delta} \\varphi) (x) := (g_t \\ast \\varphi) (x)\n\\]\nfor $\\varphi \\in L^1_{loc}(\\mathbb{R}^d),$ where $\\ast$ denotes the convolution with respect to the space variable. Let $\\mathcal{S}'(\\mathbb{R}^d)$ denotes the space of the Schwarz distributions.\nFor $\\varphi \\in \\mathcal{S}'(\\mathbb{R}^d)$, $e^{t\\Delta}\\varphi$ \nis defined by duality. \n\nIn what follows, we denote by $C_0^\\infty(\\mathbb{R}^d)$ the space of all smooth functions \nwith compact support. We also denote by $\\mathcal{L}^q_s (\\mathbb{R}^d)$ \nthe closure of $C_0^\\infty(\\mathbb{R}^d)$ with respect to the topology of $L^q_s (\\mathbb{R}^d).$ \nNext we give a definition of mild solution as follows. \n\\begin{definition}[Mild solution]\\label{def:sol-A}\nLet $T \\in (0,\\infty]$, $\\tilde s\\le s_c$ and $u_0 \\in L^q_{\\tilde s} (\\mathbb{R}^d)$. Let $Y := \\mathcal{K}^s(T)$ if $\\tilde s = s_c$ \nand $Y := \\tilde{\\mathcal{K}}^{s}(T)$ if $\\tilde s<s_c.$\nA function $u : [0,T] \\times \\mathbb{R}^d \\to \\mathbb C\\ \\text{or}\\ \\mathbb{R}$ is called an \n$L^q_{\\tilde s} (\\mathbb{R}^d)$-mild solution to \\eqref{HH} with initial data \n$u(0)=u_0$ if it satisfies \n$u\\in C([0,T]; L^q_{\\tilde s} (\\mathbb{R}^d)) \\cap Y$\nand the integral equation \n\\begin{equation}\\label{integral-eq}\nu(t,x) = e^{t\\Delta} u_0(x) \n\t+ \\int_0^t e^{(t-\\tau)\\Delta} \n\t\\left\\{ |\\cdot|^{\\gamma} |u(\\tau,\\cdot)|^{\\alpha-1}u(\\tau, \\cdot)\\right\\}(x) \\, d\\tau\n\\end{equation}\nfor any $t \\in [0,T]$ and almost everywhere $x \\in \\mathbb{R}^d$.\nThe time $T$ is said to be the maximal existence time, which is denoted \nby $T_m$, if the solution cannot be extended beyond $[0,T).$ \nMore precisely, \n\\begin{equation}\\label{d:Tm}\n\tT_m = T_m (u_0) := \\sup \\left\\{T>0 \\,;\\, \n\t\t\\left.\\begin{aligned}&\\text{There exists } \\text{ a unique solution $u$ of \\eqref{HH}} \\\\\n\t\t\t&\\text{in } C([0,T]; L^q_{\\tilde s}(\\mathbb{R}^d)) \\cap Y \n\t\t\t\\text{ with initial data $u_0$}\n\t\t\\end{aligned}\\right. \\right\\}.\n\\end{equation}\nWe say that $u$ is global in time if $T_m = + \\infty$ \nand that $u$ blows up in a finite time otherwise. \nMoreover, we say that $u$ is dissipative if $T_m = + \\infty$ and \n\\[\n\t\\lim_{t\\to\\infty} \\|u(t)\\|_{L^q_{\\tilde s}} = 0. \n\\]\n\\end{definition}\nThe following is one of our main results on local well-posedness of \\eqref{HH} in the critical space $L^q_{s_c}(\\mathbb{R}^d)$. \n\\begin{thm}[Well-posedness in the critical space]\t\\label{t:HH.LWP}\nLet $d\\in\\mathbb{N},$ $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \n\\begin{equation}\\label{t:HH.LWP.c0}\n\t\\gamma> -\\min(2,d)\n\t\t\\quad\\text{and}\\quad\n\t\\alpha> \\alpha_F(d,\\gamma).\n\\end{equation}\nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{t:HH.LWP.c1}\n\t\\alpha\\le q \\le \\infty \n\t\t\\quad\\text{and}\\quad\n\t\\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\,\n\t\t\\frac{2}{d(\\alpha-1)} + \\frac{(d-2)\\alpha - d -\\gamma}{d(\\alpha-1)^2}\\right\\}\n\\end{equation}\nand let $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{t:HH.LWP.c2}\n\ts_c - \\frac{d(\\alpha-1)}{\\alpha} \\left(\\frac{2}{d(\\alpha-1)} - \\frac1{q} \\right) \\le s\n\t< \\min \\left\\{ s_c, \\, s_c + \\frac{(d-2)\\alpha - d -\\gamma}{\\alpha(\\alpha-1)} \\right\\}. \n\\end{equation}\nThen the Cauchy problem \\eqref{HH} is locally well-posed in $L^q_{s_c}(\\mathbb{R}^d)$ for arbitrary data $u_0\\in L^q_{s_c}(\\mathbb{R}^d)$ and globally well-posed for small data $u_0\\in L^q_{s_c}(\\mathbb{R}^d)$. \nMore precisely, the following assertions hold. \n\\begin{enumerate}[$(i)$]\n\\item {\\rm (}Existence{\\rm )} \nFor any $u_0 \\in L^q_{s_c}(\\mathbb{R}^d)$ with $q <\\infty$ \n(Replace $L^\\infty_{s_c}(\\mathbb{R}^d)$ with \n$\\mathcal{L}^\\infty_{s_c}(\\mathbb{R}^d)$ when $q = \\infty$), \nthere exist a positive number $T$ and an $L^q_{s_c}(\\mathbb{R}^d)$-mild solution \n$u\n\\ to \\eqref{HH} satisfying \n\\begin{equation}\\label{t:HH.LWP.est} \n\t\\|u\\|_{\\mathcal{K}^s(T)} \n\t\t\\le 2 \\|e^{t\\Delta} u_0 \\|_{\\mathcal{K}^s(T)}. \n\\end{equation}\nMoreover, the solution can be extended to the maximal interval $[0,T_m),$ \nwhere $T_m$ is defined by \\eqref{d:Tm}. \n\\item {\\rm (}Uniqueness{\\rm )} \nLet $T>0.$ If $u, v \\in \\mathcal{K}^s(T)$ satisfy \n\\eqref{integral-eq} with $u(0) = v(0)=u_0 \\in L^q_{s_c}(\\mathbb{R}^d)$ \n(Replace $L^\\infty_{s_c}(\\mathbb{R}^d)$ with $\\mathcal{L}^\\infty_{s_c}(\\mathbb{R}^d)$ when $q=\\infty$), \nthen $u=v$ on $[0,T].$\n\\item {\\rm (}Continuous dependence on initial data{\\rm )}  \nLet $u$ and $v$ be the $L^q_{s_c}(\\mathbb{R}^d)$-mild solutions constructed in (i) with given initial data $u_0$ and $v_0$ respectively. \nLet $T(u_0)$ and $T(v_0)$ be the corresponding existence times. \nThen there exists a constant $C$ depending on $u_0$ and $v_0$ such that \nthe solutions $u$ and $v$ satisfy \n\\begin{equation}\\nonumber\n\t\\|u-v\\|_{L^\\infty(0,T;L^q_{s_c}) \\cap \\mathcal{K}^s(T)} \n\t\\le C \\|u_0-v_0\\|_{L^q_{s_c}}\n\\end{equation}\nfor some $T\\le \\min\\{T(u_0), T(v_0)\\}.$ \n\\item {\\rm (}Blow-up criterion{\\rm )} \nIf $u$ is an $L^q_{s_c}(\\mathbb{R}^d)$-mild solution constructed in the assertion $(i)$ and \n$T_m<\\infty,$ then $\\|u\\|_{\\mathcal{K}^s(T_m)}=\\infty.$\n\\item {\\rm (}Small data global existence and dissipation{\\rm )} \nThere exists $\\varepsilon_0>0$ depending only on $d,\\gamma,\\alpha,q$ and $s$ such that if $u_0 \\in \\mathcal{S}'(\\mathbb{R}^d)$ satisfies \n$\\|e^{t\\Delta}u_0\\|_{\\mathcal{K}^s}<\\varepsilon_0,$ then $T_m=\\infty$ and \n$\\|u\\|_{\\mathcal{K}^s} \\le 2\\varepsilon_0.$ Moreover, the solution $u$ is dissipative. In particular, if $\\|u_0\\|_{L^p_{s_c}}$ is sufficiently small, then \n$\\|e^{t\\Delta}u_0\\|_{\\mathcal{K}^s}<\\varepsilon_0.$ \n\\end{enumerate}\n\\end{thm}\n\\begin{rem}[Optimality of the power $\\alpha$ for the nonlinearity]\nBy the blow-up result in \\cite{Qi1998}, \nthe condition $\\alpha>\\alpha_F(d,\\gamma)$ is known to be optimal. \nIndeed, if $\\alpha \\le \\alpha_F(d,\\gamma)$, then the solutions of \\eqref{HH}\nwith positive initial data blow up in a finite time. \n\\end{rem}\n\\begin{rem}[Uniqueness $(ii)$]\nIn $(ii)$, $T$ is arbitrary and there is no restriction on the size of the quantity $\\|u\\|_{\\mathcal{K}^s(T)}.$ We note that this uniqueness result concerns a so-called {\\sl conditional} uniqueness since we can prove that $u\\in \\mathcal{K}^s(T)$ is a solution to \\eqref{HH} if and only if \n$u \\in C([0,T] ; L^q_{s_c}(\\mathbb{R}^d)) \\cap \\mathcal{K}^s(T)$ is a solution to \\eqref{HH}, \nprovided that $u_0 \\in L^q_{s_c}(\\mathbb{R}^d).$ See Remark \\ref{r:crt.est2} below. \nWe note that for the Hardy case, unconditional uniqueness has been established by \\cite{Tay2020} in the Lebesgue framework. \n\\end{rem}\n\\begin{exa}[Small data global existence $(v)$]\nWe give a typical example of the initial data $u_0$ \nsatisfying the assumptions in $(v)$ : \n$u_0 \\in L^1_{loc}(\\mathbb{R}^d)$ such that \n$|u_0(x)| \\le c |x|^{-\\frac{2+\\gamma}{\\alpha-1}}$ for almost all $x\\in\\mathbb{R}^d,$ \nwhere $c$ is a sufficiently small constant. \nThis initial data in particular generates a self-similar solution. \nSee Theorem \\ref{t:HH.self.sim} below. \n\\end{exa}\n\\begin{rem}[New contributions for $\\gamma\\neq0$]\n\\label{r:HH.LWP}\nFor the Hardy case $\\gamma > 0,$ Theorem \\ref{t:HH.LWP} is new \nconcerning sign-changing solutions for singular initial data. \nTheorem \\ref{t:HH.LWP} also gives a new result in the Hardy case ($\\gamma<0$). \nIn particular, when $s_c \\equiv 0$, that is, $q = \\frac{d(\\alpha-1)}{2+\\gamma}$, the critical space is the usual Lebesgue space \n$L^{\\frac{d(\\alpha-1)}{2+\\gamma}}(\\mathbb{R}^d).$ \nTheorem \\ref{t:HH.LWP} gives a new well-posedness result in the usual Lebesgue space $L^{\\frac{d(\\alpha-1)}{2+\\gamma}}(\\mathbb{R}^d)$ for $d\\ge2$ and $-2<\\gamma<0$. \n\\end{rem}\n\\begin{rem}\nWe note that $s_c$ is always positive when $\\gamma>0$ \nwhile $s_c$ can be either negative or non-negative. In other words, \nthe initial data $u_0$ must have a stronger decay at infinity when $\\gamma>0.$ \n\\end{rem}\n\nWe next discuss global existence of forward self-similar solutions to \\eqref{HH}. \nAs mentioned earlier, the result below is not known in the literature for large $\\gamma>0.$ \n\\begin{thm}[Existence of forward self-similar solutions]\t\\label{t:HH.self.sim}\nLet $d\\in\\mathbb{N},$ $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \\eqref{t:HH.LWP.c0}. \nLet $\\varphi(x) := \\omega(x) |x|^{-\\frac{2+\\gamma}{\\alpha-1}},$ \nwhere $\\omega\\in L^\\infty(\\mathbb{R}^d)$ is homogeneous of degree 0 and \n$\\|\\omega\\|_{L^\\infty}$ is sufficiently small so that $\\|e^{t\\Delta}\\varphi\\|_{\\mathcal{K}^s}<\\varepsilon_0$, where $\\varepsilon_0$ appears in Theorem \\ref{t:HH.LWP}.\nThen there exists a self-similar solution $u_\\mathcal{S}$ of \n\\eqref{HH} with the initial data $\\varphi$ such that $u_\\mathcal{S}(t) \\to \\varphi$ in $\\mathcal{S}'(\\mathbb{R}^d)$ as $t\\to0.$ \n\\end{thm}\n\nThe following theorem deals with the local well-posedness \nof \\eqref{HH} in the subcritical space $L^q_{\\tilde s}(\\mathbb{R}^d)$ with $\\tilde s< s_c.$ \n\\begin{thm}[Well-posedness in the subcritical space]\t\\label{t:HH.LWP.sub}\nLet $d\\in\\mathbb{N},$ $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \\eqref{t:HH.LWP.c0}. \nLet $\\tilde s\\in\\mathbb{R}$ be such that \n\\begin{equation}\\label{t:HH.LWP.sub.cs}\n\t\\max\\left\\{-\\frac{d}{\\alpha}, \\, \\frac{\\gamma}{\\alpha-1} \\right\\}\n\t\t<\\tilde s < \\frac{2+\\gamma}{\\alpha-1}.\n\\end{equation}\nLet $q\\in[1,\\infty]$ be such that \n\\begin{equation}\\label{t:HH.LWP.sub.c1}\n\t\\alpha\\le q \\le \\infty \n\t\t\\quad\\text{and}\\quad\n\t-\\frac{\\tilde s}{d} < \\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\,\n\t\t\\frac1{\\alpha} \\left(1-\\frac{\\tilde s}{d} \\right), \\, \n\t\t\\frac1{d} \\left(\\frac{2 + \\gamma}{\\alpha-1} -\\tilde s \\right)  \\right\\}\n\\end{equation}\nand let $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{t:HH.LWP.sub.c2}\n\t\\frac{\\tilde s+\\gamma}{\\alpha} \\le s\n\t\t\\quad\\text{and}\\quad\n\t- \\frac{d}{q} < s \n\t< \\min \\left\\{ \\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}, \\tilde s \\right\\}. \n\\end{equation}\nThen the Cauchy problem \\eqref{HH} is locally well-posed in $L^q_{\\tilde{s}}(\\mathbb{R}^d)$ for arbitrary data $u_0\\in L^q_{\\tilde{s}}(\\mathbb{R}^d)$. More precisely, the following assertions hold. \n\\begin{enumerate}[$(i)$]\n\\item {\\rm (}Existence{\\rm )} \nFor any $u_0 \\in L^q_{\\tilde s}(\\mathbb{R}^d),$ \nthere exist a positive number $T$ depending only on $\\|u_0\\|_{L^q_{\\tilde s}}$ and an $L^q_{\\tilde s}(\\mathbb{R}^d)$-mild solution \n$u \n\\ to \\eqref{HH} satisfying \n\\begin{equation}\\nonumber\n\t\\|u\\|_{\\tilde{\\mathcal{K}}^s(T)} \n\t\t\\le 2 \\|e^{t\\Delta} u_0 \\|_{\\tilde{\\mathcal{K}}^s(T)}. \n\\end{equation}\nMoreover, the solution can be extended to the maximal interval \n$[0,T_m),$ where $T_m$ is defined by \\eqref{d:Tm}.\n\\item {\\rm (}Uniqueness in $\\tilde{\\mathcal{K}}^s(T)${\\rm )} \nLet $T>0.$ If $u, v \\in \\tilde{\\mathcal{K}}^s(T)$ satisfy \n\\eqref{integral-eq} with $u(0) = v(0)=u_0,$ then $u=v$ on $[0,T].$ \n\\item {\\rm (}Continuous dependence on initial data{\\rm )}  \nFor any initial data $u_0$ and $v_0$ in $L^q_{\\tilde s}(\\mathbb{R}^d),$ \nlet $T(u_0)$ and $T(v_0)$ be the corresponding existence time given by $(i).$ \nThen there exists a constant $C$ depending on $u_0$ and $v_0$ such that \nthe corresponding solutions $u$ and $v$ satisfy \n\\begin{equation}\\nonumber\n\t\\|u-v\\|_{L^\\infty(0,T;L^q_{\\tilde s}) \\cap \\tilde{\\mathcal{K}}^s(T)} \n\t\\le C \\|u_0-v_0\\|_{L^q_{\\tilde s}}\n\\end{equation}\nfor some $T\\le \\min\\{T(u_0), T(v_0)\\}.$ \n\\item {\\rm (}Blow-up criterion{\\rm )} If $T_m<\\infty,$ \n\tthen $\\lim_{t\\rightarrow T_m-0}\\|u(t)\\|_{L^q_{\\tilde s}}=\\infty.$ \nMoreover, the following lower bound of blow-up rate holds: \nthere exists a positive constant $C$ independent of $t$ such that \n\\begin{equation}\\label{t:HH.LWP:Tm}\n\t\\|u(t)\\|_{L^q_{\\tilde s}} \\ge  \\frac{C}{(T_m - t)^{\\frac{s_c-\\tilde s}{2}} }\n\\end{equation}\nfor $t\\in (0,T_m)$.\n\\end{enumerate}\n\\end{thm}\n\\begin{rem}\nNote that \n\\eqref{t:HH.LWP.sub.c1} implies $\\tilde s<s_c,$ i.e., $u_0 \\in L^q_{\\tilde s}(\\mathbb{R}^d)$ \nis a scale-subcritical data.  \n\\end{rem}\n\nFinally, for the scale-supercritical case, i.e. $s>s_c$, we prove non-existence of a weak local positive solution, whose definition is given below. More precisely, we may prove that there exists a positive initial data $u_0$ in $L^q_s(\\mathbb{R}^d)$ with $s>s_c$ that does not generate a \nlocal solution to \\eqref{HH} even in the distributional sense. \n\\begin{definition}[Weak solution]\n\\label{d:w.sol}\nLet $T>0$. We call a function $u:[0,T)\\times \\mathbb{R}^d\\rightarrow \\mathbb{R}$ a weak solution to the Cauchy problem \\eqref{HH} \nif $u$ belongs to $L^{\\alpha}(0,T;L^{\\alpha}_{\\frac{\\gamma}{\\alpha},loc}(\\mathbb{R}^d))$ \nand if it satisfies the equation \\eqref{HH} in the distributional sense, i.e., \n\\begin{align}\\label{weak}\n\\notag\\int_{\\mathbb{R}^d} &u(T',x) \\eta (T',x) \\, dx-\\int_{\\mathbb{R}^d} u_0(x) \\eta (0,x) \\, dx\\\\\n\t&= \\int_{[0,T']\\times\\mathbb{R}^d} u(t ,x)(\\Delta \\eta + \\eta_t) (t ,x) \n\t+ |x|^{\\gamma} |u(t, x)|^{\\alpha-1} u(t,x) \\,\\eta(t,x)  \\, dx\\,dt\n\\end{align}\nfor all $T'\\in [0,T]$ and for all $\\eta \\in C^{1,2}([0,T]\\times \\mathbb{R}^d)$ such that\n$\\operatorname{supp} \\eta(t, \\cdot)$ is compact. \n\\end{definition}\nWe remark that our $L^q_{\\tilde s}(\\mathbb{R}^d)$-mild solutions \nare weak solutions in the above sense. See Lemma \\ref{mildweak} in Appendix.\n\\begin{thm}[Nonexistence of local positive weak solution]\n\\label{t:nonex}\nLet $d\\in \\mathbb N$ and $\\gamma \\in \\mathbb R$. \nAssume that $q\\in [1,\\infty],$ $\\alpha\\in\\mathbb{R}$ and $s\\in\\mathbb{R}$ satisfy \n$\\alpha>\\max(1, \\alpha_F(d,\\gamma))$ and $s>s_c$. \nThen there exists an initial data $u_0 \\in L^q_s (\\mathbb{R}^d)$ such that the \nproblem \\eqref{HH} with $u(0)=u_0$ has no local positive weak solution. \n\\end{thm}\n\\bigbreak\nThe rest of the paper is organized as follows: In Section 2, we \nprove the linear estimates and nonlinear ones in weighted Lebesgue spaces. \nSection 3 is devoted to the proof of Theorems \\ref{t:HH.LWP}, \\ref{t:HH.LWP.sub} \nand \\ref{t:HH.self.sim}. \nWe then give a sketch of the proof of Theorem \\ref{t:nonex} in Section 5. \nIn Appendix, we collect some elementary properties related to our function spaces and prove Lemma \\ref{mildweak}. \n\n\\section{Linear and nonlinear estimates}\nThroughout the rest of the paper, we denote by $C$ \na harmless constant that may change from line to line. \n\\subsection{Linear estimate}\nThe following estimate for the heat semigroup $\\{e^{t\\Delta}\\}_{t\\ge0}$ in weighted Lebesgue space is known except for the endpoint cases (see \\cites{Tsu2011, OkaTsu2016}).\n\\begin{lem}[Linear estimate]\n\t\\label{l:wLpLq}\nLet $d\\in\\mathbb N,$ $1\\le p \\le q \\le \\infty$ and  \n\\begin{equation}\t\\label{l:wLpLq:cs}\n\t-\\frac{d}{q} < s' \\le s < d\\left( 1-\\frac{1}{p} \\right).\n\\end{equation}\nIn addition, $s\\le 0$ when $p=1$ and $0\\le s'$ when $q=\\infty.$ \nIn particular,  \\eqref{l:wLpLq:cs} is understood as $s'=s=0$ when $p=1$ and $q=\\infty.$ \nThen there exists some positive constant $C$ depending on $d,$ $p,$ $q,$ $s$ and\n$s'$ such that \n\\begin{equation}\\nonumber\n\t\\| e^{t\\Delta} f\\|_{L^q_{s'}} \n\t\\le C t^{-\\frac{d}2 (\\frac1{p}-\\frac1{q}) - \\frac{s-s'}{2} } \n\t\t\\| f\\|_{L^p_{s}}\n\\end{equation}\nfor all $f\\in L^p_{s}(\\mathbb{R}^d)$ and $t>0$. \nMoreover, condition \\eqref{l:wLpLq:cs} is optimal. \n\\end{lem}\nWe mainly focus on the endpoint cases in the following proof.\n\\begin{proof}\nThe inequality for $1< p\\le q <\\infty$ follows from Lemma 3.2, \\cite[Proposition C.1]{OkaTsu2016} and the fact that the weight function \n$|x|^{s p}$ belongs to the Muckenhoupt class $A_p$ if and only if \n$- \\frac{d}{p} < s < d(1- \\frac1{p}).$ \n\nFor the endpoint exponents, we divide the proof into five cases : \n$(i)$ $1<p<q=\\infty,$ $(ii)$ $p=q=1,$ $(iii)$ $1=p<q<\\infty,$ \n$(iv)$ $p=q=\\infty$ and $(v)$ $(p,q)=(1,\\infty).$ \nIt suffices to prove the inequality for $e^{\\Delta} f$ and then resort to \na dilation argument as in the proof of \\cite[Proposition 2.1]{BenTayWei2017}. \n\nThroughout the proof of this lemma, \nwe write $a\\lesssim b$ if $a \\le C b$ with some constant $C.$\\\\ \n\\underline{$(i)$ $1<p<q=\\infty$}: Since $|x|^{s'} \\lesssim |x-y|^{s'} + |y|^{s'}$ if $s' \\ge 0,$ we have \n\\begin{equation*}\n\t|x|^{s'} | e^{\\Delta} f(x)| \n\t\\lesssim \\int_{\\mathbb{R}^d} |x-y|^{s'} g(x-y) |f(x)| \\, dy \n\t\t+ \\int_{\\mathbb{R}^d} |y|^{s'} g(x-y) |f(x)| \\, dy = : I_1 + I_2.\n\\end{equation*}\nFor $I_1,$ H\\\"older's inequality with $\\frac1{p}+\\frac1{p'}=1,$ $p>1,$ leads to \n\\begin{equation*}\\nonumber\n\tI_1\n\t\t \\le \\left( \\int_{\\mathbb{R}^d} (|y|^{-s} |x-y|^{s'} g(x-y) )^{p'}\\, dy \\right)^{\\frac1{p'}}  \\, \\|f\\|_{L^p_s} \n\t\t\\lesssim \\|f\\|_{L^p_s}, \n\\end{equation*}\nthanks to Lemma \\ref{l:g.unfrm.bnd} $(1)$ with $q\\equiv p',$ $a\\equiv s$ \nand $b\\equiv s',$ where $0\\le s<\\frac{d}{p'}$ and $s'\\ge0.$ \nSimilarly, H\\\"older's inequality and Lemma \\ref{l:g.unfrm.bnd} $(2)$ \nwith $q\\equiv p'$ and $c\\equiv s-s'$ yields \n\\begin{align*}\\nonumber\n\tI_2\n\t\t& \\le \\left( \\int_{\\mathbb{R}^d} (|y|^{-(s-s')} g(x-y) )^{p'}\\, dy \\right)^{\\frac1{p'}}  \\, \\|f\\|_{L^p_s} \n\t\t\\lesssim \\|f\\|_{L^p_s},\n\\end{align*}\nwhere $0\\le s-s' < \\frac{d}{p'}.$ \nThus, $\\| e^{\\Delta} f\\|_{L^{\\infty}_{s'}} \\lesssim \\| f\\|_{L^p_{s}}$ provided that \n$0 \\le s' \\le s < d\\left(1-\\frac1{p}\\right).$ \n\n\\underline{$(ii)$ $p=q=1$}: We have \n$|y|^{-s} \\lesssim |x-y|^{-s} + |x|^{-s}$ if $s\\le 0$ and thus \n\\begin{align*}\n\t\\|e^{\\Delta} f\\|_{L^1_{s'}} \n\n\t&\\lesssim \\int_{\\mathbb{R}^d} |x|^{s'} \\int_{\\mathbb{R}^d}  g(x-y) |x-y|^{-s} |y|^{s} |f(y)| \\,dy \\, dx \\\\\n\t\t&\\qquad\\qquad+ \\int_{\\mathbb{R}^d} |x|^{s'-s} \\int_{\\mathbb{R}^d}  g(x-y) |y|^{s} |f(y)| \\,dy \\, dx \\\\ \n\t&\\lesssim \\int_{\\mathbb{R}^d} \\left( \\int_{\\mathbb{R}^d} |x|^{s'} g(x-y) |x-y|^{-s} \\, dx \\right) |y|^{s} |f(y)|  \\,dy\\\\\n\t\t&\\qquad\\qquad+ \\int_{\\mathbb{R}^d} \\left( \\int_{\\mathbb{R}^d}  |x|^{s'-s}  g(x-y) \\, dx\\right) |y|^{s} |f(y)| \\,dy \\ \\lesssim \\|f\\|_{L^1_s}\n\\end{align*}\nthanks to Fubini's theorem and Lemma \\ref{l:g.unfrm.bnd} with \n$q\\equiv 1,$ $a\\equiv -s',$ $b\\equiv -s$ and $c\\equiv s-s',$ where \n$0 \\le -s' < d,$ $0 \\le -s$ and $0 \\le s-s' < d.$ \nThus, $\\| e^{\\Delta} f\\|_{L^{1}_{s'}} \\lesssim \\| f\\|_{L^1_{s}}$ provided that $-d<s'\\le s \\le 0.$\n\n\\underline{$(iii)$ $1=p < q < \\infty$}: By H\\\"older's inequality \nwith $1=\\frac1{q}+\\frac1{q'},$ $q<\\infty,$ we have  \n\\begin{align*}\n\t|e^{\\Delta} f(x)| \n\n\t&\\le \\left( \\int_{\\mathbb{R}^d} |y|^{-sq} g(x-y)^q |y|^s |f(y)| \\, dy \\right)^{\\frac1{q}} \\|f\\|_{L^1_s}^{\\frac1{q'}}\n\\end{align*}\nfor $s\\le 0.$ Taking the $L^q_{s'}(\\mathbb{R}^d)$-norm of the both sides of the above, we obtain \n\\begin{align*}\n\t\\|e^{\\Delta} f\\|_{L^q_{s'}} \n\t&\\le \\left( \\int_{\\mathbb{R}^d} |x|^{s'q} \\left( \\int_{\\mathbb{R}^d} |y|^{s(1-q)} g(x-y)^q |f(y)| \\, dy \\right) dx \\right)^{\\frac1{q}} \\|f\\|_{L^1_s}^{\\frac1{q'}}.\n\\end{align*}\nSince $|y|^{-qs} \\lesssim |x-y|^{-qs} + |x|^{-qs}$ if $s<0,$ \nFubini's theorem and Lemma \\ref{l:g.unfrm.bnd} with \n$q\\equiv q,$ $a\\equiv -s',$ $b\\equiv -s$ and $c\\equiv s-s'$ yield  \n\\begin{align*}\n\\int_{\\mathbb{R}^d} |x|^{s'q} \n\t&\\left( \\int_{\\mathbb{R}^d} |y|^{-sq} g(x-y)^q |y|^s |f(y)| \\, dy \\right) dx \\\\\n\t&\\lesssim \\int_{\\mathbb{R}^d} |x|^{s'q} \\left( \\int_{\\mathbb{R}^d} |x-y|^{-qs} g(x-y)^q |y|^{s} |f(y)| \\, dy \\right) dx \\\\\n\t&\\qquad\\qquad \n\t+ \\int_{\\mathbb{R}^d} |x|^{-(s-s')q} \\left( \\int_{\\mathbb{R}^d} g(x-y)^q |y|^{s} |f(y)| \\, dy \\right) dx \\\\\n\t&\\lesssim \\int_{\\mathbb{R}^d} |y|^{s} |f(y)| \\left( \\int_{\\mathbb{R}^d} (|x|^{s'} |x-y|^{-s} g(x-y) )^q \\, dx \\right) dy \\\\\n\t&\\qquad\\qquad \n\t+ \\int_{\\mathbb{R}^d} |y|^{s} |f(y)| \\left( \\int_{\\mathbb{R}^d} (|x|^{-(s-s')} g(x-y))^q \\, dx \\right) dy\n\\lesssim \\|f\\|_{L^1_s}, \n\\end{align*}\nwhere $0\\le -s' < \\frac{d}{q},$ $0\\le -s$ and $0 \\le s-s' < \\frac{d}{q}.$ \nThus, $\\| e^{\\Delta} f\\|_{L^{q}_{s'}} \\lesssim \\| f\\|_{L^1_{s}}$ provided that \n$-\\frac{d}{q}<s'\\le s \\le 0.$\n\n\\underline{$(iv)$ $p=q=\\infty$}: \nSince $|x|^{s'} \\lesssim |x-y|^{s'} + |y|^{s'}$ if $s' \\ge 0,$ we have \n\\begin{align*}\n\t|x|^{s'} &|e^{\\Delta} f(x)| \n\t\\le |x|^{s'} \\int_{\\mathbb{R}^d} |y|^{-s} g(x-y) \\, dy \\|f\\|_{L^\\infty_s}\\\\\n\t&\\lesssim \\left(  \\int_{\\mathbb{R}^d} |y|^{-s} |x-y|^{s'} g(x-y) \\, dy \n\t\t+ \\int_{\\mathbb{R}^d} |y|^{s'-s} g(x-y) \\, dy \\right)  \\|f\\|_{L^\\infty_s}\n\t\\lesssim \\|f\\|_{L^\\infty_s}\n\\end{align*}\nthanks to Lemma \\ref{l:g.unfrm.bnd} with \n$q\\equiv 1,$ $a\\equiv s,$ $b\\equiv s'$ and $c\\equiv s-s',$ where \n$0\\le s< d,$ $0\\le s'$ and $0\\le s-s' < d.$ \nThus, $\\| e^{\\Delta} f\\|_{L^{\\infty}_{s'}} \\lesssim \\| f\\|_{L^{\\infty}_{s}}$ provided that \n$0\\le s'\\le s <d.$\nThe case $(v)$ $(p,q) = (1,\\infty)$ is trivial. \nWe complete the proof of the endpoint estimates. \n\n\\smallbreak\nNext, we prove the optimality of \\eqref{l:wLpLq:cs} for \n$1< p \\le q < \\infty$ by contradiction. \nSuppose that the inequality holds when $s' \\le -\\frac{d}{q}.$ \nWe notice that every function $g$ in $L^q_{s'}(\\mathbb{R}^d)$ must satisfy \n$\\displaystyle \\liminf_{|x|\\to0} |x|^{\\frac{d}{q}+s'} |g(x)| =0$ thanks to \nCorollary \\ref{c:wLp.sg.dcy} in Appendix. \nIn particular, we have \n$\n\t\\Liminf_{|x|\\to0} |g(x)| =0\n$\nas $0 \\le -s'-\\frac{d}{q}.$ Since $0<\\frac{d}{p}+s < d,$ a function $f$ defined by \n\\[\nf(x) := \\left\\{\\begin{aligned}\n\t&C, &&|x|\\le 1\\\\\n\t&0, &&\\text{else},\n\\end{aligned}\\right. \n\\]\nwhere $C$ is a positive constant, belongs to $L^p_s(\\mathbb{R}^d).$ However, clearly \n\\[\n\\displaystyle \\liminf_{|x|\\to0} |e^{t\\Delta} f(x)| \\neq 0,\n\\]\nwhich implies that $e^{t\\Delta} f \\notin L^{q}_{s'}(\\mathbb{R}^d)$ and leads to a contradiction. \n\nThe optimality of the upper bound of \\eqref{l:wLpLq:cs} is based on the \nfact that the space $L^p_s(\\mathbb{R}^d)$ contains functions that are not in $L^1_{loc}(\\mathbb{R}^d),$ \nif $d\\left( 1-\\frac{1}{p} \\right)< s.$ Let \n\\[\nf(x) := \\left\\{\\begin{aligned}\n\t&|x|^{-d}, &&|x|\\le 1\\\\\n\t&0, &&\\text{else},\n\\end{aligned}\\right. \n\\]\nso that it belongs to $L^p_s(\\mathbb{R}^d)$ as $p(d-s)<d$ \n(Note that the space $L^p_s(\\mathbb{R}^d)$ is defined for all measurable functions). \nA standard argument then shows that $e^{t\\Delta} f$ does not make sense for the \nfunction $f.$ Indeed, for every $t>0$ and every $x$ such that $|x|\\le 1,$ the estimates hold:\n\\begin{equation*}\n\te^{t\\Delta} f(x)\n\t=  (4\\pi t)^{-\\frac{d}2} \\int_{|y|\\le 1} e^{\\frac{-|x-y|^2}{4t}} |y|^{-d} \\, dy \n\t\\ge  (4\\pi t)^{-\\frac{d}2} \\int_{|y|\\le 1} e^{-\\frac{1}{t}} |y|^{-d} \\, dy = \\infty, \n\\end{equation*}\nwhere we have used $|x-y|\\le 2.$ \nThus $e^{t\\Delta}$ is not well-defined in $L^p_s(\\mathbb{R}^d)$ if $d\\left( 1-\\frac{1}{p} \\right)< s.$ \nWhen $s= d(1-\\frac1{p}),$ it suffices to take \n\\[\nf(x) := \\left\\{\\begin{aligned}\n\t&|x|^{-d} \n\t\\left( \\log\\left(e+ \\frac1{|x|}\\right) \\right)^{-\\frac{a}{p}}, &&|x|\\le 1,\\\\\n\t&0, &&\\text{else},\n\\end{aligned}\\right. \n\\]\nwhere $p\\ge a>1,$ and show that $e^{t\\Delta} f$ is not well-defined for the function \nby carrying out the same argument as above. Thus, we conclude the lemma. \n\\end{proof}\n\n\n\\subsection{Nonlinear estimates}\nGiven $u_0\\in L^q_{s_c}(\\mathbb{R}^d)$ in the critical regime (resp. $L^q_{\\tilde{s}}(\\mathbb{R}^d)$ in the subcritical regime) and $T>0,$ let us define a map \n$\\Phi : u \\mapsto \\Phi(u)$ on $\\mathcal{K}^s(T)$ (resp. $\\tilde{\\mathcal{K}}^s(T)$) by \n\\begin{equation}\\label{map}\n\t\\Phi(u) (t) := e^{t\\Delta} u_0 + N(u)(t)\n\\end{equation}\nwith \n\\begin{equation}\\label{mapN}\n\tN(u)(t) :=  \\int_0^t e^{(t-\\tau)\\Delta} \n\t\\left\\{ |\\cdot|^{\\gamma} F(u(\\tau,\\cdot)) \\right\\} d\\tau\n\t\\quad\\text{and}\\quad\n\tF(u) := |u|^{\\alpha-1}u.\n\\end{equation}\n\n\\subsubsection{Critical case}\nThe following are the stability and contraction estimates in the critical regime. \nThe assertion $(2)$ below for $\\theta<1$ is not required in the proof of existence \nbut is used in the proof of uniqueness.  \n\\begin{lem\n\\label{l:Kato.est}\nLet $T \\in (0,\\infty]$ and $d\\in\\mathbb{N}.$ \nLet $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \\eqref{t:HH.LWP.c0}. \n\\begin{enumerate}[$(1)$]\n\\item \nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{l:Kato.est.c1}\n\t\\alpha\\le q \\le \\infty \\quad\\text{and}\\quad\n\t\\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\, \n\t\\frac{2}{d(\\alpha-1)} + \\frac{(d-2)\\alpha - d -\\gamma}{d\\alpha (\\alpha-1)} \\right\\}. \n\\end{equation}\nLet $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{l:Kato.est.c2}\n\t\\frac{\\gamma}{\\alpha-1}\\le s\n\t\t\\quad\\text{and}\\quad\n\t\\max\\left\\{- \\frac{d}{q}, \\, s_c - \\frac2{\\alpha} \\right\\} < s \n\t< \\min \\left\\{ s_c, \\, s_c + \\frac{(d-2)\\alpha - d -\\gamma}{\\alpha(\\alpha-1)} \\right\\},\n\\end{equation}\nwhere $s_c$ is as in \\eqref{d:sc}. \nThen there exists a positive constant $C_0$ \ndepending only on $d,$ $\\alpha,$ $\\gamma,$ $q$ and $s$ such that \nthe map $N$ defined by \\eqref{mapN} satisfies \n\\begin{equation}\\label{l:Kato.est1}\n\\|N(u)\\|_{\\mathcal{K}^s(T)} \n\t\\le C_0 \\|u\\|_{\\mathcal{K}^s(T)}^{\\alpha}\n\\end{equation}\nfor all $u \\in \\mathcal{K}^s(T).$ \n\\item\nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{l:Kato.est.c1'}\n\\begin{aligned}\n\t&\\alpha \\le q \\le \\infty, \\\\\n\t\t\\text{and}\\quad\n\t&\\frac1{q} < \\min \\left\\{ \\frac2{d(\\alpha-1)},\\, \n\t\\frac2{d(\\alpha-1)} + \\frac{\\theta(d-2)(\\alpha-1) - 2 - \\gamma}\n\t\t\t{d(\\alpha-1)(1+\\theta(\\alpha-1))} \\right\\},\n\\end{aligned}\n\\end{equation}\nwhere $\\theta \\in (0,1]$ ($\\frac1{2+\\gamma} < \\theta$ if d=1). \nLet $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{l:Kato.est.c2'}\n\\begin{aligned}\n&\ts_c - \\frac{d}{\\theta} \\left( \\frac{2}{d(\\alpha-1)} -\\frac{1}{q} \\right)\\le s \\\\\n\t\t\\text{and}\\quad\n&\t\\max\\left\\{ -\\frac{d}{q}, \\, s_c - \\frac2{1+\\theta(\\alpha-1)}  \\right\\}  \n\t< s < \\min\\left\\{ s_c, \\, s_c + \\frac{(d-2)\\alpha-d-2}{(1+\\theta(\\alpha-1))(\\alpha-1)}\n\t\t\t\t\t \\right\\}. \n\\end{aligned}\n\\end{equation}\nThen there exists a positive constant $C_1$ \ndepending only on $d,$ $\\alpha,$ $\\gamma,$ $q,$ $s$ and $\\theta$ such that \nthe map $N$ defined by \\eqref{mapN} satisfies  \n\\begin{equation}\\label{l:Kato.est2}\n\\begin{aligned}\n\\|N(u) - N(v)\\|_{\\mathcal{K}^s(T)} \n\t\\le C_1 &\\left(\\|u\\|_{\\mathcal{K}^s(T)}\n\t\t\t\t+\\|v\\|_{\\mathcal{K}^s(T)} \\right)^{\\theta(\\alpha-1)} \\\\\n\t\t&\\times\\left(\\|u\\|_{L^\\infty(0,T; L^q_{s_c}) }\n\t\t\t\t+\\|v\\|_{L^\\infty(0,T; L^q_{s_c}) } \\right)^{(1-\\theta)(\\alpha-1)} \n\t\\|u-v\\|_{\\mathcal{K}^s(T)}\n\\end{aligned}\n\\end{equation}\nfor all $u,v \\in \\mathcal{K}^s(T) \\cap L^\\infty(0,T ; L^q_{s_c}(\\mathbb{R}^d))$ \n($u,v \\in \\mathcal{K}^s(T)$ if $\\theta = 1$). \n\\end{enumerate}\n\\end{lem}\n\\begin{rem}\nNote that \\eqref{l:Kato.est.c1'} and \\eqref{l:Kato.est.c2'} for $\\theta=1$ \nare equivalent to \\eqref{l:Kato.est.c1} and \\eqref{l:Kato.est.c2}, respectively. \nThe estimate \\eqref{l:Kato.est2} fails for $\\theta=0$ as $C_1$ is divergent as \n$\\theta\\to0.$ \n\\end{rem}\n\\begin{proof}\nWe first prove \\eqref{l:Kato.est1}. We have \n\\begin{align*}\n\\|N(u)(t)\\|_{L^q_s} \n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 \\{(\\alpha-1)s - \\gamma\\}} \n\t\\| |\\cdot|^{\\gamma} F(u(\\tau)) \\|_{L^{\\frac{q}{\\alpha}}_{\\sigma}} d\\tau \\\\\n\\end{align*}\nby Lemma \\ref{l:wLpLq} with $q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s\\equiv s$ and $s'\\equiv \\sigma := \\alpha s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < s \\le \\alpha s -\\gamma < d(1-\\frac{\\alpha}{q}),$ i.e., \n\\begin{equation}\\label{l:Kato.est:pr1}\n\t\\alpha \\le q \\le \\infty, \\quad \n\t\\frac{\\gamma}{\\alpha-1} \\le s\n\t\t\\quad\\text{and}\\quad\n\t-\\frac{d}{q} < s < \\frac{\\gamma+d}{\\alpha}-\\frac{d}{q}. \n\\end{equation}\nAs $\\| |\\cdot|^{\\gamma} F(u) \\|_{L^{\\frac{q}{\\alpha}}_{\\sigma}}= \\| u \\|_{L^q_s}^{\\alpha}, $\nwe have \n\\begin{align*}\n\\|N(u)(t)\\|_{L^q_s} \n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 \\{(\\alpha-1)s - \\gamma\\}} \n\t\\tau^{-\\frac{(s_c-s)\\alpha}{2}} d\\tau \\times  \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha}, \n\\end{align*}\nwhere the last integral is bounded by \n\\begin{equation*}\n\tt^{-\\frac{s_c-s}2} B\\left(\\frac{\\alpha-1}2 (s_c -s), 1- \\frac{(s_c-s)\\alpha}2\\right),\n\\end{equation*}\nwhere $B:(0,\\infty)^2\\rightarrow \\mathbb{R}_{>0}$ is the beta function given by $B(x,y):=\\int_0^1t^{x-1}(1-t)^{y-1}dt$, which is convergent if and only if \n\\begin{equation}\\label{l:Kato.est:pr2}\n\ts_c - \\frac2{\\alpha} < s < s_c.\n\\end{equation}\nGathering \\eqref{l:Kato.est:pr1} and \\eqref{l:Kato.est:pr2}, \nwe have condition \\eqref{l:Kato.est.c2}. For such an $s$ to exist, \nit suffices to take $\\gamma,$ $\\alpha$ and $q$ so that \nconditions \\eqref{t:HH.LWP.c0}  and \\eqref{l:Kato.est.c1} are met. \n\n\t\\smallbreak\nWe next show \\eqref{l:Kato.est2}. \nSince there exists a constant $C=C(\\alpha)$ such that \n\\begin{equation}\\label{diff.pt.est}\n\t|F(u)-F(v)| \\le C (|u|^{\\alpha-1} + |v|^{\\alpha-1})|u-v| \n\t\\quad\\text{for all} \\quad u,v \\in \\mathbb{C},\n\\end{equation}\nwe have \n\\begin{align*}\n\\|N(u)(t) - N(v)(t)\\|_{L^q_s} \n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 \\{(\\alpha-1) (\\theta s + (1-\\theta) s_c) -\\gamma\\}} \\\\\n\t& \\quad \\times \\left\\| |\\cdot|^{\\gamma} \n\t(|u|^{\\alpha-1} + |v|^{\\alpha-1})|u-v| \\right\\|_{L^{\\frac{q}{\\alpha}}_{\\sigma}} \n\td\\tau,\n\\end{align*}\nthanks to Lemma \\ref{l:wLpLq} with $q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s\\equiv s$ and $s'\\equiv \\sigma :=  (\\alpha-1) (\\theta s + (1-\\theta) s_c) +s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < s \\le (\\alpha-1) (\\theta s + (1-\\theta) s_c) +s-\\gamma < d(1-\\frac{\\alpha}{q}),$ $\\theta \\in (0,1],$ i.e., \n\\begin{equation}\\label{l:Kato.est:pr1'}\n\\begin{aligned}\n&\t\\alpha \\le q \\le \\infty, \\quad \n\ts_c - \\frac{d}{\\theta} \\left( \\frac{2}{d(\\alpha-1)} -\\frac{1}{q} \\right) \\le s\\\\\n\t\t\\text{and}\\quad\n&\t-\\frac{d}{q} < s < s_c + \\frac1{1+\\theta(\\alpha-1)}\n\t\\left( d-2 -\\frac{2+\\gamma}{\\alpha-1} \\right). \n\\end{aligned}\n\\end{equation}\nBy H\\\"older's inequality with \n$\\frac{\\alpha}{q} = \\frac{\\theta(\\alpha-1)}{q} + \\frac{(1-\\theta)(\\alpha-1)}{q} + \\frac1{q},$  \nwe have \n\\begin{align*}\n& \\left\\| |\\cdot|^{\\gamma} \n\t(|u|^{\\alpha-1} + |v|^{\\alpha-1})|u-v| \\right\\|_{L^{\\frac{q}{\\alpha}}_{\\sigma}}  \\\\\n&\\le \\left( \\|u\\|_{L^q_s} + \\|v\\|_{L^q_s} \\right)^{\\theta(\\alpha-1)}\n\t\\left( \\|u\\|_{L^q_{s_c}} + \\|v\\|_{L^q_{s_c}} \\right)^{(1-\\theta)(\\alpha-1)} \n\t\t\\, \\|u-v\\|_{L^q_s}.\n\\end{align*}\nThus, \n\\begin{align*}\n&\\|N(u)(t) - N(v)(t)\\|_{L^q_s} \\\\\n&\\le C t^{-\\frac{s_c-s}2}\n\tB\\left(\\theta\\frac{\\alpha-1}2 (s_c -s), 1- \\frac{\\theta (\\alpha -1)+ 1}2 (s_c-s)\\right) \\\\\n&\\times \\left(\\|u\\|_{\\mathcal{K}^s(T)}\n\t\t\t\t+\\|v\\|_{\\mathcal{K}^s(T)} \\right)^{\\theta(\\alpha-1)} \n\t\t\\left(\\|u\\|_{L^\\infty(0,T; L^q_{s_c}) }\n\t\t\t\t+\\|v\\|_{L^\\infty(0,T; L^q_{s_c}) } \\right)^{(1-\\theta)(\\alpha-1)} \n \t\\|u-v\\|_{\\mathcal{K}^s(T)} \\\\\n\\end{align*}\nin which the last beta function is convergent if $\\theta>0$ and \n\\begin{equation}\\label{l:Kato.est:pr2'}\n\ts_c - \\frac2{\\theta(\\alpha-1)+1} < s < s_c.\n\\end{equation}\nGathering \\eqref{l:Kato.est:pr1'} and \\eqref{l:Kato.est:pr2'}, we deduce that \nthe restrictions for $s$ are \\eqref{l:Kato.est.c2'}. \nConsequently, for such an $s$ to exist, it suffices to take $q$ such that \\eqref{l:Kato.est.c1'}. \nFinally, for such a $q$ to exist, one must have \n$0<\\frac1{d(1+\\theta(\\alpha-1))} \\{ \\frac2{\\alpha-1} \n\t+ \\theta ( d- \\frac{2+\\gamma}{\\alpha-1} ) \\},$ i.e., \n$\\alpha>1+\\frac{2+\\gamma}{d} - \\frac2{\\theta d}$ and $0 < \\frac2{d(\\alpha-1)},$ \nboth of which hold thanks to \\eqref{t:HH.LWP.c0}. This concludes the proof of the lemma. \n\\end{proof}\n\nThe following is the stability estimate for the critical norm. \n\\begin{lem}\t\\label{l:crt.est}\nLet $T \\in (0,\\infty]$ and $d\\in\\mathbb{N}.$ \nLet $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \\eqref{t:HH.LWP.c0} . \nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{l:crt.est.c1}\n\t\\alpha\\le q \\le \\infty \n\t\t\\quad\\text{and}\\quad\n\t\\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\,\n\t\t\\frac{2}{d(\\alpha-1)} + \\frac{(d-2)\\alpha - d -\\gamma}{d(\\alpha-1)^2} \\right\\}\n\\end{equation}\nand let $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{l:crt.est.c2}\n\t s_c - \\frac{d(\\alpha-1)}{\\alpha} \\left(\\frac{2}{d(\\alpha-1)} - \\frac1{q} \\right) \\le s \n\t< \\min \\left\\{ s_c, \\, s_c + \\frac{(d-2)\\alpha - d -\\gamma}{\\alpha(\\alpha-1)} \\right\\}, \n\\end{equation}\nwhere $s_c$ is as in \\eqref{d:sc}. \nThen there exists a positive constant $C_2$ \ndepending only on $d,$ $\\alpha,$ $\\gamma,$ $q$ and $s$ such that \nthe map $N$ defined by \\eqref{mapN} satisfies \n\\begin{equation}\\nonumbe\n\t\\|N(u)\\|_{L^\\infty(0,T ; L^q_{s_c})} \n\t\t\\le C_2 \\|u\\|_{\\mathcal{K}^s(T)}^{\\alpha}\n\\end{equation}\nfor all $u,v \\in \\mathcal{K}^s(T).$ \n\\end{lem}\n\\begin{proof}\nLet $T>0$ and $u,v \\in \\mathcal{K}^s(T).$ We have \n\\begin{align*}\n\\|N(u)(t)\\|_{L^q_{s_c}} \n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 (\\alpha s -\\gamma- s_c)} \n\t \\| u(\\tau) \\|_{L^q_s}^{\\alpha} d\\tau \\\\\n&\\le C B\\left(\\frac{\\alpha}2 (s_c-s), 1 - \\frac{(s_c-s)\\alpha}2 \\right)  \n\t\\times  \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha}, \n\\end{align*}\nthanks to Lemma \\ref{l:wLpLq} with $q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s\\equiv s_c$ and $s'\\equiv \\alpha s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < s_c \\le \\alpha s -\\gamma < d(1-\\frac{\\alpha}{q}),$ i.e., \n\\begin{equation}\\nonumber\n\t-2<\\gamma, \\quad \n\t\\alpha \\le q \\le \\infty\t\\quad\\text{and}\\quad \n\t\\frac{s_c+\\gamma}{\\alpha} \\le s < \\frac{d+\\gamma}{\\alpha}-\\frac{d}{q}. \n\\end{equation}\nThe final beta function is convergent if \\eqref{l:Kato.est:pr2} holds. \nSince $s_c - \\frac2{\\alpha} \\le \\frac{s_c+\\gamma}{\\alpha},$\nthe restrictions on $s$ are \\eqref{l:crt.est.c2}. \nFor such an $s$ to exist, $q$ must satisfy \\eqref{l:crt.est.c1} in addition to \n$\\alpha\\le q \\le\\infty.$ \nIndeed, $\\frac{s_c + \\gamma}{\\alpha}<s_c$ \nis equivalent to $\\frac1{q} < \\frac2{d(\\alpha-1)}$ and \n$\\frac{s_c+\\gamma}{\\alpha} < s_c$ is equivalent to \n$\\frac1{q} < \\frac1{\\alpha-1} \\left(1-\\frac{2+\\gamma}{d(\\alpha-1)} \\right).$ \nThis completes the proof of the lemma. \n\\end{proof}\n\\begin{rem}\t\\label{r:crt.est2}\nNote that the above lemma along with Lemma \\ref{l:wLpLq} imply that \na solution $u \\in \\mathcal{K}^s(T)$ yields the regularity \n$u\\in C([0,T] ; L^q_{s_c}(\\mathbb{R}^d)),$ if $u_0 \\in L^{q}_{s_c}(\\mathbb{R}^d).$ Thus, \nif we allow the abuse of notation, the equivalence \n$\\mathcal{K}^s(T) = C([0,T] ; L^q_{s_c}(\\mathbb{R}^d)) \\cap \\mathcal{K}^s(T)$ \nholds as solution spaces of \\eqref{HH}. \n\\end{rem}\n\n\n\\subsubsection{Subcritical case}\nThe following are the stability and contraction estimates in the subcritical regime. \n\\begin{lem\n\\label{l:Kato.est.sub}\nLet $T \\in (0,\\infty]$ and $d\\in\\mathbb{N}.$ Let $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \n\\eqref{t:HH.LWP.c0}. \nFix $\\tilde s\\in\\mathbb{R}$ so that \n\\begin{equation}\\label{l:Kato.est.sub.c0}\n\t\\tilde s < \\frac{2+\\gamma}{\\alpha-1}.\n\\end{equation}\nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{l:Kato.est.sub.c1}\n\t\\alpha\\le q \\le \\infty\n\t\t\\quad\\text{and}\\quad\n\t\\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\,\n\t\t\\frac1{\\alpha} \\left(1 - \\frac{\\gamma}{d(\\alpha-1)}\\right), \\, \n\t\t\\frac1{d} \\left(\\frac{2+\\gamma}{\\alpha-1} -\\tilde s \\right)\\right\\}\n\\end{equation}\nand let $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{l:Kato.est.sub.c2}\n\t\\frac{\\gamma}{\\alpha-1}\\le s\n\t\t\\quad\\text{and}\\quad\n\t\\max\\left\\{\\tilde s - \\frac2{\\alpha}, \\, - \\frac{d}{q} \\right\\} < s \n\t< \\min \\left\\{ \\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}, \\, s_c \\right\\}, \n\\end{equation}\nwhere $s_c$ is as in \\eqref{d:sc}, \nThen there exist positive constants $\\tilde C_0$ and $\\tilde C_1$ \ndepending only on $d,$ $\\alpha,$ $\\gamma,$ $q,$ $\\tilde s$ and $s$ such that \nthe map $N$ defined by \\eqref{mapN} satisfies \n\\begin{equation}\\label{l:Kato.est.sub1}\n\\|N(u)\\|_{\\tilde{\\mathcal{K}}^s(T)} \n\t\\le \\tilde C_0 T^{\\frac{\\alpha-1}2(s_c-\\tilde s)} \\|u\\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha}\n\\end{equation}\nand \n\\begin{equation}\\label{l:Kato.est.sub2}\n\\|N(u) - N(v)\\|_{\\tilde{\\mathcal{K}}^s(T)} \n\t\\le \\tilde C_1 T^{\\frac{\\alpha-1}2(s_c-\\tilde s)} \\left( \\|u\\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha-1} + \\|v\\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha-1} \\right)\n\t\\|u-v\\|_{\\tilde{\\mathcal{K}}^s(T)}\n\\end{equation}\nfor all $u,v \\in \\tilde{\\mathcal{K}}^s(T).$ \n\\end{lem}\n\\begin{rem}\nNote that $\\frac1{q}<\\frac1{d} \\left(\\frac{2+\\gamma}{\\alpha-1} -\\tilde s \\right)$ \nin \\eqref{l:Kato.est.sub.c1} amounts to $\\tilde s<s_c,$ \nso the power of $T$ in \\eqref{l:Kato.est.sub1} and \\eqref{l:Kato.est.sub2} is positive. \n\\end{rem}\n\\begin{proof}[Proof of Lemma \\ref{l:Kato.est.sub}]\nWe have \n\\begin{align*}\n\\|N(u)(t)&\\|_{L^q_{s}} \n\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 ((\\alpha-1) s -\\gamma)} \n\t \\| u(\\tau) \\|_{L^q_s}^{\\alpha} d\\tau \\\\\n&\\le C t^{-\\frac12(s-\\tilde s)} t^{\\frac{\\alpha-1}2(s_c-\\tilde s)} B\\left(\\frac{(\\alpha-1)}2 (s_c-s), 1 - \\frac{(\\tilde s-s)\\alpha}2 \\right)  \n\t\\times  \\|u \\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha}, \n\\end{align*}\nthanks to Lemma \\ref{l:wLpLq} with\n$q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s\\equiv \\tilde s$ and $s'\\equiv \\alpha s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < s \\le \\alpha s -\\gamma < d(1-\\frac{\\alpha}{q}),$ i.e., \n\\begin{equation}\\nonumber\n\t\\alpha \\le q \\le \\infty, \\quad\n\t\\frac{\\gamma}{\\alpha-1} \\le s\t\\quad\\text{and}\\quad \n\t-\\frac{d}{q} < s < \\frac{d+\\gamma}{\\alpha}-\\frac{d}{q}. \n\\end{equation}\nThe final beta function is convergent if $\\tilde s-\\frac2{\\alpha}<s< s_c.$\nThus, the restrictions on $s$ are \\eqref{l:Kato.est.sub.c2}. \nFor such an $s$ to exist, $q$ must satisfy \\eqref{l:Kato.est.sub.c1}. \nFinally, for such a $q$ to exist, \nwe immediately see that $\\tilde s$ must satisfy \\eqref{l:Kato.est.sub.c0}. \n\nThe proof for the difference is similar to the above so we omit the details. \nThis completes the proof of the lemma. \n\\end{proof}\n\n\\begin{lem\n\\label{l:subcrt.est}\nLet $T \\in (0,\\infty]$ and $d\\in\\mathbb{N}.$ \nLet $\\gamma\\in\\mathbb{R}$ and $\\alpha\\in\\mathbb{R}$ satisfy \\eqref{t:HH.LWP.c0}. \nFix $\\tilde s$ so that \\eqref{t:HH.LWP.sub.cs} is satisfied. \nLet $q\\in [1,\\infty]$ be such that \n\\begin{equation}\\label{l:subcrt.est.c1}\n\t\\alpha\\le q \\le \\infty\n\t\t\\quad\\text{and}\\quad\n\t-\\frac{\\tilde s}{d} <\\frac1{q} \n\t<\\min\\left\\{ \\frac1{\\alpha} \\left(1 -\\frac{\\tilde s}{d}\\right), \\, \n\t\t\t\\frac1{d} \\left(\\frac{2+\\gamma}{\\alpha-1} -\\tilde s \\right) \\right\\}\n\\end{equation}\nand let $s \\in \\mathbb{R}$ be such that \n\\begin{equation}\\label{l:subcrt.est.c2}\n\t\\frac{\\tilde s+\\gamma}{\\alpha} \\le s \n\t< \\min \\left\\{ \\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}, \\tilde s \\right\\}, \n\\end{equation}\nwhere $s_c$ is as in \\eqref{d:sc}. \nThen there exists a positive constant $\\tilde C_2$ \ndepending only on $d,$ $\\gamma,$ $\\alpha,$ $\\tilde s,$ $q$ and $s$ such that \nthe map $N$ defined by \\eqref{mapN} satisfies \n\\begin{equation}\\nonumber\n\t\\|N(u)\\|_{L^\\infty(0,T ; L^q_{\\tilde s})} \n\t\t\\le \\tilde C_2 T^{\\frac{\\alpha-1}2(s_c-\\tilde s)} \\|u\\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha}\n\\end{equation}\nfor all $u\\in \\tilde{\\mathcal{K}}^s(T).$ \n\\end{lem}\n\\begin{proof}\nWe have \n\\begin{align*}\n\\|N(u)(t)&\\|_{L^q_{\\tilde s}} \n\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 (\\alpha s -\\gamma- \\tilde s)} \n\t \\| u(\\tau) \\|_{L^q_s}^{\\alpha} d\\tau \\\\\n&\\le C t^{\\frac{\\alpha-1}2(s_c-\\tilde s)} B\\left(\\frac12 \\{(\\alpha-1)(s_c-s) + \\tilde s-s\\}, 1 - \\frac{(\\tilde s-s)\\alpha}2 \\right)  \n\t\\times  \\|u \\|_{\\tilde{\\mathcal{K}}^s(T)}^{\\alpha}, \n\\end{align*}\nthanks to Lemma \\ref{l:wLpLq} with\n$q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s\\equiv \\tilde s$ and $s'\\equiv \\alpha s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < \\tilde s \\le \\alpha s -\\gamma < d(1-\\frac{\\alpha}{q}),$ i.e., \n\\begin{equation}\\nonumber\n\t\\alpha \\le q \\le \\infty, \\quad\n\t-\\frac{\\tilde s}{d} <\\frac1{q}\t\t\\quad\\text{and}\\quad \n\t\\frac{\\tilde s+\\gamma}{\\alpha} \\le s < \\frac{d+\\gamma}{\\alpha}-\\frac{d}{q}. \n\\end{equation}\nThe final beta function is convergent if $\\tilde s-\\frac2{\\alpha}<s<\\tilde s< s_c.$\nSince $\\tilde s-\\frac2{\\alpha}<\\frac{\\tilde s+\\gamma}{\\alpha}$ \n(by $\\tilde s < s_c < \\frac{2+\\gamma}{\\alpha-1}$), the restrictions on $s$ are \n\\eqref{l:subcrt.est.c2}. \nFor such an $s$ to exist, $q$ must satisfy \\eqref{l:subcrt.est.c1} \nand $\\frac{\\gamma}{\\alpha-1} < \\tilde s.$ \nFinally, for such a $q$ to exist, $\\tilde s$ must satisfy \\eqref{t:HH.LWP.sub.cs}\nsince \n\\[\n\t-\\frac{d}{\\alpha-1}\n\t<\\max\\left\\{ -\\frac{d}{\\alpha}, \\, \\frac{\\gamma}{\\alpha-1} \\right\\} \\quad\\text{and}\\quad\n\t\\frac{2+\\gamma}{\\alpha-1} < d.\n\\]\nThis completes the proof of the lemma. \n\\end{proof}\n\n\\subsubsection{Upgrade of regularity}\nThe following lemma is used to show the regularity of the $L^q_{\\tilde{s}}(\\mathbb{R}^d)$-mild solution. \n\\begin{lem}\t\\label{l:b.strap}\nLet $p,q \\in [1,\\infty]$ and $s,s' \\in \\mathbb{R}.$ Under condition \\eqref{t:HH.LWP.c0}, \nlet pairs $(q,s)$ and $(p,s')$ be such that either\n\\begin{equation}\t\\label{l:b.strap:c}\n\\begin{aligned}\n\t&\\alpha \\le q < \\infty, \\quad\n\t\\max\\left\\{- \\frac{d}{q}, \\, \\frac{1}{\\alpha}\\left( \\gamma- \\frac{d}{q} \\right)\\right\\} < s \n\t<\\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}, \\\\\n\t&\\max\\left\\{0, \\, -\\frac{s}{d}, \\, \\frac{\\gamma-\\alpha s}{d} \\right\\}\n\t\t<\\frac1{p} \\le \\frac{1}{q}, \\quad\n\t -\\frac{d}{p} < s' \\le \\min\\{s, \\, \\alpha s -\\gamma\\}. \n\\end{aligned}\n\\end{equation}\nor \n\\begin{equation}\t\\label{l:b.strap:p=inf}\n\\begin{aligned}\n\t&\\alpha < q \\le \\infty, \\quad\n\t\\max\\left\\{0,\\, \\frac{\\gamma}{\\alpha}\\right\\} \\le s \n\t<\\frac{d+\\gamma}{\\alpha} - \\frac{d}{q},\\\\\n\t&p= \\infty, \\quad 0\\le s' \\le \\min\\{s, \\, \\alpha s -\\gamma\\}.\n\\end{aligned}\n\\end{equation}\nLet $u$ be the $L^q_{s_c}(\\mathbb{R}^d)$-mild solution of \\eqref{HH} with initial data $u_0 \\in \\mathcal{S}'(\\mathbb{R}^d)$ on $[0,T_m)$\nsuch that \n\\begin{equation}\\nonumber\n\t\\sup_{t\\in [0,T_m)} t^{\\frac{s_c(q)-s}{2} } \\|u(t)\\|_{L^q_{s}} < \\infty.\n\\end{equation}\nThen it follows that \n\\begin{equation}\\nonumber\n\t\\sup_{t\\in [0,T_m)} t^{\\frac{s_c(p)-s'}{2} } \\|u(t)\\|_{L^p_{s'}} < \\infty.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nWe use a similar argument as in \\cite{SnoTayWei2001} (See also \\cite{BenTayWei2017}). \nLet \n\\[\n\tA := \\sup_{t\\in [0,T_m)} t^{\\frac{s_c(q)-s}2}\\|u(t)\\|_{L^{q}_{s}}<\\infty.\n\\]\nLet $t\\in (0,T_m)$. We use the integral representation \n\\begin{equation}\\nonumber\n\tu(t) = e^{\\frac{t}2 \\Delta} u(t\/2) \n\t+ \\int_{\\frac{t}{2}}^t e^{(t-\\tau)\\Delta} \\left\\{ |\\cdot|^{\\gamma} \n\t\tF(u(\\tau,\\cdot)) \\right\\} d\\tau. \n\\end{equation}\nIt follows from Lemma \\ref{l:wLpLq} with $q\\equiv p,$ $p\\equiv q,$ \n$s'\\equiv s'$ and $s\\equiv s,$ that \n\\begin{align*}\n\t\\|e^{\\frac{t}2\\Delta} u(t\/2)\\|_{L^{p}_{s'}} \n\t\\le C t^{-\\frac{d}{2}(\\frac1{q}-\\frac1{p}) - \\frac{s-s'}2} \\|u(t\/2)\\|_{L^{q}_{s}} \n\t\\le C t^{-\\frac{1}{2}(\\frac{2+\\gamma}{\\alpha-1}-\\frac{d}{p} - s')} A \n\\end{align*}\nif \n\\begin{equation}\\label{l:b.strap:pr0}\n\\begin{aligned}\n\t&1\\le q \\le p \\le \\infty\n\t\t\\quad\\text{and}\\quad \n\t-\\frac{d}{p} < s' \\le s < d\\left(1-\\frac1{q}\\right) \\quad\\left(0\\le s' \\text{ if } p=\\infty\\right). \\\\\n\\end{aligned}\n\\end{equation}\nOn the other hand, \n\\begin{equation}\\nonumbe\n\\begin{aligned}\n\\|&N(u)(t)\\|_{L^{p}_{s'}} \n\\le  C \\int_{\\frac{t}{2}}^t \n\t\t(t-\\tau)^{-\\frac{d}2(\\frac{\\alpha}{q}-\\frac1{p}) - \\frac{\\alpha s -\\gamma-s'}2} \n\t\t \\| u(\\tau) \\|_{L^q_s}^{\\alpha} d\\tau \\\\\n&\\le CA^{\\alpha} \\int_{\\frac{t}{2}}^t (t-\\tau)^{-\\frac{d}2(\\frac{\\alpha}{q}-\\frac1{p}) - \\frac{\\alpha s -s'-\\gamma}2} \n\t\\tau^{-\\frac{(s_c-s)\\alpha}{2}} d\\tau \\\\\n&= CA^{\\alpha}  t^{-\\frac12(\\frac{2+\\gamma}{\\alpha-1} - \\frac{d}{p} - s')} \n \\int_{\\frac{1}{2}}^1 (1-\\tau)^{-\\frac{d}2(\\frac{\\alpha}{q}-\\frac1{p}) - \\frac{\\alpha s -s'-\\gamma}2} \n\t\\tau^{-\\frac{(s_c-s)\\alpha}{2}} d\\tau, \n\\end{aligned}\n\\end{equation}\nthanks to Lemma \\ref{l:wLpLq} with $q\\equiv p,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s'\\equiv s'$ and $s\\equiv \\alpha s-\\gamma,$ provided that \n\\begin{equation}\\label{l:b.strap:pr1}\n\\begin{aligned}\n\t&1\\le \\frac{q}{\\alpha} \\le p \\le \\infty\n\t\t\\quad\\text{and}\\quad \n\t-\\frac{d}{p} < s' \\le \\alpha s-\\gamma < d\\left(1-\\frac{\\alpha}{q}\\right)\n\t\\quad\\left(0\\le s' \\text{ if } p=\\infty\\right).\\\\\n\\end{aligned}\n\\end{equation}\ni.e., $\\alpha \\le q \\le \\infty,$ $\\frac1{p} \\le \\frac{\\alpha}{q},$\n$-\\frac{d}{p} < s' \\le \\alpha s-\\gamma$ ($0\\le s'\\le \\alpha s-\\gamma$ if $p=\\infty$) and $s < \\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}.$ \nThe final integral is convergent if \n\\begin{equation}\\label{l:b.strap:pr2}\n\t1-\\frac{d}2\\left(\\frac{\\alpha}{q}-\\frac1{p}\\right) - \\frac{\\alpha s-s' -\\gamma}2>0, \n\t\t\\quad\\text{i.e.,}\\quad\n\t\\alpha\\left( \\frac{d}{q} + s\\right) - 2 - \\gamma - \\frac{d}{p}<s'.\n\\end{equation}\nThus, we have \n\\begin{equation}\\nonumber\n\t\\sup_{t\\in [0,T_m)} t^{\\frac{s_c(p)-s'}{2} } \\|u(t)\\|_{L^p_{s'}} \n\t\\le C (A + A^{\\alpha})\n\\end{equation}\nunder \\eqref{l:b.strap:pr0}, \\eqref{l:b.strap:pr1} and \\eqref{l:b.strap:pr2}. \nSince $s < \\frac{d+\\gamma}{\\alpha} - \\frac{d}{q}$ from \\eqref{l:b.strap:pr1}, we have \n$\\alpha( \\frac{d}{q} + s) - 2 - \\gamma - \\frac{d}{p} < - \\frac{d}{p}.$ Thus, \nthe conditions for $s'$ are that in \\eqref{l:b.strap:c}. \nBy tedious but straightforward computations, \nwe may easily see that under condition \\eqref{t:HH.LWP.c0}, \nthe necessary and sufficient conditions of \n\\eqref{l:b.strap:pr0}, \\eqref{l:b.strap:pr1} and \\eqref{l:b.strap:pr2} are \n\\eqref{l:b.strap:c} or \\eqref{l:b.strap:p=inf}. Hence, the lemma is proved. \n\\end{proof}\n\n\n\n\n\\section{Local well-posedness and self-similar solutions}\n\n\\subsection{Proof of Theorem \\ref{t:HH.LWP}}\nIn order to prove Theorem \\ref{t:HH.LWP}, we prepare the following lemma. \n\\begin{lem}\\label{l:exist.crt}\nLet positive numbers $\\rho>0$ and $M>0$ satisfy \n\\begin{equation}\\label{l:exist.crt.c0}\n\t\\rho + C_0 M^\\alpha \\le M \\quad\\text{and}\\quad\n\t2 C_1 M^{\\alpha-1} <1,\n\\end{equation}\nwhere $C_0$ and $C_1$ are as in Lemma \\ref{l:Kato.est}. \nUnder conditions \\eqref{t:HH.LWP.c0}, \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}, \nlet $T\\in (0,\\infty]$ and $u_0 \\in \\mathcal{S}'(\\mathbb{R}^d)$ be \nsuch that $e^{t\\Delta}u_0 \\in \\mathcal{K}^s(T).$ \nIf $\\|e^{t\\Delta}u_0\\|_{\\mathcal{K}^s(T)}\\le \\rho,$ \nthen a solution $u$ to \\eqref{HH} exists \nsuch that $u -e^{t\\Delta} u_0 \\in \nL^\\infty(0,T ; L^q_{s_c}(\\mathbb{R}^d)) \\cap C((0,T] ; L^q_{s_c}(\\mathbb{R}^d))$ and \n$\\|u\\|_{\\mathcal{K}^s(T)} \\le M.$ \nMoreover, the solution satisfies the following properties:\n\\begin{enumerate}[$(i)$]\n\\item $u -e^{t\\Delta} u_0 \\in L^\\infty(0,T ; L^q_{\\sigma}(\\mathbb{R}^d))$ for $\\sigma$ such that \n\\begin{equation}\t\\label{l:exist.crt.csig}\n\ts_c \\le \\sigma \\le \\alpha s -\\gamma.\n\\end{equation}\n\\item\n$u -e^{t\\Delta} u_0 \\in C([0,T) ; L^q_{\\sigma}(\\mathbb{R}^d))$ \nand $\\Lim_{t\\to0} \\|u(t) -e^{t\\Delta} u_0\\|_{L^q_{\\sigma}} = 0$ for $\\sigma$ such that \n\\eqref{l:exist.crt.csig} and $\\sigma>s_c.$ \n\\item $\\displaystyle \\lim_{t\\to0} u(t) = u_0$ in the sense of distributions. \n\\item Let $\\gamma\\ge0.$ Then the solution $u$ satisfies \n\\[\n\\sup_{0<t<T} \n\tt^{\\frac{s_c(p_{\\theta} ) - s_{\\theta}}{2}} \\|u(t)\\|_{L^{p_{\\theta}}_{s_{\\theta}}} < \\infty\n\\]\nwhere \n\\begin{equation}\\nonumber\n\tp_{\\theta} = \\frac{q}{\\theta}, \\quad \n\t\\theta s \\le s_{\\theta} \\le s \\quad\\text{and}\\quad 0\\le \\theta \\le 1.\n\\end{equation}\nIn particular, if $\\gamma\\ge0,$ $u(t) \\in L^\\infty(\\mathbb{R}^d)$ for $t>0.$\n\\end{enumerate}\n\\end{lem}\n\\begin{rem}\\label{r:exist}\nTo meet \\eqref{l:exist.crt.c0}, it suffices to take $M=2\\rho$ and \n\\[\nM < \\min\\left\\{  (2C_0)^{-\\frac{1}{\\alpha-1}}, \\, \n                        (2C_1)^{- \\frac{1}{\\alpha-1}} \\right\\}.\n\\]\n\\end{rem}\n\\begin{rem}\nIf $u_0\\in L^q_{s_c}(\\mathbb{R}^d)$, then $u_0$ satisfies the assumptions of \nLemma \\ref{l:exist.crt} with $T=\\infty.$ Indeed, letting $q\\equiv q,$ $p\\equiv q,$ \n$s\\equiv s_c$ and $s'\\equiv s$ in Lemma \\ref{l:wLpLq}, we obtain \n\\begin{equation}\\nonumber\n\t\\|e^{t\\Delta} u_0\\|_{L^q_s} \\le C t^{-\\frac{s_c-s}2} \\|u_0\\|_{L^q_{s_c}}\n\\end{equation}\nprovided that $-\\frac{d}{q} < s\\le s_c < d(1-\\frac1{q}),$ i.e., \n$\\gamma>-2$ and $\\alpha>\\alpha_F(d,\\gamma).$ \nThus, $e^{t\\Delta} u_0 \\in \\mathcal{K}^s.$ \n\\end{rem}\n\\begin{proof}[Proof of Lemma \\ref{l:exist.crt}]\nSetting the metric $d(u,v) := \\|u-v\\|_{\\mathcal{K}^s(T)}$, we may show that $(\\mathcal{K}^s(T),d)$ is a nonempty complete metric space. Let\n$X_M := \\{ u \\in\\mathcal{K}^s(T) \\,;\\,  \\|u\\|_{\\mathcal{K}^s(T)} \\le M \\}$ be the closed ball in $\\mathcal{K}^s(T)$ centered at the origin with radius $M$.\nWe prove that the map defined in \\eqref{map} has a fixed point in $X_M.$ \nThanks to Lemma \\ref{l:Kato.est} and \\eqref{l:exist.crt.c0}, we have \n\\begin{equation}\\nonumbe\n\\begin{aligned}\n\\|\\Phi (u)\\|_{\\mathcal{K}^s(T)} \n\\le \\|e^{t\\Delta} u_0 \\|_{\\mathcal{K}^s(T)} \n\t+ C_0 \\|u\\|_{\\mathcal{K}^s(T)}^\\alpha \n\\le \\rho + C_0 M^\\alpha \\le M\n\\end{aligned}\\end{equation} \nand \n\\begin{equation}\\label{t:HH.LWP.pr.Lip'}\n\\|\\Phi (u)-\\Phi (v)\\|_{\\mathcal{K}^s(T)} \n\\le C_1 \\left( \\|u\\|_{\\mathcal{K}^s(T)}^{\\alpha-1}\n\t+\\|v\\|_{\\mathcal{K}^s(T)}^{\\alpha-1} \\right) \n\t\\|u-v\\|_{\\mathcal{K}^s(T)}\n\\le 2 C_1 M^{\\alpha-1} \\|u-v\\|_{\\mathcal{K}^s(T)}\n\\end{equation}\nfor any $u, v\\in X_M,$ where $2 C_1 M^{\\alpha-1}<1.$ \nThese prove that $\\Phi(u) \\in X_M$ and that \n$\\Phi$ is a contraction mapping in $X_M.$ \nThus, Banach's fixed point theorem ensures the existence of \na unique fixed point $u$ for the map $\\Phi$ in $X_M,$ \nprovided that $q$ and $s$ satisfy \\eqref{l:Kato.est.c1} and \\eqref{l:Kato.est.c2}. \nThe fixed point $u$ also satisfies, by construction, the estimate \n$\\|u\\|_{\\mathcal{K}^s(T)} \\le M.$ \n\nHaving obtained a fixed point in $\\mathcal{K}^s(T)$ for some $T,$ \nwe have $u -e^{t\\Delta} u_0 \\in L^\\infty(0,T;L^q_{s_c}(\\mathbb{R}^d))$ by Lemma \n\\ref{l:crt.est}, provided further that \\eqref{l:crt.est.c1} and \\eqref{l:crt.est.c2} are satisfied. \nWe see that $\\frac1{q} < \\frac{2}{d(\\alpha-1)}$, \n$q>0,$ $\\alpha>1$ and $\\gamma>-2$ imply \n\\[\n\t\\max\\left\\{\\frac{\\gamma}{\\alpha-1}, \\, - \\frac{d}{q} \\right\\} \n\t\t< \\frac{s_c+\\gamma}{\\alpha} = s_c - \\frac{d(\\alpha-1)}{\\alpha} \\left(\\frac{2}{d(\\alpha-1)} - \\frac1{q} \\right) \n\\]\nso $\\frac{s_c+\\gamma}{\\alpha}$ is the stronger lower bound for $s.$ \nThus, $s$ must satisfy \\eqref{t:HH.LWP.c2}. \nCombining \\eqref{l:Kato.est.c1} and \\eqref{l:crt.est.c1}, we end up with \n\\begin{equation}\\label{t:HH.LWP:pr1}\n\t\\frac1{q} < \\min \\left\\{ \\frac{2}{d(\\alpha-1)}, \\,\n\t\t\\frac1{\\alpha} \\left(1 - \\frac{\\gamma}{d(\\alpha-1)}\\right), \\, \n\t\t\\frac1{\\alpha-1} \\left(1-\\frac{2+\\gamma}{d(\\alpha-1)} \\right)\\right\\}, \n\\end{equation}\nwhich in fact amounts to \\eqref{t:HH.LWP.c1}. \n\t\\smallbreak\nWe next prove the assertion $(i)$--$(iii).$ Fix a solution $u \\in \\mathcal{K}^s(T)$ \nwith $q$ and $s$ as in \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}. We have \n\\begin{equation}\\label{t:HH.LWP:pr2}\n\\begin{aligned}\n\\|N(u)(t)\\|_{L^q_{\\sigma}} \n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 (\\alpha s -\\gamma- \\sigma)} \n\t \\| u(\\tau) \\|_{L^q_s}^{\\alpha} d\\tau \\\\\n&\\le C \\int_0^t (t-\\tau)^{-\\frac{d(\\alpha-1)}{2q} - \\frac12 (\\alpha s -\\gamma- \\sigma)} \n\t\\tau^{-\\frac{(s_c-s)\\alpha}{2}} d\\tau \\times  \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha} \\\\\n&= C  t^{\\frac{\\sigma-s_c}2} B\\left(\\frac{(\\alpha-1)(s_c-s) + \\sigma-s}2, \\, 1 - \\frac{(s_c-s)\\alpha}2 \\right)  \n\t\\times  \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha}, \n\\end{aligned}\n\\end{equation}\nthanks to Lemma \\ref{l:wLpLq} with $q\\equiv q,$ $p\\equiv \\frac{q}{\\alpha},$ \n$s' \\equiv \\sigma$ and $s\\equiv \\alpha s-\\gamma,$ \nprovided that $1\\le \\frac{q}{\\alpha} \\le q \\le \\infty$ and \n$-\\frac{d}{q} < \\sigma \\le \\alpha s -\\gamma < d(1-\\frac{\\alpha}{q}).$ \nThe power of $t$ in the final line is non-negative if $\\sigma \\ge s_c.$ \nThe use of Lemma \\ref{l:wLpLq} along with the convergence of the beta function require, \nin addition to \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}, that $\\sigma$ satisfies\n\\eqref{l:exist.crt.csig}. For such a $\\sigma$ to exist, one needs \n$\\frac{s_c+\\gamma}{\\alpha}\\le s,$ which is assured by \\eqref{t:HH.LWP.c2}. \nIf $\\sigma>s_c,$ (i.e., if $\\frac{s_c+\\gamma}{\\alpha}< s,$) then the power of $t$ is positive, thus the right-hand side of \n\\eqref{t:HH.LWP:pr2} goes to zero as $t\\to0.$ Hence, the assertions $(ii)$ \nand $(iii)$ are proved. \n\t\\smallbreak\nFinally, we prove the assertion $(iv).$ Fix a solution $u \\in \\mathcal{K}^s(T)$ with \n$q$ and $s$ as in \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}. \nHere, we notice that under $\\gamma>0,$ the lower bound of \\eqref{t:HH.LWP.c2} \nalways satisfies \n\\begin{equation}\\nonumber\n\t\\max\\left\\{0, \\, \\frac{\\gamma}{\\alpha}\\right\\} \n\t\\le s_c - \\frac{d(\\alpha-1)}{\\alpha} \\left(\\frac{2}{d(\\alpha-1)} - \\frac1{q} \\right),\n\\end{equation}\nwhich implies that the condition \n\\eqref{l:b.strap:p=inf} of Lemma \\ref{l:b.strap} is always satisfied as well. Thus, \nLemma \\ref{l:b.strap} immediately implies \n\\begin{equation}\\nonumber\n\t\\sup_{t\\in [0,T)} t^{\\frac{s_c(\\infty)-s'}{2} } \\|u(t)\\|_{L^{\\infty}_{s'}} < \\infty\n\\end{equation}\nfor \n\\begin{equation}\\nonumber\n\t0\\le s' \\le \\min\\{s, \\, \\alpha s -\\gamma\\}.\n\\end{equation}\nWe also have $u\\in \\mathcal{K}^s(T)$ by assumption. Thus, the conclusion follows from \nProposition \\ref{p:wL.sp} $(3).$ \n\\end{proof}\nWe start by proving the uniqueness of our solution. \n\t\\subsubsection{Proof of $(ii)$}\nLet $T>0$ be given and fixed. We prove the uniqueness in $\\mathcal{K}^s(T).$ Under conditions \\eqref{t:HH.LWP.c0}, \n\\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}, \nlet $u$ and $v$ be two solutions to \\eqref{integral-eq} \nbelonging to $C([0,T] ; L^q_{s_c}(\\mathbb{R}^d)) \\cap \\mathcal{K}^s(T)$ \nwith the same initial data $u_0 \\in L^q_{s_c}(\\mathbb{R}^d)$ \n($u_0 \\in \\mathcal{L}^{\\infty}_{s_c}(\\mathbb{R}^d)$ if $q=\\infty$)\n\\footnote{We assume $u_0 \\in \\mathcal{L}^{\\infty}_{s_c}(\\mathbb{R}^d)$ if $q=\\infty$ in \norder to utilize the density, which is needed in the proof of \\eqref{id.lmt.K} and \\eqref{id.lmt.crt}.}\nsuch that \n\\begin{equation}\\nonumber\n\t\\|u\\|_{\\mathcal{K}^s(T)}+\\|v\\|_{\\mathcal{K}^s(T)} \\le K,\n\\end{equation}\nfor some positive constant $K.$ \nLet us recall that we have the following two limits at our disposal:\n\\begin{equation}\\label{id.lmt.K}\n\t\\lim_{T\\to0} \\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)} = 0\n\\end{equation}\nand \n\\begin{equation}\\label{id.lmt.crt}\n\t\\lim_{T\\to0} \\|u - e^{t\\Delta} u_0\\|_{L^\\infty(0,T;L^q_{s_c})} = 0. \n\\end{equation}\nThe former is the well-known fact stemming from the density of \n$C_0^\\infty(\\mathbb{R}^d)$ in $L^q_{s_c}(\\mathbb{R}^d)$ (See Proposition \\ref{p:wL.sp} in Appendix). \nThe latter is shown by the triangle inequality and the continuity at $t=0$ of solutions \nfor both the linear and nonlinear problems. \nLet $w :=u-v.$ By \\eqref{diff.pt.est}, we have \n\\begin{equation}\\nonumber\n\t|F(u)-F(v)| \n\t\\le C |e^{t\\Delta} u_0|^{\\alpha-1} |u-v| \n\t+ C (|u-e^{t\\Delta} u_0|^{\\alpha-1} + |v-e^{t\\Delta} u_0|^{\\alpha-1})|u-v|,\n\\end{equation}\nwhich implies that $|w| \\le C( I_1 + I_2 + I_3)$ (thanks to the maximum principle), where  \n\\begin{equation}\\nonumber\n\\begin{aligned}\n&\tI_1 := \\int_0^{t} e^{(t-\\tau)\\Delta} \n\t\t\\left\\{|\\cdot|^{\\gamma} |e^{t\\Delta} u_0|^{\\alpha-1} |w| \\right\\} \\, d\\tau, \\\\\n&\tI_2 :=  \\int_0^{t} e^{(t-\\tau)\\Delta} \n\t\t\\left\\{|\\cdot|^{\\gamma} |u-e^{t\\Delta} u_0|^{\\alpha-1} |w| \\right\\} \\, d\\tau \\\\\n\t\t\\text{and}\\quad\n&\tI_3 :=  \\int_0^{t} e^{(t-\\tau)\\Delta} \n\t\t\\left\\{|\\cdot|^{\\gamma} |v-e^{t\\Delta} u_0|^{\\alpha-1} |w| \\right\\} \\, d\\tau. \n\\end{aligned}\n\\end{equation}\nGiven $q$ and $s$ satisfying \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}, \nwe may always choose $\\theta$ so that \\eqref{l:Kato.est.c1'} and \\eqref{l:Kato.est.c2'} \nare satisfied. Indeed, \\eqref{l:Kato.est.c1'} and \\eqref{l:Kato.est.c2'} become \n\\eqref{l:Kato.est.c1} and \\eqref{l:Kato.est.c2} as $\\theta\\to1,$ respectively, \nwhich are weaker than the assumptions on $q$ and $s$ in Theorem \\ref{t:HH.LWP}. \nThe only condition that has to be considered independently is \n$s_c - \\frac{d}{\\theta} \\left( \\frac{2}{d(\\alpha-1)} -\\frac{1}{q} \\right)  \\le s$ in \n\\eqref{l:Kato.est.c2'} (as this is not a strict inequality), \nbut this causes no problem since \n$s_c - \\frac{d}{\\theta} \\left( \\frac{2}{d(\\alpha-1)} -\\frac{1}{q} \\right) \n\\le s_c - \\frac{d(\\alpha-1)}{\\alpha} \\left(\\frac{2}{d(\\alpha-1)} - \\frac1{q} \\right)$\nholds for any $\\theta \\in (0,1].$ \nThus, we may use estimate \\eqref{l:Kato.est2} freely for our $q$ and $s.$ \n\nBy the same calculation leading to \\eqref{l:Kato.est1}, we deduce that \n\\begin{equation}\\label{pr.uni.1}\n\t\\|I_1\\|_{\\mathcal{K}^s(T)} \n\t\\le C \\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}^{\\alpha-1} \\|w\\|_{\\mathcal{K}^s(T)}. \n\\end{equation}\nFor $I_2,$ estimate \\eqref{l:Kato.est2} implies \n\\begin{equation}\\label{pr.uni.2}\n\\begin{aligned}\n\\|I_2\\|_{\\mathcal{K}^s(T)} \n&\\le C \\|u-e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}^{\\theta(\\alpha-1)} \n\t\t\\|u-e^{t\\Delta} u_0\\|_{L^\\infty(0,T; L^q_{s_c}) }^{(1-\\theta)(\\alpha-1)} \n\t \\|w\\|_{\\mathcal{K}^s(T)}  \\\\\n&\\le C K^{\\theta(\\alpha-1)} \n\t\\|u-e^{t\\Delta} u_0\\|_{L^\\infty(0,T; L^q_{s_c}) }^{(1-\\theta)(\\alpha-1)} \n\t\\|w\\|_{\\mathcal{K}^s(T)}. \n\\end{aligned}\n\\end{equation}\nSimilarly, we have \n\\begin{equation}\\label{pr.uni.3}\n\\begin{aligned}\n\\|I_3\\|_{\\mathcal{K}^s(T)} \n\\le C K^{\\theta(\\alpha-1)} \n\t\\, \\|v-e^{t\\Delta} u_0\\|_{L^\\infty(0,T; L^q_{s_c}) }^{(1-\\theta)(\\alpha-1)} \n\t\\, \\|w\\|_{\\mathcal{K}^s(T)}. \n\\end{aligned}\n\\end{equation}\nGathering \\eqref{pr.uni.1}, \\eqref{pr.uni.2} and \\eqref{pr.uni.3}, we deduce that \nthere exists some positive constant $C$ \nindependent of $T,$ $u_0,$ $u$ and $v$ such that \n\\begin{equation}\\nonumber\n\\|w\\|_{\\mathcal{K}^s(T)} \n\\le C \\mathcal{N}(T, u_0, u, v)\n\t\\|w\\|_{\\mathcal{K}^s(T)} \n\\end{equation}\nwhere\n\\begin{equation}\\nonumber\n\\mathcal{N}(T, u_0, u, v) := \n \\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}^{\\alpha-1} \n\t+\\|u-e^{t\\Delta} u_0\\|_{L^\\infty(0,T; L^q_{s_c}) }^{(1-\\theta)(\\alpha-1)} \n\t+ \\|v-e^{t\\Delta} u_0\\|_{L^\\infty(0,T; L^q_{s_c}) }^{(1-\\theta)(\\alpha-1)}.  \n\\end{equation}\nSince $0<\\theta<1,$ the above quantity goes to zero as \n$T$ tends to zero, thanks to \\eqref{id.lmt.crt} and \\eqref{id.lmt.K}. \nThus, there exists some $T'$ such that \n\\begin{equation}\\nonumber\n\\|w\\|_{\\mathcal{K}^s(T')} \n\t\\le \\frac1{2} \\|w\\|_{\\mathcal{K}^s(T')}\n\\end{equation}\nfor instance, which implies the uniqueness on the interval $[0,T'].$ Set \n\\begin{equation}\\nonumber\nT^* = \\sup \\{t\\in [0,T] \\,; \\, u(\\tau) = v(\\tau) , \\ 0\\le \\tau \\le t\\}.\n\\end{equation}\nThe preceding argument shows that $T^*>0.$\nNow assume by contradiction that $T^*<T.$ By continuity of $u$ and $v,$ \nwe have $u(T^*) = v(T^*).$ \nSetting $u^*(t) = u(t+T^*)$ and $v^*(t) = v(t+T^*),$ we may express the solutions as \n\\begin{align*}\\nonumber\n&\tu^*(t) = e^{t\\Delta} u(T^*) + \\int_0^t e^{(t-\\tau)\\Delta} \n\t\\left\\{ |\\cdot|^{\\gamma} F(u(T^*+\\tau,\\cdot)) \\right\\} d\\tau \\\\\n\t\t\\text{and} \\quad\n&\tv^*(t) = e^{t\\Delta} u(T^*) + \\int_0^t e^{(t-\\tau)\\Delta} \n\t\\left\\{ |\\cdot|^{\\gamma} F(v(T^*+\\tau,\\cdot)) \\right\\} d\\tau,  \n\\end{align*}\nwhere $0\\le t < T - T^*.$ \nBy a similar calculation as above, we may show that \n\\begin{equation}\\nonumber\n\\|u^* - v^*\\|_{\\mathcal{K}^s(T)} \n\t\\le \\mathcal{N}(T, u(T^*), u, v)  \n\t\t\\|u^* - v^*\\|_{\\mathcal{K}^s(T)}, \n\\end{equation}\nwhich implies again that there exist some $T'$ such that $u^*(t) = v^*(t)$ \nfor $t\\in [0,T'],$ i.e., $u(t) = v(t)$ for $t \\in (T^*, T^* +T'),$ a contradiction. \nThus, $u(t)=v(t)$ on the whole interval $[0,T].$ \nThis completes the proof of Theorem \\ref{t:HH.LWP} $(ii).$  \n\n\t\\subsubsection{Proof of $(i)$} \nLet $u_0 \\in L^q_{s_c}(\\mathbb{R}^d)$ \n($u_0 \\in \\mathcal{L}^{\\infty}_{s_c}(\\mathbb{R}^d)$ if $q=\\infty$). \nWe recall that $C_0^\\infty(\\mathbb{R}^d)$ is dense in the space $L^q_{s_c}(\\mathbb{R}^d)$ \nby Proposition \\ref{p:wL.sp}, which ensures the property \\eqref{id.lmt.K}. \nThus, there exists some real number $T$ that is \nsmall enough so that $\\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}\\le \\rho.$ \nNow Lemma \\ref{l:exist.crt} asserts that \n\\begin{equation}\\nonumber\n\\|u\\|_{L^\\infty(0,T \\,;\\, L^q_{s_c})} \n\t\\le \\|e^{t\\Delta} u_0\\|_{L^\\infty(0,T \\,;\\, L^q_{s_c})} + C_2 \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha}\n\t\\le \\|u_0\\|_{L^q_{s_c}} + C_2 M^{\\alpha}. \n\\end{equation}\nThe time-continuity at $t=0$ follows from a well-known argument \n(see \\cites{OkaTsu2016, Tsu2011} for example). \nThus, $u$ is an $L^q_{s_c}(\\mathbb{R}^d)$-mild solution to \\eqref{HH} on $[0,T]$ \nsuch that $\\|u\\|_{\\mathcal{K}^s(T)}\\le M.$ \nTo deduce the estimate \\eqref{t:HH.LWP.est}, \nit suffices to take $\\rho=\\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}$ \nand $M$ as in Remark \\ref{r:exist}. \nGiven $u_0 \\in L^q_{s_c}(\\mathbb{R}^d),$ let the maximal existence time \n$T_m = T_m (u_0)$ be defined by \\eqref{d:Tm} with $\\tilde s = s_c.$ \nBy a standard argument, uniqueness ensures that \nthe solution can be extended to the maximal interval $[0,T_m).$ \n\n\t\\subsubsection{Proof of $(iii)$}\nGiven two initial data $u_0, v_0 \\in L^q_{s_c}(\\mathbb{R}^d),$ \nwe next show the Lipschitz continuity of the flow map. \nLet $u$ and $v$ be two solutions associated with the initial data $u_0$ and $v_0,$ \nrespectively, constructed in (i) with the estimate \n$\\|u\\|_{\\mathcal{K}^s(T)}\\le 2\\|e^{t\\Delta} u_0\\|_{\\mathcal{K}^s(T)}.$ \nLet $w := u-v$ and $w_0 := u_0-v_0.$ \nWe carry out the same calculations as before to \nsee that there exists a positive constant $C_3$ such that \n\\begin{align*}\n\\|w\\|_{L^\\infty(0,T; L^q_{s_c})\\cap \\mathcal{K}^s(T)} \n&\\le \\|e^{t\\Delta}w_0\\|_{L^\\infty(0,T; L^q_{s_c})\\cap \\mathcal{K}^s(T)} \n\t+ C_3 \\left( \\|u \\|_{\\mathcal{K}^s(T)}^{\\alpha-1}\n\t\t+\\|v \\|_{\\mathcal{K}^s(T)}^{\\alpha-1}\\right) \\|w\\|_{\\mathcal{K}^s(T)} \\\\\n&\\le \\|w_0\\|_{L^q_{s_c}} \n\t+ 2 C_3 M^{\\alpha-1} \\|w\\|_{\\mathcal{K}^s(T)}, \n\\end{align*}\nwhere $M= \\max \\{ \\|e^{t\\Delta} u_0 \\|_{\\mathcal{K}^s(T)}, \n\t\\, \\|e^{t\\Delta} v_0 \\|_{\\mathcal{K}^s(T)}\\}.$ \nBy taking $T$ smaller if necessary \n($2 C_3 M^{\\alpha-1}\\le \\frac12$ for instance), we deduce \nthe Lipschitz stability on the short time-interval $[0,T].$ \n\n\t\\subsubsection{Proof of $(iv)$}\nWe prove the blow-up criterion by a contradiction argument. \nLet $T_m<\\infty$ and suppose that $\\|u\\|_{\\mathcal{K}^s(T_m)} <\\infty$ holds.  \nLet $u$ be a maximal solution and let $t_0 \\in (0,T_m),$ to be fixed later.  \nWe aim to prove \nthere exists an $\\varepsilon>0$ such that \n\\begin{equation}\\label{buc-aim}\n\t\\|e^{t\\Delta} u(t_0)\\|_{\\mathcal{K}^s(T_m-t_0+\\varepsilon)} \\le \\rho,\n\\end{equation}\nwhere $\\rho > 0$ is the constant as in Lemma \\ref{l:exist.crt}. \nOnce \\eqref{buc-aim} is proved, \nthe solution $u$ can be smoothly extended to $T_m+\\varepsilon.$ \nMoreover, $u$ is unique in \n$C([0,T_m+\\varepsilon] ; L^q_{s_c}(\\mathbb{R}^d)) \\cap \\mathcal{K}^s(T_m+\\varepsilon)$ by $(ii),$ \nwhich contradicts the definition of $T_m.$ \nThus, $\\|u\\|_{\\mathcal{K}^s(T_m)}=\\infty$ if $T_m<\\infty.$ \n\nLet us concentrate on proving \\eqref{buc-aim}. We may express the maximal solution as follows: \n\\begin{equation}\\nonumber\n\tu(t+t_0) = e^{t\\Delta} u(t_0) + \\int_0^t e^{(t-\\tau)\\Delta} \\left\\{ |\\cdot|^{\\gamma} F(u(t_0+\\tau)) \\right\\} d\\tau, \\quad 0\\le t < T_m - t_0.\n\\end{equation}\nThus, we have \n\\begin{align*}\\nonumber\n\t\\|e^{t\\Delta}u(t_0)&\\|_{\\mathcal{K}^s(T_m-t_0)}  \\\\\n\t&\\le \\|u(\\cdot+t_0)\\|_{\\mathcal{K}^s(T_m-t_0)} \n\t+ \\left\\| \\int_0^t e^{(t-\\tau)\\Delta} \\left\\{ |\\cdot|^{\\gamma} F(u(t_0+\\tau)) \\right\\} d\\tau \\right\\|_{\\mathcal{K}^s(T_m-t_0)}. \n\\end{align*}\nFor the first term, we have \n\\begin{equation}\\label{pr.iv.1}\n\\begin{aligned}\n\t\\|u(\\cdot+t_0)&\\|_{\\mathcal{K}^s(T_m-t_0)} \n\t= \\sup_{0\\le t \\le T_m - t_0} t^{\\frac{s_c-s}2} \\|u(t + t_0)\\|_{L^q}\n\t= \\sup_{t_0\\le s\\le T_m} (s-t_0)^{\\frac{s_c-s}2} \\|u(s)\\|_{L^q} \\\\\n\t&\\le \\left(\\frac{T_m-t_0}{t_0} \\right)^{\\frac{s_c-s}2} \n\t\t\\sup_{t_0 \\le s\\le T_m} s^{\\frac{s_c-s}2} \\|u(s)\\|_{L^q} \n\t\\le \\left(\\frac{T_m-t_0}{t_0} \\right)^{\\frac{s_c-s}2} \\|u\\|_{\\mathcal{K}^s(T_m)}.\n\\end{aligned}\n\\end{equation}\nFor the second term, Lemma \\ref{l:Kato.est} yields \n\\begin{equation}\\label{pr.iv.2}\n\t\\left\\| \\int_0^t e^{(t-\\tau)\\Delta} \\left\\{ |\\cdot|^{\\gamma} \n\t\t\t\tF(u(t_0+\\tau)) \\right\\} d\\tau \\right\\|_{\\mathcal{K}^s(T_m-t_0)}\n\t\\le C_0 \\|u(\\cdot+t_0)\\|_{\\mathcal{K}^s(T_m-t_0)}^\\alpha.\n\\end{equation}\nSince the right-hand sides in \\eqref{pr.iv.1} and \\eqref{pr.iv.2} go to $0$ as $t_0 \\to T_m$,\nwe may fix some $t_0$ \nclose enough to $T_m$ so that \n\\begin{equation}\\nonumber\n\t\\|e^{t\\Delta} u(t_0)\\|_{\\mathcal{K}^s(T_m-t_0)} \n\t\\le 2^{-\\frac{s_c-s}2}  \\frac{\\rho}{2}.\n\\end{equation}\nLet $\\varepsilon\\in (0, T_m -t_0)$, to be fixed later. \nThen, we have \n\\begin{equation}\\label{pr.iv.4}\n\\begin{aligned}\n\\sup_{2\\varepsilon\\le t\\le T_m - t_0 +\\varepsilon} t^{\\frac{s_c-s}2} \\|e^{t\\Delta} u(t_0)\\|_{L^q_s} \n\t&=  \\sup_{\\varepsilon \\le s\\le T_m - t_0} \\left( \\frac{s+\\varepsilon}{s}\\right)^{\\frac{s_c-s}2} \n\t\t s^{\\frac{s_c-s}2} \\|e^{(s+\\varepsilon)\\Delta} u(t_0)\\|_{L^q_s} \\\\\n\t&\\le \\sup_{\\varepsilon \\le s\\le T_m - t_0} \\left( \\frac{s+\\varepsilon}{s}\\right)^{\\frac{s_c-s}2} \n\t\t\\|e^{t\\Delta} u(t_0)\\|_{\\mathcal{K}^s(T_m-t_0)}\\\\\n\t&\\le 2^{\\frac{s_c-s}2} \n\t\t\\|e^{t\\Delta} u(t_0)\\|_{\\mathcal{K}^s(T_m-t_0)} \\le  \\frac{\\rho}{2},\n \\end{aligned}\n \\end{equation}\nwhere we have used \n$$\\displaystyle{\\sup_{\\varepsilon \\le s\\le T_m - t_0} \\frac{s+\\varepsilon}{s} \\le 2.}$$\nOn the other hand, since $u(t_0) \\in L^q_{s_c}(\\mathbb{R}^d),$ we may fix some $\\varepsilon>0$ \nsuch that \n\\begin{equation}\\label{pr.iv.0}\n\t\\|e^{t\\Delta}u(t_0)\\|_{\\mathcal{K}^s(2\\varepsilon)} \\le \\frac{\\rho}2, \n\\end{equation}\nBy \\eqref{pr.iv.0} and \\eqref{pr.iv.4}, we deduce that \n\\[\n\\begin{split}\n\t\\|e^{t\\Delta} u(t_0)\\|_{\\mathcal{K}^s(T_m-t_0+\\varepsilon)} \n\t& \\le \\|e^{t\\Delta}u(t_0)\\|_{\\mathcal{K}^s(2\\varepsilon)} + \\sup_{2\\varepsilon\\le t\\le T_m - t_0 +\\varepsilon} t^{\\frac{s_c-s}2} \\|e^{t\\Delta} u(t_0)\\|_{L^q_s} \\\\\n\t& \\le \\frac{\\rho}2 + \\frac{\\rho}2 = \\rho,\n\\end{split}\n\\]\nwhich proves \\eqref{buc-aim}. \n\n\t\\subsubsection{Proof of $(v)$}\nTaking $T=\\infty$ in Lemma \\ref{l:exist.crt}, we deduce the global existence. \nLastly, we show that if $T_m=\\infty,$ then the solution is dissipative. \nWe sketch the proof, as most of the computations are similar to the previous ones. \nWe take $\\{u_{0n}\\}_{n\\ge0} \\subset C_0^\\infty(\\mathbb{R}^d)$ such that \n$u_{0n} \\to u_0$ in $L^q_{s_c}(\\mathbb{R}^d)$ and decompose the integral equation into \n\\begin{equation}\\nonumber\\begin{aligned}\nu(t) =  e^{t\\Delta} u_{0n} + e^{t\\Delta} (u_0-u_{0n}) \n&+ e^{(t-t')\\Delta} \\int_{0}^{t'} e^{(t'-\\tau)\\Delta} \\left( |\\cdot|^{\\gamma} F(u(\\tau)) \\right)d\\tau \\\\\n&+ \\int_{t'}^{t} e^{(t-\\tau)\\Delta} \\left( |\\cdot|^{\\gamma} F(u(\\tau)) \\right)d\\tau,\n\\end{aligned}\\end{equation}\nwhere $0 < t' < t.$ \nThe first and second linear terms obviously tend to 0 as $n\\to \\infty$ and $t\\to\\infty.$\nOn the other hand, we may let $t'$ so close to $t$ so that the fourth term is small. \nNow that $t'$ is fixed, the third term can be written as $e^{(t-t')\\Delta} f(t')$ with \n$f(t') \\in L^q_{s_c}(\\mathbb{R}^d)$, \nso we may use the semigroup property of $e^{t\\Delta}$ and an \napproximation argument again. \nThis completes the proof of the theorem. \n\n\n\n\\subsection{Proof of Theorem \\ref{t:HH.LWP.sub}}\n\\begin{lem}\\label{l:exist.sub}\nLet real numbers $T\\in (0,\\infty),$ $\\rho>0$ and $M>0$ satisfy \n\\begin{equation}\\label{l:exist.sub.c0}\n\t\\rho + \\tilde C_0 T^{\\frac{\\alpha-1}2(s_c-\\tilde s)} M^\\alpha \\le M \n\t\t\\quad\\text{and}\\quad\n\t2 \\tilde C_1 T^{\\frac{\\alpha-1}2(s_c-\\tilde s)} M^{\\alpha-1} <1,\n\\end{equation}\nwhere $\\tilde C_0$ and $\\tilde C_1$ are as in Lemma \\ref{l:Kato.est.sub}. \nUnder conditions \\eqref{t:HH.LWP.c0}, \\eqref{t:HH.LWP.c1} and \\eqref{t:HH.LWP.c2}, \nlet $u_0 \\in \\mathcal{S}'(\\mathbb{R}^d)$ be such that \n$e^{t\\Delta}u_0 \\in \\tilde{\\mathcal{K}}^s(T)$ for $T$ fixed as above.  \nIf $\\|e^{t\\Delta}u_0\\|_{\\tilde{\\mathcal{K}}^s(T)}\\le \\rho,$ \nthen a solution $u$ to \\eqref{HH} exists \nsuch that $u -e^{t\\Delta} u_0 \\in C([0,T] ; L^q_{\\tilde s}(\\mathbb{R}^d))$ and \n$\\|u\\|_{\\tilde{\\mathcal{K}}^s(T)} \\le M.$ \n\\end{lem}\n\\begin{rem}\\label{r:exist.sub}\nTo meet condition \\eqref{l:exist.sub.c0}, it suffices to take $M=2\\rho$ and \n$T$ such that \n\\[\nT < \\min\\left\\{  (2^{\\alpha}\\tilde C_0)^{-\\frac{2}{(\\alpha-1)(s_c-\\tilde s)}} , \n\t\t\t\\, (2^{\\alpha}\\tilde C_1)^{-\\frac{2}{(\\alpha-1)(s_c-\\tilde s)}} \n\t\t\t\\right\\} \\rho^{-\\frac2{s_c-\\tilde s}}.\n\\]\n\\end{rem}\n\\begin{proof}[Proof of Lemma \\ref{l:exist.sub}]\nSetting the metric $d(u,v) := \\|u-v\\|_{\\tilde{\\mathcal{K}}^s(T)}$, we may show that $(\\tilde{\\mathcal{K}}^s(T),d)$ is a nonempty complete metric space. Let\n$X_M := \\{ u \\in\\mathcal{K}^s(T) \\,;\\,  \\|u\\|_{\\tilde{\\mathcal{K}}^s(T)} \\le M \\}$ be the closed ball in $\\tilde{\\mathcal{K}}^s(T)$ centered at the origin with radius $M$.\nSimilarly to the critical case, we may prove \nthat the map defined in \\eqref{map} has a fixed point in $\\tilde X_M,$ \nthanks to Lemma \\ref{l:Kato.est.sub} and \\eqref{l:exist.sub.c0}. \nThus, Banach's fixed point theorem ensures the existence of \na unique fixed point $u$ for the map $\\Phi$ in $\\tilde X_M.$ \n\nHaving obtained a fixed point in $\\tilde{\\mathcal{K}}^s(T),$ \nwe deduce $u -e^{t\\Delta} u_0 \\in L^\\infty(0,T;L^q_{\\tilde s}(\\mathbb{R}^d))$ \nthanks to Lemma \\ref{l:subcrt.est}, provided further that \n\\eqref{t:HH.LWP.sub.cs}, \\eqref{l:subcrt.est.c1} and \\eqref{l:subcrt.est.c2} \nare satisfied. \nWe see that $s<\\tilde s < s_c$ imply \n\\[\n\t\\max\\left\\{\\frac{\\gamma}{\\alpha-1}, \\, \\tilde s - \\frac2{\\alpha} \\right\\} \n\t< \\frac{\\tilde s+\\gamma}{\\alpha} \n\\]\nso $\\frac{s_c+\\gamma}{\\alpha}$ is a new lower bound for $s.$\nIn conjunction with this stronger lower bound $\\frac{s_c+\\gamma}{\\alpha} \\le s,$ \nthere also appears a new upper bound for $\\frac1{q}.$ \nMore precisely, for such an $s$ satisfying \\eqref{l:Kato.est.sub.c2}\nand \\eqref{l:subcrt.est.c2} to exist, $q$ must satisfy, \nin addition to \\eqref{l:Kato.est.sub.c1} and \\eqref{l:subcrt.est.c1}, \n\\begin{equation}\\label{t:HH.LWP.sub:pr2}\n\t\\frac1{q} < \\frac1{d\\alpha} \\left(\\frac{2\\alpha + \\gamma}{\\alpha-1} -\\tilde s\\right). \n\\end{equation}\nIndeed, $\\frac{\\tilde s+\\gamma}{\\alpha} < s_c$ \nis equivalent to $\\frac1{q} <\\frac1{d\\alpha} (\\frac{2\\alpha + \\gamma}{\\alpha-1} -\\tilde s).$ We notice that \n$\\frac1{\\alpha} (1-\\frac{\\tilde s}{d}) < \\frac1{\\alpha} (1-\\frac{\\gamma}{d(\\alpha-1)} )$ \nand \n$\\frac1{d\\alpha} (\\frac{2\\alpha + \\gamma}{\\alpha-1} -\\tilde s) > \n\\frac1{d} (\\frac{2 + \\gamma}{\\alpha-1} -\\tilde s)$ as $\\frac{\\gamma}{\\alpha-1} < \\tilde s.$ \nThus, combining \\eqref{l:Kato.est.sub.c1} and \\eqref{t:HH.LWP.sub:pr2}, we deduce \nthat the conditions for $q$ are \\eqref{t:HH.LWP.sub.c1}\n\\end{proof}\n\nWe omit the proofs of $(i),$ $(ii)$ and $(iii)$ of Theorem \\ref{t:HH.LWP.sub} \nas they are standard. We only prove $(iv).$\n\n\t\\subsubsection{Proof of $(iv)$}\nLet $u_0 \\in L^q_{\\tilde s}(\\mathbb{R}^d)$ be such that $T_m = T_m(u_0)$ is finite and let \n$u \\in C([0,T_m) ; L^q_{\\tilde s}(\\mathbb{R}^d))$ be the maximal solution of \\eqref{HH}. \nFix $t_0 \\in (0,T_m)$ and so that we may express the maximal solution by \n\\begin{equation}\\nonumber\n\tu(t+t_0) = e^{t\\Delta} u(t_0) + \\int_0^t e^{(t-\\tau)\\Delta} \\left\\{ |\\cdot|^{\\gamma} F(u(t_0+\\tau,\\cdot)) \\right\\} d\\tau, \\quad 0\\le t < T_m - t_0.\n\\end{equation}\nWe observe that \n\\begin{equation}\\nonumber\n\t\\|u(t_0)\\|_{L^q_{\\tilde s}} \n\t\t+ \\tilde C_0 (T_m-t_0)^{\\frac{\\alpha-1}2(s_c-\\tilde s)} M^\\alpha > M\n\\end{equation}\nholds for all $M>0,$ where $\\tilde C_0$ is as in \\eqref{l:Kato.est.sub1}.\nOtherwise there exists $M>0$ such that \n\\begin{equation}\\nonumber\n\t\\|u(t_0)\\|_{L^q_{\\tilde s}} \n\t\t+ \\tilde C_0 (T_m-t_0)^{\\frac{\\alpha-1}2(s_c-\\tilde s)} M^\\alpha \\le M\n\\end{equation}\nso that one may argue as in the proof of existence to obtain a local solution such that \n$\\|u(t + t_0)\\|_{L^q_{\\tilde s}} \\le M$ for $t\\in [0,T_m -t_0]$ and in particular, \n$u(T_m)$ is well-defined in $L^q_s(\\mathbb{R}^d),$ contradicting the definition of $T_m.$ \nLet $M = 2\\|u(t_0)\\|_{L^q_{\\tilde s}}$ so that \n\\begin{equation}\\nonumber\n\t\\|u(t_0)\\|_{L^q_{\\tilde s}} + 2^{\\alpha} \\tilde C_0 \\|u(t_0)\\|_{L^q_{\\tilde s}}^\\alpha (T_m-t_0)^{\\frac{\\alpha-1}2(s_c-\\tilde s)}  > 2 \\|u(t_0)\\|_{L^q_{\\tilde s}}, \n\\end{equation}\nwhich yields \\eqref{t:HH.LWP:Tm}. \nIn particular, $\\|u(t)\\|_{L^q_{\\tilde s}} \\to \\infty$ as $t\\to T_m.$ \nThus, we conclude Theorem \\ref{t:HH.LWP.sub}. \n\n\\subsection{Proof of Theorem \\ref{t:HH.self.sim}}\nLet $\\psi (x) := |x|^{-\\frac{2+\\gamma}{\\alpha-1}}$ for $x\\ne 0$. \nWe first claim that a initial data $u_0$ given by $u_0(x):= c\\psi(x)$ with a sufficiently small $c$ satisfies the all assumptions of $(v)$ in Theorem \\ref{t:HH.LWP} with $T=\\infty,$ thereby generating a global solution to the Cauchy problem (\\ref{HH}) with the initial data $u_0$. Since $\\psi \\in L^1_{loc}(\\mathbb{R}^d)$ as $\\alpha>\\alpha_F(d,\\gamma),$ $\\psi \\in \\mathcal{S}'(\\mathbb{R}^d)$ and $e^{t\\Delta} \\psi$ is well-defined. Since $s<s_c,$ there exist $s_1, s_2 \\in\\mathbb{R}$ such that $s<s_1 < s_c <s_2.$ As in the proof of \\cite[Theorem 1.3]{BenTayWei2017}, we can prove that $\\psi$ can be decomposed into $\\psi = \\psi_1 + \\psi_2,$ \n$\\psi_1 := \\chi_{|x|>1} \\psi$ and $\\psi_2 := \\chi_{|x|<1} \\psi$ \nso that $\\psi_1 \\in L^q_{s_1}(\\mathbb{R}^d)$ and $\\psi_2 \\in L^q_{s_2}(\\mathbb{R}^d).$ \nThis implies that the estimate $\\|e^{\\Delta} \\psi\\|_{L^q_s} \n\t\\le C( \\| \\psi_1\\|_{L^q_{s_1}} + \\| \\psi_2\\|_{L^q_{s_2}})$ holds, \nthanks to Lemma \\ref{l:wLpLq}. By the homogeneity of the data, we deduce\n$\\|e^{t\\Delta} \\psi\\|_{\\mathcal{K}^s} < \\infty.$ Thus, if the constant $c$ is taken small enough so that $(v)$ in Theorem \\ref{t:HH.LWP} is satisfied, the initial data $u_0=c\\psi$ generates a unique global solution to (\\ref{HH}). \n\nLet $\\varphi := \\omega \\psi$ be as in the assumption of Theorem \\ref{t:HH.self.sim}. \nThen we note that $\\varphi$ is homogeneous of degree $-\\frac{2+\\gamma}{\\alpha-1}.$ \nWe show that the global solution $u$ to (\\ref{HH}) with the initial data \n$\\varphi$, which is obtained by $(v)$ in Theorem \\ref{t:HH.LWP}, is also self-similar. To this end, for $\\lambda>0$, \nlet $\\varphi_{\\lambda}$ be defined by \n$\\varphi_{\\lambda} (x) := \\lambda^{\\frac{2-\\gamma}{\\alpha-1}} \\varphi(\\lambda x).$  \nSince the identity $\\|\\varphi_{\\lambda}\\|_{\\mathcal{K}^s} = \\|\\varphi \\|_{\\mathcal{K}^s}$ holds for all $\\lambda>0,$ it follows that $\\varphi_{\\lambda}$ also \nsatisfies the assumptions of $(v)$ in Theorem \\ref{t:HH.LWP}. \nAs $u_\\lambda$ given by \\eqref{scale} is a solution of \\eqref{HH} with initial data \n$\\varphi_{\\lambda},$ and $\\|u_{\\lambda}\\|_{\\mathcal{K}^s} = \\|u \\|_{\\mathcal{K}^s}$ \nfor all $\\lambda>0,$ we deduce that $u$ must be self-similar since $\\varphi_{\\lambda}=\\varphi$. We denote the global self-similar solution $u$ by $u_{\\mathcal{S}}$. The fact $u_{\\mathcal{S}}(t)\\rightarrow\\varphi$ in $\\mathcal{S}'(\\mathbb{R}^d)$ as $t\\rightarrow +0$ follows from $(iii)$ in Lemma \\ref{l:exist.crt}. This completes the proof of Theorem \\ref{t:HH.self.sim}. \n\n\\section{Nonexistence of local positive weak solution}\nIn this section we give a proof of Theorem \\ref{t:nonex}. As the argument is standard, we only give a sketch of the proof. \nFor the details, we refer to \\cite[Proposition 2.4, Theorem 2.5]{II-15}. \n\\subsection{Proof of Theorem \\ref{t:nonex}}\nLet $T\\in (0,1)$. Suppose that the conclusion of Theorem \\ref{t:nonex} does not hold. \nThen there exists a positive weak solution $u$ on $[0,T)$ \n(See Definition \\ref{d:w.sol}). Let \n\\[\n\\psi_T(t,x) := \\eta\\left(\\frac{t}{T}\\right) \\phi\\left(\\frac{x}{\\sqrt{T}}\\right),\n\\]\nwhere $\\eta \\in C^\\infty_0([0,\\infty))$ and $\\phi\\in C^\\infty_0(\\mathbb R^d)$ are such that\n\\[\n\\eta (t)\n:= \n\\begin{cases}\n1,\\quad 0\\le t \\le \\frac12,\\\\\n0,\\quad t\\ge1,\n\\end{cases}\n\\quad \\text{and}\\quad \n\\phi(x)\n:= \n\\begin{cases}\n1,\\quad |x| \\le \\frac12,\\\\\n0,\\quad |x|\\ge1.\n\\end{cases}\n\\]\nLet $l\\in\\mathbb N$ with $l\\ge3$, which will be chosen later. \nWe note that $\\psi_T^l\\in C^{1,2}([0,T)\\times \\mathbb{R}^d)$ and the estimates $|\\partial_t \\{\\psi_T (t,x)\\}^l|\\le \\frac{C}{T} \\psi_T(t,x)^{l-1}$ \nand $|\\partial_{x_j}^2\\{\\psi_T(t,x)^l\\}|\\le \\frac{C}{T} \\psi_T(t,x)^{l-1}$ hold for $j=1,\\ldots, d.$ We define a function $I:[0,T)\\rightarrow \\mathbb{R}_{\\ge 0}$ given by\n\\[\n  I(T):=\\int_{[0,T)\\times \\{|x|<\\sqrt{T}\\}}|x|^{\\gamma} u(t,x)^{\\alpha} \\, \\psi_T^l \\, dtdx.\n\\]\nWe note that $I(T)<\\infty$, since $u\\in L_t^{\\alpha}(0,T;L^{\\alpha}_{\\frac{\\gamma}{\\alpha},loc}(\\mathbb{R}^d))$.\nBy using the weak form (\\ref{weak}) and the above estimates, the estimates hold:\n\\[\n\\begin{aligned}\nI(T) + \\int_{|x|<\\sqrt{T}} u_0(x) \\phi^l\\left(\\frac{x}{\\sqrt{T}}\\right)\\, dx\n& = \n\\left|\\int_{[0,T)\\times \\{|x|<\\sqrt{T}\\}}u(\\partial_t \\psi_T^l + \\Delta \\psi_T^l )\\,dt\\,dx \\right|\\\\\n& \\le \\frac{C}{T}\n\\int_{[0,T)\\times \\{|x|<\\sqrt{T}\\}}|u| \\psi_T^{\\frac{l}{\\alpha}} \\,dt\\,dx.\n\\end{aligned}\n\\]\nHere we choose $l$ as \n\\begin{equation}\\nonumber\n\t-\\frac{l}{\\alpha}+l-2>0, \\quad\\text{i.e.,}\\quad l > \\frac{2\\alpha}{\\alpha-1}.\n\\end{equation}\nBy H\\\"older's inequality and Young's inequality, we may estimate the integral \nin the right-hand side above by \n\\[\n\\begin{aligned}\nT^{-1}&\\int_{[0,T)\\times \\{|x|<\\sqrt{T}\\}}|u| \\psi_T^{\\frac{l}{\\alpha}}\\, dtdx \n\\le I(T)^\\frac{1}{\\alpha} \\cdot T^{-1}K(T)^\\frac{1}{\\alpha'}\n\\le \\frac12 I(T) + \\frac{C}{T^{\\alpha'}}K(T).\n\\end{aligned}\n\\]\nwhere $1= \\frac{1}{\\alpha} + \\frac{1}{\\alpha'}$, i.e., $\\alpha'=\\frac{\\alpha}{\\alpha-1}$, and \n\\[\nK(T) :=  \\int_{[0,T)\\times \\{|x|<\\sqrt{T}\\}}(|x|^{-\\frac{\\gamma}{\\alpha}})^{\\alpha'}\\, dtdx\n=  T\\int_{|x|<\\sqrt{T}}|x|^{-\\frac{\\gamma}{\\alpha-1}}\\, dx=CT^{1-\\frac{\\gamma}{2(\\alpha-1)}+\\frac{d}{2}}\n\\]\ndue to $\\alpha>1+\\gamma\/d$. Summarizing the estimates obtained now, we have\n\\begin{equation}\\label{ineq1}\n\\begin{aligned}\n\\int_{|x|<\\sqrt{T}} u_0(x) \\phi^l\\left(\\frac{x}{\\sqrt{T}}\\right)\\, dx\n \\le \nI(T) + 2\\int_{|x|<\\sqrt{T}} u_0(x) \\phi^l\\left(\\frac{x}{\\sqrt{T}}\\right)\\, dx\n \\le\n C T^{- \\frac{2+\\gamma}{2(\\alpha-1)} + \\frac{d}{2}}.\n\\end{aligned}\n\\end{equation}\nWe now choose the initial data $u_0$ as\n\\[\nu_0(x) := \n\\begin{cases}\n\t|x|^{-\\beta} \\quad & |x|\\le 1,\\\\\n\t0 & \\text{otherwise}\n\\end{cases}\n\\]\nwith \n\\begin{equation}\\label{beta1}\n\t\\beta< \\min\\left\\{s + \\frac{d}{q},d\\right\\}.\n\\end{equation}\nThen $u_0 \\in L^q_s (\\mathbb{R}^d)$ and by $T<1$ and $\\beta<d$, we have\n\\begin{equation}\\label{ineq2}\n\\begin{aligned}\n\t\\int_{|x|<\\sqrt{T}} u_0(x) \\phi^l\\left(\\frac{x}{\\sqrt{T}}\\right)\\, dx\n\t& = T^{-\\frac{\\beta-d}{2}}\t\\int_{|y|<1} |y|^{-\\beta} \\phi^l(y)\\, dx\n\t = C T^{-\\frac{\\beta-d}{2}}.\n\\end{aligned}\n\\end{equation}\nCombining \\eqref{ineq1} and \\eqref{ineq2}, we obtain\n\\begin{equation}\\label{contradiction}\n0< C \\le T^{\\frac{\\beta}{2} - \\frac{2+\\gamma}{2(\\alpha-1)}} \\to 0 \\quad \\text{as }T\\to 0,\n\\end{equation}\nwhere\n\\begin{equation}\\label{beta2}\n\\frac{\\beta}{2} - \\frac{2+\\gamma}{2(\\alpha-1)} >0\\quad \\text{i.e.}\\quad \\beta > \\frac{2+\\gamma}{\\alpha-1},\n\\end{equation}\nwhich leads to a contradiction. Thus the proposition holds if we take \n$\\beta$ satisfying \\eqref{beta1} and \\eqref{beta2}, which amount to $s>s_c$ and $\\alpha>\\alpha_F(d,\\gamma)$. The proof is complete.\n\n\\section{Appendix}\n\\par\nWe list basic properties of the weighted Lebesgue spaces $L^q_s(\\mathbb{R}^d)$. \n\\begin{prop}\n\\label{p:wL.sp}\nLet $s\\in\\mathbb{R}$ and $q\\in [1,\\infty].$ Then the following holds:\n\\begin{enumerate}[$(1)$]\n\\item The space $L^q_{s}(\\mathbb{R}^d)$ is a Banach space. \n\\item $C_0^\\infty(\\mathbb{R}^d)$ is dense in $L^q_{s}(\\mathbb{R}^d)$ if $q$ and $s$ satisfy \n\t\\begin{equation}\\nonumber\n\t\t1\\le q < \\infty\t\\quad\\text{and}\\quad\n\t\t-\\frac{d}{q} < s < d\\left( 1-\\frac{1}{q} \\right).\n\t\\end{equation}\n\\item For $s_1, s_2 \\in \\mathbb{R},$ $q_1, q_2 \\in [1,\\infty],$ we have \n\t\\begin{equation}\\nonumber\n\t\\|f\\|_{L^q_s} \\le \\|f\\|_{L^{q_1}_{s_1}}^{\\theta} \\|f\\|_{L^{q_2}_{s_2}}^{1-\\theta}\n\t\\end{equation}\n\tfor $s = \\theta s_1 + (1-\\theta) s_2,$ \n\t$\\frac1{q} = \\frac{\\theta}{q_1} + \\frac{1-\\theta}{q_2}$ and $\\theta \\in (0,1).$ \n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n$(1)$ The space $L^q_{s}(\\mathbb{R}^d)$ is a Lebesgue space \nwith a measure $d\\mu = |x|^{sq} \\,dx.$ See any standard textbook for the proof of \nits completeness. \\\\\n$(2)$ Recall that the weight $|x|^{sq}$ belongs to \nthe Muckenhoupt class $A_q$ if and only if $- \\frac{d}{q} < s < d(1- \\frac1{q})$ \nwhen $q\\in (1,\\infty),$ and $|x|^{s} \\in A_1$ if and only if $-d<s\\le 0$ when $q=1.$ \nNow the density follows from \\cite{NakTomYab2004}[Theorem 1.1]. \n\\\\\n$(3)$ For $s$ and $q$ as in the assumption, we have \n\\begin{align*}\\nonumber\n\t\\|f\\|_{L^q_s} \n\n\n\t&\\le \\||\\cdot|^{s_1} f\\|_{L^{q_1}}^{\\theta} \\||\\cdot|^{s_2} f\\|_{L^{q_2}}^{1-\\theta} \n\t= \\|f\\|_{L^{q_1}_{s_1}}^{\\theta}  \\|f\\|_{L^{q_2}_{s_2}}^{1-\\theta}. \n\\end{align*}\n\\end{proof}\nThe following pointwise bound is well-known in the literature. \n\\begin{lem}\\label{l:g.unfrm.bnd}\nLet $d\\in\\mathbb N,$ $q\\in[1,\\infty)$ and $a,b,c\\in\\mathbb{R}.$ \nLet $g(x):=(4\\pi)^{-\\frac{d}2} e^{-\\frac{|x|^2}4}.$ \n\\begin{enumerate}[$(1)$]\n\\item \nThere exists a constant $C$ depending only on $d,$ $q,$ $a$ and $b$ such that \n\\begin{equation}\\nonumber\n\t\\sup_{x\\in\\mathbb{R}^d} \\int_{\\mathbb{R}^d} ( |y|^{-a} |x-y|^{b} g(x-y))^q \\,dy \\le C\n\\end{equation}\nprovided that $0\\le a<\\frac{d}{q}$ and $b\\ge0.$ \n\\item \nThere exists a constant $C$ depending only on $d,$ $q$ and $c$ such that \n\\begin{equation}\\nonumber\n\t\\sup_{x\\in\\mathbb{R}^d} \\int_{\\mathbb{R}^d} ( |y|^{-c} g(x-y))^q \\, dy \\le C\n\\end{equation}\nprovided that $0\\le c<\\frac{d}{q}.$\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nIn what follows, we shall use the fact that there exists an absolute constant $C$ such that \n\\begin{equation}\\label{l:g.unfrm.bnd:pr1}\n\tg(x) \\le C \\langle x\\rangle^{-N}\n\\end{equation}\nfor any $N\\in\\mathbb N,$ where $\\langle x\\rangle :=(1+|x|^2)^{\\frac{1}2}.$ Let \n\\begin{equation}\\nonumber\n\t\\begin{aligned}\n\tI(x) &:= \\int_{\\mathbb{R}^d} ( |y|^{-a} |x-y|^{b} g(x-y))^q \\, dy \\\\\n\t&= \\int_{|y|< |x-y|} ( |y|^{-a} |x-y|^{b} g(x-y))^q \\, dy \n\t\t\t+ \\int_{|y|> |x-y|} ( |y|^{-a} |x-y|^{b} g(x-y))^q \\, dy \\\\\n\t&=: I_1 (x) + I_2 (x).\n\t\\end{aligned}\n\\end{equation}\nThanks to \\eqref{l:g.unfrm.bnd:pr1} and $0\\le b,$ we have \n\\begin{align*}\n\tI_1(x) \\le C \\int_{|y|< |x-y|} |y|^{-aq} \\langle x-y\\rangle^{-(d+1)}\\, dy \n\t\\le C  \\int_{|y|< |x-y|} |y|^{-aq} \\langle y\\rangle^{-(d+1)} \\, dy \n\t< \\infty,\n\\end{align*}\nif $aq<d.$ Moreover, we have \n\\begin{equation*}\n\tI_2(x) \\le C \\int_{|y|> |x-y|} |x-y|^{-(a-b)q}  g(x-y)^q \\, dy \n\t\\le C \\int_{\\mathbb{R}^d} |y|^{-(a-b)q}  g(y)^q \\, dy < \\infty,\n\\end{equation*}\nif $a\\ge 0$ and $(a-b)q <d.$ \nThus, $I(x) < \\infty$ uniformly with respect to $x\\in\\mathbb{R}^d.$ \nThe proof for the second inequality is similar so we omit it. \n\\end{proof}\n\nWe recall the following elementary characterization of $L^1(\\mathbb{R}^d)$-functions. \n\\begin{prop}\t\\label{p:L1sg.dcy}\nIf $f \\in L^1(\\mathbb{R}^d),$ then \n\\begin{equation}\\nonumber\n\t\\liminf_{|x|\\to 0} |x|^d |f(x)| = \\liminf_{|x|\\to \\infty} |x|^d |f(x)| =0.\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nWe show the contrapositive. Suppose that \n$\\displaystyle\\liminf_{|x|\\to0} |x|^d |f(x)| = c>0.$ \nThen there exists some positive $\\delta$ such that $\\frac{c}2 \\le |x|^d |f(x)|$ \nfor $|x|\\le \\delta.$ Thus, \n\\begin{equation}\\nonumber\n\t\\int_{|x|\\le\\delta} |f(x)| dx \\ge c \\int_0^\\delta r^{-1} \\,dr\n\t= c \\left[ \\log r\\right]_{0}^r = + \\infty,\n\\end{equation}\nwhich implies $f\\notin L^1(\\mathbb{R}^d).$ The second equality is similarly proved. \n\\end{proof}\nAs a corollary, we have the following. \n\\begin{cor}\\label{c:wLp.sg.dcy}\nLet $s\\in\\mathbb{R}$ and $p \\in [1,\\infty].$ \nIf $f \\in L^{p}_{s}(\\mathbb{R}^d),$ then \n\\begin{equation}\\nonumber\n\t\\liminf_{|x|\\to 0} |x|^{s+\\frac{d}{p}} |g(x)| \n\t= \\liminf_{|x|\\to \\infty} |x|^{s+\\frac{d}{p}} |g(x)| =0.\n\\end{equation}\n\\end{cor}\n\nFinally, we give a proof of the fact that the $L^q_{\\tilde{s}}(\\mathbb{R}^d)$-mild solutions \nalso satisfy the equation \\eqref{HH} in the distributional sense. \n\\begin{lem}\t\\label{mildweak}\nWe assume the same assumptions as in Theorem \\ref{t:HH.LWP} (resp. Theorem \\ref{t:HH.LWP.sub}). Let $u$ be a $L^q_{\\tilde{s}}(\\mathbb{R}^d)$-mild solution on $[0,T)$ in the sense of Definition \\ref{def:sol-A}. \nThen $u$ is a weak solution in the sense of Definition \\ref{d:w.sol}.\n\\end{lem}\n\\begin{proof\nWe prove the critical case only, since the subcritical case can be treated in the similar manner. Let $T>0$ and $u$ be an $L^q_{s_c}(\\mathbb{R}^d)$-mild solution on $[0,T]$. First we prove $u\\in L^{\\alpha}(0,T;L^{\\alpha}_{\\frac{\\gamma}{\\alpha},loc}(\\mathbb{R}^d))$.\nLet $\\Omega\\subset \\mathbb{R}^d$ be a compact subset of $\\mathbb{R}^d$. We also assume that $q>\\alpha$ since the case $q=\\alpha$ can be treated in the similar manner with a slight modification. Since $s_c-2\\le s<(d+\\gamma)\/\\alpha-d\/q$, by the H\\\"older inequality, the following estimates hold: \n\\begin{align*}\n   \\|u\\|_{L^{\\alpha}(0,T;L^{\\alpha}_{\\frac{\\gamma}{\\alpha}}(\\Omega))}^{\\alpha}\n   &\\le \\int_0^T\\left(\\int_{\\Omega}|x|^{\\frac{q(\\gamma-\\alpha s)}{q-\\alpha}}dx\\right)^{\\frac{q}{q-\\alpha}}\\|u(t)\\|_{L_s^q} \\, dt \\\\\n   &\\le C\\int_0^Tt^{\\frac{s-s_c}{2}}\\,dt \\, \\|u\\|_{\\mathcal{K}^s(T)}<\\infty,\n\\end{align*}\nwhich implies that $u$ belongs to $L_t^{\\alpha}(0,T;L^{\\alpha}_{\\frac{\\gamma}{\\alpha},loc}(\\mathbb{R}^d))$. Next we prove that $u$ satisfies the weak form (\\ref{weak}). Let $\\eta\\in C^{1,2}([0,T]\\times\\mathbb{R}^d)$ be such that for any $t\\in [0,T]$, $\\operatorname{supp} \\eta(t, \\cdot)$ is compact. Let $T'\\in (0,T)$. Since $C_0^{\\infty}(\\mathbb{R}^d)$ is dense in $L^q_{s_c}(\\mathbb{R}^d)$ thanks to Proposition \\ref{p:wL.sp}, there exists a sequence $\\{u_{0j}\\}\\subset C_0^{\\infty}(\\mathbb{R}^d)$ such that the following identity holds:\n\\[\n    \\lim_{j\\rightarrow\\infty}\\|u_0-u_{0j}\\|_{L^q_{s_c}}=0.\n\\]\nBy this identity and the integration by parts, we can prove the following identity:\n\\begin{align*}\n   \\int_{[0,T']\\times\\mathbb{R}^d}&(e^{t\\Delta}u_0)(x)(\\Delta\\eta+\\partial_t\\eta)(t,x)\\,dxdt \\\\\n   &=\\int_{\\mathbb{R}^d}(e^{T'\\Delta}u_0)(x)\\, \\eta(T',x)\\,dx-\\int_{\\mathbb{R}^d}u_0(x)\\eta(0,x)\\,dx.\n\\end{align*}\nThus it suffices to prove the identity\n\\begin{equation}\t\\label{weak1}\n   \\int_{[0,T']\\times\\mathbb{R}^d}N(u(t,x))(\\Delta\\eta+\\partial_t\\eta)(t,x) \\,dxdt\n   =-\\int_{[0,T']\\times\\mathbb{R}^d}|x|^{\\gamma}F(u(t,x))\\eta(t,x) \\,dxdt,\n\\end{equation}\nwhere $N$ is defined by (\\ref{mapN}). We write $G(t,x):=|x|^{\\gamma}F(u(t,x))$. Then we can express $N(u)$ as\n\\[\nN(u)=\\int_0^te^{(t-\\tau)\\Delta}G(\\tau)\\,d\\tau.\n\\]\nMoreover, the equality\n\\[\n    \\sup_{t\\in [0,T]}t^{\\frac{(s_c-s)\\alpha}{2}}\\|G(t)\\|_{L_{\\sigma}^{\\frac{q}{\\alpha}}}=\\|u\\|_{\\mathcal{K}^s(T)}^{\\alpha}<\\infty\n\\]\nis valid, where $\\sigma:=\\alpha s-\\gamma$. Since the time interval $[0,T]$ is compact, by using mollifiers with respect to the time variable and the space variables, we can find $\\{G_j\\}\\subset C_0^{\\infty}([0,\\infty)\\times \\mathbb{R}^d)$ such that \n\\begin{equation}\t\\label{appro1}\n    \\lim_{j\\rightarrow\\infty}\\sup_{t\\in [0,T]}t^{\\frac{(s_c-s)\\alpha}{2}}\\|G(t)-G_j(t)\\|_{L^{\\frac{q}{\\alpha}}_{\\sigma}}=0.\n\\end{equation}\nWe define a sequence $\\{N_j\\}$ as\n\\[\n    N_j(t,x):=\\int_0^{t}e^{(t-\\tau)\\Delta}G_j(\\tau,x) \\, d\\tau.\n\\]\nIn a similar manner as the proof of Theorem \\ref{t:HH.LWP}, we can prove that\n\\[\n    \\|N_j-N(u)\\|_{\\mathcal{K}^s(T)}\\le C\\sup_{t\\in [0,T]}t^{\\frac{(s_c-s)\\alpha}{2}}\\|G_j(t)-G(t)\\|_{L_{\\sigma}^{\\frac{q}{\\alpha}}}\\rightarrow 0\n\\]\nas $j\\rightarrow \\infty$. By this fact, we deduce that \n\\[\n   \\text{R.H.S of (\\ref{weak1})}\n   =\\lim_{j\\rightarrow\\infty}\\int_{[0,T']\\times\\mathbb{R}^d} N_j(t,x)(\\Delta\\eta+\\partial_t\\eta)(t,x) \\,dxdt.\n\\]\nSince $G_j$ is smooth, so is $N_j$ and hence, by the integration by parts, the identity\n\\[\n   \\int_{[0,T']\\times\\mathbb{R}^d}N_j(t,x)(\\Delta\\eta+\\partial_t\\eta)(t,x)\\,dx\\,dt\n   \t=\\int_{[0,T']\\times\\mathbb{R}^d}G_j(t,x)\\eta(t,x) \\,dx\\,dt.\n\\]\nholds for any $j$. By taking the limit $j\\rightarrow\\infty$ in the right-hand side and (\\ref{appro1}), we have\n\\[\n   \\lim_{j\\rightarrow\\infty}\\int_{[0,T']\\times\\mathbb{R}^d}\n   N_j(t,x)(\\Delta\\eta+\\partial_t\\eta)(t,x) \\,dxdt\n   =\\int_{[0,T']\\times\\mathbb{R}^d}G(t,x)\\eta(t,x)\\,dxdt.\n\\]\nThus we obtain (\\ref{weak1}), which completes the proof of the lemma.\n\\end{proof}\n\n\\section*{Acknowledgement}\n\\par\nThe first author is supported by Grant-in-Aid for Young Scientists (B) \n(No. 17K14216) and Challenging Research (Pioneering) (No.17H06199), \nJapan Society for the Promotion of Science. \nThe second author is supported by JST CREST (No. JPMJCR1913), Japan and \nthe Grant-in-Aid for Scientific Research (B) (No.18H01132) and \nYoung Scientists Research (No.19K14581), JSPS.\nThe third author is supported by Grant-in-Aid for JSPS Fellows \n(No.19J00206), JSPS.\n\n\\begin{bibdiv}\n \\begin{biblist}[\\normalsize]\n \n\n\\bib{BenTayWei2017}{article}{\n   author={Ben Slimene, B.},\n   author={Tayachi, S.},\n   author={Weissler, F. 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+{"text":" \n# TINY  \nHOUSES  \nBUILT WITH  \nRECYCLED  \nMATERIALS\n\n## Inspiration for Constructing  \nTiny Homes Using Salvaged  \nand Reclaimed Supplies\n\n## Ryan Mitchell, author of _Tiny House Living_\n\nAvon, Massachusetts\n\n# DEDICATION\n\n_To those who supported me in my dream to live tiny and beyond._\n\n# SPECIAL THANKS\n\n_A special thanks to Amy Annette Henion._\n\n# CONTENTS\n\n**Part I: Reclaiming Tiny Houses**\n\nChapter 1: Why Tiny? Why Reclaimed?\n\nChapter 2: Reclaiming Materials\n\n**Part II: Talking to Tiny House Owners**\n\nApril Anson\n\nJamie, Wendy, Avery, & Gabe Barrington\n\nStacey's Airstream\n\nMakayla Cavanagh\n\nKate Fox & Andy Gill\n\nJames Galletly\n\nTiny Hall House\n\nSarah Hastings\n\nZee Kesler\n\nAaron Maret\n\nMark Miller\n\nDominique Moody\n\nGreg Parham\n\nNatalie Pollard\n\nGareth Roberts\n\nConrad Rogue\n\nAlexis Stephens & Christian Parsons\n\nCherie Southwick\n\nDan & Jess Sullivan\n\nBlaine & Mackenzie Vossler\n\nClosing\n\nResources\n\nAbout the Author\n\n# PART I  \nRECLAIMING TINY HOUSES\n\n# 1  \nWHY TINY? WHY RECLAIMED?\n\nTiny houses have captured the imagination of people for more than a decade now. The promise of affordable housing is an enticing proposition in today's world. We see people struggling to find themselves in this modern age, a time defined by \"bigger is better,\" \"keeping up with the Joneses,\" and a mentality that tells us we must always strive for more, go faster, and achieve perfection.\n\nTiny houses stand in stark contrast to this outlook. Small homes, which typically range from 100 to 400 square feet, have drawn much attention as a viable alternative to traditional housing because of cost, design sense, and environmental friendliness.\n\nIn this book we profile tiny houses of all shapes, sizes, and styles, which embody the spirit of the movement. These homes are built with a purpose: to pare down on space and possessions in order to focus on the important things in life.\n\nWhile many want tiny houses to be confined in a neat box, the truth is a tiny house, at its core, is about breaking preconceived notions of what a house is.\n\nThe essence of a tiny home is not the square footage, it is not space-saving storage, it is not a particular style or design practice. Rather, a tiny house is a structure that provides everything you need to live your best life in the most efficient and cost-effective manner possible.\n\nA tiny house is a home that conforms to the following three principles:\n\n  1. It focuses on effective use of space.\n  2. It relies on good design to meet the needs of the residents.\n  3. It serves as a vehicle to a lifestyle that the resident wishes to pursue.\n\nA tiny home for a family of four might be 1,000 square feet. For one single person a house of 150 square feet might be right, while another individual might want 300 square feet. All of these are tiny houses.\n\n## Reclaimed Materials and Their Role in Tiny Houses\n\nThere is often a correlation between people who are interested in tiny homes and people who are both environmentally conscious and budget minded. Their ethos are closely aligned. It is no surprise then that we find a lot of people using reclaimed materials in tiny houses.\n\nMany looking to make the move into a tiny house are looking for a more affordable way to live. Others want to live in a home that is gentle on the earth and lessens their impact. Reclaimed materials are great in meeting both of these goals.\n\nLots of people don't have enough funds to build a home, so using reclaimed materials is the only way to make the construction of a house possible. They don't want to spend years saving, so using reclaimed materials can cut down on the time to when they can walk in the front door. Time and money are both precious commodities, and using reclaimed materials as a way to achieve your dream of living in a tiny home more quickly can have a profound impact on the trajectory of your life.\n\nThere are others who are drawn to the aesthetics of reclaimed materials, the rich tones aged wood can bring, the unique look of their house that comes from materials with a history. Often using salvaged materials can lead to innovations, creating homes with a lot of character. Many people like the fact that they are able to design and build a house that is purpose-built to their lives, their tastes, and their needs.\n\nThis is a strong contrast with modern-day tract homes, which are built \"on spec\" and optimized for an \"ideal buyer,\" a data-driven profile created by market research. That research is used to create a home that appeals to the largest segment of home buyers. What this means is today's modern homebuilding isn't really built _for_ anyone; it's built to maximize sale and resale. The person who lives in that home doesn't get a house built for her or his life but for someone else's.\n\nFinally tiny house people love reclaimed materials because they allow for a story. The story comes both from the life that the materials afford them in a tiny house and the stories from the materials themselves. Obtaining reclaimed materials is a complex process. You must interact with many different people to piece together the materials for your house. Such a process, finding everything you need, often requires a strong social network of individuals. Through all our interviews with people who used reclaimed materials in their homes, we found they had to construct a strong web of relationships for their homes to be completed. This brings a narrative of connected humans and objects to each of these houses.\n\nAll these things add up to stories about the people who helped you build your tiny home and the materials that ended up in it\u2014stories that you then share with others when they come into the tiny house. In building your home, you are writing the next chapter in that material's story.\n\n## Tiny Home Cost\n\nA huge advantage of tiny houses and, in particular, the ones that use reclaimed materials, is their cost. The expense of a tiny house pales in comparison to that of a traditional house. The cost of housing has grown dramatically over the past sixty-five years. This is particularly noticeable today when income has stagnated. In 1950 the median household income was $4,237 and the average house cost was $7,354 for an income to home cost equivalent of 1.7. Compared to today, the median household income is $51,939 and the average house cost is $343,300 for an income to home cost equivalent of 6.6.\n\nThis sharp increase means that most people can no longer afford to purchase a home. Even if they are able to earn enough to qualify for a mortgage, more and more homeowners owe more on their mortgage than they can actually earn in a standard working life. Mortgages used to have fifteen-year terms; today the norm is thirty years, but we are beginning to see a rise in forty-year mortgages and reverse mortgages just to make owning a home possible.\n\nConsidering the average home in America sold for $341,500 in 2015, tiny houses are a breath of fresh air. The average tiny home built by its owner costs an average of $20,000, a price easily achievable for many people with a few years' savings. The houses we profile in this book range in cost from $7,000 to $25,000, the average being around $13,000.\n\nPhotograph by Aaron Maret\n\n## Tiny Home Design\n\nBeyond the cost of the home, tiny houses are an attractive option because of their design. Often a great deal of care is devoted to their aesthetics and layout. They are homes that are built and designed from the ground up for the occupant, because to live in such a small space, you must have it perfectly customized to you.\n\nTiny home designers focus on both the structural and social aspects of a home. How does the design impact how its owner interacts with the space? How do the placement and design choices of storage, seating, windows, and materials impact the feeling of the space? Does the storage hide clutter to promote a clear mind? Do the windows extend the sight line to make the home feel bigger? Do the materials provide a warm sensory experience?\n\nIt is this attention to details and interactions that people find so appealing. Modern home designs largely ignore these things, so it is a breath of fresh air to many when they see quality design. Modern cookie-cutter neighborhoods are predicated on designs that maximize profits, not the owner's happiness. Large-scale builders strive to minimize lot sizes to increase the number of units per acre. They focus on formulas for optimal marketability, and they cut corners in building quality to such an extent that code enforcers must step in to keep many builders honest.\n\nGood design itself is difficult to put a number to, but the feeling it creates, a comfort within a space, is undeniable. Mass-market homebuilding can't market design, so it concentrates on telling prospective buyers about the number of rooms, square footage, granite countertops, and walk-in closets. While big house Realtors sell to the largest number of people\u2014which they define as their market\u2014they have honed their house designs to a type; anyone outside of that ideal customer has a difficult time buying, building, or selling a home. Today most homebuilders design homes for large families, with extra rooms for entertaining and accommodating guests. As a result many single millennial home buyers are forced to buy homes with multiple bedrooms that will remain mostly empty.\n\n## Reclaimed Materials\n\nThe use of reclaimed materials in housing is nothing new, but it's seen a recent rise in popularity as people look for unique ways to decorate and build their space. Reclaimed materials have a long history that is rooted in practicality. Reclaiming materials was and is an efficient way to build and maximize what you already have. At a time when hardware stores weren't on every corner selling inexpensive lumber, you saved on the labor of milling boards.\n\nWay back when, cutting trees into dimensional shapes was difficult and labor intensive. Back in the pioneer days, buildings were constructed by hand hewing logs with which to build and later hand sawing them. It was only later that modern lumber mills began processing lumber with methods that didn't require removing material by hand with an axe or hand saw.\n\nToday's mills are advanced operations that precisely monitor moisture, use lasers to scan for perfect cuts, and employ computer-controlled milling to maximize the boards yield per log. Add to this a global transport system allowing for goods to be moved quickly around the world and we now have lumber that is easily accessible and cheap to buy. We've come a long way, but the appeal of reclaimed materials is still ever-present\u2014however, for different reasons.\n\n### What Are Reclaimed Materials?\n\nIn today's world we often focus on buying new, but there has been a rise in recycled and repurposed materials in all aspects of our lives. Lumber is no different. At its simplest definition reclaimed wood is lumber that was used for one construction project, but is then salvaged and used in another.\n\nTypically when we speak of reclaimed wood, we are talking about wood that was harvested some time ago and possesses some unique attributes that new wood does not. Often people prefer to leave it in its original form, instead of milling it or chipping it. This doesn't have to be the case, but it is often implicit.\n\nBroadly speaking, we can also consider wood that has been recycled to be reclaimed as well. This is a growing trend in the building-material industry as it looks to maximize its use of materials.\n\nThere are two main types of recycling in wood products: industrial wood product waste and recaptured wood materials. With any manufacturing operation you will find some waste from the process; in the case of wood product manufacturing, this includes wood pulp, sawdust, and rejected materials. Recaptured wood products are discarded consumer wood products collected in mass, chipped, and then used to make new products; a good example is retired pallets that are chipped and turned into particle board.\n\nThen there are items besides lumber, such as fixtures, handles, lights, scrap metal sheeting, used windows and doors, along with a whole host of other materials. These items are often kept in their original form and either used again in the same way or reconfigured into new creative uses.\n\nPhotograph by Aaron Maret\n\n### Why Should You Use Reclaimed Materials?\n\nAs is the case with tiny houses, the reasons for wanting to use reclaimed materials are varied. Reclaimed materials offer many opportunities and drawbacks, so it's up to you to determine what makes sense for you to use in your tiny home. Following are some common reasons why people use them.\n\n#### THEY'RE BUDGET-FRIENDLY MATERIALS\n\nOne of the biggest reasons people use reclaimed materials is because they can save a lot of money when it comes to building their home. There are opportunities to reclaim lumber from old buildings, items and fixtures from building salvage stores, and everything and anything from those who no longer need the materials.\n\nYou may not have a lot of funds, but you do have free time. In many cases it's enough payment that you can show up and haul away the materials. In this way you're able to gain building materials for free. In many cases you'll have to act quickly so it's important to be flexible and ready to take advantage of opportunities that present themselves.\n\nPhotograph by Christian Parsons, Tiny House Expedition\n\n#### THEY'RE AN ECOFRIENDLY ALTERNATIVE\n\nThe lumber industry is a destructive one\u2014by its very nature it cuts down trees to be fashioned into wood products. We have seen a lot of progress on this front with sustainable harvesting practices, more efficient use of trees, and greater environmental protections put into place, but in the end, lumbering is a damaging practice.\n\nWith reclaimed lumber, not only are you offsetting the use of new wood, but you are preventing that used wood from entering a landfill. Upcycling materials and displacing them from waste streams make this an ecofriendly activity.\n\nBy not purchasing new wood products we reduce the number of living trees that must be harvested. Currently 900 million trees are cut down each year; this represents a huge impact on our ecosystem. To put this in perspective, the average tree will sequester 1 ton of carbon in a forty-year period. In any given year a mature tree will produce enough oxygen for two human beings.\n\nThe damage is not just to the trees themselves but to the earth's capacity to sustain life and maintain homeostasis. At the scale which trees are harvested each year, we begin to see how this can add up to create a long-term impact.\n\nGoing to places such as the Habitat for Humanity ReStore, a local salvage yard, or a similar place, capturing fixtures and other building materials before they enter into the waste stream . . . these things can have a big impact. Many of the items you can capture are still in good working order, some of them almost new, and many are available at a good price.\n\n#### YOU CAN GET QUALITY MATERIALS\n\nThe quality of older-growth woods is undeniable. With reclaimed materials that are a few decades older or more, the wood quality is much higher in terms of wood grain, strength, cure, and fiber stability.\n\nOlder-growth woods tend to have a much denser fiber structure. This compares favorably to today's lumber. The industry selects trees that grow quickly; this produces larger grains that are less tightly packed. The tighter the wood fibers, the stronger the wood, the more durable it is, and the better it is to work with.\n\nTighter fibers allow for a better-quality end product because this kind of wood tends to finish well. Woodworking is essentially the removal of material and then joining one piece to another. With a denser pack of fibers, we can better control the cuts and finish the wood in a more refined manner.\n\nAnother benefit of reclaimed wood is that it is often very stable. As with all natural materials a certain amount of warping will occur. With older wood, the grains have twisted to their full extent, and this means the wood will generally stay in its shape better than newer boards.\n\nEven if the wood is warped you can either use it as is, highlighting its character, or you can remill it. There still is a chance that it will warp depending on air moisture, but it's usually less than is the case with new wood.\n\nIt is worth noting that some people find older woods more difficult to cut because of the fiber density. In some cases it may require higher-powered tools to handle the denser material, particularly in the case of your table saw and router.\n\nWhen it comes to fixtures and other hard goods, many people like the older styles of the items. Many also appreciate the quality that you can find; older things often were made to last longer and be more durable than those we turn out today.\n\n#### IT'S GOT A UNIQUE CHARACTER\n\nThe rise in popularity of reclaimed woods is often attributed to their character or the aesthetic qualities of the wood itself. Aging older wood often contains rich colors, which are hard to replicate through artificial techniques.\n\nIn aging, wood tends to take on two major tones, depending on how it was kept. Exposed to the elements, reclaimed wood will take on a shade of gray. These grays will vary depending on the type of wood, climate, and exposure. On the other hand, with prolonged exposure to the sun wood often suffers from UV degradation. This type of wood should only be used in accent pieces, not structural pieces. Wood that is protected from sun and most moisture often cures in brown tones, and because it isn't exposed to UV rays, it often cures harder as the resins in the board dry and solidify.\n\nYou will often find unique patterns in reclaimed wood; these derive from how the pieces were processed. You'll find old nail holes, saw blade patterns, or a chipped look that comes from the wood being hand hewn. These unique looks add character to your tiny house and tell the story of the wood.\n\nPhotograph by Aaron Maret\n\n#### IT HAS UNIQUE DIMENSIONS\n\nToday's lumber is uniform: You have two-by-fours, four-by-fours, and so on. In older times, lumber often lacked uniformity in the dimensions of the wood because they were often determined by the size of the tree or resulted from the lack of precision of hand tools. Older-growth trees were generally larger. They yielded a wider board and longer beams.\n\nMuch of today's lumber is often only available in smaller widths. Larger sizes cost a lot more\u2014if you can even find them at all. The main reason for this is that most of today's harvested trees are between twelve and twenty-four inches in diameter, compared to older lumber that was often milled from much larger trees. By comparison older-growth trees can be much bigger depending on the species; old-growth poplar trees can be six feet, an oak tree can be nine feet, and a redwood tree can grow to twenty-seven feet or more in diameter.\n\n#### IT COMES WITH STORIES\n\nOne thing that has struck me while talking to people who use reclaimed materials in their tiny houses is the stories that they tell about their reclaimed items. I think these stories are an interesting and often overlooked benefit of reclaimed materials.\n\nOften the stories are about a key piece of the home, an accent feature, or focal point. When out and about collecting the materials, scrounging them from different places, you meet interesting people who tell you about themselves. The materials too have a history, a place that they came from, a life before they were part of your home.\n\nThe story could be an interesting or bizarre encounter with the previous owner, the struggle to get that material, or a past use that makes it interesting. Regardless, using reclaimed materials builds that history, adding a new chapter as the materials are gathered, worked, and reborn.\n\n### Benefits and Negatives of Using Reclaimed Materials\n\n_Pros_ | _Cons_  \n---|---  \nCan be less expensive | Can be more time intensive  \nVisual character and uniqueness | Nonstandard sizes of wood  \nDenser grain, higher quality | Requires more labor to prepare for use  \nMaterials with a story | More difficult to source materials  \nInspires unique designs | Harder to plan; materials drive design  \nEnvironmentally friendly | Inefficiencies from sourcing uses more fossil fuels  \nWood more stable, less likely to warp | Pests and mold may come with wood  \nCan be more durable from tighter grain | Unknown previous treatments (lead paint, chemicals, etc.)  \nMinimizes forest destruction | May require additional tools for processing\n\n## Building for the Future\n\nAs we move into the future, reclaimed and recycled materials are going to need to become a larger part of building structures for ourselves. We currently use recycled materials in some of our engineered wood products, but moving forward we need to increase our ability to capture more recyclable waste and find uses for it in our structures; doing so will bring our use of wood into a more sustainable future.\n\nWe are quickly approaching a point when we must grapple with the reality that harvesting wood the way we do today is not sustainable in the long term. Even with planned forestry, selective cutting, and efficiencies in our manufacturing processes we can't displace the destructive nature of the practice.\n\nFor a variety of reasons it is difficult for us to make decisions as a global community. Our world operates on an economy that is predicated on perpetual growth on a planet with finite resources. Impacts are externalized to places we do not see and motivated by the drive for immediate profits.\n\nIt is difficult to steer a new course when there is so much wealth being accumulated by those who have a vested interest in seeing the status quo maintained. That said, the tides are turning.\n\nWe need to turn our focus to harvesting from existing wood sources; we must draw materials from buildings that need to be dismantled or demolished. This is, of course, already happening but on a very small scale, one that is not making a dent in the market. What we need is larger-scale operations that harvest wood from unused buildings en masse. It is at this scale, one where we could supply chipped wood in the hundreds of thousands of tons a year, that we can begin to move from live forest harvesting to reclaimed harvesting.\n\nWith sources such as these we can then design building materials to use chipped reclaimed wood. We can shape it into wood stud equivalents, into dimensional lumber replacements, and into other wood-based products that we use every day. The technology will be developed to match these, including more sustainable and environmentally friendly resins to bind these chipped materials together. However, without the source material (large-scale reclaimed chipped wood) there is no drive to expend R & D dollars on these innovations.\n\n# 2  \nRECLAIMING MATERIALS\n\nUnlike buying new materials, there are two major considerations when it comes to using reclaimed materials: sourcing and processing. While some places are fortunate enough to have resources that reclaim materials and then sell them in one spot, many people find their only option is to scour their social networks and online sources.\n\nEven once you locate the materials, some work is often required to get them into a useable state. They may not be in a uniform dimension, the wood may have metal fasteners in it, there may be layers of paint that must be scraped off, and so on.\n\nThe challenge with reclaimed materials is that even when you do find a good source of materials, you often cannot find uniform pieces, the quantities are small, and they require some work to get them ready.\n\n## Estimating Required Materials\n\nFinding sources may be the biggest challenge when it comes to using reclaimed materials; however there are some good go-to places to look. First, though, we must determine what we are looking for.\n\nSpend some time deciding what types of parameters you are going to look for in the materials that you encounter. Ask yourself these questions:\n\n  1. What will the materials be used for in your tiny house?\n  2. Will the materials be used for a structural piece, an aesthetic trim piece, or do you need both?\n  3. What types of wood will you need: sheet goods, dimensional lumber, larger beams, etc.?\n  4. How many pieces will you need in board feet?\n  5. What is the longest span of wood in your design that you'll need to gather?\n  6. What are some things that would cause you to decline to take certain lumber?\n\nIt is important to understand how the reclaimed elements that you have gathered will find their way into your design. It can be tricky to make actual plans for your house when a lot of the materials will vary widely. You may well find you have to adjust your plans.\n\nThere are, however, some elements common to all homes, from which we can generate some rough numbers to use as goals in our search for reclaimed materials. I suggest that if you're going to be framing with reclaimed wood, you do some of these calculations ahead of time so you know how much material you'll need to collect. This will make the building process go much more smoothly.\n\nView a text version of this table\n\nAll tiny houses have framing; these are typically made up of a 2 \u00d7 4 (which is actually 11\u20442\" \u00d7 31\u20442\") studs that are spaced 16\" on center. Windows and doors are typically framed with headers, \"cripple\" plates, and sills. Finally, roof framing is built with a ridge beam and rafters.\n\nFor walls take your total length of all your walls and assume an 8' stud every 16\" of that wall. Doing this basic math, you can then estimate the number of studs you'll need to gather. For example if you have 46' of walls with studs every 16\", you know you'll need 35 boards that are around 8' long and roughly the size of a 2 \u00d7 4.\n\nA typical wall, built by framing lumber, consists in its simplest form of three main parts: a top plate, wall studs, and a bottom plate. The bottom plate is most often a 2 \u00d7 4 that lies on the floor of your house. Vertical studs then attach to the bottom plate, perpendicular to it. Finally the top plate is attached on top of the studs, going in the same direction as the bottom plate. These three parts are nailed together to form the structure of a wall. Things become more complicated and require more materials as you add things like windows and doors, but the same form persists.\n\nYou can then assume that you'll need a top and bottom plate the length of the walls. If it's a 20'-wide wall, we know that realistically we aren't going to find such a long piece. If you're unable to find a board that spans the entire distance of the wall then you'll need to create the wall out of 8' wall sections; then add a layer on top of the top plates (double top plate with offset seams) to join the 8' sections together.\n\nThis will add to the number of boards you will need, but it allows you to use short pieces where you can offset the seams of shorter boards to create a longer-spanning plate. In the previous example we estimated we'd need 6 boards for the top plate and the bottom plates of the wall framings (done in 8' sections), then another 3 boards to join all the walls together by creating a double top plate to tie it all together.\n\nFinally we estimated the number of windows, doors, and internal walls in the tiny house to give us our final required 2 \u00d7 4s estimated number of 84. We then assumed an additional 15 percent more boards will be needed because of mistakes or not being able to fully use the wood scraps in the framing. If you're a first-time builder, it might be smart to raise this to 25 percent just in case. You can always find other uses for the wood you don't use.\n\nWith this example you can begin to see how to create a list of required materials, which forms a \"shopping list\" of items you need to find when you source your materials. You can conduct similar estimates with your build. Estimate:\n\n  * Roof framing\n  * Floor framing\n  * Sheathing\n  * Siding\n  * Roofing\n  * Windows\n  * Flooring\n  * Interior trim work\n\nFrom this process\u2014which is time consuming, but valuable\u2014you create a master shopping list that allows you to more accurately estimate what you need to have on hand for your build. Having a list such as this means you know when you're ready to build, gives you an idea of how things will all fit together, and allows you to build faster.\n\n## Sourcing Materials\n\nFinding materials is usually one of the more challenging parts of using reclaimed wood. While some cities have great sources for reclaimed materials, others don't offer much. In general sources of reclaimed wood fall into three main categories: commercial operations, online listings, and personal networking.\n\nIn many cities around the country we are beginning to see businesses opening smaller-scale reclaiming operations that source, process, and sell wood (and often other reclaimed materials) to customers. These commercial operations often work closely with the building community to take advantage of materials that would otherwise be thrown away.\n\nProbably the best-known operation is the Habitat for Humanity ReStore, which takes old building materials they either received via donations or took out of houses they reclaimed. There are other smaller shops that do similar things all around the nation, providing great places to source materials.\n\nThe next major source for materials is online listings; many websites connect people who are trying to get rid of stuff (for sale or for free) with those looking for various items. The two most popular online connecting points mentioned in interviews for this book were Craigslist.org and Freecycle.org.\n\nCraigslist offers two main categories you can explore for reclaimed materials: Free and For Sale. While the For Sale section implies a cost, you can often barter for the items or find great deals.\n\nOne of our interviewees, Zee Kesler, told us about a strategy she often uses when finding things for sale online. She contacts the sellers and asks if they'll consider letting her have the items for free if she loads them up and takes them away that same day. Many times, she says, people decline, but Zee then counters and says, \"The offer still stands; I'll check back in a week.\" She waits a week, and in many cases the items have not been sold and the person has reached a point where she or he just wants the thing gone. In the end she has been able to get many things for free.\n\nIn the Free section of Craigslist you have to be very quick to move in on free materials. It is often a first-come-first-served arrangement. One tip we learned from our interviews is that you can set up a keyword search on your phone, using Craigslist's RSS feeds to alert you every time a new posting containing the keywords goes live.\n\n### There's an App for That\n\nApps such as Feeddler allow you to tie into their RSS feed and input certain keywords for notifications. Some keywords you might consider:\n\n  * Wood\n  * Scrap wood\n  * Barn wood\n  * Pallets\n  * Lumber\n\nIt might also be worthwhile to try alternate spellings, hyphenations, common misspellings, and colloquial terms. These search parameters are important because the apps look for the exact matches to what you input. So, for example, if you enter \"barn wood\" and someone listed it under \"barnwood\" or \"barn-wood\" or \"wood from barn\" you would not see these in your search results unless you included all of them as search parameters in the app.\n\nSetting up this type of system on your smartphone will instantly alert you to new stuff that may be worth checking out, and being first to know gives you an edge. Combine this with a flexible schedule and a handy way to pick up and transport the materials, and you can discover some great stuff.\n\nFreecycle is a service designed to connect people who have things that they don't want to people who are looking for those items, often for free. The website emphasizes environmental motivations, to keep things out of landfills.\n\nJust as with Craigslist, it's important to check Freecycle often because things tend to move quickly, particularly the really good items. Some communities have websites set up for their local Freecycle, while others have mailing lists that you join. In some cases you can set up RSS notifications for this as well.\n\n### Networking\n\nThe final source for reclaimed materials is from your own social network. This can be both online and offline, but it relies on your personal relationships. The best way to start is by doing the initial legwork to understand what you will need to build your tiny house.\n\nFrom there consider how you can pick up, transport, and store the items. During my build of my own tiny house, I had a very small car, so I decided to install a trailer hitch on my car and was able to find a free trailer on which I could load materials so I could transport them home. This made the task easy and kept my car clean.\n\nOnce you have a good idea of what you need and how you can transport it, start reaching out. Many tiny house builders have found it useful to make a simple poster, which they hand out to friends and family, and post throughout their local community. On this they explain why they need the materials to build a tiny house, what they are looking for, and how to get in contact with them.\n\nWhen considering who to reach out to, it's best to throw a wide net. You never know who might be a good source; materials often come from unlikely sources. Don't just ask people if they have materials\u2014find out if they know someone who has materials or would be a good lead.\n\nExploring each connection and referral is a time-consuming venture, but the connections that you build can lead to new friends. They can also lead to a great source for materials or some other positive outcome. The truth is you don't really know where it will all lead, but it's an important part of using reclaimed materials.\n\n### Talk It Out\n\nWhen talking to people who do have materials make sure that you understand what it is they are offering, what they want in return (if anything), and how you can make it easy for them to give you materials. Some people will be really excited about your tiny house because they think it's interesting. Others don't care much what you're going to do with the materials; they just want you to take them away. Time and time again I've seen situations where people just want to clear out their wood shop. All they really care about is if you're going to take it all and load it yourself. They may or may not be interested in tiny houses, but what they really care about is how you're going to help them solve the problem of having too much stuff.\n\nDo your best to anticipate their concerns and their objections, and find ways to make their lives a little bit easier when it comes to giving you these things. Offer to take all the materials; be sure to communicate that you will load the materials yourself; be flexible to best suit their schedule. Do these things, and you'll find a lot of people willing to help you out with free materials.\n\nAt the same time, know when to say \"no.\" There will be times that you arrive and discover that the materials are not as they were described. It may not be what you need, the wood may be rotted, or it isn't useable (too small, to short, not structurally sound). It can be tricky to say no to someone, particularly when you've shown up at her house to rifle through her woodpile, but you need to navigate that social situation with honesty and courtesy.\n\nOne of the biggest lessons that we learned from talking to the tiny home builders who used reclaimed materials was that in the beginning they didn't know how to say no. This meant they got a lot of materials they couldn't use, and then they were stuck with them. In many cases stuff piled up during the build and the homeowners ended up paying to have it disposed of. Many people in our interviews said that saying no to materials they couldn't use was a big shift for them, but it was a necessary one.\n\n## Evaluating Materials\n\nOnce you find sources for your materials you need to evaluate them to determine if they suit your needs. This can be a difficult thing. When evaluating materials it is important to know if the item will be a structural piece, meaning it needs to carry a load, or an aesthetic piece, for looks only and not load bearing.\n\nLoad-bearing or structural pieces of lumber are essential to the integrity of your tiny home. Many people argue that structural pieces should always be new in order to minimize risks associated with reclaimed woods. It's true that using reclaimed wood for structural pieces is a risk, one that you'll need to evaluate. Consider the repercussions or liability if something bad happens.\n\nWhile older lumber typically has a tighter grain and thus is stronger and more durable, this fact doesn't mean reclaimed wood is always the proper choice for load-bearing purposes. When considering a piece, look at how it was stored and where it existed when it was in use. You want pieces that were protected from the elements, particularly ones that were shaded from the sun and kept away from moisture.\n\nWhen it comes to integrity, sun and moisture are wood's two biggest enemies. Prolonged sun exposure can cause UV damage. The UV rays degrade the wood to the point that it can become brittle. They can also reduce its strength considerably. Water can soak into wood and begin to break it down with mold and rot.\n\nDamage caused by bugs and pests can also be a cause for concern. Termites, ants, and other insects can bore into the wood and hollow it out from the inside. Making sure that the wood isn't damaged or even worse, still infested with bugs can be difficult to determine, but this is important to know before you start using the wood in your build.\n\nExamine the materials for previous fastener infiltration. As with any wood, reclaimed or new, every time you put in a fastener or drill a hole, you weaken the integrity of the wood. With enough holes or points of infiltration, the wood can weaken so that it can be easily broken with enough force. Consider the previous fastener holes and how you plan to use them rather than making new ones.\n\nIf there are already holes near the place where you'll need to create new ones, it might make sense to use a different piece of wood. A good rule of thumb is that a nail or screw should have a 1\"-diameter circle of clear, undamaged wood around it.\n\nWhen you fasten the board in its new position, make the hole cleanly with the fastener; make sure that no cracks or other structural defects form around the fastener.\n\nWhen in doubt, find another option for structural pieces, even if it means going to a new piece. Paying a little extra for a higher-quality reclaimed material or choosing a new option can go a long way in ensuring your safety and peace of mind. In the end, you assume all responsibility for your choices.\n\n## Processing Materials\n\nOnce you locate and transport your materials back to your build site, there are several things to consider about processing the wood. This is the time- and labor-intensive part of using reclaimed materials, but establishing a system, having the right tools on hand, and doing a little planning can make the process much easier.\n\nProcessing the materials as they come in allows you to work more efficiently, store more compactly, and spread the workload evenly. From the interviews we learned that the task of processing the materials took a lot of time and effort. The following tips will help streamline that workload.\n\n### Key Tools\n\nLuckily when it comes to using reclaimed materials you don't need to purchase or borrow a lot of specialized tools above and beyond what you'd need anyway to build a tiny house.\n\nThe first main category of tools is nail-removal devices: mainly a claw hammer, a pry bar, and a pair of vice grip pliers. Whenever you reclaim materials, it is a safe bet that there will be nails and screws to remove from the wood. Having these standard tools is both useful and inexpensive.\n\nAnother tool for this task is a pallet pry bar. This is a long-handled hook designed specifically for breaking apart pallets; it hooks under the slat of a pallet, and its fulcrum rests on the main skid allowing for you to more easily remove the slat.\n\nThe next category of reclaiming tools is those used for resurfacing. When working with reclaimed materials it is often necessary to clean up and square the edges and surfaces of the wood. The go-to tools for this are a sander (orbital and belt), table saw, and surface planer.\n\nWhile many tiny house people don't want to spring for a planer, most of the people we spoke with who built their tiny houses with reclaimed materials used one extensively. The other tools, table saw and sander, are needed in order to build a tiny house and are very useful in getting materials to a useable state for the build.\n\nRe-sawing and milling are other parts of the process for which you need specialized tools. Builders use them for milling wood from a full tree into dimensional lumber and re-sawing larger pieces of dimensional lumber into smaller pieces. In many cases people don't do much if any of this because it's a bit more involved than they want.\n\nFor those of you who want to do this, though, you can use portable band saw mills for milling raw trees into lumber. These can process large logs into boards and beams. In many cases you can find mobile band saw mills that will come to your property and for a per-hour rate, mill lumber from felled trees. These portable mills are commonly found in the For Hire sections of Craigslist.\n\nSome people use a special jig for a chainsaw to mill wood, but this tends to produce a less-clean cut than that of the band saw mills; on the other hand, the chainsaw mill option is much cheaper (a few hundred dollars versus several thousand). When milling your own lumber, after cutting it to a rough dimension, you then need to let the wood cure. Air-drying green wood takes one to two years. Alternately you can dry the wood with the aid of a kiln, which typically takes one to two days, depending on the type of wood. The latter heats the wood to a certain temperature and more quickly dries the boards. Once the green wood has been dried, you can mill it into its final size.\n\nFor re-sawing, you are taking larger pieces of dried dimensional lumber and cutting them into smaller pieces. For this a shop band saw is most often the best option; however you can also do it with a table saw.\n\nThe final main category of tools is safety equipment. While most of this is the same for any woodworking or building project, one piece of equipment is unique to reclaiming materials: a metal detector wand.\n\nWhen starting on any construction project, you should have hearing protection, eye protection, and protection for your hands in the form of gloves or push sticks for your table saws. The metal detector wand is useful for finding nails that you might have missed in your processing. Hidden nails can lead to an injury when you're cutting or fastening later on.\n\n## Deconstruction\n\nIn some cases you'll be able to find materials that are already in a useable state; they are scraps or they have already been broken down or been taken apart. In other cases you need to break down the materials from their former state. This is often the case when you get the materials from a home that is being demolished.\n\nIt is important in the deconstruction and demolition phase to consider your safety. Deconstructing a structure isn't for a layman. There is a recommended process to safely dismantle a building. Even professionals with years of experience know danger still exists. When you're breaking down smaller, more manageable objects, safety concerns still exist and you should take precautions.\n\nA hardhat for larger structures is a good idea to prevent falling debris from hitting you in the head. Thick work gloves are strongly recommended because nails and screws in the wood are a major safety concern. As with any woodworking activity, you should always wear eye protection. Finally, deconstruction is dirty business and often that means a lot of dust. Wear a respirator to keep air particulates from entering your lungs.\n\nStart by identifying the pieces that meet your needs and pass your criteria. In some cases you may need to remove other parts to gain access to them. For example, you may have to strip down drywall to access the studs. Consider what is attached to the piece you want and what removing it could do to the structural integrity of the pieces around it.\n\nOnce you've identified and considered your desired piece, begin removing it by prying or cutting it out. In order to keep the item useable, consider how much damage will occur during the removal process. In some instances you'll be able to cut just below the top of the stud and just above the bottom with a Sawzall. This will gain a 7' board from a normal 8' stud. If that's all you want for your construction, two quick cuts are all you need.\n\nLonger boards are more difficult to reclaim because there are fewer of them. Those that do exist often have more points of attachment to other pieces, so you'll need to consider if it is even possible to remove a useable piece from the structure.\n\nPhotograph by Aaron Maret\n\n### De-Nailing\n\nOnce you have the pieces of lumber, an important part of processing the material is de-nailing. Removing the nails makes working with the material easier and safer. By removing nails and screws you can stack the wood more efficiently so it takes up a lot less space.\n\nRemoving the nails and screws prevents accidental injuries from sharp metal pieces in the wood, which might cause concern about tetanus. It also makes the wood safe to process with power tools. If a nail is hidden in a board and is run through a saw, it could kick back against the blade and injure you or cause your hand to slip and fall into the path of the blade.\n\nSimilarly, when using nail guns if you don't see an existing nail in the wood, the new nail could strike the old fastener and ricochet. When this happens, the nail often curves back toward the user's hand and impales it.\n\nUnfortunately there is no simple way to remove nails besides brute strength. Hammers, pry bars, and vice grip pliers are the common go-to tools to remove nails. When you pair them with some elbow grease, it can be quite effective. If you are unable to draw out nails and screws with the claw of a hammer or pry bar, consider cutting off that part of the wood that contains the nails. This may be possible if it is at the very end and you don't need the full length of the board.\n\nAs mentioned earlier, a metal detector is helpful in making sure you don't miss any metal parts in the wood you are processing. While some see this as overkill, it saves you from the dangers of unexpected metal in the wood.\n\n### Resurfacing\n\nOnce you have removed the metal parts from the wood, it's smart to resurface the wood, especially if other people are going to see it. Things such as studs and interior wall pieces don't need to look aesthetically pleasing, but those parts that will be seen by others will benefit from being cleaned up.\n\nThe main question here is whether you want to preserve the aesthetic of the old, rough, and dirty look of the wood. There are times when resurfacing can take that look away, so consider how you will use that piece.\n\nResurfacing removes a minimal amount of the material from the surfaces. You can do this in a number of ways, but the goal here is to bring about a clean edge and square the wood. When you first get the wood, it will probably have some warp to it, as do all natural materials. Some pieces are so warped that you may find they are unusable or need to be cut down into a smaller straight piece.\n\nIn an ideal situation you'd have a tool called a jointer, which allows you to process wood into perfectly square edges to allow accurate cuts and jointing. The problem is that jointers are large machines that are not within the budget of most amateur woodworkers.\n\nYou can achieve a similar result with a table-mounted router with a modified gate. There is a plethora of tutorials online about how to do this, but realize that while it does a decent job, it will never be as accurate as a jointer. Once you have achieved two perfectly square sides you can then move to a table saw that has been calibrated to make your cuts.\n\nYou can also remove small amounts of material to even out a rough or uneven surface. In some cases you'll need to mill down a board to a desired thickness. To do this you can re-saw a board with a band saw if you need to remove large portions of the material. If you only need to remove a little bit, a surface planer will allow you to clean up a wide board quickly.\n\nIt is important to note that a surface planer does not make pieces of wood flat; only jointers do that. It is a common misconception, but the rollers in the planer are very strong and actually force boards that are slightly warped into a flat position before they move the board into the path of the blades, and as the board exits the planer, the board resumes its original shape, leaving it warped.\n\nMany of the people whom we interviewed for this book found a planer to be essential to their use of reclaimed materials. A new planer is an expensive tool; quality models start at around $500, but there's no good substitute for its function, and there's a lot of demand for it. If you're able to find a used planer, we recommend changing out the blades with a new set to ensure you're making clean and true cuts.\n\nFinally if you only need to clean up small portions of smaller pieces, use a sander to take a rougher piece into a presentable state. There are two main types of sanders that we recommend for tiny house builders: a random orbital sander and a belt sander.\n\nRandom orbital sanders move in an inconsistent circular motion, which allows the sander to avoid making obvious patterns as you remove the material. Belt sanders have the advantage of removing a lot more material quickly but can create a pattern in the wood because they move in only one direction.\n\nIf you use a belt sander, finish with a random orbital sander, starting with a high-grit paper, then moving to finer-grit paper. The only downside to sanders is that they make it hard to produce a truly flat surface over a wide board because you may remove more material in one part of the wood than another. The variations may not be perceivable and if that is the case and you don't join that surface to another it will be fine. You'll run into an issue when you have to join two surfaces together; if those surfaces aren't perfectly flat, you'll be able to see gaps and the joint may not be as strong.\n\n## Creating Uniformity\n\nWhen working with reclaimed materials the variations in sizes of wood can present some unique challenges. In some cases the uneven edges or variations in the material can create a visual effect that is desirable\u2014a way to bring an organic feel to the construction or to add an element of whimsy. In other cases it is important to have crisp edges for joints, to ensure proper coverage or bonding at key structural points.\n\nWhen you create uniformity in your materials you often bring efficiencies into the equation. When the boards are all the same width, depth, and length you can more quickly use them in the build. This is particularly true whenever you are creating pieces in repetition (siding, flooring, etc.).\n\nWhen you frame your wall, all the wood must be the same size. If it already is, you can batch out the cuts quickly. When you plane your floorboards to the same thickness, you can quickly lay out and install them. If you decide to side your house with pallet wood, planing and jointing the boards to a common size will make the installation of the entire outside much easier.\n\nWhenever you need uniformity in your materials you inherently bring in efficiencies and in some cases, bring strength to the structure. Determining the uniform size of the material can be approached in one of two ways: determine what the ideal end size will be or take into consideration the most common dimension of the material you are reclaiming. If for example you are framing, it's best to use a 2 \u00d7 4 stud, and you would be smart to find lumber that size or larger and re-saw it to the appropriate measurement.\n\nCompare this to siding, which doesn't have to be a specific thickness as long as it's uniform. If you notice that most of the wood you are sourcing is a 1\u20442\" thick or thicker, it makes sense to plane all the wood to a 1\u20442\" thick. Think about the function and requirements of that material in the context of the whole house.\n\nWhen you are processing wood to a uniform state do your best to prevent human error. Use jigs to set your cuts, cut all the boards in one direction at the same time, and use guides such as a table saw fence to ensure your cuts are continuously accurate across all the boards.\n\nWhen using a planer, note that while the unit often has thickness demarcations, those are only so accurate. Ideally you'll set the planer to one setting, then run all the boards through at the same time. If you need to plane even thinner, reset the planer to a thinner setting, and run all the boards through again at one time. This allows you to maintain a uniform cut across all the boards. Keep doing this until you have achieved your desired thickness. Remove waste material put out by the planer as you go because if it builds up inside the planer you run the risk of it clogging and no longer making uniform cuts.\n\nSimilarly with jigs, guides, and your table saw fence, set them once and process everything at the same time. This will ensure that your cuts based off that guide will stay consistent.\n\nPhotograph by Aaron Maret\n\n## Storage\n\nThe number one thing we heard from all the people we talked to for this book was the importance of storage. Many of them recommended a large space with built-in racks to keep pieces out of the elements. The toughest part of using reclaimed materials is often finding them, and this usually means collecting all of them takes an extended length of time.\n\nIf you don't have a place to store materials, the items you find in the beginning may be damaged, rotted, or unusable by the time you find your last pieces. From my experience storage needs to be in a structure of some sort; tarps don't provide enough protection beyond a few days of mild weather.\n\nLonger-term storage should be in a garage, shed, or enclosed trailer, which will keep the wood out of the elements, particularly the rain. Many people noted in their interviews that they wish they had spent some time initially building racks to better sort and access the wood.\n\nWhen it comes to sorting wood, two approaches were popular. About half the people found sorting the wood by similar length was helpful because when they needed a board of a certain size, they generally knew where to look. Others preferred to stack their wood in the order in which they were going to use it. This allowed them to work from a single pile; more often than not, the top piece was what they needed.\n\nRegardless of your approach, one thing is clear: Having a large storage space is key, and unlike the house you're trying to build, bigger is generally better because it makes accessing easier and the entire process more efficient.\n\n## Common Types of Source Materials\n\nReclaimed materials often come from a few sources: pallets, flooring, and barn wood. Each of these comes with its own set of considerations and challenges.\n\n### Pallets\n\nWood from pallets is an extremely popular source of reclaimed wood. The shipping industry moves millions of pallets a year, and in many cases the receiver finds himself with a lot of leftover pallets that he would rather not pay to dispose of. This is an ideal situation for people who want to use reclaimed materials.\n\nThe best place to find pallets is at places that do a lot of shipping and receiving. Warehouses, shipping ports, manufacturers, distributors, and retail stores all use pallets in large quantities. Finding them can be as simple as driving through industrial or commercial sections of town. More often than not, you'll be able to find warehouses and other places with pallets stacked up by a dumpster near the back door.\n\nIt is important to get permission from the owner before you take the pallets. While it's usually a safe bet that pallets stacked near a dumpster aren't wanted any more, technically speaking they are still the property of the owner. Even taking things from the dumpster is a legal gray area, and people have been prosecuted for dumpster diving items.\n\nWhen using pallets you'll find that the wood you reclaim from them generally is of low quality, not something that you really want to rely on for your structural components. However, it can be used for aesthetic pieces.\n\nOne major word of warning: Pallets are often treated for bugs and biological contaminants with a variety of processes. The two most popular methods are heat and chemicals. Heat treatments bring the material up to a certain temperature that would kill most bugs. The use of chemicals to kill bugs is a common practice but raises concerns about off-gassing. If you use chemical-treated boards from pallets on the interior of your house, the off-gassing of those boards may result in health problems.\n\nThere are many opinions about how to tell heat-treated from chemical-treated wood, but in the end, after speaking to many people with real-world knowledge, I've concluded that there is no reliable method to determine how pallets were treated. I recommend instead to only use reclaimed pallet wood for outdoor applications or in limited indoor applications after an initial off-gassing period of at least six months. Even with these considerations there may still be some risk.\n\nWhen it comes to breaking down the pallets there seems to be two main schools of thought. The first says to use a pallet pry bar to pull the slats off the main frame. These pry bars are widely available online and make it much easier to break down pallets; numerous people have reported that they are worth the investment.\n\nThe second way people separate the slats from the main frame is by cutting. Some people use a reciprocating saw to cut through the nails while others use a circular saw to cut just inside the row of nails. This yields small boards, which are better suited for small woodworking projects or crafts.\n\nIn some cases people have gone so far as to use a pry bar to remove the slats, then run them through a planer to create uniform boards. This is particularly popular with tiny house construction to create siding and accent walls.\n\nRather than breaking the pallets down into boards, some people prefer to use the whole structure as it is. In some cases, builders use the pallets to create the wall structure. While this technique is not recommended, if you do consider this, you should use the pallets in conjunction with other main structural beams to ensure wall integrity. Of the people to whom we talked who used this method, most said they would not do it again; all of them indicated that the pallets were different thicknesses that created a difficult wall to work with. Many were concerned about the strength of the walls.\n\n### Flooring\n\nThe reclaiming and reuse of flooring is an obvious choice. Hardwood floors typically are made from high-quality lumber that has a clearer grain, is more durable, and generally can be easily reused. Add to this that the worn look of an old floor can bring a lot of character to a house. Using smaller pieces where the old nails at the ends have been cut off means flooring makes a lot of sense to reclaim.\n\nLuckily, reclaimed flooring is a well-recognized commodity in the building industry. There are many commercial operations that specialize in reclaimed floors, making your job of finding it easy. The scraps and leftovers from larger flooring jobs are often enough to floor an entire tiny house, but they wouldn't be useful in most other applications. This means you can often get the floor scraps for free or cheap, sometimes even brand new.\n\nIn older buildings the floors were often made from boards that are thicker than modern flooring. You can reclaim the old flooring, plane it down so it looks clean, reinstall it, and refinish it into a good-looking floor.\n\nIf you don't source from commercial reclamation operations you'll be able to find options on Craigslist, Freecycle, and by searching for \"old building renovations\" online. Because flooring is typically installed over subflooring, it isn't a structural piece, so you can focus on the aesthetic qualities, unlike other pieces that need to support major loads like studs or beams.\n\n### Barn Wood\n\nThere is something special about the wood you find in old barns. It is both beautiful and often of high quality. You will find larger pieces of lumber that were used to build the barn. This makes old barns a unique source of lumber because you can find longer pieces or find larger logs that you can saw into smaller pieces.\n\nA lot of the quality of old barns comes from their age. Barns typically come from an era when older-growth forests were harvested for wood. This means you'll find much larger logs with tighter grains in more exotic or costly species of woods. The patina of these woods is usually very rich, ranging from a deep shade of brown to the bright gray of exposed wood that has weathered years in the elements.\n\nBarn wood that has been exposed to direct sun and rain often is best suited for aesthetic pieces that don't carry much load; their color is often a gray weather look. As a rule of thumb, reclaimed wood that is colored in brown tones often has been shielded from the elements by a roof or was an interior member that was allowed to age and cure; lumber that has been allowed to cure like this often maintains its structural integrity and such a piece may be more suitable for structural parts.\n\nThe sourcing of barn wood and the quality of materials you get from both an aesthetic and structural viewpoint make it a good option for any recycled wood project. Several commercial operations exist, and barn wood can also be found through places such as Craigslist. When considering where to find wood, looking at old barns is a great place to start.\n\n# PART II  \nTALKING TO TINY HOUSE OWNERS\n\nTHE FOLLOWING SECTION CONTAINS THE STORIES OF TWENTY PEOPLE WHO HAVE BUILT THEIR OWN TINY HOUSES. EACH OF THESE HOUSES FEATURE RECLAIMED MATERIALS, SALVAGED ITEMS, AND OBJECTS THAT WERE RECYCLED, REPURPOSED, OR REUSED IN SOME MANNER.\n\nAPRIL ANSON, A PHD STUDENT IN OREGON, learned about tiny houses at an environmental conference she attended in 2012. A master's student at the time, looking toward six years of doctoral work, she knew instantly she wanted a small home of her own. A few short months later April moved into her tiny house, which allowed her to \"act on environmental ethics in a more everyday way.\"\n\nLiving in a tiny house was an adjustment for April. As she describes it, \"If I want to get into shape, I'll sign up for a race that forces me to train for it. In a tiny house, building a tiny house, forced me to take steps to live in it.\" Even today April finds that living small, while giving her a great place to live that allows her to exist on a student's budget, does have its downsides.\n\nPhotograph by April Anson\n\nLiving in a small home has \"shaped ways of living in ways I can't tell yet because I'm still in my tiny house,\" April explains. \"There are things that have been trained into me by living tiny.\" Instead of hosting others, she meets others outside of her home or goes to their houses instead. Rather than hosting dinner parties with friends and family, she more often goes out for dinner.\n\nIn the end her home was built of two-thirds reclaimed materials. She spent between six and eight hours a week during the build in what she called \"a treasure hunt every day.\" April learned that you cannot get attached to doing things in a certain way; the materials shaped how things came out.\n\nPhotograph by April Anson\n\nPhotographs by April Anson\n\nHer build included a lot of reclaimed materials, and not only because she needed to stay on budget. April loves the parts of her home that have stories. She sourced a lot of her materials from the local rebuilding center and ReStore, where she found her sheathing, light fixtures, and other parts of her house.\n\nShe chose to keep most of the wood natural and avoided paints. She likes the natural aesthetic of the raw materials, going so far as using smooth stones for handles. April treated the interior of the house by rubbing linseed oil on the wooden surfaces. Her outside shingles are cedar, and the floors were treated with a natural epoxy alternative to give them more durability. While she likes that the fact that these treatments are less toxic than others, April admits that you do need to reapply them once a year to maintain the coatings.\n\nPhotograph by April Anson\n\nOne of her favorite items in the house is the half-moon-shaped window, which came to her through a twist of fate. Initially she purchased another window for the spot, but soon realized it had some water damage. In a stroke of luck, April was able to find a replacement window that same day for only $30; it fit the rough opening perfectly. The window became one of her favorite features of her home.\n\nPhotograph by April Anson\n\nIN 2011 WENDY AND JAMIE BARRINGTON LOST THEIR 2,600-SQUARE-FOOT HOME to foreclosure due to the recession. For a family of four including their sixteen-year-old daughter and eleven-year-old son, you wouldn't think that a tiny home of 220 square feet would be enough, but for the Barringtons it was a dream.\n\nDuring the build the family had moved into a 1,600-square-foot apartment, then later a 1,000-square-foot apartment as they built their tiny home on nights and weekends. With each downsizing the family decluttered their home and minimized their possessions. They were tired of throwing away money on rent and wanted to get back to a place where they could pursue their dreams, moving to Colorado. The tiny house is the key to that.\n\nPhotograph by Wendy Barrington\n\nOver eight months the family built their tiny home, mainly led by Jamie, who has been a carpenter for twenty-five years. Because he had the skill and the tools, Jamie was able to do all the work himself except for the welding of the trailer, which they hired a welder to do.\n\nThe trailer was an old one that Jamie had from many years before. It needed to be reinforced and new components added in order to get it ready for the tiny house build. For the house, much of the wood came from two main sources: the leftover scraps of a large home Wendy's brother had built, and wood reclaimed from jobs Jamie worked on when he did remodels.\n\nPhotograph by Wendy Barrington\n\nThe flooring of their tiny home was built from the leftovers of flooring from the large home of Wendy's brother. During their build, Jamie had remodeled a nearly brand-new condo. Even though it was only a few years old, the owners wanted to gut the whole kitchen. That night Jamie came home with a flawless black marble countertop, which made its way into the tiny house for their kitchen. They reclaimed insulation from job sites and reused it in the house. However, they bought most of the framing and sheathing new.\n\nPhotographs by Wendy Barrington\n\nIn addition to job sites on which Jamie worked, the couple spent hours scouring Craigslist to find lumber and other objects that would go into their home. At the Habitat for Humanity Restore they were able to find much of the electrical components, bathroom lighting, and all six of their windows, which were bought early on and informed their design. They found some old wood fence panels on the side of the road, which they cleaned up and turned into their interior wall paneling.\n\nIn their searching, the Barringtons also relied heavily on word of mouth and talked to lots of people beyond just their friends and family. They shared what they were looking for on Facebook, and soon their work paid off.\n\nWendy commented that you have to be very persistent when it comes to sites such as Craigslist. Things get posted on the site every day; be ready to hop into the car to get it before someone else can. While shopping in ReStore or similar stores, don't be scared to really dig through bins. The objects there are often dirty, but look for the potential behind the grime. For construction sites, make sure you ask permission before taking anything. Many contractors will freely give you scraps or leftovers rather than pay to dispose of them.\n\nOnce you get the reclaimed materials, Wendy suggests, realize that there is going to be a lot of labor to clean them up. Set up an assembly line and get friends and family involved. Many hands make the work go faster. In general the process is: disassemble the items into their base parts, choose the good parts, remove (cut off) any bits that aren't useable, remove nails and screws, use a wire brush to clean up the materials, and then store them for later use.\n\nPhotograph by Wendy Barrington\n\nAfter months of scrounging materials and building their tiny house, the Barringtons moved in early 2014 and have been living there ever since. As a member of a family in only 230 square feet, Wendy shares that \"it's daunting sometimes, but I've learned it's a state of mind. You have problems in a big house just as much as in a small one. It's just your perspective and how you deal with it.\"\n\nWhile it's tough at times, Wendy and Jamie love their tiny home and how their family gets along in it. The one thing Jamie and Wendy want is for their family to be close. In a tiny house \"it comes true, because you can't help but be close. It's a wonderful thing despite all the test of the will. To do this with your family is the greatest thing.\"\n\nPhotographs by Wendy Barrington\n\nTHE FIRST TIME STACEY KNEW THAT SOMETHING HAD TO CHANGE WAS WHEN she found herself shopping on weekends for the sake of filling her large house. She realized that she \"wanted to shift, to live more simply.\" \"I'm a single person living in a three-bedroom, two-bathroom house\u2014why do I need all this space?\" she asked herself.\n\nPhotograph by rawrestorations.com\n\nSomething about Airstreams has always appealed to her. She looked forward to the day when she could travel the roads with her trailer in tow, seeking adventures. In the immediate period, though, the small space of the trailer would also provide a nice answer to the problem of living in a home that no longer met her needs.\n\nWhen Stacey first started her journey to convert her Airstream trailer into a place that she would soon call home, her friends and family became worried. As she began to tell her friends what she was doing their response was, \"Stace, call me any time if you need to talk.\" Her stepmother was almost tearful, asking, \"Why are you moving into a trailer? Why are you doing this?\"\n\nIt left Stacey unsure of her path. \"I honestly thought I was going to have a panic attack,\" she says. But she moved forward, knowing this was going to challenge her. If nothing else, she would see if she could live in the Airstream for the summer. After the first week of living in her new home things just seemed right. She began to wonder, \"How could I ever live in a big house again?\"\n\nPhotographs by rawrestorations.com\n\nPhotographs by rawrestorations.com\n\nStacey's dad and stepmother made a last-ditch effort to persuade her not to do this. They were so concerned over her new Airstream-trailer life that they called her sister, Carrie, who is Stacey's \"voice of reason.\" They told Carrie, \"You've got to get a hold of this! You've got to talk her out of this!\" Her sister came over and \"started to freak out. She said this is how you're meant to live your whole life!\" It was then Stacey began to realize how perfect it really was for her.\n\nOver time Stacey began to invite her skeptics over for dinners and to hang out. She would invite them in, and in short order they changed their minds. They went from being \"really worried for me to, oh my gosh, I totally get it!\"\n\nWhen Stacey first bought the used 1992 trailer it felt very \"sterile,\" and she instantly knew that she wanted to bring in elements of warmth to the space. She chose barn wood to ground the space, adding a cabin feel to it so it could start to feel like a place she could call home.\n\nPhotograph by rawrestorations.com\n\nShe worked with most of the existing elements of the original Airstream trailer but added accent pieces to the surfaces. Some parts she replaced altogether. The countertops of the kitchenette were replaced with bamboo plywood. She had contractors install the barn wood as flooring and accent walls, and she worked to rehab parts of the trailer that needed some attention including new upholstery and the addition of a fold-out tabletop for meals.\n\nAll of her reclaimed wood came from a friend's company called R.A.W. Restorations (rawrestorations.com); it was reclaimed from an old barn in Utah. What resonated deeply with Stacey was the company's goal to preserve and repurpose parts of history. She liked the idea that the story of past things would be carried on in her trailer, creating a new chapter for them.\n\nPhotograph by rawrestorations.com\n\nAFTER RETURNING FROM VOLUNTEERING ABROAD, Makayla Cavanagh found herself looking for direction. She was trying to figure out what was next when she landed on building a tiny house. The mobility of a tiny house meant that she could move and take her home with her, and the size meant she didn't have to work the rest of her life to pay a mortgage.\n\nTo Makayla building a house was unlike anything she had ever done before, but in her mind it was something that she just had to figure out, not something she couldn't do. She remarked that \"you can't be afraid to make mistakes,\" and for her, knowing that she had the support of her friends and family helped.\n\nPhotograph by Jessica Humiston\n\nShe comes from a family of craftspeople: Her mother is a machinist, and her father works in construction. For her house she was able to get about half her framing lumber from a job site on which her father worked. The lumber was going to be thrown out as scrap. Makayla encourages people to consider looking to construction sites for scrap wood, saying, \"It's really disappointing what construction companies will throw away, but it was great for me.\"\n\nPhotograph by Jessica Humiston\n\nHer flooring came from an old building in downtown Spokane, WA. In all, it took her three days of cleaning and resurfacing to get the wood ready for installation. In the end it was worth it; the floors are her favorite part of her house.\n\nSome of her siding and all of her roofing came from a decommissioned military building that was being torn down. She had seen an ad on Craigslist and picked up enough wood to cover her roof and provide accent pieces for the siding. The rest of her siding came from a local reclaimed store. Despite it already being de-nailed, it still took a lot of work to get ready because the boards were badly warped.\n\nPhotograph by Jessica Humiston\n\nHer kitchen and much of her storage was built from a mixture of furniture pieces from the ReStore and scraps she found online. One of her countertops was an old wood door that she spent two days resurfacing. In the end she had a beautiful kitchen nestled underneath her loft.\n\nPhotograph by Jessica Humiston\n\nThere were a few items that she wanted to purchase new. To her it was important to have a solid foundation and invest dollars where it mattered. With this in mind she purchased a brand-new trailer on which to build her house. All of her electrical and plumbing fixtures were new so she didn't have to worry about dangers resulting from used electrical components or leaky plumbing parts.\n\nNow that her build is done, Makayla is very happy living in her tiny house. In total the house took less than a year and $13,000 to build. Throughout the process Makayla learned a lot about herself and has inspired others to build tiny houses through her example. It was a lot of work, but in the end she is glad to be living tiny.\n\nPhotograph by Jessica Humiston\n\nKATE FOX AND ANDY GILL (KATEFOXMICROHOLDING.CO.UK) ARE NO STRANGERS TO LIVING IN SMALL SPACES. They traveled all over Britain in vans and campers, tending to farms, until they decided to settle down and buy a home. But the couple soon realized that owning a traditional home didn't reflect the life they wanted to live.\n\nPhotograph by Kate Fox\n\n\"We asked ourselves, why are we living in a box in a town, doing the things we're told we should do like go to work and pay bills, when what we actually did was go off in our camper van and use our house as a great big cupboard for keeping our stuff,\" Kate says. They sold the house and decided to design and build their own tiny home on wheels.\n\nThe couple took their inspiration from old railway cars and shepherd's huts, wagons used by sheepherders in the United Kingdom to follow their flocks. Then, they went to a camper showroom to step inside the small spaces and get a sense of how to lay out their new home. They took those ideas and incorporated them into their own floor plan. By keeping the design simple, Kate and Andy enjoyed a quick building process. It took them two months to get the structure to a point where they could live in it, so they could take their time working on the final details.\n\nPhotograph by Kate Fox\n\nKate has experience with tools, and Andy is a builder by trade, so between them they could tackle the project. Not that it was always easy\u2014making the curved roof was a challenge. They made their own beams using reclaimed wood from an old stable. They had to de-nail it, saw it all down into strips, and glue it all back together again in a curved shape. They spent six weeks on the rafters alone.\n\nKate and Andy wanted their house to be as sustainable as possible. To do this, they used as much free and salvaged lumber as they could, and took the time to clean it up. A friend gave them some Victorian-era floorboards, and Kate worked for four days with a drill and a wire brush attachment to get them looking like new again. Andy found some old wood from a job site where he was helping to build a barn, and they used those timbers for the studs in the walls. The kitchen is made of old wine crates from all over Europe, which adds a touch of whimsy to the hut. This is one reason why Kate loves working with reclaimed materials. \"There are so many materials around and they all have their own stories to tell.\"\n\nPhotograph by Kate Fox\n\nKate and Andy move their hut every six weeks so they can legally live in it full time, according to UK laws. Since they're always on the move, they chose not to include a toilet or shower inside the hut. There are always facilities wherever they park, whether it's a campsite or a farm. They also have a homemade shower and composting toilet they can set up to use outdoors if they're situated closer to the wilderness.\n\nPhotograph by Kate Fox\n\nThe couple loves their new home that allows them to move from place to place. They encourage people who are excited about small living to take the leap and just begin, because your energy is really what makes it fun and possible. \"If you're enthusiastic and excited, you'll find a way, and everybody's way will be different.\"\n\nPhotograph by Kate Fox\n\nJAMES GALLETLY WAS VISITING A FRIEND IN CALIFORNIA WHEN HE FIRST PICKED UP A BOOK by tiny house publishing legend Lloyd Kahn. \"The biggest part of the appeal is the DIY mentality,\" he says. \"People are building their own homes and coming up with designs that are far more original and creative and have more personality than what is normally considered a home.\" He took the tiny house idea with him back to his native Australia.\n\nPhotograph by Alicia Fox Photography, _www.aliciafoxphotography.com_\n\nJames had plenty of experience with straw bale and other natural building techniques, but he had never built a tiny house before. He approached a local rebuilding center and asked them to sponsor the tiny house as a demonstration project. They gave him first pick of the reclaimed materials in stock along with space to work right outside the building. They auctioned off the tiny house once it was finished. The goal of the project was to raise awareness of how recycled materials can be used to reduce building waste and create a beautiful living space.\n\nThe design of the tiny caravan depended on the materials at hand. \"I enjoyed the creative process. I enjoy the interaction between letting the materials dictate the outcome and working with what you're given,\" he says. James visited other salvage yards and went dumpster diving to supplement the things he found at the rebuilding center. He spent a month collecting materials before he started building and kept looking for things throughout the rest of the build.\n\nPhotograph by Alicia Fox Photography, _www.aliciafoxphotography.com_\n\nHe also reached out to the community in an unconventional way. Next to the tiny house, he had a blackboard where he could write down the things he needed for the project. People from the neighborhood would come by and drop off their extra materials for him to use. One day, James needed insulation and wrote it on the blackboard and shared a photo on social media. He came back the next morning to find piles of insulation left for him inside the half-finished building.\n\nJames found a stack of 200 flooring samples at the rebuilding center. He dug through to pick out his favorites, which are used as the interior paneling. A fold-down table and some corner shelves are made from Tasmanian oak cabinet doors. A patchwork of colorful sheets of corrugated metal, reminiscent of a quilt, make up the exterior cladding. He laid all the sheets on the ground, decided which colors looked nice together, and carefully arranged them on the wall for the most pleasing composition.\n\nThe French doors were a favorite find. All he had to do was sand them down and finish them with tung oil to bring them back to life. He had trouble finding the perfect door handles to complete the look and was about to go out and buy something new. Minutes later, someone came into the rebuilding center to donate the perfect hardware, and James snapped it up!\n\nThe few materials bought new included the house wrap, the electrical wiring, and the nails and screws. James estimates that 95 percent of the tiny house is built with reclaimed materials. He advises builders looking to use secondhand materials to keep an open mind and creatively use the things you find along the way. \"It's the most enjoyable part about secondhand materials. It's so much easier to work with the textures and the palette you're given.\"\n\nPhotographs by Alicia Fox Photography, _www.aliciafoxphotography.com_\n\nTHE HALL FAMILY HAD BEEN THINKING OF BUYING THE BOSTON-AREA TOWNHOUSE THEY WERE RENTING, but the real-estate prices were sky-high. They knew of tiny houses and loved the idea of living small with just the things they needed, so they sold their stuff, settled on a design, and dove right into the building process.\n\nPhotographs by A. Hall\n\nThey didn't have much building experience, but with a lot of research and the help of friends, they were able to build the house in six months. It was important to them that they built the house as a family, with their teenage son taking part in the build. The Internet made it easy to find solutions to problems as they arose. \"There were very few things we couldn't look up,\" they said.\n\nPhotographs by A. Hall\n\nPhotograph by A. Hall\n\nPhotographs by A. Hall\n\nThe Hall family wanted their house to be as sustainable and nontoxic as possible. They used as many reclaimed, handmade, unprocessed, and locally sourced materials as they could. \"The whole point of the tiny house was to align our political and social beliefs with how we were actually living,\" they said. Their front door is a reclaimed 100-year-old Mission-style door, which took a lot of time and money to restore\u2014but it's made of solid wood, and there's no off-gassing. Other things were free; the drawer pulls in their kitchen came from a collection of beach rocks.\n\nPhotographs by A. Hall\n\nThe family spent hundreds of hours digging through salvage yards to find diamonds in the rough. They found three handmade brass hinges with acorn details for $40, a fraction of the cost to buy them new. But they were grimy and covered in paint, and it took weeks to clean them. The stained-glass window in the bathroom door was another salvage yard find. It was in great shape, but it was stuck in a moldy wooden frame. Getting the frame off the window without breaking the glass was a challenge!\n\nPhotographs by A. Hall\n\nFor the Halls, putting in the work to get the best-quality materials was worth it, but sometimes it was difficult to find items that they needed. At one point, they bought a porch light from Home Depot because they couldn't find a light they liked at an affordable price. Then they found one that was even better on Craigslist, snatched it up, and quickly returned the original light to the store. They also found a brass bar stool footrest in a salvage bin and dug through every hardware barrel in the store until they found three more. The footrests now hold the bed in place in the loft.\n\nThe Halls were deliberate in their approach to getting rid of their excess clutter. They picked their favorite books to keep and digitized many of the rest before donating them. All of their photos, movies, and music are now stored on two hard drives. They hosted \"Craigslist days\" where they invited people to come through their house and buy the things they wanted. Each member of the family got a memory box, and whatever mementos they wanted to keep had to fit in their box. Everything else was donated or given away. They advise that you take a lot of time\u2014six months to a year\u2014to get rid of your unneeded things. \"The experience of evaluating every single object in your life really forces you to deal with what's actually important in your life.\"\n\nAfter months of hard work the house was done, and as they moved into the home the family agreed that if anyone was unhappy with living in the tiny house, they would find a different place to live and the house could be a vacation home. The house was an experiment for the family, but they were excited to live in it together. A year later everyone has loved living in the home the entire family built.\n\nPhotographs by A. Hall\n\nSARAH HAD ALWAYS DREAMED OF BUILDING HER OWN HOUSE. She discovered tiny houses online during her freshman year of college while researching sustainable homes. Her own tiny house would later become her senior year thesis project to complete her architecture degree. In July 2014 Sarah bought a twenty-seven-foot-long gooseneck trailer and dove right into the build. Working mostly on weekends with friends and family to help, she finished the house a year later.\n\nSarah built only with materials found within a 200-mile radius of her building location. Even the structural insulated panels (SIPs) she used to construct the walls were manufactured in New Hampshire. Sarah chose the SIPs because of their strength, ease of use, and high insulation value. She decided to keep the electrical components on the surface of the interior walls in case she ever needed to fix them. The exposed conduit lends the house an industrial look.\n\nMany of the fittings and finishes in Sarah's house are reclaimed. All of her windows are used\u2014two of them had a previous life in a library. The siding on the gable ends of her house came from a recent renovation of the Springfield, MA, fire station, and the siding on the long sides of the house she found at a junkyard. Her front door is a midcentury modern style from the 1960s.\n\nPhotograph by Sarah G. Hastings\n\nSarah's new bamboo flooring was left over from a friend's project, and it was just enough to cover the whole floor. Even her kitchen sink is reclaimed\u2014it came from a dentist's office.\n\nFreestanding furniture makes Sarah's house feel more like a home. Her Hoosier cabinet was a Craigslist find purchased from a woman with a house full of antiques. The piece is the first thing you see when you walk in the door. \"They're on Craigslist all the time, because they were made in high quantities in the 1920s, 1930s, and 1940s,\" Sarah says. The Hoosier cabinet is a great solution for Sarah's kitchen, because it has lots of storage and even features pull-out counter space.\n\nThe timber trusses that support the roof are the centerpiece of the house. They're crafted from 100-year-old reclaimed wood salvaged from a factory that was part of the oldest furniture-making company in the United States. Tom Musco at Royalston Oak Custom Timber Frames donated a lot of his time to help her and her father build the frames. The timbers add a lot of color and texture to the home, and they still feature the original nail holes. They add a unique character that you can't find in many tiny houses.\n\nSarah suggests assembling a support system of more experienced builders before you start your tiny house project. \"Don't overly romanticize the building process,\" she says. \"It's mentally and physically exhausting. But it's totally possible.\" She also advises prospective tiny house builders to treat all challenges and obstacles as puzzles to be solved rather than roadblocks.\n\nPhotograph by Sarah G. Hastings\n\nShe hopes that other tiny house builders will build their houses to reflect their own local culture and the places where they build. She also thinks that the merging of old and new\u2014historical inspiration and modern technology\u2014is what can make a tiny house truly special.\n\nPhotograph by Sarah G. Hastings\n\nZEE KESLER LIVES IN VANCOUVER, CANADA, A CITY THAT RECENTLY ACHIEVED the doubtful distinction of having the highest cost of living in North America. It is also in the top twenty for the most expensive places to live in the world. With the city continuing to grow, tiny houses are one solution to urban infill that could help bring some affordable options to the city.\n\nWhile most tiny houses are designed to live in, Zee saw the potential for other uses. While the house could be used as a home with a few modifications, it initially started out as a mobile classroom and mini art gallery.\n\nPhotograph by Erin Leigh Pasternak, _www.erinpasternak.com_\n\nWith the house now complete Zee admits that while it's a great focal point of learning and art, it \"brings up conversations of affordable housing without it being the mission.\" Zee says that in her own way she is able to spark conversations about the rising cost of living, explaining \"that the artist's job is not to directly address issues but divert the conversation.\"\n\nStarting out Zee didn't really know too much about building, but she was very good at enlisting help and support. She spent the first year of her journey collecting materials, often spending twenty to thirty hours a week at it. Every time she found a lead on some materials that could be reclaimed she had to borrow or rent a car because she didn't own one.\n\nPhotograph by Erin Leigh Pasternak, _www.erinpasternak.com_\n\nShe did have a place to keep all the collected items; her landlord didn't use the garage so she was able store all the materials that later would become a house there. From the beginning Zee made many key choices about her systems: heating, cooling, water, power, and appliances.\n\nOnce she had her systems selected she built a design from the outside in. Because the tiny house needed to be mobile, there were requirements for how wide it could be. This led her to design the house starting at the outermost edge and working toward the center of the home to make sure she fit with the necessary parameters.\n\nWith her design finalized Zee and her designer were able to make a list of what lumber they would need, from which she developed her \"cut list.\" This was a list of each board they'd need and how long it should be. This list proved to be very valuable when evaluating materials; if an item could be used for something on their list, then they'd take it; otherwise they'd decline.\n\nPhotograph by Erin Leigh Pasternak, _www.erinpasternak.com_\n\nSaying no was a big lesson Zee learned early on. There are times when people want you to take all their materials. Sometimes, since you drove out to check what people have, they'll guilt you into saying yes. The downside of taking things you don't need is that you'll later have to give them away or worse, pay to have them disposed of.\n\nZee quickly learned this lesson and began to collect only what she needed. Each time she found something on her cut list, she would de-nail it, cut it to rough dimensions, and stack it on her storage racks.\n\nThe storage racks were something Zee wishes she had done sooner. At first they tried piling things neatly but soon it got unwieldy. Once she erected the racks, she piled her pieces of wood in reverse order of when she would need them. The result was that as they built, the board on top was most often the next board she needed for that part of the house.\n\nPhotograph by Erin Leigh Pasternak, _www.erinpasternak.com_\n\nWith all the materials in place the house took about six months to build. Her team of volunteers and paid contractors came together around the inspiring idea that Zee had envisioned. Her team\u2014John McFarlane (designer), Josh Armstrong (carpenter), Dave Myers (carpenter), Nathyn Sanche (welder), David Heck (volunteer builder), Shauna Langfield (volunteer builder), Ben Garratt (building and assistance with electrical), Tom Ferguson (volunteer builder), and Ruth Munro (artist)\u2014all worked to build the house.\n\nIn all, the house cost $25,000 Canadian to build. It now serves as a public art piece, classroom, and gallery all in one tiny space. It's currently parked at the Trout Lake Community Centre as part of their Artists in Communities program. The build would have never happened without the community rallying around the idea of the tiny house. Zee says, \"When people see something interesting, they just want to be part of it\u2014that's art.\"\n\nPhotograph by Erin Leigh Pasternak, _www.erinpasternak.com_\n\nAARON MARET STARTED HIS CAREER AS AN ARCHITECT, DESIGNING BUILDINGS THAT WOULD LATER be built far away, and he often never saw them once they were completed. Over the years he was drawn to the construction of homes, not just their design. In the end he came across tiny houses when he was building homes. What drew him to tiny houses was the size. In his words a tiny house is \"the sweet spot in terms of scale\" when it comes to building because he can touch every part of the build. In most of his professional projects he often had help or in some cases work was subcontracted out, but with his tiny house he was able to enjoy the entire process of the build and add his craftsmanship to the project.\n\nPhotograph by Aaron Maret\n\nIn particular, Aaron loved the fact that due to the small size of the project, it was possible to use reclaimed materials. This wasn't true of larger projects because of the quantity of materials needed. Aaron says, \"The whole process\u2014the research, the design, and the actual construction of it\u2014was one of my favorite things I've ever done.\"\n\nPhotograph by Aaron Maret\n\nAaron had a large shop, which made the processing and storage of reclaimed materials much easier. As a builder, he was \"always looking for materials\" for his projects. He admits that while using reclaimed materials often results in superior finished pieces, it's a lot of work.\n\nFor his house's siding he used wood off an old barn, which he was able to purchase in a raw state from a local reclaimed store. He then spent well over 100 hours de-nailing, jointing, and planing so the boards could later be installed on the outside of the house. As with all the reclaimed materials he used in his house, this process took time, often more than he had planned for. \"Take your estimates and times it by three\" is Aaron's mantra when it comes to the time investment for reclaimed materials.\n\nPhotograph by Aaron Maret\n\nDespite all the labor, Aaron says, the attraction of working with older reclaimed wood \"is the durability, the quality, and at its heart: honesty. It has so much character, the original saw marks from the milling, and it tells a story that adds a richness. It adds meaning, but it also has warmth.\"\n\nPhotograph by Aaron Maret\n\nPhotograph by Aaron Maret\n\nIn his home about a third of the framing was reclaimed. Most of the sheathing was salvaged from an art studio and had paint drips and splashes all over it. He thought about keeping that as the siding because it made the house look neat. His door was from a 1920s farmhouse, and a few of his windows came from salvage as well.\n\nThere are a few things that he feels shouldn't be reclaimed. \"Invest where it matters,\" Aaron advises. All of his operable windows were brand new, his roof and insulation were bought at the store, and he insisted that all wiring and plumbing should be new.\n\nPhotograph by Aaron Maret\n\nLiving in the house with his partner and their child was tough at first because they didn't always have the space to be alone when they needed to be. He says that as a family \"the process of reducing was iterative. It took session after session and took a remarkable amount of energy to fit into a tiny house.\" But in the end Aaron's family settled into the tiny house very comfortably, finding nooks where each person could have his space in the small home.\n\nPhotograph by Aaron Maret\n\nMARK MILLER HAS BEEN COLLECTING AND USING RECLAIMED MATERIALS ALL HIS LIFE. In his words, \"Taking things that most people call junk and turning them into something neat looking is something I enjoy doing.\" As a contractor he is an experienced builder who loves to work the materials he finds into his build.\n\nLooking toward retirement Mark knew that he would need to be creative, so he started looking for other sources of income. He decided to build a small cabin to move into so he could rent out his larger home, a home that was now too big since his children had grown.\n\nThrough his entire build on the cabin he had to be very creative when it came to using what he found. He says, \"You really have to be able to see things for what they will be eventually.\" Much of the wood came from scraps he found while he was driving around the area. He'd stop and put them in his truck and then store them for later. More often than not he'd collect things without knowing what he'd use them for.\n\nThe flooring of his cabin came from an old office space. The floors had been installed in 1895. The fireplace was from a dumpster. Much of the dimensional lumber came from trees that had fallen over in his grandmother's yard. He cut them to length and milled them into boards.\n\nPhotograph by Mark Miller\n\nMark shared that he is always on the lookout for materials to use in various projects. He is fortunate to have a large outbuilding in which he can store everything until a need arises. When it comes to working on a project, Mark can look through what he has on hand and most often can find what he needs.\n\nWhenever Mark finds wood, he always asks permission to take it. He never assumes that it's okay to take scrap pieces even if they're in the trash. When he gets permission, Mark says, \"I always make sure to leave the site neater than it was when I got there.\" This good will has landed Mark a lot of great finds, some of which made it into his cabin.\n\nPhotographs by Mark Miller\n\nWhenever Mark finds wood for free, he takes it home and immediately de-nails. That way the wood can be stored flat. He told me, \"If you leave nails in the board, each board doesn't seat neatly into a pile.\"\n\n\"It's important,\" says Mark, \"to have a good storage space out of the elements.\" There had been times that he stored wood reclaimed from old buildings and barns under tarps, but he found that even with a good tarp, moisture proves to be a challenge. A solid structure where you can keep materials off the ground is important to prevent lumber from rotting.\n\nToday his cabin sits next to a stream on Mark's property. He loves the small space and the view from the large reclaimed windows overlooking the water. What started out as odds and ends collected over the years came together to build his little cabin.\n\nPhotograph by Mark Miller\n\nDOMINIQUE MOODY HAS BEEN AN ASSEMBLAGE ARTIST FOR MANY YEARS, making art with found materials. The Nomad is both her home and latest masterpiece. Since the eighties, she had been planning to build a house truck. However, in 2002, she discovered an article about Jay Shafer and his tiny house. After contacting him and learning more, Dominique knew that a tiny house was the perfect medium through which to tell her story.\n\nShe is no stranger to the nomadic lifestyle. Her father was in the military, and Dominique was born in Germany. Her family moved around constantly during her childhood, living in a forty-foot New Moon trailer when they returned to the United States. Her current location in Los Angeles is one of forty-five different addresses she's had throughout her life. It turns out she has wanderlust in her blood\u2014her family line can be traced back to an ancient African nomadic tribe, the largest on the planet.\n\nDominique's tiny house is truly a labor of love. The house took four years to design, and an additional three years to construct. She is legally blind, and built the house with the help of friends, family, volunteers, and hired professionals.\n\nPhotograph by Khari Scott and Dominique Moody\n\nBecause she is partially sighted, she focused on filling the house with round, organic shapes and the textures of wood and metal. The corrugated steel siding of her house isn't reclaimed, but it's striking. Dominique rusted the metal with a mixture of water, vinegar, and salt, and then colored the steel with metal stain.\n\nDominique knew she wanted round windows in her tiny house, but specialty windows are expensive. One day at the laundromat, she realized the round door on the washing machine looked just like the kind of window she wanted! She found a company that refurbishes old commercial laundry machines, and she was able to get six windows for the price of one round window bought new.\n\nReclaimed wood adds a variety of stories to the tiny house. Clear heart redwood from 1,500-year-old trees came from a demolished bridge in Bakersfield, CA, and Dominique had used it for an art installation she made for a hospital. She used the leftover redwood as the baseboard, trim for her skylight, and shower surround. Found tree branches work beautifully as hand rails on the porch. The porch itself is clad with 100-year-old barn wood, and the wooden threshold of an old stable, worn down in the center by horse hooves, serves as the lintel above the door. One woman even gifted Dominique ten boxes of cedar paneling, which she used as lining for her kitchen drawers and as the flooring in her storage loft.\n\nPhotographs by Khari Scott and Dominique Moody\n\nSince the house is an art piece reflecting her family history, Dominique made sure that family items were given a place of honor. Her father carried a globe with him wherever he traveled, and it now hangs above the entry in a half-spherical window. His apple crates that once held books became the drawers for her kitchen cabinets. A simple galvanized tub of his is now her bathroom sink, and it is the first thing you see when you walk in the door.\n\nDominique Moody's tiny house is a living, breathing work of art many years in the making. Not only does it represent the past and present of her story, but it also helps strike up conversations about what it means to create a meaningful home in this day and age. \"A place is one thing\u2014a home is a complete other kind of environment,\" Dominique says. \"It's an art form to create our shelter in reflection of who we are.\"\n\nPhotograph by Khari Scott and Dominique Moody\n\nAFTER MOVING TO DURANGO, CO, IN LATE 2011 GREG PARHAM WAS LOOKING FOR A PLACE TO CALL HOME. He began looking at small lots on which to build a cabin. While looking for land his real-estate agent showed him an article about tiny houses, and he knew he had found the answer to what he was looking for.\n\nPhotographs by Rocky Mountain Tiny Houses\n\nThe house took him two months working on it part time. During the weekdays he was a contractor working on a construction job. The tiny house he built is the one he lives in today, but he has since gone on to launch a building company called Rocky Mountain Tiny Houses.\n\nBuilding with reclaimed materials has been a challenge for Greg. He notes that \"there just aren't a lot of people around to get stuff from.\" The small town of Durango has 17,000 people, which makes for slim pickings when it comes to reclaimed materials. As a result it took a long time to gather materials to build the house. Over time he was able to gather much of what he needed, but up to that point, it proved challenging to keep piles from being an eyesore.\n\nPhotograph by Rocky Mountain Tiny Houses\n\nWorking with reclaimed materials can be much more difficult despite the amazing finished products. To get wood to a usable state, Greg says, \"you have to pull nails, you have to scrape off paint, and maybe they're warped or twisted so you have to fix that.\" It really comes down to a limited budget because \"if you're building your own house on your own time, yes, it will be cheaper. But if you have to pay somebody, in the end it's probably going to be more.\"\n\nPhotograph by Rocky Mountain Tiny Houses\n\nFor Greg, using reclaimed materials is driven by a mixture of cost and aesthetic. He loves the look and richness reclaimed lumber brings. While sometimes building as a business dictates using new wood purchased from the store, he loves it when he can bring reclaimed elements into his build. Newer lumber, he thinks, doesn't stack up when compared to reclaimed materials. The end product with those materials always turns out better.\n\nPhotographs by Rocky Mountain Tiny Houses\n\nEach of the houses he has built has been different; each of them used a variety of reclaimed materials in it. For the house he built for himself, he got the windows from a renovation in which he removed nearly brand-new windows that were being ripped out and replaced. He often uses old corrugated metal roofing panels for siding because of its durability and aesthetic.\n\nIn the end Greg has built several tiny homes using reclaimed materials, each one as unique as the parts that go into it. The homes are beautiful, with a mixture of new, used, and upcycled parts. They come together to create warm and inviting spaces.\n\nPhotograph by Rocky Mountain Tiny Houses\n\nHAVING ONCE OWNED A HOME, NATALIE POLLARD (FORVILLAGERS.COM) KNEW THAT WHEN she moved to Asheville, NC, buying a house didn't appeal to her at all. \"The weight and responsibility of owning a home,\" she says, was too much. Renting, however, had left her searching for a better option. Finally she spoke with a friend who asked, \"When were you the happiest? What living situation left you feeling the best?\" Natalie spent some time reflecting seriously on the question and came to realize \"the point I was really happy in my life was when I lived in really tiny spaces. I made the correlation that less space meant more money, more time, and more freedom.\" That realization solidified her plans, giving her a direction to move forward. She would live tiny.\n\nShe spent time planning her move to a new home, initially thinking about building a yurt. One day, though, she found herself talking with some friends who happened to be builders. Their company is called Nanostead, and they have built small homes for years. Now, though, they wanted to build a tiny house. Natalie offered to be their first client; the perfect option seemed \"to just fall into my lap,\" Natalie says.\n\nNatalie was excited to see her home take shape. The fact that it was built by friends made it feel as if \"it had a special quality to it, like a handmade gift.\" Because the home was being built for her, she had to make a number of decisions when it came to the finishing and design elements.\n\nPhotograph by Tamara Gavin\n\nPhotograph by Max Cooper Photography\n\nThe house used a lot of reclaimed elements in it, and where the components weren't reclaimed, the builders opted for locally sourced materials. The house was framed, sheathed, and sided with lumber milled by a local family-owned lumber mill. Several accent walls are sided with reclaimed materials from an old cabin built during the Civil War; the cabin had been deconstructed and milled into planks.\n\nNatalie's flooring was from a local salvage store, which captures lumber from construction sites and resells it so it doesn't enter the waste stream. Her counters, storage, and cabinets were all bought at thrift and antique stores and then combined into the various components of her kitchen and storage spaces.\n\nThroughout her house she loved using reclaimed and repurposed materials because many of them had a closeness to her. Some things came from friends and family, adding a sentimental dimension to the elements of her house. Natalie says for those things that came from people she knew she likes \"the embodied energy from the materials that comes from time.\" Some items came from her own store, Villagers, in Asheville, NC ( _http:\/\/forvillagers.com_).\n\nThe final house was exactly what Natalie was looking for. Moving into her home was an adjustment, but she notes that \"a well-designed space made it easy.\" Her new life meant that she was \"afforded so much freedom and had so little stress\" that she couldn't imagine living any other way.\n\nPhotograph by Tamara Gavin\n\nHUMMINGBIRD MICRO HOMES IS A TINY HOUSE COMPANY IN FERNIE, BRITISH COLUMBIA, founded in the spring of 2014. Gareth Roberts is one of the cofounders and head builders at the company. Gareth had built a tiny home for his sister-in-law a year prior, and this experiment in building and design turned into a thriving business. Hummingbird is a two-man building team that focuses on building tiny houses with creativity and thoughtful intention.\n\nPhotograph by Hummingbird Micro Homes\n\nPhotograph by Hummingbird Micro Homes\n\nPhotograph by Hummingbird Micro Homes\n\nGareth used to work in conventional construction, but now he likes to follow what he calls \"the emerging process\" of building. He starts with the blank slate of a tiny house frame, then designs the house as he finds secondhand materials. The form, size, and shape of the wood he uses often dictates and guides the final design. It's a unique perspective for a tiny house building company. Gareth says it's time consuming but economical to find materials this way. \"Everything's already out there. Somebody's got a pile of something in their backyard they want to get rid of,\" he says. \"You just have to commit to going around on weekends to find stuff.\"\n\nGareth mostly finds materials on the side of the road, so he's learned to keep his eyes open. He also has a \"borderline addiction\" to Kijiji, the Canadian equivalent of Craigslist\u2014and he doesn't limit himself to local listings. He once drove twenty-six hours to pick up a woodstove and also got a mini cast-iron claw-foot bathtub out of the deal. He suggests doing research and knowing how much things cost, so you can determine what's worth your time and more easily spot a deal.\n\nOne of the company's newest tiny house models, the Cowboy, features all kinds of reclaimed materials. The reclaimed lumber siding was a roadside find. Gareth was driving when he spotted a pile of lumber in someone's front yard. The wood had been left outside for years, and most of it was unusable. He had to sort through the pile for the best pieces, but the owner let him have the lumber for free. The rest of the siding is made up of recycled metal.\n\nThe kitchen cabinets in the Cowboy were converted from old found wooden boxes. The kitchen sink was found at a secondhand store (Gareth says that stainless steel sinks are great to find secondhand, because there are so many of them). The windows are reclaimed, and even though they're different shapes and sizes, they look cohesive when paired with the contemporary exterior. Most of the interior bamboo wall covering and pine tongue-and-groove ceiling are reclaimed too, and the different colors add a lot of texture to the inside of the house.\n\nGareth says it's easy to find things if you're willing to take the time. He treats the hunt for reclaimed materials like a job and dedicates a little time every day to hunt for treasure online or out in the community. \"There's so much waste with construction materials because something isn't absolutely perfect,\" he says. If you put in the work to find those slightly imperfect but completely useable secondhand materials, you can have a tiny home filled with much more beauty and character.\n\nPhotograph by Hummingbird Micro Homes\n\nCONRAD ROGUE IS NOT NEW TO BUILDING SPACES FOR HIMSELF. Early on in life he bought some land and built a small cabin, and from that he began a life-long journey of building ecofriendly homes. Today he focuses on cob homes, earthen structures that are built from a mix of clay, sand, and straw. He lives in a very tiny cob house; coming in at seventy square feet it's the tiniest home about which we interviewed the owner for this book.\n\nConrad told me with a chuckle that cob is \"the original reclaimed material\" and the one he most likes to work with. After building and living in many types of structures including a cabin, a teepee, a yurt, and a dome house, cob seemed to be just right for Conrad. He says, \"Energetically it gives a very nice feel.\"\n\nThe lack of straight lines\u2014of milled lumber with hard angles, and manufactured surfaces\u2014was something unique that came up in my discussion with Conrad. While most reclaimed houses very often rely on milling and re-sawing to get the wood into a squared shape, Conrad feels that it brought the wrong feel to homes in which he had lived in the past.\n\nIn stark contrast to homes made with traditional milled lumber, a cob house has curves, organic textures, and a softness. While there is wood in the space, it is often processed in a way to keep its original form. Conrad compared this softness to milled lumber, saying, \"There is an inherent energetic violence because of the way we harvest wood or the process it took to cut it up.\" It's an interesting notion\u2014that often the building of a home, even one that uses reclaimed materials, is inherently destructive.\n\nPhotographs by Conrad Rogue\n\nThis ethos permeates Conrad's approach to building with natural and reclaimed materials. This connection to the natural process of cob building is perhaps embodied by how cob is made. After combining clay, sand, and straw, cob builders often mix the substance with bare feet. It is hard not to feel a natural connection when you have cob squishing through your toes.\n\nPhotographs by Conrad Rogue\n\nWhen it comes to building Conrad says, \"There is always a cost to speed.\" With cob, there is an innate slowness to the process. \"You make the decision to have a building site that is safe and pleasant.\" The cost of this is the time it takes to build the house, but paying that cost results in a building that \"at the end of the day leaves you a stronger, healthier, and more vital person.\" While many people worry about how long it may take to construct such a house, Conrad suggests that misses the point. While building the home, he says, \"make sure it is fun. So if it takes five years, you at least had five years of fun.\"\n\nConrad's house is comprised of an upper and lower floor. In the loft is a large sleeping space where he has his bed and storage for clothing; downstairs he segmented the space into a sitting area, a small desk, and a tiny kitchenette. The oval house\u2014the shape gave it its name: the Jelly Bean\u2014is only seventy square feet. It is part of a small cluster of homes that have a common house where he eats most of his meals and spends time with his fellow residents. He spends a lot of time outdoors.\n\nPhotograph by Conrad Rogue\n\nBecause he has the common house to fall back on, he designed his house to accommodate 70 percent of situations (sleeping, having a quite place to read a book or make a pot of tea); the remaining 30 percent are the exceptions (hosting large dinner parties, having a guest stay), and you'll adapt to accommodate them. This mentality pervades his small home: his bed will accommodate two, but there is not a guest bedroom. Since he does most of his cooking in the common house, his kitchenette is designed to make tea and a few basics things. Finally instead of having a larger entertaining space, his living-room area has seating for just a few people in a cozy setting.\n\nThe Jelly Bean initially started out as guesthouse, but as Conrad built it, he quickly became attached to it. For Conrad the small size of his home leaves him feeling safe, makes it feel cozier, and grounds him to the space. Soon after it was finished, Conrad moved into the house and since then it has become home.\n\nPhotograph by Conrad Rogue\n\nALEXIS STEPHENS AND CHRISTIAN PARSONS ONCE TOYED WITH THE IDEA OF BUILDING a teardrop trailer, but when the thought of building a tiny house on wheels popped up, they were hooked. The pair worked with friends and family as well as a builder in Winston-Salem, NC, to bring their tiny home to life. The house was an \"all-hands-on-deck\" community endeavor that took eight months to complete.\n\nInspired by similar tiny house stories across the country, the couple was eager to get back to their filmmaking and producing roots. Christian and Alexis are the masterminds behind Tiny House Expedition, a community outreach and documentary storytelling project. For two years, they are traveling around the United States and Canada in their tiny house, documenting tiny house communities and developments. They share the stories of how tiny houses are used in eco-villages, housing for the homeless, urban infill, and everything in between. They hope the documentary will inspire discussion about sustainable and affordable housing, and encourage people to \"dream big and build small.\"\n\nPhotographs by Christian Parsons, Tiny House Expedition\n\nPhotograph by Christian Parsons, Tiny House Expedition\n\nReclaimed materials add a lot of warmth and character to Christian and Alexis's tiny home. Inspired by 1920s bungalows and Pacific Northwest lodge-style homes, they knew they wanted to use a lot of wood and unique details in their home. A friend gave them a whole pile of raw oak boards that had been sitting unused in a tobacco barn, which they used as flooring as well as for the stairs and the porch. They got permission to dumpster dive at the local company where they bought their roof, and they found some colorful striped metal for their shower walls. Even their curtains were lovingly made from mismatched fabric remnants in shades of red and orange.\n\nHabitat for Humanity of Forsyth County was a local sponsor that kept a lookout for secondhand materials the couple could use. The organization had been given a sixty-year-old farmhouse that was going to be demolished, and Habitat invited Christian and Alexis to take whatever they wanted. They brought friends and had a demolition party, pulling out nails and salvaging the interior and exterior wood paneling. They painted and lapped the boards to use as their wall cladding. It took a lot of time, but the character they gained for their house was worth it. As Alexis states, \"I couldn't be more in love with this house because it reflects our personality like no other place I've lived.\"\n\nPhotograph by Christian Parsons, Tiny House Expedition\n\nPhotograph by Christian Parsons, Tiny House Expedition\n\nChristian and Alexis kept their eyes open for reclaimed materials at all times. Aside from dumpster diving, they relied on websites such as Craigslist to find odds and ends they could use. Driving by new construction sites was a great way for them to find a lot of perfectly useable plywood that otherwise would have been thrown out. Word of mouth\u2014both in person and on social media\u2014turned out to be a big help in finding materials. They discovered that people in the community are eager to help as long as you're not afraid to ask!\n\nAlexis advises people interested in living small to do your research and dive in. She stresses how important it is to pare down your belongings. People may think that they can't live without all of their stuff but are later surprised by how quickly they adapt once they get rid of their clutter. \"You are still you with fewer things,\" she says. Most of all, Christian and Alexis love the community of the tiny house movement. Their house allows them to dedicate their time and unique skills to their documentary project, which they hope will help grow the movement even more.\n\nPhotograph by Christian Parsons, Tiny House Expedition\n\nIN HER LAST YEAR OF COLLEGE, CHERIE SOUTHWICK WAS LOOKING AHEAD TO WHAT WAS NEXT FOR HER. Knowing that she was going to have a lot of student loans and worried about finding a job, she wanted to build a tiny house so she wouldn't have to pay rent anymore. Since it was on wheels, she could take it with her if her career took her somewhere else.\n\nShe was turned on to tiny houses by her mother and sister, who were obsessed with them. At the time of writing Cherie's mother is having a tiny house built for herself and her sister is planning to build her own. Initially Cherie looked at tiny houses as an interesting concept, but it wasn't until she was facing a looming graduation date that it clicked for her.\n\nPhotograph by Laila Alamiri\n\nHer building experience was limited to shop class in high school, but she had always had an interest in alternative buildings. She built her home over the course of ten months, almost entirely on her own. Now after the build, Cherie reflected back, saying, \"Anyone who is competent enough to use their hands, do a lot of research, and willing to ask for help, they'll be able to get it done.\" While Cherie was able to figure most things out, she noted that \"asking for help was the hardest part\" for her.\n\nGoing into her build Cherie knew that she didn't want to have a lot of materials that would off-gas. Her studies in college all had a strong environmental focus. Both of these things led her to use reclaimed materials.\n\nPhotograph by Laila Alamiri\n\nAll of the lumber in her home was either reclaimed or from a sustainable lumber operation. The main framing of the house came from a program that her college, Warren Wilson, runs as part of its educational programing. Their sustainable forestry program manages land to provide lumber for local builds in a sustainable manner, focusing on selective timber cutting and ecologically minded land management.\n\nCherie ordered her framing materials from her school, and the rest of the materials were reclaimed. Her flooring, door, and other smaller items she found at her local ReStore. At a local construction salvage store she found all her windows, except for her skylight, which she bought new so she didn't have to worry about leaks. The insulation in her house is foam board that she found on Craigslist.\n\nAll of the curtains, cushions, and fabric elements were deconstructed dresses that Cherie purchased from Goodwill and then sewed into the final product. For her kitchen she found broken pottery and ceramic pieces and created a mosaic backsplash from them. While she loves her home she spoke at length about the amount of energy, time, and effort that goes into using reclaimed materials, saying, \"People don't realize that most of your time goes into preparing for installation, not the installation itself.\"\n\nAll in, the house cost her $11,000 to build, including her solar panel system. Now she has a home to call her own. While she plans to stay in or near Asheville, NC, for the time being, she has options now that she has escaped the cycle of paying rent. From the outside many find the house a curiosity, but to Cherie, \"it has just become normal. It has become home.\"\n\nPhotograph by Laila Alamiri\n\nJESS ONCE BOUGHT A 1,500-SQUARE-FOOT TRADITIONAL HOUSE THAT SHE THOUGHT WOULD BE A QUICK REMODEL, BUT IT TURNED INTO A FIXER-UPPER THAT DEMANDED ALL HER TIME. \"I was working, working on the house, or sleeping,\" she says. Coming from an environmentally conscious family, she started doing research on alternative housing that would be more practical. She soon met Dan and suggested the tiny house idea.\n\nThe couple decided to build their own tiny house in October 2012 and spent the winter designing it. They got started the following April and moved in during October of the next year with the interior details left to finish. Jess was traveling for work, and Dan was balancing his job and grad school, so they had to work on the tiny house during the weekends. Now married, the Sullivans are happily settled in their completed tiny house.\n\nJess and Dan did their best to reuse and recycle as many materials as possible in order to have an affordable, ecofriendly house. All of their windows came from Craigslist. A full-glass front door still in its packaging was another Craigslist find, worth $800 new but purchased for $125. They kept spreadsheets tallying their expenses and cost savings during the build, and they estimate that they saved more than $7,000 by using secondhand materials where they could.\n\nThey got their reclaimed insulation from a man who took apart a warehouse full of industrial coolers. All the insulation came in four-foot by eight-foot blocks, so they had to cut every piece by hand to fit them between the studs. They also had to carve grooves into the foam for all their electrical wiring. Tackling the insulation took weeks of work during one of the hottest Rhode Island summers on record.\n\nThe Sullivans used free shipping palettes for their siding. First they picked up the palettes from local manufacturing companies before other people could snatch them up for firewood. They broke the palettes apart, planed them down, cut them to the correct width, shiplapped them so they would interlock, drilled them to the exterior, and stained them all to match. It's an ongoing, time-consuming process for Dan and Jess, but it was a great way to recycle free lumber, and they love the result.\n\nReclaimed doors add a lot of family history to the tiny house. Two doors came from Dan's grandmother's house, and a third door came from Jess's father's house. The couple used the doors as wall partitions and loft support throughout the kitchen, bathroom, and closet areas. They even kept the windows intact, which brings light into the closet.\n\nOne of their favorite features of the house is their ceiling, made of reclaimed barn wood from 1776. A man in New Hampshire was demolishing the old barn and offered the wood for sale. Dan and Jess used the leftover barn wood to build their library shelves, desk, and pantry. They say that the richness and character of the wood can't be beat.\n\nPhotograph by Nat Rea Photography\n\nThe tiny house is a stepping stone that has allowed Dan and Jess to live their lives to the fullest. Within three months after they moved in, the house was fully paid off. They now have time to travel and do the things that are important to them. Jess has even been able to leave her corporate job and start her own business baking gluten- and dairy-free foods. And as Jess says of their relationship after building the tiny house: \"It is five times stronger than when we started, and it was pretty strong to begin with!\"\n\nPhotograph by Nat Rea Photography\n\nBLAINE AND MACKENZIE VOSSLER WERE LIVING IN SAN FRANCISCO, LOOKING FOR the next step to take with their business. Together they run The Local Branch (thelocalbranch.co), making and selling clothing and leather goods with a vintage Americana vibe. With high rent prices in town and no way to open a brick-and-mortar store, the pair came up with a brilliant idea\u2014take their business on the road with a custom Airstream!\n\nPhotograph by The Local Branch\n\nBlaine and Mackenzie now travel full-time in their thirty-two-foot 1978 Airstream Sovereign Land Yacht, which they use as a live-in work studio. The Airstream allows them to travel to trade shows, craft fairs, and festivals all over the United States. They also run an online store, so they can sell and ship their products from anywhere in the country. Living small has eliminated their rent payment and dramatically reduced their business costs.\n\nThe couple bought the Airstream for $3,000. The bones of the trailer were good, but the interior was rough. The fridge was moldy, a window was missing, tires were flat, and there was green slime residue on the walls from cigarette smoke. They parked the trailer by the side of the road across from the ocean and got to work. It took them a couple of days to completely clean and gut the trailer. Then they hired an Airstream specialist to bring the utilities and electrical components up to snuff.\n\nReclaimed materials and antiques let Blaine and Mackenzie get the exact look they wanted inside their trailer. Inspired by the American West and the Airstream itself, they wanted to create a cozy cabin feeling with industrial touches. Blaine guesses they spent thirty to forty hours looking for materials and that 90 to 95 percent of everything in the trailer is reclaimed.\n\nPhotograph by The Local Branch\n\nPhotograph by The Local Branch\n\nThey found reclaimed redwood boards from a wood salvage company at a great price, and used them throughout the entire trailer to build their walls, closet, and bed. The rough-hewn wood fits their style perfectly. Mackenzie also knew exactly what kind of sink she wanted, and she found the perfect one at a high-end antique store in San Francisco. But the decimal point was in the wrong place\u2014instead of $80, it cost $800! The next secondhand shop they visited had almost the same sink for a much better price, so they snapped it up.\n\nJust as with the Airstream, Blaine and Mackenzie used their creativity to give old objects a new life. Old silver gutters were turned into planter boxes. Metal grain scoops found a second life as wall cubbies. The drawers under their bed are made from old fruit crates. Vintage lockers laid on their side became the base for a tabletop. Even leftover leather scraps from their business became door handles. \"Keep an open mind and think about different ways to use something,\" Mackenzie advises. Blaine adds, \"We kept the costs down by being creative.\"\n\nPhotograph by The Local Branch\n\nAs the couple discovers new storage needs, they'll pick up new antiques during their travels. Old tobacco tins and ammo boxes make great storage for pens and utensils. Even vintage breadboxes store plugs and wires. Mackenzie says that they're very choosy about which items they bring into the trailer. \"It needs to be visually beautiful to us but also be really practical. You really can't have more than you need.\"\n\nFrom start to finish, Mackenzie and Blaine had their new home in three weeks. They have adjusted quickly to their Airstream lifestyle. \"The hardest things about living on the road aren't so hard because we're always waking up in a cool new place, and seeing parts of the country we've never been to before,\" Mackenzie says.\n\nPhotograph by The Local Branch\n\n# CLOSING\n\nThe tiny homes featured in this book are unique, much like their owners. The use of reclaimed materials in tiny houses allow for beautiful homes to be built that are both budget-friendly and do their part for the environment. The combination of diverting materials from the waste stream and the small size of the house all contribute to a lighter footprint when it comes to where we live.\n\nTiny houses have captured the imagination and attention of many, primarily because of their design. Each house is different, and the use of reclaimed materials adds to this profoundly. The stories, the history, and beauty coming from old barn wood and fixtures of years gone by all add to the next chapter in these homes.\n\n# RESOURCES\n\n## Interviewee Websites\n\n_http:\/\/aaronmaret.com_\n\n_http:\/\/aatinyhouse.wordpress.com_\n\n_http:\/\/dominiquemoody.com_\n\n_http:\/\/forvillagers.com_\n\n_www.housealive.org_\n\n_http:\/\/thelocalbranch.co_\n\n_http:\/\/tinyhallhouse.tumblr.com_\n\n_www.facebook.com\/ourtinyhouseview.barringtontinyhouse_\n\n_www.hummingbirdmicrohomes.com_\n\n_www.katefoxmicroholding.co.uk_\n\n_www.livinginatinyhouse.blogspot.com_\n\n_www.rhizhome.com_\n\n_www.tinycommunitycenter.com_\n\n_www.tinyhouseexpedition.com_\n\n_www.rockymountaintinyhouses.com_\n\n_www.theupcyclist.com.au_\n\n_http:\/\/thelittlecabinunderthebigtreehousenearthesmallriver.yolasite.com_\n\n## General Tiny House Websites\n\nTheTinyLife.com\n\nTinyHouseBlog.com\n\nTinyHouseMagazine.co\n\nMiniMotives.com\n\nTinyHouseTalk.com\n\nTiny-Project.com\n\nTinyHouseListings.com\n\nTinyHouseSwoon.com\n\nRowdyKittens.com\n\nTinyHouseBuild.com\n\nPADTinyHouses.com\n\nRelaxshacks.blogspot.com\n\nTinyHouseGiantJourney.com\n\nNicheDesignBuild.com\n\nThisIsTheLittleLife.com\n\nSolHausDesign.com\n\nABedOverMyHead.com\n\nBoneyardStudios.org\n\nChrisAndMalissa.com\n\nTheTinyHouse.net\n\n120SquareFeet.com\n\nTinyHouseOnThePrairie.net\n\nTinyHouseTeacher.com\n\nLittleYellowDoor.wordpress.com\n\nShelterwise.com\n\nTinyrEvolution.us\n\n## Tiny House Books\n\nDiedricksen, Derek. _Humble Homes, Simple Shacks, Cozy Cottages, Ramshackle Retreats, Funky Forts: And Whatever the Heck Else We Could Squeeze in Here_ (Guilford, CT: Lyons Press, 2012).\n\nDiedricksen, Derek. _Microshelters_ (North Adams, MA: Storey Publishing, 2015).\n\nKahn, Lloyd. _Tiny Homes: Simple Shelter_ (Bolinas, CA: Shelter Publications, 2012).\n\nMitchell, Ryan. _Cracking the Code: A Guide to Building Codes & Zoning for Tiny Houses_ (Asheville, NC: Ryan Mitchell, 2013).\n\nMitchell, Ryan. _The New Retirement_ (Asheville, NC: Ryan Mitchell, n.d.).\n\nMitchell, Ryan. _Shockingly Simple Electrical for Tiny Houses_ (Asheville, NC: Ryan Mitchell, n.d.).\n\nMitchell, Ryan. _Tiny House Living_ (Cincinnati, OH: Betterway Home Books, 2014).\n\nRowan, Gerald. _Compact Cabins_ (North Adams, MA: Storey Publishing, 2010).\n\nWilliams, Dee. _The Big Tiny_ (New York: Blue Rider Press, 2014).\n\n# ABOUT THE AUTHOR\n\nRyan Mitchell lives in Chaand has been active in the tiny house movement for years. You might know him as the main blogger and founder of TheTinyLife.com, a website dedicated to sharing stories of tiny living\u2014whether it's tips for simplifying your life or advice for building your own tiny residence. Because nothing beats one-on-one interactions, Ryan helps people connect with each other through community events at TinyHouseConference.com.\n\nSince Ryan built his own tinbeen able to actively pursue his passions, which include photography, backpacking, and travel. Ryan believes that alternative housing options and sustainable local agriculture are key components to meet the future needs of society.\nCopyright \u00a9 2016 by Ryan Mitchell.\n\nAll rights reserved.\n\nThis book, or parts thereof, may not be reproduced in any form without permission from the publisher; exceptions are made for brief excerpts used in published reviews.\n\nPublished by\n\nAdams Media, a division of F+W Media, Inc.\n\n57 Littlefield Street, Avon, MA 02322. U.S.A.\n\n_www.adamsmedia.com_\n\nISBN 10: 1-4405-9211-X\n\nISBN 13: 978-1-4405-9211-9\n\neISBN 10: 1-4405-9212-8\n\neISBN 13: 978-1-4405-9212-6\n\n**Library of Congress Cataloging-in-Publication Data**\n\nMitchell, Ryan, author.\n\nTiny houses built with recycled materials: inspiration for constructing tiny homes using salvaged and reclaimed supplies \/ Ryan Mitchell.\n\nAvon, Massachusetts: Adams Media, 2016.\n\nIncludes index.\n\nLCCN 2015043710 | ISBN 9781440592119 (pb) | ISBN 144059211X (pb) | ISBN 9781440592126 (ebook) | ISBN 1440592128 (ebook)\n\nLCSH: Small houses. | Building materials--Recycling. | BISAC: HOUSE & HOME \/ Design & Construction.\n\nLCC NA7533 .M58 2016 | DDC 690\/.837--dc23\n\nLC record available at _<http:\/\/lccn.loc.gov\/2015043710>_\n\nMany of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book and F+W Media, Inc. was aware of a trademark claim, the designations have been printed with initial capital letters.\n\nCover design by Erin Alexander.\n\nCover photographs by Christian Parsons, Tiny House Expedition; Rocky Mountain Tiny Houses; Alicia Fox Photography; Wendy Barrington; Mark Miller; A. Hall; Hummingbird Micro Homes.\n\nInterior photographs by Aaron Maret; Christian Parsons, Tiny House Expedition; April Anson; Wendy Barrington; A. Hall; Laila Alamiri; Conrad Rogue; Nat Rea Photography; Khari Scott and Dominique Moody; Hummingbird Micro Homes; Rocky Mountain Tiny Houses; Alicia Fox Photography, _www.aliciafoxphotography.com_; Kate Fox; The Local Branch; Jessica Humiston; Mark Miller; Max Cooper Photography; Tamara Gavin; Sarah G. Hastings; rawrestorations.com; Erin Leigh Pasternak, _www.erinpasternak.com_; Guillaume Dutilh.\n\nFrame photograph \u00a9 Leszek Glasner\/123RF; Folded paper house photograph \u00a9 Kanate Chainapong; Graph paper illustration \u00a9 Kanate Chainapong; Wood plank wall photograph \u00a9 Weerapat Kiatdumrong; Paperclip illustration \u00a9 bryljaev.\n\n# Table\n\n### Sample Wall Estimation\n\n_Estimated Wood Needs for Walls_  \n---  \n_PART_ | _NUMBER_ | _MEASUREMENTS_ | _NOTES_  \nStuds | 35 | 8' 2 \u00d7 4s | 46' of wall at 16\" on center  \nWall plates | 10 | 8' 2 \u00d7 4s | Two 8'-wide & two 20'-wide walls  \nDouble top plate | 8 | 8' 2 \u00d7 4s | To join pieces of walls  \nWindow #1 | 3 | 8' 2 \u00d7 4s | Framing for window box  \nWindow #2 | 3 | 8' 2 \u00d7 4s | Framing for window box  \nWindow #3 | 3 | 8' 2 \u00d7 4s | Framing for window box  \nWindow #4 | 3 | 8' 2 \u00d7 4s | Framing for window box  \nDoor | 4 | 8' 2 \u00d7 4s | Framing for door frame  \nL wheel well | 3 | 8' 2 \u00d7 4s | Framing to box in left trailer wheel well  \nR wheel well | 3 | 8' 2 \u00d7 4s | Framing to box in right trailer wheel well  \nInternal wall | 9 | 8' 2 \u00d7 4s | Internal wall frame for bathroom  \nSubtotal | 84 | 8' 2 \u00d7 4s | Rounded up  \nExtra 15% | 13 | 8' 2 \u00d7 4s | Extra for mistakes and unusable leftovers  \n**Total** | **97** | **8' 2 \u00d7 4s** | **Rounded up**\n\nReturn to main text\n\n# Contents\n\n  1. Cover\n  2. Title Page\n  3. Table of Contents\n  4. Part I: Reclaiming Tiny Houses\n    1. Chapter 1: Why Tiny? Why Reclaimed?\n    2. Chapter 2: Reclaiming Materials\n    3. Part II: Talking to Tiny House Owners\n    4. April Anson\n    5. Jamie, Wendy, Avery, & Gabe Barrington\n    6. Stacey's Airstream\n    7. Makayla Cavanagh\n    8. Kate Fox & Andy Gill\n    9. James Galletly\n    10. Tiny Hall House\n    11. Sarah Hastings\n    12. Zee Kesler\n    13. Aaron Maret\n    14. Mark Miller\n    15. Dominique Moody\n    16. Greg Parham\n    17. Natalie Pollard\n    18. Gareth Roberts\n    19. Conrad Rogue\n    20. Alexis Stephens & Christian Parsons\n    21. Cherie Southwick\n    22. Dan & Jess Sullivan\n    23. Blaine & Mackenzie Vossler\n  5. Closing\n  6. Resources\n  7. About the Author\n  8. Copyright\n\n# Guide\n\n  1. Cover\n  2. Table of Contents\n  3. Start of Content\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n# dedication\n\nThis book is for all quilters and sewers, for those who need some more color in their lives, and in memory of Judith Leigh Thompson (1941\u20132013).\n\n# Contents\n\nDedication\n\nIntroduction\n\nSix Steps to a Quilt\n\nWhat You'll Need\n\nFind Your People\n\nWhat Is Modern Quilting\n\n1: A Sense of Play\n\nWeeks Ringle & Bill Kerr\n\nQuilters Unite!\n\nProject: Broadband\n\nRashida Coleman-Hale\n\nAndrew Mowbray\n\nYoshiko Jinzenji\n\nAngela Walters\n\nTutorial: Free-Motion Quilting Paisleys\n\nProject: Jagged\n\nDavid Butler\n\nMiriam Blaich\n\nKathy Mack\n\nRebecca Loren\n\nGrandmother's Fan Variation\n\nAnna Williams\n\n2: Improv\n\nRossie Hutchinson\n\nCredit Where Credit Is Due\n\nProject: Fraction Quilt\n\nGee's Bend Quilts\n\nKatie Pedersen\n\nKyoung Ae Cho\n\nRayna Gillman\n\nTutorial: Improv Piecing with Scraps\n\nNancy Crow\n\nVictoria Findlay Wolfe\n\nTutorial: Curvy Dresden Improv Piecing\n\nDenyse Schmidt\n\nTutorial: Curved Piecing\n\nSherri Lynn Wood\n\nProject: Modern Memory Lane T-Shirt Quilt\n\nDanielle Krcmar\n\n3: The Personal Is Political\n\nKristin Link\n\nChawne Kimber\n\nDenise Burge\n\nMichelle Engel Bencsko & Gina Pantastico\n\nBetz White\n\nLine Bruntse\n\nAlice Webb Greer\n\nTutorial: Finishing Your Quilt\n\nJan Johnson\n\nTutorial: Bullion Stitch\n\nMuseums Discover Quilts\n\nLaurel Krynock\n\nAlexandra Ledgerwood\n\nDonating Quilts\n\nLee Heinrich\n\nBlogging Advice\n\nThomas Knauer\n\n4: Quilting from Tradition\n\nAnna Maria Horner\n\nProject: Bouquet\n\nReading Up on Quilt History\n\nPepper Cory\n\nLog Cabin Block\n\nAllison Harris\n\nAmish Quilts\n\nAneela Hoey\n\nTutorial: Hand-Quilting Basics\n\nKathreen Ricketson\n\nTacha Bruecher\n\nProject: Paper-Pieced Pillow\n\nKatie Blakesley\n\nKaty Jones\n\nVanessa Christenson\n\nCynthia Mann\n\nStarting a Business\n\n5: For the Love of Color\n\nAmy Butler\n\nElizabeth Hartman\n\nAlexis Deise\n\nAmy Keefer's Color Culture\n\nNatural Dyes\n\nElizabeth Barton\n\nKaffe Fassett\n\nMalka Dubrawsky\n\nResist Dyeing\n\nProject: Pieces-of-Eight Pillow\n\nMonica Ripley\n\nSonia Delaunay\n\nTutorial: Dyeing with Avocado Pits\n\nRed & White Quilts\n\nKim Eichler-Messmer\n\nKelle Boyd\n\nCreating Your Own Fabric\n\n6: Practicing Scale(s)\n\nJane Sassaman\n\nJacquie Gering\n\nAshley Newcomb\n\nTutorial: Paper-Pieced Heart\n\nHeather Grant\n\nThe Modern Quilt Guild\n\nDebbie Grifka\n\nStar Variations\n\nProject: Milky Way\n\nValori Wells\n\nJessica Kovach\n\nAimee Raymond\n\nCaro Sheridan\n\nTutorial: The Four-Patch Trick\n\n7: Coming Full Circle\n\nValerie Maser-Flanagan\n\nGeta Grama\n\nStacey Shrontz\n\nTutorial: Fusible Web Appliqu\u00e9\n\nProject: Quilting with Kids\n\nLaurie Matthews\n\nLisa Mason\n\nCaroline Mason\n\nJohn Q. Adams\n\nMen Who Quilt\n\nProject: Summer Twist Quilt\n\nMaritza Soto\n\nInspiration\n\nPippa Eccles Armbrester\n\nTutorial: Reverse Appliqu\u00e9ing Ovals\n\nVirginia B. Johnson\n\nOpening a Shop\n\nSarah Fielke\n\nMelody Miller\n\nAcknowledgments\n\nContributors\n\nBibliography\n\nMetric Conversion\n\nPhotography Credits\n\nCopyright\n\nShare Your Experience!\n\n# Introduction\n\nMost of the people in this book began quilting on a whim, having seen something in a book, online, or at an exhibit. They opened a blog, went to a museum, or browsed through a bookstore, and wham! Struck with the notion that they wanted to make _that_ , weeks later their houses were half-full of fabric, and they were addicted to making quilts. That's the story I heard over and over again. He saw something on Pinterest and knew he wanted to make a quilt. She had decided to stay home with the kids and needed something to fill the gap that not working had left in her life. He wanted to do something with color. She needed to make something with her hands, to have a complete project to show for her work at the end of the day.\n\nMaybe this is you.\n\nSix years ago, it was me. My sister had given my mother a book of Gee's Bend quilts for Christmas, and, stealing a peek over the holiday, I was intrigued. I'd known how to sew since I was little, but hadn't been interested enough in making the outfits that I once fantasized would fill my closet. I didn't like the careful measuring and cutting process required for making clothes, the precision needed for each seam to be in just the right place.\n\nAs I interviewed quilters, I heard over and over again that inspiration struck, and a few days later, they were making their first quilt.\n\nStill, I had a sewing machine sitting in the closet. And if I made quilts sort of freestyle, I thought, it might be fun. I went to Jo-Ann's and rummaged around in the sale basket for scraps that ended up costing a total of $25. I went home and pieced my first quilt, a wedding present for friends. I didn't measure or plan too far ahead, just one strip at a time, one piece next to the other, then the strips became rows until I had the top of a quilt. Once I finished that, I Googled how to make a sandwich. I watched a video. I bought some safety pins, tied the quilt layers, and did a hodgepodge binding job.\n\nIt was a quilt!\n\nI was thrilled, and . . . addicted. And once I learned how to do a proper binding, I saw the beauty of quarter-inch seams, gaining more complex skills, and learning from patterns.\n\nAll it took was a look at my mother's book of Gee's Bend quilts for me to get my sewing machine out of the closet.\n\n## Give It a Go\n\nIf you've never sewn a stitch before, the easiest way to start is to get a used sewing machine, find a tutorial on threading it, and go home and practice sewing straight lines. Once you can sew a straight line, you can make lots of different projects \u2014 including a quilt. It really is that easy. I hope that this book will give you some ideas to get you started. You might work with a pattern, make a mini-quilt, or improvise. Some people get a lot of satisfaction from handwork; you could start by paper-piecing hexagons. Others like to work on the machine. Follow the thread of your curiosity.\n\nThe quilters in these pages were inspired by Amish quilts, traditional patterns, Nancy Crow (page ), art quilts, Gee's Bend makers (page ), a local quilt exhibit \u2014 and on and on. I've included information about quilt history that inspired today's quilters, in the hopes that you'll explore even further. Check out the bibliography (page ) and list of these quilters' online sites (page ). Head for your local bookstore or fabric shop. Go online and look at quilt history sites and contemporary blogs and Flickr. There are endless possibilities for getting that spark of inspiration, gaining knowledge about quilt history, acquiring skills, and finding what you love.\n\nAmish quilts\n\nGee's Bend's quilts\n\nClockwise from top-left: _Blocks and strips work-clothes quilt_ (1970) by Lucy Mingo, 79\" \u00d7 69\"; _Snowball_ (1950) by Lucy T. Pettway, 83\" \u00d7 85\"; _Housetop_ (1975) by Qunnie Pettway, 82\" \u00d7 74\"; _Housetop \u2014 four-block variation_ (1965) by Mary L. Bennett, 77\" \u00d7 82\"\n\nWhat I hope you'll discover as you read are the myriad styles and processes of quilting. And I hope that, as you read these people's stories and look at their work, you'll find something that strikes you and inspires you to pick up fabric and scissors, needle and thread, and make something of your own. As you sew, you'll find what you love \u2014 and eventually, your own voice.\n\nRachel May\n\n###### How-To\n\n# Six Steps to a Quilt\n\n  1. **1.** **Piece the top.** This can be done in any style, from improv to traditional pattern. A simple way to start might be strip piecing: simply sew together long strips of fabric in rows until your quilt is long enough. Or try this easy improv log cabin (see page  for more log cabin variations).\n  2. **2.** **Make the backing.** If you're making a small quilt, you may be able to use a single piece of fabric, rather than sewing together two long strips. (You can piece more, of course: make the backing as fancy or as plain as you want.)\n  3. **3.** **Cut the batting to size.** The batting (a.k.a. wadding, filler, insulating material) should be a few inches wider on every side than your top. Lay out the batting underneath the top to make this easy.\n\n  4. **4.** **Make a quilt sandwich** in this order: backing on the bottom, batting in the middle, quilt top on the top (naturally). \n    * First, lay down your backing on the floor, right side facing down, and use painter's tape on the corners to keep it in place and wrinkle-free on the floor. It's okay if your backing is bigger than your top.\n    * Lay down your batting (which should be a little bigger than your quilt top, remember) over the backing, and smooth it out with your hands.\n    * Lay your top piece right side up, lining it up with the backing underneath the oversized batting. Spread it smooth, and then start pinning the quilt from the center out, applying pins about 4 inches apart (the holes the pins make will disappear with quilting and washing).\n\n  5. **5.** **Quilt it.** This could be as basic as tying the quilt, or anything from a simple to complicated machine-quilting design, or hand-quilting.\n\nHand-quilting\n\nMachine quilting\n\n  6. **6.** **Bind it.** See how in the tutorial, Finishing Your Quilt, on page .\n\nHandquilted by Pamela Thompson\n\n# Curved Pins\n\nCurved pins made just for basting will make your life easier, allowing you to push the pin down through every layer of the sandwich and then back up to the surface. If you don't like pins, you can also stitch-baste, or spray baste. Once your quilt is basted, you'll be able to move it through the machine (rolling it up to fit through the throat, if the quilt is large) without fear of wrinkles or shifting.\n\n# Spray Baste\n\nFor some, spray baste makes life a lot easier, saving you from long pinning or stitch-basting sessions hunched over a quilt on the floor. You'll need to use spray baste in a well-ventilated area, simply applying it to both sides of the batting. If you want your quilt to last as an heirloom, though, there's research showing that spray baste will break down fabric over time.\n\n# Stitch-Basting\n\nYou can also stitch-baste your quilt sandwich. This involves making long stitches across the quilt (with unknotted thread), starting in the middle and working your way out. Use a different colored thread that you'll spot easily after quilting. If you have a pattern you want to stitch into the quilt, baste just inside or outside of the pattern. Use an upholstery needle, and make your stitches long to save time and cut out easily after quilting.\n\n# Sew-Ins\n\nA sew-in is when you meet with friends to sew by hand or machine. More often than not, when members of our guild have sew-ins, we chat and eat together; our projects never make it out of the bag. Fun \u2014 no matter how you do it.\n\n# Fabric Lingo\n\n**Bias:** The diagonal of the fabric. Why does it matter to know about this? Fabric moves differently depending on how you cut it. Bias-cut fabric is stretchier, which is why bias-cut bindings (see page ) are so wonderful.\n\n**UFO:** Unfinished Object\n\n**WIP:** Work in Progress\n\n**Stash:** The pile o' fabric\n\n**Scrappy:** A quilt made with lots of fabric scraps\n\n**Strip piecing:** Sew together rows of fabric, then cut to re-piece in different ways.\n\n**Warp:** The long threads that run the length of the fabric\n\n**Weft:** The shorter threads that run across the fabric\n\n**Hand:** When we talk about how fabric or quilts feel or move, that's the \"hand.\" As in, \"This has a really nice hand.\"\n\n# Cutting Fabric\n\n**Stack 'n' whack:** Piling fabric and cutting through all layers at once with a rotary cutter\n\n**Fussy cut:** Cutting by hand to include prints or designs.\n\n**Subcut:** Cutting a piece of fabric that's been cut once already.\n\n# Quilt Lingo\n\n**Crazy:** Patchwork technique of irregular shaped pieces of fabric\n\n**Cutaway:** Remnants from apparel factories, usually forming irregular shapes\n\n**Drafting:** The process of drawing a quilt design\n\n**Strike-offs:** A printed sample of a fabric design\n\n# Tools\n\n## What You'll Need\n\nThe items in this list will come up again and again in projects throughout this book. I won't repeat them in the materials lists for each project; by then you'll know what you need.\n\n**Fabric.** Any fabric will do, from repurposed, outgrown clothes to muslin (super-cheap and good for practice) to a friend's scraps to high-quality quilting cottons found in a local quilt shop or online. There's something to be said for starting with cotton, only because it's the easiest to work with for quilting. Silks, corduroy, denim, and other fabrics make for beautiful quilts but are a little trickier to work with. But, hey, if that's what inspires you, go for it.\n\n**Scissors.** Fabric scissors are best because they're so sharp and cut much better than your average scissors. If you buy fabric scissors, save them only for fabric (paper will dull them) and know that you can get them sharpened over and over to make them last forever. I use my scissors for everything when I quilt, but most people just use them for fussy-cutting and other small jobs. You'll also want a pair of smaller snips, like the ones shown below, belonging to Alexis Deise (see page ).\n\n**Rotary cutter and mat.** A rotary cutter is the modern quilter's best friend. You can slice and dice like nobody's business with this sucker. You can get pretty fancy with your rotary cutter models, too.\n\n**Needle and thread.** If you plan to quilt by hand or appliqu\u00e9, invest in some basic sewing and quilting (very tiny) needles. Thread can get pretty fancy, but you can start with an all-purpose thread for piecing and quilting. The thread you use in your machine is important, because certain threads will shed less, saving your machine over time. You can start by using an all-purpose thread, but if you sew a lot, consider a brand that comes on bigger spools.\n\n**Sewing machine.** There are a myriad brands and models of machines out there from which to choose. If you already have a sewing machine, you don't need to get a new one for quilting! Just start and see how your machine handles it. If you don't have a machine or you start quilting a lot, a quality refurbished quilt-specific machine is always a safer bet than a cheap model that's made for general sewing projects. Your most basic new quilting machine will cost about $300.\n\n**Author's Choice.** I still have the smallest Janome they make, and I've been able to piece and quilt full-size quilts on it. Some claim to be able to quilt king-size quilts on machines this size, but I find that pretty hard. I've outgrown this machine, but it took me four years to do it. It's a good place to start, and a far, far cry better than the all-purpose Singer I first used, mostly because the Janome motor is made to go for the long hours I use it each time I sew and it has all the feet and stitch settings that I've needed to start quilting.\n\n**Walking foot.** You'll need this if you want to machine-quilt (a regular foot compresses the layers of fabric too much and creates creases and folds as you sew).\n\n**Free-motion foot.** This is a little trickier to use, but once you get the hang of it, you can create all sorts of quilting patterns and designs on your quilts. When you use a free-motion foot, you lower the feed dogs (those spiky circles under your needle) so the machine doesn't feed the fabric through. Then you can control the motion and direction of the stitching.\n\n**Iron and table.** This can be a small DIY table that's set right next to your cutting table (a luxury when piecing a complex project that requires lots of pressing), or you can use the old standard ironing board. Like machines, irons come in a variety of sizes and styles. I love the Cadillac irons that have more weight to them and smooth metal surfaces that just glide over the fabric, but I use a cheap iron that does the job just fine as long as I keep it filled with water to steam the cotton.\n\n**Design wall.** You can make your own design wall pretty easily by pinning a piece of felt, sized as big as you like, up on your wall. Fabric will stick to the felt, so you can place the pieces and rearrange them as you conceptualize each project. If you want to get fancier, you can stretch the felt over a wooden frame, which allows you to move it out of the way if you don't have the space to keep it up all the time.\n\n**Pins and cushion.** In addition to safety pins for basting, you'll also need straight pins to use while you're piecing. You can get them in anything from your basic metal to those with colored tops that help you keep different piles of cut fabric organized. Make sure you also have a pincushion or magnetic pin holder to stash the pins as you remove them.\n\nIt's fun to make your own pincushion, or participate in a pincushion swap. I still use a green pear cushion that Laurie Matthews (page ) gave me during our first guild sew-in.\n\n**Barrettes or binder clips.** These come in handy to hold your binding down as you sew it. You can easily shift the clips ahead as you sew.\n\n**Quilting gloves.** Save your arms and hands from getting sore as you hang on tightly to the fabric while free-motion quilting. The rubber grips on gloves will do the holding for you.\n\n**Thimble** If you're hand-sewing or quilting, save your fingertip with a rubber, metal, or (my favorite) leather thimble.\n\n# Find Your People\n\nIf you want, you can sit in your room and sew, relishing in the quiet and time away from the world. But one of the best parts of quilting is the community you'll find, people who will become your friends and from whom you can learn more. Seek out quilters at local shops, online (through blogs, Flickr, or Pinterest), or by joining a local guild. Your local fabric shop and both traditional and modern guilds offer the chance to get to know fellow quilters who share your enthusiasm. They can offer you all sorts of advice about where to buy fabric, how to perfect a new technique, the best way to store quilts, and everything else you'll wonder about the further you get into quilting. You can meet friends at one another's homes for sew-ins or get together at local shops that offer a seating area or even at a coffee shop for some sewing and chatting time. Your quilting friends will likely become some of your best friends.\n\nIn a quilting community, you will no doubt find opportunities to join challenges and bees. Challenges offer the chance to make a quilt within certain parameters, say with a certain stack of fabrics or in a certain style. You then enter your quilt into the challenge and a \"winner\" might be chosen.\n\nWhile a traditional quilting bee is a gathering of quilters who make a single project together, today's bee is slightly different and can be organized in person or online. Each person chooses the fabric and style in which she'd like her quilt to be made. She might choose a particular pattern, or ask quilters to improvise with a few guidelines (for example, curvy-piecing and no appliqu\u00e9). She'll then give or send her fabric to the quilters, enough for each one to make a block. The quilters make their blocks, send them back to their owner, and she then pieces them together for (voil\u00e0!) a finished quilt.\n\n# Start Small\n\nA great way to start quilting is by making a mini-quilt, 16\"\u201324\" square or even smaller. You could make a single block pattern, or try strip-piecing or improv-piecing some scraps together.\n\nPart of the fun is the surprise; no one knows exactly what each quilter will make with her block. You also get the chance to collaborate by working with everyone else's finished blocks. You may be forced to learn a new skill that you'd dared not try before, or to work in a way that you hadn't thought of experimenting with. The only hard-and-fast rule is that if you enter a bee, you have to follow through and make those blocks for your friends, or else you'll leave them with eternal WIPs (see next page) and hurt your credibility in quilting land.\n\nThere are lots of different ways to execute bees and challenges, so check out what's available and what you might like to try. There are many that are perfect for beginners. You can also make a collaborative quilt, like this one (see below) that Virginia Johnson made when she had friends over for a celebration and quilting fiesta. Another example: Our guild pieced a charity quilt together at a series of sew-ins, and then one person quilted and bound it.\n\n**Take a Challenge.** \"What's so cool about doing a challenge is if you follow a set of rules, that can help you grow as a quilter. It seems really counterintuitive, but rules force you to move forward because you have these skills that you didn't have before.\" \u2014heather Grant\n\nAs you move along, you'll begin to notice different sorts of quilters. While some people prefer not to define themselves, most will identify with one of three categories: traditional, art, or (the most recent addition) modern. These categories and their histories are explored in this book, so you'll have the chance to learn about them in more detail and decide for yourself if and where you might draw the boundaries between them.\n\nOne way of thinking about it is that _art_ quilters make quilts that will likely hang on the wall; _traditional_ quilters make quilts for functional purposes, relying on the techniques and patterns that have been passed on through the generations. As for _modern_ quilters, that's still up for grabs. They're essentially some hybrid of the other two categories, making quilts that perform both a design and functional purpose. (Traditional quilters would probably say the same about their quilts, though.)\n\n# What is Modern Quilting?\n\nTraditional and art quilts have been around for a long time, but modern quilts are a newer category. Most quilters agree that it has something to do with a sense of experimentation. Modern quilters might take a traditional block or pattern and innovate to turn it into something \"fresh\" (an oft-used word). They might use all solids (Kona and Moda make solid lines, and now most other quilt companies do, too). They might use modern printed fabrics, or scraps of clothing, corduroy, or traditional fabrics. They could piece words into the quilt, make a modern-painting abstraction, or improvise.\n\nImprovisation is big in modern quilting, as it allows quilters to conceptualize, make a design of their own, and adjust the concept as they go. Just as everyone has their own way of \"modern\" quilting, those who improvise have their own definition and process for that, too. So, how are modern quilts really so different from art and traditional quilts, you might be asking?\n\nWeeks Ringle and Bill Kerr (see page ) define modern quilting as quilting that \"is expressive of the time in which we live,\" noting that the way their quilts look may change from day to day.\n\nModern quilters cite the Japanese quilting tradition, Gee's Bend (page ), Nancy Crow, midcentury modern design, Marimekko, the Bauhaus, and Anna Williams as inspiration for their work. But while everyone's definition varies, what might be most important is that it is a community invested in quilting. Most quilters agree that anything that keeps the craft going is a good thing.\n\nNancy Crow (page )\n\nYoshiko Jinzenji (page )\n\nAnna Williams (page )\n\nAt one of the first meetings of the Boston Modern Quilt Guild, we spent a good half of the time on show-and-tell, and it quickly became our favorite part of gathering. Sitting in a circle, each woman went around and showed her quilt or work in progress (WIP), proudly (or shyly) explaining her process, the pattern, her inspiration, mistakes, lessons she'd learned. Some women's quilts were loud and bold, others were made with soft palettes, some traditional, some modern, some art quilts. There was a huge range of styles and skill levels. But every time someone shared a quilt, it was met with enthusiastic _oohs_ and _ahs,_ and we'd all point out different aspects of the piece that we loved. _That color is fantastic. I love that quilting \u2014 you did it all by hand? You matched your points perfectly! Where did you find the pattern, so I can make one, too?_ No matter what, we all supported one another's work and progress. I loved it.\n\nIn my first show-and-tell quilt, I combined different fabrics, using one modern fabric (Heather Bailey's _Red Poppies_ ), eyelet backed in white muslin, a blue-and-white pinstripe, and a traditional blue toile. I didn't know what they'd make of it, having seen their gorgeously pieced and quilted projects. But, of course, they were generous and kind, making the same sounds of approval for me that they'd made for the woman who had been quilting for fifteen years.\n\nI started writing this book with the hope that it would enact that same sense of inclusion, welcoming a range of quilting styles and experience levels. May you experience the same community and support that I found at those early guild meetings \u2014 and still find in Boston every time I go back! It's the cheer of the show-and-tell that I want you to feel when you read. I hope that you flip through these pages and find quilts that make you say _oooh,_ whether they are made by quilters just starting out, or by those who have been at it for decades. You might follow some of these quilters at shows, in classes, at local meetings, or online (see page  for contact information).\n\nAs I wrote, I wanted to discover where we draw the lines between traditional, modern, or art quilters, and look for what we share in common. I'm interested in learning how the \"modern\" quilt community has added to this conversation and will continue to evolve. In this book, you'll find tips and tutorials, patterns and advice, something for the new quilter, the I-haven't-even-sewn-a-block inquisitor, and the advanced quilter alike. I wanted to make readers feel like I do when I come home from a quilt guild meeting: inspired by someone else's beautiful work, encouraged by sage advice, and brimming with ideas to start or keep on creating.\n\nI love Malka Dubrawsky's advice: Be fearless, and have fun. After all, it's just fabric.\n\n###### 1\n\n# A Sense of Play\n\nFrom those who have pushed the bounds of quilting, to the first to define what a \"modern\" quilt could look like, to those who teach kids to quilt, this section celebrates all that we do to innovate and play as we sew. All art rises, at least in part, out of a sense of play, of trying something new and creating new worlds. While quilting and sewing are challenging at times (especially in the beginning, when you're just mastering basic skills), try to take it slow, be patient with yourself, and keep on trying new ideas and techniques. Playing with fabric, form, and design will lead you to find your own way of creating quilts that express who you are, as Jacquie Gering (page ) proves here with her _Broken Cogs_. You're going to have those frustrating moments when you realize you need to pull out a row of stitches, or that your seams don't match up _at all_. That's what your quilting friends are there for \u2014 the venting and encouragement! But as you sew, try to hold onto your sense of fun and exploration. Try new patterns and skills, experiment with new techniques. Just see what happens. You'll surprise yourself.\n\n# Weeks Ringle & Bill Kerr\n\n##  Expressing the Times\n\n_Unfinished Business_ by Weeks Ringle and Bill Kerr, 40\" \u00d7 40\"\n\nOur understanding of \"modern\" quilts today \u2014 the very use of the term, in fact \u2014 comes from the collaborative, vivacious team of Weeks Ringle and Bill Kerr, who were the first to apply the word \"modern\" to these new-looking quilts. Weeks called Nancy Crow \"our mitochondrial Eve,\" the one who birthed us all \u2014 art quilters, modern quilters, innovative traditional quilters, improv-piecers, and everyone in between. But if that's the case, then Weeks and Bill are quilting's great-grandparents (in metaphor only, not in age!).\n\nWeeks and Bill emphasize that modern quilting, for them, is \"a philosophy, not a style.\" As Bill says, their quilts \"are expressive of the time in which we live. That doesn't describe what they look like. What I make tomorrow may bear no resemblance to what I made yesterday.\" Their definition of modern quilting is an expansive one that allows for possibilities, with a push for _in_ clusion rather than _ex_ clusion. \"We're not ego-driven designers. What drives us is the excitement of concepts. So, if we're teaching a design workshop, what will get me so jazzed is talking one-on-one with students about the visions they have. It's a back and forth. One of the women in my workshops said that our discussions about the quilt she was working on drove many of the ideas in a book she has coming out. And that's exciting! Something is getting pushed forward.\"\n\n# Embrace Your Sense of Play\n\nWeeks Ringle and Bill Kerr told me about some quilts they've made that \"look nothing like what's in our books.\" One of Bill's quilts was made with \"eight layers of solids with hundreds of different hole punches with leathering tools, and Weeks has made quilts with aluminum rivets through gauzy fabrics \u2014 things that are explorations of ideas.\" It's this sense of play and discovery that makes their work exciting and that pushes other quilters to try something new.\n\nThis ebullience, a clear love of design and quilting, as well as their sense of acceptance and inclusion, reflects the best of the modern quilting community. This is what first drew me and many others to this world, and it's Bill and Weeks who have helped to foster what the community is today.\n\nBoth quilters hold graduate degrees in design and together have published five books, launched a magazine and fabric line with Andover Fabrics, made custom quilts, and taught workshops. They published their first book in 2002, _Color Harmony for Quilts_ , illustrating quilts made with solids and the \"fresh\" composition that has come to be known as \"modern quilting.\" When their second book, _The Modern Quilt Workshop,_ came out in 2005, this new aesthetic was thus named and had already begun to take hold; it flourished as new quilters found one another online and published books in a similar vein. It's Weeks's and Bill's drive to continue creating quilts and talking about color and composition that has pushed modern quilters to think about design in new ways and to focus on developing sewing and quilting skills.\n\n_Some Settlement May Occur_ by Weeks Ringle and Bill Kerr, 40\" \u00d7 52\"\n\nAs much as they've loved writing their books, they're clearly delighted with the process of creating a magazine. \"What's fun about the magazine and what's challenging, is that we want it not to be just about the quilts but about the whole way of communicating and teaching and the graphic presentation, understanding how people think about the making process.\n\n# Happy Accidents\n\nYears ago, Weeks designed her _Happy Accidents_ quilt after hearing a story on NPR on the social history of the color mauve, \"Mauve, the Color that Changed the World.\" The story is of William Henry Perkins experimenting with chemicals to create quinine, and accidentally inventing a purple dye, the first aniline, when he saw it spill all over the floor. Weeks loved \"the idea that the purple stain all over the floor becomes permanent. . . . I had to make an improvisationally pieced quilt because the idea behind it was that things happen that you can't plan.\" Brilliant, no? What would happen if we turned a science experiment or news story into a quilt? I'm thinking about the news of those fast-moving particles under the earth that were thought to exceed the speed of light. What would happen if that story were birthed in fabric?\n\n\"The quilt world, whether it's contemporary or traditional, has followed the lead of the publishing industry. People figured out long ago that there's a way you can make patterns legible and flow \u2014 books are pretty much templated. And we wanted to challenge that notion. If every quilt is different, each quilt needs a different approach. The magazine is a way to explore, for our own sake and for our readers', different ways of doing things. Some may work, some may not work as well. We're not doing new for the sake of new, we're doing new for the sake of discovery.\"\n\nWeeks began making quilts in Tokyo in 1987, when she was studying Japanese flower arranging. \"I didn't know anybody who quilted. There was nobody in my family, nobody among my friends, nobody I knew in Japan \u2014 I had never had a conversation with anybody who had made a quilt. And then I went to this quilt show, where there were quilts made by Japanese quilters, and they were all made of indigo fabrics. When you walked into this show, it was all indigos, and their craftsmanship was incredible \u2014 but they looked so _different_. And I realized at the time that sometimes you have to live in another country to get a perspective on your own heritage.\n\n\"I did have this preconceived notion that quilts were very old-fashioned and Sunbonnet Sue and that whole thing.\" She pauses to tell me, laughingly, about the Sunbonnet Sue spin-offs that people have created. Sunbonnet Sue gone mad. \"There's a whole series of Sunbonnet Sue Death Quilts. They're kind of like Edward Gorey meets Sunbonnet Sue, hysterical black humor.\" (They are really funny \u2014 I had to look them up.) \"Anyway, I had this vision that that's what quilts were. And, I was studying Japanese flower arranging at the time, so I was deeply involved in a craft that had already had a modern branch develop. So, when I exhibited my flower arrangements in Tokyo, there would be a traditional part of the show and a modern part of the show. And seeing these indigo quilts made me realize, 'Oh, there could be a modern approach to quilting.' The flower arranging was very helpful to me because I could see that this is how an art form evolves.\"\n\n_Lace_ by Weeks Ringle and Bill Kerr, 38\" \u00d7 46\"\n\nShe goes on to say that she was an investment banker at the time, and that she'd never seen herself as someone who was artistic. (I've heard this echoed many times when talking with other quilters.) \"I was of that era where art teachers told you, if you didn't have an artistic talent by the time you were five, you were not going to get it. I put that constraint on myself, and thought, 'Well, I couldn't paint, because that's a creative art, and I'm not that person. But, you know, quilting is just sewing. You don't have to know how to draw, you can just sew. I can sew. So, I gathered all of these modern Japanese textiles \u2014 blue and white fabrics that were contemporary Japanese designs with traditional inspirations. And, I made this quilt that was very, very contemporary, I had never seen anything like it. I was sewing without any influence.\"\n\nIt's fascinating to hear this story, since modern quilting owes such a debt to Japanese textiles and quilting, too: the use of indigos, the sense of minimalist design. It makes sense that Weeks was there, designing from her flower-arranging inspiration, and then bringing her aesthetic to the community here in the States. When Weeks and Bill met, they started making quilts together, first as gifts (one was a Hawaiian shirt quilt, which they laugh about now; one thing you should know about Weeks and Bill is that they're also very, very funny). Bill had a background in design, of course, and his mother was a weaver. So, Weeks says, he was comfortable with this kind of work. As they made quilts for friends, their interests began to evolve, and their collaboration as the designers behind their books, and the site Modern Quilt Studio (n\u00e9e Fun Quilts) was born.\n\nSubscribe to _Tattoo Artist Magazine_ , _Snowboard_ , or _Scientific American_ _._ Each of these niche publications has figured out how to do something, and has a voice, and that voice may snap you out of your rut. I think we need to look outside of the quilt world, and even the art world. That is exciting \u2014 it makes us quilt differently, it makes us design differently. \u2014 BK\n\n# Quilters Unite!\n\nThe divide between modern quilters and, well, everyone else is a concern for Weeks and Bill, who see the pitfalls of such exclusion. Weeks explains the importance of valuing not just each other's modern quilting blogs but also traditional (often older) quilters: \"There's a generation of quilters who are not on the Internet, who don't have websites or blogs, who are a wealth of information. One of the quilters I know has among the most consistently original, beautifully made quilts I've ever seen. She isn't known to anybody in the Modern Quilt Guild; but if they saw her work, they would be stunned. If they saw her, they wouldn't strike up a conversation. But she's a thinker. And the diversity of her work is stunning. I am constantly looking at her quilts thinking, I wish I could have made that. She's never going to get a book deal or a fabric deal, but there's a lot more there than people who have big names.\n\n\"The modern quilt movement is wonderful for quilting, but I hope that people will start to look outward more. . . . From the start, our approach to modern wasn't that it was going to exclude ideas and concepts and techniques, but rather it would open up to include different techniques and different fabrics and different combinations of fabrics. Our goal is to say, let's have everything available \u2014 let's not discount anything.\"\n\n###### project\n\n# Broadband\n\nby Weeks Ringle & Bill Kerr\n\nModern Quilt Studio\n\nFrom Weeks Ringle and Bill Kerr comes a pattern that exemplifies their vision. Spare, clean, fresh \u2014 use any adjectives you like; this is certainly _modern_ _._ With teeny-tiny pieced stripes, it also poses a welcome challenge in consistent piecing and corner alignment.\n\n_Broadband_ by Bill Kerr and Weeks Ringle, Modern Quilt Studio\n\nFinished size 181\u20442\" \u00d7 251\u20442\"\n\n### What You'll Need\n\n  * 1 yard of blue solid for field and backing\n  * 1 fat quarter of gray solid\n  * 1 fat quarter of white solid\n  * 191\u20442\" \u00d7 261\u20442\" piece of batting\n\n# Construction Notes\n\n  * All directions assume 44\/45\" fabric and 1\u20444\" seam allowances unless stated otherwise.\n  * To achieve uniform 1\u20444\" finished strips, cut carefully and maintain a consistent seam allowance.\n  * To distribute the bulk of the thin seams and maintain the crisp, graphic quality of the design, we press all our seams open, not to the side.\n\n### Cut Out the Fabric\n\n  1. 1. From blue fabric, cut: \n    * One piece for backing 191\u20442\" \u00d7 261\u20442\" piece\n    * **A** piece, 191\u20442\" \u00d7 123\u20444\"\n    * **B** piece, 191\u20442\" \u00d7 6\"\n    * **C** piece, 71\u20442\" \u00d7 2\"\n    * **D** piece, 41\u20444\" \u00d7 2\"\n    * **E** piece, 191\u20442\" \u00d7 16\"\n  2. 2. From white fabric, cut: \n    * **F** piece, 83\u20444\" \u00d7 15\"\n    * **G** piece, 83\u20444\" \u00d7 2\"\n  3. 3. From gray fabric, cut: \n    * **H** piece, 71\u20442\" \u00d7 15\"\n    * **I** piece, 41\u20444\" \u00d7 15\"\n\nBlue fabric cutting layout\n\n### Piece the Quilt\n\n**Note** Rather than sew narrow strips together (which would warp easily), these directions have you sew a larger piece and trim it down.\n\n  1. 1. Sew together C-G-D. Press the seams open.\n  2. 2. Sew together H-F-I. Press the seams open.\n  3. 3. Sew C-G-D to bottom of A. Press the seams open.\n  4. 4. Trim C-G-D 1\u20442\" from seam with A.\n  5. 5. Sew E splicing piece to bottom of C-G-D in step 4.\n  6. 6. Trim E 1\u20442\" from seam with C-G-D. Press the seams open.\n\nPiecing layout\n\n  7. 7. Sew C-G-D to top of B. Press the seams open.\n  8. 8. Trim C-G-D 1\u20442\" from seam with B.\n  9. 9. Sew H-F-I to bottom of A\/C-G-D\/E unit. Press the seams open.\n  10. 10. Trim H-F-I 1\u20442\" from seam with A\/C-G-D\/E unit. Press the seams open.\n  11. 11. Sew E splicing piece to bottom of H-F-I in step 10. Press the seams open.\n  12. 12. Trim E 1\u20442\" from seam with H-F-I. Press the seams open.\n  13. 13. Repeat steps 9\u201312 fourteen more times, making sure to align the white stripes as you go.\n  14. 14. Sew the top part of the wall hanging you've just pieced to C-G-D\/B. Press the seams open.\n\n### Assemble the Quilt\n\n  1. 1. Layer the top and backing, with right sides together, then align batting on top. Make sure all edges are flush.\n\n  2. 2. Stitch these three layers together 1\u20442\" from the edge on three sides, leaving the top edge open.\n  3. 3. Carefully trim batting flush with the seam on three sides and 1\u20442\" from the top edge.\n\n  4. 4. Turn right side out through the top opening, easing out the corners.\n  5. 5. Fold in the top edges 1\u20442\" and then hand-sew the opening closed.\n\n### Quilt the Layers\n\nWe quilted _Broadband_ in a dense, allover spiral pattern using light blue thread that matches the background fabric.\n\n# Rashida Coleman-Hale\n\n## Zakka Style\n\n_Left to right: Baby Washi Quilt_ , 44\" \u00d7 44\", and _Pinwheel_ , 60\" \u00d7 56\", both by Rashida Coleman-Hale\n\nRashida's first book, _I Love Patchwork: 21 Irresistible Zakka Projects to Sew,_ came out in 2009, and it's been quickly followed up by her second, _Zakka Style,_ out in 2011. She laughs about bringing her kids to her sewing space with a babysitter to help out, and how she managed to juggle everything as she wrote. She's now designing her own fabric lines and was picked up to design for Cloud9 Fabrics, which she says is a dream come true.\n\n\"My mom decided one summer when I was a teenager that I needed to know how to sew. 'I was like, oookay, Mom.' We went to a fabric store and picked out some fabric, which was hideous.\" She laughs. \"Then we picked out a pattern, which was equally hideous. I picked the craziest fabric! It looked like a clown suit. It was terrible.\" Rashida took to sewing, though, and her mother helped her step by step. It was her mother's being discovered by a fashion designer that took them to Japan. \"It's definitely helped my design eye,\" Rashida says, and thus grew her Zakka style.\n\nMy style grew to be more like that of the Japanese quilts. I found a book called _Handmade Zakka_ , and I thought, \"I LOVE this book!\" My work is very similar \u2014 I love the linens and simplicity, and the patchwork with a burst of color. I spent a lot of time in Japan as a girl, and there was always a lot of Zakka stuff there. Zakka is like chotchkes, cute little things for the house. I just took to it. \u2014 RC-H\n\n# Andrew Mowbray\n\n## Reuse, Reimagine, Redesign\n\n_Floor Plan_ by Andrew Mowbray, 18\" \u00d7 18\"\n\n\"Where we live now, there's a lot of construction, and I saw men putting Tyvek up, and it looked like a giant quilt wrapped around a building.\" From there, Andrew Mowbray began thinking about the relationship between the insulation the Tyvek provides, versus the warmth and comfort of a quilt \u2014 and \"how quilts function and what they do, in the context of history.\"\n\nThe obvious next step? To make quilts with the plastic that's used to cover houses.\n\n\"We've had the Bronze Age, the Stone Age. Today, we're in the Plastic Age. Half of what's in your house is made out of polyethylene, and you work on your computer all day and it's made of plastic.\" Andrew is interested in playing with the medium that has taken over our lives, questioning its function and place in our culture \u2014 and asking why we'll venerate art of other mediums but not necessarily plastic.\n\nMost of his quilts are about four feet square, but he's working on one right now that's eight feet square, which is about as big as he can go on his machine (it's hard to roll Tyvek as you move it through the machine). He combines the patterns and shapes of the word _Tyvek_ to form interesting geometric patterns and images. My favorite is his _Saturday Night Fever_ quilt, with a picture of John Travolta dancing in the center.\n\nThat ideal of the male as provider \u2014 the _Leave It to Beaver_ family \u2014 has changed over time. That's something that men are struggling with right now. Traditional quilting is more associated with women's craft, and Tyvek is more associated with construction, a traditionally male field. \u2014 AM\n\n# Go Heavy\n\nSewing with paper, plastic, and even heavier cloth (say, upholstery weight) can be a fun way to experiment with form. You might make cards, art quilts, place mats, or book covers. You'll need a new needle when you're done, but as long as you don't sew through too many layers at a time, your machine should withstand the play. Just beware of the possibility of a flying needle if the layers are too thick.\n\n# Yoshiko Jinzenji\n\n## The Beauty of the White Space\n\n_Hieroglyphic Quilt_ by Yoshiko Jinzenji, 96\" \u00d7 101\"\n\nWhite quilts with intricately \"embossed\" designs, or small blocks of color jumping across a cream- colored surface: these are the deceptively simple designs of Yoshiko Jinzenji, a Japanese quilter and weaver who has been working with fiber for more than 30 years. Many people are drawn to what looks \"modern\" in her designs. The spare use of color, for example. But step closer to one of her quilts, and you'll see its beautiful intricacies \u2014 the depth of the quilting (or, in some cases, what she calls embossment) and the play of small, varied shapes of color against the texture of the stitching. Her work has inspired many quilters; she was certainly mentioned many times over by other people in this book. She's published several books of her own, so you can learn more about her work and techniques there. If you want to take a class with her, you'll have to voyage to Japan or Bali.\n\nIn choosing to use innovative contemporary synthetics at times and at other times to weave and even dye her own cloth herself, Yoshiko has set a new asthetic standard as a quiltmaster and maker of cloth. \u2014 Jun'ichi Arai, Introduction to Quilt Artistry: Inspired Designs From the East\n\n# Angela Walters\n\n## Playing with the Long-Arm\n\n_Machine quilting_ by Angela Walters\n\nAngela can work in either traditional or modern modes; she doesn't identify herself with one or the other. But, modern quilters have found her quilting style and imagination to be the perfect match for her quilts. Ships at sea, spiders nesting in pieced corners, a wide range of quilting styles within one piece \u2014 what's not to love?\n\nAfter learning to quilt from her grandfather, Angela took his suggestion to buy a quilting machine; once she did that, she was sold. \"I knew pretty early on that I loved that part of quilting the best.\" She quilted traditional quilts for nine or ten years, and now that she's also quilted for modern sewers, she's \"in a good place to teach modern quilters to machine quilt.\"\n\nAngela Walters at her long-arm quilting machine\n\nIn her two fabulous books, she describes how to create different patterns, updating traditional designs, such as pebbling, by varying the sizes of the pebbles, or applying the patterns differently across the quilts. \"For years, I've been teaching classes in my mind as I sew. That was part of the conception of the first book. It's also really conversational \u2014 that's how I am when I teach, so that's how I wanted the book to be.\" She illustrates ways that quilters can \"use designs they already know to add to the composition, for example using the quilting to add movement or color. . . . I feel with some of the quilters I work with, the quilts are up here [she raises her arm up high]. I want my quilting to rise to that.\"\n\n# Long-Arm\n\nThe giant quilting machine: It has a large throat and you move the needle over the fabric (instead of moving the fabric under the needle). This way, you can quilt king- and queen-size quilts without having to scrunch them through your home machine. Look for a local long-arm shop or ask your local quilt shop for a referral. You'll need training before you can use the machine, and then you usually pay per hour.\n\nGo for it. Let what comes to you come. Even if you make a \"mistake,\" keep going \u2014 no one will notice but you, in the end. \u2014AW\n\n###### Tutorial\n\n# Free-Motion Quilting Paisleys\n\nBy Angela Walters\n\nFree-motion quilting happens to be my favorite part of making a quilt, and one of my favorite go-to designs is the paisley design. This quilting design is so versatile and will work in many different kinds of quilts, from modern to traditional. This particular design is a meander, which means you will quilt the same shape repeatedly to fill in an area on the quilt, whether it is a block or the whole quilt top!\n\n  1. 1. Start by quilting a teardrop shape and echo around to add another layer.\n\n  2. 2. Here is where you can add your own flair to the quilting design! You can make the curve to the side or make it more symmetrical. The more you quilt it, the more you will add your own flair to it.\n\n  3. 3. You can echo the design again or quilt another paisley. In this example, I have quilted another paisley and echoed around it.\n\n  4. 4. As you quilt, the most important thing is that you fill the area consistently, alternating between paisleys and echoing.\n\n  5. 5. Continue quilting until you fill the whole area with the quilting design.\n\n  6. 6. If you are struggling with quilting the design (or any design for that matter), try drawing it on a piece of paper. The most important thing in quilting is knowing where to go next. Drawing the design will help your mind figure out how the design flows.\n\n###### project\n\n# Jagged\n\nThis is a fun and easy quilt to make, especially if you have a couple of new quilting designs that you want to try out. This quilt pattern will help you make your own Jagged quilt and also give you ideas for the quilting.\n\n_Jagged_ by Angela Walters\n\nFinished size 44\" \u00d7 52\"\n\n### What You'll Need\n\n  * 11\u20444 yards of a print (dark blue)\n  * 1 yard of a solid fabric (tan)\n  * 1\u20443 yard of binding\n  * 3 yards of backing\n  * 50\" \u00d7 60\" piece of batting\n\n### Cut Out the Fabric\n\n  1. 1. From tan solid fabric: \n    * Cut 7 strips that are 41\u20442\" \u00d7 width of fabric (44\").\n    * Subcut 3 of the strips in half to measure 41\u20442\" \u00d7 22\".\n  2. 2. From the print fabric: \n    * Cut 9 strips that are 41\u20442\" \u00d7 width of fabric (44\").\n\n### Sew the Pieces\n\n  1. 1. Lay one 41\u20442\" \u00d7 22\" strip of solid (tan) fabric on a flat surface. With right sides together, place one corner of a 41\u20442\" \u00d7 22\" print (blue) strip on top, as shown.\n\n  2. 2. Carefully stitch a line diagonally as shown. (Feel free to mark the line first, if it makes you more comfortable.)\n\n  3. 3. Carefully trim 1\u20444\" from the line. Press the seam open.\n\n  4. 4. Repeat the above steps to make a total of 6 strips.\n  5. 5. Trim each strip to 44\", cutting off of the end with the print fabric.\n\n### Assemble the Pieces\n\n  1. 1. Lay out the strips together according to the diagram.\n\n  2. 2. Stitch the strips together, then trim the quilt top.\n\n### Quilt the Layers\n\nNow it's time for my favorite part, the quilting! This quilt is great for quilting because it has a lot of open space for you to practice! Of course, you can quilt this quilt anyway that you would like, but I have included suggestions.\n\n#### In the Print Sections\n\nQuilting this part is all about practice! In this particular quilt, the quilting isn't going to show up a whole lot, so you could use this as an opportunity to practice a new design.\n\nI wanted dense quilting, so I used a back-and-forth line to really smash it down. Don't worry about measuring and marking the lines for the quilting; you just want the lines to be close and straight-ish, for the most part. Starting at one side of the quilt, just work your way across the quilt.\n\n#### In the Solid Sections\n\nWhoo-hoo! This is where the quilting is really going to stand out. For this example, we are going to discuss echo quilting as well as a fun pebble design.\n\nBack of _Jagged_\n\n  1. 1. First up, the echo quilting. The term _echoing_ just refers to quilting a line that runs parallel to another line or seam. For this quilt we are going to quilt a line 1\u20442\" away from the seam in the solid fabrics. Doing this helps provide a little contrast between the different bands of the quilt, and also really adds interest to the quilt. When I quilted this, I quilted the echoing lines first and then filled in the rest of the area with pebbling.\n  2. 2. Up next, the pebbling. One thing I love to do when quilting modern quilts is to take a traditional machine-quilting design and change up the size of it. The circular quilting design in this quilt is a perfect example of just that. Instead of quilting the circles all the same smallish size, like traditional pebbling, I quilted the circles in several different sizes. You can make them as large and as small as you want! Mixing up the sizes will make the quilting interesting and keep you from getting bored.\n\nOnce you are done with the quilting, bind and enjoy! (See Finishing Your Quilt on page .)\n\n# Tip\n\nIf you want to piece a top but don't feel inspired to quilt it yourself yet, you can send it out to be quilted. You can request particular designs, collaborate with the quilter, or leave it entirely up to him or her. Costs vary, depending on size and complexity of the quilting. The best way to find a great quilter is to ask around \u2014 friends are usually happy to recommend quilters they like. You can also ask at your local quilt shop. If you search online and find a quilter whose work you love, it may be possible to ship your quilt to him or her for the job.\n\n# David Butler\n\n## Folk Modern\n\n_Curious Nature_ by David Butler, 62\" \u00d7 62\"\n\nAfter years of working as an artist and graphic designer, as well as photographing and marketing Amy Butler's fabric and designs, David finally accepted an invitation to design his own fabric line. His first line was Curious Nature, released in 2011, and his work as Parson Gray has been given a hearty welcome in the modern quilting world, resulting in richly textured gray, navy, and black quilts with accents of rust or sea green. \"I've always been interested in folk art and midcentury modern design. It has that hand-drawn, fine-art, folk-art character, but I wanted to do it in a minimalist way.\"\n\nThe Butlers keep their creativity alive by doing things their way. They create what David calls \"guerrilla photo shoots,\" shooting and styling everything themselves and capturing \"the shots between the shots.\" When it's time to get back to design, they work in their respective studios during the day and periodically consult one another for advice and feedback. They also spend time in the garden, make dinner together, and David sings lead in his band, the Black Owls.\n\nThe warm colorway from Parson Gray's Seven Wonders line.\n\nHe and Amy (page ) get reenergized and inspired by traveling the world. \"Amy and I love to travel, so we get away as much as we can. We recently did some book signings and TV appearances down in Australia. We took advantage of our time there and went to Bali, to do some of the things we wanted to do for creative fire. We take a lot of photos. Photography is everything. Amy's also a really good photographer, so we capture everything we see and bring it home with us.\"\n\nThere isn't a competitive or aggressive element that I've seen in other industries. There's a lot of support between designers and companies, and we've had great luck with professional relationships. It's an amazingly warm and welcoming community. \u2014 DB\n\n# Miriam Blaich\n\n## Sewing to Stay Grounded\n\n_Sunny Berlin_ by Miriam Blaich, 50\" \u00d7 67\"\n\nThough she says that Germany doesn't have the same quilting tradition as the United States, England, and Australia, Miriam has discovered several other German quilters in her own city of Berlin, and she's even founded (with two other quilters) the Modern Quilt Guild there.\n\nA peek of reverse, _Sunny Berlin_\n\nMiriam designs and sews her own clothing for a living, but it's making quilts that gives her the chance to unwind and relax. \"Some of my friends don't understand why I also sew in my free time, because I do it the whole day in my job. At work, I have to be fast, because the more clothing I sew, the more I can sell. But quilting is an absolutely different thing than I do at my job. I can take my time, and if a project takes months, who cares?\"\n\nAlways put your needle down through your fabric before lowering the presser foot. It eliminates the tension on the thread when starting to stitch, and it keeps the thread from breaking or slipping out of the needle. I only learned this about a year ago, and could have saved myself many years of frustration (and cursing) had I known it sooner. Plant your needles, people, for real. \u2014 Lauren Spencer Hunt ( _My Aunt June_ )\n\nQuilting is my anchor, where I can sew without worrying about making it perfect. \u2014 MB\n\n# Collaborate\n\nCollaborate with a friend, trading blocks or fabrics. Try setting up your sewing machines in the same room and co-creating a quilt. Join a bee. As Tacha Bruecher (page ) says, \"Bees have really helped me broaden my skills base and have taken away the fear of trying something new. Working in a group is tremendously inspiring, and you end up coming up with ideas that you wouldn't have had if it hadn't been for the input of others in the group.\"\n\n# Keep Someone in Mind\n\nKeep someone in mind as you make a quilt, even if it's not for them. See how your choices are driven by that person's personality, and what you'd like to express in a quilt made for them (idea from Rossie Hutchinson, page , and Rebekah May).\n\n# Kathy Mack\n\n## Virtual Community in Full Color\n\n_Line Dance_ by Kathy Mack, 41\" \u00d7 49\"\n\nKathy Mack uses her power for good, supporting up-and-coming designers, sewists, and bloggers at Pink Chalk Studio, featuring others' projects on her blog each day. Some of the patterns are theme-based, such as December holiday projects or sweet February valentines. After a career as a software designer, she began to blog, and then opened her shop. Now, she has eight employees, office space in a cool industrial-looking modern loft, and supports her family with the business.\n\n\"The whole online community is so empowering for women,\" Kathy says. \"Everything I've done, someone's helped me with it. I remember when I first met Susan Brubaker Knapp in person at Quilt Market. She ran all over to find the pattern buyer from Moda for me. She said, 'I am going to find her and introduce you; she needs to see your pattern.' And there are all kinds of examples of that; people just _really_ go out of their way to help and support colleagues in the industry. I just love being part of that. I try to do that as well. I always share knowledge, because I don't think that kind of knowledge should be kept to yourself.\"\n\nPeople ask how I got 11,000 followers on my blog. I tell them, 'I was a five-year overnight success.' I wrote a blog post every day for five years! \u2014 KM\n\nAlways express a genuine point of view \u2014 whatever your point of view is. And for me, it's always been sharing my enthusiasm about sewing. Sharing an experience I had making something has always been how I blog. I don't like the hard sell in social media, personally; it makes me cringe. Everything we do comes from a genuine feeling. We're here to make sewing fun for people. \u2014 KM\n\n# Rebecca Loren\n\n## Quilting by Palette\n\nRebecca's very first quilting project was the cathedral windows pattern \u2014 no small feat. She did it all by hand, to boot. Having learned to sew in eighth grade, Rebecca \"went back to it years later when I was feeling creatively itchy.\" Rebecca talks about her journey through quilting as an expression of the creativity that's always been there, but hadn't been let loose in a while. \"A lot of my journey as a quilter is a process of rediscovering the creative instincts I haven't had in daily use since I was very young. Having my children work side by side with me is inspirational and serves as a daily reminder to me of how much I loved exploring with art as a child.\"\n\nI met my husband for dinner after our first guild meeting, and spent the next hour screaming things across the guacamole: \"They're just like me! We like all the same websites!! You should see their quilts!! We're going to meet every month! We're going to sew at each other's houses!!!\" It's a revelation to find a true community of friends after having toiled alone with my needle for so many years. \u2014 RL\n\n# Make a Sampler\n\nLook for a quilting class at your local community center or fabric shop. Classes can help you build basic skills. And, as Rebecca explains, making a sampler quilt is a great way to begin. Sampler quilts include six to eight different blocks that you learn as you go.\n\n# Grandmother's Fan Variation by Rebecca Loren\n\n_Here, Rebecca describes how she was inspired to create her_ Crazy Grandma _, an Amish pattern that she made new, using colors inspired by interior design. Her soft palette and use of asymmetry update and reimagine this pattern._\n\nI found this palette on Design*Sponge, and I used it to choose my fabrics. I matched the grays and tans to Kona Cottons (Charcoal, Ash, and Raffia), and expanded the pink to include purple and lavender.\n\nI wasn't sure what I wanted to create until I saw a quilt created in 1899 by Lizzie Amanda Sundheimer in _America's Glorious Quilts_. Apparently, the quilting was completed in one day, in what \"one witness declared was 'the best bee ever sat.'\"\n\nThen, I started sketching my own design, trying to figure out how I'd arrange the fans to update the pattern. My first sketch was fairly close to the original.\n\nA traditional fan quilt and Rebecca's sketches on the way to her variation.\n\nThen, I found a fan __ template for free on www.quilterscache.com. I bought plastic grid material at Jo-Ann's and traced the fan onto the plastic, then cut it out so that I could cut lots of fabric at once with my plastic templates.\n\nOn one of my lunch breaks, I did a test run with these fabrics. Then I moved to my Design*Sponge\u2013inspired palette, and started cutting and piecing my fans. My kids helped me arrange the quilt on my design wall.\n\n_Crazy Grandma_ by Rebecca Loren, 90\" \u00d7 92\", cotton\n\n# Anna Williams\n\n## Scissors and Lyricism\n\n_Quilt_ (1995) by Anna Williams, 76\" \u00d7 61\"\n\n_Nancy Crow pays tribute to Anna Williams in her interview in the Smithsonian Archives of American Art (see page 59). She describes Williams's work as \"lyrical\" and remarks on the importance of realizing that quilts could be such. _\n\nAnna Williams really needs to be documented. She lives in Baton Rouge, Louisiana. She's illiterate. She's totally self-taught in her quiltmaking. She started making quilts on what I'd call a regular basis when she was 58. She's now made over 300 quilts. When I saw them for the first time, it would have been 1988, '89, and I was dumbstruck because she hadn't used a ruler. And I just thought, \"What's wrong with me?\"\n\nWhy did I think quilts had to be made with a ruler? Why did everything have to be straight? Why did I bite that off \u2014 hook, line, and sinker? You know? That's what I never understand about myself, that I am slow in some of my departments here. So, seeing her quilts, the fact that the line actually could be what I call sensuous, or lyrical, just blew me out of the water. But at the time, you know, I could see this in passing, but it hadn't really kicked in. It took another couple of years for it to kick in and really slap me up around the face. And she never used a ruler.\n\nOh, yes, that's when the bell went off in me. I knew I wasn't going to start using scissors, like Anna, but I thought, \"Man, I can take that darn rotary cutter and I can start to use it in a more expressive way.\" And I did. But it took me two years. I really believe you have to train the muscles. The muscles have to work with your eye and your heart. It all has to be a coordinated effort. \u2014 nc\n\nEverything she did was with scissors, so that's why the lines were crooked, because she just sort of cut the way she felt.\n\n###### 2\n\n# Improv\n\nFabric designer and (very funny) blogger Lauren Spencer Hunt says it best when she talks about the relationship of improvisation to quilting: \"I think of quilting improvisation as being like musical improvisation: you establish a key and time signature before you start playing, you create variations based on a predetermined riff.\" Though today's improv quilters may trace their roots to Nancy Crow (page ), Anna Williams (page ), and the Gee's Bend quilters (one of whom is pictured here), quilters have been improvising for centuries as they designed with available scraps. The level of improv varies from quilter to quilter. Maybe it's used in one section, or the whole quilt, or maybe the quilter sticks to an overall concept. Quilters make their own rules and parameters for improv, emphasizing different elements of the process. No matter what: Don't mistake improv as working willy-nilly. It requires a critical eye and careful attention to design.\n\n# Rossie Hutchinson\n\n## Copy Leftist\n\n_Coal Fraction Quilt_ (2009) by Rossie Hutchinson, 74\" \u00d7 86\", quilted by Bernie Olszewski\n\nRossie was the first to give modern quilters a place to gather online, with her creation of Fresh Modern Quilts on Flickr. She celebrates the wide range of quilters, \"those who have studied for decades under Gwen Marston, and those who have no idea who Gwen Marston is.\" She cites some of her own influences: Malka Dubrawsky's dye practices (page ), the Gee's Bend quilters (page ), and Denyse Schmidt (page ), as well as midcentury modern design.\n\nFor a long time, Rossie has been pushing quilters to think harder about what we're making and to ask, \"What is our aesthetic, and what does this community want?\" She's not interested in defining what modern quilting is for the group, but asks each person to think about what she makes.\n\nSay where you got the inspiration for your design. Explain whatever you can about your roots. In doing so, you'll let others find their own. \u2014 RH\n\n# Credit Where Credit Is Due\n\n\"My students,\" Rossie says, \"believe that they own everything that they make \u2014 that everything _should_ __ be owned.\" Instead, she argues, we should credit one another for our ideas, and not try to run \"around making everything [we] say or do into prop-erty,\" which is what many believe traditional copyright asserts.\n\nRossie is careful to always credit the source of her inspiration by including their name and a link back to their blog and quilt. And, in turn, she asks that others do the same for her. She and many other copy leftists have come to use the Attribution Share Alike license, which puts the emphasis on sharing and respect, rather than owner-ship. (For more information, see www.creativecommons.org.)\n\n\u00a9 Rivane Neuenschwander, Courtesy Tanya Bonakdor Gallery, New York\n\n_The Conversation Quilt_ (2012) by Rossie Hutchinson, quilted by Bernie Olszewski, lap-sized\n\n###### Project\n\n# Fraction Quilt\n\nby Rossie Hutchinson\n\ninspired by Cheryl Arkison\n\nWhat many modern quilters love is the chance to establish their own rules, whether they are sticking closely to tradition, working in improv, focusing on a \"modern\" surface design, or creating a conceptual piece. Rossie's project is a mix of these, with its emphasis on organic or \"imperfect\" shapes and using what Rossie calls an \"algorithm\" rather than a pattern as instructions. It was inspired by a quilt by Cheryl Arkison called _Your Parents Are Cool_ , which was in turn inspired by her friends' paint job in their nursery.\n\n_Curry Fraction Quilt_ (2012) by Rossie Hutchinson, 79\" \u00d7 86\", quilted by Bernie Olszewski\n\n### Lay the Groundwork\n\n  1. 1. In order to make this quilt, you need the following skills: \n    * Be able to sew straight seams on a sewing machine. Ideally this should be a scant 1\u20444\" seam\n    * Be able to press seams without stretching the fabric\n    * Be able to use a rotary cutter safely\n    * Be able to use fabric scissors\n  2. 2. The first step in designing your fraction quilt is deciding on a color scheme. I've presented two versions: a yellow and a gray one. The basic block for both quilts is the same, though the size of each element varies.\n\n### Organize the Fabric\n\n  1. 1. The background fabric in both quilts is a solid. Because the piecing will be improvisational, it is tricky to predict exactly how much fabric you will need. Both quilts are queen-size, and I bought 11 yards of the background fabrics. I used the same fabric to bind the quilts and happily put an extra yard or so from each in my stash.\n  2. 2. Select fabrics for the bar. In the gray quilt, the bar fabric is cut from a variety of gray prints and solids, all of which read as lighter than the background fabric. In the gold quilt, the bar fabric is Kona Cerise, which is darker than the background fabric.\n  3. 3. For the numerators and denominators pick three colors or color families, in a variety of prints and\/or solids. I like to have them be of similar value (darkness).\n\n#### Why Three Colors?\n\n  * One for the numerators (50 blocks)\n  * One for the denominators (50 blocks)\n  * One will be the \"switcher,\" which will get represented far more often as it will be used both as a numerator and denominator (100 blocks)\n\n### Make the Blocks\n\n  1. 1. **Cut numerators and denominators.** Part of what makes the fraction quilt interesting and lively is that the angles aren't perfect 90-, 45-, and 30-degree cuts. Also, the numerators and denominators vary in size throughout the quilt. The way to achieve this look is to hand-cut the numerators and denominators; this allows you to have straight lines but imperfect angles. Here are some pointers: \n    * Do not use a template or a rotary cutter. Use fabric scissors to cut quadrilaterals that are approximately 3\" wide and 4\" tall (give or take 1\u20444\"). The longer dimension will be the height.\n    * I find it useful to cut a piece of paper to the minimum and maximum sizes and keep them nearby so I can spot-check if my numerators and denominators are within range.\n    * When cutting, don't try to make the pieces perfectly rectangular. But, don't try to make them look wonky or super-slanty either. Think of them as old houses that are settling ever-so-slightly. Unforced and beautiful irregularity can be created by moving the end of the scissors just 1\u20448\" or 1\u20444\" to the left or right.\n\nYou need to cut 50 numerators, 50 denominators, and 100 switchers. You can cut some now and some later. Waiting can be a good choice if you want to get a look at how your colors play together before cutting all of your fabric.\n\nIn the gray quilt, all numerators and denominators were cut from whole cloth. In the gold quilt, some were cut from patchwork made using a charm pack of Kona pastels. If you enjoy working with scraps, you may want to make color-family patchworks and then use those.\n\n  2. 2. **Attach sides to denominators, numerators, and switchers.** In the photograph (left), A, B, C, and D mark the pieces of background fabric that you attach in this step. \n    * Cut approximately thirty 21\u20442\"-wide strips of background fabric (cut from selvage to selvage).\n    * Take one of those 21\u20442\"-wide strips and place it faceup on your sewing machine. Feed it under the presser foot, attaching one side of a numerator to it as you go. Leave about a 1\u20444\" gap after each numerator, then feed in another without stopping to cut the fabric strip or the thread. After completing this step, cut the strip between the numerators.\n    * Follow the same procedure to attach B to the other side of the numerator.\n    * Repeat for the denominators and switchers.\n\nPress all of the patchwork. I press toward the background fabric, but you may wish to press the seam open or toward the darker fabric.\n\n  3. 3. **Attach the bar to the numerators.** Use a rotary cutter and ruler to straighten the bottoms of the numerators. Cut 11\u20442\"- to 2\"-wide strips of the bar fabric with the rotary cutter. Adding a tiny slant to one side of some of the bars adds liveliness to your blocks. You can do this with a rotary cutter, but be careful to keep the angles very slight. With right sides together, stitch one side of the bar to a pieced numerator strip.\n  4. 4. **Pair up numerators and denominators.** Straighten the tops of the denominators. Pair them up with the numerator-bar pieces. With right sides together, stitch them to the remaining side of the bar.\n  5. 5. **Attach a frame.** This step consumes a bit of time and fabric, but this is what allows the fractions to look like they are floating, so hang in there! \n    * Divide your fractions into two even piles of 50.\n\n    * Take the first pile and use your rotary cutter and ruler to straighten the tops and bottoms of the blocks. Do not square them; just give them a straight edge. Cut 21\u20442\"-wide strips of your background fabric (about 10 strips) and sew to the tops and bottoms of each block in this pile. Press the blocks.\n\n    * Take the second pile and this time straighten the sides of the blocks. Do not square them; just give them a straight edge. Cut 21\u20442\"-wide strips as before and sew to the sides of each block in this pile. Press the blocks.\n    * Finally, add tops and bottoms to the blocks that have only sides, and sides to the blocks that have only tops and bottoms. Press the blocks and trim them to 9\" tall and 81\u20442\" wide. It is part of the nature of improv piecing that you may have to go back and add more fabric to reach that size. Keep the two kinds of blocks in separate stacks.\n\n### Construct the Quilt\n\n  1. 1. **Make rows.** Use a design wall or another surface to lay out your quilt. \n    * For each row, alternate between the two types of blocks. I like to begin two rows with one type of block, and the next two with the other type, and so on.\n    * The top row has 11 blocks.\n    * Before you lay out another row, cut 10 pieces of the background fabric, 41\u20442\" x 9\". Use one of these pieces to begin the second row, then place 9 blocks, alternating between the two types of blocks, and ending with another 41\u20442\" \u00d7 9\" piece of background fabric.\n    * Repeat these two types of rows for a total of 10 rows.\n    * Once you have everything in place, play around with the placement of the blocks. In the gold quilt, I found that I needed to place all the blocks with bright turquoise in them near each other. Be sure to keep the types of blocks alternating as you make changes.\n    * Once you are happy with your composition, assemble the rows. Prevent twisted seams by pinning the blocks so that the ones with unpieced sides are on the bottom of each pair. Press.\n  2. 2. **Connect the rows.** Piece them to each other. I find it useful to match the rows at their centers and pin from the center out.\n  3. 3. **Back, baste, and quilt** as you like (see Finishing Your Quilt, page ).\n\n# Gee's Bend Quilts\n\nIn 2003, the Gee's Bend quilts were exhibited at the Whitney Museum of American Art in New York. This was a stunning surprise in the art world, as women who had never had any art training nor traveled beyond their homes in rural Alabama had created these masterworks of modern art. The exhibit was incredibly popular, because the women had been able to achieve such great design under such limited circumstances.\n\nThese were quilts made of necessity by former slaves and their descendents and laborers with little means in an isolated region of Alabama. But their care in designing the quilts resulted in a powerful sense of color and improvisation. It's possible to see the living done in the clothes that were repurposed in the quilts: denim and corduroy with worn knee and elbow marks, old feed sacks, and dress scraps.\n\nMany quilters with whom I spoke cited the Gee's Bend quilts as their inspiration. These quilts opened up new possibilities of what quilts could be: beautifully designed _and_ __ functional, with improvised patterns and strong color combinations.\n\nToday, you can take quilting classes from Gee's Bend quilters, stay with the quilters, or visit with them while they sew, through the Quilters Collective. The women sew each morning at the Gee's Bend Ferry Terminal and Welcome Center.\n\nKeeping pieced quilt tops in closets, hampers, and under beds was like keeping gold bullion in the home. \u2014 Alvia Wardlaw, introduction to The Quilts of Gee's Bend.\n\n# Katie Pedersen\n\n## Building Skill Blocks\n\n_Value Quilt_ by Katie Pedersen, 75\" \u00d7 77\", machine-quilted by Angela Walters\n\nKatie's innovative quilt designs brighten Seattle's gray skies in her blog photos. She and Jacquie Gering (see page ) are the talented team who wrote _Quilting Modern: Techniques and Projects_ for _Modern Quilts_. Both women talked about how productive the collaborative process was for them. They pushed each other to try new techniques and colorways, hone their writing voices, and polish the manuscript. \"That was a step outside the box for us, to work with different techniques and colorways. There was a lot of back-and-forth communication: what works, what doesn't, how can we fit this quilt into the book, what tweak does this quilt need to make it original?\" The result is a book with techniques that guide a quilter to innovate and find her own voice, skill by skill, rather than quilt by quilt. Quilters are encouraged to apply those skills to design their own pieces.\n\nIf you're lucky enough to live in the Seattle area, you can find Katie's classes at many of the local shops or follow along on her blog. See page  for another of Katie's quilts.\n\n# Stacked Coins Quilt\n\nThis was the second type of quilt Katie Pedersen ever made, and she found it was great for beginners. It's easy to improvise, or you can follow a pattern. It features several strip-pieced rows (strips of fabric sewn together lengthwise).\n\nDetail of quilt by Monica Ripley (page )\n\nI live in a really small space, so I try to be organized and finish projects. Generally my process is to switch between an improvisational quilt design and then a modern adaptation of a more traditional design. This process of jumping back and forth keeps quilting fun for me. Improvisational design takes more thought, but sometimes it's nice just to sit down and mindlessly sew some blocks together. Both styles of quilting require planning out the fabrics, which is my favorite part. \u2014 KP\n\n# Kyoung Ae Cho\n\n## Collaborating with Nature\n\n_Fallen_ by Kyoung Ae Cho, 42\" \u00d7 42\"\n\nDiscarded natural materials, such as gathered or harvested wood, leaves, or flowers, usually appear in whatever Kyoung Ae Cho is creating. Since 1984, her work has been shown in solo and group exhibitions around the world, and anthologized in many books on art and art quilting.\n\nBefore she came from South Korea to the States for her graduate degree, Kyoung Ae was interested in working with nature, but she says she has been able to focus on that work more intensely here. \"Everyone says, 'Oh, you're Asian, you must deal with nature!'\" she laughs. \"My conceptual work had to do with nature in Korea, but it was coming to America that made me start truly dealing with natural materials because they're so abundant here. As a Korean growing up in a tiny country, we don't have that much land to have big trees in front of our house \u2014 nobody has grass in their front yard. So, it was very refreshing to find that in America everyone has a lawn. I started working with those materials.\"\n\nKyoung Ae will embed hundreds of pieces of leaves or wood into a quilt, hand-sewing around the pieces to hold them in place. The pieces will degrade over time. \"I'm trying to have a conversation with nature in order to understand it. We're all part of nature, and I honor and appreciate it. Even disaster can be a beautiful natural phenomenon. There is a rhythm in nature, and things we all want to be part of. And to be part of it, we have to understand it. That's what's behind my work, but I know what I'm doing is a baby step toward that, like a child playing in a sandbox who understands after awhile. I sew one little leaf at a time, and if someone comes along and appreciates that leaf, that's good enough. I'm playing, but with a sense of responsibility.\"\n\nKyoung Ae's grandmother taught her to sew as a child in South Korea.\n\nAs a professor at the University of Wisconsin\u2013Milwaukee, she stresses that she wants her students not only to \"play,\" but also encourages them to think about why they're making their work, and how they can carry with them a sense of responsibility for the world we live in.\n\nI have been learning about how leaves change and hold colors over time. It seems that leaves behave differently depending on when they are picked up and in what stage they are harvested. I understand that they will change over time as nature does. Those gathered leaves are cleaned and pressed in a book, and then I iron them when they're quite dry. It's like blanching vegetables; the heat seems to help to seal up the color in the leaves, and helps the color stay in a little longer. I have tried to protect leaves by sealing with beeswax and lots of other things, but I find holding the leaves between fabric works the best for me for what I do so far. Since leaves are in a natural state, they seem to react to moisture in different seasons by expanding and shrinking, which I keep in mind when I work. I call my work a collaboration with nature. \u2014 KAC\n\nIt doesn't matter how we categorize ourselves. I'm a maker, and not only for selfish reasons; I hope that I'm contributing to a better world, in some way. \u2014 kac\n\n# Rayna Gillman\n\n## Improv Harvest\n\n_Harvest_ by Rayna Gillman, 30\" \u00d7 27\"\n\nRayna Gillman hand-dyes her fabric and often pieces improvisationally. For her, the improv process is about spending time with the fabric, putting it on the design wall, and moving the pieces around. \"To find the right balance, there's a little bit about color and value, there's a little bit about composition. It's not an easy thing. Sometimes, a piece will stay on my wall for months and months and months until I'm happy with it. Sometimes, I'll take pieces down, then put them up later to see them with a fresh eye.\"\n\nDetail of _Harvest_\n\nShe works to pass on this sense of experimentation and self-critique to her students, encouraging them to look at their own designs until they've achieved that sense of the whole. \"When I teach this, there's a lot of critique and feedback from the group. We look at people's work in process, and ask, 'What if we moved this over here? What if you tried this?'\"\n\nI try to get people into the process of _not_ planning ahead. \u2014 rg\n\n**Quilt to Heal.** Machine-quilter Angela Walter's blog name is _Quilting Is My Therapy_. Many quilters agree, saying they use their quilting to fuel creativity, release tension, grieve, or celebrate. One of Rayna's favorite pieces came out of a difficult time when her husband was sick. She spent months working on _Harvest_ , and says that for her, \"There always has to be an emotional connection to a piece.\"\n\n###### Tutorial\n\n# Improv Piecing with Scraps\n\nby Rayna Gillman\n\nThis is an improvisational process, so feel free to play and experiment!\n\n  1. 1. Cut 7 to 10 strips without a ruler. Strips can be anywhere from 3\u20444. \" to 11\u20444\" wide and 10\" to 12\" long. Varied widths make it more interesting. Stitch the strips together.\n\n  2. 2. Slice the strip set in half (or thirds, depending on length and your mood). Do not use a ruler.\n\n  3. 3. Insert a free-form strip and join. Generally speaking, a solid color is better if you're using prints, though here I'm already breaking that rule.\n\n  4. 4. Sew units together.\n\n  5. 5. You can also turn one side upside down, instead, if you like.\n\n# Nancy Crow\n\n## Improv Master\n\n_Constructions #84: No!_ \u00a9 Nancy Crow (2007), 75\" \u00d7 70\", hand-dyed cottons, machine-pieced, photo \u00a9 J.Kevin Fitzsimons\n\nPerhaps no single person has influenced today's quilters more than Nancy Crow. Her quilts have changed the way people see what is possible with fabric. Though I was still a child when she was revolutionizing the quilting world in the early 1980s, I was lucky to have seen her quilts up close at a local exhibit. I loved her rich, hand-dyed colors; her zigzag compositions; and the way each quilt in the series spoke to the others in line and color.\n\nAnd, the perfect-but-imperfect hand-quilting (done by Martha Hattabaugh), accenting the straight lines of the piecing. She says that what she \"really loves are proportions,\" which is clear when you look at her quilts and see the balance of the colors and the shapes that are in harmony within the quilt, and in conversation with every other quilt on the wall.\n\nThis quilt is one of a series of 10 quilts Nancy created to memorialize the Chinese teenagers she witnessed being loaded into trucks to be driven to their execution for petty crimes. In a statement about these quilts, she said, \"The circles represent their souls and the bull's-eye embroidery and the hand-quilting represent the ropes tied around their souls. The colors of the circles represent the individuals. I have always felt there is an eerie energy that radiates out from the surface of each of these quilts.\"\n\n_Chinese Souls #2_ \u00a9 Nancy Crow (1992), 79\" x 89\", fabrics dyed by Crow and resist-dyed by Lunn Fabrics; embroidered by Crow, Marla Hattabaugh, Suzanne Keller, and Maria Magisano; machine-pieced by Crow; quilted by Hattabaugh with pattern denoted by Crow\n\nCrow learned to sew in 4-H at about 11 years old. Her father, who was apprenticed to a tailor as a teenager, taught her to love fine fabrics. After learning to quilt in the traditional modes, she began to innovate, first cutting templates that she'd fit together \"like a puzzle,\" and then finding ways to cut improvisationally. While she teaches workshops and leads groups on textile adventures around the world (wouldn't you love to explore Peruvian textiles with Nancy Crow?), she says that she doesn't teach her own method of quilting, because that's hers \u2014 her voice, her distinct way of working. Spoken like a true artist.\n\nCrow founded Quilt National, the major art quilt show where, as Jane Sassaman (see page ) says, you'll see what's cutting edge in quilting today. She's had exhibits at the Smithsonian, Boston's Museum of Fine Arts, and major galleries around the world. She's also authored several books \u2014 most recently, the award-winning _Nancy Crow_. Suffice it to say: She's done _a lot_ , and her influence on modern quilting in terms of surface design, as well as improvisational piecing, is immeasurable.\n\nWhat I love about her as well is her insistence on developing her own voice, and establishing her own space in which to work. And what a space it is: her idyllic Quilt Barn, where she creates, has specialized lighting and high ceilings.\n\nNancy Crow is often called the founder of the contemporary art quilt movement. . . . Her work is the standard against which all other art quilts are measured. \u2014 Rebecca A. T. Stevens\n\nHere is an excerpt from a December 18, 2002, interview with Jean Robertson for the Smithsonian Institution's Archives of American Art.\n\n**CROW:** I brought this magazine down here because a statement in an article really hit me. This is the interview in the current issue of the _New Yorker_ magazine, and this is about a man . . . [who] died in the Holocaust, as a young artist. . . . [H]ere's what he says . . . \"All artists spend their lives interpreting images that are stamped in their minds during childhood.\" I believe that. . . . I would lie in bed at night staring out the gridded windows in my bedroom, because we had those panes, those small panes with the wood between . . .\n\n**ROBERTSON:** True divided.\n\n**CROW:** Divided. True divided pane windows or whatever they called them. As a child, those windows just impacted me incredibly, and it just comes up over and over in my work, the fact that I was up in a room where there were a lot of windows, and all I saw were big trees out the windows. So I think those two images, the gridded window and the tree, have just impacted me.\n\n# Victoria Findlay Wolfe\n\n## Make It New, Give It Back\n\n_Rooster Tracks_ by Victoria Findlay Wolfe, 56\" \u00d7 56\"\n\nBecause Victoria endured long, cold winters as a child, quilts were a necessity. \"This was in Minnesota, so you had four or five quilts on your bed so you wouldn't freeze to death in the wintertime. My mother and grandmother made those quilts, and we had that weight of the quilts on us all winter.\" She was raised with the notion that quilts are made from scraps rather than new fabric, and she's always pieced improvisationally. It wasn't until she started quilting professionally about four years ago that she realized, \"Oh, I could go to City Quilter and buy fabric!\" and began to stumble upon the great quilt books and blogs that are in the world. Pretty soon, she started a modern quilt guild.\n\n# Play!\n\nVictoria runs the 15 Minutes of Play website (and has published a book by the same name), encouraging quilters to spend 15 minutes a day playing with fabric. And of course, as with anything, the small parts add up \u2014 those minutes build, day by day, to finished projects, and eventually to a body of work.\n\nVictoria also spends a lot of energy donating quilts to others. She started the organization Bumble Beans BASICS, which gathers quilts to donate to homeless families in New York. The project grew out of her decision to donate one quilt, making a call for blocks from blog readers. She got far more than she imagined she would; they've raised $30,000 and donated 700 quilts to homeless families waiting for transitional housing.\n\nI live in the garment district, so I have to walk down the street with blinders on. I can get drawn to the notions shop, button shop, trim shop, the luxurious fabric shop. Satin, taffeta, brocades, upholstery fabric \u2014 I always think, \"Hmm, could I use that in a quilt?\" \u2014 VFW\n\n###### Tutorial\n\n# Curvy Dresden Improv Piecing\n\nby Victoria Findlay Wolfe\n\nwww.bumblebeansinc.com\n\nFor those who have yet to try improv, this is a great way to change up your quilting. If you've been into improv for a while now, working with curves and Dresdens might lead you to something new.\n\n_Dropping a Stone_ by Victoria Findlay Wolfe, 54\" \u00d7 54\"\n\n  1. 1. Cut various width strips of fabric from 11\u20442\" to 3\" and about 9\" long.\n\n  2. 2. Overlap a light and a dark strip about halfway and use a rotary cutter to cut a simple wavy line from top to bottom through both layers.\n\n  3. 3. You can see here that you have the left mirrored cut of the right.\n\n  4. 4. Flip the dark strip with right sides together onto the light.\n\n  5. 5. Match your edges at the beginning and proceed to sew slowly, constantly keeping your raw edges together. You have to move the top layer a bit to do so.\n\n  6. 6. If you have a wave that is a bit pointy, clip your seam allowances to help them lie flat. Often these waves are mild enough that you won't need to clip them.\n\n  7. 7. Iron your seam allowances to the darker color.\n\n  8. 8. Continue wavy-cutting your other strip sets and sewing them together in this way.\n\n  9. 9. Once you have your wavy strips sewn together and pressed, lay a template of your choice on the fabric, with the wavy lines running vertically.\n\n  10. 10. Playing with different shapes can give you surprisingly fun results! Try Dresdens, tumblers, or even hexagons. (My Dresden template is 1\" wide at the bottom, 3\" wide at the top, and 6\" tall.)\n\n# Denyse Schmidt\n\n## An Artist's Vision\n\n_Hope as the Anchor of the Soul_ (Mount Lebanon series #3) by Denyse Schmidt, 85\" \u00d7 92\"\n\nWho was featured in _Martha Stewart Living_ before quilting was \"cool\"? Who was written up in the _Wall Street Journal_ as the \"Queen of the Neo-Hillbilly Quilt\"? Why yes, that's our friend Denyse Schmidt. You might be surprised to learn that before she found fame and fortune as an artist and quilter, she held jobs sewing for the Boston Ballet, was a children's book art director (she was \"an expert in Barbie pink,\" she says with laughing horror), was part of the 1980s downtown New York art world, and even made ecclesiastical vestures for priests. \"What was interesting about it from a design point of view is that those garments haven't evolved much since medieval times. Albs, stoles, bishop's miters \u2014 look at a painting from the fourteenth century and they were wearing the same thing! The fabrics were really incredible, gold brocade. It was this weird dichotomy of incredibly ancient, beautiful, simple design, and beautiful, very expensive fabrics. And then all the ladies sewing and me.\" She laughs.\n\nThis doesn't sound so different from the focus of her quilting \u2014 marrying beautiful fabrics with traditional patterns. She continues to create one-of-a-kind quilts for exhibits and sales in New York and elsewhere, and has been claimed by the modern quilting world as one of its heroes, someone making modern quilts before they'd been labeled \"modern.\"\n\nNarrowing down our choices helps us create. \u2014ds\n\n# Out of the Comfort Zone\n\nDenyse Schmidt is well known for running workshops to help quilters learn to improvise. She often teaches using her now-famous paper bag method. \"It's really hard to unlearn that blue doesn't go with brown. It stays with you for some reason. I don't know how I landed on the paper bag thing, but in some ways it was the smartest thing I've ever done. It's the best way to get people to let go \u2014 you're not responsible anymore. Maybe it was my dance background; Merce Cunningham incorporated chance into his choreography and John Cage did the same in his music.\" Narrowing down our choices helps us create, she says. \"That idea of making things from what you have on hand \u2014 this is all I have \u2014 we don't have that today. People have so much fabric and so much choice. So, to limit choice seemed like a great way to get people past what they think they know. Because I think that's when you make discoveries.\"\n\nDenyse got into quilting after spending a lot of time in the art, dance, and sewing worlds, having long loved simple, antique quilts. It was while working as an art director that she began seeking a way to express her voice in quilting. \"Before I started the business, I was looking at the work of Nancy Crow (page ), and through Nancy discovered the work of Anna Williams (page ), and that completely changed my whole perspective.\n\n\"I took a workshop with Nancy Crow for a couple of weeks at Haystack, this craft camp for adults up in Maine. It's really beautiful, and you have the opportunity to work all day, and your meals are made for you. When I returned home, I figured out how to make my schedule work and made my first samples for a trade show.\" She explains that her quilts were different from the art quilt world, or the traditional quilt world, and that she wanted to make something for which there didn't yet seem to be a place.\n\n\"I wanted to make functional quilts that harkened back to very spare and simple designs. I wanted to get people who were like me to look at quilts in a different way.\" She found a way to express her aesthetic, and it's the title of her new book that says it best: _Modern Quilts, Traditional Inspiration_. Now she's the one to whom people turn for workshops, ideas, and ways of innovating antique designs.\n\nI wanted to get people who were like me to look at quilts in a different way. \u2014 DS\n\nWhile so many in the modern quilting world see Denyse Schmidt as their, well, queen, she sees herself as slightly apart, not really a member of any particular school or group (art quilting, traditional quilting, or even modern quilting). Her touchstones and sources of inspiration are still in the art world. When we talked at Quilt Market, she was excited about heading off to see the Friends meetinghouse designed by James Turrell, \"who works with light and space.\" While all of us were milling about, networking until our feet were sore and our heads ached, Denyse was watching the sun move over a hole in the top of the chapel, observing the changing shadows and light.\n\n\"Neo-hillbilly\" is what she sometimes calls her style. While most art quilters in America push the envelope on over-the-top embellishment and dense threadwork, Ms. Schmidt strives to make simplicity a virtue. \u2014 Meg Cox, Wall Street Journal\n\n###### Tutorial\n\n# Curved Piecing\n\nBy Angela Walters\n\nPiecing curves doesn't need to be intimidating, or even frustrating. All you need are some pins and a little time. Soon you will be piecing perfect curves!\n\n### Instructions\n\n  1. 1. Find the center of the curve on each piece of the block. I do this by folding it in half and pinching the center with my thumb and finger leaving a temporary mark. However, you could use a marking utensil or a light pencil mark.\n\n  2. 2. With the right sides together, line up the middle of each curve and pin the two pieces together.\n\n  3. 3. Pin the ends together as well. Don't worry if it looks weird!\n\n  4. 4. Gently easing the sides of the block together, pin the end and the center of each side until the curve is pinned and ready to sew.\n\n  5. 5. Using a 1\u20444\" seam, carefully sew around the curve, removing the pins as you go.\n\n  6. 6. Open up and carefully press your block, being careful not to stretch out the fabric.\n\n# Sherri Lynn Wood\n\n## Quilting to Help Others Heal\n\n_RGB Modern_ by Sherri Lynn Wood, daintytime.net, 72\" \u00d7 57\"\n\nHer background in theology, art, and quilting all come together to create the quilts that are Sherri Lynn Wood's masterpieces of improvisational piecing. \"My work right now is really all about improvising, which is a process essential in all forms of art and, more important, in life. Certain African-American quilts are considered to be made in the improvisational style because of the way they shift patterns freely. I'm primarily focused on the improv process as a subset of quiltmaking.\"\n\nSherri uses improvisational quilting to help people who are grieving. She's able to interweave the creation of her own work with the workshops she runs to help others. \"I'm currently working on a king-size linen quilt for a commission. And on Saturday I facilitated an improv-quilting workshop for people who were friends with a man recently killed in a car accident. They wanted to make quilts from his clothing to give to his surviving wife and children.\" The making of these quilts isn't just about \"memorializing people; it's also a hands-on bereavement process for the participants \u2014 a kinetic, creative process for moving through grief.\"\n\nImprov is in music, theater, dancing, painting, drumming \u2014 there are all kinds of improv. \u2014 SLW\n\n###### project\n\n# Modern Memory Lane T-Shirt Quilt\n\nby Sherri Lynn Wood\n\nT-shirt quilts are all the rage, probably because T-shirts with logos of all sorts have become a staple of the American wardrobe. More often than not, the T-shirts' logos are centered and cut into large uniform squares and sewn into a grid. This approach effectively concentrates sentiment by gathering all the memories in one place, so let's take it to the next level by adding more visual interest.\n\n_Modern Memory Lane T-shirt Quilt_ by Sherri Lynn Wood, daintytime.net, 43\" \u00d7 48\"\n\nWhen I'm working with memories, I want to create a quilt that is as beautiful as it is meaningful. So when I make a quilt with lots of T-shirt logos, I like to mix it up. To do this I combine a contrasting mix of fabrics with the T-shirt to create what I call a _memory lane_. Even though this process fragments the logos, they are still readable, because they are held together in a path of color.\n\nAnother example of a T-shirt quilt\n\n### What You'll Need\n\n  * T-shirts\n  * Contrasting fabric from your stash or other meaningful clothing\n  * Iron-on interfacing for knits (optional)\n  * Spray starch (optional)\n  * Backing fabric\n  * Batting\n\n### Instructions\n\n  1. 1. **Gather and prepare your T-shirts.** If they are mostly bland, you will want to include other brightly colored or contrasting cloth to combine with your T-shirts. This will give your memory lanes some punch. Fabric from your stash will do, or you might choose other clothing and materials that carry meaning or tell your story. If your T-shirts are threadbare or extremely stretchy, use an iron-on interfacing to reinforce and stabilize. Heavy-duty spray starch is another alternative for stabilizing stretchy material. Otherwise, try not to stretch the material as you sew.\n\n  2. 2. **Cut out each T-shirt logo without a ruler.** Leave plenty of space around it.\n  3. 3. **Include neck cuffs.** Sometimes I include the neck cuff of the T-shirt along with the logo, then fill in the curve with other fabric to maintain a straight edge at the top. To do this:\n\n    * Place the neck cuff of the T-shirt over the contrasting memory lane cloth and pin in place.\n    * Use a blind appliqu\u00e9 stitch to sew the cuff in place on top of the memory lane cloth.\n\nBlind appliqu\u00e9\n\n    * Trim away excess cloth on the wrong side and across the top from shoulder to shoulder.\n\nTrim\n\n  4. 4. **Prepare a base-cloth.** Use a contrasting color and\/or value from your T-shirts to create a base-cloth that roughly matches the size of each T-shirt section. Sometimes I combine two or more fabrics to create my memory-lane base-cloth. In this instance I'm using the blank sides of my T-shirts, but feel free to include materials from your stash or other clothing that carries meaning.\n\n  5. 5. **Create your memory lanes.** Layer the T-shirt logo under the base-cloth, _right sides up_ , so that the top edge of the base-cloth overlaps where you want to make the first cut-through of your T-shirt.\n\n    * Cut the same line horizontally across and through both pieces of cloth. (See Options for Shaping Cuts, below.)\n\n    * A couple of inches below the first cut, repeat the above step, this time layering your T-shirt section on top of the memory-lane section.\n\n    * Carefully gather all your cuts, keeping them in order.\n    * Starting at the top, pin the first two strips with rights sides together and sew.\n\n    * Pin the third strip to the second strip in the same way and sew. Work your way down the strips until you're done.\n\n# Options for Shaping Cuts\n\nWhen cutting freehand, be present and move from your core. Your cutting line is the signature of your hand.\n\nYou can cut straight across. You can cut wedge shapes. You can cut curve shapes. You can cut curvy wedge shapes. You can rotate your base-cloth in the opposite direction, if made from multiple fabrics.\n\nRemember that a 1\u20444\" of the logo will disappear in the seam at each cut.\n\nIt is okay if your memory lane curves or the sides are unequal. Leave finished sections untrimmed for now.\n\n  6. 6. **Assemble your memory lanes.** Once you have created all your memory lanes, arrange them like a puzzle with right sides facing up. Pay attention to natural fits, and bleeds of colors and shapes across seamlines.\n\nOnce arranged, match up the outer edges by overlapping sections that will be next to each other and cutting the same line through both sections, so that they mirror each other. As needed, add fabric to lengthen a section (so it's equal to the piece you are sewing it to), or cut off the excess after you sew the sections together.\n\n  7. 7. **Make the quilt sandwich and quilt.** (See Six Steps to a Quilt on page .) Hand-quilting puts the body in a soothing meditative posture, perfect for quiet reflection when quilting alone, or for sharing stories, memories, and emotions with others.\n\nWhen hand-quilting, I mirror and echo a simple freehanded scallop pattern that requires very little advance marking. I use a size 006 embroidery needle and #8 DMC pearl cotton. It helps to concentrate on making straight and even stitches and not on stitch length. Smaller stitches will come naturally with repetition.\n\n**Tip:** Make a striped fabric out of your remaining T-shirt, and base-cloth scraps to use as a border.\n\n# Danielle Krcmar\n\n## Evidence of the Body\n\n_Beach_ by Danielle Krcmar, 75\" \u00d7 81\"\n\nAn artist who usually works as a figurative sculptor but also slips into other mediums, Danielle is interested in our attachment to and interest in the body, and the body's attachment to and connection with the world. \"Everything comes from and goes back to that. That's also what drew me to quilts.\" She says that it's the sense of where the \"fabric came from\" that inspires her quilts \u2014 and much of her art.\n\nI remember reading a story ages ago, about a quilt that included a rose made from somebody's bloomers; people at the time were scandalized. \u2014 DK\n\nHer _Beach_ was made from old shirts, and she keeps some of their shapes: the pocket (perfect for hiding valuables when walking), strips of buttons, and cuffs. These pieces echo their former lives as protectors of the body. \"And for both of these pieces and the other work, there's a communal sourcing \u2014 the shirts are old shirts from family, friends, or tag sales \u2014 there's a randomness and a pleasure in that. It's the opposite of going to the fabric store and getting the kits. There's this other meaning because of where the material comes from. The same for the glass, too, that I gathered from the beach.\"\n\nDanielle has gathered discarded glass, shoe leather (from bottle dumps near turn-of-the-century houses in rural Massachusetts and Pennsylvania), and cloth, among other things, to make her work. That repurposing has a lot to do with the piece's layers of meaning. She's both inspired and constrained by using found materials. \"You can't assume what it's going to be until you get the pieces and put them together \u2014 like putting together a puzzle. I'm making choices about it, but it's about working with what's been given as opposed to using some pristine material.\" Those materials hold evidence of former lives: bottles that were drunk from and discarded, shirts and shoes worn until they took the form of the body, things were thrown out or given away, to become something else.\n\nDetail of _Beach_\n\n# Finding Balance in Improv\n\nDanielle says that she responds to outsider art (work made by those who are self-taught) because of the evidence of the hand in the work. \"It's the hand-stitched quality of the older quilts that I love. And with my glass sculpture, I start with a grid that's mapped out on the wires. But as I work, the tension and the pulling changes it \u2014 I'm not a machine, so it's not a tight, perfect grid.\" This process seems very similar to the process of making a quilt: coming up with a design, finding balance in shapes and sizes, adjusting as you go.\n\n\"It's playing with a balance that appears random \u2014 small, small, big \u2014 and balancing shapes. I don't lay it all out very methodically, but it's a cumulative process. That's the way I work with the quilts, too; there's a very clear overarching vision \u2014 but within that larger structure, there's randomness that comes in. You can visualize something, but once you do it, it may be different. With the machine-stitching in some quilts, it's like it's almost too perfect \u2014 and by perfect, I mean _closed_ _._ I want the pleasure of feeling someone's hands when you look at the work.\"\n\nDetail of _Fragment House_ by Danielle Krcmar\n\n###### 3\n\n# the personal is political\n\nWe all know this phrase from feminism's second wave, and it seems apt to mention it here in the context of quilts and quiltmaking, an art (or craft, depending on your politics!) that is deeply connected to world history, the textile industry (and therefore the environment, labor, and slavery), domesticity, and the idea of \"women's work.\" Thus, the title for this section, which features those whose work is environmentally conscious, focuses on developing community, or engages with or subverts notions of women's work. How do your quilts and the choices you make in composing them speak to who you are and what you value? How does what we do when we quilt translate into the rest of our lives? This section celebrates the power of making with fabric, and what it means in each of our lives.\n\n# Kristin Link\n\n## Get Your People Together\n\n_Spring Math_ by Kristin Link, 54\" \u00d7 72\", based on the _Simple Math_ pattern by Elizabeth Hartman (page )\n\nModern quilters could not live without Kristin's website Sew Mama Sew! It started early and grew quickly, offering a much-needed site for quilters to buy modern fabrics. Along with the blog came the chance of getting involved with sew-alongs, and it was these sorts of activities that helped to establish the community. \"The first theme we ever did was skirt month \u2014 getting people involved with sewing skirts. No one was doing those sorts of sew-alongs or bees yet. We were lucky we got in early.\"\n\nFor Kristin, one of the most exciting parts of the website's success has been promoting other sewers and bloggers. Quilters and sewers are featured on the site, along with links back to their blogs and shops. \"A friend of mine called it the Sew Mama Sew bump \u2014 when she contributes to our blog, she sees a spike in her blog activity.\" Kristin says that this has been her goal from the beginning. \"It's part of my mission to help build the community and give recognition to other people. We make a conscious decision to participate in a way that's reciprocal. People support us and contribute to our blog and shop with us, and we want to give back to them by sending business to their shop or their blog. We also compensate our contributors with gift certificates to our shop.\" This is a great perspective on participating in the community. And with the blog so successful, Kristin decided to close the online shop and concentrate on what she loves best: \"collaborating, motivating, and teaching.\"\n\nI think I still have the mind-set of an educator, in that I'm always trying to think about how to involve people in different ways, and engage them, and help them learn and grow. \u2014 KL\n\n# Chawne Kimber\n\n## How Words Matter\n\nDetail of _The Big Eff_ by Chawne Kimber, 75\" \u00d7 76\", long-arm quilted by Christina Lane\n\nChawne once asked fellow quilters to make a block with the F-word in it \u2014 pieced, embroidered, appliqu\u00e9d, whatever inspired them \u2014 and then Chawne compiled all the blocks into one big collage. \"It started because I posted a question asking what color the word was, and I got so many answers back. One person said it was 'grass green,' because it was like the act, and the way you feel so alive in that act.\" Chawne laughs as we talk about it, saying that she doesn't really use expletives in her daily life.\n\nChawne made these quilts in response to some awful events. She explains how the earlier quilts in her expletives series took form. \"The big collection of quilts I made most recently was a reaction to incidents of graffiti on bathroom stalls, walls in public buildings, and on cars. And these were all ethnic slurs, slurs about sexual orientation, mean words about women, and swastikas. This happened repeatedly during a nine-month period, and the powers-that-be in the community did not react. They thought that by ignoring it, it would just stop. It became very intimidating to be here because it appeared that the people in charge were actually condoning this behavior. I've spoken out about single events in the past, when kids have written the N-word on the wall. No one was teaching this community that it shouldn't happen.\"\n\nMy work is about questioning how we communicate through quilts. \u2014 CK\n\nChawne realized that she'd have to choose whether or not to speak up. \"As one of only a few minorities in the community, it's a burden to have to speak out. I chose, in this case, not to. Instead, I made a series of quilts about how we express our identity through what we put on our beds. You can think this through if you've flipped through any kind of catalog like Garnet Hill, for instance, where you have a range of different bedcovers and you have to ask yourself, well, who would buy this one as opposed to this other one. How do you choose it? What are you trying to express by what you put on your bed? When you wrap my black quilt around yourself, you cannot hide the N-word. No matter how you wear your identity, people will always see this word.\" In posting those quilts on her blog, she asks others to confront those themes as well.\n\nAfter a lot of time spent working on the series, Chawne laughs when she says, \"I'm currently shifting into positive affirmations.\" And, she's made many other pieces that evoke emotion in different ways: the memorial quilt (below) for Trayvon Martin, the unarmed Florida teenager killed in 2012; a stunning quilt of her father's ties after his death; an incredible shot cotton quilt made from the ethically sourced fabrics of Oakshott (based in England); a graceful log cabin denim, corduroy, and khaki quilt made with family clothes; and scrappy, colorful, pieced quilts that look like impressionistic paintings or pixilated skies. What you _won't_ find in her work is a focus on designer fabrics, or the latest lines from the big companies, nor will she post patterns for what she creates (thus falling into the category, as she says, of \"more of an art quilter\"). She resists the commodification that drives the quilting world, and instead chooses to use scraps, repurposed fabrics, or solids.\n\nDetail of _Self-Study #4: The One for T_ by Chawne Kimber\n\n# Piecing and Stitching Letters\n\nChawne's idea to piece letters first came from a book she read by Tonya Riccucci, which teaches an ad-hoc method of improvisationally piecing letters. This got her into studying the history of fonts. If you want to learn more about using text on quilts, there's plenty of information on appliqu\u00e9ing letters, too. You can find templates for letters and a range of fonts. Maybe you'd start with an alphabet quilt, combining letters and images. Or tap into your subversive side and see what else you can do with words.\n\nDetail, quilt by Chawne Kimber\n\n# Denise Burge\n\n## The Land in Our Fiber\n\nDenise Burge's _Harriman_ is a quilt with a video of a Tennessee coal ash spill projected onto it. The text, Denise's notes upon visiting the site, is quilted on the white ground; all of the black and white elements are quilted. The rest is part of the projection.\n\nAn artist whose medium is often fiber, Denise Burge's quilts deal with family and local history, and she says that they do in fact _speak_ to the viewer. \"I knew that quilts often traditionally included texts, but I loved the graphic qualities of text as well. It felt like they needed to speak \u2014 they were the storyteller as opposed to being the story. I've often felt that way about quilts, that they're not just pictures, but that they actually speak. It was an intuitive choice that I made to include text in the quilt because of that.\"\n\nHer work has been shown in many galleries and anthologized in stacks of books; she's well known in the art quilt world. \"I began working in the quilt form as a solution to a narrative problem. I had been making paintings, but started thinking about the choice of materials more carefully. I was starting to make work specifically about my family history. The quilts seemed to be a natural format for that, since I have quilters in my family. The work was about a house in the family in which no one lived anymore, so it was a memorial set of works. And the quilts seemed to suit that. Knowing what quilts are about, their function as a memorial object made sense. It all came together because of storytelling.\"\n\nThis project speaks to a December 2008 coal ash spill in Tennessee. It includes quilted images and written words, and also acts as a screen upon which video from the spill site is projected. Again, the quilt is a metaphor for the layers of the land, and a voice for the environmental problems we create. She's always been interested in critiquing the nostalgia of quilts and \"commercialization of the land.\"\n\nDenise cites many others in the quilting world who \"address the nostalgia of the quilt,\" for example, her friend, Sherri Lynn Wood, who uses quilts to work through the grieving process (see page ). \"The best quilts that are made today work with the form as a subject, not just as a medium, and they take advantage of the best aspects of what quilts are. There are quilts out there made from all sorts of interesting materials.\" She talks about a quilting group, in which a woman made a wedding ring quilt out of magnolia leaves dipped in yellow rubber. \"She sewed them together in this beautiful pattern. It's not a traditional quilt, it's not an art quilt, but it uses that pattern of repetition and form.\"\n\nI'm interested in the emotional tie that we have to quilts, the nostalgia that we attach to quilts, and the physical form of the quilt, and how it's made of layers of fabric cut up from other things. It reminds me of geology and how, in the process of decay, things break up into smaller and smaller pieces. I was trying to make fabric as an analogue for mud. I would shred the fabric and sew it back together, so it created this uniform surface with bits and bits of fabric. \u2014 DB\n\n# Michelle Engel Bencsko & Gina Pantastico\n\n## Go Organic\n\n_New Wave Quilt,_ made from the Seven Seas collection by Mich\u00e9le Brummer-Everett, design by Elizabeth Hartman, pieced and quilted by Michelle Engel Bencsko, 50\" \u00d7 55\"\n\nGina and Michelle founded Cloud9 after looking for organic fabrics in the market and coming up short. Since they're both \"decisive people,\" as Gina says, they started immediately. But, she says, they did have 40 years of experience between them; they already knew \"what it takes to go from concept to product.\" Within three weeks, they were incorporated.\n\n_Midnight at the Oasis_ made from the Miscellany collection by Julia Rothman, design by Lizzy House, pieced by Jaclyn Jordan, quilted by Angela Walters, 70\" \u00d7 77\"\n\nTheir fabric is made of cotton grown on organic farms on the Indian subcontinent, and is dyed and finished throughout Asia by mills that use eco-friendly dyes. Gina travels overseas to visit the mills to make sure that the management is in compliance with good labor standards. They pick artists with whom they'd like to work, including Ed Emberly, with his fantastic animals.\n\n# Betz White\n\n## Make Time to Make\n\nPillow by Betz White, 18\" \u00d7 18\"\n\nBetz once designed clothes for a kids' apparel company and played with her own designs on the side. \"I went to school for fashion design, and I worked in the fashion industry for about 20 years. When I went to school, I learned how to knit on the knitting machine and did a lot of experiments with wool and felting and things like that. I decided it would be much faster and easier and less expensive if I went to the thrift store and bought wool sweaters and did the same thing. This was in 1990, and it wasn't an eco-conscious thing then.\"\n\nAfter years of experimenting with the felting process, she decided to do freelance work and started selling her own felted items at a local art fair. \"I kept working with the recycled felting because with my kids, I didn't have time to make it on my machine. I decided that I would do an art fair and make a bunch of things for that, and that's when I really started focusing on making felted wool items to sell, which led to my first book, _Warm Fuzzies_. When I was going to thrift stores and picking out sweaters for that book, I became aware of the multitude of stuff that was out there \u2014 jeans and T-shirts and sheets \u2014 all this thrift store stuff that was probably destined for the landfill. That's when I became in tune with reuse on the bigger level and started writing _Sewing Green_.\" She's now known as a green sewer and fabric designer, and she blogs about her projects, bag patterns, and ideas for upcycling materials.\n\n# Line Bruntse\n\n## Subverting the Form\n\n_Blanke_ t by Line Bruntse, 76\" \u00d7 91\"\n\nAn artist whose work engages with fiber and quilting, Line Bruntse has a deep knowledge of the art of sewing as women's work. She understands how women who engage with these mediums now are reengaging with and often subverting that tradition.\n\nOne of Line's pieces is a quilt that hangs from the ceiling, with long strips of cloth (Line calls them \"tentacles\") that hang down in a grid. The pattern of the hanging cloth reflects on the cross-vaulted space of the room, and the quilt defines a cross-vaulted space where the ends of the tentacles touch. Not just our sense of vision and touch is affected in this piece. There's also \"the smell of your grandmother's closet as you walk underneath it, all of those different fabrics you could imagine hanging in your grandmother's closet.\" Line was given the fabric to make the piece from someone's attic; and, in a strange twist of fate, as she made the quilt, thinking of her grandmother, her grandmother died. \"Then I went to my grandmother's house, and it smelled exactly like the closet I used to hide in as a kid.\"\n\nThe audience's engagement in her work is especially important to Line, whether she's made a rubber quilt from recycled tires, or a sculpture, or an installation of threads creating a grid through the interior of a building. \"I really want my work to touch people, not just be beautiful to look at, so they can get some of that story that inspired the work but with their own information plugged in.\"\n\nWhen you're working with this intimate process, sewing, quilting \u2014 any of those repetitive processes that are very meditative \u2014 they become natural outlets for whatever is going on in your life. No matter what you make, you take something out of yourself and put it somewhere else, whether it's something to get rid of or something to preserve. \u2014 LB\n\nHappy Drawing Giraffe by Ed Emberly\n\nIt's not always that it is organic that catches people's eye \u2014 sometimes it's the prints. At only a dollar more a yard than conventional cotton, it's a price that makes people feel good about their purchase. \u2014 MEB & GP\n\n# Alice Webb Greer\n\n## Precision Sewer\n\n_Spooks_ by Alice Webb Greer, 57\" \u00d7 68\"\n\n\"I've always been into crafty textile things. When I was eight, I went to summer camp and learned to crochet. My worm bookmark made with hot pink variegated yarn won a blue ribbon. My mother actually signed me up for my first quilting class when I was 11. I think she just wanted an hour to herself every week, but a neighbor taught me to hand-piece half-square triangles, and we put them together into a pillow that we quilted with little flowers. My mother found another neighbor to teach me to sew clothes and I made some lovely dressy overalls, the height of fashion in 1982. Yet another neighbor taught me to knit, and I knitted myself a sweater from a pattern in _Seventeen_. When I was 23, I woke up one day and decided I wanted to make a quilt and signed up for a class in Cambridge.\" Clearly, it takes a neighborhood to raise a quilter.\n\nFor her _Spooks_ quilt, Alice machine-quilted a spider's web and signed her name on a tombstone.\n\nI actually enjoy the _process_ of quilting more than the product, so picking a best project is almost impossible. Not that I don't love my finished quilts, but I'm always moving on to the next one, and most of them I will happily give away to anyone who shows even a passing interest. \u2014 AWG\n\nOnce you get in the groove, it's really fun. I like picking out fabrics because there are so many of them and you can find just the right style and they turn out so cool. Then, you can wear it to school and when somebody asks you, \"Where'd you get that shirt?\" you can say, \"I made it,\" and people are like, \"Wow!\" \u2014 Audrey Greer, Alice's daughter\n\n###### Tutorial\n\n# Finishing Your Quilt\n\nby Alice Webb Greer\n\nwww.alidiza.com\n\nYou've designed and pieced your masterpiece. You've quilted the layers together, and finally the day has come to finish. Yes! Here are the steps to follow to give your quilt a professional, long-lasting edge.\n\n### What You'll Need\n\n  * Fabric for the binding (1\u20442 yard cut at 21\u20442\" should be about 200\"), pressed\n  * General quilter's supplies (page )\n  * Barrettes or binder clips\n\n### Square Up Your Quilt\n\n  1. 1. Place your quilted top on your rotary mat with as much of the quilt supported as possible.\n\n  2. 2. Starting in one corner, straighten out your quilt and align the edges of your ruler with the corner of your quilt top. Use a rotary cutter to trim away the batting and backing from the side and bottom so that your corner is square.\n  3. 3. Move to an adjacent corner and repeat this process. You can now trim the length of the quilt between these two corners. Continue working your way around the quilt until you have trimmed all four sides.\n\n  4. 4. Measure the length and width of the quilt so that you can calculate what length of binding you need.\n\n### Make Your Bias Binding\n\n  1. 1. Fold pressed fabric in half with the selvage edges together. Square up the raw edges of your fabric and remove the selvages.\n\n  2. 2. Fold the selvage edge of your fabric diagonally across to the right edge of your fabric.\n\n  3. 3. Fold the bottom point up about halfway, matching the folded edges.\n\n  4. 4. Fold the top point down about halfway, again matching the folded edges.\n\n  5. 5. Trim away the folded edge.\n\n  6. 6. Starting at the cut edge, cut 21\u20442\" strips along the width of the fabric.\n\n  7. 7. Some of your strips will have a V in the middle. Trim these on the fold line.\n\n  8. 8. Align the diagonal ends of two strips with right sides together, leaving 1\u20444\" overhang on each side. Stitch the strips together using 1\u20442\" seam allowance.\n\n  9. 9. Continue sewing strips together until you have the desired length of binding, then press the seams of your binding strip open.\n\n  10. 10. Fold the strip in half lengthwise with wrong sides together, align the raw edges, and press.\n\n### Attach Your Binding to the Quilt\n\n  1. 1. Starting roughly in the middle of one side, align the raw edges of your folded binding with the raw edges of your quilt. Start stitching about 3\" below the starting edge of your binding, using a 1\u20444\" seam allowance.\n\n  2. 2. Stitch down the side of the quilt, stopping exactly 1\u20444\" from the corner of your quilt.\n\n  3. 3. Turn your quilt 90 degrees and sew in reverse off the edge of the quilt.\n\n  4. 4. Miter the corner by folding the binding straight up at a 45-degree angle.\n\n  5. 5. Fold the binding back down so that the raw right-hand edge of the binding is aligned with the raw right-hand edge of the quilt, and the fold is flush with the top edge of the quilt.\n\n  6. 6. Begin stitching off the edge of your quilt and stitch down the side using a 1\u20444\" seam allowance.\n\n  7. 7. Continue stitching the binding around your quilt, mitering at each corner. Stop stitching about 1\" before the starting point of your binding.\n\n  8. 8. Trim the end of your binding about 1\" past the starting point. Fold under the starting point edge 1\u20444\" and tuck the end piece inside the fold. Continue stitching until your binding is completely attached.\n\n### Hand-Tack Your Binding\n\n  1. 1. Begin on any side of your quilt and fold the binding over the raw edge of your quilt to the back. Place the folded edge just beyond your stitching line. Hold in place using barrettes or binder clips (page ). Repeat until about 5\" of your binding is prepared.\n\n  2. 2. To hand-sew, start inside the seam allowance, bringing your needle out through the back-side folded edge. Put your needle in to the backing right next to where your thread came out, being careful not to poke through to the front of your quilt. Push back into the binding and let the needle travel about 1\u20444\" through the fold before coming back out for another stitch.\n\n  3. 3. Continue stitching in this manner until you reach the corner. Stitch right up to the corner seamline, taking an extra stitch at the corner.\n\n  4. 4. Fold your binding back up to meet the quilt, making a mitered edge. Take another stitch in the corner to hold the binding in place, then stitch down the folded edge of your binding to the point.\n\n  5. 5. At the point, run your needle back up to the seamline and continue stitching your binding.\n\n# Jan Johnson\n\n## Repurposing Grandmother's Handkerchiefs\n\n_For You I Keep My Legs Apart_ by Jan Johnson, 12\" \u00d7 13\". Silk thread on vintage handkerchief.\n\n_Endless Knot Untangle_ by Jan Johnson, 12\" \u00d7 12\". Silk thread on vintage handkerchief.\n\nFor Jan Johnson, embroidery is her form of art-making, akin to drawing but with more texture. After seeing that her paintings looked like embroideries, she began playing with the traditional uses of embroidery, and challenging that association with domesticity.\n\nJan is grounded in the history of textiles and even made a project from the Antiquarian Society in Worcester, Massachusetts, which opened its doors to contemporary artists to make pieces inspired by their archives. There she saw an embroidered piece that read, \"So fade my hours, look on these flowers.\" This got her thinking about \"how lovely that was to look on the little things, the idea of the decorative being important, and also the hours spent making it.\"\n\nSo, she included the inscription in an embroidered image of flowers and a teacup with a navigational chart dipping into it, thinking about the process of her hours. The image of a boat sailing evoked the idea of men sailing off to sea, with the women back home not knowing if they'd return. The inscription, she says, expresses \"the idea of time fleeting, and the need to enjoy this beautiful moment.\" This piece is stitched on a handkerchief that belonged to her grandmother.\n\nAfter her grandmother fell ill with Alzheimer's, Jan inherited her sewing kit and handkerchiefs and says that using those is a \"way to honor my grandmother, and that way of working. And normally, you fold up a handkerchief and keep it in your pocket. It can be a keepsake and a love token; I like that about it, too. But also, some of these are so thin that they're like skin. I learned a technique called trapunto (see page ) so the fabric puckers out. I love working on paper, but I love how the cloth can fold.\"\n\n[E]mbroidery has been the means of educating women into the feminine ideal, and proving that they have attained it, but it has also provided a weapon of resistance to the constraints of femininity.\"\n\nRozsika Parker, _The Subversive Stitch_\n\nJan learned to sew from her grandmother and took it up again when she was pregnant with her daughter, Sophia, and could no longer work with toxic paints. \"My grandmother taught me a lot. She'd sit down with me and teach me stitches, like the bullion stitch (see page ), which you'll see in my work.\" In the stash of inherited supplies was a \"small stitch book\" that got Jan started with embroidery. From there, she taught herself even more and has been inspired by the work of Louise Bourgeois, and books by Betty Ring and Rozsika Parker.\n\nShe told me about a fantastic project called \"Revitalizing Historic Sites Through Contemporary Art,\" in which curator Kate Laurel Burgess-MacIntosh argues that people want to see the historic through a contemporary lens: What does the past mean to us _now?_\n\n# Embroidery Books that Inspired Jan Johnson\n\n_Drawn to Stitch: Line, Drawing, and Mark-Making in Textile Art,_ by Gwen Hedley\n\n_American Needlework Treasures: Samplers and Silk Embroideries from the Collection of Betty Ring,_ by __ Betty Ring\n\n_The Subversive Stitch,_ by Rozsika Parker\n\n###### Tutorial\n\n# Bullion Stitch\n\nby Jan Johnson\n\nThis embroidery uses several stitches. Thread in dense or sparse use becomes various types of line and visual texture on the surface. Each stitch was chosen to produce a desired effect above and below the surface: backstitch, cable stitch, satin stitch, Jacobean couching, and bullion stitch.\n\nBullion stitch is detached from the surface of the fabric, and creates a raised topographic effect. It can be used singly, or massed together, as shown in the rose, leaf, and other areas.\n\nJan Johnson embroidery, _Ring around, we all fall down and how to get up again._\n\n### Instructions\n\n  1. 1. Separate one piece of floss into its individual strands, six total, and then thread three strands together through the needle, as pictured below. Knot the ends of the strands.\n\n  2. 2. Bring the needle up through the underside of the cloth at A. Pull the thread all the way through to the knot. Insert the needle at B (the length of the intended bullion stitch), leaving most of the thread on the topside of the fabric.\n\n  3. 3. Poke the needle out just above point A without pulling the needle all the way through the fabric.\n\n  4. 4. Hold the needle between your thumb and forefinger on the underside of the fabric. Wrap the thread from point A around the needle seven or eight times, depending on the length of the stitch from A to B.\n\nHold a thumb and forefinger on the wrapped thread and pull the needle and its following thread through the coil. Take care not to twist the coils.\n\nTo complete the stitch, insert the needle again at B and pull all of the working thread of the Bullion Stitch through to the underside. The stitch may or may not lie completely flat, depending on the number of wraps made around the needle and the length from A to B.\n\nWhen grouped together, the stitch creates a massed surface texture.\n\n# Museums Discover Quilts\n\nMany quilters will cite modern art as inspiration, occasionally pop art or some other subgroup. Many will also cite midcentury modern design (from about the 1930s to the 1960s), and the Bauhaus school in Germany (which combined art and craft to make furniture and architecture).\n\nWhat made the art world start to take notice of quilts, though, was _Abstract Design in American Quilts,_ the 1971 Whitney exhibit curated by Jonathan Holstein. It was considered, back then, revolutionary to hang quilts on the wall; that presentation gave quilts new recognition as art objects. This helped to catapult art quilting into the mainstream.\n\nA few years later the bicentennial celebration in 1976 prompted a lot of people to take up the craft of quiltmaking again, as a way to commemorate our nation's 200th anniversary. Women came together to collaboratively create quilts, just as in the days of the quilting bee. Ever since then, quilting has sustained a resurgence, which continues to peak and ebb as time goes on.\n\n_Abstract Design in American Quilts_ was a groundbreaking show at the Whitney Museum of American Art in New York.\n\nOver time, I've learned to quilt what I like, not what is expected. \u2014 Christy Foltz\n\n# Laurel Krynock\n\n## A Good Thing for Kids to See\n\nPillow by Laurel Krynock, 18\" \u00d7 18\"\n\nA former engineer, Laurel Krynock stays home with her kids and spends a lot of time on her new(ish) passion. \"If someone had told me ten years ago that I would be this into quilting and sewing, that it would be this much part of my life, I would have said, 'Yeah, whatever.'\" She laughs.\n\nHaving found a passion that feeds her, she sees her love of quilting as a great example for her kids. \"I've always been a technical, detailed person and never thought I'd do something artistic at all. I needed something to keep me going because I missed that aspect of my career and my job. A lot of people like Anna Maria Horner (page ) say that it's healthy for your kids to see you working hard on something. My kids comment on that all the time. My daughter will bring her friends over and say, 'This is where my mom sews.'\" She laughs. \"I don't ignore them, but I think it's good for them to see what I do. They probably won't remember a time when I wasn't doing it.\"\n\nNo matter where I go in my life \u2014 quilting, sewing, whatever \u2014 I still want my blog to be personal. And even if I'm talking about something I made, I still want to express how it is a part of my life and my family and what I do. When I read other people's blogs, I really enjoy that. Anna Maria Horner is one; she's a beautiful writer and talks about her life and her family and how it all intertwines. Some people complain that things are too personal, but I think this is your space, write about what's important to you. \u2014 LK\n\n# Alexandra Ledgerwood\n\n## Going Zen with Scrap Improv\n\n\"Piecing improvisationally with scraps is the most relaxing thing ever,\" says Alex. \"There's no pressure. I also love to take more structured, familiar methods like curved piecing and appliqu\u00e9, and marry that with improvisational methods.\"\n\n_Furrows_ by Alexandra Ledgerwood, 25\" \u00d7 29\"\n\n# Donating Quilts\n\n**Alex made four quilts** this year to donate to Project Linus.\n\nMany modern quilters are part of such charity projects. If you're on the lookout for ways to share your lovingly made quilts with those in need, you might start local. Every city has a women's shelter, hospital, food bank, or homeless shelter that would happily pass a quilt on to someone who could use the comfort and hope. Call one of these places and ask for guidelines you might need to follow, and how to donate. Or, check out these organizations:\n\n**Project Linus** Donating blankets to children who are \"seriously ill or traumatized,\" Project Linus has branches in most states, and holds an annual blanket-making day in February. For more details about what's needed, head to projectlinus.org.\n\n**Quilts of Valor** We often think of children as quilt recipients, but you might also consider passing along love and comfort to servicemen and servicewomen and veterans. For more information about making and sending quilts to those who serve our country, head to QOVF.org.\n\n**Wrap Them in Love** These quilts are for children around the world and in the States. The founders even mention on their website that some quilts have been made as \"classroom projects\" \u2014 a great way for kids to get involved. To get involved yourself, find them at wraptheminlove.com.\n\n**BumbleBeanBasics** Victoria Findlay Wolfe's goal is to donate 700 quilts to the Bronx's Basics\/Promesa program, which provides housing and support for individuals and families. If you'd like to donate, go to bumblebeanbasics.com.\n\n# Lee Heinrich\n\n## Enjoy the Process\n\n_Cartwheels_ by Lee Heinrich, 48\" \u00d7 54\"\n\nLee says that when she saw Denyse Schmidt featured on _Martha Stewart_ , she thought, \"I want to make things like _that_!\" This is a sentiment echoed by many modern quilters. Lee took a class with Malka Dubrawsky, whom she found really inspiring. And she now loves the blogs of Elizabeth Hartman (page ), John Q. Adams (page ), and Ashley Newcomb (page ) \u2014 her \"heroes\" in the quilt world.\n\nLee used to work as a graphic designer, and she says that she always carries a pad of graph paper with her for sketching designs. \"I'm doodling a lot. Sometimes, I'll see something that will make a really cool quilt \u2014 such as the Chinese latticework that inspired the _Cross-Terrain_ quilt. I usually start with a design that I'm interested in, and then I'll choose the color palette and fabric.\" She has plenty of original designs on her blog, including _Cross-Terrain_ , with its rows of interlocking circles, and _Supernova_ , which became one of her quilt-alongs.\n\n# Blogging Advice\n\n**Take good pictures!** That's so important. People don't have much time, and they'll skim your blog to see if it interests them; they're not necessarily going to read your words. What will grab their attention is great pictures, and if they're interested, then they'll read more. I use a digital SLR, and if I can't take pictures outside, I try to get in as much natural light as possible. Also, people are interested in your projects, not stories about what you ate for dinner. So, concentrate on your sewing.\n\n**If you want a good blog,** the way to start is to sew things, and make sure you've got something to put on there. It's hard because it's a big time commitment. People want to see what's original about your work, unusual fabric combinations. Even if you're not designing patterns yourself, they want to see what your unique twist on it is. Later, after you have a following, you can post a little bit more about your kids and your life, but at first, it's just the projects that people want to see. \u2014 LH\n\n# WIP (Work in Progress) Wednesday\n\nis Lee Heinrich's brainchild and has taken hold on the blogosphere. She says it was inspired by Rossie Hutchinson (page ). \"I saw the process pledge that Rossie made, and I thought: this is what blogs should be, because it's about people seeing how you make things, step-by-step. Just showing a picture of the finished quilt doesn't tell the whole story of that project. So, that was important to me in my own blog.\" She says she started WIP Wednesday for herself, as a reminder to stay true to that goal of posting about what she's making.\n\n_Lifesavers_ by Lee Heinrich of Freshly Pieced\n\n# Thomas Knauer\n\n## Big _M_ vs. Little _m_\n\n_Lava Lamps_ by Thomas Knauer, 48\" \u00d7 60\"\n\nThis is what happens when a design professor becomes a quilter and fabric designer \u2014 first, he produces a lot of quilts and gives himself an \"apprenticeship\" period, then he applies all that he's learned in the 20 years he studied and taught design to create a line of fabrics that sells (within a week, they take him up on it, mind you) to Andover Fabrics. Then, he gets a book contract. And, as he does all this, he starts pushing everyone to think a little harder about what we're doing and why. He hopes to more clearly define Modern quilting.\n\n\"This is not about setting up strict limits and\/or excluding anyone from anything . . . Taxonomy and description are about learning. They allow us to better understand the world around us and to better describe what we see and what we are thinking about. . . . [E]xpanding the verbal vocabulary of the field leads to expanding its possibilities.\"\n\nHe calls himself a big-M Modern quilter for now, but he's out to rename the genre, and perhaps even revolutionize the quilting world.\n\n_Mindless Functioning_ by Thomas Knauer, 55\" \u00d7 65\"\n\n###### 4\n\n# Quilting from Tradition\n\nOne of the best ways to begin quilting is to learn from the long, long (long!) tradition that came before. We often think of quilts as an American art form, but the tradition extends around the world, beginning well before North America was colonized by the British. According to Spike Gillespie, author of _Quilts Around the World_ , there's evidence of Egyptians using quilts back in 1900 BCE, of Mongolians using quilted items circa 100 BCE, and of Chinese quilts harking back to 770 BCE. You can find contemporary quilts and a long tradition of the craft in Pakistan, India, Zimbabwe, Kenya, the Hmong communities of Southeast Asia, Korea, Central America, and Japan. Learn more about the worldwide history of quilting, and you'll be inspired by the myriad of forms and styles, fabrics, and colors.\n\nToday's quilters have evolved from that long line of sewers and quilters, as everyone in this book employs skills, techniques, and sometimes even patterns that have come down through the centuries. While some modern quilters love to improvise or create their own designs, others identify deeply with the tradition of quilting, and find their inspiration in the patterns and designs of their predecessors. Some people in this chapter are committed to the tradition of \"women's work\" (both engaging with that term and challenging it). Some have demonstrated their beginnings in tradition and then shifted to more innovative forms. Take \"tradition\" as you will; here, I intend it to be used as each quilter defines it for him- or herself.\n\n# Anna Maria Horner\n\n## Heritage in the Stitches\n\nQuilt by Anna Maria Horner\n\nMother of seven, celebrator of her Greek heritage, quilter, fabric designer, author, blogger, and now designer and maker of beautiful embroidered projects, Anna Maria Horner embodies the juggling and interweaving of many roles with which most modern quilters can identify. Her blog features images of her family growing up, stories about their daily lives and the joys and struggles she's faced as a mother and designer. If you've been following her since 2006, you know that Anna creates her designs in the context of a rich family life, surrounded by her husband and children. Anna's fabrics reflect her vibrant life and strong sense of design: bright colors, large-scale hand-drawn florals, and whimsical scenes on her soft, soft voiles. She's followed her own aesthetic and instincts, even when the market has been hesitant to follow her gutsy lead (as in the introduction of voiles), and the result is a business whose success relies on her singular vision.\n\nMake what you like, and don't just make something that others are making.  \u2014 Anna Maria Horner\n\nShe explains that her decision to make a line of voiles came out of a need she saw in the market: \"I was tired of seeing people walking around looking like quilts,\" she laughs. \"I saw these young, vivacious, super-stylish women making things out of fabrics that were beautiful. They had great style, but it wasn't the right fabric for the clothes. I didn't set out to shake anything up, I just wanted to make something that I really, really liked. . . . Watching the sales was really interesting because at first, people were pretty tenuous about it; but then, people sewed with it, talked about it online, and the sales had this _huge_ spike, once people had reported back in the blog world.\" Quilters who blogged about the fabric were enthusiastically in favor, and now, many other designers have followed suit with their own voiles.\n\nI would hate to lose sight of those slow, beautiful, cherished qualities that make handmade so special \u2014 special in a way that a two-hour skirt just can't replicate. Works in progress \u2014 even dozens of them \u2014 are a good thing. It's life! \u2014 AMH\n\nAnna's next book was another gutsy move, a break from her previous books, which focus on clothing and quilt patterns. Instead, this book is more of a journal of Anna's process of making needlework pieces. After seeing a video about women in Cyprus who have made lacework by hand for centuries \u2014 and the danger of that skill being lost as the women die without anyone in the younger generations taking up the craft \u2014 Anna was inspired to think (as a businesswoman) and learn (as an artist). This book records that learning process, describing how she's chosen to apply handwork in new ways. Perhaps it will reinvigorate the quilting world with the value of handwork and making something that might take a long time.\n\n\"I have learned over the years to relish the process as much as the result, and only recently am I learning to dig deeper into timeless handcrafts to absorb all that they have to offer. As a mother, I am learning that passing on _how_ to make something to my children is as important as passing on _what_ I made. So if something takes months (or more), it's okay.\"\n\nField Study Raindrops by Anna Maria Horner\n\n###### project\n\n# Bouquet\n\nby Anna Maria Horner\n\nMany quilters use embroidery to embellish their quilt tops. Anna Maria Horner encourages quilters to take on projects that require time and attention. This embroidery could be reproduced at any scale, and repeated, if you like, on your next quilt top. Or, keep it small and make a pillow, or frame it on its own for handmade wall-art.\n\nleft: Xerox and blow up to the size you would like to use for your project. (Download this image at www.storey.com\/files\/quilting-with-a-modern-slant-patterns.pdf.) right: _Bouquet_ by Anna Maria Horner\n\n**Hoops** for embroidery and quilting come in all shapes and sizes. You can even get hoops on stands, to save your wrists as you sew.\n\n# Reading up on Quilt History\n\nThere are stacks and stacks of books about the history of quilting. Why is it important? You'll know the stories behind the quilts you're making (where improv was born, who made the first crazy quilts, when the log cabin came into being), and you'll have more of a base from which to innovate and design. You'll have a wider range of styles, color combinations, and patterns in your back pocket, and you'll be one step closer to finding your own voice. Plus, as many, many quilters, including Denyse Schmidt (page ), have noted, many a quilt from the past looks awfully modern. You'd be surprised at what you turn up.\n\nJapanese kasuri quilt (19th or 20th century)\n\nAmerican Album quilt (19th century)\n\nEnglish Sawtooth Stripy Quilt (19th century)\n\n**The Quilt Index ( www.quiltindex.org).** This organization has thousands of quilts in its archives, plus historical information about quiltmaking. It's a great place to go for knowledge and inspiration for your own quilts.\n\n**Down by the Old Mill Stream: Quilts in Rhode Island,** __ by Linda Welters and Margaret Ordonez. This is one of the books that inspired the quilts in Denyse's book. You'll have to pay a pretty penny for this one, as it's out of print, but your local library might have it for you.\n\n**The American Quilt: A History of Cloth and Comfort, 1750\u20131950,** __ by Roderick Kiracofe. A great source of detailed history, and beautifully laid out as well.\n\n**Quilts around the World: The Story of Quilting from Alabama to Zimbabwe** _,_ by Spike Gillespie, Karey Bresenhan, Marsha MacDowell, and Hollis Chatelain. A fascinating look at quilts from across the world. I loved learning about quilts in the Indian tradition, in particular.\n\n**New York Beauties: Quilts from the Empire State,** __ by Jacqueline M. Atkins and Phyllis A. Tepper. This is another book focused on regional quilts, but here, as in _Down by the Old Mill Stream_ , the quilts are surprisingly diverse. This is another book that inspired Denyse's designs.\n\n**The Quilt: A History and Celebration of an American Art Form,** __ by Elise Schebler Roberts. The book includes many beautiful images along with informative text.\n\n**The Art Quilt,** __ by Robert Shaw. He's a big name in the land of quilt history, and has authored many books, including the popular _Quilts: A Democratic Art, 1780\u20132007._\n\n**International Quilt Study Center & Museum (www.quiltstudy.org).** The website of this Lincoln, Nebraska, institution features loads of information, including a series of podcasts, as well as images of quilts in their collection and a searchable database.\n\n# Pepper Cory\n\n## The Power of a Scrap\n\nSashiko by Pepper Cory\n\n\"I have known some very elderly ladies, who are probably gone now,\" Pepper Cory says, \"who, when I said I was going to the fabric store to purchase fabric, looked at me sideways and said, 'You don't _buy_ fabric for quilts.' For them, buying fabric was like cheating.\"\n\nPepper started cheating in 1972. She was privy to the resurgence in quilting in the 1970s and 1980s, the Whitney exhibit (page ) that changed it all, and the ever-evolving acceptance of quilting in the art world. \"And somewhere along there, at a garage sale, I met a quilt. I met the right quilt for me. It seemed possible. If I'd met a machine-pieced mariner's compass, it might have scared me to death. But this didn't. And within a couple of hours, I had this very strong feeling that I was going to learn to do it.\"\n\nOne of the things that keeps quilting _real_ __ is that at its roots, it's a craft of the people, a recycling kind of craft, something that everybody did, and everybody _could_ do. \u2014 PC\n\nShe laughs as she recalls her first project: a log-cabin pot holder. Since then, she's become a skilled quilter and teacher, and the author of seven books. But no matter how much she's learned and honed her skills, she stays true to quilting's roots as an art born of recycling.\n\nI didn't have any leftover scraps because I hadn't been a seamstress. So, I had to buy small amounts of fabric, because I had limited funds. I'd buy a quarter of a yard, and then I'd trade with people. \u2014 PC\n\n# Log Cabin Block\n\nOne of the earliest-known quilting patterns is the log cabin, and it's also one of the easiest to try out as a beginner, as Pepper Cory did. Start with a square or rectangle in the center, and then build out with one strip at a time, around and around the center, until your block is the right size for you. You can improv piece, or follow a pattern for each square. Then create a variety of patterns by arranging the blocks different ways. Using light colors on one side of your blocks and dark on the other allows for some interesting color play when you put them all together. Below, Alexis Deise (see page ) tries out three different patterns with her traditional log cabin squares.\n\nStreak o' lightening\n\nFurrows\n\nDark and light\n\nNext is an off-kilter variation \u2014 what modern quilters might call \"wonky\" \u2014 on the classic log cabin. This one's made in hand-dyed shibori silks and cotton, by Amy Nguyen.\n\nEvery time I want to learn a new technique, I look up a tutorial online and give it a try; there are so many good tutorials out there. Use a fabric you don't necessarily love, and it'll all be good! You have to be kind of fearless. \u2014 Lee Heinrich\n\n# Allison Harris\n\n## Fresh Traditional\n\nQuilt by Allison Harris, 56\" \u00d7 73\"\n\nLong, long ago, when modern quilting was still a relatively new phenomenon online (circa 2007), Allison Harris began blogging on her site, Cluck Cluck Sew. You may have found her then, and since seen her two boys grow and gain a sister. Allison quilts from a valley in northern Idaho, and talks about the change of heart she had about sewing years ago that brought her to her love of quilting. \"My mom always sewed, and I thought it was really dumb until I saw the cover of a magazine with a Denyse Schmidt quilt on it, made with Katie Jump Rope, her second line. I _loved_ it, I loved the fabric, and it had a different look to it than any other quilts I'd seen. So, I asked my mom to teach me, and we tracked down the fabric at a local shop.\"\n\nAllison has been quilting for over five years and has a huge blog following. \"I think of my blog as a for-fun outlet for myself, not to make money or for people to follow. When you get into that trap of trying to get followers, it's never enough. I think so far I've been pretty good about keeping it separate from work.\"\n\nWhile I think of Allison as one of the early makers in the modern quilting world, I was fascinated to hear that she doesn't define herself in those terms. Her aesthetic is \"fresh traditional,\" and she calls herself a \"scrappy quilter,\" relishing in combining fabrics and almost never using a single line in a project.\n\nAs much as she appreciates what the blogosphere has to offer, she issues a warning about becoming too deeply attached to our designs, and emphasizing what many others have explained as well: _nothing is new in quilting_. \"All quilting has really been done before \u2014 all new patterns are based on the old patterns. You can always find something that looks similar.\"\n\nAs the blogosphere and fabric companies become more and more saturated with new work and designs, Allison urges us to remember the roots of quilting, and to keep engaging with that sense of community and our love of making that started it all. \"At the end of the day, when you've been doing the same thing all day \u2014 taking care of the kids, and it's the same every day \u2014 it's really nice to have a break and sit down at the machine, and just pick up fabric.\"\n\nAll quilting has really been done before \u2014 all new patterns are based on the old patterns. \u2014 AH\n\n# Amish Quilts\n\nWhat is it so many people love about Amish quilts? The power of sophisticated simplicity: their use of rich, solid colors, and repetitive, geometric designs. Clearly.\n\nAnd also, for some, the beauty of the hand-quilting. Nancy Crow (page ) sends her quilt tops to Amish quilters, and Denyse Schmidt (page ) has had Amish women do some of her quilting as well. If you've seen one of these quilts up close, you know the stunning quality of those tiny, regular stitches, adding texture and design to the piecing.\n\nThe Amish first came to the United States in the early 1700s (before there even _were_ __ \"the States\"), after a parting of ways in the Swiss Mennonite community; Jakob Amman led the so-named Amish sect. The Amish migrated to Pennsylvania and are known for their Pennsylvania Dutch or German dialect; they now live in 20 states.\n\nHistorically, there are aesthetic differences between the quilts of the Pennsylvania and Midwestern Amish (though they may not persist today). In his book _Amish: The Art of the Quilt,_ __ Robert Hughes notes\n\nLancaster quilts have large, geometric color-fields, unlike the busier Midwestern Amish patchwork patterns. They use deep, saturated colors, but not black, a great favorite of the American heartland. They have their own . . . designs, like _Diamond in the Square_ , whereas Midwestern Amish quilts use traditional American designs like _Baskets_ and _Baby's Blocks_.\n\nThe list of differences continues, including matching versus contrasting thread, and wool versus cotton fabric. Interestingly, Hughes shuns the notion that Amish quilts might be seen as \"modern\" (and this was in 1993, before \"modern\" quilting):\n\n[T]hey are not covered with the emblems of day-to-day life one finds on some other American quilts \u2014 animals and people and tools and trees and locomotives. You cannot look at them and think, How cute. Perhaps you cannot help thinking, How modern. But that is an illusion: the truth is that Amish quilts embody the reductionism, the search for fundamentals, that modernism wanted to find in more \"primitive\" cultures, but they are no more modern than a Fang mask is cubist. In fact, they come from a culture to which modernism is anathema.\n\nIt's true that the Amish don't engage with what is \"modern,\" rejecting the vanity of contemporary clothing and the complications that technological advances would add to their lives. But even as they maintain their way of life, they have always interacted with the communities around them, as is exemplified in their employment as quilters.\n\nMany people in the modern quilting community have innovated Amish quilt designs in one way or another. For just one example of how you might play with such traditional designs, see Rebecca Loren's Grandmother's __ Fan Variation (page ).\n\n# Aneela Hoey\n\n## Embroidery-Love\n\nHexies out of various Aneela Hoey fabrics; Posy Story in Daisy by Aneela Hoey for Moda Fabrics; Signature Hoey embroidery\n\nAneela designs in the front room of her house, which she can transform from living room to studio. \"It's family life with design added into it. My girls are so used to seeing me there designing, then going off to do a bit of cooking, then coming back to make a phone call. It's working, just about,\" she says, laughing. Believe it or not, she's designed all that glorious fabric on her coffee table.\n\nShe sews and also loves to embroider. She started embroidering as a young girl (learning in school), earned a degree in design, and has since published a book on embroidery, _Little Stitches._ It's the embroidery that's inspired some of her fabric lines (like her famous Sherbet Pips). When I asked where those whimsical design ideas came from, she explains that time with her daughters inspires her. \"It's how they are when they're playing. I never went on a tree swing when I was young \u2014 but there was just something about that image that I loved. I'd have loved that when I was little. And now, if I had the land, I'd have a tree swing in the back for the girls.\"\n\nAneela remembers her mother sewing for fun, and that the sewing machine was off-limits. \"I thought, 'Someday, I'm going to have a sewing machine.'\" She laughs. Now, her daughter has her very own machine, and Aneela's teaching her to sew. \"I like being able to share that with them.\"\n\n###### Tutorial\n\n# Hand-Quilting Basics\n\nBy Alice Webb Greer and Rachel May\n\n### Instructions\n\n  1. 1. Use tiny needles when you hand-quilt (these are \u2014 no surprise \u2014 called quilting needles). When hand-quilting with a hoop, leave your fabric a little less tense than when you embroider, so that your needle can move through all three layers several stitches at a time.\n\n  2. 2. You can buy special, slightly thicker thread for hand-quilting. If you're new to it, you may want to choose a color that blends with your fabric. Or, try using pearl cotton, which is heavier weight and will draw more attention to your stitches (and will also require a bigger needle).\n\n  3. 3. Make a quilter's knot in your thread by wrapping the thread around your needle three or four times.\n\n  4. 4. Push your needle through just the top layer of fabric (not the batting or backing), and then push your needle back out the top, tugging at the thread until the knot \"pops\" under but doesn't come out the other side. This takes practice. Trim the end of the thread.\n\n  5. 5. The next stitch goes through every layer. If you can do more than one stitch at a time, this will make the quilting go faster. This is called \"rocking\" the needle. You can plan to follow the piecing of your quilt (stitching in the ditch), or quilt in any other pattern. If you prefer, freehand or stencil a quilting pattern with tailor's chalk, which washes out.\n\n  6. 6. When you get to the end of your thread, leave yourself enough thread to do the same hiding of your knot that you did in step 4.\n\n**Start Simple.** A great way to learn about the quilting process is to skip piecing altogether, and make a wholecloth quilt, with a single whole piece of fabric for the quilt top. Baby quilts are a manageable size to start with (about 40\" square) or else lap quilts (about 60\" square).\n\nCrib quilt, c. 1744, English\n\n# Kathreen Ricketson\n\n## Retro-Inspired, Remembered with Love\n\nQuilt by Kathreen Ricketson\n\nKathreen studied art for five years at university. After working in photography, she shifted to textiles because \"I wanted to use my hands and liked the idea of doing something really tactile.\" She also loved \"that feeling of being in touch with my roots as a woman.\" She said she thought about this last bit a lot because her family has always been skilled in handwork.\n\nShe defined her own aesthetic as \"Modern Retro,\" a combination of new and old. With a strong sense of art history, she was inspired by mid-century modern textiles and 1930s textiles, as well as by pop culture and punk culture \u2014 that shocking clash of colors. \"Generally, I like clean lines and unusual color combinations. I like to have a bit of surprise, like using interesting motifs and shapes. Or being inspired by some aspect of contemporary culture, but then making it my own. Or, I spotted this tiny section on an eighteenth-century quilt in a book on eighteenth-century design, and loved how they used appliqu\u00e9. There's a bit of folk in my quilting, too.\" She laughed.\n\nShe shared a glimpse of her stash: Ellen Luckett Baker's fabric, Denyse Schmidt's fabrics, a \"massive solids\" collection, Japanese fabrics, linens, and Kokka. At the time, she was working on a quilt for her son and had a couple of graphic designs from him that she hoped would inform the quilt. She was often inspired by her correspondence with Weeks Ringle (page ), who gave her advice and feedback about her work. And Kathreen, in turn, offered plenty of her own advice, as well as playful and accessible projects as she sewed and wrote from Australia.\n\nThe quilt and larger crafting community was saddened by her death in May 2013, and will miss her dearly.\n\n# Using Precut Fabric\n\nWhile quilting is rooted in making use of leftovers (fabric was expensive and precious two hundred years ago), today's fabric manufacturers know that quilters have a lot to choose from, and that most of us aren't quilting from worn-out clothes. If you're interested in making a project with a single line or fabric, or just don't feel like doing much cutting, you can try these out. Kathreen Ricketson wrote a book on quilting from precuts: _Little Bits Quilting Bee: 20 Quilts Using Charm Packs, Jelly Rolls, Layer Cakes, and Fat Quarters._\n\nJelly Roll: 21\u20442\" wide \u00d7 44\" long (the width of the fabric)\n\nLayer Cake: 10\" square\n\nCharm Pack: 5\" square\n\nFat Quarter: 18\" \u00d7 22\" (a regular quarter is 9\" \u00d7 44\")\n\nI'm constantly thinking about the history of this design, or the fabrics that I'm using, or someone else who has made this design in the 1920s and spent time on it. It has all these pieces of history. \u2014 KR\n\n# Tacha Bruecher\n\n## The Leader of the Pack\n\nTacha Bruecher is the mastermind behind the _Fat Quarterly_ magazine, the online powerhouse that brings a team of writers together to \"share projects and inspiration.\" The magazine is incredibly popular, and for good reason. It's a great place to gather and learn and chat with other quilters about what's happening now.\n\nTacha's favorite way to sew is using English paper piecing, which is the basis of her book, _Hexa Go-Go_. She prefers paper piecing because it's more precise than machine sewing. It's portable, too, for when you're on the subway or traveling in the car. I was surprised to see so many new ideas for the hexagon; I hadn't imagined that after two hundred years of use, it could be so revitalized!\n\n\"I think this technique is actually perfect for the modern-day quilter. Most people these days lead such busy lives and are always on the go. You can take paper piecing with you wherever you go, and it's so versatile. There are so many shapes you can make; precision is easier than with a machine, and you can mix it in with machine-sewn blocks.\"\n\nPaper piecing opens up so many more design possibilities that you can't accomplish with regular piecing. And I do love the fact that if you can sew on that line, you're going to get a really accurate seam. You don't have to worry about 1\u20444\" seams and cutting everything perfectly. All you have to do is sew on that line, and it'll be perfect. \u2014 Lee Heinrich (page )\n\n###### project\n\n# Paper-Pieced Pillow\n\nby Tacha Bruecher\n\nwww.FatQuarterly.com\n\nPaper piecing, otherwise known as English paper piecing, has always struck me as one of the most traditional ways of making a quilt, bringing to mind women hand-sewing hexies back in the mid-1800s. But Tacha Bruecher has updated not only the hexie (in her book _Hexa Go-Go_ ), __ but now in the following pattern, the tumbling block \u2014 another paper-pieced pattern. Enjoy her take on this traditional form, and keep on innovating with it!\n\n### Finished size: 20\" square pillow\n\n### What You'll Need\n\n  * 1\u20442 yard of a white solid\n  * 1\u20444 yard of a dark solid (dark purple solid from Art Gallery's Pure Elements range)\n  * 1\u20444 yard of print #1 (light purple print from Lotta Jansdotter's Echo range for Windham)\n  * Fat quarter of print #2 (light blue print from Paula Flight of Fancy range for Michael Miller)\n  * Fat quarter of print #3 (black on cream text print from Sweetwater's Authentic range for Moda)\n  * Fat quarter of print #4 (gray print from Denyse Schmidt's Flea Market Fancy range for Freespirit)\n  * 6\" \u00d7 7\" of print #5 (dark purple print from Joel Dewberry's Heirloom range for Freespirit)\n  * 6\" \u00d7 7\" of print #6 (yellow print from Lotta Jansdotter's Echo range for Windham)\n  * 1 yard of lining\n  * 3\u20444 yard of backing fabric\n  * 24\" square of batting\n  * 75 paper templates for 11\u20442\" 60-degree diamonds\n  * 201\u20442\" square piece of freezer paper (tape two pieces of freezer paper together)\n\n# Making Templates\n\nYou can cut the diamond templates using strips of paper and a regular ruler, turning the ruler each time to make the 60-degree diamonds. Or you could use template plastic to create a master diamond template that can be used over and over again. Create the shapes on your computer using a program such as Word, or draw them on gridded paper. Place the template plastic over the shape, trace, and cut out. Draw around the plastic diamond on paper and cut out to make the templates for the pillow. Template plastic is available at many quilt shops, bigger sewing stores, and online, and can be cut with scissors or a knife. Often template plastic is gridded, so you can draw the shape straight onto the plastic itself.\n\n### Cut Out the Fabric\n\n  1. 1. From the white solid cut: \n    * 1 strip 4\" \u00d7 30\"\n    * 2 strips 4\" \u00d7 21\"\n  2. 2. From the dark solid cut: \n    * 2 strips 1\" \u00d7 30\"\n    * 2 strips 1\" \u00d7 211\u20442\"\n    * 2 strips 1\" \u00d7 14\"\n  3. 3. From print #1 cut: \n    * 2 strips 21\u20442\" \u00d7 30\"\n    * 7 rectangles 2\" \u00d7 31\u20442\"\n  4. 4. From print #2 cut: \n    * 2 strips 21\u20442\" \u00d7 13\"\n    * 6 rectangles 2\" \u00d7 31\u20442\"\n  5. 5. From print #3 cut: \n    * 1 square 7\"\u00d7 7\", cut in half diagonally to make 2 triangles\n    * 25 rectangles 2\" \u00d7 31\u20442\"\n  6. 6. From print #4 cut: \n    * 25 rectangles 2\" \u00d7 31\u20442\"\n  7. 7. From each of print #5 and print #6 cut: \n    * 6 rectangles 2\" \u00d7 31\u20442\"\n  8. 8. From the lining fabric cut: \n    * 1 square 22\" \u00d7 22\"\n    * 1 piece 22\" \u00d7 15\"\n    * 1 piece 22\" \u00d7 12\"\n  9. 9. From the backing fabric cut: \n    * 1 piece 22\" \u00d7 15\"\n    * 1 piece 22\" \u00d7 12\"\n\n### Make the Pillow Background\n\n  1. 1. Start by drawing a line diagonally across the 201\u20442\" freezer paper square. Center the 4\" \u00d7 30\" white strip over the diagonal on the shiny side and iron in place.\n\n  2. 2. Each side of the center white strip is pieced in the same way. Start by sewing the 1\" \u00d7 30\" strips of dark solid to both sides of the white strip. To do this, lay each strip (one at a time) on one side of the white strip with right sides together, and stitch through both the fabric and the freezer paper. Then press the dark strip flat.\n\n  3. 3. Continue sewing strips to both sides of the central strips in the following order: \n    * 21\u20442\" \u00d7 30\" strip of print #1\n    * 1\"x 211\u20442\" dark solid strip\n    * 4\" \u00d7 21\" white strip\n    * 1\" \u00d7 14\" dark solid strip\n    * 21\u20442\" \u00d7 13\" strip of print #2\n    * Triangle of print #3\n  4. 4. Trim the pillow front to size by turning it over and using the freezer paper as a guide.\n\n  5. 5. Remove the freezer paper.\n\n**Tip:** Reduce your stitch length to make the removal of the paper easier when you have finished.\n\n### Make the Tumbling Blocks\n\n  1. 1. Trim all the 2\" \u00d7 31\u20442\" rectangles so that they are about 1\u20444\" larger than the diamond templates on all sides. Baste to the templates to make 75 diamonds.\n\n  2. 2. Take a print #3 and print #4 and one of the other print diamonds and sew together as shown.\n\n  3. 3. Repeat to make 25 tumbling blocks.\n\n**Tip:** Use a little starch to help the blocks keep their shape.\n\n  4. 4. Sew the tumbling blocks into 1 strip of 11 blocks and 2 strips of 7 blocks.\n\n  5. 5. Press. Turn over and press the wrong side, making sure to tuck all the ears under in preparation for machine-appliqu\u00e9ing to the pillow front. You might want to trim the ears a little bit to help with this.\n\n### Quilt the Pillow Top\n\n  1. 1. Layer the 22\" square of lining fabric, the batting, and the pillow top and baste in place (see Six Steps to a Quilt on page ). Quilt as desired. I straight-line quilted down the print strips.\n  2. 2. Remove the basting and paper templates from the longest strip of tumbling blocks and pin along the center white strip. Machine-appliqu\u00e9 the strip in place.\n  3. 3. Remove the basting and paper templates for the smaller strips of tumbling blocks and pin along each of the shorter white strips. Machine-appliqu\u00e9 the strips in place.\n  4. 4. Trim the pillow top to 201\u20442\" square.\n\n### Finish the Pillow\n\n  1. 1. Place the 15\" \u00d7 22\" pieces of lining and backing right sides together and sew along the 22\" edge. Turn right side out, press, and topstitch along the same edge.\n  2. 2. Repeat for the 12\" \u00d7 22\" pieces of lining and backing.\n  3. 3. Place the pillow top right side up and lay the 2 backing pieces on top, right sides facing, so that the backing pieces overlap. Pin in place and sew all the way around the edge.\n  4. 4. Use a serger or a zigzag stitch to sew once more around the pillow.\n  5. 5. Snip the corners and turn right side out. Press. Topstitch the whole way around the pillow.\n\n# Katie Blakesley\n\n## Women's Tradition, Today\n\n_Geese in a V_ by Katie Blakesley, 28\" \u00d7 29\"\n\nback detail\n\nWhile Katie comes from a Mormon background with a rich history of quilting, it's the community she's found at Sewing Summit and Quilt Market that helps fuel her quiltmaking. \"Having met in person people whom I admire has helped so much with collaborating. Having a face to a name is so valuable in that way. I'd never have had the courage to ask someone I admire to collaborate on a challenge, or ask them for advice, unless I'd met them in person.\"\n\nKatie's blog is called _Swim, Bike, Quilt_ , and one of her friends told her, \"If you spent as much time exercising as you did quilting, you'd be across the country by now!\" We could all be iron-women and men by now. But who wants six-pack abs when we could have an everlasting quilt to show for all this time and effort?\n\n# Look Locally\n\nKatie cites as inspiration Laurel Thatcher Ulrich's Pulitzer Prize\u2013winning book, _A Midwife's Tale_ , in which she tells stories of the midwife Martha Ballard based on her diaries. As Katie Blakesley points out, while we don't have many women's diaries, and women were not often included in newspapers or other documents that are saved from the past, we have many of their quilts and can study their lives by studying the objects they left behind.\n\nYour local historical society can be a great source of inspiration; usually, volunteers and employees will happily let you peruse letters and documents from the past. Learn the stories of the people who lived where you do, and if you're lucky, you may even come across tales of their sewing or the objects themselves. My town's historical society has a small collection of hand-sewn quilts and dolls from the eighteenth and nineteenth centuries.\n\n# Katy Jones\n\n## Dare to Be Uncool\n\n_D's Giant Granny_ by Katy Jones, 61\" \u00d7 79\"\n\nShe's ready to step in as Martha Stewart's tattooed replacement, quilts in hand, craft skills at the ready. Who could this be? Our friend Katy, of _Monkey Do_. She's hilarious, joking about replacing Martha while holding her tiny black chihuahua in her arms. \"My new book is short, so people will say, 'Do I want to buy this magazine, or this whole book that's only a couple of dollars more? Written by that girl with the crazy dog?'\"\n\nShe released a book in 2012 with the _Fat Quarterly_ crew, and her own book, which focuses on six-inch blocks, came out in May 2013. Sounds yummy. Katy says that she's not a modern quilter, but that she prefers more traditionally pieced quilts. Modern or traditional, they're beautiful. She works during the day while her children are at school, and at night, she'll often hand-sew.\n\nHer blog is wildly popular, she's addicted to Twitter and Instagram, and she says that \" . . . after working like a crazed fool for the past however many years, I can say finally this is my real job. That's a pretty amazing feeling.\"\n\n**Advice on blogging?** Just be yourself. If you prefer old lady quilts, make old lady quilts and be proud of what you do. It's way cooler if you're yourself, rather than trying to be another \"cool kid.\" I'm not cool, but people still read my blog. \u2014 KJ\n\n# Vanessa Christenson\n\n## Fashion Forward\n\n_Just Dandy_ by Vanessa Christenson, 47\" \u00d7 47\"\n\nMany people's quilting and design lives seem to have grown out of adversity. For Vanessa, that was when her husband was deployed to Afghanistan for a year, and she started her blog to stay in touch with him, posting pictures of their four kids so he could see what was happening at home. She'd post projects she was making as well. Over time, she began to notice that it wasn't just her family and close friends reading her blog, but that others were interested in what she was doing, too. So bloomed V & Co.\n\nI love to learn from the very traditional people who have been doing it for 30 years, because they have a lot of information about how to be precise. Take what you learn, and make it your own. . . . I take new colors and use them with old traditions. \u2014VC\n\n\"I'll get ideas when I'm flipping through a fashion magazine. Fashion totally inspires me. I'll think, 'This eggplant is gorgeous! I haven't thought of eggplant in years!' Or 'Emerald green!' For so many years, it was turquoise and red, those very popular colors. But now, J. Crew, for example, is all about colors. So, for me, that becomes, 'How gorgeous would it be to use those colors in a quilt?' Or, 'That blouse has such great ruffles on that one side; it would be fun to use that in a pillow. Let's try to incorporate that fashion that they're doing right there, and turn it into a quilt.' I love how the fashion industry brings classics back and adds a little bit of pow.\" Her _Make It Sew Modern_ book includes lots of sweet projects, many with fashion-inspired embellishments.\n\n# Cynthia Mann\n\n## Know Your Market\n\nFabrics from Cynthia Mann's Birch Fabrics line, featured in _Northern Exposure_ by Melissa Lunden of Lunden Designs, 70\" \u00d7 84\"\n\nAfter working in marketing, merchandizing, and production in various companies, Cynthia Mann opened the brick-and-mortar shop Birch Fabrics, in Paso Robles, California, launched the well-loved website fabricworm.com, and is now manufactoring organic cottons under the name Birch Fabrics. \"I spent weeks attached to the computer, researching and researching. Without a business plan it was risky, but I kind of winged it, and it worked out.\" She laughs, and says that's not what people want to hear. Rest assured, it did take a lot of work. She put in an average of 80 hours a week the first year. \"It took a year to get the first fabric here. The investment, and the first time working with a mill \u2014 you're shipping things back and forth, checking color. It was a slow process at first, but once two or three collections came in, then things started to pick up.\"\n\nThe idea to produce organic fabric came when she couldn't find what she wanted in the market. \"It all started when I had my first son. I was trying to find organic fabric to make his bedding, and I couldn't find it anywhere. And if I _could_ find it, it was really basic \u2014 brown check, or white \u2014 and really expensive. So, I started contacting the manufacturers of my favorite fabrics and asking if they had any organic fabrics I could sell on our site. I felt like there were new options that weren't being offered to people here. It was a big commitment for us to make financially, but I sensed that it was a market that would grow. I thought, 'If I'm passionate about organic fabric, and I'm interested in using it, there are definitely other moms out there who want it, too.'\"\n\nMy mom was part of the original crafting revolution of the 1970s. She made folk art, using electric saws and oil paints, she made dough ornaments, and she even sewed all my clothes and doll clothes, kind of similar to what's going on now. It's people my age who grew up in that environment. \u2014 CM\n\n# Starting a Business\n\nI went to SCORE (a nonprofit devoted to helping small businesses) to get business counseling, and met regularly with SCORE advisors. I still go in occasionally and meet with someone. It's been an amazing \u2014 and free \u2014 resource. It's a bunch of retired businesspeople who volunteer their time to help start-ups. They helped me write a business plan and a buying budget.\n\nYou have to find your niche, as people always say, and define yourself in terms of what you want to do. There is a lot of competition right now, but there are certain businesses that have found their niche and do really well. \u2014Kristin Link, Sew Mama Sew\n\nThe research is the most important thing that you can do in your area: going to swap meets, doing a survey, and getting as much information as you can from the people in the industry. If you only have a small amount of money to invest, start by buying a few bolts from a manufacturer; there's a minimum with most manufacturers. Maybe start at a swap meet and sell the bolts that way. Or cut fat quarters and sell them. This will give you a sense of what's trending before you break the bank. \u2014cm\n\n# Add Flourish\n\nAdding ruffles, flowers, piping, and other flourishes to your quilts is just one way to create texture and can help you develop new skills that you might use in other sewing projects. For example, you could add piping to pillows or a ruffle to a cardigan you want to fancy up.\n\n###### 5\n\n# For the Love of Color\n\nThe quilters and designers in this section have a special love of color that draws us to their work. Color is an aspect of all of our designs, but these people have a great eye for it, from which we can learn and improve our own skills. If you want to try your hand at designing fabric, turn for advice to Amy Butler (page ), the first to put what we now call \"modern\" fabric design out into the world. Or see Kaffe Fassett (page ), who leaps between the knitting and quilting worlds with his consistently vibrant designs. Maybe you want advice on how to combine colors, or new ideas for color inspiration. If you're interested in dyeing your own fabric, you can try some of the methods featured here, including dyeing with natural materials that you can find in your yard or grocery store. Color is what makes many people turn to quilting in the first place. Have fun playing with different combinations, palettes, materials, and dyes!\n\n# Amy Butler\n\n## Modern Fabric Design\n\nAmy Butler fabrics (clockwise from top left) from the Cameo collection: Angelica in Clover, Tea Rose in Silver, Harriet's Kitchen in Sugar and Flora in Berry from the Alchemy Sateen Bliss collection\n\nWhere would we be without her? Amy's been sewing since she was seven, when she learned from her grandmother. Her own mother began quilting at the age of 69. (In fact, her mother is now finishing a quilt that Amy's grandmother began.) Amy inherited a quilt that her grandmother made of Amy's childhood clothes, as well as a passion for beautiful vintage objects, art, and antiques.\n\nAmy and her husband, David (page ), live in a beautiful house in Ohio, surrounded by a garden that Amy and Dave maintain. They are the epitome of the chic design couple, spending their time painting, illustrating, designing, gardening, cooking, sewing, and traveling \u2014 and they're also both incredibly kind. They emphasize the importance of supporting others in the fabric and design community, and their lead in doing this has inspired many others to follow suit.\n\nThough Amy's designs might vary in style, motif, and scale, they're always made with rich colors. Her inspiration comes, in large part, from her journeys around the country and the world. She and David recently hiked around the lakes in Tennessee, and she's had the chance to travel to India and Bali. She also returns to England as much as possible, spending time in the countryside visiting historic gardens, and then \"of course, going to London and the Victoria and Albert Museum.\" All of these travels, and the photographs that Amy takes along the way, make their way into her work, over time. \"Travel has a cumulative effect on your subconscious and your design eye. These feelings and inspirations fuel ideas for my creative work.\"\n\nThis fueling of her design eye might be what's made her so prolific. She produces multiple new lines each year, as well as patterns for bags, quilts, housewares, and fashion. Then there are the rugs, books, maintaining the website, creating photo shoots with David . . . the list goes on. She and David work independently but often come together to bounce ideas off each other.\n\nAmy says that she's also inspired by stories. For example, her fabric line Lark was based on the story of Maxime de la Falaise. \"She was very chic and eccentric, born into a bohemian family of artists whose unapologetic passion for art and creativity surely shaped her vision of life. Lark is filled with my imaginings of what would have been the makings of her world.\"\n\n# Strike-Offs\n\nDid you know that most of our quilting fabric is produced in Korea? And that the mills send back strike-offs for approval from the designer? As Amy explains it, \"You turn in your artwork, and after about eight weeks, they send back your designs on fabric for the first time in the form of a strike-off. And that's so exciting.\" After one to three rounds of strike-off reviews, Amy decides which ones are keepers.\n\nOnce the stories begin, Amy starts designing with a collection of archival prints that she continues adding to. She works on paper, in black and white, rescaling and reimagining the designs. \"I eventually edit them down to 24 \u2014 but start with more than what I need. I look at the balance of the prints and how they relate to one another.\"\n\nOnce she's chosen the designs, she'll shift to color \u2014 another intensive process. \"Then there are the color palettes. I spend days and days making collages and building color stories. And then I add color chips to my stories. I do more color palettes than I possibly need.\" Once she's narrowed her selections, it comes down to instinct: \"It's a gut feeling.\" Finally, Amy colors her designs and sends her art to Westminster Fibers and then off to the mills.\n\n_Lark Quilt_ by Amy Butler, 72\" \u00d7 72\" (free pattern available at www.amybutlerdesign.com)\n\nIn looking at Amy's career from the outside, it's easy to think that she's always been \"Amy Butler,\" fabric success! But she tells the story of how she started out, first in school for design, then picking apples and painting at night, then (at David's behest) illustrating and doing licensing work for big companies \u2014 all the while sewing.\n\nBut her first trip to Quilt Market proved a success. \"I was a little black sheep in the show! Nobody looked like me. Which I guess turned out to be a good thing.\" It was at that show that companies approached her to design fabric. She says, modestly, \"I was at the right place at the right time, and I was doing what I love to do. When you present something from that perspective, it's always going to be understood by people, on some level.\" Today, you'll see lots of modern fabric booths at Market, but it's Amy who got the ball rolling, over a decade ago \u2014 and Amy who's still pushing modern fabric design forward, with rich color and layered stories.\n\nTravel has a cumulative effect on your subconscious and your design eye. \u2014 ab\n\n# Elizabeth Hartman\n\n## Inch by Inch, Row by Row\n\n_Tokyo Subway Map_ by Elizabeth Hartman, 80\" \u00d7 80\"; back of quilt (opposite)\n\nElizabeth Hartman, with her ongoing, ever-helpful blog, _Oh, Fransson!_ , is the one to whom everyone turns for help when learning to quilt online. Her second book helps quilters take those next steps in the learning process. _Modern Patchwork_ is more about interesting designs and quilts as entire compositions, as opposed to teaching particular skills. \"I'm teaching more challenging piecing styles, and using more focused palettes, but it's all in the service of making more interesting modern designs.\"\n\nThe idea for the book was born when she saw quilters who had been beginners a few years ago and were now in search of new, more difficult projects to try \u2014 and they didn't seem to have any luck in finding them. \"People have a misconception that modern quilting is about finishing quilts really quickly, and that it's not about craftsmanship. I saw a lot of modern quilters going to traditional quilts when they were in search of more challenging patterns, because there weren't that many more challenging modern patterns. I wanted to provide the same sort of challenges that working with a traditional quilt might, but still staying with the modern aesthetic.\" Her book is for all those quilters who are ready to take the next step in their skills and design.\n\nI keep a sketchbook of my ideas. And if I have an idea, sometimes I need to start working on it immediately, develop it, and see where it goes. I can't always sit down and come up with something right away, but I might wake up in the middle of the night and have ten ideas that I need to work on right then. I find that if you're in that creative mode, and the ideas are coming, it's so much better to sit down and work on them then. \u2014EH\n\nI usually start with solid fabrics. I use them as the base; when I'm deciding on a color scheme, I'll use them first and bring in prints to match it. Sometimes, there's a print that's so fabulous, I start with that and bring in everything around it. \u2014 EH\n\n# Alexis Deise\n\n## Geese in Full Color\n\ntop: _Flight_ by Alexis Deise, 41\" \u00d7 46\", cotton fabric, batting, and thread; bottom: back of quilt\n\nAlexis prefers to work with solids. She is very inspired by the Gee's Bend quilters (page ), and loves the idea of taking a traditional block and making it new. Midcentury modern and Bauhaus are also big sources of inspiration for her. Even though she works full-time as a lawyer, she's able to squeeze in time to quilt. One weekend, she and her husband planned a beach getaway on the South Shore of Massachusetts. \"We went there for the weekend, and I took my sewing machine and all my quilting stuff with me for our romantic weekend away, and I made a whole quilt that weekend.\"\n\nMaking quilts is almost like painting. \u2014 AD\n\n###### Color\n\n# Amy Keefer's Color Culture\n\nHave you ever thought about the long history behind each of the colors in your quilts? The way that red was once dyed with madder root, back in the days of the Egyptians? Of course, that tradition lasted centuries, and it's how red was made right into the nineteenth century.\n\nOr how about the emotions that drive your color choices? Or why we think of blue as the color of loyalty? There are so many great resources for color theory these days. You can give yourself more than an introductory education by reading quilting books written by those who specialize in the science of color. Talking with artist Amy Keefer inspired me to think differently about color, considering the cultural and folkloric implications.\n\n\"When I was at California College of Arts for grad school,\" Amy said, \"I took a class called Chromophilia, a response to the book _Chromophobia_ , which is about why certain designers used minimal colors \u2014 black or gray. We played a game in which the professor would ask people to mix particular colors based on words or feelings. She would say things like, 'anxiety,' or 'love,' or 'quiet,' and then we would each mix our own colors and make swatches and talk about why we had those associations. That's a game I play with my students now.\"\n\nMany of our color associations come from our cultural backgrounds. \"It's always really interesting. I would say the word, 'luck,' and the kids who had gone to the Mandarin school would pick bright, scarlet red. Someone else may pick green.\n\n# Color Wordplay\n\nTry Amy Keefer's color wordplay game. Ask someone to give you a list of words that have to do with feelings, states of being, or ideas. If no one's around, grab a dictionary or a book, close your eyes and open to a random page. Point to a place on the page. Open your eyes. There's your word. What colors do you associate with it? You could even get out some paints and mix up your answer. (I just tried this with the catalog that's on my desk and came up with \"bold,\" \"assembly,\" \"chevron,\" \"test.\")\n\nColor drives Amy's work, and she uses it to ask questions and challenge our assumptions about class and labor. Sometimes, those ideas emerge in interesting forms, such as soap that will turn your body blue when you use it. \"In the past, we used animal fat and lye to make soap, and you don't have to anymore. But, since lye is also one of the chemicals that breaks down indigo to make it solvent, I made a kind of soap that uses both lye and vegetable fat and indigo. When you wash with the soap, it dyes your skin blue.\"\n\n\"I was really interested in how political parties use contrasting colors. This was around the time of the 2008 election, and my partner and I are from Arizona, a really 'red' state. I had this feeling that I wanted to scrub people and change their minds.\"\n\n### True Blue\n\nWhat is so unique about indigo is the way it takes the dye (it has to be raised out of the bath and exposed to oxygen in order to turn blue) and the fact that the color is \"only removed by sunlight and abrasion.\" So, it's a long-lasting dye, which is why, Amy tells me, we associate blue with loyalty and \"being steadfast.\"\n\nBeginning in the nineteenth century, Japanese boro textiles were made of indigo scraps. Detail of a futon cover.\n\nIndigo dye has also given Amy the chance to talk not only about electoral politics but also labor and class. \"Indigo opened a door for me to talk about labor politics \u2014 the blue-collar association, denim, scrubs \u2014 I was fascinated by that relationship.\" And, indigo is also a mythical, magical color. \"When you dye with indigo, you tend to overdye, dipping cloth in over and over again. It's not the same as with other dyes, where it's the length of time leaving the cloth in dye that makes it darker. When you dye fabric with indigo and pull it out of the bath, the fabric will look like this weird green color, and then once you open the fabric up, oxygen hits the chemicals that oxidize and turn the cloth that blue color. Every time you put it in the bath and then take it out and open it again and oxidize it, the color gets richer and richer.\"\n\nThis process makes indigo unique from all other dyes, and means that it takes longer to create those deep blue fabrics that we love. Interestingly, the indigo bath and the process of dyeing is fragile, requiring just the right balance of materials. It takes \"weeks and weeks\" to make indigo, and the bath has to remain at the same temperature during that time.\n\n# Natural Dyes\n\nDyeing with natural fibers is now the cool thing to do. You can boil cottons in red cabbage, onion skins, acorns, and lots of different spices. Or even avocado pits (see page ).\n\nAmy has made soap with turmeric, which not only turned the soap gold but is also a \"natural analgesic.\" She also mixed in Saint-John's wort, \"which you can brew as tea or use as an oil extraction.\" The result? A beautiful, gold-colored soap that acts as a mood lifter. As Amy says, \"It's a natural antidepressant that's also a form of currency, in a witch-doctor kind of way,\" she laughs.\n\nCochineal, a bug that grows on different cacti in southern North America and South America, has long been used to make red. To access the color, you, well . . . _squish_ the bug, and the chrysalis makes the red dye.\n\nMadder was used to make red dye in the Middle Ages, and has been used all over the world, from India to Europe to the United States. The madder comes from the root of a plant whose Latin name begins with \"Rubia,\" which makes sense.\n\nTo learn more about natural dyes, check out this website: www.naturaldyes.org. It has a great page with links to other sites with instructions on dyeing fabric.\n\nThe spice turmeric is powerful dye.\n\n# Elizabeth Barton\n\n## Quilting from the Rooftops\n\n_The Strength of Quiet Windows_ by Elizabeth Barton, 41\" \u00d7 55\", cotton, hand- and machine-quilted, hand-embroidered\n\nElizabeth started quilting while working as a psychologist, and later had the chance to shift to making art full-time. She was very inspired by Nancy Crow's work. \"There were several others whose work I really admired. Rachel Brumer, Dorothy Caldwell \u2014 the people who seemed to put more of themselves into it, those were the ones that I loved.\"\n\nElizabeth's designs are full of color, and she works in both representational and abstract designs. She's created an underwater cities series, rooftops seen from up high, factories beautiful in oranges and blues, and abstract zigs and zags and rectangles and squares. She hand-dyes all of her fabric and sees palettes in nature, art, and even magazine pages. Although she once longed for an art degree, she's given herself an education, studying art in books, taking many surface design classes, and now taking up painting to learn more about composition and color.\n\nI grew up in England and always liked to get up on top of things \u2014 going up to church towers where you can get those views from above. My hometown of York, England, is medieval, and the very first series I did was based on the streets of that town. \u2014 EB\n\nI asked Elizabeth how she became so grounded and focused. \"I suppose it comes from years of being a psychologist, where you see people tying themselves into knots because they're trying to fit themselves into categories that other people have created. That's where it comes from. And I was trained more scientifically I need to see the data, I need to know what you mean by it before I can say if I'm part of it. I don't think there's any need to define whether or not quilting is art or craft. It's obviously both. Certainly, in medicine, there's a lot of art in medicine and there's a lot of craft, too. You want to have both of those things.\"\n\nI don't use art, or craft, or anything \u2014 if people ask me what I do, I don't use words. \"I'll show you a picture!\" I say, and I hand them a postcard of my work. They can make of it what they like. Because the word \"artist\" can mean so many different things to so many different people. \u2014EB\n\n# Kaffe Fassett\n\n## Yarn to Fabric: Saturated Color\n\n_Leafy Rosy_ , designed and made by Kaffe Fasset and Liza Lucy and quilted by Donna Laing, 68\" \u00d7 77\"\n\nKaffe is an innovator in the fiber world, creating wild, large-scale floral designs and his hugely popular shot cottons, which are gorgeous, subtle not-quite-solids. With his shift from yarn design to fabric, he helped to kick off modern quilting's design revolution, with its focus on rich, deeply saturated colors.\n\n\"I love complexity; I like very much to delve into extraordinary worlds and put them together. I notice more and more people are doing it. There are fewer simplistic quilts, and there's much more enjoyment of color. I've seen the same thing happening with knitting. When I was designing knitwear, there was very little color, and then it became quite commonplace \u2014 lots of people were doing it. That's happening in the fabric world, and we see the scale of people's fabric collections is going up and up and up. We were the most unusual with our large-scale florals. And now people are following along. It's fun. It's interesting to see people's interpretations.\"\n\nKaffe's love for color, he says, came from his mother, who was passionate about color. From very early, I was getting this wonderful enthusiasm for life's magic moments. She was extraordinarily creative. She had a terrific sense of the grandeur and the beauty of life, and she'd drag me down to the fabric shops and show me how wonderfully those fabrics used color. She was incredibly dynamic. She went by a bridge that was being torn down, and found a carpenter who agreed to use the girders to build another house on our property. She was a seer and doer.\"\n\nI love complexity; I like very much to delve into extraordinary worlds and put them together. \u2014KF\n\nThe shot cottons came to be on a visit to India. \"We went to India with Oxfam, and they took us to a little village where they did hand-weaving. We sent them an order for sample bolts, but found that they didn't have enough money for the yarns to weave our order. Eventually Oxfam supplied the yarn, and that first order launched the Rowan Patchwork and Quilting Company. Rowan asked me to do floral prints, and then the second fabric was Chard. It was the first time I'd done anything like that. When you think of what we do now \u2014 roses as big as your head \u2014 but in those days, it was calicos head to toe, so this was really revolutionary!\"\n\nThe best part of life for Kaffe is still the fact that he gets to travel, find new inspiration, work with quilters in workshops, and paint and design. \"What's most exciting is the fact that I'm not forced into retirement. At my great age, I'm seeing people drop off into retirement, or die off, and the fact that I'm still able to work and do what I want to do \u2014 and be taken seriously, paint my fabrics, and travel the world for workshops and lectures is wonderful. I get to test those waters all the time, and hear how people are using the fabrics and what turns them on.\"\n\nYou have to be bloody astute as a designer to make four colors interesting. If you use 250 colors, it will never be boring, and it will always be personal because nobody's going to combine 250 colors like you. \u2014 KF\n\nIn patchwork workshops, I'll say start with my fabrics, and throw in LOTS of other people's \u2014 Amy Butler, Thimbleberries, Civil War prints \u2014 bring it all in! Have fun and make it your own. Make it a scrappy madness. I always want quilts that look like you just went into a charity shop and found a skirt and a Hawaiian shirt and a wonderful pair of tights, and you cut them all up and had some fun. Then there's an energy, something goes off that's beyond us. \u2014KF\n\n# Malka Dubrawsky\n\n## It's Just Fabric\n\n_Lines #6_ by Malka Dubrawsky, 30\" \u00d7 40\"\n\nMany quilters in these pages cited Malka Dubrawsky as the person who got them excited about quilting.She is particularly known for her hand-dyed fabric and her books on the ease of hand-dyeing in your own home. She has also designed a line of fabrics that replicate that hand-dyed look for those who want her extraordinary touch in their work (see below). Malka was a printmaker when she was in art school, but once she graduated, she didn't have access to studio space and the tools she'd need to make prints. Thus, she started quilting: \"A lot of people labor over which fabrics will go together, and a lot of people make a baby quilt as their first quilt because it's small and manageable and you can get it done and have a feeling of success and move on to something else.\n\n\"But I come from art school, and I'm like, 'I can do anything!' So I made a double-bed-sized quilt and spent the next year and a half hand-quilting this thing. I did use it for many, many years, but I can't imagine doing that again. Dyeing fabric and manipulating the surface reminded me of printmaking, and the surprise when you pull a piece of fabric out and you don't know what it's going to be like, that's like the surprise of pulling a print off the press.\"\n\nShe encourages others to take risks in their work, reminding us that it's not a matter of life and death: \"The one thing that I know for absolutely damn sure: It's just fabric, okay? We're not curing cancer here, we're not saving the world from hunger. It's just fabric. So if you bend a rule, or you don't do it exactly as the instructions say, or you throw away the instructions completely, it's just not that big a deal.\"\n\nSimple Marks by Malka Dubrawsky\n\nFor Malka, inspiration comes from textiles from around the world: \"The number one thing that inspires me is textiles from other cultures. I own a couple of Kuba cloths, textiles from Africa. I look at a lot of textiles from other cultures, and at graphic prints for patterning, and to nature for color combinations.\"\n\nShe speaks the feeling that all of us relate to about the process and product of making. \"I love, love, love it when I'm wearing something \u2014 a sweater that I've knit \u2014 or I give someone a quilt, and then I get to use my favorite phrase in the world, 'Thank you, I made it myself.' And it's like I said I understood particle physics, because so many people say, 'Noo! How did you do that?' I'm always fascinated when people say they could never do that themselves. And I want to say, _Why? Why could you never do that yourself?_ People are discovering that they _can_ do that themselves; they're not divorced from their own ability to create, and they shouldn't be.\"\n\nOne of the advantages that going to art school gives you is the attitude, \"Yeah, I can do that.\" \u2014 MD\n\n**Resist Dyeing.** Rossie Hutchinson (page ) got into resist dyeing when she saw Malka Dubrawsky show it on her blog. \"I was completely smitten. I like the way it looks when you overdye commercial fabric. And once you realize how easy it is to dye fabric, you want to do it with everything.\"\n\nThe kind of dyeing Rossie's been interested in, itajime shibori, involves clamping a plastic shape over the fabric, so that the compressed piece of cloth emerges from the dye without having been altered, though all around it is dyed. Remember tying rubber bands around the T-shirts you tie-dyed as a kid? That's a simple method of shibori, too.\n\nBatik is an ancient process that I updated. I took those same skills and same techniques, using really common everyday objects \u2014 vegetables, card-board you can shape, brushes you can buy almost anywhere \u2014 and made contemporary, graphic prints with those batiks. I wanted to recreate what I thought were modern, fresh, contemporary prints, using this ancient process. And I wanted to emphasize that this is a totally doable technique that you could do in your house, and I knew it because I was doing it in my house \u2014 on my back porch and then in my garage, not in a fancy studio. My only concession is a little stand-up fan out there so I don't pass out in the Texas heat. \u2014Md\n\n###### project\n\n# Pieces-of-Eight Pillow\n\nBy Malka Dubrawsky\n\nwww.astichindye.blogspot.com\n\nThis pillow design really speaks to many of my current interests: simple, circular shapes, constructed in very straightforward ways and utilizing solid fabrics so as to easily play with color. The single block that makes up this pillow can be recreated as is for a contemporary pillow or repeated to craft a modern, fresh quilt. This block is super easy to construct, with no curved piecing or complicated inset seams, yet it looks complex. Eight wedgelike shapes come together to make a clean, bold design.\n\n_Pieces-of-Eight Pillow_ by Malka Dubrawsky, 15\" \u00d7 15\"\n\n### Finished size: 14\" \u00d7 14\"\n\n### What You'll Need\n\n  * Fat quarter pieces of 4 different solids for pillow front\n  * Fat quarter of coordinating solid or print for pillow backing\n  * Fat quarter of muslin\n  * Fat eighth of coordinating print or solid for pillow binding\n\n### Other Supplies\n\n  * Quilter's clear, acrylic ruler\n  * Template pieces A and B (download them at www.storey.com\/files\/quilting-with-a-modern-slant-patterns.pdf)\n\nThese templates can be downloaded for printing at www.storey.com\/files\/quilting-with-a-modern-slant-patterns.pdf\n\n  * Paper scissors\n  * Coordinating machine, hand-sewing, and quilting thread\n  * Quilt basting materials\n  * Cotton batting measuring about 18\" square\n  * Coordinating machine quilting thread\n  * 18\" coordinating plastic zipper\n\n### Cut Out the Fabric\n\n  1. 1. From one fat quarter solid, cut 8 strips measuring 23\u20444\" \u00d7 8\" (red in pillow shown).\n  2. 2. From a second fat quarter solid, cut 8 strips measuring 21\u20442\" \u00d7 8\" (orange).\n  3. 3. From a third fat quarter solid, cut 8 strips measuring 21\u20442\" \u00d7 8\" (yellow-green).\n  4. 4. From fourth fat quarter solid and using template B, cut 8 pieces (brown).\n  5. 5. From binding fabric cut the remaining fabric into 11\u20442\" strips (brown).\n\n### Make the Pillow Top\n\n**Note:** All seam allowances are 1\u20444\". Press seams to one side unless otherwise noted _._\n\n  1. 1. Pin one 21\u20442\" \u00d7 8\" (orange) strip to a second 21\u20442\" \u00d7 8\" (yellow-green) strip, right sides together. Stitch the pieces together and press the seam. These will be the two innermost colors on the pillow.\n  2. 2. Pin a 23\u20444\" \u00d7 8\" (red) strip to the sewn pair, right sides together, matching up to the 8\" edge of the outermost color (orange). Stitch the pieces together and press the seam.\n  3. 3. Position template piece A on the stitched strips; center the long narrow point of the triangle on the edge of the innermost (yellow-green) strip and align the top of the triangle on the edge of the top (brown) strip.\n  4. 4. Using a rotary cutter, cut around template piece A.\n  5. 5. Repeat steps 1\u20134 to create a total of eight sewn sections.\n  6. 6. Pin one cut-out template B section (brown) to one sewn section, right sides together, as shown. Stitch the pieces together and press the seam.\n\n  7. 7. Repeat step 6 with the remaining sewn sections and cut template B sections.\n  8. 8. Pin two sewn sections, right sides together, as shown. Stitch the pieces together and press the seam.\n\n  9. 9. Repeat step 8 to create four sewn sections. Position them together with the smallest color strips (yellow-green) forming the center octagon.\n  10. 10. Pin two sewn sections, right sides together. Stitch the pieces together and press the seam. Repeat step to create two sewn halves.\n  11. 11. Pin the sewn halves, right sides together. Stitch the pieces together and press the seam.\n\n### Quilt the Pillow Top\n\n  1. 1. Working on a flat surface, layer the muslin (wrong side facing up), batting, and pillow top (right side facing up).\n  2. 2. Baste the layers together using your preferred method.\n  3. 3. Machine-quilt layers together.\n\n**Note:** I free-motion machine-quilted my version in concentric swirls working from the center out.\n\n  4. 4. Trim layers flush.\n\n### Finish the Pillow\n\n  1. 1. Find the center of the 22\" edge of the fat quarter, and cut the piece in half to make two 11\" \u00d7 18\" pieces.\n  2. 2. Working with one piece on an 18\" edge, press under a 1\u20444\" seam allowance, then press under an additional 1\". This inch-wide folded edge will become the flap that hides the zipper when installed.\n  3. 3. Pin one side of the zipper tape, wrong side facing up, to the _inside_ pressed edge. Machine-stitch the zipper tape to the fabric.\n\n  4. 4. Working with a second backing piece, press under a 1\u20444\" seam allowance on an 18\" edge.\n  5. 5. Pin the opposite side of zipper tape, right side facing up, to wrong side of second backing piece along the pressed edge. You will need to lift the zipper flap as you machine-stitch the zipper.\n  6. 6. Working on a flat surface, layer the zippered backing (wrong side facing up) and pillow top (right side facing up), making sure that the top of the zipper tape aligns along one pillow edge.\n  7. 7. Pin the layers together and trim them flush. Set aside.\n  8. 8. Using diagonal seams, sew binding strips together into one long length. Bind the pillow and backing together as you would a quilt (see Finishing Your Quilt, page ).\n\n# Monica Ripley\n\n## Quilt for Color\n\nQuilt by Monica Ripley, 62\" \u00d7 77\"\n\nDetail of back\n\nA professional potter and instructor of art, Monica uses quilting to explore color. \"Every day in my studio, I work with form, proportion, composition, design, etc. We all have our weaknesses and mine is color. A lot of my colleagues laugh at me when I say this, but I'm serious. I watch other people design effortlessly with color while I, in turn, struggle. Quilting has given me the opportunity to investigate color and pattern in an environment where I won't feel adversely affected economically. In the end, I believe that my quilting adventures will, in some way or another, inform my professional work, while at the same time providing me with both great bed coverings for myself and wonderful, heartfelt gifts for others.\"\n\n###### Inspiration\n\n# Sonia Delaunay\n\nMaster of color and composition, designer Sonia Delaunay (and her husband, Robert) inspire Monica Ripley's quilting. Sonia's sewing and weaving projects were explorations of color and form; she's a model of radical experimentation. Check out her patchwork dress, left, and her linen coat, below, from 1925.\n\nLeft: Nancy Cunard in a Sonia Delaunay dress, photo Curtis Moffat. London, UK, c. 1925; \u00a9 Curtis Moffat\/Victoria and Albert Museum \/ V&A Images; Right: \u00a9 Les Arts D\u00e9coratifs, Paris\/Jean Tholance\/akg-images\n\n###### Tutorial\n\n# Dyeing with Avocado Pits\n\nby Sasha Duerr\n\nQuilting has a long history of creative reuse and artistic resourcefulness. Plant dyeing is an awe-inspiring way to create your own fabric designs with handmade artisan color. Natural dye materials can be sourced directly from your kitchen, garden, or even urban neighborhood. A perfect example of a natural dye that applies gorgeous color and transforms waste into wonder is the avocado pit. Without adding a mordant (a metal or plant-based additive that fixes the color), you can achieve light to deep pinks and russet reds. Or, make rich blues, purples, and mauves to dark gray-black with an iron-based after mordant. Some mordants can alter or change the plant color entirely. In the case of avocado pits, the tannin (already in the pit itself) acts as a mordant that binds well to cotton fiber. When an iron solution is added, the tannins react to the metal mordant and the color transforms from pink to inky blues, purples, and blacks.\n\n### What You'll Need\n\n  * 5 to 10 fresh, washed avocado pits per 1\u20444 pound of fiber, depending on depth of color desired (Save pits in freezer until you have enough.)\n  * Well-washed 100% natural undyed cotton fabric\n  * Nonreactive stainless steel cooking pot with lid, reserved just for dyeing (Thrift stores and garage sales are great places to find affordable stainless steel dyeing pots!)\n  * Nonreactive stainless steel tongs reserved just for dyeing\n  * pH-neutral washing soap (Eco-friendly dish soap works great!)\n  * Iron mordant solution (optional)\n  * Heat-resistant gloves\n\n### Dye the Fabric\n\n  1. 1. Prewash and soak cotton in hot to warm water in pH-neutral soap. Fabric can be left to soak overnight (for fiber to best take dye evenly) and be placed wet into dye pot.\n  2. 2. Fill your stainless steel dye pot with what will eventually be enough water to cover your cotton fiber, with enough room for the material to move evenly and freely in the pot.\n  3. 3. Add the quantity of avocado pits you desire for strength and depth of color. The more avocado pits added, the darker and richer the hue and intensity of color. Bring avocado pits to a boil and then turn down to a simmer. Simmer until the avocado pits begin to turn the water to pink and then to a deep maroon. The color change will be visible in 20 to 60 minutes. The deeper the color of the water, the more vibrant the dye will be on your fabric. For the darkest colors, the dye can be taken off the burner once the color has deepened and left to steep overnight.\n\nAdd the pits\n\n  4. 4. Immerse cotton in the dye pot, either while it is still simmering or when cold, once the dye has steeped long enough. Let your fiber soak in dye bath for a minimum of 10 minutes. Again, the longer the fabric soaks the more vibrant the pinks and russet reds.\n\nImmerse cotton\n\n  5. 5. When your avocado pit dye reaches your desired shade, remove fabric from dye pot with your stainless steel tongs. Wash fabric in warm to cool water with pH-neutral soap. Hang to dry.\n\nRemove fabric\n\n### Adding an Iron Modifier\n\nTo change the color of your avocado pit fabric from pinks to inky blues, purples, and blacks, you will need to add an iron modifier. You can buy one or make your own ahead of time. Applying an iron afterbath to your pink and russet-red dyed fiber can also allow you to resist-pattern and dip-dye your fabrics for interesting surface design.\n\nIron is very reactive to tannins, so anywhere you splash iron onto your avocado pit dye there will be an immediate change from pink to black. This makes for some interesting artistic potential: you can splatter or paint avocado-dyed fabric with a brush and iron liquid, fold and clamp patterns, or dip-dye to achieve some gorgeous transformations. Just as with the pure avocado pit pink dye bath, your iron avocado pit bath can be used as either a hot or cold bath.\n\n### Make Your Own Iron Mordant\n\n#### Additional Supplies\n\n  * Glass mason jar with lid\n  * Rusty iron scraps (nails, filings, other small rusty objects)\n  * Pure white vinegar\n\n#### Instructions\n\n  1. 1. Cover iron scraps and filings with a solution of 2 parts water and 1 part vinegar in the mason jar. Close the lid tight and let sit for 1 to 2 weeks, depending on desired strength of solution.\n\n  2. 2. Apply just enough iron solution to your dye bath that you start to see the pink dye bath change to black. Approximately 1 to 2 teaspoons of homemade iron solution will do for 8 ounces of cotton fabric.\n\nAdd iron solution\n\n  3. 3. Wearing your heat-resistant gloves and being very careful to submerge only your fabric in the iron\/avocado pit dye bath, hold fabric carefully where you would like to see the color change from pink to gray. Hold steadily in the iron dye bath for at least 1 to 5 minutes.\n\nSubmerge\n\n  4. 4. Remove your dip-dyed fabric carefully and wash with pH-neutral soap. Hang to dry.\n\n**Note:** Strong acids, such as pure lemon juice, will remove iron from tannin-based fabrics, so if you do make a mistake, the iron can be easily removed to reveal again the pink color underneath.\n\n# Red & White Quilts\n\nIn March 2011, the American Folk Art Museum presented _Infinite Variety: Three Centuries of Red and White Quilts_ at the Park Avenue Armory. The quilts were hung from the ceiling in spirals, occupying the space in a glorious variety of red-and-white patterns. That same year, the museum also celebrated _Masterworks_ and _Superstars_ (quilts employing star designs) in two separate exhibits. You can learn more on its website, www.folkartmuseum.org.\n\n\"Infinite Variety: Three Centuries of Red and White Quilts,\" presented by the American Folk Art Museum at the Park Avenue Armory March 25\u201330, 2011. Photo by Gavin Ashworth. Collection American Folk Art Museum, New York.\n\nIf I ask any of my fellow teachers what kind of quilts they like, their response would be all of them. We just happen to make certain kinds. But we love them all. When the red and white show was in New York, it blew your socks off, and it was two colors! I love that. \u2014Jane Sassaman (page )\n\n# Kim Eichler-Messmer\n\n## Hand-Dyed Landscapes\n\n_Sky.KS.Nov._ by Kim Eichler-Messmer, 63\" \u00d7 76\", hand-dyed cotton\n\nDetail back\n\nKim Eichler-Messmer's quilts look like a fading sky at sunset or during a green storm. \"A lot of the quilts I'm doing now are the landscape quilt series, inspired by photographs I take in the Midwest \u2014 in Kansas City, or when I go back to Iowa. I'm not a photographer, but I try to capture the weather, or the quality of light in the sky, or strange cloud formations. Then I try to capture that quality in my colors.\"\n\nKim's fabric is all hand-dyed on Kona cotton. Because she learned so much about the dye process in her MFA program at the University of Kansas, she can change the color and quality of each piece, creating some that fade from dark to light. \"My dye process is pretty mathematic, which changes how the dye appears in the fabric. It can take anywhere from three to ten steps, depending on what I want the fabric to do.\"\n\nShe's always been inspired to look to nature for inspiration, having \"grown up camping and being outside.\" Now, with less time to run outside as an adult, she makes quilts that celebrate the outdoors. \"Being able to play with color, that mathematical process, with my interest in the landscape, is really exciting for me. I spent a long time making quilts about my family history, but then I reconsidered why I'm making art. And I realized that having this soft, tactile piece is what I love about quilting \u2014 you don't have to know the backstory to appreciate them.\"\n\nFinding inspiration in things that are not quilts is important \u2014 whether it be nature, film, contemporary painting, Asian ceramics \u2014 anything that will inspire your quilting. \u2014 KE-M\n\n# Use High-Quality Materials\n\nWhile in grad school, Kim Eichler-Messmer learned to use high-quality materials, such as good thread that won't tangle or ruin a machine over time, good batting, well-produced cotton rather than muslin, double-thickness bindings \u2014 all things that will make the quilts last longer. This is important especially for Kim because she makes quilts that can either be hung on the wall or used on a bed. \"I think that functional quilts can be art, and art quilts can be functional \u2014 it just depends on who's making it. I think of my quilts as art, but I pay really close attention to the craft and use the best materials, so that they can stand up to being used and washed.\"\n\n# Kelle Boyd\n\n## Love the Small Things\n\n_Heart Remix_ by Kelly Bowser, 59\" x 59\"\n\nKelle Boyd's attention to the small things shines through in her fabric designs, which are often of silhouetted shapes \u2014 butterflies, hearts, whales, ice cream cones \u2014 and colorful, graphic designs, celebrating our sense of childlike wonder and happiness.\n\nShe made the shift to fiber differently than many others in the modern quilting world. After earning degrees in social work, she worked for years in the mayor's office in Nashville before deciding to follow her passion for art. \"In any spare time, I was painting; I'd be going to meetings and thinking about painting. My love of art just grew until I started to consider leaving my position in public policy for something in the art world. After a lot of time and prayer, I finally decided to make that leap.\"\n\nHer father was one of the people who encouraged Kelle to follow her passion; he took her portfolio to a gallery (unbeknownst to Kelle) and booked her first art show. \"My family and friends have been really supportive. I wanted to stay in my little shell \u2014 I'm somewhat shy. To put my artwork on something and have people see it and decide if they like it, that was hard at first.\" Believe it or not, it took a lot of encouragement for Kelle to share her work with the world. \"Once I realized that my art could possibly help in a way, that helped me get over myself and want to share my designs with other people. I thought, 'Maybe this isn't about just me having my art for me to relieve stress. Maybe I could share it.'\"\n\nMy mom and dad taught us at a young age \u2014 and it's played a role on my design and my outlook \u2014 to love the small things: good food, the color pink, a sunny day. \u2014 KB\n\n# Getting Started in Fabric Design\n\nDesign from your heart, with no boundaries and no expectations. Designing just for the fun of it yields some of my best work. \u2014 KB\n\n**Spoonflower.** If you want to make your own fabulous fabric designs, Spoonflower is a great place to start. There you can personalize fabric for a child or friend, practice your own skills as a designer, and even launch your own business on a small scale to help you make sales and gain recognition. It's very user-friendly, with a step-by-step guide to walk you through.\n\n# Creating Your Own Fabric\n\n**Get to know people online through Twitter.** There are so many of us who actively engage with each other. And it helps to talk to people \u2014 not because someone can give you an in, but because people can give you so much information about how a company operates. A lot of people seem untouchable, but there are a whole lot of us who are willing to be chatty and friendly. And Twitter is a great place to get to know people.\n\n**Follow the companies' guidelines online for submitting portfolios.** When I first started, there were a couple of other people wandering around to show their work at Market, and then the next year, there were fifty people doing it. And the companies are there to sell, not to look at designers' work.\n\n**Stay true to your own voice.** It's really tempting to get intrigued by the work other people are doing and want to do something kind of like it. But the more you do that, the less you're needed in the community. If your work is very different, it can feel very scary because you don't know if it will be accepted. But if it's very different, it has a stronger chance of being important. \u2014 Melody Miller\n\n# Fabric Story\n\nAt first, I got a lot of noes. Some people didn't even reply. But then the director at Timeless Treasures said they had just seen my book and they'd be interested to see what I'd do with a fabric line. They had seen some things that I'd put on Spoonflower and really liked them. So I sent them some designs, and they were excited. I got the contract within a week \u2014 it happened really quickly. Then, when I was designing it, I was playing with my kids one day with Washi tape, and I thought, \"Oh, it would be so great to have Washi tape ALL OVER your clothes!\" Timeless Treasures did a great job bringing my vision to life, and I'm so excited it's getting a great response. \u2014 Rashida Coleman-Hale\n\n###### 6\n\n# Practicing Scale(s)\n\nThe quilters in this section are all playing with scale, in some form or another. Whether they've enlarged plants and flowers so that the details sing out in bright color and beautiful appliqu\u00e9, or arranged large and small circles across a quilt top, they're taking that sense of play that we talked about in the last chapter, and using it to explore composition. You might be inspired to experiment with more detailed hand- or machine-appliqu\u00e9, focus in very closely on one detail from your world and recreate it in fabric. Blow it up to 50 times its size, or collage together big and small forms to see how they move together and create a new energy in your design. No matter how you choose to work, remember that, as Jane Sassaman reminds us, if you keep on practicing, you can't help but get better and better.\n\n# Jane Sassaman\n\n## Nature Study\n\n_Seeds & Blossoms_ by Jane Sassaman, 43\" \u00d7 43\", photo by Gregory Gantner\n\n_Flower Field_ by Jane Sassaman, 54\" \u00d7 54\", photo by Gregory Gantner\n\n_Ground Cover 1_ by Jane Sassaman, 57\" \u00d7 28\", photo by Gregory Gantner\n\nWhile she says she really is a \"less is more girl\" at heart, she describes her work as \"opulent,\" and indeed it is: lush green leaves folding over in symmetrical forms, moths against the moon, skeletons, geometric designs. \"Shapes are really my thing. For an appliqu\u00e9 quilt, I basically start with three contrasting shapes. I work in collage, so I'm making a stack of this kind of shape and a stack of that. And then, I 'wee gee' them around \u2014 that's my technical term, like a Ouija board,\" she says, laughing.\n\nSassaman was an art major in school, and practiced screen-printing with professors who were fabric designers. She says she always knew that she'd do something with craftsmanship. After school, she began making paper collages, but once she saw Nancy Crow's quilt on the cover of _American Craft_ in 1980, she thought, \"Look at this! I can do this!\" She laughs, explaining how much kinship she felt with Nancy's work, that this is what would marry her love of craftsmanship and collage. \"The designs were so similar, it was really uncanny. So I just started making quilts right away, not really knowing how to make a quilt. I had 5\u20448-inch seams in my quilts for quite awhile, but it didn't matter to me.\"\n\nNow a far cry from those beginner days, Jane not only makes quilts and designs fabric, she also writes books and teaches workshops. Her quilts have been shown in countless shows (single and group), and anthologized in many books, including the \"Best of\" Quilt National books. And in 1996, her quilt _Willow_ was named one of the century's one hundred best quilts. That's quite an award.\n\nA Jane Sassaman quilt is instantly recognizable. The forms, the appliqu\u00e9, the symmetry, and the colors are all distinctly hers. While she says that many quilters find inspiration in other quilts, her roots and sources of inspiration are in the decorative arts. \"My quilt book collection is really spare, but my decorative arts collection fills several rooms. I like to show those things in class, to open people's minds up to the design that's around them all the time. And nature, of course, too \u2014 all you have to do is look outside and there's lots of inspiration there.\"\n\nShe's become expert at looking closely, and it shows in the details in her work \u2014 the veiny ferns interlocking, or the barbed edge of a plant that makes it distinct from all others. She laughs and says that she's glad her filmmaker husband took pictures of the kids when they were small, because \"all my pictures are of leaves and flowers.\"\n\n_Life Totem_ by Jane Sassaman, 23\" \u00d7 71\"\n\nI quilted in the closet for many, many years before anybody even knew what I was doing, but in a way that was a real gift, because being isolated kept my voice stronger. I wasn't mimicking other people, because I didn't know what other people were doing. \u2014 JS\n\nTrain yourself to notice what you're seeing. Notice how many petals something has, or what the veins in the leaves look like, picking up on the characteristics of each plant that you look at. Once you have that in mind, then you can stylize those characteristics. \u2014 JS\n\n# Jacquie Gering\n\n## Making Bridges\n\n_Broken Cogs_ by Jacquie Gering, 66\" \u00d7 72\", machine-quilted by Angela Walters\n\nGering's _Broken Cogs_ was inspired by the old cast iron meat grinder her boys use to grind cranberries every Thanksgiving.\n\nWhen I asked Jacquie what inspired the book she and Katie Pedersen cowrote, _Quilting Modern: Techniques and Projects for Improvisational Quilts_ , she says, \"You know the phrase teach a man to fish?\" Their book focuses on the _techniques_ of quilting, rather than projects, so that people can apply those techniques to anything they do. \"We were really trying to put out into the community that if you want to duplicate this quilt just how it is, we're good with that. And if you want to take this technique wherever it will go, we're great with that, too.\"\n\nJacquie is all about making quilting accessible to all, and stripping away the misconception that we have to be masters in order to make a decent quilt. She says that she takes it as a compliment when someone calls her quilts simple, because it means that they're accessible to everyone. That's what the Gee's Bend women were doing in their quilts, Jacquie explains, and those were the quilts that inspired her to take up quilting herself, though she'd been sewing since she was a child.\n\n\"My family is Mennonite. Handmade is a legacy tradition in my family. I've been sewing forever, but I was not inspired to make a quilt until I saw a Gee's Bend exhibit. That was the spark for me, and it just happened to coincide with me quitting my job. I came home and said, 'Honey, it's time for me to quit,' and he said, 'Okay,' and then I looked around and realized I had nothing to do.\"\n\nShe did what many of us did when we first stumbled upon quilting \u2014 she started looking around online. And lo and behold, \"I found Alexander Henry fabric and thought I'd died and gone to heaven. I grew up with calicoes and really traditional stuff. Alexander Henry was the first really modern fabric I saw. I let my fingers do the clicking and the rest is history!\" She laughs. Jacquie's impulse for accessibility definitely comes from a warm heart.\n\nI'm not sure I'd use the word \"simple,\" though, to describe her eclectic, vivid range of quilts. She's designed everything from a sunburst (in a round quilt, no less) to a quilt of bridges, inspired by her new hometown of Chicago. \"I didn't realize that Chicago was a city of so many bridges \u2014 I'm walking over or under a bridge all the time! They're so inspirational with great shapes, and they're really graphic. That was my first quilt when I first moved here.\" She'd previously been president of the powerful Kansas City Modern Quilt Guild, and what with downsizing and leaving her favorite fabric shop, it was a bit of an adjustment.\n\nAs Jacquie says, \"If you love to do something, you learn to make your surroundings work for you.\"\n\nBridges. Jacquie's all about making bridges, pushing her own work and making quilting accessible for us all.\n\nMaybe improvisation is for those of us who have trouble following directions or doing what we're told. \u2014 JG\n\n# Ashley Newcomb\n\n## From Photos to Quilts\n\n_Converging Corners_ by Ashley Newcomb, 44\" \u00d7 44\"\n\nAshley's been blogging since the modern quilting world took off, and has lots and lots of followers \u2014 though she tries not to think about that most of the time. Her inspiration often comes from traditional quilting books, which she'll pick up at local thrift shops whenever she's traveling, or from taking pictures on her journeys. She jokes that everyone knows that if she's \"looking at something for too long, I'm thinking, 'Oh, it's a quilt.'\" Her pictures are of everyday things we might otherwise pass without considering, such as house or garage doors and bricks. Then she'll look at the pictures at home, and somehow \u2014 not directly, but somehow \u2014 those ideas inspire what she sews.\n\nIt was her photography that inspired her blog in the first place. Her husband thought she should start a photo blog, and he came up with the name _Film in the Fridge_. \"I didn't even sew when I started the blog, but around this time, I had started looking at Etsy for some handmade gifts, and it was then that I saw all the wonderful fabric that was out there. I wanted a market bag and didn't want to buy one, so I asked my mom to come over and teach me to sew. And it was the fabrics themselves that really got me into it.\"\n\nThis was when Flea Market Fancy (the first run, that is) was still available, and Ashley was able to get her hands on some inspiration. Once she made her way to Flickr and discovered all the modern quilts she loved, she was ready to start quilting herself. \"It's funny, because I look back and wonder why I even thought to make a quilt. I didn't have any of the tools I have now. I was using scissors and a cardboard cutout. I didn't know the first thing about it. But I think it was good that way. I just jumped in and did it little by little.\"\n\nI do what I love to do, and I don't spend a lot of time thinking about what I _should_ be doing. It's strange to me that people recognize me sometimes. But most of the time, I don't think about who might be reading my blog, and I just sew quilts in my living room. \u2014 AN\n\n# Piece by Piece\n\nAshley cuts her fabric for a quilt all at once, and has it stacked and ready to piece. So whenever she has a few free minutes, she can sit down and sew.\n\n###### Tutorial\n\n# Paper-Pieced Heart\n\nBy Jenifer Dick\n\nwww.42quilts.com\n\nPaper piecing is a method of sewing fabric pieces to a paper foundation in numerical order. Sewing on a foundation allows you to create accurate blocks when regular piecing is too cumbersome. Small blocks, sharp points or unusual angles are all attainable with perfect results.\n\n### What You'll Need\n\n  * Heart pattern (copy mine, or make your own version)\n\nNote: The image on the template represents the wrong side of the block (the image is reversed). This template can be downloaded for printing at www.storey.com\/files\/quilting-with-a-modern-slant-patterns.pdf\n\n  * Lightweight paper for paper piecing (parchment, newsprint, velum, printer paper, or phone book pages, among others)\n  * Size 14 needle\n  * Fabric\n  * Rotary cutter, mat, and ruler\n  * Iron and ironing board\n  * Tweezers\n\n### Instructions\n\n  1. 1. Blow up the template to the size you like and copy the pattern onto the paper piecing paper. Make adjustments if necessary. This template can be downloaded for printing at www.storey.com\/files\/quilting-with-a-modern-slant-patterns.pdf\n  2. 2. Put the size 14 needle in your machine and set the stitch length to 1.5 mm.\n  3. 3. Determine which fabrics you want to use and rough-cut them so they are large enough to cover the shape with 1\u20444\" seam allowance on all sides. Or, you can cut each piece precisely by measuring each shape and adding 1\u20444\" seam allowance. This will reduce fabric waste but add time. Rough-cutting the fabric \"about\" the right size takes less time and is more forgiving if you don't line up the fabrics perfectly before sewing.\n\n### Do the Piecing\n\n  1. 4. Working from the _back side_ of the pattern, lay the #1 piece over the #1 space on the pattern, right side facing up. If you have trouble seeing the lines, use a light table or hold it up to the light to make sure the lines are covered and a minimum 1\u20444\" seam allowance is visible on all sides of the shape.\n\n  2. 5. Lay the #2 fabric on top of the #1 fabric, right side facing down (so the fabrics are right sides together). Align the edges at least 1\u20444\" beyond the line between #1 and #2.\n\n  3. 6. Carefully turn over the pattern, pinning the fabrics if necessary to keep them from shifting. Stitch on the line between #1 and #2, backstitching at the beginning and end of each seam.\n\n  4. 7. Fold back the pattern along the line of stitching. Use your rotary cutter and trim the seam allowance to 1\u20444\".\n\n  5. 8. Open up the pattern and press the #2 fabric open with a hot, dry iron.\n  6. 9. Fold back and crease with your fingers the edge of the fabrics you just sewed on the next sewing line. This will show you where the sewing line is so you can place the next fabric easily.\n\n  7. 10. Using the crease as a guideline, lay down the #3 fabric, leaving a generous 1\u20444\" seam allowance beyond the crease. Turn over the pattern and sew on the line. Fold back the paper and trim the seam allowance to 1\u20444\" as you did in Step 4.\n\n  8. 11. Continue adding fabrics in this way, following the numbers in order, until you are done with the block. Make sure the fabrics along the edges of the block extend 1\u20444\" beyond the edge for trimming to the right size.\n\n  9. 12. From the back side, use your rotary cutter and mat to trim the block 1\u20444\" from the outside lines on the pattern. This will allow for the seam allowance.\n\n  10. 13. Take out the paper. Since you used a large needle and a short stitch length, the paper should come out easily. If any little pieces are left, use tweezers to get them out. Don't stress about getting every tiny bit out; leaving some in is okay.\n\n# Heather Grant\n\n## Modern Is . . .\n\n_Pods_ by Heather Grant, 42\" \u00d7 55\"\n\nIf you've had a quilt featured on Heather Grant's _Modern Day Quilts_ , you know the giddy joy of seeing one of your creations hand-picked and held up to the community.\n\nHeather says that she's \"always been a modern quilter,\" but just didn't know what to call it until she found the community as it began to emerge in books and online. She's since become deeply invested in defining modern quilts. \"The lines between modern quilting and other kinds of quilts are fluid but they're clear. I think a lot of people know what makes a quilt modern, but they don't know how to verbalize it.\" In fact, she's established a graph system to define quilts as modern.\n\nWhile she defines these boundaries for modern quilts and quilters, she admits that seeing something new could change her mind. She thought that burgundy could never be a color for modern quilts, but, \"Hot damn, they proved me wrong!\" she laughs. \"It's _how_ they did it that made it work.\" She describes Weeks Ringle and Bill Kerr (page ) as unique among quilters because of their willingness to push the boundaries and nudge the movement forward. \"They challenge themselves beyond what's typical of what modern quilters are doing. And I think that's why they are where they are. If Bill and Weeks do it, it can be done.\" She also cites art quilters as a big influence on the modern community. \"Modern quilters need to give art quilters a lot of credit for modern quilting to even exist, by getting their quilts into shows and national competitions, and getting their quilts into books.\"\n\nIn thinking about how far we've come, she refers to Rossie Hutchinson's (page ) Fresh Modern Flickr group, the first collective space for modern quilters, and asks if I've scrolled all the way back. She laughs, \"No offense to anyone else, because my quilts are back there, too!\" We joke about how this is like looking back at your outfits from the 90s. Now, she says, there's so much different, strong work being created out there. We've grown a lot already, and can evolve into who knows what? We'll have to wait and see.\n\n# Joining the Community\n\nPinterest and Flickr are online sites to post your work and see that of others. But if you don't want to go online at all, head to a local guild meeting \u2014 traditional or modern \u2014 to find others. You don't have to blog to be part of the community. \u2014 HG\n\n# The Modern Quilt Guild\n\nThe Modern Quilt Guild started in 2009, when Latifah Saafir and Alissa Haight Carlton attended the Quilt Festival in Long Beach. They were both stunned to find the festival devoid of modern quilts, when as Alissa says, \"It was such a thriving community online.\" They talked about what they could do to get involved and get modern quilting voices heard in the greater quilting community.\n\nSince the local guilds weren't accessible because of meeting hours, they decided to take matters into their own hands. Thus, the LA Modern Quilt Guild was born, and soon, people in other cities created their own guilds, too. Latifah says, \"We had no idea it would happen the way it did.\" As of this writing, there are 170 guilds around the world. Alissa says, \"We set up a hub so they could all reference each other, and it's just . . .\" Latifah jumps in and finishes for her, \" _Grown. Like. Crazy._ There were friendships already online, and people really wanted to meet.\"\n\nThe movement came of age with its first national convention in February 2013, QuiltCon, in Austin, Texas, which attracted hundreds of people from 10 countries and 80 Modern Quilt Guilds around the country.\n\n# Debbie Grifka\n\n## Give It a Shot\n\n_Spin Cycle_ by Debbie Grifka, 63\" \u00d7 79\"\n\nDebbie wants everyone to feel welcome in the quilting community. \"You don't need a new, fancy machine. I'm not saying it might not make your life easier, but you don't have to have one. I sew on a 30-year-old machine. It's not always ideal, but it works.\n\n\"People should try lots of different techniques, because there are so many different ways to do things. If you take a machine-appliqu\u00e9 class, and it doesn't work for you, there are 50 different ways to do machine appliqu\u00e9, so try another one. The same with your style \u2014 keep looking around and see what you like.\"\n\nDebbie began her own search in 2002. \"I was trying to discover, 'What kinds of quilts are _my_ kind of quilts?' And I remember picking up _Modern Quilt Workshop_ by Weeks Ringle and Bill Kerr, and thinking, ' _That's_ it! That's what I've been looking for!' Gwen Marston had a big influence on me, too.\" She's since founded her own company, Esch House Quilts.\n\nDebbie advises new quilters: \"Don't be afraid to try things and not have them work out. When I first started quilting, I was always afraid I'd waste fabric. But that's part of the process. Give it a shot, and if it doesn't work out, go on to the next one!\"\n\nHer quilts are often united by a single color, with dashes of a second or third color in an asymmetrical design. You'll see that reflected in the quilt she designed for this book, in deep blue with white stars dancing across in a diagonal (page ). The simplicity of the surface design shows off the beauty of the color and her quilting.\n\nMy extended family lives in Australia, and quilting makes me feel connected to generations before. It's not the primary reason that I do it, but I really like it that there have been women for generations who have taken the same joy out of quilts that I do. \u2014 DG\n\nQuilting is really part of my mental health. If I don't sew, I get grumpy. It's good for me to have some time to get into my sewing room, and whatever I'm doing \u2014 sewing, cutting \u2014 it calms me, and centers me, and shuts down all the to-do lists temporarily. \u2014DG\n\n# Star Variations\n\nStars are a classic motif in quilts and can be used in traditional or modern modes. Here are just a few ideas to get you playing with stars. Some of these are deconstructed stars, shattered into little triangles, or scattered stars across a blue sky. You can break each point down into smaller points, or make a hundred small stars. If you want to add some sparkle, take a cue from Alexis Deise (page ) and piece some of your stars in silk to make them shimmer.\n\nTop: Quilt (c. 1850) 140\" x 138\"; Bottom left: detail, quilt by Alexis Deise (page ); bottom right: detail, back of Katie Pedersen's Seattle Star\n\nSeattle Star, by Katie Pedersen, features a traditional design with improv stitch-and-flip triangles, a technique she and Jacquie Gering (page ) teach in their book, _Quilting Modern_. Stitch-and-flip triangles begin with a background square of fabric and a contrasting scrap that becomes the triangle.\n\n60\" \u00d7 60\", machine quilted by Krista Withers\n\n###### project\n\n# Milky Way\n\nby Debbie Grifka\n\nEsch House Quilts\n\nTry Debbie Grifka's constellation, or play with the size and placement of the stars, as she does on the back of her quilt.\n\n_Milky Way_ by Debbie Grifka, 84\" \u00d7 96\"\n\n### Finished block size: 12\"\n\n### Finished quilt size: 84\" \u00d7 96\" queen, or 60\" \u00d7 72\" throw\n\n**Note:** Directions are written for queen size (throw size in parentheses).\n\n### What You'll Need\n\n  * 3 fat quarters of silver gray\n  * 3 fat quarters of yellow gold\n  * 71\u20444 yards (4 yards) of navy\n  * 1 yard (3\u20444 yard) of binding\n  * 71\u20442 yards (33\u20444 yards) of backing\n  * 90\" \u00d7 102\" (66\" \u00d7 78\") of batting\n\n### Cut Out the Fabric\n\n  1. 1. Cut each of the following from the silver and gold fat quarters:\n\n### Piece the Stars\n\n#### Half-Square Triangle Block\n\n  1. 1. Choose a pair of squares and draw a line on the wrong side of the lightest fabric.\n  2. 2. With the squares' right sides together, sew 1\u20444\" from the drawn line on both sides of the line.\n\n  3. 3. Cut along the line.\n\n  4. 4. Press the blocks open and trim. Press the silver\/navy blocks toward the navy, the gold\/navy blocks toward gold, and the silver\/gold blocks toward gold.\n\n**Note:** Each star is made up of half-square triangle blocks.\n\n#### Block A (12\" stars)\n\n  1. 1. Using the 4\" squares, make 8 (6) silver\/navy pairs, 8 (6) gold\/navy pairs, and 8 (6) silver\/gold pairs.\n  2. 2. Make each pair into two half-square triangle blocks and trim to 31\u20442\" square.\n  3. 3. Lay these out according to the diagram, placing the 31\u20442\" solid navy squares in the corners. Stitch the squares together into rows, pressing in alternate directions. Stitch the rows together to complete the block.\n\n  4. 4. Make 4 (3) of Block A.\n\nBlock A\n\n#### Block B (8\" stars)\n\n  1. 1. Using the 3\" squares, make 14 (10) silver\/navy pairs, 14 (10) gold\/navy pairs, and 14 (10) silver\/gold pairs.\n  2. 2. Make each pair into two half-square triangle blocks and trim to 21\u20442\" square.\n  3. 3. Lay these out according to the diagram, placing the 21\u20442\" solid navy squares in the corners. Stitch the squares together into rows, pressing in alternate directions. Stitch the rows together to complete the block.\n\nBlock B\n\n  4. 4. Stitch the 21\u20442\" \u00d7 81\u20442\" strips to the top and bottom of the star. Stitch the 21\u20442\" \u00d7 121\u20442\" strips to the sides.\n  5. 5. Make 7 (5) of Block B.\n\n#### Blocks C and D (6\" stars)\n\n  1. 1. Using the 21\u20442\" squares, make 16 (14) silver\/navy pairs, 16 (14) gold\/navy pairs, and 16 (14) silver\/gold pairs.\n  2. 2. Make each pair into two half-square triangle blocks and trim to 2\" square. Lay these out according to the diagram, placing the 2\" solid navy squares in the corners. Stitch the squares together into rows, pressing in alternate directions. Stitch the rows together to complete the block.\n\nBlock C\n\n  3. 3. For Block C (one 6\" star): Stitch together two 61\u20442\" navy squares. Stitch another navy square to the 6\" star. Stitch these two rows together to form a 121\u20442\" block.\n  4. 4. Make 2 (1) of Block C.\n  5. 5. For Block D (two 6\" stars): Stitch a 61\u20442\" navy square to two 6\" star blocks. Lay out these pieces so that the stars are in opposite corners of the block. Stitch the rows together to form a 121\u20442\" block.\n\nBlock D\n\n  6. 6. Make 3 (3) of Block D.\n\n### Assemble the Quilt Top\n\n  1. 1. Follow the diagram to lay out the blocks, paying special attention to the orientation of the C and D blocks.\n\n  2. 2. Stitch the blocks into rows.\n  3. 3. Stitch the rows into the quilt top.\n\n### Finish Up\n\n  1. 1. Press the quilt top.\n  2. 2. Make the backing, as plain or fancy as you like.\n  3. 3. Pin together the layers (quilt top, batting, and backing) in a quilt sandwich (see Six Steps to a Quilt, page ).\n  4. 4. Quilt as desired.\n  5. 5. Bind with 21\u20442\" strips (see Finishing Your Quilt, page ).\n\n# Valori Wells\n\n## In Full Bloom\n\nQuilt by Valori Wells, 47\" \u00d7 31\"\n\nValori started sewing when she was young, learning from her mother, Jean, a well-known art quilter who has written many books (some with Valori) and taught for years. Jean started their quilt shop when Valori was almost two. \"I lived in it,\" Valori says, laughing.\n\nWhen she was 25, Valori started designing fabric. \"I gradually kept designing bigger and bigger. I love Marimekko's huge-scale designs. I still draw everything, but now I do it in black and white, and then scan it into the computer and use Adobe Illustrator as my paintbrush.\" This way, it's easier to communicate color more easily to the mills overseas.\n\nMy designs are inspired by nature and my surroundings. Everywhere I go, I carry a camera and sketchbook. \u2014vw\n\nThe fabric market has changed as more and more designers have joined in the fun of designing \"modern.\" Valori is proud to have been one of the first. \"I'm a sewer and a quilter (and a fabric shop owner), so I know what people are using their fabric for, and I make things that I think they'll love and find useful.\" Besides owning the Stitchin' Post, Jean and Valori organize the gigantic Sisters Quilt Show in their hometown, which brings in over 20,000 people each year (wow!). Jean and Valori bring in and connect with different facets of the quilting community. Valori says of her mother, \"She's more on the art side, and I'm definitely on the modern side.\"\n\nEvery July, my mother and I put on a week of classes, called Quilter's Affair, and bring in 30 teachers for a week. We rent out our high school and hold classes there, and then the week ends in the Sisters Quilt Show. Our show is different from the average quilt show because it's free, it's hung outside, and there's no jury. My mom started it as a sort of show-and-tell; it's a family event. \u2014 VW\n\n# Jessica Kovach\n\n## Twin Blog\n\n_Baby Medallion_ by Jessica Kovach, 44\" \u00d7 54\"\n\nWhat happens when twins start quilting and blogging together? _Twin Fibers_! What started as a way for Jessica and her twin, Jennifer, to keep in touch has become a well-loved blog among modern quilters. \"We both came across quilting blogs at the same time. Heather Bailey was an inspiration for us; we loved things she was sewing. Jennifer and I lived kind of far apart, so it was a way for us to share what we were working on. Jennifer has three young children, so she has very little time to sew. But it's a fun way to share what we're doing. I'm still surprised today that people admire what I sew. It's really nice to get positive feedback about what we make and do.\"\n\nThey now live just 20 minutes away from each other, and delight in meeting up and finding that they're wearing the same outfit (yes, it really does happen like that with twins sometimes). \"Jennifer showed me this box set of cards she bought the other day, and I said, 'Yeah, I bought that last week!'\" They can spend more time together in person now, but the blog is still going strong.\n\n**Keep it Fresh.** Go to a museum, find your favorite painting or piece of art, and make a quilt that embodies, mirrors, or speaks back to that piece.\n\n# Aimee Raymond\n\n## Piping and All\n\n_Double Hourglass_ by Aimee Raymond, 50\" \u00d7 57\"\n\nAimee recalls how the Amish quilt book that she bought at a yard sale for 25 cents is just one source of inspiration, giving her ideas about using the patterns and innovating with color and design choices. \"I get more excited about color choice and different combinations than I do necessarily about seeing other quilts online. And, I love fabric \u2014 each fabric line lends itself to something unique. So, I also get a lot of inspiration from fabrics themselves.\"\n\nShe explains her plans for the next couple of quilts she'll make, working on them at a retreat in Kennebunkport, Maine. \"I've been collecting silks and have an improv quilt in mind, using Denyse Schmidt's paper bag method. I've also been collecting Heather Ross prints and probably have 20 different ones, and those would lend themselves well to improv blocks.\"\n\nThe yellow piping adds another level of texture and design to this quilt.\n\nThere are two different types of sewers out there \u2014 people who can follow a pattern, and people who, when they try to follow a pattern, just want to blow their brains out. \u2014 AR\n\n# Caro Sheridan\n\n## Pixelated Quilts\n\n_Pixel Pusher_ by Caro Sheridan, 61\" \u00d7 61\"\n\nCaro remembers being turned off by her first encounter with quilting and making choices that no one else there could understand. \"In my first quilting class, while the rest of the class picked out forest green, brown, and burgundy calico prints, I was drawn to the black, gray, and white end of the spectrum. The teacher somewhat unkindly told me that 'nobody uses those colors for the main fabric, dear. It's so morose and depressing!' It always stuck in my craw that there were 'rules' to what colors were acceptable together and what were not.\" Caro has rejected those rules and found her own way of quilting, whether that's designing her own fabric on Spoonflower (page ) or creating quilts featuring robots and pixilated images gone large-scale. She's since found a welcome audience and quilting friends who appreciate these designs.\n\nWhen I first started quilting, I was happy to learn by sewing samplers; there's a lot of technique to be learned in those quilts. Then, I yearned for something asymmetrical and off-kilter. \u2014 CS\n\nMost of my quilts begin with a sudden, inspirational moment. Sometimes while I'm driving, I'll see something, or I'll catch a glimpse of something on TV. I generally make a quick note on my iPhone to return to later. I always draw it out on a big sheet of graph paper or in Excel so that when it comes time to cut the fabric, I know exactly what I'm doing. I'm not very improvisational at all \u2014 I plan the entire quilt before even picking colors. \u2014 CS\n\nOver and over again, I reach for Kona Cotton. The wide range of colors and quality of fabric are second to none. The best quilting project for me has been designing and finishing my _TV Party Tonight_ quilt (the TV test pattern) Once I was struck with the idea, I worked out the measurements, picked the colors, and had it under the sewing machine in what felt like no time at all. I also designed the fabric on the backing and had it printed through Spoonflower, so I felt like this quilt was more representative of _me_ than anything I'd ever made. \u2014 CS\n\nTutorial\n\n# The Four-Patch Trick\n\nBy Caro Sheridan\n\nwww.splityarn.com\n\nIf you've ever sewn a four-patch and always pressed to the dark side in both steps, you know how quickly that seam gets bulky. Five layers of soft fabric can turn into a hard nub that will break a needle when you try to quilt over it. Thankfully, I have just the trick to make it easy. Surprisingly, it isn't well known, so it seemed perfect for a tutorial. It also begs for photos to show the beauty (and simplicity) of it.\n\n**Note:** This tutorial works for 4-patches, 9-patches, and anywhere you're joining a whack of squares where you don't want a bulky seam allowance.\n\n### Instructions\n\n  1. 1. Start with the initial 2-patch with seams pressed to one side. Then make a second 2-patch; if you always press in the same direction, you can snuggle the seam folds up together and you'll feel them lock into place like a puzzle. This is the key to getting your points to match up exactly. (In this example, the seams are pressed to the light side.)\n\n  2. 2. Stitch the two 2-patch pieces together, right sides facing and center seam aligned.\n\n  3. 3. Lay the piece open and facedown on your pressing surface. Before busting out the iron, gently nudge the center of the seam open. There will be one or two stitches to undo, but since you don't need to backstitch on quilting seams, this will be easy to pry apart with your fingers.\n\n  4. 4. Finger-press the seams in opposite directions; in this example, that means pressing to the light sides. You'll see a wee checkerboard appear dead center.\n\n  5. 5. Press these seams with your iron.\n  6.\n\n  1. 6. Behold the magic of the nonbulky seam. Beautifully flat and crazy easy to quilt over because the seam is now minus two layers of fabric. Neat, eh?\n\n###### 7\n\n# Coming Full Circle\n\nToday's technology has given us the chance not only to innovate with quilts and the process of their making, but also to connect with one another in ways that we couldn't in the past. As Rossie Hutchinson (page ) said in her online lecture, \"Modern quilting and the Internet are in love.\" It's true that this new enthusiasm for quilting, and the experiments with aesthetics and process, could not have spread so quickly without blogs, Flickr, and (more recently) Pinterest.\n\nBut, in spite of our rapidly changing world, and new ideas and innovations of the quilt form, we'll always have to come back to its most primal parts: fiber, thread, needle, and our hands working the cloth, to machine- or hand-piece and sew the quilt. No matter how far afield our designs may go, no matter how much time we spend blogging and trading ideas online, it's this tactile component that keeps us quilting. Touching fabric, talking about its \"hand,\" the \"grain,\" feeling it through our hands as we stitch \u2014 that's what slows us down, what centers us, and what keeps us sewing and gathering together. We need the connections that we've forged over quilting \u2014 not just online, but in person.\n\nAnd so we'll find each other in community meeting rooms, in churches on Saturdays, in library lounges, and at our local fabric and quilting shops. Technology might help us to make these connections and to think about design differently (or even help us in the process of designing our quilts with a computer program), but it's the touch of fabric and the allure of sharing all of this in person that make this all worthwhile. This chapter celebrates traditional quilting, modern quilting, and our urge to pass on the art-craft and keep it alive.\n\n# Valerie Maser-Flanagan\n\n## The Beauty of the Seams\n\n_City Walk #2_ by Valerie Maser-Flanagan, 33\" \u00d7 33\"\n\nInspired by the trees outside her rural home, Valerie learned improvisational piecing from Nancy Crow (page ). For the past few years, she's taken two workshops a year with Nancy, and used the skills she learned to find and develop her own voice and style. She emphasizes the importance of letting the design change as you work.\n\n_Chrysalises #2_ by Valerie Maser-Flanagan, 68\" \u00d7 46\"\n\n\"You have the pieces laid out on the design wall, but once you start sewing them together, things start changing a lot. You have to be flexible the whole way through, so that you can make those changes work with your original intention. It's different than a lot of traditional quilts where you put the blocks together, and you have to use an exact 1\u20444\" seam to make the blocks come out to a certain size.\" She's able to create arched forms, curves that seem to overlap, and spare designs that reflect the trees outside, as well as more complex block-pieced quilts.\n\nTake the pressure off of making one quilt perfect, and think instead in terms of a series. \"It's exciting to think about working in a series, because you can't always think of everything in one piece, and as you keep developing it and designing it, you might move to a different zone. And you can't accomplish everything in one piece. So, you might try this much in one piece. And then maybe use different colors in another.\" \u2014 VM-F\n\n**Tip:** Nancy Crow taught Valerie Maser-Flanagan that \"the seamline is also the design line.\" Try not hiding the seams but instead making them a feature of the quilt. Even if that seam runs through the \"empty canvas\" of your negative space, it can provide interest and a new dimension to the quilt.\n\n# Geta Grama\n\n## Shadow Play\n\n_A World of Many Colors_ by Geta Grama, 55\" \u00d7 55\"\n\nA Romanian quilter who taught herself by finding tutorials and blogs online, Geta makes quilts that are masterpieces of color and geometry. Her quilt _A World of Many Colors_ uses a technique called shadow trapunto, and has gone on tour with the International Quilt Association.\n\nTrapunto adds extra texture and dimension to a quilt. Shadow trapunto differs from regular trapunto because it uses an additional layer of a sheer fabric. The white trapunto design is made from organza and batting and underneath there is an opaque fabric. That fabric could be a wholecloth top or an appliqu\u00e9d top that adds more color to the quilt.\n\n**Trapunto.** Trapunto (Italian for \"embroider\") is a centuries-old technique of using extra batting or stuffing underneath certain sections of a quilt so that those sections are raised. Modern quilters can play with the technique in a myriad of ways.\n\n# Stacey Shrontz\n\n## Gift It\n\n_You Are My Sunshine_ by Stacey Shrontz, 48\" \u00d7 66\"\n\nHailing from Michigan's Upper Peninsula, Stacey is a UP-er (pronounced \"Yooper\") at heart. \"People in the UP support wholesome living, without thinking that you're a snob or a crunchy person for doing it. We'd hike, kayak, fish from our kayaks, work at dogsled races making fires at the checkpoints, swim in Lake Superior. I worked for two years at the farmers' market, selling toddler tutus and baby shoes with tulle ribbons, as well as afghans and quilts. And I sold my photos of landscapes in Upper Michigan.\" Stacey's quilt was inspired by a photo of a quilt that she once saw ( _Dot Party_ , by Julie Herman) and adapted with smaller blocks from a charm pack (5\" squares), finished with loose, textured stitching.\n\n# More Circles\n\nAnna Maria Horner's Super Circles method involves appliqu\u00e9ing the circles with the edges turned under, rather than leaving the edges raw. You can find her tutorial on her website (she, in turn, credits Kath of Material Obsession). Or, check out the fusible web method on the next page.\n\n**Tip:** You can choose from the insert menu in word-processing programs to print out any kind and size shape: circles, diamonds, equilateral triangles. You name it! Stacey used a 3\"-diameter circle to make this quilt.\n\n###### Tutorial\n\n# Fusible Web Applique\n\nBy Jenifer Dick\n\nwww.42quilts.com\n\nFusible web appliqu\u00e9 is a fast and easy raw-edge technique. Since the edges aren't turned under, it's possible to create super-sharp points, deep Vs, or tight curves that are difficult with traditional appliqu\u00e9. The outer raw edge needs to be finished to keep it from fraying or coming off altogether. Traditionally, raw-edge appliqu\u00e9 is finished with a buttonhole stitch, but today quiltmakers use a variety of stitches and threads to get different looks for their projects. Fusible web is best for art quilts, wall hangings, and other projects that won't be laundered often.\n\n### What You'll Need\n\n  * Paper-backed fusible web\n  * Pencil\n  * Scissors\n  * Iron and ironing board\n  * Prewashed appliqu\u00e9 and background fabric\n\n### Instructions\n\n  1. 1. Trace the appliqu\u00e9 shape on the paper side of the fusible web. Cut out the shape 1\u20444\" from the drawn line.\n\n  2. 2. For larger shapes or layered patterns, cut out the shape again, 1\u20444\" from the inside of the drawn line. Discard the inner piece and use only the outer ring for fusing. This reduces stiffness in the finished quilt.\n\n  3. 3. Press the shape to the wrong side of the appliqu\u00e9 fabric, following manufacturer's instructions. Unwashed fabric can inhibit the webbing from adhering to the fabric, which is why I recommend prewashing.\n\n  4. 4. Cut out the shape on the drawn line. Be careful to make curves smooth and points sharp. The finished shape will show any imperfections in cutting.\n\n  5. 5. Cut the background 1\u20442\" larger than necessary to allow for any shrinking during the appliqu\u00e9 process. Fuse shape to background fabric.\n\n  6. 6. Sew the raw edges with your favorite stitch. Use invisible, contrasting, or matching thread, depending on the look you want. Trim the block to size to finish.\n\n###### Project\n\n# Quilting with Kids (Pass It On)\n\nfrom Virginia B. Johnson\n\nGather Here (Cambridge, Massachusetts)\n\nAges 8 to 12 . . . or older \u2014 this could work well for us adults, too! Kids don't need sewing skills as long as there are enough adults to help at the machines.\n\n# Sewing with Kids\n\nMy mom tells me what to do, cuts the fabric, and lets me sew. It's kinda like I'm doing my own thing, and I can ask \"What's my next step?\" but I can just keep going if I want to. \u2014 Audrey Greer, age 12\n\n  1. 1. Choose your favorite local fabric shop.\n  2. 2. Gather a group of three to six kids, ready to buy about 11\u20442 yards of fabric. Ask each child to choose five different fabrics that he or she loves \u2014 and set them free in the shop.\n  3. 3. Each child should have a quarter-yard of each piece cut. If the shop has pre-cuts and scrap bins, all the better. Ask the kids to choose even more of the smaller pieces.\n  4. 4. Have the kids tell each other what they love about the fabrics they chose.\n  5. 5. Then (here's the twist!) tell the kids to trade half of their fabric with one friend, and the other half with another friend. They will not like this, but you can explain that it's going to help with their creative process!\n  6. 6. Improv-piece the fabrics, sewing-as-you go, with helpers at the machine(s). (This might be a good way to get friends involved, asking them to bring their sewing machines along to help out. Or, if you're lucky enough to have a sit-and-sew shop nearby, head there.) If you want to, you can talk about the different types of fabric, balancing colors, and so on, or you can let the kids have at it.\n  7. 7. Each child should finish with a mini-quilt, which you can teach them to bind, or bind for them.\n  8. 8. Be sure to leave time for show-and-tell \u2014 everyone's favorite part! \u2014 so the kids can show off their quilts.\n\nI bought a cheap sewing machine for my girls, and my oldest has taken to it with gusto. She is currently sewing up her first quilt, and it has been so much fun talking with her about the design she wanted to make. \u2014 Tacha Bruecher\n\n# Laurie Matthews\n\n## Driven by Fabric Design\n\n_Sunflowers_ by Laurie Matthews, 82\" \u00d7 93\"\n\nBack of quilt\n\nLast year, Laurie finished twenty- _seven_ quilts. And that count doesn't include the smaller things she made, such as receiving blankets for her newest arrival, or a sweet moon pillow for her son. Maybe because she's so prolific, she can identify any fabric by designer and line (it's an impressive skill to see in action).\n\nAs she's produced each quilt, Laurie's aesthetic has evolved. \"Just recently, I've found that I really enjoy establishing a color theme, mixing prints from various collections, and using a very simple pattern that allows the fabrics to do a lot of the 'work' of a quilt.\" You can see that motif at work in this quilt that she made with fabrics from her time in Africa, combined with contemporary prints.\n\nThe sunflower print is easily my favorite souvenir I brought home after spending four months volunteering in Uganda. I knew I didn't want to cut it up too much, for fear of interfering with the scale of the flowers. So I created a design incorporating some colorful fabrics with an element of improv piecing. \u2014 lm\n\n# Lisa Mason\n\n## Get outside the Box\n\n_Straighten Up and Fly Right_ by Lisa Mason, 58\" \u00d7 58\"\n\nBack of _Straighten Up_\n\nLisa is an artist who has been sewing for years and decided about ten years ago that quilts were her medium. Seeking others who were making in the same spirit of discovery as she was, Lisa joined the Metro Modern Guild (New York City), whose president, Victoria Findlay Wolfe (page ), has hosted a 15 Minutes of Play project, encouraging improvisational piecing. Lisa loved this time experimenting with fabric. \"It's so great to be in a room full of women who are sharing their ideas and encouraging each other.\" The group has been especially supportive of Lisa since her diagnosis of breast cancer. They sent a cozy, sunny-looking yellow quilt to her home; each member had made a block. \"It was so touching, because you know that while they're making that block they're thinking about who it's going to; it's such a warm feeling of love when you get something like that.\"\n\nBeing around creative women got me interested in looking at things in different ways, and checking out more nontraditional ways of sewing. I started working with really bright colors and more modern fabrics and techniques \u2014 just thinking out of the box, rather than doing what's expected. \u2014 LM\n\n# Caroline Mason\n\n## Quilt Therapy\n\nCaroline lives about 20 minutes from her mother Lisa's house, where her sewing machine lives. \"I'll go over there and sew for a whole day. I love that I can go over to her house and we can have that time together.\"\n\nShe began to develop a love of studio art from a young age, and now makes collages. Caroline and her mother laugh when they talk about the quilt Caroline made in high school for her boyfriend; her mother's advice was that he wasn't quite quilt worthy.\n\nCaroline's work as an art therapist often involves quilts as well. As a family-bonding and legacy-building activity, Caroline has had families use their children's clothing to make quilts as a way to transition to the next stage of recovery. This is a powerful use of the medium.\n\nQuilts are often used as comfort and activism in the face of illness. Cancer survivors at one hospital make quilts for newly diagnosed patients. The _AIDS Memorial Quilt_ , which began in 1987, now spans football fields, bringing awareness of the disease and commemorating the lives of lost loved ones.\n\n# John Q. Adams\n\n## Lots of Men Do It\n\nFor John, it all started back in 2004. \"I began mixing and matching fabrics and buying fat quarters even before I knew how to sew a stitch. I got really into it \u2014 I'd think, 'This would look great with this, and this.' That was when Amy Butler's Lotus line had just come out, and all these great modern fabrics started showing up. Before long, I decided I was going to make a quilt.\" He read free tutorials online until he was comfortable with the basics, and then . . . he was hooked.\n\nIn between raising his three kids, working his day job, and cheering for the UNC Tar Heels, he started his blog, _quiltdad_ , the moniker by which he's known (and, at events like Quilt Market, sometimes hailed by admirers) in the quilting community. He's also cowritten a book with the _Fat Quarterly_ gang, been featured in several other books, and has more in the works. In short, his quilting life has taken off.\n\nAs a father who sews, John faces some challenges: \"I often get requests to make teacher gifts the week before school ends,\" he laughs. \"With three children, there's no way I can whip up six quilts in a week!\"\n\n# Men Who Quilt\n\nIn the history of quiltmaking, many men have taken up the craft. In Sue Prichard's _Quilts 1710\u20132010: Hidden Histories, Untold Stories_ _,_ she describes quilting as a tradition in soldiers' lives, with quilts and their (male) makers documented from about 1850 to the mid-1900s.\n\nSoldiers would quilt during tedious posts, or when they were recovering from injury. There are also several quilts docu-\n\nmented in _New York Beauties: Quilts from the Empire State,_ in which it's explained that many men quilted and often won awards at shows. Some of these men were tailors, who likely used their scraps for the quilts.\n\nThen there are the superb art quilters, such as Michael James, whose innovative work has, in many ways, paved the way for the modern quilting community. Kaffe Fassett (page ) is a long-loved quilter and fabric designer who has helped push design forward. Bill Kerr (page ), another important foundational member, is a design professor and quilter. Relative newbie Thomas Knauer (page ) has a big following now. The list is growing! And, as I've interviewed people for this book, I've heard many family stories of men quilting or couples quilting together. One woman said that her nine-year-old son wanted to make a quilt; he'll have lots of male and female role models to follow.\n\nI don't tell everyone that I quilt, but when I do tell someone, they think it's pretty cool. \u2014jqa\n\n###### project\n\n# Summer Twist Quilt\n\nby John Q. Adams,\n\nQuilt Dad and Fat Quarterly\n\nI used ten prints from the Heirloom line by Joel Dewberry for Free Spirit Fabrics, and four solids from the Pure Elements collection by Art Gallery Fabrics.\n\n_Summer Twist_ by John Q. Adams\n\nFinished block size: 61\u20444\" \u00d7 123\u20444\"\n\nFinished quilt size: 69\" \u00d7 761\u20442\"\n\n### What You'll Need\n\n  * 13 (1\u20442-yard) cuts and 1 (1\u20444-yard) cut assorted prints and solids\n  * 5 yards backing fabric\n  * 2\u20443 yard binding fabric\n  * General quilting supplies (page )\n\n### Cut Out the Fabric\n\n  1. 1. Cut off the selvages of all half-yard pieces (do not cut your quarter-yard fabric piece yet). From _each_ half-yard piece, cut the following: \n    * Two 5\" \u00d7 18\" strips (for a total of 26 strips)\n    * Four 75\u20448\" \u00d7 18\" strips (for a total of 52 strips)\n  2. 2. From each of your 5\" \u00d7 18\" strips, cut three 5\" squares. This will yield a total of 78 squares; you will use 71. In the illustrations, these squares will be referred to as piece A.\n\n  3. 3. Select 18 of your 75\u20448\" \u00d7 18\" strips. Cut two 75\u20448\" squares from each strip (for a total of 36 squares). Cut each square into quarters by cutting across both diagonals (as shown). This will yield a total of 144 triangles; you will use 142. In the illustrations, these triangles will be referred to as piece B.\n\n  4. 4. Trim your remaining (34) 75\u20448\" \u00d7 18\" strips down to 71\u20444\" \u00d7 18\". In addition, trim your quarter-yard cut down to 71\u20444\" \u00d7 44\". From this cut piece, cut three 71\u20444\" squares.\n  5. 5. Cut two 71\u20444\" squares from each of your 71\u20444\" \u00d7 18\" strips. This will yield a total of 68 squares. Add to this the 3 squares you cut from your quarter-yard piece, for a total of 71 squares.\n\n  6. 6. Cut each square in half on the diagonal, creating two triangles (as shown). This will yield a total of 142 triangles. You will use all 142. In the illustrations, these triangles will be referred to as piece C.\n\n### Assemble the Blocks\n\n  1. 1. The entire quilt is made from one block pattern. To construct each block, begin with a 5\" A square, set on point. Sew a B triangle to two opposite sides of the square (as shown). Press seams toward the B triangles.\n\n  2. 2. Sew a C triangle to the two other sides of the block, as shown. Press seams toward the C triangles. Your block should now measure approximately 63\u20444\" \u00d7 131\u20444\" (unfinished).\n  3. 3. Repeat steps 2 and 3 to make 71 blocks.\n\n### Assemble the Quilt Top\n\n  1. 1. You will lay out and piece your quilt top together in 11 columns. Piece together 6 columns that are 6 blocks tall each, sewing the blocks together along their 63\u20444\" edge.\n\n  2. 2. Piece together 5 columns that are 7 blocks tall each, sewing the blocks together along their 63\u20444\" edge.\n  3. 3. Sew your columns together, alternating between your 6-block and 7-block columns. When piecing your columns together, offset the placement by half of each block, as shown.\n  4. 4. Trim off the hanging block edges, as shown. Your quilt top should measure approximately 69\" \u00d7 761\u20442\".\n\n### Finishing Your Quilt\n\nBaste, quilt, and bind your quilt as desired (see Six Steps to a Quilt on page  and Finishing Your Quilt on page ).\n\n# Maritza Soto\n\n## Remember: This is Fun\n\nQuilt by Maritza Soto, 59\" \u00d7 65\"\n\nMaritza teaches at Gather Here in Cambridge (see page ), and has taught her students to make this quilt. Her best advice? \"I tell my new students to have patience with themselves, and to have fun.\" A former New Yorker, Maritza and her family moved to Cambridge because of the chance to have a better quality of life. Since she's not working full-time here, and since they have much more space, this has given her quilting the chance to grow. She jokes that her son screens the quilts for her. \"My son is the litmus test. If he likes a quilt, we know people will respond to it well.\"\n\nPeople are making **** star quilts or flying geese quilts but altering the arrangements or construction to make pieces that are almost lyrical. Or maybe someone's making a traditional quilt, like a Lone Star quilt, but they're using these fresh, bold, modern prints and arranging them in such a way that it's more a reinvention or reinterpretation of traditional design. \u2014MS\n\n# Inspiration\n\n# Finding Community Online\n\nBefore she started blogging, Laurel Krynock (page ) posted images to Flickr as a way of connecting with other quilters and sharing work. Heather Grant (page ) notes that Pinterest is another great source of quilting images and a space to share. If you don't want to commit to a blog but hope to find others and share your quilts, consider starting with one of these sites.\n\n# Old Clothes\n\nTake a cue from Gee's Bend quilters (page ), Betz White (page ), and Sherri Lynn Wood (page ) and repurpose a piece of clothing that you would otherwise donate to charity. How will you use it? Maintaining the curves of the shoulders, waist, or the embellishments and buttons? Cutting it into little pieces to work in with other fabrics? Making the color the center of a new project? Creating a usable object, such as a bag or wallet, rather than a quilt?\n\nInspiration is everywhere around you. It doesn't have to be on the Internet. I make an effort not to look at other people's designs online, just because I don't want to be influenced by someone else. For me, inspiration comes from the fashion world, or architecture. You can find design everywhere; it doesn't have to be from someone else's quilt. \u2014 Vanessa Christenson\n\n# Pippa Eccles Armbrester\n\n## Circle Wonder\n\n_Candyland_ by Pippa Eccles Armbrester, 42\" \u00d7 52\"\n\nPippa's quilts are always made of solids, and she combines circles or stripes with beautiful hand sewing: appliqu\u00e9, reverse appliqu\u00e9, hand-quilting, or embroidery. \"I love solid cotton fabrics of all colors (mainly, the bright ones) and use them in most of my quilts. I also love shot cotton fabrics, since they have such depth and dimension, even though they're solids.\"\n\nPippa found her own way of quilting as she kept sewing. \"I feel like I truly realized my aesthetic a little over a year ago. I had tried so many different approaches and different types of fabrics. Then I made a quilt with all solid fabrics and a simple geometric design of stripes and squares. It excited me more than anything else I'd made in a long time and helped me realize the power of simple geometric designs, that more complicated doesn't necessarily yield a better result.\n\n\"I also discovered that, though I love patterned fabrics in and of themselves, I really prefer to work with solids. I feel like I'm not competing with the designs in the fabric, just focusing on color and the geometry or pattern of the quilt itself. Now, my quilts generally have bold geometric designs and bright contrasting colors.\" Her book, _Solids, Stripes, Circles, and Squares: 16 Modern Patchwork Quilt Patterns,_ gives voice to her aesthetic. \"I guess you could say my love of quilting brought me to the life I want to lead.\"\n\nIn my family, I'm known for always being covered in bits of thread, for carrying my reverse appliqu\u00e9 with me so I never waste a moment of precious sewing time. \u2014 PEA\n\n**Tip:** Try listening to totally different music than you usually listen to as you design a pattern, choose your fabrics, and sew. Classical? Metal? Jazz? This can work especially well for improv piecing.\n\nTransform someone else's UFO: a family heirloom, a friend's abandoned piece. \"My grandmother started a postage-stamp quilt, and I knew I'd never finish the postage-stamp pattern. I told my mom, 'I've got to be realistic. I'm never going to finish this quilt that Grandma started.' So we cut it into thirds and then sewed the pieces together into a baby quilt for my son. That quilt has more sentimental value than anything else.\"\n\n\u2014Heather Grant\n\n###### Tutorial\n\n# Reverse Appliqueing Ovals\n\nby Pippa Eccles Armbrester\n\nwww.pippapatchwork.com\n\nLearning to reverse appliqu\u00e9 with ovals will open doors to many new design opportunities. You can combine appliqu\u00e9 and reverse appliqu\u00e9 or use it to complement your piecing, and work up to more complex patterns and designs.\n\n### Instructions\n\n  1. 1. Make two oval templates, to whatever size you wish. (See tip on page .) Trace the large oval template onto fabric, then cut it out.\n\n  2. 2. Trace and cut a small oval from the center of the large oval.\n\n  3. 3. Using a pencil, draw a line 1\u20444\" from the edge of the inner oval. Snip 1\u20444\" cuts into the inner circle up to the penciled line, every 1\u20444\" or so around the entire edge.\n\n  4. 4. Arrange the large oval on the quilt top, or whatever you are appliqu\u00e9ing onto.\n\n  5. 5. Fold the seam allowance under and hand-sew as close to the edge as possible. Hand-sew the ovals in place using regular appliqu\u00e9 for the large oval exterior, and reverse appliqu\u00e9 for the small oval interior. Reverse appliqu\u00e9 is done just the same way as appliqu\u00e9, except that you're creating a negative space instead of a positive space.\n\n  6.\n\n**Note:** Using matching color thread will make it easier!\n\n**Mini Iron.** To save herself from singeing her fingers, Pippa uses a teeny tiny iron for her appliqu\u00e9 and reverse appliqu\u00e9 projects, making it easier to get into the curves and tiny angles of the turned-under edges. Its official name is a \"mini iron,\" and it looks like a pen with a metal triangle on the end.\n\n# Virginia B. Johnson\n\n## The Third Space\n\nGather Here birthday quilt, 60\" \u00d7 74\", cotton fabric, thread, and batting\n\nVirginia and her partner, Noah, opened their Cambridge, Massachusetts, shop, Gather Here, in 2011, and business has been booming. People are drawn in by the oodles of classes offered \u2014 everything from knitting socks to sewing a dress to quilting with kids \u2014 and the spaces they've created inside for being together and making things. \"We wanted it to be about the idea of activity and industry. That's why the tag 'and make something' is there. We don't ever want this to feel stagnant.\"\n\nThe counter area is charmed with the little felted animals that Noah's mastered; a sculptor by training, it makes sense that he'd make these 3-D creatures. And Virginia sews the clothes that bedeck the front window, as well as what she wears. She was a costume designer in her past life; the dream for the shop began to grow while Virginia was working in film and spending 15 to 18 hours a day \"creating a visual story.\"\n\n\"When I was traveling, one of the things I was always doing was visiting local quilting and knitting shops, because I'd immediately have something in common with people there. It's a nice way to meet people and have conversations that went further than 'what are we going to make and shoot today?'\"\n\nWhile she was working on a not-particularly-inspiring film that had a huge budget, her friend said to her, \"What if instead of putting in 15-hour days to make someone else's dream come true, we put that much time into making our own dreams come true?\" And that was Virginia's life-changing moment. She started to invest her savings into the \"little idea that became a big idea in the course of a year.\"\n\nIn the history of quilting, garment construction, knitting, and crochet, there was always an element of group work. That's something that we have less of now in our personal spaces and our jobs; it's hard to know where to find it. We offer that third space here. This isn't work, it isn't home \u2014 it's a space you want to spend time in. \u2014 vj\n\n# Opening a Shop by Virginia B. Johnson www.gatherhereonline.com\n\n**The Small Business Association** has SCORE (a nonprofit devoted to helping small businesses), which includes really helpful forms to help you figure out how much money this will take, what does it mean to open a store, what kinds of things do you need to think about (employees and human resources, and things you may not know anything about and will eventually have to deal with).\n\n**There are also a lot of really good books** out there, such as _Handmade Nation, Craft, Inc.,_ and _The Handmade Marketplace_. They helped me to get a sense of what it means to be on your own and create your vision. My one product is the entire store. Those books about the creative person are really valid when you're opening a store that's focused on creative people.\n\n**Recognize the time commitment.** It was easy for us because we came from the film industry, in which this kind of time commitment was necessary. But the biggest difference for us is that we're responsible for the bottom line. I did not foresee being as successful early on as we were. I wish I'd had the time to devote to things that I projected needing time for a year later \u2014 like an employee handbook and payroll. I had to hire people to help sooner than I thought I would, so I was doing those things really late at night.\n\n**Noah and I have been together** for 12 years now. The store opened just after our tenth wedding anniversary. We had already gone through buying a house together, and survived that. But now we need to take care of our employees, and ourselves \u2014 and pay our own mortgage. It's easier for us to do it together in our tenth year than it would have been in our second or third years of marriage. We already know how we work and what we can take off each other's plates.\n\n**It's so easy to buy fabric** and yarn from your own home now, so the brick-and-mortar needs to offer other reasons for people to leave their homes and shop. We created that by making sure that we offered classes every single day of the week, for children as well as adults. I think that's really, really important to anyone considering doing this. Having that as a cornerstone of your business plan will make you more successful when you open.\n\n**Project-oriented classes** are so much more appealing to people. We still offer some skill clinics (a zipper clinic or binding clinic), but it's the project-based classes that are the best-selling and repeat- student based. And we keep our classes really small so that everyone gets something out of it.\n\n# Sarah Fielke\n\n## Material Culture, Material Love\n\n_Paint by Numbers_ by Sarah Fielke, 100\" \u00d7 100\"\n\nAustralian quilting has \"come and gone the same way it has in the States,\" as Sarah Fielke says, and has seen the \"same resurgence in younger people quilting.\" Her four patchwork books, including _Material Obsession One_ and _Two,_ cowritten with Kathy Doughty, plus the blog Sarah writes, have had a strong hand in that resurgence.\n\nWhile many people admire her quilting and credit her as a leader in the modern quilting movement, she's wary of taking on the label \"modern.\" As she says, \"I think quilting's quilting. I don't see why everyone has to have a tag. I've always been interested in old quilts and textiles. I have the most terrifying library of books, and I encourage people to learn more about the craft before trying to revolutionize it.\" Sarah herself has been sewing for years and years, honing her skills over time.\n\nShe describes her aesthetic as \"contemporary traditional,\" saying that she reinterprets traditional designs. \"There's all sorts of quilts in my books, just quilts I like to make that can translate into any kind of fabric. Even the really, really bright, mad quilts I make can be reinterpreted in Civil War prints if you want to. I think the fact that I don't label my quilts really helps reach a broader range of people.\" She pointed to a quilt hanging behind her as we talked, saying that it has all sorts of fabrics in it, all reds. \"A lot of my readers are quilters already, accomplished quilters, so that's why I make more complicated quilts, which I love making.\"\n\nI keep a lot of journals. There are notebooks all over the place, with bits of paper stuck in them. Sometimes, when I start a new quilt, I'll go through and look for something particular. Or, I'll go through my antique quilts and get a spark. Or, there might be a color combination that I really love. I recently saw a really colorful bird on Pinterest that inspired me. Frequently, if I start with a sketch, the finished quilt turns out nothing like the sketch. \u2014SF\n\n\"I get e-mails from people all over the world, everyone from 16-year-old students who made my quilt for their school project to 90-year-old grandmas who used hot pink and feel absolutely daring. I love that. I get such a kick out of hearing back from people.\"\n\nI have the most gorgeous 1870s hexagon quilt. I have a lot of tops, and I don't subscribe to the idea that you shouldn't finish them. Sometimes, I'll finish the quilt and write a pattern for making the quilt, so I can justify having bought them. Unless they're in a museum or something, if you don't finish them, then what happens \u2014 they go into another cupboard, and someone else buys them in a hundred years' time. Finishing them gives them some life, like they're supposed to have. \u2014SF\n\n# Melody Miller\n\n## Objects Can Live Second, Third, and Fourth Lives\n\nFabric panel by Melody Miller, 70\" \u00d7 87\"\n\nTypewriters, viewfinders, cameos of women of the 60s, handheld radios, point-and-click retro cameras, clocks, and the odd deer head. If you saw this list on someone's fabrics-to-design schedule, you might scratch your head and wonder where these ideas are coming from. But if you know Melody Miller's work, you know her retro flavor and love of the object.\n\nShe says that she's visited lots of thrift shops and found inspiration on the shelves where the objects we needed yesterday are passed along to land in someone else's hands. \"Fifty years ago, you didn't go to Target to buy a new napkin holder because you saw a cool one there. You used the same one your whole life. These objects I've seen at thrift stores were an important part of people's lives. And now that napkin holder might be on a shelf at a thrift store, and if I get that napkin holder, it has this rich past to it \u2014 if I could know what that object has seen \u2014 that's so intriguing to me.\"\n\nWe're all a part of this material culture, after all, creating objects that someone, years from now, might hold and remember. Melody is intrigued by these objects' pasts, and longs for the days when we were more \"loyal\" to our objects, not simply discarding them for the next best thing that comes along every week.\n\nAs she builds a fabric design, once she has the object in mind, she'll start drawing it and creating its color palette, building up one color at a time. \"It's about what I'm seeing or feeling while I'm working with color. I pay a lot of attention to value. I'm always looking at a whole spectrum of value. If they're all too much the same value, it doesn't have the emotion for me.\" She keeps adjusting the colors until they all start to balance one another.\n\nWhen I see something I love, instead of trying to imitate it, I ask myself, \"What do I like about that?\" Sometimes, it's an emotional reaction, or a feeling about color, or a technical difficulty. I try to figure out what it is that I'm responding to, and then set that standard for myself. I ask, \"Can I do something technically that difficult as well?\" Or, \"Can I make color that's that exciting?\" Or maybe it's speaking to another era, maybe there's a nostalgia that feels good. So, I try to identify what it is about the thing that I love, rather than trying to copy it. And I try to meet that standard for myself. \u2014MM\n\n# Acknowledgments\n\nThis book grew out of my friendships in the Boston Modern Quilt Guild. I'm especially grateful to Monica Ripley, Rebecca Loren, Pippa Eccles Armbrester, Laurie Matthews, Monet Brewerton, and Caro Sheridan, who shared their work and ideas in the early stages. Many thanks to Laurie, Rebecca, and Alice Webb Greer for their help finding quilters and their generosity in sharing what they know and love.\n\nI'm indebted to all the contributors in this book, whose thoughts and ideas inspired me; thank you for sharing your work and passion. I'm especially grateful to Weeks Ringle and Bill Kerr, for a conversation that reinvigorated me, and for their astounding generosity. Virginia Johnson and Noah Dubreuil, owners of Gather Here, in Cambridge, Massachusetts, opened the doors of their gorgeous \"third space\" for our photo shoot. Fabric for some of the projects was supplied by Dear Stella, Robert Kaufman, Ann Kelle, and Joel Dewberry. Many thanks to Pam Thompson, whose patience, insight, and wise advice in editing this book pulled it all together and brought it to life; to Deborah Balmuth, who believed in this project early on; to Carolyn Eckert for the gorgeous design; and to Alee Marsh for ushering it into the world.\n\nTo friends and family who helped in ways writerly and beyond \u2014 especially Joan Maki for your faith, friendship, and writer-talks; Amy Brown, for your friendship and feedback; Ellen Goldstein and Mike Martin, for your jokes, advice, dog walks, spare bedroom, and brunches, among other invaluable things; and to Danielle Krcmar, for your help finding quilters and artists, quilting books(!), and advice. Thank you also to Anna Brecke for the supply of gluten-free treats and laughter, Stacey and Jason Schrontz, Sara Gebhardt, Liz Hagyard, Paola Massoli, and Brinda Avandani, for various forms of encouragement and celebration. Thanks to Piper, of course. A heartfelt and forever-grateful word of thanks to Leslie and Thea, who showed me the way through. Thea, thank you for your patience, encouragement, and good dose of humor, which has helped me take on this book, among many other things.\n\nFinally, to my family \u2014 Jeff and Melissa Primeau gave me a home and family when I needed it most, and to Ryan and Owen for your joy and warm welcome. Thank you, forever. And to Michelle and Andrew Moore, and Meagan, Katelyn, and Erin, for stepping in when I needed you, and for always being there. And to the Bush family for including me in warm holiday celebrations. Thank you also to my mother, for teaching me to sew and giving me an artist's dream, to my father for his support, to Mary for my first typewriter, and to my siblings Josh and Becky. Becky, you are an inspiration in your artist's life and tenacity; thank you for being there.\n\nRachel May, 95\" \u00d7 80\"\n\n# Contributors\n\n#### Adams, John Q.\n\n  * Quilt Dad\n  * www.quiltdad.com\n\n#### Armbrester, Pippa Eccles\n\n  * Pippa Quilts\n  * www.pippapatchwork.com\n\n#### Barton, Elizabeth\n\n  * Art and Quilts, Cogitations Thereon\n  * www.elizabethbarton.blogspot.com\n\n#### Bencsko, Engel, and Michelle and Gina Pantastico\n\n  * Cloud9 Fabrics, Inc.\n  * www.cloud9fabrics.com\n\n#### Blaich, Miriam\n\n  * Berlin Quilter\n  * www.berlinquilter.blogspot.com\n\n#### Blakesley, Katie\n\n  * SwimBikeQuilt\n  * www.swimbikequilt.com\n\n#### Boyd, Kelle\n\n  * Ann Kelle\n  * www.annkelle.com\n\n#### Brown, Amy\n\n  * craftymammal\n  * www.craftymammal.wordpress.com\n\n#### Bruecher, Tacha\n\n  * Fat Quarterly\n  * www.fatquarterly.com\n\n#### Bruntse, Line\n\n  * www.linebruntse.com\n\n#### Burge, Denise\n\n  * My Original Dirt\n  * www.myoriginaldirt.com\n\n#### Butler, Amy\n\n  * Amy Butler Design\n  * www.amybutlerdesign.com\n\n#### Butler, David\n\n  * Parson Gray\n  * www.parsongray.com\n\n#### Cho, Kyoung Ae\n\n  * www.kyoungaecho.com\n\n#### Christenson, Vanessa\n\n  * V and Co.\n  * www.vanessachristenson.com\n\n#### Coleman-Hale, Rashida\n\n  * i heart linen\n  * www.iheartlinen.typepad.com\n\n#### Cory, Pepper\n\n  * www.peppercory.com\n\n#### Crow, Nancy\n\n  * www.nancycrow.com\n\n#### Deise, Alexis\n\n  * Materials & Method\n  * www.materialsandmethod. blogspot.com\n\n#### Dick, Jenifer\n\n  * www.42quilts.com\n\n#### Dubrawsky, Malka\n\n  * a stitch in dye\n  * www.stitchindye.blogspot.com\n\n#### Duerr, Sasha\n\n  * Permacouture Institute\n  * sashaduerr.tumblr.com\n\n#### Eichler-Messmer, Kim\n\n  * Kim E-M Handmade Quilts\n  * www.kimemquilts.com\n\n#### Fassett, Kaffe\n\n  * Kaffee Fassett Studio\n  * www.kaffefassett.com\n\n#### Fielke, Sarah\n\n  * www.sarahfielke.com\n  * the last piece\n  * www.thelastpiece.net\n\n#### Gering, Jacquie\n\n  * Tallgrass Prairie Studio\n  * www.tallgrassprairiestudio.blogspot.com\n\n#### Gillman, Rayna\n\n  * studio 78\n  * www.studio78.net\n\n#### Grama, Geta\n\n  * Romanian Quilt Studio\n  * www.romanianquiltstudio.com\n\n#### Grant, Heather\n\n  * Modern Day Quilts\n  * www.moderndayquilts.tumblr.com\n\n#### Greer, Alice Webb\n\n  * alidiza\n  * www.alidiza.com\n\n#### Grifka, Debbie\n\n  * Esch House Quilts\n  * www.eschhousequilts.com\n\n#### Harris, Allison\n\n  * Cluck Cluck Sew\n  * www.cluckclucksew.com\n\n#### Hartman, Elizabeth\n\n  * Oh, Fransson!\n  * www.ohfransson.com\n\n#### Heinrich, Lee\n\n  * Freshly Pieced\n  * www.freshlypieced.com\n\n#### Hooey, Aneela\n\n  * comfortstitching\n  * www.comfortstitching.typepad.co.uk\n\n#### Horner, Anna Maria\n\n  * www.annamariahorner.com\n\n#### Hutchinson, Rossie\n\n  * r0ssie blog \/ Fresh Modern Quilts\n  * www.r0ssie.blogspot.com\n\n#### Jinzenji, Yoshiko\n\n  * The World of Yoshiko Jinzenji\n  * www.yoshikoquilt.com\n\n#### Johnson, Jan\n\n  * Lowell, Massachusetts\n\n#### Johnson, Virginia B.\n\n  * gather here and make something\n  * www.gatherhereonline.com\n\n#### Jones, Katy\n\n  * Monkey Do\n  * www.imagingermonkey.blogspot.com\n\n#### Keefer, Amy\n\n  * www.amykeefer.com\n\n#### Kimber, Chawne\n\n  * Completely Cauchy\n  * www.cauchycomplete.wordpress.com\n\n#### Knauer, Thomas\n\n  * Thomas Knauer Sews\n  * www.thomasknauersews.com\n\n#### Kovach, Jessica\n\n  * Twin Fibers\n  * www.twinfibers.blogspot.com\n\n#### Krcmar, Danielle\n\n  * Babson College\n  * www.babson.edu\/faculty\/profiles\/Pages\/krcmar-danielle.aspx\n\n#### Krynock, Laurel\n\n  * Sing All You Want\n  * www.krynocks.blogspot.com\n\n#### Ledgerwood, Alexandra\n\n  * Teaginny Designs\n  * www.teaginnydesigns.blogspot.com\n\n#### Link, Kristin\n\n  * Sew Mama Sew\n  * www.sewmamasew.com\n\n#### Loren, Rebecca\n\n  * Parsley is the New Black\n  * www.parsleyisthenewblack.blogspot.com\n\n#### Mack, Kathy\n\n  * Pink Chalk Fabrics\n  * www.pinkchalkfabrics.com\n  * Pink Chalk Studio\n  * www.pinkchalkstudio.com\n\n#### Mann, Cynthia\n\n  * Birch Fabrics\n  * www.birchfabrics.com\n  * fabricworm\n  * www.fabricworm.com\n\n#### Maser-Flanagan, Valerie\n\n  * www.valeriemaserflanagan.com\n\n#### Mason, Caroline\n\n  * Greenwich, Connecticut\n\n#### Mason, Lisa\n\n  * The Red Headed Mermaid\n  * www.theredheadedmermaid.blogspot.com\n\n#### Matthews, Laurie\n\n  * Dresden Lane\n  * www.dresdenlane.com\n\n#### Miller, Melody\n\n  * www.melodymiller.net\n\n#### Mowbray, Andrew\n\n  * www.andrewmowbray.com\n\n#### Newcomb, Ashley\n\n  * Film in the Fridge\n  * www.filminthefridge.com\n\n#### Nguyen, Amy\n\n  * Amy Nguyen Textiles\n  * www.amynguyentextiles.com\n\n#### Pedersen, Katie\n\n  * Sew Katie Did\n  * www.sewkatiedid.wordpress.com\n\n#### Raymond, Aimee\n\n  * Salt Marsh Designs\n  * www.saltmarshdesigns.blogspot.com\n\n#### Ricketson, Kathreen\n\n  * Whip Up\n  * www.whipup.net\n\n#### Ringle, Weeks & Bill Kerr\n\n  * Modern Quilt Studio\n  * www.modernquiltstudio.com\n\n#### Ripley, Monica\n\n  * Somerville, Massachusetts\n\n#### Sassaman, Jane\n\n  * www.janesassaman.com\n\n#### Schmidt, Denyse\n\n  * www.dsquilts.com\n\n#### Sheridan, Caro\n\n  * Splityarn\n  * www.splityarn.com\n\n#### Shrontz, Stacey\n\n  * lucky button baby\n  * www.sshrontz.blogspot.com\n\n#### Soto, Maritza\n\n  * soto softies\n  * www.sotosofties.blogspot.com\n\n#### Spencer Hunt, Lauren\n\n  * Aunt June\n  * www.myauntjune.blogspot.com\n\n#### Wells, Valori\n\n  * Valori Wells Designs\n  * www.valoriwells.com\n\n#### White, Betz\n\n  * www.betzwhite.com\n\n#### Wolfe, Victoria Findlay\n\n  * Bumble Beans, Inc.\n  * www.bumblebeansinc.com\n\n#### Wood, Sherri Lynn\n\n  * daintytime\n  * www.daintytime.net\n\n# Bibliography\n\n## Websites\n\n#### The Quilt Index\n\n  * www.quiltindex.org\n\n#### International Quilt Study Center & Museum\n\n  * University of Nebraska, Lincoln, Nebraska,\n  * www.quiltstudy.org\n\n#### The Modern Quilt Guild\n\n  * www.themodernquiltguild.com\n\n## Quilt & Textile History\n\nArnett, Paul, Joanne Cubbs, and Eugene W. Metcalf, eds. _Gee's Bend: The Architecture of the Quilt._ Tinwood Books, 2006.\n\nAtkins, Jacqueline M., and Phyllis A. Tepper. _New York Beauties: Quilts from the Empire State_. __ Dutton Studio Books, 1992. __\n\nBeardsley, John, William Arnett, Paul Arnett, and Jane Livingston. _Gee's Bend: The Women and Their Quilts._ Tinwood Books, 2002.\n\nBowman, Doris M. _American Quilts: The Smithsonian Treasury._ Smithsonian Institution Press, 1995.\n\nBrackman, Barbara. _Quilts from the Civil War._ C&T Publishing, 1997.\n\nCameron, Dan, Richard J. Powell, Michelle Wallace, Patrick Hill, Thalia Gouma-Peterson, Moira Roth, and Ann Gibson. _Dancing at the Louvre: Faith Ringgold's French Collection and Other Story Quilts._ University of California Press, 1998.\n\nCrow, Nancy and David Hornung. _Crossroads: Constructions, Markings, and Structures._ Breckling Press, 2007.\n\n\u2014\u2014\u2014. _Improvisational Quilts._ C&T Publishing, 1995.\n\n\u2014\u2014\u2014. _Nancy Crow:_ _Quilts and Influences._ American Quilter's Society, 1990.\n\nDick, Jenifer, Carol Bohl, Linda Hammontree, and Janice Britz. _Stories in Stitches: Quilts from the Cass County Documentation Project._ Kansas City Star Books, 2012.\n\nGillespie, Spike. _Quilts Around the World._ Voyageur Press, 2010.\n\nGillow, John. _African Textiles: Color and Creativity Across a Continent._ Thames & Hudson, 2009.\n\nGordon, Beverly. _Textiles: The Whole Story._ Thames & Hudson, 2011.\n\nHanson, Marin F., and Patricia Cox Crews. _American Quilts in the Modern Age, 1870\u20131940._ University of Nebraska Press, 2009.\n\nHicks, Kyra E. _This I Accomplish: Harriet Powers' Bible Quilt and Other Pieces._ Black Threads Press, 2009.\n\nHolstein, Jonathan. _Abstract Design in American Quilts._ Kentucky Quilt Project, 1991.\n\nJames, Michael. _Michael James: Art and Inspirations_. C&T Publishing, 1998.\n\nKiracofe, Roderick, and Mary Elizabeth Johnson. _The American Quilt: A History of Cloth and Comfort, 1750\u20131950._ Clarkson Potter, 2004.\n\nLiu, Xinru. _The Silk Road in World History_. Oxford University Press., 2010.\n\nMason, Darielle. _Kantha: The Embroidered Quilts of Bengal._ Yale University Press, 2010.\n\nMcKinley, Catherine E. _Indigo: In Search of the Color that Seduced the World_. Bloomsbury, 2011.\n\nMeller, Susan, and Joost Elffers, _Textile Designs: Two Hundred Years of European and American Patterns._ Harry N. Abrams, 2002.\n\nParker, Rozsika. _The Subversive Stitch: Embroidery and the Makings of the Feminine_. I. B. Tauris, 2010.\n\nPellman, Rachel, and Kenneth Pellman. _The World of Amish Quilts,_ rev. ed _._ Good Books, 1998.\n\nPrichard, Sue. _Quilts, 1700\u20132010: Hidden Histories, Untold Stories._ V&A Publishing, 2010.\n\nQuilters Hall of Fame. _The Quilters Hall of Fame: 42 Masters Who Have Shaped Our Art_. Voyageur Press, 2011.\n\nRing, Betty. _American Needlework Treasures: Samplers and Silk Embroideries from the Collection of Betty Ring_. E. P. Dutton, 1987.\n\nRoberts, Elise Schebler. _The Quilt: A History and Celebration of an American Art Form._ Voyageur Press, 2010.\n\nSchoeser, Mary. _Textiles: The Art of Mankind._ Thames & Hudson, 2012.\n\nShaw, Madelyn, and Lynne Z. Bassett. _Homefront & Battlefield:_ _Quilts & Context in the Civil War. _American Textile History Museum, 2012.\n\nShaw, Robert. _Art Quilts: A Celebration: 400 Stunning Contemporary Designs_. Lark Books, 2005.\n\nStoddard, Patricia Ormsby. _Ralli Quilts: Traditional Textiles from Pakistan and India._ Schiffer Publishing, 2003.\n\nSumberg, Bobbie. _Textiles: Collection of the Museum of International Folk Art._ Gibbs Smith, 2010.\n\nUlrich, Laurel Thatcher. _A Midwife's Tale: The Life of Martha Ballard Based on Her Diary_. Vintage Books, 1991.\n\nWaldvogel, Merikay. _Soft Covers for Hard Times: Quiltmaking & the Great Depression. _Rutledge Hill Press, 1990.\n\nWarren, Elizabeth V. _Quilts: Masterworks from the American Folk Art Museum._ Rizzoli, 2010.\n\nWelters, Linda, and Margaret T. Ordo\u00f1ez, eds. _Down by the Old Mill Stream: Quilts in Rhode Island._ Kent State University Press, 2000.\n\n## Contemporary Quilting\n\nAdams, John Q. _Pretty in Patchwork Holidays: 30+ Seasonal Patchwork Projects to Piece, Stitch, and Love_. Lark Books, 2012.\n\nArkison, Cheryl. _A Month of Sundays: Family, Friends, Food, & Quilts: 16 Projects, Precut Friendly_. C&T Publishing, 2013.\n\nArmbrester, Pippa Eccles. _Solids, Stripes, Circles, and Squares: 16 Modern Patchwork Quilt Patterns_. Martingale, 2012.\n\nBarton, Elizabeth. _Inspired to Design_. C&T Publishing, 2013.\n\nBeal, Susan. _Modern Log Cabin Quilting._ Potter Craft, 2011.\n\nBell, Katherine. _Quilting for Peace: Make the World a Better Place One Stitch at a Time._ STC Craft, 2009.\n\nBrocket, Jane. _The Gentle Art of Quilt-making: 15 Projects Inspired by Everyday Beauty_. C&T Publishing, 2010.\n\nBruecher, Tacha. _Hexa-Go-Go: English Paper Piecing: 16 Quilt Projects_. Stash Books, 2012.\n\nButler, Amy. _Amy Butler's Midwest Modern._ Stewart, Tabori & Chang, 2007.\n\n\u2014\u2014\u2014. _Amy Butler's Little Stitches for Little Ones_. Chronicle, 2008.\n\n\u2014\u2014\u2014. _Amy Butler's Style Stitches: 12 Easy Ways to 26 Wonderful Bags_. Chronicle, 2010.\n\nCier, Emily. _Scrap Republic: 8 Projects for Those Who Love Color_. C&T Publishing, 2011.\n\nColeman-Hale, Rashida. _I Love Patchwork_. Interweave Press, 2009.\n\n\u2014\u2014\u2014. _Zakka Style: 24 Projects Stitched with Ease to Give, Use & Enjoy_. C&T Publishing, 2011.\n\nCory, Pepper. _Mastering Quilt Marking_. C&T Publishing, 1999.\n\nDoughty, Kathy, and Sarah Fielke. _Material Obsession: Modern Quilts with Traditional Roots._ Stewart, Tabori, & Chang, 2009.\n\n\u2014\u2014\u2014. _Material Obsession 2: More Modern Quilts with Traditional Roots._ STC Craft, 2010.\n\nDubrawsky, Malka. _Fresh Quilting: Fearless Color, Design & Inspiration._ Interweave, 2010.\n\n\u2014\u2014\u2014. _Color Your Cloth: A Quilter's Guide to Dyeing and Patterning Fabric_. Lark, 2009.\n\nDuerr, Sasha. _The Handbook of Natural Plant Dyes._ Timber Press, 2010. __\n\nFassett, Kaffe. _Kaffe Fassett's Simple Shapes Spectacular Quilts: 23 Original Designs._ Stewart, Tabori & Chang, 2010.\n\n\u2014\u2014\u2014. _Dreaming in Color: An Autobiography._ STC Craft, 2012.\n\nGering, Jacquie, and Katie Pedersen. _Quilting Modern: Techniques and Projects for Improvisational Quilts_. Interweave, 2012.\n\nGillman, Rayna. _Create Your Own Free-Form Quilts._ C&T Publishing, 2011.\n\n\u2014\u2014\u2014. _Create Your Own Hand-Printed Cloth._ C&T Publishing, 2008.\n\nGrama, Geta. _Shadow Trapunto Quilts: Simple Steps, Remarkable Results, 30 Elegant Projects._ C&T Publishing, 2012.\n\nHarding, Deborah. _Red & White: American Redwork Quilts and Patterns_. Rizzoli, 2000.\n\nHarris, Allison. _Growing Up Modern: 16 Quilt Projects for Babies and Kids_. Stash Books, 2013.\n\nHart, Jenny. _Embroidered Effects: Projects and Patterns to Inspire Your Stitching._ Chronicle, 2009.\n\nHartman, Elizabeth. _The Practical Guide to Patchwork._ C&T Publishing, 2010.\n\n\u2014\u2014\u2014. _Modern Patchwork: 12 Quilts to Take You Beyond the Basics_. C&T Publishing, 2012.\n\nHedley, Gwen. _Drawn to Stitch: Line, Drawing, and Mark-Making in Textile Art_. Interweave Press, 2010.\n\nHicks, Kyra. _1.6 Million African American Quilters: Surveys, Sites, and a Half-Dozen Art Quilt Blocks_. Black Threads Press, 2010.\n\nHoey, Aneela. _Little Stitches: 100+ Sweet Embroidery Designs_. C&T Publishing, 2012.\n\nHorner, Anna Maria. _Anna Maria's Needleworks Notebook_. Wiley, 2012.\n\n\u2014\u2014\u2014. _Handmade Beginnings: 24 Sewing Projects to Welcome Baby._ Wiley, 2010.\n\n\u2014\u2014\u2014. _Seams to Me: 24 New Reasons to Love Sewing._ Wiley, 2008.\n\nHouse, Cherri. _City Quilts: 12 Dramatic Projects Inspired by Urban Views._ C&T Publishing, 2010.\n\nHoverson, Joelle. _Last-Minute Patchwork & Quilted Gifts_. Stewart, Tabori & Chang, 2007.\n\nJinzenji, Yoshiko. _Quilt Artistry: Inspired Designs from the East._ Kodansha International, 2009.\n\n\u2014\u2014\u2014. _Quilting Line & Color: Techniques and Designs for Abstract Quilts._ Interweave, 2010.\n\nJones, Katy, Brioni Greenberg, Tacha Bruecher, and John Q. Adams. _Fat Quarterly Shape Workshop for Quilters: 60 Blocks Plus a Dozen Quilts and Projects_. Lark Crafts, 2012.\n\nKight, Kimberly. _A Field Guide to Fabric Design._ C&T Publishing, 2011.\n\nKoseki, Suzuko. _Natural Patchwork: 26 Stylish Projects Inspired by Flowers, Fabric & Home._ Trumpeter, 2011.\n\n\u2014\u2014\u2014. _Playful Patchwork: Happy, Colorful, and Irresistible Ideas and Instruction for Modern Piecework, Applique, & Quilting._ Creative Publishing, 2011.\n\nMarston, Gwen. _Liberated Quiltmaking II._ American Quilter's Society, 2010.\n\nMazloomi, Carolyn, and Faith Ringgold. _Spirits of the Cloth: Contemporary African American Quilts_. Clarkson Potter, 1998.\n\nMiller, Melody, and Allison Tannery. _Ruby Star Wrapping: Creating Packaging to Reuse, Regive & Relove. _Roost Books, 2012. __\n\nModa Bakeshop Designers. _Fresh Fabric Treats: 16 Yummy Projects to Sew from Jelly Rolls, Layer Cakes, and More._ C&T Publishing, 2011.\n\nNyberg, Amanda Jean, and Cheryl Arkison. _Sunday Morning Quilts: 16 Modern Scrap Projects._ C&T Publishing, 2012.\n\nOwen, Cheryl. _Simple Stitches: 18 Projects for the New Quilter._ Lark Crafts, 2011.\n\nRicketson, Kathreen. _Little Bits Quilting Bee: 20 Quilts Using Charm Squares, Jelly Rolls, Layer Cakes, and Fat Quarters_. Chronicle Books, 2011.\n\n\u2014\u2014\u2014. _Whip Up Mini Quilts: Patterns and How-To for More than 20 Contemporary Small Quilts._ Chronicle Books, 2010.\n\nRicucci, Tonya. _Word Play Quilts:_ _Easy Techniques from the UnRuly Quilter_. Martingale, 2010.\n\nRinggold, Faith, and Curlee Raven Holton. _A View from the Studio_. Bunker Hill Publishing, 2005.\n\nRingle, Weeks, and Bill Kerr. _The Modern Quilt Workshop: Patterns, Techniques, and Designs from the FunQuilts tudio._ Quarry Books, 2005.\n\n\u2014\u2014\u2014. _Quilts Made Modern: 10 Projects, Keys for Success with Color & Design, From the FunQuilts Studio. _C&T Publishing, 2010.\n\n\u2014\u2014\u2014. _Transparency Quilts: 10 Modern Projects, Keys for Success in Fabric Selection from the FunQuilts Studio_. C&T Publishing, 2012.\n\nRoss, Heather. _Heather Ross Prints: 50+ Designs and 20 Projects to Get You Started._ STC Craft, 2012.\n\nSaito, Yoko. _Japanese Quilting Piece by Piece: 29 Stitched Projects from Yoko Saito_. Interweave Press, 2012.\n\nSassaman, Jane A. _Patchwork Sassaman Style: Recipes for Dazzling Quilts._ Dragon Threads, 2012.\n\n\u2014\u2014\u2014. _The Quilted Garden: Design & Make Nature-Inspired Quilts_. C&T Publishing, 2000.\n\nSchaefer, Kim. _Cozy Modern Quilts: 23 Easy Pieced Projects to Bust Your Stash._ C&T Publishing, 2009.\n\nSchmidt, Denyse. _Denyse Schmidt Quilts: 30 Colorful Quilt and Patchwork Projects._ Chronicle, 2005.\n\n\u2014\u2014\u2014. _Denyse Schmidt: Modern Quilts Traditional Inspiration_. STC Craft, 2012.\n\nSchmidt, Kathryn. _Rule-Breaking Quilts._ American Quilter's Society, 2010.\n\nSielman, Martha. _Art Quilt Portfolio: The Natural World, Profiles of Major Artists, Galleries of Inspiring Works_. Lark Crafts, 2012\n\nWalters, Angela. _Free-Motion Quilting with Angela Walters: Choose & Use Quilting Designs for Modern Quilts_. C&T Publishing, 2012.\n\n\u2014\u2014\u2014. _In the Studio with Angela Walters: Machine-Quilting Design Concepts Add Movement, Contrast, Depth & More_. C&T Publishing, 2012.\n\nWells, Jean. _Intuitive Color & Design: Adventures in Art Quilting. _C&T Publishing, 2009.\n\n\u2014\u2014\u2014. _Journey to Inspired Art Quilting._ C&T Publishing, 2012.\n\nWells, Valori. _Simple Start Stunning Finish._ C&T, 2007.\n\n\u2014\u2014\u2014. _Stitch 'n' Flip Quilts: 14 Fantastic Projects_. C&T Publishing, 2000.\n\nWhite, Betz, and John Gruen. _Sewing Green: 25 Projects Made with Repurposed and Organic Materials_. STC Craft, 2009.\n\nWisbrun, Laurie. _Mastering the Art of Fabric Printing and Design._ Chronicle, 2011.\n\nWolfe, Victoria Findlay. _15 Minutes of Play: Improvisational Quilts._ C&T Publishing, 2012.\n\nWoods, Susanne. _We Love Color: 16 Iconic Quilt Designers Create with Kona Solids._ C&T Publishing, 2012.\n\n# Metric Conversion\n\n# Photography Credits\n\n1 \u00a9 John Polak\n\n3 \u00a9 Keller + Keller Photography Inc.\n\n4, 5 \u00a9 John Polak\n\n6 Sawtooth Diamond in a Square, c. 1910 (textile), American School, (20th century)\/Indianapolis Museum of Art, U.S.A. \/ Roger Wolcott Fund\/The Bridgeman Art Library (top); Amish Quilt, 1932 (wool & cotton), American School, (20th century)\/ University of East Anglia, Norfolk, UK \/Robert and Lisa Sainsbury Collection\/The Bridgeman Art Library (2nd from top); Amish Quilt, 1890 (wool & cotton), American School, (19th century)\/ University of East Anglia, Norfolk, UK \/Robert and Lisa Sainsbury Collection\/The Bridgeman Art Library (3rd from top); The Granger Collection, New York (bottom)\n\n7 Courtesy of Souls Grown Deep Foundation; Stephen Pitkin\/Pitkin Studio\n\n8 \u00a9 John Polak (left), \u00a9 Keller + Keller Photography Inc. (right)\n\n9 \u00a9 Keller + Keller Photography Inc. (left), \u00a9 John Polak (top right), \u00a9 Shades of Gray Photography (bottom right)\n\n10 \u00a9 Shades of Gray Photography (far right), \u00a9 Keller + Keller Photography Inc. (all others)\n\n11 \u00a9 Shades of Gray Photography (top row 2nd from left & bottom), \u00a9 Keller + Keller Photography Inc. (all others)\n\n12, 13 \u00a9 Keller + Keller Photography Inc.\n\n14 \u00a9 Keller + Keller Photography Inc. (top row); Chinese Souls 2, 1992 (cotton), Crow, Nancy (b. 1943)\/Indianapolis Museum of Art, USA\/Mary Black Fund\/The Bridgeman Art Library (left top); Courtesy of Yoshiko Jinzenji (left middle); Brooklyn Institute of Arts and Sciences\/Gavin Ashworth\/Gift in memory of Horace H. Solomon (left bottom)\n\n15, 16\u201317 \u00a9 Keller + Keller Photography Inc.\n\n17 \u00a9 John Polak (small squares)\n\n18\u201320 Courtesy of Bill Kerr & Weeks Ringle\n\n22 \u00a9 John Polak\n\n25 Courtesy of Rashida Coleman-Hale\n\n26 \u00a9 John Polak\n\n27 Courtesy of Yoshiko Jinzenji\n\n28\u201330 \u00a9 Shades of Gray Photography\n\n31\u201333 \u00a9 John Polak\n\n34, 35 \u00a9 David Butler\n\n36, 37 \u00a9 John Polak\n\n38, 39 Courtesy of Kathy Mack\n\n40 \u00a9 Keller + Keller Photography Inc. (left), \u00a9 John Polak (right)\n\n41 \u00a9 John Polak (top), The Granger Collection, New York (bottom left), \u00a9 Rebecca Loren (bottom center &right)\n\n43 Brooklyn Institute of Arts and Sciences\/Gavin Ashworth\/Gift in memory of Horace H. Solomon\n\n44\u201345 \u00a9 Buyenlarge\/UIG\/agefotostock.com\n\n45 \u00a9 John Polak (small squares)\n\n46 \u00a9 Rivane Neuenschwander, Courtesy of Tanya Bonakdar Gallery, New York (left), Courtesy of Rossie Hutchinson (right)\n\n47\u201350 \u00a9 John Polak\n\n51 Courtesy of Souls Grown Deep Foundation; Stephen Pitkin\/Pitkin Studio\n\n52, 53 \u00a9 John Polak\n\n54, 55 Courtesy of Kyoung Ae Cho\n\n56\u201358 \u00a9 John Polak\n\n59 \u00a9 J. Kevin Fitzsimons\n\n60 Chinese Souls 2, 1992 (cotton), Crow, Nancy (b. 1943)\/Indianapolis Museum of Art, USA\/Mary Black Fund\/The Bridgeman Art Library\n\n62\u201365 \u00a9 John Polak\n\n67, 68 Gamma One Conversions\n\n69 \u00a9 Shades of Gray Photography\n\n70, 71 \u00a9 John Polak\n\n72 \u00a9 John Polak (left), \u00a9 Sherri-Lynn Wood ~ daintytime.net (right)\n\n73 \u00a9 John Polak (middle row left), \u00a9 Sherri-Lynn Wood ~ daintytime.net (all others)\n\n74 \u00a9 Sherri-Lynn Wood ~ daintytime.net\n\n75 \u00a9 Sherri-Lynn Wood ~ daintytime.net (top, left & right), \u00a9 John Polak (bottom)\n\n76 \u00a9 John Polak\n\n77 \u00a9 John Polak (left), Courtesy of Danielle Krcmar (right)\n\n78\u201379 \u00a9 Keller + Keller Photography Inc.\n\n79 \u00a9 John Polak (small squares)\n\n81 Courtesy of Kristin Link\n\n82 Courtesy of C. M. Kimber\n\n83 Courtesy of C. M. Kimber (left), \u00a9 John Polak (right)\n\n84 Courtesy of Denise Burge\n\n86 \u00a9 Keller + Keller Photography Inc. (left), \u00a9 John Polak (right)\n\n87 \u00a9 John Polak\n\n88 \u00a9 John Polak (top), \u00a9 Keller + Keller Photography Inc. (bottom)\n\n89\u201391 \u00a9 John Polak\n\n92\u201395 \u00a9 Keller + Keller Photography Inc.\n\n97, 98 \u00a9 Clements Howcraft\n\n99 Photo by Chris Pullman, Whitney Museum of American Art, New York, New York. From \"Why Quilts Matter: History, Art & Politics episode 6, \"How Quilts Are Viewed and Collected,\" produced by The Kentucky Quilt Project\n\n100, 101 \u00a9 John Polak\n\n102, 103 Courtesy of Lee Heinrich, Freshly Pieced\n\n104, 105 \u00a9 John Polak\n\n106\u2013107 \u00a9 Keller + Keller Photography Inc.\n\n107 \u00a9 John Polak (small squares)\n\n108 Courtesy of Anna Maria Horner\n\n109 \u00a9 Keller + Keller Photography Inc.\n\n110 Courtesy of Anna Maria Horner (top), \u00a9 Keller + Keller Photography Inc. (bottom)\n\n111 \u00a9 Victoria and Albert Museum, London\/V&A Images (top); Album quilt with season flowers, 1844 (cotton), American School, (19th century)\/Private Collection\/\u00a9 Boltin Picture Library\/The Bridgeman Art Library (middle); Sawtooth Stripy Quilt (textile), English School (19th century) \/ Quilt Museum and Gallery, New York\/\u00a9 The Quilter's Guild of the British Isles\/The Bridgeman Art Library (bottom)\n\n112 Courtesy of Pepper Cory\n\n113 \u00a9 Keller + Keller Photography Inc. (left), Courtesy of Amy Nguyen (right)\n\n115 \u00a9 John Polak\n\n116 The Granger Collection, New York\n\n117 Courtesy of Aneela Hoey (left & right), \u00a9 Keller + Keller Photography Inc. (middle)\n\n118 \u00a9 Keller + Keller Photography Inc.\n\n119 Crib Quilt, c. 1744 (silk, satin, quilted), English School, (18th century)\/Museum of Fine Arts, Boston, Massachusetts, USA\/Gift of Miss Katharine Amory Homans\/The Bridgeman Art Library (bottom), \u00a9 Keller + Keller Photography Inc. (all others)\n\n120 \u00a9 Keller + Keller Photography Inc.\n\n121 Courtesy of Kathleen Ricketson\n\n122 Courtesy of Natasha Bruecher\n\n123, 124, 127\u2013131 \u00a9 John Polak\n\n132\u2013133 \u00a9 Keller + Keller Photography Inc.\n\n133 \u00a9 John Polak (small squares)\n\n134 \u00a9 Amy Butler\n\n135 \u00a9 Amy Butler (top), \u00a9 Keller + Keller Photography Inc. (all others)\n\n136, 137 \u00a9 John Polak\n\n138 \u00a9 Keller + Keller Photography Inc. (left), \u00a9 John Polak (right)\n\n139 \u00a9 John Polak\n\n141 \u00a9 J. Marshall-Tribaleye Images\/Alamy (top), \u00a9 John Polak (bottom)\n\n142 Courtesy of Elizabeth Barton\n\n143, 144 \u00a9 John Polak\n\n145 \u00a9 Keller + Keller Photography Inc.\n\n146 Courtesy of Malka Dubrawsky\n\n147 \u00a9 Keller + Keller Photography Inc. (left), Courtesy of Rossie Hutchinson (right)\n\n148 \u00a9 John Polak\n\n150 \u00a9 Victoria and Albert Museum, London\/V&A Images (left), \u00a9 Les Arts D\u00e8coratifs, Paris\/Jean Tholance\/akg-images (right)\n\n151 \u00a9 John Polak\n\n152, 153 \u00a9 Sasha Duerr\n\n154 \"Infinite Variety: Three Centuries of Red and White Quilts,\" presented by the American Folk Art Museum at the Park Avenue Armory March 25\u201330, 2011. Photo by Gavin Ashworth. Collection American Folk Art Museum, New York.\n\n155 \u00a9 Keller + Keller Photography Inc.\n\n156\u2013159 \u00a9 John Polak\n\n160\u2013161 \u00a9 Keller + Keller Photography Inc.\n\n161 \u00a9 John Polak (small squares)\n\n162, 163 Gregory Gantner, courtesy of Jane Sassaman\n\n164\u2013166 \u00a9 John Polak\n\n167 \u00a9 John Polak (left), \u00a9 Keller + Keller Photography Inc. (right)\n\n168, 169 \u00a9 Shades of Gray Photography\n\n170\u2013173 \u00a9 John Polak\n\n174 Quilt, c. 1850 cotton roller-printed pieced work, Philadelphia Museum of Art, Philadelphia, PA, USA\/Gift of the Friends of the Philadelphia Museum of Art, 1968\/The Bridgeman Art Library (top); \u00a9 Keller + Keller Photography Inc. (left); \u00a9 John Polak (right)\n\n175, 176, 180\u2013183 \u00a9 John Polak\n\n184 Courtesy of Caro Sheridan\n\n185 \u00a9 John Polak\n\n186, 187, 188\u2013189 \u00a9 Keller + Keller Photography Inc.\n\n189 \u00a9 John Polak (small squares)\n\n190, 191 Joe Ofria, courtesy of Valerie-Maser-Flanagan\n\n192 Courtesy of Geta Grama\n\n193 \u00a9 John Polak\n\n194 \u00a9 Shades of Gray Photography\n\n195 \u00a9 Keller + Keller Photography Inc.\n\n196\u2013199 \u00a9 John Polak\n\n200 Courtesy of John Q. Adams\n\n201 \u00a9 John Polak\n\n204 Courtesy of Maritza Soto\n\n205 \u00a9 John Polak\n\n206 \u00a9 Keller + Keller Photography Inc. (top), \u00a9 John Polak (bottom)\n\n207 \u00a9 John Polak\n\n208 \u00a9 Keller + Keller Photography Inc.\n\n209 \u00a9 John Polak\n\n210, 211 \u00a9 Keller + Keller Photography Inc.\n\n213 Photo by Sue Stubbs; quilt from \"Quilting From Little Things\" by Sarah Fielke\n\n214, 216 \u00a9 John Polak\n\n224 \u00a9 Keller + Keller Photography Inc.\nThe mission of Storey Publishing is to serve our customers by publishing practical information that encourages personal independence in harmony with the environment.\n\nEdited by Pam Thompson and Deborah Balmuth\n\nArt direction and design by Carolyn Eckert\n\nText production by Liseann Karandisecky\n\nCover photography by \u00a9 Keller + Keller Photography Inc. (front: left, top row center, bottom row center right; back: top, bottom row 5th and 7th from left) and \u00a9 John Polak (front: top row left & right, middle row all, bottom row all except center right, back: bottom row all except 5th and 7th from left)\n\nInterior photography credits appear on page\n\nDiagrams and illustrations by Missy Shepler\n\nIndexed by Nancy D. Wood\n\n\u00a9 2014 by Rachel May\n\nEbook design by Dan O. Williams\n\nEbook production by Dan O. Williams and Kristy L. Rustay\n\nEbook version 1.0\n\nDecember 18, 2013\n\nAll rights reserved. No part of this book 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 book 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 book 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.\n\nStorey books are available for special premium and promotional uses and for customized editions. For further information, please call 1-800-793-9396.\n\n**Storey Publishing**\n\n210 MASS MoCA Way\n\nNorth Adams, MA 01247\n\n_www.storey.com_\n\n# Share Your Experience!\n\nWe value your feedback, and so do your fellow readers. Reviews you leave on goodreads.com and at your ebookstore help more people find and enjoy the books you love. So spread the word, write a review, and share your experience!\n\nREVIEW ON GOODREADS\n\nREVIEW ON AMAZON.COM\n\nEMAIL STOREY\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nA violent past leads to a present danger...\n\nWhen an old college roommate invites Claire Britten to join her on an archaeological dig at a Florida peat bog, it's an offer the renowned forensic psychologist can't refuse. Claire's husband, criminal lawyer Nick Markwood, is comforted to see Claire working on a prehistoric burial site instead of an open grave for once. But Claire's investigative instincts kick in when some of The Black Bog's perfectly preserved corpses show signs of a grisly fate. What really happened to these people?\n\nWhat started as an exploration of the past soon escalates into an all-too-current danger. Someone is watching\u2014someone who _really_ doesn't want Claire digging into the past or Nick making connections to a current, violent murder case he's investigating. The bog's corpses may be long dead, but if Claire and Nick don't figure out who's gunning for a fresh kill, the next bodies to be discovered will be their own.\nPraise for the novels of Karen Harper\n\n\"Karen Harper has created top-notch characters caught up in a thrilling novel of suspense! A must-read and a keeper.\"\n\n\u2014Heather Graham, New York Times bestselling author, on Shallow Grave\n\n\"Karen Harper keeps the action and mystery going at full throttle right up to the very last chapter.\"\n\n\u2014Suspense Magazine on Chasing Shadows\n\n\"The thrilling finish takes a twist that most readers won't see coming. While intrigue is the main driver of the story, the able, well-researched plotting and sympathetic characters will keep romance readers along for the ride.\"\n\n\u2014Publishers Weekly on Broken Bonds\n\n\"Haunting suspense, tender romance and an evocative look at the complexities of Amish life\u2014Dark Angel is simply riveting!\"\n\n\u2014Tess Gerritsen, New York Times bestselling author\n\n\"A compelling story...intricate and fascinating details of Amish life.\"\n\n\u2014Tami Hoag, #1 New York Times bestselling author, on Dark Road Home\n\n\"Harper, a master of suspense, keeps readers guessing about crime and love until the very end...of this thrilling tale.\"\n\n\u2014Booklist on Fall from Pride (starred review)\n\n\"A tale guaranteed to bring shivers to the spine, Down River will delight Harper's current fans and earn her many more.\"\n\n\u2014Booklist (starred review)\n\n\"Well-researched and rich in detail... With its tantalizing buildup and well-developed characters, this offering is certain to earn Harper high marks.\"\n\n\u2014Publishers Weekly on Dark Angel, winner of the Mary Higgins Clark Award\nAlso available from New York Times bestselling author Karen Harper\n\nSouth Shores\n\nSHALLOW GRAVE\n\nFALLING DARKNESS\n\nDROWNING TIDES\n\nCHASING SHADOWS\n\nCold Creek\n\nBROKEN BONDS\n\nFORBIDDEN GROUND\n\nSHATTERED SECRETS\n\nHome Valley Amish\n\nUPON A WINTER'S NIGHT\n\nDARK CROSSINGS (featuring \"The Covered Bridge\")\n\nFINDING MERCY\n\nRETURN TO GRACE\n\nFALL FROM PRIDE\n\nNovels\n\nDOWN RIVER\n\nDEEP DOWN\n\nTHE HIDING PLACE\n\nBELOW THE SURFACE\n\nINFERNO\n\nHURRICANE\n\nDARK ANGEL\n\nDARK HARVEST\n\nDARK ROAD HOME\n\nVisit karenharperauthor.com for more titles.\n\nLook for Karen Harper's next South Shores novel _Dark Storm_ coming soon from MIRA Books.\nKAREN  \nHARPER\n\nSilent Scream\n\nI cherish our thirty winters in Naples, Florida. This novel and the others in the South Shores series are dedicated to our friends who are still there, full- or part-time. And as ever to Don for his continued support.\nContents\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\nChapter 35\n\nChapter 36\n\nChapter 37\n\nAuthor's Note\n\nSuspense in Real Life\n\n# 1\n\n## Naples, _Florida  \n_ Sunday, May 22\n\n\"Would you stop pacing and looking out the window, sweetheart?\" Nick asked Claire. \"You're making me nervous.\"\n\n\"I'm walking the baby,\" she said and smiled down at their three-month-old son, Trey, in her arms. \"Well, I am a bit on edge. I haven't seen my old college roommate for years, and that's quite an offer she's made me. Really, Nick, I'd just retire from my forensic psych work for a while, but this opportunity sounds too intriguing to pass up. Plus, it's nearby\u2014and important. I've always examined the lives of the deceased, but ones who are thousands of years old in an archaeological dig? Oh, here she is, parking out in front.\"\n\nShe put the baby into his arms. \"Give us a second, then bring Trey and Lexi in so Kris can meet them,\" she told him and pecked a kiss on his cheek.\n\n\"I just don't want you two to adopt your old 'women in jeopardy' nickname again,\" he called after her as she headed for the door.\n\n\"That only related to our problems in the old days, and we 'womjeps' are both under control now,\" she called back to him.\n\nBut her heartbeat kicked up. She and Kristen Kane, nickname Kris, had been through hard times together at Florida State as they'd studied, worked jobs and dealt with their disabilities. With her face blindness, Kris had struggled to recognize people, even those close to her, while Claire was desperately trying to navigate a life with narcolepsy and taking powerful drugs to deal with that. If she hadn't, she'd have fallen asleep in class and had terrible nightmares.\n\nShe swept open the door before her old friend could ring the bell. \"Dr. Kane, I presume! Welcome! So good to see you after all this time!\" Claire cried as they hugged each other.\n\n\"You still have your great red hair,\" Kris said as, arm in arm, they walked into the house. \"I was hoping so.\"\n\n\"Yep, not going white-haired yet, though I've been through some things that could have done me in.\"\n\n\"So I hear and read, but you know what I mean,\" Kris told her as they stopped in the tiled foyer while Claire closed the door behind them. \"Your unique hair color was how I identified you until I heard your voice. But you know what, my friend? Working with the bog bodies in Denmark\u2014even here\u2014somehow, I can recall their faces and ID them easily. And it isn't just their twisted postures either. Well, you'll see. What a great house,\" she said, looking around. \"But then since you're married to a big-deal criminal lawyer, what would I expect?\" she teased. \"I haven't found my knight in shining armor yet, but then I've moved around too much. I'm here now though, for the foreseeable future anyway.\"\n\n\"Come sit in here, and I'll introduce my family before we have lunch and get caught up,\" Claire said, leading the way. \"I can't wait to hear about this fascinating dig.\"\n\nAt age thirty-four, Kris looked much the same, maybe with a few new worry lines or squint marks at the outside edges of her blue eyes. Her Florida-girl complexion was a bit paler than Claire recalled. She probably bleached her straight blond hair now, though she still wore it blunt-cut, chin-length with bangs. As ever, Kris looked so serious when she didn't smile. Almost as tall as Claire, who was five-foot-ten, she came across as confident, though for both of them, that confidence had once been a mutual facade to get through tough times. But what a challenge for Kris to not even recall the faces of her family members without other visual clues.\n\nKristen Kane had thrown herself into a major in forensic archaeology while Claire had stayed with forensic psychology. Kris had a doctorate and studied dead people from the past, ones who no doubt weren't upset if she didn't recognize them.\n\nAs Lexi bounced into the room, Claire introduced her six-year-old daughter. \"Almost seven in ten months!\" the child told Kris. Nick followed, holding the baby whom Kris soon fussed over. Lexi was Claire's child from her first husband, Jace, but little Nicholas Markwood III, whom they called Trey, was hers and Nick's.\n\n\"I see Nick is a good catch, looking like a Mr. Mom,\" Kris said with a little laugh after introductions. She hit Nick's arm lightly with a fist. He just smiled, but Claire felt he still looked every bit the courtroom lawyer with his height of over six feet, erect posture, piercing gray eyes and dark but silvering hair. Kris was studying his face as she always did with someone new.\n\n\"I did look up your picture online, Nick, and found one on your law firm's website,\" she admitted. \"I try to do my prep work. I guess Claire explained my prosopagnosia.\"\n\n\"It certainly hasn't slowed you down, Doctor,\" Nick assured her. \"It sounds like you have a fascinating career.\"\n\n\"Important too\u2014really. We learn so much about the present and future by studying the triumphs and tragedies of the past.\"\n\n\"Are you a baby doctor?\" Lexi asked, looking up at her. \"We have one now, and Mommy said she wished he'd come to the house.\"\n\n\"No, not that kind of doctor,\" Kris told her as they sat back down, Nick still holding Trey and Lexi perched on the footstool by Claire's chair. \"I do try to find out what happened to people sometimes, though\u2014if they got sick or not when they were alive, things like that.\"\n\n\"I heard Mommy say she might help you, and she told Dad that her motto is 'The dead still talk if you know how to listen.' But I don't think you can hear what dead people say unless they are Halloween ghosts and that's all pretend. I do like secrets, though, and I keep them really quiet.\"\n\n\"Well, you have a very bright thinker here,\" Kris said, turning toward Claire. \"I can tell Nick's concerned about what I want to offer, but don't be, either of you. Not a thing to worry about except getting bog mud on yourself, a few mosquito bites and keeping a secret. And, if you accept my offer, both of you must keep our privacy agreement too.\"\n\nClaire looked at Nick, but thank heavens, he didn't frown or protest, because she wanted desperately to help with this mysterious dig into the past.\n\n* * *\n\nAfter lunch in the Florida room overlooking their fenced-in pool and backyard, Nick excused himself and went to put Trey down for a nap. He was a great father and stepfather, Claire told Kris, explaining that they were still friends with Lexi's father, Jace, and his fianc\u00e9e, Brittany.\n\n\"I'm so glad to hear that,\" Kris said, frowning. \"I was wondering how you and Jace were doing, sharing your daughter and all after the divorce. Before I saw how great she is today and how happy you are now, I was feeling guilty that I introduced Jace and you. I know you two fell for each other hard and fast, and it seemed so right at the time. Jace isn't like so many of those other handsome flyboys. I always had a soft spot for that derring-do kind of guy myself, though nothing ever came of it.\"\n\n\"When you set us up seems so long ago. Obviously, Jace and I had some great times\u2014and then some bad. But Lexi was the best, and we are sharing her well. He's engaged now. Nick and I know and like his fianc\u00e9e, and she's good with Lexi. Works at a zoo, so that goes a long way. Even Nick and Jace are friends after some of the tough times we've all been through.\"\n\nThey both looked out in back to see that Nick was now sitting in a lounge chair with some paperwork while Lexi perched by his feet staring at a picture book she had already \"read\" over and over. Claire was glad they were in the shade because the sun was hot for late May.\n\nBut Claire and Kris remained in the glassed-in, air-conditioned room, facing each other in matching rattan swivel basket chairs. At lunch, Kris had talked about her life in Denmark in general, but with Lexi there, she had obviously said little about the ancient bog people her crew had disinterred and studied. But now she leaned forward, elbows propped on her knees, gripping her hands together.\n\n\"Claire, you won't believe this amazing local archaeology find until you see it. Not to sound cute, but it is absolutely groundbreaking. Extremely significant, but we've managed to keep it top secret to all but a few in the Florida state government and museum world because support money is a challenge. The site is privately owned, funded and controlled to keep a lid on what we are uncovering there.\"\n\nClaire sat up straighter. She only nodded at Lexi when she came in and announced she was going to the bathroom before disappearing down the hall.\n\n\"Let me just give you an overview,\" Kris went on, leaning even farther forward. \"In Denmark, I've been working with Early Neolithic bodies\u2014Stone Age. Most of the ones we excavated were sixteen to twenty years old at death, so we assumed, and then found evidence that they were either human sacrifices or executed criminals. And that's what got me the job offer here.\"\n\nKris's face and voice were so intense, but Claire realized she was holding herself rigid too, hanging on each word.\n\nKris went on, \"But so far none of the bodies in Black Bog nearby show signs of violence or ritual. Claire, archaeological sites are often like crime scenes. You specialize in psychological autopsies where you don't have a body, but still examine the person's life and death to help the police make a determination of the type of death\u2014natural, accidental, suicidal or homicidal.\"\n\nRiveted, Claire just nodded. She'd recently discovered a shallow grave and the tragedy it had been connected to, but to be able to peer into an ancient grave and solve that death\u2014amazing.\n\n\"Have you heard of the Windover Culture?\" Kris went on. \"The bodies dug up in a bog quite a ways north of here near Disney World in the early 1980s?\"\n\n\"Yes, but I don't remember details. I know they found Stone Age people there.\"\n\n\"Here too, near Naples in Black Bog. I have been hired to lead the dig team that is excavating a treasure trove of ancient people, and I need you to consult. After all, your expertise of studying the lives of the dead could be invaluable to us. I'll do the science, you put the clues together to psych out how they lived and died. If you'll agree to visit the dig with me, I'll introduce you to the property owners and dig controllers, Andrea and Bradley Vance.\"\n\n\"Bradley Vance? Wasn't he a state senator or something?\"\n\n\"One and the same, recently retired, so he has some clout around here and in Tallahassee, not to mention an inheritance from his family to finance this, though I think he and his wife have gone through that already. But his wife, Andrea, is even more important. Do you remember my talking about Professor Andrea Carson after you graduated, and I went on for a masters and doctorate?\"\n\n\"You admired and worked closely with her.\"\n\n\"My mentor, oversaw my dissertation, later my orals. I did some prehistoric Anasazi tribal digs with her out West. I've worked hard for her, and she's brought me along in her impressive wake. A genius, and we'd be working directly with and for her.\"\n\n\"I'll have to convince Nick if it will take a lot of my time.\"\n\n\"Like I said in my letter, you can work three days a week, part-time, if need be, but I\u2014well, I need your help, frankly, figuring these ancient people out. Like I said, for now, it's top secret so we don't get looting or reporters tramping all over, ruining things in that delicate environment of moss, mud, water and peat. Neither you nor Nick\u2014and not even sweet, little Lexi with her secret-keeping\u2014can let on about what you and I are really doing, not until we know enough to go public with this huge find without having outsiders tell us what to do and how to do it. Deal?\"\n\n\"I will explain to Nick, tell him how much I'd love to do this. When we first got your note right after Trey was born, he said that, after all, what can go wrong when you're dealing with those who are long dead, who have no family members or even enemies to file a lawsuit or cause a problem?\"\n\n\"Right. I know you've been through a lot, but this is different. Oh, by the way, though the dig is state-of-the-art, I should tell you there are no cell towers close enough to the bog that cell phones work there, so we use two-way radio transmitters to keep in touch with each other on-site. You know, walkie-talkies, our only real concession to nontech. I know from your Clear Path website that you do consulting only online these days, but this will be hands-on. The Vances and I need your help, and it will contribute to the knowledge of all mankind.\"\n\nClaire nodded. For the first time, she would not be helping living people, but knowledge of the past was important too.\n\n\"Think about this, Claire. Black Bog bodies are seven thousand years old but only look two years dead! They were interred before the time of Christ, before the pyramids were built! Yet they're only a bit shriveled. They've been preserved under piles of peat that has tanned their bodies to a dark brown, but you can see their expressions, absolutely what they looked like. Even someone with face blindness is blown away by their individuality and the stories they have to tell.\"\n\nSo passionate, so convincing. Deeply moved, Claire could only nod yet again. Instead of hugging this time, they shook hands.\n\n* * *\n\nLater that evening, Claire was ecstatic. Maybe she could have it all\u2014a wonderful family life, an understanding husband she was madly in love with, and a safe and exciting at-least-part-time career. Nick had agreed and had even been intrigued by her consulting offer, which she would actually accept or reject after she visited the Black Bog dig, met the Vances and saw the terms of her contract. Nick, who had worked hard to build his father's criminal law firm and run his South Shores project on the side to help people determine whether the deaths of their loved ones were murders or suicides, had thought that was a good plan.\n\nThey had dropped Lexi and Trey off at Claire's sister, Darcy's, house for the early evening, because they were going to help their newlywed friends, Bronco and Nita Gates, clean up the house they had bought for an excellent price with a loan partly financed by Nick. Bronco worked as a security guard at Nick's law firm. Nita was their part-time nanny for Trey and Lexi, and the four adults were close friends.\n\nThough they were still living in an apartment, Bronco and Nita had bought their new house in East Naples, a one-floor stucco painted aqua in a middle-class development just off the Tamiami Trail. The price had been good because the house had been owned by a recently deceased elderly woman who was a hoarder, and it would need a lot of work even after being cleared out. It had been a private sale by the woman's son, Dale Braun, who was a junior partner at Nick's law firm, Markwood, Benton and Chase. The housing development covered a large piece of land where a mansion once stood at the back of the Braun estate.\n\n\"Nita loves the color of the house,\" Claire told Nick as they parked in front. \"'The color of the sea,' she says.\"\n\n\"She'll love it a lot more once they get all that stuff hauled out of here, but they're making progress.\" As they headed for the house, they passed a rented dumpster sitting in the driveway, full of various junk to be thrown out.\n\n\"We're here!\" Claire called as they knocked, then went in through the unlocked screen door.\n\nJuanita, whom they called Nita, a lovely Mexican-born woman, appeared, wiping her hands on her jeans. \"Been digging through so much and sure can use your help,\" she told them. \"I'm in the kitchen area, and Bronco's out back, Nick. There's stuff spilling out of that little shed behind the garage. Don't know how someone could live this way. Does her son have a messy desk at work?\"\n\n\"No, it's evidently not hereditary,\" he told her with a smile.\n\n\"Go ahead,\" Claire encouraged him. \"I'll pitch in with Nita here.\"\n\n\"Good thing you dressed the part,\" Nita told Claire as Nick went out back. \"That little room off the kitchen is jammed full of stuff, a lot of it piled on top of an old freezer.\"\n\n\"I'll bet that's an electricity hog if it's still plugged in,\" Claire said.\n\n\"Oh, _si_ , __ still running. The way she threw nothing out, bet it's full of old food,\" Nita said and gestured at the big chest freezer.\n\n\"You'll need a team to get that heavy thing out of here. At least you've been making good progress unloading the things on top of it,\" Claire told her, eyeing the piles of items not only on it but sitting on the floor.\n\n\"Wish I'd find some hidden money. Just kidding. I'd give it straight to Mr. Braun. Doesn't pay to do anything illegal in this life, that's for sure. Maybe there's something we can sell round here, though, 'cause Mr. Braun said all of this is ours if we want to use it, if it's not tied direct to him or his family. He's not married though, broke up with his fianc\u00e9e, a neighbor said. Mr. Braun told Bronco he's looking for another place to live now 'stead of next door since he doesn't have to worry about his elderly bats-in-the-belfry mother anymore. Honest, Claire, that's what he said.\"\n\n\"It doesn't take much looking around here to realize she was eccentric at the least. Maybe she had dementia because that increases the tendency to hoard. Let's keep clearing stuff out.\"\n\nFrom atop the old freezer, they lifted and carted out stacks of moldy newspapers and magazines, then two boxes of Christmas decorations, including a big one with a smiling plastic Santa Claus with a frayed extension cord dangling from its innards. Once they had unburied the freezer, which was still humming with power, they unlatched and lifted the heavy lid together. A cloud of moist, stale air wafted up at them.\n\n\"Quite the antique,\" Claire said, fanning her face to see better as they peered into the crowded depths. \"Maybe the manufacturer will buy it back as proof their product lasts.\"\n\n\"I'm just hoping there are steaks or lobster in here,\" Nita said with a nervous laugh. \"This house has been quite a treasure hunt.\"\n\n\"Looks mostly like frozen vegetables. Wow, I haven't seen this Birds Eye __ brand packaging for years. Even though this stuff is frozen, you'd better not eat it. There must be an expiration date on this,\" she said picking up a package of rock-hard frozen broccoli and scanning it. \"With the piles of things on the lid, she obviously didn't get in here much and not for years. What a waste, so sad.\"\n\nWhile Nita rooted around at the other end of the long freezer, Claire moved a large opaque plastic bag that looked as if it might contain whole strawberries. And under that, as if she'd unearthed her from her icy tomb, the frozen face of a young woman stared upward with her eyes and mouth wide open as if in shock or horror.\n\n# 2\n\nTheir screams brought Nick and Bronco running.\n\nNita leaned against the wall, her hands covering her face, while Claire steadied herself, gripping the open freezer and staring into it. This reminded her of standing by her mother's casket with her sister beside her. Darcy had been crying, Claire trying to hang on to sanity just before they'd closed the lid.\n\n\"Claire, what?\" Nick's voice jolted her back to reality.\n\nNita threw herself into Bronco's arms. Claire pointed and cried, \"There! A dead woman\u2014frozen!\"\n\nNick peered down and gasped. \"She's real,\" he whispered and pulled Claire hard against him. \"Nita, do you know who this is? Bronco, take a look. I'm going to call the police and the medical examiner.\"\n\nNita took another look and Bronco peered inside. Shaking her head and trembling, Nita whispered, \"Never seen her.\" Pressing her hands over her mouth, she started to cry again.\n\n\"Me neither,\" Bronco muttered and pulled Nita back into his arms. \"She is\u2014was\u2014kinda young. Boss, this gonna mean trouble for us in our new house, even if we're still in our old apartment for now?\"\n\n\"Just for a little while if we handle this right,\" he said, digging his phone out of his jeans pocket and thumbing in numbers. He put one hand on Claire's shoulder as she continued to stare down at the frozen face.\n\n\"Yes, an emergency,\" he said into the phone. \"This is attorney Nicholas Markwood. We have discovered a dead body in a newly purchased house that's been empty for about two weeks since the death of its previous owner. No\u2014it's not the previous owner's body. We don't need a squad but send officers\u2014I'm requesting Detective Ken Jensen, if he's available, and the ME. Yes, I know. I promise you we won't move or touch anything, though we did open the freezer where we found her.\"\n\nHe gave them the address, then took a photo of the woman in the freezer using a flash. Claire looked down at the woman again, her features lit by that sudden jab of light: maybe late twenties, a pretty blonde, staring up at them through boxes of frozen vegetables and clear-wrap packages of chicken breasts. How absurd. How horrible.\n\n\"You took that picture to ask around who she is, boss?\" Bronco asked. Big and burly as he was, he was trembling too. \"Won't the police do that?\"\n\n\"Probably. But since this house was recently owned by Dale Braun, one of the firm's junior partners, he may be questioned and I may be too, so I'll keep this photo. Sadly, I'm always thinking like a criminal lawyer\u2014plan ahead for every contingent.\"\n\n\"Yeah, I know who he is from checking him in at my guard post,\" Bronco said. \"A real go-getter, comes early, stays late. He's a handsome dude, acts like it too.\"\n\n\"Oh, _that_ junior partner,\" Claire said. \"The one who always seems dressed up and doesn't have a dark hair out of place. Since this was his mother's house, it ties to him and so to the firm. And Nita and my fingerprints are all over that stupid freezer.\"\n\n\"You both have a rational explanation and no ties to this corpse,\" Nick assured them. \"I just wish she didn't have her eyes and mouth open like that. And I hope it's not just her head without a body.\"\n\nNita said, \"It makes this house feel\u2014well, not good now\u2014kind of not ours.\"\n\n\"They'll clear it out good as new,\" Bronco insisted. \"Hey, I hear a siren already.\"\n\n\"Me too,\" Nick said, pulling Claire away with him as Bronco and Nita followed. They waited in the small cluttered living room. It was paramedics who pulled up in front, but what could they do for a definitely dead and frozen, no doubt murdered, woman?\n\nThey saw two police cars and an unmarked vehicle arrive in front too. Nick greeted them at the door with Bronco right behind him. The detective, Ken Jensen, with whom they had worked before, asked an officer to put police tape around the front of the house, and told the paramedics to wait to see if the ME's office brought a van.\n\n\"I'm glad you got word that I asked for you, Detective,\" Nick told the man. They hadn't seen him since the shallow grave case they'd worked six months ago. He shook hands with the tall blond detective, who also greeted Claire.\n\n\"I understand why you turned down our department's offer to consult on cases, Claire,\" Detective Ken Jensen said. \"Congrats on the new baby. But here you are discovering a case of your own.\"\n\n\"Not our case, this time,\" she insisted. \"Pure, sad chance.\" She extended her hand too, then introduced Bronco and Nita as the new homeowners. \"None of us know the victim. Nita and I were just clearing things off the top of the freezer while the men worked out back. The previous owner, an elderly woman, was a hoarder\u2014and was evidently unknowingly or knowingly hoarding a very dead, frozen body.\"\n\nNick moved his knee against hers to stop her. Damn, he thought, she sounded like she was testifying in court like she had so many times, as a forensic psychologist expert witness to help clarify cause and manner of death. She had real instincts for psyching people out, but he didn't want her any more involved in this than she was, because he was afraid he might have to be. He only hoped there would be no link between the dead woman and his junior partner who had sold Bronco and Nita this house. No\u2014no way, or Dale would have made sure the body wasn't here to be found, though stranger things had happened.\n\n\"Sorry, detective, maybe too much too soon,\" Claire added, her voice sounding steady at last. \"Back in my forensic psych mode.\"\n\n\"Well,\" Detective Jensen said, \"if this does become a case that goes to trial, you could be called to testify, but not in that capacity. Let's have a look at the body before the ME gets here. Lead the way.\"\n\nBronco went first, holding Nita's hand, then Claire, then Nick and Ken Jensen. The five of them crowded into the small room with the freezer. Frosty air still wafted upward. Jensen looked inside and jerked a bit at the sight.\n\n\"She looks like she's shocked and screaming.\" He said the obvious. \"Like she is being attacked right now.\"\n\nNo one said a word while he took several flash pictures with his phone, then used it to call for a crime scene photographer and evidence technicians. He moved them back to the living room and called one of the policemen in. \"I see there's a curious neighborhood crowd gathering outside,\" he told the officer. \"Take my phone with this photo and see if anyone can ID this woman. Send the ME in when he or his people get here while I take statements.\"\n\nThey sat on unpacked boxes in the living room, waiting to individually give their statements to Detective Jensen out back in the small Florida room. Nick dug his phone out again and texted Dale Braun, who was not at his house next door, but got no answer back. If it hadn't been a Sunday, he would have called him at the office. Dale was a dedicated worker, a little hard to get to know, very bright but not really socially adept for a guy so good-looking. He was in his early thirties, an upper-range millennial.\n\nClaire looked stunned. Nita too, even after talking to Jensen. Nick jolted when one of the cops knocked on the door near where he sat, and Jensen got up to answer it. The detective had closed the place up and turned the air-conditioning on full blast, as if to keep the already frozen corpse cold for the ME. With an assistant, he would dig the body out for a preliminary onsite investigation before they took the corpse to the Collier County morgue for an autopsy.\n\n\"Hey, Detective Jensen,\" they heard the officer at the door say as he handed Jensen's cell phone back. \"Got a positive ID of the deceased from a neighbor lady two doors down, a Mrs. Betty Richards. Says it's the former fianc\u00e9e of the guy next door whose mother owned this house, ah, a Dale Braun, a lawyer no less. Even knows the name of the victim\u2014Cynthia Lindley. We'll go next door and see if Dale Braun's there.\"\n\nNick's gaze slammed into Claire's. He'd been hoping for more time to find and question Dale. This would implicate him, pull the firm into the investigation. The husband, the boyfriend, the fianc\u00e9\u2014always the first one the cops looked at.\n\n\"I've been trying to reach Dale Braun,\" Nick told Jensen. \"He's a junior partner in my law firm. He's not home next door and not answering his phone.\"\n\n\"But you've seen him recently?\"\n\n\"At the firm on Friday.\"\n\n\"No kidding, Counselor,\" Jensen said, jotting that down in his notes. \"Hate to say it, but the plot thickens about who could have hurt that girl.\"\n\n\"All circumstantial so far, as you know,\" Nick said. He wanted to say much more, that Dale Braun seemed like a straight arrow, though Nick had not known he had a fianc\u00e9e, let alone a former one\u2014evidently a missing one. It didn't make sense that Dale would sell this house if he knew a woman's body was still in it, knowing he could be tied to it so easily. And where was Dale and would he show up to work tomorrow morning or at his house next door before that? No doubt the cops would be there in both places waiting to question him.\n\nClaire reached out to take his hand. She sat up straighter and only narrowed her eyes, unlike Nita, who had not stopped crying. Yeah, Claire looked stunned but steady. But then what a shock to open an old freezer and find a human face staring upward. A pretty face. A face contorted in a silent scream.\n\n* * *\n\nWith Jensen's permission, Nick took Nita and Bronco home with them that evening to stay in their guest room rather than letting them head home to their apartment, in case the police or the press showed up there. If they were implicated at all\u2014and he knew they weren't guilty of anything\u2014he'd assign someone from the firm to represent them\u2014or even take the case on himself. Before he and Claire headed to pick up their kids at Claire's sister's, he had wanted to get some warnings out of the way.\n\n\"No way you can hang around a crime scene,\" Nick had told them before they went out to their cars. \"But I'll bet you'll get back in a couple of days.\" _If people don't stand around staring at the \"death house,\"_ he did not tell them. \"I wish I\u2014or they\u2014could locate Dale,\" he went on. \"It's going to look like he fled, which I hope he didn't. He'd better have an airtight alibi.\"\n\n\"You'll defend him if he asks?\" Claire asked.\n\n\"If he's indicted either I, or at least the firm will, but let's not jump to conclusions any more than Ken Jensen evidently has. You just go away from all this with your friend Kris tomorrow, and Nita can stay with Trey. Bronco, you just do your security job at the firm tomorrow and don't answer questions from anyone but me when word of this breaks,\" he said, frowning out at the neighborhood gawkers. \"The newspaper and TV reporters will be here soon with a frozen dead woman for a headline. The four of us plus the kids are going to lock ourselves in our house tonight with no comment. __ That's at least until tomorrow when we see what the ME and Jensen come up with. Everybody got that?\"\n\n\"Sure, boss, Nita and me are fine with that, right honey?\"\n\n\" _Si._ I just want our new house back with no more awful surprises,\" she said and blew her nose.\n\n\"Aye, aye, Captain,\" Claire said to his orders, but he could tell her clever brain was spinning. He hoped she was thinking about visiting the Black Bog dig with her friend tomorrow\u2014dead people, sure, but ones that would cause them no trouble. He'd seen that Claire had not only texted her sister they would be there soon to pick up Lexi and the baby. By borrowing his phone, she'd also taken another look at the picture of the dead woman, Dale Braun's former fianc\u00e9e whose stare sliced right through you as she opened her mouth to scream and\u2014and then someone killed her.\n\n# 3\n\n\"I'll bet you're relieved to be going to the land of the long dead instead of the newly dead. Sorry if that came out wrong,\" Kris told Claire as she drove them in a Jeep on the Tamiami Trail toward Black Bog. Kris had told Claire their destination was beyond Seminole State Park, east of Blackwater Bay, which was only about a half-hour drive. \"I just mean after you had that terrible experience yesterday.\"\n\nKris frowned and bit her lower lip hard as if she wanted to say something else, but kept it back. Sometimes Claire wished she didn't read body language. But maybe her friend only regretted she had brought that up after Claire had recounted the horrific find in the freezer, not that it wasn't public knowledge on all the news media this morning, including some national programs.\n\n\"It's a joke among the crew,\" Kris said, still frowning and obviously trying to change the subject, \"that we're not far from Mud Bay. We do get 'boggy dirty' on this dig. At least the Everglades runoff from Lake Okeechobee doesn't flow through the bog. It's more solid peat, not water.\"\n\n\"Sounds delightfully muddy anyway. Kris, it's okay to ask about the woman in the freezer. Honestly, I've been through tough times before.\"\n\n\"I didn't mean to hit you with all that right away. The bodies we deal with are hardly frozen in this South Florida climate but they are frozen in time.\"\n\n\"The TV stations and _Naples Daily News_ picked up on the story fast, but I can see why, so I'm glad to 'get out of Dodge' today. Thanks for picking me up at my sister's where\u2014I hope\u2014reporters won't know to find me later. Nick won't escape them at the office, though, especially if Dale Braun shows up there for work this morning.\"\n\n\"Yeah, to no doubt get arrested the moment he does\u2014at least dragged in for questioning. And that poor couple who works for you\u2014to have a picture of their new house and its address all over the news. They'll get ghoulish gawkers, which is exactly why the Black Bog dig is top secret.\"\n\n\"Bronco and Nita have had it tough. They and another couple we know\u2014a tech guy who also works for Nick, and the techie's girlfriend\u2014have been through a lot with us. But let's talk about something cheery\u2014like the bog bodies you've managed to hide from the world so far.\"\n\n\"Claire,\" Kris said, turning her head briefly toward her, before looking back at the road, \" _that\u2014_ excuse me for putting it this way\u2014is deadly serious. You'll see why this has to be classified information for now. Carefully read that part of the contract the Vances will offer you. You may even have to lie to people you know. We have to keep the lid on this. I've said it before, but our findings are important to archaeology and mankind. You'll see why when Andrea and I show you around.\"\n\n\"I can't wait, and, believe me, you can trust Nick and me. The others will just know I'm working on a project that's private for now\u2014even chatterbox Lexi.\"\n\n\"She's a doll,\" Kris said, as she made the westward turn onto an unpaved road which kicked up a trail of dust behind them. The month of May was early in the rainy season: the sky was a bit gray, and rain clouds threatened. Claire was grateful that Kris seemed so easy to work with. Even when her friend had some serious, almost stressed-out moments, they had always segued easily back into their longtime trust and friendship. Claire sat up straight and pulled her big purse with touch screen tablet and good old pen and paper notebook closer to her legs. She wanted to be prepared in case her iPad didn't work out here. She had to really pay attention, take notes to go over later, since this was foreign territory in more ways than one. She'd taken things pretty easy with work since Trey's birth and wondered if her concentration and stamina would hold up. She'd seldom been in on something as unusual or fascinating as this\u2014lots of dead bodies with totally foreign lives and deaths. And she had the opportunity to psych them out, to bring them back to life in a way.\n\n\"I saw that road sign we just passed said we're heading toward Blackwater Bay,\" she observed. \"Is Black Bog along that bay?\"\n\n\"In the vicinity, but not on the water. You do know the difference between a bog, a fen and a swamp, don't you?\"\n\n\"I have a feeling I'm going to have a lot to learn here. I know a swamp is standing water and various stages of plant growth. Tell me the rest.\"\n\n\"A fen\u2014which there are so many of in the colder countries of Europe, but some here in the Southern US\u2014is an open body of water on a bog's surface. If there's a fen in a dig, we drain it, as we have here, at least as best we could during the dry seasons, so pray there's not much rain for a while,\" she said, squinting at the gray sky. \"If it rains, we cover what we have and come back another day.\"\n\n\"But this is obviously a bog, not a fen or swamp per se.\"\n\n\"Right. Bogs, which we are mostly dealing with, hold layers of dead plant material often called peat, including rotting sphagnum moss. Sphagnum releases tanning acids that preserve the corpses, almost as if they've been pickled\u2014a gross picture, I admit. Some bogs are as thick as forty feet, but fortunately, Black Bog is only about twelve feet deep, and we've drained and excavated that down to about eight to ten feet. We have to be very careful resurrecting bodies and any artifacts, of course.\"\n\n\"Do the bodies damage easily once they're out?\"\n\n\"No, they're amazingly resilient. The bones are pretty soft because the acid in peat has leached out a lot of calcium phosphate, so the bodies are rubbery and look kind of deflated. But, Claire, you can see expressions, stubble, eyelashes!\n\n\"It would make things so much easier,\" Kris went on, \"if we could use a GPS finder like we can use over digs on solid ground, but those things are heavier than a lawn mower and would sink. You'll see our system of planks laid out in grids over our finds. You'll have to learn, as we say, to 'walk the plank' to get a good look at the bodies in their graves before we disinter them. Enough said, because I'll let Andrea give you the tour. Brad may be back later, but he had to oversee their art and antique store in Naples this morning. Art For Art's Sake. Funny, huh? Their real antiques are top secret, but they sell other ones in stores on both Florida coasts.\"\n\n\"It all sounds totally intriguing. As Lexi likes to say, 'Are we there yet?'\"\n\n\"Claire, our work in this place is going to blow your mind.\"\n\n* * *\n\n\"Nick, I swear to you, I had no idea there was a body in my mother's freezer\u2014especially not Cyndi's! Yeah, we had a bad breakup, but I'll always love her and never would have hurt her!\" Dale Braun protested and threw himself into a chair in front of Nick's desk.\n\nNick had finally reached him by phone and told him to come up to the office the back way, using the janitor's entrance and stairs. Bronco had met him and led him to Nick's office because the police were waiting out by his secretary's desk as well as at the front door of the building. They both knew they had little time before, at the least, he was taken downtown for questioning.\n\nNick shoved away from leaning against his desk. He didn't want to seem intimidating, but he wanted the truth from this man in a hurry.\n\nNick had been pretty hard on Dale this morning, but he needed to get a feel for his employee's emotions as well as the truth about this mess. He wasn't putting this firm out on a limb only to have the branch cut off if this guy had murdered that poor girl.\n\n\"Now I've got the law after me,\" Dale said the obvious behind muffling hands over his mouth. \"Damn, that's a great way to say it, right? The law is after the lawyer, and, yes, I know why,\" he went on as his hands flopped to grasp his knees. \"Yeah, I broke up with her, not the other way around, so who knows what she told her friends or family.\"\n\n\"Exactly when was this? I'm sure the ME will soon rule on how long she's been dead and what killed her, but give me the basic time frame and just hope it doesn't sync with what the ME will find on time of death.\"\n\n\"Just over a week ago. Ten days, I guess. Yeah, before last weekend, shortly after my mother's funeral.\"\n\n\"Not at your house or that one, I hope. Dale?\" he prompted when the man hesitated. \"Just answer questions, don't fume and agonize or it will look like you're making things up as you go.\"\n\n\"Yeah\u2014at my house. But Nick, she's the one who was angry, not me. Will the firm stand by me, defend me if it comes to that? I did not kill her and sure as hell didn't stash her in my demented mother's freezer just after she died.\"\n\n\"We'll stand by you, but you're going to have to cooperate with the police. I'll be there with you for your statement. You know the drill. Just stick to the facts, offer nothing extra and, as my forensic psych wife would counsel, look the detective in the eye and don't fidget all over the place. Dale,\" he said, leaning down to grip the man's shoulder, \"we not only defend strangers but our own here. The firm is like a family too.\"\n\n\"Thank God,\" he said and exhaled hard. He seemed to deflate with relief.\n\n\"I'll need your complete statement, of course, but you're going to have to go with Detective Jensen before he knocks my office door down. I'm sure he wanted to question you before you lawyered up, but this is our territory.\"\n\n\"Your sticking by me will carry weight, yours and this firm's reputation,\" he said in a raspy voice, but he shook his head as if trying to convince himself. Nick didn't like how strung out Dale looked, with bloodshot eyes and dark circles under them when he had supposedly been away to relax and hadn't been hit with all this until about an hour ago, though of course\u2014hopefully\u2014it was a shock. So why hadn't the guy been sleeping and decompressing this weekend? Had his decision to break up with Cyndi done that to him or was he tormented by something more recent?\n\n\"I'll invite the detective in,\" Nick said, heading for the door. \"He'll probably insist on our going to the station. But he can't arrest and hold you for anything\u2014can he?\" he asked, turning back. \"Like your fingerprints on that old freezer?\"\n\n\"Of course, I've been all over that house for years, even recently before the sale to Bronco and Nita was finalized. Nick, my mother had stage four dementia, so who knew what she would have said if she was alive. She still thought my German great-uncle was alive and living in his big manor at the rear of the estate. She'd get everything confused.\n\n\"Anyway,\" he plunged on, \"I'm sure I left DNA if\u2014if they find that was the\u2014the crime scene. Not just where she was hidden but where she was murdered as well. I just hope to hell she wasn't put in there alive. I mean, you said it looked like she was screaming.\"\n\nA whole string of questions popped into Nick's head such as who else had keys to the place, but he'd save that all until later. Right now they had to cooperate with the police\u2014and eventually, he feared, with the prosecuting attorneys of Collier County.\n\n\"Nick, if I touched that freezer, it was long ago,\" Dale repeated, circling back to that, maybe overexplaining. \"I was hoping Bronco and his wife would get it out of there\u2014just because it's so heavy, I mean. Obviously, I would not have stashed my ex-girlfriend\u2014\"\n\n\"Your former fianc\u00e9e. Keep everything straight.\"\n\n\"Right. I would never have put her body there, then sold the house. Damn, my head hurts with all this,\" he said, gripping his skull in his hands again. \"Too much so soon after learning she's dead. I did love her...but it just wasn't going to work out.\"\n\n_Why not?_ Nick wanted to demand, but that too would come later. He wished he didn't empathize with Dale so much but he understood emotional devastation. Fear. The powerful way a woman could change everything in life, one way or the other. He hesitated with his hand on the doorknob, because he sensed something else was coming.\n\n\"I just wanted to get away this weekend, get over her. Nick, she was just too grasping, self-centered, wanted too many big things too fast. She knew my family had money, though I told her my great-uncle didn't leave me much except that deserted old mansion that's in ruins. I plan to tear down what little of it's left, but demolition costs big bucks. Listen, I thought Cyndi went to Georgia to stay with her brother's family for a while\u2014that is, I assumed\u2014\"\n\n\"Assume nothing. You're rambling. Sit up straight. Don't put your hands over your mouth and mumble. I'm opening this door to let the detective in, and we'll face him together. And you know damn well not to offer so much\u2014blurt things out like this, no matter how devastated or emotional you are. Self-preservation time, Dale. Answer questions, tell the truth, but no more than that.\"\n\nHe came back to his desk, squeezed Dale's shoulder, went over and opened the door.\n\n* * *\n\nClaire was really glad to have Kris here as friend and support. One look at the setup at Black Bog made her realize she should assume nothing and take in everything. Her senses went on alert. She felt the quickening breeze brush her skin. She heard the screech of a gull overhead and the distant roar of an alligator\u2014or was that thunder? Darn, she hadn't thought of gators or being near a bog if there was a lightning storm. She gripped her purse as she got out of the Jeep.\n\nThe first surprise was how finished, even formal, everything looked. She'd almost imagined tents, but after all, Kris had said the Vances lived here, and they were obviously wealthy.\n\nOn one side of the gated entry road stood a small guard house, where a guy in khaki shorts waved at Kris and opened the gate. The entire acreage was fenced with a see-through wire barrier. To the left on the narrow entry lane, an elevated new-looking one-story house with a wooden deck all around it and a large, screened-in porch in back dominated the area. Two vehicles, one a small truck, the other a black BMW, similar to Nick's, were parked under the first floor which was elevated in typical South Florida storm-surge style. But a storm would be a catastrophe for a bog this close to the water.\n\n\"That's where the Vances live now,\" Kris told her, pointing to the house. \"It's so shady in here the house only gets sun for a couple of hours around noon. You should have seen their other place, really gorgeous. Had a pool, right on the beach, you know, one of those places on Gordon Drive in Port Royal. It just shows you how dedicated they are to this project that they have that place up for sale. They are funding this Black Bog effort and that takes big bucks.\"\n\nThe other main edifice was really a collection of buildings. Elevated walkways connected them, one of which reminded her of a long dock, reaching out into what looked like swampland guarded by twisted ficus trees with overhanging branches. That sprawling building had several air conditioners humming.\n\n\"The forest primeval, right?\" Kris said as she led Claire toward the large spread-out wooden building. \"That's the path to Black Bog, but we'll talk to Andrea first, and she'll give you the tour if you're still interested in the assignment.\"\n\n\"So the Vances actually live here all the time?\"\n\n\"They do now\u2014total dedication to this project. There is a staff restroom and shower over there,\" she said pointing at a small building on the edge of the sprawl.\n\nThey stopped at a door under an overhang of roof that had printed letters A VANCE on it. Claire saw there was a slot for a pass card. Kris knocked.\n\n\"I saw on the cameras you're here, Kris,\" came a sharp voice through the door. \"In more ways than one, bring her in!\"\n\n# 4\n\nFirst with preliminary questions in his office, then at the police station, Nick sat with Dale, who had finally pulled himself together to face the barrage of questions from Ken Jensen. Dale left with the knowledge he'd be interrogated again soon. Nick drove him back to the office rather than taking him home where, no doubt, gawkers at the least and the press at the most awaited.\n\nNick told Dale to just lock himself in his office for a while to defuse. The younger secretaries and paralegals always fussed over him and eyed him, but today were standing clear.\n\nNick knew he had work waiting, and he wasn't expecting any word from Claire until at least the afternoon. What a day so far!\n\nHis tech genius Hector\u2014called Heck\u2014Munez was camped out on a chair by Nick's secretary's desk. \"More chaos than usual,\" Nick told Heck, who followed him in. Nick dropped his briefcase and closed the door behind them. Heck kept the firm up on various online activity, so this visit wasn't really unusual.\n\n\"If it's good news, let's hear it,\" Nick said. \"Bad\u2014save it.\"\n\n\"The firm gonna defend him?\"\n\n\"If it comes to that, absolutely. He would not have left Cynthia Lindley's body on his mother's property to be found. We don't hire idiots in this firm and that includes you, so what have you got?\"\n\n\"Facial recognition technology.\"\n\nNick sank into the chair behind his desk and put his feet up, tilting back. He closed his eyes for a moment and pinched his nose with thumb and finger. \"That's a good one. What about it? Claire is with a friend right now from her college days\u2014they're just hanging out for a while,\" he added, wishing he didn't have to hold things back from people he trusted. \"Anyway, this woman has what is commonly called 'face blindness,' which I guess isn't common, though some famous people have it. Her facial recognition is nil\u2014 _nada_.\"\n\n\"Heard of it. I'll look it up. Maybe this technology could help someone like her someday. See, I got a chance to moonlight, make some extra money with a company that's into\u2014 _way_ into\u2014producing facial recognition software and then selling it. I could work on the production part. It's big business and getting bigger. For example, companies like high-end stores use face prints to ID their top customers when they walk in so the staff can greet them and serve them better. The thing is, I could work for this company in a consulting capacity to produce even better technology.\"\n\nWhen it rained, it poured, Nick thought. Claire and now Heck, both eager to branch out, but he could see why. Amazing possibilities abounded lately.\n\n\"I'd like to take this side gig, boss. But the thing is, they're gonna need legal help on all this too, 'cause, of course, there will be pushback on it, privacy violation and all that. But I need your permission and your advice.\"\n\n\"How are the faces recognized?\"\n\nHeck sat even farther forward. \"Boss, I swear, it's the wave of the future. So a scanner picks up a face it's programmed to recognize, and immediately the owner of the scanner gets data on his screen or phone as to who the person is, their preferences, maybe their financial past, stuff like that.\"\n\n\"Whew!\" he said, lifting his feet off the desk and sitting up straight in his leather swivel chair. \"What next?\"\n\n\"Yeah, but this is already here, and we ought to get in on it. Nothing like learning from the present to see the future.\"\n\n* * *\n\n\"There is nothing like looking at the past to see the present and future better,\" Andrea Vance told Claire once they'd been introduced and everyone was seated at a round conference table with laptop stations.\n\nLooking at all the advanced tech gear made Claire realize the touch screen tablet she'd brought was almost primitive. Funny to be using such technology to record information on primitive people who probably didn't even have an alphabet.\n\n\"You come very highly recommended and trusted by my indispensable bog forewoman here,\" Andrea said, and Claire noted a little smile between the two.\n\nAndrea had brown hair with wisps of silver worn loose around a serious face. Next to Claire's five-feet-ten, she was short at about five-two, but she seemed tall\u2014in command, charismatic. Her handshake was very strong, almost too strong. The former archaeology professor was sturdy looking, and her commanding voice projected well. She wore immaculate white running shoes, and Claire saw she had several spare pairs of them as well as white jeans and blouses on a rack nearby. So much white with so much bog and peat and mud? Perhaps the elements didn't dare sully Andrea Vance.\n\n\"I'm sorry Bradley isn't here to meet you, but he will be tomorrow if you choose to come back and help us,\" Andrea went on. \"I hear you have a young family, so if we can have you for at least twenty hours a week here for starters, that will be most helpful. Flexible hours are fine, although when we uncover a new find and bring it\u2014him or her or them\u2014up, we really would need your presence and expertise on-site so you can view them both in situ and in the labs.\"\n\nShe glanced at Kris, then reached behind on her desk to produce a document, one with at least ten pages stapled together. \"I understand you are married to a criminal lawyer who knows his way in and out of these silly clauses, but please read the privacy section carefully. I am sure you will find the hourly salary section to your liking.\"\n\n\"Yes. I understand,\" Claire assured her as Andrea slid the contract toward her. Knowing she and Nick would have to go over it thoroughly, Claire slid it into her bag.\n\n\"I found it ironic,\" Andrea said with a nod, \"that you discovered that local shallow grave that made all the newscasts several months ago, when we had you on our radar to help us examine these graves. Kris and I saw it as a sort of sign you would be perfect to help with our exhumations here.\"\n\nA young woman came in with coffee and doughnuts, each one frosted with vanilla icing and decorated with a chocolate BB on it, no less, but Claire could sense Kris and Andrea were in a hurry to take her on the tour of Black Bog. She nearly burned her mouth on the coffee, but its temperature didn't seem to bother Andrea or Kris a bit.\n\n\"First, let me give you an idea of what we are speaking of here when we say bog bodies,\" Andrea said, standing after they finished their food. She gestured them through a door in the corner behind her desk, a door Claire had not noticed, but then Nick had a private back \"escape door\" as he called it in his office. Criminal lawyers always had enemies as well as friends by the nature of their careers.\n\nThey walked into a room that reminded Claire of a bunker. Under stark ceiling lights, surrounded by gray concrete walls, she saw wide filing cabinets with drawers lining both sides of the long narrow room. The metal file drawers were labeled with things like Young Man with Tooth Gap, Woman with Broken Wrist, Woman with Medicine in Stomach. The drawers seemed to stretch on and on, maybe thirty at least, some with no labels.\n\n\"My!\" was all Claire managed.\n\n\"Each drawer is climate controlled and infused with nitrogen for preservation,\" Andrea said.\n\nShe pulled out the Woman with Medicine drawer, and there lay a leathery-looking corpse with totally recognizable arthritic hands with each swollen knuckle visible. Claire's eyes widened and she sucked in a breath.\n\nThe facial features looked skewed and a bit smashed, but the face\u2014even eyelashes\u2014were visible. The head had hair that stuck out from a flat round cap made of some sort of plant, maybe palm tree fiber. For one moment, Claire's thoughts flashed to opening the freezer, to seeing the female body there, though, thank heavens, this ancient woman's eyes and mouth were closed. But for millennia she had been frowning as if in consternation or pain.\n\nClaire had to struggle to find her voice. Her gaze was riveted to the body, nearly naked but for a sort of apron that curved around her knees she had bent up to her chest in fetal fashion. \"Medicine in her stomach? But\u2014\"\n\n\"Oh, yes,\" Andrea said. \"We always check the contents of the stomach for clues of diet, location, illness. She had quickly chewed\u2014not well\u2014elderberries and willow bark, which contain some of the same alleviants as are in our aspirin today.\"\n\n\"The thing is,\" Kris spoke for the first time in a while, \"we've found that in quite a few of them\u2014as if they are drugging themselves, hoping to head off pain for something\u2014or preparing for some terrible rite or ritual.\"\n\nClaire's head snapped up and around toward her friend. \"Not such as being buried alive?\" she asked. \"Those stakes and that crude rope with her. Was she\u2014staked down?\"\n\n\"I told you, Andrea,\" Kris said, \"Claire will help to solve crimes. She's worth the training and the salary.\"\n\n\"Then let's show her the pi\u00e8ce de r\u00e9sistance,\" Andrea said, without answering her question either. She turned toward Claire with an avid look in her clear brown eyes as she slowly closed the drawer. \"I refer to the brains of the Black Bog people, like those in the Windover cemetery up near Titusville. We remove, study and store their brains in a freezer, which later will be more completely examined and maybe someday, somehow\u2014probed and, well, the results uploaded to our database.\"\n\nClaire nodded, trying to take in each new bombshell. But it was the word \"freezer\" that made her own brain ache.\n\n* * *\n\n\"So how was school today, Lex?\" Jace asked his daughter as he drove her home from school Tuesday afternoon.\n\nHe picked her up that one day each week for some sort of adventure or treat. He also took her on occasional weekend jaunts. The school knew their schedule, but he still went inside to pick her up in the office. A good precautionary practice, he figured, since his daughter had been abducted, and by someone posing as him. Once he and Brit were married and he had better than a bachelor pad, it was agreed he'd have her for longer periods sometimes.\n\n\"School was good. When I told Miss Gerald my dad was going to marry a lady who works at a zoo, she said not to tell everybody right now so they get all excited, but maybe we can take a visit there someday. Her favorite animal is butterflies, but she likes all kinds.\"\n\n\"Great,\" he said, wishing they weren't behind one of the big yellow Collier County school buses, since it would make frequent stops. \"Brit will be excited about that, and she'll understand why it has to be a secret for now. Don't want to get kids so excited they can't pay attention in school.\"\n\n\"I think there are too many secrets around here.\"\n\n\"Really? I thought you liked secrets.\"\n\n\"Well, if they don't take Mommy away into the swamp with bodies.\"\n\n\"Whoa! What? Your mother is certainly not going into any swamp with bodies.\"\n\n\"I heard so\u2014buried ones you dig up.\"\n\n\"No, honey. You're just mixing that up with something that happened earlier and turned out bad, but now everything is okay.\"\n\n\"Well, when I went to the bathroom at home I heard them talking, her and that arch lady, Kris Kane.\"\n\n\"Kristen Kane? Your mother's old friend, the archaeologist? I thought she was living in Europe.\"\n\nAs Lexi chattered on about Mommy's friend from the old days when Mommy went to school, then veered into the names of her own best friends, Jace's mind spun back. He remembered that Kris, who had fixed him up with Claire, had a problem recalling faces. She'd told him once with his blond hair, blue eyes and muscular physique that he should have been a Viking from the days of yore. That's how she'd said she remembered who he was\u2014\"the Viking guy from the days of yore that used to sail in their dragon ships.\" Funny, how over the years when he flew fighter jets in combat over desert sands or the big passenger 747s over the Pacific, he'd sometimes thought of that.\n\n\"So, Daddy, did you have a good day? You always ask me, so I should ask you too. And I know we will both have really good days when you marry Brit, and I get to be a flower girl at your wedding, 'cause I'm good at that after doing it when Mommy married Nick and Nita married Bronco.\"\n\n\"Yeah, honey, I had a good day flying my little airplane and spraying crops,\" he said with a sigh. But he didn't tell her that he and his pilot friend Mitch were considering something in addition to that undercover gig for the government, something far more dangerous but damned exciting. It was in his blood, his fictional Viking blood, but he hadn't told Brit yet.\n\nHe was almost scared to before the wedding, or she might just call it off. She didn't even like the fact that some of the local drug smugglers he helped to get arrested might want to shut him down. It had happened in Tennessee where a drug dealing cartel had brought down a Stingray plane like his with a drone, and the pilot had burned to death in the wreckage. But yeah, maybe there _were_ a lot of secrets around here, because no way was he telling Brit about a pilot's death or even what his challenging new career could mean.\n\n# 5\n\nAndrea Vance led Claire and Kris into another back room, a smaller one this time. It was evidently a lab with large electron microscopes and other instruments Claire could not name. Several suitcase-sized silver metal freezers sat on a tile countertop.\n\nMisty, icy air wafted out of the state-of-the-art freezer Andrea opened. Claire stepped back as Andrea pulled out a plastic tray, placed it carefully on the counter and removed its lid. Within were four small divided compartments, each holding a single gray-brown shrunken brain next to a laminated white card labeled with neat printing.\n\n\"These are the brains of the first four inhabitants we excavated from Black Bog,\" Andrea said, her voice almost reverent. \"If only we could learn what once passed through these convoluted lobes.\" She paused a moment, then went on, \"After removing these for study and preservation, we put the top of the skulls back on and pull up any skin, replacing any hair we can. Some of the heads have plant fiber hats we can replace. Textiles come out great at first but then shrivel, even if kept wet. With the textiles, we can use polymer coating to keep them looking fresh but not with the brains. Since their brains do not speak to us, we need you, Claire, to examine what is buried with the bodies as well as the inhabitants' body language. In short, we need you to pick the brains without psyching them out.\"\n\n\"I'm amazed they are only as large as baseballs, though they remind me more of big walnuts,\" Claire marveled, looking even closer at them. There was a faint odor, something like formaldehyde.\n\n\"Shrunk over the centuries to about one fourth their size,\" Andrea said. \"Their DNA is intact, however. But, as I said, perhaps you can glean some of their thoughts and memories, just as you would with a living\u2014or newly deceased\u2014person from observation of their bodies and, in this case, burials.\"\n\nKris put in, \"Perhaps someday, some brilliant techie will find a way to extract what is in there. For now that's where you come in, though we can contribute some hypotheses too. Let's go out to the burial site\u2014ritual site, accident site, execution site\u2014it may be all of the above.\"\n\n\"Yes, let's go on out to the bog,\" Andrea said, recapping the tray and replacing it in the freezer where Claire saw two other trays of ancient brains were stored. With a nod and intense stare at Claire, she added, \"Yes, we need your expertise, your intuition, your opinions and judgments to go beyond what archaeology and science can surmise or prove, and that is a great deal. We'll just give you a glimpse of our work today, but I must tell you there are some puzzling and strange burials here in Black Bog, and we are totally dedicated to getting answers one way or the other.\"\n\n* * *\n\n\"Detective Jensen's not the only one who needs more answers from you,\" Nick told Dale as they drove from the sheriff's office back toward the law office after Dale had been interrogated a second time. \"At least they didn't detain or arrest you, but that search warrant for your house and office\u2014taking your electronics too\u2014when they've already fine-tooth combed your mother's house means you are definitely a prime person of interest.\"\n\n\"I know\u2014I know!\" Dale said, raking his hands through his hair. \"But do you think for one minute that a trained criminal lawyer\u2014even one in his twenties\u2014would be stupid enough to put someone he murdered\u2014someone he was linked to\u2014in his mother's freezer right next door to his own house? I'm not crazy or suicidal!\"\n\nThe usually calm, stoic man generally kept his hair neatly upswept with some kind of gel product and was careful about his looks and clothes. But since getting in the passenger seat, he had raked his hair into clumps and had dried gel all over his hands and now his trousers.\n\n\"So, let me ask a question they didn't,\" Nick said, not willing to back off until he had more answers. He had the worst feeling a clock was ticking, that time was of the essence. \"You told both me and Jensen that you last saw Cyndi alive over two weeks ago.\"\n\n\"That's right. So?\"\n\n\"So\u2014have you ever seen that old Alfred Hitchcock movie, _Psycho_? __ Or the _Bates Motel_ TV series?\"\n\n\"Nick, nobody got slashed in the shower, and I'm not some nutcase who dresses like my dead mother to kill an attractive woman so her son doesn't go after her!\"\n\n\"Granted. But three weeks ago your mother was still alive. Still in her house. Could your mother have resented Cyndi, the woman you were enamored with, sleeping with, as you admitted to Detective Jensen? You said your mother had dementia and didn't like Cyndi. Your mother is so recently deceased that the timeline would work.\"\n\n\"You go for the jugular, don't you? You're supposed to be helping me, not incriminating an old woman I recently buried!\"\n\n\"Yeah, I go for the jugular,\" he said, hitting the brakes for a red light and lots of traffic. \"How do you think I got so many of my clients off on murder one charges? You damn well know your defense team, if it comes to that, will need to cover every possibility that will pop into Jensen's or the prosecutor's devious brains.\"\n\n\"I know, I know!\" he said, putting his head in his hands and messing up his hair even more. \"And now they used that warrant to search my place, my office, my online history, even though I've got nothing to hide except a few gambling debts at the Seminole Casino.\"\n\n\"Which might have made a fianc\u00e9e angry with you, and an argument ensued, things got physical\u2014an accident occurred and you didn't know how to hide a body fast.\"\n\n\"Hell, Nick! Whose side are you on?\"\n\n\"I've just learned to think like the enemy\u2014psych them out.\"\n\n\"Wish you'd leave that to your wife. Well, not really,\" he added with such a huge sigh it seemed to deflate him. \"Nick, I've seen you interrogate others, but lay off! I wanted to go into law to help the accused, not be the accused. But Cyndi\u2014it's horrible that someone killed her, and I don't know why, let alone who. Like I told Jensen, I really cared for her, even though I knew we wouldn't work out. And what I and my defense team\u2014if it comes to that\u2014need to do, is figure out not only who hated her, but who wanted to set me up as\u2014as her possible killer.\"\n\n\"We may have to shake the trees for other persons of interest. Look, we'll go in the back way at the firm in case the media is still hanging around. Duck down when we pull in. Why don't you phone Bronco at the front desk to be sure we'll have a clear path to get upstairs? You get back in your office, see if the tech team took anything other than your laptop. Then brainstorm for every\u2014I said _every_ , including your mother's\u2014possible motive people might have for harming Cynthia Lindley and then having the nerve and access to put her in that freezer. I don't know why the ME's cause of death report is taking so long, but it has to be imminent.\"\n\n\"Maybe it's taking so long,\" Dale said, his voice shaky, \"for that very reason, that she was frozen in death.\"\n\n* * *\n\nClaire stared out at the grid of planks that neatly dissected the part of Black Bog the team was digging in today. Andrea said, \"Kris, tell her what you can at this point. I need to talk to the dig team. Be right back.\"\n\nClaire asked Kris, \"'Everything you can at this point' means until I sign the contract and can actually get close to one of these digs\u2014graves?\"\n\n\"I told you this is all top secret. So, anyway,\" she said, pointing, \"what do you think of the site at first glance?\"\n\n\"This setting seems like something out of Edgar Allan Poe. Remember how I told you my mother used to read to us all the time after my dad deserted us and she became a recluse? The dark water and peat here, the sun creeping through the heavy foliage over that ancient, open grave...\"\n\n\"Better stick with calling it a dig site. I've got to admit I feel that way too, if I stand back objectively\u2014which I seldom do. Wetlands archaeology, this site especially, reminds me of those terrifying Dead Marshes from the _Lord of the Rings_ books and movies. But, the thing is, my friend,\" she said, putting a hand on Claire's shoulder as they walked on the boardwalk that stretched out over the bog, \"you just have to shut that out. We need to bring these ancient people back to life, so to speak. We can usually discover cause of death, but we need you to weigh in on what they might have been like in life. You can't question them, but you can observe and theorize. Now watch where you step on the planks, and if you come back tomorrow as part of the team, I'll show you the latest excavations and exhumations up close.\n\n\"And,\" Kris added with a swat at a mosquito, \"these voracious buggers are the monsters of this bog, though we fog for mosquitoes a lot. Sorry, I forgot to spray you before we came out. That funny smell out here comes from several camphor pots we have way back in the trees. It helps keep the mosquitoes away, but the smell is pretty acrid.\"\n\nThey stepped out on a pair of planks which shuddered slightly under their weight, but held firm, supported by two-by-fours Claire could see were pounded into the base of the bog.\n\nKris pointed outward as two men and one woman, all on their knees on a wooden platform, peered into a five-foot-square hole about fifteen feet away. Near them rested the tools of their trade. Shovels, trowels, brushes, cameras, clipboards and plastic sheeting were things Claire could recognize from here. Standing behind her team, hand propped on her knees, Andrea leaned over, looking down too, blocking Kris and Claire's view. Around the digging trio lay slivers of peat they had been slicing and scooping away to get to the corpse.\n\n\"We shall call this one 'Hunter.'\" Andrea's words floated to Claire as she and the team still stared down in apparent awe. \"He looks to be quite muscular and still holds his knife. He's dressed in some kind of pelt. And\u2014look\u2014another body must be close, because there's a hand as if reaching out toward him. Be especially careful if there are other bodies nearby. I've never seen one that close. I've got to tell Bradley. He should be here for this!\"\n\nShe pulled her two-way radio out of her shirt pocket, then turned and looked at Claire through the gray, shifting shadows. She nodded as if to say, _By tomorrow you could be here with us._\n\nBut suddenly this scene seemed so staged, so perfect a lure, for who would not want to get a better look into that grave with the ancient hunter and someone else's outstretched hand?\n\nEven as Andrea put her transceiver to her ear and began to talk, Claire thought about the words from the Robert Louis Stevenson poem, ones she could not quite recall, but went something like, \"Home is the hunter, home from the hill.\" __ Above all, she realized she had to talk to Nick about this significant but strange place the moment they were both home.\n\n* * *\n\nNick was exhausted and hungry, so he'd asked his secretary to send out for a sandwich, and it awaited him on his desk. He hadn't even thought about food for a while. Dale had been in a waiting room here until the police techs were finished with his office, and they had just handed it back.\n\nOn his desk phone console, he hit the number of the secretary who worked with two of the junior partners, including Dale. \"Tracey, Nick here. Be sure Dale gets another laptop, because I'll bet they took his.\"\n\n\"Will do. Has he\u2014well, has he been indicted or what? He just stormed into his office and shut the door.\"\n\n\"Not yet and we hope never. Keep an eye on him, since he's upset.\"\n\n\"I guess so. He looks like he's been through the wringer. We'll take care of him.\"\n\nThough Nick had an open-door policy with the firm's partners and employees, he was going to close his door to call Claire. Surely, she was out of bog land by now, though he wasn't sure if there were cell towers out that deep in the wilds. He checked his voice mail and texts. Nothing yet, except one from Jace saying he had picked up Lexi from school as he and Claire had arranged. Nick decided she'd call when she could and he shouldn't seem to be checking up on her if she was with the staff there.\n\nThough his mouth was full of his sandwich, at a knock on his office door, he called out \"Enter!\" He was expecting his secretary, even Dale again.\n\nIt was Bronco, looking jumpy and pale.\n\n\"Sorry to bother you again, Boss, but Nita called to say a couple of reporters knocked on our apartment door for a statement. She said no but one filmed her at the door. So they got us staked out both at our new house and old apartment. Nita\u2014she's real upset.\"\n\n\"As soon as I can talk to Claire, it might be good for you both to pack a bag and stay a couple days with us. It would be good to have Nita right there at the house for the kids anyway, especially if Claire takes that part-time consulting job.\"\n\n\"Oh, yeah. What's that all about?\"\n\n\"Consulting for some people who need help at a job site. Bronco, calm Nita down and tell her everything will be fine.\"\n\n\"She don't think so, boss. Doesn't want to live there anymore. I\u2014I think she believes the place might be\u2014kind of haunted.\"\n\n\"But we know better than that, right? I'll work on it as soon as I hear from Claire. If you need to go home early, go.\"\n\nCuddling little Trey in her arms, Claire paced at home. Jace had let Lexi call her to say they were at the zoo with Brit but he'd have her home in plenty of time for bed. \"We're hunting for the funniest monkey and the tallest giraffe,\" he had told her, \"now that we've seen Brit and the big cats.\"\n\nNow Claire's thoughts quickly returned to her own day. She wanted to convince Nick she should take the job. It was only the secrecy that bothered her, but she understood that too. After all, Nick had to keep a lot at work private. Jace was flying on classified government missions, not only to spray for bugs, but to pinpoint the locations of criminals through an undercover Stingray program that hunted down and traced cell phones\u2014another sort of bug. Many things in life had to be top secret. As long as she could share _some_ of what was going on at Black Bog with Nick, surely nothing could go wrong, even though she'd have to lie to some people about what she was doing.\n\nLittle Trey went beautifully, innocently to sleep on her shoulder, so she went to put him in his crib. How eerie it was to think that those long-dead human beings lying in the bog had also once had people they loved and needed and had children to protect. No doubt they had people they hated too, problems they tried to solve. And she had picked up on the fact that Kris had said the people in Black Bog were not only being exhumed but might have been executed. Since Claire had worked with murder and suicide cases before, was that why they really wanted her?\n\nShe began to pace again, in and out of Trey's nursery, back toward the front door. Nick might be late because of the mess with Dale Braun, so she hated to bother him at work. But she needed his coolheaded advice, not that she'd seen anything really amiss at Black Bog. It was just that one look Andrea had given her. Claire had psyched out that the woman was used to getting what she wanted. And herself? Yes, she wanted to jump in\u2014graves and all\u2014with both feet.\n\n# 6\n\n\"I don't think we should put off setting a date and planning our wedding any longer,\" Jace told Brit when he picked her up after taking Lexi home. \"We've weathered some bad times, but let's forge ahead.\"\n\nAs he pulled out into traffic, she turned toward him in the passenger seat, bending her left leg to face him despite her seat belt. \"You do bring up the most important things at the craziest, least romantic times, flyboy. But actually, that would make me happy too\u2014not to mention our little flower girl.\"\n\n\"Yeah, she's psyched. But I want it for us, not Lexi. I know it's pretty soon after losing your father, but\u2014\"\n\n\"But he would have been very happy for us. And if we can somehow get my dear brother _in_ the wedding party rather than playing his violin, that will suit me.\"\n\n\"I'd ask Mitch to be best man.\"\n\n\"I can hardly ask Claire to stand up with me, however close we've become. Just wouldn't do to have your ex up there with you again while you say your vows. But what triggered this all of a sudden, if it wasn't Lexi?\"\n\n\"I don't want to let you down or lose you,\" he said, hitting the steering wheel with his left fist. \"I can see how the other big cat keeper looks at you. Don't want him or some other guy swooping in.\"\n\n\"Swooping in? Says a guy whose nickname is Hawk! Don't be silly. I'm wearing your engagement ring, and everyone knows we're a couple\u2014a couple in love, sharing everything, right?\"\n\n\"Absolutely. Affirmative that,\" he insisted, though guilt bit at him again that he hadn't told her about his new, dangerous job opportunity. \"Now I'm gonna pull in to this grocery store, because I need to seal this with a _looong_ kiss. I admit this wasn't the most romantic place to bring it up, and neither is this parking lot, but we'll make up for that, I promise.\"\n\nBefore he could pull over, they got stopped not only at a red light but by a long funeral procession. Life could be short, he told himself, gripping the steering wheel and wondering if she was thinking the same thing.\n\nThe truth was, part of the reason he'd decided they should get married soon was that who knew how long either of them had. Not only was he hesitant to tell her about the dangers of hurricane flying, but what he was doing now was getting even more lethal.\n\nOnce the funeral procession passed, he pulled into the grocery store parking lot and drove into the back row under a tree to get some shade. He killed the motor and undid his seat belt while she unsnapped hers. They hugged hard over the console between them and kissed deep and long until a couple of teenage boys walked by and hooted.\n\nAs they broke their embrace and he honked the horn to make the boys move on, he felt like both a rat and a coward. He had dodged her question about why he wanted to speed things up. He'd almost worked up the nerve to tell her he was considering a dangerous assignment, one that would serve the public good. Surely, she'd understand\u2014but would she? She'd lost so much recently, he could picture her going ballistic over this.\n\nMitch had talked to him about signing on with him to become a hurricane hunter, flying into the eye of deadly storms for the National Oceanic and Atmospheric Administration. Mitch had even kidded about changing his nickname from Hawk to Hunter.\n\nJace almost blurted it out, but he didn't want to argue either about the hazards of that career or have her insist he turn it down because he could be instantly on call if the weather was bad. And he'd need special training again. But he wanted more excitement in his life, he needed more than what he was doing, more than loving her, and she wouldn't understand that.\n\n* * *\n\n\"So because of his weapon, clothing and muscular build, they named their newest find Hunter,\" Claire told Nick in another gush of words after he got home and sat on the couch to read through the Black Bog contract. She'd put out wine, crackers and cheese on the coffee table for him while she put Lexi in bed. The child had been tired and full of a chicken nugget dinner when Jace brought her home, so she had conked out already, and Trey had not awakened for a feeding yet.\n\nAfter being filled in on the latest about Dale's questioning at the sheriff's office, Claire had tried to keep calm as she told Nick about the day she'd had at Black Bog. She realized how excited she was to share what she'd seen\u2014and convince him she should take the job.\n\n\"Very professional, very complete,\" Nick told her, tossing the contract on the coffee table while she sat beside him on the couch. \"Of course, like all legal documents, it benefits the creator more than the signer, but it looks solid enough, and the pay per hour is excellent. I just hope the secrecy clauses will be pass\u00e9 soon, and they can release some of this to the public.\"\n\n\"But you do see why? In a way, they're sitting on a powder keg if some of this information were to get out.\"\n\n\"True, and we've had enough of getting things blown up in front of us. But I think, since this is really part-time, it will work out.\"\n\n\"Especially if we have Nita on-site here to help with Trey and Lexi. I'm all for your idea of letting them live here for a while until things calm down after finding that poor woman's body.\"\n\n\"Which I'd like more information on, so I told my secretary and assistant to call me here if the ME releases cause of death tonight. Meanwhile, Mrs. Markwood, what's your strategy to keep your new endeavor secret from our inquisitive daughter, let alone everyone else?\"\n\n\"The cover story will be that a fairly new business is hiring me part-time to interview prospective employees. They are developing a new product they don't wish to discuss right now. Not a lie, not quite the truth, but, after all, I've advised other companies on hiring tactics, so\u2014\"\n\nNick's phone on the coffee table sounded with the music from a lawyer TV show, then a clickety-click she'd always thought of as his brain working. Oh, darn, after seeing those shrunken brains today, maybe she had brains on the brain.\n\nHe snatched it up. \"Nick here.\" She was sitting so close to him she could hear what the voice on the other end was saying. She recognized the woman's voice as his secretary, Cheryl.\n\n\"Nick, I was about ready to go home when the ME's office called about information on the death of Cynthia Lindley. She had some bad bruising on her arms and legs, especially her neck, though those were hardly cause of death. She was fully clothed. Of course, she was frozen solid, but cause of death was strangulation. Her hyoid bone was fractured. We'll have a fax of the entire report tomorrow a.m.\"\n\n\"Thanks, Cheryl,\" Nick said, and punched off. \"Did you hear that?\"\n\n\"I admit I did.\"\n\n\"So, conclusions, my fave forensic psych?\"\n\n\"Whoever killed her and lifted her into that freezer was strong\u2014not some old woman.\"\n\n\"Good point. Unless the murderer had help. I'm thinking that supposedly one third of female deaths where the hyoid bone at the front base of the neck is broken is from rough sex\u2014temporarily cutting off the oxygen for a higher high, and then it all goes wrong.\"\n\n\"Really? So there's another piece of evidence that could point to Dale.\"\n\n\"I wouldn't figure the guy's that type, very controlled, but then still waters run deep. Anything else that comes to mind?\"\n\n\"The person hated or feared her enough to kill her, yet had feelings for her\u2014didn't just dump her body somewhere, even kind of arranged her clothing and body for the freezer. But then, he or she perhaps opened the victim's eyes and her mouth in that silent scream\u2014but why?\"\n\n\"Feared her. Never thought of that. She might have known something secret and she said she'd tell. Blackmail?\"\n\n\"Unfortunately, until we somehow look into her life, the possibilities are endless,\" she admitted.\n\n\" _We?_ If you sign that contract and don't intend to put Lexi and Trey up for adoption, no way you're on this case. Yes, I know we decided to have Nita here for a while to help. You will be busy enough, even though I would love your help. And the one thing I don't like about your lucrative Black Bog contract is that your cell phone won't work there if I need you\u2014or if the kids do.\"\n\n\"I will only be part-time there and won't stay for entire days. Meanwhile, I predict, leader of all you command, Mr. Brilliant Mind, Senior Partner of Markwood, Benton and Chase, that you will need my skills\u2014at least my opinions\u2014before this is all said and done. I intend to sign this Black Bog contract now, but I also signed on to help you through thick and thin, through hell and high water, or whatever we vowed, and we've managed to survive all that.\"\n\nHe hugged her hard, pulled her down onto the couch and then lay beside her. \"I don't know what I'd do without you in all kinds of ways. I love you, sweetheart, and always will. As obsessed as I\u2014you too\u2014get with our projects and endeavors, it's our family and our marriage that matter most. I run that law firm that seems like a little town and try to help people on the sly through South Shores, but you're at the heart of everything. I know we don't want to upstage Jace and Brit, whenever they finally set a wedding date, but I think it's about time for us to plan that long-delayed wedding reception we've talked about. We can have it at the country club I never use. What do you think?\"\n\n\"I think it's a great idea\u2014perfect. With a shotgun wedding in another country where we only had Lexi at the ceremony, we owe that much to our friends and family\u2014and ourselves. Life goes by fast, things happen. That's one truth that hit me at Black Bog today. Let's do it!\"\n\nTears stung her eyes. She tried to blink them back, but they clung to her lashes. \"I never thought things could be so perfect\u2014that I could be so happy,\" she got out before he crushed her to him in a commanding and possessive hug.\n\n\"There's another contract we are going to celebrate tonight,\" he said, sitting up and pulling her into his arms as he stood. \"No better place to plan a celebration than in bed.\"\n\n\"But you said you were hungry.\"\n\n\"Mostly for you. Lexi's out like a light, Trey's quiet. This way, my love.\"\n\nHe actually carried her out of the Florida room and down the hall to their bedroom. He lay her on the bed and, standing over her, began to strip off his shirt and pants.\n\n\"I feel like we're newlyweds again,\" she said, pulling off her top and wiggling out of her shorts. \"And we'll soon have Nita and Bronco around, and they really are just married. Then there's Jace and Brit, thinking of\u2014\"\n\n\"Enough,\" he told her as he pushed her back on the bed and lay naked beside her. \"Too much thinking. Let's just go with feeling for now.\"\n\nAnd they did.\n\n# 7\n\nDespite how tired Claire was after last night, adrenaline poured through her. Taking her signed contract, she drove to Black Bog and was welcomed by Kris and Andrea. Kris gave her a two-way radio transmitter and a quick lesson in using it.\n\n\"Obviously, only for on-site communications,\" Kris explained while Andrea nodded. \"Numbers for each staff member are pasted here on the back. Your cell phone will work when you get about four miles back toward town.\"\n\nSoon Bradley Vance joined them for what must be the daily hot coffee and BB-decorated doughnut ritual.\n\n\"Damn glad you're on board!\" he told her, shaking her hand with an overly firm grip that almost made her wince. He was an imposing man, with premature white hair over a broad forehead, sharp blue eyes and a healthy-looking tan. \"We need to ID our precious finds from all sorts of angles, and psychology is one key way. More of our brand of great women's intuition around here doesn't hurt either,\" he added with a nod at Andrea.\n\n\"Which reminds me that I wanted to ask,\" Claire said. \"So far, what has been the percentage of men you've found in bog burials compared to women\u2014or even children?\"\n\n\"No children,\" Brad\u2014Kris had said to call him either Brad or Senator\u2014told her. \"At least not easily discernible ones, depending on where you draw the line on children versus adolescents. I mean, no doubt by our standards, these people died young, but it seems to be an adult cemetery. One theory is that a burial bog was a magical place where people communicated with the dead, so young children weren't allowed to be part of that. It was perhaps a frightening place, a netherworld, so they buried children elsewhere. That's one reason you're on the staff now, to look for clues, draw conclusions and present us with hypotheses.\"\n\n\"I understand. I can't wait to get started.\"\n\nAndrea said, \"Then let's begin with the final exhumation of the man we've dubbed 'Hunter.' And then follow that outstretched hand we've uncovered to see why someone would be interred so close to him\u2014unusual. Black Bog's secrets await, so let's go.\"\n\n* * *\n\nNick had a ridiculously busy day ahead, but he'd scheduled Heck into a 9:00 a.m. appointment, or he figured he wouldn't get to him at all. The guy had been invaluable over the years. He could find about anything online and stayed up with the huge tech curve for Nick and for the firm. After all they'd been through together, Heck was as much friend and family as employee.\n\nWaiting for the familiar knock on his door, Nick stared at the lengthy ME's report which had been faxed to them this morning, since the firm was now on the record as attorneys for Dale Braun.\n\n_Although the bruises on the female victim's body_ , he read, skimming partway down, _were consistent with the subject being manually restrained and possibly held down_ , _there is no evidence of sexual activity, no rape, no bruising in the genital area._ Nick looked even lower in the long report. It had been difficult to establish time of death because the timing for the frozen state of the corpse could interrupt and throw off the estimation of time for rigor mortis.\n\nThere were no drugs in the subject's system. There were, however, the remains of a steak dinner with salad and baked potato and red wine, but no unusually high alcohol readings.\n\nThe open mouth and open eyes did not indicate the victim had been placed in the freezer alive. In the ME's best estimate, the mouth and eyes had been deliberately opened before the body was frozen, though that assumption could be contested.\n\nNick sighed. Everything could be\u2014maybe would be\u2014contested if it came to an arrest and a trial. But what had really caused the death of Cynthia Lindley, or, that is, what had led up to it? They knew how she had died. But the why, that was the thing. And, yes, Claire was right. He could use her help on this, as long as she was not taking risks.\n\nHe heard Heck's usual three quick raps on the open door. \"Enter!\"\n\n\"Morning, boss.\" He ambled in and sat down as if neither of them had a care in the world. \"Word is you're busy on a big case, so I can make this quick.\"\n\n\"First, how is Gina liking med school and living in Miami?\" Nick asked about Heck's girlfriend.\n\n\"Likes it, especially the big Cuban population there, almost like home, she said. When I told her about this face recognition stuff, she said she'd recently considered being one of those artists who does facial reconstruction\u2014you know, with clay after measuring a skull. I didn't even know she had art talent. Crazy, huh?\"\n\n\"Maybe not. She's been worried about the expense and length of med school, and that would mean she wouldn't go after that medical degree. So explain a bit more about facial recognition technology in case I want to talk to the senior partners about possibly getting involved with your idea for a sideline career.\"\n\n\"Just learned the Russians are deploying it to spy on their own citizens, so it's far-reaching, developing fast. Moscow has a network of around 170,000 surveillance cameras across the city to ID criminals and boost security. That's way beyond what I was telling you about high-end stores using it. This facial recognition tech was designed by a Russian startup called N-Tech Lab, Ltd. It cross-references images from a database against those captured by cameras at entrances to buildings.\"\n\n\"Big brother is watching you, in other words. I'd like to think that's just Russia, but it's coming here, I'll bet. I do know that even our more advanced allies are jumping in with both feet. I read that the UK uses so-called CCTV cameras too, maybe as many as 70,000 across their nation.\"\n\nHeck nodded, scooting to the edge of his seat. \"I knew that too. Boss, it's as scary as it sounds, that a government can find and arrest criminals that way. So why not the good guys too, so-called enemies of the state, or just a guy who criticizes them? I'm sure the Russians don't want us getting ahead of them in this or even catching up.\"\n\n\"You're right. A different kind of terrorist, the ones running the country.\"\n\n\"There's even an app called FindFace that was a big hit in Russia last year. Really accurate, this technology is going to take our world by storm, and this company that wants me will be in on that. _Caramba,_ the competition is really cutthroat as well as cutting edge. It will be just to consult\u2014kind of like Bronco said Claire's agreed to do for some startup company. And like I said, they're gonna need legal help 'cause there's lots of folks gunning against this new invasion of privacy.\"\n\n\"So that's where the firm would come in,\" Nick said. \"I'm all for this, Heck\u2014someone like you involved who knows the possible dangers if this isn't reined in, as long as\u2014like I told Claire about her consulting\u2014it's safe, and you still have time for what you were hired for here. And by cutthroat, you don't mean Russia could reach out here to try to harm American tech firms whom they see as competition in this software?\"\n\n\"I don't think so,\" Heck said with a shrug. \"No one said that, and my work would be pretty hush-hush.\"\n\n\"Someone like that friend of Claire's might be able to benefit from face recognition technology down the road, so keep that in mind.\"\n\nHis mind darted to last night with Claire in bed. His beautiful wife, under him, on him in the dark of their bedroom where he couldn't see the expression on her face, but he knew what it was. Trust. Joy. Ecstasy, and\u2014\n\n\"So what do you think about someone from this startup company, FindFace, contacting you, boss? I'll go ahead and sign on to work with them then.\"\n\n\"Oh, yeah,\" Nick said, jumping back to reality. \"Sounds good, at least to explore our options. Meanwhile, I need some help on this current murder case that involves Dale Braun. Look into anything you can find online about the victim, Cynthia Lindley from Atlanta. History, social media presence, whatever. Dig up anything you can.\"\n\n* * *\n\n\"It's kind of funny that you call this walking the plank,\" Claire told Kris as they followed Andrea out to the site where the team had uncovered \"Hunter\" yesterday. To Claire's surprise, Brad Vance had not come along, so had he no curiosity about a new find, or did he just leave all of that up to Andrea? No doubt, there were business matters he must oversee here, but he'd given no indication he wouldn't come out with them. She would ask Kris later, because she didn't want to overstep or seem too curious in front of Andrea about the living here\u2014just the dead.\n\nKris tried to prep her for what she would see. She explained they had slowly, carefully dug out more of Hunter's body, then had placed plastic sheeting over the site until today, in case of rain, so she'd be there for the retrieval of the body.\n\nEveryone hovering at the site looked up and stood when she joined them. Andrea made introductions. \"Just first names for now, and you can get to know each other later. Kind of like when we excavate a new grave site, right, gang, then figure out what we've found?\"\n\nClaire nodded and said hello to the same team she had seen here yesterday. The two men\u2014Doug and Aaron\u2014seemed young, maybe grad students. Yi Ling was a beautiful woman with sleek black hair held back from her face by a cord. She had also covered it with a net, no doubt to keep from dropping foreign objects, even strands of hair, into the graves. They were all obviously anxious to keep at their work. After short greetings where Andrea introduced Claire as \"part of the brain trust,\" they eagerly bent back over the site.\n\nClaire steadied herself for the first glimpse of Hunter. Kris braced her by her upper arm as they and Andrea looked in over the heads of the excavation team.\n\nAlthough she'd seen the bog bodies in lab drawers inside, Claire jolted. Unreal, but so real. All tanned to a dark brown, yet amazingly preserved with chin-length hair, beard stubble, even a facial expression, which was one of grief or maybe pain. At being buried here? Alive? What had he been thinking when he died or was laid to rest? He hardly looked at peace\u2014had violence been involved?\n\nShe also noted the hand attached to a wrist from a nearby, still bog-covered corpse, indeed reaching out toward Hunter. Beseeching him for something? Begging? Could their closeness be just coincidence or was there some relationship, shared feelings or a unity in death? Andrea had said it was unusual for two bodies to be so close. Though not quite touching, they seemed connected somehow. After all, Hunter's tormented face had been turned toward that other body for eons. And it was a small hand, a delicate hand. A woman, perhaps his wife. Claire hoped this wasn't a culture where a wife was forcefully interred with her powerful mate before her time. As shriveled as Hunter looked, he evoked strength.\n\nAndrea, Kris and Claire stood back on the wooden platform which held the tools for excavation. The team struggled to slowly, carefully slide a plastic sheet down and under the man's body.\n\nClaire whispered to Kris, \"He won't slip deeper, will he?\"\n\n\"They've already shored up the peat under him, or otherwise, yes. Everywhere off the platform and planks, it's like a swamp that can swallow a person. That's why we have to be careful where we walk, and never get off the supports. We found one modern body that seems to have fallen in and gotten sucked down, suffocated. A corpse maybe from the late 1800s or early 1900s. So never, never come out here alone. If you fall in, you can be gone in no time\u2014I'm sure a terrible death.\"\n\nClaire saw Andrea shudder. \"When I was young,\" she said quietly, still watching her team work, \"I saw a movie set in Africa where the villain was horribly sucked down to his death in a swamp. It took me a lot of courage\u2014and the desire to resurrect this people and their culture\u2014to walk out here at first. Brad still avoids it, because he did fall in once, and I had to run for help to pull him out or I would have gone down too. I laid tree limbs for him to hang on to until I brought help. But, the truth is, he's had claustrophobia from a young age, so that was double horrible for him.\"\n\nThat explained his absence, Claire thought. The three of them watched intently as the team lifted the man's body by the plastic sheet, then laid him nearly at their feet in a space on the platform.\n\nClaire stared wide-eyed. Centuries old but he could have been disinterred from a merely historic, not prehistoric grave. Although not this close-up, she'd seen an exhumation of a murder victim once, but this...\n\nShe could tell he had been an imposing man with big shoulders. His face was smashed a bit sideways by the long weight of water, mud and peat, but he still seemed\u2014well, he must have been handsome with that strong nose, high cheekbones and firm chin. One foot was shod with a sort of laced leather shoe, the other bare. His toenails were visible. In his right hand, Hunter clenched a double-edged dagger as if he would never let it go.\n\nHis only other garb besides the one crude shoe was the animal pelt wrapped around him\u2014no, it was more than one. They were crudely stitched together to hide his body, chest to thighs.\n\nAndrea broke the awed silence. \"I wonder how someone this strong, and apparently young, died. He looks\u2014I know this sounds insane\u2014healthy.\"\n\nClaire cleared her throat. She wasn't sure her voice would even come out at first. \"I observe the same. Amazingly, he seems to evoke a personality. Have you seen others buried with a dagger?\"\n\nAndrea answered, \"Not like that one. Only scraping knives with plant residue still on their dull edges. But nothing this apparently special, this\u2014commanding.\"\n\nClaire noted they were all having trouble finding words. In a glint of light, she could see that Hunter's dagger was etched with some sort of design, though peat now filled the cracks to make the surface one dark blur. It was not metal but some sort of stone.\n\nShe asked, \"Do we know what kind of animal pelts those are? If it's a rare or special animal, that could mean high status, which his dagger suggests.\"\n\nWithout looking up, Doug answered, \"Short hair. Probably white-tailed deer. Doubt if they had the means to kill bears or panthers.\"\n\nWhen Claire finally glanced away, she noted that Kris had narrowed her eyes and kept frowning at Hunter's face as if forcing herself to remember it. She had claimed to be able to recall ancient faces, but she could just be memorizing his build or clothes as she had in college days. Claire sensed something else was also bothering her, so she looked where Kris was staring\u2014at the part of the pelt over his chest.\n\n\"Is there blood on that deer skin?\" Claire asked, getting carefully on her knees to lean closer to the body. \"I know the pelt is dark because of bog and tanning acids, but I see blotches. Maybe blood on his garment is a status thing too, because blood means so much in many cultures, especially in sacred and spiritual matters. Wouldn't it be something if we could figure out some of their religious beliefs\u2014if they had any?\"\n\n\"Let's get him inside and do our usual exams,\" Andrea's voice cut in. \"I'm sure Brad is waiting to see what we found, and he oversees artifacts like that dagger.\"\n\nWhen Kris stepped forward to take the fourth corner of the plastic and the team lifted the body, the pelt slipped off his chest. Despite the brown skin and curly hair, they saw where the blood had come from\u2014much more of it.\n\nEveryone gasped. At first Claire thought Hunter had been stabbed, but then she saw his heart was missing and long-dried blood bathed his entire chest and torso. She cried out as though she'd been hit in the gut, but Kris's scream drowned that out, echoing over Black Bog.\n\n# 8\n\nThe circle of the living stared aghast at the murdered dead.\n\n\"Never seen that before,\" Aaron said, sounding breathless. \"Man! A sacrifice or an execution? And he holds that dagger, though it could have been put in his hand after.\"\n\nAndrea, her voice very quiet, said, \"That may not be the murder weapon but a sort of marker of his occupation or even his social standing. Let's get him inside, then the three of you come back out and work on whoever has\u2014had\u2014that hand that's reaching out toward him. Careful now. Let's go.\"\n\nClaire followed the strange procession. She felt as if they were going to a funeral. Her thoughts came fast and hard. Were they in a section of the bog where criminals had been buried? What about the woman next to him\u2014if it was a woman? As many people as she had advised, as many cases as she had worked to understand and help the living, this one was starting to obsess her.\n\nInside, the team laid Hunter on a stainless steel table that reminded Claire of the one she'd seen in the Collier County morgue. They kept the pelt, now askew, over his hips and, under bright lights, bent closer to examine his ravaged chest. Even now his hand had not released his dagger, as if he still needed it for protection.\n\nAndrea called Brad on an intercom system which Claire had noted as well as the various security cameras. Then Andrea put on some sort of protective glasses. They must magnify, because her eyes looked huge. \"When you go out again, be certain Hunter's heart wasn't anywhere in the excavated site,\" she told the team.\n\n\"We'll look again while we disinter the woman,\" Yi Ling said, \"unless it's a small man or a child. Should we bring a surgeon in for an opinion of what\u2014what was used to cut into Hunter like that?\"\n\n\"No outsiders!\" Brad's voice boomed from behind them as he came in. \"Claire is the last of this team until I say otherwise.\"\n\nHis tone of voice and assumption of authority surprised Claire, who figured Andrea had the last word on excavations here. As if the dig team wanted no problems from Brad, the three of them hustled back outside, though Claire noted that Yi Ling turned to smile back at Brad.\n\n\"Any further observations, Kris, Claire?\" Andrea asked, pulling on latex gloves.\n\nClaire said, \"You've found no other burials with people maimed, maybe punished or executed? And that dagger\u2014you said nothing else but crude knives have been found.\"\n\nAndrea nodded as she reached out with gloved hands to try to take the dagger from Hunter's hand. Claire held her breath, almost wanting to ask Andrea to leave it alone. He'd held it all these centuries, and it had meant something to him.\n\n\"There!\" Andrea said, wresting it away and lifting it toward the overhead light to study it.\n\n\"I can't wait to clean this and see what it reveals\u2014to hold it,\" Brad said, making Claire want to speak out again to tell them it wasn't really theirs. She not only pitied this ancient man, but, strangely, wanted to protect him.\n\nKris's gaze met Claire's over the corpse as Andrea placed the dagger in a metal tray and handed it to Brad. \"Bog and blood clings,\" she said, \"but the blood doesn't mean he hurt himself. People don't tear out their own hearts, except emotionally, and that's where you'll come in on this, Claire.\"\n\nClaire hoped Brad would let them look at the dagger closer, even before it was cleaned, but, with another quick glance at the corpse, he took the artifact from the room.\n\n* * *\n\n\"Unbelievable!\" Nick told Claire that night as she described her first working day at Black Bog. He sat at a kitchen bar stool while she put the dinner plates in the dishwasher. He had gotten home late; Lexi and Trey were both in bed, and he'd kissed Lexi good-night and stood over Trey's crib for a while before joining Claire in the kitchen. \"I'm just grateful you were in an archaeology lab and not a modern morgue.\"\n\n\"I'm certainly going right to work for the Vances. I'm anxious to know if that next body\u2014pretty sure it's a woman's\u2014will have met with violence too.\"\n\n\"I hate to bring this up now when I'd rather be relaxing, but I have sicced Heck on the frozen-woman case to figure out what Cynthia Lindley was like before she died. We need to find who could have hated her enough to kill her, who would have known where to find her\u2014and known where to store or hide her body.\"\n\n\"Heck always turns something up.\"\n\n\"I've also got to have Dale give me his version of Cyndi's life and character, and I'd like to have him over here for dinner for that. You could listen too\u2014not take notes, not seem to interrogate him, but I'd like another set of ears, not a lawyer's. And you do come at a reasonable price.\"\n\n\"Right. Your love and life. Your lovemaking, your son...\"\n\nClaire's voice drifted off. Did Hunter once have a son? A wife? Did someone wait for him to come home with deer meat, with a warm pelt for their bed? What went so wrong that he\u2014\n\n\"Sweetheart, I said is that okay? I know it puts more work on you.\"\n\n\"I'm honored you asked, and you know I love to be included. Yes, of course, as soon as you need to pursue that\u2014him. Dale still could be involved, right?\"\n\nClaire's phone on the counter rang. She walked over and glanced at it. \"It's Brit,\" she said. \"Hope nothing's happened to Jace, doing all that flying, just when things are looking good for them.\"\n\n\"I know you still worry about him,\" Nick said.\n\n\"Of course I do. Lexi would be devastated if something happened to him,\" she said as she slid the phone icon upward to answer it. \"Brit, Claire here.\"\n\n\"Hi, Claire. I know it's late, but Jace and I were having a celebration dinner, and we wanted to tell you and Nick something before you hear it elsewhere. We've set a date for the wedding, Saturday, August eleventh. Now don't get your hopes up that Lexi's going to have another new baby to fuss over, because it's not a shotgun wedding except I may have held one to his head so he'd set a date. Of course, we want you two as guests and Lexi as flower girl.\"\n\n\"Congratulations from both of us\u2014and, of course, from Lexi. No problem with setting a date, since you're both ready.\"\n\n\"It will be a small wedding, though I'm not sure we'll be able to take a honeymoon then with Jace's flying demands\u2014well, my work with the animals too.\"\n\nThey chatted a moment more, then said goodbye. Claire explained to Nick, though he'd picked up on this end of the conversation.\n\n\"So, that's good news,\" he said, smiling. \"Let's have that private reception of ours sooner than that. I called the Orange Grove Country Club today and we can have the Blue Room there this Sunday afternoon. Nothing huge, no formal invitations, just our friends, including the almost newlyweds. I suppose you'd better explain to them we had this more or less set up, so Brit doesn't think we're stealing their thunder.\"\n\n\"That's great about our belated reception,\" she told him, but she turned quickly back to wiping down the sink area. \"She'll understand. I'll let them know tomorrow.\" As she wrung out the dish rag, she glanced at her face in the window over the sink. Daylight was almost gone, and the lower pane of glass had become a dark mirror.\n\nJace had rushed his proposal to her years ago. They had rushed their wedding. It had almost been an elopement. A whirlwind romance that had too soon become regrets and separations\u2014but at least they had Lexi. With his long international flights, she and Jace had spent too much time apart. She had tried to hide her narcolepsy from him. Too late, she'd tried to reach out to him to mend things...\n\nThe memory of that hand reaching toward Hunter in the bog leaped at her. A woman he'd loved and lost or left?\n\nShe jumped as Nick appeared in the window reflection behind her and put his hands on her shoulders.\n\n\"Are you all right, sweetheart? You said you like Brit and think they are good for each other\u2014and for Lexi.\"\n\nShe turned toward him and into his embrace, pressing her cheek against his chest where she could hear his heart beating. \"Of course they are, despite being risk takers, both of them. Jace on the hunt for bad guys, Brit protecting her beloved endangered animals.\"\n\nWishing that long-gone Hunter didn't make her think of Jace, she put her arms around Nick's waist and held hard.\n\nAt the firm the next morning, Nick kept thinking about Claire's reaction to her ex-husband's coming marriage. He hoped she was just worried about things working out well when she had to share Lexi with him and Brit for longer periods, not that she was somehow sad or reluctant he was remarrying.\n\nAnd, as fascinating as her work at Black Bog was, he could already feel it was consuming her thoughts. Well, he told himself as he sat down at his full-of-work desk, that was always the way Claire was, full-steam ahead, and he recognized that workaholic drive in himself. What was that saying about birds of a feather flocking together?\n\nHe heard a man's raised voice outside, one he couldn't place, then his secretary Cheryl's steady reply, higher pitched than usual. \"Sir, I don't know how you got past our security desk, but you cannot just go into Counselor Braun's office. No, I will not tell you which it is, and I am calling security.\"\n\nNick jumped up and headed for the door. He'd told Bronco he could come in late today because he and Nita were moving in with him and Claire for a few days. Was a police officer out here, ready to arrest Dale? Why hadn't Detective Jensen advised him of that?\n\nHe yanked the door the rest of the way open. It wasn't Jensen, not a police officer or anyone he knew. It was a red-bearded man, stocky and bald, dressed in jeans and a leather jacket over a long-sleeved flannel plaid shirt\u2014totally un-Naples, especially this time of year. At least at first glance, Nick thought the guy wasn't armed.\n\n\"May I help you, sir? I'm Nick Markwood, senior partner here.\"\n\n\"I'm looking for that bastard rat lawyer works here, kilt my sister. She said he works here.\"\n\nNick almost blurted out Cyndi Lindley's name, but that might be construed as admission Dale killed her.\n\n\"No one in this firm killed anyone, but am I to assume you are Cynthia Lindley's brother from Georgia?\"\n\n\"The same. Drove down here straight from Zebulon, an hour south of Atlanta. Got a call from the county coroner, so I'm here to claim her body. Never met her so-called intended before he dumped her, but I got more'n words for him!\"\n\n\"Please come into my office and calm down. Cheryl, we won't need security, but bring in some coffee for\u2014your name, sir?\"\n\n\"Tanner Linschwartz. It's her real last name too, but she was hell bent on getting away from her former life, out of Zebulon where she was hostessing in the Down Home Restaurant. Gone first to Atlanta and ended up here, seating people in some fancy restaurant, where she musta met her killer.\"\n\n\"Step inside,\" Nick coaxed again, gesturing toward his open office door. Finally, the man's angry stance relaxed a bit; his shoulders drooped and his fists uncurled. It was totally understandable that he'd be exhausted and on edge. Vengeful too, so Nick had to watch that. Seemingly calmer now, the man shuffled past Nick into his office.\n\n\"Cheryl, I'll leave the door open. Get hold of Bronco and have him wait out here. And,\" he whispered, \"tell Dale to clear out for now, go somewhere nearby, and I'll call him when the coast is clear.\"\n\nVisibly shaken, though she'd been through a lot around there, Cheryl nodded and jumped up to head for Dale's office just down the hall.\n\n\"Very sorry for your loss, Tanner,\" Nick told the man and sat down in the chair next to him. \"You'll take her back to Georgia for a funeral?\"\n\n\"Going to get her cremated here, then spread her ashes on the beach, in the waves. She'd've liked that, even said so once, so I'll honor that.\"\n\n\"It's unusual for someone as young as your sister\u2014\"\n\n\"Near on thirty.\"\n\n\"Yes, but to think about dying, to plan for her funeral.\"\n\n\"She was scairt of someone, and now I know who, and got a big score to settle with him. She said he might just be playing her along, that he was going places, maybe where he wouldn't want to take her. Ten to one he kilt her and put her in his dead momma's freezer till he could toss her in the swamp or somewhere. Well, I'm gonna be sure he gets caught for killing her, or if not, pays the price for what he done.\"\n\n\"So,\" Nick said, realizing he should also tell Cheryl to call Detective Jensen, \"tell me about why your sister came to Naples, what she was like. As I said, I'm sorry for the loss of such a lovely, and no doubt ambitious, person.\"\n\n\"She shoulda stayed to home and married Will MacBride, not go gallivanting off to parts unknown.\"\n\n\"So Will MacBride was pretty sad or upset she left?\"\n\n\"Can't blame him. Childhood sweethearts and all that. She gave him his ring back. I was damn ticked off too.\"\n\nThis man was right on the surface, coming in here with threats and admitting a motive for being angry with his sister, though he didn't seem the type to think of putting her in anybody's freezer. But then there was this Will MacBride and what was he thinking when he heard about Cyndi?\n\nHell, who really knew what anyone was thinking or was capable of in the heat of passion? He wished this case and Claire's new endeavor could just be calm, objective business. Like she had said last night about that Black Bog body they were calling Hunter, didn't the poor prehistoric guy's chest wounds seem symbolic of life in general? Didn't some things just tear your heart out?\n\n# 9\n\nClaire thought the arm that was slowly being uncovered by the dig team was definitely stretched out toward where Hunter had been buried. She grew impatient at the slow going as they were barely to the body's elbow, so she went back inside again.\n\nShe had been given a desk in the corner of Kris's small office. Its single, long window had a view of the bog, that is of the twisted-trunk ficus trees that surrounded it. She tried to put things out of mind and get familiar with their online system of linked laptops, but the dig kept calling to her.\n\nBy standing up to look out the window, she could see Kris still waiting, so she prepared to head back out.\n\nShe passed Andrea's office down the hall. The door was open and lights on. No Andrea. Brad was in there, though he had a separate lab area, not really an office. He was not seated but was bending over Andrea's chair, stiff armed on her desk, looking closely at her large laptop screen. At a quick glance, Claire thought a series of ancient tools or utensils were on the screen.\n\nHe must have heard or sensed her presence, because he closed the screen and turned around just as Claire started to move on.\n\n\"Hey, going back out?\" he called to her.\n\nSo he knew she had come in. She went back a few steps. \"I think they'll have reached the second body soon, so I'd like to be there for that. Are you going out too?\" She knew the answer to that, but wondered what he would say. His claustrophobia and reluctance to be on scene outside obviously wasn't a secret around here.\n\nHe came a couple of steps closer and jammed his hands in his jeans pockets, almost as if he were a kid caught at something. \"As you'll see, I don't do much of the field work. My bag is PR. Contracts. Especially funding. Actually, I have claustrophobia, so I don't like to see or even think about someone being closed in the dark heavy ground that long.\"\n\n\"I can understand,\" she said, edging away, though he came closer to the door. \"So, are you able to look at the photos they take?\"\n\n\"I look at the bodies once they're in the lab.\"\n\n\"One more thing, if you don't mind. Do you prefer being called Brad, Bradley or Senator by the staff, as I've heard all three.\"\n\n\"I answer to each and any,\" he said with a shrug and a sheepish smile as he leaned against the frame of Andrea's office door. His body language was so unlike the take-charge former Florida senator as she'd pictured him. Had she caught him at something when he was looking at Andrea's screen? \"Calling me Brad is fine. So, I know who your husband is, and see you two have had your problems with the law.\"\n\nThat was a strange way to word that, she thought. \"Only because we had some unscrupulous people trying to harm us. And then, of course, finding the body in the freezer, which gave the media a field day recently. That was pretty horrible.\"\n\n\"I hope they're right that poor woman was already dead when someone stuck her in there. Talk about claustrophobic nightmares...\"\n\nClaire almost asked him if he had such nightmares, but she was probing and presuming too much. Yet she could not believe that Brad wasn't curious enough about a new, unusual discovery in the bog to go out and at least take a look, which she intended to do right now. The grave, after all, was being opened, not closed, though she had to admit those huge, heavy ficus trees with their twisted trunks and roots seemed to press in on the entire area.\n\n\"I'd best go back out, Brad,\" she told him and walked away.\n\nShe was relieved that he had seemed not only friendly but honest. And it made her feel good he had admitted knowing Nick\u2014at least who he was, though she hoped his mention of their public troubles wouldn't cause trouble here in this supersecret environment. But then she had not been hired here to psych out the living\u2014just the dead.\n\n* * *\n\nIt was late morning by the time Detective Ken Jensen responded to Nick's call about Tanner Linschwartz having left his law office and heading to the ME's morgue to take possession of his sister's body. Nick sat him in the same chair Tanner had recently vacated. Nick thought he had settled Tanner down, maybe even convinced him that Dale had really cared for Cyndi but had just realized their marriage would not work out. Since she was a hostess at a tony restaurant in town, Tanner had to admit she could have met someone there who had hurt her, especially since he'd said, \"Yeah, she was a man magnet everywhere she went.\"\n\nGreat, just great, Nick thought, in growing frustration. That probably meant lots of possible perps to check out\u2014all of which he conveyed to Jensen right away. He also explained that Bronco and Nita Gates would be at their house for a few days if he needed further statements from them.\n\n\"Hey,\" Jensen said, finally looking up from scribbling notes, \"maybe you and Claire can take me in too so we don't have to keep meeting like this.\" His expression hovered between a grin and a grimace.\n\nNick told him, \"I hope you're not focusing on Bronco and Nita. I know you and I have a silent agreement not to tell each other how to do our jobs, but shouldn't you focus on this new loose cannon from Georgia? He's volatile and aching for a confrontation at the least.\"\n\n\"I warned the ME's office after your secretary called and have an officer there, but I can understand why he's distraught. He sounds like a real Georgia good old boy with a touch of redneck in him. Thanks for defusing him here a bit\u2014though I'm sure you were thinking he'd sidetrack me off focusing on your junior partner. Irate brother finds sister at Dale's house, they have a fight, he chokes her, stashes her body next door in a freezer to make it look like Dale did it.\"\n\nNick decided to ignore that. \"Tanner told me Cyndi had big dreams about moving up in the world,\" he told Jensen. \"She dumped a repairman boyfriend named Will MacBride, and if he's anything like his buddy Tanner, he should be a person of interest. Evidently Zebulon is just a long day's drive from here. Since Cyndi worked her way first to Atlanta and then to Naples, other men she's met along the way might need looking into. She was a magnet for men, as Tanner put it.\"\n\n\"Gotcha. By the way, Nick, as I've said before, if Claire ever wants to work part-time, the department could use a forensic psych. Seriously, I like her, trust her. Keep it in mind, okay?\"\n\n\"She's working part-time as a business consultant. With the new baby, that's enough, at least for now.\"\n\n\"Sure, I understand. If she did facial reconstruction models, we'd have a full-time job for her, but I get you want her working somewhere private where she can keep out of the trouble you two seem to find. See you soon, Nick,\" Jensen said, leaning over so they could shake hands. \"Hopefully not in court.\"\n\n* * *\n\nClaire thought the extended arm and hand of the next body looked as if she\u2014or he\u2014had wanted to touch Hunter, or even shake hands, but who knew what their customs of greeting were. The team had excavated one shoulder, but to everyone's surprise, the corpse was not turned toward Hunter. Rather the body lay flat on her or his back with the other shoulder and arm evidently reaching out in the opposite direction.\n\nDoug said, \"If this burial wasn't centuries before Christ, I'd say it's almost posed like someone on a cross. 'For all this His anger is not turned away, but His hand is stretched out still.'\"\n\n\"Another Bible verse?\" Yi Ling asked as she carefully brushed bog from the arm and Doug kept picking at tiny hardened clumps of peat with what looked like a dental pick. \"Claire, Doug is our walking, talking quote man of Bible verses. What was that one you use to warn people not to get off the planks and into the bog?\"\n\n\"Oh, the one from _Jeremiah_?\" he asked.\n\n\"I think so,\" Yi Ling said. \"I'm afraid it fits even more now that we've learned Hunter was executed or at least\u2014what's that word Andrea used? Eviscerated?\"\n\n\"I don't mean to interrupt,\" Claire said, \"but that's a possibility too\u2014that Hunter was an honored man so they kept his heart when he died.\" She sighed. \"But then, I think they would have done a better job of removing it. So what's the Bible verse, Doug?\"\n\n\"Okay,\" he said. \"Jeremiah wasn't called 'the weeping prophet' for nothing. That section goes, 'Your close friends have set upon you, and prevailed against you; your feet have sunk in the mire, and they have turned away again.'\"\n\n\"That possibly fits here,\" Claire said, surprised her voice was shaky. \"Not only the sunk-in-mire part but the idea that Hunter's own friends might have turned against him. But for what?\"\n\nYi Ling said, \"How and what and why\u2014that's the name of the game around here. Well, we give you the evidence, and you come up with the answers, that's your thing here, Claire.\"\n\n\"Wish I did have answers instead of just more questions.\" She realized she was trembling. Was she just excited to see what they would turn up here, had her awkward encounter with Brad shaken her\u2014or was it because she felt so bad for Hunter? Ridiculous to get emotionally involved with a millennia-long dead man.\n\n\"Someone should go get Andrea,\" Yi Ling said, without looking up again. \"She always likes to be here when we uncover the body itself. It's her tradition to name each person, and I hate to just summon her on the walkie-talkie for something this important.\"\n\n\"I can go,\" Claire volunteered, \"but I didn't see her in her office. I just saw Brad.\"\n\n\"I think,\" Aaron put in, \"she's back in the room with the brains. As for the senator, he never likes to come out where we're working.\"\n\nAnd now she knew why, Claire thought, as she carefully walked the planks back toward the main walkway and the building. Stopping for a moment, she glanced back at the trio of archaeologists. Even though Yi Ling was so petite, she held her own with the men. Kris had said that Yi Ling, last name Chung, was Chinese, a graduate of Peking University. She had written Andrea for the opportunity to study with her, then had stayed on.\n\nClaire walked toward the building. Even in late morning, with the sun pretty much overhead, the heavy limbs and leaves of the ficus trees made it feel darkish here. Besides, clouds had covered the sun, and she thought she even heard a rumble of distant thunder out over the gulf.\n\nShe went in the bog access entryway and walked down the hall toward Andrea's lab. The door was closed, so she knocked.\n\n\"Who?\" Andrea called out.\n\n\"It's Claire. The team would like you outside since they are close to the body.\"\n\n\"Step in a minute.\"\n\nClaire opened the door and went in. On a screen, many times magnified, was an image of Hunter's dagger, at least one side of it. For the first time, Claire could see that it was flint, carefully chipped to a sharp blade and point\u2014and it had some kind of pattern carved in the hilt.\n\n\"Surely, that isn't some sort of writing,\" Claire said as she squinted to study the strange shapes.\n\n\"An attempt at artwork, I think, but I can't tell of what. The game he hunted maybe\u2014abstract, of course,\" Andrea said with a shrug.\n\n\"Stick figures. People?\"\n\n\"Some sort of message from the grave. This blow-up of it could blow up the world of archaeology,\" she added in an awed voice as she continued to stare at the image.\n\n\"The motto for my Clear Path website has been 'The dead still talk if you know how to listen.' That's true here too.\"\n\n\"I remember that, from when I was researching you,\" she admitted.\n\n\"In this case, the motto should be 'The dead still talk if you know how to see,'\" Claire said.\n\nAndrea finally looked her way. \"Somehow, some way, we will get the answers from these bog people and then be able to share all we've learned and found with others. I will do anything it takes to accomplish that as my life's work!\"\n\nStaring back at the intense woman, Claire nodded. What else was there to say, at least right now, but for one thing. \"Are there photos of the other side of the dagger?\"\n\n\"This is the decorated side. Well, time to study this later.\" She turned off her projector. \"Let's go outside and see who was reaching out to poor, mutilated Hunter, just the way we would love to reach out to all these poor, buried, long-lost souls.\"\n\n* * *\n\n\"You might know it started to rain right then,\" Claire told Nick that evening when they finally had both Trey and Lexi asleep and their guests, Bronco and Nita, had gone up to their bedroom. \"And just when they were ready to uncover the next body. Their policy is to just cover what they have and get back to it tomorrow\u2014if it isn't raining then. Andrea was as disappointed as I was. The trees that edge the bog protected us for a while, but then dripped water on us too, so that was that.\"\n\n\"It's like a treasure hunt, isn't it?\" Nick asked as he added another name to the guest list for their Sunday afternoon reception. The club was providing the catering, so things were going smoothly. Claire would extend the invitation to friends by phone tomorrow before she went to work. Nick would ask his staff at the firm.\n\n\"Yes, a treasure hunt's a good analogy,\" she admitted. \"I never thought I would get so involved with long-dead people, though that's exactly what I've been hired to do.\"\n\n\"It's a good thing you're doing that part-time, because I think you can use a break already. Today will be your last day there until Monday, so we can drop by the club and check out the room, talk to the staff. I was at a function there, but it was a couple of years ago\u2014pre-Claire.\"\n\n\"Sounds good. Let's do that in the morning because Kris is going with me to dress shops in the afternoon, probably to get an outfit for herself too, since I invited her. Lexi loves her dress from Nita and Bronco's wedding, so she'll wear that. Sorry for the fashion commentary. Any more word on poor Cynthia's brother? From what you said, he sounds like a loose cannon.\"\n\n\"Since I've been able to informally depose him, he's Jensen's problem now, but, yeah, he bears watching. I assume he'll be busy with her cremation tomorrow or whenever they release the body to him. Sad to think that beautiful woman will become ashes in the wind and water.\"\n\nBlinking back tears, Claire nodded and reached out to cover Nick's hand with hers.\n\n# 10\n\nUnder their big umbrella in the steady, gentle rain, Claire held Nick's arm as they went into the Orange Grove Country Club. They had been invited to look over the facilities for their much-belated wedding reception. Despite all Nick had to do at the office, he was planning to go in late. At least this off-and-on rainy weather was supposed to clear up soon.\n\n\"If it wasn't for this umbrella, I could still carry you over the threshold,\" he teased as he opened the door for her. He shook out the umbrella and left it with the others on a rack.\n\n\"I'm happy to say your toting me into our bedroom now and then is quite enough. Besides, it wouldn't do to have the man of the house and my favorite lawyer-at-large hurt his back two days before our party.\"\n\nThey were greeted by Mary Ann Manning, special occasions director, who showed them the Blue Room they had reserved. It had adequate seating for forty as well as one long buffet table and several other tables scattered throughout the area.\n\n\"The Blue Room comes with the air-conditioned, glassed-in verandah which runs along the side,\" Mary Ann told them, with a sweep of her hand at the long space outside which also had chairs and small tables. \"In good weather, we open the sliding doors, and it becomes one large room. We could do that, put the cake and food in here, then let your guests go either in or out. We have special laminated glass on this long window to cut the UV rays. Even golf balls hitting the glass won't make it shatter,\" she said with a little laugh. \"And, as you can see, there is a lovely view of the eighteenth green and the orange grove surrounding the area.\"\n\n\"I read there's been a dispute over the property lines,\" Nick said.\n\n\"I'm afraid so. The original orchard owners are demanding an exorbitant price for the small amount of extra land we want to lengthen the practice driving range. I'm afraid they have even pulled some dirty tricks, like smashing rotten oranges on some of the greens, even stuffing them in the holes. Someone has lobbed oranges at foursomes out by the seventh and eighth holes. But slightly higher tech, someone has buzzed golfers with a drone. The grove owners have denied doing that. Dare I say we could use some legal advice on how best to proceed?\"\n\nNick turned and smiled at the dark-haired woman. \"Dare I say, I could recommend someone to help in exchange for all the help you're giving us here? It would have to be a junior partner right now, but the senior partners ultimately oversee everything.\"\n\n\"I'll tell our board,\" she said with a nod. \"Now, please, both of you, explain how you would like things laid out on the verandah, and our caterers will have everything, including that belated wedding cake, ready for your guests at four on your big day.\"\n\nAs they spent a half hour with her, Claire realized how excited she was over all this. As much as she loved being married to Nick, their wedding had been a nightmare, forced on them by their sworn enemy. But this would all make up for it, choosing a cake, buying a dress...\n\n\"I said I've got to run, sweetheart,\" Nick told her and pecked a kiss on her cheek as he walked her to her car, since they'd driven separately. \"Have a good time at lunch and dress shopping with Kris.\"\n\n\"Will do. And I'll text you or leave you a message if we get the call that the rain has let up enough that the team will be ready to disinter the second body in Hunter's grave\u2014their grave, I mean. It's custom that everyone's there for that\u2014except for Senator Vance\u2014unless there is some real conflict.\"\n\nConflict. Was there some sort of unspoken conflict she was picking up between the senator and Andrea, or was she just reading too much into it?\n\nNick gave her another quick kiss before she got in her car. \"You do realize,\" he said, leaning down with the umbrella still in his hand to open the car door for her, \"that if someone overheard us talking graves and bodies, they would either think we were nuts or call the cops. See you, amateur archaeologist.\"\n\n\"You look great in that!\" Kris assured Claire as they met in the mirrored central aisle between their separate dressing rooms in Talbot's at the Waterside Shops in Naples. \"The color goes really well with your red hair. Good thing I'm not color-blind. Being face blind is bad enough,\" she added with a sigh.\n\n\"I know, but you do beautifully, handling it. Now, look at this. I love the cut out pattern on the shoulders and back neckline.\" Claire tried to get her back on track and she turned to glance in the mirror again. \"I have the perfect earrings and necklace for this light green. Do you think I'd need matching shoes or just white or beige?\"\n\n\"Either light color. I think I'm going with the pink-and-navy suit, but probably won't wear the jacket to the reception.\"\n\n\"This kind of grabs my hips in back,\" Claire said, still turning. \"I'm going out to see if I should go up one size, because the bust is a wee bit tight too. I guess that's what having a baby does to you. I was so much younger when Lexi came along.\"\n\nSince the saleswoman who had been helping them was ringing up a customer, Claire went to the well-lit rack by the front window to check for a larger size in the same dress. And found herself staring through the glass at someone who was looking in. The other person seemed to startle, stared, then turned quickly away and hurried off.\n\nHadn't she seen that same person more than once walking past the restaurant window earlier? They'd had lunch at Brio's before starting their shopping.\n\nThe surprise of it made Claire's heartbeat kick up. Was that a man or a woman? A Tampa Bay Buccaneers baseball cap and big sunglasses hid the forehead and eyes, but she could see chin-length hair. That mustache looked fake. What _did_ that person look like? It made her feel she was face blind too and sympathized with Kris even more.\n\nAs for clothing, the person wore loose slacks and a pullover, both in black. Well, what did it matter if someone dressed weird? she scolded herself and went back in to the dressing rooms.\n\n\"Lucky you, to find that,\" Kris greeted her. She'd changed back into her casual clothes. \"If I go up a size, I have no excuse of a new baby. Which reminds me, we never found a body of a bog woman who was pregnant or had just given birth, so I was hypothesizing that they seemed to survive childbirth\u2014sturdy women. I'm sure in general they got enough exercise. Yet so many of them seemed to chew those natural painkillers.\"\n\n\"Don't so much as say the word _killers_ ,\" Claire said from under the dress as she pulled it up over her head.\n\nShe pictured again the person looking through the windows. Well, so what? Didn't lots of people window-shop? Maybe the person was eccentric or mentally off or homeless and just roved the shopping center.\n\n\"By the way,\" she added, \"I noticed the rain has stopped outside.\"\n\n\"Hope at Black Bog too. Can't wait to see Hunter's companion.\"\n\nClaire almost told her about the person peering into the store, one she couldn't even identify by sex. That made her wonder if the body in the bog they would see next was a female.\n\n\"Look,\" she said, trying to keep her mind on the present, \"this dress fits much better.\"\n\n\"You know,\" Kris said as they finally gathered their purchases, \"I never have trouble recalling what someone wears. So why the curse of the face blindness?\"\n\nClaire took her arm. \"Do you think you can recall the bog faces so well because you recognize them by their clothing or the position of their bodies?\"\n\n\"It can't be by the artifacts with them, because Andrea always takes those to study right away, and that's one thing the senator is totally intrigued by. But as for your psyching out my coping mechanisms, why is my best longtime friend a sort of shrink, even if a forensic one?\" Kris said with a mock sigh, looking upward.\n\nThey bought their outfits and were heading out toward a shoe store in the open-air mall when Kris's cell phone and Claire's rang almost at the same time. \"Oh-oh,\" Kris said, as she produced hers from her pants pocket. \"I\u2014we\u2014may have to head for the bog.\"\n\nKris took the call fast. Claire could hear Andrea's voice on the other end as she dug her phone out of her purse. \"I'm in town so I could call you. Surely the rain stopped at the bog too. I'm heading there and expect to see you there quickly.\" They both had their cars, and Claire skimmed the parking lot for hers just two rows over from Kris's.\n\nAnd there, staring at her then moving quickly away, was that androgynous person again. Were she and Kris\u2014or just one of them\u2014being followed? But by whom and why? It was probably just a random thing and they had to get going. Her past had made her paranoid. She didn't want to alarm Kris, but if that person appeared again, she'd point him or her out.\n\n\"Let's go,\" Kris said, punching off, as Claire took her own phone call. Hers was from Yi Ling from the dig team who was also heading for the bog.\n\n\"Are you coming out, even though you're not scheduled?\" Yi Ling asked excitedly. \"I think you're right that it's a woman. The rain has quit, and we're going to excavate around her right away. I swear she has breasts. And maybe there's a string of bodies there, linked somehow.\"\n\n\"We're on our way,\" Claire told her even as Kris jumped in her car.\n\n\"Follow me, okay?\" Kris called to her as she rolled her window down.\n\n\"After you drive around to my car!\" Claire called back as she began to cut through cars in the next rows. She had no intention of meeting that person\u2014where had he or she disappeared to? Surely Nick wouldn't assign someone to keep an eye on her for safety's sake. It must be coincidence, but she'd learned long ago to expect no such thing\u2014especially if you dabbled in danger like finding a body in a freezer.\n\nA peek inside her car showed no one in the back seat, nothing disturbed. She put the box with her new dress on the passenger side. Should she show Nick her dress, or surprise him, like a bride at her wedding?\n\nOh, no. That's right! Dale Braun was coming for dinner tonight\u2014and for a conversational interrogation she was supposed to assess right along with Nick's observations. So she couldn't stay at Black Bog too long. She'd promised Nick she'd help, and family still had to come first.\n\nBut she was so certain, so hopeful after all that she'd been through, that her life was all\u2014well, almost\u2014normal now.\n\n* * *\n\nJace waited for his Stingray contact and friend, Mitch Blakeman, at the huge door of the hangar at the Marco Island Airport just as Mitch was taxiing off the runway toward him. Jace watched as \"Falcon,\" a former Marine pilot, shut down the single-engine so-called crop dusting plane inside and walked to meet him. They had both been in the local skies lately, spraying for insect pests, including Zika virus mosquitoes. But the equipment in their planes actually allowed them to troll for drug runners' locations by tracking their cell phones. Huge stashes of drugs were supposed to be coming in from Mexico. Since a recent murder of a Stingray pilot in California, their job had become a lot more tense.\n\n\"Sometimes I feel we're on a par with the Wright brothers, in the small old planes\u2014sitting ducks,\" Jace called to Mitch, with a glance back at the single-engine propeller aircraft that passed for what local farmers would use. \"I'm homesick for a fighter jet and some speed. How about you?\"\n\n\"Yeah, a baseball hat instead of a helmet's not really my style,\" Mitch said as he fell into step with Jace, twisting and nearly wadding up his billed cap. Even as they passed through the small airport, they kept their aviator sunglasses on. \"So, have you done any more thinking about my proposal to fly for the hurricane hunters? A WC-130 turboprop plane sounds good to me, even though it's not a jet. Those big babies fly right through the eyewall of the storm, so the excitement's sure there, all in a day's work. Besides, like I said, there are signs the bad boys running drugs into Southwest Florida might be onto us. Time to face a storm instead. So, did you tell your Brittany yet?\"\n\n\"Not yet. Got formally engaged, though, and set a date. Finally gave her a ring. We've had each other's back for a while, and I'd appreciate it if you'd be my best man.\"\n\n\"I'm honored,\" Mitch said, shaking his hand, then clapping him on the shoulder as they emerged from the small airport terminal. They sat at a picnic table on the other side of a fence from the runway. \"I think we're both ready to move on to new careers\u2014new lives. But will Brittany be okay with the new gig? I mean, our current flights can be dangerous if our cover gets blown, but eleven-hour-long flights into insane winds while being responsible for a crew of up to sixteen people...\"\n\n\"Yeah. On the other hand, I miss flying five hundred people over the Pacific. But you know, I have a friend named Bronco whose new wife went ballistic when he kept something back from her\u2014\"\n\n\"Are you nuts? You mean you proposed to Brittany but didn't tell her\u2014ask her\u2014about flying into hurricanes?\"\n\n\"Keep it down,\" Jace whispered as a man and wife with two small kids walked past them into the airport. \"We're going to a wedding reception\u2014my ex's, no less\u2014and Brit will be all emotional over that, so I'll bring it up when we're alone after.\"\n\nMitch heaved a sigh that seemed to deflate his always rigid military posture. He was lanky whereas Jace was more solidly built. Mitch slumped a bit, elbows on knees, staring at the grass. The sun on his close-cropped light blond hair made it seem to glow.\n\n\"I've got to tell you something else,\" Mitch said. \"About our current gig\u2014about that California pilot an explosive drone shot out of the sky. His girlfriend was with him. I didn't learn that right away\u2014didn't read the papers from there. But just out of curiosity, I looked it up online yesterday.\"\n\n\"Damn! Yeah, I'd better get out of Stingray. More than once Brit wanted to go up with me, and I said later she could. Listen, I figure once we change jobs we can get an article in the paper to announce, not that we're leaving Stingray, but that two local guys are now consulting and flying full-time to protect the entire state from hurricane dangers. Okay, Mitch?\"\n\n\"Yeah, yeah. I'm in.\"\n\n\"Doubly terrible what happened. I don't want to run or be a coward. I've risked my life plenty of times for the right reason, but I'm not going to endanger Brit\u2014or let my chance at happiness go.\"\n\n\"I wish I had someone. I'm tired of going home to an empty house after missions. Like I said, I'd be honored to be your best man, and I'm taking what you said as a yes for your becoming a hurricane hunter with me. Otherwise, protocol would have to make me move on without you so we wouldn't be seen together where the drug boys could spot us\u2014so, hell,\" he said, turning his head to look at Jace again. \"Here we are like sitting ducks at the airport we fly out of, on the chance that someone wants to knock us off. My marine sergeant would have killed me himself in the old days for something stupid like this\u2014hoo-ya, man!\"\n\nThey smacked raised hands. As Claire had said to tease him in the old days, Jace told Mitch, \"Pilots of a feather fly together. And know what? I think I might just have the girl for you, a sharp blonde I knew years ago in college\u2014no, I never dated her, so don't look at me like that, but she did introduce me to my ex, Claire. Thing is though she's into digging up bodies.\"\n\n\"What in the...\"\n\n\"She's an archaeologist. You look up, she looks down. Tell you more if you meet me at Snook Inn. Let's go get a brew to toast our new careers, and I'll explain. For once, my man, you're buying.\"\n\n* * *\n\nClaire and Kris both ignored the fact they were going to get their clothes bog-mud dirty. Even the raised planks they walked were water-and peat-speckled. Although the rain had stopped, drops of water peppered them from sodden leaves when the wind blew. But the plastic tarp that had been over the dig site had kept the grave quite dry. Doug, Aaron and Yi Ling were in their usual positions, half in and half out of the grave. Andrea was bending over them, somehow in another immaculate white outfit, even her shoes.\n\n\"Oh, I see what you mean!\" Claire cried as she sidled closer to get a better angle. \"Yes, that arm must be a woman's, and I see the swell of her breast.\"\n\n\"And,\" Kris said, \"her flat-on-her-back position and that other raised shoulder do indicate her other arm must be stretched out.\"\n\nAndrea said, \"Obviously she and Hunter were somehow related, and not just in death. Claire, a challenge for you to discern and hypothesize how. Is that\u2014Is she wearing a pelt? Perhaps she was Hunter's wife.\"\n\n\"Yeah, a pelt,\" Aaron said, \"though it's so muddy it's hard to tell. But I can see she has shoes still on her feet that look as if they might be deer hide\u2014kind of shaped like modern bedroom slippers.\"\n\nClaire's back and neck ached from leaning forward by the time they uncovered the woman's face. She looked\u2014well, young. Strangely, still pretty yet either she had died distressed or the weight of the bog had distorted her features.\n\n\"Look,\" Claire cried. \"That leather thong or whatever it is. Those stones and shells on it. It must be a necklace that came off her neck! Some smashed shells but those must have been pretty, polished stone.\"\n\n\"I see it,\" Andrea said. \"I'm going to radio Brad and tell him we have another important artifact. We'll treat it carefully and analyze it thoroughly. Put it in one of your plastic packages\u2014carefully so it doesn't come apart any more\u2014and hand it to me, Yi Ling.\"\n\nDespite the peat and mud obscuring some of the stones and shells, Claire thought it must be a real work of prehistoric art. But then, Andrea was probably the kind that had to clean it first, study it and label it before coming to that conclusion.\n\n\"All of you know I name each of our finds.\" Andrea's sharp voice cut into Claire's agonizing. \"It's their welcome to this modern world, sort of a naming ceremony. I am naming this one Reaching Woman.\"\n\nAren't all women reaching for something? Claire thought. But when such a rare artifact was found, didn't Andrea usually name the body after the artifact versus the body's position? The early bodies she had seen were Woman with Mirror, Woman with Medicine in Stomach. __ Why not Woman with Necklace here? It must be because Andrea valued, above all, the relationship among these three people, and psyching that out was now Claire's assignment\u2014and burden. All this seemed not just fascinating but almost sacred to her, and perhaps it did to Andrea too.\n\nAs they excavated Reaching Woman's head and torso carefully, Claire asked, \"She's all there, isn't she? I mean, not like Hunter with his chest hollowed out?\"\n\nWithout looking up, Aaron said, \"I can tell she's all here, not caved in like Hunter, so\u2014\"\n\nBut he stopped in midthought. Yi Ling gave a grunt that sounded as if she'd been punched in the stomach. \"Damn!\" Doug said. \"Do you believe it? What are we in here, murderer's row\u2014or murdered row?\"\n\n\"What is it?\" Andrea demanded, leaning closer.\n\nClaire saw there was a large slit in the bodice of the woman's clothing. The poor thing's heart might be there, but it had a flint dagger much like Hunter's in it, protruding upward from the left side of her chest.\n\n# 11\n\nClaire was dying to tell Nick all about the second executed or murdered person they'd uncovered today at Black Bog. But he came in late, and she couldn't unload all that on him while helping Nita prepare dinner. Besides, their guest Dale Braun pulled into their driveway just a few minutes after Nick.\n\n\"Bronco and I are gonna eat our fajitas out in the gazebo,\" Nita told Claire. \"I'll check on Trey now and then, and we'll feed Lexi, give you three time to talk.\"\n\nShe lowered her voice though Nick and Dale were out in the Florida room having a martini\u2014Claire's idea to get Dale to relax a bit so he might be more conversational. Their goal tonight, besides assuring him as a coworker and client that the firm was working hard for him, was to learn as much as they could about Cyndi and their relationship. Nick had said that if Cyndi was anything like her brother, she'd been as different from Dale in background and personality as day and night.\n\n\"Lexi will love eating out with you two,\" Claire told Nita. \"Just be sure she's got mosquito spray on her. Those little nasties are out early this year.\"\n\n\"Oh, _si_ , all three of us get sprayed, even though Bronco always says he's too orn'ry to get bites. You know, Claire, helping with your family\u2014kinda being a part of it,\" she said, turning away from the stove, \"it really makes me want children even more.\"\n\nClaire put her hand on Nita's arm. \"You will have them, my friend. So sorry your plans took a hit when we found the body in the freezer.\"\n\nNita sniffed back tears. \"It must have been so awful for her\u2014Cyndi. I\u2014I swear she was screaming, but it was a silent scream with no one to help.\"\n\nBecause, Claire almost said, whatever sort of person Cyndi was, whatever Dale or her brother or anyone else said about her, she was a woman who was in danger. Such a sad, too common story even in this modern world. And then there was Reaching Woman, newly born from the bog but stabbed in the heart. Both that ancient female and Cyndi had needed someone to reach out to them, and Dale Braun had let Cyndi go. She just prayed he hadn't hurt her more than that.\n\n* * *\n\n\"Delicious food,\" Dale said, wiping salsa off his mouth. \"If I'd known Nita was a great cook, I'd have reduced the price of their house and had her deliver a meal now and then. But, yeah, I'm out of there as soon as possible, a decision I made way before the\u2014the tragedy. Gonna move way downtown, I think, turn into metro man and let somebody else sell the houses on the land I inherited, now that my mother's gone too. But I'm not even staying there now.\"\n\nNick said, \"I asked this yesterday, but you haven't seen Cyndi's brother, Tanner, have you? At least he hasn't been back to the office.\"\n\n\"No, but that's one big reason I'm staying in the hotel downtown right now. I don't need him cornering me somewhere, especially after what you said. I'm hoping the guy scatters Cyndi's ashes on the beach and heads north. Tanner is just one more piece of proof I never should have fallen for her.\"\n\n\"Nick, I forgot to ask,\" Claire said. \"What does Tanner look like?\"\n\n\"Big and burly. Shaved head. Not to stereotype but a typical good-old Southern boy.\"\n\nDefinitely not, she thought, the person who might have been stalking her or Kris. Claire realized this was not the time to think about it, but she couldn't forget that person watching, perhaps following her. It wasn't coincidence and she wanted to tell Nick, but that, like her news of the new bog body, would have to wait. Besides, all she needed was him thinking she should stay home all the time.\n\n\"I've seen pictures of Tanner,\" Dale said, rolling his eyes. \"The guy was standing in front of the courthouse in their Georgia hometown with a statue of Jefferson Davis and the Stars and Bars flag flying behind him.\"\n\n\"Oh, great,\" Nick said, frowning and shaking his head. \"So what else made you decide to call off the engagement? I consider you to be a pretty discerning, careful guy, so why did you fall for her in the first place?\"\n\n\"Infatuation, I guess. Well, it\u2014she\u2014kind of swept me away. Obviously, she's\u2014she was\u2014beautiful, so feminine, flirty. She seemed needy, looked up to me, all that.\"\n\nClaire just nodded whenever he glanced her way. She intended to let Nick guide this conversation, but he'd asked her to weigh in if she saw a good opening to learn more. All she could sense so far was that Dale was suddenly more nervous, almost stuttering, but that was understandable. Still, she wasn't in total agreement with Nick that his junior partner had to be innocent.\n\nIn the awkward silence, Claire said, \"I'll bet there was some incident that made you realize the two of you were from different backgrounds.\"\n\n\"Yeah, no kidding. I ignored that at first and shouldn't have. Two things. We double-dated with one of the other junior partners\u2014it was Brent Atkins, Nick\u2014and that night it hit me that Cyndi had no knowledge or interest in things, like\u2014well, even current events, let alone culture, beyond Hollywood personalities. She thought the Kardashians were the royalty of America. I don't mean to sound like a snob, but I am ambitious in my important and damn serious profession. She could hardly talk about something else over dinner besides what someone at the next table was wearing.\"\n\n\"I understand,\" Nick put in.\n\n\"Of course, she was ambitious too in her own way\u2014just not my way. I know,\" Dale rushed on, gripping his hands together so hard his fingers went white, \"here we're both from the deep, deep South, but her small-town Georgia background just didn't mesh with mine. I finally came to my senses that night\u2014the double date.\"\n\nThere was a long pause. He stared at his hands, then folded his arms over his chest as if to hide or protect himself, classic \"I'm guilty\" body language. And his reference to \"that night.\" Was that also the night he broke it off with her?\n\n\"We know this is so hard, Dale,\" Claire assured him, leaning slightly forward. \"That night you told her your engagement was off\u2014when was that?\"\n\n\"I told Detective Jensen and Nick it was just before my mother died.\"\n\n\"Then she was seen after that, surely,\" Nick put in. \"You even told Jensen that Cyndi went to say goodbye to your mother.\"\n\n\"Right. Mother didn't like her, so that would have been good news. That is, if Mother didn't think she was someone else.\"\n\n\"You mean she might have thought Cyndi was someone else because of her dementia?\" Claire prompted.\n\n\"Right. The thing is,\" he plunged on after taking a breath, \"you know, that double-date night Cyndi even made a couple of comments about her family once owning slaves way back. She said she could 'use a couple of those.' Just a joke, I guess, but enlightening and objectionable. Politics\u2014I learned not to go there with her. That's what I meant by saying I realized\u2014too late\u2014it just wouldn't work out.\"\n\nHe sighed, almost seeming to deflate, still gripping himself with crossed arms as if trying to hold himself up. \"Besides all that, when I showed her the old ruined mansion my great-uncle had built when he fled here from Germany just after World War II, she said she absolutely wanted a place like that\u2014maybe rebuild our home right there. She thought my heritage was great\u2014probably thought I was rich too\u2014even though I've tried to keep that low key.\"\n\nNick visibly startled. \"You mean that ruined mansion is still in the picture, still a concern?\"\n\n\"Wait,\" Claire said, hoping to head off Nick's frustration. \"I'm evidently missing something here. You own a ruined mansion? And you've tried to keep your German ancestor low key because they fled Germany?\"\n\n\"Yeah. Right near the end of World War II. It's all water over the dam\u2014way past, so I shouldn't even bring it up. But here's the thing. My great-uncle, Wilhelm Braun, who went by the name Will Braun here, came from Germany with a lot of money near the end of the war.\"\n\n\"Go on,\" Nick prompted when he hesitated.\n\nDale nodded, and plunged on talking fast as if he had to get it all out. \"So Uncle Will, like many others, was trying to establish himself as a solid citizen here, because he had been a Nazi, pretty top ranking. The name of the game in Uncle Will's life was to bury the past, though I knew nothing about this until after he died. See, after World War II, many of Hitler's elite SS fled to South America, but Will came to South Florida, a less settled place in the 1940s. He was wily\u2014knew not to go where the others went and could be found, not safety in numbers but safety in being a loner.\"\n\nNick looked riveted. Dale paused and took a huge breath. Claire nodded. She knew there had to be more coming.\n\n\"As I said, Uncle Will had money,\" Dale went on, frowning. \"He built a sort of Southern plantation on his large estate, the land on which I now have a house, where my mother lived and where Bronco and Nita bought their dream home.\"\n\nHe cleared his throat, hunched over with his elbows on the table and rested his chin on the heels of his hands, almost as if he'd like to gag himself, though he kept talking.\n\n\"What's left of the mansion sits at the far back of the properties that were built on the site after Uncle Will died. Of course, it was a family secret he'd been a Nazi, in favor with Hitler, high up. It's amazing he escaped arrest and a trial when the Americans and Russians came crashing into Berlin and rounded up war criminals. He was just plain lucky that no Nazi hunters tracked him down later.\"\n\nClaire said, \"Of course, all that is safe with us. He's long gone. The sins of the fathers\u2014and great-uncles\u2014should not be visited upon the children.\"\n\nHe nodded. \"Before he died twenty plus years ago, he told me never to call his fortune dirty or tainted money. I didn't but I'm sure it was _Reich_ money.\"\n\nSilence reigned for a moment. Blood money, Claire would have called it, but she didn't want to stifle these revelations now. But how could any of this tie into Cyndi's murder? Could someone have known about Dale's family's Nazi ties and tried to frame him for a murder, using poor Cyndi as the sacrificial lamb? No, she was reading too much in here, creating fiction instead of finding facts.\n\n\"So he'd worked closely with Hitler?\" Nick asked. He cleared his throat. He looked and sounded stunned.\n\n\"Yes, but there's more,\" Dale admitted. \"My last name\u2014his last name, Braun. Uncle Will was related to Hitler through marriage.\"\n\nClaire's mind raced again. \"But isn't it true Hitler only married right before he and his bride killed themselves so the Allied forces wouldn't take them alive?\"\n\nDale nodded, kept nodding. \"See, I shouldn't have shared all that with Cyndi. You no doubt see it for what it was\u2014something to be hidden. To her it meant fame\u2014she thought I should write a book, make a big deal of it to get publicity, TV interviews, a movie contract, and she'd be right there with me in the limelight.\"\n\nNick just kept nodding, so Claire said, \"Tell us how he was related to Hitler's wife.\"\n\n\"So here's the deal,\" Dale said, sitting up straight at last. \"My great-uncle was the brother of the father of Hitler's longtime mistress Eva Braun whom _Der F\u00fchrer_ married just three days before their deaths. Cyndi wanted to use that connection to make hay, as she put it, wanted me to rebuild and open up the Braun mansion to the public, give it back its German name. Frankly, I was appalled, saw her for what she was. I told her not to go out there to see the place but then caught her there. But damn it, I didn't kill her, though I guess this would give Detective Jensen and his cohorts a lot more ammo. I'm not an idiot. I realize my wanting to stop her stupid ideas is another motive for murder.\"\n\nHe put his forehead in his hands, staring down at his empty plate. \"I'm just telling you two this now\u2014that seeing that side of her hurt and infuriated me, and I threatened her not to exploit my family's legacy. I told her it was over, but\u2014I repeat,\" he said, looking up at Nick and then Claire, \"I did\u2014not\u2014kill\u2014her!\"\n\n\"But maybe she told someone else about your family,\" Claire whispered. \"Which, I suppose might mean someone could try to blackmail you. Have you ever felt or noticed you were being followed?\"\n\nHe shook his head. \"I just want to be proud of my name\u2014make my name my own,\" Dale said with a sniff as Nick reached across the corner of the table to put a hand on his shoulder. \"I don't want it dragged through the mud, fame as she put it, a curse as I thought of it. I am not my great-uncle even if I'm stuck with the remnants of his haunted mansion\u2014well, not haunted, except all this haunts me.\"\n\n\"I can see why,\" Claire said. \"You're like an innocent bystander in someone else's wreck. So Cyndi went out to see the remnants of the mansion without your permission or knowledge at some point, then maybe even again later, maybe the day she was killed. Have there ever been vagrants or just nosy people there? You said it's not well known but...\"\n\n\"A lot of area people used to know about the place, but\u2014I think\u2014not its sordid past. I suppose it was once a real attraction on Halloween. The place has been vandalized more than once. I hate it myself\u2014can't bear to go there. But if you want to see it, I'll take you. In broad daylight.\"\n\n\"Why didn't you just have the rest of it torn down?\" Nick asked.\n\n\"Did you ever check out the price of having anything razed and hauled away? Besides, I have to admit, as much as the place feels like the corpse of a family curse sometimes, there's just something about it that whispers 'hands off!'\"\n\nNick said, \"How about tomorrow morning? We need to get a take on this, then consider telling Jensen about it before he learns some other way. Sorry, but he should even know about the Hitler and Eva Braun connections, because it may all come out anyway. Was there any way Cyndi could have known you were well-to-do and come on to you for your money at first?\"\n\n\"I don't think she knew, not at first,\" he said, looking and sounding hurt. \"The houses Mother and I lived in are middle-class at best, though I took Cyndi out to good restaurants, took her on a one-week cruise.\"\n\nRising, Nick said, \"I think this is important. Claire. You aren't working Saturday morning, are you?\"\n\n\"Not scheduled, but\u2014you know\u2014on call if something else turns up. But this is important too, and I want to go with you.\"\n\n\"Right. Dale, we'll meet at your house and go together to take a look at the ruins of the place. What did you say its original name was?\"\n\n\"In German, _Verdrehte Baume_. It means Twisted Trees.\"\n\nClaire pictured the ancient trees surrounding Black Bog. Twisted trunks like the twisted lives of the living and the dead.\n\nLater, they both walked Dale out to his car in the driveway. Nick was grateful that they had managed to settle him down over dessert. Dale had even asked to speak with Bronco and Nita. He'd thanked Nita for the excellent meal and tried to reassure them that their house and neighborhood would survive the scandal. He even offered them some money to help furnish the house, but Nick could tell that Bronco, and especially Nita, were reluctant to accept\u2014or to even move in there. But just wait, Nick thought, until they tried to resell a place where a woman's body had been stashed, though the cops and forensic techs had found no evidence she'd been strangled there too.\n\n\"Oh, the full moon's out,\" Claire said. \"The clear sky gives me hope the rain will stay away for a while. And look at the reflection of it on your car.\"\n\nBut it wasn't that. The silvery dust on the dark trunk, top, hood and front windshield glittered coldly.\n\n\"Dew?\" Nick asked, as if to himself. \"I smell ashes. Maybe someone had a barbeque or backyard fire, and it blew over here...\"\n\nDale pulled out his car keys and with the other hand took a long swipe of the stuff on the top of his car's hood. \"It _is_ ashes,\" he said, smelling, then trying to look at his hand in the semidarkness. \"Heavy here, but not on the driveway or grass.\"\n\nNick heard Claire gasp. It hit him then too, even as Dale seemed to realize what this might be. The poor guy sank to his knees in the grass and put his head in his hands. Bronco, who must have been watching from the house, ran out. \"Boss, what's that stuff? It smells ashy.\"\n\nNick ordered, \"Everyone, back into the house. Now! Dale, don't wipe that off your hand until we get a sample. Bronco, turn on the porch light and keep an eye out the window. Everybody, move!\"\n\nBronco helped Dale to his feet. Claire seized Nick's hand as they ran inside. Nick told her, \"Got to call Detective Jensen. This may be a trick or some kind of threat. Maybe Cyndi's brother couldn't find Dale, but could find us\u2014and then found Dale. Maybe a very bitter brother didn't scatter his dead sister's ashes in the Gulf of Mexico after all.\"\n\n# 12\n\nClaire perched on her vanity stool in the steamed-up bathroom as Nick took a shower that night. Although he hadn't let on until Dale left, he was steamed up in another way, and Claire was trying to calm him down. So unlike him, he'd been on a tirade and hadn't quit. His voice echoed from behind the frosted-glass shower door. It was nearly midnight. She'd taken her narcolepsy meds but felt dead on her feet. Jensen and his forensic evidence team had finally left with most of the ashes, which a rising breeze had not pulled away. At least it hadn't rained.\n\n\"Nick,\" she tried another tactic, \"at least the police now have another suspect besides Dale. If those ashes could be Cyndi's, that leads to Tanner. If he does something that crazy, he might have lost his temper at his sister for something else. Maybe he didn't just recently arrive from Georgia.\"\n\n\"Yeah, but put her in a freezer in a house he knew nothing about? I'll bet Tanner has an alibi that he was in good old Zebulon when Cyndi was killed\u2014even though they can't quite pinpoint when she died. But there is her wronged, one-time Georgia fianc\u00e9 who could be looked at too. Jensen said those could well have been human ashes, not my specialty,\" Nick went on as his voice echoed above the pounding of the water.\n\n\"Then as for Dale,\" he added, \"the man's a bright lawyer, for heaven's sake, and he let his hots for an icy blonde\u2014no pun intended\u2014do his thinking for him? I'll bet the moment she opened her mouth, he should have moved on, but she was probably only too glad to open her legs for him too. I swear Dale could have hired someone to dump those ashes when none of us were looking out front, so the assumption would be it was good-old-boy Tanner or an accomplice. Actually, I hope Tanner has an alibi that leaks like a sieve\u2014if the cops can find him. For all we know, he's bound for Georgia again.\"\n\n\"Nick, this isn't like you. Calm down. If the ashes are human\u2014are Cyndi's, if they can even tell\u2014they will focus on Tanner or, like you said, maybe her dumped fianc\u00e9 back in Georgia. Still, I think your loyalty to a junior partner can only go so far if you really think he's guilty of murder.\"\n\n\"Damn, but I've gotten guilty people off before. Early ignorance. Overly stupid ambition. Everyone deserves a defense and from time to time I've been too good at that. But I'm not going to court for Dale if he killed that girl and stuck her in his mother's freezer, then got someone to dump ashes on his car to implicate her redneck brother. Dale could be thinking there's no way people would believe he's that dumb, whereas Tanner is. Reverse psychology, right, Mrs. Markwood?\"\n\n\"You're tired. You're going in circles. You need a good night's sleep. And so do I.\"\n\n\"I can just see myself or one of the partners in court arguing this case we'd like to keep low-key because it involves the reputation of the firm. Oh no, Your Honor, we are not just defending Dale Braun because he's tied to Markwood, Benton and Chase. Yes, as well as possibly losing his temper at the victim, or being afraid she'd embarrass him and ruin his career path, yes, he could also have been afraid she was going to blow the Braun family secret wide open. Oh, is that another member of the international press coming in with their cameras to cover this trial because of the Nazi connection, Your Honor? Yes, that's right, the defendant's property was inherited from and belonged to a high-ranking Nazi who was even related to the defendant through marriage and to Hitler's wife\u2014and furthermore, Your Honor\u2014\"\n\n\"Nick, stop it! It will work out. If you become certain he's guilty, there surely are alternatives to defending him, to putting the firm on the line.\"\n\nThe shower stopped. His shadow moved behind the frosted glass. The door slid open. His hand reached out for his towel. That reminded her\u2014who else, she thought, was Reaching Woman trying to touch or beg from on her other side in that grave? Claire had left a message on Kris's phone that she could not come out to the bog Saturday morning but would stop by that afternoon unless it was raining, so answers must surely be coming.\n\n\"Sorry to carry on like that,\" Nick said as he stepped out with the towel wrapped around his waist. \"This whole thing is really getting to me. I'm afraid I can't see the forest for the twisted trees. And I see I've created our very own sauna.\" He wiped a circle on the foggy mirror so he could see his face in it. He frowned at himself, then ran a comb through his hair. \"Thanks for listening and bringing me back down to earth, sweetheart. It could be worse\u2014we've been through worse. What is it you wanted to tell me when I went off like that? With Dale, the ashes, then Jensen, today's been a mess.\"\n\n\"I just wanted to tell you that the second bog body I've seen disinterred is a young female whom Andrea has named Reaching Woman. Talk about needing a murder trial\u2014she was stabbed in the chest.\"\n\nHe stopped combing his wet hair and turned to her, eyes wide. \"And the first body had its heart hacked out. More tragic proof that murder is as old as mankind.\"\n\n\"Murder or an execution. I've got to examine everything in that bog grave for answers. And, I imagine, if these two somehow-related people were executed\u2014I surmise for a common crime\u2014they did not have a brave attorney to stand up for them. What you do is important, Nick.\"\n\nShe almost told him about their strange stalker\u2014but that could just be coincidence and he already had enough to keep him awake tonight when he\u2014she too\u2014desperately needed sleep. If she or Kris spotted that man or woman again, she'd tell him. Maybe she and Kris could split up in a safe place and see if the person stuck with one or the other of them.\n\nNick threw his comb down, came over and pulled her to her feet. He hugged her hard to him. \"It's just\u2014I know I said we've been through worse\u2014but I'm scared, Claire. Dale is scared too\u2014I get that\u2014but this whole thing just doesn't feel right. I think this mess is going to get worse, so after we check out the 'Twisted Tree' Nazi ruins, that's it for your being involved with this case unless it's totally online research, and Heck can cover that.\"\n\n\"You said he's really busy right now, with his new sideline\u2014facial recognition technology, of all things.\" She realized again that she and Kris could use some of that\u2014but their stalker seemed almost faceless and sexless.\n\nNick went on, \"You just stay with ancient crimes, and I'll try to handle this one, and with a lot more self-control than I showed just now. It's a blessing I have you to listen, to care. Don't know what I'd do without you.\"\n\n* * *\n\n\"So what were you doing up so early?\" Nick asked Claire with a yawn as he backed them out of their garage the next morning. They were heading for the neighborhood where Dale lived, where Nita and Bronco had bought their house\u2014and where they were going to see the ruins of the Twisted Trees mansion.\n\n\"Doing internet research,\" she told him. She waved again to Nita and Lexi at the front door, feeling a bit sad. After all, she should be the one standing there on a Saturday morning with her children, not Nita. Maybe she was trying to do too much, getting too involved with outside challenges.\n\n\"You mean Googling more about bog people?\" he asked.\n\n\"Actually, at first I was reading some about former Senator Bradley Vance. He and Andrea have an art and antiques store downtown on 5th Avenue and one on upscale Las Olas Boulevard in Fort Lauderdale, you know. But I spent most of my time reading about Nazis living in the US.\"\n\n\"Whoa! That can't be common. I mean most of them would be really old or dead by now.\"\n\n\"You're right, most are probably dead, like Dale's great-uncle. But I learned there could be hundreds alive still. The Simon Wiesenthal Center hunts them down. They recently found one living undercover for years in Minnesota. Anyway, research has found that people without a conscience have less stress and may live longer than we would expect.\"\n\n\"It's going to be bad enough that Dale's great-uncle is dead, let alone if he was still living. Damn, I wish this case would just go away so I could go back to defending people I believe are innocent and you could\u2014\"\n\n\"Right. So I could go back to dealing with ancient executed people, not recently frozen ones.\"\n\n* * *\n\n\"Look, there's Dale pacing in his driveway,\" Nick said and they pulled up by the curb when they arrived. \"At least his stress proves he has a conscience, right?\"\n\nWhen he rolled the window down, Dale asked, \"How about I just get in your car? We'll drive a little ways back in and walk the rest of the way.\"\n\n\"Sure,\" Nick said and hit the door locks open so Dale could climb in the back. \"Good to see that there's no one hanging around Bronco's place, waiting for them to show up to question or bother.\"\n\n\"Not now. Betty Richards, down the street\u2014she knows everyone else's business\u2014told me that several people have been by just to gawk and take pictures, look in the windows, stuff like that.\"\n\n\"Oh no,\" Claire said. \"We've all been trying to assure Nita that wouldn't happen anymore. Mrs. Richards is the one who ID'd Cyndi to the police, so I hope Nita doesn't phone her to see how things are going.\"\n\nBeyond the intersecting streets of the housing development, screened by those invasive melaleuca trees and several ficus like the ones surrounding the bog, Dale directed them to an overgrown lane that bent back out of sight. Nick felt closed in already. He could just imagine the trees around the bog Claire had described, but he thrust that thought away.\n\n\"Let's leave the car here and just walk,\" Dale said. \"It's rained, and we don't want to get bogged down.\"\n\nAt the word _bogged_ , Nick caught Claire's narrow gaze. Sometimes he could almost read her mind, but not now. Was she thinking there might be a body bog back here? Probably not. That this Nazi connection could screw things up for Dale?\n\nHe took her arm as they followed Dale back through the twisted trees for which the mansion had no doubt been named.\n\n* * *\n\nHis eyes still heavy with sleep, Jace rolled over in bed at his apartment and reached out for Brit. And got nothing but pillow and twisted sheets.\n\nHe propped himself up on one elbow. The bathroom door was open but the light wasn't on and he heard no sounds of movement anywhere. But hadn't some sound awakened him? Oh, he did hear banging around in the kitchen. And smelled something great\u2014or was that a burning odor?\n\nHe got up, wearing only his shorts, and tiptoed toward the kitchen. Yes, some serious banging of pots and pans\u2014and crying?\n\n\"Brit, honey?\" he said as he peeked into his apartment's small galley kitchen. \"What in the...\" was all he got out before he took in the sight of scrambled eggs\u2014or maybe an omelet\u2014all over the tile floor.\n\n\"Trying not to get burned, I spilled it and got so mad that I threw it,\" she choked out and burst into tears. \"I was\u2014was trying to make you breakfast in bed. And, it slipped. I'd just like to scream, to break things. I want things to go so well, and I just keep losing it when they don't. You've got cold feet, don't you\u2014about the wedding?\" she demanded with the empty skillet still in her hand. \"I can tell you're upset, and I am too!\" she insisted, her voice rising as she choked out sobs again.\n\nMan, had he been that transparent? And in her state of mind, she looked as if she'd like to brain him with that skillet. He might have fought the damn Taliban, but he'd take them over this right now.\n\n\"Put that down. I'll help you clean up, honey. No, I don't have cold feet, not for that.\"\n\nShe put the skillet down on the stove, not too carefully, and threw the mitt into it. \"What then?\"\n\n\"I need to tell you something I was scared to.\"\n\n\"Another woman?\" her voice rose to a near screech. He started toward her now that she wasn't armed, but he slipped barefooted in the egg mess. He went down on one knee, would have fallen flat, but she grabbed him, held him up.\n\n\"No way,\" he insisted. \"It's just I\u2014I've been offered a job that I really want to take but there is some danger involved. It's with Mitch, flying for a good cause, gathering data about hurricanes to save lives, but I thought you might have a fit if I took it.\"\n\nShe helped him get up, then grabbed a dishcloth to wipe the eggs off his knee and shin. \"Jason Britten, I will have a fit if you _don't_ tell me things, not if you do. You think my working with big feral cats after my father was killed by one isn't scary, isn't dangerous? I may not like a new job for you, but I know you're sick of flying little planes. Besides, I finally pried it out of you that the drug cartels could want to harm Stingray pilots like you and Mitch, so I've been worried about that as it is. Tell me, then help me clean up this mess, and let's eat some toast and juice and go back to bed\u2014and not just to talk. I love you, Jace, and we're a team,\" she said throwing her arms around him. \"So no more secrets!\"\n\n* * *\n\n\"So how secret is this place, really?\" Nick asked Dale as they plunged through the thick screen of trees. Claire just kept quiet, studying Dale, trying to take it all in, ignoring even the low whine of mosquitoes.\n\n\"I know some people stumble on it, then tell others. Years ago, the cover story was that the owner was a German industrialist whose factory had been bombed out in World War II, and he just wanted to leave a devastated country. After my great-uncle died, my parents lived in it for a while, but it was too damned big\u2014obviously,\" he added as they got their first look at the ruined structure.\n\nIt was a silhouette against the trees, yet its strong cedar bones still stood. Claire gasped. She could see where a grand staircase had been and the doorways, even some rooms. No windows remained; some plants grew inside as if the place had a green beard. The roof, like a moss-covered toupee had partly caved in near the chimney which had obviously served several large hearths. Most of a garage or stables still stood.\n\n\"There was a fire in the mid-1980s,\" Dale explained. \"Hurricane Andrew in 1992 did the rest. Then water damage, critters, some vagrants... It would be exorbitant to repair or replace, even to tear down. I just pay the property taxes out of my trust fund.\"\n\nThey walked through deep sawgrass that grasped their pant legs as if in a feeble attempt to keep them away. Claire had pictured that the place would have been brick, but it must have been mostly wood. Several crows flew out from under the shattered roof, cawing at them as if screaming, \"Away! Away!\"\n\n\"It must have been grand,\" she said.\n\n\"I hardly remember it that way, and remember him only from pictures. I burned his photo album as soon as my mother died. He actually dared to have photos of his 'grand days,' as he put it. SS uniforms, swastikas, like huge black spiders on blood red flags. But especially when my mother's dementia increased, she had either inked out the face of Eva Braun in the photos or actually cut them out. Said she wanted to kill Eva\u2014well, just a figure of speech, born of dementia, and anger, of course, because Eva was long dead.\"\n\nClaire's gaze slammed into Nick's. A revelation hit Claire hard, a crazy leap of logic, but who knew? When she'd researched Nazis this morning, she'd seen a picture of the blonde, pretty Eva Braun. Her smile in the photo had been so sweet and sincere, innocent. Hitler's fianc\u00e9e, later briefly his wife, was hardly screaming\u2014but\u2014but she looked a lot like Cyndi. Could an old woman with dementia who'd hated Hitler's wife have mistaken Cyndi for her and, elderly or not, had rage and strength enough to\u2014\n\n\"Look!\" Nick interrupted her agonizing as he pointed toward the desolate mansion. \"There's a blonde woman over there in the trees watching us.\"\n\n# 13\n\nFor one moment, Claire feared the woman watching them might be lifting a rifle, but it was a camera with a telephoto lens. Still, Nick pulled her behind him.\n\n\"Hello!\" Dale called out. \"Can we help you? You're trespassing, you know.\"\n\n\"It's obviously not lived in,\" the woman shouted back and came out of the trees. \"But still such a beautiful place. I can explain why I'm here.\"\n\nClaire breathed out hard. The woman was dressed in jeans and a long-sleeved navy shirt, probably to keep mosquitoes at bay. Her clothes were tight fitting, even belted. She was probably in her fifties, much too tan for safety's sake these days, with a heart-shaped face framed by hair so blond it looked white. She was extremely slender\u2014gaunt, really. And she walked with the slightest hint of a limp.\n\n\"May I ask who you are?\" Dale went on as she came closer and stopped about ten feet from the three of them. \"I'm the owner here.\"\n\n\"Then you must be a descendant of the German man. I couldn't learn a lot from the county property records.\"\n\n\"Yes, I\u2014\"\n\n\"The Nazi. You realize that makes this site even more important and unique than if it were just built and owned by a wealthy early landowner.\"\n\nDale looked speechless and shaken. His worst nightmare, Nick's too, would be for it to get out that Dale had a relative tied to Hitler. Talk about prejudicing a jury and causing a media circus. They'd been through that trauma before.\n\n\"I'm Marian James with the Endangered Properties Committee\u2014not the same as the county historical society. We're into historic preservation, though, and\u2014sorry to say\u2014I took a phone call about nine days ago from the poor woman who was murdered, your former fianc\u00e9e, Cyndi Lindley.\"\n\nDale gasped. \"Why would she call you?\"\n\n\"She wanted to let me know about the great history of this place and invited me out here. She said it could lead to magazine articles, news stories, even television promotion for our group. Well, I was away and when I returned and got her phone message at the office, I learned she was dead\u2014murdered\u2014and...frozen.\"\n\nNick said, \"But I'm sure you understand that Mr. Braun needs some privacy at this time. For himself and this property.\"\n\n\"Oh, yes,\" Ms. James plunged on. \"Please accept my deepest sympathy on your loss, though I read in the paper she was your _former_ fianc\u00e9e. But I'm hoping you'll still want this story told. Why, ties to the Nazis and Hitler just a couple of miles from the Everglades\u2014what a story. It will echo far beyond Naples. Our efforts are well endowed by some wealthy individuals, and they could help you turn this place into a protected historic area or at least have a historical plaque here. So sad that this house has died,\" she added as she came slowly closer, \"and Cyndi cared about it, and now she's dead too.\"\n\n\"The publicity\u2014not at this time,\" Dale said, almost stammering. \"I\u2014I may sell.\"\n\n\"What a pity. But one more thing,\" she said, not retreating one step when Dale tried to move past her. \"Cyndi mentioned over the phone that she thought someone might be following her, trying to scare her. I just tell you that\u2014and would be willing to testify\u2014if the police wonder who could have harmed her. I could also tell she was heartbroken and really angry about something.\"\n\nClaire doubted most people could get that much from a recorded message when there was no real conversation. Was this woman to be trusted? Had she actually talked to Cyndi, maybe met with her in Cyndi's attempt either to get publicity for herself or even to find a way to blackmail Dale? What Marian James had just said could be taken two different ways. If Cyndi had a stalker, that could help to clear Dale of suspicion. But if Cyndi was angry over their breakup, it could suggest she confronted Dale and he, accidentally or not, strangled her sometime within the last nine days. If this woman was telling the truth.\n\nNick said, \"Where did you park to walk way in here, Ms. James? And what a coincidence you're here the same time we are. If you have further comments or observations, please make them to me, Nick Markwood, Dale's attorney at Markwood, Benton and Chase,\" he said, pulling his card out of his billfold.\n\nClaire's thoughts still rampaged. Indeed they had not seen her car, though she was now telling Nick a friend dropped her off and would be back soon. And was it mere chance she showed up here when they did? The more she studied this woman the more she thought she could be the person who had followed her and Kris.\n\nStill, she knew she was getting too suspicious of absolutely everyone. She was even having a tough time liking or trusting Dale as well as Ben Vance, and he was her employer and a well-respected former senator, not that politicians were ever to be trusted.\n\nMarian James started to walk away, then turned back. \"I really should call the police with all this. If I told them about poor Cyndi's stalker\u2014if she was right about that\u2014it could change the way some people always blame the man in the victim's life, so I'm sure that could be a help to you. I see in the newspaper and TV photos she was very pretty. Maybe she attracted a stranger who killed her instead of someone else close to her.\"\n\nClaire could tell Nick had a comeback, but he kept quiet. With a nod at the still stunned Dale and a narrow-eyed look at Nick and Claire, she turned and walked away, yes, with just the slightest hitch in her step, not really a limp. As she disappeared behind the twisted trunks of the ficus trees, Claire thought it was almost like a ghost vanishing, but this one would no doubt materialize again and could spell real trouble for Dale if he didn't cooperate with the Endangered Properties Committee.\n\n* * *\n\nNick told Claire he didn't want to leave Dale right now. He wanted to calm him down and plan a strategy if Marian James went public with her story. So Claire drove Nick's car home from Dale's house and Dale would drop him off later. Besides, she knew Nick wanted to look around Dale's place since the police had searched it.\n\nAt home, she spent an hour with Lexi and Trey\u2014\"How long before a little baby can finally talk, Mommy?\" her daughter had asked\u2014then grabbed a sandwich and headed for Black Bog. What a help it was to have Nita tend the children and Bronco as extra security. Claire was in a hurry now, as Kris had left a message that they were getting close to uncovering the next body.\n\nShe kept her eyes on the road as her thoughts pounded her again. Nick didn't want her anywhere near Dale's case, but psyching things out was in her blood. Cyndi Lindley was emerging as a clever, greedy person, not entirely sympathetic, despite having been murdered. Claire had already checked the woman's Facebook page and had gone way back through the pictures\u2014mostly of herself, pouting sexily, some in a skimpy swimsuit\u2014real bait for voyeurs, so maybe she _had_ attracted a stalker. Why hadn't Dale gotten turned off if he'd checked her page\u2014or had he just gotten turned on by it? Also, there were a few pictures from two years ago that showed Cyndi with her beefy first fianc\u00e9, a real bruiser. She planned to show those to Nick to see if he could get Detective Jensen to look into that angle.\n\nShe was so wrapped up in thoughts she almost missed the turnoff toward Black Bog. She hit the brakes, and the car behind her\u2014following much too close\u2014honked and roared around. Claire slowed on the narrow approach road.\n\nThe guard at the gate opened it electronically and waved her on through. It would have been good to have such security to protect the Twisted Trees mansion, she thought, as she pulled into a parking place next to Kris's Jeep.\n\nClaire was tempted to run, not even stopping to ditch her big purse, but the planks bounced and she slowed. They had said they didn't want her or any of the team to take their own photos, except for the official ones Doug shot, so she reluctantly put her phone away, thinking again of Marian James's expensive camera at the Twisted Trees mansion.\n\nNick had assured Dale he'd get a stop on the photos with a threat of illegal trespassing if necessary. And despite the fact it was a Saturday, he'd already called a junior partner to get on contacting Ms. James and her so-called committee immediately. He said he hadn't warned her on the property because that might have made her quickly disperse or publish her photos.\n\nOut of breath, Claire joined the dig team and leaned closer to see.\n\n\"Bog body number three,\" Andrea told her, looking up, then into the grave again. \"I've named him Leader\u2014more formal than Chief or Boss. He commands attention and respect.\"\n\n\"He's tall,\" she said, \"or maybe it's just that he's not curled up like the others. And he's dressed well. That robe has some sort of faded stripes. And look at that polished stone bracelet around his wrist. It reminds me of the necklace on Reaching Woman. Whatever happened to that? I'd like to study it.\"\n\n\"All the artifacts are specially cleaned, handled and stored,\" Andrea told her. \"As for the stripes on his garment, berry juice dye and beautifully done. It's obvious that he was an important person\u2014but who and why, Claire. That's your bailiwick.\"\n\n\"What's that in his hand?\" Claire asked as the three diggers carefully removed more peat from his head so they could uncover his face.\n\n\"Some sort of rod or staff,\" Kris said. \"Maybe symbolic.\"\n\n\"It's carved,\" Doug said. It was hard to hear him because he was bent low in the grave. \"We found some other items with him, but we've already sent them inside.\"\n\nStill awed, Claire whispered, \"The staff and robe are no doubt symbols of his office, his importance. But what is his relation to the executed pair? At least he doesn't seem to be stabbed or worse\u2014his heart's not cut out, is it?\"\n\n\"He's apparently intact,\" Yi Ling said. \"Even dead and buried and muck-covered, he still has an aura of power or knowledge.\"\n\n\"I hope he wasn't judge and executioner of the other two,\" Claire said. \"But then, why would he be buried near them, especially if his death is a natural one? And Reaching Woman seems to be stretching her arm toward him, as her other one was toward Hunter. Maybe caught somehow between the two men.\"\n\n\"You're right,\" Kris said. \"If he's a kind of judge that sentenced both of them to death, why be buried with them? Maybe Reaching Woman is meant to be begging for mercy, something like that. Leader stands, the other two have bent knees. But all that's what we hope you can figure out, Claire.\"\n\nEveryone went silent as Doug, Yi Ling and Aaron brushed away the last bits of centuries-old peat and soil from Leader's face. His eyes were wide open. At least his mouth was closed\u2014stern, firm\u2014no hint of terror or a scream. Did his open eyes mean his people meant for him to be looking at the other two, looking into the future, or into the land of the dead which the bog represented?\n\nClaire suddenly felt overwhelmed with all she was expected to theorize. She was trembling uncontrollably as if she were freezing when the day was warm. It scared her even more to realize something else: Leader couldn't help but remind her of Nick.\n\n* * *\n\nDale was still shaken, and Nick couldn't calm him down. He had looked around Dale's house from which the police had confiscated items and fingerprinted a lot of places, leaving graphite powder Dale had not cleaned up.\n\n\"I think you need to stay here for a while and calm down. Maybe take a nap, clean up the remnants of the police search,\" Nick told him when Dale came out of the bathroom for the fourth time. His stomach must be really upset too. The guy was falling apart. Guilt? Fear? Suppressed anger? Damn, he wished he had Claire here to read the signs.\n\nNick went on, \"I called Bronco to come get me so you can stay here for a while. Maybe you shouldn't drive right now.\"\n\n\"I don't like to be here anymore. I'll head back to my temp place downtown. At least Bronco knows how to get here, huh? Nick, I'm sorry that he and Nita are going through this. Cyndi is kind of haunting me from the grave\u2014from her ashes, that is\u2014telling that woman all about my background, which my family did a good job hiding.\"\n\n\"I've put things in motion to muzzle her for now. Maybe one of the heavy hitter donors she mentioned will want to buy Twisted Trees from you. Just be careful if you drive, okay? And if your secret about who built the mansion goes public, don't act like you've been trying to hide it. Say you just didn't have the time to restore the place. You can't go anywhere now or it would look bad, but after we see this through\u2014and no one's made a move to arrest you\u2014you might want to clear out of the entire area.\"\n\n\"Nick,\" he said, grabbing his arm as they saw Bronco pull up in front, \"just don't throw me to the dogs. I didn't kill Cyndi!\"\n\n\"Then just be sure you level with me and your defense team. Maybe you won't even be indicted or arrested. But this Hitler connection is a big deal if it gets out, especially since Cyndi wanted to use it to get attention, money, revenge\u2014something. And you didn't want that.\"\n\n\"So a second motive for my strangling her? Damn that 'Endangered Properties' woman! I wish she'd just disappear.\"\n\n\"Be careful what you wish for,\" Nick warned. He squeezed Dale's shoulder and headed out to the car as he heard Dale close the door firmly behind him. He worried for a moment he should go back in, but he didn't think Dale, even distraught, was either a flight\u2014or a suicide\u2014risk.\n\n\"Hey, boss,\" Bronco greeted him, getting out of the car and coming around to the passenger side. \"Thought I might have to come in for you. When I told Nita where I was going, she got all shook again.\" A thin, sprightly woman was coming down the sidewalk tugging at her short-legged, perky-looking Chihuahua on a bright red leash that matched the dog's collar. She had a slight limp, but it wasn't slowing down her speed.\n\n\"Oh, is Dale moving back in?\" she asked them. \"It's all so sad. I'm Betty\u2014live right over there three doors away, the one who identified the poor dead woman for the police from her photo. Why, I've told the other neighbors and the newspaper people it was terrible to see her all frozen with her mouth open like that.\"\n\nEven without Claire, Nick knew he could nail this woman's character exactly. But rather than brushing her off, what could he learn?\n\n\"We're friends of Dale's,\" Nick told her as the little dog looked up at its mistress with the curiosity that Nick suddenly felt too. \"And that was such a help to the police that you could quickly identify Dale's friend.\"\n\n\"Yes, well, she and Dale were tight, if you know what I mean,\" she said with a nod of her white, curly head. \"But really, I was friends with his mother, Lucinda\u2014Lucy, I called her. She had him quite late in life, you know\u2014could have been his grandmother age-wise. The Brauns lived quietly, but she said they had money, and that Cyndi was a gold digger, because she probably knew the truth about their wealth\u2014mostly land wealth.\"\n\n\"I guess it's always good to know the truth,\" Nick said with a nod at Bronco who nodded back. \"So you knew Dale's mother well?\"\n\n\"Years, lots of talk time, especially after she was widowed, like me. I really regretted it when she started to get so terribly confused with dementia. It changed her from a sweet person to a crabby one, even nasty sometimes.\"\n\n\"That's very sad. So maybe that's why she really didn't like Cyndi\u2014it was just her personality with dementia.\"\n\n\"That and, I think Cyndi reminded her of someone she didn't like, someone who gave the Braun family a bad name or a bad time. I think,\" she said, lowering her voice and leaning closer, \"Cyndi reminded Lucy of a woman who had ruined the family, because she said that woman was in with the Nazis. I mean, who thinks about the Nazis anymore?\"\n\n\"Did Lucy ever mention Hitler?\"\n\n\"The more she lost common sense, the more she seemed to go back in time. Yes, she mentioned Hitler. I told her he was dead and gone. But she not only thought Dale's father was still alive, but Dale's great-uncle too, the one who came from Germany and built that mansion that's in ruins.\"\n\nShe dropped her voice to such a whisper that Nick almost had to read her lips. She was much shorter than him, so he leaned closer. \"But the one Lucy really blamed was some relative named Eva.\"\n\n\"Really?\" Nick asked as Bronco stopped leaning against his car and stood up straighter.\n\n\"Yes, it was a bit confusing, but it was a woman she hated. Anyhow, Lucy thought Cyndi looked like or acted like\u2014I don't quite know\u2014this Eva, and that's another reason she didn't approve of poor Dale's fianc\u00e9e. Why, I recall those awful days of World War II when I was young. I put things together since Lucy's last name was Braun, but she explained the Eva Braun connection to me anyway. I wish Lucy hadn't been so out of touch before she knew that Dale was splitting up with that gold digger. Why, a nice, hardworking young man like Dale ought to be married, don't you think? Just not to some gold digger!\"\n\nNick didn't know what to think and with a nod and a \"thanks for all your insights, nice to meet you\" headed for the car. Claire would have known how to get more out of this woman, but he could bring her in later if evidence led to the possibility that an old woman with dementia could have strangled a much younger, stronger Cyndi\u2014thinking she was Eva\u2014and somehow lifted her into that freezer.\n\n# 14\n\nThat evening Bronco took Nita out to dinner, so for the first time in a few days, it was just the family at the table. Claire had held Trey in her lap to feed him, burped and walked with him, then had laid him down to sleep. Sprawled on the sofa, Lexi was reading a book on butterflies because her teacher, Miss Gerald, had shown the class her butterfly collection, though Lexi didn't like that they were \"dead and stuck with pins.\"\n\n\"Finally, peace and quiet,\" Claire told Nick as she slipped back into her chair across from him to eat her dessert.\n\n\"Quiet anyway,\" he said, finishing the rest of his wine.\n\n\"Why are we whispering?\"\n\n\"'Little pitchers have big ears,' to quote my mother.\"\n\n\"Meaning?\"\n\n\"Meaning,\" he said, reaching across the table for her hand while she met him halfway, ignoring her caramel flan, \"I need to tell you I got a call from Ken Jensen before you got home. Private, so I didn't bring it up before Nita and Bronco left, but not private from you.\"\n\n\"He had the ashes tested? Were they Cyndi's or just fake?\"\n\n\"Unfortunately, the lab can't tell\u2014meaning he'd tell us if he knew. It turns out DNA is destroyed during cremation.\"\n\n\"Darn. So that ashes-on-Dale's-car threat does not necessarily point to Cyndi's brother Tanner.\"\n\n\"Could, but doesn't. Here's the kicker. I couldn't figure out why Jensen didn't pursue not only Tanner but also Cyndi's former fianc\u00e9. But he admitted, when I pressed him on it, that he was calling from Atlanta and he'd been to Zebulon. He said neither Tanner nor Cyndi's once-upon-a-time high school sweetheart are currently in what he called 'scenic' Zebulon where he went to interview them. Tanner, of course, could still be here, but maybe the former boyfriend is too. Jensen said he would never have divulged that except he'd learned both men were volatile, and we should tell Dale to be extra careful.\"\n\n\"And he meant, I'll bet\u2014nice guy that Jensen really is\u2014that we should be careful too. Whoever put those ashes on Dale's car did it in our driveway.\"\n\n\"I don't think that was aimed at us. But it does worry me if Dale's being followed.\" He nodded and heaved a heavy sigh. \"Let me put things away here, while you get Madame Butterfly over there in bed. Bronco and Nita won't be out long, and he's got instincts like a guard dog, so I'll hate to see them go. Maybe we should keep them around a little longer. Nita won't mind. And tomorrow's our big day to finally celebrate our marriage with friends and family.\"\n\nThey clasped hands together tighter over the table. \"I'm so looking forward to it,\" she said. \"Despite complications sometimes, I've never been happier.\"\n\n\"Me too,\" Nick said and blew her a kiss, a promise she knew he would make good on tonight\u2014tomorrow\u2014all their days together. She'd once thought she was happy married to Jace, yet that had gone all wrong. But had Reaching Woman reached out to the wrong man at first? And then paid some sort of terrible price\u2014her life?\n\n* * *\n\nSunday morning, while Bronco and Nita went to mass\u2014Nita was praying hard for the Virgin Mary to cleanse and bless their new house\u2014Claire and Nick took their children to join Darcy's family at their church.\n\nNothing like a real family day from beginning to end, Claire thought. Besides, she got to show off Trey to people she had known when she used to attend here with Darcy, and Lexi was only too happy to see her cousin and almost-sister, Jilly. Drew, Darcy and Steve's son, now seven, who thought it was beneath him to have to put up with two younger girls, was the only one who fidgeted at all.\n\nClaire had not realized there was going to be a full water baptism at the beginning of the service or she would have explained it to Lexi. Especially when the pastor lifted the person from the water tank behind the altar, she herself felt as riveted as Lexi was at his words:\n\n\"And from the _Book of Romans_ , I say to you these words, 'Buried in death, but raised to walk in newness of life.'\"\n\n\"What does that really mean, Mommy?\" Lexi asked in a stage whisper.\n\n\"I'll explain later. It means the person is choosing a new life.\"\n\nBut after church, even after Claire tried to explain to Lexi about changing a former life to make a new beginning, those words haunted her. The Black Bog bodies had been buried in death\u2014two of them a violent death\u2014but had now been raised from the grave. The bodies, if not the people, had a new life, to be studied, probed and puzzled over.\n\nIn a way, Claire agonized, the three ancient people she knew best and would get to know more would not let her go as if they walked beside her, clung to her, especially Reaching Woman. She surely had meant something to both Hunter and Leader. She was trapped between them, stretching out her hands to both, asking for something. Yet had one of them ordered her death or taken her life?\n\nWhy? What had she done to them? Or what, as so many times in history and even in modern life, had the men of her society done to her?\n\n* * *\n\nLater, as Claire dressed for the reception, fastening the double strand of pearls Nick had given her around her neck, she thought of Reaching Woman's broken necklace of smooth polished stones and shells. She would have to ask to see it after it was cleaned. She knew an examination of the woman's wounds might reveal if the dagger that had killed her had been similar to the one Hunter held. She would like to see and touch those, to really feel items they had used and treasured.\n\n\"Claire, you ready, sweetheart?\" Nick's voice broke into her thoughts. \"Time to go. Nita's got Trey ready, and Lexi's driving everyone crazy. Oh, yeah, and Jace just called, because Brittany had a problem with that pregnant lioness at the zoo, but they'll be there later.\"\n\n\"Coming,\" she told him, shaking off her almost-visions of long-dead people. She reached for him to keep him close a moment and to kiss him. \"Yes, I'm ready.\"\n\n* * *\n\nClaire had sent Bronco and Nita ahead with potted orchids for the buffet table and the one with the cake, while their country club hostess, Mary Ann, had brought in other plants and decorations. The Blue Room entrance led to the glassed-in verandah where everything was set up.\n\nNick lifted Lexi so she could see what was written on the large, one layer cake: Never Too Late to Celebrate Love _._ The only other thing on the table with the cake, plates and napkins was a large crystal bowl of pink punch. A coffee urn and bottles of wine were set up on an open bar at the end of the verandah, staffed by a bartender from the country club.\n\n\"That means love for all of us, right, not just grownups?\" Lexi asked, pointing at the cake. \"Even Trey?\"\n\n\"Especially you and Trey,\" Nick told her as Claire blinked back tears of joy. The cake\u2014in more ways than one\u2014was beautiful, with elaborately scrolled words and iced orchid flowers to carry out the theme.\n\n\"Well, where is everybody?\" Lexi asked as Nick put her down. \"Mommy, will you save Jilly and me good corner pieces with lots of frosting?\"\n\n\"Yes, but we'll cut the cake later,\" Claire told her, straightening the child's taffeta puff sleeves. \"We are going to have nice, polite talks with our guests, then they can have the food from the buffet table, and the cake and ice cream will come last.\"\n\n\"Pretty much like a birthday party,\" Lexi said with a nod as she scampered over to Nita who was seated and holding the sleeping baby on her lap.\n\nTo Lexi's chagrin, the earliest guests were Nick's law partners and staff. Claire was glad to share time with them under these relaxed circumstances. She hadn't seen Nick's longtime secretary, Cheryl, for quite a while and had a good chat with her before taking her place in the informal reception line.\n\nShe and Nick stood near the verandah door to greet guests. The noise level rose: chatter, laughter. It felt so good.\n\nHeck came in with Gina, home for the weekend from medical school in Miami. Claire hugged her while Nick and Heck huddled for a moment in a lull. \"How are things?\" she whispered to Gina. She still looked as beautiful as when they'd first met her in Cuba, though the lack of sleep was showing.\n\n\"Tough, but that's good. My Spanish really helps me in the clinic, though I know I got\u2014have\u2014a too-strong accent in English. Anyhow, it's great to be helping others too,\" Gina told her.\n\nShe pulled Claire a bit aside and lowered her voice. \"Please tell Nick I'm worried about Heck. He's so set on this extra facial recognition firm consulting job. But he finally admitted there's a lot of competition to control that technology in the future\u2014big money, political power. And another firm\u2014I had to pry this out of him\u2014is playing tough ball.\"\n\n\"Hardball? Meaning what?\"\n\n\"There's a rogue firm\u2014not the one Heck trusts\u2014well, the brakes on one guy's car who wanted the job Heck is up for got tampered with and he had a wreck\u2014who knows what's next? Cutthroat competition, very bad, especially if it is aimed at my man. Like he always say, _Caramba_!\"\n\n\"I'll tell Nick, and he can talk with him. Maybe he's telling him that right now.\"\n\nSpeaking of being in trouble, Claire thought, here came Dale, looking sporty as usual but nervous. He kept fooling with his pocket handkerchief and shifting his weight from one foot to the other.\n\n\"Glad you came,\" Claire heard Nick tell him as Gina snared Heck and headed for the bar. Claire joined Nick and Dale. \"You have a lot of people in this room who support you,\" Nick tried to assure him with a nod.\n\n\"And a lot who still look at me like 'There's the guy who might have killed that girl.'\"\n\n\"I know it's difficult to keep your chin up,\" Claire told him. \"But it's important for how it looks\u2014and for how it feels for you too.\"\n\n\"I keep worrying about how it's going to look if that Marian James woman blabs about my heritage on top of everything else. And she hinted she's got wealthy, important backers. Look, sorry,\" he added with a shrug. \"You've both been great to me, and I don't mean to darken this occasion. Congrats to you both on a great marriage and two darling kids. I'd love to get there someday\u2014you know what I mean.\"\n\nAs if that had summoned Lexi, she came over, dragging Nita who still held Trey. \"You said you wanted to show him off, Claire,\" Nita said. \"He's a magnet and being so very good.\"\n\n\"It's 'cause he's sleeping,\" Lexi put in. \"When he's awake, Nita, you know he's not always good, but I am.\"\n\nClaire's gaze met Nita's. Lexi had not shown a bit of jealousy before, so they'd have to work on that. Claire took Trey in her arms and he stirred. So beautiful, so innocent.\n\nSoon others arrived including Darcy, Steve, Drew and Jilly, who could barely stand still until she saw Lexi across the room and ran to meet her. Hugs all around. Steve and Nick talked off to the side, then Nick showed them the family table Claire had staked out.\n\nKris arrived in her new outfit she'd bought recently. \"Thanks for including me,\" she told them.\n\n\"Of course,\" Nick said and gave her a quick, one-armed hug. \"After all, you've known Claire longer than anybody here except her sister.\"\n\n\"I don't do crowds well,\" she whispered. \"But I will recognize your kids. Oh, dear,\" she added as she looked across the room. \"There are two little girls over there, but I'm guessing Lexi's the one on the left.\"\n\n\"You are right,\" Claire told her. \"Just remember the taffeta dress with puff sleeves. The other little girl is her cousin Jilly, the closest she has to a sister.\"\n\n\"At least so far, but you never know for the future,\" Nick put in with a little laugh and a waggle of his eyebrows as he put his finger in Trey's fist when the baby stirred awake at last.\n\nClaire and Kris both laughed. \"Wish I had that problem\u2014a man who loved me making innuendos,\" Kris said. \"You know, Nick, Claire and I used to kid about us being 'womjeps,' women in jeopardy. I think we've both found our niche, though I admit a good man is hard to find.\"\n\nWith Trey still in her arms, Claire took Kris over to meet Heck and Gina, then came back to Nick when Jace and a beaming Brittany came in. It made Claire's day to see both of them so happy, and she hoped things worked out better for them than it had for her and Jace, who kissed her cheek for the first time in ages.\n\n\"He's just so darling,\" Brit told Claire as she cooed \"hi-there-handsome\" to Trey, who thought she looked pretty interesting too. Claire let Brit hold the baby.\n\n\"She never says that to me,\" Jace deadpanned.\n\n\"Lexi's been waiting for you,\" Claire said. \"See if you can keep her from going into her flower-girl mode this afternoon and sticking her finger in the icing before we cut the cake.\"\n\n\"Speaking of the birds and the bees\u2014which we weren't,\" Jace told them, \"I see Kris Kane is here. Do you know if she's seeing anyone? I don't like to play matchmaker, but I have a friend who would like to meet her.\"\n\n\"She just said she wished she had someone. I think her face blindness has held her back in the men department, plus how dedicated she is to her work. You know, Jace, she always picked you out because she thought you looked like a Viking\u2014her focus on the buried past coming out there. But I'm sure she'd be interested. Sounds perfect.\"\n\n\"Yeah, well, he's a pilot too, and I know that can cause problems.\"\n\n\"Sometimes,\" Claire said.\n\nNick cleared his throat almost in unison with Brit. Claire realized she and Jace had been having an intense, two-way talk. She pulled her gaze away from him and glanced at the huge window behind the cake table. Because the trees of the orange grove beyond were thick and dark green, the window made a muted mirror. And there she was, with her hands lifted slightly toward not only Jace but the taller Nick. Reaching Woman flashed through her mind again, caught somehow between Hunter and Leader.\n\n\"Enjoy, you two,\" Nick said to Jace and Brit in the sudden lull as Brit handed Trey back to Claire. \"Get something to eat and talk to Kris. She really doesn't know anyone here but us and Lexi. And there's a great view, Jace, out the window to watch the golfers on the eighteenth green.\"\n\n\"Sure, good,\" Jace said, seeming suddenly uneasy as Brit took his arm and they moved away.\n\n\"Let's circulate, get something to eat, then cut the cake,\" Claire said, turning away to look at it again.\n\n\"You're a poet and don't know it,\" Nick said with a groan at his own words. \"But most importantly, above all else, you are my beloved wife and Lexi and Trey's mother.\"\n\n\"I love you, Nick,\" she said, turning back to face him and reaching out to touch his arm. \"You are so strong and supportive\u2014and\u2014and understanding.\"\n\n\"Not always.\"\n\n\"I mean we've had some tough times, but things are different now.\"\n\n\"Good,\" said the man of many, well-chosen words. \"Good.\"\n\nAs they started away from the cake table, Claire heard a boom and a strange crackling sound. Her ears popped. Trey started to cry. Oh, where was Lexi?\n\nThe long, high window behind the cake and buffet table seemed to explode, not shatter, but blast into a thousand ugly glass cobweb pieces which rained down onto the floor in thumb-sized shards.\n\nThe bottles behind the small bar shook. Shelves holding them broke, dropped in a second round of shrieking noise. The bartender ducked, then fled crunching glass.\n\nThe noise, the impact, several screams\u2014people hit the floor. Nick grabbed Claire, still holding Trey and rolled them under the white folds of the skirt of the cake table. They huddled there, shaking, waiting for what came next. But nothing but people's screams, cries and questions.\n\nA golf ball against the glass? Was someone shooting? A bomb blast somewhere outside or sonic boom? She realized just a minute or so had passed.\n\n\"Stay here with Trey!\" Nick told Claire. \"I'll get Lexi.\" Keeping low, he rolled out and scrambled away, crunching glass.\n\nClaire cuddled Trey close as his cries went to sucking sobs, his little voice blending with other shrieks. She was tempted to crawl out and find Lexi herself. Where were Darcy and her family?\n\nShe peered out to see fleeing feet and bodies on the floor. Was that blood in front of her? It glittered in the sunlight pouring through without the safety glass.\n\nShe gasped. She was looking at globs of white icing stained by red punch that had exploded onto the floor with bits of broken glass in it.\n\nClaire opened her mouth to scream, but no sound came out.\n\n# 15\n\nNick's fear fast became fury. The place looked like a small war zone, with those who hadn't fled huddled\u2014hurt?\u2014on the floor. He saw Jace and Gina had protected Lexi with their bodies. Kris, Bronco and Nita, all sprawled on the floor amidst glittering glass, were looking around, stunned. The hot, humid outside air slammed into him. The safety glass had crackled into circular, fine-line pieces, and some parts had shattered into a storm of shards.\n\nHe saw Jace's cheek was cut and bleeding. Lexi was wailing. Gina, God bless her, was pressing a napkin against Jace's cheek, then Kris took over.\n\n\"Who else is hurt?\" Gina cried.\n\nNick echoed her in a louder voice, \"Anybody else hurt or cut? We have someone to help here!\"\n\nJace held the napkin to his cheek and, one-armed, handed Lexi to Nick who took a quick look at her, then tugged her toward the ruined cake table.\n\n\"Dad, my ears hurt! Too loud a bad noise! Why did the window break?\"\n\nNo time for all that, and who had answers? \"You'll be all right, sweetie. I'll put you with Mommy and Trey out in the hall.\"\n\nThe hell with _Never Too Late to Celebrate Love_ , Nick thought when he saw the shattered cake _._ Some damn idiot must have lobbed a huge rock against the window. Surely a golf ball didn't do all this. Hadn't the hostess told them this window was unbreakable?\n\nStill on her knees under the cake table, rocking their son in her arms, Claire reached out for Lexi, but Nick hauled the child past them into the hall. \"Stay right here!\" he ordered. \"I'll bring Mommy and Trey, and you can keep them from crying.\"\n\nLexi immediately stopped sobbing and nodded. The kid had some of her gutsy mother in her. Shaking, her cheeks wet with tears and her pretty dress a mess with some sort of food, Lexi sat up straight in an armchair. Nick darted back in for Claire, lifting her to her feet.\n\n\"I hope there's no follow-up to that,\" he muttered.\n\nCheryl appeared. \"Nick, how can I help?\"\n\n\"I think the bartender ran for help, but be sure they called 9-1-1 and the police. Detective Ken Jensen if he's available.\"\n\n\"Oh, and I think Dale's left arm might have broken when he hit the floor,\" she told him. \"Heck's girlfriend is tying it up for him.\"\n\nShe darted off. Poor Dale. Poor all of them. Claire looked both stricken and angry. Nick led her out to the chair where Lexi perched and sat her down next to her.\n\n\"Darcy and her family?\" Claire asked, cuddling Trey tight as she pulled Lexi against her.\n\n\"Huddled in a corner. I'll get them.\"\n\nSeveral others came out, looking dazed, covered with food and drinks and those tiny shards of glass, though no one else looked bloodied. Jace must have been hit by flying broken glass from the bar, not this window stuff. Nick turned to go back in when their hostess, Mary Ann, came running down the hall toward him.\n\n\"A drone!\" she cried, out of breath. \"I would have come in earlier, but we thought we could spot who was flying it. A drone with an explosive on it, but it didn't detonate, we think\u2014fell below to the patio after it hit the window. Damn that Lavell bunch who own the grove! I've called the squad and the police. Is anyone in there hurt?\"\n\n\"At least one broken arm, one cut.\"\n\nShe tore inside with Nick. She started to make her rounds of the people who were getting up to brush off glass and food, and she was soon joined by two other club workers, both men who likely worked in the office. They were asking everyone if they were okay and explaining, commiserating.\n\nThough it must not have even been ten minutes since the attack, Nick could hear the screech of a siren\u2014more than one\u2014coming closer. His eyes filled with tears as he surveyed the ruin of the verandah, now an open-air room. At least Steve and Darcy looked all right and were comforting their kids. Their boy Drew looked more shook than Jilly. Nick gestured for them to come out into the hall.\n\n\"A drone!\" Nick told them as they hurried toward him. \"The club hostess said the family that owned the grove has harassed golfers with a drone before and this one went on attack, probably not aimed at us.\"\n\nOnce everyone cleared the room and were checked out by the paramedics, Nick went to the verandah entry again and looked in. The room had been roped off by the staff and stood vacant now except for a worker getting ready to board up the long, open space with plywood sheets. Hot, humid air still swept in, smothering the air-conditioning. He realized he was wet with sweat, and his heart was still thundering in his chest.\n\nBut what he was really seeing in his head were the drones that his arch enemy had sent to spy on him and Claire when they first knew each other. The bastard was dead now, but this seemed a nightmare from the past. And what if, despite what he had just told Steve and Darcy, this drone was not sent by the grove owner, but had targeted him or those he loved? Had the explosive been meant to go off and not just threaten?\n\n* * *\n\nTo Claire's disappointment, Ken Jensen was not back from his business trip, and a Sergeant O'Brien was in charge of the chaotic scene and questioning the many witnesses\u2014or targets. The afternoon and evening she had so hoped would go well was a catastrophe. At least people were not seriously hurt but for Dale's broken arm the medics had tended to, though he needed to go to the hospital to have it set and get a cast. The first responders had also tended some cuts\u2014Jace's was the worst. Everyone had ruined clothing from exploding food at the buffet table or bottles at the bar that made them look as if they'd been in a food fight. People kept picking little pieces of glass out of their hair and clothes.\n\nWhen Claire finally went to use the bathroom, she saw her red hair had reddish icing in it, like coagulated blood. She took wet paper towels and tried to pick and wash it out even though that plastered her hair to her scalp. Her head hurt, her eyes ached from crying. This was agony on many levels, but it was fast turning to fury.\n\nAt least Nita and Bronco had taken Trey and Lexi home. Claire went back out into the office Sergeant O'Brien was using to talk to witnesses before he let them leave. Nick and Claire had said goodbye to each guest, apologizing, though no one blamed them.\n\n\"So counselor,\" Sergeant O'Brien, thin and tall with a shaved head, said to Nick, \"hated to save you till last but was hoping what I learned from the others might turn up a motive we could discuss.\"\n\n\"And did it?\"\n\n\"Some things have come up, but I don't want to go into specifics until I hand this information over to Detective Jensen when he returns.\"\n\nClaire said, \"It will be easier to put that window and the room back together than to salvage our special day.\"\n\nShe sniffed hard to hold back more tears. Hadn't she cried enough? This wasn't like her. She had to regain control. Nick reached for her hand and held it.\n\n\"I'm sorry for your\u2014the loss of your celebration,\" O'Brien said. \"But the thing is, there was an explosive head on that drone.\"\n\nNick squeezed her hand so hard she winced. \"Which obviously did not detonate,\" Nick said.\n\n\"Luckily for all of you, it didn't. Our forensic team disarmed it and took it to the lab to see if it was a dud or malfunctioned. We already have a search warrant for the grove owners' house and property. The, ah, Lavell family,\" he said, glancing down at his notebook, \"the original property owners who've been harassing the club over the price of extra land. If it was Lavell, stupid of him to use a drone, since he's used them before around here, and it can probably be traced to him.\"\n\n\" _If_ it was him,\" Nick muttered. \"I'd like to think so. Then he could be locked up, and we won't have to look for worse, more personal motives.\"\n\nClaire recalled Nick saying the same about why Dale could not have killed Cyndi. Who would do something as obvious\u2014especially a trained, bright lawyer\u2014as putting his former fianc\u00e9e he'd just strangled in his mother's freezer, then let the new owners in to clean the house? Could the grove owners be that stupid, take that risk? And eons ago, would Hunter have stabbed Reaching Woman with a hunting knife that looked just like his own? She had to ask Andrea to see those artifacts they guarded so closely.\n\nShe and Nick answered the sergeant's questions and went over the guest list with him. \"No, I don't have a pending case where I've received threats,\" Nick told him in answer to the next question. \"Of course, there can always be disgruntled former clients or, usually, those our firm beat in court.\"\n\n\"Mrs. Markwood,\" the sergeant said, turning to her, \"any possible suspects who could be trying to warn or hurt you?\"\n\n\"There have been times I would have said so, but nothing I know of that's current.\"\n\n\"Then my report is complete for Detective Jensen,\" he said, snapping his small notebook closed. \"I can tell you this much to ease your minds. There are two possible targets besides the country club itself, just my observation, but this will all immediately be handed over to him.\"\n\nTwo possible targets besides the club? Those words echoed through the wall of Claire's anger and exhaustion. That made three total possibilities? Well, thank the Lord, as this man had said, for once it wasn't them. But who then?\n\n* * *\n\n\"Sad to think that everything got ruined,\" Claire said to Nick over, of all things, the comfort of hot chocolate on that warm evening. Their kitchen clock said ten, and they'd finally gotten Lexi to sleep. Trey had gone back to his crib with hardly a peep. Ah, to sleep like a baby. She prayed neither of her children would inherit her narcolepsy.\n\n\"At least no one was killed or maimed, if that was the intent,\" she went on, sounding so weary to herself. \"Or maybe that was meant as a warning\u2014that the explosive was a dud this time, but look out next.\"\n\n\"Not a dud, I'll bet, but didn't detonate. That grove owner must be a nutcase.\"\n\n\"Nick, I'm angry. Everything seemed so perfect, but I guess we've never had that. Oh, forgot to tell you, Bronco said he's going to stay up all night, sleep on the couch in the Florida room, prowl around. Poor Nita with that body in the freezer at her home\u2014then we get ashes on a car, and someone explodes our party...\"\n\n\"Two other possible targets in the group besides the poor country club that idiot keeps harassing. Let's check on our dynamic duo of kids again, then go to bed. In the morning, things will seem more\u2014well, better, if the aftermath of this publicity and your working with ancient bodies can be called normal.\"\n\nBefore they could get up, Nick's cell phone sounded\u2014the hard-driving music from the original but now defunct TV series _Law and Order_.\n\n\"Nick Markwood here.\" He listened for a moment, then mouthed to her, \"Ken Jensen.\"\n\n\"Yes, okay,\" Nick said. \"Tomorrow morning we'll wait for you here. Seven a.m., then. Drive carefully, detective, and keep your eyes out for drones, especially when you get into Naples. You probably know more than we do about who that damned thing was aiming at, because we sure can't figure it out.\"\n\n# 16\n\nNick kept pacing while Claire fixed more coffee and set out bagels for Ken Jensen.\n\n\"I'd forgotten today's Memorial Day,\" she told him, \"and you won't have to go in to work. Nothing seems real right now\u2014a big blur. I ignored my herbal stimulants and took my hard-hitting meds to get through this interview and handle the bad memories of yesterday.\"\n\n\"I know, sweetheart. Hard-hitting stamina is what we both need. At least Jensen's been straight with us before. I trust him, and I think he trusts us. Hey, there he is, pulling into the driveway in an unmarked car, right on time. I wonder if he even went home last night after driving back from Georgia.\"\n\nNick greeted him and brought him into the kitchen. He was grateful to be able to trust this man, though he had no doubt that his job would take precedence over any friendship the three of them had built.\n\n\"Thanks, Claire,\" Jensen said as he sat down with them at their kitchen table, and she put a steaming mug of coffee in front of him. \"Could smell the coffee when I hit the door. Wish I could mainline caffeine into my veins right now.\"\n\n\"I know what you mean,\" she said. \"We're running on adrenaline and anger.\"\n\n\"So how was Zebulon, Georgia?\" Nick asked. He knew they'd soon move on to the country club mess, but he hoped Jensen had found out something to incriminate Cyndi's brother or first fianc\u00e9 to take the heat off Dale.\n\n\"Scenic, if you like to see the Stars and Bars still flying over the old courthouse and statues of Confederate generals. Let's just say the two Southern gentlemen of interest were gone with the wind and haven't been back home for a while, and no one knows where they've gone to grieve for Cyndi's death. Yeah, they're in the mix of suspects for ashes on cars and even drone attacks\u2014and maybe murder one\u2014so I hope we can find them hanging out around here.\n\n\"But to business, because I need your help,\" Jensen went on. \"Once again, welcome to the law enforcement club. I got the info on the interviews from O'Brien about the incident,\" he said as he paused to down some coffee. Nick leaned forward, his chin in his hand; Claire clasped her hands in her lap. Jensen did not have a notebook as O'Brien had. How could he look so awake and seem so sharp after driving back from Georgia?\n\n\"Okay,\" he said, looking from one of them to the other, \"here's the nitty-gritty. There was an explosive on that drone, and not one damned fingerprint. The lab guys think either the explosive was not set to go off, or the pin was jostled in flight or when it hit the window.\"\n\n\"If their assumption's right,\" Nick said, \"maybe the drone was a threat rather than attempted murder\u2014so far.\"\n\n\"Maybe.\"\n\nClaire said, \"Sergeant O'Brien admitted that there were two people in the room who might have been targets, and, for once, we think, it wasn't us. Can you share what you know with us\u2014to help us\u2014to let us help?\"\n\n\"Forewarned is forearmed, right?\" he said, looking from one of them to the other. \"I agree. But first\u2014sorry, you two\u2014but looking at your track record for attracting trouble, I need to ask you both again. Has anything other than Dale Braun's fianc\u00e9e's murder come up in your lives lately\u2014anyone who could be out to get you? Anything at all suspicious, unusual, out of order?\"\n\nNick shook his head. \"Not even volatile legal cases, though I'll have a look back at the files we've always kept on possible explosive situations\u2014sorry to use the word _explosive_ \u2014that we've kept at the firm for years for that very reason.\"\n\n\"Good. Let me know. Claire? Any new dangerous assignments or kooks you're working with lately with your Clear Path online site or in general?\"\n\n\"I am on a part-time assignment but nothing unusual there,\" she told him, feeling like a total liar the way she'd worded that. Then it hit her. That person who had seemed to be watching her and Kris...in the rush and chaos, she hadn't even told Nick, and if she blurted that out now he'd have a fit. But she had to share it.\n\n\"There was something minor I haven't even had time to mention to Nick yet\u2014didn't think it was much, but it was strange. It happened when I was shopping late last week with my old friend Kris Kane\u2014who was at the party.\"\n\n\"Right. The archaeologist. She insisted there was nothing strange going on in her life.\"\n\nClaire sat up straighter. Of course Kris would say that to hide top-secret Black Bog. She hoped Nick didn't decide to breech her contract, but he only shifted in his seat and said nothing. \"Yes,\" she said. \"Anyhow, when the two of us were shopping at the Waterside Shops on the Trail, I noticed a person who might have been following us and mentioned it to Kris.\"\n\nNick sniffed in a big breath of frustration but didn't interrupt.\n\n\"Description?\" Jensen said, taking another long drink of coffee, then finally producing a notebook and pen.\n\n\"Kind of difficult to say, because I couldn't actually tell if it was a man or a woman. Loose, nondescript black clothes. Baseball cap pulled low. Big dark sunglasses. Thin.\"\n\n\"Did this person approach you or Ms. Kane?\"\n\n\"No, but when I went out of the dressing room to a clothing rack, he or she was staring close through the window. We saw the same person when we went to our cars but he\/she just walked away, and we left in a hurry.\"\n\n\"Duly noted, but then both of you are attractive ladies. Besides, that shopping area had some shoplifters lately, stupid ones who stick out like sore thumbs but we just haven't nabbed them yet. Let me know if either of you see that person again.\"\n\nNick added, \"And perhaps let your husband know too. So, Ken, can you tell us who the two guests at our reception were who might have been targets or had some link to a drone or whatever Sergeant O'Brien was alluding to?\"\n\n\"Sure, because I want your help to find out more, maybe things you and Claire can get out of them the police couldn't. Both seemed to be holding back on naming specific suspects we could talk to. I was surprised you didn't know because you've been quite close to both of the people who consider they could be in some danger\u2014Jason Britten and Hector Munez.\"\n\n* * *\n\nJace was annoyed that Mitch was late when he joined him at the back corner table of the Skillets Sunrise Caf\u00e9 on Airport Road in Naples. They both liked the place, partly for the food, partly for the sounds of planes flying over from the airport just down the road. Frowning, Jace looked at his watch\u20147:30 a.m. Damn, even frowning hurt the cut on his face.\n\n\"We've got to stop meeting like this,\" Mitch told Jace with an exaggerated waggle of his eyebrows.\n\n\"Very funny. I am starting to feel like we're hiding out, though. You think that drone attack I told you about could have been meant for me?\"\n\nMitch slid into the booth opposite Jace, who had positioned himself so he could see the restaurant. Let his former superior in the Stingray program look at the damn boring wall. He only hoped he could convince him of what he'd already set in motion. Mitch thought he held all the cards, but they were about to become equals in their next endeavor\u2014he hoped.\n\n\"Sorry I'm late,\" Mitch said. \"Before I got your text about the drone attack\u2014and the friend you want to set me up with\u2014I was out like a light last night.\"\n\n\"Yeah, well, we're both going to be out for good if we don't pull off this change of occupation and let whoever might be targeting us know we're out of Stingray. Not only me, but you could be next on a hit list.\"\n\n\"Yeah, especially if you lead the hit man of some drug czar we've been following to me at this restaurant. I took a roundabout way and watched to see if I was being tailed like you suggested. But, really, sorry everyone went through that attack\u2014and thank God no one was seriously hurt.\"\n\n\"I didn't text this but a junior lawyer from Markwood's law firm did get a broken arm when he fell to the floor in an awkward position.\"\n\n\"Not the guy suspected of killing and freezing that woman?\"\n\n\"You got it, sad to say.\"\n\n\"How's that rather obvious cut on your cheek? You texted you were going into an all night ER for stitches.\"\n\n\"Fourteen of them. The only good thing was Brit was there to hold my hand. Hope I don't look like Frankenstein's monster when I get this bandage off.\" But maybe, Jace thought, it would add to what Kris Kane had kidded was his \" _Top Gun_ and Viking allure.\"\n\n\"So,\" Mitch went on, \"you're thinking you could have been the target, because of the connection to the Stingray pilot killed by a drone in California? Tell me about this secret plan to let our enemies know to lay off\u2014and tell me about this Kris Kane you mentioned.\"\n\n\"Okay, but something else first,\" he said, leaning closer across the table and lowering his voice despite the restaurant's buzz of voices. He stopped talking entirely when the waitress came to fill their coffee cups and said she'd be right back. \"I've got a reporter from _The Naples Daily News_ joining me in about a half hour\u2014or us, if you agree to my plan. If not, I'll go solo with it.\"\n\n\"You're getting desperate,\" Mitch said, leaning forward and keeping his voice down.\n\n\"I'm getting real. I've got a daughter and a fianc\u00e9e here, and I'll have to steer clear of both of them if I don't get some life insurance\u2014make sure that explosive drone wasn't meant for me or else make sure they're not in the next line of fire. You know those drug boys don't mind wiping out related lives as a warning to others. So, yeah, I think a newspaper interview I mentioned before is the way to go\u2014life insurance. We are now flying hurricane hunter planes as full-time work...yada, yada, yada...\"\n\n\"Hell, we can't let on about Stingray to a newspaper, even if we're getting out of it.\"\n\n\"Would you just listen for once? Here's all the reporter\u2014and anyone out to get us\u2014needs to know. We are just retired Iraq war veterans who have been battling the Zika virus and citrus crop pests and blight, but now we have both been accepted to the government team that flies into hurricanes to save lives. It's our new career, and we're dedicated to that effort. The reporter already liked that slant.\"\n\n\"But do we think a public story like that will\u2014well, will fly?\"\n\n\"See, that makes us community good guys, heroes again. I'm hoping, if the drug cartel we've been spying on realizes that we're not living in the shadows but are public faces, they'll lay off. Otherwise, they'd have public opinion and the press after whoever harmed us, as well as the police on their backs.\"\n\n\"Okay, got to hand it to you,\" he said with a sigh that made his usual erect posture and square shoulders slump. \"Sounds like a plan, flyboy, and I'll hang in as your wingman, even though I don't like the 'coincidence' of the California Stingray pilot and his death by drone. But what if this reporter really wants to talk to you because you were at the site of the country club drone attack? Bet they do a story on that today or tomorrow.\"\n\n\"They did today, but it was all about harassment of the golf club by the owner of the grove. The article had some interviews with golfers he's bothered when they were on the course. The guy, Lavell, was arrested but posted bond already. He's got to have a screw loose to use rotten oranges and drones to harass the players and club.\"\n\nTheir waitress came back, all bright-eyed, which annoyed Jace. However exhausted and nervous he was, Brit had been all chatty this morning too, talking about how sad it was for Claire and Nick to have their reception ruined, how they had to be sure it didn't happen to them. Of course, she had no clue that he could have been a target of that damned drone, but he supposed he had to level with her about that. She was going to have a fit if he said she should move out and avoid him until he was sure the coast was clear of more deadly drones.\n\nThey each got their second jolt of java, then ordered pancakes and eggs. \"So,\" Mitch said, looking at the cute waitress as she sashayed away, \"I repeat, tell me more about this lady you talked to about meeting me. If this local media blitz of yours doesn't work, I don't want to endanger her.\"\n\n\"See, now you're starting to think like me. I passed you off as a vet who just never had time to settle down, not that you were a local kingpin of secret ops.\"\n\n\"Okay. What's she like?\"\n\n\"Kristen\u2014goes by Kris\u2014is probably about five-foot-five, around age thirty, good-looking. Much more petite than my ex, Claire, who's a friend of hers from way back. Educated and well traveled. An archaeologist, though I think she's just working with one of her former professors right now as a consultant\u2014something like that. She did say to tell you she has a problem though and\u2014\"\n\n\"I knew it. Twice divorced? Actually hates men? Has three legs?\"\n\nJace ignored that. Their food came. \"Well?\" Mitch demanded.\n\n\"She has what's called face blindness, which simply means she can't recall faces enough to remember who is who, so she looks for signs like hair color, clothing, voice recognition, stuff like that.\"\n\n\"Wow. Never heard of it. Tough lady if she copes with that but has done well in life,\" Mitch admitted as they tucked into their food. \"Wonder what she'll remember about my face.\"\n\n\"That it has a smart mouth and roving eyes. Claire says Kris is very bright,\" Jace went on while he poured maple syrup on his pancake stack. \"Of course, if she puts up with you after one date, I'd have to question that.\"\n\n\"Very funny.\"\n\n\"You want me to set something up?\"\n\n\"Just get me her number and ask if I can call her, okay?\"\n\n\"I just happen to have her number here,\" he said and dug out a business card Kris had given him. Yeah, he'd remembered right, he thought as he handed it over. Kristen Kane, Educational Archaeological Consultant.\n\n\"If this works out,\" Mitch added with a little laugh while he studied the card, \"maybe we could double-date. Sorry if I sound like I'm fifteen. It's been a while for me since I met a lady I really cared about. And I won't do more than phone her yet in case we're targets.\"\n\n\"Sounds good. We both need something good, right? Maybe wait to call her, though, until we see if our job-change declaration goes public. Hey, I think I see the reporter,\" he said, squinting past Mitch toward the restaurant entrance. \"At least he was willing to meet with us on Memorial Day. He's got a camera too. We ready to really step out into the light with this?\"\n\n\"It's a gamble we don't tick off the drug boys this way too, but I'm thinking it's the lesser of two evils. If we pose outside for a photo,\" he added as Jace half rose and gestured to the man, \"let's just be sure there are no damn drones in the sky.\"\n\n# 17\n\nClaire had Nick's breakfast ready when he came into the kitchen the next morning. She perched across the table from him and waited at least until he drank his orange juice.\n\n\"Nick, if Jace is being watched or targeted somehow, probably because of his Stingray flights, he shouldn't be anywhere near Lexi right now. Actually, he should steer clear of Brit, too, until this whole thing is solved and settled.\"\n\nHe put his juice glass down and narrowed his eyes at her. \"Nor should he be near you or this house.\"\n\n\"Right, but he's not going to take any of that well, at least not to stop seeing Lexi. And just when the two of us were getting along better. Tomorrow's his day to pick her up after school and take her to the zoo or on some adventure. But surely he'll see we'll have to cancel that.\"\n\n\"Explaining it to Lexi will be just as hard.\"\n\n\"She's been through a lot because of us, and I'd just die if she was ever abducted or hurt again. She'll freak out if I try to tell her that her father's in danger. Jace is going to have to go along with us\u2014say he's busy, something. Maybe he can say he had to leave to be a hurricane hunter early, since Brit says that's in his plans.\"\n\nShe thought again of the Hunter of old. What was his relationship to Reaching Woman who had lain beside him all those centuries? Or was she tied somehow to Leader\u2014or to both of them and that had led to her terrible fate?\n\n\"Earthman Nick Markwood calling spaced-out wife Claire. I said I have to take my coffee and go, sweetheart. Since Lexi's not down yet, kiss her for me before Bronco drops her off at school. I told him to come into the office late, hang around here a little longer. If you feel you need to call Jace, go ahead, but if he's being targeted, his phone could be bugged. How about I call him from the office?\"\n\n\"Yes, all right. Then let me know somehow as I'm not going out to the bog until about two. This morning I'll spend time with Trey. I plan to catch Brad Vance this afternoon since he seems to be the keeper of the keys to view the artifacts from the graves.\"\n\n\"Sounds like a plan\u2014one of them,\" he said in a teasing voice, obviously trying to lighten her mood. As if they had not a care in the world, he tossed his keys in the air and caught them in a quick move behind his back, though he was in full lawyer dress mode: striped tie, cufflinks, power watch, suitcoat over his arm and newspaper under it. \"Don't you just wonder,\" he said as he leaned down to kiss her goodbye, \"what archaeologists would think of us and our artifacts if we were buried with our favorite toys?\"\n\nShe nodded but didn't answer. Had Leader with his striped robe and staff and dagger chosen to be buried with those items or had they been selected by someone else so he would have them for his life-in-death in the bog? She would never really know, but somehow she was going to dig deep into all their lives and present the Vances\u2014and eventually the world\u2014with the very best theory she could figure out.\n\n* * *\n\nThank heavens Jace agreed with them about not seeing Lexi right now, Nick thought as he hung up his office phone after a short, intense talk with him. Nick's phone had protection on it to scan for bugs or third-party eavesdropping. Jace was taking measures, he said, to end any relationship he'd had with the Stingray program. And he didn't have to worry his contact wouldn't take it well because the guy he referred to only as \"his wingman\" was also taking a new job as a hurricane hunter.\n\n\"As a matter of fact, check _The Naples Daily News_ for a feature article announcing that to the world,\" Jace had said. \"Life insurance, though I'll give it a little time before coming near Lexi or you guys again. But you and Claire have to promise to explain to her that it wasn't my choice\u2014without upsetting her. Easier said than done with that kid, I know, but Claire's not a psych major for nothing.\"\n\nNick wanted to call Claire right away to assure her Jace would stay away, but _would_ that assure her? No, it would make her sad, and not only because Lexi couldn't see Jace. Nick knew Claire still cared for her first husband, though he didn't worry about that. Not too much, that is, especially now that Jace had Brit.\n\nA knock on his door jerked him back to reality. Cheryl stuck her head in. What would he do without her? She was almost old enough to be his mother and sometimes treated him that way, but he gladly kept her on and would miss her when she retired. Besides, she was a link to his father. She'd been a temp here at the firm when his father had died so terribly. After law school, when he'd tried to fill his father's very big, empty shoes, she had guided him through more than one complication of taking over the law practice.\n\n\"Is Heck here?\" he asked.\n\n\"On his way.\" She came clear over to his desk. \"You doing okay after that chaos yesterday?\"\n\n\"Not really, but one foot ahead of the other. I'm grateful you and the staff weren't hurt\u2014but for Dale.\"\n\n\"Not an omen for his situation, I hope. He's living in fear the police could arrest him any day. It did hurt me to see you and Claire suffering. You've both been through a lot. Oh, Dale called a while ago and insists he's coming in soon, even though you told him to rest. I understand he needs to keep busy.\"\n\n\"Yeah, I get that too. Speaking of which, Madame Secretary, I promised Detective Jensen that I would look through our files to see if there were any loose cannons from past cases who might hold a grudge against me or the firm. I can think of a few but I don't want to miss anything.\"\n\n\"You want me to look in my spare time\u2014ha, ha\u2014or just pull case files for you?\"\n\n\"All of the above.\"\n\n\"How far back, Nick?\"\n\n\"I'd say ten, fifteen years. I can't believe someone wouldn't act on a serious grudge by now if the firm or I let them down.\"\n\n\"Got it. Got your back too,\" she said, heading away. \"Don't know what you'd do without me,\" she teased.\n\n\"You know me too well!\" he called after her as she went out, but she popped right back in. He was expecting another rejoinder, but she said, \"Hector Munez here to see you, Attorney Markwood.\"\n\nHeck came in without the usual swagger. He looked as if he hadn't slept either.\n\n\"Sorry, boss, but I drove Gina back to Miami, and we left at four in the morning. Just got back. Hated to leave her there, but she's adjusting. Can't say I am without her.\"\n\n\"No more bruises or hard knocks after that mess yesterday?\"\n\n\"Only that I still worry I'll lose her\u2014to some guy, some med student or doctor over there. Anyway, I know what you want.\"\n\n\"Tell me then. What did you tell Sergeant O'Brien that made him put you on the possible hit list for that drone? Something with that new facial recognition deal?\"\n\n\"Yeah, _caramba._ I had no idea what I was getting into, the cut-throat part of it. Facial recognition is about to become the next big thing in security software. There's ways to hack into digital devices and locations and the like with fingerprints, eye or voice recognition, but facial is a lot harder to fake. Whoever controls facial rec tech is gonna be filthy rich\u2014big, big bucks. So maybe sharks are circling to scare away or knock off people who could help their competition.\"\n\n\"So dirty play competition is of concern? And someone has figured out you're good\u2014and might go with a particular firm?\"\n\n\"And maybe trying to scare me off. Or kill me and anybody nearby who might have a clue why. So sorry to tell you all this, that I might be a sitting bird on a wire.\"\n\n\"I think the phrase is a sitting duck, Heck. But I get that this is going to lead to new frontiers of privacy laws.\"\n\nHe nodded, kept nodding. \"That's why I told you about it in the first place, plus I wanted your okay to be part-time with them. But I'm feeling scared, boss, real scared. I'd rather live off Gina's doctor salary someday than have a lot of money and be in someone's crossover hairs.\"\n\nNick didn't correct his English this time. He knew damn well what his friend meant. And he knew he'd give up this law practice and his covert South Shores investigations\u2014all of it\u2014to protect the ones he loved.\n\n* * *\n\nClaire had two goals at Black Bog that afternoon. One, to view the bodies, especially the faces, of the three who had shared the latest excavated grave. Second, to get a close look at the artifacts which had been buried with each person, and which had been taken quickly away for cleaning and preservation. She thought it was rather sad that those items which had been with them so long should be separated from them now.\n\nShe felt much more relaxed and grounded after a morning holding and playing with Trey. He was getting so responsive, and despite the fact he was only a little over three months old, developing a personality.\n\n\"Hope today's been better,\" she greeted Kris as she went to her own corner desk and laptop in their shared office. \"No sore spots or bruises from the shock yesterday, I hope?\"\n\n\"I'm fine. And I won't even say your party was a blast,\" she said and hugged her, while Claire gave a dramatic groan. After all, Claire thought, what was there to do but plunge ahead and be grateful for the support of friends and family who had\u2014for the most part\u2014not been hurt.\n\n\"So,\" Claire said, putting her purse in a drawer and turning on her laptop, \"think you'll hear from Jace's pilot friend?\"\n\n\"Don't know, but it will be interesting. I need something like that, an interest in a live man instead of the other kind. What's that motto on your website, 'The dead still talk if you know how to listen'?\"\n\n\"And I'm hoping they do, especially around here. I'm going to ask Andrea for a close look at the new trio. Want to come along? And then\u2014what's the protocol for getting a glimpse of the artifacts from Brad?\"\n\nKris didn't answer at first. Had she even heard that last question?\n\n\"He does huge blow-up photos of each one,\" she finally said. \"Of course, he'll let you study those.\"\n\n\"But not see the items themselves?\"\n\n\"They're precious, priceless, and it's not good to handle them, so much as breathe on them, once they're cleaned.\"\n\nClaire sank into her swivel chair and rotated it to face Kris. \"So I can view them only on my laptop screen?\"\n\n\"On his. It's like he's keeper of the keys for the artifacts, though I can see why.\"\n\nClaire felt deflated and a bit frustrated. Brad didn't go out to the digs, had claustrophobia and guarded the artifacts even from the team? Well, everyone had their odd habits and quirks. And she should talk. She understood how important, how consuming it could be to protect precious things and precious people.\n\n\"So,\" Kris cut into her agonizing, \"let's go to Andrea and take a look at our three somehow related bog people. You can ask Brad to view the artifacts later. Maybe we can get a theory on these unusual bodies before they dig up any more.\"\n\n* * *\n\nNick was scanning some of the files Cheryl had brought him when his desk intercom rang. Cheryl said, \"Bronco called from his post downstairs. Another most interesting person is insisting on seeing you.\"\n\n\"Tanner Linschwartz again or another guy from Georgia? Call Jensen and tell him before Bronco even passes them into the lobby. We might need security.\"\n\n\"No, it's a very irate woman who is throwing her weight around\u2014which isn't much. I took a look at her on the security camera down there. A Ms. Marian James from a local group called the Endangered Properties Committee. And she says if you don't back off, she's going go public with your obstruction of rights and sue Dale and the firm.\"\n\n\"Actually, I've been expecting something from her. Is Dale here?\"\n\n\"He came in about a half hour ago, and I hear he's on pain pills and not looking good.\"\n\n\"Keep him in his office and have Bronco escort her immediately and politely all the way to my office, then hold my calls.\"\n\n# 18\n\n\"I want to really get started with my study of the Hunter, Reaching Woman and Leader today,\" Claire told Andrea and Kris as they walked through Andrea's office to the back room that stored the bog bodies in deep drawers. \"If I can just study these remains and the artifacts, I believe I can come up with a hypothesis and then consult with all of you.\"\n\nAndrea didn't so much as blink at her saying she wanted to study the artifacts, so perhaps she'd be more of an ally than Kris in getting Brad to show her those.\n\nAlthough Claire had been present at the disinterment of all three bodies, she steeled herself as Andrea opened the first drawer, now marked Hunter of Trio.\n\nThere was no gut-wrenching smell as Claire had learned to associate with a coroner's office, just an earthy odor. Hunter was still partly covered with the tarp he had been carried in, curled up almost as if he were kneeling. Even under the bright blue tarp, Claire could see his chest was hollowed out.\n\n\"We'll be working on examining him tomorrow,\" Andrea told her. \"After a thorough outer examination, we'll remove all three brains for study, then reconstruct head and hair.\"\n\nThe pelt Hunter had worn for centuries was matted with peat and blotches of blood, but Claire could still tell it was animal hair. She noted it might have been made extraheavy, two-layered, up around his big shoulders. Had someone lovingly sewn it that way to keep him warm in the winter? Or to buffer something heavy he carried over his shoulders? But it appeared to have been pulled awry when someone had stabbed him, stolen his heart and life.\n\n\"Was his heart found in the grave?\" Claire asked.\n\n\"The team looked around carefully,\" Andrea said. \"They found only that dagger in his hand, encrusted with either peat or blood. Brad's examination of it will let us know soon.\"\n\n\"If there is blood on the dagger,\" Claire said, \"Brad won't wash it off, will he?\"\n\n\"Eventually. What does it matter? We can't type their blood.\"\n\nAs accommodating as she had started out, Andrea seemed in a hurry now, even on edge. Claire wondered if she'd overstepped. To answer the questions they had hired her for, surely they knew she had to probe.\n\n\"Could I see the other side of his face?\" she asked, shifting her position so their shadows from the bright overhead lights didn't obscure him.\n\n\"Of course,\" Andrea said and slowly lifted the top of the tarp away.\n\nClaire moved around Kris to look at the other side of Hunter's face she could not see from this angle. The right side was somewhat smashed but this left side had fared better.\n\n\"Oh,\" she cried, \"he has a slashed cheek! I mean a scar, long and deep, not new. Well, definitely not new.\"\n\nBoth women gathered close and edged her out to see it themselves. It reminded her so much\u2014too much, as if it were some awful omen\u2014of the cut on Jace's face.\n\n* * *\n\n\"So you can get an article in the paper, but I can't!\" Marian James shouted and threw a copy of today's _The_ _Naples Daily News_ on Nick's desk.\n\nHe had asked both Bronco and Cheryl to stay, and they sat against the wall in the back of the room by his bookcases.\n\n\"Please sit down, Ms. James, and we will talk rather than shout,\" Nick told her, ignoring the newspaper at first. She must be referring to the article about the drone attack at the country club.\n\nShe sat, perched ramrod stiff on the edge of her chair. He slid the paper closer. Where had he put his own copy of it? He'd left the house with it. Too damned distracted. Maybe it was still in the car. He'd read it in a hurry, not much of it at that.\n\nHe glanced down at it again, a small article, thank heavens, including a picture taken of the shattered window from the eighteenth green. At least there were no photos of the sad chaos inside the verandah. And, he thought, at least the article didn't include a guest list and only quoted the club hostess. Nothing about Jace's or Dale's specific injuries, no other interviews except a brief comment by Sergeant O'Brien.\n\nBut in the second section that had slid out from under the front page was an article he hadn't seen. Jace, face bandage and all, and\u2014oh, yeah\u2014he was with his pilot friend, Mitch Blakeman. They were both smiling in front of the nose of a jet at the Naples Airport. Local Vets to Hunt New Enemy\u2014Hurricanes, the headline read. He and Claire had been so intent on the drone article, they hadn't looked beyond.\n\nHe had to tear his attention away from that back to the irate woman, but at least she seemed to have calmed down a bit.\n\n\"Despite what Cynthia Lindley told you before her death, Ms. James,\" Nick said, trying to steady his voice, \"access to information and legal entry to the grounds of the Twisted Trees mansion was not hers to give. Mr. Braun has every right to keep those ruins and information about their long dead original owner private.\"\n\n\"It's living history! I'm sure we could find a buyer for the property if he doesn't want to share it with the public. As I said, our committee is not some fly-by-night organization but a well-funded one with some influential people on its board. And the German-Nazi connection makes the Braun mansion all the more important and valuable. Why, that's of international concern. People have a right to know!\"\n\n\"Perhaps at a later time. I congratulate you on your admirable efforts but\u2014\"\n\n\"What if there are artifacts there? This could even go into the realm of the state government control at the least, national interest at most. Magazines like _The Smithsonian_ and _National Geographic_ would be all over this, provide funding for reclamation and restoration of the site. And why is that carriage house or servants' quarters or whatever it is boarded up and fairly intact when the rest of the place is in shambles?\"\n\nNick had to admit that this woman did raise some interesting questions. He had no answer for that. Did Dale?\n\n\"I know Mr. Braun has problems of his own right now,\" she went on in a calmer tone, \"but please put this to him again, or I will have to bring more pressure to bear\u2014and publicity. Yes, I know you said you'd insist I was trespassing when I took those photos. But since I'd earlier been invited by Mr. Braun's fianc\u00e9e\u2014\"\n\n\"Former fianc\u00e9e, not a family member, and now, sadly, deceased.\"\n\nMarian James shot to her feet so fast, behind her, Bronco stood too. \"I have a recording of the woman's phone call to me! I know the timing is bad for this, but time is of the essence for derelict buildings, just as it is for murder suspects like your client. Please inform him he needs to cooperate with what will be very generous offers. On the other hand, I can at least let the media know that I had a call from a soon-to-be dead woman and have every right to share what she told me. I will, however, think that over for exactly one week before I act. I assure you, I and my committee will be in touch, but perhaps you will want to be in touch with me first.\"\n\nShe seized the newspaper from his desk and made a fast exit with Bronco right behind her, hopefully to be sure she didn't try to get to Dale.\n\nCheryl closed the door behind them and leaned against it as if to block the woman out. \"Do you want Heck to research her and her committee, Nick?\"\n\n\"He's got enough on his mind. I'll have Claire check things out with me looking over her shoulder. I'm starting to think I might have missed something at Twisted Trees. But we have enough to do around here. Once you're certain she's gone, send Dale back in if he feels up to it,\" he told her. \"And even if he doesn't.\"\n\n* * *\n\nClaire stared down at the next drawer Andrea pulled out. It was labeled Reaching Woman of Trio.\n\n\"Her pelt is the same kind as that on Hunter, don't you think?\" she asked. \"Do other bodies have deer pelt clothing?\"\n\n\"No, more woven plant material, similar to Leader's robe,\" Kris said.\n\nAt least she was getting steadier and more used to seeing these naturally mummified bodies. Her voice wasn't shaking right now, nor was she trembling, and her stomach wasn't twisted in knots. She was simply left with the feeling of awe\u2014and fierce determination.\n\n\"We'll examine the crude thong stitching on both of the similar deer pelts, but that even looks the same,\" Andrea observed.\n\n\"Like Hunter, she has her legs drawn up,\" Claire said, \"as if she were kneeling\u2014or cold.\"\n\nKris added, \"Or trying to protect herself from that knife thrust\u2014almost a praying position, if they prayed.\"\n\n\"Or one of deference or submission,\" Claire said. \"Maybe she was begging for mercy. Shouldn't that knife have been left in place until she was further examined? I see it's been removed.\"\n\n\"A key artifact to be studied and preserved,\" Andrea recited her usual mantra.\n\nIn other words, Claire thought, although Andrea was the trained archaeologist here, Brad ruled on that. Dare she question him on his tactics to get his hands on those precious items so quickly? Perhaps since he could not bear to see people enclosed in their graves, he was especially eager to oversee his part of this work. Besides the dagger being gone, what had appeared to be polished stones on a necklace spilled off a broken thong from around the woman's neck was missing too. She knew better than to ask, but she planned to face Brad over that. Especially if the stone beads matched the bracelet Leader had worn.\n\n\"Let's just be certain again that her heart is there,\" Andrea said, frowning. \"Hunter's chest was so carved out, but I'm sure her heart is intact, even if she was stabbed in it.\"\n\nKris shined a light she wore around her neck. It reminded Claire to ask for one of those little flashlights. Andrea carefully lifted the hem of the stiff garment from the woman's bent knees.\n\nAll three of them peered up the woman's skirt. Claire gasped. Kris grunted as if she'd been struck, and Andrea swore. Reaching Woman's chest had not been cut open, although she had indeed been stabbed in it. But a second heart, stony, hard and dark\u2014but unmistakable\u2014lay upon her shriveled thighs.\n\n# 19\n\n\"All right, what's up?\" Brit demanded. \"Jace, why are we parking at a police station? Something else about the drone attack? What's going on?\"\n\n\"I wanted to get you here before I explained. Insurance for you and for us,\" he said, killing the engine and unhooking his seat belt. He turned toward her so she wouldn't have to see only the bandaged side of his face. He didn't want to remind her of that attack or scare her more than he had to.\n\n\"I figured you wouldn't like it\u2014would refuse,\" he explained, reaching out to cup her shoulder with one hand. It was more to steady himself than her. \"Nick has set all this up, but I asked him to. I'm going in to talk to Detective Jensen while an officer drives you to our apartment and waits until you pack your things, then follows you while you drive your car to your mother's house. Brit, you have got to stay with her, or at least somewhere else, until I'm sure I'm clear of whoever sent that damn drone. If both of us moved and I'm the target, they'd find us again.\"\n\n\"Whoa! Who would find us? You think that drone was sent to scare\u2014or even kill\u2014you? I figured Nick must be the target. You mean because of Stingray, like\u2014like some criminal found out you're trying to nail drug smugglers?\"\n\n\"That's not exactly top secret anymore. I can't take the chance that you'd be hurt, if I'm the target. If they try again, you or anyone else could be\u2014well, collateral damage.\"\n\n\"So we're going to be apart for how long? What about our wedding plans? Oh\u2014you think because the Markwoods' reception was hit...\"\n\n\"Yeah, maybe. We can't take that chance.\"\n\n\"Don't you think that write-up in the paper about you and Mitch moving on from supposedly crop and mosquito spraying to mere hurricanes is enough to keep the bad boys at bay?\"\n\n\"I'm sorry, honey,\" he said, gripping her hands in his as she turned even more toward him. \"I screwed up, miscalculated. See, there was a Stingray pilot in California who was targeted by a drone\u2014in his plane. It crashed and he died. For all Mitch and I know, a drone's the latest calling card of the Mexican drug mafia instead of a quick bullet in the head by some hit man or enforcer. I just can't take a chance you'd be hurt.\"\n\n\"Or you would be. But for how long? Can we even talk?\"\n\n\"I figure I can call your mother, and she can give you the phone.\"\n\n\"Oh, really, like I'm six years old? And I call you how? Send smoke signals? Use poor Claire as a go-between? I know darn well you still care for her, and she'd do it for you, even if it endangered her.\"\n\n\"Not that now, honey. I'm just trying to do what will keep you safe.\"\n\nShe started to cry. \"So this Detective Jensen agrees you might be the target?\"\n\n\"He doesn't know\u2014we don't know. There are other possibilities, including, as you heard, that the family who owns the orange grove wants more money for an extra piece of land. But that's too obvious.\"\n\nShe pulled one hand free and wiped tears from her face. \"Just like it's too obvious that Dale Braun would have strangled then frozen that poor girl, like she was stored chicken on ice to be devoured later. Someone running and ruining our lives from afar is horrible. I hate it, but I don't hate you. So okay, we'll be apart, for a little while.\"\n\n* * *\n\n\"I hate grilling you on this with all you're going through right now, Dale,\" Nick told him as they sat in Nick's interview chairs at the small table away from his desk. He was trying to trust this junior partner\u2014his client\u2014but there was something strange going on.\n\n\"I understand,\" Dale told him. He rested his cast on the table; in general, he looked like the walking wounded. \"I wish Marian James hadn't turned up right now with everything else falling in on me. I can remember my dad quoting some Shakespeare play\u2014'Thus do all things conspire against us,' __ something like that.\"\n\n\"I'm sure Claire's heard of it, but I do know the feeling. So, help me out with a couple of questions I have about the Twisted Trees estate. Why is what Claire called the carriage house, which I'm sure was used for a garage, so intact-looking compared to the rest of the mansion?\"\n\nDale shrugged, then winced from the movement. \"The hurricane that ripped the roof off didn't touch that. Sturdier built maybe or lower to the ground. Dad used to say that was fate, like there was some sort of reverse curse on that part of the place, after its history.\"\n\n\"I'm not following now. Explain what he meant.\"\n\nDale heaved a huge sigh. \"Sorry if I don't make sense. These pain pills are making me a little spacey. He meant\u2014I found out from my mother years later, because she was bitter and talked more once she had dementia\u2014that's where my great-uncle kept his German mistress.\"\n\nNick sat up and leaned forward. \"No kidding? In a garage?\"\n\n\"There's an apartment on the second floor, above where the cars were kept. That's all I'd need is that woman and her committee getting a look inside there. It's been sealed up for years, like a time capsule or a tomb. Mother said there's Nazi regalia in there, things that no one needs to see, but she couldn't bring herself to clean it out\u2014well, you know she was a hoarder. Now that there's pressure to see it or sell it, I've got to get in there and pack things up, clean it out, but\u2014hell, Nick\u2014I was told not to when I was young and just haven't been able to get up the nerve to do it.\"\n\n\"You're right about checking it out, maybe clearing it out. I can always claim attorney privilege if we find strange things in there, so how about I go with you? Bronco too, to keep a watch for us outside. So your great-uncle brought not a wife but a mistress with him when he fled the Third Reich in its darkest days? She's obviously long gone\u2014dead, or did she return to Germany after the war? And why didn't he marry her?\"\n\n\"Don't know, but no, he didn't bring a wife to Florida when he fled. She had died before the end of the war, some American B-17 bombing raid, so I overheard once.\" Dale grimaced as he attempted another shrug. \"I suppose it might have been secret because, in past times, a mistress was something you hid. And why didn't he marry her? Don't know that either.\"\n\n\"Yeah, you'd think with no wife, she'd be mistress in the big house, especially one so private and plush.\"\n\n\"I've thought about trying to get answers, going through the place, but it just seemed\u2014well, as both my parents used to say, _verboten_.\"\n\n\"Yeah,\" Nick said. \"When I found my father dead with his gun in his hands and half his head gone, no one wanted me to talk about it or pursue it, to prove there was no way he would have committed suicide. But I did and eventually proved it was murder. Then I finally stopped the man behind it.\"\n\n\"I heard,\" Dale said. \"I'm sorry it\u2014it haunted you like this haunts me. But I guess I'm finally ready to have answers, even if they are terrible ones. Okay, before Ms. James and her cabal of secret sponsors push us more, let's take a look at what's been locked up there for decades.\"\n\n* * *\n\nClaire had waited in the hall outside Brad Vance's office for a good ten minutes after he'd said on the in-house phone she could come to see the artifacts. She was excited, pacing. She must have misunderstood that she couldn't see them. He had opened his door a crack to say that he would be a little longer on the phone with what must be an inside-the-grounds call, then closed it again.\n\nShe scolded herself for being tempted to put her ear to the door to see if she could make out anything he was saying, but all she'd need was to get booted out of there for eavesdropping.\n\nThe muted sounds of his deep voice stopped inside. Brad opened the door. She realized that he must have some way to call out of there when no one else did. Or did he have some private line to their house nearby and linked in to that? But no way she was going to pursue that question and seem nosy. She'd just ask Kris later. Her target was getting to see those artifacts. He motioned her in, and she went, her eyes skimming the room for any of the precious grave finds he might have out to study.\n\nBut his examination table under bright lights was bare. The huge computer monitor she'd seen him looking at before was lighted but blank. An even larger screen, one that would have been good for watching movies, was suspended from the ceiling, also blank. No, there was a small caption at the bottom of it: Trio Exhibit.\n\n\"I don't usually keep myself shut in here\u2014open door policy,\" he said as he sat on a high rotating stool and gestured for her to sit in the other one about three feet from him. \"In general, I don't like closed rooms or doors.\"\n\nShe sat and put her feet on the round rail beneath. \"I've worked with people who have claustrophobia,\" she told him. \"It often, as I'm sure you know, stems from a particular frightening incident in childhood that is internalized in an older adult who suffers from that malady too.\"\n\n\"I've looked into it more than once,\" he admitted. \"I just live with it, but that's right, you're a psych major\u2014which is why Andrea and Kris thought you could help. So\u2014to the artifacts. I have them blown up here twenty times their size for comparative study.\"\n\nHer hopes fell. He was still going to have her looking at pictures, not the real items. How much should she protest? He was obviously the man with power behind Andrea's archaeological throne.\n\nBut she was intrigued by what was on the screen: two daggers, no peat or blood on them, evidently chipped out of flint. One was no doubt Hunter's and one the murder weapon that had been in Reaching Woman's heart. So strange, but at the moment, Claire felt as if something pierced her chest with pain. She scolded herself, for she'd always been too empathetic with those who were hurting. But of a long-dead people, buried for centuries in a bog?\n\n\"I suppose you heard,\" she said, \"we found what we believe is Hunter's heart nestled on Reaching Woman's thighs.\"\n\n\"Andrea told me just a few minutes ago. She's hoping you can posit the meaning of that. Punitive? Ritualistic? Sexual? Religious?\"\n\n\"Just think how we might say today, 'She had his heart.' I need time\u2014and clues, like from a close look at the artifacts\u2014to try to figure that out.\"\n\n\"So, okay,\" he said, turning his attention back to the screen as if he did not want to discuss that bizarre find further or her subtle plea to see the real items. \"Both daggers are made of chert, not flint. Chert is a lot more lustrous than flint and a bit stronger. A mixture of chalcedony, quartz and silica, probably from microfossils\u2014even bacteria\u2014over the ages. The greenish tinge may come from eons of alga or moss being pressed together. Like flint, striking a chert dagger against another stone can start a fire, so the dagger could be doubly useful, for instance to cook meat a hunter brought home.\"\n\n\"Do you mind if I take notes?\"\n\n\"Sure, but I'll keep any photos in my control for now. So you can see why I cleaned the bog mud and blood off these weapons.\"\n\n\"Chert does have a certain luster to it. Can those kinds of stones be found locally, or do you think they had to trade for them?\"\n\n\"I'd say they came from somewhere north, but did they trade for them or come from there themselves? So many questions to answer, puzzles to solve.\"\n\n\"If Andrea and her team have turned up other daggers, utensils and the like, were any of them of those substances?\" she asked.\n\n\"No. Much more, let's say, plebian, but nothing this valuable and useful. Of course, there is the fact Hunter's dagger seemed to have carvings on it.\"\n\n\"As did Leader's staff of office or whatever it is. Do you have slides of those?\"\n\n\"Of course, but let's stick to these right now.\"\n\nClaire jumped as a high voice came from right behind her: \"Brad, I have what you wanted\u2014Oh, so sorry. I did not see you had someone here. Hello, Claire.\"\n\n\"Hi, Yi Ling.\"\n\n\"No problem. I will come back later,\" she said, frowning.\n\nYi Ling had nothing in her hands. Claire read that the young woman was used to just walking in there and was rather put out that Claire had been there with Brad. It had seemed to her earlier that the digging trio reported only to Andrea, but she must be mistaken. She sensed strange vibes that bounced between Senator Vance and this petite woman who nearly scurried from the room.\n\n\"Anyway,\" he went on, after clearing his throat, \"we've found flint that would chip away, even wooden utensils, some with sharks' teeth embedded in them to give them bite, so to speak.\" He turned and smiled at her, as if he'd made a joke. His teeth, especially for a man of his age, were\u2014well, telegenic. White and even. Large.\n\n\"Could I actually see the woman's broken necklace and Leader's bracelet, at least?\" she asked, looking down at her notes to break the tension between them. She didn't know why she felt so nervous near him. He had even left the door open, so they weren't really alone.\n\n\"Locked up in a safe for now. Let's stick with these high-resolution photos.\" He moved the slides along, showing what looked like big bone needles, maybe a leather drumhead, a small clamshell cup and a turtle shell that might have been a mixing bowl.\n\n\"There,\" he said, sounding strangely triumphant as a slide appeared which showed cleaned stones with thongs through their drill holes, though the longer one\u2014Reaching Woman's broken necklace\u2014had its polished stones separate and laid out side-by-side.\n\n\"Agate is the material for these,\" Brad told her. \"Lovely patterns called banding, though these are not fine enough examples to be classed as gemstones. A rough surface here and there, because their tools were so primitive. Nothing carved on any of them, though the mud stuck in some of the crevasses made it look that way at first.\"\n\n\"I was hoping so.\"\n\n\"There is carving on Leader's staff, which I'll show you next, and that is a mystery. Nothing recognizable to modern eyes, though maybe you can copy the lines and discern some pattern\u2014primitive writing, even.\"\n\n\"Wouldn't that be a bombshell?\"\n\n\"Speaking of that, I\u2014we\u2014need you to be very certain that you are not being followed or somehow targeted, after that attack on your event at the country club. If something else happens like that, I'm afraid you might need to withdraw from this project. Kris tried to explain it all away, but, as you know, secrecy is of the utmost importance to us here until we are ready to go public with all this.\"\n\n\"I told the detectives it was not aimed at me, though, of course, my husband has a career which has some fallout at times.\"\n\n\"Yes, I should think so. Having been in politics, I can sympathize. I made a lot of friends, and, unfortunately, a few enemies too.\"\n\n\"And despite the fact that it was our event, other people attended besides us, and the country club itself is under a sort of attack by a neighbor.\"\n\n\"Yes, so I gathered from Kris and the newspaper article. Just wanted to be clear that you must bow out if there is any question of your being a chink in our armor, so to speak, a break in the dam of privacy and secrecy here.\"\n\n\"I'm aware of the concerns, Senator Vance, really.\"\n\n\"Good,\" he said with a decisive nod and what she considered to be a forced smile as he brought up the slide of the cleaned and lighted staff. Claire was disappointed. There were some carvings on it but they seemed only decorative. She'd glimpsed the staff briefly, but hadn't there been more, or had the mud and peat just made it look that way? The other side of the staff was amazingly similar but with more peat clinging to it.\n\n\"Let me run you through some earlier photos of artifacts,\" he said after she'd taken some notes without further comment. \"Nothing as grand as the ones found with our trio, that's for certain.\"\n\nClaire nodded, but what was for certain around here? She could fully grasp their need for secrecy with the outside world until they had a chance to study everything, but she had the strangest sensation that there were other secrets being kept around here too.\n\n# 20\n\n\"So is Daddy just too busy for me right now?\" Lexi asked Claire when she tried to tell her Jace couldn't pick her up after school tomorrow and might not come over to see her for a little while. She was sitting on Lexi's bed, trying to tuck her in, though the child kept throwing the covers off.\n\n\"It's just that he's changing jobs.\" Claire tried another tack.\n\n\"Then doesn't he get a little vacation between them? Or does he just want to spend time with Brit now 'cause he loves her more.\"\n\n\"No, sweetheart. He will always love you, whether or not Brit is in his life. The way he loves Brit will be like a husband and wife, but you are special, his only and very best girl.\"\n\nLeaning down, she hugged Lexi closer. \"I don't know,\" the child said with crossed arms and a big sniff, \"'cause I heard him say Brit was his girl.\"\n\nClaire knew she was doing a bad job of explaining this, but what to say? Nice to have all those psych classes years ago and a darned diploma on her wall, but she was failing miserably counseling her own daughter. The problem was she felt protective of both Lexi and Jace in this situation. Her heart went out to both of them...both of them...again she saw that horrid heart cradled in Reaching Woman's thighs.\n\nShe shook her head to clear that dreadful picture. \"You know grownups have problems sometimes,\" she told her daughter. \"But it doesn't mean that you are not dearly, dearly loved, and Daddy will make it up to you when he can.\"\n\n\"Can you pick me up at school instead?\"\n\n\"You must ride the bus like usual. Okay? Hey, how about we have mac and cheese tomorrow night and make dinner together? I have to go see the people I'm helping tomorrow afternoon.\"\n\n\"Mac and cheese is okay, but I'm kind of sad. I want to call Daddy and tell him.\"\n\n\"We'll call him later\u2014sometime. Lexi, he's feeling bad too, and he can't help some things right now.\"\n\n\"Sometimes I wish my 'maginary friend would come back, but I think she died.\"\n\nClaire held her even closer. \"Don't say that. Remember, you finally let her go off on a trip, but you don't have to let Daddy go. You know Dad Nick loves you too,\" she said, evoking Nick as Lexi had come to refer to him.\n\n\"It is funny to have two dads and you had both of them for husbands, right, Mom?\"\n\n\"It happens, my dear girl. You know, I think I'll just stay right here until you go to sleep.\"\n\nFeeling exhausted and overwhelmed\u2014and so inadequate\u2014Claire turned off the bedside table lamp and stretched out beside her. And found herself wondering if Reaching Woman had loved both of the men she had slept beside in the bog, almost for eternity.\n\n* * *\n\n\"I didn't know where you were until I looked in on Lexi,\" Nick said later that night while she reheated supper for him. \"Sorry to wake you up.\"\n\nHe hadn't wanted to get home late, but he had. Too much time spent on current cases after he'd sent Dale home early.\n\n\"Yeah, I conked out having a heart-to-heart with Lexi about why she can't see Jace for a while. I did a lousy job of it, I'm afraid.\"\n\n\"I don't mean to put this on you when you're tired, but maybe tomorrow, as soon as possible, I'm hoping you can do me\u2014and Dale\u2014a favor. I know you'd once started to research Cyndi Lindley and were on her Facebook page, but if you could turn up anything else about her online, it might help. Her phone call to Marian James is causing a mess. The woman has given me and Dale one week to let her release her photos and info on Twisted Trees or she's going to publicize the phone call, and that won't help Dale. Frankly, I'm hoping you can find something else to cast doubts on Cyndi being responsible or trustworthy. If Dale didn't kill her, we have to come up with who she got in with who could have. And Tanner and her ex-fianc\u00e9 might not be enough.\"\n\n\"Back to proving the gold digger and getting-in-over-her-head angle? I agree the Nazi connection could blow things wide open for Dale again,\" she said, putting his plate of spaghetti before him and pouring him a glass of red wine.\n\n\"Yeah,\" he said with a sigh. \"If it comes out Dale and two other generations of his family wanted Twisted Trees and the Nazi connection kept secret, that provides another motive for him murdering Cyndi, either to keep her quiet or in anger for her already going public with Marian James. Dale and I are going to take a look in that well-preserved garage\u2013carriage house on the mansion property tomorrow. Dale says his great-uncle kept a mistress in an upstairs apartment there, one he brought from Germany, but why not in the big house if there was no wife in the way?\"\n\n\"I can't believe all that. A German mistress, probably swastika wallpaper in the garage apartment. 'Curiouser and curiouser,' said Alice in Wonderland.\"\n\n\"You and your mother's literary quotes. I'll take Bronco along to Twisted Trees for security's sake.\"\n\n\"In other words,\" she said, sitting up straighter, \"I can do the online research, but not go with you to help solve the riddle of what's there? Nick, that's exactly what I'm being paid big bucks to do at Black Bog, find and prove a theory\u2014on-site. I want to go with you to look around there.\"\n\n\"I know you're busy. And I'm not sure that it's entirely safe.\"\n\n\"So I'll be stuck looking at my laptop in the den?\" she asked, her voice rising as she jumped up. \"We used to be a team!\"\n\nHe could tell she was tired, right on the edge of one of her narcoleptic meltdowns.\n\n\"At least,\" she went on, facing the sink with her hand propping her up, \"I'm allowed to theorize about age-old corpses, where little me can't get hurt. Maybe there's some way I can do all that online instead of actually being there, seeing so much as historic artifacts.\"\n\n\"What? Claire, you've been through enough, and you have responsibilities,\" he said, getting up too and turning her toward him.\n\n\"I sure do! And one of them is to be partners with you, not like some secondary secretary, kept in a corner, hiding behind a laptop. Brad Vance won't let me see the real artifacts and you want me to stay in a gilded cage, a padded room!\"\n\n\"Okay, okay. I'm sorry. Of course, I can use your expertise at Twisted Trees. You want to go with us to check the place out, that's fine,\" he said, holding up both hands as if she had drawn a gun on him and demanded his money or his life.\n\n* * *\n\nThat night, thrusting aside the blackness of sleep, Claire climbed out of her grave and went to open a drawer to get more sleeping meds. But in the drawer lay a woman reaching out to her, a woman between two men.\n\n\"I feel I know you,\" she whispered to the woman. \"Did you love them both?\"\n\nThe woman nodded. \"Caught between,\" she said. \"Caught between and died for both. An affair of the heart, so I kept one.\"\n\nClaire walked down the hall to Leader's office and took his dagger off the desk. She went back and clawed through the necklaces she kept inside the drawer. Where was that one with the polished stones, all broken now, broken...\n\nShe put one necklace around her neck and then another, another. One from Hunter years ago, one from Leader just last month. They weighed her down so much. Was she choking under the huge weight of the bog above her? Were they strangling her or stabbing her? She reached out and knocked the dagger away, no not the dagger, but something else...\n\n\"Claire. Claire, sweetheart, what is it? Why are you up?\"\n\nHer vision cleared. She stared into the dark mirror over the dresser. Nick was here, out of bed, holding her by the shoulders, pulling her to him. Leader was in the mirror, but Hunter was gone.\n\n\"Claire, it's all right. You haven't had a narcoleptic nightmare for a long time. Maybe you missed your meds.\"\n\nShe collapsed into his arms, holding tight.\n\n\"But why all the necklaces?\" he asked.\n\n\"Don't know,\" she said, shaking. \"Some dream, like you said. Maybe a nightmare.\"\n\n\"Here, let's take them off, put them back,\" he said, lifting them in one bunch from her neck and over her mussed head. \"Maybe, my love, you were planning to go out on the town without me.\"\n\n\"Can't recall, but not that. Lexi\u2014I should check on her.\"\n\n\"I'll look in on her. Here, back to bed.\"\n\nHe started to lead her that way but jolted and looked down into the grave\u2014no, down at the floor. \"What's that?\" he asked, stooping to retrieve a dagger\u2014no, of course it wasn't that. \"Oh,\" he said, \"the letter opener from my desk. You or Lexi must have brought it in, but I didn't see it here earlier.\"\n\nShe went back to bed, but she couldn't recall bringing the letter opener in here. Someone had taken it away, locked it up.\n\nUnder the sheet he pulled over her, she felt around her neck. No necklaces now, none broken or twisted. But how much that letter opener looked like a glowing, greenish dagger.\n\n* * *\n\nEarly the next morning, when the sun was barely up, though Nick had wanted her to stay home in bed, Claire made sure she took her meds and went with him and Bronco to pick up Dale at his apartment and head for Twisted Tree. They stopped at his house and waited while he went inside to get a ring of keys his mother had once told him had unlocked all the doors in the mansion. If only it would unlock its secrets.\n\nThe doors to the main house were long gone, but one key on the ring would open the door to the standing building, Dale had said. But they'd have to pry off boards securing the single entryway first since the garage doors themselves had long been broken and would not lift.\n\nDale with his broken arm was no help, but Bronco and Nick pulled off the old boards with the claw part of two hammers. Dale handed the key ring to Nick, who tried key after key on the rusted lock of the door, until one worked. But it still took Bronco banging his shoulder against the door to get it open.\n\n\"Dusty in there,\" were Bronco's last words to them as they filed in and he went back to his vantage point under a crooked tree to be sure they weren't surprised or disturbed. They could hear him sneezing as they went single file inside.\n\nThe dust of decades made all three of them sneeze too and their eyes water. \" _Heil Hitler_ ,\" Dale muttered when they came across the first of several dust-covered, framed pictures of the long-dead dictator even on the wall of the small entry room. No wonder his parents had made sure this place stayed boarded up.\n\n\"Don't joke,\" Nick told him. \"Damn, look at this!\" he marveled as he dusted off the glass of the eight-by-ten color photo with his shirttail. It was of Hitler again\u2014signed by him as simply _Adolph_. \"Your great-uncle must have been a fanatic, but I guess many were.\"\n\n\"You don't think your great-uncle's mistress was also in love with Hitler?\" Claire asked. \"Or maybe that madman just thought everyone loved him and signed pictures that way, even for his SS friends like your great-uncle.\"\n\nThey stepped out of this small entry room into what was, no doubt, a garage large enough for three cars. An old Esso gas can, a few extra tires, nothing unusual. Whatever vehicles had been parked here were as long gone as any human inhabitants.\n\n\"Looks like the stairs to the apartment above are over there,\" Dale said, pointing with his good arm.\n\nClaire could tell this was a tremendous strain on him, but then, her heart was pounding, and she reached out to hold Nick's arm. She had to be strong here, for she was the one who had insisted on coming along.\n\nDale led the way upstairs. The steps creaked in the best tradition of a haunted house. Another door blocked their way at the top of the flight. Dale tried the knob. Nick leaned forward to help. \"Locked,\" Nick said. \"Try your other keys, or we'll break it in.\"\n\nClaire watched while the men fumbled with more keys. Then she recalled she had worn the small, bright flashlight around her neck which she'd admired on Kris and now had one of her own. She leaned around between the two men to shine it on the keys and the lock.\n\n\"I think that one worked,\" Dale muttered. Nick shoved, and the door opened. Claire squealed as a bat dove out and whipped past them.\n\n\"And it isn't even Halloween,\" Nick muttered, but no one laughed.\n\nDale went up the last step into a small slant-roofed apartment with two inset dormer windows which were not boarded up but were covered by heavy dark green velvet drapes. Seeing no lamps or lights, Claire pulled one pair of draperies open to reveal a set of sheer tattered curtains.\n\nWan morning sun streamed through them, half lace, half spider webs. The sagging, torn mattress on the narrow bed was full of mice, which jumped to the floor and skittered away. An armchair, a table, a sunken settee and a single shelf of books, all frosted with gray dust, completed the bedroom apartment, though the doors to an empty closet and a small bathroom with a claw-footed tub stood open.\n\nTwo things on the wall over the bed they simply gaped at. A red-and-white Nazi flag with a black swastika. And a framed photo of Hitler with a smiling, pretty blonde woman.\n\nDale gasped. \"But that's my great-uncle's mistress!\" he whispered. \"I saw one picture of her before Mother ripped it up. She was standing with my uncle on the grounds here, not really smiling, looking sad. So she must have known Hitler, since they're in this picture together. Ha\u2014Mother missed that one. But who knows what's still in the piles of stuff in her house Nita and Bronco said they'd get rid of? After that cursed freezer, no one went through any more items, though the police gave things a once-over, looking for blood or proof that Cyndi's murder happened on the premises\u2014which they didn't find, couldn't prove. Surely she wasn't killed there.\"\n\nClaire studied Dale's face instead of the photograph. He was so nervous yet defiant, talking too much as guilty people often did, but then that didn't mean he was guilty. She reached out with a tissue from her pocket to gently dust the photo. \"Look,\" she said, leaning closer between the two men. \"Along the bottom, near the frame. They've signed it to each other, 'Eva,' and over here, 'Adolph.'\"\n\nDale looked stunned.\n\n\"Her name\u2014the mistress\u2014was Gretchen, not Eva.\"\n\n\"So this can't be your uncle's mistress,\" Nick said, his voice shaky. \"Maybe your great-uncle knew Eva before he fled Germany. Are you sure the picture of him you saw once with Hitler's Eva wasn't taken in Germany?\"\n\n\"No way\u2014the picture I saw was taken here.\"\n\n\"But this can't be your uncle's mistress,\" Nick repeated. \"I had Heck check out Eva Braun. He showed me a couple of pictures of her. The ones here are of her for sure, but Hitler finally married his mistress just a couple of days before they both committed suicide in a bunker in Berlin. Their burned bodies were found later by the Russians, who insisted the teeth in the charred remains matched.\"\n\n\"Yeah,\" Dale muttered, leaning his shoulder against the wall as if he just couldn't stand on his own anymore. \"And we've seen\u2014I think\u2014how powdery bodies are when they get burned up to ashes.\"\n\n\"Unless she had a twin,\" Nick went on, \"that picture, like the others in this place, are absolutely of Eva Braun herself, so they surely can't be your great-uncle's mistress. Or\u2014or can they?\"\n\n# 21\n\nDale sounded more panicked than ever to Claire. \"We've got to hide or destroy this stuff and get out of here!\"\n\n\"Calm down,\" Nick told him, putting a hand on his uninjured arm. \"This is priceless and important. I don't mean to sound like I'm on Marian James's side, but you can't just destroy amazing history like this.\"\n\n\"But you're my lawyer, my friend! I own all this now, and, like you said, we don't need all this hitting the media or the police so they think I was out to stop Cyndi to cover this up. I wasn't. I just wanted to get rid of\u2014I mean, get her out of my life.\"\n\nWhile the men continued to argue, Claire moved away and peeked in the bathroom. The tub had brown stains. The small mirror over the basin was dust-shrouded and cracked. The toilet lid had a crocheted cover, yellow with age, which seemed to fit with the cobwebs all over this place.\n\nWhile Nick tried to calm Dale, she sidled around to peer in the closet. Lots of old wire hangers, nothing else.\n\nShe moved on and bent to read the titles of the ten or so books on the single shelf. Every title was in German, but one book looked like a ledger. She slid it out and blew it off. Inside was page after page of large, loopy handwriting, all in German.\n\nShe flipped through it, fanning dust so she sneezed again. She noted little drawings here and there. She closed the book and stuck it in the waist of her jeans in back, then settled her shirt over it.\n\n\"I have every right just to burn this place down,\" Dale was saying. \"If that was Eva Braun in the picture here with Uncle Will, that could mean she visited here! People would be crawling all over this place. Nick, no way I can allow that, especially with the police looking for motives I could be hiding. And I don't need to be labeled some neo-Nazi in this mess of my life!\"\n\n\"Let's lock up here, you go back to your apartment and think this through, unless you'd rather come into the firm with me today. We'll make a plan. Maybe you can donate these photographs anonymously, if we can just hold Marian James and her committee off for a while. You've obviously mixed up what you remember the mistress looked like with photos of Eva.\"\n\n\"All right. It was a couple of years ago when I saw that picture, so that must be it. But if Eva died with Hitler, she can't have visited here. I\u2014my memory\u2014I'm just confused. Then, maybe my great-uncle or his mistress just brought photos of Hitler and Eva here when he fled. I am taking your advice, leaning on you, Nick.\"\n\nHe turned to look at Claire. \"And you, too, Claire. I'm grateful.\"\n\n\"We'll get to the truth,\" she told him.\n\nHe nodded, but she could sense he did not like the way she'd worded that.\n\n* * *\n\nShortly after, they watched Bronco drive Dale away to his apartment as they still sat parked in their car on the street between Dale's house and Bronco and Nita's new one.\n\nNick said, \"We're going to have to talk Bronco and Nita into coming back to their house to clear out whatever's inside. Maybe we can help. We need proof that the mistress was not Eva Braun, and Dale's mother has lots of things stashed inside to look through.\"\n\n\"But I have a theory,\" she told him, sitting forward to pull out the notebook from the waistband of her jeans. \"Nick, don't get upset, but I thought I'd better rescue this. It was wedged between what looked like some novels in German. I don't trust Dale not to go back inside and clear everything out, maybe destroy things. What if his great-uncle or his mistress wrote this?\"\n\n\"Can you tell who that belonged to?\" he asked, leaning closer.\n\n\"I can't read German, but surely we can find someone who can. Also,\" she said, pointing, \"every so often, there are little sketches, and even they might tell a story. Like knowledge about the Black Bog people, this can rewrite history. Oh, look! A few of these entries are signed.\"\n\n\"By whom? Let me see.\"\n\nShe carefully turned pages and pointed to a signature: _Eva_.\n\n\"But that's impossible if Eva Braun's burned body was found. Unless this diary was written earlier and just brought here by Dale's great-uncle\u2014if this signature is by Eva Braun.\"\n\n\"Remember, their bodies were found and burned by Hitler's enemy, the Russians. And are we even believing them these days? Maybe the Russians didn't want to admit they couldn't find her body. Here's my theory\u2014Oh, who is that?\" she cried, startling as a thin old lady with a dog on a leash knocked on their passenger-side window and waved.\n\n\"Close and hide that and stay put,\" he said, opening his driver's-side car door. He got out and shut it behind him.\n\nSince the windows were up, Claire strained to catch the conversation.\n\n\"I saw Mr. Bronco Gates, the man who bought the house, was with you,\" the woman was saying. \"Will they be moving back in after cleaning out poor Lucy's clutter?\"\n\nAh, Claire thought, that must be Betty Richards, the woman who had identified Cyndi's frozen face and then talked to Nick later about Dale's mother. She hadn't met her, but there seemed to be something familiar about her.\n\nShe went on, \"I take it the police haven't proved a thing against poor Dale, so I am happy for that, but then who hurt that poor woman? Well, I just want to tell you, Mr. Markwood, if Dale ever does go to trial, you can call on me to testify that Cyndi was not a nice person, you know, very rude to Dale's mother, not that she cared for that tart.\"\n\n\"I'm sorry to hear they didn't get along. Was there a specific instance you can recall that really showed that?\"\n\n\"Why, more than one. I overheard Cyndi tell her she shouldn't live like a heathen\u2014that was the very word she used. All her clutter should go out in the trash, Cyndi claimed, and then Lucy told her she was trash and she should be thrown out of Dale's life. Oh, is that your wife? Hello,\" she said louder with a smile and a wave through the glass. Claire was tempted to get out but she stayed put. Since Nick had evidently not wanted her in on this conversation she just waved and smiled, though it was getting warm in there with the windows up.\n\nClaire waved back a second time to be certain the woman had seen her through the reflections on the glass. A busybody and perhaps a loose cannon, but she made a mental note to suggest to Nick that she talk to the woman sometime, maybe if she came with Nita to start weeding through decades of possessions again.\n\nThe woman finally walked away with her dog\u2014actually, back to her house the way she had come, so was she really going for a walk, or had she spotted them and just wanted an excuse to talk to Nick? With her own mother and then in several of Nick's cases recently, eccentric, elderly women had inadvertently helped to solve problems. Was there something in the water in South Florida that produced Florida characters, only some of whom could be trusted?\n\nNick got back in, started the motor and air conditioner and drove away while they assessed \"nosy Betty.\" But he pulled into Sugden Park off US 41 and parked where they could see the lake.\n\n\"This was where we had a picnic before we left for St. Augustine the first time we worked together,\" Claire said. \"We are definitely not here for you to try to hire me this time, though, I believe, I have a piece of rare evidence here\u2014evidence for something. Want to sit at a picnic table and let the view calm us down?\"\n\n\"No, psych major. I'll leave the A\/C on and we'll sit in here guarding that prime piece of evidence until you explain to me your theory about that diary or whatever it is. I'm going to leave the doors locked. Tell me\u2014show me again\u2014what you've found.\"\n\n\"Found and stolen. Well, let's say borrowed. I hope you'll be willing to defend me if Dale eventually brings charges,\" she said with a little laugh. They both unhooked their safety belts and leaned closer over the console, shoulder to shoulder. \"Look, this Eva drew little doodle-like pictures, and they translate well.\"\n\nShe turned to the page that had caught her eye. It was laid out like a cartoon in three blocks. To the left in the first drawing, two figures clung together\u2014one with a black, brushy mustache\u2014crying tears yet smiling. They were sharing a bed. In the middle drawing the man waved to the woman as she boarded a crudely drawn airplane. The woman looked pregnant. In the picture on the right, the woman was alone reading a newspaper and crying so hard that\u2014well, evidently, real tears had blurred that drawing and the words beneath. And a tiny baby was floating in a cloud drawn overhead, as if it were in the sky. Or in heaven.\n\nThey were both silent as Claire flipped through the other pages, but no other drawings seemed as important. Sketched egrets on a lawn under a twisted tree, like those which grew on the mansion grounds. A woman alone with open but empty arms.\n\nAnd, at the very end of the book, the last page, though there were others left blank, a drawing of a woman drinking something and then that woman on a bed\u2014maybe the single bed in the garage apartment\u2014with her arms crossed over her chest and what looked like a framed picture clutched to her chest.\n\nWhen Nick spoke, Claire was so into her racing thoughts that she jumped.\n\n\"She could be cradling that picture that's over the bed.\" He ignored his cell phone when it sounded. \"What is your interpretation of all this?\"\n\n\"Too early to tell, but, for some reason, Hitler must have sent Eva Braun away before he died.\"\n\n\"But the Russians found two burned bodies.\"\n\n\"But like I asked before, do you believe that? Staging that would be easy enough, just kill and burn some innocent bystander or pull one out of the ruins of Berlin. But Hitler and Eva must have thought she was pregnant with his child, and the so-called _Fuhrer_ wanted his heir to live. To come back to Germany in triumph someday when all was lost at that earlier time? I know this sounds crazy, but I've been theorizing that Reaching Woman might have been married\u2014belonged somehow\u2014to one of the men she was entombed with, but became pregnant by the other. But back to Eva Braun. Perhaps the baby died\u2014a miscarriage, something. When she lost Hitler and her heir, she killed herself in that apartment. Maybe Dale's uncle had been ordered to hide and protect her and then he had to bury her. Nick, we've got to get this translated, carefully, privately.\"\n\n\"But it's Dale's property.\"\n\n\"And he'd no doubt like to destroy it. This Hitler's heir possibility is more of a bombshell than simply Eva escaping and some other woman taking her place in a suicide pact so that two burned bodies would be found. Do you think Dale could have killed Cyndi? Maybe she was pressing about getting in to see what we saw today. Maybe she was even blackmailing him about his Hitler connection, however distant and long past. Maybe he knows more about all this than he's telling. Can we get this diary locked in the safe at the firm and have someone who can read it come there?\"\n\nHe heaved a huge sigh that seemed to deflate him.\n\n\"Nick, I can take this on myself. I took the journal while you two were talking. I didn't tell you until later\u2014all true. So I can claim I'm the one who had it translated before telling you what it said. You know, I'm pretty sure Kris could wade through this. She had some high school German, and I think she picked up more when she lived in Europe those early years on digs.\"\n\n\"If you can trust her.\"\n\n\"Of course. I'd better. And she obviously knows how to keep a secret, with all her work at Black Bog.\"\n\nShe kept silent a moment, watching Nick agonize over this dilemma. Brad Vance had strongly questioned her own pledge to keep the work there secret, had even said she might have to withdraw from her contract if she was targeted with more publicity. But she didn't have to go public with this. If she and Kris could verify her theory, surely this diary could be handed over to historians, or even the government, anonymously.\n\nBut this was so important\u2014as key, in a way, as the discoveries at Black Bog. She held her breath.\n\n\"All right, ask Kris if she can come to the office tonight to have a look at this when everyone's gone. But we'll at least have to tell Dale what she finds.\"\n\n\"Isn't there some moral law about something being so important for history\u2014for the knowledge of all mankind\u2014that what someone owns really belongs to humanity?\"\n\n\"I'll look into it. Sounds like international law to me, not my forte, though, sometimes lately, I'm not sure what is. Damned if I want to defend a man who has been sitting on this for years, and who had motive, means and opportunity to murder a woman. But we're in deep, sweetheart, and getting deeper. I'm sorry I wanted to keep you out of this.\"\n\n\"Apology accepted. We're better working together on things, have been since we first sat at that picnic table over there, and you offered me a job\u2014and I found a new life.\"\n\nThey kissed, then refastened their seat belts before he drove them back out to the highway. Of course, she thought, getting in deep and deeper had caused them problems before, but together they would work through this, like resurrecting human ashes from the bombed-out ruins of Berlin years ago or bodies in a South Florida bog days ago.\n\n# 22\n\nAfter Lexi was in bed that night\u2014and had been reassured again that Jace loved her\u2014Claire took a shower, then hovered over her laptop. She wanted to do more research on Eva Braun, but she was looking into Cynthia Lindley again for Nick.\n\n\"Bingo!\" she cried after she'd gone down many a rabbit trail, rereading parts of the woman's Facebook page, then finding Cyndi on LinkedIn, talking about \"her dream job.\" _I think I will be a fabulous salesperson for luxury real estate, here in beautiful Naples! I will soon have some high-class backing from a member of a committee that preserves historic sights, maybe more than one influential sponsor!_\n\nNo doubt she was referring to Marian James, but more than one backer? So was Marian lying that she only received Cyndi's recorded phone message and could not respond? Or was Cyndi lying here?\n\nAnd\u2014amazing! Pictures accompanied the post that had to be of the overgrown lawn at Twisted Trees with ficus trees and more distant palms in the background, though the ruined mansion was not in the picture. The next photo showed Cyndi standing in front of a large modern mansion with palm trees. Behind her, on the lush lawn was a for sale sign. The implication was there, though it was a lie. The Twisted Tree grounds hardly had that stunning home on it.\n\nAnd the picture looked like a very good selfie, maybe one made with a selfie stick or one of those new little drones that took photographs, obviously a drone much smaller than the one that had crashed into the country club window.\n\nIn the photo, Cyndi stood in front of the sign. She had positioned herself perfectly to block out the name of the Realtor and the phone number, so there was no way to follow up to see if she had managed to get a job somewhere, which Claire doubted. No one had evidently come forward when she was murdered to say she'd worked for a particular firm\u2014unless Ken Jensen was not telling them everything.\n\nOr perhaps Marian had lied and had actually talked to Cyndi instead of just receiving her recorded message. Had Cyndi bargained with Marian to get an in with a posh real estate office here in town for the promise of help with her access to Twisted Trees? Or was this all just an egotistical lie from Cyndi, pie in the sky to build herself up? The woman was starting to sound more and more unbalanced and aggressive.\n\nShe ran to get Nick.\n\n* * *\n\nClaire did not go to Black Bog the next day, but stayed home with Trey and tried to bolster Nita's self-confidence. She had agreed to move back into her and Bronco's house for the coming weekend to clean the place out. That way, Bronco wouldn't be working and could be there to help, and Nick and Claire could pitch in some too. Until then, Claire had Black Bog work she could do at home, namely going through her voluminous handwritten notes, trying to puzzle out a theory about the trio. Also, she, Nick and Kris were going to meet at the law offices tonight after the staff left to take a look at what they were calling _Eva's Diary._\n\nTrey was asleep in his bassinet in the shade on the back patio table next to her while she searched for even more about Cyndi online. Nick had been astounded at what she'd found last night, though they'd decided not to blindly call any of the dozens of local real estate firms. He planned to call Ken Jensen to see if his team had turned up these clues, but she knew Nick was dragging his feet. No way he wanted her work to be more admired by the police department so they'd try to hire her as a forensic psychologist.\n\nToday, Claire was looking under both the dead woman's names, her real one, Cynthia Linschwartz, and her chosen but evidently not legal name, Cynthia Lindley.\n\nSo what else had the woman been willing to lie about? Nick had been amazed and ecstatic last night that Claire had found what must be a mention of Marian James. He had agreed with her questions: was Marian lying about whether the two women had made contact? Had something gone awry in the meeting and Cyndi ended up dead?\n\nBut now, it seemed both lines of research had come to a dead end. Maybe Heck could take this further as he was better at online detecting than she'd ever be. If Heck's dream job to work for a facial recognition tech team\u2014the rec tech team, as he called it\u2014was too dangerous, maybe he could be the one to consult with the Naples police. As for her, it was time to turn her thoughts to theories, based on facts, of course, about the trio in the Black Bog grave.\n\nBut Trey picked that very moment to wake up and fuss. She fed him a bottle and walked around the pool, holding him against her shoulder. Could there have been a child involved in the trio's lives? If there had been a love triangle between Reaching Woman and the two men, whose baby had she carried? Who belonged to whom, if their culture had recognized some form of marriage? The woman's deerskin clothing matched Hunter's, so had she fallen for Leader when her man was out hunting? But the broken necklace was more like Leader's bracelet. Or had she been Leader's mate, then had fallen for Hunter and so it was decreed she would be dressed like him and die with him as punishment?\n\nBut how and why had Leader died and been interred with them? Could it have been natural causes and so, with his eyes wide open, his people had honored him to lie with the woman he loved? But why did Hunter's executed body lie there too and why was his heart with the woman's body?\n\nShe jolted when Nita's voice came close. \"Want me to take him for a while, so you can work?\"\n\n\"I'm thinking while I walk.\"\n\n\"Okay. I will go back to making lists. I thought at first to keep some of the furniture in the house at least, but now I don't know.\"\n\n\"Did you sleep some, Nita?\" she asked as her friend turned away to go back inside.\n\nShe shook her head and shrugged. \"Some, I guess. Claire, I'll go back there, help everyone by cleaning that place out, looking for things\u2014clues\u2014like you said. But her face frozen in that silent scream still scares me. At least the freezer will be gone soon.\"\n\n\"I understand. It still scares me too,\" Claire said, wishing she could share with Nita how the trio of bog faces, especially the woman's, haunted her. And Leader's open eyes, as if he wanted a last glimpse of the woman he was buried next to, while she reached out toward him and the executed Hunter, who had paid with his life for some terrible deed or act.\n\n\"We both need to be brave and find answers,\" Claire told her friend. Nita, of course, thought she meant about their house a hoarder had left littered with a lifetime of things. But she meant so much more.\n\n* * *\n\n\"I can't believe you brought Lexi even if it's not a school night,\" Nick whispered to Claire as they got set up in his office to have Kris look at the German diary.\n\n\"She needs to be with us now that she's upset about not seeing Jace. Besides, she'll be content looking up butterfly pictures and information for a two-minute talk she has to give in school about her favorite animal.\"\n\n\"Yeah, but that has to be her horse, Scout. As soon as I can, I'll take her over to the stable stall we rented. Ever since she fell in love with that little dude when we were hiding out in Michigan, it's been like therapy for her to be near him and ride him.\"\n\n\"Right, but she talked her teacher into letting her mention two animals, so butterflies are in also. Ms. Gerald is ecstatic someone else cares about butterflies. I think she's a real fanatic.\"\n\n\"Nice they have that connection,\" Nick told her. \"So, Kris, how much German did you have?\"\n\n\"Three years in high school until I came to my senses. A very hard language with huge, long nouns. But I was glad I could speak it later and came to like it. Speaking of Hitler as she does here, it was amazing how he screamed his speeches, and an entire nation followed a man like that.\"\n\n\"Can you tell what the writer's attitude toward him is\u2014was?\" Nick asked, sitting on her other side.\n\n\"She adored him. She says she wanted to bear his prince\u2014and then\u2014right here, see,\" she said, pointing to the page with the telltale crude pictures. \"She says she's pregnant, but he says she has to leave him to save their child if the fatherland falls.\"\n\nNick and Claire stared wide-eyed and speechless and at each other.\n\n\"Something here,\" Kris said, frowning at the loopy, cursive penmanship, \"about how they could not stay together, despite their death vow, because if they were taken or killed, the baby, Hitler's heir, could be too.\"\n\n\"Worrying about the baby but not her\u2014except as the child's mother?\" Claire asked.\n\n\"Not sure. There's so much here, frenzied thoughts, on and on. Strange, isn't it?\" she said, finally looking off into the distance and blinking back tears. \"Here I've found a man who wants to date me\u2014and I him. Mitch sounded so great on the phone, so steady, even when he explained that he had some things to settle before he could ask me out and it wasn't another woman.\" The corners of her mouth lifted in a wistful smile.\n\n\"Then I saw him in the newspaper,\" she went on, \"good-looking, macho\u2014and read that article about him taking yet another dangerous job, and I understood. One has to follow one's passions no matter what the strictures or sacrifices. That's what's going on in this sad, trapped, pitiful woman's life, Eva Braun,\" she whispered, looking down at the diary again.\n\nNick shifted position, and Claire said, \"I'm wondering if that conflict is ageless. That maybe the ancient people we've been studying had exactly that sort of conflict.\"\n\n\"If so,\" Kris said, refocusing and frowning, \"it didn't go well for them, did it? Well, let me skim through the rest of this and see what else I can figure out.\"\n\nThough Claire continued to sit beside Kris, Nick, bless him, instead of hovering, went over to talk to Lexi and look at her butterfly pictures on the laptop screen. He had set it up so that she could use his printer to get pictures of \"the prettiest ten of them.\"\n\n\"What else?\" Claire prompted as Kris seemed to puzzle overlong at the last pages of increasingly shaky scribbles in the diary.\n\n\"You don't need to speak German to see it,\" Kris told her, pointing at a page Claire leaned over. \"You had to take a handwriting analysis course in your forensic tech studies, didn't you?\"\n\n\"Yes, and I still have my notes from the class.\"\n\n\"Well, see how Eva's handwriting is degrading here? See this squiggle like she might have dropped off to sleep more than once? And she says she's depressed. Life is not worth living since there is no baby, Hitler's dead and the dream of the _Reich_ is gone.\"\n\n\"Does she say the child died?\"\n\n\"I don't know,\" she said as Nick came quickly back when he saw them whispering. \"It's like\u2014like it just vanished inside her, she says.\"\n\n\"Like a false pregnancy or phantom pregnancy?\"\n\n\"Maybe. I think it's technically called a hysterical pregnancy\u2014Weird term, huh?\"\n\nNick said, \"Maybe she felt guilty she'd misled Hitler about the baby. Maybe she actually wished she'd stayed to die with him.\"\n\nSquinting at the page, Kris nodded. \"She keeps repeating this defiant phrase over and over\u2014see here: ' _Wir Schaffen das!_ ' That means 'We can do it!'\"\n\n\"But do what?\" Nick asked.\n\n\"She evidently asked\u2014begged,\" Kris said, \"her guardian here in Florida from Hitler's _Schutzstaffel_ staff\u2014his SS elite troops\u2014to get her some poison. The _Fuhrer_ had shot himself, but she wanted to take poison. Her contact here said no at first, but now\u2014I mean back then\u2014she thinks he will do it. She must mean so she can commit suicide.\"\n\nClaire saw Nick startle at that last word. He'd spent years trying to prove his own father had not shot himself. He'd finally done that, made the murderer pay, but the horror of finding his dad dead still haunted him.\n\n\"That last drawing,\" Claire said, \"must be how she envisioned her death, maybe how she staged it. See, she's cradling that monster's picture to her breasts, lying on the bed we saw, Nick.\"\n\n\"As Claire told you, Kris,\" he said, \"this all has to be top secret, but I know you understand that.\"\n\n\"I sure do. Secret for secret then.\"\n\n\"Unless this backfires and I get hit with attorney misconduct charges over mishandling this,\" Nick muttered, almost to himself. \"But I hope my client won't see our taking and reading this book as a betrayal. I'm going to level with him.\"\n\n\"Kris, we can't thank you enough,\" Claire said, giving her a one-armed hug. \"It's great to have someone we can trust in this delicate matter.\"\n\n\"No one would believe me anyway,\" she said with a little shake of her head. \"Wish our bog people would have left diaries or drawings, but we'll have to go on body language\u2014one of your strengths, Claire.\"\n\n\"Don't I wish.\"\n\n\"Mommy, do you and Kris want to see my favorite ten butterflies? But I think it's so sad people catch them in nets, and then they die or get cold in a refrigerator till their babies hatch, and they get stuck dead under a picture frame, and then people just study them.\"\n\n\"Ouch,\" Claire whispered as she got up from the table and Kris handed the diary back to Nick to lock up in his safe. \"My darling daughter doesn't know it, but she just nailed what's sad about what we do, Kris.\"\n\n\"That's partly why what we do is a huge secret, even from those we love,\" Kris whispered back with a quick glance at Nick. \"If anyone talks, and Andrea or Senator Vance find out, then we all get to take something like poor Eva's poison pill. Sorry, not funny, but you know what I mean. Sure,\" Kris said in a loud voice. \"I'd love to see your butterfly pictures. I want to see if I can tell them apart.\"\n\n\"This one here is the monarch because he is king,\" Lexi said, pointing at the bright orange-and-black butterfly. \"The men ones are way prettier than the lady ones, which isn't fair.\"\n\nClaire's thoughts blurred as Lexi displayed picture after picture. She understood why nature tried to hide the females so they would not be attacked since they carried the next generation. The males in charge again, the males like Leader, dressed more grandly than Reaching Woman, although she too had worn beautiful stones on her necklace, very much a match for his broken bracelet. A love gift? A sign of possession?\n\nAnd like the tragedy of Eva Braun's life, could Reaching Woman have chosen to join the man she loved in death? But if so, was that Hunter or Leader?\n\n# 23\n\nTo clear her head, Claire did not go out to the bog on Saturday, but went to the beach. It had been only two weeks since they had found Cyndi's body, but it seemed like an eternity. She had to cut off all her internal diversions: mourning for their ruined reception; the tragedy of Cyndi's murder and Dale's problems; Jace and possibly Heck being in danger.\n\nShe wanted to focus only on the Black Bog trio right now.\n\nIt wasn't a sunny day, which helped her find a parking place near the big tourist draw of the Naples Pier, shops and restaurants, and one of the longest stretches of sand and waves in the area. Actually, Brad and Andrea Vance's Art for Art's Sake shop was just a block away. Though they were probably both at the bog, maybe she'd pop in to the shop later. But right now she needed solitude and soul-searching, not of her own soul but of Reaching Woman's.\n\nClaire locked her car and, as was her habit, since they'd been through so much, looked around to be sure she wasn't being watched or followed. At least the thin person had not appeared again.\n\nShe walked out on the pier, glad there were some strollers and fishermen around. As ever, the pelicans sat on the slanted shingle roofs or rode the waves, hoping to be tossed pieces when the fishermen cleaned their catch.\n\nAt the end of the pier, feeling that she was almost walking the waves in the warm wind, she propped her hands on the wooden railing and stared out into the gulf.\n\nShe tried to clear her other concerns away, but were there similarities to Reaching Woman's demise in Eva Braun's tragic tale? Even in her own life, caught between Jace and Nick, though she'd made her choice and adored her husband? Or her own parents' tragedy: her father's desertion of their family and her mother's retreating into agoraphobia, buried in books, ignoring the needs of her daughters. But what was the centuries-old story of the trio?\n\nShe decided to write a short journal entry from the point of view of each ancient person. Creative writing, but informed by what she knew about the facts of the Black Bog burials. And she wouldn't do that where there were distractions at home or at her office in Black Bog. She would do it right here in the paper notebook stuffed in her purse. She'd channel them in a way from their dark past.\n\nBut she heard a rumble of thunder and saw a circle of rain out in the gulf. Well, then, she'd just sit in her car. Let the rain pound down, because, at last, thoughts were pounding in her head.\n\n\"You didn't get word they are going to arrest me, did you?\" Dale demanded the minute he entered Nick's office and closed the door. \"I just got here, and Cheryl said you wanted to see me. Thanks for coming in on a Saturday.\"\n\n\"Nothing yet about an arrest, so calm down. Actually, no news from Detective Jensen is good news. I just need to explain something I've learned about Cyndi which I thought you might be able to throw light on, but something else too about what was in the Twisted Trees apartment. Let's sit over at the table.\n\n\"I need to have you trust me on this new turn of events,\" Nick told him when they were seated. \"By the way,\" he added, \"did Bronco drive you here or\u2014\"\n\n\"Told him I'd drive myself. I'm careful even though I look like a mess and can only use one hand. What's new then?\" he asked, leaning forward, then shifting farther back in his seat as if he were afraid to hear.\n\n\"Really, two things. For one, Claire has done more research on Cyndi\u2014not Facebook this time, but LinkedIn.\"\n\n\"She spent too damn much time online. Obsessed with social media, like an unreal world she could control when she couldn't control the real one. In her mind, she had a claim to fame, right up there with the Kardashians. Never knew about LinkedIn, but I checked out her Facebook page once with all her sexy pictures. Then I was stupid enough to prefer the real thing. So what did Claire find?\"\n\nClaire was right, Nick thought. Dale seemed to put up a buffer of a lot of words sometimes, which she'd said was a tendency of people trying to hide something.\n\n\"Do you have any knowledge about Cyndi wanting to be a real estate agent?\" Nick asked. \"Or that she pursued that as a career here in Naples?\"\n\n\"Not that she actually did it, though I can recall a couple of times she brought it up. Piece of cake, she thought, showing rich people luxury real estate like houses on Gordon Drive or in Grey Oaks or Pelican Bay.\"\n\n\"But she never worked for a real estate office? There are enough of them around here.\"\n\n\"Not unless she did it secretly.\"\n\n\"I'm just trying to learn who she came in contact with who could have hurt her. We can hardly phone all the Realtors in town and ask if she worked there.\"\n\n\"What's the second thing you have to tell me?\" Dale prompted.\n\n\"Okay. Eva Braun kept a diary in German, which was wedged in with the few dusty books in the manor house apartment. Claire and I thought it should be in our possession to inform us and protect you.\"\n\n\"I\u2014I don't read German,\" he said, gripping his cast in his free hand. The fingers protruding from the cast looked bruised and swollen. \"Granted, I heard a lot of it when I was young.\"\n\n\"As your lawyer,\" Nick went on, \"I had someone read the diary. There are several crude drawings too. The gist of it is that Eva Braun and Hitler believed she was pregnant. He sent her away from Germany by plane in the last days of the war, probably right after he married her. She must have come to South Florida with your great-uncle, who had probably sworn to protect her and the child.\"\n\n\"The idea of Hitler's heir is horrible! The child\u2014it's not someone in my family?\"\n\n\"As best we can tell, there was no child.\"\n\n\"She lied? Or miscarried?\"\n\n\"Unless she made it up just to get out of Germany before it fell, which we doubt, as she adored Hitler. It might have been that she had false signs of pregnancy, which would be understandable if she\u2014they\u2014desperately wanted a child, and with all the stress they were under at the end. At some point, after she was at Twisted Trees and knew Hitler was dead and there would be no child, she evidently took poison on the bed in the apartment.\"\n\n\"I\u2014I can't believe it! I'll bet she was like\u2014like a holy Madonna to him. I\u2014This is a bigger bombshell than just knowing she lived at Twisted Trees after the war, after she'd lost everything.\"\n\nNick nodded. Tears clung to Dale's eyelashes and flew when he blinked. It touched Nick that he could care about Eva, instead of just hate her. He'd have to tell Claire that and ask if he could fake this emotion.\n\n\"You have the diary?\" Dale asked.\n\n\"Locked in a safe place for now.\"\n\n\"And it was there all that time. Dad and Mother should have gone in and destroyed it.\"\n\n\"We're not going to destroy it. We're certainly not going to announce it to the world right now, but since you have not been arrested\u2014\"\n\n\"Yet.\"\n\n\"They must not have a case against you, and Cyndi's brother and her former fianc\u00e9 being missing makes it look like they are on the run. So we're just going to wait this out, and I'll try to reason with Marian James and her committee to lay off publishing pictures of the ruined mansion right now. Maybe we can make her a deal to get some coverage later, once Cyndi's murderer is found.\"\n\n\"Nick, if Marian James put two and two together on this\u2014that Eva Braun was here\u2014there would be hell to pay.\"\n\n\"Instead of just attorney's fees, right?\" he said, putting his hand on Dale's good shoulder. \"You know, as a member of this firm, you have the right to our defense gratis. You did read the fine print in your initial contract, right?\"\n\n\"I did, but I suppose I'd be better off with an outside firm. That is, people might think you're defending me just because I'm in the family. You know, nepotism so to speak.\"\n\n\"Are you thinking of other representation?\"\n\n\"No.\"\n\n\"Then my only caveat right now is that I keep Eva's diary locked up so you can truly say you have never seen it, if it comes to that. Back to work, junior partner. I'm buried too.\"\n\nAs Dale left, Nick wished he hadn't put it quite that way. Claire was getting obsessed with the bog bodies she was studying, and he was afraid they had caused her waking nightmare last night. And the Black Bog ancients didn't write diaries to tell how they lived or died.\n\nSitting in the driver's seat of her car with her open notebook propped up on the steering wheel, Claire stared at the empty sheet of paper.\n\nThen she began to write. For Hunter, she wrote, _I was proud to hunt to help feed my woman and my people, though I had to be gone a lot. Often in danger too. I made traps. I used my dagger. I carried or dragged meat back to our village for the feasts. I gained praise from Leader. I looked up to him. My woman was proud too and happy when I made her a warm garment like mine from the hides of deer. She was my woman but when I came home, I started to see how her eyes followed him. She had a new necklace, made from fine stones like those in his bracelet._\n\nFor Leader, Claire wrote, _Because I was clever and could explain things well in stories and had a good memory, I became the leader of the village, so I never told anyone I had heart pains at times. I sent people to fetch medicinal bark, to catch fish and turtles and to hunt. I decided who could carve well, especially to make my staff of leadership. I accepted the special striped robes that told everyone who I was. I could have any woman I wanted, and I wanted Hunter's mate. I had to be certain that if she had a child, it was mine, so I sent him far away to hunt the elusive panther. I gave him a fine dagger for protection, but I sometimes wished he would not come back. But he did and found my woman and me together. And the pains in my chest and heart hit me as hard as he did._\n\nFor Reaching Woman, Claire wrote, _My father gave me to Hunter, and he was good to me. But then Leader saw me swimming and came to me in the water. He asked for me and I went. Everyone knew. Everyone but Hunter until he came home and found us. Hunter meant to stab Leader, but he cut deep into my chest. The necklace Leader had given me fell away, and I did too. I did not die right away but heard Leader and Hunter mourning me. Leader fell down from grief, because he had lost me and the child. Or did he hate himself for decreeing Hunter's terrible death for my murder? I raised my hands to hold them both when the others rushed in. And then my love for both and my sadness in death made everything I ever knew go dark._\n\n\"I can't believe this,\" Claire whispered to herself as she sat in her car, and the sun surprised her by slanting in when the rain was gone. What she had written might or might not be true, but she felt it strongly, almost as if she had channeled these ancient people's thoughts.\n\nSuddenly, she had to get out of the car. Out of this strange mood. Out of the tragedy that must have taken three lives. As fascinating as her work with the bog people was, she had to get out of this depression and obsession they pulled her into.\n\nShe jammed her notebook back in her purse. She'd just take a walk before she went home in time to meet Lexi's school bus. She'd take a look at the Vances' shop, maybe pop in to say hello to whoever staffed it.\n\nShe opened the car door and got out. The breeze was fresh, the sun warm. It was good to be alive, good to be Claire Markwood, living in the twenty-first century.\n\nShe glanced once behind her, then looked around, ready to dive back into the car if she saw anyone suspicious among strolling tourists, bicycles or cars.\n\nNothing unusual. She locked the car and headed down the street.\n\n# 24\n\nClaire's reflection on the front window of the Vances' store stared back at her as she studied one of the most attractive displays she'd ever seen. Such interesting items, so beautifully arranged on several velvet-draped levels: an elaborate bird cage, silk flowers, spills of pearl necklaces and small silk-lined jewelry boxes. The wording in a gilded arch over the top of the window read Art for Art's Sake, Antique Jewelry and Bibelots.\n\n\"Bibelots?\" she whispered. \"Do most people know what they are? Do I?\"\n\nHer mother would have known not only what that word meant but could probably cite ten historical novels where the word appeared.\n\nMaybe, she thought, the intrigue of that word alone brought some people in\u2014or scared off others. No, shoppers for this elegant, upscale store must know exactly what they wanted.\n\nAn old-fashioned bell over her head tinkled when she went in. No other customers right now, just a silver-haired man who must be a salesman. He had a neatly clipped mustache and wore a bow tie with his pinstriped, long-sleeved shirt with cufflinks and a garter on one sleeve. She decided she'd best just say she knew the Vances but not from where.\n\n\"Hello,\" the man called out to her and came closer, still behind the counter near the front display case, which looked old-fashioned with curved glass over display items. Behind that were modern, better-lit cases including some jewelry displays. What she would call antique knick-knacks made up the back three-quarters of the store's front room.\n\n\"And how may I help you?\" he asked with a smile. \"Bibelots is my bailiwick, jewelry too, unless it's one of the marked Victorian pieces in this case. In that case,\" he said with a little smile at his pun, \"I will ask our collector's expert and jewelry designer to join us.\"\n\n\"Actually, I don't have a specific item in mind,\" she told him as her eyes skimmed the amazing array of china, brass, marble, iron and other unique items. \"I'm a friend of the Vances and just wanted to pop in to see their store.\"\n\n\"Ah, both busy people. Have you met them through the university\u2014her former work\u2014or his political career?\"\n\n\"A little of this and that, which is what I see you have on display here.\"\n\n\"Our latest offerings are doing quite well among collectors,\" he said with a sweep of his arm, leading her toward the second lighted case. \"These are all authentic pieces of Victorian mourning jewelry, with miniature photos of the dearly departed hidden within as well as locks of their hair\u2014quite common in that time period. Of course, if you are really among the cognoscenti of any of our offerings, we have other items under lock and key we could share a look at.\"\n\nHe unlocked the black velvet-lined case with the mourning jewelry and reached in to hold up a chain dangling a gold locket with fancy filigree. \"See,\" he said as he clicked the locket open, \"a daguerreotype of the widow's deceased husband and a lock of his hair to remember him by.\"\n\nAn involuntary shudder wracked Claire. \"Different customs for different times. I hope you don't mind if I just look around.\"\n\n\"Of course not, and I shall tell the Vances\u2014Senator Vance comes in far more often than she\u2014that we were visited by...\"\n\n\"Claire Markwood.\"\n\nThe velvet curtain to the back room swept open, and a slender woman appeared. \"I thought I heard voices. Too much staring at glass Murano beads for now, so I popped out to show a customer our latest historical beadwork...\"\n\n\"This is Ms. Markwood, Pippa,\" the man said quickly. \"She knows the Vances.\"\n\n\"They're great, aren't they?\" she asked as her face lit up. \"They've given me the opportunity to design for them. I'm Pippa Lee,\" she said, extending her hand across the counter. It was a delicate hand, Claire thought as she shook it.\n\nPippa talked about the art, antiques and jewelry stores in the area and mentioned that the Vances' Fort Lauderdale store by this same name was much larger. \"And,\" she added, \"they have a very talented jewelry designer there.\"\n\n\"As we do here,\" the still nameless salesman put in.\n\n\"Thank you, Reggie.\"\n\nIn a way, this woman reminded Claire of her sister\u2014lively, petite, even down to her pixie-like haircut. Pippa the pixie, Claire mused, quite enjoying their chat until Pippa pointed to a small display at the very back of the shop that was labeled in flowing, old-fashioned script: Inspired by the Ancients.\n\nWithin lay several large agate bead necklaces and bracelets, not crudely done but with much more polished banded stone in more colors than the ones from the Black Bog graves. Yet they were so reminiscent of the items Claire had glimpsed before Andrea swept them away\u2014and of Brad's slides of the enlarged, priceless items. Her mind raced. Did Brad quickly hide the grave artifacts so they could be copied\u2014or even sold? Too late now, but she should have pursued a glance at whatever this store offered for the so-called cognoscenti. Was that some password to view even more expensive historic jewelry?\n\n\"What ancients were the inspiration for these?\" Claire asked.\n\n\"Just in the _style_ of the ancients,\" Pippa said. \"I take inspiration from many cultures, Inca, Egyptian, Celtic, an amalgam. I do these from study sketches and my imagination.\"\n\n\"Oh, I see,\" Claire said and forced herself to inquire about some of the metal coin banks on display. But her imagination was running rampant. Brad Vance had just become more interesting and yet more mysterious.\n\n* * *\n\n\"Can I just walk you to your car, boss?\" Heck asked, appearing so suddenly in the nearly empty parking lot behind the law firm that Nick startled. He'd been worried about Claire and Kris possibly being followed, but he'd been unable to convince his stubborn wife not to go off by herself today. She'd insisted she needed the \"think time.\" But did he have to start watching himself to see if he was being followed?\n\n\"Didn't know you were here,\" Nick said.\n\n\"I called your place, and Nita said you'd come in here, though it's Saturday. You're working late, boss, but didn't want to bother you till you were done. Been sitting over in my car. I just need to bounce an idea off you real fast, if you got a minute.\"\n\n\"For you, my friend, of course. Want to go back in?\"\n\n\"How 'bout we just sit on that picnic table people eat lunch at sometimes?\"\n\n\"And breakfast and dinner. Lots going on here as usual.\"\n\nHeck nodded and slid into the bench across the table from Nick and said, \"On second thought, is it okay with you if we don't sit out here, boss? I mean since that drone broke the windows and broke up your party, I been antsy about being in plain sight.\"\n\n\"Let's go back inside then.\"\n\nThey got up, and Nick dug his master key out again. This whole thing was making him not as much scared as angry. He'd thought he was done looking over his shoulder, but now this. Damn, if someone was after Claire or any of them, he had to be more careful. He'd been thinking too hard to be careful.\n\nHe locked the door behind them, and they sat at one of the two small tables Bronco used when he was working security. The one in the front lobby was a lot nicer than this one.\n\n\"Tell me,\" he said to Heck.\n\n\"That friend of Claire's\u2014Kris Kane. She has face blindness, right? I heard her explaining some of it and looked it up\u2014formal name is prosopagnosia.\"\n\n\"Right.\"\n\n\"I'm thinking of researching it better, maybe talking to her about it if you or Claire could set that up. And I'm thinking of steering clear of that face recognition tech firm I was interested in. Maybe I could go out on my own in a small way at first\u2014not leaving my consulting job with you. What if I could invent some kind of eyeglasses or something that read people's faces, then with an earpiece informed the wearer of that device who the person approaching them was?\"\n\n\"Interesting!\" Nick told him, leaning closer. \"And a real help to someone like Kris. Claire says, as intelligent and successful as she is, her face blindness always held her back, always made her a bit afraid of people. Who knows, maybe that's one reason she works in archaeology, with people who won't be offended, right?\"\n\n\"Right,\" he said with a sheepish smile. \"So then I'm hoping you could draw up some official resignation letter from this consulting company's job offer. I didn't sign nothing yet. _Caramba_ , __ if I'm the target of what could have been a deadly drone, I think there must be some kind of spy in that organization. Or maybe a competitor who wants to do them and me in. If I make a big deal about declining the offer, maybe it will take me off that hit list\u2014if someone's after me.\"\n\n\"Sounds like a plan, a good one.\"\n\n\"Gotta admit I took the idea from Jace and his pilot friend going public.\"\n\n\"I'll draw the letter up and have Cheryl type it up and see that it's hand delivered Monday morning after you sign it.\"\n\n\"Whew! I'm just gonna lay low till then since I can't see Gina this weekend anyway. She's working in the campus clinic.\"\n\n\"You have this tech firm's address and contact person?\" Nick asked, digging out a ballpoint pen. \"Good move to examine who might want to hurt you. I'm looking through old cases where someone was disgruntled or worse and blamed me, but nothing so far. Man, I'd like to think Claire and I are out of the line of fire for once.\"\n\n* * *\n\n\"Claire, Kris is here!\" Nita called to her.\n\nClaire had just changed Trey's diaper when the doorbell rang, but, with him in her arms, she came into the Florida room to greet her friend.\n\n\"Want me to take Big Boy?\" Nita asked. \"I'm glad we put off going over to our house until tomorrow, since Nick went in to work.\"\n\nClaire knew that Nita was especially glad to put off facing their house until after the freezer was gone. Nick had received permission from the police to get rid of it, though they were going to put it in a storehouse in case they arrested a suspect for Cyndi Lindley's murder and wanted to use it as evidence for a trial.\n\n\"Sure, thanks, Nita,\" she said, shifting Trey to her.\n\n\"He's a good baby, isn't he?\" Kris asked, putting her finger into his hand. When he grasped her finger, she bounced it a bit, and he gurgled in delight. \"Can't say I know much about babies,\" Kris said, studying his face almost wistfully. \"Maybe someday.\"\n\nClaire lowered her voice. \"Trey's even better than Lexi was, but she's going through a stage right now. Let's sit down,\" she said as Nita carried Trey out. \"You look happy.\" Claire wanted to find out what Kris knew about the Vances' store, but she didn't want to break what she sensed was her friend's really good mood.\n\n\"I just had a first date with Mitch Blakeman, and it was great.\"\n\n\"Wow! I thought he told you he'd have to call you later to be sure he's not the one that drone was aimed at. He must figure that newspaper article about moving on to hurricane hunting is insurance.\"\n\n\"We did a special ops date, he called it. Low-key. Unexpected. Out of town in a place no one would target for us. Claire, we went\u2014get ready for this\u2014ice skating at the Germaine Arena, where the Florida Everblades hockey team plays! They give ice skating lessons there and have an open rink during Saturdays if the team's not playing and they aren't converting it into a concert venue for the evening.\"\n\n\"You ice skate?\"\n\n\"I do now. Mitch once lived in Minnesota and he's good. He held me up, if you know what I mean. Holding hands, arm around my waist.\"\n\n\"Oh, that's a riot. An ice skating first date in South Florida when it's near eighty! How romantic.\"\n\n\"I thought so too. And he met me there so we weren't in a car together. He told me he'd have on a bright green shirt so I would know him. Wasn't that kind? I guess I sound like a teenager, but I don't care\u2014except about him.\"\n\nHer friend looked so starry-eyed that Claire decided to ask her about the Vances' art stores on Monday. She just couldn't bear to change the glow and the near giddiness.\n\n\"I'm going to get two glasses of wine, and we're going to drink to that,\" she told Kris. \"Those flyboys can make you fly pretty high!\"\n\nShe stood and turned to go into the kitchen and saw Nick had come in. He was just standing there, listening to them. What she'd just said\u2014she hoped he didn't take it wrong.\n\n\"Glad you're home,\" she said, trying not to sound nervous. \"Kris was just telling me she had a first date with Mitch.\"\n\n\"Great!\" He gave Kris a wave, then lowered his voice so only Claire could hear. \"Sure got to watch out for those flyboys.\"\n\nEven when Claire told Nick about her day writing short entries to psych out bog people and checking out the Vances' shop, there was an unspoken icy chill in the air. He said no more to her about Heck's idea to invent his own facial recognition solution, though he had asked Kris if she'd be on board to help him test it out, and she'd happily agreed. Surely, Nick didn't think his own wife was still in love with Jace, Claire agonized. Tension between them made her want to scream. Rather than just go to bed uneasy or hardly speaking, she decided to clear the air.\n\n\"Nick, it's just that I feel bad for Jace lately. Not seeing Lexi. More or less separated from Brit against their wishes when they were planning a wedding.\"\n\n\"I know. Don't worry, I'm just tired.\"\n\n\"And overworked. And worrying about Dale, because you're afraid he's guilty. I'm sorry if I upset you,\" she said, walking behind him and wrapping her arms around his waist. He was wearing only the shorts he slept in and, wearing only a silk nightgown herself, she could feel each tight muscle in his back.\n\nHe turned in her arms and hugged her hard to him. \"I know I get overpossessive,\" he whispered, burying his mouth in her hair. \"I've nearly lost you more than once, and I worry. Now that we're a family\u2014I honestly don't want you to just stay home but\u2014well, I guess I do, but you shouldn't. You're talented, you're helping people. Honestly, sweetheart, I'm proud of you, but I just can't stand the idea of your being hurt again.\"\n\n\"Understood. Nick, we'll see all this through just as we have bad times before. I was really shook at our belated wedding reception being blown up\u2014like it was an omen\u2014but we persevere, don't we? Our love makes us strong and\u2014\"\n\nHe had just put his hands down to cup her bottom and pull her even tighter against him when his cell phone on the dresser rang.\n\n\"I'll ignore it,\" he said. \"I need you.\"\n\n\"But with everything that's going on...just check the caller ID, and I'll be right over there in bed.\"\n\n\"No, I'm not letting you go,\" he said and, clamping her to him, he moved them a couple of steps to the dresser to lean over the phone.\n\n\"Damn,\" he muttered. \"It's Detective Jensen.\"\n\nShe clung to him as he took the call. She could clearly hear every word.\n\n\"Nick Markwood here. What's up, detective?\"\n\n\"Counselor Markwood, I thought I should inform you that your client Dale Braun is under arrest for a woman's murder.\"\n\nNick's grip on her tightened. \"The murder of Cynthia Lindley?\"\n\n\"That may come, but this is for the murder of Marian James. Murder and arson of his own property. We are at the site of the Twisted Trees mansion, and it's damn dark as pitch out here, except for the headlights of fire trucks and police cars and the remnants of the fire. It was Dale who ID'd the deceased. Somehow she was trapped inside the garage, knocked out, I don't know\u2014flames or smoke inhalation. She's badly burned and definitely dead. ME is on the way.\"\n\n\"And so am I,\" Nick said and punched off.\n\nClaire held him even tighter for a moment. \"I'm going too. I'll knock on Nita and Bronco's door and tell them.\"\n\nThey scrambled for their clothes, for calm, for courage.\n\n# 25\n\nNick and Claire hardly needed their flashlights by the time they approached the ruins of Twisted Trees mansion. They'd left their car in the same spot as when they'd been there in the daylight, then walked around a police barrier a little farther on. Through the hulking line of trees ahead, flashing light bars from at least six police cars lit a lurid scene while blinding white beams of light from two fire trucks slashed the smoking rubble.\n\nThey held hands, helping each other over the rough ground. Wet leaves smacked their faces, so they bent low under the boughs of the ficus trees. The smell of smoke bit deep into their lungs and clung to their clothes. They walked up to the barrier of yellow plastic Crime Scene Do Not Enter tape.\n\nNick told the officer who hurried toward them, \"Detective Jensen called me. Nick Markwood, Dale Braun's attorney.\"\n\n\"They're both over there,\" he said, pointing. \"Watch where you walk. Fire hoses all over. No fireplugs here, so they used what little water was stored on the trucks. The bay's too far to get water from.\"\n\nHe lifted the tape for them, and they ducked under. How different the grounds and the ruined mansion looked at night, especially with the garish lights. Nick thought of a movie trailer he'd caught Lexi watching of a zombie movie, _Night of the Living Dead_.\n\nPeople's shadows loomed huge. Hot spots still flickered from the rubble of the skeletal two-story garage and apartment. The pulsating red-blue-white strobes from the police vehicles made everything look as if it would burst into flame again.\n\nThey turned off their flashlights and stopped gripping each other's hands as they approached a dark unmarked car with Jensen standing next to it. Nick saw Dale was in the back seat; Jensen had the front passenger door open and one foot in the car, talking to him. An officer stood nearby as if guarding Dale. When Dale spotted them, he held up his wrists, either in greeting or to show Nick he was handcuffed. Nick noted his hair was a mess, his face cut and bruised, his shirt torn.\n\n\"Counselor. Claire,\" Jensen said, looking surprised to see her. \"I read him his rights.\"\n\n\"Detective Jensen,\" Nick said, and they shook hands.\n\nNick leaned down into the open car door. \"Sit tight, Dale. No comments right now. We'll get you out on bail.\" Turning back to Jensen, he asked, \"Why is he cut up and bruised? I'm sure he didn't resist arrest.\"\n\n\"That happened before we got here,\" Jensen said with a lift of his eyebrows Nick could not quite decode.\n\nDale shouted, \"It's my property! I had no idea she was up there\u2014trespassing again. I tried to pull her out of the rubble when I saw her after the upper floor collapsed, but she was unconscious or dead by that point! No comment?\" he went on, ignoring Nick's raised hand for silence. His voice broke. \"My only comment so far is I didn't kill her either. Yeah, I set the fire, but I also called it in. Someone's trying to frame me, and it's costing people's lives and mine too! She had to be unconscious in there or she would have run out\u2014at least have screamed or pounded on the door or window. The smoke wasn't that bad at first to knock her out.\"\n\n\"I said, hold it for now,\" Nick said, bending down to look at Dale. \"I'll be with you when you make your statement. Anything you say can be used against you\u2014remember that, counselor.\"\n\nJensen closed the car door and the hovering officer stepped closer. \"Over there,\" he said, pointing, \"under that tarp is Marian James's body. As I said, Dale ID'd her, though we found her burned purse and camera near where Dale says he found her body, and both items had her ID too. Hope the camera's not too melted to access its photos. That place burned hot. Arson investigators might find an accelerant.\"\n\n\"You do see a pattern here,\" Nick said, trying to steady his voice. \"However shaken he is right now, Dale Braun would not set himself up so obviously to kill two women and have the sites of their deaths and or bodies linked to him. He would not have called the fire department, but let it burn and be miles away.\"\n\n\"I get all that. It makes him look good that he hauled her body out, right? But why did he torch the place? Was he also in a struggle with her beforehand? Will there be any DNA under her nails or even in her body after that conflagration? One victim, frozen, one burned to death? Nick, I can't possibly let him walk this time until we do an investigation. He could be a flight risk. Or a serial killer.\"\n\n\"Hardly. I just hope the judge is fair enough to keep the bail bond reasonable. Can I see the body?\"\n\n\"Photos of it later. You beat the ME here. The evidence techs haven't even had time to sift through things\u2014too damned hot.\"\n\nIt annoyed Nick that what popped into his head was not only that things were getting too hot, but that Claire had been upset lately that Bradley Vance had shared only pictures of the bog artifacts with her, not the originals.\n\n\"Then can you tell us what you know of the scene?\" Nick asked. \"I need to tell you that Dale took us through the apartment above the garage two days ago. Marian James could have broken in to try to see it for herself as it's pretty much all that's left of the historic old mansion. I can testify to the fact she was trespassing here the day we visited, so no doubt again tonight.\"\n\n\"News flash, Nick. Whether or not the deceased\u2014who was evidently trapped in a burning, arson-set building\u2014invited herself here or was invited, she was incinerated, and your client admits he started the fire. Both his alibis have holes leaking like crazy, and however good a criminal defense attorney you are, I'm betting you can't dam up those holes.\"\n\n\"Granted, arson is a criminal act,\" Nick said. He could tell Claire wanted to say something but he was proud of her for letting him handle this. \"But he owns what he burned. No one else but an intruder who was trespassing was hurt, and he's not after insurance or involved in some sort of scam. Get him for not having a burn permit. If Marian James was trespassing again, her death was an accident and not first degree murder.\"\n\n\"So he says. But with the other murder charges pending, maybe he wanted to look like a hero, kind of like that Jace Britten who was at your reception and showed up in the newspaper as a valiant hurricane hunter.\"\n\nNick could tell Claire also wanted to respond to that too, but, thank heavens, she didn't. Headlights cut across them as the ME's van drove in, bouncing across the thick grass. As Jensen walked away to meet it, Nick pulled Claire back out of the glare of crisscrossed headlights.\n\n\"Are we leaving?\" she asked.\n\n\"Strange as it sounds, I think we can see clearer here in the darkness. Don't turn on your light again.\"\n\n\"You want a glimpse of the body?\"\n\n\"I do. Don't you look.\"\n\n\"Not so different maybe from studying those bog bodies. Method of death, yes, but all so wrenching, so tragic.\"\n\nBut the two men from the ME's office blocked their view when they bent over the thin, twisted and blackened form on the ground. Damn, but Nick had been hoping things would not blow up, that Dale could ride it out so the Eva Braun connection wouldn't come to light. But it sure would now, under the glare of publicity that could do nothing but hurt the case and hurt the firm\u2014and maybe cost Dale his life.\n\n* * *\n\nEvery bone in Claire's body ached the next morning, and she was certain she still smelled smoke in her hair, in the bedsheets. She could not get the sight, even distant, of the charred corpse of that woman out of her mind. Nick was gone already, heading for the station where Dale had been booked. They'd decided late last night that they'd drop Lexi and Trey off at her sister's for the day while Claire helped Nita and Bronco clean junk and trash out of their new house, and Nick would be there as soon as he could get Dale out on bail.\n\nAt least, Nick had said, the fire had happened late enough last night that the local newspaper and other outlets probably wouldn't report on it until later today when word got out about the second woman's murder in that same neighborhood\u2014with the common denominator now under arrest.\n\nClaire got up and dragged herself to the bathroom, took a stimulant pill to battle her narcolepsy on top of her exhaustion. Wait until Brad Vance heard or read about this latest publicity, not that she would be named in it. But Markwood, Benton and Chase probably would be, and that alone might be too much for him. She did not want to be terminated from her work at Black Bog. Would Andrea and Kris be on her side, or did they have to side with Brad? The man seemed in control and yet ready to explode sometimes.\n\nAnd her own state of being? She'd have to try again to question Kris on Monday morning but keep from putting a damper on her high spirits and excitement over her ice skating date with Mitch Blakeman. Hopefully, she herself would not be skating on thin ice with her friend to inquire about Brad and Andrea's two antique and jewelry stores. She wished she had entered the store as an anonymous special buyer, a cognoscente, as the shop staff called it.\n\nMaybe if she drove across the state to Fort Lauderdale and went in their other store, she could start all over, learn more. Or maybe she could get someone to do it for her. But for now, she could only hope that Andrea and former Senator Brad Vance trusted her because she wanted desperately to prove herself to them, which she hoped to do tomorrow when she shared her initial theory on the buried trio.\n\nStifling a yawn and trying not to look at herself in the mirror, Claire jammed her feet into her slippers and went to check on Lexi and Trey.\n\n* * *\n\n\"Where to start in this mess?\" Nita said. \"Okay, one room at a time, the master bedroom first, then the kitchen. Still can't picture me fixing food here where someone was put on ice. At least the guys from the police are ready to load up that freezer,\" she said with another glimpse out the living room window which Bronco was watching too.\n\nThe freezer which had been Cyndi's tomb sat in the driveway as the two men backed their truck closer to where they'd set it down. They had a lift on the back of the truck and had used a dolly to get it that far.\n\n\"Let's get busy,\" Claire suggested. \"The sooner this place is emptied, then repainted and furnished, the better you'll feel about making it your own, though I don't mean you have to leave us right away.\"\n\nBronco was still watching the men with the truck. \"This real nice neighborhood we picked, partly 'cause Dale gave us a good price for his Mom's place, is not looking so good right now,\" Bronco said. \"Swear I can smell smoke hanging in the air.\"\n\n\"No more dragging your feet about this place,\" Claire said with a one-armed hug of Nita. \"Come on, you two. Let's show Nick we got a lot done when he gets here.\"\n\n\"Right. Good riddance to that freezer,\" Bronco said and headed out the back door to the shed.\n\nInside, it was like a treasure hunt, but of worthless treasure. Though Claire supposed there was some sales value in stacks of old _Life Magazine_ s, __ and at least ten years of _People_ , __ local newspapers, clay pots, some of which still held dead plants, and sacks of grocery sacks. They carted them outside to either Bronco's truck or the dumpster they had rented. \"Let's keep these old 78 and 45 records, though,\" she told Nita when the two of them attacked that pile. \"Collectors are really getting back into vinyl.\"\n\nShe wondered if people were really getting back into Victorian mourning jewelry\u2014and if the so-called cognoscenti were actually buying jewelry inspired by the ancients. Maybe she could ask Jace and Brit to go over to Fort Lauderdale, in case Pippa Lee was over there for the day or she had to show ID and the staff there had heard about her visit.\n\nShe lugged out another load into the living room for Bronco to take to the dumpster when he finally finished cleaning out the backyard shed. She looked out a front window again to see if the freezer was gone. The two men hired by the police were surely taking their good old time, probably getting paid by the hour and not the job. They were just finishing cigarettes. At least they ground out the butts on the driveway instead of tossing them into the truck or dumpster. One lowered the lift, and they bent to slide the freezer onto it from the wheeled dolly.\n\nSuddenly two other men appeared, both big bruisers. \"That bastard Dale Braun trying to get rid of the coffin he put my sister in?\" the one shouted. \"He here? Heard he lives here'bouts.\"\n\nOh, no! Southern accent. Sounded slurred, maybe drunk. Fists clenched. One guy hit the truck driver on the shoulder, pushing him away from the freezer, while the other man opened it and peered in. Both Nick and Ken Jensen had described Cyndi's redneck brother and her first fianc\u00e9 well enough to match what Claire had just heard and seen.\n\n\"Nita!\" Claire called, lunging to lock the front door, then closing and locking the front window that was open. \"Run out back and get Bronco. I'm sure the police are sick of us, but I'm calling them if those delivery men they sent aren't able to handle this. And I'm calling Nick.\"\n\nIn the background, on the street curb, she saw Betty Richards just watching, holding her little dog in her arms. Crazy thought, but it almost looked like she'd summoned the two loud louts and now was just watching, watching.\n\n# 26\n\n\"Don't you go anywhere near those two,\" Nick insisted on the phone when she told him Tanner and his friend\u2014she could not recall the other guy's name\u2014were at the house.\n\n\"Nita went to get Bronco. Okay, he's out in front to talk to them,\" she said, still looking out the window. \"The two guys picking up the freezer are not cops, but there is safety in numbers.\"\n\n\"Lie low. I'm halfway there. See you.\"\n\nNita ran in the back door and locked it behind her. She joined Claire in the living room, looking out the window. \"Bronco and those two taking the freezer will make them leave,\" she said, out of breath.\n\nThe voices were so loud they had no trouble hearing the men.\n\n\"This here freezer was my sister's tomb!\" Tanner repeated. Together, the two men slammed the lid on it. \"Don't want it hid or made a sideshow. Gonna make that fancy lawyer bastard admit he kilt her, put him in here, show him what it's like. And hope they fingerprinted it already, 'cause ours is on there now, and you can tell the cops why.\"\n\n\"You'd better get off this private property,\" Bronco told them, not budging. At least, Claire thought, he was a physical match for Tanner\u2014except that there were two of them, and the delivery guys were just staying out of the way, though one was on his phone.\n\n\"I heard Braun's out of jail, probably hiding out, lawyer fancied hisself up by hiring those other lawyers he works with. Get him out here so I can at least tell him what I think of his kind,\" Tanner insisted.\n\nClaire told Nita, \"At least Bronco didn't tell him he has the wrong house, Dale's mother's old place next door to Dale's.\" She shuddered. She dreaded Nick hearing this blame-the-lawyer talk. Here she was worried that he'd be upset that she'd called Jace to see if he and Brit would go over to the Vances' Fort Lauderdale shop to check it out for backroom buys. He would go ballistic if she went out there after he'd said not to, but she had to get these men calmed down or away before Nick arrived.\n\n\"Mr. Braun doesn't live here anymore!\" Bronco insisted. \"He sold me this house, and that's that.\"\n\n\"Well, won't trash it or burn it down then, like I was fixin' to do, like he deserves or worse!\" Tanner shouted. \"Where's he at then?\"\n\nBurn it down? Claire thought. Maybe these guys needed a closer look.\n\n\"Moved somewhere in the area,\" Bronco said. \"Can't say I know.\"\n\nNita spoke again. \"Good how he said that\u2014not quite a lie.\" She was wringing her hands. \"Hope those two don't have a gun.\"\n\n\"I think they would have pulled it by now. They were thinking of hurting the house\u2014and that mention of wanting to burn it is interesting.\"\n\n\"Claire, you are not on this case. Nick will have your head.\"\n\n\"Not if I can get somewhere with them. Thank heavens we were here, but I can't have Nick charging in, even with Bronco to back him up, and those two workers are worthless. Besides, the neighbor lady is an eyewitness and the more people\u2014maybe us so-called weaker sex\u2014dealing with them, the better.\"\n\nIt seemed a momentary stalemate outside. Claire made a quick decision. She had to get rid of these two, and she'd handled worse before. Besides, how else was she going to assess whether they were sincere about losing Cyndi, or if they could have had something to do with her murder? Jensen had said they weren't accounted for when she died and that they hadn't been back to Zebulon when he was there. But would they have been stupid enough to come back here and make a ruckus if they were guilty? Over the years, she'd seen stupider moves by guilty parties.\n\n\"I'm going out,\" she told Nita, handing her the phone. \"I'll talk them down and find out a few things. I'm not afraid with Bronco and those two others there\u2014and Betty Richards and her tiny attack dog\u2014ha.\"\n\n\"No, Claire, you can't,\" she said and started to cry.\n\n\"It's all right. The only problem I'll have is Nick will be livid, so I've got to hurry.\"\n\nIgnoring Nita's protest, she opened the front door and went out.\n\n\"Hello,\" she called to them.\n\n\"You the missus?\" Tanner, who seemed to be the spokesman\u2014the thinker of the two, if that was possible\u2014called to her.\n\n\"No, just a friend of the couple who recently bought this house,\" she said, gesturing toward Bronco. Claire wasn't certain whether to identify herself as Nick's wife or not, but if he arrived, they'd figure it out. \"We're inside cleaning it up, trying to move on from your poor sister being found here. We will always remember her.\n\n\"I didn't mean to eavesdrop,\" she went on, \"but I take it you're Cyndi's brother and a friend of hers. I am so sorry for your loss.\" She dared to walk closer and extend her hand. The scent of alcohol hit her hard. Obviously surprised, Tanner shook her hand, his big paw engulfing hers. His fingers were dirty and calloused. Bronco edged along right beside her but at least he didn't interrupt.\n\n\"A damn tragedy,\" Tanner said, releasing her hand. \"When I called the police station, I heard the guy's finally been arrested but got out on bail. Read in the paper he might of kilt some other woman too.\"\n\n\"That's all hearsay\u2014just guessing at this point. Yet to be proved. But I'm sure whoever hurt your sister will be brought to justice.\"\n\nThe fianc\u00e9\u2014why couldn't she think of his name?\u2014stepped closer. _Always use the person's name_ , she recited to herself from her hostile interview training years ago. She would have called Tanner by his name, but she didn't want to tip them off she knew much about them and, although she'd admitted she'd been listening, neither man had given his name. Could that mean they were hiding something?\n\n\"Bad enough she got killed,\" the former fianc\u00e9 said. \"But then to be put in that freezer like deer meat or somethin'.\"\n\n\"It's terrible. Where were you two when you heard what had happened?\"\n\n\"On the road,\" Tanner said. \"Truck drivers, work together sometimes. Got a call from home.\"\n\n\"Oh, what a sad way to hear about the passing of someone you loved and cared for. Have you been here in Naples for a while?\"\n\n\"Truth is,\" he said, lowering his voice for the first time, \"we been on a binge in Key West, tryin' to drink it all away, how she deserted both of us, took off with some cash she took from me, then paid the price.\"\n\n_Bingo!_ Claire thought. He had just given her at least the hint of a motive suggesting they might have met Cyndi here, argued with her about deserting them, perhaps stealing from them and maybe more than just argued.\n\n\"Well,\" Tanner said, frowning at the freezer again, \"nothing else to do here. We're heading home but sure would like to give that Braun bastard a piece of my mind and a piece of this.\"\n\nClaire gasped and Bronco pulled her back behind him as the man drew a hunting knife from his jacket pocket. \"You ever see him, maybe he comes around for rent or whatever,\" he told Bronco, \"tell him if he don't get life in prison for what he done, he's gonna get this!\"\n\nThank God, he turned and walked away with his friend right behind.\n\n\"Get his license plate,\" Claire told Bronco, then turned to the workers. \"And you two get that freezer out of here\u2014please!\"\n\nTo her amazement, Betty had not budged. Feeling she'd done something helpful\u2014and Nick had not gotten here to blow up yet\u2014Claire figured she was on a roll. The Georgia truck pulled away, and the two workers went back to their task just as a police car pulled up in front.\n\n\"Bronco,\" Claire whispered, \"Nick should be here soon, so fill him and this officer in, but watch to be sure those two don't come back. I've been meaning to talk to Betty. And question number one is why would she be this nosy right now with two obvious brutes making a lot of noise.\"\n\n* * *\n\nJace met Brit in Wynn's Market on the Trail. He hated not being open with her as much as he hated steering clear of Lexi lately. Mitch had managed some time with Kris, so he was just going to take Claire up on the idea she'd given him on the phone, partly because it would give him time with Brit.\n\n\"We've got to stop meeting like this,\" she told him with a little laugh. She was carrying an empty plastic grocery basket, but she reached out to squeeze his hand. She looked good. Either she was getting more sleep not being in his bed, or her natural resilience was back again.\n\n\"Hey, at least you're still speaking to me,\" he told her. \"And smiling. And kidding around.\"\n\n\"This mess isn't your fault. You and Mitch were only trying to help crack down on drug czars.\"\n\n\"Want to head over to Lauderdale for a day? Get out of here?\"\n\n\"Really? Just for a day?\"\n\n\"Got meetings for the new job coming up in DC. Mitch and I are going to have to head there soon, where I know I'll be safe. But meanwhile, yeah, just for a day let's cross the state to Fort Lauderdale\u2014I need to visit a really upscale jewelry and antiques store, and you could come along.\"\n\n\"Oh, I'm sure you'll be safe in DC. Nothing bad ever happens there,\" she added with a roll of her eyes and a wry smile.\n\nFor sure this woman had bounced back from her own tragedy, and he loved her for it. Fall down, get up. Get shot at, so to speak, hunker down and shoot back.\n\n\"So what's the Lauderdale deal?\" she went on, as they strolled the aisles.\n\n\"It's actually a favor for Claire, though she couldn't tell me why. Says you should stop by the house, she'll fill you in, then you tell me. Man, I'm missing Lexi, so hug her for me.\"\n\n\"For Claire, for Lexi, will do. Claire almost lost her life helping me last year.\"\n\n\"Let's not think that. She needs someone who is not obviously tied to her to go into an arty store there and look at some pricey jewelry they might keep just for high rollers. Not buy it, just try to see it.\"\n\n\"Oh, yeah, that's us, high rollers. Meaning, I'll have to wear my one pair of designer jeans and Manolo Blahnik heels, and you can't just stroll in there in work jeans. Did she tell you more than that?\"\n\n\"Yeah. She thinks the key word to get a look at the pricey stuff might be _cognoscente_.\"\n\n\"Key word? No kidding? Is she working on something to do with jewelry theft?\"\n\n\"I think it's maybe tied somehow to knockoffs. You, know, copies of the real thing. Don't really know, and she wouldn't say.\"\n\n\"Maybe that's her consulting job,\" Brit mused as she put something in the plastic shopping basket on her arm. She took some smoked fish spread and cocktail crackers, then tossed in some granola bars. \"You know,\" she went on, \"maybe she's looking for people who are stealing old tribal jewelry, because there are all kinds of laws about returning Native American pieces, artifacts, even bones. I suppose a sting operation would be really hush-hush.\"\n\n\"That's my girl\u2014you, I mean, up for a challenge. I love you,\" he said, planting a fast kiss on her mouth. \"Believe me, I can do better than that. But here's the thing. I'm going to ask Mitch if he can fly us over in the chopper, then wait for us at the small airport I know just for a couple hours. We'd fly out of the Marco Island airport near here. The guy loves to fly what he calls helos, and this week's the last time he'll have access to a bird. We can take a taxi, have lunch, hit the jewelry store incognito.\"\n\n\"I'd like to see the Everglades from the air, and choppers fly pretty low. Maybe we'll spot a Florida panther. When?\"\n\n\"Can you get off day after tomorrow\u2014Tuesday? I know that's short notice.\"\n\n\"I think I can swing it. I've practically been living at the zoo, especially since I'm not living with you and have to move in with my second love.\"\n\n\"Yeah. Glad to know it's a tiger. Want to go out to my car and neck, like they used to call it in the old days? I'm really missing you, babe, but Mitch and I figure it won't be long before we can come out of hiding, so to speak. We really think that article in the newspaper about our new public government job seeking out hurricanes is insurance, instead of the undercover government job looking for drug runners and drug lords.\"\n\n\"Make out in a car in the store parking lot?\" she said, much too loudly, so that two elderly shoppers looked their way.\n\n\"Well, I never,\" the man said to the woman as they turned away.\n\n\"Well, I never either,\" Jace said, so only she could hear, \"but it seems like we're hitting parking lots for a lot of our special events lately.\"\n\n\"This isn't all an elaborate ruse to buy me a huge wedding ring in a chichi jewelry store in this so-fancy strip mall, is it?\" she asked, elbowing him with a little laugh.\n\n\"Naw, I already bought the wedding ring and in a real nice store.\"\n\n\"Just promise me our ceremony and reception won't be in a parking lot,\" she teased.\n\n\"I thought maybe in a helo over the Glades.\"\n\n\"Over my\u2014your\u2014dead body.\"\n\n\"Don't kid about that, honey.\"\n\n\"We're a wacky pair, you know.\"\n\n\"Just as long as we're a pair. I repeat, I love you, Brit. What would I ever do without you?\"\n\n# 27\n\nNick still wasn't in sight when Claire walked up to Betty. She held her little dog in her arms. The dog looked scared, but the old lady only looked interested. Why hadn't she scurried inside when all this started?\n\n\"Betty, remember me? Nick Markwood's wife, Claire.\"\n\n\"Oh, yes, I recognized you. All that made me wish I had a German shepherd or a pit bull the way those two were acting.\"\n\n\"You seemed brave to stand there and listen. Let me walk you home. I know how hard all of this must be for you. Bronco, I'm going next door with Betty! Tell the police and Nick I'll be back soon,\" she called and steered the woman away.\n\nShe could see Nick's car turning onto the street and she didn't want him to stop her. No, this time, she was going to use her forensic psych skills to latch on to what Betty really knew.\n\n\"I'm sure it upset you to see the freezer in the yard. Those men were rude and crude, weren't they?\" Claire asked, holding Betty's screen door open while she unlocked the inner one.\n\n\"Indeed they were, but you handled them. Dreadful to see that freezer out in the open like that and to realize what it held, and to think that poor girl was buried in frozen vegetables. Dreadful!\"\n\nClaire tried to recall if that detail had been released, but then Betty had viewed the picture of Cyndi in the freezer to ID her. Yet hadn't that photo just been of her face and not the surrounding bags of frozen veggies?\n\n\"So, did you ever see how packed that freezer was?\" Claire asked as she followed her into a Colonial kitchen, which reminded her of her grandmother's whom she'd known so briefly. \"I know Lucy was a hoarder, but even in her freezer?\"\n\n\"Oh, I've seen it a time or two. Sit here at the kitchen table, and I'll get us lemonade. I'm not sure anyone's sat right there since Lucy died.\"\n\n\"I'm sorry your neighborhood keeps getting hit with tragedies. But you seem very brave and informed about them all.\"\n\n\"Including that dreadful fire on Saturday. I read all about it in the paper, but they didn't mention the smoke and drifting ashes, and they should have. Obviously that journalist wasn't up close and personal like some of us who live here. One had to be here to realize that old ruin was back there, you know.\"\n\nSome of that seemed a mixed message. Was the subtext that Betty had been near the scene of the fire the night of the fire? But she didn't want to spook her or silence her, so she tried another tactic.\n\n\"Because knowing Lucy so well over the years, I'll bet you knew more than anyone else about that old ruined mansion, except for Dale.\"\n\nBetty set the tall glasses of lemonade on the table. \"Well, you know the name of that old TV show, _I Love Lucy._ I cared for my friend Lucy and she did me\u2014before she lost her senses, we were best friends. That's why I could forgive her for treating Dale's so-called fianc\u00e9e the way she did. I was her partner and backup in that,\" she said with a shake of her head and a strong sniff.\n\nClaire saw her grip her lemonade so hard that her fingers went white against the icy glass. The way she'd said that\u2014it was almost as if she meant she was Lucy's partner in crime. Backup in what?\n\n\"I suppose you also had words with Cyndi,\" she threw out another leading line.\n\n\"More than that.\"\n\nClaire tried to keep her relaxed posture, elbows on the table, leaning slightly forward, but she was tempted to bolt out of her chair. \"More than that?\" Claire repeated when Betty said no more, but her lower lips trembled.\n\n\"I told her never to darken Lucy's door again or she'd regret it dearly. When I said that, the smart aleck bent over in the plants outside Lucy's door, grabbed a handful of dark mud and smeared it on the door to darken it, just to be cute\u2014which she wasn't. Smarter than she looked, though, the snippy little tart.\"\n\n\"Did you see her again after that, and what did she say then?\"\n\nBetty shrugged, then looked down at her sweating glass. \"I only saw her at a distance after that. When Dale was at work one day\u2014unless he'd come home for lunch which he did sometimes, he drove right into his garage, so it was hard to tell\u2014she and a woman drove back in toward the mansion.\"\n\n\"Do you know who the other woman was?\"\n\n\"Didn't until I saw her picture in the paper today, the one died in the fire there, the historic committee woman.\"\n\nClaire tried not to gasp. She had to go next door to tell Nick that he really should hire her as a junior partner. Since she'd seen him last, she'd collected eyewitness accounts and comments that indicated Cyndi's murderer could have been Tanner and her first fianc\u00e9. A longer shot, but even perhaps the Lucy-and-Betty duo had somehow done away with Cyndi. They both had access to the freezer. Maybe when Lucy died, Betty just figured there was no way to hide Cyndi's corpse elsewhere so she let it be found.\n\nAnd then the ultimate gamble: If Marian and Cyndi had actually met, could something have gone wrong and Marian accidentally or intentionally harmed Cyndi? Dale had told Nick earlier that on her body Cyndi could well have had keys not only to Dale's place but his mother's\u2014even on her very dead, soon to be frozen, body.\n\n* * *\n\n\"I thought you'd be furious with me,\" Claire said as they finally drove away from Bronco and Nita's house after three more hours of hard work clearing out the clutter of over fifty years. She'd told him about what she'd learned from the Georgia duo and Betty. \"Or were you just waiting until we were alone?\"\n\n\"Oh, yes, Mrs. Markwood, I was scared and upset at first when Bronco said you'd talked to the Georgia boys. But he said you calmed them down, and he was right there with you. I wasn't crazy about the appearance of the knife, though.\"\n\n\"Which reminds me, I have to insist Brad Vance shows me the actual dagger buried with Hunter. It had some sort of crude carving on it, and that could be a clue to a lot of things. The Senator's close-up photos did not include pictures of the carvings for me to study.\"\n\n\"Nice try at diversion, but back to your taking over for Detective Jensen by questioning suspects,\" he told her as they turned out onto the busy Tamiami Trail and headed toward home.\n\n\"I was thinking I was taking over for you\u2014to get more suspects than Dale, at least for Cyndi's murder, if not for Marian's,\" she explained, turning toward him. \"I know it sounds crazy, but did you ever see the old Cary Grant movie _Arsenic and Old Lace_?\"\n\n\"Claire, I don't have the benefit of your amateur English major degree from your mother.\"\n\n\"It wasn't based on a book, but a play. Anyway, two dotty old ladies have a body hidden in their house. And they are the murderers.\"\n\n\"You don't really think Dale's demented mother and that character next door to her\u2014\"\n\n\"I don't know what I think. Then too, about the Georgia crackers, Tanner said that Cyndi not only deserted both of them but stole cash from him and what's-his-name\u2014\"\n\n\"Will MacBride.\"\n\n\"That's it. Anyway, Tanner seemed righteously pleased to say that Cyndi paid the price for deserting them. So maybe they were the enforcers of revenge for her betrayal. Besides, you told me after you first met Tanner, he said he was at home and the coroner called him to say Cyndi was dead. But earlier today he told me he was on the road with Will driving a truck and relatives called them with the news. Nick, he's making things up as he goes.\"\n\n\"Or, like you said, he's blown what brains he had drinking in Key West for a week, and is just confused.\"\n\n\"I'm trying to help you!\"\n\n\"And you are. All this is valuable for defending Dale.\"\n\n\"So maybe Tanner and Will then came here, met with Cyndi. Maybe it started out as an argument, she told them off, one of them in a fit of anger grabbed her around the throat. I don't know, but remember Dale told us early that Cyndi could well have had a key not only to his place but to his mother's. What if she had taken one to the Twisted Trees mansion too, and gave or sold it to Marian? Whoever killed Cyndi found the key on her and stashed her in the dead woman's freezer to make it look like Dale did it, or maybe they were planning to move her body after dark. The only flaw in that theory is that Betty would have seen something of that, though I suppose she has to leave her house sometimes.\"\n\n\"Then there's Marian James, since you said Betty claims she was heading with Cyndi toward the Twisted Trees property, so you could be right that Cyndi had a key to that too.\"\n\n\"That's what I mean. Maybe Cyndi also had or took a key which gave her access to the locked apartment above the garage. Maybe she even met with Marian and showed her around, or tried to assure her the place could be a historical site and she could act as its Realtor.\"\n\n\"I swear, sweetheart, I'm going to start taking you into court with me. You impressed the hell out of me when you testified to ruin my case the first time I saw you, and I'm glad you're on my side now.\"\n\n\"Wish I could stay around to cash in on that,\" she said, smiling and stroking his thigh. \"But I'm scheduled to present some ideas to the powers-that-be at Black Bog tomorrow morning and I've got to get some notes together. I can't just read them my little essays as if I were channeling three dead, prehistoric people. Oh, and Brit's stopping by tonight so I can fill her in on her trip to Fort Lauderdale on Tuesday with Jace.\"\n\n\"Good thing Jace isn't coming. I still don't know if he's safe\u2014to be at our house in case he's being followed or targeted, I mean.\"\n\nThey stopped at a red light. She reached for his hand, and he held hers tight. \"You're going to solve these two murders,\" she said. \"And I'm going to do everything I can to figure out who killed two of my three Black Bog people, if it's the last thing I do! Just a figure of speech,\" she added, realizing how that might sound, as the light turned green.\n\nThat evening, since Bronco and Nita hadn't come back, Nick put Trey to bed and read a story to Lexi while Claire talked with Brit.\n\n\"I'm sorry I can't fill you in on more than these details I wrote out for you,\" Claire told her, looking down again at the sheet of paper. \"I've signed a confidentiality clause with my current employers, so I'm trying to honor that, even though I'm sending you and Jace on a cloak-and-dagger mission across the state.\"\n\nShe wished she hadn't put it that way\u2014so much as mentioned a dagger\u2014but she said no more as Brit looked over her notes.\n\n\"Okay, got it,\" Brit said, frowning at the paper. \"We need to use an alias. We need to appear well-to-do and work in the word _cognoscente._ Glad you spelled it out for me.\"\n\n\"If they show you any private stock, act interested, find out as much as you can about the original or the designer. The Naples shop has a jewelry designer, and she said there's a more veteran one in Lauderdale. I'm thinking there may be jewelry, maybe even an antique dagger so old it's made of stone, not metal.\"\n\n\"Really? Kind of prehistoric?\"\n\n\"Antique. Very antique.\"\n\n\"So don't give the salesperson\u2014or designer\u2014info about how to contact us,\" Brit kept reading the notes. \"We will discuss our finances, etcetera and contact him or her later.\"\n\n\"Right. And if you get the names of the shop owner, shopkeeper, salesperson, jewelry designer\u2014anything like that\u2014don't write it down until you get out of there so it doesn't look too suspicious. And, Brit,\" she added, reaching out to grasp her wrist, \"if anything looks funny or, God forbid, dangerous, just leave calmly and quickly. I'm really grateful to both of you for doing this.\" She loosed Brit's wrist and sat back in the sofa with a sigh.\n\n\"I don't know what I would have done without your support when everything went so wrong at my beloved Backwoods Animal Adventure,\" Brit said, turning toward her. \"And for supporting Jace with me\u2014you know what I mean. Oh, I forgot that he asked me to hug Lexi for him, but I'm sure you can do that. I don't want to get her riled up if Nick's got her calmed down to go to sleep.\"\n\n\"Actually,\" Claire told her, \"she has not been a happy camper about Jace not seeing her lately. I know he'll patch things up when he can. Soon, I hope.\"\n\n\"You know, we do plan to stay in this area, even if he has to report to or fly out of Washington, DC, for his new job. I'm sure he'll find something to do here when he's not on call, maybe teach flying at the little airport on Marco Island, something like that. That's where we're flying out of, you know\u2014Mitch at the controls of a helicopter, no less. Then we'll land over there at a smaller executive airport, not the big busy one.\"\n\n\"It sounds like a good plan. I said I'd pay for the chopper gasoline and lunch and dinner for all, so here's an envelope with some cash, and don't say no. I consider the three of you are on assignment for me and please be careful above all.\" She gave Brit the envelope and, thankfully, she took it.\n\n\"Claire, whatever the point of this is, you be careful too. Jace said to tell you that you need to get out of your temp job or assignment if there's anything strange, illegal or dangerous\u2014like maybe someone ripping off antiques or jewels or whatever. He said to tell you it isn't life and death.\"\n\n\"No, of course not. I understand that. Thank him for his concern.\"\n\nBut she couldn't help but think that her attempts to solve the dilemma of what happened to her beloved Bog People did hinge on their life and death; though, of course, she wasn't going to risk her own.\n\n# 28\n\n\"I wish we had time to really enjoy the shops and a restaurant here,\" Brit told Jace as he parked their rental car down the street from the Art for Art's Sake shop in Fort Lauderdale about eleven o'clock on Tuesday morning. \"It's a beautiful area.\"\n\n\"That it is, babe, but we're on assignment.\"\n\nHe thought things were going great so far. The heliport at the Executive Airport had been right on top of a parking garage, and they had easily rented a car. The airport was just five miles from downtown, and Brit was right\u2014this neighborhood was great. Lots of restaurants, sidewalk caf\u00e9s, art stores and shops. Not an area for carryout, but he'd find something to take back to Mitch, who had stayed at the airport.\n\n\"No time for a shopping spree, no matter how much cash Claire gave you,\" he said, taking her arm as they approached their target shop. \"Okay, remember, you're using your mother's name, Ann Hoffman.\"\n\n\"All right, Ben Hoffman. And in case they ask where we live, we'll use Marco Island and just say we flew over for the day to give the impression we are rich. Best not to lie about too much, or we'll make a mistake.\"\n\nThe display windows were stunning\u2014even Jace admitted that. Old-fashioned yet very elegant. Several antique swords, or maybe they were called sabers, and several knives from the Civil War era were displayed on dark blue velvet. They were shiny steel, heavily etched, hardly the stone primitive-looking daggers Claire had told Brit to be on the lookout for.\n\nA bell jingled over their heads as they entered. A middle-aged woman with pink hair, no less, looked up with a smile. \"Lovely day for strolling from shop to shop,\" she greeted them.\n\n\"Actually,\" Brit said, \"we are fascinated by your display window. My father has a collection of old swords and daggers. He's kind of a connoisseur of those.\"\n\nWhere the hell did that father collection come from? Jace thought. They hadn't discussed that at all. Brit reminded him of Claire in some ways, which wasn't all bad. Bright, dedicated, clever on her feet, and sometimes too damn unafraid.\n\n\"Oh, is that right?\" Pink Hair said with an obvious perusal of both of them. That made Jace nervous.\n\n\"We're looking for something very special for him for a birthday gift,\" he put in, trying to stay poker-faced. \"He's been very good to us. We heard about your shop, so we flew over for the day from Marco Island. In her father's helicopter, no less.\"\n\nThe woman perked right up. \"I always wanted a ride in one of those. Then have you visited our other store in Naples?\" she asked.\n\n\"We've heard of it, but no,\" Brit put in. \"We'll have to, though, if we don't find something here.\"\n\nMan, Jace thought, she's good at this, barely hinting at the idea that something illegal might be for sale.\n\nPink Hair smiled. \"Let me summon my associate in the back room\u2014our master artisan, our jeweler\u2014and I'm sure he'd be happy to show you what collector's pieces we have. Just a moment then.\"\n\n\"In like Flynn,\" Jace whispered, hardly moving his lips because they'd talked about the possibility of security cameras monitoring them.\n\n\"Yes,\" Pink Hair said, emerging from behind the curtain, \"if you wouldn't mind stepping back into our private showroom, my associate Roger Bassett will discuss what might suit.\"\n\nJace took Brit's elbow as they went behind the curtain Pink Hair held open for them. Brit's arm was shaking, but, damn, his was too.\n\n* * *\n\nClaire actually had an extra day to prepare her initial theory about the possible relationships among the Black Bog trio of Hunter, Reaching Woman and Leader, because the Monday meeting had been canceled and moved to Tuesday late morning. More than once she glanced at the clock on the meeting room wall, one of those old schoolroom types with big numbers that went _tick-tock_ if there was a moment of silence.\n\nKris had said the reason for the rescheduling was that Andrea and Brad were closing on the sale of their house on the beach, one of the earliest mansions built in the upscale Port Royal neighborhood, though they'd done much to renovate it. Today, Yi Ling had left the bog site early because of terrible tooth pain. Brad had said he'd called their dentist to take her in right away. He himself looked the worse for wear. He said he'd cut his arm and his cheek\u2014Claire thought of Hunter's slashed face again\u2014on broken glass when he'd tried to change a window of their new house here at the bog and had dropped it.\n\n\"Which just goes to show,\" Andrea had said, \"he can fix most of the problems around here, but he should leave the Mr. Fix-It domestic work to someone we hire.\"\n\n\"Every dollar,\" Brad said, gingerly fingering the large bandage on his cheek which also showed the gauze wrap on his arm, \"needs to go to Black Bog. It's all input right now with no profit, but just wait until we can sell magazine articles, book and TV rights. We'll make it up.\"\n\n\"Speaking of making things up.\" Andrea smoothly took over the meeting again, \"Claire says she has put together some possibilities of relationships among the bog trio\u2014fiction based on fact.\"\n\n\"A good way to say that,\" Claire said. \"In a way what I'm doing is what Alex Haley, the author of _Roots_ , __ the groundbreaking book on African heritage, called _faction_. Much research but then some fictionalized accounts. I wasn't going to just read these three self-portraits of the trio, but I think I will start that way and then tie what I've written to the facts\u2014and artifacts.\"\n\nShe looked across the table at Brad. He stared back, not budging, not commenting. Finally he said, \"Slides will be available,\" as she lifted her paper to read the first person statement she'd written for Hunter.\n\n\"And, of course,\" Andrea put in, \"the bodies themselves can be studied in more detail. And someday perhaps someone will find a way to download what's in those stored brains.\"\n\nClaire thought of a smart remark about needing to use their own brains\u2014by studying the artifacts themselves. And that she couldn't tell enough just from looking at the bodies. But, due to the sudden unspoken tension at the table, she said neither and started to read.\n\n* * *\n\n\"So it went really smooth?\" Mitch asked Jace and Brit as they walked the roof of the parking garage toward the chopper for the flight back. They'd be flying into a sinking sun so they wanted to head out as soon as possible.\n\n\"It seemed to,\" Brit said. \"First we saw some chunky bead necklaces and a bracelet. And a dagger, which we'd kind of set up with the story about my father being a connoisseur of old swords and daggers. The so-called artisan jeweler called that the pi\u00e8ce de r\u00e9sistance. The dagger was kind of crude-looking but it had chiseled designs on it, strange ones. Animals and a woman holding up both hands like she was blessing them. At least that's what we thought it represented.\"\n\n\"So you sprang for how many hundred thousand dollars?\" Mitch teased as they neared the chopper.\n\nJace saw a man had emerged onto the top floor of the heliport where they had landed the chopper, but he evidently knew to keep back from the wash or the rotor blades before Mitch started the engine. The man was covering his eyes with both hands in the sunlight. Mitch turned his head to look at Jace and Brit in the back seat.\n\n\"We told them we were going to think it over, maybe even ask my father and be back soon,\" Kris said. \"Our supersalesman did leave us to make a phone call he had scheduled, but since we saw a camera and tiny microphone mounted in the room we made sure our conversation while he was gone was all about how Daddy would love to have that.\"\n\n\"Way to go,\" Mitch told them and gave both a fist bump. \"And speaking of that, the helo's gassed up, so strap in and let's get going.\"\n\n\"I loved the view of the Glades from the air on the way over,\" Brit said.\n\nIt was a windy early evening, perfect for a flight into the sunset with the island hammocks and patches of water beneath them, the sight of gators catching the last of the sun, the flocks of ibis and egrets taking flight below\u2014though the pilot had to watch out for those. Birds near airports in general were bad news.\n\nJace was content to be in the back with Brit, though he'd rather be at least in the copilot seat if she weren't here. Man, he missed flying, but jets, not helos. He couldn't wait to start training for the hurricane flights. The _whap-whap_ of the rotor overhead became a blur of noise.\n\n\"Homeward bound,\" Jace told Brit and took her hand. He had to raise his voice. \"And someday we'll say that about our own home, too. Soon. I promise.\"\n\nAs they flew westward from Big Cypress National Park, they could spot the Western Florida coastline with its mangrove islands of all sizes. The sun was sinking, throwing a blaze of crimson across the water. Mitch had notified Marco Executive Airport that they were coming in to be sure their landing would be clear of jets. He would put the chopper down near the hangar where they had left their car.\n\nMitch put the helo's lights on. As they sliced through the evening dusk, holding hands, Brit and Jace looked down at the jigsaw puzzle pieces of small islands and water. Some lights on heavily populated Marco Island were already lit. He spotted the bridge to the mainland. If she wasn't busy putting Lexi to bed, he would have insisted on stopping by Claire's to tell her about today.\n\n\"A small jet is going to land and gets precedence,\" Mitch told them, repeating what the airport control radioed back. \"Going to go around, then put down.\"\n\n\"Ever heard about the sunset explosion?\" Jace asked Brit, raising his voice over the sound of the rotor and rush of wind.\n\n\"Isn't that just a myth, a tourist look-for thing?\"\n\n\"Absolutely not. When the sun sets over the Gulf of Mexico, if conditions are just right, there's a split second when the human eye sees what appears to be an explosion of extra color, sometimes even movement.\"\n\n\"Have you seen it?\"\n\n\"Twice, but I think you have to be at beach level. Another exciting experience I promise to my beautiful fianc\u00e9e.\"\n\nJace knew Brit wasn't a nervous flier as long as she had something to distract her, but she hadn't flown that much, except with him. He'd like to do more of that, fly them off into the sunset of happiness someday. He had to grin at that poetic thought. Should he tell her that too?\n\n\"Our turn to go in,\" Mitch said, maybe to himself.\n\nJace saw they were over either Addison Bay or East Marco Cove. The water here was darker, shadowed by the low mangrove islands as the bright ball of sun slanted even lower.\n\nA jolt wracked the chopper. Brit screamed and grabbed his arm.\n\n\"Rotors!\" Mitch shouted. \"Bird strike? I saw something!\"\n\nJace flashed back to the scene at the country club reception. Chaos. Cuts. Broken glass.\n\nThey tilted almost sideways. Despite his seat belt, Jace bumped into Brit.\n\n\"No control. Going down! Brace!\" Mitch shouted. \"Still got some gas on board, so pray for water instead of land!\"\n\nThe chopper rotated, went into a spin. Jace was suddenly so dizzy he almost threw up.\n\n\"Brace,\" he told Brit, reaching for her arm. \"Feet flat on floor! Head down! I'll get you out.\"\n\n_If_ I get out, he thought, as the impact of the water crashed into the side of the chopper and their spinning downward stopped.\n\nThe last horrible sounds stabbing his brain were _bird_ and _brace_ and Brit's endless scream.\n\n# 29\n\nClaire had just put Trey down and read a story to Lexi, who was a very good guess-some-words reader for her age and now wanted to help \"read\" her own big-girl books at night. Nita and Bronco were here for another day or two, until a still-shaky Nita could feel good about moving to their new house again. Every time they went there to clean the place out, something awful happened.\n\nClaire sank on the couch, more than tired. Well, so what, because she had more Black Bog work to do. She was not allowed to bring any photos or detailed descriptions of bodies home with her but she was making some notes about the corpses she'd seen from earlier excavations. If she could see some sort of pattern, it might throw more light on her trio.\n\nShe'd just spend a few minutes on this, then check on Nick in his home office where he'd been making lists of assignments for himself and law firm associates in preparing Dale's defense. He was agonizing over whether to advise Dale to explain his Nazi connection to the police or just wait to see if it came up. Nick was also worried whether he, as lead lawyer, would be withholding evidence on that, in which case he could face disciplinary hearings before the Bar Association. But wouldn't people sympathize with someone who wanted to obliterate his great-uncle's ties to Hitler and his horrible regime by destroying Nazi memorabilia on his own property?\n\nNick had also shared with her that he was wondering who else of Marian James's associates might have been told about Twisted Trees, even perhaps the Hitler connection. Could word of that leak to the media or the police through a source, a contact of Marian's, now that she was dead? Nick had said he was tempted to research the silent partners on her Endangered Properties Committee to interview people who knew her. Claire had told him she'd gladly help\u2014but not tonight.\n\nShe kept her phone close by, waiting for Brit or even Jace to call with what they'd learned across the state today at the Vances' other Art for Art's Sake shop. Maybe Brit would even stop by, since both Jace and Heck were hoping the drone attack at the country club had been perpetrated by the wacko family in the dispute over the price of golf course land. That would mean it had nothing to do with them, and they were just innocent bystanders.\n\nNick came in stretching and yawning before she even started her project. \"Not heard from them yet?\" he asked. \"It's dark. They won't just show up here, will they?\"\n\n\"Brit said they'd call. I'd sure like to know what they found before I go to the bog tomorrow. I'm going to really press 'the Senator' for even brief looks at the artifacts I haven't seen, including one special item.\"\n\n\"Claire, if you're thinking he might be selling copies of\u2014or the real\u2014artifacts from those graves, be damn careful. I suppose it's partly finders keepers in the law, but I'd have to look into it\u2014in my spare time,\" he said with a roll of his eyes.\n\n\"Remember, I didn't tell you any of that. I'm just fascinated by all this archaeology of the ancients. It is important, and I want to keep this job.\"\n\n\"Okay, I know.\" He yawned again and flopped down on the couch beside her. \"Don't tell me any more so you don't have to shoot me, Mata Hari.\"\n\n\"Pick someone else to joke about,\" she said, leaning her shoulder against his. \"She was shot to death by the French for being a German spy. I wish Brit and Jace would check in.\"\n\nAs if she'd willed it, her cell phone sounded. She reached for it and looked at the caller ID. Marco Island Fire and Rescue?\n\n\"Hello.\"\n\n\"I would like to speak with Alexandra Britten or Claire Markwood please.\"\n\n\"Alexandra's my six-year-old daughter. I'm Claire Markwood.\" Her voice trembled. She swallowed hard. Nick leaned closer, listening.\n\n\"This is the Marco Island airport patched through from our fire and rescue unit. A helicopter has gone down on approach to the Marco Island Executive Airport. I've called the contact name of Jason Britten left by the pilot Mitchell Blakeman, who filed a Fort Lauderdale flight plan, but Jason Britten doesn't answer, and from earlier flights from here we cross-referenced his contact number to this one.\"\n\nClaire could tell by Nick's frown that he could hear.\n\nThe woman on the phone went on, \"We know the approximate site from where the blips went off the radar screen. Help is on the way.\"\n\nClaire's hand began to shake. She could hardly hold the phone. Couldn't breathe. That time they were in a plane that ditched near Cuba\u2014raw nightmare.\n\n\"We know the people,\" she said, finally in full voice. \"Three aboard. Went down where?\"\n\nNick had put his arm hard around her and was leaning close to hear, maybe to prop her up. This was all her fault. She had sent them. How could this have happened with Mitch at the controls? Jace had said Mitch was a great helicopter and jet pilot.\n\n\"Our squad has gone out to search, and we've advised local Fire and Rescue also. We don't know their status at this time. We're sending another helicopter with search lights and the beach rescue patrol. We believe it was a water landing but\u2014\"\n\n\"If we come to the airport, can we get to the site from there?\"\n\n\"Please don't come at this time. It may hamper the efforts of\u2014\"\n\nClaire ended the call. \"I'll tell Nita we're leaving,\" she told Nick, pulling from his embrace and springing to her feet. \"You heard?\"\n\n\"I'll get our jackets and flashlights, but we can't get in their way.\"\n\n\"It's my fault,\" she insisted, feeling sick to her stomach but also angry and energized. \"I have to be there, have to help. If they are hurt or gone\u2014any of them\u2014I'll never forgive myself, and Lexi won't either.\"\n\n* * *\n\nJace wasn't sure where he was. Swimming? No water near the base in Iraq. His head hurt, his soul hurt. Had he ditched his jet in the Persian Gulf?\n\nThen reality returned. Chopper crash. Had to get out. Had to save Brit.\n\n\"Mitch? Mitch?\"\n\nDamn dark here, water rushing in through shattered Plexiglas. Not in the Gulf, but one of the waterways between the mangrove islands.\n\nBrit. Where was Brit? Oh, strapped in right beside him. He unhooked his seat belt. The power of the water pouring in scared him. She was still in her seat belt, bent over in the brace position. Unconscious or dead? How long since they'd crashed? With this deluge, they were going under, going down to drown.\n\nHe unhooked her seat belt and hauled her up into his arms, sopping wet. She sucked in air, panting, clinging to him so hard her fingernails bit into his skin. Thank God, she was alive and conscious.\n\nHe tried to sound rational, calm. How large was this pocket of air? \"Brit, we're going out through that hole in the dome, but it's sideways now.\"\n\n\"Can't see. Dark.\"\n\n\"Listen to me! We have to wait until the water comes up to a good level so we can swim out, hold our breath. Hang on here. Got to find Mitch.\"\n\nHe half swam, half dove toward the pilot's seat. Mitch must have hit his head bringing them down in a somewhat upright position. Thank God, his head was above water but he wasn't responding. Was he even breathing?\n\nJace unhooked his friend's seat belt, and, still fighting the driving force of fairly warm salt water, keeping his head above the rising level of it, he dragged Mitch into the back seat, nearer to the broken window. He was terrified at how limp he was.\n\n\"Mitch? Mitch!\"\n\nHe shook him and hit his back between his shoulder blades as hard as he could manage in the water. If he stayed in here to give him breaths or CPR, the chopper would fill with too much water, and they'd all drown.\n\nBut Mitch spit out water and coughed. Two for two, Jace thought, but if they didn't move fast and just right, all three of them could be gone.\n\nAnd then, made heavier with the air bubble gone, the entire chopper shifted and settled lower.\n\n\"Gotta go now!\" Jace shouted but swallowed a mouthful of water.\n\nHe spit it out, grabbed Brit's arm and thrust her ahead of him, shoving her toward where he was certain the opening had been. Water wasn't pouring in anymore. He let go of Mitch and pushed ahead of Brit, feeling in the darkness for the jagged opening. If it had settled on the bottom of the bay, they were goners.\n\nBut no, there was a space, just at a different angle. He surfaced and told her, \"There is an opening. Get a big breath. Go down my body, feel my leg. It's where my foot is. Go! I'll be right behind you! I love you. Go!\"\n\nShe sucked in a breath and went under, grabbing his leg, kicking, fighting her way down. And then, thank God, she let go, must have gone out and surfaced. _Please, God, let her be all right. Let her make it, whatever happens to us._\n\nHe could sense the tide or the moving water was going to shift this baby again. Glad he never flew these birds, only his beloved jets. Got to go now, drag Mitch out to fly another day.\n\nHe had trouble pulling Mitch under. Were his lungs full of air? They weren't going to make it, but he sure as hell wasn't leaving Mitch behind. Never leave a fellow soldier or pilot behind, not even their bodies.\n\nHis lungs were bursting as he held his breath to shove Mitch through the opening. His arm got cut along the edge of the shattered plastic bubble. So damn dark here. He wriggled through the opening after Mitch, still holding to him. His ankle snagged something hard\u2014a broken rotor?\u2014and he shot toward the surface with it clinging to him.\n\nThe first thing he heard and thought he must be imagining was the _whap-whap_ of rotor blades. Was he remembering the wreck? But a blinding floodlight shot through the black sky to fix on them. A rescue chopper overhead?\n\n\"Jace! Jace! They've found us, they're all around!\" a female voice screamed.\n\nBrit. She'd made it. She must have seen or heard the rescue chopper, but who was all around? Mitch sucked in a breath, coughing, gagging, then another.\n\nBut in the reflected light of the search beam as the chopper circled overhead, Jace realized they likely hadn't been found by search and rescue. For in that beam, when it wasn't boring into his eyes, he saw pairs of glowing eyes, reflecting the light.\n\nHe'd been gator hunting with Bronco once, where they located their prey by shining lights into their eyes. He knew what those were.\n\n* * *\n\n\"Claire, we can't get farther into this swampy area,\" Nick insisted, as he drove the unlit road near the small airport, following the big beam of light up ahead. That must be a second chopper, searching for the wreck. And it was circling one spot.\n\n\"Then we can go on foot,\" Claire insisted, leaning toward the dashboard and looking up.\n\n\"They have professionals out here searching,\" he told her, stopping their car but reaching over to be sure she didn't get out. \"I know you're blaming yourself, but you couldn't know the chopper would go down.\"\n\n\"Poor Brit too! I asked her for this favor. What if some drug czar is still out to get Jace and Mitch? That picture in the paper may have been the wrong move. Nick, look, I see vehicle lights heading out there on the road.\"\n\n\"If it is criminals after them, we can't go closer. I'll call Jensen's police station, ask them to let him know. They may send help too.\"\n\n\"No, look! One of the vehicles out there is some sort of police car or rescue vehicle\u2014a blinking light bar, see? Reminds me of when we got to Twisted Tree the night Marian died. Hope that's\u2014that's not an omen,\" she added and burst into tears. \"You know, with her body so charred from that fire that they're taking so long with the autopsy. Nick, please just go a little closer. If we can help... I've got to help, to know...for Lexi.\"\n\nHe heaved a sigh. \"Yeah, for all of us. Jace is my friend too,\" he told her, regretting he'd not trusted him lately\u2014or Claire's feelings for him.\n\nHe turned on the car's bright lights again and drove carefully down the bumpy road behind the lighted vehicles, toward the hovering helicopter which seemed balanced atop its light beam in the black sky.\n\n* * *\n\n\"Brit, can you climb a tree?\" Jace shouted. \"There's gators!\"\n\n\"I saw one. I am up a tree. But help on the ground must be coming!\"\n\nDamn, what a hell of a brave woman! But what if those gators came in the water after him and Mitch or tried to snatch them on the bank? He'd seen those prehistoric hangovers drag their prey into the water to drown and devour it.\n\n\"I can swim,\" Mitch gasped out. \"Better off on the bank.\"\n\n\"I can hit them with something here caught on my foot,\" Jace said and reached down to free his ankle. He thought it was a piece of broken rotor blade at first, but it was something else. Shiny plastic. Black. It would have to do.\n\nThey both swam toward Brit's voice as she called encouragement. Jace crawled quickly up on the bank, then helped Mitch out. Though the chopper noise was still loud, Jace heard what he knew was a gator roar. Then, damn, here came Brit stumbling toward them to help drag Mitch to his feet.\n\nA cluster of cabbage palms loomed before them. They'd get cut climbing those, but who cared? Better than a gator's teeth. He'd hit the gators right across the eyes if they attacked, or if these were hit men to try to finish them off. His thoughts\u2014his ideas\u2014foggy\u2014floating...\n\nMosquitoes swarmed them, though that was nothing next to gators. Brit tried to swat the swarm of bugs away as men on foot with more lights burst into the area.\n\nGators or not, Jace was going to make them all hit the ground. If those men had guns, they were doomed.\n\n\"Airport search and rescue!\" a man shouted. In the beam of the man's flashlight, for the first time, Jace could see they had missed the island, though this ground was swampy.\n\n\"Alligators here!\" Jace shouted, but they could see and hear the beasts smacking into the water to get away from the noise and the men.\n\nBrit burst into tears of relief. Jace sat Mitch on the ground where the guy looked dazed but at least conscious. In the looming lights he saw his friend had a big bruise blooming on his forehead. Hell, flying through hurricanes would be a snap after this.\n\nThen\u2014amazing\u2014he heard Claire's voice. No, maybe he was hallucinating again. But there she was, crying, pushing closer, fighting her way along the narrow sandy bank on dry land that appeared to be across from a decent-sized mangrove island the chopper had just missed. Thank God, the gators had just vacated the area.\n\n\"Good landing, pal,\" Jace said to Mitch as his buddy shook his head, trying to clear it. \"Is this real? Is she really here?\"\n\n\"I'm so sorry I asked you to do this, Brit!\" Claire cried, hugging Brit who hugged her back, before Claire embraced Jace too. Nick was here, right behind Claire, and he hugged Jace too.\n\nThey all huddled, Claire and Nick now holding up the two of them, sopping wet. \"What's that you're holding, Jace?\" Nick asked.\n\nThat's right, Jace thought. He'd hauled a weapon of sorts out of the water. He stared down in shock at the dark plastic piece of chopper blade under his left arm. No, not that. It was the chopped-up rotor of a drone.\n\n# 30\n\nClaire thought it was like a reunion at Naples Community Hospital that next afternoon\u2014family and friends plus Ken Jensen visiting Jace and Mitch. The two friends were in the same hospital room, both under concussion protocol watch as well as being treated for cuts, contusions and some water intake, not to mention mosquito bites, which Claire and Nick had been scratching at too.\n\nMitch, who had some water still in his lungs, had been sedated since he'd been delirious with a fever, raving and trying to get out of bed, but by late afternoon the day after the crash, he had calmed down and his fever was gone.\n\nBrit had been treated for bruises in the ER last night, but, amazingly, was in the best shape of the three of them. More power to women, Claire thought. The stubborn Brit had camped out next to Jace's bed last night.\n\nClaire was so thankful the three of them were not seriously hurt. She'd promised Jace she'd bring Lexi to see him. After conferring with Detective Jensen, Nick had gone to pick Lexi up from school to bring her to see Jace. Claire had not gone to Black Bog as she'd planned, but had phoned to tell Andrea only that she had to visit a friend in the hospital. If the Vances could hold things back, she could too. But she had also called Kris to tell her what happened and that Mitch was going to be fine.\n\nAs Claire paced alone in the waiting room down the hall, Ken Jensen came in after interviewing both men and Brit.\n\n\"Nick's not back yet?\" he asked, closing the door behind him.\n\n\"He should be soon. He's agreed to bring our daughter here for a short visit, because Jace and Mitch are going to disappear to Washington, DC, when they're released.\"\n\n\"So I hear. Hope that means they won't get targeted again. They said they needed to go there for training anyway.\"\n\n\"We're grateful for the officer you stationed outside their door. Washington won't be exactly the witness protection program, but they probably told you that Jace notified their new government flight boss what happened.\"\n\n\"Yeah. They told me. The two new hurricane hunters sound like they've been the hunted so far.\"\n\n\"Detective, the thing is, if this mess isn't solved, they may be too big of a risk for that new assignment they're so excited about. At least Jace considers a big busy hospital\u2014with a police officer\u2014safe for a little time with Lexi. Not seeing Lexi has been hard on him\u2014on both of them.\"\n\n\"I can understand. Meanwhile, dealing with drug lords, if that's who's after them, is out of my element,\" Jensen admitted. \"Except for an occasional low-level arrest, that is. Since there doesn't seem to be any reason Brittany Hoffman would have enemies, she may just have been in the wrong place at the wrong time.\"\n\n\"My fault, since I asked them to go across the state to do a favor for me.\"\n\n\"So you said\u2014and that's all you're saying?\" he asked, looking narrow-eyed at her. \"None of the walking wounded are talking about where or why they went, and Nick said it's up to you to tell me. The thing is either someone's following them closely to know they took that helicopter, or else it was a random act by an idiot\u2014not uncommon in this new world of drones. Everyone's on edge that a drone is going to bring down a passenger jet someday.\"\n\n\"Jace\u2014my ex\u2014used to fly those.\"\n\n\"Nice try at a topic change. So he told me. Fighter jets and travel jets, he called them. Look, Claire, there were no fingerprints on the drone, since it's been scoured by water and sand, but then there were none on the one at the country club either. By the way, no one in the helicopter blames you and they were glad to run the favor for you. That's all they'll say. I repeat, want to tell me more?\"\n\n\"Not unless I have to. Nothing illegal, but my current assignment needs to remain private until\u2014well, until the project is announced. I'm afraid I can't tell you more right now.\"\n\n\"I'll bet it's that you _won't_ more than you _can't_. Let's sit down,\" he said gesturing to a round table with four chairs. He pulled out one for her and himself. He put his phone on the table, stared at something on it for a moment, then went on, \"Claire, I suppose your favorite lawyer will have a fit I'm asking this without him here, but just answer this. Could the drone attack on their helicopter have anything to do with this little mission you sent them on?\"\n\nShe startled. \"I can't see how. No one but them knew about it. The drone\u2014like you said, it must have been some idiot targeting a random helicopter or else the drug lords you mentioned. I'm sure Jace and Mitch told you there have been hits on Stingray pilots elsewhere.\"\n\n\"Yeah, and it needs to stay elsewhere. So this special advisory assignment you're on\u2014no links to Marian James and pricy, historic real estate, right? I looked at your Clear Path website, and I see in the past you've advised various local companies about interviewing staff or vetting job applicants, even detecting fraud.\" He tapped his cell phone and her website came up on it. He lifted the phone and waved it at her, before tapping her homepage off. \"Were your ex-husband and two friends over in Lauderdale on a fraud case, anything like that? I'm just groping in the dark on this.\"\n\n\"No one has hired me to look into fraud for several years,\" she protested, but her mind was racing. She had sent two men and a woman to check out the possible misuse of bog artifacts, so would that be considered fraud or theft? Had the ancient bog trio committed some similar crime and died for it?\n\nShe didn't want to lie to this man, but she could hardly confess her reasoning. Besides, just as with Dale's act of arson on his own land, the artifacts were found on private property. Even if the Vances had been told she'd been in their Naples shop asking about ancient artifacts, there was no way they could have known she was checking on them through friends across the state. She hadn't even told Kris about her suspicions.\n\n\"Did Nick have time to tell you the findings on Marian James's autopsy?\" Jensen asked, surprising her at the sudden change in topic as he pocketed his cell phone.\n\n\"No,\" she said sitting forward, elbows on the table. \"It must have just come out. Obviously, cause of death would be the fire. She must have been burned to death or died from smoke inhalation first.\"\n\n\"Neither, because the ME says she was dead by then. She had a high carbon monoxide level in her lungs. No throat damage because she wasn't breathing when the fire spread. She was choked to death\u2014strangled. Her hyoid bone at the front of her throat was broken, just like Cyndi Lindley's.\"\n\n* * *\n\n\"So, Lexi,\" Nick said once they were in his car, \"I need to explain something to you.\"\n\nIt had taken him a lot longer to get her out of school than he'd planned. He understood safety protocol, but the secretary had to consult the assistant principal and he had to show ID because Claire, Darcy and Jace were the only three people on the safe-to-go list. That annoyed him. Sure, he was always at the firm when Lexi was finished with school for the day, but Claire should have thought to put his name on the permission list anyway.\n\n\"Okay, I'm listening,\" she said, sounding very adult, as if to calm him. \"You said Mommy's okay, but I heard you say we are going to the hospital.\"\n\n\"Yes, Mommy's okay. Your dad is okay too\"\n\n\"Remember, you're Dad and my real dad is Daddy.\"\n\n\"Right. The thing is your real father is just fine but he's in the hospital because the helicopter he was in with Brit and his friend Mitch had an accident. But no one is hurt bad.\"\n\n\"How bad, though?\" she said, turning to him. \"And Brit's gonna be my stepmother. You sure she's all right?\"\n\n\"Yes, honey, she's all right. Jace\u2014your Daddy\u2014just has some cuts and bruises, but he wants to see you.\"\n\nHe saw out of the corner of his eye that she folded her arms over her chest. \"He didn't want to see me before.\"\n\n\"Yes, he really did, but he was afraid to. Darn it, Lexi, I'm going to talk to you like an adult now, so I want you to act like one, all right? Can you?\"\n\n\"Was he flying the plane? He loves planes.\"\n\n\"No, he wasn't flying, his friend Mitch was. I'll let your Daddy tell you. But the thing is, someone must have been angry with him or his friend Mitch or even Brit. I know you've been angry with him for staying away lately, but he wanted really badly to see you and does now too.\"\n\n\"Somebody bad is after him?\" she asked, frowning, almost pouting.\n\n\"We\u2014I\u2014don't know. Maybe they are after Mitch, and your Daddy was with him. You're going to see a policeman there guarding them, so don't be afraid. Your Daddy needs you to hug him and smile, not to be angry with him when it wasn't his fault. You know adults and parents sometimes have hard times too, just like kids.\"\n\nShe heaved a huge sigh and reached over to touch his arm. \"I love him, but I love you too, Dad, 'cause you love my Mommy and me too.\"\n\nHe had to blink back tears. \"I sure do, Lexi. And I want to be not only your stepdad\u2014your Dad\u2014but your friend.\"\n\nShe patted his arm as if to reassure him. \"Mommy loves you too, even if she worries about my real Daddy sometimes. I'll try to think like an adult\u2014like you said\u2014but don't tell her you said I'm grown up now, 'cause she might get upset. I want to take care of her but it's hard sometimes.\"\n\nHe was thinking _You got that right,_ but he said only, \"Then we agree on that. Good.\" Emotion swamped him so hard he almost lost his voice. He blinked back tears blurring his vision, so he wouldn't steer them off the road.\n\n* * *\n\nAfter Lexi hugged Claire, Claire took her in to see Jace. The dividing curtain was closed between his and Mitch's beds.\n\nAcross the curtain, Kris had just come with flowers for Mitch. \"I should be sending you those!\" Claire had heard Mitch tell Kris. \"So glad to see you\u2014and no ice skates this time.\"\n\nClaire had noticed that Kris had hesitated at the door to the room for a moment, staring at a sleeping Jace, before Claire had pointed and whispered to her, \"Mitch is in the bed by the window.\" With a nod and a squeeze of her arm, Kris had walked farther in. That face blindness, Claire thought. Could any man of a certain age in a hospital bed and a hospital gown pass for Mitch to Kris at first?\n\nJust before Claire stepped out, she saw Lexi squeal and make a dash for Jace. He startled, opened his eyes and embraced her in a huge hug, then pulled her up to sit on his bed. He smiled at Claire over the child's head before Claire stepped out into the hall. Nick leaned there against the wall, looking tired and upset.\n\n\"The school didn't want me to take her at first,\" he said.\n\n\"Oh\u2014sorry. I guess I forgot to put you on the list.\"\n\n\"Yeah, well, I'd like to be at the head of it.\"\n\n\"Let's go down the hall,\" she suggested, and they walked together. \"You are at the head of _my_ list, Nick. It's just that I overlooked it because you're always working when she gets out. I'll fix that right away.\" She linked her arm through his, and he pulled her closer to his side. \"We've come through a lot. I thank God we have each other.\"\n\nThey sat on a sofa next to a potted palm and a window. They could see Jace's hospital door from there.\n\n\"Lexi's amazing,\" he told her. \"She ended up comforting me in a way.\"\n\n\"She has her moments.\"\n\nStill holding her hand, he leaned back and heaved a huge sigh. \"Hopefully, all of us do. I see Jensen's gone but I'm agonizing over whether to level with him about Dale's Nazi connection, with Dale's permission, of course.\"\n\n\"Which he may not give you.\"\n\n\"I think I could convince him.\"\n\n\"And I'm agonizing about not telling him anything about what's going on at Black Bog. But what about the possible Cyndi\u2013Marian James connection? Did our favorite detective tell you the results of Marian's autopsy, at least what the ME can tell so far? Are two strangulation deaths and two broken hyoid bones enough of a connection to assume they had the same killer?\"\n\n\"Jensen thinks so. I do too. Is there any way you can do a little very private research for me into Marian's life\u2014friends, enemies? Heck is too busy with this idea he has for an invention.\"\n\n\"You're thinking especially to look at people Marian worked with?\"\n\n\"Jensen's pretty much eliminated family troubles as a reason for her death. There's an ex-husband she hasn't seen for years who lives in Hawaii. She lived alone, so that's a dead end\u2014Didn't mean it that way. Not funny. Damn, nothing's fun or funny right now.\"\n\nShe squeezed his hand tighter. \"Yes, I'll check into Marian's Endangered Properties Committee, but I think she had some silent partners or investors, so I may run into a wall there.\"\n\n\"I know you're busy with your own bog business, so I appreciate that.\"\n\nShe nodded, wishing she could tell him even more than she had about her suspicions. Brit had filled her in on everything that had happened at the Lauderdale shop, so Claire felt she had something to hold over Brad Vance's head now, but to make him admit what? Was he so desperate for funding that he ignored the fact that someone could ask what inspired them to sell copies of ancient artifacts? Their work at the bog needed funding, which was why, she thought, the Vances had just sold a longtime home they loved.\n\nSo she was not going to confront either of them but just keep her eyes open\u2014and view at least Brad's photos of those precious objects again. Brit had said the dagger they had seen in Fort Lauderdale had a woman with raised hands on the hilt, and Claire thought that figure might be Reaching Woman\u2014or was she reaching for too much on that? And why hadn't she noted that in the detailed photos? Had he showed her all he had?\n\nAfter a long pause, she told Nick, \"Jensen tried to get me to explain why I sent Jace and Brit across the state, but I stonewalled him. I don't need him crashing in at the bog to question the Vances until I'm more sure what's really going on. So are we both holding out on him?\"\n\n\"For now. I'll talk to Dale, and you just stick to keeping your eyes open, but no interrogations or confrontations.\"\n\n\"The truth is I don't want to get the Vances in trouble. I want to protect them and their work. It's important, far-reaching for historical knowledge, for mankind, really. And I want to be part of their continuing project.\"\n\nBut, she thought, wouldn't Brad's first sniff that she'd been checking up on his handling of artifacts mean that, no matter how badly Kris or even Andrea wanted her on the team, he'd terminate her?\n\n# 31\n\nThat evening Nick was glad he'd phoned Dale. The guy sounded shaky. He said he was sitting at the end of the Naples pier just staring at the water, waiting for it to get dark and that he might stay there all night.\n\nNick explained to Claire, kissed her, Lexi and Trey and headed for the pier. He didn't like the idea of one of the firm's formerly competent lawyers, his client, out on bail but accused of a double homicide, sounding damned depressed out on the end of the pier as darkness fell.\n\nHe found a parking place and jogged toward the pier, then out onto it. The sun looked huge and crimson as it bled into the watery horizon. Not too many fishermen were out this late at this time of year like during the winter season. He was careful not to get hit when they flicked their line back over their shoulders to cast. When he was a kid, he'd gotten snagged by a hook on his ear, but his dad had rushed him to a doctor friend to tend to it.\n\nHow he still missed his dad who had been gone for years\u2014his murder staged to look like suicide. Since then, Nick had helped numerous people prove cause of death with his South Shores side project, and he needed to help Dale now. But he didn't see him where he'd said he'd be.\n\nHis heartbeat kicked up. No\u2014there he was, way over in the corner of the pier, as far out as he could go, leaning against the wooden railing, staring into the sliding depths of the breaking waves.\n\n\"Dale!\" he called to him and joined him at the rail.\n\nHis former junior partner\u2014relieved of his duties at the firm for now\u2014turned toward him. Either the guy had been standing there so long that his face was streaked with salt water spray or he'd been crying. His usual suave and neat appearance had disintegrated to his wearing cut-offs and a ratty-looking sweatshirt.\n\n\"Thanks for coming out here,\" Dale said, his voice a monotone. \"I figured we needed to talk. I suppose the office would be the best place, but if I can't be there as a partner, I'd rather not come in as a client. Everybody stares, pretending not to, no doubt thinking, 'Poor Dale. Law career over, might get disbarred. Maybe he even committed those murders, lots of proof against him.'\"\n\nNick put a firm hand on Dale's shoulder. \"We would all be behind you if you were just any client, but you're more than that. You're one of us, and we're going to fight for you.\"\n\nHe nodded, still looking out over the white-capped waves. The wind was increasing and so was Nick's unease about Dale's state of mind. But he had to pursue his plan.\n\n\"Dale, I was with Ken Jensen earlier today because he's investigating a plane crash three friends of ours were in\u2014brought down by a drone right off Marco Island, though they all survived in pretty good shape.\"\n\nDale finally turned to look at him. \"That's good news for them. And Jensen brought me up?\"\n\n\"He was investigating the crash, but when I started to think about what we're holding back from him, I wanted to consult with you. You've got a good lawyer's brain. We don't want the Nazi connection curse coming out at the trial. It needs to be eased out beforehand, common knowledge by then. The fact you wanted to burn Nazi memorabilia you detest and past ties to your family could get you sympathy, not blame, from a detective, a prosecutor and a jury.\"\n\n\"Except for that woman's burned body in the mix\u2014like I was trying to stop her, as well as Cyndi, from dragging that out after all these years.\"\n\n\"Yeah, and one other problematic thing. Marian James was burned in that fire, but cause of death before that was strangulation with a broken hyoid bone\u2014sound familiar?\"\n\n\"Damn! And Jensen arrested me before he knew that, but now he'll have more evidence\u2014so he thinks\u2014to nail me for both murders. But I see what you mean. The Nazi connection needs to be communicated to the prosecutor before we go to trial. But we don't even have a date yet! That's way off in time, way off in thinking.\"\n\n\"Is it? As soon as that breaks from another source\u2014\"\n\n\"If it does.\"\n\n\"That would mean you could lose me and maybe the firm when we get slapped with professional misconduct for not admitting it earlier. And, yes, I'm worried about myself and the firm getting hit with fines or sanctions. We don't need that, but you need us!\"\n\nHe looked away, squinting out over the waves again. Nick knew it would be so easy for him to jump from here, to hit his head on the huge wooden girders holding up the pier, especially with daylight fading where no one would see to help him.\n\n\"Nick,\" Dale said, still not looking at him, \"I give you permission to tell Jensen about my great-uncle and the Nazi stuff I wanted to burn in that apartment. It gives me an even bigger\u2014hopefully sympathetic\u2014reason for the fire.\"\n\n\"That's brave of you and the right thing to do. Eva Braun's German diary suggests she was never pregnant\u2014and of her desire to kill herself, which she must have done in the upper room of the apartment. Dale, suicide is always wrong, and your state of mind right now scares me. We're going to see this through together. And that diary is yours if you want it back for proof later, but how about I just keep it for now?\"\n\n\"Yeah, that's best. So, Eva Braun speaks from the grave. Wonder where my great-uncle buried or burned her body. But for now, however damning it is to link my last name to Hitler's wife, I do need help,\" Dale whispered, pressing the palms of his hands over his eyes. \"I have nightmares I'm burning up in that apartment I torched, that the woman burned there is Eva Braun, not Marian James. I've gotten to the point I can't stand to be closed in my apartment or at the house I want to sell. I keep thinking about prison, and I couldn't stand that, couldn't face being locked up or worse. Nick, I don't know how my life went so wrong so fast when all I wanted to do was get Cyndi out of my life.\"\n\nNick shuddered. Granted, Dale was the obvious candidate for the two murders. Other possibilities he'd checked into, as had Ken Jensen, came out in weak second place. And what Dale had just admitted sounded almost like a confession he'd hurt Cyndi, at least. This guy was really losing it. If he worded his desire to move on from Cyndi like that\u2014\"get her out of my life\"\u2014that could ruin him in a courtroom. Didn't he know that?\n\n\"Dale, let's get you to a psychiatrist. There's a good one the firm's used before. The doctor's trustworthy and discreet. It sounds like you're clinically depressed, maybe claustrophobic. Okay? You have your car near here?\" he said, putting his hand on the man's elbow.\n\n\"I walked. Just wanted sky and sea.\"\n\n\"Sure, I can understand that. Let's take my car. If you don't want to come into the office as we put your defense together, we can meet outside somewhere, but you need to go home right now, see that doctor tomorrow. I'll set it up.\"\n\n\"Wish I could go home again,\" he said with a sigh as they started back down the long pier. \"Have my dad around, my mother in her right mind. That old Betty Richards next door has tried to mother me too, ever since her son hanged himself about twenty years ago, so maybe I wouldn't even mind her around. She used to be so possessive and protective of me after that. I hated it then, but could use some of that now.\"\n\nNick's mind seized on that latest bombshell as they walked off the windy, spray-spattered pier. Dale might be a flight risk, suicidal or even guilty. Betty Richards just had another motive hung around her neck for having at least killed Cyndi. And if Marian had been bothering Dale, maybe for killing her too. But could that old woman strangle someone? Betty had told Claire she'd seen the two murdered women together back when they were still alive. Could he trust that? Could he trust anyone but Claire?\n\n* * *\n\nJolted awake, Claire sat straight up in bed. Nightmare! But at least one she hadn't gotten up to act out, though she wished she didn't recall it so clearly.\n\nBeside her in bed, Nick slept on, snoring slightly. He'd come home late after getting a junior partner at the firm to stay with Dale in his apartment, then made sure he'd get him to a psychiatrist's office tomorrow for an appointment Nick had made.\n\nShe was too warm. She whipped the sheet off her side of the bed and got up carefully, quietly. She didn't have to have a psych degree to analyze where she'd cooked up that dream, at least some of it. She'd dreamed she'd crashed into the bog, going under, going down. She had fought her way upward through the black peat and mud and found Reaching Woman\u2014Yes, reaching down to help her up so she wouldn't die.\n\nBut after Reaching Woman had pulled her up, Claire had seen she raised her arms the way Brit had described on that dagger in Fort Lauderdale, the way some people did in church when they were deeply moved. It was the way referees at a football game signaled a touchdown. Was Reaching Woman giving a blessing or a curse? Was she a religious figure, one who should be pure, but who had fallen for either Hunter or Leader and paid with her life?\n\nClaire gasped. She envisioned the worst part of the nightmare, the part that had woken her up. She had seen again, just as when the dig team uncovered the woman's body, the dagger in her chest. And in the dream, Claire had clearly seen\u2014knew for sure by the angle of the weapon\u2014that the woman had stabbed herself.\n\nHer knees went weak, and she knelt by the bed with her head on the sheet, using it to blot her silent tears. Could that be true? Or had she dreamed that because Nick had told her he thought Dale might be suicidal? That had set off his memories of his own father's death, and she had comforted him.\n\nShe knew she wouldn't sleep right now. Maybe she would go in Trey's room and watch him sleep innocently, no worries in the world.\n\nShe put on a robe and tiptoed out, checked on both children and went quietly past the guest room where Bronco and Nita slept. She sat at the dining room table where she'd left her laptop and turned it on. She'd just do a little work\u2014that favor for Nick tracing Marian James's contacts\u2014then go back to bed.\n\nShe turned a lamp on. First she searched for ancient artifacts for sale in the state of Florida. Nothing but some Seminole tribe souvenirs at their casinos, but then who would be stupid enough to advertise priceless, prehistoric artifacts online? Certainly not a former Florida senator or his savvy wife, who now owned and operated a historic dig that would rock the world when it was announced.\n\nShe turned to searching Marian James's Endangered Properties Committee. Bingo! A beautifully crafted, detailed website. Marian's name was still all over it, but no one else's except the committee's treasurer, Harmon Kingsdale, with his contact information and a plea to make a donation to the cause of preserving Florida history for future generations.\n\nShe wrote down that contact information. It wouldn't hurt to make a call or visit Mr. Kingsdale to see if she could dig up any other names for Nick's defense team to question. His name sounded familiar but she couldn't quite place it.\n\nBut now the hunt was on for any possible enemies Marian might have had. And her own hunt was on to study Hunter's dagger handle again so she could decide if Reaching Woman might have stabbed herself and why. Had one of the men she might have loved forced her to do it?\n\n* * *\n\nThe next morning after Claire waited with Lexi until the school bus came and before Nick left for the office, they sat together for a cup of coffee. She held Trey in her arms. Nita was still in bed, and Bronco had headed for the law firm, since they were especially conscious of security lately with not being able to trust that Tanner and Will would be heading home as they'd said.\n\n\"Harmon Kingsdale,\" Nick said when she filled him in on her research last night. \"No kidding? So there's probably big bucks behind Marian's committee.\"\n\n\"His name is familiar, but who is he?\"\n\n\"The founder of Pine Ridge Luxury Homes. Big houses, some in Pelican Bay, some up by Estero.\"\n\n\"Well, I guess it makes sense a home builder would be interested in helping to save and preserve endangered lands or properties. Should one of us interview him?\"\n\n\"We can hardly send Jensen. But I might make him nervous as he knows I'm a criminal lawyer. Dale would be persona non grata right now, though he needs something purposeful to do. You'd have to go with a fake name, and that's no good. Who's left that we can trust?\"\n\n\"Lexi? Trey?\" she said giving the baby a little bounce which he traded for a gurgling smile.\n\n\"Very funny.\"\n\n\"I don't want to ask anything else of Brit after I nearly got her killed.\"\n\n\"That wasn't your fault, unless you've been sneaking out to master using drones as a weapon. But without alarming Kingsdale that we're trying to learn more about Marian, how to approach him to get the names of others on their committee who might have had an argument with Marian, or at least know someone who did. Maybe we could have someone approach him for a donation. It's Gina's weekend to be off duty in the clinic in Miami, and she'll be here Friday afternoon, so how about we ask her to go see Kingsdale with Heck, maybe play up the donation angle, just something to get in the door with him and not tip him off as to why.\"\n\n\"Sounds good,\" she said and they clinked coffee mugs.\n\nThey were desperate for help. But what Claire didn't say again was that the last time she asked friends to do an errand for her to find out information, they were all nearly killed.\n\n# 32\n\n\"Finally,\" Nick said, appearing from behind today's newspaper which he was reading at the breakfast table. \"Marian James's funeral announcement is here. It's tomorrow afternoon at three, calling hours one until three. Her next of kin are evidently a niece and nephew from Pensacola. Not much time for people to plan to attend, but then the ME probably kept her body for a long time for a complicated autopsy. The testing for toxins would have dragged out the release of her body.\"\n\nClaire shuddered. Although they hadn't seen the woman's corpse close-up, it had looked so dark and terribly twisted from the fire. At a distance, it had reminded her of the Black Bog corpses. And, though she tried to get it out of her mind, she kept remembering Cyndi's frozen body with that silent scream on her face. Had Marian's face\u2014had Reaching Woman's face once\u2014looked that way too?\n\n\"Maybe we should go and see who shows up,\" she said, trying to keep the tremor out of her voice. \"I've heard of cases where the murderer attends a funeral out of curiosity or sick pride.\"\n\n\"For sure, Dale won't go. I wouldn't let him even if he wanted to.\"\n\n\"I didn't mean he'd go for that reason. Of course, he shouldn't go. But should we?\"\n\n\"Let's. It's one of those long obits,\" he said, frowning at the paper he handed over. \"She was altruistic...concerned about her community...trying to preserve Florida's history, especially around her beloved Naples...formed the Endangered Properties Committee.\"\n\n\"And then _she_ ends up endangered. It's a good photograph of her.\"\n\n\"The two times I tangled with her, she hardly looked like that. Yes, let's go and keep our eyes open, not only for the committee treasurer Harmon Kingsdale we want to talk to, but any others who might show up we know.\n\n\"Going to brush my teeth and head out,\" he told her, popping up. \"I'll kiss Lexi and Trey goodbye as well as my beautiful wife. Did you say Kris is dropping by early?\"\n\n\"On her way out to the bog. I'm not going until I get my thoughts together on Monday, but remember I told you Heck is going to meet Kris here to advise him on his invention.\"\n\n\"Oh, yeah. I'm getting overprogrammed.\"\n\n\"Aren't we all and no wonder. I guess he wants to ask her style type questions.\"\n\n\"Style?\" he asked, turning back in the kitchen doorway. \"About an aid to help people with face recognition problems?\"\n\nShe got up and started to clear dishes. \"I think like whether to put the tiny computer camera in a sort of bulky necklace, some kind of pin or sunglasses, something like that.\"\n\n\"Oh, he did mention that on the phone. Got to get going, sweetheart. TGIF! Our private date night, even if we don't go out.\"\n\n\"Right\u2014and tomorrow we have a date for a funeral. But remember, we promised Lexi a Saturday walk on the beach.\"\n\n\"When I say goodbye to her, I'll tell her we'll go midmorning tomorrow. Be right back.\"\n\nAt least there were several things to look forward to when they'd been through so much, Claire thought. She glanced down at the obituary on the table, then sat to read it carefully. You just never knew when someone who was so busy and full of life, maybe someone you needed and loved, wasn't coming home.\n\n* * *\n\nHeck asked Claire to sit in with him and Kris to give her opinion too. He showed them three computer-generated drawings of where he could secrete the camera lens so that when a person with face blindness was approached by someone who was not a stranger, but had his or her image already recorded and identified, a tiny, invisible device in one ear would give the person's name. If there was another short prompt needed, that would have been recorded along with the person's identity.\n\n\"Amazing,\" Kris said, beaming. \"But can you make the camera micro-sized and make it work fast enough so there won't be a stupid pause, waiting for that message? That's the way I've always felt as I stand there like an idiot, trying to ID the person, looking for telltale signs, something familiar they might be wearing, hair color\u2014like with Claire\u2014anything.\"\n\n\"Got to admit size and speed will be the challenges,\" Heck said. \"And finding a partner company to manufacture a good-looking necklace, large pin, glasses or sunglasses. But I've got to get this computerized camera and earpiece down first.\"\n\n\"Which means,\" Kris said, \"you might not like my answer. I think it would be great if face blindness sufferers had a choice of devices to do this great spy work. Especially for a male to use this, the glasses would seem the best, but you know women like a variety of things. I suppose a chunky necklace would be my first choice, something ceramic, not a precious metal, to keep the price down, right?\"\n\nShe looked at Claire, who nodded. Was Kris also picturing Reaching Woman's broken necklace? That reminded Claire that she wanted a moment alone with Kris before she left.\n\nWhile Heck and Kris continued to confer, Claire went to get Trey up. The door to the guest room was open, and Nita was packing a suitcase on the bed. Claire gave a little _knock_ , _knock_ on the door frame, and Nita jumped.\n\n\"You're okay with moving into your house this weekend?\" Claire asked, hovering in the doorway.\n\n\"Bronco says if it's too tough, I can't sleep or whatever, just stay two nights, then talk it over. I want it to work, Claire. But if there's one more thing that happens there, I got to say the house is cursed and we somehow got to find another.\"\n\nClaire stepped in and hugged her. \"It will all work out. I'm sure it will.\"\n\n\"I know you got guests. I heard Hector's voice. I was just going to get Trey up, change him and bring him down to all of you.\"\n\n\"I'd appreciate that. I'll see if Heck and Kris are done yet.\"\n\nShe stepped back and put her hands on Nita's shoulders. \"You're a strong woman, Nita Gates. You are going to be more than okay.\"\n\nNita nodded, and as Claire went back out, she hoped that could be true for her too.\n\n* * *\n\nStill holding a very curious Trey, who seemed enamored of Kris, Claire walked her friend out to her Jeep.\n\n\"Kris, I wanted to ask you something before I ask Brad. Do you recall, either on Hunter's dagger or on Leader's staff, that there was a picture or carving of a woman with her hands uplifted? You know, kind of like a ref signaling a touchdown in a football game?\"\n\n\"Now there's a mixed message,\" Kris said with what Claire thought was a forced laugh. \"But Andrea whisks the artifacts themselves away pretty fast for preservation and cleaning. Besides, since the Senator's claustrophobia bothers him if he comes out in the bog all hemmed in by those trees, he kind of panics. With my disability, I feel for him. So I get it that Andrea snatches the artifacts up and gets them to him so he can study them in his office or lab to feel part of a new discovery. That and fundraising are his part in all this.\"\n\n\"Wait. You said Brad has an office as well as that large lab with the screen to show the photos?\"\n\n\"His office is behind the lab, through a door off it. Considering his problem with closed spaces, I don't think he uses it much. I only saw it once\u2014desk, chairs, bookcase, locked drawers of relics, as he likes to call them, I think a safe with papers, contracts, stuff like that. Yi Ling helps him with his photography of items, so maybe you could ask her if she saw that raised-arms petroglyph too.\" She frowned and looked down at the ground. \"But no, I don't recall that myself among all the other pictures or depictions.\"\n\n\"And it would have stood out. If it was a rendition of Reaching Woman, the upraised arms would mean something important. Maybe that she worshipped a higher being\u2014some sort of creator or god, good or evil\u2014or it could mean that she herself should be worshipped.\"\n\n\"What? I never thought of that. But she has a dagger in her!\"\n\n\"I know what I'm going to say now sounds insane, but you're the only one I'm telling. I want you to promise you won't even tell Andrea or the dig team, let alone the senator, until I think this out to present to them.\"\n\n\"All right, I promise. But if you're on to a new theory\u2014or some kind of suspicion\u2014you are going to have to tell me that. Now you promise me!\"\n\n\"When I can. I know we've been looking for a reason someone would have stabbed Reaching Woman. I have, anyway. But the angle of that dagger\u2014kind of upward into her chest\u2014and the feeling we have that something was wrong among the three of them... Kris, what if she stabbed herself? Decided to or was reared to be a sacrifice or was even made to because she broke some sort of rule about belonging to Leader but loving Hunter?\"\n\n\"Claire,\" she said, hands on hips, \"you're married to a criminal lawyer who gets people off from murder charges. Are you sure that isn't coloring your theory on this\u2014I mean, like, you don't want her to have been murdered by one of the men? I know Nick's new client is up for murder\u2014double murder now\u2014and I know the two of you have been through tough times, but you're not transferring some of your past into this ancient death, are you?\"\n\nClaire bounced Trey in her arms. \"I don't know. I don't think so. I'm going to ask Brad to let me see those items on Monday, and I hope you'll back me up on that.\"\n\n\"Andrea makes all those decisions, and I have to honor what she wants.\"\n\n\"Does Andrea make final decisions? Or does he?\"\n\n\"They're a team. You know that. Like you and Nick. Like I hope to be someday with a man\u2014maybe Mitch, though I don't know about falling for men with dangerous careers.\"\n\n\"Yeah. Been there, done that,\" Claire admitted, realizing Kris had shifted them off the subject of murder-suicide\u2014murdercide, Nick called it. \"And who knows that Reaching Woman didn't share that fear with us? In her world, Hunter was living a dangerous life, but maybe that made him more appealing to her, more desirable.\"\n\n\"The price we pay for the exciting, macho guys, right?\"\n\n\"Right. See you on Monday then since we're going to have a family weekend here. And let me know if you get any hint about what the Vances\u2014all for one and one for all\u2014might say about my looking at and handling the artifacts.\"\n\nKris nodded and patted Trey's back before she climbed in the Jeep. Claire lifted his little hand to wave goodbye as Kris pulled out.\n\nShe had learned a couple of interesting things. Brad probably kept the artifacts in his closed-in office he didn't like, so he wanted badly enough to hide them that he braved that room for storage. Yi Ling worked closely with him. And, although she hadn't heard this from him, he called the artifacts _relics_ , __ which connoted they were especially dear and precious\u2014almost sacred or holy\u2014to him.\n\n* * *\n\nOn the beach as they waded in the surf, the wash of waves felt so good around their bare legs. Claire and Nick held Lexi's hand for a while, then let her pick up shells. The water slid up the slick sand, depositing them and uncovering the tiny coquina, which then upended themselves and dug in again.\n\nLexi chattered on about everything she saw, but Claire was only half listening. \"Mommy, those little shells! The ones that are open look like butterflies, see?\" she said, extending an almost plaid coquina in her open palm.\n\n\"They sure do. Let Dad see them up close.\"\n\nClaire was still floating on air, or maybe it was more like riding the waves, after Nick's lovemaking last night. Tender at first, almost teasing. She replayed some of it like a recorded favorite show. They danced, then kissed, then finally couldn't wait to love each other, and she was, as ever with him, swept away with passion and abandon...\n\n\"Oooh, what's that?\" Lexi broke into her reverie, pointing at a blob in the sand that looked like a plastic bag full of air.\n\n\"Oh, oh,\" Nick said, tugging her back from it. \"That's what they call a Portuguese man-of-war. It's an animal, and those purple cords are tentacles that can sting you really bad, so we have to walk way around.\"\n\n\"You can see inside of it. Ick!\" Lexi said, but she let him lead them in a large path to avoid it. \"I think it's dead!\"\n\n\"It can still hurt you,\" he warned.\n\nClaire noted a sign up on a grassy slope above the concrete breakwater: Sold. Pine Ridge Luxury Realty. While they avoided being stung, she remembered the for sale sign that Cyndi had posed with on her LinkedIn page.\n\n\"I'll watch out for more bad man-of-war tents,\" Lexi promised as she ran a bit ahead of them.\n\n\"Pretty close to man-o'-war tentacles,\" Nick said with a little laugh.\n\n\"Nick, this house\u2014even though we're at the back of it\u2014looks familiar,\" she told him, stopping to study the mansion with its fabulous view of the Gulf of Mexico. \"It would face on Gordon Drive, right?\"\n\n\"Sure. All these along here do. Why?\"\n\n\"The Vances just sold a home along here, and this sign says sold.\"\n\n\"So, maybe it was their house. And?\"\n\n\"And when we walk back to the car, can we drive by the front of it? There's a picture on Cyndi Lindley's LinkedIn page where she's standing in front of a house with a for sale sign as if she's the Realtor. That was either wishful thinking or a con job, which she was obviously capable of. Wouldn't that be something if the house she chose to take a fake picture in front of was the Vances' place\u2014and being sold by that Harmon Kingsdale's company?\"\n\n\"A spider's web of possibilities,\" he said as Lexi ran back to them with a piece of warped driftwood with some kind of small shells clinging to it. \"But I've learned to honor what Ken Jensen likes to say: 'There are very few if any coincidences in detective work.' Sure, we can drive around.\"\n\n\"What are these?\" Lexi asked, pointing to the piece of wood.\n\n\"I think they're called limpets,\" Nick told her. \"They are a lot like little snails that just grab on with a sticky foot and won't let go.\"\n\nClaire patted Lexi's back and nodded. Her suspicions about Cyndi and Marian\u2014even Brad Vance\u2014hung on, suspicions that clung like fears. Ones that just would not let go.\n\n# 33\n\nMarian James's funeral was held at the Saint Leo Catholic Church in Bonita Springs. It was a large lovely building of Spanish decor with a bronze statue of the Virgin Mary in front as if guarding the place.\n\nClaire stared at the Madonna's slightly outstretched hands with her palms open. Were those hands supposed to be reaching out with a message of welcome? Of peace? Of help? This was another reaching woman from the past, one venerated and worshipped here. Claire could not shake the idea that Reaching Woman had been something like that to her people. She had to examine those hands-off relics Brad had evidently been hiding. Surely, he and Andrea wanted her to do what they'd hired her for.\n\nThey sat halfway back in the large sanctuary where they could watch people, since most sat closer to the front.\n\n\"There's Harmon Kingsdale,\" Nick whispered but he didn't point. \"The one with the silver hair, mustache and light blue sport coat. Let's see who he sits with.\"\n\n\"That must be his wife. Really pretty. Younger.\"\n\nOut of the side of his mouth, Nick whispered, \"People are saying that about us.\"\n\nShe elbowed him and smiled. The glow of their lovemaking last night and their fun with Lexi earlier still warmed her.\n\nThey stood with everyone as the closed casket with an arrangement of roses and carnations was rolled down the aisle with the priest following. Boys with censers led the procession, their wisps of smoke a reminder to Claire of the terrible way Marian had perished. No, she was strangled first, Claire reminded herself. Two women dead, one frozen, one burned. And one\u2014eons ago\u2014stabbed to death by an enemy or by herself.\n\nThe priest's message praised all Marian had done for people, for her community. Still, Claire thought, it didn't seem that the priest had actually known her, as he offered no memories or comments of his own about her. Had she attended here, was she active in this parish? He hadn't said so. Maybe she'd been born and baptized Catholic but had fallen away and the church was being kind to bury her.\n\nShe drew her attention away from the service and admired the high wooden beams overhead with the stained-glass window of Noah and some animals from the ark just above the large statue of Christ on the cross. She skimmed the people gathered there again and jolted so hard she punched Nick with her elbow.\n\n\"What?\"\n\n\"I didn't see him slip in. That's Brad Vance sitting with the Kingsdales.\"\n\n\"You're right. He hasn't changed much since he left office. But why are you surprised? You said Harmon must have sold their house\u2014his company did, I mean.\"\n\n\"Or Brad was one of Marian's committee's silent partners, but I doubt it since money seems tight at the bog.\"\n\n\"Claire, Bradley Vance was a state senator for years. He probably knows most movers and shakers in the state, not just around here. Maybe Harmon Kingsdale was a big donor or the senator did some favor for him or for Marion too, if you must tie her in.\"\n\n\"Right. Right,\" she repeated.\n\nBut something just didn't feel right.\n\n* * *\n\nMonday morning as she drove toward Black Bog, Claire was still agonizing. In a reversal of her plans, she had given up involving Gina and Heck and had talked Nick out of their going to Harmon Kingsdale's office to see if they could get more information about Marian's silent partners and investors in the Endangered Properties Committee. If Kingsdale mentioned the Vances were on board with that, he might tell Brad or Andrea they had been asking and she didn't want to get any more friends in trouble. And so what if the Vances did contribute to Marian's work? Or perhaps they got some financial backing from her since they always seemed concerned for money. But how much had Marian known about the Vances' secret project, and had Cyndi just picked their house at random to lie about her real estate career on LinkedIn? Cyndi could have just stumbled on the house, walking the beach, as she and Nick had earlier today.\n\nClaire waved to the guard at the gate. Of course, he took a salary too, a lot of outlay to run this operation and to keep it quiet. Her own salary was very generous; then there was Kris, the dig team, the office and labs they'd built. If Brad was trying to make extra money for their project by selling knockoff artifacts or things inspired by ancient designs, so what, the Vances would probably argue.\n\nAs she parked, she studied the Vances' new home they had built here. So modest compared to their mansion on the beach. And, although a few hours of sun hit it during midday, didn't all these ficus trees hanging over the bog make the claustrophobic former senator feel oppressed even in this house? Or was it just the bog itself here that disturbed him?\n\nThough Brad's car was there, Andrea's was not. Nor was it in their short driveway or in their carport. Yes, even that was a real comedown from the three-car garage at their beachside home, but then, as she'd picked up more than once before, when the Vances were ready to go public and sell rights to things, all of that would change, because they'd be famous and rich.\n\nClaire used her security card to get in the building. She turned on the lights in her and Kris's office, then put her purse in her desk drawer and took her two-way radio out. Kris must be out at the bog, unless she was in Andrea's or Brad's lab areas. Maybe she had asked him to see the artifacts or more complete photos, but if not, Claire was going to.\n\nShe headed for his lab. Andrea's door was closed. Claire knocked on that door then tried the knob. Locked. As for Brad's lab, the door was open as usual, but not wide, only slightly ajar. And someone was crying inside.\n\n\"I don't want to go back home. It's not really home anymore. A whole year?\"\n\nYi Ling's distinctive voice.\n\n\"I just think it's best. You've been great, but we can't risk going on like this or someone might figure it out,\" Brad told her. \"I'll pay for the round-trip ticket, of course, and your salary while you're away. We cannot have people finding out, questioning you, blaming you. We'll tell everyone your family asked you to come back for a while since your father is ill.\"\n\n\"I did it for this amazing project but for you too. Andrea still trusts me, and I've kept things secret like you said, so I'd like to stay.\"\n\nClaire silently clapped her hand over her mouth. Brad and Yi Ling had been having an affair? Of course, the girl looked up to him, but he must have been the instigator, the seducer. Kris had said the young woman took his photographs for him, so this conversation couldn't be about that, could it? Maybe Yi Ling knew too much. After all, there were petroglyphs as Kris had called them on some artifacts which Yi Ling had perhaps photographed. Or maybe Yi Ling was in on the fake jewelry and ancient weapons being sold undercover, and he didn't want her questioned or caught.\n\nThough she wanted to stay to hear more, Claire quickly did a U-turn and hurried back to her office. She hadn't told anyone on staff except Kris that she was coming in this morning. With Andrea away, and Kris evidently out at the bog, perhaps with the dig team, Brad and Yi Ling must have believed they were alone.\n\nAnd, darn it, but she really felt for Yi Ling! A girl here on a green card who wanted to make good, but she had been brought here by the woman she was betraying\u2014if she and Brad were lovers. Yet Brad Vance was the power partner in the affair, and she blamed him for this.\n\nStill, she was going to calm down and put on a good front and face him about seeing those artifacts\u2014at least a complete set of photos\u2014today.\n\nAnd here was her chance to face him alone, she thought, as Yi Ling raced past her office door, ebony hair flying. The poor girl looked distraught. Had she even stopped eating over this betrayal\u2014this love affair? She looked almost gaunt.\n\nWhen Claire thought she'd given her enough time to get outside, she went out of the office. If Brad was in a bad or sad mood, so be it. This couldn't wait. Maybe it would help to hit him when his defenses were down.\n\nBut there was Yi Ling, pressed into a corner inside the exit door to the bog, turned away, shuddering with now silent sobs.\n\nClaire meant to leave her to grieve over the double hit of the end of a love affair and Brad's insistence she go back to China for a while. But the young woman must have heard her and turned her teary face away from the wall.\n\n\"Oh, sorry,\" Yi Ling said.\n\n\"Is there anything I can do to help? Are you ill?\" First thought\u2014 _don't let her know I overheard her and Brad._\n\n\"Just that I must go back to China for a bit,\" she said with single nod. She squared her thin shoulders. \"My father\u2014he is ill.\"\n\nClaire darted back into her office and snatched two tissues from Kris's box on her desk. She hurried back and held them out to Yi Ling. \"I'm sure with your experiences\u2014all you have done here\u2014you can work on a project there, or maybe go back to school for a year.\"\n\n\"If so, I must never tell\u2014tell about this dig\u2014until they are ready for the world to know.\"\n\n\"Secrecy is hard,\" Claire said. \"Necessary at times, but sometimes not good.\"\n\nThe girl wiped under her eyes. \"So very true. I will miss it much here,\" she whispered and blew her nose. \"I am sorry.\"\n\nIt almost seemed the young woman was apologizing to her, but for what? She meant she regretted leaving.\n\n\"I'm sure you'll be able to come back.\"\n\n\"Thank you for those words\u2014for your kindness,\" the slender young woman said as she opened the door to the bog. There was something familiar in the way she walked away before the door closed behind her. She seemed so defeated, but Claire was still angry at how Brad had treated the girl, used her, then planned to just send her away. But maybe that anger would help her to face him now.\n\nClaire strode down the hall and called in his door, \"Brad, excuse me.\"\n\n\"I didn't know you were here. No one except Kris and the dig team are around.\"\n\n\"I just got here. I may go out but wanted to ask you something first.\"\n\nHe sat on one of his stools and motioned her to step in. He looked totally calm, not like a man who had just broken someone's heart. And wouldn't Andrea be very upset when he or Yi Ling told her one of her dig team was leaving\u2014for a year?\n\n\"To ask me what?\" he said.\n\n\"It's come up that there was quite a unique petroglyph, perhaps on Hunter's dagger and perhaps on the staff or whatever that was that Leader had with him. One of a woman with her hands raised. Yet I did not see it on the photos you shared with me earlier and feel it could be key in my theory of their roles and relationships to each other.\"\n\n\"I can't imagine those weren't in the pictures you saw,\" he said, frowning and looking very concerned. \"Yes, there was such a petroglyph on the artifacts of both men.\"\n\n\"I see now why you've called the artifacts relics.\"\n\n\"Did I? I try not to, though, to me, each thing we get from the bog is a relic\u2014essential, almost sacred to understanding these ancient lives. Yi Ling took the photos, so perhaps she skipped or mislaid those of, not Reaching Woman per se, but Raised Arms Woman, almost as if she was under arrest, right?\"\n\n\"That's another way of describing her positioning, perhaps better than my saying it was like a football ref calling a touchdown.\"\n\n\"Saying to whom?\" he asked, frowning, though he didn't budge from his stool, nor she from just inside the doorway.\n\n\"Saying to myself as I try to figure out what that figure means.\"\n\n\"The woman with upraised arms may not even be Reaching Woman. But yes, let me go through the photos, because I'm sure I can find ones that showcase that figure.\"\n\nShe was surprised. So easily accomplished? \"Brad, why not just let me look at the figure on the items\u2014the relics\u2014themselves? It would save you from going through the pictures.\"\n\nShe was trying to study his body language, but he seemed frozen in place. Yet his eyes darted toward the door at the back of the lab, no doubt the one Kris had said was his office.\n\n\"My rule from the first\u2014Andrea's too, of course\u2014is that we protect the bog bodies and their grave goods,\" he went on, sounding now in lecture mode. \"The photographs will have to do. And, if the ones Yi Ling took do not have that raised hands figure, I must be certain she didn't pull them out for some reason.\"\n\n\"I'm sure she would have done exactly as you ordered.\"\n\nHe stared at her a bit too long, then said, \"And I'm sure you have things to do while I check on this.\" He lifted his left arm and stared, frowning, overlong at his watch.\n\nHe went on, \"I'll have them here for you to study to your heart's content in about an hour, because I have some other business. By the way, I've been glad to see you and your high-profile husband have managed to keep your names out of the news lately, because\u2014I repeat\u2014we like to keep a low profile here, to say the least.\"\n\n\"I understand,\" she said, deciding not to rush to Nick's defense but keep on track with the relics. \"I'll be back in an hour, Brad. I really think that figure is important.\"\n\nShe went toward Kris's office, relieved no one knew of their ties to the helicopter wreck, which had received media coverage. Is that what Brad had been referring to? If so, Kris had probably told him about it, in connection with Mitch. But once Claire saw those petroglyphs, she would be clearer on her theories about the status of Reaching Woman and the men in her life. It was obsessing her, and that wasn't good.\n\nAnd now, once again, Claire had seen that human nature had not changed over the centuries. A powerful man had seduced or coerced a younger woman, one in a weaker, dependent position, into a relationship with him. No doubt poor Yi Ling would have done anything to please him. If Andrea had known, would she have protected her and blamed Brad?\n\nClaire jumped as the door to the outside banged open and Yi Ling strode in from the bog. \"I don't want to leave,\" she told Claire, \"but I must accept. My dig team friends and Kris, they are sad. I'll be gone tomorrow. I will pack my important things now,\" she added and went down the hall to the small room the dig team shared.\n\nThe door to the bog opened again. Kris, Doug and Andy stood there.\n\n\"Have you heard Yi Ling's leaving?\" Kris asked, and Claire nodded. \"She's really torn up about it, but her father's very ill in Beijing. It's quite sudden, and she's flying out of Miami tomorrow.\"\n\n\"Wait until Andrea hears and has to replace her,\" Doug said, shaking his head.\n\n\"Anyhow,\" Kris said, \"we're going to take her out to dinner to say goodbye for now, an early dinner since she has to pack. If you'd like to join us, that's great, but I suppose you'll want to head home to your family.\"\n\n\"Yes, and I have a meeting at four,\" she said, deciding not to tell Kris in front of the others that she'd gotten as far as Brad promising photos, at least.\n\n\"Do you know if Andrea will be here this afternoon?\" Claire asked.\n\n\"Not sure,\" Kris said. \"Doctor's appointment, I think. But, listen, if you have a second, could you help us with repositioning the platform and planks out in the bog? It helps to have four people, and Yi Ling's pretty upset. We've decided to branch out just a bit from the area where we found the trio to see if there are any artifacts we've missed before we move on.\"\n\nClaire certainly didn't say so, but she felt there was something missing in understanding the living people here at Black Bog, not just the dead ones.\n\n# 34\n\nNick left work early, even though it was a Monday and he had a lot to do. Bronco had said he and Nita were doing okay in the house this weekend\u2014which meant Nita had probably stopped crying\u2014so he'd given Bronco today off.\n\nThe guy had stuck with him from the first case Nick had worked with Claire, when he'd almost lost her. He hated to be so possessive of her now, but he couldn't bear to ever be in a position like that again. She had always said if anything happened to her that her sister would raise Lexi, but now there was Trey too. And he wasn't sure he could go on if she was hurt or lost.\n\nHe'd like to check with her right now, he thought, as he turned into the housing division which had once all been Twisted Trees mansion grounds. But until she drove partway back toward town on the Trail, there were no cell towers to pick up a call, and the bog staff used walkie-talkies, no less. Claire had said the place was state-of-the-art, but that sort of communication\u2014or lack thereof\u2014to the outside world seemed as primitive as the people they were digging up. He wished too that he knew more about the culture Claire was studying, because \"Stone Age\" didn't really say much to him, but he honored her contract with its secrecy clauses and just accepted whatever she chose to tell him. After all, legal contracts helped to make his world go round.\n\nHe pulled up in front of Dale's mother's old house, next to Dale's own vacated one. He had to admit Bronco and Nita's place looked better than when he'd last seen it. Bronco had not only cut the grass but had trimmed the overgrown bright pink bougainvillea bushes. The windows sparkled in the late afternoon sun, and it looked like Nita had hung fresh curtains. Claire would be thrilled.\n\nHe got out and went up to the front porch, glad he didn't see Betty and her little dog spying on him. He heard the purr of the air-conditioning. Bronco must have gotten that repaired, because Dale's mother supposedly hadn't used it. But had she used the freezer for more than hoarding frozen food? Nick used to think perhaps the old lady had offed Cyndi Lindley, but evidently not, since Lucy Braun was dead by the time Marian was strangled\u2014unless Mrs. Braun had come back as a ghost.\n\nAs he raised his hand to knock, Bronco opened the door.\n\n\"Hey, things are looking up!\" Nick greeted him as Bronco unlocked the screen door and held it open.\n\n\"They were,\" he said, his voice quiet, which was unusual for the big guy. \"Come on in. Finally got Nita to take a nap. She's been working hard.\"\n\n\"I'd say you both have. The front yard looks one hundred percent better. But what do you mean things _were_ looking up?\"\n\nThey sat in the front room, which also looked clean and rearranged from when Nick had seen it last. Bronco hunched forward, hands on his knees.\n\n\"That old busybody next door stopped by with a welcome gift\u2014pecan rolls.\"\n\n\"Sounds nice. And?\"\n\n\"And in chatting with Nita when I was out in back finally getting that shed cleaned out, the old lady evidently told her she always pictured her friend Lucy here\u2014Dale's mother.\"\n\n\"They were friends for years. But that upset Nita?\"\n\n\"When Nita asked her more about it, she said she'd seen her ghost here recently. For sure it was a woman, she said, maybe even Cyndi, all white, like covered with frost.\"\n\n\"What?\"\n\n\"Lights at night. Shadows passing the windows in the daytime.\"\n\n\"Oh, hell. That's ridiculous and not what Nita needed to hear.\"\n\n\"No kidding, boss. I'm scared it will set her back about this place. I don't want to get off to a bad start in this neighborhood by telling the old lady to stay away\u2014man, she walks by all the time\u2014but she's really screwing things up.\"\n\n\"Crazy Betty has seemed to like me from the first, lucky me. I'll just stop next door on my way home and tell her that she needs to stop that kind of stuff. If she can't support her new neighbor, be upbeat, all that, she needs to stay away.\"\n\n\"She's creepy to say the least. Anything you can do\u2014'preciate it, 'cause now I've got to settle Nita down again.\"\n\n* * *\n\nClaire knew she had to settle down. Finally, she was going to see missing photos of the petroglyph that must be key to her work here. Finally, Brad was going to trust her enough to share those with her, so she'd be fully informed in theorizing about the bog trio\u2014and who knew what bodies they would dig up here after that? She hoped, as wary and difficult as he sometimes seemed, that she could establish a good working relationship with him the way she had with Andrea.\n\nShe didn't see him in his lab. She stood in the doorway, uneasy about going in if he'd stepped out for a moment. Or maybe he was in his office at the back. She glanced at her wristwatch. Exactly four o'clock.\n\n\"Brad? Senator Vance!\"\n\nPerhaps he'd just stepped out to the bathroom or to get some of that perpetual coffee and a BB frosted doughnut.\n\nShe saw a note on the stool where he'd sat last time she was here, and she stepped in to glance down at it. Yes. To her.\n\nHer heart fell. What if he had left, made some excuse to put her off? But no, she thought, as she skimmed his angular handwriting. _Be back soon. Pics in desk drawer._\n\nWhat a relief! She moved to the desk in the corner of the lab and tried the open the center drawer. It didn't budge. Locked?\n\nShe tried to slide out each drawer, but all held fast. Maybe he didn't mean this desk, but she could hardly just start opening drawers in here. She looked around for anything else that resembled a desk. His light table for viewing negatives. Shelves. Surely not that filing cabinet.\n\nCould he have meant his desk in his office? Kris had seen it, described it. Well, if the door to that room was locked, she'd know that wasn't what he'd meant.\n\nJust to be sure he wasn't coming down the hall, she stepped out and looked up and down the corridor. Then she punched in #1 on her handheld radio, his number. Andrea was #2, and Claire had recently peeked outside again to see her car wasn't back yet. Only Brad's and Claire's were there because Kris and the dig trio had gone out to a restaurant to say goodbye to Yi Ling. She could see that the gate guard's car was gone, so perhaps he'd been invited to Yi Ling's impromptu farewell dinner.\n\nBrad didn't pick up on his radio. What if he'd fallen asleep or somehow hurt himself? Surely, he hadn't gone out to the bog, especially on his own. That was against Andrea's rules and anyway he feared the place that was the fierce focus of his attentions.\n\nShe went back into his lab, then to his office door. \"Brad!\" she shouted. \"Are you in there? Are you okay?\"\n\nFeeling it would be locked, like so much around here, Claire knocked, then reached for the knob. It turned easily in her hand, and the door opened.\n\nThe light was on. Oh, whew, there was a note on the desk with an arrow and the word _pictures_.\n\nLeaving the door ajar, she stepped into the room.\n\n* * *\n\nNick knocked on Betty's door and heard her little dog start barking. At least she was home. He needed to get her to back off. Her snooping had gotten way out of line.\n\nWearing a pink housecoat, she opened the door, holding the dog.\n\n\"Oh, my favorite criminal lawyer has come calling,\" she said with an almost coquettish smile.\n\n\"Hi, Betty. Can you step out so we can talk for a second?\"\n\n\"Any more trouble in this lunatic neighborhood?\" she asked. He thought she might balk at coming out, but she didn't, and they sat in matching pink plastic molded chairs.\n\n\"I sure hope no more trouble,\" he told her. \"I think you know that your new neighbors, the Gateses, are friends of mine.\"\n\n\"He works for you and you gave him the day off so he and Nita can finish tidying things up.\"\n\n\"Right. They will be good neighbors, but Nita's kind of shaky with all that's gone on here, so\u2014\"\n\n\"Aren't we all?\"\n\n\"Yes, well, I'm hoping you won't scare her with ghost stories about her new house. It's hard enough that Cyndi's body was found there.\"\n\n\"Not to mention that not far away from here, another woman died\u2014before a fire finished off the rest of that mansion. Talk about haunted. Besides, I think my dear friend Lucy\u2014Dale's mother\u2014probably does haunt her house. She loved living there all these years.\"\n\n\"I'd like to have your promise to be a good neighbor to\u2014\"\n\n\"I took them baked goods, pecan rolls, a touch of Southern hospitality. I have some left, if you'd like one.\"\n\nNick was starting to sweat. Not from the afternoon heat and humidity, but from having to deal with this old woman, who seemed as sticky and nutty as pecan rolls. Besides, he kept recalling the story Claire had related to him in too damn much detail about how the old ladies in _Arsenic and Old Lace_ had murdered people and hidden them around their house. Ridiculous, Nick scolded himself.\n\n\"Oh, by the way,\" she told him, popping up and holding the dog close, \"I forgot to tell you I copied down the license plate numbers of everyone who doesn't live around here who parked on this street the evening of the fire. I could work for the police or a criminal lawyer like you, really I could. I'll get it for you\u2014a pecan roll too.\"\n\nNick sighed as she disappeared into the house. That's all he needed, though maybe a list of cars could show others were around then, not just Dale. He needed everything he could get his hands on to get Dale's defense together now that he'd been indicted.\n\nShe came back out and handed him a paper plate with a roll and an index card\u2014no, a recipe card. In large handwriting, she had indeed recorded a list of license plate numbers, nine of them. No doubt some would turn out to be curious onlookers who smelled the smoke, or TV or newspaper reporters, but he would give Jensen a call to run these numbers.\n\n\"I appreciate this, Betty. And I'd sure appreciate your building up Nita and not scaring her, okay? They're still kind of newlyweds and can use the time alone.\"\n\n\"All right, my favorite real-life Perry Mason. By the way, those reruns are on TV all afternoon\u2014and, you know, that famous lawyer had a female helper named Della, his secretary, someone who knew how to dress, not like those girls in pants sitting at office desks today. Why, I told Cyndi she dressed like a tramp with those little shorts and bra-type things!\"\n\n\"Did that start an argument?\" Nick asked, still thinking that maybe there was a glimmer of a chance that this eccentric little sneak had gotten into a fight with the conniving Cyndi. He was grabbing at straws, but that's where he was at in preparing to defend Dale. And darned if he was going to eat this pecan roll since the title of that old movie had the word _arsenic_ in it.\n\n\"Cyndi called me an old bat and used that other b-word right to my face!\"\n\n\"She shouldn't have done that,\" he said as she kept nodding.\n\nBut, Nick thought, if Betty had somehow strangled and frozen Cyndi, no way she'd tell him all this\u2014although he had seen some criminals who like to boast about what they'd done, loved telling it all to a cop or psychiatrist or lawyer. Yeah, it was kind of like those old Perry Mason reruns where the guilty party no one suspected stood up in court and shouted, \"Yes, I did it! I killed so-and-so and here's how and why!\"\n\n* * *\n\nWith her heart beating hard, Claire stood still in Brad's office, holding her breath. She sucked in a huge gasp. A drone, so much like the piece of one Jace had pulled from the wreckage of the chopper, hung from the ceiling in the corner. Was it also like the drone that had exploded the glass window and ruined their reception? Of course, so what, lots of people had drones today, but\u2014\n\n\"Find the photos?\"\n\nShe spun to see Brad standing in the doorway. He seemed to fill it, block it.\n\n\"I\u2014No. I looked out there. I thought you might mean this desk. I saw the sign,\" she said pointing.\n\n\"And, of course, saw my drone. Just a hobby. Or did Kris tell you I used to use a drone to view the excavations here since I couldn't stand the closed-in feeling the bog always gave me? But with the low-hanging trees, it got snagged and sank\u2014down in the peaty depths forever, so I got a new one and don't use it for that anymore. Yi Ling borrows it sometimes.\"\n\n\"Sad she's leaving so suddenly.\"\n\n\"Some things, as sad as they are, are necessary. Well, confession time all around. Yi Ling has done several odd jobs for me.\"\n\nFor some reason, it was then that Claire's brain made the connection. That androgynous person who had followed her and Kris that time\u2014so thin, face mostly covered, not a limp, but a hesitation with one leg. And the way Yi Ling had walked away out into the bog today... Could it have been?\n\n\"Confession time all around, for what?\" she asked, edging toward the door. He did not budge.\n\n\"I hope the third time's a charm, but you do understand that this precious work for all mankind we are doing here must be protected at all costs.\"\n\n\"I've kept the secrecy agreement.\"\n\n\"It isn't enough. I was uneasy about having Andrea and Kris put you on the team, but I couldn't protest too much about your husband and your high media profile or they would have been upset. So I took a gamble, but you've rocked the boat, been in the wrong place too much, hanging out with the wrong people, lawyers, law officers. Getting in the newspapers. Damn, but people should have listened to me. Sad, but you'll be in the media this time too. Well, go ahead and open that drawer with the arrow pointing to it,\" he said.\n\n\"I'm leaving, Brad. Perhaps leaving for good.\"\n\n\"I'll second that.\"\n\nHe pulled a gun and pointed it toward her. Startled, she stepped back. The black end of the barrel loomed large.\n\n\"Open the drawer, then we'll discuss your leaving.\"\n\nStarting to shake, she stepped back and pulled the drawer open, expecting to finally see a petroglyph of an ancient woman with her arms raised. But within it lay a folded faded flag, smudged with dirt or ashes, but she could see the swastika on it. And next to that was a photograph of Hitler with Eva Braun.\n\nShe gasped and looked at Brad. He was smiling.\n\n\"No, I'm not a neo-Nazi. But I am looking for things to sell on the international black market to help keep this place afloat, and I regret that you seem to be onto me for that, with your own visit to our antique shop in town and then sending someone across the state. I got a phone call from there while they were visiting, and it didn't take much to put two and two together. I regret the drone Yi Ling used only brought them down but didn't get rid of them. Now, don't you think I know you broke your contract to send them on that little dead end hunt?\"\n\n\"There was nothing in the contract about that. Just let me out of here,\" she demanded. \"My husband knows I'm here and so do your staff.\" She suddenly felt so nauseated she could have thrown up.\n\n\"Let you out of here?\" he said. \"Oh, I plan to. As much as I hate that bog, let's just go out there to see what we can find. This won't take long and then you\u2014like two other smart-mouthed women who tried to pressure me and ruin this momentous work\u2014will be on your way. This time, no strangulation needed at all. Now let's move.\"\n\n# 35\n\n\"Hey, Detective Jensen, Nick Markwood here,\" Nick said, glad he'd picked up on the call.\n\nNick was sitting in his car in the driveway of Claire's sister Darcy's house. Claire had said she'd meet him here around four. He glanced at his watch as he killed the engine: four fifteen. Darcy had picked up both Jilly and Lexi at their schools and had brought them here, so it was probably a noisy place inside.\n\n\"Counselor,\" Jensen said, \"how goes your Monday? Got a dynamite case together to defend Mr. Braun?\"\n\n\"Speaking of that, remember the old lady who lived next door to the house where we found the woman in the freezer, the one who ID'd the deceased from your photograph? The night of the fire and second murder, Betty Richards evidently went up and down their street with a flashlight, writing down license plate numbers of cars that didn't belong. I could probably find a way to have the owners identified, but I'd like to hand that over to you. We defense attorneys like to do everything on the up-and-up.\"\n\n\"Yeah, I hear you. And we detectives who work for prosecuting attorneys aren't supposed to make friends of defense attorneys and their wives. Okay, read me the list, and I'll check those out. Too many dead ends in this case, pardon the pun.\"\n\n\"And you'll get back to me with the names, in case anything clicks? I may have to share them with Dale to see if he can place any of them.\"\n\n\"Okay. Just don't rat me out so someone thinks I'm helping the opposition when we go to trial. Give Claire my best\u2014and when she gets done with whatever private assignment she's on, I'm telling you, the department could use her for a forensic tech, part-time if she wants.\"\n\n\"You know, if it was just an office job, I'd say that sounds great\u2014if she could work from home and not go out to interview possible would-be criminals.\"\n\n\"Dream on.\"\n\n\"Yeah. At least she has a temporary gig where she's only consulting. Thanks, Ken. I'll wait for your call back.\"\n\nTaking his cell with him, Nick got out and walked toward the house. He could hear Jilly's and Lexi's voices from here. It was great being a dad as well as a husband of the woman he loved.\n\n* * *\n\n\"I'm not going out to the bog,\" Claire insisted as Brad gestured with the gun that she should precede him out his office door. \"I'm going to get my purse and leave. I'll keep this quiet, just as I have what's really going on here.\"\n\n\"Meaning what?\" He moved aside for her to pass, but when she tried to squeeze through the doorway, he yanked her back against the open door. \"I said there's something I want to show you outside. Move. Right ahead of me.\"\n\n\"No,\" she insisted, desperately trying to sound defiant and unafraid when she wanted to scream and explode in tears. This was so perfectly set up. No one was here. When would Andrea be back? But was she in on this too?\n\n\"Look, former state Senator Bradley Vance,\" she said, fighting to keep her voice from quavering. She had decided to try a different tack. \"You have evidently murdered two women so you'd better take off before the police find you. My husband and his contacts will\u2014\"\n\n\"No way they can pin this on me. I have no obvious connection to Cyndi what's-her-name. Would you believe the little slut hung around our beach house, wanted to sell the place for us instead of our real estate rep, then she actually followed me here and threatened to blow the lid off this project? She went on and on about how she could get funding for the bog work by selling passes to a mansion with a Nazi connection, so I drove her to her friend's house and had to shut her up\u2014stop her there before she went public with too damn much.\"\n\n\"I'd say strangling and freezing is one way to keep someone quiet.\"\n\nAs if he hadn't heard her, he went on, \"I did have a business relationship with Marian James, though unlike Cindi, she had the moxie and power to expose everything here\u2014take it from my control. Those two women could have blown everything, but I've covered my tracks. Haven't even tried to sell that Nazi stuff I took yet\u2014and could have paid dearly for, since that idiot Braun started a fire when I was there meeting with Marian James.\"\n\nShe almost bluffed that her husband knew of the Marian\u2013Nazi connection and would pursue him. But then\u2014whether he killed her or not\u2014would he go after Nick? And the children could be endangered. She had to stay calm. She had to find a way out of this. She feared he was going to try to stage a fatal accident.\n\n\"You know,\" he said, sounding almost wistful, \"what really matters above all is eventually sharing my American Adam and Eve burial spot with all mankind. I'll get the rights to publicly and legally sell copies of the artifacts. As for you\u2014I knew you were far too clever from the first. And now,\" he repeated as he shoved her out of the lab and down the hall with the gun barrel pressed between her shoulder blades, \"our very own Over-Reaching Woman Claire Markwood will get a close-up and personal experience of what our Black Bog people went through.\"\n\n* * *\n\n\"Claire's late,\" Darcy told him, wiping her wet hands on a dish towel. \"Probably got involved in something. I can't believe wherever she's working they don't have a phone.\"\n\n\"Too far from a cell tower,\" Nick told her, pushing Trey gently in the infant swing. \"Traffic's probably bad. She'll be here, and if she's late, she'll call when she gets in tower range.\"\n\n\"I don't see why her work has to be such a big secret,\" Darcy groused. \"You okay with Trey? I'm going to check on our angels\u2014far too quiet in Jilly's bedroom.\"\n\n\"Sure,\" Nick said with a glance at his watch. Twenty-five after four. She'd be here soon.\n\nTrey gurgled and cooed something when the music for Nick's phone went off. Hoping it was Claire, he checked the caller ID. Ken Jensen.\n\n\"Nick here.\"\n\n\"Fast work, huh? Listen, one of the plate numbers was of interest. You sitting down? Former longtime state senator Bradley J. Vance.\"\n\nNick jerked erect on the ottoman where he sat. His brain raced. His gut wrenched. He stopped swinging Trey, steadied him with one hand.\n\n\"Nick, I gotta tell you, he does have a connection to Marian James, but I can't say more.\"\n\n\"Was he one of the silent partners underwriting her Endangered Properties Committee? We thought maybe so but couldn't turn it up.\"\n\n\"All right. Yes. And he was at her funeral with another of her cronies\u2014\"\n\n\"Harmon Kingsdale. We saw them together.\"\n\n\"Damn, you and Claire have been overstepping again!\"\n\n\"I hope not. Got to go.\"\n\n\"Nick, wait! Is there some connection now? I mean, Claire's been really secretive about who she's working for. If she was advising that committee in any way, she may be subpoenaed at the trial, and that won't look good with you as defense attorney. I need to talk to both of you\u2014\"\n\n\"I'll call you back soon.\"\n\nNick punched off his phone. Brad was near or at Twisted Trees the night of Marian's murder. So he knew the area, maybe had been there before. Dale may have set the place on fire, but someone had killed Marian first, just the way someone had killed Cyndi. Could the former senator have had any connection to Cyndi? No, too far out\u2014unless, as Claire surmised, she had visited them at their beachside home to have that picture taken. No, he was thinking in circles.\n\nDamn, he had to get to Claire, even if he got in trouble driving into Black Bog, if the guard she'd mentioned would let him in.\n\n\"Darcy!\" he shouted, and Trey startled and fussed. Nick bent to hug and kiss him. \"You need your mom and I do too,\" he told the baby.\n\n\"Nick, what?\" she said, running in. \"Was that Claire who called? Is she all right?\"\n\n\"Of course she is, but I'm going to go check on why she's late. If she shows up here after I'm gone, call my cell.\"\n\nHe didn't say or do more, didn't even kiss Lexi. Something was so, so wrong.\n\n* * *\n\n\"Are you really claustrophobic?\" Claire was desperate for calm talk\u2014for distractions. Maybe she could play on his phobia. Brad shoved her toward the door to the bog. Her pulse pounded, and her mind raced. She had to find a way to knock that gun out of his hand before they walked the planks over the bog. She could not go out there with him. Should she risk taking a bullet instead, so she could maybe fight back on dry land?\n\n\"Damn, I am paranoid about being shut in,\" he said, giving her a little shove again when she hesitated. \"My older sister used to lock me in a big cedar chest when she had to stay home with me. Dark, hard to breathe. Everyone in the family thought she was Miss Perfect, pretty and blonde. But she was selfish and pushy.\"\n\nClaire almost asked him if Cyndi had reminded him of his sister.\n\n\"But what must be done must be done,\" he added with an even more trembling voice.\n\n\"No one will believe I came out here alone, against the rules, when I should be heading home.\"\n\n\"Yeah, and no one will believe I finally came out here again.\"\n\n\"Andrea called and will be back soon. Is she in on all this too?\"\n\n\"You've had too much contact with a lawyer. That's a lie about Andrea calling, because I talked to her on our house phone\u2014the only outside line that works around here\u2014just about when you went in my office. She's been diagnosed with breast cancer and is in the hospital for treatment. I'm going there when I close up\u2014end things\u2014here.\"\n\n\"I'm sorry for her. She didn't have any part in the murders, did she?\"\n\n\"Just shut up. I said you're sounding like a lawyer.\"\n\n\"And they as well as the police will be all over this when I go missing,\" she insisted, then regretted bringing Nick into this. He would be insane with grief and rage, endangering himself to learn what happened to her. Then this warped bastard would get rid of him too. Somehow, she had to save herself, turn the tables here. She'd walked these planks recently and he had not. There must be a way to knock that gun into the bog, maybe to knock him in before he did her in.\n\nLexi's sweet face floated through her panicked brain. Trey's. Nick's. Barely minutes now until this monster made his move. She knew his plan now: he was going to bury her in the bog with the ancients, and maybe someday someone would find her deep in the muddy peat, reaching out for help.\n\n* * *\n\nEarly rush hour traffic in Naples slowed Nick down. He'd taken back streets out of Darcy's neighborhood. At least she lived in the southeast part of town. Damn, but he wasn't even certain where to turn off the Trail to the bog. Claire had said it wasn't marked, but he knew about where it was and would look for an unpaved road. Hopefully Kris was with her, other members of the staff too. If Brad was a suspect in some way, was Andrea too?\n\nHe tried to settle his panic. Keep calm. So what if Senator Vance and his wife had hired Claire to keep their secret? They were public figures, dedicated, trustworthy, but who trusted politicians these days?\n\nStill, Nick berated himself, he should have demanded to know more about Claire's work. But Kris was there, others too, he told himself again. So why did he feel she was in danger just became she was late? It seemed she attracted trouble. Hell, he had too over the years. At least the kids were safe, but what would he ever do without Claire?\n\n\"Steady,\" he told himself as he had to stop at another red light. \"Steady.\" He hoped\u2014he prayed\u2014that it would turn out everything was fine, that he could give Claire the news that the senator knew Marian, had proximity, opportunity to harm her\u2014but would a probably faked photo on LinkedIn taken in front of the Vances' former home be a possible connection to Cyndi Lindley?\n\n* * *\n\nClaire knew she was doomed and there was only one way to save herself\u2014well, two. Talk Brad out of this or fight him somehow, and he outweighed her by far, not to mention he had the gun. But he was shaking, literally terrified.\n\n\"You do know that your little squeeze Yi Ling planned to tell the dig team and Kris about her affair with you\u2014about the drones you've tried to kill people with.\"\n\n\"Nice try. Been eavesdropping?\"\n\n\"No, Yi Ling was in tears and blurted it out, so I told the team to be prepared for her confession. I think they'll take her to the police to tell her story so she doesn't have to return to China. At least, you didn't seem to be planning to plant her in the bog\u2014or were you hoping to bring down her international flight with a drone?\"\n\n\"Very funny. Just shut up! You know too damn much!\"\n\n\"The police believe\u2014and will prove,\" she went on, though her voice was trembling, \"that the person who killed Marian also murdered Cyndi Lindley. They'll figure out it was you.\" He shoved her with the gun again and she had no choice but to start out on the plank walk over the bog,\n\n\"No, that would be Dale Braun,\" he insisted. His voice had gone up to almost high-pitched. Somehow she had to use his fear of this place with its hulking trees and growing darkness to feed his fears, because hers were out of control.\n\n\"Were the two women working together to interest you in buying or helping them get access to Twisted Trees with its Nazi mini-museum? Not only to preserve historic property, but as a source of money for the Black Bog project? Great name for this place with its grasping, clinging peat and twisted trees. Everything seems to just close in here.\"\n\n\"I said shut up! Maybe Andrea and Kris were right to hire you to get answers, so here's a couple. Like I said, Cyndi made the big mistake of trying to push me around for support to set up a real estate firm. She had followed Andrea here one day and figured out we were digging for something, said she was taking it to the papers. She told me she had the keys to not only two Braun houses, but the mansion. She had Marian on her side by then. I met Marian at Twisted Trees, just like I'd met Cyndi\u2014at her request\u2014at the house next to Dale Braun's. The little blackmailer threatened to go public about our project here\u2014which Marian had surmised and blabbed, though she had the details wrong\u2014and I had to stop Cyndi. Eventually, had to stop both of them.\"\n\n\"Why the freezer?\" she asked, trying hard to keep ahead of him and steady her feet on the slightly bouncing boards. \"Why didn't you just get rid of Cyndi's body or put it in Dale's house?\"\n\nShe had to keep talking and walking. Soon they'd be to the platform where the dig team worked, and there was no safety beyond that in the very center of the bog. If he pushed her in\u2014he surely didn't want his bullet in her\u2014could she grab a board or the platform to keep from sinking? But no one would be out here until maybe tomorrow, and she'd seen from buried bodies how the peat pulled a person down. And down.\n\n\"Ah, the freezer,\" he said finally. She could tell it was actually calming him down now to talk, maybe to keep his mind off the clasping, suffocating feeling here. She sensed it too.\n\nHe went on, \"I needed to make it look like Dale Braun had stashed the body _not_ in his house, but to keep it for now, move it later. The keys to both the house and the mansion's garage were in Cyndi's purse so I took them, being careful with fingerprints. I kind of liked the idea of preserving that poor, pushy little redneck in the freezer, though not the way my bog people were preserved. And Cyndi had told Marian about our bog work. Marian insisted that she and her committee come in on the action\u2014use the profits and the precious information that should have been all mine\u2014and Andrea's, of course.\"\n\nShe turned to face him at the end of the platform she had just helped to move earlier today. At least it was more stable than the pathway of planks. She had tried to keep control, but terror was taking over. She saw Nick's face and those of her beloved children, then the forms and faces of the dead\u2014of Reaching Woman caught between the two men she loved.\n\nBut Brad looked terrified too. Ashen and trembling. He lifted his hand with the gun but did not shoot. \"I'm sorry,\" he said, then lunged at her, shoved her.\n\nShe grabbed for his arm, seized it, trying to right herself, save herself. She went down, rear end first, into the muck.\n\nShe spread-eagled her legs and thrust out her arms, rather than letting herself be pulled down more easily by going feet first, but already she felt the pull of it. She took a huge breath, tried to blink mud from her eyes.\n\nAnd then she saw he had toppled in too, panicked, thrashing, going down.\n\n# 36\n\nClaire realized her attempt to spread herself out horizontally wasn't working. Neither would trying to swim to stay afloat because that's what Brad was doing, and he was going down fast, maybe because he weighed more. She let her lower body sink so she was more upright to hold her head up longer.\n\nIf she wasn't so terrified, she would have laughed because the murderer thrashing next to her, trying to reach the platform, was screaming for help. His claustrophobia would make this the ultimate just death for him. One victim strangled, enclosed in a freezer, the other strangled, consumed by flames\u2014and he would suffocate in the heavy grasp of the bog.\n\nDear God, what if they never found either of them\u2014or when they did, far in the future, would they be just more bog bodies to study? Claire stopped trying to move, held her legs and torso steady. Her arms and shoulders were still above the dark surface. Her descent to death slowed a bit.\n\nHer thoughts came scattered in her shock and panic. Surely, Reaching Woman, Hunter and Leader had been dead when they were interred. If only one of the dig team would come back. Kris could save her. Andrea had cancer when her own husband had really been the cancer in her life.\n\nShe tried to stay very still. She panted, filling her lungs with air. Lexi's smiling face floated before her. How desperately she wanted to hold Trey again. What would their lives be like without her?\n\nAnd her beloved Nick. Not enough time together. He was right to be possessive, want to keep her home... Darcy, their mother... And really, finally, no thought of Jace but that she hoped he would be happy with Brit.\n\nBrad's cries had finally stopped, but he was sucking in huge sobs, nearly up to his chin in black muck.\n\nShe looked toward the heavens, praying wordlessly. The sky was blue but clouds were frowning. The trees leaned in, but much too far away to grasp.\n\nHer chin hit the surface of the bog. She tipped her head up to keep her face free.\n\nBrad screamed once, sobbed, then gurgled. She could not look at him, only at the sky as she tried to concentrate on those she loved.\n\n* * *\n\nNick finally thought he was in the right place. An unpaved lane heading west\u2014second one he'd tried just now\u2014had a barrier and a little guard house, other buildings behind it in the growing shadows.\n\nYes! Claire's car was there. But why hadn't she headed home by now?\n\nHe jumped out, vaulted the barrier and ran up to the closest door of the sprawling building. Or should he try the house? One car was there. Kris's Jeep nowhere.\n\nThe door was locked. He saw an entry card slot.\n\nHe ran around the building toward a short boardwalk. So that was the bog beyond. It looked dark but placid and benign. Could she be out there?\n\n\"Claire!\" he shouted, cupping his hands around his mouth. \"Claire!\"\n\n* * *\n\nStrange how she thought she heard Nick's voice. It echoed, echoed. It must be just that she was thinking of him, loving him at the last. Brad had disappeared, and she would too. She was dreaming she was on dry land, in their backyard, holding Trey with one arm around Lexi and Nick touching her... She gave a silent scream and then one that had her very soul in it.\n\nHer feet hit something. Her eyes, nose and mouth were still above the surface. But surely the bottom of the bog was farther down. Brad was taller and he was gone.\n\nShe jerked alert. Had she hit bottom? She was standing on something. A bog body? It seemed to be something sturdy, something that did not suck her down, though she could feel the pull of the bog everywhere.\n\n\"Claire!\"\n\nNick? Was she dreaming or was she dead? She felt then that Reaching Woman had once belonged to Hunter but loved Leader. In the end, had she chosen wrong and killed herself? Claire knew that ancient woman did not want to die. She did not want to die either.\n\n\"Claire! Claire, are you out here?\"\n\nShe finally dared to turn her head, nearly slipped off whatever she was standing on.\n\nIt was Nick! Nick, really here, running on planks to the platform. Didn't he know to be careful? She tipped her head back farther to shout to him.\n\n\"Brad threw me in and he fell in! He's gone!\"\n\n\"Never mind him. Can you raise an arm? I'm going to swing a board out to you. If you can't, I'll sit on planks and come in.\"\n\n\"No! It sucks you down! I'm standing on something solid! Brad killed them, both women!\"\n\n\"Claire, shut up and reach for the board!\"\n\nShe feared the effort would make her slip and pull her under, her last bit of air gone. But she couldn't have Nick in this deadly bog too. She sucked in a huge breath and tried to lift her arm closest to the heavy plank he extended. He was on his stomach on the platform, grunting, straining.\n\n\"Come on. Hold it,\" he shouted. \"Now lift the other arm and grip it with that hand too!\"\n\nShe tried hard. She had to or he'd come in. She knew Nick. He'd die for her, and she had to tell him not to, to live for Trey and Lexi. She gasped for air, trying so hard to get free of this drowning destroyer and preserver of people.\n\nHer hands slipped off. Splinters in her palm. She grabbed at it again and tried to hold tight. Slippery grips. He pulled her closer, closer. Her feet left her platform. Slowly, heavily, she moved toward Nick. He was reaching out toward her because he loved her, because he wanted to save her, the way Reaching Woman had done for the ones she loved.\n\nNick had a fistful of her slick, slimy hair. Her scalp hurt, but she was so hysterically grateful for his touch she didn't care. He grabbed for her under one armpit. She slipped away, but he seized her again.\n\nHe grasped her muddy wrist so hard her hand went numb but he pulled her closer to him, closer. Finally, he had her by her T-shirt and a belt loop on her jeans. He half dragged, half lifted her onto the platform to her waist while her legs still dangled. Her chin scraped the wood. He reached down and yanked her up by her jeans, then laid her on her back like some slimy mermaid pulled from a prehistoric sea.\n\nHe laid beside her to embrace her as if they were in their very own bed, safe and so very much in love.\n\n# 37\n\n## One Month Later\n\nKris hugged Claire, then sat beside her on the couch in the Markwoods' crowded Florida room. They were hosting an open house for their friends with no worries that a drone would come smashing through the plate glass window.\n\nBrad was dead, and Yi Ling had been charged with multiple counts of attempted murder for helping him to try to bring down the helicopter. A phone call from the manager of the Fort Lauderdale antique store had tipped Brad off that someone who flew in via helicopter from Marco Island knew too much about what to ask for. The man had followed Jace and Brit back to the executive airport. According to Yi Ling, who was even more terrified of being deported back to China than of being sent to prison here, Brad had thought the couple must be Claire and Nick\u2014and he'd tried to stage an \"accident\" with a drone to eliminate them.\n\nIt turned out that Brad and Yi Ling had not been having an affair, though he'd convinced the girl that if she didn't go along with his plans, she could be deported. Andrea was insisting on testifying on Yi Ling's behalf, however weak she was from her radiation treatments for cancer. She had not been charged, as she could not be linked to Brad's actions or plans and insisted she knew nothing about anything beyond his having copies made of artifacts for sales to support their huge, important project.\n\n\"Claire found a unique way to discover more bog bodies,\" Kris announced to everyone. \"Get thrown in by a murderer, then stand on one of their most precious relics, a 'Rosetta stone' to their ancient culture. She left not only no stone unturned but no bones unturned. We have finally recovered the stone and will begin to study it soon\u2014with Claire's help. And former Senator Bradley Vance got a chance to rest in peat before his body was recovered\u2014with difficulty\u2014by the dig team while the police looked on. Sorry for the sick jokes, but Brad Vance was a very sick man.\"\n\nClaire just shook her head at the gallows\u2014that is, bog\u2014humor. But Kris wasn't done yet.\n\n\"And here\u2014Andrea said it's okay to show everyone\u2014is the large artifact-relic we finally dug out that saved you from going all the way under, Claire. She wanted to show you she will totally share all artifacts with you to study.\" Kris motioned to Mitch and Jace, hovering in the doorway, to bring something in.\n\nThe two men, who had just finished early training in DC for flying into hurricanes, came in lugging what looked like a\u2014Well, what was it?\n\nJace and Mitch put the stone carefully on the tile floor in front of the couch. It was about the size of a small bed pillow and covered with carvings. Claire leaned down and patted it. \"Well, it did save my life\u2014and of course Nick had a little something to do with that,\" she added, smiling up at him. \"And I'm so glad to see we can now study these relics in person, so to speak.\"\n\nClaire had been so busy working with the police and testifying that she hadn't heard what the new dig team had resurrected from the spot where she had almost drowned. She gasped when she saw what was carved on top.\n\nEveryone came over, huddled, looking down at the stone, now cleaned of its mud and peat. It was covered with deep primitive carvings, including one which took Claire's breath away as she reached down to touch it. For there, in stark carved relief was a woman with her arms raised. It was as if Reaching Woman had reached out to save her, held her up when she could have been sucked down by the bog. And next to her were carved two men kneeling before her in obvious deference or even worship.\n\nClaire held her breath for a moment. Her beloved Reaching Woman had lived a triumphant but tragic life. She could tell everyone was waiting for her to say something. She wanted to say _That stone was her gift to me._ But she said only, \"They really weren't prehistoric if they worshipped a woman, now were they? That makes them sound pretty advanced to me.\"\n\nNick smiled but rolled his eyes as he leaned down to touch the stone. Jace elbowed Mitch. Nita and Gina grinned.\n\n\"Actually,\" Claire went on, her voice more quiet, \"I believe the woman was some sort of goddess. I believe she belonged to both men\u2014or they belonged to her somehow. I also theorize that Hunter was forbidden to her or beneath her, and that he paid for their love by being executed by his people. Not to go into the details with everyone here now,\" she said with a glance at Lexi, \"let's just say, their form of justice was the price he paid. Yet, in more ways than one, Reaching Woman kept his heart. The tall man called Leader loved her too, and perhaps she loved him. I suppose he either died of a broken or weak heart but, when her people or herself saw her weakness and humanity, they let her do away with herself and buried her with both men\u2014or she decreed it that way. In my own heart, I believe all that. But as for in my head\u2014there's still much to be found and figured out at Black Bog.\"\n\nLexi broke into their huddle around the stone and got on her knees to study it. \"This lady reminds me of you, Mommy. Like when I do things you don't like and you just raise your hands like you are really ticked off but don't know what to do.\"\n\n\"That's my girl,\" Jace said with a little laugh. \"The small one, that is,\" he said and took Brit's hand to raise it to his lips, the hand with the diamond engagement ring.\n\n\"And my girl too,\" Nick said, and ruffled Lexi's hair. Everyone laughed, even Nita, who was living with a beaming Bronco in their new house and had finally learned to smile again. And to put up with Betty Richards who had followed every word of the national and worldwide news that she herself had helped to break by being nosy enough to read license plates at night.\n\nIt seemed the bog people had finally had their revenge for being disturbed and disinterred. Former and now deceased Senator Bradley J. Vance was guilty not only of selling ancient, protected items on the black market but of two murders and a third attempted one, besides his trying to eliminate those in the helicopter. Documents he had left behind indicated he felt vindicated to do anything to protect what he considered his great gift to mankind.\n\nNick had said earlier that it was a good thing Brad had died in Black Bog because that was far better than having to face the two revenge-bent hellions from small-town Georgia, let alone a lifetime in prison or even the death penalty\u2014which Brad had managed to dole out to himself.\n\nAndrea did have an aggressive form of breast cancer, but she had been cleared of any crime. It was punishment enough, Kris had said, not only that she'd lost her husband after learning of his terrible acts, but that the Black Bog project\u2014the Vances' secret, private obsession\u2014was now big news and in the government's domain. Kris had been named as head archaeologist, and Claire knew that she still kept Andrea up-to-date on everything they found. At least there had been only depositions and not a trial for Claire and Dale to go through in the witness box. Of course, it had helped Dale's reputation that he had donated Eva Braun's diary to the National WWII Museum in New Orleans instead of selling it to the highest bidder.\n\nThe double whammy of notoriety in this case was that Dale was so relieved to be exonerated that he had agreed to scores of interviews\u2014at the law offices, with Nick present\u2014about his great-uncle's ties to Hitler. Germany and Russia were calling the \"Hitler's Wife Lived in South Florida\" story an elaborate hoax, but the United States and the United Kingdom had the story splashed everywhere as real, not fake news.\n\nAt least all the publicity had made Ken Jensen lay off pushing for Claire to join the NPD as a forensic consultant. He hadn't been too pleased that Nick hadn't cued him in on Dale's Nazi connection or that Claire had not let him know about Brad Vance possibly selling knockoff jewelry and daggers. Ken had served a subpoena on both of the Vances' antiques stores and learned that several wealthy Europeans had recently contracted to receive stone statues or metal bangle bracelets and necklaces emblazoned with the image of an ancient woman with her arms raised in prayer.\n\nWhat Nick was calling The Black Bog Case had also made Kris and Claire famous\u2014though at least not infamous\u2014as they gave interviews about what they knew of the ancient people's lives and artifacts.\n\nFinally, when everyone had gone and Kris had left complete photos of all the latest finds for Claire to study before she could return to the bog and see the artifacts themselves, the Markwood family of four sat on the couch, Lexi holding Trey now, all quiet at last.\n\n\"Well,\" Nick said, \"that's enough of special secret assignments.\"\n\n\"Not secret anymore,\" Claire said, leaning her head on his shoulder.\n\nHe sighed heavily. \"And once you're finished with Black Bog, let's just move on to something light and lovely\u2014and safe.\"\n\n\"Like pretty butterflies,\" Lexi said, \"'cause they can't hurt anybody.\"\n\n\"Maybe,\" Claire said with a smile at Lexi, \"we can plant some flowers and put in a feeder out by the pool to draw them here.\"\n\n\"Sounds good to me,\" Nick agreed, as he put his arm around her shoulders. \"And, like Lexi says, _safe_. But I think I've stupidly said that before.\"\n\n* * *\n\nIf you enjoyed Silent Scream, don't miss the next suspenseful story in Karen Harper's South Shores series, Dark Storm.\n\nComing soon from MIRA Books.\n\n# Author's Note\n\nSeveral books I've read on forensic psychology claim that \"crime is character.\" Circumstances, motive, opportunity, proximity\u2014it all boils down to the character of the criminal, and character is a central focus of suspense writing. I may start with a fascinating or frightening setting or plot, but above all character counts.\n\nI also thought the claim that Sherlock Holmes created forensic analysis is fascinating. He solved crimes by psyching out what sort of person would murder someone and why. So the idea of studying possible \"perps\" to solve crimes came far before _CSI_ and _Law & Order: Special Victims Unit _or some of the popular fiction about not only who-dunit but why-dunit.\n\nI have read several articles about Nazi war criminals hiding in the United States, most recently the Minnesota arrest of Michael Karkoc, the \"Nazi next door.\" One article, \"There Could be 'Hundreds' of Nazi War Criminals Still Living in the United States\" by Dan Amira appeared in _NY Magazine_ on March 17, 2017, was especially helpful. Although war criminals would be in their eighties and nineties, we all must admit people are living longer these days. But I did make Dale's great-uncle dead for years so that Dale has to deal with that tragic heritage rather than the man himself.\n\nThe idea of having Eva Braun not die with Hitler, but flee to the United States, is an idea that occurred to me while reading a book about Hitler's fall in Germany. An author's imagination is always dealing with \"what if.\"\n\nOther somewhat unusual things mentioned in the book are fact, not fiction. It is difficult for most of us to understand that some people cannot recognize others' faces, even those close to them. The July 14, 2017, issue of _Time Magazine_ had a fascinating article on this written by Kate Samuelson. An expert on face blindness, she believes that as many people as 1 in 50 have some degree of prosopagnosia. So I thought it might be interesting to link the big boom in facial recognition technology to that disability. Also, the fact that Claire suffers from narcolepsy bonds her with Kris even more.\n\nAs Kris mentions in this story, the prehistoric Windover Bog Culture, a 1982 archaeological discovery near Titusville, Florida, was a shock to the sedate, touristy south. It is now on the US National Register of Historic Places. Archaeologists used forensic psychologists to help them probe the ancient culture at that dig. Researching _Florida Bog Mummies_ online __ will lead you to several articles about this site. NOVA has an official website about this on their PBS site under _Ancient America's Bog People._ There is also a video online entitled _Windover's Ancient \"Bog People\"_ _Among Most Significant Archaeological Finds in North America._\n\nI hope you will watch for _Dark Storm_ , the next book in the South Shores series. After all, Nick is finally convinced nothing bad or dangerous could possibly happen in connection with beautiful butterflies. He's been wrong before when he thought Claire's working with prehistoric dead bodies would not lead to danger. Stay tuned.\n\nSpecial thanks to my great support team at MIRA Books, especially my editor, Emily Ohanjanians, and my agent, Annelise Robey. Thanks to my brother Tom in Atlanta for the information on Zebulon, Georgia. And, as ever, to my husband Don for proofreading my stories.\n\n_Karen Harper_\n\nFebruary 2018\nSuspense in Real Life\n\nBy Karen Harper\n\nI didn't realize that, for years, my contemporary novels have been what's currently referred to as \"domestic suspense\" or \"domestic thriller.\" To use several high-profile examples, think _Gone Girl_ or _Girl on the Train._ (Hmm, should I start putting the word _girl_ in my titles? _Girl Gives Silent Scream_?) __ Obviously readers love novels with marriage or family drama as well as the solve-this-scary-case stories.\n\nLiterary domestic dramas mixed with mayhem and murder are not new. The popular 1930s\/1940s _Thin Man_ series, __ featuring Nick and Nora Charles, became a television show. Also in that era on the brink of World War II, Dorothy L. Sayers wrote the delightful Lord Peter Whimsey\/Harriet Vane series. Another domestic drama\/suspense series with cozy and humorous touches were novels under the _Mr. and Mrs. North_ umbrella which also was a television series. More recently Anne Perry's _Charlotte and Thomas Pitt_ novels featuring a historic Victorian inspector and his wife are a continually fresh, charming and suspenseful blend of mystery and domestic drama.\n\nI had written both stand-alone novels and an earlier series (The Queen Elizabeth I Mysteries) __ without categorizing them as domestic suspense. All I knew was I wanted to give my readers books with mystery, romance and complicated human relationships.\n\nIn my earlier stand-alone MIRA books, the two main characters fall in love while trying to solve a crime and stay alive. The romance evolves during the deadly suspense. The main characters have problems, but the stories end in commitment, usually an engagement or plans for a future wedding.\n\nHowever, in this South Shores series, I stirred up this pattern by creating a very real heroine who has some modern \"complications\" she must work through in her domestic life as well as in her career: forensic psychologist Claire Markwood has narcolepsy, she is divorced with a child, and her ex-husband is one of the main viewpoint characters even as her new relationship with Nick, a criminal lawyer, develops. Lexi is a troubled child for reasons obvious in the first three books. Claire is trying to balance her career with family life; Nick is too. Their commitment to their work is admirable but is also a problem.\n\nDoes anything here sound familiar when we look at \"real life\" in today's demanding world? We have all observed stressed romantic relationships and marriages through media, family ties\u2014hopefully not too close-up and personal. I know my silent study of such situations has helped me to create reality, tension and resolutions through my plots, characters and dialogue.\n\nPerhaps how my main characters work through their problems now and in the future\u2014as do the other secondary pairs of romantic characters in the series\u2014will not only interest but also encourage readers in their own domestic dramas and suspenseful lives.\nISBN-13: 9781488088520\n\nSilent Scream\n\nCopyright \u00a9 2018 by Karen Harper\n\nAll rights reserved. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this ebook 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 the publisher, Harlequin Enterprises Limited, 22 Adelaide St. West, 40th Floor, Toronto, Ontario M5H 4E3, Canada.\n\nThis is a work of fiction. Names, characters, places and incidents are either the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events or locales is entirely coincidental. This edition published by arrangement with Harlequin Books S.A.\n\n\u00ae and \u2122 are trademarks of the publisher. Trademarks indicated with \u00ae are registered in the United States Patent and Trademark Office, the Canadian Intellectual Property Office and in other countries.\n\nwww.Harlequin.com\n\n## Contents\n\n  1. Cover\n  2. Back Cover Text\n  3. Praise\n  4. Booklist\n  5. Title Page\n  6. Dedication\n  7. Contents\n  8. Chapter 1\n  9. Chapter 2\n  10. Chapter 3\n  11. Chapter 4\n  12. Chapter 5\n  13. Chapter 6\n  14. Chapter 7\n  15. Chapter 8\n  16. Chapter 9\n  17. Chapter 10\n  18. Chapter 11\n  19. Chapter 12\n  20. Chapter 13\n  21. Chapter 14\n  22. Chapter 15\n  23. Chapter 16\n  24. Chapter 17\n  25. Chapter 18\n  26. Chapter 19\n  27. Chapter 20\n  28. Chapter 21\n  29. Chapter 22\n  30. Chapter 23\n  31. Chapter 24\n  32. Chapter 25\n  33. Chapter 26\n  34. Chapter 27\n  35. Chapter 28\n  36. Chapter 29\n  37. Chapter 30\n  38. Chapter 31\n  39. Chapter 32\n  40. Chapter 33\n  41. Chapter 34\n  42. Chapter 35\n  43. Chapter 36\n  44. Chapter 37\n  45. Author's Note\n  46. Suspense in Real Life\n  47. Copyright\n\n## Landmarks\n\n  1. Cover\n  2. Title Page\n  3. Table of Contents\n  4. Chapter 1\n  5. Copyright Notice\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**CONTENTS**\n\n_Title Page_\n\n_Dedication_\n\n_Introduction_\n\n[CHAPTER ONE  \nPEOPLE](Will_9780307449351_epub_c01_r1.htm)\n\nThe Fun of William F. Buckley\n\nBuckley: A Life Athwart History\n\nDavid Brinkley: Proud Anachronism\n\nBarry Goldwater: \"Cheerful Malcontent\"\n\nJohn F. Kennedy's Thoughts on Death\n\nEugene McCarthy: The Tamarack Tree of American Politics\n\nWhat George McGovern Made\n\nDaniel Patrick Moynihan: The Senate's Sisyphus\n\nJohn Kenneth Galbraith's Liberalism as Condescension\n\nMilton Friedman: Ebullient Master of the Dismal Science\n\nAlan Greenspan: High-Achieving Minimalist\n\nThe Not-at-All Dull George Washington\n\nGeorge Washington's Long Journey Home\n\nJohn Marshall: The Most Important American Never to Have Been President\n\nJames Madison: Well, Yes, of Course\n\nLongfellow: A Forgotten Founder\n\nRonald Reagan: The Steel Behind the Smile\n\nReagan and the Vicissitudes of Historical Judgments\n\nJohn Paul II: \"A Flame Rescued from Dry Wood\"\n\nAyaan Hirsi Ali: An Enlightenment Fundamentalist\n\nHugh Hefner: Tuning Fork of American Fantasies\n\nLawrence Ferlinghetti: The Emeritus Beat as Tourist Attraction\n\nBuck Owens's Bakersfield Sound\n\nAndrew Nesbitt: Seventy-nine-Pound Master of Tourette Syndrome\n\nSimeon Wright's Grace\n\n[CHAPTER TWO  \nPATHS TO THE PRESENT](Will_9780307449351_epub_c02_r1.htm)\n\nThe Most Important American War You Know Next-to-Nothing About\n\nThe Amazing Banality of Flight\n\nThe Price of Misreading the Prairie Sky\n\n\"A Range of Mountains on the Move\"\n\nThe Emblematic Novel of the 1930s (No, It Is Not About the Joads)\n\nAll Quiet at the Overpass\n\nFDR's Transformation of Liberalism\n\nRetailers Give Thanks for Thanksgiving (and FDR)\n\nFDR's Christmas Guest from Hell\n\n\"My Place Is with My Shipmates\"\n\nAn Anthem of American Optimism\u2014in 1943\n\nWhen War _Was_ the Answer\n\nCatching Up to Captain Philip\n\nThe Most Fateful Heart Attack in American History\n\nHow Ike's Highways Helped Heal Civil War Wounds\n\nThe Short, Unhappy Life of the Edsel\n\nThe Fifties in Our Rearview Mirror\n\n2002: Superstitions Are Bad Luck\n\n2003: Lingerie and Duct Tape\n\n2004: _The Passion of the Christ_ and _The Passions of the Faculty Clubs_\n\n2005: \"In Lieu of Flowers, Please Send Acerbic Letters to Republicans\"\n\n2006: \"Go Ahead, We Will Get into One of the Other Boats\"\n\n2007: Ready, Fire, Aim\n\n[CHAPTER THREE  \nGOVERNING](Will_9780307449351_epub_c03_r1.htm)\n\nThe Two Americas: Hard and Soft\n\nAngela Jobe's Resilience\n\nConservatism's Infrastructure\n\nAgainst \"National Greatness Conservatism\"\n\nSumma Contra Reagan Nostalgia\n\nThe Left's Plea for Materialistic Politics\n\nConstitutional Monomania\n\nJudicial Activism, Wise and Not\n\nThe Hard Truth About \"Soft Rights\"\n\nOologah's\u2014and America's\u2014Slide\n\nA Fraudulent \"Fairness\"\n\nPolicing Speech in Oakland\n\nLiberalism's Itch in Minneapolis\n\nChicago: From the White City to the Green City\n\nOur Moralizing Tax Code\n\n\"Electronic Morphine\" on the Ohio River\n\nProhibition II: Interestingly Selective\n\nBeing Green at Ben & Jerry's\n\nThe Tyranny of the Small Picture\n\nDraining the Reservoir of Reverence\n\n_United 93:_ \"We've Got to Do It Ourselves\"\n\nNothing Changes Everything\n\n[CHAPTER FOUR  \nSENSIBILITIES AND SENSITIVITIES](Will_9780307449351_epub_c04_r1.htm)\n\nNarcissism as News\n\nThe Speciesism of Featherless Bipeds\n\nWhat We Owe to What We Eat\n\nThe Holocaust: Handcrafted\n\nThe \"Daring\" of the Avant-Garde Yet Again\n\nAnti-Semitism Across the Political Spectrum\n\nWhen Harry Remet Hanne\n\nCars as Mobile Sculpture\n\nHog Heaven: Happy One Hundredth, Harley\n\nRestoration at 346 Madison\n\nStarbucks, Nail Salons, and the Aesthetic Imperative\n\nManners vs. Social Autism\n\nA Punctuation Vigilante\n\nAmerica's Literature of Regret\n\nChief Illiniwek and the Indignation Industry\n\nChristmas at Our Throats\n\n[CHAPTER FIVE  \nLEARNING](Will_9780307449351_epub_c05_r1.htm)\n\nNational Amnesia and Planting Cut Flowers\n\nA Sensory Blitzkrieg of Surfaces\n\n\"Philosophy Teaching by Examples\"\n\nFascinating Contingencies\n\nEd Schools vs. Education\n\nThis Just In from the Professors: Conservatism Is a Mental Illness\n\nThe Law of Group Polarization in Academia\n\nAntioch College's Epitaph\n\nA Scholar's Malfeasance Gunned Down\n\nJuggling Scarves in the Therapeutic Nation\n\nNature, Nurture, and Larry Summers's Sin\n\nAP Harry Applies to College\n\nTeaching Minnows the Pleasure of Precision\n\n[CHAPTER SIX  \nGAMES](Will_9780307449351_epub_c06_r1.htm)\n\nRaising Michael Oher\n\nThe Man from Moro Bottom\n\n\"Rammer Jammer Yellowhammer!\"\n\nRandy Shannon's Realism\n\nThe NFL: An Intensification of Reality\n\nSpeaking SportsCenterese\n\nThe Movie, and the Truth, About Texas Western\n\n[CHAPTER SEVEN  \n _THE_ GAME](Will_9780307449351_epub_c07_r1.htm)\n\n\"Remember 1908!\"\n\nJackie Robinson: The Possible and the Inevitable\n\nTed Williams: \"I Can't Stand It, I'm So Good\"\n\nRoberto Clemente: \"We Think He Can Hit\"\n\nGreg Maddux: \"Watch This\u2014the First-Base Coach May Be Going to the Hospital\"\n\nTake Me Out to the Metric\n\nElias Knows _Everything_\n\nThe Game's Gifted Eccentrics\n\nDon't Beat a Dead Horse in the Mouth\n\nThe Golden Age\n\nPete Rose, Always Hustling\n\nThe Precious, Precarious Equipoise\n\nBarry Bonds: Enhanced and Devalued\n\nThe Methodical Mr. Aaron\n\nRealism Among the RiverDogs\n\nStriving for Motel Years\n\nSeeking Anonymous Perfection\n\n\"Where Baseball?\"\n\n[CHAPTER EIGHT  \nWONDERING](Will_9780307449351_epub_c08_r1.htm)\n\nIncest at \"a Genetically Discreet Remove\"\n\nAn Intellectual Hijacking\n\nFrom Dayton, Tennessee, to Rhode Island's Committee on Fish and Game\n\nEarth: Not Altogether Intelligently Designed\n\nIntelligent Design and Unintelligent Movies\n\nThe Pope, the Neurosurgeon, and the Ghost in the Machine\n\nHow Biology Buttresses Morality, Which Conforms to...Biology\n\nThe Space Program's Search for...Us\n\nNuclear Waste: That's Us\n\nThe Loudest Sound in Human Experience\n\nL = BB + pw + BC\/BF\n\nWonder What We Are For? Wondering\n\n[CHAPTER NINE  \nMATTERS OF LIFE AND DEATH](Will_9780307449351_epub_c09_r1.htm)\n\nGolly, What Did Jon _Do?_\n\nThe Long Dying of Louise Will\n\n_Acknowledgments_\n\n_Permissions_\n\n_Also by George F. Will_\n\n_Copyright_\n_To the memory of\n\nWilliam F. Buckley Jr.\n\n1925\u20132008_\n**INTRODUCTION**\n\n**A** mong the shortcomings of the current administration of the universe is the fact that Alistair Cooke is gone. The British-born journalist, who died in 2004 at age ninety-five, was one of the scarce bits of evidence that there really is an Intelligent Designer of the universe. Cooke lived in this country for sixty-seven years, producing a body of work of unrivaled perceptiveness, affectionateness, and elegance. One of his books, published in 1952, was titled _One Man's America._ The title of the book you are holding is one man's homage to Cooke.\n\nLiving in Manhattan and traveling around the forty-eight, and then the fifty, states, Cooke developed a thoroughly American sensibility\u2014cheerful, inquisitive, egalitarian, droll, and enthralled without being uncritical. His delicate sensibility was apparent in his description of Harold Ross, founder of _The New Yorker_ in 1925 and editor of it until his death in 1951, as a man \"who winced for a living.\" Cooke was so well-disposed toward America, and so utterly at home and so exquisitely well-mannered, that he did not wince promiscuously or ostentatiously. Still, wincing is, inevitably, what conscientious social commentators often do, not only in America, but especially in America.\n\nMatthew Arnold, for example, was a fastidious social critic and hence an accomplished complainer. When he died, an acquaintance (Robert Louis Stevenson, no less) said: \"Poor Matt, he's gone to Heaven, no doubt\u2014but he won't like God.\" American social critics wince when this country, in its rambunctious freedom, falls short, as inevitably it does, of the uniquely high standards it has set for itself. But different things make different people wince, because sensibilities differ. And nearly four decades of observing American politics and culture have convinced me that, in both, sensibility is fundamental.\n\nThat is, people embrace a conservative (or liberal) agenda or ideology, or develop a liberal (or conservative) political and social philosophy, largely because of something basic to their nature\u2014their temperament, as shaped by education and other experiences. Broadly\u2014 _very_ broadly\u2014speaking, there are, I believe, conservative and liberal stances toward life, conservative and liberal assumptions about how history unfolds, and conservative and liberal expectations about how the world works. This is one reason why we have political categories like \"liberal\" and \"conservative\": People tend to cluster. That is one reason why we have political parties.\n\nThis collection of my writings is not designed to recapitulate the large events of recent years. Consider this volume an almost entirely Iraq-free zone. Rather, it is intended to illustrate, regarding smaller (but not necessarily minor) matters, how one conservative's sensibility responds to myriad provocations and pleasures. At a moment when there is considerable doubt and rancor about what it means to be a conservative, perhaps this collection will provide a useful example.\n\nTime flies when you're having fun, and also when you're not. Time is, of course, magnificently indifferent to whether or not people enjoy what occurs as it passes. The first years of the twenty-first century have not been, on balance, enjoyable for Americans. These have been years characterized by a miasma of anxiety about a new and shadowy terrorist threat to security, and a torrent of acrimony about the dubious inception and incompetent conduct of a war that became perhaps the worst foreign policy debacle in the nation's history. (Well, I said this book would be an _almost_ entirely Iraq-free zone.)\n\nLucretius (as translated by Dryden) wrote about the enjoyment people sometimes derive from watching other people in peril:\n\n_'Tis pleasant, safely to behold from shore,_\n\n_The rolling ship, and hear the tempest roar._\n\nBut Americans have not felt safe ashore\u2014not safe from foreigners who wish them ill, not safe from unusually virulent domestic squabbles. And Americans have not suffered from any insufficiency of journalism and other hectoring. The simultaneous arrival of saturation media (broadcast, podcast, Internet, etc.) and uncivil discourse might be a matter of mere correlation, not causation. It would, however, not be rash to think otherwise.\n\nAnyway, it would be almost impertinent to ask readers to revisit commentary focused on the largest, and painfully familiar, events of these bleak years. I do not do so in this, the eighth collection of my columns, book reviews, and other writings. If, in any given year, more than a dozen of my columns were not about books, I would think that I had not done my job properly. This is because, for all the fascination with new media, I believe that books remain the most important carriers of ideas, and ideas are always the most important news. Hence books themselves are often news.\n\nWith this volume, I am taking a different approach. The essays in the first seven were selected and arranged in order to give readers a retrospective _tour d'horizon,_ a look back at the political and cultural controversies of the four or five years from which the writings were drawn. In this volume, I hope to illustrate how one conservative's sensibility responded to disparate people, stories, and events.\n\nIn the past forty or so years, conservatism has grown from a small, homogenous fighting faction in an unconverted country to a persuasion at least at parity with liberalism in terms of political muscle and intellectual firepower. In the process, conservatism has become large enough to have schisms, and hence an identity crisis. This volume makes no attempt to distill a coherent political philosophy from episodic writings in response to disparate events. Perhaps, however, the skeleton and ligaments of one conservative's philosophy can be discerned in the response of his sensibility, or temperament, to the people, events, and controversies featured herein. This is, I think, even so (perhaps especially so) when considering the ethics of competition and craftsmanship on what General Douglas MacArthur called \"the fields of friendly strife\"\u2014that is, sports.\n\nThe basic approach to writing columns and other periodic journalism resembles what used to be the unwritten but understood rules regarding Catholic confession: Be brief, be blunt, and be gone. In commentary, this approach is not optional, because print journalism is governed by two scarcities. One is a scarcity of space: Columnists who cannot get said what they want to say in 750 words should consider another vocation. The other scarcity is of time: Americans are harried, and their attention spans are not lengthening. Increasingly clamorous media, covering an always turbulent world, are constantly tugging at Americans' sleeves, urgently saying, \"Pay attention to _this!_ \"\n\nSaturation journalism, ravenous for the attention of a jaded and distracted public, ratchets up the hyperbole, like the character in a Tom Stoppard play who exclaims, \"Clufton Bay Bridge is the fourth biggest single-span double-track shore-to-shore railway bridge in the world bar none.\" Gosh. One character in the American drama, Richard Nixon, said of the first landing by men on the moon, \"This is the greatest week in the history of the world since Creation.\" A friend and supporter, the evangelist Billy Graham, thought that was a bit over the top and notified the president that there had been three bigger events: \"1. The first Christmas. 2. The day on which Christ died. 3. The first Easter.\" Nixon, not exactly chastened but certainly prudent, scrawled a note to his chief of staff, Bob Haldeman: \"H\u2014Tell Billy RN referred to a week not a day.\"\n\nThe first human step on the moon, although not quite competitive with Creation as a headline, was a grand event. But with the passage of time\u2014usually not very much time; a day often suffices\u2014the subjects of most media cacophonies turn out to seem small indeed. But from many unheralded events and obscure people, large and durable lessons can flow, as I hope the essays in this volume demonstrate. Be that as it may, the essays that follow will perhaps remind readers how endlessly entertaining and instructive the unfolding American story invariably is.\n\nThe passing American scene certainly is that, always. Still, any sensible journalist should develop the habit of periodically lifting his or her gaze from the crisis du jour in order to remind himself or herself of this: Journalism is evanescent. But, then, this, too, is true: Under the eye of Eternity\u2014or, less grandly, just given time\u2014almost everything is evanescent. Everything, that is, other than the value of the simple virtues and decencies that can make communities flourish and that have made America great and exemplary. That is what Alistair Cooke believed, and what this conservative's sensibility tells him.\n**Chapter One**\n\n**PEOPLE**\n\n# **The Fun of William F. Buckley**\n\n**I** n his fortieth anniversary toast to his Yale class of 1950, William F. Buckley said, \"Some of us who wondered if we would ever be this old now wonder whether we were ever young.\" Those who were not young forty years ago, in 1965, can have no inkling of what fun it was to be among Buckley's disciples as he ran for mayor of New York vowing that, were he to win, his first act would be to demand a recount.\n\nMurray Kempton, the wonderful liberal columnist who later joined Buckley's eclectic legion of friends, wrote after Buckley's first news conference that the candidate \"had the kidney to decline the customary humiliation of soliciting the love of the voters, and read his statement of principles in a tone for all the world that of an Edwardian resident commissioner reading aloud the 39 articles of the Anglican establishment to a conscript assemblage of Zulus.\" For conservatives, happy days were here again.\n\nBack then, espousing conservatism was regarded by polite society, then soggy with that era's barely challenged liberalism, as a species of naughtiness, not nice but also not serious. Buckley, representing New York's Conservative Party, which was just three years old, won 13 percent of the vote. When the winner, John Lindsay, limped discredited from office eight years later, Bill's brother Jim had been elected, on the Conservative line, U.S. senator from New York.\n\nBuckley, for whom the nation should give thanks, turns eighty on Thanksgiving Day, and _National Review,_ the conservative journal he founded in the belly of the beast\u2014liberal Manhattan\u2014turned fifty this month. It is difficult to remember, and hence especially important to remember, the slough of despond conservatism was in 1955.\n\nOhio senator Robert Taft, for more than a decade the leading conservative in elective office, had died in 1953. Joseph McCarthy had tainted conservatism in the process of disgracing himself with bile and bourbon. President Eisenhower had so placidly come to terms with the flaccid consensus of the 1950s that the editor of _U.S. News & World Report,_ the most conservative newsweekly, suggested that both parties nominate Eisenhower in 1956.\n\n_National Review_ demurred. When it nailed its colors\u2014pastels were not encouraged\u2014to its mast and set sail upon the choppy seas of American controversy, one novel on the bestseller list was Sloan Wilson's _The Man in the Gray Flannel Suit,_ voicing the 1950s' worry about \"conformity.\" _National Review_ 's premise was that conformity was especially egregious among the intellectuals, that herd of independent minds. The magazine is one reason why the phrase \"conservative intelligentsia\" is no longer an oxymoron.\n\nIn 1964, _National Review_ (circulation then: 100,000) did what the mighty Hearst press had never done\u2014determined a major party's presidential nomination. Barry Goldwater's candidacy was essentially an emanation of _National Review_ 's cluttered office on East 35th Street. Which is why an audience of young Goldwaterites took it so hard when, two months before the election, Buckley warned them that bliss would be a bit delayed:\n\n\"The point of the present occasion is to win recruits whose attention we might never have attracted but for Barry Goldwater; to win them not only for November the third, but for future Novembers; to infuse the conservative spirit in enough people to entitle us to look about us, on November fourth, not at the ashes of defeat, but at the well-planted seeds of hope, which will flower on a great November day in the future, if there is a future.\"\n\nThere was. It arrived sixteen years later.\n\nAuthor of more than four thousand columns, and still adding two a week; author of forty-seven books, eighteen of them novels; host of the _Firing Line_ television program for thirty-four years; a public speaker, often making as many as seventy lectures and debates a year, for almost fifty years; ocean mariner; concert harpsichordist\u2014his energy reproaches the rest of us. Married to a woman who matches his mettle, his proposal to her, made when he called her away from a card game, went like this:\n\nHe: \"Patricia, would you consider marriage with me?\"\n\nShe: \"Bill, I've been asked this question many times. To others I've said no. To you I say yes. Now may I please get back and finish my hand?\"\n\nBuckley, so young at eighty, was severely precocious at seven when he wrote a starchy letter to the king of England demanding payment of Britain's war debts. Seventy-three years on, Buckley's country is significantly different, and better, because of him. Of how many journalists, ever, can that be said? One.\n\n[NOVEMBER 24, 2005]\n\n# **Buckley: A Life Athwart History**\n\n**T** hose who think Jack Nicholson's neon smile is the last word in smiles never saw William F. Buckley's. It could light up an auditorium; it did light up half a century of elegant advocacy that made him an engaging public intellectual and the twentieth century's most consequential journalist.\n\nBefore there could be Ronald Reagan's presidency, there had to be Barry Goldwater's candidacy. It made conservatism confident and placed the Republican Party in the hands of its adherents.\n\nBefore there could be Goldwater's insurgency, there had to be _National Review_ magazine. From the creative clutter of its Manhattan offices flowed the ideological electricity that powered the transformation of American conservatism from a mere sensibility into a fighting faith and a blueprint for governance.\n\nBefore there was _National Review,_ there was Buckley, spoiling for a philosophic fight, to be followed, of course, by a flute of champagne with his adversaries. He was twenty-nine when, in 1955, he launched _National Review_ with the vow that it \"stands athwart history, yelling Stop.\" Actually, it helped Bill take history by the lapels, shake it to get its attention, and then propel it in a new direction. Bill died Wednesday in his home, in his study, at his desk, diligent at his lifelong task of putting words together well and to good use.\n\nBefore his intervention\u2014often laconic in manner, always passionate in purpose\u2014in the plodding political arguments within the flaccid liberal consensus of the post\u2013World War II intelligentsia, conservatism's face was that of another Yale man, Robert Taft, somewhat dour, often sour, wearing three-piece suits and wire-rim glasses. The word _fun_ did not spring to mind.\n\nThe fun began when Bill picked up his clipboard, and conservatives' spirits, by bringing his distinctive brio and \u00e9lan to political skirmishing. When young Goldwater decided to give politics a fling, he wrote to his brother: \"It ain't for life and it might be fun.\" He was half right: Politics became his life, and it was fun, all the way. Politics was not Bill's life\u2014he had many competing and compensating enthusiasms\u2014but it mattered to him, and he mattered to the course of political events.\n\nOne clue to Bill's talent for friendship surely was his fondness for this thought of Harold Nicolson's: \"Only one person in a thousand is a bore, and _he_ is interesting because he is one person in a thousand.\" Consider this from Bill's introduction to a collection of his writings titled _The Jeweler's Eye: A Book of Irresistible Political Reflections:_\n\n\"The title is, of course, a calculated effrontery, the relic of an impromptu answer I gave once to a tenacious young interviewer who, toward the end of a very long session, asked me what opinion did I have of myself. I replied that I thought of myself as a perfectly average middle-aged American, with, however, a jeweler's eye for political truths. I suppressed a smile\u2014and watched him carefully record my words in his notebook. Having done so, he looked up and asked, 'Who gave you your jeweler's eye?' 'God,' I said, tilting my head skyward just a little. He wrote that down\u2014the journalism schools warn you not to risk committing anything to memory. 'Well,'\u2014he rose to go, smiling at last\u2014'that settles that!' We have become friends.\"\n\nPat, Bill's beloved wife of fifty-six years, died last April. During the memorial service for her at New York's Metropolitan Museum of Art, a friend read lines from \"Vitae Summa Brevis\" by a poet she admired, Ernest Dowson:\n\n_They are not long, the days of wine and roses:_\n\n_Out of a misty dream_\n\n_Our path emerges for a while, then closes_\n\n_Within a dream._\n\nBill's final dream was to see her again, a consummation of which his faith assured him. He had an aptitude for love\u2014of his son, his church, his harpsichord, language, wine, skiing, sailing.\n\nHe began his sixty-year voyage on the turbulent waters of American controversy by tacking into the wind with a polemical book, _God and Man at Yale_ (1951), that was a lovers' quarrel with his alma mater. And so at Pat's service the achingly beautiful voices of Yale's Whiffenpoofs were raised in their signature song about the tables down at Mory's, \"the place where Louis dwells\":\n\n_We will serenade our Louis_\n\n_While life and voice shall last_\n\n_Then we'll pass and be forgotten with the rest_\n\nBill's distinctive voice permeated, and improved, his era. It will be forgotten by no one who had the delight of hearing it.\n\n[FEBRUARY 29, 2008]\n\n# **David Brinkley: Proud Anachronism**\n\n**T** o have worked alongside David Brinkley on television is to have experienced what might be called the Tommy Henrich Temptation. Henrich, who played right field for the Yankees when Joe DiMaggio was playing center field, must have been constantly tempted to ignore the game and just stand there watching DiMaggio, who defined for his generation the elegance of understatement and the gracefulness that is undervalued because it makes the difficult seem effortless.\n\nBrinkley, who died Wednesday, a month shy of his eighty-third birthday, was a Washington monument as stately, and as spare in expression, as is the original. Long before high-decibel, low-brow cable shout-a-thons made the phrase \"gentleman broadcaster\" seem oxymoronic, Brinkley made it his business to demonstrate the compatibility of toughness and civility in journalism.\n\nHe was the most famous son of Wilmington, North Carolina, until Michael Jordan dribbled into the national consciousness. Brinkley arrived in Washington in 1943, an era when a gas mask occasionally hung from the president's wheelchair and the city\u2014then hardly more than a town, really\u2014fit John Kennedy's droll description of it as a community of Southern efficiency and Northern charm.\n\nIt was a town in which the second-most-powerful person was the Speaker of the House, Sam Rayburn, a Texan whose office wall was adorned with five portraits of Robert E. Lee, all facing south, and who said he did not socialize because \"these Washington society women never serve chili.\" Washington had fifteen thousand outdoor privies and a cleaning establishment that handled white flannel suits by taking them apart at the seams, hand-washing each piece, drying the pieces in the sun, then reassembling each suit. The process took a week\u2014longer during cloudy weather\u2014and cost $10.\n\nBy the time Brinkley retired from ABC in 1996, he had covered (in the subtitle of his 1995 autobiography) \"11 Presidents, 4 Wars, 22 Political Conventions, 1 Moon Landing, 3 Assassinations, 2,000 Weeks of News and Other Stuff on Television.\" Like Walter Cronkite, the only other journalist of comparable stature from television's founding generation, Brinkley began his career in print journalism. Indeed, Brinkley began at a time when the phrase \"print journalist\" still seemed almost a redundancy.\n\nDuring the Second World War, Edward R. Murrow and his CBS radio colleagues, such as Eric Sevareid, Charles Collingwood, Robert Trout, and William Shirer, elevated broadcast journalism. But television took awhile to get the hang of it.\n\nIn 1949, John Cameron Swayze's _Camel News Caravan,_ for which young Brinkley, who had joined NBC in 1943, was a reporter, was carried for fifteen minutes five nights a week. NBC's network consisted of four stations, in Boston, New York, Philadelphia, and Washington. The sponsor required Swayze, who always wore a carnation in his lapel, to have a lit cigarette constantly in view. Not until 1963 did Cronkite's _CBS Evening News_ become the first thirty-minute newscast.\n\nIn 1981, after thirty-eight years with NBC, Brinkley became host of ABC's _This Week._ He understood a fundamental truth about television talk shows: what one does on them one does in strangers' living rooms. So mind your manners; do not make a scene. Those thoughts guided Brinkley as he provided adult supervision to others on _This Week,_ the first hour-long Sunday morning interview program.\n\nHow anachronistic the maxim \"mind your manners\" seems in the harsh light cast by much of today's television. How serene, even proud, Brinkley was about becoming somewhat of an anachronism.\n\nEvelyn Waugh's novel _Scott-King's Modern Europe_ (1947) concludes on what can be called a Brinkleyesque note. The protagonist, Mr. Scott-King, a teacher at an English boys' school, is warned by the school's headmaster that the boys' parents are only interested in preparing their boys for the modern world.\n\n\"You can hardly blame them, can you?\" said the headmaster. \"Oh, yes,\" Scott-King replied, \"I can and do,\" adding, \"I think it would be very wicked indeed to do anything to fit a boy for the modern world.\"\n\nBrinkley's backward-looking gentility made him regret, among much else, the passing of the days when it was unthinkable for a gentleman to wear other than a coat and tie when traveling by air. It is, then, an irony of the sort Brinkley savored that he was not merely present at the creation of television as a shaper of the modern world, he was among the creators of that phenomenon. Like the Founders of this fortunate Republic, Brinkley set standards of performance in his profession that still are both aspirations and reproaches to subsequent practitioners.\n\n[JUNE 13, 2003]\n\n# **Barry Goldwater: \"Cheerful Malcontent\"**\n\nIn 2007, I was asked to write the foreword for a new edition of Goldwater's book _The Conscience of a Conservative,_ the first in a series of important political books republished by Princeton University Press.\n\n**W** hen Barry Goldwater ran for president in 1964, he was Arizona's _junior_ senator. But, then, measured by length of Senate service, ninety-eight other senators also were junior to Arizona's senior senator, Carl Hayden, who was a former sheriff in Arizona territory. Hayden had entered the House of Representatives at age thirty-five when Arizona acquired statehood in 1912, and entered the Senate at age forty-nine, where he served until 1969. The Western frontier, so vivid in the national imagination and so associated with American libertarianism, _lived_ in Goldwater's Senate colleague.\n\nWhen I visited Goldwater at his home in Phoenix a few years before his death in 1998, he said he had built his house on a bluff to which, when he was young, he would ride his horse and sleep under the stars. When he was a boy, about one hundred thousand people lived in the Valley of the Sun. When Goldwater died, the population of a _suburb_ of Phoenix\u2014Mesa\u2014was larger than St. Louis, and the population of the Phoenix metropolitan area, the nation's fourteenth largest, was approaching three million.\n\nYou must remember this: Goldwater was a conservative from, and formed by, a place with precious little past to conserve. Westerners have no inclination to go through life with cricks in their necks from looking backward. When Goldwater became the embodiment of American conservatism\u2014partly by his own efforts, and partly because he was conscripted by others for the role\u2014that guaranteed that the mainstream of American conservatism would be utterly _American._ The growing conservative intelligentsia would savor many flavors of conservatism, from Edmund Burke's to T. S. Eliot's, conservatisms grounded on religious reverence, nostalgia, and resistance to the permanent revolution of conditions in a capitalist, market society. Such conservatisms would have been unintelligible, even repellent, to Goldwater, if he had taken time to notice them.\n\nIn the beginning, which is to say in the early 1950s, America's modern conservative movement was remarkably bookish. It began to find its voice with Whittaker Chambers' memoir _Witness_ (1952), Russell Kirk's _The Conservative Mind_ (1953), and the twenty-five-year-old William F. Buckley Jr.'s _God and Man at Yale_ (1951). The books most congruent with what came to be Goldwaterism included one published in London in 1944 by an Austrian and future Nobel laureate in economics\u2014Friedrich Hayek's _The Road to Serfdom._ Another book by another winner of the Nobel price for economics was Milton Friedman's _Capitalism and Freedom_ (1962). Like Hayek and Friedman, Goldwater's central preoccupation was freedom, and the natural tendency of freedom's sphere to contract as government's sphere expands.\n\nGoldwater was a man of many parts\u2014politician and jet pilot, ham radio operator and accomplished photographer\u2014but no one ever called him bookish. And if anyone ever had, Goldwater, a man of action and of the West, might have said\u2014echoing the protagonist of the novel that invented the Western, Owen Wister's _The Virginian_ (1902)\u2014\"When you call me that, _smile!_ \"\n\nThen Goldwater would have smiled, because although he could be gruff, he could not stay out of sorts. He was, as journalist Richard Rovere said, \"the cheerful malcontent.\" In that role, he also was an early symptom\u2014a leading indicator\u2014of the 1960s ferment.\n\nThe 1960s are rightly remembered as years of cultural dissent and political upheaval, but they are wrongly remembered as years stirred only from the left. Actually, they were not even stirred first, or primarily, or most consequentially from the left. By the time the decade ended, with Richard Nixon in the White House, conservatism was in the saddle, embarked on winning seven of the ten presidential elections from 1968 through 2004.\n\nBut because of the political complexion of the journalists who wrote the \"first rough draft of history,\" and because of the similar complexion of the academic historians who have written subsequent drafts, and because much of the decade's most lurid political turbulence, such as the turmoil on campuses and at the riotous 1968 Democratic Convention in Chicago, were episodes of dissent by the left\u2014because of all this, the decade is remembered as one dominated by dissent from the left. Nevertheless, it can reasonably be said that dissent in the 1960s began on the right, and it is certain that the most nation-shaping dissent was from the right.\n\nSome say we should think of the sixties as beginning on November 22, 1963, and ending in October 1973\u2014that is, as beginning with a presidential assassination that supposedly shattered the nation's sunny postwar disposition, and ending with the Yom Kippur War and the oil embargo that produced a sense of scarcity and national vulnerability. Arguably. But although it may seem eccentric\u2014or banal\u2014to say so, the sixties, understood as a decade of intellectual dissent and political insurgency, began in 1960.\n\nOn July 27, to be precise, when an Arizona senator strode to the podium of the Republican Convention in Chicago and barked: \"Let's grow up, conservatives. If we want to take the party back\u2014and I think we can\u2014let's get to work.\"\n\nBack from whom? In two words, \"moderate\" Republicans. In one word, Northeasterners. What that word denoted, to those who used it as an epithet, was the old Republican establishment that had nominated Wendell Willkie (the \"barefoot boy from Wall Street\" was from Indiana, but not really), New York's Governor Tom Dewey twice, and Dwight Eisenhower twice. (Eisenhower was from Texas and Kansas, long ago, but had sojourned in Paris and in Manhattan's Morningside Heights\u2014as Supreme Allied Commander and president of Columbia University\u2014before winning the 1952 Republican nomination by defeating \"Mr. Republican\" and the conservatives' favorite, Senator Robert Taft of Ohio.)\n\nThe Republican establishment, speaking through the _New York Herald-Tribune,_ represented what Goldwater and kindred spirits considered a flaccid postwar Republican consensus. Goldwater's complaint was that timid Republicans challenged neither the New Deal notion of the federal government's competence and responsibilities nor the policy of mere containment regarding the Soviet Union.\n\nThe GOP establishment against which Goldwater rose in rebellion is, like the _Herald-Tribune,_ which ceased publication in 1966, a mere memory. As is the subject of Goldwater's last chapter, \"the Soviet menace.\" But what makes this book of lasting interest, and what makes it pertinent to the Republicans' deepening intramural conflicts in the first decade of the twenty-first century, is this: Goldwater's primary purpose was to refute the perception that conservatism was an intellectually sterile and morally crass persuasion.\n\nIn the first sentence of his first chapter, Goldwater wrote: \"I have been much concerned that so many people today with Conservative instincts feel compelled to apologize for them.\" Nearly half a century later, people calling themselves \"progressives\" are in flight from the label \"liberal.\" It is difficult to remember, but well to remember, how rapidly and thoroughly political fashions can change: There was a time in living memory when...well, in 1950, a man was arrested for creating a public disturbance and a witness said: \"He was using abusive language, calling people conservative and all that.\"\n\nIn 1960, the common caricature was that liberals had ideas and ideals, whereas conservatives had only material interests. Goldwater set out to refute the idea that conservatism is merely \"a narrow, mechanistic _economic_ theory that may work very well as a bookkeeper's guide, but cannot be relied upon as a comprehensive political philosophy.\" Goldwater insisted that it was liberalism that had become thin intellectual gruel. He said it produced government that saw the nation as a mere aggregation of clamorous constituencies with material itches that it was Washington's duty to scratch with federal programs. The audacity of _The Conscience of a Conservative_ was its charge that the post\u2013New Deal political tradition, far from being idealistic, was unworthy of a free society because it treated citizens as mere aggregations of appetites.\n\nIn recent years, the intellectual energy in American politics has been concentrated on the right side of the spectrum, and today two kinds of conservatives are at daggers drawn with each other. The last twenty-five years or so produced the rise of \"social conservatives,\" a group generally congruent with the \"religious right.\" These conservatives, alarmed by what they consider the coarsening of the culture, believe in \"strong government conservatism.\" They argue that government can, and urgently must, have an active agenda to defend morals and promote virtue, lest freedom be lost. Other conservatives, the political descendants of Goldwater, agree that good government is, by definition, good for the public's virtue. They also believe, however, that limited government _by its limitations_ nurtures in men and women the responsibilities that make them competent for, and worthy of, freedom.\n\nHad Goldwater lived to see the republication of his book in this supposedly conservative era, he might have made some characteristically blunt remarks about the impotence of books. This edition of _The Conscience of a Conservative_ comes after a Republican president and a Republican-controlled Congress enacted in 2001 the largest federal intervention in primary and secondary education (the No Child Left Behind law) in American history. And, in 2002, enacted the largest farm subsidies. And, in 2003, enacted the largest expansion of the welfare state (the prescription drug entitlement added to Medicare) since Lyndon Johnson, the president who defeated Goldwater in 1964, created Medicare in 1965. In _The Conscience of a Conservative,_ Goldwater insisted that most Americans embraced conservative principles, and he blamed conservatives for failing to persuade the country of the \"practical relevance\" of conservatism.\n\nWas he mistaken about what most Americans believe? Are they now ideologically, meaning rhetorically, conservative, but operationally liberal?\n\nIf so, Goldwater might say this vindicates his argument: One consequence of unlimited government is unlimited dependency\u2014 _learned_ dependency, a degrading addiction of citizens to public provisions.\n\nBut that gloomy conclusion could not long withstand Goldwater's Western cheerfulness. Besides, it does not begin to do justice to the changes conservatism has wrought, or helped to bring about, since Goldwater summoned conservatives to take back the Republican Party. In 1960, the top income tax rate was 90 percent, there was a lifetime entitlement to welfare, the economy was much more regulated than it is now, and the Iron Curtain looked like confirmation of George Orwell's image of totalitarianism: \"Imagine a boot stamping on a human face\u2014forever.\" That \"forever\" expired twenty-five years after Goldwater's presidential campaign.\n\nMost historians probably think that Goldwater's 1964 run for the White House was the apogee of his public life, which began with his election to the Phoenix city council in 1949 and lasted until his retirement from the Senate in 1987. Goldwater, I suspect, thought otherwise.\n\nBefore and after 1964, he was a man of the Senate; he probably thought of his presidential run as a brief detour in a career otherwise as level as the surface of a Western mesa. The Senate suited him as a venue for taking stands and enunciating views. He was a \"conviction politician\"\u2014a term later minted to describe a soul mate, Margaret Thatcher\u2014who thought the point of public life was to advance a creed. Therefore, this book, more than his presidential candidacy, was, in a sense, the essence of the public man.\n\nWhen he delivered his acceptance speech to the 1964 Republican Convention in San Francisco's Cow Palace, and thundered that \"extremism in defense of liberty is no vice\" and \"moderation in pursuit of justice is no virtue,\" a journalist rocked back in his chair and exclaimed: \"My God, he's going to run as Goldwater!\" Indeed. Goldwater ran as the author of _The Conscience of a Conservative,_ with the book as his platform.\n\nHe had been an eager author of that book, although an author with the assistance of a professional polemicist\u2014L. Brent (\"Hell Bent\") Bozell, William Buckley's brother-in-law. Bozell helped Goldwater weave various speeches and other pronouncements into a coherent argument.\n\nGoldwater was, however, a reluctant presidential candidate, especially after the Kennedy assassination. Even before that, he did not have the monomania requisite for a successful candidate. It has been well said that anyone who is willing to do the arduous things necessary to become president probably is too unbalanced to be trusted with the office. Goldwater preferred flying himself around Arizona to photograph Native Americans to being flown around the country in pursuit of convention delegates.\n\nBefore the Kennedy assassination, however, Goldwater rather fancied the idea of challenging Kennedy's reelection effort. Goldwater liked Kennedy\u2014they had been freshmen senators in 1953\u2014and he suggested to Kennedy that they might share a plane and hopscotch around the country debating each other. After the assassination, Goldwater knew that the outcome of the 1964 election was not in doubt because, as he put it with the pungency that sometimes got him in trouble, the country was not going to assassinate two presidents in less than twelve months. But a merry band of Republican insurgents, many of them associated in one way or another with Buckley's _National Review_ \u2014which was not yet nine years old when Goldwater was nominated\u2014disregarded his reluctance and launched him on a campaign that would lose forty-four states.\n\nBut it was a spectacularly creative loss. In the process, conservatives captured the Republican Party's apparatus. And in October 1964, when Goldwater was shown a speech he was supposed to deliver to a national television audience, he said: \"This is good, but it doesn't quite sound like me. Get Ronald Reagan to give it.\" After Reagan won the presidency, conservatives liked to say that Goldwater won in 1964, but it took sixteen years to count all the votes.\n\nAnother way of understanding Goldwater's constructive defeat involves a dialect that a Marxist might relish. In 1938, a there was backlash in congressional elections against President Franklin Roosevelt's plan to \"pack\" the Supreme Court. From 1938 through 1964, there never was a reliably liberal legislating majority in Congress. A coalition of Republicans and conservative, mostly Southern, Democrats held the balance of power. But Goldwater's landslide defeat swept liberal majorities into the House of Representatives and Senate. For two years, liberalism was rampant, until the 1966 elections began to correct the partisan imbalance. During those two years, when the prestige of government was perhaps higher than ever before or since in American history, the Great Society initiatives became an exercise in political overreaching, made possible by Goldwater's defeat. Disappointment with the results laid the predicate for Reagan's victory.\n\nWhich was followed by President George Herbert Walker Bush's \"kinder and gentler\" conservatism, then William Clinton's centrism, then George W. Bush's \"compassionate conservatism.\" And so continues an American political argument about how much government we want, and how much we are willing to pay for it in the coin of constricted freedom. That argument gathered steam when Goldwater threw down a gauntlet\u2014this book.\n\nForty-seven years after the publication of _The Conscience of a Conservative,_ Goldwater, a seasoned politician and a child of the West, probably would look equably upon America as, like Phoenix\u2014today approaching four million people\u2014a work forever furiously in progress. He knew that popular government rests on public opinion, which is shiftable sand. With this book, and with his public career that vivified the principles expressed herein, he shifted a lot of sand.\n\n[FOREWORD TO _THE CONSCIENCE OF A CONSERVATIVE,_ APRIL 2007]\n\n# **John F. Kennedy's Thoughts on Death**\n\n**L** anding in New York on a speaking trip, the president impulsively decided not to have a motorcade into Manhattan, so his limousine stopped at ten traffic lights. At one, a woman ran to the car and snapped a photograph inches from his face. A policeman exclaimed, \"Oh, my God. She could have been an assassin.\" It was November 15, 1963.\n\nOn the Sunday night of October 28, 1962, at the conclusion of the Cuban Missile Crisis, John Kennedy quipped to his brother Robert, \"This is the night I should go to the theater,\" a reference to Lincoln's visit to Ford's Theatre after the Civil War was won. Thoughts of death were not new to the man whose father had medicines stored for him in banks around the world. They were to treat chronic illnesses so serious that he had been given the last rites of the Catholic Church at least three times before he became president at age forty-three.\n\nEven if he had not gone to Dallas, he probably would have died long before now. He would have been killed partly by the horrifying cocktails of pills and injections\u2014sometimes six Novocain shots in his back in a day; one drug drove his cholesterol count above 400\u2014mixed by doctors sometimes unaware of what the others were administering just to keep him ambulatory and alert.\n\nThe soaring arc of Kennedy's truncated life combined success achieved by discipline, and sexual recklessness\u2014seventy calls through the White House switchboard to a mistress he shared with a Mafia don; said another woman, Marilyn Monroe, \"I think I made his back feel better\"\u2014that risked everything.\n\nIn _President Kennedy: Profile of Power,_ much the best book on Kennedy, Richard Reeves says that Kennedy\u2014\"very impatient, addicted to excitement, living his life as if it were a race against boredom\"\u2014was well matched to his moment. He was a man in a hurry at a time when the pulse of communication was accelerating.\n\nIn seeking the presidency, Reeves wrote, \"he did not wait his turn.\" One of the elders he elbowed aside, Adlai Stevenson, said, \"That young man! He never says 'please'...\" When a friend urged Kennedy to wait beyond 1960, he said, \"No, they will forget me. Others will come along.\"\n\nAlways there was his fatalistic sense of how perishable everything was, and his ironic awareness of how nothing is what it seems\u2014least of all himself. Campaigning in 1960 as a vessel of \"vigor,\" his health often forced him to spend about half of the day in bed.\n\nThe Kennedy years had, as Reeves writes, \"an astonishing density of events,\" from the building of the Berlin Wall to the Birmingham church bombing, and the integration of the University of Mississippi a month before the Cuban Missile Crisis. Kennedy was a quick study, with much to learn.\n\nAstonishingly callow when inaugurated, he was unable to stem or even discern the intragovernmental delusions and deceits that propelled the Bay of Pigs invasion just eighty-seven days into his presidency. Much flowed from that debacle. Kennedy said that in order to reverse Nikita Khrushchev's assessment of him as weak, he had to find somewhere to show U.S. resolve: \"The only place we can do that is in Vietnam. We have to send more people there.\" Soon he was at the Vienna summit, where Khrushchev, impervious to his charm, concluded that he was \"a pygmy.\"\n\nOnly foreign affairs held Kennedy's attention. His response to the \"freedom riders\" who lit a fuse of the civil rights revolution was to ask his civil rights adviser, who was white, \"Can't you get your goddamned friends off those buses?\" But foreign affairs were plentiful enough.\n\nPlentiful, and a sure cure for boredom. When on May 30, 1961, Rafael Trujillo, dictator of the Dominican Republic, was assassinated, Kennedy asked Secretary of State Dean Rusk: \"Were we involved?\" Rusk replied: \"I don't think so. There's some confusion.\"\n\nIn 1963, too, the days were eventful. Twenty-two days after a Saigon coup encouraged by the United States\u2014it produced regime change through the assassination of South Vietnam's two principal leaders\u2014and on the day a ballpoint pen containing poison intended to kill Fidel Castro was scheduled to be delivered by CIA agent Desmond Fitzgerald to a potential assassin, Kennedy awoke in Fort Worth. He was to speak there, then fly to Dallas.\n\nLooking down from his hotel room at the platform from which he would speak, he said to an aide, \"With all these buildings around it, the Secret Service couldn't stop someone who really wanted to get you.\" It was Friday, November 22, 1963.\n\n[NOVEMBER 20, 2003]\n\n# **Eugene McCarthy: The Tamarack Tree of American Politics**\n\n_I love you so.... Gone? Who will swear you wouldn't have done good to the country, that fulfillment wouldn't have done good to you._\n\n\u2014ROBERT LOWELL, \"For Eugene McCarthy\" (July 1968)\n\n**B** y August 1968, Senator Eugene McCarthy was gone and his supporters were left to wonder how\u2014whether\u2014his fulfillment was connected to doing good to the country. When the Democratic convention nominated another Minnesotan, Hubert Humphrey\u2014who in 1964 won the vice presidential nomination McCarthy had craved\u2014McCarthy went to the south of France, then covered the World Series for _Life_ magazine. Had he campaigned for Humphrey, who narrowly lost, there probably would have been no Nixon presidency.\n\nMcCarthy died last Saturday in his ninetieth year, in this city which he sometimes seemed to include in his capacious disdain but which, for a while, he leavened with a distinctive sensibility. In 1980, he endorsed Ronald Reagan, reasoning that Reagan could not be worse than Jimmy Carter. But even in 1968 he had a sometimes ill-disguised disdain for many who flocked to his diffidently unfurled banner.\n\nDisgusted by Vietnam policy, he laconically announced himself \"willing\" to be an \"adequate\" president, and went to New Hampshire to unseat his party's president. McCarthy got 41.9 percent of the vote. Johnson got 49.6 percent\u2014all write-ins; his name was not on the ballot\u2014and three weeks later withdrew from the race.\n\nMcCarthy's 1968 achievement elevated New Hampshire's primary to the status it has subsequently enjoyed. His death occurred the day the Democratic Party gingerly suggested modifying its primary schedule in a way that might diminish New Hampshire's potency.\n\nThe sacramental status of Iowa's caucuses and New Hampshire's primary as the first two nominating events testifies to the power of the mere passage of time to sanctify the accidental, even the unreasonable. Now the Democratic Party suggests allowing one or two states to hold caucuses\u2014not primaries\u2014between Iowa and New Hampshire.\n\nThe case against caucuses is that they take hours, often at night, and thus disproportionately attract the ideologically fervid\u2014not what the Democratic Party needs. The case against New Hampshire's primary is that its power is disproportionate for a state so unrepresentative of America's demographic complexities. The case for New Hampshire can be put in a name: Gene McCarthy. The small state gives an unknown underdog challenger, practicing retail politics, a fighting chance.\n\nMcCarthy's insurgency, the most luminous memory of many aging liberals, would today be impossible\u2014criminal, actually\u2014thanks to the recent \"reform\" most cherished by liberals, the McCain-Feingold campaign regulations. McCarthy's audacious challenge to an incumbent president was utterly dependent on large early contributions from five rich liberals. Stewart Mott's $210,000 would be more than $1.2 million in today's dollars. McCain-Feingold codifies two absurdities: Large contributions are inherently evil, and political money can be limited without limiting political speech. McCain-Feingold criminalizes the sort of seed money that enabled McCarthy to be heard. Under McCain-Feingold's current limit of $2,100 per contributor, McCarthy's top five contributors combined could have given just $10,500, which in 1968 dollars would have been just $1,834.30. But, then, McCain-Feingold was written by incumbents to protect what they cherish: themselves.\n\nMcCarthy first seized national attention with a theatrical act, a gesture of elegant futility. At the 1960 convention, when John Kennedy's nomination was already certain, McCarthy delivered an eloquent philippic urging a third nomination for the man who had been trounced in 1952 and 1956, Adlai Stevenson.\n\nWitty, elegant, and problematic, Stevenson was the intelligentsia's darling and a harbinger of liberalism curdled by condescension toward ordinary Americans. When an aide assured Stevenson he had the votes of thinking people, Stevenson quipped: But I need a majority. A majority of the disdained?\n\nMcCarthy's acerbic wit sometimes slid into unpleasantness, as when, after Governor George Romney, the Michigan Republican, said that briefers in Vietnam had \"brainwashed\" him, McCarthy said that surely a light rinse would have sufficed. McCarthy's wit revealed an aptitude for condescension, an aptitude that charmed intellectuals but not Americans condescended to.\n\nA talented poet, McCarthy, in his mordant \"The Tamarack,\" surely summarized his experience of being beaten by Robert Kennedy after New Hampshire:\n\n_The tamarack tree is the saddest tree of all;_\n\n_it is the first tree to invade the swamp,_\n\n_and when it makes the soil dry enough,_\n\n_the other trees come and kill it._\n\nNever mind his subsequent lackadaisical presidential campaigns. After 1968, he adhered to the fourth of the commandments in his \"10 Commandments\":\n\n_Do not relight a candle_\n\n_whose flame has drowned_\n\n_in its own excess of wax._\n\n[DECEMBER 13, 2005]\n\n# **What George McGovern Made**\n\n**T** he former bomber pilot's spry walk belies his eighty-five years; he dresses like a _boulevardier_ \u2014gray slacks, blue blazer, shirt with bright-red stripes and a white collar\u2014and tucks into a robust breakfast. Long ago, he began shaping the Democrats' presidential nomination process into the one that has his party's two contenders locked in a long march to Pennsylvania's April primary. He has seen important aspects of American politics move in his direction in the thirty-six years since he lost forty-nine states to Richard Nixon.\n\nThe belittling of George McGovern, especially by Democrats, only waned as memory of him faded after he lost his bid for a fourth Senate term in the 1980 Reagan landslide. But his story is fascinating, and pertinent to current events.\n\nThis minister's son was raised on South Dakota's parched prairies during the Depression. He remembers hiking home to the town of Mitchell by following the railroad tracks in a blinding dust storm. He was only the second major-party nominee with a Ph.D. (Woodrow Wilson was the first), which he earned at Northwestern University under Arthur Link, Wilson's foremost biographer.\n\nLike Wilson, also a minister's son, McGovern was a political moralist. And he was a tenacious politician, who, inspired by the untenacious Adlai Stevenson's presidential campaign the year before, went to work for the South Dakota Democratic Party in 1953, when it held only 2 of 110 seats in the state legislature. Just four years later, McGovern was in Congress, where his first roll-call vote was in opposition to granting President Eisenhower broad authority for military intervention in the Middle East.\n\nIn tumultuous 1968, with the Tet Offensive and two assassinations (of Martin Luther King and Robert Kennedy) in five months, two insurgent candidates, Eugene McCarthy and Robert Kennedy, sought the Democratic nomination. It was won by Vice President Hubert Humphrey, _who competed in no primaries._ More than one-third of the delegates to the riotous convention in Chicago had been selected _in 1967,_ months before President Lyndon Johnson decided to retire.\n\nMcGovern was named chairman of a commission to reform the nomination process, which put the party on a path to the proliferation of caucuses and primaries allocating delegates proportionally rather than winner-take-all\u2014the long, winding path Obama and Clinton are on. In 1972, McGovern became the first winner under the democratized process. Then he was buried by the demos, Nixon vs. McGovern.\n\nNixon was, McGovern notes, running nationally for the fifth time (only FDR had done that) and was at his pre-Watergate apogee, fresh from the opening to China and a strategic-arms agreement with Moscow. McGovern was bitterly opposed all the way to the Miami convention by the Democratic constituencies he was displacing. He says Barry Goldwater had warned him, \"Don't get fatigued,\" but he reached Miami exhausted, lost control of the convention (he delivered his acceptance speech at 2:30 a.m.), and disastrously selected a running mate, Missouri senator Tom Eagleton, who did not disclose previous psychiatric problems and was forced off the ticket.\n\nStill, McGovern thinks he could have won with a running mate then called \"the most trusted man in America\"\u2014Walter Cronkite. Before choosing Eagleton, McGovern considered asking Cronkite, who recently indicated he would have accepted.\n\nBruce Miroff, a political scientist and an admirer of McGovern, argues in his new book, _The Liberals' Moment: The McGovern Insurgency and the Identity Crisis of the Democratic Party,_ that although McGovern's domestic proposals featured redistributions of wealth, this was Ivy League, not prairie, populism. Branded the candidate of \"acid, amnesty, and abortion\" (the Democrats' platform, adopted six months before the Supreme Court in _Roe v. Wade_ legislated a liberal abortion policy, did not mention abortion), McGovern became the first candidate since the New Deal to lose the Catholic and labor union vote. So 1972, more than 1968, was the hinge of the party's history. In 1972, Miroff writes, \"college-educated issue activists\" supplanted the \"labor\/urban machine coalition.\"\n\nGeorge Meany, head of the AFL-CIO, had dropped out of high school at age fourteen. Speaking about McGovern's 1972 convention, where 39 percent of the delegates had advanced degrees, he said: \"We heard from people who look like Jacks, acted like Jills, and had the odor of Johns about them.\" The Reagan Democrats of 1980 were incubated eight years earlier.\n\nMcGovern won only 14 percent of Southern white Protestants. This, Miroff notes, made Democrats susceptible four years later to the appeal of a pious Southerner. Thus did a disaster compound itself.\n\nIn September 1963, McGovern became the only senator who opposed U.S. involvement in Vietnam during the _Kennedy_ administration. He came by his horror of war honorably in thirty-five B-24 missions over Germany, where half the B-24 crews did not survive\u2014they suffered a higher rate of fatalities than did Marines storming Pacific islands. McGovern was awarded a Distinguished Flying Cross with three oak-leaf clusters. In his seventies, he lost a forty-five-year-old daughter to alcoholism. Losing a presidential election, he says softly, \"was not the saddest thing in my life.\" Time confers a comforting perspective, giving consolations to old age, which needs them.\n\n[FEBRUARY 25, 2008]\n\n# **Daniel Patrick Moynihan: The Senate's Sisyphus**\n\n**M** any of America's largest public careers have been those of presidents. Many, but by no means all. Chief Justice John Marshall was more consequential than all but two presidents\u2014Washington and Lincoln. Among twentieth-century public servants, General George Marshall\u2014whose many achievements included discerning the talents of a Colonel Eisenhower\u2014may have been second in importance only to Franklin Roosevelt. And no twentieth-century public career was as many-faceted, and involved so much prescience about as many matters, as that of Daniel Patrick Moynihan, who died Wednesday at seventy-six.\n\nHe was born in Tulsa but spent his formative years on Manhattan's Lower East Side, from which he rose to Harvard's faculty and the administrations of Presidents Kennedy, Johnson, Nixon, and Ford, serving as, among other things, ambassador to India and the U.S. representative at the United Nations. Then four Senate terms. Along the way he wrote more books than some of his colleagues read, and became something that, like Atlantis, is rumored to have once existed but has not recently been seen\u2014the Democratic Party's mind.\n\nHis was the most penetrating political intellect to come from New York since Alexander Hamilton, who, like Moynihan, saw over the horizon of his time, anticipating the evolving possibilities and problems of a consolidated, urbanized, industrial nation. A liberal who did not flinch from the label, he reminded conservatives that the Constitution's framers \"had more thoughts about power than merely its limitation.\"\n\nBut he was a liberal dismayed by what he called \"the leakage of reality from American life.\" When in 1994 the Senate debated an education bill, Moynihan compared the legislation's two quantifiable goals\u2014a high school graduation rate of \"at least 90 percent\" by 2000, and American students \"first in the world in mathematics and science\"\u2014to Soviet grain production quotas.\n\nThe Senate's Sisyphus, Moynihan was forever pushing uphill a boulder of inconvenient data. A social scientist trained to distinguish correlation from causation, and a wit, Moynihan puckishly said that a crucial determinant of the quality of American schools is proximity to the Canadian border. The barb in his jest was this: High cognitive outputs correlate not with high per-pupil expenditures but with a high percentage of two-parent families. For that, there was the rough geographical correlation that caused Moynihan to suggest that states trying to improve their students' test scores should move closer to Canada.\n\nFor calling attention, four decades ago, to the crisis of the African-American family\u201426 percent of children were being born out of wedlock\u2014he was denounced as a racist by lesser liberals. Today the percentage among all Americans is 33, among African-Americans 69, and family disintegration, meaning absent fathers, is recognized as the most powerful predictor of most social pathologies.\n\nAt the U.N., he witnessed that institution's inanity (as in its debate about the threat to peace posed by U.S. forces in the Virgin Islands, at that time fourteen Coast Guardsmen, one shotgun, one pistol) and its viciousness (the resolution condemning Zionism as racism). Striving to move America \"from apology to opposition,\" he faulted U.S. foreign policy elites as \"decent people, utterly unprepared for their work.\"\n\nTheir \"common denominator, apart from an incapacity to deal with ideas, was a fear of making a scene, a form of good manners that is a kind of substitute for ideas.\" Except they did have one idea, that \"the behavior of other nations, especially the developing nations, was fundamentally a reaction to the far worse behavior of the United States.\"\n\nMoynihan carried Woodrow Wilson's faith in international law, but he had what Wilson lacked\u2014an understanding that ethnicity makes the world go 'round. And bleed. The persistence of this premodern sensibility defeats what Moynihan called \"the liberal expectancy.\" He meant the expectation that the world would become tranquil as ethnicity and religion became fading residues of mankind's infancy.\n\nMoynihan's Senate campaigns were managed by as tough-minded and savvy a pol as New York's rough-and-tumble democracy has ever produced, a person who also is a distinguished archaeologist\u2014his wife, Elizabeth. In his first campaign, in 1976, Moynihan's opponent was the incumbent, James Buckley, who playfully referred to \" _Professor_ Moynihan\" from Harvard. Moynihan exclaimed with mock indignation, \"The mudslinging has begun!\"\n\nHis last home was an apartment on Washington's Pennsylvania Avenue. That \"Avenue of Presidents\" was transformed from tattiness to majesty and vibrancy by three decades of his deep reflection about, and persistent insistence on, proper architectural expressions of the Republic's spiritedness and reasonableness, virtues made wonderfully vivid in the life of Daniel Patrick Moynihan.\n\n[MARCH 27, 2003]\n\n# **John Kenneth Galbraith's Liberalism as Condescension**\n\n**J** ohn Kenneth Galbraith, the Harvard economist who died last week in his ninety-eighth year, has been justly celebrated for his wit, fluency, public-spiritedness, and public service, which extended from New Deal Washington to India, where he served as U.S. ambassador. Like two Harvard colleagues\u2014historian Arthur Schlesinger Jr. and Senator Pat Moynihan, another ambassador to India\u2014Galbraith was among liberalism's leading public intellectuals, yet he was a friend and skiing partner of William F. Buckley. After one slalom down a Swiss mountain, inelegantly executed by the six-foot-eight Galbraith, Buckley asked how long Galbraith had been skiing. Thirty years, he said. Buckley mischievously replied: About as long as you have been an economist.\n\nGalbraith was an adviser to presidents (John Kennedy, a former student, and Lyndon Johnson) and presidential aspirants (Adlai Stevenson and Eugene McCarthy). His book _The Affluent Society,_ published in 1958, was a milestone on liberalism's transformation into a doctrine of condescension. And into a minority persuasion.\n\nIn the 1950s, liberals were disconsolate. Voters twice rejected the intelligentsia's pinup, Stevenson, in favor of Dwight Eisenhower, who elicited a new strain in liberalism\u2014disdain for average Americans. Liberals dismissed the Eisenhower administration as \"the bland leading the bland.\" They said New Dealers had been supplanted by car dealers. How to explain the electorate's dereliction of taste? Easy. The masses, in their bovine simplicity, had been manipulated, mostly by advertising, particularly on television, which by 1958 had become the masses' entertainment.\n\nIntellectuals, that herd of independent minds, were, as usual, in lockstep as they deplored \"conformity.\" Fear of that had begun when the decade did, with David Riesman's _The Lonely Crowd_ (1950), which was followed by C. Wright Mills's _White Collar_ (1951), Sloan Wilson's novel _The Man in the Gray Flannel Suit_ (1955), William Whyte's _The Organization Man_ (1956), and Vance Packard's _The Hidden Persuaders_ (1957).\n\nGalbraith brought to the anticonformity chorus a special verve in depicting Americans as pathetic, passive lumps, as manipulable as clay. Americans were what modern liberalism relishes\u2014victims, to be treated as wards of a government run by liberals. It never seemed to occur to Galbraith and like-minded liberals that ordinary Americans might resent that depiction and might express their resentment with their votes.\n\nAdvertising, Galbraith argued, was a leading cause of America's \"private affluence and public squalor.\" By that he meant Americans' consumerism, which produced their deplorable reluctance to surrender more of their income to taxation, trusting government to spend it wisely.\n\nIf advertising were as potent as Galbraith thought, the advent of television\u2014a large dose of advertising, delivered to every living room\u2014should have caused a sharp increase in consumption relative to savings. No such increase coincided with the arrival of television, but Galbraith, reluctant to allow empiricism to slow the flow of theory, was never a martyr to Moynihan's axiom that everyone is entitled to his own opinion but not to his own facts.\n\nAlthough Galbraith coined the phrase \"conventional wisdom,\" and thought of himself as the scourge of groupthink, _The Affluent Society_ was the distilled essence of the conventional wisdom on campuses. In the 1960s, that liberalism became a stance of disdain, describing Americans not only as Galbraith had, as vulgar, but also as sick, racist, sexist, imperialist, etc. Again, and not amazingly, voters were not amused when told that their desires\u2014for big cars, neighborhood schools, and other things\u2014did not deserve respect.\n\nBut for liberals that was precisely the beauty of Galbraith's theory. If advertising could manufacture demands for whatever corporations wanted to supply, there was no need to respect markets, which bring supply and demand into equilibrium.\n\n_The Affluent Society_ was the canonical text of modern liberalism's disparagement of the competence of the average American. This liberalism\u2014the belief that people are manipulable dolts who need to be protected by their liberal betters from exposure to \"too much\" advertising\u2014is one rationale for McCain-Feingold. That law regulating campaigns embodies the political class's belief that it knows just the right amount of permissible political speech.\n\nOf course if advertising really could manufacture consumer wants willy-nilly, few new products would fail. But many do. _The Affluent Society,_ postulating the awesome power of manufacturers to manufacture whatever demand they find it convenient to satisfy, was published nine months after Ford Motor Company put all of its marketing muscle behind a new product, the Edsel.\n\nSmall wonder that a conservative wit has surmised that the wisdom of economists varies inversely with their heights. Milton Friedman, ninety-three, is five feet tall.\n\n[MAY 4, 2006]\n\n# **Milton Friedman: Ebullient Master of the Dismal Science**\n\n**A** t Oxford University in 1962, a small coterie of students, mostly Americans, merrily rowed against the leftist political currents predominant among intellectuals everywhere. Some of these rowers had been at the University of Chicago, others had come within the ambit of people from there, and all of us were infused with the doctrines of laissez-faire political economy prevalent in that university's economics department.\n\nA conservative member of Parliament, meeting with these free-market firebrands, began by saying, \"Well, presumably we agree that at least the roads should be owned by the government.\" The group greeted with stony silence this heresy against limited\u2014 _very_ limited\u2014government.\n\nIn one of the group's favorite periodicals\u2014the _New Individualist Review,_ published at the University of Chicago\u2014a theorist argued that the government must own lighthouses because no market mechanism could price a lighthouse's service. That provoked this spirited rebuttal: When the light sweeps the ocean's surface, it improves the surface, which becomes the property of the lighthouse owner, who can charge whatever the market will bear for ships to cross the illuminated surface.\n\nAh, but does the property right lapse when fog obscures the beam of light? Hairs were split as ideological purity hung in the balance.\n\nThese contumacious students were, as students frequently are, inebriated by ideas to the point of silliness. But they were early acolytes of the extraordinary man who, as he celebrates his ninetieth birthday this month, merits celebration as America's most consequential public intellectual of the twentieth century.\n\nBy 1962, when he published his great manifesto _Capitalism and Freedom,_ Milton Friedman, then a University of Chicago economist, had done much of the scholarly work for which he would receive the 1976 Nobel Prize in economics. He has been the foremost champion of \"monetarism,\" the theory that money supply and interest rates can do more than government fiscal policy (\"demand management\" and other Keynesian fine-tuning measures) to control business cycles. The theory that stable growth of the money supply can control inflation and moderate recessions is an important ingredient in the recipe for modest government.\n\n_Capitalism and Freedom_ inserted into political discourse such (then) novel ideas as flexible exchange rates, a private dimension of Social Security, tuition vouchers to empower parents with school choice, and a flat income tax. Gary Becker (Nobel Prize, 1992), Friedman's colleague at the University of Chicago and the Hoover Institution, notes that when Friedman began arguing the case, most nations had top tax rates of at least 90 percent (91 percent in America). Today most top rates are 50 percent or less, so the world has moved far toward Friedman's position.\n\nFriedman was a charter member of the most influential society you have never heard of\u2014the Mont Pelerin Society, named after the Swiss community where this association of laissez-faire thinkers first gathered in 1947. Its animating spirit was Friedrich Hayek, soon to be at the University of Chicago. In 1955, Hayek prompted the founding of the like-minded Institute of Economic Affairs in London, which around 1962 caught the attention of a junior member of Parliament who seventeen years later brought Friedman's monetarism and respect for markets into No. 10 Downing Street\u2014Margaret Thatcher.\n\nMany intellectuals disdain the marketplace because markets function nicely without the supervision of intellectuals. Their disdain is ingratitude: The vulgar (as intellectuals see them) people who make markets productive make the intellectual class sustainable. As another of Friedman's Chicago colleagues, George Stigler (Nobel Prize, 1982), says, \"Since intellectuals are not inexpensive, until the rise of the modern enterprise system, no society could afford many intellectuals.\" So \"we professors are much more beholden to Henry Ford than to the foundation which bears his name and spreads his assets.\"\n\nEconomics is not the \"dismal science\" when infused with Friedman's ebullient spirit and expressed in his sprightly prose. So as President George W. Bush said in May when honoring Friedman, it was fortunate for the nation and the world that Friedman \"flunked some of his qualifying exams to become an actuary and became an economist instead.\"\n\nJohn Maynard Keynes, whose preeminence among economists Friedman eclipsed, said the world is mostly ruled by the ideas of economists and political philosophers: \"Practical men, who believe themselves to be quite exempt from any intellectual influences, are usually the slaves of some defunct economist.\" But Friedman is far from defunct as he strides jauntily into his tenth decade, still an intellectual dynamo.\n\nAdam Smith, whose banner Milton Friedman has borne high, said, \"There is much ruin in a nation.\" There is much less of it in ours than there would have been were it not for Milton Friedman.\n\n[JULY 14, 2002]\n\n# **Alan Greenspan: High-Achieving Minimalist**\n\n**F** requently the fate\u2014gratifying, yet melancholy\u2014of consequential public persons is this: They so transform an ominous social landscape that, by the time they leave the public stage, the public no longer remembers the banished dangers, and hence cannot properly value the banisher. So as Alan Greenspan heads to the end, in January, of more than eighteen years as head of the Federal Reserve, recall that thirty years ago the intelligentsia worried that democracies, including this one, had become \"ungovernable.\"\n\nThe worrying was caused by inflation, then thought to be the systemic disease of democracies. The theory was that democratic electorates would reward governments that delivered the pleasure of public spending and would punish those that inflicted the pain of taxation sufficient to pay for that spending. Hence democracies would run chronic deficits. These, it was assumed, both caused inflation and gave government a powerful incentive to tolerate inflation as a means of steadily reducing the real value of its debts\u2014inflation as slow-motion repudiation.\n\nFurthermore, deficit spending\u2014giving the public a dollar's worth of government goods and services and charging the public only, say, eighty cents for them\u2014produced big government and, by making big government inexpensive, reduced public resistance to making it even bigger. And because of affluent voters' low and steadily lowering pain thresholds, democracies would not tolerate the discomforts associated with wringing inflation out of the economy.\n\nThat supposition was slain by a fact: President Reagan and Paul Volcker, Greenspan's predecessor, put the country through the rigors of wringing inflation out of the economy, and in 1984 Reagan carried forty-nine states. Between 1945 and 1982 the economy was in recession 22.4 percent of the time. In the 276 months since the recession ended in 1982, it has been in recession fourteen months\u2014just 5.1 percent of the time.\n\nBecause of Americans' low pain threshold, the Reagan-Volcker recession was considered hideous. It was the worst since the Depression, but the economy contracted less than 3 percent. In the years between 1890 and 1945, America's period of hard learning about managing an industrial economy, three times there were contractions of 5 percent, twice contractions of 10 percent, and twice contractions of 15 percent.\n\nSince 1945, and especially since 1982, we have learned the real secret of managing the economy: Do not try to manage it. If you refrain from trying to \"fine-tune\" business cycles, the cycles will be less frequent and less severe.\n\nGreenspan's tenure has illustrated an axiom to which his successor, Ben Bernanke, should subscribe: Minimalist missions by government produce maximum results. He has not defined the Fed's primary purpose as achieving this or that level of employment or economic growth. Rather, its mission is to preserve the currency as a stable store of value\u2014to control inflation. However, Greenspan's impeccable credentials as an inflation fighter have enabled him to keep inflation rates low even during very low unemployment without kindling inflationary expectations, which can be self-fulfilling.\n\nAmerica's economy is so dynamic that in any five-year period, approximately 45 percent of Americans move from one income quintile to another. Twenty percent move up from the bottom quintile in any twelve-month period, and 40 percent to 50 percent move up over ten to twenty years. Because of the constant transformation of dynamic economies, the study of economics has become a science of single instances. Its practitioners are constantly in uncharted waters, reasoning inferentially. Just as astronomers inferred the existence of Pluto from the behavior of known planets, Greenspan inferred a rate of productivity growth higher than most estimates because inflation and unemployment were falling simultaneously.\n\nThe Federal Reserve system\u2014to give the devil his due, it is one of Woodrow Wilson's unregrettable undertakings\u2014annoys some populists who think every U.S. senator and representative should write on his or her bathroom mirror, and read every morning, this thought: \"The Fed is a creature of Congress.\" Indeed, Congress made it and could dictate to it\u2014could dictate interest rates and the money supply. A terrifying thought.\n\nGreenspan's famously, at times hilariously, circumspect rhetoric has been prudent because some word, or inflection, or even arched eyebrow could have caused vast sums to slosh in this or that direction in capital markets. His rhetorical style\u2014or perhaps antistyle\u2014is a high-stakes illustration of Voltaire's idea that men use speech to conceal their thoughts.\n\nGreenspan's wife has said he had to propose marriage three times before she understood what he was saying. And he was being droll when he said\u2014if he said it; apocrypha collect around legends\u2014that \"if I have made myself clear I have misspoken.\" His achievements speak clearly for him.\n\n[OCTOBER 25, 2005]\n\n# **The Not-at-All Dull George Washington**\n\n**T** onight, after you have given up trying to get the mustard stains off the dog, and after you have treated the fingers singed by sparklers, pour a beer\u2014a Sam Adams would be apposite\u2014and settle down to watch on PBS the documentary biography of the man most responsible for there being an Independence Day. Ninety minutes later, Richard Brookhiser's _Rediscovering George Washington_ will have convinced you that its subject, whom many historians have managed to mummify into dullness, may have been the most interesting and indispensable American.\n\n\"Indispensable\"? Advanced thinkers instruct us that we are not supposed to believe anyone is more important than anyone else. Not democratic. We are supposed to prefer \"history from below,\" meaning \"history with the politics left out,\" explaining the past not with reference of event-making individuals, but in terms of the holy trinity of today's obsessions\u2014race, gender, class.\n\nBut Brookhiser begins his film standing at Yale in front of a portrait of Washington and says: \"An empire might break on that forehead.\" Then he explains why an empire did: Washington's character.\n\nCharacter, particularly that of someone dead two centuries, is difficult for a camera to capture. Besides, Washington's army, unlike Caesar's or Napoleon's, lost more battles than it won. It is said that America won the war because of its superior retreats, such as the one after the British landed an army on Long Island\u2014an army larger than the population of New York City. Still, Brookhiser's script summons Washington to life.\n\nBrookhiser's camera takes us to the Brooklyn scene\u2014now an auto body shop\u2014where brave Marylanders enabled Washington to escape to fight another day. Washington's next defeat occurred at what is now Thirty-fourth and Lexington Avenue. Brookhiser refutes the myth that Washington's troops usually fought \"Indian style,\" in forest skirmishes and ambushes. As in most eighteenth-century battles, close combat\u2014the bayonet\u2014caused most casualties.\n\nWashington understood that the creation of a nation depended on creating a regular army that could slug it out with Britain's. The tide began to turn in New Jersey, at Monmouth.\n\nThe most powerful person in American history, Washington had less formal education than any subsequent president, other than Lincoln. This six-foot-three leader of soldiers who averaged five-foot-eight was charismatic before the term was coined. Abigail Adams, no swooning teenager, described Washington with lines from Dryden: \"Mark his majestic fabric. He's a temple sacred from his birth and built by hands divine.\"\n\nHe was not the only one of that era who learned showmanship from studying theater and popular entertainments such as Punch and Judy puppet shows. Joseph Addison's play _Cato_ was the source of Nathan Hale's dying words (\"I only regret that I have but one life to lose for my country\") and Patrick Henry's \"Give me liberty, or give me death!\" In the play, Cato held mutinous officers in line by force of personality, as Washington was to do at Newburgh, New York.\n\nBrookhiser, a senior editor of _National Review_ magazine and frequent contributor to other magazines and to C-SPAN, is also known for his slender, sprightly biographies of Washington, Alexander Hamilton, the Adams dynasty. The longest, on Hamilton, is just 240 pages long. His next subject will be Gouverneur Morris. Brookhiser's capsule summary of Morris is characteristically pithy: \"Peg-legged ladies' man who polished the Constitution's language.\"\n\nBrookhiser does not write \"pathographies\"\u2014biographies that present historic figures as the sum of their pathologies. His Washington adopted a noble character, then grew into it. Intensely interested in manners, Washington pioneered a civic etiquette suitable for a democracy in which preeminence was to be based on behavior, not birth.\n\nAnd of the nine presidents who owned slaves, only Washington freed his at his death. Brookhiser visits a family reunion of descendants of those slaves\u2014one of whom is a guide at Mount Vernon. Congress wanted Washington's body to rest in a room beneath the Capitol rotunda, like Napoleon's in the Invalides. But Washington's remains are at Mount Vernon. So, Brookhiser says:\n\n\"The Capitol is not his tomb but the people's house. This reflects his wishes and the goal of his life. Washington wanted to establish a government that would prove that mankind was not made for a master\u2014not even the mastership of a hero's memory.\"\n\n\"They wanted me to be another Washington,\" whined Napoleon in his exile, as stunned as the rest of the world by Washington's voluntary yielding of power. The final component of Washington's indispensability was the imperishable example he gave by proclaiming himself dispensable.\n\n[JULY 4, 2002]\n\n# **George Washington's Long Journey Home**\n\n**H** is goal, 220 years ago, was to sleep Christmas Eve in his six-foot six-inch bed at his Virginia home on the bank of the Potomac. It would be nicer than some other recent Christmas Eves.\n\nSuch as in 1776, when he led soldiers across Delaware River ice floes to one of his greatest\u2014and, truth be told, relatively few\u2014victories, at Trent Town, as Trenton was then known. Only four Americans died that night, two\u2014probably shoeless\u2014from frostbite.\n\nGeorge Washington spent Christmas Eve 1777 with an army leaving bloody footprints in the Valley Forge snow. Six years later, he was heading to a home he had left in 1775 to lead farmers and shopkeepers against the British Empire.\n\nSince Yorktown, Washington, like his embryonic nation, had lived in a peculiar limbo as negotiators, two months' travel away in Paris, codified peace with Britain. In late November, from headquarters along the Hudson River north of Manhattan island, he began his trek from strenuous public service into a placid future of private enjoyments, or so he thought.\n\nHis journey was through a nation deep in the throes\u2014it would be in them for many years\u2014of regime change. To the extent that there was a national regime, he was it, and he was retiring.\n\nIn January 1783, Congress had fled Philadelphia, going to ground in Princeton, New Jersey, to escape a mutiny of unpaid soldiers. Congress was a place of empty palaver by representatives of states that retained virtually untrammeled sovereignty. By July 1783, with Congress sitting in Annapolis, only South Carolina\u2014seven decades later, it would be the least cooperative state\u2014had paid its full assessment to the national treasury. Of the other twelve, only Washington's Virginia had contributed half its quota. The two weightiest states, Pennsylvania and New York, had contributed one-fifth and one-twentieth, respectively.\n\nWhat united the barely united states was six feet three inches of American in a blue coat and buff trousers, carrying a sword and buckle engraved \"1757\" that testified to his frontier service for the British against the French, whose fleet, twenty-four years later, sealed the victory at Yorktown. If on his trip home this fifty-one-year-old man had caught a chill and died, as he would do sixteen Decembers later, national unity might have been unattainable.\n\nThe story of his triumphal trip home, itself an act of nation building, is well told by historian Stanley Weintraub in his new book _General Washington's Christmas Farewell: A Mount Vernon Homecoming, 1783._ It evokes the frail seedling from which the mighty American nation grew. In a seven-year (1775\u20131781) war in which fewer than forty-five hundred American soldiers died in combat, Washington lost more battles than he won. But he won the battle that mattered most\u2014the last one\u2014and adulation unlike any ever bestowed on an American.\n\nHis homeward journey paused at Harlem, a Manhattan village nine miles north of New York City, a community of twenty-one thousand on the island's southern tip that Washington had never captured. As Washington's party entered the city, Loyalist emigrants were being ferried to departing British ships in the harbor. A British officer marveled:\n\n\"Here, in this city, we have had an army for more than seven years, and yet could not keep the peace of it. Scarcely a day or night passed without tumults. Now we are gone, everything is in quietness and safety. These Americans are a curious, original people; they know how to govern themselves, but nobody else can govern them.\"\n\nThen it was four days to Philadelphia, passing along what is now U.S. Route 1 through difficult New Jersey. In 1776, Washington had urged Jerseymen in the village of Newark to join his cause. Thirty did\u2014but three hundred joined the British. In Annapolis, he surrendered his commission after a ball at which, Weintraub reports, fashionable ladies wore their hair in the _dress \u00e0 l'independence_ \u2014thirteen curls at the neck.\n\nWashington's journey to Mount Vernon, which he reached after dark, December 24, was a movable feast of florid rhetoric and baked oysters. It also was a foretaste of what was to be, for more than a century, his central place in America's civic liturgy. Abraham Lincoln wore a ring containing a sliver from the casket Washington was buried in until his body was moved to its current tomb in 1831. At his inauguration in 1897, William McKinley wore a ring containing strands of Washington's hair. Presidents no longer inspire such reverence, perhaps because America is different, perhaps because presidents are.\n\n[DECEMBER 25, 2003]\n\n# **John Marshall: The Most Important American Never to Have Been President**\n\n**A** nation's identity consists of braided memories, which are nourished by diligence at civic commemorations. It is, therefore, disappointing that at this moment of keen interest in the Supreme Court and the office of chief justice, scant attention has been paid to the 250th anniversary of the birth of the nation's greatest jurist, Chief Justice John Marshall.\n\nThe oldest of the family's fifteen children, he was born September 24, 1755, into Virginia rusticity where women pinned their blouses with thorns. Yet he developed the most urbane and subtle mind of that era of remarkable statecraft. He was a member of Virginia's ratifying convention, and in nearly thirty-five years as chief justice he founded American constitutional law. That kind of legal reasoning by Supreme Court justices is a continuous exegesis of the Constitution and is sometimes not easily distinguished from a continuing writing of the document.\n\nMarshall is the most important American never to have been president. Because of his shaping effect on the soft wax of the young republic, his historic importance is greater than that of all but two presidents\u2014Washington and Lincoln. Without Marshall's landmark opinions defining the national government's powers, the government Washington founded might not have acquired competencies\u2014and society might not have developed the economic sinews\u2014sufficient to enable Lincoln to preserve the Union.\n\nArticle I, Section 8, enumerates Congress's powers, and then empowers Congress \"to make all laws which shall be necessary and proper for carrying into execution the foregoing powers.\" Marshall's capacious construction of the \"necessary and proper\" clause shaped the law, and the nation's consciousness of itself.\n\nDid Congress have the power\u2014unenumerated but implied\u2014to charter a national bank? In 1819, forty-two years before Lincoln grappled with unprecedented exigencies, Marshall ruled:\n\n\"Throughout this vast republic, from the St. Croix to the Gulph of Mexico, from the Atlantic to the Pacific, revenue is to be collected and expended, armies are to be marched and supported. The exigencies of the nation may require that the treasure raised in the north should be transported to the south.... Is that construction of the constitution tobe preferred which would render these operations difficult, hazardous, and expensive?\"\n\nTwo years later he held that \"we are one people\" in war, in making peace, and\u2014third, but not of tertiary importance\u2014in \"all commercial regulations.\" The Framers' fundamental task was to create a federal government with powers impervious to encroachments by the states. The Framers had been frightened by the states' excesses in using political power on behalf of debtors against creditors and to limit competition by mercantilistic practices such as granting monopolies. Marshall made constitutional law a bulwark of the sanctity of contracts, the bedrock of America's enterprise culture. And by protecting the private rights essential to aspirational individualism, Marshall's court legitimized an inequality\u2014not of opportunity but of outcomes\u2014compatible with a republic's values.\n\nWhen in 1801 Marshall was nominated to be chief justice\u2014one of the last things, and much the best thing, President John Adams did\u2014the nation still largely had an Articles of Confederation mentality. Formally, it was a nation; emotionally\u2014hence, actually\u2014it was still in many ways many countries, most states being older than and more warmly embraced than the nation. Marshall's jurisprudence built the bridge to 1862, the year it became clear that many men would have to die in a protracted conflict to preserve the Union, and that many would be willing to do so.\n\nMarshall had been willing to die to help midwife the nation's birth, seeing much hard action during the Revolutionary War. Amiably sociable and broadly tolerant, he had friends of vastly different political persuasions, and the only adversary he seems to have steadily disliked was a second cousin named Thomas Jefferson, in part because of Jefferson's partisan criticisms of Washington, who Marshall celebrated, in an immense biography, as the symbol of a national identity transcending state loyalties.\n\nAmong the many recent fine biographies of America's Founders, none is finer than Jean Edward Smith's _John Marshall: Definer of a Nation_ (1996). Smith locates Marshall's greatness in this fact: Unlike Britain's constitutional documents, which are political documents that it is Parliament's prerogative to construe, the U.S. Constitution is a legal document construed by courts, not Congress. When judicial supervision of our democracy seems tiresome, consider the alternative.\n\nMarshall's life of strong, consequential prose had, Smith writes, a poetic coda. Marshall died in Philadelphia, birthplace of the Constitution into which he breathed so much strength and meaning. The Liberty Bell, while tolling his death, cracked. It never rang again.\n\n[SEPTEMBER 25, 2005]\n\n# **James Madison: Well, Yes, of Course**\n\nIn 2007, a _Princeton Alumni Weekly_ panel voted\u2014unanimously\u2014that James Madison was the university's most influential graduate. I was asked to write something about that for the magazine.\n\n**I** am writing this wee tribute to the greatest Princetonian on a morning which began, as most of my mornings do, with a predawn walk accompanied by my dog. His name is Madison. I am wearing my favorite necktie. It is blue, with silver profiles of James Madison. Later this morning, I shall work on a book I am writing. It is to be titled \"The Madisonian Persuasion.\" I am not one who needs to be persuaded that Madison merits being ranked as Princeton's greatest gift to the nation.\n\nBefore I turned to journalism\u2014or before I _sank_ to journalism, as my father, a professor of philosophy, put it\u2014I was, briefly, a professor of political philosophy. I cheerfully accepted that I would never be nearly as original and consequential as the philosophic Madison had been. Then I became a newspaper columnist, in which role I have always known that I could never be nearly as original and consequential as was Madison, America's foremost columnist.\n\nThe Federalist Papers, of which Madison wrote the two most important, were, of course, columns written to advance the ratification of the Constitution, in the drafting of which Madison was the most subtle participant. If a student of American thought fully unpacks the premises and implications of Federalist 10 and 51, that student comprehends not only this nation's political regime but also the Madisonian revolution in democratic theory.\n\nBefore Madison, almost all political philosophers who thought about democracy thought that if\u2014a huge if, for most of them\u2014democracy were to be feasible, it would be so only in a small, face-to-face society, such as Pericles' Athens or Rousseau's Geneva. This was supposedly true because the bane of democracies was thought to be self-interested factions, and only a small society could be sufficiently homogenous to avoid ruinous factions.\n\nBut America in the second half of the eighteenth century, although small compared with what it would become, was in size already a far cry from a Greek polis. Besides, Americans had spacious aspirations. A small nation? They were having none of that. At a time when 80 percent of them lived on a thin sliver of the eastern fringe of the continent, within twenty miles of Atlantic tidewater, what did they call their political assembly? The Continental Congress. They knew, more or less, where they were going: California.\n\nMadison understood the need for philosophic underpinnings for an \"extensive republic,\" a phrase that seemed oxymoronic to others. He can be said to have had a political catechism, which went approximately like this:\n\nWhat is the worst outcome of politics? Tyranny.\n\nTo what form of tyranny are democracies susceptible? The tyranny of a single, durable majority.\n\nHow can this threat be minimized? By a saving multiplicity of factions, so that majorities will be unstable and transitory.\n\nHence, in Federalist 10, he wrote that \"the first object of government\" is \"the protection of different and unequal faculties of acquiring property.\" From these differences arise different factions in their freedom-preserving multiplicity.\n\nHaving said in Federalist 10 that \"neither moral nor religious motives can be relied on as an adequate control\" of factions, Madison turned, in Federalist 51, to the institutional controls established in the Constitution\u2014\"this policy of supplying, by opposite and rival interests, the defect of better motives\":\n\n\"Ambition must be made to counteract ambition.... It may be a reflection on human nature, that such devices should be necessary to control the abuses of government. But what is government itself but the greatest of all reflections on human nature? If men were angels, no government would be necessary. If angels were to govern men, neither external nor internal controls on government would be necessary.... You must first enable the government to control the governed; and in the next place, oblige it to control itself.\" In a masterpiece of understatement, Madison said, \"Enlightened statesmen will not always be at the helm.\" No kidding. And Madison did not mince words regarding those about whom no one nowadays dares to say a discouraging word\u2014\"the people.\"\n\n\"There is,\" he said, \"a degree of depravity in mankind which requires a certain degree of circumspection and distrust.\" Let the record show that once we had a president who had spoken of the voters' depravity. Those were the days. Madison did qualify his astringent judgment about the people by acknowledging that there \"are other qualities in human nature which justify a certain portion of esteem and confidence.\" But notice the carefully measured concession to the public's sensibilities: \"a certain portion,\" indeed.\n\nIn 1976, the nation's bicentennial, the presidency was won by someone who, pandering in the modern manner, promised to deliver government \"as good as the people themselves.\" One can imagine Madison muttering, \"Good grief!\"\n\nIn Washington, the seat of the government that he did so much to design, there is no monument to Madison comparable to the glistening marble temples honoring Jefferson and Lincoln. There is, however, a splendid Madison Building, which is part of the Library of Congress. It was said of the five-foot-four Madison that he contained an astonishing ratio of mind to mass. So it is altogether right that Madison, physically the smallest of the Founders, is honored in his nation's capital by a repository of learning.\n\nMy next dog, if female, will be named Dolley.\n\n[JANUARY 23, 2008]\n\n# **Longfellow: A Forgotten Founder**\n\n**O** ne hundred years ago, February 27 was enlivened by events around the nation commemorating what had happened one hundred years before that, in 1807. But last week's bicentennial of the birth of Henry Wadsworth Longfellow passed largely unnoted, which is noteworthy. It was, naturally, a poet (Shelley) who declared that \"poets are the unacknowledged legislators of the world.\" Wishful thinking, that, but Plato took poets so seriously as disturbers of the peace that he wanted them expelled from his republic. And Longfellow was, in a sense, an American Founder, a maker of this Republic's consciousness.\n\nTime was, children learned\u2014in schools; imagine that\u2014the origins of what still are familiar phrases: \"Ships that pass in the night,\" \"Life is real! Life is earnest!\" \"footprints on the sands of time,\" \"the patter of little feet,\" \"the forest primeval,\" \"Let the dead Past bury its dead!\" \"In this world a man must either be anvil or hammer,\" \"Into each life some rain must fall.\" Even the first stanza of Longfellow's serene \"The Village Blacksmith\"\u2014\n\n_Under the spreading chestnut tree,_\n\n_The village smithy stands_\n\n\u2014has a haunting, sinister echo in George Orwell's _1984._ Winston Smith, distraught, thinks he hears a voice singing\n\n_Under the spreading chestnut-tree_\n\n_I sold you and you sold me._\n\nLongfellow was a gifted versifier, and today is dismissed as only a versifier. Well, as C\u00e9zanne supposedly said of Monet, \"He is only an eye\u2014but what an eye!\"\n\nLongfellow was very Victorian\u2014sentimental and moralistic. He in no way foreshadowed twentieth-century poetry's themes of meaninglessness (\"I have measured out my life with coffee spoons\"\u2014T. S. Eliot, 1917) and social disintegration (\"the blood-dimmed tide is loosed\"\u2014William Butler Yeats, 1921). Longfellow wrote for a young nation that was thinking \"Let us, then, be up and doing, with a heart for any fate,\" before he wrote that exhortation.\n\nHe aimed to shape the nation's identity by making Americans aware of the first European settlers (\"Why don't you speak for yourself, John?\"\u2014\"The Courtship of Miles Standish\"), the Native Americans they displaced (\"By the shore of Gitche Gumee\"\u2014\"The Song of Hiawatha\") and the nation's birth (\"Listen, my children, and you shall hear \/ Of the midnight ride of Paul Revere\").\n\n\"Paul Revere's Ride\" was written in 1860, as events were mocking Longfellow's great national poem (\"The Building of the Ship,\" 1849):\n\n_Cedar of Maine and Georgia pine_\n\n_Here together shall combine._\n\n_A goodly frame, and a goodly fame,_\n\n_And the UNION be her name!_\n\n. . .\n\n_Sail on, O UNION, strong and great!_\n\n_Humanity with all its fears,_\n\n_With all the hopes of future years,_\n\n_Is hanging breathless on thy fate!_\n\nLongfellow's civic purposes made him a public figure, the nation's first literary celebrity. His image decorated cigar boxes and beer-bottle labels. He kept a supply of autographed cards for the many strangers who made pilgrimages to his Cambridge house, where George Washington had lived during the siege of Boston.\n\nNot long ago there still were celebrity poets. Jeffrey Hart, professor emeritus of English at Dartmouth, in his _When the Going Was Good: American Life in the Fifties,_ remembers Robert Frost's receiving a standing ovation from an overflow house at Carnegie Hall, and Eliot's reading his poems to an overflow audience at Columbia University, with people outside listening to him over loudspeakers.\n\nThe audiences were intense because the issues were large, if abstruse. Frost and Eliot represented dueling sensibilities, the empirical and the transcendental. In contrast, Longfellow intended his narrative and lyric poems\u2014genres disdained by modernists\u2014as inspiriting guides to the nation's honorable past and challenging future. Yeats ascribed Longfellow's popularity to his accessibility\u2014\"he tells his story or idea so that one needs nothing but his verses to understand it.\" This angers today's academic clerisy. What use is it to readers who need no intermediary between them and the author? And what use is Longfellow to academics who \"interrogate\" authors' \"texts\" to illuminate the authors' psyches, ideologies, and social situations\u2014the \"power relations\" of patriarchy, racism, imperialism, etc.? This reduction of the study of literature to sociology, and of sociology to ideological assertion, demotes literature to mere raw material for literary theory, making today's professoriate, rather than yesterday's writers, the center of attention.\n\nDana Gioia, chairman of the National Endowment of the Arts, has written that \"Longfellow's vast influence on American culture paradoxically makes him both central and invisible.\" The melancholy fact that the two hundredth birthday of the poet who toiled to create the nation's memory passed largely unremarked is redundant evidence of how susceptible this forward-leaning democracy is to historical amnesia.\n\n[MARCH 12, 2007]\n\n# **Ronald Reagan: The Steel Behind the Smile**\n\n**O** ne measure of a leader's greatness is this: By the time he dies, the dangers that summoned him to greatness have been so thoroughly defeated, in no small measure by what he did, it is difficult to recall the magnitude of those dangers, or of his achievements. So if you seek Ronald Reagan's monument, look around, and consider what you do _not_ see.\n\nThe Iron Curtain that scarred a continent is gone, as is the Evil Empire responsible for it. The feeling of foreboding\u2014the sense of shrunken possibilities\u2014that afflicted Americans twenty years ago has been banished by a new birth of the American belief in perpetually expanding horizons.\n\nIn the uninterrupted flatness of the Midwest, where Reagan matured, the horizon beckons to those who would be travelers. He traveled far, had a grand time all the way, and his cheerfulness was contagious. It was said of Dwight Eisenhower\u2014another much-loved son of the prairie\u2014that his smile was his philosophy. That was true of Reagan, in this sense: He understood that when Americans have a happy stance toward life, confidence flows and good things happen. They raise families, crops, living standards, and cultural values; they settle the land, make deserts bloom, destroy tyrannies.\n\nReagan was the last president for whom the Depression\u2014the years when America stopped working\u2014was a formative experience. Remarkably, the 1930s formed in him a talent for happiness. It was urgently needed in the 1980s, when the pessimism of the intelligentsia was infecting people with the idea that America had passed its apogee and was ungovernable.\n\nIt also was said then that the presidency destroyed its occupants. But Reagan got to the office, looked around, said, \"This is fun. Let's saddle up and go for a ride.\" Which he did, sometimes in the middle of the afternoon. Scolds, who thought presidents were only serious when miserable, were scandalized.\n\nIn an amazingly fecund twenty-seven-month period, Margaret Thatcher, Pope John Paul II, and Reagan came to office. The pope and the president had been actors. Reagan said he wondered how presidents who have _not_ been actors could function. Certainly the last century's greatest democratic leaders\u2014Churchill, FDR\u2014mastered the theatrical dimension of politics.\n\nGood actors, including political actors, do not deal in unrealities. Rather, they create realities that matter\u2014perceptions, aspirations, allegiances. Reagan in his presidential role made vivid the values, particularly hopefulness and friendliness, that give cohesion and dynamism to this continental nation.\n\nA democratic leader's voice should linger in his nation's memory, an echo of his exhortations. Reagan's mellifluous rhetoric lingers like a melody that evokes fond memories. Because of demagogues, rhetoric has a tainted reputation in our time. However, Reagan understood that rhetoric is central to democratic governance. It can fuse passion and persuasion, moving free people to freely choose what is noble.\n\nHe understood the axiom that people, especially Americans, with their Founders' creed and vast reservoirs of decency, more often need to be reminded than informed. And he understood the economy of leadership\u2014the need to husband the perishable claim a leader has on the attention of this big, boisterous country.\n\nTo some, Reagan seemed the least complicated of men\u2014an open book that the country had completely read. However, he had the cunning to know the advantage of being underestimated. He was more inward than he seemed. And much tougher. The stricken fields of American and world politics are littered with those who did not anticipate the steel behind his smile.\n\nThe oldest person ever elected president had a sure sense of modernity, as when he told students at Moscow University that mankind is emerging from the economy of muscle and entering the economy of mind. \"The key,\" he said, \"is freedom,\" but freedom grounded in institutions such as courts and political parties. Otherwise \"freedom will always be looking over its shoulder. A bird on a tether, no matter how long the rope, can always be pulled back.\"\n\nReagan was a friendly man with one close friend. He married her. He had one other great love, for the American people, a love intense, public, and reciprocated.\n\nPresidents usually enter the White House as shiny and freshly minted dimes and leave tarnished. Reagan left on the crest of a wave of affection that intensified in response to the gallantry with which he met illness in his final years.\n\nToday Americans gratefully recall that at a turbulent moment in their national epic, Reagan became the great reassurer, the steadying captain of our clipper ship. He calmed the passengers\u2014and the sea.\n\n[JUNE 6, 2004]\n\n# **Reagan and the Vicissitudes of Historical Judgments**\n\n**R** onald Reagan, unlike all but ten or so Presidents, was a world figure whose career will interest historians for centuries, and centuries hence his greatness will be, and should be, measured primarily by what happened in Europe, as a glorious echo of his presidency, in the three years after he left the White House. What happened was the largest peaceful revolution in history, resulting in history's largest emancipation of people from tyranny\u2014a tyranny that had deadened life for hundreds of millions of people from the middle of Germany to the easternmost of Russia's eleven time zones.\n\nHowever, Reagan will also be remembered for his restoration of American confidence that resulted in a quickening tempo of domestic life. During his first term, the most remarkable run of wealth creation in the history of this or any other nation began. Arguably, it began with a seemingly unrelated event in the first year of his first term.\n\nIn 1981, when the nation's air-traffic controllers threatened to do what the law forbade them to do\u2014strike\u2014Reagan warned that if they did they would be fired. When they struck in August, Reagan announced that the strikers would be terminated in two days. By firing the controllers, Reagan, the only union man\u2014he had been head of the Screen Actors Guild\u2014ever to be president, destroyed a union, the Professional Air Traffic Controllers Organization (PATCO). This has often, and not incorrectly, been called a defining episode of the Reagan presidency because it notified foreign leaders, not least those of the Soviet Union, that he said what he meant and meant what he said.\n\nBut now, more than two astonishing decades on, it also is reasonable to conclude that Reagan's fracas with the controllers had huge economic consequences, domestic and foreign. It altered basic attitudes about relations between business and labor in ways that quickly redounded to the benefit of the nation, and not least the benefit of American workers. It produced a cultural shift, a new sense of what can be appropriate in business management: layoffs can be justifiable even when a company is profitable, if the layoffs will improve productivity and profitability. Within a few years, both AT&T and Procter & Gamble, although quite profitable at the time, implemented large layoffs, without arousing significant protests.\n\nReagan's action against the air-traffic controllers came on the eve of the explosive growth of information technologies, and some astute people, including Alan Greenspan, believe that Reagan's action facilitated that growth.\n\nSince 1981, labor in America has prospered because it is less protected. In theory, it might seem that since the showdown with PATCO, business's informal protocol about layoffs must have resulted in rising unemployment. The reverse has happened. In the post-PATCO climate of business operations, employers have been more inclined to hire because they know that if the hiring proves to be improvident, those hired can be discharged. The propensity to hire has risen much more than the propensity to fire. In all of America's post\u2013Civil War era of industrialization, unemployment has never been as low for as long as it has generally been in the years since the extraordinary expansion that began during Reagan's first term.\n\nAlthough Reagan entered politics late, running for governor of California at the age of fifty-five, and entered the White House a few weeks shy of his seventieth birthday, older than any other president when elected, he lived so long after leaving office that his tenure already has been subject to striking vicissitudes of historical judgment. Nothing is so irretrievably lost to a society as the sense of fear it felt about a grave danger that was subsequently coped with. So in measuring Reagan's greatness, insufficient attention is now given to his long-headedness and toughness in putting the Soviet Union on the path to extinction. This is particularly so because the intelligentsia likes nothing less than giving Reagan credit, and so has embraced the theory that the Soviet Union's extinction was an inevitability that just happened while Reagan was standing around.\n\nSo as memories of the Cold War fade, Reagan is remembered more for the tax cutting and deregulating that helped, with the information technologies, to shift the economy into a hitherto unknown overdrive. But the truth is that Reagan always thought that winning the Cold War and revving up the American model of wealth creation were parts of the same project. That project was to convince the watching world that the American social and political model\u2014pluralism, the rule of law, allocation of wealth and opportunity mostly by markets, and maximum diffusion of decision making\u2014is unrivaled. To the extent that anything in history can ever be said to be completed, that project has been.\n\nReagan always believed that the world was watching America. Indeed, he thought the point of America was to be watched\u2014to be exemplary. Hence the complete sincerity of his reiterated references to the City on the Hill. And when the democratic revolution against communism came, Tiananmen Square in Beijing and Wenceslas Square in Prague and points in between rang with the rhetoric of America's third and sixteenth presidents. The fortieth president was not surprised.\n\n[JUNE 14, 2004]\n\n# **John Paul II: \"A Flame Rescued from Dry Wood\"**\n\n**I** n Eastern Europe, where both world wars began, the end of the Cold War began on October 16, 1978, with a puff of white smoke, in Western Europe. It wafted over one of Europe's grandest public spaces, over Michelangelo's dome of St. Peter's, over statues of the saints atop Bernini's curving colonnade that embraces visitors to Vatican City. Ten years later, when the fuse that Polish workers had lit in a Gdansk shipyard had ignited the explosion that leveled the Berlin Wall, it was clear that one of the most consequential people of the twentieth century's second half was a Pole who lived in Rome, governing a city-state of 109 acres.\n\nScience teaches that reality is strange\u2014solid objects are mostly space; the experience of time is a function of speed; gravity bends light. History, too, teaches strange truths: John Paul II occupied the world's oldest office, which traces its authority to history's most potent figure, a Palestinian who never traveled a hundred miles from his birthplace, who never wrote a book, and who died at thirty-three. And religion, once a legitimizer of political regimes, became in John Paul II's deft hands a delegitimizer of communism's ersatz religion.\n\nBetween October 16, 1978, and January 20, 1981, the cause of freedom was strengthened by the coming to high offices of Margaret Thatcher, Ronald Reagan, and John Paul II, who, like the president, had been an actor and was gifted at the presentational dimension of his office. This peripatetic pope was seen by more people than anyone in history, and his most important trip came early. It was a visit to Poland that began on June 2, 1979.\n\nIn nine days, a quarter of that nation's population saw him. Marx called religion the opiate of the masses, but it did not have a sedative effect on the Poles. The pope's visit was the nation's epiphany, a thunderous realization that the nation was of one mind, mocking the futility of communism's thirty-five-year attempt to conquer Poland's consciousness. Between 1795 and 1918, Poland had been erased from the map of Europe, partitioned between Austria, Prussia, and Russia. This gave Poles an acute sense of the distinction between the state and the real nation.\n\nIgor Stravinsky, speaking with a Russian's stoicism about Poland's sufferings, said that if you pitch your tent in the middle of New York's Fifth Avenue, you are going to be hit by a bus. The Poland where John Paul II grew to sturdy, athletic manhood was hit first by Nazism, then communism. Then, benignly, by John Paul II.\n\nIt was said that the fin de si\u00e8cle Vienna of Freud and Wittgenstein was the little world in which the larger world had its rehearsals. In the late 1970s, the Poland of John Paul II and Lech Walesa was like that. The twentieth century's worst political invention was totalitarianism, a tenet of which is that the masses must not be allowed to mass: Totalitarianism is a mortar and pestle for grinding society into a dust of individuals. Small wonder, then, that Poland's ruler, General Wojciech Jaruzelski, visibly trembled in the presence of the priest who brought Poland to its feet in the face of tyranny by first bringing Poland to its knees in his presence.\n\nJohn Paul II almost did not live to see this glorious consummation. In 1981, three of the world's largest figures\u2014Ronald Reagan, Anwar Sadat, and John Paul II\u2014were shot. History would have taken an altered course if Sadat had not been the only one killed.\n\nOur age celebrates the watery toleration preached by people for whom \"judgmental\" is an epithet denoting an intolerable moral confidence. John Paul II bristled with judgments, including this: The inevitability of progress is a myth, hence the certainty that mankind is wiser today than yesterday is chimeric.\n\nSecular Europe is, however, wiser because of a man who worked at an altar. Europeans have been plied and belabored by various historicisms purporting to show that individuals are nullities governed by vast impersonal forces. Beginning in 1978, Europeans saw one man seize history by the lapels and shake it.\n\nOne of G. K. Chesterton's Father Brown detective stories includes this passage: \"'I'm afraid I'm a practical man,' said the doctor with gruff humor, 'and I don't bother much about religion and philosophy.' 'You'll never be a practical man till you do,' said Father Brown.\"\n\nA poet made the same point: \"A flame rescued from dry wood has no weight in its luminous flight yet lifts the heavy lid of night.\" The poet became John Paul II.\n\n[APRIL 3, 2005]\n\n# **Ayaan Hirsi Ali: An Enlightenment Fundamentalist**\n\n**W** hile her security contingent waits outside the Georgetown restaurant, Ayaan Hirsi Ali orders what the menu calls \"raw steak tartare.\" Amused by the redundancy, she speculates that it is intended to immunize the restaurant against lawyers, should a customer be discommoded by that entree. She has been in America only two weeks. She is a quick study.\n\nAnd an exile and an immigrant. Born thirty-six years ago in Somalia, Hirsi Ali has lived in Ethiopia, Kenya, Saudi Arabia, and the Netherlands, where she settled in 1992 after she deplaned in Frankfurt, supposedly en route to Canada for a marriage, arranged by her father, to a cousin. She makes her own arrangements.\n\nShe quickly became a Dutch citizen, a member of parliament, and an astringent critic, from personal experience, of the condition of women under Islam. She wrote the script for, and filmmaker Theo van Gogh directed, _Submission,_ an eleven-minute movie featuring pertinent passages from the Koran (such as when it is a husband's duty to beat his wife) projected on the bodies of naked women.\n\nIt was shown twice before November 2, 2004, when van Gogh, bicycling through central Amsterdam in the morning, was shot by an Islamic extremist who then slit his throat with a machete. Next, the murderer (in whose room was found a disk containing videos of \"enemies of Allah\" being murdered, including a man having his head slowly sawed off) used another knife to pin a long letter to van Gogh's chest. The letter was to Hirsi Ali, calling her a \"soldier of evil\" who would \"smash herself to pieces on Islam.\"\n\nThe remainder of her life in Holland was lived under guard. Neighbors in her apartment building complained that they felt endangered with her there and got a court to order her evicted. She decided to come to America.\n\nHolland evidently tolerates everything except skepticism about the sacramental nature of multiculturalism. One million of the country's 16 million residents are Islamic, and the political left has appropriated the European right's traditional celebration of identity grounded in racial and ethnic traditions and culture. But the recoil of many Dutch people from Hirsi Ali suggests that the tolerance about which Holland preens is a compound of intellectual sloth and moral timidity. She was more trouble than the Dutch evidently think free speech is worth.\n\nHer story is told in a riveting new book, _Murder in Amsterdam,_ by Ian Buruma, who is not alone in finding her\u2014this \"Enlightenment fundamentalist\"\u2014somewhat unnerving and off-putting. Having experienced life circumscribed by tribal and religious communities (as a girl she suffered the genital mutilation called female circumcision), she is a fierce partisan of individualism against collectivism.\n\nShe reminds Buruma of Margaret Thatcher's sometimes abrasive intelligence and fascination with America. He is dismissive of the idea that she is a Voltaire against Islam: Voltaire, he says, offended the powerful Catholic Church, whereas she offends \"only a minority that was already feeling vulnerable in the heart of Europe.\"\n\nShe, however, replies that this is hardly a normal minority. It is connected to Islam's worldwide adherents. Living sullenly in European \"dish cities\"\u2014enclaves connected by satellite television and the Internet to the tribal societies they have not really left behind\u2014many members of this minority are uninterested in assimilation into open societies.\n\nShe calls herself \"a dissident of Islam\" because, given what Allah supposedly enjoins and what she knows is right, \"the cognitive dissonance is, for me, too much.\" She says she is not \"a militant atheist,\" but the emphasis is on the adjective.\n\nSlender, elegant, stylish, and articulate (in English, Dutch, and Swahili), she has found an intellectual home here at the American Enterprise Institute, where she is writing a book that imagines Muhammad meeting, in the New York Public Library, three thinkers\u2014John Stuart Mill, Friedrich Hayek, and Karl Popper, each a hero of the unending struggle between (to take the title of Popper's 1945 masterpiece) \"The Open Society and Its Enemies.\" Islamic extremists\u2014the sort who were unhinged by some Danish cartoons\u2014will be enraged. She is unperturbed.\n\nNeither is she pessimistic about the West. It has, she says, \"the drive to innovate.\" But Europe, she thinks, is invertebrate. After two generations without war, Europeans \"have no idea what an enemy is.\" And they think, she says, that leadership is an antiquated notion because they believe that caring governments can socialize everyone to behave well, thereby erasing personal accountability and responsibility. \"I can't even tell it without laughing,\" she says, laughing softly. Clearly she is where she belongs, at last.\n\n[SEPTEMBER 21, 2006]\n\n# **Hugh Hefner: Tuning Fork of American Fantasies**\n\n**L** OS ANGELES\u2014Asked how it feels to have won, Hugh Hefner pauses, looks down and almost whispers, \"Wonderful.\" Then he says: \"I guess if you live long enough...\"\n\nFifty years ago, he was pecking at a typewriter on a card table in his Chicago apartment, preparing the first issue of a magazine he planned to call _Stag Party_ but, because there already was a magazine called _Stag,_ he called it _Playboy._ The first issue appeared in December 1953. It bore no date because Hefner was not sure there would be a second, such were the troubles the first issue caused with the post office and other defenders of decency.\n\nFour years later, in the nick of time, Searle pharmaceutical company introduced Enovid\u2014\"the pill.\" Back then Hefner, the tuning fork of American fantasies, said he wanted to provide \"a little diversion from the anxieties of the Atomic Age.\" But three emblematic books of the supposedly repressed 1950s\u2014 _Peyton Place, Lolita,_ and _The Kinsey Report_ (Professor Alfred Kinsey of Indiana University was another Midwestern sexual subversive)\u2014showed that more than geopolitical anxiety was on the mind of Eisenhower's America.\n\nBy 1959, the post office was delivering millions of copies of Hefner's magazine. _Playboy_ 's rabbit-head logo is now one of the world's most recognized brands, even in inscrutable China, where _Playboy_ merchandise sells well but the magazine is banned.\n\nHefner's daughter Christie, who was born thirteen months before the magazine, says _Playboy_ was \"a great idea executed well at exactly the right time.\" A no-nonsense executive, she now runs the Chicago-based business she joined twenty-seven years ago, fresh from earning a summa cum laude degree from Brandeis. When she arrived, Playboy was primarily an American magazine publisher. She has made it into an international electronic entertainment company.\n\nThe magazine, the twelfth-highest-selling U.S. consumer publication, sells 3.2 million copies monthly. That is slightly less than half its 1970s peak, but its eighteen international editions sell another 1.8 million copies a month, and it remains the world's bestselling monthly men's magazine.\n\nStill, it provides only about one-third of Playboy Enterprises' annual revenues of $277.6 million. Playboy owns six cable networks that deliver to 38 million North American households movies of a sexual explicitness that would have been instantly prosecuted in all forty-eight states in 1953.\n\nThe magazine, the mere mention of which used to produce pursings of lips and sharp intakes of breaths, is still Hefner's preoccupation, but has been overtaken by the libertarian revolution he helped to foment. In 1953, _Playboy_ magazine was pushing the parameters of the permissible, but it is hard to remain iconoclastic when standing waist-deep in the shards of smashed icons.\n\nBorn to \"puritanical\" (Hefner's words) parents in Chicago, city of broad shoulders, Hefner founded an empire based on breasts. What is it about that protean city? Chicagoan Ray Kroc, entrepreneur of McDonald's, did his Army training with Chicagoan Walt Disney\u2014two prodigies of mass marketing, the creator of the Big Mac and the creator of Mickey Mouse, in the same Army unit.\n\nThen Chicago produced the Henry Luce of the skin game\u2014Hef, as everyone, including his daughter, calls him. The Chicago boy recalls that the Sears Roebuck mail order catalog\u2014another Chicago innovation\u2014was called \"a dream book\" because it brought \"the dream of urbanity to rural communities. _Playboy,_ for young, single men, is a variation of this.\"\n\nRecently, dressed in his black pajamas and merlot-colored smoking jacket\u2014it was 1 p.m.\u2014he looked a bit tuckered, but he had been living what Teddy Roosevelt called \"the strenuous life,\" although not as TR envisioned it. Hefner's recent seventy-seventh birthday party had rambled on for more than a week, during which he took to dinner\u2014simultaneously\u2014the seven ladies he is currently dating. As F. Scott Fitzgerald, writing of Jay Gatsby, suggested, \"personality is an unbroken series of successful gestures.\"\n\nAn eleventh-generation descendant of William Bradford, who arrived on the _Mayflower_ to begin a religious errand in the wilderness, Hefner says, \"In a real sense we live in a Playboy world.\" He lives here in a thirty-room mock-Tudor mansion that sits on six acres of posh Holmby Hills decorated with wandering peacocks, among other fauna.\n\nHe says, \"I grew up in the Depression and World War II and I looked back to the Roaring Twenties and I thought I'd missed the party.\" The party turned out to be a movable feast.\n\n[MAY 29, 2003]\n\n# **Lawrence Ferlinghetti: The Emeritus Beat as Tourist Attraction**\n\n**S** AN FRANCISCO\u2014America's gauzy popular culture has the power to envelop even its perfervid critics in a tolerant, domesticating embrace. If they live long enough, these critics run the risk of winding up full not only of years, but of honors. They can, like Lawrence Ferlinghetti, eighty-three, become tourist attractions.\n\nThese tourists, he notes, are intellectually upscale. They come in a small but steady trickle, from across the country and around the world, to his City Lights Booksellers & Publishers, next door to a street named after the most famous of the many writers who have hung out there\u2014Jack Kerouac. The store, which is a short walk from the street\u2014actually, an alleyway, which seems right\u2014named Via Ferlinghetti, has been designated by this city a protected landmark. This is not because the wedge-shaped structure built in 1907 is a gem (it is not) but because of its cultural significance, which is primarily its association with Kerouac, Allen Ginsberg, and other designated voices of the Beat Generation.\n\nIt was in City Lights that San Francisco police arrested Ferlinghetti on obscenity charges for publishing Ginsberg's \"Howl.\" Ferlinghetti's and Ginsberg's acquittals helped make possible the American publication of D. H. Lawrence's _Lady Chatterley's Lover_ and Henry Miller's _Tropic of Cancer,_ which involved legal dustups that now seem quaint.\n\nFerlinghetti publishes as well as sells books. He published \"Howl\" after first rejecting it. It was after he heard Ginsberg recite it that he sent Ginsberg a telegram repeating words from the letter Emerson sent to Walt Whitman after reading _Leaves of Grass_ \u2014\"I greet you at the beginning of a great career.\"\n\nFerlinghetti was born in Bronxville, New York, spent much of his youth in France, and went to the University of North Carolina because his roommate at a prep school in Massachusetts hooked him on the novels of Thomas Wolfe. He began a four-year hitch in the Navy before Pearl Harbor (\"I was a good American boy\") and was back near France, on a U.S. Navy submarine chaser, on June 6, 1944. After earning\u2014thank you, GI Bill of Rights\u2014a master's degree in Victorian literature at Columbia, and a doctorate at the Sorbonne, and after finding that the mailroom at _Time_ magazine was not a promising rung on the ladder of journalism, he headed for here, to start a bookstore, a vocation suggested by life in Paris.\n\nCity Lights is in the North Beach district, which once was a scene of San Francisco's bohemian ferment. Now the district is mostly seedy. Visible from Ferlinghetti's cluttered office in his bookstore's second floor is an establishment with a resonant name\u2014Hungry I. It was at a nightclub called the hungry i a few hundred yards from the location of the topless bar now bearing that name\u2014bohemia isn't what it used to be, but then, what is?\u2014that Mort Sahl and Lenny Bruce performed their political riffs that were the outer edges of dissent in the 1950s and early 1960s.\n\nBruce lived across the street from City Lights. He once fell out of a window. \"No doubt he was on something\" is Ferlinghetti's safe surmise. Sahl used to browse the City Lights magazine rack for ideas for his performances.\n\nFerlinghetti looks the part of an emeritus Beat\u2014small silver earring, tatty sport coat, blue jeans\u2014and looks askance at tourists (no kidding), Republicans (of course: one of his most popular works was the long 1958 poem \"Tentative Description of a Dinner Given to Promote the Impeachment of President Eisenhower\"), George W. Bush (\"shredding the Constitution,\" \"dismantling the New Deal\"), gentrification (San Francisco has been \"dot.conned\"), automobiles (\"auto-geddon\" inflicted by SUVs), and chain stores (does he have Borders and Barnes & Noble in mind?).\n\nBut the grouchiness of San Francisco's first poet laureate seems perfunctory, even cheerful. City Lights is open until midnight seven days a week; books are shelved under ideological categories (look under \"Stolen Continents\" for American history). It retains what Ferlinghetti says San Francisco itself had when he arrived in 1951, \"an island mentality, a sort of offshore territory.\" American life has been good to the man whose _Coney Island of the Mind_ was the bestselling poetry book in the 1960s and 1970s. A million copies are in print.\n\nIt has been famously said that there are no second acts in American life. Actually, there are. And third, fourth, and fifth acts. But Ferlinghetti has happily stayed with his one act, and the world, or at least a minority steeped in literary nostalgia, is still beating a path to his door.\n\n[JUNE 14, 2002]\n\n# **Buck Owens's Bakersfield Sound**\n\n**B** AKERSFIELD, CALIFORNIA\u2014Buck Owens came to this city, a hundred miles north of Los Angeles, at the southern end of the prodigiously fertile San Joaquin Valley, to pick cotton, not a guitar. He came for the same reason lots of others came west from Texas and Oklahoma: happiness was the Dust Bowl in their rearview mirrors.\n\nThe Owens family's rearview mirror was on a 1933 Ford sedan. In 1937, when Buck was eight and John Steinbeck was just beginning to write _The Grapes of Wrath,_ ten Owens family members packed into it and headed west. His parents had been sharecroppers on the southern side of the Red River that separates Texas from Oklahoma. Because the trailer hitch broke in Phoenix, the family lived there for a few years, sometimes traveling to the San Joaquin to pick carrots in Porterville, peaches in Modesto, potatoes and cotton in Bakersfield. During such work, he got the idea that picking a guitar might be more fun.\n\nWhich he is doing at seventy-three, in his Crystal Palace nightclub, where he recently began a rollicking hour set with \"Okie from Muskogee,\" a sixties\u2014actually, an anti-sixties\u2014anthem by another Bakersfield boy, Merle Haggard. (Has there ever been a better name for a country-music singer?)\n\nBy sixteen, Owens had begun playing in Phoenix honky-tonks, sometimes with the young Marty Robbins, earning whatever change he could collect by passing a soup bowl. In 1951, he moved to Bakersfield, in Kern County, which produces more oil than Oklahoma, and had plenty of roughnecks to appreciate the country music of rising stars like Bob Wills and Ferlin Husky, who were honing what has come to be called the Bakersfield sound.\n\nThat is identified with Owens's solid-body Fender Telecaster steel guitar. It produces the sharp, twangy, driving, biting sound that seems especially suited to the subjects of what is called \"hard country music.\" Such music sometimes teeters on the brink of self-caricature, or embraces it (\"I was drunk the day my momma got out of prison\"), but its essential message is that life is difficult and so are most of the people we meet, including those we marry.\n\nThe life that drove many people down Steinbeck's road to California was hard, and so was the life Owens led chasing stardom. After touring hard\u2014sometimes three hundred nights a year\u2014Owens got off the road in 1980. And he spent too many years associated with the instant kitsch of the television program _Hee Haw._ As a result, too few fans of country music appreciate how much his Bakersfield sound helped give that music a steely integrity and propel it to the point that Owens could play a much-praised concert in President Johnson's White House in 1968.\n\nBy 1980, however, when the John Travolta movie _Urban Cowboy_ helped make country music fashionable, country music was beginning to lose its edge. More to the point, the Nashville music establishment set out to rub the edge off, to envelop it in a syrup of strings and softening production techniques, the better to appeal to a broader audience that wanted country music that was close kin to soft rock.\n\nBut some \"new traditionalists\" were, and are, having none of that. These include Randy Travis, George Strait, the Dixie Chicks, and Dwight Yoakam\u2014another nifty name for a country singer. Yoakam was born in Kentucky and spent a while in Nashville, but says, \"I was drawn to Los Angeles by my earlobes...the country-rock sound, and the Bakersfield sound.\"\n\nA related development is the recent emergence of a new category of old-style music called \"Americana,\" which is the most popular radio format for Internet listeners. Although Americana is not strictly defined (see americanamusic.org), a good sampler is the sound track of the movie _O Brother, Where Art Thou?_ a mixture of blues, bluegrass, gospel, folk, and mountain music. Sales of that CD, released in 2000, are heading toward 7 million. Americana encompasses new performers, such as Alison Krauss's Union Station and Nickel Creek, and hardy perennials like Johnny Cash. One song written years ago by Homer Joy probably would qualify as Americana:\n\n_Spent some time in San Francisco_\n\n_Spent a night there in the can_\n\n_Well, they threw this drunkard in my jail cell_\n\n_Took fifteen dollars from that man_\n\n_But I left him my best watch and my house key_\n\n_Don't like folks sayin' that I steal_\n\n_Then I headed out to Bakersfield._\n\nWhen Buck Owens is onstage, singing that song, the years fall away. He is as energized by the audience as his guitar is by electricity, and the young man in flight from the cotton fields is present again.\n\nParts of his life resemble hard-country lyrics (his fourth divorce is not going well), but he is now an icon in the community he first saw when picking cotton. Today, as he drives the streets of Bakersfield, he can steer his pickup truck down Buck Owens Boulevard.\n\nBakersfield, although prosperous, is still a place where billboards proclaim, tough times never last but tough people do. So does hard-country music, and Buck Owens's Bakersfield sound.\n\n[DECEMBER 9, 2002]\n\n# **Andrew Nesbitt: Seventy-nine-Pound Master of Tourette Syndrome**\n\n**C** OPPELL, TEXAS\u2014Even in what passes for repose, your basic eleven-year-old boy resembles the former Yugoslavia\u2014a unity of sorts, but with fidgeting and jostling elements. Andrew Nesbitt is like that, only more so, because he has Tourette syndrome.\n\nHe also has something to teach us about the power of a little information and a lot of determination. And about how life can illuminate philosophy, which is supposed to do the illuminating.\n\nHe is seventy-nine pounds of shortstop and relief pitcher\u2014a closer, no less, which is a high-stress vocation. Stress often triggers Tourette symptoms. Hitting a thrown ball with a round bat is hard enough, and so is throwing the ball over a seventeen-inch-wide plate with the game on the line. Hard enough, even if you do not have an inherited neurological disorder that causes recurrent physical and phonic tics.\n\nThe physical tics can include involuntary muscle spasms\u2014blinking, clapping, hopping, and the more or less violent twitching of shoulders and flailing of limbs. The vocalizations are usually grunts, hisses, barks, and other meaningless sounds. Rarely, and not at all in Andrew's case, there is the compulsive utterance of obscenities.\n\nAt the benighted school he attended last year, teachers could not\u2014would not\u2014understand that he did not have a mischievous penchant for bad behavior. They frequently banished him from the classroom to sit in the hall.\n\nWhen he was younger, his parents had to hold his thrashing head so he could eat. Playing soccer, he sometimes bruised his behind by kicking himself with backward leg spasms. This year, he says, Mrs. Marill Myers, his math and homeroom teacher, \"asks me if it's a tic.\" She gives him a jump rope to use to subdue unmanageable energy. Or pauses to briefly rub his back. Not complicated, really.\n\nHe was five, standing on a swimming pool diving board, when his mother first saw him jerking his head and shrugging his shoulders oddly. He is bright as a new dime\u2014at ten months he had a fifty-word vocabulary\u2014but his gross and fine motor problems became so bad that in fourth grade hip spasms would throw him out of his desk chair.\n\nA visiting columnist is Andrew's excuse for taking a break from the work part of a sixth-grader's day in Coppell Middle School West (math, English\u2014the _school_ stuff) and savoring anticipation of the good parts, such as lunch, baseball, and lacrosse practice. He is dressed conservatively, even formally, as his age cohort understands such matters: red T-shirt reaching almost to his knees and blue shorts that aren't short\u2014they reach below his knees, toward his white sneakers.\n\nNowadays, he says, \"I sometimes hold the tics in when I'm batting.\" Extreme concentration also helps Mike Johnston, a Pittsburgh Pirates reliever, contain his Tourette symptoms: \"I'll sometimes stare at something until my eyes water.\" Johnston, who was awakened in a Chicago hotel on an off-day by the thoughtless columnist, chats on the phone with Andrew, who is asking important questions, such as: \"Have you ever pitched to A-Rod?\" Johnston gets important information from Andrew: cap and jersey sizes. Pirates gear is on the way.\n\nLast year, Andrew came close to exhaustion from dread of teachers' incomprehension and from some children's cruelty. This year, Andrew's teachers and classmates are better informed. What causes his odd behavior may have caused similar behavior by some high-achievers\u2014probably Samuel Johnson, perhaps Mozart. Even more impressive, Jim Eisenreich, formerly of the Twins, Royals, Phillies, Marlins, and Dodgers, has Tourette syndrome, as does Tim Howard, current goalie for Manchester United soccer club, the world's most famous sports team.\n\nThe mind-body dichotomy is a perennial puzzlement for philosophers. Most people say, \"I have a body.\" Perhaps we should say, \"I _am_ a body.\" People who say the latter mean that the mind, the soul\u2014whatever we call the basis of individual identity\u2014is a \"ghost in the machine,\" a mysterious emanation of our physicality. They may be right. But were Andrew given to paddling around in deep philosophic water\u2014if he were, he would not be your basic boy\u2014he might reply:\n\n\"No way. Wisdom is encoded in our common language. We all have, to some extent, a complex, sometimes adversarial, relationship with our physical selves. And I more than most people know that it is correct to say, 'I have a body.' There is my body, and then there is _me,_ trying to make it behave.\"\n\nLet the philosophers contend about the mind-body distinction. If you think Andrew has it wrong, spend a day in his sneakers.\n\n[APRIL 25, 2004]\n\n# **Simeon Wright's Grace**\n\n**A** LSIP, ILLINOIS\u2014In a cemetery here, a few miles southwest of Chicago's city limits, Simeon Wright, sixty-two, a trim, articulate semiretired pipe fitter, and a deacon in the Church of God in Christ, recently attended a ceremony at an unquiet grave. The gravestone has a weatherproof locket with a photograph of a boy, and these carved words:\n\nEMMETT L. TILL\n\nIN LOVING MEMORY\n\nJULY 25, 1941 AUGUST 28, 1955\n\nWright participated in a service for the reinterment of the body of the boy with whom Wright, then twelve, was sharing a bed in the Mississippi home of Wright's father fifty years ago. It was the night that lit the fuse of the civil rights revolution.\n\nThe eulogy delivered at the reinterment\u2014Emmett's remains had been exhumed as part of the reopened investigation of his murder\u2014was by Wheeler Parker, a barber and minister in nearby Argo, Illinois. The night of August 28, 1955, Parker, then sixteen, also was sleeping in the Wright home. Two white men, one with a .45 caliber pistol, shone a flashlight in Parker's face and one of them said, \"Where's the fat boy from Chicago?\"\n\nA few weeks before, Wright's father, a preacher in the vicinity of Money, Mississippi, had come to Chicago to deliver a eulogy for a former parishioner, one of the hundreds of thousands of black Mississippians of the great migration\u2014an $11, sixteen-hour ride on the Illinois Central to Chicago. A week or so later, Mamie Till\u2014Emmett's mother\u2014put Emmett on the Illinois Central to visit his great uncle, and cousin Simeon.\n\nThree days into his visit, at the ramshackle Bryant's Grocery and Meat Market, Emmett, fourteen, whistled at a white woman, Carolyn Bryant. Three nights later, her husband Roy and his half brother J. W. Milam came for Emmett. Simeon's father pleaded, \"Why not give the boy a whipping and leave it at that?\" They beat him to an unrecognizable pulp, knocked out his right eye, shot him, tied a seventy-five-pound cotton gin fan around his neck with barbed wire, and threw his body into the Tallahatchie River.\n\nMississippi authorities had made the Chicago undertaker agree to keep Emmett's casket nailed closed, but they met their match in his mother. An estimated fifty thousand Chicagoans saw the body in the open casket. _Jet_ magazine ran a picture. \"When people saw what had happened to my son,\" Mamie Till said, \"men stood up who had never stood up before.\" And one woman refused to stand up: sixty-nine days after the acquittal of Emmett's murderers, and three hundred miles away, Rosa Parks refused to surrender her seat on a bus.\n\nTwenty-six days after murdering Emmett, Bryant and Milam were acquitted by a jury that waited sixty-seven minutes, a juror said, to \"drink a pop\" before embracing the defense argument that the body might not have been Emmett's. Bryant and Milam later told _Look_ magazine they killed Emmett. They said they took turns smashing him across the head with the .45. But the trial was the first event to turn the gaze of American journalism to the causes of the coming civil rights storm.\n\nA week after the acquittal, Simeon Wright's father, who testified against Bryant and Milam, left his car at the railroad station and went to Chicago. He never returned to Mississippi.\n\nOthers besides Bryant and Milam, both dead, may have been complicit in the killing. But beyond DNA proof that it was Emmett's body, it is unclear what forensic evidence his remains might provide to the Mississippi district attorney who sought the disinterment.\n\nMartin Luther King came to Chicago in January 1966, but Simeon Wright says: \"I didn't qualify for Dr. King's march. They told us that if bricks and things were thrown at us, we couldn't retaliate. I couldn't do that.... Now I'm back to what he was teaching.\"\n\nWright says friends who only recently discovered his connection with the Till case say, \"He's so easy going!\" He says, \"I guess they think I'd be angry all the time. You don't live long that way.\" At the reinterment, he recited the first verse to \"Taps,\" which concludes: \"All is well, safely rest, God is nigh.\" He says, \"It got a little emotional then.\"\n\nWhere do they come from, people like Simeon Wright, people of such resilience and grace? From Mississippi and Illinois. And everywhere else. They are all around us.\n\nWhat has this country done\u2014what can any country do\u2014to deserve such people? Wrong question. They _are_ this country.\n\n[JUNE 19, 2005]\n**Chapter Two**\n\n**PATHS TO THE PRESENT**\n\n# **The Most Important American War You Know Next-to-Nothing About**\n\n**F** or your Fourth of July reading, open a mind-opening book about an immensely important American war concerning which you may know next to nothing. King Philip's War, the central event in a bestseller that is one of this summer's publishing surprises, left a lasting imprint on America.\n\nAmericans in this era of sterile politics have an insatiable appetite for biographies of the Founders. But why are so many readers turning to a book\u2014 _Mayflower: A Story of Courage, Community, and War,_ by Nathaniel Philbrick\u2014that casts a cool but sympathetic eye on an era usually wrapped in gauzy sentimentality?\n\nOne reason might be that it is fun to read about one's family: Philbrick estimates that there are approximately 35 million descendants of the passengers on the _Mayflower._ (Do the math: 102 passengers; 3.5 generations in a century. But remember, 52 passengers died of disease and starvation before the first spring.) Perhaps a second answer is that the story is particularly pertinent as America is engaged abroad in a clash of civilizations, and is engaged at home in a debate about immigration and the common culture.\n\n\"In the American popular imagination,\" Philbrick writes, \"the nation's history began with the Pilgrims and then leapfrogged more than 150 years to Lexington and Concord and the Revolution.\" That version misses, among much else, the history-turning fourteen months of war in 1675 and 1676 that set in train events that led to Lexington and Concord. The war was between the English settlers and the Pokanoket Indians led by Metacom, whose English name was Philip.\n\nIn a six-decade downward spiral of mutual incomprehension and unintended consequences, the uneasy but growing coexistence of English settlers and Native Americans dissolved in mutual suspicions, conflicts, and retaliations. During the war, the colony lost 8 percent of its men (compared to the 4 percent to 5 percent of adult men killed in the Civil War). But Native Americans fared far worse. Of the 20,000 in the region at the war's beginning, 2,000 died of wounds, 3,000 of sickness or starvation, 2,000 fled west or north\u2014and 1,000 were shipped to the West Indies as slaves. Taxation and other costs of the war so injured economic life that a century passed before New England's per capita income returned to the prewar level.\n\nPhilbrick writes that after ethnic cleansing, or at least ethnic sorting out, there were no friendly Indians as buffers between the settlers and the unfriendly ones. So the settlers were forced to look to London for help. Soon a royal governor was appointed to govern New England. Then came irritating taxation\u2014of stamps, of tea\u2014and arguments about representation. Exactly one hundred years after King Philip's War ended, the United States began.\n\nBut an American frame of mind began in 1623. _Mayflower_ illustrates a timeless fact of politics everywhere\u2014the toll reality takes on ideology\u2014and a large theme of American life: the fecundity of individualism and enlightened self-interest.\n\nThe first important book-length manuscript written in America was _Of Plymouth Plantation,_ the journal of William Bradford, the colony's governor for nearly thirty-six years. Not published in full until 1856, it was then avidly read by a nation bent on westward expansion and fearing civil war.\n\nIn a section on private versus communal farming, Bradford wrote that in 1623, because of a corn shortage, the colonists \"began to think how they might raise\" more. After much debate, they abandoned their doctrine, which they brought with them on the _Mayflower,_ that all agriculture should be a collective, community undertaking. It was decided, Bradford wrote, that \"they should set corn every man for his own particular, and in that regard trust to themselves.\" That is, they \"assigned to every family a parcel of land,\" ending communal cultivation of that crop.\n\n\"This,\" Bradford reported, \"had very good success, for it made all hands very industrious, so as much more corn was planted than otherwise would have been by any means.\" Indeed, \"the women now went willingly into the field, and took their little ones with them to set corn; which before would allege weakness and inability; whom to have compelled would have been thought great tyranny and oppression.\" So began the American recoil from collectivism. Just three years after the settlers came ashore (not at Plymouth Rock, and far from their intended destination, the mouth of the Hudson), they began their ascent to individualism.\n\nSo began the harnessing, for the general good, of the fact that human beings are moved, usually and powerfully, by self-interest. So began the unleashing of American energies through freedom\u2014voluntarism rather than coercion. So began America.\n\n[JULY 4, 2006]\n\n# **The Amazing Banality of Flight**\n\n**T** he twelve-second flight one hundred years ago this morning reached a height of just 10 feet, less than the 63-foot height of a Boeing 747, and covered just 120 feet of ground, less than a 747's 195-foot wingspan. But the Wright brothers' fourth and final flight that day in North Carolina lasted fifty-nine seconds and went 852 feet. So by sunset the twentieth century's themes\u2014farther, faster, higher, _now_ \u2014were, so to speak, in the air.\n\nAlmost everything\u2014commerce, war, art\u2014would change as aviation began altering, as nothing had ever done, humanity's experience of the most basic things: time and space. Politics, too. The first important politician to campaign by air was a militant modernist, Adolf Hitler. The newsreels screamed: _\"Der F\u00fchrer fliegt \u00fcber Deutschland.\"_\n\nAviation's infancy was not for the fainthearted. In the early 1920s, an airmail pilot named Dean Smith, on the Chicago-to-Omaha route, cabled his superintendent:\n\n\"On trip 4 westbound. Flying low. Engine quit. Only place to land on cow. Killed cow. Wrecked plane. Scared me. Smith.\"\n\nAirmail was one way government subsidized aviation, which drew government into deep involvement with technology. So, of course, did the great driver of social change, war. In their new book, _Reconsidering a Century of Flight,_ Roger D. Launius and Janet R. Daley Bednarek note how rapid was the development of the airplane \"from a machine in some ways most lethal to those who used it to a machine of great lethality to those against whom it is directed.\"\n\nIn 1905, the Wright brothers testified to Congress that airplanes' military uses would be \"scouting and carrying messages.\" Forty years later, cities would be laid waste from the air. But city bombing was not as lethal as was feared. In April 1939, the British government, anticipating city bombing, issued to local authorities 1 million burial forms. The actual British casualties from aerial bombardment, 1939\u20131945, were sixty thousand.\n\nOne early theory, refuted by experience, was that strategic bombing might make wars less bloody by bypassing bloody clashes between armies, such as the First World War's trench warfare, and instead quickly inducing an enemy's surrender by disrupting his \"vital center.\" The fallacious assumption was that modern economies and societies are fragile.\n\nIt was nearly a century after Kitty Hawk, and due less to developments of aircraft than of munitions, that military aircraft really became lethal for targets smaller than whole cities. Until recently, the question about bombing was how many sorties it would take to destroy a target. Suddenly, because of precision munitions, the question is how many targets one sortie can strike. In World War II, about one bomb in four hundred landed close enough to affect\u2014not necessarily destroy\u2014its target. Now nine out of ten do.\n\nThe most astonishing consequence of aviation is not its military applications or their civilian echoes. (After World War II, Harley J. Earl, General Motors' chief stylist, turned his fascination with the twin tails of the P-38 fighter into automobile tail fins that defined the chrome-plated 1950s.) Rather, the amazing consequence was the banality of flight\u2014the routinization of mobility\u2014especially after 1958, when Boeing's 707 speeded the democratization of air travel. Unfortunately, this had some negative public health consequences because viruses\u2014HIV, for one\u2014also became mobile.\n\nFrom the first, flight expressed the essence of the modernist movement\u2014freedom understood as the absence of limits, and a future of infinite possibilities. While developing cubism, Pablo Picasso sometimes painted wearing aviator's gear. His response to the April 26, 1937, bombing by Germans of Guernica during Spain's civil war\u2014a rehearsal, or overture, for what was soon to come from Europe's skies\u2014moved Picasso to produce what may be the twentieth century's iconic painting.\n\nCubism itself was influenced by a perspective no previous generation knew, that of the earth\u2014the geometry of its urban grids and rural plots\u2014seen from above. The Eiffel Tower had provided Europeans their first downward vision of their environment. Robert Hughes, the art critic, says that what was spectacular was not the view of the tower from the ground but the view of the ground from the tower. Until then, almost everybody lived their entire lives no more than forty feet\u2014the height of an ordinary apartment building\u2014above the ground.\n\nModernism shaped another expressive activity that flourished in tandem with aviation, the competition to build the tallest skyscraper. In Manhattan, epicenter of the competition, the race was eventually won by the twin towers of the World Trade Center, where ninety-eight years after Kitty Hawk the histories of aviation and architecture intersected.\n\n[DECEMBER 17, 2003]\n\n# **The Price of Misreading the Prairie Sky**\n\n**W** hen giving thanks this year, think of Lena Woebbecke. She and many others paid a terrible price for misreading the prairie sky on the afternoon of January 12, 1888.\n\nThat day was unseasonably balmy, by prairie standards\u2014some temperatures were in the twenties\u2014and many children scampered to school without coats or gloves. Then, at about the time schools were adjourning, death, in the shape of a soot-gray cloud, appeared on the horizon of Dakota Territory and Nebraska.\n\nIn three minutes the temperature plunged eighteen degrees. The next morning hundreds of people, more than one hundred of them children, were dead beneath the snowdrifts. David Laskin, a Seattle writer, reconstructs this tragedy in a terrifying but beautifully written new book, _The Children's Blizzard._\n\nIt picks up the many threads of the story in Norway, Ukraine, Germany, Vermont, and other tributaries to the river of immigration set in motion partly by the 1862 Homestead Act. In return for an $18 filing fee and five years farming, the act conferred ownership of 160 acres. By the tens of thousands the homesteaders came, to live in sod houses, heated by burning buffalo bones and twisted hay.\n\nOf immigrants, the saying was that the cowards stayed home and the weak died on the way. One in ten crossing the Atlantic in steerage did die. But Laskin says \"the mystique of the Dakotas\" was such that the territory's population nearly quadrupled in the 1880s. Those who made it, with a trunk or two and the clothes on their backs, reached towns that were perishable scratches on the prairie. They got land, freedom, and hope.\n\nAnd prairie fires. And grasshoppers, 100 billion at a time in roaring clouds a mile high and a hundred miles across. And iron weather in which children, disoriented by horizontal streams of snow as hard as rock and fine as dust, froze to death groping their way home from a school 150 yards away.\n\nLena was five in 1882 when her father, a German immigrant, died of smallpox. Her mother remarried twice, having eleven children, eight of whom survived. In August 1887, Lena, her marriage prospects diminished by her smallpox scars, was sent to live with the Woebbeckes and their three children in a two-room house. It was half a mile from the school where she was, five months later, when a cataclysmic cold front came dropping southeast out of Canada at forty-five miles per hour.\n\n\"To those standing outside,\" Laskin writes, \"it looked like the northwest corner of the sky was suddenly filling and bulging and ripping open.\" In four and a half hours, the temperature at Helena, Montana, fell fifty degrees. The prairie air tingled with the electricity of a horizontal thunderstorm. All over the region, schoolteachers, many of them not much older or more educated than their pupils, had to make life-and-death decisions about how to get the children home.\n\n\"The fear came first,\" Laskin writes, \"but the cold followed so hard on its heels that it was impossible to tell the difference.\" In minutes, nostrils were clogged by ice. Eyelids were torn by repeated attempts to prevent them from freezing shut. Unable to see their hands in front of their faces, people died wandering a few yards from their houses, unable to hear, over the keening wind, pots being pounded a few yards away to tell them the way to safety.\n\n\"For years afterward,\" writes Laskin, \"at gatherings of any size in Dakota or Nebraska, there would always be people walking on wooden legs or holding fingerless hands behind their backs or hiding missing ears under hats\u2014victims of the blizzard.\" Lena learned to walk on a wooden foot. In 1901, at twenty-four, she married. At twenty-five, she died, perhaps in childbirth, or perhaps of a complication from the amputation necessitated by frostbite.\n\n\"Lena was laid to rest in her wedding dress in the graveyard of the Immanuel Lutheran Church near the country crossroads called Ruby. If there ever was a town called Ruby, it has disappeared, as has the Immanuel Lutheran Church. The church cemetery remains\u2014a fenced patch of rough grass studded with headstones between two farmhouses not far from the interstate. A tiny island of the dead in the sea of Nebraska agriculture.\"\n\nThis Thanksgiving, when you have rendered yourself torpid by ingesting an excess of America's agricultural bounties, summon thoughts of thanks for the likes of Lena, those whose hard lives paved the stony road to America's current comforts.\n\n[NOVEMBER 25, 2004]\n\n# **\"A Range of Mountains on the Move\"**\n\n_The soil is the one indestructible, immutable asset that the nation possesses. It is the one resource that cannot be exhausted._\n\n\u2014FEDERAL BUREAU OF SOILS, 1878\n\n**S** eventy-five years ago, America's southern plains were learning otherwise. Today, amid warnings of environmental apocalypse, it is well to recall the real thing. It is a story about the unintended consequences of technological progress and of government policies. Above all, it is an epic of human endurance.\n\nWho knew that when the Turks closed the Dardanelles during World War I, it would contribute to stripping the topsoil off vast portions of Texas, Oklahoma, Colorado, and Kansas? The closing cut Europe off from Russian grain. That increased demand for U.S. wheat. When America entered the conflict, Washington exhorted farmers to produce even more wheat, and guaranteed a price of $2 a bushel, more than double the 1910 price. A wheat bubble was born. It would burst with calamitous consequences recounted in Timothy Egan's astonishing and moving book, _The Worst Hard Time: The Untold Story of Those Who Survived the Great American Dust Bowl._\n\nAfter the war, the price plunged and farmers, increasingly equipped with tractors, responded by breaking up more prairie, plowing under ever more grassland in desperate attempts to compensate for falling wheat prices with increased volume. That, however, put additional downward pressure on the price, which was forty cents a bushel by 1930.\n\nThe late 1920s had been wet years, and people assumed that the climate had changed permanently for the better. In that decade, another 5.2 million acres\u2014equivalent, Egan says, to the size of two Yellowstone Parks\u2014were added to the 20 million acres previously in cultivation. Before the rains stopped, fifty thousand acres a day were being stripped of grasses that held the soil when the winds came sweeping down the plain.\n\nIn 1931, the national harvest was 250 million bushels, perhaps the greatest agricultural accomplishment in history. But Egan notes that it was accomplished by removing prairie grass, \"a web of perennial species evolved over 20,000 years or more.\" Americans were about to see how an inch of topsoil produced over millennia could be blown away in an hour.\n\nOn January 21, 1932, a cloud extending ten thousand feet from ground to top\u2014a black blizzard with, Egan writes, \"an edge like steel wool\"\u2014looked like \"a range of mountains on the move\" as it grazed Amarillo, Texas, heading toward Oklahoma. At the end of 1931, a survey found that of the 16 million acres cultivated in Oklahoma, 13 million were seriously eroded.\n\nOn May 10, 1934, a collection of dust storms moved over the Midwest carrying, Egan says, \"three tons of dust for every American alive.\" It dumped six thousand tons on Chicago that night. By morning, the storm was eighteen hundred miles wide\u2014\"a great rectangle of dust\" weighing 350 million tons\u2014and was depositing the surface of the Great Plains on New York City, where commerce stopped in the semidarkness.\n\nOn the southern plains, dust particles, one-fifth the size of the period at the end of this sentence and high in silica content, penetrated lungs, jeopardizing newborns and causing \"dust pneumonia\" in others. Houses were so porous that the only white part of a pillow in the morning was the profile of the sleeper. Storms in March and April 1935 dumped 4.7 tons of dust per acre on western Kansas, denting the tops of cars. During one storm, the wind blew at least forty miles per hour _for one hundred hours._ Egan reports that it would have required a line of trucks ninety-six miles long, hauling ten loads a day for a year\u201446 million truckloads\u2014to transport the dirt that had blown from western to eastern Kansas.\n\nIn Washington, in a Senate hearing room, a man was testifying to bored legislators about the need for federal aid for the southern plains. A senator suddenly exclaimed, \"It's getting dark outside.\" The sun vanished and the air turned copper-color, thanks to red dust that the weather bureau said came from the western end of Oklahoma's panhandle. The aid was approved the next day.\n\nThe southern plains got what Egan calls frenzied skies of grasshoppers\u2014sometimes 14 million per square mile\u2014because the insects' natural predators were gone. Eventually, however, rain fell on the convulsed land and on the tenacious people who never left it, and the government devised soil conservation measures. The Earth turned out to be more durable, and the people who wrested their livings from the earth more resilient, than had been thought.\n\n[APRIL 29, 2007]\n\n# **The Emblematic Novel of the 1930s (No, It Is Not About the Joads)**\n\n**C** onfined to her bed in Atlanta by a broken ankle and arthritis, her husband gave her a stack of blank paper and said, \"Write a book.\" Did she ever.\n\nThe novel's first title became its last words, \"Tomorrow Is Another Day,\" and at first she named the protagonist Pansy. But Pansy became Scarlett, and the title of the book published seventy years ago this week became _Gone With the Wind._\n\nYou might think that John Steinbeck, not Margaret Mitchell, was the emblematic novelist of the 1930s, and that the publishing event in American fiction in that difficult decade was his _The Grapes of Wrath._ Published in 1939, it captured the Depression experience that many Americans had, and that many more lived in fear of. Steinbeck's novel became a great movie and by now 14 million copies of the book have been sold.\n\nBut although the $3 price of _Gone With the Wind_ ($43.50 in today's dollars) was steep by Depression standards, it sold 178,000 copies in three weeks and 2 million by April 1938, when it ended a twenty-one-month run on the bestseller list. By now nearly 30 million have been sold. About 250,000 are still purchased in America every year, and 100,000 elsewhere.\n\nIn 1935, there had been an early indicator of the American yearning that Mitchell's novel satisfied. That year saw the publication of the final two volumes of another durable work of Southern sympathy, Douglas Southall Freeman's Pulitzer Prize\u2013winning four-volume biography of Robert E. Lee. What was afoot?\n\nBy the middle of the 1930s, with the Depression entering its second half-decade and showing no sign of succumbing to the New Deal's attempts to end it, Americans were rightly skeptical about the idea that happy days would soon be here again. Their world having been turned upside down, they saw a parallel between their plight and the story of the disappearance of the antebellum South. Hence their embrace of Mitchell's epic about a society pulverized to human dust that is blown about by history's leveling wind.\n\nParts of the novel reek of magnolia and cloying sentimentalism. But Mitchell writes sarcastic passages about the Lost Cause:\n\n\"How could anything but overwhelming victory come to a Cause as just and right as theirs?...Of course, there were empty chairs and babies who would never see their fathers' faces and unmarked graves by lonely Virginia creeks and in the still mountains of Tennessee, but was that too great a price to pay for such a Cause?\"\n\nScarlett certainly was no sentimentalist. When Rhett Butler, the embodiment of unapologetic realism, asks her if she ever thinks \"of anything but money,\" she replies with words that struck a chord with a nation that had heard quite enough of the song \"Brother, Can You Spare a Dime?\": \"No.... I've found out that money is the most important thing in the world and, as God is my witness, I don't ever intend to be without it again.\"\n\nIn 1936, the _Washington Post_ reviewer called the novel \"unsurpassed in the whole of American writing,\" which was a bit strong, considering what Hawthorne, Melville, Twain, and Wharton had produced. What could, however, accurately have been said of _Gone With the Wind_ was that it was the most cinematic novel yet written in America. A month after it was published, $50,000 was paid for the rights to turn it into the movie that has grossed (adjusted for ticket-price inflation) a record $3.8 billion worldwide.\n\nLike another Southern woman who wrote a novel about her region, a novel that is still in print nearly half a century later and that became a classic movie (Harper Lee, _To Kill a Mockingbird,_ published in 1960), Mitchell never wrote another. In 1949, at age forty-eight, she was killed by a taxi driven by a drunk in Atlanta, which was already on its way to becoming the symbol of the New South.\n\nMitchell had been born in 1900, just thirty-five years after Appomattox and twenty-three years after Reconstruction ended. Her sensibilities were not what ours are. The novel has passages that cannot be read without cringing. (\"Not trust a darky! Scarlett trusted them far more than most white people.... They still stuck with their white folks and worked much harder than they ever worked in slave times.\") But to read such passages is to be stunned, once again, by the amazing speed with which America has changed for the better. In 1936, in Mitchell's Atlanta, the pastor of the Ebenezer Baptist Church, Martin Luther King, had a son who was seven.\n\n[JUNE 25, 2006]\n\n# **All Quiet at the Overpass**\n\n**D** EARBORN, MICHIGAN\u2014A suitable venue for contemplating organized labor's current disarray is here, at the footbridge over Miller Road. In 1937, it led to the main entrance of the foremost example of America's manufacturing might\u2014the Ford Motor Company's River Rouge plant, then the world's most fully integrated car-manufacturing facility, from blast furnaces to assembly line. Five years later, the plant would exemplify America as the \"arsenal of democracy.\" It made jeeps, tanks, trucks, and engines for B-24 bombers. But on May 26, the footbridge to the plant made history.\n\n\"The Battle of the Overpass,\" a heroic event in American labor history, began when Walter Reuther, president of UAW Local 174, and three colleagues started across the footbridge to distribute leaflets as part of their campaign to unionize the plant. They were savagely beaten by perhaps forty Ford thugs and thrown down the overpass stairs. The thugs confiscated most photographers' film, but James (Scotty) Kilpatrick of the _Detroit News_ surrendered only blank film. His pictures made Reuther a national figure and aroused American opinion against tactics then used to thwart unionization.\n\nFord came to terms with the UAW in May 1941, seven months before American industry was conscripted into war. Even though women flooded into factories (Rose Will Monroe, who as \"Rosie the Riveter\" symbolized this social transformation, worked in Ford's Willow Run plant), labor was scarce and wage controls limited companies' means of competing for workers. So companies offered medical and pension benefits as untaxed compensation not covered by wage controls. Today whole industries are buckling beneath the weight of these \"legacy costs.\"\n\nBy 1955, 33 percent of the nation's nonfarm workforce was unionized. But few government employees were. As Steven Malanga of the Manhattan Institute writes in his book _The New New Left: How American Politics Works Today,_ the assumption was that because there is no competition in the delivery of government services, strikes could cripple cities. That was then.\n\nNow, Bentonville, Arkansas (population: 28,000, up 40 percent in four years), has supplanted Detroit as the home of the nation's largest company. Intense and protracted efforts by organized labor have failed to unionize a single one of Wal-Mart's 3,190 North American stores. General Motors, the largest corporation in 1955, is unionized. Two credit-rating agencies have reduced its debt, and one has reduced Ford's to junk-bond status as both companies struggle to finance medical and pension benefits for current and\u2014even more numerous\u2014retired workers. Such costs also are major reasons for the parlous condition of all the older airlines.\n\nIn 1955, when Japan was still struggling to recover from the damage done by B-24s with their Ford-built engines, the American automobile industry was riding high and the UAW was riding along. With negligible foreign competition\u2014their market share was 95 percent\u2014American car companies could pass along to consumers the costs of labor contracts. Today, while domestic carmakers are planning to shed jobs by the thousands, employment is surging in the nonunionized plants, most in the South, where foreign automakers build one-quarter of all the cars and trucks made in America.\n\nIn the 1930s, American workers were literally fighting to get into unions. Today, unions are fighting with themselves about the appropriate tactics to adopt in response to the fact that just 7.9 percent of private-sector workers are union members. But at the apogee of the American labor movement in the 1950s, the transformation of the movement began.\n\nIn 1958, the American Federation of State, County and Municipal Employees won collective-bargaining rights from New York mayor Robert Wagner\u2014son of Senator Robert Wagner, author of the Wagner Act, aka the National Labor Relations Act of 1935, the most important federal action enabling private-sector unionization. AFSCME's members rose from one hundred thousand in 1955 to 1 million in 1985. It has 1.4 million members today and is just one of many government employees' unions.\n\nThe River Rouge complex, which is still humming, is a National Historic Landmark, and \"the Battle of the Overpass\" and the UAW's successes are commemorated at a plaza\u2014built by Ford\u2014on Miller Road, by the footbridge. In the auto industry, where labor and management used to fight over the allocation of abundance, the coming showdown will be over the reduction of benefits won in palmier days.\n\nSoon, perhaps, a majority of organized labor will be government employees. The labor movement will be primarily government organized as an interest group to lobby and pressure itself. Already New York City, which has about the same size population it had forty years ago, has 30 percent more city employees. Antonio Villaraigosa, the new mayor of Los Angeles, is a former organizer for that city's teachers union. The footbridge over Miller Road led to this. The heroic era of organized labor is long gone.\n\n[JULY 18, 2005]\n\n# **FDR's Transformation of Liberalism**\n\n**S** ome mornings during the autumn of 1933, when the unemployment rate was 22 percent, the president, before getting into his wheelchair, sat in bed, surrounded by economic advisers, setting the price of gold. One morning he said he might raise it twenty-one cents: \"It's a lucky number because it's three times seven.\" His treasury secretary wrote that if anybody knew how gold was priced \"they would be frightened.\"\n\nThe Depression's persistence, partly a result of such policy flippancy, was frightening. In 1937, during the depression within the Depression, there occurred the steepest drop in industrial production ever recorded. By January 1938, the unemployment rate was back up to 17.4 percent. The war, not the New Deal, defeated the Depression. Franklin Roosevelt's success was in altering the practice of American politics.\n\nThis transformation was actually assisted by the misguided policies\u2014including government-created uncertainties that paralyzed investors\u2014that prolonged the Depression. This seemed to validate the notion that the crisis was permanent, so government must be forever hyperactive.\n\nIn his second inaugural address, Roosevelt sought \"unimagined power\" to enforce the \"proper subordination\" of private power to public power. He got it, and the fact that the federal government he created now seems utterly unexceptional suggests a need for what Amity Shlaes does in a new book. She takes thorough exception to the government he created.\n\nRepublicans had long practiced limited interest-group politics on behalf of business with tariffs, gifts of land to railroads, and other corporate welfare. Roosevelt, however, made interest-group politics systematic and routine. New Deal policies were calculated to _create_ many constituencies\u2014labor, retirees, farmers, union members\u2014to be dependent on government.\n\nBefore the 1930s, the adjective \"liberal\" denoted policies of individualism and individual rights; since Roosevelt, it has primarily pertained to the politics of group interests. So writes Shlaes, a columnist for Bloomberg News, in _The Forgotten Man: A New History of the Great Depression._ She says Roosevelt's wager was that by furiously using legislation and regulations to multiply federally favored groups, and by rhetorically pitting those favored by government against the unfavored, he could create a permanent majority coalition.\n\nIn the process, says Shlaes, Roosevelt refined his definition of the \"forgotten man.\" This man had been thought of as a general personality, compatible with the assumption that Americans were all in it together. \"Now, by defining his forgotten man as the specific groups he would help, the president was in effect forgetting the rest\u2014creating a new forgotten man. The country was splitting into those who were Roosevelt's favorites and everyone else.\"\n\nActing with what Shlaes calls \"the restlessness of the invalid,\" Roosevelt implemented the theory that (in her words) \"spending promoted growth, if government was big enough to spend enough.\" In only twelve months, just one Roosevelt improvisation, the National Recovery Administration, \"generated more paper than the entire legislative output of the federal government since 1789.\"\n\nBefore Roosevelt, the federal government was unimpressive relative to the private sector. Under Calvin Coolidge, the last pre-Depression president, its revenues averaged 4 percent of GDP, compared to 18.6 percent today. In 1910, Congress legislated height limits for Washington buildings, limits that prevented skyscrapers, symbols of mighty business, from overshadowing the Capitol, the symbol of government.\n\nIn 1936, for the first time in peacetime history, federal spending exceeded that of the states and localities combined. Roosevelt said modern \"civilization\" has tended \"to make life insecure.\" Hence Social Security, which had the added purpose of encouraging workers to retire, thereby opening jobs to younger people. Notice the assumptions of permanent scarcity, and that the government has a duty to distribute scarce things, such as work.\n\nIn 1938, when the New Deal's failure to spark recovery made Roosevelt increasingly frantic, he attempted to enlarge the Supreme Court so he could pack it with compliant justices. He said Americans had the right to \"insist that every agency of popular government\" respond to \"their will.\" He included the court among \"popular,\" meaning political and representative, institutions.\n\nRoosevelt's overreaching called forth an opponent whom Shlaes rescues from obscurity. Wendell Willkie, who would be Roosevelt's opponent in a 1940 election overshadowed by war, called upon Roosevelt to \"give up this vested interest you have in depression\" as the justification for a \"philosophy of distributed scarcity.\"\n\nWar, as has been said\u2014and as George W. Bush's assertion of vast presidential powers attests\u2014is the health of the state. But as Roosevelt demonstrated and Shlaes reminds us, compassion, understood as making the \"insecure\" securely dependent, also makes the state flourish.\n\n[JULY 8, 2007]\n\n# **Retailers Give Thanks for Thanksgiving (and FDR)**\n\n_Thirty days hath September,_\n\n_April, June and November._\n\n_All the rest have thirty-one,_\n\n_Until we hear from Washington._\n\n**T** he country heard from Washington\u2014the man, not the place\u2014when he issued a National Thanksgiving Proclamation for November 26, 1789. The new nation had much for which to be thankful, including the fact that it would be 150 years before Thanksgiving was officially made into a handmaiden of commerce and turned into the starting gun for the sprint of Christmas shopping.\n\nBy now the sprint is a marathon that seems to begin around Labor Day. Soon there will be after-Christmas sales before Halloween, such is the relentless expansion of what is called, with telling vagueness, \"the holiday season.\"\n\nThe country heard from Washington\u2014the place; the mentality\u2014in 1939 when President Franklin Roosevelt threw Thanksgiving into the battle to get happy days here again. FDR's governmental hyper-kinesia had failed to banish the Depression. Unemployment was still 17.2 percent and the ultimate cure for the Depression\u2014Admiral Yamamoto's fleet approaching Hawaii\u2014was still twenty-four months over the horizon.\n\nEven the calendar was conspiring against prosperity because in 1939, as in FDR's first year in office, 1933, November had five Thursdays, and Thanksgiving was to fall on the thirtieth. So FDR moved Thanksgiving from the last Thursday in November to the fourth.\n\nAlthough President Washington was a Virginian, the idea of a national Thanksgiving Day had seemed somehow New Englandish, tainted by Yankee sanctimony and, worse still, Federalist notions of national supremacy over states' prerogatives. Even President John Quincy Adams of Massachusetts thought a national Thanksgiving observation might be \"introducing New England manners\" where they were unwelcome.\n\nPresident Lincoln, a great affirmer of the national facets of the nation's life, was the first to set the last Thursday in November as the national day of Thanksgiving. Bill Kauffman, who has explained all this (in \"New Deal Turkey,\" the _American Enterprise_ magazine, December 2000), says Lincoln's successor, President Andrew Johnson, who had quite enough fights on his hands without picking another one\u2014he was the first president impeached\u2014shoved Thanksgiving into December. President U. S. Grant, who rarely retreated but knew how to, put it back to the last Thursday in November.\n\nBut Appomattox notwithstanding, states remained free to do as they liked about Thanksgiving, and Southern states liked to observe it when they chose. Or not at all, as in Texas during the governorship of Oran Milo Roberts, who said, \"It's a damned Yankee institution anyway.\"\n\nBut in 1939, many of the nation's larger merchants\u2014the National Retail Dry Goods Association, the presidents of Gimbel Brothers and Lord & Taylor\u2014asked FDR for relief from the fact that in 1939 Thanksgiving would arrive so late\u2014November 30\u2014that it would injure the economy by delaying the start of Christmas shopping.\n\nHowever, the class struggle erupted, pitting smaller merchants against the larger merchants. The proprietor of Arnold's Men's Shop in Brooklyn wrote to urge FDR to allow the later Thanksgiving: \"If the large department stores are overcrowded during the shorter shopping period before Christmas, the overflow will come, naturally, to the neighborhood store.... We have waited many years for a late Thanksgiving to give us an advantage over the large stores.\"\n\nFDR felt the pain of the large merchants. But some people felt pained by FDR's tampering with Thanksgiving, including Oregon's attorney general, author of the doggerel printed above. A West Virginian wrote FDR to say, while you are at it, please declare it \"strictly against the Will of God to work on Tuesday\" and \"have Sunday changed to Wednesday.\" A South Dakota real estate man admonished FDR to \"remember we are not running a Russia or communistic government,\" and he added: \"Between your ideas of running for a third term, and your changing dates of century-old holidays, we believe you have practically lost your popularity and the good will of the people of the Northwest.\" FDR lost South Dakota in 1940.\n\nBut in 1939, twenty-three states followed FDR's lead and celebrated Thanksgiving on November 23. Twenty-three stayed with November 30. Colorado and Texas celebrated on both days, Texas doing so to avoid having to reschedule\u2014speaking of things to give thanks for\u2014the Texas\u2013Texas A&M football game.\n\nFDR, who enjoyed fiddling with things, promised in 1941 to return Thanksgiving to the last Thursday in November. But history has its hold on us and Congress shoved it back to the fourth Thursday, partly because many constituents believed the Pilgrims had put it there in the first place.\n\n[NOVEMBER 27, 2003]\n\n# **FDR's Christmas Guest from Hell**\n\n**I** magine how tiresome it would be to have, at Christmas, a houseguest of whom your spouse disapproves and who you have met only twice before, the first time twenty-three years ago (annoyingly, your guest does not remember the meeting), the second time four months ago, for a few hours, out of town, on business. Imagine that the houseguest invites himself to your home, stays almost three weeks, and one morning early on during his stay he summons your butler (you don't have one? pity) and issues the following ukase:\n\n\"Now, Fields, we had a lovely dinner last night but I have a few orders for you. We want to leave here as friends, right? So I need you to listen. One, I don't like talking outside my quarters; two, I hate whistling in the corridors; and three, I must have a tumbler of sherry in my room before breakfast, a couple of glasses of scotch and soda before lunch and French champagne and 90-year old brandy before I go to sleep at night.\"\n\nFurthermore, this Guest from Hell declares that for breakfast he requires hot \"eggs, bacon or ham and toast\" and \"two kinds of cold meats with English mustard and two kinds of fruit plus a tumbler of sherry.\" You would be forgiven for asking your guest if he had been born in a palace.\n\nHe who so firmly addressed President Franklin Roosevelt's butler Alonzo Fields sixty-four Christmases ago was, in fact, born in Blenheim Palace, England's gift to the first Duke of Marlborough. And if no whistling and lots of sherry and whiskey would help the duke's great-great-great-great-great-great-grandson, Winston Churchill, function, stop whistling and pour liberally. There is a war to win.\n\nThe story of this December 1941 visit is told by two Canadians, David Bercuson and Holger Herwig, in an entertaining book with an idiotic subtitle, _One Christmas in Washington: The Secret Meeting Between Roosevelt and Churchill That Changed the World. Secret_ meeting? It was about as secret as a circus, featuring a press conference with FDR and a speech to a joint session of Congress in which Churchill said: \"I cannot help reflecting that if my father had been American and my mother British, instead of the other way round, I might have got here on my own.\" But the meeting did change the world by constructing the machinery of cooperation that led to the defeat of the Axis.\n\nHow ancient it now seems, 1941. The city of Washington had fifteen thousand outdoor privies. German U-boats sank 432 ships in the Atlantic. In August, FDR could deceive everyone, including the Secret Service, for a really secret meeting with Churchill\u2014their only previous meeting had been at a London dinner in 1918\u2014at Placentia Bay, Newfoundland. In the days after Pearl Harbor, some of the antiaircraft guns on the White House were wooden fakes\u2014real ones were scarce. On his voyage, sometimes through forty-foot waves, to his Christmas visit with FDR, Churchill watched American movies, including _Santa Fe Trail,_ starring Errol Flynn, Olivia de Havilland, and Ronald Reagan.\n\nFDR greeted Churchill in Washington in a black limousine the Treasury Department had confiscated from a tax evader named Al Capone. Churchill met here with Admiral Ernest King, commander in chief of the U.S. fleet, who had served in the Spanish-American War, and with General Henry \"Hap\" Arnold, the head of the Army Air Forces, who in 1911 received flight training in Dayton, Ohio, from the Wright brothers.\n\nWhat could have been the most important event of Churchill's almost three weeks in America was not known until his doctor published his memoirs in 1966: Churchill suffered a heart attack while straining to open a stuck window in his White House bedroom. Had it been fatal, that could have changed the world.\n\nEleanor Roosevelt disapproved of Churchill the imperialist, but on Christmas Day 1941 she, he, and the president attended Washington's Foundry Methodist Church, the second iteration of a church founded by Henry Foxall, who in 1812 vowed that he would build a church as a thanksgiving offering if the British did not destroy his cannon foundry when they took Washington and burned the White House.\n\nChristmas Day was the birthday of General Sir John Dill, chief of the Imperial General Staff, so a cake was found and adorned with a set of American and British flags which, Dill discovered when he removed them, were made in Japan. This occasioned laughter, at a time when that, like much else, was scarce.\n\n[DECEMBER 25, 2005]\n\n# **\"My Place Is with My Shipmates\"**\n\n**P** EARL HARBOR, HAWAII\u2014Sixty-one years later, it is still not over.\n\nOne by one, oil drops still seep from the submerged hull of the USS _Arizona._ And one by one, some men who survived the bomb that hit the battleship's forward magazine still return, the urns containing their ashes lowered through the water into the hole of a gun turret, because as one of them said, \"Ever since December 7, 1941, I've been living on borrowed time. My place is with my shipmates.\"\n\nHere, about as far as you can get in America from the scenes of last year's attacks, you see this difference: In 1941, a mighty empire\u2014an enemy with a serious if reckless geopolitical strategy\u2014struck at real sinews of U.S. power. In 2001, a delusional, premodern enemy lashed out at American symbols\u2014iconic buildings\u2014and instantly magnified American power by dispelling an American mood. Call it end-of-history complacency.\n\nThe attack that came here sixty-one years ago from across the broadest ocean erased forever the belief that geography\u2014wide oceans, placid neighbors\u2014confers permanent security on America. The attacks last year erased the comparably soothing belief that the logic of military technology (deterrence) and the march of modernity (the retreat of primitivism) had written an end to history, meaning the immunity of great powers to attacks.\n\nSixty-one Decembers ago, as last year, America suffered from intelligence failures. But in both instances, for the officials charged with protecting the nation's security, the attacks came not as bolts from the blue but as bolts from what they knew to be ominously darkening skies.\n\nBoth times American officials knew enough to know that the international atmosphere was charged with hidden menace. Which is why the first shots fired on December 7, 1941, were fired by America's wary military: At 6:45 a.m., the destroyer USS _Ward_ attacked one of the two-man midget submarines\u2014its wreckage was found two weeks ago\u2014lurking at the mouth of the harbor, poised to participate in the attack that was still seventy minutes over the horizon.\n\nThe attacks of December 7, 1941, like the attacks of September 11, 2001, were a curious mixture of virtuosity and primitivism. Al Qaeda skillfully used nineteen suicidal fanatics wielding box cutters and commercial airliners to attack a continent. It was deadly, and absurd.\n\nJapan's military achievement\u2014moving thirty-two warships four thousand miles undetected; designing shallow-running torpedoes; brilliantly coordinating the bomb and torpedo attacks\u2014was military sophistication of the highest order. Yet the pilots skimming thirty feet over the water used carpenters' levels in their cockpits to make sure their planes were properly positioned to insert their torpedoes into the water.\n\nThe Americans who died here\u2014on the _Arizona,_ twenty-three sets of brothers, and a father and his son\u2014were mostly military men. Those whom the terrorists targeted last September were mostly civilians.\n\nFor all Americans, being a focus of furies\u2014which a muscular nation, extending almost five thousand miles from the cavity in southern Manhattan to the _Arizona_ 's hull, will be\u2014is a dangerous destiny. It is a destiny that, in a sense, was just dawning 104 years ago when the USS _Baltimore_ sailed from here.\n\nOn March 25, 1898, that cruiser left to join Commodore Dewey's fleet in Hong Kong. It entered Manila Bay with the fleet on May 1 and participated in the destruction of the Spanish fleet. The Spanish-American War established the United States as a global power, its power projected then entirely by its Navy. In 1941, an important portion of the Navy was based here because\u2014westward the course of empire takes its way\u2014the United States had annexed these islands in that eventful year of 1898.\n\nOn a December Sunday forty-three years later, the _Baltimore,_ which had been decommissioned in 1922, was a ghost ship moored at the end of battleship row, where it escaped damage. But in a sense its career was still not over. In 1942, it was turned into scrap. No doubt some of it was sent back to war in bits and pieces.\n\nHalf a million gallons of fuel oil remain in the _Arizona_ 's hull. With leakage of a quart a day, no one now living will be alive when the surface sheen from the last drop drifts away. And no one now living will live to see a day when Americans forget the lesson now associated with September 11 as well as December 7: A powerful nation embodying a powerful idea and spanning six time zones is permanently exposed to dangers from all the other eighteen zones.\n\n[SEPTEMBER 8, 2002]\n\n# **An Anthem of American Optimism\u2014in 1943**\n\n_Every night my honey lamb and I_\n\n_sit alone and talk_\n\n_and watch a hawk_\n\n_makin' lazy circles in the sky_\n\n\u2014\"Oklahoma!\"\n\n**N** EW YORK\u2014\"Honey lamb\"? That is as corny as Kansas in August. Perhaps such lyrics please people in Manhattan, Kansas, where the waving wheat can sure smell sweet when the wind comes right behind the rain. But surely the sophisticates in this Manhattan prefer to tap their patent leather dancing pumps to the urbane lyrics of Ira Gershwin.\n\nSurely not. The show-business collaboration of Richard Rodgers and Oscar Hammerstein long ago ended, but their melodies and lyrics linger on. In fact, rather more than just linger. They reverberate. Advance ticket sales topped $12 million by the time the revival of _Oklahoma!_ opened two weeks ago\u2014in the Gershwin Theater, so there.\n\nWhoever said that America's imperishable gifts to the world are the Constitution, baseball, and jazz should have included a fourth: a distinctive kind of stage musical, the greatest of which is _Oklahoma!_ It took that top ranking away from _Show Boat_ (1927) fifty-nine years ago and has kept it ever since. _Oklahoma!_ was the first \"integrated\" musical, meaning the singing and dancing arose organically from the action, advancing rather than interrupting the narrative. Audiences were ready for this maturation of the musical.\n\nWhen _Oklahoma!_ first came to Broadway, the opening\u2014it was a sleety March night\u2014was not sold out. Soon tickets were scarce. However, servicemen in uniform were admitted to standing room without charge. It was 1943, a terrible year.\n\nBut the first words of the first song were\u2014are\u2014\"There's a bright, golden haze on the meadow.\" The song, \"Oh, What a Beautiful Mornin'!\" was an anthem of an American optimism that not even world war could dent. It is serendipitous that this revival, which played for four years in London, arrives with the nation again at war.\n\nAnd, as usual, the nation is fretting, as Oklahoma Territory fretted, about keeping the peace between disparate and sometimes rivalrous factions, such as \"The Farmer and the Cowman\" of the rousing number that opens the second act. But cultural events are always filtered through the mental lenses of the moment, so Ben Brantley of the _New York Times_ writes that this revival of _Oklahoma!_ is saturated with...well, he says:\n\nIt suggests \"the West was won on the strength of sexual hormones.\" It finds \"rushing erotic currents in the frontier spirit.\" It is \"dewy with an adolescent lustiness\" and a \"darker sexual element.\" There is \"a glistening sense of young people eagerly groping their way through an unfamiliar landscape\" and that a world parallel to \"the virgin land\" of Oklahoma Territory is the \"shadowy realm of sexual initiation.\" And the choreography\u2014as in \"The Farmer and the Cowman,\" which has \"wild, procreative energy\"\u2014vibrates \"with the sense of sensual restlessness in search of an outlet.\"\n\nWhoa. Perhaps Brantley should take a long walk or a cold shower, or the latter after the former. Yes, boys and girls in Oklahoma Territory had a keen interest in girls and boys. Otherwise there would not have been so many subsequent Oklahomans. But _Oklahoma!_ is about which boy\u2014the sunny cowboy Curly or the glowering hired hand Jud Fry\u2014should take the girl, Laurey, to the box social, for Pete's sake. How steamy can that be?\n\nSome critics whose admiration for this revival of _Oklahoma!_ is as high as an elephant's eye nevertheless emphasize its \"dark\" side. But this is not new. Curly has been fighting Jud Fry since March 31, 1943, when Fry first stabbed himself to death with his own knife while fighting with Curly. And for fifty-nine years, before every final curtain, Curly has been quickly acquitted of manslaughter.\n\nIs this \"dark\"? Ethan Mordden, a historian of Broadway musicals, notes that _Oklahoma!_ is especially American in presenting \"the unpleasant truth that evil will keep coming at you until you kill it. One piece of democracy is the harmonizing of discordant agendas. But another piece is the expunging of the wicked.\" The revival of _Oklahoma!_ is timely, Mordden says, \"because it defines Americans as a people, generous but plainspoken and tough on spoilers.\"\n\nThe first Broadway run of _Oklahoma!_ lasted five years and nine months, a record not broken until _My Fair Lady_ ran from 1956 to 1962. But in a sense _Oklahoma!_ has never closed since 1943. In a normal year, there are about six hundred new North American productions of it. It is part of the permanent music of the American people, who know that the land they belong to is grand.\n\n[APRIL 7, 2002]\n\n# **When War _Was_ the Answer**\n\n**O** MAHA BEACH, NORMANDY\u2014On a bluff above the sand and a half a mile from the ocean's edge at low tide, which was the condition when the first Allied soldiers left their landing craft, a round circle of concrete five feet in diameter provides a collar for a hole in the ground. On the morning of June 6, 1944, the hole was Widerstandsnest (nest of resistance) 62, a German machine-gun emplacement.\n\nHein Severloh had been in it since shortly after midnight, by which time U.S. aircraft were droning overhead, having dropped young American paratroopers Severloh's age behind the beaches in order to disrupt German attempts to rush in reinforcements. Severloh had been billeted near Bayeux, home of the eleventh-century tapestry depicting a cross-channel invasion that went the other way, taking William, Duke of Normandy, to become William the Conqueror, England's sovereign.\n\nSeverloh believed he killed hundreds of GIs, so long and slow was their walk to the safety, such as it was, of the five-foot embankment where the beach meets the bluff. Severloh returned here in sorrow and was consoled by survivors of the forces that waded ashore.\n\nToday, in an America understandably weary of a war of choice that has been defined by execrable choices, a frequently seen bumper sticker proclaims: \"War is not the answer.\" But here, especially, it is well to remember that whether war is the answer depends on the question.\n\nWar was the answer to what ailed Europe in 1944. \"In 1942,\" writes Timothy Garton Ash of Oxford and Stanford's Hoover Institution, \"there were only four perilously free countries in Europe: Britain, Switzerland, Sweden and Ireland.\" Twenty years\u2014a historical blink\u2014later, almost all of Western Europe was free. Twenty years after that, Spain, Portugal, and Greece had joined the liberal democracies. Today, for the first time in twenty-five hundred years, most Europeans live under such governments.\n\nGarton Ash argues that Europe cannot define itself negatively\u2014as _not_ America or _not_ Islam. \"Europe's only defining 'other' is its own previous self\"\u2014its self-destructive, sometimes barbaric past. \"This is,\" Garton Ash says, \"still a very recent past.\"\n\nIn 1951, just seven years after Severloh and some other Germans surrendered on June 7 to Americans at the village of Saint-Laurent, Europe began building the institutions that, it hoped, would keep such young men out of machine gun emplacements. It created the European Coal and Steel Community, precursor of the Common Market (1958), which led to the single market in 1993 and the common currency in 2002.\n\nThe implicit hope was that commerce could tame Europe's turbulent nations. The perennial problem of politics\u2014mankind's susceptibility to storms of passions\u2014could perhaps be solved, or at least substantially ameliorated, by getting Europe's peoples to sublimate their energies in economic activities. The quest for improved material well-being would drain away energies hitherto tapped and channeled by demagogues.\n\nReminders of Europe's problematic past were recently found a few miles from Saint-Laurent. Workers preparing a foundation for a new house overlooking Omaha Beach came upon parts of the bodies of two German soldiers. There was scant media attention to this because such discoveries have not been rare.\n\nAlso near here, 21,160 German soldiers are buried at the La Cambe Cemetery. Thirty percent\u2014more than 6,000\u2014were never identified, so some German parents conducted \"assumed burials.\" They placed metal markers bearing the names of their missing sons near the graves of unknown soldiers who were known to have died near where the parents' sons were last known to be fighting.\n\nSuch heartbreaking stories are written into Normandy's lovely landscape. At the American Cemetery overlooking this beach, amid the many rows of white marble gravestones, are two, side by side, marking the burial places of Ollie Reed and Ollie Reed Jr., a father and his son. The son, an Army first lieutenant, died in Italy on July 6. His father, an Army colonel, was killed July 30 in Normandy. Two telegrams notified the father's wife, the son's mother. The telegrams arrived in Manhattan, Kansas, forty-five minutes apart.\n\nThe nineteenth-century French scholar Ernest Renan, from a Brittany town on the English Channel, defined a nation as a community of shared memory\u2014and shared forgetting. Europe's emotional equipoise, and the transformation of \"Europe\" from a geographical to a political expression, has required both remembering and forgetting. Americans who make pilgrimages to this haunting place are reminded of their role, and their stake, in that transformation.\n\n[SEPTEMBER 2, 2007]\n\n# **Catching Up to Captain Philip**\n\n_\"Don't cheer, boys. The poor devils are dying.\"_\n\n\u2014CAPTAIN JOHN PHILIP of the USS _Texas_ to his crew as they watched the Spanish ship _Vizcaya_ burn off Santiago Bay, Cuba, in 1898\n\n**O** n March 9, 1945, 346 B-29s left the Marianas, bound for Tokyo, where they dropped 1,858 tons of incendiaries that destroyed one-sixth of Japan's capital, killing eighty-three thousand. General Curtis LeMay, then commander of the air assault on Japan, later wrote, \"We scorched and boiled and baked to death more people in Tokyo...than went up in vapor at Hiroshima and Nagasaki combined.\"\n\nThat was inaccurate\u2014eighty thousand died at Hiroshima alone. And in his new biography of LeMay, Barrett Tillman writes that the general was more empathetic than his rhetoric suggested: \"He could envision a three-year-old girl screaming for her mother in a burning house.\" But LeMay was a warrior \"whose government gave him a task that required killing large numbers of enemy civilians so the war could be won.\"\n\nIt has been hotly debated how much indiscriminate killing of civilians in the Asian and European theaters really was \"required\" and therefore was morally permissible. Even during the war there was empathy for civilian victims, at least European victims. And less than fifteen years after the war, movies (e.g., _The Young Lions,_ 1958) offered sympathetic portrayals of common German soldiers swept into combat by the cyclone of a war launched by a tyrant.\n\nBut attitudes about the Japanese soldier were especially harsh during the war and have been less softened by time than have attitudes about the German soldier. During the war, it was acceptable for a billboard\u2014signed by Admiral William F. \"Bull\" Halsey\u2014at a U.S. Navy base in the South Pacific to exhort kill japs, kill japs, kill more japs. Killing America's enemies was Halsey's trade. His rhetoric, however, was symptomatic of the special ferocity, rooted in race, of the war against Japan: \"We are drowning and burning them all over the Pacific, and it is just as much pleasure to burn them as to drown them.\" Halsey endorsed the Chinese proverb that the \"Jap race\" was the result of \"a mating between female apes and the worst Chinese criminals.\"\n\nWartime signs in West Coast restaurants announced: this restaurant poisons both rats and japs. In 1943, the Navy's representative on the committee considering what should be done with a defeated Japan recommended genocide\u2014\"the almost total elimination of the Japanese as a race.\"\n\nStephen Hunter, movie critic for the _Washington Post,_ says that of the more than six hundred English-language movies made about World War II since 1940, only four\u2014most notably _The Bridge on the River Kwai_ (1957)\u2014\"have even acknowledged the humanity\" of Japanese soldiers.\n\nPerhaps empathy for the plight of the common enemy conscript is a postwar luxury; it certainly is a civilized achievement, an achievement of moral imagination that often needs the assistance of art. That is why it is notable that Clint Eastwood's _Letters from Iwo Jima_ was one of five films nominated for Best Picture.\n\nIt is stressful viewing. An unsparing attempt to come as close as cinema can to conveying the reality of combat, specifically the fighting that killed 6,821 Americans and all but 1,083 of the 22,000 Japanese soldiers on the small (eight square miles) black lava island. Remember the searing first fifteen minutes of _Saving Private Ryan_ \u2014the carnage at Omaha Beach? In _Letters from Iwo Jima,_ it is exceeded, with harrowing permutations.\n\nThe Japanese commander on the island, Tadamichi Kuribayashi, was\u2014like the admiral who attacked Pearl Harbor, Isoroku Yamamoto\u2014a cosmopolitan warrior who had lived in, and never stopped admiring, America. In 2005, a team of Japanese archaeologists scouring the island's man-made caves for artifacts of the battle found a sack of undelivered mail from Kuribayashi and other officers and soldiers. All the writers knew they faced overwhelming force\u2014Japan had no assistance to send\u2014and were doomed to die in accordance with the Japanese military code that forbade surrender and encouraged suicide.\n\nJapanese forces frequently committed barbarities worse even than those of the German regular army, and it is difficult to gauge the culpability of conscripts commanded by barbarians. Be that as it may, the pathos of the letters humanizes the Japanese soldiers, whose fatalism was a reasonable response to the irrational. Viewers of this movie, while moved to pride and gratitude by the valor of the U.S. Marines, will not feel inclined to cheer. We are catching up to Captain Philip's sensibility.\n\n[FEBRUARY 25, 2007]\n\n# **The Most Fateful Heart Attack in American History**\n\n**T** he Supreme Court's decision fifty years ago, although an immense blessing to the nation, also carries a melancholy lesson. It is that great events\u2014the school desegregation ruling was the largest judicial event since the Dred Scott case of 1857\u2014have myriad reverberations, some beneficial, others not.\n\n_Brown v. Board of Education_ accelerated the process of bringing this creedal nation into closer conformity to its creed. But the decision also encouraged the abandonment of constitutional reasoning\u2014of constitutional law. It invested the judiciary with a prestige that begot arrogance. And it seemed to legitimize a legislative mentality among judges wielding an anticonstitutional premise. The premise is that \"unjust\" and \"unconstitutional\" are synonyms.\n\nThe board of education being sued for its segregation policies was not in the South, but in Kansas\u2014Topeka. Segregation was widely practiced, and even more widely approved. Yes, in Montgomery, Alabama, it was illegal for a white to play checkers in public with a black. But Congress was running a segregated school system in the nation's capital. In 1948, President Harry Truman could not persuade Congress to make lynching a federal crime.\n\nWhen the case was first argued in 1952, the Supreme Court was composed entirely of Democratic\u2014of Roosevelt and Truman\u2014appointees. And if the court's composition had not been soon and unexpectedly changed by the addition of a Republican nominee, the legal basis of segregation\u2014the doctrine that \"separate but equal\" public facilities are constitutional\u2014probably would have been affirmed.\n\nNo Republican nominee had served on the court since Owen Roberts, a Hoover nominee, resigned in 1945. But in 1953, eight months into Dwight Eisenhower's presidency, there occurred the most fateful heart attack in American history. It killed Chief Justice Fred Vinson, a Kentuckian who believed the \"separate but equal\" doctrine, enunciated in an 1896 decision, should remain.\n\nFour other justices were, to varying degrees, inclined to agree. Cass Sunstein of the University of Chicago Law School, writing in _The New Yorker,_ notes that the waspish Justice Felix Frankfurter said that Vinson's heart attack was \"the first indication that I have had that there is a God.\" But Frankfurter and another liberal-leaning justice, Robert Jackson, were FDR appointees who had learned the virtues of judicial modesty by watching the judicial hubris of the court as it struck down many of FDR's early New Deal measures.\n\nVinson's death preceded a rehearing of the case. His replacement, Earl Warren, former governor of California, was a post\u2013New Deal politician. He was comfortable with the premise that the federal government's responsibilities extend to the general amelioration of citizens' conditions. A man of immense charm in the court's face-to-face politics, he also was impatient with the idea that justices must go only where led by judicial reasoning about the Constitution's text as it has been illuminated by precedents based thereon.\n\nSome Northern states had segregated schools when they ratified the Fourteenth Amendment. It includes the guarantee of \"equal protection of the laws\" that in 1954 the court decided was incompatible with segregated schools. To reach this conclusion, the court cited social science evidence that segregation induced feelings inimical to young children's self-esteem, thereby injuring their capacity to learn.\n\nThat this rationale was window dressing became clear when the court invoked the Brown decision to outlaw segregated beaches, golf courses, etc. The court would have done better with this simple argument:\n\nThe \"separate but equal\" doctrine came from a correct understanding that equality for blacks was the intent of the Fourteenth Amendment. But the court in 1896 erred because, when separation is enforced on racial lines, \"separate but equal\" is inherently oxymoronic.\n\nWhen the Brown ruling was rendered, Thurgood Marshall, the NAACP's lead litigator, expected segregation to be gone in five years. But ten years later, only 1.17 percent of Southern black schoolchildren attended public schools with whites.\n\nIn 1954, the court's majesty could not compel compliance. Today, the court's reserves of prestige are immeasurably greater, partly because of what it did then. What also is much enlarged is the public's belief that judicial fiats can and should remedy many social ills, broadly defined to include the refusal of legislatures to adopt polices deemed just.\n\n\"John Marshall has made his decision, now let him enforce it.\" That supposedly was President Andrew Jackson's response to a Supreme Court decision he disliked. Then, as now, the court's power flowed largely from its prestige, which was not sufficient to bend Old Hickory. No president could act similarly today. This progress owes much to what happened on May 17, 1954.\n\n[MAY 16, 2004]\n\n# **How Ike's Highways Helped Heal Civil War Wounds**\n\n**O** n Tuesday, July 11, the United States will become more geographically stable than it has ever been. It will have been 17,126 days since the admission of Hawaii to statehood on August 21, 1959. The longest previous span between expansions of the nation was the 17,125 days between the admission of Arizona on February 14, 1912, and the admission of Alaska on January 3, 1959. Since then the nation has become, in a sense, smaller through the annihilation of distance and, to some extent, of difference.\n\nAn important part of the groundwork\u2014literally, it covered a lot of ground\u2014for today's America was begun fifty years ago this summer. A conservative Republican president, who grew up in a Kansas town where hitching posts for horses lined unpaved streets, launched what was, and remains, the largest public works project in the nation's history\u2014the Interstate Highway System. Its ribbons of concrete represent a single thread of continuity through the nation's history.\n\nWith that program, Dwight Eisenhower, the thirteenth Republican president, helped heal the wounds of the war won by another general, U. S. Grant, the second Republican president. That war was related to \"internal improvements,\" as infrastructure projects such as roads and canals used to be called.\n\nIn 1816, South Carolina representative John Calhoun\u2014then a nationalist; later, a secessionist\u2014introduced legislation for a federal program of internal improvements. The legislation passed, but President James Madison vetoed it because he thought Congress was not constitutionally empowered to do such things. So, prosperous Northern states built their own improvements while the South sank into inferiority and increasing dependence on slavery.\n\nThe military handicap of an inferior transportation system was one reason the South lost the Civil War. Another reason was the industrialization of the North. Its transportation system (the Erie Canal, railroads) cut the price of shipping a ton of wheat from Buffalo to New York City from $100 to $10, and the difference between the wholesale price of pork in Cincinnati and New York plunged from $9.53 to $1.18. Suddenly, workers flooding into the North's cities had more disposable income to spend on the North's manufactured goods.\n\nThe first Republican president began his public life as a twenty-three-year-old candidate for the Illinois General Assembly by telling voters of Sangamon County his \"sentiments with regard to local affairs,\" the first sentiment being \"the public utility of internal improvements.\" The vigor of the union also was a preoccupation of Teddy Roosevelt, the eighth Republican president, whose great internal improvement, the Panama Canal, was external, although he thought of Panama as America's private property. And Eisenhower's message to Congress advocating the IHS began, \"Our unity as a nation is sustained by free communication of thought and by easy transportation of people and goods.\"\n\nNo legislator more ardently supported the IHS than the Tennessee Democrat who was chairman of the Senate Public Works subcommittee on roads. His state had benefited handsomely from the greatest federal public works project of the prewar period, the Tennessee Valley Authority, which, by bringing electrification to a large swath of the South, accelerated the closing of the regional development gap that had stubbornly persisted since the Civil War. This senator who did so much to put postwar America on roads suitable to bigger, more powerful cars was Al Gore Sr. His son may consider this marriage of concrete and the internal combustion engine sinful, but Tennessee's per capita income, which was just 70 percent of the national average in 1956, today is 90 percent.\n\nThe IHS\u2014combined, as _Fortune_ magazine's Justin Fox writes, with another bright idea from 1956, the shipping container\u2014made America's distribution system more flexible. This benefited manufacturers, foreign and domestic, especially in America's hitherto lagging region, the South. This is one reason there is a thriving Southern-based automobile industry (BMW in South Carolina; Mercedes in Alabama; Honda in both Carolinas, Georgia, and Alabama; Toyota in Tennessee, Alabama, and Kentucky). Furthermore, the South is home to some of today's \"big-box\" retailers\u2014Wal-Mart (Bentonville, Arkansas), Home Depot (Atlanta)\u2014as well as FedEx (Memphis).\n\nAmerican scolds blame the IHS and the automobile for everything from obesity (fried food at every interchange) to desperate housewives (isolated in distant suburbs without sidewalks). Nikita Khrushchev, during his 1959 visit to America, told Eisenhower, \"Your people do not seem to like the place where they live and always want to be on the move going someplace else.\" Eisenhower knew that wherever people are going on their nation's roads, they are going where they live.\n\n[JULY 9, 2006]\n\n# **The Short, Unhappy Life of the Edsel**\n\n**L** eaving no talent untapped in its quest for perfection, the Ford Motor Company asked Marianne Moore, one of America's foremost poets in the 1950s, to suggest a name for the product it would debut in late summer, fifty years ago. She replied: \"May I submit Utopian Turtletop? Do not trouble to answer unless you like it.\"\n\nFord instead named the product for Henry Ford's late son Edsel. The Edsel would live twenty-six months.\n\nThe short, unhappy life of that automobile is rich in lessons, and not only for America's beleaguered automobile industry. The principal lesson is: Most Americans are not as silly as a few Americans suppose.\n\nNo industry boomed more in the 1950s than the manufacturing of social criticism excoriating Americans for their bovine \"conformity,\" crass \"materialism,\" and mindless manipulability at the hands of advertising's \"hidden persuaders.\" Vance Packard's _The Hidden Persuaders_ was atop the _New York Times_ bestseller list as Edsels arrived in showrooms. No consumer product in history had been the subject of so much \"scientific\" psychology-based market research.\n\nRemember the basketball coach who said of his team, \"We're short but we're slow\"? The Edsel was ugly but riddled with malfunctions. So many malfunctions that some people suspected sabotage at plants that had previously assembled Fords and Mercurys. Those two Ford divisions perhaps hoped the Edsel would bomb.\n\n\"It was,\" wrote John Brooks, a student of American business, in _The New Yorker,_ \"clumsy, powerful, dowdy, gauche, well-meaning\u2014a de Kooning woman.\" Chrome seemed to be piled upon chrome. Potential buyers recoiled from the vertical egg-shaped grille, which reminded them of a toilet seat. The transmission was worked by push buttons placed\u2014convenience sacrificed on the altar of novelty\u2014in the center of the steering column. The larger Edsels weighed more than two tons, were 219 inches long\u2014longer than the grandest Oldsmobiles\u2014and 80 inches wide. These were not the cars for a year in which the surprise success was American Motors' little Rambler.\n\nBy Sunday, October 13, barely more than a month after the Edsel's debut, anemic sales caused the company to preempt _The Ed Sullivan Show_ with a Sunday evening Edsel extravaganza featuring Bing Crosby and Frank Sinatra. But there was no sales spurt. Nine days earlier, the Soviet Union had launched its first Sputnik satellite, provoking a crisis of confidence in America's technological prowess and a reaction against chrome-laden barges as emblems of national self-indulgence. On November 27, Manhattan's only Edsel dealer gave up his franchise and switched to selling Ramblers.\n\nIn the spring of 1958, S. I. Hayakawa, a professor of semantics (and later a Republican U.S. senator from California), ascribed the Edsel's failure to the Ford executives' excessive confidence in the power of motivational research to enable them to predict\u2014and modify\u2014Americans' behavior. In their attempt to design a car that would cater to customers' sexual fantasies, status anxieties, and the like, Ford's deep thinkers had neglected to supply good transportation.\n\n\" _Only_ the psychotic and the gravely neurotic _act out_ their irrationalities and their compensatory fantasies,\" Hayakawa wrote. \"The trouble with selling symbolic gratification via such expensive items...is the competition offered by much cheaper forms of symbolic gratification, such as 'Playboy' (fifty cents a copy), 'Astounding Science Fiction' (thirty-five cents a copy), and television (free).\"\n\nIn 1958, with the Edsel already turned to ashes, John Kenneth Galbraith, with bad timing comparable to the launch of the Edsel, published _The Affluent Society._ It asserted that manufacturers, wielding all-powerful advertising, were emancipated by the law of supply and demand because advertisers could manufacture demand for whatever manufacturers wished to supply.\n\nThis theory buttressed the liberal project of expanding government in the name of protecting incompetent Americans from victimization, and having government supplant the market as the allocator of wealth and opportunity. But all of Ford's then-mighty marketing prowess could not keep the Edsel from being canceled in 1959. Brooks calculated that it would have been cheaper for Ford to skip the Edsel and give away 110,000 Mercurys.\n\nToday, the United Auto Workers union and General Motors, Ford, and Chrysler are trying to reverse the slide of the American automobile industry. Fifty Septembers ago, the country was atingle with anticipation of a new product that turned out to be a leading indicator of the slide. As Detroit toils to undo some contractual provisions that have burdened the companies with crippling health care and pension costs, it should remember the real lesson of 1957: Americans are more discerning and less herdable than their cultured despisers suppose, so what matters most is simple. Good products.\n\n[SEPTEMBER 6, 2007]\n\n# **The Fifties in Our Rearview Mirror**\n\n_There was, too, a wonderful simplicity of desire. It was the last time that people would be thrilled to own a toaster or waffle iron._\n\n\u2014BILL BRYSON\n\n**W** hat Thanksgiving is to gluttony, the three days after it are to consumerism\u2014the main event. So, with Americans launching the Christmas season by storming the stores, let us recall when consumption had an exuberance remembered now only by those who experienced the 1950s.\n\nBill Bryson remembers. The author of thirteen books (e.g., _A Walk in the Woods_ and _A Short History of Nearly Everything_ ), Bryson has most recently written _The Life and Times of the Thunderbolt Kid,_ a memoir of growing up in Des Moines in the fifties, when downtown department stores\u2014with white-gloved operators in the elevators and pneumatic tubes carrying money and receipts to and from cashiers\u2014served the pent-up demands of a nation making up for consumption missed during the Depression and World War II.\n\nIn 1951, when the average American ate 50 percent more than the average European, Americans, Bryson says, controlled two-thirds of the world's productive capacity, owned 80 percent of the world's electrical goods, produced more than 40 percent of its electricity, 60 percent of its oil, and 66 percent of its steel. America's 5 percent of the world's population had more wealth than the other 95 percent, and Americans made almost all of what they consumed: 99.93 percent of new cars sold in 1954 were U.S. brands.\n\nBy the end of the fifties, GM was a bigger economic entity than Belgium, and Los Angeles had more cars than did Asia\u2014cars for a gadget-smitten people, cars with Strato-Streak engines, Strato-Flight Hydra-Matic transmissions, and Torsion-Aire suspensions. The 1958 Lincoln Continental was nineteen feet long. And before television arrived (in 1950, 40 percent of Americans had never seen a television program; by May 1953, Boston had more televisions than bathtubs) America made almost a million comic books a month.\n\nConsider what was new or not invented then: ballpoint pens, contact lenses, credit cards, power steering, long-playing records, dishwashers, garbage disposals. And remember words now no longer heard: icebox, dime store, bobby socks, panty raid, canasta (a card game). In 1951, a Tennessee youth was arrested on suspicion of narcotics possession. The brown powder was a new product\u2014instant coffee.\n\nFifties food was, Bryson reminds us, not exotic: In Iowa, at least, folks did not eat foreign food \"except French toast,\" or bread that was not \"white and at least 65 percent air,\" or \"spices other than salt, pepper and maple syrup,\" or \"any cheese that was not a vivid bright yellow and shiny enough to see your reflection in.\"\n\nBut unlike today, when everything edible, from milk to spinach, has its moment as a menace to health, in the fifties everything was good for you. Cigarettes? Healthful. Advertisements, often featuring doctors, said smoking soothed jangled nerves and sharpened minds. \"X-rays,\" Bryson remembers, \"were so benign that shoe stores installed special machines that used them to measure foot sizes.\"\n\nIn Las Vegas, downwind from some atomic weapons tests, government technicians used Geiger counters to measure fallout: \"People lined up to see how radioactive they were. It was all part of the fun. What a joy it was to be indestructible.\" But, Bryson dryly notes, people knew without a warning label \"that bleach was not a refreshing drink.\"\n\nWhite House security precautions were so lax that on April 3, 1956, a somewhat disoriented Michigan woman detached herself from a White House tour and wandered through the building for four hours, setting small fires. When found, she was taken to the kitchen and given a cup of tea. No charges were filed.\n\nThe fifties did have worries. When a contestant on a TV game show said his wife's astrological sign was Cancer, the cigarette company sponsoring the show had the segment refilmed and her sign changed to Aries. You could get fourteen years in an Indiana prison for instigating anyone under age twenty-one to \"commit masturbation.\" And to get a New York fishing license, you had to swear a loyalty oath.\n\nNothing has changed more for the worse since the fifties than childhood. The lives of children were, Bryson remembers, \"unsupervised, unregulated and robustly\" physical. \"Kids were always outdoors\u2014I knew kids who were pushed out the door at eight in the morning and not allowed back in until five unless they were on fire or actively bleeding.\"\n\nBut as the twig is bent, so grows the tree: These children, formed by the fifties, grew up to be Olympic-class shoppers. They are indoors this Sunday, at malls.\n\n[NOVEMBER 26, 2006]\n\n# **2002: Superstitions Are Bad Luck**\n\n**O** nward and upward with homo sapiens. A 7 million-year-old skull uncovered this year in Central Africa belonged to someone the size of a chimpanzee and is the earliest\u2014by about a million years\u2014yet discovered member of the human family. In 2002, his descendants were threatened by savage primitives who, in the name of the Creator, were possibly plotting to reverse, using smallpox spores, one of Homo sapiens' recent triumphs over an infectious scourge. Much the most important event of 2002 was a nonevent\u2014the second major terrorist attack on the American homeland that did not happen. Four homegrown terrorists from the 1970s, members of the Symbionese Liberation Army, pleaded guilty to a murder committed during a 1975 bank robbery in Carmichael, California.\n\nComplex geometric carvings on a rock found this year in a South African cave suggest that complex and abstract thinking began in Africa, not in Cambridge, Massachusetts, and began twice as long ago\u2014seventy-seven thousand years\u2014as had been believed. When did it stop? Trent Lott regretted that Harry Truman rather than Strom Thurmond won the 1948 presidential election.\n\nThe perpetrators of this year's most lurid skullduggery and corruption? No, not Wall Street stock analysts\u2014Olympic figure-skating judges. Not long ago, tycoons were \"masters of the universe.\" This year they were mastering the perp walk. Enron and Arthur Andersen almost vanished. Martha Stewart wished she could. United Airlines and the Boston diocese of the Roman Catholic Church had reason to remember the aphorism of Frank Borman, who was president of Eastern Air Lines before it went bankrupt: \"Capitalism without bankruptcy is like Christianity without Hell.\"\n\nTwo Bronx teenagers, one four feet ten and the other five feet six, are suing McDonald's because they weigh 170 pounds and 270 pounds, respectively. The legal theory behind their suit derives from the Garth Brooks lyric: \"Longneck bottle, let go of my hand.\"\n\nLieutenant John Kennedy's PT 109, sunk in 1943, was found off the Solomon Islands. _Oklahoma!,_ launched in 1943, was back on Broadway. Off-Broadway, Bill Clinton starred in \"It\u2014everything\u2014is all about me.\" Campaigning for Massachusetts gubernatorial candidate Shannon O'Brien, Clinton, whose self-absorption remains one of the wonders of the world, said an O'Brien victory would be \"a wonderful way to celebrate the 10th anniversary of my victory in 1992.\"\n\nA little difference makes a big difference: A mouse's genome has been mapped. Humans and mice have about thirty thousand genes. Less than 1 percent are unique to either species.\n\nAn Italian artist, seeking to make an \"ironic statement,\" produced ninety cans of his feces. London's Tate Gallery paid $35,000 for one. If your cell phone rang in the Rising Sun pub in Brighton, England, the proprietor nailed the phone to the bar.\n\nA member of the U.S. table-tennis team was suspended for steroid use. When Mets pitcher Shawn Estes lost his no-hitter in the seventh inning, Mets manager Bobby Valentine rejected the idea that Estes was jinxed in the fifth inning when Shea Stadium's JumboTron announced that Estes had not yet given up a hit. Said Valentine: \"I don't believe in superstitions. They're bad luck.\"\n\nBaseball avoided a season-ending strike, enabling a San Francisco woman, who wanted to be artificially inseminated, to advertise a barter: World Series tickets for \"healthy sperm.\" In his first three major-league at-bats, Seattle Mariners designated hitter Ron Wright caused six outs by striking out and hitting into a double play and a triple play. After the game, he was sent to the minors. Pete Gray, a one-armed outfielder who during the Second World War played a season for the St. Louis Browns, died at eighty-seven. Joe Black, the first black pitcher to win a World Series game (for Brooklyn, in 1952), was seventy-eight.\n\nMontgomery, Alabama, bus driver James Blake died at 89, forty-seven years after he had Rosa Parks arrested because she refused to move to the back of his bus. Traudl Junge, a private secretary to Hitler and the last surviving witness to his final hours in the bunker, said he \"gave me a feeling of security, safety and being cared for.\" She died at 82, wondering: \"If he discovered he had Jewish blood in his family tree, would he have gassed himself?\" Chaike Spiegel, one of the last surviving combatants of the 1943 Warsaw ghetto uprising, was 81. Flags all across Australia were flown at half mast for Alec Campbell, 103, the last survivor of more than seventy thousand Australians and New Zealanders who fought in 1915 at Gallipoli, the ill-fated operation that almost destroyed the career of its architect, Winston Churchill. Of Gallipoli, Campbell said: \"It was a lovely place, you know, if conditions had been better...\" Queen Elizabeth the Queen Mother, born when Churchill was 25, died at 101.\n\nAnd after half a century of sultry singing, Peggy Lee, who was eighty-one, left the stage. There lingered in our minds a lyric suitable for any year: \"If that's all there is, my friends, then let's keep dancing.\"\n\n[DECEMBER 23, 2002]\n\n# **2003: Lingerie and Duct Tape**\n\n**\"W** hatever happens,\" said Lord Salisbury (1830\u20131903), a conservative in thought, word, and deed, \"will be for the worse, and therefore it is in our interests that as little should happen as possible.\" By that sensible standard, eventful 2003 was not in our interests.\n\nMake _love_ and _war,_ or else the terrorists will have won: During Valentine's week in February, with war impending and the government elevating the terrorism alert, two of Wal-Mart's hot-selling items were lingerie and duct tape. Talk about ingratitude: Terrorists struck in Saudi Arabia. The war with Iraq went well, aside from the detail that the reason for it\u2014weapons of mass destruction\u2014has been elusive. The following is a complete list of all those fired because of the intelligence failure: _______.\n\nWhile America was trying to acquaint 25 million Iraqis with democracy, 144.5 million Russians fell under President Vladimir Putin's \"managed democracy\"\u2014czarism leavened by state-manipulated plebiscites. If only the world were more like California, where the people, groaning under the wicked choices of the electorate, chose a new governor to wrestle with the people's chosen legislature over how to undo the damage done by voters in dozens of plebiscites.\n\nIn America's \"(court-) managed democracy,\" judges began discovering a right to same-sex marriage. Forty-one years after the U.S. Army helped end racial discrimination in admissions at Ole Miss, the Supreme Court found nothing amiss in the University of Michigan's racial discrimination in admissions.\n\nThat court, having said that the First Amendment protections extend to virtual child pornography and tobacco advertising, finally exclaimed \"Enough!\" In a 5\u20134 decision cowritten by Sandra Day O'Connor, it held that concern for political hygiene justified Congress's passing the McCain-Feingold legislation to restrict the amount and regulate the content of speech about members of Congress. Thus did a Ronald Reagan 1980 campaign promise (to appoint the first woman justice) result, twenty-three years later, in ratification of George W. Bush's decision to sign a bill that he, while campaigning\u2014before taking the oath to defend the Constitution\u2014called unconstitutional.\n\nThe Acad\u00e9mie Fran\u00e7aise, which never stooped to admit Flaubert or Zola, admitted Val\u00e9ry Giscard d'Estaing, author\u2014sort of\u2014of the European Union's proposed 263-page Constitution, a screamingly funny political satire. Because of Jayson Blair, heads rolled at the _New York Times,_ where all the \"news\" had not been fit to print.\n\nAmerican conservatives, controlling both elected branches of government for the first full year since 1954, used their power to...vastly expand the welfare state. The prescription-drug entitlement may cost $2.5 _trillion_ over the next twenty years. Big deal. That sum amounts to only five years of deficits at the current level. Besides, Congress showed that it could act with dispatch against a menace that _really_ annoys people: spam.\n\nHoward Dean discerned what liberals want: _attitude._ In San Francisco, ground zero of Deanism, sensitivity police stipulated that pets' owners shall also be called \"guardians.\" Kobe Bryant, impulse buyer, bought his wife a $4 million diamond ring. _Friends_ headed for oblivion, American style: ubiquitous reruns.\n\nRemember Henry Adams's jest that the succession of presidents from Washington to Grant disproved the theory of evolution? After another year watching their royals, the British could say something similar about the progress, so to speak, from Elizabeth I to the son of Elizabeth II. Senator Pat Moynihan, dead at seventy-six, was America's foremost public intellectual. Another giant of the Finance Committee, Russell Long, eighty-four, was sixteen when his father was assassinated, and not quite thirty when elected to the Senate in 1948. David Brinkley, eighty-two, the most famous son of Wilmington, North Carolina, until Michael Jordan came along, said of television, \"When there is no news, we give it to you with the same emphasis as if there were.\"\n\nThis year, the movie musical was revived with _Chicago,_ and Sam Phillips died at 80. On July 5, 1954, at his Memphis recording studio, Phillips recorded a 19-year-old singing \"That's All Right Mama.\" Elvis's blending of white and black music helped end the world defended by Lester Maddox and Strom Thurmond, dead at 87 and 100.\n\nIn 1951, the Boston Braves' Warren Spahn, en route to becoming baseball's winningest left-hander, stood on the mound, sixty feet six inches from a New York Giants rookie who was 0\u2013for\u201312. Willie Mays homered. Said Spahn, \"For the first sixty feet, that was a helluva pitch.\" Spahn was eighty-two. In the 1934 World Series, Tiger shortstop Billy Rogell's relay to first hit Cardinal runner Dizzy Dean in the forehead. The next day a headline supposedly said: x-rays of dean's head reveal nothing. Rogell died at ninety-eight.\n\nAt 114, Mitoyo Kawate was the world's oldest person. She was working on her farm six miles from Hiroshima on August 6, 1945. Jack Davis, 108, was Britain's oldest veteran of mankind's final war\u2014the war to end war, 1914\u20131918.\n\n[DECEMBER 22, 2003]\n\n# **2004: _The Passion of the Christ_ and _The Passions of the Faculty Clubs_**\n\n**I** n 2004, an IBM supercomputer set a world record with 36.01 trillion calculations per second. The U.S. electorate may have made its calculation the instant John Kerry, who is not a supercomputer, explained why Toy's restaurant in Canonsburg, Pennsylvania, \"is my kind of place\":\n\n\"You don't have to\u2014you know, when they give you the menu, I'm always struggling: ah, what do you want? He just gives you what he's got, right?...whatever he's cooked up that day. And I think that's the way it ought to work, for confused people like me who can't make up our minds.\"\n\nThis year, some paleoanthropologists reported that our cousins the Neanderthals, who disappeared thirty thousand years ago, had better minds than has been thought: On a plain in Spain there is a mass grave containing evidence of funeral ritual, which means that Neanderthals had a capacity for symbolism. This year, Democrats stressed their superior brains. (Bumper stickers: some village in texas is missing their [ _sic_ ] idiot; john kerry\u2014bringing complete sentences back to the white house.) A campaign flier in Tennessee pictured George W. Bush's face superimposed over that of a runner in the Special Olympics, and proclaimed this message: \"Voting for Bush is like running in the Special Olympics. Even if you win, you're still retarded.\"\n\nFrom an Indonesian island came evidence that as recently as eighteen thousand years ago\u2014only yesterday, as paleoanthropologists reckon\u2014there was a race of Hobbit-size (about three feet tall) semi-people. Their small brains probably were incapable of idealism of James Kilgore's sort. In 2004, Kilgore, fifty-six, was sentenced to six years in prison for his part in the murder of a mother of four during a 1975 California bank robbery that was supposed to help finance the Symbionese Liberation Army. Martha Stewart was sentenced to prison for lying about a crime she was not charged with. Scott Peterson was convicted of double murder\u2014killing his wife, and killing his unborn child, a problematic idea given the current understanding of abortion rights. Before death tardily overtook another dispenser of death, Yasir Arafat, he received a letter from People for the Ethical Treatment of Animals\u2014well, of animals other than people\u2014asking him to stop using donkeys in suicide bombings. It was said that the death of this winner of the Nobel Peace Prize might make peace possible.\n\nFifty years after _Brown v. Board of Education,_ an African-American was nominated to replace an African-American as secretary of state. Nashville, Tennessee, schools stopped displaying the honor rolls of A students because some parents complained that the displays might hurt the feelings of dimmer students. In Washington State, the Puyallup school district ended the grade-school tradition of children parading in Halloween costumes, partly because some costumes might be offensive to real witches. Said a district spokeswoman, \"Witches with pointy noses and things like that are not respective [ _sic_ ] symbols of the Wiccan religion, and so we want to be respectful of that.\" In Michigan, Jon Blake Cusack named his son Jon Blake Cusack 2.0.\n\nSome American sub-Neanderthals photographed themselves abusing Iraqi prisoners. By October 24, the war in Iraq had lasted longer than the U.S. involvement in the First World War. Two movies symptomatic of the temper of the times were _The Passion of the Christ_ and _The Passions of the Faculty Clubs_ (aka _Fahrenheit 9\/11_ ). The Massachusetts Supreme Judicial Court ignited a debate about whether homosexuals could do more damage to the institution of marriage than heterosexuals are doing. Britney Spears has been married twice this year, so far. What, other than Janet Jackson's breast, do you remember about the Super Bowl?\n\nAt the Olympics, an elite collection of NBA stars lost to Puerto Rico and Argentina, but hey, they beat the Lithuanians after losing to them. Early in 2004, Alex Rodriguez, eager to win a World Series, was courted by the Yankees and Red Sox and signed with...oh, well. Paul Hopkins, the oldest former major-league player, died at ninety-nine. The first batter he faced as a pitcher for the Washington Senators was Babe Ruth, who homered. Hopkins's career ended ten games later. In 2004, Washington was awarded another team.\n\nThe world's oldest man, Joan Riudavets Moll, died at 114 in Spain. He was born the same year\u20141889\u2014as Hitler and Charlie Chaplin. Alberta Martin, a Civil War widow, died in Alabama at 97. Born in 1906, in 1927 she married a Confederate widower born in 1845. She was poor and he had a $50-a-month Confederate pension\u2014a not uncommon aphrodisiac causing May-December marriages early in the twentieth century. Did she at 21 love her 81-year-old groom? \"That's a hard question to answer.... You know the difference between a young man and an old man.\" Two months after he died in 1931, she married his grandson.\n\nTwo great musical instruments fell forever silent\u2014the voices of Ronald Reagan and Ray Charles. But their melodies linger on.\n\n[DECEMBER 20, 2004]\n\n# **2005: \"In Lieu of Flowers, Please Send Acerbic Letters to Republicans\"**\n\n**S** eeking the serenity that a sense of history confers in testing times, Mike Cameron, a Mets outfielder in 2005, said in defense of a teammate who lost a fly ball in the sun, \"Stuff is going to happen sometimes. The sun has been there for five hundred, six hundred years.\" Stuff happened in 2005, when an obituary in the _Chicago Tribune_ advised, \"In lieu of flowers, please send acerbic letters to Republicans.\" At home, the president's, and the nation's, disagreeable year can be summarized by three female names: Terri Schiavo, Harriet Miers, and Katrina. The first involved grotesque overreaching by the federal government, undertaken by self-described conservatives whose action refuted their description. The second involved indifference to competence. The third displayed the consequences of incompetence. Abroad, Iraq illustrated one, two, and three.\n\nIn Russia, despotism continued to make a comeback, but Lenin, at least, may soon be buried: His cadaver in Red Square is said to sometimes sprout fungi. The 482-page European Union \"constitution\" was rejected, but the common currency marches on in white boots. Wearing those and a red miniskirt, Renate Dolle, sixty-three, told a Berlin newspaper she will soon end her forty-nine-year career as a prostitute (n30; $36) so she can spend more time with her husband and granddaughter.\n\nWhy was America's consumer-driven economy not derailed by higher oil prices? In 2005, Americans' housing stock increased in value $2.38 trillion more than their oil bill increased ($120 billion). _Vox populi, vox dei?_ When Katrina's disruption of supplies caused gasoline prices briefly to pass $3 a gallon, the public thought this proved that conniving oil companies control prices. When, a few weeks later, prices plunged toward, and some places below, $2, the public thought...what?\n\nRising oil prices and General Motors' declining health reflected, among other things, the success of sixty years of U.S. policies promoting free trade and globalization. India and China are slurping up oil because, having joined the international economy, they are booming. This year, upwards of sixty thousand Americans were employed manufacturing more than 3 million \"foreign\" cars. Toyota, which in 2006 may sell more cars worldwide than GM, has opened a design center in Ann Arbor, just forty miles from Detroit.\n\nOnward and upward with progressivism: In a Las Vegas suburb, the United Food and Commercial Workers union hired temp workers at $6 an hour to picket a nonunion Wal-Mart, where wages start at $6.75 an hour. A British teachers-union official proposed that instead of bad students' receiving a \"failing\" grade, their grade should be called \"deferred success.\" A Milwaukee seventeen-year-old and his father sued to end summer homework because the stress of honors precalculus assignments spoiled the lad's summer. When Jada Pinkett Smith, wife of actor Will Smith, told a Harvard audience that women \"can have it all\u2014a loving man, devoted husband, loving children, a fabulous career,\" the campus Bisexual, Gay, Lesbian, Transgender, and Supporters Alliance said its members were made \"uncomfortable\" because Mrs. Smith's words were \"extremely heteronormative.\" A majority of teachers, parents, and students at Jefferson Elementary School in Berkeley favored renaming the school Sequoia Elementary because Jefferson owned hundreds of slaves. Under Chief Sequoia, the Cherokee nation owned more than fifteen hundred black slaves. You cannot be too careful, so Timnath, Colorado, banned smoking in bars and restaurants, of which Timnath at the time had none. The often hilarious _New York Times,_ which opposes capital punishment, reported disapprovingly that a life sentence \"is death in all but name.\"\n\nAn Oklahoma judge granted the request of a criminal who wanted his thirty-year prison sentence increased three years to match Larry Bird's Celtics jersey number. Death, as it must to all, came to six-foot-ten George Mikan, eighty, who was the NBA's first superstar. Madison Square Garden's marquee once read: wed. basketball: geo. mikan vs. knicks.\n\nAsked to switch from guitar to bass, which he could not afford to buy, Eric Griffiths quit the rock group the Quarry Men in 1958 and joined the British Merchant Navy. On a radio on a ship in the Persian Gulf in 1963 he heard \"Please Please Me,\" the first hit by the former Quarry Men, by then called the Beatles. Griffiths died in 2005 at 64. On November 22, 1963, in a darkened Dallas theater, the hammer of Lee Harvey Oswald's revolver jammed on the flesh of the palm of the policeman arresting him, so Maurice McDonald lived another forty-two years. Vic Power, a Puerto Rican first baseman, was one of baseball's first Hispanic stars. _Sports Illustrated_ reports that when Power was playing in the minor leagues in the South, he was told by a waitress that the restaurant did not serve Negroes. He replied, \"That's OK, I don't eat Negroes.\" Mark Matthews, 111, was the oldest of the surviving Buffalo Soldiers, the African-Americans who fought Native Americans on behalf of Euro-Americans. Ah, multiculturalism.\n\n[DECEMBER 19, 2005]\n\n# **2006: \"Go Ahead, We Will Get into One of the Other Boats\"**\n\n**H** ow gruesome was 2006? The year's most consequential person was Iran's president, who says the Holocaust did not happen and vows to complete it. Regarding his nuclear aspirations, Mahmoud Ahmadinejad, whose manias are leavened with realism, treated the United Nations as a figment of the imagination of a fiction\u2014the \"international community.\" Democrats, given control of Congress because of Iraq, vowed to raise the minimum wage. Nimble and graceful Barack Obama became the Democrats' Fred Astaire, adored because of, well, perhaps the way he wears his hat, the way he sips his tea. And the way he isn't Hillary.\n\nThis year's civil-rights outrage was \"soaring\" and \"record\" gasoline prices, a violation of Americans' inalienable right to pay for a gallon no more than they paid twenty-five years ago. By December, the price of a gallon, adjusted for inflation, was eighty-three cents lower than in 1981. Kansas voters removed some skeptics of evolution from the state's school board. A fossil 3.3 million years old revealed that a little girl from the human lineage had arms and shoulders suited to climbing and swinging through trees.\n\nIn order to show \"tolerance of people's beliefs,\" government workers in England's West Midlands were told, after a Muslim complained, to remove from sight all pig-related items, such as a tissue box featuring Winnie the Pooh and Piglet. But tolerance was episodic in Europe in 2006: In Sweden, police said the soccer fan who wore on his clothes a Swedish flag, which features a cross, \"provoked some emotions.\" Indeed. He was beaten nearly to death by Muslim immigrants. Inspector Clouseau, call your office: French police denied that anti-Semitism was involved in the kidnapping and murder of a Jewish man by Muslim immigrants who demanded a ransom from a synagogue. Angry about those Danish cartoons depicting the prophet, Iran's bakers renamed Danish pastries \"Roses of the Prophet Muhammad pastries.\" Although no one had complained, the human-rights director for the provocatively named city of St. Paul, Minnesota, had a happy easter sign removed from city hall.\n\nTwo U.S. explorers went to the North Pole to study how global warming threatens polar bears. They had planned to go last year, but were forced to delay Project Thin Ice because of unusually heavy snow and ice. The \"emerging hurricane problem,\" which, after Katrina, the _New York Times_ identified as a consequence of global warming, did not emerge. The unusually tranquil Atlantic hurricane season was explained as a consequence of...global warming affecting the Pacific. Two senators, Jay Rockefeller of West Virginia and Olympia Snowe of Maine, warned ExxonMobil that global warming is an undeniable fact, so the corporation should desist from its \"dangerous\" support of research by persons with doubts. The senators did not explain the danger involved in doubting the indubitable. There were dangers\u2014disorder, sporadic violence\u2014among those gathered outside stores in the predawn hours before the PlayStation 3 gaming console went on sale.\n\nGreat moments in government: The Florida woman who wounded with a shotgun the alligator that entered her house and attacked her golden retriever was given a warning citation for hunting without a permit. Compassionate social democracy: The Danish government continued to pay prostitutes to service the disabled.\n\nAncient Greece pioneered philosophy and democracy. Modern Greece this year gave the world a new wrinkle in creative accounting: It became 25 percent richer after its GDP was revised to account for such booming service industries as prostitution and money laundering. The intellectual fare served at the University of Wisconsin-Milwaukee included a course called the Social Construction of Obesity. (Fatness, like beauty, is in the eye of the beholder to whom society's power structure, always eager to foment new forms of discrimination, has given false consciousness.) Elsewhere in higher education, at Bucknell's \"celebration of whore culture,\" a woman stripped on a trapeze.\n\nIn Tacoma, Washington, a judge asked those in her courtroom to cheer \"Go, Seahawks!\" Then she sentenced a man convicted of manslaughter to thirteen years. The chief executive of Eternal Image Inc., which announced caskets and urns with logos of all thirty Major League Baseball teams, called this \"a way to make team loyalty a final statement.\" Red Auerbach, whose Celtics _teams_ won seven championships without having a player among the NBA's top ten scorers, died this year at eighty-nine. Romano Mussolini, who died this year at seventy-nine, son of Il Duce, had played jazz with Dizzy Gillespie, Duke Ellington, and Chet Baker.\n\nLillian Gertrud Asplund was five when her father smiled and said, \"Go ahead, we will get into one of the other boats.\" He did not. Lillian never married, and retired early to take care of her mother, who never recovered from losing her husband. Lillian, the last American survivor of the _Titanic,_ was ninety-nine.\n\n[DECEMBER 18, 2006]\n\n# **2007: Ready, Fire, Aim**\n\n**I** n 2007 came the revolution. Determined to end the war in Iraq and begin the reign of justice in America, Democrats took over Congress and acted on the principle \"ready, fire, aim.\" They threatened to tell the Ottoman Empire (deceased 1922) that it should be ashamed of itself (about Armenian genocide) and raised the minimum wage to $5.85, which is worth _less_ than the $5.15 minimum was worth when it was set in 1997. Onward and upward with compassionate liberalism: The Democrat-controlled Senate flinched from making hedge-fund multimillionaires pay more than a 15 percent tax rate. At the year-end, there were more troops in Iraq than there were at the year's beginning. Although it was not yet possible to say the war was won, it was no longer possible to say the surge was not succeeding. The McClatchy newspapers, with the media's flair for discerning lead linings on silver clouds, offered this headline: as violence falls in iraq, cemetery workers feel the pinch.\n\nThe king of Spain told the president of Venezuela to \"shut up,\" and 51 percent of Venezuelans seconded the motion. Rudy Giuliani said, \"I took a city that was known for pornography and licked it.\" Hillary Clinton accused Barack Obama of having been ambitious in kindergarten. Disraeli once said of Lord Russell: \"If a traveler were informed that such a man was leader of the House of Commons, he may well begin to comprehend how the Egyptians worshipped an insect.\" Mike Huckabee became a leader among Republican presidential candidates.\n\nIn March, when a planned trek by two explorers to the North Pole, intended to dramatize global warming, was aborted because of temperatures 100 degrees below zero, an organizer of the consciousness-raising venture explained that the cancellation confirmed predictions of global warming because \"one of the things we see with global warming is unpredictability.\" Al Gore won the Nobel Peace Prize that should have gone to nine-time Grammy winner Sheryl Crow, who proposed saving the planet by limiting\u2014to one\u2014\"how many squares of toilet paper can be used in any one sitting.\" At the U.N. global-warming conference in Bali there was Carbon Footprint Envy\u2014the airport did not have space to park all the private jets.\n\nAs Americans debated expanding government involvement in health care, Britain's National Health Service told Olive Beal she would have to wait eighteen months to get her hearing aid. She is 108.\n\nThanks to federal supervision of K\u201312 education, when a Johnson City, New York, parent complained that cheerleaders lead cheers for the boys' basketball team but not the girls', the U.S. Department of Education, citing Title IX's requirement of sexual equality in scholastic sports, demanded equal \"promotional services.\" Two Los Angeles teachers were fired after a controversy that began when one had her class, during Black History Month, make a presentation about Emmett Till, the Chicago fourteen-year-old who was tortured and murdered in Mississippi in 1955 after his wolf whistle at a white woman. Some students and teachers charged that school officials said Till's whistle could be construed as sexual harassment. In an inexplicable (and probably temporary) spasm of good taste, public opinion sent Don Imus packing because he said on his radio program something no more tasteless than things he had been saying for years, to the delight of a large (and evidently fickle) public.\n\nA Seattle day-care center banned LEGO building blocks because the beastly children \"were building their assumptions about ownership and the social power it conveys, assumptions that mirrored those of a class-based, capitalist society.\" The center reinstated LEGOs but allowed the children to build only \"public structures\" dedicated to \"collectivity and consensus.\" In other lingering reverberations of communism, scientists unearthed what they think are remains of two more of Czar Nicholas II's children, murdered by Bolsheviks, who never played with LEGOs. A Cuban exile, former CIA operative, and Bay of Pigs veteran announced plans to auction what he says is a lock of Che Guevara's hair taken from the corpse before burial in Bolivia.\n\nWhen the Confederate monument in Montgomery, Alabama, was desecrated, was that a \"hate crime\"? Saying he wanted to bring Alabama \"into the twentieth century\"\u2014the twenty-first would be a bridge too far?\u2014a legislator, worried that \"a shower head\" might be illegal, moved to repeal the state's ban on the sale of sex toys. A mayor looked on the bright side of his city's high homicide rate: \"It's not good for us but it also keeps the New Orleans brand out there.\" Lucky Belgium has been without a government since June.\n\nIn 2007, for the first time, two Hispanic surnames, Garcia and Rodriguez, were among America's ten most common. Paul and Teri Fields of Michigan City, Indiana, named their baby boy Wrigley.\n\nDeath, as it must to all, came to Paul Tibbets, 92. Eighty years ago, 12-year-old Paul flew with a barnstorming pilot who dropped Baby Ruth candy bars over a Florida racetrack. In 1945, Tibbets was pilot of the _Enola Gay,_ the B-29 that dropped the atomic bomb on Hiroshima. \"What about the shortstop Rizzuto,\" asked Casey Stengel long ago, \"who got nothing but daughters but throws out the left-handed hitters in the double play.\" Phil Rizzuto, the oldest living Hall of Famer, was 89. Emma Faust Tillman, 114, of Hartford, Connecticut, had been the world's oldest person. She was born during the presidency of Benjamin Harrison. Robert Adler, 93, gave the modern world its most beloved invention. The TV remote, of course.\n\n[DECEMBER 22, 2007]\n**Chapter Three**\n\n**GOVERNING**\n\n# **The Two Americas: Hard and Soft**\n\n**M** ichael Barone, America's foremost political analyst, wonders why America produces so many incompetent eighteen-year-olds but remarkably competent thirty-year-olds. The answer is in his new book, _Hard America, Soft America: Competition vs. Coddling and the Battle for the Nation's Future._ It illuminates the two sensibilities that sustain today's party rivalry.\n\nOne answer to Barone's question is: schools. In 1900, only 10 percent of high-school-age Americans went to high school. Subsequently, schooling became universal and then schools became emblematic of Soft America, suffused with \"progressive\" values\u2014banning dodgeball and other games deemed too competitive, attempting personality adjustment, promoting self-esteem and almost anyone with a pulse.\n\nIn contrast, Barone says, \"Hard America plays for keeps: The private sector fires people when profits fall and the military trains under live fire.\" Soft America depends on the productivity, creativity, and competence of Hard America, which protects the country and pays its bills.\n\nFor a while, Soft America, consisting of those sectors where there is little competition and accountability, threatened to extinguish Hard America. By 1950, America had what Barone calls a Big Unit economy\u2014big business and big labor, with big government often mediating between them. This economy was, Barone says, \"inherently soft.\" Security, a concept not relevant to the Hard America of the novel _Sister Carrie_ (1900), was central to the corporatist world of conformity\u2014the world of the novel _The Man in the Gray Flannel Suit_ (1955).\n\nWith a novelist's eye for the telling detail, Barone notes that the Labor Department building, constructed in the 1960s, had two conference rooms adjacent to the secretary of labor's office, one for management, one for labor, so the secretary could shuttle between them. Together, the three big units would work out agreements, passing the costs of them along to consumers in an era much less competitive than today's deregulated and globalized era.\n\nBetween 1947 and 1968, big business got bigger: the share of assets owned by the two hundred largest industrial companies rose from 47 percent to 61 percent. Then came a hardening. Deregulation ended soft niches (e.g., airlines, trucking) protected by government-sponsored cartelization. The Interstate Commerce Commission, which encouraged cartelization, was abolished.\n\nNew financial instruments (e.g., junk bonds) fueled hostile takeovers. Capital gains taxes were cut, stimulating entrepreneurship. Between 1970 and 1990, the rate at which companies fell from the Fortune 500 quadrupled. The portion of the gross national product accounted for by the one hundred largest industrial corporations fell from 36 percent in 1974 to 17 percent in 1998.\n\nIn 1957, the Soviet Sputnik provoked some hardening of America's schools\u2014with more science and advance placement courses, and consolidation of rural schools. President Kennedy's vow to reach the moon by the end of the 1960s was an inherently hard goal, with a hard deadline measuring success or failure.\n\nBut the second half of the 1960s brought the Great Softening\u2014in schools and welfare policies, in an emphasis on redistribution rather than production of wealth and in the criminal justice system. The number of violent crimes per 100,000 people rose from 1,126 in 1960 to 2,747 in 1970 while the prison population declined from 212,000 in 1960 to 196,000 in 1970. In 2000, after the swing toward hardening, there were 1.3 million prisoners.\n\nBarone says racial preferences, which were born in the 1960s and 1970s, fence some blacks off from Hard America, insulating them in \"a Soft America where lack of achievement will nonetheless be rewarded.\"\n\nThe Detroit riot of 1967 lasted six nights before twenty-seven hundred federal troops restored order. In 1992, after the 1980s turn toward hardness, the Los Angeles riots lasted eighteen hours, ending six hours after twenty-five thousand federal troops were dispatched.\n\nIn the Soft America of 1970, the tapestry of welfare benefits had a cash value greater than a minimum wage job. In the Harder America of 1996, welfare reform repealed Aid to Families with Dependent Children, a lifetime entitlement to welfare. And in the 1990s, welfare dependency\u2014and crime\u2014were cut in half. A harder, self-disciplined America is a safer America.\n\nWhat institution is consistently rated most trustworthy by Americans? The institution that ended its reliance on conscription, that has no racial preferences and has rigorous life-and-death rules and standards: the military.\n\nBarone believes that promotion of competition and accountability\u2014hardness\u2014is the shared theme of President Bush's policies of educational standards, individual health accounts, Social Security investment accounts, and lower tax rates to increase self-reliance in the marketplace. Barone's book is a guide to electoral map reading: the blue and red states have, respectively, softer and harder sensibilities.\n\n[MAY 9, 2004]\n\n# **Angela Jobe's Resilience**\n\n**M** ILWAUKEE\u2014Angela Jobe, thirty-eight, is a grandmother who has lived most of her adult life at ground zero of the struggle to \"end welfare as we know it.\" At about the time candidate Bill Clinton was promising to do that\u2014in autumn 1991\u2014she boarded a bus in Chicago, heading for Milwaukee, lured by Wisconsin's larger benefits and lower rents. Unmarried, uneducated, and unemployed, she already had three children and eight years on welfare.\n\nToday, she is in her ninth year of employment in a nursing home, earning $10.50 an hour. How she left welfare, and how her life did and did not change, is one of the entwined stories in Jason DeParle's riveting new book, _American Dream: Three Women, Ten Kids and a Nation's Drive to End Welfare,_ the fruit of DeParle's seven years of immersion in Jobe's world.\n\nHis subject is the attempt of welfare reformers, in Wisconsin and then Washington, to end the intergenerational transmission of poverty in the chaotic lives of fractured families. His book reads more like a searing novel of urban realism\u2014Theodore Dreiser comes to Milwaukee\u2014than a policy tract. His reporting refutes the 1930s paradigm of poverty\u2014the idea that the perennially poor are strivers like everyone else but are blocked by barriers unrelated to their behavior. Angie Jobe is not Tom Joad.\n\nAfter the liberalization of welfare in the mid-1960s, the percentage of black children born to unmarried mothers reached 50 by 1976 (it is almost 70 today), and within a generation the welfare rolls quadrupled. But DeParle says people mistakenly thought people like Jobe were organizing their lives around having babies to get a check. Actually, he says, their lives were too disorganized for that.\n\nWhat can help organize lives\u2014at least those that are organizable\u2014is work. The requirements of work\u2014mundane matters such as punctuality, politeness, and hygiene\u2014are essential to the culture of freedom. The 1996 reform replaced a lifetime entitlement to welfare with a five-year limit, and called for states to experiment with work requirements. Welfare rolls have since declined more than 60 percent. DeParle writes:\n\n\"In Creek County, Oklahoma, the rolls fell 30 percent even as the Legislature was still debating the law, a decline officials largely attributed to the mere rumors of what was coming.... The late 1990s can be thought of as a bookend to the 1960s. One era, branding welfare a right, sent the rolls to sudden highs; the other, deeming welfare wrong, shrank them equally fast.\"\n\nThe mass movement from welfare rolls to employment rolls is progress. But DeParle's unsentimental reporting offers scant confirmation of the welfare reformers' highest hope, that when former welfare mothers go to work, their example will transform the culture of their homes, breaking the chain of behaviors that passes poverty down the generations. On the street where Jobe lives today, almost every house is the home of a working mother with children but no husband.\n\nWhen Jobe was thirteen, her parents divorced and she went to live with her father, who let her roam Chicago's South Side streets. The father of the child she had at seventeen is serving a sixty-five-year prison sentence. And the wheel turns: Jobe's daughter Kesha got pregnant at sixteen. Kesha told the fourteen-year-old father at his eighth-grade graduation, and hardly heard from him again. Kesha, a checkout clerk at a grocery store, has two children and lives with a boyfriend.\n\nJobe's house teems with life. During a recent visit, there were two infants in diapers, and the seventeen-year-old girlfriend who lives there with Jobe's eighteen-year-old son.\n\nMilwaukee's mandatory self-esteem classes were part of the \"hassle factor\" designed to diminish welfare's appeal. But, says Jobe, \"There's nothing wrong with my self-esteem,\" the timbre of her voice validating the assertion. She is a four-foot-nine geyser of pluck, humor, and compassion for her nursing home patients. She has no sense of entitlement. DeParle says of her and the other two women whose story he tells, \"When welfare was there for the taking, they got on the bus and took it; when it wasn't, they made other plans.\"\n\nWhat of her future? Today, she says, \"I don't think much about tomorrow.\" Complete absorption in the present is both a cause and a consequence of living a precarious and disorganized life, but so far her postwelfare story illustrates two truisms: People respond to strong social cues, as she did when she got on the bus, and later when she got off welfare. Second, poor people are more resilient\u2014and more resistant to fundamental behavior modification\u2014than their various would-be improvers suppose.\n\n[DECEMBER 30, 2004]\n\n# **Conservatism's Infrastructure**\n\n**I** f by the dawn's early light of November 3 George W. Bush stands victorious, seven of ten presidential elections will have been won by Southern Californians and Texans, all Republicans. The other three were won by Democrats\u2014a Georgian and an Arkansan.\n\nThis rise of the Sunbelt is both a cause and a consequence of conservatism's rise, which began in 1964 with, paradoxically, the landslide loss of the second post\u2013Civil War major-party presidential nominee from that region\u2014Arizona's Barry Goldwater, four years after the first, Richard Nixon. His campaign was the first stirring of a mass movement: Nixon's 1960 campaign attracted 50,000 individual contributors; Goldwater's attracted 650,000.\n\nConservatism's 40-year climb to dominance receives an examination worthy of its complexity in _The Right Nation,_ the best political book in years. Its British authors, John Micklethwait and Adrian Wooldridge of the _Economist,_ demonstrate that conservative power derives from two sources\u2014its congruence with American values, especially the nation's anomalous religiosity, and the elaborate infrastructure of think tanks and other institutions that stresses that congruence.\n\nLiberals, now tardily trying to replicate that infrastructure, thought they did not need it because they had academia and the major media. But the former marginalized itself with its silliness, and the latter have been marginalized by their insularity and by competitors born of new technologies.\n\nLiberals complacently believed that the phrase \"conservative thinker\" was an oxymoron. For years\u2014generations, really\u2014the prestige of the liberal label was such that Herbert Hoover called himself a \"true liberal\" and Dwight Eisenhower said that cutting federal spending on education would offend \"every liberal\u2014including me.\"\n\nLiberalism's apogee came with Lyndon Johnson, who while campaigning against Goldwater proclaimed, \"We're in favor of a lot of things and we're against mighty few.\" Johnson's landslide win produced a ruinous opportunity\u2014a large liberal majority in Congress, and incontinent legislating. Forty years later, only one-third of Democrats call themselves liberal, whereas two-thirds of Republicans call themselves conservative. Which explains this Micklethwait and Wooldridge observation on the Clinton presidency:\n\n\"Left-wing America was given the answer to all its prayers\u2014the most talented politician in a generation, a long period of peace and prosperity, and a series of Republican blunders\u2014and the agenda was still set by the right. Clinton's big achievements\u2014welfare reform, a balanced budget, a booming stock market and cutting 350,000 people from the federal payroll\u2014would have delighted Ronald Reagan. Whenever Clinton veered to the left\u2014over gays in the military, over health care\u2014he was slapped down.\"\n\nMicklethwait and Wooldridge endorse Sir Lewis Namier's doctrine: \"What matters most about political ideas is the underlying emotions, the music to which ideas are a mere libretto, often of very inferior quality.\" The emotions underlying conservatism's long rise include a visceral individualism with religious roots and antistatist consequences.\n\nEurope, postreligious and statist, is puzzled\u2014and alarmed\u2014by a nation where grace is said at half the family dinner tables. But religiosity, say Micklethwait and Wooldridge, \"predisposes Americans to see the world in terms of individual virtue rather than in terms of the vast social forces that so preoccupy Europeans.\" And: \"The percentage of Americans who believe that success is determined by forces outside their control has fallen from 41 percent in 1988 to 32 percent today; by contrast, the percentage of Germans who believe it has risen from 59 percent in 1991 to 68 percent today.\" In America, conservatives much more than liberals reject the presumption of individual vulnerability and incompetence that gives rise to liberal statism.\n\nConservatism rose in the aftermath of Johnson's Great Society, but skepticism about government is in the nation's genetic code. Micklethwait and Wooldridge note that in September 1935, during the Depression, Gallup polling found that twice as many Americans said FDR's administration was spending too much than said it was spending the right amount, and barely one person in ten said it was spending too little.\n\nAfter FDR's 1936 reelection, half of all Democrats polled said they wanted FDR's second term to be more conservative. Only 19 percent wanted it more liberal. In 1980, when Ronald Reagan won while excoriating \"big government,\" America had lower taxes, a smaller deficit as a percentage of GDP, and a less-enveloping welfare state than any other industrialized Western nation.\n\nAmerica, say Micklethwait and Wooldridge, is among the oldest countries in the sense that it has one of the oldest constitutional regimes. Yet it is \"the only developed country in the world never to have had a left-wing government.\" And given the country's broad and deep conservatism, it will not soon.\n\n[OCTOBER 10, 2004]\n\n# **Against \"National Greatness Conservatism\"**\n\n**I** n the 1920s and 1930s, the American left was riven by multiple factions furiously representing different flavors of socialism, each accusing the others of revisionism and deviationism. Leftists comforted themselves with the thought that \"you can't split rotten wood.\"\n\nBut you can. And the health of a political persuasion can be inversely proportional to the amount of time its adherents spend expelling heretics from the one true (and steadily smaller) church. Today's arguments about conservatism are, however, evidence of healthy introspection.\n\nThe most recent reformer to nail his purifying theses to the door of conservatism's cathedral is Michael Gerson, a former speechwriter for the current president, and now a syndicated columnist. He advocates _Heroic Conservatism_ in a new book with that trumpet-blast of a title.\n\nHis task of vivifying his concept by concrete examples is simplified by the fact that he thinks the Bush administration has been heroically conservative while expanding the welfare state and trying to export democracy. His task of making such conservatism attractive is complicated by the fact that...well, it is not just the Twenty-second Amendment that is preventing the president from seeking a third term.\n\nGerson, an evangelical Christian, makes \"compassion\" the defining attribute of political heroism. But compassion is a personal feeling, not a public agenda. To act compassionately is to act to prevent or ameliorate pain and distress. But if there is, as Gerson suggests, a categorical imperative to do so, two things follow. First, politics is reduced to right-mindedness\u2014to having good intentions arising from noble sentiments\u2014and has an attenuated connection with results. Second, limited government must be considered uncompassionate, because the ways to prevent or reduce stress are unlimited.\n\n\"We have a responsibility,\" Bush said on Labor Day 2003, \"that when somebody hurts, government has got to move.\" That is less a compassionate thought than a flaunting of sentiment to avoid thinking about government's limited capacities and unlimited confidence.\n\nConservatism is a _political_ philosophy concerned with _collective_ aspirations and actions. But conservatism teaches that benevolent government is not always a benefactor. Conservatism's task is to distinguish between what government can and cannot do, and between what it can do but should not.\n\nGerson's call for \"idealism\" is not an informative exhortation: Huey Long and Calvin Coolidge both had ideals. Gerson's \"heroic conservatism\" is, however, a variant of what has been called \"national greatness conservatism.\" The very name suggests that America _will_ be great _if_ it undertakes this or that great exertion abroad. This grates on conservatives who think America _is_ great, not least because it rarely and usually reluctantly conscripts people into vast collective undertakings.\n\nMost Republican presidential candidates express admiration for Theodore Roosevelt. A _real_ national greatness guy (\"I have been hoping and working ardently to bring about our interference in Cuba\"), he lamented that America lacked \"the stomach for empire.\"\n\nHe pioneered the practice of governing aggressively by executive orders. Jim Powell, author of _Bully Boy,_ an unenthralled assessment of TR, says that in the forty years from Abraham Lincoln through TR's predecessor, William McKinley, presidents issued 158 executive orders. In seven years, TR issued 1,007. Only two presidents have issued more\u2014TR's nemesis Woodrow Wilson (1,791) and TR's cousin Franklin Roosevelt (3,723).\n\n\"I don't think,\" TR said, \"that any harm comes from the concentration of power in one man's hands.\" That sort of executive swagger is precisely what Washington does not need more of. It needs more conservatives such as David Keene, chairman of the American Conservative Union for twenty-three years and Southern political director of Ronald Reagan's 1976 presidential campaign. Writing on \"The Conservative Continuum\" in the September\/October issue of the _National Interest,_ Keene says of Reagan:\n\n\"He resorted to military force far less often than many of those who came before him or who have since occupied the Oval Office.... After the [1983] assault on the Marine barracks in Lebanon, it was questioning the wisdom of U.S. involvement that led Reagan to withdraw our troops rather than dig in. He found no good strategic reason to give our regional enemies inviting U.S. targets. Can one imagine one of today's neoconservative absolutists backing away from any fight anywhere?\"\n\nIt is a pity that TR built the Panama Canal. If he had not, \"national greatness\" and \"heroic\" conservatives could invest their overflowing energies and vaulting ambitions into building it, and other conservatives\u2014call them mere realists\u2014could continue seeking limited government, grounded in cognizance of government's limited competences. That is an idealism consonant with the nation's actual greatness.\n\n[NOVEMBER 25, 2007]\n\n# **Summa Contra Reagan Nostalgia**\n\n**I** n this winter of their discontents, nostalgia for Ronald Reagan has become for many conservatives a substitute for thinking. This mental paralysis\u2014gratitude decaying into idolatry\u2014is sterile: Neither the man nor his moment will recur. Conservatives should face the fact that Reaganism cannot define conservatism.\n\nThat is one lesson of John Patrick Diggins's new book, _Ronald Reagan: Fate, Freedom, and the Making of History._ Diggins, a historian at the City University of New York, treats Reagan respectfully as an important subject in American intellectual history. The 1980s, he says, thoroughly joined politics to political theory. But he notes that Reagan's theory was radically unlike that of Edmund Burke, the founder of modern conservatism, and very like that of Burke's nemesis, Thomas Paine. Burke believed that the past is prescriptive because tradition is a repository of moral wisdom. Reagan frequently quoted Paine's preposterous cry that \"we have it in our power to begin the world over again.\"\n\nDiggins's thesis is that the 1980s were America's \"Emersonian moment\" because Reagan, a \"political romantic\" from the Midwest and West, echoed New England's Ralph Waldo Emerson. \"Emerson was right,\" Reagan said several times of the man who wrote, \"No law can be sacred to me but that of my nature.\" Hence Reagan's unique, and perhaps oxymoronic, doctrine\u2014conservatism without anxieties. Reagan's preternatural serenity derived from his conception of the supernatural.\n\nDiggins says Reagan imbibed his mother's form of Christianity, a strand of nineteenth-century Unitarianism from which Reagan took a foundational belief that he expressed in a 1951 letter: \"God couldn't create evil so the desires he planted in us are good.\" This logic\u2014God is good, therefore so are God-given desires\u2014leads to the Emersonian faith that we please God by pleasing ourselves. Therefore there is no need for the people to discipline their desires. So, no leader needs to suggest that the public has shortcomings and should engage in critical self-examination.\n\nDiggins thinks that Reagan's religion \"enables us to forget religion\" because it banishes the idea of \"a God of judgment and punishment.\" Reagan's popularity was largely the result of \"his blaming government for problems that are inherent in democracy itself.\" To Reagan, the idea of problems inherent in democracy was unintelligible because it implied that there were inherent problems with the demos\u2014the people. There was nothing\u2014 _nothing_ \u2014in Reagan's thinking akin to Lincoln's melancholy fatalism, his belief (see his second inaugural) that the failings of the people on both sides of the Civil War were the reasons why \"the war came.\"\n\nAs Diggins says, Reagan's \"theory of government has little reference to the principles of the American founding.\" To the Founders, and especially to the wisest of them, James Madison, government's principal function is to resist, modulate, and even frustrate the public's unruly passions, which arise from desires.\n\n\"The true conservatives, the founders,\" Diggins rightly says, constructed a government full of blocking mechanisms\u2014separations of powers, a bicameral legislature, and other checks and balances\u2014in order \"to check the demands of the people.\" Madison's Constitution responds to the problem of human nature. \"Reagan,\" says Diggins, \"let human nature off the hook.\"\n\n\"An unmentionable irony,\" writes Diggins, is that big-government conservatism is an inevitable result of Reaganism. \"Under Reagan, Americans could live off government and hate it at the same time. Americans blamed government for their dependence upon it.\" Unless people have a bad conscience about demanding big government\u2014a dispenser of unending entitlements\u2014they will get ever-larger government. But how can people have a bad conscience after being told (in Reagan's first inaugural) that they are all heroes? And after being assured that all their desires, which inevitably include desires for government-supplied entitlements, are good?\n\nSimilarly, Reagan said that the people never start wars, only governments do. But the Balkans reached a bloody boil because of the absence of effective government. Which describes Iraq today.\n\nBecause of Reagan's role in the dissolution of the Soviet Union, Diggins ranks him among the \"three great liberators in American history\"\u2014the others being Lincoln and Franklin Roosevelt\u2014and among America's three or four greatest presidents. But, says Diggins, an Emersonian president who tells us our desires are necessarily good leaves much to be desired.\n\nIf the defining doctrine of the Republican Party is limited government, the party must move up from nostalgia and leaven its reverence for Reagan with respect for Madison. As Diggins says, Reaganism tells people comforting and flattering things that they want to hear; the Madisonian persuasion tells them sobering truths that they need to know.\n\n[FEBRUARY 11, 2007]\n\n# **The Left's Plea for Materialistic Politics**\n\n**I** t has come to this: The crux of the political left's complaint about Americans is that they are insufficiently materialistic.\n\nFor a century, the left has largely failed to enact its agenda for redistributing wealth. What the left has achieved is a rich literature of disappointment, explaining the mystery, as the left sees it, of why most Americans are impervious to the left's appeal.\n\nAn interesting addition to this canon is _What's the Matter with Kansas?: How Conservatives Won the Heart of America._ Its author, Thomas Frank, argues that his native Kansas\u2014like the nation, only more so\u2014votes self-destructively, meaning conservatively, because social issues such as abortion distract it from economic self-interest, as the left understands that.\n\nFrank is a formidable controversialist\u2014imagine Michael Moore with a trained brain and an intellectual conscience. Frank has a coherent theory of contemporary politics and expresses it with a verve born of indignation. His carelessness about facts is mild by contemporary standards, or lack thereof, concerning the ethics of controversy.\n\nHe says \"the preeminent question of our times\" is why people misunderstand \"their fundamental interests.\" But Frank ignores this question: Why does the left disparage what everyday people consider their fundamental interests?\n\nHe says the left has been battered by \"the Great Backlash\" of people of modest means against their obvious benefactor and wise definer of their interests, the Democratic Party. The cultural backlash has been, he believes, craftily manufactured by rich people with the only motives the left understands\u2014money motives. The aim of the rich is to manipulate people of modest means, making them angry about abortion and other social issues so that they will vote for Republicans who will cut taxes on the rich.\n\nSuch fevered thinking is a staple of what historian Richard Hofstadter called \"the paranoid style in American politics,\" a style practiced, even pioneered, a century ago by prairie populists. You will hear its echo in John Edwards's lament about the \"two Americas\"\u2014the few rich victimizing the powerless many.\n\nFrank frequently lapses into the cartoon politics of today's enraged left, as when he says Kansas is a place of \"implacable bitterness\" and America resembles \"a panorama of madness and delusion worthy of Hieronymus Bosch.\" Yet he wonders why a majority of Kansans and Americans are put off by people like him who depict their society like that.\n\nHe says, delusionally, that conservatives have \"smashed the welfare state.\" Actually, it was waxing even before George W. Bush's prescription-drug entitlement. He says, falsely, that the inheritance tax has been \"abolished.\" He includes the required\u2014by the left's current catechism\u2014blame of Wal-Mart for destroying the sweetness of Main Street shopping. \"Capitalism\" is his succinct, if uninformative, explanation of a worldwide phenomenon of the past century\u2014the declining portion of people in agricultural employment\u2014which he seems to regret.\n\nIf you believe, as Frank does, that opposing abortion is inexplicably silly, and if you make no more attempt than Frank does to empathize with people who care deeply about it, then of course you, like Frank, will consider scores of millions of your fellow citizens lunatics. Because conservatives have, as Frank says, achieved little cultural change in recent decades, he considers their persistence either absurd or part of a sinister plot to create \"cultural turmoil\" in order to continue \"the erasure of the economic\" from politics.\n\nFrank regrets that Bill Clinton's \"triangulation\" strategy\u2014minimizing Democrats' economic differences with Republicans\u2014contributed to the erasure. Politics would indeed be simpler, and more to the liking of liberals, if each citizen were _homo economicus,_ relentlessly calculating his or her economic advantage, and concluding that liberalism serves it. But politics never has been like that, and is becoming even less so.\n\nWhen the Cold War ended, Pat Moynihan warned, with characteristic prescience, that it would be, like all blessings, a mixed one, because passions\u2014ethnic and religious\u2014that were long frozen would come to a boil. There has been an analogous development in America's domestic politics.\n\nThe economic problem, as understood during two centuries of industrialization, has been solved. We can reliably produce economic growth and have moderated business cycles. Hence many people, emancipated from material concerns, can pour political passions into other\u2014some would say higher\u2014concerns. These include the condition of the culture, as measured by such indexes as the content of popular culture, the agendas of public education, and the prevalence of abortion.\n\nSo, what's the matter with Kansas? Not much, other than it is has not measured up\u2014down, actually\u2014to the left's hope for a more materialistic politics.\n\n[JULY 8, 2004]\n\n# **Constitutional Monomania**\n\n**P** HILADELPHIA\u2014At one end of Independence Mall, at the historic center of this city where so much of America's foundational history was made with parchment and ink, stands the brick and mortar of Independence Hall. Built between 1732 and 1756, this model of what is called the Georgian style of architecture is where independence was voted and declared, and where, eleven years later, the Constitution was drafted.\n\nAt the other end of the mall sparkles a modernist jewel of America's civic life, the National Constitution Center, a nongovernment institution that opened July 4, 2003, and already has received more than 2 million visitors. It is built of gray Indiana limestone\u2014it is possible, even in Philadelphia, to have a surfeit of red brick\u2014and lots of glass. The strikingly different, yet compatible, styles of the eighteenth-century building where the Constitution was drafted and the twenty-first century building where it is explicated and studied in its third century is an architectural bow to the fact that a constitution ratified by a mostly rural nation of 4 million persons, most of whom lived within twenty miles of Atlantic tidewater, still suits an urban nation that extends twenty-five-hundred miles into the Pacific.\n\nThe center is a marvel of exhibits, many of them interactive. For example, it uses newspapers and film to give immediacy to such episodes as the Supreme Court holding in 1952 that President Truman exceeded his constitutional powers\u2014what a thought: there are limits on the commander in chief's powers\u2014when he seized the nation's steel mills to prevent a labor dispute from disrupting war production. And it shows President Eisenhower, thirteen years after sending paratroopers into Normandy, sending them to Central High School in Little Rock.\n\nThroughout, the center illustrates what Professor Felix Frankfurter\u2014before he became Justice Frankfurter\u2014was trying to express more than seventy years ago when he said, \"If the Thames is 'liquid history,' the Constitution of the United States is most significantly not a document but a stream of history.\" But it is, first and always, a document that is to be understood, as the greatest American jurist, John Marshall said, \"chiefly from its words.\"\n\nThose words\u2014which, by the way, do not include \"federal\" or \"democracy\"\u2014comprise a subtle, complicated structure that nourishes various aims and virtues. So it would be wonderful if some of the liberal groups now gearing up for a histrionic meltdown over the coming debate about the confirmation of John Roberts could spend a few hours at the National Constitution Center. Judging by the river of rhetoric that has flowed in response to the Supreme Court vacancy, contemporary liberalism's narrative of American constitutional history goes something like this:\n\n_On the night of April 18, 1775, Paul Revere galloped through the Massachusetts countryside, and to every Middlesex village and farm went his famous cry of alarm, \"The British are coming! The British are coming to menace the ancient British right to abortion!\" The next morning, by the rude bridge that arched the flood, their flag to April's breeze unfurled, the embattled farmers stood and fired the shot heard round the world in defense of the right to abortion. The Articles of Confederation, ratified near the end of the Revolutionary War to Defend Abortion Rights, proved unsatisfactory, so in the summer of 1787, fifty-five framers gathered here to draft a Constitution. Even though this city was sweltering, the Framers kept the windows of Independence Hall closed. Some say that was to keep out the horseflies. Actually, it was to preserve secrecy conducive to calm deliberations about how to craft a more perfect abortion right. The Constitution was ratified after the state conventions vigorously debated the right to abortion. But seventy-four years later, a great Civil War had to be fought to defend the Constitution against states that would secede from the Union rather than acknowledge that a privacy right to abortion is an emanation loitering in the penumbra of other rights. And so on._\n\nThe exhibits at the National Constitution Center can correct the monomania of some liberals by reminding them that the Constitution expresses the philosophy of natural rights: People have various rights, including and especially the right to property and self-government. These rights are not created by government, which exists to balance and protect the rights in their variety.\n\nAnd the center can remind conservatives of an awkward\u2014to some of them\u2014fact: The Constitution was written to correct the defects of the Articles of Confederation. That is, to strengthen the federal government.\n\n[AUGUST 14, 2005]\n\n# **Judicial Activism, Wise and Not**\n\n**U** sing arguments \"that range from the unpersuasive to the offensive\" (says the _Washington Post_ ), Senate Democrats are filibustering the nomination of Miguel Estrada to the U.S. Court of Appeals for the D.C. Circuit because he is conservative. Those senators should examine two recent writings by Judge J. Harvie Wilkinson of the U.S. Court of Appeals for the Fourth Circuit.\n\nIn \"Is There a Distinctive Conservative Jurisprudence?\" ( _University of Colorado Law Review,_ Fall 2002), he refutes the charge that there is no principled distinction between the \"activism\" of the Supreme Court under Chief Justice Rehnquist and that of the New Deal and Earl Warren courts. The Rehnquist Court has indeed invalidated many laws. However, Wilkinson says the earlier courts would \"constitutionalize freely,\" meaning \"extend constitutional rights to a point that impaired the democratic process.\" All judicial activism intrudes upon democratic processes, but many of the Rehnquist Court's invalidations have \"restructured democratic responsibilities,\" partially restoring the Founders' understanding of the proper allocation of responsibilities.\n\nIn the 1995 overturning of the federal Gun-Free School Zones Act, the Rehnquist Court acted, Wilkinson says, as \"a structural referee, not an ideological combatant.\" Congress justified preempting states and regulating guns near schools by citing the usual justification for extending its reach\u2014its power to regulate interstate commerce. But, says Wilkinson, \"the proposition that regulable commerce must mean something short of everything is hardly debatable.\" And the Rehnquist Court's ruling left states empowered to enact gun-free school zones democratically.\n\nIn the Rehnquist Court's conservative jurisprudence of balancing, federalism does not always trump competing constitutional values. The states, says Wilkinson, are not the only important \"mediative institutions\" between the individual and the national government. When state and local governments have imposed intrusive regulations on the Boy Scouts (mandating that they accept gay scoutmasters) and political parties (mandating primaries open to persons not members of the party), the Rehnquist Court has restricted states' powers in the name of the very principle that, in other contexts, caused the court to affirm states' powers\u2014to protect the intermediary institutions of civil society through which our communal, as opposed to our solitary, selves are expressed. Such protection has been, Wilkinson believes, scanted by the \"binary\" vision of liberal activism, which is committed to \"sweeping and virtually limitless national power\" and \"the recognition of new individual rights,\" but little in between.\n\nIn \"Why Conservative Jurisprudence Is Compassionate\" (to be published in the _Virginia Law Review_ ), Wilkinson argues that compassion in jurisprudence is more complicated than merely rendering judicial succor to those with poignant circumstances. Rather, compassion, as judges can properly consider it, begins by understanding this:\n\nRules\u2014rules that restrict judges' discretion to heed the promptings of poignancy\u2014have considerable virtues. They give people advance notice of what is permitted and required, they produce uniform and consistent treatment of comparable cases, and they respect whatever democratic processes have produced the rule.\n\nWilkinson says liberals and conservatives differ about \"the place of compassion in the democratic process.\" The human condition features myriad misfortunes and devastating conditions. \"Victims of social circumstance, however, are altogether distinct from victims of another's violation of a specific legal duty. It is the job of the democratic process to ameliorate the effects of the former. It is the judiciary's charge to rectify the latter.\"\n\nAnd \"modesty\"\u2014for Wilkinson, the cardinal virtue\u2014is required of judges by society's complexity. Are rent controls compassionate, or do they create a shortage of rental units and a disincentive for landlords to spend on maintenance? Does bilingual education, compassionately intended, impede the mastery of English and upward mobility? Such vexing policy questions about applied compassion are quintessentially those to which democratic rather than judicial processes should provide answers.\n\nFor judges struggling with what Wilkinson calls \"the inscrutability of compassion,\" a guiding principle should be that individual plaintiffs are not the only focus of compassion. Collective entities often are instruments of society's compassion. Judges must \"personalize\" social injury, understanding, for example, that inertial law enforcement has its victims. And that although supposedly compassionate malpractice and product-liability awards may increase patient and consumer safety, they also may drive up prices and prevent needed goods from reaching the market.\n\nWhen next there is a Supreme Court vacancy, Wilkinson's measured jurisprudence might make him the ideal nominee to silence those whose arguments against judicial conservatism range from the unpersuasive to the offensive.\n\n[MARCH 3, 2003]\n\n# **The Hard Truth About \"Soft Rights\"**\n\n**I** n contemporary American politics, as in earlier forms of vaudeville, it helps to have had an easy act to follow. Gerald Reynolds certainly did.\n\nThe U.S. Commission on Civil Rights' new chairman follows Mary Frances Berry, whose seedy career\u2014twenty-four years on the commission, eleven of them as chairman\u2014mixed tawdry peculation, boorish behavior, and absurd rhetoric. Because Reynolds represents such a bracing change, it is tempting to just enjoy the new six-to-two conservative ascendancy on the commission and forgo asking a pertinent question: Why not retire the commission?\n\nIts $9 million budget\u2014about sixty employees and six field offices\u2014is, as Washington reckons these things, negligible. So even Berry's flamboyant mismanagement of it\u2014several Government Accountability Office reports have said federal guidelines were ignored during her tenure; another report is coming\u2014was small beer, even when including the hundreds of thousands of dollars a year paid to the public relations firm that mediated her relations with the media. But although the monetary savings from closing the commission would be small, two prudential reasons for doing so are large.\n\nOne is that someday Democrats will again control the executive branch and may again stock the commission with extremists\u2014Berry celebrated Communist China's educational system in 1977, when she was assistant secretary of education; she made unsubstantiated charges of vast \"disenfranchisement\" of Florida voters in 2000\u2014from the wilder shores of racial politics. The second reason for terminating the commission is that civil rights rhetoric has become a crashing bore and, worse, a cause of confusion: Almost everything designated a \"civil rights\" problem isn't.\n\nThe commission has no enforcement powers, only the power to be, Reynolds says, a \"bully pulpit.\" And if someone must be preaching from it, by all means let it be Reynolds. Born in the South Bronx, the son of a New York City policeman, he is no stranger to the moral muggings routinely administered to African-American conservatives. But he says, \"If you think I'm conservative, you should come with me to a black barbershop. I'm usually the most liberal person there,\" where cultural conservatism\u2014on crime, welfare, abortion, schools\u2014flourishes.\n\nAfter working in some conservative think tanks, he became head of the Department of Education's Office for Civil Rights in the administration of the first President Bush. He is currently a corporate lawyer in Kansas City, where he has witnessed the handiwork of an imperial judge who, running the school system, ordered the spending of nearly $2 billion in a spectacular, if redundant, proof that increased financial inputs often do not correlate with increased cognitive outputs.\n\nBut about this commission as bully pulpit: Does anyone really think America suffers from an insufficiency of talk about race? What is in scarce supply is talk about the meaning of the phrase \"civil rights.\" Not every need is a right, and if the adjective is a modifier that modifies, not every right is a civil right\u2014one central to participation in civic life.\n\nReynolds, forty-one, says that the core function of civil rights laws is to prevent discrimination, meaning \"the distribution of benefits and burdens on the basis of race.\" But if so, today a\u2014perhaps the\u2014principal discriminator is government, with racial preferences and the rest of the reparations system that flows from the assumption that disparities in social outcomes must be caused by discrimination, and should be remedied by government transfers of wealth.\n\nReynolds rightly says that the core function of the civil rights laws, which required \"a lot of heavy lifting by the federal government,\" was to dismantle a caste system maintained by law. But that has been accomplished.\n\nIt is, as Reynolds says, scandalous that so few black seventeen-year-old males read at grade level; that so many black teenagers are not mentored to think about college as a possibility and of SAT tests as important; that many young blacks\u201468.2 percent are now born out of wedlock\u2014are enveloped in the culture that appalls Bill Cosby, a culture that disparages academic seriousness as \"acting white\" and celebrates destructive behaviors. Reynolds is right that much of this can be traced far back to discriminatory events or contexts.\n\nBut this is a problem of class, one that is both cause and effect of a cultural crisis. It is rooted in needs, such as functional families and good schools, that are not rights in the sense of enforceable claims. Civil rights laws and enforcement agencies are barely relevant. Proper pulpits\u2014perhaps including barbershops\u2014are relevant. Government pulpits are not.\n\n[MARCH 10, 2005]\n\n# **Oologah's\u2014and America's\u2014Slide**\n\n**A** mericans are not losing their minds, but they are afraid of using their minds. They are afraid to exercise judgment\u2014afraid of being sued.\n\nIn 1924, Will Rogers said Americans thought they were getting smarter because \"they're letting lawyers instead of their conscience be their guide.\" Rogers was from Oologah, Oklahoma, where in 1995 a child suffered minor injuries when playing unattended on the slide in the town park. The parents sued the town, which subsequently dismantled the slide.\n\nProducts come plastered with imbecilic warnings (on a baby stroller: \"Remove child before folding.\") for the same reason seesaws and swings are endangered species of playground equipment: fear of liability. A federal handbook morosely warns: \"Seesaw use is quite complex.\" So seesaws are being replaced with spring-driven devices used by only one child at a time. Swings? Gracious, suppose a child falls on the\u2014imagine this\u2014ground. The federal handbook again: \"Earth surfaces such as soils and hard-packed dirt are not recommended because they have poor shock-absorbing properties.\" No wonder a Southern California school district has banned running on the playground.\n\nThe early twentieth-century playground movement aimed to acquaint children with mild risks. In 1917, a movement leader said: \"It is reasonably evident that if a boy climbs on a swing frame and falls off, the school board is no more responsible for his action than if he climbed into a tree or upon the school building and falls. There can be no more reason for taking out play equipment on account of such an accident than there would be for the removal of the trees or the school building.\" Today, New York City cuts branches off trees so children will not be tempted to climb.\n\nToday, when a patient complains of a headache, a doctor, even when knowing that an aspirin is almost certainly the right treatment, may nevertheless order an expensive CAT scan. You cannot be too careful in a country in which six Mississippians have been awarded $150 million not because they are sick, but because they fear that they someday may become sick from asbestos-related illnesses.\n\nMichael Freedman reports in _Forbes_ magazine that 42 percent of obstetricians are leaving the Las Vegas area now that 76 percent of that city's obstetricians have been sued\u201440 percent of them three or more times. Pharmaceutical companies are limiting research on \"orphan drugs\" that treat serious but rare diseases because tort liability is so disproportionate to possible return on investment.\n\n\"Dismissing a tenured teacher,\" says a California official, \"is not a process, it's a career.\" Which is why in a recent five-year period only 62 of California's 220,000 tenured teachers were dismissed. The multiplication of due-process protections has turned jobs into a property right, undoing the progressive movement's dream that a civil service would end the tradition of treating public jobs as private property. In 1998, Pennsylvania reported that in the preceding forty years only thirteen teachers had been removed for incompetence. In New York State, terminating a teacher costs an average of $194,000 in legal bills\u2014the cost in time and energy of school officials is extra. Termination is a seven-year process in Detroit.\n\nBy the mid-1970s, writes Philip K. Howard, due process \"had become a kind of legal airbag inflating instantly\" to protect individuals aggrieved about any adverse encounter with authority. Howard's book, _The Collapse of the Common Good: How America's Lawsuit Culture Undermines Our Freedom,_ is a compendium of the social havoc caused by the flight from making commonsense judgments. Americans now \"tiptoe through the day,\" fearful that an angry individual with a lawyer will extort money from society while imposing irrational rules on society.\n\nOliver Wendell Holmes defined law as \"prophecies of what the courts will do.\" But Howard rightly says that \"nobody has any idea what a court will do.\" A Delaware River canoe rental company is found liable on the theory that it should have stationed lifeguards along miles of riverbank. After a church was sued\u2014unsuccessfully\u2014because a parishioner committed suicide, many churches began discouraging counseling by ministers. When any harmful event can give rise to a lawsuit, the result is \"law a la carte,\" changeable from jury to jury.\n\nThe cost of this\u2014in money, health, lives, self-government, and individual liberty\u2014is staggering. Howard, a thinking person's Quixote, has founded (with Shelby Steele, Mary Ann Glendon, John Silber, George McGovern, Newt Gingrich, and others) Common Good, an organization \"to lead a new legal revolution to restore human judgment and values at every level of society.\"\n\nTo the barricades! The address of the barricades is: ourcommon good.com.\n\n[JUNE 2, 2002]\n\n# **A Fraudulent \"Fairness\"**\n\n**S** ome illiberal liberals are trying to restore the luridly misnamed Fairness Doctrine, which until 1987 required broadcasters to devote a reasonable amount of time to presenting fairly each side of a controversial issue. The government was empowered to decide how many sides there were, how much time was reasonable, and what was fair.\n\nBy trying to again empower the government to regulate broadcasting, illiberals reveal their lack of confidence in their ability to compete in the marketplace of ideas, and their disdain for consumer sovereignty\u2014and hence for the public.\n\nThe illiberals' transparent, and often proclaimed, objective is to silence talk radio. Liberals strenuously and unsuccessfully attempted to compete in that medium\u2014witness the anemia of their Air America. Talk radio barely existed in 1980, when there were fewer than one hundred talk shows nationwide. The Fairness Doctrine was scrapped in 1987, and today more than fourteen hundred stations are entirely devoted to talk formats. Conservatives dominate talk radio\u2014although no more thoroughly than liberals dominate Hollywood, academia, and much of the mainstream media.\n\nBeginning in 1927, the government, concerned about the scarcity of radio-spectrum access, began regulating the content of broadcasts. In 1928, it decided that the programming of New York's WEVD, which was owned by the Socialist Party, was not in the public interest. The station's license was renewed after a warning to show \"due regard for the opinions of others.\" What was \"due\"? Who knew?\n\nIn 1929, the government refused the Chicago Federation of Labor's attempt to buy a station because, spectrum space being limited, all stations \"should cater to the general public.\" A decade later, the government conditioned the renewal of a station's license on the station's promise to broadcast no more anti-FDR editorials.\n\nIn 1969, the Supreme Court rejected the argument that the Fairness Doctrine violated the First Amendment protection of free speech, saying the doctrine enhanced free speech. The court did not know how the Kennedy administration, anticipating a 1964 race against Barry Goldwater, had wielded the doctrine against stations broadcasting conservative programming. The Democratic Party paid people to monitor conservative broadcasts and coached liberals in how to demand equal time. This campaign burdened stations with litigation costs and won 1,678 hours of free airtime.\n\nBill Ruder, a member of Kennedy's subcabinet, said: \"Our massive strategy was to use the Fairness Doctrine to challenge and harass right-wing broadcasters in the hope that the challenges would be so costly to them that they would be inhibited and decide it was too expensive to continue.\" The Nixon administration frequently threatened the three networks and individual stations with expensive license challenges under the Fairness Doctrine.\n\nIn 1973, Supreme Court justice and liberal icon William Douglas said: \"The Fairness Doctrine has no place in our First Amendment regime. It puts the head of the camel inside the tent and enables administration after administration to toy with TV and radio.\" The Reagan administration scrapped the doctrine because of its chilling effect on controversial speech, and because the scarcity rationale was becoming absurd.\n\nAdam Thierer, writing in the _City Journal,_ notes that today's \"media cornucopia\" has made America \"as information-rich as any society in history.\" In addition to the Internet's uncountable sources of information, there are fourteen thousand radio stations\u2014twice as many as in 1970\u2014and satellite radio has nearly 14 million subscribers. Eighty-seven percent of households have either cable or satellite television with more than five hundred channels to choose from. There are more than nineteen thousand magazines (up more than five thousand since 1993). Thierer says, consider a black lesbian feminist who hunts and likes country music:\n\n\"Would the 'mainstream media' of 25 years ago represented any of her interests? Unlikely. Today, though, this woman can program her TiVo to record her favorite shows on Black Entertainment Television, Logo (a gay\/lesbian-oriented cable channel), Oxygen (female-targeted programming), the Outdoor Life Network and Country Music Television.\"\n\nSome of today's illiberals say that media _abundance,_ not scarcity, justifies the Fairness Doctrine: Americans, the poor dears, are bewildered by too many choices. And the plenitude of information sources disperses \"the national campfire,\" the cozy communitarian experience of the good old days (for liberals), when everyone gathered around\u2014and was dependent on\u2014ABC, NBC, and CBS.\n\n\"I believe we need to reregulate the media,\" says Howard Dean. Such illiberals argue that the paucity of liberal successes in today's radio competition\u2014and the success of Fox News\u2014somehow represent \"market failure.\" That is the regularly recurring, all-purpose rationale for government intervention in markets. Market failure is defined as consumers' not buying what liberals are selling.\n\n[MAY 7, 2007]\n\n# **Policing Speech in Oakland**\n\n**M** _arriage_ is the foundation of the _natural family_ and sustains family values. That sentence is inflammatory, perhaps even a hate crime.\n\nAt least it is in Oakland, California. That city's government says those words italicized here constitute something akin to hate speech, and can be proscribed from the government's open e-mail system and employee bulletin board.\n\nWhen the McCain-Feingold law empowered government to regulate the quantity, content, and timing of political campaign speech about government, it was predictable that the right of free speech would increasingly be sacrificed to various social objectives that free speech supposedly impedes. And it was predictable that speech suppression would become an instrument of cultural combat, used to settle ideological scores and advance political agendas by silencing adversaries.\n\nThat has happened in Oakland. And, predictably, the ineffable Ninth U.S. Circuit Court of Appeals has ratified this abridgment of First Amendment protections. Fortunately, overturning the Ninth Circuit is steady work for the U.S. Supreme Court.\n\nSome African-American Christian women working for Oakland's government organized the Good News Employee Association (GNEA), which they announced with a flier describing their group as \"a forum for people of Faith to express their views on the contemporary issues of the day. With respect for the Natural Family, Marriage and Family Values.\"\n\nThe flier was distributed _after_ other employees' groups, including those advocating gay rights, had advertised their political views and activities on the city's e-mail system and bulletin board. When the GNEA asked for equal opportunity to communicate by that system and that board, they were denied. Furthermore, the flier they posted was taken down and destroyed by city officials, who declared it \"homophobic\" and disruptive.\n\nThe city government said the flier was \"determined\" to promote harassment based on sexual orientation.\" The city warned that the flier and communications like it could result in disciplinary action \"up to and including termination.\"\n\nEffectively, the city has proscribed any speech that even one person might say questioned the gay rights agenda and therefore created what that person felt was a \"hostile environment.\" This, even though gay rights advocates used the city's communication system to advertise \"Happy Coming Out Day.\" Yet the terms _natural family, marriage,_ and _family values_ are considered intolerably inflammatory.\n\nThe treatment of GNEA illustrates one technique by which America's growing ranks of self-appointed speech police expand their reach: They wait until groups they disagree with, such as GNEA, are provoked to respond to them in public debates, then they persecute them for annoying those to whom they are responding. In Oakland, this dialectic of censorship proceeded on a reasonable premise joined to a preposterous theory.\n\nThe premise is that city officials are entitled to maintain workplace order and decorum. The theory is that government supervisors have such unbridled power of prior restraint on speech in the name of protecting order and decorum that they can nullify the First Amendment by declaring that even the mild text of the GNEA flier is inherently disruptive.\n\nThe flier supposedly violated the city regulation prohibiting \"discrimination and\/or harassment based on sexual orientation.\" The only cited disruption was one lesbian's complaint that the flier made her feel \"targeted\" and \"excluded.\" So anyone has the power to be a censor just by saying someone's speech has hurt his or her feelings.\n\nUnless the speech is \"progressive.\" If GNEA claimed it felt \"excluded\" by advocacy of the gay rights agenda, would that advocacy have been suppressed? Of course not\u2014although GNEA's members could plausibly argue that the city's speech police have created a \"hostile workplace environment\" against them.\n\nA district court affirmed the city's right to impose speech regulations that are patently _not_ content neutral. It said the GNEA's speech interest\u2014the flier\u2014is \"vanishingly small.\" GNEA, in its brief asking the U.S. Supreme Court to intervene, responds that some of the high court's seminal First Amendment rulings have concerned small matters, such the wearing of a T-shirt, standing on a soapbox, holding a picket sign, and \"other simple forms of expression.\"\n\nCongress is currently trying to enact yet another \"hate crime\" law that would authorize enhanced punishments for crimes committed because of, among other things, sexual orientation. A coalition of African-American clergy, the High Impact Leadership Coalition, opposes this, fearing it might be used \"to muzzle the church.\" The clergy argue that in our \"litigation prone society\" the legislation would result in lawsuits having \"a chilling effect\" on speech and religious liberty. As the Oakland case demonstrates, that, too, is predictable.\n\n[JUNE 24, 2007]\n\n# **Liberalism's Itch in Minneapolis**\n\n**T** he campaign to deny Luis Paucar his right to economic liberty illustrates the ingenuity people will invest in concocting perverse arguments for novel entitlements. This city's taxi cartel is offering an audacious new rationalization for corporate welfare, asserting a right\u2014a _constitutional right, in perpetuity_ \u2014to revenues it would have received if Minneapolis's City Council had not ended the cartel that never should have existed.\n\nPaucar, thirty-seven, embodies the best qualities of American immigrants. He is a splendidly self-sufficient entrepreneur. And he is wielding American principles against some Americans who, in their decadent addiction to government assistance, are trying to litigate themselves to prosperity at the expense of Paucar and the public.\n\nSeventeen years ago, Paucar came to America from Ecuador, and for five years drove a taxi in New York City. _Because_ that city has long been liberalism's laboratory, many taxi drivers there are akin to, as an economist has said, \"modern urban sharecroppers.\"\n\nIn 1937, New York City, full of liberalism's itch to regulate everything, knew, just _knew,_ how many taxicab permits there should be. For seventy years, the number (about twelve thousand) has not been significantly changed, so rising prices have been powerless to create new suppliers of taxi services. Under this government-created scarcity, a permit (\"medallion\") now costs about $500,000. Most people wealthy enough to buy medallions do not drive cabs, any more than plantation owners picked cotton. They lease their medallions at exorbitant rates to people like Paucar who drive, often for less than $15 an hour, for long days.\n\nAttracted by Minneapolis\u2013St. Paul's vibrant Hispanic community, now 130,000 strong, Paucar moved here, assuming that economic liberty would be more spacious than in New York. Unfortunately, Minnesota has a \"progressive,\" meaning statist, tradition that can impede the progress of people like Paucar but who lack his knack for fighting back.\n\nThe regulatory impulse came to the upper Midwest with immigrants from northern Europe, many of whom carried the too-much-government traditions of \"social democracy.\" In the 1940s, under a mayor who soon would take his New Deal liberalism to Washington\u2014Hubert Humphrey\u2014the city capped entry into the taxi business.\n\nBy the time Paucar got here in 1999, 343 taxis were permitted. He wanted to launch a fleet of 15. That would have required him to find 15 incumbent license holders willing to sell their licenses for up to $25,000 apiece.\n\nAs a by-product of government intervention, a secondary market arose in which government-conferred benefits were traded by the cartel. In 2006, Minneapolis had only one cab for every one thousand residents (compared to three times as many in St. Louis and Boston), which was especially punishing to the poor who lack cars.\n\nThat fact\u2014and Paucar's determination and, eventually, litigiousness; he is a _real_ American\u2014helped persuade the City Council members, liberals all (twelve members of the Democratic Farmer-Labor Party, one member of the Green Party), to vote to allow forty-five new cabs per year until 2010, at which point the cap will disappear. In response, the cartel is asking a federal court to say the cartel's constitutional rights have been violated. It says the cap\u2014a barrier to entry into the taxi business\u2014constituted an entitlement to profits that now are being \"taken\" by government action.\n\nThe Constitution's Fifth Amendment says no property shall be \"taken\" without just compensation. The concept of an injury through \"regulatory taking\" is familiar and defensible: Such an injury occurs when a government regulation reduces the value of property by restricting its use. But the taxi cartel is claiming a _deregulatory_ taking: It wants compensation because it now faces unanticipated competition.\n\nWhen the incumbent taxi industry inveigled the city government into creating the cartel, this was a textbook example of rent seeking\u2014getting government to confer advantages on an economic faction in order to disadvantage actual or potential competitors. If the cartel's argument about a \"deregulatory taking\" were to prevail, modern government\u2014the regulatory state\u2014would be controlled by a leftward-clicking ratchet: Governments could never deregulate, never undo the damage that they enable rent seekers to do.\n\nBy challenging his adopted country to honor its principles of economic liberty and limited government, Paucar, assisted by the local chapter of the libertarian Institute for Justice, is giving a timely demonstration of this fact: Some immigrants, with their acute understanding of why America beckons, refresh our national vigor. It would be wonderful if every time someone like Paucar comes to America, a native-born American rent seeker who has been corrupted by today's entitlement mentality would leave.\n\n[MAY 27, 2007]\n\n# **Chicago: From the White City to the Green City**\n\n**C** HICAGO\u2014Thirty-five summers ago, in the angriest year of a boiling era, the forces of peace, love, and understanding\u2014they fancied themselves \"flower power\"\u2014clashed violently at the Democratic Convention with the police of Mayor Richard Daley. Today, the mayor of this famously muscular city\u2014big-shouldered hog butcher and stacker of wheat\u2014practices flower power. His name is Richard Daley.\n\nThe son is in his fourth four-year term. Chicago has been governed by him or his father for thirty-five of the forty-eight years since 1955. Chicago's name derives from an American Indian word meaning \"wild onion,\" and the city's motto Urbs in Horto means \"city in a garden.\" Daley's green thumb has produced a city chock-full of gardens.\n\nIncluding one on the roof of city hall. It has twenty thousand plants of more than 150 species. And three beehives, which produce sixty pounds of honey a year. The hives are emblematic of the intensely practical nature of Daley's passion to prove that \"nature can exist in an urban environment.\"\n\nA Daley aide calls him \"the Martha Stewart of mayors,\" then quickly decides there must be a more felicitous encomium to celebrate his attentions to fine touches conducive to gracious urban living. Daley explains his passion for prettification by recalling that shortly after being elected in 1989, he took a train from Washington to New York, and was struck by the ugliness travelers saw as trains entered and left cities along the way. So today the river of traffic flowing from O'Hare airport to downtown on the Kennedy Expressway passes between, as it were, landscaped riverbanks.\n\nWithin most cities, Daley says, people experience \"a canyon effect of steel and concrete.\" But cities need not mean just \"steel, concrete and dirt.\" Flowers, he says, are one way \"to change the perception of cities.\" He has created median strips on some major streets and planted them with flowers \"to slow down traffic.\" And, he says, \"Flowers calm people down.\"\n\nThis apostle of calming influences is the son of the man called\u2014it is the title of a biography of him\u2014an \"American pharaoh.\" His father would stop his limousine to pick up littered newspapers, but his passion was not for sensory pleasure but for tidiness. The father considered even Republicans litter: In his last three elections, he carried 148 of the city's 150 wards.\n\nThe pharaoh was a builder, but his most famous pyramids were not what he wanted. He favored low-rise public housing, but federal pressures forced him to build high-rises that became crime-infested pillars of poverty. The son has been dismantling them. The son, too, is a builder\u2014witness Chicago's sparkling skyscrapers. But he is primarily a planter.\n\nSince 1989, he has presided over the planting of more than three hundred thousand trees, which he says not only please the eye but \"reduce noise, air pollution and summer heat.\" Twenty-one underutilized acres around the city have been turned into seventy-two community gardens and parks. The renovation of Soldier Field on the lakefront will include seventeen acres of new parkland. The largest park project, the Calumet Open Space Reserve on the far Southeast Side, is four thousand acres of prairies, wetlands, and forests.\n\nThe city contracts with an organization called the Christian Industrial League, which hires many down-and-out persons to wash streets and water plants. And the organization is building a greenhouse that will sell flowers in winter.\n\nThe great Chicago architect Louis Sullivan (1856\u20131924) relished the city's \"intoxicating rawness,\" which was still abundant when the first Mayor Daley won the first of his six elections. He became perhaps America's most important twentieth-century mayor. Alan Ehrenhalt, executive editor of _Governing_ magazine, notes that Chicago could have become a Detroit, a symbol of urban failure.\n\nIf the son is a twenty-first-century model mayor, it is because he senses the importance of the senses: Human beings respond to aesthetic values. Daley is not given to flights of theory, so he may not realize the extent to which he is continuing a project begun on this city's South Side 110 years ago.\n\nChicago successfully competed against New York, Philadelphia, and Washington for the right to host the World's Columbian Exposition of 1893, celebrating the four hundredth anniversary of Columbus's voyage. The White City, as the exposition was called, received 27 million visitors and gave rise to the City Beautiful movement, the premise of which was that cities do not need to be grimly utilitarian, and that improvement of a city's material environment would be conducive to the moral improvement of its residents. Coarse environments would coarsen people; refined environments would help ameliorate what reformers considered the moral deficiencies of people struggling to adapt to urban life.\n\nA century ago, reforming elites thought of beautification in terms of social control: it would tame the lower orders. Daley, a Chicago chauvinist, primarily just wants his city to be second to none\u2014Second City, indeed. But he also aims at mild social control: He hopes his flowers will calm down everyone.\n\n[AUGUST 4, 2003]\n\n# **Our Moralizing Tax Code**\n\n_The taxing power of government must be used to provide revenues for legitimate government purposes. It must not be used to regulate the economy or bring about social change._\n\n\u2014PRESIDENT RONALD REAGAN, State of the Union message, February 18, 1981\n\n_(b) no portion of the proceeds of such issue is to be used to provide (including the provision of land for) any private or commercial golf course, country club, massage parlor, hot tub facility, suntan facility, racetrack or other facility used for gambling, or any store the principal business of which is the sale of alcoholic beverages for consumption off premises._\n\n\u2014Title 26, Internal Revenue Code (tax exemption requirements for qualified redevelopment bonds)\n\n**W** ell, yes, certainly no massage parlors. Or hot tubs, of course; one shudders to think what happens in those. And tanning facilities, too, are the Devil's playgrounds. As for racetracks, although state governments promoting their lotteries are America's most energetic advocates of gambling, government should err on the side of caution when protecting whatever this tax provision protects by frowning on racetracks, hot tubs, and other things.\n\nThis peculiar wrinkle in the tax code, first approved the year after President Reagan said the tax code should not be used to leverage social change, makes certain projects ineligible to be financed by industrial redevelopment bonds that are subsidized by preferential tax treatment. This provision recently popped back into the news, thanks to Katrina.\n\nThat ill wind blew some (barely) offshore casinos onto the shores of the Gulf Coast. As part of the plan to \"rebuild,\" as the saying goes, the damaged coast, such bonds are going to be issued. But not promiscuously. Some legislators do not want tax-subsidized bonds to finance the rebuilding of casinos.\n\nNot that the casinos need help: They are rebounding briskly, even expanding. Still, government has a sorry record of dispensing billions in corporate welfare for flourishing businesses.\n\nIt is mysterious why states or localities that want casinos operating nearby\u2014and providing jobs and tax revenues\u2014also want them afloat, a few feet from a riverbank or ocean shore. (Mississippi has just decided to let them come ashore.) Does the narrow band of water provide prophylactic protection against sin? The communities already have weighed the sin against the jobs and revenues and found the sin congenial.\n\nBut such awkward questions arise when government begins moralizing, especially about the minutiae of life, such as hot tubs. Which brings us to Reagan's 1981 statement about inappropriate uses of the tax code.\n\nHe disliked government using the code to conduct industrial policy, picking commercial winners and losers, which is a recipe for what is called \"lemon socialism\"\u2014tax subsidies for failing businesses that the market says should fail. Regarding the second part of Reagan's statement, any tax code is going to shape society. But he opposed manipulating the tax code to stigmatize this or that consumer preference. Which is what the code's anti-hot-tub provision does.\n\nOne wonders: Why did the social improvers who used the code to put the government, in its majesty, on record against hot tubs and tanning facilities not extend their list of disapproved choices? Their list looks morally lax.\n\nReally stern social conservatives probably favor explicitly proscribing government assistance to lots of things, most of them somehow involving sex. Government could preen about being too moral to subsidize, with tax-preferred bonds, economic projects that include bookstores that sell Judy Blume novels, or hotels that offer in-room pornography. And wouldn't it be fun to find the words _lap dance_ in the nation's tax code?\n\nAs strongly as social conservatives deplore commercialized sex, liberals deplore cigarettes, Big Macs, firearms, fur coats, SUVs, pornography not printed on recycled paper, pornographic movies produced by nonunion studios, holiday trees provocatively labeled \"Christmas trees,\" and much more.\n\nBut do we really want to march down this path paved with moral pronouncements? When government uses subsidies to moralize, as with tax preferences for bonds that can be used to finance this but not that, government is speaking. It is expressing opinions about what is and is not wholesome. And once government starts venting such opinions, how does it stop?\n\nGovernment could spare itself the stress of moralizing about so many things if it decided that the choices people make with their money is their, not its, business. And government could avoid having opinions about so many things if it would quit subsidizing so many things.\n\nWhen, for example, the valuation and allocation of money through bonds is left to the market, government can be reticent. And reticent government sounds wonderful.\n\n[JANUARY 8, 2006]\n\n# **\"Electronic Morphine\" on the Ohio River**\n\n**O** n the north bank of the Ohio River sits Evansville, Indiana, home of David Williams, fifty-two, and of a riverboat casino. During several years of gambling in that casino, Williams, a state auditor earning $35,000 a year, lost approximately $175,000. He had never gambled before the casino sent him a coupon for $20 worth of gambling.\n\nHe visited the casino, lost the $20, and left. On his second visit, he lost $800. The casino issued to him, as a good customer, a \"Fun Card,\" which when used in the casino earns points for meals and drinks, and enables the casino to track the user's gambling activities. For Williams, those activities became what he calls \"electronic morphine.\"\n\nBy the time he had lost $5,000, he said to himself that if he could get back to even, he would quit. One night he won $5,500, but he did not quit. In 1997, he lost $21,000 to one slot machine in two days. In March 1997, he lost $72,186. He sometimes played two slot machines at a time, all night, until the boat docked at 5 a.m., then went back aboard when the casino opened at 9 a.m. Now he is suing the casino, charging that it should have refused his patronage because it knew he was addicted. It did know he had a problem.\n\nIn March 1998, a friend of Williams's got him involuntarily confined to a treatment center for addictions, and wrote to inform the casino of Williams's gambling problem. The casino included a photo of Williams among those of banned gamblers, and wrote to him a \"cease admissions\" letter. Noting the \"medical\/psychological\" nature of problem gambling behavior, the letter said that before being readmitted to the casino he would have to present medical\/psychological information demonstrating that patronizing the casino would pose no threat to his safety or well-being.\n\nAlthough no such evidence was presented, the casino's marketing department continued to pepper him with mailings. And he entered the casino and used his Fun Card without being detected.\n\nThe _Wall Street Journal_ reports that the casino has twenty-four signs warning: \"Enjoy the fun...and always bet with your head, not over it.\" Every entrance ticket lists a toll-free number for counseling from the Indiana Department of Mental Health. Nevertheless, Williams's suit charges that the casino, knowing he was \"helplessly addicted to gambling,\" intentionally worked to \"lure\" him to \"engage in conduct against his will.\" Well.\n\nIt is unclear what luring was required, given his compulsive behavior. And in what sense was his will operative? The fourth edition of the _Diagnostic and Statistical Manual of Mental Disorders (DSM-IV)_ says \"pathological gambling\" involves persistent, recurring, and uncontrollable pursuit less of money than of the euphoric state of taking risks in quest of a windfall. Pathological gamblers often exhibit distorted thinking (denial, superstition, overconfidence). They lie to friends and family to conceal their behavior, resort to theft or fraud to finance it, and succumb to \"chasing\"\u2014ever more risky and high-stakes gambling in attempts to recoup losses.\n\nIt is worrisome that society is medicalizing more and more behavioral problems, often defining as addictions what earlier, sterner generations explained as weakness of will. Prodded by science, or what purports to be science, society is reclassifying what once were considered character flaws or moral failings as personality disorders akin to physical disabilities.\n\nHowever, at least several million Americans do have a disposition\u2014a \"mental disorder\"? a \"compulsive disease\"?\u2014that seems to make them as unable to gamble responsibly as an alcoholic is unable to drink responsibly. This is a small portion of the nation's population, but a large pool of misery for themselves and loved ones.\n\nGambling has been a common feature of American life forever, but for a long time it was broadly considered a sin, or a social disease. Now it is a social policy: The most important and aggressive promoter of gambling in America is government.\n\nForty-four states have lotteries, twenty-nine have casinos, and most of these states are to varying degrees dependent on\u2014you might say addicted to\u2014revenues from wagering. And since the first Internet gambling site was created in 1995, competition for gamblers' dollars has become intense. The October 28 issue of _Newsweek_ reported that 2 million gamblers patronize eighteen hundred virtual casinos every week. With $3.5 billion being lost on Internet wagers this year, gambling has passed pornography as the Web's most lucrative business.\n\nThe anonymous, lonely, undistracted nature of online gambling is especially conducive to compulsive behavior. But even if government knew how to move against Internet gambling, what would be its rationale for doing so? Government curbs on private-sector gambling enterprises look like attempts to cripple the competition\u2014to prevent others from poaching on the population of gamblers that government has done so much to enlarge.\n\nDavid Williams's suit should trouble this gambling nation. But don't bet on it.\n\n[NOVEMBER 25, 2002]\n\n# **Prohibition II: Interestingly Selective**\n\n**P** erhaps Prohibition II is being launched because Prohibition I worked so well at getting rid of gin. Or maybe the point is to reassure social conservatives that Republicans remain resolved to purify Americans' behavior. Incorrigible cynics will say Prohibition II is being undertaken because someone stands to make money from interfering with other people making money.\n\nFor whatever reason, last Friday, the president signed into law Prohibition II. You almost have to admire the government's plucky refusal to heed history's warnings about the probable futility of this adventure. This time the government is prohibiting Internet gambling by making it illegal for banks or credit-card companies to process payments to online gambling operations on a list the government will prepare.\n\nLast year, about 12 million Americans wagered $6 billion online. But after Congress, thirty-two minutes before adjourning, passed its ban, the stock of the largest online-gambling business, Gibraltar-based PartyGaming, which gets 85 percent of its $1 billion annual revenue from Americans, declined 58 percent in one day, wiping out about $5 billion in market value. The stock of a British company, World Gaming PLC, which gets about 95 percent of its revenue from Americans, plunged 88 percent. The industry, which has some twenty-three hundred websites and did half of its business last year with Americans, has lost $8 billion in market value because of the new law. And you thought the 109th Congress did not accomplish anything.\n\nSupporters of the new law say it merely strengthens enforcement; they claim that Internet gambling is illegal under the Wire Act enacted in 1961, before Al Gore, who was then thirteen, had invented the Internet. But not all courts agree. Supporters of the new law say online gambling sends billions of dollars overseas. But the way to keep the money here is to decriminalize the activity.\n\nThe number of online American gamblers, although just one-sixth the number of Americans who visit real casinos annually, doubled in the last year. This competition alarms the nation's biggest gambling interests\u2014state governments.\n\nIt is an iron law: When government uses laws, tariffs, and regulations to restrict the choices of Americans, ostensibly for their own good, someone is going to make money from the paternalism. One of the big winners from the government's action against online gambling will be the state governments that are America's most relentless promoters of gambling. Forty-eight states (all but Hawaii and Utah) have some form of legalized gambling. Forty-two states have lottery monopolies. Thirty-four states rake in part of the take from casino gambling, slot machines, or video poker.\n\nThe new law actually legalizes online betting on horse racing, Internet state lotteries, and some fantasy sports. The horse-racing industry is a powerful interest. The solidarity of the political class prevents the federal officials from interfering with state officials' lucrative gambling. And woe unto the politicians who get between a sports fan and his fun.\n\nIn the private sector, where realism prevails, casino operators are not hot for criminalizing Internet gambling. This is so for two reasons: It is not in their interest for government to wax censorious. And online gambling might whet the appetites of millions for the real casino experience.\n\nGranted, some people gamble too much. And some people eat too many cheeseburgers. But who wants to live in a society that protects the weak-willed by criminalizing cheeseburgers? Besides, the problems\u2014frequently exaggerated\u2014of criminal involvement in gambling, and of underage and addictive gamblers, can be best dealt with by legalization and regulation utilizing new software solutions. Furthermore, taxation of online poker and other gambling could generate billions for governments.\n\nProhibition I was a porous wall between Americans and their martinis, giving rise to bad gin supplied by bad people. Prohibition II will provoke imaginative evasions as the market supplies what gamblers will demand\u2014payment methods beyond the reach of Congress.\n\nBut governments and sundry busybodies seem affronted by the Internet, as they are by any unregulated sphere of life. The speech police are itching to bring bloggers under campaign-finance laws that control the quantity, content, and timing of political discourse. And now, by banning a particular behavior\u2014the entertainment some people choose, using their own money\u2014government has advanced its mother-hen agenda of putting a saddle and bridle on the Internet.\n\nGambling is, however, as American as the Gold Rush or, for that matter, Wall Street. George Washington deplored the rampant gambling at Valley Forge, but lotteries helped fund his army as well as Harvard, Princeton, and Dartmouth. And Washington endorsed the lottery that helped fund construction of the city that now bears his name, and from which has come a stern\u2014but interestingly selective\u2014disapproval of gambling.\n\n[OCTOBER 23, 2002]\n\n# **Being Green at Ben & Jerry's**\n\n**I** f you have an average-size dinner table, four feet by six feet, put a dime on the edge of it. Think of the surface of the table as the Arctic National Wildlife Refuge in Alaska. The dime is larger than the piece of the coastal plain that would have been opened to drilling for oil and natural gas. The House of Representatives voted for drilling, but the Senate voted against access to what Senator John Kerry, Massachusetts Democrat and presidential aspirant, calls \"a few drops of oil.\" ANWR could produce, for twenty-five years, at least as much oil as America currently imports from Saudi Arabia.\n\nSix weeks of desultory Senate debate about the energy bill reached an almost comic culmination in...yet another agriculture subsidy. The subsidy is a requirement that will triple the amount of ethanol, which is made from corn, that must be put in gasoline, ostensibly to clean America's air, actually to buy farmers' votes.\n\nOver the last three decades, energy use has risen about 30 percent. But so has population, which means per capita energy use is unchanged. And per capita GDP has risen substantially, so we are using 40 percent less energy per dollar output. Which is one reason there is no energy crisis, at least none as most Americans understand such things\u2014a shortage of, and therefore high prices of, gasoline for cars, heating oil for furnaces, and electricity for air conditioners.\n\nIn the absence of a crisis to concentrate the attention of the inattentive American majority, an intense faction\u2014full-time environmentalists\u2014goes to work. Spencer Abraham, the secretary of energy, says \"the previous administration...simply drew up a list of fuels it didn't like\u2014nuclear energy, coal, hydropower, and oil\u2014which together account for 73 percent of America's energy supply.\" Well, there are always windmills.\n\nSometimes lofty environmentalism is a cover for crude politics. The United States has the world's largest proven reserves of coal. But Mike Oliver, a retired physicist and engineer, and John Hospers, professor emeritus of philosophy at USC, note that in 1996 President Clinton put 68 billion tons of America's cleanest-burning coal, located in Utah, off-limits for mining, ostensibly for environmental reasons. If every existing U.S. electric power plant burned coal, the 68 billion tons could fuel them for forty-five years at the current rate of consumption. Now power companies must import clean-burning coal, some from mines owned by Indonesia's Lippo Group, the heavy contributor to Clinton, whose decision about Utah's coal vastly increased the value of Lippo's coal.\n\nThe United States has just 2.14 percent of the world's proven reserves of oil, so some people say it is pointless to drill in places like ANWR because \"energy independence\" is a chimera. Indeed it is. But domestic supplies can provide important insurance against uncertain foreign supplies. And domestic supplies can mean exporting hundreds of billions of dollars less to oil-producing nations, such as Iraq.\n\nBesides, when considering proven reserves, note the adjective. In 1930, the United States had proven reserves of 13 billion barrels. We then fought the Second World War and fueled the most fabulous economic expansion in human history, including the electricity-driven \"New Economy.\" (Manufacturing and running computers consume 15 percent of U.S. electricity. Internet use alone accounts for half of the growth in demand for electricity.) So by 1990 proven reserves were...17 billion barrels, not counting any in Alaska or Hawaii.\n\nIn 1975, proven reserves in the Persian Gulf were 74 billion barrels. In 1993, they were 663 billion, a ninefold increase. At the current rate of consumption, today's proven reserves would last 150 years. New discoveries will be made, some by vastly improved techniques of deepwater drilling. But environmental policies will define opportunities. The government estimates that beneath the U.S. outer continental shelf, which the government owns, there are at least 46 billion barrels of oil. But only 2 percent of the shelf has been leased for energy development.\n\nOpponents of increased energy production usually argue for decreased consumption. But they flinch from conservation measures. A new $1 gasoline tax would dampen demand for gasoline, but it would stimulate demands for the heads of the tax increasers. After all, Americans get irritable when impersonal market forces add twenty-five cents to the cost of a gallon. Tougher fuel-efficiency requirements for vehicles would save a lot of energy. But who would save the legislators who passed those requirements? Beware the wrath of Americans who like to drive, and autoworkers who like to make, cars that are large, heavy, and safer than the gasoline sippers that environmentalists prefer.\n\nSome environmentalism is a feel-good indulgence for an era of energy abundance, which means an era of avoided choices. Or ignored choices\u2014ignored because if acknowledged, they would not make the choosers feel good. Karl Zinsmeister, editor in chief of the _American Enterprise_ magazine, imagines an oh-so-green environmentalist enjoying the most politically correct product on the planet\u2014Ben & Jerry's ice cream. Made in a factory that depends on electricity-guzzling refrigeration, a gallon of ice cream requires four gallons of milk. While making that much milk, a cow produces eight gallons of manure, and flatulence with another eight gallons of methane, a potent \"greenhouse\" gas. And the cow consumes lots of water plus three pounds of grain and hay, which is produced with tractor fuel, chemical fertilizers, herbicides, and insecticides, and is transported with truck or train fuel:\n\n\"So every time he digs into his Cherry Garcia, the conscientious environmentalist should visualize (in addition to world peace) a pile of grain, water, farm chemicals, and energy inputs much bigger than his ice cream bowl on one side of the table, and, on the other side of the table, a mound of manure eight times the size of his bowl, plus a balloon of methane that would barely fit under the dining room table.\"\n\nCherry Garcia. It's a choice. _Bon app\u00e9tit._\n\n[MAY 6, 2002]\n\n# **The Tyranny of the Small Picture**\n\n_What good is happiness? It can't buy money._\n\n\u2014HENNY YOUNGMAN\n\n**S** ocial hypochondria is the national disease of the most successful nation. By most indexes, life has improved beyond the dreams of even very recent generations. Yet many Americans, impervious to abundant data and personal experiences, insist that progress is a chimera.\n\nGregg Easterbrook's impressive new book, _The Progress Paradox: How Life Gets Better While People Feel Worse,_ explains this perversity. Easterbrook, a Washington journalist and fellow of the Brookings Institution, assaults readers with good news:\n\nAmerican life expectancy has dramatically increased in a century, from forty-seven to seventy-seven years. Our great-great-grandparents all knew someone who died of some disease we never fear. (As recently as 1952, polio killed 3,300 Americans.) Our largest public health problems arise from unlimited supplies of affordable food. The typical American has twice the purchasing power his mother or father had in 1960. A third of America's families own _at least_ three cars. In 2001, Americans spent $25 billion\u2014more than North Korea's GDP\u2014on recreational watercraft. Factor out immigration\u2014a huge benefit to the immigrants\u2014and statistical evidence of widening income inequality disappears. The statistic that household incomes are only moderately higher than twenty-five years ago is misleading: Households today average fewer people, so real-dollar incomes in middle-class households are about 50 percent higher today. Since 1970, the number of cars has increased 68 percent and the number of miles driven has increased even more, yet smog has declined by a third and traffic fatalities have declined from 52,627 to 42,815 last year. In 2003, we spend much wealth on things unavailable in 1953\u2014a cleaner environment, reduced mortality through new medical marvels ($5.2 billion a year just for artificial knees, which did not exist a generation ago), the ability to fly anywhere or talk to anyone anywhere. The incidence of heart disease, stroke, and cancer, adjusted for population growth, is declining. The rate of child poverty is down in a decade. America soon will be the first society in which a majority of adults are college graduates.\n\nAnd so it goes. But Easterbrook says that such is today's \"discontinuity between prosperity and happiness,\" the \"surge of national good news\" scares people, vexes the news media, and does not even nudge up measurements of happiness. Easterbrook's explanations include:\n\n\u2022 \"The tyranny of the small picture.\" The preference for bad news produces a focus on smaller remaining problems after larger ones are ameliorated. Ersatz bad news serves the fund-raising of \"gloom interest groups.\" It also inflates the self-importance of elites, who lose status when society is functioning well. Media elites, especially, have a stake in \"headline-amplified anxiety.\"\n\n\u2022 \"Evolution has conditioned us to believe the worst.\" In Darwinian natural selection, pessimism, wariness, suspicion, and discontent may be survival traits. Perhaps our relaxed and cheerful progenitors were eaten by saber-toothed tigers. Only the anxiety-prone gene pool prospered.\n\n\u2022 \"Catalogue-induced anxiety\" and \"the revenge of the plastic\" both cause material abundance to increase unhappiness. The more we can order and charge, the more we are aware of what we do _not_ possess. The \"modern tyranny of choice\" causes consumers perpetual restlessness and regret.\n\n\u2022 The \"latest model syndrome\" abets the \"tyranny of the unnecessary,\" which leads to the \"ten-hammer syndrome.\" We have piled up mountains of marginally improved stuff, in the chaos of which we cannot find any of our nine hammers, so we buy a tenth, and the pile grows higher. Thus does the victor belong to the spoils.\n\n\u2022 The cultivation\u2014even celebration\u2014of victimhood by intellectuals, tort lawyers, politicians, and the media is both cause and effect of today's culture of complaint.\n\nEasterbrook, while arguing that happiness should be let off its leash, is far from complacent. He is scandalized by corporate corruption and poverty in the midst of so much abundance. And he has many commonsensical thoughts on how to redress the imbalance many people feel between their abundance of material things and the scarcity of meaning that they feel in their lives. The gist of his advice is that we should pull up our socks, spiritually, and make meaning by doing good while living well.\n\nHis book arrives as the nation enters an election year, when the opposition, like all parties out of power, will try to sow despondency by pointing to lead linings on all silver clouds. His timely warning is that Americans are becoming colorblind, if only to the color silver.\n\n[JANUARY 11, 2004]\n\n# **Draining the Reservoir of Reverence**\n\n_Why did we run? Well, those who didn't run are there yet._\n\n\u2014AN OHIO SOLDIER\n\n**C** HANCELLORSVILLE, VIRGINIA\u2014The twelve-mile march on May 2, 1863, took Stonewall Jackson from the clearing in the woods where he conferred for the last time with Robert E. Lee, to a spot from which Jackson and thirty thousand troops surveyed the rear of the Union forces. Those forces, commanded by a blowhard, Joe Hooker (\"May God have mercy on General Lee, for I shall have none\"), were about to experience one of the nastiest shocks of the Civil War.\n\nTwo hours before dusk, Federal soldiers were elated when deer, turkeys, and rabbits came pelting out of the woods into their lines. It was, however, not dinner but death approaching. By nightfall, Federal forces were scattered. When the fighting subsided four days later, Lee was emboldened to try to win the war with an invasion of Pennsylvania. The invasion's high-water mark came at the crossroads town of Gettysburg.\n\nOne hundred and thirty-nine years after the battle here, a more protracted struggle is under way. In 1863, the nation's survival was at stake. Today, only the nation's memory is at stake. \"Only\"? Without memory, the reservoir of reverence, what of the nation survives?\n\nHence the urgency of the people opposing a proposal to build, on acreage over which the struggle surged, 2,350 houses and 2.4 million square feet of commercial and office space. All this would bring a huge increase in traffic, wider highways, and the further submergence of irrecoverable history into a perpetually churned present.\n\nNorthern Virginia, beginning about halfway between Richmond and Washington, is a humming marvel of energy and entrepreneurship, an urbanizing swirl of commerce and technology utterly unlike the static rural society favored by Virginia's favorite social philosopher, Jefferson. Chancellorsville is in an east-west rectangle of terrain about fifteen miles long and ten miles wide, now divided by Interstate 95, that saw four great battles\u2014Fredericksburg, Chancellorsville, Spotsylvania, the Wilderness\u2014involving one hundred thousand killed, wounded, or missing.\n\nWhere a slavocracy once existed, Northern dynamism prevails. But Northern Virginia has ample acreage for development, without erasing the landscapes where the Army of Northern Virginia spent its valor. As for the Federals' side, it is a scandal that the federal government's cheese-paring parsimony has prevented the purchase of historically significant land\u2014twenty-thousand acres, maximum\u2014at Civil War battlefields from Maryland to Mississippi.\n\nJust $10 million annually for a decade\u2014a rounding error for many Washington bureaucracies\u2014would preserve much important battlefield land still outside National Park Service boundaries. The government's neglect can be only partially rectified by the private work of the Civil War Preservation Trust, just three years old. (You can enlist at www.civilwar.org. Also check www.chancellorsville.org.)\n\nCWPT's president James Lighthizer, a temperate, grown-up realist, stresses that CWPT's members are \"not whacked-out tree-huggers\" who hate development and want to preserve \"every piece of ground where Lee's horse pooped.\" But regarding commemorations, Americans today seem inclined to build where they ought not, and to not build where they should, as at the site of the World Trade Center.\n\nIn New York City, many people who are antigrowth commerce despisers want to exploit Ground Zero for grinding their old ideological axes. They favor making all or most of the sixteen-acre parcel a cemetery without remains, a place of perpetual mourning\u2014what Richard Brookhiser disapprovingly calls a \"deathopolis\" in the midst of urban striving.\n\nBut most who died at Ground Zero were going about their private pursuits of happiness, murdered by people who detest that American striving. The murderers crashed planes into the Twin Towers, Brookhiser says, \"in the same spirit in which a brat kicks a beehive. They will be stung, and the bees will repair the hive.\" Let the site have new towers, teeming with renewed striving.\n\nBut a battlefield is different. A battlefield is hallowed ground because those who there gave the last full measure of devotion went there because they were devoted unto death to certain things.\n\nThose who clashed at Chancellorsville did so in a war that arose from a clash of large ideas. Some ideas were noble, some were not. But there is ample and stirring evidence that many of the young men caught in the war's whirlwind could articulate what the fight was about, on both sides. See James M. McPherson's _For Cause and Comrades: Why Men Fought in the Civil War._\n\nLocal government here can stop misplaced development from trampling out the contours of the Confederacy's greatest victory. A Jeffersonian solution.\n\n[SEPTEMBER 22, 2002]\n\n# **_United 93:_ \"We've Got to Do It Ourselves\"**\n\n**I** n most movies made to convey dread, the tension flows from uncertainty about what will happen. In _United 93,_ terror comes from knowing exactly what will happen. People who associate cinematic menace with maniacs wielding chain saws will find that there can be an almost unbearable menace in the quotidian\u2014in the small talk of passengers waiting in the boarding area with those who will murder them, in the routine shutting of the plane's door prior to push-back from the gate at Newark airport on September 11.\n\nBut two uncertainties surrounded _United 93:_ Would it find an audience? Should it?\n\nIt has found one, which is remarkable, given that in 2005 most moviegoers\u201457 percent\u2014were persons twelve to twenty-nine years old. Twenty-nine percent were persons twelve to twenty-four. These age cohorts do not seek shattering, saddening experiences to go with their popcorn. In its first weekend, _United 93_ was the second most watched movie, with the top average gross per theater among major releases. It was on 1,795 screens, and 71 percent of viewers were thirty or older.\n\nTo the long list of Britain's contributions to American cinema\u2014Charles Chaplin, Bob Hope, Cary Grant, Stan Laurel, Deborah Kerr, Vivien Leigh, Maureen O'Hara, Ronald Colman, David Niven, Boris Karloff, Alfred Hitchcock, and others\u2014add Paul Greengrass, writer and director of _United 93._ He imported into Hollywood the commodity most foreign to it: good taste. This is especially shown in the ensemble of unknown character actors, and nonactors who play roles they know\u2014a real pilot plays the pilot, a former flight attendant plays the head flight attendant\u2014and several persons who play on screen the roles they played on 9\/11.\n\nGreengrass's scrupulosity is evident in the movie's conscientious, minimal, and minimally speculative departures from the facts about the flight that were painstakingly assembled for the _The 9\/11 Commission Report._ This is emphatically not a \"docudrama\" such as Oliver Stone's execrable _JFK,_ which was \"history\" as a form of literary looting in which the filmmaker used just enough facts to lend a patina of specious authenticity to tendentious political ax grinding.\n\nA _New York Times_ story on the \"politics of heroism\" deals with the question of whether the movie is \"inclusive.\" Well, perhaps _United 93_ did violate some egalitarian nicety by suggesting that probably not all the passengers were equally heroic. Amazingly, no one has faulted the movie for ethnic profiling: All the hijackers are portrayed as young, fervently devout Islamic males. Report Greengrass to the U.S. Commission on Civil Rights.\n\nIn a movie as spare and restrained as its title, the only excess is the suggestion, itself oblique, that the government responded even more confusedly that morning than was to be expected. Most government people, like the rest of us, were in the process of having their sense of the possible abruptly and radically enlarged.\n\nGoing to see _United 93_ is a civic duty because Samuel Johnson was right: People more often need to be reminded than informed. After an astonishing fifty-six months without a second terrorist attack, this nation perhaps has become dangerously immune to astonishment. The movie may quicken our appreciation of the measures and successes\u2014many of which must remain secret\u2014that have kept would-be killers at bay.\n\nThe editors of _National Review_ were wise to view _United 93_ in the dazzling light still cast by a Memorial Day address, \"The Soldier's Faith,\" delivered in 1895 by a veteran of Ball's Bluff, Antietam, and other Civil War battles. Oliver Wendell Holmes Jr. said why understanding that faith is important:\n\n\"In this snug, over-safe corner of the world...we may realize that our comfortable routine is no eternal necessity of things, but merely a little space of calm in the midst of the tempestuous untamed streaming of the world, and in order that we may be ready for danger.... Out of heroism grows faith in the worth of heroism.\"\n\nThe message of the movie is: We are all potential soldiers. And we all may be, at any moment, at the war's front, because in this war the front can be anywhere.\n\nThe hinge on which the movie turns are thirteen words that a passenger speaks, without histrionics, as he and others prepare to rush the cockpit, shortly before the plane plunges into a Pennsylvania field. The words are: \"No one is going to help us. We've got to do it ourselves.\" Those words not only summarize this nation's situation in today's war, but also express a citizen's general responsibilities in a free society.\n\n[MAY 7, 2006]\n\n# **Nothing Changes Everything**\n\n**B** efore the dust from the collapsed towers had settled, conventional wisdom had congealed: \"Everything has changed.\" But what about what matters most, the public's sensibility?\n\nIt has taken five years for 9\/11 to receive a novelist's subtle and satisfying treatment, but it was worth the wait for Claire Messud's _The Emperor's Children._ Her intimation of the mark the attacks made on the American mind is convincing because in her comedy of manners, as in the nation's life, that horrific event is, oddly, both pivotal and tangential.\n\nMessud's Manhattan story revolves around two women and a gay man who met as classmates at Brown University and who, as they turn thirty in 2001, vaguely yearn to do something \"important\" and \"serious.\" Vagueness\u2014lack of definition\u2014is their defining characteristic. Which may be because\u2014or perhaps why\u2014all three are in the media. All are earnest auditors and aspiring improvers of the nation's sensibility.\n\nMarina is a glamorous child of privilege because she is the child of Murray, a famous liberal commentator given to saying things such as, to a seminar on Resistance in Postwar America, \"once upon a time, poetry _did_ matter.\" A former intern at _Vogue,_ Marina lives with her parents, on an allowance from them, on Central Park West. She is having trouble finishing her book on \"how complex and profound cultural truths\u2014our _mores_ entire\u2014could be derived from\" analysis of changing fashions in children's clothes. \"I want to make a difference.\" But get a job? \"I worry that will make me ordinary, like everybody else.\" She is, her father recognizes, \"stymied, now, by the very lack of smallness\" in her life, \"by the absence of any limitations against which to rebel.\"\n\nDanielle, from Ohio, is a producer of documentaries who hopes to \"articulate\" an \"ethos\" into a \"movement.\" Her current project, to raise \"questions about integrity and authenticity,\" concerns women who had bad experiences with liposuction.\n\nJulius, from Michigan, is an independent book and film reviewer \"with a youthful certainty that attitude would carry him.\" His \"life of Wildean excess and insouciance seemed an accomplishment in itself.\" He is \"an inchoate ball of ambition,\" and is intermittently aware that at thirty \"some actual sustained endeavor might be in order.\" That might, however, be difficult, given his belief that \"regularity was bourgeois.\"\n\nThe problem the three share is not that their achievements, if there ever are any, will be ephemeral, but that their intentions to achieve them are ephemeral. Not solid, like those of the Australian who comes to New York \"to foment revolution.\" With a new magazine.\n\nMurray's nephew, Bootie, a morose autodidact\u2014imagine Holden Caulfield with his nose in a book of Emerson's essays\u2014rounds out Messud's central cast, each illustrating Messud's acute understanding of the Peter Pan complex now rampant among young adults who feel entitled to be extraordinary: \"To be your own person, to find your own style\u2014these were the quests of adolescence and young adulthood, pushed, in a youth-obsessed culture, well into middle age.\"\n\nNot until page 370 of Messud's delicious depiction of the quintet's tangled lives, \"torn between Big Ideas and a party,\" do the planes hit the towers. Bootie\u2014it could have been any of these people preoccupied with manufacturing interpretations of fashions and fashions of interpretations\u2014has \"a fearful thought: you could make something inside your head, as huge and devastating as this, and spill it out into reality, make it really happen.\" Imagine that.\n\nBefore 9\/11, Messud began writing a Manhattan novel about young adults living in the media hall of mirrors. After 9\/11, she abandoned it. Then returned to it. Asked if she thought she had written a \"9\/11 novel,\" she demurs: \"I wrote an August 1914 novel.\" Meaning, \"The world I had set out to describe in 2001 had become historical.\"\n\nBut what had changed? The party, scheduled for 9\/11, to launch the Australian's magazine and the revolution\u2014Ren\u00e9e Zellweger had accepted; Susan Sontag was a maybe\u2014was canceled, as was the magazine. Murray \"formulated a reasoned middle ground\": America did not deserve the attacks, but remember the West Bank. \"He wasn't opposed to the invasion of Afghanistan, but qualified about its methods.\" Danielle decides to proceed with her liposuction documentary.\n\nNothing changes everything. And even huge events that, as Messud says, make \"certain things seem particularly frivolous\" leave most of our enveloping normality largely unscathed. That truth and a heightened sense of the frivolous are conducive to national poise five years into a long war.\n\n[SEPTEMBER 10, 2006]\n**Chapter Four**\n\n**SENSIBILITIES AND SENSITIVITIES**\n\n# **Narcissism as News**\n\n**T** _ime_ magazine asked a large number of people to name the Person of the Year. They were in a populist mood and named the largest possible number of Persons of the Year: Everybody.\n\nOf course. The most capacious modern entitlement is not to Social Security but to self-esteem. So _Time_ 's cover features a mirrorlike panel. The reader\u2014but why bother to read the magazine when merely gazing at its cover gives immediate and intense gratification?\u2014can gaze at the reflection of his or her favorite person. Narcissism is news? Evidently.\n\nTo the person looking at his reflection, _Time_ 's cover announces, congratulations: \"You control the Information Age.\" By \"control,\" _Time_ means only that everyone is created equal\u2014equally entitled to create content for the World Wide Web, which is controlled by neither law nor taste.\n\nRichard Stengel, _Time_ 's managing editor, says, \"Thomas Paine was in effect the first blogger\" and \"Ben Franklin was essentially loading his persona into the MySpace of the eighteenth century, 'Poor Richard's Almanack.'\" Not exactly.\n\nFranklin's extraordinary persona informed what he wrote but was not the subject of what he wrote. Paine was perhaps history's most consequential pamphleteer. There are expected to be 100 million bloggers worldwide by the middle of 2007, which is why none will be like Franklin or Paine. Both were geniuses; genius is scarce. Both had a revolutionary civic purpose, which they accomplished by amazing exertions. Most bloggers have the private purpose of expressing themselves, for their own satisfaction. There is nothing wrong with that, but nothing demanding or especially admirable, either. They do it successfully because there is nothing singular about it, and each is the judge of his or her own success.\n\nAccording to the Pew Internet & American Life Project, 76 percent of bloggers say one reason they blog is to document their personal experiences and share them with others. And 37 percent\u2014soon, 37 million\u2014say the primary topic of their blog is \"my life and experiences.\" George III would have preferred dealing with 100 million bloggers rather than one Paine.\n\nStengel says that bloggers and the people who upload videos onto YouTube (sixty-five thousand new videos a day; 100 million watched each day) are bringing \"events\" to us in ways that are often more \"authentic\" than the services of traditional media. But authenticity is easy, and of no inherent value, if it is simply and necessarily the attribute of any bit of reality (\"event\") captured on video.\n\n_Time_ 's Lev Grossman writes that \"an explosion of productivity and innovation\" is under way as \"millions of minds that would otherwise have drowned in obscurity\" become participants in \"the global intellectual economy.\" Grossman continues:\n\n\"Who actually sits down after a long day at work and says, I'm not going to watch 'Lost' tonight. I'm going to turn on my computer and make a movie starring my pet iguana? I'm going to mash up 50 Cent's vocals with Queen's instrumentals? I'm going to blog about my state of mind or the state of the union or the steak frites at the new bistro down the street? Who has that time and that energy and that passion?\n\n\"The answer is, you do. And for seizing the reins of the global media, for founding and framing the new digital democracy, for working for nothing and beating the pros at their own game, Time's Person of the Year for 2006 is you.\"\n\nThere are, however, essentially no reins on the Web\u2014few means of control and direction. That is good, but vitiates the idea that the Web's chaos of entertainment, solipsism, and occasional intellectual seriousness and civic engagement is anything like a polity (a \"digital democracy\"). _Time_ 's bow to the amateurs who are, it strangely suggests, no longer obscure, and in the same game that _Time_ is in, is refuted by a glance\u2014which is all an adult will want\u2014at YouTube's most popular videos.\n\n_Time_ 's issue includes an unenthralled essay by NBC's Brian Williams, who believes that raptures over the Web's egalitarianism arise from the same impulse that causes today's youth soccer programs to award trophies\u2014\"entire bedrooms full\"\u2014to any child who shows up: \"The danger just might be that we miss the next great book or the next great idea, or that we will fail to meet the next great challenge...because we are too busy celebrating ourselves and listening to the same tune we already know by heart.\"\n\nThe fact that Stengel included Williams's essay proves that Stengel's _Time_ has what 99.9 percent of the Web's content lacks: seriousness.\n\n[DECEMBER 21, 2006]\n\n# **The Speciesism of Featherless Bipeds**\n\n**O** ne thinks twice, even thrice, before using in a magazine as decorous as _Newsweek_ the four-letter F word that causes so much discord. But words should not be minced. So, what is being done for British pets is just not _fair._\n\nOne wants to avoid speciesism, the moral disease of being species-centric. Still, why should British pets have more\u201425 percent more, to be precise\u2014freedoms than humans do?\n\nIn January 1941, President Franklin Roosevelt envisioned a \"world founded upon four essential human freedoms\"\u2014freedom of speech and expression, freedom of every person to worship God in his own way, freedom from want, and freedom from fear. In January 2006, Prime Minister Tony Blair's\u2014technically, Her Majesty's\u2014Department for Environment, Food and Rural Affairs said pets should have five freedoms. The Animal Welfare Bill says the five are:\n\n1. An appropriate diet.\n\n2. Suitable living conditions.\n\n3. Companionship or solitude, as the cat, canary, or gerbil prefers.\n\n4. Monitoring for abnormal behavior.\n\n5. Protection from pain, suffering, injury, and disease.\n\nWell. Politicians' jokes are usually recognizable as such because they elicit boisterous laughter from the politicians' friends, families, and employees. But there is no evidence that Blair's government is joking. The Labour Party, having recently saved the foxes from the fox hunters and their hounds, is serious\u2014not to say grim and humorless\u2014about perfecting society.\n\nBesides, it is not funny. It is Orwellian to say that when governments provide this and that benefit they are providing freedoms. As Bishop Joseph Butler (1692\u20131752) said, back when clear thinking was a British attribute, \"Everything is what it is, and not another thing.\" Appropriate diets, suitable living conditions, etc., are not \"freedoms.\" They are nice things, but they could be provided by a benevolent despot. Freedom is about the absence of some things\u2014of coercion, dependency, restraints not consented to\u2014and the presence of institutions and the habits, mores, customs, and dispositions that sustain those absences. But nowadays there is confusion arising from a non sequitur that governments encourage: Freedom is a nice thing, therefore governments that provide nice things are expanding freedom.\n\nThe _Times_ of London reports that pet owners will be supplied with lots of rules. Such as: \"Dogs should be introduced to cats very carefully.\" What would we do without government to guide us? The pet police, who will be empowered to enter houses and seize animals, will enforce rules like the one that pets must have \"mental stimulation\" sufficient to ward off boredom and frustration. A nine-point guide about cats \"going to the toilet\" mandates provisions for privacy. A British headline: GET YOUR CAT A PRIVATE LOO OR EXPECT PET POLICE. ANOTHER: LABOUR'S PET POLICE MAY POUNCE IF YOUR DOG GETS BORED. The _Times_ says a code of conduct for invertebrates, such as lobsters, may be coming.\n\nYou will not be surprised that in America, it is San Francisco (which has more dogs\u2014an estimated 110,000\u2014than children) that is especially punctilious about codifying animal entitlements. One such is that a dog's water must be changed at least once a day, and must be served in a nontipping bowl. A San Franciscan who has two Dobermans says pets \"need as much care as a child does.\" The chairman of the commission that drafted the nontipping-water-bowl ordinance says much of its language replicates that of a Los Angeles ordinance. So there.\n\nBut if British pets are going to have five freedoms, we must ask: Did FDR, who stipulated that he was enumerating four \"human\" freedoms, shortchange us? Humans, too, are animals\u2014featherless bipeds, as Plato said. It seems, however, that the ruling pigs in George Orwell's _Animal Farm_ were correct: Some animals\u2014British pets\u2014are more equal than others. So, what is to be done to erase the 25 percent freedom advantage enjoyed by lower animals\u2014is it still legal in Britain to use that locution?\u2014such as British pets?\n\nWhen Woodrow Wilson proclaimed his Fourteen Points that would guarantee a sweeter world, Georges Clemenceau complained that God had only ten. But Americans believe in going the competition one better. So when President George W. Bush\u2014who says that \"freedom is on the march\" everywhere, which it is, except where it isn't\u2014is done perfecting the switch-grass-powered automobile, he should step up to the challenge. It is time to add two freedoms to FDR's four. How about:\n\nFreedom from government attempts to codify and supervise every transaction between people, let alone those between people and their hamsters and turtles and tropical fish.\n\nAnd: Freedom from the idea that we have only as many freedoms\u2014speaking correctly, only as much freedom\u2014as governments in their graciousness choose to enumerate.\n\n[FEBRUARY 13, 2006]\n\n# **What We Owe to What We Eat**\n\n**M** atthew Scully, a former speechwriter for President George W. Bush, is the most interesting conservative you have never heard of. He speaks barely above a whisper and must be the mildest disturber of the peace. But he is among the most disturbing.\n\nIf you value your peace of mind, not to mention your breakfast bacon, you should not read Scully's essay \"Fear Factories: The Case for Compassionate Conservatism\u2014for Animals.\" It appeared in the May 23, 2005, issue of Pat Buchanan's magazine, the _American Conservative_ \u2014not where you would expect to find an essay arguing that industrial livestock farming involves vast abuses that constitute a serious moral problem.\n\nThe disturbing facts about industrial farming by the $125 billion-a-year livestock industry\u2014the pain-inflicting confinements and mutilations\u2014have economic reasons. Ameliorating them would impose production costs that consumers would pay. But to glimpse what consumers would be paying to stop, visit factoryfarming.com\/gallery.htm. Or read Scully on the miseries inflicted on billions of creatures \"for our convenience and pleasure\":\n\n\"...400-to 500-pound mammals trapped without relief inside iron crates seven feet long and 22 inches wide. They chew maniacally on bars and chains, as foraging animals will do when denied straw...The pigs know the feel only of concrete and metal. They lie covered in their own urine and excrement, with broken legs from trying to escape or just to turn...\"\n\nIt is, Scully says, difficult, especially for conservatives, to examine cruelty issues on their merits, or even to acknowledge that something serious can be at stake where animals are concerned. This is partly because some animal-rights advocates are so off-putting. See, for example, the February 3, 2003, letter that Ingrid Newkirk, president of People for the Ethical Treatment of Animals\u2014animals other than humans\u2014sent to the terrorist Yasir Arafat, complaining that an explosive-laden donkey was killed when used in a Jerusalem massacre.\n\nThe rhetoric of animal \"rights\" is ill-conceived. The starting point, says Scully, should be with our obligations\u2014the requirements for living with integrity. In defining them, some facts are pertinent, facts about animals' emotional capacities and their experience of pain and happiness. Such facts refute what conservatives deplore\u2014moral relativism. They do because they demand a certain reaction and evoke it in good people, who are good because they consistently respect the objective value of fellow creatures.\n\nIt may be true that, as has been said, the Puritans banned bearbaiting not because it gave pain to the bears but because it gave pleasure to the spectators. And there are indeed degrading pleasures. But to argue for outlawing cruelty to animals because it is bad for the cruel person's soul is to accept, as Scully does not, that man is the only concern.\n\nStatutes against cruelty to animals, often imposing felony-level penalties, codify society's belief that such cruelty is an intrinsic evil. This is a social affirmation of a strong moral sense in individuals who are not vicious. It is the sense that even though the law can regard an individual's animal as the individual's property, there nevertheless are certain things the individual cannot do to that property. Which means it is property with a difference.\n\nThe difference is the capacity for enjoyment and suffering. So why, Scully asks, is cruelty to a puppy appalling and cruelty to livestock by the billions a matter of social indifference? There cannot be any intrinsic difference of worth between a puppy and a pig.\n\nAnimal suffering on a vast scale should, he says, be a serious issue of public policy. He does not want to take away your BLT; he does not propose to end livestock farming. He does propose a Humane Farming Act to apply to corporate farmers the elementary standards of animal husbandry and veterinary ethics: \"We cannot just take from these creatures, we must give them something in return. We owe them a merciful death, and we owe them a merciful life.\"\n\nSays who? Well, Scully replies, those who understand \"Judeo-Christian morality, whose whole logic is one of gracious condescension, or the proud learning to be humble, the higher serving the lower, and the strong protecting the weak.\"\n\nYes, of course: You don't want to think about this. Who does? But do your duty: Read his book _Dominion: The Power of Man, the Suffering of Animals, and the Call to Mercy._ Scully, a conservative and hence a realist, knows that man is not only a rational creature but a rationalizing creature, putting his intellectual nimbleness in the service of his desires. But refraining from cruelty is an objective obligation. And as Scully says, \"If reason and morality are what set humans apart from animals, then reason and morality must always guide us in how we treat them.\"\n\nYou were warned not to read this. Have a nice day.\n\n[JULY 18, 2005]\n\n# **The Holocaust: Handcrafted**\n\n_Little by little we were taught all these things. We grew into them._\n\n\u2014ADOLPH EICHMANN\n\n**T** hese are the best of times for the worst of people. And for the toxic idea at the core of all the most murderous ideologies of the modern age. That idea is that human nature is, if not a fiction, at least so watery and flimsy that it poses no serious impediment to evil political entities determined to treat people as malleable clay to be molded into creatures at once submissive and violent.\n\nAll political philosophies rest on notions of human nature. And what we think human nature is\u2014indeed, whether we think there is such a thing\u2014depends somewhat on conclusions we draw from political events, such as these: A mother rejoicing that her teenage child has blown herself up in the process of blowing up other mothers' children. A Palestinian infant dressed as a suicide bomber\u2014parents will have glittering dreams for their children.\n\nThere was violence, but there were not suicide bombers with celebrating choruses, when Yasir Arafat's Palestinian Authority thugocracy began its occupation of Gaza and the West Bank. After eight years of incitements\u2014in schools, mosques, mass media\u2014to anti-Semitic genocide, those areas now need de-Nazification. Eichmann's \"little by little\" has been compressed into just eight years.\n\nHistorian Richard Rhodes's new _Masters of Death: The SS-Einsatzgruppen and the Invention of the Holocaust_ may not be the ideal beach book. But this latest contribution to the debate about the origins of Nazi behavior\u2014the processes of socialization to butchery\u2014is dreadfully timely.\n\nMuch has been made of the Nazis' \"modern\" and \"scientific\" means of \"industrializing\" mass murder\u2014railroads leading to gas chambers. But most Holocaust murders were, Rhodes says, \"handcrafted,\" one at a time, using bullets, often fired at such close range that the shooters were splattered with gore. Himmler worried that this involved suffering\u2014by his perpetrators.\n\nAfter long days of _Sardinenpackung_ (having Jews lay down on the last layer of those already murdered, in order to efficiently fill killing pits), Himmler hoped the performers of \"this burdensome duty\" would relax in civilized evenings: \"The comradely gathering must on no account, however, end in the abuse of alcohol. It should be an evening on which\u2014as far as possible\u2014they sit and eat at table in the best German domestic style, and music, lectures and the introductions to the beauties of German intellectual and emotional life occupy the hours.\"\n\nOf course there was shop talk, too. At one post-massacre dinner, an officer explained that his more experienced killers tossed children into the air to shoot them, not out of unseemly exuberance, but because bullets often passed through children's bodies, so shooting them on floors or streets could cause dangerous ricochets.\n\nRhodes's most disturbing vignette is not of the German walking with a year-old baby impaled on his bayonet, still crying weakly. More chilling, in its way, is this: One supervisor of massacres \"had the photographs taken at the executions developed at two photographic shops in southern Germany and showed them to his wife and friends.\" Were the technicians who developed the film perturbed? The wife and friends\u2014was their moral sense, which supposedly is part (a large part? a durable part?) of human nature, disturbed?\n\nCivilization's enemies attack civilization's foundational idea, the proposition that human nature is not infinitely plastic, that people cannot be socialized to accept or do anything. These enemies believe that human beings have no common nature, no shared moral sense that is a component of a universal human nature. Rather, all we have in common is a capacity to acquire an infinite variety of cultures, however vile.\n\nRhodes's book contributes evidence to the debate about the roles of nature and nurture, of ideology and peer pressure and other things in the making of people who participate in mass murder. The Palestinian Authority is also contributing much evidence. Rhodes's book, and Arafat's willing executioners of Jews, and Palestinian parents who rejoice at the suicides of their murderous children, and Palestinian street mobs drunk with delight about dismembered Jews\u2014all these point to a conclusion: Teaching (to use Eichmann's verb) such participants is disturbingly easy.\n\nWith the recent terrorist bomb planted to kill young people at an Israeli university, terrorists reached an apogee, a purity of evil, simultaneously targeting youth and learning. In Joseph Conrad's _Secret Agent_ (1907), a novel about terrorism, a theorist of terror justifies targeting England's Greenwich observatory. Mere butchery is a bit banal, so:\n\n\"The attack must have all the shocking senselessness of gratuitous blasphemy. Since bombs are your means of expression, it would be really telling if one could throw a bomb into pure mathematics. But that is impossible.\"\n\nSuch are the disappointments of modern barbarians, who otherwise are prospering.\n\n[AUGUST 19, 2002]\n\n# **The \"Daring\" of the Avant-Garde Yet Again**\n\n**N** EW YORK\u2014An iron law of avant-garde art is that theorizing expands to fill a void of talent. That law explains the execrable exhibit \"Mirroring Evil: Nazi Imagery\/Recent Art,\" now in its final days at the Jewish Museum at the corner of Ninety-second Street and Fifth Avenue.\n\nThe works by thirteen \"internationally recognized\" artists make, the museum brochure says, \"new and daring use of imagery taken from the Nazi era.\" In the cant of artists' self-puffery, the word _daring_ usually means artists are daring to strike political poses that are imbecilic and, among the avant-garde, fashionable. Here the artists daringly draw \"unnerving connections between the imagery of the Third Reich and today's consumer culture.\"\n\nExamples of the works are:\n\n\"Giftgas Giftset,\" three replicas of Zyklon B gas canisters in the colors, and bearing the logos, of Chanel, Herm\u00e8s, and Tiffany's. \"Prada Deathcamp\" is a model of a concentration camp on cardboard from a Prada hatbox. The exhibit catalog theorizes that the artist \"dares to observe Holocaust museums and their visitors from the position of a critique of consumption.\" These two works ask the \"irreconcilable\" (does the illiterate author mean unanswerable?) question of \"whether the artist is fascinated by the label-logo culture or mocking it.\"\n\n\"LEGO Concentration Camp Set\" consists of replicas of boxes of the children's building blocks, but the boxes bear photographs of models of barracks and crematoria. The catalog theorizes that this work shows \"how such seemingly harmless items may pose serious psychological and philosophical questions about gender, sexuality, and childhood.\"\n\nIn \"It's the Real Thing\u2014Self-Portrait at Buchenwald,\" the artist digitally inserts a photograph of himself, holding a Diet Coke, into the foreground of a famous photograph of emaciated Jews in their bunks shortly after the liberation of Buchenwald. The catalog theorizes that this work \"draws parallels between brainwashing tactics of the Nazis and commodification. Just as much of Europe succumbed to Nazi culture because it was the dominant paradigm, so does our contemporary culture succumb to consumerism.\"\n\nEnough. The smug narcissism and overbearing didacticism, all expressed in jargon-clotted prose about \"aesthetic strategies\" and \"transgressive images,\" is repulsive. The use of genocide as a plaything for political posturing is contemptible. What was the Jewish Museum thinking, and why did it not think again after September 11?\n\nMany of the works in \"Mirroring Evil\" are based on photographs (there are no oil paintings), which should demonstrate that photography has said almost everything that can properly be expressed graphically about the Holocaust. But, then, \"Mirroring Evil\" is evil because it really has nothing to do with the murder of 6 million Jews. Rather, it is an exhibit of the artists' exhibitionism, their fathomless fascination with their shallow selves. And \"Mirroring Evil,\" although an extreme example, is hardly the only example of the miniaturization of the Holocaust, the turning of tragedy into mere raw material for intellectuals' fads.\n\nIn the June 14 issue of the _Chronicle of Higher Education,_ William F. S. Miles, professor of Jewish historical and cultural studies at Northeastern University, reflects uneasily on his experience at a two-week course for college teachers conducted by the Center for Advanced Holocaust Studies at the United States Holocaust Memorial Museum in Washington. There he heard colleagues say how the explosive growth of Holocaust studies has turned that genocide into a \"wonderful, creative teaching opportunity.\"\n\nParticipants in the course said \"a gendered approach to the Holocaust is truly exciting\" and \"we can examine victims in terms of their class, too, or their national origins\" and \"you can tie [the Holocaust] in to dance, art, architecture. Even Web-page making.\" Miles reports: \"Repeated analogies between victims of the Holocaust and battered women in America are made.... There also is implied criticism of articulate survivors: 'We privilege them because they are eloquent.'\" Miles' mild response is:\n\n\"Experts are no longer eyewitnesses but rather clever scholars with the latest new angles, spins or hypotheses. All one can hope is that the intellectualization of the Holocaust be pursued in good faith with a modicum of sensitivity toward the survivors.\"\n\nBut what hope can there be for even minimal decency and understanding when today's intelligentsia is hospitable to trivializations of a huge tragedy? No vulgarity is unthinkable now that the Holocaust has become fodder for semi-intellectual wisecracks, the plaything of theory-weaving and ax-grinding academic and artistic mediocrities who discern a moral equivalence between commercial advertising and Nuremberg rallies.\n\nA wit once said that everything changes except the avant-garde. But it does change. It gets worse.\n\n[JUNE 20, 2002]\n\n# **Anti-Semitism Across the Political Spectrum**\n\n**I** t used to be said that anti-Catholicism was the anti-Semitism of the intellectuals. Today, anti-Semitism is the anti-Semitism of the intellectuals.\n\nNot all intellectuals, of course. And the seepage of this ancient poison into the intelligentsia\u2014always so militantly modern\u2014is much more pronounced in Europe than here. But as anti-Semitism migrates across the political spectrum from right to left, it infects the intelligentsia, which has leaned left for two centuries.\n\nHere the term _intellectual_ is used loosely, to denote not only people who think about ideas\u2014about thinking\u2014but also people who think they do. The term _anti-Semitism_ is used precisely, to denote people who dislike Jews. These people include those who say: We do not dislike Jews, we only dislike Zionists\u2014although to live in Israel is to endorse the Zionist enterprise, and all Jews are implicated, as sympathizers, in the crime that is Israel.\n\nWednesday's release of Mel Gibson's movie _The Passion of the Christ_ has catalyzed fears of resurgent anti-Semitism. Some critics say the movie portrays the governor of Judea\u2014Pontius Pilate, the Roman prefect responsible for the crucifixion\u2014as more benign and less in control than he actually was, and ascribes too much power and malignity to Jerusalem's Jewish elite.\n\nJon Meacham's deeply informed cover story \"Who Killed Jesus?\" in the February 16 _Newsweek_ renders this measured judgment: The movie implies more blame for the Jewish religious leaders of Judea of that time than sound scholarship suggests. However, Meacham rightly refrains from discerning disreputable intentions in Gibson's presentation of matters about which scholars, too, must speculate, and do disagree. Besides, this being a healthy nation, Americans are unlikely to be swayed by the movie's misreading, as Meacham delicately suggests, of the actions of a few Jews two thousand years ago.\n\nFears about the movie exacerbating religiously motivated anti-Semitism are missing the larger menace\u2014the upsurge of political anti-Semitism. Like traditional anti-Semitism, but with secular sources and motives, the political version, which condemns Jews as a social element, is becoming mainstream, and chic among political and cultural elites, mostly in Europe. Consider:\n\n\u2022 A cartoon in a mainstream Italian newspaper depicts the infant Jesus in a manger, menaced by an Israeli tank and saying \"Don't tell me they want to kill me again.\" This expresses animus against Israel rather than twisted Christian zeal.\n\n\u2022 The European Union has suppressed a study it commissioned, because the study blamed the upsurge in anti-Jewish acts on European Muslims\u2014and the European left.\n\n\u2022 An EU poll reveals that a European majority believes the greatest threat to world peace is Israel.\n\n\u2022 Nineteen percent of Germans believe what a bestselling German book asserts: The CIA and Israel's Mossad organized the September 11 attacks.\n\n\u2022 On French television, a comedian wearing a Jewish skullcap gives a Nazi salute while yelling \"Isra-Heil!\"\n\n\u2022 If Israel is not the Great Satan, it is allied with him\u2014America. European anti-American demonstrations often include Israel's blue and white flag with a swastika replacing the star of David, and signs perpetuating the myth, concocted by Palestinians and cooperative Western journalists, of an Israeli massacre in Jenin: \"1943: Warsaw \/ 2002: Jenin.\"\n\n\u2022 Omer Bartov, historian at Brown University, writes in the _New Republic_ that much of what Hitler said \"can be found today in innumerable places: on Internet sites, propaganda brochures, political speeches, protest placards, academic publications, religious sermons, you name it.\"\n\nThe appallingly brief eclipse of anti-Semitism after Auschwitz demonstrates how beguiling is the simplicity of pure stupidity. All of the left's prescriptions for curing what ails society\u2014socialism, communism, psychoanalysis, \"progressive\" education, etc.\u2014have been discarded, so now the left is reduced to adapting that hardy perennial of the right, anti-Semitism.\n\nThis is a new twist to the left's recipe for salvation through elimination: All will be well if we eliminate capitalists, or private property, or the ruling class, or \"special interests,\" or neuroses, or inhibitions. Now, let's try eliminating a people, starting with their nation, which is obnoxiously pro-American and insufferably Spartan.\n\nEurope's susceptibility to political lunacy, and the Arab world's addiction to it, is not news. And the paranoid style is a political constant. Those who believe a vast conspiracy assassinated President Kennedy say: Proof of the conspiracy's diabolical subtlety is that _no_ evidence of it remains. Today's anti-Semites say: Proof of the Jews' potent menace is that there are so few of them\u2014just 13 million of the planet's 6 billion people\u2014yet they cause so many political, economic, and cultural ills.\n\nGosh. Imagine if they were, say, 1 percent of Earth's population\u201463 million.\n\n[FEBRUARY 25, 2004]\n\n# **When Harry Remet Hanne**\n\n**A** mong the radiating effects of the United States Holocaust Memorial Museum, which opened along Washington's Mall ten years ago, is the story of how Harry remet Hanne. The friendship of Harry Ettlinger, now seventy-seven, and Hanne Hirsch, now seventy-eight, was interrupted for sixty-four years by war and genocide.\n\nIt began when he lived on the second floor and she on the fourth floor of an apartment building in Karlsruhe, Germany, where they attended the same school. The friendship was renewed last spring, thanks to two New Jersey teenagers, Jennifer Bernardes, of an immigrant family from Brazil, and Leonie Barrett, of an immigrant family from Jamaica. Leonie's sister is currently serving in the Persian Gulf.\n\nHarry, a Holocaust survivor, participates in New Jersey's \"Adopt a Survivor\" program that brings middle and high school students to the museum. Each student studies a survivor's personal history and commits to tell his or her story in 2045, the one hundredth anniversary of the liberation of the death camps.\n\nMuseum visitors are issued identity cards recounting the history of someone who was swept up in the Holocaust whirlwind. Jennifer and Leonie noticed that one card detailed the life of a girl from Karlsruhe. Harry recognized Hanne Hirsch as the girl from the fourth floor. He had not known her fate. But when he looked her up in the museum's registry of survivors, he found that her good fortune was to be sheltered by the good people of the French Huguenot village of Le Chambon, in the south of France near Lyon. Hanne Hirsch Liebmann lives in New York with her husband, Max, eighty-one.\n\nHanne's father and then her widowed mother ran a photography shop in Karlsruhe until Nazi anti-Jewish laws put them out of business in 1938. Hanne was sixteen in 1940 when she was deported to a camp in Vichy France. In the camp, she met Max Liebmann, then nineteen.\n\nHe got out of the camp and was sheltered illegally in Le Chambon until he could get into Switzerland. She received livesaving help from the villagers, whose long memories of the persecution of Huguenots fueled their resistance to German and Vichy crimes.\n\nJews still in the camp on August 1, 1942, were destined for Auschwitz. Hanne was legally removed to the village shortly before that, and in February 1943 she followed Max to Switzerland. They married, and in 1948 came to America.\n\nHarry, who says he was \"the last bar mitzvah boy in my synagogue,\" fled Germany with his family after the Munich Agreement of September 1938. In January 1945, he was in a U.S. Army truck en route to join the infantry unit that soon would seize the Remagen bridge over the Rhine. He was plucked from the truck to become an interpreter. Among the Germans he interviewed after the war was Heinrich Hoffmann, Hitler's photographer, who had been an apprentice in Munich under Hanne's uncle. Harry, Hanne, and Max had lunch together here this week while participating in the Holocaust Museum's tenth anniversary observances.\n\nIn an editorial saluting the museum as \"among the finest historical exhibits of any kind, on any subject, anywhere,\" the _Washington Post_ nevertheless recalled the \"skeptical questions\" asked when the museum was proposed. The questions concerned whether the Mall, which is the epicenter of America's civic life, is a suitable site for a museum dedicated to an event of European and Jewish history. The _Post_ said that the questions have been given \"no adequate philosophical or theoretical answers.\" Here are six answers.\n\nThe first answer has many facets: America is congenitally cheerful and hence relentlessly focused on the future, so it is susceptible to historical amnesia. And Americans, having uniquely broad and grave responsibilities in the world, must be trained to look life unblinkingly in the face. The Holocaust, the eruption of barbarism in modern Western civilization, is the black sun into which Americans, especially, must be taught to stare. The Holocaust Museum, a grim sermon in stone, is an experience of darkness amidst the Mall's glistening monuments to the success of American society. It is a mind-opening reminder of the furies beyond our shores. The Mall's welcoming geometry of openness suggests the symmetry and temperateness of America's social arrangements. The museum, a counterpoint in one of the world's most magnificent urban spaces, inflicts on visitors\u2014almost 19 million of them so far\u2014excruciating knowledge that is intensely relevant to this era of terrorism, knowledge of the hideous possibilities of human action.\n\nFive other answers to the question of why the museum is pertinent to American experience and governance, and hence is properly on the Mall, are: Harry, Hanne, Max, Jennifer, and Leonie, Americans all.\n\n[MAY 1, 2003]\n\n# **Cars as Mobile Sculpture**\n\n**D** ETROIT\u2014One car company is running ads in which its suave forty-four-year-old CEO underscores his love for the outdoors by saying, \"I won't even stay in a hotel if I can't open the windows.\"\n\nAnother car company, its tone set by its seventy-year-old vice chairman\u2014an ex-Marine aviator\u2014is putting up three billboards. One shows a 1957 Chevy's grille\u2014think of Teddy Roosevelt's grin in chrome\u2014and says: \"Proof your parents were actually cool once.\" Another shows the rear deck of a little red 1963 Corvette Sting Ray and says: \"They don't write songs about Volvos.\" The third shows the gritted-teeth grille of a 1970 Chevy Chevelle SS and says: \"Not everyone wants a car with a bud vase on the dash.\"\n\nGuess which company is doing best.\n\nBill Ford's problems at the company his great-grandfather founded are bigger than odd advertising. And there are many reasons why GM is soaring like the jet fighter Robert Lutz flies for fun. But institutions are the lengthening shadows of strong individuals, and Lutz is, in the elemental argot of this muscular city, a \"car guy.\"\n\nWhen GM lured Lutz back into the car business last summer, the _Detroit News_ headline (\"Lutz Rides In to Rev Up GM\") was of a size usually reserved for Pearl Harbors or two-game Tiger winning streaks. But are Americans still \"car people\" the way they were when Lutz was young, in the 1950s?\n\nThen they were automobile voluptuaries, Detroit was in its rococo period and its great stylist was GM's Harley Earl, \"the Cellini of chrome,\" of whom it was said that if he could have put chrome on his clothes, he would have. Cars had front bumpers that were protuberant, not to say nubile, and tail fins. Cars looked, a wit said, \"like chorus girls coming and fighter planes going.\" Indeed, Buick's LeSabre emulated the F-86 Sabre jet.\n\nLutz, tall and trim, knows that today's Americans generally have a less erotic relationship with cars. They look upon many cars, he says, \"as more or less an appliance.\" As mere transportation. Utilitarian. Boring. Furthermore, twenty years ago, a \"premium\" car meant one substantially more capable. Today, premium technologies (e.g., high-tech engines, overhead cams) are everywhere.\n\nBut, Lutz says happily, your car is still \"an extension of your psychomotor system.\" More than the other stuff we surround ourselves with\u2014do you know the brand of your refrigerator? will you replace it before it breaks down?\u2014your car \"continually makes an instant statement about you, even to complete strangers.\"\n\nSo, Lutz insists, design is still central to success in the automobile business. Art is _supposed_ to \"evoke emotional responses\" and cars are art\u2014\"mobile sculpture.\" He also believes that when everybody else is doing it, don't. Most cars today have rounded aerodynamic lines. But the new Cadillac CTS, with angular lines, is described in ads as \"edgy.\"\n\nAnd when Lutz was at Chrysler a few years ago, he pushed through the development of the popular PT Cruiser, an echo of a 1937 Ford. Why? Surely not nostalgia. Probably most of the (mostly young) people buying these cars do not know who was president in 1937. Go figure.\n\nLutz believes that \"aspirational aspects overwhelm the functional differences\" when car customers make their choices. When that happens, the \"left-analytical brain has been defeated again,\" the \"right brain\" has prevailed, and Lutz rejoices. But this does not mean people plunk down large sums merely for high-status brands. Chevrolet sells more vehicles costing more than $30,000 than do Mercedes, BMW, Lexus, and Audi combined, but this is partly due to the popularity of light trucks, a category that includes sport utility vehicles. Today, an \"extremely high-end demographic\"\u2014e.g., investment bankers and stockbrokers\u2014are buying GMC SUVs.\n\nSome Americans (let us avoid the term _liberals_ ) hate fun, such as cheeseburgers, talk radio, guns, Las Vegas, and cars that are larger than roller skates and that look more interesting than shoe boxes. They hated 1950s cars that looked\u2014as a sniffy critic said\u2014like jukeboxes on wheels. Such people love guilt, and want people to feel guilty about cars because cars have made possible suburbs, Wal-Mart, McDonald's, and emancipation from public transportation.\n\nGM's \"car guy\" knows that Americans generally keep their cars longer than they used to\u2014creeping utilitarianism\u2014and do not define automotive fun as they did in the gaudy 1950s. But he is betting that lots of them still are guilty of letting their right brains rip when purchasing a car.\n\n[APRIL 18, 2002]\n\n# **Hog Heaven: Happy One Hundredth, Harley**\n\n**M** ILWAUKEE\u2014In 1903, young men (the hyperkinetic president was just forty-five) were on the move. The Wright brothers\u2014Wilbur, thirty-six, and Orville, thirty-two\u2014left their bicycle shop in Dayton to take their twelve-second, 120-foot flight at Kitty Hawk. And William Harley, twenty-one, and Arthur Davidson, twenty, working in a ten-by-fifteen-foot shed here, built a motorcycle. On the eve of its centennial, the company born in that shed is spectacularly successful, and one of America's best-known brands. No American company has such devoted customers.\n\nThe Information Superhighway is littered with the wreckage of New Economy companies. But America's real highways are humming with the distinctive sound of an iconic Old Economy product\u2014Harley-Davidsons. Their sound (think potato-potato-potato) is so beloved by enthusiasts that the company tried to have it declared a trademark.\n\nLast weekend, the company began a fourteen-month-long one-hundredth-birthday bash. It has much to celebrate, including increases in production of more than 10 percent annually for sixteen years. And 16 consecutive years of record earnings. And average annual earnings growth of 37 percent. And a share price up 15,000 percent since 1986, about nine times as much as General Electric's, which is no slouch.\n\nIn the First World War, Harley-Davidson helped mechanize the Army\u2014the first American to enter Germany after the armistice rode in on a Harley\u2014and AMF bought the company in 1969. But by 1981, Harley-Davidson was reeling, partly because of Japanese imports. So thirteen of the company's executives\u2014including today's CEO, Jeffrey Bleustein, sixty-two, a former Yale engineering professor\u2014bought it for a highly leveraged $82 million.\n\nIn 1983, they got a five-year tariff protection, but recovery was so rapid they asked the government to end the protection a year early. The tariff\u2014Bleustein doubts that anyone paid it\u2014was a warning shot against \"dumping\" (selling at a price below the cost of manufacture) by the Japanese, who at the time had a two-year inventory in warehouses and were still producing full tilt. But the tariff exempted European manufacturers, who were not dumping, and the biggest Japanese maker, Honda, because its bikes were assembled in Ohio.\n\nToday, the tariff seems as quaint as other relics of the 1980s, such as the U.S. feeling of inferiority regarding Japanese economic prowess, and the country-music lament \"everything I buy these days has a foreign name.\" Twenty years after the $82 million buyout, Harley's sales were $3.3 billion and earnings $435 million, thanks to passionate motorcyclists of the sort who every August descend, three-hundred thousand strong, on Sturgis, South Dakota.\n\nHollywood\u2014and a few motorcyclists\u2014have made motorcyclists seem like bad boys, like Marlon Brando in _The Wild One_ (1954) and Jack Nicholson and Peter Fonda in _Easy Rider_ (1969). But most motorcyclists are middle-aged and middle class. The average Harley customer is forty-six and has a household income of $78,000. There are many riders in their seventies and eighties. Says Bleustein, himself a rider, \"Even after they stop playing golf, they can still ride.\"\n\nThe average Harley customer pays $15,000 for his machine, but some Harleys, such as those with music provided by a six-CD changer, can list for $22,000. Although output has been increased at the three factories\u2014here and in York, Pennsylvania, and Kansas City, Missouri\u2014demand so much exceeds supply (261,000 bikes this year) for some models that \"scalpers\" sell them for up to $4,000 over list. Some people sell their places on waiting lists. Others wait on the list, then sell their bikes at a markup before they even leave the dealer's parking lot.\n\nHonda sells more bikes in America than Harley does, but Harley dominates the high-profit market for heavy bikes (aka hogs), even in Japan, and has 21 percent of the Asia-Pacific market for all motorcycles. In America, 650,000 people pay $40 annual dues to be members of the Harley-owners group. They doubtless agree with the company that it is selling a \"lifestyle.\" This year bikers will spend more than $1 billion on Harley gear, from black motorcycle jackets for toddlers to Harley-Davidson fountain pens for bikers with literary aspirations.\n\nBleustein's office, in the 1920 brick building where all the bikes once were made, features a glass tabletop supported by two Harley engines. He wears a gray suit, crisp white shirt, French cuffs\u2014and a colorful Harley-Davidson necktie that is the sartorial equivalent of chrome. Many of the office workers dress as though they just got off their Harleys, which they did. Worldwide, says Bleustein, motorcycling means \"freedom, adventure, individual expression.\" As does America.\n\nHarley-Davidson's ten-city centennial tour begins this week in Atlanta and will rumble to Mexico City, Sydney, Tokyo, Barcelona, and on to Munich next July. It will culminate next August when a quarter of a million satisfied customers are expected to descend on Milwaukee to compare chrome and enjoy the music of their machines. Think: potato-potato-potato times 250,000.\n\n[JULY 22, 2002]\n\n# **Restoration at 346 Madison**\n\n_This man now\u2014surely he came from that heavenly world, that divine position at the center of things where choice is unlimited._\n\n\u2014MARY MCCARTHY, \"The Man in the Brooks Brothers Shirt\" (1941)\n\n**S** ome business stories are social parables. One such is the long, stately rise, then the swift, undignified descent, and now the resurrection of Brooks Brothers, the men's clothier that long ago became one of America's iconic brands.\n\nIt was founded in 1818 near the southern tip of a mostly rural Manhattan. The day the store opened\u2014the store that was to define American male gentility\u2014the city council was fretting about swine in the streets.\n\nAs Manhattanites moved north, so did the store, several times. By 1915, it had moved to 346 Madison, at the corner of Forty-fourth, a store with dark wood and soft lighting from Tiffany chandeliers. Not until 1928 did Brooks Brothers open a second store, on Newbury Street in Boston. But long before that, 346 Madison had become for many men the quiet definer of sartorial good taste.\n\nGenerals Grant, Sherman, Sheridan, and Hooker bought Civil War uniforms from Brooks Brothers. Lincoln wore a Brooks Brothers overcoat to his second inauguration, and to Ford's Theatre. He was buried in a Brooks Brothers suit. J. P. Morgan, when he was a boy, was taken to Brooks Brothers for his first suit, and sixty years later was still buying suits there from the same salesman.\n\nBut although Brooks Brothers catered to the carriage trade, by pioneering high-quality ready-made garments it helped democratize dress. Teddy Roosevelt and Woodrow Wilson detested each other, but each was inaugurated in a Brooks Brothers suit. And after World War I, rising men in a nation brimming with confidence made Brooks Brothers a convenient literary symbol. Fitzgerald in _This Side of Paradise,_ Hemingway in _The Sun Also Rises,_ John O'Hara, Somerset Maugham, and J. P. Marquand all used Brooks Brothers to suggest character traits\u2014not always flatteringly. Clark Gable and Fred Astaire were Brooks Brothers customers.\n\nEmblematic of Brooks Brothers' power to define classicism was\u2014is\u2014what the company's official history rightly calls \"the single most imitated item in American clothing history.\" It is the shirt with a button-down \"polo collar,\" so named because a grandson of the founder, visiting England, liked the way polo players buttoned down their collars to keep them from flapping during play.\n\nThe 1950s are disparaged by advanced thinkers as \"buttoned down,\" meaning too reticent and emotionally reserved. But the button-down shirt, a striped tie, and a \"sack suit\"\u2014unpadded shoulders, not very wide lapels\u2014became the 1950s \"Ivy League look.\" Sloan Wilson's _The Man in the Gray Flannel Suit_ bought it at Brooks Brothers. And in _The Catcher in the Rye,_ Holden Caulfield's adolescent disapproval of almost everything extended to \"guys fitting your pants all the time at Brooks.\"\n\nNever mind. In 1960, the _New York Times_ wrote of John Kennedy: \"Though visibly exhausted from a long day in the Senate, he walked to the floor, a model of Brooks Brothers perfection.\"\n\nIn 1956, the company's president said, \"Whenever we contemplate changing anything around here, a perceptible shudder goes through the store.\" In the turbulent 1960s it took seven years for the lapels of Brooks Brothers suits to grow\u2014in three increments\u2014from three inches to three and a half. Then came the male peacockery of the 1980s. It was an era in which, a fashion historian writes, clothes\u2014broad-shouldered Italian suits, suspenders, ties the size and color of Third World nations' flags\u2014became \"badges of communication,\" advertisements for the wearers.\n\nAnd Brooks Brothers\u2014sold, then sold again to a British department-store chain\u2014decided to \"get with it.\" Big mistake.\n\nBrooks Brothers had been, _Newsweek_ wrote, \"the last holdout against the 1980s 'if you've got it, flaunt it' mentality.\" As \"brand repositioning,\" it began to compete with Gap, Banana Republic, and the like. \"Casualization\" was the ugly word for the ugly trend in men's fashion. Brooks Brothers' quality declined, the company's old customers were scandalized\u2014lavender dress shirts!\u2014and the new customers were fickle. Brooks Brothers even opened a Fifth Avenue store that is all glass and stainless steel. Good grief.\n\nThen, two years ago, a white knight rode to the rescue. A knight from\u2014thank you, globalization\u2014Italy. The new owner, Claudio Del Vecchio, says nobody can be J.Crew better than J.Crew, but only Brooks Brothers can be a fixed point in a world of flux.\n\nPolitical parties decline when they alienate their core voters. The Episcopal Church, once the Brooks Brothers of American Protestantism, has lost a third of its members while courting new ones with trendy theological fashions. In the nick of time, Brooks Brothers has remembered that millions of men still think \"buttoned down\" is a phrase of praise.\n\n[SEPTEMBER 1, 2003]\n\n# **Starbucks, Nail Salons, and the Aesthetic Imperative**\n\n**C** reative thinkers do not merely answer questions that interest others, they answer questions that others have not realized are interesting, or even are questions. For example:\n\nStarbucks' coffee is not that much better than everyone else's coffee, so what is Starbucks really selling?\n\nWhat does it say about today's America that travelers changing concourses in the United terminal at Chicago's O'Hare airport pass beneath a 744-foot neon-light sculpture, the colors of which change in sync to music?\n\nWhy have the number of nail salons doubled, the number of manicurists tripled, and the number of cosmetic medical procedures almost quintupled in a decade? Why do 13 percent of middle-aged _men_ spend more than $1 billion on hair coloring, up 34 percent in five years?\n\nIf computers are just tools, why bother making them as pretty as the Sony Vaio and Apple iMac?\n\nHow much of the booming membership in gyms is about something other than\u2014more pleasurable than\u2014health maintenance?\n\nVirginia Postrel, an economics columnist for the _New York Times_ who writes perceptively about everything on which her penetrating gaze alights, answers these questions, and others you may not have asked yourself, in her new book, _The Substance of Style: How the Rise of Aesthetic Value Is Remaking Commerce, Culture, and Consciousness._ It is an appreciation of what she calls the \"aesthetic imperative\" in this expressive age.\n\nBiologically we are, she says, visual, tactile beings responsive to our sensory surroundings. And we now are\u2014thanks to such factors as travel, education, immigration, and media\u2014producing a society of aesthetic plenitude and pluralism.\n\nPeople are eager to pay Starbucks for more than mere coffee\u2014for a sensory environment that pleases more than just their palates. Demand often creates supply, but supply can create demand: Travelers do not demand O'Hare's neon light sculpture, but the supply of such aesthetic amenities raises expectations for a more pleasurable environment. And from gyms and nail salons to tattoo parlors and the emporia where people get their bodies pierced in so many interesting places, Americans are consuming design and designing themselves.\n\nTime was, Henry Ford told customers they could have cars in the color they wanted, as long as they wanted black. Time was, Walter Gropius, the minimalist architect, when asked what he would say if some students did not like the way he had arranged the furniture in a Harvard dorm he designed, replied, \"Then they are neurotic.\"\n\nBut the breakdown of cultural homogeneity in the 1960s has been followed by what Postrel says are the twin propellants of today's aesthetic abundance\u2014rising incomes and falling prices. Household income has increased about 30 percent in less than thirty years, and families have shrunk, further expanding disposable income.\n\nEconomic data does not measure the increases that aesthetics add to quality of life. In national income statistics, a $20 steak dinner in an aesthetically pleasing restaurant is indistinguishable from a $20 steak dinner in a banal environment. Which means we are exaggerating inflation and underestimating the economy's real production of value.\n\n\"Aesthetics,\" says Postrel, \"shows rather than tells, delights rather than instructs. The effects are immediate, perceptual and emotional. They are not cognitive, although we may analyze them after the fact.\" Aesthetics, Postrel stresses, is not irrational or antirational, it is pre-rational or nonrational.\n\nThat does not mean it should be distrusted, as rhetoric has come to be, as a manipulative force manufacturing synthetic desires. Aspiration, Postrel believes, is an aspect of identity, including aesthetic identity\u2014\"I like that\" means \"I am like that.\" Her cheerful analysis of all this puts her athwart a tradition of disapproving intellectuals.\n\nHalf a century ago, Adlai Stevenson, Democratic presidential nominee and darling of the intelligentsia, asked: \"With the supermarket as our temple and the singing commercial as our litany, are we likely to fire the world with an irresistible vision of America's exalted purposes and inspiring way of life?\" His question radiated what was then\u2014and still is; everything changes except \"progressives\"\u2014the intellectuals' conventional disdain of America's \"consumer society.\"\n\nToday, however, thoughtful people have more appreciation of the complex prerequisites\u2014social, political, and intellectual\u2014of a society that produces the abundance, and honors the emancipation of choice and desire, that results in supermarkets, advertising, and other things that are woven inextricably into the fabric of a free society. Those mundane things actually are related to what exalts America and makes it inspiring.\n\nUnbounded, imaginative desiring can be a problem for democratic governance. However, it certainly is both a cause and a consequence of a democratic culture.\n\n[OCTOBER 26, 2003]\n\n# **Manners vs. Social Autism**\n\n**L** et's be good cosmopolitans and offer sociological explanations rather than moral judgments about students, the _Washington Post_ reports, having sex during the day in high schools. Sociology discerns connections, and there may be one between the fact that teenagers are relaxing from academic rigors by enjoying sex in the school auditorium, and the fact that Americans in public soon will be able to watch pornography, and prime-time television programs such as _Desperate Housewives_ \u2014and, for the high-minded, C-SPAN\u2014on their cell phones and video iPods.\n\nThe connection is this: Many people have no notion of propriety when in the presence of other people, because they are not actually in the presence of other people, even when they are in public.\n\nWith everyone chatting on cell phones when not floating in iPod-land, \"this is an age of social autism, in which people just can't see the value of imagining their impact on others.\" We are entertaining ourselves into inanition. (There are websites for people with Internet addiction. Think about that.) And multiplying technologies of portable entertainments will enable \"limitless self-absorption,\" which will make people solipsistic, inconsiderate, and antisocial. Hence manners are becoming unmannerly in this \"age of lazy moral relativism combined with aggressive social insolence.\"\n\nSo says Lynne Truss in her latest trumpet blast of a book, _Talk to the Hand: The Utter Bloody Rudeness of the World Today, or Six Good Reasons to Stay Home and Bolt the Door._ Her previous wail of despair was _Eats, Shoots & Leaves: The Zero Tolerance Approach to Punctuation,_ which established her as\u2014depending on your sensibility\u2014a comma and apostrophe fascist (the liberal sensibility) or a plucky constable combating anarchy (the conservative sensibility).\n\nGood punctuation, she says, is analogous to good manners because it treats readers with respect. \"All the important rules,\" she writes, \"surely boil down to one: _remember you are with other people; show some consideration._ \" Manners, which have been called \"quotidian ethics,\" arise from real or\u2014this, too, is important in lubricating social frictions\u2014feigned empathy.\n\n\"People,\" says Truss, \"are happier when they have some idea of where they stand and what the rules are.\" But today's entitlement mentality, which is both a cause and a consequence of the welfare state, manifests itself in the attitude that it is all right to do whatever one has a right to do. Which is why acrimony has enveloped a coffee shop on Chicago's affluent North Side, where the proprietor posted a notice that children must \"behave and use their indoor voices.\" The proprietor, battling what he calls an \"epidemic\" of antisocial behavior, told the _New York Times_ that parents protesting his notice \"have a very strong sense of entitlement.\"\n\nA thoroughly modern parent, believing that children must be protected from feelings injurious to self-esteem, says: \"Johnny, the fact that you did something bad does not mean you are bad for doing it.\" We have, Truss thinks, \"created people who will not stand to be corrected in any way.\" Furthermore, it is a brave, or foolhardy, man who shows traditional manners toward women. In today's world of \"hair-trigger sensitivity,\" to open a door for a woman is to play what Truss calls Gallantry Russian Roulette: You risk a high-decibel lecture on gender politics.\n\nOne writer on manners has argued that a nation's greatness is measured not only by obedience of laws but also by \"obedience to the unenforceable.\" But enforcement of manners can be necessary. The well-named David Stern, commissioner of the NBA, recently decreed a dress code for players. It is politeness to the league's customers who, weary of seeing players dressed in \"edgy\" hip-hop \"street\" or \"gangsta\" styles, want to be able to distinguish the Bucks and Knicks from the Bloods and Crips. Stern also understands that players who wear \"in your face\" clothes of a kind, and in a manner, that evokes Sing Sing more than Brooks Brothers might be more inclined to fight on the floor and to allow fights to migrate to the stands, as happened last year.\n\nBecause manners are means of extending respect, especially to strangers, this question arises: Do manners and virtue go together? Truss thinks so, in spite of the possibility of \"blood-stained dictators who had exquisite table manners and never used their mobile phones in a crowded train compartment to order mass executions.\"\n\nActually, manners are the practice of a virtue. The virtue is called _civility,_ a word related\u2014as a foundation is related to a house\u2014to the word _civilization._\n\n[NOVEMBER 20, 2005]\n\n# **A Punctuation Vigilante**\n\n**T** he actress Margaret Anglin left this note in the dressing room of another actress: \"Margaret Anglin says Mrs. Fiske is the best actress in America.\" Mrs. Fiske added two commas and returned the note: \"Margaret Anglin, says Mrs. Fiske, is the best actress in America.\"\n\nLittle things mean a lot. That is the thesis of a wise and witty wee book, _Eats, Shoots & Leaves,_ just published by Lynne Truss, a British writer and broadcaster. She knows that proper punctuation, \"the basting that holds the fabric of language in shape,\" is \"both the sign and the cause of clear thinking.\"\n\nThe book's title comes from a joke: A panda enters a caf\u00e9, orders a sandwich, eats, draws a pistol, fires a few shots, then heads for the door. Asked by a waiter to explain his behavior, he hands the waiter a badly punctuated wildlife manual and says: \"I'm a panda. Look it up.\" The waiter reads the relevant entry: \"Panda: large black-and-white bear-like mammal. Eats, shoots and leaves.\"\n\nBehold the magical comma. It can turn an unjust aspersion against an entire species (\"No dogs please\") into a reasonable request (\"No dogs, please\"), or it can turn a lilting lyric into a banal inquiry (\"What is this thing called, love?\"). The Christmas carol actually is \"God rest ye merry, gentlemen,\" not \"God rest ye, merry gentlemen.\"\n\nHuge doctrinal consequences flow from the placing of a comma in what Jesus, when on the cross, said to the thief: \"Verily, I say unto thee, This day thou shalt be with me in Paradise\" or \"Verily, I say unto thee this day, Thou shalt be with me in Paradise.\" The former leaves little room for purgatory.\n\nCombined with a colon, a comma can fuel sexual warfare: \"A woman without her man is nothing\" becomes \"A woman: without her, man is nothing.\" But a colon in place of a comma can subtly emit a certain bark.\n\n\"President Bush said, 'Get Bob Woodward.'\"\n\n\"President Bush said: 'Get Bob Woodward.'\"\n\nBut beware the derangement known as commaphilia, which results in the promiscuous cluttering of sentences with superfluous signals. A reader once asked James Thurber why he had put a comma after the word _dinner_ in this sentence: \"After dinner, the men went into the living room.\" Thurber, a comma minimalist, blamed the _New Yorker_ 's commaphilic editor, Harold Ross: \"This particular comma was Ross' way of giving the men time to push back their chairs and stand up.\"\n\nTruss, a punctuation vigilante, says punctuation marks are traffic signals telling readers to slow down, pause, notice something, take a detour, stop. Punctuation, she says, \"directs you how to read, in the same way musical notation directs a musician how to play\" with attention to the composer's intentions regarding rhythm, pitch, tone, and flow.\n\nThe almost-always-ghastly exclamation point has been rightly compared to canned laughter. F. Scott Fitzgerald said it was like laughing at your own joke. But not always. Victor Hugo, wondering how his _Les Mis\u00e9rables_ was selling, sent this telegram to his publisher: \"?\" The publisher wired back: \"!\"\n\nThe dash can be, among other things, droll, as Byron understood:\n\n_He learned the arts of riding, fencing, gunnery,_\n\n_And how to scale a fortress\u2014or a nunnery._\n\nOr:\n\n_A little still she strove, and much repented,_\n\n_And whispering \"I will ne'er consent\"\u2014_\n\n_consented._\n\nThe humble hyphen performs heroic services, making possible compounds that would otherwise be unsightly (\"de-ice\" rather than \"deice\" \"shell-like\" rather than \"shelllike\"). And a hyphen can rescue meaning. As Truss says, \"A cross-section of the public is quite different from a cross section of the public.\" If you are a pickled-herring merchant, you will not want to be called a pickled herring merchant. The difference between extra-marital sex and extra marital sex is not to be sneezed at.\n\nThe connection between the words _punctilious,_ which means \"attentive to formality or etiquette,\" and _punctuation_ is instructive. Careful punctuation expresses a writer's solicitude for the reader. Of course punctuation, like most other forms of good manners, may yet entirely disappear, another victim of progress, this time in the form of e-mail, cell-phone text messages, and the like.\n\nNeither the elegant semicolon nor the dashing dash is of use to people whose preferred literary style is \"CU B48?\" and whose idea of Edwardian prolixity is: \"Saw Jim\u2014he looks gr8\u2014have you seen him\u2014what time is the thing 2morrow.\"\n\nOh, for the era when a journalist telephoned from Moscow to London to add a semicolon to his story!\n\n[MAY 20, 2004]\n\n# **America's Literature of Regret**\n\n_His name was George F. Babbitt. He was forty-six years old now, in April, 1920, and he made nothing in particular, neither butter nor shoes nor poetry, but he was nimble in the calling of selling houses for more than people could afford to pay._\n\n\u2014SINCLAIR LEWIS, _Babbitt_ (1922)\n\n**B** ut today his name is Warren Schmidt. He is sixty-six now as he stolidly watches the clock on his otherwise bare office wall tick the final seconds of his career as an actuary at Woodmen of the World Insurance in Omaha.\n\n_Babbitt_ begins: \"The towers of Zenith aspired above the morning mist; austere towers of steel and cement and limestone, sturdy as cliffs and delicate as silver rods. They were neither citadels nor churches, but frankly and beautifully office-buildings.\" Beautiful, Lewis intimates, if only because of the frankness of their banality.\n\n_About Schmidt,_ the new Jack Nicholson movie, begins with the camera lingering on a flat slab of a spire in Omaha, the Woodmen building, which is replicated in the cake at Schmidt's retirement party that evening. If \"party\" is applicable to so flat an affair. Flat as champagne that has lost its fizz. Flat as the Midwest landscape through which Schmidt, suddenly widowed, rolls, a depressed Jack Kerouac in a gigantic Winnebago, on the road to Denver to try to forestall yet another disappointment, the marriage of his daughter to a waterbed salesman Schmidt despises.\n\nIn the novel on which _About Schmidt_ is loosely based, Schmidt retires from a Manhattan law firm to Long Island affluence. So why (other than the fact that director Alexander Payne is from Omaha) turn Schmidt into a stereotypical Midwesterner whose taciturnity is presumably symptomatic not of still waters running deep, but only of a low emotional metabolism?\n\nBecause it is still very modern to suppose that people like Schmidt who do not \"share their feelings\" have none. And because it is very traditional to disparage life in the Midwest's small towns, such as Sherwood Anderson's _Winesburg, Ohio_ (1919).\n\nIn 1920, Sinclair Lewis, from Sauk Centre, Minnesota, who in 1930 became the first American winner of the Nobel Prize for Literature, published _Main Street,_ an unaffectionate depiction of fictional Gopher Prairie, Minnesota, where individualism is suffocated by what would later be called \"conformity.\" In England, E. M. Forster said Lewis had lodged \"a piece of a continent\"\u2014the Midwest\u2014\"in our imagination.\"\n\nAlso in 1920, another Minnesotan, F. Scott Fitzgerald, published _This Side of Paradise,_ in which Amory Blaine decides he has \"grown up to find all Gods dead, all wars fought, all faiths in man shaken.\" But whereas Fitzgerald came East to be exhilarated by Princeton and the 1920s, and kept moving east, to Europe, Lewis was unhappy at Yale and looked back in anger at the Midwest. His curdled spirit considered that region unforgivably middling\u2014no longer a heroic frontier, never likely to become more than (as another young Midwesterner, Ernest Hemingway, called his native Oak Park, Illinois) a place of \"broad lawns and narrow minds.\"\n\nSome critics insist that the portraits of Winesburg, Gopher Prairie, Zenith and Schmidt's Omaha are \"really\" sympathetic. Perhaps the recurring cows in _About Schmidt_ (in paintings on a restaurant wall, in a cattle truck) are not supposed to suggest that the people, too, are bovine. See the movie and decide for yourself if it is yet another exercise in condescension. As Evan Connell's two nuanced novels about Mr. and Mrs. Bridge of Kansas City (and the Paul Newman and Joanne Woodward movie made from them) are not.\n\n\"I don't say he's a great man,\" says Linda Loman of her husband, Willy, in Arthur Miller's _Death of a Salesman_ (1949), another depiction of disappointed American striving. But, she says, \"attention must be finally paid to such a person.\" Surely ample attention is paid. The likes of Loman, Babbitt, and Schmidt inhabit a large American literature of regret. Which may be what Schmidt is feeling in the movie's final frame, when he is reduced to tears by receiving in the mail the slender evidence of his single success in connecting with another\u2014a drawing from a six-year-old Tanzanian boy to whom Schmidt has sent hilariously inapposite, hence unconnecting, letters.\n\nBabbitt says, \"I've never done a single thing I wanted to in my whole life! I don't know's I've accomplished anything except just get along.\" However, a haunting sense of regret about time wasted is a timeless theme of literature. Timeless, and placeless. It is the human condition, not a Midwestern affliction.\n\n[JANUARY 16, 2003]\n\n# **Chief Illiniwek and the Indignation Industry**\n\n**T** he University of Illinois must soon decide whether, and if so how, to fight an exceedingly silly edict from the NCAA. That organization's primary function is to require college athletics to be no more crassly exploitative and commercial than is absolutely necessary. But now the NCAA is going to police cultural sensitivity, as it understands that. Hence the decision to declare Chief Illiniwek \"hostile and abusive\" to Native Americans.\n\nCensorship\u2014e.g., campus speech codes\u2014often are academic liberalism's preferred instrument of social improvement, and now the NCAA's censors say: The Chief must go, as must the university's logo of a Native American in feathered headdress. Otherwise the NCAA will not allow the university to host any postseason tournaments or events.\n\nThis story of progress, as progressives understand that, began during halftime of a football game in 1926, when an undergraduate studying Indian culture performed a dance dressed as a chief. Since then, a student has always served as Chief Illiniwek, who has become the symbol of the university that serves a state named after the Illini confederation of about a half-dozen tribes that were virtually annihilated in the 1760s by rival tribes.\n\nIn 1930, the student then portraying Chief Illiniwek traveled to South Dakota to receive authentic raiment from the Oglala Sioux. In 1967 and 1982, representatives of the Sioux, who had not yet discovered that they were supposed to feel abused, came to the Champaign-Urbana campus to augment the outfits Chief Illiniwek wears at football and basketball games.\n\nBut grievance groups have multiplied, seeking reparations for historic wrongs, and regulations to assuage current injuries inflicted by \"insensitivity.\" One of America's booming businesses is the indignation industry that manufactures the synthetic outrage needed to fuel identity politics.\n\nThe NCAA is allowing Florida State University and the University of Utah to continue calling their teams Seminoles and Utes, respectively, because those two tribes approve of the tradition. The Saginaw Chippewa tribe starchily denounces any \"outside entity\"\u2014that would be you, NCAA\u2014that would disrupt the tribe's \"rich relationship\" with Central Michigan University and its teams, the Chippewas. The University of North Carolina at Pembroke can continue calling its teams the Braves. Bravery is a virtue, so perhaps the 21 percent of the school's students who are Native Americans consider the name a compliment.\n\nThe University of North Dakota Fighting Sioux may have to find another nickname because the various Sioux tribes cannot agree about whether they are insulted. But the only remnant of the Illini confederation, the Peoria tribe, is now in Oklahoma. Under its chief, John Froman, the tribe is too busy running a casino and golf course to care about Chief Illiniwek. The NCAA ethicists probably reason that the Chief must go because no portion of the Illini confederation remains to defend him.\n\nOr to be offended by him, but never mind that, or this: In 1995, the Office of Civil Rights in President Clinton's Education Department, a nest of sensitivity mongers, rejected the claim that the Chief and the name Fighting Illini created for anyone a \"hostile environment\" on campus.\n\nIn 2002, _Sports Illustrated_ published a poll of 351 Native Americans, 217 living on reservations, 134 living off. Eighty-one percent said high school and college teams should _not_ stop using Indian nicknames.\n\nBut in any case, why should anyone's disapproval of a nickname doom it? When, in the multiplication of entitlements, did we produce an entitlement for everyone to go through life without being annoyed by anything, even a team's nickname? If some Irish or Scots were to take offense at Notre Dame's Fighting Irish or the Fighting Scots of Monmouth College, what rule of morality would require the rest of us to care? Civilization depends on, and civility often requires, the willingness to say, \"What you are doing is none of my business\" and \"What I am doing is none of your business.\"\n\nBut this is an age when being an offended busybody is considered evidence of advanced thinking and an exquisite sensibility. So, People for the Ethical Treatment of Animals has demanded that the University of South Carolina's teams not be called Gamecocks because cock fighting is cruel. It also is illegal in South Carolina.\n\nIn 1972, the University of Massachusetts at Amherst replaced the nickname Redmen with Minutemen. White men carrying guns? If some advanced thinkers are made miserable by this, will the NCAA's censors offer relief? Scottsdale Community College in Arizona was wise to adopt the nickname Fighting Artichokes. There is no grievance group representing the lacerated feelings of artichokes. Yet.\n\n[JANUARY 5, 2006]\n\n# **Christmas at Our Throats**\n\nORANGE CITY, FLORIDA _\u2014A mob of shoppers rushing for a sale on DVD players trampled the first woman in line and knocked her unconscious as they scrambled for the shelves at a Wal-Mart Supercenter._\n\n\u2014ASSOCIATED PRESS\n\n**I** n sorting out the sociological significance of the fact that rival shoppers, according to the trampled woman's sister, \"walked over her like a herd of elephants,\" note that elephants do not behave that way to others of their species, even when they are stampeded by a 6 a.m. siren announcing, on the famously anarchic day after Thanksgiving, open season on a finite supply of $29 DVD players. But, then, elephants do not have Christmas celebrations.\n\nConservatives, in their simplistic way, will blame the Florida trampling on facets of human nature to which the Christmas story pertains\u2014mankind's fallen condition, meaning original sin. Liberals, being less judgmental and more alert to the social causes of things, will blame Wal-Mart. They already blame it for many flaws in creation, from low wages in Asia to America's \"loss of community,\" by which liberals mean the migration of shoppers from large-hearted Main Street merchants to the superior variety and lower prices at the Wal-Mart on the edge of town.\n\nBut at the risk of sounding like Ebenezer Scrooge, who was _not_ the character in English literature who said, \"We shall soon be having Christmas at our throats,\" consider a possibility. Perhaps, as liberals like to say, the \"root cause\" of modern Christmas discontents is the ruinous success of Puritanism\u2014ruinous, that is, to Puritanism.\n\nThat Christmas-at-our-throats fellow is a character in a novel by P. G. Wodehouse, who was as sweet-tempered as Scrooge was not. If the Christmas season, as it has become, could cause the preternaturally amiable Wodehouse to pen such a dark thought, how did it come to this?\n\nThat God works in mysterious ways is not news, but it is particularly puzzling that the birth of Jesus occurred when Romans, who then set the tone of the times, were celebrating Saturnalia\u2014think of a Wal-Mart at 6 a.m., plus wine, women wearing less than those little Wal-Mart vests, and songs that are not carols. Songs that would not have been amusing to Oliver Cromwell, whose piety caused him to ban the celebration of Christmas.\n\nHe did the right thing for the wrong reason. A Puritan scold and a killjoy, he thought Christmas had become too much fun, which is _not_ our problem today, unless getting trampled at a mall is your idea of merriment.\n\nToday's problem, in addition to the toll taken on the body by seasonal wassailing and gorging, is shopping that includes stocking up on \"retaliation presents.\" They are used to counter unexpected gift giving by persons not on your list, which by now includes family, friends, the stockbroker who got you out of Enron in time, and the person who cleans your gutters.\n\nThe first Americans included a number of Cromwell's fellow travelers, who, like him, saw the long arm of the papacy behind Christmas festiveness. It was, they thought, a short slide down a slippery slope from liturgical \"smells and bells\" to jingle bells and mulled cider. But in a delicious dialectic, the modern hedonistic Christmas emerged from the cultural contradictions of Puritanism.\n\nPuritanism inculcated Scrooge-like asceticism, deferral of gratification, green-eyeshade parsimony, and nose-to-the-grindstone industriousness. But those led to accumulation, investment of surplus capital, and, in time, prodigies of production and a subversive\u2014to Puritanism\u2014cornucopia of material delights.\n\nSoon there were department stores, those cathedrals of consumption. Against their plate-glass windows\u2014prerequisites of \"window shopping\" precursors of the holiday shopping catalog\u2014were pressed the noses of the Puritans' descendants.\n\nThose noses no longer detected a sulfurous stench of damnation wafting from the stores' perfume counters. Those counters, you may have noticed, are strategically placed on the stores' first floors, to start the shoppers' pleasure synapses firing.\n\nThe Wal-Mart stampede style of Christmas was a long time coming. It was, for example, not until 1885 that federal workers were even given Christmas Day off. Which, come to think about it, is odd. Here in modern Washington, Christmas Day is one of the minority of days that are _not_ like Christmas elsewhere\u2014not devoted to the lavish disbursal of gifts.\n\nAt least a portion of the government's largesse can be considered a gift because part of the cost is debt that will be paid by others. By future generations. They are not consulted, but surely they will pay cheerfully, in the Christmas spirit.\n\n[DECEMBER 4, 2003]\n**Chapter Five**\n\n**LEARNING**\n\n# **National Amnesia and Planting Cut Flowers**\n\n**T** he 9\/11 summons to seriousness ended the nation's 1990s holiday from history, and even the National Endowment for the Humanities has enlisted in the war. Emphasizing that historical illiteracy threatens homeland security\u2014people cannot defend what they cannot define\u2014the NEH's chairman, Bruce Cole, is repairing the ravages of the 1990s, when his two immediate predecessors made the NEH frivolous.\n\nThe first was Sheldon Hackney, a former college president whose big idea was a \"national conversation\" about diversity, his peculiar theory being that there is insufficient talk about that subject. His NEH distributed instructional kits telling Americans how to converse.\n\nThe second NEH chairman of the Clinton administration was William Ferris, a Mississippi folklorist guided by today's hedonistic calculus\u2014the greatest self-esteem for the greatest number. His NEH worked to \"celebrate\"\u2014preferred verb of the warriors against intellectual elitism\u2014the quotidian. He said: \"Today the lives of ordinary American people have assumed a place beside volumes of European classics in the humanities.\" So _Middlemarch_ and the \"life\" of your mailman are equally humanities classics.\n\nBack then, the chairman of the National Endowment for the Arts was another folklorist, Bill Ivey, whose definition of art was latitudinarian. It included \"the expressive behavior of ordinary people,\" such as \"piecrust designs\" and \"dinner-table arrangements.\" He talked like this: \"Do we want to possess a confidence that the rich cultural matrix of our nation is appropriately auditioned for the world?\"\n\nThat was then. This is now:\n\nCole, author of fourteen books, many on the Renaissance, was for twenty-nine years a professor of fine arts, art history, and comparative literature at Indiana University. One of his missions is to reverse America's deepening amnesia, and especially the historical illiteracy of college students.\n\nFresh evidence of the latter came last week from the National Association of Scholars, whose members defend academic standards against the depredations of those levelers who, rigorous only in applying the hedonistic calculus (see above), are draining rigor from curricula. A survey sponsored by the NAS, using questions on general cultural knowledge originally asked by the Gallup Organization in 1955, establishes that today's college seniors score little\u2014if any\u2014higher than 1955 _high school_ graduates.\n\nIn his office in the Old Post Office on Pennsylvania Avenue, Cole says of the war on terrorism, \"What we fight for is part of what we do around here.\" What the NEH aims to help do is \"make good citizens.\" And \"scholarship should be the basis of all we do.\" Not all scholars are professors. David McCullough, the historian and biographer, is not an academic. But, then, neither were Thucydides and Gibbon.\n\n\"A wise historian,\" says McCullough, \"has said that to try to plan for the future without a sense of the past is like trying to plant cut flowers.\" Hence the three components of the NEH's \"We the People\" initiative.\n\nOne is the funding of scholarship on significant events and themes that enhance understanding of America's animating principles. Another is an essay contest for high school juniors, concerning America's defining tenets. The contest winner\u2014Cole dryly notes that there will be a winner, whatever the cost to anyone's self-esteem\u2014will be recognized at the third component of the \"We the People\" initiative, the annual \"Heroes of History\" lecture.\n\nThe very subject of this new lecture series goes against the grain of today's academic culture, which rejects the idea of heroes\u2014those rare event-making individuals who are better and more important than most people. To banish elites from the human story, many academic historians tell that story as one of vast impersonal forces, in which individuals are in the iron grip of economic, racial, or gender roles. Small wonder students turn away from history taught without the drama of autonomous individuals moved by reason, conviction, and rhetoric that appeals to the better angels of their natures.\n\nCole soon will have a worthy colleague at the National Endowment for the Arts. The fact that its new chairman will be Dana Gioia, the distinguished poet, critic, and translator, is additional evidence that cultural revival is a priority of today's president, who has so many despisers among the lettered.\n\nGeorge W. Bush is married to a librarian and his vice president is married to a former NEH chairman. Bush may have passed through Yale largely unscathed by what his professors were professing (which is just as well, considering campus conditions in the 1960s). However, he is restoring both endowments to their proper functions, defending culture as the poet Allen Tate defined it\u2014\"the study of perfection, and the constant effort to achieve it.\"\n\n[DECEMBER 26, 2002]\n\n# **A Sensory Blitzkrieg of Surfaces**\n\n**T** he first modern celebrity\u2014the first person who, although not conspicuous in church or state, still made his work and life fascinating to a broad public\u2014may have been Charles Dickens. Novelist Jane Smiley so argues in her slender life of Dickens, and her point is particularly interesting in light of _Reading at Risk,_ the National Endowment for the Arts' report on the decline of reading.\n\nA survey of 17,135 persons reveals an accelerating decline in the reading of literature, especially among the young. Literary reading declined 5 percent between 1982 and 1992, then 14 percent in the next decade. Only 56.9 percent of Americans say they read a book of any sort in the past year, down from 60.9 percent in 1992. Only 46.7 percent of adults read any literature for pleasure.\n\nThe good news is that \"literature,\" as the survey defined it, excludes serious history, for which there is a sizable audience. The bad news is that any fiction counts as literature, and most fiction, like most of most things, is mediocre. But even allowing for the survey's methodological problems, the declining importance of reading in the menu of modern recreations is unsurprising and unsettling.\n\nDickens, a volcano of words, provided mass entertainment before modern technologies\u2014electricity, film, broadcasting\u2014made mass communication easy. His serialized novels seized the attention of the British public. And America's: Ships arriving from England with the latest installment of Dickens's 1840 novel _The Old Curiosity Shop_ reportedly were greeted by American dockworkers shouting, \"Did Little Nell die?\"\n\nWhen journalists in 1910 asked an aide to Teddy Roosevelt whether TR might run for president in 1912, the aide replied, \"Barkis is willin',\" and he expected most journalists, and their readers, to recognize the reference to the wagon driver in _David Copperfield_ who was more than merely willin' to marry Clara Peggotty, David's childhood nurse.\n\nExposure to _David Copperfield_ used to be a common facet of reaching adulthood in America. But today, young adults eighteen to thirty-four, once the most avid readers, are among the least. This surely has something to do with the depredations of higher education: Professors, lusting after tenure and prestige, teach that the great works of the Western canon, properly deconstructed, are not explorations of the human spirit but mere reflections of power relations and social pathologies.\n\nBy 1995\u2014 _before_ the flood of video games and computer entertainments for adults\u2014television swallowed 40 percent of Americans' free time, up one-third since 1965. Today, electronic entertainments other than television fill 5.5 hours of the average child's day.\n\nThere have been times when reading was regarded with suspicion. Some among the ancient Greeks regarded the rise of reading as cultural decline: They considered oral dialogue, which involves clarifying questions, more hospitable to truth. But the transition from an oral to a print culture has generally been a transition from a tribal society to a society of self-consciously separated individuals. In Europe, that transition alarmed ruling elites, who thought the \"crisis of literacy\" was that there was too much literacy: readers had, inconveniently, minds of their own. Reading is inherently private, hence the reader is beyond state supervision or crowd psychology.\n\nWhich suggests why there are perils in the transition from a print to an electronic culture. Time was, books were the primary means of knowing things. Now most people learn most things visually, from the graphic presentation of immediately, effortlessly accessible pictures.\n\nPeople grow accustomed to the narcotic effect of their own passive reception of today's sensory blitzkrieg of surfaces. They recoil from the more demanding nature of active engagement with the nuances encoded in the limitless permutations of twenty-six signs on pages. Besides, reading requires two things that are increasingly scarce and to which increasing numbers of Americans seem allergic\u2014solitude and silence.\n\nIn 1940, a British officer on Dunkirk beach sent London a three-word message: \"But if not.\" It was instantly recognized as from the book of Daniel. When Shadrach, Meshach, and Abednego are commanded to worship a golden image or perish, they defiantly reply: \"Our God who we serve is able to deliver us from the burning fiery furnace, and He will deliver us out of thine hand, O king. But if not, be it known unto thee, O king, that we will not serve thy gods....\"\n\nBritain then still had the cohesion of a common culture of shared reading. That cohesion enabled Britain to stay the hand of Hitler, a fact pertinent to today's new age of barbarism.\n\n[JULY 23, 2004]\n\n# **\"Philosophy Teaching by Examples\"**\n\n**W** hen Yale awarded President Kennedy an honorary degree, he said he had the ideal combination\u2014a Yale degree and a Harvard education. Today, he might rethink that, given the Harvard faculty's tantrum that caused President Lawrence Summers' cringing crawl away from his suggestion of possible gender differences of cognition. At least the phrase \"Yale education\" does not yet seem, as \"Harvard education\" does, oxymoronic.\n\nAnd will not while Donald Kagan adorns Yale's campus, where he is professor of history and classics. Last week in Washington, he delivered the thirty-fourth Jefferson Lecture in the Humanities, \"In Defense of History,\" which was agreeably subversive of several sides in today's culture wars.\n\n\"The world we live in,\" he said, \"is a difficult place to try to make a case for the value of history.\" The essence of the historian's craft\u2014the search for truth by painstaking research unassisted by revelation or other recourse to supernatural explanations\u2014is embattled from two directions.\n\nSome intellectuals today are know-nothings\u2014literally and proudly. They argue that objectivity is a chimera, that we cannot confidently know anything of truth or virtue. These postmodernists argue, Kagan says, that all studies, including history, are \"literature\" because all knowledge is guesswork colored by politics\u2014self-interest, ideology, power relationships, etc.\n\nThose who most confidently dispute this idea derive their confidence from religious faith, arguing that only through religion can individuals, or societies, know and steadily adhere to valid standards of right and wrong. This translates into a religious litmus test in politics\u2014only devout individuals should be chosen to lead societies.\n\nHistorians, however, say to the postmodernists that the defining characteristics of postmodernism\u2014skepticism and cynicism\u2014have long histories. And the historians' riposte to those who say that religion is the only foundation for knowledge or virtue is, Kagan says, to insist that in the study of history, knowledge, far from impossible, is cumulative.\n\nHerodotus, whom Kagan calls the first true historian, said he wrote to preserve the memory of \"great and marvelous deeds\" and the reasons they were done. All writers of history have \"the responsibility of preserving the great, important and instructive actions of human beings,\" says Kagan, which is why history is \"the queen of the humanities,\" ranking above literature and even philosophy.\n\nPhilosophy, he says, is valuable for untangling sloppy thinking. But philosophy, like religion, leads to investigations of first principles and ultimate reality, and such investigations have produced profound disagreement. Hence the primacy of history as the study from which, especially today, we can take our moral bearings:\n\n\"Religion and the traditions based on it were once the chief sources for moral confidence and strength. Their influence has faded in the modern world, but the need for a sound base for moral judgments has not. If we cannot look simply to moral guidance firmly founded on religious precepts, it is natural and reasonable to turn to history, the record of human experience, as a necessary supplement if not a substitute.\"\n\nKagan's idea is not novel. Nearly three centuries ago, Lord Boling-broke said that \"history is philosophy teaching by examples.\" However, at this American moment of mutual incomprehension and even contempt between theists and their postmodernist despisers, it is \"transgressive\"\u2014to purloin a bit of the postmodernists' jargon\u2014for Kagan to insist that there is a firm middle, or perhaps higher, ground for moral confidence.\n\nThis ground, he says, is occupied by neither those who say that only theological reasoning leads to certainty nor those who say that no reasoning does. It is held by those who study and write history.\n\nThe late Daniel Boorstin, historian and librarian of Congress, said that \"trying to plan for the future without knowing the past is like trying to plant cut flowers.\" His point was that knowledge of history is conducive to practicality. Kagan's point is that history, properly studied, is conducive to virtues, of which practicality is one.\n\nModeration is the virtue of which hubris is the opposite\u2014and often ruinous\u2014political vice. Historian David McCullough says the study of history is \"an antidote to the hubris of the present\u2014the idea that everything we have and everything we do and everything we think is the ultimate, the best.\" Compare, for example, the heroic construction of the Panama Canal and the debacle of Boston's \"Big Dig\" one hundred years later.\n\nNear the Big Dig sits today's Harvard, another refutation of the theory of mankind's inevitable, steady ascent. From Yale, however, comes Kagan's temperate affirmation of the cumulative knowledge that comes from the study of history.\n\n[MAY 19, 2005]\n\n# **Fascinating Contingencies**\n\n**W** hen George Washington, in a spiffy uniform of buff and blue and sitting his horse with a grace uncommon even among Virginians vain about their horsemanship, arrived outside Boston in July 1775 to assume command of the American rebellion, he was aghast. When he got a gander at his troops, mostly New Englanders, his reaction was akin to the Duke of Wellington's assessment of his troops, many of them the sweepings of Britain's slums, during the Peninsular War: \"I don't know what effect these men will have upon the enemy, but, by God, they terrify me.\"\n\nYou think today's red state\u2013blue state antagonism is unprecedented? Washington thought New Englanders \"exceeding dirty and nasty.\" He would not have disputed the British general John Burgoyne's description of the Americans besieging Boston as \"a rabble in arms.\" A rabble that consumed, by one sober estimate, a bottle of rum per man each day.\n\nIf, in the autumn of 1775, a council of Washington's officers had not restrained him from a highly risky amphibious attack on Boston across the shallow Back Bay, there might never have been a Declaration of Independence. If a young officer, Henry Knox, had not had the ingenuity to conceive, and the tenacity to execute, a plan for dragging captured mortars, some weighing a ton, and cannon, some weighing two and a half tons, the three hundred miles from Fort Ticonderoga on Lake Champlain to the Dorchester Heights overlooking Boston, the British might have fought, and perhaps won, rather than evacuating the city. If after the disastrous Battle of Brooklyn, the first great battle of the war, a fog had not allowed nine thousand of Washington's soldiers to escape across the East River, the war might have effectively ended less than two months after the Declaration.\n\nSo says David McCullough in his new book, _1776,_ a birthday card to his country on this Independence Day. \"Ingratitude,\" he has said elsewhere, \"is a shabby failing,\" and he writes to inspire gratitude for what a few good men, and one great one, did in the nation's Year One.\n\nWhat British historian George Otto Trevelyan said of the December 1776 Battle of Trenton, which may have saved the Revolution, could be said of all the events\u2014defeats redeemed by skillful retreats, and a few victories\u2014of that year: \"It may be doubted whether so small a number of men ever employed so short a space of time with greater and more lasting effects upon the history of the world.\"\n\nWhat is history? The study of it\u2014and the making of it, meaning politics\u2014changed for the worse when, in the nineteenth century, history became History. When, that is, history stopped being the record of fascinating contingencies\u2014political, intellectual, social, economic\u2014that produced the present, and became instead a realm of necessity. The idea that History is a proper noun, denoting an autonomous process unfolding a predetermined future in accordance with laws mankind cannot amend, is called historicism. That doctrine discounts human agency, reducing even large historical figures to playthings of vast impersonal forces. McCullough knows better.\n\nSolid, unpretentious narrative history like _1776_ satisfies the healthy human thirst for a ripping good story. McCullough says E. M. Forster, the novelist, efficiently defined a story: If you are told that the king died and then the queen died, that is a sequence of events. If you are told that the king died and then the queen died of grief, that is a story that elicits empathy.\n\nUsing narrative history to refute historicism, McCullough's two themes in _1776_ are that things could have turned out very differently, and that individuals of character can change the destinies of nations. There is a thirst for both themes in this country, which is in a less-than-festive frame of mind on this birthday. It is, therefore, serendipitous that _1776,_ with 1.35 million copies already in print, sits atop the _New York Times_ bestseller list on Independence Day.\n\nBut, then, serendipity has often attended the Fourth of July. That day is the birthday of Nathaniel Hawthorne (1804), arguably the father of American literature. And of Stephen Foster (1826), arguably the father of American music. And\u2014saving the most luminous for last\u2014of the sainted Calvin Coolidge (1872), who oversaw a 45 percent increase in America's production of ice cream.\n\nSo, this Fourth read McCullough. Perhaps by the light of a sparkler.\n\n[JULY 3, 2005]\n\n# **Ed Schools vs. Education**\n\n**T** he surest, quickest way to add quality to primary and secondary education would be addition by subtraction: Close all the schools of education. Consider the _Chronicle of Higher Education_ 's recent report concerning the schools that certify America's teachers.\n\nMany education schools discourage, even disqualify, prospective teachers who lack the correct \"disposition,\" meaning those who do not embrace today's \"progressive\" political catechism. Karen Siegfried had a 3.75 grade-point average at the University of Alaska Fairbanks, but after voicing conservative views, she was told by her education professors that she lacked the \"professional disposition\" teachers need. She is now studying to be an aviation technician.\n\nIn 2002, the National Council for Accreditation of Teacher Education declared that a \"professional disposition\" is \"guided by beliefs and attitudes related to values such as caring, fairness, honesty, responsibility, and social justice.\" Regarding that last, the _Chronicle_ reports that the University of Alabama's College of Education proclaims itself \"committed to preparing individuals to\"\u2014what? \"Read, write, and reason\"? No, \"to promote social justice, to be change agents, and to recognize individual and institutionalized racism, sexism, homophobia, and classism,\" and to \"break silences\" about those things and \"develop anti-racist, anti-homophobic, anti-sexist community [ _sic_ ] and alliances.\"\n\nBrooklyn College, where a professor of education required her class on Language Literacy in Secondary Education to watch _Fahrenheit 9\/11_ before the 2004 election, says it educates teacher candidates about, among many other evils, \"heterosexism.\" The University of Alaska Fairbanks, fluent with today's progressive patois, says that, given America's \"caste-like system,\" teachers must be taught \"how racial and cultural 'others' negotiate American school systems, and how they perform their identities.\" Got it?\n\nThe permeation of education schools by politics is a consequence of the vacuity of their curricula. Concerning that, read \"Why Johnny's Teacher Can't Teach,\" by Heather Mac Donald of the Manhattan Institute (available at city-journal.org). Today's teacher-education focus on \"professional disposition\" is just the latest permutation of what Mac Donald calls the education schools' \"immutable dogma,\" which she calls \"Anything But Knowledge.\"\n\nThe dogma has been that primary and secondary education is about \"self-actualization\" or \"finding one's joy\" or \"social adjustment\" or \"multicultural sensitivity\" or \"minority empowerment.\" But is never about anything as banal as mere knowledge. It is about \"constructing one's own knowledge\" and \"contextualizing knowledge,\" but never about knowledge of things like biology or history.\n\nMac Donald says \"the central educational fallacy of our time,\" which dates from the Progressive Era of the early twentieth century, is \"that one can think without having anything to think about.\" At City College of New York, a professor said that in her course Curriculum and Teaching in Elementary Education she would be \"building a community, rich of talk\" and \"getting the students to develop the subtext of what they're doing.\" Although ed schools fancy themselves as surfers on the wave of the future, Mac Donald believes that teacher education \"has been more unchanging than Miss Havisham. Like aging vestal virgins, today's schools lovingly guard the ancient flame of progressivism\"\u2014an egalitarianism with two related tenets.\n\nOne, says Mac Donald, is that \"to accord teachers any superior role in the classroom would be to acknowledge an elite hierarchy of knowledge, possessed by some but not all.\" Hence, second, emphasis should be on group projects rather than individual accomplishments that are measured by tests that reveal persistent achievement gaps separating whites and Asians from other minorities.\n\nNumerous inner-city charter and private schools are proving that the gaps can be narrowed, even closed, when rigorous pedagogy is practiced by teachers in teacher-centered classrooms where knowledge is regarded as everything. But most ed schools, celebrating \"child-centered classrooms\" that do not \"suffocate discourses,\" are enemies of rigor.\n\nThe steady drizzle of depressing data continues. A new assessment of adult literacy shows a sharp decline over the last decade, with only 31 percent of college graduates able to read and extrapolate from complex material. They were supposed to learn how to read before college, but perhaps their teachers were too busy proving their \"professional dispositions\" by \"breaking silences\" as \"change agents.\"\n\nFewer than half of U.S. eighth graders have math teachers who majored in math as undergraduates or graduate students or studied math for teacher certification. U.S. twelfth graders recently performed below the international average for twenty-one countries on tests of general knowledge of math and science. But perhaps U.S. pupils excel when asked to \"perform their identities.\"\n\n[JANUARY 16, 2006]\n\n# **This Just In from the Professors: Conservatism Is a Mental Illness**\n\n**T** his just in: Conservatism often is symptomatic of a psychological syndrome. It can involve fear, aggression, uncertainty avoidance, intolerance of ambiguity, dogmatic dislike of equality, irrational nostalgia, and need for \"cognitive closure,\" all aspects of the authoritarian personality.\n\nActually, this theory has been floating around academic psychology for half a century. It is reprised in \"Political Conservatism as Motivated Social Cognition,\" written by four professors for _Psychological Bulletin._\n\n\"Motivated social cognition\" refers to the \"motivational underpinnings\" of ideas, the \"situational as well as dispositional variables\" that foster particular beliefs. Notice: situations and dispositions\u2014not reasons. Professors have reasons for their beliefs. Other people, particularly conservatives, have social and psychological explanations for their beliefs. \"Motivated cognition\" involves ways of seeing and reasoning about the world that are unreasonable because they arise from emotional, psychological needs.\n\nThe professors note, \"The practice of singling out political conservatives for special study began...[with a 1950] study of authoritarianism and the fascist potential in personality.\" The industry of studying the sad psychology of conservatism is booming. It began with a European mixture of Marxism and Freudianism. It often involves a hash of unhistorical judgments, including the supposedly scientific, value-free judgment that conservatives are authoritarians, and that fascists\u2014e.g., the socialist Mussolini, and Hitler, the National Socialist who wanted to conserve _nothing_ \u2014were conservatives.\n\nThe four professors now contribute \"theories of epistemic and existential needs, and socio-political theories of ideology as individual and collective rationalizations\" and \"defensive motivations\"\u2014defenses against fear of uncertainty and resentment of equality. The professors have ideas; the rest of us have emanations of our psychological needs and neuroses.\n\n\"In the post-Freudian world, the ancient dichotomy between reason and passion is blurred,\" say the professors, who do not say that _their_ judgments arise from social situations or emotional needs rather than reason. The professors usefully survey the vast literature churned out by the legions of academics who have searched for the unsavory or pathological origins of conservatism (fear of death? harsh parenting? the \"authoritarian personality\"?).\n\nBut it is difficult to take the professors' seriousness seriously when they say, in an essay responding to a critique of their paper, that Ronald Reagan's \"chief accomplishment, in effect, was to roll back both the New Deal and the 1960s.\" His \"accomplishment\"? So _that_ is why Social Security and Medicare disappeared.\n\nThe professors write, \"One is justified in referring to Hitler, Mussolini, Reagan, and Limbaugh as right-wing conservatives...because they all preached a return to an idealized past and favored or condoned inequality in some form.\" Until the professors give examples of political people who do not favor or condone equality _in any form,_ it is fair to conclude that, for all their pretensions to scientific rigor, they are remarkably imprecise. And they are very political people, who would be unlikely ever to begin a sentence: \"One is justified in referring to Stalin, Mao, Franklin Roosevelt and the editors of the _New York Times_ as left-wing liberals because...\"\n\nThe professors acknowledge that \"the same motives may underlie different beliefs.\" And \"different motives may underlie the same beliefs.\" And \"motivational and informational influences on belief formation are not incompatible.\" And no reasoning occurs in a \"motivational vacuum.\" And \"virtually all belief systems\" are embraced because they \"satisfy some psychological needs.\" And all this \"does not mean that conservatism is pathological or that conservative beliefs are necessarily false.\"\n\nNot _necessarily._ What a relief. But there is no comparable academic industry devoted to studying the psychological underpinnings of liberalism. Liberals, you see, embrace liberalism for an obvious and uncomplicated reason\u2014liberalism is self-evidently true. But conservatives embrace conservatism for reasons that must be excavated from their inner turmoils, many of them pitiable or disreputable.\n\nThe professors' paper is adorned with this epigraph:\n\n_\"Conservatism is a demanding mistress and is giving me a migraine.\"_\n\n\u2014GEORGE F. WILL\n\nA \"mistress\" who is \"demanding\"? Freud, call your office. The epigraph is from _Bunts,_ a book of baseball essays, from an essay concerning what conservatives should think about the designated hitter. Will probably _thought_ he was being lighthearted. Silly him. Actually, he was struggling with fear of ambiguity and the need for cognitive closure.\n\nConservatives, in the crippling grip of motivated social cognition, _think_ they oppose the DH because it makes the game less interesting by reducing managers' strategic choices. But they _really_ oppose that innovation because mental rigidity makes them phobic about change and intolerant of the ambiguous status of the DH. And because Mussolini would have opposed the DH.\n\n[AUGUST 10, 2003]\n\n# **The Law of Group Polarization in Academia**\n\n_Republicans Outnumbered in Academia, Studies Find_\n\n\u2014 _New York Times,_ November 18\n\n**O** h, well, if _studies_ say so. The great secret is out: Liberals dominate campuses. Coming soon: \"Moon Implicated in Tides, Studies Find.\"\n\nOne study of one thousand professors finds that Democrats outnumber Republicans at least seven to one in the humanities and social sciences. That imbalance, more than double what it was three decades ago, is intensifying because younger professors are more uniformly liberal than the older cohort that is retiring.\n\nAnother study, of voter registrations records, including those of professors in engineering and the hard sciences, found 9 Democrats for every Republican at Berkeley and Stanford. Among younger professors, there were 183 Democrats, 6 Republicans.\n\nBut we essentially knew this even before the _American Enterprise_ magazine reported in 2002 of examinations of voting records in various college communities. Some findings about professors registered with the two major parties or with liberal or conservative minor parties:\n\nCornell: | 166 liberals, 6 conservatives\n\n---|---\n\nStanford: | 151 liberals, 17 conservatives\n\nColorado: | 116 liberals, 5 conservatives\n\nUCLA: | 141 liberals, 9 conservatives\n\nThe nonpartisan Center for Responsive Politics reports that in 2004, of the top five institutions in terms of employee per capita contributions to presidential candidates, the third, fourth, and fifth were Time Warner, Goldman Sachs, and Microsoft. The top two were the University of California system and Harvard, both of which gave about nineteen times more money to John Kerry than to George Bush.\n\nBut George Lakoff, a linguistics professor at Berkeley, denies that academic institutions are biased against conservatives. The disparity in hiring, he explains, occurs because conservatives are not as interested as liberals in academic careers. Why does he think liberals are like that? \"Unlike conservatives, they believe in working for the public good and social justice.\"\n\nThat clears that up.\n\nA filtering process, from graduate school admissions through tenure decisions, tends to exclude conservatives from what Mark Bauerlein calls academia's \"sheltered habitat.\" In a dazzling essay in the _Chronicle of Higher Education,_ Bauerlein, professor of English at Emory University and director of research at the National Endowment for the Arts, notes that the \"first protocol\" of academic society is the \"common assumption\"\u2014that, at professional gatherings, all the strangers in the room are liberals.\n\nIt is a reasonable assumption, given that in order to enter the profession, your work must be deemed, by the criteria of the prevailing culture, \"relevant.\" Bauerlein says various academic fields now have regnant premises that embed political orientations in their very definitions of scholarship:\n\n\"Schools of education, for instance, take constructivist theories of learning as definitive, excluding realists (in matters of knowledge) on principle, while the quasi-Marxist outlook of cultural studies rules out those who espouse capitalism. If you disapprove of affirmative action, forget pursuing a degree in African-American studies. If you think that the nuclear family proves the best unit of social well-being, stay away from women's studies.\"\n\nThis gives rise to what Bauerlein calls the \"false consensus effect,\" which occurs when, due to institutional provincialism, \"people think that the collective opinion of their own group matches that of the larger population.\" There also is what Cass Sunstein, professor of political science and jurisprudence at the University of Chicago, calls \"the law of group polarization.\" Bauerlein explains: \"When like-minded people deliberate as an organized group, the general opinion shifts toward extreme versions of their common beliefs.\" They become tone-deaf to the way they sound to others outside their closed circle of belief.\n\nWhen John Kennedy brought to Washington such academics as Arthur Schlesinger Jr., John Kenneth Galbraith, McGeorge and William Bundy, and Walt Rostow, it was said that the Charles River was flowing into the Potomac. Actually, Richard Nixon's administration had an even more distinguished academic cast\u2014Henry Kissinger, Pat Moynihan, Arthur Burns, James Schlesinger, and others.\n\nAcademics, such as the next secretary of state, still decorate Washington, but academia is less listened to than it was. It has marginalized itself, partly by political shrillness and silliness that have something to do with the parochialism produced by what George Orwell called \"smelly little orthodoxies.\"\n\nMany campuses are intellectual versions of one-party nations\u2014except such nations usually have the merit, such as it is, of candor about their ideological monopolies. In contrast, American campuses have more insistently proclaimed their commitment to diversity as they have become more intellectually monochrome.\n\nThey do indeed cultivate diversity\u2014in race, skin color, ethnicity, sexual preference. In everything but thought.\n\n[NOVEMBER 28, 2004]\n\n# **Antioch College's Epitaph**\n\n**D** uring the campus convulsions of the late 1960s, when rebellion against any authority was considered obedience to every virtue, the film _To Die in Madrid,_ a documentary about the Spanish civil war, was shown at a small liberal arts college famous for, and vain about, its dedication to all things progressive. When the film's narrator intoned, \"The rebels advanced on Madrid,\" the students, who adored rebels and were innocent of information, cheered. Antioch College in Yellow Springs, Ohio, had been so busy turning undergraduates into vessels of liberalism and apostles of social improvement that it had not found time for the tiresome task of teaching them tedious facts, such as that the rebels in Spain were Franco's fascists.\n\nThat illustrates why it is heartening that Antioch will close after the 2007\u20132008 academic year. Its board of trustees says the decision is to \"suspend operations\" and it talks dottily about reviving the institution in 2012. There is, however, a minuscule market for what Antioch sells for a tuition, room, and board of $35,221\u2014repressive liberalism unleavened by learning.\n\nFounded in 1852\u2014its first president was Horace Mann\u2014Antioch was, for a while, admirable. One of the first colleges to enroll women and blacks, it was a destination for escaped slaves. Its alumni include Stephen Jay Gould, Coretta Scott King, and Rod Serling, whose _Twilight Zone_ never imagined anything weirder than what Antioch became when its liberalism curdled.\n\nIn 1972\u20131973, Antioch had 2,470 students. In 1973, a protracted and embittering student and employee strike left the campus physically decrepit and intellectually toxic. By 1985, enrollment was down 80 percent. This fall there may be 300 students served by a faculty of forty.\n\nIn 1993, Antioch became an international punch line when it wrote rules to ensure that all sexual conduct would be consensual, step by minute step: \"If the level of sexual intimacy increases during an interaction...the people involved need to express their clear verbal consent before moving to that new level.\" Does consent to a _touch_ cover a _caress_? Is there consent regarding _all_ the buttons?\n\nAlthough laughable, Antioch was not funny. Former public radio correspondent Michael Goldfarb matriculated at what he calls the \"sociological petri dish\" in 1968. In his first week, he twice had guns drawn on him, once \"in fun\" and once by a couple of drunken ex-cons \"whom one of my classmates, in the interest of breaking down class barriers, had invited to live with her.\" A true Antiochian still, Goldfarb says: \"I do think I was made stronger for having to deal with these experiences.\"\n\nSteven Lawry\u2014Antioch's fifth president in thirteen years\u2014came to the college eighteen months ago. He told Scott Carlson of the _Chronicle of Higher Education_ about a student who left after being assaulted because he wore Nike shoes, symbols of globalization. Another left because, she told Lawry, the political climate was suffocating: \"They all think they are so different, but they are just a bunch of conformists.\"\n\nCarlson reports that Lawry stopped the student newspaper's practice of printing \"announcements containing anonymous, menacing threats against other students for their political views.\" Antioch likes to dabble in menace: It invited Mumia Abu-Jamal to deliver its 2000 commencement speech, which he recorded on death row in a Pennsylvania prison, where he lives because twenty-six years ago he shot a Philadelphia police officer first in the back, then three times in the face. Antioch's invitation was its way of saying...what?\n\nIn an essay in the _Chronicle,_ Cary Nelson, Antioch class of 1967 and now a professor of English at the University of Illinois, waxes nostalgic about the fun he had spending, as Antioch students did, much time away from campus, receiving academic credits. What Nelson calls \"my employee resistance to injustice\" got him \"released from almost every job I had until I became a faculty member.\" But \"my little expenditure was never noticed\" when \"I used some of Lyndon Johnson's anti-poverty money\" to bus anti-Vietnam war protesters from Harlem to Washington.\n\nGiven that such was Antioch's idea of \"work experience\" in the \"real world,\" it is unsurprising that the college never produced an alumni cohort capable of enlarging the college's risible $36 million endowment. Besides, the college seems always to have considered raising money beneath its dignity, given its nobility.\n\n\"Ben & Jerry could have named a new flavor for us,\" says John Feinberg, class of 1970 and president of the alumni board, with a melancholy sense of unfulfilled destiny. His lament for a forfeited glory is a suitable epitaph for Antioch.\n\n[JULY 15, 2007]\n\n# **A Scholar's Malfeasance Gunned Down**\n\n**I** n a large event, much commented on, the Justice Department last week told the Supreme Court that the Second Amendment (\"A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed\") \"broadly protects the rights of individuals,\" not just the right of states to organize militias.\n\nThis event was pertinent to a small event two weeks earlier, noticed by almost no one. The National Endowment for the Humanities demanded a review of \"the serious charges that have been made against Michael Bellesiles' scholarship,\" which the NEH helped to finance.\n\nHe is the Emory University historian whose 2000 book _Arming America: The Origins of a National Gun Culture_ earned\u2014well, received\u2014critical acclaim, including the Bancroft Prize, the most distinguished prize in American history. But now, slowly but relentlessly, some responsible intellectuals are defending standards of scholarship.\n\nBellesiles's thesis is startling. It is that guns were not widely owned, or reliable enough to be important, at the time the Second Amendment was written. The implication is that the amendment should be read to protect only the collective rights of states, not the rights of individuals. The book pleased partisans of a cause popular in the liberal political culture of academia\u2014gun control. Reviews were rapturous: \"exhaustive research,\" \"intellectual rigor,\" \"inescapable policy implications,\" \"the NRA's worst nightmare.\"\n\nNot exactly.\n\nWhat has become Bellesiles's nightmare began when a historian, suspecting nothing and hoping to build upon Bellesiles's data, asked for more details about the eighteenth-and early-nineteenth-century probate records that Bellesiles says show that guns were infrequently listed among the estates of deceased people. He also purported to find that many of the guns that were listed were in disrepair.\n\nWhen Bellesiles's evasive response led to more tugging on the threads of his argument, it unraveled. The unraveling revealed a pattern of gross misstatements of facts and unfounded conclusions. His errors are so consistently convenient for his thesis, it is difficult to believe that the explanation is mere sloppiness or incompetence. It looks like fraud.\n\nResponding to critics, he said some of his crucial research notes had been destroyed in a flood in his office. He said he had relied on microfilm records in the federal archive in East Point, Georgia. But it has no such records. He said he had examined probate records in thirty places around the country, such as the San Francisco Superior Court. But those records were destroyed in the 1906 earthquake.\n\nWell, then, he said he had seen them in the Contra Costa County Historical Society. But the society has no such records, and no record of Bellesiles's visiting the society. Then he said he did the research somewhere else, but is not sure where. Researchers have found that he consistently misrepresents extant records in Providence, Rhode Island, and Vermont. When he tried to buttress his case by posting evidence on his website, critics found grave errors there, too, purporting to support his thesis. He blamed the errors on a hacker. A hacker who attacked his website on his behalf?\n\nThis spring, the _William and Mary Quarterly,_ the preeminent journal of early American history, undertook to referee this rumble, publishing essays from historians on aspects of Bellesiles's argument, and a response from him. The criticism is lacerating (\"nonsense\" \"boggles the mind\" \"he regularly uses evidence in a partial or imprecise way\" \"gives the impression that he has shaped his figures to suit his argument\" \"every tally of homicides Bellesiles reports is either misleading or wrong\"). Bellesiles's limping response (which begins with a jest about a French mutineer saying to his firing squad, \"I am honored by all this attention\") tries to change the subject, to the meaning of \"culture.\"\n\nThis academic scandal is still several chapters from a satisfactory resolution. Emory, having taken the unusual step of directing Bellesiles to respond to his critics, now is conducting its own investigation, which must weigh the patent inadequacies of his responses so far, and the proper penalty for what has already been proved. Furthermore, do those responsible for awarding him the Bancroft Prize believe the award should be revoked?\n\nBellesiles's malfeasance, although startling in its sweep, brazenness, and apparently political purpose, actually reveals something heartening\u2014a considerable strength in America's scholarly community. Its critical apparatus is working. Scholars and their journals are doing their duty, which is to hold works of scholarship up to the bright light of high standards.\n\nAs a result, when next the Supreme Court is required to rule on the controversy concerning which Bellesiles's book was supposed to be so decisively informative, the court's judgment will not be clouded by Bellesiles's evident attempt to misrepresent the context in which the Framers wrote the Second Amendment.\n\n[MAY 20, 2002]\n\n# **Juggling Scarves in the Therapeutic Nation**\n\n**I** t hurt her feelings, says Jane Fonda, sharing her feelings, that one of her husbands liked them to have sexual threesomes. \"It reinforced my feeling I wasn't good enough.\"\n\nIn the Scottsdale, Arizona, Unified School District office, the receptionist used to be called a receptionist. Now she is \"director of first impressions.\" The happy director says, \"Everyone wants to be important.\" Scottsdale school bus drivers now are \"transporters of learners.\" A school official says such terminological readjustment is \"a positive affirmation.\" Which beats a negative affirmation.\n\nManufacturers of pens and markers report a surge in teachers' demands for purple ink pens. When marked in red, corrections of students' tests seem so awfully judgmental. At a Connecticut school, parents consider red markings \"stressful.\" A Pittsburgh principal favors more \"pleasant-feeling tones.\" An Alaska teacher says substituting purple for red is compassionate pedagogy, a shift from \"Here's what you need to improve on\" to \"Here's what you have done right.\"\n\nFonda's confession, Scottsdale's tweaking of terminology, and the recoil from red markings are manifestations of today's therapeutic culture. The nature and menace of \"therapism\" is the subject of a new book, _One Nation Under Therapy: How the Helping Culture Is Eroding Self-Reliance,_ by Christina Hoff Sommers and Sally Satel, MD, resident scholars at the American Enterprise Institute.\n\nFrom childhood on, Americans are told by \"experts\"\u2014therapists, self-esteem educators, grief counselors, traumatologists\u2014that it is healthy for them continuously to take their emotional temperature, inventory their feelings, and vent them. Never mind research indicating that reticence and suppression of feelings can be healthy.\n\nBecause children are considered terribly vulnerable and fragile, playground games like dodgeball are being replaced by anxiety-reducing and self-esteem-enhancing games of tag where nobody is ever \"out.\" But abundant research indicates no connection between high self-esteem and high achievement or virtue. Is not unearned self-esteem a more pressing problem? Sensitivity screeners remove from texts and tests distressing references to things like rats, snakes, typhoons, blizzards, and...birthday parties (which might distress children who do not have them). The sensitivity police favor teaching what Sommers and Satel call \"no-fault history.\" Hence California's Department of Education stipulating that when \"ethnic or cultural groups are portrayed, portrayals must not depict differences in customs or lifestyles as undesirable\"\u2014slavery? segregation? anti-Semitism? cannibalism?\u2014\"and must not reflect adversely on such differences.\"\n\nExperts warn about what children are allowed to juggle: Tennis balls cause frustration, whereas \"scarves are soft, nonthreatening, and float down slowly.\" In 2001, the Girl Scouts, illustrating what Sommers and Satel say is the assumption that children are \"combustible bundles of frayed nerves,\" introduced, for girls eight to eleven, a \"Stress Less Badge\" adorned with an embroidered hammock. It can be earned by practicing \"focused breathing,\" keeping a \"feelings diary,\" burning scented candles, and exchanging foot massages.\n\nVast numbers of credentialed\u2014that is not a synonym for \"competent\"\u2014members of the \"caring professions\" have a professional stake in the myth that most people are too fragile to cope with life's vicissitudes and traumas without professional help. Consider what Sommers and Satel call \"the commodification of grief\" by the \"grief industry\"\u2014professional grief \"counselors\" with \"degrieving\" techniques. Such \"grief gurus\" are \"ventilationists\": They assume that everyone should grieve the same way\u2014by venting feelings sometimes elicited by persons who have paid $1,795 for a five-day course in grief counseling.\n\nThe \"caregiving\" professions, which postulate the minimal competence of most people to cope with life unassisted, are, of course, liberal and politics can color their diagnoses. Remember the theory that because Vietnam was supposedly an unjust war, it would produce an epidemic of \"post-traumatic stress disorders.\" So a study released in 1990 claimed that half of Vietnam veterans suffered from some PTSD\u2014even though only 15 percent of Vietnam veterans had served in combat units. To ventilationists\u2014after a flood damaged books at the Boston Public Library, counselors arrived to help librarians cope with their grief\u2014a failure to manifest grief is construed as alarming evidence of grief repressed, and perhaps a precursor of \"delayed onset\" PTSD.\n\nPredictably, 9\/11 became another excuse for regarding healthy human reactions as pathological. Did terrorist attacks make you angry and nervous? Must be PTSD. And 9\/11 gave rise to \"diagnostic mission creep\" as the idea of a \"trauma\" was expanded to include watching a disaster on television. Sommers and Satel's book is a summons to the sensible worry that national enfeeblement must result when therapism replaces the virtues on which the republic was founded\u2014stoicism, self-reliance, and courage.\n\n[APRIL 21, 2005]\n\n# **Nature, Nurture, and Larry Summers's Sin**\n\nHYSTERIA\u2014A functional disturbance of the nervous system, characterized by such disorders as anaesthesia, hyperaesthesia, convulsions, etc., and usually attended with emotional disturbances and enfeeblement or perversion of the moral and intellectual faculties.\n\n\u2014 _Oxford English Dictionary_\n\n**F** orgive Larry Summers. He did not know where he was.\n\nAddressing a conference on the supposedly insufficient numbers of women in tenured positions in university science departments, he suggested that perhaps part of the explanation might be innate\u2014genetically based\u2014gender differences in cognition. He thought he was speaking in a place that encourages uncircumscribed intellectual explorations. He was not. He was on a university campus.\n\nHe was at Harvard, where he is president. Since then he has become a serial apologizer and accomplished groveler. Soon he may be in a Khmer Rouge\u2013style reeducation camp somewhere in New England, relearning this: In today's academy, no social solecism is as unforgivable as the expression of a hypothesis that offends someone's \"progressive\" sensibilities.\n\nSomeone like MIT biology professor Nancy Hopkins, the hysteric (see above) who, hearing Summers, \"felt I was going to be sick. My heart was pounding and my breath was shallow.\" And, \"I just couldn't breathe because this kind of bias makes me physically ill.\" She said that if she had not bolted from the room, \"I would've either blacked out or thrown up.\"\n\nIs _this_ the fruit of feminism? A woman at the peak of the academic pyramid becomes theatrically flurried by an unwelcome idea and, like a Victorian maiden exposed to male coarseness, suffers the vapors and collapses on the drawing room carpet in a heap of crinolines until revived by smelling salts and the offending brute's contrition.\n\nHopkins's sufferings, although severe, were not incapacitating: She somehow found strength quickly to share them with the _Boston Globe_ and the _Today_ show, on which she confided that she just did not know whether she could bear to have lunch with Summers. But even while reeling from the onslaught of Summers's thought, she retained a flair for meretriciousness: She charged that Summers had said \"that 50 percent\" of \"the brightest minds in America\" do not have \"the right aptitude\" for science.\n\nMen and women have genetically based physical differences; the brain is a physical thing\u2014part of the body. Is it unthinkable\u2014is it even counterintuitive\u2014that this might help explain, for example, the familiar fact that more men than women achieve the very highest scores in mathematics aptitude tests? There is a vast and growing scientific literature on possible gender differences in cognition. Only hysterics denounce interest in those possible differences\u2014or, in Hopkins's case, the mere mention of them\u2014as \"bias.\"\n\nHopkins's hysteria was a sample of America's campus-based indignation industry, which churns out operatic reactions to imagined slights. But her hysteria also is symptomatic of a political tendency that manifested itself in some criticism of President Bush's inaugural address, which was a manifesto about human nature.\n\nThis criticism went beyond doubts about his grandiose aspirations, to rejection of the philosophy that he might think entails such aspirations but actually does not. The philosophy of natural right\u2014the Founders' philosophy\u2014rests on a single proposition: There is a universal human nature.\n\nFrom that fact come, through philosophic reasoning, some normative judgments: Certain social arrangements\u2014particularly government by consent attained by persuasion in a society accepting pluralism\u2014are right for creatures of this nature. Hence the doctrine of \"natural right,\" and the idea of a nation \"dedicated,\" as Lincoln said, to the \"proposition\" that all men are created equal.\n\nThe vehemence of the political left's recoil from this idea is explained by the investment political radicalism has had for several centuries in the notion that human beings are essentially blank slates. What predominates in determining individuals' trajectories\u2014nature or nurture? The left says nature is negligible, nurturing is sovereign. So a properly governed society can write what it wishes on the blank slate of humanity. This maximizes the stakes of politics and the grandeur of government's role. And the importance of governing elites, who are the \"progressive\" vanguards of a perfected humanity.\n\nThe vehemence of Hopkins's recoil from the idea that there could be gender differences pertinent to some cognition might seem merely to reflect a crude understanding of civic equality as grounded shakily on a certain identical physicality. But her hysteria actually expresses the left's ultimate horror\u2014the thought that nature sets limits to the malleability of human material. Summers should explain this to her, over lunch, when he returns from camp.\n\n[JANUARY 27, 2005]\n\n# **AP Harry Applies to College**\n\n**\"I** vies,\" \"safeties,\" \"AP prep courses,\" \"legacy,\" \"r\u00e9sum\u00e9-enhancing activity,\" \"nonbinding early acceptance,\" \"rolling admissions,\" \"single-choice early action.\" If this argot is familiar to you, poor you: You have a child in high school, and these are the days that try your soul, the spring days when many college admissions are announced, often by e-mail, which is how AP Harry learned he was deferred by Harvard.\n\nHarry is a character in Susan Coll's new novel _Acceptance,_ set in Verona County, Maryland, which is the real Montgomery County, Maryland, thinly disguised\u2014rich, liberal, full of strivers and contiguous to strivers' paradise, Washington. Harry earned the nickname AP because beginning with his freshman year he took almost every Advanced Placement course offered at Verona High School, which is so serious about placing graduates in prestigious colleges that the principal stalks the halls quizzing students on vocabulary words. For Harry, only Harvard will do.\n\nBut Harry is a white male without a legacy at Harvard, and although he got a perfect 800 on his math SAT, even with the help of private SAT prep tutoring he could boost his critical reading score only to 720. And when he got a B in an AP English course, he worried that it was the beginning of a long slide that would terminate on some skid row or, worse, at a \"safety\" school not among the Ivies.\n\nHarry, who wears starched shirts and a blazer and carries a briefcase, is a real _rara avis_ in Verona County\u2014a conservative whose heroes include Trent Lott. And he is a wee bit obsessive. He has his mother quiz him to confirm that he remembers the _U.S. News & World Report_'s list\u2014 _in order_ \u2014of the top fifty liberal-arts colleges. He subscribes to a service that each day sends an SAT-type question to his cell phone. Harry taps his phone keyboard and reads:\n\n\"'Their ideal was to combine individual liberty with material equality, a goal that has not yet been realized and that may be as [ _blank_ ] as transmutation of lead into gold.'\"\n\n\"Before Harry could continue, a small girl wearing orthodontic headgear blurted out the answer: 'A, _chimerical._ '\"\n\nAlso, Harry's sentences frequently trail off into lists of synonyms useful for the SAT vocabulary test:\n\n\"'You look kind of pale, Mom... _pale, sallow, pallid, wan_...'\n\n\"Grace forced a smile... _'Ashen?'_ she asked.\n\n\"'Very good, Mom,' Harry said, smiling adorably.\"\n\nGrace's neighbor, who walks her dog on a Burberry leash, began fretting about college admissions during the summer before their children entered eighth grade. That neighbor hired a private college counseling service at a two-year cost of $30,000, and she signed up her daughter\u2014who she insists preferred NPR to television at age four\u2014for SAT prep courses three years ahead of the normal schedule.\n\nColl writes: \"How had a test originally intended to give a smart kid stuck farming pigs in the Midwest a chance to compete with the children of the Northeastern elite morphed back into a tool to help the rich stay on top?\" How? By what Coll calls the \"snakepit of parental competition\" among the kind of parents who send holiday letters like this:\n\n\"We are ringing in the New Year in Ireland at the behest of Bree, who was so taken with her reading of 'Ulysses' in her rapid learner reading class that we are taking a self-guided tour of Joyce's Dublin...Sixth grade has proven a bit dull for Bree...An aspiring novelist (as you may have guessed!), she plans to spend the summer honing her writing skills at a workshop at Johns Hopkins...Conveniently, her little brother will also be attending Johns Hopkins this summer. Gordon has been accepted into the 'HeadsUp' program for preschoolers who show an innate predisposition for design and engineering...\"\n\nSuch parents produce children who, Coll writes, worry unhealthily as they were taught to worry in health class: \"If exchanging flirty text messages was the first step toward contracting a sexually transmitted disease, a bad decision about where to apply to college would probably lead to a life of future unemployment, then homelessness, and finally exclusion from family gatherings at holidays.\"\n\n_Acceptance_ also examines the travails of the admissions official at fictional Yates College (\"the Princeton of Upstate New York\"), which has just had the deranging experience of cracking the _U.S. News_ list at number fifty. Imagine plowing through applicants' essays \"about how Mahatma Gandhi was the single greatest inspiration in these kids' lives, or how the historical figure with whom they most closely identified was Harry Potter.\"\n\nThe mother with the Burberry leash suggests that her daughter's college application essay begin, \"Family lore has it that my first words were 'Standard Oil.'\" What happens to that daughter, to Harry and other young victims of \"the Verona madness\"? Buy Coll's book and find out. It is hilarious and dismaying... _alarming, disturbing, disquieting, agitating, perturbing_...\n\n[APRIL 9, 2007]\n\n# **Teaching Minnows the Pleasure of Precision**\n\n**L** OS ANGELES\u2014After eight years at Robert F. Kennedy Elementary School, Ethel Bojorquez knows a thing or two about teaching. She radiates calm, no-nonsense authority, and today she is watching a kindred spirit, Carole Valleskey, put Bojorquez's thirty-five fourth and fifth graders briskly through their paces.\n\nActually, the paces are Valleskey's. A former ballerina with the Joffrey, she now choreographs dance classes at eight fortunate Los Angeles elementary schools. For a few hours a week, Valleskey's students restrain their anarchic individualism in order to perform as a dance troupe. Think of training young minnows in synchronized swimming.\n\nThe children have high-energy encounters with high-quality popular culture\u2014Ellington, Gershwin, Copland\u2014that is a far cry from hip-hop. Bojorquez, whose experience has immunized her against educational fads, admiringly watches her pupils perform under Valleskey's exacting tutelage and exclaims, \"They are learning about reading _right now._ \"\n\nThey are, she marvels, learning about\u2014experiencing, actually\u2014\"sequencing, patterns, inferences.\" She explains: \"You don't only listen to language, you _do_ it.\"\n\nBojorquez and Valleskey, like all teachers, function under the tyranny of the 9\/91 formula: between ages six and nineteen, a child spends 9 percent of his life in school, 91 percent elsewhere. In contemporary America, \"elsewhere\" means immersed in the undertow of popular culture's increasingly coarse distractions. In Los Angeles, where most public school pupils are Latino (Kennedy school is almost entirely Latino), \"elsewhere\" often means homes where English is barely spoken.\n\nBojorquez's raven-haired students, their dark eyes riveted on Valleskey, mimic her motions. These beautiful children have a beautiful hunger for the satisfaction of structured, collaborative achievement.\n\nThat begins when Valleskey, a one-woman swarm, bounces into the room and immediately, without a word of command, reduces the turbulent students to silent, rapt attention. They concentrate in order to emulate Valleskey's complex syncopation of claps, finger snaps, and thigh slaps by which she sets the tone of the coming hour: This will be fun _because_ things will be done precisely right.\n\nPart Marine Corps drill instructor, part pixie, Valleskey knows that children are realists. They do not want false praise. She knows that self-esteem is result of, not a precondition for, achievement. Her credo is: Every child can do it. The antecedent of \"it\" is: learn how to learn.\n\nHer students experience a kind of freedom that is, for most children, as exhilarating as it is novel. It is not merely the absence of restraints. Rather, it is the richer freedom of a cooperative group performing to high standards within a structure of rules.\n\nValleskey's California Dance Institute is, essentially, her and a few teaching assistants and musicians, sustained by a few exceptionally discerning philanthropists. CDI is associated with, but not financially supported by, the National Dance Institute, founded by Jacques d'Amboise, for many years a leading dancer with the New York City Ballet. He was the subject of the 1983 Academy Award\u2013winning film _He Makes Me Feel Like Dancin'._ It explored his insight that dance\u2014the pleasures of precision, of a task done just right\u2014serves all the pedagogic goals of schools.\n\nVirtues, says Valleskey, are habits, and dance, as taught by CDI, is habituation in many of the skills of learning, as well as the components of good character. Dance, properly taught, is like sport, properly understood.\n\nThe ancient Greeks considered sport _serious_ play, a civic\u2014meaning moral\u2014undertaking. It is because man's noblest activity is active engagement, as talented performer or informed spectator, with worthy things such as beauty. Including the beauty of strenuous exertion in conformity to exacting rules and high standards. By using our bodies beautifully, we come to appreciate beauty and the discipline\u2014the restraint\u2014that is its prerequisite and civilization's premise.\n\nGifted teachers like Bojorquez and Valleskey master the patience required for the unending business of transmitting civilization down the generations, transforming biological facts\u2014children\u2014into social artifacts called citizens. It is wearying work, and it is a wonder teachers can summon the stamina for it. Ralph Waldo Emerson wondered:\n\n\"It must be admitted that civilization is onerous and expensive; hideous expense to keep it up;\u2014let it go, and be Indians again; but why Indians?\u2014that is costly, too; the mud turtle and trout life is easier and cheaper, and oyster, cheaper still.\"\n\nCDI is inexpensive. Operating on a financial shoestring\u2014a frayed shoestring\u2014CDI is a gift to a few of this city's public schools. It makes one marvel at what educational improvements could be achieved with small sums in the service of something much scarcer than money\u2014imagination.\n\n[MARCH 25, 2004]\n**Chapter Six**\n\n**GAMES**\n\n# **Raising Michael Oher**\n\n**E** ven if you think football consists primarily of two regrettable elements of life\u2014violence, punctuated by committee meetings, called huddles. Even if you wince at institutions of higher education engaging in the low practice of exploiting young, often black, men who emerge, after four years generating revenues for campus football factories, unscathed by education. Even if you think that if the Watergate and Iran-Contra investigations had been really thorough they would have traced both scandals to connivings by college football coaches. Even if you think all those things, you are going to enjoy Michael Lewis's book _The Blind Side: Evolution of a Game._ Your enjoyment, however, will be tempered by dismay about some of what you learn.\n\nLewis tells an amazing true story in an appropriately mordant style, some samples of which are:\n\n\"When the coaches walked into the living room of the Tuohys' lovely Memphis home, the first thing they saw was the Rebel Christmas tree: red and blue branches festooned with nothing but Ole Miss ornaments.\"\n\n\"There were a number of colleges\u2014and Ole Miss was one of them\u2014for which the expropriation of the market value of pre-professional football players was something very like a core business.\"\n\n\"Hugh was a football coach and so he tended to take an indulgent view of bad grades.\"\n\nThe Evangelical Christian School \"was as close to a church as a school could get. E.C.S. wouldn't accept kids unless both parents gave testimony of their experience of being born again\u2014and the stories better be good.\"\n\nLewis's subject is the salvation of Michael Oher, a black child virtually raised on the mean streets of Memphis. But Lewis also continues what he began with _Moneyball,_ his 2003 bestseller explaining new thinking about how to construct baseball teams. He is advancing a new genre of journalism that shows how market forces and economic reasoning shape the evolution of sports. Oher, who today plays left offensive tackle for the University of Mississippi, is a valuable commodity because of the lasting impact on football made by someone who played on the other side of the ball.\n\nAfter the 1981 regular season, Lawrence Taylor, linebacker for the New York Giants, became the only rookie ever named the National Football League's defensive player of the year. He was six-foot-three, 240 pounds and quick as a cat, running forty yards in 4.5 seconds. He was, Lewis says, \"a new kind of athlete doing a new kind of thing.\" This is how Taylor described his thing:\n\n\"I'll drive my helmet\" into the quarterback, \"or, if I can, I'll bring my arm up over my head and try to ax the sonuvabitch in two. So long as the guy is holding the ball, I intend to hurt him.... If I hit the guy right, I'll hit a nerve and he'll feel electrocuted, he'll forget for a few seconds that he's on a football field.\"\n\nTerrifying, disorienting, and injuring quarterbacks is most of a linebacker's job description. Taylor concentrated coaches' minds on the problem of protecting quarterbacks. Most of them are right-handed, which means that when they are passing, threats from their left come thundering at them from their blind side. Hence the sudden interest in large and agile left offensive tackles. This interest intensified when Bill Walsh, coach of the San Francisco 49ers, devised the West Coast offense. It floods the secondary with receivers. More receivers mean fewer pass blockers, so the left tackle has more problems to cope with.\n\nWhich is why by 2004 the average salary of an NFL left tackle was $5.5 million a year, second only to the average for quarterbacks. The five most highly paid left tackles were earning almost $3 million more than the five most highly paid right tackles. That is very good news for Oher, one of thirteen siblings from the nation's third-poorest ZIP code.\n\nHe had finished the ninth grade, but the tenth was unlikely, and he was on track to be selling drugs, en route to jail or an early grave. His father was long gone, his mother was in and out of drug rehab. In his file from the public school system\u2014he had been in eleven different schools in nine years\u2014the numbers looked, Lewis says, \"like misprints.\" His measured IQ was 80; his \"ability to learn\" placed him in the sixth percentile. He had repeated first and second grades. The school system said that through the fourth grade he had performed at \"grade level,\" which is odd because he never attended third grade. He was almost nonverbal. Essentially unparented throughout his childhood, he scavenged for clothes, slept here and there in (sometimes slightly) less disorganized households than that of his mother, who when Michael was five was caring for seven boys and three girls, all under fifteen. Caring, that is, in her fashion. And not at all for ten or so days after the first of each month, when her welfare check arrived and she disappeared to feed her addiction to crack cocaine. On one occasion, lasting for weeks, his mother and seven boys slept in an old Chevy, washing themselves in a service station bathroom. For five years he lived in various households in a housing project with about a thousand residents and \"no two-parent families: zero.\" It is unsurprising that Oher, according to Lewis, became a teenager who \"didn't know what an ocean was, or a bird's nest, or the tooth fairy.\"\n\nBut the father of a friend, seeking a Christian education for his son, took Oher along on a visit to one of the many private schools created after the policy of forced busing, intended to achieve racial balance, provoked the departure of seven thousand children from Memphis's public schools. Oher came to the attention of Sean and Leigh Anne Tuohy, parents of children at one of these schools. They are white; they are the religious right.\n\nTaco Bell franchises had made Sean a lot of money. Nature had made Leigh Anne into a human firecracker, exploding with energy. Their evangelical Christianity made them receptive to the possibility of redemption in the here and now. And attending Ole Miss had made them football-crazy. They became Oher's legal guardians. (One of Leigh Anne's cousins called late one night: \"All right, I've just had my fifth beer. Who the hell is this black kid in y'all's Christmas card?\") They got Oher, with his fifty-inch waist, into a size-58-long sport coat and into the Briarcrest Christian School.\n\nWhen Oher wandered into the school gym, a coach tossed a basketball to him, expecting him to take it to the hoop, as a six-foot-five boy might, or kick it into the stands, as a 344-pound boy might. Instead, Oher caught it, dribbled three times between his legs, spun and drained a three-point shot from the corner of the floor. A human mountain with moves like a point guard? Two words spring to mind: left tackle.\n\nGetting Oher his high school diploma became quite a project for the Tuohys. No one in Oher's family had ever had even a driver's license. When Leigh Anne told him to get his backpack from the foyer, he had no idea where that was. But with the help of tutors, by the first semester of his senior year he had risen to 162nd in a class of 163. \"He's picking them off one at a time,\" Sean crowed, \"like Sergeant York.\" The Tuohys found some interesting ways of getting high school credits off the Internet. (Lewis calls the Brigham Young University program of correspondence courses \"the great Mormon grade-grab.\" Sean, evangelical but broad-minded, says: \"The Mormons may be going to hell. But they really are nice people.\") Anyway, Oher satisfied NCAA criteria for college eligibility.\n\nOher's childhood of grinding deprivation might have put him on a path to riches. Lewis commits some perhaps dubious sociology when he declares that a miserable ghetto childhood can be excellent preparation for football: \"It made you angry, it made you aggressive, it made you want to tear someone's head off. The N.F.L. was loaded with players who had mined a loveless, dysfunctional childhood for sensational acts of violence.\"\n\nBe that as it may, there came a day when a bevy of assistant football coaches from the Southeastern Conference, the Big Ten, the Atlantic Coast Conference, and elsewhere came to a Briarcrest practice. There Oher and a wee 270-pound teammate faced off, one on one, for a demonstration collision. In an instant, Oher effortlessly shoved his teammate down the field. In a flash, the coaches fired up their cell phones. Lewis says the Clemson coach rushed up to Oher's coach, saying, \"I seen all I need to see,\" and added that Oher could have a full scholarship. Later, Tennessee's head coach watched Michael for half an hour and pronounced him the nation's best. A national recruiting frenzy had begun, from schools in states Michael could not find on a map.\n\nWhen Ole Miss introduced its new football coach, he announced that his first goal was to recruit Oher. The coach barged into the Tuohys' living room, took a gander at Oher, and exclaimed, as Lewis renders it, \"YAAAWWW BEEE BAAWWW!\" (\"You a big boy!\") Michael could not understand a word he said. That did not matter. He wanted to go to Ole Miss, where his newfound parents had gone.\n\nThere Sean Tuohy told the coach, Ed Orgeron, that Oher would have trouble learning plays from a book full of X's and O's, but could learn the plays if they were presented visually, using mustard bottles and ketchup bottles to represent players. Lewis reports:\n\n\"'Coach,' said Sean. 'My faith believes that the Lord sends down gifts for everyone and our job is to find those gifts. Michael's gift is the gift of memory. When he knows it, he knows it.'\n\n\"Coach O stopped scribbling and looked up. 'I'm going to tell you one thing, Sean,' he bellowed. 'He's got some pretty good [expletive] feet, too. You seen them feet? Now them feet: that's a [expletive] gift!'\"\n\nOher was not a whale out of water at Ole Miss. Lewis says that the typical football player in Michael's college class \"had third-grade-level reading skills. Several had never taken math. Ever.\" Michael's three closest friends among his Ole Miss teammates had children. One had become a father at fifteen. Michael brought a teammate, Quentin Taylor, to the Tuohys' home for Thanksgiving, and Taylor mentioned that he had fathered three children by two different mothers. Lewis writes:\n\n\"Leigh Anne pulled the carving knife from the turkey and said, 'Quentin, you can do what you want and it's your own business. But if Michael Oher does that I'm cutting his penis off.' From the look on Quentin's face Michael could see he didn't think she was joking.\"\n\nShe probably wasn't. She and Sean seriously, ferociously, implacably care about Oher. They were the first adults to do so.\n\nLast season Michael was named to the first-team freshman all-American team\u2014as a \"true freshman.\" That is college-football-speak for a player who is not \"redshirted.\" And that is college-football-speak for holding a player out of competition for a year while he spends time in the center of his academic life\u2014no, silly, not the library, the weight room.\n\nOn Friday nights, American high schools discover talent that universities refine on Saturdays, for the nation's eventual delight on Sundays. Oher's story is not pretty, but Lewis tells it well\u2014and against all odds, it may be heading for a happy ending.\n\n\u2014NOVEMBER 12, 2006\n\n# **The Man from Moro Bottom**\n\n**O** n January 26, 1983, phone service throughout area code 205, which then included all of Alabama, crashed from overload. Was the cause a natural disaster? Yes. Something very natural\u2014death\u2014had claimed the University of Alabama's football coach.\n\nAllen Barra's illuminating book _The Last Coach: A Life of Paul \"Bear\" Bryant_ explains why Alabamians felt so bereft. It also answers a question especially pertinent since Thomas Herrion, twenty-three, a 315-pound lineman for the San Francisco 49ers, died in August of a previously undetected heart disease: Has football become grotesque?\n\nAfter Bryant became a coach, and to his regret, football players became specialists\u2014often dangerously large specialists. In 1964, all limits on substitution ended, bringing the virtual extinction of players who played \"on both sides of the ball\"\u2014both offense and defense. Some teams swelled to more than 130 players until the NCAA cut scholarships to only\u2014only!\u201485. This was the end of the \"eleven men and sic 'em\" football favored by the man who earned his nickname when, at the age of fourteen, he wrestled a bear.\n\nFor some Alabamians, Barra says, September 11 means the day in 1913 when Bryant was born in a place\u2014it was not a town\u2014called Moro Bottom, Arkansas. He played at Alabama and got most of his then record 323 victories there, where he won six national championships\u2014as many as the top three active coaches combined. He would have won a seventh in 1966 if the country, including those who vote on team rankings, had not been so angry about the only Alabamian more famous than Bryant\u2014Governor George Wallace.\n\nBut football helped change the face of the South. Before the 1963 Orange Bowl, President Kennedy visited the locker room of the integrated Oklahoma Sooners, but not Alabama's. The 1969 Texas Longhorns were the last all-white team voted national champions.\n\nBryant\u2014\"My players are athletes first and students second\"\u2014had an agreeable aversion to hypocrisy and cant. He told players: \"Ten years from now you are going to be married with a family, your wife might be sick, your kids might be sick, you might be sick, but you will get your butt up and go to work. That's what I'm going to do for you. I'm going to teach you how to do things you don't feel like doing.\"\n\nBryant understood what football meant to the South. The Rose Bowl had been reluctant to invite Alabama to play in 1926, the era of Erskine Caldwell's novels about rural Georgia, _Tobacco Road_ and _God's Little Acre._ Alabama's victory over Washington occasioned, Barra says, \"the greatest statewide celebration since the shelling of Fort Sumter.\"\n\nThe University of Alabama's enrollment actually increased during the Depression partly because it welcomed Northeastern Jewish students who were excluded by quotas from many prestigious Northern schools. In the 1960s, the South's most turbulent decade since the 1860s, Alabama dominated college football, and because an ABC television prodigy named Roone Arledge knew charisma when he saw it, Bryant became the craggy face of college football.\n\nAlso in the 1960s, unlimited substitution began making huge players practical as offensive or defensive specialists. Barra notes that Bryant's 1966 team \"looked like an average high school team today.\" It went 11\u20130 and then won the Sugar Bowl. It had only fourteen players who weighed more than 200 pounds. The two heaviest weighed 213. The linemen averaged 195. The quarterback weighed 175.\n\nToday, Scouts, Inc., reports that nearly 40 percent of the interior linemen who will go to Division I colleges in September 2006\u2014many of these players not yet eighteen\u2014already weigh at least 300 pounds. In 1980, only one NFL player topped 300. In 1994, the year a mortality study found that linemen have a 52 percent greater risk of dying from cardiovascular disease than the general population and that the largest players have six times the risk of cardiac death than normalsize players, the number of 300-pounders was 155. Ten years later, 370 NFL players exceeded 300, and 10 exceeded 350.\n\nThis season, the offensive lines of thirty of the thirty-two NFL teams average at least 300 pounds, and one team averages 323. Of the sixty-one offensive college linemen invited to last February's NFL Scouting Combine, fifty-eight weighed at least 300. Of the three little fellows, one weighed 299 and two weighed 298.\n\nAfter eighteen college players died in 1905, President Theodore Roosevelt\u2014it took serious carnage to cause that cowboy and warrior to flinch\u2014compelled rules changes to make football safer. Today, it is unhealthy because of the kinetic energy involved in collisions between huge men\u2014and because of what they do to become huge. Not coaching football was unhealthy for Bryant. \"If I quit coaching,\" he frequently said, \"I'll croak in a week.\" He died twenty-eight days after his last game.\n\n[NOVEMBER 7, 2005]\n\n# **\"Rammer Jammer Yellowhammer!\"**\n\n**D** on Cole, aka the Heart Guy, was ailing and wore a beeper. He was a candidate for a heart transplant and was not supposed to ever be more than a two-hour drive from his Nashville hospital, in case it received a heart that could be transplanted. He said that if the hospital learned that he left the two-hour radius he would be removed from the list of recipients. So why, weekend after weekend, was he three and a half hours from Nashville, in Tuscaloosa, Alabama? \"If I can't go to Alabama football games, what's the point in living?\"\n\nThen there is the couple whose huge RV resembles the fuselage of a Boeing 737. What sacrifices have they made for their devotion to Alabama football? \"Let's see,\" muses the husband. \"We missed our daughter's wedding. We told her, just don't get married on a game day and we'll be there, hundred percent, and she went off and picked the third Saturday in October which everybody knows is when Alabama plays Tennessee, so we told her, hey, we got a ballgame to go to. We made the reception\u2014went there as soon as the game was over.\"\n\nYou can meet these folks and others of their tribe in _Rammer Jammer Yellowhammer: A Journey into the Heart of Fan Mania,_ a hilarious\u2014and a little bit scary\u2014book by Warren St. John, a New York writer who was born in Alabama. A few years ago, he returned to try to figure this out: \"Why do I care?\" About sports, that is.\n\nTo plumb the depths of the human fascination with contests, St. John went for total immersion. He spent a football season with the seriously hard-core fans. They are the ones for whom the phrase \"Roll Tide\" is an all-purpose exclamation-incantation-salutation. They are the purchasers of official Alabama coffins (red, with the school logo on the top and a white velvet \"A\" sewn into the lid\u2014$1,999). These fans travel from game to game in their $300,000-to-$1,400,000 RVs, turning game day into a three-day festival of cold beer, artery-clogging broiled bologna sticks, 'Bama Bombs (Maraschino cherries soaked in PGA\u2014pure grain alcohol) and sacramental events like the Bear Bryant Namesake Reunion at Tuscaloosa's Bryant Museum.\n\nThe museum, which contains such relics as the jacket and slacks one fan wore to his five hundredth consecutive Alabama game, is across Bryant Drive from where the RVs gather, near Bryant-Denny Stadium and Bryant Hall, a dorm, and not far from Paul William Bryant High School. The reunion is for people named for Bryant, who coached the Tide to six national championships. There are almost six hundred such people.\n\nAlabama's obsession with football began, in a sense, on January 1, 1926, five months after the Scopes trial\u2014about teaching evolution\u2014gave the South's despisers fresh ammunition. On that day Alabama's Crimson Tide became the first team from the South to play in the Rose Bowl. The Tide won, 20\u201319. The South really would rise again.\n\nBut in every region, sport can produce a collective mind, or sometimes a collective setting aside of mind. In 1895, a French psychologist published _The Crowd: A Study of the Popular Mind,_ pioneering the study of the mental unity of crowds\u2014people in the same frame of mind. \"Freud,\" writes St. John, \"disparaged crowds as neurotic on the grounds that, like neurotics, crowds 'demand illusions, and in fact can't live without them' and 'are guided not by ordinary objective reality but psychological reality.'\"\n\nHowever, St. John believes in using Occam's razor\u2014that is, in first trying the simplest explanation of a phenomenon: \"We can't paint our faces and scream like maniacs at our desks, in the classroom or at the dinner tables with our families, so...\"\n\nWell, then, why _does_ St. John care about the Tide? \"I chose Alabama the way a baby bird chooses its mother: it was the first thing I saw.\" We all acquire such allegiances, but there also is a regional twist to this. For Southerners, the myth of the Lost Cause is all very well, but winning is nice, too.\n\nSo try to think anthropologically about those 'Bama fans who fire up their RVs, break out their radar detectors, and sing \"Rammer Jammer Yellowhammer! Give 'em hell Alabama!\" They are not just emulating the RVer who said: \"We can't be young but we can be immature.\" They are pursuing what sportswriter Frank Deford called \"that curious Southern combination of eternal knighthood and childhood.\" Roll Tide.\n\n[NOVEMBER 21, 2004]\n\n# **Randy Shannon's Realism**\n\n**M** IAMI\u2014Occasionally\u2014 _very_ occasionally\u2014a football person says something that punctures the fog of George Patton\u2013style rhetoric that football people emit. Before a Super Bowl in the 1970s (the MCMLXXs, for those of you in a Super Bowl frame of mind), Dallas Cowboys running back Duane Thomas asked a subversive question about the game: \"If it's the ultimate, how come they're playing it again next year?\"\n\nBut most football people, and especially football coaches, are of the \"Football Is Not a Matter of Life and Death\u2014It's More Important Than That\" school of thought. However, when Randy Shannon, recently named head coach of the University of Miami Hurricanes, says that football can be a matter of life and death, that is not hyperbole, it is autobiography.\n\nShannon, forty, grew up in Miami's Liberty City, which is what sociologists and other refined thinkers call a challenging urban environment. Shannon was three when his father was murdered by one of his friends. \"They had an argument,\" Shannon says matter-of-factly. Two of Shannon's brothers and a sister died, from cocaine and AIDS. By age sixteen, Shannon was a father. He could easily have been on a glide path to a prison or a cemetery. Instead, because of football, he went to the University of Miami and became the first member of his family to earn a college degree.\n\nAfter a brief NFL career with the Cowboys, he went into coaching, and now he is hopscotching around the country recruiting high school seniors, many of whom think college football is a certain path to the NFL. \"That,\" says Shannon, \"is the mentality that has to change.\" Less than 2 percent of even Division I college football players will have NFL careers, and most of those who do will be out of the game by the time they are thirty\u2014the average NFL career lasts less than four years.\n\nThe _Washington Post_ 's Amy Shipley reports that the University of Miami has more players\u2014forty-two\u2014on NFL rosters than any other school. Miami's main rival\u2014the Florida State Seminoles (a T-shirt favored by Miami students reads: \"I think, therefore I am not a 'Nole\")\u2014is second with forty-one. The University of Florida ranks seventh with thirty-five. But in the last ten years, those three teams have had, combined, more than one thousand players, all of them exceptional athletes but most of them not of NFL caliber. Which is why Shannon says that when visiting the home of a potential recruit, \"I talk to the parents about everything but football.\"\n\nOn a recent day, Shannon was in the Palm Beach area recruiting a wide receiver, and then was off to Omaha to make sure that a very large lineman was still eager to be a Hurricane.\n\nBecause South Florida is the incubator of so much high school talent (skill positions, Shannon says; for linemen, look to the Midwest, hence the Nebraska trip), during the off-season many NFL players come home to train at the University of Miami's facilities. Shannon says his players \"see the fancy cars, the gold chains,\" so as he takes over Miami's football program, he plans to \"come in with a stern attitude.\"\n\nStern adults got Shannon to the peak of his profession at age forty. A fourth-grade teacher told him, \"You're very smart\u2014don't let anyone tell you different.\" A fifth-grade teacher, disapproving his choice of clothes one day, said, \"Don't ever come to school like that again.\" When he was in junior high school, his football coach took the team to play a team in a juvenile detention center, a sobering experience.\n\nShannon's rules for his players include: If you miss a class, you don't start the next game. Fall below a certain grade point average, you can't set foot off campus. A conservatively dressed man, with the elegant hands of a surgeon or pianist, Shannon wants his players to learn \"how to respect life,\" so when \"they leave the university and the football program, they will go with confidence.\" They will go, all of them, having taken a public speaking course.\n\nDuffy Daugherty, who coached Michigan State from 1954 through 1972, was an aphorist (\"Football is not a contact sport, it's a collision sport. Dancing is a contact sport\") and a realist. Because of alumni demands for football perfection, Daugherty said: \"A football coach's main problem is that he is responsible to irresponsible people.\" Shannon, who like 80 percent of his players is African-American, feels responsible to, and for, them.\n\n[FEBRUARY 4, 2007]\n\n# **The NFL: An Intensification of Reality**\n\n**A** fat lot Keats knew about autumn. \"Season of mists and mellow fruitfulness\"? Fiddlesticks. It is football season, the distilled essence of modern life.\n\nIt is sex (pneumatic cheerleaders), violence (when the 1976 Super Bowl made the Dallas Cowboys cheerleaders famous, a CBS producer said, \"The audience deserves a little sex with its violence\"), technology (quarterbacks electronically instructed by coaches wired to subordinate Merlins in the upper reaches of the stadium), committee meetings (huddles), division of labor (interior linemen specializing in third-and-short yardage situations), jargon (zone-flooding nickel packages and seam-splitting nose tackles, or something like that), and a hallmark of a commercial society\u2014strategic parsimony about time.\n\nWelcome to the National Football League, a cultural artifact that causes thinkers to commit sociology. Michael MacCambridge plumbs these depths in his fine new book, _America's Game: The Epic Story of How Pro Football Captured a Nation._ It is a rip-roaring epic of American business.\n\nIn 1920, eleven men met in a Canton, Ohio, Hupmobile showroom and assessed each other $100 franchise fees. In 1999, the fee for the Houston Texans franchise was $700 million. The eight-year, $17.6 billion TV deal signed in 1998 pays each club $84 million this year. The NFL has come a long way since the Philadelphia Eagles\u2014named after the symbol of FDR's National Recovery Administration\u2014traveled by train to New York on game day to avoid hotel expenses and ate at Horn & Hardart Automats.\n\nIn 1952, the Chicago Bears\u2013Dallas Texans Thanksgiving Day game was moved from Dallas to Akron in quest of better attendance\u2014and drew just 3,000 fans to a field where, that morning, 14,800 had attended a high-school game. But a new appliance was coming, and soon the NFL supplanted boxing as the sport whose compact action seemed most suited to television screens. Of which there were only fourteen thousand in 1947 but 26 million in 1954.\n\nBy 1980, the League of Women Voters had to beat a hasty retreat from scheduling two presidential debates on Monday nights. (Carter might have been reelected if that year's debate had been up against _Monday Night Football._ Few would have watched.) By the 1990s, the NFL had the power to transform Fox into a major player among the networks.\n\nOn the field, unlimited substitution was restored in 1950 and made most of the swarming players seem like interchangeable parts in large machines. That, and competitive balance, a product of equal team shares of the dominant source of revenue (national television contracts), led to the apotheosis of head coaches, and especially to the cult of the Packers' Vince Lombardi. Both Richard Nixon and Hubert Humphrey considered him as a running mate in 1968.\n\n\"The period of the early '70s,\" writes MacCambridge, \"brought an odd sense of cognitive dissonance to pro football's rise. At no other time in its history did the guiding ethos of football\u2014teamwork, self-sacrifice, the concerted application of mental and physical discipline toward a single, united goal\u2014seem more out of step with the larger cultural moment.\"\n\nThe NFL, with its aversion to understatement, flourished in the 1960s and 1970s, when cultural change was accelerating, and the tone setters in American sports were Muhammad Ali and Howard Cosell. Soon the NFL produced the first black celebrity featured in a national corporate advertising campaign, for Hertz. O. J. Simpson. Oh, well.\n\nThis NFL season will reach a climax with the XXXIXth Super Bowl. (Roman numerals for gladiatorial spectacles.) It will be watched by many millions more Americans than will have watched the presidential inauguration seventeen days before. The fourth Super Bowl, in January 1970, was watched by more people than had watched Neil Armstrong walk on the moon six months earlier. The ten-most-watched television programs in history are all Super Bowls. Most viewers have financial stakes in the outcomes, having bet on them. Super Bowl Sunday is second only to Thanksgiving in Americans' caloric intake. \"If Jesus Christ were alive today,\" said Norman Vincent Peale in 1974, \"he'd be at the Super Bowl.\" But surely not in a luxury suite.\n\nThe best thing about NFL teams is the purity of their professionalism. None are appendages of institutions of higher education, so there is no damned nonsense about \"student athletes.\" When in 1957 Queen Elizabeth attended a Maryland\u2013North Carolina game, she asked Maryland's governor, \"Where do you get all those enormous players?\" He replied, \"Your Majesty, that's a very embarrassing question.\"\n\nIn _Sports Illustrated_ 's recent fiftieth-anniversary issue, Jeff MacGregor wrote, \"Organized sports are the perfection of the unnecessary.\" Perhaps. But, then, most of what makes life sweet involves emancipation from necessity. The NFL is an acquired taste that Americans have acquired less as an alternative reality than as an intensification of modern reality, although why they want that is a mystery.\n\n[OCTOBER 11, 2004]\n\n# **Speaking SportsCenterese**\n\n_If we had ESPN twenty-two years ago, we wouldn't have any children._\n\n\u2014A COLLEGE COACH, 1990\n\n**Y** ou are in a ballpark with your twelve-year-old. The shortstop makes a sparkling play and your child murmurs, \"Web gem.\" As a slugger approaches the plate, your child says, with a hint of drollery, \"You can't stop him, you can only hope to contain him.\" When the slugger hits one four hundred feet, the child says, \"That'll make the Top Ten Plays.\"\n\nCongratulations: Your child is bilingual. He or she speaks SportsCenterese, the lingua franca of ESPN nation, the capital of which is Bristol, Connecticut, where twenty-seven satellite dishes scarf up forty thousand feeds a year, the best of which are sent around the clock to sports addicts, such as the viewer who, in 1987, said: \"Please show the Nebraska-UCLA game at 6:00 as I have a 5:00 Mass and would have to find a priest to replace me if you show it earlier.\"\n\nESPN will be a quarter-century old on September 7. Measurements of \"brand resonance\" show that among 138 brands, including Coca-Cola and McDonald's, ESPN ranks first among men. Each week more than 90 million people are exposed to ESPN media\u2014ESPN (there are locally produced SportsCenters in Canada, Brazil, a Spanish version for the rest of Latin America, China, India and Taiwan), ESPN2, ESPN Classic, ESPN.com (2.3 million page views in a peak hour), and _ESPN The Magazine_ (a circulation of 1.7 million in just five years).\n\nThis stunning growth reflects ways America has changed in a quarter of a century. The change can be measured in money.\n\nIn 1979, when the Entertainment and Sports Programming Network began, the average major league baseball salary was $113,558 and pitcher Nolan Ryan became the first million-dollar-a-year athlete in team sports. Today the average baseball salary ($2.55 million) has increased 2,241 percent and there are 1,702 million-dollar athletes. In 1979, broadcasters paid the National Football League $8.8 million annually; today, the fee is approximately $2.25 billion, an increase of more than 25,400 percent.\n\nAmerica is a lot richer than it was in 1979, but not _that_ much richer. Something else is afoot, turning so many eyes\u2014that is what pulls the tide of money\u2014to sports. Perhaps people are drawn to sports because they really don't mean a thing. In this politicized age, even\u2014no, _especially_ \u2014cultural arguments are political arguments. Politics is understood as a series of angry confrontations, and war (on drugs, poverty, illiteracy, etc.) is a metaphor for policy. Perhaps, then, sports delight because they are a refuge\u2014one of society's few meaning-free zones.\n\nOr not. Perhaps there is an opposite explanation for the unslakable appetite for the spectacle of sport, an appetite that has produced ESPN.\n\nMichael Mandelbaum, author of eight books on international relations, argues in his ninth book, _The Meaning of Sports,_ that sports are \"a variety of religious experience.\" Like religion, sports stand apart from the mundane and are a realm of special coherence and heroic example. The rise of team sports coincided with what Mandelbaum calls the twentieth century's \"social and political hurricanes.\" Those were urbanization\u2014people moving from countryside to town and from job to job\u2014and world wars, unprecedented confusions and traumas from which people sought diversions. The twentieth century, Mandelbaum writes, \"was the era of free verse in poetry, stream-of-consciousness writing in literature, atonal music in place of traditional harmony and melody, and abstract rather than figurative art. James Joyce succeeded Charles Dickens, Jackson Pollock filled the place Rembrandt had occupied.\"\n\nAt a time when Robert Frost was comparing free verse to playing tennis without a net, sports became cultural counterpoints because they are transparent and coherent. Transparent because spectators can see for themselves what is happening, and why. Coherent in that they are defined and governed by rationality\u2014rules\u2014and reach definitive conclusions. It is surely not mere coincidence that sports and detective novels found mass audiences simultaneously.\n\nThese clues to the mystery of ESPN's remarkable success may assuage any guilt you feel about the time you spend with the boys and girls from Bristol. But don't get carried away. There has been at least one ESPN divorce in which the wife gave to her husband an improvident ultimatum: It's ESPN or me. In at least ten harmonious marriages, the parents have named children ESPN, Espn, Espin, or Espyn. How many children have been named HBO or CNN?\n\n[SEPTEMBER 7, 2004]\n\n# **The Movie, and the Truth, About Texas Western**\n\n**A** Division I college basketball program is not the sort of enterprise easily confused with a seminary or a seminar on ethics. But according to what is currently America's most popular movie, forty years ago one such program became a nation-shaking, history-shaping moral force. The movie, although not too noble to palter with facts, is no more parsimonious with the truth than movies often are when turning history into entertainment.\n\n_Glory Road_ celebrates the 1965\u20131966 basketball team of Texas Western College (which in 1967 became the University of Texas at El Paso). The Miners included seven black players, most recruited far from mining country\u2014the South Bronx, Gary, Indiana, and other mostly urban places. The drama was that five of them started the 1966 NCAA championship game that Texas Western won, beating an all-white University of Kentucky team, 72\u201365.\n\nThe game was not quite, as the movie insists, David against Goliath. Granted, the Kentucky Wildcats, then college basketball's aristocrats, were college basketball's winningest team in the 1940s and 1950s. But Texas Western had lost only one game and was ranked third in the nation as the tournament began.\n\nThe game's racial dimension looks much larger in retrospect than it did then. In the movie, a Texas Western official urges coach Don Haskins to abide by an unwritten rule: Play one black at home and two on the road\u2014three if behind. And another white character scoffs at the idea that blacks might be \"the future\" of basketball. But Ron Rapoport of the _Chicago Sun-Times_ notes:\n\n\"A decade before the game that supposedly changed basketball, the undefeated 1955\u20131956 University of San Francisco team won the NCAA championship with a team that played four blacks\u2014Bill Russell, K. C. Jones, Hal Perry, and Gene Brown. In 1958, the coaches' all-American team was all black\u2014Wilt Chamberlain of Kansas, Oscar Robertson of Cincinnati, Bob Boozer of Kansas State, Guy Rodgers of Temple, and Elgin Baylor of Seattle. In 1962, the University of Cincinnati started four black players when it won the NCAA championship, and Loyola University of Chicago started four when it won in 1963. Frank Deford, a distinguished writer, covered the Texas Western\u2013Kentucky game for _Sports Illustrated_ and did not _mention_ the fact of five black starters. Neither did the _New York Times_ nor the _Washington Post._ Already the ascendancy of blacks in basketball was such that the four best players in the NBA were Chamberlain, Russell, Baylor, and Robertson.\"\n\nIn the movie, Haskins tells his team the day before the game that he will play only black players the next night\u2014he used all seven\u2014in order to make a social statement. But former Georgetown coach John Thompson, a black man famous for his bluntness, minced no words when talking to Eddie Einhorn for a book, _How March Became Madness,_ a history of the NCAA tournament, that Einhorn is publishing next month (with Rapoport's collaboration). Thompson told Einhorn that Haskins said his only goal was to win, so he played his best players.\n\nAnd what of the movie scene where the players' motel rooms are trashed and racist epithets are painted on the walls? One of the players, Nevil Shed, recently told _Sporting News_ columnist Dave Kindred, \"Could have happened.\" Kindred calls that Shed's way of handling \"the fiction.\"\n\nAlthough the movie shows Haskins emphasizing basketball fundamentals and telling the players that \"showboating is nothing but insecurity,\" the movie also makes much of the black players successfully seeking his permission for the more flamboyant style of play they learned on city asphalt. This much is true: Between 1967 and 1976, the NCAA banned dunk shots, even during warm-ups. What do you suppose _that_ was about?\n\nIn his just-published _At Canaan's Edge: America in the King Years, 1965\u201368,_ Taylor Branch writes that when in 1950 Kentucky lost to City College of New York's integrated team, Kentucky's legislature flew the flag at the capitol at half-staff. Two months after the 1966 championship game, a black player received an athletic scholarship from one of Kentucky's Southeastern Conference rivals, Vanderbilt. Kentucky's coach, Adolph Rupp, was born in 1901 and probably was not much different than his peers in his time and place. According to Branch, Rupp \"complained of incessant calls from his university president: 'That son-of-a-bitch wants me to get some n\u2014\u2014in here. What am I gonna do?'\" But Kentucky had no black professor until 1965.\n\nWhen Rupp retired in 1972, his team was all white. Today, Kentucky has a black coach, Tubby Smith, whose fifteen-man team includes ten blacks. They play in Rupp Arena.\n\n[JANUARY 22, 2006]\n**Chapter Seven**\n\n**_T HE_ GAME**\n\n# **\"Remember 1908!\"**\n\n**C** hicago Cub fans, that numerous and inexplicable cohort, have a weird rallying cry: \"Remember 1908!\" Not one of them really does remember that season, the last time the Cubs won the World Series. That is all the more reason for them to join Cait Murphy on her jaunty walk through that tumultuous season. All other baseball fans should tag along. So should anyone interested in the rough texture of this bumptious nation in the early twentieth century, when twenty-five cents\u2014not a piddling amount for a low-skilled factory worker making $7 a week\u2014would get you into a ballpark where whiskey, waffles, and pigs' knuckles were served.\n\n_Crazy '08_ is a walk on the wild side: Brooklyn fans on rooftops would hurl sharpened umbrella shafts at visiting players in the outfield. When only boxing and horse racing competed with baseball for the public's attention, baseball stirred tribal feelings.\n\nBaseball fans relish arguments about which was the greatest this or that\u2014greatest game, team, left-handed right fielder, right-handed left fielder, whatever. Murphy will ignite a dandy rhubarb with her subtitle: _How a Cast of Cranks, Rogues, Boneheads, and Magnates Created the Greatest Year in Baseball History._ The author, an assistant managing editor at _Fortune_ magazine, makes a powerful case for those last six words.\n\nIn 1908\u2014the year a play titled _The Melting Pot_ put that phrase into the American lexicon\u2014Americans were unmelted. When the best player in the game, Honus Wagner, came to bat, a band might break into \"Wacht am Rhein.\" When John McGraw's Giants visited Springfield, Illinois, which had recently experienced a hideous race riot\u2014the NAACP was born partly in response to it\u2014McGraw was given, as a souvenir, a piece of the rope used to lynch a black man. Murphy reports that McGraw said the rope would replace a rabbit's foot as the Giants' good-luck emblem.\n\nThat year, construction began on the first fireproof (concrete and steel) ballpark, Shibe Park in Philadelphia. With its terra-cotta casts and copper-trimmed roof, it embodied the City Beautiful movement's belief that attractive buildings would uplift the downtrodden. Furthermore, 1908 gave the world the greatest piece of music since Mozart (\"Take Me Out to the Ball Game,\" of course) and an audacious and successful bit of flapdoodle (the campaign to convince the gullible that Cooperstown was the birthplace, and Abner Doubleday the father, of baseball).\n\nBetween the white lines, baseball in 1908 also included:\n\n\u2022 Two pennant races in which a total of six teams were in contention with two days left.\n\n\u2022 The finest (Murphy says so; she does like laying down the law) pitching duel in baseball history. Ed Walsh, whose record in 1908 would be 40\u201315 with a 1.42 ERA, struck out fifteen\u2014at that time a record for a nine-inning game\u2014and allowed only one run, which was unearned. But he lost because Addie Joss used just seventy-four pitches to throw a perfect game.\n\n\u2022 What was perhaps the best season any National League player would have in the twentieth century. The Pittsburgh Pirates' Honus Wagner led the league in almost everything\u2014you can look it up. \"There ain't much to being a ballplayer,\" he said, \"if you're a ballplayer.\"\n\n\u2022 The only doubleheader in which one pitcher pitched every inning and threw two shutouts.\n\n\u2022 A steal of first base. (A runner on first stole second, hoping the runner on third would score on the throw to second. But the catcher did not throw. So on the next pitch the runner on second ran back to first. Then he stole second again.)\n\nWhen crucial games were being played, tens of thousands around the nation packed concert halls and blocked streets to watch large electric scoreboards relaying telegraph information of the games' progress, batter by batter. Murphy provides delightful samples of 1908 baseball writing for newspapers: \"There was a sharp report as Tommy caught the pellet squarely on its proboscis and sent it screeching toward the distant middle.\"\n\nIn 1908, Murphy writes, the average player's salary was $2,500\u2014three times as much as what Chicago paid a primary-school teacher with seven years' experience. In 1910, almost a quarter of major leaguers had some college education, compared with 5 percent of the population. But it was not until the late eighties that Pennsylvania, with its history of mills and mines, was surpassed by sunny California as the incubator of the most big-league players. For many men\u2014the kind who poured whiskey on spike wounds\u2014baseball a century ago was a way to avoid life sentences of hard labor, so they played with grim intensity. In 1907, a player was \"beaned so badly that he was given last rites on the field.\" (He survived.)\n\nThe umpire\u2014often there was only one\u2014was given three balls at the beginning of the game. If these did not suffice, the home team was required to supply balls. If the home team was ahead, those supplied were apt to be old and lifeless. Murphy gives this example of how, in a pennant race decided by a one-game margin, the Cubs stole a game because the umpire Cy Rigler, working alone, was calling balls and strikes standing behind the pitcher:\n\n\"In the fifth inning, the Cubs attempt a double steal with men on first and third and two out; Rigler turns to call the runner out at second. The run scores if the runner on third, Johnny Kling, touches home before the out is recorded. Rigler has no way of knowing when or even if Kling crosses the plate in time. He simply flips a mental coin and admits the run\u2014and the Cubs beat the Cardinals 4\u20133. Even the Chicago press admits that Kling was several steps short.\"\n\nBetween the Cubs, who were then a dynasty (no other team has ever won 530 games in five years, as the Cubs did from 1906 to 1910), and the New York Giants, this was, as Murphy says, an \"era of really bad feelings.\" On September 23, with the Giants leading the Cubs by half a game, the Giants' regular first baseman woke up with lumbago, so Fred Merkle, nineteen, got his first start of the season. After the tumultuous ninth inning, he would forever be known as Bonehead Merkle. Murphy's reconstruction of the jaw-dropping confusion that effectively sent the Cubs to the World Series is lucid and hilarious, and justifies her assertion\u2014baseball fans do love such judgments\u2014that the game involved \"perhaps the single most courageous act\" ever by an umpire. He ruled that on an infield covered with fans and several balls more or less in play, Merkle never touched second. All baseball fans know something about this game; few know the astonishing details Murphy supplies, including a brazen attempt to bribe the umpires before the game was replayed, because the Merkle game had been declared a tie.\n\nMurphy's book is rich in trivia\u2014not that anything associated with baseball is really trivial. Did you know, for example, that when the Yankees were still the Highlanders (they played at the highest point in Manhattan) they adopted their interlocking NY lettering \"based on the Tiffany design for the Police Department's Medal of Honor\"?\n\nReaders of _Crazy '08_ can almost smell the whiskey and taste the pigs' knuckles. This rollicking tour of that season will entertain readers interested in social history, will fascinate students of baseball, and will cause today's Cub fans to experience an unaccustomed feeling\u2014pride\u2014as their team enters the 2007 season, the ninety-ninth season of its rebuilding effort.\n\n[APRIL 1, 2007]\n\n# **Jackie Robinson: The Possible and the Inevitable**\n\n**L** ike many New Yorkers leaving home for work on April 15, 1947, he wore a suit, tie, and camel-hair overcoat as he headed for the subway. To his wife he said, \"Just in case you have trouble picking me out, I'll be wearing number 42.\"\n\nNo one had trouble spotting the black man in the Dodgers' white home uniform when he trotted out to play first base at Ebbets Field. Suddenly, only 399, not 400, major-league players were white. Which is why 42 is the only number permanently retired by every team.\n\nJackie Robinson's high school teachers suggested a career in gardening. Robinson's brother Mack had finished second to Jesse Owens in the 200-meter dash at the 1936 Berlin Olympics. Whites who won medals found careers opened for them. Mack, writes Jonathan Eig in _Opening Day: The Story of Jackie Robinson's First Season,_ wore his Olympic jacket as a Pasadena, California, street sweeper, while Owens found himself racing against horses at county fairs, \"one small step removed from a circus act.\"\n\nTo appreciate how far the nation has come, propelled by what began sixty years ago this Sunday, consider not the invectives that Robinson heard from opponents' dugouts and fans, but the way he had been praised. \"Dusky Jack Robinson,\" as the _Los Angeles Times_ called him, alerting readers to the race of UCLA's four-sport star, ran with a football \"like it was a watermelon and the guy who owned it was after him with a shotgun.\"\n\nThat cringe-inducing fact is from Eig's mind-opening book, an account of a twenty-eight-year-old man \"filled with fear and fury,\" and terribly alone. It includes unfamiliar details about familiar episodes. There is Lieutenant Robinson's 1944 refusal, eleven years before Rosa Parks, to move to the back of a bus at Fort Hood, Texas. And shortstop Pee Wee Reese, a Kentuckian who until 1947 had never shaken hands with a black person, crossing the infield to put a hand on Robinson's shoulder when Cincinnati fans were being abusive.\n\nBut Eig is especially informative about the dynamics among the Dodgers, who, like many teams, had a Southern tinge. The most popular player was nicknamed Dixie (Walker) and one of the best pitchers was the grandson of a Confederate soldier. The Dodgers' radio broadcaster, Red Barber, a Mississippian, considered resigning, then thought better. Radio presented Robinson as television cameras could not have done\u2014as, Eig shrewdly writes, \"all action,\" undifferentiated by visual differences from his teammates.\n\nAfter the opening two games against the Boston Braves, the Dodgers played the Giants at the Polo Grounds in Harlem. The president of the National League, fearing excessive enthusiasm, suggested that Robinson should develop a sprained ankle. He did not, and the crowds were large, dressed as if for church\u2014men in suits and hats, women in dresses\u2014and decorous. Soon a commentator wrote, \"Like plastics and penicillin, it seems like Jackie is here to stay.\"\n\nThe Dodgers were not. Ebbets Field's turnstiles clicked 1.8 million times in 1947, more than they ever had before or would again. But in 1947, in a Long Island potato field, Levittown was founded, offering mass-produced low-cost housing emblematic of postwar suburbanization. Dodger fans were moving east on the island. After the 1957 season, the Dodgers moved west.\n\nOnly 25,623 fans went to the game on April 15, 1947\u20144,000 fewer than on opening day 1946 and 6,000 fewer than the ballpark's capacity. Perhaps some white fans were wary of being with so many blacks. Usually blacks were no more than 10 percent of Dodger crowds but on this day they may have been 60 percent.\n\nBy 1956, Robinson's last season, he had lost his second base position to Jim Gilliam, a black man. Robinson died of diabetes-related illnesses in 1972, at fifty-three, the same age Babe Ruth was when he died. Ruth reshaped baseball; Robinson's life still reverberates through all of American life. As Martin Luther King Jr., who was eighteen in 1947, was to say, Robinson was \"a sit-inner before sit-ins, a freedom rider before freedom rides.\"\n\n\"Robinson,\" writes Eig, \"showed black Americans what was possible. He showed white Americans what was inevitable.\" By the end of the 1947 season, America's future was unfolding by democracy's dialectic of improvement. Robinson changed sensibilities, which led to changed laws, which in turn accelerated changes in sensibilities.\n\nJack Roosevelt Robinson's middle name was homage to the president who said \"speak softly and carry a big stick.\" Robinson's deeds spoke loudly. His stick weighed thirty-four ounces, which was enough.\n\n[APRIL 15, 2007]\n\n# **Ted Williams: \"I Can't Stand It, I'm So Good\"**\n\n**T** here is no joy in Red Sox nation, aka New England, or in any heart where baseball matters. When Ted Williams arrived in Boston at age twenty in 1939, a spindly six-foot-three, the Splendid Splinter said, \"All I want out of life is that when I walk down the street folks will say, 'There goes the greatest hitter that ever lived.'\" When he died Friday at age eighty-three, many people did say that, and no one said they were foolish.\n\nWhen, as a twelve-year-old in San Diego in 1930, he heard that the Giants' Bill Terry had batted .401, \"I got my little bat, ran out to our little back yard, and began to swing.\" His swing became baseball's gold standard.\n\nIn 1939, a golden moment on the eve of dark years, Bob Feller, Williams, and Joe DiMaggio were twenty, twenty-one, and twenty-four respectively. \"I can't stand it, I'm so good,\" Williams used to exclaim in his youthful ebullience.\n\nIn 1941, when DiMaggio mesmerized the nation with his still unmatched fifty-six-game hitting streak, Williams did what has not been done in six decades since\u2014batted over .400. Batting .3995 going into the season's last day, a doubleheader in Philadelphia against the Athletics, he went 6-for-8, finishing at .406.\n\nThere was no sacrifice fly rule in effect that year (today a batter is not charged with an at bat if he hits a fly that scores a runner). Had there been, his average would have been about 10 points higher. Biographer Ed Linn says that had Williams not lost the four and a half years he spent as an aviator in the Second World War and Korea, he probably would rank first or second in runs, runs batted in, total bases, extra-base hits, and perhaps home runs.\n\nAn alloy of innocence and arrogance, young Williams came to Boston when it had four morning and four evening local newspapers engaged in perpetual circulation wars. He became grist for their mills, and his wars with the sportswriters brought out the worst in him, and cost him. He won two Most Valuable Player Awards and finished second four times. Several of those times he would have won had he not had such poisonous relations with the voting press. A writer said that when Williams retired, Boston knew how Britain felt when it lost India\u2014diminished, but relieved.\n\nHe is one of only two players (the other was Rogers Hornsby) to win a triple crown (highest batting average, most home runs and runs batted in) twice, and he would have won a third if the Tigers' George Kell had not beaten him for the 1949 batting title .3429 to .34275. If the sacrifice fly rule had been in effect that year, Williams would have beaten Kell, who would have had one fewer sacrifice fly. Williams won six batting titles, including one hitting .388 in 1957, when his thirty-eight-year-old legs surely cost him five infield hits, enough to put him over .400 again.\n\nHe used a postal scale to check that humidity had not added an ounce to the weight of his bats. Challenged to find from among six bats the one that was half an ounce heavier than the others, he quickly did. He once returned to the maker a batch of his Louisville Sluggers because he sensed that the handles were not quite right. The handles were off by five-thousandths of an inch.\n\nLike many great players, he remembered, obsessively. That grand slam home run in Minneapolis before coming to the big leagues? \"Fifth inning, three-and-two count, low fastball.\"\n\nHe hit a home run in his last time at bat\u2014twice. He assumed his career was over\u2014and he homered\u2014when the Marine Corps called him to Korea (where No. 9 flew an F-9 jet as wingman for a squadron commander named John Glenn). And on September 26, 1960, in the final at bat of his final game, in Boston's gray autumnal gloom, he homered. Among the only 10,454 fans was John Updike, who wrote \"Hub Fans Bid Kid Adieu\": \"For me, Williams is the classic ballplayer of the game on a hot August weekday before a small crowd, when the only thing at stake is the tissue-thin difference between a thing done well and a thing done ill.\"\n\nNever, not even after that farewell home run, did Williams tip his hat to the cheering fans. \"Gods,\" wrote Updike, \"do not answer letters.\"\n\nLate in life, Williams said that often he fell asleep hearing in his head three songs\u2014\"The Star-Spangled Banner,\" \"The Marines' Hymn,\" and \"Take Me Out to the Ball Game.\" An American life.\n\n[JULY 7, 2002]\n\n# **Roberto Clemente: \"We Think He Can Hit\"**\n\n**M** ost biographies of great athletes are tinged with melancholy, for three reasons. Athletic greatness is often achieved by a narrowing, even infantilizing, monomania about physical things. Sport compresses life's natural trajectory of ascent, apogee, and decline. And often an athlete's life after sport is a long, dispiriting decrescendo. David Maraniss's splendid _Clemente: The Passion and Grace of Baseball's Last Hero_ is different, for three reasons. Roberto Clemente was an unusually elegant, even noble, athlete. He was emblematic of a social transformation. And he had no life after baseball.\n\nMaraniss's biography of Bill Clinton is still the best of the first president formed by the 1960s. He is also the author of one of the best books on the 1960s, _They Marched Into Sunlight: War and Peace, Vietnam and America, October 1967._ And now he has produced a baseball-savvy book sensitive to the social context that made Clemente, a black Puerto Rican, a leading indicator of baseball's future. Clemente was not the first Latino player, but as the first Latino superstar\u2014the National League's first Latino batting champion and MVP\u2014he propelled baseball's \"southern strategy\" for finding talent.\n\nBaseball has come a long way since the San Francisco Giants' manager Alvin Dark, in 1964, banned Spanish in the clubhouse. In 1989 and 1990, five of the twenty-six major-league teams had a starting shortstop from the same Dominican town, San Pedro de Macor\u00eds. In 2005, 29 percent of the players on the thirty teams' opening day rosters were born outside the United States\u201470 percent of them from the Dominican Republic, Venezuela, or Puerto Rico. Among the nearly twelve hundred players on the forty-man rosters this spring, ten of the sixteen most common surnames were Hernandez, Gonzalez, Perez, Ramirez, Rodriguez, Cabrera, Guzman, Lopez, Pena, and Sanchez.\n\nThe emblematic Pirate of the 1960 World Series\u2013winning team was, as Maraniss notes, \"ethnic\"\u2014second baseman Bill Mazeroski, son of a coal miner from nearby West Virginia. Not until the late 1980s did California supplant Pennsylvania as the state that had produced the most major leaguers. California's climate provides opportunities to develop baseball talent. Pennsylvania provided incentives\u2014escape from the mines and dark satanic steel mills. Latin America has both the climate and the incentives.\n\nThe report on young Fred Astaire's screen test said: \"Can't act. Slightly bald. Can dance a little.\" When the Pittsburgh Pirates drafted Clemente out of the Brooklyn Dodgers' farm system in 1954, they said, \"He can run and throw\u2014and we think he can hit.\" Oh, yes. In his last at bat, in September 1972, he became the eleventh player\u2014fifteen others have done it since\u2014to get three thousand hits. A bad-ball hitter with a \"nose to toes\" strike zone, he was difficult to walk. But then, as the saying goes, you can't walk off an island.\n\nIn eighteen seasons, all with the Pirates, Clemente used the whole field. Maraniss believes that two factors\u2014Clemente used an unusually heavy bat, and had chronic back and neck discomfort from a 1954 automobile accident\u2014kept Clemente from being a home-run-seeking pull hitter. Instead, he sprayed line drives into the left-and right-field power alleys.\n\nTim McCarver, the first catcher ever to lead either league in triples, once said hitting a triple is better than sex. Most of us are unable to make that comparison, but if Clemente, who hit 166 triples, agreed with McCarver, he played in two home ballparks built for ecstasy. The Pirates never led the league in home runs while playing in Forbes Field, but its spaciousness\u2014365 feet to left, 416 to right-center, 457 to the deepest part of center field, where the batting cage was kept on the field of play\u2014turned many Clemente drives into triples. So did the hard fake-grass surface of cavernous Three Rivers Stadium, to which the Pirates moved in 1970.\n\nThe first black Puerto Rican to play in the American League was Clemente's friend Vic Power, a flashy-fielding first baseman. Like Clemente, he was incensed by the 1950s contrast between Puerto Rico's easygoing race relations and America's segregation. But unlike Clemente, Power responded with wit. When a waitress said her restaurant did not serve Negroes, he replied equably, \"That's OK, I don't eat Negroes.\"\n\nClemente, playing in a city with a minuscule Latino population, said he felt like a \"double nigger.\" As late as 1971\u2014in one game that year, the Pirates became the first team ever to have nine black players in its starting lineup\u2014some sportswriters still quoted him in phonetic English: \"Eef I have my good arm thee ball gets there a leetle quicker.\"\n\nArrestingly handsome, at five feet eleven inches and 185 pounds he was about the size of today's smaller middle infielders. But the smoldering energy of his national and professional pride and resentments seemed transmuted into energy at bat and in the field. Right fielders need the strongest arms, to give runners second thoughts about going from first to third on singles. In the golden age of right fielders (Hank Aaron, Frank Robinson, Reggie Jackson, Al Kaline), Clemente's arm was the best.\n\nA Clemente line drive broke the leg of one Hall of Fame pitcher (the Cardinals' Bob Gibson, who pitched to three more batters before collapsing) and, Maraniss believes, hastened the retirement of another, the Dodgers' Don Drysdale. In August 1969, after a Clemente shot whistled into the outfield, Drysdale flicked what felt like an insect off his neck, but discovered blood on his fingers. Clemente's drive had torn the skin off the top of Drysdale's ear. Shaken, Drysdale walked Clemente the next time up and retired six days later.\n\nThe last act of Clemente's life was in character. On December 23, 1972, a severe earthquake devastated Nicaragua. Clemente, making use of his heroic status, threw himself into organizing Puerto Rico's charitable response. Incensed by reports that agents of Nicaragua's dictatorship were diverting, and profiting from, the shipment of aid, he chartered an ancient DC-7. He did not know it was a ramshackle contraption operated by a shady pilot with an untrained crew. Hoping his presence could force the Nicaraguan officials to distribute his materials, he boarded the plane. Overloaded and unbalanced, it plunged into the sea a few thousand yards from the end of the San Juan airport. His body was not recovered.\n\n\"The mythic aspects of baseball,\" Maraniss concludes, \"usually draw on cliches of the innocent past, the nostalgia for how things were. Fields of green. Fathers and sons. But Clemente's myth arcs the other way, to the future, not the past, to what people hope they can become. His memory is kept alive as a symbol of action and passion, not of reflection and longing. He broke racial and language barriers and achieved greatness and died a hero.\"\n\nIn 2005, thirty-three years after Clemente's death, Ozzie Guillen, the Venezuelan manager of the world champion Chicago White Sox, said his home has a shrine to the player he most reveres, Roberto Clemente. Now, thanks to Maraniss, Clemente's legacy is suitably defined and explained.\n\n[MAY 7, 2006]\n\n# **Greg Maddux: \"Watch This\u2014the First-Base Coach May Be Going to the Hospital\"**\n\n**B** aseball's almost seamless history has had only one stark disjunction, the one about 1920, between the dead-ball and lively ball eras. But within the lively ball era there has been the steroids parenthesis\u2014the era of some synthetically lively players\u2014which now is closing.\n\nGreg Maddux has thrived throughout it. Only three pitchers in the lively ball era have had four consecutive seasons with an earned run average under 2.40\u2014Maddux, Pedro Martinez, and Sandy Koufax, who threw from a mound five inches higher than today's. In Maddux's four seasons (1992\u20131995) his ERA was an astonishing 1.98, two runs per game lower than the National League's 3.99 over the same period. Today, as he prepares to win at least fifteen games for a record eighteenth consecutive year, he represents physical normality in baseball.\n\nJust six feet tall and 180 pounds, Maddux is a reminder that, as Bill Veeck said, you do not need to be seven feet tall or seven feet wide to play baseball. When Maddux, now thirty-nine, enters the Hall of Fame five years after he retires, he will be the smallest major-league pitcher inducted since Whitey Ford (five feet ten, 181), who retired in 1967.\n\nWhen baseball is cleansed of steroids there will be fewer lurid records, like those of Barry Bonds in 2001. But numerous factors, from the strength training of hitters to the proliferation of hitter-friendly ballparks, will keep home runs plentiful. Besides, pitchers, too, have probably used steroids. Maddux says steroids made some track stars' legs move faster, so they probably increased some pitchers' arm speeds. Steroid testing began in 2003. In 2004, only one-eighth as many players (twelve) tested positive as in 2003 (ninety-six). Yet home runs per game and slugging percentages increased.\n\nIn Maddux's first full season, 1988, the major-league-leading home-run total was Jose Canseco's forty-two. But of the fifty-homer seasons in baseball history\u2014there have been thirty-six of them\u2014nineteen have occurred since 1990. This power explosion has not perturbed Maddux, who last year methodically became the twenty-second three-hundred-game winner, and perhaps the last for a long time. This year, his eighty-fourth strikeout will be his three thousandth, making him the ninth pitcher with three thousand Ks and three hundred wins.\n\nHe is proof\u2014redundant proof\u2014that ballplayers can perform well late in their careers without performance-enhancing drugs. Ty Cobb, who batted .316 in 1906 at age nineteen, batted .323 at age forty-one in 1928. Warren Spahn, the winningest left-handed pitcher ever, got 158 of his 363 wins after turning thirty-six and was 23\u20137 at age forty-two. Henry Aaron\u2014currently and, we may hope, for many years to come, baseball's all-time home-run hitter\u2014had his best year at age thirty-seven.\n\nMaddux says laconically that when he came to the big leagues he threw between eighty and ninety miles per hour, and today he throws the same four pitches\u2014fastball, change, slider, curve\u2014seventy-five to eighty-five today. But he throws them with uncanny control: Among three-hundred-game winners since 1900, his walks-per-nine-innings ratio (1.87) is fourth best, behind only Cy Young (1.49), Christy Mathewson (1.59), and Grover Cleveland Alexander (1.65), who played all or most of their careers in the dead-ball era. And the key to his success has been less the speed of his arm than that of his mind.\n\nOne year in spring training, facing a Met who had hit him hard the previous season, Maddux told teammates he would throw dinky sliders to encourage the Met to hit a home run. Maddux figured that hitters remember, and subsequently look for, what they crush. The Met homered\u2014then, always looking for the same pitch, went hitless against Maddux in the regular season.\n\nLeading 8\u20130 in a regular-season game against the Astros, Maddux threw what he had said he would never throw to Jeff Bagwell\u2014a fastball in. Bagwell did what Maddux wanted him to do: He homered. So two weeks later, when Maddux was facing Bagwell in a close game, Bagwell was looking for a fastball in, and Maddux fanned him on a changeup away.\n\n_Sports Illustrated_ 's Tom Verducci collects such stories demonstrating Maddux's knowledge of hitters. Four times in one season, Maddux, while in the dugout, warned the man sitting next to him that the batter would line a foul into the dugout. Three times the batter did. Another time Maddux said on the bench: \"Watch this\u2014the first-base coach may be going to the hospital.\" The batter lined the next pitch off the coach's chest. Once with runners on second and third and two outs, Maddux's manager suggested an intentional walk. \"Don't worry,\" said Maddux, explaining that on the third of his next pitches the batter would pop out foul to third. Maddux was wrong: The pop was a few feet fair.\n\nMaddux, who grew up in Las Vegas, is a formidable poker player. Amarillo Slim, former winner of the World Series of Poker, once said: \"The results of one particular game doesn't mean a damn thing, and that's why one of my mantras has always been 'decisions, not results.' Do the right thing enough times and the results will take care of themselves in the long run.\" Maddux has had a long run pitching the way Slim played. But all runs end, so this year pay particular attention to the most artistic pitcher of the lively-ball era.\n\n[APRIL 25, 2005]\n\n# **Take Me Out to the Metric**\n\n**M** ichael Bourn needs to get out more. A database programmer in Nashua, New Hampshire, he created the website plunkbiggio. blogspot.com that tells everything\u2014really, everything\u2014about the 273 times that Craig Biggio of the Astros has been hit by a pitch, the modern major-league record.\n\nOn average, Biggio's plunks have occurred 493 feet above sea level, up 36 feet after two plunkings last year in Denver. The shortest pitcher to hit him? Byung-Hyun Kim (five feet nine inches). The average age and weight of the plunking pitchers are 28.5 and 200.22. He has been hit most often by pitchers whose astrological sign is Sagittarius, but more Leos have hit him. He has been hit fifteen times while Tiger Woods was on _Sports Illustrated_ 's cover. In 1997, the Dow rose an average of 28.63 on trading days after Biggio was hit. And on, and on.\n\nWhy does Bourn do this? \"It is better than following Ruben Sierra's approach to the sacrifice-fly record.\" (Sierra is 9 short of Eddie Murray's 128. Feel the excitement.) An obsessive-compulsive fascination with numbers is an occupational hazard of baseball fans. Baseball, unlike games of flow such as hockey, soccer and basketball, is a series of episodes that encourage quantification. This week, baseball resumes its prodigious production of numbers in another season of 2,430 games with 21,870 innings and approximately 700,000 pitches during 166,000 at bats. The rage to quantify\u2014to reduce reality to measurable units\u2014is an impulse in modern societies. In baseball, it produces illuminating metrics. For example:\n\nThe objective is to win, which means scoring runs while efficiently getting the other team to make twenty-seven outs. Every three outs, you must start over. Until recently, most people assumed that the key to runs was hits. Hence a misplaced emphasis on batting averages. But counting all hits alike is as foolish as counting different denominations of currency as identical. Nowadays, more emphasis is placed on not making outs. Hence the importance of on-base percentage, which is (hits + walks + hits by pitch)\/(at bats + walks+ hits by pitch + sacrifice flies). That led to the statistic OPS, which is on-base percentage + slugging percentage (which is total bases\/at bats).\n\nBut Bill James, a pioneer of novel metrics (see _The Mind of Bill James,_ by Scott Gray), says OPS takes the elements of run creation and puts them together incorrectly. \"They shouldn't be added together, they should be multiplied. A team with a .400 on-base percentage and a .400 slugging percentage would score more runs than a team with.350 and .450, although both add up to .800 OPS.\" James suggests calculating \"runs created\": (hits + walks+ caught stealing) + (total bases + .7 steals)\/(at bats + walks + caught stealing).\n\nYikes. One reason we were so glad to get out of school was to get away from math homework. For fans more fond of John Kruk's mind than Isaac Newton's, here are some more accessible numbers, pertaining to baseball's health as life resumes this week:\n\nCompetitive balance is getting better: Baseball has had six different World Series winners in the past six years. The NFL has not had six different winners of Super Bowls since 1968\u20131973. One moral of this story is that the Yankees, with their $202 million payroll, have learned the declining marginal utility of the last $80 million.\n\nMajor-league baseball's long history is divided into just two eras\u2014the dead ball and, beginning about 1920, the lively ball. But the latter contained a steroid parenthesis. It is closing because baseball now has the severest steroid penalties in professional sports. Last year there were 434 fewer home runs than in 2004\u2014and the game became more interesting. _Sports Illustrated_ 's Tom Verducci reports:\n\n\"Baseball captivates us so deeply that the anticipation of action is as compelling as action itself. The 20 seconds between pitches with the bases loaded, two outs, down a run in the eighth are Agatha Christie chapters unto themselves. With the powerball version of the game subsiding, fans are getting more of these worth-the-price-of-admission moments. For instance, 47.9 percent of games last season were decided by one or two runs, up 9 percent from the slugfest apex in 2001 and the highest such percentage since 1993.\"\n\nDuring the slugfest era, it was said, \"Chicks dig the long ball.\" Maybe. But as home runs fly away less frequently, ballpark turnstiles spin faster. Last year, baseball set an attendance record that it will break this year. Already, five teams\u2014Angels, Cardinals, Cubs, Yankees, Red Sox\u2014have essentially sold out their seasons. On February 24, the Cubs, who are in the ninety-eighth year of their rebuilding effort (they last won the World Series in 1908), put single-game tickets on sale. They set a major-league record, selling more than 597,000 tickets that day, just 38,000 fewer than they sold in the entire 1966 season.\n\nThis season will include more plunkings of Biggio, who is one of 1,563 players whose names begin with B who have been hit 8,601 times in 1,380,366 plate appearances. You can look it up. But if you do, you need to get out more.\n\n[APRIL 10, 2006]\n\n# **Elias Knows _Everything_**\n\n**L** ast Monday, Nancy and Henry Kissinger arrived at a Manhattan restaurant at 8:10 p.m. and excitedly recounted what they had just listened to in their car: a Yankee rookie in his first major league at bat had hit a home run off a fearsome pitcher\u2014the Diamondbacks' Randy Johnson, who is six feet ten and looks like a giant praying mantis with an attitude.\n\nBefore the Kissingers had time to examine their menus, some baseball commentators were reporting that this was the first time since 1986 that a player in his first major league at bat had homered against a likely future Hall of Famer (Will Clark off Nolan Ryan) and the first time ever that a player homered in his first at bat off a pitcher who the previous season won the Cy Young Award as the best pitcher in his league.\n\nWho tells us such things lickety-split? The busy beavers at the Elias Sports Bureau.\n\nOn a Saturday evening last month the Devil Rays scored four runs in the bottom of the ninth to beat the Orioles, 6\u20134, thereby snapping a fifteen-game losing streak. The game ended just after ESPN's 10 p.m. _Baseball Tonight_ went on the air. Soon Elias sent a message to reporter Tim Kurkjian on the _Baseball Tonight_ set: This was the first time ever that an American League team had snapped a double-digit losing streak by scoring more than two runs in the ninth.\n\nHow do such nuggets of baseball history get mined? Here is how. The Hirdt brothers, Steve, fifty-one, and Peter, forty-eight, both Fordham graduates, are the heart of Elias's batting order, which never sleeps, at least not all at once. This is a twenty-four-hour-a-day business whose approximately thirty employees, when not in the office, are logged on and talking to one another at all hours from their homes. Elias, whose clients now include all the major professional sports leagues, was begun in 1961 by Seymour Siwoff, who is still a bundle of energy at an age he thinks is nobody's business.\n\nElias's business is to examine the statistical histories of the major professional sports using custom-written software that will retrieve the answers to the kind of questions Peter put to it when he returned home from dinner and saw what the Devil Rays had just done to the Orioles. Peter wondered: In baseball\u2014sport of the long history and long seasons\u2014has _this_ ever happened before?\n\nLearning from the Elias computer that it had not, Peter e-mailed the news to a researcher at the nation's central cultural institution. No, silly, not to the Library of Congress. To ESPN, an Elias client. The researcher sitting on the set of _Baseball Tonight_ instantly e-mailed back: \"Tim will cry when he sees this.\" Tim didn't. There really are thoughts too deep for tears.\n\nSteve Hirdt says that when he and Peter were growing up they were \"the only boys in New York City who, when our mother said, 'How many times do I have to ask you to clean up your room?' would tell her.\" A statistical literacy is part of being a fan of any sport, but is especially important to baseball fans. Big league baseball, now in its third century, produces a steadily thickening sediment of numbers, pitch by pitch, inning by inning. Elias sifts the sediment.\n\nToday, Elias is located in a building overlooking an almost-as-impressive storehouse of knowledge, the New York Public Library. But just as there was a McDonald's brothers restaurant in San Bernardino, California, before Ray Kroc came along and had a bright idea, there was something called Elias, a dormant sports-information bureau run by two brothers, before Siwoff had his brainstorm, while shaving one morning, about putting a (then) newfangled gadget, the computer, to work deepening our understanding and enjoyment of sports.\n\nWhen Braves pitcher Tom Glavine recently went to 100 games over .500 in his career (as this is written, he is 101 over\u2014235\u2013134) he joined teammate Greg Maddux in that category, and it was the first time since 1908 that two teammates (the Giants' Christy Mathewson and Joe McGinnity) were at least 100 wins over .500.\n\nThe top three American League home-run hitters in May were all Yankees (Jason Giambi, ten; Alfonso Soriano and Bernie Williams, nine each). This was the first time since September 1950 that three teammates had led either league in homers in a month (the Yankees' Yogi Berra, Joe DiMaggio, and Johnny Mize, all now Hall of Famers).\n\nIt is incessantly said that pitchers do not pitch inside as aggressively as they did in the rough-and-tumble past. Elias says: Oh? In 1941, 1 in every 309 batters was hit by a pitch. In 1951, 1 in 214. In 1971, 1 in 179. In 2001, 1 in 99. Mickey Mantle, a power hitter, was hit 13 times _in his career._ The Astros' Craig Biggio was hit 28 times _last year._ When asked if any pitcher faced both Babe Ruth and Mantle, Elias reported: Al Benton pitched against Ruth for the 1934 Philadelphia Athletics and against Mantle for the 1952 Red Sox.\n\nElias knows _everything_ worth knowing.\n\n[JUNE 24, 2002]\n\n# **The Game's Gifted Eccentrics**\n\n**I** f you are the sort who wants to know how many doubles A-Rod hit on 2\u20131 counts in 2003 or Nolan Ryan's ground-ball-to-fly-ball ratio in 1974, you are living at the right time. Advances in medicine, communication, transportation, and plumbing since the middle of the nineteenth century are all very well, but what really makes this a golden age to live in is the multiplication and refinement of baseball statistics.\n\nAnd if you do not even know who A-Rod is, you will still enjoy the story of baseball's progress to today's information abundance as it is told by Alan Schwarz, a senior writer for _Baseball America,_ in _The Numbers Game._ Its lessons extend beyond baseball to politics and much else. It is an Information Age story about how new abilities to measure things beget new behaviors. The evolution of that cornucopia of information from its birth in 1845 to today's iterations is, in Schwarz's lively telling, a history of the game's path to the present.\n\nThat has been a path blazed by some gifted eccentrics whose apparently unslakable thirst for baseball numbers is shared by many millions of Americans, like Al Munro Elias and his brother Walter. A friend said of Al, there were only two ways to deal with him when he was talking baseball: \"One was to listen to what he had to say. The other was to kill him.\" Eighty years later, the Elias Sports Bureau is still the Spindletop of sports statistics\u2014the great original gusher.\n\nOne of Schwarz's subjects, Bill James, began his statistical assault on baseball's conventional wisdom while working as a night watchman at the Stokely-Van Camp canning plant in Lawrence, Kansas. That plant is, in modern baseball lore, akin to the Swiss patent office where Einstein began revolutionizing physics. (James has recently risen to the glory of the Boston Red Sox front office.) Schwarz also bestows honors on lesser-known luminaries, like Dick Cramer, who was driving home in St. Louis one night listening to a Cardinals game. \"Ozzie Smith,\" Schwarz writes, \"hit a ground ball that advanced a runner from second to third, to which announcer Mike Shannon reflexively commented, 'You won't see that in tomorrow's box score.' That's all Cramer needed to hear. 'I can fix that!' he said to himself.\" So now, in the blizzard of agate type that is part of a good sports section, you may see the result of Cramer's eureka moment\u2014a notation for \"runners advanced.\" But, then, we have not yet caught up to 1884, when some box scores listed each pitcher's number of called strikes.\n\nBaseball, unlike games of flow like hockey, soccer, and basketball, is a series of episodes\u2014of what Schwarz calls \"measurable states of combat.\" Box scores illustrate the symmetry\u2014what Schwarz calls baseball's \"double-entry personality\"\u2014that perhaps helps to explain baseball's peculiar hold on its fans. Every hitting event is \"part of a pitcher's record and every pitching event part of a hitter's record.\" No other team sport leaves such a satisfying statistical residue of coherence. \"A 10-yard run by a halfback or a point guard's breakaway layup cannot be assigned against any particular defensive player.... Baseball, however, is the most individual of team sports: in perfectly discernible packets the game reduces to one batter versus one pitcher, with each assuming responsibility for the other.\"\n\nThe arrival of statistical fluency has changed the way baseball is played. The arrival can be dated from Branch Rickey's hiring of Allan Roth in 1947 to wield his pencil\u2014baseball numeracy is not a gift recently conferred by computers\u2014on behalf of the Brooklyn Dodgers. In 1947, the Dodgers' Dixie Walker hit for a fancy average\u2014.306\u2014but was then traded. Why? Partly because he objected to playing with a rookie, Jackie Robinson. But also because Roth's charts of the pattern of Walker's hits showed that he was pulling the ball less, a leading indicator of aging. By 1950, Roth, whose nimbleness with numbers did not extend beyond baseball (he did not do his own taxes), sat at Ebbets Field in coat and tie, dispensing statistical tidbits to a twenty-two-year-old broadcaster, Vin Scully. Fifty-four years later, Scully's voice fills the Los Angeles Basin with Dodgers' numbers. During a 1987 National League Championship Series telecast, Scully cited a statistic showing the sort of thing managers now like to know: the Cardinals' pitcher, Danny Cox, held opponents to a .268 batting average with his first seventy pitches in games, but opponents hit .345 against pitches number seventy-one and higher. Moments after Scully's statistic was broadcast, the San Francisco Giants' Kevin Mitchell doubled on Cox's seventy-third pitch and Jeffrey Leonard homered on the seventy-fourth.\n\nBaseball took a long and winding road to reach Allan Roth and to travel far beyond him to today's sophisticated uses of the statistics generated by 2,430 games a year\u2014by more than 11 million at bats in more than 150,000 games since the major leagues began. In 1938, the Cincinnati Reds' Johnny Vander Meer pitched two consecutive no-hitters. In May, the Arizona Diamondbacks' Randy Johnson pitched just the seventeenth perfect game in major-league history. Statisticians, Schwarz writes, could not say when or by whom such things would be done, but they could predict that they would be done about as often as they have been.\n\nIn 1920, a Yankee outfielder hit more home runs (fifty-four) than fourteen of the fifteen other teams hit, ushering in the decade of ballyhoo\u2014the interrelated births of broadcasting, public relations, and sports superstars like Jack Dempsey, Red Grange, Bill Tilden, Man o' War, and, especially, that outfielder. His legacy includes an adjective, \"Ruthian,\" meaning prodigious. The new arithmetic of sports statistics was both a cause and an effect of superstardom. \"Spotlighting players' statistics in greater detail than ever began a tectonic shift in sports,\" Schwarz writes, \"as intrigue that once focused mostly on teams began to go to individual players and their statistics lines.\"\n\nLately, much, but not inordinate, attention has been focused on baseball executives who know how to tickle marginal insights from numbers\u2014insights like the fact that inexpensive players trapped in the high minor leagues are often satisfactory replacements for expensive but mediocre major leaguers. Because of the success of Billy Beane, the Oakland A's general manager, and of his epigones who run several teams' front offices, their interest in on-base percentage has become, Schwarz says, \"baseball's version of rock 'n' roll, scaring the old and galvanizing the young.\" He explains that before there was Beane there was the man who hired him, Sandy Alderson. A graduate of Dartmouth, Harvard Law School, and a Vietnam tour as a Marine officer, Alderson was in a San Francisco bookstore when he came upon a volume by Eric Walker, the most important baseball thinker you have never heard of. Alderson reading Walker was, Schwarz says, like Martin Luther King Jr. reading Gandhi, sparking a revolution. Scoring runs has always been the point of baseball, but Walker's epiphany was that when you make three outs you have to start over from scratch. Hitherto, the assumption was that runs\u2014and wins\u2014were achieved by hits. Nowadays the stress is on avoiding outs.\n\nBaseball was a long time awakening from its dogmatic slumbers to the realization that not all hits are created equal. Hence a high batting average can be overrated. The difference between a .275 hitter and a .300 hitter is, essentially, one hit every two weeks. Besides, as the baseball writer Ferdinand Cole Lane fulminated one hundred years ago, measuring a player's value by his batting average, which ignores the difference between singles and extra-base hits, is akin to measuring a man's financial worth by a system that treats different denominations of currency as identical.\n\nToday the \"\u00fcbermeasure of hitting,\" Schwarz writes, is OPS\u2014on-base percentage added to slugging percentage. Someday baseball statistics may be so sophisticated that they will be what James Joyce said his work was, something we should devote our lives to mastering. But if human beings have, as Schwarz believes, a \"compulsion to count, to quantify the world around them,\" then they are hardwired to be baseball fans.\n\nIf so, that fact lifts a load of guilt off this Puritan nation's shoulders. All those hours\u2014years, actually\u2014we have spent watching games when we should have been reading _Finnegans Wake_? Not our fault. Nature has made us do it. Which means that baseball is, as we chauvinists of the sport have long suspected, not merely the national pastime but the species' pastime. So there.\n\n[AUGUST 15, 2004]\n\n# **Don't Beat a Dead Horse in the Mouth**\n\n* * *\n\nOPENING DAY QUIZ\n\n_(answers on Don't Beat a Dead Horse in the Mouth)_\n\na. Name the only player to get at least five hundred hits with four teams.\n\nb. The first time a Brewer swung a bat in the game with the Rockies last June 29, the result was a sacrifice fly. How?\n\nc. In a 1965 game in Yankee Stadium, with the score tied, two outs in the bottom of the ninth, a runner on first and a 3\u20131 count on the batter, Yankee manager Johnny Keane ordered the batter to take the pitch\u2014even though the pitcher was sure to throw a strike rather than walk the potential winning run into scoring position at second. Why did Keane do that?\n\n* * *\n\n**T** he rule of thumb is that every team\u2014we are talking baseball today, so if you really want to read about stuff like Social Security reform, look elsewhere on this page\u2014will win 60 of its 162 games, will lose 60, and will play the season to settle the other 42. But the 2004 Arizona Diamondbacks did not get the memo explaining this.\n\nThey were epochally awful, losing 111 of 162 games. Yet _Sports Illustrated_ 's Tom Verducci notes that Diamondbacks attendance\u20142.5 million\u2014was larger than the attendance of each of the Yankee teams that won twenty-one world championships between 1923 and 1977. Major League Baseball's 2004 attendance was a record 73,022,969. Per game attendance was 30,075, compared to 14,106 in 1950, 16,110 in 1960, 14,788 in 1970, 20,434 in 1980, and 26,045 in 1990. All this indicates that the fans have not received the memo explaining that the game is going to hell in a handcart.\n\nWell, you ask, what about steroids? According to ESPN, twenty years ago five NFL players weighed more than three hundred pounds. The number of three-hundred-pounders on teams' current rosters? 433. Could chemistry as well as cheeseburgers be involved? Baseball is held to higher standards than other sports, and receives more intense and often unjust criticism, as it has regarding its supposed \"inaction\" regarding steroids. Testing for steroids began in the major leagues in 2003, and in 2004, 98.3 percent of players passed their tests. This is news to Congress but, then, what isn't?\n\nBaseball's competitive balance is much improved and compares favorably with the NFL and NBA. Three of the last four Super Bowls were won by one team (the New England Patriots), but the last five World Series have been won by five different teams. The National League has sent seven different teams to the last seven World Series. The worst winning percentage in baseball last year (the Diamondbacks' .315) was not as embarrassing as those (as of Thursday morning) of the NBA's Atlanta Hawks (.155), Charlotte Bobcats (.214) New Orleans Hornets (.229), and Utah Jazz (.310).\n\nIn the ten seasons since baseball included two wild-card teams in its postseason, twenty-two of baseball's thirty teams have played into October. And as Verducci says, the three most storied franchises\u2014the Yankees, Red Sox, and Cubs\u2014have each won at least 88 games in two consecutive seasons for the first time ever.\n\nBaseball also is thriving because it is a bargain (average MLB ticket, $19.82; average NBA ticket, $44.68; average NFL ticket, $54.75) and because of the flood of Spanish-speaking talent. The U.S. population (296 million) is more than thirty-three times that of the Dominican Republic (8.8 million), but bet on this all-Dominican lineup against the rest of the world:\n\nC........ . Miguel Olivo, Mariners\n\n1B........ Albert Pujols, Cardinals\n\n2B........ Alfonso Soriano, Rangers\n\n3B........ Aramis Ramirez, Cubs; Adrian Beltre, Mariners\n\nSS........ Miguel Tejada, Orioles\n\nOF........ Sammy Sosa, Orioles\n\nOF........ Vladimir Guerrero, Angels\n\nOF........ Manny Ramirez, Red Sox\n\nDH........ David Ortiz, Red Sox\n\nSP........ Pedro Martinez, Mets; Bartolo Colon, Angels\n\nRP........ Armando Benitez, Giants\n\nMANAGER . . Felipe Alou, Giants\n\nConservatives are forever being lectured that \"you can't turn the clock back\"\u2014and shouldn't want to. Oh? This season, for the first time since the Astrodome opened in 1965, every National League game will be played on real grass. What a concept. There are many other reasons why this is baseball's golden age but, in the words of former Phillies manager Larry Bowa, \"I don't want to beat a dead horse in the mouth.\"\n\n* * *\n\nQUIZ ANSWERS\n\na. Rusty Staub (Houston, Montreal, Detroit, New York Mets).\n\nb. The first eighteen pitches from the Rockies' pitcher were fourteen balls and four called strikes.\n\nc. Because with a 3\u20132 count and two outs, the runner on first would be running with the pitch and could be almost certain to score on a double. Which he did.\n\n* * *\n\n[APRIL 4, 2005]\n\n# **The Golden Age**\n\n**S** ins can be such fun. Of the seven supposedly deadly ones, only envy does not give the sinner at least momentary pleasure. And an eighth, schadenfreude\u2014enjoyment of other persons' misfortunes\u2014is almost the national pastime.\n\nSpeaking of baseball, two Saturdays ago, old Dodger Stadium was reverberating with fans' excitement. It might seem odd to call \"old\" a ballpark that opened in 1962, but it is tied with the Nationals' RFK Stadium as the National League's second oldest, behind only the Cubs' Wrigley Field (1914). Anyway, shortly before their Dodgers were beaten by the Mets in the National League Division Series, Angelenos emitted animal roars of approval as they watched, on the giant screen in left-center field, the Tigers defeat the Yankees in the ALDS.\n\nSome Dodgers fans still nurse a grudge they inherited from Brook-lynites when the Dodgers decamped for California after the 1957 season. But rooting against the Yankees is as American as a microwaved wedge of frozen apple pie topped with a slice of processed cheese. Such rooting often is the unlovely underside of the democratic ethos\u2014envy of excellence. But there also is resentment of the Yankees' financial advantage that has been inimical to baseball's competitive balance.\n\nThat, however, is a diminishing problem, for two reasons: Major League Baseball has implemented more redistribution of resources, and a new breed of general managers (e.g., Oakland's Billy Beane and Minnesota's Terry Ryan) are using new player-evaluation metrics to wring more baseball value from fewer dollars.\n\nThe Yankees' payroll of $206.4 million (not including the almost $30 million tax paid to MLB on the portion of the payroll over $136.5 million) is 2.4 times the Tigers' payroll. The Yankees' third baseman earns 68.7 times the salary of the Mets' all-star third baseman (Alex Rodriguez, $25.7 million; David Wright, $374,000). The shortstop makes approximately what the Marlins' team makes (Derek Jeter, $20.6 million; Marlins, $20.68 million). But the 2006 Yankees did baseball\u2014and the rest of America, if it learns the larger social lesson of the story\u2014the favor of demonstrating the steeply declining utility of the last $100 million of payroll.\n\nNew York, the world's financial capital, takes money _very_ seriously. And New York has been the intellectual epicenter of political liberalism, which has consistently preached, and has consistently disproved, the efficacy of pitching large sums of money at social problems. In the city where America's welfare state was first imagined and implemented, the entitlement mentality bred by the welfare state includes the assumption that the Yankees are entitled to be in the World Series, which they have not been since\u2014gasp\u20142003.\n\nThere still are revenue and spending disparities between baseball teams that are impossible between NFL and NBA teams because those leagues have salary caps and more centralized revenue sources. Nevertheless, when the Tigers dispatched the Yankees that Saturday, baseball was guaranteed its seventh different World Series winner in seven years. There _never_ have been seven consecutive Super Bowls, or seven consecutive NBA championships, won by seven different teams.\n\nBaseball's supposed \"golden age\" of the 1940s and 1950s was not so golden outside New York. In 1947, the Yankees won the American League pennant and beat the Dodgers in the World Series. In 1949, 1950, 1951, 1952, and 1953 the Yankees were World Series winners over the Dodgers, Phillies, Giants, Dodgers and Dodgers, respectively. If the Phillies had not beaten the Dodgers in the tenth inning of the last game of the 1950 season, every World Series game for five years would have been played in New York. And if 103 wins, which usually are enough to win the pennant, had sufficed in 1954 (the Indians won 111, an American League record for a 154-game season), the Yankees would have won ten pennants in a row, because they also won in 1955, 1956, 1957, and 1958.\n\nGreat Yankee teams have been good for baseball. In the 1930s, one of every four tickets sold to an American League game was for a game involving the Yankees. And this year, when the Yankees were drawing 4,200,518 fans to Yankee Stadium, they also played in front of 3,080,290 million on the road. But improved competitive balance is one reason why, for the third consecutive year, MLB set an attendance record (76,043,902), and why today is MLB's golden age, even west of the Hudson River.\n\n[OCTOBER 15, 2006]\n\n# **Pete Rose, Always Hustling**\n\nHUSTLE _verb. to work or act rapidly or energetically._\n\n\u2014 _W EBSTER'S NEW WORLD DICTIONARY_\n\n**B** ut _hustle_ also is a noun: \"A way of making money, esp. a dishonest way.\"\n\nPete Rose, who walked 1,566 times in his major league career but never walked to first base, always sprinting, was called Charlie Hustle. His new hustle is his book, for which he reportedly received a $1 million advance, in anticipation of sales generated by his coming clean about having bet on baseball, which no one seriously interested in the subject doubted. No one, that is, other than professional contrarians, or commentators emancipated from facts by not having read the 1989 report that caused Rose to accept \"permanent\" banishment from baseball.\n\nRose's coming clean is the most soiled conversion of convenience since...well, August 17, 1998, when DNA evidence caused Bill Clinton to undergo a memory clarification. On the diamond, no one ever wrung more success from less natural talent than Rose did. But his second autobiography\u2014which refutes the first\u2014makes worse the mess he has made.\n\nThe supposedly truth-telling book contains this patent lie: \"During the times I gambled as a manager, I never took an unfair advantage. I never bet more or less based on injuries or inside information.\" But he also says\u2014does he even _read_ his autobiographies?\u2014\"I began betting regularly on the sport I knew best\u2014baseball.\" Managing the Reds, he knew\u2014he _decided_ \u2014when a tired or injured star would be played or rested. And the network of bookies handling his bets knew that he knew.\n\nWhile saying \"it's time to take responsibility,\" he cunningly exploits the Zeitgeist of today's therapeutic society. He is, he insists, a victim.\n\nA victim of an addiction\u2014gambling while managing the Reds substituted for the \"high\" he had gotten when competing as a player. And he is a victim of a double standard: He would have been treated more leniently\u2014more therapeutically\u2014had his problem been drugs rather than gambling. But baseball has especially severe sanctions about gambling because competitive integrity is baseball's raison d'\u00eatre.\n\nAmericans, a forgiving people, are forever refuting the proposition that there are no second acts in American life. Almost anyone can recover from almost anything by convincingly saying \"I'm sorry.\"\n\nRose lied\u2014and charmed the gullible\u2014for fourteen years. Now, with the clock running out on his eligibility to election by baseball writers to the Hall of Fame, he pugnaciously says: I lied but \"I'm just not built\" to \"act all sorry or sad or guilty\" about it. \"Act\"?\n\nRose's critics have said that repentance is a necessary\u2014not a sufficient\u2014prerequisite for restoring his eligibility to the Hall of Fame. Many, probably most, of Rose's critics are revolted by the moral obtuseness of his synthetic repentance.\n\nHis dwindling band of defenders responds that it is unfair to judge Rose not by what he does but by the way he does it. Yet regarding repentance, the way you do it _is_ what you do.\n\nCooperstown primarily honors players for, in players' parlance, the \"numbers they put up.\" Hence it is widely believed that selection to the Hall is exclusively about the statistical residue of players' careers and should not involve a \"morals clause\"\u2014consideration of character.\n\nBut the rules for election by members of the Baseball Writers' Association of America include: \"Voting shall be based upon the player's record, playing ability, integrity, sportsmanship, character, and contributions to the team(s) on which the player played.\" The rules for voting by the Veterans Committee similarly mention \"integrity, sportsmanship, character.\"\n\nSome will say that if admittance to the Hall were limited by a strict calculus of character, the Hall would be much smaller. Yes, Babe Ruth might have hit even more home runs if he had gone to bed earlier, and more often with Mrs. Ruth. But not all character issues are equally pertinent to the proper criteria for honoring athletes' achievements. The crucial criteria concern the integrity of the competition.\n\nRose has said, \"I was raised, but I never did grow up.\" He is not the only ballplayer who will be forever a boy. But what distinguishes him is not mere boyish roguishness but a hard, calculating adult amorality. There is a constancy to it that goes beyond recidivism, which implies episodes of recovery between relapses.\n\nOn the evidence of his book, he should never be back in a major-league uniform as a manager or coach. And he should not be admitted to the Hall of Fame unless its character criterion is declared irrelevant, which is not what the nation needs from the national pastime.\n\n[JANUARY 8, 2004]\n\n# **The Precious, Precarious Equipoise**\n\n**C** hicago baseball fans, who are composites of scar tissue and mortifying memories, instantly drew upon one of those memories for their response\u2014\"Say it ain't So-sa\"\u2014to Sammy Sosa's ejection from Tuesday night's game for the rule infraction of using a corked bat. Their words echoed the boy who supposedly exclaimed, \"Say it ain't so, Joe\" to Shoeless Joe Jackson, one of the White Sox players accused of throwing the 1919 World Series.\n\nThe Sox have been to only one Series since then, in 1959, which they lost. The Cubs have not been to a Series since 1945, which they lost, and have not won one since 1908, two years before Tolstoy died. But even Cub fans, although inured to pain, winced Tuesday night.\n\nSosa drove in a run with an infield out, but his bat shattered, revealing cork in the barrel. So the run was disallowed. Sosa says a corked bat he used to produce fan-pleasing fireworks during batting practice and home run hitting contests mistakenly got mixed in with his game bats. Corking a bat reduces its weight, enabling a batter to increase bat speed and to drive some pitches he might not otherwise be able to hit hard.\n\nBefore Tuesday's game was over, Major League Baseball took possession of seventy-seven of Sosa's bats. None of them is corked, which lends powerful support to Sosa's explanation.\n\nThis is good news for baseball. The ebullient Sosa is the game's most marketable star. His competition with Mark McGwire for the 1998 National League home-run title, which McGwire won seventy to sixty-six, was a crucial ingredient in baseball's recovery from the fan-alienating strike that truncated the 1994 season on August 12 and canceled the World Series.\n\nFurthermore, baseball produces\u2014inning-by-inning, game-by-game, season-upon-season\u2014a rich sediment of statistics that sustain the arguments that nourish interest in the game with the longest history. If Sosa's slugging\u2014he is the only player to hit sixty or more home runs in three seasons\u2014was assisted by cheating, he will be diminished, as will the game's ongoing narrative. And all other players will come under a lowering cloud of cynicism about the authenticity of their achievements.\n\nMajor League Baseball will decide the seriousness of what Sosa did Tuesday\u2014whether it was an accident arising from injudicious showmanship (actually, fans deserve to know that Sosa's prodigious achievements in home-run hitting exhibitions are unassisted by illegal bats) or whether he has repeatedly cheated in games. The stakes are high. Bart Giamatti knew why.\n\nIn 1987, pitcher Kevin Gross of the Philadelphia Phillies was caught with a small patch of sandpaper affixed to his glove, and a sticky substance on his glove. Sandpaper can be used to scuff a ball's surface, changing its wind resistance and hence its movement when pitched. Foreign substances also can alter the movement of a thrown ball, and it is no defense to say, as a pitcher (a Cub, of course) said when indignantly denying that he put a foreign substance on the ball: \"Everything I use on it is from the good ol' USA!\"\n\nGross was suspended for ten days by Giamatti, then National League president. A former president of Yale and a professor of Italian and comparative literature, Giamatti died in 1989 shortly into his five-month tenure as baseball commissioner, after imposing a lifetime suspension from baseball on Pete Rose for gambling on games. Giamatti knew exactly why \"boys will be boys\" is not a satisfactory response to paltering with the rules of the game.\n\nMost of baseball's punishable offenses involve fighting or other violence that arises from the heat of competition. While such acts cannot be tolerated, Giamatti wrote, \"It must be recognized that they grow often out of impulse, and the aggressive, volatile nature of the game and of those who play it.\"\n\nSuch offenses, he said, are less execrable than acts \"of a cool, deliberate, premeditated kind\"\u2014acts that have \"no organic basis in the game and no origins in the act of playing.\" They are acts of cheating that are \"intended to alter the very conditions of play to favor one person.\" Such acts \"are secretive, covert acts that strike at and seek to undermine the basic foundation of any contest declaring the winner\u2014that all participants play under identical rules and conditions.\"\n\nGiamatti understood that a team sport, like democratic society itself, involves a precious and precarious equipoise of individual striving and collective endeavor. In sport or society, break the rules that govern that equipoise and hark! what discord follows.\n\n[JUNE 5, 2003]\n\n# **Barry Bonds: Enhanced and Devalued**\n\n**W** ould that Barry Bonds had retired after the 1998 season. He might be happier than he seems to be in his long trudge toward tainted glory. Certainly everyone who cares about baseball, and about the integrity of athletic competition generally, would be spared the disturbing spectacle of his unlovely approach to Henry Aaron's career record of 755 home runs.\n\nThe numbers Bonds had put up before the 1999 season were luminous enough to have guaranteed him first-ballot election to the Hall of Fame. He had 411 home runs, 445 stolen bases\u2014he is now the only \"500-500\" player in history\u2014eight All-Star selections, and eight Gold Glove awards. He had won three MVP awards and should have won a fourth that was given to a lesser, but less-obnoxious, player.\n\nSince 1998, his gaudy numbers have earned him four more MVP awards. From his 1986 rookie season through 1998, he averaged a home run every 16.1 at bats (Babe Ruth averaged one every 11.8 at bats), and his season high was 46. Since 1999, when he turned thirty-five, an age by which most players are past their peak production, he has averaged one every 8.9 at bats, and in the 2001 season he hit 73. If Bonds, even as he aged, had continued to average one home run every 16.1 at bats, he would have entered this season at age 42 with 590 home runs, not 734, and Aaron's record would have been beyond his reach.\n\nEqually startling are these numbers: According to Mark Fainaru-Wada and Lance Williams, the _San Francisco Chronicle_ reporters who wrote _Game of Shadows: Barry Bonds, BALCO, and the Steroids Scandal That Rocked Professional Sports,_ Mike Murphy, equipment manager of the San Francisco Giants, testified that since Bonds became a Giant in 1993, the size of his uniform jersey has gone from 42 to 52. His cap size has expanded from 71\/8 to 71\/4, even though while it was expanding he shaved his head. (Bonds reportedly shaved his head because his hair was falling out as a result of steroid use.) And Fainaru-Wada and Williams also say Murphy testified that Bonds's baseball shoe size has changed from 101\/2 to 13.\n\nSteroids, human growth hormone (HGH), and other performance-enhancing drugs (PEDs) can cause gradual enlargement of bones in the feet, hands, face, jaw, and skull. Bonds has never failed a steroid test, but there is no reliable test for HGH, and chemists concocting PEDs also devise masking ingredients to defeat tests.\n\nVarious PEDs can increase muscle mass (and the speed of hitters' bats and pitchers' arms). They can hasten recovery from the exertions of training or competing, and can reduce pain and increase the sort of concentration needed when a ninety-six-mile-per-hour fastball is coming at you during a day game after a night game. George Vecsey, in his short new history of baseball, quotes a player: \"The funniest thing I ever saw in baseball was Pete Rose's greenies kicking in during a rain delay.\" Greenies\u2014amphetamines, a booster fuel for a 162-game season that is played across four time zones\u2014were for years as openly available as sunflower seeds in teams' clubhouses.\n\nThe fascinating history of PEDs runs back into history's mists, to potions concocted to increase soldiers' aggressiveness in battle. This history is recounted in Will Carroll's _The Juice: The Real Story of Baseball's Drug Problems,_ an indispensable guide to today's controversies.\n\nIn 1898, a Welsh cyclist in a Paris-to-Bordeaux race died after drinking an alcohol-based product designed to increase stamina and control pain. In 1921, a University of Chicago chemist ground up tons of bulls' testicles and used chemicals to isolate testosterone. By 1932, Carroll writes, sprinters were experimenting with nitroglycerine to dilate their coronary arteries. In 1936, at the Berlin Olympics, injectable testosterone, developed the year before by Nazi doctors for military use, probably helped propel German athletes to eighty-nine medals, more than any other team. In 1945, some German scientists involved in synthesizing testosterone moved to the Soviet Union, which soon became dominant in weight lifting. At the 1976 Montreal Olympics, large East German women with deep voices, body hair on their torsos, and severe acne won eleven of thirteen possible gold medals in swimming.\n\nNow, the caffeine in your coffee is a PED. Some major-league ballparks feature advertisements for another widely used PED\u2014Viagra. Testosterone and HGH, which the body produces naturally, are components of some potent PEDs. Distinguishing legitimate from illegitimate enhancements, in the context of competition, is not always easy. One should begin by understanding the temptation.\n\nAlthough dangerous, steroids and other PEDs can tempt two kinds of ballplayers. One is the superior athlete for whom mere superiority is insufficient when immortality might be injected from a syringe. The other is the marginal player, a category that includes most major leaguers at some point in their careers, and many throughout their careers. If such a player knows or suspects that competitors for his roster spot or playing time are getting illegal and hazardous chemical assistance, he must choose between jeopardizing his career or his health.\n\nAside from certain grossly anomalous achievements by a few individuals such as Bonds, it is difficult to measure the extent to which PEDs have distorted baseball. Carroll stresses that their increased use has coincided with other changes that have tended to increase offensive production. The changes have included more sophisticated strength training unrelated to ingested or injected substances; sixteen new, mostly hitter-friendly ballparks; contraction of the strike zone; expansion of the number of teams, which diluted the quality of pitching; and maple bats that are more durable than those made from ash and so can have thinner barrels that increase bat speed.\n\nAnd PEDs are, of course, not a problem only in baseball. Track\u2014it was a track coach who blew open the BALCO lab scandal that brought Bonds before a grand jury\u2014might be the sport most distorted by PEDs. And it requires the willful suspension of disbelief to think that diet and strength training are the only reasons why the average NFL offensive lineman weighs 307 pounds. But baseball is held to higher standards, for several reasons.\n\nOne is that baseball, unlike football, has a statistical measure of players' strength\u2014the home run. For thirty-four years, sixty homers was the season record. Then, for thirty-seven years, the record was sixty-one. Then, in four seasons, 1998 to 2001, that total was surpassed six times. In football, Carroll writes, \"the players most likely to use steroids are offensive and defensive linemen. If these players get stronger via steroids, their gains in strength will merely cancel each other out, and there will be no noticeable difference in the statistics.\" Furthermore, Carroll says, football's premier players\u2014quarterbacks\u2014achieve greatness by recognizing defenses and throwing accurately, not by the strength that steroids can augment.\n\nAnother reason baseball is held to higher standards than are other sports is that fans relate to baseball players differently. This is partly because, as Bill Veeck said, players do not need to be seven feet tall or seven feet wide. Players are generally much bigger than they used to be: Mickey Mantle (5 feet 111\/2, 195 pounds) was smaller than most of today's middle infielders. But last year's World Series MVP, the Cardinals' David Eckstein, is 5 feet 7 and 165 pounds.\n\nAlso, Tim Marchman, who writes about baseball for the _New York Sun,_ notes that last year Shawne Merriman, linebacker for the San Diego Chargers, was suspended for four games for steroid use\u2014then was selected for the Pro Bowl. And few cared. Marchman detects a \"soft bigotry of low expectations\": Most NFL and NBA players are black; most of the paying customers are white and, Marchman argues, do not expect better behavior. Baseball, however, is, Marchman says, \"culturally white and middle class\" and its players \"are more widely expected to conform to ethical norms.\"\n\nPerhaps. Certainly those norms are under pharmacological attack, and the attack will be a protracted contest between the chemists who devise PEDs and the testers who try to detect the use of the substances. In any case, the norms need to be explained and affirmed, as follows.\n\nDrugs enhance performance by devaluing it when they unfairly alter the conditions of competition. Lifting weights and eating spinach enhance the body's normal functioning; many chemical intrusions into the body can jeopardize the health of the body and mind, while causing both to behave abnormally.\n\nAthletes who are chemically propelled to victory do not merely overvalue winning, they misunderstand why winning is properly valued. Professional athletes stand at an apex of achievement, but their achievements are admirable primarily because they are the products of a lonely submission to a sustained discipline of exertion. Such submission is a manifestation of good character. The athlete's proper goal is to perform unusually well, not unnaturally well. Drugs that make sport exotic, by radical intrusions into the body, drain sport of its exemplary power by making it a display of chemistry rather than character. In fact, it becomes a display of some chemists' virtuosity and some athletes' bad character.\n\nSport is play, but play has a serious side. It can elevate both competitors and spectators. But cold, covert attempts to alter unfairly the conditions of competition subvert the essence of sport, which is the principle that participants shall compete under identical rules and conditions.\n\nHarvey Mansfield, a Harvard philosopher, says we need our athletes and their integrity because excellence is always endangered in democracies that often cherish equality indiscriminately. PEDs, he says, do not merely expand the limits of human nature, they erase those limits as a standard: \"Perfection disappears as the upper limit, and is replaced by an indefinite, indefinable perfectibility.\"\n\nMansfield's colleague Michael Sandel, in his new book, _The Case Against Perfection,_ acknowledges that \"the line between cultivating natural gifts and corrupting them with artifice may not always be clear.\" In 1999, Tiger Woods, whose eyesight was so poor he could not read the large E on the eye chart, had LASIK eye surgery\u2014then won his next five tournaments. This was not a corrupting artifice. It enabled his eyes to do what normal eyes naturally do, not what unnatural eyes would do. But, Sandel says, when the role of chemical enhancement increases, our admiration for the achievement decreases. An athlete who succumbs to the \"Promethean aspiration to remake nature, including human nature\" ceases to be the agent of his achievements, which are drained of merit and moral responsibility.\n\nIt is a truism that baseball involves a lot of failure. Babe Ruth struck out 1,330 times\u2014that is the equivalent of three seasons of at bats without ever putting the ball in play. Ty Cobb, whose .366 career batting average is the highest in history, failed more than 63 percent of the time. In a sense, most Americans are failed ballplayers. That is one reason for the sport's unique grip on the nation's imagination and affections.\n\nPEDs make baseball less of a shared activity. Because of them, a few excel but everyone loses\u2014everyone in the stands and on the field, and Bonds more than anyone.\n\n[MAY 21, 2007]\n\n# **The Methodical Mr. Aaron**\n\n**M** OBILE, ALABAMA\u2014This city has belonged to five nations\u2014France, Britain, Spain, the United States, and the Confederate States of America. Or four, if you think, as Lincoln did, that the Southern states never succeeded in seceding, so the CSA never existed. In any case, Mobile has done much for the national pastime of the country to which it currently belongs.\n\nMobile has incubated tremendous major-league talent. In a few games in 1969, the \"Miracle Mets\" had an all-Mobile outfield. Five Hall of Famers were raised here\u2014Satchel Paige, Willie McCovey, Ozzie Smith, Billy Williams, and the man whose achievements gain luster from the contrast between him and the man who may soon surpass one of those achievements. As Barry Bonds continues his gimpy, joyless pursuit of such glory as he is eligible for, consider the odyssey of Mobile's greatest native son.\n\nHenry Aaron's parents had moved south from Selma, drawn by work in the shipyards during World War II. So many blacks came here that Davis Avenue\u2014named for Jefferson Davis\u2014became known as Little Harlem.\n\nYou think _that_ is incongruous? Try this. Grip a bat as a right-hander\u2014but with your left hand on top. That is how the man who would hit 755 home runs in twenty-three major-league seasons gripped his bat when, as an utterly uncoached seventeen-year-old, he signed his first professional contract, with the Indianapolis Clowns of the Negro Leagues, who recognized an uncut diamond.\n\nWhen he boarded the train to his future, he had $2 in his pocket. He had never had his own bed, and with the Clowns often slept six nights a week in a bus. He remembers sitting with teammates in a Washington restaurant \"hearing them break all the plates in the kitchen after we were finished eating.\"\n\nAaron's signing bonus with the Milwaukee Braves was a cardboard suitcase. In his first spring training, during a game against the Red Sox, Ted Williams came running from the clubhouse to see whose bat was making that distinctive sound. The bat had a slender handle and was whipped by wrists developed hitting dipping and floating bottle caps, pitched by Aaron's playmates when, as was usual, baseballs were scarce.\n\nHe was 0-for-5 in his first regular season game, which was the first day in which players were no longer allowed to toss their gloves on the field when coming in to bat. Soon, however, _Time_ magazine was heralding \"The Talented Shuffler\" who \"is not as dumb as he looks when he shuffles around the field.\" Misperceiving, through the lens of race, economy of motion for lethargy, sportswriters called him \"uncomplicated\" and \"a child of nature.\" Lonnie Wheeler, who helped Aaron write his autobiography, _I Had a Hammer,_ notes that Joe DiMaggio's similar understated manner was characterized as dignified and graceful.\n\nIn 1973, as Aaron approached Babe Ruth's record of 714 home runs\u2014he would break it in April 1974\u2014he received, according to the U.S. Postal Service, about 930,000 letters, more than any nonpolitician in America. Dinah Shore was second with 60,000. Much of his mail was hateful. He took out his anger on baseballs. The 1973 season was the last in which horsehide balls were used. Aaron's 714th was the first home run ever hit with a cowhide ball.\n\nWhen Aaron retired, he was Major League Baseball's last link to the Negro Leagues. Today, he is baseball's link to the era when home runs did not cause fans, suspecting steroids, to view sluggers with a moral squint. Aaron became baseball's most methodical\u2014and, properly measured by total bases, most effective\u2014hitter after being raised in a household where, he remembers, \"we almost never ate anything that was store-bought. I've gone many, many weeks with just cornbread, butter beans and collard greens.\"\n\nMobile's public library, writes Wheeler, \"opened its doors to blacks before other Southern cities encouraged them to read.\" Spring Hill College here, which integrated\u2014by conscience, not coercion\u2014in 1954, was praised by Martin Luther King in his \"Letter from Birmingham Jail.\" Today, if you turn onto Satchel Paige Drive, then onto Bolling Brothers Boulevard (Frank and Milt, nephews of a major leaguer, played a combined nineteen seasons), you reach Hank Aaron Stadium, home of the Mobile BayBears.\n\nWhen Bonds hits his 756th, real fans, who know how to read the record book, will yawn, confident that Aaron's record will remain the real one until Alex Rodriguez, who has 175 more home runs than Bonds did when he was Rodriguez's age, breaks it.\n\n[MAY 6, 2007]\n\n# **Realism Among the RiverDogs**\n\n**C** HARLESTON, SOUTH CAROLINA\u2014Realism is overrated. Putting it aside makes possible some sweet things, such as the idea of Santa Claus. And the fact of minor-league baseball.\n\nThis city's RiverDogs play at The Joe\u2014the Joseph P. Riley Jr. Park. Their rivals in the Sally\u2014actually, South Atlantic\u2014League include the Delmarva Shorebirds and Hagerstown Suns from Maryland; the Lake County Captains from Ohio; the Lakewood BlueClaws from New Jersey; the Greensboro Grasshoppers, Hickory Crawdads, Asheville Tourists, and Kannapolis Intimidators from North Carolina; the West Virginia Power from another Charleston; the Lexington Legends from Kentucky; the Columbus Catfish, Rome Braves, Augusta Green-Jackets, and Savannah Sand Gnats from Georgia; and the Greenville Bombers (reborn as the Greenville Drive in 2005) from just up the road. Such small cities and towns that are incubators of big dreams.\n\nTalk to the players, most of them under age twenty-three, and you will find few, if any, who do not believe they are bound for glory\u2014for Yankee Stadium, the RiverDogs being a Yankee Single-A affiliate. Actually, the RiverDogs are the Yankee's low Single-A club, and by this point in the season, many of the best prospects have been promoted to the high-A club in Tampa, or up to Double-A Trenton.\n\nThe RiverDogs play 140 games in 151 days, traveling by bus, living at least two to a room in motels, some earning as little as $1,050 a month\u2014and only during the season\u2014with a $20 per diem for food. \"Sometimes,\" says a player touchingly grateful for life's little blessings, \"the motel is near an Outback.\" A young man from West Texas says, \"I had a brother working in the oil fields. So if I wake up tired one day, I think, 'I could be doing that.'\"\n\nMost of today's Sally Leaguers will be doing something like that sooner than they can bring themselves to imagine. But for now they are delighting some of the 40 million fans who will see minor-league baseball this summer. The RiverDogs, averaging about thirty-eight hundred fans a game, are one of five teams partly owned by Mike Veeck, a third-generation baseball man\u2014his father put the ivy on Wrigley Field's outfield walls\u2014whose management doctrine is: \"Treat people as if they're coming into your home. Nothing is too much trouble.\"\n\nThe minor leagues reflect the nation's durable regional differences. South Carolinians, for example, are feisty\u2014they fired on Fort Sumter from places not far from The Joe\u2014so french fries are still called freedom fries at the ballyard. The real delicacies, however, are grilled turkey legs. A week's worth of protein for $5, they are not much smaller than the players' bats, and about as tender.\n\nThe Joe is almost in the backyard of The Citadel, a military school, and on game nights the patriotism is as warm as the beer is cold. Just before the first pitch on a recent evening, the teenager selling hot dogs and sodas at a concession stand out on the concourse behind the seats suddenly said, politely but firmly: \"One moment, please.\" Turning his back to the line of waiting customers, he took off his cap and faced the brick wall at the back of the stand, in the direction of the flagpole in center field. He stood ramrod straight with his hand over his heart while the National Anthem was sung. Even people in The Joe's parking lot come to attention when they faintly hear the distant sound of \"Oh, say can you see...\"\n\nAbout 40 percent of the players on the forty-man rosters of the thirty major-league clubs each spring are Sally League alumni, including, last April, Derek Jeter, Curt Schilling, Ivan Rodriguez, Luis Gonzalez, Scott Rolen, Andruw Jones, and John Smoltz. But nowhere near 40 percent of Sally League players get to the majors. Most were the best on their high school teams and are slow\u2014mercifully so\u2014to understand the severity of professional baseball's meritocracy.\n\nThe buses will not carry most of the RiverDogs to Trenton, let alone to Triple-A Columbus\u2014never mind the big leagues. But don't try to tell that to the pitcher who, when asked if his curve is as good as the Oakland A's Barry Zito's, confidently replies, \"Not yet.\" Says another, \"I want to be the best center fielder that ever came out of the Yankees' organization.\" Better than Joe DiMaggio and Mickey Mantle? \"Sure.\"\n\nSuch unrealism, and the reality of the oil fields, keeps young men getting on buses for late night rides to Motel 6s, which sometimes\u2014a major benefit for minor leaguers\u2014are near Outbacks.\n\n[AUGUST 18, 2005]\n\n# **Striving for Motel Years**\n\n**K** ISSIMMEE, FLORIDA\u2014It is past 10 p.m., dark, cold and damp. Traffic hisses on the highway in front of the nondescript motel that for five weeks is home to about 155 young adults. In back, in the gloom beneath the parking lot's dim lights, a dozen of them seem to have lost their minds. Actually, they are finding their dream.\n\nAnd doing their homework, far from home. Their choreography on the asphalt is simulating situations in baseball games\u2014runners on first and second, single to left; runner on second, ground ball to the pitcher. And on and on. The participants are pretending to be infielders, outfielders, a batter, base runners and\u2014this is the point of it all\u2014two umpires.\n\nThey have come to Jim Evans's Academy of Professional Umpiring. For six long days a week\u2014on the manicured infields of the Houston Astros' spring-training complex, and on the asphalt\u2014they are learning the craft of baseball's judicial branch.\n\nFor the few who will land jobs in the low rung of baseball's ladder\u2014say, Class A\u2014starting pay will be $1,900 a month, six months a year, plus $22 per diem. The home team finds motel rooms. For umpires, there are no home games.\n\nThe process of becoming an umpire is as severely meritocratic as the process of becoming a player. In the lower minor leagues, where only two umpires work each game, umpiring is as physically strenuous and mentally stressful as playing.\n\nOn a recent morning, Evans, who was a big-league umpire for twenty-eight seasons, showed a rapid-fire tape of twenty-five pitcher's moves with runners on base. His students had to instantly identify the balks. There were sixteen. Did the pitcher's knee bend? Did his shoulder turn, his glove twitch, his heel land improperly?\n\nThere are thirteen criteria for the correct stance of the home-plate umpire calling a pitch. Get only twelve right and Evans's instructors\u2014mostly minor-league umpires\u2014will correct you, vigorously. They reject the theory that a student's self-esteem is indispensable to learning.\n\nWhen umpiring the bases, the rule is \"angle over distance\"\u2014having the correct angle to see the play is more important than being close to the play. Students practice umpiring first base blindfolded, distinguishing the sounds of the fielder's foot hitting the bag and the ball hitting his glove. The final exam's 200 questions are like these:\n\n_Two outs, bases loaded. The batter hits a home run. He rounds first base and passes the runner who was on first. The runner from third touched home plate before the batter passed the runner from first, but the runner from second had not touched the plate at that time. (a) Four runs count because the infraction occurred during a dead ball situation. (b) No runs score because the batter made the third out. (c) This is a time play. One run scores. (d) Because the runner from first did not advance one base, the third out is considered a force out. No run scores.\"_ [The answer is c.]\n\n_No outs, runner on first. A hot grounder is hit up the middle. The shortstop fields the ball but throws wildly trying to retire the runner approaching second base. At the time the ball rolled into the first base dugout, the runner from first had just rounded second and the batter had touched first. Place the runners. (a) Both runners score. (b) Runner scores; batter is awarded third. (c) Runner scores; batter is awarded second. (d) Runner is awarded third; batter is awarded second.\"_ [The answer is d.]\n\nBaseball combines fame and failure. The best batters fail more than 60 percent of the time. But umpires, baseball's designated grown-ups, aspire to anonymous perfection. For an umpire, success is not being noticed. A Randy Johnson slider slides across a corner of a seventeen-inch-wide plate at ninety-four miles an hour. Imagine trying to be perfect on 260 pitches a game.\n\nSport\u2014strenuous competition structured and restrained by rules\u2014replicates the challenge of freedom and satisfies the human hunger for coherence. If players are mediocre, the result is mediocre baseball. If umpires are mediocre, the result is chaos.\n\nBaseball, national pastime of a litigious nation, allows arguments, within reason. So Evans's students learn how to manage rhubarbs. He teaches that strong voices and vigorous gestures\u2014body language is _language_ \u2014buttress authority. You especially need that if, like twenty-year-old Susan Reed, you are about the size of a bunt.\n\nPetite and laconic and just now wrapped in the armor of a home-plate umpire, she was a college student until, she says blandly, \"I forgot my major.\" Say what? \"My head broke open.\" Six months ago in the Missouri Ozarks, she was given the last rites of her church after being thrown through the sunroof of her SUV as it rolled over about a dozen times. Her painfully unhealed ankle slows her slightly gimpy run ninety feet up the line to call the play on a runner reaching third.\n\nWhat pulls her painfully down the third-base line? What pulls all of Evans's pupils to central Florida for Spartan living in a quest, against steep odds, for jobs that will mean many motel years of an endless road trip? One student says what most of them feel: \"I get chills every time I walk onto a ball field.\" Do you, reader, feel that way when you go to work?\n\n[FEBRUARY 16, 2004]\n\n# **Seeking Anonymous Perfection**\n\n**P** HILADELPHIA\u2014On a recent night here, as on most summer nights for thirty-seven years, Bruce Froemming went to work. He performed for about three hours in front of a large, attentive, and opinionated audience. His job involves about 290 snap judgments, any of which might infuriate thousands of people. He has done his job well if no one notices him doing it. His goal is anonymous perfection.\n\nAt less than five-foot-eight and more than 250 pounds, Froemming, sixty-seven, looks like he might have siblings at Stonehenge. But in this summer of dismal developments in sports\u2014a left fielder suspected of better hitting through chemistry; an NFL quarterback accused of dog fighting; an NBA referee guilty in a betting scandal; the Tour de France ruined by failed drug tests\u2014Froemming is a sight for sore eyes.\n\nNow in his thirty-seventh and final major-league season\u2014after thirteen in the minors\u2014he holds the record for most consecutive seasons of big league umpiring. His 5,127 games, through Sunday, are second only to Bill Klem (5,374), who did not have in-season vacations, which umpires did not get until 1979. If Froemming had not had twenty-eight days off each of the last twenty-eight summers, by now he would have worked nearly 6,000 games. He has spent more than forty-six thousand innings and approximately one and a half years on baseball diamonds, a well-spent life.\n\nPitch by pitch, baseball produces a rich sediment of numbers, such as: Every fourth day, Froemming is behind the plate. Over his career, the average game has involved about 290 pitches, so he has been behind the plate for more than 370,000 pitches. Has he given strict scrutiny\u2014a Supreme Court concept is apposite when discussing baseball's judicial branch\u2014to every one of them? Yes, he says.\n\nReally? His attention never flags during, say, a late inning in an August game in front of a small crowd in Tampa Bay? Never, he insists. \"Every pitch is important to someone.\"\n\nBaseball now has an electronic system for grading home plate umpires' performances. Froemming says it shows that umpires are right 94 percent of the time, but \"you get a lot of crap for the other 6 percent.\"\n\nEarly in his career, working behind the plate in a game involving Bob Gibson, the Cardinals' regal and ferocious Hall of Fame pitcher, Froemming made some calls that displeased Gibson. At the end of an inning, he walked past Froemming and quietly said, \"You're better than that.\" Froemming says, \"I remember that like it was yesterday.\"\n\nA story for Froemming: Rogers Hornsby, who _averaged_ .400 over five years, was facing a rookie pitcher who threw three pitches that he thought were strikes but that the umpire called balls. The rookie shouted a complaint to the umpire, who replied: \"Young man, when you throw a strike, Mr. Hornsby will let you know.\"\n\nSo, a question for Froemming: Is it true, as is said, that umpires give great hitters and pitchers the benefit of the doubt on close pitches? \"Not one bit,\" he says.\n\nOK, then, another question: Suppose it is the bottom of the ninth in the seventh game of a World Series, two outs, the potential tying run on third, two strikes on a right-handed batter. He starts to swing, tries to stop his bat, and the home-plate umpire calls the pitch a ball. But the catcher asks the home-plate umpire to ask the first-base umpire, who has a better vantage point, to say if the batter swung. The home-plate umpire accedes to this request. You, Froemming, are at first. You think the batter did swing. But seriously: Are you going to end a seven-game World Series on a check-swing appeal call? \"Yes.\"\n\nHe might. Consider September 2, 1972, when Froemming was behind the plate and the Cubs' Milt Pappas was one strike from doing what only fifteen pitchers have done\u2014pitch a perfect game, twenty-seven up, twenty-seven down.\n\nWith two outs in the ninth, Pappas quickly got an 0\u20132 count on the twenty-seventh batter. Then Froemming called the next three pitches balls. An agitated Pappas started walking toward Froemming, who said to the Cubs' catcher: \"Tell him if he gets here, just keep walking\"\u2014to the showers.\n\nPappas's next pitch was low and outside. Although he did get his no-hitter, the greater glory\u2014a perfect game\u2014was lost. Another kind of glory\u2014the integrity of rules\u2014was achieved.\n\nThe photographer Edward Steichen said that when God created his brother-in-law, the poet Carl Sandburg, God didn't do anything else that day. When the Intelligent Designer designed Froemming, He spent the rest of the day at a ballpark because He had done a good day's work by producing an archetype: The Umpire.\n\n[AUGUST 19, 2007]\n\n# **\"Where Baseball?\"**\n\n**O** n his first day of school after coming to Missouri from the Dominican Republic, Albert Pujols, then sixteen, went to the school office and in two words expressed everything on his mind: \"Where baseball?\" He found the field, and soon professional baseball found him.\n\nNow twenty-four, Pujols, the only player ever to finish in the top four in MVP voting in each of his first three seasons, is tied with Ralph Kiner and Mark McGwire for the most home runs (114) in his first three years. Here, from ESPN's Peter Gammons, are three players' stats after three seasons:\n\n* * *\n\n---\n\nBatting avg.: | .331 | .359 | .334\n\n* * *\n\nDoubles: | 111 | 120 | 138\n\n* * *\n\nHome runs: | 107 | 91 | 114\n\n* * *\n\nRBIs: | 432 | 378 | 381\n\n* * *\n\nThose are the numbers of Joe DiMaggio, Ted Williams, and Pujols.\n\nPujols is one of the remarkable Hispanic players who are among the more than 25 percent of major leaguers (and almost 50 percent of all professional baseball players) from outside the United States. This geyser of talent is one reason why the answer to the question \"Where baseball?\" is: soaring. This, in spite of the steroid crisis.\n\nIn a nation committed to better living through chemistry\u2014where Viagra-enabled men pursue silicone-contoured women\u2014the national pastime has a problem of illicit chemical enhancement. Steroids threaten the health of the 5 percent to 7 percent of players proved, by a mild regime of scheduled tests, to be using them. Steroids also endanger emulative young people. Further, steroids subvert what baseball is selling\u2014fair competition. And they strike at the pleasure of engagement with America's team sport with the longest history.\n\nThat pleasure is the comparison of players across many generations. Until now, comparisons have been complicated by only one substantial discontinuity in the game's nature\u2014that between the dead-and lively ball eras. Steroids threaten to define a second discontinuity\u2014a parenthesis\u2014in baseball's narrative.\n\nThe parenthesis opened in the 1990s. It must be closed to remove the cloud of suspicion that hovers over all players. Americans standing in stockings while their shoes and luggage are X-rayed at airports doubt that privacy considerations should prevent random, year-round testing, backed by serious sanctions, for illegal drugs that traduce baseball's integrity. The Players Association is too democratic, and its head, Don Fehr, is too intelligent, to continue to countenance the damage the status quo is doing.\n\nMeanwhile, fans are flocking to ballparks. Preseason ticket sales are at record levels\u2014the Cubs and Red Sox are essentially sold out for the season. Attendance probably will top 73 million, a record. In 1950, average attendance was 14,105 per game\u2014in 1970, just 15,130. Last year? 28,013.\n\nNinety-eight percent of NFL fans have _never_ been to a game. Many, probably most, baseball fans are made by going to ballparks. But baseball this year probably will be played in front of 50 million empty seats. So baseball is working on giveaways\u2014e.g., tickets to reward scholastic achievements\u2014to start young people on the baseball habit.\n\nSixteen of the thirty clubs have opened new parks since 1991. Fourteen new parks, counting San Diego's and Philadelphia's coming this month, have opened since Bud Selig became baseball's ninth commissioner in 1992. Aside from Wrigley Field (1914), Dodger Stadium (1962) is now the National League's oldest park.\n\nSelig has been\u2014baseball is a game of inches, but this is not a close call\u2014the greatest commissioner. His achievements include a quickened pace of games (in three seasons, twelve minutes have been shaved from the average game length), interleague play, the unbalanced schedule, three divisions, partial realignment, wild-card teams (the last two World Series winners were wild cards), increased revenue sharing ($250 million this year; $20 million in 1992) and the competitive balance tax on the highest payrolls. That tax and revenue sharing will cost the Yankees $81 million this year.\n\nCompetitive balance is improving: twenty-two different teams have made it to postseason play since 1995. In the last three years, eleven different teams (of a possible twelve) have played in League Championship Series. As late as September 7 last season, fourteen of fifteen games had play-off implications.\n\nThe special task force for the Commissioner's Initiative: Major League Baseball in the 21st Century, on which this columnist serves, knows that the national pastime, like the nation, is prone to hypochondria. But as Sparky Anderson, a greater manager than grammarian, once said, \"We try every way we can do to kill the game, but for some reason nothing nobody does never hurts it.\"\n\nOh, it gets hurt, but its recuperative powers are _Ruthian._ That American adjective means: prodigious.\n\n[APRIL 4, 2004]\n**Chapter Eight**\n\n**WONDERING**\n\n# **Incest at \"a Genetically Discreet Remove\"**\n\n**I** nvited by the University of Miami to address members of the class of 2005, the columnist repaid this courtesy by telling them that even though they surely had showered before donning their caps and gowns, each of them had about a trillion bacteria feeding on the 10 billion flakes of skin each of us sheds in a day. If each 2005 graduate were disassembled into his or her constituent atoms, each graduation gown would contain nothing but atomic dust. But as currently assembled, this stardust\u2014really: we are all residues of the big bang\u2014is living stuff, capable of sublime emotions like love, patriotism, and delight in defeating Florida State.\n\nThe body of every Miami graduate has about 10 thousand trillion cells, each containing a strand of DNA that, uncoiled, would extend about six feet. If that person's DNA were spliced into a single strand, it would extend 20 million kilometers\u2014enough to stretch from Miami to Los Angeles and back 2,270 times.\n\nSo says Bill Bryson, author of the delightful _A Short History of Nearly Everything._ According to him, everyone now alive contains some Shakespeare. That is, some of the physical stuff he was made of. And Julius Caesar's stuff, and Genghis Khan's and Charlemagne's. And Charlemagne's cook's. There are trillions of trillions of atoms in each of us, so lots\u2014probably billions\u2014of atoms have been recycled in each of us from Beethoven. In that sense we all are, as Bryson says, reincarnations.\n\nIndeed, each member of Miami's class of 2005 is related to every other member and to\u2014facts must be faced\u2014every graduate of Florida State. It took 2 parents to produce each of us, and 4 people to produce our parents. If we look back eight generations, to Lincoln's day, Bryson says that more than 250 people contributed to the creation of each of us. Look back to Shakespeare's day, and we are directly descended from 16,384 ancestors. Look back 64 generations, to the era of the Roman Empire, and we have a thousand trillion ancestors.\n\nBut wait. A thousand trillion people is thousands of times more than the number of human beings who have ever lived. So everyone is the product of a lot of incest\u2014but incest at what Bryson calls \"a genetically discreet remove.\" This extended single family\u2014humanity\u2014inhabits the little planet Earth, whose continents are wandering.\n\nBryson says Europe and North America are moving away from each other at about the speed that a fingernail grows\u2014about two yards in a normal human lifetime. The African continent is creeping northward and someday will squeeze the Mediterranean Sea out of existence and will shove up a chain of mountains as high as the Himalayas extending from Paris to Calcutta.\n\nThe Earth is restless partly because its molten core retains heat amazingly well: It has lost only about two hundred degrees in the 4 billion years since the planet coalesced. Not that we have come close to that core: Bryson says that if the planet were an apple, our underground exploration would not yet have broken the skin.\n\nThe sun around which Earth orbits is one of perhaps 400 billion stars in the Milky Way, which is a piddling galaxy next door to nothing much. There are perhaps 140 billion galaxies in the still-unfolding universe. If all the stars in the universe were only the size of the head of a pin, they still would fill Miami's Orange Bowl to overflowing _more than 3 billion times._\n\nWe should by now be used to strange thoughts. It has been one hundred years since June 1905, when Albert Einstein began publishing the scientific papers that taught us that gravity bends light, that space and time are warped, that matter and energy are interchangeable, that the mass of an object increases the faster it moves and that the experience of time is a function of speed.\n\nBut there is a not-at-all-strange reason that a Washington columnist would belabor Miami graduates with strange facts. It is this: The more they appreciate the complexity and improbability of everyday things\u2014including themselves\u2014the more they can understand the role that accidents, contingencies, and luck have played in bringing the human story to its current chapter. And the more they understand the vast and mysterious indeterminacy of things, the more suited they will be to participate in writing the next chapter.\n\nThis is so because the greatest threat to civility\u2014and ultimately to civilization\u2014is an excess of certitude. The world is much menaced just now by people who think that the world and their duties in it are clear and simple. They are certain that they know what, or who, created the universe and what this creator wants them to do to make our little speck in the universe perfect, even if extreme measures\u2014even violence\u2014are required.\n\nAmerica is currently awash in an unpleasant surplus of clanging, clashing certitudes. That is why there is a rhetorical bitterness absurdly disproportionate to our real differences. It has been well said that the spirit of liberty is the spirit of not being too sure that you are right. One way to immunize ourselves against misplaced certitude is to contemplate\u2014even to savor\u2014the unfathomable strangeness of everything, including ourselves.\n\n[MAY 23, 2005]\n\n# **An Intellectual Hijacking**\n\n**N** ot since the medieval church baptized, as it were, Aristotle as some sort of early\u2014very early\u2014church father has there been an intellectual hijacking as audacious as the attempt to present America's principal founders as devout Christians. Such an attempt is now in high gear among people who argue that the founders were kindred spirits with today's evangelicals, and that they founded a \"Christian nation.\"\n\nThis irritates Brooke Allen, an author and critic who has distilled her annoyance into _Moral Minority: Our Skeptical Founding Fathers._ It is a wonderfully high-spirited and informative polemic that, as polemics often do, occasionally goes too far. Her thesis is that the six most important Founders\u2014Franklin, Washington, Adams, Jefferson, Madison, and Hamilton\u2014subscribed, in different ways, to the watery and undemanding Enlightenment faith called deism. That doctrine appealed to rationalists by being explanatory but not inciting: it made the universe intelligible without arousing dangerous zeal.\n\nEighteenth-century deists believed there was a God but, tellingly, they frequently preferred synonyms for him\u2014\"Almighty Being\" or \"Divine Author\" (Washington) or \"a Superior Agent\" (Jefferson). Having set the universe in motion like a clockmaker, Providence might reward and punish, perhaps in the hereafter, but does not intervene promiscuously, or perhaps at all, in human affairs. (Washington did see \"the hand of Providence\" in the result of the Revolutionary War.) Deists rejected the Incarnation, hence the divinity of Jesus. \"Christian deist\" is an oxymoron.\n\nAllen's challenge is to square the six founders' often pious public words and behavior with her conviction that their real beliefs placed all six far from Christianity. Her conviction is well documented, exuberantly argued, and quite persuasive.\n\nWhen Franklin was given some books written to refute deism, the deists' arguments \"appeared to me much stronger than the refutations; in short, I soon became a thorough deist.\" Revelation \"had indeed no weight with me.\" He believed in a creator and the immortality of the soul, but considered these \"the essentials of every religion.\"\n\nWhat Allen calls Washington's \"famous gift of silence\" was particularly employed regarding religion. But his behavior spoke. He would not kneel to pray, and when his pastor rebuked him for setting a bad example by leaving services before communion, Washington mended his ways in his austere manner: he stayed away from church on communion Sundays. He acknowledged Christianity's \"benign influence\" on society, but no ministers were present and no prayers were uttered as he died a Stoic's death.\n\nAdams declared that \"phylosophy looks with an impartial Eye on all terrestrial religions,\" and told a correspondent that if they had been on Mount Sinai with Moses and had been told the doctrine of the Trinity, \"We might not have had courage to deny it, but We could not have believed it.\" It is true that the longer he lived, the shorter grew his creed, and in the end his creed was Unitarianism.\n\nJefferson, writing as a laconic utilitarian, urged his nephew to inquire into the truthfulness of Christianity without fear of consequences: \"If it ends in a belief that there is no god, you will find incitements to virtue in the comforts and pleasantness you feel in its exercise, and the love of others which it will procure you.\"\n\nMadison, always commonsensical, briskly explained\u2014essentially, explained away\u2014religion as an innate appetite: \"The mind prefers at once the idea of a self-existing cause to that of an infinite series of cause & effect.\" When Congress hired a chaplain, he said \"it was not with my approbation.\"\n\nIn 1781, the Articles of Confederation acknowledged \"the Great Governor of the World,\" but six years later the Constitution made no mention of God. When Hamilton was asked why, he jauntily said, \"We forgot.\" Ten years after the Constitutional Convention, the Senate unanimously ratified a treaty with Islamic Tripoli that declared the United States government \"is not in any sense founded on the Christian religion.\"\n\nAllen neglects one argument for her thesis that the United States is a \"secular project\": The Constitution mandates the establishment of a political truth by guaranteeing each state the same form of government (\"republican\"). It does so because the Founders thought the most important political truths are knowable. But because they thought religious truths are unknowable, they proscribed the establishment of religion.\n\nAllen succumbs to what her six heroes rightly feared\u2014zeal\u2014in her prosecution of today's religious zealots. In a grating anachronism unworthy of her serious argument, she calls the founders \"the very prototypes, in fact, of the East Coast intellectuals we are always being warned against by today's religious right.\" (Madison, an NPR listener? Maybe not.) When she says, \"Richard Nixon and George W. Bush, among other recent American statesmen,\" have subscribed to the \"philosophy\" that there should be legal impediments to an atheist becoming president, she is simply daft. And when she says that Bible study sessions in the White House and Justice Department today are \"a form of potential religious harassment that should be considered as unacceptable as the sexual variety,\" she is exhibiting the sort of hostility to the free exercise of religion that has energized religious voters, to her sorrow.\n\nTwo days after Jefferson wrote his letter endorsing a \"wall of separation\" between church and state, he attended, as he occasionally did, religious services in the House of Representatives. Jefferson was an observant yet unbelieving Anglican\/Episcopalian throughout his public life. This was a statesmanlike accommodation of the public's strong preference, which then as now was for religion to have ample space in the public square.\n\nChristianity, particularly its post-Reformation ferments, fostered attitudes and aptitudes associated with popular government. Protestantism's emphasis on the individual's direct, unmediated relationship with God, and the primacy of individual conscience and choice, subverted conventions of hierarchical societies in which deference was expected from the many toward the few. But beyond that, America's founding owes much more to John Locke than to Jesus.\n\nThe founders created a distinctly modern regime, one respectful of preexisting rights\u2014natural rights, not creations of the regime. And in 1786, the year before the Constitutional Convention constructed the regime, Jefferson, in the preamble to the Virginia Statute for Religious Freedom, proclaimed that \"our civil rights have no dependence on our religious opinions, any more than our opinions in physics or geometry.\"\n\nSince the founding, America's religious enthusiasms have waxed and waned, confounding Jefferson's prediction, made in 1822, four years before his death, that \"there is not a young man now living in the United States who will not die an Unitarian.\" In 1908, William Jennings Bryan, the Democrats' presidential nominee, said his Republican opponent, William Howard Taft, was unfit because, being a Unitarian, he did not believe in the Virgin Birth. The electorate yawned and chose Taft.\n\nA century on, when the most reliable predictor of a voter's behavior is whether he or she regularly attends church services, it is highly unlikely that Republicans would nominate a Unitarian. In 1967, when Governor George Romney of Michigan evinced interest in the Republican presidential nomination, his Mormonism was of little interest and hence was no impediment. Four decades later, the same may not be true if his son Mitt, also a Mormon, seeks the Republican nomination in 2008.\n\nIn 1953, the year before \"under God\" was added to the Pledge of Allegiance, President Dwight D. Eisenhower declared July 4 a day of \"penance and prayer.\" That day he fished in the morning, golfed in the afternoon, and played bridge in the evening. Allen and others who fret about a possibly theocratic future can take comfort from the fact that America's public piety is more frequently avowed than constraining.\n\n[OCTOBER 22, 2006]\n\n# **From Dayton, Tennessee, to Rhode Island's Committee on Fish and Game**\n\n**J** ohn Scopes attended high school in Salem, Illinois, where his commencement speaker was the town's most famous native son, William Jennings Bryan. Their paths would cross again.\n\nEighty years ago, Scopes, twenty-four, a high school football coach and general-science teacher, attended a meeting in Robinson's drugstore in Dayton, Tennessee. There, to the satisfaction of community leaders who thought that what was to come would be good for business, Scopes agreed to become the defendant in a trial testing Tennessee's law against teaching \"any theory that denies the story of the divine creation of man as taught in the Bible, and to teach instead that man has descended from a lower order of animals.\"\n\nSo began \"the most widely publicized misdemeanor case in American history.\" That is Edward J. Larson's description of the \"monkey trial\" in his 1997 Pulitzer Prize\u2013winning _Summer for the Gods: The Scopes Trial and America's Continuing Debate over Science and Religion._ With that debate again at a rolling boil, that book by Larson, professor of history and law at the University of Georgia, demonstrates that the trial pitted a modernism with unpleasant dimensions against a religious fundamentalism that believed, not without reason, that it was faithful to progressive values.\n\nBy 1925, many Christian geologists were comfortable with the fact that Earth has a long geologic history. They saw God immanent in the dynamic of appearance and disappearance of life forms. What most distressed some Christians was not the fact of evolution but the postulated mechanism\u2014a nature-red-in-tooth-and-claw randomness that erased God's purposefulness and benevolence.\n\nSince the publication of Charles Darwin's _Origin of Species_ in 1859, religiously motivated critics of the theory of evolution by natural selection had stressed the supposed failure of paleontology to supply the \"missing link\" that would establish continuity in the descent of man.\n\nDarwinism did not ignite a culture war until the 1920s, when high school education became common in the rural South, where Christian fundamentalism was strong. When school seemed to threaten children's souls, fundamentalists sought and found a champion in Bryan, a three-time Democratic presidential nominee and star of the prosecution team in Scopes's trial.\n\nScopes's defense, led by Clarence Darrow, stressed individual rights\u2014academic freedom. The prosecution stressed the community's right to control the curriculum of public schools. As a young man, Bryan had been a force for progressivism understood as, Larson says, a \"sunny faith in the curative power of majoritarian reforms,\" such as popular election of U.S. senators. So the vocabulary of progressivism served Bryan's argument that the issue was not what should be taught, but who should decide.\n\nHe, like many antievolutionists, believed that the idea of natural selection fueled merciless social Darwinism in domestic policies and militarism and imperialism among nations, justifying the survival of the fittest nations or races, and their dominion over lesser breeds. Modernists considered World War I a progressive crusade. Bryan resigned in protest as President Wilson's secretary of state.\n\nMany scientists at the time were, Larson says, receptive to the idea that we could channel human evolution through selective breeding. Some believed that acquired human characteristics could be inherited, hence improvement of the human race could be engineered. And many evolutionary biologists embraced eugenics. By 1935, thirty-five states had laws compelling the sexual segregation and sterilization of people considered unfit\u2014the mentally ill and retarded, habitual criminals, and epileptics.\n\nToday's proponents of \"intelligent design\" theory are doing nothing novel when they say the complexity of nature is more plausibly explained by postulating a designing mind\u2014aka God\u2014than by natural adaptation and selection. By 1925, Larson's book notes, \"Christian apologists had long regarded the intricate design of the eye as a 'cure for atheism.'\"\n\nThe problem with intelligent-design theory is not that it is false but that it is not falsifiable: Not being susceptible to contradicting evidence, it is not a testable hypothesis. Hence it is not a scientific but a creedal tenet\u2014a matter of faith, unsuited to a public school's science curriculum.\n\nThe Dayton jurors were eager to get on with their lives\u2014\"The peach crop will soon be coming in,\" one said\u2014and did not even sit down before deciding that Scopes was guilty. But Bryan did not believe penalties should be attached to antievolution laws\u2014\"We are not dealing with a criminal class\"\u2014and offered to pay Scopes's $100 fine.\n\nBryan died five days after the trial. Scopes left to study geology\u2014how fitting\u2014at the University of Chicago and became a petroleum engineer. The argument about science, religion, the rights of communities' majorities, and academic freedom rolled on, but not everywhere. When an antievolution bill was introduced in the Rhode Island Legislature, it was referred to the Committee on Fish and Game.\n\n[JULY 4, 2005]\n\n# **Earth: Not Altogether Intelligently Designed**\n\n**E** arth, that living, seething, often inhospitable and not altogether intelligently designed thing, has again shrugged, and tens of thousands of Pakistanis are dead. That earthquake struck ten months after the undersea quake that caused the December 2004 tsunami that killed 285,000 in Asia. Americans reeling from Katrina, and warned of scores of millions of potential deaths from avian flu, have a vague feeling\u2014never mind the disturbing rest of the news\u2014of pervasive menace from things out of control. Too vague, according to Simon Winchester.\n\nHis timely new book, _A Crack in the Edge of the World: America and the Great California Earthquake of 1906,_ teaches\u2014reminds, really\u2014that we should have quite precise worries about the incurably unstable ground on which scores of millions of Americans live. This almost certainly will result in a huge calamity, probably in the lifetime of most people now living.\n\nBefore the study of plate tectonics revolutionized geology just forty years ago, that science, Winchester writes, was concerned with \"rocks, fossils, faults and minerals that were scattered around simply and solely _on the surface of the earth._ \" But the surface consists of between\u2014depending how they are defined\u2014six and thirty-six floating plates, which Winchester calls \"rafts of solid rock.\" The plates' slow movements are powered by earth's molten innards, the boiling and bubbling radioactive residue of the planet's formation 4.5 billion years ago.\n\nThe plates grind against\u2014and slide up on, or plunge below\u2014one another. But not smoothly, which is the lethal problem. When friction freezes them for a while, stupendous energy builds up until, suddenly, plates unlock and the energy is released, sometimes in ways that seem to involve related spasms around the world.\n\nOn the last day of January 1906, that seismically dangerous year, an earthquake in Ecuador and Colombia of perhaps 8.8 magnitude on the Richter scale killed about 2,000. Sixteen days later there was a large Caribbean quake, followed five days later by one in the Caucusus, and on March 17 by one that killed 1,228 on the island of Formosa. On April 6, a ten-day eruption of the volcano Vesuvius began with rocks blown forty thousand feet into the air over Naples. Two days after Vesuvius subsided, San Francisco was knocked down, and twenty-six hundred acres of it were then devoured by three days of fires. About 3,000 San Franciscans died then, four months before a Chilean quake killed 20,000.\n\nSan Francisco's quake was smaller than the series of shocks around New Madrid, Missouri, over a few winter weeks in 1811\u20131812. They were strong enough to ring the bells in a Charleston, South Carolina, church that was later destroyed in that city's 1886 quake. Scores of millions of Americans now live on the unstable faults that shook mid-America in 1811\u20131812.\n\nFor San Francisco, the bad news is that the quake that killed sixty-three in 1989 (6.9 magnitude, compared to 8.3 in 1906) was caused not by the San Andreas fault, but by a neighboring one. So the big menace, the San Andreas, has not recently lurched, as it surely will because it is moving, sporadically, in grinding concert with the Pacific Plate. Since 1906 there have been only five major earthquakes along the 750 miles of the San Andreas, and none in Northern California. The U.S. Geological Survey estimates a 62 percent probability of a quake in that area of _at least_ 6.7 magnitude before 2032. Pondering the prosperous town of Portola Valley, south of San Francisco, exactly astride two of the most active strands of the San Andreas, Winchester, like many geologists who have warned the town, is fascinated by \"humankind's insistent folly in living in places where they shouldn't.\"\n\nAfter Earth's heavings subside, they reverberate in people's minds. Winchester says that when the 1755 Lisbon earthquake killed sixty thousand, \"priests roved around the ruins, selecting at random those they believed guilty of heresy and thus to blame for annoying the Divine, who in turn had ordered up the disaster. The priests had them hanged on the spot.\"\n\nThe 1883 eruption of Krakatoa in what is now Indonesia fueled the growth of an extremist strain of Islam, bent on purging society of impurities displeasing to God. That strain has twice recently been heard from in Bali terrorist attacks.\n\nSan Francisco's 1906 disaster prompted the explosive growth of a Pentecostal movement based in Los Angeles, a movement then embryonic but now mighty. Yet when A. P. Hotaling's whiskey warehouse survived San Francisco's postquake inferno, a wit wondered:\n\n_If, as some say, God spanked the town_\n\n_For being over frisky,_\n\n_Why did He burn the churches down_\n\n_And save Hotaling's whiskey?_\n\nGood question.\n\n[OCTOBER 11, 2005]\n\n# **Intelligent Design and Unintelligent Movies**\n\n**T** his summer's movie stars are not the usual bipeds, but other animals\u2014emperor penguins and grizzly bears. Their performances are pertinent to some ongoing arguments.\n\n_March of the Penguins_ raises this question: If an Intelligent Designer designed nature, why did it decide to make breeding so tedious for those penguins? The movie documents the seventy-mile march of thousands of Antarctic penguins from the sea to an icy breeding place barren of nutrition. These perhaps intelligently but certainly oddly designed birds march because they cannot fly. They cannot even march well, being most at home in the sea.\n\nIn temperatures of eighty below and lashed by one hundred mile per hour winds, the females take months to produce an egg while the males trek back to the sea to fatten up. Returning, the males are entrusted with keeping the eggs warm during foodless months while the females march back to the sea to fill their stomachs with nutriments they will share with the hatched chicks.\n\nThe penguins' hardiness is remarkable, as is the intricate choreography of the march, the breeding, and the nurturing. But the movie, vigorously anthropomorphizing the birds, invites us to find all this inexplicably amazing, even heroic. But the penguins are _made_ for that behavior in that place. What made them? Adaptive evolution. They have been \"designed\" for all that rigor\u2014meaning they have been shaped by adapting to many millennia of nature's harshness.\n\nSpeaking of harshness, Timothy Treadwell, college dropout, drug abuser, and failed actor, became a Southern California beach bum, had a heroin overdose, and then an epiphany: He must spend summers in Alaska \"protecting\" the grizzlies. The idea that these huge, robust carnivores need protection provided by this mentally wobbly narcissist\u2014a developmentally arrested adolescent in his forties\u2014would be funny, had not Alaska officials \"hauled four garbage bags of people out of the bear\" that devoured Treadwell and his girlfriend at the end of his fifth summer filming grizzly bears to which he gave cute names like Mr. Chocolate and Sgt. Brown.\n\nAbout half of _Grizzly Man,_ a documentary about Treadwell, is his film. The rest consists of interviews with, among others, a dry-eyed Alaskan who says \"he got what he was asking for.\" Although Treadwell has been described as an \"animal lover,\" the grandiosity of his self-praise as he preens and waxes metaphysical in front of his camera reveals that his great love was himself. His cooing of \"I love you\" at magnificently indifferent bears and his swooning over the warmth of bear feces (\"This was just inside of her!\") is as repulsive as his weeping over evidence that nature really is red in tooth and claw.\n\nEvidence such as bear cubs killed by mature male bears so the mother will stop lactating and be sexually available. Call that the Summer of Love, Alaska-style.\n\nTreadwell was not far from mental illness, or from a social stance\u2014nature is sweet, civilization is nasty\u2014not easily distinguished from mental illness. Call it Sixties Envy. So, see _Grizzly Man,_ then read T. C. Boyle's 2003 novel _Drop City._\n\nIt is about a bunch of Treadwells who, having dropped out and dropped acid, are addled but able to see that their California commune, based on \"voluntary primitivism,\" has become overrun with inane philosophy and the communards' sewage. Also, the county sheriff is angry. So a few of them decide to found Drop City North in Alaska. As one of these pioneers explains, in Alaska there are \"no rules,\" but there are food stamps.\n\nThere they plan \"to live the vegetarian ideal,\" but where will the cheese come from, now that a wolverine has eaten the communal goats? When an Alaskan explains that \"we eat bear and anything else we can get our hands on,\" a nature worshipper is horrified:\n\n\"'But to kill another creature, another living soul, a soul progressing through all the karmic stages to nirvana'\u2014she paused to slap a mosquito on the back of her wrist with a neat slash of her hand\u2014'that's something I just couldn't do.'\n\n\"'You just did.'\n\n\"'What? Oh, that. All right...I shouldn't have...but a bug is one thing...and like a bear is something else. They're almost human, aren't they?'\"\n\nThe movies and novel prompt a thought: Reality's swirling complexity is sometimes lovely, sometime brutal; its laws propel the comings and goings of life forms in processes as impersonal as Antarctica is to the penguins or the bears were to Treadwell or Alaska was to Drop City North. It is so grand that nothing is gained by dragging an Intelligent Designer into the picture for praise. Or blame.\n\n[AUGUST 28, 2005]\n\n# **The Pope, the Neurosurgeon, and the Ghost in the Machine**\n\n**A** concatenation of three events last week\u2014two protracted deaths and one literary birth\u2014was, as a stimulus to reflection, remarkable. Or, some will say, providential.\n\nIn a utilitarian, if humane, place, a hospice in Florida, a woman tangled in some toils of modernity\u2014medical technology, and the machinery of litigation and legislation\u2014died because, after fifteen years in a vegetative state, and supposedly out of respect for her autonomy, nutrition and hydration were withdrawn from her. In any other age\u2014even a generation or two ago\u2014she would not have become an appendage of devices that can sustain the body, or most of it, while a part of it, the brain, has stopped performing the functions essential to personhood, as normally understood.\n\nThere was another death, in a place of purposeful splendor, the Vatican, which was built as a defiant assertion of confident authority in response to the tempest of the Reformation. The foremost contemporary steward of an ancient faith, Pope John Paul II did more than anyone in our lifetimes to make vivid the task of defining respect for life in the context of modern science.\n\nIt used to be said matter-of-factly that a person who died \"gave up the ghost.\" But are we still confident there is, in the language of a modern philosopher, a \"ghost in the machine\"?\n\nLast week a gifted novelist published a new work that is, among other things, a materialist's manifesto. Ian McEwan lives in London, where _Saturday_ is set. His protagonist, Henry Perowne, has the sensibility of today's post-Christian Europe. Perowne believes we are, in a sense, machines\u2014matter and nothing more. He thinks as those people do who say, \"I do not have a body, I am a body.\" Perowne is a neurosurgeon.\n\nWith sharpened steel a neurosurgeon slices and splices and pares physical matter to palliate injuries to minds\u2014to consciousness. Pharmacology also can do that. McEwan writes:\n\n\"A man who attempts to ease the miseries of failing minds by repairing brains is bound to respect the material world, its limits, and what it can sustain\u2014consciousness, no less. It isn't an article of faith with him, he knows it for a quotidian fact, the mind is what the brain, mere matter, performs.\"\n\nPerowne, the voice of scientifically sophisticated secularism, and presumably of McEwan, almost lyrically, and rightly, exhorts us to appreciate the \"wonder of the real.\" One can, however, imagine a faint, droll smile flickering on the strong, intelligent face of John Paul II were he to have read those almost casually appended three words\u2014\"consciousness, no less.\" He might think to himself: The materialist must not tarry, he must hurry on, because as Emerson said, when skating on thin ice, safety lies in speed.\n\nThis pope might have read Emerson, and it is easy to imagine him, before frailty conquered his body, keeping abreast of contemporary literature, including McEwan. Before he was John Paul II he was Karol Wojtyla, a skiing poet, playwright, and philosopher. And a defining theme of his papacy was the compatibility of faith and science, the explainer of reality. The explainer, but only up to a point, so far.\n\nPerowne reads an arresting sentence from Darwin\u2014\"There is a grandeur in this view of life\"\u2014and says, yes, indeed:\n\n\"Endless and beautiful forms of life, such as you see in a common hedgerow, including exalted beings like ourselves, arose from physical laws, from war or nature, famine and death. This is the grandeur. And a bracing kind of consolation in the brief privilege of consciousness.\"\n\nBut a bemused John Paul II, no stranger to materialism, dialectical and otherwise, might have responded: There you go again\u2014that word _consciousness._ What is the grandeur in the spectacle, however interesting, of the blind, brute, violent necessity of physical laws at work? Is consciousness of an existence supposedly governed by such laws really much of a privilege?\n\nIt is, John Paul II might have responded to Perowne, one thing to say that a cosmic sneeze, aka the big bang (never mind what, or who, or Who, produced that) set all this physical-law-governed necessity in motion, and this process resulted in matter, including the cooling, unstable and sometimes lethally violent lump of it called Earth. But how much progress has science really made in explaining how some matter came to be conscious of itself?\n\nSuch arguments are not just hardy perennials in philosophy, they are part of today's political arguments. Arguments about school curricula (evolution, intelligent design). And about conundrums that modern science confronts us with concerning the end of life and the waning of consciousness. And about aborting what some people call \"fetal material\" as it grows toward consciousness.\n\nAlas, death has removed from the unending and probably unendable debates about respect for consciousness and life the intellectual pope who, one imagines, appreciated the profundity, and perhaps the finality, of the philosophic jest: \"Of course I believe in free will\u2014I can't help it.\"\n\n[APRIL 11, 2005]\n\n# **How Biology Buttresses Morality, Which Conforms to...Biology**\n\n**S** cience is reshaping the argument about whether nature or nurture is decisive in determining human destinies, and about what the answer means for social policy. Consider a fascinating new report arguing the scientific evidence for the importance of \"authoritative communities\"\u2014groups, religious or secular, devoted to transmitting a model of the moral life.\n\nThe report is from the thirty-three research scientists, children's doctors and mental health and youth services professionals comprising a commission jointly sponsored by the Dartmouth Medical School, the Institute for American Values, and the YMCA of the USA. The report's conclusion is in its title: human beings are _Hardwired to Connect._\n\nIn an era of increasing prosperity, the evidence of children's failures to thrive\u2014depression, anxiety, substance abuse, conduct disorders\u2014is also increasing. Pharmacological and psychotherapeutic responses to such deteriorating mental and social health are necessary but insufficient. Also needed is recognition of how environmental conditions\u2014the _social_ environment\u2014contribute to childhood suffering.\n\nThe problem is a deficit of connectedness. The deficit is the difference between what the biological makeup of human beings demands and what many children's social situations supply in the way of connections to other people, and to institutions that satisfy the natural need for moral and spiritual meaning.\n\nThe need expresses itself in religious cravings\u2014the search for moral meaning and an openness to the possibility of a transcendent reality. The need is natural in that it arises from \"our basic biology and how our brains develop.\" The report draws upon the science of infant attachment, and of brain development, particularly during adolescence, when the brain changes significantly.\n\nThe report argues that our understanding of children's difficulties is thwarted by the assumption that each child's problems are exclusively personal and individual, thereby ignoring social and communal factors. In fact the report argues that we are \"biologically primed\" for finding meaning through attachments to others.\n\nThe need for meaning is increasingly discernible in the basic structure of the brain. \"The idea,\" says Allan N. Schore of the UCLA School of Medicine, \"is that we are born to form attachments, that our brains are physically wired to develop in tandem with another's, through emotional communication, beginning before words are spoken.\"\n\nFurthermore, the report says, social environments that meet\u2014or defeat\u2014this need \"affect gene transcription and the development of brain circuitry.\" And \"a social environment can change the relationship between a specific gene and the behavior associated with that gene.\" A child's \"relational context,\" says Schore, \"imprints into the developing right brain either a resilience against or a vulnerability to later forming psychiatric disorders.\"\n\n\"The biochemistry of connection\" will seem too, well, deflating for some people's comfort. The report cites, for example, another study that says oxytocin, a hormone, enters a woman's bloodstream during sexual intercourse, childbirth, and lactation, promoting, the report says, \"emotional intimacy and bonding (also sometimes known as 'love').\" In men, marriage\u2014sexual and emotional intimacy with a spouse\u2014seems to lower testosterone levels, thereby lowering the \"biological basis for violent male behavior and male sexual promiscuity.\"\n\nSo biology, it seems, buttresses important moral conventions. And they may have evolved in conformity to biological facts.\n\nThe scientific fact, if such it is, that religious expression is natural to personhood does not vindicate any religion's truth claims. A naturalistic hypothesis is that the emotions of religious experience have neurobiological origins: The brain evolved that way to serve individual and group survival.\n\nIn any case, the social utility of religion remains. And there may be a biological basis for religious affiliation reducing the risk of certain pathologies, and even enhancing immune systems.\n\nThe most basic authoritative community, the family, is the most crucial. Its decline weakens the other institutions of civil society. The result is a thinness of social connectedness, and what Tocqueville warned was a risk of American individualism\u2014each person confined \"entirely within the solitude of his own heart.\"\n\n_Hardwired to Connect_ suggests that there is no simple \"versus\" in \"nature versus nurture.\" There is a complex interaction, which means, among many other important things, that IQ is not a simple genetic inheritance, it is a function of that inheritance and the influence on it of a context of connections.\n\nThe implication for governance is that social policies should foster the health of authoritative communities, especially given the fact that the yearning for such communities among adolescents often takes the form of gang membership. And evidently the Bush administration's belief in the wisdom of delivering social services through faith-based institutions is not just a matter of faith.\n\nBut then, the utility of faith does not establish its validity.\n\n[SEPTEMBER 21, 2003]\n\n# **The Space Program's Search for...Us**\n\n**P** ASADENA, CALIFORNIA\u2014On September 8, a spacecraft will insert into the atmosphere over Utah a glider that will spiral to Earth carrying a sealed canister containing gusts of solar wind captured over the past two years. The wind contains dust, gases, and other possible evidence of the dynamics of the solar system, dynamics that have somehow given rise to the splendor of...us. NASA's name for the canister project: the Genesis mission.\n\nSeptember 8 will be just another day at the office here on the campus of the Jet Propulsion Laboratory, where office work is not mundane but otherworldly. JPL\u2014an appendage of, but not contiguous to, Caltech\u2014may be the only place on the planet where you can gather around a lunch table with people who, in a sense, work on another planet.\n\nOn a recent day, some of them were behind a laboratory at a pile of sand that resembles the surface of Mars. They were trying to drive a rover, like the two currently on Mars. The JPL scientists were trying to operate one with the kind of mechanical defect\u2014an inoperative drive wheel\u2014that is giving a slight limp to one of the rovers on the red planet 110 million miles away.\n\nOne Mars rover was landed in a crater\u2014what one scientist calls \"an interplanetary hole-in-one.\" Both were expected to rove about six hundred meters, but they have covered three thousand. Their batteries are recharged daily by solar panels; they already have lasted twice as long as had been expected and may last ten times longer.\n\nEarth, which is constantly changing, became home for life 4 billion years ago. We know neither the conditions then nor the processes by which life ignited. However, Mars may have had an early history like Earth's. One question the rovers may answer is: Were there, long ago, pools of standing water\u2014standing for hundreds of millions of years\u2014where life could have developed?\n\nThe rovers' arms, manipulated at JPL, put instruments in contact with rocks and read their mineral contents. By drilling into rocks, through several billion years worth of settled dust, the rovers have found sediments that were formed in bodies of water. Within the next decade, samples may be put robotically in canisters and launched off Mars to rendezvous with an orbiting spacecraft for a six-month trip to Earth.\n\nIt would be understandable if the people at JPL were jubilant, having just\u2014after a seven-year flight\u2014precisely inserted the Cassini spacecraft into the rings of Saturn. This multinational project will include putting instruments on Titan, Saturn's largest moon, which may have some tantalizing similarities to Earth of 4 billion years ago.\n\nBut people here know that all their marvels\u2014JPL's deep-space control center is monitoring thirty-five space ventures\u2014are performed against a backdrop of deepening public indifference. And cosmology's human capital is declining as young scientists choose other career paths.\n\nThe public's diminishing capacity for astonishment is astonishing. Perhaps second only to Einstein's question (Did God have a choice in the creation of the world?) is this one: How did matter, which is what we are, become conscious, then curious? Not all clues can be found on Earth.\n\nCuriosity is why a Voyager spacecraft launched in 1977 is now 8.5 billion miles away. It is, in the scheme of things, just next door: traveling now at 1 million miles per day, it would have to continue for forty thousand more years just to be closer to another star than to our sun. Still, here in our wee solar system\u2014our little smudge on the skies of uncountable billions of galaxies\u2014Voyager's and JPL's other undertakings must be measured against Einstein's axiom: \"All our science, measured against reality, is primitive and childlike\u2014and yet it is the most precious thing we have.\"\n\nAll space programs search for...us. For, that is, understanding of how we came to be. Does that mean space exploration amounts to species narcissism? Yes, and that is an excellent thing. It is noble to strive to go beyond the book of Genesis and other poetry, to scientific evidence about our origins, and perhaps destiny.\n\nThe Scottish physicist James Clerk Maxwell (1831\u20131879), an early authority on Saturn's rings, had, as cosmologists should, a poetic bent:\n\n_At quite uncertain times and places,_\n\n_The atoms left their heavenly path,_\n\n_And by fortuitous embraces,_\n\n_Engendered all that being hath._\n\nThe phrase \"fortuitous embraces,\" although lovely, is not explanatory. Knowledge, tickled from the heavens, is the business of a small band of possible explainers\u2014the people of JPL and NASA, government at its best.\n\n[AUGUST 26, 2004]\n\n# **Nuclear Waste: That's Us**\n\n**T** he Department of Transportation deals with the movement of things, which is important. The Department of Agriculture deals with food, which is vital. However, the National Aeronautics and Space Administration deals with the origin, nature, and meaning, if any, of the universe. Attention should be paid.\n\nSpace lost its hold on America's imagination after the last lunar expedition in 1972. But the really exciting research had just begun, with the 1965 discovery that the universe is permeated with background radiation which confirmed that a big bang had indeed set what are now distant galaxies flying apart.\n\nA famous aphorism holds that the most incomprehensible thing about the universe is that it is comprehensible. It is remarkably so because of advances in particle physics and mathematics. And because of magnificent telescopes, like the Hubble, which is now eleven years old and due to cease functioning in 2010. Operating above the filter of Earth's atmosphere, it \"sees\" the past by capturing for analysis light emitted from events perhaps\u2014we cannot be sure how fast the universe is expanding\u201412 billion to 14 billion years ago.\n\nAstronomy is history, and NASA's Next Generation Space Telescope, coming late in this decade, will see even nearer the big bang of 13 billion to 15 billion years ago. That was when, in a trillionth of a trillionth of a trillionth of a second, the big bang inflated from a microscopic speck to all that now can be seen by NASA's wondrous instruments.\n\nMankind is being put in its place, but where is that? Mankind felt demoted by Copernicus's news that this cooled cinder, Earth, is not the center of the universe. Now Martin Rees, Britain's Astronomer Royal, in his new book, _Our Cosmic Habitat,_ adds insult to injury: \"particle chauvinism\" must go. All the atoms that make us are, it is truly said, stardust. But Rees puts it more prosaically: They are nuclear waste from the fuel that makes stars shine.\n\nSo, is life a cosmic fluke or a cosmic imperative? Because _everything_ is a reverberation from the big bang, what is the difference between fluke and imperative?\n\nRees says our universe is \"biophilic\"\u2014friendly to life\u2014in that molecules of water and atoms of carbon, which are necessary for life, would not have resulted from a big bang with even a slightly different recipe. That recipe was cooked in the universe's first one-hundredth of a second, when its temperature was a hundred thousand million degrees centigrade. A biophilic universe is like Goldilocks' porridge, not too hot and not too cold\u2014just right.\n\nHere cosmology is pressed into the service of natural theology, which rests on probability\u2014actually, on the stupendous improbability of the emergence from chaos of complexity and then consciousness. Natural theology says: A watch implies a watchmaker, and what has happened in the universe\u2014the distillation of the post-big-bang cosmic soup into particles, then atoms, then, about a billion years ago, the first multicellular organisms that led, on Earth, to an oxygen-rich atmosphere and eventually to us\u2014implies a Creator with a design so precise.\n\nPerhaps. But not necessarily, unless you stipulate that no consequential accident is an accident. \"Biological evolution,\" says Rees, \"is sensitive to accidents\u2014climatic changes, asteroid impacts, epidemics and so forth\u2014so that, if Earth's history were to be rerun, its biosphere would end up quite different.\" There is a lot of stuff in the universe\u2014the estimated number of stars is 10 followed by 22 zeros. But as to whether there are other planets with life like Earth's, Rees says the chance of two similar ecologies is less than the chance of two randomly typing monkeys producing the same Shakespearean play.\n\n\"Eternity,\" says Woody Allen, \"is very long, especially toward the end.\" The end of our universe\u2014long after our sun has died, 5 billion years from now\u2014is certain to be disagreeable.\n\nIn his book on the universe's infancy _(The First Three Minutes),_ Steven Weinberg concludes that \"there is not much of comfort\" in cosmology. It indicates that Earth, \"a tiny part of an overwhelmingly hostile universe,\" is headed for \"extinction of endless cold or intolerable heat,\" either an unending expansion or a fiery collapse backward\u2014a big crunch.\n\nYet research like NASA's is its own consolation. \"The effort to understand the universe is,\" says Weinberg, \"one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy.\" Not a negligible mission for NASA.\n\n[MARCH 24, 2002]\n\n# **The Loudest Sound in Human Experience**\n\n**I** ra Gershwin didn't know the half of it. He said the Rockies may crumble, Gibraltar may tumble. But terra firma itself is far from firm.\n\nEven the continents are wandering, half an inch to four inches a year. Earth is a work in violent progress. The engine of its evolution is heat\u2014boiling gas, molten rock, and other stuff\u2014left over from the planet's formation 4.5 billion years ago. The heat frequently bursts through Earth's crust, although rarely as catastrophically as it did 120 years ago on the island of Krakatoa.\n\nIf Simon Winchester is correct in his new book\u2014 _Krakatoa: The Day the World Exploded, August 27, 1883_ \u2014the current trial in Indonesia of accused perpetrators of last year's terrorist bombing in Bali may be part of the lingering reverberation of the volcanic eruption\u2014the loudest sound in modern human experience, heard three thousand miles away\u2014that made an island disappear.\n\nBillions of tons of material\u2014six cubic miles of it\u2014were hurled 120,000 to 160,000 feet in the air. They filtered sunlight, lowering Earth's temperature and creating spectacular sunsets that for months inspired painters and poets.\n\nAnd in the East Indies outpost of the Dutch empire, where a notably relaxed and tolerant Islamic faith had long flourished, Krakatoa, by terrifying and dispossessing people, may have catalyzed the much fiercer form of Islam that fused with anticolonialism. It is alive and dealing death today.\n\nAlthough the people of the East Indies will be forgiven for not appreciating this at the time, Winchester says volcanoes are part of what makes this planet hospitable to humans. They do not erupt so promiscuously as to render the planet unfit for life. And by churning Earth's mantle, they bring fertile soil and useful minerals to the surface, thereby sustaining the outer earth and the biosphere. For a while.\n\nAs Earth heads for frigid lifelessness, the leakage of heat from Earth's interior causes currents of matter to flow\u2014movements measured in millimeters a year\u2014above the molten core and below the crust. Just as, writes Winchester, \"one sees working in a vat of vegetable soup simmering on the stovetop.\"\n\nScience in the 1960s at least explained what had long pricked curiosity\u2014the matching concavity of Africa's west coast and the convexity of South America's east coast. According to the study of plate tectonics, there are, depending on how they are defined, between six and thirty-six rigid plates on Earth's surface. In \"subduction zones,\" where one plate slips beneath another, the descending plate pulls down untold billions of tons of material and water. This fuels white-hot seas of soup in immense chambers, from which energy seeks to break through Earth's surface.\n\nWhich is what happened in 1883 in the archipelago that now is Indonesia. Krakatoa's eruption resulted in the destruction of 165 villages and the death of 36,417 people. Most died not from the searing ash, pumice, and gas but from giant sea waves produced by Earth's spasm.\n\nThe shock wave circled Earth seven recordable times. Sea surges were detectable in the English Channel. Three months after the eruption, firemen in Poughkeepsie, New York, scrambled in search of what they thought was an immense conflagration that caused the sky to glow. Actually, the glow was light refracted by Krakatoa's debris.\n\nThe first major catastrophe to occur after the invention of the telegraph and undersea cables, Krakatoa produced an intimation of the \"global village\" seventy-seven years before Marshall McLuhan coined that phrase to describe the world-contracting effect of television. Krakatoa was, Winchester argues, \"the event that presaged all the debates that continue to this day: about global warming, greenhouse gases, acid rain, ecological interdependence.\" Suddenly the world seemed to be less a collection of isolated individuals and events and more \"interconnected individuals and perpetually intersecting events.\"\n\nAs an epigraph for his book, Winchester chose this from a W. H. Auden poem written in 1944, when the world was in agony and, unbeknownst to Auden, potentially world-shattering knowledge was being acquired at Los Alamos, New Mexico:\n\n_At any given instant_\n\n_All solids dissolve, no wheels revolve,_\n\n_And facts have no endurance\u2014_\n\n_And who knows if it is by design or pure inadvertence_\n\n_That the Present destroys its inherited self-importance?_\n\nGeology has joined biology in lowering mankind's self-esteem. Geology suggests how mankind's existence is contingent on the geological consent of the planet. Although the planet is hospitable for the moment, it is indifferent\u2014eventually it will be lethally indifferent\u2014to its human passengers.\n\n[MAY 22, 2003]\n\n# **L = BB + pw + BC\/BF**\n\n**O** ne hundred years ago, a minor Swiss civil servant, having traveled home in a streetcar from his job in the Bern patent office, wondered: What would the city's clock tower look like if observed from a streetcar racing away from the tower at the speed of light? The clock, he decided, would appear stopped because light could not catch up to the streetcar, but his own watch would tick normally.\n\n\"A storm broke loose in my mind,\" Albert Einstein later remembered. He produced five papers in 1905, and for physicists, the world has never been the same. For laypeople, it has never _felt_ the same.\n\nIn his book _Einstein's Cosmos,_ Michio Kaku, professor of theoretical physics at the City University of New York, makes Einstein's genius seem akin to a poet's sensibility. Einstein, says Kaku, was able to \"see everything in terms of physical pictures\"\u2014to see \"the laws of physics as clear as simple images.\"\n\nHitherto, space and time were assumed to be absolutes. They still can be for our everyday business, because we and the objects we deal with do not move at the speed of light. But since Einstein's postulate of relativity, measurements of space and time are thought to be relative to speed.\n\nOne implication of Einstein's theories did have thunderous practical implications: Matter and energy are interchangeable, and the mass of an object increases the faster it moves. In the most famous equation in the history of science, energy equals mass multiplied by the speed of light squared. A wee bit of matter can be converted into a city-leveling amount of energy.\n\nIn the 1920s, while people were enjoying being told that space is warped and it pushes things down (that is the real \"force\" of gravity), Einstein became an international celebrity of a sort not seen before or since. Selfridges department store in London pasted the six pages of an Einstein paper on a plate-glass window for passersby to read. Charlie Chaplin said to him, \"The people applaud me because everyone understands me, and they applaud you because no one understands you.\"\n\nThe precision of modern scientific instruments makes possible the confirmation of implications of Einstein's theories\u2014e.g., the universe had a beginning (the big bang) and its expansion is accelerating; time slows in a large gravitational field and beats slower the faster one moves; the sun bends starlight from across the sky and there are black holes so dense that they swallow light. Does all this bewilder you? The late Richard Feynman, winner of the Nobel Prize in physics, said, \"I think I can safely say that nobody understands quantum mechanics.\"\n\nThree years ago we learned that the Milky Way galaxy, which is next door, contains a black hole weighing as much as 2 million suns. \"Thus,\" says Kaku, \"our moon revolves around the earth, the earth revolves around the sun, and the sun revolves around a black hole.\" Can this story have a happy ending?\n\nScience offers no guarantees. Astronomy evicted us from our presumed place at the center of the universe many centuries before we learned that \"center\" is unintelligible in an expanding universe where space and time are warped. And before nineteenth-century biology further diminished our sense of grandeur by connecting us with undignified ancestors, eighteenth-century geology indicated that seashells unearthed on mountain tops proved that Earth has a longer, more turbulent and unfinished history than most creation stories suggest. December 26, 2004, brought another geological challenge to the biblical notion of an intervening, caring God.\n\nEinstein's theism, such as it was, was his faith that God does not play dice with the universe\u2014that there are elegant, eventually discoverable laws, not randomness, at work. Saying \"I'm not an atheist,\" he explained:\n\n\"We are in the position of a little child entering a huge library filled with books in many different languages. The child knows someone must have written those books. It does not know how. It does not understand the languages in which they are written. The child dimly suspects a mysterious order in the arrangement of the books but doesn't know what it is.\"\n\nA century on from Einstein's \"miracle year,\" never mind _E = mc_ 2. Try this: L = BB + pw + BC\/BF. Meaning: Life equals the big bang, followed by lots of paperwork, ending with either a big crunch, as the universe collapses back on itself, or a big freeze as it expands forever.\n\nA bad ending? Compared to what? Everything, as has been said since Einstein, is relative.\n\n[JANUARY 6, 2005]\n\n# **Wonder What We Are For? Wondering**\n\n**A** TOP MAUNA KEA, ON HAWAII'S BIG ISLAND\u2014On a clear day, you can see almost forever. With the help of adaptive optics, almost back to the beginning of this universe. And it is usually clear here at 13,796 feet above sea level, and above half of the atmosphere's oxygen. That is why the W. M. Keck Observatory's two telescopes, primarily operated by the University of California and the California Institute of Technology, are here, far from urban lights and above much of the atmosphere, which, although it makes the stars twinkle prettily, does so by distorting light.\n\nHence the need for adaptive optics. This technology became available for civilian science when the end of the Cold War led to the declassification of some devices developed for the Strategic Defense Initiative (SDI).\n\nThe Keck telescopes\u2014the world's largest\u2014are gathering light produced shortly (as these things are reckoned; about 800 million years) after the big bang, just under 14 billion years ago. Analysis of the light, which can be done by astronomers working anywhere, yields information about the life cycle of stars. (Grim news: our star, the sun, is doomed, so we are, too, in less than 2 billion years.) The Keck telescopes have detected more than sixty planets orbiting other stars. Ten years ago, the only planets we knew of were those orbiting our own sun.\n\nIt is axiomatic that not only is the universe stranger than we know, it is stranger than we can know. But one reason the Keck telescopes are significantly augmenting our store of knowledge is the application to astronomy of adaptive optics developed for SDI. SDI's challenge is to target, from space, ballistic missiles launched on Earth. This requires making ultraprecise measurements from space, through the distortions of Earth's atmosphere. Astronomy's challenge involves looking outward\u2014analyzing light that is distorted by the atmosphere before it reaches telescopes on Earth.\n\nThe Keck telescopes each weigh three hundred tons, stand eight stories tall, and involve operations of more precision than those of the finest wristwatch. They can gather 13-billion-year-old light that is 500 million times fainter than the naked eye can see. They gather the light using a primary mirror ten meters (thirty-three feet) in diameter, composed of thirty-six hexagonal segments, each engineered to conform to within a millionth of an inch of single continuous surface.\n\nBut the _really_ remarkable device is a mirror about the size of a man's hand. Distortions in the gathered light are removed by bouncing the light off this mirror, which has four hundred pistons operated by tiny, computer-driven motors that make adjustments in the mirror's surface 642 times _a second._\n\nFrom 1609, when Galileo built a refracting telescope (a lens assisting the naked eye), until the Hubble space-based telescope was launched in 1990, the atmosphere complicated astronomy. However, Hubble, which cost more than all other telescopes in history combined, does not make Earth-based telescopes anachronistic.\n\nHubble and its successors\u2014next comes the Next Generation Space Telescope\u2014operate in the cold vacuum that is space. But a multinational consortium has proposed an Overwhelmingly Large Telescope that would gather light on Earth with two thousand panels of mirrors in an apparatus the size of a football field. Ever-better land-and space-based telescopes will find tantalizing hints about how the expansion of the universe (actually, _this_ universe; there may be many others) began\u2014a big bang?\u2014and whether it will continue to expand or will collapse back on itself in a big crunch.\n\nIn any case, Earth's fate is not going to be pretty, so what's the use in wondering? Because wondering is what we are for.\n\nMauna Kea is a dormant volcano. About forty miles southeast of here, lava from the Kilauea volcano, boiling with heat left over from Earth's formation 4.35 billion years ago, has recently been spilling across a highway and into the ocean. To stand a few hundred feet from the stream of lava plunging into the Pacific, amid the searing heat and sulfurous fumes, is to sense what the Keck Observatory, in its very different setting, explores\u2014the violent impermanence that permeates the entire universe.\n\n\"We are curious people,\" says Keck Observatory director Frederic Chaffee matter-of-factly. \"And the universe is an amazing place.\" The most amazing things in it are the curious creatures. They have evolved literally from stardust, becoming conscious beings capable of building\u2014indeed, their glory is that they are, in a sense, incapable of not building\u2014mountaintop telescopes, silhouetted against the edge of the atmosphere, searching for clues as to how all this started and how it will end.\n\n[SEPTEMBER 1, 2002]\n**Chapter Nine**\n\n**MATTERS OF LIFE AND DEATH**\n\n# **Golly, What Did Jon _Do?_**\n\n**W** hat did Jon Will and the more than 350,000 American citizens like him do to tick off the American College of Obstetricians and Gynecologists? It seems to want to help eliminate from America almost all of a category of citizens, a category that includes Jon.\n\nBorn in 1972, Jon has Down syndrome. That is a congenital condition resulting from a chromosomal defect that causes varying degrees of mental retardation and some physical abnormalities, such as low muscle tone, small stature, a single crease across the center of the palms, flatness of the back of the head, and an upward slant to the eyes (when Jon was born, Down syndrome people were still commonly called Mongoloids). There also is increased risk of congenital heart defects, childhood leukemia, and Alzheimer's disease. Down syndrome, although not common, is among the most common congenital anomalies\u201447.9 per 100,000 births (compared with 77.7 with cleft lips or palates, which also can be diagnosed in utero, and which sometimes result in abortions).\n\nAs women age, their risk of having a Down syndrome baby increases. It has become standard practice for women older than thirty-five to be offered genetic counseling and diagnostic testing. But because of the higher fertility rates of women under thirty-five, such women have 80 percent of Down syndrome babies. So new ACOG guidelines recommend that all pregnant women, regardless of age, be offered such counseling and testing.\n\nThe ACOG guidelines are formally neutral concerning what decisions parents should make on the basis of the information offered. But what is antiseptically called \"screening\" for Down syndrome is, much more often than not, a search-and-destroy mission: At least 85 percent of pregnancies in which Down syndrome is diagnosed are ended by abortions.\n\nMedicine now has astonishing and multiplying abilities to treat problems of unborn children in utero, but it has no ability to do anything about Down syndrome (the result of an extra twenty-first chromosome). So diagnosing Down syndrome can have only the purpose of enabling\u2014and, in a clinically neutral way, of encouraging\u2014parents to choose to reject people like Jon as unworthy of life. And as more is learned about genetic components of other abnormalities, search-and-destroy missions will multiply.\n\nNothing\u2014 _nothing_ \u2014in the professional qualifications of obstetricians and gynecologists gives them standing to adopt policies that predictably will have, and seem intended to have, the effect of increasing abortions in the service of an especially repulsive manifestation of today's entitlement mentality\u2014every parent's \"right\" to a perfect baby. Happily, that mentality is not yet universal: 214 American families are looking for Down syndrome children to adopt.\n\nJon, a sweet-tempered man, was born the year before _Roe v. Wade_ inaugurated this era of the casual destruction of preborn babies. And he was born just as prenatal genetic tests were becoming routine. Since then, it has become routine to abort babies like Jon, because they are like Jon. Without this combination of diagnostic advances and moral regression, there would be more people like Jon, and the world would be a sweeter place.\n\nAmerica has, however, become a more congenial, welcoming place for its Down syndrome citizens who have escaped \"screening.\" On the second day of Jon's life, the hospital's geneticist asked his parents if they intended to take him home. Nonplussed, they answered that taking a baby home seemed like the thing to do.\n\nJon was born at the end of the era in which institutionalization of the retarded was considered morally acceptable, but in what was still an era of gross ignorance: In the first year of Jon's life, a network-television hospital drama featured a doctor telling parents of a Down syndrome newborn that their child would probably never be toilet-trained. But ignorance lingers. There are doctors who still falsely counsel parents that a Down syndrome person will never read, write, or count change. Such doctors should not try to get between Jon and his _USA Today_ sports section.\n\nIn 1972, the odds were heavily against Jon's living as long as he already has lived. Just twenty-five years ago, the life expectancy of Down syndrome people was twenty-five. Today, because of better health care, better mental stimulation in schools and homes, and better community acceptance, their life expectancy is fifty-six.\n\nJon has a disability, but he also has some things most men would like to have\u2014season tickets for Nationals and Orioles baseball, Redskins football, Capitals hockey, and Georgetown University basketball. He gets to and from games (and to his work three days a week for the Nationals at RFK Stadium) by himself, taking public transportation to and from his apartment.\n\nJon experiences life's three elemental enjoyments\u2014loving, being loved, and ESPN. For Jon, as for most normal American males, the rest of life is details.\n\n[JANUARY 29, 2007]\n\n# **The Long Dying of Louise Will**\n\n**N** EWPORT BEACH, CALIFORNIA\u2014The long dying of Louise Will ended here recently. It was time. At ninety-eight, her body was exhausted by disease and strokes. Dementia, that stealthy thief of identity, had bleached her vibrant self almost to indistinctness, like a photograph long exposed to sunlight.\n\nIt is said that God gave us memory so we could have roses in winter. Dementia is an ever-deepening advance of wintery whiteness, a protracted paring away of personality. It inflicts on victims the terror of attenuated personhood, challenging philosophic and theological attempts to make death a clean, intelligible, and bearable demarcation.\n\nIs death the soul taking flight after the body has failed? That sequence\u2014the physical extinguished, the spiritual not\u2014serves our notion of human dignity. However, mental disintegration mocks that comforting schemata by taking the spirit first.\n\nIn the very elderly, the mind can come and go, a wanderer in time, and a disintegrating personality can acquire angers and jagged edges that are, perhaps, protests against a growing lightness of being. No one has come back from deep in that foreign country to report on life there. However, it must be unbearably frightening to feel one's self become light as a feather, with inner gales rising.\n\nDementia slowly loosens the sufferer's grip on those unique tokens of humanity, words. An early sign is a forgetfulness that results in repetitiveness, and fixation on the distant past.\n\nFor a while, one of Louise's insistently recurring memories was of spring 1918, a war year, and eastbound troop trains passing through Greenville, Pennsylvania. When the trains stopped, residents offered candy and magazines to the soldiers\u2014but not to black units. That infuriated Louise's father, whose fury was a fine memory for Louise to have among those of a father who died at age forty-four.\n\nTo the end, even when virtually without speech, Louise could recognize her children, could enjoy music and being read from love letters written seventy-five years ago by Fred, her future husband. She could even laugh, in spite of the tormenting chasm between her remaining cognition and the prison of her vanished ability to articulate.\n\nIn 1951, in Champaign, Illinois, for her ten-year-old son, she made a mother's sacrifice: She became a White Sox fan so she could converse with the argumentative Cubs fan who each evening dried the dishes as she washed. Even after much of her stock of memories had been depleted, she dimly knew that the name Nellie Fox (a second baseman) once meant something playful.\n\nThe aging that conquered Louise was, like war, a mighty scourge, and, like war, elicited nobility from those near its vortex. The nearest was Fred Will, who died eight years ago, at the end of his ninth decade.\n\nA few years before his death, Fred, a reticent romantic, whose reticence may have been an effect of his tinge of melancholy, shared with his children some poetry he had written for Louise, including this from 1933:\n\n_The warm sun_\n\n_beams through the clear air_\n\n_Upon glistening leaves._\n\n_And the birds_\n\n_sweep in long arcs_\n\n_Over the green grass._\n\n_They seem to say,_\n\n_\"This might last forever!\"_\n\n_But it doesn't._\n\nBut it lasted more than six decades, which is forever, as foreverness is allotted to us.\n\nA retired professor of philosophy, Fred probably knew what Montaigne, quoting Cicero, meant when he said that to study philosophy is to prepare to die. Fred was, strictly speaking, philosophic about his wife's affliction. A common connotation of \"philosophic\" is placid acceptance of what can be comprehended but not altered. However, Fred's philosophic response to the theft of his wife by aging was much richer than mere stoicism grounded in fatalism. It was a heroic act of will, arising from clear-sightedness about the long trajectory of Louise's life.\n\nHe understood this stern paradox: Families seared by a loved one's dementia face the challenge of forgetting. They must choose to achieve what dementia inflicts on its victims\u2014short-term memory loss. They must restore to the foreground of remembrance the older memories of vivacity and wit.\n\n\"All that we can know about those we have loved and lost,\" Thornton Wilder wrote, \"is that they would wish us to remember them with a more intensified realization of their reality. What is essential does not die but clarifies. The highest tribute to the dead is not grief but gratitude.\" Louise, released from the toils of old age and modern medicine, is restored to clarity.\n\n[JULY 13, 2006]\n**ACKNOWLEDGMENTS**\n\n**T** his collection of my writings, like the previous seven, reflects the assistance I have received from superb professionals. As an undergraduate at Georgetown University, Jed Donahue worked part-time for me in my Washington office. Now that he has become a senior editor at the Crown Publishing Group, I am, in a sense, working for him, and pleased to be. I am now in my fourth decade of writing for _Newsweek_ 's back page. The magazine has never been in better hands than it is under its current editor, Jon Meachem. Alan Shearer, editorial director of the Washington Post Writers Group, makes all of my columns, and most of my paragraphs, better. Seth Meehan was an invaluable assistant in my office during most of the period covered by this volume. He has gone on to graduate school at Boston College, where he is preparing to be a professor of history. My loss is academia's gain. My current assistants\u2014colleagues, really\u2014are Sarah Walton and Greg Reed. They make it a pleasure to come to work in the morning and make me reluctant to leave in the evening. Mary Longnecker is now in her third decade of making my business run smoothly. Finally, Schuyler Hawkins and Daniel Phillips are the latest in a long line of very able interns I have found at Georgetown University, which has earned my gratitude for bringing such young men to Washington and to my attention.\n**PERMISSIONS**\n\nGrateful acknowledgment is made to the following for permission to reprint previously published material:\n\n**BasicBooks:** Excerpts from _The First Three Minutes_ by Steven Weinberg, copyright \u00a9 1977, 1988 by Steven Weinberg. Afterword to second edition copyright \u00a9 1993 by Steven Weinberg. Reprinted by permission of BasicBooks, a member of the Perseus Books Group.\n\n**Broadway Books:** Excerpts from _The Life and Times of the Thunderbolt Kid_ by Bill Bryson, copyright \u00a9 2006 by Bill Bryson. Reprinted courtesy of Broadway Books, a division of Random House, Inc.\n\n**Doubleday:** Excerpts from _Other Things and the Aardvark_ by Eugene J. McCarthy, copyright \u00a9 1970 by Eugene J. McCarthy. Reprinted courtesy of Doubleday, a division of Random House, Inc.\n\n**The Free Press:** Excerpt from _General Washington's Christmas Farewell_ by Stanley Weintraub, copyright \u00a9 2003 by Stanley Weintraub. Reprinted by permission of The Free Press, a division of Simon & Schuster, Inc.\n\n**Gotham Books:** Excerpts from _Eats, Shoots & Leaves_ by Lynne Truss, copyright \u00a9 2003 by Lynne Truss. Reprinted courtesy of Gotham Books, a division of Penguin Group (USA) Inc.\n\n**Henry Holt and Company, LLC.:** Excerpts from _What's the Matter with Kansas: How Conservatives Won the Heart of America_ by Thomas Frank, copyright \u00a9 2004 by Thomas Frank. Reprinted by permission of Henry Holt and Company, LLC.\n\n**_Hoover Digest_ and Professor Timothy Garton Ash:** Excerpt from \"Weaving a New Identity\" by Professor Timothy Garton Ash ( _Hoover Digest,_ 2007:3). Reprinted by permission of _Hoover Digest_ and Professor Timothy Garton Ash.\n\n**Alfred A. Knopf and John Updike:** Excerpt from \"Hub Fans Bid Kid Adieu\" by John Updike ( _The New Yorker,_ October 22, 1960). Reprinted in _Assorted Prose_ by John Updike, copyright \u00a9 1965, copyright renewed 1993 by John Updike. Reprinted by permission of Alfred A. Knopf, a division of Random House, Inc., and John Updike.\n\n**Little, Brown and Company:** Excerpt from \"Scott-King's Modern Europe\" from _The Complete Stories of Evelyn Waugh,_ copyright \u00a9 1998 by the Estate of Evelyn Waugh. Reprinted courtesy of Little, Brown and Company, a division of Hachette Book Group USA.\n\n**William F. S. Miles:** Excerpts from \"Intellectualizing the Holocaust\" by William F. S. Miles ( _The Chronicle of Higher Education,_ June 14, 2002). Reprinted by permission of William F. S. Miles.\n\n**Michael Pack:** Excerpted quotes from _Rediscovering George Washington_ (Public Broadcasting System, July 4, 2002), copyright \u00a9 2002 by Manifold Productions, Inc. Reprinted by permission of Michael Pack.\n\n**Palgrave Macmillan:** Excerpt from _The General Theory of Employment, Interest, and Money_ by John Maynard Keynes; excerpts from _LeMay_ by Barrett Tillman, copyright \u00a9 2007 by Barrett Tillman. Reprinted by permission of Palgrave Macmillan, United Kingdom.\n\n**Princeton University Press:** Excerpt from _Our Cosmic Habitat_ by Martin Rees, copyright \u00a9 2001 by Princeton University Press. Reprinted by permission of Princeton University Press.\n\n**Random House, Inc.:** \"For the Time Being,\" copyright \u00a9 1944, copyright renewed 1972 by W. H. Auden, from _Collected Poems_ by W. H. Auden; excerpt from _The Progress Paradox: How Life Gets Better While People Feel Worse_ by Gregg Easterbrook, copyright \u00a9 2003 by Gregg Easterbrook. Reprinted by permission of Random House, Inc.\n\n**Simon & Schuster Adult Publishing Group:** Excerpts from _At Canaan's Edge: America in the King Years, 1965\u20131968_ by Taylor Branch, copyright \u00a9 2006 by Taylor Branch; excerpts from _William F. Buckley, Jr.: Patron Saint of the Conservatives_ by John B. Judis, copyright \u00a9 1988 by John Judis. Reprinted by permission of Simon & Schuster Adult Publishing Group.\n\n**Simon & Schuster, Inc.:** Excerpts from _Opening Day: The Story of Jackie Robinson's First Season_ by Jonathan Eig (New York: Simon & Schuster, Inc., 2007); excerpts from _President Kennedy: Profile of Power_ by Richard Reeves (New York: Simon & Schuster, Inc., 1993). Reprinted by permission of Simon & Schuster, Inc.\n\n**Sony\/ATV Music Publishing:** Excerpt from \"Streets of Bakersfield\" by Homer Joy, copyright \u00a9 1973 by Sony\/ATV Songs LLC. All rights administered by Sony\/ ATV Music Publishing, 8 Music Square West, Nashville, TN 37203. All rights reserved. Reprinted by permission of Sony\/ATV Music Publishing.\n\n**Viking Penguin:** Excerpts from _American Dream_ by Jason DeParle, copyright \u00a9 2004 by Jason DeParle; excerpt from \"Conscience\" from _Mayflower_ by Nathaniel Philbrick, copyright \u00a9 2006 by Nathaniel Philbrick. Reprinted by permission of Viking Penguin, a division of Penguin Group (USA) Inc.\n\n**The _Washington Post:_** Excerpt from \"Museum with a Message,\" copyright \u00a9 2003 by the _Washington Post._ Reprinted by permission of the _Washington Post._\n\n**W. W. Norton & Company, Inc.:** Excerpts from _Ronald Reagan: Fate, Freedom, and the Making of History_ by John Patrick Diggins, copyright \u00a9 2007 by John Patrick Diggins; excerpts from _Einstein's Cosmos_ by Michio Kaku, copyright \u00a9 2004 by Michio Kaku; excerpts from _The Blind Side: Evolution of a Game_ by Michael Lewis, copyright \u00a9 2006. Reprinted by permission of W. W. Norton & Company, Inc.\n**Other Books by George F. Will**\n\nWITH A HAPPY EYE BUT...  \n _America and the World, 1997\u20132002_\n\nBUNTS  \n _Curt Flood, Camden Yards, Pete Rose, and Other Reflections on Baseball_\n\nTHE WOVEN FIGURE  \n _Conservatism and America's Fabric, 1994\u20131997_\n\nTHE LEVELING WIND  \n _Politics, Culture, and Other News, 1990\u20131994_\n\nRESTORATION  \n _Congress, Terms Limits, and the Recovery of Deliberative Democracy_\n\nSUDDENLY  \n _The American Idea Abroad and at Home, 1986\u20131990_\n\nMEN AT WORK  \n _The Craft of Baseball_\n\nTHE NEW SEASON  \n _A Spectator's Guide to the 1988 Election_\n\nTHE MORNING AFTER  \n _American Successes and Excesses, 1981\u20131986_\n\nSTATECRAFT AS SOULCRAFT  \n _What Government Does_\n\nTHE PURSUIT OF VIRTUE AND OTHER TORY NOTIONS\n\nTHE PURSUIT OF HAPPINESS, AND OTHER SOBERING THOUGHTS\nAll essays with the exception of the seven credited below were originally published in _Newsweek_ and the _Washington Post,_ and are reprinted courtesy of their publishers.\n\n\"The Game's Gifted Eccentrics\" (August 15, 2004), \"An Intellectual Hijacking\" (October 22, 2006), \"Raising Michael Oher\" (November 11, 2006), \"Remember 1908!\" (April 1, 2007), and \"Roberto Clemente: 'We Think He Can Hit'\" (May 7, 2006) were originally published in _The New York Times Book Review,_ and are reprinted here courtesy of the _New York Times._\n\n\"James Madison\" originally appeared in the _Princeton Alumni Weekly_ (January 23, 2008).\n\n\"Barry Goldwater: Cheerful Malcontent\" originally appeared as the foreword to _The Conscience of a Conservative_ by Barry Goldwater, edited by C. C. Goldwater (Princeton University Press, 2007). Reprinted by permission of Princeton University Press.\n\nCopyright \u00a9 2008 by George F. Will\n\nAll rights reserved.\n\nPublished in the United States by Crown Forum, an imprint of the Crown Publishing Group, a division of Random House, Inc., New York.\n\nwww.crownpublishing.com\n\nCrown Forum and the Crown Forum colophon are trademarks of Random House, Inc.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nWill, George F.\n\nOne man's America : the pleasures and provocations of our singular nation \/ George F. Will.\u20141st ed.\n\n1. United States\u2014Civilization. 2. National characteristics, American. 3. Popular culture\u2014United States. 4. Political culture\u2014United States. 5. United States\u2014Social conditions. 6. United States\u2014Biography. 7. Will, George F. 8. Journalists\u2014United States\u2014Biography. I. Title.\n\nE169.1.W4895 2008\n\n973\u2014dc22 2007051281\n\neISBN: 978-0-307-44935-1\n\nv3.0\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nAlso by Lloyd Clark\n\n_The Battle of the Tanks: Kursk, 1943_\n\n_Crossing the Rhine: Breaking into Nazi Germany 1944 and 1945_\n\n_Anzio: Italy and the Battle for Rome\u20141944_\n\n# **BLITZKRIEG**\n\nMyth, Reality, and Hitler's Lightning War\u2013France, 1940\n\n## LLOYD CLARK\n\nCopyright \u00a9 Lloyd Clark 2016\n\nCover design by Gretchen Mergenthaler  \nCover photograph: WWII Stuka (Hans Schaller), 1940. \u00a9 AKG images  \nAuthor photo \u00a9 Adam Hollier\n\nFirst published in hardback in Great Britain in 2016 by Atlantic Books, an imprint of Atlantic Books Ltd.\n\nThe moral right of Lloyd Clark to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act of 1988. All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher, except by a reviewer, who may quote brief passages in a review. Scanning, uploading, and electronic distribution of this book or the facilitation of such without the permission of the publisher is prohibited. Please purchase only authorized electronic editions, and do not participate in or encourage electronic piracy of copyrighted materials. Your support of the author's rights is appreciated. Any member of educational institutions wishing to photocopy part or all of the work for classroom use, or anthology, should send inquiries to Grove Atlantic, 154 West 14th Street, New York, NY 10011 or permissions@groveatlantic.com.\n\nFirst published by Grove Atlantic, September 2016\n\n_Published simultaneously in Canada  \nPrinted in the United States of America_\n\nISBN 978-0-8021-2513-2  \neISBN 978-0-8021-9034-5\n\nAtlantic Monthly Press  \nan imprint of Grove Atlantic  \n154 West 14th Street  \nNew York, NY 10011\n\nDistributed by Publishers Group West\n\ngroveatlantic.com\n\n16 17 18 19 10 9 8 7 6 5 4 3 2 1\n_For my parents \u2013 John and Pauline Clark \u2013 and Jasper Clark_\n\n# Contents\n\n_List of Illustrations_\n\n_List of Maps_\n\nPrologue\n\nIntroduction\n\nChapter 1: Ingredients\n\nChapter 2: Plans\n\nChapter 3: Final Preparations\n\nChapter 4: 10 May \u2013 Forward\n\nChapter 5: 11\u201312 May \u2013 To the Meuse\n\nChapter 6: 13 May \u2013 Crossing the Meuse\n\nChapter 7: 14\u201315 May \u2013 Counter-Attacks and Exploitation\n\nChapter 8: 16\u201320 May \u2013 Crisis of Command and the Coast\n\nChapter 9: 21\u201324 May \u2013 Arras, Weygand and the Halt Order\n\nChapter 10: 25 May\u20134 June \u2013 Withdrawal and Evacuation\n\nChapter 11: 5\u20138 June \u2013 _Fall Rot_ and Resilience\n\nChapter 12: 9\u201322 June \u2013 Driving South, Paris and Armistice\n\nEpilogue\n\nConclusion\n\n_Order of Battle_\n\n_Endnotes_\n\n_Select Bibliography_\n\n_Acknowledgements_\n\n_Index_\n\n# Illustrations\n\n**Section one**\n\nAdolf Hitler arrives in Poland ( _Bundesarchiv, Bild 101I-013-0060-20_ )\n\nHitler at OKH headquarters in Berlin ( _Bundesarchiv, Bild 146-1971-070-61_ )\n\nFranz Halder and von Brauchitsch ( _Bundesarchiv, Bild 183-H27722_ )\n\nGeneral Fedor von Bock ( _Bundesarchiv, Bild 183-1986-0226-500_ )\n\nGeneral Gerd von Rundstedt and Maximilian von Weichs ( _Bundesarchiv, Bild 101I-056-1643-29A_ )\n\nHeinz Guderian with Georg-Hans Reinhardt ( _Bundesarchiv, Bild 146-1988-120-10_ )\n\nGeneral Erich Hoepner ( _Bundesarchiv, Bild 146-1971-068-10_ )\n\nGeneral Ewald von Kleist ( _Bundesarchiv, Bild 183-1986-0210-503_ )\n\nGeneral Friedrich Kirchner ( _Bundesarchiv, Bild 146-1997-018-33)_\n\nGeneral Erich von Manstein ( _Bundesarchiv, Bild 183-H01757_ )\n\nFrench prime minister Paul Reynaud ( _\u00a9 IWM, HU70629_ )\n\nGeneral Maurice Gamelin and General Lord Gort (\u00a9 _IWM, O158_ )\n\nWinston Churchill talks to General Lord Gort ( _\u00a9 IWM, O190_ )\n\nGeneral Alphonse Joseph Georges inspects men from the BEF's Royal Inniskilling Fusiliers with General Lord Gort ( _\u00a9 IWM, F3959_ )\n\nA German armoured fighting vehicle factory ( _Bundesarchiv, Bild 183-L04352_ )\n\nFrench troops prepare defences (\u00a9 _IWM, GSA Album 17 850, Image 145_ )\n\nGerman pilots' briefing at an airfield near Arras ( _Bundesarchiv, Bild 101I-383-0338-03_ )\n\nFrench fighter patrol (\u00a9 _IWM, GSA Album 17 850, Image 76_ )\n\nJoachim Meissner, Rudolf Witzig and Walter Koch listen to Hitler ( _Bundesarchiv, Bild 146-1993-033-26_ )\n\nA Bren Carrier passes across the Belgian border (\u00a9 _IWM, F4348_ )\n\n**Section two**\n\nTwo light German tanks in the Ardennes ( _Bundesarchiv, Bild 101I-382-0248-33A_ )\n\nThe pontoon bridge over the Meuse at Gaulier ( _Bundesarchiv, Bild 146-1978-062-24_ )\n\nA French army signals post ( _\u00a9 IWM, GSA Album 15 848, Image 219_ )\n\nSergeant Walther Rubarth ( _Bundesarchiv, Bild 183-L05202_ )\n\nHermann Balck ( _Bundesarchiv, Bild 183-L05202_ )\n\nGerman tanks advance across Belgian terrain ( _Bundesarchiv, RH 82 Bild-00058_ )\n\nHeinz Guderian in his command and communications halftrack ( _Bundesarchiv, 101I-769-0229-11A_ )\n\nRommel with General Hermann Hoth (\u00a9 _IWM, RML182_ )\n\nFrench tank crews run to their Renault D2 tanks (\u00a9 _IWM, HU75654_ )\n\nRommel and a group of officers from 7th Panzer Division ( _Bundesarchiv, 146-1972-045-02_ )\n\nHeinz Guderian receives the remains of a captured French Army standard (\u00a9 _IWM, MH10935_ )\n\nBritish troops march through Dunkirk (\u00a9 _IWM, MH24141_ )\n\nMajor General Victor Fortune glowers at Rommel ( _\u00a9 IWM, RML342_ )\n\nSurrendered Belgian troops being goaded by Wehrmacht troops (\u00a9 _IWM, HU75891_ )\n\nFrench refugees heading south (\u00a9 _IWM, GSA Album 16 849, Image 270_ )\n\nTwo Schutzpolizisten visit a damaged bunker ( _Bundesarchiv, 121-0486_ )\n\nAn RAF Fairey Battle attacks a German transport column ( _\u00a9 IWM, C1737_ )\n\nGerman infantry march through Paris ( _Bundesarchiv, 183-L11769_ )\n\nGerman infantry and armour pushing deeper into France ( _Bundesarchiv, 101I-769-0228-22_ )\n\nMarshal Philippe P\u00e9tain greets Hitler ( _\u00a9 IWM, HU76027_ )\n\n# Maps\n\nThe Invasions of Poland, Denmark and Norway, 1939\u201340\n\nDefences, Deployments and Plans, May 1940\n\n10\u201311 May 1940\n\nThe Ardennes, 10\u201312 May 1940\n\nApproach and Crossing of Meuse by Panzer Corps Guderian, 13 May 1940\n\nGerman Advances and French Counter Attack, 14 May 1940\n\nAdvance to the Coast, 15\u201320 May 1940\n\nThe Weygand Line and _Fall Rot,_ 5\u201318 June 1940\n\nThe Advance to the Armistice, 25 June 1940\n\nThe Invasions of Poland, Denmark and Norway, 1939\u201340\n\nDefences, Deployments and Plans, May 1940\n\n10\u201311 May 1940\n\nThe Ardennes, 10\u201312 May 1940\n\nApproach and Crossing of Meuse by Panzer Corps Guderian, 13 May 1940\n\nGerman Advances and French Counter Attack, 14 May 1940\n\nAdvance to the Coast, 15\u201320 May 1940\n\nThe Weygand Line and _Fall Rot,_ 5\u201318 June 1940\n\nThe Advance to the Armistice, 25 June 1940\n\n# Prologue\n\nHORDES OF PARISIANS choked the streets on their journey to the Gare de l'Est. A mother used the time to tell her thirty-seven-year-old son Andr\u00e9 how proud she was of him and, looking into his tired eyes, told him that his decision to fight was the correct one. The son felt no fear, no excitement, just a desire to get to the front now that he had made his decision. On arriving at the station, he was keen to be on his way. He scythed through the masses which blocked his route to the platform from where his train was due to depart. He made slow progress through fretful soldiers who embraced ashen-faced women. Both were on the cusp of an entirely new and shocking reality of war.\n\nWhat would Andr\u00e9's fate be? His mother could not help but recall a similar farewell to her brother, who had left her to fight the Prussians in 1870 and returned with a mortal illness. Andr\u00e9 had enlisted as a private soldier \u2013 just another _poilu_ with a wife and young children \u2013 ignoring the opportunity to take a comfortable staff job in a headquarters away from the fighting. In fact, the only influence he had used as under-secretary of state for war was to ensure that he was assigned to a unit defending his beloved Lorraine. Consequently, on 1 August 1914, Andr\u00e9 Maginot slipped off to war without public fanfare or farewell.\n\nAndr\u00e9's destination was Verdun, a city which lay in a vulnerable salient created by the German thrust through the heavily wooded Ardennes. The French had fallen back towards Sedan, Stenay and Verdun, but only Verdun held, protected by enhancements to Vauban's original seventeenth-century fortifications. The totemic city remained in French hands that autumn, an achievement assisted in a small way by Andr\u00e9's reconnaissance patrols. At six foot three inches tall, he was not an inconspicuous figure on the battlefield clad in his horizon-blue jacket and red pantaloons but, unlike many others, he remained unharmed after fifty patrols. On 9 November 1914, however, the newly promoted sergeant's good fortune came to an end. The day after he had been awarded the M\u00e9daille Militaire for his personal courage and leadership, Andr\u00e9 was leading a patrol when he was struck by a great blow to his left leg and then another to his knee. Lying in a pool of blood and in agony, he expected to die.\n\nSix years later, on 10 November 1920, walking with the assistance of canes, his left leg useless, Andr\u00e9 Maginot, who was now minister of pensions, found himself in Verdun's cold, damp underground citadel. In exquisite pain, he watched as Auguste Thien, a young soldier of the 123rd Regiment of Infantry who had been invited to make a choice, pointed to the sixth of eight caskets containing the remains of an unknown soldier. Three years later, on 11 November 1923, the casket was interred below the Arc de Triomphe in Paris in a ceremony during which Maginot, now minister of war, ignited an eternal flame on the tomb.\n\nIt was five years since the armistice that had ended the First World War. France was still in mourning, traumatized by the hideous destructiveness of protracted industrial warfare and fractured, just like the bodies and minds of so many of its surviving veterans. It was an experience that Maginot was determined to spare the next generation and he supported Marshal Philippe P\u00e9tain, France's senior soldier and the Victor of Verdun, in his desire to fortify the common border with Germany. Completed after his death, the defences took Andr\u00e9's name \u2013 the Maginot Line \u2013 and so it was that a brave and principled man became forever associated with his nation's defeat.\n\n# Introduction\n\nThey were conquerors, and for that you want only brute force \u2013 nothing to boast of when you have it, since your strength is just an accident arising from the weakness of others.\n\nJoseph Conrad, _Heart of Darkness_\n\nTHE FALL OF FRANCE and the Low Countries in May and June 1940 was one of the most remarkable military campaigns in Western history and, arguably, one of the most distinguished victories ever won. The fighting in those weeks shocked the world, not for its protracted horror but for the lack of it. The Germans managed to conjure up a decisive victory over a first-class military power that was predicted by very few and which proved to be a significant step towards an increasingly global and destructive conflict. As such, it is an event that has retained a great vibrancy to all of those who have recognized its military and historical importance. There are few other campaigns that have raised more questions about its origins, conduct and outcome, and offered so many contrasting answers in response. Part of its fascination \u2013 and complexity \u2013 stems from the fact that its main participants had been fighting each other over much of the same terrain a little over two decades before. It is, of course, almost impossible not to contrast images of the Stuka dive-bomber and dust-coated panzer in a war of great movement with others of the heavy artillery and mud-encrusted rifles of that war of position fought earlier in the century. Whether either pair of images is an accurate representation of either conflict is a matter of debate, but the fact that they have seeped into Western consciousness is irrefutable and challenges us to remember that the two conflicts were separated by just a single generation and, moreover, that the 1940 campaign took place within living memory of the seminal French defeat by Prussia in 1870.\n\nThe First World War was but the latest episode in a long history of Franco-German enmity. It was an antagonism that \u2013 together with Germany's chronic anxieties about the East \u2013 was central to the history and development of Europe. It was an hereditary enmity nourished by territorial rivalry, the aspiration to dominate continental Europe, national pride and shame, and a desire for revenge. It was marked by centuries of threats, invasion, aggressive diplomacy and, on occasion, simmering antipathy which helped forge national identities. It led to bloodshed, and to twenty-three Franco-German wars from the mid-fourteenth century to the early sixteenth century alone. As one commentator has written: 'For centuries, the history of France and Germany has been a continual effort of the two nations to get closer, to understand, to unite, to merge. Indifference was never possible for them; they had to hate or love one another, fraternize or go to war. France's and Germany's destiny will never be established nor assured individually.' The Treaty of Versailles \u2013 signed in late June 1919 in the same building where in 1871 a conceited Prussian king had proclaimed himself Kaiser \u2013 was an attempt by France to put a full stop to its old rival's ambitions. But the Germans only perceived the terms of the treaty as a humiliation and determined to take their revenge.\n\nFranco-German enmity provides some useful context for an understanding of Hitler's campaign against France, but that task is made much more difficult not only by the pain, humiliation and division caused by the French defeat in 1940, but also by the eventual German defeat in 1945. As a result, despite voluminous publications by academics and authors, attempts to hit the bullseye of Leopold von Ranke's insistence that historians establish 'what really happened' have, in this instance, largely failed. Such attempts have been further hampered by the fact that any campaign is a fast-moving target, for with each passing decade events are reinterpreted to reflect changing times, themes and attitudes, writing styles and methods of research. Consequently, the 1940 campaign is no longer the province of the military historian alone for, in the words of Sir Michael Howard:\n\nThe roots of victory and defeat often have to be sought far from the battlefield, in political, social, and economic factors which explain why armies are constituted as they are, and why their leaders conduct them in the way they do... Without some knowledge of the broader background to military operations one is likely to reach totally erroneous conclusions about their nature and the reasons for their failure and success.\n\nThis book provides some new ideas about why events happened as they did in May and June 1940 from the perspective of a military historian who has been studying the subject for over a quarter of a century and has walked the battlefield many times. It offers new interpretations and challenges some enduring myths \u2013 of which this campaign has more than its fair share \u2013 about how Germany was able to achieve in six and a half weeks in 1940 what it could not achieve in over four years of fighting between 1914 and 1918. It will do so by asking three key questions: what preparations did the belligerents make for war and why? What were the belligerents' fighting capabilities in 1940? And how well did the belligerents conduct operations during the campaign? In doing so, it seeks to establish how well those nations involved managed to blend their ability to conduct operations with their strategic ambitions, whether the Allies ever had a chance of defeating the Germans and, likewise, whether the Germans could have failed. It rejects the erroneous, lazy and dangerous argument that Hitler's victory was somehow inevitable due to qualitatively and quantitatively superior forces using new, carefully honed and mechanized fighting methods that combined air and ground operations within a unifying plan. Yet it also hopes to cut through the complexity to see whether the campaign is an illustration of General James Mattis's argument that 'Ultimately a real understanding of history means that we face _nothing_ new under the sun.'\n\nIt is, of course, for readers to draw whatever lessons they think pertinent from what follows, although as they do so it should be remembered that an historian's conclusions are made with the benefit of hindsight \u2013 and often with the luxury of source material which was not available at the time. Yet while an historian making criticisms from the comfort of an armchair and a commander making decisions in the heat of battle might not seem to have too much in common, both share the responsibility to make sense when there seems to be none and to create some degree of order out of chaos. And there was chaos aplenty on and over the battlefields of the Netherlands, Belgium and France in 1940.\n\n# CHAPTER ONE\n\n# Ingredients\n\nThere are always antecedent causes. A beginning is an artifice, and what recommends one over another is how much sense it makes of what follows.\n\nIan McEwan, _Enduring Love_\n\nYou live in interesting times. Interesting times are always enigmatic times that promise no rest, no prosperity or continuity or security. [In our age] there coexist a number of incompatible forces, none of which can either win or lose... Never has humanity joined so much power and so much disarray, so much anxiety and so many playthings, so much knowledge and so much uncertainty.\n\nPaul Val\u00e9ry, poet and philosopher, in a  \nprize-giving speech at a Paris lyc\u00e9e in 1932\n\nIT WAS THE END FOR WARSAW and Poland. After days and nights of pounding by the Germans, the nation's leadership reviewed the situation. The city's brave civilians could, and would, continue their resistance if necessary, but to what ends? Not for the first time in Polish history, the enemy was grinding Warsaw's buildings into dust, its barely recognizable streets filled with thousands of dead. Further fighting was deemed to be futile and there seemed to be no hope of salvation. On 28 September 1939, a spokesman of the High Command announced in a clear, cold voice that 'After twenty days of heroic defence... [and] practically the destruction of half the city' an armistice had been agreed and conditions for capitulation were being discussed. It was a remarkable military achievement for Germany, which was now reaping the benefits of Hitler's investment in the nation's military machine. Swift and effective, the campaign filled the excited German chancellor with a confidence that was evident to senior officers at a F\u00fchrer Conference on 27 September at which he announced his intentions to invade the West. A long-held ambition was to be realized: France was to be invaded and the humiliation of 1918 and 1919 avenged. Yet Hitler's audience, still digesting events in Poland, was stunned by the news. Neither the army (Heer) nor the air force (Luftwaffe) believed that they could possibly be ready in time: disengagement and post-surrender duties would vie with the need to relocate, rearm and reorganize, to learn lessons and train and plan accordingly \u2013 the list was endless. Hitler had anticipated their concerns \u2013 in his view this was the apprehension of the feeble \u2013 but his attempt to remind the military professionals of victories over the old enemy in 1815 and 1870 was not relevant in the circumstances. Further attempts to reassure them with ideological mumbo-jumbo about the German soldier's 'natural superiority' which, when combined with 'experience and aggressiveness', made a German division 'worth more than a French division' also failed to hit their mark. General Franz Halder, Chief of Staff to the Army High Command (OKH), could not hide his misgivings and expressed his belief that any invasion plans for the West would have to be delayed by months or even years if the necessary preparations were to be made. Hitler dismissed the opinion. 'The relative strength will not improve in our best interests,' he said testily. 'The enemy strength will gradually improve....' The meeting was left under no illusion that Hitler, as Supreme Commander of the German Armed Forces, sought invasion before the year was out.\n\nHitler's F\u00fchrer Directive No. 6, dated 9 October 1939, provided a little more detail about his ambitions towards France and spoke of a 'swift and shattering blow in the West' in an attack through Luxembourg, Belgium and Holland. The aim was to:\n\n[D]efeat as much as possible of the French Army and of the forces of the allies fighting on their side, and at the same time to win as much territory as possible in Holland, Belgium, and Northern France to serve as a base for the successful prosecution of the air and sea war against England and as a wide protective area for the economically vital Ruhr.\n\nKeen to instil a sense of urgency into proceedings and to guide OKH's planning of the invasion, Hitler followed the directive with a F\u00fchrer Conference the very next day. Here he asserted that an attack against France would prove 'no more difficult than Poland' although, he conceded, a decision might only be achieved after Germany had been 'forced into positional' warfare. Such a situation was deemed a distinct possibility because it was accepted that, despite significant improvements to the German military over recent years, any invasion would still need to outflank the Maginot Line to the north and so run into a waiting enemy in central Belgium. Altogether, it was an undertaking that horrified the generals as they considered the damage which was likely to be done to their still-developing military machine, and one they thought could well end in defeat.\n\nGeneral Wilhelm Keitel, the chain-smoking head of Hitler's strategic decision-making body Oberkommando der Wehrmacht (OKW) who was 'temperamentally unequipped to deal with Hitler' and trusted his brilliance unreservedly, offered his resignation over the issue, but it was refused. General Walther von Brauchitsch, the commander-in-chief of the army who was becoming increasingly deferential to the Supreme Commander, nonetheless shared his field army commanders' concerns that the operational challenges involved in the scheme were too great. One of their number was the dour General Wilhelm von Leeb, of whom a fellow officer said if he 'ever tried to smile, it would crack his face'. He had recently been called out of retirement to command Army Group C, had lost a son in the Polish campaign and was one of several senior officers who used the word 'insane' to describe Hitler's proposal. In the wake of the new directive, private conversations were had between old professional colleagues, but although they agreed that an invasion was ill-advised, it did not take them long to recognize the futility of encouraging an argument with a man whose mind was already made up.\n\nDisagreement and friction between Hitler and his senior generals was not a new phenomenon and gained an edge as the F\u00fchrer's demands became more exacting and risky. Hitler felt threatened by the traditional and aristocratic Prussian officer class, with their influence, easy confidence, sense of entitlement and conservatism \u2013 didn't these men want to fight? In return, the generals despised his lack of self-control, his political fanaticism, his cult of personality and the power he had accumulated, even if they did share his vision of a Germany returning to greatness. Most found their F\u00fchrer and his associates boorish, stained by the politics of the gutter, and believed him ill-suited to be their country's leader, let alone their Supreme Commander. Hitler had assumed this position in 1938 and exploited the officer corps' abiding sense of honour by having them all swear a personal oath of loyalty and obedience to him: he needed to control these leaders of men and to harness their professionalism in order to achieve his political aims in Europe. To which end, he used OKW as his central military staff; it had been formed in 1935 to replace the War Ministry and direct the service commands and was comprised of both loyal and competent men. These officers took Hitler's strategic ideas and translated them into orders that were passed on to the subordinate service commands, whose latitude of action was increasingly confined solely to operational issues. Further undermining their influence, Hitler rarely invited service chiefs to the same F\u00fchrer Conferences and ruled by creating division. Disunited, the services posed less of a threat to OKW's strategic plans.\n\nHitler was particularly keen to neuter OKH since Germany was a great land power and the army had always directed strategy and acted as the politicians' principal adviser on war policy. By succeeding in this, he made the General Staff less a focal point for military creativity and more an organization which turned his strategic whim into operational reality. Nevertheless, as the autumn of 1939 set in and winter loomed, Hitler believed that OKH was deliberately and unnecessarily vacillating over the production of its own plans. Frustrated and sensing treachery, he decided that it was time to reassert his authority and on 23 November held a command conference at the imposing new Reich Chancellery on Berlin's Vossstrasse at which he reaffirmed: 'My decision is unalterable. I will attack France and England at the best and soonest moment... I will stop at nothing and I will annihilate anybody who is against me.' OKH's reaction was immediate, the tipping point had been reached, and staff officers \u2013 led by Halder \u2013 did their utmost to produce a plan that would satisfy the Supreme Commander.\n\nOKH's concerns about Germany's ability to emerge victorious in a campaign in the West remained undiminished, despite the successful outcome of the Poland campaign. The recent strengthening of the armed forces had been welcomed by the service chiefs, but it had been a long and arduous road to recover from the strict military provisions of the Treaty of Versailles. During the 1920s, a number of talented officers had seen the armed forces through intensive care, waiting patiently until strength could be regained. Nurturing both the physical and intellectual abilities of the new Reichswehr, a quasi-military organization, early in this period was General Hans von Seeckt. Intelligent and shrewd, as head of the 'defence force' from 1920 to 1926 he laid the foundations for a new, modern German military ready to contest industrialized warfare. It was von Seeckt who put the German military on a course that his successors duly followed, and Hitler took full advantage of this gift and, in full defiance of Versailles, openly rearmed.\n\nJust weeks after coming to power in January 1933, Hitler announced to his ministerial colleagues that 'billions of marks are necessary for German rearmament... the future of Germany depends exclusively and alone on the rebuilding of the armed forces. Every other task must take second place to rearmament....' This, together with the reintroduction of conscription in 1935, was part of a process of mass mobilization for war. The German people, keen to right the wrongs of 1919, were carefully conditioned for war and came to believe their destiny was closely tied to the military and personal sacrifice. Meanwhile, the armed forces prospered in the knowledge that they were an essential part of Hitler's vision. A glimpse of the future was provided in March 1936 with the reoccupation of the Rhineland by German troops. Although this risky operation involved just three battalions of infantry and two squadrons of aircraft, Hitler believed it essential if Germany was to confront the status quo successfully. As it was, France blinked first, and Germany stepped back into the international sunlight. The confidence that the episode gave Hitler further fuelled the nation's preparations for war. Within a year, almost every area of economic life had been brought under the control of the Nazi state and the speed of rearmament was increased. By 1938, Germany was spending 17 per cent of its GNP on the military (it had been 3 per cent in 1914). This was twice the figure for both Britain and France, and in 1939 it rose to 23 per cent. Yet the tempo of rearmament was sometimes impossible to maintain. In 1936, for example, there was a plan to increase the army's forty divisions to over 100 and to have them ready for action three years later. Within twelve months, however, the schedule was thrown into disarray as half of the 1937 target was missed. Hitler, though, displayed a stubborn unwillingness to accept reality, and he refused to acknowledge the army's advice that they would not be ready for a general war until 1943. Such, in fact, was the senior officers' pessimism that they argued that the _Westwall_ \u2013 nearly 400 miles of defences on the border with France which sought to deter an attack in the West, and particularly one undertaken to aid France's allies in the East \u2013 would not be completed until 1953.\n\nThe military consistently challenged the demanding schedule required by Hitler's unfurling foreign policy. In the wake of the _Anschluss_ with Austria during May 1938, for example, while the army had concerns about aspects of their performance which demanded attention, Hitler looked to use force in Czechoslovakia. Major General Alfred Jodl, Chief of the Operations Staff of OKW, noted in his diary at the time: 'Sharp controversy between Hitler and army leaders. Hitler says: we must go ahead this year, the army says we cannot.' The invasion of Czechoslovakia was eventually replaced by a negotiated settlement, but Hitler was furious that his clenched fist had been prised open into a handshake. As he eschewed anything that smacked of German weakness, the military had to fight a constant battle to make Hitler recognize the practicalities involved in the application of force. Yet while the military wanted to develop its strength and capability, Hitler was keen to strike as early as possible in order not to give his enemies more time to prepare. He could not help being vexed by the military's persistent hand-wringing, even if he also knew that the Germans were still not ready for another war. His response was characteristic: the service chiefs would be brought into line and the population's mental conditioning would be completed through a renewed propaganda effort. The armed forces were given another year to develop muscle \u2013 and, in Hitler's eyes, a spine \u2013 and the populace to become hardened to the idea of war. Both would be tested at the end of the summer of 1939 by the invasion of Poland. But while the German people were relatively keen to see territory lost at Versailles regained, the generals recognized that the invasion would very likely lead to war with France and Britain.\n\nAs it was, by August 1939 the armed forces were on standby once again and a sense of anticipation gripped the nation. Uniforms were to be seen everywhere while barracks, training areas and new airfields sprang up across the country. By this time, Germany's armed forces had grown to over 4,564,000 personnel \u2013 3,737,104 in the army, 677,000 in the Luftwaffe and 150,000 in the navy \u2013 out of a population of 80 million. Yet such figures hint more at potential capability, and the generals were very aware that only 1.31 million troops were in active units, 647,000 were fully trained in reserve, around 1.8 million were partly trained and 808,000 had received no training at all. This meant that the field army of 103 divisions that had been raised in four waves was of widely varying quality. The first wave consisted of fifty-two active divisions \u2013 fifteen of which were armoured and motorized \u2013 with 78 per cent regulars and the remainder being reservists recently released from active duty. These divisions were fully trained and equipped. In contrast, the third wave was comprised of fourteen divisions containing men up to the age of forty-five who had not been trained since 1918, and a supply train that contained more horses than trucks. Indeed, equipment and weapons were in such short supply that the field army had thirty-four divisions that were only half-equipped. To this end, von Leeb informed Halder that the third-wave divisions were only fit for static defence and the fourth wave needed more training to be capable of anything.\n\nBut while past German military success had been firmly rooted in its army, air power was now a factor too, which meant that the Luftwaffe also had to be carefully developed. Central to this was the identification of whether it would be most effective at the strategic, operational or tactical level, or a mix of all three. Reichsminister of aviation and former fighter pilot Hermann G\u00f6ring was keen on the strategic potential of heavy bombers to bring about a rapid and decisive victory. Even so, by 1935 Germany had proved unable to produce the 450 aircraft required and changed tack towards a more flexible air fleet. Evidence of this can be seen in August 1939, for by this date the Luftwaffe's operational strength included 788 fighters, 431 long-range fighters, 361 dive-bombers, 1,542 medium bombers, 630 reconnaissance aircraft and 488 transport aircraft. It was an impressive force and certainly offered more potential than the navy, which became relatively neglected because, other than its U-boats, it was deemed to offer few strategic advantages. Indeed, as Admiral Erich Raeder, its commander-in-chief, noted: 'The navy is not ready for the great fight with England. The only thing the fleet can do is to prove that it can sink honourably.'\n\nYet although both the army and air force had mass by August 1939, their potential for offensive success depended on the application of the relevant fighting methods and how well those methods could be integrated into a suitable operational plan. Throughout the rapid development of air and ground forces under the Nazis, staff officers had to remain keenly aware of what Hitler might ask the military to achieve, against whom and where. It was these considerations, therefore, that shaped German doctrine and thence, procurement, organization and training. That doctrine was built on the foundations laid by von Seeckt and his successors which sought not only to provide the basis for a large army but also, through the hard work of a general staff \u2013 the brain of the army \u2013 an intelligent one. It demanded excellence in every facet of the Reichswehr in order to make it the best small force anywhere. Its officers, from whom the general staff were chosen, were rigorously and competitively selected, trained and educated. It was a system that produced a doctrine that was to remain influential throughout the opening years of the war, but also a general staff that was instinctively deferential to history and tradition, and therefore uncomfortable with the unorthodox and radical.\n\nGerman fighting methods were therefore developed in a conservative context, even if directed by von Seeckt's belief that 'The goal of modern strategy will be to achieve a decision with highly mobile, highly capable forces, before the masses have even begun to move.' As a result, by 1939 the German army had a doctrine emphasizing speed, aggression, combined arms and initiative. Yet while modern, it looked back in as many ways as it looked forward as that same doctrine incorporated traditional German concepts \u2013 for example, striking first, hard and deep \u2013 which were given a modern edge through the embrace of new technology and organizations. Even so, von Seeckt also advised that such methods were 'useless if they are not guided by a grand idea and if they are not combined toward and aimed at the purpose of the campaign, yes, the entire war.' To him, local tactical advances were for nothing if not linked to wider operational success which would have a decisive strategic impact. Indeed, Prussian and later German military leaders had historically sought decisive battles of annihilation using swift and skilful troops \u2013 Frederick the Great in the eighteenth century, Helmut von Moltke in 1870, and Alfred von Schlieffen in his 1905 plan to outflank Paris, to name but three. Thus, during the inter-war years the operational concept that dominated was a war of movement ( _Bewegungskrieg_ ) followed by a battle of annihilation at the operational level ( _Vernichtungsschlacht_ ) which might include the battle of encirclement ( _Kesselschlacht_ ) _._ Doctrine facilitated this concept, and so it is worth exploring it in a little more detail.\n\nArmy Regulation 487, _Leadership and Battle with Combined Arms,_ was published in the early 1920s and, despite the restrictions of Versailles, referred to infantry assault battalions, artillery preparation and support, combined arms and the use of tanks ( _panzers_ ) in an offensive setting. It emphasized surprise, initiative and infiltration tactics to penetrate weak points in the enemy's defences as well as the requirement to find the point where a decisive action could be achieved ( _Schwerpunkt_ ) _._ It discussed the importance of the continuous battle, by-passing centres of resistance, maintaining the initiative and creating a psychological blow to create panic as breaches were widened by follow-up forces which then reduced the centres of resistance. It was a method that demanded rapid, 'on the spot' decision-making with subordinate commanders using their initiative to achieve their superiors' intent. It was founded on the German tradition of _Auftragstaktik_ (mission-orientated tactics) and sought to take advantage of fleeting opportunities. As a consequence, doctrine emphasized the requirement for excellent communications forwards, rearwards and laterally. Although _Leadership and Battle with Combined Arms_ was duly revised to incorporate lessons learned and new technology, it was the foundation of German inter-war doctrine. Its heavy influence, therefore, can be found in Army Regulation 300, _Troop Leadership_ ( _Truppenf\u00fchrung_ ) _,_ which was published in 1933 and remained in use until 1945. The major difference between the two regulations lay in the weight given to motorized (wheeled transport without protection) and mechanized (armoured and armed transport \u2013 armoured fighting vehicles) commands, and armoured formations, which included motorized and mechanized units along with other elements. The 1933 doctrine recognized that all vehicles were now more reliable and possessed greater speed and range. This meant that, when married with the improved communications offered by high-powered and portable radios, armoured warfare demanded serious consideration. As one Wehrmacht general later observed: 'The decisive breakthrough into modern military thinking consisted not of a breakthrough in armour weapons, but a breakthrough in the communications weapons.' Suddenly there was the possibility of restoring the operational opportunities for Germany that had proved so difficult to achieve in 1914.\n\nThe potential for motorization and mechanization to have a tactical _and_ operational impact in a war of movement that could achieve decision was vigorously debated. Several schools of thought emerged among its advocates. Oswald Lutz, Walter Spannenkrebs and Heinz Guderian, for example, argued for tanks to be massed in all-tank armies with independent missions to intensify and fully utilize their firepower and mobility. Ernst Volckheim and Ludwig von Eimannsberger, however, believed that tanks should form part of larger combined arms formations with \u2013 in most cases \u2013 an emphasis on their tactical utility, particularly for infantry support. The massed tank school was far too radical for the senior generals to embrace comfortably. This was partly because they saw the inherent risks and difficulties in operations which aimed to strike at the enemy in depth, but also because of the threat this approach posed to the traditional arms, with tanks playing the lead and the rest of the field army in a supporting role. As a result, it was the combined arms school \u2013 a concept at the heart of German doctrine \u2013 that emerged victorious and led to a powerful yet flexible formation being developed. Following successful experimental manoeuvres with a full combined arms armoured division in 1934, the first three panzer divisions came into existence in October of the following year. It was a remarkable development.\n\nA German panzer division was comprised of a tank brigade of two regiments and, commonly by the end of 1939, two motorized infantry regiments. Supporting arms included a regiment of artillery and motorcycle, reconnaissance, anti-tank, pioneer and signals battalions. Thus, despite their title, panzer divisions were not tank divisions but all-arms divisions which were also developed to cooperate with air power. It was a division, its supporters contended, that was stronger than the sum of its individual parts and, rather than operating independently, would work with the rest of the army to improve mobility. The massed tank school, however, saw the potential of the panzer divisions to enjoy independent operations if given the responsibility, for they had the ability to manoeuvre, and to assault and pursue a defeated enemy to his destruction. This was because the striking power of the formation was comprised of tanks \u2013 each division had a notional establishment of 561 \u2013 and these were generally swift and mobile.\n\nThe _PanzerKampfwagen_ (PzKpfw) I was the most common tank and had a top speed of 22 mph, but was only lightly armoured and had but two machine guns. The similarly numerous PzKpfw II was also lightly armoured, although it was armed with a 2cm gun as well as a machine gun and had a top speed of 35 mph. The PzKpfw III and IV were far less common but more heavily armed and armoured. The Mk III boasted a 3.7mm gun and two machine guns and a top speed of 20 mph and armour up to 30mm thick. The Mk IV was similarly armoured, had the same top speed, a 7.5cm gun and two or three machine guns. The preponderance of PzKpfw Is and IIs made the panzer divisions far more inclined towards high mobility and far less so towards heavy contact and slugging it out against powerful enemy positions, especially if these included anti-tank guns. Such considerations were not immaterial to the debate about how these divisions might be used in an operational context. Indeed, having been given command of one of the new panzer divisions in October 1935, Guderian remained a passionate advocate of independent operations and, as more of these formations became available, argued for them to be massed together in panzer corps. To him, the panzer division, if correctly used, was a campaign-winning asset.\n\nOne of the reasons that men like Guderian were convinced that panzer divisions were so useful was because they could fight at a distance from the main body, and particularly if supported by air power. Although it would seem reasonable that the Germans' move away from strategic bombing, coinciding as it did with the birth of the panzer division, would see the Luftwaffe naturally gravitate towards the support of ground units, this was not the reality. Far from being a mere adjunct of the army, the Luftwaffe had evolved a doctrine that focused on the broader role of defeating the enemy's armed forces through the breaking of his will and of supporting ground forces where required. _The Conduct of the Air War,_ a regulation published in 1935, advocated gaining air superiority and air supremacy at the decisive point. This would allow for bombing strikes against enemy assets in depth, but also provide close air support when the circumstances of its impact might be decisive. The air power-panzer division combination therefore offered considerable potential and was brought to a relatively high degree of tactical effectiveness through trials in 1937 which included the largest manoeuvres undertaken since Versailles. During one exercise held in northern Germany during September, 3rd Panzer Division scored a stunning success. Moving rapidly, its armour, supported by air power, broke through the enemy, drove into his rear, cut his supply lines and created an encirclement while reinforcements were kept at bay by air power. By the fourth day of a seven-day exercise, the attackers had secured victory. Hitler made a personal visit with Mussolini to watch proceedings, moving from site to site by car and cheered by troops. A new era was dawning.\n\nBy the end of 1937, the panzer division had made its mark in German military thinking and became the face of the army to the people. It was a period marked by Hitler's support of the Nationalists in the Spanish Civil War with both air and armoured units, but not panzer divisions. It allowed the Germans to blood a small number of troops and test some concepts which saw, for example, the Luftwaffe increasingly called on to perform close air support missions. It was here that practical solutions to challenges were tested and some lessons learned \u2013 including the vital importance of air-to-ground communications and vice versa. Even so, the experience did not lead to a major Luftwaffe reorganization and although close air support was hardly neglected, in general terms the war was deemed to be of limited value in preparation for a war against first-class opponents. But Spain did offer valuable experience as regards planning and staff work, fighting experience with important weapons systems, and minor tactics; in other words those involving formations of company size and below. The same was also true of the March 1938 annexation of Austria \u2013 the _Anschluss._ The success of Guderian's 2nd Panzer Division in racing 420 miles in just two days was remarkable, although not achieved without a third of its tanks breaking down, logistic frailty and horrendous traffic jams. The mistakes, however, demanded a solution and so subsequently mobile workshops were integrated into the panzer divisions for repair and recovery, lead elements of the divisions carried up to five days' worth of vital supplies with them, and traffic control became better organized.\n\nThe German military machine had come a great distance in a short time by the summer of 1938, but it remained untested in the sort of conflict and against the sort of enemy that Hitler was hurtling towards. This is not, of course, an uncommon position for the military to be in but senior officers remained concerned by the risks inherent in such rapid evolution without the requisite time for reflection, adjustment and training. What the Germans had in fact developed was a two-speed army since they were unable to put such a large number of divisions on wheels. Even the flagship panzer divisions were under-strength, while the rest of the field army remained very largely unmotorized and unmechanized, instead relying on boot leather and beasts of burden. This proved a great challenge to those with responsibility for developing plans that were to incorporate both the motorized and mechanized elements of the army \u2013 there were six panzer divisions by the summer of 1939 \u2013 and the standard infantry divisions. Should operations be limited to the speed and ability of the traditional majority, or should the potential of the minority be unleashed in a vanguard of risky independent operations by massed panzer divisions in panzer corps? Put another way, should the Germans fight conservatively and with little hope of a decisive outcome, or radically by utilizing their costly mobile divisions in search of a decisive result?\n\nThe fact that no decision had been made about massing panzer divisions to seek such a decisive operational impact was partly due to a lack of relevant formations, but more to the conservative approach to fulfilling Hitler's strategic ambitions. There was no _Blitzkrieg_ (lightning war) concept in existence to shape a campaign and it certainly was not formalized in doctrine. Although the German army had used the word _Blitzkrieg_ in military articles since 1935 to describe a swift and decisive operation or strategy, there was no unifying definition of the term at this time. Indeed, far from preparing for a _Blitzkrieg_ , Germany was in the midst of preparing for a protracted, attritional war and it was the failure to properly prepare for such a war that was seen by the High Command as a reason why any attack on a first-class military opponent should be delayed. It was in this context that the debate between the traditionalists and radicals about how Germany should conduct operations developed. Might not, argued some, German military assets be used in such a way as to allow Germany to swiftly defeat the likes of France? It was a question that had not been answered when, in the summer of 1939, Germany planned her invasion of Poland and remained only partially answered eight months later when Hitler launched his attack in the West.\n\nHitler's confidence in eventually overcoming France was based partly on his belief in 'natural' German superiority, but also because he considered his enemy to be 'decadent', 'tired' and 'divided' after a prolonged period of political and social division. France was, he believed, broken \u2013 its inter-war governments lasting, on average, a mere seven months \u2013 and thus ripe for the picking. There was more than a kernel of truth in this appraisal as the country's leaders had failed to ensure its armed forces were sufficiently strong and possessed a robust strategy: France lacked a vital self-awareness and was consequently vulnerable to Germany's arrogant aggressiveness. French historian and reserve officer Marc Bloch argued in his memoir _Strange Defeat: A Statement of Evidence Written in 1940_ that the armed forces reflected the weaknesses of France:\n\nIn no nation is any professional group ever entirely responsible for its own actions. The solidarity of society as a whole is too strong to permit the existence of the sort of moral autonomy, existing in isolation, which any such total responsibility would seem to imply. The staffs worked with tools that were put into their hands by the nation at large. The psychological conditions in which they lived were not altogether of their own making, and they themselves, through their members, were as their origins had moulded them. They could only be what their totality of the social _fact,_ as it existed in France, permitted them to be.\n\nThe situation was not helped by the poor relationship between France's politicians and the military. This meant that the executive's ideas regarding defence could not always be critically or properly evaluated by those who would put them into action and it led to mutual suspicion and mistrust. Such friction was not new as the French officer corps had feared the revolutionary ambitions of the political left for 150 years, while the left feared a large standing army's potential as an instrument of repression. As a consequence, ever since Versailles a series of leftist governments had attempted to emasculate the army and reaffirm political mastery over those in uniform. The Superior War Council (SWC or Conseil Sup\u00e9rieur de la Guerre), the supreme body controlling military affairs in both peacetime and wartime, was consequently placed under civilian control. It consisted of senior army generals and members of the executive and was headed by the minister of war. The designated wartime army commander, the chief of staff, was its vice-president. The SWC became, de facto, a general staff under close civilian control and it lacked autonomy and dynamism and was slow to react. Even so, it played an important role as an advisory and consultative body across nearly all military matters but left a great deal of discussion about defence and military matters to councils, committees and the labyrinthine routines of bureaucracy.\n\nThe French military decision-making system wanted for both efficiency and effectiveness, but it reflected a national inclination to hold the armed forces in check. The population was weary of war, guns and uniforms. Losses of 1.4 million men in the First World War (one in ten of the country's population of military age) were greater \u2013 and proportionally higher \u2013 than those of any other western European belligerent. Furthermore, over half of the 6.5 million that served and survived the war had sustained injuries. The most visible were the 1.1 million _mutil\u00e9s \u2013_ those that were maimed, disfigured or invalided \u2013 but there were millions of others who suffered from deafness or nightmares or were psychologically damaged. The First World War was an experience that had a fundamental impact on France during the inter-war years. As Omer Bartov has written: '[I]t is impossible to understand any of the major political, cultural, military or popular trends and attitudes without realizing that visions of war, memories of past massacres and fears of their recurrence dominated the minds of the French.' The result was that, when there were competing claims on dwindling national resources during exacting economic times, it was only the political right that stood against swingeing cuts to the military budget. This led directly to the cutting of the length of military service from three years to two in 1921, then to eighteen months in 1923 and finally to just twelve months in 1928. In line with this, the peacetime army was reduced from forty-one to thirty-one and then just twenty infantry divisions. It was a situation that was partially justified by the argument that, with a declining population, France just could not countenance a large standing army. In 1919, its population had been 39 million, with the prospect of 184,000 men being of military age by the mid-1930s. Germany, however, had a population of 59 million and 464,000 men of military age in 1933. It was only in 1935, when this disparity became dangerously obvious, that two-year military service was reintroduced in France.\n\nThe latent German threat, although hardly a popular concern, remained a worry for the country's leaders during the 1920s. As a consequence, despite appalling economic difficulties and a diminishing army, plans were put in place to ensure that France was prepared for renewed aggression from its angry and frustrated neighbour to the east. Thus in 1926 the country began to develop a national defence policy based on 'total war'. In the whirlpool of short-term French decision-making, this was instead an attempt to take a long-term view in order to defend the population. The problem with the policy, though, was its lack of flexibility. An all-or-nothing strategy, one which demanded that the nation mobilize totally before committing the army, meant that France became unable to use military force in a more restrained manner. Yet while the policy at least offered some sort of plan, the military continued to express their concern that upon mobilization any mass national army needed to be a large, modern and flexible force capable of offence as well as defence. There was little to be gained, it was argued, from neglecting the armed forces and then dramatically expanding them only when absolutely necessary.\n\nThe difficulty was, of course, that maintaining a highly capable army meant considerable investment as this force would require the latest weaponry and equipment, a carefully constructed doctrine and relevant training and education, not to mention high morale. This, it seems, was far too much to ask and instead the army atrophied. It was simply directed to use the surplus weapons of the First World War, while the quality of its regular troops was diminished by poor opportunities for promotion in a shrinking force, increasingly uncompetitive rates of pay, unenticing living conditions and unedifying social standing. It was an army that looked hopefully to tomorrow while, unlike its German equivalent, it did little to prepare for it. Indeed, many decisions thoroughly undermined its future capability, such as that which directed that each active division be divided in three upon mobilization, with each part acting as the 'hub' for a new division which would be two thirds reservist. Thus it would become the job of the active personnel to train the reservists and act as a cadre for some of the mobilized units, although there was not the opportunity for reserve units to train with their wartime brother units or parent division. 'Having accepted the concept of total warfare,' argues Allan Doughty, 'the French \u2013 including the High Command \u2013 thus viewed the military as a relatively rough and blunt instrument, not as a highly tuned, highly flexible force.'\n\nThe choice of a defensive strategic stance with armed forces of a limited capability saw the French naturally gravitate towards the building of defensive positions. The High Command had been ruminating over the development of defensive works since Versailles and in May 1920 the SWC examined future German threats to France. The Council decided that it was essential to defend the heavily industrialized territory in the north-east of the country which contained the majority of French natural resources and the workforce critical to the fighting of an industrialized war. This area had been overrun by the Kaiser's troops in 1914, but there was a desire not to fight on French soil again and an equally fervent wish to keep the Germans out of artillery range of the border. There was, however, also the threat of a surprise attack ( _attaque brusqu\u00e9e_ ) across the historically sensitive and relatively long common border of Alsace-Lorraine. As the defence of this area would require more men to defend it than the frontier with Belgium, the recommendation from the SWC was to fortify the Alsace-Lorraine border. Although the industrial north-east was vital territory, the strategic depth offered by Belgium would give time for France to mobilize and race to meet the invaders, fighting them on another nation's soil with its own forces in pre-prepared positions. This would be possible because since September 1920 there had been a Franco-Belgian military agreement which not only diluted French isolation, but added a prop to her strategy and morale. It was also a consideration, of course, that building fortifications on the French border _behind_ the Belgians would not have been diplomatically astute. France needed all the friends it could get, and it was also hoped that German violation of Belgian territory would bring the otherwise isolationist Britain back into the great alliance as this would threaten that country's own interests. It was then anticipated that after a period of attritional warfare, the Allies, their troops now seasoned by combat, could then \u2013 just as in 1918 \u2013 exploit the enemy's weakness before attacking to victory.\n\nThe form that the defences of Alsace-Lorraine were to take was not quickly decided. In the early 1920s, a special commission under the SWC was established to look into the issue. The generals feared that strong, expensive, permanent defences were the thin end of a defensive wedge which would further fracture the military budget, severely compromise French strategic and operational flexibility, undermine capability and limit the potential for creative military thinking. Yet while the generals cogently argued for temporary and inexpensive defences which did not make offence impossible, their words made little impression on one of the best-known and most respected soldiers in the country, the victor of Verdun, Marshal Henri Philippe P\u00e9tain. In 1922, while minister of war, Andr\u00e9 Maginot inserted P\u00e9tain as Inspector General between himself and the High Command with the power to review all military decisions. It was an appointment which sought to ensure that the Marshal's belief in a strong defence would be influential. And so it proved, for in October 1927, after protracted discussion, a decision was taken to construct deep underground fortifications in key areas of the Alsace-Lorraine front, and smaller blockhouses connected by other obstacles along the other parts of the common frontier with Germany. It took two years for the figures and plans to be drawn up and so it was not until January 1930, at the start of a new period of financial instability in Europe, that a very strong majority of the National Assembly voted in favour of the scheme. It was a massive capital investment at the start of what was to prove a desperate decade economically, but was seen as an efficient use of public finances considering the alternative of massive rearmament. The work was to be spread over four years and construction was to start immediately. The first phase would cover the frontier with fortifications along the joint front and continue along the border with Luxembourg to a point just beyond the Belgian frontier, stopping where the Ardennes began. This densely forested region consisted of hills and winding roads, rivers, streams and narrow bridges, forming a front facing the French border of around seventy-five miles in length. The nature of the terrain, together with the fact that here the River Meuse offered a considerable obstacle just within French territory, led to a decision not to fortify the region. It was argued that the Ardennes was too difficult for a motorized and mechanized force to negotiate successfully. Indeed, P\u00e9tain appeared before the Senate Army Commission in March 1934 to say that the sector was 'not dangerous'.\n\nThe initial Maginot Line construction phase was completed by 1935. It cost far in excess of the estimate and led to further defence cuts. Just as the generals had foreseen, when married with a national defence policy of total war, the 'Great Wall' marked a further shift away from a large, flexible and capable force and towards a defensively minded one which could not be used for a pre-emptive strike, or even limited offensive action across the German border. It was a stance defended by P\u00e9tain's disciple, General Louis Maurin, who, when minister of war, said to loud applause in the Chamber of Deputies on 3 March 1935: 'How could it be that we are still thinking of the offensive when we have spent billions to establish a defensive barrier? Would we be mad enough to advance beyond this barrier upon goodness knows what adventure?' The impact of the defensive total war strategy was starkly illustrated by France's inaction in response to the German reoccupation of the Rhineland in 1936. The French had decided to dig in physically and mentally, leading to an oppressive 'Maginot mentality', eschewing the offensive, rejecting the initiative and reflecting a sense of vulnerability, fear, defensiveness, conservatism, feebleness and self-delusion. This mindset was 'designed to guarantee France the greatest chance of surviving an unwanted war from which it had nothing to gain. Not a formula for decisive victory, it was meant to be proof against catastrophic failure.' However, events moved fast and soon after the completion of the Maginot Line and the reoccupation of the Rhineland, Belgium lost its confidence and, fearing that it might get drawn into a war with Germany due to its alliance with France, declared its neutrality. No longer could the French expect to be allowed easy access into Belgium; they would have to be invited, and any limited staff planning between the two states' militaries would have to be secret. With Belgium taking a path away from France, Britain playing hard to get and the rise in German military expenditure threatening to catch Europe wrongfooted, French rearmament made its way back up the political agenda in 1936.\n\nThe remarkable aspect of the French undertaking rearmament at the end of the 1930s was that it was done in the teeth of continued popular anti-war feeling. Yet Edouard Daladier, who from December 1932 to May 1940 held the position of minister of war for all but twenty-eight months, provided vital stability and continuity of decision-making. Although a former pacifist, Daladier had come to recognize that fascism in Europe could not be removed without resorting to war. He therefore backed rearmament as part of a left-of-centre Popular Front government elected in May 1936 under L\u00e9on Blum \u2013 the first socialist prime minister of France \u2013 although it had ostensibly been elected to deliver social reform. Daladier managed to achieve increased defence spending and in September 1936 a programme of 14 billion francs was announced, to be divided evenly between the three services. With this new funding the Maginot Line could now be extended with defences running up to the North Sea coast \u2013 an insurance policy now that the Belgians had abandoned the French. Even so, the new defences that were to be built were lightest in the area where the French hoped to advance to contact with the Germans in central Belgium, and heavier \u2013 but not as heavy as the Maginot Line \u2013 across the Ardennes front. Here they merely consisted of 'special dispositions' \u2013 roadblocks, earthworks, demolitions and the like. This Maginot Line extension was barely completed by the end of 1939. Nevertheless, it seems that the French had forgotten the old military maxim: he who decides to defend everything, defends nothing. After visiting the Line in 1939, British corps commander Alan Brooke, who had been brought up and educated in France, wrote that the defences were 'a masterpiece in its way and there is no doubt that the whole concept of the Maginot Line is a stroke of genius. And yet! It gave me but little feeling of security, and I consider that the French would have done better to invest the money in the shape of mobile defences such as more and better aircraft and more heavily armoured divisions than to sink all this money into the ground.' It was a point of view with which many of Brooke's French colleagues agreed.\n\nRearmament could not come soon enough for the army and air force, even if the navy had soaked up a larger proportion of the defence budget, for it was a strategic force, vital for the maintenance of communications with the country's colonies and the supply of raw materials and manpower from them. Yet renewed investment in ground and air assets did not mean that France immediately became transformed into a nation ready for war. Daladier recognized this during the 1938 Czechoslovakian Crisis. Prime minister for the third time from April 1938 to March 1940, he saw that time was needed for the military to bulk up, and for the population to share his belief in the need for war. Indeed, explaining the government's pursuit of appeasement in 1938, French foreign minister Georges Bonnet later said:\n\n[The] Czechoslovak army would have been quickly overrun. England could do nothing, France hardly more. France did not have an air force, and General Gamelin [France's Commander-in-Chief] admitted the impossibility of a strong ground offensive 'within at least two years'.\n\nRecognizing that time was not on France's side and that it did not hold the initiative, Daladier, supported by finance minister Paul Reynaud, provided the armed forces with what he expected to be a final financial boost at the end of 1938. Funding had averaged a little over 13 billion francs between 1931 and 1935, but by 1938 had risen to 29.1 billion francs and was increased to 93.6 billion francs in 1939. Yet finance was not the only issue; there was a difficulty turning francs into firepower after so many years of economic hardship in which there had been little investment in modern industrial technology, while there was also a lack of raw resources and skilled labour. The system groaned and creaked into action, however, and industry eventually started producing new weapons \u2013 weapons which the armed forces would require time to familiarize themselves and train with. Nevertheless, when in the wake of the capitulation of the Czech army and of Prague in March 1939 the French government offered guarantees to Poland, a turning point was reached and the population recognized not only that war was necessary, but that in all likelihood it would start soon over Poland. From this point on, the government would stand up to Germany and there was evidence that the people, albeit reluctantly, would stand shoulder to shoulder with it in using war as a political tool.Hitler managed to do what a generation of politicians in France had failed to do: unite the country in a common cause. As Richard Overy has written: 'The ordinary Frenchman did not welcome war, but he welcomed Hitler less.'\n\nThis left the French army and air force in a position where they were likely to face the Germans in a general war, relying on a defensive strategy and a doctrine based on lessons drawn from the First World War. Indeed, convinced as they were that Germany was the once and future enemy, largely reliant on those stockpiles of First World War weaponry and heavily influenced by P\u00e9tain, the French were more likely than not to adopt a conservative approach to operations. As it was, they sought mobility, but only inasmuch as it allowed for the wearing down of the enemy in a protracted attritional struggle which placed strength against strength. This was reflected in doctrine which was first formulated in the _Provisional Instructions for the Tactical Employment of Larger Units_ in 1921 and continued in various forms through to 1939. Its supporting pillars were the Methodical Battle ( _Bataille Conduite_ ) and supremacy of firepower ( _'Le feu tue',_ literally 'Firepower kills') in defence and attack. From the outset, regulations focused on the centralized use of massive artillery fire to dominate the battlefield, either fragmenting and destroying enemy attacks, or supporting the infantry in the offensive. As in the First World War, there was a central strand to this thinking: the artillery destroys and the infantry conquers. Thus in the attack, the infantry were limited to the effective range of the artillery and would advance in successive bounds of around three miles so as not to overstretch logistics and communications. The aim was not the prompt exploitation of an enemy's disintegration but instead the physical destruction of his troops and equipment. Moreover, any rapidity of movement was stifled by a highly centralized organization of the battle itself in which movement of all units and fire from all weapon systems were fully synchronized and strictly executed in accordance with prepared timetables and written orders. Such a system of control aimed to minimize casualties, marshal resources carefully, guide decision-making and ensure that officers played their part in a carefully choreographed battle. It all made for a very unresponsive chain of command, and it discouraged commanders from using their own initiative or seizing fleeting opportunities on the battlefield.\n\nThe Methodical Battle was, in essence, an offensive doctrine until the early 1930s. Armour was to be used to support the infantry in small groups. Yet despite technological developments in the 1920s, tanks remained tied to the speed and requirements of the infantry and were certainly not massed into formations for an operational impact. The 1921 _Instructions_ had reflected this by clearly stating, 'Tanks are not able to conquer or occupy terrain by themselves alone.' Aircraft, meanwhile, retained their First World War roles for reconnaissance, acting as the eyes of the artillery and harassing enemy positions. More radical ideas for the employment of tanks, in particular, gained little traction. General Jean-Baptiste Estienne's 1919 _Study of the Mission of Tanks in the Field_ argued for the massing of motorized and mechanized vehicles supported by aircraft. Despite Estienne having established the French tank force and its fighting methods in co-operation with other arms during the First World War, his theory was derided as impractical, not least because he advocated a 100,000-man army with 4,000 tanks and 8,000 other vehicles, and was quashed by the traditionally minded High Command. Thus, by 1929, the manner in which the French used tanks, as outlined in _Instructions on the Employment of Combat Tanks_ , remained essentially backward-looking: they were to be nothing more than penny-packeted assets in support of the infantry.\n\nAlthough the French did not embrace new operational roles for armour, the 1930s saw tanks acquire a better blend of firepower, protection and mobility that enabled them to undertake different tactical tasks, even within the limitations of infantry support. The heaviest was the Char B1 \u2013 for infantry breakthrough \u2013 with a 47mm gun in the turret and a 75mm howitzer in the hull which could kill any other tank, and armour which could not be penetrated by enemy tank guns or most towed anti-tank guns. It was also the only French tank routinely equipped with radio. The Char B1 possessed only limited range and there were significant weaknesses in its design: like other French tanks, it had a singleman turret, an arrangement which required the commander both to load and aim the gun mounted there. Its complicated design and high cost, moreover, meant that few Char B1s were produced. The SOMUA S35, meanwhile, was a medium tank which specialized in exploitation and was, perhaps, the best tank of its type in Europe. It also had the 47mm gun and, for reasons of economy, the same turret as the B1, a fair turn of speed and armour that was impervious to German shells in most areas. The French also had a number of light tanks which, being less expensive and easier to manufacture, were most numerous. The Renault R35 was designed for reinforcing the infantry in breakthrough battles, but although it was heavily armoured, its 37mm gun lacked killing power. The cavalry also used armour for its role at the head of the main body and deployed the Hotchkiss H35\/39, a tank very similar to the R35, while reconnaissance units used lightly armoured and armed armoured vehicles such as the AMR 35 ZT and the admirable Panhard 178 armoured car.\n\nThe French therefore had a variety of tanks and other armoured vehicles but, although they were undertaking gradual mechanization, their conservative outlook ensured that they lagged behind the Germans in terms of the development of a balanced all-arms division. Nevertheless, by the mid-1930s, the French were planning on a move into Belgium which required mobility and thus at first five and ultimately seven infantry divisions became motorized, while one brigade in each of four light cavalry divisions was equipped with half-tracks and armoured cars. In 1934 France's first Light Mechanized Division (Division L\u00e9g\u00e8re M\u00e9canique or DLM) was introduced. A combination of reconnaissance units together with 220 light and medium tanks, truck-mounted infantry and towed artillery, it was a move towards a more combined arms formation. Four DLMs were eventually formed and were assigned the cavalry missions of reconnaissance and security which would lead the French forces into Belgium when the Germans attacked. Even so, the majority of French armour continued to be diluted through its dispersal \u2013 the numbers varied between forty-five and sixty tanks \u2013 in battalions for infantry support. The ideas of Estienne and his follower, Lieutenant Colonel Charles de Gaulle, whose 1934 book _Towards the Professional Army_ called for a huge armoured army in all-arms divisions with a manoeuvre capability, were simply too expensive and radical for the taste of politicians and the High Command. As a result, the 1936 Regulations \u2013 summarized in _Instructions on the Tactical Employment of Large Units_ , the doctrine with which the French army went to war \u2013 introduced some new concepts yet continued to acknowledge the dominance of the previous maxims. The Methodical Battle remained the model, firepower was 'the preponderant factor of combat' and where static positions were involved, defence was preferred over offence.\n\nMeanwhile, the French relied on radios only when there was no other means of communication available. The Methodical Battle did not require flexible communications as battles with a slow tempo meant that commanders could rely on field telephones and couriers. These would connect the various elements of the defensive system. First, an Outpost Line would alert the main position and endeavour to delay the enemy. The Principal Position of Resistance, the strongest and most important line, would then corral the enemy between natural and staggered, man-made obstacles to force him onto routes that were pre-surveyed artillery killing zones and covered by fire from mutually supporting positions. These positions, which included machine guns and anti-tank weapon systems, were to slow the enemy's momentum and sap his strength. If the attacker was not destroyed in the principal line, he would then have to make his way past a number of strongpoints before being engaged at a Stopping Line, where he would be halted and counter-attacked. If the Stopping Line was broken, the French would then endeavour to _colmater_ or plug a gap as a mobile reserve was moved smartly into the threatened area.\n\nIn theory, this was a sensible and well-judged defensive system, but it relied on completed and well-sited positions as well as well-trained and prepared troops who knew those positions well. The first phase of the Maginot Line was certainly more in keeping with these requirements than the second phase extension, which had yet to be completed. Moreover, upon mobilization the majority of divisions would be partially trained non-regulars with limited \u2013 if any \u2013 acquaintance with the area they were to defend and particularly so if it was deemed low-risk. In addition, it was a system that anticipated slow-moving German infantry with perhaps fifty supporting tanks concentrated in just over half a mile of front. However, since French anti-tank weapons were in theory to be deployed one every 110 yards and with only fifty-eight guns available per division, commanders had to consider carefully where to concentrate them. There were also too few anti-tank and anti-personnel mines and so the artillery \u2013 the old 75mm guns still packed a punch \u2013 had a crucial role in targeting, fragmenting and destroying enemy penetrations.\n\nOf course, the great unanswered question \u2013 unanswered because it was not asked loudly, if at all \u2013 was how the system would cope if the German attack did not fit in with preconceived French ideas about its location, focus and strength. The French were just not organized and trained to react to the unexpected and to regain the initiative quickly. In such circumstances, the flexibility and mobility of the French air force would prove critical. Although in the 1920s the air force had been an appendage of the army, by the early 1930s it had been tempted by the independence offered by becoming a heavy bomber fleet and the politicians were keen on such a force's ability to deter. In these circumstances, the air force gradually lost the ability to support the army although by 1937, four years after it had gained independence, it was still not entirely committed to strategic bombing.\n\nAs the air force drifted, caught between strategic and tactical pressures on its resources, air minister Pierre Cot had sought to add clarity in 1936 by favouring a massive enlargement of the bomber force. In keeping with the fast-moving times and constant political change in France, two years later the new air minister, Guy La Chambre, recognized that it would take too long to build Cot's vision and, acknowledging the ever-increasing German threat to French airspace, changed tack to a plan that would see the air force fight for air superiority and provide support to ground forces. This too, however, ran into problems for, as with tank production, despite some good designs and technology, there was insufficient industrial capacity, too little skilled labour and too much poor organization to meet targets. Modern aircraft were only slowly built and, along with trained ground and air crews, drip-fed into the system. By the outbreak of war, along with 250 bombers, the French had 826 fighters. Inevitably, some of these were obsolete, although modern types like the Bloch MB 151 and precious Morane-Saulnier MS 406 could challenge the German Bf 109. The upshot of all this was that by the end of 1939 the air force, commanded by an elderly former bomber pilot, General Joseph Vuillemin, was, despite its size, the least effective and most confused military service. Its valuable aircraft were \u2013 like the tanks \u2013 to be parcelled out to units in small quantities to provide localized air cover across the front and were caught in a command and control maze.\n\nIt was therefore the case that by the end of the 1930s, whether on the ground or in the air, the French had undermined their own ability to be in the right place at the right time and with the required numbers, weaponry and skills. To make matters worse, senior officers, although aware of flaws and foibles in the system, believed that they had a winning formula. Focused on their own defence, lacking contrary military experience and rejecting lessons from Spain, Austria and Czechoslovakia as not pertinent to their situation, the French took a tack and stuck to it no matter in what direction the wind blew. A generation of officers held on to the hard-won knowns \u2013 the lessons of the First World War \u2013 for dear life and would not be swayed from them as they settled into a velvet-lined rut. This was partly due to the unexceptional quality of the officer corps during this period, but also to the education of those officers, which emphasized an understanding of doctrine and procedures rather than the development of critical faculties and the application of initiative. It was a dangerous culture, claustrophobic and unquestioning, one in which officers sought reassurance and avoided creative thinking. Obedience was a virtue that was extolled and nowhere was this more clearly seen than in the Methodical Battle. It fostered a lack of imagination that was dangerous not only because French methods were not challenged by new ideas but also because the enemy's options and how he might use his own initiative were not usefully explored. Thus, in short, the French failed to develop a capacity which allowed its military to anticipate the enemy and make adequate preparations to apply their will to the battlefield.\n\nIn a system that so heavily leant on the past, that was chary of the radical, denuded battles of boldness and personality, and favoured procedure over flexibility and dynamism, it was hardly surprising that competent but uninspiring officers with good staff skills \u2013 men such as Gamelin and Vuillemin \u2013 rose to the top rather than charismatic leaders. They were not the sort of men to shake their services out of an unhealthy complacency because, as products of the French system and promoted for certain organizational abilities, they could not identify that complacency. There was a belief that, as in 1914, the outcome of a war would eventually favour France. General Weygand, for example, said in July 1939, the month before he was called out of retirement: 'The French army is stronger than ever before in its history; its equipment is the best, its fortifications are first-rate; its morale excellent, and it has an outstanding High Command. Nobody wants war but if we are forced to win a new victory then we will win it.' Foreign observers tended not to demur from this view, clearly seeing impressive adult plumage from a distance, rather than the less impressive specimen close up. Indeed, _Time_ magazine commented in August 1939, '[Gamelin] is head of what, by almost unanimous acclaim, is today the world's finest military machine.'\n\nThe British also proclaimed French military strength. Winston Churchill, seeking to give reassurance to the British people at a time of rearmament efforts in 1937, remarked that the French had an 'incomparable military machine' and in the following year that the French army was 'the most perfectly trained and faithful mobile force in Europe'. The plaudits became commonplace after London drew closer to Paris in the wake of Munich and continued even when staff talks and joint military planning began in early 1939 and the British general staff began to furrow its collective brow about French preparations. Nevertheless, Britain wanted to believe in French military strength as much as France did and so the nation bought into the popular French slogan 'We shall win because we are stronger'. All the time, however, France adopted a swagger that was based on little more than a 'mixture of faith, hope [and] resignation'.\n\nAs was to be expected, it was the Germans that took the most objective view of French military capability. Berlin had identified weaknesses through agents and intelligence work by the late 1930s: the limitations imposed by France's lack of manpower, the plethora of poor equipment, and the military's lack of flexibility and grasp of modern mobile warfare. Indeed, Halder and his team had studied Germany's latest enemy in detail and in August 1939 he wrote that the French army:\n\n[R]esembles a weak man trying to carry machine guns, guns etc., on his back. Age classes of conscripts are small, and service for a long time has been only one year. Armament, too, is not in the best condition. Potential of army on the whole limited.\n\nYet even such an assessment did not lead the Germans to feel that France was anything other than a first-class power. The First World War had taught them to respect defensive preparations and not to underestimate their enemy. The German ambassador to Paris from 1936 to 1939, German Count Johannes von Welczeck, was aware that agents in France were sending optimistic reports back to Berlin, '[telling] their masters what they wish to hear and not what they ought to know'. Von Welczeck, however, urged caution. It remained to be seen whether or not he was right.\n\n# CHAPTER TWO\n\n# Plans\n\nON THE AFTERNOON OF 26 AUGUST 1939, Albert Winkler was in a Bavarian wood engaged in a tactical leadership exercise. It was the last phase of his officer training which, apart from an NCO's mistake during a live-firing exercise which had killed one of his comrades and a gash on his forearm caused by a grenade malfunction that required twenty stitches, had all gone well. He returned to his barracks that evening, a wooden hut infested with mosquitoes and lacking hot water, to be told that the commanding officer would be holding a briefing in ten minutes. Having stored his kit, Winkler immediately headed for an old gymnasium, which, unusually, was being patrolled by guards, and he noted that traffic had been held at the entry gate. Winkler's heart began to beat just a little faster, his instincts telling him that something was afoot. The young men, some 150 in total, stood to attention as the lieutenant colonel limped onto a stage. He was in his early fifties, grey at the temples, his right upper lip curled into a permanent smile by a deep scar. Russia 1915. His stories had held the potential officers entranced, but each one of them had a point, lessons to learn. Their respect for him was undoubted and many said that they would gladly be led by him 'into the mouth of hell'. Once he had been stood easy, Winkler retrieved a notebook and pencil from his breast pocket to take notes, as he always did when the colonel spoke:\n\nGentleman, although your training has another two days to run, I can inform you that from midnight tonight, by order of General Brauchitsch, you will become commissioned officers with the rank of leutnant. My congratulations. You may well be wondering why this is the case. The reason for this is simple: the army needs officers and at 0400 hours tomorrow morning you will parade outside this building before being transported to the railway station. From there you will take a train to M\u00fcnchen and thence make your way to your units. War is about to break out in Europe. You will be heading east into Poland. It will, I am sure, be the first in a series of campaigns which the F\u00fchrer has deemed necessary and I have little doubt that the war will be long. Remember your training. Trust your instincts. Rely on your comrades. Do your duty. *\n\nThe Polish campaign gave the German High Command an occasion to test their capability against a stronger enemy than previously, and it was still deemed to be winnable inexpensively and quickly. It was a stepping-stone to a war with the West, but it was not one that Hitler thought would lead to confrontation in the very near future as he believed France and Britain were too militarily weak and their populations too unsupportive to risk a war over Poland. Emphasizing the point to his senior generals, Hitler told them in August 1939: 'Our enemies are little worms. I saw them at Munich.' With the West seemingly impotent and a non-aggression pact signed with Stalin by the end of August to ensure the passivity of the Soviet Union, Germany had created the chance to strike at Poland unfettered by fighting on another front. Hitler had, however, significantly underestimated the rapidly growing sense of outrage and fear generated latterly in Paris and London by the Munich Crisis and the later German occupation of the rest of Czechoslovakia. Although there was a desire in both capitals to win more time to strengthen their armed forces, both \u00c9douard Daladier, the French prime minister, and Neville Chamberlain, his British counterpart, believed that the time had come to make a stand and had agreed in the spring that they would declare war on Germany if Hitler invaded Poland. Partial French mobilization took place in the last week of August as tensions began to rise, and full mobilization followed on 1 September when Germany attacked eastwards. The next day France and Britain issued an ultimatum demanding that Germany withdraw from Poland, or there would be war. Hitler, described by Halder at this time as 'Worn, haggard, creaking voice, preoccupied', had the ultimatum read to him by interpreter Paul Schmidt in the Reich Chancellery. 'When I finished,' recalled Schmidt, 'there was complete silence. Hitler sat immobile, gazing before him... after an interval which seemed an age he turned to [Foreign Minister] Ribbentrop, who had remained standing at the window. \"What now?\" asked Hitler with a savage look.'\n\nThe Allied stance had taken Hitler by surprise. They had torpedoed his loose schedule for an invasion of the West \u2013 he would now have to act sooner than planned because he did not want France and Britain to benefit from any delay to gain strength. Thus, from these declarations of war on Germany, which were announced on 3 September, the clock was ticking for both sides. Yet while Hitler believed that Germany was ready enough for general war, the High Command was extremely concerned that the combined strengths of France and Britain were better placed to win an attritional conflict. It was a belief articulated by Franz von Papen, a former general staff officer who had been briefly chancellor in 1933: 'Germany can never win this War. Nothing will be left but ruins.' It was the 'ruins' of war that were still so fresh in the minds of the German people, who did not greet the onset of hostilities with Poland enthusiastically. There were no cheering masses in Berlin as there had been in 1914, only anxious, silent men and women looking mournful.\n\nThe French and British also failed to greet Hitler's refusal to respond to the Allied ultimatum with unbridled joy. Sending the sons of the soldiers of 1914\u201318 off to war was a trauma that they had desperately hoped to avoid, and yet Hitler's prediction that such a state of affairs would lead to mobs bringing down their governments proved wide of the mark. Indeed, some in France felt that for the first time in a generation the country appeared united against a common foe and were confident of success, even if there was likely to be a cost. Reserve Captain Daniel Barlone, a man in his forties, was called up on 23 August and wrote in his diary that his colleagues in Paris 'fully believe that I shall be back in the office by next week' and opined: 'We do not doubt our victory, but wonder what price we shall have to pay.'\n\nWhile Hitler may have misunderstood the strength of feeling in France and Britain against his recent actions, OKH had nonetheless correctly advised him that the French were incapable of launching a major offensive against Germany while its forces were engaged in Poland. On 14 August Halder had noted: '[I]n view of what we know about French operational doctrine and the political difficulties which would first have to be overcome, such a thrust would be unlikely'. His forecast was accurate, for Gamelin's nine-division-strong show of force against the Saar which began on 7 September, was half-hearted, ponderous and retreated not long after running into the defences of the _Westwall._ The British, meanwhile, did nothing of note and their planned bombing campaign of the Ruhr was reduced to a lame propaganda leaflet-dropping exercise because the French feared that the Luftwaffe would retaliate by raiding their cities. What might have followed had the British struck with bombs and the French been bolder can only be guessed at, but Halder was relieved that Allied reaction had been so timid as the German presence in the Saar was a scratch force of second-line divisions with no tanks or supporting aircraft.\n\nFor their part, the Germans needed operations in Poland \u2013 _Fall Weiss_ (Case White) \u2013 to be completed as quickly as possible in order not to expend valuable resources and to ensure the invasion of the West could now be launched without delay. With German or German-occupied territory on three of Poland's four sides, Army Group North's Third and Fourth Armies under Generaloberst Fedor von Bock formed one pincer while Army Group South's Eighth, Tenth and Fourteenth Armies under Generaloberst Gerd von Rundstedt, who had recently been called out of retirement, formed the other. Between them, the aim was to catch the Polish army in a great _Kesselschlacht_ \u2013 a task in which they would be assisted by the Soviet invasion of eastern Poland, which had been agreed as part of the non-aggression pact and began on 17 September. All the German armies involved were a mix of infantry, panzer and motorized divisions. Von Bock and von Rundstedt commanded three and four panzer divisions respectively, with one army in each Army Group containing a corps strike force consisting of a panzer division and two motorized divisions. In von Bock's Fourth Army, for example, XIX Corps was the strike force. Commanded by Heinz Guderian, it consisted of 3rd Panzer Division, 2nd and 20th Motorized Divisions with the attached Panzer Lehr (Panzer) Battalion and a Reconnaissance Battalion. All thirty-four of Germany's tank units \u2013 divisions, independent brigades and regiments \u2013 went to war in Poland: 2,820 tanks with 1,051 PzKpfw Is and IIs, 361 PzKpfw IIIs and IVs, and 301 35(t) and 38(t) tanks commandeered from the Czechs.\n\nIt was a campaign designed around the tenets of strike first, hard and deep, and operations were conducted in line with the doctrinal emphasis on speed, aggression, combined arms and initiative with air power. Although panzer divisions did not have independent missions, violent assaults by massed panzer columns made successful penetrations by driving reserves of tanks and motorized infantry deep into the rear of the Polish positions, preventing the defenders from re-forming their line or bringing up their reserves. The Luftwaffe, meanwhile, won air superiority and then supported ground forces through tactical missions by Ju 87 Stuka dive-bombers and operational missions by Heinkel He 111s and Dornier Do 17s. The Poles could not cope with the mobility, tempo and intensity of German operations; their defences were unhinged within four days, and Warsaw fell on the twenty-seventh day of the campaign. It was little wonder Daniel Barlone was so shaken by the news that he felt compelled to write in his diary, 'We refuse to believe in the smashing advance announced by the Germans. Besides, things will change.' The Poles lost 65,000 killed in action, 144,000 wounded and 587,000 prisoners of war while the Germans lost less than 16,500 killed, 30,000 wounded and 3,500 missing.\n\nThe sheer speed and scale of German operations were such that strategic victory, albeit with the assistance of the Soviets, was assured. It had achieved OKW's aims without a protracted fight and provided Hitler with a solid platform from which to launch an offensive in the West before the end of the year. OKW did, however, want time to relocate its troops and to identify and then apply the lessons learned in Poland: the French and British would pose a sterner challenge. While von Brauchitsch and Halder did what they could to forestall an early offensive in the West, the general staff and field commands collaborated closely and their conclusion was that yet more work was needed to improve logistics and to create the necessary tactical and operational momentum before any invasion of France could be attempted. Panzer divisions were big, hungry, thirsty organizations which occupied seventy miles of road space or eighty trains of fifty-five wagons on rail, and if they were moving fast, their supply was of the essence. The result of study and discussion about how this could be best facilitated led to the sharing of best practice as well as some innovation. Older ideas such as the attachment of mobile field workshops to panzer divisions became standard, as did the _Rucksack-Prinzip_ (backpack principle), which saw each formation carry as many supplies as was practical to make them as self-sustaining as possible. Such developments were further enhanced by using the Luftwaffe to deliver fuel and ammunition to forward airfields and the panzer divisions then using trucks, which were increased in number, to retrieve the supplies and deliver them direct to the spearhead units. Highly mobile Luftwaffe units were also to keep in close contact with the leading elements of the divisions in order that they might open their own temporary airfields close to the front, which would allow short-range German aircraft to fly timely close air-support and interdiction missions. 'This,' Robert M. Citino argues, 'was the true birth of the _Air-Land battle._ '\n\nWherever the army and air force could better their performance in both old and new roles, they were willing to do so. Improving what was already in place was critical as time was short. There was a plethora of initiatives to achieve this. The infantry, for example, having been taken to task for being too slow to achieve objectives, found its leadership coming under intense scrutiny. The result was the establishment of a two-week 'fundamentals of practical leadership' course for junior officers and NCOs and something similar for battalion and regimental commanders but pitched at a higher level. OKH was keen to increase the army's striking power, and so changes were instigated. With no resources to establish new armoured divisions from scratch, a practical alternative was found by adding a panzer regiment to light divisions in order to create four more panzer divisions. More motorized formations were also formed by each established division of this type losing a motorized infantry regiment, which then became the basis for a new motorized formation. Meanwhile, concerns over the vulnerability of light tanks in combat after the loss of 217 tanks in Poland were dealt with in two ways: first, by the ordering of more PzKpfw IIIs and IVs and second, by panzer commanders, who were aware of Germany's own difficulties in tank production, developing new tactics that would protect their assets. Yet while a welter of influential adjustments were made over the autumn of 1939, one important organizational issue of potentially great importance was not grasped: the massing of panzer divisions for independent operations. It was an issue that went to the heart of German campaign-fighting methods, but it also challenged tradition and therefore the conservative nature of many in senior command. As the planning for the invasion of the West took shape, this was a confrontation that became increasingly inevitable and one which the Wehrmacht would ignore at its peril.\n\nFranz Halder was well aware of the various debates swirling around the army about how tanks might best be used in future operations, but they were just one influence on his planning for Hitler's offensive in the West. It was his job as chief of the general staff to draft plans that would fulfil OKW's ambitions and so he had to have a firm grasp of what the army was capable if it was to be effective. His initial attempts revealed much about what he and von Brauchitsch believed the capability of the field armies to be in the immediate wake of Poland \u2013 mediocre. During the period termed the Phoney War or _Sitzkrieg_ before the invasion of France, Halder did all that he could to ensure that his plans did not ask too much of the armed forces and were produced quickly enough, but not so quickly as to give Hitler the opportunity to launch a snap offensive. The first task of his staff officers was to give shape to Hitler's broad scheme as outlined on 9 October in F\u00fchrer Directive No. 6, which required an attack through Luxembourg, Belgium and Holland and into France in order to destroy the French army. Halder's first draft plan, known as the First Deployment Directive, was produced ten days later. Army Group B would carry out two separate but related thrusts: the main force was to advance through central Belgium in two armies, including six panzer divisions, to converge on Brussels before arcing towards the River Somme, while a subsidiary attack would also be launched into the Netherlands with two more armies. Army Group A, meanwhile, would strike out to protect the left or southern flank of the advance into central Belgium. One of its armies would traverse the Ardennes with a single panzer division while, just to the south, the second would make a minor push through Luxembourg to the northern end of the Maginot Line. Army Group C's part in the plan was to develop pressure on the Maginot Line in order to pin and fix French divisions in Alsace-Lorraine. The bear-like Keitel was underwhelmed by Halder's offering, saying that it was 'just the old Schlieffen Plan, with the strong right along the Atlantic coast' and adding caustically, 'You won't get away with an operation like that twice running.' This was not entirely true, because the First Deployment Directive lacked the strategic ambition of Schlieffen's projected encirclement of Paris as carried out in 1914 (the 1914 version of the plan was, of course, a bastardised version of the original) and merely advanced German forces to make contact with the enemy in Belgium. Indeed, this was exactly the sort of plan envisioned by the French themselves, and even Halder considered the plan a rushed 'tangle' without a _Schwerpunkt_ because the whole concept lacked emphasis on achieving a decisive result.\n\nHitler, who dismissed OKW's early plans as being akin to the 'ideas of a military cadet', sent the first draft back, demanding something that produced more 'breakthroughs'. To focus the generals' attention, Hitler announced that he wanted the invasion to commence on 12 November. Halder felt that his options were severely limited, and not only by the time constraint imposed by his Supreme Commander. At heart a conservative staff officer but with a good sense of what was and was not achievable, he struggled with the restrictions on his creativity imposed by the Maginot Line, the Ardennes and an army still in development. Nevertheless, the Second Deployment Directive was delivered on 29 October 1939, just ten days after his first attempt. In it, all three Army Groups kept their previous roles, but Army Group B was enhanced to give it the extra power to make a breakthrough at the expense of Army Group A's Ardennes advance. Hitler received it with a coolness that made his dissatisfaction only too clear. He ordered Halder to produce a third version of the plan. Frustrated by Hitler's lack of practical understanding of what could be achieved with the resources available and in the area selected, the general lamented in his journal: 'None of the higher Hq. thinks that the offensive ordered by OKW has any prospect of success. No decisive success can be expected on ground operations.' Fedor von Bock, the serious and single-minded commander of Army Group B, agreed and confided to his diary: 'The F\u00fchrer knows that the bulk of the generals do not believe that attacking now will produce a decisive success.'\n\nIt was in early November, just after the rejection of the Second Deployment Directive, that Hitler began to show a heightened interest in the wooded Ardennes region of Belgium and Luxembourg, a sector which abutted the German border and was assumed to be impossibly difficult ground to cross quickly. Halder was directed to enhance the push through the Ardennes in order to protect the southern flank of the main effort and, in so doing, take Sedan, where Napoleon III had been captured in 1870. Halder, however, saw Sedan as militarily unimportant and a stronger push through the region as an unwarranted dilution of resources. OKW was well aware, however, that vital supplies remained scarce and on 5 November von Brauchitsch met with Hitler and not only pointed out the army's lack of readiness and the challenges imposed by the weather, but also invented mutinies, widespread ill-discipline and a lack of fighting spirit at the front. Hitler's rage at such allegations led to him accusing von Brauchitsch of a lack of loyalty towards his own army and branding the general a liar as he spat, 'Not one frontline commander mentioned any lack of attacking spirit to me!' Von Brauchitsch had committed professional suicide, while his falsehoods had only diverted attention from what were very real resource difficulties.\n\nVon Bock's diary is full of references on the resources theme: on 30 October he wrote, '[R]eplacement of materiel is far from adequate'; on 3 November, 'Today the 4th Army reported that its divisions are nowhere near up to strength and ready for operations!' and on 12 November, '[Army Group B is] capable of a brief offensive; major operations only when timely replacement of everything, especially motor vehicles, is assured. Additional training time desirable in any event.' Added to these concerns was von Bock's continuing unease about the plan's lack of a clear focus. Thus, on hearing about the new emphasis that Hitler was seeking to put on the Army Group A area of operations, the humourless, arrogant and ambitious officer was moved to write: 'Originally the point of main effort was supposed to lie with me, now there are three attack spearheads! I was right about my concern over the blurring of the point of main effort.' It seemed, as Halder had been complaining in recent weeks, that Hitler wanted a 'breakthrough everywhere'. But although Hitler's interference in the operational realm was not welcomed by OKH, the Ardennes sector was heavily laden with potential \u2013 a potential that had not been missed by Generalleutnant Erich von Manstein, the chief of staff to Army Group A.\n\nVon Manstein was a fifty-two-year-old Prussian aristocrat who had been beaten to the coveted appointment of chief of the general staff by Halder but not through any lack of self-confidence. He had dismissed his rivals' early plans as 'intellectually bankrupt' and, encouraged by Gerd von Rundstedt, the Prussian commander of Army Group A who had been an important figure in the reorganization of the Wehrmacht in the mid-1930s, immediately set about producing an alternative. Having had previous contact with his fellow officers and armoured warfare proponents Oswald Lutz and Heinz Guderian, von Manstein believed that Germany should seek a bold operational plan that offered the possibility for a decisive strategic result through the daring use of armoured forces and yet was in keeping with German military tradition. His ideas were given time to develop when in mid-November, with no agreed plan, deteriorating weather and concerns over resources, Hitler reluctantly postponed the launch of the offensive until 'conditions improved'. The start date was subsequently postponed on a further twenty-eight occasions \u2013 well into 1940. It was early during this period, on 23 November, that Hitler held a command conference at the Reich Chancellery, the attendees of which included his three Army Group commanders: von Rundstedt, von Bock and Generaloberst Wilhelm Ritter von Leeb, the commander of Army Group C. Here he asserted his military authority, re-emphasized his aims and further detailed the deficiencies he found in Halder's Second Deployment Directive. In doing so, he further undermined the authority of OKH, a point not missed by von Brauchitsch, who offered his resignation to Hitler that day over the issue, although it was not accepted and the army's subservience only grew thereafter. This was the F\u00fchrer's final warning to stop dithering, cease whingeing and instead to act decisively to fulfil his will. It was an admonition that, despite Hitler's clear ambition to subvert the traditional relationship between OKH and the head of state, and despite their own mutinous feelings, von Brauchitsch, Halder and the field commanders felt poorly placed to ignore.\n\nWhile the Germans struggled through their preparations, the Allies embarked on their own. The central figure in this was General Maurice Gamelin, who upon mobilization had been made both commander-in-chief of land forces and Supreme Commander of the Allied forces. The diminutive sixty-six-year-old had been born into a Parisian family that had produced many generals over the centuries and from an early age had absorbed some of the skills required for high command. One of them was a calm assuredness that had been noted when he stood as a staff officer beside Joffre when the Germans were denied Paris in 1914, and in his appointments before being promoted to chief of the general staff in 1931. Since then his knowledge, confidence and tact had made him well-respected across the military, in political circles and by the populace at large. _L'\u00c9poque_ opined in July 1939 that in war Gamelin would 'more than fulfil the high expectations the nation has of him' and that same year _Time_ described him as being exactly what the French like their generals to be \u2013 'mute and professional... the good grey little General leads a good grey little life'. Yet there were men of influence, including Paul Reynaud, the finance minister who was to become prime minister in March 1940, with a much lower opinion of the general's abilities and who believed him to be devoid of bold ideas, fusty and too set in his ways. There was, though, little doubt that Gamelin was the personification of just the sort of officer that the French army sought to produce: professional, orthodox and systematic.\n\nOn the day that Germany invaded Poland, Gamelin had established his war headquarters at Ch\u00e2teau de Vincennes in eastern Paris with a staff directed by Colonel Jean Petibon. It was from here that he sought to oversee the final developments of a trap that had been set for the Germans across central Belgium into which, he believed, France's mortal enemy could not fail to fall. His army would then, with the assistance of Britain, destroy the German armies in an attritional \u2013 but thoroughly methodical \u2013 manner. It was a confrontation for which the French had been preparing themselves for the last twenty years but which Britain had studiously sought to avoid. There was no desire on the part of the latter's politicians or people to go to war or to finance a hungry military machine, and as a result Britain was psychologically and materially late in readying itself for war. It had consistently reduced defence spending in the wake of the First World War, and it was not until Hitler's Germany began to threaten Britain's vital interests in 1934 that there was a reassessment of such expenditure. Initially, the Royal Air Force was the main beneficiary of this increase in funding, but Hitler's increasing menace meant that spending was more broadly spread across all three services. It was a period during which Britain relied heavily on France to provide a 'great army' on the continent, the RAF to protect her islands and the Royal Navy to protect her empire. London also depended on a policy of appeasement towards Germany to provide the time in which to rearm, but by the _Anschluss_ in March 1938 defence spending accounted for 38 per cent of all government expenditure. It was only after Munich, however, that the financial brakes on defence spending were finally and fully released.\n\nBritish rearmament was a compromised, extemporized and rushed process. Preparation for war was poorly co-ordinated as the services still scrapped for funding, took self-serving decisions and eschewed co-operation in what amounted to a strategic vacuum. By the outbreak of war, the RAF boasted a total of 832 fighters and 2,520 bombers. Bomber Command would undertake strategic bombing and attack targets of significance in any ground campaign, while Fighter Command would defend British airspace. Support for the army was not considered to be important; indeed, the Air Staff stated in Regulations that it would only happen 'in exceptional circumstances'. There was also little emphasis on the need to win air superiority over the battlefield and Fighter Command was determined not to see its capability diluted by attempts to draw off its aircraft to fight over the continent. In the end, just fifteen squadrons were sent across the English Channel for dedicated army support, along with the light and medium bombers of the Advanced Air Striking Force (AASF) which would bomb targets in Germany from airfields across the English Channel while preparations were made to facilitate operations direct from British soil by aircraft with greater range. It was a decision that was to lead Lieutenant General Alan Brooke, the commander of the British Expeditionary Force's (BEF) II Corps, to make a typically insightful entry in his diary:\n\nThe feeling the [War Office] give me is that whilst concentrating on ensuring that they are going to win the war in 3 years from now they neglect to realize the danger of losing it this year!... [W]e are courting disaster against an enemy who adheres to the doctrine of concentrating of effort at the vital point at the right time. To contemplate bombing the Ruhr at a time when the Germans are using their combined army and air force effort in one mighty uniform attempt to crush the French and British forces to clear their way into France, is in my mind sheer folly.\n\nThe BEF of which Brooke was a part had been formed after the start of joint staff talks with the French in mid-February 1939. Initially consisting of a general headquarters and two corps, each of two regular divisions together with its small RAF air component, by May 1940 it comprised ten infantry divisions, half of which were regular, and an armoured brigade, with the whole organized into three corps. Its fighting methods were an evolution of those used in the later stages of the First World War, but there had been some experimentation with mechanization and motorization since then. The result was a doctrine which bore all the hallmarks of a compromise between the old, limited, closely controlled battles and new experimental thinking.\n\nSuch thinking could be seen in the establishment of the Experimental Mechanical Force \u2013 later called the Armoured Force \u2013 on Salisbury Plain in 1927. It was a small, prototype mechanized division consisting primarily of armoured fighting vehicles designed to co-operate with RAF bomber and fighter squadrons, but without infantry. During exercises over the next couple of years, this formation provided a number of important lessons, chief among them that it could achieve little without strong infantry support. Within the confines of what financial resources and deference to tradition would allow, the British army endeavoured to modernize in an attempt to avoid the stalemate of the First World War and, in the words of Field Marshal Sir George Milne, Chief of the Imperial General Staff and an advocate of mechanization, to aim for 'mobility, activity and quickness from the beginning...'. Its doctrine in the late 1930s thus touched on armoured units attacking enemy centres of gravity and emphasized combined arms but this was not reflected in the development of balanced formations cut loose from the artillery. Instead, Britain motorized all of its infantry and offered support with quite separate tank brigades while the ability to exploit any opportunities or gains made on the battlefield lay with mobile divisions \u2013 little more than cavalry in light tanks \u2013 which, by the spring of 1940, took the shape of 1st Armoured Division. It was therefore an army that, due to its massive recent expansion, entered the war bursting with untrained and poorly equipped personnel. The BEF was its most capable component: brave, tenacious and well organized, but lacking relevant fighting experience and a rooted doctrine. Here was a force in transition which, although mobile, was probably better suited to defence than to high-tempo manoeuvre warfare.\n\nNeither the BEF nor the French army had any great concern about what they were likely to be asked to do as part of a coordinated Allied action to meet the German offensive. Both had the ability, if required, to move towards the enemy, settle into defensive positions and then fight a series of battles designed to grind the opposing forces down. This facility suited Gamelin's nascent planning for countering a German invasion. The plan was founded on the 450-mile-long line of defences from the Swiss border to the North Sea incorporating seventy-two divisions. Second Army Group was in the Maginot Line, with First Army Group occupying the line north to the coast. With Allied intelligence indicating that the Germans were massing 116 divisions along their western border, Gamelin assumed that their main offensive effort would be north of the Ardennes and so decided to deploy the most mobile Allied troops to advance towards them. Although Belgium's neutrality would preclude Allied troops from preparing defensive positions on its soil, Gamelin hoped that its own troops would make the necessary preparations so that, in the event of an invasion, the Allied troops could advance to occupy them. The entire plan hung on the ability of the Belgians to hold forward defensive positions long enough to give the French and British troops the time they needed to set themselves. Second-rate troops were deemed good enough to man both the Maginot Line and the Ardennes sector. Any attack through the difficult wooded Ardennes was, the High Command concurred, likely to be minor but, even if it happened to be more significant, they believed that it would take nine days for the enemy to penetrate the area and amass sufficient troops to launch an attempt to cross the Meuse. In such circumstances, reinforcements would immediately be moved into position in time to foil German plans. Map exercises conducted in May and June 1938 by General Andr\u00e9-Gaston Pr\u00e9telat, at that time the commander of Second Army with responsibility for the Sedan-Montm\u00e9dy region, concluded that the Germans could reach the Meuse with three corps in just sixty hours, but these findings were dismissed as unduly pessimistic. Indeed, there was a growing feeling that the High Command wished for war games to be used only to refine existing plans and not challenge decisions that had already been made. This stance was in keeping with the uncritical ethos that had permeated the French army, but it meant that no alternatives to a mechanized version of the 1914 Schlieffen Plan were considered as viable German invasion options.\n\nWith French minds made up about where the enemy would place his main effort in an invasion, the question remained as to how far forward the Allies should push into Belgium before establishing a defensive line. Too far to the east and the Allies would struggle to reach it before the Germans arrived, but too close to the French frontier and the Belgian 'buffer zone' would have been unnecessarily surrendered to the enemy. During the autumn of 1939, while defences were prepared along the Franco-Belgian border by General Gaston Billotte's First Army Group, a decision was made to advance to a line from the French border at Givet, up to Namur, across the Gembloux Gap \u2013 an ancient invasion route due to its lack of natural obstacles \u2013 and the Dyle River to Antwerp. This scheme, known as Plan D, demanded a challenging yet achievable advance for Billotte's formations \u2013 which included the mobile BEF, the First Army with its attached Cavalry Corps and Seventh Army in reserve \u2013 while the Belgians fought on the Albert Canal, and it allowed key battles to be fought on Belgian soil away from the French border. By the first days of January, as the details of Plan D were being worked out and the necessary preparations made, OKW was still waiting for what it considered a suitable offensive plan by OKH. The hours of daylight were short, but for Halder's discouraged staff officers the hours of work were long as they sought to produce an offensive that met Hitler's requirements. It was while they were looking into how Hitler's requirement for an enhanced push through the Ardennes might be incorporated into a wider operational plan, that events took a turn which resulted in Halder developing a plan that made the Ardennes thrust the main effort.\n\nOn the freezing morning of 10 January, Hitler received news for which he had been waiting since the late autumn: two weeks of clear weather would start in five days' time. It took him just moments to decide that the invasion would be launched on 12 January. Time was of the essence and even though the Second Deployment Directive lacked finesse, it would catch the enemy cold and considerably weaker than if the attack was launched in the spring. However, when a light aircraft on an unauthorized flight from M\u00fcnster to Cologne made an emergency landing at Vucht, just north of Maastricht on neutral Belgian territory by the Meuse, the Germans needed to think again. The aircraft contained fifty-year-old Major Helmuth Reinberger, a general staff officer in the airborne force who was carrying secret documents which provided enough detail for the enemy to deduce the context of the overall campaign. The capture of Reinberger, his pilot, the incompetent fifty-two-year-old reserve officer Major Erich Hoenmanns, and the documents by Belgian frontier guards on 10 January, was a potential disaster for the campaign. The Belgians inevitably asked themselves whether the event was an elaborately staged ruse to widely mislead and provoke the government into calling on British and French support, thereby giving the Germans a pretext to invade Belgium. After secretly recording conversations between the two officers and German air and army attach\u00e9s, however, the Belgians had no reason to believe the documents to be fakes. The Germans, meanwhile, had to assume that the enemy now had the basic outline of their current offensive plans. Jodl's diary for 12 January has a brief entry: 'If the enemy is in possession of all the files, situation catastrophic!' Hitler, it was said, was simply furious. The offensive was postponed once more, and the opening offered by the fine weather was wasted by the idiocy of two junior officers.\n\nYet while the inadvertent death of the Second Deployment Directive might have caused a lesser general staff to panic, OKH did not and became more receptive to radical ideas. This coincided with Hitler's own interest in developing the advance through the Ardennes even further than he had demanded in late November. His new focus had been stimulated by war games on 27 December and 7 January which, using intelligence that the French were developing a strong mobile reserve in the Verdun area, suggested that this posed a significant threat to the southern flank of the proposed German offensive. Thus, the Third Deployment Directive of 30 January, although not radically different from the previous plan, included two panzer divisions in Twelfth Army's advance through the Ardennes to fall on Sedan and so protect the vulnerable flank. This left six panzer divisions for Army Group B's main effort and one in reserve. Even so, the third iteration of OKH's plan remained predictable and conservative and was still unlikely to achieve a decisive victory. It was in this barren landscape that von Manstein's fertile imagination began to find a receptive audience.\n\nThe advancement of von Manstein's ideas came about very largely due to fortunate happenstance. Army Group A's headquarters was at the fashionable Hotel Riesen-F\u00fcrstenhof overlooking the Rhine in Koblenz, and for its day-to-day work its staff took over the headquarters building of 34th Division close to Deutsches Eck six miles to the north. It was here that von Manstein was to write seven memorandums \u2013 the first on 31 October 1939 and the last on 12 January 1940 \u2013 which provided alternatives to OKH's plans. He had a good working relationship with his commander, the tough sixty-five-year-old von Rundstedt who, despite his rank, had the odd habit of dressing in his uniform as honorary colonel of the 18th Infantry Regiment and relaxed during the day with paperback detective novels. Von Rundstedt rated his chief of staff's skills highly, liked to encourage his ideas and was content to leave details to him. Although essentially an old-school, traditional officer, von Rundstedt at least agreed with von Manstein that OKW's plans 'contained no clear-cut intention of fighting the campaign to a victorious conclusion'. Furthermore, Halder's plan ran contrary to German military history, tradition and understanding of war while von Manstein's scheme ran with them as he advocated Army Group A becoming the campaign's armoured main effort to achieve a decisive result. Army Group B was to advance as the Allies expected in order to pin and fix Gamelin's best formations in central Belgium while von Rundstedt's panzer divisions sped to the Channel coast to effect an encirclement. Other formations would plunge simultaneously south having crossed the Meuse in order to thwart a Verdun counter-attack, pierce any developing defensive protecting Paris on the Aisne, and unhinge the Maginot Line position. It was a plan, von Manstein argued, that avoided protracted attritional warfare and offered the prospect of strategic victory in the next phase of operations because the Allies would already have lost the decisive battle.\n\nVon Manstein was an arrogant officer who did not suffer fools gladly, but he was not so self-confident as to offer his scheme up without having ensured that it was a practical proposal. Accordingly, by early 1940 he had discussed his idea with several other officers, but none were as enthusiastic or as important as Heinz Guderian. The most experienced armoured commander in the Wehrmacht and commander of XIX Panzer Corps was accommodated in a hotel adjacent to Army Group A's headquarters. The two men knew each other from the War Academy in 1907, had met periodically since then and now discussed the feasibility of large tank forces passing through the Ardennes, where both men had served during the First World War. By the end of their detailed discussions, both Manstein and Guderian could see no reason why, with careful preparation, panzer divisions could not reach the Meuse within four days. Both men subsequently sensed a great opportunity: von Manstein to trump OKW's plan and Guderian to command the spearhead formation in a campaign that could alter the course of history. Halder, however, had continually rejected his rival's increasingly detailed memorandums not just because they contained proposals that were too radical for his taste, but because he believed them to be a form of insubordination. Indeed, by the new year Halder had become so piqued by von Manstein's persistence that he convinced von Rundstedt that his chief of staff \u2013 whose ideas the commander of Army Group A now considered to be flights of fancy \u2013 should be 'moved on' to make better use of his talents and, on 27 January, von Manstein learned that he had been given command of a paper corps in a military backwater. His ideas, however, were not lost, for by the time he was preparing to move to a small town 120 miles north-east of Berlin, Hitler had already expressed his dissatisfaction with Halder's Third Deployment Directive.\n\nDespite his innate conservatism and his distaste for von Manstein's methods, Halder nonetheless found himself increasingly enthusiastic about the main effort forging its way through the Ardennes and across the Meuse. It was a position confirmed by the outcome of an Army Group A war game held on 7 February. Organized by von Manstein to take place two days before he left for his new post, it convinced him that Halder was now coming round. As a consequence, OKH began to feed OKW the outline of the spurned staff officer's plan. This process can be followed in Jodl's diary entry for 13 February, which states:\n\nI hand over to the F\u00fchrer a summarized report, from which the great possibility for formation of a centre of gravity south of the line L\u00fcttich [Li\u00e8ge]-Namur becomes obvious (at least five times the strength of the forces committed north of that line). I bring to his attention that the thrust against Sedan is a tactical secret path, where one can be surprised by the God of War. If the French attack from the south flank, we must veer to the south.\n\nIn fact, Hitler was already aware of the existence of von Manstein's scheme after Rudolf Schmundt, his adjutant, had come across it while visiting the headquarters of Army Group A at the end of January. Schmundt subsequently arranged for the plan's author to discuss his proposals with Hitler at the Reich Chancellery on 17 February. At the meeting the F\u00fchrer revealed his enthusiasm for Army Group A's strengthened attack. There was an important difference of focus, however, between the two men: while von Manstein looked for von Rundstedt's spearheads to push west all the way to the coast as well as south behind the Maginot Line, Hitler was fixated on Sedan. The differences led the banished officer to write one final detailed memorandum on the subject which he sent to Army Group A:\n\nThe _aim of the offensive in the West_ must be to _bring about a decision on land_ [emphasis in the original]. For the limited objectives given in the present deployment order, the defeat of [the] largest possible enemy grouping in Belgium and the seizure of parts of the Channel coastline, the political and military stakes are too high. The goal must be the final victory on land. Operations must therefore be directed [immediately] towards achieving a final decision in France, and the destruction of French resistance.\n\nHitler, meanwhile, spoke to Jodl and asked him to order Halder to develop a new directive based on von Manstein's concept. In fact, Halder was already working out the practical details of just such a plan and, having met with Hitler and von Brauchitsch on 18 February, produced the Fourth \u2013 and last \u2013 Deployment Directive six days later. It was this document that was to be enacted as Germany's offensive against France and the Low Countries as _Fall Gelb_ (Case Yellow).\n\n_Fall Gelb_ was a remarkably bold blend of 'experience, intuition and understanding'.  Army Group B with three panzer divisions would attack into the Netherlands using Eighteenth Army and central Belgium using Sixth Army. Everything about von Bock's advance was designed to make Gamelin believe that this was the main effort, and so the Luftwaffe was to initially focus on supporting Sixth Army while airborne forces would be deployed against the position known as Fortress Holland and the key bridges over the Albert Canal to aid the ground advance. The attack on the Netherlands had been questioned due to the country's challenging terrain and the resources that would be required to force its capitulation. G\u00f6ring, however, demanded the country as a base from which to launch future Luftwaffe operations against Britain while also noting that its occupation would deny the Allies a base from which they could bomb Germany. It was also believed that Fortress Holland might be reinforced by the Allies in the midst of their offensive and then used to threaten the Wehrmacht's northern flank. The invasion of the Netherlands therefore became a subsidiary offensive for von Brauchitsch's troops, but one of considerable import to _Fall Gelb_ and German strategy. Meanwhile, as Army Group C demonstrated against the Maginot Line in order to contain its defenders, Army Group A was to move across Luxembourg and Belgian territory to reach the Meuse centred on Sedan. On the right, Fourth Army would be led by a panzer corps of two panzer divisions to cross the Meuse around Dinant while on the left, Twelfth Army would fall on Sedan using five panzer divisions and the motorized corps. Advancing through Luxembourg, the non-armoured Sixteenth Army was to seal off the left flank of the penetration south of Sedan and, by engaging the northern end of the Maginot Line, feign a turning movement to its rear in order to tie up French resources in the area.\n\nAt this stage, it had not been decided whether the crossing of the Meuse was to be accomplished by the panzer divisions before the arrival of the infantry, or a pause to be taken for the infantry to arrive and assume the lead. Moreover, mindful of the old military dictum that 'no plan survives first contact', the Germans only formalized theirs as far as the Meuse for Army Group A in order to retain a degree of flexibility: much would depend on how the enemy reacted to the offensive. For now, the plan only speculated about what might happen after the Meuse had been crossed, but highlighted the possibility for 'an operational probing attack by armoured and motorized elements and of pushing toward the mouth of the Somme river'. There was no mention of a simultaneous thrust south behind the Maginot Line. Halder had embraced a high degree of boldness, but he was not willing to put the entire operation in jeopardy for what he considered to be a non-essential manoeuvre when resources were tight and the co-ordination of the attacking formations was challenging enough already.\n\nThe plan was divisive. While von Manstein and Guderian lamented the fact that the Fourth Deployment Directive had clipped the radical wings of the original scheme and in so doing had preferred operational success to strategic victory, many seem to have regarded the plan as far too risky. Similarly, while some advocates of armoured warfare dubbed Halder the 'gravedigger of the Panzer force', some infantry generals who were uneasy at the potential of mechanization and motorization deemed the plan 'crazy and foolhardy'. Having had command of the main effort taken away from him, von Bock wrote despairingly in his diary: 'I can't warm to this operation, because it has to bog down if the French haven't taken leave of their senses... [and] Army Group B will be too weak to carry out the missions left to it.' There were also concerns expressed by a number of senior officers \u2013 particularly those from the traditional arms \u2013 that OKH was placing the relatively untried panzer divisions centre-stage while the tried and tested infantry languished behind as a support player. All of the old arguments about the vulnerability and weaknesses of armoured forces were brushed down and now given a specific operational context: Were armoured forces capable of penetrating the Ardennes quickly? Was it possible for panzer divisions to establish themselves across the Meuse without close infantry support? If armoured and motorized elements did exploit the river crossing, wouldn't they risk dangerous exposure if they advanced far ahead of the infantry? The sceptical von Bock seemed to sum up the concerns of many when he wrote to OKH:\n\nYou will be creeping ten miles from the Maginot Line with the flank of your breakthrough and hope the French will watch inertly! You are cramming the mass of the tank units together into the sparse roads of the Ardennes mountain country, as if there were no such thing as air power! And, you then hope to be able to lead an operation as far as the coast with an open southern flank 200 miles long, where stands the mass of the French Army\n\nBy this stage, von Rundstedt had completed his transformation from a supporter of von Manstein's original idea to a doubter of the Halder plan. Recognising that his success depended on the achieving of operational surprise followed by speed and momentum, he had understandable concerns considering he needed to cross the Ardennes, the Meuse and then, most likely, conduct an extended dash to the coast. It was at this point that von Rundstedt's traditional leaning came to the fore because he could not help but think that the operation's risk could be significantly mitigated by the panzer divisions and infantry working together as they had in Poland. It was not for nothing that Halder noted in his diary that the commander of his main effort was 'sceptical about the effectiveness of the armoured wedge. Is afraid that the second and third echelons would not be able to catch up.'\n\nJust such issues were raised on 15 March 1940, when a meeting to discuss the Army Group A attack was held at the Reich Chancellery. The gathering included Hitler, Halder, von Rundstedt and his competent but utterly conventional new chief of staff, Georg von Sodenstern. The latter had taken an immediate dislike to OKH's plan and was to have a considerable influence on his commander's deepening reservations about it. On 22 February, in a formal submission to von Rundstedt, von Sodenstern wrote:\n\nI am not convinced that even the reinforced panzer and motorized units will manage to force the crossing over the Meuse in the kind of breadth that is necessary for operational purposes. Yes, I doubt, to begin with, that they will be in a position to cross the Meuse River even only here and there, holding the bridgeheads thus gained until the following infantry division would be able to make room for an operational exploitation featuring the necessary breadth and depth... But even if that should come off successfully, the panzer and motorized units by that time will be so 'exhausted' that sending them deep into enemy rear areas will no longer offer any chances of success.\n\nAlso present at the meeting were the senior commanders of Army Group A's formations: Generaloberst G\u00fcnther von Kluge commanding 4th Army, which included General Hermann Hoth's XV Panzer Corps consisting of 5th and 7th Panzer Divisions and an infantry division; 12th Army's General Wilhelm List; and 16th Army's General Ernst Busch. They were joined by General Paul von Kleist, the fifty-eight-year-old commander of a new formation, a panzer group, who was an upstanding Prussian cavalryman and an officer who 'never compromised with the Nazis or anyone else'. It was Panzer Group Kleist that would lead Twelfth Army \u2013 to which it was subordinated \u2013 to the _Schwerpunkt_ on the Meuse, and the formation was put in the hands of a traditionalist such as von Kleist because the High Command wanted the formation to be used with restraint and for Guderian to be controlled. The panzer group itself consisted of the 1st, 2nd, and 10th Panzer Divisions in Guderian's XIX Panzer Corps, 6th and 8th Panzer Divisions in Major General Georg Hans Reinhardt's XXXXI Panzer Corps, and 2nd, 13th and 29th Motorized Divisions in General Gustav Anton von Wietersheim's XIV Motorized Infantry Corps. Formed on 5 March to provide a distinct organization for the armoured spearhead, it consisted of 134,370 men and 41,140 vehicles including 1,222 tanks (which was half the total German number and were painted with a 'K' for Kleist for identification). If von Kleist's vehicles had been placed nose to tail on a single road, they would have extended over 600 miles. It was just the sort of powerful concentration of panzer divisions that the traditionalists had feared, for at a stroke it made independent panzer operations far more likely and more viable.\n\nWhile von Rundstedt, von Sodenstern and Busch, who did not understand tank warfare, all raised various concerns at the Reich Chancellery meeting with Hitler about the armour-led offensive, it was left to Guderian to represent the views of von Manstein and to argue that there should be no dilution of the plan's boldness. Halder recognized the concerns of both parties, but told the conference that, given the operation had been successfully war-gamed several times, 'We must resort to extraordinary means and bear the attendant risk.' He added that the armoured spearhead should not only lead the infantry to the Meuse, but cross it and create a bridgehead before the infantry arrived. This, Halder opined, would make best use of the surprise appearance of the panzer divisions on the river, and help maintain the momentum of the spearhead's advance. In this way, he concluded, more opportunities would be created than if the panzer divisions were forced to pause. It was an opinion that won Hitler over and so it was decided that elements of Panzer Group Kleist would bounce the Meuse in a single push rather than waiting for the infantry to arrive. Halder noted in his diary on 17 March: 'The F\u00fchrer now approves the preparations made and is manifestly confident of success... Decision reserved on further moves after the crossing of the Meuse.'\n\nThe detailed planning of the attack could now continue on a firm footing, although the conflicting demands of the panzer divisions and those of the infantry would be manifest throughout the process. In this vein, while List respected the decision that von Kleist's panzer and motorized divisions would lead his infantry army up to and over the Meuse, he did everything in his power to ensure that his subordinate's panzer group was kept on a short leash. This caused von Kleist's highly capable chief of staff, Oberst Kurt Zeitzler, endless difficulties. Perhaps the best example of this was the way in which List denied von Kleist the space that his divisions needed to advance by limiting Guderian's vanguard corps to just four roads in an attempt to slow its advance to the speed of the infantry. In fact, the decision had the opposite effect, because Zeitzler merely demanded that Guderian advance hard and fast from the outset, moving ahead of the infantry and thereby allowing them to use whatever roads they required. Reinhardt and von Wietersheim, denied the space they needed to attack abreast of Guderian, would follow immediately behind and then fan out as they reached the Meuse. It was anything but ideal yet Guderian recognized the importance of traversing the 105 miles from the German border to the river as quickly as possible, proclaiming to his officers: 'In three days the Meuse, on the fourth day across the Meuse!'\n\nThe Allies had no idea that the Germans intended to send their main effort through the Ardennes and across the Meuse, and Gamelin was resolute in his belief that the enemy would attack on the traditional invasion route. They failed to seriously consider that the Germans might change their plans after the capture of their secret documents on 10 January largely because senior officers could not comprehend what an alternative plan might look like, and since the French were only primarily interested in the detail of what faced them north of the Ardennes, the picture of German movements and deployment they obtained was necessarily incomplete. Intelligence also helped to confirm their assumptions because it identified increasingly large German concentrations of tanks and motorized units east of Holland and Belgium. Gamelin therefore read the numerous reports about the developing strength of Army Group B and was more confident than ever that the enemy was about to fall into his trap. Such was his optimism that he sought to augment Plan D with the so-called Breda Variant by pushing formations beyond the Dyle, past the Scheldt and into the Netherlands to Breda (and no further than Tilburg) to link up with ten Dutch divisions. This 'enhancement' sought to ensure that the Dutch forces were not isolated, safeguard the important port of Antwerp and retain valuable territory and resources. To achieve this, Gamelin strengthened the French First Army Group on the Allied left wing with Seventh Army \u2013 the most highly mobile and capable French army \u2013 from the General Reserve near Reims. This was the very formation that the Germans feared might be in the ideal position to strike at the exposed southern flank of Army Group A after it had crossed the Meuse. On 12 March 1940 a directive to General Alphonse Georges, the pedestrian commander of the North-East Front which included First and Third Army Groups, formally adopted the Breda Variant as part of Plan D. Georges already harboured concerns about the Allies' ability to occupy defensive positions on the Dyle, and now this amendment demanded that Seventh Army advance up to 110 miles to Tilburg while the Germans needed to move only fifty-five miles. It was also a move that would have the effect of significantly reducing the threat to von Rundstedt's flank while significantly increasing the number of first-class formations that the Germans could encircle if his formation reached the Channel coast.\n\nSuddenly, the Breda Variant cast doubt on a basic French assumption. The British did not like it and Georges was moved to pronounce: 'This is an adventure. If the enemy should be feigning in Belgium, then he can manoeuvre elsewhere. We should therefore not employ our reserves for this operation! This is nothing more than a dream!' Gamelin ignored this jibe and put it down to the signs of the strain that Georges was beginning to show. The commander of the North-East Front had been hit by four bullets during the assassination of King Alexander I of Yugoslavia in Marseilles in 1934 and latterly there had been questions raised about his ability to concentrate, his patience, and his nerves. Colonel Jean Petibon, the head of Gamelin's staff at Vincennes, thought that Georges should have been replaced while the British military attach\u00e9 in Paris, Colonel William Fraser, opined that he was 'not a fit man and that the weight of responsibility which would be his in war might be too much for him'. Why Gamelin did not replace him may have had something to do with the lack of better alternatives or the fact that, with every passing day, the replacement of the man who had overseen the preparations of his Front with one who had not might have proved counter-productive. It may just be that the French Supreme Commander simply avoided his responsibility. Indeed, when he was asked in January 1940 about Georges's competency, he replied, 'We shall not know until the first clash comes. In 1914 there were many officers and men who failed, but old Joffre handled the situation with great firmness. Will the _Blitzkrieg,_ when it comes, allow us to rectify things if they are the same? I must say I don't know.'\n\nAs it was, the final Allied plan, one which assumed the Belgians and Dutch would invite the French and British to enter their countries if and when the Germans invaded, involved Seventh Army advancing into the southern Netherlands by the coast while, on its right, the BEF and French First Army rushed to the Dyle. Having held the Germans for the requisite period, the Belgian defenders would fall back from the Albert Canal, which ran close to their border with Germany, to take up a position on the Dyle from Antwerp south to Louvain. On their right flank, nine divisions of the BEF were to defend 12.5 miles of front down to Wavre, while fifty divisions of First Army covered twenty-two miles to Namur. This French sector included the area that Gamelin considered the most dangerous avenue of approach \u2013 the Gembloux Gap. From Namur, Ninth Army's left flank was to move into Belgium to take up a position along the Meuse while its right remained stationary as the river ran through French territory. On Ninth Army's right, Second Army was also dug in on the Meuse from Pont \u00e1 Bar, four miles west of Sedan, to Longuyon and acted as a hinge for the movement of First Army Group into Belgium and the Netherlands.\n\nBy the beginning of spring 1940, both the Germans and the Allies had plans that were being refined, with staff officers working on their details to ensure that they could be applied as efficiently and effectively as possible. Yet on both sides there were senior figures who questioned the ability of those plans to deliver what was required and were frustrated that decision-makers were failing to ask the right questions and look in the right places for answers. Such concerns, however, were trampled over by those in key appointments, the men of influence who seemed to retain a confidence in their schemes. They, of course, had anxieties of their own, private concerns of a type that always bedevil commanders on the eve of operations. Were the right personnel in critical posts and would they work well with their peers and subordinates? Were the right formations and resources assigned to the right sectors? How would the enemy react? And how would both plan and personnel survive the strain of contact with the enemy? Planning dealt with some of these issues, but other preparations were also needed to ensure that both sides were in the best possible condition for the campaign.\n\n* In Poland, Winkler lost his left foot during a Polish artillery barrage. Lucky to survive, he was taken by train back to Germany where he spent three months in hospital. He married his nurse.\n\n# CHAPTER THREE\n\n# Final Preparations\n\nTHE PHONEY WAR WAS A LONG TEST for France's devotion to making the necessary preparations to fight Germany. As Daniel Barlone wrote in November 1939: 'No one talks of the war now. Winter is on its way and we are convinced that Hitler will not attack before the spring... The whole world seems to be unruffled. Shall we really have this war? Or shall we remain facing each other, for years?' Some argued that the war would not necessarily lead to the trading of blows and the prospects of confrontation would 'die away', but generally the population just wanted to 'get on with it'. At first, this caused anxiety across the nation and particularly in the north-east which, once again, was going to be at the forefront of any action. The government and military hierarchy nonetheless went about their business with an admirable briskness and poise as they enacted their plans. The system worked tolerably well once some early issues had been solved, and war production increased while mobilization produced more troops. The government also worked hard to promote calmness in the population by reassuring them that, despite the challenges ahead, France _would_ win, as she had earlier that century. Anything that upset the continuing preparation for war was avoided; it was to be business as usual. Arrangements were, however, made for those living along the Franco-German frontier to be moved to departments in the south of the country. Civil servants also drew up contingency plans for the evacuation of the population from the capital, but these would be put into action only _in extremis_ and were not publicized. In fact, some 500,000 people left Paris of their own volition in the days after the declaration of war: most were frightened by the prospect of German air raids and had no concerns about jobs and money, but the majority heeded official advice and stayed put. France settled into a quiet groove: it was at war but as yet there were only short, local skirmishes while commercial, political and everyday life continued largely as before. Indeed, during December 1939 the French Cabinet spent a considerable amount of time discussing the intricacies of family law.\n\nBy the early spring of 1940, however, a dangerous lassitude had embraced France. Even though military commanders knew that better weather made a German attack more likely, the population seems to have been lulled into a state of false security. Journalist Alexander Werth wrote that 'Life in Paris became quite normal' with the political merry-go-round still turning and on 20 March Paul Reynaud replacing Daladier as prime minister, although the latter remained minister of defence. Reynaud was a decisive man and had serious reservations about Gamelin's suitability for the job of Supreme Commander and, by extension, his plans and preparations. He immediately began to make arrangements to replace him when the time was right. In the meantime, he hoped that the German invasion would be further delayed so that France was strengthened not only by the production of more war resources but also by the perspicacity of his leadership. The troops at the front knew nothing of this and some were concerned by yet another change in government, with Daniel Barlone writing, 'The moral effect on the troops is bad... Our enemies do not waste their time over parliamentary manoeuvres which are both unintelligible and disconcerting. Spring is now here, the time of trial is probably at hand and look what a government we have!'\n\nThe troops commanded by Barlone, in common with many others along the French eastern border, found life extremely tedious during the Phoney War. Although the way in which an individual's time was filled very much depended on his unit, formation, sector and role, days tended to be dreary. For those deployed at the northern end of the Maginot Line to the North Sea coast, the critical task was to develop the Line's second phase of light defence works. This manual construction absorbed time and energy but was hardly designed to make the troops better soldiers or any military organization better at its primary defensive task. Initially, the novelty of the situation and fine weather allowed morale and motivation to remain robust. Even so, Barlone, who had been given command of a 2nd North African Division horse transport company just behind the Maginot Line in Lorraine, had observed in late August 1939: 'The men of the reserve come in without undue haste, without the enthusiasm of 1914, or even that of last year, being fully persuaded that they will be sent home in two weeks' time, and annoyed at not being able to complete the harvest.' Unlike in 1914, when there were Alsace and Lorraine to retrieve, in 1939 there was no such motivation, leading Barlone to write: '[T]oday we desire nothing... We know our land is safe from invasion, thanks to the Maginot Line.'\n\nThe fighting spirit in Barlone's unit was not high to start with and soon, as in many comparable French outfits, it was eroded by monotony and chronic equipment shortages and, when winter came, by rain and darkness, frost and snow. Attempts to raise morale by allowing access to local estaminets, and arranging entertainment in the larger towns and cities, village film shows, games of football and the like were appreciated, but could do little more than only temporarily warm and lighten the hearts of damp, cold and bored troops. Recently mobilized reservist Pierre Roussel, serving five miles south of Sedan, noted in his diary during a particularly foul spell of weather in December 1939:\n\nLife is bad at the moment. It is very cold. There is mud everywhere and even simple tasks are difficult in such conditions. We smoke and yarn when the weather is too poor to work, but most often we toil with frozen hands... We have almost forgotten why we are here. I think that prison might be preferable to this routine.\n\nMen who had joined up to fight found that there was, in fact, very little fighting during this period of stand-off, with any military activity often being limited to patrols. 'We are forced labourers rather than soldiers,' wrote Roussel. 'I hardly ever see a rifle and only have the vaguest notion how our key weapons work and should be maintained.'\n\nThe winter months were a difficult time for all troops living in the field who had to work long hours in poor conditions to ensure that their section of the line was fit for purpose. As the official history of the British 5th Division notes:\n\nDuring the appalling weather conditions of the 1939\u201340 winter, first wet and muddy, then cold and hard, the division worked strenuously to improve their positions; digging deeper, revetting and wiring. Field and anti-tank guns, machine guns and other support weapons were cunningly concealed in the then fashionable disguise of houses, haystacks and chicken sheds... It was difficult, however, to believe that there was a war around the corner.\n\nThe impact of the worsening conditions at the front, however, seems to have affected the second-rate troops far more than those in Billotte's First Army Group. Lieutenant Colonel Sumner Waite, an American observer touring French First Army, found morale 'splendid everywhere' and he noted that there was a clear desire throughout the ranks 'to do away with the threat of German domination for good and all'. He also made a point of emphasizing that the reservists gave the impression 'that they had been soldiering for years', and praised the 'keenness and enthusiasm of reserve officers as well as regulars', noting 'a gallant and pugnacious spirit was met at every turn. They're ready to meet the best the enemy has'. Morale and motivation closely reflected a formation's fighting capability. Billotte's formations retained their fighting spirit over the hard winter months, while those further south struggled to maintain what little they had.\n\nGamelin's headquarters was aware that certain formations of the Ninth and Second Army seemed to lack the _esprit des corps_ of their sister formations towards the coast, but would have been far more concerned had the reverse been true. Of course, the troops disliked the hard physical labour in such conditions, but the spring would surely see a change when the back of the work had been broken and the weather improved. Indeed, over the winter of 1939\u201340, far from being worried about conditions at the front or, more specifically, at that part of the front they assumed would meet the main German effort, the French High Command became increasingly confident. Continued improvements in rearmament, the progress on the second phase of the Maginot Line, good news about the improvements made to the Belgian and Dutch armies and their defences, all led to a sense of wellbeing at Vincennes \u2013 and also ushered in the refinement of the Breda Variant to Plan D.\n\nUnderlying everything was the increased size of the army, which had grown to ninety-four divisions by the spring of 1940, including five that were partly mechanized cavalry divisions, three that were light mechanized divisions (DLMs) and seven that were motorized infantry divisions. There were also three new heavily armoured divisions or Divisions de Cuirassiers de R\u00e9serve (DCRs), which were established for armoured breakthrough in the wake of the German victory in Poland. Smaller than the DLMs, they had 158 medium and heavy tanks and a single battalion of motorized infantry. Yet just as the capability of France's infantry divisions varied wildly, a note of caution must be expressed here: the DCRs only contained half the number of tanks that each was established to receive and the divisions lacked the balance in other arms that would enable them to be flexible and of high utility in a defensive confrontation. By May 1940, only 1st and 2nd DCRs were at anything like operational strength and, with no likelihood of an armoured breakthrough being required in the short term, they were held in reserve.\n\nThere is little doubt that the High Command drew assurance from the improvements that the French army had made since the outbreak of war, but digging in was not cost-free. Every hour that a soldier spent moving earth from one place to another was an hour not spent in training and an hour nearer to the German onslaught. Stringing barbed wire across your company's front was all very well, but it encouraged a mindset which relied more on the defensive properties of field works and less on a soldier's own military skills. Indeed, by the spring of 1940 Gamelin was presiding over an army that desperately needed shaping and emboldening. The training that would have helped to provide this, so often relegated to a subsidiary activity during the development of defences and commonly postponed or cancelled due to inclement winter weather, began in earnest during late March. Seventh Army, for example, undertook exhaustive training in terrain similar to that north of Antwerp where they were scheduled to meet the enemy, and the two DLMs under General Prioux practised engagements in territory similar to that they would find at the Gembloux Gap.\n\nYet in those areas where more substantial defences were required, such as that held by General Charles Huntziger's unimpressive Second Army before the Ardennes, the digging and building continued. Consisting of five divisions in two corps, Second Army held a sector of forty-seven miles on French soil, partly along the Meuse around Sedan but then along the far less formidable River Chiers heading east through La Fert\u00e9 towards Longwy. The front included the Sedan sector, where developing defences on the west bank looked down on the open slope to the river on the opposite side. Yet it was not Sedan that Huntziger believed to be most at threat \u2013 on 7 May he declared, 'I do not believe that the Germans will ever consider attacking in the region of Sedan' \u2013 but the area between the end of the Maginot Line and Sedan, the area known as the Stenay Gap. This large, flat, open valley, he perceived, could well be a route taken by the Germans in order to outflank the main defensive wall. Huntziger therefore positioned the competent XVIII Corps in the Stenay Gap while the less impressive X Corps defended the Sedan sector. Although 55th Division at Sedan itself was an unimpressive formation, General Pierre-Paul-Charles Grandsard, the X Corps commander, deemed it to be considerably stronger than 71st Division on its right flank. Indeed, Grandsard became so concerned at the latter's leadership and preparations in April that he took it out of the line and into reserve for remedial training and replaced it with 3rd North African Division. Thus X Corps relied for its counter-attacking ability on four independent tank battalions and a division that did not have its senior commander's confidence.\n\nIt was no coincidence that Huntziger's first exercise scenario in April 1940 was 71st Division leading a counter-attack by X Corps towards Sedan after the Germans had made a crossing from the town. The conclusion drawn from this valuable piece of work was that 55th Division had to significantly slow the enemy if the corps' counter-attacking force were to have the time that they needed to assemble, form up and strike back. If the 16,000 men of this division were merely rolled over, Grandsard's ability to assert his will on the battlefield would be extremely limited. A great deal was therefore expected of the four regiments of 55th Division, a Category B formation comprised largely of older reservists and almost wholly officered by them. In fact, 96 per cent of 55th Division's officers were reservists, although its regimental commanders were regulars. Commanded by General Pierre Lafontaine, it had arrived on the west bank of the Meuse at Sedan in October 1939, joining 147th Fortress Infantry Regiment and working assiduously thereafter to improve its defences which, to a large extent, were seen as a means by which to ameliorate its lack of fighting ability. Sedan was built largely on a loop in the Meuse which cuts deeply into the east bank. That loop also has the river flowing across its base, thus creating a large island in the middle of the river, surrounded on three sides by the east bank of the Meuse. As this would make the island vulnerable to the enemy if they reached the east bank, the French decided in November 1939 to withdraw their main line of resistance away from the river bank in this area. The line was eventually built as part of the second-phase Maginot Line constructions which commenced during the autumn of 1939.\n\nUp to the end of April 1940, Second Army used enough concrete to cover thirteen acres to a depth of one foot. At Sedan, Lafontaine oversaw building which included dozens of blockhouses intended to deprive the enemy of momentum. The defences were developed along the lines of X Corps orders, which stated: '[T]he defence will be organized into circular fighting positions or centres of resistance, capable of defending themselves when isolated, even if bypassed by enemy infantry and tanks. As a consequence, the fighting positions or centres of resistance will be interlocked with obstacles of terrain, woods, villages etc.' Second Army also worked on a second line of defences some nine miles behind the principal line of resistance. The result was that Second Army spent twice as much time in construction work as it did in training. Although in terms of priorities for resources the Sedan sector continued to rank below the needs of the Stenay Gap, the ground opposite Sedan was a construction site throughout the winter and into the spring. While Lafontaine's large command bunker at Fond Dagot, five miles south of the Sedan defences, was completed by early May, 55th Division continued to be engaged in work completing their defensive works and was still waiting for vital items such as bunker doors and steel shutters for the embrasures.\n\nWhen the weather improved and hours of daylight increased in March, 55th Division had begun to benefit from more training as Lafontaine established an intensive three-week programme instigated for regiments behind the lines. Yet even this rather late attention to a pressing need came with its own set of problems. When on 6 May 331st Regiment completed its training, for example, it replaced in the line 213th Regiment, which then went for training. The latter had been in the same position since October 1939 and knew its ground intimately, but 331st Regiment took over its sector with little more than an hour-long briefing, a file of important information and whisper of _'Bonne chance'._ To complicate matters even more, although every regiment had its own clear sector of command responsibility, each of the three sectors contained units from other regiments. This system did provide an element of continuity in each sector when a regiment was removed for training, particularly as units of 147th Fortress Infantry Regiment remained resolutely in occupation of the forward positions across the division's front, but it also fragmented the regiment when in the line and was a potential recipe for command and control chaos under pressure. All regimental commanders were well aware of this fact and also that they and their units would have to rely on field telephones and runners to communicate during battle. During the first week of May, field telephone communications cables still lay above ground and looped over tree boughs waiting to be put underground, where they would be safer from enemy bombs and shells.\n\nThe regiments were also only too aware of the deficit in their defensive firepower. Along 55th Division's front there were a mere fifty-six modern anti-tank weapons \u2013 approximately one third of what it should have had \u2013 and three light anti-aircraft batteries. They also lacked mines \u2013 Second Army had only 2,000 anti-personnel mines available for its formations' principal line of resistance \u2013 and X Corps had only 1,972 anti-tank weapons with which to cover both banks of the Meuse. 55th Division received 422 of them, forty-two per mile of front. According to doctrine, the deficiencies were to be made up by artillery firepower. The division was directly supported by seven battalions of guns and could also call on a further four if required, as well as two heavier batteries of X Corps artillery. By May this amounted to 140 guns, which was more than double the usual support for a division. Lafontaine's satisfaction at having such artillery support would have been tempered, however, by Grandsard's confession to him that the X Corps' artillery was 'without the capacity and without the will for command': 110 out of 280 of its NCOs had been relieved of their duties by May.\n\nDespite the clear deficiencies in Second Army, and particularly X Corps, the formation gave Gamelin no cause for concern. The Supreme Commander visited Huntziger twice in nine months and on both occasions found nothing to be terribly awry. He made suggestions for improvements to Second Army as he did to every formation he inspected, but they were suggestions rather than orders and, as such, were not always acted on. This was Gamelin's command style \u2013 he argued that at 'certain levels of responsibility, it is no longer a matter of giving orders but of persuading'. It was a technique totally out of step with what was required and as a result the French army \u2013 and particularly Ninth and Second Armies \u2013 atrophied during the winter of 1939\u201340. Thus, although observing in mid-September, 'When the war really begins [in the west], it will come as a very rude awakening,' Gamelin used the time available to temper the likely shock of a German attack extremely poorly. He was far too satisfied with merely increasing the size of the army rather than with developing the fighting capability of what he had, and he was guilty of na\u00efve assumptions about German intentions which left his defences vulnerable. In short, the French Supreme Commander failed to provide the leadership and direction that was central to his role and which his army desperately needed.\n\nSuperficially, there was nothing wrong with the French army, and Sir Edmund 'Tiny' Ironside,* the bulky, British, 6' 4\" tall Chief of the Imperial General Staff, wrote after a visit to the front in January 1940:\n\nI must say that I saw nothing amiss with it on the surface. The Generals are all tried men, if a bit old from our point of view. None of them showed any lack of confidence... In 1914 there were many officers and men who failed, but old Joffre handled the situation with great firmness. Will the _Blitzkrieg,_ when it comes, allow us to rectify things if they are the same? I must say I don't know. But I say to myself that we must have confidence in the French Army. It is the only thing in which we can have confidence.\n\nAlan Brooke had also made an age-related entry in his diary, having met Gamelin a few months earlier and written, '[He] struck me as looking old and rather tired.' The British had their concerns about Plan D, the Breda Variant and the ability of the French to stand up to a German onslaught, with Brooke complaining, 'I am not happy about our general attitude, we are facing war in a half-hearted way... I feel that the Germans would have been tackling the situation very differently', but they were also acutely aware that they were the junior partner in France and had challenges enough of their own without getting involved in the business of their ally. It was therefore left to the civilian French deputy Pierre Taittinger to raise concerns about the military state of play. Visiting the Ardennes front in his capacity as a member of the Parliamentary Army Committee during March 1940, he was shocked to find the critical sector of the 1870 campaign so poorly prepared for a German onslaught. On the day of his return, Taittinger submitted a report to the war minister and to Gamelin which stated:\n\nIn this region, we are entirely too much taken with the idea that the Ardennes woods and the Meuse River will protect Sedan and we assign entirely too much significance to these natural obstacles. The defences in this sector are rudimentary, not to say embryonic.\n\nHe was also unimpressed with the divisions defending the area, noting that the Sedan sector was a particular 'weak point', that he 'trembled' at the thought of it being attacked and going as far as to say: 'This place spells misfortune for our troops.'A riled Gamelin informed Georges of the report, and the latter, somewhat taken aback by Taittinger's criticisms, asked Huntziger to reply. Huntziger was furious that a civilian had cast aspersions on the professionalism of the army \u2013 and no doubt also because these criticisms were being taken so seriously by his superiors \u2013 and perceived Taittinger's remarks as a personal attack. In a measured response, however, the commander of Second Army endeavoured to put the politician's mind at rest, stating that Belgian resistance and the natural obstacles of the Ardennes, together with the Meuse, needed to be taken into consideration when assessing the security of the sector. He concluded by saying, 'I believe that there are no urgent measures to take for the reinforcement of the Sedan sector.' Everything was in hand.\n\nLike the French, the BEF spent the period of the Phoney War putting on muscle, developing their sector of the border for defence and undertaking what training time and the weather allowed. Its commander, the fifty-three-year-old Lord Gort \u2013 John Standish Surtees Prendergast Vereker, sixth Viscount Gort \u2013 had been Chief of the Imperial General Staff at the outbreak of war but, having been replaced by Ironside, took to his new field appointment a robust working knowledge of the capabilities of the British army together with youth and vitality. Yet despite a distinguished First World War in which he had been wounded four times, mentioned in despatches on nine occasions and awarded the Military Cross, the Distinguished Service Order with two bars and the Victoria Cross, some doubted Gort's ability to command a large organization and believed him to be limited 'in mind and personality'. II Corps's Alan Brooke for one was concerned by his lack of wider focus and his inability to delegate, and wrote in his diary on 22 November:\n\nGort is a queer mixture, perfectly charming, very definite personality, full of vitality, energy and joie de vivre, and gifted with great powers of leadership. But he fails to see the big picture and is continually returning to those trivial details that counted a lot when commanding a battalion, but which should not be the concern of the Commander-in-Chief.\n\nBrooke himself dealt with his superior with all the tact that he could muster, but continued to be frustrated by details about training schedules when there was a desperate need to sort out the command and control hierarchy with the French. Brooke's divisions and units were also constantly asking him when vital equipment was going to arrive. 15\/19 King's Royal Hussars (KRH) was a cavalry regiment which would lead 3rd Infantry Division forward to the Dyle Line when the time came. The regiment's history, written by Guy Courage, who was a lieutenant in 1940, later stated: 'As far as equipment was concerned, the harvest of the fruit of twenty years' parsimony would find a scarce and rotten crop. Our only real assets were the sterling qualities of the officers and men, who, confronted with a desperate situation, willingly gave of their best.' It was positioned near Lille, just a few miles from the frontier with Belgium, and 'settled down to a steady round of training and maintenance; time might be short but there was much to do'.\n\nThe French High Command had a high degree of confidence in the BEF, something which was reflected in its position on the left flank of First Army, but although encouraging noises were made by the Belgians and Dutch about their own successful preparations, Gamelin did not have any real sense of their capabilities. Belgium, with a population of just 8.2 million, had experienced many invasions in its history, which was why there were so many fortifications along its eastern border. Many had been restored and modernized by May 1940, by which time the country had eighteen infantry divisions, two partly motorized divisions and two mechanized cavalry divisions: a total of 650,000 men and 270 largely obsolete light tanks commanded by General Raoul van Overstraeten. Yet despite Belgium's attempts to improve its armed forces over the past one or two years, the quality of training and equipment was poor. The air force consisted of 377 aircraft, half of which were also obsolete, and, to make matters worse, the country's air bases lacked the space in which to disperse the aircraft, leaving them clustered and vulnerable. The Belgian plan for defence, despite the lack of joint staff talks, anticipated Gamelin's moves: the Germans would be delayed first along the frontier, then along a line ten miles to the rear before the main position along the Albert Canal from Antwerp to Maastricht and then down the Meuse to Dinant. Both the Albert Canal and the Meuse were studded with casemates and fortresses which, it was hoped and believed, would take days for the Germans to overcome, thus providing the French and British forces with the minimum of seventy-two hours to reach the Dyle position.\n\nAs for the Ardennes, the Belgians did not consider them as a priority given their limited resources, while both they and the French made the dangerous assumption that the other would take responsibility for the area's defence. As it was, van Overstraeten covered the area with a single army corps known as 'K' Group \u2013 named after its commander, Lieutenant General Maurice Keyaerts \u2013 which largely consisted of 1st Ardennes Light Infantry (Chasseurs Ardennais) Division, 1st Cavalry Division and some engineers. Once 'K' Group had delayed any German attack, it would withdraw not towards the French border but north behind the Meuse between Li\u00e8ge and Namur to reinforce the units defending that sector. Behind 'K' Group there were no substantial defences for sixty miles on an eighty-mile-wide front before the French border. Into that void, the French were preparing to push nothing more than their own delaying force, which consisted of five cavalry divisions, three cavalry brigades and light advance units from the divisions defending the border. The defence of the Ardennes in neutral Luxembourg was even less well-defined, with a mere thirteen officers, 255 armed gendarmes and 425 soldiers. Their role was to man the Schuster Line \u2013 named after its architect, Joseph Schuster \u2013 which consisted of frontier posts and to use iron gates at the bridges on the border with Germany together with barriers to block roads heading west and cause delay to the German advance. The area identified by the Germans for their main armoured attack was, therefore, defended by a collection of mobile light French and Belgian formations, some Luxembourg policemen and a number of road blocks. Even so, Gamelin maintained that the complex terrain was the equivalent of several heavy divisions and it would take the enemy five to nine days to penetrate the region and at least another week before he would be in a position to cross the Meuse.\n\nThe Netherlands did not offer a buffer zone to France as Belgium did, but the Allied High Command recognized that the more German troops that could be absorbed by the country, the easier it would be to hold central Belgium. The difficulty of defending the Netherlands, however, was its long border with Germany and lack of strategic depth, which left both the capital city of Amsterdam and The Hague, the seat of government and the Dutch royal family, just sixty miles from the frontier. The likelihood of a protracted fight was further undermined by a military left weak due to the nation's long-term neutrality. Despite the country's population of nine million, which was slightly larger than that of Belgium, just 300,000 men were called up and formed into eight infantry divisions in four corps, together with a motorized division, two independent brigades (A and B) and twenty-four frontier defence battalions. The force had just forty aged tanks, 656 mostly horse-drawn artillery pieces, a few dozen anti-tank guns and an air force of 124 aircraft, of which half were modern. By the spring of 1940, the professional head of the army, General H. G. Winkelman, and the commander of the field army, Lieutenant General Jan Joseph Godfried Baron van Voorst tot Voorst, were extremely concerned about the severe limitations of their small, poorly organized and trained force. As Dutch historian L. de Jong later wrote, reflecting on this period: 'The fact that the previous war [Great War] had spared us, had filled many of us with a curious belief that history would repeat itself... So we hoped that the storm once again would not touch us, and if so, that our army would give a good account of itself.' Plans were made, nonetheless, to use the advantages that the Netherlands had for its defence: a plethora of water obstacles and mechanisms for deliberate inundation.\n\nAlthough the Netherlands lacked large fortresses, the countryside was peppered with around 2,000 machine gun-armed pillboxes which were incorporated into a defensive plan based on a 300-year-old system. With frontier troops holding a line on the border, the ground behind them east of Utrecht was to be flooded while troops manned defensive lines to the west. These included the forward Ijssel Line, with the Grebbe Line behind it in the north and Meuse Line, itself backed by the Peel-Raam Line, in the south. Along the coast, the key Dutch cities were protected by a final fallback position, known as Fortress Holland, from which it was planned, if necessary, to fight for three months while awaiting reinforcement. However, just like Belgium, the Netherlands made its plans more in the hope than the expectation that they suited Allied thinking. Although Gamelin's Breda push by Seventh Army sought to link up with Dutch divisions in the south of the country, this had not been co-ordinated with them. Indeed, even as the French were making plans for this controversial advance, the Dutch were becoming concerned that their southern flank might be exposed by any Belgian withdrawal from the Albert Line. As a result, Winkelman decided to abandon the Peel-Raam position when the Germans invaded, and tuck the wing of his force back to protect the southern part of Fortress Holland. This meant that if Gamelin's Seventh Army advanced to link up with the Dutch in the vicinity of Breda-Tilburg, it would in fact find itself vulnerable in a defensive void.\n\nGiven that they had handed Germany the initiative, there was little France and Britain and the neutral countries on Hitler's western frontier could do but prepare and await events. In many ways, however, although the strategic decision as to when to order the launching of the offensive was OKW's, the Wehrmacht's predicament was broadly the same as the Allies': it made preparations and waited for its orders to attack. By late March, formations began to be moved into position, resources gathered, plans were refined and final training took place. It was a delicate business for all the power that the Wehrmacht was husbanding, as Halder was keen to ensure that the correct balance was struck between Army Groups A and B. Although von Rundstedt's formation was the main effort, von Bock's formation was to facilitate its success. Indeed, Halder noted that the Army Group B attack must be 'staged in sufficient strength' and that the operation 'designed to contain enemy forces must be launched concurrently with our big offensive'. Ensuring that balance between the two army groups was therefore essential, but throughout the spring the Germans continued to struggle to provide both with what their generals required. From November 1939, divisional commanders were asked to indicate the level of combat effectiveness of their formations in a monthly evaluation report so as to properly assess German fighting power. This revealed a far from uniform picture, despite a gradual improvement, with peaks and troughs depending on new cohorts and the arrival of equipment. Thus, although heavy infantry gun ammunition production rose by 86 per cent during the Phoney War, there remained great shortages and none more so than in all things required for motorization. As a consequence, only 10 per cent of German divisions were motorized and this remained an army highly dependent on horses.* The strength of a typical infantry division was 17,000 men, along with 5,375 horses requiring fifty-three tons of oats and hay daily and the requisite amounts of water.\n\nAs Karl-Heinz Frieser comments, 'In terms of its structure, the German army resembled a lance whose point consisted of hardened steel; but the wooden shaft looked all the longer and therefore ever more brittle.' Indeed, only half of the 157 German divisions were combat-ready by the spring of 1940 as Germany struggled to replenish depleted ammunition stocks and losses of tanks, trucks and aircraft incurred in Poland. A mere 1,000 trucks a month were being produced by the spring of 1940 as the tonnage of German imports fell by three-quarters which in turn drastically undermined industry's access to raw materials. As a consequence, the military had to use whatever vehicles they could. In 1938 the Wehrmacht already had 100 different types of lorries and 150 different types of motorcycle, and by the following year this number was added to by Czech and Polish vehicles as well as a further 16,000 civilian vehicles that had been commandeered. This papered over cracks but made securing replacement parts a logistical nightmare. Indeed, on 13 March 1940, von Bock noted of one of his armoured formations:\n\nI could not characterize 4th Panzer Division as fully attack-capable as it did not have a single serviceable command vehicle and all requests for additional vehicles and spare parts had come to nothing. Many of the panzer division's junior company commanders could never lead their companies in practice, because there is no way to train at the front. Training opportunities absolutely have to be provided.\n\nMeanwhile, by April fuel stocks were also causing concern as they had dropped two thirds since the beginning of the war. This situation had a dramatic effect on the formations preparing to go to war, with von Bock confiding to his diary: 'The fuel situation is alleged to be so tight that the motorized divisions aren't receiving enough fuel to train their drivers.' The commander of Army Group B became so agitated that his concerns were not being taken seriously that he sent a senior member of his staff, Oberst Karl Gerber, to Berlin to find out whether his reports were being read and what was being done about them. The officer arrived in the German capital on 8 April to find that both OKW and the high commands for all three services were so busy with the launch of another invasion that nobody could afford the time to meet with him.\n\nAt a vital time in the run-up to _Fall Gelb,_ Hitler launched a campaign in Scandinavia. Although it aimed to be completely independent of _Fall Gelb_ 's schedule and resources, its shadow fell heavily across preparations for it. _Fall Weser\u00fcbung_ opened against neutral and poorly armed Norway on 9 April. Its aim, aside from increasing German control of the North Sea, was to ensure that Britain was not able to stop the supply of iron ore reaching Germany from Sweden via the Norwegian port of Narvik. And with the Luftwaffe interested in establishing airfields in Denmark, it too was invaded. The Norwegian episode was a humiliating failure for the British and French forces, which sought to land their troops to reinforce the defenders. Denmark capitulated within a few hours, and although some fighting continued in Norway until 10 June, its defeat had been all but sealed five weeks earlier. It was an operation that Hitler called 'not only bold, but one of the sauciest undertakings in the history of modern warfare'. Yet although a painful kick to the Allied solar plexus, _Fall Weser\u00fcbung_ highlighted a number of German problems. Chief among these was Hitler's inability to recognize that in the chaos of war plans rarely survive contact with the enemy. His persistent anxiety and his desire to meddle in operational details was a source of considerable frustration to experienced military commanders who looked on at their Supreme Commander's lack of self-control with growing incredulity. General Walter Warlimont observed Hitler at close quarters at this time and later pointed to 'the impression of truly terrifying weakness of character on the part of the man who was at the head of the Reich', and Jodl noted 'a striking picture of agitation and lack of balance'. When success had been gained, however, Hitler immediately became calm and basked in the glory of another victory as he turned his full attention \u2013 and apprehension \u2013 to offensive operations in the West.\n\nAlthough it might be argued that the resources required by _Fall Weser\u00fcbung_ were a drain on those required for _Fall Gelb,_ the campaign did help to further refine staff procedures and provided a useful boost to German morale at an important time. Von Bock, however, continued to receive reports from his subordinates bemoaning a lack of vital equipment and no doubt wondered what impact the fighting in Norway and Army Group A's elevation to becoming the main effort in the plan were having on his own formation. Yet he had to accept the fact that von Rundstedt was receiving the best troops and the most modern equipment, and that his panzer divisions were voracious consumers of highly prized assets and resources. One of the newest was Major General Erwin Rommel's 7th Panzer Division, which was established in mid-February. A bold infantry officer, Rommel believed that lessons he had learnt in the First World War about the need for speed, momentum and excellent communications made him well qualified to command an all-arms armoured division. His consisted of 218 tanks, of which half were Czech and the other half PzKpfw IIIs and IVs. It also had an armoured reconnaissance battalion, two rifle regiments (each of three battalions), a motorcycle battalion, an engineer battalion, an artillery component of thirty-six guns and seventy-five anti-tank guns. While deftly picking up the technical knowledge required to command armour, Rommel immediately made it his business to impress his personality on his new formation and took no time to dismiss one battalion commander who did not meet his high standards. By the spring, he was putting the formation through its final paces on exercise and remedying any errors by further training. Rommel particularly emphasized blanketing the enemy with fire, the impact of shock through rapid movement, traffic control drills, all-arms co-operation and communications. His slim figure, rapid movements and penetrating Swabian tones became well known around the division, where he turned up at unexpected times and in unexpected places to gain knowledge and raise performance. One biographer has described his personality thus:\n\n[B]risk, incisive, intolerant of slackness or infirmity of purpose, inventive, questioning, essentially business-like, enormously energetic. Rommel believed with passion in physical fitness, going for early-morning runs himself and [being] intolerant of flabbiness or inertia in his subordinates. He had always been impatient of others, whatever their rank, who failed to meet his own standards, and with the authority of a divisional commander his impatience was fortified... Rommel was by instinct a humane man, but his kindness never extended to toleration of inadequacy.\n\nHe was above all a man of action, instinctive and not prone to over-analysis, and he was disparaging of 'unnecessary academic nonsense'.\n\nCorps commanders kept a watchful eye on their divisions, trying to ensure that, while they retained their own identity, they co-operated with their fellow divisions fully in their endeavour to achieve the wider aim. Rommel's corps commander Hoth found this difficult as the commander of 7th Panzer Division was so consumed with the preparation of his division and, Hoth found, had ambitions and a personality that were not always conducive to the team effort. Guderian would have had no truck with such a prima donna, partly because there was only room for one such individual in his corps and that place was already taken by himself. But both Guderian, at corps level, and Rommel, at division level, used the force of their characters and leadership to drive their formations on. Of the three divisions under Guderian's command, it was Major General Friedrich Kirchner's 1st Panzer Division that had been selected to lead the corps and, in so doing, lead Panzer Group Kleist, Twelfth Army and Army Group A. The formation moved into its assembly area near Cochem, twenty-five miles south-west of Koblenz, in March and began its final, rigorous training. Oberstleutnant Hermann Balck, the forty-six-year-old commander of the division's 1st Rifle Regiment, had great respect for Guderian, saying later that: 'He was very hard to get along with, and it's a tribute to the German army, as well as to Guderian's own remarkable abilities, that he was able to rise as high as he did within the German army.' He recognized the energetic corps commander's hand in their training schedule, which sought to identify faults and correct them, and remarked:\n\nTraining was intensive and geared toward the future mission. Up to the breakthrough at Sedan, everything was practised down to the last detail in map exercises and every situation was run through in similar terrain also with live firing and air support. The Mosel [Moselle] River had to stand in for the Meuse River and I did not let up until every last man in my regiment knew how to handle rubber boats like an engineer. I let the exercises run their course, completely and freely, to get everybody accustomed to independent action. This was the best preparation for an offensive that I ever saw.\n\nMechanized infantry, meanwhile, practised attacking fortified locations while tanks refined their movement and both were supported by the Luftwaffe. Units rehearsed their tasks until these became second nature, and there was distinct training for units with special roles such as the combat engineers, who would be responsible for dismantling barriers and destroying bunkers. Like Rommel, Guderian was mindful not to break the morale of his formations, but he too pushed them hard. By early May, every panzer division had been put through a systematic training regime which would enable them to reach the limits of what a formation could achieve on the modern battlefield. They were, however, just one small part of the huge forces that had been drawn up by both sides for the coming campaign.\n\nOn the eve of battle, the German army was 4.2 million men strong, with a further one million in the Luftwaffe and 100,000 in the Waffen SS. Of this total, some three million were designated for the war in the West in 135 of 157 divisions: forty-five in Army Group A, twenty-nine in Army Group B, nineteen in Army Group C and forty-two in reserve. They were armed with 7,378 guns and 2,439 tanks, of which nearly 1,500 were Mk I and IIs, 349 were Mk IIIs and 278 were Mk IVs. The French army stood at 5.5 million men in 117 divisions, of which 2.24 million were in ninety-five North-East Front divisions with eleven in reserve. With the British, Belgian and Dutch contingents included, there were a total of 151 Allied divisions. Of these, fifteen French divisions, ten British divisions, twenty-two Belgian and ten Dutch divisions \u2013 a total of fifty-seven \u2013 faced the twenty-nine German divisions of Army Group B, while against the forty-five divisions of Army Group A there were a mere eighteen Allied divisions, with two Belgian divisions providing a delaying force in the Ardennes. Altogether, the Allies boasted 14,000 guns and 4,204 tanks, of which 3,254 were French, including 300 SOMUA S35s and 274 Char B1s. The Germans had 2,589 operational aircraft in the west by May (including 1,090 bombers, 316 Stukas and 923 fighters) but while the French had 2,562 aircraft, a mere 879 were operational and at the front, with a further 1,687 held back in reserve. The RAF had some 384 operational combat aircraft in France on 10 May 1940, with another 1,500 held back in Britain. Thus, together with Belgian and Dutch aircraft, the Allies could muster 1,453 machines at the front on the eve of battle, while the Germans could call on 2,589.\n\nThe time was coming when both sides would finally play their hands in what the great Prussian military theorist Karl von Clausewitz argued was an event that 'closely resembled a game of cards'. War is indeed an unpredictable business. Chance and luck play their part, as do factors as various as morale and the weather. But the most effective armed forces are those which can respond quickly to the unexpected, deploy their resources skilfully and exploit opportunities. The fact that the Allies outnumbered the Germans in every key area except aircraft was not in itself important since _Fall Gelb_ sought to avoid precisely the sort of drawn-out, attritional contest where such statistics might carry weight. Instead, the German aim was to take the initiative and hold on to it until a decision was reached, and from the late winter onwards Admiral Wilhelm Canaris, the experienced and capable chief of the Abwehr, the military intelligence service, prepared the ground. Rumours were carefully spread, intentionally indiscreet conversations were staged and some lines of communication were left insecure, all in an attempt to mislead the French. German intelligence, meanwhile, learned a good deal that was to its benefit from its agents and captured enemy personnel, and from monitoring radio traffic, counter-espionage and its secret listening posts. Gradually the Germans learned that the Allies were falling into their trap by expecting a mechanized Schlieffen Plan and had acquired a sound idea of their organization, troop concentrations and even of Plan D. The French put remarkably little emphasis on security at any level, with Daniel Barlone writing, '[C]ountry folk, inquisitive folk, and spies wander about continuously. No one makes any attempt to preserve any secrecy about our defences.' The Germans consequently built up an accurate and detailed intelligence picture of the enemy and his deployments, and this was updated with the latest information gleaned from patrols and Luftwaffe reconnaissance.\n\nThe French made various attempts to secure information about enemy intentions which included material from Enigma, the principal German cipher machine which was used for encrypting and decrypting messages. With the assistance of Polish cryptanalysts in Paris and British code-breakers at Bletchley Park, the Allies learned a good deal about the Luftwaffe's plans and deployments for the coming campaign. It was thoroughly misleading information, however, not because the Germans knew that their code had been cracked \u2013 they did not \u2013 but because they had decided to favour Army Group B with the vast majority of air support in the initial stages of _Fall Gelb_ in an attempt to mislead the Allies into thinking von Bock's army group was to be the main effort once the offensive began. The ruse was successful, for Enigma decrypts seemed only to confirm Gamelin's assumptions about German operational intentions. Indeed, along with the information recovered from the captured German plans in January, the French High Command might well have felt confident about its predictions and been surprised to learn that its enemy was in fact extremely security-conscious. For despite appearances, the Germans put great effort into maintaining the integrity of their plans, which is why on 11 January OKW issued an order that 'No officer may learn of something to be kept secret if they don't absolutely have to have knowledge of it for official reasons'. Officers were to learn what they needed in order to perform their duties, but nothing more and no earlier than absolutely necessary. Moreover, as an extension of this, the time between putting the army on alert for the offensive and it starting was gradually reduced from a week in the autumn of 1939 to just twenty-four hours by late winter.\n\nBreaches of misleading Enigma traffic, Abwehr disinformation and Gamelin's belief that the Germans were limited in their choice of routes of attack, all combined to bamboozle the French as to the truth of Hitler's intentions. Indeed, such was the complacency that began to reveal itself within the upper echelons of the French army during the spring of 1940 that solid intelligence which ran contrary to received wisdom was unceremoniously rejected. German counter-intelligence officer Oberst Hans Oster, who opposed Hitler, provided considerable amounts of secret information to the Dutch military attach\u00e9, Colonel Gijsbertus Jacobus Sas. Although in and of itself, no single piece of information by Oster indicated that the French were wrongly deployed, when this material was properly assessed and cross-referenced with other intelligence, an unsettling picture began to emerge. Belgian reports indicated that the enemy was increasing his strength south of Li\u00e8ge, while one of the few aerial reconnaissance missions to return intact and with significant information identified German armoured and motorized formations just east of the border with Luxembourg. The French were also told that the intelligence services of Belgium and Switzerland had both received information which convinced them that the Ardennes would be the route for the main German effort. This was followed by a reliable French agent reporting in mid-April that German forces were gathering for an attack on the Sedan-Charleville-St. Quentin axis and that the offensive would open in early May and would seek to reach the Seine in less than one month.\n\nFrench intelligence chiefs endeavoured to persuade Gamelin and Georges to listen, but failed. Georges's intelligence officer, Colonel Louis Baril, did show enough concern to induce his superior to correspond with Gamelin in mid-April about his worries that the Allied advance to the Dyle might leave them open to being outflanked to the south, and proposed an advance as far as the Escaut as a safer option. Such anxieties, however, were rejected by Gamelin as incomplete, inconclusive and the result of misleading information fed by the Germans. There was, it was thought, simply too much detailed intelligence indicating that the enemy's main attack would be through central Belgium. During the first week of May, information that the German invasion was imminent came from more than one source. But even a summary received on 1 May 1940 from the French military attach\u00e9 in Bern which stated quite categorically, 'The German Army will attack between 8 and 10 May along the entire front, including the Maginot Line. Point of main effort: Sedan', was dismissed by Gamelin and Georges: any attack in the Sedan sector would surely be a diversion. The two men did become convinced, however, that the German attack was imminent for, although failing to identify the point of main effort, over the next eight days a host of French agents reported that the offensive would begin 'within days'.\n\nIn the final days before he gave the order for _Fall Gelb_ to be launched, Hitler was understandably nervous. The staff of OKW sensed that, far from having total confidence in the invasion plan, he still hoped that the political fragility he had identified in his enemies would finally cause the downfall of their governments when the offensive began and that their populations would demand that an agreement be reached with him to end the war. As it was, had Hitler known what was happening in London and Paris on 9 May, he might have felt even more convinced that his prediction would come true since both the British prime minister and the French Supreme Commander were on the verge of relinquishing their appointments. Having been so badly damaged by the outcome of the Scandinavian campaign, Neville Chamberlain was considering his position and would resign the next day, while in the French capital Reynaud had called a cabinet meeting to discuss the 'state of command'. At the meeting, the French prime minister spoke for an hour against Gamelin. It was, according to one of those present, like 'witnessing an execution' and only Minister of War Daladier spoke up in Gamelin's support and even then just to say, 'I disagree'. Reynaud's intention was quite simple \u2013 to force former premier Daladier out of the government and to sack Gamelin \u2013 and the meeting had started what he hoped would be a swift process for all concerned. Hitler, meanwhile, informed that the weather was due to be fine on the following day and for as far ahead as the meteorologists were willing to forecast, decided to launch _Fall Gelb._ At around noon, Keitel signed the order specifying A-Day as 10 May and X-hour as 0345 hours. Having received the high-alert signal warning that an attack was imminent, Rommel wrote a brief letter to his wife:\n\nDearest Lu,\n\nWe're packing up at last. Let's hope not in vain. You'll get all the news for the next few days from the papers. Don't worry yourself. Everything will go alright.\n\nGeneral Kirchner at 1st Panzer Division received the same signal and within minutes his headquarters was a hive of activity as orders were received for units to prepare for a move to the border and its jumping-off points. Everything worked just as it had been rehearsed, with the advanced elements of the staff at the head of the division moving off at 1630 hours and the main body following thirty minutes later. Hitler, meanwhile, had boarded a special armoured train, _Amerika,_ at a small, out-of-the-way station just outside Berlin and headed north. Such was the secrecy surrounding the journey that even his press chief, Otto Dietrich, thought he was going to visit shipworks in Hamburg, and his secretaries believed he was travelling to Norway and Denmark to visit the troops. The train travelled slowly northwards as the forward elements of the three army groups moved towards their attack positions and, just after 2100 hours, received the code word 'Danzig' confirming that the attack was to be unleashed at 0345 hours. One officer from 1st Panzer Division, Major von Kielmansegg, later recalled:\n\nThe long columns of Panzer Group Kleist moved forward on three large routes, with the infantry primarily using trucks. The night enveloped us; not a single light could be seen. The march was slow and difficult. It was also very difficult to establish an itinerary permitting the forward movement of the column and the simultaneous joining of it by small detachments coming from all directions.\n\nHauptmann Helmuth Mahlke was at his airfield at Hennweiler, just south of Koblenz, that evening. He commanded a Luftlotte 3 Stuka _staffel_ (squadron) and had just been briefed in preparation for his first operational mission against the West. At 0600 hours on 10 May, his _gruppe_ was to attack a French air base at Metz-Frescaty. Mahlke memorized the details of his target and went through a mental checklist. He later recalled:\n\nThe preparations had been made. With nothing more to be done, we turned in early. But on this particular night it took me a very long time to get off to sleep. Thoughts of what tomorrow might bring were going round and round inside my head... Had I done everything in my power to justify the responsibility placed on me? What was going to happen tomorrow, our first day of action? What did the future hold? What else could I have done \u2013 what else _must_ I do \u2013 to make sure that all crews entrusted to my care got home safely?\n\nJust after midnight, _Amerika_ changed tracks for the south-west and arrived at the station of Eifel near Euskirchen shortly before dawn. Cars drove Hitler and his party, which included Keitel, Jodl, Julius Schaub, his chief military aide, Schmundt, Dr Karl Brandt, his personal physician, and various adjutants, through hilly, wooded countryside to a new F\u00fchrer Headquarters near M\u00fcnstereifel called the _Felsennest_ (Rock eyrie). As the party gathered in front of the main bunker, a rumble of distant shellfire broke the silence. Pointing to the sound, Hitler said simply: 'Gentlemen, the offensive against the western powers has just started.'\n\n* John Buchan is said to have based his character Richard Hannay from _The Thirty-Nine Steps_ on Ironside's early career in German South-West Africa.\n\n* In the First World War Germany used 1.4 million horses and in the Second World War 2.7 million.\n\n# CHAPTER 4\n\n# 10 May \u2013 Forward\n\nIN THE EARLY HOURS OF 10 MAY, a proclamation was issued by Hitler which was to be read by officers to all personnel on the Western Front. Nervous and already tired, their minds no doubt racing, the Reich's warriors listened intently. The last sentences were poignant:\n\nThe battle which begins today will decide the fate of the German nation for the next thousand years. Now do your duty. The German people give you their blessing.\n\nMost remained silent when the message ended, but those in the SS cheered. The opening of _Fall Gelb_ was epochal. After seven months of the Phoney War, the posturing, planning and preparation had come to an end. Europe would never be the same again and those involved felt it \u2013 some who were privy to _Fall Gelb_ because they had no faith in it, others because they did. The opening moves, they understood, would be critical. Would the Allies fall into the trap or was Germany inviting disaster?\n\nGefreiter Otto Gull, an engineer in 1st Panzer Division, was at the vanguard of the attack and later recalled:\n\nI had no real sense of the general plan \u2013 that was for the generals \u2013 and only an outline of what our division had to achieve. We had been told the night before that all our training had been done to push us through the Ardennes and across the Meuse. That was our objective. We were given a vague idea that what we were doing was vital to the operation and that speed was vital. No time-wasting\n\nGull would only learn later that the advance of his formation within Panzer Group Kleist was so dependent on the ability of Army Group B, as he described it, 'to lure the enemy bear from his cave and distract him'. Von Bock's intention was to move as fast as his formations would allow him in order to undermine the opposition's defensive preparations and create an irresistible momentum. Although he lacked the mobile armoured assets of Army Group A, his task would be assisted by significant air power, which would undertake roles such as the insertion of airborne forces just minutes before the ground troops crossed the border at 0535 hours. Faced with significant fortified positions and water obstacles close to his start line, the commander of Army Group B was intent on making a major psychological impact on the enemy; if he managed as much, he might unhinge his opponent's defences and so avoid having to lay siege to his forts and waste time wearing down his positions.\n\nYet even if von Bock's primary objective was to engage the enemy in central Belgium using Sixth Army, his formations also had the tricky task of forcing the capitulation of the Netherlands. This would be conducted by Eighteenth Army under General Georg von K\u00fcchler. In the north, the main bulk of the Dutch forces would be engaged by 1st Cavalry Division while a simultaneous landing from barges near Enkhuizen turned the enemy. In the south, Lieutenant General Kurt Student's 7th Fliegerkorps (parachute and air-landing infantry) would strike initially to occupy a number of airfields to facilitate the arrival of more troops, and also attempt to capture the government, the Dutch High Command and Queen Wilhelmina. It was a move designed, according to Student, 'to severe the head from the snake' by removing the nation's decision-making capability and which, it was believed, would lead to the collapse of its defences, undermine its will and create the conditions for Dutch capitulation. If the operation failed to achieve the desired result, other airborne troops were also tasked with securing bridges at Rotterdam, Dordrecht and Moerdijk to allow 9th Panzer Division to penetrate Fortress Holland on the third day of the offensive.\n\nThis attack south of Rotterdam, the main effort of Eighteenth Army, had little hope of achieving success unless the airborne forces could seize critical bridges before they were destroyed by the defenders. These troops, with their ability to be inserted into areas that were otherwise inaccessible, were central to the German ability to overcome the Netherlands quickly. Even so, the risks inherent in airborne operations, a very new way of applying force at specific points, were considerable: the method of insertion, the necessarily light armaments, the vulnerability to encirclement, the requirement for resupply and rapid relief \u2013 the list was a long one, reflected _Fall Gelb._ The plan was the embodiment of a carefully considered high-risk, high-gain solution to Germany's strategic dilemma and so the use of airborne troops reflected the character of the offensive. On 27 October 1939, Hitler had summoned Student to Berlin for consultations about how airborne forces might best be used in an invasion of the West. The officer was quick to stress to his audience 'that one had to realize that paratroopers and the airborne arm were a completely new, untried and \u2013 so far as Germany was concerned \u2013 still secret weapon. The first airborne operation had to employ every resource available, and be delivered boldly at a decisive time and place.' At this time, Hitler was merely looking for any means of dislocating the enemy and so invigorating Halder's early schemes. But airborne operations became more critical to the final version of his plan, for they became the key which would unlock von Bock's front, the means of adding early momentum and avoiding the laborious business of having to kick the enemy's barriers down.\n\nThe German attack on the Netherlands caught the Dutch by surprise despite the fact that the previous evening the Abwehr's treacherous Hans Oster had telephoned an anxious Gijsbertus Sas at the Dutch Embassy in Berlin and said simply: 'Tomorrow, at dawn.' Oster's information had helped keep the military authorities on high alert since the previous autumn, but the repeated German postponements of their invasion had made the Dutch wary of his warnings. As it was, they were caught unnecessarily unawares when the Luftwaffe began bombing their airfields in the early hours of 10 May. These raids were the first part of the German attempt to achieve superiority over the battlefield in order to safeguard the insertion of the airborne forces, protect the ground forces and allow the Luftwaffe to support the wider operation. Oberst Martin Fiebig's _Kampfgeschwader_ 4 (KG4), for example, attacked the naval airfield at De Kooy north of Amsterdam, destroying thirty-five aircraft, and the air bases at Schiphol, where one third of the Dutch medium bombers were rendered inoperable, and at The Hague, where twenty-one fighters were accounted for. KG 4 then assisted KG 30 and KG 54 in attacks against Dutch ports and communications. These raids, together with losses to German fighters in aerial combat, led to the Dutch having just seventy dispersed aircraft operational by the end of the day.\n\nThe raids in the Netherlands were replicated in Belgium and France, where surprise was also gained and, in the face of insignificant anti-aircraft fire, considerable damage was done to facilities and aircraft before the latter could get airborne. With no radar or early-warning systems, those on the ground generally only began to realize what was happening when the bombs started to fall. Ordered to attack an aerodrome just south of Metz, the ground and air crew of Helmuth Mahlke's Stuka _staffel_ were up well before dawn. The airfield at Hennweiler, some forty-five miles east of the border with Luxembourg, came to life at 0400 hours, the beams of the mechanics' shaded torches dancing around in the dark as the armourers attached 250 kg bombs onto the bellies of machines set out in ranks. After completing a visual check of his Stuka, Mahlke eased himself into the cockpit, was strapped in and tested the intercom with his rear gunner. Within minutes, the starter motors were jerked into life by a rapid turn of a mechanic's handle and Mahlke fired up the engine. Across the airfield, other pilots of the _gruppe_ did the same and, he later recalled, 'suddenly the nocturnal quiet of Hennweiler was shattered by the thunderous roar of engines as thirty Ju 87s started up almost as one'.\n\nThe _gruppe_ flew in formation to the target area as dawn broke and crossed the border. 'Like a delicate veil,' Mahlke later wrote, 'the milky morning haze still shrouded the landscape below us. Nothing stirred. It was as if the whole countryside was in a deep slumber.' Mahlke, however, was as alive as he had ever been, his senses heightened as his Stukas closed in on the target and manoeuvred themselves into line astern as soon as the aerodrome was spotted. Then, having circled the area once, the aircraft began to dive on their individual targets like taxis peeling off from a rank. Mahlke hardly needed to think, his actions had become automatic:\n\nA brief check: bomb switches in, radiator flaps closed, dive brakes extended, close throttle, drop the left wing and put the machine into an 80-degree dive. And suddenly the large airship hangar that was my target \u2013 and which had drifted into my bombsight with such apparent slowness only a moment ago \u2013 now came rushing up to meet me at a terrifying rate... 2,000 metres, 1,500 metres \u2013 is the enemy flak still shooting at us? No time to worry about that now \u2013 1,000 metres. At the height of 500 metres I pressed the bomb button as briefed. There was a slight jolt... then the bomb was on its way. As we pulled out of the dive into a climbing left-hand turn, we watched it strike almost dead centre near the front of the hangar roof. My two wingmen also scored direct hits.\n\nOn 10 May, the Luftwaffe managed to destroy over 140 French, Belgian and Dutch aircraft on the ground and in aerial combat but it recognized that the task of securing the skies was not over. For the rest of the day, the air forces of all the belligerent nations endeavoured to assert themselves in order to disrupt the opposition's plans and gain information about its movements. The tenacity of the Dutch air force was to pay dividends in the immediate aftermath of the carefully choreographed insertion of 7th Flieger (Air) Division and 22nd Luftlande-Infanterie (Airlanding) Division into the Netherlands. Near The Hague, German paratroopers successfully secured three airfields but plans started to go awry with the arrival of the air-landing infantry. At Valkenburg, the first Ju 52 transport aircraft to land became stuck on the soft grass runway, which forced a further twenty-six to land on a nearby beach \u2013 many crashed and others were shot down by fighters. One Dutch pilot, Conrad Houtkooper, recalls:\n\nMy ancient aircraft was a match for the slow transporters. I could not believe my luck when I managed to latch on to one of these beasts and was not followed by one of the Me 109s buzzing around to protect them. I managed to fire off all my ammunition in a couple of long bursts. One hit the starboard engine of the Ju 52 as it began to land. It burst into flames and crumpled and exploded in front of my eyes.\n\nBy that evening, none of the three targeted Dutch airfields was under close German control and the attempt to take The Hague in a surprise attack had failed in a bloody vortex of burning aircraft and heavy casualties. It was recognized that the airborne troops would be overrun if left in situ, and so they were ordered to make their way to Rotterdam twenty miles to the south and there to reinforce colleagues engaged in the undermining of Fortress Holland's defences.\n\nIn Rotterdam itself, a remarkable _coup de main_ had seen twelve seaplanes land in the middle of the city to allow two platoons of infantry to take the Wilhelms Bridge over the Nieuwe Maas. Further to the south, a series of bridges along a sixteen-mile road leading from Moerdijk \u2013 over the wide estuary and over the Rhine and Meuse rivers \u2013 to the city had also been seized following the capture of the airfield at Waalhaven. It was now up to German ground forces to advance up the road before the lightly armed airborne troops were overrun and the bridges destroyed. Dutch counter-attacks began that morning against the Wilhelms Bridge, but failed to dislodge the tenacious German defenders while Blenheim bombers from 2 Group RAF and Dutch artillery tried to stop their reinforcement by respectively attacking and shelling the airfield at Waalhaven, yet with only limited success.\n\nMeanwhile, German ground forces pushed forward across the border with the support of specially trained commandos of the Brandenburg Battalion to capture bridges over the Ijssel in the north and Meuse in the south. Infiltrating over the border in small teams after dark on 9 May, these special forces sought to ensure that the break-in forces did not stall even as their offensive began. Most of the bridges were destroyed before the Brandenburgers had even reached the rivers \u2013 although in two instances they were blown up while the German troops were standing on them \u2013 but a railway bridge over the Meuse was captured intact and this severely compromised the forward Dutch defences. The crossing at Gennep was taken by an eight-man team led by Leutnant Wilhelm Walther. Dressed as Dutch military policemen, the Germans acted as if they were escorting enemy prisoners when they launched their assault and successfully overcame the defenders while removing the charges on the bridge. With the bridge cleared, an armoured train crossed the Meuse and, followed by a troop train, penetrated ten miles in the direction of Breda and off-loaded a battalion of infantry.\n\nThe German advance caught the enemy in the midst of General Winkleman's pre-planned abandonment of the Peel-Raam Line but the attempt by 256th Infantry Division to link up with the forward battalion kept the Dutch forces in situ. The division lacked artillery support as it struggled across the Meuse via a single pontoon bridge, and the decision was made to attack without waiting for the guns. In confused hand-to-hand fighting with stoic defenders, the rather nervous German assault only began to make progress when a fortuitously timed Stuka raid was followed by the shells of a 105mm battery which had been hurriedly brought up. The Dutch line did not collapse, however, and it was not until mid-evening when the Dutch commander, Colonel Leonard Johannes Schmidt, ordered the position to be abandoned.\n\nFurther north, the elite SS Leibstandarte Adolf Hitler, a regiment of Hitler's personal bodyguard, led 227th Division's assault on the Dutch defences. An initial punch forward found them at Zwolle by midday, but there the regiment stalled as the bridges over the Ijssel had been destroyed. Its 3rd Battalion eventually managed to force a crossing more than twenty miles to the south at Zutphen which allowed elements of the regiment to reach the Grebbe Line before the end of the day, but no breakthrough had been achieved. By nightfall, a smaller than anticipated number of Eighteenth Army formations had managed to establish themselves along the main Dutch defensive line, with all finding the enemy's tenacity in the inundated and obstacle-strewn terrain a considerable challenge to their progress. In the far north of the country, for example, 1st Cavalry Division was frustrated by 236 destroyed bridges. Indeed, some of von K\u00fcchler's divisions decided to temporarily abandon their attempts to cross the Meuse and Ijssel until the front became more stable. Von Bock had always recognized that operations in what he called the 'Dutch water-park' would be challenging and was not overtly concerned by the day's numerous setbacks. He had achieved sufficient success along the line to give hope for prospects on the second day of the offensive and, in particular, for making progress towards the southern end of the airborne corridor at Moerdijk.\n\nWhile the northern part of Army Group B's offensive staggered into the Netherlands, the southern part sought to advance into central Belgium. Conducted by Sixth Army, it crossed a narrow corridor of Dutch territory defended by nothing more than border guards until running into the most southerly part of the Meuse Line, which incorporated the city of Maastricht close to the Belgian frontier. The roads through Maastricht offered an important route towards the Gembloux Gap for Sixth Army, but rather than concentrating on the city alone, the plan was to unhinge the defences in the area by attacking a broad sector of the Meuse and Albert Canal. Central to the sector was the fort at Eben-Emael. Standing six miles south of Maastricht, it acted as a strongpoint which, in conjunction with others in the vicinity, provided protection for the canal defences. It had been built in response to the German threat and stood proudly at the cutting edge of defensive technology. Completed in 1935 atop a low hill less than 100 yards from the west bank of the canal, it was a reinforced artillery position designed as much to deter the Germans as to hold out against an attack for any significant period of time. If attacked, its primary purpose was to provide artillery support for 2nd Grenadier Regiment of 7th Infantry Division, whose 2,600 men were stretched across a 5.6-mile front defending the canal. The fort was roughly the shape of an equilateral triangle, consisting of two subterranean levels and one above ground, with each separated by hermetically sealed armoured doors to protect them from infiltration. The fort boasted a 1,200-man garrison commanded by Major Jean Fritz Lucien Jottrand and thirty artillery pieces. These pieces included two 120mm guns with a range of 10.5 miles, sixteen rapid-firing 75mm guns with a range of five miles and a dozen 60mm anti-tank guns with a range of 1.8 miles. Its 360\u00b0 defences were made up of ditches, trenches, minefields and barbed wire swept by thirty machine guns. Gamelin confidently expected Eben-Emael to hold out for five days but Jottrand was not so sure as his men were poorly trained and he had no confidence in the ability of the other forts along the front to mutually support each other.\n\nAlthough Eben-Emael was supposedly impregnable, the Germans had obtained information about its defences which revealed that it was nothing of the sort. Chief among the chinks in its armour was a large, flat meadow on its concrete roof which, because the garrison used it as a football pitch, contained no obstacles. This, together with the fact that the position had a mere four anti-aircraft machine guns, provided the Germans with an opportunity to silence the position as part of a wider airborne operation to seize bridges over the Meuse and Albert Canal. Hauptmann Walther Koch's battalionsized _Sturmabteilung_ (Assault Detachment) was to use paratroopers in gliders to take three bridges over the canal which would be crossed by 61st Infantry Division while a mixture of eighty-five specially selected airborne infantry and combat engineers \u2013 Group _Granite \u2013_ were to land on Eben-Emael's roof, again in gliders. Commanded by twenty-three-year-old Oberleutnant Rudolf Witzig, who later called his men 'superior soldiers in every way', the combat engineers were to break into the fort using a hollow charge, a new weapon which could blow through any known steel and ballistic concrete fortification, and then clear its three levels with the infantry.\n\nTraining had commenced in October 1939 without any real sense of when the operation would be launched. The airborne troops lived in a gymnasium at an airfield at Hildesheim, where they practised glider landings as well as methods and tactics specifically designed for their mission by Witzig. They trained twelve hours a day, six days a week. The _Granite_ force was confined to camp, their letters censored and their cherished airborne insignia removed from their uniforms. Security was so tight that when three men decided to sneak out of camp for a cup of coffee at a nearby caf\u00e9, they were caught, tried and sentenced to death. 'Although the sentence was never carried out,' Witzig later explained, 'it does indicate how seriously the authorities took our security measures.' Delays in the launching of _Fall Gelb_ allowed the airborne soldiers more time to learn everything they needed to make their mission a success from lectures, sand tables, analysis of intelligence including aerial photographs, detailed engineering drawings of the fort \u2013 inside and out \u2013 to information gleaned from deserters from other Belgian forts. Each man was given a map of the fort in a small case, and the group even travelled to casemates in the Czech Bene\u0161 Line to practise their assault and so hone their techniques. By the time the attack came, each member of Witzig's unit could find his way around the fort blind-folded and knew the job of every member of his squad as well as he knew his own.\n\nGroup _Granite_ took off in eleven DFS 230 fabric-covered gliders towed by Ju 52 transports at 0430 hours on 10 May. Nine landed on top of Fort Eben-Emael fifty minutes later after being released from their tugs eighteen miles from the objective. One of the two gliders that failed to reach its objective contained Witzig \u2013 it had been forced to land after its tow rope had snapped. Even as the young officer was making a remarkable effort to return to his airfield to restart his journey and join up with his men, the attack began. While the gliders were still skidding to a halt, the exit doors were opened and the paratroopers poured out. Having surprised the enemy, the eleven squads destroyed the fort's protective infantry weaponry and then fell on the guns \u2013 and particularly those facing the oncoming 61st Infantry Division. Within minutes, the fort's communications had been cut, its machine guns neutralized and some of its guns had been destroyed. With the garrison having fled to the sealed heart of the fort to lay fire on Walter Koch's units as they assaulted the bridges, Witzig's men began to blast open the protective steel doors and grapple with the fort's defenders and to silence the guns. Witzig himself had managed to return to his home airfield, where he ordered a reserve Ju 52 to fly to his stricken glider, reattach it and then tow it over the battlefield. Witzig eventually joined the battle at Eben-Emael three hours late and was told that the Belgian infantry had tried and failed to counter-attack the fort while the German paratroopers continued to fight to suppress its occupants. It eventually took until noon on 11 May to finally neutralize the remaining guns in the fort and so allow Koch's force, which had successfully taken two of its three allocated bridges, to relieve Witzig's men \u2013 twenty-four hours behind schedule. Over 1,000 men of the Belgian garrison were eventually extracted from the fort's bowels, including Jottrand, who had performed as well as he could under the circumstances; a further twenty-three had been killed during the battle and another sixty-two wounded. Six members of Group _Granite_ lost their lives in the attack, and a further thirty had been wounded.*\n\nThe battle at Eben-Emael was, of course, just one tactical engagement occupying the German forces seeking to plunge into Belgium, but it was not the only one to fall behind schedule. Although Maastricht had surrendered quickly after being overwhelmed by German troops on the morning of 10 May, its bridge over the Meuse had been destroyed and so 4th Panzer Division's advance was delayed until 11 May while a replacement was built. Von Bock recognized the need to push forward as quickly as possible so as not to give the enemy an opportunity to settle into the Dyle Line, but he was satisfied with what had been achieved that first day and wrote in his diary on the evening of 10 May: 'I pushed where and however I could... It is going well!' The Allies, on the other hand, were equally keen not to delay their advance to the Dyle after the Belgians had sent the French government a request for assistance at 0650 hours that morning. Within a couple of hours, French and British reconnaissance units were pushing forward 'to the unbridled encouragement of the Belgian population' and these fast-moving units reached the Dyle and conducted an initial reconnoitre of the area that evening, leaving the main body of the Allied force to link up with them over the coming days. Those divisions were moving forward by the early afternoon.\n\nFor Gamelin, Georges and Billotte, it was essential that the six divisions and several independent tank battalions of General Georges Blanchard's First Army moved into position swiftly in order to defend the Gembloux Gap. It was the task of Prioux's Cavalry Corps, with its two DLMs, to advance eighteen miles ahead of the Gap towards Li\u00e8ge to hold the Germans until at least 14 May between Tirlemont (Tienen), Hannut and Huy. Behind Prioux, in recognition of the critical importance of the Gap to the enemy, the Belgians had built some anti-tank obstacles consisting of steel bars and concrete blocks before the village of Perwez and protected them with mines and barbed wire. Meanwhile, to the north, the lead elements of General Henri Giraud's Seventh Army sped across Belgium parallel to the coast, heading towards Antwerp in an attempt to link up with Dutch forces in the Breda-Tilburg area. Their advance was swift and by 2200 hours reconnaissance units in armoured cars from the French 1st DLM arrived at the Dutch border to find total confusion. The Dutch commander responsible for the area, Colonel Schmidt, could not be found or raised on the radio and, the French were distressed to hear, the defenders were already in retreat.\n\nSandwiched between the two French armies, General Gort's BEF also moved east with three divisions forward and five in support to hold the line east of Brussels. The British lurched into action after something of a slow start. One army liaison officer, Philip Gribble, serving on the staff of Air Marshal Sir A. S. Barratt, the commander-in-chief of British Air Forces in France, noted:\n\nG.H.Q. seem to have been caught off their guard... Quite a lot of people away on leave. The contrast between the real thing and the opening phase of an exercise is very forcible.... [In the exercises] Intelligence summary trumps intelligence summary. There is not a second to spare. And now the fact, the real thing in practice. There is no news of our own troops or aircraft. No messages, no intelligence summaries, no telephone.\n\nYet despite the lack of information, the BEF set off and made good progress with the II Corps commander, Alan Brooke, finding time to make a positive diary entry: 'Everything so far has been running like clockwork and with less interference from bombing than I'd anticipated \u2013 3rd Division started off at 2.30 pm this afternoon and by now its advance elements should be approaching the Dyle.' Providing reconnaissance for Major General Bernard Montgomery's 3rd Division was 15\/19 King's Royal Hussars (KRH). The regiment had been engaged in some physical training that morning when they learned from the BBC News on the wireless that the Germans had invaded the Netherlands and Belgium. Lieutenant Guy Courage later wrote: 'At first we could not believe our ears; but it was true and, as we rushed out of our billets, questioning and wondering, we knew that we should soon be in action and we prayed that victory might be ours.' The next few hours saw the men make their preparations with maps issued, kit packed and stowed in lorries, ammunition belts filled, equipment checked and orders given. As they crossed the border that afternoon, Courage noted that 'Excitement was on every face.' A little later, the look had turned to relief. Just after the regiment had passed through Alost on the Escaut, the town was attacked by sixty Stukas.\n\nAlthough its _modus operandi_ was different in many ways to Army Group B's due to the Maginot Line, von Leeb's Army Group C had a role which also demanded that it exert pressure on the enemy and pin and fix as many French divisions along its front as possible. Thus on 10 May Seventh Army began to simulate an offensive against Switzerland which, Gamelin thought, might develop into an attempt to envelop the Maginot Line from the south. During the spring the Germans had fed their enemy a plethora of misleading information about the arrival of various elite formations in the sector in preparation for this deception and had made easily observable troop movements to emphasize its likelihood. The result was that, at the start of the campaign in the West, thirty-six French divisions were concentrated in the area of the heavily fortified Maginot Line facing only nineteen divisions of Army Group C on the German side. In this way, Army Groups B and C fixed the French where the Germans wanted them while Army Group A undertook its tacit move through the Ardennes and to the Meuse. It was to be led on a twelve-mile front by three divisions: Major General Rudolf Veiel's 2nd Panzer Division on the right, Major General Friedrich Kirchner's 1st Panzer Division in the centre and Major General Ferdinand Schaal's 10th Panzer Division on the left. These formations, von Kleist recognized, would have to be given the ability to break the rules of accepted operational behaviour if _Fall Gelb_ were to succeed. Consequently, Guderian's leash sat loosely in von Kleist's hand. This, however, was a liberty not offered to von Kleist by his own superior, von Rundstedt. Anything that smacked of unnecessary risk and threatened to leave his three armies isolated would, the commander of Army Group A promised, bring about a prompt and aggressive reaction in order to bring the panzer group back under control.\n\n1st Panzer Division's attack began at 0535 hours when a whistle blew and a squad of German soldiers rushed across the Sauer to disarm the Luxembourg border gendarmes. Engineers then began to deal with the concrete barriers on two bridges, one a little downstream over the Our, and to build three additional bridges across the rivers. It took just forty minutes for the bridges to be constructed, the Schuster Line of obstacles to be overcome and for the formation to start moving forward. Acting as Kirchner's trailblazers were a motorcycle company and three reconnaissance platoons in armoured cars supported by combat engineers, followed by the infantry of 1st Rifle Regiment and some artillery. The division advanced thirty miles and did not encounter any opposition before reaching the Belgian border at Martelange after three hours. Here, there was a thin line of Belgian border guards and behind them the light covering force offered by the two reinforced divisions of 'K' Group. Lieutenant General Keyaerts had established two thinly held and fragmented lines of resistance \u2013 amounting to a series of strongpoints \u2013 across the Ardennes: one forward and centred on Bastogne and running close to the border with Luxembourg, and the other behind along the Semois river close to the border with France. 1st Panzer Division's aim was to penetrate both on the first day and reach the high ground west of Neufch\u00e2teau to complete an advance on 10 May of around forty miles. The French, meanwhile, sent their own delaying force into the Ardennes consisting of the equivalent of five cavalry divisions, three cavalry brigades and some advance units from the divisions defending the border. Ninth Army sent 1st and 4th Light Cavalry Divisions and 3rd Brigade of Spahis while Second Army sent 2nd and 5th Light Cavalry Divisions and 1st Cavalry Brigade. Added to these forces were the reconnaissance squadrons of each of the divisions and corps in the two field armies. How the Belgian and French forces were to work together, what their boundaries would be, who would be in command and a host of other questions had not been answered because Belgian neutrality had meant there had been no co-ordination between the two forces. The efficiency or effectiveness of this important delaying force had been severely compromised by circumstances and now, unbeknown to its commanders, its component parts were spread out over a wide front \u2013 outnumbered and outgunned \u2013 facing the _Schwerpunkt_ of the main German effort. Guderian's corps, meanwhile, was expecting some opposition, somewhere and at some time and had been left in no doubt that there was no time for a gentle probing of the difficult terrain through which the formations were moving. Any delay, the three division commanders knew, would be deleterious to the achievement of the operation's aim and thus they kept in mind Guderian's adage: 'Punch with the fist rather than feel with the fingers.'\n\nThe combined power of the panzer divisions surging through the Ardennes notwithstanding, the German planners were well aware that in the close terrain relatively few defenders armed with explosives, guile and tenacity could make life extremely difficult. There was the possibility, particularly while Guderian's corps was endeavouring to arrange itself on a broad front due to its less than ideal echeloning, that the vanguard of the leading 1st Panzer Division advancing down a critical road could find itself thwarted by a few well-positioned Belgian defences. It was for this reason that airborne troops were to be inserted behind the border village of Bodange. Expecting Belgian resistance in the area, Operation NiWi (as in the towns of N\u00eeves and Witry) was to cause chaos in the rear area, stop the defenders from being reinforced and also to thwart any attempts they might make to conduct a fighting withdrawal. Lacking Ju 52s for the mission, two companies of Oberstleutnant Eugen Garski's specially trained 3rd Battalion Gross Deutschland Regiment, an elite unit of volunteers drawn from every province of the Reich, and some combat engineers were instead to be delivered in two lifts by nearly 100 tiny Fieseler Fi 156 Storch liaison aircraft. These airborne troops were to capture two towns: a force led by Garksi was to take Witry, some 3.5 miles behind Bodange on 1st Panzer Division's route, and Hauptmann Kr\u00fcger was to seize N\u00eeves, 3.5 miles north of Witry on 2nd Panzer Division's route.\n\nAfter crossing Luxembourg at an altitude of under 100 feet, the swarm of light aircraft began to land in Belgium at 0600 hours. At Witry, however, just five aircraft, including Garski's, landed where they were supposed to. A mixture of fog and small arms fire had sent the majority of the Witry-bound force off-course and it had mistakenly joined Kr\u00fcger's group, only for the enlarged group to land in completely the wrong place at L\u00e9glise, 5.5 miles south-west of Witry. Kr\u00fcger immediately endeavoured to link up with Garski at Witry but, having requisitioned a number of vehicles, ran into a Belgian patrol and some armoured cars. Eventually arriving in Witry at 1300 hours on foot, Kr\u00fcger found that Garski and his nine men had successfully blocked the route out of the town leading to Neufch\u00e2teau. The successful arrival of this second wave had enlarged Garski's force just in time to beat off Belgian counterattacks, which had for a time threatened to wipe them out. With his company now complete, Garski subsequently pushed forward to the rear of the Belgian defences at Bodange, where they arrived at around 1730 hours. Kr\u00fcger's second lift, meanwhile, had arrived at N\u00eeves earlier that morning to find themselves alone. Leutnant Obermeier took charge and blocked the Bastogne-Neufch\u00e2teau road before beating off several counter-attacks from Belgian and French troops.\n\nDespite its chaotic application, Operation NiWi managed to successfully thwart enemy attempts in the area to move up important roads towards the Belgian border. Assisted by the scattering of Garski's force which suddenly turned up behind the front line, panicked reports reached French and Belgian commanders variously that 'a battalion', 'a regiment' and even a 'a division' had been inserted in the middle of the Ardennes. The effect of NiWi was to sow doubt into the minds of the defenders about where and in what numbers their adversaries might be found. In the context of events that were unfolding on 10 May, those defenders' confusion and concern were understandable. The operation did have a negative effect on the advance of 1st Panzer Division, however, since the Belgian units defending Martelange and Bodange could now not withdraw.\n\nSituated just behind the Belgian border, 2nd Battalion Chasseurs Ardennais was put on high alert immediately after the first Luftwaffe attacks on the airfields. Their task was to block the route to Neufch\u00e2teau in complex terrain with narrow, curving roads and set themselves in two fortified villages. 4th Company was positioned in the border village of Martelange while 5th Company lay behind on the same road at Bodange. Both had covered the approaches to their villages with minefields in the fields and demolitions which made the roads immediately impassable to vehicles. Nevertheless, the preparations did little to stop 1st Panzer Division's motorcyclists first reaching Martelange, where they were immediately engaged by the enemy. Oberstleutnant Hermann Balck, the commander of 1st Rifle Regiment following on, was always grateful for the efforts of the motorcycle infantry and later testified that the combination of men and machines were able to 'get very, very quickly to the decisive point... [and so were] often out far in front of the tanks'. He added that, although 'restricted to paved roads and noisy, which made it difficult to do reconnaissance while on the motorcycle. [the machines would] move out very quickly in advance of the main forces in order to grab a bridge before it could be blown up. This happened time after time in the advance through France, and had a good deal to do with greatly speeding up the rate of movement of our main forces.' Although on this occasion the bridge at Martelange had been destroyed before the arrival of the motorcycle troops, the reports that they provided Balck on his arrival allowed him to make important decisions immediately. First, the company-sized group of motorcyclists were to dismount and attack across the river in order to drive the enemy off a hill overlooking the village. It was a success. Using fire and manoeuvre \u2013 a German tactical procedure in which one squad of men moved under the covering fire of another, then reversing the roles \u2013 the speed and aggression of the attack was decisive and as the motorcyclists stormed up the slope towards the enemy, a number of the defenders began to flee. With this, the defences of Martelange quickly began to disintegrate and the men of 1st Panzer Division seized the opportunity to move forward without delay.\n\nTwo miles beyond Martelange lay Bodange, defended by Major Maurice Bricart's 5th Company Chasseurs Ardennais. Now aware that the Germans were behind him at Witry, Bricart recognized that his situation was extremely precarious. He expected orders to withdraw from 2nd Battalion, but they never came as Operation NiWi had cut communications between regimental and battalion headquarters, and then the company's explosion of a demolition charge had destroyed a field telephone line between battalion headquarters and Bricart. Thus, with no orders to withdraw and with the Germans to their rear in any case, the company was forced into a position where it would have to fight for its survival. The vanguard of 1st Panzer Division approached them after first pushing aside a screen of Belgians en route \u2013 thus announcing their imminent arrival \u2013 and immediately drew fire at around 1330 hours. The Germans were stopped first by a demolition which had cratered the road, and then by a destroyed bridge over a large stream. From here, the road led just to the south of the small village from where 5th Company overlooked it. The slope up from the stream was occupied by a platoon in some stout buildings, while another was in position just to the south of the road on some high ground \u2013 thus creating a killing zone between the two small units. The Germans later described the Belgian defences at Bodange as 'well-camouflaged and carefully fortified positions of fire. The defence was strong. Everywhere there were dense and deep rolls of barbed wire; the roads and trails were all blocked. It was not possible to go around them: that would accomplish nothing. It was necessary that we go directly over them.'\n\nAs in Martelange, the Germans did not waste any time taking the decision to attack. While a company of infantry were to cross the stream and attack the village from the north, three companies of motorcyclists were to clear the high ground to the south and then use the road to link up with the NiWi troops as soon as possible. The motorcyclists took to the hill and, coming under withering fire from the village which had yet to be silenced, moved speedily on foot towards Garski's force. Meanwhile, the defence of Bodange by a depleted light infantry company continued to hold up the advance of an impatient panzer division. Although a battalion of artillery managed to get forward along the choked road and some field howitzers laid fire on the village's solid buildings, it was not until powerful 88mm guns were deployed that the defenders' grip began to loosen on the position and a battalion of infantry managed to close in.\n\nBricart's last message was at 1600 hours, when he reported that he had taken heavy casualties and that his ammunition was running low. Even so, it took another two hours for the Germans to start clearing the village and some bitter hand-to-hand fighting ensued. In the end, there were just twenty-six Belgian defenders left to surrender and Major Bricart died endeavouring to evade capture. The village finally fell at 1900 hours, five and a half hours after 1st Panzer Divsion had made its first approach. Even so, the formation was stuck fast for, as it later reported, 'all possible crossing points and tracks were completely destroyed. Detours were rarely found.' Thus it was not until a new bridge had been built and mines cleared from the surrounding marshy ground at 2115 hours that vehicles finally began their progress forward again. Although the men of the motorcycle battalion had successfully linked up with Garski's men at 1730 hours, two companies of Chasseurs Ardennais had unhinged Guderian's schedule as they had contrived to stop 1st Panzer Division from reaching Neufch\u00e2teau on 10 May as planned.\n\nIt was a long, hot day for the German troops and while those relatively few involved in the fighting had obvious challenges, those stuck sweltering in cramped vehicles had problems of their own. For hour after hour, drivers, radio operators, gunners, loaders and commanders were tied to their tanks, trucks, self-propelled guns, armoured cars and a panoply of other motorized and mechanized vehicles. The corps had been pushed to its limits during training, and now it was being pushed to its limits on operations. Headaches and eye strain were common by the evening after a day of concentrating on the often tortuously complicated routes ahead and the instructions being given over the intercom to avoid demolitions and soft ground, to negotiate narrow bridges and evade other vehicles. Some men took breaks sitting on the outside of their tanks or perched precariously on bonnets, wheel arches or any other surface that gave them respite, a change of scene, an opportunity to stretch and to feel fresh air on their faces. One tank driver, Gefreiter M\u00f6llman, recalls: 'They clench their teeth. Stay awake at all costs! Roads, roads, roads \u2013 all the same. The men at their side talk to them, telling them anything that comes into their heads. Anything to stay awake.' Guderian told his officers: 'I demand that you go sleepless for at least three nights if that should be necessary.' As a consequence, the divisions were issued with 20,000 Pervitin tablets \u2013 known by panzer troops as 'tank chocolates' \u2013 which were an early version of crystal meth and used to help tank crews stay awake as they carved through the Ardennes and beyond. Adrenaline, however, would keep them wide awake until they needed to resort to drugs. As Otto Gull later explained:\n\nI find it difficult to believe that any man during those opening days was not anxious about what lay before him. There we were, often sitting waiting and vulnerable. We lived on our wits and on our nerves and disliked having to be idle while the infantry dealt with the enemy just in front of us. We just wanted to keep moving. It was frustrating.\n\nIt was a frustrating day for Guderian as well, for all three of his divisions struggled to achieve any of the necessary momentum that day. While 1st Panzer Division encountered difficulties at Bodange, 2nd Panzer Division was also held up by a handful of defenders and it failed to take its first-day objective of Libramont. 10th Panzer Division, with two battalions of the Gross Deutschland Regiment attached to strengthen it for the protection of Guderian's flank, also fell short of obtaining its first-day objective. Advancing on two routes into Belgium, it came up against strong opposition offered by the French 2nd Light Cavalry Division in Etalle. Both 5th and 2nd Light Cavalry Divisions had pushed forward behind the Belgian 'K' Group that morning, their main body holding a line just beyond the Semois River with a focus on the main roads heading west through the area. Schaal's division took several hours to overcome the dogged resistance in Etalle which, among others, cost the lives of the commanders of both 2nd Gross Deutschland Regiment and 69th Infantry Regiment.\n\nAll Guderian could do was look on at events, encourage and hope that the second day of the offensive would be more kind. If his corps was to be able to bounce the Meuse on the move, he knew that the achievement and maintenance of momentum was essential. Without it, he feared that his divisions would stall on the east bank of the river and that all hope of mounting independent operations towards the coast would have been lost. As a result, he was driven to take risks and so, having identified a potentially weak spot in the enemy's line from intelligence reports just to the north of Etalle, sought permission from von Kleist for units on the southern of 10th Panzer Division's two routes to disengage, cut across the rear of the units on the northern route, which would pin and fix the enemy in that area, and infiltrate through the weak spot in terrain that the division's log called 'barely passable'. It put great pressure on the unsung heroes of the Ardennes thrust, the traffic police, who had to quickly find ingenious new routes for hundreds of vehicles, ensure that they were clearly marked and then manage the consequences. The complications of the proposal, together with the fact that the movement would remove the corps' flank protection units, meant that it took until dawn on 11 May for von Kleist to agree to the reorientation. He was won over by Guderian's argument that a static flank was far more vulnerable than a moving one and also by his insistence that it was the only way to ensure all three of his divisions would hit the Meuse at the same time. Guderian had also inadvertently ensured that, by shifting the southern part of Schaal's division to the north, all three of his divisions were now faced by 5th Light Cavalry Division alone.\n\nThe sluggish advance of Guderian's panzer corps on 10 May had implications for Army Group A's offensive. These were most evident at the front edge of the attacking divisions, but there were other, wider repercussions. By evening, a massive armoured and motorized tailback had built up from behind the German border across Luxembourg and into Belgium. A good deal of the problems had been caused by the decision made during planning by conservative forces in Army Group A to limit von Kleist's room for manoeuvre. As it was, he had to echelon his division with its lines of communication running back through the infantry armies, and so there were bound to be issues of bunching up and disagreements over which formation had priority over others on the very limited routes. These certainly surfaced on 10 May, for it took ten hours for a single panzer division to cross the German border, a situation that was not helped by Guderian's slow progress and saw 6th Panzer Division of Reinhardt's corps in the second echelon run into the tail of 2nd Panzer Division in the first. This meant that neither Reinhardt's corps, which was due to take up a position on Guderian's right, nor von Wietersheim's, which was to advance on Guderian's left, had room to deploy forward. Not only did this potentially affect the forward mass of Army Group A, it also left Guderian's flanks wide open to counter-attacks.\n\nYet despite the obvious and understandable anxiety caused by the enemy on the first day of the offensive through the Ardennes, such difficulties were not wholly unexpected and the Germans did not panic. Guderian and his commanders understood that the terrain favoured the defender and that the power of the panzer divisions could not be fully unleashed until they were properly massed and deployed in more favourable country. Halder had noted this in the Army Group A war games he had attended and, moreover, saw how the inability of Guderian to 'show his full hand' during this period would mean that the panzer divisions' infantry and not their armour would play the key role until a bridgehead had been established on the Meuse. He also recognized that in such circumstances, when the Belgian and French commanders facing von Kleist submitted their reports up the chain of command during the course of the day, the word 'armour' was unlikely to appear in them. This is just what Halder wanted, for the appearance of German infantry rather than armour in the Ardennes was far less likely to lead the Allies to question whether Army Group B was really the main enemy effort.\n\nIndeed, at Gamelin's Vincennes headquarters on 10 May there was considerable relief that the German offensive had begun and that it appeared to be taking place just where it had been expected, with what looked like the main attack coming north of Maastricht. To this end, General Pierre Jacomet, the Secretary-General in the War Ministry, later noted: 'If you had seen, as I have done this morning, the broad smile of General Gamelin when he told me the direction of the enemy attack, you would feel no uneasiness. The Germans have provided him with just the opportunity he was awaiting.' There were nonetheless reports of German forces advancing into and through the Ardennes. A 'large motorized column' was spotted by Third Army approaching Luxembourg city and Second Army identified a fifty-vehicle motorized column moving towards Bastogne, while aerial reconnaissance found about forty motorized vehicles moving west from Arlon, although this too was what the French had expected and certainly did not indicate a worryingly significant attack in the area. It was a belief reinforced by a Second Army intelligence summary received by Gamelin's headquarters at end of the day which concluded:\n\n[E]nemy motorized elements have been pushed back by the Chasseurs Ardennais... and have come in contact with our covering [force] elements. Belgian obstacles seem to have been sufficient to halt the progression of the enemy. THERE HAVE BEEN NO INDICATIONS OF ARMOURED VEHICLES ON THE ARMY'S FRONT.\n\nOther reports from the air force did nothing to dissuade Gamelin that this pr\u00e9cis of the situation was not accurate. Although the information received from French aerial reconnaissance was severely limited by the 1,500 Luftwaffe machines maintaining a defensive air umbrella over Panzer Group Kleist and the formidable anti-aircraft assets of the panzer divisions themselves, it suggested that there was 'a secondary but extremely strong effort to the west of Luxembourg'. But Guderian's spearhead had not been identified and so the French continued to believe that the 'principal enemy effort would be in the region between Maastricht and Nijmegen'.\n\nThe Allies thus continued to commit themselves to Plan D and threw themselves into the German trap. From the outset, therefore, the Germans out-thought their enemies, even if at this stage they were not necessarily out-fighting them. Although the Allies did score some local tactical successes on 10 May by causing the German airborne operations in the Netherlands some difficulties and slowing Guderian in the Ardennes, operationally OKH's plan to draw the enemy into central Belgium while masking the main effort through the Ardennes was unfolding tolerably well. Halder believed that a relatively encouraging first step had been taken, although he was also aware that many more such steps were needed before success could be anticipated.\n\nTaking the events of 10 May as a whole, therefore, neither the Allies nor the Germans were particularly concerned by the way in which the campaign had opened. Both sides had made a solid start, with their plans being rolled out and the enemy reacting in a way that caused neither any great concern. Both Gamelin and von Brauchitsch were experienced enough to recognize that, while few campaigns are won in the first twenty-four hours, great strides can be made towards losing them. The opening shots of the German invasion of the West had been fired and no side had obviously acquired the upper hand. The fate of the campaign was still in the balance, but senior commanders in both Paris and Berlin believed that the picture would become much clearer within the next seventy-two hours. They were right.\n\n* On returning to Germany, all the officers involved in the Eben-Emael assault attended a special ceremony at their base, where Hitler addressed them and awarded each a Knight's Cross. Other ranks were awarded the Iron Cross by General Albert Kesselring and promoted one rank \u2013 save for one man who, it was found, had carried a canteen full of rum into battle.\n\n# CHAPTER 5\n\n# 11\u201312 May \u2013 To the Meuse\n\nTHE BAUWENS FAMILY had lived near the small Belgian village of Zammelen a dozen or so miles west of Maastricht for as far back as anybody could remember. They were not wealthy, but had a little land that fifty-two-year-old Maarten Bauwens farmed while his young wife, Margot, made some extra money teaching part-time in the local primary school. Their teenage children, Arnaud and Emma, were just getting out of bed and making their way to the kitchen for breakfast on the morning of 10 May when their father called for the family to gather around the wireless. The German invasion had begun. Maarten had noticed a good deal of air activity over the village that morning but had thought little of it. There had been plenty of unusual military comings and goings over the past few weeks and he was becoming accustomed to such activity. Now, however, the news that everybody had been dreading had been announced. The Germans had crossed the border and were engaged with the Belgian army. The air force was lending support but its resources had been badly damaged by earlier Luftwaffe raids.\n\nLiving so close to the border, Maarten knew that the time had come to make a decision about fleeing their comfortable home and, with it, the animals that so relied on them. In the end, it was Margot who said that they should leave: the Germans were capable of vile acts \u2013 she remembered hearing about what they had done to Belgians in 1914 \u2013 and she would not abandon Emma to these Nazi monsters. Even as she made her case, a stream of people passed by the kitchen window on the road heading west and carrying whatever they were able in suitcases, boxes and wheelbarrows, and on bicycles. An exodus had begun and within the hour the Bauwens had packed a bag each, said farewell to their farm and livestock, and joined a growing number of people fleeing from an unseen enemy.\n\nBy the following morning, the Bauwens had walked over twenty miles. Footsore, tired and without any firm destination in mind other than to reach the French border, they tramped on. Maarten was already concerned that they had made a mistake and wondered what was to become of the home and animals that meant so much to him. Margot tried not to listen and instead attempted to keep the spirits of Arnaud and Emma as high as a few songs and the occasional joke would allow. By the evening of 11 May, the family had reached the Dyle at Louvain and there were ushered across a bridge by soldiers who, they later found out, were British. Now there were soldiers with lots of guns and what looked like little tanks between them and the Germans. For the first time since they had left Zammelen, the Bauwens felt a degree of safety.*\n\nAlthough most of the French population living close to the Maginot Line had been removed from the area prior to the German invasion, elsewhere they had not. While only some of those from southern Holland decided to flee their homes, there was a massive displacement of people from Belgium, Luxembourg and the French border region after the announcement that the German offensive had begun. Most, invariably without men of military age, used whatever means of transport was available to move themselves and essentials away from whatever danger they foresaw. Accurate information was scant, but stories of bombers, tanks, airborne troops, fifth columnists and atrocities were rife and gave urgency to the civilian movement which clogged the roads and impeded the movement of the belligerents in the fine, hot weather that was to last throughout the campaign. Guy Courage, a junior officer in the 15\/19 KRH, noted that on 11 May the trickle of Belgian refugees coming through the lines on the Dyle had become a constant stream, demoralized and disorganized. Their number included the Bauwens. Courage recalled:\n\nIt was a most pathetic sight, which could not fail to move the most hardened heart \u2013 carts drawn by weedy horses, hand carts, wheelbarrows, a few cars, and all sorts of 'soapbox vehicles', piled high with precious belongings, were pushed or pulled along the roads to the west by refugees. The people were, of course, all women, old men and children: on they plodded, mile after mile along the straight, dusty road, going they knew not whither and caring little what became of them.\n\nThe governments of the Netherlands, Belgium and France had made no plans for the evacuation of such vast numbers of people and so their movement was unco-ordinated and unplanned. Gradually, some order was established, with certain roads being kept solely for the use of the military and routes being signposted to ease the congestion, but no official information was provided, although the refugees were increasingly directed towards Paris.\n\nAt Sedan, there had been an orderly evacuation of civilians just before dark on 10 May after it became clear that the town could well be a German objective. Those living in the town left by train, soon followed on the roads by the rural population. As elsewhere, once the inhabitants of one town or village left, there was a domino effect as those of the towns and villages they passed through were encouraged to do the same. Many left in haste with a few warm clothes and a little food and drink, but without any serious thought about what would be required. One fourteen-year-old girl, who left Amiens in the Somme department of France with her mother, sister and her sister's baby, recalls: 'We did not take much, just what we could carry. We just dropped everything and took off. We were so scared.' They shut the front door and began walking along with thousands of others in the opening scenes of a human tragedy that is often overlooked in the light of the dramatic military events that were unfolding around them.\n\nBy and large, the troops ignored the refugees \u2013 there was little that they could do for them in any case \u2013 but there were some in uniform from regions most obviously affected by the German invasion who could not help worrying about their families. Most, however, were more concerned about what these people were fleeing from. The displaced offered little information, and as one soldier later wrote: '[T]he one constant statement on their lips was, \"No one could stand the bombing on the Albert Canal pillboxes.\" We drew our own conclusions and anxiously awaited the coming Hun.'\n\nThe tide of German field grey and black had begun moving west again by dawn on 11 May, having had little rest. The period after dark on the 1940 battlefield was not a time that the belligerents commonly sought to fight in, but it did provide an opportunity for other activities. On both sides these undertakings were organized, managed and overseen by officers and noncommissioned officers and completed during and after some food, and before snatching a couple of hours' sleep. Perimeter defence, sentry duty, reconnaissance, refuelling, rearming, weapon and vehicle maintenance, writing reports, orders groups, briefings \u2013 there was a host of essential undertakings that, unless properly catered for and carried out, rapidly undermined a unit's fighting capability and increased its vulnerability. One of Balck's 1st Rifle Regiment men, a twenty-two-year-old rifleman named Gerd Ritter, later wrote:\n\nThere was little sleep \u2013 we slept where we fell, on our packs \u2013 little time for anything really. Often we had to be told to eat or else we could drift off standing up. The intensity was unremitting from the first day to the last. A blur of marching, fighting, weapon maintenance \u2013 we didn't know where we were or when it would end.\n\nThe experience of an airborne soldier of Student's, Peter Meier, was much the same. He got no sleep at all for three days while fighting in Rotterdam and recalled:\n\nWe were prepared for a lack of rest and constant night-time activity. As a paratrooper I was trained for it because our situation often involved being surrounded and waiting for reinforcement or relief... In Rotterdam on the night 10-11 May, the fighting was sporadic but it could be intense. After a tiring night it was a boost to see the sun rise in a clear blue sky.\n\nThe job of those airborne forces in the Netherlands on 11 May was essentially a straightforward one \u2013 to hold on until the arrival of the ground forces. General Winkelman's task as Dutch Commander-in-Chief was similarly unambiguous \u2013 to eliminate the potentially fatal German penetration of Fortress Holland. The counter-attacks that were launched by the Dutch against the German airborne forces that day, however, all failed in the face of fierce resistance from a technically superior enemy. In Rotterdam, despite being reinforced by an infantry regiment, the Dutch made several unsuccessful attempts to oust Meier and his colleagues from their perimeter around the Wilhelms Bridge and a bombing raid mounted to destroy the structure also failed. At the other bridges further south, a number of gallant Dutch actions were also beaten off with the assistance of waves of Stukas. Similar outcomes also wrecked Dutch attempts to overwhelm the airborne troops on the three Dutch airfields near The Hague who had failed to extract themselves and head towards Rotterdam. The situation was, Peter Meier later explained, less difficult for the German airborne soldiers to contend with than for the Dutch fighting them:\n\nWe had been mentally and physically trained for such difficult scenarios. Being encircled was something that we had practised time and time again. How often had the Dutch tried to oust an enemy fighting for his life from an objective that he would die for? Not once, I would think. So, I looked on at the growing number of enemy corpses surrounding our position with rather less surprise than the enemy did. We were supremely motivated while the Dutch found that the defence of their country was beginning to collapse around them.\n\nFor the Dutch and the Allies, the news from the Netherlands on 11 May was indeed disheartening. Not only had the defenders withdrawn from the Peel-Raam Line, thus leaving the French Seventh Army without a host when it arrived in the Breda-Tilburg sector that morning, but the Grebbe Line to the north was also crumbling under intense pressure from troops who had advanced to the Zwolle-Zuphten area the previous day. Before first light on 11 May, a ferocious German artillery bombardment was followed by an attack by two battalions of the SS Leibstandarte Adolf Hitler. Leaving 227th Infantry Division to attack towards Amersfoort, the regiment was sent south-west to link up with 9th Panzer Division, which crossed the Meuse on the morning of 11 May and then threw itself towards the Moerdijk Bridge and Rotterdam. In so doing, the panzer division's reconnaissance force ran into Seventh Army around Tilburg. The resultant contact led the vulnerable yet important French formation to begin a withdrawal back to Antwerp under intense Luftwaffe pressure; it was an ignominious end to the Breda Variant, which had been \u2013 and would continue to be \u2013 of far greater benefit to the Germans than to the Allies. The extraction of Seventh Army ended any possibility of a French counter-attack south of Rotterdam, and with the SS Leibstandarte Adolf Hitler having broken the Grebbe Line defences during that afternoon \u2013 the Dutch troops only just preventing a rout \u2013 and 1st Cavalry Division forcing the Dutch back in Friesland to the west of the Ijsselmeer, by the end of the day Fortress Holland was being squeezed from all landward sides.\n\nThe dominance of the Luftwaffe over the Netherlands proved to be of great importance to Eighteenth Army on 11 May. It facilitated close air support of both the defensive efforts of the airborne troops and the offensive efforts of those formations looking to dismantle the Dutch defences. It also brought home the threat posed by the German overwhelming of Fortress Holland. As one Rotterdam resident has testified: 'We feared German bombing raids above all else. The idea of our city being crushed like a Dutch Guernica was a real worry. Our eyes were constantly looking skywards for the first signs of any sort of attack.' Across the front, the Luftwaffe sought to further establish its dominance of the skies. While there were some repeats of missions to destroy aircraft and airfield facilities, most of the action now took the form of aerial combat.\n\nAdolf Galland was a German fighter pilot who had served in Spain and Poland. Although a non-flying _geschwader_ adjutant at this time, he talked himself into an Bf 109E and took to the skies around the Maastricht sector. He later said that his first kill, a Belgian Hurricane, 'was child's play'. Attacking his enemy from above, Galland was not spotted and recalled:\n\nI was neither excited nor did I feel any hunting fever. 'Come on, defend yourself!' I thought as soon as I had one of the eight [Hurricanes] in my gun sight. I closed in more and more without being noticed. 'Someone ought to warn him!'... I gave him my first burst from a distance which, considering the situation, was still too great. I was dead on the target. The poor devil at last noticed what it was all about. He took rather clumsy avoiding action which brought him into the fire of my companion. The other seven Hurricanes made no effort to come to the aid of their comrade in distress but made off in all directions. After a second attack my opponent spun down in spirals minus his rudder. Part of the wing came off. Another burst would have been a waste of ammunition.\n\nThe Allied air forces also took heavy losses in their attempts to stem the German flow of men, vehicles and materiel by targeting the bridges over the Albert Canal and Meuse. The Germans had anticipated this and ensured that each was well-protected with anti-aircraft guns. A Blenheim squadron of 2 Group RAF attacked the bridges at Maastricht causing little damage. One pilot who took part in the raid that day recalled: 'Running in, we were in the middle of the biggest barrage I could possibly imagine. I do not know to this day how some of us ever got through it for there didn't seem to be an inch of sky that was not covered with flak.' Later in the day, one Belgian squadron lost eleven of its twelve aircraft in a single mission against the same bridge.\n\nEven so, the Germans did not have it all their own way, for on the morning of 11 May ten Stukas of Sturzkampfgeschwader 2 were shot down while attacking ground forces between Namur and Dinant despite the protective presence of fighters. Generally, however, the Luftwaffe quickly became a constant menace to the Allied ground forces, who wondered how the enemy was being allowed such freedom. Henry de la Falaise, who was serving with the 12th Lancers in the vanguard of the BEF, felt extremely vulnerable and noted that 'Everyone is beginning to feel uneasy and even mortified about the unexpected and total absence of British or French planes' and, after being targeted, was angered when 'Some of them insultingly make barrel rolls and loops over our heads'.\n\nAll along the line, troop concentrations and movements were targeted and the Stukas made a considerable psychological impact on those unfortunate enough to be attacked by them. One such soldier later wrote, 'If anybody said he wasn't frightened when the Stukas were around, he was a liar \u2013 anybody!' And as Second Lieutenant John Dixon explained:\n\nTo be dive-bombed by Stukas was a nerve-shattering experience. With their cranked wing shape, they could drop like a stone from the sky, position their bomb accurately and with the howling mechanism in operation on the dive, those on the ground were forced to scatter in terror.*\n\nBut the Luftwaffe was not simply involved in close air support on 11 May. On just the second day of _Fall Gelb_ attempts were begun to stifle Allied reaction to the Germans' offensive moves, and plenty of these deeper interdiction missions were conducted against the British and French formations moving up to their Dyle positions; they not only sobered the enthusiasm of the troops but also caused considerable destruction, disruption and delay. Charles-Michel L\u00e9p\u00e9e was a truck driver supporting an infantry regiment in First Army and had just crossed the Senne twenty miles south of Brussels when his column was attacked. As he later explained:\n\nThe aircraft came from nowhere and before I knew what was happening a vehicle was thrown up into the air in a massive explosion right before my eyes. Debris fell onto my own vehicle, smashing the windscreen and causing me to swerve off the road in order to avoid the burning wreck. The truck hit a drainage ditch and immediately fell onto its side. I broke my shoulder and was to have fifty stitches in two ugly face wounds. I counted myself lucky, however, as I later heard that six drivers had been killed in that one incident.\n\nReports from both the French formations and the Luftwaffe during this period only confirm the success the Stukas were having in both targeting enemy columns moving towards the front and making those moves more tentative and dispersed and, in some cases, forcing them to be undertaken only during the hours of darkness.\n\nDespite being attacked from the air, the BEF and French divisions pushed forward against a growing tide of refugees and through a series of small, indistinguishable Belgian towns and villages. News had reached various units about the scale and early success of the German invasion, but details remained scant and rumours were rife. It was with some relief, therefore, that the main bodies of the divisions began to meet up with their forward elements already on the Dyle during the evening of 11 May. They immediately began to prepare their defences in a process that would take days rather than hours: guns were sited, minefields laid, trenches dug, observation points established, troops deployed, and headquarters established in a frenzy of activity. Montgomery's 3rd Division began to arrive in force at its planned sector at Louvain to the east of Brussels after a thirty-six-hour journey. 15\/19 Hussars immediately pushed forward to set up an outpost on a ridge five miles to the east of the river and a standing patrol was sent another six miles forward to make contact with any Belgian troops that might be in the vicinity, and also to warn of any approaching Germans. Before they could move off, however, contact was made with Belgian 10th Division, which was heading towards Montgomery's formation.\n\nRather than holding the Albert Canal for several days as Gamelin had anticipated, the Belgians had managed around thirty hours since, after the fall of Eben-Emael, the German Sixth Army had advanced beyond Tongeren, south-west of Maastricht. During the afternoon, General Walter von Reichenau's formations looked to push on to the Dyle by attacking north-west towards Antwerp and south-west towards the Gembloux Gap. The 10th Division had therefore endured a torrid twenty-four hours \u2013 it was tired after its withdrawal and had been shaken by numerous Luftwaffe attacks \u2013 and the discombobulated Belgians initially mistook Montgomery's men for German parachutists and opened fire on them. It was a situation typical of the confusion that existed at this time, with one set of troops pulling back, another set pushing forward and neither knowing about the presence of the other until contact was made.\n\nAlthough the Belgian fusillade stopped when the British made it clear that they were friendly forces, Alan Brooke, the astute commander of II Corps, soon became so concerned about 10th Division's disinclination to co-operate with him that he went to see the Belgian commander-in-chief, the thirty-eight-year-old King Leopold, at Belgian GHQ in St. Andr\u00e9, just outside Bruges. He received little support from the monarch or his military aide General van Overstraeten and was tartly informed that, as far as they were concerned, the 10th was a Belgian division, on Belgian soil defending the Belgian capital and would not be dictated to by the army of a foreign power \u2013 no matter how well-meaning. Brooke left the meeting not only concerned about the integrity of 3rd Division's sector in the light of what he'd been told, but also the competence of the Belgian High Command.\n\nGuderian, meanwhile, having fallen behind schedule on the first day of the offensive, was keen to make up for lost time on 11 May and to ensure that his corps crossed the Semois River, which was close to the French border and the last natural obstacle before the Meuse. With the corps' divisions also manoeuvring themselves into a position whereby they would be able to arrive on the Meuse simultaneously, there was now considerable pressure on 2nd Panzer Division to move particularly quickly on the right of the corps as the Semois, the French border and the Meuse were all located to the front of Veiel's formation. By dawn the forward screen of French 5th Light Cavalry Division had begun to make contact with the three German divisions and to undertake a fighting withdrawal back into the main body of the formation. By mid-morning there was intense infantry fighting along the Libramont-Neufch\u00e2teau Line. 2nd Panzer Division looked to take Libramont as quickly as possible in order to surge on, but its tussle in the town lasted for several hours and was only concluded after some vicious house-to-house fighting forced the French to withdraw back to the Semois. It was a delay that the division could ill afford and by the end of the day it remained several miles short of the river.\n\nIn contrast, 1st Panzer Division proved too powerful for its adversaries on 11 May. The formation had its tank regiment up front, the only one of Guderian's divisions to do so, but although this gave Kirchener's attack a sense of menace and immense firepower, it was far from ideal for the battle it faced in Neufch\u00e2teau that morning. Nevertheless, in a sensible attempt to utilize the strength of his armour rather than showcase its weaknesses, the commander decided to encircle the town rather than try to blast his way through it. It was his great good fortune to strike at a weak point just south of Neufch\u00e2teau where the 5th Light Cavalry Division's 11th Cuirassier (Dragoon) Regiment and 60th Reconnaissance Battalion met. Tearing a hole in the defences, the tanks broke the French line after an hour of fighting and pushed deep into the enemy's rear while Balck's 1st Rifle Regiment, which was following up, moved into the town to clear it. As they did so, the armour forged on, rolling over the French artillery before threatening the headquarters commanding Neufch\u00e2teau garrison in the village of Petitvoir. Panic ensued.\n\nThe speed of the German attack had caught the French off guard and now vehicles hastily withdrew alongside fleeing, riderless horses, the sight of them alone being enough to ensure that pandemonium gripped the French rear area. The German armour then faced the challenge of how to attack the village housing the enemy's headquarters without resorting to the infantry. A plan was quickly formulated and put into action: 2nd Panzer Regiment continued to draw the attention of the defenders in the village while 1st Panzer Regiment outflanked them. It worked and shortly after 1500 hours, with the defenders' withdrawal route blocked, Petitvoir surrendered and in that same moment the co-ordinated defence of Neufch\u00e2teau became untenable. One last message from the headquarters informing the garrison that they had been outflanked by panzers was all that was required to spark the mass capitulation of the French garrison in Neufch\u00e2teau to Balck's men.\n\nThe men of 5th Light Cavalry Division began to cross back across the Semois in small parties during the late afternoon of 11 May and continued to arrive after dark. Here the remnants of the division joined with a single battalion from French 55th Infantry Division commanded by Captain Picault. Moving up behind them quickly, 1st Panzer Division arrived on the Semois at Bouillon before dark and immediately attempted to take two bridges that remained intact because the French had not expected the enemy's arrival so soon. Both were blown just as the armour approached them but, keen to cross the river before dark, Kirchner ordered Oberst Johannes Nedwig's 1st Panzer Regiment, elements of 1st Rifle Regiment and a wave of Stukas to plaster the opposite bank with fire. Under this cover, an infantry company led by Hauptmann Friedrich Freiherr von Kressenstein used a ford to get to the west bank, where his men then began to infiltrate the French positions. A misdirected Stuka attack against Nedwig's tanks together with a French bombardment from heavy artillery located to the rear of Sedan forced von Kressenstein's withdrawal, but his infantry moved back into Bouillon under the cover of darkness to find that the French had already left and at that moment were falling gratefully back across the Meuse. At the end of a successful day, 1st Panzer Division's log read:\n\nA feeling of superiority over our western opponents arose in the division on this day \u2013 a day in which it encountered for the first time French forces along the second Belgian resistance line. The division has reached its goals planned for it on the second day \u2013 far in advance of its sister divisions \u2013 without regard for its flanks and in spite of the slow progress of the day.\n\nIn fact, 1st Panzer Division had taken more than its own objectives on 11 May for, having spotted an opportunity to seize another crossing point over the Semois at Mouzaive, this too was taken. Mouzaive lay in the sector designated for 2nd Panzer Division but as Veiel's division had been delayed at Libramont and then by 2nd Moroccan Spahi Regiment, Oberst Karl Keltsch, the commander of 1st Panzer Brigade, used his initiative and threw a reinforced 1st Motorcycle Battalion into the gap between the Semois and the rear of the Moroccans. The bridge at Mouzaive was taken just after midnight and from there the motorcyclists pressed across the river to create a bridgehead. Kirchner immediately contacted corps headquarters for permission to use the site as a second crossing point and Guderian readily agreed. The bridge at Mouzaive was to have massive operational significance for, unbeknown to Keltsch, Kirchner or Guderian, it lay on the boundary between the French Second and Ninth Armies. In the immediate term, however, Keltsch's quick thinking had led directly not only to the panic and unauthorized withdrawal of 2nd Moroccan Spahi Regiment and its parent 3rd Spahis Brigade, but also to that of 5th Light Cavalry Brigade \u2013 and thus its troops at Bouillon \u2013 after it realized that its left flank was being exposed. The Semois front collapsed early on 12 May, therefore, not because it had been prodded by 1st Panzer Division several hours earlier, but because of the speedy appreciation and exploitation of an opportunity by Oberst Keltsch.\n\nThe third of Guderian's formations, 10th Panzer Division, successfully completed its re-routing on 11 May. Although the drive of the left flank through the rear of the right flank and out the other side had slowed the westward drive of Schaal's units, the fact that the Semois River was closer to the formation in this sector, allowing it to reach the obstacle more quickly, was a reassurance to all concerned. In the end, the complicated movement proved to be unnecessary due to the withdrawal of 5th Light Cavalry Brigade, but a crossing was taken at Mortehan by the early hours of 12 May \u2013 albeit it with the tanks lagging ten miles behind. It was at this point that the Gross Deutschland Regiment \u2013 a formation that had been temporarily attached to the division \u2013 was moved back to corps reserve in preparation for its attachment to 1st Panzer Division for the crossing of the Meuse. With Guderian beginning to reorder his corps for this undertaking and content with how events were unfolding, it was with some anger therefore that he learned from von Kleist that Berlin was concerned the corps was not paying enough attention to the protection of its southern flank. Although managing to diffuse the situation by saying that 29th Motorized Division would soon arrive to assume that very role, Guderian became acutely aware that his superiors might already be losing their nerve. Indeed, he fully expected that, as his corps' advance developed and its flank became more exposed, so they would become increasingly anxious. He was prepared, however, to do whatever was within his power to protect his corps from what he saw as any unwarranted interference.\n\nGuderian knew that his corps was vulnerable while it was still deploying but, even though the opposition had managed to cause some local difficulties, he was relieved that these had not been exploited by the arrival of Allied aircraft, which could have done great damage to his strung-out divisions. The reality was, though, that Allied air power was being directed at what was believed to be the main German ground effort \u2013 itself still supported by the main Luftwaffe effort \u2013 to the north. What aircraft there were across the Ardennes and protecting French Second Army came from Air Group 23, which had received orders on the morning of 10 May to bomb any German columns that were spotted, but not to hit any buildings. Thus, towns and villages \u2013 just the places where enemy vehicles could be seen after they had emerged from cover amongst the trees and which also acted as choke points \u2013 could not be targeted. When the Allied aircraft did manage to engage the enemy, they were swatted away with contempt. One attack by light bombers of the British Advanced Air Striking Force was launched against a German column in the Ardennes. Eight sluggish and vulnerable Battles managed to avoid the protective air cover and even the anti-aircraft fire, but were met by a hail of machine-gun and small arms fire which brought three down. Of the thirty-two Battles that took to the air that day against a variety of targets, thirteen were shot down and the rest were damaged.\n\nBut it was not only strike aircraft the Germans were careful to defend themselves against; it was also the reconnaissance machines that could have revealed the true extent of the Wehrmacht's commitment to the Ardennes. As one German officer later recalled:\n\nAgain and again, I cast a worried look up at the bright blue sky; my division now presents an ideal attack target because it is not deployed and it is forced to move slowly forward on a single road. But we could not spot a single French reconnaissance aircraft.\n\nIndeed, the aerial unit supporting the French Second Army had maintained a weak presence over the battlefield and raised no concerns in its daily report on 11 May. It identified four three-mile-long enemy columns in Belgium, but this was not seen to be anything other than what was expected in the area. Even so, it was information which, when seen alongside other intelligence gathered that day, could have provided an early warning of German intentions. In the end, documents from a dead German soldier identifying his unit as the elite Gross Deutschland Regiment and a radio intercept detecting the presence of 1st Panzer Division were regarded as perplexing rather than evidence of anything more sinister. The reports from French commanders at the front were confused, contradictory and spasmodic. While the vast majority referred to actions against enemy infantry, only the battle at Neufch\u00e2teau involved German tanks. Second Army informed First Army Group of a 'panzer presence' on the afternoon of 11 May, and followed it with a detailed report to Billotte that evening saying that it had been 'heavily pressed at Neufch\u00e2teau by heavy tanks in the beginning of the afternoon... Lost one [artillery] battery which destroyed three tanks armed with large cannons'.  Yet although this threat was not replicated across the Ardennes front, largely because only 1st Panzer Division deployed its tanks on 11 May, the French High Command signally failed to react to the evidence with which it had been presented: its blinkered focus remained on Army Group B.\n\nThat attention on von Bock's formation continued into 12 May as the situation in the Netherlands deteriorated significantly south of Rotterdam due to the advance of 9th Panzer Division. Reconnaissance units from the formation punched forward from dawn and by noon had linked up with cheering airborne troops at the Moerdijk bridgehead. This achievement successfully ended any opportunity that the Allies had to reinforce Fortress Holland from the south without a contest. General Winkelman hoped that the British might send three divisions by sea with which he might launch a counter-attack, but London refused, stating a 'lack of reserves'. The Netherlands was to be unceremoniously abandoned, and Winkelman knew it. With pressure mounting in Belgium and the French border threatened, he accepted that the British and French had other pressing demands on their resources. He also understood that in such a situation, the time would soon come when a decision would have to be made about whether the Netherlands continued to fight and thus tied down German resources in Holland, or whether the nation capitulated and allowed the enemy to concentrate his fighting power against the Dyle.\n\nFor the time being, the Dutch continued to fight. If 9th Panzer Division could be stopped from advancing to Rotterdam, then there might yet be a hope of conducting a protracted defence from Fortress Holland. The day, however, was Germany's. Despite continued counter-attacks against the Moerdijk Bridge and efforts to make it impassable by turning the artillery on the airborne troops, these remained intransigent and elements of the panzer division gradually established themselves in force. Compounding the situation was the collapse of the Grebbe Line after SS Leibstandarte Adolf Hitler had rolled through a series of enemy positions on a two-mile front. With this the Germans looked to make further inroads into the Dutch defences and to make continued resistance from Fortress Holland untenable. To facilitate this, OKW formed 9th Panzer Division, SS Leibstandarte Adolf Hitler and 254th Infantry Division into a new corps which would strike at Rotterdam and then become the manoeuvre group which would pursue the French Seventh Army as it withdrew towards Antwerp. Rotterdam would be forced into submission by some of 9th Panzer Division's tanks and infantry with close air support. Time was, however, of the essence if Seventh Army was to be pressurized by ground forces and so the Luftwaffe began to make preparations for bombing raids against Rotterdam, which was still full of civilians. If the enemy continued to be stubborn, high explosives would blow the fighting spirit out of him.\n\nThroughout 12 May the Dutch government continued to discuss its limited options, which all agreed looked 'bleak in the extreme'. The Allied High Command also recognized the fact and prepared themselves for the reorientation of Eighteenth Army towards central Belgium within the next couple of days. That morning King Leopold, van Overstraeten, Daladier, Georges, Billotte and Gort's chief of staff, General Henry Pownall, met near Mons. The mood was tense, businesslike and frank. The Belgian contingent were keen to assert that its army had fulfilled the demand to delay the Germans for long enough for French and British divisions to move forward. The French and British could hardly agree, but such a meeting was not the place for recriminations as Gamelin wanted Leopold to assent to his forces being brought under French command. The king agreed and with that details were confirmed for their part in the defensive line running from Antwerp, along the Dyle and down to Namur.\n\nNone of those present openly expressed whatever concerns they may have had about the campaign thus far, although the difficult position of the Netherlands was acknowledged, and the German movement through the Ardennes was mentioned only in passing. Meanwhile, with the first phase of operations completed, King Leopold issued the following proclamation to his troops:\n\nSoldiers,\n\nThe Belgian Army, brutally assailed by an unparalleled surprise attack, grappling with forces that are better equipped and have the advantage of a formidable air force, has for three days carried out difficult operations, the success of which is of the utmost importance to the general conduct of the battle and to the result of the war. These operations require from all of us \u2013 officers and men \u2013 exceptional efforts, sustained day and night, despite a moral tension tested to its limits by the sight of the devastation wrought by a pitiless invader. However severe the trial may be, you will come through it gallantly. Our position improves with every hour; our ranks are closing up. In the critical days that are ahead of us, you will summon up all your energies, you will make every sacrifice, to stem the invasion. Just as they did in 1914 on the Yser, so now the French and British troops are counting on you: the safety and honour of the country are in your hands.\n\nLeopold\n\nOn the afternoon of 12 May, a series of critical engagements began in central Belgium. The Battle of Hannut was part of a wider Sixth Army attack against the Allied line and was to last for two days. Its progress was tracked closely by Gamelin and the High Command because it was fought at the gateway to the Gembloux Gap. Leading the way were 3rd and 4th Panzer Divisions of General Erich Hoepner's XVI Panzer Motorized Corps which ran into a twenty-mile-wide defensive position occupied by Prioux's Cavalry Corps some eighteen miles beyond the main defensive line. Prioux's corps lacked the number of formations to provide a depth that would hold two concentrated panzer divisions for long, but it had to hold them long enough for First Army to strengthen the defences behind them and for 1st and 2nd DCRs to move into position to act as a final stopping force if required. The panzer divisions were resupplied for their attack by air as the lines of communications over the rivers to their rear were congested with all manner of units and the bridges were under frequent Allied air attack. It was a testament to the work of OKH's chief logistician, Major General Eduard Wagner, whose flexible planning and preparations across the front allowed for resupply up to 190 miles beyond railheads. As a consequence, Hoepner's formations were in better condition to fight the battle than the French, who were still busy preparing their defences and whose lines of communication \u2013 regularly attacked by the Luftwaffe \u2013 stretched all the way back to the French border.\n\nHoepner concentrated his two divisions against 2nd and 3rd Light Mechanized Divisions (DLM). The Germans had a quantitative superiority of 623 tanks against 415, but 498 of the German total were PzKpf Mk Is and IIs, leaving only 125 Mk IIIs and IVs to be used against the French SOMUA and Hotchkiss tanks. Lieutenant Robert La Bel of 11th Regiment, 3DLM observed the massing German armour through field glasses from his Hotchkiss H39 parked beneath an apple tree:\n\nI saw an extraordinary show which was played out about three kilometres away: a panzer division shaping itself for battle. The massive gathering of this armoured armada was an unforgettable sight, the more so that it appeared even more terrifying through the glasses. How many were there? It was not possible to tell from so far away but they were numerous and their guns seemed to be potent... Some men, probably officers, walked to and fro gesticulating in front of the tanks. They were probably giving last-minute orders to the tank commanders, the head and shoulders of whom I could see between the two open parts of the turret hatches. Suddenly, as if swept away by a magic trick, they all disappeared. No doubt 'H'-hour was approaching. A dust cloud soon appeared on the skyline, disclosing the enemy move.\n\nLa Bel's desire 'to give the Boche a kick in the face that he will remember and which will send him reeling back to his homeland' was not, however, matched by the French ability to provide it.\n\nHannut was the largest armoured clash the world had seen to date \u2013 and was to be the largest of the campaign \u2013 and was a great test of the belligerents' fighting methods, leadership, technology, command, control and training. Initially the firepower of Prioux's tanks caused the enemy considerable losses. It did not take long, however, for the Germans to realize that the Mk Is and IIs could not be used to take on the French armour and instead the panzer regiments tried to avoid engagements and to out-manoeuvre the enemy. The French proved themselves to be sluggish, slow to react and predictable against such tactics, while the Germans were mobile, lithe and unpredictable. The confrontation seemed to be a microcosm of the two sides' operational capabilities, with the Germans achieving armoured mass where it was most needed and so managing to overwhelm a large number of qualitatively superior French tanks.\n\nSergeant Major Georges Hillion of 4th Squadron Cuirassiers was in a Hotchkiss H39: 'My first target!' he later recalled, 'I fired and saw the direct hit.' However, his tank then stalled. 'A shell burst through my turret,' Hillion continued, 'I was wounded in the face and left arm; blood covered my face and I could not see any more from my left eye.' The tank fought on, shell after shell crashing into it. But the 40mm armour protected Hillion and his driver until an anti-tank round pierced it and before long further resistance became impossible as the interior filled with smoke. Blinded in his left eye, Hillion clambered out of the tank dragging its machine gun, was thrown to the ground by the impact of another shell and came to in great pain. 'I opened my right eye,' he later said, 'and saw a tank rolling over both my legs, the edge of the track just under my knees.' His driver had been killed in the exchange and his regiment lost twenty-four Hotchkiss tanks that day.\n\nAltogether, the panzers showed better tactical awareness, focusing on obtaining a critical mass and then winning each individual fire-fight, only firing when stationary, zigzagging to make difficult targets, keeping the sun behind them when manoeuvring into the flanks and rear of the enemy, and linking their efforts together with first-class radio proficiency. One German tank commander remarked that the French armour at Hannut was 'leaderless, purposeless, poorly led, tactically inferior'. Doctrine, the design of their tanks and a lack of radios severely undermined the ability of the French to cope with the enemy's speed of thought and action, while their failure to concentrate themselves against the most likely attack routes allowed the Germans to develop significant local superiority. The latter's strengths, especially when married with excellent co-operation with other arms, allowed them to penetrate the French positions, although not to break through them. Hauptman Ernst von J\u00fcngenfeld, commander of a 4th Panzer Division tank company, later wrote: '[The day was] hard and bloody, many a brave _Panzermann_ had to lay down his life for the Fatherland, many were wounded and a large number of our tanks was lost, in part to shells and in part to breakdowns.'\n\nThe first day of the Battle of Hannut was a tumultuous confrontation that held the French transfixed for Gamelin believed that its outcome was a critical precursor to Gembloux and the continuing integrity of the Dyle defences. The attention of the German High Command was also held by the battle, for if the French defeated Hoepner, then Army Group B's attempt to fix the Allied formations in central Belgium was threatened, whereas if Hoepner succeeded in pushing on to the Gembloux Gap, German prospects remained good. Because of this, events in the Ardennes were of at least equal importance to those taking place west of Maastricht. The continued advance of von Rundstedt's Army Group through the Ardennes was critical, but this was not a task that was the sole preserve of Panzer Group Kleist.\n\nTo the south of Hannut, Panzer Corps Hoth led Fourth Army forward not only to protect the north flank of the main effort, but also to break the French Ninth Army line as it moved forward to take up a position on the Meuse. Hoth thus ordered his 5th and 7th Panzer Divisions to advance as quickly as possible in order to reach the river before the enemy's defences were set. This was just the sort of direction that Erwin Rommel liked. In the days and weeks that followed, the maverick general was to gain himself the reputation as one of the finest and boldest divisional commanders in the Wehrmacht while his formation, much to Rommel's delight, became dubbed 'The Ghost Division'. 7th Panzer Division made rapid progress, making short work of the Belgian defences and the delaying efforts of 1st and 4th Light Cavalry Divisions in a manner very similar to Guderian's divisions. Rommel was always energetic, ensuring that forward momentum was achieved by avoiding unnecessary contact and being on the spot to make important decisions to ensure that no opportunity was missed. It was an exhausting style of command and on just the second day of battle he wrote to his wife:\n\nI've come up for breath for the first time today and have a moment to write. Everything wonderful so far. Am way ahead of my neighbours. I'm completely hoarse from orders and shouting. A bare three hours' sleep and an occasional meal. Otherwise I'm absolutely fine. Make do with this, please, I'm too tired for more.\n\nRommel had indeed outstripped 5th Panzer Division and during the morning of 12 May Hoth put Oberst Paul Hermann Werner's 31st Panzer Regiment under 7th Panzer Division's command in order to broaden its attacking front. That afternoon, Werner's Detachment was sent to Yvoir on the east bank of the Meuse where aerial reconnaissance had identified an intact bridge. It had been prepared for demolition but it had not been destroyed as civilian refugees were using it. It was protected by a platoon of Ardennes Light Infantry and part of a French infantry battalion when Leutnant Heinz Zobel's armoured assault team consisting of two armoured cars and a panzer platoon raced forward and reached it before the defenders knew what was happening. Lieutenant de Wispelaere pushed the button of an electronic detonation ignition system, but there was no explosion. Despite coming under fire from German tanks and approaching infantry, he managed to detonate the explosives manually. The bridge heaved and crumpled into the river below a cloud of dust.\n\nAlthough thwarted at Yvoir, Werner despatched a number of motorcycle infantry patrols to look for an alternative crossing site. Two miles to the south one patrol found a weir and lock system which connected the island of Houx to both banks of the Meuse. Although it was an obvious crossing point \u2013 indeed, it was the place where the Germans had crossed the river in August 1914 \u2013 responsibility for the area fell on a boundary between French formations which were only just arriving. IX Corps' 18th Infantry Division and II Corps' 5th Motorized Divisions, joined by 1st Light Cavalry Division as it withdrew from the Ardennes, were in a poor position to fend off a major attempt to cross the river and could hardly believe that the Germans had managed to advance so far, so fast. At dusk, the defenders were still confirming their boundaries and moving units into position along the front.\n\n18th Infantry Division took command of the Houx sector, but it was a 5th Motorized Division battalion that was used to defend the area around the island, although its commander failed to occupy the island itself. After dark, therefore, a few of the German motorcycle infantrymen managed to cross the river, the first soldiers from Army Group A to reach the west bank of the Meuse. It was only as this small group probed forward to ascertain the enemy's strength that they came under fire and were forced to ground. Reinforcements sent by Werner, however, managed to expand the tiny bridgehead, assisted by a French decision to wait until daylight to counter-attack. In the meantime, a panzer company led Rommel's 7th Panzer Division to Dinant, just a couple of miles south of the island and where both bridges were found to have been destroyed. As Rommel prepared his division for an assault crossing the following morning, he was informed of the pin-prick that had been made in the French defences at Houx. Encouraged, he ordered 7th Panzer Division to begin its attack at dawn.\n\nYet despite Rommel's not inconsiderable success on the evening of 12 May, it was the progress of Guderian's corps at the _Schwerpunkt_ that most occupied Halder's mind. Having crossed the Semois in two places, 1st Panzer Division led the charge south with Kirchner, encouraged by his corps commander, keen not to give the French time to settle into the Meuse defences. Making use of their two crossing points while new bridges over the Semois were constructed by the engineers, Keltsch's 1st Panzer Brigade and a battalion of infantry led the attack from Mouzaive, while a group commanded by Oberst Kr\u00fcger containing the majority of Balck's regiment and supported by two tank platoons led the main body of the division out from Bouillon.\n\nSouth of the Semois, the terrain was tortuous for the attacking forces \u2013 so much so that, in trying to find a route south from Mortehan, 10th Panzer Division had moved onto the route designated for Kirchner's advance. Kr\u00fcger found a different but more meandering road which also contained several large demolitions requiring attention by the engineers. While waiting for the advance to proceed, the column came under accurate artillery fire, which caused casualties among the staff of 1st Rifle Regiment. When Kr\u00fcger's attack continued, it was greatly advantaged by Keltsch, who was already putting pressure along the road to Sedan and forced the French cavalry back and into a line of blockhouses and pillboxes camouflaged as residential homes approximately three miles before the Meuse. The German divisions spent much of the late morning and afternoon fighting through these mutually supporting positions in close terrain. The most stubborn proved to be centred on a pillbox at La Hatrelle which contained an anti-tank gun and machine guns and was secured by mines. Kr\u00fcger's group attacked it several times during the mid-afternoon with various combinations of tanks and infantry but failed to make an impression. After nearly four hours of obstruction, a single panzer succeeded in firing a round through the pillbox's gun embrasure. This was followed by an infantry assault which finally silenced the occupants. That evening, the last French defenders gave up on their attempt to delay the advancing Germans and fled into Sedan to cross the bridges that had been kept intact for their withdrawal at around 1900 hours. As the leading elements of 1st Panzer Division entered the town that evening, they heard a series of explosions which told them that these bridges had finally been destroyed.\n\nBy nightfall, Kirchner's division was establishing itself in Sedan and preparing for its assault crossing of the town which, at that time, was expected to be the following morning. It did so in close proximity to the river because of the protection offered by the town's buildings, a luxury not afforded to 10th Panzer Division, whose crossing point was out of the town in more open countryside. As a consequence and in order to thwart observation by the enemy, its assembly area was one or two miles from the Meuse in the wooded fringes of the Ardennes. The forward units of Schaal's division reached their destination, the village of Givonne two miles north-east of Sedan, just as Sedan's bridges crashed into the Meuse. The last division to arrive in position was 2nd Panzer Division which, finding its crossing point at Mouzaive being used by 1st Panzer Division when its lead elements arrived on the Semois just after dawn and that a bridge at Membre had been destroyed, set about constructing a new bridge at Vresse. This was completed in the mid-afternoon and Veiel immediately sent his motorcycle infantry forward to the industrialized area of Donchery on the east bank of the Meuse from where his division would launch its assault. The rest of the division was held up in awful traffic jams which led back many miles east of the Semois and meant that by nightfall the majority of the division was still four miles short of the Meuse.\n\nBehind Guderian's corps there continued to be a hideous congestion of formations which were struggling to wriggle free of the confines imposed on them by List and the terrain. It included Reinhardt's 6th and 8th Panzer Divisions and attached 2nd Motorized Division which, rather than being mere follow-up forces, had the task of crossing the Meuse in the Montherm\u00e9 sector on 13 May between Guderian's and Hoth's corps. 6th Panzer Division's war diary for that day reported:\n\nThe Division was ripped apart during the attack by parts of the 2nd Panzer Division, along with 16th, 23rd, 24th and 32nd Infantry divisions that pushed in between. In the afternoon and evening there was little clarity about where the advancing groups and their individual formations were located. It was also impossible to get accurate information by radio because such communications were limited due to the great distances and radio interference.\n\nThe corps strained every sinew to get into position, but as soon as its units struggled free of the traffic jams and onto its own routes, it then had to overcome enemy obstacles and the terrain. At one point it was even bombed by some errant He 111s which left twenty dead and twenty-six wounded. Thus, although elements of 6th Panzer Division were to arrive on the Meuse during the early hours of 13 May, 2nd Motorized Infantry Division did not arrive at Nouzonville four miles south of Montherm\u00e9 until 14 May and 8th Panzer Division did not reach the Meuse until 16 May. The echeloning of Panzer Group Kleist had led to 2nd Panzer Division and Corps Reinhardt falling behind schedule too and so, rather than seven panzer divisions being on the Meuse on the morning of 13 May, there were only five and there was a gap in their centre. Halder watched and waited, applauding Guderian's unconventional and unprecedented 170-mile advance but concerned about whether he would have the power to make a successful crossing on the following day and whether two panzer corps would be enough to punch through the enemy's defences and into the open ground beyond.\n\nOKW hoped that the French remained unaware that what was building on the Meuse was the main German effort. They did. Reports from Ninth and Second Armies to Billotte and Gamelin on 12 May continued to point at the enemy's advance south of Maastricht as being a secondary offensive and suggested the planned delaying operations should hold them. Intelligence officers at both army headquarters continued to receive information from various sources that ran contrary to the main thrust of the submissions they were receiving from the front, but these continued to be treated as unexplained anomalies. One aerial reconnaissance mission conducted during the early hours of 12 May, for example, identified columns of enemy vehicles advancing with their lights blacked out. Another aircraft was sent up during daylight hours to verify the sighting. It limped back to base full of holes after running into a hail of fire from the ground as the pilot confirmed the presence of the German columns. Second Army reported enemy tanks advancing towards Sedan along with vehicles carrying vast amounts of bridging equipment. There were even aerial photographs of the ensuing bridge-building and of vehicles driving across the Semois at Bouillon. But all such intelligence was ignored, partly because of the weight of evidence suggesting that nothing was amiss in the Ardennes and partly because the armies did not want to alarm an already tense High Command, which at that time was preoccupied by the Battle of Hannut. Indeed, although Gamelin's chief of staff noted on the evening of 12 May that German forces in the Ardennes were 'making contact in some places with the forward elements of our fortified line', he added, '[but in] contrast, to the west of Maastricht, his advance has noticeably decreased.'\n\nSedan itself had been quiet for the first two days and the French troops there had assisted with the evacuation of civilians from the area. Although they had been put on alert, there had been no air raids and nothing to unduly concern them. The men of 55th Infantry Division spent their time in preparation, making sure that their positions were as strong as possible, checking communications and weapons, ensuring they had the necessary supplies and running through likely scenarios. There was a feeling abroad they were soon going to taste the war at first hand, and they were pleased to learn that they would do so while grappling with the enemy's secondary effort. Their first sight of the Germans came during the evening of 12 May shortly after the last units of 5th Cavalry Division had run across Sedan's bridges and into friendly lines. A little later David Boyer, a young soldier manning the division's forward defences, had an unsettling experience. He later recalled:\n\nI was already extremely anxious at the sight and sound of the Germans just across the river and was at my post waiting for orders when a man from 295th Infantry Regiment ran past. He was stopped by my officer, who asked him what was going on. The man stood to attention and said that he had just returned from fighting the Germans with the cavalry and was looking for his company's lines. The Lieutenant immediately asked him what he had learned of the enemy. The man spoke just two sentences, saluted and left. He said: 'There are many of them, they have tanks and flamethrowers. We are dead men.'\n\n* The Bauwens were eventually overtaken by advancing German troops and, taking the advice of a kindly officer, made a 100-mile return journey back to Zammelen. They arrived home on 1 June 1940 to find that it had been ransacked by German troops and that every single animal had gone.\n\n* The 'howling mechanism' was the so-called Jericho Trumpet, a propeller-driven siren attached to the aircraft's undercarriage legs which produced a scream as the Stuka dived in order to undermine enemy morale.\n\n# CHAPTER 6\n\n# 13 May \u2013 Crossing the Meuse\n\nDAVID BOYER DID NOT SLEEP AT ALL during the night of 12\u201313 May. He felt sick and could not dislodge from his mind the image of that man from the 295th nor his dire warning. Boyer's officer deliberately sought him out later that evening with the intention of reassuring him. In his experience on exercise, he said, soldiers always over-exaggerated and no matter what, the defences that they had built together could withstand the attack of an entire army. He advised his charge to get a couple of hours' rest and not to worry. All would be well.\n\nLess than half a mile away, 1st Panzer Division was planning, preparing and positioning its various units. During the night Guderian had produced an outline corps plan for the divisions so that they might begin their own more detailed planning. A mere three pages long, it was immediately recognized by the division staffs as being very similar to a corps exercise run in late March and provided the divisions with plenty of scope. At 1600 hours 1st Panzer Division would attack from Gaulier, 2nd Panzer Division at Donchery, and 10th Panzer Division from the eastern fringe of Sedan, a front of just five miles and containing some 60,000 men and 22,000 vehicles in its depth. 1st Panzer Division had a particular advantage because after one such exercise run less than two weeks before, the division's operations officer, Major i.G. Walther Wenck, had ordered twenty-five copies of the final attack orders he had drafted to be run off. On 13 May, Wenck issued those copies with changes merely to the timings and locations.\n\nBy daybreak, Kirchner and Schaal's divisions were making good, if somewhat frantic, progress with their preparations while their units continued to arrive in the assembly areas. The fact that it was still the leading units of Veiel's formation that were arriving in Donchery during the morning was of some concern to Guderian. As one French officer wrote after the war: 'What a chance to acquire imperishable glory... by smothering at birth the German offensive and transforming all these mechanized and armoured units into scraps of burnt and twisted metal.' Unfortunately, it was a chance the French did not take. Launching an assault on a major river obstacle without an operational pause was always going to be difficult, but the French had no idea that this is what the Germans were going to attempt and as a consequence made no major effort to undermine their preparations. The French artillery lobbed shells into the divisional areas and at anything that moved, but their weight of fire merely inconvenienced the Germans and there were no air strikes or any spoiling attacks. As 1st Panzer Division engineer Otto Gull later commented:\n\nIt was as if the enemy had not expected our arrival. In training we were told to expect the worst and had learned to do our work under the most appalling conditions. Now, here we were on the Meuse and other than some nasty artillery concentrations there was nothing to suggest that the enemy were ready for us. I drew great strength from this.\n\nAfter receiving overnight reports about the arrivals at Sedan, Second Army informed superior headquarters that 'the enemy had arrived in force' on the Meuse and that they included 'significant numbers of tanks'. In mid-morning this was followed by a French air reconnaissance mission which informed Georges's headquarters that a significant enemy movement up to the Meuse had occurred during the night of 12\u201313 May and clearly stated that 'Everything confirms a large German effort on the Bouillon-Sedan axis.' Andr\u00e9 Beaufre, an operations officer at Gamelin's headquarters, collated the latest information from the Ardennes sector and transferred it onto a map. To his horror, he could only conclude that the main German effort was not, in fact, towards the Gembloux Gap but towards Sedan. Georges's headquarters came to the same, stultifying deduction at about the same time. There was no panic, however, because it was believed that the Germans would take several days to gather what they required to attempt a crossing of the Meuse. It was an assumption that was not immediately challenged and re-tested because Gamelin, Georges and Billotte were focused on the continuation of the Battle of Hannut and the establishment of the Allied formations along the Dyle Line.\n\nAlthough inexcusable in the circumstances of a developing wider campaign, the French generals' preoccupation with events east of the Gembloux Gap is at least understandable because, having withstood Hoepner's attack on the previous day, on 13 May Prioux's line cracked. With the assistance of Wolfram Freiherr von Richthofen's VIII Air Corps providing close air support, the French were sent into withdrawal. It was an appalling setback for Gamelin, who had anticipated such an eventuality only after the Belgians had held the enemy for several days on the Albert Canal and Prioux had managed the same behind them. Instead, the Sixth Army was motoring _en masse_ towards the Gembloux Gap and incomplete Dyle defences on just the fourth day of the offensive. Billotte took some comfort from the fact that Prioux's divisions had inflicted a loss of 160 tanks on Hoepner at a cost of 105 of his own machines. His comfort was misplaced, however, for within forty-eight hours mobile workshops had repaired 111 of the 'lost' German tanks and they immediately rejoined their divisions.\n\nThe situation in central Belgium was concerning for the Supreme Commander and it looked set to deteriorate further after news arrived of setbacks in the Netherlands. On the morning of 13 May, Winkelman advised the Dutch government that he considered the general situation to be critical. With German tanks poised to advance on Rotterdam from Moerdijk and infantry divisions pressing beyond the Grebbe Line, the general was not confident of prolonged successful resistance. The Cabinet's reaction was to flee to England and there, joining the Dutch royal family, establish a government in exile. A dismayed Winkelman was to remain in the Netherlands, co-ordinating the nation's resistance, avoiding unnecessary casualties, until such time as he believed the situation hopeless, whereupon he was to surrender. It was a huge responsibility for a military man, but Winkelman understood that if the Netherlands was to have an independent future, then he must assist the Allies by ensuring that the German forces were tied up in and around Fortress Holland for as long as possible.\n\nAs it was, 9th Panzer Division had restarted its offensive from the south side of the Moerdijk bridgehead just after dawn. Leading with four tank companies, the Germans pressed on to the Moerdijk Bridge itself, swatting aside all Dutch attempts to thwart them. As the first panzers rolled across, the last operational medium Dutch bomber \u2013 a Fokker T.V. \u2013 dropped two bombs on the bridge. In another forlorn scene of Dutch courage against the odds, one of the bombs hit a supporting pillar but failed to explode and the bomber was immediately shot down. Subsequent attempts to destroy the bridge by artillery fire from guns whose crews remained at their posts even when under intense Stuka attack, also foundered. To add insult to injury, last-ditch attempts to inundate the area also miscarried as the inlet sluices were not only too small but failed to open.\n\nThe German armour moved steadily north, to the very great relief of some airborne troops who were at the end of their supplies, and took Dortrecht. The Dutch continued to throw themselves at the corridor that the armour was creating but were kept under intense aerial bombardment and failed to achieve anything more than a temporary tactical impact. Lacking any useful information about the broader operational situation, the Dutch were forced to act in an unco-ordinated fashion which severely undermined their efforts. As one Dutch soldier has remarked, 'We were not always sure who was friend and who was foe. Often we had no idea who was on our right and left flank, all we knew is where the enemy was and that we had to attack. There was little co-ordination, but we did what we could \u2013 but it was not enough.' Yet sometimes the Dutch did not even know where the enemy was, and at one point the defenders mistook the red air recognition cloths strapped to the German tanks as orange Dutch flags and were mercilessly gunned down as they approached the vehicles.\n\nAs 9th Panzer Division pushed on to Barendrecht Bridge \u2013 the last before reaching the Rotterdam suburbs \u2013 the Dutch defence became increasingly desperate and more fractured. Even so, an initial attack by a platoon of German tanks saw the leading three fall to 47mm anti-tank fire, but such was the obvious strength of the units following on behind that the defenders abandoned the bridge. In Rotterdam, the Wilhelms Bridge was held by just fifty remaining airborne troops ensconced in nearby buildings and they were being extremely hard-pressed. An attempt by two companies of Dutch marines to oust the defenders and destroy the bridge lasted several hours. The armour arrived just in time to stop the perimeter being overrun. Peter Meier had been wounded in his left arm by a Dutch grenade when the tanks opened fire on the marines. 'I sank to my knees and thanked God,' he later recalled, 'I had twenty rounds left and had already unsheathed my combat knife in preparation for one last stand... It had been a real ordeal at that bridge.' But the bridge was now secured and an entry point had been made by 9th Panzer Division into the very heart of Fortress Holland.\n\nThe advance to Rotterdam was accompanied by further progress elsewhere in the country: in the far north 1st Cavalry Division had managed to work its way through to the mouth of the Ijsselmeer while along the rear of the Grebbe Line infantry supported by Stukas tried to fight its way through difficult inundated terrain against the Dutch rearguard. Some success was achieved east of Utrecht and the defenders fell back yet again to the outer defensive lines of Fortress Holland protecting The Hague. As one Dutch infantryman later testified:\n\nWhile we held our line, we always believed that there was hope. During the first three days of the invasion, we fought well even if we knew that without reinforcement we could not hold out indefinitely... We were subjected to heavy artillery bombardments, merciless infantry attacks and when we retreated back towards Utrecht on the 13th, murderous Stuka attacks. It was then that we began to lose hope. In fact, it flooded away.\n\nThe battle for the Netherlands was fast reaching its denouement by the afternoon of 13 May, but even as Winkelman considered whether a final defensive effort was in the best interests of the people of the Netherlands, a new phase in _Fall Gelb_ was opening on the Meuse south of Maastricht. Having created a very small bridgehead at Houx overnight, three rifle battalions from 5th Panzer Division crossed the weir in single file at 0530 hours to reinforce and expand it. By this time the French had woken up to the threat posed by the link that the Germans had found to the west bank and covered it with fire, causing heavy casualties. An after-action report stated: 'The dead are floating in the water. The wounded desperately clinging to the lock gates. They call for help but nobody can help them.' Commanders on the spot ordered their men to continue the attack and get to the opposite bank of the river as quickly as possible in order to deal with the defenders from the rear. Gradually, the defenders' fire diminished and the attacking infantry pushed on to take the village of Grange, which acted as an anchor for the bridgehead. From this firm base, the Germans pushed a further 2.5 miles, plunging deeper between the two French corps, and approached the village of Haut-le-Wastia, which stood on high ground. The French fought tenaciously throughout 13 May and into the next day to hold onto the tactically important position while their armour tried, and failed, to push the men of 5th Panzer Division back into the Meuse. With increasing numbers of attackers with heavy weapons filing into the bridgehead during the course of the day, it became increasingly secure, but the French had resisted well after a poor start and had not collapsed. The intensity of the fighting at Houx, however, undoubtedly advantaged Rommel's assault across the Meuse as he would otherwise have found the enemy concentrating purely against his division.\n\n7th Panzer Division had two crossing sites. The main one at Leffe on the northern edge of Dinant had good road communications on the west bank, while a secondary one south of Houx not only lacked adequate roads but was also overlooked by high, rocky cliffs which provided the defenders with a superb view. Even so, Rommel was determined that the enemy should not be able to concentrate all his fighting power against a single crossing site and so at dawn 6th Rifle Regiment and 7th Motorcycle Rifle Battalion began their crossing from the significantly inferior location. Fortuitously, the site was shrouded in mist as the first rubber boats started their 100-yard crossing but, noticing movement, the defenders raked the river with fire and were soon joined by artillery sited at a castle at Bouvignes. The Germans returned fire with two light artillery battalions and some spasmodic Luftwaffe support, but the assault teams knew that success depended on the infantry themselves clearing the defenders from their rocky eyries.\n\nRommel strode along the bank and later called the scene before him 'none too pleasant' and watched helpless as a damaged rubber boat drifted past with a badly wounded man clinging to it, screaming for help and near to drowning. He did what he could, however, to provide encouragement and leadership to his troops in their endeavours. As the mist lifted, he immediately ordered that the house on the east bank be set on fire to provide smoke cover and so the assault continued. Rommel left in a PzKpfw Mk IV to visit his primary crossing point just as soon as the infantry had a critical mass of troops on the west bank and were beginning to infiltrate forward.\n\nAt Leffe, Rommel arrived to find the crossing being undertaken by Oberst Georg von Bismarck's 7th Rifle Regiment under heavy fire. As the general observed the scene, his aide-de-camp, Hauptmann Hans-Joachim Schraepler, was wounded at his side by shell splinters. Rommel was unharmed, but the incident was a reminder of the cost that had to be borne if commanders were to conduct their business in the front line. Yet Rommel, like many others, believed that to be able to make swift and informed decisions an officer had to be at the point of the action, and that the alternatives were unthinkable. In fact, as if wishing to provide an immediate illustration of his indispensability in the line of fire, he took personal command of 2nd Battalion for a short time. With the infantry having managed to get a company onto the west bank but the troops following faltering, the general screamed orders in an attempt to kick-start the reinforcement. He even crossed the river himself in one of the first boats, partly to be able to give direction to the assault that was developing on the other side of the Meuse, but also to show that he would not ask his men to do something that he himself was not prepared to do.\n\nFew who were at Leffe on 13 May would disagree that Rommel did not play an important personal role in the establishment of 7th Rifle Regiment's bridgehead that morning. For the remainder of the day, he remained on the eastern river bank \u2013 largely at the crossing point south of Houx where enemy artillery fire continued its disruption \u2013 encouraging his troops. On one occasion he waded waist-deep into the river to assist engineers building a bridge and, when the bridge was completed, Rommel was one of the first across in his command tank.\n\nBy the evening, 7th Panzer Division had established two bridgeheads which were now being connected and also linked to that achieved at Houx. It was an extremely impressive achievement as any opposed crossing, particularly one made on the move, was a high-risk enterprise. The fact that the French were poorly set counterbalanced some of those challenges but in lesser hands the assault could easily have stalled. A significant counter-attack before the Germans were well established across the river could have had a momentous impact, but the French were not only slow to react to the initial crossing at Houx, they were also tardy in informing the chain of command about their deteriorating situation while they established exactly what was happening where and to whom.\n\nAt noon the IX Corps commander, General Julien-Fran\u00e7ois-Ren\u00e9 Martin, was finally in a position to order a counter-attack consisting of an infantry regiment, a company of tanks, some scout cars and four artillery battalions. As the infantry was slow to arrive, the armour attacked 5th Panzer Division without them. They made good progress and advanced to within a mile of the crossing point before a precipitate decision was made to suspend the thrust when darkness fell due to the tanks' vulnerability without infantry support. A languid counter-attack towards Dinant by II Corps also ended in failure after slow reaction times were, once again, compounded by sluggish assembly and a timely Stuka raid. Although the intention was for the enterprise to be restarted the following morning, the Germans were careful to ferry ever more troops, armour and heavy weapons across the Meuse during the night via their newly constructed bridges and so by dawn the French had missed their opening.\n\nThis meant that they had lost two very important opportunities to strike back at the Germans while they were in an unusually vulnerable situation. These were to prove to be just the first of several similar episodes, all stemming from a chronic French inability to react in a timely fashion to German initiatives. To make matters worse, the situation at Dinant and Houx was not properly communicated up the Allied chain of command. With each successive recipient of such information wary of the tendency of subordinates to over-exaggerate situations, Georges eventually told Gamelin that 'a battalion had got into trouble' near Houx. In reality, for the cost of 182 casualties, Rommel had a firm footing on the west bank of the Meuse and, along with 5th Panzer Division, provided Hoth with an excellent springboard from which to launch operations to the west.\n\nThe breaching of the Meuse line on the right flank of Army Group A was just one of three that were planned. Another was to be attempted forty-five miles south of Dinant by Reinhardt's second echelon corps which, because of the difficulties that its formations had getting to the Meuse, was represented on the river in the early hours of 13 May by Kempf's 6th Panzer Division alone. By dawn it had been decided that the assault crossing to Montherm\u00e9 would take place at 1600 hours, the same time as Guderian's attack at Sedan, thus giving the formation the time that it needed to make its preparations. It was a tricky proposition, for the Meuse in this area is set in a deep gorge and Montherm\u00e9 occupies a tight loop in the river, one which made it easy for the enemy to be cut off, and it was defended by a well-organized colonial infantry battalion from 102nd Fortress Division. Reinhardt later wrote:\n\nThe terrain really scares you. The only road leading to Montherm\u00e9 snakes down... into the deeply cut Meuse valley... The riflemen are having trouble climbing down the steep slopes leading down to the Meuse river while manhandling their heavy weapons. The rubber boats have to be taken down to the Meuse in the armoured personnel carriers because it was impossible to carry them down.\n\nKempf would ideally have liked the Luftwaffe to provide some close air support and interdiction to assist his division, but he had been told little was available as Guderian's divisions had priority. When during the course of the battle some Stukas did turn up, the division was horrified to watch as the dive-bombers mistakenly attacked their own troops, leading to the destruction of several guns and thirty casualties. For the most part, however, Kempf was forced to rely on the division's own guns both for the preparation of the battlefield and also in support of its assault forces. Some of those guns belonged to the eighteen tanks of 1st Company 11th Panzer Regiment, which was commanded by Oberleutnant Dr Franz Bake and covered the assault crossing of 4th Rifle Regiment.\n\nThe first two attempts to get the infantry across the Meuse ended up in a bloody mess, but the third crossing was a success after a key French machine-gun position in a caf\u00e9 was spotted and destroyed by accurate tank fire. With infantry on the west bank and moving into the town, tanks immediately began to cross at a weir that had been spotted a little earlier by an eagle-eyed reconnaissance patrol. By placing himself in one of the first groups of vehicles that crossed the Meuse \u2013 one of which moved slightly off the narrow route and sank up to its turret \u2013 Kempf was able to organize his units quickly and give orders to his command team on the east bank to release specific assets to aid the advance. Engineers, meanwhile, set about making one of the poorly demolished bridges useable again. The clearing of the town was an infantry task, but the tanks were on hand to offer firepower where it was needed to remove stubborn positions. Meanwhile, the French sealed off the town by occupying a pre-prepared defensive line running across the neck of the urban salient which had been occupied by the Germans. What was needed now was the arrival on the Meuse of one of the corps' other divisions in order to dilute the French concentration against 6th Panzer Division and provide a threat to their rear area.\n\nIt was not without reason that Reinhardt's post-war account of his campaign in France was titled _In Guderian's Shadow_ for in the Sedan sector, even as Montherm\u00e9 was being attacked, Guderian's three divisions were crossing the Meuse at the _Schwerpunkt_ centred on Sedan. Von Kleist had wanted a crossing several miles further west, in the broad Sedan sector but where there were fewer river obstacles, to make progress beyond the Meuse while pushing the corps closer to Hoth and facilitating an attack on the boundary between the French Second and Ninth Armies. Guderian, however, ignored direct orders and presented his superior with a _fait accompli_ by heading for and finally arriving in Sedan itself. Guderian's argument for his choice was that it offered 1st Panzer Division more cover for its assault crossing than the alternative further west, but thinking ahead he also recognized that ten miles south of Sedan was some high ground \u2013 Mont Dieu \u2013 which, if captured, would make an ideal hard protective edge to his southern flank as his corps swung west.\n\nIronically, having told von Kleist on 12 May that, if he wanted his corps to reorientate to the west, he would have to postpone his crossing by twenty-four hours, on the morning of 13 May Guderian reflected on the fact that a delay might have been useful to him as his artillery was emerging from the woods at a worryingly slow rate. The 141 guns contained within his three divisions were a vital component in the corps plan for the break-in and break-through battle as he expected the French to enjoy a 3:1 gun superiority. As such, his scheme demanded that 1st Panzer Division take the majority of the heavy guns from 2nd and 10th Panzer Divisions as they arrived in order to concentrate them and so enhance the likelihood of a break-in by Kirchner. In preparation for this, Guderian had put a huge effort into developing a precise sequence of firing on a detailed target map which aerial reconnaissance would keep updated right through to the end of the day before the assault. That fire plan was then to be applied in co-ordination with the Luftwaffe for the attack.\n\nThe level of air-ground co-operation that Guderian sought was unprecedented, but it had been trialled by him in Poland and he had learned a good many lessons doing so. Consequently in early May, Guderian and Generalleutnant Bruno Loerzer, commander of II. Fliegerkorps, had developed the 'rolling raid' in which only a few Luftwaffe formations would attack at any one time, but small raids were repeated throughout the day in order to permanently neutralize the enemy artillery, target bunkers and create such chaos as might break the will of the defenders. Guderian was therefore furious to learn at his meeting with von Kleist on 12 May that his superior had intervened and ordered the commander of Luftflotte 3, General der Flieger Hugo Sperrle, to change his plans. His orders were to provide a long preparatory raid to prepare the battlefield and then a conventional brief and concentrated bombing attack immediately before the infantry crossed the river to force the enemy deep into his defences. It was a fundamental alteration to Guderian's carefully considered arrangement and matters were further jeopardized by the fact that, just hours before his assault was due to be launched, his guns were still struggling to get into position.\n\nGuderian could only hope that the guns which did arrive, when combined with von Kleist's new air plan, would be enough to allow the 1st Panzer Division, reinforced with Gross Deutschland Regiment, to crack the French defences open and make a breakthrough by seizing high ground immediately south of the river and, with 2nd and 10th Divisions on its flanks, push to a line running between Ch\u00e9h\u00e9ry and Chaumont, then punch through the final French defences and pivot west with the corps's left on Mont Dieu. The first but critical step was for the divisions to get across the river and break into the enemy's defences. After some superb traffic control, the guns of 1st and 10th Panzer Divisions arrived in enough time to be positioned and sighted, although those of 2nd Panzer Division were still arriving as the attack began. In the final hours before 1600 hours, plans were briefed, unit preparations were completed and troops checked their equipment. The French shelling remained relatively light, but the ground running down to the river became increasingly full of burning vehicles, nearby buildings were set on fire too, and casualties were taken by troop concentrations close to the river in both Sedan and Donchery. It made for an anxious wait until the attack began.\n\nThe French too were apprehensive at dawn on 13 May, although they had no idea that the Germans intended to attempt an assault crossing that day. The defenders of the Sedan sector needed every minute available to prepare because they were lacking both manpower and weaponry. With 147th Fortress Infantry Regiment and 11th Machine Gun Battalion included, 55th Infantry Division should have been 16,712 men strong but had lost nearly 25 per cent of its strength through Pentecostal holiday leave and only a small number had returned when Sedan was threatened. The division was further hampered by its unfinished defences, lack of anti-tank guns and mines and also a gap in its defences across a 2-mile stretch of river opposite Gaulier at Glaire. Thus, at the most obvious crossing place at Sedan \u2013 the place where 1st and 2nd Panzer Divisions looked to attack the west bank of the Meuse \u2013 there was no interlocking fire and there were no mines following the decision to move the principal line of resistance back in this sub-sector in order to shorten the line overall. The commander at Sedan, General Lafontaine, believed that he had several days in which to enhance his defences and welcomed the news that the corps reserve, 71st Infantry Division, would arrive on his right flank during the night of 13\u201314 May and so reduce his area of front by one third. His pleasure at this increase of manpower, despite the thoroughly disruptive impact the arrival of this formation would have after months of careful organization in the area, gives a good indication of Lafontaine's concerns.\n\nThe German commanders had all put themselves in the place of their French counterparts. Their understanding of their enemy's doctrine, his army and circumstances, led them to conclude that the French would be slow to react to the nascent threat emerging from the Ardennes and that an immediate attempt to assault the Meuse, despite its inherent risks, was essential if their own vision was to be fulfilled. Halder had therefore watched developments with great interest and knew that the campaign had reached an important and delicate point on 13 May. Having read the morning reports, he noted: 'Group Kleist bunched together near Sedan, intends attacking at 1600. Plan of attack rather complicated... Outcome of Meuse drive will decide if, and when, and where we would be able to take advantage of [our] superiority. The enemy has no substantial mobile forces in the rear of this front, no more than three armoured divisions at best, but he does have a very well integrated railroad system.'\n\nGuderian was under huge pressure, but if there was a German general who was suited to excelling under such a burden, it was the commander of XIX Corps at Sedan. Even as Halder was writing, on a clear, warm and sunny day, the Luftwaffe suddenly vanished over Army Group B and concentrated almost totally against the Sedan sector. With the guns of the three divisions still arriving and being positioned, the air attack began. It was a devastating surprise to the French, who not only learned during its course that the Germans were about to make an early assault across the Meuse but also found themselves under the largest concentration of aircraft so far mustered in the war. In total around 1,500 machines, largely from Luftflotte 3, supported Panzer Group Kleist, with the vast majority of them at Sedan. The German aerial force included 600 bombers, 250 dive-bombers, 500 single-engine and 120 twin-engine fighters, most of which flew several sorties on 13 May. The impact of the Luftwaffe's attack was better than Guderian could have imagined for, as he watched the opening raids from Hill 266 south of Givonne that afternoon, he realized that it was his plan that was being put into action after all. Loerzer, Guderian found out when he called to thank him later that day, had decided that von Kleist's order had reached him from Luftflotte 3 'too late' not to have caused confusion and so stuck with the original plan. Good fortune, not for the first or last time in the campaign, was on Guderian's side.\n\nThe rolling raid that Guderian and Loerzer had designed for Sedan focused on zones which were sub-divided into areas of between 275 and 550 yards and contained specific targets along the Meuse and then the defences immediately behind them \u2013 the depth of 55th Infantry Division. It came in five phases. During the morning, there were harassing missions carried out along the Sedan sector front, then from noon dive-bombing and the bombing of specific targets until twenty minutes before the river assault, when there was a huge raid against the loop in the Meuse at Sedan. At 1600 hours and for ninety minutes after, the raids moved away from the river bank to deeper targets, with Stukas hitting rear-area positions and the artillery, and then, finally, from 1730 hours until dusk, there were interdiction operations to seal off the rear area by targeting reserves approaching the battlefield. The day was to see 1,215 bomber and dive-bomber sorties flown against the area to be attacked by 1st Panzer Division alone.\n\nLieutenant Colonel Pinaud, commanding 147th Fortress Infantry in the main attack area of the Fr\u00e9nois sub-sector, later testified:\n\nThe bombardment was particularly heavy on the principal line of resistance, Sedan railway station, and on Torcy where numerous fires were started... [At noon] the aerial bombardment intensified and was interspersed with lulls until [1800 hours]... [T]he entire area, but particularly the principal line of resistance, was shrouded by a thick cloud of smoke. The attacks were implemented by successive waves, each including around forty bombers while fighters strafed the ground.\n\nThe deployment of wave after wave of steeply diving and screaming Stukas against second-rate troops so affected a significant portion of the men cowering under the onslaught that they were rendered militarily useless. Only the brave and foolhardy and the psychologically unhinged discharged their rifles and machine guns at the aircraft. There was just a single, unsuccessful attempt by seven French aircraft to become involved in the battle and only one light anti-aircraft battalion at Sedan. One of its gunners, Hugo Novak, later said: 'All hell seems to have broken loose... The enormous air pressure causes glass panes to rattle and crack.' It was during the second and third phases of the Luftwaffe blitz that the bombing increasingly focused on 1st Panzer Division's area of main effort and the defenders' will and ability to resist really began to erode. Deafened, winded and disorientated by the explosions, the French struggled to know what to do among the devastation. Some bombs created craters forty-five feet across and twenty-one feet deep.\n\nDavid Boyer of 55th Infantry Division found himself in the middle of the maelstrom:\n\nThe air attack was far worse than my imagination could have ever have invented. After a night of worrying about what might happen, I found that I could not escape the bombs. I was in a reinforced trench linked to a bunker and felt as unprotected from the bombs as any man that day. I remember a bell being rung when the first aircraft approached and we were supposed to take cover. This meant cowering in our trench and praying. The first attack lasted around five minutes, and I thought that it was all over, but moments later another raid began, and then another and another until they merged into one nightmare lasting hours. One bomb landed by my trench and collapsed a section of it burying my sergeant... The moments of silence were as frightening as the explosions as I could hear screams of pain, but no movement or sense that we were doing something to defend ourselves. We were defenceless... In the minutes before the [German] attack, the explosions were so frequent that one could not distinguish between the individual blasts. I was deaf and could hardly breathe. I tried to make my body as small as possible, my hands over my head, my world shaking and swaying. I sobbed.\n\nThe Luftwaffe's attack was so long and so intense that those who might well have been able to withstand a raid of conventional duration eventually succumbed to its power. One staff officer testified: 'When the dive-bombers came down on them they stood the noise \u2013 there were hardly any casualties \u2013 for only two hours, and then they bolted out with their hands over their ears.' In the final minutes before the river assault, the aerial bombardment rose to a crescendo. Watching the scene from the east bank, Unteroffizier Pr\u00fcmer of 3rd Battalion 1st Rifle Regiment later reported:\n\nWhat we now got to see during the next twenty minutes is one of the most tremendous impressions of this war. Squadron after squadron approaches at high altitude, deploys into a line formation and then the first aircraft comes diving down perpendicularly, followed by the second, the third, and so on, ten, twelve aircraft pouncing simultaneously like predatory birds on their quarry, releasing their bomb load right over the target... The explosion is immense each time and the noise is deafening. Everything is blurred... The enemy is hit here by an enormous annihilation strike and still more squadrons keep coming on, climbing to high altitudes in order to dive steeply down with the same objective, that is, to bust the Sedan invasion gateway wide open. We stand there and watch events as if transfixed.\n\nDavid Boyer remained physically uninjured, even through the final twenty minutes, but when the raids suddenly moved on to targets behind the front line, he had been stripped of his ability to act as a soldier, later recalling: 'When it ended, it took me several minutes to stand. My world was silent and my view obscured by smoke which, as it cleared, looked like I had been transported to 1916 Verdun \u2013 and then they came at us. I raised my hands hoping more to be shot than taken prisoner.'*\n\nIn common with the great German 'hurricane' artillery preparations of 1918, the greatest impact of the Luftwaffe's attack was psychological rather than physical. Indeed, only fifty-six men were listed as casualties during the raid, very few bunkers were destroyed and few guns were put out of action. As Guderian and Loerzer knew, German aircraft were not capable of the sort of reliable precision bombing required to consistently destroy small targets but their plan recognized the impact of intense and unrelenting bombing on a soldier's fighting effectiveness. Nevertheless, the air attacks, gradually intensifying artillery bombardment and then tank fire from across the river in the twenty minutes before the river assault did enough physical damage to severely undermine French co-ordination and defensive capabilities. Several bunkers and machine-gun positions were destroyed before the first German rubber boats touched the water and the field telephone lines of many 55th Infantry Division units were cut, which was of great advantage to the attackers.\n\n1st Panzer Division's 1st Rifle Regiment was to attack with two battalions up-front, crossing the Meuse and landing at Glaire. On its left, Gross Deutschland Regiment with one battalion up-front would attack to Torchy. The regiments' tasks were to create a wedge into the French defences and establish a bridgehead. Supporting Balck's Rifle Regiment was a battalion of engineers which not only controlled their movement across the river in forty-four large and sixty-six small rubber assault boats, but also provided specialists in bunker destruction. They had worked together on numerous exercises, during which time they had practised crossing similar rivers against a similar defence on many occasions. Thus, when at 1600 hours the engineers crewing 1st Rifle Regiment's boats had failed to arrive, it took just ten minutes for Balck take stock of the situation and order the infantry to do the job themselves.\n\nAs the first wave of boats started their short journey across the Meuse \u2013 supported by fire from their tanks, anti-tank guns and assault guns \u2013 they came under fire from those French positions that had managed to recover from the mauling meted out by German bombs and shells. Machine-gun rounds splashed into the water around the vulnerable attackers and some hit their mark, sinking the light craft and making casualties of many on board. But the attackers had inadvertently found what might be called the Glaire Gap, and they could not be stopped. Encouraged by their junior leaders, the German infantry and engineers moved quickly on French positions and began silencing them with a mixture of small arms fire, smoke bombs, hand grenades and flamethrowers. Some positions fought to the death and some fought until their fate was sealed, but many saw the Germans coming towards them and fled to the rear. The strong-point at Les Forges, for example, fought up to the moment it was overwhelmed before its commander, Lieutenant Lamay, ordered a withdrawal 650 yards back to Bellevue and occupied a bunker left empty by its fleeing occupants.\n\nThe initial attack around Glaire had made a successful lodgement on the west bank of the Meuse by around 1700 hours and the three battalions involved immediately fought to develop it further. 1st Rifle Regiment was rallied by Balck, who was soon joined by Guderian, who had crossed with the second wave. The general's presence was noted by a captured French junior officer, Second Lieutenant Verron, whose forward bunker had been taken by surprise after hand grenades had rattled down the air vents and were followed by an infantry attack. Fearing the enemy's next move, Verron surrendered and was taken by a sergeant to a high-ranking German officer \u2013 probably Guderian \u2013 who was crouching over a map and talking into a radio. The general scanned Verron from head to foot and barked out an order before motioning for the confused prisoner to be led away.\n\nThe Germans pushed on ruthlessly, seeking to maintain their tempo, the initiative and advantage they had gained by their preparation of the battlefield. The 1st Panzer Division journal says: 'One bunker after the other, anti-tank guns, machine guns, and field fortifications are taken in individual combat and through the personal example of all the leaders who advance in front of their men.' One such leader was Oberleutnant G\u00fcnther Korthal, who commanded the assault platoon of 3rd Company of the engineer battalion. Failing to find either his company commander, battalion commander or Balck, he linked up with another platoon and led the group forward, fighting tenaciously throughout the day as he worked methodically through the complex defensive system of the principal line of resistance. Other units did the same. Although charging at the enemy had its place at certain times and in certain places, a degree of finesse was required to overcome the French defences, one that necessitated close, quick verbal orders, equally close co-ordination and mutual support, all of which had been developed to a pitch of excellence over months of hard training. It was not until 1830 hours, therefore, that Korthal destroyed two bunkers at the Bellevue intersection a mere one and a half miles from their crossing point at the loop in the river, and turned west along the west bank of the Meuse.\n\nThe Germans now began to engage the French positions on high ground that confronted the crossing of 2nd Panzer Division at Donchery and dominated the flat, coverless launch point for Veiel's units. This was the place where Lafontaine expected any attempt to cross the river to take place, and so the defence here was particularly strong and included artillery support from the neighbouring 102nd Infantry Division. 2nd Panzer Division had started its assault at 1730 hours due to the late arrival of their units, but had faltered in the face of French firepower in the hours since. Without the benefit of an intense air attack on its front and deprived of its heavy artillery \u2013 the division was left with just twenty-four light guns which arrived an hour after the assault began \u2013 the formation encountered more opposition opposite Donchery than had been offered south of Sedan. Yet the advance of Korthal's group so undermined the French defences by silencing several bunkers and casemates that the division was still able to cross the Meuse on 13 May. Indeed, by 2030 hours, the first of Veiel's infantry and engineers clambered onto the west bank, thus creating a vital lodgement that was rapidly reinforced after dark, and began pushing forward at around midnight.\n\nAs Korthal went about his vital work, Gross Deutschland Regiment attempted to fulfil its role as the eastern part of 1st Panzer Division's wedge into the French defences. With Balck's men on their right flank, the men from Gross Deutschland sought to make progress into and around the extension of Sedan on the west bank of the Meuse known as Torchy. After traffic jams had forced the regiment to march the final six miles on foot under French artillery fire (which killed one of the battalion commanders), its leading battalion was then ordered to begin its assault with very little time to organize itself. Even so, the first companies grabbed their boats, only to be prevented from putting them into the water by a wall of French fire. Despite the supporting fire of 150mm cannon from the Heavy Weapons Company, it was not until the powerful and extremely accurate 88mm anti-aircraft guns were brought forward that two critical French bunkers were destroyed by shells entering their embrasures and the infantry found it possible to start across the river. Some managed to get within twenty yards of the west bank, but a cleverly placed machine gun fired into the flank of the assault and thwarted any landings.\n\nSeeing the problem, the engineer in charge of the crossings on the east bank sent as many boats across the river as he could in a single wave and this ploy allowed some men from 2nd Battalion's 7th Company to land and then silence the machine-gun position. They made their way forward and the strongpoints at Pont Neuf and Cimeti\u00e8re fell before the attack stalled again in front of the western edge of Torchy as fire from the area ripped into the German left flank. 3rd Battalion was therefore detailed to cross the Meuse and conduct a building-clearance operation that was to last for much of the rest of the day while 2nd Battalion endeavoured to move south. By 1800 hours, 6th and 7th Companies were supporting each other as they took on the French principal line of resistance.\n\nIt was most often the courage, aggression and tactical awareness of very small groups, however, that contributed to the success of the final attack. At Bunker 104, for example, a diversionary frontal assault fixed the attention of its two machine guns, while another attack outflanked the position by moving quickly through the cover provided by an orchard and several bomb craters. The final approach was made by two men who moved carefully towards the bunker to create a second, closer diversion, while a third man ran forward to drop a grenade into one of the bunker's six ports. After the ensuing explosion, the occupants ran out and surrendered, with several yelling, 'Shoot!' The attackers refused and when the French were asked by an officer why they were asking to be shot, he was informed that they had expected to be tortured before being killed.\n\nBunker 104 was found to have fired 10,000 rounds before falling to Gross Deutschland Regiment at 1830 hours. It was a linchpin in the strongest French line in the sector and other bunkers began to fall once it had been captured. This was not achieved without considerable German casualties being taken, including those sustained from a French anti-tank gun in a bunker that was disguised as a barn. Gradually, however, the line began to lose its cohesion as gaps were created in the defenders' interlinked arcs of fire, which allowed for infiltration. Such successes further undermined the defenders' ability to co-ordinate their defence as the Germans sought to make each bunker deaf, dumb and blind. Indeed, Captain Carribou, who had responsibility for the defences south of Torchy from his headquarters on the northern edge of the Bois de la Marf\u00e9e, found his sub-sector in a state of collapse at around 2000 hours. Having personally spotted the fact that his left flank had become exposed as the advance of Gross Deutschland Regiment had caused the infantry to flee, he was then informed that the Germans were also breaking through on his right by Second Lieutenant Loritte, who had been captured by the Germans but managed to escape despite being badly wounded. Carribou immediately ordered that all official documents be destroyed and, while repositioning his headquarters to the rear, ordered Loritte and the other troops that had converged on the area to join the defence of this wooded area under Captain Vitte. These troops were soon involved in the defence of Hill 247, which overlooked the woods, but could not stop it falling and were forced into a fighting withdrawal and also swept up Carribou's headquarters. They found the defences there incomplete and that the men in place lacked information about what was happening. A firm position was not established until the village of Chaumont had been reached two miles to the south of Hill 247 at dawn on 14 May.\n\nAfter a difficult start, Gross Deutschland Regiment had managed to make an important breakthrough by the end of 13 May, but its ability to achieve this owed much to 10th Panzer Division's attack towards Wadelincourt. It was this advance by Schaal's men that had undone Carribou's right flank and then helped to put Hill 247 in a pincer. In common with its sister divisions, however, 10th Panzer's crossing of the Meuse had not been trouble-free. Just getting their boats onto the Meuse at their two crossing points had been tortuous because there was 800 yards of open ground leading to the river and the assaulting forces suffered heavy casualties as they moved towards the water. 86th Rifle Regiment attacked close to the eastern edge of Sedan while 69th Rifle Regiment tried to make a crossing two miles along the river towards Pont Maugis. Indeed, a huge barrage hit the engineers just as they were unloading their equipment from trucks for the latter assault and destroyed eighty-one of the regiment's ninety-six rubber boats. Like 2nd Panzer, Schaal's division was not as well served in either air or artillery support as 1st Panzer Division and this had important consequences for a formation which also drew fire from both 71st Infantry Division as it moved into position on Lafontaine's right and the guns of X Corps. Having suffered such a loss of assault craft and attracting such a fusillade of French shells, 69th Rifle Regiment abandoned its attempts to cross the Meuse at its designated point at 1700 hours and joined its sister regiment closer to Sedan.\n\nThe initial crossing of the Meuse by 86th Rifle Regiment was conducted under intense machine-gun fire and failed everywhere except for a team of five assault engineers and six infantrymen led by Walther Rubarth. An engineer like Korthal, Feldwebel Rubarth had a vital tactical role to play on a shoulder of the main effort that was to have operational significance. Having put a small bunker out of action soon after landing on the French side of the river, the team cut through some barbed-wire entanglements and seized the next bunker from the rear. Rubarth later wrote: 'I use an explosive charge. In a moment, the force of the detonation tears off the rear part of the bunker. We use the opportunity and attack the occupants with hand grenades.' As a white flag appeared, cheering was heard from the beleaguered troops waiting to cross from the opposite bank and the group pushed on to silence another three bunkers before running into the principal line of resistance. There was now a 300-yard gap for the infantry following on behind, and these men duly moved forward and attacked. Rubarth's team joined a company of infantry and he first stormed a machine-gun position and then, dropping over a railway embankment covered in wire, his team attacked a bunker from both sides. Rubarth later recalled: 'One Frenchman, who had left the bunker and aimed [his weapon] directly at me, was rendered harmless with a hand grenade.' Having blasted the bunker doors open and forced the surrender of its occupants, the group moved on to take another two bunkers and in so doing cracked open the main line of defences.\n\nThe dismantling of the French defences allowed the 10th Panzer Division infantry to lunge forward into Wadelincourt and forced the defenders commanded by Captain Leflon to withdraw to a pre-planned stopping line which he found empty. These French troops continued to move south and only stopped when they linked up with Carribou's in Chaumont. While 86th Regiment took their objective, Hill 246 just to the south of Wadelincourt, which fell to them at 2100 hours, Rubarth and his team had been working their way down the principal line of resistance towards Fr\u00e9nois. The last bunker he secured was No. 8 _ter,_ halfway towards the village and well into the 1st Panzer Division's area of operations. With this, the company of 86th Rifle Regiment following Rubarth's team stormed up Hill 247 to assist Gross Deutschland Regiment in taking the position and the French principal line of resistance fell. For his excellent work on that day, Rubarth was to receive a commission and he also received the Knighthood of the Iron Cross.*\n\nBalck's 1st Rifle Regiment were also the grateful recipients of good work on their flanks, for without it they might well have stalled at Fr\u00e9nois. Korthal's destruction of the two bunkers at the Bellevue intersection at around 1830 hours opened a vital area of the front, allowing Fr\u00e9nois to be by-passed by the forward companies. The village was eventually taken at around 2010 hours by follow-on troops after the French fire support provided from atop Hill 247 was ended by the Gross Deutschland Regiment and 86th Rifle Regiment's pincer on the position. The battalion commander of the 331st Infantry Regiment defending Fr\u00e9nois, Lieutenant Colonel Foucault, later complained that he had no idea what was happening because his communication had been cut during the air raids. This situation had been replicated across the platoons, companies, battalions and regiments of the division while Lafontaine's communications with Grandsard had also been degraded by the Luftwaffe. Throughout the day, 55th Infantry Division units felt isolated and lacked information, which encouraged poor decision-making, rumour and fear. As a result, there were both unauthorized local withdrawals and swift capitulations. By the evening, the cracks that had been created in the French line had been expanded to create potential for Guderian's corps not only to secure a significant bridgehead by consolidating their gains, but also to follow up on their success and deepen their lodgement. Balck's regiment had an important role in this in the centre of the corps and he intended to capture La Boulette (Hill 301) \u2013 the site of von Moltke's 1870 command post \u2013 and the Bois de la Marf\u00e9e (Hill 336) to complete his breakthrough.\n\nBy 2100 hours and with dusk approaching, the exertions of the day \u2013 and the previous three or more days before that \u2013 began to affect all three German divisions. It was a natural reaction to the success that had been achieved and had partly been fuelled by the adrenaline of such a challenge. Yet while the troops may well have been encouraged to learn that the enemy was in withdrawal \u2013 one or two units were even beginning to dig in for the night and eat some food \u2013 commanders recognized that the battle was at a critical stage. Overnight it was essential not only for the divisions to construct the bridges that would allow tanks, vehicles and heavy weapons to cross the Meuse, but there also needed to be the space on the west bank to deploy them. Commanders also recognized the need to exploit the French withdrawal. The enemy had to be put under unremitting pressure and prevented from establishing strong defences, and the front line had to be pushed as far away from the river as possible prior to the inevitable French counterattack. As a result, German officers and NCOs had to try and motivate their men to get to their feet and continue their attack in a southerly direction in the gathering gloom. In 1st Rifle Regiment, when the omnipresent Balck was informed by subordinates that their men were too exhausted to move, he replied: 'Fine. Whoever wants to stay here can stay here. I'm leading the attack on the next village.' With that, every man got to his feet and Balck led them forward towards Hill 301.\n\nThe battle that followed has been described by Captain Litalien, the 331st Infantry Regiment company commander defending the high ground:\n\nThe [enemy's] fire progressively increased in intensity, and we had great difficulty containing the German advance [which was] very superior in numbers, and which finally progressed even more rapidly despite the unleashing of final protective fire by every automatic weapon. At [2230 hours] the Germans were thirty metres from the centre of resistance... [and] completely overran the left and began to move on the right...\n\nLitalien's withdrawal was immediately followed up by men of 1st Rifle Regiment's 2nd Battalion, who took forty-one prisoners before signalling to division headquarters:\n\nAt 2240, 1st Rifle Regiment took commanding heights just north of Cheveuges. Last enemy bunker in our hands. Complete breakthrough. Elements 1st Rifle Regiment sent towards Ch\u00e9h\u00e9ry and high ground to the east of the village.\n\nThe onrushing infantry subsequently developed a salient approximately one mile wide to the south and had reached the outskirts of Ch\u00e9h\u00e9ry by midnight. Soon after Hill 301 had fallen, a company-strong unit defending Hill 336 just to the south-east feared being outflanked and had withdrawn to a position between Ch\u00e9h\u00e9ry and Bulson by dawn but lost sixty men in doing so. By this time, 2nd Panzer Division's 2nd Rifle Regiment had grafted itself onto the right flank of 1st Panzer Division and advanced to take much of the high ground overlooking Donchery and in so doing created a narrow shoulder on Guderian's right. 10th Panzer Division created something similar on the German left flank by taking Hills 246, 247 and the village of Wadelincourt. It was a move that left Guderian's corps in possession of all the high ground immediately south of Sedan. Kirchner's satisfaction at the day can be seen in the formation's log that night which noted that it was the 'best day' so far for 1st Panzer Division which:\n\n[S]ucceeded in being the first to cross the Meuse, to punch a hole in the defensive line of the French and to tear through a defensive zone believed by the French to be impenetrable... The importance of this penetration had been pounded into every man from the moment the division was attached to the XIXth Army Corps in Cochem.\n\nThe achievement of Guderian's corps which allowed 1st Panzer Division to fight through 5.5 miles of French defences in just eight hours should not be underestimated. It was a great vindication of German preparation, planning, training, doctrine, leadership and fighting spirit. The disaster that had befallen X Corps and 55th Division was equally immense and revealed French weaknesses in all the areas where their enemy had shown strength. The differences between the two sides can be seen in the actions of Kirchner and Lafontaine during the afternoon and evening of 13 May. While the commander of 1st Panzer Division was a model general officer going about his business of command and control, organizing and co-ordinating the various elements of his division through radio communication and, where necessary, personal visits, the commander of 55th Infantry Division was completely out of touch with rapidly developing events. Shielded in his command post at Fond Dagot five miles from Fr\u00e9nois, its communication with his regimental commanders fractured, his own knowledge of the unfolding battle woefully incomplete and his ability to command, control and assert his will on the battle fatally weakened, Lafontaine was impotent and virtually useless. In a report he wrote soon after the battle, he explained that although he had a broad understanding that the enemy had managed to 'infiltrate rapidly, bypassing and taking from the rear our strongpoints', he had also heard 'alarmist information coming from the front and rear' while 'orders from unqualified cadres or coming from an unknown source' led to some infantry units making disorganized withdrawals, some artillery batteries putting their guns out of service and some columns withdrawing 'far to the rear, creating bottlenecks on the roads and a feeling of panic in the rear'.\n\nThe 'panic' to which Lafontaine refers had been a very real phenomenon during the evening of 13 May at the time the principal line of resistance was broken. From around 1900 hours, when some of the French troops that had been resisting the enemy's onslaught began a semi-orderly withdrawal, others fled the battlefield and caused great alarm from Bulson southwards with their overblown stories of battle and warnings of imminent annihilation. An officer was despatched from Fond Dagot to Bulson to ascertain what was happening further forward and returned to tell Lafontaine that the division had broken and that troops \u2013 on foot, in vehicles and many without weapons \u2013 were in retreat. They included gunners from north of Chaumont who had been swept along in a tide of southbound troops, their will already undermined by shells that had exploded near their positions and had been incorrectly identified as having been fired by tanks. Both the headquarters of Lieutenant Colonel Dourzal, commander of a group of X Corp's heavy artillery at Bulson, and Colonel Baudet, commander of 55th Division artillery just to the south, believed the rumours that panzers were close by and, rather than attempting to verify the information, withdrew and so lent credence to it. The panic manifested itself to the division commander and his staff at Fond Dagot and left them King Canute-like as they attempted to stop the inundation. Lafontaine himself stood, revolver in hand, before two trucks blocking the road down which the troops were withdrawing. His men tried not to meet his eye or those of other officers who stood shouting at the hundreds of troops to stop and make a stand. Very few, if any, obeyed and most dissolved into the darkness, with some eventually being spotted sixty miles to the rear.\n\nWith no knowledge of any troops remaining between him and 1st Panzer Division, Lafontaine ordered his headquarters to withdraw to Ch\u00e9mery and its documents to be destroyed, leaving him incommunicado and without any control over his units at 2000 hours when Hill 247 fell. The impact of 55th Division's reverses was great and led to the recently arrived 71st Division withdrawing for fear that its left flank would become exposed along with Colonel Poncelet's headquarters of X Corps' heavy artillery. It was an error of judgement that Poncelet immediately regretted and, shamed by his part in the disintegration of the French defences, he took his own life on 24 May. It was a mass withdrawal which left those 55th Infantry Division units still in place without any means of communicating their existence to the chain of command and to their fate. Lieutenant Colonel Pinaud's headquarters just south of Chaumont did not withdraw until dawn on the following day. Throughout the night he and Lieutenant Colonel Lafont commanding 331st Regiment to the west were left to their own devices as they endeavoured to try to pull together a defensive line running from Cheveuges to Bulson. The meagre troops under these officers did their best to resist the Germans until the next morning, but were finally forced to withdraw after they found that the enemy had seeped through the holes in their line during the night.\n\nThe stand taken by Pinaud and Lafontaine raises the question of what might have been achieved had Lafontaine been able to provide the two officers with a regiment's worth of infantry. While there is little doubt that 1st Panzer Division would undoubtedly have been successful in breaking through, this would most likely have been more slowly and more expensively achieved. Indeed, the shoulders of Guderian's bridgehead remained narrow during the night of 13\u201314 May and, had 71st Division remained in situ on Lafontaine's right flank, the prospects for a counter-attack within a couple of miles from the German crossing points would have been reasonable. As it was, the X Corps front fell back under moderate German infantry pressure, making any attempt to strike back much more difficult. The key question by midnight on 13\u201314 May was whether Guderian could strengthen his hold on the west bank quickly enough before the French attempted to push his divisions back into the Meuse.\n\nBut while Guderian drove his troops on to even greater efforts at Sedan, Gamelin still seemed to be working under the misapprehension that it was the developing challenge to the Dyle Line that demanded his concentration and focus. His report concluded: 'It is not yet possible to determine the zone in which the enemy will make his main attack.' The summary ended with a handwritten line \u2013 probably by General Louis Koeltz, the head of Gamelin's operations staff \u2013 which read, '[Overall] impression [is] very good.' Georges hardly helped the situation for, despite all the information made available to him, he described the afternoon's events at Sedan as nothing more than a 'rather serious pin prick'. Von Bock, meanwhile, noted in his diary, 'The French really do appear to have taken leave of their senses.'\n\n* As it was, Boyer was captured and spent the rest of the war as a PoW.\n\n* Rubarth was later killed in action in Russia on 26 October 1941.\n\n# CHAPTER 7\n\n# 14\u201315 May \u2013 Counter-Attacks and Exploitation\n\nWHILE THE FIGHTING south of Sedan on 13 May became less intense after dark, the rear areas of the two opposing corps were thrown into a frenzy of activity. The French were responding to a completely unexpected situation, while the Germans, having created that situation, were moving into a series of well-rehearsed drills. Gefreiter Otto Gull, an engineer with 1st Panzer Division, had slept an average of three hours a night since 9 May. He later recalled:\n\nSleep just was not an option, there was always something to do and we were exhorted to ever greater feats of endurance. We were constantly reminded how important we were to the offensive and that the lives of comrades were in our hands. It was a huge responsibility, we thought, but in the end we just did what we were told.\n\nGull was well aware that the huge opportunity created on the Meuse could not be allowed to expire. Central to this was Guderian's plan to get his panzers across the river in order to add valuable protection to the infantry but also, when the time was right, to be in a position to thrust out of the bridgehead. 1st Panzer Division's tanks were waiting in their tree-covered assembly area at Corbion eight miles north-west of Sedan. During 13 May their crews and supporting mobile workshops undertook some much-needed vehicle maintenance, rested, rearmed, refuelled, reorganized and waited for orders to start moving forward in anticipation of the completion of the bridges. In the meantime, a pontoon ferry was constructed at Gaulier during the evening of 13 May so that vital weapons and supplies could be received on the west bank. The cargo included anti-tank guns and their ammunition in anticipation that the French might try to drive an armoured wedge through to Sedan against the lightly armed infantry. This ferry was absorbed into a heavier bridge, which became operational at midnight. Priority was initially given to the remaining anti-tank guns and artillery interspersed with vehicles containing resupplies of ammunition and other essentials for the infantry. The construction of the bridges for 10th and 2nd Panzer Divisions, however, began considerably later due to the difficulties that they had encountered getting a foothold across the Meuse and as a result their bridges were not open to traffic until dawn on 14 May and first light on 15 May respectively.\n\n1st Panzer thus had an important role in protecting the corps while the other two divisions created their own arteries with the west bank. Pushing their anti-tank guns forward and positioning their light artillery pieces on the reverse slopes of the high ground south of Sedan, Kirchner remained offensively minded but ensured that he had made adequate defensive preparations should they become necessary \u2013 and he was certain they would. As his infantry made a concerted effort to push forward again at dawn on 14 May, they found the remnants of 55th Infantry Division which amounted to two battalions with three anti-tank platoons on a 4.5-mile front from Ch\u00e9r\u00e9y across to Chaumont, then to Noyers-Pont Maugis and from there down to the Meuse. On the division's left flank across the narrow River Bar was the remains of 5th Light Cavalry Division, while on their right was 71st Infantry Division.\n\nBy this time, the French High Command had a far better understanding of the threat posed by Guderian after more detailed reports had been received and absorbed. Although still awaiting evidence that would confirm that events at Sedan were the German main effort, Gamelin and Georges immediately saw the implications if it was. The Supreme Commander retained his composure but Georges, it seems, did not. The commander of the North-East Front spent most of his time at a 'personal command post' in Ch\u00e2teau des Bondons at La-Fert\u00e9-sous-Jouarre thirty-seven miles east of Vincennes. There, in the early hours of 14 May, a staff officer, Andr\u00e9 Beaufre, and Major General Aim\u00e9 Doumenc, the French Army Chief of Staff, found a sombre scene. Beaufre later wrote:\n\nThe room was barely half-lit. Major Navereau was repeating in a low voice the information that was coming in. Everyone else was silent. General Roton, the Chief of Staff, was stretched out in the armchair. The atmosphere was that of a family in which there has been a death. Georges got up quickly and came to Doumenc. He was terribly pale. 'Our front has been broken at Sedan! There has been a collapse...' He flung himself into a chair and burst into tears. He was the first man I had seen weep in this campaign. Alas, there were to be others. It made a terrible impression on me.\n\nFrom this point on, Gamelin spent a good deal of time at Georges's headquarters providing support and reassurance and taking the decisions that he could not rely on his subordinate to take. It was a hugely inefficient and awkward arrangement that could and should have been avoided considering the weakness that Georges had shown before 10 May. As it was, from 14 May the Supreme Commander had to shoulder the burden of a senior general suffering a nervous breakdown and, despite his other responsibilities, he was concerned not to leave Georges for too long. Gamelin later wrote of his incapacitated subordinate: 'Right from the start of the crisis he was overwhelmed. He did not know how to organize his work, became submerged in details and exhausted himself to no avail.'\n\nWhile the High Command endeavoured to conjure with the broader implications for the German emergence at Sedan, local initiatives were left to Grandsard's X Corps headquarters at La Berli\u00e8re. Over the course of the winter, the corps had practised a table-top response to a major river crossing at Sedan on several occasions and just three weeks before had held a map exercise which mirrored the situation faced on 13\u201314 May in many respects. In these rehearsals the X Corps reserve was to occupy a position from Ch\u00e9h\u00e9ry through Bulson to the edge of the woods south of Haraucourt and launch a counter-attack in the centre to drive the enemy into the Meuse, giving time for Second Army's XXI Corps, which was released during the afternoon of 13 May, to move up from the south to add more weight: a classic _colmater._\n\nGrandsard's written orders were sent to the two infantry regiments and two tank battalions by motorcycle despatch riders and informed them that they were to attack at dawn on 14 May. Both groups struggled to get forward and into position through fleeing troops from 55th Division during the evening of 13 May. On the left, Lieutenant Colonel Labarthe's 213th Infantry Regiment got to a position between Ch\u00e9mery and Maisoncelle by 2300 hours and dug in as 7th Tank Battalion drove through the night to reach them. On the right, Lieutenant Colonel Montvignies-Monnet's 205th Infantry Regiment and its already attached 4th Tank Battalion, meanwhile, dug in to the west of Flaba. Both groups, therefore, would need to make their final moves to the start line just north of Ch\u00e9mery-Maisoncelle-Villers during the last hours of darkness. The great difficulty for these attacking units, however, is that they were not being managed and lacked any information after having received Grandsard's orders.\n\nThe X Corps commander had telephoned Lafontaine to order him verbally to take command of the counter-attack \u2013 with written orders to follow \u2013 just before the 55th Division commander moved his headquarters to Ch\u00e9mery. Indeed, Lafontaine later argued that he moved his headquarters to better co-ordinate the counterattack, but this was refuted by X Corps, which recognized that the divisional commander was incommunicado just when the counter-attacking groups needed encouragement, orders and direction to ensure that they were in the planned positions to attack at dawn. Yet even though he arrived at Ch\u00e9mery that night, Lafontaine failed to take charge of the counter-attack because he demanded written orders from Grandsard. In search of them, he tried to reach Grandsard's headquarters through the choked roads, only to find when he arrived that the corps commander was not there and that his deputy, Colonel Cachou, was en route to Ch\u00e9mery with Lafontaine's written orders. Lafontaine arrived back at his headquarters before Cachou arrived and continued to wait for the written orders, even though they had already been received verbatim over the telephone in his absence. Time was running out if the counter-attack was to face only German infantry lacking firepower and still the French general vacillated, a slave to a procedure, the product of a system that was just not configured for rapid decision-making on a mobile battlefield, and a man of responsibility who feared applying his own common sense and rejected the pressing need for immediate action.\n\nAfter finally receiving Grandsard's one or two sentences of written orders, Lafontaine at last issued his own at 0500 hours, some nine hours after he had first been placed in charge of the counter-attack. Unbeknown to the French, their push from Bulson though the Bois de la Marfe\u00e9 and to the Meuse would now be a race against the German armour to claim the high ground. The first of Kirchner's tanks had moved to Gaulier through traffic jams and French shelling during the night, and they crossed the bridge at 0720 hours, spearheaded by 2nd Regiment of 1st Panzer Brigade. Gull had worked on the construction of the bridge and later testified:\n\nOur eyes stung with the effort of staying awake... There were one or two mishaps along the way with heavy equipment due to fatigue, but seeing the panzers cross was a wonderful sight. We cheered, shouted good luck to the tank crews, who waved an acknowledgement back... The division now felt that it was really on its way.\n\nThe tanks' appearance on the west bank was a great relief to the German infantry, who since first light had begun to penetrate the weakly held French line north of Bulson. As part of this move 1st Rifle Regiment had taken Ch\u00e9h\u00e9ry and pressed on to the River Bar at Omicourt while Gross Deutschland Regiment followed them past Cheveuges in order to by-pass some stubborn opposition at the Bois de la Marf\u00e9e and sought to attack towards Bulson. Orchestrating these movements was Kirchner standing atop Hill 301, keeping his eye on the infantry moving below him to his front, eagerly awaiting the arrival of the armour from his rear and watching the high ground around Bulson for signs of enemy armour. He and Guderian both knew from aerial reconnaissance that two mobile divisions were heading towards the area \u2013 these were from Second Army reserve \u2013 but also expected a more local counter-attack of the sort that Grandsard and Lafontaine were endeavouring to develop.\n\nHad the French launched their X Corps initiative at dawn as planned, they would have caught their enemy reorienting his infantry and without any armour. As it was, 7th Tank Battalion was spotted organizing at Ch\u00e9mery at 0700 hours. Once again, the Germans reacted immediately and sent an advance detachment of two Gross Deutschland Regiment anti-tank platoons and two armoured scout cars under the command of Oberleutnant Helmut Beck-Broichsitter to delay them. At 0730 hours, while 205th Infantry Regiment and 4th Tank Battalion continued their lamentably organized movement into their forming-up area, 7th Tank Battalion and 213th Infantry Regiment began its counterattack. It was launched just as the leading units of 1st Panzer Regiment arrived on the west bank of the Meuse. The trepidation with which this under-strength push was mounted can be seen in Labarthe's response to Lafontaine's orders: 'This is a mission of sacrifice that you ask of my regiment.'\n\nAs a single company of French infantry and another of tanks were sent to seize Connage, two tank companies were sent in support of Labarthe's infantry but were tied to the speed of its reluctant trudge towards Bulson. They were all observed by German aerial reconnaissance aircraft, which reported their findings back to Kirchner and Guderian. It was this information that led the first German panzer unit, a company of 2nd Panzer Regiment commanded by Oberleutnant Krajewski, to advance with the 'greatest possible speed' on the road to Ch\u00e9h\u00e9ry before branching off to the south-east to Bulson. The panzer company won the race to the ridge by just a couple of minutes. Krajewski's after-action report said:\n\nCautiously, we drove through Bulson, which had been evacuated by the enemy, and slowly, riding single file, we approached the Hill [322] a few hundred yards south-west of the village. Heavy fire commenced the moment the first panzers reached the hill. Our two lead panzers took several direct anti-tank gun hits and burned.\n\nKrajewski's own tank was one of those that took a French shell on its front armour and he was forced to scramble out and find some cover. The enemy had caught Krajewski's armour in a pincer, and Second Lieutenant Penissou has offered a French perspective on this opening engagement:\n\nThree German tanks move towards Fond Dagot. They are taken under fire by the 25mm's at around 500 meters. The first is hit and bursts into flames: the second in turn is hit and bursts into flames: the third is immobilized and hit with about fifteen 25mm shells.\n\nThe outnumbered German tanks fought bravely until just one tank was left mobile but held on long enough for Oberleutnant Friedrich von Grolman's panzer company to arrive. These tanks plugged the gap and, when another company arrived along with units from the Gross Deutschland Regiment, the Germans struck back. While the panzers lent fire support, the German infantry attacked 213rd Infantry Regiment in positions they had taken up around Fond Dagot. Penissou said of the situation by noon:\n\nWe are surrounded and engage the enemy from all sides. The small arms fire is heavy. The crew of the 25mm [gun] fall dead or wounded beside their weapon. We are completely encircled and the ammunition is almost exhausted.\n\nMeanwhile, the French infantry and tanks heading for Connage managed to advance through a screen of German defenders just north of Ch\u00e9m\u00e9ry, but then ran into Beck-Broichsitter's anti-tank team at the road intersection just east of the village at 0900 hours. The position consisted of a hedgehog of six 37mm anti-tank guns which offered all-round protection, with the two armoured scout cars offering an element of mobility. The French attack was led by their thirteen tanks, with the infantry following immediately behind. The anti-tank guns had considerable success after waiting for the enemy armour to approach to within a couple of hundred yards, and then aiming at each machine's weak spot. Even so, part of the infantry-tank group managed to work its way into Connage as the remainder endeavoured to outflank the hedgehog position and was on the verge of success when two companies of 43rd Assault Engineer Battalion arrived and hit back. This critical holding operation provided the time for a company of German tanks to arrive and begin to push the French out of Connage and back towards Ch\u00e9mery and then beyond just as Labarthe's attack was in withdrawal from Fond Dagot.\n\nIt was a success that had been so swift that the fall of Ch\u00e9mery had not been communicated to the Luftwaffe, which launched a Stuka attack on the village just as German commanders were meeting by the church to discuss their next moves. It killed the commander of 43rd Assault Engineer Battalion and three officers from 2nd Panzer Regiment and seriously wounded several others including the commander of the panzer brigade. But other German air raids hindered the advance of French reinforcements and the withdrawal of Labarthe's troops throughout 14 May, and led the regiment's commander to later write:\n\nDuring the course of the day, squadron after squadron of enemy dropped their bombs... without being harassed by one of our own aircraft... This was a baptism of fire for the 213th Infantry Regiment. The troops were very affected by the whistling bombs and the sight of the casualties.\n\nThe dive-bombers also targeted Lieutenant Colonel Montvignies-Monnet's 205th Infantry Regiment and supporting 4th Tank Battalion which had finally attacked east of Maisoncelles at 1045 hours, just minutes before Labarthe began his withdrawal from Fond Dagot. The forward movement of this secondary advance lasted just fifteen minutes before it too was forced to withdraw and by noon Montvignies-Monnet's men were facing a German reaction which demanded that they fight for their very lives. They eventually managed to extract themselves from this precarious situation and link up in the mid-afternoon with 213th Infantry Regiment and the sorry remnants of 55th Division in a line two miles south of Ch\u00e9mery protecting the high ground around Mont Dieu. Labarthe and his entire staff were taken prisoner, his regiment having suffered 50 per cent casualties as had 7th Tank Battalion, along with three-quarters of its machines. Montvignies-Monnet, meanwhile, had completely lost control of his formation and was later picked up by the military police in Verdun, initially on suspicion of desertion, supposedly seeking orders from superiors.\n\nWith the first, limp French counter-attack having been beaten back, Guderian, standing on Hill 301 with Kirchner, thought the success 'almost a miracle'. Throughout the activity south of Bulson, 1st and 2nd Panzer Divisions used their completed bridges to fill the expanding footprint south of Sedan with more armour, artillery, combat support units and vital supplies. The Allies, understanding how important these crossing points were to the Germans, spent much of the day trying to destroy them. General Billotte appealed to General d'Astier de la Vigerie, the commander of the air force in the northern operations zone, for support in this task, saying to him: 'Victory or defeat will depend on these bridges.' Indeed, with no more bridging materials to replace destroyed bridges, Guderian's corps would have been placed in a precarious position had their arteries across the Meuse been severed. Anticipating the need for their defence, the Germans moved anti-aircraft teams crewing 303 guns into position during the night of 13\u201314 May and the following morning. These would provide a considerable defence, but with a heavy enough concentration of aircraft the Allies could still hope to achieve an important success. It was to Billotte's great dismay, therefore, that although 152 bombers were available (of which 109 were from the RAF), problems in their co-ordination led to them being diluted in twenty-seven separate attacks of between ten and twenty aircraft, with Allied fighters flying 250 supporting sorties.\n\nThe attacks began at 0530 hours and continued until midnight on 14\u201315 May. The German anti-aircraft gunners coped well with them and were supported by fighters flying 814 sorties, which meant that only a small proportion of the bombers reached the Sedan area. One German officer who watched the Allied attempts to punch through to the bridge at Gaulier later wrote:\n\nAgain and again an enemy aircraft crashes out of the sky dragging a long plume of smoke behind it, which after the crash of the succeeding explosion remains for some time perpendicular in the warm air. Occasionally from the falling machines one or two white parachutes release themselves and float slowly to earth. In the short time that I am at the bridge, barely an hour, eleven planes were brought down.\n\nThe largest raid of the day took place in the late afternoon by the RAF when seventy-three Battles and Blenheims attacked, protected by twenty-seven French fighters. It was a disaster, with forty of the bombers shot down and leading the Official RAF History to note: 'No higher rate of loss in an operation of comparable size has ever been experienced by the Royal Air Force.' One British pilot, Flying Officer Sarll, recalled: 'I do not think anyone who did not experience what we were called upon to perform this day could ever visualize the tremendous courage of our people, so many of whom died.' That courage was also noted by the Germans, with Oberst von Hippel, the commander of a flak regiment, later recalling:\n\nWe could not help admiring the guts displayed by our opponents in attacking the bridges again and again despite all their losses and the fierce flak fire, and even shooting from planes already on fire, wounding some of our artillerymen.\n\nThe total Allied aircraft losses on 14 May at Sedan were fifty-three bombers and fifty fighters. Many more aircraft were badly damaged, and some of these could not be returned to action. The backbone of the Allied bomber fleet, therefore, had been broken in a defensive action which was to prove of considerable benefit to the Germans in the days and weeks of the campaign that followed.\n\nThroughout the Allied air raids, whatever delays there were to the continued use of the bridges were short. Guderian made a point of standing on various bridges during the attacks, at one point with a rather dubious von Rundstedt, in order to encourage continued traffic crossings. As a consequence, the bulk of Guderian's corps \u2013 600 German tanks, 60,000 men and an impressive 22,000 vehicles \u2013 passed into the bridgehead on 14 May. This put the formation in an excellent position to exploit the offensive movements it had also made that day as it partially reorientated itself westwards. While Gross Deutschland Regiment and part of its armour was dealing with the X Corps counter-attack, the remainder of 1st Panzer Division attacked west and crossed the Bar and Ardennes Canal at Omincourt at 0730 hours. Had the French managed to outflank Connage with the left wing of their counter-attack, this development would have been imperilled, but they had been pushed out of the village and then out of Ch\u00e9mery, and so Kirchner's troops were also able to make crossings towards Malmy at 1230 hours.\n\nTogether, these successes allowed 1st Panzer Division to push 6.5 miles with the infantry-dominated Kampfgruppe Kr\u00fcger to the village of Singly on the left and three miles with the armour-heavy Kampfgruppe Nedtwig to Vendresse on the right. Further north, 2nd Panzer Division made a similar movement across the Bar and Ardennes Canal for, although its own bridge at Donchery was not due for completion until the following morning, ferries and the Gaulier bridge allowed its units to cross the river during the afternoon of 14 May. Splitting into two parts in a manner very similar to Kirchner's division, one group pushed four miles to Flize on the Meuse while another on its left flank thrust three miles to Sapogne. Neither division's attacks were without difficulty. Both formations were tired, their lines of communication stretched and they had taken accumulative losses which affected their immediate offensive outlook: Nedtwig, for example, had lost a quarter of the tanks under his command by the end of the day.\n\nAlthough the defence by 3rd Spahi Brigade, 5th Light Cavalry Division and 53rd Infantry Division was tenacious and had some tactical value, Guderian's move to the west had wrongfooted the French. They were not anticipating a breakout from the Sedan bridgehead so early and, when it came, they assumed that it would head east to outflank the Maginot Line. As a result, only 10th Panzer Division's advance conformed to their expectations. Schaal managed to get a single panzer regiment into the bridgehead on 14 May which, together with his infantry, managed to overcome some stiff French resistance at Noyers during the morning, and continued its advance as the French began their wider withdrawal during the afternoon. At this point the division still needed confirmation that Guderian wanted it to push on to take the high ground of Mont Dieu half a dozen miles to the south in order to provide protection for the bridgehead and the rear of 1st and 2nd Panzer Divisions. Although an assessment of intelligence concerning the potential threats to the corps in the short and medium term initially pointed to the possibility of Schaal's division being able to turn west itself and leave the defence of the southern flank to 29th Motorized Division, within hours the risk posed by the two divisions from Second Army that had been spotted heading to the area proved too great for Guderian to ignore. Thus, in the early hours of 15 May he ordered 10th Panzer Division to seize the high ground of Mont Dieu at Stonne assisted by Gross Deutschland Regiment.\n\nDespite the reports he was now receiving from the sector, Gamelin still failed to fully grasp the immensity of the situation that was confronting the Allies at Sedan, referring to it as 'merely a local interlude' at noon on 14 May. Throughout the day, he continued to battle roads clogged up with refugees and military vehicles to be with Georges. Meanwhile staff officers at both Vincennes and La-Fert\u00e9-sous-Jouarre continued to rely on motorcycle messengers to send and receive messages of all types. It was extremely common for these messengers to be held up by the choked roads and by Luftwaffe attacks, which were increasingly targeting anything that moved along them. Scores of these motorcyclists were killed or seriously wounded in accidents, and particularly at night, as they attempted to deliver messages as quickly as possible. Yet despite their best efforts, it was not uncommon for the required \u2013 and time-sensitive \u2013 written orders of commanders to take two days to reach them and then be carried out. As officer and historian Marc Bloch argued, 'From the beginning to the end of the war, the metronome at headquarters was always set at too slow a beat.'\n\nSuch communication problems exacerbated the doctrinal requirement for written orders and the close control of operations, but the reality was that commanders had a poor 'grip' of the battle and that the French were drifting irrevocably from being able to influence events at Sedan. Yet with Huntziger confirming to his superiors that General Jean-Adolphe-Louis-Robert Flavigny's two counter-attacking divisions were on their way to the sector, Gamelin and Georges felt confident the Germans could be contained. Flavigny was commonly regarded to be France's leading armoured warfare commander and his 3rd Motorized Infantry and 3rd Armoured Divisions were prepared to counter what had always been considered to be the main threat offered by an enemy crossing at Sedan: a turning of the Maginot Line. In this sense, 10th Panzer Division had been offered by Guderian as bait in a trap that the French were only too willing to take, and so Second Army's counter-attack was consequently drawn away from his westerly thrust. Huntziger even moved his headquarters \u2013 including 2,638 staff \u2013 from Senuc thirty miles to Verdun in order to be better placed to organize the defences against the expected German turn east. But as soon as appropriate forces had arrived south of Sedan, Schaal's division would be relieved and could join its sister divisions in their attack towards the Channel coast.\n\nFlavigny's counter-attack was planned to be conducted along with 1st Cavalry Brigade, 5th Light Cavalry Division on the left and 2nd Light Cavalry Division, 1st Colonial Infantry Division and elements of 71st Infantry Division on their right. Although there were question marks about the military capability and strength of some of these formations, together the force concentrated some 300 French tanks against the bridgehead and Flavigny recognized that the earlier he attacked, the greater the likelihood he would be able to push Guderian's force back into the Meuse. The counter-attack was initially to have started in the late morning of 14 May in the immediate wake of X Corps' own, over the same ground and at a time when only 1st and some of 10th Panzer Divisions had crossed to the west bank. Yet there remained a question over whether the French force was capable of striking a blow against the Germans for Flavigny was not only concerned about the co-ordination and co-operation of the divisions but also the fact that his main strike force, General Antoine Brocard's 3rd Armoured Division, lacked most of its combat support, twenty of its 158 tanks and all of its anti-tank weapons, and had no radios. Even so, it was an essential element in the French scheme, but when the division arrived at its Le Chesne assembly area at 0600 hours, Brocard then needed time to refuel and reorganize before moving the final six miles to the start line. This laborious but essential process forced a frustrated Flavigny to postpone his attack until 1400 hours, a less favourable start time as 2nd Panzer Division's tanks had begun to cross the Meuse by the early afternoon and Guderian's troops had already crossed the Ardennes Canal in force. Yet even this later hour asked too much of the attacking divisions, for orders only made slow progress through the immense command bureaucracy, which meant that some units did not receive them until 1300 hours. So the attack was postponed again, this time until the evening, and then, with Brocard's force still tinkering and vacillating in the wake of X Corps' failure and with dusk approaching, it was put off until the next day.\n\nBy this stage, Flavigny had moved on from being irritated by the delays to infuriated. He recognized during the evening of 14 May that he had missed the opportunity that was so beguiling twelve hours earlier as Guderian's force had split and wondered whether a counter-attack might prove to be a futile gesture in the circumstances and that failure might leave Mont Dieu and the Maginot Line extremely vulnerable. The day had taught him that 3rd Armoured Division very likely lacked the morale and capability to take on the responsibility that it had been given and that German speed of thought and action had allowed them to retain the initiative. Having discussed the situation with Huntziger and in recognition of the threat that the Germans posed whether they continued their attack west or began to push east, Flavigny dispersed his force in defensive positions. There would be no Second Army counter-attack at Sedan, there was no major counter-attacking force in the General Reserve and 14 May slipped by as a day of missed opportunity. This was the apogee of French feebleness, revealing the cost of divisions whose quality varied widely, incoherent command, sluggish movement and lack of offensive spirit. While Flavigny may have seen the need for rapid action, he had no means to achieve it and this was fatal on the 1940 battlefield.\n\nBy nightfall, there was a considerable bulge in the French line twelve miles wide and nine miles deep. 71st Division defended the eastern side of the bridgehead, with 1st Colonial Division and 2nd Light Cavalry Division around La Besace together with the remnants of 55th Infantry Division, while 3rd Armoured Division and 3rd Motorized Division looked to protect Mont Dieu and Stonne. To the west of the Ardennes Canal, the Second Army's 1st Cavalry Brigade and 5th Light Cavalry Division took the front to a point just short of Omont from where 152nd Infantry Regiment, the first part of General Jean de Lattre de Tassigny's 14th Infantry Division to arrive, held the line. This formation was part of a new Sixth Army that was being formed to strengthen the join between Second and Ninth Armies to protect the route to the crucial communications hub of Rethel. Beyond 152nd Infantry Regiment, and taking the line in a westerly facing bulge north to Charleville M\u00e9zieres on the Meuse, were Ninth Army's 3rd Spahis Brigade, 53rd Infantry Division and 102nd Infantry Division.\n\nThis was a defensive front which, for want of a better plan, would see the French formations contain the Germans, wear them down and then, as the enemy's expedition withered on the vine, strike back. Guderian, however, had every reason to avoid being contained and, recognizing that his actions would determine the shape of German plans now that the Meuse had been crossed, sought to conduct an independent operation. While 10th Panzer Division provided protection by fighting for the high ground south of Sedan, 1st and 2nd Panzer Divisions would continue to drive west towards Rethel. Thus, while his superiors argued about whether von Kleist's spearhead should wait for the marching infantry to catch up, Guderian's intention was to have made such progress as to present them with a fait accompli. In the early hours of 15 May, a circumspect von Kleist approved the move.\n\nThe continuance of Guderian's attack would not only see his corps take another large step forward towards his longer-term goal of pushing through to the Channel coast, but would also close the gap between his divisions and Reinhardt's corps crossing the Meuse at Montherm\u00e9 and then Hoth's corps at Dinant. Together, this mass of panzer divisions, he believed, would have far greater ability to burst through behind the Allied troops pinned in central Belgium than if they fought their own isolated battles while waiting for the infantry armies. On 14 May, having managed to cross the Meuse and occupy Montherm\u00e9, Kempf's 6th Panzer Division \u2013 its sister 8th Panzer Division was still closing on the river \u2013 endeavoured to fight out of the town and surge forward. However, being trapped in a built-up area in a loop of the river drastically reduced the division's ability to concentrate its force and retain its mobility while, effectively acting as the cork in a bottle, a battalion of the competent 102nd Fortress Infantry Division's 42nd Colonial Infantry Regiment plugged the exit.\n\nThe deployment of French air power against the trapped Germans at this point could have been devastating for Kempf, but no aircraft were available due to the attacks on the Sedan sector. Left to apply its infantry to loosen the French defences, 6th Panzer Division gradually made some important infiltrations which by dawn on 15 May meant that two battalions were working hard to expand and would lead to a breakthrough later that morning. Further north in the Dinant sector, Hoth's 5th and 7th Panzer Divisions were not quite so hemmed into their conjoined bridgehead and both probed forward on the morning of 14 May. Max von Hartlieb's 5th Panzer Division finally secured the fiercely defended Haut-le-Wastia two miles from the river while 7th Panzer Division's 7th Rifle Regiment commanded by Georg von Bismarck advanced to take Onhaye. It was at this stage that the two division commanders clashed when, with just one military bridge having been constructed at Leffe due to a lack of bridging equipment, Hartlieb asked Rommel if he could pass his tanks over the 7th Panzer Division bridge to reinforce his formation. Rommel responded that because 5th Panzer Division lacked boldness and its panzers always lagged behind, he would agree only if he was authorized to take those tanks under his command. Hartlieb was furious but when he informed Hoth about the situation, the corps commander recognized the offensive advantage that this would give 7th Panzer Division and acceded to Rommel's petulant demand. He liked and even encouraged Rommel's boldness, but that did not mean that he approved of his subordinate's vainglorious approach.\n\nThe family squabble within Hoth's corps was unsavoury but inevitable considering Rommel's character and ambition, but it was soon overshadowed by an apparent emergency in his division. Shortly after 0800 hours Rommel was informed that a radio message had been received \u2013 'Bismarck encircled in Onhaye' \u2013 just before all contact with him had been lost. The situation bore all the signs of a French counter-attack and worried not only Rommel, who had not anticipated such a swift and strong enemy reaction, but also Fourth Army headquarters when it was informed. Rommel responded to the situation without hesitation and immediately ordered all his armour across the Meuse to break the encirclement. Rommel's personal group followed the spearhead column as it rushed towards Onhaye so that he might be on the spot to command a response to the German encirclement, but his PzKpfw Mk III was targeted by a French anti-tank gun en route. Hit several times, the tank slipped down a slope as the driver took evasive action. Bleeding heavily from a shell splinter wound on his cheek, Rommel tried to engage an enemy gun with the tank's main armament but the turret would not move as it was wedged into the soil. The general ordered that the vulnerable tank be abandoned and, once he was outside it, it became clear that the column had suffered a number of losses, with one subaltern reporting to him, 'Herr General, my left arm has been shot off.'\n\nIt was another close escape for the cavalier Rommel, who shortly after was told that Bismarck had not in fact radioed 'encircled [ _eingeschlossen_ ] in Onhaye' but 'arrived [ _eingetroffen_ ] in Onhaye'. Rommel nonetheless continued his attack, advancing to Sommi\u00e8re with Hartlieb's panzers, breaking the main French defensive line in the sector and pushing his armour to Morville, eight miles west of the Meuse. Yet although Hoth had succeeded in opening up the French line at Dinant, the very possibility that the French had seemed to be launching a swift counter-attack unsettled some senior officers and particularly those who were concerned about the panzer divisions mounting their own independent operations. The 'Bismarck Fright' had revived some long-standing fears and was used as evidence of the folly in allowing the panzer divisions to bolt forward 'off the leash'.\n\nThe French had meanwhile tried to respond to the Dinant crossings, but in common with the response at Sedan, their counterattack was plagued by a mixture of incompetence and practical difficulties. At midnight on 13\u201314 May, 1st DCR, commanded by General Bruneau, had been dispatched from Charleroi, just twenty-five miles away from Rommel's penetration. The forward elements of the division, however, only reached an assembly area north of Flavion, some ten miles west of Dinant, at around 2100 hours on 14 May due to the congested roads and some units getting lost. Nevertheless, the Ninth Army commander, Corap, sent a message to Bruneau: 'You must counterattack this evening with all that you have. That's a formal order.' Unbeknown to both men, and to Rommel, the leading elements of 1st DCR and 7th Panzer Division's 25th Panzer Regiment at Morville were just three miles apart. Had Bruneau been able to carry out Corap's order, he would have been able to cause a great deal of damage to Rommel's relatively flimsy armour, but 1st DCR's Char B1s could not move anywhere because they had run out of fuel and needed to reorganize. With the attack postponed until Bruneau's force was ready, another fleeting but important opportunity for the French had gone begging.\n\nIn retrospect, Rommel's breakthrough on 14 May, when considered together with that achieved by Guderian and the growing threat at Montherm\u00e9, could not have looked more like the main German effort. Yet intense action to the north continued to supply what seemed to be compelling evidence to Gamelin that this was not necessarily the case. On the morning of 14 May, the Allied High Command felt that a critical point had been attained in the defence against Army Group B with the Germans reaching the Gembloux Gap and the Netherlands on the point of being overrun. The Dutch were, indeed, under unrelenting pressure: in the north the Germans were threatening to break through and envelop Amsterdam while in the east there was a withdrawal from the Grebbe Line back to the incomplete defences of Fortress Holland, this pursued by German ground and air forces with a tenacity and ferocity which belied their tiredness. It was in the south, however, that the invaders posed the greatest immediate threat as 9th Panzer Division prepared to cross the Meuse into the centre of the city over the Wilhelms Bridge.\n\nYet the continued resistance of the Dutch was becoming a source of great frustration to the Germans, particularly when it was compared with the remarkable developments on the Meuse and the success in central Belgium. On 13 May, therefore, the commander of Eighteenth Army, General Georg von K\u00fcchler, set out his intent to his subordinates: 'Resistance in Rotterdam should be broken with all means, if necessary threaten with and carry out the annihilation of the city.' Hitler agreed and in F\u00fchrer Directive No. 11 stated: '... the speedy conquest of the Fortress Holland is to be facilitated through a deliberate weakening of the [air] power operated by Sixth Army.' As a result, Heinkel He 111 bombers of Kampfgeschwader 54 were shifted north away from their support of operations in central Belgium in order to force a Dutch capitulation.\n\nThere was considerable discussion about what K\u00fcchler meant by 'annihilation' as the ground forces were concerned by the thought of having to fight through piles of Rotterdam rubble containing innumerable civilian corpses against an enraged enemy. Even so, the word played on the worst Dutch fears and so it was employed when a messenger was sent to the Dutch commander of Rotterdam, Colonel Pieter Scharroo, demanding the city's capitulation or promising that the 'severest means of annihilation' would be employed. Although the deadline for a decision passed on 14 May, negotiations were still continuing when nearly sixty Heinkels were not called back and dropped 1,308 bombs on the city which destroyed its centre, killed 814 civilians and made 80,000 homeless in the resultant fires. A second attack was cancelled when Scharroo capitulated just minutes before its aircraft dropped their bombs.\n\nThe Germans were in no mood to enter into protracted dialogue with their enemies for they had no time to waste as commanders attempted to turn operational success into final victory. Whatever stood in the way would be destroyed. It was a ruthlessness that became increasingly obvious to those that sought to deny the Germans and was immediately recognized by their enemies. Indeed, the loss of Rotterdam and the rapidly deteriorating situation across the country left Winkelman with precious few options. He could order that the fight continue even though the enemy were in the process of advancing into the body of Fortress Holland, but he also appreciated the ultimate futility of such attempts. Thus, having heard that the Germans were threatening the destruction of Utrecht and Amsterdam too and with a mandate to avoid unnecessary suffering, Winkelman decided that the Netherlands would surrender the next morning \u2013 15 May. The Dutch troops west of Breda \u2013 approximately two battalions of army and navy personnel \u2013 were exempt from the surrender as they were fighting under the command of the French and were gradually withdrawn onto Belgian soil and to Ostend. Although a tiny force, their presence was welcomed by the Allies as the Germans swiftly followed the Dutch capitulation with a reorientation that saw Eighteenth Army's XXVI Corps attack into northern Belgium.\n\nThe collapse of resistance in the Netherlands was hardly greeted with great enthusiasm by Gamelin and the High Command, but it did at least allow them to focus more intently on the struggle for the Gembloux Gap, which had been at the heart of French defensive planning ever since the building of the Maginot Line. This critical area was protected not only by Blanchard's French First Army but also the two weakened and withdrawing DLMs of Prioux's cavalry corps. The Battle of Gembloux began on the afternoon of 14 May when 4th Panzer Division mounted an attack even before 3rd Panzer Division had arrived as Hoepner wished to make the most of the formation's momentum and so exploit French disorganization. What followed was a brutal confrontation that reflected the importance of the Gembloux sector to both sides. Although the eventual arrival of 3rd Panzer Division assisted its sister division, there was no immediate breakthrough since 2nd and 3rd DLMs held the enemy five miles in front of Blanchard's men, who used the opportunity to dig in faster and deeper. The appearance of the three divisions of Sixth Army's IV Corps, however, gave the Germans such a superiority of numbers that Prioux had no option other than to withdraw behind Blanchard's main positions that evening and to leave the approaching enemy to the artillery. Prioux's Corps was one of the few Allied success stories during the first week of the German campaign: it successfully delayed the advance of Hoepner's two panzer divisions, inflicted significant damage on them and allowed a critical position in the Dyle Line to be reached and prepared. Although the corps was undoubtedly assisted by excellent preparation for its semi-independent role, its application and capability was rarely, if ever, seen in any other French formation during the campaign.\n\nOn 15 May, however, the engagement of the Dyle Line infantry by Army Group B would be complete and a new phase in the campaign would begin. Indeed, increasingly undermined by the advance of Army Group A's outflanking panzer divisions south of Gembloux, Billotte's headquarters had on the evening of 14 May ordered Blanchard to prepare for a retreat. The reality of the German trap that the French had marched into had finally dawned. For the Allied soldiers and civilians whose knowledge of unfolding events was limited to what they could see and embellished with stories and wild rumours, the end of the fifth day of the campaign was as confused as it was distressing.\n\nAlthough Captain Daniel Barlone of 2nd North African Division wrote in his diary about the fate of the Netherlands, 'Fortunately our _poilus_ have no more idea where Rotterdam is than Peking, so this news does not worry them,' there is little doubt that tales of collapse and the fall of great cities was percolating through to soldier and civilian alike. Government radio broadcasts and press briefings were solemn, but rarely gave a faithful sense of the implications of what was befalling a relatively small corner of north-west Europe. Alexander Werth, a Russian-born British journalist living in Paris, was not taken in by the broadcasts of Ludovic-Oscar Frossard, the French minister of information, who asked that the population be patient for success: '\"Verdun lasted six months,\" he says, \"the Somme as long...\". Are these valid arguments? Haven't the French Ministers realized yet that this type of war is _different_?'\n\nMeanwhile, for those senior officers, politicians and other interested parties who were following events across France and Belgium, what was happening was both confusing and distressing. Georges, the commander of the North-East Front, was showing obvious signs of strain, but he was not the only one. The British army liaison officer at the headquarters of British Air Forces in France made a diary note of a meeting held at this time with attendees who included Air Marshal Barratt and Generals Vuillemin and Billotte:\n\nThe conference was a tragic affair. Most of the French officers were in tears, some quite openly sobbing at having to admit the shame they felt in acknowledging the appalling fact that the French had walked out of their fortified positions without any attempt at genuine resistance. This is a very bad affair.\n\nYet even if the tension of the situation was permeating through the upper echelons of the French High Command, this was not translated into the political collapse in Paris and London that Hitler had thought likely. Although Neville Chamberlain was replaced as British prime minister by Winston Churchill on 10 May, this was an event that coincided with the German invasion but was not caused directly by it, and the change not only stabilized the country but also made it more resilient and tenacious. Something had almost happened in France, where Reynaud had revealed a clear desire to remove Daladier from his cabinet and Gamelin from command, but the German offensive had temporarily forced the prime minister to postpone his manoeuvrings and place what faith he had in the incumbent Supreme Commander.\n\nAs it was, by 15 May Churchill had already shown signs of being a strong and pugnacious leader and he had coped well with the early pressure put upon him. But he did find that he spent increasing amounts of time trying to extract information from the French and reassuring them. Indeed, on 14 May Reynaud had sent a remarkably jittery telegram to him, saying: 'The situation is really very serious. The German Army has pierced our fortified lines south of Sedan... Between Sedan and Paris there are no more fortifications comparable to those.' This revealed just the sort of pessimism that Churchill could not abide and it was compounded the next morning when he was woken to take a telephone call from the French leader, who told him, 'We have been defeated... We are beaten; we have lost the battle... The front is broken near Sedan; they are pouring through in great numbers with tanks and armoured cars.' Although Churchill's military knowledge led him to respond that the Germans were extremely unlikely to be able to sustain such an advance, it was clear that the situation was grave, and in an attempt to test the temperature in France, learn the truth about events and buoy up Britain's ally, he prepared to make a trip to Paris on 16 May.\n\nIf it had been left to Guderian, the situation about which Churchill would be briefed when he met Reynaud would have been one of unremitting woe. The panzer commander, however, was as aware as anybody that even if the fate of France rested in the hands of many, the decisions of the few would have a disproportionate impact on the outcome. The few that most concerned Guderian included von Kleist, von Rundstedt and Hitler. This triumvirate simply would not countenance a carefree thrust to the west: like Churchill, they did not think that panzer forces had the ability to sustain such an advance, and they retained long-standing concerns about French counter-attacks and the turning of the Maginot Line. As a result, Guderian had reluctantly split his force in two but continued his attack with the support of Halder, who saw the very great benefit of maintaining the initiative rather than waiting in a small bridgehead for the arrival of the infantry. Both men were aware, however, of the nervousness in some quarters that mounted every extra mile the panzer divisions advanced from the Meuse.\n\n10th Panzer Division had the significant challenge of pinning and fixing a numerically superior enemy south of Sedan by attacking and holding the high ground at Mont Dieu. Guderian's aim was for Schaal's division, backed by the arriving 29th Motorized Division, to distract the French long enough to hand over to the advancing foot infantry and then rejoin the offensive. In the meantime, its objective on the morning of 15 May was to seize a long line from the Ardennes Canal west, on to the heights of Mont Dieu \u2013 including Stonne \u2013 then down to a bend in the Meuse south of Villemontry. The next two days saw 10th Panzer Division and the attached Gross Deutschland Regiment involved in some extremely intense fighting as the French sought to deny the Germans the high ground. The epicentre was the elevated village of Stonne, a small collection of farm buildings which was protected by units of 3rd Armoured Division and won and lost several times by both sides before the matter was concluded.\n\nThe Germans attacked at dawn on 15 May, using a battalion of Gross Deutschland Regiment, supported by a company of anti-tank guns, a battalion of tanks from 8th Panzer Regiment and men from 43rd Assault Engineer Battalion on the flanks. Stonne fell to the Germans after some frenetic close-quarters fighting which led to all units taking casualties, seven tanks among them. Fearful that the Germans could now observe their lines, the French were ordered to strike back immediately and, using their heaviest tanks, briefly retook the village. Unwilling to allow the French time to settle, grizzled Gross Deutschland units took Stonne for a second time in the late morning \u2013 and so the die was cast for a battle of tenuous holds.\n\nThe French attacked again at 1100 hours, using three companies of tanks to lead several hundred infantry into the village. 1st Battalion Gross Deutschland Regiment and twelve anti-tank guns were waiting. Three guns commanded by Oberfeldwebel Hans Hindelang had just arrived and were being sighted in the southwest corner of the village by a water tower when three Char B1s fell on the position. The guns opened fire and saw their rounds bounce off the advancing armour which continued to fire their machine guns. One of the anti-tank guns was destroyed, its commander and gunner were wounded and the remainder of the crew killed. The lack of penetration offered by the German guns frightened the defenders and led to some of the infantry withdrawing, but the gunners held their positions, hoping to penetrate a weak spot on the tanks. Describing the first success for the anti-tank guns, a participant later testified:\n\nThe fire of the three heavy tanks threatens to wipe out the anti-tank section. But it remains in position. One moment one of the colossuses crosses the front. The left gun commander, Corporal Giesemann, discovers in the middle of its right side a small-ribbed surface; apparently it is the radiator! It is not much bigger than an ammunition box. He aims at it. A tongue of flame shoots out of the tank.\n\nA second tank turned and destroyed Giesemann's gun, but the weakness in the armour had been noted and the final gun, withdrawn by Hindelang to better cover by some buildings, struck the other two tanks in the same area. The three burning tanks were a beacon of hope for the German defenders, and their destruction forced the French to withdraw. They returned, however, and retook the village at noon.\n\nThese were the first blows in a fight that was to last three days and was to see Stonne change hands seventeen times. The village became a running sore for Flavigny, who continued to attack along Mont Dieu to deny the Germans the high ground, but was in no position to attempt a breakthrough, not least because of the increasing German strength that was flooding south of the Meuse. The increasing solidity of the Sedan bridgehead also gave Guderian firmer foundations for his attempt on 15 May to turn his advance into something of critical operational significance. Standing in the way was Sixth Army under General Robert-Auguste Touchon, an effective if rather hidebound commander, with XXIII Corps on the right consisting of 14th Infantry Division, 36th Infantry Division and with 2nd DCR moving into position, and XXXXI Corps on the left largely made up of the remnants of 5th Light Cavalry Division, 3rd Spahis Brigade and 53rd, 102nd and 61st Infantry Divisions. These two hastily organized corps were weakly joined fifteen miles west of Charleville M\u00e9zi\u00e8res and had the task of holding Reinhardt as 6th Panzer Division emerged from Montherm\u00e9 and Guderian and his two divisions pushed towards Rethel. Touchon's ad hoc formation was in an invidious position as it had the important job of being the glue that held together two failing armies but was stretched even as it was being inserted and long before its front had hardened.\n\n15 May was a day of heavy fighting for both 1st and 2nd Panzer Divisions in extremely close terrain. Kirchner's remained the main effort as he continued with his two-pronged advance with the infantry of Kampfgruppe Kr\u00fcger having to struggle hard for La Horgne. Here elements of 3rd Spahis Brigade had done good work fortifying the village and for the first time since 10 May, 1st Rifle Regiment just could not find a way through the defences. It led to one of the most ferocious contested engagements of the entire campaign between two aggressive and well-led formations. After a series of attacks had failed to make any progress despite some desperate hand-to-hand fighting, Balck came forward to lead his men and encourage them to new feats of endurance and achievement. In an immense effort, the German infantry rallied and, despite lacking the supporting fire of tanks and artillery due to the nature of the terrain, gradually began to infiltrate and outflank the Spahis' positions. La Horgne fell that afternoon, but the defenders had done a first-class job in slowing 1st Rifle Regiment, which suffered more casualties in the struggle for the village than they were to receive on any other day of the campaign. Balck later declared:\n\nI fought against all foes in both wars and I was always in the thick of it. Rarely did anyone fight as outstandingly as the 3rd Spahis Brigade. Its commander, Colonel Mar\u00e9, was captured having been wounded... Out of the brigade's twenty-seven officers, twelve were killed in action, seven were wounded, along with 610 Spahi dead or wounded. The brigade had ceased to exist. It had sacrificed itself for France. I gave orders that prisoners we took be given especially good treatment.\n\nThe _kampfgruppe_ then headed to the last defended villages before more open ground. Ba\u00e2lons posed little problem for Balck's 3rd battalion, but that evening his 1st battalion strained every weary sinew to try and overcome the strong defence at Bouvellemont. Having taken heavy officer casualties in La Horgne, units bounced off the French positions and clearly lacked the drive to take the enemy on. Balck observed this weariness with understanding but dismay and feared that their preference was to eat, sleep and reorganize before launching a full-blooded attack the next morning. Dressed in a windproof smock, a grubby scarf tight to his neck and carrying a knotted stick, he wandered around his units, talking to his commanders and explaining that there was nothing to be gained by waiting until dawn and a great deal to lose because the enemy would only be reinforced. Grimy-faced and with red-rimmed eyes, he exhorted his men to attack before nightfall, saying, 'Gentlemen, we are attacking or we are simply giving victory away... if you do not want to go on, I will take the village myself.' Once again, he obtained the desired reaction for, as he led the attack alone, he turned after 100 yards to see his officers right behind him, and they turned to see their men right behind them. Balck later recalled:\n\nMen and officers, who just a few seconds earlier were at the end of their tether, passed by me. Nobody was thinking of seeking cover. Everybody rushed on ahead. Our fixed bayonets glistened in the setting sun. There was no stopping us. From everywhere came the sound of wild cheering as a thin line of completely exhausted riflemen pushed into the village. Bouvellemont was ours.\n\nThe infantry were joined that evening by men from Kampfgruppe Nedtwig who had advanced well that morning through Terron and Omont. The fight for Chagny \u2013 less than two miles from Bouvellemont and defended by cavalrymen and a battalion of 14th Infantry Division \u2013 came next. Here the Germans were tank-heavy and the unit had been informed that a French counter-attack was possible from the south. Both mitigated against swift success and, despite efforts throughout the afternoon, Nedtwig was forced to withdraw his tanks back to Omont for the night. The prospect of attacking Chagny again the next day was not a welcome one for anyone concerned, not least because they feared that the enemy would have been reinforced and, with so little opportunity to manoeuvre in such dense terrain, they would again be forced into a frontal attack. It was, therefore, only as the result of a stupendous effort by the drained infantry infiltrating during the night of 15\u201316 May \u2013 and doubtless assisted by the fall of Bouvellemont \u2013 that Chagny was taken.\n\nAlthough 1st Panzer Division's two _kampfgruppen_ had failed to burst the front open and reach Rethel on 15 May, Guderian was aware of the significance of what had been achieved by exhausted troops in difficult terrain. When he visited Kirchner early the following morning, he was keen to emphasize his belief that the hole which had been punched in the French line would release a great amount of pressure. 2nd Panzer Division had made better ground to the north in terrain more conducive to movement and the deployment of its firepower, and it had quickly established bridgeheads beyond the River Vence at Boulzicourt and Poix-Terron, just three miles from Kirchner's right wing. Despite some desperate defence by elements of 14th, 53rd and 102nd Divisions, by the middle of the afternoon 2nd Panzer Division had also taken the last high ground. In so doing they forced the enemy off the best defensive terrain and into a withdrawal back towards Rethel. It was this that had inadvertently assisted 1st Panzer Division's capture of Bouvellemont and Chagny that evening. Yet although a new defensive position was being formed by 14th Infantry Division around Rethel to contain Guderian, the northern front was collapsing. 2nd DCR would have added considerable ballast to that position, but significant transport problems meant that it was struggling forward and so arrived in its positions piecemeal north of Rethel between Guderian's corps on its right and Reinhardt's corps on its left.\n\nIt was Reinhardt's advance that caught the eye on 15 May. Kempf's 6th Panzer Division managed not only to fight its way out of the tiny Montherm\u00e9 bridgehead that morning, but then stormed forward thirty-five miles to advance via Liart towards Montcornet. Once again, it had been the good work of engineers and infantry taking French bunkers in a manner very similar to that seen on the west bank of the Meuse at Sedan on 13 May that pierced the defences after around four and a half hours of intense and close-quarters fighting. The torch was then handed by Kempf to an ad hoc unit, Pursuit Detachment von Esebeck, an all-arms grouping including battalions of tanks, motorcyclists and engineers supported by artillery, anti-tank and anti-aircraft guns, to exploit the success and push into the enemy's rear. Having crossed the Meuse on a new bridge, von Esebeck pushed forward in the middle of the afternoon and so paralysed the enemy that his troops had only to engage in skirmishes as they went. Indeed, one staff officer's overriding memory of the advance was that of the bellowing of unmilked cows belonging to farmers that had left the beasts to their fate.\n\nIt was a thrust that threw the French into a spin and certainly influenced the decision of XXIII Corps to withdraw to Rethel as the rampaging panzer division tore between the two corps of Sixth Army that had been brought in to strengthen the front between Second and Ninth Armies. The defenders were sent scuttling back and the bridgeheads created by Guderian and Reinhardt were merged on 2nd Panzer Division's extreme right flank. With this thrust, Reinhardt ensured the survival of Panzer Group Kleist as an operational independent organization. The continued bottling up of 6th Panzer Division at Montherm\u00e9 would undoubtedly have been used as evidence to end the experiment, and with this in mind von Kleist had ordered the division to do whatever it could to stay ahead of the infantry army that had come up to the east bank of the Meuse. This was why Reinhardt, having received orders from Twelfth Army to assist 3rd Infantry Division across the Meuse just to the south of Montherm\u00e9, then said that it was impossible to do so as Kempf had already, and instinctively, set off on his exploitation. Consenting to orders and then evading them was to become increasingly common among panzer division and corps commanders in the coming days. It was not just the Allies that were being out-manoeuvred, it was the conservative forces within the Wehrmacht that would take any opportunity to rein in the opportunistic panzer forces. For the time being, however, von Rundstedt saw merit in retaining the independent role of Panzer Group Kleist \u2013 but always with one eye on threats to its flanks.\n\nIt was just such a threat that had blundered towards Panzer Group Hoth, and specifically Rommel's 7th Panzer Division near Flavion, on the night of 14 May. Although any success it achieved would only be tactical, General Bruneau's 1st DCR represented the last French hope to thwart Hoth's attempts to strike west and so significantly broaden the attacking front of Army Group A. While 25th Panzer Regiment rested and reorganized south of Flavion during the night of 14\u201315 May, just a few miles to the north French 28th and 25th Tank Battalions \u2013 comprised of Char B1s and Hotchkiss H39s respectively \u2013 waited for fuel and made preparations to strike the next morning. Then, at 1000 hours on 15 May, the predatory 25th Panzer Regiment came across its still supine enemy and attacked. It drove a wedge formation into the middle of the group and then fanned out. Within moments, French troops were scattered, vehicles were ablaze and black smoke filled the air. The German tanks were outnumbered, but keeping calm was something that had been drummed into their crews, with one of those involved, Hans Becker, later saying: 'When you are involved in combat and the banging starts around you to the right and to the left, then it's as if the human mechanism is switched off. All you can think of is \u2013 watch out  Rommel's tanks did not fully commit to the battle, but by the time that 5th Panzer Division's 31st Tank Regiment arrived on the scene from the north-east at 1100 hours, considerable damage had already been inflicted. Seizing the opportunity to move on, Rommel immediately withdrew his armour, leaving 5th Panzer Division to complete the rout while he pushed on to Philippeville.\n\nRommel's departure left Werner's 31st Panzer Regiment considerably outnumbered, out-gunned and out-armoured by the French division \u2013 170 Char B1s and H39s against 30 Mk IIIs and IVs and 90 Mk Is and IIs \u2013 but over the course of the afternoon, the heavier tanks were out-thought and out-fought by the nimble German armour. Well led and once again closely co-ordinated through first-class radio communication, the more powerful of the German tanks combined well with anti-tank guns which had been rushed into the area to expose and target weak spots. Stuka raids, meanwhile, stopped more French tanks from entering the battlefield and also those that sought to leave it. By the evening, with scores of his tanks blazing wrecks and others running out of fuel, Bruneau ordered a withdrawal of his thirty-six remaining machines. The carnage they left behind was numbing, as one of the survivors recalled:\n\nI could not think or feel that evening. We drove into the darkness in more ways than one. I was never the same man again. I left myself on that battlefield, among the glowing hulks of broken machines, the charred bodies of men that I knew well.\n\nBy the next morning, 1st DCR had been diminished to a mere sixteen tanks. It had been eviscerated in less than ten hours by two German tank battalions.\n\nEven as 31st Panzer Regiment was in the throes of completing its task \u2013 which meant that Hartlieb's division only managed to advance a handful of miles on 15 May \u2013 7th Panzer Division was speeding ahead with its tanks in the lead, followed by lorried infantry and artillery. Rommel later wrote: 'It was my intention to ride with 25th Panzer Regiment so that I could direct the attack from up forward and bring in the artillery and dive-bombers at the decisive moment.' The line of advance was agreed with his chief of staff and artillery commander and briefed to officers during an Orders Group at which they all marked their maps, whereupon the advance began at an average speed of 40 mph. The division moved forward with little thought for its flanks because it sought to shatter the enemy's cohesion \u2013 and it did, Rommel noting that the division passed:\n\n... fully armed French motorcyclists coming in the opposite direction, and picked them up as they passed. Most of them were so shaken at suddenly finding themselves in a German column that they drove their machines into a ditch and were in no position to put up a fight.\n\nHad the Allies been able to get sufficient aircraft into the air over the sector at this time, they would have had little difficulty locating the column for, as Rommel wrote, 'the dust cloud behind us was enormous', but the sky was as clear of enemy aircraft as it was of clouds. By the end of the day, 7th Panzer Division had advanced so successfully that it was several miles beyond its objective, having reached the outskirts of Froid-Chapelle. It was a plunge deep into the enemy's rear area that was fully in keeping with Rommel's aim of avoiding unnecessary fighting in order to further both the cause of Hoth's corps and his burgeoning reputation as a panzer commander. Which was more important to him was, at times, difficult to discern.\n\nIt was only on 15 May that Gamelin truly understood that he had been caught in the Germans' trap as their panzer divisions began to break out to the south of his main force. Along the Franco-German border, nineteen divisions of Army Group C continued to attack and pin thirty-six French divisions to the Maginot Line, but although the situation now tempted Gamelin to redeploy them, he could not for fear of a German attack through Switzerland on 16 or 17 May. This particular, and brilliant, ruse had been developed by Hitler's propaganda minister Goebbels with the Abwehr and succeeded in stretching the enemy's mental and physical resources while enabling the Germans to keep the strategic initiative.\n\nMeanwhile, the great deception of central Belgium continued to play out. But by this stage, Gamelin could not simply disengage from Army Group B in order to deal with the developing threat of Army Group A. By 15 May, the critical task that fell to the French High Command was to act quickly to ensure that their forces were not so fixed by von Bock that those of von Rundstedt were free to outflank them. All along the Dyle, French, British and Belgian forces were being engaged by strong but not overwhelming German forces. The BEF had three divisions on the Dyle and all came under extreme pressure on 15 May. 2nd Division was on Gort's right and was hit particularly hard by the attacks launched in von Bock's main effort emanating from Maastricht. The fighting was intense as the Germans endeavoured to cross the Dyle and establish themselves on the west bank.\n\nAll along the line, in battles reminiscent of those fought by their fathers during the First World War, the German infantry attacked the British trenches after brief but powerful artillery barrages. 2nd Battalion Durham Light Infantry, for example, had a torrid time denying German 31st Infantry Division a bridgehead at Gastuche. Twenty-five-year-old Second Lieutenant Dick Annand's 16 Platoon of B Company took the brunt of the attack that morning, suffering heavy casualties but remaining in place as Annand and his NCOs kept order and led by example. Annand led two counter-attacks personally before being wounded. The battalion came under increasing pressure during the afternoon, however, and after dark the Germans made another attempt to oust the defenders with a strong assault. Annand personally endeavoured to disrupt them and, armed with hand grenades, went forward to cause mayhem despite his wounds. Remarkably, he survived, and for his leadership and personal gallantry \u2013 which included retrieving his wounded batman from the battlefield in a wheelbarrow \u2013 on 15 and 16 May he was awarded the first British Victoria Cross of the Second World War.\n\nDuring the course of 15 May, the Allied line on the Dyle was nonetheless infiltrated after the river was crossed in several sectors, forcing the defenders into a withdrawal while the Germans constructed bridges and established themselves on the west bank. Von Bock's aim at this point, and which he had made clear to subordinates, was to make steady progress forwards by threshing the opposition like a combine harvester in a summer field, not allowing the enemy to avoid contact and certainly not attempting a breakthrough. The ambition continued to be the pinning, fixing and destruction of the enemy in situ, so allowing Army Group A to affect its outflanking movement. For this reason, the Battle of Gembloux was less about Hoepner bursting through the French line, and more about an attritional push to remove vital enemy assets from the battlefield while retaining the integrity of the wider German front.\n\nThe French frontline divisions of First Army had been well served by Prioux's Cavalry Corps, having been given vital time to prepare their defences, including digging in and sighting their artillery, but now the fight was theirs. Hoepner attacked on the morning of the 15th, throwing his weakened panzer divisions across a four-mile front without waiting for the support of the infantry divisions marching up behind him and in the hope of catching the enemy less well prepared. In the van of the ensuing storm General Albert Mellier's elite Moroccan Division performed excellently, defending the village of Ernage and flanked by motorized infantry divisions.* The Stuka attacks preparing the battlefield had less impact here than they did in Sedan as the defenders were more resolute and understood that such raids had a bark significantly stronger than their subsequent bite. This meant that when the panzer divisions advanced, they were caught in a killing zone of artillery fire and interlocking defences which remained manned and co-ordinated. With field telephone wires buried deeply and communications as good as such a system would allow, the French felt far more comfortable in their defences than they had facing Guderian on 13 May. The attack stalled and Hoepner's decision to attack without extra infantry support was seen to be misguided after continued attempts to throw metal and flesh against the French were rebuffed, causing heavy casualties and tank losses. The attack was suspended that afternoon with the prospect of resuming it the next morning with the support of two infantry divisions. The order, however, was not received by all 3rd Panzer Division units and, against the odds, 3rd Rifle Regiment managed to achieve a lodgement in the French line and by evening had broken the main line of resistance.\n\nThe French had nothing with which to respond as Prioux's remaining armour had been dispersed within the front line to reinforce the individual tank battalions there. Yet even if Hoepner had wished to complete his success by turning behind the enemy to undermine the integrity of the line, the recent battle had so weakened his two divisions that they were no longer strong enough to do it. Hannut and Gembloux had combined to reduce the corps' strength by well over one third, with 4th Panzer Division boasting a mere 137 armoured fighting vehicles, including just four Mk IVs, by the end of the day. The corps was in dire need of the attentions of its mobile field workshops and of resupply, reorganization and rest. Its achievement had been central to both Army Group B's success and to the continuing success of _Fall Gelb,_ but now that the Dyle Line had been engaged, the load would be shared more evenly along the front.\n\nThe Battle of Gembloux was _Bataille Conduite_ epitomized \u2013 a battle that the French could control because they were prepared and expecting the attack. It showed exactly why the Wehrmacht had been so keen to avoid such frontal attacks and was concerned that they had seemed to be _de rigueur_ for the main effort during early invasion planning. It also revealed that in certain places the French did have the wherewithal to confront German methods and that in different circumstances, if it had been forced to undertake frontal attacks, the Wehrmacht may well have struggled to gain any momentum. As it was, the French fought along the Dyle with more than one eye on what was happening on their southern flank and, rather than launching strong counter-attacks to restore positions on the Dyle and at Gembloux, the Allies were forced to withdraw from the river in order to maintain the cohesion of their line.\n\nThis move, in fact, coincided with Gamelin being informed, 'Very strong [German] motorized column advancing in the direction of Montcornet, with nothing much remaining between Paris and the panzer divisions.' The news confirmed his worst fears that the Germans were outflanking Billotte's force, and reports suggested that attempts to thwart that thrust had failed, with Ninth Army struggling to cope. Its commander, Corap, had been sacked by Georges that morning \u2013 he was the most senior of the thirty-five officers who were relieved of their duties during this period \u2013 and replaced by General Henri Giraud. The unfortunate Giraud arrived at his new headquarters on the afternoon of the 15th to find that his army was disintegrating, with the strength of the German thrust from Maastricht threatening his left flank and that from Sedan smashing his right. The state of Ninth Army was confirmed to Gamelin that evening when a Lieutenant Colonel Guillaut, despatched from Gamelin's personal staff to ascertain the situation, reported to him the 'lamentable state in which he had found the [Ninth] Army headquarters \u2013 like a lifeless corpse \u2013 entirely ignorant of the situation of the army's divisions'. Guillaut added that he was aware of positions where whole units were missing because 'from the moment the Germans appeared, everyone had run away'.\n\nIn such circumstances, Gamelin directed Billotte to order a phased disengagement from the Dyle position and he did so more in reaction to the threat offered by Army Group A than the pressure exerted by Army Group B. It was to take place under the cover of darkness over the next four nights, and would see a complicated withdrawal by troops of four nations while they remained in contact with the enemy and would use the Rivers Senne, Dendre and Scheldt as phase lines. It was a decision that grasped the reality of the situation for the first time. It recognized that Billotte's force had to be protected, that a hard front had to be formed south of von Rundstedt to protect Paris, and looked to future counter-attacks into the flanks of the well-extended German formations to restore the situation.\n\nYet Gamelin was thoroughly pessimistic; the unexpected and unthinkable had happened, and his forces were on the back foot and barely coping with the enemy's offensive. The Supreme Commander's plans and French strategy had been unhinged by _Fall Gelb_ and there was no doubt that the Germans held the initiative. That evening, Gamelin admitted defeat to Daladier in a telephone call outlining the situation across the front that left the minister of war 'stupefied' and shouting, 'No! What you tell me is not possible! You must be wrong! It's impossible!' He demanded an immediate counter-attack, only to receive a terrible response from Gamelin: 'With what? I don't have the reserves.' There was a pause. 'Then the French Army is finished?' asked Daladier. 'It's finished,' the Supreme Commander replied.\n\n* That day the Moroccan Division lost 2,000 men, with one of its frontline battalions suffering 90 per cent casualties and another reduced to four officers and thirty-one men out of 700.\n\n# CHAPTER 8\n\n# 16\u201320 May \u2013 Crisis of Command and the Coast\n\nTHE PARIS THAT WINSTON CHURCHILL ARRIVED IN on the morning of 16 May was emptier of people than usual but filled with rumours about the character and nature of Germany's invasion. There were stories of fifth columnists engaged in sabotage, German parachutists in rural districts, atrocities committed by ideological zealots, panzers rampaging around Soissons, broken troops fleeing the front, the government quitting Paris \u2013 a mixture of truth and fiction that cloaked the French capital in a dark anxiety that suffocated its sun-drenched streets. The British prime minister and his military advisers, Generals John Dill and Hastings Ismay, felt the claustrophobia and they were shocked by what they learned from their visit.\n\nAs bundles of documents were being incinerated on the lawns outside the French Foreign Ministry, Gamelin provided an erudite summary of the military situation to his foreign guests as they sat, furrow-browed, with Reynaud and Daladier. Churchill later recalled: 'Utter dejection was written on every face. In front of Gamelin, on a student's easel, was a map about two yards square, with a black ink line purporting to show the Allied front. In this there was drawn a small sinister bulge at Sedan.' Gamelin said that it was not clear whether the Germans would push to the coast at Abbeville or turn towards Paris, but what was apparent was that the panzer divisions were moving fast. Churchill then asked a question: 'Where is the strategic reserve?' Gamelin's reply was no surprise to Daladier. There was, he said, no strategic reserve. A long silence followed while the British tried to take in the information. Churchill later wrote:\n\nI was dumbfounded. What were we to think of the great French Army and its highest chiefs? It had never occurred to me that any commanders would have left themselves unprovided with a mass of manoeuvre... I admit this was one of the greatest surprises I have had in my life.\n\nAs Churchill flew back to England, he began to discuss with his colleagues the options that Britain had available to assist the French and, of course, the British Expeditionary Force. Should more troops be sent across the Channel, and what about more aircraft? But there was also an understanding that the primary role of the government was the defence of its own people \u2013 a responsibility not lost on Reynaud and Gamelin \u2013 and that, in the final analysis, Churchill had to ensure that if France fell, he had the necessary assets to protect the British mainland. Being able to continue the fight was essential, and the French government recognized this as well because they had already taken steps to ensure that, if and when the Germans arrived in the capital, they would find no official documents and that the vital organs of government had moved out of harm's way.\n\nIt was a lonely and anxious Reynaud who wondered how much longer he would continue to attract the support required to prolong his leadership of a nation in such peril. The French prime minister had attempted to assert his authority but found that his ability to do so was consistently undermined by fickle colleagues and fast-moving events. In no area was this truer than in his attempt to maintain calm in Paris. On the afternoon of 16 May, he said in the Chamber:\n\nThere are absolutely absurd rumours going round. People are saying that the government wanted to leave Paris; this is not true! People are saying that the enemy has new irresistible weapons when our aviators are covering themselves in glory, and our tanks outclass those of the Germans in the same category.\n\nYet the people of Paris increasingly lacked faith in the words of politicians and officials \u2013 weren't these the same people who had told them that French defences were 'impenetrable' and that France was too strong for Germany? \u2013 and began to think about taking responsibility for their own destiny. Paris itself remained quiet and although thousands had left the city, the majority stayed despite the arrival of the first bedraggled refugees from the Netherlands, Belgium, Luxembourg and the border regions. The authorities were quick to try and move the foreigners on, for they feared that their demoralizing stories would affect Parisians, but they need not have worried: these particular refugees had left their homes well before they could have had contact with either the Germans or dispirited French troops. To this end, Eug\u00e8ne D\u00e9pigny, who worked in the _mairie_ of Paris, wrote: 'In spite of their destitution, [the refugees'] evident anxiety and suffering, these lonely individuals did not look hunted in May.' Those who were to leave their homes in the middle of the month, however, would take weeks to reach Paris and when they arrived, their stories were terrifying.\n\nDefending Paris was a central part of the developing French plan now that Billotte's First Army Group was undertaking a withdrawal while Second and Ninth Armies desperately sought to work with Sixth Army to block the German main effort. To stabilize the situation west of the German breakthrough while ensuring that the capital was protected required Touchon's force to fall back on the Oise and Aisne to establish a blocking position from Rethel west to the Somme and link up there with Seventh Army. This was a campaign-affecting decision as it would break the link between First Army facing Army Group B and Sixth Army facing Army Group A, leaving just a few fragmented units before the Germans and the Channel coast, and the only hope of reestablishing that link lay in attacking the open flanks of Panzer Group Kleist as it advanced.\n\nBut the defence of Paris was not Gamelin's sole concern because the invaders had also created an option to outflank the Maginot Line with a pivot from Mont Dieu. To deal with this potentially fatal blow, the left wing of Second Army fell back. This allowed Huntziger to shorten his line while maintaining contact with the enemy around Stonne \u2013 where the battle raged until 25 May \u2013 and concentrate his Second Army forces at the end of the Maginot Line around La Fert\u00e9. In the days that followed, the French were put under extreme pressure here as the Germans made a big show of concentrating around 250 artillery pieces in this narrow sector to feign an important breakthrough attempt. As a result, troops from the Maginot Line who could have been transferred to defend against the main German effort were kept in place to counter the threat of the Line being outflanked at both ends, and the La Fert\u00e9 sector defending the front around Sedan was reinforced.\n\nThe success of this German deception reduced the potential pressure on Guderian's westward advance on an eighteen-mile front, leading Halder to write, 'Our breakthrough wedge is developing in a positively classic manner. West of the Meuse our advance is sweeping in, smashing tank counterattacks in its path...' Even so, the wedge looked extremely vulnerable as it began to achieve a head of steam and pull away from the infantry divisions traipsing behind. It was happening so fast that even some of the more sceptical German commanders found themselves reacting instinctively to the situation. Von Rundstedt and Hitler, for example, were concerned about the southern flank but so amazed by the advance and its potential that they had initially been willing to see it develop. Even so, they were not so anesthetized by success that they would not react adversely to suspected threats and their imaginations were increasingly febrile. Guderian, Hoth and Reinhardt were aware of this but continued to push as hard and as fast as possible in the belief that it was the only way to dislocate the enemy and protect the wedge. This was reflected in the log of 1st Panzer Division for 15 May which stated: 'Now it is important to thrust forward without consideration of casualties and exhaustion before the French have had the opportunity to set up again. There is no time for half-stepping now.' It did not take long, however, before von Rundstedt began to feel uneasy at the build-up of French forces on the Aisne. Thus, despite the sixty-mile gap that had opened up before Army Group A, on 16 May its commander stopped anything other than reconnaissance troops crossing a line from Beaumont to Montcornet and ordered that the Sambre-Oise Line fifteen miles further west could only be broken with his personal permission.\n\nTo the north, von Bock was keen to maintain the tempo of his advance and co-ordinate it with von Rundstedt's. With the Allied armies having begun their withdrawal, Hoepner's 3rd and 4th Panzer Divisions moved through the Gembloux Gap while a squadron of 2 Group RAF dropped 250lb bombs in an attempt to slow them. The Germans were ready and waiting for, as the twelve Blenheims approached their target in close formation, they ran into a flak barrage that immediately turned one aircraft into a falling ball of flames. The barrage stopped and moments later fifteen Bf 109s dived on their prey. One of the RAF crew, Sergeant T. J. Watkins, who was himself shot down in the attack, later recalled:\n\n[The fighters] were having a field day shooting bombers out of the sky. Sergeant Grierson's Blenheim belched flames from its mainplane and two others were falling out of the sky, a mass of brilliant yellow and red fire and acrid smoke. In a matter of seconds almost the entire formation was shattered, fluttering to earth in a welter of fragments.\n\nEleven aircraft and thirty-three crew were shot down, leaving just one Blenheim able to return to base. It took a little while for those waiting on the ground to realize what must have happened and then, in the words of the official history of 2 Group, 'Mesmerized, the ground crews made their stunned way back to the hangars and bore the shock for the rest of their lives.'\n\nMeanwhile, the two panzer divisions continued their advance, gradually dropping south to eventually come up on Hoth's right flank and under Army Group A's command for the final push to the coast. That corps continued to observe Rommel with a mixture of admiration and apprehension as he did all that he could to lead the German advance as far and as fast as possible, and such was his mindset that he was not afraid of ignoring or deliberately avoiding direct orders to do so. Indeed, this period of the campaign was later dubbed by Hitler as 'Rommel's private offensive', one which, he told the general, caused him a sleepless night or two as corps, army and army group had little idea where the 'ghost division' was and what it was doing.\n\nCrossing the border into France at 1800 hours on 16 May, Rommel's formation prepared to take on those defences that had been prepared during the second phase of Maginot Line construction, some of which Guderian's corps had already passed through after their crossing of the Meuse farther to the east. Despite orders from Fourth Army to attack 'only if the weakness of the garrison promises a sure success', Rommel did not wait for detailed orders from Hoth and advanced towards the defences in the direction of Avesnes. He later justified such actions by saying: 'The officers of a panzer division must learn to think and act independently within the framework of the general plan and not wait until they receive orders.' On this occasion, his boldness was rewarded.\n\nRommel's aim was to pierce the outer French defences by surprise in daylight, before mounting a division-sized attack on the bunkers of the main line of resistance under the cover of darkness. The first attack was launched at 1830 hours on 16 May and the arrival of the Germans was a surprise to French units from 84th Regiment of 101st Fortress Infantry Division. Rommel later recalled: 'Suddenly we saw the angular outlines of the French fortification about 100 yards ahead. Close behind it were a number of fully-armed French troops, who, at the first sight of the tanks, at once made as if to surrender.' However, once the defenders realized what was happening they fought well from their bunkers, knocking out two tanks and causing casualties in the supporting infantry. 7th Panzer Division was not in the least disconcerted by this robust response and, showing excellent all-arms co-operation and formidable aggression, attacked under the cover of a smoke barrage. In the murk, engineers set to work with their flamethrowers, grenades silenced pill-boxes and high explosives blasted a route through obstacles. Tanks offered direct covering fire, some managing to shoot shells into the embrasures of casemates, while the infantry stormed forward to lend the sappers support and the artillery laid fire down into the enemy's immediate rear area.\n\nAs dusk turned to night, 7th Panzer Division fought their way into and then through the main fortified zone and cleared significant steel hedgehogs that were blocking the road west of Clairfayts. Once those roads were open, the panzers led the way, firing their machine guns as they moved, closely followed by the motorcycle infantry and then the reconnaissance battalion. By 2300 hours the leading group was a mile south of Solre-le-Ch\u00e2teau and Rommel later wrote:\n\nThe flat countryside lay spread out around us under the cold light of the moon. We were through the Maginot Line! It was hardly conceivable. Twenty-two years before we had stood for four and a half long years before this self-same enemy and had won victory after victory and yet finally lost the war. And now we had broken through the renowned Maginot Line and were driving deep into enemy territory. It was not just a beautiful dream. It was reality.\n\nThe only position that had to be cleared by this time was that occupied by the French artillery, which began to target the armoured column. Unlike guns supporting Lafontaine's division at Sedan, they stood firm, emboldened by the defence offered by the infantry, and forced Rommel to storm them, the guns of his tanks firing on the move 'from all barrels' despite the lack of accuracy this would entail. The intended psychological effect was achieved, the artillery line crumbled quickly, some of its crews gunned down and run over by the rampaging armour, and Rommel now had the opportunity to drive deep into the enemy's rear. It was at this point, drunk with his night-time success and feeling invulnerable, that the 7th Panzer Division commander ignored orders stating 'No breakthrough!' and sped on towards Avesnes.\n\nRommel was anxious to avoid contact with superior headquarters during this 'unauthorized phase'. He later explained that his lack of communication was due to 'radio failure', conveniently ignoring the division's superior equipment. It seems that while Rommel could receive communications from his command post back at Froid-Chapelle, he could not contact it. He was also keen to avoid contact with the enemy, but this was sometimes unavoidable as, for instance, on the occasion his leading units ran into French 5th Motorized Division, which was resting along the road. The panzer division seized the opportunity, its tireless commander standing in his command vehicle barking orders into the radio, and wreaked destruction, chaos and havoc over six miles. The attack lasted minutes before the panzers in the vanguard continued their advance and entered Avesnes at midnight, pushed through it to some high ground, and then paused for the waylaid armour and motorcycle infantry to catch up.\n\nWhen the vanguard of the panzer division heard the sound of engines, they initially believed that their colleagues had made better progress towards them than expected, but then realized that these were enemy tanks. They were, in fact, the last sixteen machines belonging to 1st DCR and these were swiftly taken on by the German armour that had just emerged from Avesnes a mile or two from their leading group. A bloody engagement followed which continued until dawn while Rommel's small group, consisting of two tank battalions and three motorcycle platoons, pushed on to successfully take an important bridge over the Sambre at Landrecies eleven miles away. The speed of the advance so confused the French that they were unsure whose tanks these were. While administering a fuel depot in Landrecies, Marc Bloch recalled a brother officer's interest in some tanks after noticing that they were painted an unusual colour and moving away from the front. It was only when he made a move to find out what was happening from their commander that he was told by a comrade that they were German tanks.\n\nThose Frenchmen who did approach the armour were told to lay down their weapons and walk east. Most did so without a murmur and problems were caused by whole units wishing to be taken prisoner. There was one French lieutenant colonel, however, who approached the German command echelon and was in no mood to take orders. Rommel later wrote: 'His eyes glowed with hate and impotent fury and he gave the impression of being a thoroughly fanatical type.' The colonel began to walk east, but Rommel thought that he might try to escape and so ordered him into his tank. He refused and so 'after summoning him three times to get in, there was nothing for it but to shoot him'. As it was, the spearhead group crossed the bridge at Landrecies, leaving a unit behind to protect the valuable structure, and finally stopped its advance when it reached the hills east of Le Cateau at around 0630 hours.\n\nIt was only when a satisfied Rommel had his eyes on Le Cateau with his fragmented division spread out over twenty miles that he reestablished radio communications with his furious corps commander. Rommel claimed not to have received the order to advance no further than Avesnes, despite the operations officer having radioed 'Stop immediately' on several occasions, and said in mitigation that he had endeavoured to gain permission to advance from corps but, on receiving no reply, had used his initiative. Having spoken to Hoth, Rommel suddenly wondered where the rest of his division was, not least because his leading group needed fuel and ammunition. Heading back in a signals vehicle with a PzKpfw Mk III to Avesnes to re-establish contact, he became aware of how long, narrow and vulnerable the corridor he had created was. On his journey he came across around a dozen French lorries. They did not try to escape, and nor did their drivers try to destroy their vehicles but instead allowed themselves to be led back to Avesnes by the general. On his arrival in the town, he handed over his prisoners and ordered that the division move forward without delay to support the isolated group just east of Le Cateau.\n\nBy late morning, the division was moving forward once again. Rommel had broken through the French defences for the cost of 115 casualties and had taken 10,000 prisoners, captured 200 tanks, thirty armoured cars and twenty-seven guns. He had destroyed the ragged remnants of 2nd Corps of Ninth Army, completed the demolition of 1st DCR, run over several other formations and secured a bridge over the Sambre. In an extraordinarily audacious advance of forty-five miles in twenty-four hours, Rommel had pursued a high-risk, high-gain policy and it had come off. At the end of it, one of the general's biographers explains: 'He clearly felt that nothing could stop him, that 7th Panzer Division was irresistible, could win the war single-handed.' Meanwhile, Rommel ensured that Hitler's adjutant, Schmundt, was kept well briefed about events and the fact that both a representative from the Propaganda Ministry and the editor of the Nazi newspaper _Der St\u00fcrmer_ were attached to the division is revealing. This whirl of vainglorious self-publicity had caught the attention of Berlin, and Rommel's superiors were placed in the difficult position of not knowing whether to reprimand him or reward him. In the end, Rommel received a Knight's Cross \u2013 the highest order of the Iron Cross \u2013 for his personal bravery and an advance 'of decisive significance for the whole operation'.\n\nThe noteworthy aspect of the German advance at this stage in the campaign was not just that 7th Panzer Division achieved outstanding success, but that it was replicated elsewhere. Although both Corps Reinhardt and Corps Guderian did not advance all of their divisions to the Oise, by 0700 hours on 17 May 1st and 6th Panzer Divisions had both created bridgeheads on the west bank of the river and had therefore advanced as far west as Rommel. This had been achieved after the two divisions had converged on Montcornet during the morning of 16 May and set the agenda for a meeting between Guderian and Kempf in the town. They decided to continue their attacks and organize how these would proceed using their on-the-spot knowledge of what was a developing opportunity to side-step the orders of von Kleist and, in so doing, thoroughly undermine him.\n\nBy the time that von Kleist had approved von Rundstedt's order prohibiting a push beyond Montcornet, two of his divisions were already well beyond it, with the corps commander travelling with 1st Panzer Division, and they kept on moving westwards. It seems that the Germans were suffering considerable 'communication difficulties' as the exploitation phase began since Guderian, despite his emphasis on excellent communications, did not receive the order. It took a staff officer until just before 0100 hours on 17 May to deliver by hand this important information to the corps commander, by which time he was creating a bridgehead west of the Oise. This task was completed six hours later, when 2nd Panzer Division was twelve miles behind on his right flank. 6th Panzer Division also created an Oise bridgehead some thirteen miles to the north of Kirchner's while 8th Panzer Division was finally crossing the Meuse some twenty-five miles to the rear. Von Kleist's pleasure at this success was certainly tempered by his understandable indignation at Guderian for not only failing to use the chain of command to organize the advance to the Oise \u2013 particularly as it included the division of another corps \u2013 but also, he suspected, wilfully ignoring orders. To von Kleist it was not just the old way of fighting that was being stood on its head; the generals who favoured the new methods seemed to revel in disobedience masquerading as the application of initiative while they failed to pay due respect to the time-honoured command hierarchy.\n\nBoth von Rundstedt and von Kleist had tried and failed to keep the panzer divisions under control on 16 May and had come away with a distinct feeling their fears were being realized and that individual division and corps commanders were putting themselves before the team. This was a critical issue, they argued, that did not just affect the efficiency and effectiveness of the main effort but put _Fall Gelb_ in jeopardy because they believed that a significant French counter-attack was in the offing. Halder did not feel that such concerns were viable and fought hard to avoid an instinctive and conservative reaction. His own anxiety at the situation can be seen in his diary entry for 17 May:\n\nRather unpleasant day. The F\u00fchrer is terribly nervous. Frightened by his own success, he is afraid to take any chance and so would rather pull the reins on us. Puts forward the excuse that it is all because of his concern for the left flank\n\nHalder tried to put Hitler's mind at rest, saying that Luftwaffe reconnaissance revealed no such threat, and he was supported in this view by Oberst Ulrich Liss, the chief of the Foreign Armies West section of Army Intelligence. Even so, the German Supreme Commander could not bring himself to believe that the French were impotent and wished for the infantry to be brought up. Von Brauchitsch and Halder were writhing at Hitler's interference in such operational matters, with the latter arguing that to 'tie up any of its strength on the southern flank' would be a mistake as this would undermine the striking power of Army Group A, and advocating building a defensive front there later with following troops. Far from thinking about defence, Halder wished to develop the offensive and, now that the breakthrough had been achieved, to take a step closer to von Manstein's original plan by conducting a limited advance south-west in order to stop the French from establishing a defensive position on the Somme and Aisne towards Compi\u00e8gne 'with the possibility of subsequently wheeling the right wing in a south-eastern direction past Paris'. Halder believed that 'A great decision must be taken now!' Yet the mood was not conducive to the development of such a radical idea: quite the opposite.\n\nIssued on 17 May, F\u00fchrer Directive No. 12 placed Hitler at the heart of operational decision-making and stated: 'The bulk of the Panzer formations will not cross the line Le Cateau-Laon to the West.' Hitler directed that, for the southern flank to be strengthened, von Kleist's thrust would stop and move into defensive positions while Twelfth Army moved forward to create a protective barrier. Field officers throughout the chain of command thought that this decision was not merited considering the situation that had been achieved and those in 1st and 2nd Panzer Divisions believed that this 'flank psychosis' was contrary to everything that Guderian had said about the ability to ignore threats on the flanks by maintaining the initiative and disrupting the enemy with a hard, fast and deep advance.\n\nHalder's reaction to Hitler's decision is recalled in his diary:\n\nThe F\u00fchrer unaccountably keeps worrying about the south flank. He rages and screams that we are on the best way to ruin the whole campaign and that we are leading up to a defeat. He won't have any part of continuing the operation in a westward direction, let alone to the south-west, and still clings to the plan of a north-western drive. This is the subject of a most unpleasant discussion at F\u00fchrer Hq. between the F\u00fchrer on one side and [von Brauchitsch] and myself on the other.\n\nHalder recognized that a hard-won operational opportunity which could have a strategic impact was being spurned by those who lacked the military sense to see that continued exploitation was required rather than stultification. Just as the French were developing a defensive front along the Aisne and the Somme, OKH lamented, so Hitler had aspirations to do the same. Nevertheless, the Supreme Commander had justified his decision during a visit to von Rundstedt's headquarters at Bastogne on 17 May, details of which were noted in the Army Group A War Diary by von Sodenstern in which he emphasized 'the significance the southern flank has not only for the operations of the whole Army, but also politically and psychologically'. He continued:\n\nUnder no circumstances must a setback occur at this moment anywhere, a setback that would give a fateful rise [in spirit] to our adversaries, not alone to the military, but above all to their political leadership. Thus, the decision, for the moment, rests not so much in the rapid forward push to the Channel, but much more... in the speediest establishment of absolutely _defensive_ readiness on the Aisne... and later on the Somme... Toward this purpose all measures are directed even if time is lost temporarily in the push toward the West.\n\nIt was a decision that handed the Allies the time and the space they needed and, if only temporarily, it gave away the initiative.\n\nThe pressure of the situation began to spill over into the relationships between commanders. At a meeting between von Kleist and Guderian at a Montcornet airstrip at 0700 hours on the morning of 17 May, the panzer group commander's anger spilled out. Such was his fury that he neglected to even greet his corps commander and instead immediately admonished him for disobeying orders. Guderian, as cool as only a man who believed that he was in the presence of an inferior could be, immediately asked to be relieved of his command. Von Kleist accepted without hesitation in unsavoury circumstances which were not conducive to level-headed decision-making. Recognizing that the situation was unravelling, von Rundstedt characteristically despatched List, the commander of Twelfth Army to whom Panzer Group Kleist had been subordinate for two days, to act as peace-maker. Guderian was quickly reinstated and duly informed by List that he had been authorized to launch a 'reconnaissance in force' on the condition that he would not move his corps' command post forward from its current location at Soize, just east of Montcornet.\n\nIt appeared, however, that the lead that had been attached to Guderian's corps was of the extendable type and he took this flexibility as a personal victory. As a result, he would not kowtow to senior officers who, it seemed, could not be trusted with what he had achieved, and he even decided that in future he would issue orders using field telephones back to his corps command so that they could not be tracked by superiors as they would have been over an open radio net. What was more, 10th Panzer Division was returning back into the corps area after it had been released from the fighting at Stonne by the advance of Twelfth Army infantry divisions. The team was back together, and Guderian had every intention of moving forward whenever the opportunity presented itself. His attack would slip neatly into Halder's clarified campaign plan, which had been accepted by Hitler: Army Group B would continue to push the Allied forces back to the coast while Army Group A, its southern front having been protected, would endeavour to prevent the Allied formations dropping south by advancing to the mouth of the Somme. The resultant pocket would then be squeezed while preparations were made for a thrust south-west towards Paris. For this, von Rundstedt would not only receive Hoepner's corps but also XXXIX Motorized Corps commanded by Generalleutnant Rudolf Schmidt. Both were to be placed under Hoth's command to create Panzer Group Hoth.\n\nBy the end of 17 May, Guderian's 'reconnaissance in force' was working west of the Oise while the remainder of his panzer formations complied with Hitler's directive and took a much-needed opportunity to rest, rearm, refuel and reorganize. At this time, Guderian's lines of communication were stretched, vulnerable and an obvious target. The difficulty for the Allies, however, was as it had been since 13 May: getting what resources were available to where they were needed in a timely fashion to do something useful with them. It was a situation that was made more difficult with every passing day as French units were being chewed up and confidence \u2013 from Gamelin downwards \u2013 ebbed away. It was thus left to a young, dynamic officer to see the opportunity presented by the enemy's operational pause and the need to act quickly and decisively. Colonel Charles de Gaulle, who had been given command of the newly formed 4th DCR on 15 May, sought to succeed where 1st, 2nd and 3rd DCRs had failed. Even though only three and a half battalions of tanks had arrived and he lacked supporting infantry, the ambitious officer, who had been a long-term advocate of armoured warfare, spotted another fleeting opportunity and attacked two days later from exactly the area that had been the cause of so much angst among senior German commanders, the southern flank. De Gaulle had scant information but nonetheless ordered his units to strike at the enemy's stretched and poorly defended lines of communication.\n\nOn the morning of 17 May, a single French Char B1 emerged unseen from Cr\u00e9cy \u2013 eight miles from the Oise \u2013 and simply drove up to 1st Panzer Regiment, which was surprised but reacted quickly to halt the enemy's bold probe. Meanwhile, as Guderian was on the verge of resigning during his meeting with von Kleist, de Gaulle's 100-tank main attack was put in from a position several miles to the south-west of Montcornet. The French armour shot up some German trucks in an assault that was wholly unexpected by Kirchner, who was in the process of trying to establish exactly where each of his units were. For a while it looked as if de Gaulle's tanks would sever 1st Panzer Division's line of communications but the situation was brought under control by the prompt action of Johann Graf von Kielmansegg, a junior supply staff officer, who personally organized a defensive position using local troops, some tanks undergoing repair and a handful of anti-tank and anti-aircraft guns that were moving through the area at the time.\n\nA significant engagement ensued in which the French tanks endeavoured to break through even though they lacked infantry and artillery support and had no radios with which to co-ordinate their movements. They had nonetheless made a noteworthy impression on their harassed enemy, who struggled to bring the requisite firepower to bear on the roving machines before the arrival of a hastily arranged Stuka strike seemed to lead directly to a French withdrawal. Close air support was not used as a matter of course during a panzer division's exploitation phase as the Luftwaffe concentrated on missions designed to undermine the enemy's operational mobility and will, but it could still be requested when required, as it was near Montcornet. Yet despite the French breaking off their attack in what seemed to be a clear reaction to the arrival of the German dive-bombers, de Gaulle's decision to withdraw had instead been prompted by the threat posed to his right flank by units from 10th Panzer Division speeding towards the battlefield.\n\nDe Gaulle's sally towards Montcornet became little more than a mere tactical irritation to Kirchner, one which led to twenty-three French tanks being destroyed and one or two hours of disruption for 1st Panzer Division. In the mind of Guderian, it did not change anything. Although those few hours on the morning of 17 May revealed what might have been had the French been able to mount a larger counter-attack, it was in fact the diminutive size of de Gaulle's attack which meant that it was not spotted but gave no cause for operational alarm when its true potential became clear. Although the naysayers leapt on the French infraction as further evidence of the threat to the southern flank, Guderian believed that its lack of strength and the way in which it was contained was evidence that movement trumped immobility. There was, of course, a strong element of ego in what Guderian sought to achieve, but he never lost sight of the potential fragility of _Fall Gelb_ and this is why he felt that the Montcornet halt order was such a mistake.\n\nOn 18 May, Hoth, Reinhardt and Guderian used strong advance combat teams to tidy up their line along the Escaut as the limitations on their advance continued to be enforced. 7th Panzer Division took Cambrai on an advance which had covered 175 miles since 10 May and 6th Panzer Division captured the entire staff of Ninth Army in a Le Catelet hotel and just missed seizing its commander, Corap. 2nd Panzer Division, meanwhile, had taken St. Quentin while 1st Panzer Division had also crossed the Oise and was moving on Peronne. In this way, as the infantry sought to close the gap with the forward armoured formations, all three panzer corps created positions from which they could immediately launch their advances over the old Somme battlefields just as soon as OKW gave the order. For Halder, that release could not come soon enough with Army Group B having achieved the major success of having captured both the vital port of Antwerp and Brussels, the Belgian capital, on 18 May. Now, OKH believed, the time was ripe for the panzers to effect the encirclement of those Allied forces that could be trapped between the Somme and the Escaut, using the massed armour to strike between the two rivers.\n\nIndeed, it was from the sides of this developing corridor that Gamelin saw the opportunity of a counter-attack in what he considered was likely to be the final opportunity for the Allies to salvage something from their precarious position. The French Supreme Commander's ability to act, however, was not only undermined by the difficulty in finding the resources for such an endeavour, but also by Georges's decline into ever greater ineffectiveness and the French prime minister's revitalized attempts to remove him. Matters came to a head on the afternoon of 18 May when Reynaud made a personal visit to Georges to assess the situation for himself. It was a decision that thoroughly undermined Gamelin's position and, to make matters worse, the prime minister was accompanied by his new deputy, the eighty-four-year-old Marshal Philippe P\u00e9tain, who had replaced the sacked Daladier.\n\nReynaud had been given a sense of what he might find at the headquarters, and he found the place struggling to function effectively under the joint strains of enemy action and an ineffectual commander. He and P\u00e9tain returned to Paris convinced that there needed to be changes at the top of the army, and the prime minister himself was determined to take hard decisions before it was too late \u2013 what, after all, did he have to lose? He knew that he would have to act fast as even at this stage his generals could not tell him whether the main threat was to the turning of the Maginot Line, a southward plunge towards Paris, a race to the Channel coast or a combination of all three.\n\nThe success of the withdrawal through Belgium back to the Scheldt, however, at least gave a little breathing space for the Allied High Command. It was a complicated procedure requiring excellent co-ordination under great pressure and one which stretched even the first-class formations under Billotte's command to the limits of their capability and endurance. A junior staff officer at First Army headquarters, Maxime Dufour, had the testing job of compiling reports that provided the chief of staff with an idea of what formations were where. He later recalled:\n\nThe situation was changing hour on hour, and sometimes minute by minute... Information arrived by despatch riders and field telephone. It was only when there was a lull in communication from a division that we began to get concerned for its situation. No message for a couple of hours often meant that it was under heavy attack. By the time we knew what was happening and reacted, several hours had often passed... Generally, however, the divisions withdrew in good order.\n\nAlan Brooke's II Corps also coped admirably but he noted how difficult the situation was for all concerned, writing on the evening of 18 May:\n\nI was too tired to write last night and now can barely remember what happened yesterday. The hours are so crowded and follow so fast on each other that life becomes a blur and fails to cut a groove on one's memory.... Michael Barker [the commander of I Corps] [is] in a very difficult state to deal with, he is so overwrought with work and the present situation that he sees dangers where they don't exist and cannot make up his mind on any points. He is quite impossible to cooperate with. He has been worse than ever today and whenever anything is fixed he changes his mind shortly afterwards.\n\nLike H. C. Lloyd, the commander of 2nd Division, Barker was, in fact, in the grips of a nervous breakdown that sleep deprivation only served to hasten. Tiredness, no matter what arm a soldier was in or what army's uniform he wore, was ever present on the ninth day of the campaign. Henry de la Falaise recalled his Sergeant Ditton 'fast asleep, all crumpled up in a heap at the bottom of the [Morris armoured] car and wedged somehow between all the paraphernalia which clutters it, his head on an ammunition case'. Some were so tired that they fell asleep while eating. One British troop commander did so into his plate of eggs, leading an observer to scribble in his small diary, 'Young Andrew is still asleep in the same position, his face smeared with egg yolk.'\n\nIn such situations, as the Germans also understood, the training, leadership and discipline of formations came to the fore and none, according to Brooke, performed better than Bernard Montgomery's 3rd Division. Its King's Royal Hussars Regiment had a particularly difficult time as they endeavoured to make a fighting withdrawal from the Dyle and across the Senne to allow the infantry the time they needed to reach the Dendre. Under constant aerial reconnaissance which brought down accurate artillery fire and finding their radios jammed by German interference, the regiment was then attacked by enemy tanks. Guy Courage later wrote, '[We] never stood a chance. The enemy was behind and in front of us with vastly greater numbers, with artillery; his speed of movement and his use of anti-tank guns, many of them horse-drawn, well up in front impressed themselves on every mind.' By the end of the day, the regiment had two thirds of its strength. Courage continued, 'That evening we gathered in a field... our thoughts were grim and black. It was so difficult to realize the extent of our losses... [M]any troops had vanished without trace and those of us who had managed to cross the river gradually knew little more than our own small part of the tale. Wild rumours were rife.' The entire division found itself under great pressure which did not let up after the Dendre had been crossed. In a brief note made in his diary about the process of his 3rd Division's withdrawal to the Escaut on 19 May, Montgomery wrote: 'Disengaged in daylight... Used artillery to effect a break-away... Used a lot of shells... Whole DIV back on L'Escaut line by dark. A very tricky operation.'\n\nFedor von Bock was keen to keep the Allies under unremitting pressure as their formations slipped back towards several Channel ports, but Halder recognized that the advantages of this would be undermined if Army Group A failed to make its own advance to the coast in order to complete the encirclement. It was with great relief, therefore, that on the morning of 19 May, OKW gave its permission for the continuance of the panzers' advance westwards the next morning. As 10th Panzer Division continued to move up on the southern flank to take over responsibility for its defence from Kirchner's men and as 3rd and 4th Panzer Divisions completed their own move into line from the north, in between them the other six armoured divisions of the three panzer corps established themselves on the line Cambrai-P\u00e9ronne-Ham ready for the new advance. That evening, Halder wrote in his diary as he mused on plans for 20 May:\n\nThe breakthrough wedge is developing in a satisfactory manner. Follow-up proceeds as planned. Bulk of enemy forces lately operating in Belgium probably straddle Belgium border. Our big tank drive in direction Arras (starting 0700) consequently will squarely hit the bulk of the retreating enemy. It will be a big battle, lasting several days, in which we have the advantage of initiative, while the enemy has that of heavy concentration. But since all the psychological factors work in our favour and we have the benefit of a superior and tremendously effective air force, I am certain of success.\n\nThe big battle in the Arras area which Halder was clearly expecting would be conducted by Panzer Group Hoth in order to protect Reinhardt and Guderian as they entered the twenty-four-mile-wide corridor between the city and French Seventh Army, which was establishing itself on the Somme. During the 19th, de Gaulle had made another attempt to use his 4th DCR to strike at 1st Panzer Division's flank and it was swatted away with little difficulty, but Army Group A remained determined to avoid the enemy slicing across von Kleist's divisions and reconnecting the two parts of the Allied army.\n\nWinston Churchill had also seen the opportunity that had presented itself, writing to Georges on 19 May: 'The tortoise has protruded its head very far from its shell. Some days must elapse before their main body can reach our lines of communication. It would appear that powerful blows struck from north and south of this drawn-out pocket could yield surprising results.' Gamelin, however, finding on the morning of 19 May that Georges was no longer capable of command, drafted the orders for the counterattack himself. Known as _Instruction Personnelle et Secr\u00e8te No.12,_ it detailed a push south by Billotte, who could call on BEF and First Army divisions around Arras, and Touchon's extremely thinly stretched Sixth Army attacking north from the Somme. Gamelin ended his _Instruction_ with the words 'Everything now is a question of hours', and although he was clearly endeavouring to imbue his scheme with a sense of urgency, he too was sacked by Reynaud shortly after its promulgation.\n\nGamelin's replacement, the charismatic General Maxime Weygand, had impressed Reynaud on their first meeting and it was no secret that the prime minister favoured the seventy-four-year-old over Gamelin. As a member of the French negotiations delegation, Weygand had been present at the signing of the armistice by the Germans in a railway carriage at Compi\u00e8gne in 1918. By 1931 he was Chief of Staff of the French Army and Vice President of the Supreme War Council before retiring in 1935. Even so, he was recalled to active service at the outbreak of war and returned to France from Syria to assume command of the Orient Theatre of Operations. As it was, on his return from a visit to see Georges on the afternoon of 19 May the fatally weakened Supreme Commander was not entirely surprised to find Weygand at his headquarters or to receive official notification of his replacement later that evening. Gamelin cleared his desk and left his office for the last time the next morning. The new man had refused a handover and, having indicated to Gamelin that he would cancel _Instruction No.12,_ closed the door behind his predecessor and set to work making his own assessment of the situation.\n\nThe British looked on incredulously as the French High Command seemed to implode. At the start of what was to prove to be a spiralling decline in the relationship between the two allies, Ironside had said on 17 May that the situation was shaping up to be 'the biggest military catastrophe in history' and three days later he had no reason to alter his opinion. In such circumstances, contingency planning for the evacuation of the BEF was entirely sensible. Gort's headquarters had begun the process on 18 May and since then had become convinced that it was time well spent. Even so, Churchill was not willing to authorize the lifting of the British divisions from French soil until it became a military imperative and he still had hopes of a successful attack from Arras by the BEF. To which end on 20 May Ironside travelled to France to meet with Gort and, bypassing the French chain of command, he ordered him to launch a major attack in order to link up with the French forces on the Somme. Gort immediately said that he thought such an attack to be impossible in the circumstances: seven of his nine divisions were on the Escaut and he believed the French to be in no position to lend any support. In the end, however, recognizing the sense in trying to do something, Gort agreed to a limited attack from Arras using two reserve divisions and a tank brigade but emphasized again the precarious position of the French.\n\nWith Gamelin sacked and Georges useless, Ironside immediately went on to see Billotte and Blanchard to find out what they were capable of. The imposing British general later wrote of the meeting:\n\nI found Billotte and Blanchard all in a state of complete depression. No plan, no thought of a plan. Ready to be slaughtered. Defeated at the head without casualties... I lost my temper and shook Billotte by the button of his tunic. The man is completely beaten.\n\nA furious Ironside had to goad the commander of First Army Group into action and made him promise that he would support Gort's attack by ordering an attack towards Cambrai. Billotte agreed to ensure that the two divisions of General Ren\u00e9-F\u00e9lix Altmeyer's V Corps, supported by Prioux's Cavalry Corps consisting of the remnants of 2nd and 3rd DLM and the recently added 1st DLM from Seventh Army, would play their part. Ironside left France unconvinced that the French would act and made it his business to ensure that any plans for a counter-attack did nothing to destabilize the work that was being done on the BEFs preparations for evacuation. He was wise to do so, because, unbeknown to Ironside, Altmeyer was also in a fragile mental state. He told Major Vautrin, a French First Army liaison officer to the BEF, that he believed his troops had no fighting spirit and was willing to take the consequences of disobeying an order by refusing to attack as he 'would not continue to sacrifice his army corps, which had already lost half its personnel strength'.\n\nEven as the British were attempting to extract themselves from a tight corner, the Germans were seeking to make it even tighter. On the morning of 20 May, while Hoth advanced towards a confrontation around Arras, Guderian sought to reach the coast along the Somme and Reinhardt pushed between the two in the centre of the corridor. Hoth made a careful advance, because like Halder he recognized that a major confrontation was likely at Arras, a vital anchor to the Allies' line, as the enemy defended his open flank. Thus, as Rommel moved towards the city from Cambrai he had to retain his self-control. With 3rd and 4th Panzer Divisions having only just crossed the French border and the Sambre, and 5th Panzer Division being attacked by elements of First Army as it crossed the Scarpe, the general knew how vulnerable his division was. As a result, when enemy raids began to affect his lines of communication, he was cautious and ensured that they had been contained and cleaned up before progressing. As he approached Arras from the south, the division was engaged by British troops defending the city, three battalions of Major General Roderick Petre's 12th Division, which ensured that Rommel's advance was a difficult one. On 20 May, 7th Panzer Division suffered more casualties than it had crossing the Meuse and Rommel understood that he needed to be watchful as his division continued to skirt round the city.\n\nBetter protected and moving through the central channel of the corridor was Reinhardt's corps. 6th Panzer Division ran into some British troops from 36th Brigade of Petre's 12th Division who provided it with some considerable resistance at Doullens. Ultimately, however, Brigadier George Roupell's formation could not hold the Germans and Kempf broke through that evening with 8th Panzer Division on his right ready to outflank Arras to the west. The advance of Guderian's corps in the southern sector of the corridor also ran into scattered elements of Petre's 12th Division which fought hard as well but were ultimately unable to do no more than temporarily hinder the advance. 1st Panzer Regiment of 1st Panzer Division raced thirty-five miles and seized Amiens and a bridgehead over the Somme from a battalion of the Royal Sussex Regiment in the early afternoon. 2nd Panzer Division on Kirchner's right flank made an even more remarkable drive and thrust of fifty-five miles \u2013 along the route taken by Henry V to Agincourt, across the old Somme battlefield and into the Allies' rear.\n\nAlthough there were few enemy to cause 2nd Panzer difficulties, the combination of sun and dry ground did. Standing in the turrets of their tanks, commanders were plastered with dust and their eyes became red and painfully swollen as, unlike the motorcyclists, they were not issued with goggles. One of the men in Leutnant Wold-Max Ostwald's reconnaissance troop wrote of his officer: '[He] cools his driver's eyes with a wet handkerchief, although he can hardly keep his own eyes open, and tears are rolling down his blackened, dirt-streaked face.' Anything, in fact, to ensure that his vehicle could keep moving, since a stop, no matter how short, was prohibited. The only limit on the division's advance was its logistics and, as Oberleutnant Dietz noted in his diary, there was concern about this as the leading units pushed on:\n\nThe Battle Group Commander is travelling behind the lead tanks. His expression is grave. There must be something in his calculations that is not quite right. Every now and again he measures with his thumb the distance travelled against what is still to be covered. The Panzer regiments have reported their fuel will last just as far as Abbeville. What then? What could he do with immobilized tanks outside a heavily fortified town?\n\nIn the end, the commander need not have worried. The division did not run into any enemy as they closed in on the mouth of the Somme and reached Abbeville that evening with just enough fuel to spare at 2030 hours, while at 0200 hours, 3rd Battalion 2nd Rifle Regiment reached the coast west of Noyelles.\n\nNow that 2nd Panzer Division had reached the Channel, the door had been closed on an Allied withdrawal across the Somme. Jodl reported in his diary entry for 20 May: 'F\u00fchrer is beside himself with joy. Talks in words of highest appreciation of the German Army and its leadership... Return of territory robbed over the past 400 years from the German people, and other values.' The Netherlands had capitulated, Belgium would follow, fifteen Allied divisions had been lost and the entire northern wing of the Allied army, forty-five divisions including its elite formations, had been encircled between the two German army groups in a giant pocket some 120 miles long and eighty-five miles deep. It contained the remnants of the Belgian Army, the BEF, French First Army Group and some elements of French Seventh and Ninth Armies. Yet although the Germans had achieved a significant and substantial military success in just eleven days, there was satisfaction across the Wehrmacht but no celebration \u2013 the job was still unfinished.\n\nBy midnight on 20\u201321 May, Hitler's forces had a tenuous hold on the coast, but his panzer divisions' lines of communication were stretched and exposed while there remained a threat to them from Arras and the French Seventh Army on the Somme. There was also the huge pocket of Allied troops to be reduced and decisions had to be taken about how this task should be balanced with the need to turn through 90 degrees in order to cross the Somme and Aisne in the next phase of operations. That afternoon, Halder had at least secured agreement at a conference that there would be a new operation \u2013 _Fall Rot_ (Case Red) \u2013 which would thrust into the French interior. Planned by the General Staff, the necessary preparations for the offensive were made without delay. In the meantime, the immediate threats posed by the Allies would be countered while the wedge that had been driven to the coast was strengthened and von Bock exerted pressure from Belgium.\n\nThat night, while the German generals moved into position for a new day of offensive action and planned their _coup de gr\u00e2ce,_ their Allied counterparts were left dazed but increasingly aware that, with their backs against the sea, only courage could bolster their now limited optimism. At the headquarters of the British Air Forces in France, the army liaison officer despaired:\n\nIt looks like the end already. Can this really be true? The thing is farcical; it is an anti-climax without equal. I find it impossible to believe, but we cannot get away from the facts. As instructed in orders, I have destroyed all my private and official papers. Everything is burned. It is one of those acts we read of in tragedies \u2013 something that we never expect to undertake ourselves. It is more than the papers we have burned, for with them we have burned nearly everything that remains of hope.\n\nFor the French army, which was now in the ponderous hands of Weygand, whose findings on 20 May were rapidly making him as pessimistic as Gamelin had been, the activities of the British had suddenly taken on greater significance. Would Churchill stick by the French to the bitter end and send reinforcements, or would they fall back on the Channel ports and evacuate? On the morning of 20 May, the British prime minister had finally recognized that an evacuation might become necessary and reluctantly directed that '... as a precautionary measure the Admiralty should assemble a large number of small vessels in readiness to proceed to the ports and inlets on the French coast'. The responsibility of planning and organizing an operation capable of plucking the BEF from the Channel coast fell to Bertram Ramsay. The fifty-seven-year-old vice admiral, who had been lured out of retirement by Churchill the previous year and given command of the Dover area of operations, now faced a challenge that would draw deeply on his calm and methodical personality. Working from rooms carved into the famous white cliffs under Dover Castle during the night of 20\u201321 May, Ramsay and his team started to gather the vessels required for _Operation Dynamo._ As he did so, eighty miles away across the Channel at his headquarters near Lille, Gort was desperately endeavouring to find enough troops to launch a counter-attack from Arras.\n\n# CHAPTER 9\n\n# 21\u201324 May \u2013 Arras, Weygand and the Halt Order\n\nDURING THE EARLY HOURS of 21 May, Emmanuel Marcel, an infantryman in 143rd Regiment of First Army's 32nd Infantry Division, was nursing wounds that had been inflicted on him by 5th Panzer Division. He could not remember anything about the incident which had left him with a raging headache and an open wound the size of his palm that revealed part of his skull above his closed eye. He was told that he had been breaking from cover with his section when a German shell landed among them, throwing Marcel into the air. He was the only survivor. As Marcel would later explain:\n\nThe man who described to me what had happened was from a neighbouring platoon and had viewed the scene while providing covering fire from a copse of trees. This man was wounded minutes later in the same bombardment after a shell exploded in the trees and left him with large chunks of wood in his shoulders and lower neck. He looked like a ghost but we tried to chat to keep our minds from our predicament.\n\nA dazed Marcel and his comrade, who drifted in and out of consciousness as the night wore on, were at a casualty clearing station in a large farmhouse where their wounds had been assessed and, after some superficial treatment, they were waiting transportation to take them to a field hospital. It was a traumatic wait with the moaning of the delirious, the cries of pain from several and, Marcel remembered, the names of unknown women \u2013 presumably the loved ones of the wounded \u2013 drifting across the stinking, airless room. The man opposite Marcel, his shattered right leg bound in blood-soaked bandages, sobbed 'Mama' quietly to himself. Marcel was desperate to leave the claustrophobic cell, testifying:\n\nI thought that I had been left in the room to die and so, despite my wounds, I tried to stand. Before my legs had straightened, I slumped to the floor. Somebody came over and gave me a canteen of water and told me to stay put as the transport had arrived. My head was in a whirl when the orderlies came in the room and started to remove the stretchers. When it was my turn, I laid back and asked the man holding the handles near my head, 'What's the news?' He replied: 'Gamelin is gone. Weygand is in command. It's all over.' I said nothing. It was over? So soon? Where was the counter-attack that we were promised?*\n\nThe attack to which Marcel was referring had been the talk of his regiment in recent days and they had anticipated a 'great armoured advance south towards the Somme' by a concentration of French tanks that were supposedly gathering around Arras. It is impossible to know whether the regiment had been told about Gamelin's intentions, its officers had made certain assumptions or the attack was merely a rumour, but what is beyond doubt is that Weygand had immediately postponed the counter-attack outlined in _Instruction No. 12._ While Gort had been prompted by Ironside to make an offensive effort out of Arras with his very limited assets on 21 May, the new French Supreme Commander continued to vacillate as he remained wedded to the operational tempo of 1918, and this despite all the evidence to the contrary provided by the German advance since 10 May.\n\nAfter breakfast on 21 May and as Guderian consolidated his southern flank and lodgement on the coast, Weygand flew north for a meeting with Allied commanders. He landed at Calais, which was considered a suitably safe distance from the fighting front, and took a waiting car that was to drive him to his destination in the Belgian town of Ypres. With time of the essence but the roads from the coast filled with troops and traffic, Weygand did not manage to start his discussions with King Leopold until mid-afternoon, but while Billotte arrived later, Gort did not arrive at all. The absence of the British commander at the conference mattered not only because Weygand immediately questioned the loyalty of France's ally, but also because he had decided that there would be a counter-attack across the Somme corridor with the BEF acting as its spearhead. In order for the British to provide the required concentration of troops at the vital point, a contraction of the front was required and this, Weygand proposed, could be achieved if the Belgian army abandoned most of the remainder of Belgium by withdrawing to the Yser. Leopold refused, saying this would be an act of betrayal to his people, but he did offer the compromise of moving back to the Lys. Weygand accepted and, in the absence of Gort, began to work on the details of the attack with Billotte.\n\nAlthough Weygand waited until around 2000 hours in the hope of seeing the British commander, he must have left Ypres feeling that there was a tension between his plan, which leant heavily on the BEF, and what he believed to be an increasing lack of British commitment to continuing operations. As it transpired, the French Supreme Commander was driving back to the coast \u2013 at the start of a return journey to his headquarters after a crucial twenty-six hours away \u2013 when Gort arrived in Ypres. He had not snubbed Weygand but had spent the day trying to find out where the meeting was taking place after a telegram had failed to reach him. Billotte was still in the town, however, and so briefed Gort about Weygand's scheme. His response was non-committal: the plan made sense, but he considered that, with the situation changing with every hour, it might not be possible. This was a view partly shaped by reports that Gort had received about the counter-attack that had been launched at Arras that day.\n\n7th Panzer Division had the task of completing their advance around the south and west of Arras while the remainder of Panzer Group Hoth sought to break through French First Army to the east. The aim was to isolate and then unseat the Allies from the city before turning north into what was, on 21 May, an open flank between Arras and the coast. In this task Rommel's division was to be supported on his left flank by the untried SS Totenkopf Division. Formed in October 1939 from concentration camp guards, the division brought ideological zeal and fanaticism to France and was keen to prove itself. As it was, by the time that the British attacked at 1500 hours on 21 May, the main body of both German divisions were advancing on a three-mile front with its right flank just two miles south of the city. Rommel was advancing warily, but not tentatively, and had ordered his commanders to move quickly in order to provide as fleeting a target to the enemy as possible. Accordingly, he ordered that 25th Panzer Regiment would lead the attack, followed by his armoured reconnaissance regiment to hold the road open and create a link with the two rifle regiments that followed. The formation's right flank was screened by artillery but no close air support was planned.\n\nThe British aim was to reach the River Cojeul six miles south of the city and so decapitate the advancing enemy force. With Billotte having informed Gort that Altmayer's V Corps force would not be ready to lend any support until 22 May, it was left to a pitiably small British force to take on two German divisions. This force, commanded by Major General Harold Franklyn, was divided into two columns: 4th Royal Tank Regiment (RTR) led by 6th Battalion Durham Light Infantry (DLI) on the left and 8th DLI led by 7th RTR on the right. Thus, eighty-eight British tanks were to act as the spearhead against 7th Panzer Division, while sixty tanks from the battered remains of 3rd DLM were to engage SS Totenkopf Division on Franklyn's right flank. No air cover was available, the attack had been hastily developed and there had been no time for reconnaissance. Gort did not have much hope for the endeavour: the tiny force, the lack of preparation time and the failure of every French counter-attack since 14 May did not bode well. Even so, the plucky thrust at Arras on 21 May was destined to have considerable \u2013 and unexpected \u2013 operational significance.\n\nThe attack by 7th RTR and 8th DLI did not start well and revealed the lack of co-ordination inherent in the endeavour when, on crossing the Scarpe, it was shot up by the French tanks to its right. Then, shortly after, the British tanks lost contact with the infantry when they mistakenly split down parallel routes, only for 7th RTR to drift into the left-hand column's sector. Throughout the attack the armour struggled to ensure that it was where it was supposed to be because, as Trooper 'Butch' Williams has recalled, 'We had no map of the area, nor any idea of the objective, neither had we been allowed to net in the radio.' 8th DLI was not in fact reunited with its tanks again until they reached the village of Wailly, three miles south-west of Arras, which had just been occupied by a battalion of Rommel's 7th Rifle Regiment. Much of the German armour, accompanied by the division commander himself, had already moved through to the west of the city and so the arrival of the British tanks initially caused some considerable concern. Rommel was immediately informed of the attack and with Leutnant Most, his ADC, drove back to find that the British had 'created chaos and confusion among our troops in the villages and they were jamming up the roads and yards with their vehicles, instead of going into action with every available weapon to fight the oncoming enemy. We tried to create order.'\n\nThe British tanks had been held up initially outside Wailly by a howitzer battery and Rommel surveyed the scene from Hill 111 in the middle of his rapidly developed defensive line. He later wrote:\n\nWith Most's help, I brought every available gun into action at top speed against the tanks. Every gun, both anti-tank and anti-aircraft, was ordered to open rapid fire immediately and I personally gave each gun its target. With the enemy tanks so perilously close, only rapid fire from every gun could save the situation. We ran from gun to gun... All I cared about was to halt the enemy tanks by heavy gunfire.\n\nSeveral British Matilda Mk Is were knocked out, but rounds bounced off the Matilda Mk IIs, which continued to creep forward, destroying the anti-tank guns. Leutnant Most fell at his general's side, mortally wounded, with Rommel later writing, 'Blood gushed from his mouth, [he] was beyond help and died before he could be carried into cover beside the gun position. The death of this brave man, a magnificent soldier, touched me deeply.' The attempted breakthrough was eventually stopped by a mixture of the guns of a nearby artillery battery, 7th Rifle Regiment's reinforcement of Wailly, and the motorized element of SS Totenkopf, which had closed up to the west of the village.\n\nThe British left flank of the attack, 4th RTR and 6th DLI, made better progress and engaged the 6th Rifle Regiment a little over a mile south of Arras either side of Agny. East of the village, the Germans had hurriedly established an anti-tank gun blocking position, but the Matildas rolled right over this and, leaving the DLI to tackle the German infantry, linked up with tanks advancing to the west of Agny. Together, this armoured battering ram advanced a further three miles towards Mercatel and on a three-quarter mile front thumped into the rear of 7th Rifle Regiment.\n\nMeanwhile, on the far left of 4th RTR's attack east of the village of Beaurains, reconnaissance officer Second Lieutenant Peter Vaux was leading the advance. Three hours after crossing the start line, they attacked a group of 6th Rifle Regiment lorries, infantry and anti-tank guns. It was so successful, Vaux later recalled, 'We really didn't see why we shouldn't go all the way to Berlin at the rate we were going.' The advance was soon stopped, however, by a line of infantry, artillery and 88mm anti-aircraft guns Rommel had ordered to form behind Tilloy, Beaurains and Agny and which successfully prevented a breakthrough by 4th RTR along its attacking front.\n\nPeter Vaux's account of the line's fortunes east of Beaurains is chilling. Having been sent by his commanding officer, Lieutenant Colonel Fitzmaurice, to encourage a lone French tank that had appeared to their rear to join the group \u2013 which its commander refused to do \u2013 Vaux endeavoured to rejoin the advance. He approached a hayfield containing around twenty British tanks, including that of his colonel, which were being engaged by those powerful 88mm guns. Moving through 'A' Squadron, Vaux instinctively felt that something was not right and later said:\n\nI thought it very odd that they weren't moving and they weren't shooting, and then I noticed that there was something even odder about them \u2013 because their guns were pointing at all angles; a lot of them had their turret hatches open and some of the crews were half in and half out of the tanks, lying wounded and dead \u2013 and I realized then, suddenly, with a shock, that all these twenty tanks had been knocked out, and they had been knocked out by these big guns and they were, in fact, all dead \u2013 all these tanks. In the grass I could see a number of black berets as the crews were crawling through the grass... and getting away \u2013 those who were not dead.\n\nRecognizing that he was now a target, Vaux needed to act quickly. While reversing to allow the tank's machine gunner to depress the weapon enough to engage the treeline in which the enemy were ensconced, a German rifleman took aim at Vaux's head, resting his weapon on a kitbag. The British Officer's adjutant, Captain Cracroft, in another tank spotted this and, Vaux later recalled, 'pulled out his revolver and, quick as a flash, he shot the chap in the throat. It must have been a jolly good revolver shot, and it saved my life.' Having raked the woods with its machine gun, Vaux's tank withdrew under heavy fire, leaving its commander to ruminate on what he had just experienced:\n\n[M]y heart sank because I realized what had happened: there were all those tanks that I knew so well \u2013 the familiar names \u2013 Dreadnought, Dauntless, Demon, Devil; there were the faces of these men with whom I had played games, swum, lived with for years \u2013 lying there dead; and there were these tanks \u2013 useless, very few of them burning but most of them smashed up in one way or another. And as the Adjutant and I drove back up to the top of the hill, one realized that this really was it. This was a tragedy. This was the end of the 4th Tanks as we knew it. In that valley, the best of crews, our tanks, our soldiers, our officers were left behind.\n\nBy the evening, the Allies were in withdrawal, hounded by 300 Stukas. Having failed to break through, Franklyn wanted to disengage his force before nightfall while he still had the semblance of a force to disengage. His attempt, however, was undermined by 25th Panzer Regiment, which lunged towards Arras from its advanced position near the 3rd DLM start line. While the German tanks failed to block the withdrawal routes due to the sterling work of a handful of anti-tank guns and the courage of some British and French tanks in the gathering gloom, it certainly upset the Allied equilibrium at a sensitive time. By the time the fighting petered out at midnight on 21\u201322 May, Franklyn was fraught but pleased to have completed a 'short-armed jab'* at the enemy without being wiped out.\n\nConsulting their maps, both Franklyn and Rommel would have seen that the front line at the end of the battle closely resembled what it had been at the beginning \u2013 but with the miserable addition of corpses, burning vehicles and piles of twisted metal. The British lost 100 infantry casualties and forty tanks, the French twenty tanks and the Germans 400 men, scores of vehicles, twelve tanks and a similar number of guns. The counter-attack was a chastening experience for both sides, but while the Anglo-French effort might only be said to have stalled one or two German divisions for a few hours and temporarily protected the Allied right flank, the day was soon to have more far-reaching implications for the invasion. During the afternoon of 21 May and over the days that followed, Hitler and some of his more circumspect generals also pored over maps and entered into conjectural discussions about what the Allies might have achieved. One of those generals was Gerd von Rundstedt, who later wrote:\n\nA critical moment in the drive came just as my forces had reached the Channel. It was caused by a British counterstroke southward from Arras toward Cambrai, on 21 May. For a short time it was feared that our armoured divisions would be cut off before the infantry divisions could come up to support them.\n\nWhy von Rundstedt was quite so concerned can be found in Rommel's desire to protect his reputation \u2013 and even develop it \u2013 by reporting to his superiors that he had been attacked by 'hundreds of British tanks and five divisions' as opposed to considerably less than 100 tanks and two battalions of infantry.\n\nArras offered a salutary warning that immediately began to fester in the minds of some German officers because they overestimated what the Allies were capable of at this point in the campaign. While it was true that Churchill, Ironside and Gort, Weygand and Billotte were all endeavouring to take the opportunity that the Somme corridor presented, by the end of 21 May the chances of mounting a well-resourced and co-ordinated counter-attack had significantly diminished as the Allies sought to maintain their cohesion while they continued their withdrawal and to establish a defensive line from Arras to the coast. With Guderian's divisions trying to turn the open flank, Panzer Group Hoth exerting great pressure on the British contingent at Arras and against First Army, while Army Group B pushed against the BEF and the remains of Leopold's army in Belgium, the Allies were stretched and barely coping. Churchill himself was in a quandary for, although he wished to see the Allies strike back before the Germans reached the Channel ports, he was concerned that he was waiting for something that was impossible and imperilled the BEF. In an attempt to ascertain what was and what was not possible, the British prime minister flew to Paris on 22 May, to be greeted by unfounded rumours that Gamelin had shot himself. Here he agreed to the 'Weygand Plan', which was very similar to that cancelled two days earlier but now had a clear form: a pincer attack against the Somme corridor. Formations from the newly formed Third Army Group under General Antoine-Marie-Beno\u00eet Besson and consisting of Fr\u00e8re's Seventh Army on the left and Touchon's much diminished Sixth Army on the right would strike north to Amiens, while First Army struck south from Arras.\n\nChurchill could not ignore the fact that the situation remained unpropitious for any Allied attempt to deal the Germans a blow. Indeed, on 22 May the delayed counter-attack by Altmeyer's V Corps and Prioux's Cavalry Corps at Cambrai turned out to be a damp squib. Although it managed to struggle to the outskirts of the town in Hoth's rear, the French formations were ejected by rapidly concentrated Luftwaffe attacks \u2013 including some by squadrons of obsolete Henschel Hs 123 biplane dive-bombers \u2013 and 88mm anti-tank guns firing at point-blank range. Britain had been asked to provide more RAF support over France for several days, but now the French became insistent that it was essential. Weygand endeavoured to prise Fighter Command from Churchill's hands at their meeting on 22 May, claiming that its presence would make all the difference. The British understood the impact this transfer might have, but they were far from convinced that it would be decisive, and Churchill could not help remembering his Gallipoli experience and the lessons that military history taught about not reinforcing defeat. As a result, he agreed to nothing, despite being aware that the French would see it as yet more evidence of British selfishness. Indeed, British aircraft had become a symbol of hope to the French, with Alexander Werth noting after conversations with his Parisian neighbours: 'They all agree that the R.A.F. is magnificent. Curious how during the last few weeks the R.A.F. has become more popular in France than the French Army.'\n\nThe RAF had indeed become more visible as General d'Astier de la Vigerie husbanded his own aircraft for the protection of Paris while 2 Group struck operational targets which mirrored those being targeted by the Luftwaffe: roads, railways and troop concentrations. There was not a single British squadron that did not understand how much the ground forces were relying on them to slow the enemy's advance and consequently the intensity of operations increased with every passing day. Flying a Blenheim, Flight Lieutenant Webster received orders on 22 May to attack a column of German vehicles that had been spotted approaching the battlefield:\n\nWe were leading and flying really low \u2013 literally climbing over hedges and passing between trees... Ahead of us and running directly across our flight path was a road crammed with AFVs upon whom we unloaded our 250-pounders before they saw us coming... At the same time we spotted a similar convoy on a parallel road to the first, a few fields farther on, so we continued straight on, intending to let them have our 40-lb anti-personnel bombs. Whereas the first lot seem to have been taken completely by surprise, the second were wide awake and gave us a right old pasting as we came in. It was a very hectic and noisy few seconds with one particularly loud bang right under my seat accompanied by a strong smell of hot, angry cordite.\n\nWebster duly landed his burning aircraft in a French field and joined a group of refugees on the road and headed towards the coast. By 22 May, 2 Group had lost forty-four Blenheims and 108 crew killed, with many more wounded or missing in action. Ground crews worked in shifts around the clock to get badly damaged aircraft patched up and airborne as quickly as possible as the pressure mounted on squadrons to fly increasing numbers of sorties each day. As another pilot, Peter Mansfield, later testified:\n\nWe had little idea what impact we were having on the campaign \u2013 if any \u2013 but we knew that things were not good on the ground and we did our bit. We were told that this railway and that column had been put out of action, which was good for our morale, but what we really wanted to hear is that the army had counter-attacked and turned the situation round. Sadly, that was not to be.\n\nAlthough initially postponed for several days in order to prepare and deploy its strike force, the Weygand Plan was never launched as the Allied situation continued to deteriorate. Although Arras had been identified as the springboard for the northern pincer of the counter-attack, Gort found it impossible to hold the city and protect the vulnerable flank back to the coast. Moreover, co-ordination of the Allies' effort was becoming increasingly difficult as the time, space and resources that they had at their disposal were reduced hour on hour. It was further undermined by the death of General Billotte, who on leaving Ypres on the evening of 21 May had been involved in a traffic accident when his car hit a refugee lorry in the darkness, leaving him in a coma from which he never emerged. Yet despite the significant hole that Billotte's accident left at the heart of the command structure in this vital sector, with a lethargy that was representative of the pedestrian French response to any crisis, the new First Army Group commander, General Georges-Marie-Jean Blanchard, was not appointed Billotte's successor for three days.\n\nWhile the French continued to immerse themselves in a dangerous mixture of indecisiveness and hope and the sceptical British looked more to the salvation of evacuation with every step they took back towards the coast, the battlefield shrank and became increasingly congested. One of the many challenges that the belligerents were faced with was trying to go about their business with civilians in their sights. A mobile circus was caught in an artillery exchange near the Belgian town of Tournai which injured many of the animals, causing the elephants to charge around and roar amongst the blasts and smoke. They were joined by the mentally ill who had been released into the town, where they roamed vulnerable and frightened but barely noticed by hundreds of locals desperately trying to escape a scene from perdition. This was the fate potentially waiting for all those who had decided not to leave their homes and who now found themselves in the middle of a battlefield. Douglas Manning, a British gunner in support of 3rd Division, recalls being given an order to stop firing despite the fact that the enemy could clearly be seen advancing towards them:\n\nOur officer said that the village that we were gradually demolishing still contained civilians, that women and children had been seen by our forward observer running from the buildings. The Germans did not seem to have the same ethical concerns as we did in such matters and I personally witnessed the sad sight of the bodies of women and children lying under piles of rubble. They had been caught in a hell that they could not escape, like mice in a trap.\n\nElsewhere, refugees continued to trudge west, where Hans Becker of 7th Panzer Division spotted a large group waiting to cross a dusty track in the wake of the fighting at Arras and was sympathetic to their plight:\n\nI thought to myself, what if you had to leave your house and farm and didn't know when you would return, and ended up looking like this. This had an effect on me. _'\u00c7'est la guerre,'_ as the French would say. But the sadness was if someone returned later and saw that his house was destroyed. What must this person be thinking? He must be really angry with the Germans\n\nThe Reynaud government was in fact endeavouring to stem the tide of refugees for their own safety, but also to reduce the burden of responsibility that it felt to assist them and to stop their numbers from clogging up the same roads that the military were relying on. As a result, an official government communiqu\u00e9 told the population to stay in their homes, saying:\n\nThe Germans... wish to create panic among populations to incite them to leave towns and villages \u2013 even those which are situated far from the combat zones. Their aim is in this way to block roads and hinder the manoeuvre of the Allied forces. Our military operations are being slowed down as a result. German planes are taking advantage of this to bomb and gun down both military personnel and civilian evacuees indiscriminately. Such bomb attacks are considerably aggravating the sense of panic in the population.\n\nThere was a good deal of truth in this, for with the shift of the Luftwaffe to operational targets had come deliberate attacks on refugees. Initially, the German raids sought out military columns, which were attacked despite the awareness that refugees shared the roads, but before long some pilots began to target civilians in an attempt to cause chaos and fear. The Lebeouf family \u2013 a mother and her two young children \u2013 had left their home in Reims on 19 May and were attacked on 21, 22 and 23 May. Diane Lebeouf, who was just twenty in 1940, later recalled:\n\nWe left in a hurry with just a suitcase between us and a bag with some food. We spent the first night with another family, sheltering in a barn but were back on the road early the next morning. We joined hundreds of others shuffling along the dusty road heading west... Then up ahead we saw people running, and then screaming. They ran into the woods or just threw themselves onto the verge as bullets ripped into the earth around them. There were no casualties, but some would not come out of the woods while others \u2013 often old men \u2013 shook their fists at the sky... It was a lone aircraft, but on the second day it was a group of three and they swept the roads twice. Five people were killed \u2013 two were children, and several others were wounded. We did what we could for them. I thought about taking to the country lanes, but in the end thought that there was safety in numbers, particularly as we heard that German troops were getting close.\n\nAnother family was undertaking a similarly fraught journey having left Albert on the Somme. Julia Lapointe noted in her diary: 'Children are tired. We are thirsty and finding it difficult to get food... Everybody is finding the march very difficult and are concerned to see soldiers with no rifles walking with us. Still, we have been told that P\u00e9tain is now in government. There is hope still!' Indeed, P\u00e9tain's appointment as deputy prime minister had been well received in France, although London was cynical as he had long been known to be against the war and 'defeatist'. But the victor of Verdun had already managed to deliver a triumph out of a seemingly hopeless situation in the First World War and now Reynaud said that the Marshal was 'putting all his wisdom and strength at the service of our country. He will remain there until we are victorious.'\n\nIn spite of everything, France maintained a sense of hope, and in the wake of the Arras counter-attack OKW seemed intent on giving the Allies the vital breathing space that they yearned for to establish the defensive pocket forming around the Channel ports. Hitler used events on 21 May to chide Halder and argue that as a politician he read the operational situation better than the experienced military professionals. He also quoted von Kleist, von Kluge and von Rundstedt, who all remained concerned that the panzer divisions remained extremely vulnerable. As a result, the panzers were paused for twenty-four hours after the commencement of Franklyn's counter-attack because, von Rundstedt ordered, the priority was 'first of all to straighten out the situation at Arras' before any attack might be mounted on this particularly vulnerable sector of the line. In the meantime, 2nd Panzer Division prepared to attack Boulogne, 1st Panzer Division readied itself for a lunge towards Calais, while 10th Panzer Division, which might have been used to strike into the soft Allied flank towards Dunkirk, was instead ordered into reserve. Reinhardt's divisions, meanwhile, were to continue their attack on Guderian's right, while Hoth's dealt with Arras and the surrounding area during the pause to the west.\n\nHalder and Guderian were extremely frustrated by what they believed to be an unnecessary pause over exaggerated fears about a counter-attack at a time when von Kleist's divisions were poised to strike behind the Allies to deny them the ports that offered the possibility of reinforcement and evacuation. The intelligence that OKH was reading indicated that the enemy was barely keeping his front together, that his troops were demoralized and, as a consequence, they were ripe for the plucking. Hitler, however, revelled in his intuition and squandered the chance to unhinge the Allied defences. The survival of the Allied pocket depended on a successful defence of the Channel ports. While Ostend was seen as expendable in view of the likely Belgian withdrawal back to the Yser, Boulogne and Calais were most immediately in danger but regarded as vital support to the central port of Dunkirk during this period. Allied plans, therefore, were to send formations out to the western ports to prepare their defence while a defensive line was developed along the line of the La Bass\u00e9e and Aa Canals emanating from Gravelines, less than ten miles west of Dunkirk and stretching south-east to St Omer, B\u00e9thune and then, after Arras had been evacuated, down to Douai. Thus, for as long as Arras continued to offer resistance and the panzers were paused to the west, the Allies could rush troops into defensive positions.\n\nOn 22 May, Rommel was still licking his wounds and reorganizing while 5th Panzer Division came up on his right flank and 20th Motorized Division had arrived and was poised to attack through the city itself. The aim was to widen the corridor to make an Allied counter-attack to link up with French troops on the Somme far less likely. It worked, for on the following day Hoth's divisions pushed north heading towards B\u00e9thune and forced British 5th and 50th Divisions in what one officer called 'a nightmare withdrawal' towards Ypres. They nonetheless resisted with great fortitude to slow the panzer divisions, knowing that their colleagues needed every hour that could be provided to firm up the west flank and allow the BEF and French First Army to withdraw out of a salient, its nose resting on Valenciennes. This bulge in the line had been formed by a synchronized advance by several German divisions with von Bock's Sixth Army pushing from the east. Its aim was to encircle seven enemy corps and promote the rapid collapse of the Allied pocket.\n\nEven as Hoth was pushing north through Arras, Gort was using the diminishing length of his front line to release formations to strengthen the line north of the city to the coast. This included 2nd and 48th Divisions, which were moved to the La Bass\u00e9e and Aa Canals, while units were also sent to Boulogne and Calais with orders to hold them to the end: Gort recognized the likelihood that the Germans would isolate them and deal with the towns piecemeal and that there was merit in tying up as many of the enemy forces for as long as possible. If the Allies had one unifying intention during this period, it was to sacrifice the lesser for the greater. Thus, if a unit had to fight to the point of annihilation in order to benefit the mass of formations elsewhere, it was done.\n\nBoulogne was garrisoned by 1,100 French naval personnel commanded by Commandant Dutfoy de Mont de Benque, who, having lost control of his men, a significant proportion of whom were wandering the streets drunk on 21 May, decided to withdraw back to Dunkirk. Meanwhile two Guards battalions \u2013 the _cr\u00e8me de la cr\u00e8me_ of the BEF and recently arrived in France \u2013 had been sent to reinforce him. Known as much for their discipline and tenacity as their military skills, the units of Brigadier William Fox-Pitt's 20th Guards Brigade arrived at the port on the morning of 22 May just after the last French sailors had left and found the town being bombed and burning. The Guardsmen immediately set about establishing just the sort of defences that Guderian feared they would. The Irish and Welsh Guards together amounted to 1,962 men supported by the brigade anti-tank regiment and an attached anti-tank battery. The town had also seen the recent arrival of around 1,500 men of the Auxiliary Military Pioneer Corps (AMPC) who had been sent to Boulogne for evacuation. Although largely unacquainted with fighting, being an organization that handled stores and engaged in road-building and construction projects, they immediately became a reserve, along with some similarly untrained French and Belgian units, for the 7.5-mile perimeter around the town.\n\nFox-Pitt's force was attacked by 2nd Panzer Division even as they were digging in. Now that von Rundstedt's pause order to Guderian had expired, the formation had worked its way through a regiment of the French 21st Infantry Division, which had rushed into position as a screen, and then fell on Boulogne. The Irish Guards were the first to be engaged in a two-mile sector to the south-west of the town. After a careful reconnaissance, the Germans probed forward in the early evening. Sergeant Arthur Evans of the Irish Guards' anti-tank platoon remembers an artillery bombardment preceding the arrival of the first tanks:\n\nI could clearly see the tank commander's head above the open turret with his field glasses to his eyes. We opened fire and the tank rocked as we scored two direct hits. The crew baled out and abandoned it. Soon a second appeared and that too, was effectively disposed of.\n\nNevertheless, the brave defence of the Irish Guards was not enough to stop the Germans reaching Boulogne. Although a few units from 21st Infantry Division managed to fight their way through to the town during the evening, the Germans infiltrated and probed throughout the night and the battle continued to rage. Early the next morning, 23 May, the attack continued with a concentration along the coast from the south in an attempt to create an encirclement. One hundred and seven men of the Irish Guard's 4th Company took the brunt of the attack and held the line, but at the cost of ninety-eight casualties. The Germans were more successful with a push against the eastern part of the town held by the Welsh Guards during which they managed to open up a gap between the two British battalions. This was filled by 800 nervous men from the AMPC, but the pressure exerted on the perimeter that morning had caused considerable casualties and forced the defenders into tighter positions around the port, which immediately led to bitter, close-quarters fighting. One German infantryman, eighteen-year-old Marc Leicht, recalled a bloody and passionate struggle, with heavy casualties on both sides:\n\nWe fought with bayonets and rifle butts and elbowed our way into positions, often to lose them again in mad counterattacks. The enemy came at us screaming and battered us down. We did the same \u2013 and so the battle continued.\n\nThat afternoon, the First World War destroyer HMS _Vimy_ arrived at the quayside as it was being shelled to evacuate the wounded and also disembarked a naval demolition team tasked with disabling the port facilities. The vessel passed orders to Fox-Pitt that he was to hold Boulogne at all costs. In this task, he was now supported by several other Royal Navy destroyers, which lobbed 4.5-inch shells into the columns supporting the German troops fighting in Boulogne. The co-driver of one ammunition truck fled when the first of them exploded in a field one hundred yards from his vehicle. The driver, Gerd Reismann, a twenty-three-year-old from Hamburg who had already been injured in an accident with a Mk III panzer a week earlier, recalled:\n\nIt was dangerous and, as a replacement, I am not surprised he ran. It was his first time in action and the power of the explosions so close could be felt. The truck was showered with dirt, and some pieces of the shell casing ripped through the canvas covering the load area, and peppered my door. Why didn't I run? I was too tired and was convinced sooner or later that I would die. Why not now, honourably? At least it would be quick.\n\nReismann drove past several burning vehicles and others that had been reduced to barely recognizable mangled metal to where his MG 34 ammunition and hand grenades were off-loaded. He later said:\n\nWhile I was pleased to see the back of those deadly boxes, I knew that they were gratefully received by the infantry... Looking at the smoke over Boulogne and [hearing the] rat-a-tat of small arms fire, I was glad to be turning away. It looked and sounded like hell.\n\nThe Germans continued to make progress which, although it could be measured in yards, began to fragment the defences and rendered the British ability to withstand continued assault highly unlikely. In a decision that was based more on compassion than sound military thinking, Churchill made the decision to evacuate as many of 20th Guards Brigade as could be embarked before the enemy overran the remaining defenders. During a lull in the fighting \u2013 one instigated by the German infantry withdrawing a safe distance in anticipation of a Luftwaffe raid to render the port unuseable for an evacuation attempt \u2013 commanders organized their units for a withdrawal in a way that would allow as many of their men as possible to go down to the quayside.\n\nThat evening, while more than sixty German bombers attacked, the evacuation began. A modern French destroyer was sunk, but the arrival of twelve Spitfires overhead managed to protect two British vessels that were tied up while their crews urged Fox-Pitt's men aboard. Even so, the commander of HMS _Vimy_ was mortally wounded and the commander of HMS _Keith_ was killed during the air raid and eventually a decision was taken to depart for Dover while the ships were still able to do so. Although some 4,368 men \u2013 including French and Belgian troops \u2013 were lifted from the port, some 400 British troops and nearly all who had arrived from the French 21st Infantry Division were left behind. In a remarkable defence, those who remained in Boulogne continued to fight until they were overwhelmed on the morning of 25 May.\n\nThe action at Boulogne was representative of many that were taking place across the front as the defenders tied up German resources, caused casualties and took the initiative from the enemy. All were courageous and, while they seemed like futile attempts to thwart an opponent of great strength, should be seen more as parts of a co-ordinated action that allowed for a perimeter to form around Dunkirk and for evacuation plans to be advanced. The very idea of lifting troops off the French coast rather than reinforcing them was an anathema to many of the French decision-makers, who declared the British self-interested and seemed unable to see the sense in salvaging as many troops as possible from the pocket, not only to protect Britain but also to act as a cadre to strike back at the Germans elsewhere in France. The British were, perhaps, as selfish as the French were illogical, but at least London's position had the virtue of ensuring there could be long-term resistance to Hitler, no matter what the outcome of the current German offensive.\n\nNevertheless, Churchill later regretted his decision to evacuate Boulogne and, when it came to Calais, he was determined that 'fighting to the finish' would mean exactly that. By 24 May, 10 Panzer Division was bearing down on the ancient coastal town while other formations exerted pressure on the newly established British canal-line defences between Gravelines and B\u00e9thune whose creation had only been allowed by the panzer pause of 21\u201322 May. Panzer Group Kleist, including Schaal's division, took on the Allied west flank with Guderian attacking north of St. Omer and Reinhardt either side of Aire. Panzer Group Hoth used Hoepner and General Rudolf Schmidt \u2013 now commanding a corps that included 5th and 7th Panzer Divisions \u2013 in an advance either side of B\u00e9thune. Both Guderian and Reinhardt seized crossings over the Aa Canal on 24 May which left 1st Panzer Division less than ten miles south-west of Dunkirk, the last port with the facilities for a major evacuation. But as they were on the verge of completing the encirclement of approximately one million Allied troops, the panzers were halted once again.\n\nThe reason for this decision can be explained by the continuing concern about the vulnerability of the thirteen divisions in the two Panzer Groups north of Arras. Although it was Hitler who formally ordered the infamous Halt Order, it was von Kleist who on the morning of 23 May had expressed his concerns to the commander of Fourth Army, General von Kluge. Von Kleist wrote:\n\nThe Group now lacks the strength to mount an attack to the east against a strong enemy after losses taken after fourteen days of fighting which include half of all tanks. It should be noted that if the enemy launches a strong attack the Panzer Divisions are poorly suited to defence.\n\nWith this ammunition to hand, it did not take much to persuade von Kluge and von Rundstedt to conclude that the campaign had moved on from an armoured pursuit to an infantry struggle and, as such, required another pause to bring forward more divisions for both attack and defence. The result was an order issued to the mobile elements of Army Group A to wait on the canal line until 25 May while the infantry closed up. Only von Kluge's II and VIII Corps, pitched against the nose of the Allied salient protruding forward from Lille, were allowed to attack. OKH and the panzer commanders vehemently disagreed with the decision, not least because their own information and instincts could not be reconciled with the belief that there were 'strong enemy forces' ready to upset any attack against the Allied west flank. Indeed, Halder commented in his diary, 'The enemy's fighting power probably does not amount to more than local resistance,' and 6th Panzer Division's war diary noted:\n\nIn contrast to combat operations in the past, the tempo of the advance, dictated by operational considerations, has become slower than the tactical conditions would have permitted. The Division could have attacked out of the bridgehead it had been able to win with full force at dawn on 24 May against an inferior enemy to the east.\n\nGuderian himself could hardly understand what might have led to such an order, later reflecting:\n\nWe were utterly speechless. But since we were not informed of the reasons for this order, it was difficult to argue against it. The panzer divisions were therefore instructed: Hold the line of the canal. Make use of the period of rest for general recuperation.\n\nMeanwhile, the OKH looked on incredulous and, fearing that the continuation of such a conservative approach to operations might cripple the ability to attain a strategic victory, intervened. Halder and von Brauchitsch no longer had faith in von Rundstedt and this led to von Brauchitsch acting in a most uncharacteristic fashion. At midnight on 23\u201324 May, von Rundstedt was informed that Fourth Army was being transferred from his command to Army Group B the following evening. This was officially justified as being required to unify the direction of operations under Army Group B for the last phase of the encircling battle while Army Group A, with one panzer division and both Twelfth and Sixteenth infantry armies, would provide flank protection along the Somme.\n\nHalder did not agree with this decision: he was concerned that von Bock would not be able to exert his will over the panzer commanders at vital times in the battle as he had not had the time to establish full command and control over his new army and communications with it. Halder also believed that von Brauchitsch should have bitten the bullet and, rather than causing an inevitable friction between the two army group commanders, 'coordinate[d] the efforts of the various elements converging on the pocket, under his own command'. The chief of the general staff believed that von Brauchitsch had been weak and wrote, 'He seems to be glad to let someone else take the responsibility. But with that he also foregoes the honours of victory.' As a protest, Halder let the written order for the transfer of Fourth Army to Army Group B be promulgated without his signature.\n\nHalder was not the only one angered by von Brauchitsch's actions. When Hitler found out about the order the next morning, he was determined to make the commander-in-chief pay for his lack of respect for the chain of command. He not only rescinded the order but visited von Rundstedt's headquarters at Charleville in the late morning of 24 May and thoroughly undermined von Brauchitsch by asking the commander of Army Group A to decide for how long the panzer divisions should be paused. The famous Dunkirk Halt Order was issued in Directive No.13 by OWK and followed by details from headquarters Army Group A. The result was to stamp Hitler's authority on the military \u2013 the men who had told him that his ambitions in the West were unobtainable and yet now were trying to sideline their superior \u2013 and to give von Rundstedt a free hand.\n\nIn the meantime, faith was placed in the Luftwaffe to apply pressure on those Allied troops in the Dunkirk pocket, and particularly around the ports, where the ground forces could not. Despite the increase in the RAF presence and the vagaries of the weather, Goring \u2013 so very eager to please \u2013 had convinced Hitler that the air force could do what the armoured forces, limited by the sandy terrain and waterways, would find difficult after such a long period in action. It was an opportunity that Hitler leapt at as an opportunity to preserve his force prior to the turn south-west in the next phase of operations, but the Luftwaffe itself was not convinced, despite G\u00f6ring's blandishments. General Albert Kesselring, commander of Luftlotte 2 supporting Army Group B, and Generalmajor Wolfram Freiherr von Richthofen, commander of Fliegerkorps VIII, both believed that the mission that had been assigned to their aircraft looked like a classic task for ground forces and that the over-stretched Luftwaffe, which had lost 810 aircraft totally destroyed between 10 and 24 May, was not well placed to carry it out and was actually in more need of respite than the panzers. As for von Bock, he could not understand why he was now being asked to push to the coast when other panzer divisions were better placed to have a decisive impact and noted in his diary that the Halt Order would 'have very unpleasant consequences for the outcome of the battle I am fighting!' Even von Kluge, who had wanted time for his infantry formations to close up, argued that the pause was misguided.\n\nMilitarily, the Halt Order certainly made little sense, but it was about far more than operations and strategy. According to Hitler's Luftwaffe adjutant, Nicolaus von Below, 'the English army had no significance for him' at Dunkirk and his motives for stopping the panzers were entirely to do with asserting his dominance. Von Brauchitsch met Hitler during the afternoon of 24 May and was admonished for his actions and left the presence of the Supreme Commander in no doubt that he had lost the right to direct the employment of his own field armies and that alterations to the command structure were in Hitler's gift alone. The commander-in-chief of the army had been brutally and publicly emasculated. Despite his recent disagreement with von Brauchitsch, Halder was livid not only at his colleague's treatment by OKW but also by the way in which the headquarters was now interfering in operations. Reflecting on the Halt Order, he wrote:\n\nThis is a complete reversal of the elements of the plan. I wanted to make AGp.A the hammer and AGp.B the anvil in this operation. Now B will be the hammer and A the anvil. As AGp.B is confronted with a consolidated front, progress will be slow and casualties high. The air force, on which all hopes are pinned, is dependent on the weather. The divergence of views results in a tug-of-war which cost more nerves than does the actual conduct of operations.\n\nTactically, the pause as Army Group A transformed into an unwilling anvil led to some reversals as the bridgeheads across the Aa Canal had to be evacuated since they had become so vulnerable. SS-Obergruppenf\u00fchrer Sepp Dietrich, despite being the commander of the SS Leibstandarte Adolf Hitler, one of the most loyal formations in the German order of battle, was so infuriated at the illogicality of what was being done that he took unauthorized action and occupied a hill on the east bank of the canal as an act of defiance. Dietrich's direct superior, Guderian, took no action.\n\nA humiliated von Brauchitsch returned to his own headquarters that evening to find that Halder and the Chief of the Operations Staff, i.G. Hans von Greiffenberg, had been busy. In a marked escalation in the private \u2013 but increasingly public \u2013 war that OKH was fighting against OKW, von Brauchitsch's headquarters had sent a radio message to Army Groups A and B which gave their commanders some flexibility within the Halt Order. The deliberately vague communication informed von Rundstedt and von Bock that '... the go-ahead is hereby given for the continuation of the attack up to the line Dunkirk-Cassel-Estaires-Armenti\u00e8res-Ypres-Ostende'. It acknowledged that, unless there was immediate action, the BEF and French First Army would have an opportunity to withdraw from the Lille salient and begin an evacuation from the coastline. Both army group commanders immediately recognized what was happening but while von Bock acted on it, von Rundstedt refused to pass it on to Fourth Army \u2013 whose operations he alone handled \u2013 considering that more time was required for 'closing up'. Old tensions that had remained submerged in less exacting circumstances had come to the surface at a vital time and were now threatening to tear the German command apart just at a point when they needed to exploit their hard-won gains.\n\nFor their part, Weygand and Gort recognized that they needed some sort of sensation in order for their frail position not to become a disastrous one \u2013 and the Germans seemed intent on handing it to them. At the front, Alan Brooke noted in his diary: 'Nothing but a miracle can save the BEF now and the end cannot be very far off!... It is a fortnight since the German advance started and the success they have achieved is nothing short of phenomenal. There is no doubt that they are most wonderful soldiers.' Counterattack plans had not been able to turn the situation and it looked increasingly likely that evacuation from the Dunkirk sector was the only realistic proposition for the Allies' beleaguered troops. But although the relationship between the French and the British was deteriorating as quickly as that between OKH and OKW, the two armies were still working together and, with the RAF lending increasing support, withdrawal and defence continued. By the end of 24 May, the Lille salient was being emptied of Allied troops, leaving those who remained there to fight on until they were killed, wounded or overrun. While their commanders did what they could to facilitate men's safe removal from an extremely precarious situation, Gort and Blanchard were at a loss to understand why the German panzer divisions had not moved in for the kill.\n\n* Marcel was taken prisoner at the field hospital on 30 May and was found to have a fractured skull. He recovered from his wounds in a German PoW camp, where he remained until April 1945.\n\n* A description used by Franklyn to Peter Vaux \u2013 letter to author 20 February 2009. Author's Archive.\n\n# CHAPTER 10\n\n# 25 May\u20134 June \u2013 Withdrawal and Evacuation\n\nBY THE THIRD WEEK of the offensive, despite Hitler's interference in operations and internal command frictions within the Wehrmacht, there was no doubt who was winning the campaign. Even so, the awful and worsening situation of the BEF, French and Belgian forces in the coastal pocket did not mean that there was nothing left to fight for. On the contrary, the surrounded Allied forces were increasingly motivated by their predicament to fight for every yard of ground, with the British further spurred on by the prospects of an evacuation that Churchill now deemed vital 'to the furtherance of the defence of the British mainland', and to allow the French time to strengthen their defences along the Somme and Aisne. It would be na\u00efve to think, however, that, regardless of any strategic or operational factors, troops were not also fighting for their survival.\n\nFrench R35 tank driver David Schneider, who had been in Breda on 11 May as part of Seventh Army but by 24 May was armed with a rifle as part of a composite unit in First Army, remembered:\n\nSurrender was not an option. Before we had properly contacted the Germans, there was fear of the unknown, but ever since we had been fighting them \u2013 and particularly as we got closer to the sea \u2013 there was an unspoken agreement that we would fight until we could fight no more. It was clear to us that along the line, we all relied on each other. Any breach would spell disaster. Every soldier I knew would not be the man to cause that breach. Verdun was won by stubbornness, not flight.\n\nFor his part, a British soldier in the 10th Hussars remarked that he was very aware that the heavy hand of the past was on his shoulder as he fought, explaining, 'We had to measure up to our fathers of 1914\u201318.'\n\nIt was not uncommon for units to have little idea of the broader context of their situation, only to know that what they were doing was of value. Andrew Hardcastle, a rifleman with 5th Division's 2nd battalion Wiltshire Regiment, has said: 'Evacuation was spoken about. We did not know whether it was possible and we certainly did not know whether our battalion would be one of the ones lucky enough to sail off to safety. We fought because we were told to do so, because we had our pride and our mates to consider... In the end, our withdrawal was brilliantly organised, if tough \u2013 a great achievement.' Indeed, the movement back towards the coast was a remarkably accomplished affair considering the circumstances. The retreat from the Dyle Line had been extremely well handled and now the Dunkirk pocket was consolidated, this phase in the Allied campaign stood in stark contrast to the ineffectiveness of the French forces defending against Army Group A that had been either shattered or pushed aside to the south. With the Royal Navy offering an exit from the perimeter and the BEF's command and control remaining robust, Gort began to calmly grip the situation in a way that other senior commanders had failed to do. Although Brooke's opinion of the BEF's commander had been poor during the 'Phoney War', now under the pressure of the German attack, Gort's instinctive leadership, fighting spirit and resilience came to the fore.\n\nStoicism and leadership were critical to the survival of the troops in the Allied pocket, who had suffered days of hardship. A new fighting resolve emerged just in time, for the Germans were on the brink of operations aimed at squeezing the pocket out of existence. Although those panzer commanders ordered to discontinue their attacks were displeased to do so, by the second day of their pause they could not but agree that their formations had benefited from some rest, resupply and attention to their vehicles. Yet even though von Kleist had reported that half of his tanks were out of service before the Halt Order, few tank commanders believed that their situation was so poor as to require any sort of postponement to their operations. Reflecting on this, von Bock wrote in his diary: 'The English continue their evacuation from Dunkirk, even from the open beaches! By the time we finally get there they will be gone! The halting of the panzer units by the Supreme Command is proving to be a serious mistake!'\n\nAs it was, the Allies had a fifty-six-mile front from the Dutch border and along the Lys to the French frontier manned by seven Belgian corps, albeit of greatly different strength and capability. These were struggling and 'beginning to break' after being made to give up their capital and then driven back towards the coast. The two divisions of Belgian I Corps which were on the right of their line and linked up with the BEF were particularly stretched. Gort reacted immediately, but without seeking the authority of the French, to plug the gap with his 5th and 50th Divisions, fresh from Arras. Even so, the pressure on the Belgians was unremitting and led the British to fear a collapse on their left.\n\nMeanwhile, the bulge in the Allied line had been reduced in depth during 25 May as three French corps withdrew behind the River Scarpe and the west flank of the perimeter gained strength. By this time, British 2nd, 44th and 48th Infantry Divisions were in position from La Bass\u00e9e northwards to where units of the French 21st Infantry Division held the line at the coast around Gravelines. The units of these dispersed divisions established themselves on the terrain that was best suited to defence, but although attempts were made to ensure that gaps in the line were covered by fire, this was not always possible and, indeed, not every formation knew exactly what ground was being held on its flanks, far less the unit responsible for its defence. As a result, the holding of various strongpoints behind the forward defences was essential because Gort recognized that enemy infiltrations would eventually lead to a withdrawal. In preparation for this, a series of fortified positions had been established. These were often based on towns, as was the case at Cassel and Hazebrouck, and the aim was to slow the enemy's advance while preventing the collapse of a flank that was precariously close to Dunkirk. By 25 May, however, Panzer Groups Kleist and Hoth were poised to knock that flank in and four Allied divisions faced seven panzer divisions, four motorized divisions, two of which were SS, an SS motorized regiment and an infantry division.\n\nIt should be remembered, though, that elements of two of Guderian's divisions received no rest during this period. Although Boulogne fell to 2nd Panzer Division on 25 May, Calais had only just been surrounded by 10th Panzer Division. The town had been bombed and continued to be so as the first British troops from 1st Armoured Division sailed into the port on 22 May. This force, commanded by Brigadier Claude Nicolson, included the tanks of 3RTR, an under-strength battalion of the Queen Victoria Rifles, 1st Battalion The Rifle Brigade and 2nd Battalion King's Royal Rifle Corps, but it lacked artillery and important equipment. The northernmost part of the town was defended by a small force of French naval reservists and volunteers commanded by Capitaine de Fr\u00e9gate Carlos de Lambertye and this left the rest to the British. The 3RTR tanks (twenty-one light tanks and twenty-seven cruisers) had not been prepared to go straight into battle, which meant that their guns were still coated in mineral jelly and had not been calibrated, and their radios had not even been tested. To make matters worse, there were no maps of the area and no orders for commanders other than to 'hold the town'. Lieutenant Colonel Reginald Keller, the commander of 3RTR, later said: 'A great deal of valuable time was wasted owing to my not knowing who or what Brigadier Nicolson was and why I had come under his orders and what I was supposed to be doing.' 10th Panzer Division would have made short work of such a poorly organized enemy in open countryside, but Calais was of great advantage to the defender with its seventeenth-century Vauban fortifications, which could absorb shelling and bombing and dissuaded frontal attacks. Having withdrawn into the town on 24 May and been surrounded, Nicolson's force prepared for an evacuation that had been agreed in principle for the next day, but the French protest in the wake of Boulogne's early abandonment and the prospect of Nicolson's force being able to hold their ground, encouraged Churchill to change his mind. By the time that an 'open-mouthed' Keller had been informed about the change of plan, he was halfway through the demoralizing task of destroying his tanks.\n\nAfter a rapid change in plans, organization and deployments, the Anglo-French force realigned themselves for the sort of siege warfare that was more redolent of a bygone age. Perhaps this was just as well, since it was at this point that Philip Gribble, the army liaison officer at the headquarters of the British Air Forces in France, could only note that the French were 'helpless in the face of the unorthodox black magic of the German High Command'. Either way, the result was a noisy, destructive struggle which saw the German heavy artillery blast the defenders and occasionally follow up with engineer and infantry assaults. Nicolson refused to surrender, despite his impossible position, and in keeping with Churchill's requirement, replied to the suggestion he do so, 'No, I shall not surrender. Tell the Germans that if they want Calais they will have to fight for it.'\n\nThe RAF lent what support it could by attacking targets of opportunity. 2 Group's Order of the Day for 25 May read:\n\nIt must be impressed upon leaders that risks must be taken in this emergency to find the really important targets, then attack them. It must now be accepted that the day has passed when attacks can be launched at definite targets as a result of previous reconnaissance. This is due to the rapidity of movement of enemy forces. In view of the critical situation of the B.E.F. it is essential that all attacks are pressed home with vigour.\n\nWhile the bombers grappled with the Luftwaffe and the panzer divisions' anti-aircraft units to reach their targets, Royal Navy destroyers were brought into position to lend some fire support. Indeed, Oberstleutnant von Jaworski was so incensed by the rounds falling among the panzers outside Calais that he ordered the driver of his Mk III into a firing position and, after several rounds, landed one that eventually sank the vessel in question. From then on his tank sported a large, childlike image of a boat applied to the turret in white paint to record and publicize the kill. The episode neatly encapsulated the British difficulty in engaging the enemy because the Germans maintained the initiative despite their ill-timed pause in offensive operations against Dunkirk. The Allies were constantly trying to fend off their attackers and found that they rarely had the resources they needed, or where they wanted them. Captain David McCullen, who was working on the staff of 48th Division, confirmed as much in his diary:\n\nChaos is usual in war, but we are finding it difficult to do some of the most basic things because of a combination of the enemy artillery, air attack and [the division's] own dispersed and fractured structure. Moving and communicating are terribly difficult and I spend most of my time trying to find out who and what are where.\n\nThis might be compared to the testimony of a transport officer with 10th Panzer Division, Hauptmann Jan Hoch, who wrote:\n\nHaving moved to the coast we were well set. Our units were organized and well-supplied but we were frustrated by Calais, which I believed could be screened and left alone. However, I believe that our division was not seen as essential in the push into the British because we had such strength already in place... The defence of Calais was nothing less than heroic. We had great admiration for those defending the old town against our power. Had we seen more of such devotion to duty, our division may not have made it as far as it did, as quickly as it did.\n\nLate on 26 May, Calais finally succumbed to the power of 10th Panzer Division and the German bombers. De Lambertye died that day from a heart attack while touring his positions and the resilience of the defending troops had, like the gallant French commander, been broken. Winston Churchill later wrote of the battle:\n\nCalais was the crux. Many other causes might have prevented the deliverance of Dunkirk, but it is certain that the three days gained by the defence of Calais enabled the Gravelines waterline to be held and without this, even in spite of Hitler's vacillations and Rundstedt's orders, all would have been cut off and lost.\n\nThis overstates the case because, as Hoch reflected, the Germans already enjoyed numerical superiority against the west flank of the Allied pocket, but there is no doubt that Calais revealed an intent and a stoicism that would be reflected elsewhere around the perimeter. Indeed, David McCullen noted in his diary during the battle: 'The flames of Calais are a beacon that invites us to ever greater efforts.'\n\nResilience, wherever it could be found and however it could be promoted, was in great demand around the perimeter because the Germans were on the cusp of restarting their armoured offensive. In the early afternoon of 26 May, OKH received notice that, after a visit to the front, von Rundstedt had decided to recommence the offensive by von Kleist and Hoth. That same afternoon, von Brauchitsch met with Hitler to discuss the military situation and the commander-in-chief boldly outlined his belief that von Rundstedt's protracted pause had done considerable damage to the prospects for subsequent operations and the German ability to achieve a strategic victory. Von Brauchitsch returned 'beaming', Halder notes in his diary, having heard at first hand from Hitler that he believed that von Rundstedt had paused for too long and that the Luftwaffe had not achieved what it had said it would achieve. As a consequence, an order was promulgated which stated: 'The F\u00fchrer has authorized the left wing to be moved within artillery firing range of Dunkirk in order to cut off, from the land side, the continuous flow of transport (evacuations and arrivals).' It was not, therefore, an order without limitations, but it did at least give the panzer commanders the opportunity to move forward again. The relief felt in OKH was palpable, for the situation was still advantageous if operations were now moved on quickly and aggressively.\n\nThe potential for the Allies to be overwhelmed within a matter of days was not lost on Churchill, who, having kept a keen eye on Ramsay's preparations, decided that the time was right to evacuate the main body of the BEF. During the morning of Sunday 26 May minister of war Anthony Eden informed Gort of the verdict in a message which said: 'In such conditions only course open to you may be to fight your way back to west where all beaches and ports east of Gravelines will be used for embarkation.' Operation Dynamo, the evacuation of British and Allied troops \u2013 twelve British, twenty-eight French and twenty-two Belgian divisions \u2013 was to start without delay. Few now believed that there would be a major French counter-attack that would fundamentally alter the operational picture. Philip Gribble wrote in his diary:\n\nThe confusion is indescribable. The situation must resolve itself in the course of the next few days. Personally I feel that the psychological moment for the attack has been passed. I hope that this apparently unjustified pause will ultimately prove to have been due to the foresight, patience and self-control of great generalship.\n\nHe was to be disappointed, for the French were barely coping with the situation and the British were focusing on the need to extract as much of the BEF as possible. John Colville, Churchill's assistant private secretary, returned from a much-needed weekend break to find that the mood had changed in his absence:\n\nAt Downing Street I was distressed to find the situation much blacker than when I left on Friday. It appears a grave deterioration has taken place in the last forty-eight hours: the BEF, unable to force their way southwards, have got to retreat to the coast as best they can and re-embark for England from whatever Channel ports remain open to them. The French seem demoralized and there is now serious fear that they may collapse. The Cabinet are feverishly considering our ability to carry on the war alone in such circumstances.\n\nOn 26 May, the Belgian left flank withdrew to the outskirts of Bruges just half a dozen miles from the coast while their right flank in the Ypres-Menin sector had been broken. The British were in an invidious situation on the right flank for they had to hold the shoulders of the Lille salient in order for its formations to escape the closing jaws of the German trap and in doing so shorten their front and prepare a new defensive line behind the Belgians in anticipation of their capitulation. It was an operation that had to be closely co-ordinated with the formations of French First Army for, if the shoulders of the salient collapsed, it would lose its head. In such circumstances, the prospect of a British evacuation order filled the French with a sense of dread. Blanchard did not hear about Churchill's decision until 28 May, but in the meantime the first BEF units withdrew back towards Dunkirk. Although bombing had put the docks out of action, the East Mole, a 1,400-yard-long and five-foot-wide wooden sea wall which protected the harbour, could still be used for embarkation, as could the outer harbour wall. Ramsay's emissaries were further aware that there were sixteen miles of coastline stretching east from Dunkirk that also could be used for evacuation as its gently sloping beaches could be used by shallow-draught craft to transfer troops to vessels waiting in deeper water. The area also boasted a defendable hinterland which incorporated the towns of Nieuport, Furnes and Bergues, which were connected by a canal that was between forty-five and sixty feet wide. Other waterways, a central flooded area just behind the defences and dunes provided a further impediment to any attacker, while small resorts along the coast provided ideal bases for command and control. This perimeter was approximately thirty-five miles long and, while just three miles deep at its eastern end, was more than twice that at its western end around Dunkirk. As Private Alexander Wilkins, an infantryman in the 1st Division's 4th Battalion Green Howards, later said:\n\nIt was gratifying to be told that preparations were being made for us to be evacuated. It did spur us on through some very dark hours, but we all knew that there was plenty of hard fighting to do before we set foot on a boat \u2013 the Germans would make sure of that.\n\nThe two German panzer groups south of Gravelines resumed their attack at 0800 hours on 27 May, three days and eight hours after they had been ordered to stop. Having crossed the British-manned canal defences with little difficulty, von Kleist and Hoth both then encountered stout defence as they approached the BEF's strongpoint line. These British divisions protected the eastern flank of a corridor used by I Corps to move towards Dunkirk during the night of 27\u201328 May and continued to do so during the following night. Meanwhile, on the western flank, the more dispersed British positions began to fall back as they were overwhelmed by superior enemy numbers. At Hazebrouck, for example, which was defended by a battalion of the Oxfordshire and Buckinghamshire Light Infantry, there was an intense battle on 27 May. As the enemy fell on the town intent on removing the defenders as quickly as possible, the battalion's orders were simply to 'Hold at all costs'. A German bombardment that morning was quickly followed by an infantry attack supported by light armour and air support. The under-gunned Ox and Bucks defenders stood their ground in fortified houses using whatever they could to slow the enemy. Medic Les Allan later recalled: 'The Germans were infuriated that they were being held back. They were anxious to push past us. So the bombing was terrific. [The Stuka was] deadly when used against men armed with just rifle and bayonet.'\n\nIt was just this sort of fighting that led Halder to note 'stronger resistance than expected' in his diary after reading the morning reports, but he added, 'We must bear in mind that a total of four enemy armies are packed into this pocket and that there is nothing left for them but to fight back as long as there is any ammunition; it must give out eventually.' That afternoon the house-to-house fighting in Hazebrouck continued and British casualties grew at an alarming rate. Les Allan did what he could to help, which meant putting himself in great danger since he was armed with nothing more than a Red Cross-emblazoned armband and a haversack, but he had to make some difficult choices with his dwindling medical supplies. He came across one man in the entrance to a cellar whom he knew to be a Private Johnson and noticed that the young man had suffered 'awful head wounds. His face was all torn open. There was blood everywhere. His jaw and skull were bashed in. I just kicked him out of the way. You soon realize you have to help those who can be saved, not those who are virtually dead.' (As it transpired, Johnson survived both his injuries and the war, and he and Les Allan later met at a reunion.)\n\nOfficers also put themselves in harm's way as they tried to visit positions for vital information about enemy progress and their own defences, but also to encourage their men. One company commander, Major Elliott Viney, not wishing for two boxes of his precious cigars to fall into enemy hands, handed out their contents to everybody he met on one of his regular rounds. Small comforts were much appreciated as the day wore on and it became clear that reinforcements were very unlikely to appear while ammunition stocks diminished and casualties continued to mount. One of those mortally wounded by a German shell was the battalion's thirty-three-year-old commanding officer, Major Brian Heyworth, which left Viney to organize the unit's diminishing defences. By the early evening the situation for the British in Hazebrouck was critical and with the Germans pressing towards it, Viney ordered the evacuation of the blazing battalion headquarters, which left around 100 men with two Bren guns and very little ammunition trapped in its large walled garden. Although a portion of the battalion managed to evacuate the town and head for Dunkirk \u2013 some twelve officers and 200 men eventually making it back to England \u2013 Viney and his men, after a superlative defensive effort, had no option other than to surrender.\n\nWithout room for manoeuvre, the Germans resorted to a grinding attritional method rather than concentrating their forces to create a local breakthrough at Gravelines. Indeed, this First World War approach was reflected in Hitler's order to move within 'artillery firing range of Dunkirk', which was achieved that day by 1st Panzer Division. The fighting on the west flank of the pocket was particularly intense and bloody during the first two days of the renewed offensive as the Allied divisions recognized that, if they broke, the pocket would simply collapse. Between Bergues and Cassel, for example, 48th Division resisted bravely against SS Leibstandarte Adolf Hitler. The fighting at Wormhoudt was particularly vicious, but eventually a group from 4th Battalion Cheshire Regiment, 2nd Battalion Royal Warwickshire Regiment, and a group of gunners were overwhelmed and surrendered. The SS rounded up their prisoners, stripped them of their possessions \u2013 including their identification papers \u2013 and ordered them to a nearby barn. Those wounded that could not walk were shot on the spot. Around some 100 apprehensive British soldiers were eventually crammed inside the small, airless wooden building but when Captain James Lynn-Allen, the senior officer present, protested, he was told: 'Where you are going there will be a lot more room.' Understanding their fate, the condemned were taking their first drags on what they expected to be their last cigarettes when hand grenades were thrown into the barn followed by submachine gun and rifle fire. Lynn-Allen was unhurt, but standing immediately behind him, nineteen-year-old Private Bert Evans had suffered a catastrophic injury to his right shoulder. The barn doors were then wrenched open and the forlorn survivors pulled out and executed. Rather than waiting for his turn, Lynn-Allen grabbed Evans and ran towards an open field, but only managed to reach the edge of a pond before Evans stumbled. Lynn-Allen stopped to help him, but as he did so a pursuer caught up with the pair and opened fire: Lynn-Allen was shot at point-blank range in the head and Evans in the neck. Meanwhile, the remaining prisoners were ordered out of the barn in groups of five and summarily shot by the SS troops waiting for them in the yard. Anti-tank gunner Lance-Bombardier Brian Fahey \u2013 who had already been shot in the leg during the fighting earlier in the day \u2013 also tried to escape. Dragging his wounded leg, Fahey did not get far before being shot in the back. As the young soldier lay unconscious, his lungs hissing and bubbling, the SS troops turned their attention to finishing off their captives by opening fire on them for a final time while they stood in the barn. They only left when there was no sign of movement in the mound of blood-drenched khaki which stood testament to their barbarity.*\n\nSouth of Wormhoudt, Major General Noel Irwin's 2nd Division continued to hold the west shoulder of the Lille salient on a fifteen-mile front either side of B\u00e9thune against several divisions which included SS Totenkopf and 7th Panzer Division. Rommel was determined to break through and trap several French divisions before they could withdraw, but standing in his way was 1st Battalion Queen's Own Cameron Highlanders in the Le Bass\u00e9e sector. The unit held out for two days before being overrun, but not before one company had been reduced to just six men. Meanwhile, SS Totenkopf, having been thwarted at the village of Le Paradis by the Royal Norfolks, took revenge against a particularly stubborn company after it had surrendered. Signaller Bert Polley, already wounded when beaten by his guards, was forced to his feet and was marched with the other prisoners to a farm. Here, two machine guns opened fire on the British group and Polley, nicked by a bullet, acted dead. The SS troopers used pistols and bayonets to kill the wounded but they missed Polley and left.\n\nRumours of such atrocities as this and that at Wormhoudt against unarmed men had already washed across the Allied forces and while they undoubtedly had a negative effect on the fighting spirit of some, far more became determined to fight to the death. As 2nd Battalion Wiltshire Regiment passed through a nameless Belgian village during the evening of 28 May, Andrew Hardcastle later recalled, a 'loud-mouthed mechanic' informed them that the Germans were shooting prisoners and the wounded. 'I was furious,' Hardcastle says, 'and at that moment decided that I would not surrender. We all agreed. We would fight to the final man and save the last bullet for ourselves.' In a similar vein later that same night, David McCullen wrote in his diary: 'Tales reach us of prisoners being badly handled by the enemy. Most agree that this will fortify our resolve to fight to the last round.'\n\nOn 28 May, the shoulders of the salient began to fall in, having held the Germans just long enough for 1st, 3rd and 4th Divisions to withdraw. It was an organized and carefully co-ordinated process in which divisions were released at just the right time to either head for Dunkirk or establish a new defensive position in the rear. That evening, for example, 5th Division, which was holding the eastern shoulder, disengaged from the enemy and then, using the protection offered by 3rd and 4th Divisions between Ypres and Poperinghe, withdrew behind them. This 'leap-frogging' under the cover of darkness was extremely difficult to control but was successful enough to allow the British formations to fall back into the emerging perimeter around Dunkirk in good order. The withdrawal was assisted by some excellent work by rearguard units of French First Army deep in the diminishing bulge in the Allied line, the first bad weather of the campaign, which thwarted the Luftwaffe, and the lacklustre performance of some tired German formations which, unlike their Allied counterparts, were not 'fighting for their lives'.\n\nEven so, there was no opportunity for the Allied divisions to lower their guard and General John Dill, the new Chief of the Imperial General Staff who had until April been a corps commander in France and subsequently become Ironside's deputy, brought all his knowledge and experience of the BEF to ensure that the force was extracted as quickly as the situation would allow. Reflecting on the events, John Colville wrote:\n\nFortunately Ironside has gone and Dill, who inspires great confidence, has taken his place; but I do not envy him, confronted as he is with the problem of salvaging the wreckage of the BEF. I still feel convinced that we shall win, even if the French collapse; but we have reached all but the last ditch and a timely miracle would be acceptable.\n\nThe replacement of Ironside as CIGS was not a surprise but, as Colville notes, took place at a very difficult time. Even as Dill was taking up the job on the evening of 27 May, the Belgian army's will was finally shattered and King Leopold, observing his troops' appalling situation, decided to capitulate. This was hardly a surprise either, and the British and French had made contingency plans for just such a decision, even though Churchill tried to dissuade Leopold by writing him a note stating that the Belgian surrender would have 'disastrous consequences to the Allies and to Belgium'. Once again, Churchill was overstating the case, but the British were contemptuous of the Belgian decision as well as being, quite fairly considering their collapse, sneeringly dismissive of the nation's military performance since 10 May. Indeed, when Gort's Chief of Staff, Henry Pownall, was asked whether the Belgian troops in France might be evacuated from Dunkirk, the general replied: 'We don't care a bugger what happens to the Belgians.' Nevertheless, when Leopold surrendered to the Germans on 28 May, the British were already in position behind them to protect the eastern half of the Dunkirk perimeter which was on Belgian soil.\n\nIn the meantime, Rommel managed to push through the BEF's 2nd Division holding the western shoulder of the Lille salient in a critical advance which carved out yet another opportunity for the Germans. With the withdrawal of 4th, 3rd and 1st Divisions behind 5th Division at Wytschaete on the eastern shoulder producing a gap into which the two German infantry divisions could penetrate, the Wehrmacht's pincers were now in a position to pinch out the Allied bulge. Rommel had recognized the importance of a swift, short push into the enemy rear area and, still leading from the front in order to ensure that the enemy could not concentrate against him, was again nearly killed, this time by 150mm fire from his own guns. He was in the village of Lommel just to the west of Lille with the commander of the Reconnaissance Battalion, Major Erdmann, when the bombardment began. Both men ran back to the signals vehicle to send an urgent message to get the artillery to stop firing. Rommel later wrote of the incident:\n\nErdmann [was] running a few yards in front, when a heavy shell landed close by the house door near which the vehicle was standing. When the smoke cleared, Major Erdmann... lay face to the ground, dead, with the back of his head shattered. He was bleeding from the head and from an enormous wound in his back. His left hand was still grasping his leather gloves. I had escaped unscathed, though the same shell had wounded several other officers and men.\n\nA shaken Rommel subsequently ensured that the Lille pocket was sealed and it was slowly reduced, despite continued attempts by French units to fight their way out of their incarceration. On 29 May, Halder wrote:\n\nThe enemy pocket has again shrunk. It will indeed be interesting to see how much of the enemy did get caught in this pocket, 45 km [28 miles] in length and 30 km [18 miles] in width. Even now, with Lille, Roubaix, Tourcoing in our hands, the enemy is still fighting desperately against our troops pressing on his flanks.\n\nAs it was, the remnants of five First Army divisions were trapped in the pocket and seven generals, 350 officers, 34,600 enlisted men, 300 guns and 100 armoured fighting vehicles were eventually captured when the French surrendered at midnight on 31 May-1 June. Although their attempts to break out had been doomed to failure, they were of benefit to the Allied defences at the coast because they preoccupied seven German divisions for four vital days as the Dunkirk perimeter was completed. Von Brauchitsch was livid at the delay and told Halder that he believed not only that the Lille salient could have been eliminated far earlier had the panzer Halt Order not been in place but also that it could have been sealed at the coast, thereby ensnaring the entire Allied force.\n\nThe final formation of the defensive line around Dunkirk and its beaches saw, at least from the operational perspective, the BEF and the French working in tandem as if blind to nationality. Although in the aftermath there was a plethora of accusations of selfishness as formations tumbled back towards a modicum of safety and the prospect of being taken off by boat, on the ground there was evidence of some solid teamwork. David Blundell, a Guardsman fighting near Furnes, has recalled that his company and a similarly sized French unit worked as one as they entered the perimeter. 'Our Major had a little Franglais,' Blundell recalled, 'but the officer with the French troops had not one word of English, but they both knew their jobs and with hand signals, pointing and diagrams on scraps of paper led us to our destination.' The common bond felt by soldiers of different units, different arms and different nations drove them instinctively \u2013 and often unquestioningly \u2013 to assist others. Lance Corporal Tom Reed of 5th Battalion Green Howards recalls slumping down exhausted behind a wall in a small village while German machine-gun rounds crashed into the masonry above his head:\n\nI was quite alone in my own world and had become inadvertently detached from my unit. I wondered whether I could go on but then, out of the shadows, a hand offered me a canteen of water. It was a sergeant from the East Yorks with half a dozen other men. He didn't smile, he didn't talk, but his small kindness probably saved my life. I joined that small band of men and fought with them all the way back to the beaches.\n\nThe Allies fought on. Gefreiter Hans Liebler, an infantryman in 253rd Infantry Division, noted 'a distinct increase in resistance from 30 May onwards'. While he recognized that increasing numbers of Allied troops were being pushed into an ever-decreasing pocket, he also witnessed 'units and individuals fighting harder and longer than they had a few days before. This was frustrating for us because at the same time we were becoming very tired. I can only imagine that the enemy's survival instinct became extremely strong and so gave them an edge during this period.' Indeed, Halder's diary contains a similar observation from an operational perspective:\n\nDisintegration of the bottled-up enemy forces is continuing. Some of the British units in there are still fighting stubbornly and with determination; others are streaming back to the coast and trying to get across the sea in anything that floats. _La D\u00e9b\u00e2cle._\n\nBy 30 May, the Dunkirk perimeter had been formed. French regiments were clustered around the western end of the perimeter holding approximately one third of the line up to the town of Bergues while British divisions held the remainder from Bergues, through Furnes to Nieuport. 15\/19 Hussars arrived in the perimeter 'weary and footsore' after a final twenty-five-mile movement, with Lieutenant Guy Courage writing:\n\nThis march was a most severe test, coming as it did after three weeks of fighting in the most exhausting and disheartening conditions of retreat. The roads were congested with traffic of all sorts and there was continual bombing... [I]t was a harrowing business, mile after mile in the dust and heat, with enemy planes circling overhead waiting to drop their bombs on any suitable target.\n\nSome of the British divisions arrived intact but others had been smashed beyond recognition: 3rd Division, for example, was still 13,000-strong, while 5th Division amounted to a mere 1,200 men and 2nd Division had ceased to exist. It is estimated that some 300,000 Allied troops \u2013 although not all of them were fighting men \u2013 were crammed into the Dunkirk pocket. Exhausted, disorganized, stretched and lacking heavy weaponry, the defenders had to rely on their last reserves of energy and their advantageous defensive positions. They faced eight German divisions that included 9th Panzer and 20th Motorized Divisions, which were attacking the western defences barely three miles from Dunkirk. Well organized, with offensive momentum and with extremely strong reserves, the Germans had every reason to feel confident that the last flickering light of Allied hope could be extinguished quickly. Outside the perimeter, the roads were lined for miles with abandoned and disabled vehicles and heavy equipment which marked the end of the BEF's retreat.\n\nBy the time that the Dunkirk perimeter was fully formed, evacuations were entering their fourth day. Alan Brooke had written in his diary on 26 May that the withdrawal was 'going to be a very hazardous enterprise and we shall be lucky if we save 25 per cent of the BEF!' but his prediction was eventually proved to be a very considerable underestimation. By 30 May, 77,000 troops had been lifted from the harbour and a further 48,000 from the beaches, and with every hour, embarkation techniques were further refined, to the point where 600 men were habitually loaded onto a destroyer in just twenty minutes. Although the Germans' attacks were undoubtedly weakened by their preparations for operations after the Dunkirk perimeter had been reduced, a combination of excellent defensive work, calm seas and first-class discipline and organization allowed for more men to be lifted from Dunkirk and its beaches than most thought possible. Eschewing rest until the job was done, Captain William Tennant, twelve officers and 160 men supervised the embarkation while Rear Admiral Wake-Walker commanded the shipping off the coast of Dunkirk. A collection of vessels were assembled for the task including a single cruiser, fifty-six destroyers and torpedo boats, minesweepers, coastal launches, yachts, fishing boats, barges, passenger ferries and a miscellany of other craft. In the end, the evacuation fleet, which included boats from the Netherlands, Belgium and France, grew to 861 vessels. Small headquarters along the coast liaised with the fighting formations to ensure that units were released at the appropriate time and those troops then made their way to join the queues waiting patiently on the beaches or at the port for their turn to embark under the direction of Royal Navy teams. These men generally showed excellent discipline as they waited for the order to move forward and scramble aboard any craft that would take them away.\n\nTo Captain David Barlone, who led a column of French North Africans to the water's edge along roads 'strewn with dead and hundreds of wounded', the beaches were 'An impression of awful and irretrievable disaster...,' Meanwhile, Guy Courage later wrote of the scene that greeted 15\/19 Hussars:\n\nSo to the beaches we came: it was an astounding spectacle, the like of which had not been seen before. Away to the left was Dunkirk itself, shrouded in a thick, black pall of smoke from its burning oil tanks and with the light of many fires kindled by the bombing. On the beaches were many thousands of troops waiting patiently to embark. Out to sea lay destroyers and some little ships. Overhead the sky was filled with German bombers, diving low over Dunkirk and the beaches, wave after wave of Stukas, diving, dropping their bombs with a scream among the densely packed mass of soldiers, and circling away again, only to be replaced by more.*\n\nThe Luftwaffe certainly remained active throughout the battle, even though Hitler and his commanders had given up hope of the ground forces being required to merely 'mop up'. Helmuth Mahlke had been flying missions against the port for one or two days before he wrote on 29 May:\n\nToday, it seemed as if everybody who was anybody in the Luftwaffe was heading for Dunkirk. The airspace around us was crowded with German aircraft... The target area [Dunkirk harbour] was already clearly visible in the far distance, marked by large fires and clouds of black smoke... Shortly before 2000 hours we were over the target. Standing on our wings, we dived almost vertically into the attack. We saw our bombs explode close to the lock gates... As we flew back to base I couldn't get the horrific spectacle that was Dunkirk out of my mind. An absolute inferno! Those poor beggars down there desperately fighting for survival. Just as well that at our height we were spared the gory details.\n\nEven so, the Luftwaffe's impact on the Dunkirk pocket was not as great as it might have been. There were many reasons for this, not least the fact that forward airfields in Belgium and France were still in the process of being constructed, which meant that aircraft enjoyed little time over their targets as they were at the limits of their effective range. It also proved difficult to concentrate bombers and their escorts over the coast at a time when they were also needed elsewhere to confront the French in preparation for the next phase of German operations across the Somme and the Aisne to the south-west. Those that did manage to spend any time over Dunkirk and its beaches, however, were also often hampered in their targeting by low cloud and the black smoke from those burning oil storage tanks and tyres that veiled the embarkation.\n\nIn addition, the Luftwaffe was increasingly troubled by RAF fighters over Dunkirk as the battlefield was now relatively close to Air Chief Marshal Sir Keith Park's 11 Group airfields in southern England. In sanctioning the release of these precious assets from Air Chief Marshal Hugh Dowding's Fighter Command, which had been developed for home defence, Churchill was looking to the longer-term security of the British Isles and its need for manpower, rather than the short-term preservation of the valuable aircraft at all costs. To the French High Command, however, the withholding of British fighter squadrons until they appeared over Dunkirk to provide air cover for the evacuation smacked of betrayal. Weygand, who had shown little trust in the British since taking command from Gamelin, remarked: 'Apart from his distinguished qualities, the Englishman is motivated by instinctive selfishness.' For his part, Gort felt that the BEF had been so affected by French military incompetence that the British were merely doing what they could to extract themselves from 'a mess that was not of their own making'. Even so, from the time of the Halt Order onwards, the Germans had noted a change in the skies above them, with the War Diary of von Kluge's Fourth Army recording:\n\nFor two days, the enemy enjoys air superiority over the Group von Kleist and partially above Group Hoth. This is something completely new to us in this campaign and the reason is that the base of the English Air Force is in England and therefore now quite close, while our units on the whole are still based on German airfields.\n\nThis new reality had been sustained over Dunkirk and, despite the commonly heard complaint from BEF troops that the RAF kept a very low profile during the withdrawal and evacuation, British aircraft played a significant part in the protection of the ground and naval forces. By flying intensively, RAF fighter squadrons were able to undertake 2,739 sorties during the evacuation, with German pilots testifying that these Spitfire and Hurricane pilots were much better than those they had faced up to that point. There were, though, many who flew into action for the very first time during those heady days of late May and there was precious little time to learn vital lessons. Flying Sergeant Rab Riley piloted a Hurricane and watched two young officers make basic mistakes on their first sorties, for which they paid with their lives. Other 'green sticks', however, managed to survive and quickly gained knowledge and confidence. Riley later testified:\n\nIt was never easy to leave the safety of an airfield, form a Vic \u2013 usually of three aircraft \u2013 and be in action all within thirty minutes. The speed at which one left safety and was hurtled into mortal danger was so quick that it took a pilot a little time to 'acclimatize' to his new condition, to ensure that his mind and body were ready for battle. A sluggish response or a loss of concentration could very easily lead to death. Air combat is a lonely business, but the more you do it, the better you get.\n\nThe tenacity of the RAF pilots, as well as their flying ability, soon won the respect of their adversaries, but there were a variety of responses to the stress of battle. Squadron Leader J. E. McComb, who at the time was flying a Spitfire of 611 Squadron, later recalled:\n\nI saw above me a Spitfire [flown by Flight Sergeant Sadler]... dive into five circling Bf 110s [which were twin-engined and lacking in manoeuvrability] and then get trapped in the circle. They could out-dive him but he could just turn inside them so that their shells and bullets went under his tail. Each time they fired and missed, Sadler leaned out of his cockpit and made a vulgar gesture with his two fingers.\n\nStuka pilot Helmuth Mahlke, however, had a rather more genteel encounter with the RAF. He came wingtip to wingtip with a Spitfire, and its pilot 'looked across at us and raised his hand to his flying helmet in salute! Dumbfounded, I automatically returned the compliment. Then he banked away in a graceful arc and was quickly lost to sight... [T]here _was_ such a thing as the brotherhood of the air... That unknown British pilot was a remarkable man.' If there was indeed a 'brotherhood of the air', it was one built of the common experience of hard fighting \u2013 the Germans lost 132 aircraft over the coast.\n\nIn the end, although at certain times and in certain places German air power had significance at Dunkirk, just as the Luftwaffe had not been able to stop the withdrawal into the perimeter, it was unable to stop the evacuation. That defensive perimeter held firm, even as it shrank whenever units left the line for the port or the beaches, and Operation Dynamo continued \u2013 against all expectation \u2013 into the first days of June. The fighting was often fierce, with clashes in the shallow eastern sector of the perimeter having particular importance. Here German 256th Infantry Division had an opportunity to break through to the beaches and then roll the British line up towards Dunkirk. The defence offered by Montgomery's 3rd Division, however, was particularly intense and successful. Anchored at the small fortified town of Furnes, two battalions of Grenadier Guards rebuffed numerous assaults.\n\nGuardsman Ernie Smith found himself alone overlooking the canal after the rest of his section had been blasted out of their position by German artillery. 'I shouted to them,' recalls Smith, 'but my ears were ringing so loudly from the explosions that I could not hear my own voice, let alone a response.' Smith continued to fire on the approaching enemy until midnight on 31 May\u20131 June when his ammunition ran out and he went in search of his colleagues. Smith found two dead and three others wounded and was helping them to a first-aid point when a withdrawal was ordered. The non-walking wounded were to be left behind.\n\nSmith's fellow Guardsman Douglas Rothery was part of the same action and later wrote:\n\nAt a pre-arranged time, which was at about 2 or 3am [on 1 June], we were told that we were to tie blankets around our boots and creep away from our positions so as not to alert the enemy... We were clambering over debris in pitch darkness, getting tangled up in telegraph wires, cursing as we tried to extricate ourselves, glass was being shattered and tin cans clattered as we fell over them and I would have thought that we could have been heard in Berlin... The enemy shelling was getting worse, screaming in from all directions as we struggled on for about [three miles], past burning vehicles of all types including ambulances, trapped victims in demolished buildings were calling for help, some whimpering in the darkened rubble, whilst other wounded could be seen by the light of the burning vehicles crawling in the direction of the sea... Eventually we arrived at the sand dunes of La Panne, where we (wait for it), were formed up and marched to attention down to the water's edge, Halted, Turned to the front, Ordered arms, Stood at ease, Stand easy. We were then informed it was every man for himself...\n\nBy this time, the evacuation was centred on the port at Dunkirk as the perimeter shrank further and an eleven-mile long inner line was formed to defend it. As a consequence, the Grenadiers were ushered along the beach towards the town but, pursued by Stukas and noting that their destination was also being bombed, a forty-man group, including Rothery, decided on a different course of action. Seeing that vessels were waiting offshore to receive evacuees, the men clambered aboard a large wooden boat they found in the dunes and paddled out to sea using their tin hats. After they had scrambled up the rope ladders of one of the ships and landed on the deck, Rothery recalls, a crew member greeted them. 'Welcome aboard _Little Audrey,'_ he said, 'the last of the Minesweeper Flotilla.'*\n\nThe inner perimeter was now occupied by units from the French 68th and 12th Infantry Divisions during the afternoon of 2 June and they defended the last evacuations selflessly and with great stoicism. That night, the final British troops embarked at Dunkirk and General Harold Alexander, the commander of 1st Division who had also been given responsibility for I Corps while he was still in France, searched the port and beaches with a megaphone, calling out into the darkness: 'Is anyone there? Is anyone there?' Soon after, with Alexander safely on a waiting destroyer, Captain William Tennant signalled Ramsay a message that few thought conceivable ten days before: 'BEF EVACUATED'.\n\nThe next morning, as the Germans sensed that Dunkirk would be in their hands by the end of the day, the French launched an unexpected counter-attack which temporarily rocked 254th Division. During the course of the day, the remnants of the French 68th Division were destroyed, but resistance throughout 3 June created the time and space required for one final embarkation of French troops that night. Two hundred and seventy-two vessels had been lost during Operation Dynamo, and the last ship departed from the East Mole on the morning of 4 June. It carried the final contingent of some 338,226 Allied troops to be evacuated from the Channel ports and Dunkirk beaches. A few hours later, the Germans captured the port and took the surrender of the remaining 40,000 French troops who had formed the rearguard and had sacrificed themselves for the majority. Amidst the disaster that had befallen the Allies since 10 May, Dunkirk was a remarkable achievement, one appropriately described by Winston Churchill in a speech to the House of Commons on 4 June as a 'miracle of deliverance achieved by valour, by perseverance, by perfect discipline, by faultless service, by resource, by skill, by unconquerable fidelity...'.\n\nThe taking of Dunkirk and the capitulation of the remaining French forces there marked the end of _Fall Gelb._ Hitler was elated, calling the offensive 'the greatest battle in world history' in his Order of the Day for 5 June, and directed that bells be rung across the Reich for three days in celebration. In just twenty-six days Germany had defeated the Netherlands and Belgium, seen off the BEF and destroyed thirty French divisions, including the most mobile and well-trained part of their army's order of battle. Yet the removal of formations from the battlefield, no matter how important, was just the most visible impact of the German offensive. The damage done to the Allied nations psychologically was immense, and while it may not have created a political collapse in Paris and London, it had led to the replacement of the head of the British army and, more significantly, a change in the French Supreme Commander and the sacking of dozens of field commanders. The Allies had been thoroughly humiliated and events had shamed those Frenchmen charged with the defence of their nation.\n\nEven so, Hitler's 24 May direction 'to prevent the escape of English forces across the Channel' had clearly miscarried in the wake of an unnecessary Halt Order that was itself the product of a power struggle \u2013 and breakdown in relations \u2013 in the upper echelons of the German command. Moreover, France was not defeated, and even though it had lost the vital north-eastern portion of its territory and suffered a catastrophic shock, it had not collapsed and was preparing to do its utmost to ensure that German attempts to attack into the main body of France were formidably resisted. It was a challenge to the invaders, therefore, to attack the rapidly forming Weygand Line running from the coast along the Somme and the Aisne before it reached optimum strength. This required the Battle of Dunkirk to be fought even as OKH's focus and vital fighting resources were being transferred to a new phase of operations. It was a process that was so demanding that von Rundstedt protested, 'You bring _one_ operation _to a conclusion_ before you can think of _the next_ _one_!' It was hardly an ideal situation, but Halder recognized that if a strategic victory was to be attained quickly, much depended on OKH's ability to produce a viable new plan and get what was needed where it was needed in both robust order and good time for that plan to be carried out. Although there was evidence for great optimism, Halder reminded the overconfident that even a severely weakened enemy could conceal a sting in his tail.\n\n* Just six men survived the Wormhoudt massacre, including Brian Fahey and Bert Evans. Fahey remained in the yard, where he was eventually found by a regular Wehrmacht unit. He recovered from his wounds and was sent to a PoW camp, where he worked on his musical skills. After the war, he worked as an arranger and composer for big bands, for the BBC and in film and theatre. Evans managed to crawl from the pond and across a field, where he lay for several hours. He was captured by a German ambulance unit. His arm was eventually amputated and he spent the rest of the war in a German PoW camp.\n\n* The regiment was evacuated on 30 May, by which time, out of 455 officers and men, over 36 per cent had become casualties \u2013 and 80 per cent of the officers. (See Courage, p. 43.)\n\n* _Little Audrey_ was a reference to an American cartoon character who laughed in the face of great adversity and catastrophe.\n\n# CHAPTER 11\n\n# 5\u20138 June \u2013 _Fall Rot_ and Resilience\n\nON 3 JUNE 1940, Paris experienced an event that it had feared for the past twenty years \u2013 a German bombing raid. The senses of the city's inhabitants had become heightened to the sound of aircraft engines and air-raid sirens during the previous autumn as they grappled with what it meant to be at war, but procedures were soon forgotten as the Luftwaffe kept its distance during the winter. The French authorities used every available means to remind the population how dangerous it was to become complacent in the face of the threat, but it was not until 10 May that Parisians really took notice. Since then, the capital's population had had their nerves tested by numerous false air-raid alarms which had led to some leaving the city for the safety offered by the country's more southern provinces. More had left when the enemy crossed the Meuse, and more still during the Dunkirk evacuation.\n\nAll the same, the period was marked by a lack of alarm and a confidence which was fortified by the old slogans: 'We shall win because we are the strongest', 'Remember the Marne' and 'From Verdun to victory!' Reynaud also took to the airwaves to reassure the nation, proclaiming, 'France has been invaded a hundred times and never beaten... our belief in victory is intact' \u2013 even though, as the British government well knew, this public confidence masked deep private concerns about the unfolding situation. By early June, a growing number of people in Paris began to share those concerns as an influx of refugees continued to arrive from the Netherlands, Belgium, Luxembourg and the borders. The air raid, though, was to make doubters of the majority.\n\nWith the next stage in the ground offensive looming and wishing to cause as much chaos as possible in the French capital, G\u00f6ring's aircraft flew a terrorizing mission. Although their targets had been selected for their value to the military and included railway stations, warehouses, factories, bridges and communications centres, the German aircraft lacked any precision-bombing capability and their accuracy was further hampered by the capital's anti-aircraft guns, which would have forced them to fly at a greater altitude. As it was, the damage, although random, was considerable. Alexander Werth's journalistic instinct was triggered by the raid and he immediately took a taxi to become one of the first witnesses to the destruction done by German bombs near the Mirabeau Bridge over the Seine, where he found the Citro\u00ebn factory ablaze and the Avenue de Versailles covered in broken glass. Over 1,000 bombs were dropped on Paris that day, killing 254 and injuring a further 652, and coming as it did just as the Allied military disaster culminating at Dunkirk became clear, the raid unleashed a wave of panic through Paris.\n\nBetween the French interior and the burgeoning German forces, a dam of troops had been hastily cobbled together during the interlude offered by the Halt Order and the time it took to erode the Dunkirk pocket. The Weygand Line stretched 225 miles from the mouth of the Somme to the north end of the Maginot Line and sought to hold back the Germans in a positional battle that had previously been denied them as a consequence of Panzer Group Kleist's _Sichelschnitt_ or scythe cut. The great problem for the French was not just the quantity and quality of resources available to them but, with the memory of their recent Plan D misdeployment so fresh, where to place them for the best effect. As Halder noted, while the French 'cannot but realize that we have considerable forces available for new operations... he will not have a clear picture of our new offensive plans'. The French High Command was confronted by three likely courses of German action: an attack on Paris, most likely through an encirclement, a push south behind the Maginot Line towards the Swiss border, or a combination of the two. Weygand remained pessimistic about the army's ability to defend for long against any of the options. Having taken over a situation that was already desperate, he quickly found that his ability to influence events in a positive manner was severely limited as France had long since relinquished the initiative. His Weygand Line was the physical manifestation of his need to react to the enemy, and the best that he could hope for was that the Germans would so dilute their attack as to provide the opportunity for a protracted attritional struggle. In such circumstances, there was the possibility, albeit faint, that France would be able to regain some of the strength that it had lost in the first phase of fighting, and that it might be further enhanced by British and, ideally, an American commitment to the Allied cause.\n\nWeygand and senior French generals also drew a crumb of comfort from the fighting performance shown by various Allied formations at the beginning of the offensive, when the enemy had been forced to attack their prepared positions frontally. Thus, it was anticipated that the divisions of the Weygand Line might be able to hold out against an over-stretched enemy if he attacked, as expected, without delay. Indeed, even as the French defensive line began to take coherent form during the last week of May, senior officers encouraged their subordinates to take every opportunity to ensure that their charges understood their own, individual importance to the future of France. Gone were the days when a combination of the Maginot Line and the defence of Belgium was regarded as a guarantee of the nation's security, for now this rested on the fighting effectiveness, tenacity and courage of the divisions strung out from the Meuse to the Channel coast.\n\nArtillery officer Lieutenant Michel Sigaut arrived at Bray on the Somme just as news arrived that Calais had fallen to the enemy, but was encouraged not to speak in negative terms about the military situation. On the evening of 1 June, Sigaut and around thirty fellow officers were ordered to attend a briefing held in a dark barn, where they were addressed by their regimental commander. He later recalled:\n\nWe were told to ensure that our men understood the importance of the sector, to remind them of their duty and to emphasize how much everybody relied on each other. All along the front the line divisions were putting pressure on officers to encourage their men to stand and fight... I was sent on a liaison task to another division and walked in on a regimental briefing to troops that was identical in tone to one that I had given just hours before: 'We can hold the Germans and it will lead to victory. Remember your history, remember your loved ones, do your duty. _Vive La France!_ '\n\nThe impact of motivational speeches on the troops is difficult to assess, but despite their forlorn position there is no evidence of the divisions in the Weygand Line suffering from a lack of morale and motivation during the first days of June. In fact, morale was resilient during this period: despite the undeniably poor strategic situation, the basic needs of the troops were met, assisted by shorter lines of communication and supply, and the majority of the formations along the Somme and Aisne had yet to encounter the Germans in battle.\n\nA total of just sixty French and the equivalent of five other divisions from Poland, Czechoslovakia and Britain were available to Weygand, including a few \u2013 such as the incomplete and partially ravaged 1st British Armoured Division \u2013 that had been moved into the defensive line at the end of May during the formation of the Dunkirk perimeter. They included nine newly generated divisions and nineteen others that were formed from Maginot Line garrison fortress units which had been drawn north as the result of one of Gamelin's last acts as Supreme Commander. Although few, if any, could be said to have a first-class fighting capability, these formations went some way towards replacing those lost since 10 May and were bolstered by a further 30,000 men that had returned to France after their evacuation from Dunkirk. Another three infantry divisions, two British and one Canadian, were due to be entered into battle through Cherbourg and St. Malo later in the month. The continued British involvement was partly a reaction to the French dismay at the recent evacuations, but was also regarded by Churchill as the vanguard of a new force that would be sent across the Channel if there was any healthy prospect of long-term French resistance against Germany. London realized, nonetheless, that the likelihood of any extended fighting depended on the ability of Weygand's defences to withstand the next enemy onslaught.\n\nAlthough it is impossible to say what might have happened had _Fall Gelb_ incorporated von Manstein's drive south behind the Maginot Line to stop the development of a Weygand Line, the Germans would have to face the consequences of not adopting this line of attack in their new offensive. Nevertheless, they had managed to take crossing points over the Somme at P\u00e9ronne, Amiens, Picquigny and Abbeville during their advance to the coast which provided useful stepping stones for the offensive on the right. For their part, the French could only assume that these bridgeheads would be used for the German main effort, which would be followed by an attempt to encircle Paris with a strong punch from their right wing in the manner of von Schlieffen, and as a result they expended considerable effort in the last days of May to eliminate the enemy's lodgements over the Somme. Their failure to do so was a considerable setback for the defenders, who used a combination of bombers, tanks and infantry to dislodge the Germans, but ran into opposition which reflected the importance that the enemy attached to the sites. Meanwhile, Weygand's formations continued to make their preparations. General Antoine-Marie-Beno\u00eet Besson's Third Army Group protected the Somme sector with its Tenth and Seventh Armies while Fourth Army Group, consisting of Second, Fourth and Sixth Armies and commanded by the recently promoted General Huntziger, was in position along the Aisne. The twenty divisions in these two army groups were spread more thinly: each held a frontage of around ten miles, double the figure on 10 May. They also had to take greater account of the Germans' 'new' methods.\n\nThe defences were developed in line with an order from Weygand promulgated on 24 May and the divisions worked frenetically to prepare aggressive, deep and forbidding defensive positions to receive the next German onslaught; every available pair of hands toiled in the task, often during German artillery bombardments and air attacks. A large number of lessons had been extracted from defending against the German spring offensives back in 1918, and these were augmented by conclusions drawn from the recent experience on the Meuse, then distilled into orders for implementation by commanders. The aim was to deny the enemy momentum by presenting him with obstacles in depth which slowed him, and then to confront the attackers with firepower which destroyed them. It was to be a chequerboard defence of interlocking positions with fire support from well-camouflaged anti-tank and machine-gun posts. Local populations were evacuated as a defensive zone of at least five miles in depth was rapidly constructed behind the Somme and the Aisne. Along the length of the front, hamlets and villages were militarized: some buildings were demolished for better observation, obstacles were positioned, trenches were dug, buildings had their windows knocked out, entrances were sand-bagged, and machine guns were sighted to create individual strongpoints that could fight on their own but also supported each other in a system of all-round defence that became known as 'Weygand's Hedgehogs'. The whole region became a construction site, noisy and imbued with a sense of urgency.\n\nCommanded by General Paul-Henry G\u00e9rodias, the 29th Alpine Division had recently arrived in the Somme sector from Lorraine by rail and road and was greeted with orders to develop its defences. The formation was well-led, well-equipped and competent, but in common with many divisional commanders along the front, G\u00e9rodias was only too well aware that on his left flank was the less capable 19th Infantry Division, commanded by General Toussaint, which weakened his position. That division included 22nd R\u00e9giment de Marche de Volontaires Etrangers (RMVE or Foreign Volunteers Regiment), which was led by the languid reserve officer Lieutenant Colonel Pierre Villiers-Moriam\u00e9 and was positioned on the flank adjoining G\u00e9rodias's sector. A mixture of anti-fascists from Spain, Italy, Czechoslovakia, Poland, Germany and Austria, the regiment was not easy to control and still lacked some basic training and equipment. On 31 May Toussaint reported to his corps commander, General Th\u00e9odore Sciard, that his formation was 'absolutely incapable in its present state of fulfilling its mission in battle'. Sciard was not impressed, and so with no reserves to play with, he shuffled his pack of commanders and replaced Toussaint with General Fernand Lenlud and replaced Villiers-Moriame with Major Paul Hermann, a regular officer from 41st Infantry Regiment. The timing was not ideal, but Sciard did what he could to revitalize the flagging division, believing the risk worth taking considering the impending challenge it faced.\n\nIn common with every division along the Somme and the Aisne, 29th Alpine and 19th Infantry Divisions made the best of their situation and made their preparations in order to fulfil Weygand's direction that each hedgehog 'must hold out at all costs while it awaits relief'. This was to be provided by a collection of infantry, half-mechanized 'light' divisions, DLMs and DCRs which were positioned to the rear of the static infantry division to strike back \u2013 as they had been earlier in the campaign \u2013 but were still vulnerable to communication breakdown, congested roads and resupply challenges. As a result, and as before, French counter-attacks would be a complicated race against time in order to get a strong force where it was needed when it was needed. Once in position, the divisions were to drive into the over-extended German spearheads while the main defensive zone remained in position and continued to fight in order to isolate the anticipated German armoured penetration. The heavier divisions providing the 'relief' were best configured for this task and were in varying states of repair. 1st and 2nd DCRs had new, more capable commanders, although 4th DCR had lost the dynamic de Gaulle, who had taken up a position in government, and all armoured divisions had been re-equipped. Even so, the four DCRs still only had 150 tanks between them and, even when combined with the three remaining DLMs, could muster only 324 operational machines. Weygand hoped, however, that with the French aviation industry reaching full production in May, the air force might be in a stronger position to help suppress the German offensive than it had been less than a month before. Yet, although the service possessed 2,350 aircraft by early June, the number that were operational had declined from 879 on 10 May to a mere 600 on 5 June. The army was not, however, alone in learning lessons from their recent experiences, and General Vuillemin had done what he could in a very short time to improve the reaction time of his ground-attack squadrons by concentrating them directly under a more centralized air force command, rather than parcelling them out in small numbers to provide local cover and placing them under army command.\n\nThe French tried to tackle some of the failings and frailties that had been shown up since the Meuse crossings, but there was only so much that could be done in the time available and with the resources at their disposal. Transforming the armed forces into the modern, mobile, free-thinking, initiative-taking force that was required to cope with German fighting methods was something that would have to wait. Moreover, commanders were continuing to work in desperate strategic and operational contexts in which the enemy retained the initiative while they had no meaningful sense of what they were facing or where the enemy's main effort might fall. Requests for aerial reconnaissance were more often than not refused because there were too few aircraft to meet the demand and those that took to the skies were quickly targeted either by German anti-aircraft fire or patrolling fighters.\n\nAs a consequence, the French High Command could give very little direction to the frustrated formation commanders, who sought to react as quickly as they could to whatever befell them, despite the systemic difficulties in doing so. It was a situation which led General Jules-Georges-Jacques Baudouin, the commander of 13th Infantry Division, to write, 'Personally, the absence of intelligence and firm orders from the high command caused me dreadful anguish.' This made the positioning of the counter-attacking formations, particularly the valuable DCRs, little more than an educated guess. Units used raiding parties to obtain what information they could from the enemy's forward positions and used high ground to try and observe German movements, but the result was superficial, partial and generally highly inferior intelligence from which little of use was learned. In the circumstances, Baudouin believed that he and his fellow divisional commanders were in a 'covering and sacrificial role, to allow time to be won and a defence organized to the rear on the Oise, the Seine, the Loire perhaps'. There was, in fact, a good deal of truth in this, for there were discussions between the government and the French High Command about what might be done if the Germans broke through. Various positions \u2013 usually rivers \u2013 had been identified as places where further resistance might be attempted, but the Weygand Line was always considered far more than a mere speed bump; it was the Line where the enemy really _had_ to be caught and held.\n\nWeygand was under no illusion: the Germans could not be allowed to break through the new defensive line and yet he knew that their immense power made just such a breakthrough extremely likely. There were 104 German divisions available for _Fall Rot,_ the new offensive, with another nineteen divisions in reserve. In this part of the campaign the panzer divisions served the infantry armies \u2013 albeit in different ways depending on circumstances and the commanders involved \u2013 for what the Germans recognized was a set-piece attack. The forty-seven divisions of Army Group B were pitted against the Somme and the western section of the Aisne, with three armies each led by a panzer corps. Fourth Army was in the coastal sector on the right with Hoth's 5th and 7th Panzer Divisions, Sixth Army in the central sector with 9th and 10th Panzer Divisions, while Ninth Army was on the left with 3rd and 4th Panzer Divisions. Eighteenth Army was in the second echelon. Other than Hoth's corps, the panzer divisions were part of Panzer Group Kleist and were to use the pre-existing crossings at Amiens and P\u00e9ronne to launch their attacks and then surge forward against the Seine north-west of Paris followed by the infantry.\n\nOne of the infantry formations was XXXVIII Corps, commanded by Erich von Manstein, who had been promoted to General of Infantry on 1 June and, according to his biographer, was to see his experience 'hardly taxed by organizing what was essentially an administrative road movement exercise'. To the east, Army Group A was to attack the remainder of the Aisne up to the end of the Maginot Line. Here von Rundstedt ordered that the infantry in the form of Second and Twelfth Army were to lead the attack around Rethel. Having crossed the river and broken the line, the infantry would then mop up while a new Panzer Group commanded by a promoted Heinz Guderian consisting of 1st, 2nd, 6th and 8th Panzer Divisions in two corps, was to conduct an exploitation across the Marne and then, at least in part, push to the rear of the Maginot Line. Once Guderian was making good progress, Army Group C was then to undertake an offensive which would see First Army attacking in a southerly direction from east of Metz and Seventh Army crossing the Upper Rhine south of Strasbourg. Von Leeb's aim was to dislodge the Maginot Line defenders as Guderian, under his command, outflanked them in order to avoid the area becoming a last bastion of protracted French resistance. The neutralization of the 'Great Wall' was seen by OKH as an essential part of the wider offensive because, as Halder wrote, 'The Maginot Line represents everything that stands for security in the French people's mind. The line cannot be given up without risking the moral collapse of France.' As a result, von Rundstedt's attack was made the main German effort.\n\nThe role of air power in the offensive revealed the change in the Luftwaffe's priorities during this stage of the campaign. Increasingly, despite the recent focus on operations over the Dunkirk perimeter, German aircraft \u2013 assisted by the relocation of airfields to sites in France and Belgium \u2013 became more concerned with conducting operations deeper into the French interior. Although some Stukas remained available for ground support, bombers and their escorting fighters were used to target ports, hamper road and rail communications and generally undermine the enemy's ability and desire to pursue the war. As part of this, on 3 June the Luftwaffe launched Operation Paula, which sought not only to undermine enemy morale and destroy factories around Paris \u2013 as noted by Alexander Werth by the Seine \u2013 but also destroy airfields and the remaining French air force. That day, the Germans not only insisted that their attack had produced panic in Paris but also claimed 400 enemy aircraft destroyed on the ground and a further seventy-five in the air for minimal losses themselves. Indeed, the Luftwaffe was so sure that it had succeeded in many of its aims on that single day that thereafter it moved back to its operational targets. In reality, however, it had destroyed a mere twenty operational aircraft on the ground and fifteen in the air and the damaged French airfields were all returned to action within forty-eight hours. Where Paula had succeeded, however, was, as we have seen, in taking the campaign to Paris, where it had caused damage to the capital's fabric and significant casualties as well as panic. Thus, although the operation had failed to significantly damage the French air force, this was not of any great importance for it had dented French morale and the Luftwaffe remained dominant as Germany sought to bring France to her knees.\n\nHalder's plan for _Fall Rot_ was relatively simple, sensible and achievable: an attempt through three co-ordinated offensives to strip France of its last defenders, isolate Paris and, through the widening of the fissures created since 10 May in France's ability to resist, bring about collapse. Coming so soon after _Fall Gelb,_ however, and requiring rapid planning and movement of assets, the offensive posed a considerable challenge to the General Staff and the staffs of von Bock and von Rundstedt. As a result, Halder focused on detail and paid due respect to the enemy, an attitude he believed to be more in keeping with what the situation required and stood in stark contrast to the blas\u00e9 over-confidence displayed by some members of OKW. He was also anxious that Hitler's headquarters remained one step removed from a true understanding of how the campaign was being fought for, while the General Staff was concerned only with those objectives that would lead to the military collapse of the enemy, Hitler wanted to grasp as much territory as possible before the French capitulation for reasons of prestige, the longer-term subjugation of the country and protection of his new Continental acquisitions. This difference in priority was something that Halder recognized he would have to face once the Weygand Line had been broken, for while OKH sought for elements of Army Group A and B to encircle French troops in north-eastern France \u2013 including Paris \u2013 behind the Somme and Aisne, OKW was much more set on the full weight of Army Group A plunging south to contain any threat of a counter-attack from the region and to seize the rich resources of the Lorraine ore basin before considering a drive south-west. The difference of opinion was relatively minor and lay in emphasis rather than main focus, but it was another example of the growing friction between the two headquarters and was a frustration for OKH, whose _raison d'\u00eatre_ was, after all, the facilitation of military victory through its direction of operations.\n\nAs the Dunkirk evacuations came to an end, it was decided that _Fall Rot_ would commence on 5 June with Army Group B, which was already in position, while Army Group A would attack several days later on the completion of its relocation and preparations. Both Army Groups did what they could to maintain as much operational and tactical surprise as they could and, in the immediate prelude to the attack, sought to add to the thickness of the fog that was already enveloping the Allies. Shooting down French reconnaissance aircraft was merely the most obvious sign of an attempt to deny the enemy any useful information, but the Germans also went to great lengths to ensure that only disinformation was transmitted by radio and that anything of importance was communicated by field telephone. The change of Enigma codes, coincidentally, denied the enemy valuable intelligence at a vital time, and the German divisions also put considerable effort into making the French think that troops and vehicles were on the move at night when in fact they remained in place and under camouflage. Daniel Le Roy, a young French officer in the line near Rethel, recalls the confusion caused by such methods:\n\nI was kept awake night after night listening to the sound of German vehicle movements and could even hear the shouts of the enemy who goaded us. The comings and goings were very confusing because they did not follow any pattern... We sent a raiding party across to try and capture a prisoner, but they returned only with a fresh corpse. We stripped him of any information that we thought important and passed it on to the Intelligence section. We thought that we had found out a great deal of information from the body. Later we learned that it had probably been planted there and contained false unit badges and papers.\n\nThe Germans, meanwhile, continued to take advantage of poor French security and managed to capture prisoners, written orders and marked maps all along the line. This intelligence, when added to aerial reconnaissance photographs, provided a relatively detailed picture of French strength and deployments and led one German staff officer to boast later that he knew the names of every French officer in the regiment facing his division as well as 'a full inventory of their weaponry, equipment, ammunition supplies and stores'. With the information supplied to him, on 4 June Halder surmised that the French had some sixty divisions in France, 'enough for a weakly held front and some reserve groups' and correctly identified that Weygand believed that his main effort would come across the Somme. Even so, OKH believed it vital to remain flexible in the face of the enemy's response to its opening moves but hoped that by drawing Third Army Group into battle first, the task of Army Group A's main effort across the Aisne would become that much easier when it was launched a few days later. Rommel, personally decorated by Hitler just days before, wrote to his wife hours before the start of the new offensive:\n\nWe're off again today. The six days' rest has done a lot of good and helped us to get our equipment more or less back into shape. The new move won't be so very difficult. The sooner it comes, the better for us. The country here is practically untouched by war. It all went too fast. Would you cut out all the newspaper articles about me, please? I've no time to read at the moment, but it will be fun to look at them later.\n\n_Fall Rot_ began with von Bock's attack at first light on 5 June accompanied by a huge air and artillery bombardment along the length of Tenth and Seventh Army's front. It was an event more reminiscent of the loud and brutal preliminary to the Meuse crossing on 13 May than the insertion of airborne troops and the deception that had accompanied the opening of _Fall Gelb._ As at Sedan, the French were waiting and cowered within defences which, when they emerged through the smoke and dust, they discovered had only suffered minimal damage from the enemy bombs and shells. This time, however, the Germans did not have to wait for bridges to be constructed and the panzers, anti-tank guns and artillery were hard on the heels of the leading infantry and engineers. Nevertheless, this was more in keeping with the sort of warfare for which the French had prepared and the troops were not so traumatized by the onslaught. Rommel himself was supremely confident in his own abilities and those of his division and, despite the challenges facing both, had no doubt that 7th Panzer Division would prevail and, moreover, that the campaign would be won within a fortnight. With a bridgehead already established across the Somme, the formation fell on the French defences and set to work as it had done during the break-in and breakthrough of the extended Maginot Line at Soire-le Ch\u00e2teau, but without the benefit of surprise. The division displayed well-practised all-arms co-operation, benefited from superb leadership and drew heavily on its recent experiences to avoid the French hedgehogs by keeping off major roads and ignoring important features of the terrain. In this way, units identified and opened weaknesses in the French defences so that the division could advance and not become sucked into the static, positional battle that was their enemy's preference.\n\nWatching from his command vehicle just behind the leading panzers, Rommel later wrote that 'The various arms worked in such perfect co-ordination that it might have been a peacetime exercise' although more objective participants said that the defences were the most troublesome they had faced since the beginning of the invasion. Even so, 7th Panzer Division had advanced eight miles by the evening of the first day and in doing so had carved its way through the Weygand Line far more quickly than Besson would have liked. It was now up to the French counter-attacking forces to stop the fast-moving division before it caused chaos in the rear while the defenders on the Somme attempted to close the breaches made by Rommel and hold the infantry regiments that were following in his wake to mop up. These German troops, less well-trained, with limited experience and with fewer resources, were made to fight hard for every yard of ground. As a result the French defences did not burst open on 5 June, and Besson's divisions generally did a good job of soaking up the initial momentum of von Bock's attack along the majority of the Line. The commander of Army Group B wrote at the end of the day:\n\nThe French are defending themselves stubbornly. Their tactics in the face of our panzer attacks is to establish themselves in towns and forests, placing the main weight of their defence in 'tank-proof' terrain. This has caused some local discomfort, however; since they are leaving the intervening spaces between strongpoints free, this tactic will do them little good in the long run. For the moment, however, it provides them with an advantage, for while near Amiens and P\u00e9ronne the panzer divisions can advance between the occupied towns, their riflemen are unable to follow under the effects of flanking fire from the towns. Consequently, moving motorized divisions forward today was not yet possible.\n\nVon Bock was correct in his analysis. The French could not resist the Germans indefinitely and in the days that followed the infantry divisions continued to fight among the French defences while 7th Panzer Division struck out alone. Rommel's intention was to drive deep into the French rear, undermining the enemy's cohesion and breaking any subsequent French defensive lines while the infantry completed their breakthrough. Thus, while the fighting on the Somme remained intense and congested, Rommel advanced through more open countryside towards the Lower Seine. On 6 June, 7th Panzer Division advanced a further thirteen miles and on the next day another sixteen miles. To achieve this, Rommel put into practice a method he had rehearsed before 10 May for just such a scenario. For a _Flachenmarsch_ (area march) the division formed up on a 2,000-yard front and twelve miles in depth to advance as one block which avoided villages and main roads and fired on the move into places that concealed the enemy. Clearly impressed by what his formation had achieved during this period, Rommel later wrote:\n\nThe advance went straight across country, over roadless and trackless fields, uphill, downhill, through hedges, fences and high cornfields... We met no enemy troops, apart from a few stragglers... Fleeing civilians, and also troops, were on all roads. Sometimes we even surprised refugee lorries in open country, their occupants, men, women and children, underneath the vehicles, where they had crawled in mortal fear. We shouted to them, as we passed, to go back home.\n\nRommel's thrust was so rapid and concentrated that it undid French attempts to counter-attack it. Those valuable forces that might have mounted such an attack were just too widely spread and slow to respond in the face of his speed and aggression. The timely and accurate application of German air power also managed to stifle the enemy's reaction and generally undermined its ability to reinforce the fighting front and supply it with vital resources. In such circumstances, movement by train or road during daylight hours was extremely dangerous and so mobile units either split up and tried to keep off major roads to make themselves more difficult targets or were forced to move at night. Thus, 1st and 2nd DCR, which had been positioned around thirty miles south of the Somme, found themselves in fragmented, un-coordinated groups lacking clear orders and in no position to engage Rommel's advance. As a consequence, by 8 June 7th Panzer Division was moving rapidly towards Rouen on the Seine. The river was the last major barrier that could thwart either the encirclement of Paris or a push towards Normandy and Brittany. An attempt was made by the division to cross the river that night but having been rebuffed \u2013 much to Rommel's anger at his motorcycle infantry \u2013 an operational pause was made to allow for von Manstein's infantry divisions to catch up.\n\nAs it was, after managing to reduce the enemy's defences on the Somme, von Manstein sent his corps forward with remarkable speed. Although it did indeed include the motorized 1st Cavalry Division, he treated his three infantry divisions as if they were all motorized and drove them hard towards the Seine. It was a move which was to have major implications for the integrity of the Weygand Line because its breakthrough forced the French Seventh Army into withdrawal but at a slower speed than the advancing German formations. As a result, 7th Panzer Division on the Seine found that a corps-sized group of French and British troops came tumbling back towards it, looking, perhaps, to reinforce the river line or for evacuation from Le Havre, and Rommel was sent to stem the flow on 10 June. In the meantime, 5th Panzer Division was designated to take Rouen and von Manstein's 6th and 46th Infantry Divisions to forge a crossing of the Seine. By the time that this had been successfully achieved, 7th Panzer Division had made another huge contribution to the campaign. The Armoured Reconnaissance Regiment followed by the Panzer Regiment first made for the seaside town of F\u00e9camp with the division commander, as usual, in the lead group. Reaching the coast, Rommel later wrote, 'thrilled and stirred every man of us' but although there was a temptation to feel that a culminating point had been reached in the formation's offensive, officers were quick to remind the men of the important task that was before them.\n\nThe appearance of elements of 7th Panzer Division at F\u00e9camp had scuppered the plans of the Allied divisions withdrawing along the coast and forced them to try and expedite a rapid evacuation from St. Val\u00e9ry on 11 June. Recognizing what was happening, Rommel stalked his enemy and, when it was bottled up in and around the town, went for the kill. Within one or two hours of his guns starting to bombard the concentrated Allied group, around 1,000 French troops surrendered but the vast majority continued fighting as they sought to avoid being encircled. Caught in an unedifying situation without heavy weaponry, the ability to manoeuvre or any chance of relief, the Allied divisions fought bravely throughout 11 June but as 7th Panzer Division arrived in increasing strength, the encirclement was complete and its firepower took a heavy toll. By 12 June, the commanders in the pocket recognized that they could ask their men to engage in a futile defence of a militarily unimportant French town, or they could surrender. Gradually, groups of Allied troops began to capitulate \u2013 most often when they had expended all their available ammunition \u2013 and the position collapsed. One German section was surprised to find that a large French group that surrendered to it included General Marcel Ihler, the commander of French Ninth Corps. Ihler, however, was just one of twelve generals taken prisoner that day and Rommel was delighted to hear one remark, 'You are too fast, much too fast for us. That's all there is to it!' The 46,000 prisoners 7th Panzer eventually took at St. Val\u00e9ry included the remnants of five divisions and 8,000 men of the BEF's 51st (Highland) Division. Having ensured that photographers were on the spot to capture his latest triumph, that evening Rommel wrote to his wife: 'The battle is over here. Today one corps commander and four division commanders presented themselves before me in the market square of St. Val\u00e9ry, having been forced by my division to surrender. Wonderful moments.'\n\nAlthough held back by its brief siege of the St Val\u00e9ry pocket on the Channel coast, Rommel's division had carved a deep wedge in the enemy's defences, one which allowed 5th Panzer Division and von Manstein's divisions a freedom of movement that unlocked the coastal sector of the French defences. No formation enjoyed similar success on the remainder of von Bock's front where, despite the best efforts of von Kleist's two panzer corps at Amiens and P\u00e9ronne, the French held firm. Progress was made, but although highlighting both solid French defence and impressive German all-arms co-operation, it was grinding and relied on what limited Luftwaffe support was made available. At P\u00e9ronne, for example, Hoepner's XVI Panzer Corps found that, despite pre-battle concerns about their adjoining flanks, 29th Alpine Division and 19th Infantry Division withstood the pounding meted out by the artillery and the Stukas and fought the ensuing ground attack with considerable composure. At the fortified hamlets of St. Christ and Mis\u00e9ry, Lieutenant Albert Besson, a company commander in 112th Alpine Regiment, recorded 'six wave assaults' on 5 June alone. Another defender ensconced in St. Christ, eighteen-year-old Serge Winter, later wrote:\n\nMy team had made a strong position for our machine gun in a farm building. There was another [500 yards] to our right and together we covered an obvious approach route up a small valley to our piece of higher ground... The enemy probed us to try and get us to give our positions away. We resisted the temptation to open fire, and some men in trenches [200 yards] in front of our building dealt with them... Twenty minutes later, those trenches were hit by a mortar bombardment and the enemy immediately followed up screaming. There were hundreds of them and we opened fire. We cut the enemy down, but more took their place and there were too few of our machine guns to stop them. We desperately called for artillery support, but none came... We were overrun after about an hour, our lives saved by an NCO who ordered one of his men not to shoot but to take us for interrogation.\n\nWinter could not have known at the time, but elsewhere in the sector some captured French troops were used at the head of a German attack as a human shield. Some were shot by their capturers when they refused to advance, while others threw themselves to the ground when a fire-fight began. Meanwhile, 19th Division's 22nd RMVE also fought bravely and with some skill at Fresnes-Mazancourt, March\u00e9lepot and Mis\u00e9ry on 5 and 6 June, and used its 3rd battalion to co-operate closely with 112th Alpine Regiment's 2nd Battalion to ensure a strong seam between the two divisions. Attempts to infiltrate the French line were consistently rebuffed with excellent artillery support, all of which led to eighty German armoured fighting vehicles being lost on the first day alone.\n\nSimilar difficulties were faced by XIV Panzer Corps and it was only when the Stukas, having been forced to wait for a fighter escort due to a stronger than expected French air presence, turned their attention to the destruction of the French guns that any forward momentum could be made. By the end of 5 June, a six-mile-deep bridgehead had been won at Amiens and P\u00e9ronne which, although insignificant considering what the Germans had achieved for much of _Fall Gelb,_ was more in line with the amount of ground taken immediately after the Meuse crossing \u2013 albeit that there were bridges already in place across the Somme. As the Germans later admitted, the battles in this area were 'hard and costly in lives, the enemy putting up severe resistance, particularly in the woods and tree lines continuing the fight when our troops had pushed past the point of resistance'. As a consequence, von Kleist's divisions managed to break into the French defences, where they had to fight hard ground, rather than break through them and over the next couple of days struggled in a defensive web while Rommel raced towards the Seine.\n\nIt was not uncommon for French units to fight until they could fight no more. Indeed, the remnants of 112th Alpine Regiment \u2013 100 men, of whom half were wounded \u2013 only surrendered after three days when their ammunition ran out. The Germans found this difficult to cope with, for the situation allowed the French to commit their limited reserves to plug gaps in the line while the attackers were forced to commit poorly trained second-rate troops who had not been intended to fight in such conditions. To the east of 19th and 29th Alpine Divisions, for example, General Louis-\u00c9mile Noiret's 7th Colonial Division reported that, by the fourth day of the offensive, the enemy was 'starting to show undeniable signs of fatigue'. To support this assertion, Noiret quoted a German infantryman from a regiment in reserve which had been sent into battle on the afternoon of 7 June and by daylight the next morning had already lost its cohesion. The German soldier was brought into division headquarters after his capture on 8 June and Noiret reported that he 'was sobbing with fatigue and dead on his feet' as he told French officers:\n\nWe haven't slept more than three hours a night for three weeks and we've marched on foot over 260 kilometres (162 miles) in a week... My comrades are keeling over, they're so tired, our regiment could only defend itself now not attack any more... and the regiment has had three commanding officers in the past twelve hours. Your artillery has caused us some losses, but what's really stopping us is your dug-in entrenched infantry.\n\nIn fact, the French defences so impeded the advance of XIV and XVI Panzer Corps that on 8 June a decision was taken for both to disengage and shift further east to exploit Ninth Army's crossing of the Aisne towards the Oise against the French Sixth Army. In itself, this was a clear success for the divisions facing the power of Panzer Group Kleist and was one of just two occasions during the entire campaign \u2013 the defence of the Dunkirk perimeter being the other \u2013 when operational initiative was wrested from the Germans and they were forced to deviate from their preferred course of action. However, just like at Dunkirk, the attackers were only temporarily inconvenienced. In the case of von Kleist's panzer divisions, the disruption was considerable as the moving of such a force could not be done without the requisite planning and preparation, but the fact that the Germans managed it was further evidence of the benefits that can be reaped from slick staff work and decisive action. Yet even as the two corps were redeploying, leaving Eighteenth and Sixth Army to reduce the defences south of Amiens and P\u00e9ronne, the French front began to crumble. Assisted by Rommel's success to the east, which had forced Seventh Army to withdraw, Tenth Army also began to give ground as its left flank became exposed. Thus, even as von Kleist's panzers began to reach the Aisne, Weygand gave the order for a general withdrawal from the Somme.\n\nDuring the opening days of _Fall Rot,_ the French had shown a great deal of fighting spirit, not to say impressive tactical competence, and made good use of the benefits of a defensive system which was much more conducive to upsetting German ambitions. There were enough weaknesses in Weygand's operational circumstances, however, to render his line little more than the fleeting encumbrance to the enemy that General Baudouin believed it to be. To some, therefore, the fighting that cost the Allies so dear in lives and resources during the first three days of the new German offensive was a futile gesture. This line of reasoning is understandable considering the outcome, but one has to recognize the impact that _Fall Gelb_ had had on the decision-making system of the French and their ability to move troops and equipment where they might do the most damage. Considering the recent change in Supreme Command and the fact that French formations were already forming up on the Somme and the Aisne as von Kleist swept towards the Channel, the Weygand Line was a natural and logical defensive option. In such circumstances, alternatives to it were not readily available and, incidentally, the fighting on the Somme eventually offered a good insight into what might have befallen the Germans had they not taken up the essence of von Manstein's scheme. As it was, a lack of time, too few resources, poor command and control and a severely limited counter-attacking ability, undid the defenders. Despite their courage, the bridgeheads over the Somme and the power of the Germans when married with another remarkable advance by Rommel led to the need for a withdrawal from the Somme after a mere four days. As one French staff officer in 47th Infantry Division's headquarters later said:\n\nThe situation had not changed fundamentally. We had managed to hold the line for a while, which gave us some renewed sense of worth, but ultimately we were forced back by a superior enemy... We were like that little Dutch boy trying to plug holes in the Dyke. The Germans were a force of nature and we soon recognized that we were going to be washed away. It was a matter of _when_ the dam burst, not _if._\n\nAs von Bock's troops occupied the enemy's former positions, Army Group A completed its final preparations for the launching of the German main effort. It was an attack that sought to sweep away the last foundations of French resilience and, in co-operation with von Bock's own continued operations, to bring the nation to its knees.\n\n# CHAPTER 12\n\n# 9\u201322 June \u2013 Driving South, Paris and Armistice\n\nTAKEN PRISONER BY THE GERMANS at St. Christ on the first day of _Fall Rot_ , a nervous Serge Winter joined other members of his 112th Alpine Regiment who were herded across a devastated landscape which was still under artillery fire, across the River Somme and into the enemy's rear area. The group of approximately forty prisoners were escorted by four German infantrymen who, Winter later said, 'looked even more exhausted than we did, but were friendly enough even if they did keep saying, \"France is finished\".' After being handed over to the military police and marched for another hour, each man waited in a queue by a church in a hamlet where they were searched for a second time and their papers and personal effects were placed in a large box. On reaching a small table manned by an enormous German officer, Winter was asked his name and service number, an attendant NCO checked the remaining papers that were laid out on a larger table before him and mumbled a few words to his colleague. 'The officer barely looked up,' Winter recalled, 'and when the sergeant confirmed that the information that I had given matched those on my papers, the officer scribbled a few notes on some official-looking papers and I was waved through.'\n\nThere was little opportunity to escape, although the thought barely crossed the minds of Winter and his colleagues: they hardly wished to run the gauntlet through so many German troops to rejoin a battle that they were glad to have left behind them. After being ushered into a large _salle de f\u00eates_ and offered 'water and some vile German sausage', Winter and his comrades felt safe and even managed to crack a joke or two among themselves to help raise their spirits and pass the time. 'We had heard of prisoners being roughed up by the enemy and even stories of them being murdered, but we experienced good treatment,' Winter explained, 'and within a couple of hours we were in a truck heading east.' Four days later, Winter and his colleagues had crossed the Rhine and had entered a former army barracks that was to act as their temporary prisoner of war camp. Having been shown to a hut and to his bunk by a Belgian prisoner who had already been at the camp for ten days, Winter wrote the only entry he was to make in his little diary, 'I am alive, but there is little hope for France.'*\n\nWinter's solitary diary entry was made after he had seen a little of the immense German military machine in action and its workings behind the lines. Although he had no more knowledge about the unfolding situation than he was able to glean on his journey into captivity, it was enough to convince him that France was doomed. Although _Fall Rot_ suffered some early operational setbacks, they were significant only in relation to the speed with which the Allies had been defeated in previous battles and were quickly overcome. During the opening days of the offensive, Halder and von Brauchitsch revealed no great concern about how it was progressing and instead immersed themselves in the area of higher operations on the assumption that the Weygand Line would be broken within a matter of days. As they contemplated how best to bring about the military collapse of the French, they were more convinced than ever that OKW should approve the opportunity that existed to encircle the vast bulk of the French force that remained in north-eastern France. Halder continued to argue that sending just _part_ of Army Group A behind the Maginot Line would be enough to contain and encircle any latent threat from the region, and would also allow the inner wing of von Rundstedt's formation to swing westwards to assist in a wider encirclement somewhere south of Paris. Hitler, however, would not agree to the change as he demanded the containment of the Maginot Line forces \u2013 his flank psychosis had not abated \u2013 before developing operations. It was a position that led Halder to write on 6 June:\n\nThe F\u00fchrer thinks that changing the direction of the offensive, as proposed by me, is still too hazardous at this time. He wants to play absolutely safe. First, he would like to have a sure hold on the Lorraine iron ore basin... After that he believes it would be the time to consider a drive in a westerly direction... There we have the same old story again. On top there just isn't a spark of the spirit that would dare put high stakes on a single throw. Instead, everything is done in cheap piecemeal fashion, but with the air that we don't have to rush at all.\n\nAs it was, on the morning of 9 June, the fourth day of _Fall Rot,_ Army Group A began its offensive. Attacking either side of Rethel, Second and Twelfth Armies sought to cross the Aisne and prepare a way for Guderian's two panzer corps to crack the enemy's defences open and exploit to the south. Here the French Fourth Army fought as tenaciously as their colleagues had done on the Somme and with the advantage gained from the fact that the Germans had to make assault crossings of the river. They were met with considerable French firepower, which led to some heavy casualties, but the attackers threw increasing numbers of infantry and engineers across the river and eventually some made a lodgement on the opposite bank and bludgeoned their way forward. This unsophisticated technique stood in stark contrast to the careful preparation, planning and all-arms fighting that had marked out Guderian's crossing of the Meuse at Sedan, yet the attack across the Aisne suffered not only from a rushed prelude and a lack of surprise but also from troops of lower quality and less intensive air support. Guderian's attitude to the struggle gave every impression of a gloating general who felt that his own troops could have done better, and reflected a brashness that was not missed when List visited him during that tumultuous day and found his troops relaxing in the sun and some even bathing in a stream. When List asked Guderian why his troops were not ready for action, the panzer commander replied that it was the infantry's job to establish a bridgehead. Under the cover of darkness that night, however, Guderian did begin to filter 1st and then 2nd Panzer Divisions across the Aisne as soon as pontoon bridges had been constructed and their infantry immediately reinforced the small force that were found to be rebuffing local French counter-attacks.\n\nThe building of military bridges and the subsequent enlargement of the German bridgeheads by armour on 10 June was an important step for Army Group A but there was no dissolution of the French defences. List read numerous reports that left him in no doubt that the opposition was stronger than had been expected and he developed a grudging respect for the enemy's resolve, concluding: 'The French are putting up strong opposition. No signs of demoralization are evident anywhere. We are seeing a new French way of fighting.' General Lattre de Tassigny's 14th Division even managed to produce a counter-attack which took 800 prisoners. The French also threw their armour against Guderian's massing panzers in an attempt to stop them from gaining an irresistible mass. While 3rd DCR hit the flank of General Rudolf Schmidt's 1st and 2nd Panzer Divisions with eighty-six tanks from Juniville on the afternoon of 10 June and a further attack was mounted by 7th DLM on the following day, Reinhardt's 6th and 8th Panzer Divisions were forced to 'beat off several counter-attacks by French armoured and mechanized brigades'. Yet despite valiant attempts, French Fourth Army could not hold Guderian's Panzer Group indefinitely. By 10 June, the forward elements of 2nd Panzer Division had already reached the outskirts of Rheims as Schmidt, encouraged by Guderian, sought to complete the breakthrough and by 12 June all four panzer divisions were flooding south through the Champagne region's open countryside, leaving the infantry to mop up in their wake. Halder could not help but see huge possibilities now that a breakthrough had been achieved and, anticipating a battle of encirclement and annihilation, he wrote: 'A \"Battle of Cannae\" is in the making.'\n\nOKH simply would not let go of its desire to complete a victory in the shortest possible time. Halder was in no mood for protracted fighting and revealed his concern at the cumulative, grinding effect on the army of operations that were now in their second month by writing on 11 June: 'The troops are said to have reached the limits of their physical endurance. They are fit now for pursuit fighting, but not for heavy attacks (battle fatigue). Striking power of the rifle units of the divisions is down to 50 per cent...'  In this context, the question of how the campaign was to develop was a critical one now that the Weygand Line had been broken. Indeed, the decision taken by OKW on 13 June not only to occupy Paris but to authorize a giant encirclement north of the Loire to match the attempt to encircle the troops of the Maginot Line represented a considerable shift towards defeating the enemy through movement rather than physical fighting. Although Paris had little military importance, its loss would very likely undermine the morale of the French and weaken their will. As a consequence, Halder gave verbal orders to the headquarters of Army Group B that evening: 'Order Fourth Army to start offensive tomorrow. Secure Paris with strong forces (five divisions)...' On the following day, 14 June, a Directive from Hitler gave further clarity to the aims of von Bock's army group by stating:\n\nEnemy forces on the lower Seine and in the Paris area will be vigorously pursued by the advance of the right flank of the Army [Group B] along the coast towards the Loire estuary and by a turning movement from the Ch\u00e2teau-Thierry area towards the Loire above Orl\u00e9ans. Paris will be occupied in force as soon as possible.\n\nA pincer comprised of the two wings of von Bock's army was, therefore, to use its crossings of the Marne east of Paris at Chateau-Thierry and the Lower Seine west of Paris near Rouen to launch a massive encirclement aiming at the Loire while Guderian continued towards Switzerland. To Halder and von Brauchitsch, the decisions were a victory for common sense, but they were also aware that the enemy would still be likely to have plenty of time to withdraw behind the Loire while Army Group B advanced and so felt that an opportunity had been missed to force an earlier French capitulation by means of a less deep envelopment.\n\nAs it was, the prompt withdrawal of the French armies to the south by Weygand ensured that as far as possible, the formations did not collapse and retained the ability to resist. It was a fighting withdrawal which was designed not only to cover those troops moving away from the front but also to significantly delay the enemy. As one German soldier later explained: 'There were pockets of resistance where they held out when our infantry were already thirty kilometres [eighteen miles] to the rear... the French regiments fought as if they felt they were defending the last street in France in a battle that would decide the very existence of their country.' The problem for the French, however, was their ability to retain a defensive coherence against an enemy attacking on such a broad front. As a result, they withdrew into the interior of France in an attempt to hold a line from the Atlantic coast, along the Loire and then along the Rh\u00f4ne to the Swiss border with Third Army Group in the west, Fourth Army Group in the centre and Second Army Group in the east. Their ability to get into position depended as much on the speed of the enemy as it did on their own ability to disengage and redeploy quickly and, as Halder had said, the German forces were still capable of 'pursuit fighting'. In such circumstances, the question of whether France could and should continue to fight was one that became increasingly difficult for the country's decision-makers to ignore.\n\nSince the end of the First World War, the defence of Paris had never been considered as something that would be undertaken in its suburbs or along its avenues. The development of the Maginot Line and a force that _would_ stop the Germans in Belgium was part of a strategy designed to allow the capital city to continue to function while its rich heritage and cultural significance was protected. So the threat that emerged to the city as the result of _Fall Gelb_ and then, more specifically, the success of _Fall Rot,_ did not lead to a decision to fight amongst the ruins of the Arc de Triomphe while the government struggled on in the capital \u2013 quite the opposite. After the destruction of non-essential documents that might assist the Germans and the transport of others out of the city, arrangements were made for the critical working parts of the national decision-making machinery to be evacuated to a temporary administrative centre on the Loire. Meanwhile, Parisians watched and waited. Over the previous week, the bombing of Paris, defeat at Dunkirk, the opening of the new German offensive, the withdrawal of the Weygand Line and then the appearance of French troops from the front in the city had encouraged increasing numbers to quit the city and head south. Few retained confidence in official sources of information in a country that was increasingly befuddled and led by a government which seemed to be saying that the population should do nothing but await further announcements. The capital began to disassemble while smoke continued to rise from the government quarter as officials incinerated official papers.\n\nOn 9 June, Alexander Werth wrote a cable to his newspaper:\n\nParis in its anguish strangely calm and beautiful these days. [Stop] During day numerous luggage-laden cars seen leaving town with passengers having tears in eyes. [Stop] At night streets almost deserted excepting rifled guards outside Government buildings and underground stations. [Stop]... At night distant gunfire hearable also occasional bombs dropping closer by and in night air there's faint sweet scent of resin and burning trees. [Stop] It may be wood burning somewhere near front. [Stop] Government's decision in case Paris immediately threatened being awaited. [Stop] But it's still too early to dwell on darker possibilities of coming week.\n\nThat same day, the government left the capital for Tours on the Loire. An official announcement was made in their wake: 'The government has been obliged to leave the capital for compelling military reasons.' Writer Andr\u00e9 Maurois took this as a statement of defeat, noting: 'At that moment I knew everything was over. France, deprived of Paris, would become a body without a head. The war had been lost.' Marie-Madeleine Fourcade, who would become a prominent Resistance leader, said that 'Paris was a city in mourning', clinging to the past but recognizing that the future had changed for ever. She added that there was a 'sad frenzy of departure' but despite the government's evacuation, no official arrangements had been made for that of the wider population and this caused suffering, chaos and despair.\n\nEug\u00e8ne D\u00e9pigny, who worked in the _mairie_ of Paris, later recalled:\n\nPopulation movements increased hourly. The objectiveness of the exodus developed and a sort of pathological panic-filled nervous anxiety overtook the crowds. People left, abandoning everything, more concerned to escape their fear of war than to escape the war itself.\n\nRailway stations were overwhelmed by the numbers arriving and the demand for trains massively outstripped supply. On 10 June, Aren Arnstraum arrived at Montparnasse in the hope of boarding a train to Bordeaux and noted in his diary: 'About 20,000 people are massed in front of the station, most of them seated on their belongings. It is impossible to move and the heat is unbearable.' For most, the only route out of the city was on roads clogged with cars, trucks, carts, bicycles and all manner of improvised wheeled devices carrying the possessions of thousands as they trudged away from the advancing Germans. Most were under-prepared both mentally and physically for what they encountered on their hot, dusty journeys. Alexander Werth was one of those who left Paris in a car with four others and wrote: 'Evacuation is like a funeral. All the bother of the practical arrangements blunts you during a large part of the time to the tragedy, to the loss of something you loved.'\n\nHaving arrived in Tours, the government was faced with having to make a number of hugely important decisions, but in conditions that were hardly conducive to the task. Its various ministries were scattered across the town and wider district due to the lack of suitable accommodation, which meant that all manner of buildings had to be utilized. Some did not even have a telephone and so messages had to be relayed by runners and despatch riders who had to try and negotiate their way through the congested streets. Weygand himself, who left Vincennes on 9 June for the Ch\u00e2teau du Muguet near Briare on the Loire, had access to just one telephone which could not be used for two hours in the afternoon when the lady manning the switchboard at the local exchange went on her lunch break.\n\nIn spite of all this, a decision still had to be made about France's continued resistance against Germany. Strategy was reviewed at a Supreme Allied War Council meeting held on 11 June at Weygand's headquarters. Reynaud, P\u00e9tain, Weygand and de Gaulle, the new under secretary of state for national defence, were joined by Churchill, British foreign secretary Anthony Eden, General Ismay and General Edward Spears, who was liaison officer with the French government. The atmosphere was tense. Recent German advances and the government's relocation had deepened France's humiliation while the British contingent recognized that the evacuations from Boulogne and Dunkirk had been regarded as an act of selfish betrayal by many around the table. Weygand lost no time in making his position clear, saying: 'I am helpless, I cannot intervene as I have no reserves.'\n\nChurchill was deeply concerned by the negativity that exuded from the French at the meeting, one which left him in little doubt that a strong faction wished for an armistice. He had tried to encourage France to try and save herself and had travelled back and forth across the English Channel to France six exhausting times \u2013 and at the risk of being shot down \u2013 to build some fighting spirit. As one commentator has said: 'At the cost of an enormous injection of energy \u2013 physical and emotional \u2013 he did all he could for France, the country he loved so much and whose history he knew so well, and in which he chose to spend so much of his time...' As a result, the British prime minister, who was happy to converse in either English or French, suggested that there was hope if they continued to resist the German advance and created the time for more resources to be brought to bear against Nazi Germany. P\u00e9tain dismissed the idea, but Churchill was not put off his stride and argued for a withdrawal into the north-west of the country in preparation to fight from a 'Breton Redoubt'. The conference was well aware that it was to Britain's advantage for the French to prolong their struggle but Weygand \u2013 supported by P\u00e9tain \u2013 believed that this should not be at the expense of the best possible terms which could be negotiated from the conclusion of an early armistice. Nevertheless, Reynaud was interested enough in Churchill's idea to direct de Gaulle to look into its prospects without delay.\n\nAfter spending the night at the Ch\u00e2teau de Muguet, Churchill flew back to England on the morning of 12 June and saw Le Havre in flames. He was in a gloomy mood on returning to London and, having received a briefing from him on the situation, Colville wrote:\n\nApparently the Generals have no reserves left. Their divisions are decimated and every man and weapon has been thrown into the line; but nothing can stop the German steamroller. Weygand and Georges talk about the impossibility of carrying on the struggle, at any rate on co-ordinated lines, but Reynaud is as indomitable as P\u00e9tain is defeatist.\n\nThe meeting had not come to a definite conclusion about whether the French should continue their fight, but it did prepare the ground for the official declaration on 13 June that Paris was now an 'open city' in an attempt to avoid it becoming a battleground. That morning, posters appeared on the walls of official buildings:\n\nParis having been declared an open city, the military government asks the population to abstain from all hostile acts and hopes that the people will remain calm and dignified in a manner which appears appropriate in the circumstances.\n\nThe announcement was a relief for those who remained in the capital but angered those who had recently left the city in fear of it becoming another Warsaw. The defence of the city was abandoned, leaving the Arm\u00e9e de Paris to fall back on the Loire east of Nantes. It was a decision that also angered some of the military in the capital, including Colonel G. A. Groussard, who immediately demanded to be relieved of his duties. He fell into line, however, after being told by General Henri Dentz, the commander of the Paris military region, that he could be shot for desertion if he did not carry out his orders. Even so, Groussard later wrote: 'I have never felt so humiliated and full of anger. The military should fight not to hand itself over. I sincerely believe that I would have given my life to save myself from having to obey these orders.'\n\nChurchill returned to France for another session of the Supreme War Council on 13 June. He travelled to Hendon airport with his defence secretariat, Colonel Leslie Hollis, who later recalled: 'I remember the morning well. It was a warm day with the sun shining. I marvelled at the calmness and serenity everywhere, and then realized with a shock that hardly anyone in the crowds of people out in the sunshine \u2013 the clerks, the typists in their summer frocks, the shoppers \u2013 realized what fearful danger faced Britain.' As bad weather was forecast later in the day, Churchill was advised not to fly when he reached Hendon, where he was met by other members of the British delegation, but he ignored the suggestion with a 'To hell with that...' and boarded the aircraft \u2013 which raised the eyebrows of some in the party. The two aircraft landed in thunderstorms and heavy rain and then taxied between bomb craters made during a raid the previous night. There was nobody at the airfield to meet them, but Churchill personally managed to secure a car from two French airmen, and headed to Tours.\n\nIt was another upsetting day for the British prime minister who, by the end of it, was convinced that 'the possibility of France remaining in the war had almost disappeared'. At the meeting he was told by Reynaud that the High Command believed that it was too late for the Breton Redoubt idea to work and that Weygand believed that France's military was a spent force. Churchill's tone, full of encouragement and conciliation at the previous meeting in Briare, became stern. He made a point of saying that he would hold the French to an agreement which had been made in March and forbade either side to engage in unilateral armistice negotiations, and that the French government should immediately appeal to Washington to establish whether the United States would enter the war on the Allied side. Churchill's apprehensions were somewhat alleviated, however, by his impression that, despite the High Command's nihilism, Reynaud was not ready to enter into armistice negotiations until he had exhausted all other options. The British prime minister was nonetheless aware that those other options were on the verge of being exhausted. As it was, at a cabinet meeting held after the British delegation had left, Reynaud found that only six out of twenty-seven colleagues agreed with P\u00e9tain and Weygand that France should seek an early armistice agreement and so, bolstered by this fact, he made his approach to Washington. He did so knowing that if the Americans rejected his entreaties, Paris fell and the Germans continued to exploit their advantage, P\u00e9tain and Weygand would win over a majority of the cabinet. As if to underline the urgency of the situation, the government vacated Tours and headed further south to the Atlantic coastal town of Bordeaux. Here Reynaud would wait for President Roosevelt's response and the cabinet would continue to discuss the nation's rapidly diminishing options.\n\nAlthough the majority of France was as yet physically untouched by the German invasion and those living in the rural south and west would have felt quite removed from the war, it should be remembered that French troops were drawn from all corners of the nation \u2013 the French colonies too \u2013 and that the outcome of the conflict would affect everyone living under the Tricolour, whether in metropolitan France or elsewhere. In such circumstances, what was best for France was immensely complicated, but while Reynaud was sympathetic to the need to embrace attempts to defeat Nazism over several years with the government decamping to North Africa, the influential P\u00e9tain opined: 'I personally refuse to leave French soil: I will stay among the French people and share its pains and its miseries.' The old soldier, whose mind had been made up around the time of Dunkirk, had consistently argued that French suffering should be ended as soon as possible by reaching terms with the Germans. In Paris, on their journey to Tours and then as they moved to Bordeaux, P\u00e9tain and his colleagues would have seen the refugees struggling along the roads or clamouring for space on trains, and would have heard the stories of them being machine-gunned by the Luftwaffe. The politicians were not left unaffected by the experience nor by the reports that they were reading about the unsettling effect of events from Normandy to the C\u00f4te d'Azur.\n\nGeorgette Guillot, a secretary in the ministry of the interior, noted the 'terrible spectacle of anxious faces, nervously apprehensive about what future calamities they may face'. After leaving a roof and hearth far behind them, the refugees that Guillot observed had found hunger, thirst and uncertainty but precious little safety and many began to regret their decision to flee. One Parisian, Julianne Deros, who was sixteen years old in 1940, later recalled:\n\nOn hearing that the city would not be destroyed, my mother began to cry. I asked her why and she said, 'I have made a terrible mistake.' With my only pair of shoes long since disintegrated with all the walking we had done, an empty stomach and many more miles to some other place to be covered, I felt inclined to agree. But we were very frightened living in the middle of Paris, very frightened. The road was a hard place to be, but at least we were doing something. I just wish I had understood that at the time, and told my poor mother.\n\nThe Deros family had been machine-gunned on the road out of Chartres, having walked nearly sixty miles in four days. It was 'a brief affair that left no casualties' according to Julianne, but the fear that it engendered in those stuck on the roads or in railway carriages without protection was hugely disturbing. Indeed, in their continued attempt to cause congestion and undermine the Allied armies' ability to move while causing panic among the population, the Germans bombed and strafed anything that moved. Metal worker Georges Adrey wrote that, even as he left Paris, he and those with him were on several occasions obliged to 'lie flat on our stomachs in the ditch or on the grass or even hide in the woods in order to escape death'. Parents would throw themselves on their children to protect them, screaming at the aircraft as they swooped down on the hapless refugees. Those in vehicles often struggled to extract themselves in time. Andr\u00e9 Morize later recalled:\n\nI saw and touched the lacerated cars. I saw blood on the cushions of the seats... I saw a ten-year-old boy whose shoulder was fractured by a bullet. I saw a woman who hardly knew how to drive at the wheel of an old Renault taking her three children away with her because her husband was killed on the road in the Pas-de-Calais.\n\nThe Luftwaffe's murderous marauding gave many refugees the impression that their quest for safety was futile. To make matters worse, large numbers were overtaken by German ground units, and even those that made it to the Loire Valley \u2013 which many believed to be far enough from the front to keep them out of harm's way \u2013 found the 'Boche' on their heels and aircraft bombing the bridges over the river to hamper the army's withdrawal.\n\nDuring the afternoon of 13 June, Paris lay quiet, its streets largely deserted, an anxious city awaiting its fate. Four million people had left the city, 80 per cent of its population. The last trains had departed, the last troops had withdrawn. Journalist and soldier Georges Sadoul wrote in his diary on that day after he had participated in the army's extraction from the capital: 'The Eiffel Tower passes by my left shoulder little by little... We are abandoning Paris, where, from the Eiffel Tower, tomorrow the swastika will fly.' Parisians stayed at home, since factories, shops and offices were shut. There was no information, but the sound of gunfire could be heard in the far distance, which made the city's inhabitants wonder whether the invader, who they had grown up learning was untrustworthy, would honour the open status of Paris or endeavour to subjugate the population by terror and destruction.\n\nVon Bock had ensured that his commanders were in no doubt about what was to be done and wrote:\n\nThe most vital parts of Paris are to be occupied by just one division at first. I called special attention to the rigorous sealing off and monitoring of through roads, so that no man lags behind in Paris and the march through the city results in no delaying of the general movement... Now even those 'up above' are thinking of turning south-west.\n\nAs it happened, the troops of 87th Infantry Division merely walked into the city and occupied it without fuss or ceremony on 14 June. Lieutenant Colonel Dr Hans Speidel took the surrender of the French capital. Ralf Buckholtz, a nineteen-year-old Sch\u00fctze whose father had died from wounds received at Verdun before he was born, recalled:\n\nI was barely trained and fresh into uniform and yet here I was, craning my neck at the famous buildings I had only dreamed of seeing. I had never left Cologne before September 1939 and here I was, in Paris, making history... I felt emotional knowing how proud my mother would be had she known where I was that day.\n\nVon Bock himself arrived in Paris to review troops marching past the Arc de Triomphe, and it must have been a proud moment indeed for the fifty-nine-year-old general. 'Afterward,' he wrote, 'I then drove to see Napoleon's tomb in the Invalides Church, then had a very good breakfast at the Ritz.' He found the city quiet and numbed by the experience, noting: 'Paris is rather empty; apparently only the poorer elements of the population stayed there. They stand around curiously by the roads leading into the city and voluntarily provide information when asked. The reception from and behaviour of the police is also courteous.' Meanwhile, Halder recognized that the taking of Paris was a turning point in the campaign, calling it 'A great day in the history of the German army'. With the arrival of the enemy, the city of lights drifted into darkness.\n\nThe fall of Paris was a significant step towards German victory. Although the city had ceased to be of any great military significance, its occupation \u2013 as the Germans had intended \u2013 was a deep wound to French pride, morale and fighting spirit. It was now left for the army to apply the _coup de gr\u00e2ce_ by occupying great swathes of territory, encircling the remaining troops heading to the Loire and out of the Maginot Line, forestalling evacuations and forcing the capitulation of as many French formations as possible. One last effort was required by the Wehrmacht to remove the final vestiges of the enemy's military capability and in so doing leave the French without any room for negotiation when Reynaud finally requested armistice talks.\n\nThe final flush of military activity across France therefore saw the mobile German formations drive hard and deep into the interior. By 14 June, von Kleist's panzers were across the Seine south-east of Paris and plunging south with their right heading towards the Loire south of Nevers and their left to Dijon. Five days later, they were at Vichy west of the Loire and Lyon on the Rh\u00f4ne. Hoth's Corps, meanwhile, swept westwards with 7th Panzer Division advancing towards Cherbourg on the Normandy coast and 5th Panzer Division moving towards ports on the Brittany coast, while Manstein's corps pushed down to the Loire to complete the encirclement. Resistance occurred in a few pockets but was quickly overcome and had little or no bearing on the forward momentum of the German divisions. Rommel's 150-mile advance from the Seine to the Cotentin peninsula was achieved in a single day. As a driver of one of 7th Panzer Division's PzKpfw Mk IIIs commented about his advance: 'The only real constraints on our advance were the wear and tear on our vehicles. The mobile workshops were kept extremely busy, but the weather was good and we knew that the end of the campaign was in sight. I personally felt that we had just won the war.'\n\nHoth's aim was to stop more Allied troops leaving French shores for England. Alan Brooke had been unwillingly ordered back to England by CIGS during the Battle of Dunkirk, as he noted in his diary, 'to form a new BEF!' and returned through Cherbourg on 12 June to command it from Brittany. This 'second BEF' consisted of those troops that had not been evacuated and included part of 1st Armoured Division, an ad hoc collection of BEF units known as Beauman Division, 150,000 line of communication troops together with the newly arrived 52nd (Lowland) and 1st Canadian Divisions. On Brooke's arrival, however, he was told that the French had rejected the idea of a Breton Redoubt and, recognizing the hopelessness of the Allied situation, he asked for Churchill's permission to evacuate the force. The prime minister responded by reminding the general that he been sent back to France to make the French feel that Britain was supporting them, but as Brooke later wrote, 'I replied that it was impossible to make a corpse feel.' In the end, the prime minister conceded that the needless sacrifice of Brooke's force ran contrary to all logic considering the threat that Germany would soon pose to Britain, and so evacuation commenced on 15 June \u2013 Operation Ariel \u2013 and took place from increasingly southerly ports as the Germans drew forward. Embarkations from Cherbourg were completed on the afternoon of 18 June, just as Rommel's men began to enter the city and 5th Panzer Division reached Brest. Thousands were also evacuated from the Brittany port of St. Malo and the Atlantic coast ports. At St. Nazaire, the sinking of the liner RMS _Lancastria_ in a Luftwaffe raid led to the loss of around 4,000 lives, nearly twice the number of BEF troops killed in the campaign to that date and the largest loss of life in British maritime history. Further evacuations took place at La Pallice, Bordeaux, Le Verdon, Bayonne and St. Jean de Luz and only ended on 25 June. When all Allied troops evacuated during this period are taken into account \u2013 British, Canadian, French, Polish, Czech and Belgian \u2013 the total number was nearly 192,000 men, over 300 guns, 2,300 vehicles and a paltry twenty-four tanks.\n\nTwo panzer divisions were never likely to have much of an impact on the broadly based Operation Ariel, but when combined with the failure to prevent the earlier evacuations during Operation Dynamo, their inability to disrupt Allied arrangements had longer-term strategic implications. These, though, were lost in the dust and euphoria of Germany's budding triumph. More emphasis was placed by OKW on the encirclement of General Gaston Pr\u00e9telat's second-rate Second Army Group formations still holding the Maginot Line. There had been a limited German attack against the French fortress at Le Fert\u00e9 in the Maginot Line extension twelve miles south-east of Sedan by Army Group A on 16 May. Its success not only opened a route round the main Maginot positions, but also protected von Rundstedt's flank as his divisions surged towards the Channel coast. Yet the Germans were mindful that Le Fert\u00e9 had only fallen after three days' hard fighting which revealed that Pr\u00e9telat's forces could well prove difficult to dislodge from their strong, well-protected forts. Yet as General Wilhelm von Leeb's Army Group C prepared itself for a major attack against the Maginot Line, its senior commanders drew confidence from the fact that French formations seemed to be leaving their positions. Indeed, of the fifty infantry divisions that had supported Pr\u00e9telat's static defences a month earlier, the requirements of the Weygand Line meant that just seventeen now remained. These same officers were further encouraged by the intelligence gleaned from a captured messenger that the French High Command, recognizing that Guderian's thrust had left those manning the Maginot defences increasingly isolated and vulnerable, had on 12 June ordered Pr\u00e9telat to withdraw them. The first troops that pulled back during the night of 13\u201314 June were the Second Army Group's immobile infantry and artillery formations which had been tasked with supporting the main defensive positions. The plan demanded that the fort garrisons would be the last Maginot Line troops to leave, and that their withdrawal would take place on 18 June.\n\nVon Leeb's offensive, its start date advanced in order to take advantage of the turmoil caused by Pr\u00e9telat's withdrawal and the increasing speed of Guderian's advance, was to concentrate its attack against the Maginot Line in two areas. In the Lorraine, Operation Tiger required First Army to attack the Sarre Gap \u2013 a thirty-one-mile sector with lesser defences due to its proximity to the Saarland, which was in French hands during the construction of the defences. This area had first been attacked by the Germans between 12 and 15 May to divert French attention away from the Ardennes and during this time they had gathered valuable information which a month later was to shape First Army's plan. The second attack would take place simultaneously in the Alsace sector; it would see Seventh Army launch Operation Kleiner B\u00e4r and required its divisions to cross the Upper Rhine before grappling with the French defences. Even so, von Leeb was confident that in the circumstances the offensive spirit of his troops would overcome an isolated and apprehensive enemy and looked forward to the creation of two pincers which would snap shut behind the French as Guderian's Panzer Group raced south to the Swiss border to seal the region from the French interior.\n\nGuderian's advance had, in fact, picked up considerable momentum since its initial breakthrough and his troops were soon dashing through the Champagne region. Little resistance was encountered during this period, as the commander of an armoured reconnaissance unit, Leutnant Wolf-Max Ostwald, noted: 'There is no sign of life anywhere; the Frenchies have decamped en masse once again. Fine, we'll just go on keeping them moving!' During this stage of the offensive, despite having access only to poorly copied 1:300,000 scale maps, Guderian's leading formations were advancing sixty miles a day and jostling each other for the limited road space. Ostwald wrote:\n\n... Sweat is running down the faces of the wireless operators. The dispatch riders, roaring along to find the commander, the fluttering report pads in their mouths making them look like hunting hounds, are hardly recognizable with their swollen eyes in faces furrowed by dust, heat and sweat beneath the dust-white helmets.\n\nIndeed, such was the speed at which the Germans were moving south across the front that Helmuth Mahlke, the Stuka pilot who was now flying in support of von Kleist's drive, said that he nearly bombed German troops at Ch\u00e2tillon-sur-Seine, where a pocket of resistance was being mopped up. He later wrote: 'I was already committed to the dive when I saw German troops feverishly laying out air recognition panels and German flags on the edge of the town.' Mahlke just managed to pull out in time and flew over Ch\u00e2tillon to see that German troops had already occupied the town and were waving up at the dive-bombers in relief.\n\nWith German formations rolling forward towards central France as energetically as their lines of communication would allow, von Leeb began his offensive against the Maginot Line. Although a supporting manoeuvre by Army Group A divisions around the end of the Maginot Line at Longuyon quickly began to outflank the French defences, the seven-division-strong Operation Tiger was rebuffed on 14 June. Yet although the strong resistance offered by the defenders on the first day stalled the Germans' attack, on the second von Leeb's divisions exploited the diminishing strength of the French defences between the fortifications caused by Pr\u00e9telat's withdrawal order and made vital penetrations which led to the encirclement of French strongpoints. Even so, despite the advance and concentrated artillery bombardments, the much anticipated collapse did not occur and the forts continued to fight back.\n\nEventually, the static French defences could not stop the German outflanking formations from making progress and on 16 June two Operation Tiger divisions, which had worked their way behind the Maginot Line heading west, managed to link up with formations from Army Group A coming in the other direction. In Operation Kleiner B\u00e4r, however, Seventh Army could not benefit from any such outflanking manoeuvre and was further disadvantaged by having to cross the Rhine. This, together with the dispersed nature of the attack and some resilient French defence, which included dogged resistance by the Marckolsheim casemate, meant that by nightfall on 15 June the Germans held only a shallow bridgehead. Even so, they persisted and, having constructed two pontoon bridges over the river and narrowed their attacking front to increase their concentration, managed to crack the sector open on the next day with the support of Stuka raids and the power of some 88mm guns. As Dominic Royer, a young engineer in the defending 42nd Fortress Infantry Regiment, later confided:\n\nThere is only so much pressure any one position can take... By 16 June we were finding it impossible to stop the infiltration of German units between our positions and we were isolated by incessant bombing raids and heavy bombardment. My blockhouse was about [1.5 miles] from the river and our communications had been cut with neighbouring units and battalion. Then we ran out of ammunition. We surrendered when a German grenade was pushed through an embrasure \u2013 but we only capitulated when there was no other option. That was important to us.\n\nAs the fighting in Alsace and Lorraine began to gain some momentum, on 17 June Guderian's 29th Motorized Division reached the Swiss frontier in an advance so swift that OKW failed initially to believe a signal informing it of this success. It was a critical event in the later phase of the campaign for in reaching the border the Germans had once again blocked their enemy's escape route and this time they had trapped half a million French troops. That evening, the division was ordered to turn north-east and advance forty miles to take the fortress town of \u00c9pinal.\n\n\u00c9pinal is situated on the Moselle forty-two miles west of Colmar and sixty miles south of Metz, and since German intelligence had identified French troops withdrawing to the town, Guderian sought to seize it before the area became a rallying point for a last stand. 6th Panzer Division's Pursuit Detachment von Esebeck was given a similar role to the one that it had enjoyed on the west bank of the Meuse and sped to \u00c9pinal in order to catch its defenders by surprise. The town capitulated quickly, but the fortress held out despite the pounding it took from the German guns. In the end, a team from 65th Panzer Battalion was sent to parley with the enemy, but it was only when the fortress's commander recognized the hopelessness of the strategic situation and was reassured by the German contingent that honour had been served that the garrison lay down their weapons on 22 June.\n\nThe bravery shown by the French troops at \u00c9pinal was mirrored in a host of forts along the Maginot Line \u2013 the last hurrah of the French Army protected by the concrete in which they had placed so much faith. Fort Fermont's 500-man garrison at the end of the line in Lorraine, for example, stood firm despite a series of attacks. First the Germans cut the fort's electricity supply as they encircled their objective, which forced its occupants to rely on a diesel generator for power while the position was smashed by heavy artillery, then Stukas and finally 88mm guns which fixed on a single point in the gun casemate before firing a shell at it every three minutes for several hours. When the generator caught fire and belched out toxic fumes which engulfed the interior and lingered because the bombardment had blocked the ventilation system, the fort's commander, Captain Aubert, ordered his men to fight on in their respirators. The conflagration was gradually brought under control, only for the French to find that the Germans had succeeded in piercing the gun casemate. Just one more round fired at this juncture might well have entered and exploded in the French magazine, but the 88mm guns instead stopped firing because their teams were unaware of their success. That night \u2013 17\/18 June \u2013 the French manoeuvred steel and iron sections into the breach and poured concrete over them to make a seal to deceive the enemy. The ruse worked, for the next German assault did not take place until 21 June and came in the form of a regiment of 183rd Division. It was resolutely rebuffed, however, and cost the formation so many casualties that its corps commander decided to cancel all further attacks on this section of the Maginot Line.\n\nMeanwhile, at the 'elbow' of the Line, in between the two main operations, a lesser German advance sought to penetrate the Fortified Sector of Haguenau and encircle several forts including that at Schoenenbourg. According to Colonel Pierre Stroh, the Commander of Engineers at the fort, on the day that von Leeb began his attack on the Maginot Line on 14 June, Schoenenbourg was visited by the sector commander, the impressively moustachioed Lieutenant Colonel Schwartz. A Great War veteran who had been wounded in action and relied on a stick to walk, Schwartz toured the fort, partly to give encouragement to the garrison by his presence and partly to check on its fighting spirit. He need not have worried for during a brief 'Council of War' held with the fort's officers it became clear that they were not interested in withdrawing at any time and in keeping with the role, traditions and honour of the fortress infantry, wanted to fight to the last man. When, therefore, the fort came under direct attack five days later by a massive bombardment from the artillery and then from the air \u2013 an attack which eventually cracked the walls of the fort just as the garrison heard word of P\u00e9tain's peace overtures to the Germans \u2013 the position continued to fight and was eventually hit by approximately 3,000 shells and 160 bombs. Reflecting on these events after the war, Stroh wrote:\n\n[W]e had a universally high level of esprit to prevail over our attackers... The six thousand men of the Fortified Sector of Haguenau tied up two German divisions with a strength five or six times greater than their own, supported by a squadron of dive-bombers, and by the largest artillery of the Wehrmacht. These means of attack gave Schoenenbourg the sad privilege of becoming the most heavily bombarded fort on the Maginot Line.\n\nThe tenacity of the French troops in the Maginot Line was revealed in the fact that most forts held out until the cessation of hostilities and, along with several others, Schoenenbourg continued to do so for a week beyond that. In the end, just ten out of the fifty-eight fortresses fell in German assaults. Indeed, it is entirely possible that had Pr\u00e9telat's troops not undertaken a withdrawal, Second Army Group could have beaten off von Leeb's offensive on the Maginot Line. It is very unlikely, however, that any such event would have dramatically altered the strategic landscape for on 19 June, the day that Metz fell, Halder wrote triumphantly:\n\nThe German flag is flying from the cathedral of Strasbourg. Toul is in our hands. Belfort was taken yesterday. On the Channel coast we have reached Cherbourg. Almost everywhere the front of our armies has reached the Loire and in some places even crossed it.\n\nThe successes of the German forces as they poured irresistibly south and west was set against the tragic backdrop of the Third Republic in its death throes. In a Bordeaux crammed with refugees, staff officers from the services mixed with a mass of civil servants as the government tried once more to co-ordinate its activities and make some difficult choices. Clustered around the centre of the town, ministers again found communications difficult and accommodation cramped while parliament was reduced to 100 members and had to meet on the benches of a school classroom. Here the nation's representatives wanted answers to two questions: Was France going to seek an armistice and who was responsible for the catastrophe? Although the subject of blame was natural in the circumstances, such debate would have to wait because a decision needed to be taken about the opening of negotiations with Germany and, as a consequence, how to handle the British. A cabinet meeting on 15 June was torrid. Half of its members believed that the war should be continued from abroad and the remainder that an armistice should be sought. Discussion centred on the prerequisites of any prolonging of the war. Prime among these was the ability of Britain to hold out against Germany, which, it was thought, depended on the United States joining the war effort. It was agreed, however, that the practical difficulties posed by the government and remnants of the armed forces decamping to North Africa would be immense and that their departure was hardly likely to garner the support of a population that might suffer the bombing of its towns and cities as the enemy rolled towards southern France. Even so, it was agreed that the positions of the United States and Britain had to be confirmed before a decision could be made. Indeed, Reynaud continued to hang on to a slim hope that a positive message from Washington and a hard line by the British would be enough to ensure that France would continue to fight.\n\nThe French prime minister was to be disappointed. The next day \u2013 16 June \u2013 the United States officially refused to intervene in the war and the cabinet also voted against a bizarre proposal that France and Britain should join together to become a single nation in the face of the German Reich. This deal, brokered by de Gaulle in London, would have seen France and Britain develop 'joint organs of defence, foreign, financial and economic policies' while 'Every citizen of France will enjoy immediately citizenship of Great Britain, every British subject will become a citizen of France'. Most cabinet ministers \u2013 who had met four times over the previous twenty-four hours and were now impatient \u2013 thought the idea smacked of desperation and could not suppress their belief that it would lead to Britain taking French colonies while reducing France herself to dominion status. Alexander Werth saw Weygand at this time and wrote: 'He looked extraordinarily unperturbed \u2013 almost pleased with himself... Paul Reynaud looked badly harassed.'\n\nReynaud's last hopes of continuing the war hung on the Declaration of Union and so, when the idea failed to gain support, he resigned as prime minister in the belief that the cabinet wished for an armistice. His assumption was incorrect \u2013 nine were for an armistice and fourteen against \u2013 for he had misinterpreted the desires of those who wanted to propose what were in reality impossible armistice terms to the Germans and favoured fighting on until this could be achieved. Nevertheless, Reynaud recommended that P\u00e9tain should be appointed as prime minister. The diminutive, wheezing, skeletal figure, once the victor of Verdun but now devoid of fight and fit only for capitulation, readily agreed to take the appointment. He immediately established a new cabinet which included Weygand, who replaced de Gaulle as minister for national defence. A day or two earlier, Alan Brooke had met with Weygand and was unimpressed when the Frenchman had confided that he felt that the situation was a blot on what had been an otherwise unblemished service record. Brooke later wrote: 'I remained dumb and unable to make any adequate remark; it seemed impossible that the man destined to minister to France in her death agonies should be thinking of his military career.'\n\nP\u00e9tain also approached the Spanish government to act as an intermediary between the French and Germans with a view to opening negotiations about the cessation of hostilities between the two countries. In the meantime, Churchill, recognizing that trying to hold the French to their unilateral armistice agreement was futile in the circumstances, released P\u00e9tain from it, although he did so on the somewhat optimistic condition that the French navy sailed to British harbours. Watching events develop from F\u00fchrerhauptquartier _Wolfsschlucht_ at Br\u00fbly-de-Pesche in southern Belgium, an exultant Hitler slapped his thigh and even laughed on being told that P\u00e9tain had sued for peace. That evening, the German forces heard the momentous news and it led to instinctive celebrations. Oberleutnant Dr Hans Berenbrock, fighting with Kempf's 6th Panzer Division, later wrote: 'We stood and sang, sang the songs of our homeland.'\n\nAs it was, the new French prime minister did not wait for a response from the Germans before announcing to the French people on 17 June that, despite the 'magnificent resistance' offered by the military, such was the suffering of the country that his aim was to bring the fighting to an end. It was a speech that sought to reassure the French people and its armed forces, but it also caused confusion. Although P\u00e9tain was making the case for an armistice request, the population was unsure what he meant: was he saying that France had lost the war and that the military would lay down their arms, or was he informing them that France had requested a ceasefire in order to enter negotiations with a view to striking a political deal with Germany?\n\nHowever it was interpreted, P\u00e9tain's broadcast stunned France. After decades of preparation and years of being told that 'France was strongest', the government seemed to be set on capitulation after just five weeks of fighting. 'It was unfathomable,' recalled Lieutenant Daniel Le Roy, who had escaped death by inches fighting south of Rethel when a shard of shell casing passed through his torso but missed his vital organs. Languishing in a German-occupied field hospital when he was told about P\u00e9tain's speech, Le Roy later said: 'I cried. I cried for me, I cried for my men and I cried for France. We had sacrificed so much and yet someone, somewhere had let us down.' Julianne Deros, meanwhile, was approaching the Loire with her family at the time of the broadcast. She later recalled:\n\nWe heard the news on a radio that had been brought out onto a small table in a village square... There were hundreds of people listening, all crammed into a tiny space, but there was complete silence for P\u00e9tain's words. It was not a long speech, but it was greeted with shock, not joy. For a few seconds after he had finished speaking, the silence continued \u2013 but then everybody erupted into animated conversation. Some sobbed while others wandered off to sit on the grass, lost in their own thoughts.\n\nFor a number of Deros's fellow refugees, P\u00e9tain's words made the continuation of their uncomfortable and often terrifying quest for security unnecessary. Some immediately turned back north and headed for home, others sat on sun-scorched verges to weigh up their options, and a few continued their journey, saying that the way had not yet ended and that 'Safety meant south'. Indeed, the bombing and strafing of refugees on the roads did not stop and raids continued, even on towns and cities such as Le Mans, Rennes, Nantes, Tours and Bordeaux, despite the fact that conurbations of more than 20,000 inhabitants had been declared open cities. The military, meanwhile, were largely relieved that an armistice was being sought but continued to fight, although their spirit was broken and few felt moved to die for France at a time when, in the words of one soldier, 'the nation was in discussions about its own funeral'.As a result, the Germans took huge numbers of prisoners from 17 June onwards while French officers tried to maintain control of their men and implored them to 'stand firm, do their duty, remain honourable and await events'.\n\nThere were, though, numerous valiant stands made by disparate troops as the Germans sought to cross the Loire. Some of these actions lasted hours, but one led by Colonel Daniel Michon, the commandant of the Cavalry School at Saumur, in the defence of a cluster of bridges around Saumur and Gennes was particularly noteworthy. In a briefing to his 780 cadets and their instructors, Michon sought to motivate them despite the pitiful situation and said, 'Gentlemen, for the school it is a mission of sacrifice. France is depending on you.' Not all were convinced but, reinforced by some mixed units of withdrawing troops, a few hundred Algerian Regiment trainees and a handful of tanks, Michon's men took up defensive positions ready to fight for their lives. They first came under fire at midnight on 18 June from the advance guard of General Kurt Feldt's 1st Cavalry Division, and after unsuccessful attempts to obtain a French surrender through parley, the Germans attempted a breakthrough. Their artillery had already ascertained that the bridges had been destroyed and so Feldt's men were forced into an assault river crossing in small boats, only to be repelled on 19 June and then again on the following day. In such circumstances, 1st Cavalry Division was forced to rely on other formations on their flanks making crossings of the Loire to encircle the defenders, but even then Michon's men continued to fight until they were overrun. This stoicism cost the French nearly 130 casualties and the Germans almost double that number.\n\nAcross the Channel, de Gaulle was frantic about the way in which France seemed to be haemorrhaging its spirit and will and slipping into oblivion. He had made his base in London since he feared being arrested as a strong advocate for France's continued participation in the war against Germany without the government's approval. Churchill had been impressed with the young officer, recently promoted to general, since their first meeting, and on 18 June provided him with a platform at the BBC from which to broadcast a message to the French people. Although few in France would hear his words, a passionate de Gaulle said:\n\nThe leaders who, for many years, have been at the head of the French armies have formed a government. This government, alleging the defeat of our armies, has made contact with the enemy in order to stop the fighting. It is true, we were, we are, overwhelmed by the mechanical, ground and air forces of the enemy... It was the tanks, the aeroplanes, the tactics of the Germans that surprised our leaders to the point of bringing them to where they are today. But has the last word been said? Must hope disappear? Is defeat final? No! Believe me, I speak to you with full knowledge of the facts, and tell you that nothing is lost for France. The same means that overcame us can bring us victory one day. For France is not alone! She is not alone! She is not alone! She has a vast Empire behind her. She can align with the British Empire that holds the sea and continues the fight. She can, like England, use without limit the immense industry of the United States... Whatever happens, the flame of the French resistance must not be extinguished and will not be extinguished.\n\nIt was an announcement which argued that, although P\u00e9tain was prime minister and his government retained sovereign power, their decisions did not represent the best course of action for France and there was indeed another way. De Gaulle's words were transcribed and re-broadcast several times in the days that followed, as well as being published in the press. The general's aim was undoubtedly to encourage and give some form to what he believed to be the latent French desire to resist the Germans, and his broadcast was also designed to bolster British hopes in the face of a German menace that was now prowling just twenty-two miles from Dover. But although his broadcast may well have given his host country an indirect boost, there was no doubting that, despite its desolate courage, de Gaulle's rallying cry hardly registered in a defeated France that seemed to accept its fate with a sense of angry relief.\n\nMeanwhile, P\u00e9tain waited for a response from Germany. As he did so, the enemy overran more departments, took more prisoners and pushed harder on the nation's windpipe. Although a contingent of largely Jewish deputies and ex-ministers left for North Africa, the prime minister refused to authorize parliament and the government to leave France. Along with P\u00e9tain, the nation's political leaders would face the Germans and share the consequences of any armistice with the French people. The F\u00fchrer, meanwhile, was enjoying France's excruciating predicament and orchestrated the armistice negotiations with a heavy nod to history. The timing of the talks allowed for the Wehrmacht to reach the end of their lines of communications, but not for France to recover its poise.\n\nThe site that Hitler chose to meet the French armistice negotiation delegation was the Forest of Compi\u00e8gne in Picardy, just forty miles north of Paris. It was here, in Napoleon III's railway carriage, that in November 1918 the Germans had signed the armistice agreement that led to the end of the First World War. In the 1920s, the place had become a memorial to the French fallen of the conflict which, in the great tradition of Franco-German enmity, embraced the concepts of victory and revenge. Now, in June 1940, the woods would once again host armistice talks and the old railway carriage would be retrieved from a museum and placed exactly where it had stood a generation before.\n\nThe terms that were to be offered by the Germans had been carefully drafted to ensure that, although leaving the French in no two minds about the nature of their defeat, they were not overtly humiliating. Hitler was already thinking about his next step in Europe and did not want to encourage any French desire to reject the package and continue the fight from overseas. As a result, the terms did not refer to the annexation of large areas of territory, colonies were not to be seized and the French fleet was merely to be disarmed. France was to be divided into two parts: the occupied north, which included a corridor down to the Spanish border and gave Germany access to all Channel and Atlantic ports, and the south, which comprised three fifths of the country and would become a _Zone Libre_ (Free Zone) that retained a government to administer all of metropolitan France except Alsace-Lorraine, which was to be placed under German administration but not formally annexed.\n\nIt was at this point that Italy launched an offensive after the opportunistic Italian prime minister Benito Mussolini had declared war on France and Britain on 10 June. Mussolini had told his Army Chief of Staff, Marshal Pietro Badoglio, that he needed several thousand dead Italian troops to strengthen his hand with the Germans in peace negotiations and also remarked to General Rodolfo Graziani, the Commander-in-Chief of the General Staff, that 'It would be a grave blow to our prestige to receive territory from the hands of our ally without having occupied it.' While the obsequious Graziani said that 'the morale of the troops was of the highest' and they 'yearned' to cross the French border, senior officers were relying more on a French collapse than the preparedness of their 450,000 troops for a successful attack in difficult conditions against strong pre-prepared defences through complicated terrain and along routes known to be protected by fortresses and other obstacles. But General Ren\u00e9 Olry, the commander of the 190,000-strong French Army of the Alps, was in no mood to give ground to either the Germans advancing to his rear along the Rh\u00f4ne Valley to Grenoble and Chamb\u00e9ry to assist Mussolini's attack or to the Italians facing him: he would fight to the last man.\n\nThe pitiful attack launched by the Italians on the Alpine front by their Army Group West had one thrust in the north near the Swiss border conducted by Fourth Army, but its main effort was carried out by First Army along the Mediterranean coast to capture Marseilles. At 0300 hours on 21 June, both Italian Armies crossed the French border but immediately found that their ability to manoeuvre was extremely limited and so their momentum was severely compromised in ground that one German officer called 'perhaps the most unsuitable of all conceivable theatres of operation.' In one sector of General Alfredo Guzzoni's Fourth Army attack, the Italian Alpine Corps advanced on a twenty-five-mile front with the aim of taking Albertville, which was itself twenty-five miles into French territory. They advanced a mere five miles before the offensive ground to a halt. In another sector, First Corps struggled on a similar frontage but, despite enjoying local success in outflanking some French positions, the attack was stopped just ten miles from the border. To the south, the main effort by First Army fared even worse and failed even to get as far as the French main line of resistance and only managed penetrations of one mile or two along the coast itself.\n\nIn the Alps, seven French divisions halted more than 30 Italian divisions while also temporarily stalling the Germans approaching from the rear along the Rh\u00f4ne Valley. The offensive enterprise cost Olry's Army around 500 men, over half of whom were prisoners, but cost the Italians around 900 casualties with another 616 missing in the snow, a further 2,151 suffering frostbite and some 1,100 taken prisoner. Despite Mussolini's bravado in public, his alpine fiasco left the Italian prime minister humiliated in the face of Germany's military prowess for he had contributed precious little to the defeat of France and taken a trifling amount of enemy territory. On the evening of 24 June \u2013 the day Halder noted that Mussolini wanted to send several battalions by air to the vicinity of Lyon so they could negotiate for rights to occupied French territory, a move he described as '... the cheapest kind of fraud' \u2013 an armistice was signed in Rome by the Italians and the French.\n\nIn the meantime, Germany had signed a separate agreement. On 21 June Hitler had greeted the French delegation at the Compi\u00e8gne armistice negotiations in silence and then sat in the same seat that had been occupied by Ferdinand Foch, the Allied generalissimo, in 1918 and listened \u2013 again in silence \u2013 to the preamble of the armistice terms. Upon its completion, Hitler and his entourage left the railway carriage to make the point that there was nothing to discuss, that French agreement was a foregone conclusion and that the German leader had already moved on to more important things. The head of the French delegation, General Huntziger, was left to discuss the terms with General Wilhelm Keitel, the chief of OKW. Von Brauchitsch watched proceedings carefully and returned from the historic event to brief Halder, who wrote that evening:\n\nThe French... had no warning that they would be handed the terms at the very site of the negotiations in 1918. They were apparently shaken by this arrangement and at first inclined to be sullen... [In the] negotiations, presided over by Keitel, there seems to have been a great deal of wrangling, and [von Brauchitsch] is worried that the French might not accept. I don't understand his apprehension. The French must accept and, with P\u00e9tain at the helm, will do so. Moreover, our terms are so moderate that sheer common sense ought to make them accept.\n\nThe discussions lasted for just one day before, on the evening of 22 June, Huntziger, having failed to soften the terms, was given authority by the government in Bordeaux to sign the armistice document on its behalf. The agreement was to come into effect on 25 June 1940 at 0135 hours \u2013 the same day and time as the French armistice with the Italians would be scheduled to begin. On the announcement that the armistice had been signed, von Bock issued the following order to his armies:\n\nThe war with France is over!\n\nIn forty-four days the Army Group has overrun the Dutch and Belgian land fortifications, forced the armies of both states to surrender after a tough struggle, and in a bloody battle inflicted a destructive defeat on the Anglo-French armies hastening to the scene. Paris and the coasts of France are in our hands.\n\nThe number of prisoners taken on the army group's front has exceeded 1\u00bd million men. The booty in tanks, guns and weapons of all kinds is huge.\n\nShoulder to shoulder with the Luftwaffe, we have won a victory whose beauty and scope is unsurpassed in history.\n\nThe young German Army has passed its severest test before the whole world, before history and before itself. The distress and disgrace that descended over our people after the World War has been blotted out by your loyalty and bravery!\n\nThe main burden of bloody sacrifice was borne by the German Army. We will bear it proudly and willingly in firm belief in ultimate and total victory! Long live the F\u00fchrer.\n\nWith the armistice signed, an historical disgrace extinguished and Germany reborn, Hitler ordered the Compi\u00e8gne site to be destroyed.\n\n* Winter remained a German prisoner until the end of the war.\n\n# Epilogue\n\nON 28 JUNE 1940, Hitler became a tourist in Paris. He had wished to visit the city ever since his youthful blossoming into an aesthete. He was an untalented artist, yet he had a great interest in architecture and later said that he would have liked to have become an architect. His early years had shaped his taste in art and design as much as his political and ideological preferences and they had barely developed since. The relationship between art and power was not lost on him for, as Robert Hughes argues, 'Architecture is the only art that moulds the world directly. Of all the arts, it is the supreme expression of politics and ideology. It marshals resources and organizes substance in a way that music, painting and literature cannot. It is an art that lives from power.' Hitler had grand designs and saw architecture as a way of embodying and reflecting them. He sought to create buildings that would stand for endless generations as a physical manifestation of the indomitable strength and influence of the German Reich just as ancient buildings provided a touchstone for the empires of antiquity. 'His vision of the world he wanted to make to commemorate himself and his ideology,' Hughes says of Hitler, 'was complete, overwhelming and Pharaonic.'\n\nThe F\u00fchrer's entourage on his tour of Paris included his two favoured architects, the middle-aged, experienced and accomplished Hermann Giesler and the twenty-five-year-old Albert Speer, who, although having built nothing of note, was already trusted to give some shape to Hitler's more megalomanic architectural fantasies. Having just completed a sentimental tour of his old First World War battlefields, Hitler entered Paris in a convoy of three black Mercedes sedans with his mind as much on the future as the past. The city shimmered in the early morning sun, beguiling but still grieving. The first stop was at L'Op\u00e9ra, the one place above all others that Hitler wished to visit since he had studied the building's plans. A deferential white-haired attendant accompanied the group and confirmed that Hitler's assertions about the building's renovations were correct. At the end of their visit, the Frenchman refused a fifty-mark tip. Minutes later, the party of field grey-clad sightseers drove past the neo-classical La Madeleine, a church whose modern incarnation was as a memorial to the successes of Napoleon's army and which had been built on ground initially seized from Parisian Jews in the twelfth century. The building pleased Hitler, who nodded his approval.\n\nThe next stop was at the Eiffel Tower, a photo opportunity for the F\u00fchrer that was simply too good to miss, before a brief pause at the Tomb of the Unknown Soldier beneath the Arc de Triomphe and a longer one at Les Invalides. After cogitating in silence before the tomb of Napoleon, Hitler ordered the remains of Napoleon's son to be moved from Vienna to be placed by the side of his father. Shortly after, he also directed that two First World War monuments in Paris be destroyed: that erected in memory of General Charles Mangin, the leader of France's sizeable contingent of colonial troops, and included the images of four black soldiers, and the monument to the British nurse Edith Cavell, who had been executed by the Germans in 1915.\n\nThe group pressed on, visiting the Panth\u00e9on, which boasted a neoclassical fa\u00e7ade that impressed Hitler but an interior that was 'a terrible disappointment'. The last stop was at the Sacr\u00e9 Coeur on Montmartre, the highest point in Paris. A working church with Romano-Byzantine features and nationalist themes, it had been built to remind the nation of the French defeat by Prussia in 1871 in the wake of a speech by Bishop F\u00e9lix Fournier which had stated that its cause had been the moral decline of France. The construction of Sacr\u00e9 Coeur's basilica had been completed in 1914 as Germany gripped France and consecrated in 1919 as France gripped Germany. The building was not to Speer's taste, but Hitler took considerable interest in its motifs and stood for a while pointing at various parts of the edifice, recognized but ignored by local churchgoers. Before heading back to the airport, Hitler took one last look at Paris from his magnificent viewpoint and nodded in silent appreciation.\n\nThe tour had lasted barely three hours but had thrilled Hitler. That evening, he summoned Speer, one of those men of education and training who were necessary to fulfil his ambitions. His interview with the architect was short: 'Wasn't Paris beautiful? But Berlin must be made far more beautiful. In the past I often considered whether we would not have to destroy Paris, but when we are finished in Berlin, Paris will be only a shadow. So why should we destroy it?' Hitler had seriously considered razing to the ground what he saw as a prodigious symbol of German humiliation, but now it had been conquered, he no longer felt the need. By the end of 28 June, France was already a fading memory, the idea of a formal ceremony in Paris dismissed as irrelevant. 'I am not in the mood for a victory parade,' Hitler declared. 'We aren't at the end yet.'\n\n# Conclusion\n\nThe Germans make everything difficult, both for themselves and for everyone else.\n\nJohann Wolfgang von Goethe\n\nWILSON: The Germans have reached Holland.\n\nMAINWARING: Good lord! How on earth did they manage to do that? I could've sworn that they'd never break through the Maginot Line.\n\nWILSON: Quite right, sir \u2013 they didn't.\n\nMAINWARING: Ah-ha! I thought not. I'm a pretty good judge of these matters, you know, Wilson.\n\nWILSON: They went round the side.\n\nMAINWARING: I see. ( _He does a double take_ ) They _what_?\n\nWILSON: They went round the side.\n\nMAINWARING: That's a shabby Nazi trick! You see the sort of people we're up against, Wilson?\n\nWILSON: Most unreliable, sir.\n\n_Dad's Army: The Movie,_ screenplay by Jimmy Perry & David Croft\n\nThe Fall of France and the Low Countries in 1940 is a staple on military history courses in universities and military academies around the world. Its size, scope, audacity and clinical outcome beg obvious questions, not least how it was possible for a powerful nation such as France to be defeated so quickly. It is an event that writers, commentators and historians have grappled with ever since, producing a wide range of conclusions. Yet among all the excellent and insightful work that has added a depth and richness to our understanding of the campaign, there are the inevitable, misleadingly simplistic offerings which have gained popularity precisely because they explain events in an easily digestible fashion. At their heart is the argument \u2013 one promulgated by Hitler and his cronies during the summer of 1940 in an effort to alarm enemies and promote the perception of German invincibility \u2013 that the victory over France and the Low Countries had come about through the application of a radically new and all-conquering fighting method based on an irresistible panzer army closely supported by the Luftwaffe. The truth, of course, is more complex \u2013 and considerably more interesting. As with many myths and oversimplifications, there is a kernel of truth in the idea that German armour and air power in the guise of something commonly known as _Blitzkrieg_ steamrollered an inept enemy, but it needs to be put in its correct context with important corrections made and nuances identified.\n\nGerman operational methods and the war-fighting machine that applied them were the product of careful and often secret work done to keep the armed forces healthy, despite their diminished size and capability, by a general staff that was not supposed to exist in the wake of the Versailles settlement. Through the years of trauma in the deep, dark shadow of the 1914\u201318 war and the humiliation of a defeat wrapped in a toxic peace treaty, the German state, its people and generals looked to a brighter future. The wish for revenge was an understandable, if unsavoury, reaction to events and circumstances. Its voracity also meant that for two decades before Hitler's offensive in the West, although the German people may not have _wanted_ another war, they were sympathetic to the development of a military that might help lead the nation to what Germany was 'meant to be'.\n\nBy the early 1930s, the German armed forces were a small, well-trained, thinking elite. Although severely limited in its capabilities by a lack of physical and mechanical muscle, that elite's moral and conceptual requirements had been carefully nurtured by General Hans von Seeckt and his successors and made it ripe for expansion and employment by Hitler. Under the Nazis, the army developed strength while continuing to evolve its doctrine and looked to harness the potential of new technology and give old German army battlefield ambitions \u2013 encirclement and the battle of annihilation \u2013 the mobility, speed and co-ordination that could provide significant operational impact and, possibly, a decisive strategic result. Thus, along with the bulking out of the army with conscripts who were largely sent to the foot infantry divisions came the advance of a refined cutting edge in the shape of panzer divisions. In this way, Germany developed armed forces that could, given the correct circumstances, resources and planning, avoid an attritional war which Germany had neither the resources to win nor the inclination to fight.\n\nBy 1939, the French had been preparing for and were content to fight a total, defensive, attritional war. They could see no other way to defeat a German offensive; this was, after all, how they had emerged victorious from that terrible conflict twenty years earlier. While there was no public appetite for war, having lanced the German boil once but then failed to successfully treat the subsequent infection, the population would not hesitate in protecting French soil again. Its ability to do so was, nevertheless, shaped by a political, social and economic turbulence which was itself rooted in those same bloody and costly five years of fighting and had consequences that fragmented France. Although Germany remained a threat, albeit a muzzled one, to the country's eastern border, France had little desire to see a large, powerful and expensive military soak up valuable resources that were so desperately needed elsewhere. In any case, there was neither the strength of leadership nor the political stability to indulge in the sort of long-term thinking that was required for a bespoke, flexible military machine that perfectly fitted the country's strategic requirements. As a result, by the early 1930s France leant heavily on the construction of the Maginot Line, its preparations for total mobilization and a relatively small professional military cadre which was limited to defensive action to protect the nation's security. There were consequences, however, for this meant that France was rendered impotent in the face of the challenge posed by Germany from the mid-1930s onwards, and the subsequent collapse was the direct consequence of its military failure.\n\nAfter the Second World War, a French parliamentary committee which was investigating the French defeat in 1940 concluded that the army had 'retired to its Mount Sinai [after World War I] and sat among its revealed truths and remnants of past glories...'. Yet such words unhelpfully implicate the French army as an organization which actively sought to be manacled to a limited capability and revelled in the recent past, whereas the truth of the matter was that circumstances dictated that the politicians ushered the army into a trench where it all too readily settled and atrophied. With the Maginot Line in place and an alliance with Belgium, the French leadership felt confident that they had sufficiently limited Germany's attacking option to something that approximated a mechanized rerun of the 1914 Schlieffen Plan which could be stopped and drawn into positional warfare.\n\nHad the Germans followed through with a broadly similar plan and in so doing given the initiative to the French and let them fight the type of campaign they wanted to fight and on ground dictated by Gamelin, they may very well have found success \u2013 and victory \u2013 exponentially more difficult to achieve than they did. Indeed, it was just such potentially terminal complications that senior German generals recognized but felt that Hitler might have overlooked in his desire to attack the West. Von Manstein's plan came, therefore, at a juncture when few were inspired by OKH's schemes for the invasion of France, although at the same time its radical character caused some of those generals to feel that it could lead to disaster. Halder, however, was quickly seduced by the possibility of placing concentrated German strength against known French weakness. OKH became convinced that it would make best use of the light mobile forces that Germany possessed and that there was far more value in avoiding the enemy than in deliberately initiating a clash with his best formations. Halder's ambition was to use surprise to start a sequence of events that would be psychologically damaging to the enemy's ability and will to resist, with breakthroughs aggressively and ruthlessly exploited by mobile troops to create operational success that could be converted into strategic victory. It was the antithesis of the French outlook and, as Allan Doughty has argued, 'German strategy was riskier than French strategy. Yet, it sought swift victory, while French strategy initially sought only to avoid defeat.'\n\nThe plan that became _Fall Gelb_ would not have been possible without the careful nurturing of the German armed forces before 1940 and that included the vital role of the general staff, who kept fighting traditions alive and allowed for them to be updated to make best use of new technology. The fact that this process had not been completed and the Wehrmacht and Luftwaffe were still flexible, allowed tactical doctrine, when added to that new technology, strong leadership, spirit and aggression to become the basis for a new operational fighting method. In this respect, and as the thinking which led to the formation of the mighty panzer groups bore fruit, the German military machine was not so much intentionally unleashing a doctrine of _Blitzkrieg_ upon France and the Low Countries in 1940 as 'doing little more than following its own nose'. It was a situation which both before and during the campaign understandably worried some conservative generals who were worried that German armed forces were not yet the finished article despite their success in Poland. This created a tension within the German High Command that was never fully resolved, although the proponents of the more radical aspects of the plan had faith that its speed and momentum would render detractors as helpless as they expected the enemy to be as the panzers raced across northern France.\n\nThis German improvisation was remarkable considering the scale of the military challenges posed not only by their enemies in the West, but also by the growing frictions within the High Command and the nation's unfinished preparations for war. Yet it was these very impediments that led directly to the inception of one of the most famous plans in military history and powered what was a barely formed fighting method because they forced the Germans to think and act radically if the episode was not to end in disaster. With Halder providing the foundation for success with his offensive scheme and officers such as Guderian, Rommel and Balck seeking to exploit the opportunity afforded to them with high-risk, high-gain actions on the ground, Army Group A in particular provided just the momentum and initiative that was required to pull the otherwise conservative German army and air force forward while befuddling its adversaries. The achievement of these same officers was nonetheless rooted in all the advantages provided by a thinking military that had a strong tradition and benefited from relevant training, a sound doctrine, robust leadership and a formidable fighting spirit. It was a virtuous circle. Eventual German success, of course, was not due to either the quality or quantity of their military hardware \u2013 and certainly not their armour \u2013 but where and how their resources were used.\n\nAlthough it was understandable that potential enemy threats to the plan were aired and one has to sympathize with those who were startled by the audacity of what _Fall Gelb_ sought to achieve and how it sought to achieve it, the plan was designed to uncover and then ruthlessly exploit French weaknesses. In the words of historian Andr\u00e9 Beaufre, 'The French Army was nothing more than a vast, inefficient tool, incapable of quick reaction or adaptation, quite incapable of taking the offensive or of any mobility.' French strategic decision-making had so limited the armed forces' abilities to coping with a rerun of the Schlieffen Plan but not with anything else. As things stood, there was little wrong with French doctrine, which was well suited to the nation's strategy. Unfortunately, however, it was not so well suited to what the Germans had in mind. For the war that the French expected, where they expected it and with the resources they had available, the Methodical Battle was a good fit. In the years following the First World War, the now small, poorly equipped and under-resourced French army did not need to look further than Verdun for inspiration about how it might fight a great defensive battle. Its conclusions were necessarily reflected in training, procurement, promotion and other preparations and they led to the creation of an army physically unprepared to cope with _Fall Gelb_ and also ensured that its troops, leaders and commanders were psychologically and intellectually ill-equipped for anything other than the slow tempo of positional warfare. In short, the High Command rejected the need for flexibility and diligently prepared the French armed forces for a type of warfare in which they were undoubtedly competent. But the Germans had rather different ideas, and in this light charges of French incompetence need to be qualified. As Faris Kirkland has submitted, 'French military leaders were neither traitors, pacifists nor fools.'\n\nHalder's plan recognized French frailties and used manoeuvre as the means by which to expose them. Indeed, as von Manstein's biographer Mungo Melvin has written: 'The plan worked _because_ it was so daringly unconventional, exploiting inherent systematic weaknesses in French force structures, command and control, and above all, in will and morale.' But while it is easy to see the role played by the panzer corps in the success, one should not forget the vital role played by Army Group B pinning and fixing the Allied Dyle force, the airborne forces' novel operations, the foot infantry divisions' disconcerting mass, the non-tactical endeavours of the Luftwaffe, and all the supporting troops and personnel that allowed operations to unfold as they did.\n\nFar from being a mere panzer and Stuka victory, _Fall Gelb_ was a sophisticated inter-service, all-arms team effort in which the infantry played a central role. With excellent communications enabling first-class co-ordination and the application of _Auftragstaktik,_ the Germans could react quickly and effectively to most of the situations with which the spearhead divisions were confronted and could exploit fleeting opportunities quickly and without fuss. The great combined arms success of the Meuse crossing, the breaking of the French defences and the establishment of a bridgehead was also a triumph for German leadership and mission command with the likes of men such as Rubarth and Balck making vital contributions which had a significant operational impact. The Germans' methods, along with their ability to apply them consistently and professionally, constantly gave them the edge, enabled them to get inside the Allied decision-making loop, maintain tempo and retain the initiative. In such circumstances, the panzer divisions \u2013 sometimes enhanced by tactical air support \u2013 became irresistible and remained so as their excellent logistical support meant that there was no need for them to pause in order to be resupplied. Indeed, an after-action report for Group von Kleist stated: 'Between 10 May and the capture of Calais, there was not one single supply crisis that could not be resolved with the resources of the Group von Kleist, without in any way interfering with command functions.' Although the ability to sustain independent operations of considerable distance was open to abuse \u2013 as exemplified by the actions of the ambitious and headstrong Rommel and Guderian \u2013 these same operations were also central to the campaign developing in a certain way and, although they exacerbated the worries of the more risk-averse naysayers who sought to slow the tempo down, they also revealed what could be achieved by the well-timed faits accomplis achieved by the panzer commanders. To the leaders in the spearhead formations, chaos was something to be created and embraced because it was so potent with possibilities, while to the French chaos was to be avoided at all costs because it was incompatible with their mindset and fighting methods.\n\nThe success that was eventually created by the panzer corps led to the largest encirclement in military history, one of 1.7 million men, and the capture of the best of the Allied forces and much of their equipment. But the fact that so many escaped south between the advancing panzer divisions and the following foot infantry and that this was then followed by the evacuation of 370,000 Allied troops from the Channel coast reveals that the operation was not unblemished nor unfettered by flaws and errors. Some of these had their basis in factors common to all complex human endeavours and included personality clashes, resource challenges and over-ambition. But others were specific to the German plan and included serious disagreements about independent panzer operations which led to echeloning issues, flank psychosis, Halt Orders and further encouraged OKW's smothering of OKH as some senior officers became extremely worried about those same vulnerable flanks, the gaps between panzers and foot infantry and undefended lines of communication. The inability of the Allies to exploit these frailties was indicative of their own wider capability issues and their failure to regain the initiative. As a result, the most senior French generals could not impose their will on events, partly because they remained fixated on the attack by Army Group B with its mesmeric air support and airborne operations as von Rundstedt got into position and then began to rip into Georges's formations. Despite the opportunity, there was no timely intervention by a significant Allied counter-attacking force either before or after the Meuse crossing, or against the Somme corridor or after the Germans had broken the Weygand Line.\n\nGamelin's fundamental role in the disaster can hardly be ignored. Although he was not personally responsible for French strategy and the limitations put on the military, he signally failed to challenge them and allowed politics to affect his military judgement. Balanced against this, however, it should be remembered that the head of the French army was a political appointment and that Gamelin had achieved his position because he would be a light hand at the tiller. The 'Grey General' was rather the product of a system that encouraged mediocrity and rejected the mavericks and freethinkers who might have shaken France out of her self-enforced malaise. Like the Maginot Line and Methodical Battle, Gamelin and the wider High Command were the symptom of the French approach to defence and not the cause of it. As it was, the Supreme Commander's lack of grip, his presiding over a sluggish army, his faith in the Maginot Line, his decision not to sack Georges, his ill-advised Breda Variant to Plan D, the lack of an alternative to defensive operations along the Franco-Belgian border or central Belgium, and the failure to see the possibility of a threat emerging from the Ardennes, were all, to a large extent, the consequence of poor strategic choices.\n\nIt should also perhaps be remembered that while Gamelin is judged, quite understandably and correctly, by what the Germans did, it is difficult to think of anybody who could have avoided what befell France in the circumstances. To call Gamelin incompetent is fair only in the context of wider criticisms of French political and military leadership, and it is possible that had the Germans attacked with a rerun of the Schlieffen Plan, as they might have done, history and events may well have been kinder to a 'new Foch'. After all, the sort of French defensive capabilities which Gamelin had developed and overseen were on show in the Weygand Line. Although it was Weygand who was responsible for the more appropriate and specific measures adopted in early June, their basis was fully Gamelin's and was far more in line with what he expected on the Dyle. As Martin Alexander has argued: 'Extending analysis of the fighting in 1940 beyond the usual preoccupations \u2013 the combats along the Meuse and around the Dunkirk perimeter \u2013 provides a more complex, richly textured canvas of France's military performance.' It challenges the view that _Fall Rot_ was a mere 'epilogue' and that for the French the campaign was without 'even the smallest redeeming feature'. Indeed, on the few occasions that the French were given the opportunity to fight as they had planned to, they often did so with great courage and tenacity. For the Allies, however, the defence of the Weygand Line was too brief, was not decisive and is merely instructive: it was too little, too late.\n\nThe ability of the Germans to achieve in some six weeks what they had failed to achieve in the four years and a few months of the First World War was reflected in their casualties, just 49,000 killed and missing and 111,000 wounded. The character of the German campaign contrasted with an attritional grind which meant that even those formations charged with making the breakthrough took relatively light losses: Grossdeutschland Regiment, for example, suffered 221 men killed out of 3,900, 1st Panzer Division accumulated only 267 deaths, and 7th Panzer Division lost 682 men killed. The Luftwaffe, however, lost over a quarter of its operational strength, with around 1,284 aircraft destroyed and some 6,653 personnel killed or wounded. French casualties were around 85,000 dead, 120,000 wounded, 12,000 missing and another 1.54 million becoming prisoners of war. The French air force lost 1,274 aircraft destroyed. The British lost around 10,000 dead \u2013 including those killed in the _Lancastria_ \u2013 with another 58,000 becoming casualties. The RAF lost 931 aircraft, of which 477 were valuable fighters. Belgian casualties were around 23,000 and Dutch 9,779.\n\nWhat had been achieved for such small cost was immediately apparent to Hitler, who by mid-June had taken the victory as his own and refused to share it with von Brauchitsch, Halder and OKH. During the summer of 1940, the German Supreme Commander explored what the operational methods that had been used by his armed forces to such acclaim in France and the Low Countries might achieve elsewhere and as he did so, those methods began to gain the standing of doctrine. It was a mistake, for the transplanting of _Blitzkrieg_ elsewhere took little account of the achievability of objectives, proper resourcing, and the uniqueness of different enemies, terrains, weather and a host of other factors that affect the application of force. As Hew Strachan has written: 'Germany lost the Second World War in part because... it made operational thought do duty for strategy, while tactical and operational success were never given the shape which strategy could have bestowed.' Nevertheless, the German victory over France and her allies had a huge influence not just on the form and future conduct of the developing Second World War, but on warfare more generally. The way in which the Germans took some of the timeless principles of war-fighting and remodelled them so successfully in a modern context to achieve strategic results so efficiently, effectively and swiftly with such minimal losses has consistently caught the attention of politicians and the military since 1940. The influence of the German _Blitzkrieg_ on doctrine and military thinking can still be seen around the world to this day: it changed warfare for ever.\n\n# Order of Battle (ORBAT) \u2013 Ground Forces (10 May 1940)\n\n**Germany**\n\n**Chief of OKW** \u2013 Generaloberst Wilhelm Keitel\n\n**Commander in Chief OKH \u2013** Generaloberst Walter von Brauchitsch\n\n**Chief of the General Staff** \u2013 General der Artillerie Franz Halder\n\n**German Army Group A \u2013** Generaloberst Gerd von Rundstedt\n\n**Chief of Staff** \u2013 Generalleutnant Georg von Sodenstern\n\n**Fourth Army** \u2013 Generaloberst G\u00fcnther von Kluge\n\n**II Corps** \u2013 General der Infanterie Infantry Adolf Strau\u00df\n\n12th Infantry Division\n\n32nd Infantry Division\n\n**V Corps** \u2013 General der Infanterie Richard Ruoff\n\n28th Infantry Division\n\n251st Infantry Division\n\n**VIII Corps** \u2013 General der Artillerie Walter Heitz\n\n8th Infantry Division\n\n**XV (Motorized) Corps** \u2013 General der Infanterie Hermann Hoth\n\n5th Panzer Division\n\n7th Panzer Division\n\n62nd Infantry Division\n\n**Reserves**\n\n87th Infantry Division\n\n211st Infantry Division\n\n263rd Infantry Division\n\n**Twelfth Army** \u2013 Generaloberst Wilhelm List\n\n**III Corps** \u2013 General der Artillerie Curt Haase\n\n3rd Infantry Division\n\n23rd Infantry Division\n\n**VI Corps** \u2013 General der Pioniere Otto-Wilhelm F\u00f6rster\n\n16th Infantry Division\n\n24th Infantry Division\n\n**XVIII Corps** \u2013 General der Infanterie Eugen Beyer\n\n5th Infantry Division\n\n21st Infantry Division\n\n25th Infantry Division\n\n1st Mountain Division\n\n**Reserves**\n\n9th Infantry Division\n\n27th Infantry Division\n\n**Sixteenth Army** \u2013 General der\n\nInfanterie Ernst Busch\n\n**VII Corps** \u2013 General der Infanterie\n\nEugen von Schobert\n\n36th Infantry Division\n\n68th Infantry Division\n\n**XIII Corps** \u2013 Generalleutnant Heinrich\n\nvon Vietinghoff\n\n17th Infantry Division\n\n34th Infantry Division\n\n**XXIII Corps** \u2013 Generalleutnant Albrecht\n\nSchubert\n\n58th Infantry Division\n\n76th Infantry Division\n\n**Reserves**\n\n6th Infantry Division\n\n15th Infantry Division\n\n26th Infantry Division\n\n33rd Infantry Division\n\n52nd Infantry Division\n\n71st Infantry Division\n\n73rd Infantry Division\n\n197th Infantry Division\n\n**Panzer Group Kleist** \u2013 General der\n\nKavallerie Paul Ludwig Ewald von\n\nKleist\n\nUnder direct command\n\nInfantry Regiment (Motorized)\n\nGrossdeutschland\n\n**XIV Corps** \u2013 General der Infanterie\n\nGustav von Wietersheim\n\n13th (Motorized) Infantry Division\n\n29th (Motorized) Infantry Division\n\n2nd (Motorized) Infantry Division\n\n**XIX Corps** \u2013 General der Panzertruppen\n\nHeinz Guderian\n\n1st Panzer Division\n\n2nd Panzer Division\n\n10th Panzer Division\n\n**XXXXI Corps** \u2013 Generalleutnant Georg-Hans Reinhardt\n\n6th Panzer Division\n\n8th Panzer Division\n\n**Army Group A Reserves**\n\n4th Infantry Division\n\n267th Infantry Division\n\n**German Army Group B \u2013** Generaloberst Fedor von Bock\n\n**Chief of Staff** \u2013 Generalleutnant Hans von Salmuth\n\n**Sixth Army** \u2013 Generaloberst Walter von Reichenau\n\n**IV Corps** \u2013 General der Infanterie Viktor von Schwedler\n\n18th Infantry Division\n\n35th Infantry Division\n\n**IX Corps** \u2013 General der Infanterie Hermann Geyer\n\n19th Infantry Division\n\n30th Infantry Division\n\n56th Infantry Division\n\n**XI Corps** \u2013 Generalleutnant Joachim von Kortzfleisch\n\n7th Infantry Division\n\n14th Infantry Division\n\n31st Infantry Division\n\n**XXVII Corps** \u2013 General der Infanterie Alfred W\u00e4gner\n\n4th Panzer Division\n\n253rd Infantry Division\n\n269th Infantry Division\n\n**Reserve**\n\n3rd Panzer Division\n\n20th (Motorized) Infantry Division\n\n61st Infantry Division\n\n216th Infantry Division\n\n**I Corps** \u2013 Generalleutnant Kuno-Hans von Both\n\n1st Infantry Division\n\n11th Infantry Division\n\n223rd Infantry Division\n\n**Eighteenth Army** \u2013 General der Artillerie Georg von K\u00fcchler\n\nUnder Direct Army Command\n\n9th Panzer Division\n\n1st Cavalry Division\n\n**X Corps** \u2013 Generalleutnant Christian Hansen\n\n207th Infantry Division\n\n227th Infantry Division\n\nSS 'Verf\u00fcgungstruppe' (Motorized) Division\n\n**XXVI Corps** \u2013 General der Artillerie\n\nAlbert Wodrig\n\n254th Infantry Division\n\n**Reserves**\n\n208th Infantry Division\n\n225th Infantry Division\n\n256th Infantry Division\n\n**Army Group B Reserve**\n\n**XVI Corps** \u2013 General der Kavallerie Erich Hoepner\n\n**XXXIX Corps** \u2013 Generalleutnant Rudolf Schmidt\n\n**German Army Group C \u2013** Generaloberst Wilhelm von Leeb\n\n**Chief of Staff** \u2013 Generalleutnant Hans Felber\n\n**First Army** \u2013 Generaloberst Erwin von Witzleben\n\n**XII Corps** \u2013 General der Infanterie Gotthard Heinrici\n\n75th Infantry Division\n\n93rd Infantry Division\n\n258th Infantry Division\n\n**XXIV Corps** \u2013 General der Kavallerie Leo Geyr von Schweppenburg\n\n252nd Infantry Division\n\n257th Infantry Division\n\n262nd Infantry Division\n\n268th Infantry Division\n\n**XXX Corps** \u2013 General der Artillerie Otto Hartmann\n\n79th Infantry Division\n\n95th Infantry Division\n\n**XXXVII Corps** \u2013 Generalleutnant Alfred Boehm-Tettelbach\n\n215th Infantry Division\n\n246th Infantry Division\n\n**Reserves**\n\n94th Infantry Division\n\n98th Infantry Division\n\n**Seventh Army** \u2013 General der Artillerie Friedrich Dollmann\n\n**XXV Corps** \u2013 General der Infanterie Karl von Prager\n\n555th Infantry Division\n\n557th Infantry Division\n\n**XXXIII Corps** \u2013 Generalleutnant Georg Brandt\n\n554th Infantry Division\n\n556th Infantry Division\n\n**Reserves**\n\n96th Infantry Division\n\n22nd Airlanding Division\n\n**France**\n\n**Supreme Commander** \u2013 G\u00e9n\u00e9ral Maurice Gamelin\n\n**Chief of Staff** \u2013 G\u00e9n\u00e9ral Joseph Doumenc\n\n**Supreme Headquarters Reserve**\n\n3rd Division d'infanterie motoris\u00e9e (DIM)\n\n10th Infantry Division\n\n14th Infantry Division\n\n23rd Infantry Division\n\n28th Infantry Division\n\n29th Infantry Division\n\n36th Infantry Division\n\n43rd Infantry Division\n\n5th Colonial Infantry Division\n\n7th Colonial Infantry Division\n\n1st North African Infantry Division\n\n2nd Polish Infantry Division Division\n\n**XXI Corps** \u2013 G\u00e9n\u00e9ral de Corps d'Arm\u00e9e Jean Flavigny [No organic divisions, supporting corps units only]\n\n**XXIII Corps** \u2013 G\u00e9n\u00e9ral de Division Maxime Germain [No organic divisions, supporting corps units only]\n\n**1st Armoured Group** \u2013 G\u00e9n\u00e9ral de Division Louis Keller\n\n2nd Division Cuirass\u00e9e de R\u00e9serve (DCR)\n\n3rd Division Cuirass\u00e9e de R\u00e9serve (DCR)\n\n**North East Front** \u2013 G\u00e9n\u00e9ral Alphonse Georges\n\n**Chief of Staff** \u2013 G\u00e9n\u00e9ral Gaston Roton\n\n**First Army Group** \u2013 G\u00e9n\u00e9ral Gaston Billotte\n\n**Chief of Staff** \u2013 G\u00e9n\u00e9ral Jacques Humbert\n\n**First Army** \u2013 G\u00e9n\u00e9ral Georges Blanchard\n\n**III Corps** \u2013 G\u00e9n\u00e9ral de Fornel de la Laurencie\n\n1st Division d'infanterie motoris\u00e9e (DIM)\n\n2nd North African Infantry Division\n\n**IV Corps** \u2013 G\u00e9n\u00e9ral de Corps d'Arm\u00e9e Boris\n\n15th Division d'infanterie motoris\u00e9e (DIM)\n\n1st Moroccan Division\n\n**V Corps** \u2013 G\u00e9n\u00e9ral de Corps d'Arm\u00e9e Bloch\n\n12th Division d'infanterie motoris\u00e9e (12 DIM)\n\n5th North African Division\n\n101st Fortress Infantry Division\n\n**Cavalry Corps** \u2013 G\u00e9n\u00e9ral de Corps d'Arm\u00e9e Ren\u00e9 Prioux\n\n2nd Divisions L\u00e9g\u00e8res M\u00e9canis\u00e9es (DLM)\n\n3rd Divisions L\u00e9g\u00e8res M\u00e9canis\u00e9es (DLM)\n\n**First Army Reserves**\n\n1st Division Cuirass\u00e9e de R\u00e9serve (1 DCR)\n\n32nd Infantry Division\n\n**Second Army** \u2013 G\u00e9n\u00e9ral Charles Huntziger\n\n**Chief of Staff** \u2013 G\u00e9n\u00e9ral Jean Langlois\n\n**X Corps** \u2013 G\u00e9n\u00e9ral de Corps d'Arm\u00e9e Pierre Grandsard\n\n3rd North African Division\n\n55th Infantry Division\n\n**XVIII Corps** \u2013 G\u00e9n\u00e9ral de Corps d'Arm\u00e9e Eugene Rochard\n\n1st Colonial Infantry Division\n\n3rd Colonial Infantry Division\n\n41st Infantry Division\n\n**Second Army Reserve**\n\n2nd Light Cavalry Division\n\n5th Light Cavalry Division\n\n71st Infantry Division\n\n**Seventh Army** \u2013 G\u00e9n\u00e9ral Henri Giraud\n\n**Chief of Staff** \u2013 G\u00e9n\u00e9ral Jean Baur\u00e8s\n\n**I Corps** \u2013 G\u00e9n\u00e9ral Th\u00e9odore Sciard\n\n25th Division d'infanterie motoris\u00e9e (DIM)\n\n**XVI Corps** \u2013 G\u00e9n\u00e9ral Alfred Fagalde\n\n9th Division d'infanterie motoris\u00e9e (DIM)\n\n**Seventh Army Reserve**\n\n4th Infantry Division\n\n21st Infantry Division\n\n60th Infantry Division\n\n1st Division l\u00e9g\u00e8re m\u00e9canis\u00e9e (DLM)\n\n**Ninth Army** \u2013 G\u00e9n\u00e9ral Andr\u00e9 Corap\n\n**Chief of Staff** \u2013 G\u00e9n\u00e9ral Olivier Thierry d'Argenlieu\n\n**II Corps** \u2013 G\u00e9n\u00e9ral Jean Bouffet\n\n5th Division d'infanterie motoris\u00e9e (DIM)\n\n**XIth Corps** \u2013 G\u00e9n\u00e9ral de Corps d'Arm\u00e9e Julien Martin\n\n18th Infantry Division\n\n22nd Infantry Division\n\n**XLI Corps** \u2013 G\u00e9n\u00e9ral Emmanuel Libaud\n\n61st Infantry Division\n\n102nd Fortress Infantry Division\n\n**Ninth Army Reserve**\n\n1st Light Cavalry Division\n\n4th Light Cavalry Division\n\n4th North African Division\n\n53rd Infantry Division\n\n3rd Spahis Brigade\n\n**Second Army Group** \u2013 G\u00e9n\u00e9ral Andr\u00e9 Pr\u00e9telat\n\nChief of Staff \u2013 G\u00e9n\u00e9ral Louis B\u00e9rard\n\n**Third Army** \u2013 G\u00e9n\u00e9ral Charles Cond\u00e9\n\n**Colonial Corps** \u2013 G\u00e9n\u00e9ral Henry\n\nFreydenberg\n\n2nd Infantry Division\n\n56th Infantry Division\n\n51st (British) Division\n\n**VI Corps** \u2013 G\u00e9n\u00e9ral Georges Loizeau\n\n26th Infantry Division\n\n42nd Infantry Division\n\n**XXIV Corps** \u2013 G\u00e9n\u00e9ral Loiseau\n\n36th Infantry Division\n\n51st Infantry Division\n\n**XLII Corps** \u2013 G\u00e9n\u00e9ral D\u00e9sir\u00e9 Sivot\n\n20th Infantry Division\n\n58th Infantry Division\n\n**Third Army Reserve**\n\n6th North African Infantry Division\n\n3rd Light Cavalry Division\n\n6th Infantry Division\n\n7th Infantry Division\n\n8th Infantry Division\n\n1st Spahis Brigade\n\n**Fourth Army** \u2013 G\u00e9n\u00e9ral Pierre Requin\n\nChief of Staff \u2013 G\u00e9n\u00e9ral Pierre Dame\n\n**IX Corps** \u2013 G\u00e9n\u00e9ral \u00c9mile Laure\n\n11th Infantry Divison\n\n47th Infantry Division\n\n**XX Corps** \u2013 G\u00e9n\u00e9ral de Corps d'Arm\u00e9e Louis Hubert\n\n52nd Infantry Division\n\n82nd African Infantry Division\n\n**Fourth Army Reserve**\n\n45th Infantry Divisions\n\n1st Polish Infantry Division\n\n**Fifth Army** \u2013 G\u00e9n\u00e9ral Victor Bourret\n\n**Chief of Staff** \u2013 G\u00e9n\u00e9ral Jean de Lattre de Tassigny\n\n**VIII Corps** \u2013 G\u00e9n\u00e9ral Aubert Fr\u00e8re\n\n24th Infantry Division\n\n31st Infantry Division\n\n**XII Corps** \u2013 G\u00e9n\u00e9ral Henri Dentz\n\n16th Infantry Division\n\n35th Infantry Division\n\n70th Infantry Division\n\n**XVII Corps** \u2013 G\u00e9n\u00e9ral de Corps d'Arm\u00e9e On\u00e9sime-Paul No\u00ebl\n\n62nd Infantry Division\n\n103rd Fortress Infantry Division\n\n**XLIII Corps** \u2013 G\u00e9n\u00e9ral Fernand Lescanne\n\n30th African Infantry Division\n\n**Fifth Army Reserve**\n\n44th Infantry Division\n\n**Second Army Group Reserve**\n\n4th Colonial Infantry Division\n\n87th African Infantry Division\n\n**Third Army Group** \u2013 G\u00e9n\u00e9ral Antoine Besson\n\n**Eighth Army** \u2013 G\u00e9n\u00e9ral Marcel Garchery\n\n**Chief of Staff** \u2013 G\u00e9n\u00e9ral Louis Koeltz\n\n**VII Corps** \u2013 G\u00e9n\u00e9ral Pierre Champon\n\n13th Infantry Division\n\n27th Infantry Division\n\n2nd Spahis Brigade\n\n**XIII Corps** \u2013 G\u00e9n\u00e9ral de Corps d'Arm\u00e9e George Misserey\n\n19th Infantry Division\n\n54th Infantry Division\n\n**XLIV Corps** \u2013 G\u00e9n\u00e9ral de Brigade Julien Tenc\u00e9\n\n67th Infantry Division\n\n**Third Army Group Reserve**\n\n**XLV Corps** \u2013 G\u00e9n\u00e9ral Daille\n\n57th Infantry Division\n\n63rd Infantry Division\n\n**North East Front Reserve**\n\n68th Infantry Division\n\n**British Expeditionary Force**\n\n**Commander-in-Chief** \u2013 General John Vereker, 6th Viscount Gort\n\n**Chief of the General Staff \u2013** Lieutenant General Henry Pownall\n\n**GHQ Troops**\n\nUnder command:\n\nFirst Army Tank Brigade\n\n1st Light Armoured Reconnaissance\n\nBrigade\n\n5th Infantry Division\n\n**I Corps** \u2013 Lieutenant General M. G. H. Barker\n\n1st Infantry Division\n\n2nd Infantry Division\n\n48th Infantry Division\n\n**II Corps** \u2013 Lieutenant General A. F. Brooke\n\n3rd Infantry Division\n\n4th Infantry Division\n\n50thInfantry Division\n\n**III Corps** \u2013 Lieutenant General Sir R. F. Adam, Bt.\n\n42nd Infantry Division\n\n44th Infantry Division\n\nDivisions undertaking training and carrying out labour duties:\n\n12th Infantry Division\n\n23rd Infantry Division\n\n46th Infantry Division\n\nN.B. the BEF's 51st Division was attached to French Third Army's Colonial Corps (see French Order of Battle above).\n\n**Belgian Army**\n\n**Commander-in-Chief** \u2013 King Lieutenant\n\n**Chief of the General Staff** \u2013 General \u00c9douard van den Bergen\n\n**Belgian I Corps**\n\n1st Infantry Division\n\n4th Infantry Division\n\n7th Infantry Division\n\n**Belgian II Corps**\n\n6th Infantry Division\n\n11th Infantry Division\n\n14th Infantry Division\n\n**Belgian III Corps**\n\n1st Chasseurs Ardennais\n\n2nd Infantry Division\n\n3rd Infantry Division\n\n**Belgian IV Corps**\n\n9th Infantry Division\n\n15th Infantry Division\n\n18th Infantry Division\n\n**Belgian V Corps**\n\n12th Infantry Division\n\n13th Infantry Division\n\n17th Infantry Division\n\n**Belgian VI Corps**\n\n5th Infantry Division\n\n10th Infantry Division\n\n16th Infantry Division\n\n**Belgian Cavalry Corps**\n\n1st Cavalry Division\n\n2nd Cavalry Division\n\n**Dutch Army**\n\n**Commander-in-Chief** \u2013 General Henri Winkelman\n\n**Chief of Staff** \u2013 Major General H. F. M. Van Voorst tot Voorst\n\n**II Corps**\n\n2nd Infantry Division\n\n4th Infantry Division\n\n**III Corps**\n\n5th Infantry Division\n\n6th Infantry Division\n\nDutch Light Division\n\n**Dutch IV Corps**\n\n7th Infantry Division\n\n8th Infantry Division\n\nFormations not in a corps:\n\nIndependent Brigade Group A\n\nIndependent Brigade Group B\n\n**Reserve \u2013 I Corps**\n\n1st Infantry Division\n\n3rd Infantry Division\n\n# Endnotes\n\nPrologue\n\n Written with reference to Marguerite Joseph-Maginot _He Might Have Saved France \u2013 The Biography of Andr\u00e9 Maginot_ Doubleday, Doran & Co. Inc. New York 1941 and _Time: France: Death & Crisis_ Monday, Jan. 18, 1932.\n\nIntroduction\n\n Quoted in G. Germond and H. T\u00fcrk (eds.) _A History of Franco-German Relations in Europe: from 'Hereditary Enemies' to Partners_ Palgrave Macmillan, New York 2008, unpaginated.\n\n2 Michael Howard 'The Use and Abuse of Military History' in _Royal United Services Institute Journal,_ No. 107 February 1962 p. 8.\n\n Quoted in Allan R. Millett and Williamson Murray _Military Effectiveness Volume 3: The Second World War_ Cambridge University Press, New York, 2010 p. xi.\n\n See Howard RUSI p. 6.\n\nChapter 1\n\n Quoted in Eugen Weber _The Hollow Years \u2013 France in the 1930s_ Sinclair-Stevenson, 1995 p. 9.\n\n _The War Illustrated_ 21 October 1939 Vol.1 No. 6 pp. 162\u201363.\n\n General Franz Halder _The Halder War Diary 1939\u20131942_ Burdick, Charles and Jacobsen, Hans-Alfred (eds.) Greenhill Books, 1988 pp. 64\u201365. The Private War Journal of General Oberst Franz Halder, Chief of the General Staff of the Supreme Command of the German Army (OKH) has also been used in the preparation of this book. The journal runs from 14 August 1939 to 10 September 1942, when Halder was dismissed by Hitler. Sections cited include: Vol. I The Polish Campaign Part I (N-16845-A); Vol. II The Polish Campaign Part II (N-16845-B); Vol. III The First Winter 7 December 1939 to 9 May 1940 (N-16845-C); and Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D), all Combined Arms Research Library, Digital Library, Fort Leavenworth, Kansas, USA. On occasion the typescript translation of Halder's original lacks the detail of the later published version. Where this extra information has been used, the published version of the diary is referenced rather than the original typescript version.\n\n _The Halder War Diary 1939\u20131942_ p. 63.\n\n5 F\u00fchrer Directives were instructions and outline strategic plans that were legally binding for all Germans and overruled all other instructions and laws. The German army leadership was very uneasy with these promulgations as they were sufficiently detailed to bind them to certain military strategies, no matter how impractical they were. There were several relating to the campaign in France and the Low Countries.\n\n F\u00fchrer Directive No. 6 dated 9 October 1939 Author's Archive.\n\n _The Halder War Diary 1939\u20131942_ pp. 65\u201366. At a F\u00fchrer Conference on 22 August 1939, Hitler had stated: 'Germany can be counted on to make a better showing of herself in a long war than she did in 1914.' 22 August 1939 The Private War Journal of General Oberst Franz Halder, Chief of the General Staff of the Supreme Command of the German Army (OKH): 14 August 1939 to 10 September 1942 Vol. I: The Polish Campaign Part I (N-16845-A) p. 21.\n\n Samuel W. Mitcham, Jnr _Hitler's Field Marshals and their Battles_ Grafton Books, 1989 p.167.\n\n Quoted in Samuel W. Mitcham, Jnr _Hitler's Field Marshals and their Battles_ Grafton Books, 1989 p.125.\n\n Karl-Heinz Frieser _The Blitzkrieg Legend \u2013 The 1940 campaign in the West_ Naval Institute Press, Annapolis, 2005 p. 55 and p. 57.\n\n Quoted in Frieser _The Blitzkrieg Legend_ p. 55.\n\n See Messerschmidt chapter in Allan R. Millett and Williamson Murray _Military Effectiveness Volume 2: The Interwar Years_ Cambridge University Press, New York, 2010 pp. 220-21.\n\n Quoted in Richard Overy with Andrew Wheatcroft _The Road to War_ Vintage, 2009 p. 53.\n\n Figures from Overy _The Road to War_ p. 67.\n\n Quoted in Messerschmidt chapter Allan R. Millett and Williamson Murray _Military Effectiveness Volume 2: The Interwar Years_ p. 236.\n\n For all these figures see Frieser _The Blitzkrieg Legend_ p. 23, James S. Corum, _The Roots of Blitzkrieg: Hans von Seeckt and German Military Reform_ University of Kansas Press, Lawrence, Kansas 1994 p. 200 and F\u00f6rster chapter in Millett and Murray Vol. 3 pp. 187\u201388.\n\n See F\u00f6rster chapter in Millett and Murray Vol. 3 pp. 187\u201388.\n\n Quoted by Messerschmidt in Millett and Murray Vol. 2 p. 227.\n\n Figures from Messerschmidt chapter in Millett and Murray Vol. 2 p. 232 and pp. 247\u201348.\n\n See Messerschmidt in Millett and Murray Vol. 2 p. 235.\n\n See John Andreas Olsen and Martin van Creveld (eds.) _The Evolution of Operational Art: From Napoleon to the Present_ Oxford University Press, Oxford 2010.\n\n For all this see Jo\u00e3o Resende-Santos _Neorealism, States, and the Modern Mass Army_ Cambridge University Press, Cambridge, 2007 pp. 97\u201399.\n\n Hans von Seeckt 'Grunds\u00e4tzemoderner Landesverteidigung' in _Gedankeneines Soldaten_ K.F. Koehler, Leipzig 1935 p. 77.\n\n Quoted in Frieser _The Blitzkrieg Legend_ p. 330.\n\n For more on this see Corum _Roots of Blitzkrieg._\n\n Translation of taped conversation with General Hermann Balck, 12 January 1979 p. 19.\n\n General Ludwig von Eimannsberger was Germany's best-known armoured warfare writer in the 1930s, but of all armoured warfare theorists, Heinz Guderian is most remembered as the 'father of German armoured forces'. This is due largely to Guderian's later success as a practitioner of panzer warfare and his gift for self-publicity rather than his originality of thinking.\n\n Most panzer divisions taking part in the invasion of Poland had under 350 tanks, and some considerably fewer.\n\n For all this see Buckley chapter in B. Bond and M. Taylor _Battle for France and Flanders Sixty Years On_ Leo Cooper, Barnsley, 2001.\n\n Westerners first became aware of the word after it appeared in a 1939 issue of _Time_ magazine to describe the lightning speed of the German army in Poland: 'This was no war of occupation, but a war of quick penetration and obliteration \u2013 _Blitzkrieg,_ lightning war.' _Time_ 25 September 1939. For more on this see Frieser _The Blitzkrieg Legend_ and Robert M. Citino _Quest for Decisive Victory \u2013 From Stalemate to Blitzkrieg in Europe, 1899\u20131940_ University Press of Kansas, Lawrence, Kansas, 2002 p. 181.\n\n See Halder Diary various entries during autumn 1939 and winter 1939\/40.\n\n Marc Bloch _Strange Defeat \u2013 A Statement of Evidence Written in 1940_ W. W. Norton, New York, 1968 p. 126 Emphasis in the original. This book was not published until 1946, two years after Bloch had been executed by firing squad along with twenty-seven other members of the Resistance.\n\n Weber _The Hollow Years_ p. 11. See also Thomas R. Christofferson with Michael S. Christofferson _France During World War II: From Defeat to Liberation_ Fordham University Press, New York, 2006 p. 2.\n\n Omer Bartov's chapter in Joel Blatt (ed.) _The French Defeat of 1940 \u2013 Reassessments_ Berghan Books, New York, 2006 pp. 56\u201357.\n\n See George Q. Flynn _Conscription and Democracy: The Draft in France, Great Britain and the United States_ Contributions to Military Studies, Nos 210, Westport, Conn. Greenwood, 2002.\n\n See Millett and Murray _Military Effectiveness_ Vol. 2 p. 52.\n\n Doughty chapter in Millett and Murray Vol. 2 pp. 42\u201343.\n\n The Dunkirk-Strasbourg-Paris triangle contained 75 per cent of the country's coal and 95 per cent of its iron ore production, and nearly all of its heavy industry. The Paris-Lille-Rouen triangle contained 90 per cent of its factories producing French cloth and 80 per cent of its woollen goods, the majority of its chemical products and a good deal of its vehicle and aircraft production.\n\n See Col. J. J. O'Connell _The Maginot Line: Dream or Reality?_ in _Studies: An Irish Quarterly Review,_ Vol. 29, No.116, Dec. 1940 p. 607.\n\n For more on this see Judith M. Hughes _To The Maginot Line \u2013 The Politics of the French Military Preparation in the 1920s_ Harvard University Press, Cambridge, Masschusetts, 2006 pp. 199\u2013219.\n\n Quoted in Robert Allan Doughty _The Breaking Point \u2013 Sedan and the Fall of France, 1940_ Archon Books, Hampden, Connecticut, 1990 p. 11. See also General Maurice Gamelin _Servir_ Vol. 2 Librairie Plon, Paris, 1946 p. 128.\n\n Quoted in Alistair Horne, _To Lose a Battle \u2013 France 1940_ Penguin Books, 1990 p. 80.\n\n Eugenia C. Kiesling _Arming Against Hitler \u2013 France and the Limits of Military Planning_ University of Kansas, Lawrence, Kansas, 1996 p. 174.\n\n Figures from Overy _The Road to War_ p. 164. See also Martin Thomas _French Economic Affairs and Rearmament: The First Crucial Months, June-September 1936_ Journal of Contemporary History Vol. 27, No. 4 (Oct. 1992).\n\n The French referred to the building of the Maginot Line up to 1935 as the First Phase, and these lighter defences as the Second Phase. It should be noted that this Second Phase had been discussed and anticipated, but not been built earlier due to the cost and importance of finishing the fortifications along the common border with Germany first.\n\n A. Danchev and D. Todman (eds.), _War Diaries of Field Marshal Lord Alanbrooke_ Weidenfeld and Nicolson, 2001. Entry dated 20 December 1939 p. 26.\n\n For more on Alan Brooke see David Fraser's chapter in Keegan, J. (ed.) _Churchill's Generals_ Weidenfeld and Nicolson, 1991.\n\n Quoted in Doughty chapter in Millett and Murray Vol. 2 p. 48.\n\n Figures from Overy _The Road To War_ Appendices Table 1 Government Expenditure for Defence in the Major Powers, 1931\u201340 p. 429.\n\n See Weber _The Hollow Years_ pp. 23\u201324.\n\n Overy _The Road to War_ p. 180\n\n See Colonel Igors Rajevs _The French Army in the Interwar Period_ in Baltic Security & Defence Review, Vol. 11, Issue 2, 2009 p. 186 and Robert Allan Doughty, _The Evolution of French Army Doctrine 1919\u20131939_ University of Kansas PhD, 1979.\n\n _Instruction provisoire de 6 Octobre 1921 sur l'emploi tactique des grandes unit\u00e9s_ Charles-Lavauzelle et Cie, impr.-\u00e9diteurs, 1928 p. 16.\n\n Doughty _The Breaking Point_ p. 27.\n\n For more on all this see Buckley chapter in B. Bond and Michael D. Taylor (eds.) _Battle for France and Flanders Sixty Years On_ Leo Cooper, Barnsley, 2001.\n\n For a description of military education in this period see General Andr\u00e9 Beaufre, _1940: The Fall of France_ Cassell, London, 1967, transl. by Desmond Flower, pp. 36\u201359 and Faris Russell Kirkland, _The French Officer Corps and the Fall of France: 1920-1940_ University of Pennsylvania, PhD 1982, pp. 276\u201388.\n\n See Elizabeth Kier _Imagining War: French and British Military Doctrine between the Wars_ Princeton, N. J., Princeton University Press, 1997.\n\n See Douglas Porch _Military 'Culture' and the Fall of France in 1940: A Review Essay_ in _International Security,_ Vol. 24 No. 4 Spring 2000 p. 179, Doughty's chapter in Millett and Murray Vol. 2 p. 46 and Martin S. Alexander 'The Fall of France' in _Journal of Strategic Studies_ Vol. 13 No. 1 1990 p. 16.\n\n Quoted in Karl-Heinz Frieser _The Blitzkrieg Legend_ p. 1.\n\n France: The Good Grey General _Time_ Monday, August 14, 1939 p. 31.\n\n Quoted in Horne p. 25.\n\n Quoted in Kiesling _Arming Against Hitler_ p. 186.\n\n Kiesling _Arming Against Hitler_ p. 175.\n\n The Private War Journal of General Oberst Franz Halder Vol. I: The Polish Campaign Part I (N-16845-A). Entry dated 14 August 1939. p. 8.\n\n Quoted in Kiesling _Arming Against Hitler_ p. 186.\n\nChapter 2\n\n Albert Winkler Interview 16 March 2013 and Email 20 December 2014.\n\n Quoted by Richard Overy in _The Origins of the Second World War_ Routledge, 2008 p. 122.\n\n The Private War Journal of General Oberst Franz Halder Vol. I: The Polish Campaign Part I (N-16845-A). Entry dated 28 August 1939. p. 37.\n\n Quoted in Overy _The Road to War_ p. 81.\n\n Quoted in Overy _The Road to War_ p. 82.\n\n Daniel Barlone _A French Officer's Diary 23 August 1939 To 1 October 1940_ Cambridge: At the University Press New York: The Macmillan Company, 1943 p. 1.\n\n The Private War Journal of General Oberst Franz Halder Vol. I: The Polish Campaign Part I (N-16845-A). Entry dated 14 August 1939. p. 14.\n\n The Private War Journal of General Oberst Franz Halder Vol. I: The Polish Campaign Part I (N-16845-A) p. 13 and Citino _Quest for Decisive Victory._ Entry dated 14 August 1939. p. 258.\n\n Samuel W. Mitcham Jnr _Panzer Legions \u2013 A Guide to the German Army Tank Divisions and their Commanders_ Stackpole Books, Mechanicsburg, PA, 2007 pp. 14\u201315.\n\n Barlone p. 13.\n\n Figures from Citino _Quest for Decisive Victory_ p. 257.\n\n Citino _Quest for Decisive Victory_ p. 257\n\n J. E. Kaufmann and H. W. Kaufmann _Hitler's Blitzkrieg Campaigns \u2013 The Invasion and Defense of Western Europe, 1939\u20131940_ Da Capo Press, Cambridge, MA, 1993 p. 9.\n\n _The Memoirs of Field Marshal Wilhelm Keitel_ Walter Gorlitz (ed.) First Cooper Square Press, New York, 2000 p. 102.\n\n The Private War Journal of General Oberst Franz Halder Vol. II The Polish Campaign Part II (N-16845-B). Entry dated 22 October 1939. p. 36.\n\n Quoted in I. Kershaw _Hitler 1936\u20131945 Nemesis_ Allen Lane, 2000 p. 291.\n\n See Doughty _Breaking Point_ p. 17.\n\n The Private War Journal of General Oberst Franz Halder Vol. II The Polish Campaign Part II (N-16845-B). Entry dated 3 November 1939. p. 43.\n\n Generalfeldmarschall Fedor von Bock _The War Diary 1939\u20131945._ Schiffer Publishing Ltd, Atglen, PA, 1996 Entry dated 23 November. p. 88.\n\n Quoted in Samuel W. Mitcham Jnr _Hitler's Field Marshals and their Battles_ Grafton Books, 1989 p. 64.\n\n Von Bock _The War Diary 1939\u2013_ 1945. Entry dated 30 October. p. 78.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 3 November. p. 80.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 12 November. p. 84.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 18 November. p. 86.\n\n The Private War Journal of General Oberst Franz Halder Vol. II The Polish Campaign Part II (N-16845-B) Halder. Entry dated 26 October p. 39.\n\n Quoted in Citino _Quest for Decisive Victory_ p. 260.\n\n Quoted in Alexander chapter in Brian Bond (ed.) _Fallen Stars: Eleven Studies of Twentieth Century Military Disasters_ Brassey's, 1991 p. 119.\n\n France: 'The Good Grey General' _Time_ Monday, August 14, 1939 p. 22.\n\n For more on all this see Alexander chapter in Bond (ed.) _Fallen Stars._\n\n See David French Raising _Churchill's Army \u2013 The British Army and the War against Germany 1919\u20131945_ Oxford University Press, Oxford, 2000.\n\n Figures from Overy _The Road to War_ p. 111.\n\n See Murray chapter in Millett and Murray vol. 3 p. 94 and Overy _The Road to War_ p. 132.\n\n See Barry R. Posen _The Sources of Military Doctrine: France, Britain and Germany Between the Wars_ Cornell University Press, New York, 1984 and Elizabeth Kier _Imagining War: French and British Military Doctrine between the Wars_ Princeton, N.J., Princeton University Press, 1997.\n\n Gavin J. Bailey _The Arsenal of Democracy: Aircraft Supply and the Evolution of the Anglo-American Alliance, 1938\u20131942_ Edinburgh University Press, Edinburgh, 2013 pp. 278\u201380.\n\n Murray chapter in Millett and Murray Vol. 2 p. 120.\n\n See Buckley chapter in Bond, B. and Taylor, Michael D. (eds.) _Battle for France and Flanders Sixty Years On_ p. 122.\n\n A. Danchev and D. Todman (eds.) Lord Alanbrooke _War Diaries of Field Marshal Lord Alanbrooke._ Entry dated 28 January 1940. pp. 34\u201335.\n\n See French _Churchill's Army_ and also his _Doctrine and Organisation in the British Army, 1919\u201332_ The Historical Journal Vol. 44, No. 52 (June 2001) p. 504. See also J. P. Harris _Men, Ideas and Tanks_ \u2013 _British Military Thought and Armoured Forces, 1903\u20131939_ Manchester University Press 1995.\n\n French _Churchill's Army_ pp. 15\u201316.\n\n See French _Raising Churchill's Army_ p. 25.\n\n See Kiesling _Arming Against Hitler_ p. 178.\n\n _Kriegsspiel_ (war game) had been used since the Napoleonic Wars to train Prussian officers. It simulated operations on a specially designed table marked with a grid system with specifically made terrain types laid on top. The 'game' used special gaming pieces and a dice to add an element of controlled randomness. There was also the ability to simulate the fog of war and communication difficulties (friction), and for an impartial umpire to calculate and assess the moves. Rules developed over the years to provide various conventions which continue to be used to the present day.\n\n For all this see Sebag-Montefiore, Hugh _Dunkirk \u2013 Fight To The Last Man_ Viking, 2006 Chapter 3 'The Mechelen Affair'.\n\n General Jodl's Diary (Armed Forces Operational Staff) from 1 February to 26 May 1940 Document 1809-PS. Entry dated 12 January 1940.\n\n Mungo Melvin _Manstein \u2013 Hitler's Greatest General_ Weidenfeld & Nicolson, London, 2010 p. 143.\n\n See Russell A. Hart _Guderian: Panzer Pioneer or Myth Maker?_ Pentagon Press, New Delhi, 2010 p. 60.\n\n General Jodl's Diary (Armed Forces Operational Staff) from 1 February to 26 May 1940 Document 1809-PS. Entry dated 13 February 1940.\n\n Quoted in Melvin p. 135.\n\n Melvin p. 135.\n\n Heeresgruppenkommandos: Heeresgruppe A: RH 19I\/38 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n Heeresgruppenkommandos: Heeresgruppe A: RH 19I\/38 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 95.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 24 February 1940. p. 114.\n\n Quoted in Horne p. 153.\n\n The Private War Journal of General Oberst Franz Halder Vol. III The First Winter 7 December 1939 to 9 May 1940 (N-16845-C). Entry dated 24 February 1940. p. 103.\n\n Quoted in Melvin p. 159.\n\n Samuel W. Mitcham, Jnr _Hitler's Field Marshals and their Battles_ Grafton Books, 1989 p.75.\n\n See Doughty _Breaking Point_ p. 34.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 97.\n\n The Private War Journal of General Oberst Franz Halder Vol. III The First Winter 7 December 1939 to 9 May 1940 (N-16845-C) Entry dated 17 March 1939. p. 106.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 114.\n\n See Doughty _Breaking Point_ p. 15 and also Don W. Alexander 'Repercussions of the Breda Variant' in _French Historical Studies,_ Vol. 8 No. 3 Spring 1974.\n\n Quoted in A. Goutard _The Battle for France 1940_ Washburn Inc., New York, 1959 p. 147.\n\n See Alexander chapter in Bond (ed.) _Fallen Stars_ p. 121.\n\n TNA Col. W Fraser Despatches, 16\u201330 June 1939 FO 371, 22917, C8681\/130\/17 and C9363\/130\/17.\n\n General Sir Edmund Ironside _Time Unguarded: The Ironside Diaries, 1937\u20131940_ ed. Col Roderick Macleod and Denis Kelly Constable, New York, 1962. Entry dated 10 January 1940. p. 204.\n\nChapter 3\n\n Barlone p. 21.\n\n Hanna Diamond _Fleeing Hitler \u2013 France 1940_ Oxford University Press, Oxford, 2007 p. 24.\n\n See Alexander chapter in Bond (ed.) _Fallen Stars_ p. 117.\n\n Alexander Werth _The Last Days of Paris: A Journalist's Diary_ Hamish Hamilton, 1940 p. 15.\n\n Barlone pp. 34\u201335.\n\n Barlone p. 1.\n\n Barlone p. 4.\n\n Pierre Roussel diary 1939. Entry dated 20 December 1939. Roussel family archive.\n\n George Aris _The British Fifth Division 1939 to 1945_ The Fifth Division Benevolent Fund, 1959 p. 3.\n\n Quoted in Kiesling _Arming Against Hitler_ p. 183.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 145.\n\n See Doughty _Breaking Point_ p. 105.\n\n 30 N 92 10\u00e8me Corps d' Arm\u00e9e, 3\u00e8me bureau, Service Historique de la D\u00e9fense, Vincennes, Paris.\n\n 30 N 92 10\u00e8me Corps d' Arm\u00e9e, 3\u00e8me bureau and 32 N 251 55\u00e8me Division d'Infanterie, de l'Arm\u00e9e de Terre, Vincennes, Paris.\n\n Quoted in Doughty _Breaking Point_ p. 111.\n\n See Alexander chapter in Bond (ed.) _Fallen Stars_ pp. 114\u201315.\n\n Quoted by Alexander in Bond (ed.) _Fallen Stars_ p. 117.\n\n 18 September 1939 1K 224, 9 Cabinet du G\u00e9n\u00e9ral Gamelin \u2013 Journal de marche Fonds Gamelin, SHAT.\n\n General Sir Edmund Ironside _Time Unguarded: The Ironside Diaries, 1937\u20131940._ Entry dated 10 February 1940. pp. 203\u20134.\n\n _War Diaries of Field Marshal Lord Alanbrooke._ Entry dated 16 October 1939. pp. 7\u20138.\n\n _War Diaries of Field Marshal Lord Alanbrooke._ Entry dated 9 October 1939. pp. 5\u20136.\n\n The founder of the Taittinger champagne house who later collaborated with the Germans as chairman of the municipal council of Paris.\n\n 29 N 27 2\u00e8me Arm\u00e9e Historique des operations. Service Historique de la D\u00e9fense, Vincennes, Paris.\n\n 29 N 27 2\u00e8me Arm\u00e9e Historique des operations. Service Historique de la D\u00e9fense, Vincennes, Paris.\n\n Quoted in Doughty _Breaking Point_ p. 148.\n\n For all this see Wesley chapter in Bond (ed.) _Fallen Stars_ p. 144.\n\n See Brian Bond's chapter on Gort in John Keegan (ed.) _Churchill's Generals_ Weidenfeld and Nicolson, 1991.\n\n _War Diaries of Field Marshal Lord Alanbrooke._ Entry dated 22 September 1939. p. 18.\n\n Major G. Courage _The History of the 15\/19 The King's Royal Hussars 1939\u20131945_ Gale & Polden Ltd, Aldershot, 1949 pp. 7\u20138.\n\n Courage p. 6.\n\n See Spencer Tucker (ed.) _Encyclopedia of World War II \u2013 A Political, Social and Military History_ Santa Barbara, California, 2005 Vol. I p. 200.\n\n For all this see M.B. 'The Strategic Position of Holland and Belgium' _Bulletin of International News_ Vol. 16 No. 24 (Dec 2 1939) pp. 1304\u20136.\n\n Figures from P. L. G. Doorman _Military Operations of the Dutch Army 10-17th May 1940_ Helion, Solihull, 2005 p. 10 and Spencer Tucker (ed.) _Encyclopedia of World War II \u2013 A Political, Social and Military History_ p. 1063.\n\n Figure from Walter B. Maass _The Netherlands at War: 1940\u20131945_ Abelard\u2013Schuman, Amsterdam, 1970 p. 16.\n\n L. De Jong _Holland Fights the Nazis_ The Right Book Club, 1941 pp. 5\u20136.\n\n The Private War Journal of General Oberst Franz Halder Vol. III The First Winter 7 December 1939 to 9 May 1940 (N-16845-C). Entry dated 26 March 1940. p. 131.\n\n See Williamson Murray's article 'The German Response to Victory in Poland: A Case Study in Professionalism' in _Armed Forces and Society,_ Vol. 7 1981.\n\n Frieser _Blitzkrieg Legend_ p. 22.\n\n Figure from Frieser _Blitzkrieg Legend_ p. 29.\n\n Frieser _Blitzkrieg Legend_ p. 32.\n\n Von Bock Diary _1939\u20131945._ Entry dated 13 March 1940. p. 118.\n\n Von Bock Diary 1939\u20131945. Entry dated 26 April 1940. p. 131.\n\n Email from Peter Gerber (1 May 2014) whose father worked in von Bock's headquarters during this period.\n\n See Niall Cherry _Doomed from the Start: The Allied Intervention in Norway 1940_ Helion & Co., Solihull, 2016 and Anthony Dix _The Norway Campaign and the Rise of Churchill 1940_ Pen and Sword, Barnsley, 2014.\n\n Gruppe XXI, Notiz fuer das Kriegstagebuch, 1.4.40, in Anlagenband 1 zum K.T.B. Nr. k, Anlagen 1\u201352, AOK 20 E 180\/7.\n\n Ian Kershaw, _Hitler 1936\u20131945 Nemesis_ p. 289.\n\n David Fraser _Knight's Cross \u2013 A Life of Field Marshal Erwin Rommel_ HarperCollins, 1993 pp. 160\u201361.\n\n Fraser _Rommel_ p. 159.\n\n Taped Conversation with Balck, 12 January 1979 p. 19.\n\n Hermann Balck _Ordnung im Chaos. Erinnerungen 1893\u20131948_ Osnabr\u00fcck Biblio Verlag, 1980 p. 267.\n\n All figures from Frieser _Blitzkrieg Legend_ pp. 35\u201348.\n\n Carl Clausewitz _On War_ (Translated and edited by Michael Howard and Peter Paret) Princeton, N.J., Princeton University Press, 1976 p. 86.\n\n Barlone p. 26.\n\n See Doughty _Breaking Point_ p. 74; Ernest R. May _The Intelligence Process: The Fall of France, 1940_ Unpublished paper Kennedy School of Government n.d.; Robert J. Young chapter in Ernest R. May (ed.) _Knowing One's Enemies_ Princeton, N.J., Princeton University Press, 2014 and Douglas Porch _French Intelligence and the Fall of France 1930-40 in_ Intelligence and National Security 4, No. 1 Jan 1989.\n\n Ian Kershaw _Hitler 1936\u20131945 Nemesis_ pp. 290-91.\n\n Doughty _Breaking Point_ pp. 74\u201376.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 142.\n\n See The Private War Journal of General Oberst Franz Halder Vol. III The First Winter 7 December 1939 \u2013 9 May 1940 (N-16845-C). Entry dated 5 May 1940. p. 190.\n\n Quoted in Alexander chapter in Bond (ed.) _Fallen Stars_ p. 127.\n\n See Julian Jackson _The Fall of France: The Nazi Invasion of 1940_ Oxford University Press, Oxford, 2003 pp. 128\u201329.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ Collins, 1953 p. 6.\n\n Quoted in Doughty _Breaking Point_ p. 43.\n\n Helmut Mahlke _Memoirs of a Stuka Pilot_ Frontline Books, Barnsley, 2013 p. 81.\n\n Quoted in Ian Kershaw, _Hitler 1936\u20131945 Nemesis_ p. 294.\n\nChapter 4\n\n Gilberto Villahermosa _Hitler's Paratrooper \u2013 The Life and Battles of Rudolf Witzig_ Frontline Books, 2010 p. 51.\n\n Otto Gull Interview 18 July 2011.\n\n Otto Gull Interview 18 July 2011.\n\n VIII. Fliegerkorps: RL 8\/43, 45.\n\n See Chapter 3 Lloyd Clark _Arnhem_ \u2013 _Jumping the Rhine 1944 and 1945 \u2013 The Greatest Airborne Battle in History_ Headline 2008.\n\n Quoted in Villahermosa p. 41.\n\n Mahlke p. 82.\n\n Mahlke p. 84.\n\n E. R. Hooton _Luftwaffe at War: Blitzkrieg in the West 1939\u20131940_ Ian Allan, Hersham, Surrey, 2007 p. 15.\n\n Letter from Conrad Houtkooper 21 May 2013.\n\n See James J. Weingartner _Hitler's Guard: The Story of the Leibstandarte SS Adolf Hitler 1933\u20131945_ Southern Illinois University Press, Carbondale and Edwardsville, 1968 pp. 38\u201339.\n\n Quoted in Villahermosa p. 43.\n\n Quoted in Villahermosa p. 46.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 30 October 1939 p. 135.\n\n Aris p. 27.\n\n Philip Gribble _The Diary of a Staff Officer, Air Intelligence Liaison Officer, at Advanced Headquarters North B.A.F.F., 1940_ Methuen & Co., 1941 p. 3.\n\n 10 May 1940 _War Diaries of Field Marshal Lord Alanbrooke_ p. 60.\n\n Major G. Courage _The History of the 15\/19 The King's Royal Hussars 1939\u20131945_ Gale & Polden Ltd, Aldershot, 1949 p. 16.\n\n Courage p. 16.\n\n See Doughty _Breaking Point_ p. 81.\n\n Translation of Taped Conversation with General Hermann Balck p. 28.\n\n Quoted in Doughty _Breaking Point_ p. 48.\n\n 1.PzDiv: RH 27-1 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n Robert Kershaw _Tank Men \u2013 The Human Story of Tanks at War_ Hodder and Stoughton, 2008 p. 102.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 118.\n\n Gull Interview 2011.\n\n Horne p. 215.\n\n Frieser _Blitzkrieg Legend_ p. 142.\n\n Quoted in Doughty _Breaking Point_ p. 95.\n\nChapter 5\n\n Arnaud Bauwens Interview 21 November 2014.\n\n Courage p. 20.\n\n Courage p. 20.\n\n Diamond p. 32.\n\n Courage p. 20.\n\n Gerd Ritter Interview 16 April 2013.\n\n Peter Meier Interview 21 April 2013.\n\n Peter Meier Interview 21 April 2013.\n\n See Weingartner p. 40.\n\n Email from Lisa Hanson 8 April 2013.\n\n Adolf Galland _The First and Last_ Methuen and Co., 1955 pp. 12\u201313.\n\n Michael J. F. Bowyer _2 Group RAF \u2013 A Complete History 1936\u20131945_ Faber and Faber, 1974 p. 85.\n\n Quoted in Robert Kershaw _Tank Men_ p. 102.\n\n Quoted in Robert Kershaw _Tank Men_ p. 103.\n\n Quoted in Robert Kershaw _Tank Men_ p. 103.\n\n Charles-Michel L\u00e9p\u00e9e Interview 16 November 2014.\n\n See Preface to Emile Cammaerts _The Prisoner at Laeken: King Leopold \u2013 Legend and Fact_ Cresset Press, 1941.\n\n Quoted in Doughty _Breaking Point_ p. 68.\n\n Doughty _Breaking Point_ p. 286.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 118.\n\n Quoted in Doughty _Breaking Point_ p. 96.\n\n Belgium Ministry of Foreign Affairs _Belgium: The Official Account of What Happened 1939\u201340_ Evans Brothers Ltd, London, 1941 p. 38.\n\n Figures from Frieser _Blitzkrieg Legend_ p. 241.\n\n Robert Kershaw _Tank Men_ pp. 108\u20139.\n\n Robert Kershaw _Tank Men_ pp. 110\u201311.\n\n 4.PzDiv: RH 27-4 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n Geoffrey Stewart _Dunkirk and the Fall of France_ Pen and Sword, Barnsley, 2008 p. 35.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 7\n\n 6.PzDiv: RH 27-6\/1D Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n Quoted in Doughty _Breaking Point_ p. 99.\n\n David Boyer Email 18 February 2015.\n\nChapter 6\n\n David Boyer Email 18 February 2015.\n\n For more see Frieser _Blitzkrieg Legend_ p. 156.\n\n John Williams _The Ides of May: The Defeat of France May-June 1940_ Constable, 1968 p. 130.\n\n Otto Gull Interview 18 July 2011.\n\n 29 N 27 2\u00e8me Arm\u00e9e Historique des Operations, Service Historique de la D\u00e9fense, Vincennes, Paris.\n\n Quoted in Doughty _Breaking Point_ p. 99.\n\n Max Meers Email 20 June 2012.\n\n Peter Meier Interview 28 July 2013.\n\n Joop van Praag Email 1 August 2013.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 230.\n\n See Telp chapter in Ian Beckett (ed.) _Rommel \u2013 A Reappraisal_ Pen and Sword, Barnsley, 2013 p. 48.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 233.\n\n Quoted in Frieser _Blitzkrieg Legend_ pp. 219\u201320.\n\n See Frieser _Blitzkrieg Legends_ p. 219.\n\n 32 N 251 55\u00e8me Division d'Infanterie, Service Historique de la D\u00e9fense, Vincennes, Paris.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 13 May 1940 p. 9.\n\n Heinz Guderian _Panzer Leader_ Penguin, 2000 p. 101.\n\n 34 N 145 147\u00e8me R\u00e9giment d'Infanterie de Fortresse Service Historique de la D\u00e9fense, Vincennes, Paris.\n\n Robert Kershaw _Tank Men_ pp. 103\u2013104.\n\n David Boyer Interview 18 February 2015.\n\n Quoted in Citino _Quest for Decisive Victory_ p. 269.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 159.\n\n David Boyer Interview 18 February 2015.\n\n See Frieser _Blitzkrieg Legend_ p. 161.\n\n Quoted in Doughty _Breaking Point_ p. 143.\n\n See Doughty _Breaking Point_ pp. 149\u2013150.\n\n 10.PzDiv: RH 27-10\/9 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n Quoted in Doughty _Breaking Point_ p. 155.\n\n Quoted in Doughty _Breaking Point_ pp. 156\u201357.\n\n Quoted in Doughty _Breaking Point_ p. 190.\n\n Quoted in Doughty _Breaking Point_ p. 190.\n\n 1.PzDiv: RH 27-1\/5 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n 1.PzDiv: RH 27-1\/4 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n 32 N 251 Rapport de G\u00e9n\u00e9ral Lafontaine Service Historique de la D\u00e9fense, Vincennes, Paris.\n\n Quoted in Doughty _Breaking Point_ p. 100.\n\n Quoted in Horne p. 273.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 13 May 1940. p. 140.\n\nChapter 7\n\n Otto Gull Interview 19 July 2011.\n\n Jackson _The Fall of France 1940_ p. 47.\n\n Quoted in Alexander chapter in Bond (ed.) _Fallen Stars_ p. 127.\n\n Otto Gull Interview 19 July 2011.\n\n Quoted in Doughty _Breaking Point_ p. 255.\n\n Krajewski in Guderian, H. (ed.) _Mit den Panzern in Ost und West_ Volk und Reich Verlag, Berlin, Prague and Vienna, 1942 p. 137.\n\n Rapport de l'Aspirant Penissou S.H.A.T. 34N165.\n\n Rapport de l'Aspirant Penissou S.H.A.T. 34N165.\n\n Rapport de l'Aspirant Penissou S.H.A.T. 34N165.\n\n See Rapport de Lieutenant Colonel P. Labarthe S.H.A.T. 34N165.\n\n Heinz Guderian _Panzer Leader_ Penguin, 2000 p. 84.\n\n De la Vigerie, General Fran\u00e7ois d'Astier _Le ciel n'\u00e9tait pas vide 1940_ Ren\u00e9 Julliard, Paris, 1952 p. 107.\n\n Figures from Frieser _Blitzkrieg Legend_ p. 180.\n\n Figures from Frieser _Blitzkrieg Legend_ p. 180.\n\n Quoted in Stewart p. 54.\n\n Denis Richards, _Royal Air Force Vol. I, The Fight At Odds_ HMSO, 1953 p. 120.\n\n Quoted in Michael J. F. Bowyer _2 Group RAF \u2013 A Complete History 1936\u20131945_ Faber and Faber, 1974 p. 89.\n\n Quoted in Stewart p. 54.\n\n Gerd Brausch 'Sedan 1940. Deuxi\u00e8me Bureau and Strategische \u00dcberraschung' _Milit\u00e4rgeschichtliche Zeitschrift_ Vol. 2. No. 2 1967 p. 84.\n\n Marc Bloch _Strange Defeat \u2013 A Statement of Evidence Written in 1940_ W. W. Norton, New York, 1968 p. 43.\n\n 7.PzDiv: RH 27-7\/2; 7\/9; 7\/11 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 13.\n\n 7.PzDiv: RH 27-7\/2; 7\/9; 7\/11 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 235.\n\n Quoted in Nick Shepley _Darkest Hour, Finest Hour: Norway, Dunkirk and the Battle of Britain \u2013 Britain at War: Part One_ Andrews UK, Luton 2013 p. 134.\n\n F\u00fchrer Directive No.11 dated 14 May 1940 Author's Archive.\n\n Figures from Stewart p. 38.\n\n Barlone p. 47.\n\n Werth p. 34.\n\n Gribble p. 10.\n\n Quoted in William L. Shirer _The Collapse of The Third Republic: An Inquiry into the Fall of France in 1940_ Simon and Schuster, New York 1960 p. 679.\n\n Winston S. Churchill _The Second World War \u2013 Volume II: Their Finest Hour_ Cassell, 1949 pp. 38\u201339.\n\n Quoted in Stewart p. 56.\n\n Hermann Balck _Ordnung im Chaos. Erinnerungen 1893\u20131948_ Osnabr\u00fcck Biblio Verlag, 1980 p. 276.\n\n Balck _Ordnung im Chaos. Erinnerungen_ p. 276.\n\n Balck _Ordnung im Chaos. Erinnerungen_ p. 276.\n\n Quoted in Robert Kershaw _Tank Men_ p. 113.\n\n Bruno Cantel Letter dated 12 June 1995.\n\n Figures from Frieser p. 239.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 13.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 16.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 16.\n\n _The Guardian_ Obituary 17 January 2005.\n\n For all this see J. A. Gunsburg 'The Battle of the Belgian Plain, 12\u201314 May 1940' _Journal of Military History 56,_ April 1992 and 'The Battle of Gembloux 14\u201315 May 1940', _Journal of Military History 64,_ January 2000.\n\n Frieser _Blitzkrieg Legend_ p. 246.\n\n Quoted in Alexander chapter in Bond (ed.) _Fallen Stars_ p. 124.\n\n Major General Sir Edward Spears _Assignment to Catastrophe, Vol. 1 Prelude to Dunkirk July 1939- May 1940_ A. A. Wyn Inc., New York, 1954 p. 148.\n\n Quoted in Horne p. 373.\n\nChapter 8\n\n Churchill _Finest Hour_ p. 42.\n\n Churchill _Finest Hour_ pp. 42\u201343.\n\n Quoted in Diamond p. 35.\n\n Quoted in Diamond p. 37.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 16 May 1940. p. 13.\n\n 1 Pz. Div. Kriegstagebuch Nr. 3 9.5.1940 \u2013 2.6.1940 p. 29.\n\n Bowyer p. 27.\n\n Quoted in Bowyer p. 27.\n\n Bowyer p. 28.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 266.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 17.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 18.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 20.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 266.\n\n Bloch pp. 47\u201348.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 22.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 26.\n\n Fraser _Rommel_ p. 178.\n\n Quoted in Fraser p. 180.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 17 May 1940. p. 17.\n\n See B. H. Liddell Hart _The German Generals Talk_ New York, W. Morrow, 1948 p. 128.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 17 May 1940. p. 16.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 17 May 1940. p. 16.\n\n F\u00fchrer Directive No. 12 dated 18 May 1940 Author's Archive.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 18 May 1940. p. 18.\n\n Quoted in Charles Messenger _The Last Prussian: A Biography of Field Marshal Gerd von Rundstedt 1875\u20131953_ Pen and Sword, Barnsley, 2012 p. 110.\n\n Maxime Dufour Interview 19 April 2013.\n\n A. Danchev and D. Todman (eds.) _War Diaries of Field Marshal Lord Alanbrooke._ Entry dated 18 May 1940. pp. 64\u201365.\n\n Quoted in Robert Kershaw _Tank Men_ pp. 115\u201316.\n\n Quoted in Robert Kershaw _Tank Men_ p. 16.\n\n Courage p. 34.\n\n Quoted in Nigel Hamilton _Monty: The Making of a General 1887\u20131942_ Hamish Hamilton, 1981 p. 347.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 19 May 1940. p. 19.\n\n Quoted in Horne p. 472.\n\n Gamelin _Servir_ pp. 3\u20134.\n\n See Alexander chapter in Bond (ed.) _Fallen Stars_ p. 126.\n\n _The Ironside Diaries, 1937\u20131940._ Entry dated 17 May. p. 317.\n\n _The Ironside Diaries, 1937\u20131940._ Entry dated 17 May. p. 321.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 283.\n\n Quoted in Robert Kershaw _Tank Men_ p. 106.\n\n Quoted in Stewart p. 67.\n\n General Jodl's Diary (Armed Forces Operational Staff) from 1 February to 26 May 1940 Document 1809-PS. Entry dated 20 May 1940.\n\n Gribble p. 27.\n\n Churchill _Finest Hour_ p. 52.\n\nChapter 9\n\n Emmanuel Marcel Interview 20 May 2012.\n\n Emmanuel Marcel Interview 20 May 2012.\n\n Emmanuel Marcel Interview 20 May 2012.\n\n Quoted in Robert Kershaw _Tank Men_ p. 119.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 32.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 32.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 32.\n\n Brigadier Peter Vaux _Arras, An EyeWitness Account._ Author's Archive. Undated and upaginated.\n\n Brigadier Peter Vaux _Arras, An EyeWitness Account_. Author's Archive. Undated and unpaginated.\n\n Brigadier Peter Vaux _Arras, An EyeWitness Account_. Author's Archive. Undated and unpaginated.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 33.\n\n Quoted B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 34.\n\n Bowyer p. 69.\n\n Bowyer p. 95.\n\n Figures from Bowyer p. 97.\n\n Peter Mansfield Interview 30 August 2014.\n\n Douglas Manning Interview 12 November 2014.\n\n Quoted by Robert Kershaw _Tank Men_ p. 107.\n\n Diamond p. 40.\n\n Diane Lebeouf Interview 2 June 1996.\n\n Julia Lapointe Diary. Entry dated 22 May 1940. Author's Archive.\n\n Diamond p. 39.\n\n Heeresgruppe A: RH 19I\/37 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n See The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 22 May 1940. p. 25.\n\n Aris p. 37.\n\n Quoted in Stewart p. 82.\n\n Marc Leicht Letter 10 November 2014.\n\n Gerd Reismann Interview 6 May 2011.\n\n Gerd Reismann Interview 6 May 2011.\n\n Heeresgruppe A: RH 19I\/38 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n 6.PzDiv: RH 27-6\/1D Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n Guderian _Panzer Leader_ p. 117.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 23 May 1940. p. 33.\n\n Figure quoted in Frieser _Blitzkrieg Legend_ p. 310.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 24 May 1940. p. 152.\n\n Ian Kershaw _Hitler: Nemisis_ p. 295.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 25 May 1940. p. 35.\n\n Heeresgruppe A: RH 19I\/38 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n A. Danchev and D. Todman (eds.) _War Diaries of Field Marshal Lord Alanbrooke._ Entry dated 23 May 1940. p. 68.\n\nChapter 10\n\n David Schneider Interview 18 June 2011.\n\n Quoted in Robert Kershaw _Tank Men_ p. 128.\n\n Andrew Hardcastle Interview 2 July 2013.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 26 May 1940. p. 158.\n\n Quoted in Robert Kershaw _Tank Men_ p. 126.\n\n Gribble p. 47.\n\n Quoted in Jon Cooksey _Calais: Fight to the Finish \u2013 May 1940_ Pen and Sword, Barnsley, 1999 p. 115.\n\n Quoted in Bowyer p. 99.\n\n David McCullen Diary. Entry dated 25 May 1940. Author's Archive.\n\n Jan Hoch Letter 13 January 2013.\n\n Churchill _Finest Hour_ p. 73.\n\n David McCullen Diary. Entry dated 25 May. Author's Archive.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 26 May 1940. p. 37.\n\n Quoted in Mike Rossiter _I Fought at Dunkirk_ Random House, 2012 p. 147.\n\n Figures quoted in Frieser _Blitzkrieg Legend_ p. 308.\n\n Gribble p. 44.\n\n John Colville _Fringes of Power: Downing Street Diaries 1939\u20131955_ Weidenfeld and Nicolson, 2004. Entry dated 27 May 1940. p. 113.\n\n Alexander Wilkins Interview 12 August 2011.\n\n Quoted in Sean Longden, _Dunkirk \u2013 The Men They Left Behind_ Constable, 2008 p. 74.\n\n Quoted in Longden p.75.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 27 May 1940 p. 38.\n\n Quoted in Longden pp. 75\u201376.\n\n Quoted in www.telegraph.co.uk\/history\/world-war-two\/7753912\/Wormhoudt\/Every-day-I-thank-God-we-did-our-duty.html Accessed 12 December 2014.\n\n For more on this atrocity, see Longden; Julian Thompson _Dunkirk \u2013 Retreat To Victory_ Sidgwick & Jackson 2008 and Hugh Sebag-Montefiore, _Dunkirk \u2013 Fight To The Last Man_ Viking, 2006.\n\n Andrew Hardcastle Interview 2 July 2013.\n\n David McCullen Diary. Entry dated 6 March to 20 August 1940. Author's Archive.\n\n John Colville Diary. Entry dated 27 May 1940. p. 113.\n\n Churchill _Finest Hour_ p. 81.\n\n See Brian Bond _France and Belgium 1939\u20131940_ Davis-Poynter, 1975 p. 92.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 41.\n\n _The Halder War Diary 1939\u20131942._ p. 63.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 31 May 1940. p. 163.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 31 May 1940. p. 172.\n\n David Blundell Telephone Interview 1 May 2014.\n\n Tom Reed Interview 12 May 2014.\n\n Hans Liebler Interview June 2014.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 30 May 1940. p. 42.\n\n Courage p. 41.\n\n A. Danchev and D. Todman (eds.) _War Diaries of Field Marshal Lord Alanbrooke._ Entry dated 30 May 1940. p. 73.\n\n A. Danchev and D. Todman (eds.) _War Diaries of Field Marshal Lord Alanbrooke_. Entry dated 26 May 1940. p. 70.\n\n Figures quoted in Frieser _Blitzkrieg Legend_ p. 301.\n\n Barlone p. 58.\n\n Courage pp. 42\u201343.\n\n Mahlke pp. 99\u2013100.\n\n Cairns chapter in B. Bond and Michael D. Taylor (eds.) _Battle for France and Flanders Sixty Years On_ Leo Cooper, Barnsley, 2001 p. 90.\n\n Von Kluge Diary. Entry dated 25 May 1940 quoted in Messenger p. 118.\n\n Figure quoted in Frieser _Blitzkrieg Legend_ p. 299.\n\n Rab Riley Interview 12 October 1991\n\n Quoted in Stewart p. 117.\n\n Mahlke p. 102.\n\n Ernie Smith Telephone Interview 28 November 2014.\n\n <http:\/\/www.bbc.co.uk\/history\/ww2peopleswar\/stories\/40\/a2447840.shtml> Accessed 12 November 2014.\n\n <http:\/\/www.bbc.co.uk\/history\/ww2peopleswar\/stories\/40\/a2447840.shtml> Accessed 12 November 2014.\n\n Quoted in Longden p.1.\n\n Quoted in W. J. R. Gardner, _The Evacuation from Dunkirk: Operation Dynamo 26 May to 4 June 1940_ Frank Cass, 2000 p. 108.\n\n Figures from Julian Thompson _Dunkirk \u2013 Retreat To Victory_ Sidgwick & Jackson 2008 p. 306.\n\n Hansard 4 June 1940 <http:\/\/hansard.millbanksystems.com\/commons\/1940\/jun\/04\/warsituation> Accessed 10 December 2014.\n\n 5 June Order of the Day Quoted in Patrick Wilson 'Dunkirk: Victory or Defeat?' in _History Today_ History Review Issue 37 September 2000.\n\n F\u00fchrer Directive No.13 dated 24 May 1940. Author's Archive.\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 307.\n\nChapter 11\n\n Quoted in Diamond p. 8.\n\n Figures quoted in Werth p. 127.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 2 June 1940. p. 46.\n\n Michel Sigaut Interview 15 January 2013.\n\n See Martin S. Alexander 'After Dunkirk: The French Army's Performance against \"Case Red\", 25 May to 25 June 1940' in _War in History_ Vol. 14 No. 2 April 2007 p. 225.\n\n See Bruce I. Gudmundsson, _On Armor_ Greenwood Publishing, Westport, Conn., 2004.\n\n See Alexander _After Dunkirk_ p. 230.\n\n Quoted in Alexander _After Dunkirk_ p. 246.\n\n See Alexander _After Dunkirk_ p. 238.\n\n Figures quoted in Kaufmann and Kaufmann _Hitler's Blitzkrieg Campaigns_ p. 267.\n\n Quoted in Alexander _After Dunkirk_ p. 235.\n\n Quoted in Alexander _After Dunkirk_ p. 236.\n\n Quoted in Melvin p. 162.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 5 June 1940. p. 52.\n\n Figures quoted in E. R. Hooton, _Luftwaffe at War: Blitzkrieg in the West 1939\u20131940_ Ian Allan, Hersham, Surrey, 2007 p. 84.\n\n Daniel Le Roy Telephone Interview 28 March 2013.\n\n Hauptmann Friedrich Treitler Diary. Entry dated 16 May 1940.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). 4 June 1940. p. 49.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 43.\n\n See Williams _Ides of May_ p. 274.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 48.\n\n See Telp chapter in Ian Beckett (ed.) _Rommel \u2013 A Reappraisal_ Pen and Sword, Barnsley, 2013.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 5 June 1940. pp. 165\u201366.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 52.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p. 60\n\n Quoted in Frieser _Blitzkrieg Legend_ p. 223.\n\n B. H. Liddell Hart (ed.) _The Rommel Papers_ p.66.\n\n Serge Winter Letter 4 May 2013.\n\n See Alexander _After Dunkirk_ p. 248.\n\n Quoted in Alexander p. 248.\n\n Quoted in Alexander _After Dunkirk_ p. 248.\n\n Dominic Lemaire Letter 1 May 1994.\n\nChapter 12\n\n Serge Winter Letter 4 May 2013.\n\n Serge Winter Letter 4 May 2013.\n\n Serge Winter Letter 4 May 2013.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). 6 June 1940. p. 55.\n\n Quoted in Stewart p. 122.\n\n See Major General F. W. von Mellenthin _Panzer Battles: A Study of the Employment of Armor in the Second World War_ Ballantine Books, New York 1971 p. 25.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 10 June 1940. p. 67.\n\n 11 June 1940 The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D) p. 70.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 13 June 1940. p. 76.\n\n F\u00fchrer Directive No. 15 dated 14 June 1940. Author's Archive.\n\n Quoted in Alexander _After Dunkirk_ p. 256.\n\n Werth p. 144.\n\n Diamond p. 48.\n\n Quoted in Rupert Butler _Hitler's Death's Head Division: SS Totenkopf Divisions_ Pen and Sword, Barnsley, 2004 p. 76.\n\n Quoted in Diamond p. 1.\n\n Quoted in Diamond p. 43.\n\n Quoted in Diamond p. 58.\n\n Werth p. 160.\n\n See Frieser _Blitzkrieg Legend_ p. 326.\n\n Quoted in Roy Jenkins _Churchill_ Pan Macmillan 2012 p. 616.\n\n Walter Reid _Churchill 1940-1945: Under Friendly Fire_ Birlinn, Edinburgh, 2008 p.4.\n\n Colville p. 124.\n\n Quoted in Diamond p. 50.\n\n Quoted in Diamond p. 199.\n\n Quoted in Martin Gilbert _Churchill \u2013 A Life_ Heinemann, 1991 p. 659.\n\n Major General Sir Edward Spears _Assignment to Catastrophe, Vol. 2 The Fall of France June 1940_ A. A. Wyn Inc., New York, 1954 p. 231.\n\n Quoted in Diamond p. 96.\n\n Quoted in Diamond p. 90.\n\n Julianne Deros Letter 16 November 2012.\n\n Julianne Deros Letter 16 November 2012.\n\n Quoted in Diamond p. 4.\n\n Quoted in Diamond p. 71.\n\n See E. R. Hooton _Phoenix Triumphant: The Rise and Rise of the Luftwaffe_ Arms and Armour Press, London, 1994 p. 266.\n\n Georges Sadoul _Journal de Guerre_ Paris, Les Editeurs Fran\u00e7ais R\u00e9unis 1977. Entry dated 13 June 1940. p. 314.\n\n Von Bock Diary _The War Diary 1939\u20131945._ Entry dated 13 June 1940. p. 175.\n\n Ralf Buckholtz Interview 16 April 2011.\n\n Von Bock _The War Diary 1939\u20131945._ Entry dated 14 June 1940. p. 176.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 14 June 1940. p. 76.\n\n Paul Henkelmann Letter 1 May 2013.\n\n A. Danchev and D. Todman (eds.) _War Diaries of Field Marshal Lord Alanbrooke_ later note to entry dated 30 May 1940. p. 74.\n\n A. Danchev and D. Todman (eds.) _War Diaries of Field Marshal Lord Alanbrooke_ later note to entry dated 14 June 1940. p. 81.\n\n Figures quoted in L. F. Ellis _War in France and Flanders, 1939\u20131940_ HMSO, 1954 p. 327.\n\n For more on this see M. Romanych and M. Rupp _Maginot Line 1940: Battles on the French Frontier_ Osprey, Oxford, 2010 pp. 36\u201345.\n\n For more on the withdrawal plan and the subsequent German operations against the Maginot Line see M. Romanych, and M. Rupp pp. 52\u201391.\n\n _Blitzkrieg In Their Own Words: First Hand Accounts from German Soldiers 1939\u20131940_ Amber Books, 2011 p. 210.\n\n _Blitzkrieg In Their Own Words_ p. 211.\n\n Malke p. 184.\n\n Dominic Royer Letter 8 June 2012\n\n See www.lignemaginot.com\/accueil\/indexen.htm Accessed on 15 December 2015.\n\n Jean-Yves Mary, Alain Hohnadel and Jacques Sicard _Hommes et Ouvrages de la Ligne Maginot, Tome_ 3. Paris, Histoire & Collections, 2003, p. 201.\n\n See www.lignemaginot.com\/accueil\/indexen.htm Accessed on 15 December 2015.\n\n See Romanych and Rupp pp. 90\u201391.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 19 June 1940. p. 84.\n\n <http:\/\/www.ibiblio.org\/pha\/policy\/1940\/1940-06-16d.html> Accessed 3 November 2013.\n\n Werth p. 192.\n\n A. Danchev and D. Todman (eds.) _War Diaries of Field Marshal Lord Alanbrooke_ later note to entry dated 14 June 1940 p. 80.\n\n _Blitzkrieg In Their Own Words_ p. 237.\n\n Quoted in Julian Jackson _The Fall of France: The Nazi Invasion of 1940_ Oxford University Press, Oxford, 2003 p. 143.\n\n Daniel Le Roy Telephone Interview 28 March 2013.\n\n Julianne Deros Letter 23 November 2012.\n\n Sergeant John Rigson Email 2 December 2012. Rigson had not managed to be evacuated from Nantes and joined refugees heading south.\n\n Sergeant John Rigson Email 2 December 2012.\n\n Quoted in Max Hastings _All Hell Let Loose_ HarperPress, 2012 p. 71.\n\n For more on his action see Hastings pp. 71\u201373.\n\n www.lehrmaninstitute.org\/history\/index.html Accessed 12 July 2013.\n\n Quoted in Knox MacGregor _Mussolini Unleashed, 1939\u20131941: Politics and Strategy in Fascist Italy's Last War_ Cambridge University Press, Cambridge, 1982 p.128.\n\n Quoted in MacGregor p.129.\n\n Quoted in Gerhard Schreiber, Bernd Stegemann and Detlef Vogel _Germany and the Second World War, Volume III: The Mediterranean, South-East Europe, and North Africa 1939\u20131942_ Clarendon Press, Oxford, 1995 p. 246.\n\n For more on the fighting in the Alps see Emanuelle Sica 'June 1940: The Italian Army and the Battle of the Alps' in _Canadian Journal of History,_ [S.l.], v. 47, n. 2, May. 2014 p. 374, and Douglas Porch _The Path to Victory: The Mediterranean Theater in World War II_ New York, Farrar, Straus and Giroux, New York, 2005.\n\n Figures quoted in Sica p. 374 and Porch p. 43.\n\n The Private War Journal of General Oberst Franz Halder Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D). Entry dated 21 June 1940. p. 89.\n\n Von Bock _The War Diary 1939\u20131945._ Diary entry dated 25 June 1940. p. 181.\n\nEpilogue\n\n Robert Hughes _Of Gods and Monsters_ in _The Guardian_ 1 February 2003.\n\n Robert Hughes _Of Gods and Monsters_ in _The Guardian_ 1 February 2003.\n\n Albert Speer _Inside the Third Reich_ New York, Simon and Schuster, 1997 p. 172.\n\nConclusion\n\n Quoted in Gordon A. Craig _The Germans_ Penguin, New York, 1991 p. 15.\n\n _Dad's Army: The Movie_ Screenplay by Jimmy Perry and David Croft. Norcon\/Columbia, 1971.\n\n See Doughty chapter in Millett and Murray Vol. 2 p. 66.\n\n Quoted in Doughty PhD p. 294.\n\n Doughty _Breaking Point_ p. 26.\n\n See Hew Strachan _The Direction of War_ p. 40.\n\n A. Beaufre _1940 The Fall of France_ Cassell, 1967 p. 197.\n\n See Douglas Porch's criticisms in _Military 'Culture' and the Fall of France in 1940: A Review Essay_ in _International Security,_ Vol. 24 No. 4 Spring 2000 p. 179.\n\n See Faris Russell Kirkland _The French Officer Corps and the Fall of France 1920\u20131940_ PhD Thesis, University of Pennsylvania, 1982 p. 242, also Robert Allan Doughty PhD, 1979. pp. 32\u201367.\n\n See Doughty chapter in Millet and Murray Vol. 2 pp. 58\u201359.\n\n Kirkland PhD p. 26. On the poor performance of French staffs and commanders see Beaufre, pp. 179\u2013212 and also Jean Delmas, _'La manoeuvre general, surprise allemande, d\u00e9fense fran\u00e7aise'_ in Christine Levisse-Touz\u00e9 (ed.) _La Campagne de 1940_ Paris, Tallandier, 2001 pp. 117\u201325.\n\n Melvin p. 160.\n\n Gruppe von Kleist: RH 21-1\/320 Bundesarchiv-Milit\u00e4rarchiv Freiburg, Germany.\n\n See Martin Alexander _After Dunkirk._\n\n Alexander _After Dunkirk_ p. 258.\n\n See Frieser _Blitzkrieg Legend_ p. 315.\n\n E. A. Cohen and J. Gooch _Military Misfortunes. The Anatomy of Failure in War_ New York, Free Press, 1990 p. 220.\n\n Figures quoted in Horne p. 666.\n\n Figures quoted in Frieser _Blitzkrieg Legend_ pp. 318\u201319 and Horne p. 667.\n\n Strachan p. 40.\n\n# Select Bibliography\n\nThis bibliography contains a variety of sources that have been used in the preparation of this book. Not all are quoted from, but those that are can be found in the endnotes.\n\n**Books**\n\n(Place of publication is London unless otherwise stated)\n\n_Blitzkrieg in the West_ After The Battle, Battle of Britain, 1990\n\n_Blitzkrieg In Their Own Words: First-Hand Accounts from German Soldiers 1939\u20131940_ Amber Books, 2011\n\n_The Rise and Fall of the German Air Force (1933 to 1945)_ Air Ministry Pamphlet No. 248 Public Record Office, Kew, 2001\n\nAllcorn, William _The Maginot Line 1928\u201345_ Oxford, Osprey Publishing, 2003\n\nAlexander, Martin S. _The Republic in Danger: Maurice Gamelin and the Politics of the French Defence 1933\u20131940_ Cambridge University Press, Cambridge, 1992\n\nAmersfoort, Herman and Kamphuis, Piet (eds.) _Mei 1940 \u2013 De Strijd op Nederlands grondgebied_ Den Haag, Sdu Uitgevers, 2005\n\nAris, George _The British Fifth Division 1939 to 1945_ The Fifth Division Benevolent Fund, 1959\n\nAshton, H. S. _The Netherlands At War_ George Routledge & Sons Ltd, 1941\n\nAtkin, R. _Pillar of Fire 1940_ Sidgwick & Jackson, 1990\n\nBailey, Gavin J. _The Arsenal of Democracy: Aircraft Supply and the Evolution of the Anglo-American Alliance, 1938\u20131942_ Edinburgh University Press, Edinburgh, 2013\n\nBaldoli, Claudia and Knapp, Andrew _Forgotten Blitzes: France and Italy Under Air Attack 1940\u20131945_ Continuum, 2012\n\nBankwitz, Philip Charles Farwell _Maxime Weygand and Civil-Military Relations_ _in Modern France_ Cambridge, Mass., 1971\n\nBarnett, C. (ed.) _Hitler's Generals_ Weidenfeld and Nicolson, 1989\n\nBarnett, C. et al. _Old Battles and New Defenses: Can We Learn From Military History?_ Brassey's Defence Publishers, 1986\n\nBattistelli, Pierre Paolo _Panzer Divisions: The Blitzkrieg Years, 1939\u201340_ Osprey, Oxford, 2007\n\nBattistelli, Pierre Paolo _Heinz Guderian_ Oxford, Osprey, 2010\n\nBattistelli, Pierre Paolo _Erwin Rommel_ Oxford, Osprey, 2010\n\nBeaufre, A. _1940 The Fall of France_ Cassell, 1967\n\nBeckett, Ian (ed.) _Rommel \u2013 A Reappraisal_ Pen and Sword, Barnsley, 2013\n\nBelgium Ministry of Foreign Affairs _Belgium: The Official Account of What Happened 1939\u201340_ Evans Brothers Ltd, 1941\n\nBell, P. M. H. _A Certain Eventuality... Britain and the Fall of France_ Saxon House, 1974\n\nBelow, N. von, _At Hitler's Side:_ _The Memoirs of Hitler's Luftwaffe Adjutant 1937\u20131945_ Greenhill Books, 2001\n\nBidwell, Shelford and Graham, Dominick _Fire-Power \u2013 British Army Weapons and Theories of War 1904\u20131945_ George Allen & Unwin, 1982\n\nBlatt, Joel (ed.) _The French Defeat of 1940 \u2013 Reassessments_ Berghan Books, New York, 2006\n\nBloch, Marc Strange Defeat \u2013 _A Statement of Evidence Written in 1940_ W.W. Norton, New York, 1968\n\nBond, Brian _France and Belgium 1939\u20131940_ Davis-Poynter, 1975\n\nBond, Brian _British Military Policy between the Two World Wars_ Oxford University Press, Oxford, 1980\n\nBond, Brian (ed.) _Fallen Stars: Eleven Studies of Twentieth Century Military Disasters_ Brassey's, 1991\n\nBond, B. and Taylor, Michael D. (eds.) _Battle for France and Flanders Sixty Years On_ Leo Cooper, Barnsley, 2001\n\nBowyer, Michael J. F. _2 Group RAF \u2013 A Complete History 1936\u20131945_ Faber and Faber, 1974\n\nBrongers, Lieutenant Colonel E. H. _The Battle for the Hague \u2013 1940: The First Great Airborne Operation in History_ Aspekt, Amsterdam, 2006\n\nBruce, Colin John _War on the Ground_ Constable and Company, 1995\n\nBruge, Roger _Faites Sauter la Ligne Maginot!_ Fayard, Paris, 1973\n\nBruge, Roger _On a Livr\u00e9 la Ligne Maginot_ Fayard, Paris, 1973\n\nBryant, Arthur _The Turn of the Tide 1939\u20131943_ Collins, 1957\n\nBuckley, John _Air Power in the Age of Total War_ UCL Press, 1999\n\nBurleigh, Michael _Moral Combat: A History of World War II_ HarperCollins, 2010\n\nButler, Rupert _Hitler's Death's Head Division: SS Totenkopf Divisions_ Pen and Sword, Barnsley, 2004\n\nCaffery, K. _Combat Report \u2013 The RAF and the Fall of France_ Crowood Press, Swindon, 1990\n\nCammaerts, Emile _The Prisoner at Laeken: King Leopold \u2013 Legend and Fact_ Cresset Press, 1941\n\nChapman, Guy _Why France Collapsed_ Cassell and Co., 1968\n\nCherry Niall _Doomed from the Start: The Allied Intervention in Norway 1940_ Helion & Co., Solihull, 2016\n\nChristofferson, Thomas R. with Christofferson, Michael S. _France During World War II: From Defeat to Liberation_ Fordham University Press, New York, 2006\n\nChurchill, Winston S. _The Second World War \u2013 Volume II: Their Finest Hour_ Cassell, 1949\n\nCitino, Robert M. _Quest for Decisive Victory \u2013 From Stalemate to Blitzkrieg in Europe, 1899\u20131940_ University Press of Kansas, Lawrence, Kansas, 2002\n\nCitino, Robert M. _Blitzkrieg to Desert Storm: The Evolution of Operational Warfare_ University of Kansas, Lawrence, Kansas, 2004\n\nCitino, Robert M. _The Path to Blitzkrieg: Doctrine and Training in the German Army, 1920\u201339_ Stackpole Books, Mechanicsburg, PA, 2008\n\nClark, Lloyd _Arnhem \u2013 Jumping the Rhine 1944 and 1945 \u2013 The Greatest Airborne Battle in History_ Headline, 2008\n\nClausewitz, Carl _On War_ (Translated and edited by Michael Howard and Peter Paret) Princeton, NJ, Princeton University Press, 1976 Cohen-Portheim, Paul _The Spirit of Paris_ Batsford, 1937\n\nCohen, E. A. and Gooch, J. _Military Misfortunes. The Anatomy of Failure in War_ New York, Free Press, 1990\n\nCole, Alistair _Franco-German Relations_ Longman, Harlow, Essex, 2001\n\nColville, John _Fringes of Power: Downing Street Diaries 1939\u20131955_ Weidenfeld and Nicolson, 2004\n\nCondell, Bruce and Zabecki, David T. (eds.) _On the Art of War Tuppenf\u00fchrung \u2013 German Army Manual for Unit Command in World War II_ Stackpole Books, Mechanicsburg, PA, 2009\n\nCooksey, Jon _Calais: Fight to the Finish \u2013 May 1940_ Pen and Sword, Barnsley, 1999\n\nCooper, Matthew _The German Air Force 1933\u20131945: An Anatomy of Failure_ Jane's, 1981\n\nCooper, Matthew _The German Army 1933\u20131945: Its Political and Military Failure_ Jane's, 1978\n\nCorbett, Anne and Johnson, Douglas (eds.) _A Day in June: Essays to Commemorate the Sixtieth Anniversary of de Gaulle's appel de 18 juin 1940_ Franco-British Council, 2000\n\nCorum, James S. _The Roots of Blitzkrieg: Hans von Seeckt and German Military Reform_ University of Kansas Press, Lawrence, Kansas, 1994\n\nCorum, James S. _The Luftwaffe: Creating the Operational Air War 1918\u20131940_ University of Kansas Press, Lawrence, Kansas, 1997\n\nCourage, Major Guy _The History of the 15\/19 The King's Royal Hussars 1939\u20131945_ Gale & Polden Ltd, Aldershot, 1949\n\nCrahay, Lieutenant General Albert _L'Arm\u00e9e belge entre les deux guerres_ Louis Musin, Brussels, 1978\n\nCraig, Gordon A. _The Germans_ Penguin, New York, 1991\n\nCr\u00e9mieux-Brilhac, Jean Louis _Les Fran\u00e7ais de l'an 40, Tome I_ Editions Gallimard, Paris, 1990\n\nCr\u00e9mieux-Brilhac, Jean Louis _La Guerre oui ou non? Tome II Ouvriers et soldats_ Editions Gallimard, Paris, 1990\n\nCrozier, Brian _De Gaulle: The Warrior_ Eyre Methuen, 1973\n\nCull, B., Lande, B. and Weiss, H., _Twelve Days in May \u2013 The Air Battles for Northern France and the Low Countries, May 1940_ Grub Street, 1995\n\nD'Astier de la Vigerie, General Fran\u00e7ois _Le ciel n'\u00e9tait pas vide 1940_ Ren\u00e9 Julliard, Paris, 1952\n\nDeighton, Len _Blitzkrieg \u2013 From the Rise of Hitler to the Fall of Dunkirk_ Triad Grafton Books, 1981\n\nDeist, Wilhelm (ed.) _The German Military in the Age of Total War_ Berg, New York, 1985\n\nDe la Gorce, Paul-Marie _The French Army: A Political-Military History_ Weidenfeld and Nicolson, 1963\n\nDe Jong, L. _Holland Fights The Nazis_ The Right Book Club, 1941\n\nDiamond, Hanna _Fleeing Hitler \u2013 France 1940_ Oxford University Press, Oxford, 2007\n\nDildy, Douglas C. _Fall Gelb 1940 (1) Panzer Breakthrough in the West_ Osprey, Oxford, 2014\n\nDildy, Douglas C. _Fall Gelb 1940 (2)_ _Airborne Assault on the Low Countries_ Osprey, Oxford, 2015\n\nDiNardo, R. L. _Germany's Panzer Arm in WWII_ Stackpole Books, Mechanicsburg, PA, 2006\n\nDivine, David The _Nine Days of Dunkirk_ Faber and Faber, 1959\n\nDix, Anthony _The Norway Campaign and the Rise of Churchill 1940_ Pen and Sword, Barnsley, 2014\n\nDoorman, P. L. G. _Military Operations of the Dutch Army 10\u201317th May 1940_ Helion & Co., Solihull, 2005\n\nDoughty, Robert Allan _The Breaking Point \u2013 Sedan and the Fall of France, 1940_ Archon Books, Hampden, Connecticut, 1990\n\nDunstan, Simon _Fort Eben-Emael,_ Osprey, Oxford, 2005\n\nDuroselle, J-B. _Politique Etrang\u00e8re de la France. L'Ab\u00eeme, 1939\u201345_ Imprimerie Nationale, Paris, 1982\n\nDutailly, Lieutenant Colonel Henry _Les probl\u00e8mes de l'arm\u00e9e de terre fran\u00e7aise (1935\u20131939)_ Imprimerie Nationale, Paris, 1980\n\nEllis, L. F. War in France and Flanders, _1939\u20131940_ HMSO, 1954\n\nEvans, M. M. _Fall of France: Act of Daring_ Osprey, Oxford, 2000\n\nEvans, Richard J. _The Third Reich at War: How the Nazis Led Germany from Conquest to Disaster_ Penguin Books, 2009\n\nFalkenau, Rene Paul _Of Eagles and Falcons_ Falcon Press, Newcastle upon Tyne, 2011\n\nFowler, Will _France, Holland, and Belgium 1940_ Ian Allan, Hersham, Surrey, 2002\n\nFraser, David _And We Shall Shock Them: The British Army in the Second World War_ Hodder and Stoughton, 1983\n\nFraser, David _Alanbrooke_ HarperCollins, 1982\n\nFraser, David _Knight's Cross \u2013 A Life of Field Marshal Erwin Rommel_ HarperCollins, 1993\n\nFrench, David _Raising Churchill's Army \u2013 The British Army and the War against Germany 1919\u20131945_ Oxford University Press, Oxford, 2000\n\nFrieser, Karl-Heinz _The Blitzkrieg Legend \u2013 The 1940 Campaign in the West_ Naval Institute Press, Annapolis, 2005\n\nGalland, Adolf _The First and Last_ Methuen and Co., 1955\n\nGamelin, General Maurice _Servir_ 3 Vols Librairie Plon, Paris, 1946\u20137\n\nGardner, W. J. R. _The Evacuation from Dunkirk: Operation Dynamo 26 May to 4 June 1940_ Frank Cass, 2000\n\nGermond, G. and T\u00fcrk, H. (eds.) _A History of Franco-German Relations in Europe: from 'Hereditary Enemies' to Partners_ Palgrave Macmillan, New York, 2008\n\nGilbert, Adrian _PoW: Allied Prisoners of War 1939\u20131945_ John Murray, 2006\n\nGilbert, Martin _Churchill \u2013 A Life_ Heinemann, 1991\n\nGooch, John _Decisive Campaigns of the Second World War_ Frank Cass, 1990\n\nGoutard, A. _The Battle for France 1940_ Washburn Inc., New York, 1959\n\nGribble, Philip _The Diary of a Staff Officer, Air Intelligence Liaison Officer, at Advanced Headquarters North B.A.F.F., 1940_ Methuen & Co., 1941\n\nGuderian, H. _Achtung \u2013 Panzer!_ Cassel, 1992\n\nGuderian, H. _Panzer Leader_ Penguin, 2000\n\nGuderian, H. (ed.) _Mit den Panzern in Ost und West_ Volk und Reich Verlag, Berlin, Prague and Vienna, 1942\n\nGudmundsson, Bruce I. _On Armor_ Greenwood Publishing, Westport, Conn., 2004\n\nGunsburg, Jeffrey A. _Divided and Conquered: The French High Command and the Defeat of the West, 1940_ Greenwood Publishing, Westport, Conn., 1979\n\nHallion, Richard P. _The History of Battlefield Air Attack 1911\u20131945_ Airlife, Shrewsbury, 1989\n\nHamilton, Nigel _Monty: The Making of a General 1887\u20131942_ Hamish Hamilton, 1981\n\nHarris, J. P. _Men, Ideas and Tanks \u2013 British Military Thought and Armoured Forces, 1903\u20131939_ Manchester University Press, Manchester, 1995\n\nHart, Russell A. _Guderian: Panzer Pioneer or Myth Maker?_ Pentagon Press, New Delhi, 2010\n\nHaskew, Michael E. _De Gaulle: Lessons in Leadership from a Defiant General_ Palgrave Macmillan, New York, 2011\n\nHastings, Max _All Hell Let Loose_ HarperPress, 2012\n\nHauteclet, Georges _Rommel and Guderian against the Belgian Chasseurs Ardennais: The Combats at Chabrehez and Bodange, 10 May 1940_ West Chester, Ohio, 2003\n\nHayward, J. _Myths and Legends of the Second World War_ Sutton, Stroud, 2003\n\nHinsley, F. H. _British Intelligence in the Second World War Vol. I_ HMSO, 1979\n\nHistorique de l'Arm\u00e9e (Official History) _Les Grandes Unit\u00e9s fran\u00e7aises: Historiques succincts_ 5 Vols Imprimerie Nationale, Paris, 1967\u20131975\n\nHittle, J. D. _The Military Staff \u2013 Its_ _History and Development_ Stackpole, Harrisburg, Pennsylvania, 1961\n\nHooton, E. R. _Luftwaffe at War: Blitzkrieg in the West 1939\u20131940_ Ian Allan, Hersham, Surrey 2007\n\nHooton, E. R. _Phoenix Triumphant: The Rise and Rise of the Luftwaffe_ Arms and Armour Press, London, 1994\n\nHorne, Alistair _To Lose a Battle \u2013 France 1940_ Penguin Books, 1990\n\nHorne, Alistair _The French Army and Politics, 1870\u20131970_ Macmillan Press, 1984\n\nHouse, Jonathan M. _Combined Arms Warfare in the Twentieth Century_ University Press of Kansas, Lawrence, Kansas, 2001\n\nHughes, Judith M. _To The Maginot Line \u2013 The Politics of French Military Preparation in the 1920s_ Harvard University Press, Cambridge, Massachusetts, 2006\n\nJackson, Julian _The Fall of France: The Nazi Invasion of 1940_ Oxford University Press, Oxford, 2003\n\nJackson, Julian _Dunkirk: The British Evacuation, 1940_ Arthur Barker, 1976\n\nJenkins, Roy _Churchill_ Pan Macmillan, 2012\n\nJoseph-Maginot, Marguerite _He Might Have Saved France: The Biography of_ _Andr\u00e9 Maginot_ Doubleday, Doran and Co., New York, 1941\n\nKaufmann, J. E. and Kaufmann, H. W. et al _The Maginot Line \u2013 History and Guide_ Pen and Sword Military, Barnsley, 2011\n\nKaufmann, J. E. and Kaufmann, H. W. _Fortress France: The Maginot Line and French Defenses in World War II_ Stackpole Books, Mechanicsburg, PA, 2006\n\nKaufmann, J. E. and Kaufmann, H. W. _Hitler's Blitzkrieg Campaigns \u2013 The Invasion and Defense of Western Europe, 1939\u20131940_ Da Capo Press, Cambridge, MA, 1993\n\nKeegan, J. (ed.) _Churchill's Generals_ Weidenfeld and Nicolson, 1991\n\nKeegan, J. _Intelligence in War \u2013 Knowledge of the Enemy From Napoleon to Al-Qaeda_ Hutchinson, 2003\n\nKershaw, I. _Hitler 1889\u20131936 Hubris_ Allen Lane, 1998\n\nKershaw, I. _Hitler 1936\u20131945 Nemesis_ Allen Lane, 2000\n\nKershaw, Robert _Tank Men \u2013 The Human Story of Tanks at War_ Hodder and Stoughton, 2008\n\nKier, Elizabeth _Imagining War: French and British Military Doctrine between the Wars_ Princeton University Press, Princeton, NJ, 1997\n\nKiesling, Eugenia C. _Arming Against Hitler \u2013 France and the Limits of Military Planning_ University of Kansas, Lawrence, Kansas, 1996\n\nKing, Anthony _The Combat Soldier: Infantry Tactics and Cohesion in the Twentieth and Twenty-First Centuries_ , Oxford University Press, Oxford, 2013\n\nKnox, MacGregor _Mussolini Unleashed, 1939\u20131941: Politics and Strategy in Fascist Italy's Last War_ Cambridge University Press, Cambridge, 1982\n\nKossmann, E. H. _The Low Countries 1780\u20131940_ Oxford, Clarendon Press, 1978\n\nKurowski, Franz _Panzer Aces \u2013 German Tank Commanders of WWII_ Stackpole Books, Mechanicsburg, PA, 2004\n\nLevine, Joshua _Forgotten Voices \u2013 Dunkirk_ Ebury, 2010\n\nLevisse-Touz\u00e9, Christine (ed.) _La Campagne de 1940_ Paris, Tallandier, 2001\n\nLiddell Hart, B. H. _The Other Side of the Hill_ Cassell, 1948\n\nLiddell Hart, B. H. _The German Generals Talk_ New York, W. Morrow, 1948\n\nLiddell Hart, B. H. (ed.) _The Rommel Papers_ Collins, 1953\n\nLongden, Sean _Dunkirk \u2013 The Men They Left Behind_ Constable, 2008\n\nLuck, Hans von _Panzer Commander \u2013 The Memoirs of Colonel Hans von Luck_ Cassell, 1989\n\nLukacs, John _Five Days in London May 1940_ Yale, Nota Bene, 2001\n\nLyet, Commandant Pierre _La Bataille De France (Mai-Juin 1940)_ Payot, Paris, 1947\n\nMaass, Walter B. _The Netherlands at War: 1940\u20131945_ Abelard\u2013Schuman, Amsterdam, 1970\n\nMahlke, Helmut _Memoirs of a Stuka Pilot_ Frontline Books, Barnsley, 2013\n\nMallinson, Vernon _Belgium_ Ernest Benn, 1969\n\nMartin, Benjamin F. _France and the Apr\u00e8s Guerre, 1918\u201324: Illusions and Disillusionment_ Louisiana State University Press, Baton Rouge, 1999\n\nMary, Jean-Yves, Hohnadel, Alain and Sicard, Jacques _Hommes et Ouvrages de la Ligne Maginot, Tome 3_ Paris, Histoire & Collections, 2003\n\nMaurois, Andr\u00e9 _The Battle of France_ Bodley Head, 1940\n\nMay, Ernest R. _Strange Victory: Hitler's Conquest of France_ Hill and Wang, New York, 2000\n\nMay, Ernest R. (ed.) _Knowing One's Enemies_ Princeton University Press, Princeton, NJ, 2014\n\nMead, Richard _Churchill's Lion: A Biographical Guide to the Key British Generals of World War II_ Spellmount, Stroud, Gloucestershire, 2007\n\nMearsheimer, John J. _Conventional Deterrence_ University Press, Ithaca, New York and London: Cornell, 1983\n\nMellenthin, Major General F. W. _von Panzer Battles: A Study of the Employment of Armor in the Second World War_ Ballantine Books, New York, 1971\n\nMelvin, Mungo _Manstein \u2013 Hitler's Greatest General_ Weidenfeld and Nicolson, 2010\n\nMessenger, Charles _The Last Prussian: A Biography of Field Marshal Gerd von Rundstedt 1875\u20131953_ Pen and Sword, Barnsley, 2012\n\nMichel, Henri (ed.) _La Lib\u00e9ration de la France_ CNRS, Paris, 1976\n\nMillett, Allan R. and Murray, Williamson _Military Effectiveness Volume 2: The Interwar Years_ Cambridge University Press, New York, 2010\n\nMillett, Allan R. and Murray, Williamson _Military Effectiveness Volume 3: The Second World War_ Cambridge University Press, New York, 2010\n\nMitcham, Samuel W. Jnr _Hitler's Field Marshals and their Battles_ Grafton Books, 1989\n\nMitcham, Samuel W. Jnr _Panzer Legions \u2013 A Guide to the German Army Tank Divisions and their Commanders_ Stackpole Books, Mechanicsburg, PA, 2007\n\nMoore, Mortimer William _Free France's Lion \u2013 The Life of Philippe Leclerc, de Gaulle's Greatest General Casemate_ , Philadelphia US and Newbury UK, 2011\n\nMore, Charles _The Road to Dunkirk \u2013 The British Expeditionary Force and the Battle of the Ypres-Comines Canal, 1940_ Frontline Books, Barnsley, 2013\n\nMurray, Williamson, Knox, Macgregor and Bernstein, Alvin (eds.) _The Making of Strategy: Rulers, States and War_ Cambridge University Press, Cambridge, 1994\n\nMurray, Williamson _Luftwaffe: Strategy for Defeat_ Quantum Publishing, 2000\n\nNaveh, Shimon _In Pursuit of Military Excellence: The Evolution of Operational Theory_ Frank Cass, Portland, Oregon, 1997\n\nNeave, Airey _The Flames of Calais \u2013 A Soldier's Battle 1940_ Leo Cooper, Hodder and Stoughton, 1972\n\nNeitzel, S\u00f6nke and Wezer, Harald _Soldaten \u2013 On Fighting and Killing and Dying_ Simon and Schuster, 2012\n\nNewton, Gerald _The Netherlands: An Historical and Cultural Survey 1795\u20131977_ Ernest Benn, 1978\n\nNord, Philip _France 1940: Defending the Republic_ Yale University Press, New Haven and London, 2015\n\nOusby, Ian _Occupation \u2013 The Ordeal of France 1940\u20131944_ Pimlico, 1999\n\nOvery, Richard _Why the Allies Won_ W. W. Norton and Co., New York, 1996\n\nOvery, Richard _The Origins of the Second World War_ Routledge, 2008\n\nOvery, Richard with Andrew Wheatcroft _The Road to War_ Vintage, 2009\n\nPeden, G. C. _British Rearmament and the Treasury, 1932\u20131939_ Scottish Academic Press, Edinburgh, 1979\n\nPettibone, Charles D. _The Organization and Order of Battle of Militaries in World War II: Vol. I \u2013 Germany_ (2006); _Vol. II \u2013 The British Commonwealth_ (2006); _Vol. VI \u2013 Italy and France_ (2010) and _Vol. X \u2013 The Overrun and Neutral Nations of Europe and Latin American Allies_ (2014) Trafford Publishing, Victoria, Canada\n\nPorch, Douglas _The Path to Victory: The Mediterranean Theater in World War II (1st ed.)_ Farrar, Straus and Giroux, New York, 2005\n\nPosen, Barry R. _The Sources of Military Doctrine: France, Britain and Germany Between the Wars_ Cornell University Press, New York, 1984\n\nPowaski, Ronald E. _Lightning War: Blitzkrieg in the West, 1940._ John Wiley, Hoboken, NJ, 2003\n\nPreston, Adrian (ed.) _General Staffs and Diplomacy Before the Second World War_ Croom Helm, 1978\n\nRamsden, John _Don't Mention the War: The British and the Germans Since 1890_ Little, Brown, 2006\n\nReid, Walter _Churchill 1940\u20131945: Under Friendly Fire_ Birlinn, Edinburgh, 2008\n\nResende-Santos, Jo\u00e3o _Neorealism, States, and the Modern Mass Army_ Cambridge University Press, Cambridge, 2007\n\nReynaud, Paul _In the Thick of the Fighting 1930\u20131945_ Cassell, 1955\n\nRiboud, Jacques _Souvenirs D'Une Bataille Perdue (1939\u201340)_ Fran\u00e7ois-Xavier de Guibert, Paris, 2006\n\nRichards, Denis _Royal Air Force Vol. I, The Fight At Odds_ HMSO, 1953\n\nRisser, Nicole Dombrowski _France Under Fire: German Invasion, Civilian Flight, and Family Survival during World_ _War II_ Cambridge University Press, Cambridge, 2012\n\nRobb, Graham _The Discovery of France_ Picador, 2007\n\nRomanych, M. and Rupp, M. _Maginot Line 1940: Battles on the French Frontier_ Osprey, Oxford, 2010\n\nRosbottom, Ronald _When Paris Went Dark: The City of Light Under German Occupation 1940\u201344_ John Murray, 2014\n\nRossiter, Mike _I Fought at Dunkirk_ Random House, 2012\n\nRothbrust, Florian K. _Guderian's XIXth Panzer Corps and The Battle of France \u2013 Breakthrough in the Ardennes, May 1940_ Praeger, Westport, CT, 1990\n\nSadoul, Georges _Journal de Guerre_ Les Editeurs Fran\u00e7ais R\u00e9unis, Paris 1977\n\nSartre, Jean-Paul _War Diaries: Notebooks from a Phoney War 1939\u201340_ Verso Editions, 1984\n\nSchneider, Wolfgang _Panzer Tactics \u2013 German Small Unit Armor Tactics in World War II_ Stackpole Books, Mechanicsburg, PA, 2005\n\nSchreiber, Gerhard, Stegemann, Bernd and Vogel, Detlef _Germany and the Second World War, Volume III: The Mediterranean, South-East Europe, and North Africa 1939\u20131942_ Oxford, Clarendon Press, 1995\n\nSchwarzschild, Leopold _World in Trance_ Hamish Hamilton, 1943\n\nSebag-Montefiore, Hugh _Dunkirk \u2013 Fight To The Last Man_ Viking, 2006\n\nShay, Robert Paul Jnr. _British Rearmament in the Thirties_ Princeton University Press, Princeton, NJ, 1977\n\nShepley, Nick _Darkest Hour, Finest Hour: Norway, Dunkirk and the Battle of Britain \u2013 Britain at War: Part One_ Andrews UK, Luton, 2013\n\nShirer, William L. _The Collapse of The Third Republic: An Inquiry into the Fall_ _of France in 1940_ New York, Simon and Schuster, 1960\n\nShort, Neil _The F\u00fchrer's Headquarters: Hitler's Command Bunkers 1939\u201345_ Osprey, Oxford, 2010\n\nShowalter, Dennis _Hitler's Panzers \u2013 The Lightning Attacks that Revolutionized Warfare_ Penguin, New York, 2009\n\nSmalley, Edward _The British Expeditionary Force, 1939\u201340_ Palgrave Macmillan, 2015\n\nSmith, M. A. _British Air Strategy Between the Wars_ Oxford University Press, Oxford, 1984\n\nSpeer, Albert _Inside the Third Reich_ Simon and Schuster, New York, 1997\n\nSteiger, Rudolf _Panzertaktik im Spiegel deutscher Kriegstagerb\u00fccher 1939 bis 1941_ Verlag Robach, Freiburg, 1973\n\nStewart, Geoffrey _Dunkirk and the Fall of France_ Pen and Sword, Barnsley, 2008\n\nStone, David _Twilight of the Gods: The Decline and Fall of the German General Staff in World War II_ Conway, 2011\n\nStrachan, Hew _European Armies and the Conduct of War_ Unwin Hyman, 1983\n\nStrachan, Hew _The Direction of War: Contemporary Strategy in Historical Perspective_ Cambridge University Press, Cambridge, 2013\n\nStrohn, Matthias _The German Army and the Defence of the Reich: Military Doctrine and the Conduct of the Defensive Battle 1918\u20131939_ Cambridge University Press, Cambridge, 2011\n\nThomas, Nigel _Foreign Volunteers of the Allied Forces, 1939\u201345_ Osprey, 1991\n\nThompson, Julian _Dunkirk \u2013 Retreat To Victory_ Sidgwick & Jackson 2008\n\nTombs, Robert and Tombs, Isabelle _That Sweet Enemy: Britain and France, The History of a Love-Hate Relationship_ Heinemann, 2006\n\nTombs, Robert and Chabal, Emile (eds.) _Britain and France In Two World Wars \u2013 Truth, Myth and Memory_ Bloomsbury, 2013\n\nTooze, Adam _Wages of Destruction_ Allen Lane, 2006\n\nTrevor-Roper, H. R. (ed.) _Hitler's War Directives_ Sidgwick & Jackson, 1964\n\nTucker, Spencer (ed.) _Encyclopedia of World War II \u2013 A Political, Social and Military History_ Santa Barbara, California, 2005\n\nTurner, E. S. _The Phoney War on the Home Front_ Michael Joseph, 1961\n\nVillahermosa, Gilberto _Hitler's Paratrooper \u2013 The Life and Battles of Rudolf Witzig_ Frontline Books, 2010\n\nWarmbrunn, Werner _The Dutch under German Occupation_ Stanford University Press, California, 1963\n\nWarmbrunn, Werner _The German Occupation of Belgium_ Peter Lange, New York, 1993\n\nWarner, Philip _Battle for France, 1940_ Simon and Schuster, 1990\n\nWeber, Eugen _The Hollow Years \u2013 France in the 1930s_ Sinclair-Stevenson, 1995\n\nWeingartner, James J. _Hitler's Guard: The Story of the Leibstandarte SS Adolf Hitler 1933\u20131945_ Southern Illinois University Press, Carbondale and Edwardsville, 1968\n\nWerth, Alexander _The Last Days of Paris: A Journalist's Diary_ Hamish Hamilton, 1940\n\nWeygand, J. _The Role of General Weygand: Conversations with his Son_ Eyre and Spottiswoode, 1948\n\nWeygand, M. _Recalled to Service_ Heinemann, 1952\n\nWilliams, John _The Ides of May: The Defeat of France May-June 1940_ Constable, 1968\n\nWilliamson, Gordon _Loyalty is My Honour_ BCA, 1995\n\nWiser, William _The Twilight Years \u2013 Paris in the 1930s_ Carroll and Graf Publishers Inc., New York, 2000\n\nWright, Patrick Tank \u2013 _The Progress of a Monstrous War Machine_ Faber and Faber, 2000\n\nYoung, R. J. _In Command of France: French Foreign and Military Planning 1933\u20131940_ Harvard University Press, Cambridge, Mass., 1978\n\nYoung, Desmond _Rommel_ Collins, 1950\n\nZaloga, Steven J. and Madej, Victor _The Polish Campaign 1939_ Hippocrene Books Inc., New York, 1985\n\n**Chapters and Articles**\n\nAlexander, Don W. 'Repercussions of the Breda Variant' in _French Historical Studies_ Vol. 8 No. 3 Spring 1974\n\nAlexander, Martin S. 'After Dunkirk: The French Army's Performance against \"Case Red\", 25 May to 25 June 1940' in _War in History_ Vol. 14 No. 2 April 2007\n\nAlexander, Martin S. 'The Fall of France' in _Journal of Strategic Studies_ Vol. 13 No. 1 1990\n\nAusems, Andr\u00e9 'The Netherlands Military Intelligence Summaries 1939\u201340 and the Defeat in the Blitzkrieg of May 1940' in _Military Affairs,_ Vol. 50 No. 4 October 1986\n\nBrausch, Gerd 'Sedan 1940. Deuxi\u00e8me Bureau and Strategische \u00dcberraschung' _Milit\u00e4rgeschichtliche Zeitschrift_ Vol. 2. No. 2 1967\n\nCairns, J. C. 'Some Recent Historians and the \"Strange Defeat\" of 1940' in _Journal of Modern History_ Vol. 46 No. 1 March 1974\n\nCinto, Robert 'The War Hitler Won: The Battle for Europe, 1939\u20131941' in _Journal of Military and Strategic Studies_ Vol. 14 Issue 1 Fall 2011\n\nDoughty, Robert A. 'French Anti-tank Doctrine, 1940 The Antidote that Failed' in _Military Review_ 56 May 1976\n\nDoughty, Robert A. 'Myth of the Blitzkrieg' in _Strategic Studies Institute of the US Army War College_ 1998\n\nFodor, M. W. 'The Blitzkrieg in the Low Countries' in _Foreign Affairs_ Vol. 19 No. 1 October 1940\n\nFrench, David 'Doctrine and Organization in the British Army, 1919\u201332' in _The Historical Journal_ Vol. 44 No. 52 June 2001\n\nFrieser, Karl-Heinz 'Panzer Group Kleist and the Breakthrough in France, 1940' in Krause, Michael D. and Phillips, Cody R. (eds.), _Historical Perspectives of the Operational Art_ Center for Military History, Washington D.C. 2005\n\nGunsburg, J. A. 'The Battle of the Belgian Plain, 12\u201314 May 1940' in _Journal of Military History_ 56 April 1992\n\nGunsburg, J. A. 'The Battle of Gembloux, 14\u201315 May 1940' _Journal of Military History_ 64 January 2000\n\nHarvey, A. D. 'The French Arm\u00e9e de l'Air in May-June 1940: A Failure of Conception' in _Journal of Contemporary History_ Vol. 25 No. 4 October 1990\n\nHoward, Michael 'The Use and Abuse of Military History' in _Royal United Services Institute Journal_ No. 107 February 1962\n\nJackson, Peter 'Recent Journeys Along The Road Back to France, 1940' in _The Historical Journal_ Vol. 39 No. 2 June 1996\n\nJackson, Peter 'French Intelligence and Hitler's Rise to Power' in _The Historical Journal_ Vol. 41 No. 3 September 1998\n\nJacobson, Jon 'Strategies of French Foreign Policy after World War I' in _The Journal of Modern History_ Vol. 55 No. 1 March 1983\n\nKirkland, Lieutenant Colonel Faris R. 'The French Air Force in 1940: Was It Defeated by the Luftwaffe or by Politics?' in _The Air University Review_ September-October 1985\n\nLochner, Louis P. 'The Blitzkrieg in Belgium: A Newsman's Eyewitness Account' in _The Wisconsin Magazine of History_ Vol. 50 No. 4 Summer 1967\n\nM.B. 'The Strategic Position of Holland and Belgium' in _Bulletin of International News_ Vol. 16 No. 24 December 1939\n\nMarks, Sally 'The Myths of Reparations' in _Central European History_ , Vol. 11 No. 3 September 1978\n\nMason, Henry L. 'War Comes to the Netherlands: September 1939 to May 1940' in _Political Science Quarterly_ Vol. 78 No. 40 December 1963\n\nMurray, Williamson 'The German Response to Victory in Poland: A Case Study in Professionalism' in _Armed Forces and Society_ Vol. 7 1981\n\nO'Connell, Colonel J. J. 'The Maginot Line: Dream or Reality?' in _Studies: An Irish Quarterly Review_ Vol. 29 No. 116 December, 1940\n\nPorch, D. 'French Intelligence and the Fall of France, 1930\u201340' in _Intelligence and National Security_ Vol. 4 No. 1 January 1989\n\nPorch, Douglas 'Military \"Culture\" and the Fall of France in 1940': A Review Essay in _International Security_ Vol. 24 No. 4 Spring 2000\n\nPorch, Douglas 'Why Did France Fall?' in _Quarterly Journal of Military History_ Spring, 1990\n\nPugsley, Christopher 'We Have Been Here Before: The Evolution of Decentralized Command in the British Army 1905\u20131989' in _Sandhurst Occasional Papers_ No. 9 Central Library Royal Military Academy Sandhurst, Camberley, Surrey 2001\n\nRajevs, Colonel Igors 'The French Army in the Interwar Period' in _Baltic Security & Defence Review_ Vol. 11 Issue 2 2009\n\nSica, Emanuelle 'June 1940: The Italian Army and the Battle of the Alps' in _Canadian Journal of History,_ [S.l.], v. 47, n. 2, May 2014\n\nStrachan, Hew 'Operational Art and Britain, 1909\u20132009', in Olsen, John Andreas and van Creveld, Martin (eds.) _The Evolution of Operational Art: From Napoleon to the Present_ Oxford University Press, Oxford 2011\n\nThomas, Martin 'French Economic Affairs and Rearmament: The First Crucial Months, June to September 1939' in _Journal of Contemporary History_ Vol. 27 No. 4 October 1992\n\nWilson, Patrick 'Dunkirk: Victory or Defeat?' in _History Today_ History Review Issue 37 September 2000\n\nYoung, Robert J. 'The Strategic Dream: French Air Doctrine in the Inter-War Period, 1919\u20131939' in _Journal of Contemporary History,_ January 1974\n\n**Magazine Articles**\n\nFrance: Death & Crisis _Time_ Monday, January 18, 1932\n\nFrance: The Good Grey General _Time_ Monday, August 14, 1939\n\n'Blitzkriegen' _Time_ Monday, September 25, 1939\n\n**Published Documents, Memoirs, Journals and Diaries**\n\nAlanbrooke, Lord _War Diaries of Field Marshal Lord Alanbrooke_ Danchev, A. and Todman, D. (eds.), Weidenfeld and Nicolson, 2001\n\nBalck, Hermann _Ordnung im Chaos. Erinnerungen 1893\u20131948_ Osnabr\u00fcck Biblio Verlag, 1980\n\nBarlone, D. _A French Officer's Diary 23 August 1939 to 1 October 1940_ Cambridge: At the University Press New York: The Macmillan Company, 1943\n\nBock, Generalfeldmarschall Fedor von _The War Diary 1939\u20131945_ Schiffer Publishing Ltd, Atglen, PA, 1996\n\nD'Astier de la Vigerie, General Fran\u00e7ois _Le ciel n'\u00e9tait pas vide_ Paris, Ren\u00e9 Julliard, 1952\n\nDe Gaulle, Charles _M\u00e9moires de Guerre_ (3 Vols) Plon, Paris, 1954\n\nDetwiler, Donald S. (ed.) _World War II German Military Studies: A Collection of 213 Special Reports on the Second World War Prepared by Former Officers of the Wehrmacht for the United States Army Vol. 11 Part IV_ Garland Publishing, New York and London, 1979\n\nGrandsard, General C. _Le 10\u00e8me Corps d'Arm\u00e9e dans la Bataille, 1939\u20131940_ Paris, Berger-Levrault, 1949\n\nHalder, General Franz _The Halder War Diary 1939\u20131942_ Burdick, Charles and Jacobsen, Hans-Alfred (eds.) Greenhill Books, 1988\n\nIronside, Sir Edmund _Time Unguarded: The Ironside Diaries, 1937\u20131940_ Macleod, Colonel Roderick and Kelly, Denis (eds.) Constable, London, 1962\n\n_The Memoirs of Field Marshal Wilhelm Keitel_ Gorlitz, Walter (ed.) Cooper Square Press, New York, 2000\n\nManstein, General Erich von _Lost Victories_ Chicago, Henry Regnery Co., 1958\n\nPownall, Lieutenant General Sir Henry, _Diaries of Lieutenant General Sir Henry Pownall, Vol. I 'Chief of Staff'_ Bond, B. (ed.) Leo Cooper, 1972\n\nPr\u00fcller, Wilhelm _Diary of a German Soldier_ Faber and Faber, London, 1963\n\nSpears, Major General Sir Edward _Assignment to Catastrophe, Vol. 1 Prelude to Dunkirk July 1939 to May 1940_ A. A. Wyn Inc., New York, 1954\n\nSpears, Major General Sir Edward _Assignment to Catastrophe, Vol. 2 The Fall of France June 1940_ A. A. Wyn Inc, New York, 1954\n\n**Unpublished Sources**\n\nDoughty, Robert Allan _The Evolution of French Army Doctrine 1919\u20131939_ PhD Thesis, University of Kansas, 1979\n\nKirkland, Lieutenant Colonel Faris R. _The French Officer Corps and the Fall of France 1920\u20131940_ PhD Thesis, University of Pennsylvania, 1982\n\nParker, Robert _'O\u00f9 est la Masse de Manoeuvre?': Maurice Gamelin and the Lessons of Blitzkrieg in Poland_ MA Dissertation, Concordia University Montreal, Quebec, Canada, 2013\n\nMay, Ernest R. _The Intelligence Process: The Fall of France, 1940_ Kennedy School of Government\n\nPierre Roussel diary 1939 and 1940. Roussel family archive.\n\n**Interviews and Correspondence**\n\nGermany\n\nRalf Buckholtz Interview 16 April 2011\n\nPeter Gerber for information about his father, Karl Gerber 1 May 2014\n\nOtto Gull Interview 18 and 19 July 2011\n\nPaul Henkelmann Letter 1 May 2013\n\nJan Hoch Letter 13 January 2013\n\nMarc Leicht Letter 10 November 2014\n\nHans Liebler Interview June 2014\n\nPeter Meier Interview 21 April 2013\n\nGerd Reismann Interview 6 May 2011\n\nGerd Ritter Interview 16 April 2013\n\nAlbert Winkler Interview 16 March 2013 and Email 20 December 2014\n\nThe Netherlands\n\nLisa Hanson Email 8 April 2013\n\nConrad Houtkooper Letter 21 May 2013\n\nMax Meers Email 20 June 2012\n\nPeter Meier Interview 28 July 2013\n\nJoop van Praag Email 1 August 2013\n\nBelgium\n\nArnaud Bauwens Interview 21 November 2014\n\nFrance\n\nDavid Boyer Interview 18 February 2015\n\nBruno Cantel Letter 12 June 1995\n\nJulianne Deros Letter 16 and 23 November 2012\n\nMaxime Dufour Interview 19 April 2013\n\nDiane Lebeouf Interview 2 June 1996\n\nDominic Lemaire Letter 1 May 1994\n\nCharles-Michel L\u00e9p\u00e9e Interview 16 November 2014\n\nDaniel Le Roy Telephone Interview 28 March 2013\n\nEmmanuel Marcel Interview 20 May 2012\n\nDominic Royer Letter 8 June 2012\n\nDavid Schneider Interview 18 June 2011\n\nMichel Sigaut Interview 15 January 2013\n\nSerge Winter Letter 4 May 2013\n\nBritain\n\nDavid Blundell Telephone Interview 1 May 2014\n\nAndrew Hardcastle Interview 2 July 2013\n\nDouglas Manning Interview 12 November 2014\n\nPeter Mansfield Interview 30 August 2014\n\nTom Reed Interview 12 May 2014\n\nRab Riley Interview 12 October 1991\n\nJohn Rigson Email 2 December 2012\n\nErnie Smith Telephone Interview 28 November 2014\n\nPeter Vaux Letter 20 February 1996\n\nAlexander Wilkins Interview 12 August 2011\n\n**Author's Archive**\n\n_The War Illustrated_ 21 October 1939 Vol.1 No. 6\n\n_The Guardian_ Obituary 17 January 2005\n\nBrigadier Peter Vaux _Arras: An EyeWitness Account_\n\nJulia Lapointe Diary 1940\n\nDavid McCullen Diary 6 March to 20 August 1940\n\nFriedrich Treitler Diary Extracts (in a letter from Paul Treitler, his son, dated 16 May 2013)\n\nF\u00fchrer Directive 2: Hostilities in the West (September 3, 1939)\n\nF\u00fchrer Directive 6: Plans for Offensive in the West (October 9, 1939)\n\nF\u00fchrer Directive 7: Preparations for Attack in the West (October 18, 1939)\n\nF\u00fchrer Directive 8: Further Preparations for Attack in the West (November 20, 1939)\n\nF\u00fchrer Directive 10: Concentration of Forces for 'Undertaking Yellow' (January 19\u2013February 18, 1940)\n\nF\u00fchrer Directive 11: The Offensive in the West (May 14, 1940)\n\nF\u00fchrer Directive 12: Prosecution of the Attack in the West (May 18, 1940)\n\nF\u00fchrer Directive 13: Next Object in the West (May 24, 1940)\n\nF\u00fchrer Directive 14: Continuation of the Offensive in France (June 8, 1940)\n\nF\u00fchrer Directive 15: Advance on the Loire (June 14, 1940)\n\n'Die deutsche Mai offensive in den Tagen von 17. Bis 23. Mai 1940', in _Milit\u00e4r-Wochenblatt_ 124, No. 48, May 31, 1940\n\nHans von Seeckt 'Grunds\u00e4tzemoderner Landesverteidigung' in _Gedankeneines Soldaten_ Leipzig K.F. Koehler, 1935\n\nTranslation of Taped Conversation with General Hermann Balck, 12 January 1979 Battelle Colombus Laboratories, Tactical Technology Center, 505 King Avenue, Columbus, Ohio 43201\n\n**Bundesarchiv-Milit\u00e4rarchiv, Freiburg, Germany**\n\nWehrmachtf\u00fchrungsstab (Wehrmachtf\u00fchrungsamt) RW 4 Series\n\nAllgemeines Wehrmachtamt: RW 6 Series\n\nWehrwirtschafts \u2013 und R\u00fcstungsamt (Wehrwirtschaftsamt): RW 19 Series\n\nGeneralstab des Heeres: RH 1 Series\n\nHeeresgruppe A; B and C: RH 19I; 19II and 19III Series\n\n4. Arm\u00e9e; 12. Arm\u00e9e and 18. Arm\u00e9e: RH 20-4, 20-12 and 20-18 Series\n\nGruppe von Kleist; Panzerkorps Guderian; Gruppe Hoth; Panzerkorps Hoepner; Panzerkorps Reinhardt: RH 21-1; 21-2; 21-3; 21-4 and 21-41 Series\n\n1; 2; 3; 4; 5; 6; 7; 10.PzDivs: RH 27-1; 27-2; 27-3; 27-4; 27-5; 27-6; 27-7 and 27-10 Series\n\nInfanterieregiment Grossdeutschland: RH 37 Series\n\nVIII. Fliegerkorps: RL 8 Series\n\n**Central Library, The Royal Military Academy Sandhurst, Camberley, Surrey**\n\n_The Belgian Tragedy: The Royal Question_ Report Drawn Up By Commission of Information Constituted By H.M. King Leopold III on The 14th July 1946\n\n**Combined Arms Research Library, Digital Library, Fort Leavenworth, Kansas, USA**\n\nThe Private War Journal of General Oberst Franz Halder \u2013 Chief of the General Staff of the Supreme Command of the German Army (OKH): 14 August 1939 to 10 September 1942 Vol. I: The Polish Campaign Part I (N-16845-A); Vol. II The Polish Campaign Part II (N-16845-B); Vol. III The First Winter 7 December 1939 to 9 May 1940 (N-16845-C); Vol. IV The Campaign in France 10 May 1940 to 30 October 1940 (N-16845-D)\n\n**Liddell Hart Centre for Military Archives (LHCMA)**\n\nBRIDGEMAN 1\/1 Lord Gort's Despatches\n\n**The National Archives, Kew, London**\n\nTNA, CAB 53\/37, Memoranda Nos. 698 (Revise) CID, COS Sub-committee, _Military Implications of German Aggression against Czechoslovakia_ 28 March 1938\n\nTNA, CAB 85\/16, MR (J) (40) (5) 2, Allied Military Committee, _The Major Strategy of the War. Note by the French Delegation_ , 11 April 1940\n\nTNA, FO 371\/22915, C 1503\/30\/17, COS 833, _The Strategic Position of France in a European War_\n\nTNA, CAB 16\/209 SAC\/4th Meeting, CID, Strategic Appreciations Sub-Committee\n\nTNA, CAB 53\/37, COS 698 (revise), _Military Implications of German Aggression against Czechoslovakia_ , 28 March 1938\n\nTNA, FO 371\/22922, C 1545\/281\/17 minute by Sir Robert Vansittart, on Cabinet Paper (CP) 40 (38), _Staff conversations with France and Belgium_ , 10 February 1938\n\nTNA, CAB 23\/100, Cab 36 (39), Meeting of the Cabinet, 5 July 1939\n\nTNA, CAB 21\/903, Memorandum by the Air Staff, _Bomber Support for the Army_ , 18 November 1939\n\nTNA, CAB 66\/4, WP (YO) 1, _Air Operations and Intelligence_ , 1 January 1940\n\nTNA FO 371\/22917, C8681\/130\/17 and C9363\/130\/17 Colonel W. Fraser despatches, 16 June and 30 June 1939\n\nWO 33\/1305 1933 Notes on Certain Lessons of the Great War\n\nWO 277\/36 The Official History of Training 1939\u20131945\n\nWO 231\/161 MTP No.23, Operations Part 1: General Principles, Fighting Troops and Their Characteristics\n\nWO 167\/169 III Corps War Diary\n\nWO 167\/148 II Corps War Diary September 1939 to June 1940\n\nWO 167\/124 I Corps War Diary\n\nWO 167\/218 3rd Division War Diary, September 1939 to June 1940\n\nWO 167\/203 2nd Division War Diary, September 1939 to June 1940\n\nWO 167\/362 8th Infantry Brigade War Diary, September 1939 to June 1940\n\nWO 167\/459 4th Royal Tank Regiment War Diary, September 1939 to May 1940\n\nWO 167\/721 2nd Cameronians Regiment War Diary\n\nWO 167\/839 2nd Royal Warwickshire Regiment War Diary, September 1939 to June 1940\n\n**National Archives of the United States, Washington DC**\n\nT314\/613-615 War Records of the XIXth Panzer Corps\n\nT314\/666-669 War Records of Panzer Group von Kleist\n\n**Service Historique de la D\u00e9fense, Vincennes, Paris**\n\n29 N 27 2\u00e8me Arm\u00e9e Historique des operations\n\n29 N 84 2\u00e8me Arm\u00e9e, Commandant des Chars\n\n30 N 92 10\u00e8me Corps d' Arm\u00e9e, 3\u00e8me bureau\n\n32 N 251 55\u00e8me Division d'Infanterie\n\n32 N 318 71\u00e8me Division d'Infanterie\n\n34 N 95 84\u00e8me R\u00e9giment de Fortresse\n\n34 N 145 147\u00e8me R\u00e9giment d'Infanterie de Fortresse\n\n34 N 174 295\u00e8me R\u00e9giment d'Infanterie\n\n34 N 178 331\u00e8me R\u00e9giment d'Infanterie\n\n1K 224, 9 Cabinet du G\u00e9n\u00e9ral Gamelin \u2013 Journal de Marche Fonds Gamelin\n\n34 N 165 Rapport de l'Aspirant Penissou\n\n34 N 165 Rapport de Lieutenant Colonel P. Labarthe\n\n32 N 251 Rapport de G\u00e9n\u00e9ral Lafontaine\n\n_Instruction provisoire de 6 Octobre 1921 sur l'emploi tactique des grandes unit\u00e9s_ Charles-Lavauzelle et Cie, impr.-\u00e9diteurs, 1928\n\n**US Army Center of Military History, Fort McNair, Washington DC**\n\nCentre of Military History _Airborne Operations \u2013 A German Appraisal_ CMH Pub 104-13, 1951\n\n**Yale Law School, Lillian Goldman Law Library, The Avalon Project, USA**\n\nGeneral Jodl's Diary (Armed Forces Operational Staff) from 1 February to 26 May 1940 Document 1809-PS\n\n**Websites**\n\nwww.bbc.co.uk\/history\/ww2peopleswar\/stories\/40\/a2447840.shtml Accessed 12 November 2014\n\nwww.hansard.millbanksystems.com\/commons\/1940\/jun\/04\/warsituation Accessed 10 December 2014\n\n<http:\/\/www.ibiblio.org\/pha\/policy\/1940\/1940-06-16d.html> Accessed 3 November 2013\n\nwww.lehrmaninstitute.org\/history\/index.html Accessed 12 July 2013\n\nwww.lignemaginot.com\/accueil\/indexen.htm Accessed on 15 December 2015\n\nwww.telegraph.co.uk\/history\/world-war-two\/7753912\/Wormhoudt\/Everyday-I-thank-God-we-did-our-duty.html Accessed 12 December 2014.\n\nwww.lightbobs.com\/1940-1buckshazebrouck.html Accessed 13 November 2014\n\n# Acknowledgements\n\nWithout the assistance and support of a large number of individuals and organizations, this book would never have been published and all deserve my very great thanks. At the Royal Military Academy: Sarina Davison, my colleagues in the Department of War Studies and the Director of Studies, Sean McKnight. At the University of Buckingham: the Research Fellows of the Humanities Research Institute together with Professors Martin Ricketts, John Drew, John Adamson and Saul David and also Claire Prendergast and Maria Floyd. I also owe a debt of gratitude to the post-graduates on the university's London-based MA Modern War Studies and Contemporary Military History course, as well as my MPhil and DPhil students. Many have listened patiently as I've discussed the 1940 campaign in the West with them, and on several occasions they made me rethink my arguments, or led me to useful source material. I am also grateful for enlightening conversations with and advice from General the Lord Richards, General the Lord Dannatt, Admiral the Lord West, General Sir Nick Carter, Lieutenant General Sir John Kiszely, Major General Mungo Melvin, Professor Sir Hew Strachan, Professor S\u00f6nke Neitzel, Dr Peter Leib and Sir Max Hastings.\n\nI have been studying and teaching the Battle of France for more than a quarter of a century and over that period have always taken the opportunity to interview and correspond with its veterans and, latterly, those civilians that lived through the momentous events. As always, I felt honoured and humbled to learn from them: in Germany, Peter Gerber, Otto Gull, Peter Meier, Gerd Ritter, Albert Winkler, Marc Leicht, Gerd Reismann, Jan Hoch, Hans Liebler, Ralf Buckholtz and Paul Henkelmann; in the Netherlands, Lisa Hanson, Conrad Houtkooper, Max Meers, Peter Meier and Joop van Praag; in Belgium, Arnaud Bauwens; in France, the Roussel family, Charles-Michel L\u00e9p\u00e9e, David Boyer, Bruno Cantel, Maxime Dufour, Emmanuel Marcel, Diane Lebeouf, Dominic Royer, David Schneider, Michel Sigaut, Daniel Le Ro, Serge Winter, Dominic Lemaire and Julianne Deros; and in the UK, Peter Vaux, Peter Mansfield, Douglas Manning, Andrew Hardcastle, Alexander Wilkins, David Blundell, Tom Reed, Rab Riley, Ernie Smith and John Rigson. I am also indebted to Toby Rees and Jenny Friend for tracking down many of these extraordinary people, and for their assistance with some interviewing, research and translation.\n\nThe following archives and libraries have also been unfailingly helpful in my researches, and often provided assistance that went beyond the call of duty: in Germany, the Bundesarchiv-Milit\u00e4rarchiv Freiburg; in France, the Service Historique de la D\u00e9fense, Vincennes, Paris; in the US, the National Archives of the United States, Washington DC and the US Army Center of Military History, Fort McNair, Washington DC; and in the UK, the Liddell Hart Centre for Military Archives, London, the National Archives, London, the Central Library, Royal Military Academy Sandhurst, the British Library, London, the London Library, the RAF Club library, London, the Naval and Military Club library, London, the RUSI library, London and the Imperial War Museum, London. I should also like to thank the military historians Lieutenant Colonel Dr Karl-Heinz Frieser and Dr Robert A. Doughty along with all those that have conducted research and written about the fall of France and the Low Countries in 1940 for their endeavours have had a profound influence on this book. Their names and works can be found in the bibliography. Thanks are also due to Duncan Mason, Dr Paul Harding, Pierre Galliard and Dr Josef Leinberger for reading draft sections and chapters of this book as they spotted numerous slip-ups and offered immensely helpful constructive criticism. I should emphasize, however, that all the errors and omissions that remain in this book are mine and mine alone.\n\nI also greatly appreciate the work done on my behalf by Charlie Viney, my magnificent agent, and at my publishers, Morgan Entrekin and Jamison Stolz at Grove Atlantic in the US, and James Nightingale at Atlantic in the UK. All have been remarkably patient and supportive. I would also like to offer my undying gratitude to my supremely talented editor, Angus MacKinnon. Without Angus's knowledge, good sense, eye for detail and literary skills, this book would have been very much the poorer. I would also like to thank Keith Chaffer for the maps that appear in this volume.\n\nFinally, I should like to acknowledge the love, support and sacrifice provided to me by my family: Catriona, Freddie, Charlotte and Henry \u2013 thank you.\n\nWigginton Bottom and Camberley, February 2016\n\n# Index\n\nAdrey, Georges\n\nAlexander, General Harold, commander of BEF I Corps and 1st Infantry Division\n\nAlexander I of Yugoslavia, King, assassination of (1934)\n\nAllan, Les \u2013\n\nAllied High Command , , ,\n\nAltmeyer, General Ren\u00e9-F\u00e9lix, V Corps , ,\n\nAMR 35 ZT\n\nAnnand, Second Lieutenant Dick\n\nappeasement, political use of \u2013\n\nArmy Regulation 300 ( _Troop Leadership_ ) (1933)\n\nArmy Regulation 487 ( _Leadership and Battle with Combined Arms_ ) \u2013\n\nAubert, Captain, commander of Fort Fermont\n\n_Auftragstaktik_ ,\n\nAustria ,\n\n_Anschluss_ (1938) , ,\n\nBadoglio, Marshal Pietro, Italian Army Chief of Staff\n\nBake, Oberleutnant Dr Franz, 1st Company 11th Panzer Regiment\n\nBalck, Oberstleutnant Hermann , , ,\n\n1st Rifle Regiment , , , \u2013, \u2013\n\nBaril, Colonel Louis \u2013\n\nBarker, Michael, commander of BEF I Corps\n\nBarlone, Reserve Captain Daniel , , \u2013,\n\n2nd North African Division , \u2013\n\nBarratt, Air Marshal Sir A. S., commander-in-chief of British Air Forces in France\n\nBartov, Omer\n\nBauwens, Arnaud \u2013\n\nBauwens, Emma \u2013\n\nBauwens, Maarten \u2013\n\nBauwens, Margot \u2013\n\nBeaufre, Andr\u00e9 , ,\n\nBeck-Broichsitter, Oberleutnant Helmut \u2013\n\nBecker, Hans ,\n\nBelgium , \u2013, , , , , , , \u2013, \u2013, , , , , , \u2013, , , , , , , , ,\n\nAlbert Canal , , \u2013, , \u2013, , ,\n\nAntwerp , , , , ,\n\nArdennes region of , \u2013, , \u2013\n\nArmenti\u00e8res\n\nBodange \u2013,\n\nBruges ,\n\nBrussels , , ,\n\nDinant ,\n\nDyle, River , , , , \u2013, \u2013, , , , , , ,\n\nattempted crossing of \u2013\n\nDyle Line , , , ,\n\nEtalle\n\nFurnes , \u2013,\n\ngovernment of\n\nHannut , ,\n\nHuy\n\nLa Horgne\n\nL\u00e9glise\n\nLibramont\n\nLi\u00e8ge ,\n\nLouvain ,\n\nMartelange \u2013\n\nMenin\n\nmilitary of \u2013, , , , , , , ,\n\n1st Ardennes Light Infantry (Chasseurs Ardennais) Division\n\n1st Cavalry Division ,\n\nI Corps\n\n10th Infantry Division \u2013\n\n2nd Battalion Chasseurs Ardennais \u2013\n\n5th Light Cavalry Division , , \u2013,\n\n7th Infantry Division\n\nHigh Command\n\nMons\n\nMouzaive\n\nNamur ,\n\nYvoir\n\nNazi Invasion of (Battle of Belgium) (1940) , , \u2013, \u2013, ,\n\nBattle for Fort Eben-Emael \u2013,\n\nBattle of Gembloux , \u2013\n\nBattle of Hannut \u2013, , ,\n\nOperation NiWi \u2013, \u2013\n\nNazi Occupation of (1940\u20134) , ,\n\nNeufch\u00e2teau , \u2013, ,\n\nNieuport ,\n\nN\u00eeves\n\nOstend ,\n\nPerwez\n\nPoperinghe\n\nrefugees from\n\nSenne, River\n\nSt Andr\u00e9\n\nTirlemont (Tienen)\n\nTongeren\n\nVresse\n\nVucht\n\nWavre\n\nWitry \u2013\n\nWytschaete\n\nYpres \u2013, , , ,\n\nYser, River ,\n\nZammelen \u2013\n\nvon Below, Nicolaus, Luftwaffe Adjutant to Adolf Hitler\n\nde Benque, Commandant Dutfoy de Mont\n\nBerenbrock, Oberleutnant Dr Hans\n\nBesson, General Antoine-Marie-Beno\u00eet\n\ncommander of French Third Army Group ,\n\nBf 109E\n\nBillotte, General Gaston , , \u2013, , \u2013, , ,\n\ncommander of French First Army Group , \u2013,\n\ndeath of (1940)\n\nvon Bismarck, Oberst Georg\n\n7th Rifle Regiment \u2013\n\nBlanchard, General Georges\n\ncommander of French First Army\/First Army Group , ,\n\n_Blitzkrieg_ , , , ,\n\nBloch MB 151 , ,\n\nBloch, Marc, _Strange Defeat: A Statement_ _of Evidence Written in 1940_ \u2013\n\nBlum, L\u00e9on, leader of Popular Front\n\nBlundell, David\n\nvon Bock, Generaloberst Fedor , \u2013, , \u2013, , \u2013, , , , , , , , , , \u2013, , \u2013\n\ncommander of Sixth Army\n\nBonnet, Georges, French Foreign Minister \u2013\n\nBoyer, David \u2013, \u2013\n\nBrandt, Dr Karl\n\nvon Brauchitsch, General Walther , , , , , \u2013, , , ,\n\nBricart, Major Maurice\n\n5th Company Chasseurs Ardennais\n\ndeath of (1940)\n\nBristol Blenheim , , , \u2013\n\nBritish Broadcasting Corporation (BBC)\n\nBBC News\n\nBrocard, General Antoine\n\n3rd Armoured Division\n\nBrooke, Alan , , , , , , ,\n\ncommander of BEF II Corps , ,\n\nBruneau, General\n\ncommander of 1st DCRs ,\n\nBuckholtz, Ralf\n\nCanaris, Admiral Wilhelm, chief of Abwehr\n\nCarribou, Captain \u2013\n\nCavell, Edith\n\nChamberlain, Neville \u2013\n\nBritish Prime Minister ,\n\nChar B1 \u2013, , , , ,\n\nChiers, River\n\nChina, Republic of, Peking\n\nChurchill, Winston , , , , \u2013, , , , , , , , \u2013, ,\n\nBritish Prime Minister\n\nmeeting with Supreme War Council \u2013\n\norder to evacuate Boulogne ,\n\nvisits to Paris \u2013,\n\nCitino, Robert M.\n\nvon Clausewitz, Karl\n\nCojeul, River\n\nColville, John\n\nAssistant Private Secretary to Winston Churchill\n\n_Conduct of the Air War, The_ (1935) \u2013\n\nCorap, Andr\u00e9\n\ncommander of Ninth Army\n\nsacked from ,\n\nCot, Pierre, French Air Minister \u2013\n\nCourage, Lieutenant Guy , , , , \u2013\n\nCracroft, Captain, adjutant to Second Lieutenant Peter Vaux\n\nCzechoslovakia , , ,\n\nBene\u0161 Line\n\nNazi Occupation of (1938\u201345) , \u2013,\n\nPrague\n\nDaladier, Edouard , , , , \u2013,\n\nFrench Minister of War \u2013,\n\nFrench Prime Minister\n\nDendre, River\n\nDenmark \u2013,\n\nDentz, General Henri\n\nD\u00e9pigny, Eug\u00e8ne , \u2013\n\nDeros, Julianne ,\n\nDietrich, Otto, Press Chief for Adolf Hitler\n\nDietrich, SS-Obergruppenf\u00fchrer Sepp\n\nDietz, Oberleutnant\n\nDill, General John\n\nChief of Imperial General Staff \u2013\n\nDistinguished Service Order, recipients of\n\nDixon, Second Lieutenant John\n\nDornier Do 17s\n\nDoughty, Allan ,\n\nDoumenc, Major General Aim\u00e9, French Army Chief of Staff\n\nDourzal, Lieutenant Colonel \u2013\n\nDufour, Maxime\n\nEden, Anthony\n\nBritish Foreign Secretary\n\nBritish War Minister\n\nErdmann, Major\n\nEscault, River ,\n\nEstienne, General Jean-Baptiste \u2013\n\n_Study of the Mission of Tanks in the Field_ (1919)\n\nEvans, Private Bert\n\nFahey, Lance-Bombardier Brian \u2013\n\nFairey Battle\n\nde la Falaise, Henry \u2013, \u2013\n\nFeldt, General Kurt, 1st Cavalry Division\n\nFiebig, Oberst Martin, commander of KG4\n\nFieseler Fi 156 Storch\n\nFirst World War (1914\u201318) \u2013, , \u2013, \u2013, , , , , , , \u2013, , , , , , , , ,\n\nArmistice (1918) ,\n\nBattle of Verdun (1916) , , , , , ,\n\nTreaty of Versailles (1919) , \u2013, , , , ,\n\nFitzmaurice, Lieutenant Colonel\n\nFlavigny, General Jean-Adolphe-Louis-Robert \u2013\n\n3rd Armoured Division \u2013,\n\n3rd Motorized Infantry Division ,\n\nFoch, Ferdinand\n\nFokker T.V.\n\nFoucault, Lieutenant Colonel\n\nFourcade, Marie-Madeleine\n\nFox-Pitt, Brigadier William \u2013\n\n20th Guards Brigade\n\nFrance , , , , \u2013, , , \u2013, \u2013, , , \u2013, , \u2013, , , \u2013, , \u2013, \u2013\n\nAbbeville ,\n\nAgny \u2013\n\nAire\n\nAisne, River \u2013, \u2013, , , , ,\n\nAlbertville\n\nAlsace-Lorraine \u2013, , , ,\n\nFort Fermont \u2013\n\nAmiens , , ,\n\nArdennes , , , ,\n\nArras \u2013, , \u2013, , \u2013, \u2013\n\nAvesnes , \u2013\n\nBar and Ardennes Canal , \u2013,\n\nBastogne\n\nBeaumont\n\nBeaurains\n\nBergues , ,\n\nB\u00e9thune ,\n\nBois de la Marf\u00e9e , \u2013\n\nBordeaux \u2013, , ,\n\nBouillon\n\nBoulogne \u2013, , ,\n\nBoulzicourt\n\nBray\n\nBrest\n\nBriare\n\nBulson , , ,\n\nCalais , \u2013, , \u2013, , ,\n\nCambrai ,\n\nCassel , ,\n\nChagny\n\nChamber of Deputies\n\nChamb\u00e9ry\n\nCharleville M\u00e9zieres , ,\n\nChartres\n\nCh\u00e2tillon-sur-Seine\n\nChaumont , , ,\n\nCh\u00e9h\u00e9ry , , ,\n\nCh\u00e9mery \u2013, \u2013\n\nCherbourg , \u2013\n\nCheveuges\n\nCimeti\u00e8re\n\nClairfayts\n\nConnage \u2013\n\nCorbion\n\nC\u00f4te d'Azur\n\nCr\u00e9cy\n\nDijon\n\nDonchery , , ,\n\nDouai\n\nDunkirk , \u2013, \u2013, \u2013, , \u2013, , \u2013, \u2013, , , , \u2013, , ,\n\nEstaires\n\nFirst Army Group \u2013, \u2013, , ,\n\n101st Fortress Division\n\n32nd Infantry Division\n\nFirst Army \u2013, , , , , , \u2013, , , , \u2013, , \u2013, , , ,\n\nFlavion\n\nFond Dagot , , ,\n\nFourth Army , ,\n\n47th Infantry Division\n\nFr\u00e9nois \u2013\n\nFresnes-Mazancourt\n\nFroid-Chapelle ,\n\nGaulier\n\nGembloux Gap , , , , , , , , \u2013,\n\nGlaire \u2013\n\ngovernment of , , , , \u2013\n\nGravelines , ,\n\nAa Canal \u2013, ,\n\nLa Bass\u00e9e Canal \u2013,\n\nGrenoble\n\nHaguenau\n\nHazebrouck , \u2013\n\nHoux\n\nLa-Fert\u00e9-sous-Jouarre ,\n\nLandrecies\n\nLe Cateau \u2013,\n\nLe Chesne\n\nLe Fert\u00e9\n\nLe Mans\n\nLes Forges\n\nLibramont-Neufch\u00e2teau Line\n\nLille , , \u2013, , \u2013\n\nLoire, River , \u2013, , ,\n\nLyon ,\n\nMaginot Line , , \u2013, \u2013, , , , , , , , \u2013, \u2013, \u2013, , , , , , , , \u2013, , , , , \u2013, , ,\n\nMaisoncelle ,\n\nMarch\u00e9lepot\n\nMarseilles ,\n\nMetz , ,\n\nmilitary of \u2013, \u2013, \u2013, , , , , \u2013, , , , , \u2013, \u2013,\n\n11th Machine Gun Battalion\n\n19th Alpine Regiment\n\n102nd Infantry Division\n\n112th Alpine Regiment \u2013,\n\n147th Fortress Infantry Regiment , \u2013,\n\n152nd Infantry Regiment \u2013\n\n2nd Battalion \u2013\n\n25th Tank Battalion\n\n28th Tank Battalion\n\n205th Infantry Regiment\n\n213rd Infantry Regiment \u2013\n\n295th Infantry Regiment\n\n331st Infantry Regiment \u2013\n\n4th Tank Battalion , ,\n\n42nd Fortress Infantry Regiment\n\n5th Motorized Division\n\n7th Tank Battalion\n\nIX Corps\n\nDivisions de Cuirassiers de R\u00e9serve\n\n(DCRs) \u2013, , , , , , \u2013, \u2013, , , \u2013, \u2013,\n\nGeneral Reserve\n\nHigh Command , , , , , \u2013, , , , , \u2013, , , , , , \u2013, , ,\n\nLight Mechanized Divisions (DLMs) , \u2013, , , , , , \u2013,\n\n_mutil\u00e9s_\n\nofficer corps\n\nrearmament \u2013\n\nSenate Army Commission\n\nSixth Army , , , ,\n\nSuperior War Council (SWC) , \u2013\n\nSupreme War Council ,\n\nTenth Army , ,\n\nXXI Corps\n\nXXIII Corps ,\n\nXXXXI Corps\n\nMis\u00e9ry\n\nMont Dieu , , ,\n\nMontcornet , , ,\n\nMontherm\u00e9 , ,\n\nMorville\n\nNamur ,\n\nNantes ,\n\nNational Assembly\n\nNeufch\u00e2teau\n\nNevers\n\nNinth Army , , , , , , , , , , , , , ,\n\n3rd Spahis Brigade , , \u2013\n\n53rd Infantry Division ,\n\n61st Infantry Division\n\nNormandy ,\n\nNoyers ,\n\nOnhaye \u2013\n\nParis \u2013, \u2013, \u2013, \u2013, , , , , , , \u2013, \u2013, , , , \u2013, , \u2013, , \u2013, , \u2013, \u2013, , , \u2013, , \u2013\n\nP\u00e9ronne , , ,\n\nPicardy, Forest of Compi\u00e8gne ,\n\nPicquigny\n\nPoix-Terron\n\nPont Maugis \u2013\n\nPont Neuf\n\nReims ,\n\nRennes\n\nRethel , , ,\n\nRh\u00f4ne Valley \u2013\n\nRouen\n\nSambre-Oise Line ,\n\nSarre Grap\n\nScarpe, River ,\n\nSchoenenbourg \u2013\n\nSecond Army Group , , \u2013,\n\n1st Colonial Division\n\n2nd Light Cavalry Division \u2013\n\n5th Light Cavalry , , , ,\n\n55th Infantry Division , , \u2013, \u2013, , ,\n\n71st Infantry Division \u2013, , ,\n\nSecond Army , , \u2013, , , , , \u2013, , \u2013, \u2013, , , \u2013, \u2013, , \u2013\n\nSedan , , , , \u2013, , , , , , , , , \u2013, , \u2013, , , , , , \u2013, \u2013, \u2013, , ,\n\nSeine, River , \u2013, , \u2013\n\nLower Seine\n\nSeventh Army , , , \u2013, , , , , , , , , , ,\n\n102nd Fortress Division \u2013,\n\n21st Infantry Division , \u2013\n\nSingly\n\nSomme\n\nSomme, River , , , \u2013, , , \u2013, , , , , , , , \u2013, , , , \u2013, ,\n\nSt Malo ,\n\nSt Nazaire\n\nSt Omer\n\nSt Quentin\n\nSt Val\u00e9ry \u2013\n\nStenay\n\nStenay Gap\n\nStonne \u2013\n\nStrasbourg\n\nThird Army Group , , , ,\n\n68th Infantry Division\n\nTilloy\n\nTorchy\n\nTours , ,\n\nVerdun \u2013, \u2013, ,\n\nVichy\n\nVincennes ,\n\nVillers\n\nWadelincourt\n\nWailly\n\nHill 111\n\nWormhoudt \u2013\n\nFranco-Prussian War (1870\u20131) ,\n\nFranklyn, Major General Harold\n\n4th RTR \u2013\n\n6th Battalion DLI \u2013\n\n7th RTR\n\n8th Battalion DLI\n\nFraser, Colonel William\n\nFr\u00e8re, General Aubert, commander of French Seventh Army\n\nFrieser, Karl-Heinz\n\nFrossard, Ludovic-Iscar, broadcasts made by\n\nGalland, Adolf \u2013\n\nGamelin, General Maurice , , \u2013, \u2013, , , , , , \u2013, , , , , , \u2013, , \u2013, , , \u2013,\n\nBreda push (1940)\n\n_Instruction Personelle et Secr\u00e8te No. 12_ \u2013,\n\nstaff of\n\nSupreme Commander of Allied Forces , , , , , \u2013, , , ,\n\ndismissal of \u2013,\n\nGarski, Oberstleutnant Eugen \u2013,\n\n3rd Battalion Gross Deutschland Regiment , \u2013\n\nde Gaulle, Charles \u2013, , , \u2013\n\nFrench Undersecretary of State for National Defence\n\npush towards Montcornet \u2013\n\n_Towards the Professional Army_ (1934) \u2013\n\nGeorges, General Alphonse , , , , , \u2013, , ,\n\ncommander of North-East Front\n\nshortcomings of , \u2013\n\nGermany , , , \u2013, \u2013, \u2013, \u2013, \u2013, , , ,\n\nBerlin , , , , , \u2013, ,\n\nCochem\n\nCologne\n\nGNP of\n\nHamburg\n\nHildesheim\n\nKoblenz , ,\n\nDeutsches Eck \u2013\n\nM\u00fcnstereifel\n\nG\u00e9odias, General Paul-Henry, 29th Alpine Division , ,\n\nGiesler, Hermann\n\nGiraud, General Henri\n\ncommander of French Seventh Army\n\ncommander of French Ninth Army\n\nGoebbels, Joseph\n\nG\u00f6ring, Hermann ,\n\nGort, Lord (John Standish Surtees Prendergast Vereker) , , , \u2013, , , , \u2013,\n\nBritish Chief of Imperial General Staff \u2013\n\nGrandsard, General Pierre-Paul-Charles\n\ncommander of X Corps , \u2013\n\nGraziani, General Rodolfo, Italian Commander-in-Chief of the General Staff\n\nvon Greiffenberg, I. G. Hans, Chief of Operations Staff\n\nGribble, Philip \u2013, ,\n\nvon Grolman, Oberleutant Friedrich\n\nGroussard, Colonel G. A., demand for relief of duty \u2013\n\nGuderian, Heinz , , \u2013, \u2013, , , \u2013, , \u2013, \u2013, \u2013, , \u2013, \u2013, , , , , , , , , \u2013, \u2013, , ,\n\nPanzer Group , \u2013\n\n1st Panzer Division ,\n\n2nd Panzer Division ,\n\n6th Panzer Division ,\n\n8th Panzer Division\n\nrolling raid designed by \u2013\n\nsupport for mechanization \u2013,\n\nXIX Corps , \u2013, , , , 202 , ,\n\n1st Panzer Division , , \u2013, , \u2013, \u2013, \u2013, , , , \u2013, , , , , , , , , , , \u2013, , 10th Panzer Division , \u2013, \u2013, , \u2013, \u2013, \u2013, , , , , , \u2013,\n\n2nd Motorized Division\n\n2nd Panzer Division , , , , , , , , , , , , \u2013, \u2013, , , ,\n\n20th Motorized Division , ,\n\n29th Motorized Division\n\n3rd Panzer Division ,\n\nPanzer Lehr Battalion\n\nReconnaissance Battalion\n\nGuillaut, Lieutenant Colonel\n\nGuillot, Georgette\n\nGull, Gefreiter Otto \u2013, , , \u2013\n\nGuzzoni, General Alfredo\n\ncommander of Italian Fourth Army\n\nHalder, General Franz , , \u2013, , , , , , , \u2013, , , , , \u2013, \u2013, , \u2013, , , , , \u2013, , \u2013,\n\ncampaign plan of \u2013\n\nChief of Staff to OKH\n\nFirst Deployment Directive (1939) \u2013\n\nFourth Deployment Directive (1940) \u2013\n\nSecond Deployment Directive (1939) \u2013, , \u2013\n\nThird Deployment Directive (1940) ,\n\nHardcastle, Andrew ,\n\nvon Hartlieb, Max, 5th Panzer Division \u2013, ,\n\nHawker Hurricane ,\n\nHeinkel He 111s ,\n\nHenschel Hs 123\n\nHermann, Major Paul\n\nHeyworth, Major Brian\n\nHillion, Sergeant Major Georges\n\nHindelang, Oberfeldwebel Hans\n\nvon Hippel, Oberst\n\nHitler, Adolf \u2013, \u2013, , , , , , , , , \u2013, , , , , , , , , , , , \u2013, \u2013, , , \u2013, , , , , , , , \u2013, , , \u2013, ,\n\nF\u00fchrer Directive No. 6 (1939) ,\n\nF\u00fchrer Directive No. 11 (1940)\n\nF\u00fchrer Directive No. 12 (1940) \u2013\n\nrearmament demands of \u2013\n\nrise to power (1933)\n\nHoch, Hauptmann Jan\n\nHoenmanns, Major Erich\n\nHoepner, General Erich \u2013, ,\n\nXVI Panzer Motorized Corps , ,\n\n3rd Panzer Division , \u2013, , , ,\n\n4th Panzer Division , \u2013, , ,\n\nHollis, Colonel Leslie\n\nHotchkiss H35\/39\n\nHotchkiss H39 \u2013,\n\nHoth, General Hermann , , \u2013, , , , , , , ,\n\nPanzer Group Hoth , , , , ,\n\nXV Corps , , , \u2013\n\n5th Panzer Division , \u2013, , , , , , , \u2013\n\n7th Panzer Division , , , , , , , , ,\n\nHoutkooper, Conrad\n\nHoward, Sir Michael\n\nHughes, Robert\n\nHuntziger, General Charles , ,\n\ncommander of Second Army Group \u2013\n\n55th Division \u2013\n\n71st Division\n\nX Corps \u2013, , , , \u2013, \u2013, ,\n\nXVIII Corps\n\nrole in armistice discussion\n\nIhler, General Marcel, Commander of French Ninth Corps\n\n_Instructions on the Employment of Combat Tanks_\n\n_Instructions on the Tactical Employment of Large Units_ (1936)\n\nIron Cross\n\nKnight's Cross recipients of ,\n\nIronside, Sir Edmund 'Tiny' ,\n\nBritish Chief of Imperial General Staff ,\n\nIrwin, Major General Noel, 2nd Infantry Division\n\nIsmay, General Hastings ,\n\nItaly\n\nmilitary of Alpine Corps \u2013\n\nJacomet, General Pierre, Secretary-General in French War Ministry\n\nJodl, Major General Alfred , \u2013, ,\n\nChief of the Operations Staff of OKW\n\nJottrand, Major Jean Fritz Lucieb\n\ncommander of garrison at Fort Eben-Emael \u2013,\n\nJudaism\n\nvon J\u00fcngenfeld, Hauptman Ernst\n\nJunkers Ju 52 , \u2013,\n\nKampfgruppe Kr\u00fcger ,\n\nKampfgruppe Nedtwig\n\nKeitel, General Wilhelm ,\n\nHead of OKW ,\n\nrole in armistice discussion\n\nKeller, Lieutenant Colonel Reginald, commander of 3RTR\n\nKeltsch, Oberst Karl, commander of 1st Panzer Brigade , ,\n\nKempf, General Werner , , ,\n\n6th Panzer Division \u2013, \u2013, , , , , ,\n\nKeyaerts, Lieutenant General Maurice, commander of K Group \u2013,\n\nvon Kielmansegg, Johann Graf\n\nvon Kielmansegg, Major\n\nKirchner, Major General Friedrich , , , \u2013, , , , ,\n\n1st Panzer Division , \u2013, , , , \u2013, , \u2013,\n\nvon Kleist, General Paul \u2013, , , , , , , , , \u2013, , , , , ,\n\nair plan of \u2013\n\nPanzer Group Kleist , , , \u2013, , , , , , , , \u2013, ,\n\nvon Kluge, Generaloberst G\u00fcnther\n\ncommander of Fourth Army , \u2013,\n\nII Corps\n\nVIII Corps\n\nKoch, Hauptmann Walther\n\nGroup _Granite_ \u2013\n\nKoeltz, General Louis\n\nKorthal, Oberleutnant G\u00fcnther \u2013,\n\nKrajewski, Oberleutnant\n\nvon Kressenstein, Hauptmann Friedrich Freiherr\n\nKr\u00fcger, Hauptmann\n\nKr\u00fcger, Oberst \u2013\n\nvon K\u00fcchler, General Georg ,\n\n9th Panzer Division ,\n\nLabarthe, Lieutenant Colonel\n\n213th Infantry Regiment\n\nLa Bel, Lieutenant Robert\n\nLa Chambre, Guy, French Air Minister \u2013\n\nLafontaine, General Pierre , , , , , \u2013,\n\ncommander of 55th Division \u2013, ,\n\nwithdrawal order\n\nwritten orders of \u2013\n\nde Lambertye, Capitaine de Fr\u00e9fate Carlos\n\nLapointe, Julia \u2013\n\nLebeouf, Diane\n\nvon Leeb, Generaloberst Wilhelm Ritter , , , \u2013\n\ncommander of Army Group C\n\nLeflon, Captain\n\nLeicht, Marc\n\nLenlud, General Fernand\n\nLeopold, King , \u2013,\n\nL\u00e9p\u00e9e, Charles-Michel\n\n_L'\u00c9poque_\n\nLe Roy, Lieutenant Daniel \u2013,\n\nLiebler, Gefreiter Hans\n\nLiss, Oberst Ulrich, Chief of Foreign Armies West section of Army Intelligence\n\nList, General Wilhelm ,\n\ncommander of Twelfth Army\n\nLitalien, Captain\n\nLloyd, H. C.\n\nLoerzer, Generalleutnant Bruno\n\ncommander of II. Fliegerkorps\n\nrolling raid designed by \u2013\n\nLoritte, Second Lieutenant\n\nLow Countries ,\n\nLutz, Oswald\n\nsupport for mechanization\n\nLuxembourg , \u2013, \u2013, , , , \u2013, , \u2013,\n\nArdennes region of\n\nMartelange\n\nNazi Invasion of (1940) \u2013\n\nNazi Occupation of (1940\u20134)\n\nrefugees from\n\nSchuster Line ,\n\nLynn-Allen, Captain James\n\nLys, River ,\n\nMaginot, Andr\u00e9, French Minister of War\n\nmilitary service of \u2013\n\nMahlke, Hauptmann Helmuth , , ,\n\nStuka _staffel_ of \u2013\n\nMangin, General Charles\n\nManning, Douglas\n\nvon Manstein, Generalleutnant Erich , , , , , ,\n\nadvancement plan of \u2013, ,\n\nChief of Staff to Army Group A\n\nXXXVIII Corps ,\n\n46th Infantry Division\n\n6th Infantry Division\n\nMarcel, Emmanuel \u2013\n\nMartin, General Julien-Fran\u00e7ois-Ren\u00e9 commander of IX Corps\n\nMatilda \u2013\n\nMk I\n\nMk II\n\nMattis, General James\n\nMaurois, Andr\u00e9\n\nMcComb, Squadron Leader J. E. \u2013\n\nMcCullen, Captain David \u2013,\n\nMeier, Peter \u2013\n\nMellier, General Albert, Moroccan Division\n\nMelvin, Mungo\n\nMethodical Battle (military doctrine) , , , ,\n\ncommunication methods\n\nfocus of \u2013\n\nMeuse, River , , , \u2013, , , , , , , , , , , \u2013, \u2013, \u2013, , , , , , ,\n\nattempts to cross , , , \u2013, \u2013, , , \u2013, , , , , \u2013, , , \u2013, \u2013, , , , ,\n\nbridges across , , ,\n\nrailway\n\nBunker 104 \u2013\n\neast bank of ,\n\nHill 246 ,\n\nHill 247 \u2013, ,\n\nHill 301 \u2013,\n\nHill 336\n\n_Schwerpunkt_\n\nwest bank of , , , , , \u2013, ,\n\nMichon, Colonel Daniel\n\nMilitary Cross, recipients of\n\nMilne, Field Marshal Sir George\n\nM\u00f6llman, Gefreiter\n\nMolotov-Ribbentrop Pact (1939)\n\nvon Moltke, Helmut\n\nMontgomery, Major General Bernard, commander of British 3rd Infantry Division , \u2013, ,\n\nMontviginies-Monnet, Lieutenant Colonel\n\n205th Infantry regiment ,\n\nMorane-Saulnier MS 406\n\nMorize, Andr\u00e9\n\nMost, Leutnant, ADC to Major General Erwin Rommel\n\nMunich Conference (1938) , , \u2013\n\nMussolini, Benito , \u2013\n\nNapoleon III ,\n\ncapture of (1870)\n\ntomb of\n\nNedwig, Oberst Johannes, 1st Panzer Regiment ,\n\nNetherlands , , , \u2013, \u2013, \u2013, , , , , , , ,\n\nAmersfoort\n\nAmsterdam , , ,\n\nBreda , , , ,\n\nDe Kooy airfield\n\nDinant , \u2013\n\nDordrecht ,\n\nFiresland\n\nGennep\n\ngovernment of , ,\n\nGrebbe Line , , , , , ,\n\nHague, The , , ,\n\nHaut-le-Wastia\n\nHolland , , \u2013, , , ,\n\nHoux ,\n\nIjssel Line\n\nLeffe\n\nMaastricht , , , , \u2013, \u2013, , , , ,\n\nMeuse Line , ,\n\nmilitary of , , , , ,\n\nHigh Command\n\nMoerdijk , ,\n\nMoerdijk Bridges\n\nNazi Invasion of (1940) , , \u2013, , \u2013, \u2013,\n\nNazi Occupation of (1940\u20134) ,\n\nNijmegen\n\nPeel-Raam Line \u2013, ,\n\nrefugees from\n\nRotterdam , , \u2013, , \u2013, ,\n\nWaalhaven\n\nTilburg ,\n\nUtrecht ,\n\nZutphen ,\n\nZwolle ,\n\nNicolson, Brigadier Claude\n\n1st Battalion The Rifle Brigade\n\n2nd Battalion King's Royal Rifle Corps\n\n3RTR \u2013\n\nQueen Victoria Rifles\n\nNoiret, General Louis-\u00c9mile, 7th Colonial Division\n\nNorway ,\n\nNarvik\n\nNazi Invasion of (1940) \u2013\n\nNovak, Hugo\n\nOise, River , \u2013, , , ,\n\nOlry, General Ren\u00e9, commander of French Army of the Alps \u2013\n\nOster, Oberst Hans\n\nwarning of Nazi Invasion of Netherlands (1940)\n\nOstwald, Leutnant Wold-Max ,\n\nOur, River\n\nvan Overstraeten, General Raoul , ,\n\nuse of K Group \u2013\n\nOvery, Richard\n\nPanhard 178 armoured car\n\n_PanzerKampfwagen_ (PzKpfw) 301 35(t) 45 38(t)\n\nMk I , , , \u2013,\n\nMk II , , \u2013,\n\nMk III , , , , , , , , , ,\n\nMk IV , , , , ,\n\nvon Papen, Franz\n\nPark, Air Chief Marshal Sir Keith, 11 Group airfields\n\nPenissou, Second Lieutenant \u2013\n\nP\u00e9tain, Marshal Henri Philippe , , , , , , \u2013,\n\ndiplomatic efforts of , \u2013\n\nInspector General\n\nPetibon, Colonel Jean ,\n\nPetre, Major General Roderick, 12th Infantry Division\n\nPinaud, Lieutenant Colonel\n\ncommander of 147th Fortress Infantry Regiment\n\nPoland , , ,\n\nAxis Invasion of (1939) \u2013, \u2013, , \u2013, , , , ,\n\nWarsaw , ,\n\nPolley, Signaller Bert\n\nPoncelet, Colonel\n\nPopular Front, electoral victory of (1936)\n\nPownall, General Henry, Chief of Staff to Lord Gort\n\nPr\u00e9telat, General Andr\u00e9-Gaston , \u2013,\n\nPrioux, General , ,\n\nCavalry Corps , , \u2013, , ,\n\n_Provisional Instructions for the Tactical Employment of Larger Units_ (1921) \u2013\n\nPr\u00fcmer, Unteroffizier \u2013\n\nRaeder, Admiral Erich\n\nRamsay, Vice-Admiral Bertram\n\nrole in Dunkirk Evacuation (1940) ,\n\nvon Ranke, Leopold\n\nReed, Lance Corporal Tom \u2013\n\nvon Reichenau, General Walter\n\nReinberger, Major Helmuth\n\nReinhardt, Major General Georg Hans \u2013, , ,\n\n_In Guderian's Shadow_\n\nXXXXI Corps ,\n\n2nd Motorized Division \u2013\n\n6th Panzer Division , , \u2013, , \u2013, \u2013\n\n8th Panzer Division , , , , \u2013\n\nReischswehr \u2013,\n\nReismann, Gerd \u2013\n\nRenault R35 ,\n\nReynaud, Paul , \u2013, , , , \u2013,\n\nattempted restructuring of cabinet \u2013\n\nattempts to limit refugee numbers \u2013\n\ndiplomatic efforts of \u2013\n\nFrench Finance Minister\n\nRhine, River , ,\n\nIjssel, River \u2013\n\nUpper Rhine\n\nRhineland, German reoccupation of (1936) \u2013\n\nRh\u00f4ne, River ,\n\nvon Richthofen, Wolfram Freiherr Fliegerkorps VIII ,\n\nRiley, Flying Sergeanr Rab\n\nRitter, Gerd\n\nRMS _Lancastria,_ sinking of (1940) ,\n\nRommel, Major General Erwin \u2013, , , , \u2013, \u2013, \u2013, , \u2013, , , \u2013, , \u2013, \u2013, \u2013, , , ,\n\n6th Rifle Regiment ,\n\n7th Motorcycle Rifle Battalion\n\n7th Panzer Division \u2013, , , , , , \u2013, \u2013, , , \u2013, ,\n\n7th Rifle Regiment , ,\n\n25th Panzer Regiment , ,\n\nRothery, Guardsman Douglas \u2013\n\nRoupell, Brigadier George\n\nRoussel, Pierre \u2013\n\nRoyer, Dominic\n\n_Rucksack-Prinzip_\n\nvon Rundstedt, Generaloberst Gerd , , , \u2013, , , , , , , \u2013, , , , \u2013\n\norder prohibiting push beyond Montcornet ,\n\nRussian Empire\n\nSadoul, Georges \u2013\n\nSarll, Flying Officer\n\nSas, Colonel Gijsbertus Jacobus ,\n\nSchaal, Major General Ferdinand ,\n\n10th Panzer Division , , ,\n\nScharoo, Colonel Pieter, negotiation efforts in Rotterdam\n\nSchaub, Julius, Chief Military Aide to Adolf Hitler\n\nScheldt, River\n\nvon Schlieffen, Alfred \u2013\n\nSchlieffen Plan (1905\/1914) , , , , ,\n\nSchmidt, Colonel Leonard Johannes\n\nSchmidt, General Rudolf\n\n1st Panzer Division\n\n2nd Panzer Division\n\nSchmidt, Paul\n\nSchmundt, Generealleutnant Rudolf, adjutant to Adolf Hitler , , \u2013\n\nXXXIX Motorized Corps\n\nSchraepler, Hauptmann Hans-Joachim, aide-de-camp to Major General Erwin Rommel\n\nSchuster, Joseph\n\nSchwartz, Lieutenant Colonel\n\nSciard, General Th\u00e9odore\n\nSecond World War (1939\u201345) , ,\n\nbelligerents of \u2013, , , \u2013, , , \u2013, , , \u2013, , , , , , ,\n\nDunkirk Evacuation (Operation Dynamo) (1940) , \u2013, \u2013, \u2013, , \u2013, , \u2013, , , \u2013,\n\nFall of France (1940) \u2013, \u2013, , , , \u2013, , , , \u2013, , \u2013,\n\narmistice , \u2013, \u2013, \u2013, \u2013\n\nBattle of Arras , \u2013,\n\nBattle of Gembloux , \u2013\n\nBattle of Netherlands , , \u2013, , \u2013, \u2013,\n\ncivilian evacuation efforts during \u2013,\n\ndefence of Paris \u2013, , , , \u2013\n\n_Fall Gelb_ (Case Yellow) , \u2013, \u2013, \u2013, , , , , , , , , , , , , , , , , \u2013\n\nFall of Paris \u2013\n\n_Fall Rot_ (Case Red) , \u2013, , \u2013, \u2013,\n\nInvasion of Luxembourg \u2013\n\nPlan D \u2013, , , , , ,\n\nInvasion of Poland (1939) \u2013, \u2013, , \u2013, , , , ,\n\n_Fall Weiss_ (Case White)\n\nNorwegian Campaign (1940) \u2013\n\n_Fall Weser\u00fcbung_ \u2013\n\nOperation Ariel \u2013\n\nOperation Kleiner B\u00e4r ,\n\nOperation Tiger ,\n\nPhoney War (1939\u201340) , , , , , ,\n\nWestern Front\n\nBattle of Belgium (1940) , , , , \u2013, \u2013, \u2013, \u2013, ,\n\nvon Seekt, General Hans \u2013, \u2013\n\nSemois, River , \u2013,\n\nSenne, River\n\nSigaut, Lieutenant Michel\n\nSmith, Guardsman Ernie \u2013\n\nvon Sodenstern, Georg\n\nSOMUA S35 \u2013, ,\n\nSoviet Union (USSR) \u2013,\n\nSpain , , ,\n\nCivil War (1936\u20139)\n\nSpannenkrebs, Walter, support for mechanization\n\nSpears, General Edward\n\nSpeer, Albert \u2013\n\nSperrle, General der Flieger Hugo, commander of Luftlotte 3\n\nSpiedel, Lieutenant Colonel Dr Hans\n\nStalin, Josef\n\nStudent, General Kurt\n\n7th Fliegerkorps\n\nStuka (Junkers Ju 87)\n\nairstrikes conducted by , , , , \u2013, , , \u2013, , , \u2013, , , , \u2013,\n\nSupermarine Spitfire , \u2013\n\nSweden\n\nSwitzerland , , , , , ,\n\nBern\n\nSyria\n\nde Tassigny, General Jean de Lattre, 14th Infantry Division , ,\n\nTattinger, Pierre \u2013\n\nTennant, Captain William role in Dunkirk Evacuation (1940) ,\n\nThien, Auguste\n\nThird Reich (1933\u201345) , \u2013, , , , , , , ,\n\nAbwehr , , ,\n\nBrandenburg Battalion\n\nArmy Group A \u2013, , \u2013, , , , \u2013, , , \u2013, , , , \u2013, , , \u2013, \u2013, , , \u2013, , \u2013, , \u2013\n\nFourth Army , , ,\n\nSixteenth Army ,\n\nWar Diary \u2013\n\nArmy Group B \u2013, \u2013, \u2013, , , , , , , , \u2013, , , , , \u2013, , , \u2013, , , , , \u2013, , \u2013, \u2013\n\nEighteenth Army , , , , , ,\n\nFourth Army , ,\n\nThird Reich, Army Group B ( _cont._ )\n\nSixth Army , , , , , , , ,\n\nTwelfth Army , \u2013, , , ,\n\nArmy Group C , , , \u2013, , \u2013,\n\nFirst Army\n\nSeventh Army ,\n\nArmy Group North\n\nFourth Army\n\nThird Army\n\nArmy Group South\n\nEighth Army\n\nFourteenth Army\n\nTenth Army\n\nArmy High Command (OKH) \u2013, , , \u2013, , , \u2013, , , , , \u2013, , , \u2013, , , , \u2013, , , ,\n\n_Der St\u00fcrmer_\n\neconomy of\n\nOberkommando der Wehrmacht (OKW) , \u2013, , \u2013, , , , , \u2013, , , , \u2013, \u2013, \u2013, , , , ,\n\nDirective No. 13 (Dunkirk Halt Order) \u2013, , , \u2013,\n\nformation of (1935)\n\nLuftwaffe , \u2013, , \u2013, , , , , , , , , , , , \u2013, , , , \u2013, , , , , \u2013, , , , , , , \u2013, , ,\n\nII. Fliegerkorps\n\n7th Flieger Division\n\nFliegerkorps VIII\n\nKampfgeschwader 30 (KG30)\n\nKampfgeschwader 4 (KG4)\n\nKampfgeschwader 54 (KG54) ,\n\nLuftlotte 2\n\nLuftlotte 3 , , \u2013\n\nOperation Paula\n\nSturzkampfgeschwader 2\n\nPropaganda Ministry\n\nReich Chancellery , , , ,\n\nReoccupation of Rhineland (1936)\n\nSchutzstaffel (SS) , ,\n\nLeibstandarte Adolf Hitler , \u2013,\n\nTotenkopf Division \u2013,\n\nWaffen SS\n\nWar Academy\n\nWar Ministry\n\nWehrmacht , , , , , , , \u2013, , , , , ,\n\n1st Cavalry Division , , , ,\n\n1st Light Cavalry Division\n\n1st Rifle Regiment 117 \u2013\n\n183rd Infantry Division\n\n2nd Gross Deutschland Regiment , , , , , , \u2013, \u2013,\n\n22nd Luftlande-Infanterie Division\n\n227th Infantry Division ,\n\n253rd Infantry Division\n\n254th Infantry Division ,\n\n256th Infantry Division ,\n\n3rd Rifle Regiment\n\n4th Light Cavalry Division\n\n61st Infantry Division \u2013\n\n65th Panzer Battalion\n\n69th Infantry Division\n\n69th Rifle Regiment\n\n86th Rifle Regiment \u2013\n\nXIV Panzer Corps \u2013\n\n_Time_ (magazine) ,\n\nTouchon, General Robert-Auguste\n\ncommander of French Sixth Army ,\n\nToussaint, General 19th Infantry Division ,\n\n22nd R\u00e9giment de Marche de Volontaires Etrangers (RMVE) ,\n\nUnited Kingdom (UK) , , , , , ,\n\n1st Armoured Division\n\nBletchley Park\n\nBritish Expeditionary Force (BEF) , , \u2013, , , , , , , \u2013, \u2013, , \u2013, \u2013, \u2013, , , ,\n\nI Corps , ,\n\nII Corps \u2013, , , , , , ,\n\n1st Infantry Division , , ,\n\n12th Lancers\n\n2nd Infantry Division , \u2013,\n\n3rd Infantry Division , , \u2013, , , ,\n\n4th Infantry Division ,\n\n44th Infantry Division\n\n48th Infantry Division\n\n50th Infantry Division ,\n\n51st Highland Division\n\nBeaumon Division\n\nplanned evacuation of\n\nDover\n\nExperimental Mechanical Force\n\ngovernment of\n\nImperial General Staff\n\nLondon , , \u2013, , , , , , , , , \u2013\n\nmilitary of \u2013\n\n15\/19 King's Royal Hussars (KRH) , , , , , \u2013\n\n1st Armoured Division , \u2013,\n\n1st Battalion Queen's Own Cameron\n\nHighlanders\n\n1st Canadian Division\n\n10th Hussars\n\n2nd Battalion Wiltshire Regiment\n\n20th Guards Brigade\n\n5th Battalion Green Howards\n\n5th Infantry Division , \u2013, ,\n\n52nd (Lowland) Division\n\nAuxiliary Military Pioneer Corps (AMPC) \u2013\n\nBuckinghamshire Light Infantry\n\nDurham Light Infantry (DLI)\n\nGrenadier Guards \u2013\n\nIrish Guards \u2013\n\nOxfordshire Light Infantry\n\nRoyal Norfolks\n\nRoyal Tank Regiment (RTR) ,\n\nWelsh Guards \u2013\n\nParliament, House of Commons\n\nParliamentary Army Committee\n\nRoyal Air Force (RAF) , , \u2013, , , , , \u2013,\n\n2 Group , , \u2013, \u2013,\n\n611 Squadron \u2013\n\nAdvanced Air Strike Force (AASF) ,\n\nBomber Command\n\nFighter Command ,\n\nOfficial RAF History\n\nRoyal Navy ,\n\nHMS _Keith_\n\nHMS _Vimy_ \u2013\n\nSalisbury Plain\n\nUnited States of America (USA) \u2013\n\nWashington DC \u2013\n\nVautrin, Major, French First Army Liaison to BEF\n\nVaux, Second Lieutenant Peter \u2013\n\nvan Voorst tot Voorst, Lieutenant General Jan Joseph Godfried Baron\n\nVeiel, Major General Rudolf ,\n\n2nd Panzer Division , ,\n\nVenice, River\n\nVerron, Second Lieutenant\n\nVictoria Cross, recipients of ,\n\nde la Vigerie, General d'Astier ,\n\nVilliers-Moriam\u00e9, Lieutenant Colonel Pierre\n\nViney, Major Elliott\n\nVitte, Captain\n\nVuillemin, General Joseph \u2013, ,\n\nWagner, Major General Eduard, role in OKH logistics\n\nWaite, Lieutenant Colonel Sumner\n\nWake-Walker, Rear Admiral Frederic role in Dunkirk Evacuation (1940)\n\nWalther, Leutnant Wilhelm\n\nWarlimont, General Walter\n\nWatkins, Sergeant T. J.\n\nWebster, Lieutenant \u2013\n\nvon Welczeck, German Count Johannes, German Ambassador to Paris \u2013\n\nWerner, Oberst Paul Hermann, 31st Panzer Regiment , \u2013\n\nWerth, Alexander , , , \u2013,\n\nWeygand, General Maxime , \u2013, \u2013, \u2013, , \u2013, , , , , ,\n\nSupreme Commander of Allied Forces , \u2013, , ,\n\nWeygand's Hedgehogs \u2013\n\nWeygand Plan , \u2013\n\nWeygand Line , \u2013, , , \u2013, ,\n\nattacks on \u2013,\n\nstructure of \u2013\n\nvon Wietersheim, General Gustav Anton\n\nXIV Motorized Infantry\n\n13th Motorized Division\n\n2nd Motorized Division\n\n29th Motorized Division\n\nWilhelm I, Kaiser, proclaimed Emperor (1871)\n\nWilhelmina, Queen\n\nWilkins, Private Alexander\n\nWilliams, Trooper 'Butcher'\n\nWinkelman, General H. G. ,\n\ncall for Dutch surrender\n\nCommander-in-Chief of Dutch military\n\nWinkler, Albert \u2013\n\nWinter, Serge \u2013, \u2013\n\nWitzig, Oberleutnant Rudolf, commander of Group _Granite_ \u2013\n\nZeitzler, Oberst Kurt\n\nZobel, Leutnant Heinz\n\nSeptember 1939: Adolf Hitler arrives in Poland. The officers behind him include Halder, von Rundstedt, von Brauchitsch, Keitel, Rommel and von Manstein.\n\nHitler at OKH headquarters in Berlin with three of his generals ( _left to right_ ): Keitel, von Brauchitsch and Halder. Keitel placed great trust in what he perceived to be Hitler's genius; von Brauchitsch and Halder did not.\n\n_left:_ Franz Halder and von Brauchitsch at OKH headquarters having just heard of Hitler's plans to undertake a campaign in the West. They were horrified. _right:_ General Fedor von Bock, the commander of Army Group B, was originally tasked with the main effort in the West, but his role had been reduced to secondary one by the time this photograph was taken.\n\nGeneral Gerd von Rundstedt, commander of Army Group A ( _left_ ) _,_ and his Second Army commander Maximilian von Weichs ( _right_ ) _._ Von Rundstedt allowed his subordinates as much freedom of action as possible during the campaign.\n\nHeinz Guderian ( _left_ ) with Georg-Hans Reinhardt, photographed in 1938. Both were panzer corps commanders in May\u2013June 1940.\n\n_left:_ General Erich Hoepner, whose corps fell on Li\u00e8ge, fought at Gembloux and then pushed on to Dunkirk. _right:_ General Ewald von Kleist: a conservative officer, but the first commander of a German panzer group that contained five panzer divisions during _Fall Gelb._\n\n_left:_ General Friedrich Kirchner, the commander of 1st Panzer Division, whose leadership drove his formation forward at the spearhead of Guderian's panzer corps. _centre:_ General Erich von Manstein, whose offensive concept captured Hitler's imagination. _right:_ French prime minister Paul Reynaud, who came to office just a couple of months before the German invasion.\n\nGeneral Maurice Gamelin, the French Commander-in-Chief ( _centre_ ), and General Lord Gort, the BEFs Commander-in-Chief ( _left_ ) _,_ on 13 October 1939.\n\nChateau de Couroy, France, 5 November 1939: Winston Churchill, the First Lord of the Admiralty, talks to General Lord Gort. Lieutenant General Henry Pownall stands on the right.\n\nBethune, France, 23 April 1940: General Alphonse Joseph Georges, the commander of North East Front, inspects men from the BEF's Royal Inniskilling Fusiliers accompanied by General Lord Gort.\n\nA German armoured fighting vehicle factory. A prototype Neubaufahrzeug B is being refitted, with a Pz Kpfw.III Ausf. A. and several Sd.Kfz. 251 halftracks in the background.\n\nFrench troops prepare defences along the French border during the long, cold winter of 1939\u201340.\n\nAt an airfield near Arras, a pilots' briefing takes place before a Junkers Ju 87 'Stuka'.\n\nOne of the best fighters in the French air force, the American designed Curtiss H75C-1 (or P-36 'Hawk'). It accounted for a third of French air-to-air kills during the campaign.\n\nLeutnant Joachim Meissner, Oberleutnant Rudolf Witzig and Hauptmann Walter Koch \u2013 the victors of Eben Emael \u2013 listen to Hitler.\n\n10 May 1940: a Bren Carrier passes across the Belgian border from France at Herseaux as locals look on apprehensively.\n\nThe Ardennes: two light and under-armed German tanks emerge from the woods. Both the PzKpfw II, in the foreground, and the PzKpfw I following on behind were obsolete.\n\nSedan, 14 May 1940: the pontoon bridge over the Meuse at Gaulier being used by 1st Panzer Regiment, with a PzKpfw II and III patrolling the west bank.\n\nA French army signals post in an underground shelter. The doctrine of 'Methodical Battle' allowed commanders to rely on the telegraph, field telephones and couriers.\n\n_left:_ Sergeant Walther Rubarth, who led a small team of assault engineers at the head of 10th Panzer Division's 86th Rifle Regiment during the afternoon of 13 May 1940. _right:_ Hermann Balck. Although pictured here later in the war when a general, in 1940 Balck was the outstanding commander of 1st Panzer Division's 1st Rifle Regiment.\n\n15 May 1940: German tanks advance across featureless Belgian terrain before Gembloux.\n\n_left:_ Heinz Guderian in his command and communications halftrack (Sd.Kfz 251\/3), equipped with an Enigma machine. _right:_ Rommel (in the foreground) pictured with his corps commander, General Hermann Hoth, on 7 June 1940 at Eplessier, south-west of Amiens.\n\nFrench tank crews run to their Renault D2 (model 1935) tanks. It is likely that this unit is part of de Gaulle's 4th DCR.\n\n_left:_ Rommel and a group of officers from 7th Panzer Division study their maps at an 'Orders Group' during _Fall Rot._ _right:_ A poignant moment as Heinz Guderian receives the charred remains of a captured French Army standard during the latter stages of the Battle for France.\n\n1 June 1940: British troops march through Dunkirk during a German bombardment.\n\nSt. Val\u00e9ry, 12 June 1940: having just been taken prisoner along with 46,000 Allied troops, Major General Victor Fortune, the commander of 51st (Highland) Division, glowers at a preening Rommel.\n\nSurrendered Belgian troops being goaded by Wehrmacht troops on horseback.\n\n_left:_ French refugees heading south in June 1940. Although accompanied by their dog and some blankets piled on top of the bicycle, the family would appear to have few other provisions for their journey. _right:_ In the wake of the fighting in the Maginot Line, two Schutzpolizisten (uniformed police) visit a damaged bunker.\n\nAn unescorted low-level attack by an RAF Fairey Battle on a German transport column.\n\n14 June 1940: German infantry march through Paris near the Arc de Triomphe. Civilians watched, some in tears, as the occupiers began to take control of the 'City of Lights'.\n\nGerman infantry and armour pushing deeper into the heart of France during June 1940.\n\nAfter the campaign had concluded, Marshal Philippe P\u00e9tain greets Hitler, watched by Hitler's interpreter, Paul Schmidt, and Foreign Minister Joachim von Ribbentrop.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\nIf God Doesn't Show\n\nR. Thomas Riley & John Grover\n\nPublished by Permuted Press at Smashwords.\n\nCopyright 2012 R. Thomas Riley & John Grover\n\nCover art by Zach McCain\n\nwww.PermutedPress.com\n\nTABLE OF CONTENTS\n\nPart I \u2013 Thaddeus Archer\n\nPart II \u2013 Gibson\n\nPart III \u2013 Endgame\n\nAbout the Authors\n\n#  Part I\n\n# Thaddeus Archer\n\n#\n\n\"Everyone carries a shadow, and the less it is embodied in the individual's conscious life, the blacker and denser it is. One does not become enlightened by imagining figures of light, but by making the darkness conscious.\"\n\n\u2014Carl Gustav Jung\n\n#  Chapter One\n\nSeventeen Months Ago\n\nWashington, D.C.\n\nThaddeus Archer sighed as the alarm blared in the dark bedroom. It felt as if he'd just closed his eyes minutes before. He rolled over and reached out a hand to discover his wife's side of the bed empty. He felt the depression of her body and found the covers were cold. Sitting up in bed, he rubbed the sleep from his eyes. He peered about the darkened bedroom, and saw her silhouette rocking in the chair in the corner.\n\n\"Honey? Everything all right?\"\n\n\"I'm fine. Just couldn't sleep.\"\n\n\"How long have you been up?\"\n\n\"For a few hours.\"\n\nThaddeus slid from beneath the covers and padded over to her, laying a comforting hand on her shoulder. He grimaced at the frailness beneath his hand. She was losing weight, and it worried him. Thaddeus was a proud man, and it pained him to see his wife in this condition, where he was powerless to help in any way.\n\n\"The dreams again?\"\n\n\"Yeah.\" Tracy sighed.\n\n\"Want to talk about it?\"\n\n\"No, I'm fine. You need to get moving or you're going to be late for work.\"\n\nWith that, she brightened and rose from the chair, embracing Thaddeus and breathed deeply of his morning scent. The embrace caught him by surprise. She hadn't been intimate with him for months now, even cringing at his touch if he initiated contact. He hugged her close to him, once again noting the unsubstantial feel of her body against him. He sucked in a breath to steady his nerves and ventured, \"Maybe you need to talk to someone...\"\n\nTracy pulled back and eyed him like a cornered animal. Thaddeus tried to offer her a reassuring smile, but it felt more like a terrified grimace.\n\n\"I'm fine,\" she growled, eyes flashing with anger. \"I'm just tired, that's all.\"\n\n\"OK, OK.\" Thaddeus retreated. \"Get some rest. I should be home early today. The president is staying in-house.\"\n\nHe disengaged from the embrace and moved towards the bathroom to start his day. At the doorway, he paused and glanced back at his wife. She stood where he'd left her. Her right hand twitched as she watched him, but her eyes were vacant. Her look scared him more than any armed assailant's stare ever had.\n\nHe spent longer than he wanted in the bathroom, but he didn't want to face his wife just yet. He eyed himself in the mirror as he shaved, wrestling with a decision he was quickly coming to. Unbeknownst to Tracy, he had used his power as a Secret Service agent to pull her medical records. The general public was under the impression that medical records were private, but in this new age of The Patriot Act that was no longer the case. With all the disturbed individuals writing angry letters to the president, the service had the ability to secretly look into anyone's medical history without their knowledge or consent.\n\nHe'd been loath to do this to his own wife, but Thaddeus had to know. What he found made matter even worse. Tracy's family had a long history of mental disorders. He'd never known any of this, nor had Tracy ever mentioned her family's history. With research, he had discovered that mental illness sometimes skipped a generation or two, and he rested his hope on this. He thought of Casey, their daughter, and hoped this was the case.\n\nStill, knowing about this history was one thing, dealing with and approaching it would be an entirely different issue. The moral quandary of violating his wife's private medical records would wound her if she found out. They'd always been open and honest with each other from the first time they'd met, or so he'd thought. She'd be furious he'd used his Secret Service powers to look into her past as if she were a common criminal.\n\nHe heard her moving around in the bedroom and quickly finished up. After exiting the bathroom, he watched as she made the bed. As he watched her move, he sighed with happiness. She still had the ability to make his heart race just by looking at her. Pausing, she glanced over her shoulder.\n\n\"What?\"\n\n\"Just watching. And enjoying the view.\"\n\n\"You're going to be late,\" she admonished.\n\nThaddeus smiled as she blushed. \"I love you, you know that?\"\n\n\"You've mentioned it a few times,\" she said playfully.\n\n\"I just want you to be OK.\"\n\n\"I am. Trust me. I'm just tired, like I said.\"\n\n\"Baby? If there was something wrong, you'd tell me, right?\"\n\nShe turned to fully face him and crossed her arms. Thaddeus tried not to read into her posture. He'd been trained to intuit people's body language, and hers was screaming defensiveness. He would have to tread carefully. He hated treating her like a suspect, but that was exactly what he was doing.\n\n\"I always tell you everything,\" she said. \"I always have.\"\n\nThaddeus sighed, but plunged ahead. \"I'm just worried about you, that's all. Maybe you should go see a doctor...\"\n\nTracy's eyes narrowed at this, but she held her peace. Her body language became more pronounced, more defensive.\n\n\"Whatever it is, I'm here for you,\" he said. \"You know that. We'll get through it together.\"\n\n\"I'm fine.\" Her voice went tight.\n\n\"No, you're not.\" He'd already come this far. He wasn't going to retreat now. \"Something is wrong. I'm worried about you, honey.\"\n\n\"Stop treating me like a damn suspect.\"\n\nThe change in her demeanor was instantaneous. She rushed him, and he barely had time to react. He caught her wrist and scarcely avoided the slap. She gasped as he squeezed her wrist in his powerful hand.\n\n\"You're hurting me, Thaddeus.\"\n\n\"Mom? Dad?\" Both of them started and glanced at the doorway, where Casey watched with wide eyes. \"What's going on?\"\n\nThaddeus quickly released Tracy's wrist and retreated a few steps. \"Nothing, sweetie,\" he said, slightly out of breath from the adrenaline rush. \"We're fine. Everything is OK.\"\n\n\"It's OK, baby.\" Tracy eyed Thaddeus and nodded. \"Fine. I'll go see the doctor.\"\n\n\"Doctor?\" Casey echoed.\n\nTracy came over and embraced their daughter. \"It's fine, baby. It's probably nothing, but your father is worried, and I really should have a checkup. It's been a few years.\"\n\nAs she comforted Casey, Tracy gave Thaddeus a cold look. The love he'd seen earlier was completely gone from her face.\n\nThaddeus wanted to be anywhere but that bedroom. He raced over and grabbed his jacket and attached his gun to his hip. His wife and daughter were blocking the doorway, so he waited, rather than push past them.\n\n\"Dad only wants the best for you.\" She looked at Thaddeus, and offered a sweet smile as the embrace broke.\n\n_She's good_ , he thought.\n\nCasey had been bugging him for permission to go into the city with friends, and he'd been holding fast on saying no, but she was using this fight between him and Tracy to score some points by taking his side. He loved her for that. Yes, it was manipulative, but it was pure him as well.\n\nThe tension in the room dissipated as quickly as it had formed. Tracy broke the hug and motioned for Thaddeus to join the embrace. He gladly joined in, embracing the two most important women in his life close to his massive chest. Breathing both their scents in, he wanted to cement this moment in his mind as best he could. It was moments like these that allowed him to wade through the human filth he dealt with at work. It let him know there was still some good in life.\n\n#  Chapter Two\n\nFifteen Months Ago\n\nWashington, D.C.\n\nThaddeus pawed through the magazines on the table before him. He hated hospitals, had always hated them, and their reading materials sucked. He tossed the copy of Good Housekeeping aside with a sigh.\n\nCasey glanced up from the latest issue of Rolling Stone and grinned. Thaddeus frowned as he saw the boy on the cover. Really? _SUPER BOY_! the headline exclaimed. What had happened to music? he thought with disgust. This kid was everywhere, and teenage girls went wild for him, but he had no talent that Thaddeus could see. He was just a cute boy with a ridiculous haircut, and he was on the cover because he cut that hair. Really? Thaddeus missed the old days\u2014the days of Motown and the classics\u2014where singers actually wrote their own songs and possessed real talent.\n\n\"Take it easy, Dad. You're gonna explode.\"\n\nHe got to his feet and paced. He hated everything about hospitals\u2014the smell, the death and sadness.\n\nA door opened and Tracy emerged. Thaddeus stopped pacing and tried to gauge how the meeting had gone, but Tracy's face offered nothing.\n\nCasey leapt up from the couch and rushed her mom. She hugged her, but Tracy didn't return the embrace. Thaddeus saw Casey sensed something was the matter. The girl pulled back and eyed her mother.\n\n\"What's wrong?\"\n\nTracy ignored her and looked at Thaddeus. \"The doctor would like to speak with you.\" With that, she shrugged Casey away and walked out of the room.\n\n\"Dad?\" Casey's eyes brimmed with tears.\n\n\"It'll be OK, honey.\" Thaddeus made his way to the door of the doctor's office. He paused and hugged his daughter to his chest. \"Why don't you go find your mother? Get some food from the cafeteria or something. I'll be along shortly.\"\n\nThaddeus faced the door and steeled his nerves. He wasn't this nervous serving a search warrant. As he entered, the doctor glanced up from a medical file and smiled warmly. He came out from behind his desk and briskly shook Thaddeus's hand.\n\n\"Have a seat, Mr. Archer, please.\"\n\nHe sat in the chair, and wasn't impressed it was lower than the doctor's desk. He'd used the same tactic with interrogations.\n\nThe doctor returned to his chair and sat, peering down at the file before him. Thaddeus let the silence lengthen\u2014his one advantage in this loss of control. The doctor finally glanced up and closed the file before him.\n\n\"Mr. Archer.\"\n\n\"Just cut to the chase, Doc.\"\n\nDr. Klein, according to the brass nameplate on his desk, sat back in his chair and tented his fingers. He looked to be about forty years old, but Thaddeus figured he was closer to thirty. He wondered how many people the kid (he thought of him as a kid) had actually given bad news to before. Thaddeus shifted in the chair and tried to sit up straighter, but it was no use.\n\n\"Are you aware of Tracy's drug use?\"\n\nThis was unexpected. He blinked in confusion, and Klein eyed his reaction.\n\n\"I can see you weren't. OK, first things first. The drug use is a direct result of an underlying condition. Address that condition, and the drug use should be easily reconciled.\"\n\n\"Underlying condition?\"\n\n\"Your wife is ill, Mr. Archer,\" Klein said. \"Look, you want me to give it to you straight, so I am. Tracy is suffering from the onset of early dementia, made more serious by attendant circumstances of hereditary schizophrenia.\"\n\nThaddeus felt as if the chair would swallow him whole. He wrestled his bulk from the lush leather and stood. Klein leaned back in his chair slightly, but Thaddeus waved him off.\n\n\"Don't worry, Doc, I'm not going to do anything...crazy.\"\n\nThaddeus was the only one to laugh at this attempt at humor. \"So, let me see if I know what I'm talking about. Symptoms of schizophrenia include, but are not limited to, paranoia, hallucinations, trouble understanding or speaking a native language, religious mania, sudden bizarre behaviors, such as setting a room on fire and sitting to watch the flames, inability to take care of oneself, and problems with long-term and short-term memory. How am I doing so far?\"\n\nKlein nodded. \"I see you've been doing some research on your own. How long have you suspected?\"\n\n\"About a month.\" Thaddeus sighed. \"But Tracy hasn't exhibited\u2014\"\n\n\"She's in the early stages,\" Klein said. \"We will still have to run a few tests, MRIs and the like. Schizophrenia is profoundly difficult to diagnose, as it mimics many other mental illnesses. But after having a brief discussion with Tracy, I feel she is indeed suffering from it, based on my experience.\"\n\n\"The dreams...\"\n\n\"Yes, she's told me a bit about them\u2014reluctantly, of course. Mr. Archer, your wife is scared, confused, and feeling alone at the moment. The illness is making her paranoid and unsettled. Her reactions to me, to you and your daughter, are not really hers\u2014it's the disease. The faster we treat this with medication and therapy, the better chance we have of managing it.\"\n\nThaddeus stopped his pacing and placed his hands on the desk, leaning in close to Klein. \"You're saying put her in a facility?\"\n\n\"No, not at all. There's no need for that\u2014at least not yet. I know what you do for a living. I know you have an idea of what schizophrenics do, and how they act. You deal with them at their worst in your protection of the president. This is not your wife. This can be managed if we begin treatment right away.\"\n\n\"OK. Lay it out for me, Doc. What are the options?\"\n\n#  Chapter Three\n\nTwelve Months Ago\n\nWashington, D.C.\n\n\"Casey! Casey, have you seen my wallet!\" Thaddeus called, as he rushed about the kitchen. He was already going to hit traffic if he left now. Any more delays and fifteen minutes late would bleed into an hour. Just as his daughter drifted into the kitchen, he spied the wallet on the far counter. With a sheepish grin, he held it up as she rolled her eyes.\n\nShe'd been out all night again, he noticed. Eyes blood-shot and half-lidded, she fumbled for a bowl and cereal. Thaddeus opened his mouth to say something, but thought better of it. He couldn't afford to start a fight now, and he loathed leaving for work with each in their respective corners.\n\n\"Any plans today?\" He tried to sound as innocent as he could.\n\nCasey fixed him with a blank stare as she reached behind her for the milk on the counter. \"I think I'm just going to sleep for a bit.\"\n\n\"Rough night?\" Thaddeus cringed as soon as the retort left his mouth. \"Sorry.\" He sighed. \"I'm just worried about you, that's all.\"\n\n\"I'm fine.\" Casey gave him a sullen look, then burped and grimaced.\n\n\"Hey, kiddo... I know things are hard. They're hard for me too. I miss your mom\u2014really I do. Maybe we can go see her this weekend?\"\n\nAt first, Tracy had responded well to the treatments. She had seemed to be getting back to normal, or what passed for it. She still had her bad days and nights, but they had appeared less frequent. Once they'd figured out the correct dosages of meds, Thaddeus thought they'd be on the road back to regularity, but then Tracy relapsed and returned to drug use. Thaddeus tried to deal with the issue himself, but eventually admitted defeat, and checked Tracy into a rehab facility against her will. He still cringed at his wife's reaction\u2014the scathing things she'd said to him and Casey.\n\nCasey had had to watch the woman who given birth to her, bathed, fed, and clothed her, disintegrate right before her eyes. It'd been extremely hard on their daughter, and he'd done his best to comfort her, but in her eyes, he was the one who'd taken her mother away, shutting Tracy away in some facility.\n\nCasey sighed and turned her back to him, hunched over the bowl, and shoveled cereal into her mouth.\n\nThaddeus wanted to say more, but instead shoved the wallet into his pocket and walked out through the door into the garage. In a better time, he would've pecked his daughter on the cheek, but he hadn't been able to do that for months. She bristled at the mere attempt these days\u2014ever since he'd put Tracy in the facility.\n\nAs soon as her dad disappeared through the door, Casey pushed the cereal bowl back and quickly got to her knees, opening the cabinet below the sink. She removed the rolls of paper towels and flung them behind her. Next came the cleaning supplies, flung with just as much disregard. A sigh escaped her lips as her fingers closed on the small, cellophane-wrapped package.\n\n#  Chapter Four\n\nTwelve Months Ago\n\nRugby Rock, North Dakota\n\nThe candle flames wavered in the wind, but didn't go out. Any normal candles would have done just that, but these weren't normal circumstances. A small group, numbering seven, stood about the rough circle of flames and hummed a single, mournful note. The tone was somehow pleasant and terrifying at the same time. On the rock, in the middle of the circle of candles, a young girl struggled against her bonds, but to no avail. Her eyes glistened with terror in the glow of the purple light. She fixed each person around her with a pleading look that would've melted any sane person's heart. They merely stared at her and continued their chant.\n\nThe girl's name was Mountain Spring Runs Deep, if any of her captors had bothered to ask. She was on her way home from school on the Turtle Mountain Reservation, when a dust-covered van rushed up beside her and opened its doors to certain death. She knew no more until she awoke on this piece of bone chilling rock in the middle of the plains.\n\nAfter what felt like a lifetime, one of the figures stepped forward and pulled back the hood of her robe. The woman's features were striking, with deep, piercing eyes and a noble chin. She raised her hands and made some gesture, which sent fresh terror through the small Indian girl. It was obscene and gruesome, but if one were to ask the small girl why, she wouldn't have been able to put into words how the gesture could fill her with such dread.\n\n\"Tonight, we gather,\" the woman said, over the monotone drone of her coven. \"To beseech the Old One to return and free mankind of its cancer and wanton ways. Tonight begins the journey to our rightful place at his left hand. TONIGHT. WE. WILL. BECOME. GODS.\"\n\nThe girl saw the knife too late, as it came plunging down out of the darkness.\n\n#  Chapter Five\n\nTwelve Months Ago\n\nWashington, D.C.\n\nThaddeus sighed as the answering machine came on. \"Casey, I know you're there. Please pick up.\"\n\n\"Lemme guess. You're working late again.\" His daughter's tone held an obvious taunt.\n\n\"Look\u2014\" Thaddeus bit back the retort on the tip of his tongue with great effort. \"I'm sorry, but it can't be helped.\"\n\n\"So I guess we won't be seeing Mom today. Bet that makes you happy. Another day you don't have to see her and deal with what you've done. I'll be fine.\"\n\nClick.\n\nA blaring dial tone assaulted his ear, and he yanked the phone away. He sighed, thought about calling back, but knew she wouldn't answer if he tried.\n\nThaddeus leaned back in his chair and looked over the mountain of paperwork on his desk\u2014months and months of investigation notes he still had to go over. He'd be there for hours. Sighing, he stood, shrugged into his rumpled jacket, and walked out of the office. As he left, he glanced at the rest of his team, hard at work\u2014all had families.\n\nHe'd kept Tracy's condition as quiet as he could. Still, due to his sensitive position, he'd had to inform his bosses about it, but they'd been understanding so far. He wondered if they'd dug into Tracy's records yet, as he had. Eventually, they would, if he allowed his personal trials to affect his work life. Keeping the two separate was becoming quite difficult.\n\n\"What up, boss?\" Bill Cavalari asked as he noticed Thaddeus.\n\n\"Can you guys handle things for a bit?\"\n\n\"Sure.\"\n\n\"Good. I have to go see my wife and daughter.\"\n\nThe other agents nodded, but no one said anything. They all needed to get away from the office sometime. They'd been trained to read people, and they'd read Thaddeus. The unspoken elephant in the room wasn't entirely in his head.\n\nAn hour later, he pulled into the driveway. He'd rehearsed his speech to Casey all the way home, but now that he was there, all of it sounded trite and useless. Sighing, he got out of the car and trudged up the walkway to the house.\n\n\"Casey? I'm home,\" he called as he entered. \"You're right. Your mom is more important than work. I'm sorry.\"\n\nWhen he received no response, he made his way to Casey's room. The door was closed, and he could hear music blaring inside. After knocking gently, he waited. No response. Upon trying the door, he found it locked.\n\n\"Casey? Open up.\"\n\nThe music's volume lowered and the door opened a crack. He saw she'd been crying, and it broke his heart. In vain, she tried to hide the tears.\n\n\"Let's go see your mom. Get ready.\"\n\nCasey flashed a small smile, but she was still mad at him, he could tell. She wouldn't give that easily. \"Why? You don't want to see her.\"\n\nThaddeus sighed and held his tongue. \"I do want to see her. I miss her as badly as you do, baby. I know you know that. Truce?\"\n\nShe considered his offer and nodded. \"I'll be ready in five.\"\n\nReturning to the kitchen, he grabbed a beer from the fridge. They hadn't been to see Tracy in weeks, due to his being so busy with work, and he was nervous about seeing his wife. Would it be a good visit, or another bad one? Would Tracy be lucid like her old self, or would she be raving about demons, monsters, and the end of the world again? She'd become convinced, in the last few months, that a cult was after Casey, wanting their daughter for some diabolical purpose. Dr. Klein had warned Thaddeus that the religious delusions were a possible outcome of the dementia, but neither of them anticipated they'd be so severe.\n\nCasey appeared, as promised, five minutes later. The drive to the hospital was quiet, but Thaddeus let it be, as it wasn't a completely uncomfortable silence. He was lost in his own thoughts, and he wondered what Casey was thinking about.\n\nThey parked and made their way to the entrance. The smells assaulted him, reminding him why he hated hospitals once more. Casey slipped her hand in his and squeezed. He was surprised at the gesture, but he didn't question it.\n\n\"You're always so tense,\" she said. \"Why do you hate hospitals so much?\"\n\nShe could be so perceptive. Thaddeus smiled and shrugged. \"I guess hospitals remind me of how weak I really am.\" He was surprised at the honesty of his admission.\n\n\"Wow, that's deep,\" Casey teased good-naturedly.\n\n\"Your dad still has some surprises left.\" He grinned\n\nThey both fell silent as they approached the nurse's station. From beyond the secure doors, the screams and shouts of patients drifted out.\n\nThe nurse checked them in and led them to a visitation room, then left without a word. Thaddeus waited nervously for the orderlies to retrieve Tracy.\n\nCasey sat on the couch and played with her hair. It was her nervous twitch, Thaddeus knew. He wondered what his was, and grimaced as he found himself pacing like a caged animal.\n\nThe door opened, and the orderlies ushered Tracy in. She looked from him to Casey, but showed no recognition.\n\nCasey rushed from the couch and hugged her mom. \"Mom! I'm so happy to see you!\"\n\nTracy did not return the embrace. She merely stared at Thaddeus. \"They're coming. They're close now. Soon, the Old One will rise and take our daughter.\"\n\nCasey pulled away from her mother and smiled through her tears. \"It's OK, Mommy,\" she soothed. \"I'm safe. Dad won't let anything happen to me.\"\n\nIt broke Thaddeus's heart to see his daughter struggle to be so strong for Tracy. He approached with a hesitant step. Tracy smiled and glanced down at their daughter. \"Casey! I'm so happy to see you!\"\n\nHer demeanor changed, and the \"other\" Tracy vanished. Thaddeus unleashed a sigh of relief as his wife hugged Casey close.\n\nThe orderlies shrugged and left the room.\n\nThe rest of the visit went well. Tracy showed no signs of retreating to her delusions. They caught up. Thaddeus told Tracy about his job, and Casey talked about school and boys.\n\nThaddeus was taken aback at the mention of boys. He hadn't known Casey was seeing anyone, and he felt a rush of protectiveness. When had she started seeing this boy? While he was at work? He made a mental note to discuss this with her on the drive home.\n\n\"Casey? Why don't you go get us some sodas?\" Thaddeus wasn't thirsty, but he wanted some private time with Tracy out of Casey's earshot.\n\nShe gladly agreed and left the room. As soon as the door shut, he leaned forward and grabbed his wife's hands in his own. \"Tracy.\" A sob threatened to escape, but he shoved it down. \"I miss you so much, baby.\"\n\n\"I miss you too, honey.\" She smiled and patted his hand. \"They treat me well in here. I'm happy.\"\n\n\"Are you sure?\"\n\n\"I'm sure. I am getting better, you know.\"\n\n\"I'm glad. I'm glad, baby. I can't wait to take you...\"\n\nTracy leaned forward and pressed her forehead to his. \"Listen to me,\" she hissed. \"They're talking to me in my dreams. They want Casey. They want her badly. You have to protect her. They'll be coming for her soon. You can't trust anyone. That boy is trouble...\"\n\nThaddeus looked into her eyes and sensed she was still coherent. \"Please, Tracy, enough.\"\n\n\"You have to listen to me. There isn't much time left. They're already making preparations in the sacred place. The plan is already in motion at the highest levels of government. Soon, you will be tested.\"\n\n\"Please stop.\" Thaddeus sighed.\n\n\"The shadows are coming.\"\n\nTracy pulled back as Casey returned. The look of fear on her face vanished, as if it had never been there. Thaddeus remained quiet for the rest of the visit as Tracy and Casey caught up, giggling like two school girls. He was glad he'd come for Casey's sake, but for his own, it was torture to see his wife in this condition. Her warnings ate at his very being and unsettled him. Sure, they were delusions, but it still scared him thinking about Casey being in any type of danger. He'd lost Tracy, but he wasn't going to lose Casey, no matter what he had to do to protect her.\n\n#  Chapter Six\n\nTwelve Months Ago-\n\nRugby Rock, North Dakota\n\nThe woman pulled the robe over her head and carefully folded the sacred garment, ensuring the bloodied fabric didn't touch the ground. She placed it in a plastic-lined briefcase and snapped the lid closed. Across the way, the others did the same. The sound of trunks and doors closing echoed across the deserted prairie. Soon, they would all be back in their respective positions of power across America and the rest of the world, awaiting the rise of the Old Ones.\n\nShe shivered with excitement as she contemplated the decades, perhaps centuries, of work culminated on this night.\n\nShe checked the luminous dial on her watch. She would be expected back in Washington by morning. Her protective detail was under the assumption she was still there, and she didn't want to give them any reason not to think just that. She flipped open an encrypted SAT phone. \"Have the jet powered up, I'll be there shortly.\"\n\nTheir next target was important. There could be no mistakes, no failures. They were so close. Soon, the cult would have all the pieces in place, decades of planning all coming to fruition. Pushing the excitement down, she concentrated on the next stage of the plan. The mother was taken care of, locked away in a facility, considered crazy. Tracy had almost become a problem, but Tracy had made her deal long before. Her marriage to Thaddeus Archer was no mistake. She'd been groomed to be his bride, long before they'd ever met. The child was the final step.\n\nNow, Archer was lead agent for the president\u2014his rise to such a position orchestrated by those in the cult who were in places of great power. The machinations of the cult were all set in motion.\n\nWith one last look at the blood-spattered stone altar, she got into the SUV and waited for what was to come.\n\n#  Chapter Seven\n\nTwelve Months Ago\n\nWashington, D.C.\n\nCasey planted a kiss on her father's cheek. He was working night detail for a benefit the president was attending. She could barely contain her excitement at the night to come. Her dad broached the subject of Steve on their ride home from the hospital the previous day. She'd been vague, but her explanations seemed to satisfy his concerns. No, Steve had never been in the house when Thaddeus was at work. A small white lie. He'd come over a few times, but Casey and he sat in his car in the driveway. She didn't trust herself to be alone with him in the house. At least, not yet.\n\nCasey definitely had feelings for Steve, but she wanted to take it slow. He made her feel all fluttery inside\u2014this she didn't tell her dad either. They were friends, met at school. She was interested in him, but hadn't said anything to Thaddeus until she was sure of where the relationship might be headed. He pressed her for more details, and the details she gave him were mostly the truth.\n\nShe'd been so excited to see her mother, the subject of Steve had just slipped out. As soon as it did, Casey had seen her father stiffen, and she realized her mistake. Casey felt comfortable she'd assured him enough on the drive home.\n\nShe considered calling her mother, but realized she probably wouldn't be in any shape to talk. She didn't mean to lash out at her father. After all, it wasn't his fault, and she didn't even know why she had gotten angry with him. Her mother had become a danger to herself at home, and the hospital was a safe place for her. Plus, Mom seemed to be getting better.\n\nStill, her father had taken her mother away against her will. The night he'd admitted her was terrible. Her mother had clawed at the doorjamb as the cops dragged her away, while she'd screamed obscenities at both Casey and Thaddeus. Things she didn't mean, couldn't mean. There wasn't a vicious bone in her mother's body, but that night had rocked Casey to the core. She'd never seen her mother in such a state, and the look of helplessness on her father's face was even worse. He'd always been her rock, her constant. Dad could take care of anything, no matter what it was. But that night, she'd seen him broken, unsure, deconstructed.\n\nA knock came at the door. She got up from the couch and made her way through the debris left from her night alone: an empty pizza box, a half-full bowl of cold popcorn, and empty soda cans.\n\n\"Yeah?\" she called through the door.\n\n\"It's your knight in shinning armor,\" came the playful reply.\n\nCasey smiled and flung the door open. The happy look on her face stayed in place even as she focused on the masked man standing in her doorway.\n\n\"Steve?\" she managed, before the intruder forced his way in and clubbed her in the face with a massive, gloved fist.\n\nThe man knelt down beside her unconscious form, checking to ensure she was merely knocked out. He gained his feet and called back into the night: \"She's taken care of. I want my money now.\"\n\nA flash of light exploded in front of the darkened doorway, and Steve collapsed to the ground, with a neat, smoking hole in the middle of the black ski mask.\n\n#  Chapter Eight\n\nTwelve Months Ago\n\nWashington, D.C.\n\nIt took a special type of individual to be a cop, and an even more specialized person to protect the Leader of the Free World. Thaddeus pulled into his driveway that early Friday morning, and as soon as he turned the car off, his instincts began blaring.\n\nEvery light in his house was aglow\u2014the first indication something was amiss. Thaddeus drew his service weapon before he exited the car. He checked the perimeter out of habit, but noticed nothing odd. Had his wife checked herself out of the facility? No, he'd made sure that wouldn't happen again.\n\nHe'd rather face a cadre of armed terrorists than Tracy. He paused as he noticed the front door wide open. The light from inside flooded out onto the walkway, and he spotted the legs just inside.\n\nQuickly, he circumvented the side of the house and made his way to the back door. He hated entering the house with all the lights glowing, but he had no choice. Entering low, he swept the kitchen, but found no intruders. He cleared the rest of the house and upstairs, before he forced himself to check the foyer, where he'd spied the pair of legs.\n\nBlood had pooled and dried on the tile. Archer had seen plenty of blood over the years, and from the texture, he realized the scene was a few hours old. He pulled his cell from his belt and pressed speed dial #1.\n\n\"Archer here. Code 7310. I've got an issue,\" he said calmly, though he felt anything but.\n\nWhere was Casey?\n\nThe neighborhood was crawling with Secret Service, FBI, and locals within minutes.\n\n#  Chapter Nine\n\nTwelve Months Ago\n\nWashington, D.C.\n\nThaddeus watched the CSI unit as they dusted the door for prints. He felt helpless. He'd tried calling Casey's cell for hours, but it had kept going to voice mail, which meant the phone was switched off. No luck trying to GPS it then, either.\n\nHe turned from the house and saw Cavalari approaching. The last man he wanted to deal with at the moment.\n\n\"We're doing everything we can, boss. Think Tracy had anything to do with this?\"\n\n\"NO!\"\n\n\"Easy, calm down. We're just covering all the bases.\"\n\n\"I know you are.\" Thaddeus sighed. \"I'm just on edge because I want to find out what happened to Casey. Anything on the dead guy at my front door?\"\n\n\"We ran prints. No record, but we got a hit with DMV.\" Cavalari paused to check his notebook. \"His name is Steve Booth...\"\n\n\"Steve?\"\n\n\"Yeah? You know the kid?\"\n\n\"No, but Casey mentioned him. Never met him.\"\n\nCavalari smirked at that. \"Kids these days. Who really knows what they're up to unless you ankle bracelet them, right? Sorry, I didn't mean it that way...\"\n\n\"Focus,\" Thaddeus ordered. \"I'm still your boss. You don't have my job yet, Cavalari.\"\n\n\"Right, right. I'm sorry, Thad.\"\n\n\"Does he know yet?\"\n\n\"Of course he does. You're his lead agent. RAVEN was notified as soon as you let us know.\" Agent Bill Cavalari smirked again.\n\nThaddeus resisted the urge to throttle the man. He stalked over to a uniform who nervously eyed the massive blood pool and snatched a smoke from the guy's hand, taking a long drag while he studied the ceiling.\n\n\"For the time being,\" Cavalari continued, \"I'll be taking lead until this whole thing is cleared up.\"\n\nArcher glanced at the man, a healthy dose of 'shutthehellup' in the stare. He wanted to do something, anything, but protocol dictated he not be a part of the investigation, as he was so intimately involved. Roadblocks were at every key location across the state, and having worked closely with the Staties and Highway Patrol, Archer knew they were doing their jobs. They couldn't do anymore than he could've, had he been out there with them.\n\n\"I'm going upstairs,\" he said over his shoulder.\n\nThaddeus sat on his daughter's bed and glanced across the pictures on Casey's dresser. A story of a life could be seen there. Casey a few hours after being brought home from the hospital, in her mother's arms. Tracy looked happy. They'd had their whole lives in front of them. Thaddeus fresh out of the service academy, in stiff pose, practically daring the camera to capture his image. Their first house, so small and cramped, but they were happy. His gaze fast forwarded to pictures taken in this very room\u2014Casey at probably six or seven.\n\nJust then, something caught Thaddeus' eye. He got up from the bed and pulled the dresser drawer fully open. A baggie with some kind of white substance. He sucked in a breath as he realized what it was.\n\n\"Oh god.\" He moaned.\n\nTracy was one thing, but his daughter following in her footsteps was something he resisted contemplating with all of his being. A piece of paper was wrapped around the bottom of the baggie. He pulled it off and carefully smoothed it out on the dresser.\n\nIf you ever want to see your daughter again, you will answer her cell at precisely six a.m.\n\nBelow the typed lines, some type of symbol was stamped in what appeared to be wax. Thaddeus brought the note to his nose and sniffed. Yes, it was wax. He wasn't sure what the symbol was, but it filled him with an irrational dread. It looked like a goat's head. What the hell had Casey stumbled into? He saw her phone and grabbed it, powered it up, and raced downstairs.\n\n\"Cavalari! Start a tap on this phone now,\" he shouted.\n\n\"Whose?\"\n\n\"It's Casey's. And I have a note from the kidnappers...\"\n\n#  Chapter Ten\n\nTen Months Ago\n\nArlington, VA\n\nTwo months later, Archer was still waiting for his daughter's phone to ring. Someday it would. It just had to.\n\n\"Is she in there?\" Thaddeus shouted as he tried to push past the police barricade.\n\n\"Easy now,\" Detective James Catz urged. He placed a hand on Archer's chest and refused to budge until Thaddeus looked him in the eye.\n\nThaddeus realized Detective Catz had worked tirelessly on the case, and all their man hours might have just paid off. Catz wasn't about to let a hysterical father muck it all up at the last second. Thaddeus calmed himself and squared his shoulders. It'd been a grueling two months.\n\nHe'd given up his position as lead agent, relinquishing it to Cavalari, who'd been quite happy to take the position. Technically, Thaddeus was still in the employ of the Secret Service, but all his time was consumed with finding Casey. President Wendell had given Thaddeus all the help he could, and his blessing. Thaddeus discovered a new found respect for the man in the following months. Thaddeus tracked down every lead, no matter how innocuous it seemed, a driven man, with one sole mission.\n\nHe'd dug into Steve Booth's past and discovered the kid had been into some pretty weird stuff. He'd been a member of a cult, and Thaddeus soon came to realize Casey had been targeted. After protecting the president from so many threats over the years, to be targeted himself sobered Thaddeus. He'd always been so paranoid about his family paying for his job, but he hadn't been paranoid enough. The ordeal with Tracy had distracted him, dulled his edge. He'd talked with Booth's parents, but hadn't gotten anywhere. They'd been as unaware of their son's obsession as he'd been about Casey's secret life with Steve.\n\nAfter turning Steve's life inside out, Thaddeus discovered Hediam McBain. At first glance, it'd appeared the man was a nut with a religious obsession, but the more Thaddeus delved, the more it became apparent the man was someone to take seriously. Thaddeus did what he knew\u2014treated the cult like a terrorist cell, and tracked all the threads until he found the epicenter of the cult and its leader.\n\n\"Look, Thad, I know how you feel. Some of the cult may be holed up inside that warehouse, but we're not sure how many, or even if Casey is in there.\"\n\nTaking a deep breath, Thaddeus backed up a few paces. He attempted to smooth his rumpled suit, but it was a useless effort. He realized it was more an excuse to keep his hands busy than anything else.\n\n\"I'm fine,\" Thaddeus mumbled as he eyed the warehouse just down the lot. \"They're not going anywhere?\"\n\nCatz caught the meaning loud and clear. \"No, it's locked down. Only way they're coming out is in cuffs...or body bags\u2014\"\n\n\"It's really them?\"\n\n\"Yes, we're almost certain Hediam McBain is in there. Listen, the next few hours are going to be tense. Maybe you should...\" Thaddeus could tell Catz was considering asking him to stand down, but his next words proved he chose a different track. He glanced around to ensure he wouldn't be overheard, then leaned into Thaddeus. \"If this goes sideways, you're on the entry team.\"\n\nThaddeus nodded and walked back behind the barricade, leaned against a police car, and lit a cigarette. Mentally, he went over the past two months, and all he and Catz had managed to assemble about the weird cult who'd kidnapped his daughter. What frustrated him the most was the promised call had never come. Sure, they knew a lot about this group of religious psychos, but nothing as to why he'd been targeted. It would've been one thing if they'd tried to blackmail him in order to get close to the president, but they hadn't asked for anything.\n\nA print had been lifted from the note in Casey's drawer, and that print had led them to a Hediam McBain in North Dakota. He'd admitted to knowing what the sigil on the note stood for, even admitted to having been the one who created it, but his story was so far out there Thaddeus and Catz had really started to doubt the guy's involvement. He was a three-time loser\u2014a former priest, who fell from grace after he started spouting about Old Ones and Lovecraft from the pulpit. When the Church removed him from his parish, Hediam moved to a desolate section of North Dakota\u2014a place called Rugby Rock. His reasoning? Rugby Rock was the geographical center of North America.\n\n\"There's movement inside!\" a voice rang out.\n\n\"Report!\" Catz ordered into his hand mike.\n\n\"This is tactical one leader. We have visual of the occupants inside. There's a young girl. She's tied up and McBain is armed...oh Jesus...he's going to stab her!\"\n\n\"Move, move, move,\" Catz shouted.\n\nThaddeus was already in motion, racing towards the warehouse. He pulled his weapon and crashed through the door. Everything slowed to a crawl as he entered the cavernous warehouse. It was the size of a small aircraft hanger, devoid of anything but the altar at the center.\n\nA group, clad in blood-red robes and numbering close to twenty, stood in a circle, their attention focused inward on the girl strapped to the altar. None of them turned as doors all around the warehouse were breached by the tactical teams. Shouting echoed about the space, but the group acted as if they were in a trance.\n\nThaddeus raced towards McBain as the gleaming knife in the leader's hand began its deadly descent.\n\n\"Drop the knife! Drop the knife!\"\n\nThe deranged cult leader ignored him.\n\nThaddeus was on his third shot before he realized he'd started firing. The bullets ripped into McBain, causing him to jolt about like a puppet on a string. Thaddeus found himself flung off his feet, as a deep coldness settled into his shoulder. The group turned as one and produced machine guns from beneath their robes, firing in all directions. When the smoke cleared, five agents were dead and another fifteen were injured, including Thaddeus, who'd taken two rounds to the vest.\n\nCasey hadn't been the girl on the altar.\n\n#  Chapter Eleven\n\nNine Months Ago\n\nArcher Household, Washington, D.C.\n\nWith empty whiskey bottles cluttered at his sides, Thaddeus sat on the living room floor. The phone cord was torn from the wall, and empty pizza boxes were stacked on the sofa. The sofa where Casey had fallen asleep so many times, waiting for him to come home from work. The shades were drawn, and the TV appropriately smashed. There would be no distraction, no disruptions.\n\nThaddeus Archer shoved his service pistol into his mouth and closed his eyes. The sharp taste of gun oil and metal filled his mouth. It was oddly comforting, rather than distasteful. His finger caressed the trigger, while he envisioned the bullet exiting the barrel in an explosion of gas and cordite, entering the roof of his mouth, then tearing a tunnel of relief through the top of his skull.\n\nThe cell phone vibrated in his pocket. He'd forgotten to turn the freaking thing off. Gasping, he realized it was Casey's phone.\n\nHe jerked the gun from his mouth and took the cell from his pocket.\n\nA feminine voice greeted him on the other end. \"Giving up so soon? Now, now, what would Casey think of that?\" The woman made only the slightest attempt to disguise herself.\n\n\"Damn you! If I ever get my hands on...\"\n\n\"Patience, Thaddeus. All in good time. She is safe in our hands. It's the safest place in the world, in fact.\"\n\n\"What in Christ's name do you want?\"\n\n\"Hell, Thaddeus. We want Hell. Hell's coming. Thank the Gods Casey will be with us when it comes.\"\n\nIn the background, he heard a faint voice call out. It was Casey\u2014she was alive!\n\n\"Do you hear that? See, she's still safe and sound.\"\n\n\"Casey!\"\n\n\"That's right. Casey. We're counting on you. When the time comes, we know you won't let us down.\"\n\n\"What do you want? Give her back to me. Give her back or\u2014\"\n\n\"We should make you pay for McBain...\"\n\n\"He had it coming, bitch! You're next!\"\n\nThe caller hung up. Archer threw his pistol across the room and stood up in angry defiance. He went to the bathroom, relieved himself, and splashed water on his face. \"Get yourself together,\" he snapped at his reflection. \"You can't help anyone like this. She needs you. She needs you now.\"\n\nHe made a vow to himself, and to Casey, right then and there to never give up.\n\n# Chapter Twelve\n\n**Present Day**\n\n0600 - New Orleans, Louisiana\n\n\"Hey, Archer! Fancy seeing _you_ here.\" The voice was familiar, but Thaddeus couldn't quite place it. He turned to see who called to him.\n\n\"Bill.\" His tone remained neutral, with no indication the intrusion was welcome.\n\n\"So, you're Park Pooolice, now?\" Bill Cavalari gave his signature smirk.\n\nThaddeus couldn't read if the man was being condescending or genuinely interested. Even though it was still dark out, sleek, black sunglasses covered Cavalari's eyes. Archer bit back a retort eager to leap from his lips, and merely smiled. \"Bill.\" This time, a slight uplift in his tone. \"Figured we'd run into each other eventually.\"\n\n\"Hey, I'm really sorry about...you know... What happened?\" Cavalari seemed to wait for Archer to step in, finish the thought, but when he didn't, the man faltered. \"Good to see you landed on your feet...\"\n\nThaddeus flashed a cold smile and held up a clipboard. \"Wish I could chat, but got a job to do, you know? Shouldn't _you_ be doing something?\" With that, Archer turned and stalked off.\n\nThe headache that'd been building blossomed behind his left eye. The headaches weren't new, but the chattering in his ears was. Archer had never been the type to run to a doctor at every ache or pain, but this seemed different, and the beginnings of concern flittered about his mind. That chattering was going to drive him crazy if it didn't let up soon.\n\nOnce around the corner and out of sight, Archer paused and leaned against the wall. He struggled to keep his breathing in check as he shook with barely repressed anger. _The bastard actually had the nerve to try and talk to me?_ He straightened up, smoothed his jacket with a shaky hand, and took a deep breath. \"Let it pass, buddy. Nothing you can do about it.\"\n\nAll he wanted to do was get this function behind him and get out of there. There were too many memories. Casey's face flashed through his mind and brought all the pain back. It felt like he was being shot all over again, and the old wound in his shoulder started to throb once more. Thaddeus knew the wound wasn't actually throbbing; it was a phantom sensation the doctors couldn't figure out. The bullet hadn't penetrated his shoulder, for his vest stopped it, but his mind still thought the bullet pierced his flesh. Guilt was tricky.\n\n\"Archer? You all right?\" Roger Rios asked.\n\n\"Uh, yeah, yeah, I'm OK,\" Archer replied in a rush of breath. \"Just...nothing. It's nothing. How are we on the arrival protocols?\" His voice resumed its strength. A warning look told Rios to keep his mouth shut.\n\nRios wisely changed subjects. He still wasn't sure what to make of his new boss yet, Archer could tell, probably thanks to the rumors that followed him from his prior job. Rios knew Archer had once been Secret Service, and he knew something forced him to leave the service. The kid, he suspected, knew he was carrying some pretty bad juju.\n\nDespite the past, Archer knew his business, and was very good at what he did. This was the reason he wasn't completely booted from federal service. There'd been strings pulled and people in high places cajoled. A lesser agent wouldn't have bounced back from what Archer had been through. He knew Rios knew next to nothing about his ordeal, and the little that leaked to the press a year ago left much to speculation.\n\n\"Secretary Carling will arrive at 0910 and will enter through the south entrance of the hotel,\" Rios said. \"No problems there, as SS is using that entrance as well for RAVEN's approach. So, that's pretty much locked down.\"\n\nArcher cringed as he heard his officer use the president's identifier. Now, that was one encounter he most assuredly hoped to avoid. True, President Wendell had saved his hide and his career, but Archer didn't feel like having to face the man he'd failed a year before.\n\n\"This heat is killer.\" Archer shifted uncomfortably in his suit.\n\n\"Yeah, can't wait to get back to D.C. We're not too popular down here as it is.\"\n\nArcher walked over to a window and peered down onto the promenade, where the protesters were already gathering to greet the president.\n\n\"This is going to be interesting.\" He sighed. \"Freedom of speech sucks.\"\n\n0900 \u2013 Prep Area\n\n\"Sharpshooters in place?\" Agent Bill Cavalari spoke into his hand mike. He glanced over at Archer and grinned. At that moment, Archer wanted more than anything to smash that smug idiot in the face with the butt of his Glock. Instead, he nodded stiffly and ignored his former underling. After he left the service, Cavalari stepped into the vacated position as easily as changing socks. Was he the most qualified for the job? No, but he was an expert ass kisser. Archer suspected it was Cavalari who'd turned him in, but now he wasn't so sure. Cavalari was a lap dog, subservient, not aggressive on his own. He would've been pushed into the betrayal.\n\nThaddeus made a lot of enemies during his time in the service. In Washington, it was easy to cross someone and not realize it until it came 'favor' time. Even now, he still watched himself, as some were none to happy he landed on his feet after what happened a year previous. From the outside, sure, dead to rights, Archer should've been rotting at some retirement villa in Florida. That much was true, but somehow, despite the incident, the president had his back, and he made sure Archer was punished, but not too severely.\n\nThere were only four people who knew the truth about what happened\u2014not the cover story crafted by the best spin masters in Washington. The truth probably wouldn't have been believed anyway. Archer was thankful for the help, but he'd never worked up the nerve to ask why he'd been spared. Maybe he didn't want to hear the answer. Even if he'd gotten off relatively easy, he made up for it by not forgiving himself. He'd come to realize that his own loathing was worse than anything anyone else thought of him.\n\n\"You still carrying that cell phone,\" Cavalari said.\n\nThaddeus self-consciously adjusted Casey's punk-pink compact phone on his belt. He gave the man a look and chose to ignore him. It probably was crazy to keep the phone, but he did it anyway. He faithfully kept it charged and checked to ensure service. One day it would ring again. He had to keep telling himself that.\n\nArcher brought his radio to his lips, keyed the mike and said, \"Look alive, people. Package is inbound, minus two minutes.\"\n\nHe cringed as the cheap radio squelched feedback. Cavalari shot a disgusted look in his direction and Archer flipped him off. He walked away from the agent before he could retort, rushing down the hotel's grand staircase to personally meet Secretary of the Interior, Patricia Carling. Carling was an interesting woman. Once you got past the frosty exterior and figured out how the woman operated, she wasn't too bad of a boss. Not to say she wasn't personable, but in Washington, one had to be aloof to a certain extent. The running joke around the Mall was the walls had eyes, and if they didn't, they sure had ears. One unguarded moment could haunt a person for years when it came to a career.\n\n\"What? Say again?\"\n\nArcher paused on the staircase and glanced up at Cavalari. The tone in the agent's voice was what stopped him in his tracks. Cavalari rushed away from the window and past the stunned Archer. \"Code Black, Code Black,\" Cavalari screamed into his mike.\n\nArcher's radio chattered, \"Lead One, Lead One! We are under attack. I repeat, the motorcade is under attack!\"\n\nArcher returned, \"Evade, Evade. Use alternate route 207.\" He raced after Calavari.\n\n0904 \u2013 Presidential Motorcade\n\n\"This will be a quick meet and greet, Mr. President,\" the aide rattled off, as she read from the clipboard-sized PDA resting on her lap. \"SS have indicated there are protestors present, so we want to keep it short and sweet.\"\n\n\"You know, whenever you refer to the Secret Service as 'SS,' I cringe, Melissa.\" President Neil Wendell laughed. Beside him, Secretary of the Interior, Patricia Carling, chuckled on cue. He brought the glass of scotch to his lips and sipped. \"Remind me to give my wife a call soon as we're finished here.\"\n\nHe placed a too-friendly hand on the secretary's knee, Melissa noted, as she dutifully jotted down the reminder and tapped the stylus on the screen. \"You're meeting with the police chief, and this will be in front of media, so keep it short. We don't want Chief Dryer getting cute...He's a Democrat.\" She winked.\n\n\"You hear that?\" Carling asked abruptly.\n\n\"What?\"\n\n\"That chattering sound?\"\n\nThe car lurched to the left and threw its occupants soundly against the thick, unyielding bulletproof glass. In the rear, Air Force Major Darren Woodrow, the man with the 'Football,' clutched the black leather satchel to his chest in an effort to protect its contents. An explosion rocked the heavy car.\n\n\"What the\u2014\" President Wendell managed, before a Secret Service agent smothered him and Carling beneath him.\n\n\"Stay down, Mr. President, Ms. Secretary!\" Into his mike, Agent Jason Augustson said, \"Status, what's the status?\"\n\nAt the front of the motorcade, the lead SUV disappeared in a ball of fire as the RPG found its mark. The occupants never had a chance. The next vehicle in the convoy slid to a stop and spilled its tactical team onto the roadway with frightening precision. Agent Derrick Wells was in the second SUV. As he leaped from the vehicle, the scent of burning flesh, rubber, and steel struck him full in the face. He faltered, as images of a war-torn Iraq, a scene similar to this one, replayed in his head. He heard his Marine buddies screaming as their Humvee burned, entwined with the screams of his fellow agents in the SUV in front of him. The past and present bled together in his head.\n\nThe vehicle behind the SUV exploded into flames, and the presidential limousine shuddered as it was struck by another projectile, but it stayed intact. A single gunshot rang out, and the shooter reloading the RPG slumped from sight on the adjacent rooftop. The launcher plummeted to the ground. The impact caused the launcher to discharge a wild missile directly at Derrick. He screamed for everyone to take cover and dove for some himself. The massive explosion rocked the street, and as Derrick checked the rooftops for a second attacker, he felt as if he were hearing everything from underwater. He realized either one or both of his eardrums were perforated. Ignoring the pain, he focused on the surrounding rooftops.\n\nSatisfied there were no targets present, he gained his feet to rush back towards the burning limo. He turned to urge his fellow agents on. What he found caused his stomach to clench. Agent Eric Peebles lay slumped up against the door of the SUV, trying to put his guts back into his body. The steaming entrails looked like uncooked sausage links to Derrick. The only thing left of Agent Kara Garner was a bloody, ragged stump. He recognized the body part because one of her shoes was still on the blackened foot.\n\nAt the rear of the motorcade, the up-armored chase SUV roared up the road towards the limo. It screeched to a halt beside the door, and Augustson emerged with the president and Secretary Carling.\n\n* * *\n\n\"Report!\" Lead Agent Allen Melendez shouted, as he scanned the surrounding rooftops for more bogies. The massive automatic rifle in his hands was primed to find a target. He focused on someone stumbling in his direction from the front of the convoy and almost squeezed off a shot. Agent Wells was almost unrecognizable.\n\n\"Agent!\" Wells announced himself loudly. \"They're all dead. They're all dead, Allen!\"\n\n\"Take the Package!\" Melendez ordered.\n\nIn the space of a few seconds, a perimeter was established around the burning wreckage. Satisfied his people, those still alive at least, were accounted for, Melendez gave the order to move RAVEN to the up-armored SUV. As he finished the order, someone popped up on the roof. Or rather, something popped up.\n\n\"What the\u2014-\" Melendez muttered.\n\nThere was something wrong with the form. Rather than being solid, it appeared to be flowing, undulating in the distance, and it was black as midnight. A perfect rendition of a shadow, the figure did a header off the roof and crashed to the street. No sooner had the form hit the ground, but it shambled to its feet, dragging a man's body with it. The agents opened fire on the approaching man. Bullets slammed into the attacker and gave him the appearance of a broken marionette. Yet, _he still kept coming_ , even after his left arm was severed at the elbow by a well-placed, high velocity round.\n\nMelendez didn't take time to wonder how the attacker was still standing, much less alive. He leapt into the SUV and, as they sped off, he watched in horror as more men approached from the north of the motorcade. Wells cursed as he saw one of them looked like Agent Peebles. They were shambling, their shadows leading rather than following.\n\nSomething else would trouble him later. He'd seen a bullet ricochet off one of the vehicles, and it nicked the ground where Peebles's shadow was cast. When it did, the agent collapsed and remained motionless.\n\n#  Chapter Thirteen\n\nEight Hours Ago\n\n0907 \u2013 Safe Haven\n\nThe alternate safe haven was three miles to the east of the hotel, at a private compound the FBI used from time to time to house transient witnesses in their WIT-PRO. As the SUV fled from the decimated convoy, Archer and Agent Cavalari made their way to the safe haven with their respective people.\n\nArcher drove as fast as he dared. Many of the streets in New Orleans were still all but impassable, since the hurricane three years prior. As he navigated the debris, Rios scanned the police band and their own radio net, trying to determine exactly what was going on. There were a lot of confused voices stumbling over each other on the police band.\n\n\"Patrol 751, repeat your last message,\" a frantic sounding dispatcher ordered. \"You're getting stepped on by another relay.\"\n\n\"I repeat, something's going on,\" an officer shouted into the mike, the first part of his message distorted. \"They're everywhere. Holy\u2014\"\n\nRios glanced over at Archer. \"What do you make of that? What's going on, man?\"\n\n\"We'll find out soon enough,\"\n\nUp ahead, the gates of the compound rose into view. As they pulled off to the side of the long driveway, Cavalari stalked about screaming orders. He locked gazes with Archer, seemed to want to say something, then thought better of it. \"Thad, I need you.\"\n\nDespite the circumstances, Archer had to admit the request for help was vindicating and satisfying. He filed away the feelings for later analysis. He had a job to do, and Cavalari recognized he was the guy to handle the crisis.\n\n\"All right, people,\" Archer called out. \"Details are sketchy, but when they do come roaring in here, they may have bogies on their tail. Our job is to protect the package and its contents. Let's move, people. They'll be here any minute.\"\n\nAn obvious change came over Archer as he took the reins. Thaddeus slid into command like putting on an old, familiar suit. On the far side of the compound, a HH-60G Pave Hawk roared to life.\n\n\"Rios, interface with Secret's COM,\" he ordered. \"Make sure this is the only attack going down. Make sure our men are secure with the secretary.\"\n\nRios nodded and rushed off to talk with the communications guys.\n\nArcher noticed that Rios broke out into a cold sweat after he gave the younger man his orders. The full impact of Archer's words had hit him and, despite his training and terrorism classes, the kid was now faced with the real thing. Archer knew that felt a lot different than the classroom.\n\nThe kid was probably thinking of his wife and children right now, just as Archer had back when he was a newcomer to the service. He bet Rios wished he could contact them. He knew that feeling well.\n\nArcher watched Rios pull a cell phone from his jacket, dial a number rapidly, then shut the phone, looking more worried. \"Damn it,\" the kid said through gritted teeth.\n\n0910\n\n\"Topaz One, I have movement, quadrant four,\" a voice squawked.\n\nCavalari tapped Archer on the shoulder to ensure he'd heard the transmission. They both made their way to the perimeter wall and climbed the steps. The wall resembled a maximum security prison's wall\u2014complete with observation posts at all four corners, inconspicuously masked by blooming bougainvillea vines. If one looked closely, they would see the dull razor wire that slithered through the pretty blooms.\n\nOnce on the wall, Cavalari expressed what they were all feeling. A horde of protesters emerged from the perimeter tree line, about seventy-five yards in front of them.\n\n\"How many?\" Archer asked the sniper to his left.\n\n\"Approximately forty, sir.\" The sniper faltered in his answer. \"Shit! Sir, there's something not right about these guys.\"\n\nCavalari handed Archer a pair of binoculars. \"You'd better take a look.\" His voice was barely above a whisper, and his face turned ashen.\n\nArcher took the binoculars and brought them to his face. He didn't say anything for a long time, then: \"Gentlemen, we need to get RAVEN secure and appraise him of the situation.\"\n\nThaddeus tried to raise his detail on their freq, but again to no avail. He hoped Carling and his men had managed to remain safe. The sketchy amount of information transmitted from the convoy painted a dire picture, and Archer hated not knowing the exact status of his men and his boss.\n\n0912 \u2013\n\nThe SUV accelerated up the drive. Inside, the president clutched the armrest. To his right, Melissa was on the SAT phone with D.C. briefing them on the situation. Across Washington phones began ringing, beepers chirped, and cell phones vibrated. Power breakfasts, closed-door power plays, and secret trysts were interrupted as news of the NOLA attack spread like wildfire.\n\nThe SUV stuttered to a stop.\n\n\"What's going on up there?\" Melendez asked.\n\nThe window on the right of the passenger's side turned red, as if a water balloon, ripe with blood, had collided with it. It took a few seconds for the passengers to reconcile what they were seeing. Outside, a man flung himself against the glass once more. Fresh blood splattered the window from his shattered fists. Bits of bones and flesh were all that were left of his hands after the third blow.\n\n\"Jesus!\" President Wendell said, as another attacker came from the left.\n\nSecretary Carling flew forward and collided against the back window, bumping her head. It made a sound like a wet melon, and she slumped, unconscious.\n\nThe president was aghast. \"Give me that phone,\" he demanded, snatching the phone from his aide. \"Who is this?\" He listened for a few seconds, then said, \"General, this is your President...\"\n\nMelendez leaned forward and hit the driver, Park Policeman Albert Jossart, in the shoulder. \"Drive the _fuck_ through them.\" His tone left no room for argument.\n\nJossart floored the gas and cringed as the heavy SUV slammed into a woman who refused to move out of their path. There was barely a noticeable bump as the woman disappeared beneath the heavy vehicle. Something dark splashed across the windshield. At first, Jossart thought it was blood, but there was something different about the substance. It was more tar-like, almost syrupy. He shouted in surprise as a visage formed in the black liquid and pressed against the glass with an open-mouthed scream, then the substance drifted off the window like wind-blown ash.\n\nThe sharpshooters on the walls fired into the pursuing crowd behind the SUV. Jossart focused his attention forward on his goal: the gate opening slowly ahead. Decorative foliage lined both sides of the drive and, from these shadowy recesses, men and women, even some children, jumped out by the twos and threes. Jossart didn't slow.\n\nAhead, agents fired their automatic weapons at the ones that attempted to breach the open gate. Everything went so fast, no one had time to really think about what was happening and how surreal it all became. A chattering that lurked amongst the chaos grew.\n\nJossart expertly maneuvered through the narrow gate opening. Behind him, one of the protesters managed to snake an arm between the gaps in the fence. The gate ground to a halt as it crushed the woman's arm, but she didn't seem to notice. Jossart looked in his mirror, watching Agent Darren Gibbs toss aside his rifle and pull his service pistol. He extended his weapon and rested the barrel against the slobbering monstrosity's forehead. Jossart was a huge Romero fan and, for a brief moment, before Agent Gibbs pulled the trigger, the woman resembled a zombie. No, that was all fiction. What was out there was real\u2014not a monster from some movie. Still...\n\nThe woman opened her mouth, and something dark and fluid erupted from it. This shadow slid between the bars and took a chunk out of Gibbs' forearm. Jossart's mouth dropped open. What the hell? He couldn't believe what he was seeing.\n\nAgent Gibbs screamed in agony and stumbled back, firing his pistol wildly in the air. The woman wrenched her arm back, dislocating it at the elbow. She pulled her body back and forth, trying to dislodge her trapped arm. With a sickening rending, her arm came free at the elbow.\n\nShe stumbled away a few steps, glanced at the bloody stump, and then charged, clambering up the fence with unnatural speed. Agent Gibbs was almost to his feet when the woman fell on him with such savageness the other agents froze in hesitation and horror.\n\n\"Someone shoot that bitch!\" Jossart heard a voice ring out from above.\n\nThe agents glanced up like deer caught in headlights. Jossart saw Archer come into view, pull his service pistol, and fire rapidly. The woman writhed and screamed as the hollow points ripped into her, but they didn't cease her attack. If anything, the bullets seemed to energize her actions. Somehow, Gibbs managed to roll free, and as she leapt to attack once more, he brought up his pistol.\n\nThe woman screamed in agony and rage as her teeth shattered on the barrel that disappeared deep into her mouth. Her head jerked as the gun's slide cycled each shot. The woman's face and head was an unrecognizable red and black mass when he was done. Gibbs scrambled back, disentangling himself from the corpse.\n\n* * *\n\nArcher hurried down the steps to the man's side. He was almost there when Gibbs glanced around erratically then shoved the gun into his own mouth. The man yelled in frustration as he realized there were no more rounds in the gun.\n\n\"No!\" Archer sped up.\n\nGibbs flung the empty gun aside and reached down to his ankle, pulling his backup. With a gleeful smile of triumph, he stuffed the compact gun in his mouth, and the back of his head disappeared in a pink mist. Archer stuttered to a stop as his head passed through the splatter. He grimaced and closed his eyes as he felt blood, bits of brain, and skull spray his chest and face. It smelled like black licorice and spoiled milk.\n\nDisorientated, Thaddeus barely sidestepped the woman as she reached out for him. His attention had been fixed on the shadow that had appeared to guide Gibbs's hand to his mouth. He fired four rapid shots into what was left of her face, but it seemed to have no effect.\n\nRearing up directly behind her, the shadow figure was nearly seven feet tall. What he was seeing was impossible, Thaddeus knew, but he was seeing it all the same. It was more than a shadow, he noticed, close up. It was like an absence of light\u2014a black cut out surrounding the woman. He could see something moving in the darkness, like ghost images on a snowy TV set, but darker.\n\nTentacles of dark matter wove about the vaguely humanoid figure (like coral on a reef). Where the trailing wisps touched his wrist, they engulfed it in an icy coldness that took his breath away. The thing pulled back its hand as if to strike, and the woman followed suit. Without thinking, Archer fired into the black mass. It exploded into black ash and flew off with the breeze.\n\nThe roar of the onrushing crowd brought Archer back to the present. They slammed into the iron-gate as one force. The gate groaned in its moorings, and decades-old cement puffed into the air when the hinges broke free of their casements.\n\nArcher reloaded his pistol as he backed away. He slid the magazine of unused ammo into his coat. He had a feeling he'd need it shortly. Any second the gates would give way, and they'd be overrun.\n\nHe had never seen a crowd so fierce in all his time in the service. He'd studied mob theory, and was familiar with how one acted and reacted, but this\u2014this was something he'd never seen or read about. The ones at the front of the throng were being crushed against the gate, but they didn't seem to notice or care. The expression on their faces chilled his blood. They frothed at the mouth and sputtered obscenities at him.\n\nThe wind changed and the stench of vomit, blood, and excrement wafted over him. Archer's eyes welled at the smell, and it took everything he had not to vomit. With the smell, came the crowd's guttural moaning. The sound of so many people groaning the same, low note made his insides feel as if they were vibrating.\n\nHe squinted as something caught his eye. He looked up behind him at the sky, noted the position of the sun, and then glanced back at the mob. Their shadows weren't right.\n\n\"What the\u2014\"\n\nArcher realized the sharpshooters had ceased firing. Confused, he glanced up at the walls. The agents were now facing inward, and their rifles were aimed right at him. He raised the pistol in his left hand in a non-threatening manner (at least, he hoped it was non-threatening), and he said, in as clear and strong a tone as he could manage, \"Men, the bad guys are out there.\"\n\nOne of the agents, Chris Prestin, turned to his fellow agent of seven years, raised his high-powered rifle, and casually blew the man's head into a red oblivion. Prestin's features contorted. His movements were stuttered, like those rooms in carnival funhouses with the rapidly flashing lights, where it made the person appear to be moving in spurts. Prestin's shadow seemed to move before he did. _Like stop motion_ , Archer thought.\n\nSomething bad was happening to Agent Prestin\u2014something utterly beyond Archer's experience\u2014and it scared the crap out of him. Along the wall, the other agents turned on one another. Some shot each other, while others grappled then fell over the side, into the waiting mob below. As their limbs were ripped from their bodies, the torn joints snapped and popped, sounding like a dog worrying a bone.\n\nArcher turned and ran for the house as chaos erupted behind him. Something terrible was going on, and he knew time was short, if it had not already run out. He had to get the president and the secretary off the ground to safety.\n\n0918 \u2013 Parking Garage\n\nJossart pulled into the underground garage at the back of the house. The turn was tight, and as the SUV slid around the corner, the front fender and side kissed the concrete bollards on the left side. Melendez cursed as he struggled to maintain his balance in his seat.\n\nThe kid was a driver\u2014that much was obvious. President Wendell turned to Melendez. \"Put that kid on the driving squad when all this was over. He doesn't know it, but he'd just passed the ultimate driver's test.\"\n\nMelendez chuckled and nodded.\n\nThen Melendez glanced over and fixed the president with a puzzled look. From the phone plastered against Wendell's ear, a voice wavered in and out.\n\n\"Say that again,\" Wendell ordered, concentrating on what the man on the other end of the secure line had just told him. He'd heard him wrong, because what the man just said made absolutely no sense. A land mass had appeared in the middle of the Pacific, out of nowhere?\n\nThe general took a breath and repeated his statement.\n\n\"Who else is aware of this?\" Wendell asked. He nodded a few times, grunted, and then sighed. \"OK, is Secretary of Defense there? Right...when he arrives...apprise him of the situation. He has my full approval to do whatever it takes to secure the nation, General Greenland.\"\n\nWendell glanced over at Major Woodrow. \"We may be needing that soon.\"\n\nWoodrow tapped a nervous finger against one of the buckles on the briefcase. President Wendell saw the stress on the major's face. Of course he was trained for this moment. They all were, but the president never dreamed they'd be called upon to do it while he was in office.\n\nWoodrow opened the leather case and pulled out the specially modified Zero-Halliburton case within, then showed it to the president. Inside were a few items only a select number of people had ever laid eyes on: a secure SATCOM radio and handset, the nuclear launch codes, (called the 'Gold Codes'), and the president's decision book. Commonly known as the 'Football,' it was referred to as something less than comforting by those in the know, who called it the 'Doomsday Yankee.'\n\nThe president wasn't ready to make these decisions. How could he choose the fate of the entire world\u2014of everyone he loved, of everyone that depended on him to make the right choice? No one really knew what this burden was like. There was no one else in the country that had to make this decision. He was one man. He had faults, flaws. Who was he to call upon the destruction of mankind? Kill thousands to save millions? Did that even ring true in this situation? What the hell were they dealing with here?\n\nHe looked at Woodrow solemnly, who adjusted the black cable around his wrist, playing with it as he would a rosary, if he had one. Both men wore the longest faces of their careers.\n\nJossart brought the SUV to a halt. The tires squeaked in the deep confines of the sub-basement garage. Next to the president, Secretary Carling's head rolled slightly. Spittle escaped the corner of her mouth. Her eyes popped open briefly, and she stared, almost as if in deep thought, then closed them again.\n\nThe president put a gentle hand on her shoulder then glanced around at the terrified faces staring back at him for leadership. The SUV was still in drive,\n\nMelendez reached forward and squeezed the kid's shoulder. \"Good driving, Jossart.\"\n\n\"Sir, we need to sweep the garage before Raven can disembark,\" Augustson said. Without waiting for a response, the agent exited the vehicle and began sweeping the area. Jossart glanced at Melendez, and then exited the vehicle as well.\n\nA few moments later, Augustson tapped on the window and signaled the all clear.\n\n\"Mr. President, we need to get you to the chopper,\" Melendez said as he slid from the vehicle.\n\nHe tried his COM, but all he got was static. There was no telling the status of the rest of his men. They were blind down here. The president knew that bothered Melendez to no end. He was a good man who followed the rulebook to the letter. He hated to lose anyone, but there were no guarantees right now.\n\n\"Should the chopper be compromised,\" Melendez told the president, \"we may have to drive back through that mob above.\n\nThe president shuddered at the thought. Through the static of Melendez's COM, he heard a weak voice waver.\n\n\"Say again?\" Melendez said.\n\n\"This is Papa Hotel 36,\" the voice answered.\n\nRelief washed across Melendez's face. \"Papa Hotel, good to hear a friendly voice out there. We are en route with Raven plus six. What's your status, over?\"\n\n\"Pretty hairy and deteriorating rapidly. Let's get the heck out of here. How's that sound, over?\"\n\n\"Roger that. We're in the parking structure. ETA to your location is three mike, acknowledge?\"\n\n\"Acknowledged. Will keep the rotors hot for you, out.\"\n\nMelendez swung his pistol to the left as something boomed at the far side of the garage.\n\n\"Hold your fire!\"\n\n\"Identify yourself!\" Melendez countered, as the slamming door echoed through the garage.\n\n\"Park Police! Thaddeus Archer.\"\n\nBoss!\" Jossart called out the turned to the others. \"He's good. He's my boss.\"\n\n* * *\n\nArcher came out of the darkness and shook the proffered hand. \"Agent Melendez, Secret Service.\"\n\n\"Good to see a friendly face.\" Archer glanced over Melendez's shoulder. \"That my Secretary in there? Oh, Mr. President.\" Thaddeus' voice faltered then trailed off.\n\nHis hand went to the cell on his belt. When he realized he'd done so, Archer abruptly snatched his hand away.\n\n\"Archer.\" Wendell nodded. If he noticed the phone, he chose to ignore it. Both men knew what the phone symbolized. He dusted off his suit and stretched. \"This goes no further than here, understood?\" After everyone nodded their understanding, the President of the United States took a breath and fixed the agents with a grim look. \"Whatever is happening outside...it's not confined to just here.\"\n\nArcher leaned in the vehicle, nodded at the stoic air force guy, and gently leaned Carling back. She moaned softly, but didn't open her eyes.\n\nDespite the nasty goose egg popping up on her forehead, the secretary looked no worse for wear, considering the circumstances. She mumbled something Archer didn't quite catch. He leaned in close, grimacing as he smelled something sickly sweet coming from Carling's mouth. Definite head injury.\n\n\"Ma'am?\"\n\n\"It has begun.\" She grinned, eyes still closed. \"Oh my god, it has finally begun.\"\n\n\"She's delirious,\" Archer said. He made sure she was comfortable, and turned to face the president, trying to shake a feeling of d\u00e9j\u00e0 vu.\n\n\"What?\" Melendez blurted.\n\n\"Now,\" Wendell continued as he held up a hand to stop Melendez. \"It seems there is an island that has mysteriously appeared in the middle of the Pacific. It's got everyone in an uproar. China is on High Alert, as well as Iran and North Korea. We all know they've been looking for an excuse to start something for some time now\u2014this may be it. Whatever it is, when the land mass came up out of the water, it created a massive tsunami, and from the early reports I got from Defense on the exciting ride over here, this wave is due to hit California, Alaska, and Mexico within the next hour.\"\n\n\"How big of a mass are we talking?\" Jossart spoke up.\n\n\"Why? What does it matter?\" Augustson scoffed.\n\n\"It's approximately sixty percent the mass of Africa,\" President Wendell replied.\n\n\"Sweet Mary,\" Jossart muttered and began pacing.\n\n\"What's his problem?\" Augustson said.\n\n\"Quiet,\" Melendez ordered. To Jossart, \"What's the matter?\"\n\n\"It's nothing. A silly idea, really.\" Jossart wore a bemused look.\n\n\"I think anything will help, including silly,\" Archer encouraged.\n\n\"I'm a gamer,\" Jossart said, slight embarrassment coloring his voice, as if he'd just admitted he liked diddling youngsters. \"And, well, there's this idea I've heard is being used for a new game coming out later next year...about an island coming up out of the Atlantic, and there's some pretty bad things that live on the thing...\"\n\n\"Yeah, I know what you're talking about,\" Melissa spoke up.\n\n\"You've got to be kidding me,\" Augustson exploded and stalked off.\n\n\"Agent, get back here,\" Melendez shouted, then to the president, \"I apologize for my agent.\"\n\n\"You're honestly going to listen to this?\" Augustson asked.\n\nArcher discreetly pulled his pistol and rested it against his thigh. He carefully maneuvered the door to provide a barrier between Augustson and himself. Thaddeus didn't like the wild look in the agent's eyes just then. The guy was close to cracking. He had the look of someone who had just peered into his own personal hell. Augustson's face twitched.\n\nMelendez glanced over at Archer and moved closer to the president, placing a hand on his elbow. \"Sir, I believe you'd be more comfortable in the SUV until we figure out our POA.\"\n\nArcher knew Melendez was starting to get it too. Smart man.\n\nWendell peered at Melendez in confusion and opened his mouth to protest, but at that moment Augustson lunged at Jossart. With a cry of surprise, Jossart went down to the concrete. He bit his lip, as if to silence a scream, as his ribs cracked beneath the weight of the man straddling him.\n\nWells stepped forward and trained his pistol on Augustson.\n\nArcher noticed the raging agent's fingers twitching now. He couldn't have been the only one seeing this. They contracted in spurts until they formed a clutching vise around Jossart's throat. Jossart remained calm until Auguston's face started to morph into a hideous, twitching gray mass. Archer aimed his gun alongside Wells. This was nuts.\n\n\"Let him go, Augustson,\" Wells ordered. \"Let him go, NOW!\"\n\nHis commands had no effect on the agent. Wells glanced up at Melendez, and Archer knew the agent looked to the man for guidance.\n\nMelendez stepped forward and hit Augustson across the back of the head with the butt of his pistol, but the blow failed to faze the man. Archer stepped forward and aimed at the black, ill-formed mass attached to Augustson, manipulating the man's every move. Just as it took on a humanoid shape, he fired.\n\nAugustson slumped forward and smacked his head soundly against the concrete, next to Jossart's head. Jossart sucked in precious oxygen and started hacking, free of Auguston's grip. He rolled over to his side and clutched his battered throat, groaning in agony.\n\n\"What was that?\" Melendez muttered as the mass disintegrated into a fine black mist and drifted to the concrete.\n\nTesting a theory, Archer knelt down and traced a finger over the concrete. Not a nick. The bullet left no indication it'd struck the concrete. He stood up, a troubled look on his face.\n\n\"Where'd the bullet go?\" Wendell asked.\n\n\"Exactly.\" Thaddeus toed the black residue with the point of his shoe.\n\n\"Mr. President,\" Melissa said. The SAT phone in her hand was flashing. She held it out, and Wendell took it from her.\n\nHe glanced at the black powder on the ground, then turned and walked a few feet away as he brought the receiver to his ear. \"Go.Travis, good to hear your voice. What are we dealing with? I'm ordering COGCON 1 and spool up JEEP, just in case. Yes...we're headed to the airport here shortly. Just taking stock of the situation. Hold on, I want everyone to hear it from you.\" Wendell smiled and laughed at something SecDef Travis Wayteck said. \"I'm the president. They're cleared as of now.\"\n\nWith that, Wendell activated the speaker function and motioned for everyone to gather in closer. \"You're on speaker, Travis. Go ahead.\"\n\n\"Hello everyone,\" the deep baritone rumbled. The first time Archer met Wayteck, he'd thought the man's voice greatly resembled that of James Earl Jones. \"From what our experts can tell, this outbreak of riots is happening simultaneously across the globe... without rhyme or reason.\"\n\n\"Could it be terror related?\" Wendell interjected.\n\n\"Nothing is certain at this point, so I can't rule it out, but we don't know of anything that would affect so many people in so short of a time, much less at the exact same time in so many different areas. There are early reports of massive riots in Bangladesh, but communication is difficult at the moment, as I'm sure you're aware. After discussing our Nation status, I directed the military to go to DEFCON 1, and the Guard is already being recalled. NORAD is monitoring the situations in China, North Korea, and Iran. There's something else. Really weird, but...\"\n\n\"Go ahead,\" the president urged. He glanced at all of them gathered around the phone. What the Secretary of Defense might possibly mention next all passed through their minds like a psychic fire.\n\n\"There's been reports of shadows,\" Wayteck said, hesitation evident in his tone. \"Um, not sure what is going on with that, but some, um, people are on that.\"\n\nArcher caught the tone and glanced meaningfully at the president. The man grimaced and averted his eyes. He ran a hand over his face and sighed. \"Travis, those people\u2014any word?\"\n\n\"Not yet.\"\n\n\"We've seen the shadows, sir,\" Archer spoke up. \"This is Thaddeus Archer, sir, Park Police.\"\n\n\"Ah, yes, Mr. Archer. I recall you. You said you've seen these shadows?\"\n\nArcher glanced at Wendell once more. This time, the man maintained his gaze. With an almost imperceptible nod, something passed between the two men.\n\n\"That's correct, Mr. Wayteck. This is just my best guess, but they\u2014the shadows, I mean\u2014are connected somehow. I don't know how, but I've got this gut feeling.\" He realized how lame it sounded. \"I shot one of the attackers, and it didn't affect him at all, but when I shot his shadow, the man collapsed as if his strings were cut.\"\n\nWayteck took a deep breath. His voice lowered conspiratorially. \"I understand, Mr. Archer. We're thinking the same thing, actually. You're one of the first outside of Black Rock to come to this conclusion.\"\n\n\"Travis\u2014\" Wendell's tone was clearly a warning.\n\n\"Hold on a second...\" Wayteck blurted.\n\nPapers rustled and a few voices could be heard, but they were muffled. Finally, Wayteck came back on the line. The difference in his tone was obvious as soon as he began to speak. His voice sounded haunted, hollow, and extremely disturbed. \"Mr. President...the DPRK has just launched...\"\n\nThe garage abruptly filled with howls and groans. The daylight previously apparent at the far entrance was blotted out.\n\n#  Chapter Fourteen\n\nEight Hours Ago\n\n0923 \u2013 Heliport-Roof\n\n\"H, E, double hockey sticks,\" Samantha Veleska muttered, as the crowd pushed down the gate and flowed into the compound like water from a broken dam. The wind from the rotating blades of the helicopter buffeted the few attempting to climb the side of the building. She reached over and checked the safety on her pistol, ensuring it was ready to fire. Samantha had never fired her weapon in the course of duty, and she hoped now she wouldn't have to.\n\nShe nervously readjusted her headset and spoke into the mike. \"Melendez, you need to speed it up. Perimeter's just been breached.\"\n\nVeleska waited for a response, but none came. She tightened her grip on the collective, and the helicopter jolted slightly as it rose a few inches off the tarmac. She relaxed her grip and the skids settled back down. \"Come on, come on, people.\"\n\nShe'd seen what'd happened with the agents on the walls, but she wasn't sure exactly what was happening below. She'd experienced some of the madness firsthand as her co-pilot, Todd Westenhaver, ripped his headset off and pulled his pistol. He'd leveled the gun at her and started babbling in some kind of broken Spanish, which she'd realized was more like Latin (Catholic school hadn't been a total waste, after all).\n\nIn the middle of all the weird words, Todd had kept saying, \"The chattering, the chattering. Get it outta my head, please Sweet Mary, GET IT OUT!\"\n\nVeleska, paralyzed with fear, watched her friend, one time lover, and partner of five years proceed to rip his ears off and stuff them in his mouth as he blubbered and moaned about the chattering in his head. She reached over and grabbed the gun, prepared to wrestle it from his hands, but he relinquished it readily. As she pulled back to her side of the cockpit, Westenhaver lunged at her and gripped her hands. She abruptly noticed the shadow, because it was so cold. It took her breath away as it came into contact with her hands. A split second later, Todd's physical hands grasped hers. He rested the barrel against his forehead in mock supplication and said, \"Please, Sam, just shoot me. I can't take it anymore.\"\n\nSamantha shook her head and tried to pull her hands from his grip, but it was like iron. She couldn't move. Amidst the pleading, she felt herself losing her purchase, as her hands became a separate part of her. All feeling ended at her wrist. She could see her hands through the black swirling mist, but it was as if they didn't belong to her anymore. There was a definite texture to the mist, as if it were syrup. Images filled her mind, and she screamed as they wrecked havoc. It felt like she was being raped, only the attack was occurring in her head and not between her legs.\n\nShe saw a city laid to waste by marauding black figures. They flew, crawled, and walked about the city, killing everything in their paths. They were black cutouts, and she could recognize some of the shapes among the chaos.\n\nA tiger, but not like anything she'd ever seen at the zoo. This only projected a tiger's shape. There were other features that didn't belong on the animal. A scorpion's tail lashed about, while its mane flowed with tentacles. She gasped as she recalled where she'd seen something similar. Her brother's card collection. Corry always had weird tastes in books, movies and music, but the cards succeeded in completely freaking her out. The cards featured creatures of a fantasy world, but with major artistic privileges. Animals morphed with other animals to create these horrendous and freakish monsters. What she saw now was just like what she'd seen on the playing cards, except in shadowy outlines.\n\nTodd slid his finger up behind the trigger guard, and Samantha screamed. The vision had been brief. The hammer on the 9MM slowly cocked to the rear.\n\n\"Don't do this, Todd. Please don't do this.\"\n\nHe relinquished his grip and tore open the cabin door, leaping out then lurching about the tarmac. His movements were erratic, and Sam had the feeling if he'd been on fire, he would move and writhe in the same frenetic manner. The roar of the blades filled the cabin and made it difficult for her to hear the radio, so she leaned over and swung the door shut. She started to latch the door, hesitated, then decided it was better to have Todd outside, rather than inside with her.\n\nTodd stopped stumbling about and stared right at her. His smile was bloody as he ran towards the helicopter. She sucked in a breath as she realized what he planned to do. It all happened so fast, there wasn't any time for her to react. Todd leapt up on the side of the railing surrounding the helipad and perched there like some insane primate. In the flash of an eye, the shadow shoved him off the railing and directly into the whirring blades. The chopper shuddered as he disappeared in a cloud of blood and gore. The blades caught him just beneath the chin and flung his body backwards, as if an invisible chain yanked it.\n\nAcross the way, a door swung open, and Jossart and Wells moved out onto the tarmac. They only just made it out as Todd leaped to his death. The back spray of his suicide spattered across the two men, bits of bluish white matter spackling Well's suit. Sam watched Well's face exhibit downright disgust as tried to brush the stuff off, but he succeeded in merely smearing the carnage across his chest. To their right, Sam saw something meaty smack into the wall, creating a weird kaleidoscope of bone, blood, and internal organs.\n\nHer attention was drawn to the rainbow that hovered in the air near the helicopter. She shuddered as she realized it was blood particles producing the colorful effect.\n\nThe two men exchanged some communication, the younger tapping the older man on the shoulder and pantomiming he was going to approach the waiting helicopter. Wells nodded his understanding and disappeared back into the stairwell.\n\nSam watched the man approaching and tensed in preparation to lift off. He seemed normal looking enough, but she wasn't going to take any chances. She realized Todd's body probably hadn't damaged the blades, but she wasn't anxious to fly the helicopter and find out if her hunch was correct. She wracked her brain. Sure, there had been instances of blades killing someone, but had the blades been damaged to the point of making the craft unable to fly? She wasn't sure.\n\nThe man indicated he was friendly, and Sam decided to take the chance. As the man pulled open the canopy door, though, she leveled her pistol on him.\n\n\"What's your name?\" she called over the roar of the helicopter.\n\n\"Jossart, Park Police. I'm cool, if that's what you're worried about.\"\n\n\"Where's everyone else?\"\n\nJossart turned and waved to Wells. \"We wanted to make sure you were kosher before everyone came out. Where's your copilot?\"\n\n\"You're wearing him.\"\n\n\"Oh.\" Jossart grimaced.\n\nArcher helped Carling up the stairs. She was still a bit woozy and the going was slow.\n\nIn front of them, President Wendell glanced back. \"Patricia, just a few more feet. We'll be in the air soon, back to the airport and Air Force One.\"\n\nShe glanced up and smiled. Archer didn't like how shaky the smile seemed on her face. Her eyes were glazed, in addition to the nice goose egg on her forehead. Probably from the shock, but it could be more serious. She could have a concussion. Archer would rest easier when they got back to Air Force One and medical could check her out.\n\n0928 \u2013 Stairwell\n\n\"I heeeeaaaar yooooouuuu!\"\n\nArcher glanced over the railing. A few flights down, Augustson lumbered up the stairs. Only his hand was visible in the dimly lit stairwell. The red emergency lights cast an eerie glow. Jossart appeared at the top of the stairs. Wells took hold of Wendell's arm and led him out of the stairwell, while a misshapen shadow crawled on the stairs. Archer caught it moving and his heart fluttered. He grit his teeth.\n\n\"Jossart! Take care of Carling. I've got to take care of something else,\" he ordered, his voice cold _. How the hell is he still alive? Must be another one of those shadowy things. They don't give up, do they?_\n\nHe turned and started back down the stairs as he pulled his pistol. \"Augustson, you come any farther and I will be forced to fire on you.\"\n\nHis warning was greeted with jeering laughter. What sinister force was at work here? Archer thought fiercely about what was going on, but try as he might, he wasn't coming up with an explanation to fit the current circumstances. If this was terror related, why were some people affected and not others. Why wasn't he affected?\n\nHe shuddered as he recalled what Wayteck had said. At first, when the Secretary of Defense referred to the DPRK, it hadn't registered, but now it did\u2014Democratic People's Republic of Korea, North Korea. Archer had taken a few political science classes at night, but he didn't need those to know what horror North Korea had just unleashed. By now, China may have leapt at the opportunity to launch, and who knew what other countries would gleefully join the party. Had whatever affected this side of the world caused the Koreans to launch?\n\n_It's the shadows. This isn't man made._ He stopped and considered the thought. He wasn't a religious man now, but had been raised as one. Thaddeus had a difficult time associating the world with any type of sentient God, much less the one from the Bible. After God let his wife's brain slowly eat away her sanity, which forced her to leave him, Thaddeus and the Man upstairs hadn't been much on speaking terms.\n\nArcher glanced at his watch. Any minute now, nukes could be striking the coasts. He'd heard Wendell give the order for COGCON 1: Continuity of Government Alert Conditions. This program emulated the DEFCONs of the national security community, and was achieved through a series of alerts via BlackBerry and pager to key officials. COGCON 1, the highest level of preparedness, required each department and agency to have at least one person in its chain of command, and sufficient staffing at alternate operating facilities to perform essential functions.\n\nMore clattering came from below as the immediate threat approached. Archer loathed the possibility of having to fire on a fellow agent, but Augustson posed a clear and present danger to the president.\n\nIt felt good stepping back into his old job. A pang of regret stabbed his gut as he realized this was only temporary. It was funny all the random thoughts that went through his head in the space of a few seconds. He'd likely have to fire on a fellow agent, and here he was thinking about his job and the family he lost.\n\nThe stairwell went silent.\n\n_Melendez!_ Thaddeus had completely forgotten about the man. The lapse in memory troubled him. It came back now, like a feather blown along with the breeze. As they'd been preparing to head for the helicopter pad, the garage filled with the sounds of the approaching mob. They'd all headed for the stairway, when Melendez stopped. \"I can't leave him for them,\" he'd called, and headed back towards the parked SUV.\n\nThaddeus would've stopped the man, but the Secretary leaned on him, and the president was already in the stairwell. He had figured Melendez could take care of himself. Thaddeus carefully peered over the side.\n\n\"Melendez?\"\n\nA bullet whistled by mere millimeters from his face. Stunned, he jerked his head back and cursed. \"Melendez can't come to the phone at the moment, if you'd kindly leave a message.\" Again the chilling laughter drifted up. \"My, my, that sure was close, wasn't it?\"\n\n_So the dead can operate weapons?_ Archer thought. _How the hell are we supposed to fight against something like that? Think I'd prefer terrorists right now._\n\nJossart called out from above. \"Everyone's secure, Arch.\"\n\nAnother shot rang out. The sound was deafening in the confined space. Archer heard the distinctive snick of a gun sliding from leather with a clarity that was uncanny, despite the ringing in his ears from the percussive blast of Augustson's wild shots. He chanced a glimpse over the railing. Augustson was closer than he realized, but he wasn't looking in Archer's direction.\n\nA hulking shadow drifted a few feet in front of the agent, pulling the man up the stairs. The air in the stairway seemed to contract. The chattering, just below the cusp of hearing, exploded into loud chaos. Augustson cried out as blood leaked from his ears.\n\nAn intense rage engulfed Archer's senses. He wanted to shoot something, anything. The shadow opened its mouth and screamed, and the sound was unlike anything Archer had ever heard. A cross between a chicken and a tiger was as close as he could come to describing it. What he saw in the thing's mouth was a whole other world. It was like looking through an open window. The shadow expanded, and then seemed to swallow itself.\n\nAugustson cried out as the shadow exploded, rather than dissipating like the shadows before. The explosion flung the man back down the stairs. The crack of his neck ruptured the silence.\n\n\"What was that?\" Archer reached up and wiped the blood from his nose and ears.\n\nAs quickly as the rage passed over him, it dissipated, as if it were never there.\n\n* * *\n\nSomething felt different, but Archer couldn't put a finger on it. First, there was the silence. No place was ever this silent. He glanced up the stairs and saw Jossart approaching him. Blood leaked from both his ears and from his nose as well. He reached up and felt wetness on his own face. His hand came back dark with blood. Slowly, the sound returned. It was like someone was raising the volume on a TV.\n\n\"You, OK?\" Jossart placed a hand on Archer's shoulder.\n\n\"I think so. How about you? What happened?\"\n\n\"You know, I used to work on a carrier,\" Jossart said. \"What we just felt? That's what a sonic boom feels like.\"\n\n\"Really?\"\n\n\"Yep, but now? It's too quiet. You feel it too?\"\n\nArcher nodded, and peered down over the railing into the darkness of the stairwell. It was empty. \"I've got to go check.\"\n\n\"I'll tag along.\" Jossart smiled. \"Ain't like I'm doing anything else.\"\n\n\"You got any family?\" Thaddeus asked.\n\n\"None I still talk to.\" Jossart grunted as he trained his weapon down the stairwell and began his descent. He paused and his shoulders slumped. \"Things are different now...\"\n\n\"Yeah,\" Archer agreed, and then sucked in a breath. The door at the bottom of the stairwell was propped open a few inches. He could only see a sliver of basement, but it was enough. The bodies were stacked against the door like cordwood. It looked like someone tossed a grenade amongst the crowd. No, more like a rocket launcher exploded. Splattered against the walls and ceiling like black paint was what was left of the shadows.\n\n\"Oh, man,\" Jossart muttered as he looked over Archer's shoulder.\n\n\"They can be killed.\" Thaddeus started back up the stairwell as fast as he could take the steps. \"Let's get out of here.\"\n\n0945 \u2013 Heliport-Roof\n\nNeil Marlstone Wendell, Leader of the Free World, grimaced as the percussive sound wave washed over him. He glanced over at Carling, and wanted to straddle her lap and choke the life out of her. He curled his hands around her throat, and then paused in confusion.\n\n\"What are you doing?\" Carling tried to free herself from Wendell's grip. She coughed and rubbed her neck as he sat back in his seat.\n\nOne second he'd been checking her head to make sure she was fine. The next, he'd put his hands around her neck and started choking her. He looked at her. No, he looked through her, as if she wasn't there.\n\n\"Neil! What's the matter with you?\"\n\nHe pulled his hands back as if her neck were hot to the touch.\n\n\"I-I-I...I don't know.\"\n\nMelissa sat across from them. She'd brought her hand to her mouth in horror as she'd witnessed the president lunge at the secretary and begin to throttle her. Wendell moved to comfort her, and she cringed from his touch.\n\nThey all glanced towards the door, as what sounded like shots rang out from the stairwell where Jossart disappeared a few moments before. The shots were hard to hear over the roar of the helicopter, but they all were fairly certain they'd been shots.\n\nIn front, Sam turned around and fixed Wendell with a hard stare. \"Mr. President, no disrespect, but what just happened?\"\n\n\"You all felt that? That blast?\"\n\nThe SAT phone rang in Melissa's hand. She quickly brought it to her ear. \"Yes? One moment... It's for you.\"\n\nWendell took the phone. \"Yes? What was that, Wayteck? It felt sonic or something. What? Oh my god. Are you sure?\"\n\n\"He's close,\" Wendell heard Carling whisper. \"He's so close...\"\n\nWendell held up a hand. \"Who else has launched? And they hit the island? Travis, I...a few minutes ago, about that time, something happened. We felt a blast\u2014like a sonic boom. Right, you felt it there as well...was it related to the strike?\"\n\nMelissa screamed.\n\n#  Chapter Fifteen\n\nSeven Hours Ago\n\nAirborne\n\nFires dotted the landscape below them as Sam made for the airport. She hadn't been able to reach anyone at the airport, but President Wendell and Archer still made the decision to head for it. What choice did they really have? If things didn't look right, they wouldn't land, but would continue on as far as they could. After that? Nobody wanted to think things through that far just yet.\n\nThe thing making Sam nervous was the lack of people on the ground. She'd been flying for five minutes now, low enough to read street signs on the overpasses, but not a single person was to be found. At least, not a living one. Bodies littered the streets like toys in a rebellious child's play room. Strewn about this way and that, the placement of the corpses had no rhyme or reason. It was as if they'd dropped where they stood and exploded. The air was thick with black dust.\n\nAt first, Sam worried the stuff might clog the turbines, but it seemed it was too fine to do so. She recalled her father telling her stories about his time in the Philippines, back in the early 90s, when Mt. Pinatubo erupted. The ash had wreaked havoc with the planes in the air, and he'd worked ten hours to redirect all the planes on his scope, at Clark Air Base, to safety elsewhere. The black dust in the air reminded her of that. Her father was the reason she was a pilot, and he'd instilled the love of flight early. She choked back a sob as she wondered where he was at this moment. Had he survived? As with most parents and children, the years had created a distance between them, and she hadn't noticed until it was too late and the gap was too big. They'd talked when they could, but it wasn't the same closeness as before.\n\n\"You OK?\" Wells asked from beside her.\n\nShe glanced over quickly, as if she'd been caught doing something untoward, and offered a weak smile. \"I'm fine.\" She sighed with relief. For a second, she was unsure if her voice would serve her.\n\n\"Didn't think I'd be fighting for my life when I got out of bed this morning,\" Wells hesitated and then pressed on, \"Or sitting next to such a pretty woman.\"\n\nSam fixed him with a stare. It was neither warm nor cold\u2014just there. Wells quickly avoided eye contact with her and turned toward the window. She noticed the slight blush to his face. She smirked then thought it was a bit sweet if not vulnerable.\n\nHe flinched as Sam reached over and placed a warm hand on one of his. His hand was dangerously close to his crotch, which meant _her_ hand was as well. Immediately, he began to harden. Sam laughed and removed her hand.\n\n\"No harm, no foul?\" she asked. \"If we get out of this in one piece, how's dinner sound?\"\n\nWells turned to face her, and Sam was glad to see he made an effort to look at her face and not anywhere else. \"You serious?\"\n\n\"No.\" Sam kept her face blank. \"I'm just getting your hopes up.\" She laughed. \"Seriously, you will take me out to dinner after this is all over.\"\n\n\"Here we go,\" Wells said and pointed out the window.\n\nAhead, the airport came into view.\n\n#  Chapter Sixteen\n\nSix Hours Ago\n\nAirport\n\nThe SAT phone chirped inside the case that also contained the Football. Woodard removed the outer black, leather covering and Thaddeus, who'd never seen the actual case until now in person, thought the silver inner case was the most ominous object he'd ever laid eyes on. Not just because of what was contained within (the power to destroy most of the world) but how it looked. Cold, lifeless steel reflected Woodard's stony expression. Major Woodard calmly opened it, and offered the phone to President Wendell.\n\nArcher pulled the cell from his belt and checked its reception. Two bars.\n\n\"You think it'll ever ring again?\"\n\nArcher jumped at Wendell's voice. He quickly replaced the phone and sighed. \"Someday.\"\n\nThe president stared at Thaddeus for a long moment, and then spoke into the SAT phone. \"Go.\" A sheaf of papers rested in his lap. A plastic privacy divider could be lifted like a laptop's lid, and this smoked, plastic piece prevented Thaddeus from seeing what was on the papers. \"Alpha, Mark, X, 2-9Black, November 79800,\" Wendell read off the sheet. \"Confirm and acknowledge.\"\n\nThe phone had rang every few minutes ever since Woodard opened the case, and Thaddeus and the others were extremely nervous every time the president read off the letters and numbers. They all knew what was happening. Confirms and the reconfirms. Wendell was initiating the process to bring the nation's nuclear arsenal to full alert. Knowing nukes were already launched, and America hadn't been the first to do so, everyone else appeared eager to contribute.\n\nSweat poured off Wendell as he held the phone to his ear and recited yet another sequence of numbers from the paper in his lap. Melissa leaned over with a handkerchief and dabbed at his forehead. He smiled his thanks, but the smile didn't travel to his eyes, Thaddeus noticed.\n\nThe helicopter touched down with a slight bump, and he turned his attention outside. Air Force One was a few hundred feet in front of them, waiting to pick up the president and the others. Thaddeus could see the engines were turning, and it brought him some hope. He glanced over at Jossart. The man nodded his readiness.\n\nArcher waited for a break in the president's conversation. \"I'm leaving Wells with you, Mr. President. Jossart and I will recon the area, ensure it's safe for you, and then we get out of here.\"\n\nWith that, Thaddeus looked forward. \"Veleska, keep the bird hot, just in case.\"\n\nCarling asked, \"Neil, what's with the phone?\"\n\n\"Archer's?\"\n\n\"Yes. I noticed he carries the thing religiously...\" Realization lit up her face. \"It's _that_ phone?\"\n\nWendell nodded gravely. \"Yes, it's... _it was_...his daughter's phone.\"\n\n* * *\n\nKeeping low, Archer and Jossart rushed away from beneath the whirring blades. They took up a position by an abandoned cargo transport. The suitcases and bags, previously in the trailers, were now strewn about the tarmac, and some of the bags had come open. The driver was dead, but Archer checked him anyways. His fingers came back covered with the weird black dust. Tire marks zigzagged all over the tarmac.\n\n\"Looks like he went on a joyride,\" Jossart said.\n\nArcher moved to the front of the trolley and noticed something strange. He bent down and ran a finger across the grill. It came back tacky. \"He was running people over.\"\n\nMingled with the drying blood and hair, bits of plastic were also apparent. Archer stood back up and looked out at the runway. \"It strike you as funny we're not hearing any planes?\"\n\n\"You're right,\" Jossart said, glancing up into the sky and shielding his eyes with a hand. The sky was empty. \"Maybe they managed to get the planes out of the air. Could be a no-fly in effect, like after 9\/11?\"\n\n\"There'd still be planes needing to land, though,\" Archer insisted. He checked his watch. \"It's only been about fifty minutes since everything started happening.\"\n\n\"Oh crap.\"\n\n\"What?\"\n\n\"Some of those fires we passed over...well, they looked like crash sites, but I dismissed that as impossible.\"\n\nSomething moved, and Archer's caught it just at the corner of his vision. He whirled and trained his weapon to his right.\n\nJossart followed suit. \"What is it?\"\n\n\"Not sure. Thought I caught some movement.\"\n\nArcher motioned for Jossart to provide covering fire from his position, if needed. He crouched and rushed over to a fuel truck on the far side of the tarmac in an attempt to flank whatever he'd seen. He came around the side of the fuel truck at a run and skidded to a stop. A woman stared back at him. Cradled protectively in her arms, two small children, a boy and a girl, peered out. The children appeared to be twins and about five years old. Shivering, the woman glanced fearfully at Archer.\n\nHe held up his gun and said, in as soothing a tone as he could, \"Hey, it's all right, now. You're safe. I'm a cop.\" He pointed at the badge at his belt.\n\n\"So was the last guy, and he killed my husband.\"\n\nArcher saw the gun too late. As the woman raised the weapon, he dove to the ground. He hissed in pain as the hot concrete bit into his knees and palms. The bullets whined close to his head as he frantically rolled beneath the fuel truck. Bits of tarmac careened off his forehead and stung the breath from him.\n\nJossart didn't open fire. It was too risky with the fuel truck in the way. \"Archer!\"\n\n\"Hold your fire! It's a woman and some kids.\" To the woman, Archer said, \"Seriously, we're good guys. Keep the gun even\u2014just don't shoot at us anymore.\" When the woman didn't respond, he attempted a different tactic, \"Now, tell us what happened.\" Mentally, he recalled what the gun looked like. He was fairly certain it'd been a Glock, and if it were police issue, it would be a mag of fifteen rounds. He counted the shots. She'd fired at least ten at him as he rolled. Or was it twelve?\n\nHe peered beneath the truck in an attempt to see what she was doing, and found she pointed the pistol right at him. The pistol wavered in her hands, and he could see the hesitation in her eyes.\n\n\"My name's Thaddeus Archer. I'm a Park Policeman. Right over there, in that helicopter\u2014see it? The President of the United States is in there. We're the good guys.\"\n\n\"Did you see them?\" she asked.\n\n\"Who?\"\n\n\"The mercenaries.\"\n\n\"Mercenaries?\"\n\n\"They were dressed like National Guard,\" the woman said. \"The mercenaries turned on the Guardsmen and killed them. We\u2014my family and I\u2014were flying from Galveston when all this weird stuff started happening. My husband landed our plane here. They killed him.\" She choked back a sob and struggled on. \"They just killed him. Didn't even say anything\u2014just shot him right in front of us.\"\n\n\"Archer!\" Wells shouted.\n\n\"We're OK,\" Jossart said. \"Just a misunderstanding. We're working on it.\"\n\nThaddeus stiffened as he caught a glint coming off one of the far buildings. He knew what it was. There was no mistaking it. He resisted the urge to look directly at the flash of light, trying not to give any indication he'd seen it.\n\n\"Jossart, we've got company. Ten o'clock. Sniper.\"\n\nThe woman gasped as she heard Archer. \"Oh no. It's them!\"\n\n\"Easy now,\" he said. \"He's on my side. You're out of the line of fire. I want you to follow Officer Jossart's instructions. He'll lead you to somewhere safe. OK?\"\n\nOnce the woman and her two children were at a safe distance, Archer tried to raise anyone on Air Force One. This time he got a response.\n\n\"Identify yourself,\" came the harsh reply. It was so clear, it startled him for a second.\n\n\"Park Police\u2014Thaddeus Archer. I've got some friendlies here that'd like to come aboard.\"\n\n\"Anybody from Blackwater with you?\"\n\nThe question took Archer by surprise. It took him a few seconds to place where he'd heard 'Blackwater' before. Contractors was the politically correct term. Mercenaries was the not so popular one.\n\n\"There may be someone over on the far side on one of the buildings.\"\n\n\"That's one of them,\" the voice agreed. \"Name's PFC Andy Hurt\u2014Army National Guard. We can't extract the president until the threat is eliminated. Much obliged if you'd take care of that one out there.\"\n\nArcher smiled. Who said 'obliged' these days? \"Consider it done.\"\n\nHe let Sam and the others know what was going on. Sam decided to lift the chopper off and fly to a neutral section of the airport as Wells, Jossart, and Archer dealt with the possible rouge sniper.\n\n* * *\n\nArcher and Wells slid from wreckage to body to abandoned truck. Just another war zone in the United States. Wells spotted the man crouching with a rifle fixed on Air Force One, and he gestured to Archer to divide and conquer. Archer saw Wells reach the man first, as if he couldn't wait to catch him.\n\n\"Hi there,\" Wells greeted, shoving the pistol's muzzle painfully into the back of the man's ear. \"Thought you contractors were trained better than this.\" He chuckled. \"Now, get up slowly, play nice, and maybe I won't end you right here.\"\n\nThe mercenary slowly gained his feet and raised his hands in the air. He turned and faced Wells, then broke out into a wide grin. \"Well, I'll be.\"\n\nWells did a double take and started to lower his weapon, then caught himself. \"Travis Muldoon.\" There wasn't any warmth in his tone. \"What are the chances I'd find you in the middle of this shitstorm?\"\n\n\"You still feeling sore about Fallujah? That was business\u2014nothing personal.\"\n\n\"You left us to die,\" Wells said with barely contained rage.\n\n\"Easy now,\" Archer called as he gained the roof. \"You two know each other?\"\n\n\"You could say that,\" Travis replied.\n\n\"We were in Iraq together,\" Wells said. \"There was a safe house raided, and the marines found some contraband hidden in the floor\u2014\"\n\n\"A cool eight million in crisp hundreds,\" Travis said. \"All we had to do was split it and keep our mouths shut.\"\n\n\"My unit and I saw things differently. We confiscated the money, and were in the process of transporting it back to headquarters, when Travis here tipped off the local insurgents and they attacked us.\"\n\n\"I just told them to detain you. They weren't supposed to use friggin' RPG's!\"\n\n\"But, they did!\"\n\n\"Is there anyone else?\" Archer asked Travis. When Travis shook his head, Archer said into his radio, \"Sam, we're clear here. Get the president on the plane.\"\n\n\"Where's the rest of your crew?\" Wells asked. \"You creeps don't usually work alone.\"\n\n\"Things got messy.\"\n\n\"I'll say,\" Archer said. \"Probably gonna get a lot worse.\"\n\nAirport, Air Force One\n\nAs soon as Sam received the all clear from Archer, she sat the helicopter down. A Guardsman disembarked Air Force One and rushed over, stuttering to a halt as he saw the president emerge.\n\n\"M-M-Mr. President,\" the young Private First Class Hurt stammered, saluting Wendell. In the process, the poor, nervous soldier almost lost his rifle.\n\nAs he scrambled to keep from dropping the weapon, Wendell smiled good-naturedly. \"Good to see a friendly face, PFC Hurt. Who's in charge here? Where's my service detail?\"\n\n\"I, uh, I guess I'm in charge,\" Hurt managed. \"The rest of my unit...they didn't make it, sir. The plane was abandoned when we showed up. We were traveling with some Blackwater guys, and things kinda got complicated.\" It was obvious Hurt didn't wish to elaborate.\n\nSam kept the helicopter running. After securing the door, she strolled over. She kept her pistol discreetly at her thigh and surveyed the horizon. This was too easy. For some reason, she felt getting to the plane should be harder, like they were missing something. The feeling was completely irrational, but it was with her all the same. It clung about her like an ill-fitting shirt. She paced and unconsciously tapped the gun against her thigh, while looking over to where Archer and the other men disappeared. The steady roar of the plane's engines, along with the chopper's, soothed her a bit, but not much. Something wasn't right. The feeling wouldn't leave her.\n\nA whisper on the breeze. She whirled and looked behind her, certain someone or something was right there. The whisper came again.\n\nSammmmm...\n\n\"What the?\" She'd heard the voice before, but she couldn't place it.\n\nSammmmm...Surrender. There is no hope of resisting the Old One. He rises...He is Alllll...\n\nThe radio crackled to life. \"Sam, we're clear here. Get the president on the plane.\"\n\nTo Wendell, she said, \"No offense, but let's get out of the open.\"\n\nShe started to ask if anyone had heard the voice, but resisted. No, it was more like she wasn't able to give voice to the question, she realized. A sensation flitted through her head, unpleasant and pleasant all at the same time.\n\nHe is coming...\n\nThey walked over to the plane. \"Can you fly this?\" Secretary Carling spoke up as they started up the staircase.\n\nSam smiled and nodded confidently. \"I've got a thousand hours. Not in this particular model, but close enough.\" However, she failed to mention those hours were all in a simulator. Sam knew the information wouldn't help lighten their mood. She whispered under her breath, \"You can do this. Piece of cake.\"\n\nPractically every American had an idea of what the interior of Air Force One looked like from movies and TV, but none of that did any justice to the real thing. \"Oh, wow.\" Sam said as she stepped aboard.\n\nWoodard shoved past her and headed for the main cabin, carefully placing the suitcase on the mahogany conference table. Next to it, he arranged the playbook for easy reference. Satisfied all was in its proper place, he relaxed, but only slightly. He moved to sit down, but caught himself at the last moment, grimacing as he saw the pile of ash on the chair. He took in the entire room and gasped. \"Sweet Moses!\"\n\nBody parts decorated the ceiling\u2014dried, tacky blood the only substance holding them precariously in place. It looked like someone took a weed whacker, bathed in blood, to the room. The black ash was thick on the walls and floor. When Woodard turned to look back at the doorway, he could see his footprints smeared in the muck.\n\nWendell paused in the doorway. \"Well, it's going to cost the taxpayers to get this cleaned.\" His joke sounded uneasy.\n\nWoodard looked disgusted as he checked to ensure the next room was safe. Sam thought he seemed rather soft for a major in the military. Wouldn't he be used to seeing the aftermath of war and the carnage it brought? Sam gave the room only the briefest of glances, but it was imprinted in her mind like a tattoo one regretted. It was funny how the mind worked. How one forgot the things one wanted to remember, and remembered (clearly, in stark, haunting Technicolor glory) the things one wanted to forget with all one's being.\n\nBehold his handiwork...\n\nThis time, she did turn, and found Hurt directly behind her. \"You heard that... Tell me you heard that?\" she pleaded.\n\n\"What?\"\n\n\"That voice. Are you religious, Hurt?\"\n\n\"What?\"\n\nIf he said 'What' one more time... She struggled to maintain the fury threatening to overwhelm her. \"Are you religious?\"\n\n\"Used to be.\"\n\n\"I've got this feeling we haven't seen anything yet,\" Sam muttered. \"Something bad is coming.\" She turned and stalked out. \"Want to tag along?\"\n\nShe didn't wait for an answer, but she knew he'd follow. The door to the cockpit was closed. Sam paused and lightly touched the handle, glancing back at Hurt. He shrugged his sympathy. \"Here goes nothing,\" she muttered and pushed the lever inwards. The door gave way with a soft click.\n\nRunway 7\n\nArcher ordered the whole group to start back to Air Force One and the chopper. Wells relieved Travis of his gun and pushed him ahead of them.\n\nArcher kept his gaze fixed on Travis, but a strange flicker of light caught his attention. He sighted movement in his peripheral vision, and glanced up as something darkened the sky above them. It was like a cloud passing over the sun, but Thaddeus realized it was no cloud. The creature wavered in and out of tangible existence. A black, smoky shape one moment\u2014the next, a fully realized being.\n\n\"What the\u2014\" Travis managed.\n\nThe creature was bird-like, but that's where any similarities to something of this world ended. The creature passed over them and, in the draft made by its inky wings, a carrion stench drifted. It extended its snake-like neck and unleashed a scream. The men slapped their hands to their ears to ward off the sonic assault. Peering into the swirling mass, Thaddeus could just make out internal organs. It struck him, then, the creature was somehow not fully in this world. Like a weak radio signal, the body fluttered in and out of focus.\n\n\"It's from the island,\" Jossart cried out. \"It has to be!\"\n\n\"What's he talking about?\" Travis scoffed.\n\n\"We need to go,\" Thaddeus said, his voice low and tense. \"No sudden movements.\"\n\nWells cursed as the creature changed directions and started towards Air Force One.\n\nThaddeus got on the radio. \"Sam, something's headed in your direction. Get out of here. I repeat, get out of here!\"\n\nArcher felt the air contract around him, as if he were in a pressurized cabin. He dove to the right as the creature filled the space where he'd been just a second before. Jossart screamed as the talons pierced his shoulder. With a sickening crunch, the talons locked in place, and all Archer saw were the man's dangling legs as he was lifted from sight. A few seconds later, a shoe plopped to the tarmac in front of him. He glanced up and traced the area with his weapon. It was as if the creature vanished back to wherever it came from.\n\nSomething caught Thaddeus's eye. At first glance, he wasn't sure what he was looking at. The air above him seemed to shimmer in a circular pattern and, for a brief second, Archer could see into the swirling mass. Vertigo almost overtook him, as the vortex tried to suck him towards it. He pried his attention away and the sensation vanished.\n\nThe swirling mass vomited a figure onto the tarmac. It roared and launched itself through the air at Wells. He managed to raise his weapon and fire rapidly at the creature. The bullets had no effect on the thing, merely passing through the smoky form. However, when the creature collided with Wells, it was certainly tangible enough. Archer trained his weapon on the creature, but held his fire for fear of hitting Wells. As the two struggled, the creature began to morph into something more concrete. Archer cursed as he began to recognize what the creature was. It resembled a gorilla, but its fur wasn't black. It was a deep purple, and intermingled amongst the fur, Archer could see razor sharp scales. The creature had Wells pinned to the ground by his shoulders. A tail sprouted from the thing's back, and the appendage whipped forward and impaled Wells, silencing his screams like flicking off a switch.\n\nTravis leaned down and picked up the massive rifle, firing at the creature. The .50 caliber round took a noticeable chunk out of the creature's abdomen.\n\n\"He was already dead,\" Travis muttered, as Archer whirled and pointed his weapon on him. He rested the rifle on his shoulder and dug in his pocket for a smoke. After lighting up, he squinted through the smoke. \"I'm not the enemy.\"\n\nThe creature screamed as it tried to put its guts back inside its body. Travis took another drag on his cigarette and tossed it aside. He leveled the rifle on the creature, waited for a clear shot, then fired. The creature's head became a red mist, and it slumped over Wells' still form.\n\nAcross the way, the woman and her children screamed. Archer started to run towards them, but came to a halt as the phone on his belt began to beep insistently. He glanced down at his side and stared dumbstruck at the phone.\n\n\"You going to answer that?\" Travis said as he took aim. Calmly, he began firing. The horde o f creatures chasing after the woman and her kids fell one by one. \"Well, this is about to get real interesting. Holy shit.\"\n\nAbout a dozen people shambled towards the runway's fenced outer perimeter. The shadows passed through the fence with ease, but their hosts had some difficultly scaling the fence. Travis gasped.\n\nArcher looked up from his phone at Travis's alarm. The people were obviously dead, and had been for quite some time he realized. Their clothes were tattered and dirt stained. Some of them were still dressed in their Sunday best, or burial clothes.\n\nAnother thought struck Archer, deepening his dread: Katrina had left a lot of bodies in her wake.\n\nThe phone nagged him to no end. It wasn't going to stop. Archer fumbled it from his belt and flipped it open. He wanted to say something into the receiver, but words abandoned him. Rather, a choked sound scurried up his throat.\n\n\"Daddy?\" The voice on the other end sounded hesitant.\n\n\"Casey?\" he said. \"Is that you?\"\n\n\"Daddy!\"\n\nJust then, Thaddeus noticed Muldoon wasn't firing any longer, and he followed the man's gaze to the outer fence. A few of the dead had managed to navigate up the fence, and they were currently falling over the top, crashing to the ground. The way some of them were landing would've killed most people, but they struggled to their feet and continued shuffling towards them. The shadows in front of these people were more pronounced than the previous incarnations, he noticed.\n\n\"Baby, is that you?\" He jerked the phone back from his ear as a horrendous keening erupted.\n\n\"Daddy, I miss you so,\" a voice, not his daughter's any longer, jeered. \"Come and rescue me, Daddy. Just like last time, Daddy!\" It disintegrated into maniacal laughter.\n\n\"Casey! Where are you? I'm coming to get you, baby!\"\n\n\"I'm at Grandpa's. Hurry. They're going to get in.\"\n\nThe last words were in his daughter's voice, and it turned his stomach cold. She sounded so small, so fragile, and so scared. Grandpa's? There was no way she was in North Dakota. Impossible.\n\n\"Are you going to save her this time, Thaddeus?\" the other voice hissed. A voice he recognized from a year ago. \"Or will you lose her again? You'd better hurry. She won't be safe from me much longer. When I get her, oh, the things I'll do to her!\"\n\nThe line went dead, and it took a few moments for Archer to realize he was listening to nothing.\n\nAir Force One\n\nAs the door swung outward, Sam dreaded what might wait for them in the cockpit. A rush of relief escaped her lips, as nothing hideous was revealed when it swung fully open. The cockpit was deserted. She leaned down and quickly blew the residual ash from the seat and sat in the pilot's chair, running her fingers over the yoke. She grimaced, as her butt was soaked once it settled in the seat. Peering down between her legs, she sucked in a breath. The pilot had somehow managed to cram his frame into the small recess beneath the controls, and he'd died horribly.\n\nHurt squeezed into the co-pilot's chair, awkwardly shoving his M4 between his legs. When he glanced over at her, Sam noted the concern on his face. As she became still, he fidgeted, showing his concern. \"Everything OK?\"\n\n\"Just dandy.\" Sam clenched her teeth, forcing herself to study the controls. There were so many buttons and gauges. She got dizzy just trying to take it all in.\n\nWendell asked from the doorway, \"We going to be able to get this puppy off the ground?\" His appearance jolted Sam from her concentration.\n\nShe offered a small, weak smile and nodded. \"Sure thing. Just familiarizing myself with the controls.\"\n\nAlmost nothing looked familiar. What little she did recognize, like the fuel status (the gauge read full) and the hydraulics (all green), did nothing to comfort her. The controls were really quite simple, she knew, but now faced with them in all their glory, knowing and believing were in two vastly different ballparks. At least the engines were idling, so there was a God.\n\n\"Sam, something's headed in your direction. Get out of here. I repeat, get out of here!\"\n\nHurt leaned forward and peered out the windscreen. \"You see that?\"\n\n\"Yep.\" Sam pushed forward on what she thought was the fulcrum. The plane lurched and then held fast. The engines screamed, and the plane began to shudder, but they weren't moving forward.\n\n\"What's wrong?\" Wendell asked.\n\n\"I'm not sure,\" Sam said through gritted teeth.\n\nShe pushed the lever farther forward. The engines were at full thrust now, and it felt like the plane was trying to rip itself apart. Sam pulled the lever back. As the engines powered down, the awful shuddering subsided.\n\nSomething flashed across the tarmac and caught her attention. \"Hand me those.\"\n\nHurt glanced down at the binoculars around his neck and looked at her quizzically.\n\n\"Those.\" Sam grasped the strap and yanked it roughly over his head. She brought them to her eyes and sighted in. \"It's Archer and some other guy. Where are the others? What the hell is that?\"\n\nThe men were running all out, a huge mob in hot pursuit. Something else appeared in the air. The thing swooped down, intent on the kill. Sam could barely make it out because its smoky form wasn't concrete.\n\nThe creature swiped at them, but the men hit the ground. After missing its prey, the creature soared back up into the sky, and Sam realized what it was. \"Oh God.\"\n\n\"The reflection of Satan is in my eye,\" Hurt said, his voice guttural.\n\n\"Get out there,\" Sam cried. \"Give them cover or they're as good as dead.\"\n\nShe heard Hurt jump from Air Force one and open fire. His shots ricocheted in her ears, crackling like thunder. She looked out of the cockpit and saw the hordes of dead falling at the hands of Hurt's assail.\n\n\"Hurry up, you idiots!\" Sam heard Hurt yell.\n\nArcher, Travis and Hurt rushed into the plane. Hurt took his seat back beside Sam, and Archer slammed the plane's hatch shut. \"Let get out of here.\"\n\n* * *\n\n\"We're still chalked,\" Sam said.\n\n\"What?\" the president asked.\n\n\"The chalks are still behind the wheels. Damn it. How could I have been so stupid? We're not going anywhere until those chalks are removed.\"\n\n\"We're not going to have time,\" Hurt said as he got back out of his seat, pointing to the far side of the runway at more advancing crowds. They were everywhere. \"Something tells me they're not stopping by for a friendly visit.\"\n\nA few seconds later, Sam heard Hurt firing from the entryway.\n\n\"Mr. President,\" Woodard said, as he passed the SAT phone over. \"It's Wayteck.\"\n\n\"Go ahead,\" Wendell answered.\n\n\"Sir, I dispatched a Blackhawk chopper from Moody about an hour ago. Should be there any time.\"\n\n\"Good decision. We're dead in the water here. What's the status?\"\n\n\"We attempted to move every essential out of Washington using COGCON alerts, but there was mass confusion. The roads jammed quickly and...well...things happened. The tsunami just struck California...\" Wayteck's voice cracked, but he pressed on. \"It was worse than we anticipated, sir.\"\n\n\"Go on.\"\n\n\"It struck the entire west coast, as far inland as Sacramento and Seattle, down through Mexico and most of South America.\"\n\n\"Dear God.\"\n\n\"It gets worse...\"\n\n\"Go on.\"\n\n\"The Presidential line of succession is in jeopardy. The Speaker of the House and Secretaries of State and Treasury and the Attorney General didn't make it out of Washington\u2014\"\n\n\"Wayteck? Wayteck?\" Wendell shouted, but no answer was forthcoming.\n\n\"Sir,\" Major Woodard said. \"You'd better take a look at this.\" He held out the laptop. Across the screen, red dots began appearing. They indicated positive nuclear strikes across the map. D.C. was one of those dots.\n\n\"Major, launch 'em,\" Wendell ordered. He took a deep breath as Woodard turned the key that would start nuclear winter across the globe.\n\nJust outside, a Blackhawk roared across the runway, its heavy machine guns laying waste to the crowd to the north. A few passes and there was nothing left standing in the immediate vicinity.\n\nWendell emerged from the aircraft with a cold void in the pit of his stomach. Any second now, nukes would begin launching. He caught Archer's eye as the man raced up to the stairs.\n\n\"It's done,\" Wendell called down. He didn't need to elaborate any further.\n\nThe captain emerged from the Blackhawk. \"Sir! We need to get you out of here. There's more of those things coming. The streets are packed with them!\"\n\nSam followed Wendell down the stairs. For a few seconds, the survivors just stood around and stared at each other, unsure of what to do next.\n\nArcher broke the silence. \"Sam, I need you and the chopper.\"\n\n\"Thaddeus,\" Wendell interrupted. \"Secretary Carling is with you. We need to separate. Succession rests with us. Patricia, I need you to go with Archer and Ms.Velaska. Take the chopper to Outlook and find agent Palmer. He and the others will be expecting you.\"\n\nCarling appeared less than thrilled, but at the same time the expression on her face was odd\u2014somehow knowing and smug. Archer studied her momentarily then turned to the president.\n\nHis stomach churned with stress. \"Mr. President, with all due respect...you don't mean\u2014\n\n\"\n\n\"That's exactly what I fucking mean,\" Wendell snapped. \"There is no more America, and if we hope to rise from this, our survival depends on one of us surviving.\"\n\nWith that, he motioned for Hurt and Woodard to follow him. He gave Carling a peck on the cheek and headed for the awaiting Blackhawk.\n\nWoodard placed a phone in Archer's hand. \"This will keep us in contact even through the fallout. Get to Montana as quickly as you can. Hell's coming.\"\n\n* * *\n\nWith a heavy heart, President Wendell watched the other helicopter disappear in the opposite direction. He hoped they made it to Outlook in one piece.\n\nHe sighed and rubbed his eyes. The decisions made over the previous few minutes weighed heavily on his mind. He'd ordered the launch of nuclear weapons, and had become the second American President to do so. A dubious honor to be sure. What now? How would the country recover, and would he have the fortitude to make even more harsh decisions in the coming hours?\n\nHis thoughts were interrupted as the SAT phone began to chirp. Woodard glanced at it and frowned. After the severed connection with Wayteck, they'd assumed Washington had been obliterated, and this incoming designation number was definitely not from that location. He looked to Wendell for guidance, and the man nodded his consent.\n\n\"Might as well see who's calling.\" He gave a wry, tired smile.\n\nWoodrow keyed the speaker. \"Major Darren Woodrow. Identify yourself.\"\n\n\"Major Woodrow. Is the president with you?\"\n\n\"Identify yourself,\" Woodrow insisted.\n\n\"Name's Gibson Blount. I am an operative of Black Rock, sir. I have urgent news for the president. Is he still alive? Is he with you?\"\n\n\"Black Rock?\" Woodrow's eyes widened. \"How on Earth did you get this number? Only a few people left alive would have this\u2014\"\n\n\"I have inside connections. Look, I don't have time to explain. People are dying. Goddamnit, get me the president.\n\n\"What's all the yelling? Woodrow, just give me the phone. I'll get to the bottom of it.\" Wendell cleared his throat. \"Who is this? Blount? The name is somewhat familiar, but I don't know how...what the hell is going on?\"\n\n\"I apologize Mr. President. I'm a special agent with certain intel at my disposal. I can explain it to you when the planet isn't about to go to hell. I know things are about to get real ugly.\"\n\nThe president went silent, recognizing Blount knew much more than he should. If this _specia_ l agent was on his side, he wanted to keep it that way. It was best to play along. \"Get to the point, Agent Blount.\"\n\n\"First off, sir, are you safe? Out of the thick of it?\"\n\n\"Well, we made it out of New Orleans in one piece, and we're currently airborne. It's good to know someone out there gives a damn about my well being.\" The president let out a nervous chuckle.\n\n\"Of course, Mr. President. How much have you been briefed on Black Rock?\"\n\n\"I know the highlights.\"\n\nThere was a pause as Blount decided how to proceed. He sighed and soldiered on. \"That will have to do. A couple of girls, sisters if you will, located your number for me.\"\n\nThe president had an inkling of what he meant, but really just humored him so he could hear more. The more he could learn in order to limit his chances of dying, the better.\n\n\"Go on.\"\n\n\"Here's the situation: We have landed on the island and have encountered stiff psychic resistance. There is definitely something here, sir. It's bad, and it's a mean son of a bitch...I apologize for the profanity, but I need you to understand the gravity of this situation.\"\n\n\"No apology necessary, Mr. Blount. It's a nightmare out this way as well.\"\n\n\"This island appears to be the source of the shadow creatures. They are hot and heavy here. We've suffered casualties. Whatever this is, it's definitely of supernatural origin.\"\n\nHearing Blount use terms like _psychic_ and _supernatural origin_ would have been difficult to digest under normal circumstances, but after all he'd witnessed so far, Wendell was definitely ripe to accept anything. He'd been briefed on Black Rock when he took office, but he'd dismissed the program as NSA\/CIA pipe dreams when he'd read over the psychic nature of Black Rock's missions.\n\nWendell came to the quick realization that Black Rock was designed to be nearly impossible for any sitting President to dismantle or discontinue. Black Rock's activities were also highly compartmentalized, to provide him with plausible deniability. He could probably have found out more than he was briefed on, but he had been faced with many more pressing issues than chasing ghosts through a labyrinth bureaucracy.\n\n\"This is real, isn't it?\" Wendell said.\n\n\"Excuse me?\" Blount sounded confused.\n\n\"What you do, all that you see\u2014it's all real, isn't it?\"\n\n\"You better believe it. I apologize for making you play catch up, but we are pressed for time. My team geologist has informed me that much of the flora on this island should be extinct, and that tells me we are dealing with something very old. In my experience, the older something is, the more powerful and pissed off it tends to be.\"\n\n\"Mr. Blount. In your honest opinion, do we have a chance?\"\n\n\"There's always a chance, sir.\"\n\n\"Blount...\" Wendell stopped to clear the lump in his throat.\n\n\"Yes?\" Blount could tell by the change in the president's tone what the man was about to say.\n\n\"North Korea is launching. All of us are.\"\n\n\"Where?\"\n\n\"The island.\"\n\n#  Part II\n\n# Gibson Blount\n\n#\n\n\"At any moment several millions of human beings may be smitten with a new madness, and then we shall have another world war or devastating revolution. Instead of being at the mercy of wild beasts, earthquakes, landslides, and inundations, modern man is battered by the elemental forces of his own psyche.\"\n\n\u2014Carl Gustav Jung\n\n#  Chapter One\n\nSomewhere Over the Pacific Ocean\n\nDeep in thought, Gibson Blount looked out the Blackhawk's bay door. The ocean below glistened in the sunlight, but its beauty was lost on him. He shifted in his seat and glanced at his team. They'd been handpicked for this mission. Each possessed a skill set that would likely come in handy in the next few hours. Exactly what was to come, he wasn't sure. That was the question. Blount had seen much over his lifetimes. The things he knew, the things he'd seen, would make the average citizen run screaming for the hills.\n\nYet, despite everything he'd seen, everything he'd fought, this was something new. He knew much about the Old Ones, but had never come across any evidence they'd really existed until now. Monsters went by many names. It was man's nature to attempt to name the unknown. The problem was, with so many names floating around out there for so many different gods, he was never really sure which one he was dealing with.\n\nChristianity, and a host of other religions, stated there was only one God, but Blount knew better. It was all right there in the text, if only someone would read it. Why else would God command to have no other Gods before Him, if He were the only God? Blount had met a few of these gods over the course of his lifetimes, and he'd come to the conclusion that the majority of them were ill tempered and sorely misbehaved. So it stood to reason that if God, as in the 'Bible God,' existed, He was one pissed off entity.\n\nBlount had fought minions and angels, demons and sirens. Even though all of these creatures were as different as the next, the one thing they all seemed to crave was absolute power\u2014the attainment of their own godhood.\n\nOne entity in particular he'd faced long ago, when he'd been a gunslinger, still haunted him from time to time. He'd battled Azazel and prevailed, but the victory had come at a great cost. He'd lost his best friend, Father Nathan Carpenter, and his wife, Sarah.\n\n\"ETA, three mike,\" a voice squawked in his ear.\n\nBlount adjusted the headphones and repositioned his mic in front of his mouth. \"Roger that.\" He lifted one side of his headset and slipped a cell to his ear. \"Talk to me.\"\n\n\"Normal instruments can't penetrate the island's interference, Viper, but we've been using ELDON and have some preliminary intel...\"\n\nBlount nodded as he listened. He'd utilized Project ELDON a few times while working for Black Rock, and had been quite pleased with their results. ELDON was composed of indigo children who were endowed with psychic gifts used to remote view. It got its name from its founder, Eldon Shepherd. Known to most in the organization as simply \"The Shepherd,\" he was one of the most powerful psychics Black Rock had ever known, and he loved the children, nurturing their gifts and protecting them from misuse until the day of his death. Some in the organization wondered if Eldon was a real person. Few had seen him when he was alive\u2014some rumored that he wasn't even flesh and blood. The outside world knew nothing of him. Actually, to the outside world, Black Rock and its pet project ELDON didn't exist.\n\nA few minutes later, Blount brought the conversation to a close. \"So ELDON is not seeing anything? Am I getting this correct? There's nothing on that island? Great, guess we'll find out when we get there.\" In frustration, he flipped the cell closed and shoved it back into his cargo pocket. \"Well, twins, guess you two are up.\"\n\nTwo red-haired women sat across from him, identical in every way, wearing the exact same black jumpsuits. They were in their early twenties, but their souls were much older than that. From the age of three, they'd been able to see and feel things other humans could not. Paired together, they were two of the most powerful psychics in the world, and they were good friends of Blount. Actually, they had saved his life more than once, in a past life.\n\nHe remembered that well. He remembered all of his past lives, at least this time around. He was aware that, in some lives, he didn't retain all the memories. In this particular one where the twins had saved him, he'd just finished a stay at a room above the Red Oak Saloon. Downstairs, three jackasses had been waiting to ambush him. The twins came to him in a dream and warned him. Good thing he took all his dreams seriously, never ignoring his gut or intuition.\n\nBlount got the drop on those suckers but good. He slipped out his bedroom window, climbed down the nearby cactus, and came right through the damn front door with guns blaring. That was that. Or so he thought.\n\nHe surveyed the damage to the bar\u2014smashed glasses and whiskey bottles, broken bodies on broken tables, blood everywhere. Feeling a bit guilty, he went back into the Red Oak Saloon and walked up to the bar, tossed a stack of bills at the bartender, pulled up a chair, and ordered a drink. He swigged half the whiskey down then took a rest. Wiping the sweat from his face, he sighed.\n\nNext time you need to be a little quicker. Another moment or two, and they would have started up the stairs for you.\n\nBlount drew his pistol and spun around. No one was behind him. No one stood outside the saloon doors either. All seemed quiet, since his gunfire cleared the place. He scratched his head and shrugged. Turning back around, he picked up his drink.\n\nWe know you're a bit rusty. Your new body must be strange.\n\n\"Get out of my head!\" He realized the voices were inside his mind. The bartender kept his distance, hands trembling, barely able to wash the glass he clutched in his fat fingers.\n\n\"We're sorry,\" someone said from the corner.\n\n\"We speak without using our mouths, so often we forget.\"\n\nIn the corner of the barroom, two young, strawberry-haired girls sat at a table. The shadows of the room danced around them as if alive. Blount did not remember seeing them there before. He was pretty sure he'd searched the entire room.\n\nBlount blinked and rubbed his eyes. He got up from his seat and approached the girls slowly, eying them up and down. His fingers brushed against his guns. \"I'm seeing double. What kind of witchery is this?\"\n\n\"It is not witchery.\"\n\n\"No magic. We are sisters. Twin sisters.\"\n\nHe studied them a bit more and stepped up to their table. The shadows cleared and he looked at their faces. \"You're the ones I see in my dreams. You warned me about this ambush?\"\n\n\"Yes, we did,\" they said in unison.\n\n\"Stop that. It's unnatural.\" He sat down with them. \"What's going on? Why are you saving me?\"\n\n\"We were sent to you. To help you adjust. We understand it has only been a hundred years since your last life, but we need you to remember who you are. We need to catch you up.\"\n\n\"Catch me up? What for? And how do you know about my last life? I remember a great fight. Losing those I love. Again. I gotta say, it sucks. Having to start over again and again. Not knowing why or what happened. Who was I? Why was I? It's a raw deal, girls.\"\n\n\"We understand your frustration. It is not an easy existence when the universe claims you. You must understand, it is for a greater purpose. It is not just this world that needs you but many\u2014many beings, many realities.\"\n\n\"Let's cut the jabbering. Who sent you?\"\n\nThe twins looked at one another. Uncertainty registered in their eyes. One started to speak then hesitated. Finally, they both answered. \"Abdiel sent us.\"\n\nBlount strained to remember, to find meaning in this name. He crinkled his forehead, rubbed his face. \"Should I know that name?\"\n\n\"He was there when you defeated Azazel.\"\n\n_Azazel. AZAZEL._ That name he knew. At least, he thought he did. The very uttering of it sent shivers up his spine. It made him uncomfortable. His stomach turned at the thought of the fiend, the murderer, the pain in the ass.\n\n\"All of this is filling up my head, and it hurts, girls. Too much, too fast. Enough.\" He stood up and pushed in his chair. \"Sorry. Tell Abdiel I'm adjusting just fine. I don't need your help.\" He started toward the door.\n\n\"We need each other's help.\"\n\nHe waved them off and kept walking.\n\n\"Azazel will return. He is weak and formless, but growing stronger.\"\n\nBlount stopped in the doorway and looked over his shoulder.\n\n\"His minions are looking for you. They're quite angry at you, as you might imagine. They're walking the planet while he's gone.\"\n\n\"I'll be sure to watch my back.\"\n\nBlount let the saloon doors swing behind him. He mounted his horse and road off, kicking up clouds of dust behind him.\n\n* * *\n\nBlount rode fifty miles before he found another town, yet another dust bowl with barely a population to speak of. There were a few downtrodden farmers, more saloons than the town needed, and a cavalry presence. For what reason, Blount couldn't be sure.\n\nJust on the outskirts, he was lucky enough to find a small boarding house\u2014Bessie's boarding house. Bessie had long since died, and her son and his young daughter now ran the place. The owner was a widower, his wife having left this world due to small pox that had hit the town a few years back. Probably explained the lack of population.\n\nHe paid the board up front, and turned to head upstairs to his room. The owner's daughter, Abigail, sat on the foot of the stairs reading her bible. She still wore her Sunday best dress and bonnet, and was the picture of serenity. He stared at the book in her hands and shook his head. He sometimes wondered how anyone kept there faith these days. Seemed their Heavenly Father had jumped ship long ago. If He was out there watching, why did he allow it, any of it\u2014the disease, the demons, the world brought to near destruction\u2014to happen again and again. He was tired of it. Blount hoped she didn't preach the word to him today, for he wasn't in the mood.\n\nHe settled into his room and took to the bed. It has been a long day. The springs creaked and the mattress was uncomfortable, but he didn't care. Soon enough, he caught himself snoring, as the light of the full moon streamed through the window and illuminated the room.\n\nGet up. Get up. You are in danger. It is nearly on you.\n\nIn the dead of night, Blount heard the words in his head. He strained to open his eyes, but his lids felt as if they weighed a ton. A blurry haze greeted him as he tried to focus.\n\n\"Girls?\"\n\nNow. Get out of bed now.\n\nHe lifted his head to see someone standing at the foot of the bed. He recognized the bonnet. \"Little girl? What are you...\"\n\nShe moved toward him, and the moonlight drifted across her face. Her eyes glimmered in the light\u2014enflamed, fierce. She foamed at the mouth, and her chest heaved rapidly. Howling at him, she leapt on the bed like a cat.\n\nBlount tried to reach for his guns on the nightstand ,but he was seconds too late. The girl grabbed his arm and yanked him off the bed. His back hit the nearest wall, and he crashed to the floor. The girl crouched on the bed and growled, while he eyed his guns, which were too far out of his reach now.\n\nHe began to pull himself up, but the girl sprung from the bed and collided with him. They crashed to the floor, and she crawled onto his chest, sitting on it. Blount felt as if an elephant was sitting on him. She looked down at him, smiled, and licked his face.\n\n\"Azazel sends his regards.\"\n\n\"Tell him to kiss my ass!\"\n\nThe door to his room burst open, and the girl's father appeared. \"Abigail! My Holy God, no!\"\n\nAbigail turned and leapt from Blount to her father. She forced him against the wall and bit into his throat. Blood splattered the walls.\n\nBlount crawled across the floor, toward his guns. Closer...closer...he could see them now. His fingers came within reach.\n\nThe father's lifeless body came crashing down on the night table, burying the guns. Blount looked up and saw the girl descending on him again. He braced for her attack, curled his hands into fists, and steadied his balance.\n\nHe heard creaking on the stairs outside the room. Over the girl's shoulder, he saw the twins walk into the doorway. They joined hands and closed their eyes, then an aura of light surrounded them, and a beam of energy shot out of the twins, striking the possessed girl. Glass shattered as she went out the window, her cackler echoing in the night.\n\nThe light around the twins faded and they opened their eyes. \"We told you that you were in danger.\"\n\n\"That you did.\" Blount sighed. He pulled himself to his feet and rolled the body off of his guns.\n\n\"It is not safe here,\" one twin said.\n\n\"There will be more,\" said the other. \"Come with us.\"\n\n\"Guess I better start listening to you two, seeing as how this is the second time you've saved my hide. Lead on girls.\"\n\nThe twins led him outside of town to a circle of covered wagons, some tents, and campfires. A dog barked at their approach. Blount read the words on the wagons. \"Traveling sideshow. You mean...you girls are in a freak show?\"\n\n\"Yes, for now.\"\n\n\"No. You deserve better. This is no life for you.\"\n\n\"It's not so bad,\" Maya said.\n\n\"It is where we are meant to be at this time,\" said Nina.\n\n\"Listen Maya, Nina, you are not freaks....wait... You never told me your names. How did I know them?\" He looked at the twins, and they smiled. \"Ah, you two are good. Like I was saying, you're not freaks. If we agree to help each other, you have to leave this show.\"\n\nThey smiled again. \"Our hero.\"\n\nBlount laughed.\n\nThey joined a naked tattooed man, a bearded woman, and a dwarf for dinner around the campfire. A cook with one arm made them all a stew with buffalo meat.\n\n\"So, other than being exact copies of each other, what else do you do in the show?\" Blount asked the twins.\n\n\"We levitate objects.\" They pointed at the stew pot and it lifted off the fire, hovering in the air for a few moments before lowering back down.\n\n\"Very neat trick.\"\n\n\"It's a living.\" The twins stared into the fire and frowned. \"We will have to move on again in the morning. More of the possessed are coming. This town will be overrun soon.\"\n\n\"I don't understand it. Any of it. What's the point? Demons, angels, other realities, other universes, the end of times again and again. Is it some kind of joke? God almighty, what is the point? Speaking of our Lord and savior...where is He? All of this shit goes on for hundreds of years, and he can't even show his face.\"\n\n\"He is out there,\" the twins said.\n\n\"Really? You could have fooled me.\"\n\n\"God will show. It's just a matter of which God.\"\n\n\"What?\"\n\n\"We are unsure what it means at this time.\"\n\n\"You two sure are a puzzle.\"\n\n\"Not everything we receive is always clear. Sometimes it is only in bits and pieces.\"\n\n\"I guess you can't have everything, even when you're special.\"\n\nThe dogs in camp started barking, setting everyone on edge. Blount cocked his head and heard what might have been thunder, then he realized it was the sound of galloping horses.\n\n\"Horses! Ambush!\" He jumped to his feet and drew his guns.\n\nScreams filled the camp and chaos ensued. Cavalry roared into the camp, firing on the sideshow, throwing torches onto the covered wagons. Blount caught their raging faces in the firelight, eyes enflamed, mouths frothing. They were all possessed.\n\nHe shot two down as they rode close to him. The twins locked hands and backed out of the light. The stew pot hurled into the air and caught another horseman in the head. Blount shot randomly as the scene grew out of control. He ducked behind a wagon while the cavalry searched for him.\n\n\"Come out, come out, wherever you are!\" one called with menacing laughter.\n\nThe twins huddled together, preparing their escape, when a horse streaked out of the shadows. It lifted its hooves and battered the girls. The two went down hard. Blount heard their screams and leapt from his hiding place.\n\nHe ran across camp, as a rider slashed the guns out of his hands with a cutlass. The soldier slashed again, but Blount caught his arm and dragged him from the horse. He wrestled the cutlass away and swung hard, taking the soldier's head off. Speeding across the camp, he spied the twins about to be trampled. They looked drained, as if their power were at its limit.\n\nBlount climbed the burning wagon closest to him, flames encircling him, and leapt into the air. He came down on a horse and its rider, all three collapsing to the ground. A few good slashes and Blount gained the advantage, dispatching the soldier. He pulled the twins up and moved them into the woods.\n\n\"We've got to keep moving,\" he said as they stumbled.\n\n\"We can't keep running like this. Our physical selves are not as strong as our minds.\"\n\n\"Just gotta find a safe place.\"\n\n\"These are not safe times. You must go on without us.\"\n\n\"No! You can do it. Just keep moving.\"\n\nHe dragged the girls miles away, until they reached the bed of a river. It was there that they put up the white flags.\n\n\"We need...\" Maya began.\n\n\"To stop,\" Nina finished. \"We can go no farther. We must rest and recharge.\"\n\n\"Then we set up camp,\" Blount said.\n\n\"No, you must go on without us.\"\n\n\"I'm not leaving you here.\"\n\n\"You must. You are the target\u2014not us. There is a town up the river. Look for a hill. Once you scale it, you will see the town. Abdiel will be there. Seek him out.\"\n\n\"This feels wrong. Will you be alright?\"\n\n\"We do not sense the dark ones any longer. We have escaped them. We will be alright. Now go. God is not the only savior of this world.\"\n\n\"I hope you know what you're doing, girls.\"\n\nAn aura glowed around them and they smiled. \"One day, you will see that we do.\"\n\nHe took their hands into his. \"Thank you for saving my life.\" He gave them a wink and started down the river.\n\n\"Blount,\" they called to him and he looked back. \"We will see God again. He is out there.\"\n\n* * *\n\nHe is out there.\n\nThe twins' words from all those lives ago still rang in his head. He wondered if what they told him would ever come true. He was still waiting. He glanced out the window at the lush green island coming into view, then he turned back to the girls.\n\n\"Are you getting anything now, ladies?\"\n\n\"We need to be closer,\" they said in unison. Each time they spoke at the same time, or finished each other's sentences, it creeped the others on the team out. To Blount it was just par for the course. \"We need to be on land. Touch and feel. Something is trying to block us. It knows we're here.\"\n\n\"What knows you're here?\"\n\n\"We do not know its name yet. We cannot see its face or its eyes. It blocks us with a very strong will.\"\n\n\"OK, well, we'll be down in two minutes, and the floor will be yours.\"\n\n* * *\n\nThe Blackhawk settled down in a clearing on the strange island. Outside the window, a strong wind lashed and shook the copter, and branches and brush smacked up against the sides with a clatter.\n\nBlount stood and addressed the other members of his team.\n\nGeological expert Nick Larson was brought on to examine the land, the plant life, the organic make up of the island. Perhaps he could determine how old it was, and if it was safe to stay on the island. Blount wasn't sure if the island would sink as fast as it had rose, or break apart due to volcanic activity they'd been picking up on their sensors during the flight over.\n\nMax Anders was Blount's theologian, versed in different culture's religious beliefs and spiritual folklore. If there were any relics or ruins on the island, he might be able to identify them. What if the theories on sunken cities weren't that far off the mark?\n\nThere was also the usual collection of grunts. Blount's firepower, just in case things got rough. The man in charge of this nine member ragtag tactical team, known only to the Black Rock organization, was Simon Baxter. Baxter was one tough mother. Almost as tough as Blount, but Blount had been in more combat situations. After all, he'd lived many more lives than Baxter. Finally, Doctor Graham Saylors, military-trained medical officer, would serve as doctor for the entire team.\n\n\"We all ready? This is for real people. We don't know what the hell awaits us out there, but something does. I don't care what ELDON says. The twins have been picking something up for days, even before the thing surfaced. So stay alert. Stay alive. And somebody wake that jerk up.\" Blount pointed at a sleeping soldier in the back.\n\n\"Brooks, you clown!\" Simon yelled. \"Get the hell up. This isn't a damn siesta!\"\n\nThere was no response from the man. A fellow trooper shook him, and his head bobbed to one side.\n\n\"He hates flying sir. Was complaining about some chattering in his ears,\" Private Amanda Fletcher volunteered. \"He takes sleeping pills on every flight to sleep through it. Wakes when we land. He wanted to make the noise stop.\"\n\n\"Say what?\" Simon looked perplexed.\n\nBlount made his way down the aisle, grabbing a hold of the slumbering soldier. He shook him hard, and an empty pharmaceutical bottle fell out of his pack. Blount eyeballed it, then looked up at the others. \"He's not sleeping. He's dead. The bottle is empty. Your boy's overdosed, and I doubt it was an accident. Something got to him.\"\n\nBlount let the fallen soldier gently slip back into the seat. The soldier trembled all over, arms and legs flailing before he belted Blount across the chin, sending him to the floor.\n\nThe rest of the team scrambled, withdrawing from the dead soldier as he twitched and jumped up like an out of control marionette. He stumbled forward and made another lunge for Blount, who was not about to get caught off guard twice. He rolled out of the dead soldier's way and pulled out his pistol.\n\nOne round hit the man's chest, but all he did was roll over. He crawled on his hands and knees and hissed at Blount, drool escaping his lips, his eyes enflamed.\n\nSimon Baxter shot him in the back of the head, blood splattering an arc across the walls, and yet the soldier kept on crawling. He was determined to reach Blount, hands dragging him along, his blood leaving a glistening trail down the aisle.\n\nBlount heard fingers snapping as the dead man closed in on him, then he noticed something stranger yet. A shadow crawled ahead of the soldier\u2014a dark carbon copy that was illusive and cunning. It was barely noticeable until the lights overhead picked it up. Something wasn't right. This shadow mimicked the soldier's every move, yet seemed intelligent. At the same time, it was not of this reality. It was composed of little substance, made of translucent blackness that used the soldier's body. Blount saw it stretch before him, parts of it splintering when it touched the seats and walls. He put two and two together, aimed at this new target, and fired.\n\nHe blew holes in the shadow, watching it explode into pieces before dissolving and vanishing. In the distance, he heard a faint screech. The soldier fell to the floor, lifeless. Getting to his feet, Blount wiped the sweat from his brow and holstered his gun. He looked at the stunned team, their eyes wide and faces pale.\n\n\"Like I said, stay alert. Stay alive. Right, let's open the doors.\"\n\nWhat the hell have we gotten ourselves into here?\n\n#  Chapter Two\n\nOutlook, Montana\n\n\"We'll touch down in Outlook in about eight minutes,\" Sam Veleska said to Archer, as he stared at his watch.\n\nThey'd left Air Force One on the ground back in New Orleans, and they'd hauled ass in the Pave Hawk not a moment too soon. Looking out over the bleak moonlit landscape below, he was beginning to regret leaving the big plane with all its supplies and weapons. They'd fit what they could in the chopper, to the point of almost overloading it.\n\n\"Great,\" Sam said. Archer followed her gaze. The low fuel level light flashed on the remaining reserve tank. Sam ignored it. \"Why are we in Montana again?\"\n\n\"It's one of many classified fallout shelters for POTUS,\" Archer said. \"It's just outside the town. Besides, the population is about eighty people in Outlook. The more isolated we are, the less likely we run into those things. This town is totally off anyone's radar, and I'll bet more than half the adult population works for the government.\"\n\nArcher felt Sam's unease about the whole thing, but she did her best to keep it under wraps. He appreciated it. He didn't need anymore head cases right now. \"You OK?\"\n\n\"Just thinking about Hurt. Where he might be heading.\"\n\n\"That's a good question.\" Archer wondered the same thing. To maintain security, President Wendell, refrained from divulging his group's destination, only telling them they were headed to another secure facility.\n\nHe glanced out at the western horizon and squinted against the glow. All those people dead, all those cities now silent and covered with radioactive dust. The eastern horizon wasn't much better.\n\nThe phone Major Woodard handed Archer would only blast static at him, and he wondered if the SAT system had been taken out, what with hell now hitting the planet. He didn't know what was worse: the tsunamis, the nuclear launches, or the damn shadow creatures in every shape and size trying to overrun the friggin' world.\n\nSam set the chopper down a few miles outside of town in some of the flattest land Archer had ever seen. He stepped out first, surveyed the area with pistol in hand, and when he was convinced it was clear, gestured for the others to join him.\n\nSecretary of the Interior Patricia Carling emerged next, followed by Sam. Both women looked around anxiously as the wind blew a thin film of dirt across their faces. Travis Muldoon brought up the rear. There was an unnatural heat to the breeze.\n\nTravis glanced at him and confirmed his fears. \"It's from the blasts. That's why it's so hot out here.\"\n\n\"OK, Archer, what do we need to do next?\" Carling asked.\n\n\"We need to walk. Head for town and locate our contact.\"\n\n\"Are you sure that's a good idea? Anyone in town might be one of those shadow creatures gone mad. Remember those back in New Orleans?\"\n\n\"I understand that, but standing out in the middle of nowhere isn't going to cut it either. We have no way of contacting anyone at the fallout shelter with the SAT phone on the fritz, and we just can't wait out here, hoping someone will pick us up. Trust me, we need to head to town. I can get us to the base from there.\"\n\n\"Fine. Whatever you think is best.\"\n\nArcher glanced at the others. \"We walk. Let's fall out.\"\n\nHe wasn't expecting such resistance from Carling. Her tone was changing, becoming more argumentative, and there was something about her voice. _Why does she sound familiar? Where have I heard her before?_ It had nagged him for some time now, but he couldn't put his finger on it. He pushed it aside for the time being.\n\nArcher, Sam, Travis, and Secretary Carling walked from the middle of nowhere into the great unknown. Archer felt eyes on him. Something watched them from above\u2014something shadowy, illusive and perhaps...massive? He hoped to God it wasn't ready to start something with them. He doubted his service pistol could handle the job.\n\nAll Archer could dwell on as they walked was the fact it had finally happened. Nuclear weapons had been launched and cities were gone\u2014wiped off the map. Even in his most sweat-soaked nightmares, he never thought the idiotic leaders of the world had the rocks to actually do it. He was wrong.\n\nWhat now? What did this mean? Were there survivors in those cities? How long would a nuclear winter last? What did this mean for the world?\n\nHe found his thoughts turning to Casey. It was all he could do to keep himself from falling apart. Knowing she was alive out there helped. Though he couldn't explain it, he sensed they needed her to stay alive. Surely, the cult sheltered themselves and Casey from the nuclear fallout. They needed her for their plans, and they would have seen this coming. Right? But for how long was she safe? How long would they wait? _Old ones...shadows...the island._ He was convinced the cult knew something about how this all fit together, but what did it have to do with Casey?\n\nA glimmer of light appeared on the horizon. The town proper of Outlook was only a couple miles away. They picked up their pace with anxious excitement, needing to find answers, needing to make sense of a world that was quickly going to hell.\n\nTravis muttered something about leading the lambs to the slaughter. Archer ignored him. He could stay with them or not. He couldn't care less. He had more important things on his mind.\n\n* * *\n\nOutlook looked like a ghost town. There were no running cars in the street, no pedestrians out and about. Darkness clung to the buildings in patches. The occasional streetlight or bar sign provided the only light.\n\nA few houses that dotted the horizon sat devoid of any signs of life. No lights or sound\u2014nothing. Archer began to wonder about this government fallout center.\n\nSecretary Carling continued to walk toward the horizon. Sam walked beside her, one hand on her pistol. Archer thought Sam was acting more and more like him in these last few hours.\n\nArcher turned his gaze to Travis, who cracked a few jokes and toyed with his rifle. It was obvious he avoided the reality of the situation at all cost. Archer still didn't trust him. He was not coping well with what he had seen. It wouldn't be long before his sanity cracked.\n\nThe group crossed an intersection and listened to the wind howl. Carling somewhat took the lead, as she moved the team down a narrow street with decrepit apartment buildings sprawling on either side. Archer detected a rank smell in the air, mixed with that of motor oil and dirt.\n\nAmerican flags hung outside some of the buildings, rippling in the wind. Archer stared at them in surreal disbelief, before he heard the familiar sound of someone tapping cigarettes. He swiveled his head and saw Carling lighting a smoke. She noticed his gaze and offered the pack. He hesitated for a brief second, and then took it. He'd quit a year ago, but the act of lighting the cigarette calmed him, and he inhaled deeply.\n\n\"Thanks.\" The smoke felt like magic in his chest. Old habits never died.\n\nThe group stopped to rest and smoke when the wind picked up again. Archer dragged and expelled smoke one more time, before catching movement on one of the rooftops of the adjacent apartment buildings.\n\nA person came crashing down on Travis, sending him onto the street. His rifle slid out of his hands as a wild man grabbed handfuls of his hair and bashed his head into the pavement. Howls of agony filled the night air as the group reacted, spreading out and reaching for their weapons in white-knuckled shock.\n\nThe crazed undead man drove Travis's head into the street over and over, until it was nothing but a pulpy mess. Archer saw the black shadow figure above the man, mimicking the actions of its host. The form was darker than night and here, too, as Archer had noticed before, it seemed more solid, as if gaining ground in this world.\n\n\"You son of a bitch!\" Sam screamed, drawing her pistol and running to Travis. She took aim at the stranger's head, but it was too late. Travis was gone, his hands twitching slightly until falling still against the street.\n\nThe shadow figure turned and swiped its arm towards Sam before she got off a shot. Its host followed suit, as the dead man leapt to his feet and smashed Sam across the jaw, sending her reeling.\n\nArcher whipped his service pistol up and fired at the shadowy form several times. It buckled under its spasms, shattering into thousands of pieces and dissolving into mist the air carried away. Its host collapsed to the ground motionless.\n\nSam rolled around on the street favoring her jaw. Archer pulled her to her feet, noticing the blood that seeped from her lips. \"You all right?\"\n\nArcher was pissed. Just as Sam opened her mouth to respond, they heard moans rising in the distance. They turned and saw a group of people lumbering out of a nearby alley. Their faces were drained of all color and their eyes were blank. Some of them had mortal wounds still pumping blood out of their bodies. Ahead of them, their shadows strolled like apparitions floating on fetid air.\n\n\"This is crazy,\" Sam said after she spit up blood.\n\nArcher agreed. They lifted their guns.\n\nArcher pushed Carling behind him. What good that would do, he didn't know. As the group of about fifteen advanced on them, something came roaring down the street. Seconds later, a pick-up truck plowed into the oncoming dead. Bodies sailed into the air, and a few were crushed beneath the wheels.\n\nThe truck screeched as it completed a semi-circle, wheels tossing dirt and grit, a body rolling over its hood. It raced over to Archer and his team. The door flew open and a young man shouted, \"I'm Agent Palmer. Get the hell in!\"\n\nThere was no time for arguments or verifying credentials. Archer had to trust he was the man the president mentioned. He and the others piled into the truck and its bed.\n\nThe truck sped away. As Archer looked up, he saw a hole rip into the star-lit sky, and watched in disbelief as a giant shadow bird soared through it. Darkness fell over the entire area as the thing swooped towards them to make an attack with its talons. A shrill screech filled their ears, threatening to drive them all to the brink of madness.\n\n\"Hold on!\" Palmer called as he slammed the accelerator. The truck leapt like a wild stallion and burned through the streets, narrowly missing streetlights and mailboxes, passing over unpaved roads and tossing its occupants around like pin balls.\n\nAbove, the winged beast made of sentient darkness cut through the air, attempting to follow the truck as it escaped at break-neck speed. At last, the truck ditched the hellish thing by squeezing in between tenement buildings. Sparks flickered as the truck pushed its way in, side mirrors clattering to the ground. For the moment, they waited. There was no sign of the thing, now.\n\nPalmer turned to them with a boyish grin. \"Welcome to Outlook!\"\n\nThe group said nothing.\n\n\"Is all of your team here and accounted for?\" Palmer's tone turned serious.\n\n\"We lost one of us back there,\" Archer replied. \"But yeah, we're all here.\"\n\n\"Very good. I'll be taking you to Dust Bowl. It's underground a mile south of town. Don't worry. We're all tight and by the book down there. Safe from the world's issues for now. There's a lot of debriefing needed, so let's get you there.\"\n\n\"You got it,\" Archer said, looking at Carling.\n\nShe smiled knowingly. \"Good to see you agent. Secretary of the Interior Patricia Carling.\"\n\n\"Ma'am, Agent Eli Palmer. Glad to be of service. We were expecting you. President Wendell said you were headed our way.\"\n\n\"President Wendell? Then he's still alive?\"\n\n\"Yes, ma'am.\"\n\n\"I see. We must meet with your commanding officer.\"\n\n\"No problem, ma'am. We're on our way.\"\n\nArcher simply shook his head and watched the skies, as the truck headed south out of town.\n\n# Chapter Three\n\nRugby Rock, North Dakota\n\nFlickering torches illuminated the cold, stone walls as the six of them gathered, their faces partially hidden beneath the red hoods of their robes. There was a collection of whispering among them, fraught with tension and worry. Two separated from the others.\n\n\"Catastrophes have hit the planet,\" the man said. \"We're lucky to be alive.\"\n\n\"It is just as it should be,\" the woman answered. \"It was foretold to us. Of course we are spared. _He_ protects us. There will be terrible disasters laying waste to mankind, preparing the way for the return of the Old One. We will be rewarded beyond our wildest imaginations.\"\n\n\"Why don't we get on with it then?\"\n\n\"You know we must wait for our leader. She must bring him to us. The cycle cannot be closed until he is here.\"\n\n\"The Old One is not the only thing we've brought into our world.\"\n\n\"We are aware of them.\"\n\n\"There are many in number.\"\n\n\"They will be dealt with. Remain in hiding and use the skills you have learned.\"\n\n\"Why has she not contacted us?\"\n\n\"She will in good time. She has left me in charge until her return. We have been faithful. Patricia will not let us down. We must have patience.\"\n\n\"And what of the _One_?\"\n\n\"Come...\"\n\nThe two took up torches and made their way to the back of the room. Hidden in the shadows was a passage. The torch flames revealed a set of fungus-encrusted stairs slithering down into the darkness. They descended the twisting passages that was ripe with stalactites. Finally they entered a chamber of etched stone. Inscribed in its walls were ancient hieroglyphics and texts\u2014inscriptions from cultures long dead.\n\nA pedestal stood in the room, and upon it was the Ponape Scriptures\u2014the most sacred of writings to their cult. The binding was said to be made of an ancient, now-extinct cycadean wood, and written in the language of a people who vanished at the dawn of human history.\n\nPassed down from generation to generation, it had been guarded by the lives of those in the dark faith. It crossed from inhuman to human hands, and had been kept a secret for centuries. Now, there it sat\u2014the most dangerous pile of brittle paper ever to be read. It taught them all they needed to know, and brought the very planet to its knees. At last it had made its way into skilled hands, and without killing themselves or going insane, they had managed to learn its darkest of teachings and knowledge.\n\nThe book was said to have been in the hands of many cults before theirs. Supposedly discovered in the mists of the sea, it gave explicit instructions on bringing back the Old Ones, especially their emissary Cthulhu. He would pave the way for the rest of them, to reform the world and give the planet back to them, the rightful rulers of the universe. It was they who came out of the sky when the world was young, before there were men.\n\nAmong other things, the scripture taught those in the faith magical rites for gaining personal power, vanquishing enemies, and stopping anyone from getting in the way of bringing back the Old Ones. There were many ways of destroying available to the faithful, all diagramed with great detail and fervor.\n\nAt the foot of the pedestal sat an onyx altar. More strange symbols trimmed it in what appeared to be gold. On the altar lay Casey Archer, naked except for a tattered loincloth. Her body glistened with scented oils, and her face was decorated with ink forming more archaic symbols. She slept in a supernatural-induced sleep, unaware of those around her, unable to do anything but dream terrible dreams of fire and dark clouds, plagues, and serpents. Many arms flailed about her, stroking her mind as she dreamt, preparing her for a new destiny. Her eyelids fluttered but she slept deeply, even as her body grew gooseflesh in the icy air.\n\n\"We should do it now,\" the man said.\n\n\"Now is not the time.\"\n\n\"To hell with waiting. We are wasting time. There won't be a planet left if we don't\u2014\"\n\n\"Fool!\" the woman cut him off. \"Do you want us all to be obliterated? Stand down. We can do nothing until our leader brings us the girl's father.\"\n\n# Chapter Four\n\nIsland in the Middle of the Pacific\n\n_Old Ones_. It was a blip in Blount's thoughts, and it popped into his mind upon stepping onto the island. He held tight to his pistol and beckoned for the others to follow him.\n\nSimon Baxter led his tactical team off first. Weapons drawn, they cleared the way for the rest of the Blount's special team\u2014the doctor, geologist, and theologian.\n\nThe twins stepped out last. One took the other's hand automatically. \"Death is with us,\" they said in unison. \"Death is all around us. It is here. It is upon us.\"\n\n\"Everyone stay sharp,\" Blount said as they walked on, the ground feeling soft and malleable under foot. There was a stench in the air, and, for a fleeting moment, Blount thought he recognized it. He spoke into his ear mic and told the pilot to shut the chopper down and sit tight.\n\n\"If this is a tropical island, why am I cold?\" Baxter said.\n\nBlount looked up at him. \"We can't trust anything. Not with our eyes or ears. The island may not obey the natural laws as we know them.\"\n\n\"Well, that's just great. How the hell is my team supposed to fight something like that?\"\n\nBlount did his best to contain his temper. He inhaled then exhaled before getting closer to the man. \"Let's get one thing straight. This is my team. All of you are my team.\"\n\nHe glanced around at all them. Private Amanda Fletcher looked more frightened than anyone on the team. Geologist Nick Larson paid no attention, for he was in awe of the foliage around him. All others had their attention fixed on him.\n\n\"Baxter, you were recommended to me. You were not my first choice, but I was told you were one of the best and didn't mess around. Was I wrong to have you head up my security team?\"\n\nBaxter grit his teeth. Blount could tell the guy wanted to slug him. \"No, sir.\"\n\n\"Just keep it tight. No room for heroes. No second guesses. Be ready for anything.\"\n\nThe team resumed walking and the two men put a bit of space between them. A shade of gray coated the sky, while a small breeze picked up. Behind the rolling clouds, a bright light glowed.\n\nFlurries fell softly around them. Amanda stuck out her tongue to taste the snow. It melted as soon as it hit the ground or their bodies. Unmoved by this event, the team pushed on.\n\nThey made their way through thick, overgrown plants and foliage. Rocks jutted from the ground like carnivorous teeth. The trees in the distance were huge and towering, masking a structure ahead of them. There was something massive there, but it would not yet reveal itself to them. Curtains of vines hung in their way, and they resorted to cutting through with knives and machetes.\n\nAn electric charge surged in the air. Blount looked up and saw black clouds swallowing up the gray. He didn't like the looks of it.\n\nThe twins screamed and went down, taking their heads in their hands. \"Pain!\" they cried. \"I-I-It is trying to d-destroy us. Oh...it rips and tears, it burns and devours. W-w-we...a-a-are losing-g-g...\"\n\nBlount rushed to their sides. Bending, he checked their faces and noticed blood trickling from their noses. \"Fight...fight it!\"\n\nHe watched as they shrieked and wailed. Their bodies twitched as their eyes rolled back in their heads, whites flashing briefly.\n\nBaxter froze and aimed his gun at the surrounding jungle. Their perimeter was nothing but thick trees and vines. Amanda got beside him and mimicked him, even though no one knew what to aim at.\n\nThunder bellowed inside the clouds and the ground shook. The grunts shuffled a bit but stood their ground, appearing a little disheveled. A gust of wind howled and a snowstorm whipped onto the island. Icy winds lashed them as a near whiteout dumped inches of snow.\n\n\"The hell?\" Baxter said, as he held out his hand in disbelief and caught the white powder. \"It can't snow! This is a goddamned tropical island! Somebody wake me the hell up.\"\n\n\"Calm down,\" Blount called. \"Stay together.\" He felt angst in them, and didn't want them to cut and run into different directions. Anything was possible at this moment.\n\nA scream cut above the howling winds, as something came out of the snow and ripped one of the soldiers limb from limb. Scarlet ran through the white and christened the snow blanketing the ground.\n\nShots went off. A chaotic scene unfolded as something slithered through the snow. They were huge, white, scaled and attacked randomly. Soldiers shot in different directions and chased something that moved fast, striking without warning.\n\nOne leapt out at Baxter, catching him off guard. Blount slid himself beneath this monster. He landed on his back and shot upwards, striking the giant snake in the underbelly. It bled black pus, as he shoved his pistol underneath its jaws and blew its head off.\n\n\"Thanks,\" Baxter yelled as more snakes slithered up behind Blount. The men turned and opened fire as ear-piercing screeches rose above the icy winds.\n\nFeral, azure eyes glinted in the snow as the white serpents advanced on them. M4 rifles blazed, cutting the beasts down while the skies rained white, black, and crimson.\n\nIt only brought more. Snakes coiled everywhere, surrounding them, fangs dripping with venom. Savage grins curled as they hissed and shrieked. One soldier attempted to run, but was quickly snatched up in powerful jaws, his legs torn from his body.\n\n\"Looks like this is going to be one short investigation,\" Blount said, as he braced for the unavoidable. He tapped his headset mike. \"Lift off and evac...\" Nothing happened. All radio contact was gone. \"Awwww hell.\" _I'll just have to remember this in my next life. If there is one this time, Blount... The world is in it bad this time._\n\n\"No!\" the twins screamed, as a kaleidoscope of colored energy burst from their bodies.\n\nThe snakes were obliterated, and the raging snowstorm ceased instantaneously. The dark clouds evaporated as if never there. The entire team was knocked off its feet and sent reeling. They looked up with astonished expressions on their faces.\n\nThe island became hot and humid. The snow melted quickly and the ground dried.  \nBlount laughed out loud, a hearty belly laugh, as he got to his feet. A hell of a headache pounded him like a sledgehammer. \"I knew you could do it!\" he called to the twins.\n\nEveryone else remained mute, visibly shaken by the ordeal. This was not something they had trained for.\n\n\"You should...\" one twin began.\n\n\"...know us by now,\" the other finished.\n\n\"...that may be a...\"\n\n\"...one time deal.\"\n\n\"What about the island?\" Blount asked. \"Or who sent the dandy little snowfall our way?\"\n\n\"We are still being blocked,\" they answered together. \"It is too powerful for us to penetrate, and most likely knows our every move. We are trying to construct a wall against it invading our thoughts. It's taking all of our strength.\" They stumbled as they tried to gain their footing.\n\n\"Maya, Nina, are you alright?\" Blount went to them, letting them lean on his broad shoulders and strong back.\n\n\"We will recover.\"\n\nThe soldiers rubbed their heads and shook the cobwebs out, as Doctor Graham Saylors checked them.\n\n\"Hey Doc, take a look at the girls for me would ya?\" Blount said. \"They need a bit more attention then the good ole boys.\"\n\n\"Of course.\" Doctor Saylors hurried over to their side.\n\n\"You heard the man,\" Baxter yelled at his team. \"Stand tall! This won't be the first fucked up thing we encounter. You are the best of the best. You've already seen some heinous crap.\" He referred to the shadow creatures and their hosts back at the SUV. \"This is just more of it. Get it? Got it? Good!\"\n\nBlount conferred with Nick Larson, his team geologist, as the theologian Max Anders looked on and trembled from head to toe.\n\n\"What do you make of this place, Nick?\"\n\n\"It's an anomaly. I haven't encountered anything like it in my whole career. There's plant life here that should be extinct. The air seems OK. I would need some soil samples to get a feel for what else we're dealing with.\"\n\n\"How is it even possible?\"\n\n\"Could be many things: underwater volcanic activity, explosions at the earth's core, a shift in the planet's crust or fault lines. Or...\"\n\n\"Or the return of something very old and very pissed off.\"\n\nBlount recalled his conversation with President Wendall.\n\nNorth Korea is launching. All of us are.\n\nThe president's words played over and over in his head. With all that he'd encountered over his lifetimes, all the monsters, demons, and gods he'd faced, none of it compared to the coldness he felt in his gut as he'd heard nuclear weapons would be used. He just couldn't predict how long they had left, and he needed to tell the team.\n\nMaya and Nina approached him, and Maya tenderly ran a hand over his cheek.\n\n\"The Earth cries out,\" Nina said somberly.\n\n\"She cries for mercy,\" Maya added.\n\n\"There was no choice.\" Nina traced the tears on his cheeks.\n\nBlount pulled away and stalked off a few feet. \"There's always a choice.\"\n\n\"Yes...\" Nina said.\n\n\"But not this time,\" Maya finished. \"The missiles are on their way.\"\n\nBlunt nodded, and knew this was the time.\n\nAfter regrouping, Blount informed the rest of the Black Rock team about the launches and the impending missiles headed their way. The team took the news quietly and merely nodded. They knew the risks and the dangers. They gathered their thoughts, some prayed to whatever entity they subscribed to, while others merely smoked what might be their last cigarettes.\n\nA short time later, they cut their way through to a path that twisted into many directions and sloped down out of view. Reluctantly, they took it, as Blount felt drawn to it. He started down first. They chose a singular route branching off from the many, where it ascended a small ridge. When they reached the top, they were all struck with awe at the sight that awaited them.\n\nThe horizon was filled with approaching missiles. However, as they approached, one by one they fell to the ocean, as if they'd lost power. The blast obliterated the sky in a white explosion of water and debris. The group shielded their eyes as the blast wave rammed over the island. They were thrown to the ground from the force.\n\nBlount opened his eyes a short time later and painfully gained his feet. He stared down at the valley below in awe.\n\nDark spires sprung out of the ground. Huge and towering, they were made of smooth stone that was covered with odd and disturbing hieroglyphs. Some were formed with grotesque, cold spheres that sat unnaturally, unnerving the onlookers. In the center of this cluster of towers rose a monolithic citadel. Its surface stretched at impossible angles. Its shape was abnormal and frightening, stopping the group cold.\n\n\"It waits dreaming...\" the twins said ominously. \"It waits dreaming, it waits dreaming...\"\n\nThe team grew more unsettled, and Blount stood in front of them. He soaked in the panorama, realizing that it was really there. It really existed. A part of him didn't want to believe it. It had only been a legend\u2014a whisper on superstitious lips\u2014but was never really a part of reality. That was just it. This place was real, but not a part of reality as any human being understood it. The city was here, and the Old Ones were real. The evidence was staring him right in the face.\n\nBlount searched the farthest reaches of his memories. His past lives held something that hinted at this place. A city, a city not fully raised, still working its way into the human world. Yes, the memory was there, buried, difficult to recall, or perhaps the recollection of that past life was just being blocked by whatever waited for them in the city. Bits and pieces of it came to him now, a ripple in the pool of his mind.\n\nHe saw himself standing in the dark halls of this city, perhaps millions of years ago, with the smell of rot in his nostrils, the taste of blood in his mouth. Half naked, his face painted and weapon in hand, he led an army into the city to destroy it and them once and for all. The sun sank into black just as they raged into oblivion...\n\n\"What the hell is that?\" Baxter brought Blount out of his flashback and pointed at the black clouds that circled above the towers and citadel.\n\nA flash of lightening streaked from the clouds as Blount looked up and saw shadowy, black forms emerge from the towers. Huge shadow birds flocked around the spires and encircled them. Moments later, they set sight on the team and screeched, racing towards them.\n\n\"Here they come!\" Blount yelled, aiming his pistol as the soldiers behind him readied their M4s.\n\nThe twins joined hands once again. A translucent green bubble sheltered the team, and the creatures bounced off the girls' newly formed psychic shield. The sound of crushing bones pealed through the air. The shadow birds yelped as the flock crashed, and then they scattered across the skies.\n\nSpeechless, Blount turned to the twins, who were on the verge of passing out. He wasn't sure how long they could keep up their inner war with the sleeping Old One before it killed them both.\n\n# Chapter Five\n\nOutlook, Montana\n\nPalmer led them down a winding stairway illuminated by blinking, domed lights nestled high above. Hollow, dripping sounds filled their ears as they reached a metal blast door at the end of the stairs. Archer hoped they could trust this Palmer character. With all that was going on, anything was possible. He watched as the agent inserted a gold key card. There was a clank of metal, a whir as locks disengaged, and then a scuffing as the door slowly eased open. The hum of computer equipment met them, and a tall bearded man waited on the other side.\n\n\"Eli,\" the man said. \"I'm glad you made it back safely. Bring them all in. Quickly.\"\n\nPalmer ushered Secretary Carling, Archer, and Sam into the shelter and into a deceptively wide room with a high ceiling. Computers and radar equipment filled the room. The ceiling beams hung with nests of wires and cables.\n\nFluorescent lights glowed all over the room, creating a freakishly near-sterile environment. A multitude of doors\u2014some open, some closed\u2014networked in every direction, leading into hallways that traveled off in different directions.\n\nArcher spied a pantry and kitchen area off to his left at the nexus of one of the hallways. It appeared stocked with canned and dried foods and bottles of water. Standard protocol. He'd been in similar bunkers\u2014one of them just below the White House.\n\nMen and women hustled back and forth, still doing their jobs despite the insanity that had rocked the world. Some were seated in front of computers, while others were shuffling paperwork back and forth. They barely looked up at the group as they walked to the center of the room. A large plasma screen dominated the far wall. Blue lines traveled from Russia, Korea, China, Iran, and many other countries, terminating at multiple red dots on the world map. Archer sucked in a breath as he recognized what he was seeing.\n\n\"This way.\" The tall bearded man directed them to an open door in the hallway in front of them. \"We need to get you all caught up to speed. We'll debrief you as fast as we can.\"\n\nThey followed him into a large conference room, while Palmer departed into the crowds of other agents and workers. \"I'll be with you all shortly.\"\n\nAfter they'd filed in, the tall man closed the door behind them and asked them to sit around the table. \"I'm Dan Waters. I'm commander of our little survivor base. We've a good handle on what the hell is going on here.\"\n\nThey stared up at the world map, stuck with dozens of red pins, on the wall behind Waters. There was a radio system in the corner, a chalkboard to the right. A file cabinet system occupied another wall, and scattered around were globes, flashlights, blueprints, and some stones.\n\n_Onyx and crystal_ , Archer thought.\n\nThe chalkboard held scribbled brainstorming: groups of names, locations, and times. There were a couple of strange symbols drawn there, too. Archer thought they looked familiar, as he soaked everything in.\n\n\"Secretary, I'm sure this won't come as a big surprise to you. But for the rest of you, you're about to become privy to information the general public knows nothing of. It doesn't exist. This information is code: Majestic.\"\n\nA somber silence fell over the room.\n\n\"Majestic\" jogged something in Archer's memory. He'd heard the name before, when he'd been assigned to the president, but he couldn't remember what type of information it signified exactly. What he _did_ know was that the \"Majestic\" moniker was the highest designator in the government's repertoire of secrets.\n\n\"First of all, most of the general public isn't even aware of the gravity of the situation.\" His voice was smooth and even. He seemed at peace with all that was happening, and, for some reason, it unsettled Archer. \"However, there are those of the conspiracy fringe, whom we monitor closely, who have been putting much of this together and coming to some pretty interesting conclusions that resemble our own experts' deductions. Those of us in this shelter are the only ones who know the truth of what's really going on out there. Yes, the world knows that nuclear strikes have plunged most the planet into devastation, and they even know of the tsunami that has entire nations under water right now, but only a few of the governments still functioning are striking against the real enemy. The world at large doesn't even know the half of it.\n\n\"What I'm going to tell you will cause you to question everything you know of the world\u2014your faith, your fortitude. You may not believe it\u2014\"\n\n\"Sir,\" Sam said, to the surprise of Archer. \"There isn't much I wouldn't believe right now. What I've just seen and had to do...well nothing can shock me now. So spit it the fuck out. Don't hold anything back.\"\n\n\"Alright, Miss Velaska. Point taken.\"\n\nSam smirked for the first time in days. Evidently, Archer realized, she was surprised this man knew her name.\n\n\"First off, let me inform you that I am _not_ an agent of the government as you know it. I work for an agency you've never heard of, an agency that doesn't exist. Since the world powers have gone to hell, there's no point in all these secret, cloak and dagger games any longer. No more pretenses. I am\u2014was\u2014a higher up for the Black Rock agency, and Palmer is one of my most trusted men. We've worked in this facility for years with everyone else, but we are the only two Black Rock agents assigned here. Now, you know, along with everyone else out there.\" Waters pointed to the closed conference room door.\n\nThey stared at each other, not fully comprehending what he was saying, or what Black Rock really was.\n\n\"We're a twilight organization,\" he went on. \"Specializing in other worldly cases and unexplained phenomena. I don't have to tell you _this_ is one of those cases\u2014the ultimate case. We have teams of psychics and spiritualists on this. The ones that are still alive, that is.\"\n\n\"You mean that psychic crap is for real?\" Sam said. It seemed anything that popped into her head came right out. Archer wondered why she was acting so erratic.\n\n\"More than real, Miss Velaska. It's right here, right now. Always has been. There is a delicate balance out there. A thin veil between our reality and many others\u2014realities more terrifying than imagination. There is not just one reality. There are many, with many life forms and many deities\u2014things that have been waiting to be awakened, waiting for their chance to get into our world.\"\n\n\"What else do you know?\" Archer said. \"Those things, the shadows, the island... What in the hell...\"\n\n\"Yes, Thaddeus,\" Waters said. \"I was just about to get to that. A very powerful collective has learned how to use negative energy to wake and bring something back into our world, something old and sleeping, something still without its consciousness. This group of dark psychics raised the island out in the Pacific. It houses the incarnation of the Old One. Chthulu himself.\"\n\nArcher knew exactly whom he was talking about. _The cult\u2014that blasted cult! They're who he's talking about. The ones that took Casey. All of this is their fault. What the hell do they want with Casey?_\n\nWaters continued, \"However, when they raised the island, there were side effects. Through their misuse of power, they created rifts in the fabric of the veil\u2014the delicate walls that separate our realities. Through these rifts came the shadow beings\u2014the ones that take hosts, the ones that fly, the ones that look like human and animal monstrosities. They are everywhere, and many in number.\n\n\"My people have been working hard on closing those rifts, and we've managed to do just that to almost all of them. The shadows need hosts. They prefer to take the dead, reanimate the body, and use it to carry out their will. But they will take the living when they get desperate. It doesn't give them a secure grounding in our world, however. They have a very loose grip on the living. We still don't know how they choose these living hosts. It might be those with weaker wills, or who are a bit unstable. That's why there are so many suicides. The battle between shadow and hosts breaks down the human mind\u2014erodes their sanity after a while. Those with stronger constitution seem to be immune.\n\nSo in any event, we've been destroying the dead. No body, no host. We've informed everyone to automatically burn the body of anyone who dies or is killed. It's keeping most of those shadows in line. You may have noticed you can destroy them while they animate a corpse, because they merge with it in our world and become tangible. The longer they control the body, the more solid they become in our world. They also become more vulnerable to our attack. So they do have their weaknesses.\n\n\"But without a host, they can't be touched. North Korea and China launched nukes at the island around 0930 this morning. We're not sure what exactly happened, but something did. There was a psychic black blast from the detonations. We're not entirely certain the nukes struck the island, but we are fairly certain there was some type of reaction to their detonations. Whatever the wave was, it wiped out the initial wave of shadow puppeteers. Now the other ones...\"\n\n\"The giant birds,\" Archer said.\n\n\"Yes. They are a lot tougher. Don't need hosts, and attack without warning. We have to always be on guard. One last thing. What we've been able to detect is the group responsible for all this isn't finished. They want to wake their god, bring it to life on that island. If that happens, we are as good as done. We can't let that happen. Our remote viewers are working on some leads, trying to get a pinpoint on where they are.\"\n\nArcher was very interested in this as well. He grew agitated, felt his face flush red, and his hands grew damp. _Tell me...tell me where they are so I can kill each and every one of them._\n\n\"We're being blocked right now,\" Waters said. \"But we're getting close. Once we find them, we will stop them.\"\n\n\"What makes you think you can stop them once you find them?\" Secretary Carling spoke up for the first time.\n\n_Her voice_. Archer thought. _What is it that's so familiar about it?_\n\n\"We have some methods that we will employ,\" Waters replied.\n\n\"Methods?\" Carling said, as if she were savoring the word.\n\n\"I'd like to help with this, and be kept informed up to the minute,\" Archer demanded.\n\n\"Hold on, Thaddeus. We all have a job to do. I'm assigning\u2014\"\n\n\"They have my little girl!\" Archer said.\n\n\"What?\"\n\n\"My daughter, Casey. The cult took my daughter.\"\n\n\"Jesus!\" Waters was beside himself. \"You mean she's still alive? All the other girls taken were said to be found murdered. Sacrificed to their God. If they're keeping her alive, there must be a reason. We've got to find out what they\u2014\"\n\nThere was a knock at the door. Waters went over and yanked it open. Palmer was on the other side, his eyes wide as he breathed heavy. \"The viewers have picked up something on the cult. You need to come to the COM link and talk to them.\"\n\nThe two hurried out of the room and down the hall, leaving Archer, Sam, and Carling on their own.\n\n\"None of this makes any sense,\" Carling said. \"It's ridiculous. Rifts in reality, sleeping gods, a cult... What is all this?\"\n\n\"Madame Secretary,\" Archer said. \"It's the explanation that makes the most sense. The cult is behind all of this, and they have my Casey. I've tracked and hunted them. I came close to losing my life to them. I killed their leader, McBain. I will not rest until I get my daughter back. They're real, and they're gonna pay.\" Archer stormed out of the room, following Palmer and Waters down the hall.\n\nSam and Carling stared at each other, then they went after Archer. Sam noticed Carling expression morph as they left. Her eyes dimmed, and her skin darkened with red. Veins throbbed in her temples.\n\n\"Are you alright?\" Sam asked as they went down the hall.\n\n\"Yes, it's nothing. Just a headache.\"\n\n* * *\n\nThey stood in a room that was more like a storage closet then an actual room. Waters had an old-fashioned microphone in his hand. Palmer sat in a folding chair, tuning dials and adjusting the volume. It seemed they were relying on old technology to get them through the end of the world. Archer wondered how long before they were reduced to building telegraph technology again. If the world wanted to communicate again, they'd better think of something.\n\n\"Say again?\" Waters said. \"Somewhere in North Dakota.\"\n\nArcher thought he noticed Secretary Carling's eyes widen.\n\n\"Yes,\" Waters continued. \"That's all they can find. What about a town or a landmark of any kind? Nothing. OK, keep pushing. No, nothing here either. Well, at least we know a state.\"\n\nThe COM began to buzz and click. \"We may be losing the signal, Dan.\" Palmer tried to keep the line alive.\n\n\"It's OK, Eli. Let it go. That's all the info we're gonna get for now.\"\n\nPalmer released his controls and let the machine go quiet. A slight hum droned through the air. He looked up at Waters, who simply shrugged.\n\n\"At least North Dakota is relatively free of nuclear fallout.\" Waters turned to the others. \"There hasn't been much activity there, but my contact says the remote viewers have seen the cult on the road in North Dakota. The cult was trying to delude them to throw them off the trail. They were unable to clearly make out a town or any solid descriptions.\"\n\n\"Will they keep on trying?\" Archer asked.\n\n\"Of course. It's tough. We're dealing with a real powerful group here.\"\n\n\"How do they know North Dakota isn't some sort of a trick?\"\n\n\"They don't, but they've been picking up a large energy source there, and it isn't nuclear. It's negative, and causing a lot of disturbance in the balance of nature there. There's definitely something. What we can do next is go there, meet up with some white witches or shamans, and get to the bottom of it.\"\n\n\"Mister Waters?\" Carling said. \"This has gone far enough. I must be informed of President Wendell's location, and I must speak with him. He is still the leader of our country, such as it is. It is of great urgency that I be able to reach him and know his whereabouts.\"\n\n\"Of course, I've been meaning to address that. The president is well and safe. I can debrief you all if you want to step back to\u2014\"\n\n\"I insist on holding a private conference with him and you. This is a matter of National Security, and the proper clearance is needed.\" Carling eyed Archer and Sam.\n\n\"Of course,\" Waters said, hurrying her back to the conference room.  \n\"What's her problem?\" Palmer got out of his seat and chuckled.\n\n\"She's been under a lot of stress,\" Archer said. \"I don't think she wants to believe any of this is real. Back in New Orleans, she acted as if nothing was wrong. She might feel lost, being separated from the president. I think we need to cut her some slack.\"\n\n\"If you say so,\" Sam said. \"She's starting to get on my nerves.\"\n\nMoments later, Waters stepped out of the conference room and closed the door behind him. He gave a strange look to Palmer and the others, almost one of distress, and then headed toward them.\n\n\"Secretary Carling is in touch with President Wendell now. She'll be done shortly. Palmer, I need to take care of some pressing business in my quarters. I don't wish to be disturbed.\" He rubbed his head frequently.\n\nBefore any of them could say a word, Waters started swiftly down the hall, tapped his head a few times, muttered to himself, then turned right toward where private quarters were housed.\n\n\"That's weird,\" Palmer said.\n\nArcher looked at him.\n\n\"I've never seen Dan behave like that before. He doesn't usually keep anything private from me.\"\n\nOutlook Fallout Shelter, Main Hub\n\nIt had been almost an hour since Waters had retired to his quarters, and he had yet to return. His colleagues began to grow concerned. They needed their leader, they needed answers, and they needed hope. Waters gave them all that, and now, he refused to come out of his room.\n\nSecretary Carling had long since had her meeting, and enjoyed a meal of instant potatoes and canned ham at a table with the resident cook, known only as Jesse. Jesse was an obese man who was beginning to lose his hair. He was a former military cook who could do wonders with dried food and powdered eggs. Archer and Palmer didn't like the situation one bit. Something smelled funny, and it wasn't Jesse's cooking.\n\nPalmer was a loyal agent, and he followed protocol to the letter, but this was unlike Waters. It wasn't his normal behavior. Dan Waters was the most put together person he knew, even more so in times of stress. Something felt wrong about his absence, and the longer Palmer sat and did nothing the more he sweated bullets.\n\n\"Alright.\" Palmer broke the eerie silence. \"I think I should check on Dan. Something isn't right.\"\n\n\"I'll go with you,\" Archer volunteered.\n\nThe two walked down the hall to the private quarters and stopped in front of Waters' door. Palmer knocked. \"Dan, you OK in there?\" Only silence answered them. \"Dan, it's me Eli. We're concerned about you. Can we come in?\"\n\nStill nothing.\n\nPalmer took hold of the knob, but the door was locked from the inside. \"Stand back.\" He drew his gun. \"I'm gonna have to break it down.\"\n\nPalmer stepped back and made a charge at the door. He kicked the door in, and it bashed off the back wall. The two went in, and a foul stench hit them head on.\n\n\"Ah God...\" Palmer said.\n\nArcher felt his stomach turn as they noticed the humidity in the air. There was an attached bedroom in the back of the room. Its door was closed, but light glowed faintly beneath it.\n\nAgain, Palmer insisted on going first. He drew his pistol and made his way slowly to the door. He tried the knob and it eased open. \"My God...\" As he stopped cold, all color drained from his face.\n\nThe bed was stripped to the mattress, with the blankets and sheets piled neatly in the corner. On top of the bed, Waters' naked body sat, his vacant eyes staring directly at them. Both of his wrists and his throat were cut open. Globs of crimson pooled in the center of the mattress, turning it black. On the floor, beneath Waters' blood-stained feet, lay a straight razor.\n\n\"No!\" Palmer yelled. \"No way. I don't buy it! He would never check out like this\u2014not now. No way!\"\n\nArcher stared blankly as rage filled him. He felt as Palmer did. Waters didn't seem like the type. Something was wrong with the whole situation, and it unnerved him. It scared the crap out of him, actually. There was more going on here\u2014something big. Waters knew everything that was going on, and was getting closer to the cult...closer...\n\nIt waits dreaming... it waits dreaming...\n\nA crash signaled the bed collapsing as Waters stood up, leapt from it, and toppled onto the dumbfounded Palmer.\n\nStartled, Archer unloaded a round into Waters, shattering his shoulder. It didn't stop the naked corpse's rampage, as it tried to choke the life out Palmer, who sprawled on the floor, fighting for breath.\n\n\"Damn it.\" Archer chastised himself for letting the body get the jump on them. He searched the room swiftly for the shadow. Against the white walls, he spotted it writhing its way across the room. It exploded as bullets tore into the wall behind it. A shriek filled the air as it dissolved. Waters' corpse went limp, settling on top of Palmer.\n\nPalmer coughed, strained, and rolled the body off of him. \"Jesus Christ.\"\n\n\"Are you alright?\"\n\n\"I think so. Motherfu\u2014\" Palmer's curse drew Archer's attention to the crowd that gathered at the doorway, drawn by the sound of the gunfire. \"Back off!\" Palmer called. \"It's over. Get back to your posts!\"\n\nThe crowd dispersed as Palmer got to his feet. His gaze met Archer's. \"Someone did this. This was no suicide. I want to know who.\"\n\n\"I think I might know.\" Something donned on Archer. \"She was the last one with him.\" He raced from the room.\n\nPalmer followed Archer back to the shelter's main room, to the galley where Secretary Carling was still calmly eating her meal.\n\n\"What the hell did you do to him?\" Archer yelled. His nostrils flared as he seethed. \"You were the last one with him!\"\n\nCarling stood up and edged herself out of Archer's path. \"I don't know what you're talking about.\"\n\n\"The hell you don't.\" Archer reached out to take hold of her, but big Jesse stepped in his way.\n\n\"This isn't how you treat a woman, bud.\"\n\nOut of nowhere, Palmer busted the side of Jesse's head with the butt of his pistol, leveling the big man. Gasps filled the room as all heads turned toward them.\n\n\"Now, Madam Secretary,\" Archer said as he looked down the barrel of his service pistol, pointed directly at Carling. \"Start talking.\"\n\nApproximately Two Hours Ago\n\nSomewhere Over the United States\n\nPresident Wendell and his entourage arrived at Mount Pleasant only to find the bunker overrun with shadows. The crowd of dead leered up at the hovering Blackhawk, and their shadow puppeteers roared with frustration.\n\n\"Sir?\" The pilot sought direction from Wendell.\n\nPresident Wendell gasped as he recognized a few of the faces in the mob.\n\n\"Sir?\"\n\nWendell looked over the soldiers. He knew if he ordered the pilot to land, they would gladly fight to the last man. They wanted to kill something, anything, if only to feel like they were doing something, anything. Wendell knew the other bunkers would most likely be in the same state as this one, if survivors had made it to them.\n\nWendell turned to Woodrow. \"What do you think, Major?\"\n\nMajor Woodrow was busy studying a file that he'd retrieved from the Football. \"If anyone's at the other designated bunkers, they'll probably be in the same state.\"\n\n\"Yes, I figured as much.\"\n\n\"There is one other option. There's a bunker just north of here. It was decommissioned in 91, but there's still a caretaker crew in place.\"\n\n\"How many?\"\n\n\"No more than twenty.\"\n\nOne of the soldiers spoke up, \"Them's good odds.\"\n\n\"How far?\"\n\n\"About a hundred miles.\"\n\nWendell addressed the pilot. \"Can we make it that far?\"\n\n\"Maybe. We'll be landing on fumes. If we get there, and it's like this one...\" The pilot shrugged, as if merely commenting on the weather. \"We'll have no choice but to land.\"\n\nWendell weighed his options. They were pretty slim, but he had to make a decision. He glanced around the interior of the cabin. These men were counting on him, and he could be leading them to their deaths. So many had died today. He wasn't going to lose any more, if he could help it. He nodded, and said in an icy tone, \"Head for Raven Rock. We'll take our chances.\"\n\nThe SAT phone chirped, and Major Woodrow and the rest of the men all froze, staring at the pulsing light on the phone. Wendell motioned for his major to answer and leaned in closer.\n\n\"Major Darren Woodrow.\"\n\n\"Major, this is Secretary Carling. We've made it to Outlook safely. Aback Denoug Maki Nin M'faly...\"\n\nWoodrow went rigid and his face drained of all its color.\n\n\"What's wrong?\" Wendell was startled by the change washing over the man.\n\nWoodrow turned and looked at Wendell, but it was if he didn't recognize the president. Slowly, he reached for his gun and slid it from its holster.\n\nWendell recoiled in his seat and shouted, \"What the hell are you doing?\" Something large and dark passed by his peripheral vision. Wendell jerked his head in that direction, gasping when he saw the shadow creature swoop towards the helicopter.\n\nShots rang out in the cabin as Woodrow opened fire on the soldiers. Caught unaware, the soldiers didn't have a chance. The pilot cursed as he caught sight of the shadow creature and jerked the stick. Wendell's stomach leapt into his throat as the helicopter dove. The sudden course correction saved Wendell, as Woodrow turned the gun in his direction and fired. Wendell fell to the floor. The bullet shredded the headrest where his head had been just a few seconds prior.\n\n\"Please, stop me!\" Woodrow screamed as he fired shot after shot.\n\nThe helicopter lurched once more, and the pilot slumped forward as a ricochet blew out the back of his skull. As the chopper went into an uncontrolled tailspin, Woodrow turned the pistol on himself. He grasped his right arm with his left hand and struggled to pull the gun from his mouth. Wendell watched in horror as the man fought for his life against a body that was no longer his own. \"Please,\" Woodrow said around a mouthful of barrel. \"Please, no. I'm sorry.\"\n\nPatches of sky, then ground, flashed by the glass as the helicopter rocketed downwards. Wendell managed to grab the leg of one of the seats and wrapped his arm around it. He braced for the impending impact as best he could. The cabin was splattered with bone and brain as Woodrow fired into his mouth. The helicopter hit the ground at an inverted angle, and the blades dug into the ground with a shriek, the cabin cart-wheeling fifty feet. Helpless, Wendell flew about the cabin like a rag doll.\n\nBlack silence filled his mind.\n\n* * *\n\nHours later President Wendell regained consciousness with a scream. The pain engulfed his body. Something leapt back and cooed hungrily. The small child eyed him warily as it licked its bloody lips. Wendell gingerly reached a hand to his face and whimpered as it came away slick with his blood. He glanced around and noticed he was in the middle of a field. A few yards away the mangled helicopter smoldered in the gloom.\n\nWendell pulled himself into a seated position and stared at the multitude of eyes glittering in the gloom all about him. Something glittered in the grass a few feet to his right, and Wendell willed himself across the dirt, towards the object. A journey of mere feet took what seemed like hours as he dragged himself and fought off unconsciousness. He reached the gun and greedily clutched it to his chest, sliding the magazine out to see how much ammo was left. Ten rounds. He pointed the gun at the mass of dead and aimed for their shadow masters, then fired. He saved the last round for himself. A prayer on his lips, he pulled the trigger. There was no way he'd become one of these monsters.\n\n# Chapter Six\n\nRugby Rock, North Dakota\n\nCasey Archer slept the sleep of the dead on the cold altar. Her breathing was shallow as she hovered on the edge of life and death.\n\nHe stared down at her. The voice in his head relentlessly pushed him, taunted him, yet warned him, too _. In the city, it waits dreaming. In the city, it waits dreaming. In the city, it waits dreaming._\n\nHe set his flashlight down and reached into his satchel. In his hands appeared a dagger with a bejeweled handle. The blade glimmered in the faint, artificial light. His hands trembled as he held it. His skin crawled, and he was cold. So cold.\n\nDo it. Do it now.\n\n\"I must wake our God,\" he whispered. \"We can wait no longer. What good is ruling a kingdom that's dead? The planet is being destroyed. I must wake him before there's nothing left.\"\n\nHe raised the dagger with both hands, holding it high above her body. Sweat soaked his face.\n\nDo it. Do it.\n\nThe dagger felt as if it weighed a hundred pounds. He paused, debating with himself as he stared down at the prone, young woman beneath him. The dark slithered all around him. His flashlight flickered. Shadows wriggled in his peripheral vision. His heart beat faster and his throat went dry.\n\nLooking up at the dagger, he saw a reflection in its blade.\n\nThe steel of a sword plunged through his chest.\n\nHis eyes went wide and he tasted blood in his mouth, as he let the dagger fall to the floor. He looked down at the sword coated with his blood. A warm flood of sensations filled his body as he fell to his knees, and he gasped once before going lifeless.\n\n\"Fool,\" she said, pulling the sword from his body and letting him slump to the floor.\n\nThe other cult members circled around her and stared down at the fallen one.\n\n\"He stole the ceremonial dagger,\" someone said.\n\n\"I know. I was summoned to stop his traitorous act.\" She picked up the dagger and handed to it to someone at her left. \"Return it to its resting place. We will need it when our leader returns with the sacrifice's father.\"\n\n\"Yes...\"\n\nShe nudged the lifeless body of her comrade with her foot, to make sure the job was finished, and then addressed the others. \"Take the Ambassador's body away and burn it. Quickly, before one of those shadow creatures possesses it.\"\n\nWith a black cloth, the second in command wiped the blood from the blade of her sword. She walked over to Casey and checked her. All seemed well. A sigh escaped her before turning back to the others.\n\n\"Secure the doors to this chamber. No one else is to come down here until our leader returns. Everything must be perfect. We must prepare the way for the God to awake and be restored.\"\n\nThe group started back up the stairs as darkness crawled in around them. Before exiting the room, a sound caught their attention. The second in command turned and surveyed the room.\n\nA whisper slithered past her. She wasn't sure if it was in her mind or if she had actually heard it. She wondered if the others heard it too. Before long, it would be chatter. Before long, it might drive her insane.\n\nShe glanced around one last time and left the room, slamming the door behind her.\n\n# Chapter Seven\n\nIsland in the Middle of the Pacific\n\nThe twins grimaced and shrieked as they worked hard keeping the psychic bubble above the team. Blount knew it wouldn't be long. He could sense their pain. Shadow creatures lay broken on the ground around them, but still more gathered in the skies above. They were moments from oblivion.\n\nMaya and Nina turned and locked gazes with Blount. A brief image flashed in his head, and, as it did, he looked over his shoulder at the half buried city with its massive citadels and horrific towers. He knew what he needed to do.\n\n\"OK, people. Listen up!\" He yelled at his team members, over the chaos and nonsensical chatter that filled the inside of their protection. \"This bubble is about to burst. As soon as it does, we've got to move. Everyone, when it burns away, run to the city.\"\n\nBaxter looked at Blount like he was crazy, his eyes wide with disbelief. Blount continued ignoring the man.\n\n\"We got seconds when this thing comes down, or we're all dead. Get to the city. There is a crack in the farthest tower to the left. It's enough for us to climb into. Ready...\"\n\nThe twins collapsed, and the energy bubble vanished in a flash of light.\n\n\"Move!\"  \nThe team raced toward the city, while Blount ran to Maya and Nina. He roused them as best he could as more winged creatures soared through the air. The others were already ahead of him, heading for the tower. He pulled Maya up by her arm, and Nina followed. \"Girls, don't leave me now.\"\n\nThey pushed forward, Blount guiding them to the city and the awaiting tower. They climbed through a jagged gash across the ancient stone flesh. Crumbling debris surrounded them, and dust rained as they fell into the engulfing darkness.\n\n* * *\n\nPoints of light illuminated the area. Baxter and his soldiers pulled flashlights out of their gear.\n\nBlount coughed, spit, and looked around. There were rows of hieroglyphs lining the walls. He walked up to one of the walls and examined it, then found Max Anders in the fluttering light. \"Can you read or identify any of this?\"\n\n\"I can't make heads or tails of it. It's older than any known hieroglyphs I've seen record of. It's older than the Egyptian hieroglyphs. I'm sorry. I have no idea what this is.\"\n\nBlount sighed and looked around. He watched as Amanda Fletcher walked towards a patch of darkness out of the light's reach.\n\n\"Stop right there,\" Blount called to her.\n\n\"What?\" She froze. \"I smell something. It's over here...there's something here.\"\n\nBlount walked to her side, took a dust-coated stone from the floor, and tossed it into the darkness. It made no sound. They waited and waited. Still the stone had yet to hit anything solid. \"There's no floor there. It's just a drop. Keep your heads up. This city is still partially underground. We're still hundreds of feet up.\"\n\nAmanda swallowed and stepped back.\n\n\"Fletcher!\" Baxter said. \"Back here with the rest of the team.\" He checked his M4 and addressed his troops. \"Everyone check your gear and your ammo. There is something here. Fletcher's right. I smell it too. We are not alone. Now, let's find a way out.\"\n\n\"Hold on,\" Blount called as he went to Maya and Nina, who were huddled on the floor, breathing shallowly. \"We need to tend to the twins and give them time to recuperate.\"\n\n\"We can't stay here,\" Baxter protested. \"We're freaking sitting ducks.\"\n\n\"Without the twins, your asses wouldn't even be here!\"\n\nThe two men stared each other down. Silence permeated the area. Baxter's fingers twitched over his M4 as Blount's hand rested on his pistol.\n\nEveryone in the room shifted weight and eyed each other.\n\n\"We rest here,\" Blount said at last. That was final. Baxter dropped his stare and turned away. He muttered under his breath.\n\n\"Doc,\" Blount called. \"I need your help here.\"\n\nDoctor Saylors made his way through the soldiers and joined Blount. He opened his satchel and began rummaging through his supplies.\n\n* * *\n\nAfter some time, the twins were strong enough to move under their own locomotion. The group searched for a way out of the tower from the platform they communed on.\n\n\"From what I can tell,\" Max Anders began. \"This place is a network of catacombs and corridors. Just one of the possible thousands in the city. There are myriad stairways that seem to defy any real laws of architecture. They bend at odd angles and split off into sections. Some lead up, some down. One could get really hurt trying to navigate here. I have no idea what kind of people would build this. There is no logic to it, or comfort, or convenience. It makes the human body bend and twist in unnatural ways.\"\n\n\"Max,\" Blount said. \"Can we find a stairway leading safely away from here?\"\n\n\"I've narrowed it down to one possibility, but we'll have to crawl under that overhang.\" He pointed to their right, and their flashlights illuminated a twisted mass of jagged stone that resembled teeth.\n\n\"Not a problem,\" Baxter said, with his usual cocky attitude.\n\n\"Well, I guess we move out then.\" Blount gave the guy a look. Baxter was getting on his nerves more with every passing moment, and the situation they were in wasn't helping.\n\nIt waits dreaming...it waits dreaming...\n\nThe twins whispered this in Blount's mind, and now it haunted him. He knew they tried hard to block the Old One out, but it was quickly becoming a losing battle. Before long, it would break through and all Hell would break loose with it.\n\nIt hurts...it rips...it burns! The pain! The beast...it is The One.\n\nHang in there girls. Keep him back...\n\nThe group crawled on their bellies beneath the overhang and found a chamber of water\u2014a black pool that shimmered like satin. In order to climb the stairs, they had to dive under the black waters to reach them. Faint dripping sounds echoed all around them.\n\nBlount was a little uneasy about the whole thing. The soldiers were one thing, but expecting Doc Saylors, Max, and Nick to swim in unknown and potentially dangerous water was quite another. He wanted to stay right with them and make sure they could handle the swim. He sent the soldiers under first, observing some strange behavior in Baxter.\n\nThe man tapped his head before diving under, and swatted the air as if flies buzzed around his face. It was just one more thing that made Blount feel unsure about this hotheaded leader of his squad. Blount shook his head and sucked in his breath, before diving under with his team and the twins.\n\nIt was nearly pitch black underneath, though some dim light fluttered around him as the soldiers made their way. The cold bit into them like daggers and the muck clung to their faces, slithering between their fingers.\n\nA strange sensation brushed past Blount as he saw something huge looming out of the corner of his eye. A shadow rippled around them\u2014long and massive, stretching beyond the limits of their perception. Concern set in, and Blount urged them through, guiding them in the direction of the soldiers' lights.\n\nMore shadows gathered around them. From the black, a tentacle, much like that of an octopus except covered in barbs, lashed out at them. It twisted around Nick Larson's body and pulled him into the darkness. Gargling and waves erupted, and a cloud of red bloomed in the water.\n\n\"Dagon!\" the twins cried. \"The sea-god is trying to break into our world.\" _He knows Cthulhu is close to awakening_.\n\nThe others swam frantically as Blount turned, drew his gun, and fired. The bullets streamed through the water as if in slow motion, but in the end accomplished nothing. He watched as more tentacles headed towards him.\n\nHe bolted out of the water as hell followed on his heels. Blount spied the opening ahead of him. Inside he found a set of stone steps, and nearly slipped on the glowing fungus that coated them.\n\nThree tentacles soared out of the water behind him. M4s roared to life, cutting them down and sending the tentacles back into the murky depths.\n\n\"What the hell?\" Max cried. The experience obviously got to the soft-spoken theologian. \"My God...I\u2014\"\n\n\"Don't even look for an explanation, Max.\" Blount caught his breath. \"We just lost Nick to those things. Let's keep moving.\"\n\nAs he followed them up the winding steps, Blount stared out a random window. He saw the horizon on fire. Mushroom clouds nearly blotted out the sun, and devastation swept the globe. How many people were actually still alive out there, he wondered. With the coming unknown, he wondered how many would survive in here with him.\n\n_Many have perished_. The twins' thoughts came to him. _It is the end of times. The perfect time for its return._\n\nHe hung his head and pressed on. The thought of it weighed heavy on him.\n\nA foul smell drifted past him as they climbed up the stairs and into a courtyard full of horrid statues and bizarre monuments to decay and abomination.\n\nThe group spread out and scanned the area, traversing floors of cracked marble while half-human, half sea-creature visages glared at them from carved walls and murals.\n\nAgain, the smell caught Blount.\n\nThe twins contacted his mind once more. _The offspring smell us. They know we're here. Tread carefully._\n\nA sound reverberated in the room, and movement was spotted in the host of darkened doorways that littered the area. The sound resembled dragging, then a strange cracking joined it. The ground beneath them began to rumble.\n\n\"What the...\" Blount said. \"Don't even!\" He drew his gun.\n\nA quake rocked the room. The floor cracked, yawning open like a hungry mouth. Soldiers tumbled off their feet as M4s went off, shattering glass and crystal that spanned the ceiling.\n\nTentacles slithered out of the floor and wrapped around some of the soldiers. Hulking beasts followed, pouring out of the collection of doorways.\n\nThe beasts\u2014children of the Old One\u2014were a brood of flabby, slimy pale green flesh. Bodies writhing with seaweed and barnacles, and they stood as big as elephants, even as they dragged themselves with their tentacle limbs, which were covered in razor sharp barbs.\n\n_Acolytes,_ the twins informed Blount. _Non-human worshipers of Cthulhu. They will stop at nothing to protect him, the High Priest of the Outer Gods. They are the followers of Dagon the sea-god, Nyarlathotep cosmic terror of the air, Lord of the shadow birds. They must stop us. They must assist their priest in bringing back the Outer Gods. These gods hate their human worshipers. The cult does not know this. They will not survive the rising._\n\nScreams and wails filled the courtyard as the team clashed with the unreal. A thunderous hail of bullets rained down on the room as bones cracked, limbs ripped from bodies, and blood and slime spilled across the floors.\n\nBlount leapt into action, while the twins held back. He rolled into the midst of human and beast and fired. A monstrous aberration squealed as he shot its gossamer eyes out, black fluid running thick down its face. It tumbled, and Blount helped it along with a swift kick, sending it crashing down in a cloud of stone, dust, and muck.\n\nHe turned to see Baxter spray a stream of fire with his M4, nailing a beast as it stormed towards him. It staggered, slowed down, trying to resist Baxter's attack, but in the end the hot lead proved to be effective and it collapsed. Baxter continued firing anyway. He ran up to its head and turned his weapon point blank, then shot and shot.\n\n\"Baxter!\" Blount yelled, realizing the man was losing it and letting down his guard.\n\nWith his back turned, another creature slung its tentacle around Baxter's legs and pulled him down hard. Its barbs bit into his flesh and ripped, blood spraying through the air. Despite the pain, Baxter did not give up. He pulled a serrated hunting knife from his belt and slashed the tentacle until it let go. A shriek pierced the battle, as the creature recoiled and Baxter followed up by blowing the thing away.\n\nBlount watched in bewilderment as Baxter jumped up and ignored his wounds. The solider punched himself in the head and began to babble nonsensically. \"Get it out of there! Get it outta my head!\"\n\nHe turned and blew Private Amanda Fletcher away while she was holding off a beast of her own. The bullets ripped through her back and spine, and she crumbled into the tentacles of the creatures, which promptly began to devour her.\n\n\"Baxter, what the _hell_ are you doing?\" Blount cried as he raced towards the man.\n\nHe launched himself through the air and plunged into Baxter. The two men went rolling across the floor. Baxter began laughing maniacally, his gaze burning with insanity. The two scrapped on the floor, their guns flying from their grasps, but that didn't stop Baxter. He wrapped his hands around Blount's throat and squeezed with all of his strength. A few punches to Baxter's face got him off Blount and leveled him to the floor.\n\nRecovering, Blount made it to his feet and reached for his gun. Baxter was on him again. This time, with his hunting knife drawn. It slashed across Blount's chest, leaving a gash. The Black Rock agent howled as he collapsed to one knee. Baxter laughed hysterically as he licked the blood off his blade.\n\n\"Sick bastard.\" Blount spat as he raised his gun and fired.\n\nBaxter stumbled backwards and fell through one of the cracks in the floor. His laughter still echoed as he fell to his death.\n\nBlount stood erect and caught his breath. All around him, his team fell to the hordes of monsters as they continued to pour out of the floor and doors. Screams deafened his ears, and he shot blindly at anything that moved.\n\nHe turned to see Saylors and Anders being devoured, bodies ensnared in tentacles, flesh pulled from their bones to be stuffed into salivating maws filled with teeth. He also saw that the twins were next.\n\n\"No!\" He ran to their side, firing upon the creatures as they surrounded them. The girls had become too weak to even summon a bubble of protective energy. \"Get away from them. Get away!\" He bashed through the wall of sea-creatures, using a running kick to crack the tentacle legs of the nearest one, crippling it enough for him and the twins to make an escape.\n\n\"Blount,\" the twins said. \"The stairs trimmed with gold. They lead up from the courtyard.\"\n\nThe three ran up a stairway into a pocket of darkness that stopped at a wall. There was no doorway.\n\n\"This is,\" Maya began.\n\n\"An illusion,\" Nina finished. \"We must believe.\"\n\n\"It is creating this to confuse us.\"\n\n\"We can pass.\"\n\nAt the bottom of the stairs, a crowd of hungry beasts gathered and dragged themselves up, slowly.\n\n\"I trust you,\" Blount said, as he closed his eyes and pushed on.\n\n* * *\n\nThey rested, but heard the howls and calls of their pursuers not far behind.\n\n\"We can't stay girls,\" Blount warned. \"They'll just keep coming, and there's an endless supply.\"\n\n\"We know,\" they replied. \"Come, we have sensed an exit from this place. Your destiny awaits.\"\n\n\"What do you mean? Have you broken through...\"\n\n\"Not exactly. Trust us. This way.\" They started down a dark tunnel as Blount followed. He fished a flashlight from his gear and shed some light on their journey.\n\nThe tunnel glittered with phosphorus fungi as it twisted downward, somehow growing tighter and smaller as they went. He hoped they knew where they were going. Claustrophobia set in, and his chest tightened as his breathing grew short.\n\nIt was nearly too small for the three of them to fit into now, and Blount was just about to speak up when the twins screamed. Aghast, he watched as they fell to the ground, blood seeping from their nose and lips.\n\n\"What's happening?\" He went to them, taking hold of their hands.\n\n\"It is over!\" they cried. \"We are l-l-losing t-t-this fight. We know all we can now, and must share with you before we leave this life.\"\n\n\"No! Fight it. You are the best of the best. You can fight this thing.\"\n\n\"We can no longer fight....it is the Old One. It is Cthulhu. He is waiting to return. His consciousness is not here...not in this city. It is in another place...waiting to join its body. The body sleeps here, the soul is...is...is...\"\n\nThey placed their hands upon Blount's face and the scene flashed in his mind. He caught a brief glimpse of the most massive, monstrous form he had ever seen. The entire city rested upon the beast itself\u2014a mountain of bloated flesh and tentacles, a patchwork of gray and green flesh rippling with networks and colonies of sea life and organisms forming one body, one shell. It slept...soulless.\n\nThe scene shifted and time melted, returning him to an ancient era where he witnessed a group of primitive people. They gathered around a bonfire, writhing and dancing. Above the fire, staring down at them, stood a great statue of Cthulhu himself\u2014a grotesque figure resembling a cross between an octopus and a dragon. A pulpy head with a face full of feelers crowned an abominable body with frayed wings.\n\nThe dancers reveled in the presence of the statue. They worshipped it and sacrificed to it. Madness took each one of them, their eyes enflamed, their faces twisted with maniacal grins. Their ritual culminated with them dragging a naked woman into their circle. They wasted no time in shedding her blood and tearing her open with a multitude of daggers.\n\nBlount felt the bonfire's heat, smelled the burning flesh. He felt their bloodlust frenzy. It was palpable to him, as if he were right there. The fear of their victim filled him, along with a chattering that invaded his head, just as it filled the dancers' minds.\n\nLike every cult after them, they craved the same thing\u2014the return of their Lord Cthulhu. The very definition of the Old One, for Blount understood now that this beast truly was an old one. He ruled the universe, and this planet before man even existed. All of these monstrosities did. Called down by Lord Cthulhu to rule and plunder. A city was built for Cthulhu to rule, after calling down his brethren from the cosmos, but something went wrong. The city was unstable. It sank beneath the ocean, trapping the old one in a watery tomb.\n\nAll of his followers knew it by name. R'lyeh, the corpse city. When at last the city rose from the ocean's depths, Cthulhu would live again and pave the way for the original rulers of the universe to return.\n\n\"No wonder the world is being turned on its ass,\" Blount said. \"The cult that has survived all these centuries is getting it ready for them.\"\n\nOne final scene formed in the ripples of his mind, showing him a young woman, dormant, naked. The vessel. She carried the spirit, the consciousness, of the Old One itself.\n\n_You cannot harm the body while it rests here_. They informed him. _It would mean nothing. It cannot be harmed. You must destroy the spirit. You must go to the vessel. Find and destroy it before it awakes and joins the body with its soul. If this is allowed to pass, our world will be no more._\n\nBlount saw the vessel of the Old One's soul, and it was none other than Casey Archer. The name floated into his consciousness. He'd never laid eyes on the girl, but he knew her full history in a flash. She slept waiting, waiting for the final sacrifice that would unleash the Hell inside her and return Cthulhu to its body.\n\nYou must go, Blount. Leave this island, find this vessel, and destroy it. It is your destiny. We have finally penetrated the mind while the vessel was distracted, but now...we are done.\n\n_I will not leave without you._ Blount said _. I need you both. I can't do this without you._\n\nYou must do this, or we are all lost. We are done in this life. Do not worry. We will return in another. Our spirits will continue to guide you on this journey. We can go no farther. It has finished...us...\n\nTheir hands slipped from his face, as they seemed to collapse into themselves and close their eyes.\n\n\"Maya! Nina!\" There were tears in Blount's eyes. \"Girls...\"\n\nA white energy rose off their lifeless bodies, shimmering as it floated towards the ceiling and burst a hole in it when it reached the top. Rock and debris rained down upon Blount while the sunlight streamed in. He realized now why the tunnel was so small. He got off his rump and climbed out the opening the twins created for him.\n\nHe crawled out onto his belly and realized he was in the island's wilderness again. A little ways behind him was the outskirts of the city, with its hellish walls and towers.\n\n\"Thanks, girls,\" he whispered as he raced his way back to the waiting chopper, a new mission burning inside him.\n\n# Chapter Eight\n\nFallout Shelter, Main Hub\n\nOutlook, Montana\n\nSam Veleska gaped as shock washed over her. She couldn't believe what Archer was doing, especially to someone so high up among the remaining members of the government.\n\n\"Madame Secretary, you've got 'til about the count of five.\" Archer said sternly as he stared down the barrel of his pistol, right into Carling's widened eyes.\n\n\"This is insane!\" Patricia Carling yelled. \"Have you lost your mind?\" She used the table to protect herself.\n\nA hush fell over the room.\n\n\"No, not yet. We know you did something to him, and we know you are in on all of this somehow. We can do this the easy way or the hard way.\"\n\nShe looked around the room, staring at the crowds of shocked and distraught people, as if looking for someone to come to her aid.\n\n\"Waters would never do something like that,\" Palmer said. \"Never! Something did it to him. I know the man. He was the best. He would never have given up. Not on us\u2014not on the world.\"\n\n\"He knew too much, didn't he, Carling?\" Archer said. \"How'd you do it? Some sort of power of suggestion? That's quite a trick you got there.\"\n\n\"This is crazy. You've all gone mad. I haven't done anything to Waters. I am a member of the US government. Thaddeus, listen to me...\"\n\n_Thaddeus_. _Thaddeus!_\n\nThe way she said his name. The inflection in it. He recognized that voice. Through the panic, through the fear, there was that smooth and smug tone. It was _her!_ She was the voice that had taunted him in last year's calls, when Casey first disappeared and when he thought about ending everything. God he could never forget that call. She was behind all of it...the cult, the island...Casey.\n\n\"You!\" Archer leapt over the table at her and crashed upon the floor. His pistol slid from his hands as he clasped her throat. \"Where is my little girl? Where is Casey?\"\n\nCarling flailed and squirmed under Archer's weight, her face reddening as the air cut off from her lungs. Finally, Archer realized his actions and pulled her up.\n\n\"You are responsible for all of this. The cult, the kidnapping... Where is she?\"\n\nA twisted smiled curled on Carling's face, as Big Jesse rose behind Archer and smashed the table across him. Archer crashed to the floor as bits of table sailed through the air. A seething Jesse began kicking him, and Carling used this chance to run. She made a dash towards one of the hallways, before Palmer tackled her to the floor.\n\nJesse followed up by grabbing hold of Archer and tossing him against the nearest wall. He felt his back crack, and watched as the big man rubbed his head and muttered to himself.\n\n\"The chatter...\" Jesse howled. \"What the hell is that noise?\"\n\nHe stormed toward Archer, who thrust out a leg and caught the obese man in the gut. Jesse stumbled backwards into his cooking area, crashing into a collection of pots and pans. A cleaver found its way into his hand and he jumped up with it.\n\nBy the time the big man reached his feet, brandishing the cleaver, Sam quickly pumped a few slugs into Jesse's gut, stopping him cold. Shrieks pealed from the crowd as Jesse fell dead. Sam went to Archer and extended her hand.\n\n\"Thanks,\" he muttered, as he turned to see Palmer restraining a frantic Carling.\n\n\"It's over Carling. You hear me?\" Archer stormed over to her and put his recovered pistol to her head. \"You assumed leadership of the cult when I killed McBain, didn't you?\"\n\n\"So smart aren't you, Thaddeus!\" Carling screeched. \"You're already too late. Our God has returned, and you will all be nothing but dust. The final sacrifice is at hand. It's already too late.\"\n\nHe took her by the throat. \"You'll never get to see your God if you don't tell me where Casey is. Take me to her.\"\n\nArcher turned to the crowd. \"Everyone! This is who is responsible for bringing the end of the world. This woman and her group have unleashed the evil that has destroyed your families and friends. She is why our country is burning down around us!\"\n\nCarling gasped as the crowds of people focused on her. Their hatred coalesced in a mob mentality, and the hatred burned in their eyes. Archer whispered into her ear, \"You tell me where she is, or I turn you over to them.\"\n\nAn eerie silence filled the room. Carling looked into his eyes. \"Alright, I'll take you to her. Don't let them take me, please.\"\n\n\"Fine.\" He turned to Palmer. \"I need you to drive us back to the chopper. We need to be in the air ASAP.\"\n\n\"You got it, but one request?\"\n\n\"What's that?\"\n\n\"I'm with you all the way. You go, I go. I owe it to Waters to see this to the end.\"\n\n\"As you wish. Bring lots of firepower. There's bound to be lots of dead out there, and who knows what her cult has in store. Isn't that right, Carling?\"\n\n\"They will not interfere. This is between you and Casey now.\"\n\n\"I'll bet it is. If one hair on her head is harmed, no God in your twisted pathetic mind will be able to save you from me.\"\n\nThey pushed her past the crowd, to the secured doors. Palmer paused to give instructions to a few of his trusted comrades.\n\n\"Sam,\" Archer called her to attention. \"I want you to stay here and hold down the fort.\"\n\n\"I thought we were in this together, big guy?\"\n\n\"You know we are. But I need someone who knows the story to stay safe. In case we don't return, someone needs to tell the surviving world just what happened. You know as much as I do. Besides, whatever other intelligence comes through, it's bound to come through here first. You should stay.\"\n\n\"No way, Archer.\"\n\n_It's too late Sam... The God is here_. _He's here. Might as well kill yourself...and take Archer with you._ Samantha did her best to ignore that strange nagging voice in her head.\n\nArcher pursed his lips and stared at Sam for a few moments. \"I'm not talking you out of this, am I?\"\n\nSam smirked and shrugged. \"Nope. Besides you need a pilot.\"\n\n\"Fine,\" Archer relented.\n\nWith that, Palmer pulled open the door. It gave a groaning clank, and they vanished into the night.\n\n#  Part III\n\n# \"Endgame\"\n\n#\n\n# Chapter One\n\nIsland in the Pacific\n\nBlount fought his way through the jungle, arrived at their landing sight, and discovered the Blackhawk waiting silently for him.\n\n\"Yeah, right...\" he said as he started for it. Pulling the door open, he found the cockpit empty. Drops of blood spattered the console, and a pool of crimson formed on the floor.\n\nA dark shadow formed in the cockpit then disappeared. Blount stepped out of the way as the pilot's body fell towards him, plunging to the floor. The corpse's hand trembled, so Blount placed his foot on the pilot's back, holding him down while he looked for the shadow. The corpse squirmed underneath his boot, moaning and scratching with stiff fingers.\n\nBlount spotted the shadow rippling in the air in front of him. He lifted his gun and shot with a steady hand. A shriek filled his ears as the shadow dissolved. He released his hold on the pilot and climbed into the cockpit, then looked over the console and its controls, checking to see that all was intact. After pulling himself into the seat, he closed the door.\n\n\"Want something done right you gotta do it yourself.\"\n\nBlount powered up the Blackhawk. As the chopper's rotor whirred to life, identical voices whispered in his head: _Rugby Rock, North Dakota._ The twins came to him as they always did. Their spirits guided him just as they said they would, letting him know the next step of his mission.\n\nThe vessel sleeps in Rugby Rock. Hurry, Blount. Time is fleeting.\n\n\"You got it.\" Pulling away from the island, he soared over the waters of the Pacific. They were right. There was no time to waste.\n\nHe noticed the fuel gauge and frowned. There were some tankers out there in the ocean, but he was unsure of their location. He hoped some form of communication became available when he got farther away from the island.\n\n* * *\n\nBlount checked his SAT phone again and, as if in answer to his prayers, it relayed that he had a weak signal, so he used his speed dial.\n\n\"Yeah, it's Agent Blount. I need a location on all available tankers in the Pacific...yep you bet. I also need a friendly port hidden or not near the borders of New Mexico.\"\n\nWith the world going to hell, Blount was sure a lot of tankers and destroyers would be out in the oceans right now, on high alert. He would have to show a damn lot of clearance to get his chopper fueled time and again, but he had no choice.\n\n\"Roger that. Say again? Mexico is under water, huh? What about New Mexico? Okay, well gotta hope I find the part that's dry before I run aground. Over and out.\"\n\nThe Blackhawk approached the first of the four tankers, but he couldn't raise anyone on his radio. \"To hell with it.\" He landed any way.\n\nLaced with the smell of soot and death, the ocean breeze slapped him across the face as he stepped out. Not a sound could be heard on the tanker. It was a ghost town. He walked around the deck, searched the immediate area, and found no signs of life. The surface of the tanker leaned at an uncomfortable angle. A tsunami must have rocked the platform, but miraculously it had withstood the assault. The platform yawed with a stomach-clenching sound.\n\nGet fuel and get out of there.\n\n\"Alright girls, calm down.\"\n\nHe started for the generator and the pumps. A clanking noise resounded as he worked. Lifting his head, he watched as the tanker's cabin door slowly opened. A chorus of moans filled the air.\n\nSwarms of dead men came lumbering out of the cabin. Shadow beings wriggled ahead of them, leading their hosts on a bloodthirsty rampage toward the only living human onboard.\n\n\"Great freakin' timing guys!\"\n\nBlount ripped the hose from the fuel tank just as it finished. He didn't bother to fire at the oncoming crowd of dead, as one stray shot would end his mission right then and there. He gave the first two men that reached for him the boot as he climbed back into his chopper.\n\nWithin moments he was back in the air, where he swerved about and launched two missiles at the tanker, turning it into a fireball that ignited the horizon in an orange glow.\n\nHe headed to New Mexico, hoping for the best.\n\n* * *\n\nBlount landed for fuel one last time at what remained of an air force base. He turned his back on the smoldering bodies all around him and hurried over to the refueling station. As the chopper fueled, he kept his attention on the sky and surrounding horizon for any signs of the shadow creatures.\n\n\"Human,\" a gravelly voice spoke from the darkened confines of one of the aircraft hangars.\n\nBlount whirled and trained his pistol on the opening, but no target readily presented itself. \"Who's there? Identify yourself!\"\n\nA laugh drifted from the darkness and something massive shifted within. Sweat sprouted on his forehead and ran into his eyes, but Blount didn't attempt to wipe it away. He was about to challenge again when the voice spoke once more.\n\n\"It's been a long time, Blount. A long time.\"\n\n\"Well, come on out and show yourself, so we can catch up.\"\n\nThe creature emerged from the hangar and spread its wings in the fading light.\n\nHe sucked in a breath. \"Abdiel.\"\n\n\"Adrian Blake.\" Abdiel sighed.\n\nBlount shuddered at the memories washing over him. It had been many years since he'd dwelled on what happened so long ago in the town of Amber Creek. It'd been even longer since he'd thought of Sarah, or the person he used to be.\n\n* * *\n\nThe town was covered in a black haze. Yet, it was more than that. It was more like the cloud inhabited the town, as Adrian and Sarah topped the hill. Finger-thin shafts, mingled with log-sized tendrils, caressed the town like a familiar lover. Adrian held his tongue. He waited for his wife to scream out at the sight of the anomaly, but she sat beside him, humming a church hymn, oblivious to the black cloud. Thoughts of cancer and rot came to his mind when he looked at the monstrosity.\n\nHe tightened his grip on the reins. The horses snorted against the pressure and came to a halt. Something was at the apex of the black, swirling cloud. Just a hint of something huge, but its features were obscured. It faded in and out of existence as it hovered over the town.\n\nAs Adrian looked on in horror, flames engulfed the town, obliterating it right before his eyes. Figures traveled down from the black miasma, obscured in its darkness. Adrian was thankful they were shrouded in the swirling mist. To see more would mean madness, he sensed.\n\n\" _Darling?\"_\n\nAs soon as his wife's voice invaded his thoughts, the entire illusion dissipated. Not all at once, like a hallucination might, but gradually, like watching time flow backwards. Now, the anomalous mist was all that remained.\n\n\" _Don't you see that?\" He hoped the horror remained invisible to her._\n\n\" _See what, silly goat?\"_\n\n\" _Nothing, love. Just feeling a bit queer, I reckon.\"_\n\n_Adrian was not a religious man, but what he'd just witnessed couldn't be described in any other terms. What he'd seen was pure evil. It was lying in wait for his town. There was only one person who might be able to help him with what he witnessed. The priest, Father Nathan Carpenter_.\n\nHe pulled to a halt in front of the general store, waving to the flock of children that erupted from the doors, each clutching a stick of rock candy. Softly he soothed the startled horses with calm words.\n\n\" _You run along, Sarah. I've got some business to discuss with Nathan. I'll join you shortly.\"_\n\nHe blew a kiss to his wife as she turned to glance back at him. She appeared troubled, but as soon as Adrian blew the kiss, her face lit up. She made as if to grab it, pretending to pocket it for later. Her laugh drifted off, as he clicked to the horse and the wagon lurched away.\n\nAdrian glanced up as he neared the middle of town. The shape was directly above him. It blotted out the sun, casting a graveyard shadow across the town. No one else seemed to notice the phenomena. For a moment he questioned his sanity, but he knew he wasn't crazy...yet.\n\nHe watched as two small girls jump roped, oblivious as their feet splashed in a puddle of blood. Their shoes were painted red-black, and splotches clung to their bare legs. One of the girls carelessly pushed a hand through her long blonde tresses, staining the locks a dusty red. As he rolled past, the girl glanced his way. Half of her face was gone. A ragged hole leered at him. He gasped and closed his eyes in horror. When he reopened them, the girl was normal. She giggled and called to the horses.\n\nMovement caught his eye from the rooftop to his left, and he saw the first one. It was hunched, clothed in fire, leathery wings wrapped around its hairy, muscular frame. A disturbing amber glow obscured most of the creature's face, but its eyes were what struck fear in Adrian's heart. They were oval, vacuous wells of malevolence. One's massive wolf head swiveled in his direction, and the creature looked right at him.\n\nAs the past faded, one last scene entered his mind. It appeared a memory but it couldn't have been his, for it seemed to occur before the dawn of man. In a sea of black waters, he witnessed a monstrous creature rise with a tempest. Its tentacles writhed in lashing winds. Tattered wings unfurled against a black, star-lit sky.\n\nAcross the sky a great army of angels raced. Armed with magic and maces, they descended on the beast with a fury. Agonizing screams pealed through the air as heavenly beings fell into the waters, while others ripped swollen flesh from the monstrosity and spilled its ichor. Inhuman carnage like nothing Blount had ever seen waged before him, and leading the attack on the beast, brothers in arms, were Abdiel and Azazel. No this was not Blount's memory\u2014this was someone else's. The voice of the angel pulled him out of the vision and back...back...\n\n* * *\n\n\"I haven't gone by that name in a long time,\" Blount said.\n\nThe angel smiled and nodded after the shared memory. \"You haven't changed much.\"\n\n\"What are you doing here?\"\n\n\"We have been summoned. My brothers and I.\" Abdiel shrugged.\n\n\"I banished you and your brethren.\"\n\n\"Indeed, you did.\"\n\n\"Where have you been since?\"\n\n\"Where you banished us.\" The bitterness in Abdiel's tone was strong, potent. \"In the void. You have no idea.\"\n\nIt all clicked for Blount, and the pieces fell into place with a heavy thud. \"What do you know of the island?\"\n\n\"Island?\"\n\n\"Don't play stupid with me. What do you know of the island?\"\n\n\"All I am aware of is being yanked from the void by a powerful spell. One moment there was blackness, nothing. The next, I was here.\"\n\n\"And your brethren?\"\n\n\"The spell only called forth the most prominent of us. The rest? They are somehow stuck in the in-between. This world is riddled with holes, so to speak. Just like a house has a space between its walls, so does reality.\"\n\nBlount's tongue resisted, but he pushed the name past his lips, \"Azazel?\"\n\nAzazel, formerly Asiz. The Metatron. A being necessary because it was impossible for True God to communicate or associate with man. So high-ranking and so holy was he, he was authorized to go by YHVH's very name, could be treated, regarded, received, even _worshiped_ as True God himself. Lucifer's greatest victory came when Azazel sided with him, and it was God's greatest loss. His heartbreak at Azazel's defection was said to have filled the ocean.\n\n\"He is near,\" Abdiel said. \"As well as Camael. They have joined forces and ruled what they could in the void. There was something else in there.\" Abdiel's voice grew soft. \"Something we never saw, but sensed. There was much debate as to what this creature might be.\"\n\n\"And?\"\n\n\"I believe it was the Ouroboros.\"\n\nThe name stirred some primal memory in Blount. He said the name, as if savoring its meanings. \"I was on the island. The presence I felt there was not the Ouroboros. The being was not unlike the Ouroboros, but somehow not the same. I sensed something god-like.\"\n\n\"God is dead,\" Abdiel said. \"Azazel told me so himself.\" The angel's voice faltered at this.\n\n\"And you believed that liar and murderer?\"\n\n\"What choice did I have? There is no evidence of HIM. The Ourboros moved to stabilize the fractures. The snake that eats itself. When Father All and his abode vanished, time as we know it fractured into parallel. I know this, as you do. We were both present.\"\n\n\"Now, we are all servants to the Ouroboros. Whatever I am, I have seen much during my lifetimes, and though some of the Dark Ones and the Old Ones I've faced claim He was slain, I tend to doubt that. Existence is His Being. The wheel has continued to turn, so I believe Father All has not been slain. Our remnant holds to this hope. Many times we have failed. Destined? Perhaps...yet we persevere. Father All is watching, and when things fall into perfect harmony, He will return and reward the faithful.\"\n\nAbdiel unleashed jeering laughter. \"Humans. You have always been so full of faith. Where I was banished? No God would have ever created such a place. The void, and what lurks there unseen, is no God.\" The angel leaned down until he was almost nose-to-nose with Blount. \"What lurks there...is more than a god...it is a devourer of planets, of universes, of galaxies.\"\n\n\"Cthulhu.\"\n\nAbdiel shrank back as if Blount had physically struck him. He hissed and glanced around warily.\n\n\"That name means something to you?\"\n\n\"I know the name,\" Abdiel said. \"I wish I did not.\"\n\n\"The island rests upon his form. His consciousness is elsewhere.\"\n\n\"The two cannot be joined.\"\n\n\"Why?\"\n\n\"They MUST not be joined,\" Abdiel said. \"All will be lost if this happens. Even the Ouroboros will not be able to repair the damage if this occurs.\"\n\n\"So the shadows manipulating the dead are Azazel's minions,\" Blount said.\n\n\"Not exactly. He is teaching them how to possess the living now. At first they were only adept at controlling corpses.\"\n\n\"That's why all the suicides,\" Blount said. \"They wanted the bodies.\"\n\n\"Yes, but now Azazel has taught them to go straight inside the living human form.\"\n\n\"Nice. So no one is safe.\"\n\n\"Not everyone is susceptible. It's a weak hold at best. It all depends on the human's will or enlightenment, or mental stability. The shadows can control any corpse they wish. It's a strong hold that grounds them. But a living entity is a constant internal struggle\u2014a war of wills so to speak. A battle that wages in the mind.\"\n\n\"What is this all about?\"\n\n\"Azazel's ego. He is using the shadows for his own personal war on Earth, against the coming Old Ones. They've promised to fight for him in exchange for the knowledge of possessing the human body.\"\n\n\"Fight who?\"\n\n\"Anyone he wishes. You don't understand\u2014not even half of it. There is something else I must tell you.\"\n\nBlount was filled with curiosity and apprehension. Somehow, he knew he wasn't going to like what he was about to hear. \"I'm all ears.\"\n\n\"This is bigger than all of us. Azazel wants this world. He will do anything to get it. _We_ overthrew the Old Ones so long ago. At the dawning of time, when this world was nothing but darkness and without form. With the help of God, we attacked Cthulhu and the others and defeated them. Then God brought the light. They wanted this space for themselves, but we stole it. Cthulhu and the Old Ones are enraged. They will crush the entire universe for our crimes. Now that God is gone, there is nothing to stop them. You cast us into the void with their spirits. Now that the void has been ripped open, the island raised, Cthulhu's spirit has been placed into a vessel by the ignorant cult.\"\n\n\"Casey Archer.\"\n\n\"Daughter of a mortal. The joining must be stopped. We cannot do this alone. God has been missing for some time. Without Him we have no hope of defeating Cthulhu and the Old Ones he will resurrect. Ironic...we shared the void with Cthulhu and all his rage, but he could not touch us. Not without his body. Now...\"\n\n\"What of Azazel?\"\n\n\"He is a fool. He is going to try to stop Cthulhu anyway. He will fail and we will perish with him. He must be stopped as well.\"\n\n\"What are these shadow creatures Azazel is trying to command?\"\n\n\"Many forms inhabit the void,\" Abdiel said. \"The spell the cult cast unleashed the shadows to wander in the in-between\u2014the grids. It was like a floodgate opening. Some entered your world, some got lost in the in-between. As they find their way out, they will continue to ravage this world. Cthulhu is still in the in-between, latched onto Casey.\"\n\n\"So, I'm dealing with two different scenarios.\"\n\n\"It would seem so.\" The angel shrugged.\n\n\"Great. Just great.\"\n\nBlount turned his back on the angel. It was the first time in all his lives that Abdiel was so upfront with him, without all the riddles and posturing. He felt the angel's eyes on him as he hauled himself back into the Blackhawk. Abdiel was scared, Blount sensed, and feeling human for the first time in a millennium. Blount actually felt the fear in him, and that's what frightened him most.\n\nSomewhere Over North Dakota\n\n_The spirit is here_ , the twins' voices filled his mind.\n\nBlount cried out and clutched his head. Once the pain subsided, he spoke.\"Where is Azazel?\"\n\nHe just left Outlook, Montana. The base there is overrun with shadows. Azazel slaughtered all the people within. He is heading for Rugby Rock. Cthulhu grows stronger.\n\nBlount sensed the time was close. He had a decision to make, but either decision held its own set of consequences. He lowered his altitude and studied the vast, empty stretch of land below. Somewhere down there a god waited to be born in human form.\n\nA sobering thought struck him, and he gasped as he considered the ramifications. What if Azazel was actually serving Cthulhu, not warring with him? The demon would have his ulterior motives, but both entities desired the same thing: the destruction of God's creation.\n\nHe peered below and scoured the moonlit landscape for any signs of life. \"What do I do, girls?\"\n\n_Cthulhu is more important_.\n\nWith a sigh, Blount nodded. The twins were right, of course. Still, he longed to battle Azazel, and the urge was strong\u2014a deep boring in his head.\n\n\"Where? Show me the way, girls.\"\n\nA pinpoint of light shot up from the prairie and shone brightly into the sky. Blount maneuvered the chopper in that direction and prepared his mind for the coming battle. He sensed the cameras hidden around the entrance to the bunker and avoided approaching from the front. Instead, he circled the entrance and came up from behind. He recognized the configuration as an underground missile silo, and knew from previous experience there was an alternate escape\/entry access located a few hundred yards to the west. After making his way there, he stood before a much smaller door set into a boulder. Behind this door would be a rapid elevator.\n\nLooking at the key access panel, he whispered, \"A little help girls?\"\n\nGhostly hands played across the keypad. The pad turned from red to green, and Blount entered the elevator then descended.\n\n# Chapter Two\n\nRugby Rock, North Dakota \u2013\n\nSam set the chopper on the ground gently and powered down. She was frustrated and exhausted, and the stresses of the past hours were taking their toll.\n\nArcher nudged Carling and she jerked awake with a start. Her gaze came to rest on Archer, and if looks could've killed he would've dropped dead on the spot.\n\n\"Where to?\" he said.\n\nCarling looked out the window and pointed. \"That way. We'll have to go the rest of the way on foot.\"\n\n\"If this is a trap...\" Archer let the threat hang in the air between them. He turned to Palmer and Sam and addressed them. \"Sam, you take lead. Palmer, take up the rear, if you don't mind. We'll stagger it out a bit, just in case. I'll stay with our guest here.\"\n\nThey exited the chopper and began walking\n\nArcher and Palmer escorted Carling at gunpoint as she led them across a vast stretch of land. Sam was now in the rear, covering their progress. The cold crept up on them like second skins. They walked and walked into the night, forever it seemed, as the stars glittered down on them.\n\nA structure appeared in the distance. It looked like an abandoned war bunker\u2014something left over from World War I, or perhaps even older. Scorching marred the old stone, and there was graffiti here and there. A massive blast door, crafted to look like stone to the casual observer, covered the entrance.\n\n\"What is this?\" Archer said, squeezing Carling's arm.\n\nShe pulled her arm free and shot him a spiteful look. \"Patience, Thaddeus.\" Carling muttered a few arcane words, unrecognizable to Archer and Palmer, and then placed her hand against the door. A bluish light glowed beneath her palm, and then the door slid away from the entrance with a groan.\n\nA stench wafted out from the pitch-black opening and Palmer covered his nose.\n\n\"Let's go,\" Archer said as they filed in.\n\nTorches ignited at Carling's arrival. Rows of them on the walls added flickering light to a corridor that stretched into the void.\n\nThe four continued until they reached a large, open room filled with symbols and carvings on the walls. Giant symbols glowed on the floor. Sam had seen documentaries on the inner workings of missile silos and realized they were in an abandoned one.\n\n\"Where is she?\" Archer demanded. \"I want to see her now.\"\n\n\"She is here...have no fear.\"\n\n\"You best stop dicking us around,\" Palmer said. \"You're messing with a pissed-off father. I'd be a little more forthcoming if it were me, y'know?\"\n\n\"Listen to the man,\" Archer said.\n\n\"Casey is safe. She is the One. The One we have waited a lifetime for.\"\n\n\"What in Christ's name are you talking about?\"\n\n\"He cannot help you now. Only our God exists. His time has come.\"\n\nArcher raised his pistol, his grip tightening on it. \"Bitch, so help me I will bash your brains in...don't think I won't. I'm done playing games.\"\n\nCarling broke into maniacal cackling, while her eyes lit with cruel madness.\n\n\"Come,\" she beckoned, leading them to a passage in the back of the room that ended where stairs crawled farther down into the darkness.\n\nTo the right of the stairs, Archer noticed a small alcove. The glow of monitors bathed the room in purple light. He glanced in and realized the monitors were for the perimeter cameras covering the blast door, the corridor beyond, and the surrounding area outside the entrance. The lack of people troubled him. They could've been observed during their approach here, and even now the rest of the cult could be farther below, lying in ambush. He motioned for Palmer to keep an eye on Carling and pulled Sam aside so their conversation wouldn't be overheard.\n\n\"I want you to stay here and watch these monitors\u2014just in case we run into a trap below.\"\n\nSam nodded and wiped sweat from her forehead. \"Good idea.\" She passed into the alcove and grabbed something small and compact. She pressed the object into Archer's hand, and he glanced down, realizing it was a compact radio. He pressed the transmit button and squelch echoed from Sam's receiver.\n\nSam offered him her hand. \"Good luck. I hope you find your daughter down there.\"\n\nThaddeus ignored the hand and pecked Sam on the cheek. \"Thanks for all your help. See you soon.\"\n\nArcher and Palmer took up torches and followed the stairs down into another room. A pedestal with a book sitting on it, and an altar at its base, were the centerpiece of this place. There upon the altar, still sleeping, lay Casey Archer, her naked, painted body glistening in the torchlight.\n\n\"Casey!\" Archer rushed to her side and took her hand. \"I'm here baby, I'm here...Casey!\"\n\nAs Archer nearly succumbed to his emotions, two robed figures emerged out of the dark and grabbed Palmer unawares. They gagged him and dragged him back into the shadows. Laughter erupted from Carling, once she and Archer were left alone in the room.\n\n\"Shut up!\" Archer said, his eye welling with tears. \"What have you done to her? Wake her up or so help me...\"\n\nCarling stopped laughing. \"She isn't ready to wake just yet, Thaddeus. She is the One.\"\n\n\"What are you talking about?\"\n\n\"She brings us back the Old One. Our God. Casey Archer will begin a whole new world.\"\n\n\"The hell she will! I'm taking her out of here.\"\n\n\"It's too late. Much, much too late for that. The new world is already here. Just one last sacrifice! One last _sacrifice_!\"\n\n\"How's this for a sacrifice!\" Archer raised his pistol and fired. The shot hit Carling square between the eyes, and she went down hard, her neck snapping back, her stare frozen in shock.\n\n\"You'll never get to meet your God,\" Archer said as he tried to slide his arms beneath his daughter's body, but found the girl as stiff as petrified wood, and as heavy as a hundred tons.\n\n* * *\n\nSam jumped as she heard the shot below. She quickly scanned the monitors, but none presented a view of the situation. As her gaze came to rest on a monitor at the top left, she sucked in a breath.\n\nThe front entrance was jammed with the dead. There must have been hundreds of them. Where had they all come from?\n\n\"We're OK. That's two feet of steel. No way can they breach that.\"\n\nJust as she finished speaking, the large door moved inward. She pulled her weapon and racked a round, then keyed her radio. \"We're about to have company.\"\n\nThe mass of bodies surged and the door moved a few more inches. Through the cracks, shadows slithered. The dead abruptly stopped their assault and moved back, assembling in two neat rows on either side of the blast doors. Something massive moved through their midst. Sam peered at the monitor in confusion as the creature stalked forward and placed its enormous hands against the steel.\n\nWhat stood before the blast doors was difficult for Sam's brain to process. She could feel madness taking root in her being. The creature's bulk was mammoth, and the confines of the corridor could barely contain its girth. Seven viper heads writhed in and out of its numerous, oily wings. There were two faces on each head. They cackled madly until the laughter became like a spasm. The faces jostled for supremacy, until one of the heads slithered forward and fixed its gaze on the security camera. The sensation of seeing two faces with discordant expressions made Sam's head throb, but she forced herself to keep looking at the monitor.\n\n\"What the _hell_ is that?\"\n\nThe steel beneath the creature's hands turned to liquid at its touch. Massive fingers slid through and appeared on the opposite side of the blast door.\n\n\"That's impossible.\"\n\nShe pulled her pistol and ensured there was a round ready, keeping her focus on the screen as the creature _walked_ through the steel and emerged on the opposite side.\n\n\"Archer, we've got a major problem topside.\" Sam got nothing but static in response. She tried once more with the same result. \"I'll go and hold it off as long as I can...\"\n\nShe didn't wait for a response. A few moments later she peered around the corner of the corridor at the creature, as it was just about to open the barrier.\n\n\"Hey, ugly!\" she called out and stepped into the middle of the hallway.\n\nThe monster paused, but didn't turn around. It chuckled from deep in its chest, sounding like pebbles in a tin can. \"Samantha Veleska.\"\n\nThe fact the monster knew her name chilled her deeply, but she managed to force her voice to hold strong. \"I'm at a disadvantage. What's your name, big fella?\"\n\nThe creature turned and faced her full on. The heads swirled about, and Sam had difficulty trying to keep them all in sight. She settled on the two most prominent faces and sighted the pistol. It studied her for a long while. On the other side of the blast doors, she could hear the dead pressing greedily against the steel.\n\n\"My children long to taste your flesh.\"\n\n\"Yeah, well, not if I have any say,\" Sam said with a bluster she didn't feel.\n\n\"Just lay down your arms.\" The creature spread its wings regally. \"I promise it won't hurt...for long.\"\n\n\"See, that's the thing about us humans,\" Sam said. \"We don't give up so easily.\"\n\nHer mind raced as she tried to come up with some sort of a plan. The gun would merely piss off the creature, and she had no other heavy weapons. Strangely, she was at peace just then. If this were the end for her, she'd go out swinging, at least.\n\n\"My name is Azazel.\"\n\n\"Means jack to me, pal.\" Sam shrugged.\n\nAzazel turned and placed his palm against the door. The door gave way with a hiss of released air then swung inward. The sounds of the dead filled the corridor, and Sam fired as they poured through the opening. The shadows disintegrated as the bullets struck them, and the fallen dead quickly blocked the entryway.\n\nThe dead ceased their advance and stood still, so Sam ceased firing. The smell of cordite and blood was sharp in the confines of the corridor, and it bit into her lungs and eyes like acid.\n\nSam wiped her watery eyes. \"What the hell's going on? Why did they just stop?\"\n\nAzazel cried out with fury as the shadows controlling the dead puffed into clouds of debris, and the dead fell as one to the steel floor.\n\n\"No!\" the creature shouted. He withered, and he didn't seem as formidable as he'd just been a moment before. A change had overcome the creature, Sam realized, and she opened fire with whoops of triumph and hope.\n\nSam rushed the demon, shoving her pistol into one of its faces then firing. Azazel reared back and screamed in pain as that face disintegrated in a splash of gore. He swiped at Sam, and she leapt back as the claws slashed through the space she'd just occupied. The creature went to one knee and shuddered as Sam pumped the last of her bullets into its body. Azazel tried to rise, but collapsed once more. His body became transparent for a moment, and then solidified fleetingly. He roared and struggled to his feet, then turned and began flinging the dead to free up the entrance to the bunker, so that he could retreat.\n\n\"Oh, no you don't,\" Sam said, launching herself onto the demon's back.\n\nShe dropped her pistol in the process, but refused to relinquish her grip. After fumbling for the K-bar sheathed to her ankle, she plunged the blade into Azazel's trunk over and over. Black blood bathed her face and blinded her as it spurted from the jagged wounds. Azazel collapsed to the ground, and Sam cried out as her arms were crushed beneath his substantial weight. It felt as if her bones had been ground to shards of crushed glass. At this point, she was no longer in control of her limbs, and darkness threatened at the edge of her vision. The pain was excruciating. She'd broken her collarbone as a child, and that had been the worst pain she'd ever experienced, but this held no comparison.\n\nAzazel gained his feet and Sam slid off him, collapsing to the ground. The demon turned and stood over her. He leered down, and blood dripped onto her face. She gagged and struggled to breathe. Azazel raised his heads and seemed to listen to something. The roar in her head drowned out her screams, but Sam concentrated on staying conscious as the blackness washed over her in waves of numbing agony.\n\nAzazel turned his attention back to her. \"You've proven to be a worthy adversary, but I have other things to attend to. If you're not dead when I return, then I will save your death for last.\" He ran a clawed hand down the side of her cheek and stuck the tear-soaked nail into his mouth. \"Human suffering has such a succulent taste.\"\n\nWith that, Azazel turned and lumbered down to the lower level.\n\n* * *\n\nBlount heard voices as he rushed down the stairs. He saw Archer lift his smoking pistol. The lifeless body of a woman lay in front of him.\n\n\"Drop it!\" Blount yelled as he entered the room, his own gun pointed at Archer.\n\n\"You drop it!\"\n\n\"Federal Agent! Drop your weapon and get away from the girl.\"\n\n\"This is my daughter. I have no intention of leaving her side ever again.\"\n\nThe two were at a standstill, neither lowering their weapon, both as determined as anyone on the planet, both as hard and stubborn as rabid animals. There was no way either was going to budge. Blount's heart pounded, his pulsed raced. He sensed Archer's did as well. He could nearly hear the other man's heartbeat. Sweat soaked Blount's face, and it dripped from Archer's chin.\n\nHere Blount was, a dedicated and loyal man, on a mission to save the world on one side, while facing a fiercely protective parent on the other. Archer's pain had taken a physical toll on his face. On some level Blount felt it, identified with it. He knew what losing family was like. He'd experienced that loss over and over again, but the fact remained: the fate of the world was right here. Right at his fingertips. He had to make the father understand.\n\n\"I'm sorry about your daughter,\" Blount said. \"But you don't understand what's inside her, what she's carrying. She's going to end the world. You can't move her.\"\n\n\"Bullshit! We're going home. Casey, it's Daddy...come on baby, wake up. Daddy's taking you home.\"\n\n\"Stand down! I can't let you do this. Stand down.\"\n\n\"Never!\"\n\n\"Back away or I will kill you. I swear I will kill you.\"\n\nArcher slowly shook his head while watching Blount. \"What's wrong with her? Why can't I move her?\"\n\nBlount pointed his weapon in the air and held out his other hand in a placating gesture. \"Listen, let's talk about this for a second. You can't move her from the altar because she's been placed under a very powerful spell. I work for an organization called Black Rock. I'm here to help.\"\n\n\"Black Rock?\"\n\n\"You've heard of us?\"\n\n\"Yeah. I came here with one of your men.\" He looked around and shrugged. \"Well, he _was_ here.\"\n\nSounds of a struggle caused both men to peer off into the darkness to their left. Two shots rang out, followed by something heavy collapsing to the stone floor.\n\n\"One coming out,\" a voice called from the shadows.\n\n\"Palmer? That you?\" Blount said.\n\n\"Blount,\" Palmer said as he came into the pool of torchlight. \"Good to see a friendly face.\"\n\nThe two men shook hands and smiled at each other. Archer felt relief when Palmer greeted the newcomer, and he lowered his weapon.\n\nBoth men looked at Blount as he cocked his head and listened to something off to his right. They looked at each other in confusion as Blount began a conversation with the air around him. He nodded a few times and started to pace as he spoke.\n\n\"Who you talking to?\" Palmer said.\n\nBlount held up a silencing hand. \"The girls. I'm talking to the twins.\"\n\nPalmer nodded as if that explained it, crossing his arms as he watched Blount.\n\nArcher continued to search the shadows for these \"twins,\" but he couldn't see anyone. He looked closely at Blount's ears, confused.\n\nPalmer spoke in a hushed voice. \"No he's not wearing any earpiece. The twins are psychic. Two of our best operatives.\"\n\nBlount wandered over to where Casey lay and gently stroked her arm. The girl moaned at his touch, and Blount hissed then pulled back his hand swiftly. He pulled something from one of the Velcro pockets of his flight suit and knelt on the floor, then began drawing symbols and words on the ground surrounding the altar, in a widening circular pattern. Palmer looked on as the man worked feverishly.\n\nArcher walked over to him. \"What are you doing? Get away from her!\" He attempted to stop Blount, but Palmer stopped him.\n\n\"It's OK,\" Palmer said. \"He knows what he's doing. I know this man. Trust him.\"\n\n\"How can I? She's the only thing I have left in this world. She's the only reason I haven't checked out of this God forsaken universe. Christ, what is that mumbo jumbo?\"\n\n\"He's trying to help. It's gonna be OK. We'll figure it out.\" Palmer continued to hold Archer back.\n\nBlount could hear the man shifting his weight from foot to foot. Archer breathed with a slight wheeze as he waited. A little while later, Blount stood and faced the two men.\n\nPalmer asked, \"Blount? Where's your team?\"\n\n\"They're dead. They're all dead. They died on the island.\"\n\n\"I'm so sorry.\" Palmer moved to place a comforting hand on his fellow agent's shoulder, but Blount shrugged away.\n\nBlount knelt once more and drew with the chalk. He stood back up to survey his work. Quickly he stalked over to a particular cluster of symbols, scratching out one set then adding a different collection of circles, squares, and hook-like drawings. He stood once more and took a deep breath.\n\n\"OK, this should work.\"\n\n\"What?\"\n\n\"Conveyance.\"\n\nPalmer sucked in a breath and shook his head. \"Are you sure?\"\n\n\"As sure as I'm going to be.\"\n\n\"What's going on?\" Archer said.\n\n\"He's going into the in-between,\" Palmer said.\n\nBlount noticed Archer glance from him to Palmer. He knew Archer felt the unease between him and Eli. The tension oozed off him and Palmer, and it was obvious Archer wondered what was up.\n\n* * *\n\nBlount dug into another of his pockets in the flight suit and produced a dagger. The hilt was jeweled and the blade was crooked and black. The torchlight seemed to repel the blade as Blount handled the dagger.\n\n\"I don't know about this,\" Palmer whispered as he eyed the blade, like it would fly at his jugular at any moment.\n\n\"You have any better ideas?\" Blount said. \"I'm fighting a war on two fronts here. We bring it all together in one place, maybe we have a chance.\"\n\n\"What are you talking about?\" Archer asked. \"My daughter...\"\n\n\"Your daughter is the least of my worries at the moment, Agent Archer.\"\n\n\"What?\"\n\n\"If I don't stop this, there will be no world left. Get that through your thick skull. If Cthulhu enters this realm through your daughter, it won't matter whether I save her or not.\"\n\n\"But putting the body and the consciousness in the same proximity? Isn't that what the cultists wanted?\" Palmer said.\n\n\"They're not going anywhere,\" Blount said. \"I am.\"\n\nWith that, he turned and slashed the dagger through the air. Reality split open, and darkness spilled from the gash in space. Horrible sounds drifted from within, and something massive moved in the inky blackness. The darkness reached out with syrupy tendrils, and Blount allowed it to encircle his wrists and ankles. \"If I don't make it back...\"\n\n* * *\n\n\"Blount!\" Palmer and Archer both shouted as the man was yanked violently through the rip in reality. The gash closed with a painful popping, then the cavern was silent once more.\n\n\"Where'd he go?\" Archer asked.\n\nPalmer sighed and paced. He holstered his gun and ran a tired hand across his face. \"You really want to know?\"\n\n\"Yes.\"\n\n\"Our planet is covered with geometrically constructed morpho genetic grids. These grids extend from about sixty feet under the earth's surface to about sixty miles above the earth, arranged in geometric patterns...\"\n\n\"OK,\" Archer said.\n\n\"The idea has something to do with sacred geometry,\" Palmer continued. \"You've heard the expression, 'math solves everything?' Well, this is something well beyond math.\"\n\n\"So what does that have to do with where Blount just disappeared to?\"\n\nPalmer smiled and nodded. \"A man by the name of Drunvalo Melchizadek, who worked for us for a time, discovered that each species has its own grid, which supports life and connects and unifies the consciousness of that particular species. Before any species can come into existence or make a new evolutionary leap, a new grid must be completed. On the flip side, when a species becomes extinct, that particular species' grid dissolves. These old ones, Cthulhu in particular, are not part of our grid. They're not from our world. They're not supposed to have any strands in it. They're an abomination, and one reason the planet is falling apart. The whole universe could be next. Blount thinks he can find out how Cthulhu's latched onto Casey, then perhaps cut the beast loose. I hope he's right.\"\n\n\"Sixty feet below the earth's surface...\" Archer said. Understanding dawned on him. \"The in-between is the grid itself.\"\n\n\"Close enough.\" Palmer nodded. \"These grids are countless, and each one is like a silk thread connecting all the others. Blount is about to pluck some of them and see what answers.\"\n\nThe In-Between\n\nBlount drifted in the darkness and concentrated on the glowing blade. The silence was complete, enclosing his body like a hot blanket. Slowly, shapes bled into his vision. He saw Archer first. The man was surrounded by shades of red, amber, and rust. Blount concentrated on the colors and they grew pronounced.\n\nNext, Palmer's essence flitted into his field of vision. His colors were blue, violet, and dirty yellow. Blount felt himself drifting away and he gritted his teeth, reaching out to grab a tendril wafting off Archer. He grasped the filament and grimaced as the hate and fury consumed his being. He fought off the rage and fear until it was manageable.\n\nStabilized, Blount turned his focus on the altar he knew was there but couldn't sense yet. A wirework of strands encompassed his vision, and Blount shrank back from the power emanating off the muddy darkness. He glanced around and saw the cavern as it really was. No wonder the cult had chosen this location. Rugby Rock was the geographical center of the Americas.\n\nA slab of darkness hemorrhaged into existence, and Blount knew he was looking at the altar. The body on the altar was even darker than the blackness upon which it rested. In this realm, the body was a mere shape, and what rested within was the true form. The creature's main body was situated on Casey's chest, and tentacles reached off in both directions, head to foot, entangling themselves about the girl's arms, hands, and feet. In that moment, Blount caught a hint of what the body of the creature resting beneath R'yleh looked like.\n\nHe willed himself to drift towards the altar, avoiding the strands as best he could but bumping into a few. Each time he did he shuddered and fought to maintain his very being. The trek took its toll on him, and by the time he reached the foot of the altar, he was exhausted beyond measure. The creature reared its head and fourteen eyes opened all over the body. He was awash in light the color of a bruise.\n\n\"Cthulhu,\" Blount said.\n\nThe creature cringed as its name was spoke aloud, for names held power.\n\n\"Cthulhu. You are not welcome in this realm.\"\n\n_I was summoned_. Its voice was as ancient as time itself. Filled with chattering, it buzzed in Blount's head painfully.\n\n\"What do you desire?\"\n\nDomination.\n\n\"Of what?\"\n\n_Of all._ The creature strengthened its grip on the girl and slithered deeper into her chest. It roared defiantly.\n\nBlount floated closer and brandished the dagger. \"You are not welcome in this realm. Return from whence you came, demon.\"\n\nThe creature chuckled. It sounded like marbles being fed through a paper shredder. Blount drifted as close as he dared. \"Take me instead. Let the girl rest in peace.\"\n\nBlount let his gaze trail about the creature, idly following the strands of grid that emanated from its body. He would have to move fast when the time presented itself. He'd only have one chance.\n\n\"What of Azazel?\"\n\nCthulhu fixed Blount with a steely gaze, and Blount chose his next words carefully. He knew mocking a creature that could crush him with a mere thought was risky, but if there was one thing he'd learned in all his lifetimes it was that everything possessed an ego, and that could be provoked, manipulated.\n\n\"I battled him once upon a time,\" Blount said with a shrug. \"He was a great warrior, a powerful enemy. I miss adversaries like him.\" He gave a forlorn shake of his head.\n\nCthulhu reared up on the girl's chest and roared anew.\n\n\"I'm right here,\" Blount said, tensing as the creature rose up off Casey's face.\n\nThe girl inhaled abruptly and her eyes fluttered. Cthulhu paused, settled back onto Casey. It wouldn't be provoked so easily, Blount realized.\n\n\"Your followers have fallen by my hand,\" he said. \"The ritual isn't finished. There is no one to complete your awakening.\"\n\nThe creature hissed and shifted once more. With that, it sank into Casey's body and bonded with the girl's flesh. Blount rushed forward and plunged the dagger into the receding mass.\n\n* * *\n\nCasey sat up on the altar and screamed as her chest rippled in the torchlight.\n\n\"Casey!\" Archer said, as he watched something push and stretch his daughter's skin. \"He's killing her!\"\n\nArcher reached out and touched the writhing form beneath his daughter's chest then collapsed in a daze. The images that filled his head exploded like fireworks, and Archer cried out in despair as he witnessed the end of time itself.\n\nPalmer reached out to pull Archer away then reconsidered. He clenched his teeth and balled his fists in frustration. \"What have you done, Blount?\"\n\nCasey grew beyond anything humanly possible, yet her skin continued to stretch, becoming flesh-colored putty, and didn't split or tear. Four more limbs sprouted from between her legs and more sprouted from beneath her arms. As the limbs grew, they resembled tentacles. The girl's mouth opened impossibly wide, and Palmer heard her jaw dislocate.\n\nArcher flew across the room as a bright spark engulfed his daughter's body. Palmer whirled to face the new arrival.\n\nThe air was close, ripe, like a feral animal was lurking nearby. Palmer backed up until his back pressed against the earthen wall. The creature filled the cavern as it spread its wings wide. Blood and viscera coated its body like a second skin, and one of its many faces was a shredded monstrosity. It looked down at the altar and Archer next to it. It nudged Archer with its foot, but got no response. Archer stared blankly and mumbled disjointed words and phrases under his breath.\n\n\"Who might you be?\" the creature asked.\n\nPalmer recovered from his initial shock and pulled his sidearm. His aim wavered while sweat poured into his eyes, but he maintained his ground. Palmer had heard Blount speak of Azazel, and he knew this was the arrival's name. But that was impossible, wasn't it? Blount had banished the demon long ago.\n\n* * *\n\nThe air contracted about them and a hole ripped into reality like a bleeding wound. Blount tumbled out and staggered to his feet. He fixed his gaze on Azazel, then threw the dagger without hesitation. Azazel grunted , looking down in surprise and rage. The hilt shuddered in the middle of his massive chest, then the light went out of his eyes, one by one. He collapsed and clawed at the hilt, but couldn't find any purchase.\n\nThe darkness writhing in the fissure behind Blount whistled and howled, and wisps of murkiness flowed forth, entangling Azazel's limbs. He cried out and struggled against the bonds, but they slowly dragged him towards the hole. He dug his clawed feet into the floor, gouging out jagged chunks of stone. Still, he moved closer and closer to the perforation. Blount saw the twins' faces appear briefly in the gap.\n\nAzazel unleashed one last cry, disappearing through the hole as it closed with a sound that resembled a newborn suckling a teat. The cavern went silent and Blount's ears popped painfully. He collapsed on the ground and groaned. Palmer raced over to turn him onto his back.\n\nBlount felt cold and had trouble opening his eyes. They fluttered beneath his lids, detecting movement in the darkness. He could hear Palmer's voice.\n\n\"Are you OK?\"\n\nBlount's offered a weak smile as he focused on the man. \"We did it.\"\n\n\"No, _you_ did it.\" Palmer let out a relieved laugh, which he quickly stifled.\n\nThey both looked at the altar as Casey coughed and sat up. She blinked and took in her surroundings, saw Palmer and Blount and realized she was mostly naked, so she crossed her arms self-consciously then shivered.\n\n\"Who are you?\" she said through chattering teeth. \"Where am I?\"\n\nBlount wearily gained his feet and shuffled over to her side. He smiled. \"You're safe now, Casey. You're safe.\"\n\n\"Where am I?\" She recoiled from his approach.\n\n\"My name's Gibson Blount\u2014-\"\n\n\"Dad? Dad!\"\n\nArcher gave no indication he'd heard his daughter. He was rocking back and forth, cradling his head in his hands now. His fingers clutched his hair and pulled it out in bloodied clumps. Casey tried to get off the altar, but her legs collapsed beneath her. Blount caught her in his arms and stroked her hair, trying to calm her down.\n\n\"Palmer! Give me your jacket,\" he said, wrapping the jacket about the girl's shoulders.\n\n\"She's in shock,\" Palmer said. He whirled and pointed his gun at the stairs as a figure stumbled down them. \"Samantha!\"\n\nHe rushed over and she collapsed into his arms. Palmer grimaced as he took in the extent of her injuries. Her arms flopped helplessly and bits of jagged bone jutted from various points along her shattered limbs. She rolled her eyes up at him and coughed blood. Palmer gently lowered her to the floor and cradled her head in his lap. \"Shhhh,\" he said as she moaned incoherently. \"It's all over.\"\n\nBlount disentangled himself from Casey, but she reached for him with a wild look in her glassy eyes. \"It's OK, I'm just going to check on her. I'm not leaving you.\"\n\nHe walked over to Palmer and knelt down beside Sam. \"Who's this?\"\n\n\"She was with Archer and I. The pilot. Azazel did this to her.\"\n\n\"Hey,\" Blount said to Sam. \"You really did a number on him, didn't you?\"\n\n\"I did my best.\"\n\nBlount placed his hands on hers. \"This is going to sting a bit, but it'll be over quickly.\" He didn't wait for a response before blue light crackled between his fingers and melted into Sam's body. She stiffened and gritted her teeth as her bones knit themselves in the space of seconds. They receded into her skin and Palmer gasped in awe as he watched.\n\nSam sat up and examined her arms in wonder. \"H-h-how did you do that?\"\n\nBlount winked. \"Magic.\"\n\n* * *\n\nCasey blinked as she came out of the bunker. She squinted in the glare of the early morning light. Palmer came next. He supported her father as he shuffled alongside. Sam and Blount stood next to Casey as she took in the horizon. She stood for a long while, unable to say anything, then:\n\n\"What happened?\"\n\nBlount looked at her for a moment, then out at the blazing, nuclear-scarred horizon. \"Some bad things happened while you were away, Casey, but we'll be OK. The worst of it is over. Mankind is resilient. Trust me, I know.\"\n\n\"Where will we go?\" Sam asked.\n\n\"I've got a few places in mind. There'll be no fresh shadow creatures now that the void is sealed, but there's still a lot of them out there. I've got a feeling we'll be busy for awhile yet.\"\n\n\"What's going on? Please, someone tell me.\" Casey said, exasperation saturating her voice. \"I feel like I've been in a coma for years. I dreamed someone was suffocating me.\" She looked back at Archer. \"What's wrong with my father?\"\n\nPalmer spoke. \"He's been through a lot, honey. We all have. But he never stopped looking for you. My God... this is going to take a while to explain.\"\n\n\"Let's start with: the world has changed,\" Blount said. \"It's nothing like you remember it.\"\n\nHe turned and studied Archer closely. The man had ceased his ranting and looked in Blount's direction. His gaze was clear and aware. Blount thought he saw something there, lurking just beneath the awareness, something else that oozed darkness and cunning. The sensation passed as quickly as it had appeared. Blount's eyes narrowed and his heart skipped a beat. He wasn't sure he'd seen anything, but nagging doubt filled his head. He looked at his little band of rag-tag survivors.\n\n\"Let's move out.\"\n\n# Part IV\n\n# \"Oblivion Blooms\"\n\n#\n\n# Chapter One\n\nNebraska\n\nIt was four days before they came across the first survivor, and Palmer almost shot the man. They'd landed at a rural airport to scavenge for fuel and supplies. Palmer and Sam roved the perimeter while Blount and Casey fueled the chopper. Archer had improved in increments, but he was far from recovered. Sam had mentioned that his symptoms resembled Alzheimer's, and Blount had agreed. Blount had tried using some shamanic magic, but it'd had no noticeable effect. Archer had moments of clarity, but those moments had been few and far between.\n\nBlount heard Palmer shout, and he pulled his gun, spotting the man a few hundred yards away at the doorway of one of the outbuildings. He saw another man emerge from the building with his hands raised high in the air. Blount finished filling the chopper and the reserve cans.\n\n\"I'm going to see what's up with Palmer,\" he told Casey.\n\nShe looked from Blount to her semi-catatonic father, and Blount could see the indecision in her face.\n\n\"It's going to be OK,\" he said. \"You can see me from here and you're armed. Shout if something happens, and I'll be right by your side in a flash.\"\n\nIn the preceding days Blount and Sam had become like second parents to Casey. She looked to them for guidance and comfort, and they were happy to provide it. Human contact took on a new meaning since the world was plunged into chaos. Blount was thankful to have companions. He'd gone it alone before and it'd been hell. At the same time, though, he wished he _were_ alone. Being responsible for someone else weighed heavily on his mind, and his decisions were second-guessed in spite of his best efforts to be cold and calculating.\n\nHe approached Palmer, Sam, and the man warily. The man seemed normal enough, but he was taking no chances. \"Palmer.\"\n\nPalmer nodded in Blount's direction and said, \"Blount. This here is Mr. Paul Tubule.\"\n\n\"Mr. Tubule,\" Blount said. \"What's your story?\"\n\nThe old man eyed them suspiciously. \"I'd be a might touch more comfortable if your man would stop pointing that gun at me.\"\n\n\"I'm sorry about that, but I think it's necessary for the moment.\"\n\nThe man sniffed and crossed his arms over his narrow chest. \"Fine. I used to be in charge of this little airport, well, before, you know...\" He shrugged. \"Guess I still am. I had to kill my best friend, so I'm in a foul mood, to say the least.\"\n\n\"I'm sorry to hear that,\" Blount said. \"We've all had to do some things we'll regret.\"\n\n\"You can say that again.\" The man's laugh held no mirth. \"I'm friendly enough, though. And I don't mind saying it's nice to see some people with no shadows tagging along.\"\n\nBlount nodded at Palmer and he lowered his weapon. He kept it ready at his side.\n\n\"Thank you,\" Tubule said. \"I ain't got much, but you're welcome to what I have, Mr...\"\n\n\"Gibson Blount. This is Palmer and Veleska.\"\n\n\"Glad to make your acquaintances.\"\n\nBlount tensed as he heard voices drift from the building's doorway, behind Tubule. Tubule noticed the tension and said, \"Nothing to worry about. That's just my ham radio. Seems to be the only line of communication left now. There's been a lot of chatter. People been reaching out just to make sure they're not the only ones left alive.\"\n\nPalmer perked up at this, and moved closer. \"We haven't heard any news for awhile now.\"\n\n\"I'm feeling pretty exposed,\" Blount said. \"How 'bout we secure the chopper and get inside?\"\n\nSam nodded and started for the chopper. Casey held a hand over her eyes to shield them from the late afternoon sun. She smiled as she saw Sam approaching. Sam waved and returned the smile. Casey was a good kid. It was amazing to see how fast she'd bounced back from being kidnapped by a psychotic cult and possessed by an ancient god. Sam wasn't sure if she would've been able to cope as well. She'd nearly been driven mad by that dreadful chattering. At the time she hadn't been aware just how close she'd come to cracking. Sam's smile slipped as her thoughts darkened.\n\n\"What's wrong?\" Casey asked.\n\nSam forced another smile and shook her head. Kid was observant. \"Nothing.\" She began to secure the chopper. \"We're heading inside. That guy over there has some news, and Blount wants to stay and chat for a bit.\"\n\n\"Samantha,\" Archer called as he stepped from the chopper.\n\nIt took Sam a few moments to recover her poise. \"Archer.\"\n\nArcher encircled his daughter in his arms and Sam saw that Casey tensed, but allowed her father to hug her from behind. Archer leaned his head down and kissed the top of her head. He must've noticed his daughter's rigidness, because a frown creased his face. He looked over at Sam. \"What?\"\n\n\"How are you feeling?\"\n\n\"Tired.\" He sighed. \"Drained.\" Frowning, he released his daughter. \"To be honest, I'm missing some time, I think.\"\n\nCasey turned and looked at her father. Sam's heart clutched as she saw the concern and fear on the girl's features. \"Dad, you've been out of it for almost four days. W-w-we...I was so worried about you.\"\n\n\"Oh, baby.\" Archer choked as he pulled her into a hug once more. \"I'm so glad you're all right. I was so worried about you.\"\n\nSam walked over and placed a hand on Archer's shoulder. \"Good to have you back.\"\n\nAs Archer and Casey walked away, Sam crossed her arms and frowned. Archer's recovery felt too convenient. She'd noticed the haunted glaze come over his eyes when he claimed to be missing time. He was hiding something\u2014she could feel it deep in her bones. What he was hiding she wasn't sure, but she made a mental note to discuss her suspicions with Blount as soon as the two could be alone.\n\nOn the subject of Blount, her mind was a battlefield of emotions. Blount was an enigma and that troubled her too.\n\n* * *\n\nPaul Tubule led Blount and Palmer into the building and over to a spot hidden behind some crates and boxes. With a groan, he bent down and pulled up a trap door notched in the floor. A light flickered on as the door was raised, and Blount could see a set of stairs leading down. Tubule led the way, while Palmer cleared his throat uncomfortably. He muttered something under his breath, and Blount looked back.\n\n\"What was that?\"\n\n\"Nothing,\" Palmer said. \"Just didn't think we'd be heading back underground so soon.\"\n\nTubule laughed loudly and called over his shoulder. \"It isn't much, but it's safe. There's a front coming, and we're going to appreciate this cellar when that fallout shows up.\"\n\nAs they approached the bottom of the stairs, Blount heard the voices coming from the ham radio much more clearly. He had very little experience with this type of communication, so he wasn't sure what to expect. \"Wouldn't the EMP's from the blasts have knocked out radio communications?\"\n\n\"Yes and no,\" Paul said. \"Call me paranoid, but I put my ham in a lead-lined safe before the launches.\"\n\nHe pushed open a steel door and waved them through.\n\nBlount was impressed with the room. It was about the size of a spacious bedroom, and the entire far wall was stacked floor to ceiling with supplies of can goods and other dry goods. He saw five assorted firearms against another wall and boxes of ammo as well.\n\nA well-worn couch was against another wall and a small refrigerator hummed next to it. A side table was stacked high with books. On the opposite wall, the ham radio rested on a table.\n\n\"I'm impressed,\" Blount said as he surveyed the room. \"Where's the electricity coming from?\"\n\n\"Generator up top.\"\n\nPalmer glanced around, appearing satisfied there were no surprises waiting for them, and that Tubule was on the up and up. \"I'll go and check on the girls.\" He headed back up the stairs.\n\n\"What have you been hearing on the radio?\"\n\n\"Lots.\" Paul grunted. \"It isn't pretty out there, Mr. Blount. Whatever happened was very, very bad. I talked to a guy in Canada who told me he saw fires burning so hot that the asphalt in the streets melted and burst into flames. He was in a secure, clean room at a traffic control building and watched it all on the security monitors, safe from the airbursts and initial nuclear blasts. People were trying to run, but they were literally melting into the pavement as they ran. There was a guy, on fire, that jumped into a fountain, only to catch fire again when he surfaced for air.\"\n\nTubule settled himself on the couch and sighed with relief. Blount walked over and looked at the books on the table. He scanned the titles and realized they were all about nuclear radiation, fallout, and survival. There were a few zombie novels as well. Blount glanced at a few of the titles and grimaced. _Elements of the Apocalypse, Valley of the Dead, The Undead, The Rising, Dead Sea, Entombed_.\n\nTubule must have noticed his reaction to the books. \"I grabbed those from the library.\"\n\n\"And the zombie novels?\"\n\n\"Those are for research as well...\" Paul grinned.\n\n\"Did you hear which cities were hit?\"\n\n\"All the major ones...for the most part.\"\n\n\"What about D.C.? The government?\"\n\n\"Ain't no leadership left. The evacuation was a disaster. Streets were gridlocked, and nobody knew what the heck was happening. Think Katrina times twenty.\" Paul gave a disgusted sigh. \"Ain't no government to put anything back together, but then, did you really expect them to know their elbows from their asses? A few days ago, I came across some chatter on the ham. A group of militia guys were claiming to have found the president in some helicopter wreck. Said he shot himself. Tell you the truth, I don't blame the guy if that's what he did.\"\n\n\"Where?\"\n\n\"What?\"\n\n\"Where did they find the wreck?\"\n\n\"They didn't say, but it sounded pretty legit, if you ask me.\" Paul shrugged.\n\nHe glanced up as Sam, Casey, and Archer entered the room. Blount stared at Archer with raised eyebrows. Archer crossed the room and clasped Blount's hands in his own.\n\n\"I want to thank you for all you've done.\"\n\nBlount took his hand from the other man's crushing grip and recovered from the shock of seeing Archer healed and coherent. He glanced down at his hand and was sure he'd see it encased in ice, as cold as it was. Blount stared hard at Archer. Something wasn't right, but he couldn't put a finger on it. He glanced over at Sam and she shrugged. So she'd felt it too, Blount realized.\n\nPaul spoke up, \"Agent?\"\n\nBlount turned and faced the man and nodded. \"I work...worked...for a branch of the NSA.\"\n\n\"I see.\" Silence hung heavily in the room for a while, and it appeared Paul expected Blount would elaborate. When he didn't, Paul continued. \"How much of what's happened did you know about?\"\n\nBlount fixed the man with a cold stare. \"I know a bit.\"\n\n\"Look, I've opened my home to you. Offered you what I have. The least you can do is tell me what the hell happened these past few days.\"\n\n\"Fine.\" Blount sighed. \"A cult kidnapped this man's daughter.\" He pointed to Casey. \"And used her to call forth an ancient god. An island that's been buried beneath the sea for millennia rose, and its displacement of the ocean flooded most of the west coasts of the Americas, and pretty much every other island across the globe. In this process, the cult accidentally opened a portal to a void and those creatures came out.\n\n\"Nukes were launched at us, either under the control of the shadow creatures or panicked nations like North Korea and China, and we responded by launching our own nukes.\"\n\nPaul looked at Blount for a long moment. \"You serious?\"\n\n\"Yes.\"\n\n\"Well, I'll be. I ain't much of a spiritual man, but I'm pretty sure I came face to face with evil the past few days.\"\n\n* * *\n\nBlount decided to forgo a watch, since they were all well past the brink of exhaustion. None of them would've been much use even if they'd tried to keep a lookout. They all bedded down for the night in Paul's cellar and slept the sleep of the dead.\n\nBlount should've fallen right to sleep. He'd been up for nearly six days now, but he found sleep elusive. He ran the last few days' events over and over in his mind, and he couldn't help but feel that he was missing something\u2014something vital. He was sure of it. No matter how he shifted the pieces in his mind, nothing seemed to fit, or produce that flash bulb of recognition.\n\nArcher's speedy recovery was the main thing troubling him. What had snapped him out of his stupor? And why? When Archer had shaken his hand, Blount had felt a deep coldness coming off the man. He'd made a pretense to get next to the guy earlier and brushed his hand on Archer's arm, but there was nothing there to glean. He should've been able to read something off the man, but it was as if there was nothing there\u2014a void of nothingness. Blount tried to dismiss it as the after effects of what Archer had been through, but still, it didn't sit entirely right with him.\n\nHe slipped from his sleeping bag as quietly as he could and eased out of the room. After trudging up the stairs, he pulled a rumpled pack of stale cigarettes from his flight suit. He hadn't smoked in years, but when he'd discovered Paul had cartons upon cartons stashed away, the urge was back.\n\nHe stood just inside the doorway of the building and lit his cigarette. The smoke hung about his head in the still, night air. He scanned the outskirts for any movement or threat, but all appeared to be calm.\n\n\"Got another one?\" a voice asked behind him.\n\nBlount whirled and pulled his weapon. The act was so fast and fluid it caught the new arrival by complete surprise. Archer stumbled back and swallowed with difficulty as he stared down the barrel of Blount's pistol. He raised his hands in placation.\n\n\"You just about got shot yourself, friggin' idiot.\" Blount eased up on the trigger and lowered his weapon. He was pissed at himself. How could he have not heard Archer coming up from behind him?\n\n\"I'm sorry. I didn't mean to startle you.\"\n\nBlount grunted and shook another cigarette from the pack, then offered it to Archer. The man took it and leaned in while Blount flicked the lighter. Archer inhaled deeply and sighed smoke out his mouth and nostrils.\n\n\"Ahhhh, now that is nice. It's been awhile since I quit, but don't see no point in stopping now, uh?\"\n\n\"What do you remember?\" Blount asked.\n\n\"What?\"\n\n\"What do you remember?\"\n\n\"What do you mean?\"\n\n\"When you touched Casey on the altar...something happened. What was it?\"\n\nArcher took a deep breath and dragged deeply on his cigarette. The glow played tricks with his face. He locked eyes with Blount and shrugged. \"I don't remember a whole hell of a lot, actually.\"\n\n\"Tell me.\"\n\n\"I can't tell you what I don't remember.\"\n\n\"Something's been bothering me about you, and I can't put my finger on it. I don't like that feeling.\"\n\n\"Look, I appreciate what you've done for my daughter...she'd be dead right now if weren't for you...but I think it might be best if we part ways.\"\n\n\"You're not going anywhere.\" Blount still had the gun down by his side, but he brandished it to show his seriousness.\n\n\"Look, man...\"\n\n\"What's going on?\" Sam said as she came up the stairs. \"Something wrong?\"\n\nThe two men started as if they'd been caught doing something untoward. Archer flicked his cigarette out into the night and shoved past Sam as he headed back down the stairs. She watched him go and frowned back up at Blount. \"What'd I just walk into?\"\n\nBlount's brooding gaze was his only response.\n\n# Chapter Two\n\nThe twins came to him in a dream. He found himself standing at the edge of a precipice. There was darkness all around and it writhed and cavorted. The darkness possessed a tangible presence that pressed against his face and body with a changeableness that felt wet and squirmy. A childhood memory welled up, and he recalled playing with night crawlers\u2014the way they'd felt as they'd slid through his fingers. What pressed against his body and face now felt the same. He felt his disgust rise and he quickly swallowed.\n\nThe twins stood on either side of him. Nina held his right hand and Maya his left. They gripped him tightly as an unseen wind welled up from below and buffeted him. His shirt flapped in the gusts, sounding like mini-gun shots in the silence. He looked at the twins on either side of him and they offered smiles of reassurance\u2014though the smiles didn't reach their eyes, where concern and fear was clearly evident.\n\n_You're not finished_ , Maya whispered in his ear.\n\n_There's still work to be done_ , Nina said.\n\nBlount pondered what they were telling him. He felt it too. Something else was coming. \"Tell me what to do, girls.\" His words seemed to be ripped from his mouth then flung out over the chasm in front of them. The twins began to fade, and he clutched at their hands as they became less corporeal. Then they were gone.\n\nBlount stepped back from the edge and strained to see in the darkness. He found he could see vague shapes and a scarred landscape, as if it were a cloudy night. He began to walk. The act of placing one foot in front of the next was made purely on faith and hope, because he couldn't see the ground. His next step could be off the other side of the cliff for all he knew. Yet, he felt compelled to keep walking. On an intellectual level, Blount understood this was a dream, but he was filled with doubt, believing it was completely of this world.\n\nHe gradually became aware of a hammering coming from directly in front of him. It sounded like something or someone was smashing two rocks together. Eventually he saw the vague outline of a figure hunched over. Rhythmically, the figure raised a fist-size rock and brought it crashing against the side and top of its head. _Smack, smack, smack_.\n\n\"Hey!\" Blount called out as he approached.\n\nThe figure paused, but didn't turn to see who had called out. Once more, it continued to smash its head with the rock. Blount circled, keeping his distance, until he realized it was Simon Baxter. The man's eyes were blood-filled holes, and specks of skull flashed in the darkness as he raised his head to peer at Blount. The man offered a gore-stained smile.\n\n\"Blount,\" he said as he hit his head with the rock. \"Nice to see you again.\"\n\n\"Baxter. What are you doing?\"\n\n\"Oh, this?\" The man chuckled and hit himself in the mouth with the rock. He spat teeth onto the ground and swallowed, then groaned with pleasure. \"You're not finished. You left us all there to die.\"\n\n\"I'm sorry.\" Blount sighed.\n\n\"Sorry? Sorry!\" Blood spattered from his mouth and dripped onto his chest and arms. \"I had a premonition that day, but I ignored it because I believed you would keep us safe.\"\n\n\"I did what I could.\"\n\nBaxter hit himself with the rock and laughed. \"But it wasn't enough.\"\n\n\"No!\" Blount cried and moved to stop the man as he raised the rock once more, gripping it with both hands.\n\nHe brought his head forward as he brought the rock up. The blow was strong, and Baxter's head caved in like an eggshell. The man slumped over and sighed one last time.\n\nBlount dropped to his knees and sobbed as he watched the body begin to liquefy, decomposing before his eyes. In a matter of minutes, nothing was left but slivers of bones and a few scraps of clothing.\n\nSounds of smashing drifted towards Blount. He rose to his feet and turned to walk back the way he'd come. He knew what he would find if he continued and he had no stomach for it.\n\n\"Yeah, keep walking,\" a voice called out. \"Leave us like you did the last time!\"\n\nBlount ducked his head and trudged on. He was back at the precipice a few minutes later. He could see farther now, and he saw the island in the distance. A mass of water was between him and the island, and the surface was a churning mess. Abruptly, someone grabbed his hand and he whirled to face them. Max Anders, the theologian, his face a study of anguish and horror, stared back at him.\n\n\"It's all wrong, Blount,\" he cried. \"It's all wrong.\"\n\n\"What's wrong?\"\n\n\"Everything. There is no heaven, no hell, only this place.\"\n\nBlount hugged the shivering man to his chest. \"You're absolutely right.\"\n\nThe man sobbed and shook against him. \"You're not finished. There's still work to be done.\"\n\n* * *\n\nBlount woke up violently. He stumbled to his feet and looked around in a panic. He was bathed in a cold sweat and his body shook as if he were in the fits of a powerful fever. His heart rate gradually returned to normal as he took in his surroundings, realizing he'd fallen asleep in the doorway of the shelter.\n\n\"That sounded like some dream,\" someone said.\n\nBlount whirled and saw Archer smoking. He relaxed a bit and sighed heavily. \"I get them sometimes.\"\n\nArcher nodded. The sun was just beginning to peek over the horizon. The haze in the air colored it a deep purple. The sky was a maze of cracked glass and smoke. It resembled an angry storm front in some ways.\n\n\"That fallout's getting closer,\" Archer said. \"Day or two and it'll be here.\"\n\n\"What do you remember?\"\n\n\"This again?\" Archer sighed. \"I told you: I can't remember anything. It's a gap in my memory. A blank.\"\n\nArcher shifted uncomfortably under Blount's intense gaze. He ran a hand over his closely shaved scalp and shuffled his feet. He tugged at a sleeve and averted his eyes. The man was holding something back he was certain now.\n\n\"Morning, gentlemen,\" Paul said as he lumbered up the stairs.\n\nHis voice sliced through the tension between the two men, and the air perceptively eased. Paul slipped past them and walked over to the side of another building. \"Gotta empty the bladder, excuse me.\"\n\nJust after he unzipped, he slumped forward, and his head struck the building with a crash. His pants slipped around his ankles and his brains slid down the wall, where it mingled with his piss.\n\nBlount dropped to a crouch and pulled his weapon. Archer was less than a second behind him, crouching as well. A report echoed across the air a second later. They both retreated farther into the building.\n\n\"Where'd the hell that come from?\" Archer said.\n\nBlount stilled his mind and reached out with his senses. He closed his eyes and concentrated. A flurry of voices, very faint, washed over him. The sounds were coming from the east. He concentrated harder and sensed the soft footfalls of at least a dozen men. They moved in groups. One would move forward, while the other group stilled. This was repeated a few times. They were moving to flank their position.\n\n\"There's at least a dozen of them,\" Blount said as he opened his eyes and looked at Archer. \"From the sounds of it, they were trained. Maybe military.\"\n\n\"How the hell do you know that?\" Archer appeared suspicious.\n\n\"I just do, damn it.\" Blount gained his feet and hurried to the stairway, then looked back. \"You coming?\"\n\nThey moved down the stairs and Blount slipped down the trapdoor. It would take whoever was coming a few minutes to discover the secret entrance\u2014he hoped. He swung the steel door shut and sealed them in.\n\n\"What's going on?\" Sam asked as they entered the room. She noticed the looks on their faces and she sucked in a breath. \"Where's Tubule?\"\n\n\"He's dead,\" Blount replied.\n\n\"What happened?\" Palmer asked as he rushed forward.\n\n\"'Bout a dozen of them at least,\" Blount said. \"Sniper got Paul. It was far off. He was dead before we even heard the shot.\"\n\nCasey rushed forward and embraced her father, burying her head against his chest and weeping softly. They heard the scrape of boots on the floor above them. They tensed and looked up at the ceiling. The voices were muffled and indistinct, but frustration and anger could be heard in the inflections. This went on for quite sometime, and then all was quiet once more.\n\nBreathing softly, the group waited for something more to happen. After an hour of silence, Sam whispered, \"Think they left?\"\n\nBlount shook his head. \"No, they're waiting us out.\"\n\n\"Who do you think they are?\" Palmer asked.\n\n\"Militia, rouge military unit? Could be anybody,\" Blount said with a shrug. He stood and surveyed their supplies. \"There's a fallout front headed our way. We've got enough supplies for a few weeks at least...if it comes to that...and we ration it carefully.\" He paused and sniffed the air. \"You smell that?\"\n\nArid smoke drifted into the room from the air intake in the corner of the ceiling.\n\n\"Crap,\" Archer said. \"They're trying to smoke us out.\"\n\n\"It's what I'd do,\" Blount said.\n\nThe smoke grew denser and they began to cough in the haze.\n\n\"Palmer, Archer: on me.\" Blount took up position at the door. He spun the latch and pushed the door open a few inches. Fresh air wafted in and Blount gulped at it greedily. When no shots came careening down the stairs, he pushed the door open farther, slipping through then motioning for Archer and Palmer to follow.\n\nThe top of the stairs was empty, but a hearty glow filled the space. Heat coursed down and slapped Blount in the face like a physical presence. Embers floated in the air, and some landed on his arm. He hissed in pain and patted them out. He powered on and ignored the pain and burns. Flames licked up the building's walls as he made it topside.\n\nThe roof was crackling, and, at any moment, the entire structure would collapse. He felt Palmer's hand touch his shoulder, signaling he was right behind. Blount rushed out of the burning inferno. He stayed low and scanned the area as quickly as he could. In front of him was clear, so he made for the next building. He slammed up against the wall and covered Palmer and Archer as they exited the building.\n\nArcher slid in next to Blount. \"I saw bodies in the building.\"\n\n\"What?\"\n\n\"There were at least two, maybe three, bodies in the far corner.\"\n\nThe building collapsed with a _whumping_ sound. They shielded their faces with their arms as the searing heat slammed into them.\n\n\"Casey!\" Archer cried.\n\n\"They'll be fine,\" Blount said. \"The door seal is airtight.\"\n\nBlount whirled and faced outward as a man came sprinting in their direction. He was screaming and covered in blood and gore. He fired his automatic weapon wildly, bullets chewing up the ground in front of him. Blount scrambled as bullets clanged into the building. Bits of metal tore into his face as he dove out of the way. Archer fired and dropped the man.\n\nBlount gained his feet and turned, spotting Palmer leaning against the building. He looked down at his stomach in surprise and confusion. Blood poured through his clenched fingers, and the wall behind him was awash in red. Palmer coughed and bright red blood filled his mouth. Blount saw the starkness of the blood and air bubbles, and he knew Palmer had taken rounds to the lungs.\n\nSomething stalked around the side of the adjacent building. Archer yelled out a warning as the creature pounced on the man that had been firing. The creature resembled a cougar, except it was much, much larger. It was covered in scales and its eyes were all wrong\u2014black and shimmering. The monster clutched the man's head in its jaws. Blount saw the terrified man looking at him, and realized the man was still alive. The man screamed as the creature flexed its jaws. His head began to elongate with the pressure, and then it split. His eyes bulged then popped out in a gush of blood.\n\nArcher fired at the creature, but the rounds bounced off its tough scales. Blount rushed forward and grabbed the automatic rifle from the ground, opening up on full auto. The 5.56 mm rounds dug into the creature, and it roared in fury and pain as black blood spewed from its sides and legs. It collapsed beneath the hail of gunfire then went still.\n\nBlount scanned the area and didn't see any more threats, so he tossed the gun at Archer, who caught it deftly. Archer swept the area as Blount rushed over to Palmer's side.\n\nHe knelt and gently shook the man. Palmer's head lolled to the side, and a stream of blood leaked out from between his lips. His eyes stared vacantly ahead, while Blount let out a cry of frustration as he realized Palmer was dead. The senseless loss of life enraged Blount, and he took it out on the building by kicking and punching the wall. As his fists pounded the steel siding, he reveled in the pain and broken skin.\n\nHe slammed his bloodied fists into the wall until he lost all feeling in them. Archer stood by and remained silent. He obviously shared Blount's frustration, and knew to allow the fit to continue uninterrupted. If he tried to intervene, Blount would've turned his rage on him instead of the wall.\n\nBlount finished and wiped his mangled hands on his chest. He looked over at Archer and nodded. \"Let's go dig them out.\"\n\n* * *\n\nIt was several hours until the heat dissipated enough for them to get close. Blount spent this time in deep mediation while Archer kept watch. Blount reached out for the twins, but didn't get a response.\n\nHe went deeper and began to construct the temple on the island. Walls appeared in the darkness and, one by one, they fell into place as he reconstructed from memory. Next, he moved inside the structure, where details began to solidify. When he finished, he walked around, surveying what he'd made.\n\nHe'd learned the technique of deep constructional meditation from a tribe in the Amazon. The process was dangerous because it took great mental power, and if the person was interrupted during the later stages, it was possible to become trapped in the memory construct. The mind would permanently separate from the body. He'd warned Archer of this, and the man was under the strictest of orders to ensure Blount wasn't disturbed, no matter what happened. Blount was certain he'd missed something while on the island, something that would defeat Cthulhu once and for all. This was the only way to go back and be sure.\n\nAs he stood in the middle of the room he'd been in a mere ninety-six hours before, he slowly turned in a circle. This was the place he'd lost most of his team. He avoided picturing their faces, but their screams echoed in his head. He worked to block those out as he remembered what he'd seen.\n\nNoticing a chasm in the middle of the spired room, he walked towards it. The crack in the floor was about eight feet in length and four or five feet in width. He peered over the edge, careful of his balance even in the memory place, and a stench wafted up. Grimacing, he held his breath. The darkness below was absolute. Blount thought a torch into being and tossed it down the hole.\n\nThis was where things would get tricky. He was so deep in his mind now that there was no safe way of coming back quickly. He was present on the island now in spirit. No longer a memory, his being was in the actual location. Astral travel.\n\nThe torch careened off the sides of the chasm and he lost sight of its glow. He moved out over the gap and began to float downwards in a controlled descent. Reaching out his hands, he found he couldn't feel the sides as he went lower and lower into the pit. Feeling was a relative term in this state, but he could feel just the same as if he were physically present. He could be hurt, even possibly killed in this state, though he'd never actually seen proof of this.\n\nHe'd only attempted this form of meditation a few times, and this time was no different. The uneasiness settled over him like a wet blanket. When he used the dagger to travel along the grid in the in-between, he felt more at ease. This was something different, because his body was left vulnerable and unprotected. It all came down to control and Blount loathed giving it up. Though he still harbored doubts about Archer, Blount had felt had a little choice but to deploy this method. Put boots on the ground, so to speak.\n\nBlount sensed something massive just below. He slowed his descent and strained his eyes to discern the dark shape that was denser than the nearly complete blackness.\n\nThe vision he'd experienced before of the giant, monstrous form flashed through his mind once more. The entire city rested upon the beast itself; a mountain of bloated flesh and tentacles; a patchwork of gray and green flesh rippling with networks and colonies of sea life and organisms forming one body, one shell. It slept...soulless.\n\nThe sides of the pit fell away from him, and Blount sensed that he had entered an endless cavern. Here and there smaller versions of the bloated mountain of flesh scurried about, as if they were ants disturbed from their hill. At the extreme edges of the form he could hear water lapping. The air was ripe with decay and death. Blount grimaced as an unseen wind changed and he was hit full on with the gruesome taint. The smaller creatures emitted a faint bluish glow, and from this height he could just make out the outer edges of the creature. The tentacles were the sizes of the biggest Redwoods in existence. The power that lurked within this creature's form was incalculable. Abdiel had been right. This _was_ a destroyer of galaxies.\n\nThe enormity of his task hit him full force like a speeding freight train. If Cthulhu's soul had managed to reconvene with this form...Blount shuddered at the thought. He flew along the form, which took nearly an hour to transverse end to end. Blount found himself wondering why God would've created such a creature. He recalled a mental exercise he'd had with a skeptic once: Could God create a stone so big even He wouldn't be able to move it? Blount had laughed at the absurdity of the question. Why would God do something so human? he'd countered.\n\nBlount had seen much during his time, and he was sure there was something out there, something that started it all, whatever name mankind decided to call it. He'd encountered many gods, demons, and angels, and one thing they all had in common was the belief that there was some Mega Being somewhere in the multi-verse. Myth? Perhaps. Or maybe it was the boogeyman created to keep the weaker in their place; a story to tell around the campfire to ensure no one being thought they had achieved ultimate power?\n\nWhat Blount couldn't reconcile was if God did indeed still exist, why would He create such a creature, or allow it to be brought into being, if He didn't create it? What if this was God? The thought struck him as ludicrous, but still it lingered.\n\nHe was but one mere man that had been blessed with ultimate, almost unlimited, knowledge. Being next to this creature below him made him realize just how insignificant he really was. How insignificant mankind really was in the scheme of it all. If this were God, then he'd have no qualms destroying it, once and for all. An idea began to slowly take root in his mind as he traversed the massive form once again. There was only one thing that would destroy an entity of this magnitude.\n\n# Chapter Three\n\nArcher gingerly tossed shards of debris as he cleared the wreckage. The wood was warm to the touch, and he wished he had a pair of work gloves. He turned and glanced back at Blount then scanned the horizon. Blount looked at him thoughtfully.\n\n\"Welcome back,\" Archer said.\n\nBlount slowly gained his feet and stretched the kinks from his limbs. He surveyed the progress Archer had made while he was in trance and nodded approval. Blount walked over and helped clear the debris. With both of them working they made swift progress. Shortly, the trapdoor emerged from the smoldering rubble. Banging sounded from beneath the charred wood.\n\nArcher dug faster. Clouds of soot and ash puffed up into the air, but he held his breath and persevered.\n\n\"I'm coming, Casey!\" he called as he threw fire-kissed ruins to his right.\n\nThey removed the final pieces and the trapdoor swung up. Blount smiled as he saw Sam's face peering up at them.\n\n\"What the heck happened?\" she asked in way of greeting.\n\nCasey and Sam emerged from the bunker and blinked in the dying afternoon haze. The two surveyed the damage and Sam cursed softly at the ruined remains of the building. She saw Palmer and Paul Tubule laid out side by side, which made her bring a hand to her mouth and gasped.\n\nCasey ignored the corpses and wandered over to a clearing. She shrugged her father's comforting hand from her shoulder and shook her head. Archer's shoulder's slumped at the rejection, but he remained nearby, watching his daughter's silent, quivering breakdown.\n\n\"What happened?\" Sam repeated once she and Blount were away from the others.\n\n\"I think the fire was set by humans, but they were interrupted and attacked by something else.\" Blount gestured at the black creature lying a few feet away.\n\nSam cautiously approached the slain monster and studied it with awe and fear. She could only imagine the terror the creature presented when it was alive. \"This thing is hideous.\" She walked over to the two bodies, then shot Blount another questioning glance.\n\n\"Tubule got taken out by a sniper...\" Blount nodded at Palmer, took a few deep breaths, and studied his hands.\n\nSam placed a gentle hand on Blount's shoulder then squeezed softly. Blount appeared uncomfortable at first with the contact, but something in his eyes let Sam know that, even though he was tensing, he welcomed the contact just the same.\n\n\"Palmer was eliminated during the ensuing crossfire.\"\n\n\"I'm so sorry, Blount,\" Sam whispered. She moved to hug him, and he returned the hug with a fierceness that temporarily crushed the breath from her.\n\n\"So, what now?\" Archer said.\n\nThe two broke the embrace and established a small gap between them. Archer smirked at their awkwardness.\n\n\"We're going back to the island,\" Blount replied.\n\n\"What?\" Sam said.\n\n\"What?\" Archer echoed.\n\n\"But we're going to make a slight detour first,\" Blount said.\n\n* * *\n\nA few hours later, Sam and Archer had placed the last of the supplies into the chopper. Blount stood a few feet away and stared into the distance. Sam had pressed for the details of his plan, but he'd remained tight-lipped. He still had that gut feeling about Archer, and felt the less they all knew the better.\n\n\"We're all done.\" Sam's voice intruded on his thoughts.\n\nBlount nodded and turned to face her. He offered a reassuring smile, which she didn't return. She merely stared back, showing no emotion.\n\n\"Where we headed, Blount?\"\n\nHe noted the use of his last name, then sighed and ran a hand through his hair. \"Trust me. That's all I can say.\"\n\nSam placed her hands on her hips and blew her bangs out of the way in frustration. \"Fine!\" Her face softened as she saw the look of hurt pass over Blount's features. \"Look... You've kept us safe so far and I trust you, but...\"\n\n\"But?\"\n\n\"But trust works both ways. Give a little, get a little.\" She grinned. \"Ya know?\"\n\n\"You're not going to like it.\"\n\n\"Try me.\"\n\n\"We're going to California. There's something we need. The island has to be destroyed for good, and there's something there that'll do it.\"\n\n\"You're right. I don't like it. California's under water, in case you've forgotten.\"\n\n\"Yep. Told you that you wouldn't like it.\"\n\nArcher fussed with Casey's seatbelt then strapped himself in as Sam powered up the helicopter. Blount glanced over at the two freshly dug graves before nodding for Sam to take off. As the ground receded below them, Blount went over his plan once more in his head, trying all the angles, all the outcomes, and all the options.\n\n# Chapter Four\n\n\"Dear God,\" Sam muttered as they surveyed the destruction below.\n\nArcher pulled Casey's head towards his chest as she whimpered. He looked to where she'd glanced, grimacing at the tangle of bodies in the overturned school bus. Archer could see where the water level had been on the buildings. It was receding, albeit at a slow pace. Every few minutes, they flew over survivors huddled on rooftops. When they saw the approaching helicopter, they would leap up and frantically wave their arms.\n\n\"We can't do anything for them,\" Blount muttered when Sam glanced over at him.\n\n\"We have supplies.\"\n\nBlount pointed down at a group of survivors. \"We land and give them what we have, and then what?\"\n\nSam pursed her lips and bowed her head. She saw the logic in Blount's response, but she hated the reality of it. One particular group opened fire when they realized the helicopter wasn't coming to their rescue. Sam expertly invaded the incoming fire. Blount gave her a look as if to say, \"I told you so.\"\n\nBlount spied a collection of boats and yachts in between two towering office buildings. He saw no movement on any of their decks, and from the looks of their hulls they'd been pushed inland by the tsunami and battered terribly. He pointed to them. \"Get me as close as you can to one of those boats.\"\n\nSam nodded and began to descend. She kept her eyes peeled for power lines as she maneuvered down into the manmade canyon of steel and glass. Blount unbuckled and opened his door. Hissing wind and rotor echoes engulfed the cockpit. Sam hovered above the biggest of the boats until one of the skids gently rested on its bow. The boat rocked slightly as Blount clambered out and dropped the few feet to its deck. He waved Sam away and she rose, hovering a few hundred feet above.\n\nBlount let his body acclimate to the rocking of the boat, while he scanned the surrounding area for any threats. Satisfied, he walked over to the door leading below deck. His boots squeaked on the wet fiberglass floor.\n\nThe boat rocked violently, and he cried out as he crashed to the floor. He scrambled for purchase as the boat heaved to the right.\n\n\"What the hell?\" The boat was struck once again, this time from the rear.\n\nThe boat yawed and water flooded up the stern. Blount started sliding down the steep incline, but he reached out a hand and grabbed the railing. He hissed with pain as his palm was slit open by the jagged rail, but his slide was halted for the time being. Glancing down between his feet, he sucked in a breath as he saw a massive shark take another chunk out of the rear of the boat.\n\nBlount raised himself up and looked over the railing. What he saw nearly stopped his heart. The water was swirling with gray fins of varying sizes. He watched in fascination as blood welled up in his clinched fist and fell into the water. The water grew agitated as the sharks caught scent of the blood, and he could see more sharks headed his direction. He vaguely recalled an article in National Geographic years back. Sharks could smell blood, no matter how small the amount, from up to two miles away.\n\nThe presence of the sharks dumbfounded Blount. He was in the middle of a California metropolis. What the hell were sharks doing here? Then the pieces fell into place like dominos. The tsunami! Of course, the sharks would be attracted inland with the ocean, drawn by the scent of so much blood and meat.\n\nThe shark at the end of the boat lost its purchase and slid back into the churning water. When the boat righted, Blount seized the chance to gain his feet. He glanced around for an escape route as water lapped at his knees. After rushing to the stern, he gauged the distance to the next boat. He looked down at the dark shapes circling in the murk, and then looked at the other boat. In ideal circumstances he might be able to make the jump, but this wasn't ideal. The boat was listing now, stern climbing into the air, reaching for the sky.\n\nBlount felt the air around him turn into a hurricane, and he glanced up to see Sam hovering. Her eyes were wide as she watched the sharks. Blount motioned for her to lower farther, but the helicopter shuddered in the wind and jerked upwards as an updraft scooped it higher. Blount shouted obscenities as the helicopter whirled towards him and the tail rotor careened close to his position.\n\nHe leapt into the water before he thought about the action. The water sucked the breath from him and embraced his limbs in its icy coldness. Something rubbery sandpapered against his torso and pushed him against the side of the office building. His lungs burned as he tried not to panic, and he focused on swimming to the surface. It was right there! He resisted the urge to inhale as his body fought against his efforts and urged him to do just that very thing. Two more dark shapes rushed at him from the gloom. They hit him, one in the chest, and the other in the legs. He was slammed into the glass of the building with such force that he lost all sense of time and direction. The need to breathe abruptly left him and silence settled.\n\nHe pulled his pistol and beat the first shark on the snout as hard as he could. The shark shook its enormous head and swam off, but a second shark snapped at his legs as he scissored wildly. He turned in desperation and fired into the glass. Cracks spidered through it and the glass turned opaque with fissures. With a _whoosh_ it gave way and Blount was sucked through the jagged hole.\n\nWater poured through the window and carried him down the hallway. His ears popped as the pressure changed in the building. He slammed into the end of the hall with a grunt, and his ribs cried out as he took his first breath. It felt like liquid fire down his gullet, but he gulped greedily anyway.\n\nHe wiped the stinging saltwater from his eyes and barely dodged to the right as a smaller shark came powering up the hallway, its tail lashing the walls and doors. The shark ran headfirst into the wall then slowly rolled over onto its back as blood poured from its snout and eyes.\n\nBlount sludged through the waist deep water to a door marked with the universal symbol for stairway. The water rose rapidly, and it was up to his chest now. He tried the door, but realized, with a sinking sensation in his gut, it opened inward. The pressure of the water proved too great to break the door's seal.\n\nHis head bumped against the ceiling tile and he looked up. Moving a tile aside, he hauled himself up into the ceiling crawlspace then crawled towards an air vent that led into the stairway.\n\nThe steel screen bulged outward, but didn't give when he pushed. The water filled the crawlspace as he lay on his back, kicking the vent with all his might. It clattered into the stairway with a crash, and he wiggled through the tight opening. Water gushed from the vent like a broken water main as he hurried up the stairs.\n\nHe reached the top and slammed through the door, finding himself on the roof. Four people turned their heads in his direction as he stumbled into the sunlight. The distinct sound of weapons being cocked greeted him over the distant roar of the helicopter.\n\n\"Well, what do we have here?\" a scrawny teen drawled as he leveled his hunting rifle at Blount.\n\nThe other teens giggled at their leader and circled Blount. He felt for his gun, realizing he'd lost it at some point. He held up his hands and offered an easy smile.\n\n\"Take it easy, guys.\"\n\n\"Who are you?\" the lead teen said with a jab of his rifle. \"And what the hell are _you_ doing on my roof?\"\n\nBlount slowly closed the distance between him and the teen, all the while keeping a stream of chatter going.\n\n\"Stop moving, pops,\" one of the other teens warned.\n\n\"Check his shadow, Nick!\" another said.\n\nBlount continued inching closer until he was less than two feet away from the boy. He peered into the boy's eyes to gauge his resolve. The tip of the rifle wavered slightly and a wild look glazed the kid's stare. The kid swallowed, his Adam's apple rippling beneath his sunburned skin.\n\n\"Take it easy,\" Blount said.\n\nThe boy glanced over at his pals and Blount made his play. He lunged forward and grabbed the barrel, yanking the kid towards him. In one swift motion, he twirled him around and applied a half nelson. He squeezed just enough to let the kid know he had no qualms about snapping a bastard's neck.\n\n\"Drop your weapons. Now!\" Blount ratcheted his hold on the leader's neck and the boy's eyes bulged. He whispered in the teen's ear, \"Nick? Is it? Tell them to do it. Nice and easy.\" Easing up on the pressure, he let the kid take a few shuddering breaths.\n\nAs Blount held the kid, something flashed through his mind. A brief glimpse of a carnival, a mountain, and lots of snow. Blount sucked in a breath. He'd encountered this kid before...somewhere else. \"Nick Billings.\" The kid stiffened in his grasp at the mention of his name. \"Tell your friends to drop their guns. Do it now.\"\n\n\"Drop 'em. Drop 'em damnit!\"\n\nThe guns clattered to the roof and they slowly back away. When the helicopter's backwash buffeted them, they shielded their eyes against the swirling debris and gravel peppering them like miniature missiles. Blount inched the boy forward until he was standing over the guns.\n\n\"I don't know when or how we'll meet, but I know you've been a part of my life. Good luck, until we meet again.\" He shoved the boy, who screamed as he pitched face first into the tar and gravel.\n\nBlount raised a hand above his head without looking, motioning for Sam to come in at a hover. He felt the skid press into his hand a few seconds later, and he looped an arm over it. Sam gained altitude and Blount held on tightly. She moved a few building over before landing.\n\nBlount collapsed on the roof, catching his breath as he stared up at the fading sky.\n\nArcher jumped out of the helicopter and scanned the rooftop. Satisfied there were no immediate threats, he knelt down beside Blount. \"Jesus! We thought you were dead!\"\n\n\"Sorry to disappoint,\" Blount said with a tired smile. He heard Sam powering down the helicopter, but he was too exhausted to sit up and tell her not to. He must've blacked out for a few seconds, because the next thing he was aware of was Sam peering down at him and smoothing his forehead. He sat up and groaned as dizziness overwhelmed him.\n\n\"Take it easy,\" she said.\n\nHearing those words made Blount flash back to the group of teens. He looked over at the other building but the rooftop appeared deserted for the time being. The boy, what'd he'd sensed, troubled him. It wasn't often that he sensed his past, or future, for that matter. He wondered which it was\u2014past life or future life? He shook those thoughts from his head. That was another story for another time.\n\n\"How long was I out?\"\n\n\"About ten minutes,\" Sam said. \"You've been going non-stop...\"\n\n\"You shouldn't have shut the chopper down. Everyone knows we're here, and if they're anything like those boys over there...\"\n\n\"What the hell were you thinking?\" Sam muttered. \"I saw you go in the water with all those sharks.\" She halted and swallowed back emotion. \"I-I-I thought we lost you.\"\n\nBlount attempted to sit up once again. This time the dizziness was less severe. He gained his feet and shuffled over to the edge of the roof, then glanced down at the remaining boats.\n\nSam stood beside him and followed his gaze. \"What's the deal with the boats?\"\n\n\"I need a scuba tank.\"\n\n\"What for?\"\n\n\"The place we're going is an underground facility. I figure it'll be underwater, too, if it's been breached.\"\n\n\"You're exhausted,\" Sam said. \"I'll go get one. No arguments.\" She gave a dismissive wave of her hand.\n\nBlount looked over at Archer and motioned with his head. \"Go with her.\"\n\n* * *\n\nArcher slammed his shoulder into the stairway access door and entered the stairwell. Sam followed him in, and the smell of mildew and rot filled their nostrils, while Sam fought back a sneeze. Archer flicked on the flashlight at the end of the M4 and started down.\n\nSam glanced up after a few flights and could still see the rectangle of light, which comforted her a bit. Archer was subdued as they descended, glancing back every few steps as if he were keeping an eye on the doorway.\n\nThey continued down and Sam sensed they were close, because the smell of saltwater grew more pungent. She halted as what seemed like thousands of tiny pinpoints of light reflected back at her at the next switchback. The rats squealed as the light passed over their masses.\n\n\"Great! Freakin' rats!\" Archer muttered. \"I _hate_ rats.\"\n\nSam thanked her stars for the heavy-duty work boots and thick flight suit. She looked around the stairwell and saw some chunks of concrete in one of the corners, so she hefted a good-sized piece, tossing it into the middle of the rats. As they scattered she and Archer hurried down into the void that'd been created. She quickly pushed through the door and into a dark hallway.\n\nThey were assaulted by the stench immediately. Sam placed a gloved hand over her mouth and nose and breathed shallowly.\n\n\"Jesus! That reeks.\" Archer grunted.\n\nSam played the light about the dark hallway, and the carnage was revealed in increments. A boot here, a gloved hand there. A leg, a torso, an arm, an empty SWAT riot helmet. Her light reflected off the POLICE insignia on one of the bodies. The corpses were bloated and black with decay and numerous eyes reflected from their body cavities.\n\nSam rushed forward as quickly as she could, keeping her attention fixed on the far end of the hallway. She sobbed softly and struggled to maintain her crumbling composure. This was too much. Even after everything she'd seen, everything she'd been forced to do to survive, this was too much. Death had touched the earth and turned its surface black.\n\nThe closeness of death, the stench of rot, and the reality that the world would never be the place it was before, with all its superficial safety and certainness, settled over her. This was the new way of life for humankind. Sam brushed against the left side of the hallway and puffs of black billowed into the light. She held her breath, walking even faster.\n\nArcher took his time checking the weapons strewn about the floor. Most were empty, but he did find a few more 9mm round clips, which he slipped in his pockets.\n\nSam reached the far end of the hallway. She glanced back into the gloom and saw Archer's light playing about the bodies. He was farther back than she was comfortable with, so she considered heading back to his side, but she was loath to repeat the nightmarish journey through all those ravaged corpses. She kept her back to the wall and waited as Archer finished his searches and headed towards her.\n\nSam tried the first door on her left. It opened to reveal water lapping at the windowsill. She was relieved to find the window would be fairly simple to open.\n\nArcher shut the door behind them. He noticed her looking quizzically at him. \"I feel more comfortable having something solid at our backs.\"\n\n\"Me, too.\"\n\n\"Hold up a minute.\"\n\n\"What?\" Sam tensed.\n\n\"What's the deal with you and Blount?\"\n\n\"What do you mean?\"\n\n\"Don't be coy,\" Archer said with a chuckle. \"I've seen the way you two look at each other.\"\n\n\"We don't have time for this.\"\n\n\"Hit a nerve, did I?\"\n\n\"There's nothing there. Drop it.\"\n\nBut, there was something there. Sam felt something for Blount, though she wasn't sure exactly what it was. She'd seen the looks Blount had given her, had picked up on his non-verbal cues, but she'd shoved all that to the back of her mind. She'd read once that relationships forged under the pressures of extreme stress were natural, but they had a tendency to fail after those stressors were non-existent. Who was she kidding? The world was different and would never be the same again. Still, she wondered what it would be like to be held in Blount's arms. To feel secure in the knowledge there was someone else that cared about her. Once Blount did what he planned, what would they do next? Would they stay together as a group? She found herself wondering about the future. Hope? Was that what she was feeling?\n\nSam opened the window to the sound of the bow of the boat rocking up against the building's side. She killed her flashlight and set it on the floor. Archer glanced out the window and frowned.\n\n\"Maybe you should go,\" he said. \"My feet are too big for that ledge.\"\n\nSam eased herself out onto the ledge. \"Wish me luck,\" she called back over her shoulder with a bravado she didn't feel, and made her way to the boat. She refused to glance down at the water a few inches from her boots. Sam was certain that, at any moment, one of the sharks would lunge from the water and snatch her from the ledge. When she reached the boat, she let out a huge breath of relief.\n\nThe boat rocked gently beneath her weight as she made her way over to the hatch. Darkness seemed to reach up for her, and she loathed going down into its eager arms. She scanned the rest of the boat's deck, glimpsing three scuba tanks secured in their racks. Whispering a prayer of thanks, she rushed over to them, but the surge of hope came crashing down when she noticed the tanks were secured behind a thick strand of cable with a padlock snug against said cable. She looked around for something to pry the lock loose.\n\nSam froze as she heard a moan come for the darkness of the hatch. She pulled her gun and whirled to face the rectangular void. The shadow poured out of the hatch so fast it took Sam a few seconds to realize just what it was. The body followed a moment later. Sam stumbled backwards and slipped, slamming against the side of the boat. Stars flashed through her vision as her head smacked into the fiberglass hull.\n\nThe shadow reared up in front of her and the body mimicked the action, only with a raised hatchet. She rolled out of the way as the blade came down, sending pieces of wood and fiberglass into the air. Sam fired five shots into the shadow before it burst into a black cloud. After a horrendous scream, the body collapsed to the floor with a thud.\n\nThe sharks nudged the boat, making Sam curse as she gained her feet. She grabbed the hatchet and ran over to the tanks. The cable gave way in a shower of sparks when she hit home, then she grabbed the first tank. She heaved it, fully expecting it to have weight, and was thrown off balance as the lightness surprised her. The tank went flying behind her, splashing into the water.\n\nShe tried the second tank and found it heavy, so she dragged it across the boat to the ledge then slipped her arms through the straps. The boat rocked violently once more, and she waited until it steadied before climbing back onto the ledge. She moved as quickly as she dared. With the added weight of the tank on her back, she felt awkward and unsure. The tank tugged her back with each step across the meager ledge. She reached the window and flung herself through, where she collapsed with relief.\n\n* * *\n\nBlount jerked awake as he heard the shots echo across the roof. He was so exhausted, he hadn't even realized he'd drifted off. Casey was already standing at the edge of the roof looking down as he rushed over.\n\n\"What happened?\"\n\n\"Not sure. Just heard the shots,\" Casey said.\n\nBlount looked around wildly in search of Sam then spotted her back on the ledge. \"There she is!\" He saw the tank on her back and willed her to make it safely into the building, watching as she disappeared from sight.\n\nA few moments later, the door slammed open and Sam grinned over at them. Archer appeared behind her, holding the tank out to Blount as he spoke. \"I hope that was worth it.\"\n\nBlount took the tank and set it down gently, then lunged at Sam and engulfed her in a tight bear hug. \"It was worth it,\" he whispered into her hair.\n\nSam allowed the embrace to linger. The look of smug satisfaction on Archer's face as he looked on was not lost on her, so she pulled back and flashed a tired smile. \"I don't know about you, but I'm drained. I'm in no shape to fly right now, and you don't look to hot yourself, mister.\"\n\nBlount nodded. Blushing, he released Sam then turned to Archer. \"What's it look like down there?\"\n\n\"One floor down isn't bad, but any farther is a nightmare.\"\n\n\"Let's secure the helicopter and bed down for the night.\"\n\n\"Best idea I've heard in a long while.\" Sam laughed.\n\n\"Seconded,\" Archer said.\n\n# Chapter Five\n\nBlount was nearly asleep when he felt someone maneuver into his sleeping bag. He tensed and reached for his gun at his side. A soft, warm hand closed over his and Sam whispered into his ear, \"Relax, it's just me.\" A million thoughts passed through his head as she snuggled up to his side.\n\nSam must've felt the rigidness of his body, because she hesitated. \"You cool with this?\"\n\n\"Yeah.\"\n\n\"Good.\"\n\nBlount could hear the smile in her voice. Sam rested her head on his chest and he inhaled her scent, feeling his body respond to her closeness. He didn't trust himself to talk, so he just held her close.\n\nA few minutes later, Sam raised her head and squinted up at him in the gloom. Blount looked back and waited.\n\nSam broke the silence. \"What is this?\"\n\n\"You tell me.\"\n\nShe frowned and sighed. \"Help me out here.\"\n\n\"You barely know anything about me. I'm twice your age...\"\n\n\"So?\"\n\nBlount sighed and looked away from her. She reached up and ran a hand over his scruffy face.\n\n\"So?\" she repeated.\n\n\"I don't need to worry about you more than I already do. I've lost so much. More than you can ever understand.\"\n\nFlashes of Sarah streaked through his mind's eye. The way the right side of her lips were slightly crooked when she smiled. The vivid color of her eyes when she'd told him they were pregnant. So many memories, so many lives, long buried, all rushed to the surfaces of the graves he'd tossed them into.\n\nSam's face fell as he said those words. \"Let me try to understand. At least give me that much of a chance.\"\n\n* * *\n\nCasey watched Sam and Blount from the darkness of her corner. She didn't dare move. Though the two were whispering, their voices still carried quite clearly. She found herself smiling as she followed the conversation. There had been so much death, so much despair, the past year. Yet even in these dire circumstances, love bloomed like fresh hope.\n\nHer mother intruded on her thoughts and the tinge of happiness she'd felt dulled. Casey wondered what had happened to her mother. The last time her father had taken her to visit, Mom hadn't even recognized her.\n\nCasey knew, deep down, her father had resisted hospitalizing Mom until there was no choice but to put Tanya in protective care. The madness had eaten away at her mother for years. It'd been there, like a brooding monster, before they'd married, dad had speculated, and it had revealed its existence just after Casey had been born. The stress of new motherhood had ripped and gnawed at the flimsy mental walls Tanya had erected.\n\nHer father saw the signs, sensed something was not quite right, but he'd soldiered on, hoping against hope that things would get better. Casey had to hand it to him. A lesser man would've left after all the incidents and close calls, but he'd stuck by his wife's side through it all.\n\nCasey didn't remember much of her early childhood. There were no major incidents seared into her earliest memories, but as she reached adolescence, it became clear something was off with her mother.\n\nOnce Casey became aware of this, it was easy to pinpoint how Tanya didn't act like other mothers Casey knew. Confirmation came on Casey's thirteenth birthday. Her father had, as was then routine, been working late with the service, and Casey came home to a dark, still house. This wasn't unusual, yet as Casey dropped her school bag on the kitchen table and grabbed an apple from the fruit basket, alarms went off in her head.\n\nThe kitchen wall was darker than it should've been, and as Casey flicked on the lights, she realized why. Blood covered the once white plaster. It slid down the wall in places, and in others it had dried and clumped.\n\nCasey caught a low moaning drifting from the darkness of the dining room. The hair stood up on her neck and arms as she slowly approached the dining room.\n\n\"Mom\"? She'd called as she peered into the gloom.\n\n\"The tide's all wrong. The tide's not right,\" her mother had whispered as she clutched her knees and rocked back and forth against the wall. \"The tide's all wrong. The tide's not right! The only hope we have is to contain the key, find the key, the tide's all wrong! _\"_\n\nShe remembered what the responding EMT's had told her father after they'd managed to subdue her frantic mother. \"Next time, it'll be your daughter, Mr. Archer, not the cat.\"\n\nCasey wondered if her mother was still alive out there, somewhere.\n\n* * *\n\nArcher's mind was restless. A barrage of thoughts paraded through his head at break-neck speed. Try as he might, he couldn't latch onto any one of them. These images that flashed through his brain troubled him, because they were scenes he knew he hadn't witnessed. They were images of death, destruction, blood, refuse, and fire. Picture after picture seared across his inner vision, relentless in their progression. Each new image was a hammer-punch to his consciousness. He tossed and turned and willed himself to open his eyes, but they refused to respond. Swimming up from the nightmare, he strived to break the surface, but each time he was just about to emerge in the waking world, he was yanked back down into the depths.\n\nWhen his eyes opened, at first, he wasn't sure where he was. He blinked rapidly and was startled as he focused on Blount, Sam, and Casey all huddled around him, their faces shrouded with concern.\n\n\"You're burning up,\" Sam said, the coolness of her hand caressing his forehead.\n\n\"Are you all right, Dad?\"\n\n\"I'm-I'm fine,\" Archer croaked. \"Water?\"\n\nBlount produced a bottle and guided it to Archer's parched lips. Archer drank greedily. The water felt like heaven as it flowed down his throat. He coughed violently but managed to catch his breath. He could feel the suspicion oozing off Blount, who observed him with a detached expression.\n\n\"What?\" Archer asked.\n\nBlount looked like he was about to say something, then thought better of it. \"Nothing.\" He gave a dismissive wave, stood, and busied himself with inventorying his ruck. \"We move out in fifteen.\"\n\nArcher wearily gained his feet and swayed. Gratefully, he reached out as Casey moved to offer a steady shoulder.\n\n\"You sure you're OK?\"\n\nIt felt as if he hadn't slept in years. His body was sore in places he didn't even know existed. Everything felt and sounded like he was moving beneath deep water. His ears roared with white noise, and he found he had to concentrate on his daughter's lips as she spoke. He just wanted to sleep, just wanted to fall down and never get up again.\n\n\"I'm fine,\" he said. \"Just go help Blount and Velska load up the chopper. I'll be fine. Just need to catch my breath for a second.\"\n\nCasey lowered Archer into a mildewed chair and looked at her father with apprehensiveness. He waved her away and leaned his head back, closing his blood-shot eyes.\n\nCasey grabbed her bag and her father's then made her way up the stairs to the roof, where she just missed Sam and Blount sharing an embrace. Smiling, she called out, \"'Bout time you two quit beating around the bush!\"\n\nSam blushed and cast a furtive glance at Blount. He surprised them both by offering a genuine smile. \"Yeah, 'bout time.\"\n\nSam leaned against him and grinned.\n\n\"When's the wedding?\" Casey teased.\n\n# Chapter Six\n\nBlack Rock Facility, Code Name: Hellfire \u2013\n\n\"That's it?\" Sam asked as they circled the non-descript building. Water lapped at the third story windows, agitated by the hovering aircraft.\n\n\"The bulk of it is underground,\" Blount said. \"Take us down to the heliport over there.\"\n\nSam set the helicopter down with barely a jolt. Blount pushed open the cockpit door as the rotors powered down. He heaved the scuba tank and wrestled into the straps.\n\nWhen he caught a look at Archer, his concern and unease tripled. The man looked like death warmed over. For a brief moment, Blount thought he saw something on Archer's chest. He blinked and the vision passed. _What the hell?_\n\nArcher's eyes flashed open, as if he knew he was being observed. A sensation crashed over Blount, rocking him back on his heels with its force. Those eyes. He'd seen those eyes before. Archer blinked and his normal eyes returned as Blount's vision once more faded. Blount opened his mouth to say something, but his mind was blank. He knew he was supposed to remember something, but whatever it'd been had been plucked from his brain. He frowned and struggled to clear his head, but it was a churning mess.\n\n_It's not finished_ , the twins whispered.\n\n\"Gibson?\" Sam said.\n\n\"Yeah?\" _She used my first name_. A not-unpleasant warmth passed through his limbs.\n\n\"Be careful.\"\n\n* * *\n\nBlount entered through the ventilation shaft. On the surface, the building did not appear to have the myriad of surveillance and security it actually possessed. He was thankful for the loss of power, as that would have taken care of the more stringent systems. The backup generators had ceased to work after being flooded. Still, it didn't mean this would be easy.\n\nHe didn't risk a look at Sam and the others, because he didn't want this to be the last image he held of her. Already his actions were being influenced by Sam. On one hand, he savored the emotions, but on the other he shoved them deep within, trying to ignore them. Distractions would get him killed. He knew this only too well from experience.\n\nCarefully he lowered himself into the shaft and felt for the ladder in the darkness. His boot clanged against one of the rungs. He sucked in a breath as his feet disappeared beneath the frigid water. The water sucked the heat from his legs as he descended. Blount had never been particularly claustrophobic, but the tightness in the ducts was wearing on him. Having the bulky tank on his back didn't help matters either.\n\nHe came to an intersection and paused to check the dial attached to his tank. Twenty minutes of air left. The thought of running out while trapped beneath all this water caused a twinge of panic to flair in his chest. He forced his breathing to remain steady as he floated in the gloom. Flashing his light to the right, he peered down a passageway. His light only reached about ten feet until the blackness swallowed it up. Next, he directed his light down the left fork of the corridor. A few feet farther ahead, he spied an access hatch into the main building. He swam left and worked the latch on the steel grating.\n\nThe hatch was small, but he struggled through. A few times he was sure he'd become stuck, but he managed to slip through the small opening. Blount had been in this complex many times, but seeing the hallways in this unfamiliar setting threw his bearings off. He swam over to a sign on the wall and studied the diagram affixed there.\n\nHis flashlight dimmed and fluttered, so Blount banged it against the wall. The beam steadied as he got his bearings, and he pushed off the wall to propel himself down the hallway, towards his goal.\n\n* * *\n\nSam checked her watch for the tenth time in as many minutes. Blount should've been back by now. She peered down the shaft and willed him to break the surface.\n\n\"Where are you?\"\n\nCasey ventured over and slipped a hand into Sam's. She squeezed softly and offered a comforting smile. \"He'll be fine.\"\n\nThey both shouted with relief as Blount broke the surface. He inhaled deeply and coughed. Grinning up at them, he began to climb the ladder.\n\nThe girls helped him from the shaft and he collapsed on the rooftop. He held up a hand as they both spoke, then he gulped in a lungful of precious air. \"I'm fine.\"\n\nSam sat on the roof and guided Blount's head into her lap. She smoothed his hair from his damp forehead. \"That feels nice,\" he murmured with a weak smile.\n\nHe reached into his pocket, and Sam frowned as she saw the small object in his palm. It was the size of an egg and black. When it shifted in Blount's palm, it emitted flashes of red light.\n\n\"What the hell is that?\"\n\nBlount closed his fist and slipped the object back in his pocket. \"It's a vortex enclosure.\"\n\n\"And?\" Sam said. \"You practically killed yourself for _that_?\"\n\nBlount shrugged. \"We're going to need it.\"\n\n\"What's it do?\" Casey asked.\n\n\"It's dark matter,\" Blount said. \"Encased in a multidimensional carbonate shell.\"\n\nSam looked from Casey to Blount with a confused expression on her face.\n\n\"It's a mini-black hole,\" Casey said.\n\n\"Basically.\"\n\n\"Oooo-kaaaay,\" Sam muttered. \"This just gets weirder and weirder.\"\n\n\"Wait a minute,\" Casey said. \"Don't black holes destroy matter?\"\n\n\"Yes.\"\n\n\"And keep destroying matter?\"\n\n\"Normally, yes,\" Blount said. \"This is a last resort weapon. I think we're at our last resort, don't you?\"\n\n\"Won't it keep growing, though?\"\n\n\"It shouldn't.\"\n\n\"It _shouldn't_?\"\n\nBlount had to remind himself that Casey and Sam weren't used to his world\u2014a world of magic, monsters, and destruction. It reminded him just how different each of his lives had been. He sat up and wearily gained his feet, wanting to just be normal at that moment, but he couldn't ever be normal. Once he knew what was really out there, in the dark, what was waiting to engulf mankind with the first opportunity it got, he had to do something about it\u2014whatever it took. Knowledge sucked everything out of him\u2014love, hate, wonder\u2014and left darkness in its place. The only thing that kept Blount going was hope that maybe, this time, things would be different, work out better. He won some, he lost some, but he kept fighting, because maybe this time, he would win.\n\n\"Remember what I told you guys...about me, my life? I've done this before, or something very much like it, I'm sure of it. Whatever has to be done, that's the point. We keep moving forward until we can't any longer.\"\n\n\"I'm with you,\" Sam said.\n\n\"Count me in,\" Casey said. \"I had that thing in me, remember? I don't want anyone to ever feel that kind of darkness, that emptiness, ever again.\"\n\n\"Sounds like we're going somewhere,\" Archer said.\n\nAll three of them whirled, startled, as his raspy voice broke their conversation. Blount sucked in a breath when he took in Archer's condition. The man was pale and sweating profusely. His eyes were sunken and bloodshot, and he was missing tufts of his hair. It'd only been about thirty minutes since he'd last seen Archer, but this transformation, or rather this decrepitude, looked like years of decay has occurred instead of minutes. Something shifted in Blount's mind and he blinked. Now Archer looked merely tired and worn out, not practically near death.\n\nThen something shifted in Blount's mind again.\n\n# Chapter Seven\n\nR'yleh\n\n\"We're not going to make it,\" Sam said through gritted teeth.\n\nThe helicopter shuddered as it sucked the last of their fuel. Something groaned beneath his feet, and Blount felt his stomach lurch as they dropped in altitude.\n\n\"What the hell?\" Sam shouted as they were snatched back into the air. The rotors ground to a slow halt, yet they were still airborne. The water flashed by beneath them like liquid glass.\n\nBlount glanced out and tried to see below the lip of the window. He caught flashes of white that stood out starkly against the deep blue of the ocean.\n\n\"Abdiel,\" he whispered. To Sam, he said, \"Don't worry. He's a friend.\"\n\nSam shot him an incredulous look and tightened her shoulder harness. They flew towards the beach and soon they were deposited on the sand with a bump.\n\nSam scrambled out of the chopper and stared in disbelief at a ten-foot tall man with massive white wings. \"Jesus.\"\n\nAbdiel smiled and shook his head. \"Not quite, but it's not the first time I've been mistaken for him.\"\n\nCasey stared in awe at the angel, and Abdiel smiled in her direction. \"It's good to see you, child.\"\n\n\"Me?\" Casey was taken aback by the angel's words.\n\n\"We've met before. A long time ago.\"\n\nA memory flashed through Casey's mind. She was two years old and had toddled out of the house. A butterfly had caught her attention, and she'd gleefully chased the fluttering wonder across the lawn, into the busy street at the corner. A nice man had taken her hand and guided her back safely to her front porch. The man had possessed wings.\n\n\"You've been following me,\" Blount said.\n\nAbdiel nodded. \"I figured you'd need my help eventually.\" He spread his arms and shrugged.\n\n\"Thanks.\"\n\nAbdiel's wings fluttered as Archer eased himself from the chopper. The angel's eyes narrowed and his hand moved to rest on the enormous sword strapped to his side. He crouched slightly and looked at Blount. When Blount failed to react, Abdiel frowned. The sense of uneasiness passed as quickly as it had come. He shook his head, and Blount was sure it was to clear the murkiness of his thoughts. Archer ignored the angel, leaning heavily against the fuselage with one hand.\n\nThe ground felt as if it were vibrating beneath Blount's boots. He shifted his stance and looked around to see if the others were feeling the same sensation.\n\nAs the surf crashed in, he walked off by himself, leaving the others for a moment. Foamy water washed over his boots as he stood staring out at the horizon. Abdiel came and stood beside him.\n\n\"Makes you wonder where He's at,\" Blount said.\n\nAbdiel remained silent. Blount had had this conversation with the angel many times, and Abdiel's response had always been the same. Silence. Blount often wondered what it was like being one of God's creatures, void of true free will. While Abdiel was prohibited from interfering directly in the affairs of man, he helped out where he was able.\n\nAbdiel spoke. \"All of this has happened before, and will happen again.\"\n\nBlount turned in surprise and looked at the angel. \"But, why?\"\n\n\"It is what must be. It is what has always been.\"\n\n\"Well, I think it's bull.\"\n\nAbdiel smiled. \"As do I. It's been eons since the war, but everything happens according to His plan, as it must. That's all I can offer you. There is a plan, and one day all of this,\" the angel spread his arms, \"suffering...it will end.\"\n\n\"So we just endure.\"\n\n\"Yes. We endure and fight until we have no fight left.\"\n\n\"What happens after?\"\n\n\"This world will recover. Evil has left its mark, but man will recover, as they always have before.\"\n\n\"Let's do this then.\"\n\n* * *\n\nThey moved single file\u2014Blount, Archer, Casey, Sam, and Abdiel bringing up the rear\u2014along the same path Blount's original team had followed through overgrown plants and foliage. The vines had crept back over the original path, as if to bar their progress. They stumbled over the rocks that jutted from the ground like carnivorous teeth. The trees in the distance towered, masking the structure ahead of them.\n\nBlount struggled with a multitude of feelings as they trekked. He knew what was ahead, and he waited for the others to gasp once they caught sight of the citadel. What he dreaded the most was coming upon the bodies of his slain team.\n\nThey came to the place where the freak snowstorm had erupted and Blount tensed, waiting for it to happen once more.\n\n\"This place smells of death,\" Abdiel said.\n\n\"We were attacked here.\" Blount cringed as his voice echoed. He knelt and touched the ground, and it vibrated at his touch.\n\nCasey moaned as she caught sight of a severed leg along the trail. She moved closer to Sam, who offered a comforting hand. The girl gripped Sam's hand gratefully as she whispered encouragement in the girl's ear.\n\n\"Keep moving,\" Blount called over his shoulder. He pressed on amidst the phantom cries of his fallen comrades.\n\nA short time later, they arrived at the path that twisted into many directions and sloped down out of view. Once more, Blount felt drawn to the one that led downwards.\n\n\"It draws us closer,\" Archer murmured.\n\nBlount glanced behind him and paused. It been quite awhile since Archer had spoken, and Blount was intrigued by the man's statement. He held up a closed fist. \"Hold up.\" He made his way back, carefully easing around Sam and Casey on the narrow ledge, and approached Archer.\n\n\"You feel it too?\" He studied Archer's eyes as they shifted and fluttered.\n\n\"It calls out. It hungers for sustenance, for completion.\"\n\nThe uneasiness that had clothed Blount like an ill-fitting suit made its self known once more. His hands shot to his head as an image exploded in his mind's eye. He cried out when it felt like his head would shatter in his clenched hands. The vision passed as quickly as it had come, leaving Blount drained and weak. He'd seen the future and knew what would face them inside the citadel.\n\n\"Are you OK?\" Sam helped Blount back to his feet.\n\nHe waved her away and nodded \"I'm fine.\" Blount stared at Archer and steeled his resolve. The man stared back through hooded eyes and offered nothing. \"You take the lead.\"\n\nArcher merely nodded and started past Sam and Casey.\n\n\"What just happened?\" Sam asked.\n\n\"I got a message,\" Blount said.\n\n\"A message? From whom?\"\n\nBlount leaned in close. \"When the time comes...follow my lead.\"\n\n\"What?\"\n\n\"Just trust me. Please. Follow my lead.\"\n\nBlount followed at a distance as Archer chose a singular route, as if by some unseen hand, from the many then ascended the small ridge. When he reached the top, he stopped. The rest of them gathered around and looked to where he was staring, and they were all struck with awe at the sight awaiting them. Even though Blount had seen it prior, he was still humbled by the obscene wrongness of it.\n\nDark spires sprung out of the ground, huge and towering, made of smooth stone that was covered with disturbing hieroglyphs. When Sam focused on the hieroglyphs, she felt like ants were crawling around inside her head. Some spires were formed with grotesque, cold spheres that sat unnaturally atop one another, unnerving the onlookers. In the center of this cluster of towers rose a single, monolith-shaped citadel. Its surface stretched at impossible angles that made its shape abnormal and frightening, stopping the group cold.\n\n\"This is wrong,\" Abdiel whispered. His face was slack and he breathed shallowly.\n\nBlount studied the angel's reaction and it troubled him greatly. He'd never seen Abdiel more scared.\n\n\"This shouldn't exist,\" the angel said.\n\n\"It's beautiful,\" Archer said.\n\nThe others stared at Archer in disbelief, while he continued to stare at the panorama and lick his lips. He was kneading his crotch, Sam noticed, with disgust. When he looked back and noticed they were watching him, he abruptly stopped. \"Let's go.\"\n\nHe led them without hesitation, Blount noted, to the gash in the side of the nearest tower. Archer clambered up the debris and disappeared into the darkness of the opening.\n\n\"There's something wrong with him,\" Abdiel said.\n\n\"Yeah, I know,\" Blount said.\n\n\"What's wrong with my dad?\"\n\nBlount grimaced as Casey spoke. He'd been hoping to avoid her questions. He steeled himself and plastered what he hoped was a reassuring smile on his face, turning to Casey. \"Don't worry. I'll take care of it.\"\n\n\"What's wrong with him?\"\n\nBlount looked to Sam for help, and she led Casey aside. Abdiel approached the opening and cringed, then he stepped back and looked at Blount. \"I can go no farther.\"\n\nBlount looked at the opening and the glyphs. They were worn with age, barely noticeable at first glance, but they were there. He recognized a few of them\u2014binding and blocking spells. He pulled his knife and scraped away at one of them. The glyph crumbled beneath his blade, but as soon as it was obliterated, it morphed back into existence in the clay.\n\n\"I'll try and find another way in,\" Abdiel said, lifting off with a whoosh of his wings.\n\n\"Stay close,\" Blount said as he climbed up the ruble.\n\nThe others flicked on their flashlights and looked about the corridor.\n\n\"What's that?\" Sam asked.\n\n\"My team.\"\n\n\"But, I thought they were dead,\" Casey whispered.\n\n\"They are. It's their death echoes.\"\n\nBlount pressed on. A short time later, his light revealed Archer peering down the crevice. He turned and nodded to Blount. \"There is no logic, or comfort, or convenience to this place. This place feels wrong...all wrong.\"\n\n\"How'd you know that was there? You don't have a light,\" Blount said.\n\nArcher merely shrugged. \"I just knew.\"\n\nSomething about Archer struck Blount just then, but he had trouble intuiting exactly what had changed about the man. Then it clicked. The man looked healthier, more full, more aware. Blount noted the change, but didn't mention it aloud.\n\n\"If you're being guided, then I won't interfere,\" Blount said. \"Lead the way. We can use all the help we can get.\"\n\nArcher maneuvered around the pit and paused in front of the overhang Blount and his team had entered previously. Blount held his breath. He was dreading swimming back in that dark water, and what waited on the other side of the swim. After checking his gun, he motioned for Sam to do the same. She shot him a questioning look, but complied.\n\nArcher moved away from the overhang, tilting his head as if listening to something. He stood this way for a short time, and then moved off along a path that was barely noticeable in the gloom. It was barely ten inches in width, but Archer's boots were sure as he began to ascend. With his flashlight, Blount followed Archer's progress along the narrow shelf of rock, until Archer disappeared. Blount cursed and searched the area for any sign of the man.\n\n\"Where'd he go?\" Sam searched for Archer, too.\n\n\"There,\" Blount said. \"It's a passageway between the two cliff faces.\"\n\nBlount started up the path and slipped through the camouflaged opening. He found himself in a large, empty space, where water dripped all around him. His light reflected his image back at him, and the sight was disorientating at first.\n\nShielding his light with a cupped hand, he discovered the floor of the chamber was clear as glass. He could see the pool of dark water in the next chamber below, where his team had been attacked. Piles of bones littered the rest of the crystal floor. He tentatively stepped onto it, and it held his weight.\n\nDirecting his light to the far side of the chamber, he caught a glimpse of Archer as he disappeared through yet another opening in the wall's face.\n\n\"Archer! Hold position!\" To Sam and Casey, he said, \"The floor's solid. We can't lose track of him.\" He hurried to catch up with Archer.\n\nBlount squeezed through the opening and almost ran into Archer, but he maneuvered around the man then sucked in a breath. They were in the spired chamber he'd astral traveled to back in Nebraska. Archer knelt and peered through the crack in the floor. The stench Blount had smelt in his astral state was even more powerful here. His eyes watered and his lungs burned, even though he breathed as shallowly as he could. The stench didn't seem to affect Archer. He huddled over the chasm as if warming his hands over an inviting fire.\n\nBlount stood back as Archer began to glow from within. His skeleton was outlined by the glow, and Blount watched Archer's heart beating in tune with the hum the crevice emitted. Blount steadied himself as he felt the floor's vibration intensify.\n\nHe knew he had time, not much, but time to go back to the opening and tell Sam to keep Casey away from what was about to transpire. Blount hurried back over and stopped Sam as she was preparing to step through.\n\n\"Take Casey to a safe distance.\"\n\n\"Why? What's happening?\"\n\n\"Just do it.\" His heart thudded as he saw the hurt look pass over Sam's face, so he softened his tone. \"Trust me, she doesn't need to see this.\"\n\n\"What are you about to do, Gibson?\"\n\n\"What must be done. It's the only way. I don't like it, but it's necessary. Please, just start heading back to the entrance. I'll follow as soon as it's done.\"\n\nStaring intently at Blount, Sam held her ground. He could see she was torn between arguing or accepting him at his word.\n\n\"Please, go.\" Blount glimpsed Casey peering over Sam's shoulder.\n\n\"Why'd you drag us all the way in here if you were only going to send us back?\" Sam said.\n\nBlount sighed and cast a glance back over his shoulder at Archer. His glowing form had morphed from white to a deep red now. With not much time left, Blount weighed his options. He knew Casey would go berserk if she witnessed what he'd have to do to her father, and he had his doubts about Sam's reaction, as well. How would she feel about him after?\n\n\"Fine,\" Blount said, coming to a decision. The world was more important than his happiness. He'd made these choices before and would continue to make them. It was his curse, his mission, his existence. \"Keep her back...whatever happens.\"\n\nWith that, he marched back over and stood directly behind Archer. He heard the girls gasp as they caught sight of the man.\n\n\"Daddy!\" Casey cried.\n\n\"Keep her back.\"\n\nSam cringed at Blount's warning. She'd obviously never heard anybody sound that deadly. She was seeing a new side to Blount, and he could tell it scared her greatly. She did what he instructed, holding Casey in her arms as the girl struggled to get free.\n\n\"Archer,\" Blount called out. \"Fight it.\"\n\n\"But I don't want to.\" Archer moaned, as if deep in the throes of ecstasy.\n\n\"Fight it, or I'll do what I must.\"\n\nArcher turned and faced Blount. Red light leaked from his eyes and his hands glowed brightly. When he smiled, light bled from his mouth, superimposing his teeth. Archer's voice grew deeper than humanly possible. \"We will be one.\"\n\n\"You still have a choice,\" Blount said. \"You can choose to step away. Whatever it's promised you, whatever it's whispered in your ear, the reality will be much different.\"\n\n\"Silence! I will have this body. My existence will become reality.\"\n\nBlount sensed something behind him, but he couldn't risk breaking eye contact with Archer. What happened next felt like it happened in extreme slow motion. Casey ran past him in a rush of air. Blount reached out to grab the girl, but his fingers slid through her long, blonde hair and closed on emptiness. He managed to grab Sam as she ran after the girl, wrapping his arms around her and holding her tightly.\n\n\"Don't,\" he said.\n\n\"Daddy!\" Casey howled as she raced to Archer.\n\nArcher grinned and held his arms open wide as his daughter approached. Casey crashed into nothingness, as if the emptiness were as solid as a brick wall. She went down in a heap, moaning painfully.\n\nArcher's expression of triumph exploded into fury as his daughter was flung backwards. He stalked forward and hissed as he put his hand out and hit solidness. The air surrounding him crackled, as if infused with electricity.\n\n\"What's going on?\" Sam asked.\n\n\"A little insurance I took out,\" Blount said. \"I wasn't sure about his condition, but I wasn't about to take any chances. That's a containment spell...very old and unbreakable.\" Blount pointed out the sigils he'd drawn in a circle around the crevice.\n\n\"When did you do that?\"\n\n\"When I traveled here earlier.\"\n\n\"You knew there was something wrong with Archer and still brought him here?\"\n\n\"I had to be sure. Back at Rugby Rock when Archer touched Casey, the entity transferred to him. It's been growing inside, getting stronger, and clouding my mind. It wanted me to bring Archer here so the soul and body could reunite. I wasn't aware of this on the surface, but I've suspected it.\"\n\nBlount said to Archer, \"Thaddeus, you can still come out of this. It's still your choice. It doesn't have the power to bend free will. Just tell it to leave.\"\n\n\"It's too strong. I have no choice!\"\n\n\"There's always a choice. Now choose.\"\n\n\"You're too late, human,\" Archer growled.\n\nBlount pulled the vortex enclosure from his pocket and brandished it. \"I also have a choice...I'm sorry.\"\n\nBlount rushed the circle, screaming as he broke the boundary of the sigils. Archer gripped Blount's right hand and cringed away from the vortex enclosure. Blount brought a knee crashing into Archer's gut and the man stumbled backwards. Seizing this opportunity, Blount savagely kicked at Archer's left knee. The man screamed in agony as the knee bent at an unnatural angle. Blount shoved the vortex enclosure into Archer's mouth and pushed him backwards.\n\nArcher teetered on the edge of the crevice, waving his arms wildly to maintain his balance. Then the edge was empty. Archer's scream filled the chamber as he fell into the pit.\n\nBlount threw himself out of the way, sliding across the dirt floor as oblivion bloomed within the protective dome of the containment spell's barriers. No sound escaped the dome, and Sam watched in horror as the air folded in on its self then shrank to a pinprick of darkness. The pinprick lasted seconds, then exploded in a blinding flash of matter and reality. She shielded her eyes with her arms and stumbled backwards into the darkness of unconsciousness.\n\n# Chapter Eight\n\nAbdiel froze as he sensed the explosion. He was hovering at the top of one of the spires, still not having any luck gaining entry. After touching down on the dome of the spire, he wept. Wept for man, wept for the weight of the choice Blount had made, wwept for a long time. God hadn't shown, and Abdiel wondered if He ever would.\n\nHe raised his head as someone stumbled from the opening far below. His wings fluttered in the breeze as he stood, tilting his head.\n\nSomething was different. The world had changed. The pall of death and destruction that had clothed the earth was gone. He lifted off and glided down to Blount's side.\n\nBlount glanced up and offered a tired smile. \"It's finished.\"\n\n\"It's finished.\" The angel took Blount into his arms and embraced him.\n\n_This time_ , Abdiel thought.\n\n# About the Authors\n\nJohn Grover is a fiction author residing in Massachusetts. He completed a creative writing course at Boston's Fisher College and is a member of the New England Horror Writers Association. Some of his more recent credits include stories in _Best New Werewolf Tales Vol 1_ by Books of the Dead Press, _The Epitaphs Anthology_ by The New England Horror Writers, _The Northern Haunts Anthology_ by Shroud Publishing, and _The Zombology_ series by Library of the Living Dead Press. He is the author the new online post-apocalyptic series _Aftermath_ and of several collections, including the recently released _Creatures and Crypts_ as well as various chapbooks, anthologies, and more. Please visit his website ShadowTales.com for more information.\n\nR. Thomas Riley, an active member of International Thriller Writers Association, is an internationally published author living in the distant lands of North Dakota. His short story collection _The Monster Within Idea_ was published by Hugo Nominated Apex Publications and re-released as an eBook. _The Flesh of Fallen Angels:A Gibson Blount Novella_ (co-written with Roy C. Booth) was published by Grand Mal Press in 2012. _Diaphanous_ (co-written with Roy C. Booth) is available now. _The Day Lufberry Won It All_ was adapted to short film by Frosty Moon Omnimedia in 2010. You can visit his web home at RThomasRiley.com.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n## PLAN OF THE COLLECTED WORKS  \nEdited by Bruce Caldwell\n\nVolume I: The Fatal Conceit: The Errors of Socialism (1988)\n\nVolume II: The Road to Serfdom: Text and Documents (2007)\n\nVolume III: The Trend of Economic Thinking: Essays on Political Economists and Economic History (1991)\n\nVolume IV: The Fortunes of Liberalism and the Austrian School: Essays on Austrian Economics and the Ideal of Freedom (1992)\n\nVolume V: Good Money, Part I: The New World (1999)\n\nVolume VI: Good Money, Part II: The Standard (1999)\n\nVolume VII: Business Cycles, Part I\n\nVolume VIII: Business Cycles, Part II\n\nVolume IX: Contra Keynes and Cambridge: Essays, Correspondence (1995)\n\nVolume X: Socialism and War: Essays, Documents, Reviews (1997)\n\nVolume XI: Capital and Interest\n\nVolume XII: The Pure Theory of Capital (2007)\n\nVolume XIII: Studies on the Abuse and Decline of Reason: Text and Documents (2010)\n\nVolume XIV: The Sensory Order and Other Essays\n\nVolume XV: The Market and Other Orders\n\nVolume XVI: John Stuart Mill and Harriet Taylor\n\nVolume XVII: The Constitution of Liberty: The Definitive Edition\n\nVolume XVIII: Essays on Liberty\n\nVolume XIX: Law, Legislation, and Liberty\n\n_Supplement_ : Hayek on Hayek: An Autobiographical Dialogue (1994)\n\nThe plan is provisional. Minor alterations may occur in titles of individual books, and several additional volumes may be added.\n\n# THE COLLECTED WORKS OF F. A. Hayek\n\nVOLUME XVII\n\n# THE CONSTITUTION OF LIBERTY\n\n_The Definitive Edition_\n\nEDITED BY RONALD HAMOWY\n\nThe University of Chicago Press\nRONALD HAMOWY is professor of history emeritus at the University of Alberta and affiliate professor of economics at George Mason University. He is a fellow in social thought at the Cato Institute and has written numerous articles and books on social theory, including _The Scottish Enlightenment and Spontaneous Order_ and _The Political Sociology of Freedom: Adam Ferguson and F. A. Hayek._ He is the editor of the Liberty Press edition of _Cato's Letters_.\n\nThe University of Chicago Press, Chicago 60637  \nThe University of Chicago Press, Ltd., London  \n\u00a9 1960, 2011 by the University of Chicago  \nAll rights reserved. Published 2011  \nPrinted in the United States of America\n\n20 19 18 17 16 15 14 13 12 11 1 2 3 4 5\n\nISBN - 13: 978-0-226-31537-9 (cloth)  \nISBN - 13: 978-0-226-31539-3(paper)  \nISBN - 10: 0-226-31537-1 (cloth)  \nISBN - 10: 0-226-31539-8 (paper)\n\nLibrary of Congress Cataloging- in- Publication Data\n\nHayek, Friedrich A. von (Friedrich August), 1899\u20131992.  \nThe constitution of liberty : the definitive edition \/ edited by Ronald Hamowy.  \np. cm. \u2014 (The collected works of F. A. Hayek ; v. 17)  \nISBN - 13: 978-0-226-31537-9 (cloth : alk. paper)  \nISBN - 10: 0-226-31537-1 (cloth : alk. paper)  \nISBN - 13: 978-0-226-31539-3 (pbk. : alk. paper)  \nISBN - 10: 0-226-31539-8 (pbk. : alk. paper) 1. Liberty. 2. Rule of law.  \n3. Social policy. I. Hamowy, Ronald, 1937\u2013 II. Title. III. Series: Hayek, Friedrich A. von (Friedrich August), 1899\u20131992. Works. 1989 ; v. 17.  \nHB171 .H426 1989 vol. 17  \n[JC585.H29]  \n320.01'1\u2014dc22  \n2010020835\n\n The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences\u2014 Permanence of Paper for Printed Library Materials, ansi z39.48- 1992.\n\nISBN 978-0-226-32051-9 (electronic)\n\n## THE COLLECTED WORKS OF F. A. HAYEK\n\nFounding Editor: W. W. Bartley III  \nGeneral Editor: Bruce Caldwell\n\n_Published with the support of_\n\nThe Hoover Institution on War, Revolution, and Peace  \nStanford University  \nThe Cato Institute  \nThe Earhart Foundation  \nThe Pierre F. and Enid Goodrich Foundation  \nThe Heritage Foundation  \nThe Morris Foundation, Little Rock\n\n## CONTENTS\n\nEditorial Foreword\n\nIntroductory Essay\n\n_The Constitution of Liberty_ : Editions and Translations\n\nA Note on the Notes\n\nEditor's Acknowledgments\n\nLiberty Fund Editions Cited\n\n_THE CONSTITUTION OF LIBERTY_\n\nPreface\n\nAcknowledgments\n\nBibliographical Abbreviations\n\nIntroduction\n\n**_PART I: The Value of Freedom_**\n\nOne: Liberty and Liberties\n\nTwo: The Creative Powers of a Free Civilization\n\nThree: The Common Sense of Progress\n\nFour: Freedom, Reason, and Tradition\n\nFive: Responsibility and Freedom\n\nSix: Equality, Value, and Merit\n\nSeven: Majority Rule\n\nEight: Employment and Independence\n\n**_PART II: Freedom and the Law_**\n\nNine: Coercion and the State\n\nTen: Law, Commands, and Order\n\nEleven: The Origins of the Rule of Law\n\nTwelve: The American Contribution: Constitutionalism\n\nThirteen: Liberalism and Administration: The _Rechtsstaat_\n\nFourteen: The Safeguards of Individual Liberty\n\nFifteen: Economic Policy and the Rule of Law\n\nSixteen: The Decline of the Law\n\n**_PART III: Freedom in the Welfare State_**\n\nSeventeen: The Decline of Socialism and the Rise of the Welfare State\n\nEighteen: Labor Unions and Employment\n\nNineteen: Social Security\n\nTwenty: Taxation and Redistribution\n\nTwenty-one: The Monetary Framework\n\nTwenty-two: Housing and Town Planning\n\nTwenty-three: Agriculture and Natural Resources\n\nTwenty-four: Education and Research\n\n**_POSTSCRIPT_**\n\nPostscript: Why I Am Not a Conservative\n\nAnalytical Table of Contents\n\nIndex of Authors Cited\n\nIndex of Subjects\n\n## EDITORIAL FOREWORD\n\nMany scholars view _The Constitution of Liberty_ to be F. A. Hayek's greatest work. It is a great pleasure to present here, as volume 17 in the series, _The Collected Works_ version of the book.\n\nI was delighted when Ronald Hamowy agreed to serve as the editor of the volume. Hamowy did his Ph.D. under Hayek at the Committee on Social Thought at the University of Chicago and was on the scene in 1960 when _The Constitution of Liberty_ was published. Indeed, he wrote one of the first critical reviews of the book, one that so impressed Hayek that he penned a reply.1 Because of his intimate knowledge of the material, Hamowy is in many respects the ideal choice as editor.\n\nThose familiar with the original 1960 version of _The Constitution of Liberty_ will notice some differences between it and _The Collected Works_ edition. The most prominent of these is that the endnotes of the 1960 volume have been trans-formed into footnotes in the present one. The decision to make such a dramatic change was not made lightly. The endnotes ran to over one hundred pages, and there was some fear that when set as footnotes they might overwhelm the text. As I read through the manuscript that Hamowy had prepared, though, it quickly became apparent how useful it was to have the notes immediately available. Hayek's text typically does not provide any clues as to what one is going to find in the endnotes. One would never try to check every one, and because of that, much is missed. The problem was remedied by turning them into footnotes. I have read _The Constitution of Liberty_ a number of times. In looking over Hamowy's manuscript, I learned a number of things I never knew before, simply because I had Hayek's notes right there before me. It greatly enhanced my reading experience and my engagement with Hayek's ideas.\n\nAs he indicates in his \"A Note on the Notes,\" editor Hamowy checked Hayek's notes for accuracy, making additions when Hayek omitted material and silently correcting any bibliographical errors that Hayek may have made.  But Hamowy did much more than this\u2014translating passages, adding more than two hundred citations from the 1971 German edition of the book, and providing explanatory information when appropriate.\n\nIn his notes, Hayek quoted from sources in many different languages, including German, French, Italian, Spanish, Russian, Greek, and Latin. A number of scholars contributed at the copyediting stage, and at very short notice, to ensure that any typos or other errors that crept into the manuscript were corrected. I am indebted to Professors Marina Bianchi, Linda Dan-ford, Hansjoerg Klausinger, Susan Shelmerdine, and Pedro Schwartz for their invaluable and timely assistance.\n\nThere are a number of others who contributed. Kevin Welding and Nicolas Venditti prepared an initial version of the master text, and Chandran Kukathas did some early work on the volume prior to passing on the job of editor to Ronald Hamowy. David Pervin of the University of Chicago Press oversees the whole _Collected Works_ series and has been a frequent source of assistance and sound advice. His counterpart at Routledge, Thomas Sutton, has managed the distribution of the volume outside of North America. Perhaps my greatest debt, however, is to the meticulous and unflappable Rhonda Smith, who brilliantly coordinated and executed the immensely complicated task of copyediting the manuscript.\n\nGiven the new placement of the notes, and the immense amount of work that so many people have put into this volume, we have decided to label _The Collected Works_ version of Hayek's great book _The Definitive Edition_.\n\nBruce Caldwell  \nGreensboro, North Carolina\n\n* * *\n\n1 See Ronald Hamowy, \"Hayek's Concept of Freedom: A Critique,\" _New Individualist Review_ , 1 (April 1961): 28\u201331; F. A. Hayek, \"Freedom and Coercion: Some Comments and Mr. Hamowy's Criticism,\" _New Individualist Review_ , 1 (Summer 1961): 28\u201330, reprinted in _Studies in Philosophy, Politics, and Economics_ (Chicago: University of Chicago Press, 1967), pp. 348\u201350.\n\n## INTRODUCTORY ESSAY\n\nIn September 1989 the Solidarity party, an arm of the Polish anticommunist labor movement, took control of the government in Poland after the party had earlier won all parliamentary seats. In the same month, Hungary opened its borders with Austria, thus permitting huge numbers of refugees to flee Eastern Europe and particularly East Germany. Two months later the Berlin Wall was opened and the East German government collapsed. Also, in the same month that Solidarity achieved a massive election victory in Poland, Alexander Dub\u010dek, who had been taken into custody by occupying Soviet forces in Czechoslovakia in 1968, addressed a rally of 300,000 in Prague. Mounting protests against the communist regime throughout Czechoslovakia finally led to the resignation of its Communist government in late December. These events throughout Eastern Europe soon spread to the Soviet Union where pressures for reform had been building. Finally, in December 1991, the Soviet Union was officially abolished and Russia, the Ukraine, and Byelorussia created the Commonwealth of Independent States, thus bringing to an end seventy-four years of Communist control. Despite the appearance of impregnability, the swiftness with which these governments collapsed is testimony to how corrupt and diseased their internal structures were.\n\nFew Western social theorists foresaw just how feeble the economic frame-work of communist nations in fact was. It had been assumed by millions that planned economies could somehow put an end to the depredations associated with capitalism and could open the way to a more just and fair distribution of wealth and, while it might require temporary sacrifice and hardship, would in the end result in a better world. Nor was this view limited to those living in Eastern Europe. Most Western intellectuals were equally convinced that socialism offered a realistic, and in many way superior, alternative to the free market.\n\nWhile most intellectuals were prepared to accept the fact that there was nothing inherent in socialist economies that prevented this outcome, F. A. Hayek, in a series of penetrating analyses, had demonstrated that such planning was impossible in the absence of a price system such as only free markets could provide. In the absence of prices that accurately reflect people's preferences  for various goods and services, government direction of the economy can only lead to increasing malinvestment and disorder. This constituted a crucial failing that made the ultimate disintegration of communist societies inevitable. Hayek had been preceded in his analysis by his mentor at the University of Vienna, Ludwig von Mises, whose seminal work on socialism was first published in the early 1920s. To those persuaded by the arguments put forward by Mises and Hayek, the collapse of the Communist governments of Eastern Europe came as less of a surprise than to many others. Indeed, the great contribution of these two thinkers is that they demonstrated that government attempts to plan the economy were inevitably doomed to fail.\n\nMises had argued in a seminal article published in 19201 that productive efficiency was contingent on knowing the real prices of the factors of production, since without such prices it would be impossible to know how to rationally allocate resources. With all productive resources owned by a central authority and in the absence of market-generated prices, the calculation of real costs would be impossible and thus render production essentially random. To these conclusions Hayek added the notion that the market was itself essentially a discovery process providing information that would otherwise not exist on the relative value of goods. This information, he contended, could only be supplied by free markets since it was impermanent and widely dispersed among a host of individuals, many of whom were not even aware that they possessed any relevant knowledge, knowledge that emerged only as a product of the market process itself. As one economist has written of Hayek's conclusions: \"Persons embedded in a competitive process can, by virtue of their very rivalry with one another, impart information to the system of relative prices that in the absence of competition they would have no way of obtaining.\"2 Without a price system socialist economies lacked the ability to coordinate the actions of consumers and producers and were thus doomed to substantial misallocations of resources. These insights, together with Hayek's conclusions regarding the business cycle, were on the verge of dominating academic economics when, in the early 1930s, the world found itself in the midst of the  Great Depression. In 1936 John Maynard Keynes published his _General Theory of Employment, Interest, and Money_. Released at the Depression's height, the academic world found in Keynes's recommendations regarding deficit spending and vigorous government activity a formula that had far more appeal than did Hayek's analysis of the causes of the business cycle and the need to allow the market to correct itself without more monetary intervention. The result was that Keynes's theory of underinvestment and underconsumption during periods of slow or negative economic growth came to dominate economic theory for several decades.3\n\nHayek's analysis of the role of the price system and its effect on the operation of socialist societies, however, was not limited to economic issues. Alarmed by the spectacular growth of government involvement in the economy in Great Britain and the United States, in part as a reaction to the Great Depression and the Second World War, Hayek published _The Road to Serfdom_ in 1944, his first work aimed at an audience broader than academic economists. The prevailing orthodoxy during the period held that National Socialism was, in every crucial respect, the antithesis of welfare socialism. Welfare statism had captured the imagination of most intellectuals during the Depression and remained popular during the struggle against Nazi Germany. This view was exacerbated by the barrage of propaganda issued by the allied governments during the war, when it was felt necessary to paint England, the United States, and Stalinist Russia as similar in their approach to economic and social problems, in contrast to Nazi Germany and Fascist Italy. It was generally thought that only through vigorous government intervention was it possible to forestall the more destructive aspects of unbridled capitalism, which, if left unchecked, would bring privation and misery to the great mass of people. Equally important, only government direction could galvanize and coordinate the productive facilities of a nation so as to minimize waste and maximize wealth creation.\n\nReaction to the essay was, with few exceptions, both hostile and swift, both in Britain and in the United States.4 Most of the book's readers were appalled  that Hayek could suggest that any approach to social problems as benign as welfare socialism was similar to a movement as pernicious as was National Socialism. Hayek had contended that distrust of the market and the disdain that was felt for individual decision making were common to both fascism and welfare-statism, which destroyed the spontaneous order inherent in free and undirected markets and led to a wide array of unforeseen and undesired consequences. These, in turn, led to more controls on people's actions and increasingly greater limits on freedom. Public response to _The Road to Serfdom_ doubtless contributed to Hayek's decision to devote more of his time and energies to discussing why socialist societies, by their nature, rested on coercion and to lay bare the principles of a free and open society. The upshot of this decision was _The Constitution of Liberty_ , which was published in 1960, wherein he sought more fully to examine the demarcation between the amount and area of government intervention that he regarded as consistent with a free society and governmental actions that illegitimately encroached on personal liberty.\n\nBruce Caldwell, in his excellent study of Hayek's social and economic thought,5 has suggested that _The Constitution of Liberty_ most likely constituted a part of Hayek's broader project to respond to the increasingly fashionable view that the application of the methodology of the natural sciences to social phenomena, in the form of social planning by a team of experts, could in theory solve all problems of human organization. This conclusion was predicated on the assumption that the laws of human interaction were analogous to the laws of physics, which, once uncovered, would permit the engineering  of social relationships with the same predictability of outcome as obtained with respect to the physical world. To this view Hayek gave the name scientism.6 In addition to being subject to the classic arguments against reductionism, Hayek contended that scientism disregards the fundamental fact that coherent patterns in human affairs are often the result of the interaction of numerous individuals, none of whom sought to achieve the resulting overall end. Or, as Adam Ferguson noted two hundred years earlier, complex social arrangements, while indeed the product of the action of human beings, are not the result of any conscious plan.7\n\nThis insight into the nature of social organization, that the level of complexity of institutions put them beyond the ability of any one mind or group of minds to comprehend and design, pervades all of Hayek's social theory and plays a crucial role in shaping the political conclusions he draws in _The Constitution of Liberty_. What he attempts in this work is nothing less than laying bare the political machinery necessary for a free society, treated in both its historical and philosophical dimensions. This is a monumentally ambitious project and if, in the end, Hayek occasionally falters and slips, as he indeed does, these failures are more reflections of the complexity of his enterprise than of weaknesses in his reasoning.\n\nAt no point in his autobiographical writings does Hayek indicate when he originally conceived of writing _The Constitution of Liberty_. Caldwell suggests the possibility that Hayek intended it to serve as a response to a challenge laid down by the socialist economist H. D. Dickinson in 1940 that those who opposed a collectivist economic system and embraced free markets were incapable of offering a positive program that would \"guarantee the ordinary man a reasonable security of livelihood and prevent the accumulation of wealth (and, what is still more important, the concentration of power over wealth) in the hands of a minority of the community.\"8 The central problem faced  by proponents of an economic system based on private property wherein the \"ballot of the market\" prevails, Dickinson maintains, \"involves not only outrageous plural voting, but also the wholesale intimidation of the smaller voters by a few great pluralists.\"9 It is a tribute to Dickinson's obtuseness, and to those who shared his sympathies, that they appeared oblivious to how these problems would be avoided in an economy where all decisions concerning the production and distribution of wealth would be centralized in the hands of a few bureaucrats acting under the direction of a central committee. In addition, as Hayek was keenly aware, there is something distastefully na\u00efve in the view that political power is invariably more benign than is economic power.\n\nDespite the problems implied by Dickinson's collectivist alternative, Hayek was determined to reply to the critics who claimed that a free market economy would, if left unchecked, turn the great majority of the population into helots, forced to act at the mercy of a few plutocrats. Caldwell notes that Hayek's interests in political and social theory were part of a broader concern with a larger enterprise that Hayek came to refer to as the Abuse of Reason project. While never completed, the project served to direct him into investigating new areas of thought.10\n\nR. M. Hartwell, a close friend of Hayek's and one of Great Britain's leading economic historians, records, in his history of the Mont P\u00e8lerin Society, that by the time the Society was founded in 1947, Hayek had already \"moved towards the writing of _The Constitution of Liberty_.\"11 And in his autobiographical notes Hayek recounts that the structure of the work occurred to him during a car trip through southern Europe that he and his wife made in 1954\u201355. During that fall and winter, the Hayeks had the opportunity to motor through France, Italy, and Greece following the route taken by John Stuart Mill one hundred years earlier.12 The book had its genesis, according to Hayek's biographer, in early 1953; in November of that year Hayek wrote to the economist Fritz Machlup that he was \"beginning to have definite plans for that positive complement to _The Road to Serfdom_ which people have so long [been] asking me to do.\"13 While touring southern Europe, he had taken this occasion to make a side trip to Cairo to deliver the Commemoration Lectures at the National  Bank of Egypt, and these lectures later formed a segment of _The Constitution of Liberty_. The book's detailed organization, however, took shape during 1954\u201355.\n\nIn the preface to the Cairo lectures, Hayek notes that he was invited to deliver them some time earlier and his comments \"were the tentative results of a study on which I have been engaged for some time but which is not yet concluded. I have availed myself of the opportunity to give an outline of conclusions at which I have arrived though they may still require modification in some respect.\"14 The National Bank of Egypt lectures were to serve as Hayek's treatment of the nature and history of the rule of law. Indeed, Hayek himself noted in his application to the Guggenheim Foundation, to whom he had applied for funding for the 1954\u201355 trip, that one of its purposes was to investigate how spontaneously generated rules and customs developed in more traditional societies, which, he added, was a crucial element of a more extensive study he had been engaged in for years.\n\nThe Cairo lectures, four in number, served as earlier drafts of chapters 11, 12, 13, 14, and 16 of _The Constitution of Liberty_ and are, in most respects, quite close to, albeit truncated versions of, what was finally published in 1960. In fact, the published version of these lectures even contained a good number of footnotes. Lecture one concerns itself with a historical survey of the rule of law and its relation to a free society in both Great Britain and America, topics that received far fuller discussion in chapters 11 and 12. One of the lectures' earliest paragraphs harks back to Hayek's concerns when writing _The Road to Serfdom_. He notes in the lecture that:\n\nThe main reason why I have decided to approach my subject historically is to make you aware how greatly the whole framework of governmental power already has changed, how little the legal position even in the freest countries still corresponds to the ideals and concepts to which we still pay lip-service. I want to draw your attention to a silent revolution, which during the past two or three generations, has proceeded in the sanctuaries of the law largely unobserved by the general public. This revolution has gradually whittled away most of the guarantees of individual liberty for which at one time those people had been willing to fight. It is a peculiar kind of revolution in which what is often regarded as the most conservative of professions, in working out the implications of the popular will, have more completely changed the legal framework of governmental power than either the sovereign people or its representatives ever comprehended. The crucial steps were changes in the juridical attitude on issues which to the laymen must  have appeared nice legal points which only the lawyer could understand or care for, but on which in fact the foundations of their liberty depended.15\n\nHayek then discusses the classical notion of isonomy. He there argues that the claim that individual liberty, as it is understood in the modern world, was unknown to ancient Greece has no merit, certainly not when one looks at Athens during the period of its greatness. This Greek conception was particularly significant in shaping seventeenth-century English political thought, by which it entered Locke's treatises on government and ultimately eighteenth-century British and American theories of liberty.\n\nLecture two is devoted to an analysis of the English tradition of law, which spawned personal freedom, as it was transmitted to readers on the Continent, especially through the works of Montesquieu and Kant. While the German liberal tradition, Hayek noted, crystallized in the notion of the _Rechtsstaat_ , French liberalism was crucially shaped by the events of the French Revolution, which appears to have concluded that \"since at least the control of all power had been placed in the hands of the people, all safeguards against any abuse of that power had become unnecessary.\"16 With certain exceptions, most notably Alexis de Tocqueville and Benjamin Constant, French liberal thinkers abandoned the British notion that social and political institutions should be grounded in those traditional arrangements that were compatible with individual autonomy. In its place it substituted the idea that a free society could, and should, be constructed _de novo_ , without reference to existing social formations. This distinction, which reflects the primary concern of Hayek's _The Counter-Revolution of Science_ ,17 plays a critical role in Hayek's understanding of the nature of freedom.\n\nIn lecture three, Hayek considers the attributes of a legal system consistent with a society operating under the rule of law. \"The end,\" he noted, \"is to limit coercion by the power of the state to instances where it is explicitly required by general abstract rules which have been announced beforehand and which applied [ _sic_ ] equally to all people, and refer to circumstances known to them.\"18 These criteria, Hayek claims, generality, equal applicability, and certainty, constitute the underlying structure of any free society and occupy a crucial role in Hayek's discussion of freedom in _The Constitution of Liberty_. In the same lecture, Hayek introduced the notion of spontaneously generated orders, arguing that not every social arrangement, despite the consistency of its elements, requires that it be the product of a designing intelligence. Indeed,  the knowledge necessary to produce such arrangements \"can exist only dispersed among all the different members of society and can never be concentrated in a single head, or be deliberately manipulated by any man or group of men.\"19 Law particularly is an institution of such complexity that in a free society its particulars can take their form only through an evolutionary process, the result of which is a consistent set of rules that is the product of each of us pursuing our individual ends. Both these themes, the formal nature of the rule of law and the claim that law in a free society results from a spontaneously generated order, were central to Hayek's discussion of the constitution of liberty.\n\nAfter laying bare the elements of a society in which people can be said to be free, Hayek turns to a discussion of the arguments put forward by the opponents of the rule of law in lecture four. Socialists, he notes, have always objected to the principle of equality of legal treatment inasmuch as treating people, who are inherently unequal, equally will invariably result in inequality. What socialists have sought ever since the French Revolution is not equality before the law but rather equality of outcome. Their contempt, he argues, is succinctly encapsulated in Anatole France's reference to \"the majestic equality of law, which prohibits both rich and poor from sleeping under bridges, begging in the streets, and stealing bread.\"20 This attitude pervades modern social and political theory and is a logical extension of the view that sufficient proof of the unfairness of any society is that the outcomes that befall the individuals that comprise it are unequal.\n\nIn addition, particularly in Great Britain and the United States, the transformation of governmental edicts from general rules enacted by the legislature to the _ad hoc_ directives of regulatory agencies, whose decisions in any particular instance are unpredictable, enlarges and intensifies the arbitrary nature of our interactions with the state. In particular, large areas of the economy are increasingly shaped by the decisions of these tribunals, whose outcomes cannot be foreseen. The result is not only the deterioration of the rule of law but also the erosion of the spontaneous order of the market that makes rational investment possible.\n\nThe arguments that Hayek put forward in the Cairo lectures in 1955 served as a _pr\u00e9cis_ of the theoretical portions of _The Constitution of Liberty_ , which he completed four years later. The purpose of both works is, as Hayek himself maintains, \"to assist the formation of a spontaneous order and to restrict the use of coercion as much as possible\" and thus \"to adjust our rules so as to make the spontaneous forces of society work as beneficially as possible.\"21 In  the period between 1955 and 1959, Hayek applied himself to expanding the Cairo lectures into a systematic presentation of the nature of free societies, taken in both their historical and theoretical dimensions. During that period he published a number of articles in both English and German that were to serve, in expanded and somewhat revised form, as chapters in _The Constitution of Liberty_. In 1956 Hayek's comments on progressive taxation, sections of which were reprinted as chapter 20, appeared in a collection of essays issued in honor of Ludwig von Mises22 and in the following year a slightly longer version of chapter 3, on \"the common sense of progress,\" was published in the German journal _Ordo_.23 In 1958 three more chapters were released, the first a version of chapter 4, on \"freedom, reason, and tradition,\" in _Ethics_ ; the second, incorporated into chapter 6, on \"equality, value, and merit,\" which was published in _Ordo_ ; and the third, which was to serve as a draft of chapter 2, on \"the creative powers of a free civilization,\" as part of an anthology edited by the noted journalist Felix Morley.24 Finally, in 1959, the year in which Hayek completed the manuscript of _The Constitution of Liberty_ , two further sections, chapter 5 on \"responsibility and freedom\" and chapter 8 on \"employment and independence,\" were published, the first in an anthology and the second in the periodical _Schweizer Monatshefte_.25 These essays, together with the material that earlier appeared in _The Political Ideal of the Rule of Law_ , constitute approximately one half of the larger work. Finally, on May 8, 1959, his sixtieth birthday, Hayek completed the preface to what was to prove his most important work in social theory and submitted the manuscript of _The Constitution of Liberty_ to the University of Chicago Press.\n\nHayek divides his monograph into three parts. The first is devoted to a discussion of the value of freedom, the second to the connection between freedom and law, and the third to personal liberty and its relation to the welfare  state. Sections one and two are primarily philosophical and, to a lesser degree, historical, while the arguments put forward in section three are, in the main, economic. To this discussion Hayek appends a massive scholarly apparatus, comprising no less than 1,900 citations in eight languages, the results of a lifetime of scholarship.\n\nHayek begins his analysis of the nature of a free society by attempting to define personal freedom. One is free, he maintains, when one is not coerced. And coercion, he continues, \"occurs when one man's actions are made to serve another man's will, not for his own but for the other's purpose,\"26 but only when the possibility of alternative action is open and only when that alternative action serves the other person's desires. As Hayek puts it, coercion implies \"that I still choose but that my mind is made someone else's tool, because the alternatives before me have been so manipulated that the conduct that the coercer wants me to choose becomes for me the least painful one.\"27 It is clear that what Hayek wishes to do is to construct a definition of freedom that precludes only those actions that aim at placing others in positions where they feel forced to act in a specific way and where there exists no effective alternative to their so acting. While there are theoretical problems with this formulation if interpreted in its strictest sense,28 it can at the least suffice as a rough guide to determining whether people are for the most part free.\n\nThis definition of coercion serves as the basis of Hayek's extensive discussion of the rule of law. As he previously argued in _The Political Ideal of the Rule of Law_ , the legal rules of any society said to be operating under the rule of law must meet three criteria, that they be general, equally applicable, and certain. \"Law in its ideal form,\" Hayek writes, \"might be described as a 'once-and-for-all' command that is directed to unknown people and that is abstracted from all particular circumstances of time and place and refers only to such conditions as may occur anywhere and at any time.\"29 In a free society, government edicts must take the form of general rules that prohibit specific action, rather than _ad hoc_ commands, and once having been formulated in this way they largely deprive such rules of their coercive nature. \"Insofar as the rules providing for coercion are not aimed at me personally but are so framed as to apply equally to all people in similar circumstances,\" he maintains, \"they are no different from any of the natural obstacles that affect my plans.\"30\n\nMuch has been written regarding Hayek's description of the rule of law and of its intimate connection with a free society, most taking issue with his conclusion that general rules are, at least in their effect, noncoercive.31 The logic of Hayek's argument is such that freedom is a consequence of a certain set of formal restrictions on legislative activity. He writes: \"The conception of freedom under the law . . . rests on the contention that when we obey laws, in the sense of general abstract rules laid down irrespective of their application to us, we are not subject to another man's will and are therefore free.\"32 This clearly freights the rule of law with far too extended a function. For example, it is not too difficult to imagine the rule of law, as Hayek understands it, as perfectly consistent with a regime operating under, say, Islamic religious law, a society which in almost every respect is coercive and lacking in all spontaneity. Indeed, this is the gravamen of most of the reviews of _The Constitution of Liberty_ , which point out that Hayek makes too much of the rule of law as a guardian of personal freedom.\n\nWhy Hayek chose to define freedom under law so broadly is something of a mystery. He had been more guarded in the Cairo lectures where he noted that it was still possible that the rule of law, no matter how carefully one crafts its provisions, was still compatible with \"a great deal of silly and harmful legislation.\"33 And at another point he conceded that it alone is hardly a sufficient condition for a free society. \"The Rule of Law,\" he asserts, \"gives us only a necessary and not a sufficient condition of individual freedom: within the scope that it leaves for legislation and administration these might still become very irksome and harmful. But it still seems to me not only an essential minimum condition of freedom but in practice also to secure what is most important.\"34 However, he removed these qualifications four years later by logically linking personal liberty with the rule of law.35\n\nIn the year following publication of _The Constitution of Liberty_ , Hayek had the occasion to clarify these rather far-reaching claims in response to one of the earliest critiques of his definition of coercion and its logical relation to a society operating under the rule of law. Writing in the _New Individualist Review_ , a publication founded by several of his graduate students at the University of Chicago, Hayek maintained:\n\nIt was not the main thesis of my book that \"freedom may be defined as the absence of coercion.\" Rather, as the first sentence of the first chapter explains, its primary concern is \"the condition of men in which coercion of some by others is reduced as much as is possible in society.\" I believe I am etymologically correct in describing such a state as one of liberty or freedom. But this is a secondary issue. The reduction of coercion appears to me an objective of the first importance in its own right and it is to this task that the book addresses itself.\n\nI sympathize with those who are disappointed with my admission that I know of no way of preventing coercion altogether and that all we can hope to achieve is to minimize it or rather its harmful effects. The sad fact is that nobody has yet found a way in which the former can be achieved by deliberate action. Such a happy state of perfect freedom (as I should call it) might conceivably be attained in a society whose members strictly observed a moral code prohibiting all coercion. Until we know how we can produce such a state all we can hope for is to create conditions in which people are prevented from coercing each other. But to prevent people from coercing each other is to coerce them. This means that coercion can only be reduced or made less harmful but not entirely eliminated.36\n\nHayek's concept of the rule of law is predicated on his belief that rights are neither abstract nor do they exist prior to the establishment of government. Rights, at least as they are understood in the Anglo-Saxon world, are essentially procedural and, as Burke earlier maintained, the product of the evolution of political institutions whose current constitution reflects the growth and arrangement most consistent with our understanding of the nature of a free society. A central aspect of Hayek's conception of rights is that their most crucial element is the manner by which government chooses to intervene  in our lives, which, Hayek suggests, is more significant than the extent of its intervention.\n\nA central premise of _The Constitution of Liberty_ is the distinction Hayek draws between, on the one hand, Anglo-American notions of a liberal society and, on the other, what he describes as the French conception of a free society. The first has as its models the political insights of, among others, Adam Smith, Adam Ferguson, David Hume, and Edmund Burke, while the second is indebted to the French Enlightenment thinkers, particularly Rousseau and the physiocrats. As he had earlier discussed in his 1945 lecture, \"Individualism: True and False,\"37 and which he touches on in his _Political Ideal of the Rule of Law_ ,38 we are heir to two distinct theories of liberty, one of which traces its roots to an empirical, evolutionary approach to politics, the other to a rationalist conception of social life. Hayek offers a succinct analysis of the distinction between these two notions of individualism:\n\nIt is the contention [of the true individualist tradition] that, by tracing the combined effects of individual actions, we discover that many of the institutions on which human achievements rest have arisen and are functioning without a designing and directing mind; that, as Adam Ferguson expressed it, \"nations stumble upon establishments, which are indeed the result of human action but not the result of human design\"; and that the spontaneous collaboration of free men often creates things which are greater than their individual minds can ever fully comprehend. This is the great theme of Josiah Tucker and Adam Smith, of Adam Ferguson and Edmund Burke, the great discovery of classical political economy which has become the basis of our understanding not only of economic life but of most truly social phenomena.\n\nThe difference between this view, which accounts for most of the order which we find in human affairs as the unforeseen result of individual actions, and the view which traces all discoverable order to deliberate design is the first great contrast between the true individualism of the British thinkers of the eighteenth century and the so-called \"individualism\" of the Cartesian school.39\n\nThis emphasis on the historical development of procedural rules is in some ways an extension of Hayek's broader insights into the issue of the dispersal  of knowledge within society. The function of social, and particularly of political, rules is to create the conditions within which the discrete knowledge we each possess regarding how a free society consistent with social order can best flourish. In this project abstract notions of freedom serve no purpose. Hayek is particularly disturbed by the conception of \"social justice,\" the notion that a just society obtains only when society's advantages and assets are evenly distributed among all its members. Most advocates of social justice regard this condition, in which each of us is possessed of our \"fair share\" of the wealth of the community, as the only one consistent with a truly just social order. Thus, the benefits to which we are entitled are as much a function of our existence within a community as of our own efforts. Conversely, the theory embraces the view that should a member of the group be handicapped in some manner or another, efforts should be made to socialize this disadvantage so that all share its burden equally. This view is currently embraced by most social theorists who see in it the overarching standard by which political action should be guided. As Hayek was later to point out in his extensive treatment of the subject, \"it is perhaps not surprising that men should have applied to the joint effects of the actions of many people, even where these were never foreseen or intended, the conception of justice which they developed with respect to the conduct of individuals towards each other.\"40\n\nSocial justice implies nothing less than that the government be given plenary powers to control the distribution of all wealth, of all that is good in society. Rather than providing the same circumstances for all, the state \"should aim at controlling all conditions relevant to a particular individual's prospects and so adjust them to his capacities as to assure him of the same prospects as everybody else.\"41 Previously it had been a central element of our understanding of justice that only those responsible for a particular outcome should be held to account. \"Social justice,\" on the other hand, holds that the whole group of which the victim is a member should be recompensed, while the group to which the perpetrator belongs should all be equally penalized. This is a particularly pernicious aspect of current views of justice, that it can as easily be accomplished should rewards and punishments be visited on collectivities as on individuals. This constitutes a reversion to the most primitive aspects of the Old Testament, prior to the introduction of the idea of personal responsibility, in which the sins of certain individuals issued in punishment of the whole community. It is the antithesis of the idea of justice based on a theory of individual rights that holds that only those responsible for a wrong should be held to account. Doubtless that is why the idea of punishing  hostages is so abhorrent to our sense of fairness and equity and why we have traditionally regarded personal innocence as an absolute bar to punishment.\n\nHayek undertakes his examination of social and political arrangements, including legal rules, by investigating their origins and evolutionary development and their interactions such that when taken together allow a community to adhere. Despite the complexity of these institutions and rules, which become increasingly elaborate as societies evolve, they are both internally consistent and to great extent compatible one with the other. Were they not, societies would prove inherently unstable and would soon disintegrate. These consistencies and compatibilities intuitively suggest that these social arrangements must have been the product of a designing intelligence. However, Hayek argues, it is exactly their complexity that points to the fact that they did not come about through conscious deliberation inasmuch as their level of intricacy is such that they are beyond the capacity of any mind or group of minds to design. Rather, complex social structures originate as a result of numerous discrete individual actions, none of which aims at the formation of coherent social arrangements. Their shape and function, which in the aggregate form ordered arrangements, are the end result of countless individual actions each of which seeks ends distinct from the social patterns that emerge. This is the evolutionary dynamic that makes for viable social and political institutions and creates the conditions for a society governed by the rule of law and not the commands of men.\n\nThe third section of _The Constitution of Liberty_ is devoted to the implications Hayek's theoretical analysis has for issues of public policy, all of especially significant contemporaneous importance. More specifically, Hayek's discussion centers on the rise of the welfare state, labor unions and employment, social security, taxation and redistribution, the monetary framework, housing and town planning, agriculture and natural resources, and education and research. In each of these chapters he examines the effects of existing governmental interventions both in terms of its effects on the principles of the rule of law enunciated in his earlier chapters and its general economic effects. In each instance Hayek evaluates the impact of each of the many incursions into free markets and the effects of their replacement by private arrangements. When Hayek originally wrote these sections in 1959, it was thought unimaginable by many that, for example, rent controls could be dismantled without serious negative effects on the supply of housing for the less affluent, nor was it thought that progressive taxation could have a profound effect on incentives and on the total production of wealth in society. We, reading these chapters fifty years later, are the beneficiaries of several of Hayek's insights into the dangers that follow the erosion of market forces, including the baleful effects in cost and efficiency of agricultural subsidies, the dire consequences that follow from the exceptional immunities accorded labor unions and the long-term dangers of a system of unfunded entitlements.\n\nHayek concludes _The Constitution of Liberty_ with a postscript entitled \"Why I Am Not a Conservative,\" in which he explicates the distinction between the political position he embraces and modern conservatism, with which it might easily be confused. Until the early twentieth century and especially during the nineteenth century, when it most fully flourished as a doctrine governing political life in Europe, the views that coincided with those put forward by Hayek were called \"liberal.\" Liberalism had little in common with contemporaneous conservatism, which was marked by opposition to rapid changes in political and social life, by a propensity to support traditional institutions, and by a distrust of any spontaneously generated arrangements as opposed to those directed by a supervising authority. Most importantly, conservatism as a political philosophy lacks any principles to guide its adherents regarding in which direction society should move. In sum, conservatism tends to favor authority over liberty and over the free interaction of individuals. In the first decades of the twentieth century, as Hayek points out, radicals and socialists in the United States usurped the name \"liberal\" from those who supported free markets and minimal government and by doing so left true liberals without a recognizable designation. The effect has been to mistakenly attach the label \"conservative\" to those who are in fact precursors to nineteenth century liberalism, namely \"Old Whigs,\" which is how Hayek regarded himself.\n\nUpon completion of the printing of _The Constitution of Liberty_ in December 1959, Hayek directed that the University of Chicago Press send out advance copies of the book to some fifty-five fellow academics, journalists, and heads of foundations. In addition, he personally presented copies to another two dozen colleagues at the University of Chicago and to close personal friends. Its official publication date was February 9, 1960. Despite efforts by both Hayek and the Press, the book was reviewed in only a dozen or so journals and periodicals, a shockingly small number given the reception accorded by the public to Hayek's earlier essay, _The Road to Serfdom_. Intellectuals in both Europe and the United States appear to have remained wedded to the view that an extensive welfare state was necessary to insure economic stability and the public's social welfare and that any defense of free markets bordered on the crackpot, unworthy of comment. In the spring of 1960 Hayek wrote a friend that \"so far the response has not been encouraging. If it were not for a few friends like Henry Hazlitt [columnist for _Newsweek_ magazine] and John Davenport [the assistant managing editor of _Fortune_ ], the book would not yet have had a decent review and I shall have to get help of all interested in the principles I have been trying to expound if it is to become widely known.\"42\n\nAmong the reviews that Hayek mentions in his correspondence is that by Sidney Hook, at one time a prominent Marxist and later a staunch defender of the Cold War. For some reason, the editor of the _New York Times Book Review_ selected Hook, a firm supporter of \"intelligent social control,\" to review Hayek's monograph, despite Hook's vigorous bias. The review, as predictable, was quite negative and, given that it appeared so soon after the publication of the book and in such an influential publication, doubtless played a role in the poor reception _The Constitution of Liberty_ originally received. For these reasons it is worth quoting sections of it:\n\nIt is demonstrable that Hayek suffers from the defects of the very rationalism he condemns. His antitheses between tradition and reason, experience and experiment, are analytically untenable and historically unjustifiable. Intel ligent social control always learns from experience and history. It no more need take the form of a Utopian blueprint than concern for history need make a fetish of the past. . . .\n\nThe conception of \"self-regulating forces\" in history and society is largely mythical. We would still be living in a state of slavery had we relied on them. . . .\n\nAs a cautionary voice Mr. Hayek is always worth listening to. He is an intellectual tonic. But in our present time of troubles, his economic philosophy points the road to disaster.43\n\nHayek was particularly upset because Hook's review was in such sharp contrast to several of the letters he received from academics with whose work Hayek was familiar but whom he had not personally met. For example, the eminent mathematician H. B. Phillips, at one time chairman of the mathematics department at the Massachusetts Institute of Technology, whose interests extended to political theory, wrote Hayek a glowing letter, maintaining that \"without doubt this is the greatest book ever written on liberty and liberal views in politics, and its value is greatly increased by the objective form of presentation, entirely free from the emotionalism which usually spoils discussions of this kind.\"44 In mid-March the University of Chicago arranged that _All Things Considered_ , which was broadcast on WTTW, the PBS television station  in Chicago, would devote its program to a discussion of _The Constitution of Liberty_. Among the discussants was Warner Wick, professor of philosophy at the University and an authority on metaphysics and logic, who wrote Hayek: \"I offer this [note] in thanks and appreciation for your book and the occasion which led to my being pushed into reading it. In my humble judgment, it deserves to become a classic statement of a position that few people now think of as even relevant to our times. I hope it shakes many from their dogmatic slumbers.\"45\n\nHayek's essay received only a handful of reviews in the more scholarly journals, but these were mixed. Jacob Viner, possibly the most distinguished historian of economic thought in the United States, wrote in the _Southern Economic Journal_ that _The Constitution of Liberty_ had shown the author to be in principle capable of a major contribution to social theory \"by virtue of his learning, his analytical skills, and his dialectical virtuosity.\"46 However, while Hayek's arguments in favor of limited government are impressive, he is far too doctrinaire in his conclusions. \"He writes,\" Viner maintains,\n\nwith every appearance that in reaching his conclusions he has taken adequately into consideration all the values that are relevant, and all the conjectures that are actually or potentially important except major emergency situations such as war or danger of war. He manages also to reach his conclusions without giving evidence that to do so he had found it necessary to labor with the weighing of competing values. Great as are the merits of his case, they are not overwhelming enough, I think, to explain how Hayek succeeded in reaching substantially unconditional conclusions and in avoiding what is, in social thought, the generally unavoidable and troublesome necessity of coping with major conflicts between values.47\n\nReviewers generally could not help having been impressed by Hayek's erudition and cogency but had serious reservations about his conclusions and largely because of this found them unconvincing.48 Several were essentially negative49 while others\u2014whose comments reduced to ad hominem attacks\u2014  were clearly motivated by the worst kind of ideological bias. Thus, Leslie Lipson, professor of political science at the University of California at Berkeley, ostensibly an authority on the history of political theory, wrote of Hayek: \"His basic principles are a strange medley. They are a blend of the utopian, the nostalgic, and the inhumane\u2014all of which is oddly described as liberalism. . . . There is a strange unrealism in this book. Its author clings to a never-never-world of illusion and doctrine, which not only cannot be created now, but in fact never did exist. He argues for a dream and is a slave to fantasy.\" Lipson concludes that \"it is sad to discover that _The Constitution of Liberty_ , as one man sees it, becomes the institution of private selfishness.\"50\n\nBritish academics were more favorably disposed towards Hayek's book than were those in America. In February 1961, Lord Robbins offered an on the whole quite positive review of _The Constitution of Liberty_ in one of the leading British economics journals. He wrote in part:\n\nThis is a book which certainly rises to the high plane of the matters with which it elects to deal and which, by reason both of the depth of its analysis and the width of its learning, must surely take an honourable place among the standard works on the subject. Nor is the tone less impressive than the content. . . .\n\n[ _The Constitution of Liberty_ ] is a work which surely no one with even a bare minimum of magnanimity and sense of what is fine can read with-out gratitude and admiration\u2014gratitude for a splendid contribution to the great debate, admiration for the moral ardour and intellectual power which inspired it and made it possible.51\n\nThis more positive public reception of _The Constitution of Liberty_ , especially in Great Britain, was encouraged by the appearance, in the early summer of 1961, of a collection of essays centering on Hayek's book. The work was published by the Institute of Economic Affairs in London52 and comprised ten essays edited by Arthur Seldon, whose aim was to make Hayek's monograph more easily accessible to a larger audience. Each essay is devoted to a particular aspect of _The Constitution of Liberty_ , explicating and expanding on Hayek's discussion: the individual in society, philosophy, law, objectives, the monetary framework, taxation, social welfare, the legal status of trade unions, human rights in industry, and agriculture. While the authors, scholars of some standing, were, for the most part, sympathetic to Hayek's conclusions, they were not uncritical where they regarded Hayek's arguments as flawed  or as warranting greater precision. Possibly the book's most interesting essay is that on Hayek's political philosophy, written by J. W. N. Watkins, reader in the history of philosophy at the University of London. In a tightly argued article Watkins points out the theoretical problems with Hayek's notion of coercion and with the relationship between freedom and the rule of law. But despite these telling criticisms, Watkins was enthusiastic about Hayek's monograph. He writes: \"In any circumstances _The Constitution of Liberty_ would have been an important book. Given the condition of political philosophy in the English-speaking world today, it is outstandingly important.\"53\n\nWhile the initial reception accorded _The Constitution of Liberty_ , with a few notable exceptions, was somewhat tepid, its reputation as a major treatise on law and politics grew in the years following its publication, in part because of the recognition accorded Hayek's other writings and in part because of the ingenuity of Hayek's arguments. Over the course of the next fifteen years Hayek's views were often cited in discussions in books and articles in economics, political theory, and law. The originality and insights that he brought to bear on crucial questions in these fields were increasingly appreciated as his analysis took on an authority with which serious scholars were expected to contend. Between 1960 and 1974, several hundred articles were published that referred to Hayek's published work. In addition, several books centering on Hayek's thought appeared during that period, among them _Agenda for a Free Society_ , a critical evaluation of the arguments put forward in _The Constitution of Liberty_ , surely a signal honor accorded a recently published book. As the _Times Literary Supplement_ pointed out: \"Few writers earn the distinction of having one of their books singled out as the subject of a compilation of laudatory criticisms within a year of its publication.\"54 All this, together with the growing respectability of Hayek's economic and political conclusions eventuated in his being awarded the Nobel Prize in 1974 for his \"penetrating analysis of the interdependence of economic, social, and institutional phenomena.\"\n\nHayek's reputation continued to grow between 1974 and his death in 1992, accelerated both by the international recognition accorded him and by the events in eastern Europe that offered startling proof of Hayek's claims regarding the impossibility of rational calculation in socialist economies. Within a few years of his being awarded the Nobel Prize, a number of books were published that offered an overview of Hayek's work, the best of which are those by Norman Barry, released in 1979, and by John Gray, which appeared in 1984,  both academics sympathetic to Hayek's conclusions.55 Since he received the Nobel Prize, Hayek's international reputation has reached a level unmatched by all but a few scholars. Over the course of the next two decades _The Constitution of Liberty_ has been translated into all the major European languages in addition to Chinese and Japanese.56 At present there are no less than 250 books and separately bound shorter monographs in twelve languages devoted to Hayek's work.\n\nWhen, in the spring of 1959 Hayek completed the manuscript of the _Constitution of Liberty_ , he must have realized that it was and would continue to be his most ambitious and important work, in which he set out to sketch the structural outlines of a free society that would provide for the greatest amount of individual liberty consistent with the complexities of modern life. He would have been the first to concede that there were flaws in his analysis, some of them perhaps grave. However, inasmuch as the history of the first half of the century had offered every indication that the nations of the world had settled on a path that would eventuate in the victory of the total state, the need to offer some alternative to this alarming course appeared especially acute. At this point, fifty years later, there seems little doubt that this book, if it did not reverse that process, at least slowed it considerably. We are, all of us who value freedom and an open society and who believe in the primacy of the individual over the state, greatly indebted to Professor Hayek for _The Constitution of Liberty_ , thus bringing to bear his tremendous erudition in the cause of liberty.\n\nRonald Hamowy  \nRockville, Maryland\n\n* * *\n\n1 \"Die Wirtschaftsrechnung im sozialistischen Gemeinwesen,\" _Archiv f\u00fcr Sozialwissenschaft und Sozialpolitik_ , 47 (1920): 86\u2013121. The article was translated into English in 1935 and published under the title \"Economic Calculation in the Socialist Commonwealth,\" published in _Collectivist Economic Planning: Critical Studies of the Possibilities of Socialism_ , F. A. Hayek, ed. (London: Routledge and Kegan Paul), pp. 87\u2013130. Mises's conclusions were expanded two years later in _Die Gemeinwirtschaft: Untersuchungen \u00fcber den Sozialismus_ ( Jena: Gustav Fischer, 1922), translated into English by Jacques Kahane in 1936 as _Socialism: An Economic and Sociological Analysis_ (London: Jonathan Cape).\n\n2 Don C. Lavoie, \"Economic Calculation and Monetary Stability,\" _Cato Journal_ , 3, no. 1 (Spring 1983): 164. Hayek first discusses the question of economic calculation in \"The Nature and History of the Problem,\" pp. 1\u201340, and \"The Present State of the Debate,\" pp. 201\u201343, in _Collectivist Economic Planning_ , F. A. Hayek, ed.\n\n3 The claim that Hayek's writings in political and social theory reflected a rigidity that fatally compromised his conclusions is without merit. Nor were his arguments in the field of economics \"muddled.\" To contend, as does Robert Skidelsky (\"Hayek versus Keynes: The Road to Reconciliation,\" in _The Cambridge Companion to Hayek_ , Edward Feser, ed. [Cambridge: Cambridge University Press, 2006], pp. 82\u2013110), that these failings are what account for the success of Keynesianism while Austrianism was relegated to the margins of the discipline is to misconstrue the political history of the1930s, when massive government intervention in all aspects of social and economic life became fashionable, and the attractions of Keynesianism to professional economists who saw in Keynes's conclusions an opportunity to henceforth play prominent roles in shaping fiscal policy.\n\n4 Indeed, a new low in academic discourse was probably set by Herman Finer, university professor of political science at the University of Chicago, whose venomous book, _The Road to Reaction_ (Boston: Little, Brown and Co., 1945), was written as a response to Hayek's book. The following quotation from chapter 2, entitled \"The Reactionary Manifesto,\" will give some idea of how scurrilous Finer's essay is. He writes: \"Here is a joy for all conservatives. In spite of the world's desperate travail to overthrow Hitler and Mussolini and what they stood for, many conservatives need the new joy because secretly they have just lost the old one.\n\n\"We now live in a world without Hitler. His removal has swept away the inhibition against open avowal of his doctrines of contempt for the majority and equality and popular sovereignty. There will be a babel of antidemocratic statements within a few months; murmurings can already be heard. For a time the bitterness of the reactionaries has been merely bridled, out of expediency, while the power and repute of the majority have been magnified, because it is the majority that fights world wars\" ( pp. 15\u201316). There follow another 212 pages containing a seemingly endless series of ad hominem assaults on Hayek's scholarship and motives in writing _The Road to Serfdom_. Despite the unscholarly nature of Finer's attack, his colleague at Chicago, Charles E. Merriam, in his review of Finer's essay, referred to it as \"highly skilled\" and to Hayek's book as \"an overrated work of little permanent value.\" (Review of Barbara Wootton, _Freedom Under Planning_ , and Herman Finer, _The Road to Reaction_ , in _American Political Science Review_ , 60 [1946]: 133, 135.) It is interesting that almost three-quarters of a century after Finer's diatribe first saw print, this mediocre academic is remembered solely because of the malevolence of his condemnation of Hayek's essay.\n\n5 _Hayek's Challenge: An Intellectual Biography of F. A. Hayek_ (Chicago: University of Chicago Press, 2004), pp. 288\u201389.\n\n6 F. A. Hayek, \"Scientism and the Study of Society,\" _Economica_ , n.s., 9 (1942): 267\u201391; n.s., 10 (1943): 34\u201363; n.s., 11 (1944): 27\u201339; reprinted in _The Counter-Revolution of Science: Studies on the Abuse of Reason_ (2nd ed.; Indianapolis, IN: Liberty Fund, 1979), pp. 19\u2013363, and _Studies on the Abuse and Decline of Reason_ , _Collected Works_ edition, pp. 75\u2013166.\n\n7 As Ferguson put it in 1767: \"The establishments of men, like those of every animal, are suggested by nature. . . . [They] arose from successive improvements that were made, without any sense of their general effect; and they bring human affairs to a state of complication, which the greatest reach of capacity with which human nature was ever adorned, could not have projected.\" _Essay on the History of Civil Society_ , Fania Oz-Salzberger, ed. (new ed.; Cambridge: Cam-bridge University Press, 1996), p. 174.\n\n8 H. D. Dickinson, \"Book Review: _Freedom and the Economic System_ ,\" _Economica_ , n.s., 7 (November 1940): 437. In the year prior to writing this review of Hayek's essay, Dickinson had published _The Economics of Socialism_ (Oxford: Oxford University Press, 1939), in which he argued that socialist economies were quite capable of replicating the economic calculations that are required to establish a price system.\n\n9 Dickinson, \"Book Review,\" p. 436.\n\n10 Caldwell, _Hayek's Challenge_ , p. 181.\n\n11 R. M. Hartwell, _The History of the Mont P\u00e8lerin Society_ (Indianapolis, IN: Liberty Fund, 1995): p. 16.\n\n12 F. A. Hayek, _Hayek on Hayek: An Autobiographical Dialogue_ , Stephen Kresge and Leif Wenar, eds. (Chicago: University of Chicago Press, 1994), pp. 129\u201330.\n\n13 Quoted in Ebenstein, _Hayek's Journey: The Mind of Friedrich Hayek_ (New York: Palgrave Macmillan, 2003), pp. 141\u201342. \"The work that eventually would become _The Constitution of Liberty_ would be titled, Hayek wrote in the 1953 letter to Machlup, 'Greater than Man: The Creative Powers of a Free Civilization.' It would be composed of parts titled 'The Role of Reason,' 'The Role of Morals,' 'The Role of Force,' and 'The Role of Material Resources.'\"\n\n14 F. A. Hayek, _The Political Ideal of the Rule of Law_ , (Cairo: National Bank of Egypt, Fiftieth Anniversary Commemorative Lectures, February 1955), preface.\n\n15 Ibid., p. 3.\n\n16 Ibid., p. 17.\n\n17 F. A. Hayek, _Studies on the Abuse of and Decline of Reason_ , _Collected Works_ edition, pp. 75\u2013166.\n\n18 _The Political Ideal of the Rule of Law_ , p. 34.\n\n19 Ibid., p. 31.\n\n20 \"La majestueuse \u00e9galit\u00e9 des lois, qui interdit aux riches comme aux pauvres de coucher sous les ponts, de mendier dans les rues et de voler du pain.\" _Le Lys Rouge_ (Paris: Calmann-L\u00e9vy, 1894), pp. 117\u201318; quoted in _Political Ideal of the Rule of Law_ , p. 50.\n\n21 _Political Ideal of the Rule of Law_ , p. 58.\n\n22 \"Progressive Taxation Reconsidered,\" in _On Freedom and Free Enterprise: Essays in Honor of Ludwig von Mises_ , Mary Sennolz, ed. (Princeton, NJ: D. Van Nostrand Company, 1956), pp. 265\u201384.\n\n23\"Grundtatsachen des Fortschritts\" [The Fundamental Facts of Progress], _Ordo: Jahrbuch f\u00fcr die Ordnung von Wirtschaft und Gesellschaft_ , 9 (1957): 19\u201342. _Ordo_ , in which Hayek frequently published, was founded in 1948 by Walter Eucken and Franz B\u00f6hm, both of the University of Freiburg, and was Germany's leading neoliberal journal, a consistent opponent of central planning and defender of unhindered markets. See Henry R. Oliver, Jr., \"German Neoliberalism,\" _Quarterly Journal of Economics_ , 74 (1960): 117\u201349.\n\n24 \"Freedom, Reason, and Tradition,\" _Ethics_ , 68 (1958): 229\u201345; \"Gleichheit, Wert und Verdienst\" [Equality, Value, and Merit], _Ordo: Jahrbuch f\u00fcr die Ordnung von Wirtschaft und Gesellschaft_ , 10 (1958): 5\u201329; and \"The Creative Powers of a Free Civilization,\" in _Essays on Individuality_ , Felix Morley, ed. (Philadelphia: University of Pennsylvania Press, 1958), pp. 183\u2013204 [Liberty Fund edition, pp. 261\u201389].\n\n25\"Verantwortlichkeit und Freiheit\" [Responsibility and Freedom], in _Erziehung zur Freiheit_ , Albert Hunold, ed. (Erlenbach-Z\u00fcrich: Eugen Rentsch, 1959), pp.147\u201390, and \"Freiheit und Unabh\u00e4ngigkeit\" [Freedom and Independence], _Schweizer Monatshefte_ , 39 (May 1959): 89\u2013103.\n\n26 This volume, p. 199.\n\n27 Ibid.\n\n28 See my \"Freedom and the Rule of Law in F. A. Hayek,\" _Il Politico_ , 36 (1971): 349\u201376, in which I argue that Hayek's claim that the rule of law is a sufficient condition for a free society is untenable inasmuch as the criteria he posits for the rule of law are, if analyzed, consistent with a regime that even Hayek would regard as clearly unfree.\n\n29 This volume, p. 218.\n\n30 This volume, p. 210.\n\n31 See especially Professor Bruno Leoni's penetrating study _Freedom and the Law_ (Princeton, NJ: D. Van Nostrand Company, 1961). At one point Leoni writes: \"We cannot help admitting that general rules, precisely worded (as they can be when written laws are adopted), are an improvement over the sudden orders and unpredictable decrees of tyrants. But, unfortunately, all this is no assurance that we shall be actually 'free' from interference by the authorities\" ( p. 75).\n\n32 This volume, p. 221.\n\n33 _Political Ideal of the Rule of Law_ , p. 47.\n\n34 Ibid., p. 46.\n\n35 J. W. N. Watkins has suggested one reason why Hayek made such far-reaching claims for the rule of law. He writes: \"Hayek has been over-impressed by the following logical consideration: a prohibition leaves an agent free to act in any of the indefinitely large number of ways compatible with not acting in the prohibited way, whereas a positive command leaves him unfree to act in any of the indefinitely large number of ways incompatible with acting in the commanded way. This seems to suggest that a prohibition is infinitely less coercive than a command. But we must not be dazzled by the largeness of the number of alternative courses left open by a prohibition. After all, the agent can select only one of them. To measure the degree of penalisation which a prohibition involves, what we have to weigh against the prohibited alternative is not the whole class of unprohibited alternatives but just the unprohibited alternative which he dislikes least.\" \"Philosophy,\" in _Agenda for a Free Society: Essays on Hayek's \"The Constitution of Liberty,\"_ Arthur Seldon, ed. (London: Published for the Institute of Economic Affairs by Hutchinson, 1961), p. 39.\n\n36 \"Freedom and Coercion: Some Comments on a Critique by Mr. Ronald Hamowy,\" in the _New Individualist Review_ , 1 (1961): 28; reprinted in _Studies in Philosophy, Politics, and Economics_ (Chicago: University of Chicago Press, 1967), pp. 348\u201349.\n\n37\"Individualism: True and False,\" in _Individualism and Economic Order_ (London: University of Chicago Press, 1948), pp. 1\u201332. The essay, originally delivered as the twelfth Finlay Lecture at University College, Dublin, on December 17, 1945, and published by Hodges, Figgis, and Company, Dublin, was later reprinted as the prelude to _Studies on the Abuse and Decline of Reason_ , pp. 46\u201374.\n\n38 _Political Ideal of the Rule of Law_ , pp. 11\u201318.\n\n39 \"Individualism: True and False,\" in _Individualism and Economic Order_ , pp. 6\u20138, and _Studies on the Abuse and Decline of Reason_ , pp. 52\u201354.\n\n40 _Law, Legislation, and Liberty_. Vol. 2, _The Mirage of Social Justice_ (Chicago: University of Chicago Press, 1976), p. 62.\n\n41 This volume, p. 155.\n\n42 Hayek to William H. Brady, Jr., Chicago, March 22, 1960. Three weeks earlier Hayek had written to John Davenport: \"What had made me despondent was particularly that the _Wall Street Journal_ , on which I had placed much hope, has so far taken no notice. Is there a chance that _Time_ or perhaps even _Life_ may do something? . . . Apart from the discussions [by my friends at _Newsweek_ and the _Journal of the American Medical Association_ ] and a few flattering notices by local dailies there has so far been only Sidney Hook in the _New York Times_.\" Chicago, March 4, 1960. Friedrich von Hayek Collection. Hoover Institution Archives, box 62, folder 7. Some two weeks following the date of this letter, on March 16 and 17, the _Wall Street Journal_ reprinted two excerpts from Hayek's book.\n\n43 \"Of Tradition and Change: _The Constitution of Liberty_ by F. A. Hayek,\" _New York Times Book Review_ , February 21, 1960, p. 6.\n\n44 H. B. Phillips to F. A. Hayek, South Lincoln, MA, March 30, 1960. Friedrich von Hayek Collection. Hoover Institution Archives, box 62, folder 7.\n\n45 Warner Wick to F. A. Hayek, Chicago, March 16, 1960. Friedrich von Hayek Collection. Hoover Institution Archives, box 62, folder 7.\n\n46 \"Hayek on Freedom and Coercion,\" _Southern Economic Journal_ , 27 (1961): 236.\n\n47 Ibid., p. 230.\n\n48 See, e.g., Pierre Bauchet, \" _The Constitution of Liberty_ ,\" _Revue \u00e9conomique_ , 13 (1962): 143\u201344; Charles S. Hyneman, \" _The Constitution of Liberty_ ,\" _Midwest Journal of Political Science_ , 5 (1961): 86\u201387; H. B. Mayo, \" _The Constitution of Liberty_ ,\" _Canadian Journal of Economics and Political Science_ , 27 (1961):115\u201316; and Edmund L. Pincoffs, \" _The Constitution of Liberty_ ,\" _Philosophical Review_ , 70 (1961): 433\u201334.\n\n49 Sidney C. Sufrin, \"Some Reflections on Hayek's _The Constitution of Liberty_ ,\" _Ethics_ , 71 (1961): 201\u20134.\n\n50 Leslie Lipson, \" _The Constitution of Liberty_ ,\" _American Political Science Review_ , 54 (1960): 1008.\n\n51Lord Robbins [Lionel Robbins], \"Hayek on Liberty,\" _Economica_ , n.s., 28 (1961): 66, 81.\n\n52 Arthur Seldon, ed., _Agenda for a Free Society: Essays on Hayek's \"The Constitution of Liberty.\"_\n\n53 \"Philosophy,\" _Agenda for a Free Society_ , 31. Watkins was not alone in praising Hayek. Writing in the same anthology, Arthur Shenfield, the economic director of the Federation of British Industries, maintained that \" _The Constitution of Liberty_ is one of the great books of our time, profound in analysis, ample in scholarship, noble in spirit\" (\"Law,\" _Agenda for a Free Society_ , 51).\n\n54 \"The Rule of Law,\" _Times Literary Supplement_ , October 20, 1961.\n\n55 Norman P. Barry, _Hayek's Social and Economic Philosophy_ (London: Macmillan, 1979), and John Gray, _Hayek on Liberty_ (Oxford: Basil Blackwell, 1984). See also Gray's extended essay \"F. A. Hayek and the Rebirth of Classical Liberalism,\" _Literature of Liberty_ , 5 (1982): 19\u2013101. Alas, since writing his essay on Hayek, Gray appears to have abandoned his earlier libertarian leanings and has embraced a somewhat dimwitted mix of fashionable and often contradictory views on the need for brotherhood directed by an authoritarian state. His attacks on what he calls the \"Enlightenment Project\" are, in the end, simply recapitulations of the old arguments that individuals cannot be trusted to make their own decisions and that a kind and beneficent government, directed by right-thinking bureaucrats, is far more likely to establish a livable and sustainable society than are individuals left to their own devices.\n\n56 See _The Constitution of Liberty_ Editions and Translations section of this volume.\n\n## _THE CONSTITUTION OF LIBERTY:_ EDITIONS AND TRANSLATIONS\n\n**English**\n\n_The Constitution of Liberty_. Chicago: University of Chicago Press, 1960. x + 568 pages.\n\n_The Constitution of Liberty_. Phoenix edition, paperback. Chicago: University of Chicago Press, paperback, 1978. x + 568 pages.\n\n_The Constitution of Liberty_. Gateway edition, paperback. Chicago: Regnery, 1972. x + 568 pages.\n\n_The Constitution of Liberty_. Legal Classics Library, leather- bound edition for sale by subscription. New York: Gryphon Editions, 1999. x + 568 pages.\n\n_The Constitution of Liberty_. London: Routledge and Kegan Paul, 1960. x +568 pages.\n\n_The Constitution of Liberty_. London: Routledge and Kegan Paul, 1963. x + 568 pages.\n\n_The Constitution of Liberty_. London: Routledge and Kegan Paul, 1976. x + 568 pages.\n\n_The Constitution of Liberty_. Paperback. London: Routledge, 1990. x + 568 pages.\n\n_The Constitution of Liberty_. New edition, Routledge Classics. London: Routledge, 2006. xiii + 498 pages.\n\n_The Constitution of Liberty_. Beijing: Zhongguo she hui ke xue chu ban she, 1999. viii + 567 pages.\n\n_Why I Am Not a Conservative_. CIS Occasional Papers, No. 41 (from the postscript to _The Constitution of Liberty_ ). St. Leonards, N.S.W.: Center for Independent Studies, 1992. 18 pages.\n\n_Responsibility and Freedom_. President's Essay series (Chapter 5 of _The Constitution of Liberty_ ). Washington, DC: Heritage Foundation, 1993. 33 pages.\n\n_Freedom: A Philosophical Anthology_. Ian Carter, Matthew H. Kramer, and Hillel Steiner eds. Contains a selection from Chapter 9: \"Coercion and the State,\" pp. 104\u20139. Oxford: Blackwell Publishers, 2007. 6 pages.\n\n**Chinese**\n\n_Zi you de xian zhang_. Translated by Dewei Zhou. Taipei: Taiwan yin hang jing ji yan jiu shi, 1973. xxii + 618 pages.\n\n_Zi you zhi xu yuan li_. Translated by Zhenglai Deng. 2 volumes. Beijing: Sheng huo, du shu, xin zhi san lian shu dian, 1997.\n\n_Zi you xian zhang_. Beijing: Zhongguo she hui ke xue chu ban she, 1998. xxiii + 597 pages.\n\n**French**\n\n_La Constitution de la libert\u00e9_. Translated by Raoul Audouin and Jacques Garello with the collaboration of Guy Milli\u00e8re. Paris: Litec, 1993. xxvii + 530 pages.\n\n**German**\n\n_Der Verfassung der Freiheit_. 1st edition. T\u00fcbingen: J. C. B. Mohr and P. Siebeck, 1971. xvi + 530 pages.\n\n_Der Verfassung der Freiheit_. 2nd edition. T\u00fcbingen: J. C. B. Mohr and P. Siebeck, 1983. xvi + 530 pages.\n\n_Der Verfassung der Freiheit_. 3rd edition. T\u00fcbingen: J. C. B. Mohr and P. Siebeck, 1991. xvi + 530 pages.\n\n_Der Verfassung der Freiheit_. 4th edition, volume 3 of the Gesammelte Schriften in deutscher Sprache. Translated by Alfred Bosch. T\u00fcbingen: J. C. B. Mohr and P. Siebeck, 2005. xx + 575 pages.\n\n**Italian**\n\n_La societ\u00e0 libera_. Translated by Marcella Bianchi and Lavagna Malagodi. Firenze: Vallechi, 1969. 603 pages.\n\n_La societ\u00e0 libera_. Translated by Marcella Bianchi and Lavagna Malagodi. Formello: SEAM, 1998. 647 pages.\n\n_Perch\u00e9 non sono un conservatore_. [Postscript: \"Why I Am Not a Conservative.\"] Rome: Ideazione, 1971. 85 pages.\n\n**Japanese**\n\n_Jiyu no kachi_. [Of Part 1: The Value of Freedom]. Translated by Kenzo Kiga and Katsujiro Koga. Tokyo: Shunjusha, 1986. 241 pages.\n\n_Jiyu no kachi_. [Of Part 1: The Value of Freedom]. Translated by Kenzo Kiga and Katsujiro Koga. Tokyo: Shunjusha, 1997. 241 pages.\n\n_Jiyu no kachi_. [Of Part 1: The Value of Freedom]. Translated by Kenzo Kiga and Katsujiro Koga. Tokyo: Shunjusha, 2007. 250 pages.\n\n_Jiyu to ho_. [Of Part 2: Freedom and Law.] Translated by Kenzo Kiga and Katsujiro Koga. Tokyo: Shunjusha, 1987. 254 pages.\n\n_Jiyu to ho_. [Of Part 2: Freedom and Law.] Translated by Kenzo Kiga and Katsujiro Koga. Tokyo: Shunjusha, 1997. 254 pages.\n\n_Jiyu to ho_. [Of Part 2: Freedom and Law.] Translated by Kenzo Kiga and Katsujiro Koga. Tokyo: Shunjusha, 2007. 273 pages.\n\n_Fukushi kokka ni okeru jiyu_. [Of Part 3: Freedom in the Welfare State.] Translated by Kenzo Kiga and Katsujiro Koga. Tokyo: Shunjusha, 1987. xxxvi + 266 pages.\n\n_Fukushi kokka ni okeru jiyu_. [Of Part 3: Freedom in the Welfare State.] Translated by Kenzo Kiga and Katsujiro Koga. Tokyo: Shunjusha, 1997. xxxvi + 266 pages.\n\n_Fukushi kokka ni okeru jiyu_. [Of Part 3: Freedom in the Welfare State.] Translated by Kenzo Kiga and Katsujiro Koga. Tokyo: Shunjusha, 2007. xxxvi + 277 pages.\n\n**Polish**\n\n_Konstytucja wolnos\u00b4ci_. Translated by H. Lapszyc. Warsaw, Wroclaw: Wers, 1987. 337 pages.\n\n_Konstytucja wolnos\u00b4ci_. Translated by Janusz Stawin\u00b4ski. Warsaw: Wydawnnictwo Naukowe PWN, 2006. 526 pages.\n\n**Spanish**\n\n_Los fundamentos de la libertad_. Translated by Jos\u00e9 Vicente Torrente. 2 volumes. Valencia: Fomento de Cultura Ediciones, 1961.\n\n_Los fundamentos de la libertad_. 2nd edition. Translated by Jos\u00e9 Vicente Torrente. Madrid: Uni\u00f3n Editorial, 1975. 510 pages.\n\n_Los fundamentos de la libertad_. 3rd edition. Translated by Jos\u00e9 Vicente Torrente. Madrid: Uni\u00f3n Editorial, 1978. 548 pages.\n\n_Los fundamentos de la libertad_. 4th edition. Translated by Jos\u00e9 Vicente Torrente. Madrid: Uni\u00f3n Editorial, 1982. 548 pages.\n\n_Los fundamentos de la libertad_. 5th edition, volume 18 of the Obras completas. Translated by Jos\u00e9 Vicente Torrente. Madrid: Uni\u00f3n Editorial, 1991. 510 pages.\n\n_Los fundamentos de la libertad_. 5th edition, 2 volumes. Translated by Jos\u00e9 Vicente Tor-rente. Barcelona: Folio, 1997. 510 pages.\n\n_Los fundamentos de la libertad_. 6th edition. Translated by Jos\u00e9 Vicente Torrente. Madrid: Uni\u00f3n Editorial, 1998. 541 pages.\n\n_Los fundamentos de la libertad_. 7th edition, 2 volumes. Translated by Jos\u00e9 Vicente Tor-rente. Madrid: Uni\u00f3n Editorial, 2006. 541 pages.\n\n**Swedish**\n\n_Frihetens grundvalar_. Translated by Barbro Ahlstrom and Carl Johan Ljungberg. Stockholm: Ratio, 1983. 521 pages.\n\n_Frihetens grundvalar_. Translated by Barbro Ahlstrom and Carl Johan Ljungberg. Stockholm: Timbro, 1999. 611 pages.\n\n## A NOTE ON THE NOTES\n\nHayek's _Constitution of Liberty_ is unquestionably his most important work in social philosophy, the one for which he is most likely to be remembered. I have therefore made every attempt to ensure the accuracy of his citations and quotations and to include crucial bibliographical material missing from the 1960 edition. In addition to the footnotes that appear in this edition, Hayek added approximately 200 citations to the first German edition of _The Constitution of Liberty_ , published in 1971. I have included these here.\n\nIn the German edition, Hayek duplicated the footnote numbering found in the 1960 English edition, adding new material either to existing footnotes or interpolating new notes by situating them between existing notes and affixing a letter after the number of the previous note. Thus, if a note were added between footnotes 12 and 13, it would be indicated as footnote 12a. I have renumbered these new notes so all notes are now numbered consecutively. Thus, the footnote numbering in this edition will not necessarily correspond to that in the 1960 edition. All material Hayek added to the German edition is shown here in sans serif font.\n\nThe footnoting style to which Hayek adhered is that which predominates in Britain, where surnames are prefixed solely by initials and where no publisher or subtitles are given. I have always found the use of initials infuriating since, _inter alia_ , it makes it extremely tedious to uncover other publications by the same author (consider, for example, an author cited solely as \"D. Brown\"). As a consequence, I have added first and middle names where these were available, even in those instances where they are rarely if ever used, as in \"Wystan Hugh Auden\" or \"Herbert Lionel Adolphus Hart.\" In addition, the titles of the monographs Hayek cites have been given in full, including subtitles. I realize that in a number of cases this might well present the reader with superfluous knowledge, but I thought it best for consistency's sake to provide this information, especially in light of its utility in consulting online library catalogs. Finally, publishers' names have been added to all books, as have the page numbers on which articles in journals and anthologies fall. Unfortunately, Hayek's notes contain a large number of bibliographical errors\u2014page and volume numbers of his quotations, titles, and even, in some instances, the  names of journals. All 1,900 citations have been checked for accuracy and all errors and omissions have been silently corrected. Hayek was in the habit, here and elsewhere in his writings, of employing the abbreviation \"cf.\" as a synonym for \"see\" in instances where the citation or quotation supports the contention made in the text. This has been left unaltered.\n\nWhere Hayek quotes in a foreign language, I have added a translation of the quotation. In instances where he has quoted from a source other than in English but has himself translated the material, I have included the quotation in the original language. Hayek occasionally quotes from a work that itself contains a quotation; I have tried to check the original quotation for accuracy and indicate its source. Lastly, in those instances where Hayek cites an author whose work is currently available in an edition published by the Liberty Fund of Indianapolis (e.g., Lord Acton, Adam Smith, Edmund Burke), I have keyed Hayek's references to these readily available, inexpensive editions.\n\nRonald Hamowy  \nProfessor Emeritus of History, University of Alberta  \nAffiliate Professor of Economics, George Mason University\n\n## EDITOR'S ACKNOWLEDGMENTS\n\nThe editing of Hayek's _Constitution of Liberty_ has proved a major undertaking, involving, as it has, checking each of Hayek's 1,900 citations in eight languages, the accumulated research of many decades of reading in a wide variety of fields, among them economics, history, political theory, theoretical psychology, sociology, and law. I would not have been able to accomplish this task without the help of a number of people\u2014librarians, scholars, and students\u2014and would be remiss if I failed to acknowledge their assistance. I wish to thank Ms. Silvia Brandolin, of the Banca Nazionale del Lavoro; Ms. Hannah Chandler, Ms. Christine Mason, and Mr. Andrew Milner, of the Bodleian Library, Oxford; Mr. Michael Chambers and Ms. Lalitha Fernando, of the British Library; Ms. Naomi Woodburn and Mr. Colin T. Clarkson, of the Cambridge University Library; Ms. Ingemar Schmart, of the Deutsche Zentralbibliothek f\u00fcr Wirtschaftswissenschaften; Ms. Carol Leadenham, of the Hoover Institution Library and Archives; Ms. Emily Howie, of the Library of Congress; Ms. Anna Plattner, of the \u00d6sterreichische Nationalbibliothek; Mr. Robert Nef, of the _Schweizer Monatshefte;_ Ms. Sabine Tolksdorf, of the Abteilung Historische Drucke, Staatsbibliothek zu Berlin; the staff of the _Times Literary Supplement;_ Ms. Emma Taylor, of the University of Glasgow Library; Ms. Amy Tomaszewski, of the College of Law, and Ms. Julia Dolinnaya, of the Slavic Research Service, both of the University of Illinois Library; Ms. Kate Wilcox- Jay, of the Institute for Historical Research, University of London; the staff of the University of Michigan Library; and Mr. Geoffrey P. Williams, of the State University of New York at Albany Library.\n\nAdditionally, I am indebted to Professor William Bartley and Mr. Joel Fonstad, of the Department of English, University of Saskatchewan; Mr. Alex Biscaro, of the Swiss Embassy, Washington, DC; Professor Gerhard Ens, of the Department of History, University of Alberta; Professor Ralph Lerner, of the Committee on Social Thought, University of Chicago; Professor Roderick Long, of the Department of Philosophy, Auburn University; Professor Sam Peltzman, of the Business School, University of Chicago; Professor Donald Rutherford, of the Department of Economics, Edinburgh University; Professor Stephen Stigler, of the Department of Statistics, University of Chicago;  Professor Viktor J. Vanberg of the Institut f\u00fcr Allgemeine Wirtschaftsforschung, University of Freiburg; and Professor Gerald Stourzh, of the Department of History, University of Vienna, for their generous help.\n\nMy debt is greatest to Professors Doris Bergen, of the History Department, University of Toronto, and Daniel Gargola, of the History Department, University of Kentucky, who were unstinting in their assistance in uncovering and translating many of Hayek's German and classical citations. In addition, I must express my special thanks to the staff of the American University Library in Washington, DC, and particularly to its interlibrary loan department, for their patience and unfailing help in providing me the bibliographical assistance needed to complete an examination of Hayek's innumerable citations. I am deeply grateful to the Cato Institute, and especially to Joey Coon, Ryan Hagemann, Susanna Morrison, Daniel Robinson, David Robinson, Charlotte Rommerskirchen, Marian Wirth, and Jessica Wright, for supplying much- needed editorial assistance. I should also like to thank the Department of Economics at George Mason University and particularly its chairman, Professor Donald J. Boudreaux, for granting me affiliation with the department, thus permitting me the use of the university's facilities. Above all, I am indebted to the Earhart Foundation, for its generosity in underwriting the costs of this project, thus making it possible for me to devote my full time to its completion.\n\nFinally, this task would have been impossible without the constant support, encouragement, and patience of my dear friend and companion, Mr. Clement Ho.\n\n## LIBERTY FUND EDITIONS CITED\n\nWhere the books Hayek cited have been issued in editions published by the Liberty Fund, I have added that information in the notes. What follows is a listing of Liberty Fund editions referred to.\n\nLord Acton\n\n_Lectures on the French Revolution_. Foreword by Stephen J. Tonsor. Indianapolis, IN:\n\nLiberty Fund, 2000.\n\nChapter 24, n. 14\n\n_Selected Writings of Lord Acton_. Edited by J. Rufus Fears. 3 volumes. Indianapolis, IN: Liberty Classics, 1985.\n\nVol. 1. _Essays in the History of Liberty_.\n\nChapter 4, n. 21\n\nChapter 9, n. 9\n\nChapter 11, n. 4\n\nChapter 11, n. 18\n\nChapter 11, n. 90\n\nChapter 12, epigraph n.\n\nChapter 12, n. 5\n\nChapter 12, n. 7\n\nChapter 12, n. 8\n\nChapter 12, n. 36\n\nChapter 16, epigraph n.\n\nPostscript, epigraph n.\n\nPostscript, n. 17\n\nVol. 2. _Essays in the Study and Writing of History_.\n\nChapter 1, n. 25\n\nChapter 9, n. 4\n\nJacob Burckhardt\n\n_Reflections on History_. Introduction by Gottfried Dietze. Indianapolis, IN: Liberty Classics, 1979. (This edition is similar to the edition published in 1943 by Pantheon Books under the title _Force and Freedom: Reflections on History_.)\n\nChapter 9, n. 4\n\nChapter 9, n. 20\n\nEdmund Burke\n\n_The Selected Works of Edmund Burke_. Foreword and notes by Frances Canavan. 3 volumes. Indianapolis, IN: Liberty Fund, 1960.\n\nVol. 1. _Thoughts on the Cause of the Present Discontents and The Two Speeches on America_\n\nChapter 12, n. 2\n\nVol. 2. _Reflections on the Revolution in France_\n\nIntroduction, n. 7\n\nChapter 4, n. 20\n\nVol. 3. _Letters on a Regicide Peace_\n\nChapter 19, n. 37\n\nChapter 23, epigraph n.\n\n_A Vindication of Natural Society_. Edited by Frank N. Pagano. Indianapolis, IN: Liberty Fund, 1982.\n\nChapter 4, n. 42\n\nChapter 9, n. 4\n\n_Miscellaneous Writings_. Foreword and notes by Frances Canavan. Indianapolis, IN: Liberty Fund, 1999.\n\nChapter 4, n. 29\n\n_Further Reflections on the Revolution in France_. Edited by Daniel E. Ritchie. Indianapolis, IN: Liberty Fund, 1992.\n\nChapter 4, n. 36\n\nChapter 12, n. 4\n\nSir Edward Coke\n\n_Selected Writings of Sir Edward Coke_. Edited by Steve Sheppard. 3 volumes. Indianapolis, IN: Liberty Fund, 2005.\n\nVol. 1.\n\nChapter 4, n. 20\n\nChapter 11, n. 37\n\nChapter 11, n. 43\n\nChapter 11, n. 45\n\nChapter 11, n. 46\n\nEdward S. Corwin\n\n_The \"Higher Law\" Background of American Constitutional Law_. Amagi series. Indianapolis, IN: Liberty Fund, 2008.\n\nChapter 11, n. 6\n\nChapter 11, n. 35\n\nChapter 12, n. 11\n\nChapter 12, n. 42\n\nAlbert Venn Dicey\n\n_Introduction to the Study of the Law of the Constitution_. Introduction and appendix by A. V. Dicey; foreword by Roger E. Michener. Indianapolis, IN: Liberty Fund, 1982.\n\nChapter 11, n. 7\n\nChapter 13, n. 4\n\nChapter 14, n. 4\n\nChapter 14, n. 8\n\nChapter 16, n. 36\n\n_Lecturers on the Relation between Law and Public Opinion in England during the Nineteenth Century_. Edited by Richard VandeWeting. Indianapolis, IN: Liberty Fund, 2008\n\nChapter 3, epigraph n.\n\nChapter 4, n. 47\n\nChapter 7, n. 13\n\nChapter 7, n. 15\n\nChapter 18, n. 3\n\nAlexander Hamilton, John Jay, James Madison\n\n_The Federalist, or the New Constitution_. Gideon Edition. Edited by George W. Carey and James McClellan. Indianapolis, IN: Liberty Fund, 2001\n\nChapter 12, n. 10\n\nChapter 12, n. 32\n\nChapter 15, epigraph n.\n\nWilhelm von Humboldt\n\n_The Limits of State Action_. Edited by John Wyon Burrow. Indianapolis, IN: Liberty Fund, 1993. (Originally published in English translation as _The Sphere and Duties of Government_.)\n\nChapter 13, n. 21\n\nChapter 24, n. 3\n\nDavid Hume\n\n_Essays_. Edited by Eugene Miller. Indianapolis, IN: Liberty Fund, 1987. (Unlike the 2-volume Green-Grose edition used by Hayek, the Liberty Fund edition does not include either Hume's _Enquiry Concerning Human Understanding_ or his _Enquiry Concerning the Principles of Morals_.)\n\nIntroduction, n. 9\n\nChapter 3, epigraph n.\n\nChapter 3, n. 10\n\nChapter 3, epigraph n.\n\nChapter 4, n. 26\n\nChapter 4, n. 30\n\nChapter 7, epigraph n.\n\nChapter 11, n. 34\n\nChapter 11, n. 81\n\nChapter 11, n. 81\n\nChapter 12, n. 30\n\n_History of England from the Invasion of Julius Caesar to the Revolution of 1688_. Foreword by William B. Todd. 6 volumes. Indianapolis, IN: Liberty Fund, 1983.\n\nChapter 11, n. 83\n\nChapter 12, n. 10\n\nBertrand de Jouvenel\n\n_The Ethics of Redistribution_. Introduction by John Gray. Indianapolis, IN: Liberty Fund, 1990.\n\nChapter 8, n. 13\n\n_On Power: The Natural History of Its Growth_. Foreword by D. W. Brogan. Indianapolis, IN: Liberty Fund, 1993.\n\nChapter 6, n. 13\n\nChapter 9, n. 3\n\n_Sovereignty: An Inquiry into the Political Good_. Translated by J. F. Huntington. Indianapolis, IN: Liberty Fund, 1997.\n\nChapter 1, n. 14\n\nWilliam Edward Hartpole Lecky\n\n_Democracy and Liberty_. Introduction by William Murchison. 2 volumes. Indianapolis, IN: Liberty Classics, 1981.\n\nChapter 20, n. 17\n\nJean Louis de Lolme\n\n_The Constitution of England_. Edited and with an introduction by David Lieberman. Indianapolis, IN: Liberty Fund, 2007.\n\nChapter 1, n. 11\n\nFrederic William Maitland\n\n_A Historical Sketch of Liberty and Equality_. Indianapolis, IN: Liberty Fund, 2000.\n\nChapter 2, n. 10\n\nChapter 9, epigraph n.\n\nCharles Howard McIlwain\n\n_Constitutionalism: Ancient and Modern_. Indianapolis, IN: Liberty Fund, 2007.\n\nChapter 10, n. 21\n\nChapter 11, n. 75\n\nChapter 12, n. 11\n\nJohn Stuart Mill\n\n_The Collected Works of John Stuart Mill_. 8 volumes. Indianapolis, IN: Liberty Fund.\n\nVols. 2 and 3. _Principles of Political Economy, with Some of Their Application to Social Philosophy_. Edited by V. W. Bladen and J. M. Robson, 1965.\n\nChapter 3, n. 9\n\nChapter 7. n. 7\n\nChapter 15, n. 3\n\nChapter 15, n. 4\n\nChapter 20, n. 6\n\nVol. 4. _Essays on Economics and Society_. Edited by J. M. Robson, 1965.\n\nChapter 4, n. 31\n\nVol. 10: _Ethics, Religion, and Society_. Edited by J. M. Robson, 1965.\n\nChapter 7, n. 14\n\nJohn Milton\n\n_Areopagitica and Other Political Writings of John Milton_. Foreword by John Alvis. Indianapolis, IN: Liberty Fund, 1999.\n\nChapter 5, n. 10\n\nChapter 9, n. 4\n\nLudwig von Mises\n\n_Anti-capitalistic Mentality_. Edited by Bettina Bien Greaves. Indianapolis, IN: Liberty Fund, 2006.\n\nChapter 8, n. 10\n\n_Human Action: A Treatise on Economics_. Edited by Bettina Bien Greaves. 4 volumes. Indianapolis, IN: Liberty Fund, 2007.\n\nVol. 1.\n\nChapter 7, n. 9\n\nVol. 2.\n\nChapter 8, n. 5\n\nChapter 19, n. 10\n\nChapter 21, n. 1\n\nVol. 3.\n\nChapter 20, n. 28\n\nVol. 4.\n\nChapter 17, n. 21\n\n_Nation, State, and Economy: Contributions to the Politics and History of Our Time_. Translated by Leland B. Yeager. Indianapolis, IN: Liberty Fund, 2006.\n\nChapter 24, n. 4\n\n_Planning for Freedom: Let the Market System Work_. Edited by Bettina Bien Greaves. Indianapolis, IN: Liberty Fund, 2008.\n\nChapter 20, n. 4\n\n_Socialism: An Economic and Sociological Analysis_. Translated by Jacques Kahane; foreword by F. A. Hayek. New edition. Indianapolis, IN: Liberty Fund, 1960.\n\nChapter 1, n. 6\n\nChapter 4, n. 15\n\nChapter 9, n. 14\n\nChapter 10, n. 21\n\nChapter 18, n. 6 (German edition)\n\nChapter 19, n. 28\n\nChapter 23, n. 11\n\nFelix Morley\n\n_Essays on Individuality_. Edited by Felix Morley. Indianapolis, IN: Liberty Fund, 1977.\n\nChapter 2, epigraph n.\n\nChapter 8, n. 10\n\nMichael Polanyi\n\n_The Logic of Liberty: Reflections and Rejoinders_. Foreword by Stuart D. Warner. Indianapolis, IN: Liberty Fund, 1998.\n\nChapter 2, n. 2\n\nChapter 3, n. 8\n\nChapter 9, epigraph n.\n\nChapter 10, n. 25\n\nChapter 24, n. 14\n\nDavid Ricardo\n\n_The Works and Correspondence of David Ricardo_. Edited by Piero Sraffa with the collaboration of M. H. Dobb. 11 volumes. Indianapolis, IN: Liberty Fund, 2004.\n\nVol. 9. _Letters 1821\u20131823_\n\nChapter 21, n. 16\n\nAlgernon Sidney\n\n_Discourses Concerning Government_. Edited by Thomas G. West. Indianapolis, IN: Liberty Fund, 1996.\n\nTitle page, epigraph n.\n\nChapter 11, n. 59\n\nAdam Smith\n\n_The Glasgow Edition of the Works and Correspondence of Adam Smith_. 7 volumes. Indianapolis, IN: Liberty Fund.\n\nVol. 1. _The Theory of Moral Sentiments_. Edited by David Daitches Raphael and Alec Lawrence Macfi e, 1977.\n\nChapter 6, n. 11\n\nVols. 2a and 2b. _An Inquiry into the Nature and Causes of the Wealth of Nations_. Edited by R. H. Campbell and A. S. Skinner, 1982.\n\nChapter 3, n. 9\n\nChapter 4, n. 26\n\nChapter 10, n. 15\n\nChapter 11, n. 84\n\nChapter 15, n. 5\n\nChapter 24, n. 13\n\nVol. 5. _Lectures on Jurisprudence_. Edited by Ronald Lindley Meek, David Daitches Raphael, and Peter Gonville Stein, 1982\n\nChapter 22, n. 2\n\nHerbert Spencer\n\n_The Principles of Ethics_. Introduction by Tibor R. Machan. 2 volumes. Indianapolis, IN: Liberty Classics, 1978. (The section on justice appears in vol. 2, pp. 19\u2013279.)\n\nChapter 15, n. 11\n\nJames Fitzjames Stephen\n\n_Liberty, Equality, Fraternity_. Edited by Stuart D. Warner. Indianapolis, IN: Liberty Fund, 1993.\n\nChapter 7, n. 9\n\nEmer de Vattel\n\n_The Law of Nations, Or, Principles of the Laws of Nature, Applied to the Conduct and A_ _ff_ _airs of Nations and Sovereigns_. Edited and with an introduction by B\u00e9la Kapossy and Richard Whatmore. Indianapolis, IN: Liberty Fund, 2008.\n\nChapter 12, n. 9\n\n## Friedrich A. Hayek  \nTHE CONSTITUTION OF LIBERTY\n\nOur inquiry is not after that which is perfect, well knowing that no such thing is found among men; but we seek that human Constitution which is attended with the least, or the most pardonable inconveniences.\n\n\u2014 _Algernon Sidney_\n\nThis quotation is taken from Algernon Sidney, _Discourses Concerning Government_ (London. Printed and are to be sold by booksellers of London and Westminster, 1698), chap. 3, sec. 18, p. 142 [ _The Works of Algernon Sidney_ , Thomas Hollis, ed. (London. Printed for W. Strahan Iun., 1772), p. 151 (Liberty Fund edition, p. 180).]\n_To the unknown civilization that is growing in America._\n\n## PREFACE\n\nThe aim of this book is explained in the Introduction, and my chief obligations are acknowledged in the few paragraphs that follow this preface. All that remains for me to do here is to issue a warning and to present an apology.\n\nThis book is not concerned mainly with what science teaches us. Though I could not have written it if I had not devoted the greater part of my life to the study of economics and had not more recently endeavored to acquaint myself with the conclusions of several other social sciences, I am not concerned here exclusively with facts, nor do I confine myself to statements of cause and effect. My aim is to picture an ideal, to show how it can be achieved, and to explain what its realization would mean in practice. For this, scientific discussion is a means, not an end. I believe I have made honest use of what I know about the world in which we live. The reader will have to decide whether he wants to accept the values in the service of which I have used that knowledge.\n\nThe apology concerns the particular state at which I have decided to submit the results of my efforts to the reader. It is perhaps inevitable that the more ambitious the task, the more inadequate will be the performance. On a subject as comprehensive as that of this book, the task of making it as good as one is capable of is never completed while one's faculties last. No doubt I shall soon find that I ought to have said this or that better and that I have committed errors which I could myself have corrected if I had persisted longer in my efforts. Respect for the reader certainly demands that one present a tolerably finished product. But I doubt whether this means that one ought to wait until one cannot hope to improve it further. At least where the problems are of the kind on which many others are actively working, it would even appear to be an overestimate of one's own importance if one delayed publication until one was certain that one could not improve anything. If a man has, as I hope I have, pushed analysis a step forward, further efforts by him are likely to be subject to rapidly decreasing returns. Others will probably be better qualified to lay the next row of bricks of the edifice to which I am trying to contribute. I will merely claim that I have worked on the book until I did not know how I could adequately present the chief argument in briefer form.\n\nPerhaps the reader should also know that, though I am writing in the United  States and have been a resident of this country for nearly ten years, I cannot claim to write as an American. My mind has been shaped by a youth spent in my native Austria and by two decades of middle life in Great Britain, of which country I have become and remain a citizen. To know this fact about myself may be of some help to the reader, for the book is to a great extent the product of this background.\n\nF. A. Hayek  \nChicago  \nMay 8, 1959\n\n## ACKNOWLEDGMENTS\n\nSo much of what I have been trying to say in this book has been said before in a manner on which I cannot improve, but in places widely dispersed or in works with which the modern reader is not likely to be familiar, that it seemed desirable to expand the notes beyond mere references into what is in part almost an anthology of individualist liberal thought. These quotations are meant to show that what today may often seem strange and unfamiliar ideas were once the common heritage of our civilization, but also that, while we are building on this tradition, the task of uniting them into a coherent body of thought directly applicable to our day is one which still needed to be undertaken. It is in order to present the building stones from which I have tried to fashion a new edifice that I have allowed these notes to run to this length. They nevertheless do not provide a complete bibliography of the subject. A helpful list of relevant works can be found in Henry Hazlitt, _The Free Man's Library: A Descriptive and Critical Bibliography_ (Princeton, NJ: Van Nostrand, 1956).\n\nThese notes are also far from being an adequate acknowledgment of my indebtedness. The process in which I formed the ideas expressed in this book necessarily preceded the plan of stating them in this form. After I decided on this exposition I read little of the work of authors with whom I expected to agree, usually because I had learned so much from them in the past. In my reading I rather aimed at discovering the objections I had to meet, the arguments I had to counter, and at finding the forms in which these ideas have been expressed in the past. In consequence, the names of those who have contributed most to shaping my ideas, whether as my teachers or as fellow strugglers, appear rarely in these pages. If I had regarded it as my task to acknowledge all indebtedness and to notice all agreement, these notes would have been studded with references to the work of Ludwig von Mises, Frank H. Knight, and Edwin Cannan; of Walter Eucken and Henry C. Simons; of Wilhelm R\u00f6pke and Lionel Robbins; of Karl R. Popper, Michael Polanyi, and Bertrand de Jouvenel. Indeed, if I had decided to express not my aim but my indebtedness in the dedication of this book, it would have been most appropriate to dedicate it to the members of the Mont P\u00e8lerin Society  and in particular to their two intellectual leaders, Ludwig von Mises and Frank H. Knight.\n\nThere are, however, more specific obligations which I wish to acknowledge here. Edward C. Banfield, Chester I. Barnard, W. H. Book, John Davenport, Pierre F. Goodrich, Walter Fr\u00f6hlich, David Grene, Floyd A. Harper, David G. Hutton, Arthur Kemp, Frank H. Knight, William L. and Shirley Letwin, Fritz Machlup, Laurence W. Martin, Ludwig von Mises, Alexander Morin, Felix Morley, Sylvester Petro, J. H. Reiss, Gerald Stourzh, Ralph Turvey, C. Y. Wang, and Richard Ware have read various parts of an earlier draft of this book and assisted me with their comments. Many of them and Aaron Director, Victor Ehrenberg, Duncan Forbes, Milton Friedman, Morris Ginsberg, Claude W. Guillebaud, Bruno Leoni, John U. Nef, Margaret G. Reid, Max Rheinstein, Hans Rothfels, Helmut Schoeck, Irene Shils, T. F. T. Plucknett, and Jacob Viner have supplied me with important references or facts, though I hesitate to mention their names since I am almost bound to forget some of the many who have helped me in this way.\n\nIn the final stages of the preparation of the book I have had the invaluable benefit of the assistance of Mr. Edwin McClellan. It is mainly due to his and (I understand) Mrs. McClellan's sympathetic efforts to straighten out my involved sentences if the book is more readable than I could ever have made it. It has received further polish from the hands of my friend Henry Hazlitt, who was good enough to read and comment upon part of the final typescript. I am also indebted to Mrs. Lois Fern for checking all the quotations in the notes and to Miss Vernelia Crawford for preparing the Subject Index.\n\nThough the book is not the product of the now common kind of collective effort\u2014I have never learned even to avail myself of the aid of a research assistant\u2014it has in other ways greatly benefited from opportunities and facilities which various foundations and institutions have provided. To the Volker, Guggenheim, Earhart, and Relm foundations I owe in this connection a great debt. Lectures given at Cairo, Zurich, Mexico City, Buenos Aires, and Rio de Janeiro and at various American universities and colleges have provided an opportunity not only to try out on audiences some of the ideas expounded in the book, but also to gain experiences that were important in writing it. Places of publication of earlier drafts of some of the chapters are mentioned in the notes, and I am grateful to the various editors and publishers for permission to reprint them. I also wish to acknowledge the help of the University of Chicago Library, on which I have relied almost exclusively in the work on this book, and whose Inter- Library Loan Service has invariably procured whatever I needed; and to the Social Science Research Committee and the typing staff of the Social Science Division of the University of Chicago who have provided the funds and the labor for typing successive drafts of this book.\n\nMy greatest debt, however, is to the Committee on Social Thought of the University of Chicago and to its chairman, Professor John Ulrich Nef, who made it possible for me for some years to regard as my main task the completion of this book, which was facilitated rather than hindered by my other duties on the Committee.\n\n## BIBLIOGRAPHICAL ABBREVIATIONS\n\nActon, _Historical Essays_\n\nJohn Emerich Edward Dalberg Acton, Baron Acton. _Historical Essays and Studies_. Edited by John Neville Figgis and Reginald Vere Laurence. London: Macmillan and Co., 1907.\n\nActon, _History of Freedom_\n\nJohn Emerich Edward Dalberg Acton, Baron Acton. _The History of Freedom and Other Essays_. Edited by J. N. Figgis and R. V. Laurence. London: Macmillan and Co., 1907.\n\nThe Liberty Fund has published a three-volume collection of Acton's essays under the general title _Selected Writings of Lord Acton_ , J. Rufus Fears, ed. (Indianapolis, IN: Liberty Classics, 1985). The individual volumes are titled: Vol. 1: _Essays in the History of Liberty;_ Vol. 2: _Essays in the Study and Writing of History;_ and Vol. 3: _Essays in Religion, Politics, and Morality_\n\nBagehot, _Works_\n\nWalter Bagehot. _The Works and Life of Walter Bagehot_. Edited by [Mrs.] Russell Barrington. 10 vols. London: Longman, Green, and Co., 1910.\n\nBurke, _Works_\n\nEdmund Burke. _The Works of the Right Honourable Edmund Burke_. New edition. 16 vols. London: Rivington, 1815\u201327.\n\nThe Liberty Fund has published a number of essays by Burke in several volumes. They are, in three volumes, _The Selected Works of Edmund Burke_ , foreword and notes by Frances Canavan (Indianapolis, IN: Liberty Fund, 1960); another volume containing _A Vindication of Natural Society_ , Frank N. Pagano, ed. (1982); yet another volume entitled _Miscellaneous Writings_ , foreword and notes by Frances Canavan (1999); and, finally, a volume titled _Further Reflections on the Revolution in France_ , Daniel E. Ritchie, ed. (1992).\n\nDicey, _Law of the Constitution_\n\nAlbert Venn Dicey. _Introduction to the Study of the Law of the Constitution_ , 9th ed. Introduction and appendix by E. C. S. Wade. London: Macmillan and Co., 1939.\n\nThe ninth edition, published in 1939, some 25 years after the eighth edition had appeared, was edited by Emlyn Capel Stewart Wade and contains a new introduction and appendix that replaced those written by Dicey and that appeared in earlier editions. The Liberty Fund has issued a reprint of the eighth edition containing Dicey's original material (Indianapolis, IN: Liberty Fund, 1982) with a foreword by Roger E. Michener.\n\nDicey, _Law and Public Opinion_\n\nAlbert Venn Dicey. _Lectures on the Relation between Law and Public Opinion in England during the Nineteenth Century_ , 2nd ed. London: Macmillan and Co., 1914.\n\nThe Liberty Fund republished the 1917 edition, with a new introduction by Richard VandeWeting, in 2008.\n\n_Encyclopedia of the Social Sciences_\n\nEdwin R. A. Seligman and Alvin Johnson, eds. _Encyclopedia of the Social Sciences_. 15 vols. New York: Macmillan Company, 1930\u201335.\n\nThe _Encyclopedia_ was reissued in 1937 and reprinted in 1951.\n\nHume, _Essays_\n\nDavid Hume. _Essays: Moral, Political, and Literary_. Edited by Thomas Hill Green and Thomas Hodge Grose. 2 vols. London: Longmans, Green, and Co., 1875. Vol. 2 contains _inter alia_ the _Enquiry Concerning Human Understanding_ and the _Enquiry Concerning the Principles of Morals_.\n\n_Essays_ (not including either the _Enquiry_ or the _Principles_ ) was republished by the Liberty Fund in a new edition by Eugene Miller in 1987.\n\nHume, _Treatise of Human Nature_\n\nDavid Hume. _A Treatise of Human Nature_. Edited by Thomas Hill Green and Thomas Hill Grose. 2 vols. London: Longmans, Green, and Co., 1890.\n\nLocke, _Second Treatise_\n\nJohn Locke. _The Second Treatise of Civil Government_ and _A Letter Concerning Toleration_ , 3rd ed. Edited by John Wiedhofft Gough. Oxford: Basil Blackwe11, 1946.\n\nMenger, _Untersuchungen_\n\nCarl Menger. _Untersuchungen \u00fcber die Methode der Socialwissenschaften, und der Politischen Oekonomie insbesondere_. Leipzig: Duncker und Humblot, 1883.\n\nThis work has been translated into English by Francis J. Nock and published under the title of _Problems of Economics and Sociology_. Edited by Louis Schneider. Urbana: University of Illinois Press, 1963.\n\nMill, _Principles_\n\nJohn Stuart Mill. _Principles of Political Economy, with Some of Their Applications to Social Philosophy_. Edited by Sir William James Ashley. London: Longmans, Green, and Co., 1909.\n\nA new edition of Mill's _Principles_ constitutes volumes 2 and 3 of the _Collected Works of John Stuart Mill_ , Vincent Wheeler Bladen and John Mercel Robson, eds., published by the University of Toronto Press in 1965 and reprinted by the Liberty Fund in 2006.\n\nMontesquieu, _Spirit of the Laws_\n\nCharles de Secondat, Baron de Montesquieu. _The Spirit of the Laws_. Translated by Thomas Nugent. 2 vols. in one. Hafner Library of Classics. New York: Hafner Publishing Co., 1949.\n\nThe French edition used is that contained in volume 2 of the Pl\u00e9iade complete works: \"De l'Esprit des Lois,\" in _\u0152uvres Compl\u00e8tes_ , Roger Caillois, ed. Biblioth\u00e8que de la Pl\u00e9iade. 2 vols. Paris: \u00c9ditions Gallimard, 1951.\n\nSmith, _Wealth of Nations_\n\nAdam Smith. _An Inquiry into the Nature and Causes of the Wealth of Nations_. Edited by Edwin Cannan. 2 vols. London: Methuen and Co., 1904.\n\nThe Liberty Fund republished the Glasgow edition of the works and correspondence of Adam Smith in paperback in 1960. _The Wealth of Nations_ , edited by R. H. Campbell and A. S. Skinner, constitutes volumes 1 and 2.\n\nTocqueville, _Democracy in America_\n\nAlexis de Tocqueville. _Democracy in America_. Translated by Henry Reeve. Edited by Phillips Bradley. 2 vols. New York: Alfred A. Knopf, 1945.\n\nThe French edition used is that contained in volume 2 of the Pl\u00e9iade complete works: \"De la d\u00e9mocratie en Am\u00e9rique,\" in _\u0152uvres_ , Andr\u00e9 Jardin, ed. Biblioth\u00e8que de la Pl\u00e9iade. 2 vols. Paris: \u00c9ditions Gallimard, 1992.\n\n[vol.] _U.S._ [pp.]\n\n_United States Reports: Cases Adjudged in the Supreme Court_. Washington: Government Printing Office. (According to American standard legal practice, references to this and such earlier reports of federal cases as \"Dallas,\" \"Cranch,\" \"Wheaton,\" and \"Wallace,\" and to reports of cases in state courts, are _preceded_ by the number of the volume and followed by the number of the page on which the report of the case begins and, where necessary, the page to which reference is made.)\n\n## INTRODUCTION\n\nWhat was the road by which we reached our position, what the form of government under which our greatness grew, what the national habits out of which it sprang? . . . If we look to the laws, they afford equal justice to all in their private differences. . . . The freedom which we enjoy in our government extends also to our ordinary life. . . . But all this ease in our private relations does not make us lawless as citizens. Against this fear is our chief safeguard, teaching us to obey the magistrates and the laws, particularly such as regard the protection of the injured, whether they are actually on the statute book, or belong to that code which, although unwritten, yet cannot be broken without acknowledged disgrace. \u2014Pericles\n\nIf old truths are to retain their hold on men's minds, they must be restated in the language and concepts of successive generations. What at one time are their most effective expressions gradually become so worn with use that they cease to carry a definite meaning. The underlying ideas may be as valid as ever, but the words, even when they refer to problems that are still with us, no longer convey the same conviction; the arguments do not move in a context familiar to us; and they rarely give us direct answers to the questions we are asking.1 This may be inevitable because no statement of an ideal that is likely to sway men's minds can be complete: it must be adapted to a given climate of opinion, presuppose much that is accepted by all men of the time, and illustrate general principles in terms of issues with which they are concerned.\n\nIt has been a long time since that ideal of freedom which inspired modern Western civilization and whose partial realization made possible the achievements of that civilization was effectively restated.2 In fact, for almost a century the basic principles on which this civilization was built have been falling into increasing disregard and oblivion. Men have sought for alternative social orders more often than they have tried to improve their understanding or use of the underlying principles of our civilization.3 It is only since we were confronted with an altogether different system that we have discovered that we have lost any clear conception of our aims and possess no firm principles which we can hold up against the dogmatic ideology of our antagonists.\n\nIn the struggle for the moral support of the people of the world, the lack of firm beliefs puts the West at a great disadvantage. The mood of its intellectual leaders has long been characterized by disillusionment with its principles, disparagement of its achievements, and exclusive concern with the creation of \"better worlds.\" This is not a mood in which we can hope to gain followers. If we are to succeed in the great struggle of ideas that is under way, we must first of all know what we believe. We must also become clear in our own minds as to what it is that we want to preserve if we are to prevent ourselves from drifting. No less is an explicit statement of our ideals necessary in our relations with other peoples. Foreign policy today is largely a question of which political philosophy is to triumph over another; and our very survival may depend on our ability to rally a sufficiently strong part of the world behind a common ideal.\n\nThis we shall have to do under very unfavorable conditions. A large part of the people of the world borrowed from Western civilization and adopted Western ideals at a time when the West had become unsure of itself and had largely lost faith in the traditions that have made it what it is. This was a time when the intellectuals of the West had to a great extent abandoned the very belief in freedom which, by enabling the West to make full use of those forces that are responsible for the growth of all civilization, had made its unprecedented quick growth possible. In consequence, those men from the less advanced nations who became purveyors of ideas to their own people  learned, during their Western training, not how the West had built up its civilization, but mostly those dreams of alternatives which its very success had engendered.\n\nThis development is especially tragic because, though the beliefs on which these disciples of the West are acting may enable their countries to copy more quickly a few of the achievements of the West, they will also prevent them from making their own distinct contribution. Not all that is the result of the historical development of the West can or should be transplanted to other cultural foundations; and whatever kind of civilization will in the end emerge in those parts under Western influence may sooner take appropriate forms if allowed to grow rather than if it is imposed from above. If it is true, as is sometimes objected, that the necessary condition for a free evolution\u2014the spirit of individual initiative\u2014is lacking, then surely without that spirit no viable civilization can grow anywhere. So far as it is really lacking, the first task must be to waken it; and this a regime of freedom will do, but a system of regimentation will not.\n\nSo far as the West is concerned, we must hope that here there still exists wide consent on certain fundamental values. But this agreement is no longer explicit; and if these values are to regain power, a comprehensive restatement and revindication are urgently needed. There seems to exist no work that gives a full account of the whole philosophy on which a consistent liberal view can rest\u2014no work to which a person wishing to comprehend its ideals may turn. We have a number of admirable historical accounts of how \"The Political Traditions of the West\" grew. But though they may tell us that \"the object of most Western thinkers has been to establish a society in which every individual, with a minimum of dependence on the discretionary authority of his rulers, would enjoy the privileges and responsibility of determining his own conduct within a previously defined framework of legal rights and duties,\"4 I know of none that explains what this means when applied to the concrete problems of our time, or whereupon the ultimate justification of this idea rests.\n\nIn recent years valiant efforts have also been made to clear away the confusions which have long prevailed regarding the principles of the economic policy of a free society. I do not wish to underrate the clarification that has been achieved. Yet, though I still regard myself as mainly an economist, I have come to feel more and more that the answers to many of the pressing social questions of our time are to be found ultimately in the recognition of principles that lie outside the scope of technical economics or of any other single discipline. Though it was from an original concern with problems of  economic policy that I started, I have been slowly led to the ambitious and perhaps presumptuous task of approaching them through a comprehensive restatement of the basic principles of a philosophy of freedom.\n\nBut I tender no apologies for thus venturing far beyond the range where I can claim to have mastered all the technical detail. If we are to regain a coherent conception of our aims, similar attempts should probably be made more often. One thing, in fact, which the work on this book has taught me is that our freedom is threatened in many fields because of the fact that we are much too ready to leave the decision to the expert or to accept too uncritically his opinion about a problem of which he knows intimately only one little aspect. But, since the matter of the ever recurring conflict between the economist and the other specialists will repeatedly come up in this book, I want to make it quite clear here that the economist can not claim special knowledge which qualifies him to co- ordinate the efforts of all the other specialists.5 What he may claim is that his professional occupation with the prevailing conflicts of aims has made him more aware than others of the fact that no human mind can comprehend all the knowledge which guides the actions of society and of the consequent need for an impersonal mechanism, not dependent on individual human judgments, which will co- ordinate the individual efforts. It is his concern with the impersonal processes of society in which more knowledge is utilized than any one individual or organized group of human beings can possess that puts the economists in constant opposition to the ambitions of other specialists who demand powers of control because they feel that their particular knowledge is not given sufficient consideration.\n\nIn one respect this book is, at the same time, more and less ambitious than the reader will expect. It is not chiefly concerned with the problems of any particular country or of a particular moment of time but, at least in its earlier parts, with principles which claim universal validity. The book owes its conception and plan to the recognition that the same intellectual trends, under different names or disguises, have undermined the belief in liberty throughout the world. If we want to counter these trends effectively, we must understand the common elements underlying all their manifestations. We must also remember that the tradition of liberty is not the exclusive creation of any single country and that no nation has sole possession of the secret even today. My main concern is not with the particular institutions or policies of the United States or of Great Britain but with the principles that these countries have developed on foundations provided by the ancient Greeks, the Italians of the early Renaissance, and the Dutch, and to which the French and  the Germans have made important contributions. Also, my aim will not be to provide a detailed program of policy but rather to state the criteria by which particular measures must be judged if they are to fit into a regime of freedom. It would be contrary to the whole spirit of this book if I were to consider myself competent to design a comprehensive program of policy. Such a program, after all, must grow out of the application of a common philosophy to the problems of the day.\n\nWhile it is not possible to describe an ideal adequately without constantly contrasting it with others, my aim is not mainly critical.6 My intention is to open doors for future development rather than to bar others, or, I should perhaps say, to prevent any such doors being barred, as invariably happens when the state takes sole control of certain developments. My emphasis is on the positive task of improving our institutions; and if I can do no more than indicate desirable directions of development, I have at any rate tried to be less concerned with the brushwood to be cleared away than with the roads which should be opened.\n\nAs a statement of general principles, the book must deal mainly with basic issues of political philosophy, but it approaches more tangible problems as it proceeds. Of its three parts, the first endeavors to show why we want liberty and what it does. This involves some examination of the factors which determine the growth of all civilizations. The discussion in this part must be mainly theoretical and philosophical\u2014if the latter is the right word to describe the field where political theory, ethics, and anthropology meet. It is followed by an examination of the institutions that Western man has developed to secure individual liberty. We enter here the field of jurisprudence and shall approach its problems historically. Yet it is neither from the point of view of the lawyer nor from that of the historian that we shall chiefly regard that evolution. Our concern will be with the growth of an ideal, only dimly seen and imperfectly realized at most times, which still needs further clarification if it is to serve as a guide for the solution of the problems of our times.\n\nIn the third part of the book those principles will be tested by the application of them to some of today's critical economic and social issues. The topics I have selected are in those areas where a false choice among the possibilities  before us is most likely to endanger freedom. Their discussion is meant to illustrate how often the pursuit of the same goals by different methods may either enhance or destroy liberty. They are mostly the kind of topics on which technical economics alone does not provide us with sufficient guidance to formulate a policy and which can be adequately treated only within a wider framework. But the complex issues which each of them raises can, of course, not be treated exhaustively in this volume. Their discussion serves mainly as an illustration of what is the chief aim of this book, namely, the interweaving of the philosophy, jurisprudence, and economics of freedom which is still needed.\n\nThis book is meant to help understanding, not to fire enthusiasm. Though in writing about liberty the temptation to appeal to emotion is often irresistible, I have endeavored to conduct the discussion in as sober a spirit as possible. Though the sentiments which are expressed in such terms as the \"dignity of man\" and the \"beauty of liberty\" are noble and praiseworthy, they can have no place in an attempt at rational persuasion. I am aware of the danger of such a cold- blooded and purely intellectual approach to an ideal which has been a sacred emotion to many and which has been stoutly defended by many more to whom it never constituted an intellectual problem. I do not think the cause of liberty will prevail unless our emotions are aroused. But, though the strong instincts on which the struggle for liberty has always nourished itself are an indispensable support, they are neither a safe guide nor a certain protection against error. The same noble sentiments have been mobilized in the service of greatly perverted aims. Still more important, the arguments that have undermined liberty belong mainly to the intellectual sphere, and we must therefore counter them here.\n\nSome readers will perhaps be disturbed by the impression that I do not take the value of individual liberty as an indisputable ethical presupposition and that, in trying to demonstrate its value, I am possibly making the argument in its support a matter of expediency. This would be a misunderstanding. But it is true that if we want to convince those who do not already share our moral suppositions, we must not simply take them for granted. We must show that liberty is not merely one particular value but that it is the source and condition of most moral values.7 What a free society offers to the individual is much  more than what he would be able to do if only he were free. We can therefore not fully appreciate the value of freedom until we know how a society of free men as a whole differs from one in which unfreedom prevails.\n\nI must also warn the reader not to expect the discussion to remain always on the plane of high ideals or spiritual values. Liberty in practice depends on very prosaic matters, and those anxious to preserve it must prove their devotion by their attention to the mundane concerns of public life and by the efforts they are prepared to give to the understanding of issues that the idealist is often inclined to treat as common, if not sordid. The intellectual leaders in the movement for liberty have all too often confined their attention to those uses of liberty closest to their hearts, and have made little effort to comprehend the significance of those restrictions of liberty which did not directly affect them.8\n\nIf the main body of the discussion is to be as matter of fact and unemotional as possible throughout, its starting point will of necessity have to be even more pedestrian. The meaning of some of the indispensable words has become so vague that it is essential that we should at the outset agree on the sense in which we shall use them. The words \"freedom\" and \"liberty\" have been the worst sufferers. They have been abused and their meaning distorted until it could be said that \"the word liberty means nothing until it is given specific content, and with a little massage it will take any content you like.\"9 We shall therefore have to begin by explaining what this liberty is that we are concerned with. The definition will not be precise until we have also examined such other almost equally vague terms as \"coercion,\" \"arbitrariness,\" and \"law\" which are indispensable in a discussion of liberty. The analysis of these concepts has, however, been postponed to the beginning of Part II, so that the arid effort at clarification of words should not present too great an obstacle before we reach the more substantial issues.\n\nFor this attempt at restating a philosophy of men's living together which has slowly developed through more than two thousand years, I have drawn  encouragement from the fact that it has often emerged from adversity with renewed strength. During the last few generations it has gone through one of its periods of decline. If to some, especially those in Europe, this book should appear to be a kind of inquest into the rationale of a system that no longer exists, the answer is that if our civilization is not to decline that system must be revived. Its underlying philosophy became stationary when it was most influential, as it had often progressed when on the defensive. It has certainly made little progress during the last hundred years and is now on the defensive. Yet the very attacks on it have shown us where it is vulnerable in its traditional form. One need not be wiser than the great thinkers of the past to be in a better position to comprehend the essential conditions of individual liberty. The experience of the last hundred years has taught us much that a Madison or a Mill, a Tocqueville or a Humboldt, could not perceive.\n\nWhether the moment has arrived when this tradition can be revived will depend not only on our success in improving it but also on the temper of our generation. It was rejected at a time when men would recognize no limits to their ambition, because it is a modest and even humble creed, based on a low opinion of men's wisdom and capacities and aware that within the range for which we can plan, even the best society will not satisfy all our desires. It is as remote from perfectionism as it is from the hurry and impatience of the passionate reformer, whose indignation about particular evils so often blinds him to the harm and injustice that the realization of his plans is likely to produce. Ambition, impatience, and hurry are often admirable in individuals; but they are pernicious if they guide the power of coercion and if improvement depends on those who, when authority is conferred on them, assume that in their authority lies superior wisdom and thus the right to impose their beliefs on others. I hope our generation may have learned that it has been perfectionism of one kind or another that has often destroyed whatever degree of decency societies have achieved.10 With more limited objectives, more patience, and more humility, we may in fact advance further and faster than we have done while under the guidance of \"a proud and most presumptuous confidence in the transcendent wisdom of this age, and its discernment.\"11\n\n* * *\n\nThe quotation at the head of this section is taken from Pericles' Funeral Oration as reported by Thucydides ii.36.4 to ii.37.3, Richard Crawley, trans., _The Complete Writings: The Peloponnesian War_ (New York: Modern Library, 1951), pp. 103\u20134.\n\n1 There are sayings which gain currency because they express what at one time seemed an important truth, continue to be used when this truth has become known to everybody, and are still used when, through frequent and mechanical use, they have ceased to carry a distinct meaning. They are finally dropped because they no longer provoke any thought. They are rediscovered only after they have been dormant for a generation and then can be used with new force to convey something like their original meaning\u2014only to go through the same cycle once more if they are successful.\n\n2 The last comprehensive attempt to restate the principles of a free society, already much qualified and in the restrained form expected of an academic textbook, is Henry Sidgwick, _The Elements of Politics_ (London: Macmillan, 1891). Though in many respects an admirable work, it scarcely represents what must be regarded as the British liberal tradition and is strongly tainted with that rationalist utilitarianism which led to socialism.\n\n3 In England, where the tradition of liberty lasted longer than in other European countries, as early as 1885 a writer whose work was then widely read among liberals could say of these liberals that \"the reconstruction of society, not the liberation of individuals, is now their most pressing task\" (Francis Charles Montague, _The Limits of Individual Liberty_ [London: Riving-tons, 1885], p. 16. [Montague (1858\u20131935) was regarded as one of the leading liberals of his period.\u2014Ed.]\n\n4 Frederick Mundell Watkins, _The Political Tradition of the West: A Study in the Development of Modern Liberalism_ (Cambridge, MA: Harvard University Press, 1948), p. x.\n\n5On the problem of \"expertocracy\" see Manfred Kuhn, _Herrschaft der Experten? An den Grenzen der Demokratie_ [Beitr\u00e4ge zur politischen Bildung No. 4] (W\u00fcrzburg: Werkbund-Verlag, 1961), and Kuhn's ear-lier writings mentioned there.\n\n6 I also hope that I shall not lay myself open to the reminder addressed to Edmund Burke by Samuel Taylor Coleridge, particularly important in our time, that \"it is bad policy to represent a political system as having no charm but for robbers and assassins, and no natural origin but in the brains of fools or madmen, when experience has proved that the great danger of the system consists in the peculiar fascination it is calculated to exert on noble and imaginative spirits; on all those who, in the amiable intoxication of youthful benevolence, are apt to mistake their own best virtues and choicest powers for the average qualities and attributes of the human character.\" ( _The Political Thought of Samuel Taylor Coleridge_ , Reginald James White, ed. [London: Jonathan Cape, 1938], pp. 235\u201336.)\n\n7 Cf. Wystan Hugh Auden in his \"Introduction,\" to Henry James, _The American Scene: Together with Three Essays from \"Portraits of Places\"_ (New York: Charles Scribner's Sons, 1946), p. xviii; and see also Christian Bay, _The Structure of Freedom_ (Stanford, CA.: Stanford University Press, 1958), p. 19: \"Freedom is the soil required for the full growth of other values.\" (This latter work became available too late to admit of more than occasional references in the notes.) See also Edmund Burke, _Ref ections on the Revolution in France_ in _Selected Works_ , Edward John Payne, ed. (3 vols.; Oxford: Clarendon Press, 1874), vol. 2, p. 122 [Liberty Fund edition, vol. 2, p. 201]: \"The world on the whole will gain by liberty without which virtue cannot exist.\"\n\n8 Cf. Alfred North Whitehead, _Adventure of Ideas_ (Mentor Books; New York: New American Library, 1955), p. 73. \"Unfortunately the notion of freedom has been eviscerated by the literary treatment devoted to it. . . . The concept of freedom has been narrowed to the picture of contemplative people shocking their generation. When we think of freedom, we are apt to confine ourselves to freedom of thought, freedom of the press, freedom of religious opinions. . . . This is a thorough mistake. . . . The literary exposition of freedom deals mainly with [the] frills. . . . In fact, freedom of action is a primary human need.\" [Hayek's footnote is to the first paperback edition of Whitehead's essay published by the New American Library in 1955 and now long out of print and extremely difficult of access. The original hardbound edition was published in New York by Macmillan in 1933. The quotation can be found on pp. 83\u201384 of the 1933 edition.\u2014Ed.]\n\n9 Carl Lotus Becker, _New Liberties for Old_ (New Haven: Yale University Press, 1941), p. 4.\n\n10 David Hume, who will be our constant companion and sage guide throughout the following pages, could speak as early as 1742 ( _Essays_ , \"Of Moral Prejudices,\" [Essay 2], vol. 2, pp. 371 and 373 [Liberty Fund edition, pp. 539 and 542]) of \"that grave philosophic Endeavour after Perfection, which, under Pretext of reforming Prejudices and Errors, strikes at all the most endearing Sentiments of the Heart, and all the most useful Byasses and Instincts, which can govern a human Creature.\"(vol. 2, p. 371) \"not to depart too far from the receiv'd Maxims of Conduct and Behaviour, by a refin'd Search after Happiness or Perfection\" (vol. 2, p. 373).\n\n11 William Wordsworth, _The Excursion: Being a Portion of The Recluse; A Poem_ (London: Printed for Longman, Hurst, Rees, Orme, and Brown, 1814), pt. 2, p. 62.\n\n## PART I\n\nTHE VALUE OF FREEDOM\n\nThroughout history orators and poets have extolled liberty, but no one has told us why liberty is so important. Our attitude towards such matters should depend on whether we consider civilization as fixed or as advancing. . . . In an advancing society, . . . any restriction on liberty reduces the number of things tried and so reduces the rate of progress. In such a society freedom of action is granted to the individual, not because it gives him greater satisfaction but because if allowed to go his own way he will on the average serve the rest of us better than under any orders we know how to give.\n\n\u2014H. B. Phillips\n\n* * *\n\nThis quotation is taken from Henry Bayard Phillips, \"On the Nature of Progress,\" _American Scientist_ 33 (1945): p. 255.\n\n## ONE\n\nLIBERTY AND LIBERTIES\n\nThe world has never had a good definition of the word liberty, and the American people, just now, are much in want of one. We all declare for liberty; but in using the same word, we do not mean the same thing. . . . Here are two, not only different, but incompatible things, called by the same name, liberty.\n\n\u2014Abraham Lincoln\n\n1. We are concerned in this book with that condition of men in which coercion of some by others is reduced as much as is possible in society. This state we shall describe throughout as a state of liberty or freedom.1 These two words have been also used to describe many other good things of life. It would therefore not be very profitable to start by asking what they really mean.2 It  would seem better to state, first, the condition which we shall mean when we use them and then consider the other meanings of the words only in order to define more sharply that which we have adopted.\n\nThe state in which a man is not subject to coercion by the arbitrary will of another or others3 is often also distinguished as \"individual\" or \"personal\" freedom, and whenever we want to remind the reader that it is in this sense that we are using the word \"freedom,\" we shall employ that expression. Some-times the term \"civil liberty\" is used in the same sense, but we shall avoid it because it is too liable to be confused with what is called \"political liberty\"\u2014 an inevitable confusion arising from the fact that \"civil\" and \"political\" derive, respectively, from Latin and Greek words with the same meaning.4\n\nEven our tentative indication of what we shall mean by \"freedom\" will have shown that it describes a state which man living among his fellows may hope to approach closely but can hardly expect to realize perfectly. The task of a policy of freedom must therefore be to minimize coercion or its harmful effects, even if it cannot eliminate it completely.\n\nIt so happens that the meaning of freedom that we have adopted seems to be the original meaning of the word.5 Man, or at least European man, enters history divided into free and unfree; and this distinction had a very definite meaning.6 The freedom of the free may have differed widely, but only in the degree of an independence which the slave did not possess at all. It meant always the possibility of a person's acting according to his own decisions and plans, in contrast to the position of one who was irrevocably subject to the will of another, who by arbitrary decision could coerce him to act or not to act in specific ways. The time-honored phrase by which this freedom has often been described is therefore \"independence of the arbitrary will of another.\"\n\nThis oldest meaning of \"freedom\" has sometimes been described as its vulgar meaning; but when we consider all the confusion that philosophers have  caused by their attempts to refine or improve it, we may do well to accept this description. More important, however, than that it is the original meaning is that it is a distinct meaning and that it describes one thing and one thing only, a state which is desirable for reasons different from those which make us desire other things also called \"freedom.\" We shall see that, strictly speaking, these various \"freedoms\" are not different species of the same genus but entirely different conditions, often in conflict with one another, which therefore should be kept clearly distinct. Though in some of the other senses it may be legitimate to speak of different kinds of freedom, \"freedoms from\" and \"freedoms to,\" in our sense \"freedom\" is one, varying in degree but not in kind.\n\nIn this sense \"freedom\" refers solely to a relation of men to other men,7 and the only infringement on it is coercion by men. This means, in particular, that the range of physical possibilities from which a person can choose at a given moment has no direct relevance to freedom. The rock climber on a difficult pitch who sees only one way out to save his life is unquestionably free, though we would hardly say he has any choice. Also, most people will still have enough feeling for the original meaning of the word \"free\" to see that if that same climber were to fall into a crevasse and were unable to get out of it, he could only figuratively be called \"unfree,\" and that to speak of him as being \"deprived of liberty\" or of being \"held captive\" is to use these terms in a sense different from that in which they apply to social relations.8\n\nThe question of how many courses of action are open to a person is, of course, very important. But it is a different question from that of how far in acting he can follow his own plans and intentions, to what extent the pattern of his conduct is of his own design, directed toward ends for which he has been persistently striving rather than toward necessities created by others  in order to make him do what they want. Whether he is free or not does not depend on the range of choice but on whether he can expect to shape his course of action in accordance with his present intentions, or whether somebody else has power so to manipulate the conditions as to make him act according to that person's will rather than his own. Freedom thus presupposes that the individual has some assured private sphere, that there is some set of circumstances in his environment with which others cannot interfere.\n\nThis conception of liberty can be made more precise only after we have examined the related concept of coercion. This we shall do systematically after we have considered why this liberty is so important. But even before we attempt this, we shall endeavor to delineate the character of our concept somewhat more precisely by contrasting it with the other meanings which the word liberty has acquired. They have the one thing in common with the original meaning in that they also describe states which most men regard as desirable; and there are some other connections between the different meanings which account for the same word being used for them.9 Our immediate task, however, must be to bring out the differences as sharply as possible.\n\n2. The first meaning of \"freedom\" with which we must contrast our own use of the term is one generally recognized as distinct.10 It is what is commonly called \"political freedom,\" the participation of men in the choice of their government, in the process of legislation, and in the control of administration. It derives from an application of our concept to groups of men as a whole which gives them a sort of collective liberty. But a free people in this sense is not necessarily a people of free men; nor need one share in this collective  freedom to be free as an individual. It can scarcely be contended that the inhabitants of the District of Columbia, or resident aliens in the United States, or persons too young to be entitled to vote do not enjoy full personal liberty because they do not share in political liberty.11\n\nIt would also be absurd to argue that young people who are just entering into active life are free because they have given their consent to the social order into which they were born: a social order to which they probably know no alternative and which even a whole generation who thought differently from their parents could alter only after they had reached mature age. But this does not, or need not, make them unfree. The connection which is often sought between such consent to the political order and individual liberty is one of the sources of the current confusion about its meaning. Anyone is, of course, entitled to \"identify liberty . . . with the process of active participation in public power and public law making.\"12 Only it should be made clear that, if he does so, he is talking about a state other than that with which we are here concerned, and that the common use of the same word to describe these different conditions does not mean that the one is in any sense an equivalent or substitute for the other.13\n\nThe danger of confusion here is that this use tends to obscure the fact that a person may vote or contract himself into slavery and thus consent to give up freedom in the original sense. It would be difficult to maintain that a man who voluntarily but irrevocably had sold his services for a long period of years to a military organization such as the Foreign Legion remained free thereafter in our sense; or that a Jesuit who lives up to the ideals of the founder of his order and regards himself \"as a corpse which has neither intelligence nor will\" could be so described.14 Perhaps the fact that we have seen millions voting themselves into complete dependence on a tyrant has made our generation understand that to choose one's government is not necessarily to secure freedom. Moreover, it would seem that discussing the value of freedom would be pointless if any regime of which people approved was, by definition, a regime of freedom.\n\nThe application of the concept of freedom to a collective rather than to individuals is clear when we speak of a people's desire to be free from a foreign yoke and to determine its own fate. In this case we use \"freedom\" in the sense of absence of coercion of a people as a whole. The advocates of individual freedom have generally sympathized with such aspirations for national freedom, and this led to the constant but uneasy alliance between the liberal and the national movements during the nineteenth century.15 But though the  concept of national freedom is analogous to that of individual freedom, it is not the same; and the striving for the first has not always enhanced the second. It has sometimes led people to prefer a despot of their own race to the liberal government of an alien majority; and it has often provided the pretext for ruthless restrictions of the individual liberty of the members of minorities. Even though the desire for liberty as an individual and the desire for liberty of the group to which the individual belongs may often rest on similar feelings and sentiments, it is still necessary to keep the two conceptions clearly apart.\n\n3. Another different meaning of \"freedom\" is that of \"inner\" or \"meta-physical\" (sometimes also \"subjective\") freedom.16 It is perhaps more closely related to individual freedom and therefore more easily confounded with it. It refers to the extent to which a person is guided in his actions by his own considered will, by his reason or lasting conviction, rather than by momentary impulse or circumstance. But the opposite of \"inner freedom\" is not coercion by others but the influence of temporary emotions, or moral or intellectual weakness. If a person does not succeed in doing what, after sober reflection, he decides to do, if his intentions or strength desert him at the decisive moment and he fails to do what he somehow still wishes to do, we may say that he is \"unfree,\" the \"slave of his passions.\" We occasionally also use these terms when we say that ignorance or superstition prevents people from doing what they would do if they were better informed, and we claim that \"knowledge makes free.\"\n\nWhether or not a person is able to choose intelligently between alternatives, or to adhere to a resolution he has made, is a problem distinct from whether or not other people will impose their will upon him. They are clearly not without some connection: the same conditions which to some constitute coercion will be to others merely ordinary difficulties which have to be overcome, depending on the strength of will of the people involved. To that  extent, \"inner freedom\" and \"freedom\" in the sense of absence of coercion will together determine how much use a person can make of his knowledge of opportunities. The reason why it is still very important to keep the two apart is the relation which the concept of \"inner freedom\" has to the philosophical confusion about what is called the \"freedom of the will.\" Few beliefs have done more to discredit the ideal of freedom than the erroneous one that scientific determinism has destroyed the basis for individual responsibility. We shall later (in chap. 5) consider these issues further. Here we merely want to put the reader on guard against this particular confusion and against the related sophism that we are free only if we do what in some sense we ought to do.\n\n4. Neither of these confusions of individual liberty with different concepts denoted by the same word is as dangerous as its confusion with a third use of the word to which we have already briefly referred: the use of \"liberty\" to describe the physical \"ability to do what I want,\"17 the power to satisfy our wishes, or the extent of the choice of alternatives open to us. This kind of \"freedom\" appears in the dreams of many people in the form of the illusion that they can fly; that they are released from gravity and can move \"free like a bird\" to wherever they wish, or that they have the power to alter their environment to their liking.\n\nThis metaphorical use of the word has long been common, but until comparatively recent times few people seriously confused this \"freedom from\" obstacles, this freedom that means omnipotence, with the individual freedom that any kind of social order can secure. Only since this confusion was deliberately fostered as part of the socialist argument has it become dangerous. Once this identification of freedom with power is admitted, there is no limit to the sophisms by which the attractions of the word \"liberty\" can be used to support measures which destroy individual liberty,18 no end to the tricks by which  people can be exhorted in the name of liberty to give up their liberty. It has been with the help of this equivocation that the notion of collective power over circumstances has been substituted for that of individual liberty and that in totalitarian states liberty has been suppressed in the name of liberty.\n\nThe transition from the concept of individual liberty to that of liberty as power has been facilitated by the philosophical tradition that uses the word \"restraint\" where we have used \"coercion\" in defining liberty. Perhaps \"restraint\" would in some respects be a more suitable word if it was always remembered that in its strict sense it presupposes the action of a restraining human agent.19 In this sense, it usefully reminds us that the infringements on liberty consist largely in people's being prevented from doing things, while \"coercion\" emphasizes their being made to do particular things. Both aspects are equally important: to be precise, we should probably define liberty as the absence of restraint and constraint.20 Unfortunately, both these words have come also to be used for influences on human action that do not come from other men; and it is only too easy to pass from defining liberty as the absence of restraint to defining it as the \"absence of obstacles to the realization of [our] desires\"21 or even more generally as \"the absence of external impediments.\"22 This is equivalent to interpreting it as effective power to do whatever we want.\n\nThis reinterpretation of liberty is particularly ominous because it has penetrated deeply into the usage of some of the countries where, in fact, individual  freedom is still largely preserved. In the United States it has come to be widely accepted as the foundation for the political philosophy dominant in \"liberal\" circles. Such recognized intellectual leaders of the \"progressives\" as J. R. Commons23 and John Dewey have spread an ideology in which \"liberty is power, effective power to do specific things\" and the \"demand for liberty is the demand for power,\"24 while the absence of coercion is merely \"the negative side of freedom\" and \"is to be prized only as a means to a freedom which is power.\"25\n\n5. This confusion of liberty as power with liberty in its original meaning inevitably leads to the identification of liberty with wealth;26 and this makes it  possible to exploit all the appeal which the word \"liberty\" carries in the support for a demand for the redistribution of wealth. Yet, though freedom and wealth are both good things which most of us desire and though we often need both to obtain what we wish, they still remain different. Whether or not I am my own master and can follow my own choice and whether the possibilities from which I must choose are many or few are two entirely different questions. The courtier living in the lap of luxury but at the beck and call of his prince may be much less free than a poor peasant or artisan, less able to live his own life and to choose his own opportunities for usefulness. Similarly, the general in charge of an army or the director of a large construction project may wield enormous powers which in some respects may be quite uncontrollable, and yet may well be less free, more liable to have to change all his intentions and plans at a word from a superior, less able to change his own life or to decide what to him is most important, than the poorest farmer or shepherd.\n\nIf there is to be any clarity in the discussion of liberty, its definition must not depend upon whether or not everybody regards this kind of liberty as a good thing. It is very probable that there are people who do not value the liberty with which we are concerned, who cannot see that they derive great benefits from it, and who will be ready to give it up to gain other advantages; it may even be true that the necessity to act according to one's own plans and decisions may be felt by them to be more of a burden than an advantage. But liberty may be desirable, even though not all persons may take advantage of it. We shall have to consider whether the benefit derived from liberty by the majority is dependent upon their using the opportunities it offers them and whether the case for liberty really rests on most people wanting it for them-selves. It may well be that the benefits we receive from the liberty of all do not derive from what most people recognize as its effects; it may even be that liberty exercises its beneficial effects as much through the discipline it imposes on us as through the more visible opportunities it offers.\n\nAbove all, however, we must recognize that we may be free and yet miser-able. Liberty does not mean all good things27 or the absence of all evils. It is  true that to be free may mean freedom to starve, to make costly mistakes, or to run mortal risks. In the sense in which we use the term, the penniless vagabond who lives precariously by constant improvisation is indeed freer than the conscripted soldier with all his security and relative comfort. But if liberty may therefore not always seem preferable to other goods, it is a distinctive good that needs a distinctive name. And though \"political liberty\" and \"inner liberty\" are long-established alternative uses of the term which, with a little care, may be employed without causing confusion, it is questionable whether the use of the word \"liberty\" in the sense of \"power\" should be tolerated.\n\nIn any case, however, the suggestion must be avoided that, because we employ the same word, these \"liberties\" are different species of the same genus. This is the source of dangerous nonsense, a verbal trap that leads to the most absurd conclusions.28 Liberty in the sense of power, political liberty, and inner liberty are not states of the same kind as individual liberty: we cannot, by sacrificing a little of the one in order to get more of the other, on balance gain some common element of freedom. We may well get one good thing in the place of another by such an exchange. But to suggest that there is a common element in them which allows us to speak of the effect that such an exchange has on liberty is sheer obscurantism, the crudest kind of philosophical realism, which assumes that, because we describe these conditions with the same word, there must also be a common element in them. But we want them largely for different reasons, and their presence or absence has different effects. If we have to choose between them, we cannot do so by asking whether liberty will be increased as a whole, but only by deciding which of these different states we value more highly.\n\n6. It is often objected that our concept of liberty is merely negative.29 This is  true in the sense that peace is also a negative concept or that security or quiet or the absence of any particular impediment or evil is negative. It is to this class of concepts that liberty belongs: it describes the absence of a particular obstacle\u2014coercion by other men. It becomes positive only through what we make of it. It does not assure us of any particular opportunities, but leaves it to us to decide what use we shall make of the circumstances in which we find ourselves.\n\nBut while the uses of liberty are many, liberty is one. Liberties appear only when liberty is lacking: they are the special privileges and exemptions that groups and individuals may acquire while the rest are more or less unfree. Historically, the path to liberty has led through the achievement of particular liberties. But that one should be allowed to do specific things is not liberty, though it may be called \"a liberty\"; and while liberty is compatible with not being allowed to do specific things, it does not exist if one needs permission for most of what one can do. The difference between liberty and liberties is that which exists between a condition in which all is permitted that is not prohibited by general rules and one in which all is prohibited that is not explicitly permitted.\n\nIf we look once more at the elementary contrast between freedom and slavery, we see clearly that the negative character of freedom in no way diminishes its value. We have already mentioned that the sense in which we use the word is its oldest meaning. It will help to fix this meaning if we glance at the actual difference that distinguished the position of a free man from that of a slave. We know much about this so far as the conditions in the oldest of free communities\u2014the cities of ancient Greece\u2014are concerned. The numerous decrees for the freeing of slaves that have been found give us a clear picture of the essentials. There were four rights which the attainment of freedom regularly conferred. The manumission decrees normally gave the former slave, first, \"legal status as a protected member of the community\"; second, \"immunity from arbitrary arrest\"; third, the right to \"work at whatever he desires to do\"; and, fourth, the right to \"movement according to his own choice.\"30\n\nThis list contains most of what in the eighteenth and nineteenth centuries were regarded as the essential conditions of freedom. It omits the right to own property only because even the slave could do so.31 With the addition of  this right, it contains all the elements required to protect an individual against coercion. But it says nothing about the other freedoms we have considered, not to speak of all the \"new freedoms\" that have lately been offered as substitutes for freedom. Clearly, a slave will not become free if he obtains merely the right to vote, nor will any degree of \"inner freedom\" make him anything but a slave\u2014however much idealist philosophers have tried to convince us to the contrary. Nor will any degree of luxury or comfort or any power that he may wield over other men or the resources of nature alter his dependence upon the arbitrary will of his master. But if he is subject only to the same laws as all his fellow citizens, if he is immune from arbitrary confinement and free to choose his work, and if he is able to own and acquire property, no other men or group of men can coerce him to do their bidding.\n\n7. Our definition of liberty depends upon the meaning of the concept of coercion, and it will not be precise until we have similarly defined that term. In fact, we shall also have to give a more exact meaning to certain closely related ideas, especially arbitrariness and general rules or laws. Logically, we should therefore now proceed to a similar analysis of these concepts. We cannot altogether avoid this. But before asking the reader to follow us further in what may appear to be the barren task of giving precise meaning to terms, we shall endeavor to explain why the liberty we have defined is so important. We shall therefore resume our effort at precise definition only at the beginning of the second part of this book, where we shall examine the legal aspects of a regime of freedom. At this point a few observations anticipating the results of the more systematic discussion of coercion should be sufficient. In this brief form they will necessarily seem somewhat dogmatic and will have to be justified later.\n\nBy \"coercion\" we mean such control of the environment or circumstances of a person by another that, in order to avoid greater evil, he is forced to act not according to a coherent plan of his own but to serve the ends of another. Except in the sense of choosing the lesser evil in a situation forced on him by another, he is unable either to use his own intelligence or knowledge or to follow his own aims and beliefs. Coercion is evil precisely because it thus eliminates an individual as a thinking and valuing person and makes him a bare tool in the achievement of the ends of another. Free action, in which a person pursues his own aims by the means indicated by his own knowledge, must be based on data which cannot be shaped at will by another. It presupposes the existence of a known sphere in which the circumstances cannot be so shaped by another person as to leave one only that choice prescribed by the other.\n\nCoercion, however, cannot be altogether avoided because the only way to prevent it is by the threat of coercion.32 Free society has met this problem by  conferring the monopoly of coercion on the state33 and by attempting to limit this power of the state to instances where it is required to prevent coercion by private persons. This is possible only by the state's protecting known private spheres of the individuals against interference by others and delimiting these private spheres, not by specific assignation, but by creating conditions under which the individual can determine his own sphere by relying on rules which tell him what the government will do in different types of situations.\n\nThe coercion which a government must still use for this end is reduced to a minimum and made as innocuous as possible by restraining it through known general rules, so that in most instances the individual need never be coerced unless he has placed himself in a position where he knows he will be coerced. Even where coercion is not avoidable, it is deprived of its most harmful effects by being confined to limited and foreseeable duties, or at least made independent of the arbitrary will of another person. Being made impersonal and dependent upon general, abstract rules, whose effect on particular individuals cannot be foreseen at the time they are laid down, even the coercive acts of government become data on which the individual can base his own plans. Coercion according to known rules, which is generally the result of circumstances in which the person to be coerced has placed himself, then becomes an instrument assisting the individuals in the pursuit of their own ends and not a means to be used for the ends of others.\n\n* * *\n\nThe quotation at the head of the chapter is taken from Abraham Lincoln, _The Writings of Abraham Lincoln_ , Arthur Brooks Lapsley, ed. (Federal ed.; 8 vols.; New York: G. P. Putnam's Sons, 1905), vol. 7, p. 121. Cf. the similar remark by Montesquieu, _Spirit of the Laws_ , bk. 11, chap. 2, vol. 1, p. 149: \"there is no word that admits of more various significations, and has made more varied impressions on the human mind, than that of _liberty_. Some have taken it as a means of deposing a person on whom they had conferred a tyrannical authority; others for the power of choosing a superior whom they are obliged to obey, others for the right of bearing arms, and of being thereby enabled to use violence; others, in fine, for the privilege of being governed by a native of their own country, or by their own laws.\" [\"Il n'y a point de mot qui ait re\u00e7u plus de diff\u00e9rentes significations, et qui ait frapp\u00e9 les esprits de tant de mani\u00e8res, que celui de _libert\u00e9_. Les uns l'ont pris pour la facilit\u00e9 de d\u00e9poser celui \u00e0 qui ils avoient donn\u00e9 un pouvoir tyrannique; les autres, pour la facult\u00e9 d'\u00e9lire celui \u00e0 qui ils d\u00e9voient ob\u00e9ir; d'autres, pour le droit d'\u00eatre arm\u00e9s, et de pouvoir exercer la violence; ceuxci pour le privil\u00e8ge de n'\u00eatre gouvern\u00e9s que par un homme de leur nation, ou par leurs propres lois.\"(vol. 2, p. 394)\u2014Ed.]\n\n1 There does not seem to exist any accepted distinction in meaning between the words \"freedom\" and \"liberty,\" and we shall use them interchangeably. Though I have a personal preference for the former, it seems that \"liberty\" lends itself less to abuse. It could hardly have been used for that \"noble pun\" ( Joan Robinson, _Private Enterprise or Public Control_ [Handbook for Discussion Groups, No. 11; London: Association for Education in Citizenship, 1943], p. 13) of Franklin D. Roosevelt's when he included \"freedom from want\" in his conception of liberty.\n\n2 The limited value of even a very acute semantic analysis of the term \"freedom\" is well illustrated by Maurice William Cranston, _Freedom: A New Analysis_ (New York: Longmans, Green, and Co., 1953), which will be found illuminating by readers who like to see how philosophers have tied themselves in knots by their curious definitions of the concept. For a more ambitious survey of the various meanings of the word see Mortimer Jerome Adler, _The Idea of Freedom: A Dialectical Examination of the Conceptions of Freedom_ (Garden City, NY: Doubleday, 1958), which I have been privileged to see in draft, and an even more comprehensive work by Harald Ofstad, _An Inquiry into the Freedom of Decision_ (Oslo: Norwegian University Press; Stockholm: Svenska bokf\u00f6rlaget, 1961).\n\n3 See Aristotle, _Metaphysics_ , 1.2.8 [982b]: \"As man is free w e say he exists for his own sake and not for another's.\" Gottfried Wilhelm Leibniz, \"\u00dcber die Freiheit,\" in _Philosophische Werke: Hauptschriften zur Grundlegung der Philosophie_ , Artur Buchenau and Ernst Cassirer, eds. (Leipzig: Verlag der D\u00fcrr'schen Buchhandlung, 1906), vol. 2, p. 497: \"Daher kam ich der Meinung derer nahe, es sei f\u00fcr die Freiheit genug, da\u00df das Geschehen dem _Zwange_ nicht unterworfen ist, wenngleich es der _Notwendigkeit_ untersteht.\" [\"Thus, I approached the opinion of those it would suffice for liberty that events are not subordinate to _coercion_ , albeit, they are subject to _necessity_.\"\u2014Ed.]; Cf. Jeremy Bentham, _The Limits of Jurisprudence Defined: Being Part Two of an Introduction to the Principles of Morals and Legislation_ , Charles Warren Everett, ed. (New York: Columbia University Press, 1945), p. 59: \"Liberty then is of two or even more sorts, according to the number of quarters from whence coercion, which it is the absence of, may come.\" See also Georg Jellinek, _System der subjektiven \u00f6ffentlichen Rechte_ (2nd ed.; T\u00fcbingen: Verlag von J. C. B. Mohr, 1905), chap. 8, pp. 94\u2013114: \"Der negative Status (status liberatatis)\"; Moritz Schlick, _Fragen der Ethik_ (Vienna: J. Springer, 1930), p. 110: \"Freiheit bedeutet ja den Gegensatz zum Zwang, der Mensch ist frei, wenn er nicht gezwungen handelt\" [\"Freedom means the opposite of compulsion; a man is free if he does not act under compulsion.\" _Problems of Ethics_ , David Rynin, trans. (New York: Prentice Hall, 1939), p. 150\u2014Ed.]; Frank Hyneman Knight, \"The Meaning of Freedom,\" in _The Philosophy of American Democracy_ , Charner Marquis Perry, ed. (Chicago: University of Chicago Press, 1943), p. 75: \"The primary meaning of freedom in society . . . is always a negative concept . . . and coercion is the term which must really be defined\"; and the fuller discussion by the same author in his review article \"The Meaning of Freedom,\" a review of _Freedom: Its Meaning_ , Ruth Nanda Anshen, ed. _Ethics_ , 52 (1941): 86\u2013109, and \"Conflict of Values: Freedom and Justice,\" in _Goals of Economic Life_ , Alfred Dudley Ward, ed. (New York: Harper, 1953); also Franz Leopold Neumann, _The Democratic and the Authoritarian State: Essays in Political and Legal Theory_ (Glencoe, IL: The Free Press, 1957), p. 202: \"The formula, freedom equals absence of coercion, is still correct. . . . [F]rom this formula there follows fundamentally the whole rational legal system of the civilized world. . . . It is _the_ element of the concept of freedom that we can never give up\"; and Christian Bay, _The Structure of Freedom_ (Stanford, CA: Stanford University Press, 1958), p. 94: \"Among all the freedom goals, the goal of maximizing everyone's freedom from coercion should take first priority.\"\n\n4 Currently the expression \"civil liberty\" seems to be used chiefly with respect to those exercises of individual liberty which are particularly significant for the functioning of democracy, such as freedom of speech, of assembly, and of the press\u2014and in the United States particularly with reference to the opportunities guaranteed by the Bill of Rights. Even the term \"political liberty\" is occasionally used to describe, especially in contrast to \"inner liberty,\" not the collective liberty for which we shall employ it, but personal liberty. But though this usage has the sanction of Montesquieu, it can today only cause confusion.\n\n5 Cf. Sir Ernest Barker, _Reflections on Government_ (London: Oxford University Press, 1942), pp. 1\u20132: \"Originally liberty signified the quality or status of the free man, or free producer, in contradistinction to the slave.\" It seems that, etymologically, the Teutonic root of \"free\" described the position of a protected member of the community (cf. Gustav Neckel, \"Adel und Gefolgschaft: Ein beitrag zur germanischen altertumskunde,\" _Beitr\u00e4ge zur Geschichte der deutschen Sprache und Literatur_ 41 [1916], esp. 403: \"'Frei' hiess urspr\u00fcnglich derjenige, der nicht schutzund rechtlos war\" [\"Originally the term 'free' referred to those who had neither legal protection nor rights.\"\u2014Ed.]. See also Otto Schrader, _Sprachvergleichung und Urgeschichte. Linguistisch-historische Beitr\u00e4ge zur Erforschung des indogermanischen Altertums_. Vol. 2, part 2: _Die Urzeit_. (3rd ed.; Jena: H. Costenoble, 1907), p. 294, and Adolf Waas, _Die alte deutsche Freiheit. Ihr wesen und ihre geschichte_ (Munich and Berlin: R. Oldenburg, 1939), pp. 10\u201315. Similarly, Latin _liber_ and Greek _eleutheros_ seem to derive from words denoting membership in the tribe. The signifcance of this will appear later when we examine the relation between law and liberty. See also Ruth Fulton Benedict, \"Primitive Freedom,\" _Atlantic Monthly_ , 169 (1942): 760: \"So too in pr imitive societies there are civil liberties, the crux of which is that they are guaranteed to all men without discrimination. Wherever these privileges and protections to which all members have an inalienable right are important privileges in the eyes of that tribe, people regard themselves, whatever their form of government, as free men enjoying the blessings of liberty.\"\n\n6 Max Pohlenz, _Griechische Freiheit: Wesen und Werden eines Lebensideals_ (Heidelberg: Quelle und Meyer, 1955), p. 7: \"Historisch ist die Begriffsentwicklung aber so verlaufen, da\u00df erst das Vorhandensein von Unfreien, von Sklaven, bei den anderen das Gef\u00fchl der F reiheit weckte.\" [\"Historically, it was the existence of the unfree, the slaves, that first gave the others the feeling that they themselves were free.\"\u2014Ed.]\n\n7 Cf. Thomas Hill Green, _Lectures on the Principles of Political Obligation_ [1895] (new imprint; London: Longmans, Green, and Co., 1911), p. 3: \"As to the sense given to 'freedom,' it must of course be admitted that every usage of the term to express anything but a social and political relation of one man to others involves a metaphor. Even in the original application its sense is by no means fixed. It always implies indeed some exemption from compulsion by others, but the extent and conditions of this exemption, as enjoyed by the 'freeman' in different states of society, are very various. As soon as the term 'freedom' comes to be applied to anything else than an established relation between a man and other men, its sense fluctuates much more.\" Also, Ludwig von Mises, _Socialism_ (new ed.; New Haven: Yale University Press, 1951), p. 191: \"Freedom is a sociological concept. It is meaningless to apply it to conditions outside society\" [pt. 2, chap. 9, sec. 3]; and p. 194: \"This, then, is freedom in the external life of man\u2014that he is independent of the arbitrary power of his fellows\" [pt. 2, chap. 9, sec. 3] [Liberty Fund edition, pp. 169 and 171].\n\n8 Cf. Knight, \"Review: The Meaning of Freedom,\" p. 93: \"If Crusoe fell into a pit or became entangled in jungle growth, it would certainly be correct usage to speak of his freeing himself or regaining his liberty\u2014and this would apply to an animal as well.\" This may well be established usage by now, but it nevertheless refers to a conception of liberty other than that of absence of coercion which Professor Knight defends.\n\n9 The linguistic cause of the transfer of \"free\" and of the corresponding nouns to various uses seems to have been the lack in English (and apparently in all Germanic and Romance languages) of an adjective which can be used generally to indicate that something is absent. \"Devoid\" or \"lacking\" are generally used only to express the absence of something desirable or normally present. There is no corresponding adjective (other than \"free\" of ) to describe the absence of something undesirable or alien to an object. We will generally say that something is free of vermin, of impurities, or of vice, and thus freedom has come to mean the absence of anything undesirable. Similarly, whenever we want to say that something acts by itself, undetermined, or uninfluenced by external factors, we speak of its being free of influences not normally connected with it. In science we speak even of \"degrees of freedom\" when there are several possibilities unaffected by the known or assumed determinants (cf. Cranston, _Freedom: A New Analysis_ , p. 5). And see also the excellent essays by Stanley Isaac Benn and Richard Stanley Peters, _Social Principles and the Democratic State_ (London: Allen and Unwin, 1959), p. 212: \"any condition can be described as the absence of its opposite. If health is 'freedom from disease,' education 'freedom from ignorance,' there is no conceivable object of social organization and action that cannot be called 'freedom.' But the price of making 'freedom' all-embracing as a social end is to drain it of all prescriptive meaning, and to leave only the prescriptive overtones, to make it synonymous with terms of approval like 'good' and 'desirable.'\"\n\n10 This sharp differentiation between \"freedom,\" in the sense of alternately ruling and obeying, and \"liberty,\" in the sense that we may live as we choose, occurs as early as Aristotle, _Politics_ , 6.3 [1317b]. [\"One factor of liberty is to govern and be governed in turn.\"\u2014Ed.]\n\n11 All these would have to be described as unfree by Harold Joseph Laski, who contended ( _Liberty in the Modern State_ [new ed.; London: Allen and Unwin, 1948], p. 48.) that \"the right . . . to the franchise is essential to liberty; and a citizen excluded from it is unfree.\" By similarly defining freedom, Hans Kelsen (\"The Foundations of Democracy,\" _Ethics_ , 66, no. 1, pt. 2, [1955]: 94) triumphantly reaches the conclusion that \"the attempts at showing an essential connection between freedom and property . . . have failed, though all those who have asserted such a connection have been speaking of individual and not political freedom.\"\n\n12 Edwin Mims, Jr., _The Majority of the People_ (New York: Modern Age Books, 1941), p. 170.\n\n13 Cf. Montesquieu, _Spirit of the Laws_ , bk. ix., sec. 2 (vol. 1, p. 150) [French edition, vol. 2, p. 394]: \"In fine, as in democracies the people seem to act almost as they please, this sort of government has been deemed the most free, and the power of the people has been confounded with their liberty.\" [\"Enfn, comme dans les d\u00e9mocraties le peuple paro\u00eet \u00e0 peu pr\u00e8s faire ce qu'il veut, on a mis la libert\u00e9 dans ces sortes de gouvernements; et on a confondu le pouvoir du peuple avec la libert\u00e9 du peuple.\"\u2014Ed.]\n\nSee also Jean Louis de Lolme, _The Constitution of England, or, An Account of the English Government: In Which It Is Compared Both with the Republican Form of Government, and the Other Monarchies in Europe_ [1784] (new ed.; London, G. G. and J. Robinson, 1800), bk. 2, chap. 5, p. 240 [Liberty Fund edition, p. 170]: \"To concur by one's suffrage in enacting laws, is to enjoy a share, whatever it may be, of power: to live in a state where the laws are equal for all, and sure to be executed . . . is to be free.\" [\"Contribuer, par son suffrage, \u00e0 la sanction des lois, c'est avoir une portion quelconque de puissance, mais donc l'exercice de laquelle, encore une fois, on est tr\u00e8s \u00e9loign\u00e9 de voir toujours sa volont\u00e9 r\u00e9ussir. Vivre dans un \u00e9tat o\u00f9 les lois sont \u00e9gales pour tous, et s\u00dbrement ex\u00e9cut\u00e9es . . . c'est \u00eatre libre.\" Jean Louis de Lolme, _Constitution de l'Angleterre, ou \u00e9tat du gouvernement anglois, compar\u00e9 avec la forme r\u00e9publicaine & avec les autres monarchies de l'Europe_ (2 vols.; London: G. Robinson, J. Murray, 1785), vol. 1, p. 218.\u2014Ed.]\n\nCf. also the passages quoted in nn. 2 and 5 to chap. 7. [The two passages to which Hayek refers appear in two footnotes to chapter 7 of book 2 of de Lolme's work. The footnotes are not numbered but Hayek is apparently referring to the following two quotations. The first is from Valerius Maximus, _Memorable Doings and Sayings_ (bk. 3, sec. 7) and reads: \"Valerius Maximus relates that the tribunes of the people having offered to propose some regulations in regard to the price of corn, in a time of great scarcity, Scipio Nasica over-ruled the assembly merely by saying: 'Silence, Romans! I know better than you what is expedient for the republic'\u2014which words were no sooner heard by the people, than they showed by a silence full of veneration, that they were more affected by his authority, than by the necessity of providing for their own subsistence.\" (de Lolme, vol. 1, p. 256; Liberty Fund edition, p. 179). The second, from Livy, (6.16.3\u2013 4), reads: \"The tribunes of the people,\" says Livy, who as a great admirer of the aristocratical power, \"and the people themselves, durst neither lift up their eyes, nor even mutter, in the presence of the dictator.\" (de Lolme, vol. 1, pp. 257\u201358; Liberty Fund edition, p. 180)\u2014Ed.]\n\n14 The full description of the proper state of mind of a Jesuit, quoted by William James from one of the letters of Ignatius Loyola ( _Varieties of Religious Experience: A Study in Human Nature_ [New York: Longmans, Green, and Co., 1902], p. 314) runs as follows: \"In the hands of my Superior, I must be a soft wax, a thing, from which he is to require whatever pleases him, be it to write or receive letters, to speak or not to speak to such a person, or the like; and I must put all my fervor in executing zealously and exactly what I am ordered. I must consider myself as a corpse which has neither intelligence nor will; be like a mass of matter which without resistance lets itself be placed wherever it may please anyone; like a stick in the hand of an old man, who uses it according to his needs and places it where it suits him. So must I be under the hands of the Order, to serve it in the way it judges most useful.\" [ James gives the source of Loyola's letter as Danielo Baroli, _Histoire de Saint Ignace de Loyola et de la Compagnie de J\u00e9sus, d'apr\u00e8s les documents originaux_ , translated from the Italian by P. L. Michel (2 vols.; Paris: Vaton, 1844), vol. 2, p. 13.\u2014Ed.]\n\n15This is the view that prevailed in Germany at the beginning of the century, despite being historically incorrect. Consider the comments of Friedrich Naumann, _Das Ideal der Freiheit_ (Berlin-Sch\u00f6neberg: Hilfe, 1908), p. 5. He writes: \"Freiheit ist in erster Linie ein nationaler Beg riff. Das soll hei\u00dfen: Lange ehe man \u00fcber die Freiheit des einzelnen Volksgenossen stritt und nachdachte, unterschied man freie und unfreie V\u00f6lker und St\u00e4mme.\" [\"Liberty is primarily a term associated with the nation. That is to say that long before it was conceived and discussed in terms of the individual liberty of one's countrymen, it was employed to distinguish free and unfree peoples and tribes.\"\u2014Ed.] It is signifcant, however, that this entailed that \"Die Geschichte lehrt, da\u00df der Gesamtfortschritt der Kultur gar nicht anders m\u00f6glich ist als durch Zerbrechung der nationalen Freiheit kleinerer V\u00f6lker,\" [\"History instructs us that cultural progress is possible solely by crushing the national liberty of lesser peoples.\"\u2014Ed.] and \"Es ist kein ewiges Recht der Menschen, von Stammesgenossen geleitet zu werden. Die Geschichte hat entschieden, da\u00df es f\u00fchrende Nationen gibt und solche , die gef\u00fchrt werden, und es ist schwer, liberaler sein zu wollen, als die Geschichte selber ist\" [\"It is not an eternal human right to be led by fellow tribesmen. History has decided that there are leading nations as well as such that are led, and it is difficult to wish to be more liberal than history itself.\" p. 13.\u2014Ed.]\n\n16 The difference between this concept of \"inner liberty\" and liberty in the sense of absence of coercion was clearly perceived by the medieval Scholastics, which distinguished between _libertas a necessitate_ [liberty to choose] and _libertas a coactione_ [liberty from external compulsion].\n\n17 Barbara Wootton, _Freedom under Planning_ (London: Allen and Unwin, 1945), p. 10. The earliest explicit use of freedom in the sense of power which is known to me occurs in Voltaire, _Le Philosophe ignorant_ , quoted by Bertrand de Jouvenel, _De la souverainet\u00e9, \u00e0 la recherche du bien politique_ (Paris: M. T. G\u00e9nin, 1955), p. 315: \"\u00c9tre v\u00e9ritablement libre, c'est pouvoir. Quand je peux faire ce que je veux, voil\u00e0 ma libert\u00e9.\" [\"To be really free, is (to possess) power. When I can do what I wish to do, therein my liberty lies.\" Voltaire's essay appears in _M\u00e9langes_ , J. van den Heuvel, ed. (Paris: Gallimard, 1961), p. 887. There is a Liberty Fund edition of Jouvenel's work: _Sovereignty: An Inquiry into the Political Good_ , J. F. Huntington, trans. (Indianapolis, IN: Liberty Fund, 1997). The reference is on p. 248.\u2014Ed.] It seems ever since to have remained closely associated with what we shall later (chap. 4) have to distinguish as the \"rationalist,\" or French, tradition of liberty. [The 1971 German edition reads: \"The term's meaning has since then been linked with the tradition which we shall later describe (chap. 4) as the French or the \"rational\" tradition. It seems, however, that the notion that freedom is power can be traced back, as can so many modern anti-liberal views, to Francis Bacon.\"\u2014Ed.]\n\n18 Cf. Peter Ferdinand Drucker, _The End of Economic Man: A Study of the New Totalitarianism_ (London: William Heinemann, 1939), pp. 74\u201375: \"The less freedom there is, the more there is talk of the 'new freedom.' Yet this new freedom is a mere word which covers the exact contradiction of all that Europe ever understood by freedom. . . . The new freedom which is preached in Europe is, however, the right of the majority against the individual.\" That this \"new freedom\" has been preached equally in the United States is shown by Woodrow Wilson, _The New Freedom: A Call for the Emancipation of the Generous Energies of a People_ (New York: Doubleday, Page, and Co., 1913), esp. p. 26. A more recent illustration of this is an article by Allen Garf eld Gruchy, \"The Economics of the Natural Resources Committee,\" _American Economic Review_ , 29 (1939): 70, where the author observes approvingly that \"for the economists of the National Resources Committee economic freedom is not a question of the absence of restraint upon individual activities, but instead it is a problem of collective restraint and direction imposed upon individuals and groups to the end that individual security may be achieved.\"\n\n19 A definition in terms of absence of restraint in which this meaning is stressed, such as that of Edward Samuel Corwin, _Liberty Against Government: The Rise, Flowering, and Decline of a Famous Juridical Concept_ (Baton Rouge: Louisiana State University Press, 1948), p. 7: \"Liberty signifies the absence of restraints imposed by other persons upon our own freedom of choice and action.\"\n\n20 _The Shorter Oxford English Dictionary on Historical Principles_ (2 vols.; Oxford: Clarendon Press, 1933), s.v. \"coerce,\" gives as the word's first definition: \"To constrain, or restrain by force, or by authority resting on force.\" [This definition is essentially the same as the one published in the complete _Oxford English Dictionary_ (2nd ed.; 20 vols.; Oxford: Oxford University Press, 1989). It reads: \"To constrain or restrain (a voluntary or moral agent) by the application of superior force, or by authority resting on force.\"\u2014Ed.]\n\n21 Bertrand Russell, \"Freedom and Government,\" in _Freedom: Its Meaning_ , Ruth Nanda Anshen, ed. (New York: Harcourt, Brace, 1940), p. 251.\n\n22 Thomas Hobbes, _Leviathan; or, The Matter, Forme, and Power of a Commonwealth, Ecclesiasticall and Civil_ , Michael Joseph Oakeshott, ed. (Oxford: B. Blackwell, 1946), p. 84.\n\n23 John Rogers Commons, _Legal Foundations of Capitalism_ (New York: Macmillan, 1924), esp. chaps. 2\u20134 [chap. 2, \"Property, Liberty, and Value,\" pp. 11\u201346; chap. 3, \"Physical, Economic, and Moral Power,\" pp. 47\u201364; chap. 4, \"Transactions,\" pp. 65\u2013142].\n\n24 John Dewey, \"Liberty and Social Control,\" _The Social Frontier_ , 2 (November 1935): 41\u201342. [The full quotation reads: \"Liberty is not just an idea, an abstract principle. It is power, effective power to do specific things. There is no such thing as liberty in general; liberty, so to speak, at large. If one wants to know what the condition of liberty is at a given time, one has to examine what persons _can_ do and what they _cannot_ do. The moment one examines the question from the standpoint of effective action, it becomes evident that the demand for liberty is a demand for power.\"\u2014Ed.] Cf. also his article \"Force and Coercion,\" _International Journal of Ethics_ , 23 (1916): 359\u201367: \"Whether the use of force is justified or not . . . is, in substance, a question of efficiency (including economy) of means in the accomplishing of ends\" ( p. 362). \"The criterion of value lies in the relative efficiency and economy of the expenditure of force as a means to an end\" ( p. 364). Dewey's jugglery with the concept of liberty is indeed so appalling that the judgment of Dorothy Fosdick, _What Is Liberty? A Study in Political Theory_ (New York: Harper and Brothers, 1939), p. 91, is hardly unjust: \"The stage, however, is fully set for this [identification of liberty with some principle, such as equality] only when the definitions of liberty and of equality have been so juggled that both refer to approximately the same condition of activity. An extreme example of such sleight-of-hand is provided by John Dewey when he says 'If freedom is combined with a reasonable amount of equality and security is taken to mean cultural and moral security and also material safety, I do not think that security is compatible with anything but freedom.' After redefining two concepts so that they mean approximately the same condition of activity he assures us that the two are compatible. There is no end to such legerdemain.\"\n\n25 John Dewey, _Experience and Education_ (New York: Macmillan, 1938), p. 74. [The full quotation reads: \"There can be no greater mistake . . . than to treat such freedom as an end in itself. It then tends to be destructive of the shared cooperative activities which are the normal source of order. But, on the other hand, it turns freedom which should be positive into something negative. For freedom from restriction, the negative side, is to be prized only as a means to a freedom which is power: power to frame purposes, to judge wisely, to evaluate desires by the consequences which will result from acting upon them; power to select and order means to carry chosen ends into operation ( pp. 73\u201374).\u2014Ed.] Cf. also Werner Sombart, _Der moderne Kapitalismus_ (2 vols.; Leipzig: Duncker und Humblot, 1902), vol. 2 _Die Theorie der kapitalistischen Entwicklung_ , p. 43, where it is explained that \"Technik\" is \"die Entwicklung zur Freiheit\" [the development towards freedom]. This idea is developed at length in Eberhard Zschimmer, _Philosophie der Technik. Vom Sinn der Technik und Kritik des Unsinns \u00fcber die Technik_ ( Jena: Eugen Diederichs, 1914), pp. 86\u201391.\n\n26 Cf. Ralph Barton Perry, \"Liberty in a Democratic State,\" in _Freedom: Its Meaning_ , Ruth Nanda Anshen, ed. (New York: Harcourt, Brace, 1940), p. 269: \"The distinction between 'welfare' and liberty breaks down altogether, since a man's effective liberty is proportional to his resources.\" This has led others to the contention that \"if more people are buying automobiles and taking vacations, there is more liberty\" (for reference, see chap. 16, n. 72 [Dwight Waldo, _The Administrative State: A Study of the Political Theory of American Public Administration_ (New York: Ronald Press Co., 1948), p. 73]); and Robert Lee Hale, _Freedom through Law: Public Control of Private Governing Power_ (New York: Columbia University Press, 1952), p. 385: \"Inequalities of fortune . . . are inequalities in individual liberty.\"\n\n27 An amusing illustration of this is provided by Denis Gabor and Andr\u00e9 Gabor, \"An Essay on the Mathematical Theory of Freedom,\" _Journal of the Royal Statistical Society_ , Series A (General), 117 (1954): 32. The authors begin by stating that freedom \"means the absence of undesirable restraints, hence the concept is almost coextensive with everything which is desirable\" and then, instead of discarding this evidently useless concept, not only adopt it but proceed to \"measure\" freedom in this sense.\n\n28 Cf. Lord Acton, _Lectures on Modern History_ , John Neville Figgis and Reginald Vere Laurence, eds. (London: Macmillan, 1906), p.10 [The essay is from Acton's inaugural lecture on the study of history, delivered at Cambridge in June 1895 (Liberty Fund edition, _Essays in the Study and Writing of History_ , p. 516)\u2014Ed.]: \"There is no more proportion between liberty and power than between eternity and time.\" Also Bronislaw Malinowski, _Freedom and Civilization_ (New York: Roy Publishers, 1944), p. 47: \"If we were carelessly to identify freedom with power, we obviously would nurse tyranny, exactly as we land into anarchy when we equate liberty with lack of any restraint.\" See also Frank Hyneman Knight, \"Freedom as Fact and Criterion,\" in _Freedom and Reform: Essays in Economics and Social Philosophy_ , Frank Hyneman Knight, ed. (New York: Harper and Brothers, 1947), p. 4ff.; Joseph Cropsey, _Polity and Economy: An Interpretation of the Principles of Adam Smith_ (The Hague: M. Nijhoff, 1957), p. xi; and Martin Bronfenbrenner, \"Two Concepts of Economic Freedom,\" _Ethics_ , 65 (1955): 157\u201370.\n\n29 The distinction between \"positive\" and \"negative\" liberty has been popularized by Thomas Hill Green, \"Lecture on 'Liberal Legislation and Freedom of Contract,'\" [1880] in _The Works of T._ _H. Green_ , Richard Lewis Nettleship, ed. (3 vols.; London: Longmans, Green, and Co., 1888), vol. 3, _Miscellanies and Memoir_ , pp. 365\u201386. The idea which is there connected mainly with \"inner freedom\" has since been put to many uses. Cf. Sir Isaiah Berlin, _Two Concepts of Liberty: An Inaugural Lecture Delivered Before the University of Oxford on 31 October 1958_ (Oxford: Clarendon Press, 1958), and, for a characteristic taking-over of the socialist arguments by the conservatives, Clinton Rossiter, \"Toward an American Conservatism,\" _Yale Review_ , 44 (1955): 361, who argues that \"the conservative should give us a definition of liberty that is positive and all-embracing. . . . In the new conservative dictionary, _liberty_ will be defined with the help of words like _opportunity_ , _creativity_ , _productivity_ , and _security_.\"\n\n30 William Linn Westermann, \"Between Slavery and Freedom,\" _American Historical Review_ , 50 (1945): 216.\n\n31 This was at least the case in practice, if perhaps not in strict law (cf. John Walter Jones, _The Law and Legal Theory of the Greeks: An Introduction_ [Oxford: Clarendon Press, 1956], p. 282).\n\n32 Cf. Frank Hyneman Knight, _Freedom and Reform: Essays in Economics and Social Philosophy_ (New York: Harper and Brothers, 1947), pp. 193\u201394: \"The _primary_ function of government is to _prevent coercion_ and so guarantee to every man the right to live his own life on terms of _free_ association with his fellows.\" See also his discussion of the topic in the article quoted in n. 3 above [\"The Meaning of Freedom,\" a review of _Freedom: Its Meaning_ , Ruth Nanda Anshen, ed., _Ethics_ , 52 (1941): 86\u2013109].\n\n33 Cf. Rudolph Von Ihering, _Law as a Means to an End_ , Isaac Husik, trans. (Boston: Boston Book Co., 1913), pp. 241\u201342; Max Weber, _From Max Weber: Essays in Sociology_ , C. Wright Mills, ed. and trans. (New York: Oxford University Press, 1946), p 78: \"A State is a human community that (successfully) claims the _monopoly of the legitimate use of physical force_ \"; Bronislaw Malinowski, _Freedom and Civilization_ (New York: Roy Publishers, 1944), p. 265: the state \"is the only historic institution which has the monopoly of force\"; also John Maurice Clark, _Social Control of Business_ (2nd ed.; New York: Whittlesey House, McGraw-Hill, 1939), p. 115: \"Forcible coercion is supposed to be the monopoly of the state\"; and Edward Adamson Hoebel, _The Law of Primitive Man: A Study in Comparative Legal Dynamics_ (Cambridge, MA: Harvard University Press, 1954), chap. 2 [\"What Is Law?\" ( pp. 18\u201328)].\n\n## TWO\n\nTHE CREATIVE POWERS OF A FREE CIVILIZATION\n\nCivilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle\u2014they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.\n\n\u2014A. N. Whitehead\n\n1. The Socratic maxim that the recognition of our ignorance is the beginning of wisdom has profound significance for our understanding of society. The first requisite for this is that we become aware of men's necessary ignorance of much that helps him to achieve his aims. Most of the advantages of social life, especially in its more advanced forms which we call \"civilization,\" rest on the fact that the individual benefits from more knowledge than he is aware of. It might be said that civilization begins when the individual in the pursuit of his ends can make use of more knowledge than he has himself acquired and when he can transcend the boundaries of his ignorance by profiting from knowledge he does not himself possess.\n\nThis fundamental fact of man's unavoidable ignorance of much on which the working of civilization rests has received little attention. Philosophers and students of society have generally glossed it over and treated this ignorance as a minor imperfection which could be more or less disregarded. But, though discussions of moral or social problems based on the assumption of perfect knowledge may occasionally be useful as a preliminary exercise in logic, they are of little use in an attempt to explain the real world. Its problems are dominated by the \"practical difficulty\" that our knowledge is, in fact, very far from perfect. Perhaps it is only natural that the scientists tend to stress what we do know; but in the social field, where what we do not know is often so much more important, the effect of this tendency may be very misleading. Many  of the utopian constructions are worthless because they follow the lead of the theorists in assuming that we have perfect knowledge.\n\nIt must be admitted, however, that our ignorance is a peculiarly difficult subject to discuss. It might at first even seem impossible by definition to talk sense about it. We certainly cannot discuss intelligently something about which we know nothing. We must at least be able to state the questions even if we do not know the answers. This requires some genuine knowledge of the kind of world we are discussing. If we are to understand how society works, we must attempt to define the general nature and range of our ignorance concerning it. Though we cannot see in the dark, we must be able to trace the limits of the dark areas.\n\nThe misleading effect of the usual approach stands out clearly if we examine the significance of the assertion that man has created his civilization and that he therefore can also change its institutions as he pleases. This assertion would be justified only if man had deliberately created civilization in full understanding of what he was doing or if he at least clearly knew how it was being maintained. In a sense it is true, of course, that man has made his civilization. It is the product of his actions or, rather, of the action of a few hundred generations. This does not mean, however, that civilization is the product of human design, or even that man knows what its functioning or continued existence depends upon.1\n\nThe whole conception of man already endowed with a mind capable of conceiving civilization setting out to create it is fundamentally false. Man did not simply impose upon the world a pattern created by his mind. His mind is itself a system that constantly changes as a result of his endeavor to adapt himself to his surroundings. It would be an error to believe that, to achieve a higher civilization, we have merely to put into effect the ideas now guiding us. If we are to advance, we must leave room for a continuous revision of our present conceptions and ideals which will be necessitated by further experience. We are as little able to conceive what civilization will be, or can be, five hundred or even fifty years hence as our medieval forefathers or even our grandparents were able to foresee our manner of life today.2\n\nThe conception of man deliberately building his civilization stems from an erroneous intellectualism that regards human reason as something standing outside nature and possessed of knowledge and reasoning capacity independent of experience. But the growth of the human mind is part of the growth of civilization; it is the state of civilization at any given moment that determines the scope and the possibilities of human ends and values. The mind can never foresee its own advance. Though we must always strive for the achievement of our present aims, we must also leave room for new experiences and future events to decide which of these aims will be achieved.\n\nIt may be an exaggeration to assert, as a modern anthropologist has done, that \"it is not man who controls culture but the other way around\"; but it is useful to be reminded by him that \"it is only our profound and comprehensive ignorance of the nature of culture that makes it possible for us to believe that we direct and control it.\"3 He suggests at least an important corrective to the intellectualist conception. His reminder will help us to achieve a truer image of the incessant interaction between our conscious striving for what our intellect pictures as achievable and the operations of the institutions, traditions, and habits which jointly often produce something very different from what we have aimed at.\n\nThere are two important respects in which the conscious knowledge which guides the individual's actions constitutes only part of the conditions which enable him to achieve his ends. There is the fact that man's mind is itself a product of the civilization in which he has grown up and that it is unaware of much of the experience which has shaped it\u2014experience that assists it by being embodied in the habits, conventions, language, and moral beliefs which are part of its makeup. Then there is the further consideration that the knowledge which any individual mind consciously manipulates is only a small part of the knowledge which at any one time contributes to the success of his action. When we reflect how much knowledge possessed by other people is an essential condition for the successful pursuit of our individual aims, the magnitude of our ignorance of the circumstances on which the results of our action depend appears simply staggering. Knowledge exists only as the knowledge of individuals. It is not much better than a metaphor to speak of the knowledge of society as a whole. The sum of the knowledge of all the individuals exists nowhere as an integrated whole. The great problem is how we can  all profit from this knowledge, which exists only dispersed as the separate, partial, and sometimes conflicting beliefs of all men.\n\nIn other words, it is largely because civilization enables us constantly to profit from knowledge which we individually do not possess and because each individual's use of his particular knowledge may serve to assist others unknown to him in achieving their ends that men as members of civilized society can pursue their individual ends so much more successfully than they could alone. We know little of the particular facts to which the whole of social activity continuously adjusts itself in order to provide what we have learned to expect. We know even less of the forces which bring about this adjustment by appropriately co-ordinating individual activity. And our attitude, when we discover how little we know of what makes us co-operate, is, on the whole, one of resentment rather than of wonder or curiosity. Much of our occasional impetuous desire to smash the whole entangling machinery of civilization is due to this inability of man to understand what he is doing.\n\n2. The identification of the growth of civilization with the growth of knowledge would be very misleading, however, if by \"knowledge\" we meant only the conscious, explicit knowledge of individuals, the knowledge which enables us to state that this or that is so-and-so.4 Still less can this knowledge be confined to scientific knowledge. It is important for the understanding of our argument later to remember that, contrary to one fashionable view,5 scientific knowledge does not exhaust even all the explicit and conscious knowledge of which society makes constant use. The scientific methods of the search for knowledge are not capable of satisfying all society's needs for explicit knowledge. Not all the knowledge of the ever changing particular facts that man continually uses lends itself to organization or systematic exposition; much of it exists only dispersed among countless individuals. The same applies to that important part of expert knowledge which is not substantive knowledge but merely knowledge of where and how to find the needed information.6 For  our present purpose, however, it is not this distinction between different kinds of rational knowledge that is most important, and when we speak of explicit knowledge, we shall group these different kinds together.\n\nThe growth of knowledge and the growth of civilization are the same only if we interpret knowledge to include all the human adaptations to environment in which past experience has been incorporated. Not all knowledge in this sense is part of our intellect, nor is our intellect the whole of our knowledge. Our habits and skills, our emotional attitudes, our tools, and our institutions\u2014all are in this sense adaptations to past experience which have grown up by selective elimination of less suitable conduct. They are as much an indispensable foundation of successful action as is our conscious knowledge. Not all these non-rational factors underlying our action are always conducive to success. Some may be retained long after they have outlived their usefulness and even when they have become more an obstacle than a help. Nevertheless, we could not do without them: even the successful employment of our intellect itself rests on their constant use.\n\nMan prides himself on the increase in his knowledge. But, as a result of what he himself has created, the limitations of his conscious knowledge and therefore the range of ignorance significant for his conscious action have constantly increased. Ever since the beginning of modern science, the best minds have recognized that \"the range of acknowledged ignorance will grow with the advance of science.\"7 Unfortunately, the popular effect of this scientific  advance has been a belief, seemingly shared by many scientists, that the range of our ignorance is steadily diminishing and that we can therefore aim at more comprehensive and deliberate control of all human activities. It is for this reason that those intoxicated by the advance of knowledge so often become the enemies of freedom. While the growth of our knowledge of nature constantly discloses new realms of ignorance, the increasing complexity of the civilization which this knowledge enables us to build presents new obstacles to the intellectual comprehension of the world around us. The more men know, the smaller the share of all that knowledge becomes that any one mind can absorb. The more civilized we become, the more relatively ignorant must each individual be of the facts on which the working of his civilization depends. The very division of knowledge increases the necessary ignorance of the individual of most of this knowledge.\n\n3. When we spoke of the transmission and communication of knowledge, we meant to refer to the two aspects of the process of civilization which we have already distinguished: the transmission in time of our accumulated stock of knowledge and the communication among contemporaries of information on which they base their action. They cannot be sharply separated because the tools of communication between contemporaries are part of the cultural heritage which man constantly uses in the pursuit of his ends.\n\nWe are most familiar with this process of accumulation and transmission of knowledge in the field of science\u2014so far as it shows both the general laws of nature and the concrete features of the world in which we live. But, although this is the most conspicuous part of our inherited stock of knowledge and the chief part of what we necessarily know, in the ordinary sense of \"knowing,\" it is still only a part; for, besides this, we command many tools\u2014in the widest sense of that word\u2014which the human race has evolved and which enable us to deal with our environment. These are the results of the experience of successive generations which are handed down. And, once a more efficient tool is available, it will be used without our knowing why it is better, or even what the alternatives are.\n\nThese \"tools\" which man has evolved and which constitute such an important part of his adaptation to his environment include much more than material implements. They consist in a large measure of forms of conduct which he habitually follows without knowing why; they consist of what we call \"traditions\" and \"institutions,\" which he uses because they are available to him as a product of cumulative growth without ever having been designed by any one mind. Man is generally ignorant not only of why he uses implements of one shape rather than of another but also of how much is dependent on his actions taking one form rather than another. He does not usually know to what extent the success of his efforts is determined by his conforming to habits of which he is not even aware. This is probably as true of civilized man  as of primitive man. Concurrent with the growth of conscious knowledge there always takes place an equally important accumulation of tools in this wider sense, of tested and generally adopted ways of doing things.\n\nOur concern at the moment is not so much with the knowledge thus handed down to us or with the formation of new tools that will be used in the future as it is with the manner in which current experience is utilized in assisting those who do not directly gain it. So far as it is possible to do so, we shall leave the progress in time for the next chapter and concentrate here on the manner in which that dispersed knowledge and the different skills, the varied habits and opportunities of the individual members of society, contribute toward bringing about the adjustment of its activities to ever changing circumstances.\n\nEvery change in conditions will make necessary some change in the use of resources, in the direction and kind of human activities, in habits and practices. And each change in the actions of those affected in the first instance will require further adjustments that will gradually extend throughout the whole of society. Thus every change in a sense creates a \"problem\" for society, even though no single individual perceives it as such; and it is gradually \"solved\" by the establishment of a new over-all adjustment. Those who take part in the process have little idea why they are doing what they do, and we have no way of predicting who will at each step first make the appropriate move, or what particular combinations of knowledge and skill, personal attitudes and circumstances, will suggest to some man the suitable answer, or by what channels his example will be transmitted to others who will follow the lead. It is difficult to conceive all the combinations of knowledge and skills which thus come into action and from which arises the discovery of appropriate practices or devices that, once found, can be accepted generally. But from the countless number of humble steps taken by anonymous persons in the course of doing familiar things in changed circumstances spring the examples that prevail. They are as important as the major intellectual innovations which are explicitly recognized and communicated as such.\n\nWho will prove to possess the right combination of aptitudes and opportunities to find the better way is just as little predictable as by what manner or process different kinds of knowledge and skill will combine to bring about a solution of the problem.8 The successful combination of knowledge and aptitude is not selected by common deliberation, by people seeking a solution to  their problems through a joint effort;9 it is the product of individuals imitating those who have been more successful and from their being guided by signs or symbols, such as prices offered for their products or expressions of moral or aesthetic esteem for their having observed standards of conduct\u2014in short, of their using the results of the experiences of others.\n\nWhat is essential to the functioning of the process is that each individual be able to act on his particular knowledge, always unique, at least so far as it refers to some particular circumstances, and that he be able to use his individual skills and opportunities within the limits known to him and for his own individual purpose.\n\n4. We have now reached the point at which the main contention of this chapter will be readily intelligible. It is that the case for individual freedom rests chiefly on the recognition of the inevitable ignorance of all of us concerning a great many of the factors on which the achievement of our ends and welfare depends.10\n\nIf there were omniscient men, if we could know not only all that affects the attainment of our present wishes but also our future wants and desires, there would be little case for liberty. And, in turn, liberty of the individual would, of course, make complete foresight impossible. Liberty is essential in order to leave room for the unforeseeable and unpredictable; we want it because we have learned to expect from it the opportunity of realizing many of our aims. It is because every individual knows so little and, in particular, because we rarely know which of us knows best that we trust the independent and competitive efforts of many to induce the emergence of what we shall want when we see it.\n\nHumiliating to human pride as it may be, we must recognize that the advance and even the preservation of civilization are dependent upon a maximum of opportunity for accidents to happen.11 These accidents occur in the combination of knowledge and attitudes, skills and habits, acquired by individual men and also when qualified men are confronted with the particular circumstances which they are equipped to deal with. Our necessary ignorance of so much means that we have to deal largely with probabilities and chances.\n\nOf course, it is true of social as of individual life that favorable accidents usually do not just happen. We must prepare for them.12 But they still remain chances and do not become certainties. They involve risks deliberately taken, the possible misfortune of individuals and groups who are as meritorious as others who prosper, the possibility of serious failure or relapse even for the majority, and merely a high probability of a net gain on balance. All we can do is to increase the chance that some special constellation of individual endowment and circumstance will result in the shaping of some new tool or the improvement of an old one, and to improve the prospect that such  innovations will become rapidly known to those who can take advantage of them.\n\nAll political theories assume, of course, that most individuals are very ignorant. Those who plead for liberty differ from the rest in that they include among the ignorant themselves as well as the wisest. Compared with the totality of knowledge which is continually utilized in the evolution of a dynamic civilization, the difference between the knowledge that the wisest and that which the most ignorant individual can deliberately employ is comparatively insignificant.\n\nThe classical argument for tolerance formulated by John Milton and John Locke and restated by John Stuart Mill and Walter Bagehot rests, of course, on the recognition of this ignorance of ours. It is a special application of general considerations to which a non-rationalist insight into the working of our mind opens the doors. We shall find throughout this book that, though we are usually not aware of it, all institutions of freedom are adaptations to this fundamental fact of ignorance, adapted to deal with chances and probabilities, not certainty. Certainty we cannot achieve in human affairs, and it is for this reason that, to make the best use of what knowledge we have, we must adhere to rules which experience has shown to serve best on the whole, though we do not know what will be the consequences of obeying them in the particular instance.13\n\n5. Man learns by the disappointment of expectations. Needless to say, we ought not to increase the unpredictability of events by foolish human institutions. So far as possible, our aim should be to improve human institutions so as to increase the chances of correct foresight. Above all, however, we should provide the maximum of opportunity for unknown individuals to learn of facts that we ourselves are yet unaware of and to make use of this knowledge in their actions.\n\nIt is through the mutually adjusted efforts of many people that more knowledge is utilized than any one individual possesses or than it is possible to synthesize intellectually; and it is through such utilization of dispersed knowledge that achievements are made possible greater than any single mind can foresee. It is because freedom means the renunciation of direct control of individual efforts that a free society can make use of so much more knowledge than the mind of the wisest ruler could comprehend.\n\nFrom this foundation of the argument for liberty it follows that we shall not achieve its ends if we confine liberty to the particular instances where we know it will do good. Freedom granted only when it is known beforehand that its effects will be beneficial is not freedom. If we knew how freedom would be used, the case for it would largely disappear. We shall never get the benefits of freedom, never obtain those unforeseeable new developments for which it provides the opportunity, if it is not also granted where the uses made of it by some do not seem desirable. It is therefore no argument against individual freedom that it is frequently abused. Freedom necessarily means that many things will be done which we do not like. Our faith in freedom does not rest on the foreseeable results in particular circumstances but on the belief that it will, on balance, release more forces for the good than for the bad.\n\nIt also follows that the importance of our being free to do a particular thing has nothing to do with the question of whether we or the majority are ever likely to make use of that particular possibility. To grant no more freedom than all can exercise would be to misconceive its function completely. The freedom that will be used by only one man in a million may be more important to society and more beneficial to the majority than any freedom that we all use.14\n\nIt might even be said that the less likely the opportunity to make use of freedom to do a particular thing, the more precious it will be for society as a whole. The less likely the opportunity, the more serious will it be to miss it when it arises, for the experience that it offers will be nearly unique. It is also probably true that the majority are not directly interested in most of the important things that any one person should be free to do. It is because we do not know how individuals will use their freedom that it is so important. If it were otherwise, the results of freedom could also be achieved by the majority's deciding what should be done by the individuals. But majority action is, of necessity,  confined to the already tried and ascertained, to issues on which agreement has already been reached in that process of discussion that must be preceded by different experiences and actions on the part of different individuals.\n\nThe benefits I derive from freedom are thus largely the result of the uses of freedom by others, and mostly of those uses of freedom that I could never avail myself of. It is therefore not necessarily freedom that I can exercise myself that is most important for me. It is certainly more important that anything can be tried by somebody than that all can do the same things. It is not because we like to be able to do particular things, not because we regard any particular freedom as essential to our happiness, that we have a claim to freedom. The instinct that makes us revolt against any physical restraint, though a helpful ally, is not always a safe guide for justifying or delimiting freedom. What is important is not what freedom I personally would like to exercise but what freedom some person may need in order to do things beneficial to society. This freedom we can assure to the unknown person only by giving it to all.\n\nThe benefits of freedom are therefore not confined to the free\u2014or, at least, a man does not benefit mainly from those aspects of freedom which he himself takes advantage of. There can be no doubt that in history unfree majorities have benefited from the existence of free minorities and that today unfree societies benefit from what they obtain and learn from free societies. Of course the benefits we derive from the freedom of others become greater as the number of those who can exercise freedom increases. The argument for the freedom of some therefore applies to the freedom of all. But it is still better for all that some should be free than none and also that many enjoy full freedom than that all have a restricted freedom. The significant point is that the importance of freedom to do a particular thing has nothing to do with the number of people who want to do it: it might almost be in inverse proportion. One consequence of this is that a society may be hamstrung by controls, although the great majority may not be aware that their freedom has been significantly curtailed. If we proceeded on the assumption that only the exercises of freedom that the majority will practice are important, we would be certain to create a stagnant society with all the characteristics of unfreedom.\n\n6. The undesigned novelties that constantly emerge in the process of adaptation will consist, first, of new arrangements or patterns in which the efforts of different individuals are co-ordinated and of new constellations in the use of resources, which will be in their nature as temporary as the particular conditions that have evoked them. There will be, second, modifications of tools and institutions adapted to the new circumstances. Some of these will also be merely temporary adaptations to the conditions of the moment, while others will be improvements that increase the versatility of the existing tools and usages and will therefore be retained. These latter will constitute  a better adaptation not merely to the particular circumstances of time and place but to some permanent feature of our environment. In such spontaneous \"formations\"15 is embodied a perception of the general laws that govern nature. With this cumulative embodiment of experience in tools and forms of action will emerge a growth of explicit knowledge, of formulated generic rules that can be communicated by language from person to person.\n\nThis process by which the new emerges is best understood in the intellectual sphere when the results are new ideas. It is the field in which most of us are aware at least of some of the individual steps of the process, where we necessarily know what is happening and thus generally recognize the necessity of freedom. Most scientists realize that we cannot plan the advance of knowledge, that in the voyage into the unknown\u2014which is what research is\u2014we are in great measure dependent on the vagaries of individual genius and of circumstance, and that scientific advance, like a new idea that will spring up in a single mind, will be the result of a combination of conceptions, habits, and circumstances brought to one person by society, the result as much of lucky accidents as of systematic effort.\n\nBecause we are more aware that our advances in the intellectual sphere often spring from the unforeseen and undesigned, we tend to overstress the importance of freedom in this field and to ignore the importance of the freedom of _doing_ things. But the freedom of research and belief and the freedom of speech and discussion, the importance of which is widely understood, are significant only in the last stage of the process in which new truths are discovered. To extol the value of intellectual liberty at the expense of the value of the liberty of doing things would be like treating the crowning part of an edifice as the whole. We have new ideas to discuss, different views to adjust, because those ideas and views arise from the efforts of individuals in ever new circumstances, who avail themselves in their concrete tasks of the new tools and forms of action they have learned.\n\nThe non-intellectual part of this process\u2014the formation of the changed material environment in which the new emerges\u2014requires for its understanding and appreciation a much greater effort of imagination than the factors  stressed by the intellectualist view. While we are sometimes able to trace the intellectual processes that have led to a new idea, we can scarcely ever reconstruct the sequence and combination of those contributions that have not led to the acquisition of explicit knowledge; we can scarcely ever reconstruct the favorable habits and skills employed, the facilities and opportunities used, and the particular environment of the main actors that has favored the result. Our efforts toward understanding this part of the process can go little further than to show on simplified models the kind of forces at work and to point to the general principle rather than the specific character of the influences that operate.16 Men are always concerned only with what they know. Therefore, those features which, while the process is under way, are not consciously known to anybody are commonly disregarded and can perhaps never be traced in detail.\n\nIn fact, these unconscious features not only are commonly disregarded but are often treated as if they were a hindrance rather than a help or an essential condition. Because they are not \"rational\" in the sense of explicitly entering into our reasoning, they are often treated as irrational in the sense of being contrary to intelligent action. Yet, though much of the non-rational that affects our action may be irrational in this sense, many of the \"mere habits\" and \"meaningless institutions\" that we use and presuppose in our actions are essential conditions for what we achieve; they are successful adaptations of society that are constantly improved and on which depends the range of what we can achieve. While it is important to discover their defects, we could not for a moment go on without constantly relying on them.\n\nThe manner in which we have learned to order our day, to dress, to eat, to arrange our houses, to speak and write, and to use the countless other tools and implements of civilization, no less than the \"know-how\" of production and trade, furnishes us constantly with the foundations on which our own contributions to the process of civilization must be based. And it is in the new use and improvement of whatever the facilities of civilization offer us that the new ideas arise that are ultimately handled in the intellectual sphere. Though the conscious manipulation of abstract thought, once it has been set in train, has in some measure a life of its own, it would not long continue and develop without the constant challenges that arise from the ability of people to act in a new manner, to try new ways of doing things, and to alter the whole structure of civilization in adaptation to change. The intellectual process is in effect  only a process of elaboration, selection, and elimination of ideas already formed. And the flow of new ideas, to a great extent, springs from the sphere in which action, often non-rational action, and material events impinge upon each other. It would dry up if freedom were confined to the intellectual sphere.\n\nThe importance of freedom, therefore, does not depend on the elevated character of the activities it makes possible. Freedom of action, even in humble things, is as important as freedom of thought. It has become a common practice to disparage freedom of action by calling it \"economic liberty.\"17 But the concept of freedom of action is much wider than that of economic liberty, which it includes; and, what is more important, it is very questionable whether there are any actions which can be called merely \"economic\" and whether any restrictions on liberty can be confined to what are called merely \"economic\" aspects. Economic considerations are merely those by which we reconcile and adjust our different purposes, none of which, in the last resort, are economic (excepting those of the miser or the man for whom making money has become an end in itself ).18\n\n7. Most of what we have said so far applies not only to man's use of the means for the achievement of his ends but also to those ends themselves. It is one of the characteristics of a free society that men's goals are open,19 that new ends of conscious effort can spring up, first with a few individuals, to become in time the ends of most. It is a fact which we must recognize that even what we regard as good or beautiful is changeable\u2014if not in any recognizable manner that would entitle us to take a relativistic position, then in the sense that in many respects we do not know what will appear as good or beautiful to another generation. Nor do we know why we regard this or that as good or who is right when people differ as to whether something is good or not. It is not only in his knowledge, but also in his aims and values, that man is the creature of civilization; in the last resort, it is the relevance of these individual wishes to the perpetuation of the group or the species that will determine whether they will persist or change. It is, of course, a mistake to believe that we can draw conclusions about what our values ought to be simply because we realize that they are a product of evolution. But we cannot  reasonably doubt that these values are created and altered by the same evolutionary forces that have produced our intelligence. All that we can know is that the ultimate decision about what is good or bad will be made not by individual human wisdom but by the decline of the groups that have adhered to the \"wrong\" beliefs.\n\nIt is in the pursuit of man's aims of the moment that all the devices of civilization have to prove themselves; the ineffective will be discarded and the effective retained. But there is more to it than the fact that new ends constantly arise with the satisfaction of old needs and with the appearance of new opportunities. Which individuals and which groups succeed and continue to exist depends as much on the goals that they pursue, the values that govern their action, as on the tools and capacities at their command. Whether a group will prosper or be extinguished depends as much on the ethical code it obeys, or the ideals of beauty or well-being that guide it, as on the degree to which it has learned or not learned to satisfy its material needs. Within any given society, particular groups may rise or decline according to the ends they pursue and the standards of conduct that they observe. And the ends of the successful group will tend to become the ends of all members of the society.\n\nAt most, we understand only partially why the values we hold or the ethical rules we observe are conducive to the continued existence of our society. Nor can we be sure that under constantly changing conditions all the rules that have proved to be conducive to the attainment of a certain end will remain so. Though there is a presumption that any established social standard contributes in some manner to the preservation of civilization, our only way of confirming this is to ascertain whether it continues to prove itself in competition with other standards observed by other individuals or groups.\n\n8. The competition on which the process of selection rests must be understood in the widest sense. It involves competition between organized and unorganized groups no less than competition between individuals. To think of it in contrast to cooperation or organization would be to misconceive its nature. The endeavor to achieve certain results by co-operation and organization is as much a part of competition as individual efforts. Successful group relations also prove their effectiveness in competition among groups organized in different ways. The relevant distinction is not between individual and group action but between conditions, on the one hand, in which alternative ways based on different views or practices may be tried and conditions, on the other, in which one agency has the exclusive right and the power to prevent others from trying. It is only when such exclusive rights are conferred on the presumption of superior knowledge of particular individuals or groups that the process ceases to be experimental and beliefs that happen to be prevalent at a given time may become an obstacle to the advancement of knowledge.\n\nThe argument for liberty is not an argument against organization, which is  one of the most powerful means that human reason can employ, but an argument against all exclusive, privileged, monopolistic organization, against the use of coercion to prevent others from trying to do better. Every organization is based on given knowledge; organization means commitment to a particular aim and to particular methods, but even organization designed to increase knowledge will be effective only insofar as the knowledge and beliefs on which its design rests are true. And if any facts contradict the beliefs on which the structure of the organization is based, this will become evident only in its failure and supersession by a different type of organization. Organization is therefore likely to be beneficial and effective so long as it is voluntary and is imbedded in a free sphere and will either have to adjust itself to circumstances not taken into account in its conception or fail. To turn the whole of society into a single organization built and directed according to a single plan would be to extinguish the very forces that shaped the individual human minds that planned it.\n\nIt is worth our while to consider for a moment what would happen if only what was agreed to be the best available knowledge were to be used in all action. If all attempts that seemed wasteful in the light of generally accepted knowledge were prohibited and only such questions asked, or such experiments tried, as seemed significant in the light of ruling opinion, mankind might well reach a point where its knowledge enabled it to predict the consequences of all conventional actions and to avoid all disappointment or failure. Man would then seem to have subjected his surroundings to his reason, for he would attempt only those things which were totally predictable in their results. We might conceive of a civilization coming to a standstill, not because the possibilities of further growth had been exhausted, but because man had succeeded in so completely subjecting all his actions and his immediate surroundings to his existing state of knowledge that there would be no occasion for new knowledge to appear.\n\n9. The rationalist who desires to subject everything to human reason is thus faced with a real dilemma. The use of reason aims at control and predictability. But the process of the advance of reason rests on freedom and the unpredictability of human action. Those who extol the powers of human reason usually see only one side of that interaction of human thought and conduct in which reason is at the same time used and shaped. They do not see that, for advance to take place, the social process from which the growth of reason emerges must remain free from its control.\n\nThere can be little doubt that man owes some of his greatest successes in the past to the fact that he has _not_ been able to control social life. His continued advance may well depend on his deliberately refraining from exercising controls which are now in his power. In the past, the spontaneous forces of growth, however much restricted, could usually still assert themselves against  the organized coercion of the state. With the technological means of control now at the disposal of government, it is not certain that such assertion is still possible; at any rate, it may soon become impossible. We are not far from the point where the deliberately organized forces of society may destroy those spontaneous forces which have made advance possible.\n\n* * *\n\nThe quotation at the head of the chapter is taken from Alfred North Whitehead, _An Introduction to Mathematics_ (London: Williams and Norgate, 1911), p. 61. An earlier version of this chapter appeared as \"The Creative Powers of a Free Civilization,\" in _Essays on Individuality_ , Felix Morley, ed. (Philadelphia: University of Pennsylvania Press, 1958), pp. 183\u2013204 [Liberty Fund edition, pp. 261\u201389].\n\n1 Cf. Adam Ferguson, _An Essay on the History of Civil Society_ (Edinburgh: Printed for A. Millar and T. Caddel in the Strand, and A. Kincaid and J. Bell, Edinburgh, 1767), p. 279: \"The artifices of the beaver, the ant, and the bee, are ascribed to the wisdom of nature. Those of polished nations are ascribed to themselves, and are supposed to indicate a capacity superior to that of rude minds. But the establishments of men, like those of every animal, are suggested by nature, and are the result of instinct, directed by the variety of situations in which mankind are placed. Those establishments arose from successive improvements that were made, without any sense of their general effect; and they bring human affairs to a state of complication, which the greatest reach of capacity with which human nature was ever adorned, could not have projected; nor even when the whole is carried into execution, can it be comprehended in its full extent.\"\n\n2 Cf. Michael Polanyi, _The Logic of Liberty: Reflections and Rejoinders_ (London: Routledge and Kegan Paul, 1951), p. 199 [Liberty Fund edition, p. 245]: \"The conceptions by the light of which men will judge our own ideas in a thousand years\u2014or perhaps even in fifty years\u2014are beyond our guess. If a library of the year 3000 came into our hands to-day, we could not understand its contents. How should we consciously determine a future which is, by its very nature, beyond our comprehension? Such presumption reveals only the narrowness of an outlook uninformed by humility.\"\n\n3 Leslie Alvin White, \"Man's Control over Civilization: An Anthropocentric Illusion,\" _Scientific Monthly_ , 66 (1948): 238; also his _The Science of Culture: A Study of Man and Civilization_ (New York: Farrar, Straus, and Co., 1949), pp. 337 and 342.\n\n4 See Gilbert Ryle, \"Knowing How and Knowing That,\" [The Presidential Address] _Proceedings of the Aristotelian Society_ , n.s., 46 (1946): 1\u201316; and now compare also Michael Polanyi, _Personal Knowledge: Towards a Post-critical Philosophy_ (Chicago: University of Chicago Press, 1958).\n\n5 Cf. the often quoted observation by Frank Plumpton Ramsey, _The Foundations of Mathematics and Other Logical Essays_ (London: Routledge and Kegan Paul, 1931), p. 287: \"There is nothing to know except science.\" [The statement does not appear in _The Foundations of Mathematics_ , as Hayek indicates, but in the Epilogue to Ramsey's collected essays, of which _The Foundations_ is the central article and which gives its name to the anthology.\u2014Ed.]\n\n6 On these different kinds of knowledge see my article \"\u00dcber den 'Sinn' sozialer Institutionen\" [On the Meaning of Social Institutions], _Schweizer Monatshefte_ , October 1956, pp. 512\u201324, and, on the application of the whole argument of this chapter to the more specifically economic problems, the two essays on \"Economics and Knowledge\" and \"The Use of Knowledge in Society\" reprinted in my _Individualism and Economic Order_ (Chicago: University of Chicago Press, 1948), pp. 33\u201356 and 77\u201391. See also Samuel Johnson's remark: \"Knowledge is of two kinds: we know a subject ourselves or we know where we can find information upon it.\" (James Boswell, _The Life of Samuel Johnson, LL.D.: Comprehending an Account of His Studies and Numerous Works_ [3 vols.; 2nd ed., rev. and aug.; London: Printed by Henry Baldwin, 1793], vol. 2, pp. 237\u201338).\n\n7 Giorgio de Santillana, _The Crime of Galileo_ (Chicago: University of Chicago Press, 1955), pp. 34\u201335. Herbert Spencer also remarks somewhere: \"In science the more we know, the more extensive the contact with nescience.\" [The quotation, as Hayek has it, is somewhat different from that written by Spencer. The wording as quoted by Hayek, in fact comes from the article on Herbert Spencer in the eleventh edition of the _Encyclopedia Britannica_ (New York: The Encyclopedia Britannica Co., 1911) s.v. \"Spencer, Herbert\" by Ferdinand Canning Scott Schiller. Spencer's actual wording reads: \"Regarding Science as a gradually increasing sphere, we may say that every addition to its surface does but bring it into wider contact with surrounding nescience.\" ( _First Principles_ [London: Williams and Norgate, 1862], pp. 16\u201317.)\u2014Ed.]. See also Sir Karl Raimund Popper, \"On the Sources of Knowledge and Ignorance,\" _Proceedings of the British Academy_ , 46 (1960): 69: \"The more we learn about the world, and the deeper our learning, the more conscious, specific, and articulate will be our knowledge of what we do not know, our knowledge of our ignorance\"; and Warren Weaver, \"A Scientist Ponders Faith,\" _Saturday Review_ , 3 (January 1959): 9: \"[is] science really gaining in its assault on the totality of the unsolved? As science learns one answer, it is characteristically true that it also learns several new questions. It is as though science were working in a great forest of ignorance, making an ever larger circular clearing within which, not to insist on the pun, things are clear. . . . But as that circle becomes larger and larger, the circumference of contact with ignorance also gets longer and longer. Science learns more and more. But there is an ultimate sense in which it does not gain; for the volume of the appreciated but not understood keeps getting larger. We keep, in science, getting a more and more sophisticated view of our essential ignorance.\"\n\n8 Cf. Homer Garner Barnett, _Innovation: The Basis of Cultural Change_ (New York: McGraw-Hill, 1953): \"Every individual is an innovator many times over\" ( p. 19) and \"There is a positive correlation between individualism and innovative potential. The greater the freedom of the individual to explore his world of experience and to organize its elements in accordance with his private interpretation of his sense impressions, the greater the likelihood of new ideas coming into being\" ( p. 65).\n\n9 Cf. Sir William Arthur Lewis, _The Theory of Economic Growth_ (London: Allen and Unwin, 1955), p. 148: \"These innovators are always a minority. New ideas are first put into practice by one or two or very few persons, whether they be new ideas in technology, or new forms of organization, new commodities, or other novelties. These ideas may be accepted rapidly by the rest of the population. More probably they are received with scepticism and unbelief, and make their way only very slowly at first if at all. After a while the new ideas are seen to be successful, and are then accepted by increasing numbers. Thus it is often said that change is the work of an elite, or that the amount of change depends on the quality of leadership in a community. This is true enough if it implies no more than that the majority of people are not innovators, but merely imitate what others do. It is, however, somewhat misleading if it is taken to imply that some specific class or group of people get all the new ideas.\" Also p. 172: \"Collective judgement of new ideas is so often wrong that it is arguable that progress depends on individuals being free to back their own judgement despite collective disapproval. . . . To give a monopoly of decision to a government committee would seem to have the disadvantage of both worlds.\"\n\n10 One of the few authors who have seen clearly at least part of this was Frederic William Mait land, who stresses ( _The Collected Papers of Frederic William Maitland, Downing Professor of the Laws of England_ [3 vols.; Cambridge: Cambridge University Press, 1911], vol. 1, p. 107) that \"the most powerful argument is that based on the ignorance, the necessary ignorance, of our rulers.\" [Maitland's quotation appears in _A Historical Sketch of Liberty and Equality_ , Liberty Fund edition, p. 133.\u2014Ed.] See, however, Bennett E. Kline and Norman H. Martin, \"Freedom, Authority, and Decentralization,\" _Harvard Business Review_ , 36 (1958), esp. 70: \"the chief characteristic of the command hierarchy, or any group in our society, is not knowledge but ignorance. Consider that any one person can know only a fraction of what is going on around him. Much of what that person knows or believes will be false rather than true. . . . At any given time, vastly more is not known than is known, either by one person in a command chain or by all the organization. It seems possible, then, that in organizing ourselves into a hierarchy of authority for the purpose of increasing efficiency, we may really be institutionalizing ignorance. While making better use of what the few know, we are making sure that the great majority are prevented from exploring the dark areas beyond our knowledge.\" See also William Graham Sumner, \"Speculative Legislation,\" _The Challenge of Facts and Other Papers_ (New Haven: Yale University Press, 1914), p. 215:\"It is characteristic of speculative legislation that it very generally produces the exact opposite of the result it was hoped to get from it. The reason is that the elements of any social problem which we do not know so far exceed those which we do know, that our solutions have a greater chance to be wrong than right.\"\n\nThere is one important respect in which the term \"ignorance\" is somewhat too narrow for our purposes. There are occasions when it would probably be better to speak of \"uncertainty\" with reference to ignorance concerning what is right, since it is doubtful whether we can meaningfully speak about something being right if nobody knows what is right in the particular context. The fact in such instances may be that the existing morals provide no answer to a problem, though there might be some answer which, if it were known and widely accepted, would be very valuable. I am much indebted to Mr. Pierre F. Goodrich, whose comment during a discussion helped to clarify this important point for me, though I have not been persuaded to speak generally of \"imperfection\" where I stress ignorance.\n\n11 Cf. John Archibald Wheeler, \"A Septet of Sibyls: Aids in the Search for Truth,\" _American Scientist_ , 44 (1956): 360: \"Our whole problem is to make the mistakes as fast as possible.\"\n\n12 Cf. the remark of Louis Pasteur: \"In research, chance only helps those whose minds are well prepared for it,\" quoted by Ren\u00e9 Taton, _Reason and Chance in Scientific Discovery_ (London: Hutchinson, 1957), p. 91. [Pasteur appears to have originally made the statement in a lecture at the University of Lille on December 7, 1854. The original reads: \"Dans les champs de l'observation le hasard ne favorise que les esprits pr\u00e9par\u00e9s.\"\u2014Ed.]\n\n13 Cf. Abba Ptachya Lerner, \"The Backward-leaning Approach to Controls,\" _Journal of Political Economy_ , 65 (1957): 441: \"The free-trade doctrines are valid as _general rules_ whose general use is generally beneficial. As with all general rules, there are particular cases where, if one knew all the attendant circumstances and the full effects in all their ramifications, it would be better for the rule not to be applied. But that does not make the rule a bad rule or give reason for not applying the rule where, as is normally the case, one does not know all the ramifications that would make the case a desirable exception.\"\n\n14 Cf. Rev. Hastings Rashdall, \"The Philosophical Theory of Property,\" in _Property; Its Duties and Rights: Historically, Philosophically, and Religiously Regarded_ , Charles Gore and Leonard Trelawney Hobhouse, eds. (new ed.; New York: Macmillan, 1915), pp. 61\u201362: \"The plea for liberty is not sufficiently met by insisting, as has been so eloquently and humorously done by Mr. Lowes Dickinson ( _Justice and Liberty: A Political Dialogue_ , e.g. pp. 129 and 131), upon the absurdity of supposing that the propertyless labourer under the ordinary capitalistic regime enjoys any liberty of which Socialism would deprive him. For it may be of extreme importance that _some_ should enjoy liberty\u2014that it should be possible for some few men to be able to dispose of their time in their own way\u2014although such liberty may be neither possible nor desirable for the great majority. That culture requires a considerable differentiation in social conditions is also a principle of unquestionable importance.\" [The full citation of the book quoted by Rashdall is: Goldsworthy Lowes Dickinson, _Justice and Liberty: A Political Dialogue_ (London: J. M. Dent, 1908).\u2014Ed.] See also Bennett E. Kline and Norman H. Martin, \"Freedom, Authority, and Decentralization,\" p. 69: \"If there is to be freedom for the few who _will_ take advantage of it, freedom must be offered to the many. If any lesson is clear from history, it is this.\"\n\n15 For the use of the term \"formation,\" more appropriate in this connection than the usual \"institution,\" see my study on _The Counter-Revolution of Science: Studies on the Abuse of Reason_ (Glencoe, IL: Free Press, 1952), p. 83. [ _Collected Works_ edition, vol. 13, p. 145.] [Hayek there writes of human institutions: \"Though in a sense man-made, i.e., entirely the result of human actions, they may yet not be designed, not be the intended product of these actions. The term 'institution' itself is rather misleading in this respect, as it suggests something deliberately instituted. It would probably be better if this term were confined to particular contrivances, like particular laws and organizations, which have been created for a specific purpose, and if a more neutral term like 'formations' (in a sense similar to that in which the geologists use it, and corresponding to the German _Gebilde_ ) could be used for those phenomena, which, like money or language, have not been so created.\"\u2014Ed.]\n\n16 Cf. my article \"Degrees of Explanation,\" _British Journal for the Philosophy of Science_ , 6 (1955): 209\u201325, reprinted in _Studies in Philosophy, Politics, and Economics_ (Chicago: University of Chicago Press, 1967), pp. 3\u201321 [Also reprinted in _Studies in Philosophy, Politics, and Economics_ , pp. 22\u201342.\u2014Ed.], and my \"The Theory of Complex Phenomena,\" in _The Critical Approach to Science and Philosophy: Essays in Honor of Kar l R. Popper_ , Mario Augusto Bunge, ed. (New York: Free Press of Glencoe, 1964), pp. 332\u201349.\n\n17 See Aaron Director, \"The Parity of the Economic Market Place,\" in _Conference on Freedom and the Law, May 7, 1953: Fiftieth Anniversary Celebration_ , Thuman Welsey Arnold, et al. (University of Chicago Law School Conference Series, no. 13, Chicago: University of Chicago Press, 1953), pp. 16\u201325.\n\n18 Cf. my book _The Road to Serfdom_ (Chicago: University of Chicago Press, 1944), chap. 7, pp. 88\u2013100; reprinted as vol. 2 of _The Collected Works of F. A. Hayek_ , Bruce Caldwell, ed. (Chicago: University of Chicago Press, 2007), pp. 124\u201333.\n\n19 See Sir Karl Raimund Popper, _The Open Society and Its Enemies_ (American ed.; Princeton: Princeton University Press, 1950), esp. p. 195: \"If we wish to remain human, there is only one way, the way into the open society. We must go into the unknown, the uncertain and insecure, using what reason we may have to plan for both, security _and_ freedom.\" be no other opportunity to refer to Vico, it should be mentioned here that he and his great disciple, Ferdinando Galiani, constitute the only important parallel on the Continent to the anti-rationalist British tradition, which we shall consider more fully in the next chapter.\n\n## THREE\n\nTHE COMMON SENSE OF PROGRESS\n\nMan never mounts higher than when he knows not where he is going.\n\n\u2014Oliver Cromwell\n\n1. Writers nowadays who value their reputation among the more sophisticated hardly dare to mention progress without including the word in quotation marks. The implicit confidence in the beneficence of progress that during the last two centuries marked the advanced thinker has come to be regarded as the sign of a shallow mind. Though the great mass of the people in most parts of the world still rest their hopes on continued progress, it is common among intellectuals to question whether there is such a thing, or at least whether progress is desirable.\n\nUp to a point, this reaction against the exuberant and na\u00efve belief in the inevitability of progress was necessary. So much of what has been written and talked about it has been indefensible that one may well think twice before using the word. There never was much justification for the assertion that \"civilization has moved, is moving, and will move in a desirable direction,\"1 nor was there any ground for regarding all change as necessary, or progress as certain and always beneficial. Least of all was there warrant for speaking about recognizable \"laws of progress\" that enabled us to predict the conditions toward which we were necessarily moving, or for treating every foolish thing men have done as necessary and therefore right.\n\nBut if the fashionable disillusionment about progress is not difficult to explain, it is not without danger. In one sense, civilization is progress and progress is civilization.2 The preservation of the kind of civilization that we know depends on the operation of forces which, under favorable conditions, produce progress. If it is true that evolution does not always lead to better things, it is also true that, without the forces which produce it, civilization and all we value\u2014indeed, almost all that distinguishes man from beast\u2014would neither exist nor could long be maintained.\n\nThe history of civilization is the account of a progress which, in the short space of less than eight thousand years, has created nearly all that we regard as characteristic of human life. After abandoning hunting life, most of our direct ancestors, at the beginning of neolithic culture, took to agriculture and soon to urban life perhaps less than three thousand years or one hundred generations  ago. It is not surprising that in some respects man's biological equipment has not kept pace with that rapid change, that the adaptation of his non-rational part has lagged somewhat, and that many of his instincts and emotions are still more adapted to the life of a hunter than to life in civilization. If many features of our civilization seem to us unnatural, artificial, or unhealthy, this must have been man's experience ever since he first took to town life, which is virtually since civilization began. All the familiar complaints against industrialism, capitalism, or overrefinement are largely protests against a new way of life that man took up a short while ago after more than half a million years' existence as a wandering hunter, and that created problems still unsolved by him.3\n\n2. When we speak of progress in connection with our individual endeavors  or any organized human effort, we mean an advance toward a known goal.4 It is not in this sense that social evolution can be called progress, for it is not achieved by human reason striving by known means toward a fixed aim.5 It would be more correct to think of progress as a process of formation and modification of the human intellect, a process of adaptation and learning in which not only the possibilities known to us but also our values and desires continually change. As progress consists in the discovery of the not yet known, its consequences must be unpredictable. It always leads into the unknown, and the most we can expect is to gain an understanding of the kind of forces that bring it about.6 Yet, though such a general understanding of the character of this process of cumulative growth is indispensable if we are to try to create conditions favorable to it, it can never be knowledge which will enable us to make specific predictions.7 The claim that we can derive from such insight necessary laws of evolution that we must follow is an absurdity. Human reason can neither predict nor deliberately shape its own future. Its advances consist in finding out where it has been wrong.\n\nEven in the field where the search for new knowledge is most deliberate, i.e., in science, no man can predict what will be the consequences of his work.8 In fact, there is increasing recognition that even the attempt to make science  deliberately aim at useful knowledge\u2014that is, at knowledge whose future uses can be foreseen\u2014is likely to impede progress.9 Progress by its very nature cannot be planned. We may perhaps legitimately speak of planning progress in a particular field where we aim at the solution of a specific problem and are already on the track of the answer. But we should soon be at the end of our endeavors if we were to confine ourselves to striving for goals now visible and if new problems did not spring up all the time. It is knowing what we have not known before that makes us wiser men.\n\nBut often it also makes us sadder men. Though progress consists in part in achieving things we have been striving for, this does not mean that we shall like all its results or that all will be gainers. And since our wishes and aims are also subject to change in the course of the process, it is questionable whether the statement has a clear meaning that the new state of affairs that progress creates is a better one. Progress in the sense of the cumulative growth of knowledge and power over nature is a term that says little about whether the new state will give us more satisfaction than the old. The pleasure may be solely in achieving what we have been striving for, and the assured possession may give us little satisfaction. The question whether, if we had to stop at our present stage of development, we would in any significant sense be better off or happier than if we had stopped a hundred or a thousand years ago is probably unanswerable.\n\nThe answer, however, does not matter. What matters is the successful striving for what at each moment seems attainable. It is not the fruits of past success but the living in and for the future in which human intelligence proves itself. Progress is movement for movement's sake, for it is in the process of learning, and in the effects of having learned something new, that man enjoys the gift of his intelligence.\n\nThe enjoyment of personal success will be given to large numbers only in a society that, as a whole, progresses fairly rapidly. In a stationary society  there will be about as many who will be descending as there will be those rising. In order that the great majority should in their individual lives participate in the advance, it is necessary that it proceed at a considerable speed. There can therefore be little doubt that Adam Smith was right when he said: \"It is in the progressive state, while the society is advancing to the further acquisition, rather than when it has acquired its full complement of riches, that the condition of the labouring poor, of the great body of people, seems to be happiest and the most comfortable. It is hard in the stationary, and miserable in the declining state. The progressive state is in reality the cheerful and hearty state to all the different orders of the society. The stationary is dull; the declining melancholy.\"10\n\nIt is one of the most characteristic facts of a progressive society that in it most things which individuals strive for can be obtained only through further progress. This follows from the necessary character of the process: new knowledge and its benefits can spread only gradually, and the ambitions of the many will always be determined by what is as yet accessible only to the few. It is misleading to think of those new possibilities as if they were, from the beginning, a common possession of society which its members could deliberately share; they become a common possession only through that slow process by which the achievements of the few are made available to the many. This is often obscured by the exaggerated attention usually given to a few conspicuous major steps in the development. But, more often than not, major discoveries merely open new vistas, and long further efforts are necessary before the new knowledge that has sprung up somewhere can be put to general use. It will have to pass through a long course of adaptation, selection, combination, and improvement before full use can be made of it. This means that there will always be people who already benefit from new achievements that have not yet reached others.\n\n3. The rapid economic advance that we have come to expect seems in a large measure to be the result of this inequality and to be impossible without it. Progress at such a fast rate cannot proceed on a uniform front but must take place in echelon fashion, with some far ahead of the rest. The reason for this is concealed by our habit of regarding economic progress chiefly as an accumulation of ever greater quantities of goods and equipment. But the rise of our standard of life is due at least as much to an increase in knowledge which  enables us not merely to consume more of the same things but to use different things, and often things we did not even know before. And though the growth of income depends in part on the accumulation of capital, more probably depends on our learning to use our resources more effectively and for new purposes.\n\nThe growth of knowledge is of such special importance because, while the material resources will always remain scarce and will have to be reserved for limited purposes, the uses of new knowledge (where we do not make them artificially scarce by patents of monopoly) are unrestricted. Knowledge, once achieved, becomes gratuitously available for the benefit of all. It is through this free gift of the knowledge acquired by the experiments of some members of society that general progress is made possible, that the achievements of those who have gone before facilitate the advance of those who follow.\n\nAt any stage of this process there will always be many things we already know how to produce but which are still too expensive to provide for more than a few. And at an early stage they can be made only through an outlay of resources equal to many times the share of total income that, with an approximately equal distribution, would go to the few who could benefit from them. At first, a new good is commonly \"the caprice of a chosen few before it becomes a public need and forms part of the necessities of life. For the luxuries of today are the necessities of tomorrow.\"11 Furthermore, the new things will often become available to the greater part of the people only _because_ for some time they have been the luxuries of the few.\n\nIf we, in the wealthier countries, today can provide facilities and conveniences for most which not long ago would have been physically impossible to produce in such quantities, this is in large measure the direct consequence of the fact that they were first made for a few. All the conveniences of a comfortable home, of our means of transportation and communication, of entertainment and enjoyment, we could produce at first only in limited quantities; but it was in doing this that we gradually learned to make them or similar things at a much smaller outlay of resources and thus became able to supply them to the great majority. A large part of the expenditure of the rich, though not intended for that end, thus serves to defray the cost of the experimentation with the new things that, as a result, can later be made available to the poor.\n\nThe important point is not merely that we gradually learn to make cheaply on a large scale what we already know how to make expensively in small quantities but that only from an advanced position does the next range of desires and possibilities become visible, so that the selection of new goals and the effort toward their achievement will begin long before the majority can  strive for them. If what they will want after their present goals are realized is soon to be made available, it is necessary that the developments that will bear fruit for the masses in twenty or fifty years' time should be guided by the views of people who are already in the position of enjoying them.\n\nIf today in the United States or western Europe the relatively poor can have a car or a refrigerator, an airplane trip or a radio, at the cost of a reasonable part of their income, this was made possible because in the past others with larger incomes were able to spend on what was then a luxury. The path of advance is greatly eased by the fact that it has been trodden before. It is because scouts have found the goal that the road can be built for the less lucky or less energetic. What today may seem extravagance or even waste, because it is enjoyed by the few and even undreamed of by the masses, is payment for the experimentation with a style of living that will eventually be available to many. The range of what will be tried and later developed, the fund of experience that will become available to all, is greatly extended by the unequal distribution of present benefits; and the rate of advance will be greatly increased if the first steps are taken long before the majority can profit from them. Many of the improvements would indeed never become a possibility for all if they had not long before been available to some. If all had to wait for better things until they could be provided for all, that day would in many instances never come. Even the poorest today owe their relative material well-being to the results of past inequality.\n\n4. In a progressive society as we know it, the comparatively wealthy are thus merely somewhat ahead of the rest in the material advantages which they enjoy. They are already living in a phase of evolution that the others have not yet reached. Poverty has, in consequence, become a relative, rather than an absolute, concept. This does not make it less bitter. Although in an advanced society the unsatisfied wants are usually no longer physical needs but the results of civilization, it is still true that at each stage some of the things most people desire can be provided only for a few and can be made accessible to all only by further progress. Most of what we strive for are things we want because others already have them.12 Yet a progressive society, while it relies on this process of learning and imitation, recognizes the desires it creates only as a spur to further effort. It does not guarantee the results to everyone. It disregards the pain of unfulfilled desire aroused by the example of others. It appears cruel because it increases the desire of all in proportion as it increases its gifts to some. Yet so long as it remains a progressive society, some must lead, and the rest must follow.\n\nThe contention that in any phase of progress the rich, by experimenting with new styles of living not yet accessible to the poor, perform a necessary service without which the advance of the poor would be very much slower will appear to some as a piece of far-fetched and cynical apologetics. Yet a little reflection will show that it is fully valid and that a socialist society would in this respect have to imitate a free society. It would be necessary in a planned economy (unless it could simply imitate the example of other more advanced societies) to designate individuals whose duty it would be to try out the latest advances long before they were made available to the rest. There is no way of making generally accessible new and still expensive ways of living except by their being initially practiced by some. It would not be enough if individuals were allowed to try out particular new things. These have their proper use and value only as an integral part of the general advance in which they are the next thing desired. In order to know which of the various new possibilities should be developed at each stage, how and when particular improvements ought to be fitted into the general advance, a planned society would have to provide for a whole class, or even a hierarchy of classes, which would always move some steps ahead of the rest. The situation would then differ from that in a free society merely in the fact that the inequalities would be the result of design and that the selection of particular individuals or groups would be done by authority rather than by the impersonal process of the market and the accidents of birth and opportunity. It should be added that only those kinds of better living approved by authority would be permissible and that they would be provided only for those specially designated. But, in order for a planned society to achieve the same rate of advance as a free society, the degree of inequality that would have to prevail would not be very different.\n\nThere is no practicable measure of the degree of inequality that is desirable here. We do not wish, of course, to see the position of individuals determined by arbitrary decision or a privilege conferred by human will on particular persons. It is difficult to see however, in what sense it could ever be legitimate to say that any one person is too far ahead of the rest or that it would be harmful to society if the progress of some greatly outstripped that of others. There might be justification for saying this if there appeared great gaps in the scale of advance; but, as long as the graduation is more or less continuous and all the steps in the income pyramid are reasonably occupied, it can scarcely be denied that those lower down profit materially from the fact that others are ahead.\n\nThe objections spring from the misconception that those in the lead claim the right to something that otherwise would be available to the rest. This would be true if we thought in terms of a single redistribution of the fruits of past progress and not in terms of that continuous advance which our unequal society fosters. In the long run, the existence of groups ahead of the rest is  clearly an advantage to those who are behind, in the same way that, if we could suddenly draw on the more advanced knowledge which some other men on a previously unknown continent or on another planet had gained under more favorable conditions, we would all profit greatly.\n\n5. The problems of equality are difficult to discuss dispassionately when members of our own community are affected. They stand out more clearly when we consider them in their wider aspect, namely, the relation between rich and poor countries. We are then less apt to be misled by the conception that each member of any community has some natural right to a definite share of the income of his group. Although today most of the people of the world benefit from one another's efforts, we certainly have no reason to consider the product of the world as the result of a unified effort of collective humanity.\n\nAlthough the fact that the people of the West are today so far ahead of the others in wealth is in part the consequence of a greater accumulation of capital, it is mainly the result of their more effective utilization of knowledge. There can be little doubt that the prospect of the poorer, \"undeveloped\" countries reaching the present level of the West is very much better than it would have been, had the West not pulled so far ahead. Furthermore, it is better than it would have been, had some world authority, in the course of the rise of modern civilization, seen to it that no part pulled too far ahead of the rest and made sure at each step that the material benefits were distributed evenly throughout the world. If today some nations can in a few decades acquire a level of material comfort that took the West hundreds or thousands of years to achieve, is it not evident that their path has been made easier by the fact that the West was not forced to share its material achievements with the rest\u2014that it was not held back but was able to move far in advance of the others?\n\nNot only are the countries of the West richer because they have more advanced technological knowledge, but they have more advanced technological knowledge because they are richer. And the free gift of the knowledge that has cost those in the lead much to achieve enables those who follow to reach the same level at a much smaller cost.13 Indeed, so long as some countries lead, all the others can follow, although the conditions for spontaneous progress may be absent in them. That even countries or groups which do not possess freedom can profit from many of its fruits is one of the reasons why the importance of freedom is not better understood. For many parts of the world the advance of civilization has long been a derived affair, and, with modern  communications, such countries need not lag very far behind, though most of the innovations may originate elsewhere. How long has Soviet Russia or Japan been living on an attempt to imitate American technology! So long as somebody else provides most of the new knowledge and does most of the experimenting, it may even be possible to apply all this knowledge deliberately in such a manner as to benefit most of the members of a given group at about the same time and to the same degree. But, though an egalitarian society could advance in this sense, its progress would be essentially parasitical, borrowed from those who have paid the cost.\n\nIt is worth remembering in this connection that what enables a country to lead in this world-wide development are its economically most advanced classes and that a country that deliberately levels such differences also abdicates its leading position\u2014as the example of Great Britain so tragically shows. All classes there had profited from the fact that a rich class with old traditions had demanded products of a quality and taste unsurpassed elsewhere and that Britain, in consequence, came to supply to the rest of the world. British leadership has gone with the disappearance of the class whose style of living the others imitated. It may not be long before the British workers will discover that they had profited by being members of a community containing many persons richer than they and that their lead over the workers in other countries was in part an effect of a similar lead of their own rich over the rich in other countries.\n\n6. If on an international scale even major inequalities may be of great assistance to the progress of all, can there be much doubt that the same is also true of such inequalities within a nation? Here, too, the over-all speed of advance will be increased by those who move fastest. Even if many fall behind at first, the cumulative effect of the preparation of the path will, before long, sufficiently facilitate their advance that they will be able to keep their place in the march. Members of a community containing many who are rich enjoy, in fact, a great advantage not available to those who, because they live in a poor country, do not profit from the capital and experience supplied by the rich; it is difficult to see, therefore, why this situation should justify a claim to a larger share for the individual. It seems indeed generally to be the case that, after rapid progress has continued for some time, the cumulative advantage for those who follow is great enough to enable them to move faster than those who lead and that, in consequence, the long-drawn-out column of human progress tends to close up. The experience of the United States at least seems to indicate that, once the rise in the position of the lower classes gathers speed, catering to the rich ceases to be the main source of great gain and gives place to efforts directed toward the needs of the masses. Those forces which at first make inequality self-accentuating thus later tend to diminish it.\n\nTherefore, there must be two different ways of looking at the possibility  of reducing inequality and abolishing poverty by deliberate redistribution\u2014 that is, from a long-term or a short-term point of view. At any given moment we could improve the position of the poorest by giving them what we took from the wealthy. But, while such an equalizing of the positions in the column of progress would temporarily quicken the closing-up of the ranks, it would, before long, slow down the movement of the whole and in the long run hold back those in the rear. Recent European experience strongly confirms this. The rapidity with which rich societies here have become static, if not stagnant, societies through egalitarian policies, while impoverished but highly competitive countries have become very dynamic and progressive, has been one of the most conspicuous features of the postwar period. The contrast in this respect between the advanced welfare states of Great Britain and the Scandinavian countries, on the one hand, and countries like Western Germany, Belgium, or Italy, is beginning to be recognized even by the former.14 If a demonstration had been needed that there is no more effective way of making a society stationary than by imposing upon all something like the same average standard, or no more effective way of slowing down progress than by allowing the most successful a standard only a little above the average, these experiments have provided it.\n\nIt is curious that, while in the case of a primitive country every detached observer would probably recognize that its position offered little hope so long as its whole population was on the same low dead level and that the first condition for advance was that some should pull ahead of the others, few people are willing to admit the same of more advanced countries.15 Of course, a society in which only the politically privileged are allowed to rise, or where those who rise first gain political power and use it to keep the others down, would be no better than an egalitarian society. But all obstacles to the rise of some are, in the long run, obstacles to the rise of all; and they are no less harmful to the true interest of the multitude because they may gratify its momentary passions.16\n\n7. With respect to the advanced countries of the West it is sometimes contended that progress is too fast or too exclusively material. These two aspects are probably closely connected. Times of very rapid material progress have rarely been periods of great efflorescence of the arts, and both the greatest appreciation and the finest products of artistic and intellectual endeavor have often appeared when material progress has slackened. Neither western Europe of the nineteenth century nor the United States of the twentieth is eminent for its artistic achievements. But the great outbursts in the creation of non-material values seem to presuppose a preceding improvement in economic condition. It is perhaps natural that generally after such periods of rapid growth of wealth there occurs a turning toward non-material things or that, when economic activity no longer offers the fascination of rapid progress, some of the most gifted men should turn to the pursuit of other values.\n\nThis is, of course, only one and perhaps not even the most important aspect of rapid material progress that makes many of those who are in its van skeptical of its value. We must also admit that it is not certain whether most people want all or even most of the results of progress. For most of them it is an involuntary affair which, while bringing them much they strive for, also forces on them many changes they do not want at all. The individual does not have it in his power to choose to take part in progress or not; and always it not only brings new opportunities but deprives many of much they want, much that is dear and important to them. To some it may be sheer tragedy, and to all those who would prefer to live on the fruits of past progress and not take part in its future course, it may seem a curse rather than a blessing.\n\nThere are, especially, in all countries and at all times groups that have reached a more or less stationary position, in which habits and ways of life have been settled for generations. These ways of life may suddenly be threatened by developments with which they have had nothing to do, and not only the members of such groups but often outsiders also will wish them to be preserved. Many of the peasants of Europe, particularly those in the remote mountain valleys, are an example. They cherish their way of life, though it has become a dead end, though it has become too dependent on urban civilization, which is continually changing, to preserve itself. Yet the conservative peasant, as much as anybody else, owes his way of life to a different type of person, to men who were innovators in their time and who by their innovations forced a new manner of living on people belonging to an earlier state of culture; the nomad probably complained as much about the encroachment of enclosed fields on his pastures as does the peasant about the encroachments of industry.\n\nThe changes to which such people must submit are part of the cost of progress, an illustration of the fact that not only the mass of men but, strictly speaking, every human being is led by the growth of civilization into a path that is not of his own choosing. If the majority were asked their opinion of all the changes involved in progress, they would probably want to prevent many of its necessary conditions and consequences and thus ultimately stop progress itself. And I have yet to learn of an instance when the deliberate vote of the majority (as distinguished from the decision of some governing elite) has decided on such sacrifices in the interest of a better future as is made by a free-market society. This does not mean, however, that the achievement of most things men actually want does not depend on the continuance of that progress which, if they could, they would probably stop by preventing the effects which do not meet with their immediate approval.\n\nNot all the amenities that we can today provide for the few will sooner or later be available to all; with such amenities as personal services, it would be clearly impossible. They are among the advantages which the wealthy are deprived of by progress. But most of the gains of the few do, in the course of time, become available to the rest. Indeed, all our hopes for the reduction of present misery and poverty rest on this expectation. If we abandoned progress, we should also have to abandon all those social improvements that we now hope for. All the desired advances in education and health, the realization of our wish that at least a large proportion of the people should reach the goals for which they are striving, depend on the continuance of progress. We have only to remember that to prevent progress at the top would soon prevent it all the way down, in order to see that this result is really the last thing we want.\n\n8. We have so far concerned ourselves mainly with our own country or with those countries which we consider to be members of our own civilization. But we must take into account the fact that the consequences of past progress\u2014 namely, world-wide extension of rapid and easy communication of knowledge and ambitions\u2014have largely deprived us of the choice as to whether or not we want continued rapid progress. The new fact in our present position that forces us to push on is that the accomplishments of our civilization have become the object of desire and envy of all the rest of the world. Regardless of whether from some higher point of view our civilization is really better or not, we must recognize that its material results are demanded by practically all who have come to know them. Those people may not wish to adopt our entire civilization, but they certainly want to be able to pick and choose from it whatever suits them. We may regret, but cannot disregard, the fact that even where different civilizations are still preserved and dominate the lives of the majority, the leadership has fallen almost invariably into the hands of those  who have gone furthest in adopting the knowledge and technology of Western civilization.17\n\nWhile superficially it may seem that two types of civilization are today competing for the allegiance of the people of the world, the fact is that the promise they offer to the masses, the advantages they hold out to them, are essentially the same. Though the free and the totalitarian countries both claim that their respective methods will provide more rapidly what those people want, the goal itself must seem to them the same. The chief difference is that only the totalitarians appear clearly to know how they want to achieve that result, while the free world has only its past achievements to show, being by its very nature unable to offer any detailed \"plan\" for further growth.\n\nBut if the material achievements of our civilization have created ambitions in others, they have also given them a new power to destroy it if what they believe is their due is not given them. With the knowledge of possibilities spreading faster than the material benefits, a great part of the people of the world are today dissatisfied as never before and are determined to take what they regard as their rights. They believe as much and as mistakenly as the poor in any one country that their goal can be achieved by a redistribution of already existing wealth, and they have been confirmed in this belief by Western teaching. As their strength grows, they will become able to extort such a redistribution if the increase in wealth that progress produces is not fast enough. Yet a redistribution that slows down the rate of advance of those in the lead must bring about a situation in which even more of the next improvement will have to come from redistribution, since less will be provided by economic growth.\n\nThe aspirations of the great mass of the world's population can today be satisfied only by rapid material progress. There can be little doubt that in their present mood a serious disappointment of their expectations would lead to grave international friction\u2014indeed, it would probably lead to war. The peace of the world and, with it, civilization itself thus depend on continued progress at a fast rate. At this juncture we are therefore not only the creatures but the captives of progress; even if we wished to, we could not sit back and enjoy at leisure what we have achieved. Our task must be to continue to lead, to move ahead along the path which so many more are trying to tread in our  wake. At some future date when, after a long period of world-wide advance in material standards, the pipelines through which it spreads are so filled that, even when the vanguard slows down, those at the rear will for some time continue to move at an undiminished speed, we may again have it in our power to choose whether or not we want to go ahead at such a rate. But at this moment, when the greater part of mankind has only just awakened to the possibility of abolishing starvation, filth, and disease; when it has just been touched by the expanding wave of modern technology after centuries or millennia of relative stability; and as a first reaction has begun to increase in number at a frightening rate, even a small decline in our rate of advance might be fatal to us.\n\n* * *\n\nThe quotation at the head of the chapter is taken from Jean Fran\u00e7ois Paul de Gondi de Retz, _M\u00e9moires du Cardinal de Retz, de Guy-Joli, et de la duchesse de Nemours, contenant ce qui s'est pass\u00e9 de remarquable en France pendant les premi\u00e8res ann\u00e9es du r\u00e8gne de Louis XIV_ (6 vols. in 8; Nouvelle \u00e9dition; Paris: Chez \u00c9tienne Ledoux, 1820), vol. 2, p. 497, where President Belli\u00e8vre is recorded as having said that Cromwell once told him \"on ne montait jamais si haut que quand on ne sait o\u00f9 l'on va.\" [Pomponne de Belli\u00e8vre (1606\u201357), grandson of two chancellors of France and the first president of the Parlement of Paris, at one point served as French ambassador to England.\u2014Ed.] The phrase apparently made a deep impression on eighteenth century thinkers, and it is quoted by David Hume ( _Essays_ , vol. 1, p. 124) [The essay in which Hume's reference falls originally appeared under the title \"Whether the British Government Inclines More to Absolute Monarchy, or to a Republic,\" (Essay 9) in _Essays, Moral and Political_ (Edinburgh: Printed by R. Fleming and A. Alison for A. Kincaid, 1741), p. 98n. (Liberty Fund edition, p. 50). In a footnote quoting Cromwell's statement to Belli\u00e8vre he there notes that \"a Man, possess'd of usurp'd Authority, can set no Bounds to his Pretensions.\" Hume's footnote appears in all editions of his _Essays_ until that of 1774, at which point it was dropped.\u2014Ed.], Adam Ferguson ( _An Essay on the History of Civil Society_ [Edinburgh: Printed for A. Millar and T. Caddel in the Strand, London, and A. Kincaid and J. Bell, Edinburgh, 1767], p. 187), and (according to Duncan Forbes, \"Scientific Whiggism,\" _Cambridge Journal_ , vol. 7 [1954]: 654) also by Anne Robert Jacques Turgot. [Turgot comes very close to the idea in his _Plan de Deux Discours sur l'Histoire Universelle_ , where he writes of men that \"leur passions, leurs fureurs m\u00eame, les conduits sans qu'ils sussent o\u00f9 ils allaient.\" The essay appears in Turgot's _\u0152uvres de Turgot et Documents le Concernant_ , Gustave Schelle, ed. (5 vols.; Paris: Librairie F\u00e9lix Alcan, 1913\u201323), vol. 1, p. 283. Ronald Meek in his edition of Turgot's essays, _Turgot on Progress, Sociology, and Economics_ (Cambridge: Cambridge University Press, 1973), p. 69, translates Turgot's words as \"Their passions, even their fits of rage, have led them on their way without their being aware of where they were going.\"\u2014Ed.] It appears once more, appropriately, in Albert Venn Dicey, _Law and Public Opinion_ , p. 231 [Liberty Fund edition, p. 164]. A slightly modified version occurs in Johann Wolfgang von Goethe's posthumously published _Maximen, Reflexionen: Goethe's Aufs\u00e4tze zur Kultur- , Theater- und Literatur- Geschichte_ , Grossherzog Wilhelm Ernst Ausgabe (2 vols.; Leipzig: Inselverlag, 1913\u201314), vol. 2, p. 626: \"Man geht nie weiter, als wenn man nicht mehr weiss, wohin man geht.\" [\"One never goes so far as when one doesn't know where one is going.\"\u2014Ed.] Cf. in this connection also Giambattista Vico, _Opere di Giambattista Vico, ordinate ed illustrate coll'analisi storica della mente di Vico in relazione alla scienza della civilt\u00e0_ , Giuseppe Ferrari, ed. (6 vols.; Milan: Societ\u00e0 Tipog. de' Classici Italiani, 1852\u201354), vol. 5, p. 183. \"Homo non intelligendo fit omnia.\" [The Latin should read: \"Homo non intelligendo facit omnia,\" which translates as \"Man unknowingly makes all things.\"\u2014Ed.] Since there will be no other opportunity to refer to Vico, it should be mentioned here that he and his great disciple, Ferdinando Galiani, constitute the only important parallel on the Continent to the antirationalist British tradition, which we shall consider more fully in the next chapter.\n\nA German translation of an earlier and somewhat longer version of the present chapter has been published as \"Grundtatsachen des Fortschritts,\" [The Fundamental Facts of Progress] in _Ordo: Jahrbuch f\u00fcr die Ordnung von Wirtschaft und Gesellschaft_ , 9 (1957): 19\u201342.\n\n1 John Bagnell Bury, _The Idea of Progress: An Inquiry into its Origin and Growth_ (London: Macmillan and Co., 1920), p. 2.\n\n2 Cf. John Stuart Mill, \"Representative Government,\" in _On Liberty and Considerations on Representative Government_ , Ronald Buchanan McCallum, ed. (Oxford: B. Blackwell, 1946), p. 121.\n\n3 Cf. Adam Ferguson, _History of Civil Society_ , p. 12: \"If the palace be unnatural the cottage is so no less: and the highest refinements of political and moral apprehension, are no more artificial in their kind, than the first operations of sentiment and reason.\" Wilhelm Roscher, _Ansichten der Volkswirthschaft aus dem geschichtlichen Standpunkte_ (Leipzig and Heidelberg: C. F. Winter, 1861), pp. 408\u20139, gives, as illustrations of the \"pernicious refinements\" against which austere moralists have thundered at one time or another, forks, gloves, and glazed windows; Plato in his _Phaedo_ makes one of the speakers fear that the invention of writing, by weakening memory, would lead to degeneration! [Hayek is here confusing the _Phaedo_ with the _Phaedrus_. It is in the _Phaedrus_ that Socrates recounts the following: \"The story is that in the region of Naucratis in Egypt there dwelt one of the old gods of the country, the god to whom the bird called Ibis is scared, his own name being Theuth. He it was that invented number and calculation, geometry and astronomy, not to speak of draughts and dice, and above all writing. Now the king of the whole country at that time was Thamus, who dwelt in the great city of upper Egypt which the Greeks called Egyptian Thebes, while Thamus they called Ammon. To him came Theuth, and revealed his arts, saying that they ought to be passed on to the Egyptians in general. Thamus asked what was the use of them all, and when Theuth explained, he condemned what he thought the bad points and praised what he thought the good. On each art, we are told, Thamus had plenty of views both for and against; it would take too long to give them in detail. But when it came to writing Theuth said, 'Here, O king, is a branch of learning that will make the people of Egypt wiser and improve their memories; my discovery provides a recipe for memory and wisdom.' But the king answered and said, 'O man full of arts, to one it is given to create the things of art, and to another to judge what measure of harm and of profit they have for those that shall employ them. And so it is that you, by reason of your tender regard for the writing that is your offspring, have declared the very opposite of its true effect. If men learn this, it will implant forgetfulness in their souls; they will cease to exercise memory because they rely on that which is written, calling things to remembrance no longer from within themselves, but by means of external marks. What you have discovered is a recipe not for memory, but for reminder. And it is no true wisdom you offer your disciples, but only its semblance, for by telling them of many things without teaching them you will make them seem to know much, while for the most part they know nothing, and as men filled, not with wisdom, but with the conceit of wisdom, they will be a burden to their fellows.\" (Plato, _Phaedrus_ , 274cde-275ab). The translation is that of Reginald Hackforth and appears in _The Collected Dialogues of Plato, Including the Letters_ , Edith Hamilton and Huntington Cairns, eds. (Bollingen Series; New York: Pantheon Books, 1961), p. 520.\u2014Ed.]\n\n4 If it were still possible to change an established usage, it would be desirable to confine the word \"progress\" to such deliberate advance toward a chosen goal and to speak only of the \"evolution of civilization.\"\n\n5 See Alfred Louis Kroeber, _Conf gurations of Cultural Growth_ (Berkeley: University of California Press, 1944), p. 839: \"Progress . . . is something that makes itself. We do not make it.\" [Hayek is in error in citing this work. The quotation in fact appears in Kroeber's _Anthropology_ (New York: Harcourt, Brace and Company, 1923), p. 133.\u2014Ed.]\n\n6 Cf. John Bagnell Bury, _The Idea of Progress_ , pp. 236\u201337: \"Theories of Progress are thus differentiating into two distinct types, corresponding to two radically opposed political theories and appealing to two antagonistic temperaments. The one type is that of constructive idealists and socialists, who can name all the streets and towers of 'the city of gold,' which they imagine as situated just round a promontory. The development of man is a closed system; its term is known and is within reach. The other type is that of those who, surveying the gradual ascent of man, believe that by the same interplay of forces which have conducted him so far and by a further development of the liberty which he has fought to win, he will move slowly towards conditions of increasing harmony and happiness. Here the development is indefinite: its term is unknown, and lies in the remote future. Individual liberty is the motive force, and the corresponding political theory is liberalism.\"\n\n7 See Sir Karl Raimund Popper, _The Poverty of Historicism_ (London: Routledge and Kegan Paul, 1957), and my _The Counter-Revolution of Science: Studies in the Abuse of Reason_ (Glencoe, IL: Free Press, 1952).[ _Collected Works_ edition, vol. 13.]\n\n8 It has been well put by Irving Langmuir, \"Freedom, the Opportunity to Profit from the Unexpected,\" reprinted in the [General Electric] _Research Laboratory Bulletin_ , (Fall 1956): 4. \"In research work, you cannot plan to make discoveries but you can plan work which would probably lead to discoveries.\" [These comments were originally made at a commencement address delivered before the Pratt Institute on June 1, 1956.\u2014Ed.]\n\n9 Cf. Michael Polanyi, _The Logic of Liberty: Reflections and Rejoinders_ (London: Routledge and Kegan Paul, 1951), p. 76 [Liberty Fund edition, p. 93], and the remarkable early discussion of these issues in Samuel Bailey, _Essays on the Formation and Publication of Opinions, and on Other Subjects_ (London: Printed for R. Hunter, 1821), especially the observation in the Preface ( pp. iv\u2013v): \"It seems to be a necessary condition of human science, that we should learn many useless things, in order to become acquainted with those which are of service; and as it is impossible, antecedently to experience, to know the value of our acquisitions, the only way in which mankind can secure all the advantages of knowledge is to prosecute their inquiries in every possible direction. There can be no greater impediment to the progress of science than a perpetual and anxious reference at every step to palpable utility. Assured that the general result will be beneficial, it is not wise to be too solicitous as to the immediate value of every individual effort. Besides, there is a certain completeness to be attained in every science, for which we are obliged to acquire many particulars not otherwise of any worth. Nor is it to be forgotten, that trivial and apparently useless acquisitions are often the necessary preparatives to important discoveries.\"\n\n10 Adam Smith, _The Wealth of Nations_ , vol. 1, p. 83 [Liberty Fund edition, vol. 1, p. 99]. See by way of contrast John Stuart Mill, who in 1848 ( _Principles_ , IV, vi, 2, p. 749; Liberty Fund edition, _Collected Works_ , vol. 3, p. 755) seriously contended that \"it is only in the backward countries of the world that increased production is still an important object: in those most advanced, what is economically needed is a better distribution.\" He appears to have been unaware that an attempt to cure even extreme poverty by redistribution would in his time have led to the destruction of all of what he regarded as cultured life, without achieving its object.\n\n11 Gabriele de Tarde, _Social Laws: An Outline of Sociology_ , Howard Crosby Warren, trans. (New York: Macmillan, 1907), p. 194.\n\n12 Regarding the curious conclusion reached by John Kenneth Galbraith from the fact that most of our needs are culturally determined, see my \"The Non Sequitur of the Dependence Effect,\" _The Southern Economic Journal_ , 27 (April 1961): 346\u201348.\n\n13 See David Hume, \"The Rise and Progress of the Arts and Sciences,\" _Essays: Moral, Political, and Literary_ [Essay 14], vol. 1, p. 184 [Liberty Fund edition, p. 124]: _\"That though the only proper Nursery of these noble plants be a free state, yet may they be transplanted into any government.\"_\n\n14 Cf. the two important articles in the _Times Literary Supplement_ : February 24, 1956, pp. 109\u2013 11, \"The Dynamic Society,\" and December 28, 1956, pp. 773\u201375, \"The Secular Trinity.\" The articles have also been published in pamphlet form by the _Times Literary Supplement_.\n\n15 See Kenneth Ewart Boulding, _Principles of Economic Policy_ (Englewood Cliffs, NJ: Prentice-Hall, 1958), p. 94: \"Equality . . . is a luxury of rich societies. If a poor society is to achieve anything at all it must develop a high degree of inequality\u2014the small economic surplus must be concentrated in a few hands if _any_ high-level achievements are to be made.\"\n\n16 Cf. Henry Christopher Wallich, \"Conservative Economic Policy,\" _Yale Review_ , 46 (1956): 67: \"From a dollars-and-cents point of view, it is quite obvious that over a period of years, even those who find themselves at the short end of inequality have more to gain from faster growth than from any conceivable income redistribution. A speedup in real output of only one extra percent per year will soon lift even the economically weakest into income brackets to which no amount of redistribution could promote them. . . . For the economist, economic inequality acquires a functional justification thanks to the growth concept. Its ultimate results benefit even those who at first seem to be losers.\"\n\n17 Cf. on these effects in one of the most remote parts of the world John Clark, _Hunza: Lost Kingdom of the Himalayas_ (New York: Funk and Wagnalls, 1956), p. 266: \"Contact with the West, either directly or second-hand, has reached the outermost nomad, the deepest jungle village. More than a billion people have learned that we live happier lives, perform more interesting work, and enjoy greater physical comforts than they do. Their own cultures have not given them these things, and they are determined to possess them. Most Asians desire all of our advantages with as little change as possible in their own customs.\"\n\n## FOUR\n\nFREEDOM, REASON, AND TRADITION\n\nNothing is more fertile in prodigies than the art of being free; but there is nothing more arduous than the apprenticeship of liberty. . . . Liberty . . . is generally established with difficulty in the midst of storms; it is perfected by civil discords; and its benefit cannot be appreciated until it is already old.\n\n\u2014A. de Tocqueville\n\n1. Though freedom is not a state of nature but an artifact of civilization, it did not arise from design. The institutions of freedom, like everything freedom has created, were not established because people foresaw the benefits they would bring. But, once its advantages were recognized, men began to perfect and extend the rein of freedom and, for that purpose, to inquire how a free  society worked. This development of a theory of liberty took place mainly in the eighteenth century. It began in two countries, England and France. The first of these knew liberty; the second did not.\n\nAs a result, we have had to the present day two different traditions in the theory of liberty:1 one empirical and unsystematic, the other speculative and rationalistic2\u2014the first based on an interpretation of traditions and institutions which had spontaneously grown up and were but imperfectly understood, the second aiming at the construction of a utopia, which has often been tried but never successfully. Nevertheless, it has been the rationalist, plausible, and apparently logical argument of the French tradition, with its flattering assumptions about the unlimited powers of human reason, that has progressively gained influence, while the less articulate and less explicit tradition of English freedom has been on the decline.\n\nThis distinction is obscured by the facts that what we have called the  \"French tradition\" of liberty arose largely from an attempt to interpret British institutions and that the conceptions which other countries formed of British institutions were based mainly on their description by French writers.3 The two traditions became finally confused when they merged in the liberal movement of the nineteenth century and when even leading British liberals drew as much on the French as on the British tradition.4 It was, in the end, the victory of the Benthamite Philosophical Radicals over the Whigs in England that concealed the fundamental difference which in more recent years has reappeared as the conflict between liberal democracy and \"social\" or totalitarian democracy.5\n\nThis difference was better understood a hundred years ago than it is today. In the year of the European revolutions in which the two traditions merged, the contrast between \"Anglican\" and \"Gallican\" liberty was still clearly described by an eminent German American political philosopher. \"Gallican Liberty,\" wrote Francis Lieber in 1848, \"is sought in the _government_ , and according to an Anglican point of view, it is looked for in a wrong place, where it cannot be found. Necessary consequences of the Gallican view are, that the French look for the highest degree of political civilization in _organization_ , that is, in the highest degree of interference by public power. The question whether this interference be despotism or liberty is decided solely by the fact _who_ interferes, and for the benefit of which class the interference takes place, while according to the Anglican view this interference would always be either absolutism or aristocracy, and the present dictatorship of the _ouvriers_ would appear to us an uncompromising aristocracy of the _ouvriers_.\"6\n\nSince this was written, the French tradition has everywhere progressively displaced the English.7 To disentangle the two traditions it is necessary to look at the relatively pure forms in which they appeared in the eighteenth century. What we have called the \"British tradition\" was made explicit mainly by a group of Scottish moral philosophers led by David Hume, Adam Smith, and Adam Ferguson,8 seconded by their English contemporaries Josiah Tucker, Edmund Burke, and William Paley, and drawing largely on a tradition rooted in the jurisprudence of the common law.9 Opposed to them was the tradition of the French Enlightenment, deeply imbued with Cartesian rationalism: the Encyclopedists and Rousseau, the Physiocrats and Condorcet, are their bestknown representatives. Of course, the division does not fully coincide with national boundaries. Frenchmen like Montesquieu and, later, Benjamin Constant  and, above all, Alexis de Tocqueville are probably nearer to what we have called the \"British\" than to the \"French\" tradition.10 And, in Thomas Hobbes, Britain has provided at least one of the founders of the rationalist tradition, not to speak of the whole generation of enthusiasts for the French Revolution, like Godwin, Priestley, Price, and Paine, who (like Jefferson after his stay in France11) belong entirely to it.\n\n2. Though these two groups are now commonly lumped together as the ancestors of modern liberalism, there is hardly a greater contrast imaginable than that between their respective conceptions of the evolution and functioning of a social order and the role played in it by liberty. The difference is directly traceable to the predominance of an essentially empiricist view of the world in England and a rationalist approach in France. The main contrast in the practical conclusions to which these approaches led has recently been well put, as follows: \"One finds the essence of freedom in spontaneity and the absence of coercion, the other believes it to be realized only in the pursuit and attainment of an absolute collective purpose\";12 and \"one stands for organic, slow, half conscious growth, the other for doctrinaire deliberateness; one for trial and error procedure, the other for an enforced solely valid pattern.\"13 It is the second view, as J. L. Talmon has shown in an important book from which this description is taken, that has become the origin of totalitarian democracy.\n\nThe sweeping success of the political doctrines that stem from the French tradition is probably due to their great appeal to human pride and ambition. But we must not forget that the political conclusions of the two schools derive from different conceptions of how society works. In this respect the British philosophers laid the foundations of a profound and essentially valid theory, while the rationalist school was simply and completely wrong.\n\nThose British philosophers have given us an interpretation of the growth of civilization that is still the indispensable foundation of the argument for liberty. They find the origin of institutions, not in contrivance or design, but in the survival of the successful. Their view is expressed in terms of how \"nations stumble upon establishments which are indeed the result of human action but not the execution of any human design.\"14 It stresses that what we call political order is much less the product of our ordering intelligence than is commonly imagined. As their immediate successors saw it, what Adam Smith and his contemporaries did was \"to resolve almost all that has been ascribed to positive institution into the spontaneous and irresistible development of certain obvious principles,\u2014and to show with how little contrivance or political wisdom the most complicated and apparently artificial schemes of policy might have been erected.\"15\n\nThis \"anti-rationalistic insight into historical happenings that Adam Smith shares with Hume, Adam Ferguson, and others\"16 enabled them for the first time to comprehend how institutions and morals, language and law, have evolved by a process of cumulative growth and that it is only with and within this framework that human reason has grown and can successfully operate. Their argument is directed throughout against the Cartesian conception of an independently and antecedently existing human reason that invented these institutions and against the conception that civil society was formed by some wise original legislator or an original \"social contract.\"17 The latter idea of  intelligent men coming together for deliberation about how to make the world anew is perhaps the most characteristic outcome of those design theories. It found its perfect expression when the leading theorist of the French Revolution, Abb\u00e9 Siey\u00e8s, exhorted the revolutionary assembly \"to act like men just emerging from the state of nature and coming together for the purpose of signing a Social Contract.\"18\n\nThe ancients understood the conditions of liberty better than that. Cicero quotes Cato as saying that the Roman constitution was superior to that of other states because it \"was based upon the genius, not of one man, but of many: it was founded, not in one generation, but in a long period of several centuries and many ages of men. For, said he, there never has lived a man possessed of so great a genius that nothing could escape him, nor could the combined powers of all men living at one time possibly make all the necessary provisions for the future without the aid of actual experience and the test of time.\"19 Neither republican Rome nor Athens\u2014the two free nations of the ancient world\u2014could thus serve as an example for the rationalists. For Descartes, the fountainhead of the rationalist tradition, it was indeed Sparta that provided the model; for her greatness \"was due not the pre-eminence of each of its laws in particular . . . but to the circumstance that, originated by a single individual, they all tended to a single end.\"20 And it was Sparta which became  the ideal of liberty for Rousseau as well as for Robespierre and Saint-Just and for most of the later advocates of \"social\" or totalitarian democracy.21\n\nLike the ancient, the modern British conceptions of liberty grew against the background of a comprehension, first achieved by the lawyers, of how institutions had developed. \"There are many things specially in laws and governments,\" wrote Chief Justice Hale in the seventeenth century in a critique of Hobbes, \"that mediately, remotely, and consequentially are reasonable to be approved, though the reason of the party does not presently or immediately and distinctly see its reasonableness. . . . Long experience makes more discoveries touching conveniences or inconveniences of laws than is possible for the wisest council of men at first to foresee. And that those amendments and supplements that through the various experiences of wise and knowing men have been applied to any law must needs be better suited to the convenience of laws, than the best invention of the most pregnant wits not aided by such a series and tract of experience. . . . This adds to the difficulty of a present fathoming of the reason of laws, because they are the production of long and iterated experience which, though it be commonly called the mistress of fools, yet certain it is the wisest expedient among mankind, and discovers those defects and supplies which no wit of man could either at once foresee or aptly remedy. . . . It is not necessary that the reasons of the institution should be evident unto us. It is sufficient that they are instituted laws that give a certainty to us, and it is reasonable to observe them though the particular reason of the institution appear not.\"22\n\n3. From these conceptions gradually grew a body of social theory that showed how, in the relations among men, complex and orderly and, in a very definite sense, purposive institutions might grow up which owed little to design, which were not invented but arose from the separate actions of many men who did not know what they were doing. This demonstration that something greater than man's individual mind may grow from men's fumbling efforts represented in some ways an even greater challenge to all design theories than even the later theory of biological evolution. For the first time it was shown that an evident order which was not the product of a designing human intelligence need not therefore be ascribed to the design of a higher, supernatural intelligence, but that there was a third possibility\u2014the emergence of order as the result of adaptive evolution.23\n\nSince the emphasis we shall have to place on the role that selection plays in this process of social evolution today is likely to create the impression that we are borrowing the idea from biology, it is worth stressing that it was, in fact, the other way round: there can be little doubt that it was from the theories of social evolution that Darwin and his contemporaries derived the suggestion for their theories.24 Indeed, one of those Scottish philosophers who first developed these ideas anticipated Darwin even in the biological field;25 and the later application of these conceptions by the various \"historical schools\" in law and language rendered the idea that similarity of structure might be accounted for by a common origin,26 a commonplace in the study of social  phenomena long before it was applied to biology. It is unfortunate that at a later date the social sciences, instead of building on these beginnings in their own field, re-imported some of these ideas from biology and with them  brought in such conceptions as \"natural selection,\" \"struggle for existence,\" and \"survival of the fittest,\" which are not appropriate in their field; for in social evolution, the decisive factor is not the selection of the physical and inheritable properties of the individuals but the selection by imitation of successful institutions and habits. Though this operates also through the success of individuals and groups, what emerges is not an inheritable attribute of individuals, but ideas and skills\u2014in short, the whole cultural inheritance which is passed on by learning and imitation.\n\n4. A detailed comparison of the two traditions would require a separate book; here we can merely single out a few of the crucial points on which they differ.\n\nWhile the rationalist tradition assumes that man was originally endowed with both the intellectual and the moral attributes that enabled him to fashion civilization deliberately, the evolutionists made it clear that civilization was the accumulated hard-earned result of trial and error; that it was the sum of experience, in part handed from generation to generation as explicit knowledge, but to a larger extent embodied in tools and institutions which had proved themselves superior\u2014institutions whose significance we might discover by analysis but which will also serve men's ends without men's understanding them. The Scottish theorists were very much aware how delicate this artificial structure of civilization was which rested on man's more primitive and ferocious instincts being tamed and checked by institutions that he neither had designed nor could control. They were very far from holding such na\u00efve views, later unjustly laid at the door of their liberalism, as the \"natural goodness of man,\" the existence of a \"natural harmony of interests,\" or the beneficent effects of \"natural liberty\" (even though they did sometimes use the last phrase). They knew that it required the artifices of institutions and traditions to reconcile the conflicts of interest. Their problem was how \"that the _universal_ Mover in human Nature, self-love, may receive such a Direction in this Case (as in all others) as to promote the Public Interest by those Efforts it shall make towards pursuing its own.\"27 It was not \"natural liberty\" in any literal sense, but the institutions evolved to secure \"life, liberty, and  property,\" which made those individual efforts beneficial.28 Not Locke, nor Hume, nor Smith, nor Burke, could ever have argued, as Bentham did, that \"every law is an evil for every law is an infraction of liberty.\"29 Their argument was never a complete laissez faire argument, which, as the very words show, is also part of the French rationalist tradition and in its literal sense was never defended by any of the English classical economists.30 They knew better  than most of their later critics that it was not some sort of magic but the evolution of \"well-constructed institutions,\" where the \"rules and principles of contending interests and compromised advantages\"31 would be reconciled, that had successfully channeled individual efforts to socially beneficial aims. In fact, their argument was never antistate as such, or anarchistic, which is the logical outcome of the rationalistic laissez faire doctrine; it was an argument that accounted both for the proper functions of the state and for the limits of state action.\n\nThe difference is particularly conspicuous in the respective assumptions of the two schools concerning individual human nature. The rationalistic design theories were necessarily based on the assumption of the individual man's propensity for rational action and his natural intelligence and goodness. The evolutionary theory, on the contrary, showed how certain institutional arrangements would induce man to use his intelligence to the best effect and how institutions could be framed so that bad people could do least harm.32 The antirationalist tradition is here closer to the Christian tradition of the fallibility and sinfulness of man, while the perfectionism of the rationalist is in irreconcilable conflict with it. Even such a celebrated figment as the \"economic man\" was not an original part of the British evolutionary tradition. It would be only a slight exaggeration to say that, in the view of those British philosophers,  man was by nature lazy and indolent, improvident and wasteful, and that it was only by the force of circumstances that he could be made to behave economically or would learn carefully to adjust his means to his ends. The _homo oeconomicus_ was explicitly introduced, with much else that belongs to the rationalist rather than to the evolutionary tradition, only by the younger Mill.33\n\n5. The greatest difference between the two views, however, is in their respective ideas about the role of traditions and the value of all the other products of unconscious growth proceeding throughout the ages.34 It would hardly  be unjust to say that the rationalistic approach is here opposed to almost all that is the distinct product of liberty and that gives liberty its value. Those who believe that all useful institutions are deliberate contrivances and who cannot conceive of anything serving a human purpose that has not been consciously designed are almost of necessity enemies of freedom. For them freedom means chaos.\n\nTo the empiricist evolutionary tradition, on the other hand, the value of freedom consists mainly in the opportunity it provides for the growth of the undesigned, and the beneficial functioning of a free society rests largely on the existence of such freely grown institutions. There probably never has existed a genuine belief in freedom, and there has certainly been no successful attempt to operate a free society, without a genuine reverence for grown institutions, for customs and habits and \"all those securities of liberty which arise from regulation of long prescription and ancient ways.\"35 Paradoxical as it may appear, it is probably true that a successful free society will always in a large measure be a tradition-bound society.36\n\nThis esteem for tradition and custom, of grown institutions, and of rules whose origins and rationale we do not know does not, of course, mean\u2014as Thomas Jefferson believed with a characteristic rationalist misconception\u2014 that we \"ascribe to the men of the preceding age a wisdom more than human, and . . . suppose what they did to be beyond amendment.\"37 Far from assuming that those who created the institutions were wiser than we are, the evolutionary view is based on the insight that the result of the experimentation of many generations may embody more experience than any one man possesses.\n\n6. We have already considered the various institutions and habits, tools and  methods of doing things, which have emerged from this process and constitute our inherited civilization. But we have yet to look at those rules of conduct which have grown as part of it, which are both a product and a condition of freedom. Of these conventions and customs of human intercourse, the moral rules are the most important but by no means the only significant ones. We understand one another and get along with one another, are able to act successfully on our plans, because, most of the time, members of our civilization conform to unconscious patterns of conduct, show a regularity in their actions that is not the result of commands or coercion, often not even of any conscious adherence to known rules, but of firmly established habits and traditions. The general observance of these conventions is a necessary condition of the orderliness of the world in which we live, of our being able to find our way in it, though we do not know their significance and may not even be consciously aware of their existence. In some instances it would be necessary, for the smooth running of society, to secure a similar uniformity by coercion, if such conventions or rules were not observed often enough. Coercion, then, may sometimes be avoidable only because a high degree of voluntary conformity exists, which means that voluntary conformity may be a condition of a beneficial working of freedom. It is indeed a truth, which all the great apostles of freedom outside the rationalistic school have never tired of emphasizing, that freedom has never worked without deeply ingrained moral beliefs and that coercion can be reduced to a minimum only where individuals can be expected as a rule to conform voluntarily to certain principles.38\n\nThere is an advantage in obedience to such rules not being coerced, not only because coercion as such is bad, but because it is, in fact, often desirable that rules should be observed only in most instances and that the individual should be able to transgress them when it seems to him worthwhile to incur the odium which this will cause. It is also important that the strength of the social pressure and of the force of habit which insures their observance is variable.  It is this flexibility of voluntary rules which in the field of morals makes gradual evolution and spontaneous growth possible, which allows further experience to lead to modifications and improvements. Such an evolution is possible only with rules which are neither coercive nor deliberately imposed\u2014 rules which, though observing them is regarded as merit and though they will be observed by the majority, can be broken by individuals who feel that they have strong enough reasons to brave the censure of their fellows. Unlike any deliberately imposed coercive rules, which can be changed only discontinuously and for all at the same time, rules of this kind allow for gradual and experimental change. The existence of individuals and groups simultaneously observing partially different rules provides the opportunity for the selection of the more effective ones.\n\nIt is this submission to undesigned rules and conventions whose significance and importance we largely do not understand, this reverence for the traditional, that the rationalistic type of mind finds so uncongenial, though it is indispensable for the working of a free society. It has its foundation in the insight which David Hume stressed and which is of decisive importance for the antirationalist, evolutionary tradition\u2014namely, that \"the rules of morality are not the conclusions of our reason.\"39 Like all other values, our morals are not a product but a presupposition of reason, part of the ends which the instrument of our intellect has been developed to serve. At any one stage of our evolution, the system of values into which we are born supplies the ends which our reason must serve. This givenness of the value framework implies that, although we must always strive to improve our institutions, we can never aim to remake them as a whole and that, in our efforts to improve them, we must take for granted much that we do not understand. We must always work inside a framework of both values and institutions which is not of our own making. In particular, we can never synthetically construct a new body  of moral rules or make our obedience of the known rules dependent on our comprehension of the implications of this obedience in a given instance.\n\n7. The rationalistic attitude to these problems is best seen in its views on what it calls \"superstition.\"40 I do not wish to underestimate the merit of the persistent and relentless fight of the eighteenth and nineteenth centuries against beliefs which are demonstrably false.41 But we must remember that the extension of the concept of superstition to all beliefs which are not demonstrably true lacks the same justification and may often be harmful. That we ought not to believe anything which has been shown to be false does not mean that we ought to believe only what has been demonstrated to be true. There are good reasons why any person who wants to live and act successfully in society must accept many common beliefs, though the value of these reasons may have little to do with their demonstrable truth.42 Such beliefs will also be based on some past experience but not on experience for which anyone can produce the evidence. The scientist, when asked to accept a generalization in his field, is of course entitled to ask for the evidence on which it is based. Many of the beliefs which in the past expressed the accumulated experience of the race have been disproved in this manner. This does not mean, however, that we can reach the stage where we can dispense with all beliefs for which such scientific evidence is lacking. Experience comes to man in many more forms than are commonly recognized by the professional experimenter or the seeker after explicit knowledge. We would destroy the foundations of  much successful action if we disdained to rely on ways of doing things evolved by the process of trial and error simply because the reason for their adoption has not been handed down to us. The appropriateness of our conduct is not necessarily dependent on our knowing why it is so. Such understanding is one way of making our conduct appropriate, but not the only one. A sterilized world of beliefs, purged of all elements whose value could not be positively demonstrated, would probably be not less lethal than would an equivalent state in the biological sphere.\n\nWhile this applies to all our values, it is most important in the case of moral rules of conduct. Next to language, they are perhaps the most important instance of an undesigned growth, of a set of rules which govern our lives but of which we can say neither why they are what they are nor what they do to us: we do not know what the consequences of observing them are for us as individuals and as a group. And it is against the demand for submission to such rules that the rationalistic spirit is in constant revolt. It insists on applying to them Descartes' principle which was \"to reject as absolutely false all opinions in regard to which I could suppose the least ground for doubt.\"43 The desire of the rationalist has always been for the deliberately constructed, synthetic system of morals, for the system in which, as Edmund Burke has described it, \"the practice of all moral duties, and the foundations of society, rested upon their reasons made clear and demonstrative to every individual.\"44 The rationalists of the eighteenth century, indeed, explicitly argued that, since they knew human nature, they \"could easily find the morals which suited it.\"45 They did not understand that what they called \"human nature\" is very largely the result of those moral conceptions which every individual learns with language and thinking.\n\n8. An interesting symptom of the growing influence of this rationalist conception is the increasing substitution, in all languages known to me, of the word \"social\" for the word \"moral\" or simply \"good.\" It is instructive to consider briefly the significance of this.46 When people speak of a \"social conscience\" as against mere \"conscience,\" they are presumably referring to an awareness of the particular effects of our actions on other people, to an endeavor to be guided in conduct not merely by traditional rules but by explicit consideration of the particular consequences of the action in question. They are in effect saying that our action should be guided by a full understanding of the functioning of the social process and that it should be our aim, through conscious assessment of the concrete facts of the situation, to produce a foreseeable result which they describe as the \"social good.\"\n\nThe curious thing is that this appeal to the \"social\" really involves a demand that individual intelligence, rather than rules evolved by society, should guide individual action\u2014that men should dispense with the use of what could truly be called \"social\" (in the sense of being a product of the impersonal process of society) and should rely on their individual judgment of the particular case. The preference for \"social considerations\" over the adherence to moral rules is, therefore, ultimately the result of a contempt for what really is a social phenomenon and of a belief in the superior powers of individual human reason.\n\nThe answer to these rationalistic demands is, of course, that they require knowledge which exceeds the capacity of the individual human mind and that, in the attempt to comply with them, most men would become less useful members of society than they are while they pursue their own aims within the limits set by the rules of law and morals.\n\nThe rationalist argument here overlooks the point that, quite generally, the reliance on abstract rules is a device we have learned to use because our reason is insufficient to master the full detail of complex reality.47 This is as true when we deliberately formulate an abstract rule for our individual guidance  as when we submit to the common rules of action which have been evolved by a social process.\n\nWe all know that, in the pursuit of our individual aims, we are not likely to be successful unless we lay down for ourselves some general rules to which we will adhere without reexamining their justification in every particular instance. In ordering our day, in doing disagreeable but necessary tasks at once, in refraining from certain stimulants, or in suppressing certain impulses, we frequently find it necessary to make such practices an unconscious habit, because we know that without this the rational grounds which make such behavior desirable would not be sufficiently effective to balance temporary desires and to make us do what we should wish to do from a long-term point of view. Though it sounds paradoxical to say that in order to make ourselves act rationally we often find it necessary to be guided by habit rather than reflection, or to say that to prevent ourselves from making the wrong decision we must deliberately reduce the range of choice before us, we all know that this is often necessary in practice if we are to achieve our long-range aims.\n\nThe same considerations apply even more where our conduct will directly affect not ourselves but others and where our primary concern, therefore, is to adjust our actions to the actions and expectations of others so that we avoid doing them unnecessary harm. Here it is unlikely that any individual would succeed in rationally constructing rules which would be more effective for their purpose than those which have been gradually evolved; and, even if he did, they could not really serve their purpose unless they were observed by all. We have thus no choice but to submit to rules whose rationale we often do not know, and to do so whether or not we can see that anything important depends on their being observed in the particular instance. The rules of morals are instrumental in the sense that they assist mainly in the achievement of other human values; however, since we only rarely can know what depends on their being followed in the particular instance, to observe them must be regarded as a value in itself, a sort of intermediate end which we must pursue without questioning its justification in the particular case.\n\n9. These considerations, of course, do not prove that all the sets of moral beliefs which have grown up in a society will be beneficial. Just as a group may owe its rise to the morals which its members obey, and their values in consequence be ultimately imitated by the whole nation which the successful group has come to lead, so may a group or nation destroy itself by the moral beliefs to which it adheres. Only the eventual results can show whether the ideals which guide a group are beneficial or destructive. The fact that a society has come to regard the teaching of certain men as the embodiment of goodness is no proof that it might not be the society's undoing if their precepts were generally followed. It may well be that a nation may destroy itself by following  the teaching of what it regards as its best men, perhaps saintly figures unquestionably guided by the most unselfish ideals. There would be little danger of this in a society whose members were still free to choose their way of practical life, because in such a society such tendencies would be self-corrective: only the groups guided by \"impractical\" ideals would decline, and others, less moral by current standards, would take their place. But this will happen only in a free society in which such ideals are not enforced on all. Where all are made to serve the same ideals and where dissenters are not allowed to follow different ones, the rules can be proved inexpedient only by the decline of the whole nation guided by them.\n\nThe important question that arises here is whether the agreement of a majority on a moral rule is sufficient justification for enforcing it on a dissenting minority or whether this power ought not also to be limited by more general rules\u2014in other words, whether ordinary legislation should be limited by general principles just as the moral rules of individual conduct preclude certain kinds of action, however good may be their purpose. There is as much need of moral rules in political as in individual action, and the consequences of successive collective decisions as well as those of individual decisions will be beneficial only if they are all in conformity with common principles.\n\nSuch moral rules for collective action are developed only with difficulty and very slowly. But this should be taken as an indication of their preciousness. The most important among the few principles of this kind that we have developed is individual freedom, which it is most appropriate to regard as a moral principle of political action. Like all moral principles, it demands that it be accepted as a value in itself, as a principle that must be respected without our asking whether the consequences in the particular instance will be beneficial. We shall not achieve the results we want if we do not accept it as a creed or presumption so strong that no considerations of expediency can be allowed to limit it.\n\nThe argument for liberty, in the last resort, is indeed an argument for principles and against expediency in collective action,48 which, as we shall see, is equivalent to saying that only the judge and not the administrator may order coercion. When one of the intellectual leaders of nineteenth-century liber  alism, Benjamin Constant, described liberalism as the _syst\u00e8me de principes_ ,49 he pointed to the heart of the matter. Not only is liberty a system under which all government action is guided by principles, but it is an ideal that will not be preserved unless it is itself accepted as an overriding principle governing all particular acts of legislation. Where no such fundamental rule is stubbornly adhered to as an ultimate ideal about which there must be no compromise for the sake of material advantages\u2014as an ideal which, even though it may have to be temporarily infringed during a passing emergency, must form the basis of all permanent arrangements\u2014freedom is almost certain to be destroyed by piecemeal encroachments. For in each particular instance it will be possible to promise concrete and tangible advantages as the result of a curtailment of freedom, while the benefits sacrificed will in their nature always be unknown and uncertain. If freedom were not treated as the supreme principle, the fact that the promises which a free society has to offer can always be only chances and not certainties, only opportunities and not definite gifts to particular individuals, would inevitably prove a fatal weakness and lead to its slow erosion.50\n\n10. The reader will probably wonder by now what role there remains to be played by reason in the ordering of social affairs, if a policy of liberty demands so much refraining from deliberate control, so much acceptance of the undirected and spontaneously grown. The first answer is that, if it has become necessary to seek appropriate limits to the uses of reason here, to find these limits is itself a most important and difficult exercise of reason. More-over, if our stress here has been necessarily on those limits, we have certainly not meant to imply thereby that reason has no important positive task. Reason undoubtedly is man's most precious possession. Our argument is intended to show merely that it is not all-powerful and that the belief that it can become its own master and control its own development may yet destroy it. What we have attempted is a defense of reason, against its abuse by those who do not understand the conditions of its effective functioning and continuous growth. It is an appeal to men to see that we must use our reason intelligently and that, in order to do so, we must preserve that indispensable matrix of the uncontrolled  and non-rational which is the only environment wherein reason can grow and operate effectively.\n\nThe antirationalistic position here taken must not be confounded with irrationalism or any appeal to mysticism.51 What is advocated here is not an abdication of reason but a rational examination of the field where reason is appropriately put in control. Part of this argument is that such an intelligent use of reason does not mean the use of deliberate reason in the maximum possible number of occasions. In opposition to the na\u00efve rationalism which treats our present reason as an absolute, we must continue the efforts which David Hume commenced when he \"turned against the enlightenment its own weapons\" and undertook \"to whittle down the claims of reason by the use of rational analysis.\"52\n\nThe first condition for such an intelligent use of reason in the ordering of human affairs is that we learn to understand what role it does in fact play and can play in the working of any society based on the co-operation of many separate minds. This means that, before we can try to remold society intelligently, we must understand its functioning; we must realize that, even when we believe that we understand it, we may be mistaken. What we must learn to understand is that human civilization has a life of its own, that all our efforts to improve things must operate within a working whole which we cannot entirely control, and the operation of whose forces we can hope merely to facilitate and assist so far as we understand them. Our attitude ought to be similar to that of the physician toward a living organism: like him, we have to deal with a self-maintaining whole which is kept going by forces which we cannot replace and which we must therefore use in all we try to achieve. What can be done to improve it must be done by working with these forces rather than against them.53 In all our endeavor at improvement we must always work  inside this given whole, aim at piecemeal, rather than total, construction,54 and use at each stage the historical material at hand and improve details step by step rather than attempt to redesign the whole.\n\nNone of these conclusions are arguments against the use of reason, but only arguments against such uses as require any exclusive and coercive powers of government; not arguments against experimentation, but arguments against all exclusive, monopolistic power to experiment in a particular field\u2014power which brooks no alternative and which lays a claim to the possession of superior wisdom\u2014and against the consequent preclusion of solutions better than the ones to which those in power have committed themselves.\n\n* * *\n\nThe quotation at the head of the chapter is taken from Tocqueville, _Democracy in America_ , vol. 1, chap. 14, pp. 246\u201347. [The French text, in full, reads: \"On ne saurait trop le dire: il n'est rien de plus f\u00e9cond en merveilles que l'art d'\u00eatre libre; mais il n'y a rien de plus dur que l'apprentissage de la libert\u00e9. Il n'en est pas de m\u00eame du despotisme. Le despotisme se pr\u00e9sente souvent comme le r\u00e9parateur de tous les maux soufferts; il est l'appui du bon droit, le soutien des opprim\u00e9s et le fondateur de l'ordre. Les peuples s'endorment au sein de la prosp\u00e9rit\u00e9 momentan\u00e9e qu'il fait na\u00eetre; et lorsqu'ils se r\u00e9veillent, ils sont mis\u00e9rables. La libert\u00e9, au contraire, na\u00eet d'ordinaire au milieu des orages, elle s'\u00e9tablit p\u00e9niblement parmi les discordes civiles et ce n'est que quand elle est d\u00e9j\u00e0 vielle qu'on peut conna\u00eetre ses bienfaits\" (bk. 1, pt. 2, chap. 6). \"De la d\u00e9mocratie en Am\u00e9rique,\" in _\u0152uvres_ , Andr\u00e9 Jardin, ed. Biblioth\u00e8que de la Pl\u00e9iade (2 vols.; Paris: \u00c9ditions Gallimard, 1992), vol. 2, p. 275.\u2014Ed.] Cf. also vol. 2, chap. 2, p. 96: \"The advantages that freedom brings are shown only by the lapse of time, and it is always easy to mistake the cause in which they originate.\" [\"Les biens que la libert\u00e9 procure ne se montrent qu'\u00e0 la longue, et il est toujours facile m\u00e9conna\u00eetre la cause qui les fait na\u00eetre\" (bk. 2, sec, 2, chap. 1, pp. 609\u201310). \"De la d\u00e9mocratie en Am\u00e9rique,\" in _\u0152uvres_ , vol. 2, pp. 609\u201310.\u2014Ed.] An earlier and slightly longer version of this chapter has appeared as \"Freedom, Reason, and Tradition\" in _Ethics_ , 68 (1958): 229\u201345. See also David Hume, \"The Rise and Progress of the Arts and Sciences,\" _Essays_ [Essay 14], vol. 1, p. 185 [Liberty Fund edition, p. 124]: \"To balance a large state or society , whether monarchical or republican, on general laws, is a work of so great difficulty, that no human genius, however comprehensive, is able, by the mere dint of reason and refection, to effect it. The judgments of many must unite in this work: Experience must guide their labour. Time must bring it to perfection: and the feeling of inconveniencies must correct the mistakes, which they inevitably fall into, in their first trials and experiments\" and \"law, the source of all security and happiness, arises late in any government, and is the slow product of order and liberty.\" See also Cicero, _De re publica_ , ii.1.2 (n. 19 below).\n\n1 Tocqueville remarks somewhere: \"Du dix-huiti\u00e8me si\u00e8cle et de la r\u00e9volution, \u00e9taient sortis deux fleuves: le premier conduisait les hommes aux institutions libres, tandis que le second les menant au pouvoir absolu.\" [The exact quotation reads: \"Du dix-huiti\u00e8me si\u00e8cle et de la r\u00e9volution, comme d'une source commune, \u00e9taient sortis deux fleuves: le premier conduisait les hommes aux institutions libres, tandis que le second les menait au pouvoir absolu.\" (\"From the eighteenth century and the Revolution, as from a common source, two streams issued. The first led men to free institutions, the second to absolute power.\") The quotation appears in the \"Discours de M. De Tocqueville Prononc\u00e9 dan la S\u00e9ance Publique du 21 avril 1842, en venant prendre s\u00e9ance \u00e0 la place de M. le comte de Cessac,\" _\u0152uvres_ , vol. 1, pp. 1208\u201309.\u2014Ed.] Cf. the observation by Sir Thomas Erskine May, _Democracy in Europe: A History_ (London: Longmans, Green, and Co., 1877), vol. 2, p. 334: \"The history of the one [France], in modern times, is the history of Democracy, not of liberty: the history of the other [England] is the history of liberty, not of Democracy.\" See also Guido de Ruggiero, _The History of European Liberalism_ , Robin George Collingwood, trans. (Oxford: Oxford University Press, 1927), esp. pp. 12, 71, and 81. [Ruggiero at one point notes that: \"(English) Liberalism was now confronted by the new Liberalism of France; genuinely new, because instead of basing itself upon the privileged liberties of the Middle Ages, it arose from their ashes. It was far more akin in spirit to the absolute monarchy which had already begun to destroy the old feudal world and had given to its subjects the feeling of equality. The new Liberalism, like the monarchy, was egalitarian; but its egalitarianism was inspired and ennobled by a broader rationalistic consciousness attributing to all men one identical spiritual and human value. But the love of equality which gave its peculiar tone to the new freedom was so overwhelming that it ended by overthrowing and crushing it\" ( pp. 81\u201382). \u2014Ed.] On the absence of a truly liberal tradition in France see Emile Faguet, _Le Liberalisme_ (Paris: Soci\u00e9t\u00e9 fran\u00e7aise d'imprimerie et de librairie, 1902), esp. p. 307, and Heinrich von Treitschke, _Die Freiheit_ (1861) (Leipzig: Insel B\u00fccherei, 1912), p . 12: \"Da\u00df die Franzosen trotz aller Begeisterung f\u00fcr die Freiheit doch immer nur die Gleichheit gekannt haben, nie die F reiheit.\" [\"The French, despite all the excitement regarding liberty, have known only equality, never liberty.\"\u2014Ed.]\n\n2 \"Rationalism\" and \"rationalistic\" will be used here throughout in the sense defined by Bernard Groethuysen in \"Rationalism,\" _Encyclopedia of the Social Sciences_ , vol. 13, p. 113, as a tendency \"to regulate individual and social life in accordance with principles of reason and to eliminate as far as possible or to relegate to the background everything irrational.\" Cf. also Michael Oakeshott, \"Rationalism in Politics,\" _Cambridge Journal_ , 1 (1947): 81\u201398.\n\n3 The author of the most detailed monograph on one of the first French economic theorists, Pierre de Boisguillebert (Hazel van Dyke Roberts, _Boisguilbert: Economist of the Reign of Louis XIV_ [New York: Columbia University Press, 1935] has noted: \"His positive theory may be said in a very real sense to have been simply a rationale of what he thought to be the English w ay of life, a way of life he would have the French adopt\" (p. 327n).\n\n4 See Elie Hal\u00e9vy, _The Growth of Philosophic Radicalism_ (London: Faber and Gwyer, 1928), pp. 17\u201318. See also Gerhard Ritter, _Vom sittlichen Problem der Macht_ (Bern: A. Francke AG. Verlag, 1948), p. 132.\n\n5 Cf. Jacob Lieb Talmon, _The Origins of Totalitarian Democracy_ (London: Secker and Warburg, 1952). Though Talmon does not identify \"social\" with \"totalitarian\" democracy, I cannot but agree with Hans Kelsen (\"The Foundations of Democracy,\" _Ethics_ , 66, part 2 [1955]: 95, n. 14) that \"the antagonism which Talmon describes as tension between liberal and totalitarian democracy is in truth the antagonism between liberalism and socialism and not between two types of democracy.\"\n\n6 Francis Lieber, \"Anglican and Gallican Liberty,\" originally published in a South Carolina newspaper in 1849 and reprinted in _The Miscellaneous Writings of Francis Lieber_ (Philadelphia: J. B. Lippincott, 1881), vol. 2, pp. 382\u201383. See also vol. 2, p. 385: \"The fact that Gallican liberty expects everything from organization, while Anglican liberty inclines to development, explains why we see in France so little improvement and expansion of institutions; but when improvement is attempted, a total abolition of the preceding state of things\u2014a beginning _ab ovo_ \u2014a rediscussion of the first elementary principles.\" See also J\u00f3zsef E\u00f6tv\u00f6s , _Der Einf u\u00df der herrschenden Ideen des 19. Jahrhunderts auf den Staat_ (2 vols.; Leipzig: F. A. Brockhaus, 1854), particularly vol. 1, p. 38, and James Mill, _Considerations on Representative Government_ (London: Parker, Son, and Bourn, 1861), pp. 82\u201384.\n\n7 One of the reasons why the French notion of liberty was so attractive was offered by Friedrich Naumann in his previously cited treatise ( _Das Ideal der Freiheit_ , p. 16\u201317): He wrote: \"Die L\u00e4nder, wo der Sieg der Freiheit, das hei\u00dft in diesem F alle der gleichen Rechte (!) am v ollkommensten ist, sind vom Standpunkt liberaler Romantik die langweiligsten, denn in ihnen gibt es keine Freiheitsk\u00e4mpfer mehr, h\u00f6chstens noch einen gewissen pharis\u00e4ischen Stolz denen gegen\u00fcber, die noch nicht so weit sind, und ein gewisses erhabenes Mitleid f\u00fcr die Opfer zur\u00fcckgebliebener Zust\u00e4nde. So etwa erscheint bisweilen der englische Liberalismus.\" [\"Those countries in which the victory of liberty, that is to say, equality of rights, was most complete, were, from the point of view of liberal romanticism, the most boring, since there were no longer any freedom fighters, while what remained, for the most part, was a certain Pharisaic pride towards those nations that had not yet advanced to this point and a species of pity for those people subject to these backward circumstances. This is, more or less, how English Liberalism appears.\"\u2014Ed.]\n\nOne of the most amusing episodes of World War I was the dispute, which extended beyond the battlefield, between French and German intellectuals over the question of which country had discovered the secret of social organization. See Jean Labadi\u00e9, ed., _L'Allemagne: A-t-elle le secret de l'organisation?_ (Paris: Biblioth\u00e8que de l'Opinion, 1916). It would have been diff cult for an Englishman to make this claim for his country. In this conte xt, see the discussion about the role of \"organization\" during the Napoleonic era.\n\n8 An adequate account of this philosophy of growth which provided the intellectual foundations for a policy of freedom has yet to be written and cannot be attempted here. For a fuller appreciation of the Scottish-English school and its differences from the French rationalist tradition see Duncan Forbes, \"Scientific Whiggism: Adam Smith and John Millar,\" _Cambridge Journal_ , 7(1954): 643\u201370, and my own lecture, _Individualism: True and False_ (Dublin: Hodges Figgis, 1946), reprinted in _Individualism and Economic Order_ (Chicago: University of Chicago Press, 1948), pp. 1\u201332 (the latter particularly for the role played by Bernard Mandeville in this tradition which I am passing over here). [Now see _Studies on the Abuse and Decline of Reason_ , pp. 46\u201374.\u2014Ed.] For further reference see the earlier version of this article, \"Freedom, Reason, and Tradition,\" in _Ethics_ , pp. 229\u201345.\n\n9 See especially the work of Sir Mathew Hale, \"Sir Mathew Hale's Criticism on Hobbes' Dialogue on the Common Law,\" reprinted as an appendix to William Searle Holdsworth, _A History of English Law_ (London: Methuen, 1924), vol. 5, pp. 504\u20135. [While Holdsworth spells Hale's Christian name with one \"t,\" Hale's biographer, Gilbert Burnet, shows it as \"Matthew.\"\u2014Ed.]\n\n10 Montesquieu, Constant, and Tocqueville were often regarded as Anglo-maniacs by their compatriots. Constant was partly educated in Scotland, and Tocqueville could say of himself that \"So many of my thoughts and feelings are shared by the English that England has turned into a second native land of the mind for me.\" [The Simpson translation renders Tocqueville's French thus: \"So many of my opinions and feelings are English, that England is to me almost a second country intellectually\" (Tocqueville to Nassau Senior, Versailles, 27 July 1851, _Correspondence and Conversations of Alexis de Tocqueville and Naussau William Senior, from 1834 to 1959_ , Mary Charlotte Mair Simpson, ed. and trans. [2 vols.; 2nd ed.; London: Henry S. King and Co., 1872], vol. 1, pp. 1, 264). Tocqueville's original reads: \"J'ai d'ailleurs tant de sentiments et d'id\u00e9es qui me sont communs avec les Anglais, que l'Angleterre est devenue pour moi comme une seconde patrie intellectuelle.\" Alexis de Tocqueville, _\u0152uvres compl\u00e8tes_. Tome 6: _Correspondance Anglaise_ , 3 vols.; part 2: _Correspondance et Conversations d'Alexis de Tocqueville et Nassau William Senior_ (Paris: Gallimard, 1991), p. 132.\u2014Ed.] A fuller list of eminent French thinkers who belonged more to the evolutionary \"British\" than to the rationalistic \"French\" tradition would have to include the young Turgot and Etienne Bonnot de Condillac.\n\n11 On Jefferson's shift from the \"British\" to the \"French\" tradition as a result of his stay in France see the important work by Otto Vossler, _Die amerikanischen Revolutionsideale in ihrem Verh\u00e4ltnis zu den europ\u00e4ischen: untersucht an Thomas Je_ _ff_ _erson_ (Munich: Oldenbourg, 1929).\n\n12 Talmon, _Origins of Totalitarian Democracy_ , p. 2.\n\n13 Talmon, _Origins of Totalitarian Democracy_ , p. 71. Cf. also Lewis Mumford, _Faith for Living_ (New York: Harcourt, Brace and Co., 1940), pp. 64\u201366, where a contrast is drawn between \"ideal liberalism\" and \"pragmatic liberalism,\" and William Montgomery McGovern and David S. Collier, _Radicals and Conservatives_ (Chicago: H. Regnery Co., 1958), pp. 9\u201320, where \"conservative liberals\" and \"radical liberals\" are distinguished. See also Carl Menger, _Untersuchungen_ , p. 207, regarding \"einseitigen rationalistischen Liberalismus,\" [\"one-sided rationalistic Liberalism,\"] of which he wrongfully accuses Adam Smith.\n\n14 Ferguson, _An Essay on the History of Civil Society_ , p. 187.\n\n15 [Francis Jeffrey], \"Craig's Life of Millar,\" _Edinburgh Review_ , 9 (1806): 84. Frederic William Maitland much later spoke similarly somewhere of \"the stumbling forward in our empirical fashion, blundering into wisdom.\" [The quotation appears in Maitland's _Collected Papers of Frederic William Maitland: Dowling Professor of the Laws of England_ (3 vols.; Cambridge: Cambridge University Press, 1911), in the section devoted to a discussion of \"English Law Under Norman and Angevin,\" which forms a section of his essay on \"The Outlines of English Legal History, 560\u2013 1600,\" vol. 2, p. 439.\u2014Ed.]\n\n16 Forbes, \"Scientific Whiggism: Adam Smith and John Millar,\" p. 654. The importance of the Scottish moral philosophers as forebears of cultural anthropology has been handsomely acknowledged by Edward Evan Evans-Pritchard, _Social Anthropology_ (London: Cohen and West, 1951), pp. 23\u201325.\n\n17 Ludwig von Mises, _Socialism_ , Jacques Kahane, trans. (new ed.; New Haven: Yale University Press, 1951), p. 43 [Liberty Fund edition, p. 33], writes with reference to the social contract: \"Rationalism could find no other possible explanation after it had disposed of the old belief which traced social institutions back to divine sources or at least to the enlightenment which came to man through divine inspiration. Because it led to present conditions, people regarded the development of social life as absolutely purposeful and rational; how then could this development have come about except through conscious choice in recognition of the fact that it was purposeful and rational?\"\n\n18 Quoted by Talmon, _Origins of Totalitarian Democracy_ , p. 73.\n\n19 M. Tullius Cicero, _De re publica_ , ii.1.2; cf. also ii.21.37 [wherein Cicero notes \"Now we have further proof of the accuracy of Cato's statement that the foundation of our State was the work neither of one period nor of one man; for it is quite clear that every kind contributed many good and useful institutions.\"\u2014Ed.] I am indebted to Prof. Bruno Leoni's lectures, now published as _Freedom and the Law_ ([Princeton, NJ: D. van Nostrand, 1961], p. 89), for calling this source to my attention. Neratius, a later Roman jurist quoted in the _Corpus Iuris Civilis_ [Samuel Parsons Scott, _The Civil Law, including the Twelve Tables, the Institutes of Gaius, the Rules of Ulpian, the Opinions of Paulus, the Enactments of Justinian, and the Constitutions of Leo_ (17 vols. in 7; Cincinnati: Central Trust Co., 1932), vol. 2, p. 224], even went so far as to exhort lawyers: \"Rationes eorum quae constituuntur inquiri non oportet, alioquin multa ex his quae certa sunt subvertuntur\" (\"We must avoid inquiring about the rationale of our institutions, since otherwise many that are certain would be overturned\"). [The translation is Hayek's. The quotation appears in the _Digest_ , 1.3.2. The original source is noted as Neratius, _Parchments_ , book VI.\u2014Ed.] Although in this respect the Greeks were somewhat more rationalistic, a similar conception of the growth of law is by no means absent. See, e.g., the Attic orator Antiphon, _On the Choreutes_ , par. 2 (in _Minor Attic Orators_ , Kenneth John. Maidment, ed. [Loeb Classical Library, 2 vols.; Cambridge, MA: Harvard University Press, 1941)], vol. 1, p. 247), where he speaks of laws having \"the distinction of being the oldest in this country, . . . and that is the surest token of good laws, as time and experience show mankind what is imperfect.\"\n\n20 Ren\u00e9 Descartes, _A Discourse on the Method of Rightly Conducting One's Reason and of Seeking Truth in the Sciences_ (Everyman ed.; London: Dent, 1912), pt. 2, p. 11. [The French reads: \"Je crois que si Sparte a \u00e9t\u00e9 autrefois tr\u00e8s florissante, ce n'a pas \u00e9t\u00e9 \u00e0 cause de la bont\u00e9 de chacune de ses loisen particulier . . . mais \u00e0 cause que, n'ayant \u00e9t\u00e9 invent\u00e9es que par un seul, elles tendoient toutes \u00e0 m\u00eame fin.\"\u2014Ed.]\n\n21 Cf. Talmon, _Origins of Totalitarian Democracy_ , p. 142. [Talmon notes that \"throughout the centuries of uninterrupted tyranny and crime, history knows only of one brief spell of liberty in a tiny corner of the earth\u2014Sparta: 'brille comme un \u00e9clair dans les t\u00e9n\u00e8bres immenses.' This is the key to the understanding of Robespierre and Saint-Just: Sparta as the ideal of liberty.\" \u2014Ed.] On the influence of the Spartan ideal on Greek philosophy and especially on Plato and Aristotle see Fran\u00e7ois Ollier, _Le Mirage spartiate: \u00c9tude sur l'id\u00e9alisation de Sparte dans l'antiquit\u00e9 grecque, de l'origine, jusqu'aux Cyniques_ (Paris: E. de Boccard, 1933), and Sir Karl Raimund Popper, _The Open Society and Its Enemies_ (London: G. Routledge and Co., 1945).\n\n22 \"Sir Mathew Hale's \"Criticism on Hobbes Dialogue on the Common Law,\" in Holdsworth, _A History of English Law_ , vol. 5, pp. 504\u20135 (the spelling has been modernized). Holdsworth rightly points out the similarity of some of these arguments to those of Edmund Burke. [With reference to this passage in Hale, Holdsworth (vol. 5, p. 504, n. 1) quotes Burke's _Reflections on the Revolution in France_ (London: Printed for J. Dodsley, in Pall-Mall, 1790, p. 90) (Liberty Fund edition, _Selected Works_ , vol. 2, p. 153), to the effect that: \"The science of government being therefore so practical in itself, and intended for such practical purposes, a matter which requires experience, and even more experience, than any person can gain in his whole life, however, sagacious and observing he may be, it is with infinite caution that any man ought to venture upon pulling down an edifice which has answered in any tolerable degree for ages the common purposes of society, or on building it up again, without having models and patterns of approved utility before his eyes.\"\u2014Ed.] They are, of course, in effect an attempt to elaborate ideas of Sir Edward Coke (whom Hobbes had criticized), especially his famous conception of the \"artificial reason.\" His _Seventh Report_ ( _The_ _Reports of Edward Coke, Knt.: In thirteen parts_ , John Henry Thomas and John Farquhar Fraser, eds. [13 parts in 6 vols.; London: J. Butterworth and Son, 1826] pt. 7, vol. 4, p. 6). [The quotation appears in \"Calvin's Case, or the Case of the Postnati\" (1608) Trinity Term, 6 James I.\u2014Ed.] Coke explains as follows: \"Our days upon earth are but a shadow in respect of the old ancient days and times past, wherein the laws have been by the wisdom of the most excellent men, in many succession of ages, by long and continual experience (the trial of light and truth) fined and refined, which no one man, (being of so short a time) albeit he had the wisdom of all the men in the world, in any one age could ever have effected or attained unto\" [Liberty Fund edition, vol. 1, p. 173.]. Cf. also the legal proverb: \"Per varios usus experientia legem fecit.\" [The phrase carries the following sense: Experience made law through diverse customs.\u2014Ed.] See J. G. A. Pocock, _The Ancient Constitution and the Feudal Law_ (New York: Cambridge University Press, 1957), as well as Sir John Davies, _Les Reports des Cases en La y_ (commonly cited as _Irish Reports_ [London, 1612]), Preface: \"As it is said of every Art and Science which is brought to perfection, _Per varios usus artem experiential fecit_ , so may it properly be said of our Law _Per varios usus Legem experiential fecit_. Long experience, and many trials of what w as best for the common good, did make the Common Law.\"\n\n23 A thorough examination of these problems from Bernard Mandeville's paradox to its first cogent expression by David Hume in his \"Dialogues Concerning Natural Religion,\" (see _Treatise of Human Nature_ , vol. 2, pp. 380\u2013468) has yet to be under taken. The best discussion of the character of this process of social growth known to me is still Carl Menger, _Untersuchungen_ , bk. 3 [Das organische Verst\u00e4ndnis der Socialerscheinungen] and app. 8 [\u00dcber den \"organischen\" Ursprung des Rechtes und das exacte Verst\u00e4ndnis desselben], esp. pp. 163\u201365, 203\u20134n, and 208. Cf. also the discussion in Alexander Macbeath, _Experiments in Living: A Study of the Nature and Foundation of Ethics or Morals in the Light of Recent Work in Social Anthropology_ (London: Macmillan, 1952), p. 120 and 120, n. 1, of \"the principle laid down by Frazer [Sir James George Frazer, _Psyche's Task: A Discourse Concerning the Influence of Superstition on the Growth of Institutions_ (London: Macmillan, 1909), p. 4] and endorsed by Malinowski and other anthropologists, that no institution will continue to survive unless it performs some useful function\" and the remark added in a footnote: \"But the function which it serves at a given time may not be that for the sake of which it was originally established\"; and the following passage, in which Lord Acton indicates how he would have continued his brief sketches of freedom in antiquity and Christianity (\"Freedom in Christianity,\" _History of Freedom_ , p. 58 [Liberty Fund edition, _Essays in the History of Liberty_ , vol. 1, p. 56]): \"I should have wished . . . to relate by whom and in what connection, the true law of the formation of free States was recognised, and how that discovery, closely akin to those which, under the names of development, evolution, and continuity, have given a new and deeper method to other sciences, solved the ancient problem between stability and change, and determined the authority of tradition on the progress of thought; how that theory, which Sir James Mackintosh expressed by saying that Constitutions are not made, but grow; the theory that custom and the national qualities of the governed, and not the will of the government, are the makers of the law.\"\n\n24 I am not referring here to Darwin's acknowledged indebtedness to the population theories of Thomas Malthus (and, through him, of Richard Cantillon) but to the general atmosphere of an evolutionary philosophy which governed thought on social matters in the nineteenth century. Though this influence has occasionally been recognized (see, e.g., Henry Fairfield Osborn, _From the Greeks to Darwin: An Outline of the Development of the Evolution Idea_ [New York: McMillan and Co., 1894], p. 87), it has never been systematically studied. I believe that such a study would show that most of the conceptual apparatus which Darwin employed lay ready at hand for him to use. One of the men through whom Scottish evolutionary thought reached Darwin was probably the Scottish geologist James Hutton.\n\n25 See Arthur Osborne Lovejoy, \"Monboddo and Rousseau,\" _Modern Philology_ , 30 (1933): 275\u2013 96, reprinted in _Essays in the History of Ideas_ (Baltimore: Johns Hopkins University Press, 1948), pp. 38\u201361.\n\n26 It is perhaps significant that the first clearly to see this in the field of linguistics, Sir William Jones, was a lawyer by training and a prominent Whig by persuasion. Cf. his celebrated statement in the \"Third Anniversary Discourse\" delivered February 2, 1786, in _Asiatick Researches_ , vol. 1, p. 422, and [as \"The Third Anniversary Discourse, on the Hindus\"] reprinted in _The Works of Sir William Jones_ (13 vols.; London: Printed for John Stockdale, Piccadilly, and John Walker, Paternoster-Row, 1807), vol. 2, p. 34: \"The _Sanscrit_ language, whatever be its antiquity, is of a wonderful structure; more perfect than the Greek, more copious than the Latin, and more exquisitely refined than either, yet bearing to both of them a stronger affinity, both in the roots of verbs and in the forms of grammar, than could possibly have been produced by accident: so strong indeed, that no philologer could examine them all three, without believing them to have sprung from some common source, which, perhaps, no longer exists.\" The connection between speculation about language and that about political institutions is best shown by one of the most complete, though somewhat late, statements of the Whig doctrine by Dugald Stewart, _Lectures on Political Economy_ (delivered 1809\u201310), printed in _The Collected Works of Dugald Stewart_ (11 vols.; Edinburgh: T. Constable, 1854\u201360), vol. 9, pp. 422\u201324, and quoted at length in a note to the earlier version of this chapter in _Ethics_ , 68 (1958): 243. It is of special importance because of Stewart's influence on the last group of Whigs, the _Edinburgh Review_ circle. Is it an accident that in Germany her greatest philosopher of freedom, Wilhelm von Humboldt, was also one of her greatest theorists of language?\n\n[Hayek's footnote in _Ethics_ , quoting Stewart's _Lectures_ , reads: \"The English government (it is said) has been the gradual offspring of circumstances and events, and its different parts arose at different times; some of them from acts of the legislature prompted by emergencies, and some of them from long established customs or usages, of which it is not always possible to trace the origin, so that no part of it is sanctioned by an authority paramount to that which gives force to every other law by which we are governed. It is pretended, therefore, that there are no fundamental or essential principles in our government, which fix a limit to the possibility of legislative encroachment, and to which an appeal could be made, if a particular law should appear to be hostile to the rights and liberties of the people. But surely the conclusion in this argument does not follow from the premises. For do we not every day speak of laws being _constitutional_ or _unconstitutional;_ and do not these words convey to men of plain understanding a very distinct and intelligible meaning, a meaning which no person can pretend to misapprehend, who is not disposed to cavil about expressions?\n\n\"It appears to me, that what we call the _constitution_ differs from our other laws, not in its _origin_ , but in the _importance of the subject to which it refers, and in the systematical connexion of its di_ _ff_ _erent principles_. It may, I think, be defined to be that form of government, and that mode of administering it, which is agreeable to the _general_ spirit and tendency of our established laws and usages.\n\n\"According to this view of the subject, I apprehend that the constitution, _taken as a whole_ , ought to modify every new institution which is introduced, so that it may accord with its general _spirit_ ; although every part of this constitution taken _separately_ , arose from no higher authority than the common acts of our present legislature.\n\n\"To illustrate this proposition it may be proper to remark, that although the Constitution was the gradual result of circumstances which may be regarded as accidental and irregular, yet that the very mode of its formation necessarily produced a certain consistence and analogy in its different parts, so as to give to the whole a sort of systematic appearance. For unless every new institution which was successively introduced has possessed a certain reference or affinity to the laws and usages existing before, it could not possibly have been permanent in its operation. Wherever a Constitution has existed for ages, and men have enjoyed a tranquility under it, it is a proof that its great and fundamental principles are all animated by the same congenial spirit. In such a constitution, when any law contrary to the spirit of the rest is occasionally introduced, it soon falls into desuetude and oblivion; while those which accord in their general character and tendency, acquire additional stability from the influence of time and from the mutual support which they lend to each other. Of such a law we may say with propriety that it is _unconstitutional_ , not because we dispute the authority from which it proceeds, but because it is contrary to the spirit and analogy of the laws which we have been accustomed to obey.\n\n\"Something similar to this obtains with respect to languages. _These_ , as well as governments, are the gradual result of time and experience, and not of philosophical speculation: yet every language, in process of time, acquires a great deal of systematical beauty. When a new word, or a new combination of words, is introduced, it takes its raise from the same origin with every other expression which the language contains; the desire of an individual to communicate his own thoughts or feelings to others. But this consideration alone is not sufficient to justify the use of it. Before it is allowed by good writers or speakers to incorporate itself with those words which have the sanction of time in their favour, it must be shewn that it is not disagreeable to the general analogy of the language, otherwise it is soon laid aside as an innovation, revolting, anomalous, and _ungrammatical_. It is much in the same manner that we come to apply the epithet _unconstitutional_ to a law.\n\n\"The zeal, therefore, which genuine patriots have always shewn for the maintenance of the Constitution, so far from being unreasonable, will be most strongly felt by the prudent and intelligent, because such men know that political wisdom is much more the result of experience than of speculation: and that when a Constitution has been matured by such slow steps as ours has been, in consequence of the struggle of able and enlightened individuals, jealous of their liberties, and anxious to preserve them, it may be considered as the result of the accumulated experience and wisdom of ages; possessing _on that very account_ the strongest of all possible recommendations, an experimental proof of its excellence, of its fitness to perpetuate itself, and to promote the happiness of those who live under it.\"\u2014Ed.]\n\n27 Josiah Tucker, \"The Elements of Commerce and Theory of Taxes\" (1755) in _Josiah Tucker: A Selection from his Economic and Political Writings_ , Robert Livingston Schuyler, ed. (New York: Columbia University Press, 1931), p. 92.\n\n28 That for Adam Smith in particular it was certainly not \"natural liberty\" in any literal sense on which the beneficial working of the economic system depended, but liberty under the law, is clearly expressed in _Wealth of Nations_ , bk. 4, chap. 5, vol. 2, pp. 42\u201343 [Liberty Fund edition, vol. 1, p. 540]: \"That security which the laws in Great Britain give to every man that he shall enjoy the fruits of his own labour, is alone sufficient to make any country flourish, notwithstanding these and twenty other absurd regulations of commerce: and this security was perfected by the revolution, much about the same time that the bounty was established. The natural effort of every individual to better his own condition, when suffered to exert itself with freedom and security, is so powerful a principle, that it is alone, and without any assistance, not only capable of carrying on the society to wealth and prosperity, but of surmounting a hundred impertinent obstructions with which the folly of human laws too often incumbers its operations.\" Cf. Colin Arthur Cooke, \"Adam Smith and Jurisprudence,\" _Law Quarterly Review_ , 51 (1935): 328: \"The theory of political economy that emerges in the _Wealth of Nations_ can be seen to be a consistent theory of law and legislation . . . the famous passage about the invisible hand rises up as the essence of Adam Smith's view of law\"; and also the interesting discussion in Joseph Cropsey, _Polity and Economy: An Interpretation of the Principles of Adam Smith_ (The Hague: M. Nijhoff, 1957). It is of some interest that Smith's general argument about the \"invisible hand\" \"which leads man to promote an end which was no part of his intention\" already appears in Montesquieu, _Spirit of the Laws_ , bk. 3. chap. 7, vol. 1, p. 25 [French edition: vol. 2, p. 257], where he says that \"thus each individual advances the public good, while he only thinks of promoting his own interest.\" [\"Et il se trouve que chacun va au bien commun, croyant aller \u00e0 ses int\u00e9r\u00eats particuliers.\"\u2014Ed.] See also David Hume, \"That Politics May Be Reduced to a Science ,\" _Essays_ [Essay 3], vol. 1, p. 99 [Liberty Fund edition, pp. 15\u201316]: \"[But a] republican and free government would be an obvious absurdity, if the particular checks, and controuls, provided by the constitution, had really no infuence, and made it not the interest, even of bad men, to act for the public good\"; and Immanuel Kant, _Zum ewigen Frieden: Ein philosophischer Entwurf_ , in _Werke_ , vol. 6: _Schriften zur Anthropologie, Geschichsphilosophie, Politik und P\u00e4dagogik_ , Wilhelm Weischedel, ed. (Wiesbaden: Insel-Verlag, 1956\u201364), pp. 223\u201324: \"Aber nun kommt die Natur dem verehrten, aber zur Praxis ohnm\u00e4chtigen allgemeinen, in der Vernunft gegr\u00fcndeten Willen, und zwar gerade durch jene selbsts\u00fcchtigen Neigungen, zu H\u00fclfe, so, da\u00df es nur auf eine gute Organisation des Staats ankommt (die allerdings im Verm\u00f6gen der Menschen ist), jener ihre Kr\u00e4fte so gegen einander zu richten, da\u00df der Erfolg f\u00fcr die Vernunft so ausf\u00e4llt, als wenn beide gar nicht da w\u00e4ren, und so der Mensch, wenn gleich nicht ein moralisch-guter Mensch, dennoch ein guter B\u00fcrger zu sein gezwungen wird.\" [\"But precisely with these inclinations nature comes to the aid of the general will established on reason, which is revered even though impotent in practice. Thus it is only a question of a good organization of the state (which does lie in man's power), whereby the powers of each selfish inclination are so arranged in opposition that one moderates or destroys the ruinous effect of the other. The consequence for reason is the same as if none of them existed, and man is forced to be a good citizen even if not a morally good person.\"\u2014Ed.]\n\n29 Jeremy Bentham, _Theory of Legislation_ (5th ed.; London: Tr\u00fcbner, 1887), p. 48.\n\n30 See David Hutchison MacGregor, _Economic Thought and Policy_ (Oxford: Oxford University Press, 1949), chap. 3, \"The Laissez-Faire Doctrine,\" pp. 54\u201389, and Lionel Robbins, _The Theory of Economic Policy in English Classical Political Economy_ (London: Macmillan, 1952), pp. 42\u201346.\n\n31 Edmund Burke, _Thoughts and Details on Scarcity_ , in _Works_ , vol. 7, p. 398 [Liberty Fund edition: _Miscellaneous Writings_ , p. 73]. [The full quotation reads: \"No slave were ever so beneficial to the master as a freeman that deals with him on an equal footing by convention, formed on the rules and principles of contending interests and compromised advantages.\"\u2014Ed.]\n\n32 Cf., e.g., the contrast between David Hume, _Essays_ , bk. 1, pp. 117\u201318, Essay 6: \"On the Independency of Parliament\" [Liberty Fund edition, p. 42]: \"Political writers have established it as a maxim, that, in contriving any system of government, and fixing the several checks and controuls of the constitution, every man ought to be supposed a _knave_ , and to have no other end, in all his actions, than private interest.\" (The reference is presumably to Niccol\u00f2 Machiavelli, _Discorsi_ , bk. 1, chap. 3 [ _Discourses on the First Ten Books of Titus Livy_ , in _The Works of the Famous Nicolas Machiavel, Citizen and Secretary of Florence_ (3rd ed., carefully corrected; London: Printed for Thomas Wood for A. Churchill, 1720), p. 272], where Machiavelli notes that the lawgiver must assume for his purposes that all men are bad. [Machiavelli writes: \"It is necessary to whoever will establish a government and prescribe laws to it to presuppose all men naturally bad.\" The Italian reads: \"\u00c9 necessario a chi dispone una republica, ed ordina leggi in quella, presupporre tutti gli uomini rei.\"\u2014Ed.]\n\nCf. Richard Price, _Two Tracts on Civil Liberty, the War with America, and the Debts and Finances of the Kingdom_ (2 vols. in 1; London: T. Caddell, 1778), p. 11: \"Every man's will, if perfectly free from restraint, would carry him invariably to rectitude and virtue.\" [This work contains Price's _Observations on the Nature of Civil Liberty, the Principles of Government, and the Justice and Policy of the War with America. 8th ed., with Corrections and Additions, 1778. Additional Observations on the Nature and Value of Civil Liberty, and the War with America_ , bound together and paginated separately. The quotation appears in _Additional Observations_ , p. 11.\u2014Ed.]\n\nSee also my _Individualism and Economic Order_ (Chicago: University of Chicago Press, 1948), pp. 11\u201312. [ _Collected Works_ edition, vol. 13, p. 57.]\n\n33 See John Stuart Mill, \"On the Definition of Political Economy; and on the Method of Investigation Proper to It,\" [Essay 5] in _Essays on Some Unsettled Questions of Political Economy_ (London: J. W. Parker, 1844), pp. 120\u201364 [Liberty Fund edition, _Collected Works_ , vol. 4, pp. 309\u201339].\n\n[Joseph Persky (\"The Ethology of Homo Economicus\"), writing in the _Journal of Economic Perspectives_ , 9 (1995): 221\u201331, maintains that \"while John Stuart Mill is generally identified as the creator of economic man, he never actually used this designation in his own writings. But the term did emerge in reaction to Mill's work. In its first appearances in the late nineteenth century, 'economic man' carried a pejorative connotation reflecting the widespread hostility of the historical school toward Mill's theoretical abstractions. Economic man also raised the indignation of Victorian moralists shocked at the postulation of such blatant selfishness.\n\n\"For example, the earliest explicit naming of economic man that I have identified is in John Kells Ingram's _A History of Political Economy_ (1888). Ingram, an advocate of a broad sociology in the tradition of Auguste Comte, took considerable pains to disparage John Stuart Mill's political economy, which 'dealt not with real but with imaginary men\u2014\"economic men\" . . . conceived as simply \"money-making animals\"' (p. 218). Two years later, John Neville Keynes (1890) picked up (and singularized) the phrase in his much more extensive methodological treatment. Keynes' efforts, though considerably less hostile than Ingram's, still painted 'an \"economic man\" whose activities are determined solely by the desire for wealth' and ascribed the origins of this tightly drawn abstraction to John Stuart Mill\" ( p. 222). And, \"The first use of the Latin 'homo \u0153conomicus' I turned up is in Vilfredo Pareto's _Manual_ (1906, pp. 12\u201314).\"\u2014Ed.]\n\n34 Ernest Renan, in an important essay [\"M. De Sacy et l'\u00e9cole lib\u00e9rale\"] on the principles and tendencies of the liberal school, first published in 1858 and later included in his _Essais de morale et de critique_ (now in _\u0152uvres compl\u00e8tes_ , Henriette Psichari, ed. [10 vols.; Paris: Calmann-L\u00e9vy, 1947], vol. 2, pp. 45\u201346) observes: \"Le lib\u00e9ralisme, ayant la pr\u00e9tention de se fonder uniquement sur les principes de la raison, croit d'ordinaire n'avoir pas besoin de tradition. L\u00e0 est son erreur . . . L'erreur de l'\u00e9cole lib\u00e9rale est d'avoir trop cru qu'il est facile de cr\u00e9er la libert\u00e9 par la r\u00e9flexion, et de n'avoir pas vu qu'un \u00e9tablissement n'est solide que quand il a des racines historiques . . . Elle ne vit pas que de tous ses efforts ne pouvait sortir qu'une bonne administration, mais jamais la libert\u00e9, puisque la libert\u00e9 r\u00e9sulte d'un droit ant\u00e9rieur et sup\u00e9rieur \u00e0 celui de l'\u00c9tat, et non d'une d\u00e9claration improvis\u00e9e ou d'un raisonnement philosophique plus ou moins bien d\u00e9duit.\" [\"Liberalism, claiming as it does to be uniquely founded on the principles of reason, ordinarily holds that tradition need play no role. This is its error. . . . The liberal school errs in thoroughly embracing the view that it is easy to create a liberal regime solely through reflection, not seeing that its establishment has no solidity without historical roots. . . . Liberalism failed to see that all such efforts can only result in an efficient administration, but not in liberty, since liberty is the result of law that is both anterior to and superior to the State and not a determination of the moment or the result of philosophical reasoning that has been more or less deduced.\"\u2014Ed.] Cf. also the observation by Ronald Buchanan McCallum in the Introduction to his edition of John Stuart Mill, _On Liberty and Considerations on Representative Government_ (Oxford: B. Blackwe11, 1946), p. xv: \"While Mill admits the great power of custom, and within limits its uses, he is prepared to criticise all those rules which depend upon it and are not defended by reason. He remarks, 'People are accustomed to believe, and have been encouraged in the belief by some who aspire to the character of philosophers, that their feelings on subjects of this nature, are better than reasons and render reasons unnecessary.' This was that position which Mill, as a utilitarian rationalist, was bound never to accept. It was the 'sympathy-antipathy' principle which Bentham considered was the basis of all systems of other than the rationalist approach. Mill's primary contention as a political thinker is that all these unreasoning assumptions should be weighed and considered by the reflective and balanced judgment of thinking men.\"\n\n35 Joseph Butler, _The Works of Joseph Butler_ , William Ewart Gladstone, ed. (2 vols.; Oxford: Clarendon Press, 1896), vol. 2, pp. 278. [The quotation in fact appears in Butler's \"Sermon Preached Before the House of Lords, in the Abbey-Church of Westminster, on Friday, January 30, 1740\u201341. Being the Day Appointed to be Observed as the Day of Martyrdom of King Charles I\" (Sermon III; sec. 15).\u2014Ed.]\n\n36 Even Professor Herbert Butterfield, who understands this better than most people, finds it \"one of the paradoxes of history\" that \"the name of England has come to be so closely associated with liberty on the one hand and tradition on the other hand\" ( _Liberty in the Modern World_ [Toronto: Ryerson Press, 1952], p. 21).\n\n37 Thomas Jefferson, _The Works of Thomas Je_ _ff_ _erson_ , Paul Leicester Ford, ed. (12 vols.; New York: G. P. Putnam's Sons, 1904), vol. 12, p. 11.\n\n38 See, e.g., Edmund Burke, _A Letter to a Member of the National Assembly_ , in _Works_ , vol. 6, p. 64 [Liberty Fund edition, _Further Reflections on the Revolution in France_ , p. 69]: \"Men are qualified for civil liberty, in exact proportion to their disposition to put moral chains upon their appetites; in proportion as their love of justice is above their rapacity; in proportion as their soundness and sobriety of understanding is above their vanity and presumption; in proportion as they are more disposed to listen to the council of the wise and good, in preference to the flattery of knaves.\"\n\nAlso James Madison in the debates during the Virginia Ratifying Convention, June 20, 1788 (in _The Debates in the Several State Conventions on the Adoption of the Federal Constitution_ , Jonathan Elliot, ed. [2nd ed.; 5 vols.; Philadelphia: J. B. Lippincott, 1863\u201391], vol. 2, p. 537): \"To suppose that any form of government will secure liberty or happiness without any virtue in the people, is a chimerical idea.\" And Tocqueville, _Democracy in America_ , vol. 1, p. 12: \"Liberty cannot be established without morality, nor morality without faith\" [\"On ne peut \u00e9tablir le r\u00e8gne de la libert\u00e9 sans celui des m\u0153urs, ni fonder les m\u0153urs sans les croyances.\" \"Introduction,\" _\u0152uvres_ , vol. 2, p. 13.\u2014Ed.]; also vol. 2, p. 235: \"No free communities ever existed without morals.\" [\"On n'y a jamais eu de soci\u00e9t\u00e9s sans m\u0153urs.\" _\u0152uvres_ , vol. 2, p. 712.\u2014Ed.]\n\n39 Hume, _Treatise of Human Nature_ , bk. 3, pt. 1, sec. 1 (vol. 2, p. 235), the paragraph headed \"Moral Distinctions Not Deriv'd from Reason\": \"The rules of morality, therefore, are not conclusions of our reason.\" The same idea is already implied in the scholastic maxim, \"Ratio est instrumentum non est judex.\" [\"Reason is the tool, not the judge.\"\u2014Ed.] Concerning Hume's evolutionary view of morals, I am glad to be able to quote a statement I should have been reluctant to make, for fear of reading more into Hume than is there, but which comes from an author who, I believe, does not look at Hume's work from my particular angle. In _The Structure of Freedom_ (Stanford, CA: Stanford University Press, 1958), p. 33, Christian Bay writes: \"Standards of morality and justice are what Hume calls 'artifacts'; they are neither divinely ordained, nor an integral part of original human nature, nor revealed by pure reason. They are an outcome of the practical experience of mankind, and the sole consideration in the slow test of time is the utility each moral rule can demonstrate toward promoting human welfare. Hume may be called a precursor of Darwin in the sphere of ethics. In effect, he proclaimed a doctrine of the survival of the fittest among human conventions\u2014fittest not in terms of good teeth but in terms of maximum social utility.\"\n\n40 Cf. Harry Burrows Acton, \"Prejudice,\" _Revue internationale de philosophie_ , 21(1952): 323\u2013 36, with the interesting demonstration of the similarity of the views of Hume and Burke; also the same author's address, \"Tradition and Some Other Forms of Order,\" [The Presidential Address], _Proceedings of the Aristotelian Society_ , n.s., 53 (1953): 1, especially the remark at the beginning that \"liberals and collectivists join together against tradition when there is some 'superstition' to be attacked.\" See also Lionel Robbins, _The Theory of Economic Policy_ (London: Macmillan, 1952), p. 196n.\n\n41 Perhaps even this is putting it too strongly. A hypothesis may well be demonstrably false and still, if some new conclusions follow from it which prove to be true, be better than no hypothesis at all. Such tentative, though partly erroneous, answers to important questions may be of the greatest significance for practical purposes, though the scientist dislikes them because they are apt to impede progress.\n\n42 Cf. Edward Sapir, _Selected Writings in Language, Culture, and Personality_ , David Goodman Mandelbaum, ed. (Berkeley: University of California Press, 1949), pp. 558\u201359: \"It is sometimes necessary to become conscious of the forms of social behavior in order to bring about a more serviceable adaptation to changed conditions, but I believe it can be laid down as a principle of far-reaching application that in the normal business of life it is useless and even mischievous for the individual to carry the conscious analysis of his cultural patterns around with him. That should be left to the student whose business it is to understand these patterns. A healthy unconsciousness of the forms of socialized behavior to which we are subject is as necessary to society as is the mind's ignorance, or better unawareness, of the workings of the viscera to the health of the body.\" See also p. 26.\n\n43 Descartes, _Discourse on Method_ , pt. 4, p. 26. [The French reads: \"que je rejetasse comme absolument faux tout ce en quoi je pourrois imaginer le moindre doute.\"\u2014Ed.]\n\n44 Edmund Burke, _A Vindication of Natural Society_ , Preface, in _Works_ , vol. 1, p. 7 [Liberty Fund edition, p. 9].\n\n45 Paul Henri Thiry, Baron d'Holbach, _Syst\u00e8me social, Ou Principes naturels de la morale et de la politique_ (3 vols.; London [ Rouen], 1773), vol. 1, p. 55, quoted in Talmon, _Origins of Totaltarian Democracy_ , p. 270. [The full French statement reads: \"En partant de l'homme lui-m\u00eame on trouvera facilement la morale qui lui convient. Cette morale sera vraie, si l'on voit l'homme tel qu'il est . . . principes . . . evidents . . . capables d'\u00eatre aussi rigoureusement demonstr\u00e9s que l'arithm\u00e9tique ou la g\u00e9ometrie.\"\u2014Ed.] Similarly na\u00efve statements are not difficult to find in the writings of contemporary psychologists. Burrus Frederic Skinner, e.g., in _Walden Two_ (New York: Macmillan, 1948), p. 85, makes the hero of his utopia argue: \"Why not experiment? The questions are simple enough. What's the best behavior for the individual so far as the group is concerned? And how can the individual be induced to behave in that way? Why not explore these questions in a scientific spirit?\n\n\"We could do just that in Walden Two. We had already worked out a code of conduct\u2014subject, of course, to experimental modification. The code would keep things running smoothly if everybody lived up to it. Our job was to see that everybody did.\"\n\n46 Cf. my article \"Was ist und was heisst 'sozial'?\" in _Masse und Demokratie_ , Albert Hunold, ed. (Zurich: Erlenbach-Zurich: E. Rentsch, 1957), pp. 71\u201384, reprinted as \"What is 'Social'?\u2014 What Does it Mean?\" in _Studies in Philosophy, Politics, and Economics_ (Chicago: University of Chicago Press, 1967), pp. 237\u201347. [An unauthorized translation of this essay appeared in _Freedom and Serfdom: An Anthology of Western Thought_ , Albert Hunold, ed., Richard Henry Stevens, trans. (Dordrecht: D. Reidel, 1961), pp. 107\u201318. Hayek writes of this translation that \"in parts (it) gravely misrepresented the meaning of the original.\"\u2014Ed.] See also the attempted defense of the concept in Hermann Jahrreiss, _Freiheit und Sozialstaat_ (K\u00f6lner Universit\u00e4tsreden. No. 17; Krefeld, 1957), now reprinted in the same author's _Mensch und Staat. Rechtsphilosophische, staatsrechtliche und v\u00f6lkerrechtliche Grundfragen in unserer Zeit_ (Cologne and Berlin: Carl Heymann, 1957), pp. 69\u201388.\n\n47 Cf. Tocqueville's emphasis on the fact that \"general ideas are no proof of the strength, but rather of the insufficiency of the human intellect\" [\"Les id\u00e9es g\u00e9n\u00e9rales n'attestent point la force de l'intelligence humaine, mais plut\u00f4t son insuffisance.\"\u2014Ed.], _Democracy in America_ , vol. 2, p. 13; _\u0152uvres_ , vol. 2, p. 523.\n\n48 It is often questioned today whether consistency is a virtue in social action. The desire for consistency is even sometimes represented as a rationalistic prejudice, and the judging of each case on its individual merits as the truly experimental or empiricist procedure. The truth is the exact opposite. The desire for consistency springs from the recognition of the inadequacy of our reason explicitly to comprehend all the implications of the individual case, while the supposedly pragmatic procedure is based on the claim that we can properly evaluate all the implications without reliance on those principles which tell us which particular facts we ought to take into account.\n\n49 Benjamin Constant, \"De l'arbitraire,\" in _\u0152uvres politiques de Benjamin Constant_ , Charles Louandre, ed. (Paris: Charpentier et Cie., 1874), pp. 91\u201392.\n\n50 See Dicey, _Law and Public Opinion_ , pp. 257\u201358: \"The benef cial effect of State inter vention, especially in the form of legislation, is direct, immediate, and, so to speak visible, whilst its evil effects are gradual and indirect, and lie out of sight. . . . Hence the majority of mankind must almost of necessity look with undue favour upon governmental intervention. This natural bias can be counteracted only by the existence in a given society . . . of a presumption or prejudice in favour of individual liberty, that is, _laissez faire_ \" [Liberty Fund edition, p. 182]; and Carl Menger, _Untersuchungen_ , p. 208, where he speaks of \"Pragmatismus, der gegen die Absicht seiner Vertreter unausweichbar zum Sozialismus f\u00fchrt\" [\"pragmatism, that contrary to the intention of its representatives inexorably leads to socialism.\"\u2014Ed.]; also see my essay, \"Die Ursachen der st\u00e4ndigen Gef\u00e4hrdung der Freiheit,\" in _Ordo_ , 12 (1961): 103\u20139.\n\n51 It must be admitted that after the tradition discussed was handed on by Burke to the French reactionaries and German romanticists, it was turned from an antirationalist position into an irrationalist faith and that much of it survived almost only in this form. But this abuse, for which Burke is partly responsible, should not be allowed to discredit what is valuable in the tradition, nor should it cause us to forget \"how thorough a Whig [Burke] was to the last,\" as Frederic William Maitland ( _Collected Papers_ , vol. 1, p. 67) has rightly emphasized.\n\n52 Sheldon Sanford Wolin, \"Hume and Conservatism,\" _American Political Science Review_ , 48 (1954): 1001. [The quotation in full reads: \"Hume was something more than the Enlightenment incarnate, for his significance is that he turned against the Enlightenment its own weapons. And herein lies his importance as a conservative thinker. His starting-point is to be found in _A Treatise of Human Nature_ (1739\u201340) which bears the subtitle 'An attempt to introduce the experimental method of reasoning into moral subjects.' The first book illustrates Hume's tactic: to whittle down the claims of reason by the use of rational analysis.\"\u2014Ed.] Cf. also Ernest Campbell Mossner, _Life of David Hume_ (Oxford: Clarendon Press, 1954), p. 125: \"In the Age of Reason, Hume set himself apart as a systematic anti-rationalist.\"\n\n53 See Dietrich Schindler, _Recht, Staat, V\u00f6lkergemeinschaft: ausgew\u00e4hlte Schriften und Fragmente aus dem Nachlass_ (Zurich: Schulthess and Co., 1948), p. 86: \"Der Gesetzgeber gleicht eher dem G\u00e4r tner, der mit dem vorhandenen Erdreich und mit den Wachstumsbedingungen der Pf anzen zu rechnen hat, als dem Maler, der seiner Phantasie freies Spiel l\u00e4\u00dft.\" [\"The legislator more closely resembles a gardener, who has to assess the soil and the conditions necessary for his plants' growth, than a painter who gives free rein to his imagination.\"\u2014Ed.]\n\n54 Sir Karl Raimund Popper, _The Open Society and Its Enemies_ [1945 edition], passim.\n\n## FIVE\n\nRESPONSIBILITY AND FREEDOM\n\nIt is doubtful that democracy could survive in a society organized on the principle of therapy rather than judgment, error rather than sin. If men are free and equal, they must be judged rather than hospitalized.\n\n\u2014F. D. Wormuth\n\n1. Liberty not only means that the individual has both the opportunity and the burden of choice; it also means that he must bear the consequences of his actions and will receive praise or blame for them. Liberty and responsibility are inseparable. A free society will not function or maintain itself unless its members regard it as right that each individual occupy the position that results from his action and accept it as due to his own action. Though it can offer to the individual only chances and though the outcome of his efforts will depend on innumerable accidents, it forcefully directs his attention to those circumstances that he can control as if they were the only ones that mattered. Since the individual is to be given the opportunity to make use of circumstances that may be known only to him and since, as a rule, nobody else can know whether he has made the best use of them or not, the presumption is that the outcome of his actions is determined by them, unless the contrary is quite obvious.\n\nThis belief in individual responsibility, which has always been strong when people firmly believed in individual freedom, has markedly declined, together with the esteem for freedom. Responsibility has become an unpopular concept, a word that experienced speakers or writers avoid because of the obvious boredom or animosity with which it is received by a generation that dislikes all moralizing. It often evokes the outright hostility of men who have been taught that it is nothing but circumstances over which they have no control that has determined their position in life or even their actions. This denial of responsibility is, however, commonly due to a fear of responsibility, a fear that necessarily becomes also a fear of freedom.1 It is doubtless because the  opportunity to build one's own life also means an unceasing task, a discipline that man must impose upon himself if he is to achieve his aims, that many people are afraid of liberty.\n\n2. The concurrent decline in esteem for individual liberty and individual responsibility is in a great measure the result of an erroneous interpretation of the lessons of science. The older views were closely connected with a belief in the \"freedom of the will,\" a conception that never did have a precise meaning but later seemed to have been deprived of foundation by modern science. The increasing belief that all natural phenomena are uniquely determined by antecedent events or subject to recognizable laws and that man himself should be seen as part of nature led to the conclusion that man's actions and the working of his mind must also be regarded as necessarily determined by external circumstances. The conception of universal determinism that dominated nineteenth-century science2 was thus applied to the conduct of human beings, and this seemed to eliminate the spontaneity of human action. It had, of course, to be admitted that there was no more than a general presumption that human actions were also subject to natural law and that we actually did  not know how they were determined by particular circumstances except, perhaps, in the rarest of instances. But the admission that the working of man's mind must be believed, at least in principle, to obey uniform laws appeared to eliminate the role of an individual personality which is essential to the conception of freedom and responsibility.\n\nThe intellectual history of the last few generations gives us any number of instances of how this determinist picture of the world has shaken the foundation of the moral and political belief in freedom. And many scientifically educated people today would probably agree with the scientist who, when writing for the general public, admitted that freedom \"is a very troublesome concept for the scientist to discuss, partly because he is not convinced that, in the last analysis, there is such a thing.\"3 More recently, it is true, physicists have, it would seem with some relief, abandoned the thesis of universal determinism. It is doubtful, however, whether the newer conception of a merely statistical regularity of the world in any way affects the puzzle about the freedom of the will. For it would seem that the difficulties that people have had concerning the meaning of voluntary action and responsibility do not at all spring from any necessary consequence of the belief that human action is causally determined but are the result of an intellectual muddle, of drawing conclusions which do not follow from the premises.\n\nIt appears that the assertion that the will is free has as little meaning as its denial and that the whole issue is a phantom problem,4 a dispute about  words in which the contestants have not made clear what an affirmative or a negative answer would imply. Surely, those who deny the freedom of the will deprive the word \"free\" of all its ordinary meaning, which describes action according to one's own will instead of another's; in order not to make a meaningless statement, they should offer some other definition, which, indeed, they never do.5 Furthermore, the whole suggestion that \"free\" in any relevant or meaningful sense precludes the idea that action is necessarily determined by some factors proves on examination to be entirely unfounded.\n\nThe confusion becomes obvious when we examine the conclusion generally drawn by the two parties from their respective positions. The determinists usually argue that, because men's actions are completely determined by natural causes, there could be no justification for holding them responsible or praising or blaming their actions. The voluntarists, on the other hand, contend that, because there exists in man some agent standing outside the chain of cause and effect, this agent is the bearer of responsibility and the legitimate object of praise and blame. Now there can be little doubt that, so far as these practical conclusions are concerned, the voluntarists are more nearly right, while the determinists are merely confused. The peculiar fact about the dispute is, however, that in neither case do the conclusions follow from the alleged premises. As has often been shown, the conception of responsibility rests, in fact, on a determinist view,6 while only the construction of a metaphysical \"self \" that stands outside the whole chain of cause and effect and  therefore could be treated as uninfluenced by praise or blame could justify man's exemption from responsibility.\n\n3. It would be possible, of course, to construct, as illustration of an alleged determinist position, a bogey of an automaton that invariably responded to the events in its environment in the same predictable manner. This would correspond, however, to no position that has ever been seriously maintained even by the most extreme opponents of the \"freedom of the will.\" Their contention is that the conduct of a person at any moment, his response to any set of external circumstances, will be determined by the joint effects of his inherited constitution and all his accumulated experience, with each new experience being interpreted in the light of earlier individual experience\u2014a cumulative process which in each instance produces a unique and distinct personality. This personality operates as a sort of filter through which external events produce conduct which can be predicted with certainty only in exceptional circumstances. What the determinist position asserts is that those accumulated effects of heredity and past experience constitute the whole of the individual personality, that there is no other \"self \" or \"I\" whose disposition cannot be affected by external or material influences. This means that all those factors whose influence is sometimes inconsistently denied by those who deny the \"freedom of the will,\" such as reasoning or argument, persuasion or censure, or the expectation of praise or blame, are really among the most important factors determining the personality and through it the particular action of the individual. It is just because there is no separate \"self \" that stands outside the chain of causation that there is also no \"self \" that we could not reasonably try to influence by reward or punishment.7\n\nThat we can, in fact, often influence people's conduct by education and example, rational persuasion, approval or disapproval, has probably never been seriously denied. The only question that can be legitimately asked is, therefore, to what extent particular persons in given circumstances are likely to be influenced in the desired direction by the knowledge that an action will raise or lower them in the esteem of their fellows or that they can expect reward or punishment for it.\n\nStrictly speaking, it is nonsense to say, as is so often said, that \"it is not a man's fault that he is as he is,\" for the aim of assigning responsibility is to  make him different from what he is or might be. If we say that a person is responsible for the consequences of an action, this is not a statement of fact or an assertion about causation. The statement would, of course, not be justifiable if nothing he \"might\" have done or omitted could have altered the result. But when we use words like \"might\" or \"could\" in this connection, we do not mean that at the moment of his decision something in him acted otherwise than was the necessary effect of causal laws in the given circumstances. Rather, the statement that a person is responsible for what he does aims at making his actions different from what they would be if he did not believe it to be true. We assign responsibility to a man, not in order to say that as he was he might have acted differently, but in order to make him different. If I have caused harm to somebody by negligence or forgetfulness, \"which I could not help\" in the circumstances, this does not exempt me from responsibility but should impress upon me more strongly than before the necessity of keeping the possibility of such consequences in mind.8\n\nThe only questions that can be legitimately raised, therefore, are whether the person upon whom we place responsibility for a particular action or its consequences is the kind of person who is accessible to normal motives (that is, whether he is what we call a responsible person) and whether in the given circumstances such a person can be expected to be influenced by the considerations and beliefs we want to impress upon him. As in most such problems, our ignorance of the particular circumstances will regularly be such that we will merely know that the expectation that they will be held responsible is likely, on the whole, to influence men in certain positions in a desirable direction. Our problem is generally not whether certain mental factors were operative on the occasion of a particular action but how certain considerations might be made as effective as possible in guiding action. This requires that the individual be praised or blamed, whether or not the expectation of this would in fact have made any difference to the action. Of the effect in the particular instance we may never be sure, but we believe that, in general, the knowledge that he will be held responsible will influence a person's conduct in a desirable direction. In this sense the assigning of responsibility does not involve the assertion of a fact. It is rather of the nature of a convention intended to make people observe certain rules. Whether a particular convention of this kind is effective may always be a debatable question. We shall rarely know more than that experience suggests that it is or is not, on the whole, effective.\n\nResponsibility has become primarily a legal concept, because the law requires clear tests to decide when a person's actions create an obligation or make him liable to punishment. But it is, of course, no less a moral concept, a conception which underlies our view of a person's moral duties. In fact, its scope extends considerably beyond what we commonly consider as moral. Our whole attitude toward the working of our social order, our approval or disapproval of the manner in which it determines the relative position of different individuals, is closely tied up with our views about responsibility. The significance of the concept thus extends far beyond the sphere of coercion, and its greatest importance perhaps lies in its role in guiding man's free decisions. A free society probably demands more than any other that people be guided in their action by a sense of responsibility which extends beyond the duties exacted by the law and that general opinion approve of the individuals' being held responsible for both the success and the failure of their endeavors. When men are allowed to act as they see fit, they must also be held responsible for the results of their efforts.\n\n4. The justification for assigning responsibility is thus the presumed effect of this practice on future action; it aims at teaching people what they ought to consider in comparable future situations. Though we leave people to decide for themselves because they are, as a rule, in the best position to know the circumstances surrounding their action, we are also concerned that conditions should permit them to use their knowledge to the best effect. If we allow men freedom because we presume them to be reasonable beings, we also must make it worth their while to act as reasonable beings by letting them bear the consequences of their decisions. This does not mean that a man will always be assumed to be the best judge of his interests; it means merely that we can never be sure who knows them better than he and that we wish to make full use of the capacities of all those who may have something to contribute to the common effort of making our environment serve human purposes.\n\nThe assigning of responsibility thus presupposes the capacity on men's part for rational action, and it aims at making them act more rationally than they would otherwise. It presupposes a certain minimum capacity in them for learning and foresight, for being guided by a knowledge of the consequences of their action. It is no objection to argue that reason in fact plays only a small part in determining human action, since the aim is to make that little go as far as possible. Rationality, in this connection, can mean no more than some degree of coherence and consistency in a person's action, some lasting influence of knowledge or insight which, once acquired, will affect his action at a later date and in different circumstances.\n\nThe complementarity of liberty and responsibility means that the argument for liberty can apply only to those who can be held responsible. It cannot apply to infants, idiots, or the insane. It presupposes that a person is capable  of learning from experience and of guiding his actions by knowledge thus acquired; it is invalid for those who have not yet learned enough or are incapable of learning. A person whose actions are fully determined by the same unchangeable impulses uncontrolled by knowledge of the consequences or a genuine split personality, a schizophrenic, could in this sense not be held responsible, because his knowledge that he will be held responsible could not alter his actions. The same would apply to persons suffering from really uncontrollable urges, kleptomaniacs or dipsomaniacs, whom experience has proved not to be responsive to normal motives. But so long as we have reason to believe that a man's awareness that he will be held responsible is likely to influence his actions, it is necessary to treat him as responsible, whether or not in the particular instance this will have the desired effect. The assigning of responsibility is based, not on what we know to be true in the particular case, but on what we believe will be the probable effects of encouraging people to behave rationally and considerately. It is a device that society has developed to cope with our inability to look into other people's minds and, without resorting to coercion, to introduce order into our lives.\n\nThis is not the place to enter into a discussion of the special problem raised by all those who cannot be held responsible and to whom the argument for liberty therefore does not or cannot wholly apply. The important point is that being a free and responsible member of the community is a particular status that carries with it a burden as well as a privilege; and if freedom is to fulfill its aim, this status must not be granted at anybody's discretion but must automatically belong to all who satisfy certain objectively ascertainable tests (such as age), so long as the presumption that they possess the required minimum capacities is not clearly disproved. In personal relations the transition from tutelage to full responsibility may be gradual and indistinct, and those lighter forms of coercion which exist between individuals and with which the state should not interfere can be adjusted to degrees of responsibility. Politically and legally, however, the distinction must be sharp and definite and be determined by general and impersonal rules if freedom is to be effective. In our decisions as to whether a person is to be his own master or be subject to the will of another, we must regard him as being either responsible or not responsible, as either having or not having the right to act in a manner that may be unintelligible, unpredictable, or unwelcome to others. The fact that not all human beings can be given full liberty must not mean that the liberty of all should be subject to restrictions and regulations adjusted to individual conditions. The individualizing treatment of the juvenile court or the mental ward is the mark of unfreedom, of tutelage. Though in the intimate relations of private life we may adjust our conduct to the personality of our partners, in public life freedom requires that we be regarded as types, not as unique individuals,  and treated on the presumption that normal motives and deterrents will be effective, whether this be true in the particular instance or not.\n\n5. There is much confusion of the ideal that a person ought to be allowed to pursue his own aims with the belief that, if left free, he will or ought to pursue solely his selfish aims.9 The freedom to pursue one's own aims is, however, as important for the most altruistic person, in whose scale of values the needs of other people occupy a very high place, as for any egotist. It is part of the ordinary nature of men (and perhaps still more of women) and one of the main conditions of their happiness that they make the welfare of other people their chief aim. To do so is part of the normal choice open to us and often the decision generally expected of us. By common opinion our chief concern in this respect should, of course, be the welfare of our family. But we also show our appreciation and approval of others by making them our friends and their aims ours. To choose our associates and generally those whose needs we make our concern is an essential part of freedom and of the moral conceptions of a free society.\n\nGeneral altruism, however, is a meaningless conception. Nobody can effectively care for other people as such; the responsibilities we can assume must always be particular, can concern only those about whom we know concrete facts and to whom either choice or special conditions have attached us. It is one of the fundamental rights and duties of a free man to decide what and whose needs appear to him most important.\n\nThe recognition that each person has his own scale of values which we ought to respect, even if we do not approve of it, is part of the conception of the value of the individual personality. How we value another person will necessarily depend on what his values are. But believing in freedom means that we do not regard ourselves as the ultimate judges of another person's values, that we do not feel entitled to prevent him from pursuing ends which we disapprove so long as he does not infringe the equally protected sphere of others.\n\nA society that does not recognize that each individual has values of his own which he is entitled to follow can have no respect for the dignity of the individual and cannot really know freedom. But it is also true that in a free society an individual will be esteemed according to the manner in which he uses his freedom. Moral esteem would be meaningless without freedom: \"If every action which is good or evil in [a] man of ripe years were to be under pittance and prescription and compulsion, what were virtue but a name, what praise could  be due then to well-doing, what gramercy to be sober, just, or continent?\"10 Liberty is an opportunity for doing good, but this is so only when it is also an opportunity for doing wrong. The fact that a free society will function successfully only if the individuals are in some measure guided by common values is perhaps the reason why philosophers have sometimes defined freedom as action in conformity with moral rules. But this definition of freedom is a denial of that freedom with which we are concerned. The freedom of action  that is the condition of moral merit includes the freedom to act wrongly: we praise or blame only when a person has the opportunity to choose, only when his observance of a rule is not enforced but merely enjoined.\n\nThat the sphere of individual freedom is also the sphere of individual responsibility does not mean that we are accountable for our actions to any particular persons. True, we may lay ourselves open to censure by others because we do what displeases them. But the chief reason why we should be held wholly responsible for our decisions is that this will direct our attention to those causes of events that depend on our actions. The main function of the belief in individual responsibility is to make us use our own knowledge and capacities to the full in achieving our ends.\n\n6. The burden of choice that freedom imposes, the responsibility for one's own fate that a free society places on the individual, has under the conditions of the modern world become a main source of dissatisfaction. To a much greater degree than ever before, the success of a man will depend not on what special abilities he possesses in the abstract but on these abilities being put to the right use. In times of less specialization and less complex organization, when almost everybody could know most of the opportunities that existed, the problem of finding an opportunity for putting one's special skills and talents to good use was less difficult. As society and its complexity extend, the rewards a man can hope to earn come to depend more and more, not on the skill and capacity he may possess, but on their being put to the right use; and both the difficulty of discovering the best employment for one's capacities and the discrepancy between the rewards of men possessing the same technical skill or special ability will increase.\n\nThere is perhaps no more poignant grief than that arising from a sense of how useful one might have been to one's fellow men and of one's gifts having been wasted. That in a free society nobody has a duty to see that a man's talents are properly used, that nobody has a claim to an opportunity to use his special gifts, and that, unless he himself finds such opportunity, they are likely to be wasted, is perhaps the gravest reproach directed against a free system and the source of the bitterest resentment. The consciousness of possessing certain potential capacities naturally leads to the claim that it is somebody else's duty to use them.\n\nThe necessity of finding a sphere of usefulness, an appropriate job, ourselves is the hardest discipline that a free society imposes on us. It is, however, inseparable from freedom, since nobody can assure each man that his gifts will be properly used unless he has the power to coerce others to use them. Only by depriving somebody else of the choice as to who should serve him, whose capacities or which products he is to use, could we guarantee to any man that his gifts will be used in the manner he feels he deserves. It is of the essence of a free society that a man's value and remuneration depend not on capacity in  the abstract but on success in turning it into concrete service which is useful to others who can reciprocate. And the chief aim of freedom is to provide both the opportunity and the inducement to insure the maximum use of the knowledge that an individual can acquire. What makes the individual unique in this respect is not his generic but his concrete knowledge, his knowledge of particular circumstances and conditions.\n\n7. It must be recognized that the results of a free society in this respect are often in conflict with ethical views that are relics of an earlier type of society. There can be little question that, from the point of view of society, the art of turning one's capacity to good account, the skill of discovering the most effective use of one's gift, is perhaps the most useful of all; but too much resourcefulness of this kind is not uncommonly frowned upon, and an advantage gained over those of equal general capacity by a more successful exploitation of concrete circumstances is regarded as unfair. In many societies an \"aristocratic\" tradition that stems from the conditions of action in an organizational hierarchy with assigned tasks and duties, a tradition that has often been developed by people whose privileges have freed them from the necessity of giving others what they want, represents it as nobler to wait until one's gifts are discovered by others, while only religious or ethnic minorities in a hard struggle to rise have deliberately cultivated this kind of resourcefulness (best described by the German term _Findigkeit_ )\u2014and are generally disliked for that reason. Yet there can be no doubt that the discovery of a better use of things or of one's own capacities is one of the greatest contributions that an individual can make in our society to the welfare of his fellows and that it is by providing the maximum opportunity for this that a free society can become so much more prosperous than others. The successful use of this entrepreneurial capacity (and, in discovering the best use of our abilities, we are all entrepreneurs) is the most highly rewarded activity in a free society, while whoever leaves to others the task of finding some useful means of employing his capacities must be content with a smaller reward.\n\nIt is important to realize that we are not educating people for a free society if we train technicians who expect to be \"used,\" who are incapable of finding their proper niche themselves, and who regard it as somebody else's responsibility to ensure the appropriate use of their ability or skill. However able a man may be in a particular field, the value of his services is necessarily low in a free society unless he also possesses the capacity of making his ability known to those who can derive the greatest benefit from it. Though it may offend our sense of justice to find that of two men who by equal effort have acquired the same specialized skill and knowledge, one may be a success and the other a failure, we must recognize that in a free society it is the use of particular opportunities that determines usefulness and must adjust our education and ethos accordingly. In a free society we are remunerated not for our skill but for  using it rightly; and this must be so as long as we are free to choose our particular occupation and are not to be directed to it. True, it is almost never possible to determine what part of a successful career has been due to superior knowledge, ability, or effort and what part to fortunate accidents; but this in no way detracts from the importance of making it worthwhile for everybody to make the right choice.\n\nHow little this basic fact is understood is shown by such assertions, made not only by socialists, as that \"every child has a natural 'right,' as citizen, not merely to 'life, liberty, and the pursuit of happiness,' but to that position in the social scale to which his talents entitle him.\"11 In a free society a man's talents do not \"entitle\" him to any particular position. To claim that they do would mean that some agency has the right and power to place men in particular positions according to its judgment. All that a free society has to offer is an opportunity of searching for a suitable position, with all the attendant risk and uncertainty which such a search for a market for one's gifts must involve. There is no denying that in this respect a free society puts most individuals under a pressure which is often resented. But it is an illusion to think that one would be rid of such pressure in some other type of society; for the alternative to the pressure that responsibility for one's own fate brings is the far more invidious pressure of personal orders that one must obey.\n\nIt is often contended that the belief that a person is solely responsible for his own fate is held only by the successful. This in itself is not so unacceptable as its underlying suggestion, which is that people hold this belief because they have been successful. I, for one, am inclined to think that the connection is the other way round and that people often are successful because they hold this belief. Though a man's conviction that all he achieves is due solely to his exertions, skill, and intelligence may be largely false, it is apt to have the most beneficial effects on his energy and circumspection. And if the smug pride of the successful is often intolerable and offensive, the belief that success depends wholly on him is probably the pragmatically most effective incentive to successful action; whereas the more a man indulges in the propensity to blame others or circumstances for his failures, the more disgruntled and ineffective he tends to become.\n\n8. The sense of responsibility has been weakened in modern times as much by overextending the range of an individual's responsibilities as by exculpating him from the actual consequences of his actions. Since we assign responsibility to the individual in order to influence his action, it should refer only to such effects of his conduct as it is humanly possible for him to foresee and to such as we can reasonably wish him to take into account in ordinary circumstances.  To be effective, responsibility must be both definite and limited, adapted both emotionally and intellectually to human capacities. It is quite as destructive of any sense of responsibility to be taught that one is responsible for everything as to be taught that one cannot be held responsible for anything. Freedom demands that the responsibility of the individual extend only to what he can be presumed to judge, that his actions take into account effects which are within his range of foresight, and particularly that he be responsible only for his own actions (or those of persons under his care)\u2014not for those of others who are equally free.\n\nResponsibility, to be effective, must be individual responsibility. In a free society there cannot be any collective responsibility of members of a group as such, unless they have, by concerted action, all made themselves individually and severally responsible. A joint or divided responsibility may create for the individual the necessity of agreeing with others and thereby limit the powers of each. If the same concerns are made the responsibility of many without at the same time imposing a duty of joint and agreed action, the result is usually that nobody really accepts responsibility. As everybody's property in effect is nobody's property, so everybody's responsibility is nobody's responsibility.12\n\nIt is not to be denied that modern developments, especially the development of the large city, have destroyed much of the feeling of responsibility for local concerns which in the past led to much beneficial and spontaneous common action. The essential condition of responsibility is that it refer to circumstances that the individual can judge, to problems that, without too much strain of the imagination, man can make his own and whose solution he can, with good reason, consider his own concern rather than another's. Such a condition can hardly apply to life in the anonymous crowd of an industrial city. No longer is the individual generally the member of some small community with which he is intimately concerned and closely acquainted. While this has brought him some increase in independence, it has also deprived him of the security which the personal ties and the friendly interest of the neighbors provided. The increased demand for protection and security from the impersonal power of the state is no doubt largely the result of the disappearance of those smaller communities of interest and of the feeling of isolation of the  individual who can no longer count on the personal interest and assistance of the other members of the local group.13\n\nMuch as we may regret the disappearance of those close communities of interest and their replacement by a wide-flung net of limited, impersonal, and temporary ties, we cannot expect the sense of responsibility for the known and familiar to be replaced by a similar feeling about the remote and the theoretically known. While we can feel genuine concern for the fate of our familiar neighbors and usually will know how to help when help is needed, we cannot feel in the same way about the thousands or millions of unfortunates whom we know to exist in the world but whose individual circumstances we do not know. However moved we may be by accounts of their misery, we cannot make the abstract knowledge of the numbers of suffering people guide our everyday action. If what we do is to be useful and effective, our objectives must be limited, adapted to the capacities of our mind and our compassions. To be constantly reminded of our \"social\" responsibilities to all the needy or unfortunate in our community, in our country, or in the world, must have the effect of attenuating our feelings until the distinctions between those responsibilities which call for our action and those which do not disappear. In order to be effective, then, responsibility must be so confined as to enable the individual to rely on his own concrete knowledge in deciding on the importance of the different tasks, to apply his moral principles to circumstances he knows, and to help to mitigate evils voluntarily.\n\n* * *\n\nThe quotation at the head of the chapter is taken from Francis Dunham Wormuth, _The Origins of Modern Constitutionalism_ (New York: Harper, 1949), pp. 212\u201313.\n\n1 This old truth has been succinctly expressed by George Bernard Shaw: \"Liberty means responsibility. That is why most men dread it\" (from \"Maxims for Revolutionists\" in _Man and Superman: A Comedy and a Philosophy_ [Westminster: Archibald Constable, 1903], p. 229. [This and most subsequent editions contain, in addition to the play's text, the \"Revolutionist's Handbook\" and \"Maxims for Revolutionists.\"\u2014Ed.]). The theme has, of course, been treated fully in some of the novels of Fyodor Dostoevsky (especially in the Grand Inquisitor episode of _The Brothers Karamazov_ ), and there is not much that modern psychoanalysts and existentialist philosophers have been able to add to his psychological insight. But see Erich Fromm, _Escape from Freedom_ (New York: Farrar and Rinehart, Inc., 1941; English edition entitled _The Fear of Freedom_ [London: Routledge, 1942]); Marjorie Glicksman Grene, _Dreadful Freedom_ (Chicago: University of Chicago Press, 1948); and Otto Veit, _Die Flucht vor der Freiheit: Versuch zur geschichtsphilosophischen Erhellung der Kulturkrise_ (Frankfurt am Main: Vittorio Klostermann, 1947). The converse of the belief in individual responsibility and connected respect for the law which prevail in free societies is the sympathy with the lawbreaker which seems to develop regularly in unfree societies and which is so characteristic of nineteenth-century Russian literature.\n\n2 For a careful examination of the philosophical problems of general determinism see Karl Raimund Popper, _The Logic of Scientific Discovery\u2014Postscript: After Twenty Years_ (London, 1959) [Popper originally intended the Postscript to which Hayek refers as a series of appendices to the _Logik der Forschung (The Logic of Scientific Discovery)_ to cover the period from 1934 to 1954. The postscript was set in galley proofs during the period 1956\u201357 and given the title _Postscript\u2014After Twenty Years_ ; it circulated in this form for several years and is doubtless the work to which Hayek is referring. Sections of it were incorporated in the 1959 translation of Popper's book as a series of appendices ( pp. 311\u2013464). However, Popper continued to revise and expand his discussion and it was not until 1982 and 1983 that William W. Bartley III undertook the task of bringing this material together, which eventually appeared in three volumes: (1) _Realism and the Aim of Science_ (Totowa, NJ: Rowman and Littlefield, 1983); (2) _The Open Universe: An Argument for Indeterminism_ (Totowa, NJ: Rowman and Littlefield, 1982); and (3) _Quantum Theory and the Schism in Physics_ (Totowa, NJ: Rowman and Littlefield, 1982).\u2014Ed.]; cf. also my essay \"Degrees of Explanation,\" _British Journal for the Philosophy of Science_ , 6 (1955): 209\u201325; reprinted in _Studies in Philosophy, Politics, and Economics_ (Chicago: University of Chicago Press, 1967), pp. 3\u201321. See also my \"The Theory of Complex Phenomena,\" in _The Critical Approach: Essays in Honor of Karl R. Popper_ , M. Bunge, ed. (New York: Free Press, 1964), pp. 332\u201349.[Also appears in _Studies in Philosophy, Politics, and Economics_ , pp. 22\u201342.\u2014Ed.]\n\n3 Conrad Hal Waddington. _The Scientific Attitude_ (Pelican Books; Harmondsworth: Penguin Books, 1941), p. 110. See also Burrhus Frederic Skinner, _Walden Two_ (New York: Macmillan, 1948), p. 257: \"I deny that freedom exists at all. I must deny it, or my program would be absurd. You can't have a science about a subject matter which leaps capriciously about.\" In addition, see Skinner's _Science and Human Behavior_ (New York: Macmillan, 1953), which stands as the most extreme example of anti-liberal attitudes embraced by the modern \"behavioral scientist.\"\n\n4 This was already clearly seen by John Locke ( _An Essay Concerning Human Understanding_ [London: Printed for Thomas Basset and sold by Edward Mory, 1690], bk. 2, chap. 14, sec. 14, p. 119, where he speaks of the \"unreasonable because unintelligible Question, viz. _Whether Man's Will be free, or no?_ For if I mistake not, it follows from what I have said, that the Question itself is altogether improper\"), and even by Thomas Hobbes, _Leviathan; or, The Matter, Forme, and Power of a Commonwealth, Ecclesiasticall and Civil_ , Michael Joseph Oakeshott, ed. (Oxford: B. Blackwell, 1946), pp. 137\u201338. For more recent discussions see Heinrich Gomperz, _Das Problem der Willensfreiheit_ ( Jena: Diederichs, 1907); Moritz Schlick, _Problems of Ethics_ , David Rynin, trans. (New York: Prentice-Hall, 1939); Charles Dunbar Broad, _Determinism, Indeterminism, and Libertarianism: An Inaugural Lecture_ (Cambridge: Cambridge University Press, 1934); Richard Merwyn Hare, _The Language of Morals_ (Oxford: Clarendon Press, 1952); Herbert Lionel Adolphus Hart, \"The Ascription of Responsibility and Rights,\" _Proceedings of the Aristotelian Society_ , n.s., 49 (1949): 171\u201394, reprinted in _Logic and Language_ , Anthony Flew, ed. (1st ser.; Oxford: Blackwell, 1951), pp. 145\u201366; Patrick Horace Nowell-Smith, \"Freewill and Moral Responsibility,\" _Mind_ , 17 (1948): 45\u201361, and the same author's _Ethics_ (Pelican Books; London: Penguin Books, 1954); John David Mabbott, \"Freewill and Punishment,\" in _Contemporary British Philosophy: Personal Statements, 3rd Series_ , Howell David Lewis, ed. (London: Allen and Unwin, 1956), pp. 287\u2013309; Charles Arthur Campbell, \"Is Free Will a Pseudo-Problem?\" _Mind_ , 60 (1951): 441\u201365; D. M. MacKay, \"On Comparing the Brain with Machines\" (British Association Symposium on Cybernetics), _Advancement of Science_ , 10 (1954): 402\u20136, esp. 406; Sidney Hook, ed., _Determinism and Freedom in the Age of Modern Science: A Philosophical Symposium_ [New York University Institute of Philosophy] (New York: New York University Press, 1958); Hans Kelsen, \"Causality and Imputation,\" _Ethics_ , 61 (1950\u201351): 1\u201311; Arthur Pap, \"Determinism and Moral Responsibility,\" _Journal of Philosophy_ , 43 (1946): 318\u201327; and Austin Marsden Farrer, _The Freedom of the Will: The Gifford Lectures Delivered in the University of Edinburgh, 1957_ (London: Adam and Charles Black, 1958).\n\n5 Cf. David Hume, _An Enquiry Concerning Human Understanding_ , in _Essays_ , vol. 2, p. 78: \"By liberty, then, we can only mean a power of acting or not acting, according to the determinations of the will.\" See also the discussion in my book, _The Sensory Order: An Inquiry into the Foundations of Theoretical Psychology_ (Chicago: University of Chicago Press, 1952), secs. 8.93\u20138.94, p. 193. [A _Collected Works_ edition is anticipated.\u2014Ed.]\n\n6 Though this contention still has the appearance of a paradox, it goes back as far as David Hume and apparently even Aristotle. Hume stated explicitly ( _Treatise of Human Nature_ [bk. 2, pt. 3, sec. 2], vol. 2, p. 192): \"'Tis only upon the principles of necessity, that a person acquires any merit or demerit from his actions, however the common opinion may incline to the contrary.\" On Aristotle see Yves Simon, _Trait\u00e9 du libre arbitre_ (Li\u00e8ge: Sciences et lettres, 1951), pp. 93\u201399; and Carl Friedrich Heman, _Des Aristoteles Lehre von der Freiheit des menschlichen Willens_ (Leipzig: Fues's R. Riesland, 1887), esp. pp. 168\u201394, quoted by Simon. For recent discussions see R. E. Hobart, \"Free Will as Involving Determination and Inconceivable without It,\" _Mind_ , 43 (1934): 1\u201327; and Philippa Foot, \"Free Will Involving Determinism,\" _Philosophical Review_ , 66 (1957): 439\u201350.\n\n7 The most extreme deterministic position tends to deny that the term \"will\" has any meaning (the word has indeed been banned from some kinds of superscientific psychology) or that there is such a thing as voluntary action. Yet even those who hold that position cannot avoid distinguishing between the kinds of actions that can be influenced by rational considerations and those that cannot. This is all that matters: Indeed, they will have to admit, what is in effect a _reductio ad absurdum_ of their position, that whether a person does or does not believe in his capacity to form and carry out plans, which is what is popularly meant by his will being free or not, may make a great deal of difference to what he will do.\n\n8 We still call a man's decision \"free,\" though by the conditions we have created he is led to do what we want him to do, because these conditions do not uniquely determine his actions but merely make it more likely that anyone in his position will do what we approve. We try to \"influence\" but do not determine what he will do. What we often mean in this connection, as in many others, when we call his action \"free,\" is simply that we do not know what has determined it, and not that it has not been determined by something.\n\n9 Cf. Thomas Nixon Carver, _Essays in Social Justice_ (Cambridge, MA: Harvard University Press, 1922); [see especially Carver's introductory chapter, \"What Is Justice?\" ( pp. 3\u201334) and chapter two, \"Ultimate Basis of Social Conflict\" ( pp. 35\u201358)] and the first essay [\"Individualism: True and False\"] in my _Individualism and Economic Order_ (Chicago: University of Chicago Press, 1948), pp. 1\u201332. [ _Collected Works_ edition, vol. 13, pp. 46\u201374.]\n\n10 John Milton, _Areopagitica_ (Everyman ed.; London: J. M. Dent and Sons, 1927), p. 18 [Liberty Fund edition, p. 23]. The notion that moral worth required free action was already known in ancient Greece. See Euripides, _Heracleidae_ , 551: \"Frei verlasse ich das Leben, nicht gezwungen. Denn nur die freie Tat hat Wert.\" [Hayek's German translates as: \"I give my life freely, not under compulsion. For only the free deed has value.\"\u2014Ed.] The conception of moral merit depending on freedom was already emphasized by some of the Scholastic philosophers and again especially in the German \"classical\" literature (cf., e.g., Friedrich Schiller, _On the Aesthetic Education of Man: In A Series of Letters_ [New Haven: Yale University Press, 1954], p. 74: \"Man must have his freedom to be ready for morality.\" [While these words do not appear in Schiller's _On the Aesthetic Education of Man_ , there is no question that Schiller believed that man's freedom and his moral disposition are intimately linked and that it is only when men are both physically and spiritually free that they can become fully moral. Consider: \"(Man's) culture consists of two things: first, providing the receptive faculty with the most multifarious contacts with the world, and as regards feeling, pushing passivity to its fullest extent; secondly, securing for the determining faculty the fullest independence from the receptive, and as regards reason, pushing activity to its fullest extent. Where both qualities are united, Man will combine the greatest fullness of existence with the utmost self-dependence and freedom and instead of abandoning himself to the world he will rather draw it into himself with the whole infinity of its phenomena, and subject it to the unity of reason.\" ( p. 69; \"Seine Kultur wird also darin bestehen: erstlich: Dem empfangenden Verm\u00f6gen die vielf\u00e4ltigsten Ber\u00fchrungen mit der Welt zu verschaffen und auf seiten des Gef\u00fchls die Passivit\u00e4t aufs H\u00f6chste zu treiben; zweitens dem bestimmenden Verm\u00f6gen die h\u00f6chste Unabh\u00e4ngigkeit von dem empfangenden zu erwerben und auf seiten der Vernunft die Aktivit\u00e4t aufs H\u00f6chste zu treiben. Wo beide Eigenschaften sich vereinigen, da wird der Mensch mit der h\u00f6chsten F\u00fclle von Dasein die h\u00f6chste Selbst\u00e4ndigkeit und Freiheit verbinden, und, anstatt sich an die Welt zu verlieren, diese vielmehr mit der ganzen Unendlichkeit ihrer Erscheinungen in sich ziehen und der Einheit seiner Vermunft unterwerfen.\" (\"Uber die \u00e4sthetische Erziehung des Menschen\" [Letter 13, _S\u00e4mtliche Werke in zehn B\u00e4nden_ , vol. 8: _Philosophische Schriften_ , Hans-G\u00fcther Thalheim, ed. (Berlin: Aufbau-Verlag, 2005), p. 345]); and \"If in the dynamic state of rights man encounters man as force and restricts his activity, if in the ethical state of duties he opposes him with the majesty of the law and fetters his will, in the sphere of cultivated society, in the aesthetic state, he need appear to him only as shape, confront him only as an object of free play. To grant freedom by means of freedom is the fundamental law of this kingdom.\" ( p. 137; \"Wenn in dem dynamischen Staat der Rechte der Mensch dem Menschen als Kraft begegnet und sein Wirken beschr\u00e4nkt\u2014wenn er sich ihm in dem ethischen Staat der Pflichten mit der Majest\u00e4t des Gesetzes entgegenstellt und sein Wollen fesselt, so darf er ihm im Kreise des sch\u00f6nen Umgangs, in dem \u00e4sthetischen Staat, nur als Gestalt erscheinen, nur als Objekt des freien Spiels gegen\u00fcberstehen. Freiheit zu geben durch Freiheit ist das Grundgesetz dieses Reichs.\") Letter 27; p. 406.)\u2014Ed.] Also Alexis de Toccqueville, _Voyage en Angleterre et en Ir lande de 1835_ , in _\u0152uvres compl\u00e8tes_ , Jacob Peter Mayer, ed. (18 vols.; Paris: Gallimard, 1951), vol. 5, pt. 2, p. 91: \"qu'estce que la vertu, sinon le _choix libre_ de ce qui est bien?\" [\"What is virtue, if not the free choice of what is good?\"\u2014Ed.]\n\n11 Charles Anthony Raven Crosland, _The Future of Socialism_ (London: Jonathan Cape, 1956), p. 208.\n\n12 Cf. also the observation by Johan Huizinga, _Incertitudes: Essai de diagnostic du mal dont su_ _ff_ _re notre temps_ (Paris: Librairie de M\u00e9dici, 1939), p. 216: \"Dans chaque groupe collectif une partie du jugement de l'individu est absorb\u00e9e avec une partie de sa responsibilit\u00e9 par le mot d'ordre collectif. Le sentiment d'\u00eatre tous ensemble responsables de tout, accro\u00eet dans le monde actuel le danger de l'irresponsabilit\u00e9 absolue de l'action des masses.\" [\"In each group, a portion of one's judgment, and with it a part of one's responsibility, gets swallowed up in a series of catchwords. In today's world, the view that we are all responsible for each other results in the danger that all actions of the masses are completely lacking in responsibility.\"\u2014Ed.]\n\n13 See David Riesman, _The Lonely Crowd: A Study of the Changing American Character_ (New Haven: Yale University Press, 1950).\n\n## SIX\n\nEQUALITY, VALUE, AND MERIT\n\nI have no respect for the passion for equality, which seems to me merely idealizing envy.\n\n\u2014Oliver Wendell Holmes, Jr.\n\n1. The great aim of the struggle for liberty has been equality before the law. This equality under the rules which the state enforces may be supplemented by a similar equality of the rules that men voluntarily obey in their relations with one another. This extension of the principle of equality to the rules of moral and social conduct is the chief expression of what is commonly called the democratic spirit\u2014and probably that aspect of it that does most to make inoffensive the inequalities that liberty necessarily produces.\n\nEquality of the general rules of law and conduct, however, is the only kind of equality conducive to liberty and the only equality which we can secure without destroying liberty. Not only has liberty nothing to do with any other sort of equality, but it is even bound to produce inequality in many respects.1 This is the necessary result and part of the justification of individual liberty: if the result of individual liberty did not demonstrate that some manners of living are more successful than others, much of the case for it would vanish.\n\nIt is neither because it assumes that people are in fact equal nor because it attempts to make them equal that the argument for liberty demands that government treat them equally. This argument not only recognizes that individuals are very different but in a great measure rests on that assumption. It insists that these individual differences provide no justification for government to treat them differently. And it objects to the differences in treatment by the  state that would be necessary if persons who are in fact very different were to be assured equal positions in life.\n\nModern advocates of a more far-reaching material equality usually deny that their demands are based on any assumption of the factual equality of all men.2 It is nevertheless still widely believed that this is the main justification for such demands. Nothing, however, is more damaging to the demand for equal treatment than to base it on so obviously untrue an assumption as that of the factual equality of all men. To rest the case for equal treatment of national or racial minorities on the assertion that they do not differ from other men is implicitly to admit that factual inequality would justify unequal treatment; and the proof that some differences do, in fact, exist would not be long in forthcoming. It is of the essence of the demand for equality before the law that people should be treated alike in spite of the fact that they are different.\n\n2. The boundless variety of human nature\u2014the wide range of differences in individual capacities and potentialities\u2014is one of the most distinctive facts about the human species. Its evolution has made it probably the most variable among all kinds of creatures. It has been well said that \"biology, with variability as its cornerstone, confers on every human individual a unique set of attributes which give him a dignity he could not otherwise possess. Every newborn baby is an unknown quantity so far as potentialities are concerned because there are many thousands of unknown interrelated genes and gene patterns which contribute to his makeup. As a result of nature and nurture the newborn infant may become one of the greatest men or women ever to have lived. In every case he or she has the making of a distinctive individual. . . . If the differences are not very important, then freedom is not very important and the idea of individual worth is not very important.\"3 The writer justly adds that the widely held uniformity theory of human nature, \"which on the surface appears to accord with democracy . . . would in time undermine the very basic ideals of freedom and individual worth and render life as we know it meaningless.\"4\n\nIt has been the fashion in modern times to minimize the importance of congenital differences between men and to ascribe all the important differences to the influence of environment.5 However important the latter may  be, we must not overlook the fact that individuals are very different from the outset. The importance of individual differences would hardly be less if all people were brought up in very similar environments. As a statement of fact, it just is not true that \"all men are born equal.\" We may continue to use this hallowed phrase to express the ideal that legally and morally all men ought to be treated alike. But if we want to understand what this ideal of equality can or should mean, the first requirement is that we free ourselves from the belief in factual equality.\n\nFrom the fact that people are very different it follows that, if we treat them equally, the result must be inequality in their actual position,6 and that the only way to place them in an equal position would be to treat them differently. Equality before the law and material equality are therefore not only different but are in conflict with each other; and we can achieve either the one or the other, but not both at the same time. The equality before the law which freedom requires leads to material inequality. Our argument will be that, though where the state must use coercion for other reasons, it should treat all people alike, the desire of making people more alike in their condition cannot be accepted in a free society as a justification for further and discriminatory coercion.\n\nWe do not object to equality as such. It merely happens to be the case that a demand for equality is the professed motive of most of those who desire to impose upon society a preconceived pattern of distribution. Our objection is against all attempts to impress upon society a deliberately chosen pattern of distribution, whether it be an order of equality or of inequality. We shall indeed see that many of those who demand an extension of equality do not really demand equality but a distribution that conforms more closely to human conceptions of individual merit and that their desires are as irreconcilable with freedom as the more strictly egalitarian demands.\n\nIf one objects to the use of coercion in order to bring about a more even or a more just distribution, this does not mean that one does not regard these as desirable. But if we wish to preserve a free society, it is essential that we recognize that the desirability of a particular object is not sufficient justification for the use of coercion. One may well feel attracted to a community in which there are no extreme contrasts between rich and poor and may welcome the fact that the general increase in wealth seems gradually to reduce those differences. I fully share these feelings and certainly regard the degree of social equality that the United States has achieved as wholly admirable.\n\nThere also seems no reason why these widely felt preferences should not guide policy in some respects. Wherever there is a legitimate need for government action and we have to choose between different methods of satisfying such a need, those that incidentally also reduce inequality may well be preferable. If, for example, in the law of intestate succession one kind of provision will be more conducive to equality than another, this may be a strong argument in its favor. It is a different matter, however, if it is demanded that, in order to produce substantive equality, we should abandon the basic postulate of a free society, namely, the limitation of all coercion by equal law. Against this we shall hold that economic inequality is not one of the evils which justify our resorting to discriminatory coercion or privilege as a remedy.\n\n3. Our contention rests on two basic propositions which probably need only be stated to win fairly general assent. The first of them is an expression of the belief in a certain similarity of all human beings: it is the proposition that no man or group of men possesses the capacity to determine conclusively the potentialities of other human beings and that we should certainly never trust anyone invariably to exercise such a capacity. However great the differences between men may be, we have no ground for believing that they will ever be so great as to enable one man's mind in a particular instance to comprehend fully all that another responsible man's mind is capable of.\n\nThe second basic proposition is that the acquisition by any member of the community of additional capacities to do things which may be valuable must always be regarded as a gain for that community. It is true that particular people may be worse off because of the superior ability of some new competitor in their field; but any such additional ability in the community is likely to benefit the majority. This implies that the desirability of increasing the abilities and opportunities of any individual does not depend on whether the same can also be done for the others\u2014provided, of course, that others are not thereby deprived of the opportunity of acquiring the same or other abilities which might have been accessible to them had they not been secured by that individual.\n\nEgalitarians generally regard differently those differences in individual capacities which are inborn and those which are due to the influences of environment, or those which are the result of \"nature\" and those which are the result of \"nurture.\" Neither, be it said at once, has anything to do with moral merit.7 Though either may greatly affect the value which an individual has for his fellows, no more credit belongs to him for having been born with desirable  qualities than for having grown up under favorable circumstances. The distinction between the two is important only because the former advantages are due to circumstances clearly beyond human control, while the latter are due to factors which we might be able to alter. The important question is whether there is a case for so changing our institutions as to eliminate as much as possible those advantages due to environment. Are we to agree that \"all inequalities that rest on birth and inherited property ought to be abolished and none remain unless it is an effect of superior talent and industry?\"8\n\nThe fact that certain advantages rest on human arrangements does not necessarily mean that we could provide the same advantages for all or that, if they are given to some, somebody else is thereby deprived of them. The most important factors to be considered in this connection are the family, inheritance, and education, and it is against the inequality which they produce that criticism is mainly directed. They are, however, not the only important factors of environment. Geographic conditions such as climate and landscape, not to speak of local and sectional differences in cultural and moral traditions, are scarcely less important. We can, however, consider here only the three factors whose effects are most commonly impugned.\n\nSo far as the family is concerned, there exists a curious contrast between the esteem most people profess for the institution and their dislike of the fact that being born into a particular family should confer on a person special advantages. It seems to be widely believed that, while useful qualities which a person acquires because of his native gifts under conditions which are the same for all are socially beneficial, the same qualities become somehow undesirable if they are the result of environmental advantages not available to others. Yet it is difficult to see why the same useful quality which is welcomed when it is the result of a person's natural endowment should be less valuable when it is the product of such circumstances as intelligent parents or a good home.\n\nThe value which most people attach to the institution of the family rests on the belief that, as a rule, parents can do more to prepare their children for a satisfactory life than anyone else. This means not only that the benefits which particular people derive from their family environment will be different but also that these benefits may operate cumulatively through several generations. What reason can there be for believing that a desirable quality in a person is less valuable to society if it has been the result of family background than if it has not? There is, indeed, good reason to think that there are some  socially valuable qualities which will be rarely acquired in a single generation but which will generally be formed only by the continuous efforts of two or three. This means simply that there are parts of the cultural heritage of a society that are more effectively transmitted through the family.9 Granted this, it would be unreasonable to deny that a society is likely to get a better elite if ascent is not limited to one generation, if individuals are not deliberately made to start from the same level, and if children are not deprived of the chance to benefit from the better education and material environment which their parents may be able to provide. To admit this is merely to recognize that belonging to a particular family is part of the individual personality, that society is made up as much of families as of individuals, and that the transmission of the heritage of civilization within the family is as important a tool in man's striving toward better things as is the heredity of beneficial physical attributes.\n\n4. Many people who agree that the family is desirable as an instrument for the transmission of morals, tastes, and knowledge still question the desirability of the transmission of material property. Yet there can be little doubt that, in order that the former may be possible, some continuity of standards, of the external forms of life, is essential, and that this will be achieved only if it is possible to transmit not only immaterial but also material advantages. There is, of course, neither greater merit nor any greater injustice involved in some people being born to wealthy parents than there is in others being born to kind or intelligent parents. The fact is that it is no less of an advantage to the community if at least some children can start with the advantages which at any given time only wealthy homes can offer than if some children inherit great intelligence or are taught better morals at home.\n\nWe are not concerned here with the chief argument for private inheritance, namely, that it seems essential as a means to preserve the dispersal in the control of capital and as an inducement for its accumulation. Rather, our concern here is whether the fact that it confers unmerited benefits on some is a valid argument against the institution. It is unquestionably one of the institutional causes of inequality. In the present context we need not inquire whether liberty demands unlimited freedom of bequest. Our problem here is merely whether people ought to be free to pass on to children or others such material possessions as will cause substantial inequality.\n\nOnce we agree that it is desirable to harness the natural instincts of parents to equip the new generation as well as they can, there seems no sensible ground for limiting this to non-material benefits. The family's function of passing on standards and traditions is closely tied up with the possibility of transmitting material goods. And it is difficult to see how it would serve the true interest of society to limit the gain in material conditions to one generation.\n\nThere is also another consideration which, though it may appear somewhat cynical, strongly suggests that if we wish to make the best use of the natural partiality of parents for their children, we ought not to preclude the transmission of property. It seems certain that among the many ways in which those who have gained power and influence might provide for their children, the bequest of a fortune is socially by far the cheapest. Without this outlet, these men would look for other ways of providing for their children, such as placing them in positions which might bring them the income and the prestige that a fortune would have done; and this would cause a waste of resources and an injustice much greater than is caused by the inheritance of property. Such is the case with all societies in which inheritance of property does not exist, including the Communist. Those who dislike the inequalities caused by inheritance should therefore recognize that, men being what they are, it is the least of evils, even from their point of view.\n\n5. Though inheritance used to be the most widely criticized source of inequality, it is today probably no longer so. Egalitarian agitation now tends to concentrate on the unequal advantages due to differences in education. There is a growing tendency to express the desire to secure equality of conditions in the claim that the best education we have learned to provide for some should be made gratuitously available for all and that, if this is not possible, one should not be allowed to get a better education than the rest merely because one's parents are able to pay for it, but only those and all those who can pass a uniform test of ability should be admitted to the benefits of the limited resources of higher education.\n\nThe problem of educational policy raises too many issues to allow of their being discussed incidentally under the general heading of equality. We shall have to devote a separate chapter to them at the end of this book. For the present we shall only point out that enforced equality in this field can hardly avoid preventing some from getting the education they otherwise might. Whatever we might do, there is no way of preventing those advantages which only some can have, and which it is desirable that some should have, from going to people who neither individually merit them nor will make as good a use of them as some other person might have done. Such a problem cannot be satisfactorily solved by the exclusive and coercive powers of the state.\n\nIt is instructive at this point to glance briefly at the change that the ideal  of equality has undergone in this field in modern times. A hundred years ago, at the height of the classical liberal movement, the demand was generally expressed by the phrase _la carri\u00e8re ouverte aux talents_. It was a demand that all man-made obstacles to the rise of some should be removed, that all privileges of individuals should be abolished, and that what the state contributed to the chance of improving one's conditions should be the same for all. That so long as people were different and grew up in different families this could not assure an equal start was fairly generally accepted. It was understood that the duty of government was not to ensure that everybody had the same prospect of reaching a given position but merely to make available to all on equal terms those facilities which in their nature depended on government action. That the results were bound to be different, not only because the individuals were different, but also because only a small part of the relevant circumstances depended on government action, was taken for granted.\n\nThis conception that all should be allowed to try has been largely replaced by the altogether different conception that all must be assured an equal start and the same prospects. This means little less than that the government, instead of providing the same circumstances for all, should aim at controlling all conditions relevant to a particular individual's prospects and so adjust them to his capacities as to assure him of the same prospects as everybody else. Such deliberate adaptation of opportunities to individual aims and capacities would, of course, be the opposite of freedom. Nor could it be justified as a means of making the best use of all available knowledge except on the assumption that government knows best how individual capacities can be used.\n\nWhen we inquire into the justification of these demands, we find that they rest on the discontent that the success of some people often produces in those that are less successful, or, to put it bluntly, on envy. The modern tendency to gratify this passion and to disguise it in the respectable garment of social justice is developing into a serious threat to freedom. Recently an attempt was made to base these demands on the argument that it ought to be the aim of politics to remove all sources of discontent.10 This would, of course, necessarily mean that it is the responsibility of government to see that nobody is healthier or possesses a happier temperament, a better-suited spouse or more prospering children, than anybody else. If really all unfulfilled desires have a claim on the community, individual responsibility is at an end. However human, envy is certainly not one of the sources of discontent that a free society can eliminate. It is probably one of the essential conditions for the preservation  of such a society that we do not countenance envy, not sanction its demands by camouflaging it as social justice, but treat it, in the words of John Stuart Mill, as \"that most anti-social and odious of all passions.\"11\n\n6. While most of the strictly egalitarian demands are based on nothing better than envy, we must recognize that much that on the surface appears as a demand for greater equality is in fact a demand for a juster distribution of the good things of this world and springs therefore from much more creditable motives. Most people will object not to the bare fact of inequality but to the fact that the differences in reward do not correspond to any recognizable differences in the merits of those who receive them. The answer commonly given to this is that a free society on the whole achieves this kind of justice.12 This, however, is an indefensible contention if by justice is meant proportionality of reward to moral merit. Any attempt to found the case for  freedom on this argument is very damaging to it, since it concedes that material rewards ought to be made to correspond to recognizable merit and then opposes the conclusion that most people will draw from this by an assertion which is untrue. The proper answer is that in a free system it is neither desirable nor practicable that material rewards should be made generally to correspond to what men recognize as merit and that it is an essential characteristic of a free society that an individual's position should not necessarily depend on the views that his fellows hold about the merit he has acquired.\n\nThis contention may appear at first so strange and even shocking that I will ask the reader to suspend judgment until I have further explained the distinction between value and merit.13 The difficulty in making the point clear is due to the fact that the term \"merit,\" which is the only one available to describe what I mean, is also used in a wider and vaguer sense. It will be used here exclusively to describe the attributes of conduct that make it deserving  of praise, that is, the moral character of the action and not the value of the achievement.14\n\nAs we have seen throughout our discussion, the value that the performance or capacity of a person has to his fellows has no necessary connection with its ascertainable merit in this sense. The inborn as well as the acquired gifts of a person clearly have a value to his fellows which does not depend on any credit due to him for possessing them. There is little a man can do to alter the fact that his special talents are very common or exceedingly rare. A good mind or a fine voice, a beautiful face or a skilful hand, and a ready wit or an attractive personality are in a large measure as independent of a person's efforts as the opportunities or the experiences he has had. In all these instances the value which a person's capacities or services have for us and for which he is recompensed has little relation to anything that we can call moral merit or deserts. Our problem is whether it is desirable that people should enjoy advantages in proportion to the benefits which their fellows derive from their activities or whether the distribution of these advantages should be based on other men's views of their merits.\n\nReward according to merit must in practice mean reward according to assessable merit, merit that other people can recognize and agree upon and not merit merely in the sight of some higher power. Assessable merit in this sense presupposes that we can ascertain that a man has done what some accepted rule of conduct demanded of him and that this has cost him some pain and effort. Whether this has been the case cannot be judged by the result: merit is not a matter of the objective outcome but of subjective effort. The attempt to achieve a valuable result may be highly meritorious but a complete failure, and full success may be entirely the result of accident and thus without merit. If we know that a man has done his best, we will often wish to see him rewarded irrespective of the result; and if we know that a most valuable achievement is almost entirely due to luck or favorable circumstances, we will give little credit to the author.\n\nWe may wish that we were able to draw this distinction in every instance. In fact, we can do so only rarely with any degree of assurance. It is possible only  where we possess all the knowledge which was at the disposal of the acting person, including a knowledge of his skill and confidence, his state of mind and his feelings, his capacity for attention, his energy and persistence, etc. The possibility of a true judgment of merit thus depends on the presence of precisely those conditions whose general absence is the main argument for liberty. It is because we want people to use knowledge which we do not possess that we let them decide for themselves. But insofar as we want them to be free to use capacities and knowledge of facts which we do not have, we are not in a position to judge the merit of their achievements. To decide on merit presupposes that we can judge whether people have made such use of their opportunities as they ought to have made and how much effort of will or self-denial this has cost them; it presupposes also that we can distinguish between that part of their achievement which is due to circumstances within their control and that part which is not.\n\n7. The incompatibility of reward according to merit with freedom to choose one's pursuit is most evident in those areas where the uncertainty of the outcome is particularly great and our individual estimates of the chances of various kinds of effort very different.15 In those speculative efforts which we call \"research\" or \"exploration,\" or in economic activities which we commonly describe as \"speculation,\" we cannot expect to attract those best qualified for them unless we give the successful ones all the credit or gain, though many others may have striven as meritoriously. For the same reason that nobody can know beforehand who will be the successful ones, nobody can say who has earned greater merit. It would clearly not serve our purpose if we let all who have honestly striven share in the prize. Moreover, to do so would make it necessary that somebody have the right to decide who is to be allowed to strive for it. If in their pursuit of uncertain goals people are to use their own knowledge and capacities, they must be guided, not by what other people think they ought to do, but by the value others attach to the result at which they aim.\n\nWhat is so obviously true about those undertakings which we commonly regard as risky is scarcely less true of any chosen object we decide to pursue. Any such decision is beset with uncertainty, and if the choice is to be as wise as it is humanly possible to make it, the alternative results anticipated must be labeled according to their value. If the remuneration did not correspond to  the value that the product of a man's efforts has for his fellows, he would have no basis for deciding whether the pursuit of a given object is worth the effort and risk. He would necessarily have to be told what to do, and some other person's estimate of what was the best use of his capacities would have to determine both his duties and his remuneration.16\n\nThe fact is, of course, that we do not wish people to earn a maximum of merit but to achieve a maximum of usefulness at a minimum of pain and sacrifice and therefore a minimum of merit. Not only would it be impossible for us to reward all merit justly, but it would not even be desirable that people should aim chiefly at earning a maximum of merit. Any attempt to induce them to do this would necessarily result in people being rewarded differently for the same service. And it is only the value of the result that we can judge with any degree of confidence, not the different degrees of effort and care that it has cost different people to achieve it.\n\nThe prizes that a free society offers for the result serve to tell those who strive for them how much effort they are worth. However, the same prizes will go to all those who produce the same result, regardless of effort. What is true here of the remuneration for the same services rendered by different people is even more true of the relative remuneration for different services requiring different gifts and capacities: they will have little relation to merit. The market will generally offer for services of any kind the value they will have for those who benefit from them; but it will rarely be known whether it was necessary to offer so much in order to obtain these services, and often, no doubt, the community could have had them for much less. The pianist who was reported not long ago to have said that he would perform even if he had to pay for the privilege probably described the position of many who earn large incomes from activities which are also their chief pleasure.\n\n8. Though most people regard as very natural the claim that nobody should be rewarded more than he deserves for his pain and effort, it is nevertheless based on a colossal presumption. It presumes that we are able to judge in every individual instance how well people use the different opportunities and talents given to them and how meritorious their achievements are in the light  of all the circumstances which have made them possible. It presumes that some human beings are in a position to determine conclusively what a person is worth and are entitled to determine what he may achieve. It presumes, then, what the argument for liberty specifically rejects: that we can and do know all that guides a person's action.\n\nA society in which the position of the individuals was made to correspond to human ideas of moral merit would therefore be the exact opposite of a free society. It would be a society in which people were rewarded for duty performed instead of for success, in which every move of every individual was guided by what other people thought he ought to do, and in which the individual was thus relieved of the responsibility and the risk of decision. But if nobody's knowledge is sufficient to guide all human action, there is also no human being who is competent to reward all efforts according to merit.\n\nIn our individual conduct we generally act on the assumption that it is the value of a person's performance and not his merit that determines our obligation to him. Whatever may be true in more intimate relations, in the ordinary business of life we do not feel that, because a man has rendered us a service at a great sacrifice, our debt to him is determined by this, so long as we could have had the same service provided with ease by somebody else. In our dealings with other men we feel that we are doing justice if we recompense value rendered with equal value, without inquiring what it might have cost the particular individual to supply us with these services. What determines our responsibility is the advantage we derive from what others offer us, not their merit in providing it. We also expect in our dealings with others to be remunerated not according to our subjective merit but according to what our services are worth to them. Indeed, so long as we think in terms of our relations to particular people, we are generally quite aware that the mark of the free man is to be dependent for his livelihood not on other people's views of his merit but solely on what he has to offer them. It is only when we think of our position or our income as determined by \"society\" as a whole that we demand reward according to merit.\n\nThough moral value or merit is a species of value, not all value is moral value, and most of our judgments of value are not moral judgments. That this must be so in a free society is a point of cardinal importance; and the failure to distinguish between value and merit has been the source of serious confusion. We do not necessarily admire all activities whose product we value; and in most instances where we value what we get, we are in no position to assess the merit of those who have provided it for us. If a man's ability in a given field is more valuable after thirty years' work than it was earlier, this is independent of whether these thirty years were most profitable and enjoyable or whether they were a time of unceasing sacrifice and worry. If the pursuit of a hobby produces a special skill or an accidental invention turns out to be  extremely useful to others, the fact that there is little merit in it does not make it any less valuable than if the result had been produced by painful effort.\n\nThis difference between value and merit is not peculiar to any one type of society\u2014it would exist anywhere. We might, of course, attempt to make rewards correspond to merit instead of value, but we are not likely to succeed in this. In attempting it, we would destroy the incentives which enable people to decide for themselves what they should do. Moreover, it is more than doubtful whether even a fairly successful attempt to make rewards correspond to merit would produce a more attractive or even a tolerable social order. A society in which it was generally presumed that a high income was proof of merit and a low income of the lack of it, in which it was universally believed that position and remuneration corresponded to merit, in which there was no other road to success than the approval of one's conduct by the majority of one's fellows, would probably be much more unbearable to the unsuccessful ones than one in which it was frankly recognized that there was no necessary connection between merit and success.17\n\nIt would probably contribute more to human happiness if, instead of trying to make remuneration correspond to merit, we made clearer how uncertain is the connection between value and merit. We are probably all much too ready to ascribe personal merit where there is, in fact, only superior value. The possession by an individual or a group of a superior civilization or education certainly represents an important value and constitutes an asset for the community to which they belong; but it usually constitutes little merit. Popularity and esteem do not depend more on merit than does financial success. It is, in fact, largely because we are so used to assuming an often non-existent merit wherever we find value that we balk when, in particular instances, the discrepancy is too large to be ignored.\n\nThere is every reason why we ought to endeavor to honor special merit where it has gone without adequate reward. But the problem of rewarding  action of outstanding merit which we wish to be widely known as an example is different from that of the incentives on which the ordinary functioning of society rests. A free society produces institutions in which, for those who prefer it, a man's advancement depends on the judgment of some superior or of the majority of his fellows. Indeed, as organizations grow larger and more complex, the task of ascertaining the individual's contribution will become more difficult; and it will become increasingly necessary that, for many, merit in the eyes of the managers rather than the ascertainable value of the contribution should determine the rewards. So long as this does not produce a situation in which a single comprehensive scale of merit is imposed upon the whole society, so long as a multiplicity of organizations compete with one another in offering different prospects, this is not merely compatible with freedom but extends the range of choice open to the individual.\n\n9. Justice, like liberty and coercion, is a concept which, for the sake of clarity, ought to be confined to the deliberate treatment of men by other men. It is an aspect of the intentional determination of those conditions of people's lives that are subject to such control. Insofar as we want the efforts of individuals to be guided by their own views about prospects and chances, the results of the individual's efforts are necessarily unpredictable, and the question as to whether the resulting distribution of incomes is just has no meaning.18 Justice does require that those conditions of people's lives that are determined by government be provided equally for all. But equality of those conditions must lead to inequality of results. Neither the equal provision of particular public facilities nor the equal treatment of different partners in our voluntary dealings with one another will secure reward that is proportional to merit. Reward for merit is reward for obeying the wishes of others in what we do, not compensation for the benefits we have conferred upon them by doing what we thought best.\n\nIt is, in fact, one of the objections against attempts by government to fix income scales that the state must attempt to be just in all it does. Once the principle of reward according to merit is accepted as the just foundation for the distribution of incomes, justice would require that all who desire it should be rewarded according to that principle. Soon it would also be demanded that the same principle be applied to all and that incomes not in proportion to recognizable merit not be tolerated. Even an attempt merely to distinguish between those incomes or gains which are \"earned\" and those which are not  will set up a principle which the state will have to try to apply but cannot in fact apply generally.19 And every such attempt at deliberate control of some remunerations is bound to create further demands for new controls. The principle of distributive justice, once introduced, would not be fulfilled until the whole of society was organized in accordance with it. This would produce a kind of society which in all essential respects would be the opposite of a free society\u2014a society in which authority decided what the individual was to do and how he was to do it.\n\n10. In conclusion we must briefly look at another argument on which the demands for a more equal distribution are frequently based, though it is rarely explicitly stated. This is the contention that membership in a particular community or nation entitles the individual to a particular material standard that is determined by the general wealth of the group to which he belongs. This demand is in curious conflict with the desire to base distribution on personal merit. There is clearly no merit in being born into a particular community, and no argument of justice can be based on the accident of a particular individual's being born in one place rather than another. A relatively wealthy community in fact regularly confers advantages on its poorest members unknown to those born in poor communities. In a wealthy community the only justification its members can have for insisting on further advantages is that there is much private wealth that the government can confiscate and redistribute and that men who constantly see such wealth being enjoyed by others will have a stronger desire for it than those who know of it only abstractly, if at all.\n\nThere is no obvious reason why the joint efforts of the members of any group to ensure the maintenance of law and order and to organize the provision of certain services should give the members a claim to a particular share in the wealth of this group. Such claims would be especially difficult to defend where those who advanced them were unwilling to concede the same rights to those who did not belong to the same nation or community. The recognition of such claims on a national scale would in fact only create a new kind of collective (but not less exclusive) property right in the resources of the nation that could not be justified on the same grounds as individual property. Few people  would be prepared to recognize the justice of these demands on a world scale. And the bare fact that within a given nation the majority had the actual power to enforce such demands, while in the world as a whole it did not yet have it, would hardly make them more just.\n\nThere are good reasons why we should endeavor to use whatever political organization we have at our disposal to make provision for the weak or infirm or for the victims of unforeseeable disaster. It may well be true that the most effective method of providing against certain risks common to all citizens of a state is to give every citizen protection against those risks. The level on which such provisions against common risks can be made will necessarily depend on the general wealth of the community.\n\nIt is an entirely different matter, however, to suggest that those who are poor, merely in the sense that there are those in the same community who are richer, are entitled to a share in the wealth of the latter or that being born into a group that has reached a particular level of civilization and comfort confers a title to a share in all its benefits. The fact that all citizens have an interest in the common provision of some services is no justification for anyone's claiming as a right a share in all the benefits. It may set a standard for what some ought to be willing to give, but not for what anyone can demand.\n\nNational groups will become more and more exclusive as the acceptance of this view that we have been contending against spreads. Rather than admit people to the advantages that living in their country offers, a nation will prefer to keep them out altogether; for, once admitted, they will soon claim as a right a particular share in its wealth. The conception that citizenship or even residence in a country confers a claim to a particular standard of living is becoming a serious source of international friction. And since the only justification for applying the principle within a given country is that its government has the power to enforce it, we must not be surprised if we find the same principle being applied by force on an international scale. Once the right of the majority to the benefits that minorities enjoy is recognized on a national scale, there is no reason why this should stop at the boundaries of the existing states.\n\n* * *\n\nThe quotation at the head of the chapter is taken from _The Holmes-Laski Letters: The Correspondence of Mr. Justice Holmes and Harold J. Laski, 1916\u20131935_ , Mark DeWolfe Howe, ed. (Cambridge, MA: Harvard University Press, 1953), vol. 2, p. 942. A German translation of an earlier version of this chapter, \"Gleichheit, Wert und Verdienst,\" [Equality, Value, and Merit] has appeared in _Ordo: Jahrbuch f\u00fcr die Ordnung von Wirtschaft und Gesellschaft_ , 10 (1958): 5\u201329.\n\n1 See Gerhard Leibholz, \"Die Bedrohung der Freiheit durch die Macht der Gesetzgeber,\" in _Die Freiheit der Pers\u00f6nlichkeit: eine Vortragsreihe_ , Ludwig von Bertalanffy, ed. (Stuttgart: A. Kr\u00f6ner, 1958), p. 80: \"Freiheit erzeugt notwendig Ungleichheit und Gleichheit notwendig Unfreiheit.\" [\"Freedom necessarily creates inequality and equality necessarily creates unfreedom.\"\u2014Ed.]\n\n2 See, e.g., Richard Henry Tawney, _Equality_ [Halley Stewart Lectures, 1929 (London: Allen and Unwin, 1931)], pp. 47\u201350.\n\n3 Roger John Williams, _Free and Unequal: The Biological Basis of Individual Liberty_ (Austin: University of Texas Press, 1953), pp. 23 and 70; cf. also John Burdon Sanderson Haldane, _The Inequality of Man, and Other Essays_ (London: Chatto and Windus, 1932), and Peter Brian Medawar, _The Uniqueness of the Individual_ (London: Methuen, 1957).\n\n4 Williams, _Free and Unequal_ , p. 152.\n\n5 See the description of this fashionable view in Horace Mever Kallen's article \"Behaviorism,\" _Encyclopedia of the Social Sciences_ , vol. 2, p. 498: \"At birth human infants, regardless of their heredity, are as equal as Fords.\"\n\n6 Cf. Plato, _Laws_ , vii. 757a: \"To unequals equals become unequal.\" [The Alfred Edward Taylor translation renders the passage as follows \"Indeed, equal treatment of the unequal ends in inequality when not qualified by due proportion.\" _The Collected Dialogues of Plato, Including the Letters_ , Edith Hamilton and Huntington Cairns, eds. (Bollingen Series; New York: Pantheon Books, 1961), pp. 1336\u201337.\u2014Ed.]\n\n7 Cf. Frank Hyneman Knight, _Freedom and Reform: Essays in Economics and Social Philosophy_ (New York: Harper and Brothers, 1947), p. 151: \"There is no visible reason why anyone is more or less entitled to the earnings of inherited personal capacities than to those of inherited property in any other form\"; and the discussion in Wilhelm R\u00f6pke, _Mass und Mitte_ (Erlenbach and Zurich: Eugen Rentsch, 1950), pp. 65\u201375.\n\n8 This is the position of Richard Henry Tawney as summarized by John Petrov Plamenatz, \"Equality of Opportunity,\" in _Aspects of Human Equality_ , from the Fifteenth Conference on Science, Philosophy, and Religion in Their Relation to the Democratic Way of Life, Columbia University, 1955, Lyman Bryson, ed. (New York: Distributed by Harper, 1956),p. 100. [Tawney explicates his views on equality at some length in his _Equality_ (London: Allen and Unwin, 1931).\u2014Ed.]\n\n9 See William Graham Sumner, _Andrew Jackson_ (Standard library ed.; Boston: Houghton Miff in, 1899), pp. 24\u201325: \"True honor, truthfulness, suppression of individual personal feeling, self-control and courtesy are inculcated best, if not exclusively, by the constant precept and example, in earliest childhood of highbred parents and relatives. There is nothing on ear th which it costs more labor to produce than a high-bred man. It is also indisputable that home discipline and training ingrain into the character of men the most solid and valuable elements and that, without such training, more civilization means better food and clothes rather than better men.\"\n\n10 Charles Anthony Raven Crosland, _The Future of Socialism_ (London: Jonathan Cape, 1956), p. 205.\n\n11 John Stuart Mill, \"On Liberty,\" in _On Liberty and Considerations on Representative Government_ , Ronald Buchanan McCallum, ed. (Oxford: B. Blackwell, 1946), p. 70. See particularly Helmut Schoeck, _Der Neid: Eine Theorie der Gesellschaft_ (Freiburg, Munich: Albers, 1966). [English translation by Michael Glenny and Betty Ross, published under the title _Envy: A Theory of Social Behavior_ (New York: Harcourt, Brace and World, 1969.)\u2014Ed.]\n\n12 Cf. Walter Bryce Gallie, \"Liberal Morality and Socialist Morality,\" in _Philosophy, Politics, and Society_ , Peter Laslett, ed. (Oxford: Blackwe11, 1956), pp. 123\u201325. The author represents it as the essence of \"liberal morality\" that it claims that rewards are equal to merit in a free society. This was the position of some nineteenth-century liberals which often weakened their argument. A characteristic example is William Graham Sumner, _What Social Classes Owe to Each Other_ (New York: Harper and Brothers, 1883), p. 164. [Hayek notes that this essay appears in \" _Freeman_ , vol. 6 (Los Angeles, n.d.), p. 141.\" The publication to which Hayek is referring is in fact vol. 4, no. 1 (n.d.) of a journal with that title and of very limited circulation, published irregularly by a group calling itself the Los Angeles Pamphleteers.\u2014Ed.] Sumner argued that if all \"have equal chances so far as chances are provided or limited by society,\" this will \"produce inequal results\u2014that is results which shall be proportioned to the merits of individuals.\" This is true only if \"merit\" is used in the sense in which we have used \"value,\" without any moral connotations, but certainly not if it is meant to suggest proportionality to any endeavor to do the good or right thing, or to any subjective effort to conform to an ideal standard.\n\nBut, as we shall presently see, Mr. Gallie is right that, in the Aristotelian terms he uses, liberalism aims at commutative justice and socialism at distributive justice. But, like most socialists, he does not see that distributive justice is irreconcilable with freedom in the choice of one's activities: it is the justice of a hierarchic organization, not of a free society.\n\nJohn Stuart Mill, _Principles of Political Economy, with Some of Their Applications to Social Philosophy_ , vol. 2 of _The Collected Works of John Stuart Mill_ (Toronto: University of Toronto Press, 1965), bk. 2, chap. 1, p. 210 [similarly, Liberty Fund edition]: \"The proportioning of remuneration to work done is really just only in so far as the more or less of the work is a matter of choice: when it depends on natural differences of strength and capacity, this principle of remuneration is in itself an injustice: it gives to those who have; assigning most to those already most favoured by nature.\" See Gustav Radbruch, _Rechtsphilosophie_ (5th ed.; Stuttgart: Koehler, 1956), e.g., p. 187: \"Auch das sozialistische Gemeinwesen wird also ein Rechtsstaat sein, ein _Rechtsstaat_ freilich, der statt von der ausgleichenden von der austeilenden Gerechtigkeit beherrscht wird.\" [\"The socialist polity will therefore also be a state of law, a _Rechtsstaat_ ruled by distributive instead of retributive justice.\"\u2014Ed.]\n\n13 Although I believe that this distinction between merit and value is the same as that which Aristotle and Thomas Aquinas had in mind when they distinguished \"distributive justice\" from \"commutative justice,\" I prefer not to tie up the discussion with all the difficulties and confusions which in the course of time have become associated with these traditional concepts. That what we call here \"reward according to merit\" corresponds to the Aristotelian distributive justice seems clear. The difficult concept is that of \"commutative justice,\" and to speak of justice in this sense seems always to cause a little confusion. Cf. Max Salomon, _Der Begri_ _ff_ _der Gerechtigkeit bei Aristoteles: nebst einem Anhang \u00fcber den Begri_ _ff_ _des Tauschgesch\u00e4ftes_ (Leiden: Sijthoff, 1937); and for a survey of the extensive literature Giorgio del Vecchio, _Die Gerechtigkeit_ (2nd ed.; Basel: Verlag f\u00fcr Recht und Gesellschaft, 1950).\n\nAlso see David Hume, _Enquiry Concerning the Principles of Morals_ , in _Essays_ , vol. 2, p. 187: \"were mankind to execute such a law [to assign the largest possessions to the most e xtensive virtue, and give every one the power of doing good, proportioned to his inclination]; so great is the uncertainty of merit, both from its natural obscurity, and from the self-conceit of each individual, that no determinate rule of conduct would ever result from it; and the total dissolution of society must be the immediate consequence\"; Hugo Grotius, _The Jurisprudence of Holland_ , Robert Warden Lee, trans. (Oxford: Clarendon Press, 1926), vol. 1, p. 3: \"Of the justice which has regard to right, narrowly understood, the kind which takes account of merit is called 'distributive justice'; the other kind which givies heed to property is called 'commutative justice'; the first commonly employs the rule of proportion, the second the rule of simple equality\"; Adam Smith, _The Theory of Moral Sentiments_ (London: Printed for A Miller, 1759), pt. 2, sec. 1, pp. 141\u201369, \"Of the Sense of Merit and Demerit\"[Liberty Fund edition, pp. 67\u201378];Edwin Cannan, _The History of Local Rates in England_ (2nd ed.; London: P. S. King and Son, 1912), pp . 160\u201361: \"the existing system proportions command over economic goods to the value of services rendered and property possessed. . . .As against this established state of things, we find that many people, perhaps most people, have somewhere in their minds two inconsistent and somewhat nebulous ideals . . . according to [the first], command over economic goods ought to be in proportion to moral merit. . . . The second ideal is the comm unist one of equal distribution\"; Kenneth Ewart Boulding, _Principles of Economic Policy_ (Englewood Cliffs, NJ: Prentice Hall, 1958), p. 85, remarks that \"justice as a situation in which everybody gets what he deserves, may be called the _merit standard_ ,\" and points out that \"it would be a dangerous ethical fallacy to equate desert with contribution.\" Also see Arthur William Hope Adkins, _Merit and Responsibility: A Study in Greek Values_ (Oxford: Clarendon Press, 1960), who, unfortunately, employs the term \"merit\" for what I call \"value\" and \"responsibility\" for what I have called \"merit.\"\n\n14 The terminological difficulties arise from the fact that we use the word merit also in an objective sense and will speak of the \"merit\" of an idea, a book, or a picture, irrespective of the merit acquired by the person who has created them. Sometimes the word is also used to describe what we regard as the \"true\" value of some achievement as distinguished from its market value. Yet even a human achievement which has the greatest value or merit in this sense is not necessarily proof of moral merit on the part of him to whom it is due. It seems that our use has the sanction of philosophical tradition. Cf., for instance, David Hume, _Treatise of Human Nature_ [bk. 3, pt. 2, sec. 1], vol. 2, p. 252: \"The external performance has no merit. We must look within to find the moral quality. . . . The ultimate object of our praise and approbation is the motive that produced them.\"\n\n15 Cf. the important essay by Armen Albert Alchian, \"Uncertainty, Evolution, and Economic Theory,\" _Journal of Political Economy_ , 58 (1950): 211\u201321, esp. 213\u201314, sec. 2, headed \"Success Is Based on Results, Not Motivation.\" It probably is also no accident that the American economist who has done most to advance our understanding of a free society, Frank Hyneman Knight, began his professional career with a study of _Risk, Uncertainty, and Profit_ (Boston: Houghton Mifflin Company, 1921). Cf. also Bertrand de Jouvenel, _Power: The Natural History of its Growth_ (London: Hutchinson, 1948), p. 298. [There is a Liberty Fund edition of this work, released under the title _On Power._ ]\n\n16 It is often maintained that justice requires that remuneration be proportional to the unpleasantness of the job and that for this reason the street cleaner or the sewage worker ought to be paid more than the doctor or office worker. This, indeed, would seem to be the consequence of the principle of remuneration according to merit (or \"distributive justice\"). In a market such a result would come about only if all people were equally skilful in all jobs so that those who could earn as much as others in the more pleasant occupations would have to be paid more to undertake the distasteful ones. In the actual world those unpleasant jobs provide those whose usefulness in the more attractive jobs is small an opportunity to earn more than they could elsewhere. That persons who have little to offer their fellows should be able to earn an income similar to that of the rest only at a much greater sacrifice is inevitable in any arrangement under which the individual is allowed to choose his own sphere of usefulness.\n\n17 Cf. Crosland, _The Future of Socialism_ , p. 235: \"Even if all the failures could be convinced that they had an equal chance, their discontent would still not be assuaged; indeed it might actually be intensified. When opportunities are known to be unequal, and the selection clearly biased towards wealth or lineage, people can comfort themselves for failure by saying that they never had a proper chance\u2014the system was unfair, the scales too heavily weighted against them. But if the selection is obviously by merit, this source of comfort disappears, and failure induces a total sense of inferiority, with no excuse or consolation; and this, by a natural quirk of human nature, actually increases the envy and resentment at the success of others.\" Cf. also chap. 14 at n. 8 below. [Note 8 refers to several articles dealing with the notion of \"Nulla poena sine lege\" and its central importance to the rule of law.\u2014Ed.] I have not yet seen Michael Dunlop Young, _The Rise of the Meritocracy, 1870\u20132023: An Essay on Education and Equality (_ London: Thames and Hudson, 1958) which, judging from reviews, appears to bring out these problems very clearly. [The American edition carries the title _The Rise of the Meritocracy, 1870\u20132023: The New Elite of Our Social Revolution_ (New York: Random House, 1959).\u2014Ed.]\n\n18 See the interesting discussion in Robin George Collingwood, \"Economics as a Philosophical Science,\" _International Journal of Ethics_ , 36 (1926), who concludes ( p. 174): \"A just price, a just wage, a just rate of interest, is a contradiction in terms. The question what a person ought to get in return for his goods and labor is a question absolutely devoid of meaning. The only valid questions are what he _can_ get in return for his goods or labor, and whether he ought to sell them at all.\"\n\n19 It is, of course, possible to give the distinction between \"earned\" and \"unearned\" incomes, gains, or increments a fairly precise legal meaning, but it then rapidly ceases to correspond to the moral distinction which provides its justification. Any serious attempt to apply the moral distinction in practice soon meets the same insuperable difficulties as any attempt to assess subjective merit. How little these difficulties are generally understood by philosophers (except in rare instances, as that quoted in the preceding note) is well illustrated by a discussion in Lizzie Susan Stebbing, _Thinking to Some Purpose_ (Pelican Books; Harmondsworth, UK: Penguin Books, 1939), p. 184, in which, as an illustration of a distinction which is clear but not sharp, she chooses that between \"legitimate\" and \"excess\" profits and asserts: \"The distinction is clear between 'excess profits' (or 'profiteering') and 'legitimate profits,' although it is not a sharp distinction.\"\n\n## SEVEN\n\nMAJORITY RULE\n\nThough men be much governed by interest, yet even interest itself, and all human affairs, are entirely governed by opinion.\n\n\u2014David Hume\n\n1. Equality before the law leads to the demand that all men should also have the same share in making the law. This is the point where traditional liberalism and the democratic movement meet. Their main concerns are nevertheless different. Liberalism (in the European nineteenth-century meaning of the word, to which we shall adhere throughout this chapter) is concerned mainly with limiting the coercive powers of all government, whether democratic or not, whereas the dogmatic democrat knows only one limit to government\u2014 current majority opinion. The difference between the two ideals stands out most clearly if we name their opposites: for democracy it is authoritarian government; for liberalism it is totalitarianism. Neither of the two systems necessarily excludes the opposite of the other: a democracy may well wield totalitarian powers, and it is conceivable that an authoritarian government may act on liberal principles.1\n\nLike most terms in our field, the word \"democracy\" is also used in a wider and vaguer sense. But if it is used strictly to describe a method of government\u2014namely, majority rule\u2014it clearly refers to a problem different from that of liberalism. Liberalism is a doctrine about what the law ought to be, democracy a doctrine about the manner of determining what will be the law. Liberalism regards it as desirable that only what the majority accepts should in fact be law, but it does not believe that this is therefore necessarily good law. Its aim, indeed, is to persuade the majority to observe certain principles. It accepts majority rule as a method of deciding, but not as an authority for what the decision ought to be. To the doctrinaire democrat the fact that the majority wants something is sufficient ground for regarding it as good; for him the will of the majority determines not only what is law but what is good law.\n\nAbout this difference between the liberal and the democratic ideal there exists widespread agreement.2 There are, however, those who use the word  \"liberty\" in the sense of political liberty and are led by this to identify liberalism with democracy. For them the ideal of liberty can say nothing about what the aim of democratic action ought to be: every condition that democracy creates is, by definition, a condition of liberty. This seems, to say the least, a very confusing use of words.\n\nWhile liberalism is one of those doctrines concerning the scope and purpose of government from which democracy has to choose, the latter, being a method, indicates nothing about the aims of government. Though \"democratic\" is often used today to describe particular aims of policy that happen to be popular, especially certain egalitarian ones, there is no necessary connection  between democracy and any one view about how the powers of the majority ought to be used. In order to know what it is that we want others to accept, we need other criteria than the current opinion of the majority, which is an irrelevant factor in the process by which opinion is formed. It certainly provides no answer to the question of how a man ought to vote or of what is desirable\u2014unless we assume, as many of the dogmatic democrats seem to assume, that a person's class position invariably teaches him to recognize his true interests and that therefore the vote of the majority always expresses the best interests of the majority.\n\n2. The current undiscriminating use of the word \"democratic\" as a general term of praise is not without danger. It suggests that, because democracy is a good thing, it is always a gain for mankind if it is extended. This may sound self-evident, but it is nothing of the kind.\n\nThere are at least two respects in which it is almost always possible to extend democracy: the range of persons entitled to vote and the range of issues that are decided by democratic procedure. In neither respect can it be seriously contended that every possible extension is a gain or that the principle of democracy demands that it be indefinitely extended. Yet in the discussion of almost any particular issue the case for democracy is commonly presented as if the desirability of extending it as far as possible were indisputable.\n\nThat this is not so is implicitly admitted by practically everybody so far as the right to vote is concerned. It would be difficult on any democratic theory to regard every possible extension of the franchise as an improvement. We speak of universal adult suffrage, but the limits of suffrage are in fact largely determined by considerations of expediency. The usual age limit of twenty-one and the exclusion of criminals, resident foreigners, non-resident citizens, and the inhabitants of special regions or territories are generally accepted as reasonable. It is also by no means obvious that proportional representation is better because it seems more democratic.3 It can scarcely be said that equality before the law necessarily requires that all adults should have the vote; the principle would operate if the same impersonal rule applied to all. If only persons over forty, or only income-earners, or only heads of households, or only literate persons were given the vote, this would scarcely be more of an infringement of the principle than the restrictions which are generally accepted. It is also possible for reasonable people to argue that the ideals of democracy would be better served if, say, all the servants of government or all recipients of public charity were excluded from the vote.4 If in the Western  world universal adult suffrage seems the best arrangement, this does not prove that it is required by some basic principle.\n\nWe should also remember that the right of the majority is usually recognized only within a given country and that what happens to be one country is not always a natural or obvious unit. We certainly do not regard it as right that the citizens of a large country should dominate those of a small adjoining country merely because they are more numerous. There is as little reason why the majority of the people who have joined for some purposes, be it as a nation or some supernational organization, should be regarded as entitled to extend the scope of their power as far as they please. The current theory of democracy suffers from the fact that it is usually developed with some ideal homogeneous community in view and then applied to the very imperfect and often arbitrary units which the existing states constitute.\n\nThese remarks are meant only to show that even the most dogmatic democrat can hardly claim that every extension of democracy is a good thing. However strong the general case for democracy, it is not an ultimate or absolute value and must be judged by what it will achieve. It is probably the best method of achieving certain ends, but not an end in itself.5 Though there is a strong presumption in favor of the democratic method of deciding where it is obvious that some collective action is required, the problem of whether or not it is desirable to extend collective control must be decided on other grounds than the principle of democracy as such.\n\n3. The democratic and the liberal traditions thus agree that whenever state action is required, and particularly whenever coercive rules have to be laid down, the decision ought to be made by the majority. They differ, however, on the scope of the state action that is to be guided by democratic decision. While the dogmatic democrat regards it as desirable that as many issues as possible be decided by majority vote, the liberal believes that there are definite limits to the range of questions which should be thus decided. The dogmatic democrat feels, in particular, that any current majority ought to have the right to decide what powers it has and how to exercise them, while the liberal regards it as important that the powers of any temporary majority be limited  by long-term principles. To him it is not from a mere act of will of the momentary majority but from a wider agreement on common principles that a majority decision derives its authority.\n\nThe crucial conception of the doctrinaire democrat is that of popular sovereignty. This means to him that majority rule is unlimited and unlimitable. The ideal of democracy, originally intended to prevent all arbitrary power, thus becomes the justification for a new arbitrary power. Yet the authority of democratic decision rests on its being made by the majority of a community which is held together by certain beliefs common to most members; and it is necessary that the majority submit to these common principles even when it may be in its immediate interest to violate them. It is irrelevant that this view used to be expressed in terms of the \"law of nature\" or the \"social contract,\" conceptions which have lost their appeal. The essential point remains: it is the acceptance of such common principles that makes a collection of people a community. And this common acceptance is the indispensable condition for a free society. A group of men normally become a society not by giving themselves laws but by obeying the same rules of conduct.6 This means that the power of the majority is limited by those commonly held principles and that there is no legitimate power beyond them. Clearly, it is necessary for people to come to an agreement as to how necessary tasks are to be performed, and it is reasonable that this should be decided by the majority; but it is not obvious that this same majority must also be entitled to determine what it is competent to do. There is no reason why there should not be things which nobody has power to do. Lack of sufficient agreement on the need of certain uses of coercive power should mean that nobody can legitimately exercise it. If we recognize rights of minorities, this implies that the power of the majority ultimately derives from, and is limited by, the principles which the minorities also accept.\n\nThe principle that whatever government does should be agreed to by the  majority does not therefore necessarily require that the majority be morally entitled to do what it likes. There can clearly be no moral justification for any majority granting its members privileges by laying down rules which discriminate in their favor. Democracy is not necessarily unlimited government. Nor is a democratic government any less in need of built-in safeguards of individual liberty than any other. It was, indeed, at a comparatively late stage in the history of modern democracy that great demagogues began to argue that since the power was now in the hands of the people, there was no longer any need for limiting that power.7 It is when it is contended that \"in a democracy right is what the majority makes it to be\"8 that democracy degenerates into demagoguery.\n\n4. If democracy is a means rather than an end, its limits must be determined in the light of the purpose we want it to serve. There are three chief arguments by which democracy can be justified, each of which may be regarded as conclusive. The first is that, whenever it is necessary that one of several conflicting opinions should prevail and when one would have to be made to prevail by force if need be, it is less wasteful to determine which has the stronger support by counting numbers than by fighting. Democracy is the only method of peaceful change that man has yet discovered.9\n\nThe second argument, which historically has been the most important and which is still very important, though we can no longer be sure that it is always valid, is that democracy is an important safeguard of individual liberty. It was once said by a seventeenth-century writer that \"the good of democracy is liberty, and the courage and industry which liberty begets.\"10 This view recognizes, of course, that democracy is not yet liberty; it contends only that it is more likely than other forms of government to produce liberty. This view may be well founded so far as the prevention of coercion of individuals by other individuals is concerned: it can scarcely be to the advantage of a majority that some individuals should have the power arbitrarily to coerce others. But the protection of the individual against the collective action of the majority itself is another matter. Even here it can be argued that, since coercive power must in fact always be exercised by a few, it is less likely to be abused if the power entrusted to the few can always be revoked by those who have to submit to it. But if the prospects of individual liberty are better in a democracy than under other forms of government, this does not mean that they are certain. The prospects of liberty depend on whether or not the majority makes it its deliberate object. It would have little chance of surviving if we relied on the mere existence of democracy to preserve it.\n\nThe third argument rests on the effect which the existence of democratic institutions will have on the general level of understanding of public affairs. This seems to me the most powerful. It may well be true, as has been often maintained,11 that, in any given state of affairs, government by some educated elite would be a more efficient and perhaps even a more just government than one chosen by majority vote. The crucial point, however, is that, in comparing the democratic form of government with others, we cannot take the understanding of the issues by the people at any time as a datum. It is the burden of the argument of Tocqueville's great work, _Democracy in America_ , that democracy is the only effective method of educating the majority.12 This is as true today as it was in his time. Democracy is, above all, a process of forming opinion. Its chief advantage lies not in its method of selecting those who govern but in the fact that, because a great part of the population takes an active part in the formation of opinion, a correspondingly wide range of persons is available from which to select. We may admit that democracy does not put power in the hands of the wisest and best informed and that at any given moment the decision of a government by an elite might be more beneficial to the whole; but this need not prevent us from still giving democracy the preference. It is in its dynamic, rather than in its static, aspects that the value of democracy proves itself. As is true of liberty, the benefits of democracy will show themselves only in the long run, while its more immediate achievements may well be inferior to those of other forms of government.\n\n5. The conception that government should be guided by majority opinion makes sense only if that opinion is independent of government. The ideal of democracy rests on the belief that the view which will direct government emerges from an independent and spontaneous process.13 It requires, therefore, the existence of a large sphere independent of majority control in which the opinions of the individuals are formed. There is widespread consensus that for this reason the case for democracy and the case for freedom of speech and discussion are inseparable.\n\nThe view, however, that democracy provides not merely a method of settling differences of opinion on the course of action to be adopted but also a standard for what opinion ought to be has already had far-reaching effects. It has, in particular, seriously confused the question of what is actually valid law and what ought to be the law. If democracy is to function, it is as important that the former can always be ascertained as that the latter can always be questioned. Majority decisions tell us what people want at the moment, but not what it would be in their interest to want if they were better informed; and, unless they could be changed by persuasion, they would be of no value. The argument for democracy presupposes that any minority opinion may become a majority one.\n\nIt would not be necessary to stress this if it were not for the fact that it is sometimes represented as the duty of the democrat, and particularly of the democratic intellectual, to accept the views and values of the majority. True, there is the convention that the view of the majority should prevail so far as collective action is concerned, but this does not in the least mean that one should not make every effort to alter it. One may have profound respect for that convention and yet very little for the wisdom of the majority. It is only because the majority opinion will always be opposed by some that our knowledge and understanding progress. In the process by which opinion is formed, it is very probable that, by the time any view becomes a majority view, it is no longer the best view: somebody will already have advanced beyond the point which the majority have reached.14 It is because we do not yet know which of the many competing new opinions will prove itself the best that we wait until it has gained sufficient support.\n\nThe conception that the efforts of all should be directed by the opinion of  the majority or that a society is better according as it conforms more to the standards of the majority is in fact a reversal of the principle by which civilization has grown. Its general adoption would probably mean the stagnation, if not the decay, of civilization. Advance consists in the few convincing the many. New views must appear somewhere before they can become majority views. There is no experience of society which is not first the experience of a few individuals. Nor is the process of forming majority opinion entirely, or even chiefly, a matter of discussion, as the overintellectualized conception would have it. There is some truth in the view that democracy is government by discussion, but this refers only to the last stage of the process by which the merits of alternative views and desires are tested. Though discussion is essential, it is not the main process by which people learn. Their views and desires are formed by individuals acting according to their own designs; and they profit from what others have learned in their individual experience. Unless some people know more than the rest and are in a better position to convince the rest, there would be little progress in opinion. It is because we normally do not know who knows best that we leave the decision to a process which we do not control. But it is always from a minority acting in ways different from what the majority would prescribe that the majority in the end learns to do better.\n\n6. We have no ground for crediting majority decisions with that higher, superindividual wisdom which, in a certain sense, the products of spontaneous social growth may possess. The resolutions of a majority are not the place to look for such superior wisdom. They are bound, if anything, to be inferior to the decisions that the most intelligent members of the group will make after listening to all opinions: they will be the result of less careful thought and will generally represent a compromise that will not fully satisfy anybody. This will be even more true of the cumulative result emanating from the successive decisions of shifting majorities variously composed: the result will be the expression not of a coherent conception but of different and often conflicting motives and aims.\n\nSuch a process should not be confused with those spontaneous processes which free communities have learned to regard as the source of much that is better than individual wisdom can contrive. If by \"social process\" we mean the gradual evolution which produces better solutions than deliberate design, the imposition of the will of the majority can hardly be regarded as such. The latter differs radically from that free growth from which custom and institutions emerge, because its coercive, monopolistic, and exclusive character destroys the self-correcting forces which bring it about in a free society that mistaken efforts will be abandoned and the successful ones prevail. It also differs basically from the cumulative process by which law is formed by precedent, unless it is, as is true of judicial decisions, fused into a coherent  whole by the fact that principles followed on earlier occasions are deliberately adhered to.\n\nMoreover, majority decisions are peculiarly liable, if not guided by accepted common principles, to produce over-all results that nobody wanted. It often happens that a majority is forced by its own decisions to further actions that were neither contemplated nor desired. The belief that collective action can dispense with principles is largely an illusion, and the usual effect of its renouncing principles is that it is driven into a course by the unexpected implications of former decisions. The individual decision may have been intended only to deal with a particular situation. But it creates the expectation that wherever similar circumstances occur the government will take similar action. Thus principles which had never been intended to apply generally, which may be undesirable or nonsensical when applied generally, bring about future action that few would have desired in the first instance. A government that claims to be committed to no principles and to judge every problem on its merits usually finds itself having to observe principles not of its own choosing and being led into action that it had never contemplated. A phenomenon which is now familiar to us is that of governments which start out with the proud claim that they will deliberately control all affairs and soon find themselves beset at each step by the necessities created by their former actions. It is since governments have come to regard themselves as omnipotent that we now hear so much about the necessity or inevitability of their doing this or that which they know to be unwise.\n\n7. If the politician or statesman has no choice but to adopt a certain course of action (or if his action is regarded as inevitable by the historian), this is because his or other people's opinion, not objective facts, allow him no alternative. It is only to people who are influenced by certain beliefs that anyone's response to given events may appear to be uniquely determined by circumstances. For the practical politician concerned with particular issues, these beliefs are indeed unalterable facts to all intents and purposes. It is almost necessary that he be unoriginal, that he fashion his program from opinions held by large numbers of people. The successful politician owes his power to the fact that he moves within the accepted framework of thought, that he thinks and talks conventionally. It would be almost a contradiction in terms for a politician to be a leader in the field of ideas. His task in a democracy is to find out what the opinions held by the largest number are, not to give currency to new opinions which may become the majority view in some distant future.\n\nThe state of opinion which governs a decision on political issues is always the result of a slow evolution, extending over long periods and proceeding at many different levels. New ideas start among a few and gradually spread until they become the possession of a majority who know little of their origin. In  modern society this process involves a division of functions between those who are concerned mainly with the particular issues and those who are occupied with general ideas, with elaborating and reconciling the various principles of action which past experience has suggested. Our views both about what the consequences of our actions will be and about what we ought to aim at are mainly precepts that we have acquired as part of the inheritance of our society. These political and moral views, no less than our scientific beliefs, come to us from those who professionally handle abstract ideas. It is from them that both the ordinary man and the political leader obtain the fundamental conceptions that constitute the framework of their thought and guide them in their action.\n\nThe belief that in the long run it is ideas and therefore the men who give currency to new ideas that govern evolution, and the belief that the individual steps in that process should be governed by a set of coherent conceptions, have long formed a fundamental part of the liberal creed. It is impossible to study history without becoming aware of the \"lesson given to mankind by every age, and always disregarded\u2014to show that speculative philosophy, which to the superficial appears a thing so remote from the business of life and the outward interests of men, is in reality the thing on earth which most influences them, and in the long run overbears every other influence save those which it must itself obey.\"15 Though this fact is perhaps even less understood  today than it was when John Stuart Mill wrote, there can be little doubt that it is true at all times, whether men recognize it or not. It is so little understood because the influence of the abstract thinker on the masses operates only indirectly. People rarely know or care whether the commonplace ideas of their day have come to them from Aristotle or Locke, Rousseau or Marx, or from some professor whose views were fashionable among the intellectuals twenty years ago. Most of them have never read the works or even heard the names of the authors whose conceptions and ideals have become part of their thinking.\n\nSo far as direct influence on current affairs is concerned, the influence of the political philosopher may be negligible. But when his ideas have become common property, through the work of historians and publicists, teachers and writers, and intellectuals generally, they effectively guide developments. This means not only that new ideas commonly begin to exercise their influence on political action only a generation or more after they have first been stated16 but that, before the contributions of the speculative thinker can exercise such influence, they have to pass through a long process of selection and modification.\n\nChanges in political and social beliefs necessarily proceed at any one time at many different levels. We must conceive of the process not as expanding over one plane but as filtering slowly downward from the top of a pyramid, where the higher levels represent greater generality and abstraction and not necessarily greater wisdom. As ideas spread downward, they also change their character. Those which are at any time still on a high level of generality will compete  only with others of similar character, and only for the support of people interested in general conceptions. To the great majority these general conceptions will become known only in their application to concrete and particular issues. Which of these ideas will reach them and gain their support will be determined not by some single mind but by discussion proceeding on another level, among people who are concerned more with general ideas than with particular problems and who, in consequence, see the latter mainly in the light of general principles.\n\nExcept on rare occasions, such as constitutional conventions, the democratic process of discussion and majority decision is necessarily confined to part of the whole system of law and government. The piecemeal change which this involves will produce desirable and workable results only if it is guided by some general conception of the social order desired, some coherent image of the kind of world in which the people want to live. To achieve such an image is not a simple task, and even the specialist student can do no more than endeavor to see a little more clearly than his predecessors. The practical man concerned with the immediate problems of the day has neither the interest nor the tune to examine the interrelations of the different parts of the complex order of society. He merely chooses from among the possible orders that are offered him and finally accepts a political doctrine or set of principles elaborated and presented by others.\n\nIf people were not at most times led by some system of common ideas, neither a coherent policy nor even real discussion about particular issues would be possible. It is doubtful whether democracy can work in the long run if the great majority do not have in common at least a general conception of the type of society desired. But even if such a conception exists, it will not necessarily show itself in every majority decision. Groups do not always act in accordance with their best knowledge or obey moral rules that they recognize in the abstract any more than individuals do. It is only by appealing to such common principles, however, that we can hope to reach agreement by discussion, to settle conflict of interests by reasoning and argument rather than by brute force.\n\n8. If opinion is to advance, the theorist who offers guidance must not regard himself as bound by majority opinion. The task of the political philosopher is different from that of the expert servant who carries out the will of the majority. Though he must not arrogate to himself the position of a \"leader\" who determines what people ought to think, it is his duty to show possibilities and consequences of common action, to offer comprehensive aims of policy as a whole which the majority have not yet thought of. It is only after such a comprehensive picture of the possible results of different policies has been presented that democracy can decide what it wants. If politics is the art of the  possible, political philosophy is the art of making politically possible the seemingly impossible.17\n\nThe political philosopher cannot discharge his task if he confines himself to questions of fact and is afraid of deciding between conflicting values. He cannot allow himself to be limited by the positivism of the scientist, which confines his functions to showing what is the case and forbids any discussion of what ought to be. If he does so, he will have to stop long before he has performed his most important function. In his effort to form a coherent picture he will often find that there are values which conflict with one another\u2014a fact which most people are not aware of\u2014and that he must choose which he should accept and which reject. Unless the political philosopher is prepared to defend values which seem right to him, he will never achieve that comprehensive outline which must then be judged as a whole.\n\nIn this task he will often serve democracy best by opposing the will of the majority. Only a complete misapprehension of the process by which opinion progresses would lead one to argue that in the sphere of opinion he ought to submit to majority views. To treat existing majority opinion as the standard for what majority opinion ought to be would make the whole process circular and stationary. There is, in fact, never so much reason for the political philosopher to suspect himself of failing in his task as when he finds that his opinions are very popular.18 It is by insisting on considerations which the majority do not wish to take into account, by holding up principles which they regard as inconvenient and irksome, that he has to prove his worth. For intellectuals to bow to a belief merely because it is held by the majority is a betrayal not only of their peculiar mission but of the values of democracy itself.\n\nThe principles that plead for the self-limitation of the power of the majority are not proved wrong if democracy disregards them, nor is democracy proved undesirable if it often makes what the liberal must regard as the wrong decision. He simply believes that he has an argument which, when properly understood, will induce the majority to limit the exercise of its own powers  and which he hopes it can be persuaded to accept as a guide when deciding on particular issues.\n\n9. It is not the least part of this liberal argument that to disregard those limits will, in the long run, destroy not only prosperity and peace but democracy itself. The liberal believes that the limits which he wants democracy to impose upon itself are also the limits within which it can work effectively and within which the majority can truly direct and control the actions of government. So long as democracy constrains the individual only by general rules of its own making, it controls the power of coercion. If it attempts to direct them more specifically, it will soon find itself merely indicating the ends to be achieved while leaving to its expert servants the decision as to the manner in which they are to be achieved. And once it is generally accepted that majority decisions can merely indicate ends and that the pursuit of them is to be left to the discretion of the administrators, it will soon be believed also that almost any means to achieve those ends are legitimate.\n\nThe individual has little reason to fear any general laws which the majority may pass, but he has much reason to fear the rulers it may put over him to implement its directions. It is not the powers which democratic assemblies can effectively wield but the powers which they hand over to the administrators charged with the achievement of particular goals that constitute the danger to individual freedom today. Having agreed that the majority should prescribe rules which we will obey in pursuit of our individual aims, we find ourselves more and more subjected to the orders and the arbitrary will of its agents. Significantly enough, we find not only that most of the supporters of unlimited democracy soon become defenders of arbitrariness and of the view that we should trust experts to decide what is good for the community, but that the most enthusiastic supporters of such unlimited powers of the majority are often those very administrators who know best that, once such powers are assumed, it will be they and not the majority who will in fact exercise them. If anything has been demonstrated by modern experience in these matters, it is that, once wide coercive powers are given to governmental agencies for particular purposes, such powers cannot be effectively controlled by democratic assemblies. If the latter do not themselves determine the means to be employed, the decisions of their agents will be more or less arbitrary.\n\nGeneral considerations and recent experience both show that democracy will remain effective only so long as government in its coercive action confines itself to tasks that can be carried out democratically.19 If democracy is a  means of preserving liberty, then individual liberty is no less an essential condition for the working of democracy. Though democracy is probably the best form of limited government, it becomes an absurdity if it turns into unlimited government. Those who profess that democracy is all-competent and support all that the majority wants at any given moment are working for its fall. The old liberal is in fact a much better friend of democracy than the dogmatic democrat, for he is concerned with preserving the conditions that make democracy workable. It is not \"antidemocratic\" to try to persuade the majority that there are limits beyond which its action ceases to be beneficial and that it should observe principles which are not of its own deliberate making. If it is to survive, democracy must recognize that it is not the fountainhead of justice and that it needs to acknowledge a conception of justice which does not necessarily manifest itself in the popular view on every particular issue. The danger is that we mistake a means of securing justice for justice itself. Those who endeavor to persuade majorities to recognize proper limits to their just power are therefore as necessary to the democratic process as those who constantly point to new goals for democratic action.\n\nIn Part II of this book we shall consider further those limits on government which seem to be the necessary condition for the workability of democracy and which the people of the West have developed under the name of the rule of law. Here we will merely add that there is little reason to expect that any people will succeed in successfully operating or preserving a democratic machinery of government unless they have first become familiar with the traditions of a government of law.\n\n* * *\n\nThe quotation at the head of the chapter is taken from David Hume, _Essays_ [\"Whether the British Government inclines more to Absolute Monarchy, or to a Republic\" (Essay 9)], vol. 1 p. 125 [Liberty Fund edition, p. 51]. The idea apparently derives from the great debates of the preceding century. William Haller reprints as the Frontispiece to vol. 1 of the _Tracts on Liberty in the Puritan Revolution, 1638\u20131647_ (3 vols.; New York: Columbia University Press, 1934), a broadside with an engraving by Wenceslas Hollar, dated 1641 and headed \"The World Is Ruled and Governed by Opinion.\"\n\n1 On the origin of the conception of the \"total' state and on the opposition of totalitarianism to liberalism, but not to democracy, see the early discussion in Heinz Otto Ziegler, _Autorit\u00e4rer oder totaler Staat_ [Reich und Staat in Geschichte und Gegenwart, no. 90] (T\u00fcbingen: Mohr, 1932), esp. pp. 6\u201314; cf. Franz Leopold Neumann, _The Democratic and the Authoritarian State: Essays in Political and Legal Theory_ (Glencoe, IL: Free Press, 1957). The view of what throughout this chapter we shall call the \"dogmatic democrats\" may be clearly seen in Edwin Mims, Jr., _The Majority of the People_ (New York: Modern Age Books, 1941), and Henry Steele Commager, _Majority Rule and Minority Rights_ [ James W. Richards Lectures in American History, the University of Virginia] (New York: Oxford University Press, 1943\n\n2 Cf., e.g., Jos\u00e9 Ortega y Gasset, _Invertebrate Spain_ , Mildred Adams, trans. (New York: W. W. Norton, 1937), p. 125: \"Liberalism and Democracy happen to be two things which begin by having nothing to do with each other, and end by having, so far as tendencies are concerned, meanings that are mutually antagonistic. Democracy and Liberalism are two answers to two completely different questions.\n\n\"Democracy answers this question\u2014'Who ought to exercise the public power?' The answer it gives is\u2014the exercise of public power belongs to the citizens as a body.\n\n\"But this question does not touch on what should be the realm of the public power. It is solely concerned with determining to whom such power belongs. Democracy proposes that we all rule; that is, that we are sovereign in all social acts.\n\n\"Liberalism, on the other hand, answers this other question,\u2014'regardless of who exercises the public power, what should its limits be?' The answer it gives\u2014'Whether the public power is exercised by an autocrat or by the people, it cannot be absolute: the individual has rights which are over and above any interference by the state.'\"\n\n[Volume 2 of the _Obras Completas (Sobre la raz\u00f3n hist\u00f3rica)_ , Paulino Garagorri, ed. (Madrid: Alianza 1979), shows this quotation as appearing not in _Espa\u00f1a Invertebrada_ but rather in one of Ortega's articles that was published in _El Espectador_ in 1927 entitled \"Ideas de los Castillos: Liberalismo y Democracia.\" The Spanish reads:\n\n\"Liberalismo y democracia son dos cosas que empiezan por no tener nada que ver entre s\u00ed, y acaban por ser, en cuanto tendencias, de sentido antag\u00f3nico.\n\n\"Democracia y liberalismo son dos respuestas a dos cuestiones de derecho pol\u00edtico completamente distintas.\n\n\"La democracia responde a esta pregunta: \u00bfQui\u00e9n debe ejercer el Poder p\u00fablico? La respuesta es: el ejercicio del Poder p\u00fablico corresponde a la colectividad de los ciudadanos.\n\n\"Pero en esa pregunta no se habla de qu\u00e9 extensi\u00f3n deba tener el Poder p\u00fablico. Se trata s\u00f3lo de determinar el sujeto a quien el mando compete. La democracia propone que mandemos todos; es decir: que todos intervengamos soberanamente en los hechos sociales.\n\n\"El liberalismo, en cambio, responde a esta otra pregunta: ejerza quienquiera el Poder p\u00fablico, \u00bfcu\u00e1les deben ser los l\u00edmites de \u00e9ste? La respuesta suena as\u00ed: el Poder p\u00fablico, ej\u00e9rzalo un aut\u00f3crata o el pueblo, no puede ser absoluto, sino que las personas tienen derechos previos a toda injerencia del Estado. Es, pues, la tendencia a limitar la intervenci\u00f3n del Poder p\u00fablico.\" (pp. 416\u201317).\u2014Ed.]\n\nSee also the same author's _The Revolt of the Masses_ (London: Allen and Unwin, 1932), p. 83. No less emphatic, from the dogmatic democratic position, is Max Lerner, \"Minority Rule and the Constitutional Tradition,\" in _The Constitution Reconsidered_ , Conyers Read, ed. (New York: Columbia University Press, 1938), p. 199: \"When I speak of _democracy_ here, I want to distinguish it sharply from _liberalism._ There is no greater confusion in the layman's mind today than the tendency to identify the two.\" Cf. also Hans Kelsen, \"Foundations of Democracy,\" _Ethics_ , 66, no. 1, pt. 2 (1955): 3: \"It is of importance to be aware that the principle of democracy and that of liberalism are not identical, that there exists even a certain antagonism between them.\" Also Ruth Fulton Benedict, \"Primitive Freedom,\" _Atlantic Monthly_ , 30 (1942): 760: \"But being a democracy has not itself guaranteed the blessings of liberty.\"\n\nOne of the best historical accounts of the relation is to be found in Franz Schnabel, _Deutsche Geschichte im neunzehnten Jahrhundert_ , Vol. 2: _Monarchie und Volkssouver\u00e4nit\u00e4t_ (4 vols.; Freiburg im Breisgau: Herder, 1933), p. 98: \"Liberalismus und Demokratie waren also nicht sich ausschlie\u00dfende Gegens\u00e4tze, sondern handelten von zwei verschiedenen Dingen: der Liberalismus sprach vom Umfang der staatlichen Wirksamkeit, die Demokratie vom Inhaber der staatlichen Souver\u00e4nit\u00e4t.\" [\"Liberalism and democracy were not, then, mutually exclusive nor opposites; rather they addressed two different things. Liberalism spoke to the extent of the government's potency while democracy had reference to those in whose hands the state's sovereignty rested.\"\u2014Ed.] Cf. also Abbott Lawrence Lowell, \"Democracy and Liberty,\" in _Essays on Government_ (Boston: Houghton Mifflin, 1889), esp. pp. 50\u2013117; Carl Schmitt, _Die geistesgeschichtliche Lage des heutigen Parlamentarismus_ (Munich: Duncker und Humblot, 1926); Gustav Radbruch, _Rechtsphilosophie_ (4th ed.; Stuttgart: K. F. Koehler, 1950), pp. 137ff., esp. p. 156\u201362; Benedetto Croce, \"Liberalism as a Concept of Life,\" _Politics and Morals_ (New York: Philosophical Library, 1945); and Leopold von Wiese, \"Liberalismus and Demokratismus in ihren Zusammenh\u00e4ngen und Gegens\u00e4tzen,\" _Zeitschrift f\u00fcr Politik_ , 9 (1916): 407\u201325; Heinrich Ritter von Srbik, _Deutsche Einheit: Idee und Wirklichkeit vom Heiligen Reich bis K\u00f6niggr\u00e4tz_ (4 vols.; Munich: F. Bruckmann KG, 1940), vol. 1, p. 245; Gerhard Ritter, _Vom sittlichen Problem der Macht_ (Bern: A. Francke AG. Verlag, 1948), pp. 106 and 118; Peter Rossi, \"Liberismo e regime parlamentare in Gaetano Mosca,\" _Giornale degli Economisti e Annali di Economia_ , n.s., 8 (1949): 621\u201334; Werner K\u00e4gi, \" _Rechtsstaat_ und Demokratie: Antinomie und Synthese,\" in _Demokratie und Rechtsstaat: Festgabe zum 60. Geburstag von Zaccaria Giacometti_ (Z\u00fcrich: Polygraphischer Verlag, 1953), pp. 107\u201342. A useful survey of some of the literature is Josef Th\u00fcr, _Demokratie und Liberalismus in ihrem gegenseitigen Verh\u00e4ltnis_ (dissertation; Bischofszell: Buchdruckerei L. Hildebrand, 1944). See also Konrad von Orelli, _Das Verh\u00e4ltnis von Demokratie und Gewaltenteilung und seine Wandlungen insbesondere im schweizerischen Bunde_ (dissertation; W\u00e4denswil: Villiger, 1947).\n\n3 See Ferdinand Aloys Hermens, _Democracy or Anarchy? A Study of Proportional Representation_ (Notre Dame, IN: Review of Politics, Notre Dame University, 1941).\n\n4 It is useful to remember that in the oldest and most successful of European democracies, Switzerland, women are still excluded from the vote and apparently with the approval of the majority of them. It also seems possible that in primitive conditions only a suffrage confined, say, to landowners would produce a legislature sufficiently independent of the government to exercise effective control over it. [Women's suffrage in Switzerland was introduced at the cantonal level in 1971 and enshrined in the Swiss Constitution in 1985.\u2014Ed.]\n\n5 Cf. Frederic William Maitland, _The Collected Papers of Frederic William Maitland, Downing Professor of the Laws of England_ (3 vols.; Cambridge: Cambridge University Press, 1911), vol. 1, p. 84: \"Those who took the road to democracy to be the road to freedom mistook temporary means for an ultimate end.\" Also Joseph Alois Schumpeter, _Capitalism, Socialism, and Democracy_ (New York: Harper and Brothers, 1942), p. 242: \"Democracy is a political _method_ , that is to say, a certain type of institutional arrangement for arriving at political\u2014legislative and administrative\u2014 decisions and hence incapable of being an end in itself, irrespective of what decisions it will produce under given historical conditions.\"\n\n6 Cf. Edward Adamson Hoebel, _The Law of Primitive Man: A Study in Comparative Legal Dynamics_ (Cambridge, MA: Harvard University Press, 1954), p. 100, and Fritz Fleiner, _Tradition, Dogma, Entwicklung als aufbauende Kr\u00e4fte der schweizerischen Demokratie_ (Zurich: O. F\u00fcssli, 1933), reprinted in the author's _Ausgew\u00e4hlte Schriften und Reden_ (Zurich: Polygraphischer Verlag, 1941), pp. 288\u2013302; also Carl Menger, _Untersuchungen_ , p. 277: \"Das Volksrecht in seiner urspr\u00fcnglichsten Form ist solcherart allerdings nicht das Ergebnis eines Vertrags oder einer auf die Sicherung des Gemeinwohls hinzielenden Ref exion. Es ist aber auch nicht, wie die histor ische Schule behauptet, mit dem Volke zugleich gegeben; es ist vielmehr \u00e4lter, als die Erscheinung dieses letztern, ja es ist eines der st\u00e4r ksten Bindemittel, durch welches die Bev\u00f6lkerung eines Territoriums zu einem Volke wird und zu einer staatlichen Organisation gelangt.\" [\"National law in its most original form is thus, to be sure, not the result of a contract or a reflection aiming at the assurance of the common welfare. Nor is it, indeed, given with the nation, as the historical school asserts. Rather, it is older than the appearance of the latter. Indeed, it is one of the strongest ties by which the population of a territory becomes a nation and achieves state organization.\" (Carl Menger, _Investigations into the Method of the Social Sciences_ , Francis J. Nock, trans. [Grove City, PA: Libertarian Press, Inc., 1996], p. 215.)\u2014Ed.]\n\n7 Cf., e.g., Joseph Chamberlain's speech to the \"Eighty\" Club, April 28, 1885 (reported in the _Times_ [London], April 29, 1885): \"When government was represented only by the authority of the Crown and the views of a particular class, I can understand that it was the first duty of men who valued their freedom to restrict its authority and to limit its expenditure. But all that is changed. Now, government is the organized expression of the wishes and the wants of the people and under these circumstances let us cease to regard it with suspicion. Suspicion is the product of an older time, of circumstances which have long since disappeared. Now it is our business to extend its functions and to see in what way its operations can be usefully enlarged.\" [The Eighty Club comprised a group of liberal MPs who were first returned to Parliament in the 1880 election.\u2014Ed.] But see John Stuart Mill, in 1848 already arguing against this view in _Principles_ , bk. 5, chap. 11, sec. 3, p. 944 [Liberty Fund edition, _Collected Works_ , vol. 3, pp. 944\u2013 45], and also in \"On Liberty,\" in _On Liberty and Considerations on Representative Government_ , Ronald Buchanan McCallum, ed. (Oxford: Blackwell, 1946), p. 3. See also the statement made by Thomas Jefferson noted in chap. 16, n. 79, below (Jefferson's draft of the Kentucky Resolutions, in Ethelbert Dudley Waterf eld, _The Kentucky Resolutions of 1798: An Historical Study_ [2nd ed.; New York: Putnam, 1894], pp. 157\u201358). Indeed, this idea can be tr aced back to Rousseau; see his _Du contrat social_ [Biblioth\u00e8que Philosophique; Paris: Aubier, Editions Montaigne, 1943], bk. 1, chap. 7 (\"Du Souverain\", p. 106) who offers the view that when a people form a legislature with plenary powers \"le puissance Souveraine n'a nul besoin de garant envers les sujets, parce qu'il est impossible que le corps veuille nuire \u00e0 tous ses membres, et nous verrons ci-apr\u00e8s qu'il ne peut nuire \u00e0 aucun en particuli\u00e8re.\" [\"the sovereign power need offer no guarantee to its subjects, because it is impossible for the body to wish to hurt all its members. And we shall later see that it cannot hurt any in particular.\"\u2014Ed.]\n\n8 Herman Finer, _Road to Reaction_ (Boston: Little, Brown and Co., 1945), p. 60.\n\n9 See James Fitzjames Stephen, _Liberty, Equality, Fraternity_ (London: Smith, Elder, and Co., 1873), p. 27\u201328 [Liberty Fund edition, p. 21]: \"We agree to try strength by counting heads instead of breaking heads. . . . It is not the wisest side which wins, but the one which for the time being shows its superior strength (of which no doubt wisdom is one element) by enlisting the largest amount of active sympathy in its support. The minority gives way, not because it is convinced that it is wrong, but because it is convinced that it is a minority.\" Cf. also Ludwig von Mises, _Human Action_ (New Haven: Yale University Press, 1949), p. 150 [Liberty Fund edition, vol. 1, p. 150]: \"For the sake of domestic peace, liberalism aims at democratic government. Democracy is therefore not a revolutionary institution. On the contrary, it is the very means of preventing revolutions and civil wars. It provides a method for the peaceful adjustment of government to the will of the majority.\" Similarly, Sir Karl Raimund Popper, \"Prediction and Prophecy and Their Significance for Social Theory,\" _Proceedings of the 10th International Congress of Philosophy, Amsterdam, August 11\u201318, 1948_ , Evert Willem Beth, ed. (2 vols.; Amsterdam: North Holland Publishing Co., 1948), vol. 1, esp. p. 90: \"I personally call the type of government which can be removed without violence 'democracy,' and the other 'tyranny.'\" See also Learned Hand, \"Democracy: Its Presumptions and Realities,\" _The Spirit of Liberty: Papers and Addresses of Learned Hand_ , Irving Dillard, ed. (New York: Alfred A. Knopf, 1952), p. 98: \"For abuse it as you will, it gives a bloodless measure of social forces\u2014bloodless, have you thought of that?\u2014a means of continuity, a principle of stability, a relief from the paralyzing terror of revolution.\"\n\n10 _An Exact Collection of All the Remonstrances, Declarations, Votes, Orders, Ordinances, Proclamations, Petitions, Messages, Answers, and Other Remarkable Passages between the King's Most Excellent Majesty, and His High Court of Parliament, Beginning at His Majesties Return from Scotland in December, 1641, and Continued until March the 21, 1643_ (London: Printed for E. Husbands, T. Warren, R. Best, 1643), p. 320. [These words were uttered by none other than King Charles I in 1642. By the beginning of June 1642 most of the Royalist supporters in Parliament had returned to their home districts, at which point the members of the Parliamentary party who remained at Westminster took the occasion to draw up a petition to Charles I in the form of a series of demands that would have substantially circumscribed the King's powers. Known as the Nineteen Propositions, these proposals were passed by Parliament on June 1, 1642, and presented to the King, who responded on June 18. The response was in fact written by Sir John Culpeper and Lucius Cary, Viscount Falkland. The passage to which Hayek refers was almost certainly written by Culpeper.\u2014Ed.]\n\n11 How fascinated the rationalistic liberals were by the conception of a government in which political issues were decided not \"by an appeal, either direct or indirect, to the judgment or will of an uninstructed mass, whether of gentlemen or of clowns, but by the deliberately formed opinions of a comparatively few, specially educated for the task,\" is well illustrated by John Stuart Mill's early essay on \"Democracy and Government\" from which this fragment is taken ( _London Review_ , 31 [(October) 1835]:85\u2013129, reprinted in _Early Essays_ [London: G. Bell and Sons, 1897], p. 384). [Mill's essay in the _London Review_ is from a review of Tocqueville's _Democracy in America_.\u2014Ed.] He goes on to point out that \"of all governments, ancient or modern, the one by which this excellence is possessed in the most eminent degree, is the government of Prussia\u2014 a most powerfully and skillfully organized aristocracy of the most highly-educated men in the kingdom.\" Cf. also the passage in _On Liberty_ , p. 9. With respect to the applicability of freedom and democracy to less civilized people, some of the old Whigs were considerably more liberal than the later radicals. Thomas Babington Macaulay, for example, says somewhere: \"Many politicians of our time are in the habit of laying it down as a self-evident proposition, that no people ought to be free till they are fit to use their freedom. The maxim is worthy of the fool in the old story, who resolved not to go into the water till he had learned to swim. If men are to wait for liberty till they become wise and good in slavery, they may indeed have to wait forever.\" [The quotation is from Macaulay's essay on Milton, included in his _Critical and Historical Essays_ (Everyman's Library; 2 vols.; London: J. M. Dent and Sons, 1907), vol. 2, p. 180.\u2014Ed.]\n\n12 This seems also to explain the puzzling contrast between Tocqueville's persistent faultfinding with democracy on almost all particular points and the emphatic acceptance of the principle which is so characteristic of his work.\n\n13 See Kenneth Ewart Boulding, _The Organizational Revolution: A Study in the Ethics of Economic Organization_ [Federal Council of the Churches of Christ in America] (New York: Harper and Bros., 1953), p. 250: \"Increasingly, therefore, the state becomes an entity separate from its citizens even in democratic societies, making decisions of which they are not aware, maneuvering them into positions from which they cannot retreat, itself creating the public opinion on which its power ultimately rests, until the state is now in danger of becoming the greatest enemy of man instead of his wisest friend.\"\n\n14 Cf. the passage by Dicey, _Law and Public Opinion_ , p. 33 [Liberty Fund edition, p. 25], quoted in n. 16 below.\n\n15 John Stuart Mill, \"Bentham,\" _London and Westminster Review_ , 39 (August 1838): 327, reprinted in _Dissertations and Discussions Political, Philosophical, and Historical: Reprinted Chiefly from the Edinburgh and Westminster Reviews_ (3rd ed.; 4 vols.; London: Longmans, Green, Reader, and Dyer, 1875), vol. 1, pp. 330\u201331 [Liberty Fund edition, _Collected Works_ , vol. 10, p. 77]. [Mill's essay on Bentham has also been reprinted in _Early Essays by John Stuart Mill_ , J. W. M. Gibbs, ed. (London: George Bell and Sons, 1897), pp. 327\u201328.\u2014Ed.] The passage continues: \"The writers of whom we speak [i.e., Bentham and Coleridge] have never been read by the multitude; except for the more slight of their works, their readers have been few: but they have been the teachers of the teachers; there is hardly to be found in England an individual of any importance in the world of mind, who (whatever opinions he may have afterwards adopted) did not first learn to think from one of these two; and though their influences have but begun to diffuse themselves through these intermediate channels over society at large, there is already scarcely a publication of any consequence addressed to the educated classes, which, if these persons had not existed, would not have been different from what it is.\" Cf. also the frequently quoted passage by Lord [ John Maynard] Keynes, himself the most eminent example of such influence in our generation, in which he argues, at the end of _The General Theory of Employment, Interest, and Money_ (London: Macmillan, 1936), pp. 383\u201384, that \"the ideas of economists and political philosophers, both when they are right and when they are wrong, are more powerful than is commonly understood. Indeed the world is ruled by little else. Practical men, who believe themselves to be quite exempt from any intellectual influences, are usually the slaves of some defunct economist. Madmen in authority, who hear voices in the air, are distilling their frenzy from some academic scribbler of a few years back. I am sure that the power of vested interests is vastly exaggerated compared with the gradual encroachment of ideas. Not, indeed, immediately, but after a certain interval; for in the field of economic and political philosophy there are not many who are influenced by new theories after they are twenty-five or thirty years of age, so that the ideas which civil servants and politicians and even agitators apply to current events are not likely to be the newest. But, soon or late, it is ideas, not vested interests, which are dangerous for good or evil.\"\n\n16 The classical description of the manner in which ideas at a long interval affect policy is still that by Dicey, _Law and Public Opinion_ , pp. 28ff. and esp. p. 33 [Liberty Fund edition, p. 25]: \"The opinion which changes the law is in one sense the opinion of the time when the law is actually altered; in another sense it has often been in England the opinion prevalent some twenty or thirty years before that time; it has been as often as not in reality the opinion not of to-day but of yesterday.\n\n\"Legislative opinion must be the opinion of the day, because, when laws are altered, the alteration is of necessity carried into effect by legislators who act under the belief that the change is an amendment; but this law-making opinion is also the opinion of yesterday, because the beliefs which have at last gained such hold on the legislature as to produce an alteration in the law have generally been created by thinkers or writers, who exerted their influence long before the change in the law took place. Thus it may well happen that an innovation is carried through at a time when the teachers who supplied the arguments in its favour are in their graves, or even\u2014 and this is well worth noting\u2014when in the world of speculation a movement has already set in against ideas which are exerting their full effect in the world of action and of legislation.\"\n\n17 Cf. Helmut Schoeck, \"What Is Meant by 'Politically Impossible'?\" _Pall Mall Quarterly_ , 1 (1958): 48\u201353; see also Clarence Philbrook, \"'Realism' in Policy Espousal,\" _American Economic Review_ , 43 (1953): 846\u201359.\n\n18 Cf. Marshall's observation (\"In Memoriam: Alfred Marshall,\" _Memorials of Alfred Marshall_ , Arthur Cecil Pigou, ed. [London: Macmillan, 1925], p. 89) that \"students of social science must fear popular approval: evil is with them when all men speak well of them. If there is any set of opinions by the advocacy of which a newspaper can increase its sale, then the student, who wishes to leave the world in general and his country in particular better than it would be if he had not been born, is bound to dwell on the limitations and defects and errors, if any, in that set of opinions: and never to advocate them unconditionally even in an _ad hoc_ discussion. It is almost impossible for a student to be a true patriot and to have the reputation of being one in his own time.\"\n\n19 See the fuller discussion of these issues in chap. 5 [pp. 56\u201371] of my book _The Road to Serfdom_ (Chicago, University of Chicago Press, 1944) [reprinted as vol. 2 of _The Collected Works of F. A. Hayek_ , Bruce Caldwell, ed. (Chicago: University of Chicago Press, 2007), pp. 100\u2013111] and in Walter Lippmann, _An Inquiry into the Principles of the Good Society_ (Boston: Little, Brown and Co., 1937), esp. p. 267: \"[The people] can govern only when they understand how a democracy _can_ govern itself; that it can govern only by appointing representatives to adjudicate, enforce, and revise laws which declare the rights, duties, privileges, and immunities of persons, associations, communities, and the officials themselves, each in respect to all others.\n\n\"This is the constitution of a free state. Because democratic philosophers in the nineteenth century did not clearly see that the indispensable corollary of representative government is a particular mode of governing, they were perplexed by the supposed conflict between law and liberty, between social control and individual freedom. These conflicts do not exist where social control is achieved by a legal order in which reciprocal rights are enforced and adjusted. Thus in a free society the state does not administer the affairs of men. It administers justice among men who conduct their own affairs.\"\n\n## EIGHT\n\nEMPLOYMENT AND INDEPENDENCE\n\nNot for to hide it in a hedge,  \nNot for a train attendant,  \nBut for the glorious privilege  \nOf being independent.\n\n\u2014Robert Burns\n\n1. The ideals and principles restated in the preceding chapters were developed in a society which in important respects differed from ours. It was a society in which a relatively larger part of the people, and most of those who counted in forming opinion, were independent in the activities that gave them their livelihood.1 How far, then, are those principles which operated in such a society still valid now, when most of us work as employed members of large organizations, using resources we do not own and acting largely on the instructions given by others? In particular, if the independents now constitute a so much smaller and less influential portion of society, have their contributions for this reason become less important, or are they still essential to the well-being of any free society?\n\nBefore we turn to the main issue, we must free ourselves from a myth concerning the growth of the employed class which, though believed in its crudest form only by Marxists, has gained wide enough acceptance to confuse opinion. This is the myth that the appearance of a propertyless proletariat is the  result of a process of expropriation, in the course of which the masses were deprived of those possessions that formerly enabled them to earn their living independently. The facts tell a very different story. Until the rise of modern capitalism, the possibility for most people of establishing a family and of rearing children depended on the inheritance of a home and land and the necessary tools of production. What later enabled those who did not inherit land and tools from their parents to survive and multiply was the fact that it became practicable and profitable for the wealthy to use their capital in such a way as to give employment to large numbers. If \"capitalism has created the proletariat,\" it has done so, then, by enabling large numbers to survive and procreate. In the Western world today, the effect of this process is, of course, no longer the increase in a proletariat in the old sense but the growth of a majority of employed who in many respects are alien and often inimical to much that constitutes the driving force of a free society.\n\nThe increase in population during the last two hundred years has been made up mostly of employed workers, urban and industrial. Though the technological change that has favored large-scale enterprise and helped to create the new large class of clerical workers has undoubtedly assisted this growth of the employed section of the population, the increasing number of propertyless that offered their services has probably in turn assisted the growth of large-scale organization.\n\nThe political significance of this development has been accentuated by the fact that, at the time when the dependent and propertyless were growing most rapidly in numbers, they were also given the franchise, from which most of them had been excluded. The result was that in probably all countries of the West the outlook of the great majority of the electorate came to be determined by the fact that they were in employed positions. Since it is now their opinion that largely governs policy, this produces measures that make the employed positions relatively more attractive and the independent ones ever less so. That the employed should thus use their political power is natural. The problem is whether it is in their long-term interest if society is thereby progressively turned into one great hierarchy of employment. Such a state seems to be the likely outcome unless the employed majority come to recognize that it would be in their interest to ensure the preservation of a substantial number of independents. For if they do not, we shall all find that our freedom has been affected, just as they will find that, without a great variety of employers to choose from, their position is not as it once was.\n\n2. The problem is that many exercises of freedom are of little direct interest to the employed and that it is often not easy for them to see that their freedom depends on others' being able to make decisions which are not immediately relevant to their whole manner of life. Since they can and have to live without  making such decisions, they cannot see the need for them, and they attach little importance to opportunities for action which hardly ever occur in their lives. They regard as unnecessary many exercises of freedom which are essential to the independent if he is to perform his functions, and they hold views of deserts and appropriate remuneration entirely different from his. Freedom is thus seriously threatened today by the tendency of the employed majority to impose upon the rest their standards and views of life. It may indeed prove to be the most difficult task of all to persuade the employed masses that in the general interest of their society, and therefore in their own long-term interest, they should preserve such conditions as to enable a few to reach positions which to them appear unattainable or not worth the effort and risk.\n\nIf in the life of the employed certain exercises of liberty have little relevance, this does not mean that they are not free. Every choice made by a person as to his manner of life and way of earning a living means that, as a result, he will have little interest in doing certain things. A great many people will choose employment because it offers them better opportunities to live the kind of life they want than would any independent position. Even with those who do not especially want the relative security and absence of risk and responsibility that an employed position brings, the decisive factor is often not that independence is unattainable but that employment offers them a more satisfying activity and a larger income than they could earn as, say, independent tradesmen.\n\nFreedom does not mean that we can have everything as we want it. In choosing a course of life we always must choose between complexes of advantages and disadvantages, and, once our choice is made, we must be prepared to accept certain disadvantages for the sake of the net benefit. Whoever desires the regular income for which he sells his labor must devote his working hours to the immediate tasks which are determined for him by others. To do the bidding of others is for the employed the condition of achieving his purpose. Yet, though he may find this at times highly irksome, in normal conditions he is not unfree in the sense of being coerced. True, the risk or sacrifice involved in giving up his job may often be so great as to make him continue in it, even though he intensely dislikes it. But this may be true of almost any other occupation to which a man has committed himself\u2014certainly of many independent positions.\n\nThe essential fact is that in a competitive society the employed is not at the mercy of a particular employer, except in periods of extensive unemployment. The law wisely does not recognize contracts for the permanent sale of a person's labor and, in general, does not even enforce contracts for specific performance. Nobody can be coerced to continue to work under a particular boss, even if he has contracted to do so; and, in a normally operating competitive  society, alternative employment will be available, even though it may often be less remunerative.2\n\nThat the freedom of the employed depends upon the existence of a great number and variety of employers is clear when we consider the situation that would exist if there were only one employer\u2014namely, the state\u2014and if taking employment were the only permitted means of livelihood. And a consistent application of socialist principles, however much it might be disguised by the delegation of the power of employment to nominally independent public corporations and the like, would necessarily lead to the presence of a single employer. Whether this employer acted directly or indirectly, he would clearly possess unlimited power to coerce the individual.\n\n3. The freedom of the employed therefore depends on the existence of a group of persons whose position is different from theirs. Yet in a democracy in which they form the majority, it is their conception of life that can determine whether or not such a group can exist and fulfill its functions. The dominant conceptions will be those of the great majority, who are members of hierarchic organizations and who are largely unaware of the kind of problems and views that determine the relations between the separate units within which they work. The standards which such a majority develops may enable them to be effective members of society, but they cannot be applied to the whole of society if it is to remain free.\n\nIt is inevitable that the interests and values of the employed should differ somewhat from those of men who accept the risk and responsibility of organizing the use of resources. A man who works under direction for a fixed salary or wage may be as conscientious, industrious, and intelligent as one who must constantly choose between alternatives; but he can hardly be as inventive or as experimental simply because the range of choice in his work is more limited.3 He is normally not expected to perform actions which cannot be  prescribed or which are not conventional. He cannot go beyond his allotted task even if he is capable of doing more. An assigned task is necessarily a limited task, confined to a given sphere and based on a predetermined division of labor.\n\nThe fact of being employed will affect more than a man's initiative and inventiveness. He has little knowledge of the responsibilities of those who control resources and who must concern themselves constantly with new arrangements and combinations; he is little acquainted with the attitudes and modes of life which the need for decisions concerning the use of property and income produces. For the independent there can be no sharp distinction between his private and his business life, as there is for the employed, who has sold part of his time for a fixed income. While, for the employed, work is largely a matter of fitting himself into a given framework during a certain number of hours, for the independent it is a question of shaping and reshaping a plan of life, of finding solutions for ever new problems. Especially do the employed and the independent differ in their views of what one can properly regard as income, what chances one ought to take, and what manner of life one should adopt that is most conducive to success.\n\nThe greatest difference between the two, however, will be found in their opinions of how appropriate remunerations for various services are to be determined. Whenever a person works under instruction and as a member of a large organization, the value of his individual services is difficult to ascertain. How faithfully and intelligently he has obeyed rules and instructions, how well he has fitted himself into the whole machinery, must be determined by the opinion of other people. Often he must be remunerated according to assessed merit, not according to result. If there is to be contentment within the organization, it is most important that remuneration be generally regarded as just, that it conform to known and intelligible rules, and that a human agency be responsible for every man's receiving what his fellows regard as being due to him.4 However, this principle of rewarding a man according to what others think he deserves cannot apply to men who act on their own initiative.\n\n4. When an employed majority determines legislation and policy, conditions will tend to be adapted to the standards of that group and become less favorable to the independent. The position of the former will, in consequence,  become steadily more attractive and its relative strength even greater. It may be that even the advantages which the large organization has today over the small are in part a result of policies that have made employed positions more attractive to many who in the past would have aimed at independence.\n\nThere can be little doubt, at any rate, that employment has become not only the actual but the preferred position of the majority of the population, who find that it gives them what they mainly want: an assured fixed income available for current expenditure, more or less automatic raises, and provision for old age. They are thus relieved of some of the responsibilities of economic life; and quite naturally they feel that economic misfortune, when it comes as a result of a decline or failure of the employing organization, is clearly not their fault but somebody else's. It is not surprising, then, that they should wish to have some higher tutelary power watch over the directing activities which they do not understand but on which their livelihood depends.\n\nWhere this class predominates, the conception of social justice becomes largely adjusted to its needs. This applies not only to legislation but also to institutions and business practices. Taxation comes to be based on a conception of income which is essentially that of the employee. The paternalistic provisions of the social services are tailored almost exclusively to his requirements. Even the standards and techniques of consumers' credit are primarily adjusted to them. And all that concerns the possession and employment of capital as part of making one's living comes to be treated as the special interest of a small privileged group which can justly be discriminated against.\n\nTo Americans this picture may still seem exaggerated, but to Europeans most of its features are all too familiar. The development in this direction is generally much accelerated, once the public servants become the most numerous and influential group among the employed, and the special privileges which they enjoy come to be demanded as a matter of right by all employees. Privileges such as security of tenure and automatic promotion by seniority that the public servant is given, not in his interest but in the interest of the public, then tend to be extended beyond this group. Also, it is even more true of government bureaucracy than of other large organizations that the specific value of an individual's services cannot be ascertained and that he must therefore be rewarded on the basis of assessable merit rather than result.5 Such standards that prevail in the bureaucracy tend to spread, not least through the influence of public servants on legislation and on the new institutions catering to the needs of the employed.6 In many European countries the bureaucracy  of the new social services in particular has become a very important political factor, the instrument as well as the creator of a new conception of need and merit, to whose standards the life of the people is increasingly subject.\n\n5. The existence of a multiplicity of opportunities for employment ultimately depends on the existence of independent individuals who can take the initiative in the continuous process of re-forming and redirecting organizations. It might at first seem that multiplicity of opportunities could also be provided by numerous corporations run by salaried managers and owned by large numbers of shareholders and that men of substantial property would therefore be superfluous. But though corporations of this sort may be suited to well-established industries, it is very unlikely that competitive conditions could be maintained, or an ossification of the whole corporate structure be prevented, without the launching of new organizations for fresh ventures, where the propertied individual able to bear risks is still irreplaceable. And this superiority of individual over collective decisions is not confined to new ventures. However adequate the collective wisdom of a board may be in most instances, the outstanding success even of large and well-established corporations is often due to some single person who has achieved his position of independence and influence through the control of large means. However much the institution of the corporation may have obscured the simple distinction between the directing owner and the employee, the whole system of separate enterprises, offering both employees and consumers sufficient alternatives to deprive each organization from exercising coercive power, presupposes private ownership and individual decision as to the use of resources.7\n\n6. The importance of the private owner of substantial property, however, does not rest simply on the fact that his existence is an essential condition for the preservation of the structure of competitive enterprise. The man of independent means is an even more important figure in a free society when he is not occupied with using his capital in the pursuit of material gain but uses it in the service of aims which bring no material return.8 It is more in the support of aims which the mechanism of the market cannot adequately take care of than in preserving that market that the man of independent means has his indispensable role to play in any civilized society.9\n\nThough the market mechanism is the most effective method for securing those services that can be priced, there are others of great importance that the market will not provide because they cannot be sold to the individual beneficiary. Economists have often given the impression that only what the public can be made to pay for is useful or have mentioned the exceptions only as an argument for the state's stepping in where the market has failed to provide whatever is desired. But, though the limitations of the market provide a legitimate argument for some kinds of government action, they certainly do not justify the argument that only the state should be able to provide such services. The very recognition that there are needs which the market does not satisfy should make it clear that the government ought not to be the only agency able to do things which do not pay, that there should be no monopoly here but as many independent centers as possible able to satisfy such needs.\n\nThe leadership of individuals or groups who can back their beliefs financially is particularly essential in the field of cultural amenities, in the fine arts, in education and research, in the preservation of natural beauty and historic treasures, and, above all, in the propagation of new ideas in politics, morals, and religion. If minority views are to have a chance to become majority views, it is necessary not only that men who are already highly esteemed by the majority should be able to initiate action but that representatives of all divergent views and tastes should be in a position to support with their means and their energy ideals which are not yet shared by the majority.\n\nIf we knew of no better way of providing such a group, there would exist a strong case for selecting at random one in a hundred, or one in a thousand, from the population at large and endowing them with fortunes sufficient for the pursuit of whatever they choose. So long as most tastes and opinions were represented and every type of interest given a chance, this might be well worth while, even if, of this fraction of the population, again only one in a  hundred or one in a thousand used the opportunity in a manner that in retrospect would appear beneficial. The selection through inheritance from parents, which in our society, in fact, produces such a situation, has at least the advantage (even if we do not take into account the probability of inherited ability) that those who are given the special opportunity will usually have been educated for it and will have grown up in an environment in which the material benefits of wealth have become familiar and, because they are taken for granted, have ceased to be the main source of satisfaction. The grosser pleasures in which the newly rich often indulge have usually no attraction for those who have inherited wealth. If there is any validity in the contention that the process of social ascent should sometimes extend through several generations, and if we admit that some people should not have to devote most of their energies to earning a living but should have the time and means to devote themselves to whatever purpose they choose, then we cannot deny that inheritance is probably the best means of selection known to us.\n\nThe point that is so frequently overlooked in this connection is that action by collective agreement is limited to instances where previous efforts have already created a common view, where opinion about what is desirable has become settled, and where the problem is that of choosing between possibilities already generally recognized, not that of discovering new possibilities. Public opinion, however, cannot decide in what direction efforts should be made to arouse public opinion, and neither government nor other existing organized groups should have the exclusive power to do so. But organized efforts have to be set in motion by a few individuals who possess the necessary resources themselves or who win the support of those that do; without such men, what are now the views of only a small minority may never have a chance of being adopted by the majority. What little leadership can be expected from the majority is shown by their inadequate support of the arts wherever they have replaced the wealthy patron. And this is even more true of those philanthropic or idealistic movements by which the moral values of the majority are changed.\n\nWe cannot attempt to recount here the long story of all good causes which came to be recognized only after lonely pioneers had devoted their lives and fortunes to arousing the public conscience, of their long campaigns until at last they gained support for the abolition of slavery, for penal and prison reform, for the prevention of cruelty to children or to animals, or for a more humane treatment of the insane. All these were for a long time the hopes of only a few idealists who strove to change the opinion of the overwhelming majority concerning certain accepted practices.\n\n7. The successful performance of such a task by the wealthy is possible, however, only when the community as a whole does not regard it as the sole task of men possessing wealth to employ it profitably and  to increase it, and when the wealthy class consists not exclusively of men for whom the materially productive employment of their resources is their dominant interest. There must be, in other words, a tolerance for the existence of a group of idle rich\u2014idle not in the sense that they do nothing useful but in the sense that their aims are not entirely governed by considerations of material gain. The fact that most people must earn their income does not make it less desirable that some should not have to do so, that a few be able to pursue aims which the rest do not appreciate. It would no doubt be offensive if, for that reason wealth were arbitrarily taken from some and given to others. There would also be little point if the majority were to grant the privilege, for they would select men whose aims they already approved. This would merely create another form of employment, or another form of reward for recognized merit, but not an opportunity to pursue aims that have not yet been generally accepted as desirable.\n\nI have nothing but admiration for the moral tradition that frowns upon idleness where it means lack of purposeful occupation. But not working to earn an income does not necessarily mean idleness; nor is there any reason why an occupation that does not bring a material return should not be regarded as honorable. The fact that most of our needs can be supplied by the market and that this at the same time gives most men the opportunity of earning a living should not mean that no man ought to be allowed to devote all this energy to ends which bring no financial returns or that only the majority, or only organized groups, should be able to pursue such ends. That only a few can have the opportunity does not make it less desirable that some should have it.\n\nIt is doubtful whether a wealthy class whose ethos requires that at least every male member prove his usefulness by making more money can adequately justify its existence. However important the independent owner of property may be for the economic order of a free society, his importance is perhaps even greater in the fields of thought and opinion, of tastes and beliefs. There is something seriously lacking in a society in which all the intellectual, moral, and artistic leaders belong to the employed class, especially if most of them are in the employment of the government. Yet we are moving everywhere toward such a position. Though the freelance writer and artist and the professions of law and medicine still provide some independent leaders of opinion, the great majority of those who ought to provide such a lead\u2014the learned in the sciences and humanities\u2014are today in employed positions, in most countries in the employment of the state.10 There has been a great change in this  respect since the nineteenth century, when gentlemen-scholars like Darwin11 and Macaulay, Grote and Lubbock, Motley and Henry Adams, Tocqueville and Schliemann, were public figures of great prominence and when even such a heterodox critic of society as Karl Marx could find a wealthy patron who enabled him to devote his life to the elaboration and propagation of doctrines which the majority of his contemporaries heartily detested.12\n\nThe almost complete disappearance of this class\u2014and the absence of it in most parts of the United States\u2014has produced a situation in which the propertied class, now almost exclusively a business group, lacks intellectual leadership and even a coherent and defensible philosophy of life. A wealthy class that is in part a leisured class will be interspersed with more than the average proportion of scholars and statesmen, literary figures and artists. It was through their intercourse in their own circle with such men who shared  their style of life, that in the past the wealthy men of affairs were able to take part in the movement of ideas and in the discussions that shaped opinion. To the European observer, who cannot help being struck by the apparent helplessness of what in America is still sometimes regarded as its ruling class, it would seem that this is largely due to the fact that its traditions have prevented the growth of a leisured group within it, of a group that uses the independence which wealth gives for purposes other than those vulgarly called economic. This lack of a cultural elite within the propertied class, however, is also now apparent in Europe, where the combined effects of inflation and taxation have mostly destroyed the old and prevented the rise of a new leisured group.\n\n8. It is undeniable that such a leisured group will produce a much larger proportion of _bons vivants_ than of scholars and public servants and that the former will shock the public conscience by their conspicuous waste. But such waste is everywhere the price of freedom; and it would be difficult to maintain that the standard by which the consumption of the idlest of the idle rich is judged wasteful and objectionable is really different from that by which the consumption of the American masses will be judged wasteful by the Egyptian fellaheen or the Chinese coolie. Quantitatively, the wastes involved in the amusements of the rich are indeed insignificant compared with those involved in the similar and equally \"unnecessary\" amusements of the masses,13 which divert much more from ends which may seem important on some ethical standards. It is merely the conspicuousness and the unfamiliar character of the wastes in the life of the idle rich that make them appear so particularly reprehensible.\n\nIt is also true that even when the lavish outlay of some men is most distasteful to the rest, we can scarcely ever be certain that in any particular instance even the most absurd experimentation in living will not produce generally beneficial results. It is not surprising that living on a new level of possibilities at first leads to much aimless display. I have no doubt, however\u2014even though to say so is certain to provoke ridicule\u2014that even the successful use of leisure needs pioneering and that we owe many of the now common forms of living to people who devoted all their time to the art of living14 and that many of the toys and tools of sport that later became the instruments of recreation for the masses were invented by playboys.\n\nOur evaluation of the usefulness of different activities has in this connection become curiously distorted by the ubiquity of the pecuniary standard.  Surprisingly often, the same people who complain most loudly about the materialism of our civilization will admit of no other standard of usefulness of any service than that men should be willing to pay for it. Yet is it really so obvious that the tennis or golf professional is a more useful member of society than the wealthy amateurs who devoted their time to perfecting these games? Or that the paid curator of a public museum is more useful than a private collector? Before the reader answers these questions too hastily, I would ask him to consider whether there would ever have been golf or tennis professionals or museum curators if wealthy amateurs had not preceded them. Can we not hope that other new interests will still arise from the playful explorations of those who can indulge in them for the short span of a human life? It is only natural that the development of the art of living and of the non-materialistic values should have profited most from the activities of those who had no material worries.15\n\nIt is one of the great tragedies of our time that the masses have come to believe that they have reached their high standard of material welfare as a result of having pulled down the wealthy, and to fear that the preservation or emergence of such a class would deprive them of something they would otherwise get and which they regard as their due. We have seen why in a progressive society there is little reason to believe that the wealth which the few enjoy would exist at all if they were not allowed to enjoy it. It is neither taken from the rest nor withheld from them. It is the first sign of a new way of living begun by the advance guard. True, those who have this privilege of displaying possibilities which only the children or grandchildren of others will enjoy are not generally the most meritorious individuals but simply those who have been placed by chance in their envied position. But this fact is inseparable from the process of growth, which always goes further than any one man or group of men can foresee. To prevent some from enjoying certain advantages first may well prevent the rest of us from ever enjoying them. If through envy we make certain exceptional kinds of life impossible, we shall all in the end suffer material and spiritual impoverishment. Nor can we eliminate the unpleasant manifestations of individual success without destroying at the same time those forces which make advance possible. One may share to the full the distaste for the ostentation, the bad taste, and the wastefulness of many of the new rich and yet recognize that, if we were to prevent all that we disliked, the unforeseen good things that might be thus prevented would probably outweigh the bad. A world in which the majority could prevent the appearance of all that they did not like would be a stagnant and probably a declining world.\n\n* * *\n\nThe quotation from Robert Burns at the head of the chapter is borrowed from Samuel Smiles, _Self Help: With Illustrations of Character and Conduct_ (London: John Murray, 1859), where it is used similarly at the head of chap. 9, p. 215. [The quotation is from his \"Epistle to a Young Friend,\" verse 7. In later editions of _Self Help_ , it appears at the head of chap. 10.\u2014Ed.]\n\n1 Cf. Charles Wright Mills, _White Collar: The American Middle Class_ (New York: Oxford University Press, 1951), p. 63: \"In the early nineteenth century, although there are no exact figures, probably four-fifths of the occupied population were self-employed enterprisers; by 1870, only about one-third, and in 1940, only about one-fifth were still in this old middle class.\" See also _White Collar_ , p. 65, on the extent to which this development is largely an effect of the decreasing proportion of the agricultural population, which, however, does not alter its political significance.\n\n2 It is important to remember that even those who, because of age or the specialized character of their abilities, individually cannot seriously contemplate a change in position are protected by the need of the employer to create working conditions which will secure him the necessary flow of new recruits.\n\n3 Cf. the interesting discussion of these problems in Ernst Bieri, \"Kritische Gedanken zum Wohlfahrtsstaat,\" _Schweizer Monatshefte_ , 35 (1956): esp. 575: \"Die Zahl der _Unselbstst\u00e4ndigerwerbenden_ hat stark zugenommen, sowohl absolut wie prozentuell zu den Besch\u00e4ftigten. Nun ist das Gef\u00fchl der Verantwortung f\u00fcr sich und die Zukunft bei den Selbstst\u00e4ndigerwerbenden aus naheliegenden Gr\u00fcnden lebhafter entwickelt; sie m\u00fcssen auf lange Sicht planen und haben auch die M\u00f6glichkeit, durch Geschick und Initiative f\u00fcr schlechtere Zeiten vorzusorgen. Die Unselbstst\u00e4ndigerwerbenden hingegen, die in regelm\u00e4\u00dfigen Abst\u00e4nden ihren Lohn erhalten, haben ein anderes, [ein] statisches Lebensgef\u00fchl; sie planen selten auf lange Sicht, und erschrecken bei der geringsten Schwankung. Ihr Sinnen und Trachten ist auf _Stabilit\u00e4t und Sicherheit_ gerichtet.\" [\"The number of non-self-employed has sharply increased, both in absolute terms and as a percentage of the workforce. Among the self-employed, a sense of responsibility for oneself and for the future is more strongly developed for the following reasons. They have to plan for the long term and the possibility of preparing for future difficulties through skill and initiative is thus open to them. In contrast, the non-self-employed, who receive their wages at regular intervals, have a different, more static sense of life. They rarely plan for the long term and are frightened by the slightest change. Their thoughts and their efforts are geared toward _stability and security_.\"\u2014Ed.]\n\n4 Cf. the discussion in Chester Irving Barnard, _The Functions of the Executive_ (Cambridge: Harvard University Press, 1938).\n\n5 On the connection between bureaucratic organization and practices and the impossibility of a profit-and-loss calculation see especially Ludwig von Mises, _Human Action_ (New Haven: Yale University Press, 1949), pp. 300\u2013307 [Liberty Fund edition, vol. 2, pp. 303\u201311].\n\n6 Lowell Mason, _The Language of Dissent_ (Cleveland: World Publishing Co., 1959), particularly p. 19: \"No one acquires a love of liberty working for government.\"\n\n7 Cf. on all this Joseph Alois Schumpeter, _Capitalism, Socialism, and Democracy_ (New York: Harper and Bros., 1942), p. 242, and the further discussion of the character of large organizations below, chap. 17, sec. 8.\n\n8 William Henry Beveridge, Baron Beveridge, _Power and Inf uence_ (London: Hodder and Stoughton, 1955), p. 70. The Webbs \"owed both things\u2014time for thought and social contact with the powerful\u2014to Beatrice's possession of \u00a31,000 a year inherited from her father. Where will the next generation of reformers find their Webbs?\"\n\n9 I wish I could command the eloquence with which I once heard the late Lord Keynes expatiate on the indispensable role that the man of independent means plays in any decent society. It came to me somewhat as a surprise that this should have come from the man who at an earlier date had welcomed the \"euthanasia of the rentier.\" I would have been less surprised if I had known how acutely Keynes himself had felt that for the position to which he aspired the foundation of an independent fortune was necessary and how successful he had been in acquiring this fortune. As his biographer tells us, at the age of thirty-six, Keynes \"was determined not to relapse into salaried drudgery. He must be financially independent. He felt that he had that in him which would justify such independence. He had many things to tell the nation. And he wanted a sufficiency.\" Thus he went deeply into speculation and, starting with practically nothing, made half a million pounds in twelve years (Sir Roy Forbes Harrod, _The Life of John Maynard Keynes_ [London: Macmillan, 1951], p. 297. [Harrod notes that at the beginning of 1937, the year in which he turned 54, Keynes's assets, exclusive of his paintings and books, were valued at \u00a3506,450.\u2014Ed.]). It ought not have surprised me, therefore, that to my attempt to draw him out on the subject he responded by an enthusiastic eulogy of the role played in the growth of civilization by the educated man of property; and I can only wish that this account, with the rich illustrations, had seen the light of print.\n\n10 I certainly do not object to a due influence being exerted by the intellectual classes to which I myself belong, i.e., by the employed professor, journalist, or public servant. But I recognize that, being an employed group, they have their own professional bias which on some essential points is contrary to the requirements of a free society and which needs to be countered, or at least modified, by an approach from a different position, by the outlook of men who are not members of an organized hierarchy, whose position in life is independent of the popularity of the views which they express, and who can mix on equal terms with the wealthy and powerful. Occasionally in history this role has been performed by a landowning aristocracy (or the Virginia country gentlemen in the late eighteenth century). There is no need for hereditary privilege to create such a class, and the patrician families of many republican commercial cities have probably earned more credit in this respect than all the titled nobility. Yet, without a sprinkling of men who can devote their lives to whatever values they choose without having to justify their activities to superiors or customers and who are not dependent on rewards for recognized merits, some channels of evolution will be closed which have been very beneficial. If this \"greatest of earthly blessings, independence\" (as Edward Gibbon called it in his _Autobiography_ , \"as originally edited by Lord Sheffield [ John Holroyd, Earl of Sheffield],\" World Classics [London: Oxford University Press, 1950], p. 176) is a \"privilege\" in the sense that only few can possess it, it is no less desirable that some should enjoy it. We can only hope that this rare advantage is not meted out by human will but will fall by accident on a few lucky ones. [Gibbon actually refers to independence as \"the first of earthly blessings.\"\u2014Ed.]\n\n11 Charles Darwin himself was very much aware of this; see _The Descent of Man_ ( _The Origin of Species By Means of Natural Selection; or, The Preservation of Favored Races in the Struggle for Life_ and _The Descent of Man and Selection in Relation to Sex_ [New York: Modern Library, 1960]), p. 502: \"The presence of a body of well-instructed men, who have not to labour for their daily bread, is important to a degree which cannot be over-estimated; as all highly intellectual work is carried on by them, and on such work material progress of all kinds mainly depends, not to mention other and higher advantages.\"\n\n12 On the important role that rich men have played in present-day America in spreading radical opinions see Milton Friedman, \"Capitalism and Freedom,\" in _Essays on Individuality_ , Felix Morley, ed. (Pittsburgh: University of Pennsylvania Press, 1958), p. 178 [Liberty Fund edition, p. 253] [Friedman's essay appeared in somewhat altered form under the title \"The Relation Between Economic Freedom and Political Freedom,\" which forms chap. 1 of his _Capitalism and Freedom_ (Chicago: University of Chicago Press, 1962), pp. 7\u201321.\u2014Ed.]; cf. also Ludwig von Mises, _The Anti-capitalistic Mentality_ (Princeton, NJ: Van Nostrand, 1956) [there is a Liberty Fund edition of this work]; and my essay, \"The Intellectuals and Socialism,\" _University of Chicago Law Review_ , 16 (1949): 417\u201333; reprinted in _Studies in Philosophy, Politics, and Economics_ (Chicago: University of Chicago Press, 1967), pp. 178\u201394. [ _Collected Works_ edition, vol. 10, pp. 43\u201365.]\n\n13 The expenditure on tobacco and drink alone of the population of the United States runs to about $120 per annum per each adult! [By 2003 the amount expended had increased to $415 per annum per adult.\u2014Ed.]\n\n14 A study of the evolution of English domestic architecture and living habits has even led a distinguished Danish architect to assert that \"in English culture idleness has been the root of all good\" (Steen Eiler Rasmussen, _London, the Unique City_ [New York: Macmillan, 1937], p. 294).\n\n15 Cf. Bertrand de Jouvenel, _The Ethics of Redistribution_ (Cambridge: Cambridge University Press, 1951), esp. p. 80 [Liberty Fund edition, pp. 78\u201379].\n\n## Part II\n\nFREEDOM AND THE LAW\n\nAt the first when some certain kind of regiment was once approved, it may be that nothing was then further thought upon for the manner of governing, but all permitted unto their wisdom and discretion which were to rule; till by experience they found this for all parts very inconvenient, so as the thing which they had devised for a remedy did but increase the sore which it should have cured. They saw that to live by one man's will became the cause of all men's misery. This constrained them to come unto laws, wherein all men might see their duties beforehand, and know the penalties of transgressing them.\n\n\u2014Richard Hooker\n\nThis quotation is taken from Richard Hooker, _Of the Laws of Ecclesiastical Polity_ (1593) (2 vols.; London: J. M. Dent, 1907),1, p. 192; the passage is instructive despite the rationalistic interpretation of historical development implied in it.\n\n## NINE\n\nCOERCION AND THE STATE\n\nFor that is an absolute villeinage from which an uncertain and indeterminate service is rendered, where it cannot be known in the evening what service is to be rendered in the morning, that is where a person is bound to whatever is enjoined to him.\n\n\u2014Henry Bracton\n\n1. Earlier in our discussion we provisionally defined freedom as the absence of coercion. But coercion is nearly as troublesome a concept as liberty itself, and for much the same reason: we do not clearly distinguish between what other men do to us and the effects on us of physical circumstances. As a matter of fact, English provides us with two different words to make the necessary distinction: while we can legitimately say that we have been compelled by circumstances to do this or that, we presuppose a human agent if we say that we have been coerced.\n\nCoercion occurs when one man's actions are made to serve another man's will, not for his own but for the other's purpose. It is not that the coerced does not choose at all; if that were the case, we should not speak of his \"acting.\" If my hand is guided by physical force to trace my signature or my finger pressed against the trigger of a gun, I have not acted. Such violence, which makes my  body someone else's physical tool, is, of course, as bad as coercion proper and must be prevented for the same reason. Coercion implies, however, that I still choose but that my mind is made someone else's tool, because the alternatives before me have been so manipulated that the conduct that the coercer wants me to choose becomes for me the least painful one.1 Although coerced, it is still I who decide which is the least evil under the circumstances.2\n\nCoercion clearly does not include all influences that men can exercise on the action of others. It does not even include all instances in which a person acts or threatens to act in a manner he knows will harm another person and will lead him to change his intentions. A person who blocks my path in the street and causes me to step aside, a person who has borrowed from the library the book I want, or even a person who drives me away by the unpleasant noises he produces cannot properly be said to coerce me. Coercion implies both the threat of inflicting harm and the intention thereby to bring about certain conduct.\n\nThough the coerced still chooses, the alternatives are determined for him by the coercer so that he will choose what the coercer wants. He is not altogether deprived of the use of his capacities; but he is deprived of the possibility of using his knowledge for his own aims. The effective use of a person's intelligence and knowledge in the pursuit of his aims requires that he be able to foresee some of the conditions of his environment and adhere to a plan of action. Most human aims can be achieved only by a chain of connected actions, decided upon as a coherent whole and based on the assumption that the facts will be what they are expected to be. It is because, and insofar as, we can predict events, or at least know probabilities, that we can achieve anything. And though physical circumstances will often be unpredictable, they will not maliciously frustrate our aims. But if the facts which determine our plans are under the sole control of another, our actions will be similarly controlled.\n\nCoercion thus is bad because it prevents a person from using his mental  powers to the full and consequently from making the greatest contribution that he is capable of to the community. Though the coerced will still do the best he can do for himself at any given moment, the only comprehensive design that his actions fit into is that of another mind.\n\n2. Political philosophers have discussed power more often than they have coercion because political power usually means power to coerce.3 But though the great men, from John Milton and Edmund Burke to Lord Acton and Jacob Burckhardt, who have represented power as the archevil,4 were right  in what they meant, it is misleading to speak simply of power in this connection. It is not power as such\u2014the capacity to achieve what one wants\u2014that is bad, but only the power to coerce, to force other men to serve one's will by the threat of inflicting harm. There is no evil in the power wielded by the director of some great enterprise in which men have willingly united of their own will and for their own purposes. It is part of the strength of civilized society that, by such voluntary combination of effort under a unified direction, men can enormously increase their collective power.\n\nIt is not power in the sense of an extension of our capacities which corrupts, but the subjection of other human wills to ours, the use of other men against their will for our purposes. It is true that in human relations power and coercion dwell closely together, that great powers possessed by a few may enable them to coerce others, unless those powers are contained by a still greater power; but coercion is neither so necessary nor so common a consequence of power as is generally assumed. Neither the powers of a Henry Ford nor those of the Atomic Energy Commission, neither those of the General of the Salvation Army nor (at least until recently) those of the President of the United States, are powers to coerce particular people for the purposes they choose.\n\nIt would be less misleading if occasionally the terms \"force\" and \"violence\" were used instead of coercion, since the threat of force or violence is the most important form of coercion. But they are not synonymous with coercion, for the threat of physical force is not the only way in which coercion can be exercised. Similarly, \"oppression,\" which is perhaps as much a true opposite of liberty as coercion, should refer only to a state of continuous acts of coercion.\n\n3. Coercion should be carefully distinguished from the conditions or terms on which our fellow men are willing to render us specific services or benefits.  It is only in very exceptional circumstances that the sole control of a service or resource which is essential to us would confer upon another the power of true coercion. Life in society necessarily means that we are dependent for the satisfaction of most of our needs on the services of some of our fellows; in a free society these mutual services are voluntary, and each can determine to whom he wants to render services and on what terms. The benefits and opportunities which our fellows offer to us will be available only if we satisfy their conditions.\n\nThis is as true of social as of economic relations. If a hostess will invite me to her parties only if I conform to certain standards of conduct and dress, or my neighbor converse with me only if I observe conventional manners, this is certainly not coercion. Nor can it be legitimately called \"coercion\" if a producer or dealer refuses to supply me with what I want except at his price. This is certainly true in a competitive market, where I can turn to somebody else if the terms of the first offer do not suit me; and it is normally no less true when I face a monopolist. If, for instance, I would very much like to be painted by a famous artist and if he refuses to paint me for less than a very high fee, it would clearly be absurd to say that I am coerced. The same is true of any other commodity or service that I can do without. So long as the services of a particular person are not crucial to my existence or the preservation of what I most value, the conditions he exacts for rendering these services cannot properly be called \"coercion.\"\n\nA monopolist could exercise true coercion, however, if he were, say, the owner of a spring in an oasis. Let us say that other persons settled there on the assumption that water would always be available at a reasonable price and then found, perhaps because a second spring dried up, that they had no choice but to do whatever the owner of the spring demanded of them if they were to survive: here would be a clear case of coercion. One could conceive of a few other instances where a monopolist might control an essential commodity on which people were completely dependent. But unless a monopolist is in a position to withhold an indispensable supply, he cannot exercise coercion, however unpleasant his demands may be for those who rely on his services.\n\nIt is worth pointing out, in view of what we shall later have to say about the appropriate methods of curbing the coercive power of the state, that whenever there is a danger of a monopolist's acquiring coercive power, the most expedient and effective method of preventing this is probably to require him to treat all customers alike, i.e., to insist that his prices be the same for all and to prohibit all discrimination on his part. This is the same principle by which we have learned to curb the coercive power of the state.\n\nThe individual provider of employment cannot normally exercise coercion, any more than can the supplier of a particular commodity or service. So long as he can remove only one opportunity among many to earn a living,  so long as he can do no more than cease to pay certain people who cannot hope to earn as much elsewhere as they had done under him, he cannot coerce, though he may cause pain. There are, undeniably, occasions when the condition of employment creates opportunity for true coercion. In periods of acute unemployment the threat of dismissal may be used to enforce actions other than those originally contracted for. And in conditions such as those in a mining town the manager may well exercise an entirely arbitrary and capricious tyranny over a man to whom he has taken a dislike. But such conditions, though not impossible, would, at the worst, be rare exceptions in a prosperous competitive society.\n\nA complete monopoly of employment, such as would exist in a fully socialist state in which the government was the only employer and the owner of all the instruments of production, would possess unlimited powers of coercion. As Leon Trotsky discovered: \"In a country where the sole employer is the State, opposition means death by slow starvation. The old principle, who does not work shall not eat, has been replaced by a new one: who does not obey shall not eat.\"5\n\nExcept in such instances of monopoly of an essential service, the mere power of withholding a benefit will not produce coercion. The use of such power by another may indeed alter the social landscape to which I have adapted my plans and make it necessary for me to reconsider all my decisions, perhaps to change my whole scheme of life and to worry about many things I had taken for granted. But, though the alternatives before me may be distressingly few and uncertain, and my new plans of a makeshift character, yet it is not some other will that guides my action. I may have to act under great pressure, but I cannot be said to act under coercion. Even if the threat of starvation to me and perhaps to my family impels me to accept a distasteful job at a very low wage, even if I am \"at the mercy\" of the only man willing to employ me, I am not coerced by him or anybody else. So long as the act that has placed me in my predicament is not aimed at making me do or not do specific things, so long as the intent of the act that harms me is not to make me serve another person's ends, its effect on my freedom is not different from that of any natural calamity\u2014a fire or a flood that destroys my house or an accident that harms my health.\n\n4. True coercion occurs when armed bands of conquerors make the subject people toil for them, when organized gangsters extort a levy for \"protection,\" when the knower of an evil secret blackmails his victim, and, of course, when the state threatens to inflict punishment and to employ physical force to make us obey its commands. There are many degrees of coercion, from the extreme  case of the dominance of the master over the slave or the tyrant over the subject, where the unlimited power of punishment exacts complete submission to the will of the master, to the instance of the single threat of inflicting an evil to which the threatened would prefer almost anything else.\n\nWhether or not attempts to coerce a particular person will be successful depends in a large measure on that person's inner strength: the threat of assassination may have less power to turn one man from his aim than the threat of some minor inconvenience in the case of another. But while we may pity the weak or the very sensitive person whom a mere frown may \"compel\" to do what he would not do otherwise, we are concerned with coercion that is likely to affect the normal, average person. Though this will usually be some threat of bodily harm to his person or his dear ones, or of damage to a valuable or cherished possession, it need not consist of any use of force or violence. One may frustrate another's every attempt at spontaneous action by placing in his path an infinite variety of minor obstacles: guile and malice may well find the means of coercing the physically stronger. It is not impossible for a horde of cunning boys to drive an unpopular person out of town.\n\nIn some degree all close relationships between men, whether they are tied to one another by affection, economic necessity, or physical circumstances (such as on a ship or an expedition), provide opportunities for coercion. The conditions of personal domestic service, like all more intimate relations, undoubtedly offer opportunities for coercion of a peculiarly oppressive kind and are, in consequence, felt as restrictions on personal liberty. And a morose husband, a nagging wife, or a hysterical mother may make life intolerable unless their every mood is obeyed. But here society can do little to protect the individual beyond making such associations with others truly voluntary. Any attempt to regulate these intimate associations further would clearly involve such far-reaching restrictions on choice and conduct as to produce even greater coercion: if people are to be free to choose their associates and intimates, the coercion that arises from voluntary association cannot be the concern of government.\n\nThe reader may feel that we have devoted more space than is necessary to the distinction between what can be legitimately called \"coercion\" and what cannot and between the more severe forms of coercion, which we should prevent, and the lesser forms, which ought not to be the concern of authority. But, as in the case of liberty, a gradual extension of the concept has almost deprived it of value. Liberty can be so defined as to make it impossible of attainment. Similarly, coercion can be so defined as to make it an all-pervasive and unavoidable phenomenon.6 We cannot prevent all harm that a person  may inflict upon another, or even all the milder forms of coercion to which life in close contact with other men exposes us; but this does not mean that we ought not to try to prevent all the more severe forms of coercion, or that we ought not to define liberty as the absence of such coercion.\n\n5. Since coercion is the control of the essential data of an individual's action by another, it can be prevented only by enabling the individual to secure for himself some private sphere where he is protected against such interference. The assurance that he can count on certain facts not being deliberately shaped by another can be given to him only by some authority that has the necessary power. It is here that coercion of one individual by another can be prevented only by the threat of coercion.\n\nThe existence of such an assured free sphere seems to us so much a normal condition of life that we are tempted to define \"coercion\" by the use of such terms as \"the interference with legitimate expectations,\" or \"infringement of rights,\" or \"arbitrary interference.\"7 But in defining coercion we cannot take for granted the arrangements intended to prevent it. The \"legitimacy\" of one's expectations or the \"rights\" of the individual are the result of the recognition of such a private sphere. Coercion not only would exist but would be much more common if no such protected sphere existed. Only in a society that has already attempted to prevent coercion by some demarcation of a protected sphere can a concept like \"arbitrary interference\" have a definite meaning.\n\nIf the recognition of such individual spheres, however, is not itself to become an instrument of coercion, their range and content must not be determined by the deliberate assignment of particular things to particular men. If what was to be included in a man's private sphere were to be determined by the will of any man or group of men, this would simply transfer the power of coercion to that will. Nor would it be desirable to have the particular contents of a man's private sphere fixed once and for all. If people are to make the best use of their knowledge and capacities and foresight, it is desirable that they themselves  have some voice in the determination of what will be included in their personal protected sphere.\n\nThe solution that men have found for this problem rests on the recognition of general rules governing the conditions under which objects or circumstances become part of the protected sphere of a person or persons. The acceptance of such rules enables each member of a society to shape the content of his protected sphere and all members to recognize what belongs to their sphere and what does not.8\n\nWe must not think of this sphere as consisting exclusively, or even chiefly, of material things. Although to divide the material objects of our environment into what is mine and what is another's is the principal aim of the rules which delimit the spheres, they also secure for us many other \"rights,\" such as security in certain uses of things or merely protection against interference with our actions.\n\n6. The recognition of private or several9 property is thus an essential condition for the prevention of coercion, though by no means the only one. We are rarely in a position to carry out a coherent plan of action unless we are certain of our exclusive control of some material objects; and where we do not control them, it is necessary that we know who does if we are to collaborate with others. The recognition of property is clearly the first step in the delimitation of the private sphere which protects us against coercion; and it has long been recognized that \"a people averse to the institution of private property is without the first element of freedom\"10 and that \"nobody is at liberty to attack several property and to say at the same time that he values civilization. The history of the two cannot be disentangled.\"11 Modern anthropology confirms the fact that \"private property appears very definitely on primitive levels\" and that \"the roots of property as a legal principle which determines the physical relationship between man and his environmental setting, natural or artificial, are the very prerequisite of any ordered action in the cultural sense.\"12\n\nIn modern society, however, the essential requisite for the protection of the  individual against coercion is not that he possess property but that the material means which enable him to pursue any plan of action should not be all in the exclusive control of one other agent. It is one of the accomplishments of modern society that freedom may be enjoyed by a person with practically no property of his own (beyond personal belongings like clothing\u2014and even these can be rented)13 and that we can leave the care of the property that serves our needs largely to others. The important point is that the property should be sufficiently dispersed so that the individual is not dependent on particular persons who alone can provide him with what he needs or who alone can employ him.\n\nThat other people's property can be serviceable in the achievement of our aims is due mainly to the enforcibility of contracts. The whole network of rights created by contracts is as important a part of our own protected sphere, as much the basis of our plans, as any property of our own. The decisive condition for mutually advantageous collaboration between people, based on voluntary consent rather than coercion, is that there be many people who can serve one's needs so that nobody has to be dependent on specific persons for the essential conditions of life or the possibility of development in some direction. It is competition made possible by the dispersion of property that deprives the individual owners of particular things of all coercive powers.\n\nIn view of a common misunderstanding of a famous maxim,14 it should be mentioned that we are independent of the will of those whose services  we need because they serve us for their own purposes and are normally little interested in the uses we make of their services. We should be very dependent on the beliefs of our fellows if they were prepared to sell their products to us only when they approved of our ends and not for their own advantage. It is largely because in the economic transactions of everyday life we are only impersonal means to our fellows, who help us for their own purposes, that we can count on such help from complete strangers and use it for whatever end we wish.15\n\nThe rules of property and contract are required to delimit the individual's private sphere wherever the resources or services needed for the pursuit of his aims are scarce and must, in consequence, be under the control of some man or another. But if this is true of most of the benefits we derive from men's efforts, it is not true of all. There are some kinds of services, such as sanitation or roads, which, once they are provided, are normally sufficient for all who want to use them. The provision of such services has long been a recognized field of public effort, and the right to share in them is an important part of the protected sphere of the individual. We need only remember the role that the assured \"access to the King's highway\" has played in history to see how important such rights may be for individual liberty.\n\nWe cannot enumerate here all the rights or protected interests which serve to secure to the legal person a known sphere of unimpeded action. But, since modern man has become a little insensitive on this point, it ought perhaps to be mentioned that the recognition of a protected individual sphere has in times of freedom normally included a right to privacy and secrecy, the conception that a man's house is his castle16 and that nobody has a right even to take cognizance of his activities within it.\n\n7. The character of those abstract and general rules that have been evolved to limit coercion both by other individuals and by the state will be the subject of the next chapter. Here we shall consider in a general way how that threat of coercion which is the only means whereby the state can prevent the coercion of one individual by another can be deprived of most of its harmful and objectionable character.\n\nThis threat of coercion has a very different effect from that of actual and unavoidable coercion, if it refers only to known circumstances which can be avoided by the potential object of coercion. The great majority of the threats  of coercion that a free society must employ are of this avoidable kind. Most of the rules that it enforces, particularly its private law, do not constrain private persons (as distinguished from the servants of the state) to perform specific actions. The sanctions of the law are designed only to prevent a person from doing certain things or to make him perform obligations that he has voluntarily incurred.\n\nProvided that I know beforehand that if I place myself in a particular position, I shall be coerced and provided that I can avoid putting myself in such a position, I need never be coerced. At least insofar as the rules providing for coercion are not aimed at me personally but are so framed as to apply equally to all people in similar circumstances, they are no different from any of the natural obstacles that affect my plans. In that they tell me what will happen _if_ I do this or that, the laws of the state have the same significance for me as the laws of nature; and I can use my knowledge of the laws of the state to achieve my own aims as I use my knowledge of the laws of nature.\n\n8. Of course, in some respects the state uses coercion to make us perform particular actions. The most important of these are taxation and the various compulsory services, especially in the armed forces. Though these are not supposed to be avoidable, they are at least predictable and are enforced irrespective of how the individual would otherwise employ his energies; this deprives them largely of the evil nature of coercion. If the known necessity of paying a certain amount in taxes becomes the basis of all my plans, if a period of military service is a foreseeable part of my career, then I can follow a general plan of life of my own making and am as independent of the will of another person as men have learned to be in society. Though compulsory military service, while it lasts, undoubtedly involves severe coercion, and though a lifelong conscript could not be said ever to be free, a predictable limited period of military service certainly restricts the possibility of shaping one's own life less than would, for instance, a constant threat of arrest resorted to by an arbitrary power to ensure what it regards as good behavior.\n\nThe interference of the coercive power of government with our lives is most disturbing when it is neither avoidable nor predictable. Where such coercion is necessary even in a free society, as when we are called to serve on a jury or to act as special constables, we mitigate the effects by not allowing any person to possess arbitrary power of coercion. Instead, the decision as to who must serve is made to rest on fortuitous processes, such as the drawing of lots. These unpredictable acts of coercion, which follow from unpredictable events but conform to known rules, affect our lives as do other \"acts of God,\" but do not subject us to the arbitrary will of another person.\n\n9. Is the prevention of coercion the only justification for the use of the threat of coercion by the state? We can probably include all forms of violence under coercion or at least maintain that a successful prevention of coercion  will mean the prevention of all kinds of violence. There remains, however, one other kind of harmful action which it is generally thought desirable to prevent and which at first may seem distinct. This is fraud and deception. Yet, though it would be straining the meaning of words to call them \"coercion,\" on examination it appears that the reasons why we want to prevent them are the same as those applying to coercion. Deception, like coercion, is a form of manipulating the data on which a person counts, in order to make him do what the deceiver wants him to do. Where it is successful, the deceived becomes in the same manner the unwilling tool, serving another man's ends without advancing his own. Though we have no single word to cover both, all we have said of coercion applies equally to fraud and deception.\n\nWith this correction, it seems that freedom demands no more than that coercion and violence, fraud and deception, be prevented, except for the use of coercion by government for the sole purpose of enforcing known rules intended to secure the best conditions under which the individual may give his activities a coherent, rational pattern.\n\nThe problem of the limit of coercion is not the same as that concerning the proper function of government. The coercive activities of government are by no means its only tasks. It is true that the non-coercive or purely service activities that government undertakes are usually financed by coercive means. The medieval state, which financed its activities mainly with the income from its property, might have provided services without resorting to coercion. Under modern conditions, however, it seems hardly practicable that government should provide such services as the care for the disabled or the infirm and the provision of roads or of information without relying on its coercive powers to finance them.\n\nIt is not to be expected that there will ever be complete unanimity on the desirability of the extent of such services, and it is at least not obvious that coercing people to contribute to the achievement of ends in which they are not interested can be morally justified. Up to a point, most of us find it expedient, however, to make such contributions on the understanding that we will in turn profit from similar contributions of others toward the realization of our own ends.\n\nOutside the field of taxation, it is probably desirable that we should accept only the prevention of more severe coercion as the justification for the use of coercion by government. This criterion, perhaps, cannot be applied to each single legal rule, but only to the legal system as a whole. The protection of private property as a safeguard against coercion, for instance, may require special provisions that do not individually serve to reduce coercion but serve merely to insure that private property does not unnecessarily impede action that does not harm the owner. But the whole conception of interference or non-interference by the state rests on the assumption of a private  sphere delimited by general rules enforced by the state; and the real issue is whether the state ought to confine its coercive action to enforcing these rules or go beyond this.\n\nAttempts have often been made, notably by John Stuart Mill,17 to define the private sphere that should be immune from coercion in terms of a distinction between actions that affect only the acting person and those which also affect others. But, as there is hardly any action that may not conceivably affect others, this distinction has not proved very useful. It is only by delimiting the protected sphere of each individual that the distinction becomes significant. Its aim cannot be to protect people against all actions by others that may be harmful to them18 but only to keep certain of the data of their actions from the control of others. In determining where the boundaries of the protected sphere ought to be drawn, the important question is whether the actions of other people that we wish to see prevented would actually interfere with the reasonable expectations of the protected person.\n\nIn particular, the pleasure or pain that may be caused by the knowledge of other people's actions should never be regarded as a legitimate cause for coercion. The enforcement of religious conformity, for instance, was a legitimate object of government when people believed in the collective responsibility of the community toward some deity and it was thought that the sins of any member would be visited upon all. But where private practices cannot affect anybody but the voluntary adult actors, the mere dislike of what is being done by others, or even the knowledge that others harm themselves by what they do, provides no legitimate ground for coercion.19\n\nWe have seen that the opportunities of learning about new possibilities that the growth of civilization constantly offers provide one of the main arguments  for freedom; it would therefore make nonsense of the whole case for freedom if, because of the envy of others20 or because of their dislike of anything that disturbs their ingrained habits of thought, we should be restrained from pursuing certain activities. While there is clearly a case for enforcing rules of conduct in public places, the bare fact that an action is disliked by some of those who learn about it cannot be a sufficient ground for prohibiting it.\n\nGenerally speaking, this means that the morality of action within the private sphere is not a proper object for coercive control by the state. Perhaps one of the most important characteristics that distinguish a free from an unfree society is indeed that, in matters of conduct that do not directly affect the protected sphere of others, the rules which are in fact observed by most are of a voluntary character and not enforced by coercion. Recent experience with totalitarian regimes has emphasized the importance of the principle \"never [to] identify the cause of moral values with that of the State.\"21 It is indeed probable that more harm and misery have been caused by men determined to use coercion to stamp out a moral evil than by men intent on doing evil.\n\n10. Yet the fact that conduct within the private sphere is not a proper object for coercive action by the state does not necessarily mean that in a free society such conduct should also be exempt from the pressure of opinion or disapproval. A hundred years ago, in the stricter moral atmosphere of the Victorian era, when at the same time coercion by the state was at a minimum, John  Stuart Mill directed his heaviest attack against such \"moral coercion.\"22 In this he probably overstated the case for liberty. At any rate, it probably makes for greater clarity not to represent as coercion the pressure that public approval or disapproval exerts to secure obedience to moral rules and conventions.\n\nWe have already seen that coercion is, in the last resort, a matter of degree and that the coercion which the state must both prevent and threaten for the sake of liberty is only coercion in its more severe forms\u2014the kind which, when threatened, may prevent a person of normal strength from pursuing an object important to him. Whether or not we wish to call coercion those milder forms of pressure that society applies to nonconformists, there can be little question that these moral rules and conventions that possess less binding power than the law have an important and even indispensable role to perform and probably do as much to facilitate life in society as do the strict rules of law. We know that they will be observed only generally and not universally, but this knowledge still provides useful guidance and reduces uncertainty. While the respect for such rules does not prevent people from occasionally behaving in a manner that is disapproved, it limits such behavior to instances in which it is fairly important to the person to disregard the rules. Sometimes these noncoercive rules may represent an experimental stage of what later in a modified form may grow into law. More often they will provide a flexible background of more or less unconscious habits which serve as a guide to most people's actions. On the whole, those conventions and norms of social intercourse and individual conduct do not constitute a serious infringement of individual liberty but secure a certain minimum of uniformity of conduct that assists individual efforts more than it impedes them.\n\n* * *\n\nThe quotation from Henry Bracton at the head of the chapter is borrowed from Michael Polanyi, _The Logic of Liberty: Reflections and Rejoinders_ (London: Routledge and Kegan Paul, 1951), p. 158 [Liberty Fund edition (1998), p. 194]. [The quotation is from Henry Bracton's _De Legibus et Consuetudinibus Angliae_ ( _On the Laws and Customs of England)_ , George Edward Woodbine, ed. (4 vols.; Cambridge, MA: Published in association with the Selden Society by the Belknap Press of Harvard University Press, 1968\u201377), vol. 2, p. 89. The translation of Bracton's Latin original is that of Samuel Edmund Throne.\u2014Ed.] The chief idea of the chapter has also been well expressed by Frederic William Maitland in his \"Historical Sketch of Liberty and Equality as Ideals\" (1875), in _Collected Papers of Frederic William Maitland, Downing Professor of the Laws of England_ (3 vols.; Cambridge: Cambridge University Press, 1911), vol. 1, p. 80: \"The exercise of power in ways which cannot be anticipated causes some of the greatest restraints, for restraint is most felt and therefore is greatest when it is least anticipated. We feel ourselves least free when we know that restraints may at any moment be placed on any of our actions, and yet we cannot anticipate these restraints. . . . Known general laws, however bad, interfere less with freedom than decisions based on no previously known rule.\" [Liberty Fund edition of the _Historical Sketch_ , pp. 109\u201310.]\n\n1 Cf. Frank Hyneman Knight, \"Conflict of Values: Freedom and Justice,\" in _Goals of Economic Life_ , Alfred Dudley Ward, ed. (New York: Harper and Bros., 1953), p. 208: \"Coercion is 'arbitrary' manipulation by one of another's terms or alternatives of choice\u2014and usually we should say an 'unjustified' interference.\" See also Robert Morrison MacIver, _Society: A Textbook of Sociology_ (New York: Farrar and Rinehart, 1937), p. 342.\n\n2 Cf. the legal maxim \"etsi coactus tamen voluit,\" deriving from _Corpus juris civilis, Digesta_ , 50. 4. 21 [Samuel Parsons Scott, _The Civil Law, Including the Twelve Tables, the Institutes of Gaius, the Rules of Ulpian, the Opinions of Paulus, the Enactments of Justinian, and the Constitutions of Leo_ (17 vols. in 7; Cincinnati: Central Trust Co., 1932), vol. 3 \/ 4, p. 65]. [The phrase translates as: \"Although compelled, he nevertheless wished it.\" The original source is noted as Paulus, _On the Edict_ , bk. 11.\u2014 Ed.] For a discussion of its significance see Ulrich von L\u00fcbtow, _Der Ediktstitel \"Quod metus causa gestum erit\"_ (Greifswald: Bamberg, 1932), pp. 61\u201371. [The Latin phrase in the title of the book carries the meaning \"When an act was performed because of fear.\"\u2014Ed.]\n\n3 Cf. Friedrich von Wieser, _Das Gesetz der Macht_ (Vienna: Julius Springer, 1926); Bertrand Russell, _Power: A New Social Analysis_ (London: Allen and Unwin, 1930); Guglielmo Ferrero, _The Principles of Power_ (New York: G. P. Putnam's Sons, 1942) [This is the third volume of a trilogy. The first two volumes were originally published in Paris (Vol. 1: _The Gamble_ , Vol. 2: _The Reconstruction of Europe_ ). The third appeared in New York because it could not be published on the Continent due to increased censorship.\u2014Ed.]; Bertrand de Jouvenel, _Power: The Natural History of Its Growth_ (London: Hutchinson, 1948) [The Liberty Fund reissued Jouvenel's book with a foreword by Denis William Brogan in 1993.\u2014Ed.]; Gerhard Ritter, _Vom sittlichen Problem der Macht: F\u00fcnf Essays_ (Bern: Francke, 1948); and the same author's _Machtstaat und Utopie: vom Streit um die D\u00e4monie der Macht seit Machiavelli und Morus_ (Munich: Oldenburg, 1940); Cyril John Radcliffe, Viscount Radcliffe of Werneth, _The Problem of Power_ [Reith Lectures] (London: Secker and Warburg, 1952); and John Clark MacDermott, Baron MacDermott, _Protection from Power under English Law_ , The Hamlyn Lectures (London: Stevens, 1957).\n\n4 The complaints about power as the archevil are as old as political thinking. Herodotus had already made Otanes say in his famous speech on democracy that \"even the best of men raised to such a position [of irresponsible power] would be bound to change for the worst\" ( _Histories_ , iii, 80); John Milton considers the possibility that \"long continuance of Power may corrupt sincerest Men\" (\"The Ready and Easy Way to Establish a Free Commonwealth, and the Excellence thereof, Compared with the Inconveniences and Dangers of Readmitting Kingship in this Nation\" [1660], in _Milton's Prose_ , Malcolm William Wallace, ed. [World's Classics; London: Oxford University Press, 1925], p. 459 [Liberty Fund edition, p. 428]); Montesquieu asserts that \"constant experience shows us that every man invested with power is apt to abuse it, and to carry his authority as far as it will go\" [\"Mais c'est une exp\u00e9rience \u00e9ternelle que tout homme qui a du pouvoir est port\u00e9 \u00e0 en abuser; il va jusqu\u00e0 ce qu'il trouve des limites.\"\u2014Ed.] ( _Spirit of the Laws_ , bk. 11, chap. 4, vol. 1, p. 150; French edition: vol. 2, p. 395); Immanuel Kant maintains that \"the possession of power invariably debases the free judgment of reason\" [\"Der Besitz der Gewalt das freie Urteil der Vernunft unvermeidlich verdirbt.\" ( _Zum ewigen Frieden: Ein philosophischer Entwurf_ [1795], Karl Kehrbach, ed. [Leipzig: Philipp Reclam jun., 1881], p. 36.) The essay appears in English under the title _Perpetual Peace: A Philosophical Essay._ \u2014Ed.]; Edmund Burke writes that \"many of the greatest tyrants on [ _sic_ ] the records of history have begun their reigns in the fairest manner. But the truth is, this unnatural power corrupts both the heart and the understanding\" ( _Thoughts on the Cause of Our Present Discontents_ , in _Works_ , II, p. 307 [Hayek is in error in locating the quotation in Burke's _Thoughts on the Cause of Our Present Discontents._ The quotation in fact appears in _A Vindication of Natural Society; or, A View of the Miseries and Evils Arising to Mankind from Every Species of Artificial Society_ (3rd ed., with a new preface; Dublin: Printed by and for Sarah Cotter, 1766), p. 38; Liberty Fund edition, p. 46.\u2014Ed.]); John Adams observes that \"power is always abused when unlimited and unbalanced\" ( _Works: With a Life of the Author_ , Charles Francis Adams, ed. [10 vols.; Boston: Charles C. Little and James Brown, 1851], vol. 6, p. 73), and that \"absolute power intoxicates alike despots, monarchs, aristocrats, and democrats, and jacobins and _sans culottes_ \" (vol. 6, p. 477) [The first quotation appears in chap. 1 of Adams's \"A Defence of the Constitutions of Government of the United States of America.\" The second is taken from Adams's Letters to John Taylor.\u2014Ed.]; James Madison asserts [in his letter to Thomas Ritchie dated 18 December 1825] that \"all power in human hands is liable to be abused\" and [in an unsent letter to Thomas Lehre dated 2 August 1828] that \"power, wherever lodged, is liable, more or less, to abuse\" ( _The Complete Madison: His Basic Writings,_ Saul Kussiel Padover, ed. [New York: Harper, 1953], p. 46); Jacob Burckhardt never ceases to reiterate that power in itself is evil ( _Force and Freedom: Reflections on History_ , James Hastings Nichols, trans. [New York: Pantheon Books, 1943], e.g., p. 115 [Liberty Fund edition, p. 102]); and there is, of course, Lord Acton's maxim \"power tends to corrupt, and absolute power corrupts absolutely\" ( _Historical Essays_ , p. 504 [Liberty Fund edition, _Essays in the Study and Writing of History_ , p. 383]). [Letter from Acton to Bishop Mandell Creighton (April 3,1887) regarding Acton's review of vols. 3 and 4 of Creighton's _History of the Papacy, The Italian Princes, 1464\u20131518_ , contributed to the _English Historical Review_ in 1887.\u2014Ed.] See also Carl von Rotteck, \"Absolutismus,\" in _Staatslexikon oder Encyklop\u00e4die der Staatswissenschaften_ , Carl von Rotteck and Carl. T. Welcker, eds. (Altona: Hammrich, 1834), vol. 1, p. 155: \"Es liegt in der unumschr\u00e4nkten Gewalt eine so schauerliche Macht der b\u00f6sen Versuchung, da\u00df nur die alleredelsten Menschen ihr widerstehen k\u00f6nnen.\" [\"Absolute power contains within itself the awful temptation toward evil that only the most noble can resist.\"\u2014Ed.]\n\n5 Leon Trotsky, _The Revolution Betrayed: What Is the Soviet Union and Where is it Going?_ (Garden City, NY: Doubleday, Doran and Co., 1937), p. 283.\n\n6 A characteristic instance of this which happened to come to my notice as I was writing occurs in a review by Bertram Francis Willcox, \" _The Labor Policy of a Free Society_ by Sylvester Petro,\" _Industrial and Labor Relations Review_ , 9 (1957\u201358): 273: In order to justify \"peaceful economic coercion\" by unions, the author argues that \"peaceable competition, based on free choice, fairly reeks of coercion. A free seller of goods or services, by setting his price, coerces one who wants to buy\u2014coerces him into paying, doing without, or going elsewhere. A free seller of goods or services, by setting a condition that no one may buy from him who buys from X, coerces one who wants to buy\u2014coerces him into doing without, going elsewhere, or refraining from buying from X\u2014and in the last case he coerces X as well.\" This abuse of the term \"coercion\" derives largely from John Rogers Commons (cf. his _Institutional Economics_ [New York: Macmillan, 1934]), esp. pp. 336\u201337; see also Robert Lee Hale, \"Coercion and Distribution in a Supposedly Non-coercive State,\" _Political Science Quarterly_ , 38 (1923): 470\u201394, and his _Freedom through Law: Public Control of Private Governing Power_ (New York: Columbia University Press, 1952).\n\n7 Cf. the passage by Frank Hyneman Knight, \"Conflict of Values: Freedom and Justice,\" p. 208.\n\n8 On the role of property in the American tradition of liberty, see Paul Abraham Freund, _On Understanding the Supreme Court: A Series of Lectures Delivered under the Auspices of the Julius Rosenthal Foundation at Northwestern University School of Law, in April 1949_ (3rd ed.; Boston: Little, Brown and Co., 1951), pp. 14ff.\n\n9 The expression \"several property\" used by Sir Henry Maine ( _Village Communities in the East and West: Six Lectures Delivered at Oxford to which are added Other Lectures, Addresses, and Essays_ [New York: H. Holt and Co., 1880], p. 230), is in many respects more appropriate than the more familiar one \"private property,\" and we shall occasionally employ it in place of the latter.\n\n10 Acton, \"Nationality,\" _History of Freedom_ , p. 297 [Liberty Fund edition, _Essays in the History of Liberty_ , p. 431].\n\n11 Sir Henry Maine, _Village Communities_ , p. 230.\n\n12 Bronislaw Malinowski, _Freedom and Civilization_ (New York: Roy Publishers, 1944), pp. 132\u201333.\n\n13 I do not mean to suggest that this is a desirable form of existence. It is of some importance, however, that today a not inconsiderable portion of the men who largely influence public opinion, such as journalists and writers, often live for long periods with a minimum of personal possessions and that this undoubtedly affects their outlook. It seems that some people even have come to regard material possessions as an impediment rather than a help, so long as they have the income to buy what they need.\n\n14 Immanuel Kant, \"Foundations of the Metaphysics of Morals,\" _Critique of Practical Reason and Other Writings in Moral Philosophy_ , Lewis White Beck, ed. (Chicago: University of Chicago Press, 1949), p. 87: \"Act so that you treat humanity, whether in your own person or in that of another, always as an end and never as a means only.\" [\"Handle so, da\u00df du die Menschheit sowohl in deiner Person als in der Person eines jeden andern jederzeit zugleich als Zweck, niemals blo\u00df als Mittel brauchst.\" _Grundlegung zur Metaphysik der Sitten_ , in _Kants Werke_ , (Akademie Textausgabe; 9 vols.; Berlin: Walter de Gruyter, 1968), vol. 4, p. 429.\u2014Ed.] So far as this means that no man should be made to do anything that serves only other people's purposes, it is just another way of saying that coercion should be avoided. But if the maxim is interpreted to mean that when we collaborate with other men, we should be guided not only by our own but also by their purposes, it soon comes into conflict with their freedom when we disagree with their ends. For an example of such an interpretation see John Maurice Clark, _The Ethical Basis of Economic Freedom_ (Kazanjian Foundation Lecture; Westport, CT: C. K. Kazanjian Economics Foundation, 1955), p. 26, and the German literature discussed in the work quoted in the next note. [The two works cited in Mises's _Socialism_ on the pages to which Hayek refers are: Friedrich Engels, _Ludwig Feuerbach und der Ausgang der klassischen deutschen Philosophie_ (5th ed.; Stuttgart: J. H. W. Dietz, 1910), and Hermann Cohen, _Ethik des reinen Willens_ (Berlin: B. Cassirer, 1904), pp. 303 et seq.\u2014Ed.]\n\n15 Cf. Ludwig von Mises, _Socialism_ , pp. 194 and 430\u201341 [Liberty Fund edition, pp. 171 and 388\u2013389].\n\n16 In view of the often alleged lack of individual liberty in classical Greece, it deserves mention that in the Athens of the fifth century B.C. the sanctity of the private home was so fully recognized that even under the rule of the Thirty Tyrants a man \"could save his life by staying at home\" (see John Walter Jones, _The Law and Legal Theory of the Greeks: An Introduction_ [Oxford: Clarendon Press, 1956], p. 91, with reference to Demosthenes xxiv, 52).\n\n17 John Stuart Mill, \"On Liberty,\" in _On Liberty and Considerations on Representative Government_ , Ronald Buchanan McCallum, ed. (Oxford: Blackwell, 1946), chap. 4, pp. 66\u201383.\n\n18 Cf. Mill, _On Liberty_ , p. 84: \"In many cases, an individual, in pursuing a legitimate object, necessarily and therefore legitimately causes pain or loss to others, or intercepts a good which they had a reasonable hope of obtaining.\" Also the significant change from the misleading formulation of art. 4 in the French Declaration of the Rights of Man and of the Citizen of 1789, \"La libert\u00e9 consiste a pouvoir faire tout ce qui ne nuit pas \u00e0 autrui,\" [\"Liberty consists in the freedom to do everything not injurious to others.\"\u2014Ed.] to the correct formulation of art. 6 of the Declaration of 1793: \"La libert\u00e9 est le pouvoir qui appartient \u00e0 l'homme de faire tout ce que ne nuit pas aux droits d'autrui\" [\"Liberty is the power that man possesses to do whatever is not injurious to the rights of others.\"\u2014Ed.].\n\n19 The most conspicuous instance of this in our society is that of the treatment of homosexuality. As Bertrand Russell has observed (\"John Stuart Mill,\" _Proceedings of the British Academy_ , 41 [1955]: 55): \"If it were still believed, as it once was, that the toleration of such behaviour would expose the community to the fate of Sodom and Gomorrah, the community would have every right to intervene.\" But where such factual beliefs do not prevail, private practice among adults, however abhorrent it may be to the majority, is not a proper subject for coercive action for a state whose object is to minimize coercion.\n\n20 Charles Anthony Raven Crosland, _The Future of Socialism_ (London: Jonathan Cape, 1956), p. 206.\n\n21 The statement quoted has been ascribed to Ignazio Silone. [The quotation comes from a speech made in Italian by Silone before the International PEN Club Conference held at Basle in 1947. It is reprinted, in a translation made by Eric Mossbacher, in \"On the Place of the Intellect and the Pretensions of the Intellectual,\" _Horizon: A Review of Literature and Art_ , 16 (December 1947): 323, reprinted in George Barnard de Huszar, ed, _The Intellectuals: A Controversial Portrait_ (Glencoe, IL: The Free Press, 1960), p. 264.\u2014Ed.] Cf. also Jacob Burckhardt, _Reflections on History_ , p. 118 [Liberty Fund edition, p. 70]: \"It is a degeneration, it is philosophical and bureaucratic arrogance, for the State to attempt to fulfil moral purposes directly, for only society can and may do that.\" See also Harold Stearns, _Liberalism in America_ : _Its Origins, Its Temporary Collapse, Its Future_ (New York: Boni and Liveright, 1919), p. 69: \"Coercion for the sake of virtue is as repugnant as coercion for the sake of vice. If American liberals are unwilling to fight the principle of coercion in the case of the Prohibition Amendment simply because they personally are not much interested in whether the country is dry or not, then they are discredited the moment they fight coercion in those cases where they _are_ interested.\" The typical socialist attitude on these problems is most explicitly stated in Robert Lowe Hall, _The Economic System in a Socialist State_ (London: Macmillan, 1937), pp. 202\u20133, where it is argued (with regard to the duty of increasing the capital of the country) that \"the fact that it is necessary to use such words as 'moral obligation' and 'duty' shows that there is no question of accurate calculation and that we are dealing with decisions which not only may be, but ought to be, taken by the community as a whole, that is to say with political decisions.\" For a conservative defense of the use of political power to enforce moral principles see Walter Berns, _Freedom, Virtue, and the First Amendment_ (Baton Rouge: Louisiana State University Press, 1957).\n\n22 Mill, _On Liberty_ , chap. 3, pp. 49\u201366.\n\n## TEN\n\nLAW, COMMANDS, AND ORDER\n\nOrder is not a pressure imposed upon society from without, but an equilibrium which is set up from within.\n\n\u2014J. Ortega y Gasset\n\n1. \"The rule whereby the indivisible border line is fixed within which the being and activity of each individual obtain a secure and free sphere is the law.\"1 Thus one of the great legal scholars of the last century stated the basic  conception of the law of liberty. This conception of the law which made it the basis of freedom has since been largely lost. It will be the chief aim of this chapter to recover and make more precise the conception of the law on which the ideal of freedom under the law was built and which made it possible to speak of the law as \"the science of liberty.\"2\n\nLife of man in society, or even of the social animals in groups, is made possible by the individuals acting according to certain rules.3 With the growth of intelligence, these rules tend to develop from unconscious habits into explicit and articulated statements and at the same time to become more abstract and general. Our familiarity with the institutions of law prevents us from seeing how subtle and complex a device the delimitation of individual spheres by abstract rules is. If it had been deliberately designed, it would deserve to rank among the greatest of human inventions. But it has, of course, been as little invented by any one mind as language or money or most of the practices and conventions on which social life rests.4\n\nA kind of delimitation of individual spheres by rules appears even in animal societies. A degree of order, preventing too frequent fights or interference with the search for food, etc., here arises often from the fact that the individual, as it strays farther from its lair, becomes less ready to fight. In consequence, when two individuals meet at some intermediate place, one of them will usually withdraw without an actual trial of strength. Thus a sphere belonging to each individual is determined, not by the demarcation of a concrete boundary, but by the observation of a rule\u2014a rule, of course, that is not known as such by  the individual but that is honored in action. The illustration shows how even such unconscious habits will involve a sort of abstraction: a condition of such generality as that of distance from home will determine the response of any individual on meeting another. If we tried to define any of the more truly social habits that make possible the life of animals in groups, we should have to state many of them in terms of abstract rules.\n\nThat such abstract rules are regularly observed in action does not mean that they are known to the individual in the sense that it could communicate them. Abstraction occurs whenever an individual responds in the same manner to circumstances that have only some features in common.5 Men generally act in accordance with abstract rules in this sense long before they can state them.6 Even when they have acquired the power of conscious abstraction, their conscious thinking and acting are probably still guided by a great many such abstract rules which they obey without being able to formulate them. The fact that a rule is generally obeyed in action therefore does not mean that it does not still have to be discovered and formulated in words.\n\n2. The nature of these abstract rules that we call \"laws\" in the strict sense is best shown by contrasting them with specific and particular commands. If we take the word \"command\" in its widest sense, the general rules governing human conduct might indeed also be regarded as commands. Laws and commands differ in the same way from statements of fact and therefore belong to the same logical category. But a general rule that everybody obeys, unlike a  command proper, does not necessarily presuppose a person who has issued it. It also differs from a command by its generality and abstractness.7 The degree of this generality or abstractness ranges continuously from the order that tells a man to do a particular thing here and now to the instruction that, in such and such conditions, whatever he does will have to satisfy certain requirements. Law in its ideal form might be described as a \"once-and-for-all\" command that is directed to unknown people and that is abstracted from all particular circumstances of time and place and refers only to such conditions as may occur anywhere and at any time. It is advisable, however, not to confuse laws and commands, though we must recognize that laws shade gradually into commands as their content becomes more specific.\n\nThe important difference between the two concepts lies in the fact that, as we move from commands to laws, the source of the decision on what particular action is to be taken shifts progressively from the issuer of the command or law to the acting person. The ideal type of command determines uniquely the action to be performed and leaves those to whom it is addressed no chance to use their own knowledge or follow their own predilections. The action performed according to such commands serves exclusively the purposes of him who has issued it. The ideal type of law, on the other hand, provides merely additional information to be taken into account in the decision of the actor.\n\nThe manner in which the aims and the knowledge that guide a particular action are distributed between the authority and the performer is thus the most important distinction between general laws and specific commands. It can be illustrated by the different ways in which the chief of a primitive tribe, or the head of a household, may regulate the activities of his subordinates. At the one extreme will be the instance where he relies entirely on specific orders and his subjects are not allowed to act at all except as ordered. If the chief prescribes on every occasion every detail of the actions of his subordinates, they will be mere tools, without an opportunity of using their own knowledge and judgment, and all the aims pursued and all the knowledge utilized will be those of the chief. In most circumstances, however, it will better serve his purposes if he gives merely general instructions about the kinds of actions to be performed or the ends to be achieved at certain times, and leaves it to the different individuals to fill in the details according to circumstances\u2014that is,  according to their knowledge. Such general instructions will already constitute rules of a kind, and the action under them will be guided partly by the knowledge of the chief and partly by that of the acting persons. It will be the chief who decides what results are to be achieved, at what time, by whom, and perhaps by which means; but the particular manner in which they are brought about will be decided by the individuals responsible. The servants of a big household or the employees of a plant will thus be mostly occupied with the routine of carrying out standing orders, adapting them all the time to particular circumstances and only occasionally receiving specific commands.\n\nIn these circumstances the ends toward which all activity is directed are still those of the chief. He may, however, also allow members of the group to pursue, within certain limits, their own ends. This presupposes the designation of the means that each may use for his purposes. Such an allocation of means may take the form of the assignment of particular things or of times that the individual may use for his own ends. Such a listing of the rights of each individual can be altered only by specific orders of the chief. Or the sphere of free action of each individual may be determined and altered in accordance with general rules laid down in advance for longer periods, and such rules can make it possible for each individual by his own action (such as bartering with other members of the group or earning premiums offered by the head for merit) to alter or shape the sphere within which he can direct his action for his own purposes. Thus, from the delimitation of a private sphere by rules, a right like that of property will emerge.\n\n3. A similar transition from specificity and concreteness to increasing generality and abstractness we also find in the evolution from the rules of custom to law in the modern sense. Compared with the laws of a society that cultivates individual freedom, the rules of conduct of a primitive society are relatively concrete. They not merely limit the range within which the individual can shape his own action but often prescribe specifically how he must proceed to achieve particular results, or what he must do at particular times and places. In them the expression of the factual knowledge that certain effects will be produced by a particular procedure and the demand that this procedure be followed in appropriate conditions are still undifferentiated. To give only one illustration: the rules which the Bantu observes when he moves between the fourteen huts of his village along strictly prescribed lines according to his age, sex, or status greatly restrict his choice.8 Though he is not obeying another  man's will but impersonal custom, having to observe a ritual to reach a certain point restricts his choice of method more than is necessary to secure equal freedom to others.\n\nThe \"compulsion of custom\" becomes an obstacle only when the customary way of doing things is no longer the only way that the individual knows and when he can think of other ways of achieving a desirable object. It was largely with the growth of individual intelligence and the tendency to break away from the habitual manner of action that it became necessary to state explicitly or reformulate the rules and gradually to reduce the positive prescriptions to the essentially negative confinement to a range of actions that will not interfere with the similarly recognized spheres of others.\n\nThe transition from specific custom to law illustrates even better than the transition from command to law what, for lack of a better term, we have called the \"abstract character\" of true law.9 Its general and abstract rules specify that in certain circumstances action must satisfy certain conditions; but all the many kinds of action that satisfy these conditions are permissible. The rules merely provide the framework within which the individual must move but within which the decisions are his. So far as his relations with other private persons are concerned, the prohibitions are almost entirely of a negative character, unless the person to whom they refer has himself, by his actions, created conditions from which positive obligations arise. They are instrumental, they are means put at his disposal, and they provide part of the data which, together with his knowledge of the particular circumstances of time and place, he can use as the basis for his decisions.\n\nSince the laws determine only part of the conditions that the actions of the individual will have to satisfy, and apply to unknown people whenever certain conditions are present, irrespective of most of the facts of the particular situation,  the lawgiver cannot foresee what will be their effect on particular people or for what purposes they will use them. When we call them \"instrumental,\" we mean that in obeying them the individual still pursues his own and not the lawgiver's ends. Indeed, specific ends of action, being always particulars, should not enter into general rules. The law will prohibit killing another person or killing except under conditions so defined that they may occur at any time or place, but not the killing of particular individuals.\n\nIn observing such rules, we do not serve another person's end, nor can we properly be said to be subject to his will. My action can hardly be regarded as subject to the will of another person if I use his rules for my own purposes as I might use my knowledge of a law of nature, and if that person does not know of my existence or of the particular circumstances in which the rules will apply to me or of the effects they will have on my plans. At least in all those instances where the coercion threatened is avoidable, the law merely alters the means at my disposal and does not determine the ends I have to pursue. It would be ridiculous to say that I am obeying another's will in fulfilling a contract, when I could not have concluded it had there not been a recognized rule that promises must be kept, or in accepting the legal consequence of any other action that I have taken in full knowledge of the law.\n\nThe significance for the individual of the knowledge that certain rules will be universally applied is that, in consequence, the different objects and forms of action acquire for him new properties. He knows of man-made cause-andeffect relations which he can make use of for whatever purpose he wishes. The effects of these man-made laws on his actions are of precisely the same kind as those of the laws of nature: his knowledge of either enables him to foresee what will be the consequences of his actions, and it helps him to make plans with confidence. There is little difference between the knowledge that if he builds a bonfire on the floor of his living room his house will burn down, and the knowledge that if he sets his neighbor's house on fire he will find himself in jail. Like the laws of nature, the laws of the state provide fixed features in the environment in which he has to move; though they eliminate certain choices open to him, they do not, as a rule, limit the choice to some specific action that somebody else wants him to take.\n\n4. The conception of freedom under the law that is the chief concern of this book rests on the contention that when we obey laws, in the sense of general abstract rules laid down irrespective of their application to us, we are not subject to another man's will and are therefore free. It is because the lawgiver does not know the particular cases to which his rules will apply, and it is because the judge who applies them has no choice in drawing the conclusions that follow from the existing body of rules and the particular facts of the case, that it can be said that laws and not men rule. Because the rule is laid down in ignorance of the particular case and no man's will decides the coercion used  to enforce it, the law is not arbitrary.10 This, however, is true only if by \"law\" we mean the general rules that apply equally to everybody. This generality is probably the most important aspect of that attribute of law which we have called its \"abstractness.\" As a true law should not name any particulars, so it should especially not single out any specific persons or group of persons.\n\nThe significance of a system in which all coercive action of government is confined to the execution of general abstract rules is often stated in the words of one of the great historians of the law; \"The movement of the progressive societies has hitherto been a movement _from Status to Contract_.\"11 The conception of status, of an assigned place that each individual occupies in society, corresponds, indeed, to a state in which the rules are not fully general but single out particular persons or groups and confer upon them special rights and duties. The emphasis on contract as the opposite of status is, however, a little misleading, as it singles out one, albeit the most important, of the instruments that the law supplies to the individual to shape his own position. The true contrast to a reign of status is the reign of general and equal laws, of the rules which are the same for all, or, we might say, of the rule of _leges_ in the original meaning of the Latin word for laws\u2014 _leges_ that is, as opposed to the _privi-leges_.\n\nThe requirement that the rules of true law be general does not mean that sometimes special rules may not apply to different classes of people if they refer to properties that only some people possess. There may be rules that can apply only to women or to the blind or to persons above a certain age. (In most such instances it would not even be necessary to name the class of people to whom the rule applies: only a woman, for example, can be raped or got with child.) Such distinctions will not be arbitrary, will not subject one group to the will of others, if they are equally recognized as justified by those inside and those outside the group. This does not mean that there must be unanimity as to the desirability of the distinction, but merely that individual  views will not depend on whether the individual is in the group or not. So long as, for instance, the distinction is favored by the majority both inside and outside the group, there is a strong presumption that it serves the ends of both. When, however, only those inside the group favor the distinction, it is clearly privilege; while if only those outside favor it, it is discrimination. What is privilege to some is, of course, always discrimination to the rest.\n\n5. It is not to be denied that even general, abstract rules, equally applicable to all, may possibly constitute severe restrictions on liberty. But when we reflect on it, we see how very unlikely this is. The chief safeguard is that the rules must apply to those who lay them down and those who apply them\u2014 that is, to the government as well as the governed\u2014and that nobody has the power to grant exceptions. If all that is prohibited and enjoined is prohibited and enjoined for all without exception (unless such exception follows from another general rule) and if even authority has no special powers except that of enforcing the law, little that anybody may reasonably wish to do is likely to be prohibited. It is possible that a fanatical religious group will impose upon the rest restrictions which its members will be pleased to observe but which will be obstacles for others in the pursuit of important aims. But if it is true that religion has often provided the pretext for the establishing of rules felt to be extremely oppressive and that religious liberty is therefore regarded as very important for freedom, it is also significant that religious beliefs seem to be almost the only ground on which general rules seriously restrictive of liberty have ever been universally enforced. But how comparatively innocuous, even if irksome, are most such restrictions imposed on literally everybody, as, for instance, the Scottish Sabbath, compared with those that are likely to be imposed only on some! It is significant that most restrictions on what we regard as private affairs, such as sumptuary legislation, have usually been imposed only on selected groups of people or, as in the case of prohibition, were practicable only because the government reserved the right to grant exceptions.\n\nIt should also be remembered that, so far as men's actions toward other persons are concerned, freedom can never mean more than that they are restricted only by general rules. Since there is no kind of action that may not interfere with another person's protected sphere, neither speech, nor the press, nor the exercise of religion can be completely free. In all these fields (and, as we shall see later, in that of contract) freedom does mean and can mean only that what we may do is not dependent on the approval of any person or authority and is limited only by the same abstract rules that apply equally to all.\n\nBut if it is the law that makes us free, this is true only of the law in this sense of abstract general rule, or of what is called \"the law in the material meaning,\" which differs from law in the merely formal sense by the character of the  rules and not by their origin.12 The \"law\" that is a specific command, an order that is called a \"law\" merely because it emanates from the legislative authority, is the chief instrument of oppression. The confusion of these two conceptions of law and the loss of the belief that laws can rule, that men in laying down and enforcing laws in the former sense are not enforcing their will, are among the chief causes of the decline of liberty, to which legal theory has contributed as much as political doctrine.\n\nWe shall have to return later to the manner in which modern legal theory has increasingly obscured these distinctions. Here we can only indicate the contrast between the two concepts of law by giving examples of the extreme positions taken on them. The classical view is expressed in Chief Justice John Marshall's famous statement: \"Judicial power, as contradistinguished from the power of the laws, has no existence. Courts are the mere instruments of the law, and can will nothing.\"13 Hold against this the most frequently quoted statement of a modern jurist, that has found the greatest favor among socalled progressives, namely, Justice Holmes's that \"general propositions do not decide concrete cases.\"14 The same position has been put by a contemporary political scientist thus: \"The law cannot rule. Only men can exercise power over other men. To say that laws rule and not men may consequently signify that the fact is to be hidden that men rule over men.\"15\n\nThe fact is that, if \"to rule\" means to make men obey another's will, government has no such power to rule in a free society. The citizen as citizen cannot be ruled in this sense, cannot be ordered about, no matter what his position may be in the job he has chosen for his own purposes or while, in accordance with the law, he temporarily becomes the agent of government. He can be ruled, however, in the sense in which \"to rule\" means the enforcement of general rules, laid down irrespective of the particular case and equally applicable to all. For here no human decision will be required in the great majority of cases to which the rules apply; and even when a court has to determine how the general rules may be applied to a particular case, it is the implications of the whole system of accepted rules that decide, not the will of the court.\n\n6. The rationale of securing to each individual a known range within which he can decide on his actions is to enable him to make the fullest use of his knowledge, especially of his concrete and often unique knowledge of the particular  circumstances of time and place.16 The law tells him what facts he may count on and thereby extends the range within which he can predict the consequences of his actions. At the same time it tells him what possible consequences of his actions he must take into account or what he will be held responsible for. This means that what he is allowed or required to do must depend only on circumstances he can be presumed to know or be able to ascertain. No rule can be effective, or can leave him free to decide, that makes his range of free decisions dependent on remote consequences of his actions beyond his ability to foresee. Even of those effects which he might be presumed to foresee, the rules will single out some that he will have to take into account while allowing him to disregard others. In particular, such rules will not merely demand that he must not do anything that will damage others but will be\u2014or should be\u2014so expressed that, when applied to a particular situation, they will clearly decide which effects must be taken into account and which need not.\n\nIf the law thus serves to enable the individual to act effectively on his own knowledge and for this purpose adds to his knowledge, it also embodies knowledge, or the results of past experience, that are utilized so long as men act under these rules. In fact, the collaboration of individuals under common rules rests on a sort of division of knowledge,17 where the individual must take account of particular circumstances but the law ensures that their action will be adapted to certain general or permanent characteristics of their society. This experience, embodied in the law, that individuals utilize by observing rules, is difficult to discuss, since it is ordinarily not known to them or to any one person. Most of these rules have never been deliberately invented but have grown through a gradual process of trial and error in which the experience of successive generations has helped to make them what they are. In most instances, therefore, nobody knows or has ever known all the reasons and considerations that have led to a rule being given a particular form. We must thus often endeavor to _discover_ the functions that a rule actually serves. If we do not know the rationale of a particular rule, as is often the case, we  must try to understand what its general function or purpose is to be if we are to improve upon it by deliberate legislation.\n\nThus the rules under which the citizens act constitute an adaptation of the whole of society to its environment and to the general characteristics of its members. They serve, or should serve, to assist the individuals in forming plans of action that they will have a good chance of carrying through. The rules may have come to exist merely because, in a certain type of situation, friction is likely to arise among individuals about what each is entitled to do, which can be prevented only if there is a rule to tell each clearly what his rights are. Here it is necessary merely that some known rule cover the type of situation, and it may not matter greatly what its contents are.\n\nThere will, however, often be several possible rules which satisfy this requirement but which will not be equally satisfactory. What exactly is to be included in that bundle of rights that we call \"property,\" especially where land is concerned, what other rights the protected sphere is to include, what contracts the state is to enforce, are all issues in which only experience will show what is the most expedient arrangement. There is nothing \"natural\" in any particular definition of rights of this kind, such as the Roman conception of property as a right to use or abuse an object as one pleases, which, however often repeated, is in fact hardly practicable in its strict form. But the main features of all somewhat more advanced legal orders are sufficiently similar to appear as mere elaborations of what David Hume called the \"three fundamental laws of nature, _that of the stability of possession, of its transference by consent, and of the performance of promises_.\"18\n\nOur concern here cannot be, however, the particular content but only certain general attributes which these rules ought to possess in a free society. Since the lawgiver cannot foresee what use the persons affected will make of his rules, he can only aim to make them beneficial on the whole or in the majority of cases. But, as they operate through the expectations that they create, it is essential that they be always applied, irrespective of whether or not the con  sequences in a particular instance seem desirable.19 That the legislator confines himself to general rules rather than particular commands is the consequence of his necessary ignorance of the special circumstances under which they apply; all he can do is to provide some firm data for the use of those who have to make plans for particular actions. But in fixing for them only some of the conditions of their actions, he can provide opportunities and chances, but never certainties so far as the results of their efforts are concerned.\n\nThe necessity of emphasizing that it is of the essence of the abstract rules of law that they will only be likely to be beneficial in most cases to which they apply and, in fact, are one of the means by which man has learned to cope  with his constitutional ignorance, has been imposed on us by certain rationalist interpretations of utilitarianism. It is true enough that the justification of any particular rule of law must be its usefulness\u2014even though this usefulness may not be demonstrable by rational argument but known only because the rule has in practice proved itself more convenient than any other. But, generally speaking, only the rule as a whole must be so justified, not its every application.20 The idea that each conflict, in law or in morals, should be so decided as would seem most expedient to somebody who could comprehend all the consequences of that decision involves the denial of the necessity of any rules. Only a society of omniscient individuals could give each person \"complete liberty to weigh every particular action on general utilitarian grounds.\"21 Such an \"extreme\" utilitarianism leads to absurdity; and only what has been called \"restricted\" utilitarianism has therefore any relevance to our problem. Yet few beliefs have been more destructive of the respect for the rules of law and of morals than the idea that a rule is binding only if the beneficial effect of observing it in the particular instance can be recognized.\n\nThe oldest form of this misconception has been associated with the (usually misquoted) formula \"salus populi suprema lex esto\" (\"the welfare of the people ought to be\u2014not 'is'\u2014the highest law\").22 Correctly understood, it means that the end of the law ought to be the welfare of the people, that the general rules should be so designed as to serve it, but _not_ that any conception of a particular social end should provide a justification for breaking those general rules. A specific end, a concrete result to be achieved, can never be a law.\n\n7. The enemies of liberty have always based their arguments on the contention  that order in human affairs requires that some should give orders and others obey.23 Much of the opposition to a system of freedom under general laws arises from the inability to conceive of an effective co-ordination of human activities without deliberate organization by a commanding intelligence. One of the achievements of economic theory has been to explain how such a mutual adjustment of the spontaneous activities of individuals is brought about by the market, provided that there is a known delimitation of the sphere of control of each individual. An understanding of that mechanism of mutual adjustment of individuals forms the most important part of the knowledge that ought to enter into the making of general rules limiting individual action.\n\nThe orderliness of social activity shows itself in the fact that the individual can carry out a consistent plan of action that, at almost every stage, rests on the expectation of certain contributions from his fellows. \"That there is some kind of order, consistency and constancy, in social life is obvious. If there were not, none of us would be able to go about our affairs or satisfy our most elementary needs.\"24 This orderliness cannot be the result of a unified direction if we want individuals to adjust their actions to the particular circumstances largely known only to them and never known in their totality to any one mind. Order with reference to society thus means essentially that individual action is guided by successful foresight, that people not only make effective use of their knowledge but can also foresee with a high degree of confidence what collaboration they can expect from others.25\n\nSuch an order involving an adjustment to circumstances, knowledge of which is dispersed among a great many people, cannot be established by central direction. It can arise only from the mutual adjustment of the elements and their response to the events that act immediately upon them. It is what M. Polanyi has called the spontaneous formation of a \"polycentric order\": \"When order is achieved among human beings by allowing them to interact with each other on their own initiative\u2014subject only to the laws which uniformly apply to all of them\u2014we have a system of spontaneous order in society. We may then say that the efforts of these individuals are co-ordinated by exercising their individual initiative and that this self-co-ordination justifies this liberty on public grounds. The actions of such individuals are said to be free, for they are not determined by any _specific_ command, whether of a superior or a public authority; the compulsion to which they are subject is impersonal and general.\"26\n\nThough people more familiar with the manner in which men order physical objects often find the formation of such spontaneous orders difficult to comprehend, there are, of course, many instances in which we must similarly rely on the spontaneous adjustments of individual elements to produce a physical order. We could never produce a crystal or a complex organic compound if we had to place each individual molecule or atom in the appropriate place in relation to the others. We must rely on the fact that in certain conditions they will arrange themselves in a structure possessing certain characteristics. The use of these spontaneous forces, which in such instances is our only means of achieving the desired result, implies, then, that many features of the process creating the order will be beyond our control; we cannot, in other words, rely on these forces and at the same time make sure that particular atoms will occupy specific places in the resulting structure.\n\nSimilarly, we can produce the conditions for the formation of an order in society, but we cannot arrange the manner in which its elements will order themselves under appropriate conditions. In this sense the task of the lawgiver is not to set up a particular order but merely to create conditions in which an orderly arrangement can establish and ever renew itself. As in nature, to induce the establishment of such an order does not require that we be able to predict the behavior of the individual atom\u2014that will depend on the unknown particular circumstances in which it finds itself. All that is required is a limited regularity in its behavior; and the purpose of the human laws we enforce is to secure such limited regularity as will make the formation of an order possible.\n\nWhere the elements of such an order are intelligent human beings whom  we wish to use their individual capacities as successfully as possible in the pursuit of their own ends, the chief requirement for its establishment is that each know which of the circumstances in his environment he can count on. This need for protection against unpredictable interference is sometimes represented as peculiar to \"bourgeois society.\"27 But, unless by \"bourgeois society\" is meant any society in which free individuals co-operate under conditions of division of labor, such a view confines the need to far too few social arrangements. It is the essential condition of individual freedom, and to secure it is the main function of law.28\n\n* * *\n\nThe quotation at the head of the chapter is taken from Jos\u00e9 Ortega y Gasset, _Mirabeau o El politico_ (1927), in _Obras completas_ (Madrid: Revista de Occidente, 1947), vol. 3, p. 603: \"Orden no es una presi\u00f3n que desde fuera se ejerce sobra la sociedad, sino un equilibrio que se suscita en su interior.\" Cf. John Corrie Carter, \"The Ideal and the Actual in the Law,\" Annual address delivered at the _Thirteenth Annual Meeting of the American Bar Association_ (Philadelphia: Dando Publishing Co., 1890), p. 21. [Reprinted from the _American Law Review_ , 24 (1890): 768\u201369]: \"Law is not a body of commands imposed upon society from without, either by an individual sovereign or superior, or by a sovereign body constituted by representatives of society itself. It exists at all times as one of the elements of society springing directly from habit and custom. It is therefore the unconscious creation of society, or in other words, a growth.\" Regarding Carter, who was infuenced by Luther Stearns Cushing and Frederick Carl von Savigny, see M. J. Aronson, \"The Juridical Evolutionism of James Coolidge Carter,\" _University of Toronto Law Journal_ , 10 (1953): 1\u201353. The stress on the law being prior to the state, which is the organized effort to create and enforce it, goes back at least to David Hume (see his _Treatise of Human Nature_ , bk. 3, pt. 2, vol. 2, pp. 252\u2013333).\n\n1 Friedrich Karl von Savigny, _System des heutigen r\u00f6mischen Rechts_ (Berlin: Veit und Comp., 1840), vol. 1, pp. 331\u201332. The passage quoted in translation is a condensation of two sentences which deserve to be quoted in their context: \"Der Mensch steht inmitten der \u00e4ussern Welt, und das wichtigste Element in dieser seiner Umgebung ist ihm die Ber\u00fchrung mit denen, die ihm gleich sind durch ihre Natur und Bestimmung. Sollen nun in solcher Ber\u00fchrung freie Wesen neben einander bestehen, sich gegenseitig f\u00f6rdernd, nicht hemmend, in ihrer Entwicklung, so ist dieses nur m\u00f6glich durch Anerkennung einer unsichtbaren Grenze, innerhalb welcher das Dasein und die Wirksamkeit jedes Einzelnen einen sichern, freien Raum gewinne. Die Regel, wodurch jene Grenze und durch die dieser freie Raum bestimmt wird, ist das Recht. Damit ist zugleich die Verwandtschaft und die Verschiedenheit zwischen Recht und Sittlichkeit gegeben. Das Recht dient der Sittlichkeit, aber nicht indem es ihr Gebot vollzieht, sondern indem es die freie Entfaltung ihrer, jedem einzelnen Willen inwohnenden, Kraft sichert. Sein Dasein aber ist ein selbst\u00e4ndiges, und darum ist es kein Widerspruch, wenn im einzelnen Fall die M\u00f6glichkeit unsittlicher Aus\u00fcbung eines wirklich vorhandenen Rechts behauptet wird.\" (The spelling of this passage has been modernized.) [\"We exist in the external world and the most important element in our surroundings is our contact with those who have similar natures and destinies. If these contacts are of free beings, supporting and not hindering each other in our development, then we must recognize an invisible border line surrounding each one of us within which our essential nature and effectiveness finds a secure and unconstrained space. The arrangements by which the rules governing these boundaries and these spaces are determined is the law. Here too we see how law and morality are related and distinguished. The law serves morality, not in that it fulfills her commands, but rather in that it secures the free development of the moral power as it resides in each individual will. The existence of law, however, is independent of that of morality inasmuch as it is not a contradiction when, in any specific case, the immoral implementations of an existing law is claimed.\"\u2014Ed.] See also John William Salmond, _Salmond on Jurisprudence_ , Glanville Llewelyn Williams, ed. (11th ed.; London: Sweet and Maxwell, 1957), p. 63: \"the rule of justice determines the sphere of individual liber ty in the pursuit of individual welfare, so as to conf ne that liberty within the limits which are consistent with the general welfare of mankind.\"\n\n2 Charles Beudant, _Le Droit individuel et l'\u00c9tat: Introduction \u00e0 l'\u00e9tude du droit_ (Paris: A. Rousseau, 1891), p. 5: \"Le Droit, au sens le plus general du mot, est la science de la libert\u00e9.\" [\"Law, in the most general sense of the term, is the science of liberty.\"\u2014Ed.]\n\n3 A number of particularly insightful explanations about the role of rules in determining social structures can be found in the writings of Richard Stanley Peters, especially _The Concept of Motivation_ (London: Routledge and Kegan Paul, 1958), as well as the book he wrote in conjunction with Stanley Isaac Benn, _Social Principles and the Democratic State_ (London: Allen and Unwin, 1959).\n\n4 Cf. Carl Menger, _Untersuchungen_ , app. 8, pp. 271\u201387.\n\n5 \"Abstraction\" does not appear only in the form of verbal statements. It manifests itself also in the way in which we respond similarly to any one of a class of events which in most respects may be very different from one another, and in the feelings which are evoked by these events and which guide our action, be it a sense of justice or of moral or aesthetic approval or disapproval. Also there are probably always more general principles governing our minds which we cannot formulate, yet which guide our thinking\u2014laws of the structure of the mind which are too general to be formulated within that structure. Even when we speak of an abstract rule guiding decisions, we need not mean a rule expressed in words but merely one which could be so formulated. On all these problems compare my book, _The Sensory Order: An Inquiry into the Foundations of Theoretical Psychology_ (Chicago: University of Chicago Press, 1952), and my article \"Rules, Perception, and Intelligibility,\" _Proceedings of the British Academy_ , 48 (1962): 321\u201344 [Reprinted in _Studies in Philosophy, Politics, and Economics_ , pp.43\u201365.]; also Adam Ferguson, _An Essay on the History of Civil Society_ , (London: A. Millar and T. Caddel, 1767), 38\u201346.\n\n6 Cf. Edward Sapir, _Selected Writings of Edward Sapir in Language, Culture, and Personality_ , David Goodman Mandelbaum, ed. (Berkeley: University of California Press, 1949), p. 548: \"It is easy for an Australian native, for instance, to say by what kinship term he calls so and so or whether or not he may undertake such and such relations with a given individual. It is exceedingly difficult for him to give a general rule of which these specific examples of behavior are but illustrations, though all the while he acts as though the rule were perfectly well known to him. _In a sense it is well known to him_. But this knowledge is not capable of conscious manipulation in terms of word symbols. It is, rather, a very delicately nuanced feeling of subtle relations, both experienced and possible.\"\n\n7 The treatment of law as a species of command (deriving from Francis Bacon, Thomas Hobbes, and John Austin) was originally intended to stress the logical similarity of these two kinds of sentences as distinguished from, say, a statement of fact. It should not, however, obscure, as it has often done, the essential differences. Cf. Karl Olivecrona, _Law as Fact_ (Copenhagen: E. Munksgaard, 1939), p. 43, where laws are described as \"independent imperatives\" which are \"nobody's commands, though they have the form of language that is characteristic of a command\"; also Richard Wollheim, \"The Nature of Law,\" _Political Studies_ , 2 (1954): 128\u201341.\n\n8 I have borrowed this illustration from Jos\u00e9 Ortega y Gasset, _Del imperio romano_ (1940), in _Obras completas_ (6 vols.; Madrid: Revista de Occidente, 1947), vol. 6, p. 76, who presumably derives it from some anthropologist. [Ortega writes: \"El lector no sonreir\u00eda tan absolutamente si conociese un poco mejor la historia de la circulaci\u00f3n humana, las angustias y luchas que ha ocasionado y si yo tuviese espacio libre para dibujar aqu\u00ed un gr\u00e1fico de las l\u00edneas rigurosamente prescritas que tiene que seguir hoy mismo el africano bant\u00fa para moverse, seg\u00fan su edad, sexo y condici\u00f3n, entre las catorce chozas de su aldea.\" (\"The reader would not smile so broadly were he better acquainted with the history of the movement of human beings, the anxieties and problems to which it has given rise and, had I sufficient space to describe it, the rigorously prescribed lines along which the African Bantu were today required to follow depending on his age, his sex, and his status, among the fourteen huts of his village.\")\u2014Ed.]\n\n9 If there were no danger of confusion with the other meanings of those terms, it would be preferable to speak of \"formal\" rather than of \"abstract\" laws, in the same sense as that in which the term \"formal\" is used in logical discussion. (Cf. Sir Karl Raimund Popper, _Logik der Forschung: Zur Erkenntnistheorie der modernen Naturwissenschaft_ [Vienna: Julius Springer, 1935], pp. 85 and 29\u201332.) Unfortunately, \"formal\" is also applied to everything that is enacted by the legislature, while only if such an enactment takes the form of an abstract rule, such a law in the formal sense is a law also in the substantive or material sense. For example, when Max Weber, in _Max Weber on Law in Economy and Society_ , Max Rheinstein, ed. (Cambridge, MA: Harvard University Press, 1954), pp. 226\u201329, speaks of \"formal justice,\" he means justice determined by law, not merely in the formal but in the substantive sense. On this distinction in German and French constitutional law see chap. 14, n. 10, below.\n\n10 Cf. George Cornewall Lewis, _An Essay on the Government of Dependencies_ (London: John Murray, 1841), p. 16n.: \"When a person voluntarily regulates his conduct according to a rule or maxim which he has previously announced his intention of conforming to, he is thought to deprive himself of _arbitrium_ , free will, discretion, or _Willk\u00fcr_ , in the individual act. Hence when a government acts in an individual case, not in conformity with a pre-existing law or rule of conduct, laid down by itself, its act is said to be arbitrary.\" Also, (p. 24): \"Every government, whether monarchical, aristocratical, or democratical, may be conducted arbitrarily, and not in accordance with general rules. There is not, and cannot be, anything in the form of any government, which will afford its subjects a legal security against an improper arbitrary exercise of the sovereign power. This security is to be found only in the influence of public opinion, and the other moral restraints which create the main difference in the goodness of supreme governments.\"\n\n11 Sir Henry Sumner Maine, _Ancient Law_ : _Its Connection with the Early History of Society and Its Relation to Modern Ideas_ (London: John Murray, 1861), p.141; cf. Ronald Harry Graveson, \"The Movement from Status to Contract,\" _Modern Law Review_ , 4 (1940\u201341): 261\u201372.\n\n12 Cf. n. 9 above and the later discussion to which it refers.\n\n13 Chief Justice John Marshall in _Osborn v. Bank of United States_ , 22 _U.S_. (9 _Wheaton_ ) 738, at 866 (1824).\n\n14 Oliver Wendell Holmes, Jr., _Lochner v. New York_ , 198 _U.S._ 45, at 76 (1905).\n\n15 Franz Leopold Neumann, \"The Concept of Political Freedom,\" _Columbia Law Review_ , 53 (1953): 910, reprinted in his _The Democratic and the Authoritarian State_ : _Essays in Political and Legal Theory_ (Glencoe, IL: The Free Press, 1957), p. 169.\n\n16 Cf. Smith, _Wealth of Nations_ [bk. 4, chap. 2], vol. 1, p. 421 [Liberty Fund edition, vol. 1, p. 456]: \"What is the species of domestic industry which his capital can employ, and of which the produce is likely to be of the greatest value, every individual, it is evident, can, _in his local situation_ , judge much better than any statesman or lawgiver can do for him.\" (Italics added.)\n\n17 Cf. Lionel Robbins, _The Theory of Economic Policy_ : _Essays in Political and Legal Theory_ (Glencoe, IL: The Free Press, 1957), p. 193: The classical liberal \"proposes, as it were, a division of labor: the state shall prescribe what individuals shall not do, if they are not to get in each other's way, while the citizen shall be left free to do anything which is not so forbidden. To the one is assigned the task of establishing formal rules, to the other responsibility for the substance of specific action.\"\n\n18 David Hume, _Treatise of Human Nature_ [bk. 3, pt. 2, sec. 6], ( _Works_ , vol. 2, p. 293); cf. also John Walter Jones, _Historical Introduction to the Theory of Law_ (Oxford: Clarendon Press, 1940), p. 114: \"In looking through the French Code and leaving out of account the law of the family, Duguit finds only three fundamental rules and no more\u2014freedom of contract, the inviolability of property, and the duty to compensate another for damage due to one's fault. All the rest resolve themselves into subsidiary directions to some State agent or other.\" [L\u00e9on Duguit (1859\u20131928), noted French jurist and dean of the law school at Bordeaux. He was strongly opposed to the juridical theories of Georg Jellinek, whom he regarded as too \"individualistic.\" Instead, he wished to incorporate the relationship of individuals to collectivities in legal thinking. More importantly, he rejected the notion that law was a creation of the State but posited the theory that it took its shape from the social needs of men.\u2014Ed.]\n\n19 Cf. David Hume, _Treatise of Human Nature_ , bk. 2, pt. 2, secs. 2\u20136 ( _Works_ , vol. 2, pp. 258\u2013300), which still contains perhaps the most satisfactory discussion of the problems considered here, esp. vol. 2, p. 269: \"A single act of justice is frequently contrary to public interest; and were it to stand alone, without being follow'd by other acts, may, in itself, be very prejudicial to society. . . . Nor is every single act of justice, consider'd apart, more conducive to private interest than to public; . . . But however single acts of justice may be contrary, either to public or private interest, 'tis certain, that the whole plan or scheme is highly conducive, or indeed absolutely requisite, both to the support of society and the well-being of every individual. 'Tis impossible to separate the good from the ill. Property must be stable, and must be fix'd by general rules. Tho' in one instance the public be a sufferer, this momentary ill is amply compensated by the steady prosecution of the rule, and by the peace and order, which it establishes in society.\" See also Hume's _Enquiry Concerning the Principles of Morals_ , in _Essays_ , vol. 2, p. 273: \"The benefit, resulting from [the social virtues of justice and fidelity] is not the consequence of every individual single act; but arises from the whole scheme or system, concurred in by the whole, or the greater part of the society. . . . The result of the individual acts is here, in many instances, directly opposite to that of the whole system of actions; and the former may be extremely hurtful, while the latter is, to the highest degree, advantageous. Riches, inherited from a parent, are, in a bad man's hand, the instrument of mischief. The right of succession may, in one instance, be hurtful. Its benefit arises only from the observance of the general rule; and it is sufficient, if compensation be thereby made for all the ills and inconveniencies, which flow from particular characters and situations.\" Also see the _Enquiry_ , p. 274: \"All the laws of nature, which regulate property, as well as all civil laws, are general, and regard alone some essential circumstances of the case, without taking into consideration the characters, situations, and connexions of the person concerned, or any particular consequences which may result from the determination of these laws, in any particular case which offers. They deprive, without scruple, a beneficent man of all his possessions, if acquired by mistake, without a good title; in order to bestow them on a selfish miser, who has already heaped up immense stores of superfluous riches. Public utility requires, that property should be regulated by general inflexible rules; and though such rules are adopted as best serve the same end of public utility, it is impossible for them to prevent all particular hardships, or make beneficial consequences result from every individual case. It is sufficient, if the whole plan or scheme be necessary to the support of civil society, and if the balance of good, in the main, do thereby preponderate much above that of evil.\" See my \"The Legal and Political Philosophy of David Hume,\" _Il Politico_ , 28 (1963): 691\u2013704; reprinted in Friedrich August Hayek, _Studies in Philosophy, Politics, and Economics_ (New York: Simon and Schuster, 1967), pp. 106\u201321. I would like in this connection to acknowledge my indebtedness to Sir Arnold Plant, who many years ago first drew my attention to the importance of Hume's discussion of these issues.\n\n20 See John Stuart Mill, \"On Liberty,\" in _On Liberty and Considerations on Representative Government_ , Ronald Buchanan McCallum, ed. (Oxford: Blackwell, 1946), p. 68.\n\n21 See John Rawls, \"Two Concepts of Rules,\" _Philosophical Review_ , 64 (1955): 3\u201332; John Jamieson Carswell Smart, \"Extreme and Restricted Utilitarianism,\" _Philosophical Quarterly_ , 6 (1956): 344\u201354; Henry John McCloskey, \"An Examination of Restricted Utilitarianism,\" _Philosophical Review_ , 66 (1957): 466\u201385; James Opie Urmson, \"The Interpretation of the Moral Philosophy of J. S. Mill,\" _Philosophical Quarterly_ , 3 (1953): 33\u201339; John David Mabbott, \"Interpretations of Mill's Utilitarianism,\" _Philosophical Quarterly_ , 6 (1956): 115\u201320; and Stephen Edelston Toulmin, _An Examination of the Place of Reason in Ethics_ (Cambridge: Cambridge University Press, 1950), esp. p. 161.\n\n22 John Selden in his _The Table Talk of John Selden_ ([Oxford: Clarendon Press, 1892], p. 131) observes: \"There is not anything in the world so much abused as this sentence, _Salus populi suprema lex esto_.\" [The phrase originates in Cicero _De Legibus_ , bk. 3, chap. 3, sec. 8.\u2014Ed.] Cf. Charles Howard McIlwain, _Constitutionalism: Ancient and Modern_ (rev. ed.; Ithaca, NY: Cornell University Press, 1947), p. 149, n. 6 [Liberty Fund edition, p. 4 n. 6], and, on the general issue, Friedrich Meinecke, _Die Idee der Staatsr\u00e4son in der neueren Geschichte_ (Munich: R. Oldenburg, 1924), p. 265, now translated as _Machiavellism: the Doctrine of Raison d'\u00e9tat and its Place in Modern History_ , Douglas Scott, trans. (London: Routledge and Kegan Paul, 1957); see also Mises, _Socialism_ , p. 400 [Liberty Fund edition, p. 359].\n\n23 Cf., e.g., the opinion of James I, quoted by Francis Dunham Wormuth, _The Origins of Modern Constitutionalism_ (New York: Harper, 1949), p. 51, that \"order was dependent upon the relationship of command and obedience. All organization derived from superiority and subordination.\" [ James makes this point in his \"Triplici nodo, triplex cuneus. Or an apologie for the oath of allegiance. Against the two breves of pope pavlvs qvintvs, and the late letter of cardinall bellarmine to g. blackwel the arch-priest,\" in _The Political Works of James I_ (reprinted from the edition of 1616), Charles Howard McIlwain, ed. (Cambridge, MA: Harvard University Press, 1918).\u2014Ed.]\n\n24 I apologize to the author whose words I quote but whose name I have forgotten. I had noted the passage with a reference to Edward Evan Evans-Pritchard, _Social Anthropology_ (London: Cohen and West, 1951), p. 19, but, though the same idea is expressed there, it is not in the words quoted. [ The quotation that Hayek cannot place does indeed come from Evans Pritchard's monograph _Social Anthropology_ (Glencoe, IL: The Free Press, 1954), p. 49. The passage on p. 19, which expresses the same idea, reads: \"It is evident that there must be uniformities and regularities in social life, that a society must have some sort of order, or its members could not live together.\"\u2014Ed.]\n\n25 Cf. Hermann Jahrreiss, \"Gr\u00f6\u00dfe und Not der Gesetzgebung,\" in _Mensch und Staat: Rechtsphilosophische, staatsrechtliche und v\u00f6lkerrechtliche Grundfragen in unserer Zeit_ , Hermann Jahrreiss, ed. (Cologne: C. Heymann, 1957), p. 22: \"Sozial-Ordnung ist Sozial-Berechenbarkeit.\" [\"Social order is social predictability.\"\u2014Ed.]\n\n26 Michael Polanyi, _The Logic of Liberty: Reflections and Rejoinders_ (London: Routledge and Kegan Paul, 1951), p. 159 [Liberty Fund edition, p. 195].\n\n27 Max Weber, _Theory of Social and Economic Organization: Being Part I of Wirtschaft und Gesellschaft_ , Talcott Parsons, ed. (London: W. Hodge, 1947), p. 386, tends to treat the need for \"calculability and reliability in the functioning of the legal order\" as a peculiarity of \"capitalism\" or the \"bourgeois phase\" of society. This is correct only if these terms are regarded as descriptive of any free society based on the division of labor.\n\n28 Cf. Emil Brunner, _Justice and the Social Order_ (New York: Harper, 1945), p. 22: \"Law is order by foresight. With regard to human beings, that is the service it renders; it is also its burden and its danger. It offers protection from the arbitrary, it gives a feeling of reliability, of security, it takes from the future its ominous darkness.\"\n\n## ELEVEN\n\nTHE ORIGINS OF THE RULE OF LAW\n\nThe end of the law is, not to abolish or restrain, but to preserve and enlarge freedom. For in all the states of created beings capable of laws, where there is no law there is no freedom. For liberty is to be free from restraint and violence from others; which cannot be where there is no law: and is not, as we are told, a liberty for every man to do what he lists. (For who could be free when every other man's humour might domineer over him?) But a liberty to dispose, and order freely as he lists, his person, actions, possessions, and his whole property, within the allowance of those laws under which he is, and therein not to be the subject of the arbitrary will of another, but freely follow his own.\n\n\u2014John Locke\n\n1. Individual liberty in modern times can hardly be traced back farther than the England of the seventeenth century.1 It appeared first, as it probably  always does, as a by-product of a struggle for power rather than as the result of deliberate aim. But it remained long enough for its benefits to be recognized. And for over two hundred years the preservation and perfection of individual liberty became the guiding ideal in that country, and its institutions and traditions the model for the civilized world.2\n\nThis does not mean that the heritage of the Middle Ages is irrelevant to modern liberty. But its significance is not quite what it is often thought to be. True, in many respects medieval man enjoyed more liberty than is now commonly believed. But there is little ground for thinking that the liberties of the English were then substantially greater than those of many Continental peoples.3 But if men of the Middle Ages knew many liberties in the sense of  privileges granted to estates or persons, they hardly knew liberty as a general condition of the people. In some respects the general conceptions that prevailed then about the nature and sources of law and order prevented the problem of liberty from arising in its modern form. Yet it might also be said that it was because England retained more of the common medieval ideal of the supremacy of law, which was destroyed elsewhere by the rise of absolutism, that she was able to initiate the modern growth of liberty.4\n\nThis medieval view, which is profoundly important as background for modern developments, though completely accepted perhaps only during the early Middle Ages, was that \"the state cannot itself create or make law, and of course as little abolish or violate law, because this would mean to abolish justice itself, it would be absurd, a sin, a rebellion against God who alone creates  law.\"5 For centuries it was recognized doctrine that kings or any other human authority could only declare or find the existing law, or modify abuses that had crept in, and not create law.6 Only gradually, during the later Middle  Ages, did the conception of deliberate creation of new law\u2014legislation as we know it\u2014come to be accepted. In England, Parliament thus developed from what had been mainly a law-finding body to a law-creating one. It was finally in the dispute about the authority to legislate in which the contending parties reproached each other for acting arbitrarily\u2014acting, that is, not in accordance with recognized general laws\u2014that the cause of individual freedom was inadvertently advanced. The new power of the highly organized national state which arose in the fifteenth and sixteenth centuries used legislation for the first time as an instrument of deliberate policy. For a while it seemed as if this new power would lead in England, as on the Continent, to absolute monarchy, which would destroy the medieval liberties.7 The conception of limited government which arose from the English struggle of the seventeenth century was thus a new departure, dealing with new problems. If earlier English doctrine and the great medieval documents, from Magna Carta, the great \"Constitutio Libertatis,\"8 downward, are significant in the development of the modern, it is because they served as weapons in that struggle.\n\nYet if for our purposes we need not dwell longer on the medieval doctrine, we must look somewhat closer at the classical inheritance which was revived at the beginning of the modern period. It is important, not only because of the great influence it exercised on the political thought of the seventeenth century, but also because of the direct significance that the experience of the ancients has for our time.9\n\n2. Though the influence of the classical tradition of the modern ideal of liberty is indisputable, its nature is often misunderstood. It has often been said that the ancients did not know liberty in the sense of individual liberty. This is true of many places and periods even in ancient Greece, but certainly not of Athens at the time of its greatness (or of late republican Rome); it may be true of the degenerate democracy of Plato's time, but surely not of those Athenians to whom Pericles said that \"the freedom which we enjoy in our government extends also to our ordinary life [where], far from exercising a jealous surveillance over each other, we do not feel called upon to be angry with our neighbour for doing what he likes\"10 and whose soldiers, at the moment  of supreme danger during the Sicilian expedition, were reminded by their general that, above all, they were fighting for a country in which they had \"unfettered discretion in it to all to live as they pleased.\"11 What were the main characteristics of that freedom of the \"freest of free countries,\" as Nicias called Athens on the same occasion, as seen both by the Greeks themselves and by Englishmen of the later Tudor and Stuart times?\n\nThe answer is suggested by a word which the Elizabethans borrowed from the Greeks but which has since gone out of use.12 \"Isonomia\" was imported  into England from Italy at the end of the sixteenth century as a word meaning \"equality of laws to all manner of persons\";13 shortly afterward it was freely used by the translator of Livy in the Englished form \"isonomy\" to describe a state of equal laws for all and responsibility of the magistrates.14 It continued in use during the seventeenth century15 until \"equality before the law,\" \"government of law,\" or \"rule of law\" gradually displaced it.\n\nThe history of the concept in ancient Greece provides an interesting lesson because it probably represents the first instance of a cycle that civilizations seem to repeat. When it first appeared,16 it described a state which Solon had earlier established in Athens when he gave the people \"equal laws for the noble and the base\"17 and thereby gave them \"not so much the control of  public policy, as the certainty of being governed legally in accordance with known rules.\"18 Isonomy was contrasted with the arbitrary rule of tyrants and became a familiar expression in popular drinking songs celebrating the assassination of one of these tyrants.19 The concept seems to be older than that of  _demokratia_ , and the demand for equal participation of all in the government appears to have been one of its consequences. To Herodotus it is still isonomy rather than democracy which is the \"most beautiful of all names of a political order.\"20 The term continued in use for some time after democracy had been achieved, at first in its justification and later, as has been said,21 increasingly in order to disguise the character it assumed; for democratic government soon came to disregard that very equality before the law from which it had derived its justification. The Greeks clearly understood that the two ideals, though related, were not the same: Thucydides speaks without hesitation about an \"isonomic oligarchy,\"22 and Plato even uses the term \"isonomy\" in deliberate contrast to democracy rather than in justification of it.23 By the end of the fourth century it had come to be necessary to emphasize that \"in a democracy the laws should be masters.\"24\n\nAgainst this background certain famous passages in Aristotle, though he no longer uses the term \"isonomia,\" appear as a vindication of that traditional ideal. In the _Politics_ he stresses that \"it is more proper that the law should govern than any one of the citizens,\" that the persons holding supreme power  \"should be appointed to be only guardians and the servants of the law,\" and that \"he who would place the supreme power in mind, would place it in God and the laws.\"25 He condemns the kind of government in which \"the people govern and not the law\" and in which \"everything is determined by a majority vote and not by a law.\" Such a government is to him not that of a free state, \"for, where government is not in the laws, then there is no free state, for the law ought to be supreme over all things.\" A government that \"centers all power in the votes of the people cannot, properly speaking, be a democracy: for their decrees cannot be general in their extent.\"26 If we add to this the following passage in the _Rhetoric_ , we have indeed a fairly complete statement of the ideal of government by law:27 \"It is of great moment that well drawn laws should themselves define all the points they possibly can, and leave as few as possible to the decision of the judges, [for] the decision of the lawgiver is not particular but prospective and general, whereas members of the assembly and the jury find it their duty to decide on definite cases brought before them.\"28\n\nThere is clear evidence that the modern use of the phrase \"government by laws and not by men\" derives directly from this statement of Aristotle. Thomas Hobbes believed that it was just \"another error of Aristotle's politics, that in a well-ordered commonwealth not men should govern, but the laws,\"29 whereupon James Harrington retorted that the \"art whereby a civil society of men is instituted and preserved upon the foundation of common right or interest . . . [is], to follow Aristotle and Livy, the empire of laws, not of men.\"30\n\n3. In the course of the seventeenth century the influence of Latin writers largely replaced the direct influence of the Greeks. We should therefore take a brief look at the tradition derived from the Roman Republic. The famous Laws of the Twelve Tables, reputedly drawn up in conscious imitation of Solon's laws, form the foundation of its liberty. The first of the public laws in them provides that \"no privileges, or statutes, shall be enacted in favour of private persons, to the injury of others contrary to the law common to all citizens, and which individuals, no matter of what rank, have a right to make use of.\"31 This was the basic conception under which there  was gradually formed, by a process very similar to that by which the common law grew,32 the first fully developed system of private law\u2014in spirit very different from the later Justinian code, which determined the legal thinking of the Continent.\n\nThis spirit of the laws of free Rome has been transmitted to us mainly in the works of the historians and orators of the period, who once more became influential during the Latin Renaissance of the seventeenth century. Livy\u2014whose translator made people familiar with the term \"isonomia\" (which Livy himself did not use) and who supplied Harrington with the distinction between the government of law and the government of men33\u2014 Tacitus and, above all, Cicero became the chief authors through whom the classical tradition spread. Cicero indeed became the main authority for modern liberalism,34 and we owe to him many of the most effective formulations of freedom under the law.35 To him is due the conception of general rules or  _leges legum_ , which govern legislation,36 the conception that we obey the law in order to be free,37 and the conception that the judge ought to be merely the mouth through whom the law speaks.38 No other author shows more clearly that during the classical period of Roman law it was fully understood that there is no conflict between law and freedom and that freedom is dependent  upon certain attributes of the law, its generality and certainty, and the restrictions it places on the discretion of authority.\n\nThis classical period was also a period of complete economic freedom, to which Rome largely owed its prosperity and power.39 From the second century ad, however, state socialism advanced rapidly.40 In this development the freedom which equality before the law had created was progressively destroyed as demands for another kind of equality arose. During the later empire the strict law was weakened as, in the interest of a new social policy, the state increased its control over economic life. The outcome of this process, which culminated under Constantine, was, in the words of a distinguished student of Roman law, that \"the absolute empire proclaimed together with the principle of equity the authority of the imperial will unfettered by the barrier of law. Justinian with his learned professors brought this process to its conclusion.\"41 Thereafter, for a thousand years, the conception that legislation should serve to protect the freedom of the individual was lost. And when the art of legislation was rediscovered, it was the code of Justinian with its conception of a prince who stood above the law42 that served as the model on the Continent.\n\n4. In England, however, the wide influence which the classical authors enjoyed during the reign of Elizabeth helped to prepare the way for a different  development. Soon after her death the great struggle between king and Parliament began, from which emerged as a by-product the liberty of the individual. It is significant that the disputes began largely over issues of economic policy very similar to those which we again face today. To the nineteenth-century historian the measures of James I and Charles I which provoked the conflict might have seemed antiquated issues without topical interest. To us the problems caused by the attempts of the kings to set up industrial monopolies have a familiar ring: Charles I even attempted to nationalize the coal industry and was dissuaded from this only by being told that this might cause a rebellion.43\n\nEver since a court had laid down in the famous Case of Monopolies44 that the grant of exclusive rights to produce any article was \"against the common law and the liberty of the subject,\" the demand for equal laws for all citizens became the main weapon of Parliament in its opposition to the king's aims. Englishmen then understood better than they do today that the control of production always means the creation of privilege: that Peter is given permission to do what Paul is not allowed to do.\n\nIt was another kind of economic regulation, however, that occasioned the first great statement of the basic principle. The Petition of Grievances of 1610 was provoked by new regulations issued by the king for building in London and prohibiting the making of starch from wheat. This celebrated plea of the House of Commons states that, among all the traditional rights of British subjects, \"there is none which they have accounted more dear and precious than this, to be guided and governed by the certain rule of law, which giveth to the head and the members that which of right belongeth to them, and not by any uncertain and arbitrary form of government. . . . Out of this  root has grown the indubitable right of the people of this kingdom, not to be made subject to any punishments that shall extend to their lives, lands, bodies, or goods, other than such as are ordained by the common laws of this land, or the statutes made by their common consent in parliament.\"45\n\nIt was, finally, in the discussion occasioned by the Statute of Monopolies of 1624 that Sir Edward Coke, the great fountain of Whig principles, developed his interpretation of Magna Carta that became one of the cornerstones of the new doctrine. In the second part of his _Institutes of the Laws of England_ , soon to be printed by order of the House of Commons, he not only contended (with reference to the Case of Monopolies) that \"if a grant be made to any man to have the sole making of cards, or the sole dealing with any other trade, that grant is against the liberty and freedom of the subject, that before did, or lawfully might have used that trade, and consequently against this Great Charter\";46 but he went beyond such opposition to the royal prerogative to warn Parliament itself \"to leave all causes to be measured by the golden and straight metewand of the law, and not to the incertain and crooked cord of discretion.\"47\n\nOut of the extensive and continuous discussion of these issues during the Civil War, there gradually emerged all the political ideals which were thenceforth to govern English political evolution. We cannot attempt here to trace their evolution in the debates and pamphlet literature of the period, whose extraordinary wealth of ideas has come to be seen only since their re-publication in recent times.48 We can only list the main ideas that appeared more and  more frequently until, by the time of the Restoration, they had become part of an established tradition and, after the Glorious Revolution of 1688, part of the doctrine of the victorious party.\n\nThe great event that became for later generations the symbol of the permanent achievements of the Civil War was the abolition in 1641 of the prerogative courts and especially the Star Chamber which had become, in F. W. Maitland's often quoted words, \"a court of politicians enforcing a policy, not a court of judges administering the law.\"49 At almost the same time an effort was made for the first time to secure the independence of the judges.50 In the debates of the following twenty years the central issue became increasingly the prevention of arbitrary action of government. Though the two meanings of \"arbitrary\" were long confused, it came to be recognized, as Parliament began to act as arbitrarily as the king,51 that whether or not an action was arbitrary depended not on the source of the authority but on whether it was in conformity with pre-existing general principles of law.52 The points most frequently emphasized were that there must be no punishment without a previously existing law providing for it,53 that all statutes should have only prospective and not retrospective operation,54 and that the discretion of all magistrates  should be strictly circumscribed by law.55 Throughout, the governing idea was that the law should be king or, as one of the polemical tracts of the period expressed it, _Lex, Rex_.56\n\nGradually, two crucial conceptions emerged as to how these basic ideals should be safeguarded: the idea of a written constitution57 and the principle of the separation of powers.58 When in January, 1660, just before the Restoration,  a last attempt was made in the \"Declaration of Parliament As sembled at Westminster\" to state in a formal document the essential principles of a constitution, this striking passage was included: \"There being nothing more essential to the freedom of a state, than that the people should be governed by the laws, and that justice be administered by such only as are accountable for mal-administration, it is hereby further declared that all proceedings touching the lives, liberties and estates of all the free people of this commonwealth, shall be according to the laws of the land, and that the Parliament will not meddle with ordinary administration, or the executive part of the law: it being the principal part of this, as it hath been of all former Parliaments, to provide, for the freedom of the people against arbitrariness in government.\"59 If thereafter the principle of the separation of powers was perhaps not quite \"an accepted principle of constitutional law,\"60 it at least remained part of the governing political doctrine.\n\n5. All these ideas were to exercise a decisive influence during the next hundred years, not only in England but also in America and on the Continent, in the summarized form they were given after the final expulsion of the Stuarts in 1688. Though at the time perhaps some other works were equally and perhaps even more influential,61 John Locke's _Second Treatise on Civil Government_ is so outstanding in its lasting effects that we must confine our attention to it.\n\nLocke's work has come to be known mainly as a comprehensive philosophical justification of the Glorious Revolution;62 and it is mostly in his wider speculations about the philosophical foundations of government that his original contribution lies. Opinions may differ about their value. The aspect of his work which was at least as important at the time and which mainly concerns us here, however, is his codification of the victorious political doctrine, of the practical principles which, it was agreed, should thenceforth control the powers of government.63\n\nWhile in his philosophical discussion Locke's concern is with the source which makes power legitimate and with the aim of government in general, the practical problem with which he is concerned is how power, whoever exercises it, can be prevented from becoming arbitrary: \"Freedom of men under  government is to have a standing rule to live by, common to every one of that society, and made by the legislative power erected in it; a liberty to follow my own will in all things, where that rule prescribes not: and not to be subject to the inconstant, uncertain, arbitrary will of another man.\"64 It is against the \"irregular and uncertain exercise of the power\"65 that the argument is mainly directed: the important point is that \"whoever has the legislative or supreme power of any commonwealth is bound to govern by established standing laws promulgated and known to the people, and not by extemporary decrees; by indifferent and upright judges, who are to decide controversies by those laws; and to employ the forces of the community at home only in the execution of such laws.\"66 Even the legislature has no \"absolute arbitrary power,\"67 \"cannot assume to itself a power to rule by extemporary arbitrary decrees, but is bound to dispense justice, and decide the rights of the subject by promulgated standing laws, and known authorized judges,\"68 while the \"supreme executor of the law . . . has no will, no power, but that of law.\"69 Locke is loath to recognize any sovereign power, and the _Treatise_ has been described as an assault upon the very idea of sovereignty.70 The main practical safeguard against the abuse of authority proposed by him is the separation of powers, which he expounds somewhat less clearly and in a less familiar form than did some of his predecessors.71 His main concern is how to limit the discretion of \"him that has the executive power,\"72 but he has no special safeguards to offer. Yet his ultimate aim throughout is what today is often called the \"taming of power\": the end why men \"choose and authorize a legislative is that there may be laws made, and rules set, as guards and fences to the properties of all the members of society, to limit the power and moderate the dominion of every part and member of the society.\"73\n\n6. It is a long way from the acceptance of an ideal by public opinion to its full realization in policy; and perhaps the ideal of the rule of law had not yet  been completely put into practice when the process was reversed two hundred years later. At any rate, the main period of consolidation, during which it progressively penetrated everyday practice, was the first half of the eighteenth century.74 From the final confirmation of the independence of the judges in the Act of Settlement of 1701,75 through the occasion when the last bill of attainder ever passed by Parliament in 1706 led not only to a final restatement of all the arguments against such arbitrary action of the legislature76 but also to a reaffirmation of the principle of the separation of powers,77 the period is one of slow but steady extension of most of the principles for which the Englishmen of the seventeenth century had fought.\n\nA few significant events of the period may be briefly mentioned, such as the occasion when a member of the House of Commons (at a time when Dr. Johnson was reporting the debates) restated the basic doctrine of _nulla poena sine lege_ , which even now is sometimes alleged not to be part of English law:78 \"That where there is no law there is no transgression, is a maxim not only established by universal consent, but in itself evident and undeniable; and it is, Sir, surely no less certain that where there is no transgression there can be no punishment.\"79 Another is the occasion when Lord Camden in the Wilkes case made it clear that courts are concerned only with general rules and not with the particular aims of government or, as his position is sometimes interpreted, that public policy is not an argument in a court of law.80 In other respects progress was more slow, and it is probably true that, from the point of view of the poorest, the ideal of equality before the law long remained a somewhat doubtful fact. But if the process of reforming the laws in the spirit of those ideals was slow, the principles themselves ceased to be a matter of dispute: they were no longer a party view but had come to be fully accepted by the Tories.81 In some respects, however, evolution moved away rather than toward the ideal. The principle of the separation of powers in particular, though regarded throughout the century as the most distinctive feature of the British constitution,82 became less and less a fact as modern cabinet government developed. And Parliament with its claim to unlimited power was soon to depart from yet another of the principles.\n\n7. The second half of the eighteenth century produced the coherent expositions  of the ideals which largely determined the climate of opinion for the next hundred years. As is so often the case, it was less the systematic expositions by political philosophers and lawyers than the interpretations of events by the historians that carried these ideas to the public. The most influential among them was David Hume, who in his works again and again stressed the crucial points83 and of whom it has justly been said that for him the real meaning of the history of England was the evolution from a \"government of will to a government of law.\"84 At least one characteristic passage from his _History of England_ deserves to be quoted. With reference to the abolition of the Star Chamber he writes: \"No government, at that time, appeared in the world, nor is perhaps to be found in the records of any history, which subsisted without the mixture of some arbitrary authority, committed to some magistrate; and it might reasonably, beforehand, appear doubtful, whether human society could ever arrive at that state of perfection, as to support itself with no other control, than the general and rigid maxims of law and equity. But the parliament justly thought, that the King was too eminent a magistrate to be trusted with discretionary power, which he might so easily turn to the destruction of liberty. And in the event it has been found, that, though some inconveniencies  arise from the maxim of adhering strictly to law, yet the advantages so much overbalance them, as should render the English forever grateful to the memory of their ancestors, who, after repeated contests, at last established that noble principle.\"85\n\nLater in the century these ideals are more often taken for granted than explicitly stated, and the modern reader has to infer them when he wants to understand what men like Adam Smith86 and his contemporaries meant  by \"liberty.\" Only occasionally, as in Blackstone's _Commentaries_ , do we find endeavors to elaborate particular points, such as the significance of the independence of the judges and of the separation of powers,87 or to clarify the meaning of \"law\" by its definition as \"a rule; not a transient sudden order from a superior or concerning a particular person; but something permanent, uniform and universal.\"88\n\nMany of the best-known expressions of those ideals are, of course, to be found in the familiar passages of Edmund Burke.89 But probably the fullest statement of the doctrine of the rule of law occurs in the work of William Paley, the \"great codifier of thought in an age of codification.\"90 It deserves quoting at some length: \"The first maxim of a free state,\" he writes, \"is, that the laws be made by one set of men, and administered by another; in other words; that the legislative and the judicial characters be kept separate. When  these offices are united in the same person or assembly, particular laws are made for particular cases, springing oftentimes from partial motives, and directed to private ends: whilst they are kept separate, general laws are made by one body of men, without foreseeing whom they may affect; and, when made, must be applied by the other, let them affect whom they will. . . . When the parties and the interests to be affected by the law were known, the inclinations of the law makers would inevitably attach on one side or the other; and where there were neither any fixed rules to regulate their determinations, nor any superior power to control their proceedings, these inclinations would interfere with the integrity of public justice. The consequence of which must be, that the subjects of such a constitution would live either without any constant laws, that is, without any known pre-established rules of adjudication whatever; or under laws made for particular cases and particular persons, and partaking of the contradictions and iniquity of the motives to which they owed their origin.\n\n\"Which dangers, by the division of the legislative and judicial functions, are in this country effectually provided against. Parliament knows not the individuals upon whom its acts will operate; it has no cases or parties before it; no private designs to serve: consequently, its resolutions will be suggested by the considerations of universal effects and tendencies, which always produces impartial, and commonly advantageous regulations.\"91\n\n8. With the end of the eighteenth century, England's major contributions to the development of the principles of freedom come to a close. Though Macaulay did once more for the nineteenth century what Hume had done for the eighteenth,92 and though the Whig intelligentsia of the _Edinburgh Review_ and economists in the Smithian tradition, like J. R. MacCulloch and N. W. Senior, continued to think of liberty in classical terms, there was little further development. The new liberalism that gradually displaced Whiggism came more and more under the influence of the rationalist tendencies of the philosophical radicals and the French tradition. Bentham and his Utilitarians did  much to destroy the beliefs93 which England had in part preserved from the Middle Ages, by their scornful treatment of most of what until then had been the most admired features of the British constitution. And they introduced into Britain what had so far been entirely absent\u2014the desire to remake the whole of her law and institutions on rational principles.\n\nThe lack of understanding of the traditional principles of English liberty on the part of the men guided by the ideals of the French Revolution is clearly illustrated by one of the early apostles of that revolution in England, Dr. Richard Price. As early as 1778 he argued: \"Liberty. Therefore, is too imperfectly defined when it is said to be 'a Government Of LAWS and not by MEN.' If the laws are made by one man, or a junto of men in a state, and not by common CONSENT, a government by them is not different from Slavery.\"94 Eight years later he was able to display a commendatory letter from Turgot: \"How comes it that you are almost the first of the writers of your country, who has given a just idea of liberty, and shown the falsity of the notion so frequently repeated by almost all Republican Writers, 'that liberty consists in being subject only to the laws?'\"95 From then onward, the essentially French concept of political liberty was indeed progressively to displace the English ideal of individual liberty, until it could be said that \"in Great Britain, which, little more than a century ago, repudiated the ideas on which the French Revolution was based, and led the resistance to Napoleon, those ideas have triumphed.\"96 Though in Britain most of the achievements of the seventeenth century were preserved beyond the nineteenth, we must look elsewhere for the further development of the ideals underlying them.\n\n* * *\n\nThe quotation at the head of the chapter is taken from John Locke, _Second Treatise_ , sec. 57, p. 29. The substance of this chapter as well as of chapters 13\u201316 have been used in my lectures _The Political Ideal of the Rule of Law_ , National Bank of Egypt Fiftieth Anniversary Commemoration Lectures, (Cairo: National Bank of Egypt, 1955).\n\n1 The more I learn about the growth of these ideas, the more I am convinced of the important role which the example of the Dutch Republic played. But, though this influence is fairly clear in the later seventeenth and early eighteenth centuries, its earlier operation still needs investigation. In the meantime, see Sir George Norman Clark, \"The Birth of the Dutch Republic\" [The Raleigh Lecture on History], _Proceedings of the British Academy_ , 32 (1946): 189\u2013218; John Neville Figgis, _Studies of Political Thought from Gerson to Grotius 1414 \u20131625_ (Cambridge: Cambridge University Press, 1907), pp. 191, 198, 209; John Lothrop Motley, _The Rise of the Dutch Republic_ (3 vols.; New York: Harper and Brothers, 1856); Pieter Geyl, \"Liberty in Dutch History,\" _Delta_ : _A Review of Arts, Life, and Thought in the Netherlands_ , 1 (1958): 11\u201322; Pieter Geyl, _The Revolt of the Netherlands, 1555\u20131609_ (London: Williams and Norgate Ltd., 1932); and John Selwyn Bromley and Ernest Heinrich Kossman, eds. _Britain and the Nether lands, Papers Delivered to the Oxford-Netherlands Historical Conference, 1959_ (London: Chatto and Windus, 1960).\n\nIgnorance also compels me to pass over the important discussions and the development of similar ideas in Renaissance Italy, especially in Florence. (For a brief discussion see the introduction to the notes to chap. 20.) Cf. also Rudolf von Albertini, _Das f orentinische Staatsbewu\u00dftsein im \u00dcbergang von der Republik zum Prinzipat_ (Bern: Francke, 1955), particularly p. 294, and Giovanni Battista Guarini, _Trattato della pubblica libert\u00e0_ (ca. 1599), (Venice: Andreola, 1818). And I cannot speak with any competence about the interesting fact that the one great non-European civilization, that of China, appears to have developed, about the same time as the Greeks, legal conceptions surprisingly similar to those of Western civilization. According to Feng Youlan, _A History of Chinese Philosophy_. Vol. 1: _The Period of the Philosophers (from the Beginnings to Circa 100 B.C.)_, Derk Bodde, trans. (Peiping: H. Vetch, 1937), p. 312: \"the great political tendency of the time [the seventh to third centuries B.C.] was a movement from feudal rule toward a government by rulers possessing absolute power; from government by customary morality ( _li_ ), and by individuals, to government by law\" (p. 321). The author quotes as evidence from the _Kuantzu_ , a treatise attributed to Kuang Chung (ca. 715\u2013645 B.C.), but probably composed in the third century B.C.: \"When a state is governed by law, things will simply be done in their regular course. . . . If the law is not uniform, there will be misfortune for the holder of the state. . . . When ruler and minister, superior and inferior, noble and humble all obey the law, this is called having Great Good Government.\" He adds, however, that this is \"an ideal which has never yet been actually attained in China\" (p. 322). See also Herrlee Glessner Creel, \"The Fa-chia: 'Legalists' or 'Administrators,'\" in _Bulletin of the Institute of History and Philology_ , Academia Sinica, 4 (1961): 607\u201336, and T'ung-Tsu Ch'\u00fc [Tongzu Qu], _Law and Society in Traditional China_ (Paris: Mouton, 1961), particularly pp. 242\u201344; Bertrand de Jouvenel, _On Power_ [Liberty Fund edition available], maintains that the Chinese employed the terms \"government of laws\" and \"government of men\" 2,500 years ago.\n\n2 Cf. Montesquieu's remark in _The Spirit of the Laws_ (vol. 1, p. 151; French edition: vol. 2, p. 396): \"One nation there is also in the world that has for the direct end of its constitution political liberty.\" [\"Il y a aussi une nation dans le monde qui a pour objet direct de sa constitution la libert\u00e9 politique.\"\u2014Ed.] See also Rudolph Henne, _Der englische Freiheitsbegriff_ (dissertation; Zurich: R. Sauerl\u00e4nder, 1927). A careful study of the discovery of English liberty by the Continental people and of the influence of the English model on the Continent has yet to be made. Important early works are Guy Mi\u00e8ge, _L'\u00c9tat pr\u00e9sent de la Grande-Bretagne apr\u00e8s son heureuse union en 1707, sous le r\u00e8gne glorieux d'Anne_ (2 vols. in 1; Amsterdam: Chez les Wetsteins, 1708), also in an enlarged German edition as _Geist-und weltlicher Staat von Gross-Britannien und Irrland nach der gegenw\u00e4rtigen Zeit_ (Leipzig: Weidmanns, 1718); Paul de Rapin-Thoyras, _Dissertation sur les Whigs et les Torys, or an Historical Dissertation upon Whig and Tory_ , John Ozell, trans. (London: Printed for E. Curll, 1717); and August Adolph Friedrich von Hennings, _Philosophische und statistische Geschichte des Ursprungs und des Fortgangs der Freyheit in Engeland nach Hume, Blackstone und andern Quellen ausgearbeitet_ (Copenhagen: Proft, 1783).\n\n3 Cf. particularly Sir Frederick Pollock and Frederic William Maitland, _History of English Law Before the Time of Edward I_ (2nd ed.; 2 vols.; Cambridge: Cambridge University Press, 1911); Robert von Keller, _Freiheitsgarantien f\u00fcr Person und Eigentum im Mittelalter: eine Studie zur Vorgeschichte moderner Verfassungsgrundrechte_ (Heidelberg: C. Winter, 1933); Hans Planitz, \"Zur Ideengeschichte der Grundrechte,\" in _Die Grundrechte und Grundpflichten der Reichsverfassung. Kommentar zum zweiten Teil der Reichsverfassung_ , Hans Carl Nipperdey, ed. (3 vols.; Berlin: Verlag Reimar Hobbing, 1929\u201330), vol. 3, pp. 597\u2013623, esp. 601; and Otto Friedrich von Gierke, _Johannes Althusius und die Entwicklung der naturrechtlichen Staatstheorien: zugleich ein Beitrag zur Geschichte der Rechtssystematik_ (2nd ed.; Bres lau: Verag Marcus, 1902).\n\n4 See Charles Howard McIlwain, \"The English Common Law Barrier against Absolutism,\" _American Historical Review_ , 49 (1934): 27. The extent to which even the most famous and later most influential clause of Magna Carta merely expressed ideas common to the period is shown by a decree of the Emperor Conrad II, dated May 28, 1037 (given in William Stubbs, _Germany in the Early Middle Ages, 476\u20131250_ , Arthur Hassall, ed. [London: Longmans, Green, and Co., 1908], p. 147), which states: \"No man shall be deprived of a fief . . . but by the laws of the empire and the judgment of his peers.\"\n\nWe cannot examine in any detail here the philosophical tradition handed down from the Middle Ages. But in some respects Lord Acton was not being altogether paradoxical when he described Thomas Aquinas as the first Whig (see \"Freedom in Christianity,\" _History of Freedom_ , p. 37 [Liberty Fund edition, p. 34], and cf. John Neville Figgis, _Studies of Political Thought from Gerson to Grotius_ , p. 7). On Thomas Aquinas see Thomas Gilby, _Principality and Polity_ : _Aquinas and the Rise of State Theory in the West_ (London: Longmans, Green, and Co., 1958); and on his influence on early English political theory, especially Richard Hooker, see Sheldon Sandford Wolin, \"Richard Hooker and English Conservatism,\" _Western Political Quarterly_ , 6 (1953): 28\u201347. A fuller account would have to give special attention to Nicolas of Cusa in the thirteenth and Bartolus in the fourteenth centuries, who carried on the tradition. See particularly Nicholas of Cusa, _De concordantia catholica_ (1514), Gerhard Kallen, ed. (4 vols.; Hamburg: Felix Meiner, 1963) [vol. 14 of _Opera omnia_ ], bk. 3, chap. 1, pp. 327\u201328; Marsilius of Padua, _Defensor Pacis_ , Richard Scholtz, ed. (2 vols.; Hanover: Hahnsche Buchhandlung, 1933), v ol. 1, Dictio 2, chap . 2, sec. 4, pp. 146\u201347); Franz Anton von Scharpff, _Der Cardinal und Bischof Nicolaus von Cusa als Reformator in Kirche, Reich und Philosophie des f\u00fcnfzehnten Jahrhunderts_ (T\u00fcbingen: Laupp, 1871), esp. p. 22; John Neville Figgis, \"Bartolus and the Development of European Political Ideas,\" _Transactions of the Royal Historical Society_ , n.s., 19 (1905): 147\u201368; and Cecil Nathan Sidney Woolf, _Bartolus of Sassoferato: His Position in the History of Medieval Political Thought_ (Cambridge: Cambridge University Press, 1913); and, on the political theory of the period generally, Sir Robert Warrand Carlyle and Alexander James Carlyle, _A History of Medieval Political Theory in the West_ (6 vols.; Edinburgh: Blackwood, 1903).\n\n5 Cf. Otto Vossler, \"Studien zur Erkl\u00e4rung der Menschenrechte,\" _Historische Zeitschrift_ , 142 (1930): 518 [\"F\u00fcr das Mittelalter ist nicht der Staat souver\u00e4n, sondern das Recht, die Gerechtigkeit . . . Der Staat ist nur die Sch\u00f6pfung, der Diener des Rechts und der Gerechtigkeit, seine Aufgabe ist es, das Recht zu sch\u00fctzen und zu schirmen gegen das immer drohende Unrecht. Er kann aber nicht selbst Recht schaffen, setzen, ebensowenig nat\u00fcrlich Recht abschaffen oder verletzen, denn das hie\u00dfe die Gerechtigkeit selbst abschaffen und verletzen, es w\u00e4re absurd und S\u00fcnde und w\u00e4re eine Auflehnung gegen Gott, der allein Recht schafft.\"\u2014Ed.]; also Fritz Kern, _Kingship and Law in the Middle Ages_. Vol. 1: _The Divine Right of Kings and the Right of Resistance in the Early Middle Ages_. Vol. 2: _Law and Constitution in the Middle Ages: Studies_ , Stanley Bertram Chrimes, trans. (Oxford: B. Blackwell, 1939); Edward Jenks, _Law and Politics in the Middle Ages: With a Synoptic Table of Sources_ (London: J. Murray, 1898), pp. 24\u201325; Charles Howard McIlwain, _The High Court of Parliament and Its Supremacy: An Historical Essay on the Boundaries Between Legislation and Adjudication in England_ (New Haven: Yale University Press, 1910); John Nevelle Figgis, _The Divine Right of Kings_ (2nd ed., with three additional essays; Cambridge: Cambridge University Press, 1914); Charles Victor Langlois, _Le R\u00e8gne de Philippe III, le Hardi_ (Paris: Hachette et cie., 1887), p. 285; and, for a correction concerning the situation in the later Middle Ages, Theodore Frank Thomas Plucknett, _Statutes and Their Interpretation in the First Half of the Fourteenth Century_ (Cambridge: Cambridge University Press, 1922), and _Legislation of Edward I_ (Oxford: Clarendon Press, 1949). On the whole issue see John Wiedhofft Gough, _Fundamental Law in English Constitutional History_ (Oxford: Clarendon Press, 1955).\n\n6 Cf. Bernhard Rehfeldt, _Die Wurzeln des Rechtes_ [ _Rechts_ ] (Berlin: Duncker und Humblot, 1951), p. 67: \"Das Auftauchen des Ph\u00e4nomens der Gesetzgebung . . . bedeutet in der Menschheitsgeschichte die Erfindung der Kunst, Recht und Gesetz zu machen. Bis dahin hatte man ja geglaubt Recht nicht setzen, sondern nur anwenden zu k\u00f6nnen als etwas, das seit jeher war. An dieser Vorstellung gemessen ist die Erfindung der Gesetzgebung vielleicht die folgenschwerste, die je gemacht worden\u2014folgenschwerer als die des Feuermachens oder des Schie\u00dfpulvers\u2014denn am st\u00e4rksten von allen hat sie das Schicksal des Menschen in seine eigene Hand gelegt.\" [\"The appearance of the phenomenon of legislation in the history of humanity marks the discovery of the art of determining right from wrong. Up to that point people believed that they were not able to create law but could only apply the rules that were already in place. Measured by this idea, the discovery of legislation is perhaps the most significant of all discoveries that have been made\u2014more significant than the ability to make fire or gunpowder\u2014because more than any other discovery, it placed man's destiny in his own hands.\"\u2014Ed.]\n\nSimilarly in an as yet unpublished paper contributed to a symposium on \"The Expansion of Society\" organized by the Oriental Institute of the University of Chicago in December 1958, Max Rheinstein observes: \"The notion that valid norms of conduct might be established by way of legislation was peculiar to later stages of Greek and Roman history; in western Europe it was dormant until the rediscovery of Roman law and the rise of absolute monarchy. The proposition that all law is the command of a sovereign is a postulate engendered by the democratic ideology of the French Revolution that all law had to emanate from the duly elected representatives of the people. It is not, however, a true description of reality, least of all in the countries of the Anglo-American Common Law.\" [Since Hayek wrote this, Rheinstein's article has been published. The symposium to which Hayek refers in fact carries a title substantially different from that given by him. The actual citation is: \"Process and Change in the Cultural Spectrum Coincident with Expansion: Government and Law,\" in _City Invincible: A Symposium on Urbanization and Cultural Development in the Ancient Near East, held at the Oriental Institute of the University of Chicago, December 4\u20137, 1958_ , Carl Hermann Kraeling and Robert McCormick Adams, eds. (Chicago: University of Chicago Press, 1960), pp. 405\u201318.\u2014Ed.] It may be added that a philosopher trained in the German approach to the philosophy of right (John Austin) proved a necessary prerequisite to the success of legal positivism in England.\n\nHow profoundly the traditional view that laws are found and not made still influenced English opinion in the late eighteenth century is shown by Edmund Burke's statement in the _Tracts Relative to the Laws against Popery in Ireland_ , in _Works_ , vol. 9, pp. 352\u201353: \"It would be hard to point to any errour more truly subversive of all the order and beauty, of all the peace and happiness, of human society, than the position, that any body of men have a right to make what Laws they please; or that Laws can derive any authority from their institution merely and independent of the quality of the subject matter. No arguments of policy, reason of State, or preservation of the Constitution, can be pleaded in favour of such a practice. . . . All human Laws are, properly speaking, only declaratory; they may alter the mode and application, but have no power over the substance of original justice.\" For other illustrations see Edward Samuel Corwin, _The \"Higher Law\" Background of American Constitutional Law_ (Great Seal Books; Ithaca, NY: Cornell University Press, 1955), p. 6, n. 11, reprinted from the _Harvard Law Review_ , 42 (1928\u201329): 153\u201354, n. 11; and in _Selected Essays on Constitutional Law_ , Committee of the Association of American Law Schools and Douglas Blound Maggs, et al., eds. (5 vols. in 4; Chicago: Foundation Press, 1938), vol. 1, p. 5, n. 1 [Liberty Fund edition, p. 5, n. 2].\n\n7 Cf. Dicey, _Law of the Constitution_ , p. 370 [Liberty Fund edition, p. 242]: \"A lawyer, who regards the matter from an exclusively legal point of view, is tempted to assert that the real subject in dispute between statesmen such as Bacon and Wentworth on the one hand, and Coke or Eliot on the other, was whether a strong administration of the continental type should, or should not, be permanently established in England.\"\n\n8 This is how Henry Bracton describes Magna Carta in _De legibus et consuetudinus Angliae_ , George Edward Woodbine, ed. (4 vols.; New Haven: Yale University Press, 1915\u201342), fol. 168b. [The reference to Magna Carta as \"the constitution of liberty\" occurs in vol. 3, p. 35 of this edition.\u2014 Ed.] On the consequences of what was in effect a seventeenth-century misinterpretation of Magna Carta see William Sharp McKechnie, _Magna Carta: A Commentary on the Great Charter of King John, With an Historical Introduction_ (2nd ed., rev. and in part re-written; Glasgow: J. Maclehose and Sons, 1914), p. 133: \"If the vague and inaccurate words of Coke have obscured the bearing of many chapters [of Magna Carta], and diffused false notions of the development of English Law, the service these very errors have done to the cause of constitutional progress is measureless.\" This view has since been expressed many times (see particularly Sir Herbert Butterfield, _The Englishman and His History_ [Cambridge: Cambridge University Press, 1944], p. 7).\n\n9 Cf. Thomas Hobbes's description of how \"one of the most frequent causes of it [the rebellious spirit of his period] is the reading of books of policy, and histories of the ancient Greeks and Romans\" (p. 214) and that for this reason \"there was never any thing so dearly bought, as these western parts have bought the learning of the Greek and Latin tongues\" ( _Leviathan; or, The Matter, Forme, and Power of a Commonwealth, Ecclesiasticall and Civil_ , Michael Joseph Oakeshott, ed. [Oxford: B. Blackwell, 1946], p. 141). Hobbes shares this hostility to wards the ancient classics with Francis Bacon, who wished to see the works of Aristotle banned. [But see p. 243, n. 29, below.\u2014 Ed.] See also Aubrey's remark that the roots of Milton's \"zeal for the liberty of mankind\" lay in his \"being so conversant in Livy and the Roman authors, and the greatness he saw done by the Roman Commonwealth\" ( John Aubrey, \"John Milton,\" _Aubrey's Brief Lives_ , Oliver Lawson Dick, ed. [Ann Arbor: University of Michigan Press, 1957], p. 203). On the classical sources of the thought of Milton, Harrington, and Sidney see Zera Silver Fink, _The Classical Republicans: An Essay in the Recovery of a Pattern of Thought in Seventeenth Century England_ , Northwestern University Studies in [the] Humanities, No. 9 (Evanston, IL: Northwestern University Press, 1945).\n\n10 Thucydides ii.37.2 ( _Complete Writings: The Peloponnesian War_ , translated by Richard Crawley, introduction by John Finley, Jr. [New York: Modern Library, 1951], p. 104.) The most convincing testimony is probably that of the enemies of the liberal democracy of Athens who reveal much when they complain, as Aristotle did ( _Politics_ , 1317b [bk. 6, chap. 1, sec. 7]), that \"in such democracies each person lives as he likes.\" (Also see Plutarch, _Lycurgus_ , 24.) The Greeks may have been the first to confuse personal and political freedom; but this does not mean that they did not know the former or did not esteem it. The Stoic philosophers, at any rate, preserved the original meaning and handed it on to later ages. Zeno, indeed, defined freedom as the \"power of independent action, whereas slavery is privation of the same\" (Diogenes Laertius, _Lives of Eminent Philosophers_ , vii.121, Robert Drew Hicks, trans. [Loeb Classical Library; London: W. Heinemann, 1925], vol. 2, p. 227). Philo of Alexandria, _Quod omnis probus liber sit_ [\"Every good man is free.\"\u2014 Ed.] 452.45 ( _Works_ [published in 10 volumes plus two supplementary volumes], F. A. Colson, trans. [Loeb Classical Library; London: W. Heinemann, 1941], vol. 9, p. 36), even offers a thoroughly modern conception of liberty under the law: _hosoi de meta nomou zo\u00afsin, eleutheroi_ [The English reads: \"All whose life is regulated by law are free.\"\u2014Ed.]. Incidentally, he reports an inscription from Priene that reads: \"there is nothing greater for Hellenic men than freedom\" (vol. 9, 19, 13a). [This quotation does not appear in Philo's treatise \"Quod Omnis Probus Liber Sit,\" certainly not in the form attributed. Philo does, however, write at some length on the importance of freedom to the Greeks. He notes that \"this doctrine that freedom is glorious and honourable, slavery execrable and disgraceful, is attested by cities and nations, which are more ancient, more permanent, and, as far as mortals may be, immortal, and for immortals it is a law of their being that their every word is true. The senates and national assemblies meet almost every day to discuss more than anything else how to confirm their freedom if they have it, or to acquire it if they have it not.\" (20.137\u201338; Loeb Classical Library, vol. 9, p. 89).\u2014Ed.]; and Strabo in his _Geography_ (H. L. Jones, ed.; Loeb Classical Library, vol. 5, p. 145 [10.4.16]) writes of the Cretans: \"As for their constitution, which is described by Ephorus, it might suffice to tell in a way its most important provisions. The lawgiver, he says, seems to take it for granted that liberty is a state's greatest good, for this alone makes property belong specifically to those who acquire it, whereas in a condition of slavery everything belongs to the rulers and not to the ruled.\" See Eric Alfred Havelock, _The Liberal Temper in Greek Politics_ (New Haven: Yale University Press, 1957). It is also no longer possible to deny the existence of freedom in ancient Athens by the assertion that its economic system was \"based\" on slavery, since recent research has clearly shown that it was comparatively unimportant; see William Linn Westermann, \"Athenaeus and the Slaves of Athens,\" _Athenian Studies Presented to William Scott Ferguson_ (London: H. Milford, Oxford University Press, 1940), pp. 451\u201370, and Arnold Hugh Martin Jones, \"The Economic Basis of the Athenian Democracy,\" _Past and Present_ , 1 (1952): 13\u201331, reprinted in _Athenian Democracy_ (Oxford: B. Blackwell, 1957), pp. 3\u201320.\n\n11 Thucydides vii. 69 (p. 440). The misrepresentation of Greek liberty traces back to Thomas Hobbes and became widely known through Benjamin Constant, _De la libert\u00e9 des anciens compar\u00e9e \u00e0 celle des modernes_ , reprinted in _Cours de politique constitutionnelle; ou Collection des ouvrages publi\u00e9s sur le gouvernement repr\u00e9sentatif_ (2 vols.; Paris: Guillaumin et cie., 1861), vol. 2, pp. 539\u201360, and Numa Denis Fustel de Coulanges, _La Cit\u00e9 antique_ (Paris: Durand, 1864). About this whole discussion see Georg Jellinek, _Das Recht des modernen Staates_. Vol. 1: _Allgemeine Staatslehre_ (2nd ed.; Berlin: O. H\u00e4ring, 1905), pp. 285\u2013305. It is difficult to understand how, as late as 1933, Harold Joseph Laski (\"Liberty,\" _Encyclopedia of the Social Sciences_ , vol. 9, p. 442) could still argue, with explicit reference to the Periclean period, that \"in such an organic society the concept of individual liberty was virtually unknown.\"\n\n12 Cf. Johan Huizinga, _Wenn die Waffen schweigen: die Aussichten auf Genesung unserer Kultur_ (Basel: Burg-Verlag, 1945), p. 95: \"Man muss eigentlich bedauern, dass die Kulturen, die sich auf der Grundlage der griechischen Antike aufbauten, nicht an Stelle des Wortes Demokratie jenes andere \u00fcbernommen haben, das in Athen auf Grund der geschichtlichen Entwicklung besondere Achtung erweckte und ausserdem den hier wesentlichen Gedanken einer guten Regierungsform besonders rein zum Ausdruck brachte: das Wort 'Isonomia,' Gleichheit der Gesetze. Dies Wort hatte sogar einen unsterblichen Klang. . . . Aus dem Worte 'Isonomia' spricht weit deutlicher und unmittelbarer als aus 'Demokratia' das Ideal der Freiheit; auch ist die in der Bezeichnung 'Isonomia' enthaltene These nichts Unerf\u00fcllbares wie dies bei 'Demokatia' der Fall ist. Das wesentliche Prinzip des Rechtsstaates ist in diesem Wort b\u00fcndig und klar wiedergegeben.\" [\"It is unfortunate that the cultures that were to arise on the foundations of Greek antiquity appropriated the word 'democracy' rather than 'isonomia,' that other word employed in Athens that, for reasons of historical development, expressed in its purest form the essential notion of a good government, that is, equality before the law. The word 'isonomia' carries the sense of an ideal of freedom far more clearly and more directly than does 'democratia.' In addition, the term 'isonomia' does not suggest something that is ultimately unattainable, as does 'democratia.' The notion 'isonomia' best and most concisely reflects the essential principle of the _Rechtsstaat_.\"\u2014Ed.]\n\n13 In the Italian dictionary by John Florio, _Worlde of Wordes, or Most Copious and Exact Dictionarie in Italian and English_ (London: Printed by Arnold Hatfield for Edw. Blount, 1598), p. 195.\n\n14 Titus Livius, _Romane Historie_ [also containing the _Breviaries_ of Lucius Annaeus Florus], Philemon Holland, trans. (London: Printed by Adam Islip, 1600), pp. 114, 134, 1016.\n\n15 The _Oxford English Dictionary_ , S.V. \"Isonomy,\" gives instances of use in 1659 and 1684, each suggesting that the term was then in fairly common use. [The references are to: \"1659 _Quaeries on Proposalls Officers Armie to Parlt._ 8 Every one pretending to equality and Isonomy, lifteth up and advanceth himself whilst he shoveth at, and thrusteth down others. 1684 tr. _Agrippa's Van. Arts_ lv. 155 They who prefer a Popular State have dignifi'd it with the most agreeable and specious Title of Isonomie.\"\u2014Ed.]\n\n16 The earliest preserved use of the word \"isonomia\" seems to be that by Alcmaeon about 500 B.C. (Hermann Diels, _Die Fragmente der Vorsokratiker: Griechisch und Deutsch_ (4th ed.; 3 vols.; Berlin: Wiedmannsche Buchhandlung, 1922), vol. 1, p.136, Alkmaion, frag. 4). As the use is metaphorical, describing isonomy as a condition of physical health, it suggests that the term was well established by then. [The fragment to which Hayek refers is taken from the doxography of _A\u00ebtius_ (5.30.1) quoting Alkmaion and was later reconstructed by Diels.\u2014Ed.]\n\n17 Ernst Diehl, _Anthologia lyrica Graeca_ (3rd ed.; 3 vols.; Leipzig: Teubner, 1949\u201352), frag. 24, line 18, vol. 1, p. 45. [The passage is from Solon, frag. 24, line 18: \" _thesmous d' homoio\u00afs to\u00afi kako\u00afi te kagatho\u00afi._ \"\u2014Ed.] Cf. Erik Wolf, \"Ma\u00df und Gerechtigkeit bei Solon,\" in _Gegenwartsprobleme des internationalen Rechtes und der Rechtsphilosophie: Festschrift f\u00fcr Rudolf Laun zu seinem 70. Geburtstag_ , Demetrios S. Konstantopoulos and Hans Wehberg, eds. (Hamburg: Girardet, 1953), pp. 449\u201364; Kathleen Freeman, _The Work and Life of Solon_ : _With a Translation of His Poems_ (London: Milford, 1926); William John Woodhouse, _Solon, the Liberator: A Study of the Agrarian Problem in Attika in the Seventh Century_ (London: Oxford University Press, 1938); and Karl H\u00f6nn, _Solon, Staatsmann und Weiser_ (Vienna: L. W. Seidel, 1948), p. 107.\n\n18 Sir Ernest Barker, _Greek Political Theory: Plato and His Predecessors_ (2nd ed.; London: Methuen, 1925), p. 44. [Barker footnotes the source of the statement as Sir Alfred Eckhard Zimmern, _The Greek Commonwealth: Politics and Economics in Fifth-Century Athens_ (Oxford: Clarendon Press, 1911), pp. 130\u201331.\u2014Ed.] Cf. Lord Acton, \"Freedom in Antiquity,\" _History of Freedom_ , p. 7 [Liberty Fund edition, _Essays in the History of Liberty_ , p. 10], and Paul Vinogradoff, _Collected Papers of Paul Vinogradoff_ (2 vols.; Oxford: Clarendon Press, 1928), vol. 2, pp. 41\u201342.\n\n19 Cf. Max Pohlenz, _Griechische Freiheit: Wesen und Werden eines Lebensideals_ (Heidelberg: Quelle und Meyer, 1955); Georg Busolt, _Griechische Staatskunde_. Part 1 of _Allgemeine Darstellung des griechischen Staates_ (3rd rev. ed.; Munich: Beck, 1920), p. 417\u201318; Jakob Aall Ottesen Larsen, \"Cleisthenes and the Development of the Theory of Democracy at Athens,\" in _Essays in Political Theory Presented to George H. Sabine_ , Milton Ridvas Konvitz and Arthur Edward Murphy, eds. (Ithaca, NY: Cornell University Press, 1948), pp. 1\u201316; _Pauly's Real-Encyclop\u00e4die der classischen Altertumswissenschaft_ , August Friedrich Pauly, ed., Supplement 7, Georg Wissowa, ed. (Stuttgart: J. B. Metzler, 1940), S.V. \"Isonomia,\" by Victor Ehrenberg (pp. 293\u2013301); and Ehrenberg's articles \"Origins of Democracy,\" _Historia: Zeitschrift f\u00fcr Alte Geschichte_ [Wiesbaden] 1 (1950): 515\u201348, esp. 535, and \"Das Harmodioslied,\" _Wiener Studien. Zeitschrift f\u00fcr klassische Philologie_ ( _Festschrift Albin Lesky_ ), 69 (1956): 57\u201369; Gregory Vlastos, \"Isonomia,\" _American Journal of Philology_ , 74 (1953): 337\u201366; and John Walter Jones, _The Law and Legal Theory of the Greeks: An Introduction_ (Oxford: Clarendon Press, 1956), chap. 4 [\"Eunomia, Homonomia, Isonomia,\" pp. 73\u201392].\n\nThe Greek _skolion_ mentioned in the text will be found in two versions in Diehl, _Anthologia lyrica Graeca_ , vol. 2, _skolia_ 10 (9) [vol. 2, pp. 184\u201385] and 13 (12) [p. 185]. A curious illustration of the appeal of these songs celebrating _isonomia_ to late eighteenth-century English Whigs is the \"Ode in Imitation of Callistratus\" by Sir William Jones (whom we mentioned earlier as the link between the political views of the Whigs and the evolutionary tradition in linguistics; see _The Works of Sir William Jones_ [13 vols.; London: Printed for J. Stockdale and J. Walker, 1807], vol. 4, p. 574), which is headed by the Greek text of the _skolion_ and, after twenty lines in praise of Harmodios and Aristogiton, continues:\n\n\"Then in _Athens_ all was Peace;\n\nEqual Laws and Liberty:\n\nNurse of Arts, and eye for _Greece_!\n\nPeople valiant, firm, and free!\n\nNot less glorious was thy deed,\n\n_Wentworth_ , fix'd in Virtue's cause;\n\nNot less brilliant be thy meed,\n\n_Lenox_ , friend to _Equal Laws_!\n\nHigh in Freedom's temple rais'd,\n\nSee _Fitz Maurice_ beaming stand,\n\nFor collected Virtues prais'd,\n\nWisdom's voice, and Valour's hand!\n\nNe'er shall fate their eyelids close:\n\nThey, in blooming regions blest,\n\nWith _Harmodius_ shall repose,\n\nWith _Aristogiton_ rest.\"\n\n[The verses refer to Harmodius, a youth, and his older lover, Aristogiton. The brother of the tyrant of Athens, Hipparchus, continued making advances towards Harmodius, publicly mocking the lovers when Harmodius refused to share his bed. Outraged, Aristogiton plotted the assassination both of Hipparchus and his tyrant brother Hippias. The plot failed but Hipparchus was killed while organizing a procession marking a festival. Harmodius was instantly killed by Hipparchus' bodyguard and Aristogiton was captured and tortured by Hippias. Rather than reveal the names of those involved in the plot, Aristogiton hurled insults at the tyrant, which so enraged him that he stabbed him to death. Their example was hailed by Athenians as a model of resistance to tyranny and as a model for young men.\u2014Ed.]\n\nCf. also _Works_ , vol. 4, p. 572, the \"Ode in Imitation of Alcaeus,\" where Jones says with reference to the \"Empress Sovereign Law\":\n\n\"Smit by her sacred frown\n\nThe fiend _Discretion_ like a vapour sinks.\"\n\n20 Herodotus _Histories_ iii. 80; cf. also iii. 142, and v. 37.\n\n21 Busolt, _Allgemeine Darstellung des griechischen Staates_ , p. 417, and Ehrenberg, in _Pauly_ , Sup., S.V. \"Isonomia,\" p. 299.\n\n22 Thucydides iii. 62.3\u20134 and contrast this use of the term in its legitimate sense with his reference to what he describes as its specious use, Thucydides iii. 82.8; cf. also Isokrates _Areopagiticus_ vii. 20, and _Panathenaicus_ xii. 178.\n\n23 Plato _Republic_ viii. 557bc, 559d, 561e.\n\n24 Hyperides, _In Defence of Euxenippus_ 5 ( _Minor Attic Orators_ , John Ormiston Burtt and Kenneth John Maidment, eds. [Loeb Classical Library; 2 vols.; Cambridge, MA: Harvard University Press, 1960], vol. 2, p. 468): \" \u03c0\u03c9\u03c2  \u03bd \u03b4\u03b7\u03bc\u03bf\u03ba\u03c1\u03b1\u03c4\u03af\u03b1\u03b9 \u03ba\u03cd\u03c1\u03b9\u03bf\u03b9 \u03bf\u03b9 \u0301\u03bd\u03cc\u03bc\u03bf\u03b9  \u03c3\u03bf\u03bd\u03c4\u03b1\u03b9\" (\" _hopo_ \u0312\u0312\u0301 _\u00afs en demokratiai kurioi hoi nomoi esontai_ \"). The phrase about the law being king ( _nomos basileus_ [\"the law as king\"]) already occurs much earlier.\n\n25 Aristotle, _Politics_ 1287a [bk. 3, chap. 16, sec. 10]. The translation used, in preference to the more familiar renderings by Benjamin Jowett, is that by William Ellis, _The Politics of Aristotle: A Treatise on Government_ , William Ellis, trans. (Everyman library; New York: Dutton, 1947).\n\n26 Aristotle, _Politics_ , 1292a [bk. 4, chap. 4].\n\n27 How fundamental these conceptions remained for the Athenians is shown by a law to which Demosthenes refers in one of his orations ( _Against Aristocrates_ 86; cf. _Against Timocrates_ 59) as a law \"as good as ever law was.\" [The edition Hayek employed is Demosthenes, _Orations 21\u201326: Against Meidias. Against Androtion. Against Aristocrates. Against Timocrates. Against Aristogeiton 1 and 2_ , James Herbert Vince, trans. (7 vols.; Loeb Classical Library; Cambridge, MA: Harvard University Press, 1935), vol. 3, pp. 275, 411.\u2014Ed.] The Athenian who had introduced it had been of the opinion that, as every citizen had an equal share in civil rights, so everybody should have an equal share in the laws; and he had proposed, therefore, that \"it should not be lawful to propose a law affecting any individual, unless the same applied to all Athenians.\" This became the law of Athens. We do not know when this happened\u2014Demosthenes referred to it in 352 B.C. But it is interesting to see how, by that time, democracy had already become the primary concept superseding the older one of equality before the law. Although Demosthenes no longer uses the term \"isonomia,\" his reference to the law is little more than a paraphrase of that old ideal. On the law in question cf. Justus Hermann Lipsius, _Das attische Recht und Rechtsverfahren_ (3 vols. in 1; Leipzig: Reisland, 1905\u201315), vol. 1, p. 388, and Egon Weiss, _Griechisches Privatrecht auf rechtsvergleichender Grundlage_. Vol. 1: _Allgemeine Lehren_ (Leipzig: F. Meiner, 1923), p. 93 (n. 186a); cf. also Arnold Hugh Martin Jones, \"The Athenian Democracy and Its Critics,\" _Cambridge Historical Journal_ , 9 (1953): 10, and reprinted in his _Athenian Democracy_ , p. 52: \"At no time was it legal [in Athens] to alter a law by a simple decree of the assembly. The mover of such a decree was liable to the famous 'indictment for illegal proceedings' which, if upheld by the courts, . . . exposed the mover to heavy penalties.\"\n\n28 Aristotle, _Rhetoric_ 1354ab [bk. 1, chap. 1] in _The Works of Aristotle_ , William Rhys Roberts, trans., William David Ross, ed., vol. 11 (Oxford: Clarendon Press, 1924), [p. 2]. I do not quote in the text the passage from _Politics_ 1317b.5 [bk. 6, chap. 1, sec. 8] where Aristotle mentions as a condition of liberty that \"no magistrate should be allowed any discretionary power but in a few instances, and of no consequence to public business,\" because it occurs in a context where he does not express his own opinion but cites the views of others. An important statement of his views on judicial discretion is to be found in _Nicomachean Ethics_ 1137b.5 [bk. 5, chap. 10] where he argues that the judge should fill a gap in the law \"by ruling as the lawgiver himself would rule were he there present, and would have provided by law had he foreseen the case would arise\"\u2014 thus anticipating a famous clause of the Swiss Civil Code.\n\n29 Hobbes, _Leviathan_ , p. 448. It is characteristic that Francis Bacon started with this animosity against Aristotle, whose books he wished to see banned. See the introduction to Francis Bacon, _Instauratio Magna_. [The edition to which Hayek is referring is most likely _Francisici Baconi Baronus de Verulamio . . . opera omnia_ (4 vols.; London: R. Gosling, 1730), with an introduction by Bacon's contemporary and friend, the Rev. Dr. William Rawley. If so, Hayek is in error in suggesting that Bacon wished to ban the work of Aristotle. Rawley writes: \"Though there was bred in Mr. Bacon so early a dislike of the physiology of Aristotle, yet he did not despise him with that pride and haughtiness, with which youth is wont to be puffed up. He had a just esteem of that great master in learning, and greater than that of Aristotle himself expressed towards the philosophers that went before him. For he endeavoured (some say) to stifle all their labours, designing to himself an universal monarchy over opinions, as his patron Alexander did over men\" (vol. 1, p. 20).\u2014Ed.]\n\n30 James Harrington, _The Common-wealth of Oceana_ (London: Printed by J. Streater for Livewell Chapman, and are to be sold at his shop, 1656), p. 2. The phrase occurs soon afterward in a passage in _The Leveller_ of 1659, quoted by Gough, _Fundamental Law in English Constitutional History_ , p. 137. [ _The Leveller; or, the Principles and Maxims Concerning Government and Religion, Which are Asserted by those that are commonly called Levellers_ (London: Printed, for Thomas Brewster, at the Three Bibles, at the West-End of Pauls, 1659), p. 5, which maintains that \"they assert it as Fundamental, that the Government of England ought to be by Laws, and not by Men.\"\u2014Ed.]\n\n31 See Samuel Parsons Scott, _The Civil Law, Including the Twelve Tables, the Institutes of Gaius, the Rules of Ulpian, the Opinions of Paulus, the Enactments of Justinian, and the Constitutions of Leo_ (17 vols. in 7; Cincinnati: Central Trust Co., 1932), vol. 1, p. 73 [Table 9, 'Concerning Public Law,\" Law 1].\n\nOn the whole of this section see, in addition to the works of Theodor Mommsen [particularly _The History of Rome_ , William Purdie Dickson, trans. (new ed.; 4 vols.; New York: Charles Scribner's Sons, 1885), bk. 2, chap. 8.\u2014Ed.]; Chaim Wirszubski, _Libertas as a Political Idea at Rome during the Late Republic and Early Principate_ (Cambridge: Cambridge University Press, 1950); and Ulrich von L\u00fcbtow, _Bl\u00fcte und Verfall der r\u00f6mischen Freiheit: Betrachtungen zur Kultur-und Verfassungsgeschichte des Abendlandes_ (Berlin: Blaschker 1953), which came to my knowledge only after the text was completed.\n\n32 See William Warwick Buckland and Arnold Duncan McNair, Baron McNair, _Roman Law and Common Law: A Comparison in Outline_ (Cambridge: Cambridge University Press, 1936).\n\n33 Titus Livius, _Ab Urbe Condita_ ii.1.1: \"imperia legum potentiora quam hominum.\" [\"The commands of the laws (are) more powerful than the commands of men.\"\u2014Ed.] The Latin phrase is quoted (inexactly) by Algernon Sidney (in _Discourses Concerning Government_ , in _The Works of Algernon Sydney_ , Thomas Hollis, ed., with additions and corrections by Joseph Robertson (new ed.; London: Printed for W. Strahan Iun. for T. Becket and Co. and T. Cadell in the Strand, T. Davies in Russel Street, and T. Evans in King Street, 1772), p. 10 [Liberty Fund edition of the _Discourses_ , p. 17]. See also John Adams, _Works: With a Life of the Author_ , Charles Francis Adams, ed. (10 vols.; Boston: Charles C. Little and James Brown, 1851), vol. 4, p. 403. In Holland's translation of Livy of 1600 [ _The Romane Historie, Written by T. Livius of Padua_ (also containing the _Breviaries_ of Lucus Annaeus Florus), Philemon Holland, trans. (London: Printed by Adam Islip, 1600), p. 44] these words are rendered as \"the authoritie and _rule of laws_ , more powerfull and mightie than those of men.\" The words I have italicized provide the earliest instance known to me in which \"rule\" is used in the sense of \"government\" or \"dominion.\"\n\n34 Cf. Walter R\u00fcegg, _Cicero und der Humanismus: formale Untersuchungen \u00fcber Petrarca und Erasmus_ (Zurich: Rheinverlag, 1946), and George Holland Sabine and Stanley Barney Smith, \"Introduction,\" to Marcus Tullius Cicero, _On the Commonwealth_ (Columbus: Ohio State University Press, 1929), pp. 74\u201399. On Cicero's influence on David Hume in particular see David Hume, _My Own Life_ , in _Essays_ , vol. 1, p. 2 [Liberty Fund edition, p. xxxiii], and John Laird, _Hume's Philosophy of Human Nature_ (London: Methuen and Co., Ltd., 1932), pp. 241\u201343, who speaks of \"the Ciceronian favour of Hume's ethical thinking.\" [Hume writes: \"I found an unsurmountable aversion to everything but the pursuits of philosophy and general learning; and while I fancied I was poring upon Voet and Vinnius, Cicero and Virgil were the authors I was secretly devouring.\" Johannes Voet was an eminent eighteenth-century Dutch legal theorist, who attempted to systematize Roman-Dutch civil law; Arnoldus Vinnius (1588\u20131657) was also a Dutch legal theorist and a contemporary of Grotius.\u2014Ed.]\n\n35 Marcus Tullius Cicero, _De Domo Sua_ , 13.33 (Loeb edition, pp . 172\u201373), where he speaks of \"the peculiar mark of a free community\u2014the right, I mean, in accordance with which it is unlawful for any abatement of civil privilege or private property to be made without a verdict of the senate, of people, or of the courts constituted to deal with each type of offense.\"\n\n36 Marcus Tullius Cicero, _De legibus_ ii.7.18. [\"Leges legum\" translates literally as \"laws in the legal style.\"\u2014Ed.] These \"higher laws\" were recognized by the Romans, who inscribed in their statutes a provision stating that they were not intended to abrogate what was sacrosanct or _jus_ (see Corwin, \" _Higher Law\" Background_ , pp. 12\u201318 [ _Harvard Law Review_ , 42 (1928\u201329): 157\u201364; _Selected Essays_ , vol. 1, pp. 8\u201314; Liberty Fund edition, pp. 11\u201317], and the literature there quoted).\n\n37 Marcus Tullius Cicero, _Pro Cluentio_ 53.146: \"omnes legum servi sumus ut liberi esse possimus.\" [Cicero writes: \"Legum ministri magistratus, legum interpretes iudices, legum denique idcirco omnes servi sumus, ut liberi esse possimus.\" (\"The magistrates who administer the law, the jurors who interpret it\u2014all of us in short\u2014obey the law to the end that we may be free.\")\u2014 Ed.] Cf. Montesquieu, _Spirit of the Laws_ [bk. 26, chap. 20 (vol. 2, p. 76); French edition, vol. 2, p. 772]: \"Liberty consists, principally in not being forced to do a thing where the laws do not oblige: people are in this state only as they are governed by civil laws; and because they live under those civil laws they are free.\" [\"La libert\u00e9 consiste principalement \u00e0 ne pouvoir \u00eatre forc\u00e9 \u00e0 faire une chose que la loi n'ordonne pas; et on n'est dans cet \u00e9tat que parce qu'on est gouvern\u00e9 par des lois civiles: nous sommes donc libres, parce que nous vivons sous les lois civiles.\"]\n\nVoltaire, _Pens\u00e9es sur le gouvernement_ (1752) in _\u0152uvres compl\u00e8tes_ , Louis Moland, ed. (52 vols.; Paris: Garnier fr\u00e8res, 1877\u201385), vol. 23, p. 526: \"La libert\u00e9 consiste \u00e0 ne dependre que de lois.\" [\"Liberty consists of depending only upon the law.\"] Jean Jacques Rousseau, _Lettres \u00e9crites de la Montagne_ , Letter 8, in _The Political Writings of Jean Jacques Rousseau_ , Charles Edwyn Vaughan, ed., edited from the Original Manuscripts and Authentic Editions (2 vols.; Cambridge: Cambridge University Press, 1915), vol. 2, p. 235. \"There is no liberty without laws, nor where someone is above the laws; even in the state of nature, man is free only because of the natural law, which enjoins everyone.\" [The original French reads: \"Il n'y a donc point de libert\u00e9 sans Loix, ni o\u00f9 quelqu'un est au dessus des Loix: dans l'\u00e9tat m\u00eame de nature l'homme n'est libre qu'\u00e0 la faveur de la Loi naturelle qui commande \u00e0 tous.\" \"Lettres \u00c9crites de la Montagne,\" (Huiti\u00e8me Lettre), _\u0152uvres Compl\u00e8tes_ , Bernard Gagnebin and Marcel Raymond, eds. Biblioth\u00e8que de la Pl\u00e9iade (5 vols.; Paris: \u00c9ditions Gallimard, 1964), vol. 3, p. 842.]\n\n38 Marcus Tullius Cicero, _De legibus_ iii.122: \"Magistratum legem esse loquentem.\" [\"The magistrate is the law speaking.\"\u2014Ed.] Cf. Sir Edward Coke [\"Seventh Report,\" in _The Reports of Edward Coke, Knt.: In Thirteen Parts_ , John Henry Thomas and John Farquhar Fraser, eds. (13 parts in 6 vols.; London: J. Butterworth and Son, 1826), pt. 7, 6 (vol. 4, p. 6); Liberty Fund edition, vol. 1, p. 174] in Calvin's Case (as quoted in chapter 4, note 18); \"Judex est lex loquens,\" [\"The judge is the spoken law.\"\u2014Ed.] and the eighteenth-century legal maxim, \"Rex nihil alius est quam lex agens\" [\"The king is nothing other than the law in action.\"\u2014Ed.]; also Montesquieu, _Spirit of the Laws_ , bk. 11, chap. 6 (vol. 1, p. 159; French edition: vol. 2, p. 404) \"The national judges are no more than the mouth that pronounces the word of the law, mere passive beings, incapable of moderating either its force or rigor.\" [\"Mais les juges de la nation ne sont, comme nous avons dit, que la bouche qui prononce les paroles de la loi; des \u00eatres inanim\u00e9s qui n'en peuvent mod\u00e9rer ni la force ni la rigueur\"\u2014Ed.] The phrase was still repeated in the United States by Chief Justice John Marshall ( _Osborn v. Bank of United States_ , 22 _U.S_. [9, _Wheaton_ ] 738 at 866), when he spoke of judges as \"the mere mouthpieces of the law\" and \"capable of willing nothing.\" [In fact, what Marshall wrote was: \"Courts are the mere instruments of the law, and can will nothing.\"\u2014Ed.]\n\n39 See Michael Ivanovitch Rostovtzeff, _Gesellschaft und Wirtschaft im r\u00f6mischen Kaiserreich_ , Lothar Wickert, trans. (2 vols.; Leipzig: Quelle und Meyer, 1931), vol. 1, pp. 49 and 140.\n\n40 Cf. Friedrich Oertel, \"The Economic Life of the Empire,\" in _Cambridge Ancient History_. Vol. 12: _The Imperial Crisis and Recovery, A.D. 193\u2013324_ (Cambridge: Cambridge University Press, 1939), pp. 232\u201381, esp. 270ff., and the \"Appendix\" to Robert von P\u00f6hlmann, _Geschichte der sozialen Frage und des Sozialismus in der antiken Welt_ (3rd ed.; 2 vols.; Munich: C. H. Beck, 1925), vol. 2, pp. 511\u2013 85; also von Lubtow, _Bl\u00fcte und Verfall der r\u00f6mischen Freiheit_ , pp. 87\u2013107; Michael Ivanovitch Rostovtzeff, \"The Decay of the Ancient World and Its Economic Explanation,\" _Economic History Review_ , 2 (1930): 196\u2013214; Tenney Frank, \"Epilogue: The Economic Decay of the Roman World,\" in _Economic Survey of Ancient Rome_ (6 vols.; Baltimore: Johns Hopkins Press, 1933\u201340), vol. 5, pp. 296\u2013304; Henry Joseph Haskell, _The New Deal in Old Rome: How Government in the Ancient World Tried to Deal with Modern Problems_ (New York: A. Knopf, 1939); and Luigi Roberto Einaudi, \"Greatness and Decline of Planned Economy in the Hellenistic World,\" _Kyklos_ 2 (1948) [part 1, pp. 193\u2013210; part 2, pp. 289\u2013316], reprinted as _Greatness and Decline of Planned Economy in the Hellenistic World_ (Berne: A. Francke, 1950).\n\n41 Fritz Pringsheim, \"Jus aequum und jus strictum,\" _Zeitschrift der Savigny-Stiftung f\u00fcr Rechtsgeschichte, Romanistische Abteilung_ 42 (1921): 668. [The original German reads: \"Das absolute Kaisertum verk\u00fcndet mit der aequitas zugleich die von der Schranke des jus befreite Autorit\u00e4t des kaiserlichen Willens. Justinian mit seinen gelehrten Professoren bringt die Entwicklung zum Abschlusse.\"\u2014Ed.] Cf. also the same author's _Hohe und Ende der R\u00f6mischen Jurisprudenz: Vortrag gehalten bei der Freiburger wissenschaftlichen Gesellschaft am 3. Dezember 1932_ (Freiburg in Baden: Speyer und Kaerner, 1933).\n\n42 See Adh\u00e9mar Esmein, \"La Maxime _Princeps legibus solutus est_ dans l'ancien droit public fran\u00e7ais,\" _Essays in Legal History Read Before the International Congress of Historical Studies Held in London in 1913_ , Sir Paul Vinogradoff, ed. (London: Oxford University Press, 1913), pp. 201\u201314.\n\n43 Cf. John Ulric Nef, _Industry and Government in France and England: 1540\u20131640_ (Philadelphia: American Philosophical Society, 1940), p. 114. An interesting account of how later \"the freedom of the Press thus came to England all but incidentally to the elimination of a commercial monopoly\" is given by Maurice William Cranston, _John Locke, a Biography_ (London: Longmans, 1957), p 387.\n\n44 _Darcy v. Allein_ [also spelled \"Allen\"] (\"The Case of Monopolies\") 11 Co. Rep. 84 b, 77 Eng. Rep. 1260 (1603); Moore 671, 72 Eng. Rep. 830 (1603); Noy 173, 74 Eng. Rep. 1131 (1603). [The case also appears in the Liberty Fund edition, vol. 1, pp. 394\u2013404.] [The phrase \"against the common law because it was against the liberty of the subject\" appears\u2014in law French\u2014 in _Davenant v. Hurdis_ and is quoted in English by Coke in _Darcy v. Allen_ (77 Eng. Rep. 1263).\u2014 Ed.] The principle seems to have been stated first four years earlier in _Davenant v. Hurdis_ [(\"The Merchant Tailors' Case\") Moore 576, 72 Eng. Rep. 769 (King's Bench 1599)], when it was said that \"prescription of such nature, to induce sole trade or traffic to a company or person, and to exclude all others is against the law.\" See William Lewis Letwin, \"The English Common Law Concerning Monopolies,\" _University of Chicago Law Review_ , 21 (1953\u201354): 355\u201385, and the two articles by Donald Owen Wagner, \"Coke and the Rise of Economic Liberalism,\" _Economic History Review_ , 6 (1935\u201336): 30\u201344, and \"The Common Law and Free Enterprise: An Early Case of Monopoly,\" _Economic History Review_ , 7 (1936\u201337): 217\u201320.\n\n45Great Britain, Public Record Office, _Calendar of State Papers, Domestic Series, of the Reign of James I_ [Preserved in the State Paper Department of Her Majesty's Public Record Office], Mary Anne Everett Green, ed. (5 vols.; London: Longman, Brown, Green, Longmans, and Roberts, 1857\u2013 72), vol. 5, July 7, 1610. [ _The Calendar of State Papers_ does not generally include the full text of the documents to which it refers, but rather to documents held in the British National Archives. The citation should presumably therefore be to the original document, details of which should be available from the National Archives. A transcript of the petition can be found in _Proceedings in Parliament, 1610_. Vol. 2: _House of Commons_ , Elizabeth Read Foster, ed. (New Haven: Yale University Press, 1966), pp. 257\u201371. The passage quoted is located at p. 258.\u2014Ed.]\n\n46 Sir Edward Coke, _The Second Part of the Institutes of the Laws of England. Containing the Exposition of Many Ancient and Modern Statutes_ (London: Printed for E. and R. Brooke, 1797), p. 47 [Liberty Fund edition, vol. 1, p. 851]. [The quotation can also be found in _The Golden Passage in the Great Charter of England, Called Magna Carta; or, the Charter of English Liberties Granted by King John to His Subjects in the 17th Year of his Reign in Running-Mead_ (London: Printed for the use of the London Association, 1776), p. 10.\u2014Ed.]\n\n47 Sir Edward Coke, _The Second Part of the Institutes_. [Hayek is in error in ascribing this quotation to the Second Part of Coke's Institutes. It is, in fact, to be found in the _Fourth Part of the Institutes of the Laws of England, Concerning the Jurisdiction of Courts_ (London: Printed for E. and R. Brooke, 1797), pp. 40\u201341; Liberty Fund edition, vol. 1, p. 1143.\u2014Ed.]\n\n48 See Sir William Clarke, _The Clarke Papers: Selections from the Papers of William Clarke, Secretary to the Council of the Army, 1647\u20131649, and to General Monck and the Commanders of the Army in Scotland, 1651\u20131660_ , Charles Harding Firth, ed. (4 vols.; London: Printed for the Camden Society, 1891\u2013 1901); George Peabody Gooch, _English Democratic Ideas in the Seventeenth Century_ (Cambridge: Cambridge University Press, 1898); Theodore Calvin Pease, _The Leveller Movement: A Study in the History and Political Theory of the English Civil War_ (Washington, DC: American Historical Association, 1916); William Haller, ed., _Tracts on Liberty in the Puritan Revolution, 1638\u20131647_ (3 vols.; New York: Columbia University Press, 1934); Arthur Sutherland Pigott Woodhouse, ed., _Puritanism and Liberty: Being the Army Debates (1647\u20139) from the Clarke Manuscripts, with Supplementary Documents_ (London: J. M. Dent and Sons, 1938); William Haller and Godfrey Davies, eds., _The Leveller Tracts: 1647\u20131653_ (New York: Columbia University Press in cooperation with the Henry E. Huntington Library and Art Gallery, 1944); Don Marion Wolfe, _Leveller Manifestoes of the Puritan Revolution_ (New York: T. Nelson and Sons, 1944); William Haller, _Liberty and Reformation in the Puritan Revolution_ (New York: Columbia University Press, 1955); Perez Zagorin, _A History of Political Thought in the English Revolution_ (London: Routledge and Paul, 1954).\n\n49 Frederic William Maitland, _The Constitutional History of England: A Course of Lectures_ (Cambridge: Cambridge University Press, 1909), p. 263.\n\n50 Cf. Charles Howard McIlwain, \"The Tenure of English Judges,\" in _Constitutionalism and the Changing World: Collected Papers_ (Cambridge: Cambridge University Press, 1939), p. 300.\n\n51 See Gough, _Fundamental Law in English Constitutional History_ , pp. 76ff. and 159.\n\n52 Samuel Rawson Gardiner, _History of the Great Civil War:1642\u20131649_ (new ed.; 4 vols.; London: Longmans, Green, and Co., 1898), vol. 3, pp. 392\u201394.\n\n53 This is one of the main topics of the recorded part of the Army Debates (see Arthur Sutherland Pigott Woodhouse, ed., _Puritanism and Liberty: Being the Army Debates (1647\u201349) from the Clarke Manuscripts, With Supplementary Documents_ [London: J. M. Dent and Sons, 1938], pp. 336, 345, 354\u201355, and 472).\n\n54 This recurring phrase apparently derives from Sir Edward Coke, _The Second Part of the Institute_ , p. 292: \"Nova constitutio futuris formam imponere debet, non praeteritis.\" [\"A new law ought to regulate what is to follow, not the past.\" The quotation carries the meaning that any new law that is made ought to affect future transactions, not past ones. The quotation appears in chap. 4 of the _Second Part of the Institutes_ marked \"Merton,\" which is not included in the Liberty Fund edition.\u2014Ed.]\n\n55 See Woodhouse, _Puritanism and Liberty_ , pp. 154ff. and 353ff.\n\n56 Samuel Rutherford, _Lex, Rex: The Law and the Prince_ : _A Dispute for the Just Prerogative of King and People_ (London: Printed for John Field, 1644); excerpts are given in Woodhouse, _Puritanism and Liberty_ , pp. 199, 212. The phrase of the title goes back to the ancient Greek _nomos basileus_. The issue of law versus arbitrariness was not used only by the Roundheads; it also appears frequently in the Royalist argument, and Charles I [in his _King Charls, his Speech Made upon the Scaffold at Whitehall-Gate, Immediately before his Execution, on Tuesday the 30 of Jan. 1648. With a Relation of the Maner of His Going to Execution_ (London: Printed by P. Cole, 1649), p. 6] could assert that \"Their Liberty and their Freedom consists in having of government those Laws, by which their Life and their Goods may be most their own: It is not for having share in Government.\"\n\n57 See Samuel Rawson Gardiner, _The Constitutional Documents of the Puritan Revolution, 1625\u20131660_ (3rd ed. rev.; Oxford: Clarendon Press, 1906). Much the best brief account is now to be found in Francis Dunham Wormuth, _The Origins of Modern Constitutionalism_ (New York: Harper, 1949); see also Walther Rothschild, _Der Gedanke der geschriebenen Verfassung in der englischen Revolution_ (T\u00fcbingen: J. C. B. Mohr, 1903); Margaret Atwood Judson, _The Crisis of the Constitution: An Essay in Constitutional and Political Thought in England, 1603\u20131945_ (New Brunswick, NJ: Rutgers University Press, 1949); and the work by Gough, _Fundamental Law in English Constitutional History_ ; also cf. Oliver Cromwell, _Oliver Cromwell's Letters and Speeches_ , Thomas Carlyle, ed. (2nd ed. enlarged; 3 vols.; London: Chapman and Hall, 1846), vol. 3, p. 67 [Speech delivered before the First Protectorate Parliament, 12 September 1654]: \"In every Government there must be somewhat [ _sic._ ] fundamental, somewhat like a _Magna Carta_ , which must be standing, be unalterable.\"\n\n58 The idea of the separation of powers seems first to have appeared in 1645 in a pamphlet by John Lilburne (see Pease, _The Leveller Movement_ , p. 114) [The pamphlet to which Hayek refers is entitled _England's Birth-right Justified_.\u2014Ed.], and soon after that it occurs frequently, for instance in John Milton's _Eikonoklastes_ (1649), in _The Prose Works, With a Preface, Preliminary Remarks, and Notes_ , James Augustus St. John, ed. (5 vols.; London: H. J. Bohn, 1884), vol. 1, p. 363: \"In all wise nations the legislative power, and the judicial execution of that power, have been most commonly distinct, and in several hands; but yet the former supreme, the other subordinate\"; and in John Sadler, _Rights of the Kingdom_ (1649), quoted in Wormuth, _The Origins of Modern Constitutionalism_ , p. 61: \"It may be much disputed, that the legislative, judicial, and executive power should be in distinct subjects by the law of nature.\" [The original source is Anonymous ( John Sadler), _Rights of the Kingdom; or, Customs of our Ancestors: Touching the Duty, Power, Election, or Succession of our Kings and Parliaments_ (London: Printed by Richard Bishop, 1649), p. 92.\u2014Ed.] The idea was very fully elaborated by George Lawson, _An Examination of the Political Part of Mr. Hobbs, His Leviathan_ (London: Printed by R. White for Francis Tyton, 1657). (See A. H. Maclean, \"George Lawson and John Locke,\" _Cambridge Historical Journal_ 9 [1947]: 69\u201378). Additional references will be found in Wormuth, _Origins of Modern Constitutionalism_ , pp. 59\u201372, and, for the later development, pp. 191\u2013206. One particularly useful guide to the literature of the eighteenth-century English Whigs is the work of Caroline Robbins, _The Eighteenth Century Commonwealthmen_ (Cambridge, MA: Harvard University Press, 1959).\n\n59 Wormuth, _Origins of Modern Constitutionalism_ , p. 71.\n\n60 Ibid., p.72.\n\n61 The two main authors whom a fuller account would mainly have to consider are Algernon Sidney and Gilbert Burnet. The chief points relevant to us in Sidney's _Discourses concerning Government_ (first published in 1698) are that \"liberty solely consists in an independency upon the will of another\" which connects with the maxim \"potentiora erant legum quam hominum imperia\" ( _Works_ , p. 10 [Liberty Fund edition, p. 17]), that \"laws that aim at the public good make no distinction of persons\" ( _Works_ , p. 150 [Liberty Fund edition, p. 150]), that laws are made \"because nations will be governed by rule, and not arbitrarily\" ( _Works_ , p. 338 [Liberty Fund edition, p. 392]), and that laws \"ought to aim at perpetuity\" ( _Works_ , p. 492 [Liberty Fund edition, p. 559]). Of Gilbert Burnet's numerous writings, see particularly his anonymously published _Enquiry into the Measures of Submission to the Supreme Authority and of the Grounds upon which it may be Lawful or Necessary for Subjects to Defend Their Religion, Lives, and Liberties_ (London, 1688); quoted in the _Harleian Miscellany; or, A Collection of Scarce, Curious, and Entertaining Pamphlets and Tracts_ , William Oldys, ed. (12 vols.; London: Printed for R. Dutton, 1808\u201311), vol. 9, p. 204: \"The plea for liberty always proves itself, unless it appears that it is given up, or limited by any special agreement. . . . In the management of this civil society, great distinction is to be made between the power of making laws for the regulating the conduct of it, and the power of executing those laws; the supreme authority must still be supposed to be lodged with those who have the legislative power reserved to them; but not with those who have only the executive, which is plainly a trust when it is separated from the legislative power.\" Also vol. 9, pp. 205\u20136: \"The measures of power, and, by consequence, of obedience, must be taken from the express laws of any state, or body of men, from the oaths that they swear; or from immemorial prescription, and a long possession, which both give a title, and, in a long tract of time, make a bad one become good; since prescription, when it passes the memory of man, and is not disputed by any other pretender, gives, by the common sense of all men, a just and good title. So, upon the whole matter, the degrees of all civil authority, are to be taken either from express laws, immemorial customs, or from particular oaths, which the subjects swear to their princes; this being still to be laid down for a principle, that, in all the disputes between power and liberty, power must always be proved, but liberty proves itself; the one being founded upon positive law, and the other upon the law of nature.\" Vol. 9, p. 209: \"The chief design of our whole law, and all the several rules of our constitution, is to secure and maintain our liberty.\" It was to this tract that a contemporary Continental discoverer of English liberty such as Guy Mi\u00e8ge ( _L'\u00c9tat pr\u00e9sent de la Grande-Bretagne_ , pp. 512\u2013 13), primarily referred to in his writings: Mi\u00e8ge contended that \"no subjects in the world enjoyed so many fundamental and inheritable liberties as the people of England\" and that \"their state was therefore most happy and preferable to that of all European subjects.\" [Hayek is here quoting from the German translation of 1718: _Geist-und weltlicher Staat von Gro\u00df-Britannien und Irrland nach der gegenw\u00e4rtigen Zeit_ , Johann Bernhard Heinzelmann, trans. (Leipzig: Verlag Moritz George Weidmanns, 1718), pp. 512\u201313. The German reads: \"keine Unterthanen in der Welt, die so viel _fundamentale_ und erbliche Freyheiten genie\u00dfen, wie das Volck in England\" and \"sein Stand (sey) gl\u00fcckseelig und er allen Europ\u00e4ischen Unterthanen hierin . . . vorzuziehen.\"\u2014Ed.]\n\n62 This may still be said even though it now appears that the _Treatise_ was drafted before the revolution of 1688. (See Cranston, _John Locke: A Biography_ , p. 326, and especially Peter Laslett's introductory essay in John Locke, _Two Treatises of Government: A Critical Edition_ , Peter Laslett, ed. [Cambridge: Cambridge University Press, 1964], which we are here quoting.)\n\n63 Cf. John Wiedhofft Gough, _John Locke's Political Philosophy: Eight Studies_ (Oxford: Clarendon Press, 1950). The extent to which Locke in dealing with the points here discussed merely summarized views long expressed by lawyers of the period still deserves study. Especially important in this connection is Sir Mathew Hale, who, in a manuscript reply to Hobbes which was written about 1673 and which Locke is likely to have known (see Aubrey's letter to Locke quoted in Cranston, _John Locke: A Biography_ , p. 152), argued that \"to avoid that great uncertainty in the application of reason by particular persons to particular instances; and so to the end that men might understand by what rule and measure to live and possess; and might not be under the unknown arbitrary uncertain reason of particular persons, has been the prime reason, that the wiser sort of the world have in all ages agreed upon some certain laws and rules and methods of administration of common justice, and these to be as particular and certain as could be well thought of \"(\"Sir Mathew Hale's Criticisms on Hobbes's Dialogue of the Common Law,\" reprinted as an appendix to William Searle Holdsworth, _A History of English Law_ [London: Methuen, 1924], vol. 5, p. 503). See also John Greville Agard Pocock, _The Ancient Constitution and the Feudal Law_ (New York: Cambridge University Press, 1957).\n\n64 John Locke, _Second Treatise_ , sec. 22, p. 13.\n\n65 Ibid., sec. 127, p. 63.\n\n66 Ibid., sec. 131, p. 64.\n\n67 Ibid., sec. 137, p. 69.\n\n68 Ibid., sec. 136, p. 68.\n\n69 Ibid., sec. 151, p. 75.\n\n70 See John Neville Figgis, _The Divine Rights of Kings_ (2nd ed., with three additional essays; Cambridge: Cambridge University Press, 1914), p. 242; William Searle Holdsworth, _Some Lessons from Our Legal History_ (New York: Macmillan, 1928), pp. 126\u201327; and Charles Edwyn Vaughan, _Studies in the History of Political Philosophy before and after Rousseau_ (2 vols.; Manchester: Manchester University Press, 1939), vol. 1, p. 134.\n\n71 John Locke, _Second Treatise_ , chap. 13, pp. 74\u201379. Compare n. 58, above, on the separation of powers.\n\n72 John Locke, _Second Treatise_ , sec. 159, p. 80.\n\n73 John Locke, _Second Treatise_ , sec. 222, p. 107.\n\n74 Cf. George Macaulay Trevelyan, _English Social History: A Survey of Six Centuries, Chaucer to Queen Victoria_ (London: Longmans, Green, and Co., 1942), pp. 245 and 350ff., esp. 351: \"The specific work of the earlier Hanoverian epoch was the establishment of the rule of law; and that law, with all its grave faults, was at least a law of freedom. On that solid foundation all our subsequent reforms were built.\"\n\n75 On the significance of this event see particularly Holdsworth, _History of English Law_ , vol. 10, esp. p. 647: \"As the result of all these consequences of the independence of the courts, the doctrine of the rule or supremacy of the law was established in its modern form, and became perhaps the most distinctive, and certainly the most salutary, of all the characteristics of English constitutional law.\"\n\n76 Its influence was revived in the nineteenth century by the dramatic account given of the episode in Thomas Babington Macaulay, Baron Macaulay, _History of England from the Accession of James II_ (Everyman ed.; 4 vols.; London: J. M. Dent, 1953), chap. 22, vol. 4, pp. 272\u201392. [The events described concern the attainting and execution of Sir John Fenwick in 1696. The last bill of attainder was in fact enacted by Parliament in 1798 against Lord Edward Fitzgerald for his participation in the Irish Rebellion of that year. Bills of attainder were finally abolished in Great Britain in 1870.\u2014Ed.]\n\n77 Cf. also Daniel Defoe, _The History of the Kentish Petition_ (London, 1701), and his so-called _Legion's Memorial_ (1701) [\"A Memorial from the Gentlemen, Freeholders, and Inhabitants of the Counties of \u2014\u2014\u2014, in Behalf of Themselves and Many Thousands of the Good People of England\"] in _The Works of Daniel Defoe_ (3 vols.; London: John Clements, 1843), vol. 3, p. 5 [no continuous pagination; p. 5 of the pamphlet in question], where he writes that \"Englishmen are no more to be slaves to Parliaments, than to Kings.\" See on this also McIlwain, _Constitutionalism: Ancient and Modern: A Constitutional Interpretation_ (rev. ed.; Ithaca, NY: Cornell University Press, 1947), p. 150, n. 6 [Liberty Fund edition, p. 5, n. 6].\n\n78 Cf., for instance, Sir Alfred Denning, Baron Denning, _Freedom under the Law_ (London: Steven, 1949), p. 41, where he says with respect to the Continental doctrine _Nullum crimen, nulla poena sine lege_ : \"In this country, however, the common law has not limited itself in that way. It is not contained in a code but in the breast of the judges, who enunciate and develop the principles needed to deal with any new situations which arise.\" See also Stefan Glaser, \"Nullum crimen nulla poena sine lege,\" _Journal of Comparative Legislation and International Law_ , 3rd ser., 24 (1942): 29\u201341. In the form quoted, the Latin maxim dates only from the end of the eighteenth century (see chap. 13, n. 22, below), but there was current in eighteenth-century England the similar expression: \"Ubi non est lex ibi non est transgressio.\" [\"Where there is no law, there is no transgression.\"\u2014Ed.]\n\n79 Samuel Johnson, _The Works of Samuel Johnson, LL.D._ (14 vols.; London: Printed for J. Buckland [vols. 12\u201314 printed by J. Stockdale], 1787), vols. 12 and 13 published separately as _Debates in Parliament_ , vol. 12, p. 22, reporting a speech of Mr. Campbell in the Corn Bill Debate of the House of Commons on November 26, 1740. [The debate on the corn bill reported by Johnson was held on November 25, 1740, not November 26.\u2014Ed.]. Cf. Edward Lippincott McAdam, _Dr. Johnson and the English Law_ (Syracuse, NY: Syracuse University Press, 1951), p. 17. Also see Donald Johnson Greene, _The Politics of Samuel Johnson_ (New Haven: Yale University Press, 1960).\n\n80 Thus Lord Camden's opinion is sometimes quoted. The only statement of his expressing substantially the same view that I can find occurs in _Entick v. Carrington_ (1765), in Thomas Bayly Howell, _A Complete Collection of State Trials and Proceedings for High Treason and Other Crimes and Misdemeanors from the Earliest Period [1163] to the Present Time [1820]_ (34 vols.; London: T. C. Hansard for Longman, Hurst, Rees, Orme, and Brown, 1809\u201328), vol. 19: _A.D. 1753\u20131771_ (1813), p. 1073: \"With respect to the argument of state necessity, or a distinction that has been aimed at between states offences and others, the common law does not understand that kind of reasoning, nor do our books take notice of any such distinctions.\"\n\n81 What finally decided this incorporation into Tory doctrine was probably Henry Saint-John Bolingbroke, Letter 10 (1734) in _A Dissertation upon Parties in Several Letters to Caleb d'Anvers_ (5th ed., carefully revised and corrected; London: Printed for R. Francklin, 1739), p. 111, with its acceptance of the contrast between a \"government by constitution\" and a \"government by will.\"\n\n82 Cf. Holdsworth, _A History of English Law_ , vol. 10, p. 713: \"If a lawyer, a statesman, or a political philosopher of the eighteenth century had been asked what was, in his opinion, the most distinctive feature of the British constitution, he would have replied that its most distinctive feature was the separation of the powers of the different organs of government.\" Yet even at the time that Montesquieu popularized the conception on the Continent, it was true of the actual situation in England only to a limited degree.\n\n83 In addition to the passage quoted later on in the text, see particularly David Hume, \"Of the Origin of Government,\" vol. 1, p. 117 [Liberty Fund edition, p. 41]; \"Of Civil Liberty,\" vol. 1, p.161 [Liberty Fund edition, p. 94]; \"Of the Rise and Progress of the Arts and Sciences, vol. 1, p.178 [Liberty Fund edition, p. 116] in _Essays_ , where he argues: \"All general laws are attended with inconveniencies, when applied to particular cases; and it requires great penetration and experience, both to perceive that these inconveniencies are fewer than what results from full discretionary powers in every magistrate; and also to discern what general laws are, upon the whole, attended with fewest inconveniencies. This is a matter of so great difficulty, that men have made some advances, even in the sublime arts of poetry and eloquence, where a rapidity of genius and imagination assist their progress, before they arrived at any great refinement in their municipal laws, where frequent trial and diligent observation can alone direct their improvements.\" Cf. also _Enquiry Concerning the Principles of Morals, Essays II_ , pp. 179\u201396, 256, and 272\u201378. [The sections of the _Enquiry_ to which Hayek refers are: sec. 2, \"Of Justice\" (pp. 179\u2013 96), and app. 3: \"Some farther Considerations with regard to Justice\" (pp. 272\u201378).\u2014Ed.] As Hume is often represented as a Tory, it deserves notice that he himself stated that \"my views of things are more conformable to Whig principles; my representations of persons to Tory prejudices\" (quoted in Ernest Campbell Mossner, _Life of David Hume_ [Oxford: Clarendon Press, 1954], p. 311; see also _Life_ , p. 179, where Hume is described as a \"'Revolution Whig,' though not of the dogmatic variety\"). Thomas Carlyle, \"Boswell's Life of Johnson,\" _Critical and Miscellaneous Essays_ (5 vols.; London: Chapman and Hall, Ltd., 1899), vol. 3, p. 133, even calls Hume \"the father of all succeeding Whigs.\" See also David Hume, \"Liberty of the Press,\" _Essays_ [Essay 2], vol. 1, p. 96 [Liberty Fund edition, p. 12]: \"As the republican part of the government prevails in England, though with a great mixture of monarchy, it is obliged, for its own preservation, to maintain a watchful _jealousy_ over the magistrates, to remove all discretionary powers, and to secure every one's life and fortune by general and infexible laws. No action must be deemed a crime but what the law has plainly determined to be such.\"\n\n84 Friedrich Meinecke, _Die Entstehung des Historismus_ (2 vols.; Munich and Berlin: R. Oldenbourg, 1936), vol. 1, p. 234.\n\n85 David Hume, _History of England from the Invasion of Julius Caesar to the Revolution of 1688_ (6 vols.; new ed. corr.; London: Printed for A. Miller, 1762), vol. 5, p. 280 [Liberty Fund edition, vol. 5, p. 329].\n\n86 For the manner in which Adam Smith accepts the separation of powers and its justification as a matter of course see _Wealth of Nations_ [bk. 5, chap. 1, pt. 2], vol. 2, pp. 213\u201314 [Liberty Fund edition, vol. 2, pp. 708\u201323]. An earlier incidental reference to these problems ( _Wealth of Nations_ , vol. 2, p. 201) [Liberty Fund edition, vol. 2, p. 707], in which Smith briefly explains that in England \"the public safety does not require, that the sovereign is trusted with any discretionary power,\" even for suppressing \"the rudest, the most groundless, and the most licentious remonstrances,\" because he is \"secured by a well-regulated army\" has provided the occasion for an important discussion of this unique situation by one of the acutest foreign students of the British Constitution: Jean Louis de Lolme in his _Constitution of England; or, An Account of the English Government_ [1794] (new ed., corr.; London: G. G. and J. Robinson, 1800), represents it as \"the most characteristic circumstance in the English government, and the most pointed proof that can be given of the true freedom which is the consequence of its frame\" (p. 441) [Liberty Fund edition, p. 295] [\"le plus particuli\u00e8re la mani\u00e8re dont l'Angleterre est gouvern\u00e9e, et est la preuve la plus convaincante de la libert\u00e9 r\u00e9elle qui resulte de tous son gouvernement.\" _Constitution d'Angleterre_ , vol. 2, p.178.\u2014Ed.], that in England \"all the individual's actions are supposed to be lawful, till that law is pointed out which make them to be otherwise\" (p. 436) [Liberty Fund edition, p. 292] [\"Toutes les actions de l'individu passent pour l\u00e9gitimes jusqu'\u00e0 ce qu'on nomme la loi que leur donne une autre d\u00e9nomination.\" _Constitution d'Angleterre_ , _ou \u00e9tat du gouvernement anglois_ (2 vols.; London: G. Robinson, J. Murray, 1785), vol. 2, p. 174.\u2014Ed.]. He then goes on to say: \"The foundation of that law principle, or doctrine, which confines the exertion of the power of the government to such cases only as are expressed by a law in being\" (p. 439) [Liberty Fund edition, p. 294] [\"Le fondement de cette maxime du droit, qui borne l'exercice du pouvoir supr\u00eame aux cas seuls exprim\u00e9s par une loi \u00e9crite.\" _Constitution d'Angleterre_ , vol. 2, p. 176.\u2014Ed.] and which, though tracing back to Magna Carta, was put into actual force only by the abolition of the Star Chamber, with the result that \"it has appeared by the event, that the very extraordinary restriction upon the governing authority we are alluding to, and its execution, are no more than what the intrinsic situation of things, and the strength of the constitution, can bear\" (p. 440) [Liberty Fund edition, pp. 294\u201395] [\"Il parut par l'\u00e9v\u00e9nement que cette restriction m\u00eame qui paroit singuli\u00e8re quant \u00e0 l'autorit\u00e9 supr\u00eame et ses fonctions, n'est que ce que les choses en elles-m\u00eames et la force de la constitution, peuvent supporter.\" _Constitution d'Angleterre_ , vol. 2, p. 178.\u2014Ed.]. (Note how this passage is evidently influenced by the exposition of Hume quoted in the text.)\n\nMany similar statements from the period could be quoted, but two particularly characteristic ones must suffice. The first is from John Wilkes's _The North Briton_ 64 ([Saturday], September 3, 1768) [p. 1]; quoted by Sir Carleton Kemp Allen, _Law and Orders: An Inquiry into the Nature and Scope of Delegated Legislation and Executive Powers in England_ (London: Stevens, 1945), pp. 5\u20136: \"In a _free government_ , these three powers ever have been, at least ever ought to be, kept separate: because, were all the three, or any two of them, to be united in the same person, the liberties of the people would be, from that moment, ruined. For instance, were the _legislative_ and _executive powers_ united in the same magistrate, or in the same body of magistrates, there could be no such thing as liberty, inasmuch as there would be great reason to fear lest the same monarch, or senate, should enact tyrannical laws in order to execute them in a tyrannical manner. Nor could there, it is evident, be such a thing as liberty, were the _judiciary power_ united either to the _legislative_ or to the _executive_. In the former case, the life and liberty of the subject would be necessarily exposed to the most imminent danger, because then the same person would be both judge and legislator. In the latter, the condition of the subject would be no less deplorable, for the very same person might pass a cruel sentence in order, perhaps, to execute it with still greater cruelty.\"\n\nThe second passage occurs in Junius [William Petty-Fitzmaurice, Earl of Shelburne], _The Letters of Junius_ , Charles Warren Everett, ed. (London: Faber and Gwyer, 1927), Letter 47, dated May 25, 1771, p. 208: \"The government of England is a government of law. We betray ourselves, we contradict the spirit of our laws, and we shake the whole system of English jurisprudence, whenever we intrust a discretionary power over the life, liberty, or fortune of the subject, to any man or set of men whatsoever upon a presumption that it will not be abused.\"\n\n87 Sir William Blackstone, _Commentaries on the Laws of England_ (4 vols.; Oxford: Clarendon Press, 1765), vol. 1, p. 259: \"In this distinct and separate existence of the judicial power in a peculiar body of men, nominated indeed, but not removable at pleasure, by the Crown, consists one main preservative of public liberty; which cannot subsist long in any state, unless the administration of common justice be in some degree separated both from the legislative and also from the executive power. Were it joined with the legislative, the life, liberty, and property of the subject would be in the hands of arbitrary judges, whose decisions would be then regulated only by their own opinions, and not by any fundamental principles of law; which, though legislatures may depart from them, yet judges are bound to observe.\"\n\n88 Sir William Blackstone, _Commentaries on the Laws of England_ , p. 44.\n\n89 See particularly Edmund Burke, _Speech on the Motion Made in the House of Commons, the 7th of February, 1771, Relative to the Middlesex Elections_ , in _Works_ , vol. 10, pp. 63\u201371.\n\n90 Sir Ernest Barker, _Traditions of Civility: Eight Essays_ (Cambridge: Cambridge University Press, 1948), p. 216. Note also the interesting account at pp. 245 and 248 of the same work. regarding Albert Venn Dicey's admiration for Paley. [William Paley (1743\u20131805); His _Principles of Moral and Political Philosophy_ (1785), was based on his Cambridge lectures of 1766\u201376. The work reflects the beginnings of the transformation of Whiggism into what later became liberal doctrine.\u2014Ed.]\n\n91 William Paley, _The Principles of Moral and Political Philosophy_ (1785) (London: Printed for T. Tegg, 1824), p. 308.\n\n92 Macaulay's success in making the achievement of the constitutional struggles of the past once more a living possession of every educated Englishman is now rarely remembered. But see \"The Literary Historian,\" in the _Times Literary Supplement_ , January 16, 1953, p. 40, col. 5: \"He did for our history what Livy did for the history of Rome; and he did it better.\" Cf. also Lord Acton's remark, _Historical Essays_ , p. 482, that Macaulay \"had done more than any writer in the literature of the world for the propagation of the Liberal faith, and he was not only the greatest, but the most representative Englishman then [1856] living.\" [Liberty Fund edition, _Essays in the History of Liberty_ , p. 170. Acton's comments originally appear in his review of _A History of England, 1837\u20131880_ by the Rev. J. Franck Bright, originally published in the _English Historical Review_ , vol. 3 (1888).\u2014Ed.]\n\n93 In some respects even the Benthamites could not but build on and improve the old tradition which they did so much to destroy. This applies certainly to John Austin's efforts to provide sharp distinctions between true general \"laws\" and \"occasional or particular commands\" (see Austin's _Lectures on Jurisprudence; or, The Philosophy of Positive Law_ [Lecture One], Robert Campbell, ed. [5th ed. rev.; 2 vols.; London: J. Murray, 1885]; vol. 1, p. 92).\n\n94 Richard Price, _Two Tracts on Civil Liberty: The War with America and the Debts and Finances of the Kingdom_ (2 vols. in 1; London: T. Cadell, 1778), p. 7. [In fact, Price first wrote these words in his _Observations on the Nature of Civil Liberty, the Principles of Government, and the Justice and Policy of the War with America_ (London: T. Cadell, 1776), which appeared two years earlier than the eighth edition of these _Observations_ , which appeared in the _Two Tracts_.\u2014Ed.]\n\n95 Richard Price, _Observations on the Importance of the American Revolution and the Means of Making It a Benefit to the World, to Which is Added a Letter from M. Turgot_ [dated March 22, 1778] (London: T. Caddell, 1785), p.111. [The French reads: \"Comment se fait-il que vous soyez presque le premier parmi vos \u00e9crivains qui ayez donn\u00e9 des notions justes de la libert\u00e9, et qui ayez fait sentir la fausset\u00e9 de cette notion rebattue par presque tous les \u00e9crivains les plus r\u00e9publicains, que la libert\u00e9 consiste \u00e0 n'\u00eatre soumis qu'aux loix, comme si un homme opprim\u00e9 par une loi injuste \u00eatoit libre.\"\u2014Ed.]\n\n96 William Searle Holdsworth, _A History of English Law_ , vol. 10, p. 23.\n\n## TWELVE\n\nTHE AMERICAN CONTRIBUTION: CONSTITUTIONALISM\n\nEurope seemed incapable of becoming the home of free States. It was from America that the plain ideas that men ought to mind their own business, and that the nation is responsible to Heaven for the acts of State\u2014ideas long locked in the breast of solitary thinkers, and hidden among Latin folios\u2014 burst forth like a conqueror upon the world they were destined to transform, under the title of the Rights of Man.\n\n\u2014Lord Acton\n\n1. \"When in 1767 this modernised British Parliament, committed by now to the principle of parliamentary sovereignty unlimited and unlimitable, issued a declaration that a parliamentary majority could pass any law it saw fit, it was greeted with an out-cry of horror in the colonies. James Otis and Sam Adams in Massachusetts, Patrick Henry in Virginia, and other colonial leaders along the seaboard screamed 'Treason!' and 'Magna Carta!' Such a doctrine, they insisted, demolished the essence of all their British ancestors had fought for, took the very savor out of that fine Anglo-Saxon liberty for which the sages and patriots of England had died.\"1 Thus one of the modern American enthusiasts for the unlimited power of the majority describes the beginning of the movement that led to a new attempt to secure the liberty of the individual.\n\nThe movement in the beginning was based entirely on the traditional conceptions of the liberties of Englishmen. Edmund Burke and other English sympathizers were not the only ones who spoke of the colonists as \"not only devoted to liberty, but to liberty according to English ideas, and on English principles\";2 the colonists themselves had long held this  view.3 They felt that they were upholding the principles of the Whig revolution of 1688;4 and as \"Whig statesmen toast[ed] General Washington, rejoiced that America had resisted, and insist[ed] on the acknowledgment of independence,\"5 so the colonists toasted William Pitt and the Whig statesmen who supported them.6\n\nIn England, after the complete victory of Parliament, the conception that no power should be arbitrary and that all power should be limited by higher law tended to be forgotten. But the colonists had brought these ideas with them and now turned them against Parliament. They objected not only that they were not represented in that Parliament but even more that it recognized no limits whatever to its powers. With this application of the principle of legal limitation of power by higher principles to Parliament itself, the initiative  in the further development of the ideal of free government passed to the Americans.\n\nThey were singularly fortunate, as perhaps no other people has been in a similar situation, in having among their leaders a number of profound students of political philosophy. It is a remarkable fact that when in many other respects the new country was still so very backward, it could be said that \"it is in political science only that America occupies the first rank. There are six Americans on a level with the foremost Europeans, with Smith and Turgot, Mill and Humboldt.\"7 They were, moreover, men as much steeped in the classical tradition as any of the English thinkers of the preceding century had been and were fully acquainted with the ideas of the latter.8\n\n2. Until the final break, the claims and arguments advanced by the colonists in the conflict with the mother country were based entirely on the rights and privileges to which they regarded themselves entitled as British subjects. It was only when they discovered that the British constitution, in whose principles they had firmly believed, had little substance and could not be successfully appealed to against the claims of Parliament, that they concluded that the missing foundation had to be supplied.9 They regarded it as fundamental doctrine that a \"fixed constitution\"10 was essential to any free government  and that a constitution meant limited government.11 From their own history they had become familiar with written documents which defined and circumscribed the powers of government such as the Mayflower compact and the colonial charters.12\n\nTheir experience had also taught them that any constitution that allocated and distributed the different powers thereby necessarily limited the powers of any authority. A constitution might conceivably confine itself to procedural matters and merely determine the source of all authority. But they would hardly have called \"constitution\" a document which merely said that whatever such and such a body or person says shall be law. They perceived that, once such a document assigned specific powers to different authorities, it would also limit their powers not only in regard to the subjects or the aims to be pursued but also with regard to the methods to be employed. To the colonists, freedom meant that government should have powers only for such action as was explicitly required by law, so that nobody should possess any arbitrary power.13\n\nThe conception of a constitution thus became closely connected with the conception of representative government, in which the powers of the representative body were strictly circumscribed by the document that conferred upon it particular powers. The formula that all power derives from the people referred not so much to the recurrent election of representatives as to the fact that the people, organized as a constitution-making body, had the exclusive  right to determine the powers of the representative legislature.14 The constitution was thus conceived as a protection of the people against all arbitrary action, on the part of the legislative as well as the other branches of government.\n\nA constitution which in such manner is to limit government must contain what in effect are substantive rules, besides provisions regulating the derivation of authority. It must lay down general principles which are to govern the acts of the appointed legislature. The idea of a constitution, therefore, involves not only the idea of hierarchy of authority or power but also that of a hierarchy of rules or laws, where those possessing a higher degree of generality and proceeding from a superior authority control the contents of the more specific laws that are passed by a delegated authority.\n\n3. The conception of a higher law governing current legislation is a very old one. In the eighteenth century it was usually conceived as the law of God, or that of Nature, or that of Reason. But the idea of making this higher law explicit and enforceable by putting it on paper, though not entirely new, was for the first time put into practice by the Revolutionary colonists. The individual colonies, in fact, made the first experiments in codifying this higher law with a wider popular basis than ordinary legislation. But the model that was profoundly to influence the rest of the world was the federal Constitution.\n\nThe fundamental distinction between a constitution and ordinary laws is similar to that between laws in general and their application by the courts to a particular case: as in deciding concrete cases the judge is bound by general rules, so the legislature in making particular laws is bound by the more general principles of the constitution. The justification for these distinctions is also similar in both cases: as a judicial decision is regarded as just only if it is in conformity with a general law, so particular laws are regarded as just only if they conform to more general principles. And as we want to prevent the judge from infringing the law for some particular reason, so we also want to prevent the legislature from infringing certain general principles for the sake of temporary and immediate aims.\n\nWe have already discussed the reason for this need in another connection.15 It is that all men in the pursuit of immediate aims are apt\u2014or, because of the limitation of their intellect, in fact bound\u2014to violate rules of conduct which they would nevertheless wish to see generally observed. Because of the restricted capacity of our minds, our immediate purposes will always loom large, and we will tend to sacrifice long-term advantages to them. In individual  as in social conduct we can therefore approach a measure of rationality or consistency in making particular decisions only by submitting to general principles, irrespective of momentary needs. Legislation can no more dispense with guidance by principles than any other human activity if it is to take account of effects in the aggregate.\n\nA legislature, like an individual, will be more reluctant to take certain measures for an important immediate aim if this requires the explicit repudiation of principles formally announced. To break a particular obligation or a promise is a different matter from explicitly stating that contracts or promises may be broken whenever such and such general conditions occur. Making a law retroactive or by law conferring privileges or imposing punishments on individuals is a different matter from rescinding the principle that this should never be done. And a legislature's infringing rights of property or the freedom of speech in order to achieve some great objective is quite a different thing from its having to state the general conditions under which such rights can be infringed.16\n\nThe stating of those conditions under which such actions by the legislature are legitimate would probably have beneficial effects, even if only the legislature itself were required to state them, much as the judge is required to state the principles on which he proceeds. But it will clearly be more effective if only another body has the power to modify these basic principles, especially if the procedure of this body is lengthy and thus allows time for the importance of the particular objective that has given rise to the demand for modification to be seen in the proper proportion. It is worth noting here that, in general, constitutional conventions or similar bodies set up to lay down the most general principles of government are regarded as competent to do only this, and not to pass any particular laws.17\n\nThe expression an \"appeal from the people drunk to the people sober,\" which is often used in this connection, stresses only one aspect of a much wider problem and, by the levity of its phrasing, has probably done more to veil than to clarify the very important issues involved. The problem is not merely one of giving time for passions to cool, though this on occasion may be very important, as that of taking into account man's general inability to consider explicitly all the probable effects of a particular measure and his dependence on generalizations or principles if he is to fit his individual decisions into a coherent whole. It is \"impossible for men to consult their interest in so effectual a manner, as by an universal and inflexible observance of rules of justice.\"18\n\nIt need hardly be pointed out that a constitutional system does not involve an absolute limitation of the will of the people but merely a subordination of immediate objectives to long-term ones. In effect this means a limitation of the means available to a temporary majority for the achievement of particular objectives by general principles laid down by another majority for a long period in advance. Or, to put it differently, it means that the agreement to submit to the will of the temporary majority on particular issues is based on the understanding that this majority will abide by more general principles laid down beforehand by a more comprehensive body.\n\nThis division of authority implies more than may at first be apparent. It implies a recognition of limits to the power of deliberate reason and a preference for reliance on proved principles over _ad hoc_ solutions; furthermore, it implies that the hierarchy of rules does not necessarily end with the explicitly stated rules of constitutional law. Like the forces governing the individual mind, the forces making for social order are a multilevel affair; and even constitutions are based on, or presuppose, an underlying agreement on more  fundamental principles\u2014principles which may never have been explicitly expressed, yet which make possible and precede the consent and the written fundamental laws. We must not believe that, because we have learned to make laws deliberately, all laws must be deliberately made by some human agency.19 Rather, a group of men can form a society capable of making laws because they already share common beliefs which make discussion and persuasion possible and to which the articulated rules must conform in order to be accepted as legitimate.20\n\nFrom this it follows that no person or body of persons has complete freedom to impose upon the rest whatever laws it likes. The contrary view that underlies the Hobbesian conception of sovereignty21 (and the legal positivism deriving from it) springs from a false rationalism that conceives of an autonomous and self-determining reason and overlooks the fact that all rational thought moves within a non-rational framework of beliefs and institutions. Constitutionalism means that all power rests on the understanding that it will be exercised according to commonly accepted principles, that the persons on whom power is conferred are selected because it is thought that they are most likely to do what is right, not in order that whatever they do should be right. It rests, in the last resort, on the understanding that power is ultimately not a physical fact but a state of opinion which makes people obey.22\n\nOnly a demagogue can represent as \"antidemocratic\" the limitations which long-term decisions and the general principles held by the people impose upon the power of temporary majorities. These limitations were conceived to protect the people against those to whom they must give power, and they are the only means by which the people can determine the general character of the order under which they will live. It is inevitable that, by accepting general principles, they will tie their hands as far as particular issues are concerned. For only by refraining from measures which they would not wish to be used on themselves can the members of a majority forestall the adoption of such measures when they are in a minority. A commitment to long-term principles, in fact, gives the people more control over the general nature of the political order than they would possess if its character were to be determined solely by successive decisions of particular issues. A free society certainly needs permanent means of restricting the powers of government, no matter what the par  ticular objective of the moment may be. And the Constitution which the new American nation was to give itself was definitely meant not merely as a regulation of the derivation of power but as a constitution of liberty, a constitution that would protect the individual against all arbitrary coercion.\n\n4. The eleven years between the Declaration of Independence and the framing of the federal Constitution were a period of experimentation by the thirteen new states with the principles of constitutionalism. In some respects their individual constitutions show more clearly than the final Constitution of the Union how much the limitation of all governmental power was the object of constitutionalism. This appears, above all, from the prominent position that was everywhere given to inviolable individual rights, which were listed either as part of these constitutional documents or as separate Bills of Rights.23 Though many of them were no more than restatements of the rights which the colonists had in fact enjoyed,24 or thought they had always been entitled to, and most of the others were formulated hastily with reference to issues currently under dispute, they show clearly what constitutionalism meant to the Americans. In one place or another they anticipate most of the principles that were to inspire the federal Constitution.25 The principal concern of all was, as the Bill of Rights preceding the constitution of Massachusetts of 1780 expressed it, that the government should be \"a government of laws and not of men.\"26\n\nThe most famous of these Bills of Rights, that of Virginia, which was drafted and adopted before the Declaration of Independence and modeled on English and colonial precedents, largely served as the prototype not only for those of the other states but also for the French Declaration of the Rights of Men and  Citizens of 1789 and, through that, for all similar European documents.27 In substance, the various Bills of Rights of the American states and their main provisions are now familiar to everybody.28 Some of these provisions, however, which occur only occasionally, deserve mention, such as the prohibition of retroactive laws, which occurs in four of the state Bills of Rights, or that of \"perpetuities and monopolies,\" which occurs in two.29 Also important is the emphatic manner in which in some of the constitutions the principle of the separation of powers is laid down30\u2014no less so because in practice this was honored more in the breach than in the observance. Another recurring feature which to present readers will appear to be no more than a rhetorical flourish but to the men of the time was very important is the appeal to \"the fundamental principles of a free government\" which several of the constitutions contain31 and the repeated reminder that \"a frequent recurrency to fundamental principles is absolutely necessary to preserve the blessing of liberty.\"32\n\nIt is true that many of these admirable principles remained largely theory and that the state legislatures soon came as near to claiming omnipotence as the British Parliament had done. Indeed, \"under most of the revolutionary constitutions the legislature was truly omnipotent and the executive correspondingly weak. Nearly all of these instruments conferred upon the former body practically unlimited power. In six constitutions there was nothing whatever to prevent the legislature amending the constitution by ordinary legislative process.\"33 Even where this was not so, the legislatures often highhandedly disregarded the text of the constitution and still more those unwritten rights of the citizens which these constitutions had been intended to protect. But the development of explicit safeguards against such abuses required time. The main lesson of the period of Confederation was that the mere writing down  on paper of a constitution changed little unless explicit machinery was provided to enforce it.34\n\n5. Much is sometimes made of the fact that the American Constitution is the product of design and that, for the first time in modern history, a people deliberately constructed the kind of government under which they wished to live. The Americans themselves were very conscious of the unique nature of their undertaking, and in a sense it is true that they were guided by a spirit of rationalism, a desire for deliberate construction and pragmatic procedure closer to what we have called the \"French tradition\" than to the \"British.\"35 This attitude was often strengthened by a general suspicion of tradition and an exuberant pride in the fact that the new structure was entirely of their own making. It was more justified here than in many similar instances, yet still essentially mistaken. It is remarkable how different from any clearly foreseen structure is the frame of government which ultimately emerged, how much of the outcome was due to historical accident or the application of inherited principles to a new situation. What new discoveries the federal Constitution contained either resulted from the application of traditional principles to particular problems or emerged as only dimly perceived consequences of general ideas.\n\nWhen the Federal Convention, charged \"to render the constitution of the federal government more adequate to the exigencies of the Union,\" met at Philadelphia in May, 1787, the leaders of the federalist movement found themselves confronted by two problems. While everybody agreed that the powers of the confederation were insufficient and must be strengthened, the  main concern was still to limit the powers of government as such, and not the least motive in seeking reform was to curb the arrogation of powers by the state legislatures.36 The experience of the first decade of independence had merely somewhat shifted the emphasis from protection against arbitrary government to the creation of one effective common government. But it had also provided new grounds for suspecting the use of power by the state legislatures. It was scarcely foreseen that the solution of the first problem would also provide the answer to the second and that the transference of some essential powers to a central government, while leaving the rest to the separate states, would also set an effective limit on all government. Apparently it was from Madison that \"came the idea that the problem of producing adequate safeguards for private rights and adequate powers for national government was in the end the same problem, inasmuch as a strengthened national government could be a make-weight against the swollen prerogatives of state legislatures.\"37 Thus the great discovery was made of which Lord Acton later  said: \"Of all checks on democracy, federalism has been the most efficacious and the most congenial. . . . The federal system limits and restrains the sovereign power by dividing it, and by assigning to Government only certain defined rights. It is the only method of curbing not only the majority but the power of the whole people, and it affords the strongest basis for a second chamber, which has been found essential security for freedom in every genuine democracy.\"38\n\nThe reason why a division of powers between different authorities always reduces the power that anybody can exercise is not always understood. It is not merely that the separate authorities will, through mutual jealousy, prevent one another from exceeding their authority. More important is the fact that certain kinds of coercion require the joint and co-ordinated use of different powers or the employment of several means, and, if these means are in separate hands, nobody can exercise those kinds of coercion. The most familiar illustration is provided by many kinds of economic control which can be effective only if the authority exercising them can also control the movement of men and goods across the frontiers of its territory. If it lacks that power, though it has the power to control internal events, it cannot pursue policies which require the joint use of both. Federal government is thus in a very definite sense limited government.39\n\nThe other chief feature of the Constitution relevant here is its provision guaranteeing individual rights. The reasons why it was at first decided not to include a Bill of Rights in the Constitution and the considerations which later persuaded even those who had at first opposed the decision are equally significant. The argument against inclusion was explicitly stated by Alexander Hamilton in the _Federalist:_ \"[Bills of rights are] not only unnecessary in the proposed constitution, but would even be dangerous. They would contain various exceptions to powers not granted, and on this very account would afford a colourable pretext to claim more than were granted. For why declare that things shall not be done which there is no power to do? Why, for instance, should it be said, that the liberty of the press shall not be restrained, when no power is given by which restrictions may be imposed? I will not contend that such a provision would confer a regulating power; but it is evident that  it would furnish, to men disposed to usurp, a plausible pretence for claiming that power. They might urge with a semblance of reason that the constitution ought not to be charged with the absurdity of providing against the abuse of an authority, which was not given, and that the provision against restraining the liberty of the press afforded a clear implication, that a right to prescribe proper regulations concerning it, was intended to be vested in the national government. This may serve as a specimen of the numerous handles which would be given to the doctrine of constructive powers, by the indulgence of an injudicious zeal for bills of rights.\"40\n\nThe basic objection thus was that the Constitution was intended to protect a range of individual rights much wider than any document could exhaustively enumerate and that any explicit enumeration of some was likely to be interpreted to mean that the rest were not protected.41 Experience has shown that there was good reason to fear that no bill of rights could fully state all the rights implied in \"the general principles which are common to our political institutions\"42 and that to single out some would seem to imply that the others were not protected. On the other hand, it was soon recognized that the Constitution was bound to confer on government powers which might be used to infringe individual rights if these were not specially protected and that, since some such rights had already been mentioned in the body of the Constitution, a fuller catalogue might with advantage be added. \"A bill of rights,\" it was  later said, \"is important, and may often be indispensable, whenever it operates, as a qualification upon powers actually granted by the people to the government. This is the real ground of all the bills of rights in the parent country, in the colonial constitutions and laws, and in the state constitutions,\" and \"A bill of rights is an important protection against unjust and oppressive conduct on the part of the people themselves.\"43\n\nThe danger so clearly seen at the time was guarded against by the careful proviso (in the Ninth Amendment) that \"the enumeration of certain rights in this Constitution shall not be construed to deny or disparage others retained by the people\"\u2014a provision whose meaning was later completely forgotten.44\n\nWe must at least briefly mention another feature of the American Constitution, lest it appear that the admiration that the protagonists of liberty have always felt for the Constitution45 necessarily extends to this aspect also, particularly as it is a product of the same tradition. The doctrine of the separation of powers led to the formation of a presidential republic in which the chief executive derives his power directly from the people and, in consequence, may belong to a different party from that which controls the legislature. We shall see later that the interpretation of the doctrine on which this arrangement rests is by no means required by the aim it serves. It is difficult to see the expediency of erecting this particular obstacle to the efficiency of the executive, and one may well feel that the other excellencies of the American Constitution would show themselves to greater advantage if they were not combined with that feature.\n\n6. If we consider that the aim of the Constitution was largely to restrain legislatures, it becomes evident that arrangements had to be made for applying such restraints in the way that other laws are applied\u2014namely, through courts of justice. It is therefore not surprising that a careful historian finds  that \"judicial review, instead of being an American invention, is as old as constitutional law itself, and without it constitutionalism would never have been attained.\"46 In view of the character of the movement that led to the design of a written constitution, it must indeed seem curious that the need for courts which could declare laws unconstitutional should ever have been questioned.47 The important fact, at any rate, is that to some of the drafters of the Constitution judicial review was a necessary and self-evident part of a constitution, that when occasion arose to defend their conception in the early discussions after its adoption, they were explicit enough in their statements;48 and that through a decision of the Supreme Court it soon became the law of the land. It had already been applied by the state courts with respect to the state constitutions (in a few instances even before the adoption of the federal Constitution49),  although none of the state constitutions had explicitly provided for it, and it seemed obvious that the federal courts should have the same power where the federal Constitution was concerned. The opinion in _Marbury v. Madison_ , in which Chief Justice Marshall established the principle, is justly famous also for the masterly manner in which it summed up the rationale of a written constitution.50\n\nIt has often been pointed out that for fifty-four years after that decision the Supreme Court found no further occasion to reassert this power. But it must be remarked that the corresponding power was frequently used during the period by the state courts and that the non-use of it by the Supreme Court would be significant only if it could be shown that it did not use it in cases where it ought to have used it.51 Moreover, there can be no question that it was in this very period that the whole doctrine of the Constitution on which judicial review was based was most fully developed. There appeared during these years a unique literature on the legal guaranties of individual liberty which deserves a place in the history of liberty next to the great English debates of the seventeenth and eighteenth centuries. In a fuller exposition the contributions of James Wilson, John Marshall, Joseph Story, James Kent, and Daniel  Webster would deserve careful consideration. The later reaction against their doctrines has somewhat obscured the great influence which this generation of jurists had on the evolution of the American political tradition.52\n\nWe can consider here only one other development of constitutional doctrine during this period. It is the increasing recognition that a constitutional system based on the separation of powers presupposed a clear distinction between laws proper and those other enactments of the legislature which are not general rules. We find in discussions of the period constant references to the conception of \"general laws, formed upon deliberation, under the influence of no resentment, and without knowing upon whom they will operate.\"53 There was much discussion of the undesirability of \"special\" as distinguished from \"general\" acts.54 Judicial decisions repeatedly stressed that laws proper ought to be \"general public laws equally binding upon every member of the community under similar circumstances.\"55 Various attempts were made to  embody this distinction in state constitutions,56 until it came to be regarded as one of the chief limitations upon legislation. This, together with the explicit prohibition of retroactive laws by the federal Constitution (somewhat unaccountably restricted to criminal law by an early decision of the Supreme Court),57 indicate how constitutional rules were meant to control substantive legislation.\n\n7. When in the middle of the century the Supreme Court again found occasion to reassert its power of examining the constitutionality of congressional legislation, the existence of that power was hardly questioned. The problem had become rather one of the nature of the substantive limitations which the Constitution or constitutional principles imposed upon legislation. For a time judicial decisions appealed freely to the \"essential nature of all free governments\" and the \"fundamental principles of civilization.\" But gradually, as the ideal of popular sovereignty grew in influence, what the opponents of an explicit enumeration of protected rights had feared happened: it became accepted doctrine that the courts are not at liberty \"to declare an act void, because in their opinion it is opposed to a _spirit_ supposed to pervade the constitution _but not expressed in words_.\"58 The meaning of the Ninth Amendment was forgotten and seems to have remained forgotten ever since.59\n\nThus bound to the explicit provisions of the Constitution, the judges of the Supreme Court in the second half of the century found themselves in a somewhat peculiar position when they encountered uses of legislative power which, they felt, it had been the intention of the Constitution to prevent but which the Constitution did not explicitly prohibit. In fact, they at first deprived themselves of one weapon which the Fourteenth Amendment might have provided. The prohibition that \"no state shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United  States\" was, within five years, reduced to a \"practical nullity\" by a decision of the Court.60 But the continuation of the same clause, \"nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws,\" was to achieve altogether unforeseen importance.\n\nThe \"due process\" provision of this amendment repeats with explicit reference to state legislation what the Fifth Amendment had already provided and several state constitutions similarly stated. In general, the Supreme Court had interpreted the earlier provision according to what was undoubtedly its original meaning of \"due process for the enforcement of law.\" But in the last quarter of the century, when it had, on the one hand, become unquestioned doctrine that only the letter of the Constitution could justify the Court's declaring a law unconstitutional, and when, on the other hand, it was faced with more and more legislation which seemed contrary to the spirit of the Constitution, it clutched at that straw and interpreted the procedural as a substantive rule. The \"due process\" clauses of the Fifth and Fourteenth Amendments were the only ones in the Constitution that mentioned property. During the next fifty years they thus became the foundation on which the Court built a body of law concerning not only individual liberties but government control of economic life, including the use of police power and of taxation.61\n\nThe results of this peculiar and partly accidental historical development do not provide enough of a general lesson to justify any further consideration here of the intricate issues of present American constitutional law which they raise. Few people will regard as satisfactory the situation that has emerged. Under so vague an authority the Court was inevitably led to adjudicate, not on whether a particular law went beyond the specific powers conferred on  the legislatures, or whether legislation infringed general principles, written or unwritten, which the Constitution had been intended to uphold, but whether the ends for which the legislature used its powers were desirable. The problem became one of whether the purposes for which powers were exercised were \"reasonable\"62 or, in other words, whether the need in the particular instance was great enough to justify the use of certain powers, though in other instances there might be justification. The Court was clearly overstepping its proper judicial functions and arrogating what amounted to legislative powers. This finally led to conflicts with public opinion and the Executive in which the authority of the Court suffered somewhat.\n\n8. Though to most Americans this is still familiar recent history, we cannot altogether ignore here the climax of the struggle between the Executive and the Supreme Court, which from the time of the first Roosevelt and the anti-Court campaign of the progressives under the elder La Follette had been a standing feature of the American scene. The conflict of 1937, while it induced the Court to retreat from its more extreme position, also led to a reaffirmation of the fundamental principles of the American tradition which is of lasting significance.\n\nWhen the most severe economic depression of modern times was at its peak, the American presidency came to be occupied by one of those extraordinary figures whom Walter Bagehot had in mind when he wrote: \"some man of genius, of attractive voice and limited mind, who declaims and insists, not only that the special improvement is a good thing in itself, but the best of all things, and the root of all other good things.\"63 Fully convinced that he knew best what was needed, Franklin D. Roosevelt conceived it as the function of democracy in times of crisis to give unlimited powers to the man it trusted, even if this meant that it thereby \"forged new instruments of power which in some hands would be dangerous.\"64\n\nIt was inevitable that this attitude, which regarded almost any means as legitimate if the ends were desirable, should soon lead to a head-on clash with a Supreme Court which for half a century had habitually judged on the \"reasonableness\" of legislation. It is probably true that in its most spectacular decision, when the Court unanimously struck down the National Recovery Administration Act, it not only saved the country from an ill-conceived measure but also acted within its constitutional rights. But thereafter its small conservative majority proceeded to annul, on much more questionable grounds, one after another of the measures of the President until he became convinced that his only chance of carrying them out was to restrict the powers or alter the personnel of the Court. It was over what became known as the \"Court Packing Bill\" that the struggle came to a head. The re-election of the President by an unprecedented majority in 1936, however, which sufficiently strengthened his position to attempt this, also seems to have persuaded the Court that the President's program had wide approval. When, in consequence, the Court withdrew from its more extreme position and not only reversed itself on some of the central issues but in effect abandoned the use of the due process clause as a substantive limit on legislation, the President was deprived of his strongest arguments. In the end his measure was completely defeated in the Senate, where his party held the overwhelming majority, and his prestige suffered a serious blow at the moment when he had reached the pinnacle of his popularity.\n\nIt is mainly because of the brilliant restatement of the traditional role of the Court in the report of the Senate Judiciary Committee that this episode forms a fitting conclusion to this survey of the American contribution to the ideal of freedom under the law. Only a few of the most characteristic passages from that document can be quoted here. Its statement of the principles starts from the presumption that the preservation of the American constitutional system is \"immeasurably more important . . . than the immediate adoption of any legislation however beneficial.\" It declares \"for the continuation and perpetuation of government and rule by law; as distinguished from government and rule by men, and in this we are but re-asserting the principles basic to the Constitution of the United States.\" And it goes on to state: \"If the Court of last resort is to be made to respond to a prevalent sentiment of a current hour, politically imposed, that Court must ultimately become subservient to the pressure of public opinion of the hour, which might at the moment embrace mob passion abhorrent to a more calm, lasting, consideration. . . .  No finer or more durable philosophy of free government is to be found in all the writings and practices of great statesmen than may be found in the decisions of the Supreme Court when dealing with great problems of free government touching human rights.\"65\n\nNo greater tribute has ever been paid by a legislature to the very Court which limited its powers. And nobody in the United States who remembers this event can doubt that it expressed the feelings of the great majority of the population.66\n\n9. Incredibly successful as the American experiment in constitutionalism has been\u2014and I know of no other written constitution which has lasted half as long\u2014it is still an experiment in a new way of ordering government, and we must not regard it as containing all wisdom in this field. The main features of the American Constitution crystallized at so early a stage in the understanding of the meaning of a constitution, and so little use has been made of the amending power to embody in the written document the lessons learned, that in some respects the unwritten parts of the Constitution are more instructive than its text. For the purposes of this study, at any rate, the general principles underlying it are more important than any of its particular features.\n\nThe chief point is that in the United States it has been established that the legislature is bound by general rules; that it must deal with particular problems in such a manner that the underlying principle can also be applied in other cases; and that, if it infringes a principle hitherto observed, though perhaps  never explicitly stated, it must acknowledge this fact and must submit to an elaborate process in order to ascertain whether the basic beliefs of the people really have changed. Judicial review is not an absolute obstacle to change, and the worst it can do is to delay the process and make it necessary for the constitution-making body to repudiate or reaffirm the principle at issue.\n\nThe practice of restraining government's pursuit of immediate aims by general principles is partly a precaution against drift; for this, judicial review requires as its complement the normal use of something like the referendum, an appeal to the people at large, to decide on the question of general principle. Furthermore, a government which can apply coercion to the individual citizen only in accordance with pre-established long-term general rules but not for specific, temporary ends is not compatible with every kind of economic order. If coercion is to be used only in the manner provided for in the general rules, it becomes impossible for government to undertake certain tasks. Thus it is true that, \"stripped of all its husks, liberalism is constitutionalism, 'a government of laws and not of men'\"67 if by \"liberalism\" we mean what it still meant in the United States during the Supreme Court struggle of 1937, when the \"liberalism\" of the defenders of the Court was attacked as minority thinking.68 In this sense Americans have been able to defend freedom by defending their Constitution. We shall presently see how on the European Continent in the early nineteenth century the liberal movement, inspired by the American example, came to regard as its principal aim the establishment of constitutionalism and the rule of law.\n\n* * *\n\nThe quotation at the head of the chapter is taken from Lord Acton, \"Freedom in Christianity,\" _History of Freedom_ , p. 55 [Liberty Fund edition, _Essays on the History of Liberty_ , p. 49].\n\n1 Edwin Mims, Jr., _The Majority of the People_ (New York: Modern Age Books, 1941), p. 71.\n\n2 Edmund Burke, \"Speech on Conciliation with America\" (1775), in _Works_ , vol. 3, p. 49 [Liberty Fund edition, _Selected Works_ , vol. 1, p. 237]. The predominant influence of English ideals on the American Revolution seems even more striking to the Continental European student than to contemporary American historians; cf. particularly Otto Vossler, _Die amerikanischen Revolutionsideale in ihrem Verh\u00e4ltnis zu den europ\u00e4ischen: untersucht an Thomas Jefferson_ [Beiheft 17 to the _Historische Zeitschrift_ ] (Munich: Oldenbourg, 1929); but see also Charles Howard McIlwain, _The American Revolution: A Constitutional Interpretation_ (New York: Macmillan, 1923), esp. pp. 148\u201359 and 181\u201393.\n\n3 Cf., e.g., the reply given by the Massachusetts House of Representatives to Governor Sir Francis Bernard on June 19,1769 (quoted by Andrew Cunningham McLaughlin, _A Constitutional History of the United States_ [New York: D. Appleton-Century Co., 1935], p. 67, from _Speeches of the Governors of Massachusetts, 1765\u20131775; and the Answers of the House of Representatives to the same; with Their Resolutions and Addresses for that Period, and Other Public Papers Relating to the Dispute between This Country and Great Britain which Led to the Independence of the United States_ , Alden Bradford, ed. [Boston: Printed for Russell and Gardner, 1818], p.173): \"no time can better be employed, than in the preservation of the rights derived from the British constitution, and insisting upon points, which, though your Excellency may consider them as nonessential, we esteem its best bulwarks. No treasure can be better expended, than in securing that true old English liberty, which gives a relish to every other enjoyment.\"\n\n4 Cf. Anonymous [Arthur Lee], _The Political Detection; or, the Treachery and Tyranny of Administration Both at Home and Abroad; Displayed in a Series of Letters Signed Junius Americanus_ (London: Printed by J. and W. Oliver, 1770), pp. 73\u201374. \"In principle, this dispute is essentially the same with that which subsisted in the last Century between the people of this Country and Charles the First. . . . The King and the House of Commons may differ in name, but unlimited power makes them in effect the same, except that it is infinitely more to be dreaded in _many_ than in _one_ \"; and Edmund Burke, _An Appeal from the New to the Old Whigs_ (1791), in _Works_ , vol. 6, p. 123 [Liberty Fund edition, _Further Reflections_ , p. 107], where he speaks of the Americans standing at the time of the Revolution \"in the same relation to England, as England did to King James the Second, in 1688.\" On the whole issue see George Herbert Guttridge, _English Whiggism and the American Revolution_ (Berkeley: University of California Press, 1942).\n\n5 Lord Acton, \"The Rise of the Whigs,\" _Lectures on Modern History_ , John Neville Figgis and Reginald Vere Laurence, eds. (London: Macmillan, 1906), p. 217 [Liberty Fund edition, _Essays in the History of Liberty_ , p. 107]. [The actual quotation reads: \"The charter of Rhode Island is worth more than the British Constitution, and Whig statesmen toast General Washington, rejoice that America has resisted, and insist on the acknowledgement of independence.\"\u2014Ed.]\n\n6 See Clinton Rossiter, _Seedtime of the Republic: The Origin of the American Principle of Political Liberty_ (New York: Harcourt, Brace, 1953), p. 360, where he quotes from the _Newport Mercury_ of May 19, 1766, a toast of \"A Son of Liberty in Bristol County, Mass.\": \"Our toast in general is,\u2014 _Magna Charta_ , the _British Constitution_ ,\u2014PITT and Liberty forever!\" [The quotation does not appear in Rossiter's book. However, the original quotation does indeed appear, on p. 3, of the _Newport Mercury_ of May 12 to May 19, 1766.\u2014Ed.]\n\n7 Acton, \"James Bryce's _The American Commonwealth_ ,\" _History of Freedom_ , p. 578 [Liberty Fund edition, _Essays in the History of Liberty_ , p. 398].\n\n8 See Charles Frederic Mullett, _Fundamental Law and the Amer ican Revolution, 1760\u20131776_ (dissertation; New York: Columbia University, 1933), pp. 13\u201378, quoting Randolph Greenf eld Adams, _Political Ideas of the American Revolution: Britannic-American Contributions to the Problem of Imperial Organization, 1765\u20131775_ (3rd ed.; New York: Barnes and Noble, 1958) [esp. pp. 193\u201399.\u2014Ed.], who notes that Jefferson referred to the works from Aristotle to Locke as \"elementary books\" for Americans of the eighteenth century.\n\n9 An excellent brief summary of the influence of these ideas is given in Robert Arthur Humphreys, \"The Rule of Law and the American Revolution,\" _Law Quarterly Review_ , 53 (1937): 80\u201398. See also John Walter Jones, \"Acquired and Guaranteed Rights,\" in _Cambridge Legal Essays Written in Honour of and Presented to Doctor_ [Henry] _Bond, Professor_ [William Warwick] _Buckland, and Professor_ [Courtney Stanhope] _Kenny_ , Sir Percy Henry Winfield and Arnold Duncan McNair, Baron McNair, eds. (Cambridge: W. Heffer and Sons Ltd., 1926), pp. 223\u201342; Charles Frederic Mullett, _Fundamental Law and the American Revolution, 1760\u20131776_ (New York: Columbia University Press, 1933) [Also issued in the form of a Ph.D. thesis, which, apparently, was Hayek's source]; and Alice Mary Baldwin, _The New England Clergy and the American Revolution_ (Durham, NC: Duke University Press, 1928); and cf. Lord Acton's remark, \"Freedom in Christianity,\" _History of Freedom_ , p. 56 [Liberty Fund edition, pp. 49\u201350], that the Americans \"did more; for having subjected all civil authorities to the popular will, they surrounded the popular will with restrictions that the British legislature would not endure.\"\n\n10 The expression \"fixed constitution,\" constantly used by James Otis and Samuel Adams, apparently derives from Emer de Vattel, _Law of Nations; or, the Principles of Natural Law_ (new ed., rev., corr., and enriched with many valuable notes; London: Printed for G. G. and J. Robinson London, 1797) [bk. 1, chap. 3, sec. 34], p. 11 [Liberty Fund edition, p. 95]. [The term \"fixed constitution\" appears neither in French nor in the early English translations of Vattel's work. However, in bk. 1, chap. 3, sec. 34, vol. 1, p. 37, of the 1758 French edition ( _Le Droit des Gens, ou Principes de la Loi Naturelle_ [2 vols.; London (probably Paris), 1758]), Vattel remarks that \"la Constitution de l'\u00e9tat doit \u00eatre stable.\" (\"For the constitution of the state ought to possess stability.\")\u2014Ed.] The best-known statement of the conceptions discussed in the text occurs in the \"Massachusetts Circular Letter of February 11, 1768 [the author of which was Samuel Adams\u2014Ed.] (quoted in William MacDonald, _Documentary Source Book of American History, 1606\u2013 1926_ [3rd ed., rev. and enl.; New York: Macmillan, 1929], pp. 146\u201350), the most significant paragraph of which is as follows: \"The House have humbly represented to the ministry, their own sentiments, that His Majesty's high court of Parliament is the supreme legislative power over the whole empire: that in all free states the constitution is fixed, and as the supreme legislative derives its power and authority from the constitution, it cannot overleap the bounds of it, without destroying its own foundation; that the constitution ascertains and limits both sovereignty and allegiance, and, therefore, his Majesty's American subjects, who acknowledge themselves bound by the ties of allegiance, have an equitable claim to the full enjoyment of the [fundamental rules of the British constitution; that it is an essential, unalterable] right, in nature, engrafted into the British constitution, as a fundamental law, and ever held sacred and irrevocable by the subjects within the realm, that what a man has honestly acquired is absolutely his own, which he may freely give, but cannot be taken from him without his consent; that the American subjects may, therefore, exclusive of any consideration of charter rights, with a decent firmness, adapted to the character of free men and subjects, assert this natural and constitutional right.\"\n\n11 The phrase most commonly used was \"limited constitution,\" into which form the idea of a constitution limiting the powers of government had been contracted. See especially Alexander Hamilton, James Madison, and John Jay, _The Federalist, or the New Constitution_ , Alexander Hamilton, \"The Judiciary Department\" (No. 78), Max Beloff, Baron Beloff, ed. (Oxford: B. Blackwell, 1948), p. 397 [Liberty Fund edition, p. 403], where Hamilton gives the following definition: \"By a limited constitution, I understand one which contains certain specified exceptions to the legislative authority; such, for instance, as that it shall pass no bills of attainder, no _ex post facto_ laws, and the like. . . . Limitations of this kind can be preserved in practice no other way than through the medium of the courts of justice; whose duty it must be to declare all acts contrary to the manifest tenor of the constitution void. Without this, all the reservations of particular rights or privileges would amount to nothing.\" The term \"limited constitution\" as applied to Greece and Rome already appears in David Hume, _History of England: From the Invasion of Julius Caesar to the Revolution of 1688_ (new ed.; 6 vols.; London: A. Millar, 1762), vol. 5, p. 14 [Liberty Fund edition, pp. 18\u201319].\n\n12 Cf. Jones, \"Acquired and Guaranteed Rights,\" pp. 229ff.: \"By the time of the dispute with the Mother Country the colonists were therefore well acquainted with two ideas more or less strange to the general trend of English legal thought\u2014the doctrine of the rights of man, and the possibility or even necessity (for they were now struggling against a Parliament) of limiting legislative power by a written constitution.\"\n\nFor the whole of the following discussion I am indebted mainly to two American authors, Charles Howard McIlwain and Edwin Samuel Corwin, whose chief works may be listed here instead of many detailed references:\n\nCharles Howard McIlwain, _The High Court of Parliament and Its Supremacy: An Historical Essay on the Boundaries Between Legislation and Adjudication in England_ (New Haven: Yale University Press, 1910); _The American Revolution: A Constitutional Interpretation_ (New York: Macmillan, 1923); \"The English Common Law, Barrier against Absolutism,\" _American Historical Review_ 49 (1943): 23\u201331; _Constitutionalism and the Changing World: Collected Papers_ (Cambridge: Cambridge University Press, 1939) [chap. 10, \"The Transfer of the Charter to New England and Its Significance in American Constitutional History,\" (pp. 231\u201343), and chap. 11, \"The Fundamental Law behind the Constitution of the United States,\" (pp. 244\u201358)]; _Constitutionalism, Ancient and Modern: A Constitutional Interpretation_ (rev. ed.; Ithaca, NY: Cornell University Press, 1947), esp. pp. 1\u201322 [Liberty Fund edition, pp. 22\u201338].\n\nEdwin Samuel Corwin, _The Doctrine of Judicial Review: Its Legal and Historical Basis, and Other Essays_ (Princeton, NJ: Princeton University Press, 1914); _The Constitution and What It Means Today_ (1920) (11th ed.; Princeton, NJ: Princeton University Press, 1954); \"The Progress of Constitutional Theory between the Declaration of Independence and the Meeting of the Philadelphia Convention,\" _American Historical Review_ , 30 (1925): 511\u201336; \"Judicial Review in Action,\" _University of Pennsylvania Law Review_ , 74 (1926): 639\u201371; _The \"Higher Law\" Background of American Constitutional Law_ (Ithaca, NY: Great Seal Books, Cornell University Press, 1955), first published in the _Harvard Law Review_ , 42 (1929): 149\u201385 and 365\u2013409, and reprinted in _Selected Essays on Constitutional Law_ , Committee of the Association of American Law Schools and Douglas Blound Maggs, et al. eds. (5 vols. in 4; Chicago: Foundation Press, 1938), vol. 1, pp. 1\u201367 [A Liberty Fund edition is available.]; _Liberty Against Government: The Rise, Flowering, and Decline of a Famous Juridical Concept_ (Baton Rouge: Louisiana State University Press, 1948); and his edition of _The Constitution of the United States: Analysis and Interpretation; Annotation of Cases Decided by the Supreme Court of the United States to June 30, 1952_ (Washington, DC: Government Printing Office, 1953). Several of the articles mentioned and some still to be quoted are conveniently collected in vol. 1 of the _Selected Essays on Constitutional Law_. See also Charles Lund Black, _The People and the Court: Judicial Review in a Democracy_ (New York: Macmillan, 1960).\n\n13 Cf. Humphreys, \"The Rule of Law and the American Revolution, p. 90: \"The very definition of liberty was freedom from arbitrary rule.\"\n\n14 On the derived character of the power of all representative assemblies in the process of constitution-making see particularly McLaughlin, _Constitutional History_ , p. 109.\n\n15 See chap. 4, sec. 8, and chap. 7, sec. 6, above; and cf., on the whole subject, David Hume, _Treatise of Human Nature_ [bk. 3, pt. 2, sec. 7], vol. 2, pp. 300\u2013304.\n\n16 This provision is explicitly recognized in art. 19, par. 1, of the Basic La w ( _Grundgesetz_ ) of the Federal Republic of Germany, noteworthy in that it requires that laws that infringe a basic right not only must be \"general and not solely applicable to an individual case\" but that it name the basic law which is being infringed, citing the specif c article. [The provision reads: \"Insofar as under this Basic Law a basic right may be restricted by or pursuant to a law, the law must apply generally and not solely to an individual case. Furthermore the law must name the basic right, indicating the Article.\"\u2014Ed.]\n\n17 Cf. Zaccaria Giacometti, _Allgemeine Lehren des rechtsstaatlichen Verwaltungsrechts: Allgemeines Verwaltungsrecht des Rechtsstaates_ (Z\u00fcrich: Polygraphischer Verlag, 1960), vol. 1, p. 24, n. 4. See John Lilburne's _Legal Fundamental Liberties_ (partially reprinted in _Puritanism and Liberty_ , Arthur Sutherland Pigott Woodhouse, ed. [Chicago: University of Chicago Press, 1951], p. 344), where, in providing for what we would call a constitutional convention, he explicitly stipulated that \"those persons ought not to exercise any legislative power, but only to draw up the foundations of a just government, and to propound them to the well-affected people in every County, to be agreed to: Which agreement ought to be above law; and therefore the bounds, limits, and extent of the people's legislative deputies in parliament, contained in the Agreement, [ought] to be drawn up into a formal contract to be mutually signed.\" [Woodhouse's source is Lilburne's _Legal Fundamental Liberties of the People of England, Revived, Asserted, and Vindicated; or an Epistle, Written the 8th of June 1649_ (2nd ed., corrected and amended; London: \"Reprinted in the grand year of Hipocritical and abominable Dissimulation,\" 1649), p. 34.\u2014Ed.] Significant in this connection is also the resolution of the Concord, Massachusetts, town meeting of October 21, 1776 (reprinted in _Sources and Documents Illustrating the American Revolution, 1764\u20131788, and the Formulation of the Federal Constitution_ , Samuel Eliot Morison, ed. (Oxford: Clarendon Press, 1923), p. 177), which declared that the legislative was no proper body to form a constitution, \"first, because we conceive that a Constitution in its proper idea intends a system of principles established to secure the subject in the possession of enjoyment of their rights and privileges against any encrouchment of the governing part. Second, because the same body that forms a Constitution have of consequence a power to alter it. Third, because a Constitution alterable by the Supreme Legislative is no security at all to the subject against any encrouchment of the governing part on any, or on all the rights and privileges.\" It was, of course, largely the wish to prevent the ultimate authority from concerning itself with particulars, much more than its technical impracticability, that led the fathers of the American Constitution unanimously to reject direct democracy of the kind that had existed in ancient Greece.\n\n18 David Hume, _Treatise of Human Nature_ [bk. 3, pt. 2, sec. 7], vol. 2, p. 300; cf. also p. 301.\n\n19 Cf. chap. 4, above, especially nn. 5 and 8.\n\n20 On the conception of legitimacy cf. Guglielmo Ferrero, _The Principles of Power: The Great Political Crises of History_ , Theodore R. Jaeckel, trans. (New York: G. P. Putnam's Sons, 1942), esp. pp. 131\u2013276.\n\n21 This is not true of the original concept of sovereignty as introduced by Jean Bodin. Cf. McIlwain, _Constitutionalism and the Changing World_ , chap. 2.\n\n22 As has been stressed by David Hume and a long line of theorists down to Friedrich von Wieser and his fullest elaboration of the idea in _Das Gesetz der Macht_ (Vienna: Julius Springer, 1926).\n\n23 See Roscoe Pound, _The Development of Constitutional Guarantees of Liberty_ (New Haven: Yale University Press, 1957). There exists an important German literature on the origin of the Bills of Rights, of which the following may be mentioned here: Georg Jellinek, _Die Erkl\u00e4rung der Menschen-und B\u00fcrgerrechte. Ein Beitrag zur modernen Verfassungsgeschichte_ , Walter Jellinek, ed. (3rd ed.; Munich and Leipzig: Duncker und Humblot, 1919), pp. iii\u2013xiv, which contains a survey of the discussions since the first publication of the work in 1895; Richard Karl Schmidt, \"Die Vorgeschichte der geschriebenen Verfassung,\" in _Festgabe f\u00fcr Otto Mayer: Zum siebzigsten Geburtstag dargebracht von Freunden, Verehrern und Sch\u00fclern. 29. M\u00e4 rz 1916_ , Otto Mayer, ed. (Leipzig: F, Meiner, 1916), pp. 81\u2013191; Justus Hashagen, \"Zur Entstehungsgeschichte der nordamerikanischen Erkl\u00e4rungen der Menschenrechte,\" _Zeitschrift f\u00fcr die gesamte Staatswissenschaft_ , 78 (1924): 461\u201395; Gustav Adolf Salander, _Vom Werden der Menschenrechte: ein Beitrag zur modernen Verfassungsgeschichte unter Zugrundelegung der virginischen Erkl\u00e4rung der Rechte von 12. Juni 1776_ (Leipzig: T. Weicher, 1926); Otto Vossler, \"Studien zur Erkl\u00e4rung der Menschenrechte,\" _Historische Zeitschrift_ , 142 (1930): 516\u201345.\n\n24 William Clarence Webster, \"A Comparative Study of the State Constitutions of the American Revolution,\" _Annals of the American Academy of Political and Social Science_ , 9 (1897): 415.\n\n25 Webster, \"Comparative Study of State Constitutions,\" p. 418.\n\n26 Constitution of Massachusetts (March 2, 1780): part 1, art. 30. Though this clause does not yet appear in the original draft by John Adams, it is entirely in the spirit of his thinking.\n\n27 For a discussion of the relationship see the works cited in n. 23 above.\n\n28 Cf. Webster, \"Comparative Study of State Constitutions,\" p. 386: \"Each of these instruments declared that no one should be deprived of his liberty except by law or by judgment of his peers; that every one, when prosecuted, should be entitled to a copy of the indictment brought against him, as well as to the right of procuring counsel and evidence; and that no one should be compelled to give evidence against himself. They all carefully guarded the right of trial by jury; guaranteed freedom of the press and free elections; forbade general warrants and standing armies in time of peace, forbade the granting of titles of nobility, hereditary honors and exclusive privileges. All of these instruments, except those of Virginia and Maryland, guaranteed the rights of assembly, petition, and instruction of representatives. All except those of Pennsylvania and Vermont forbade the requirement of excessive bail, the imposition of excessive fines, the infliction of unusual punishments, the suspension of laws by any other authority than the legislature, and taxation without representation.\"\n\n29 Constitution of North Carolina, art. 23. Cf. Constitution of Maryland, \"Declaration of Rights,\" art. 41: \"That monopolies are odious, contrary to the spirit of a free government and the principles of commerce, and ought not to be suffered.\" [Art. 23 of the Declaration of Rights of the Constitution of North Carolina of December 18, 1776, reads: \"That perpetuities and monopolies are contrary to the genius of a free State, and ought not to be allowed.\" The provision of the Declaration of Rights of Maryland's Constitution of November 11, 1776, that Hayek quotes is actually art. 39.\u2014Ed.]\n\n30 See especially the Constitution of Massachusetts (1780), part 1, \"Declaration of Rights,\" art. 30: \"In the government of this Commonwealth, the legislative department shall never exercise the executive and judicial powers, or either of them; the executive shall never exercise the legislative and judicial powers, or either of them; . . . to the end it may be a government of laws, and not of men.\"\n\n31 Constitution of Massachusetts (1780), part 1, art. 24. [The text of the article reads: \"Laws made to punish for actions done before the existence of such laws, and which have not be declared crimes by preceding laws, are unjust, oppressive, and inconsistent with the fundamental principles of a free government.\"\u2014Ed.]\n\n32 The phrase occurs first in the draft of the Virginia Declaration of Rights of May 1776, by George Mason (see Kate Mason Rowland, _The Life of George Mason, 1725\u20131792_ (2 vols.; New York: G. P. Putnam's Sons, 1892), vol. 1, pp. 435ff., and then as sec. 15 of the declaration as adopted. [Some two years following its passage by the Virginia Convention, Mason prepared a copy of the first draft of the Declaration from memory. The provision to which Hayek refers, art. 13, reads almost identically to its final passage on June 12, 1776. The phrase, as it occurs in Mason's first draft of May 20\u201326, 1776, remains essentially unchanged in the Committee's draft of May 27. It reads \"That no free Government, or the Blessings of Liberty can be preserved by any People, but by a firm adherence to Justice, Moderation, Temperance, Frugality, and Virtue and by frequent Recurrence to fundamental Principles.\"\u2014Ed.]\n\nSee also the Constitution of New Hampshire, art. 16 [art. 16 of the New Hampshire Constitution of June 2, 1784, reads: \"A frequent recurrence to the fundamental principles of the constitution, and a constant adherence to justice, moderation, temperance, industry, frugality, and all the social virtues, are indispensably necessary to preserve the blessings of liberty and good government.\"\u2014Ed.]; and that of Vermont, art. 16 [chap. 1, art. 16 of the Vermont Constitution of July 8, 1777, reads: \"That frequent recurrence to fundamental principles, and a firm adherence to justice, moderation, temperance, industry and frugality, are absolutely necessary to preserve the blessings of liberty, and keep government free.\"\u2014Ed.].\n\n(Since there seems to exist no collection of the state constitutions in force in 1787, I am using _The Constitutions of All the United States, According to the Latest Amendments: To Which are Prefixed, the Declaration of Independence and the Federal Constitution_ (Lexington, KY: Printed and sold by Thomas T. Skillman, 1817), which does not in all instances give the dates of the texts printed. In consequence, some of the references given in this and in the last few notes may refer to amendments later than the federal Constitution.) [In fact, all of Hayek's references to the various provisions of these early state constitutions were enacted before passage of the United States Constitution.\u2014Ed.] On the origin of this clause see Gerald Stourzh's forthcoming book, _The Pursuit of Greatness_. [Professor Stourzh did not publish a work with this title. However, he was kind enough to explain Hayek's footnote. He writes: \" _The Pursuit of Greatness_ was the title of the first draft of my book which became _Alexander Hamilton and the Idea of Republican Government_ (finally published in 1970). 'The Pursuit of Greatness' remains as the title to the last chapter, chap. 5, of the Hamilton book.\" The passage Hayek referred to, can be found in chap. 1, on pp. 34 to 37, of _Alexander Hamilton and the Idea of Republican Government_.\u2014Ed.] See also David Hume, \"Idea of a Perfect Commonwealth,\" _Essays_ [Essay 16], vol. 1, p. 482 [Liberty Fund edition, p. 516]: \"A government, says Machiavel, must often be brought back to its original principles.\"\n\n33 Webster, \"A Comparative Study of State Constitutions,\" p. 398.\n\n34 Cf. James Madison at the end of _The Federalist_ [\"These Departments Should Not Be So Far Separated as to Have No Constitutional Control Over Each Other,\" (No. 48)], p. 256 [Liberty Fund edition, p. 260]: \"A mere demarcation on parchment of the constitutional limits of the several departments, is not a sufficient guard against those encroachments which lead to a tyrannical concentration of all the powers of government in the same hands.\"\n\n35 John Jay is quoted by Michael Joseph Oakeshott (\"Rationalism in Politics,\" _Cambridge Journal_ , 1 [1947]: 151) as saying in 1777: \"The Americans are the first people whom Heaven has favoured with an opportunity of deliberating upon, and choosing the forms of government under which they should live. All other constitutions have derived their existence from violence or accidental circumstances, and are therefore probably more distant from their perfection.\" [The quotation is taken from _The Correspondence and Public Papers of John Jay_ , Henry Phelps Johnston, ed. (4 vols.; New York: P. Putnam's Sons, 1890), vol. 4, p. 365.\u2014Ed.] But compare John Dickinson's emphatic statement in the Philadelphia Convention (of August 13, 1787, quoted in _The Records of the Federal Convention of 1787_ , Max Farrand, ed. [New Haven: Yale University Press, 1911], vol. 2, p. 278): \"Experience must be our only guide. Reason may mislead us. It was not Reason that discovered the singular and admirable mechanism of the English Constitution. It was not Reason that discovered . . . the odd and in the eye of those who are governed by reason, the absurd mode of trial by Jury. Accidents probably produced these discoveries, and experience has given a sanction to them. This is then our guide.\"\n\n36 James Madison in the Philadelphia Convention mentioned as the chief objects of national government, \"the necessity of providing more effectively for the security of private rights and the steady dispensation of justice. Interference with these were evils which had more, perhaps, than anything else produced this convention\" ( June 6, 1787, in _Records of the Federal Convention_ , Max Farrand, ed., vol. 1, p. 134). Cf. also the famous passage quoted by Madison in the _Federalist_ [\"These Departments Should Not Be So Far Separated as to Have No Constitutional Control Over Each Other,\" (No. 48), pp. 254\u201355, from Thomas Jefferson's _Notes on the State of Virginia_ in _Writings_ , Merrill Daniel Peterson, ed., Library of America (New York: Literary Classics of the United States, 1984), pp. 245\u201346]: \"All the powers of government, legislative, executive, and judiciary, result to the legislative body. The concentrating these in the same hands, is precisely the definition of despotic government. It will be no alleviation that these powers will be exercised by a plurality of hands, and not by a single one. One hundred and seventy-three despots would surely be as oppressive as one. Let those who doubt it, turn their eyes on the republic of Venice. As little will it avail us, that they are chosen by ourselves. An _elective despotism_ was not the government we fought for; but one which should not only be founded on free principles, but in which the powers of government should be so divided and balanced among several bodies of magistracy, as that no one could transcend their legal limits, without being effectually checked and restrained by the others. . . . [The branches other than the legislature] have accordingly, in many instances _decided rights_ , which should have been left to _judiciary controversy_ , and _the direction of the executive, during the whole time of their session, is becoming habitual and familiar_.\"\u2014R. A. Humphreys' conclusion (\"The Rule of Law and the American Revolution,\" p. 98) is true, therefore, even of Jefferson, the idol of the latter doctrinaire democrats: \"Such was the republic which the authors of the Federal Constitution tried to build. They were concerned not to make America safe for democracy, but to make democracy safe for America. From Lord Chief Justice Coke to the Supreme Court of the United States is a long way, but a clear one. The controlling rule of law which the seventeenth century set above King or Parliament, which the Puritans exalted in matter both civil and ecclesiastical, which the philosophers saw as the governing principle of the universe, which the colonists invoked against the absolutism of Parliament, this 'was now made the essential principle of federation.'\"\n\n37 Edward Samuel Corwin, \"The Progress of Constitutional Theory between the Declaration of Independence and the Meeting of the Philadelphia Convention,\" _American Historical Review_ , 30 (1925): 536; the passage continues: \"It remained for the Constitutional Convention, however, while it accepted Madison's main idea, to apply it through the agency of judicial review. Nor can it be doubted that this determination was assisted by a growing comprehension in the Convention of the _doctrine_ of judicial review.\"\n\n38 Lord Acton, \"Sir Erskine May's _Democracy in Europe_ ,\" _History of Freedom_ , p. 98 [Liberty Fund edition, _Essays in the History of Liberty_ , p. 84].\n\n39 Cf. my essay on \"The Economic Conditions of Inter-State Federalism,\" _New Commonwealth Quarterly_ , 5 (1939): 131\u201349, reprinted in Hayek, _Individualism and Economic Order_ (Chicago: University of Chicago Press, 1948), pp. 255\u201372.\n\n40 \"Certain General and Miscellaneous Objections to the Constitution Considered and Answered\" (No. 84), _The Federalist_ , pp. 439\u201340 [Liberty Fund edition, pp. 445\u201346].\n\n41 An even clearer statement of this view than in the passage by Hamilton quoted in the text is that by James Wilson in the debate on the Constitution in the Pennsylvania convention ( _The Debates in the Several State Conventions on the Adoption of the Federal Constitution, as Recommended by the General Convention at Philadelphia in 1787_ , Jonathan Elliot, ed. [5 vols.; 2nd ed., with considerable additions; Philadelphia: J. B. Lippincott and Co., 1836\u201359], vol. 2, p. 436): He described a bill of rights as \"highly imprudent\" because \"in all societies, there are many powers and rights which cannot be particularly enumerated. A bill of rights annexed to a constitution is an _enumeration of the powers_ reserved. If we attempt an enumeration, everything that is not enumerated is presumed to be given.\" James Madison, however, seems from the beginning to have held the view which ultimately prevailed. In an important letter to Jefferson, dated October 27, 1788 (quoted here from _The Complete Madison: His Basic Writings_ , Saul Kussiel Padover, ed. [New York: Harper, 1953], p. 253), too long to reproduce here in full, he wrote: \"My own opinion has always been in favor of a bill of rights; provided it be so framed as not to imply powers not meant to be included in the enumeration. . . . The invasion of private rights is _chiefly_ to be apprehended, not from acts of Government contrary to the sense of its constituents, but from acts in which the Government is the mere instrument of the major number of the Constituents. This is a truth of great importance but not yet sufficiently attended to. . . . What use then it may be asked can a bill of rights serve in popular Governments? . . . The political truths declared in that solemn manner acquire by degrees the character of fundamental maxims of free Government, and as they become incorporated with the national sentiment, counteract the impulses of interest and passion. . . .\"\n\n42 John Marshall in _Fletcher v. Peck_ , 10 _U.S_. (6, _Cranch_ ) 87 at 139 (1810).\n\n43 Joseph Story, _Commentaries on the Constitution of the United States, With a Preliminary Review of the Constitutional History of the Colonies and States, Before the Adoption of the Constitution_ (Boston: Hilliard, Gray, and Co., 1833), vol. 3, pp. 718, 720.\n\n44 Cf. Leslie Wallace Dunbar, \"James Madison and the Ninth Amendment,\" _Virginia Law Review_ , 42 (1956): 627\u201345. It is significant that even the leading authority on the American Constitution misquotes in a well-known essay (Edward S. Corwin, \"The 'Higher Law' Background of American Constitutional Law\" [1955 reprint], p. 5 [Liberty Fund edition, p. 4]) the text of the Ninth Amendment and reprints the misquotation twenty-five years later, apparently because nobody had noticed the substitution of a phrase of six words for one of eleven in the authentic text!\n\n45 This admiration was widely shared by nineteenth-century liberals such as W. E. Gladstone, who once described the American Constitution as \"the most wonderful work ever struck off at a given time by the brain and purpose of men.\" [The quotation reads: \"As the British Constitution is the most subtle organism which has proceeded from progressive history, so the American Constitution is the most wonderful work ever struck off at a given time by the brain and purpose of man.\" William Ewart Gladstone, \"Kin Beyond Sea,\" _North American Review_ , 264 (September\u2013 October 1878): 185.\u2014Ed.]\n\n46 McIlwain, _Constitutionalism and the Changing World_ , p. 278 [The quotation in fact reads: \"Judicial review, instead of being an American invention, is really as old as constitutionalism itself, and without its constitutionalism could never have been maintained.\"\u2014Ed.]; cf. Edward S. Corwin, \"Basic Doctrine of American Constitutional Law,\" in the _Michigan Law Review_ , 12 (February 1914): 252; reprinted in _Selected Essays on Constitutional Law_ , vol. 1, p. 105: \"The history of judicial review is, in other words, the history of constitutional limitations.\" See also Gottfried Dietze, \"America and Europe: Decline and Emergence of Judicial Review,\" _Virginia Law Review_ , 44 (1958): 1233\u201372.\n\n47 All the arguments supporting the denial have recently been marshaled in detail in William Winslow Crosskey, _Politics and the Constitution in the History of the United States_ (2 vols.; Chicago: University of Chicago Press, 1953).\n\n48 See mainly _The Federalist_ , Alexander Hamilton, \"The Judiciary Department\" (No. 78), p. 399 [Liberty Fund edition, p. 405]: \"Whenever a particular statute contravenes the constitution it will be the duty of the judicial tribunals to adhere to the latter, and disregard the former\"; also James Madison, _Debates and Proceedings in the Congress_ [ _Annals of Congress_ ], 1st Congress, 1st Session, Senate: June 8, 1789, vol. 1, p. 457, where he declares that the courts would \"consider themselves in a peculiar manner the guardians of those rights; they will be an impenetrable bulwark against every assumption of power in the Legislative or Executive: they will be naturally led to resist every encroachment upon rights expressly stipulated for in the Constitution by the declaration of rights,\" and his later statement in a letter to George Thompson, dated June 30, 1825 (quoted in _The Complete Madison_ , p. 344): \"No doctrine can be sound that releases a Legislature from the controul of a constitution. The latter is as much a law to the former, as the acts of the former are to individuals and although always liable to be altered by the people who formed it, is not alterable by any other authority; certainly not by those chosen by the people to carry it into effect. This is so vital a principle, and has been so justly the pride of our popular Government, that a denial of it cannot possibly last long or spread far.\" Further, Senator Mason's and Gouverneur Morris's statements in the congressional discussion of the repeal of the judiciary act of 1801 quoted in McLaughlin, _Constitutional History of the United States_ , p. 291, and James Wilson's Lectures delivered in 1792 to students of the University of Pennsylvania ( _The Works of James Wilson, Associate Justice of the Supreme Court of the United States and Professor Law in the College of Philadelphia: Being His Public Discourses Upon Jurisprudence and the Political Science, Including Lectures as Professor of Law, 1790\u201392_ , James DeWitt Andrews, ed. [2 vols.; Chicago: Callaghan and Co., 1896], vol. 1, p.417) in which he presents judicial review as \"the necessary result of the distribution of power, made, by the constitution, between the legislative and the judicial departments.\"\n\n49 Even the most critical recent survey by Crosskey, _Politics and the Constitution in the History of the United States_ , vol. 2, p. 943, sums up the situation by saying that \"some evidence has been found, that the basic notion of judicial review had some acceptance in America, in the Colonial period.\"\n\n50 _Marbury v. Madison_ , 5 _U.S._ (1 _Cranch_ ) 137 (1803); only a few passages from this famous decision can be quoted here: \"The Government of the United States has been emphatically termed a government of laws, and not of men. It will certainly cease to deserve this high appellation if the laws furnish no remedy for the violation of a vested legal right. . . . The question, whether an Act, repugnant to the constitution can become the law of the land, is a question deeply interesting to the United States, but, happily, not of an intricacy proportioned to its interest. It seems only necessary to recognize certain principles supposed to have been long and well established, to decide it. . . . The powers of the legislature are defined and limited; and that those limits may not be mistaken or forgotten, the constitution is written. To what purpose are powers limited, and to what purpose is that limitation committed to writing, if these limits may, at any time, be passed by those intended to be restrained? The distinction between a government with limited and unlimited powers is abolished if those limits do not confine the persons on whom they are imposed and if Acts prohibited and Acts allowed are of equal obligation. . . . It is emphatically the province and duty of the judicial department to say what the law is. Those who apply the rule to particular cases must of necessity expound and interpret that rule. If two laws conflict with each other, the courts must decide on the operation of each.\"\n\n51 Cf. Robert Houghwout Jackson, _The Struggle for Judicial Supremacy: A Study of a Crisis in American Power Politics_ (New York: Alfred A. Knopf, 1941), pp. 36\u201337, where he suggests that \"this may have been the result not merely of judicial abstinence but of the fact that there was little Congressional legislation at least that would offend conservative minds: _Laissez faire_ , to some degree, was the philosophy of the legislature, as it was of the Court. It is partly this fact which obscured the potentialities of _Marbury v. Madison_ and even more of _Dred Scott_ \" [ _Dred Scott v. Sanford_ , 19 _Howard_ 393 (March 1857)].\n\n52 On the great influence of legal thought on American politics during the period see particularly Tocqueville, _Democracy in America_ [bk. 1, sec. 2, chap. 16], vol. 1, pp. 272\u201380 [French edition, vol. 2, pp. 303\u201310]. Few facts are more indicative of the change of atmosphere than the decline of the reputation of men like Daniel Webster, whose effective statements of constitutional theory were once considered classic but are now largely forgotten. See particularly his arguments in the Dartmouth Case [ _The Trustees of Dartmouth College v. Woodward_ , 17 _U.S_. 518; 4 _Wheat_ ; 4 _L. Ed._ 629 (February 1819)] and in _Luther v. Borden_ [ _Rachel Luther et al. v. Luther M. Borden et al_., 48 _U.S_. 1; 12 _L. Ed._ 581; 7 _How._ 1 ( January 1849)] in _The Writings and Speeches of Daniel Webster_ , Edward Everett, ed. (18 vols.; National ed.; Boston: Little, Brown, 1903), vols. 10 and 11, esp. vol. 10, p. 219: \"By the law of the land is most clearly intended the general law; a law which hears before it condemns; which proceeds upon inquiry, and renders judgment only after trial. The meaning is, that every citizen shall hold his life, liberty, property, and immunities under the protection of the general rules which govern society. Everything which may pass under the form of an enactment is not therefore to be considered the law of the land.\" Also Webster, vol. 10, p. 232, where he stresses that the people \"have most wisely, chosen to take the risk of occasional inconvenience from the want of power, in order that there might be a settled limit to its exercise, and a permanent security against its abuse.\" See also Webster, vol. 11, p. 224: \"I have said that it is one principle of the American system, that the people limit their governments, National and State. They do so, but it is another principle, equally true and certain, and, according to my judgment of things, equally important, that the people often _limit themselves_. They set bounds to their own power. They have chosen to secure the institutions which they establish against the sudden impulses of mere majorities. All our institutions teem with instances of this. It was their great conservative principle, in constituting forms of government, that they should secure what they had established against hasty changes by simple majorities.\"\n\n53 _Ex parte Bollman_ and _Ex parte Swartwout_ 8 _U.S. Reports_ 75 (4 _Cranch_ 750); 2 _L. Ed._ 554 (February 1807) at 127.\n\n54 See Corwin, \"The Basic Doctrine of American Constitutional Law,\" p. 258 [reprinted in _Selected Essays on Constitutional Law_ , vol. 1, p. 111], as quoted in n. 46 above.\n\n55 See Corwin, \"The Basic Doctrine of American Constitutional Law,\" p. 259 [reprinted in _Selected Essays on Constitutional Law_ , p. 112]. [The statement is not Corwin's but forms part of the decision in a case heard before the Tennessee Supreme Court, _Vanzant v. Waddell_ (2 Yerg. [10 Tenn.] 259 [1829]) that is quoted by Corwin. The Court there held that the legislature was created to enact \"general public law equally binding upon every member of the community . . . under similar circumstances.\" (Ellipses part of Corwin's quote.)\u2014Ed.]\n\n56 See the constitutions of _Arkansas_ (1874), art. 5, sec. 25; _Georgia_ (1877), art. 1, sec. 4, par. 1 [Identical language appears in the 1945 Constitution (art. 1, sec. 4, par. 1) and the 1976 Constitution (art. 1, sec. 4, par. 7), while there are similar provisions in the 1968 Constitution (art. 1, sec. 26) and in art. 3, sec. 6, par. 4 of the Georgia Constitution of 1983.\u2014Ed.]; _Kansas_ ( July, 1859), art. 2, sec. 17; _Michigan_ (1863), art. 4, sec. 29 [A similar provision appears in the Michigan Constitution of 1908 (art. 5, sec. 30).\u2014Ed.]; and _Ohio_ (1851), sec. 2, art. 26. For a discussion of this feature see Hermann von Mangoldt, _Rechtsstaatsgedanke und Regierungsformen in den Vereinigten Staaten von Amerika: Die geistigen Grundlagen des amerikanischen Verfassungsrechts_ (Essen: Essener Verlagsanstalt, 1938), pp. 315\u201318, esp. 316.\n\n57 _Calder v. Bull_ , 3 U.S. (3 _Dall_ ) 386, 388 (1798); cf. Corwin, \"The Basic Doctrine of American Constitutional Law,\" pp. 248\u201358 [reprinted in _Selected Essays on Constitutional Law_ , pp. 102\u201311].\n\n58 Thomas McIntyre Cooley, _A Treatise on Constitutional Limitations Which Rest Upon the Legislative Power of the States of the American Union_ (1st ed.; Boston: Little, Brown, and Co., 1868), p. 171. [The phrase \"but not expressed in words\" italicized by Hayek.\u2014Ed.]\n\n59 Cf. Jackson, _The Supreme Court in the American System of Government_ , p. 74.\n\n60 The \"Slaughter House Cases,\" 83 _U.S._ (16 _Wallace_ ) 36 (1873). Cf. Edward S. Corwin, _Liberty against Government_ , p. 122. [In the Slaughterhouse Cases, the Court held that the original intent of the Fourteenth Amendment was to guarantee the freedom of black slaves that had been freed by virtue of the Thirteenth Amendment. While the Fourteenth Amendment was not to be construed as referring solely to black slaves the Amendment's scope did not cover the issues raised in this case, that is, that the slaughterhouse operators who had been barred from engaging in their trade were deprived of their property without due process of law. The Court ruled that it was necessary to draw a distinction between United States citizenship and the citizenship of a state and that the Amendment did not seek to deprive the state of its legal jurisdiction over the civil rights of its citizens. It was therefore held that the restraints placed by the state of Louisiana on slaughterhouse operators did not deprive them of their property without \"due process\" nor of the \"equal protection of the laws.\"\u2014Ed.]\n\n61 In E. S. Corwin's standard annotated edition of the Constitution of the United States (1953), 215 out of 1,237 pages are devoted to the jurisdiction on the Fourteenth Amendment as against 136 pages devoted to the \"commerce clause\"! [In the 2002 edition of the annotated edition, the difference in the number of pages devoted to commentary on these two sections increased; analysis of the commerce clause occupies 113 pages while 379 pages are devoted to the Fourteenth Amendment.\u2014Ed.]\n\n62 Cf. the comment in Ernest Freund, _Standards of American Legislation: An Estimate of Restrictive and Constructive Factors_ (Chicago: University of Chicago Press, 1917), p. 211: \"The only criterion that is suggested is that of reasonableness. From the point of view of legal science it would be difficult to conceive of anything more unsatisfactory.\"\n\n63 Walter Bagehot, \"The Metaphysical Basis of Toleration,\" (1874) in _The Works and Life of Walter Bagehot_ , Mrs. Russell Barrington, ed. (London: Longman, Green, and Co., 1915), vol. 6, pp. 232\u201333.\n\n64 Quoted by Dorothy Thompson, _Essentials of Democracy: The American Scene_ [the first of three \"Town Hall Pamphlets\" published under the title \"Essentials of Democracy,\" on the basis of lectures delivered by Dorothy Thompson at Town Hall] (New York: The Town Hall, 1938), p. 21. [Dorothy Thompson, a prominent political commentator and columnist for the _New York Herald Tribune_ , is here paraphrasing Roosevelt. In actuality, in the course of his Annual Message to Congress of January 3, 1936, Roosevelt made the following remarks: \"Our resplendent economic autocracy does not want to return to that individualism of which they prate, even though the advantages under that system went to the ruthless and the strong. They realize that in thirty-four months we have built up new instruments of public power. In the hands of a people's Government this power is wholesome and proper. But in the hands of political puppets of an economic autocracy such power would provide shackles for the liberties of the people.\" (reported in the _Washington Post_ , January 4, 1936, p. 4, cols. 5\u20136).\u2014Ed.]\n\n65 _Reorganization of the Federal Judiciary: Adverse Report from the [Senate] Committee on the Judiciary Submitted to Accompany S. 1392_ (75th Congress, 1st Session; Senate Report 711, June 7, 1937), pp. 8, 15, 19, and 20. Cf. also p. 19: \"The courts are not perfect, nor are the judges. The Congress is not perfect, nor are Senators and Representatives. The Executive is not perfect. These branches of government and the office under them are filled by human beings who for the most part strive to live up to the dignity and idealism of a system that was designed to achieve the greatest possible measure of justice and freedom for all the people. We shall destroy the system when we reduce it to the imperfect standards of the men who operate it. We shall strengthen it and ourselves, we shall make justice and liberty for all men more certain when, by patience and selfrestraint, we maintain it on the high plane on which it was conceived.\n\n\"Inconvenience and even delay in the enactment of legislation is not a heavy price to pay for our system. Constitutional democracy moves forward with certainty rather than with speed. The safety and the permanence of the progressive march of our civilization are far more important to us and to those who are to come after us than the enactment now of any particular law. The Constitution of the United States provides ample opportunity for the expression of the popular will to bring about such reforms and changes as the people may deem essential to their present and future welfare. It is the people's charter of the powers granted those who govern them.\"\n\n66 I shall not easily forget how this feeling was expressed by the taxi driver in Philadelphia in whose cab we heard the radio announcement of President Roosevelt's sudden death. I believe he spoke for the great majority of the people when he concluded a deeply felt eulogy of the President with the words: \"But he ought not to have tampered with the Supreme Court, he should never have done _that_!\" The shock had evidently gone very deep.\n\n67 C. H. McIlwain, _Constitutionalism and the Changing World_ (New York: Macmillan, 1939), p. 286; cf. also Franz Leopold Neumann, _The Democratic and the Authoritarian State: Essays in Political and Legal Theory_ (Glencoe, IL: The Free Press, 1957), p. 31.\n\n68 See Max Lerner, \"Minority Rule and the Constitutional Tradition,\" in _The Constitution Reconsidered_ , Conyers Read, ed. (New York: Columbia University Press, 1938), pp. 199ff.\n\n## THIRTEEN\n\nLIBERALISM AND ADMINISTRATION: THE _RECHTSSTAAT_\n\nHow can there be a definite limit to the supreme power if an indefinite general happiness, left to its judgment, is to be its aim? Are the princes to be the fathers of the people, however great be the danger that they will also become its despots?\n\n\u2014G. H. von Berg\n\n1. In most countries of the European Continent, two hundred years of absolute government had, by the middle of the eighteenth century, destroyed the traditions of liberty. Though some of the earlier conceptions had been handed on and developed by the theorists of the law of nature, the main impetus for a revival came from across the Channel. But, as the new movement grew, it encountered a situation different from that which existed in America at the time or which had existed in England a hundred years earlier.\n\nThis new factor was the powerful centralized administrative machinery which absolutism had built, a body of professional administrators who had become the main rulers of the people. This bureaucracy concerned itself much more with the welfare and the needs of the people than the limited government of the Anglo-Saxon world either could or was expected to do. Thus, at an early stage of their movement, the Continental liberals had to face problems which in England and in the United States appeared only much later and so gradually that there was little occasion for systematic discussion.\n\nThe great aim of the movement against arbitrary power was, from the beginning, the establishment of the rule of law. Not only those interpreters of English institutions\u2014chief of whom was Montesquieu\u2014represented a government of law as the essence of liberty; even Rousseau, who became the main source of a different and opposed tradition, felt that \"the great problem in politics, that I compare to squaring the circle in geometry, [is] to find a form of government which places the law above men.\"1 His ambivalent concept  of the \"general will\" also led to important elaborations on the conception of the rule of law. It was to be general not only in the sense of being the will of all but also in intent: \"When I say that the object of laws is always general, I mean that the law always considers the subject in the round and actions in the abstract and never any individual man or one particular action. For instance, a law may provide that there shall be privileges, but it must not name the persons who are to enjoy them: the law may create several classes of citizens and even designate the qualifications which will give entry into each class; but it must not nominate for admission such and such persons; it may establish a royal government with a hereditary succession, but it must not select the king or nominate a royal family; in a word, anything that relates to a named individual is outside the scope of legislative authority.\"2\n\n2. The revolution of 1789 was therefore universally welcomed, to quote the memorable phrase of the historian Michelet, as \"l'av\u00e8nement de la loi.\"3 As  A. V. Dicey wrote later: \"The Bastille was the outward and visible sign of lawless power. Its fall was felt, and felt truly, to herald in for the rest of Europe that rule of law which already existed in England.\"4 The celebrated \"D\u00e9claration des droits de l'homme et du citoyen,\" with its guaranties of individual rights and the assertion of the principle of the separation of powers, which it represented as an essential part of any constitution, aimed at the establishment of a strict reign of law.5 And the early efforts at constitution-making are full of painstaking and often even pedantic endeavors to spell out the basic conceptions of a government of laws.6\n\nHowever much the Revolution was originally inspired by the ideal of the rule of law,7 it is doubtful whether it really enhanced its progress. The fact that the ideal of popular sovereignty gained a victory at the same time as the ideal of the rule of law made the latter soon recede into the background. Other aspirations rapidly emerged which were difficult to reconcile with it.8 Perhaps  no violent revolution is likely to increase the respect for the law. A Lafayette might appeal to the \"reign of law\" against the \"reign of the clubs,\" but he would do so in vain. The general effect of \"the revolutionary spirit\" is probably best described in the words which the chief author of the French civil code used when submitting it to the legislature: \"This ardent resolve to sacrifice violently all rights to a revolutionary aim and no longer to admit any other consideration than an indefinable and changeable notion of what the state interest demands.\"9\n\nThe decisive factor which made the efforts of the Revolution toward the enhancement of individual liberty so abortive was that it created the belief that, since at last all power had been placed in the hands of the people, all safeguards against the abuse of this power had become unnecessary. It was thought that the arrival of democracy would automatically prevent the arbitrary  use of power. The elected representatives of the people, however, soon proved much more anxious that the executive organs should fully serve their aims than that the individual should be protected against the power of the executive. Though in many respects the French Revolution was inspired by the American, it never achieved what had been the chief result of the other\u2014 a constitution which puts limits to the powers of legislation.10 Moreover, from the beginning of the Revolution, the basic principles of equality before the law were threatened by the new demands of the precursors of modern socialism, who demanded an _\u00e9galit\u00e9 de fait_ instead of a mere _\u00e9galit\u00e9 de droit_.\n\n3. The one thing which the Revolution did not touch and which, as Tocqueville has so well shown,11 survived all the vicissitudes of the following decades was the power of the administrative authorities. Indeed, the extreme interpretation of the principle of the separation of powers that had gained acceptance in France served to strengthen the powers of the administration. It was used largely to protect the administrative authorities against any interference by the courts and thus to strengthen, rather than to limit, the power of the state.\n\nThe Napoleonic regime which followed the Revolution was necessarily more concerned with increasing the efficiency and power of the administrative machine than with securing the liberty of the individual. Against this tendency, liberty under the law, which once more became the watchword during  the short interval of the July Monarchy, could make little headway.12 The republic found little occasion to make any systematic attempts to protect the individual against the arbitrary power of the executive. It was, in fact, largely the situation which prevailed in France during the greater part of the nineteenth century that gave \"administrative law\" the bad name it has had so long in the Anglo-Saxon world.\n\nIt is true that there gradually evolved within the administrative machine a new power which increasingly assumed the function of limiting the discretionary powers of administrative agencies. The Conseil d'\u00c9tat, originally created merely to assure that the intentions of the legislature were carried out faithfully, has in modern times developed in a way which, as Anglo-Saxon students have recently discovered with some surprise,13 gives the citizen more protection against discretionary action by administrative authorities than is available in contemporary England. These French developments have attracted much more attention than the similar evolution that took place in Germany at the same time. Here the continuance of monarchic institutions never allowed a na\u00efve confidence in the automatic efficacy of democratic control to cloud the issue. Systematic discussion of the problems therefore produced an elaborate theory of the control of administration which, though its practical  political influence was of short duration, profoundly affected Continental legal thought.14 And as it was against this German form of the rule of law that the new legal theories were mainly developed which have since conquered the world and everywhere undermined the rule of law, it is important to know a little more about it.\n\n4. In view of the reputation which Prussia acquired in the nineteenth century, it may surprise the reader to learn that the beginning of the German movement for a government of law is to be found in that country.15 In some  respects, however, the rule of enlightened despotism of the eighteenth century had been surprisingly modern there\u2014indeed, one might say almost liberal, so far as legal and administrative principles were concerned. It was by no means a meaningless assertion when Frederick II described himself as the first servant of the state.16 The tradition, deriving mainly from the great theorists of the law of nature and partly from Western sources, during the later part of the eighteenth century was greatly strengthened by the influence of the moral and legal theories of the philosopher Immanuel Kant.\n\nGerman writers usually place Kant's theories at the beginning of their accounts of the movement toward the _Rechtsstaat_. Though this probably exaggerates the originality of his legal philosophy,17 he undoubtedly gave those ideas the form in which they exerted the greatest influence in Germany. His chief contribution is indeed a general theory of morals which made the principle of the rule of law appear as a special application of a more general principle. His celebrated \"categorical imperative,\" the rule that man should always \"act only on that maxim whereby thou canst at the same time will that  it should become universal law,\"18 is in fact an extension to the general field of ethics of the basic idea underlying the rule of law. It provides, as does the rule of law, merely one criterion to which particular rules must conform in order to be just.19 But in emphasizing the necessity of the general and abstract character of all rules if such rules are to guide a free individual, the conception proved of the greatest importance in preparing the ground for the legal developments.\n\nThis is not the place for a full treatment of the influence of Kantian philosophy on constitutional developments.20 We shall mention here merely the  extraordinary essay of the young Wilhelm von Humboldt on _The Sphere and Duties of Government_ ,21 which, in expounding the Kantian view, not only gave currency to the much used phrase \"the certainty of legal freedom\" but in some respects also became the prototype for an extreme position; that is, he not merely limited all the coercive action of the state to the execution of previously announced general laws but represented the enforcement of the law as the _only_ legitimate function of the state. This is not necessarily implied in the conception of individual liberty, which leaves open the question of what other non-coercive functions the state may undertake. It was due mainly to Humboldt's influence that these different conceptions were frequently confused by the later advocates of the _Rechtsstaat_.\n\n5. Of the legal developments in the Prussia of the eighteenth century, two became so important later that we must look at them more closely. One is the effective initiation by Frederick II, through his civil code of 1751,22 of that movement for the codification of all the laws which spread rapidly and achieved its best-known results in the Napoleonic codes of 1800\u20131810. This whole movement must be regarded as one of the most important aspects of the endeavor on the Continent to establish the rule of law, for it determined to a large extent both its general character and the general direction of the advances that were made, at least in theory, beyond the stage reached in the commonlaw countries.\n\nThe possession of even the most perfectly drawnup legal code does not, of course, insure that certainty which the rule of law demands; and it therefore provides no substitute for a deeply rooted tradition. This, however, should  not obscure the fact that there seems to exist at least a prima facie conflict between the ideal of the rule of law and a system of case law. The extent to which under an established system of case law the judge actually creates law may not be greater than under a system of codified law. But the explicit recognition that jurisdiction as well as legislation is the source of law, though in accord with the evolutionary theory underlying the British tradition, tends to obscure the distinction between the creation and the application of law. And it is a question whether the much praised flexibility of the common law, which has been favorable to the evolution of the rule of law so long as that was the accepted political ideal, may not also mean less resistance to the tendencies undermining it, once that vigilance which is needed to keep liberty alive disappears.\n\nAt least there can be no doubt that the efforts at codification led to the explicit formulation of some of the general principles underlying the rule of law. The most important event of this kind was the formal recognition of the principle \"nullum crimen, nulla poena sine lege,\"23 which was first incorporated into the Austrian penal code of 178724 and, after its inclusion in the French Declaration of the Rights of Man, was embodied in the majority of Continental codes.\n\nThe most distinctive contribution of eighteenth-century Prussia to the realization of the rule of law lay, however, in the field of the control of public administration. While in France the literal application of the ideal of the separation of powers had led to an exemption of administrative action from judicial control, the Prussian development proceeded in the opposite direction. The guiding ideal which profoundly affected the liberal movement of the nineteenth century was that all exercise of administrative power over the person or property of the citizen should be made subject to judicial review. The most far-reaching experiment in this direction\u2014a law of 1797 which applied  only to the new eastern provinces of Prussia but was conceived as a model to be generally followed\u2014went so far as to subject all disputes between the administrative authorities and private citizens to the jurisdiction of the ordinary courts.25 This was to provide one of the chief prototypes in the discussion on the _Rechtsstaat_ during the next eighty years.\n\n6. It was on this basis that in the early part of the nineteenth century the theoretical conception of the state of law, the _Rechtsstaat_ , was systematically developed26 and became, together with the ideal of constitutionalism, the  main goal of the new liberal movement.27 Whether it was mainly because, by the time the German movement had started, the American precedent was  already better known and understood than it had been at the time of the French Revolution, or because the German development proceeded within the framework of a constitutional monarchy rather than that of a republic and was therefore less subject to the illusion that the problems would be automatically solved by the advent of democracy, it was here that the limitation of all government by a constitution, and particularly the limitation of all administrative activity by law enforceable by courts, became the central aim of the liberal movement.\n\nMuch of the argument of the German theorists of the time was indeed explicitly directed against \"administrative jurisdiction\" in the sense in which this term was still accepted in France\u2014that is, against the quasi-judicial bodies inside the administrative machinery which were primarily intended to watch over the execution of the law rather than to protect the liberty of the individual. The doctrine, as one of the chief justices of a south German state expressed it, that \"whenever a question arises whether any private rights are well founded or have been violated by official action, the matter must be decided by ordinary courts,\"28 enjoyed fairly rapid progress. When the Frankfort parliament of 1848 attempted to draft a constitution for all Germany, it inserted into it a clause that all \"administrative justice\" (as then understood) was to cease, and all violations of private rights were to be adjudicated by courts of justice.29\n\nThe hope, however, that the achievement of constitutional monarchy by the individual German states would effectively realize the ideal of the rule of law was soon disappointed. The new constitutions did little in that direction, and it was soon discovered that, though \"the constitution had been given, the _Rechtsstaat_ proclaimed, in fact the police state continued. Who was to be the guardian of public law and its individualistic principle of fundamental rights? Nobody else than that very administration against whose drive for expansion and activity those fundamental laws had been meant to protect.\"30 It was, in fact, during the next twenty years that Prussia acquired the reputation of a police state, that in the Prussian parliament the great battles over the principle of the _Rechtsstaat_ had to be fought,31 and that the final solution of the problem took form. For some time the ideal remained, at least in northern Germany, of intrusting the control of the lawfulness of the acts of administration to the ordinary courts. This conception of the _Rechtsstaat_ , usually referred to later as \"justicialism,\"32 was soon to be superseded by a different conception, advanced mainly by a student of English administrative practice, Rudolf von Gneist.33\n\n7. There are two different reasons why it may be contended that ordinary jurisdiction and the judicial control of administrative action should be kept separate. Though both considerations contributed to the ultimate establishment of a system of administrative courts in Germany and though they are  frequently confused, they aim at quite different and even incompatible ends, and thus should be kept clearly distinct.\n\nOne argument is that the kind of problems which are raised by disputes over administrative acts requires a knowledge both of branches of law and of fact which the ordinary judge, trained mainly in private or criminal law, cannot be expected to possess. It is a strong and probably a conclusive argument, but it does not support a greater separation between the courts adjudging private and those adjudging administrative disputes than often exists between courts dealing with matters of private law, commercial law, and criminal law, respectively. Administrative courts separated from ordinary courts only in this sense could still be as independent of government as the latter and be concerned only with the administration of the law, that is, with the application of a body of pre-existing rules.\n\nSeparate administrative courts, however, may also be thought necessary on the altogether different ground that disputes about the lawfulness of an administrative act cannot be decided on as a pure matter of law, since they always involve issues of governmental policy or expediency. Courts established separately for this reason will always be concerned with the aims of the government of the moment and cannot be fully independent: they must be part of the administrative apparatus and be subject to direction at least by its executive head. Their purpose will be not so much to protect the individual against encroachments on his private sphere by governmental agencies as to make sure that this does not happen against the intentions and instructions of the government. They will be a device to insure that the subordinate agencies carry out the will of the government (including that of the legislature) rather than a means of protecting the individual.\n\nThe distinction between these tasks can be drawn neatly and unambiguously only where there exists a body of detailed legal rules for guiding and limiting the actions of the administration. It inevitably becomes blurred if administrative courts are created at a time when the formulation of such rules is a task yet to be attempted by legislation and jurisdiction. In such a situation one of the necessary tasks of these courts will be to formulate as legal norms what, so far, have been merely internal rules of the administration; and in doing so they will find it very difficult to distinguish between those internal rules which possess a general character and those which express merely specific aims of current policy.\n\nThis very situation existed in Germany in the 1860s and 1870s when an attempt was finally made to translate into practice the long-cherished ideal of the _Rechtsstaat_. The argument which in the end defeated the long-maintained argument for \"justicialism\" was that it would be impracticable to leave to ordinary judges not specially trained for it the task of handling the intricate issues which would arise from disputes over administrative acts. As a consequence,  separate new administrative courts were created, which were meant to be completely independent courts, concerned exclusively with questions of law; and it was hoped that in the course of time they would assume a strictly judicial control over all administrative action. To the men who devised the system, especially to its main architect, Rudolf von Gneist, and to most of the later German administrative lawyers, this creation of a system of separate administrative courts therefore appeared as the crowning piece of the _Rechtsstaat_ , the definite achievement of the rule of law.34 The fact that there were still left open a large number of loopholes for what in effect were arbitrary administrative decisions appeared merely as minor and temporary defects, made inevitable by the then existing conditions. They believed that, if the administrative apparatus was to continue to function, it had for a time to be given wide discretion until a definite body of rules for its actions had been laid down.\n\nThus, though organizationally the establishment of independent administrative courts seemed to be the final stage of the institutional arrangement designed to secure the rule of law, the most difficult task still lay in the future. The superposition of an apparatus of judicial control over a firmly entrenched bureaucratic machinery could become effective only if the task of rule-making was continued in the spirit in which the whole system had been conceived. Actually, however, the completion of the structure designed to serve the ideal of the rule of law more or less coincided with the abandonment of the ideal. Just as the new device was introduced, there commenced a major reversal of intellectual trends; the conceptions of liberalism, with the _Rechtsstaat_ as its main goal, were abandoned. It was in the 1870s and 1880s, when that system of administrative courts received its final shape in the German states (and also in France), that the new movement toward state socialism and the welfare state began to gather force. There was, in consequence, little willingness to implement the conception of limited government which the new institutions had been designed to serve by gradually legislating away the discretionary powers still possessed by the administration. Indeed, the tendency now was to widen those loopholes in the newly created system by explicitly exempting from judicial review the discretionary powers required by the new tasks of government.\n\nThus the German achievement proved to be more considerable in theory than in practice. But its significance must not be underrated. The Germans were the last people that the liberal tide reached before it began to recede. But they were the ones who most systematically explored and digested all the  experience of the West and deliberately applied its lessons to the problems of the modern administrative state. The conception of the _Rechtsstaat_ which they developed is the direct result of the old ideal of the rule of law, where an elaborate administrative apparatus rather than a monarch or a legislature was the chief agency to be restrained.35 Even though the new conceptions which they developed never took firm root, they represent in some respects the last stage in a continuous development and are perhaps better adapted to the problems of our time than many of the older institutions. As it is the power of the professional administrator that is now the main threat to individual liberty, the institutions developed in Germany for the purpose of keeping him in check deserve more careful examination than they have been given.\n\n8. One of the reasons why these German developments did not receive much attention was that, toward the end of the last century, conditions that prevailed there and elsewhere on the Continent showed a strong contrast between theory and practice. In principle the ideal of the rule of law had long been recognized, and, though the effectiveness of the one important institutional advance\u2014the administrative courts\u2014was somewhat limited, it constituted an important contribution to the solution of new problems. But, in the short time that the new experiment was given to develop its new possibilities, some of the features of former conditions never quite disappeared; and the advance toward a welfare state, which began on the Continent much earlier than in England or in the United States, soon introduced new features which could hardly be reconciled with the ideal of government under the law.\n\nThe result was that, even immediately preceding the first World War, when the political structure of the Continental and the Anglo-Saxon countries had become most similar, an Englishman or an American who observed the daily practice in France or Germany would still feel that the situation was  very far from reflecting the rule of law. The differences between the powers and the conduct of the police in London and those in Berlin\u2014to mention an often quoted example\u2014seemed nearly as great as ever. And though signs of developments similar to those which had already taken place on the Continent began to appear in the West, an acute American observer could still describe the basic difference at the end of the nineteenth century as follows: \"In some cases, it is true, [even in England] an officer of the [local] board is given by statute power to make regulations. The Local Government Board (in Great Britain) and our boards of health furnish examples of this; but such cases are exceptional, and most Anglo-Saxons feel that this power is in its nature arbitrary, and ought not to be extended any further than is absolutely necessary.\"36\n\nIt was in this atmosphere that in England A. V. Dicey, in a work that has become a classic,37 restated the traditional conception of the rule of law in a manner that governed all later discussion and proceeded to contrast it with the situation on the Continent. The picture he drew was, however, somewhat misleading. Starting from the accepted and undeniable proposition that the rule of law prevailed only imperfectly on the Continent and perceiving that this was somehow connected with the fact that administrative coercion was still in a great measure exempt from judicial review, he made the possibility of a review of administrative acts by the ordinary courts his chief test. He appears to have known only the French system of administrative jurisdiction (and even that rather imperfectly)38 and to have been practically ignorant of German developments. With regard to the French system, his severe strictures may then have been somewhat justified, although even at that time the Conseil d'\u00c9tat had already initiated a development which, as a modern observer has suggested, \"might in time succeed in bringing all discretionary powers of the administration . . . within the range of judicial control.\"39 But they were certainly inapplicable to the _principle_ of the German administrative courts; these had been constituted from the beginning as independent judicial bodies with the purposes of securing that rule of law which Dicey was so anxious to preserve.\n\nIt is true that in 1885, when Dicey published his famous _Lectures Introductory to the Study of the Law of the Constitution_ , the German administrative courts were only just taking shape, and the French system had only recently received its definitive form. Nevertheless, the \"fundamental mistake\" of Dicey, \"so fundamental that it is difficult to understand or excuse in a writer of his  eminence,\"40 has had the most unfortunate consequences. The very idea of separate administrative courts\u2014and even the term \"administrative law\"\u2014 came to be regarded in England (and to a lesser extent in the United States) as the denial of the rule of law. Thus, by his attempt to vindicate the rule of law as he saw it, Dicey in effect blocked the development which would have offered the best chance of preserving it. He could not stop the growth in the Anglo-Saxon world of an administrative apparatus similar to that which existed on the Continent. But he did contribute much to prevent or delay the growth of institutions which could subject the new bureaucratic machinery to effective control.\n\n* * *\n\nThe quotation at the head of the chapter is taken from G\u00fcnther Heinrich von Berg, _Handbuch des teutschen Policeyrechtes_ [i.e., deutschen Polizeirechts] (7 vols.; Hanover: Hahn, 1799\u20131804), vol. 2, p. 3. The German text is: \"Wo bleibt eine bestimmte Grenze der h\u00f6chsten Gewalt, wenn eine unbestimmte, ihrem eigenen Urtheile \u00fcberlassene allgemeine Gl\u00fcckseligkeit ihr Ziel sein soll? Sollen die F\u00fcrsten V\u00e4ter des Volks seyn, so gross auch die Gefahr ist, dass sie seine Despoten seyn werden?\" See also Johann Christoph, Freiherr von Aretin (continued by Carl von Rotteck), _Staatsrecht der konstitutionellen Monarchie: Ein Handbuch f\u00fcr Gesch\u00e4ftsm\u00e4nner, studirende J\u00fcnglinge und gebildete B\u00fcrger_ (2 vols.; Altenburg: Literatur-Comptoir, 1824\u201327), vol. 2, p. 179: \"Die Rechtsherrschaft vom Staate verlangen, hei\u00dft ihn v erpf ichten, die Rechte jedes Einz elnen zu sch\u00fctzen und zu achten, die Wohlfahrt von ihm verlangen, hei\u00dft ihn auffordern, die Rechte der Einzelnen zu verletzen, weil die Mittel, wodurch die Gr\u00f6\u00dfe, die Macht, der Ruhm, der Wohlstand einer Nation bef\u00f6rder t werden soll, in der Regel von der Art sind, da\u00df jene Rechte dadurch beeintr\u00e4chtigt werden.\" [\"To demand that the state institute the rule of law means that it is obliged to protect and respect the rights of the individual. To demand that the state provide for its citizens' welfare is to ask that it abuse the rights of the individual, inasmuch as the promotion of the size, power, glory, and prosperity of a nation impairs these rights.\"\u2014Ed.] All of this seems to be rooted in Immanuel Kant, _S\u00e4mmtliche Werke_ , Gustav Hartenstein, ed. (8 vols.; Leipzig: Leopold Voss, 1868), \"Vom Verh\u00e4ltniss der Theorie zur Praxis im Staatsrecht\" [\"On the Relationship of Theory to Practice in Public Law\"], vol. 6, p. 327; \"Von dem Kampf des guten Princips mit dem b\u00f6sen und die Gr\u00fcndung eines Reichs Gottes auf Erden\" [\"Struggle between the Good and the Evil Principle and Founding the Kingdom of God\"], vol. 6, pp. 220\u201329; and \"Metaphysische Anfangsgr\u00fcnde der Rechtslehre\" [\"Metaphysical Elements of the Theory of Law\"], vol. 7, p. 130.\n\nHow little the problems have changed in a century and a half is shown when we compare this with the observation by Alfred Wilhelm von Martin, _Ordnung und Freiheit. Materialien und Reflexionen zu Grundfragen des Soziallebens_ (Frankfurt am Main: Josef Knecht, 1956), p. 177: \"Denn es kann auch bei aller revolution\u00e4r-demokratischen Ideologie-keinen weiterreichenden Freibrief fur die Macht geben, als wenn sie lediglich an den ( jeder jeweiligen 'Generallinie' nachgebenden) Kautschukbegriff des Gemeinwohls gebunden ist, der unter dem Deckmantel des Moralischen, jeder politischen Beliebigkeit freie Bahn gibt.\" [\"All revolutionary-democratic ideology notwithstanding, there can be no more wide-ranging license for power than when it is solely bound to the elastic notion (which invariably yields to whatever happens to be the 'general line' of the moment) of the general good, which under cover of morality, gives free reign to political arbitrariness.\"\u2014Ed.]\n\nFor reference to an earlier publication of the substance of this and the three following chapters see the note at the beginning of chapter 11.\n\n1 Jean Jacques Rousseau, _Lettre \u00e0 Mirabeau_ [from Rousseau to Mirabeau, Trye, 26 July 1767], in _\u0152uvres compl\u00e8tes de J. J. Rousseau; Avec des \u00e9clairissements et des notes historiques_ (2nd ed.; 25 vols.; Paris: Baudouin fr\u00e8res, 1826), vol. 24, p. 175. [The full quotation in French reads: \"Le grand probl\u00e8me en politique, que je compare \u00e0 celui de la quadrature du cercle en g\u00e9om\u00e9trie, et \u00e0 celui des longitudes en astronomie: _Trouver une forme de gouvernement qui mette la loi au-dessus de l'homme._ \"\u2014Ed.] Cf. also the passage from his _Lettres \u00e9crites de la montagne_ (no. 8), in _The Political Writings of Jean Jacques Rousseau_ , Charles Edwyn Vaughan, ed., from the Original Manuscripts and Authentic Editions (2 vols.; Cambridge: Cambridge University Press, 1915), vol. 2, p. 235, quoted above in n. 37, chap. 11 [\"There is no liberty without laws, nor where someone is above the laws; even in the state of nature, man is free only because of the natural law, which enjoins everyone.\" (The original French reads: \"Il n'y a donc point de libert\u00e9 sans Loix, ni o\u00f9 quelqu'un est au dessus des Loix: dans l'\u00e9tat m\u00eame de nature l'homme n'est libre qu'\u00e0 la faveur de la Loi naturelle qui commande \u00e0 tous.\" \"Lettres \u00c9crites de la Montagne,\" [Huiti\u00e8me Lettre], _\u0152uvres Compl\u00e8tes_ , Bernard Gagnebin and Marcel Raymond, eds. Biblioth\u00e8que de la Pl\u00e9iade [5 vols.; Paris: \u00c9ditions Gallimard, 1964], vol. 3, p. 842).\u2014Ed.], and the discussion in Hans Nef, \"Jean Jacques Rousseau und die Idee des Rechtsstaates,\" _Schweizer Beitr\u00e4ge zur allgemeinen Geschichte \/ \u00c9tudes suisse d'histoire g\u00e9n\u00e9rale \/ Studi svizzeri di storia generale_ , 5 (1947): 167\u201385. Cesare Bonesana, Marchese de Beccar ia, _Dei Delitti e delle Pene edizione rivista, corretta, e disposta secondo l'ordine della tr aduzione francese approuato dall'autore coll'aggiunta del commentario alla detta opera di Mr. de Voltaire tradotto da celebre autore_ [An Essay on Crimes and Punishments] (London [Venice]: Presso la Societ\u00e0 dei Filosof , 1774). First published anonymously in 1764 in Italy, Beccaria notes that the \"legislatore\" or the \"sovrano\" can only institute \"leggi generali, chi obblighino tutti i membri\" [\"general laws that are equally applicable to all.\"\u2014Ed.] and holds that legislation must be interpreted syllogistically. \"In ogni delitto si deve fare dal giudice un sillogismo perfetto; la maggiore dev'essere la legge generale, la minore l'azione conforme o no alla legge, la conseguenza la libert\u00e0 o la pena\" (p. 9). [\"In every criminal cause the judge should reason syllogistically. The major should be the general law; the minor, the conformity of the action, or its opposition to the laws; the conclusion, liberty, or punishment.\"\u2014Ed.]\n\n2 Jean Jacques Rousseau, _Du contrat social_ [bk. 2, chap. 6], in _\u0152uvres compl\u00e8tes de J. J. Rousseau; Avec des \u00e9clairissements et des notes historiques_ (2nd ed.; 25 vols.; Paris: Baudouin fr\u00e8res, 1826), vol. 6, p. 72. [The original reads: \"Quand je dis que l'objet des loix est toujours g\u00e9n\u00e9ral, j'entends que la loi consid\u00e8re les sujets en corps et les actions comme abstraites, jamais un homme comme individu ni une action particuli\u00e8re. Ainsi la loi peut bien statuer qu'il y aura des privileges, mais elle n'en peut donner nomm\u00e9ment \u00e0 personne; la loi peut faire plusieurs Classes de Citoyens, assigner m\u00eame les qualit\u00e9s qui donneront droit \u00e0 ces classes, mais elle ne peut nommer tels et tels pour y \u00eatre admis; elle peut \u00e9tablir un Gouvernement royal et une succession h\u00e9r\u00e9ditaire, mais elle ne peut \u00e9lire un roi ni nommer une famille royale; en un mot toute fonction qui se rapporte \u00e0 un objet individuel n'appartient point \u00e0 la puissance l\u00e9gislative.\"\u2014Ed.]\n\n3 [The phrase carries the meaning \"the advent of the law.\"\u2014Ed.] Jules Michelet, _Histoire de la r\u00e9volution fran\u00e7aise_ (5 vols.; Paris: Chamerot, 1847\u201350), vol. 1, p. xxiii. See also Fran\u00e7ois-Auguste-Marie-Alexis Mignet, _Histoire de la R\u00e9volution fran\u00e7aise, depuis 1789 jusqu'en 1814_ (2 vols.; Paris: Didot, 1824), p. 2. [In writing of the old regime, Mignet notes: \"Le peuple ne poss\u00e8dait aucun droit, la royaut\u00e9 n'avait pas de limites et la France \u00e9tait livr\u00e9e \u00e0 la confusion de l'arbitraire minist\u00e9riel, des r\u00e9gimes particuliers et des privil\u00e8ges des corps. A cet ordre abusif la r\u00e9volution en a substitu\u00e9 un plus conforme \u00e0 la justice et plus appropri\u00e9 \u00e0 nos temps. Elle a remplac\u00e9 l'arbitraire par la loi [et] le privil\u00e8ge par l'\u00e9galit\u00e9.\" (\"The people possessed no rights, the Crown was unrestrained, and France had submitted to the confusion of ministerial arbitrariness, perverse administration, and the privileges of authority. To this abusive order the Revolution substituted one more in conformity with justice and more appropriate to our time. It replaced the arbitrary with law and privilege with equality.\")\u2014Ed.]\n\n4 Albert Venn Dicey, _Introduction to the Study of the Law of the Constitution_ (1st ed.; London: Macmillan, 1885), p. 177 [Liberty Fund edition, p. 113]. The first edition of Dicey's book was published under the title _Lectures Introductory to the Study of the Law of the Constitution_ and appeared in 1885.\n\n5 See point 16 of the _D\u00e9claration des droits d l'homme et du citoyen_ , of August 26, 1789: \"Toute soci\u00e9t\u00e9 dans laquelle la garantie des droits n'est assur\u00e9e, ni la s\u00e9paration des pouvoirs d\u00e9termin\u00e9e, n'a point de Constitution.\" [\"A society in which the observance of the law is not assured, nor the separation of powers defined, has no constitution whatever.\"\u2014Ed.]\n\n6 Especially the writings and various constitutional drafts of Jean-Antoine-Nicholas de Caritat, Marquis de Condorcet are concerned with such fundamental distinctions which go right to the heart of the matter as that between true laws in the sense of general rules and mere orders. See particularly the \"Projet girondin\" in _Archives parlementaires de 1787 \u00e0 1860: Recueil complet des d\u00e9bats l\u00e9gislatifs et politiques des chambres fran\u00e7aises_ [imprim\u00e9 par ordre du Corps l\u00e9gislatif sous la direction de J\u00e9rome Mavidal and \u00c9mile Laurent, ed. Series 1 (1787\u201399), vol. 58 (Paris, 1900; vols. 52\u201370 (September 20, 1792\u2013August 10, 1793)]: 1st series, vol. 58, title 7, sec. 2, arts. 1\u20137, vol. 58, pp. 617\u201318; and Condorcet, _\u0152uvres de Condorcet_ , Arthur Condorcet O'Connor and Fran\u00e7ois Arago, eds. (12 vols.; Paris: Firmin Didot fr\u00e8res, 1847\u201349), vol. 12, pp. 356\u201358 and 367. [The pages Hayek cites are part of Condorcet's \"Exposition des Principes et des Motifs du Plan de Constitution,\" pp. 335\u2013415.\u2014Ed.] See also the passage quoted without reference by Joseph Hippolyte Jean Baptiste Barth\u00e9lemy, _Le R\u00f4le du pouvoir ex\u00e9cutif dans les r\u00e9publiques modernes_ (Paris: Giard et Bri\u00e8re, 1906), p. 489. See also Alfred Stern, \"Condorcet und der girondistische Verfassungsentwurf von 1793,\" _Historische Zeitschrift_ , 141 (1930): 479\u201396.\n\n7 Cf. Jean Ray, \"La R\u00e9volution fran\u00e7aise et la pens\u00e9e juridique: l'id\u00e9e du r\u00e8gne de la loi,\" _Revue philosophique_ , 128 (1939): 364\u2013425; and Jean Belin, _La Logique d'une id\u00e9e-force: l'id\u00e9e d'utilit\u00e9 sociale et la R\u00e9volution fran\u00e7aise_ (7 vols.; Paris: Hermann and Cie., 1939).\n\n8 Cf. Ray, \"La R\u00e9volution fran\u00e7aise,\" p. 372. It is of some interest that one of the clearest statements of the English conception of liberty occurs in a work published in Geneva in 1792 by Jean-Joseph Mounier in protest against the abuse of the word \"liberty\" during the French Revolution. It bears the significant title _Recherches sur les causes qui ont emp\u00each\u00e9 les Fran\u00e7ois de devenir libres, et sur les moyens qui leur restent pour acqu\u00e9rir la libert\u00e9_ , and its first chapter, headed \"Quels sont les caract\u00e8res de la libert\u00e9?\" begins: \"Les citoyens sont libres, lorsqu'ils ne peuvent \u00eatre constraints ou emp\u00each\u00e9s dans leurs actions ou dans le jouissance de leurs biens et de leur industrie, si ce n'est en vertu des loix ant\u00e9rieures, \u00e9tablies pour l'int\u00e9r\u00eat public, et jamais d'apr\u00e8s l'autorit\u00e9 d'aucun homme, quels que soient son rang et son pouvoir.\n\n\"Pour qu'un peuple jouisse de la libert\u00e9, les lois, qui sont les actes plus essentiels de la puissance souveraine, doivent \u00eatre dict\u00e9es par des vues g\u00e9n\u00e9rales, et non par des motifs d'int\u00e9r\u00eat particulier; elles ne doivent jamais avoir un effet r\u00e9troactif, ni se rapporter \u00e0 [des circonstances pass\u00e9s, ou \u00e0] certaines personnes.\"\n\n[\"Citizens are free inasmuch as they cannot be constrained or obstructed in their actions or in the enjoyment of their possessions and their industry unless by laws previously enunciated, established in the public interest, and never as a consequence of the authority of one man, regardless of his rank or power.\n\n\"For a people blessed with liberty, the laws, which are crucial to the sovereign power, must be determined by the general good and not those of particular interests. They must never have retroactive effect, nor have reference to prior circumstances or to particular persons.\"\u2014Ed.] Mounier is fully aware that what he is defending is the English concept of liberty, and on the next page he explicitly says: \"Suret\u00e9, propri\u00e9t\u00e9, disent les Anglois, quand ils veulent caract\u00e9riser la libert\u00e9 civile ou personelle. Cette d\u00e9finition est en effet tr\u00e8s exacte: tous les avantages que la libert\u00e9 procure sont exprim\u00e9s dans ces deux mots.\" [\"'Security,' 'Property,' say the English, when they wish to characterize personal or civil liberty. This definition is especially accurate; all of the advantages liberty allows are expressed in these two words.\"\u2014Ed.] On Mounier and generally on the initial influence and gradual receding of the English example in the course of the French Revolution see Gabriel Bonno, _La Constitution britannique devant l'opinion fran\u00e7aise de Montesquieu \u00e0 Bonaparte_ (Paris: H. Champion, 1931), esp. chap. 6 [\"La p\u00e9riode r\u00e9volutionnaire\"], pp. 191\u2013272.\n\n9 Jean Portalis in an address on the occasion of the submission of the third draft of the French civil code to the Council of the Five Hundred in 1796, quoted in P. Antoine Fenet, _Recueil complet des travaux pr\u00e9paratoires du code civil, suivi d'une \u00e9dition de ce code, \u00e0 laquelle sont ajout\u00e9s les lois, d\u00e9crets et ordonnances formant le compl\u00e9ment de la l\u00e9gislation civile de la France, et ou se trouvent indiqu\u00e9s, sous chaque article s\u00e9par\u00e9ment, tous les passages du recueil qui s'y rattachent_ (15 vols.; Paris: Ducessois, 1827), vol. 1, pp. 464\u201367. [The French reads: \"L'esprit r\u00e9volutionnaire se glisse dans toutes. Nous appelons _esprit r\u00e9volutionnaire_ , le d\u00e9sir exalt\u00e9 de sacrifier violemment tous les droits \u00e0 un but politique, et de ne plus admettre d'autre consid\u00e9ration que celle d'un myst\u00e9rieux et variable int\u00e9r\u00eat d'\u00e9tat.\" Jean Etienne Marie Portalis (1746\u20131807) drew up the Code Napol\u00e9on.\u2014Ed.]\n\n10 For an account of how France failed ever to achieve a real constitution in the American sense and how this gradually led to a decline of the rule of law see Louis Auguste Paul Rougier, _La France \u00e0 la recherche d'une constitution_ (Paris: Recueil Sirey, 1952).\n\n11 In addition to Alexis de Tocqueville, _L'ancien r\u00e9gime_ (1856), Melville Watson Patterson, trans. (Oxford: B. Blackwell, 1952), bk. 2, particularly chap. 2 [\"Administrative Centralization an institution of the 'Old Order' and not, as some have said, the Work of the Revolution and the Empire\" (pp. 37\u201346)] and chap. 4 [\"Administrative Justice and Indemnity of Officials were Institutions of the 'Old Order' of Society\" (pp. 58\u201362)], see particularly the _Recollections of Alexis de Tocqueville_ , Alexander Teixeira de Mattos, trans. (London: H. Henry, 1896), p. 238: \"When, therefore, people assert that nothing is safe from revolutions, I tell them they are wrong, and that centralization is one of those things. In France there is only one thing we can't set up: that is, a free government; and only one institution we can't destroy: that is, centralization. How could it ever perish? The enemies of government love it, and those who govern cherish it. The latter perceive, it is true, from time to time that it exposes them to sudden and irremediable disasters; but this does not disgust them with it. The pleasure it procures them of interfering with every one and holding everything in their hands atones to them for its dangers.\" [\"Lors donc qu'on pr\u00e9tend qu'il n'y a rien parmi nous qui soit \u00e0 l'abri des r\u00e9volutions, je dis qu'on trompe, et que la centralisation s'y trouve. En France, il n'y a gu\u00e8re qu'une seule chose qu'on ne puisse faire: c'est un gouvernement libre, et qu'une seule institution qu'on ne puisse d\u00e9truire: la centralisation. Comment pourrait-elle p\u00e9rir? Les ennemis des gouvernements l'aiment et les gouvernants la ch\u00e9rissent. Ceux-ci s'aper\u00e7oivent, il est vrai, de temps \u00e0 autre, qu'elle les expose \u00e0 des d\u00e9sastres soudains et irr\u00e9m\u00e9diables, mais cela ne les en d\u00e9go\u00dbte point. Le plaisir qu'elle leur procure de se m\u00ealer de tout et de tenir chacun dans leurs mains leur fait supporter ses p\u00e9rils.\" _Souvenirs d'Alexis de Tocqueville_ , Luc Monnier, ed. (nouv. ed. Paris: Gallimard, 1942), pp.163\u201364.\u2014Ed.]\n\n12 King Louis Philippe himself is reported to have said in a speech to the National Guard [quoted in an essay by Hugues F\u00e9licit\u00e9 Robert de Lamennais in _L'Avenir_ of May 23, 1831, reprinted in _Troisi\u00e8mes m\u00e9langes_ (Paris: P. Daubr\u00e9e et Cailleux,1835), p. 266]: \"La libert\u00e9 ne consiste que dans le r\u00e8gne des lois. Que chacun ne puisse pas \u00eatre tenu de faire autre chose que ce que la loi exige de lui, et qu'il puisse faire tout ce que la loi n'interdit pas, telle est la libert\u00e9: C'est vouloir la d\u00e9truire que de vouloir autre chose.\" [\"Liberty consists solely in the rule of law. That no one may be held to do other than what the law demands of him and that he may act in any manner not prohibited by law, therein lies one's liberty. To desire other than this is tantamount to destroying it.\"\u2014Ed.] A fuller account of French developments during this period would have to give considerable space to some of the leading political thinkers and statesmen of the period, such as Benjamin Constant, Guizot, and the group of \"doctrinaires,\" who developed a theory of _garantisme_ , a system of checks designed to protect the rights of the individual against the encroachment of the state. On them, see Guido de Ruggiero, _The History of European Liberalism_ , Robin George Collinwood, trans. (London: Oxford University Press, 1927); Luis D\u00edez el Corral, _El Liberalismo doctrinario_ (Madrid: Instituto de estudios pol\u00edticos, 1945). On the doctrinal development of French administrative law and jurisdiction during the period compare particularly Achille-L\u00e9on-Victor, Duc de Broglie, \"De la jurisdiction administrative,\" in _\u00c9crits et discours_ (3 vols.; Paris: Didier et cie, 1863), vol. 1, pp. 249\u2013331; and Louis-Marie de Lahaye, Vicomte de Cormenin, _Questions de droit administratif_ (2 vols.; Paris: M. Ridler, 1822). Regarding Tocqueville's intense interest in bringing bureaucratic decisions under judicial control as a essential condition f or liberty, see especially Jack Lively, _The Social and Political Thought of Alexis de Tocqueville_ (Oxford: Clarendon Press, 1962), pp. 166\u201382.\n\n13 See Bernard Schwartz, _French Administrative Law and the Common-Law World_ (New York: New York University Press, 1954); Charles John Hamson, _Executive Discretion and Judicial Control: An Aspect of the French Conseil d'\u00c9tat_ (London: Stevens, 1954); and Marguerite A. Sieghart, _Government by Decree_ (London: Stevens, 1950).\n\n14 On the importance of the German theoretical developments cf. Paul Alex\u00e9ef, \"L'\u00c9tat\u2014 le droit\u2014et le pouvoir discr\u00e9tionnaire des autorit\u00e9s publiques,\" _Revue internationale de la th\u00e9orie du droit_ , 3 (1928\u201329): 216; Charles Howard McIlwain, _Constitutionalism and the Changing World: Collected Papers_ (Cambridge: Cambridge University Press, 1939), p. 270; and L\u00e9on Duguit, _Manuel de droit constitutionnel, th\u00e9orie g\u00e9n\u00e9rale de l'\u00e9tat, le droit de l'\u00e9tat, les libert\u00e9s publiques, organisation politique_ (3rd ed.; Paris: Fontenoing and Cie., 1918), which is a good example of how one of the Continental treatises on constitutional law most widely known in the Anglo-Saxon world derives its argument at least as much from German as from French predecessors.\n\n15 See esp. Hermann Conrad, _Rechtsstaatliche Bestrebungen im Absolutismus Preu\u00dfens und \u00d6sterreichs am Ende des 18. Jahrhunderts_ [Arbeitsgemeinschaft f\u00fcr Forschung des Landes Nordrhein-Westfalen, bk. 95] (Cologne: Westdeutscher Verlag, 1961), and the ear lier studies by the same author mentioned there. Cf. the perceptive observation in Abbott Lawrence Lowell, _Governments and Parties in Continental Europe_ (2 vols.; New York: Houghton, Mifflin, 1896), vol. 2, p. 86: \"In Prussia, the bureaucracy was so ordered as to furnish a better protection of individual rights and a firmer maintenance of law. But this broke down with the spread of French ideas after 1848, when the antagonistic interests in the state, taking advantage of the parliamentary system, abused the administrative power and introduced a veritable party tyranny.\" [This quotation does not seem to appear in Lowell's book, although he does discuss the relationship of the Prussian bureaucracy at some length. At one point he writes: \"Notwithstanding the excellent organization of the bureaucracy, its enormous power could hardly be endured without restraint exercised by the administrative courts. Before the present century the elaborate system of administrative appeals, and the permanence of traditions that prevailed in the bureaucracy, many of whom were learned in the law, preserved the great uniformity in the administration, and furnished a real guarantee against arbitrary conduct on the part of officials. But with the spread of new ideas after the French Revolution, a marked change took place. The sharp distinction drawn between justice and administration deprived administrative procedure of its judicial character, and made the decisions of the officials turn less on law and more on expediency.\" (vol. 1, p. 294); and again: \"The present constitution of Prussia (1896), which dates from January 31, 1850, was granted by the King after the revolutionary movement of 1848 had begun to subside, and is far less democratic than the Liberals would have liked. In some ways it is even less liberal than the text would lead one to suppose; for although it contains quite an elaborate bill of rights, Professor Gneist spoke of it as a _lex imperfecta_ , owing to the absence of machinery for giving effect to its provisions. It purports, for example, to guarantee the liberty of instruction; but as no statute has been passed to carry this out, the previous laws remain in force, whereby no school can be opened without permission from the government. Again, it declares that the right to assemble without arms, except in the open air, shall be free; but in fact notice of every meeting held to discuss public affairs must be given to the police, who have a right to be present, and a very extensive power of breaking it up. The result of such a state of things is that neither the parliament nor the citizens have sufficient means of defending their rights; and although the recent increase of local self-government and the establishment of administrative justice have done something towards remedying this defect, personal and political liberty are still far from enjoying the same protection as in Anglo-Saxon countries. The constitution was clearly not intended as a restraint on legislation, for it can be changed by simple majority vote of both chambers, sanctioned by the King\" (vol. 1, pp. 286\u201387).\u2014Ed.]\n\n16 The conception of the power of law that prevailed in eighteenth-century Prussia is well illustrated by an anecdote known to every German child. Frederick II is said to have been annoyed by an old windmill standing close to his palace of Sans-Souci, impairing the view, and, after various unsuccessful attempts at buying it from the owner, is said to have threatened him with eviction; to which the miller is supposed to have answered: \"We still have courts of justice in Prussia\" ( \"Es gibt noch ein Kammergericht in Berlin!\" is the phrase usually quoted). For the facts, or rather absence of factual basis in the legend, see Reinhold Koser, _Geschichte Friedrichs des Gro\u00dfen_ (4th ed.; 4 vols.; Stuttgart: Cotta, 1912\u201314), vol. 3, pp. 413ff. The story suggests limits to kingly power which at the time probably existed in no other country on the Continent and which I am not sure apply today to the heads of democratic states: a hint to their town planners would quickly lead to the forcible removal of such an eyesore\u2014although, of course, solely in the public interest and not to please anybody's whim! [The actual quote, as Koser has it, is \"Es gibt noch Richter in Berlin!\"\u2014Ed.]\n\n17 For Immanuel Kant's legal philosophy see particularly his _Die Metaphysik der Sitten_ (1785). Vol. 1: _Anfangsgr\u00fcnde der Rechtslehre_ , part 2: \"Das Staatsrecht,\" secs. 45\u201349. See also Kant's two essays \"\u00dcber den Gemeinspruch: Das mag in der Theorie richtig sein, taugt aber nicht f\u00fcr die Praxis\" (1793) and \"Zum ewigen Frieden\" (1795). [For an English translation of these works, see: Immanuel Kant, _Practical Philosophy_ , Mary J. Gregory, ed. and trans. (Cambridge: Cambridge University Press, 1999), which contains the _Groundwork of the Metaphysics of Morals_ ( _Grundlegung zur Metaphysik der Sitten_ ) and the two essays to which Hayek refers: (1) \"On the Common Saying: That May Be Correct in Theory, but It Is of No Use in Practice\" (1793) and (2) \"Toward Perpetual Peace\" (1795).\u2014Ed.] Cf. Werner Haensel, _Kants Lehre von Widerstandsrecht. Ein Beitrag zur Systematik der Kantischen Rechtsphilosophie_ [Kant-Studien No. 60] (Berlin: Pan-Verlag Rolf Heise, 1926), and Friedrich Darmst\u00e4dter, _Die Grenzen der Wirksamkeit des Rechtsstaates. Eine Untersuchung zur gegenw\u00e4rtigen Krise des liberalen Staatsgedankens_ (Heidelberg: C. Winter, 1930).\n\n18 Immanuel Kant, _Fundamental Principles of Morals_ , A. D. Lindsay, trans., p. 421. [The \"Exhaustive Bibliography of English Translations of Kant\" lists no translation by Lindsay of the _Grundlegung zur Metaphysik der Sitten_ (variously translated as the _Fundamental Principles of the Metaphysics of Morals_ , the _Foundations of the Metaphysics of Morals_ , and the _Groundwork of the Metaphysics of Morals_ ). The standard translation, and the one to which Hayek is probably referring, is by Thomas Kingsmill Abbott made in 1873 and reprinted numerous times. Abbott's translation of the categorical imperative reads: \"I am never to act otherwise than so that I could also will that my maxim should become a universal law.\" ( _Kant's Theory of Ethics or Practical Philosophy: Comprising: 1. Fundamental Principles of the Metaphysics of Morals; 2. Dialectic and Methodology of Practical Reason; 3. On the Radical Evil in Human Nature_ , Thomas Kingsmill Abbott, trans. [4th rev. ed.; London: Longmans, Green, Reader, and Dyer, 1889], p. 18.) The categorical imperative is further discussed in Kant's _Kritik der praktischen Vernunft_ (1788), translated by Thomas Kingmill Abbott in 1873 as the _Critical Examination of Practical Reason_ and published as part 2 of _Kant's Theory of Ethics or Practical Philosophy_ (pp. 87\u2013262).\u2014Ed.]\n\nIt is in agreement with this transfer of the concept of the rule of law to the field of morals when for Kant the conception of freedom as depending only on the law becomes \"independence of anything other than the moral law alone.\" [The full quotation reads: \"We should also see not merely the possibility, but even the necessity, of the moral law as the supreme law of rational beings, to whom we attribute freedom of causality of their will; because both concepts are so inseparably united that we might define practical freedom as independence of the will on anything but the moral law.\" The German reads: \"Wenn man die M\u00f6glichkeit der Freiheit einer wirkenden Ursache eins\u00e4he, man auch nicht etwa blo\u00df die M\u00f6glichkeit, sondern gar die Notwendigkeit des moralischen Gesetzes als obersten praktischen Gesetzes vern\u00fcnftiger Wesen, denen man Freiheit der Causalit\u00e4t ihres Willens beilegt, einsehen w\u00fcrde: weil beide Begriffe so unzertrennlich sind, da\u00df man praktische Freiheit auch durch Unabh\u00e4ngigkeit des Willens von jedem anderen, ausser allein dem moralischen Gesetze, definieren k\u00f6nnte.\" ( _Kritik der praktischen Vernunft_ , in _Kants Werke_ [Akademie Textausgabe; 9 vols.; Berlin: Walter de Gruyter, 1968], vol. 5, 93).\u2014Ed.]\n\n19 Cf. Karl Menger, _Moral, Wille und Weltgestaltung. Grundlegung zur Logik der Sitten_ (Vienna: J. Springer, 1934), pp. 13\u201317.\n\n20 A fuller account would have to consider particularly the early work of the philosopher Johann Gottlieb Fichte, _Grundlage des Naturrechts nach Principien der Wissenschaftslehre_ (1796). Vol. 3 of _S\u00e4mmtliche Werke_ , Immanuel Hermann Fichte, ed. (8 vols.; Berlin: Veit, 1845), and the writings of the poet Friedrich Schiller [See his _Werke und Briefe_ , Klaus Harro Hilzinger, ed. (12 vols.; Frankfurt: Deutscher Klassiker Verlag, 1988\u20132004).] who did probably as much as any man to spread liberal ideas in Germany. See Erich Eyck, \"Freiheit und Demokratie, 1848\u20131948,\" in Accademia Nazionale dei Lincei (Fondazione Alessandro Volta), _Convegno di scienze morali, storiche e f lologiche. 4\u201310 ottobre 1948_ [Tema: Il 1848 Nella Storia d'Europa] (Rome: Accademia Nazionale dei Lincei, 1949), p. 31. On these and the other German classics see Gustav Falter, _Staatsideale unserer Klassiker_ (Leipzig: C. L. Hirschfeld, 1911), and Wilhelm Metzger, _Gesellschaft, Recht und Staat in der Ethik des deutschen Idealismus mit einer Einleitung: Prolegomena zu einer Theorie und Geschichte der sozialen Werte_ (Heidelberg: C. Winter, 1917). I am, however, under the impression that nobody seems to recognize the central role played by Christian Garve, who was an authority on Hume, Smith, Ferguson, Paley, and Burke, who translated Aristotle and Cicero, was a friend of Schiller, and exchanged letters with Kant and Gentz.\n\n21 Wilhelm von Humboldt, _Ideen zu einem Versuch, die Gr\u00e4nzen der Wirksamkeit des Staats zu bestimmen_ (Breslau: E. Trewendt, 1851). [The English translation was published under the title _The Sphere and Duties of Government_ , Joseph Coulthard, Jr., trans. (London: John Chapman, 1854).\u2014Ed.] Only part of this work was published soon after its composition in 1792, and the whole appeared only in the posthumous edition quoted, rapidly followed by an English translation, when it profoundly affected not only John Stuart Mill but also \u00c9douard Laboulaye in France. See the latter's _L'\u00c9tat et ses limites: suivi d'essais politiques_ (Paris: Charpentier, 1863). [The phrase used by Humboldt is \"Gewi\u00dfheit der gesetzm\u00e4\u00dfigen Freiheit.\" See chap. 9, \"On Security,\" in _\u00dcber die Grenzen der Wirksamkeit des Staates_ (Nuremberg: Verlag Hans Carl, 1946), p. 132. There is a Liberty Fund edition of this essay published under the title _The Limits of State Action_.\u2014Ed.]\n\n22 It had been preceded by a Swedish code in 1734 and an even earlier Danish code. [ _The Encyclopedia Britannica_ notes that the Prussian code ( _Code Fr\u00e9deric_ ), published by Frederick the Great in 1751, \"was intended to take the place of Roman, common Saxon, and other foreign subsidiary laws and statutes.\" Earlier, in 1683, Christian V promulgated a civil code for Denmark, which was later extended to Norway and Iceland. In 1734, the Swedish Parliament approved a new enactment for the Realm of Sweden, actually a collection of codes.\u2014Ed.]\n\n23 The principle seems to have been first stated in this form by Paul Johann Anselm Feuerbach, _Lehrbuch des gemeinen in Deutschland g\u00fcltigen peinlichen Rechts_ (Giessen: G. H. Heyer, 1801), p. 20. But see chap. 11, n. 78, above.\n\n24 \"Art. 8: \"La loi ne doit \u00e9tablir que des peines strictement et \u00e9videmment n\u00e9cessaire, et nul ne peut \u00eatre puni qu'en vertu d'une loi \u00e9tablie et promulgu\u00e9e ant\u00e9rieurement au d\u00e9lit, et l\u00e9galement appliqu\u00e9e.\" [Hayek is here quoting not the Austrian criminal code of 1787 but art. 8 of the Declaration of the Rights of Man and of the Citizen of 1789. It reads: \"The law shall provide for such punishments only as are strictly and obviously necessary, and no one shall suffer punishment except it be legally inflicted in virtue of a law passed and promulgated before the commission of the offense.\" The Austrian Criminal Code of 1787, promulgated by the Emperor Joseph II and known as the Josephine Code, provides that \"no action contrary to law, shall be considered as criminal, but such as shall have been determined by the present criminal code\" and that \"punishment follows a criminal offense discovered and proved. It cannot be decreed by a judge, appointed to discharge the functions of criminal jurisdiction\" (Articles 1 and 10). See _The Emperor's New Code of Criminal Laws, Published at Vienna, the 15th of January 1787_ , translated from the German by an officer (London: Printed for G. G. J. And J. Robinson, 1787), sec. 1, p. 1; sec. 10, p. 6.\u2014Ed.]\n\n25 Cf. Edgar Loening, _Gerichte und Verwaltungsbeh\u00f6rden in Brandenburg-Preu\u00dfen: Ein Beitrag zur preu\u00dfischen Rechts-und Verfassungsgeschichte_ (Halle: Waisenhaus, 1914), and particularly the extensive review article on this work by Otto Hintze, \"Preu\u00dfens Entwicklung zum _Rechtsstaat_ ,\" in _Geist und Epochen der preu\u00dfichen Geschichte_ (Leipzig: Koehler and Amelang, 1943) [chap. 4], pp.105\u201371.\n\n26 We cannot enter here into a further examination of the earlier history of this German concept and especially of the interesting question of how far it may have derived from Jean Bodin's conception of a \"droit gouvernement.\" On the more specific German sources see Otto Friedrich von Gierke, _Johannes Althusius und die Entwicklung der naturrechtlichen Staatstheorien: zugleich ein Beitrag zur Geschichte der Rechtssystematik_ (2nd ed.; Breslau: W. Koebner, 1880). According to Carl Schmitt, \"Was bedeutet der Streit um den' _Rechtsstaat_?\" _Zeitschrift f\u00fcr die gesamte Staatswissenschaft_ , 95 (1935): 190, the term _Rechtsstaat_ appears for the f rst time in Adam Heinrich M\u00fcller ( _Elemente der Staatskunst: \u00d6ffentliche Vorlesungen vor Sr. Durchlaucht dem Prinzen Bernhard von Sachsen-Weimar und einer Versammlung von Staatsm\u00e4nnern und Diplomaten, im Winter von 1808 auf 1809, zu Dresden, gehalten_ [Berlin: Sander, 1809]) with reference to \"a truly organic legal system.\" Later, however, this sense was seldom meant.\n\nThe word _Rechtsstaat_ seems to appear for the first time, but hardly yet with its later meaning, in Karl Teodor Welcker, _Die letzten Gr\u00fcnde von Recht, Staat und Strafe: Philosophisch und nach den Gesetzen der merkw\u00fcrdigsten V\u00f6lker rechtshistorisch entwickelt_ (Giessen: Heyer, 1813), where three types of government are distinguished: despotism, theocracy, and _Rechtsstaat_. On the history of the conception see Reimund Asanger, _Beitr\u00e4ge zur Lehre vom Rechtsstaat im 19. Jahrhundert_ (unpublished doctoral dissertation; Westf\u00e4lische Wilhelms-Universit\u00e4t, M\u00fcnster, 1938). The best account of the role of the ideal in the German liberal movement is to be found in Franz Schnabel, _Deutsche Geschichte im neunzehnten Jahrhundert_ (4 vols.; Freiburg im Breisgau: Herder, 1933\u201359), vol. 2, pp. 99\u2013109. See also Thomas Ellwein, _Das Erbe der Monarchie in der deutschen Staatskrise: Zur Geschichte des Verfassungsstaates in Deutschland_ (Munich: Isar, 1954).\n\nIt is probably no accident that the beginning of the theoretical movement that led to the development of the ideal of the _Rechtsstaat_ came from Hanover, which, through its kings, had had more contact with England than the rest of Germany. See Franz Rosin, _Gesetz und Verordnung: Gesetz und Verordnung nach badischem Staatsrecht_ (Freiburger Abhandlungen aus dem Gebiet des \u00f6ffentlichen Rechts, Heft 18, dissertation; Karlsruhe, G. Braun, 1911), p. 30\u201347, who refers particularly to Justi and Justus M\u00f6ser and shows how Freiherr vom Stein founded this tradition. See also Hermann Christern, _Friedrich Christoph Dahlmanns politische Entwicklung bis 1848: ein Beitrag zur Geschichte des deutschen Liberalismus_ (Leipzig: Haessel, 1921). During the later part of the eighteenth century there appeared here a group of distinguished political theorists who built on the English Whig tradition; among them E. Brandes, A. W. Rehberg, and later F. C. Dahlmann were the most important in spreading English constitutional ideas in Germany. [Ernest Brandes (1768\u20131810), August Wilhelm Rehberg (1757\u20131836), and Frederich C. Dahlmann (1785\u20131860), known as Hanoverian Whigs, who traced the origins of English constitutionalism to Anglo-Saxon institutions and rejected the notion that Norman political notions served as the basis of English liberty.\u2014Ed.] See on these men Hermann Christern, _Deutscher St\u00e4ndestaat und englischer Parlamentarismus am Ende des 18. Jahrhunderts_ (Munich: C. H. Beck'sche Verlagsbuchhandlung, 1939). For our present purposes the most important figure of the group is, however, G\u00fcnther Heinrich von Berg, whose work was quoted at the beginning of this chapter (see esp. the _Handbuch_ , vol. 1, pp. 158\u201360 and vol. 2, pp. 1\u20134 and 12\u201317). The influence of his work is described in Gustav Marchet, _Studien \u00fcber die Entwickelung der Verwaltungslehre in Deutschland von der zweiten H\u00e4lfte des 17. bis zum Ende des 18. Jahrhunderts_ (Munich: R. Oldenbourg, 1885), pp. 419\u201328.\n\nThe scholar who later did most to propagate the theory of the _Rechtsstaat_ , Robert von Mohl, had been a close student of the American Constitution; see his _Das Bundes-Staatsrecht der Vereinigten Staaten von Nord-Amerika_ (Stuttgart: J. G. Cotta, 1824), which appears to have earned him a considerable reputation in the United States and led to his being asked to review Judge Story's _Commentaries_ (\"German Criticism of Mr. Justice Story's _Commentaries on the Constitution of the United States_ ,\" in the _American Jurist and Law Magazine_ , 14 (October 1835): 330\u201345). The main works in which he elaborated the theory of the _Rechtsstaat_ include his _Staatsrecht des K\u00f6nigreiches W\u00fcrttemberg_ (2 vols.; T\u00fcbingen: H. Laupp, 1829\u201331), in which he embraces Welcker's distinction between despotism, theocracy, and a constitutional state, while adding a fourth type, the patriarchal state (see vol. 1, pp. 6\u20139), in the course of which he gives what is probably the most accurate defnition of the constitutional state found in German literature, a state in which \"the citizen is apprised of any changes in the purpose or conditions of every law not by virtue of the arbitrary commands of a higher human or spiritual power, but only through laws equally applicable to all (p. 182). See also Mohl's _Das Recht der Steuerverwilligung nach den Grunds\u00e4tzen der w\u00fcrttembergischen Verfassung, mit R\u00fccksicht auf entgegenstehende Bestimmungen des deutschen Bundes_ (Stuttgart: Liesching, 1836), which contains one of the earliest differentiations between substantive and formal law. [The author of this work is, in fact, Paul Achatius Pfizer.\u2014Ed.] Further, see Mohl's _Die Polizei-Wissenschaft nach den Grunds\u00e4tzen des Rechtsstaates_ (3 vols.; T\u00fcbingen: Laupp, 1832\u201334) and _Die Geschichte und Literatur der Staatswissenschaften_ (3 vols.; Erlangen: Ferdinand Enke, 1855\u201358). The best-known formulation of the conception of the _Rechtsstaat_ as it ultimately emerged is that by one of the conservative theorists of the period, Friedrich Julius Stahl. In _Die Philosophie des Rechts_. Vol. 2: _Rechts-und Staatslehre_ , part 2 (1837) (5th ed.; 2 vols. in 3; T\u00fcbingen and Leipzig: J. C. B. Mohr, 1878), pp. 137\u201338), he defines it as follows (p. 352): \"The State should be a State of law, this is the watchword and, in truth, also the tendency of recent times. It should exactly and irrevocably determine and secure the _directions_ and the _limits_ of its activity and the free sphere of the citizen, and not enforce on its own behalf or directly any moral ideas beyond the sphere of law. This is the conception of the _Rechtsstaat_ and not that the state should confine itself to administering the law and pursue no administrative purpose or _only_ protect the rights of the individual. It says nothing about the _content_ or _aim_ of the state but defines only the manner and method of achieving them.\" [The original German reads: \"Der Staat soll _Rechtsstaat_ seyn, das ist die Losung und ist auch in Wahrheit der Entwickelungstrieb der neueren Zeit. Er soll die Bahnen und Gr\u00e4nzen seiner Wirksamkeit wie die freie Sph\u00e4re seiner B\u00fcrger in der Weise des Rechts genau bestimmen und unverbr\u00fcchlich sichern und soll die sittlichen Ideen von Staats wegen also direkt, nicht weiter verwirklichen (erzwingen), als es der Rechtssph\u00e4re angeh\u00f6rt, d.i. nur bis zur nothwendigsten Umz\u00e4unung. Die\u00df ist der Begriff des _Rechtsstaates_ , nicht etwa da\u00df der Staat blo\u00df die Rechtsordnung handhabe ohne administrative Zwecke, oder vollends blo\u00df die Rechte der Einzelnen sch\u00fctze, er bedeutet \u00fcberhaupt nicht Ziel und Inhalt des Staates, sondern nur Art und Charakter, dieselben zu verwirklichen.\"\u2014Ed.] (The last sentences are aimed at the extreme position represented, for example, by Wilhelm von Humboldt.)\n\n27 Cf., e.g., Paul Achatius Pfizer, \"Liberal, Liberalismus,\" _Staats-Lexicon oder Enzyklop\u00e4die der Staatswissenschaften_ , Karl Wenceslaus von Rotteck and Karl Theodor Welcker, eds. (new ed.; 12 vols.; Altona: J. F. Hammerich, 1845\u201348), vol. 8, p. 534. \"Noch m\u00e4chtiger und unbesiegbarer muss aber der Liberalismus dann erscheinen, wenn man sich \u00fcberzeugt, dass er nichts Anderes ist als der auf einer gewissen Stufe menschlicher Entwickelung nothwendige \u00dcbergang des Naturstaats in den _Rechtsstaat_.\" [\"Liberalism must seem even more powerful, more invincible, when one is convinced that it constitutes nothing more than the transition, required at a particular stage of human history, from the state of nature to a state of law.\"\u2014Ed.]\n\n28 Ludwig Minnigerode, _Beitrag zur Beanwortung der Frage: Was ist Justiz-und was ist Administrativ-Sache?_ (Darmstadt: Meyer, 1835), p. 8. [The German reads: \"So oft Streit entsteht oder die Frage ist, ob der vorkommende Fall unter ein vorhandenes allgemeines oder spezielles Gesetz subsumiert werden m\u00fcsse oder nicht,\u2014mu\u00df die Justizbeh\u00f6rde entscheiden.\"\u2014Ed.] See also Paul Achatius Pf zer's work _Das Recht der Steuerverwilligung_ , where, as remarked in n. 27, he points out of the difference between material and formal laws.\n\n29 It deserves notice that there was a significant difference of opinion between south Germany, where French influences predominated, and north Germany, where a combination of old Germanic tradition and the influence of the theorists of the law of nature and of the English example seems to have been stronger. In particular, the group of south German lawyers who, in the political encyclopedia quoted above (n. 27), provided the most influential handbook of the liberal movement, were distinctly more influenced by Frenchmen like Benjamin Constant and Fran\u00e7ois Pierre Guillaume Guizot than by any other source. On the importance of the _Staatslexikon_ see Hans Zehntner, _Das Staatslexicon von Rotteck und Welcker: eine Studie zur Geschichte des deutschen Fr\u00fchliberalismus_ (List Studien, No. 3; Jena: G. Fischer, 1924), and on the predominantly French influences on south German liberalism see Artur Fickert, _Montesquieus und Rousseaus Einfluss auf den vorm\u00e4rzlichen Liberalismus Badens_ (Leipziger historische Abhandlungen, vol. 37; Leipzig: Quelle und Meyer, 1914). Cf. Theodor Wilhelm, _Die englische Verfassung und der vorm\u00e4rzliche deutsche Liberalismus: eine Darstellung und Kritik des Verfassungsbildes der liberalen F\u00fchrer_ (Stuttgart: W. Kohlhammer, 1928). The difference in the tradition manifested itself later in the fact that, while in Prussia judicial review was extended, at least in principle, to questions on which the administrative agencies possessed discretionary powers, in south Germany such questions were explicitly excluded from judicial review.\n\n30 Gerhard Ansch\u00fctz, \"Verwaltungsrecht,\" in _Systematische Rechtswissenschaft_ , Rudolf Stammler, ed. (Kultur der Gegenwart, vol. 2, no. 7; Leipzig and Berlin: B. G. Taeubner, 1906), p. 352. [The German reads: \"Die Verfassung war gegeben, der _Rechtsstaat_ proklamiert, der Polizeistaat dauerte fort. Denn wer war der H\u00fcter des neuen \u00f6ffentlichen Rechtes, seiner individualistischen Grunds\u00e4tze und Grundrechte? Niemand anderes als die Verwaltung selbst, eben sie, gegen deren Expansions-und Bet\u00e4tigungsdrang jene Grunds\u00e4tze gegeben, jene Grundrechte verliehen worden waren.\"\u2014Ed.]\n\n31 See Eduard Lasker, \"Polizeigewalt und Rechtsschutz in Preu\u00dfen,\" _Deutsche Jahrbucher f\u00fcr Politik und Literatur_ , 1 (1861): 27\u201348; reprinted in Eduard Lasker, _Zur Verfassungsgeschichte Preu\u00dfens_ (Leipzig: F. A. Brockhaus, 1874), pp. 179\u2013212. The essay is significant also for showing how far the English example guided north German developments.\n\n32 The representative work stating this view is Otto B\u00e4hr, _Der Rechtsstaat: Eine publicistische Skizze_ (Cassel: Wigand, 1864).\n\n33 Heinrich Rudolf von Gneist, _Der Rechtsstaat_ (Berlin: Julius Springer, 1872), and especially the second and enlarged edition of the same work, _Der Rechtsstaat und die Verwaltungsgerichte in Deutschland_ (2nd ed., enlarged; Berlin: Julius Springer, 1879). The significance which was attached to Gneist's work at the time may be gathered from the title of an anonymous pamphlet of the period: _H. Prof. Gneist, oder der Retter der Gesellschaft durch den \"Rechtsstaat\"_ (Berlin: Schoppmeyer, 1873).\n\n34 See, for example, Gustav Radbruch, _Einf\u00fchrung in die Rechtswissenschaft_ (2nd ed.; Leipzig: Quelle und Meyer, 1913); Fritz Fleiner, _Institutionen des deutschen Verwaltungsrechts_ (8th ed.; T\u00fcbingen: Mohr, 1928), and Ernst Forsthoff, _Lehrbuch des Verwaltungsrechts_. Vol. 1: _Allgemeiner Teil_ (Munich: C. H. Beck, 1950), p 394 [see chap. 7, \"Der Rechtsschutz in Verwaltungssachen,\" pp. 386\u2013420].\n\n35 It is certainly not correct to maintain with regard to the earlier phase of this German development, as did Franz Leopold Neumann, \"The Concept of Political Freedom,\" _Columbia Law Review_ , 53 (1953): 910; reprinted in the same author's _The Democratic and Authoritarian State: Essays in Political and Legal Theory_ (Glencoe, IL: The Free Press, 1957), p. 169 (also the conflicting statement in the latter vol., p. 22), that \"the English rule of law and the German _Rechtsstaat_ doctrines have nothing in common.\" This may be true of the emasculated concept of the merely \"formal\" _Rechtsstaat_ which became dominant at the end of the century, but not of the ideals which inspired the liberal movement of the first half of the century or of the theoretical conceptions which guided the reform of administrative jurisdiction in Prussia. Rudolph von Gneist, in particular, quite deliberately made the English position his model (and was, incidentally, the author of an important treatise on English \"administrative law,\" a fact which ought to have prevented A. V. Dicey, if he had known of it, from so completely misunderstanding the use of the term on the Continent). The German translation of \"rule of law,\" _Herrschaft des Gesetzes_ , was in fact frequently used in place of _Rechtsstaat_. [The treatise on administrative law by Gneist to which Hayek is referring is _Das englische Verwaltungsrecht mit Einschluss des Heeres, der Gerichte und der Kirche_. Vol 1: _Geschichte des englischen Verwaltungsrechts_ ; Vol 2: _Das heutige englische Verwaltungsrecht_ (2 vols.; Berlin: Julius Springer, 1867).\u2014Ed.]\n\n36 Lowell, _Governments and Parties in Continental Europe_ , vol. 1, p. 44.\n\n37 Dicey, _Law of the Constitution_ , originally delivered as lectures in 1884.\n\n38 Dicey later became at least partly aware of his error. See his article \" _Droit Administratif_ in Modern French Law,\" _Law Quarterly Review_ , 17 (1901): 302\u201318.\n\n39 Sieghart, _Government by Decree_ , p. 221.\n\n40 Sir Carleton Kemp Allen, _Law and Orders: An Inquiry into the Nature and Scope of Delegated Legislation and Executive Powers in England_ (London: Stevens, 1945), p. 28.\n\n## FOURTEEN\n\nTHE SAFEGUARDS OF INDIVIDUAL LIBERTY\n\nAt this little gap every man's liberty may in time go out.\n\n\u2014John Selden\n\n1. It is time to try to pull together the various historical strands and to state systematically the essential conditions of liberty under the law. Mankind has learned from long and painful experience that the law of liberty must possess certain attributes.1 What are they?\n\nThe first point that must be stressed is that, because the rule of law means that government must never coerce an individual except in the enforcement  of a known rule,2 it constitutes a limitation on the powers of all government, including the powers of the legislature. It is a doctrine concerning what the law ought to be, concerning the general attributes that particular laws should possess. This is important because today the conception of the rule of law is sometimes confused with the requirement of mere legality in all government action. The rule of law, of course, presupposes complete legality, but this is not enough: if a law gave the government unlimited power to act as it pleased, all its actions would be legal, but it would certainly not be under the rule of law. The rule of law, therefore, is also more than constitutionalism: it requires that all laws conform to certain principles.\n\nFrom the fact that the rule of law is a limitation upon all legislation, it follows that it cannot itself be a law in the same sense as the laws passed by the legislator. Constitutional provisions may make infringements of the rule of law more difficult. They may help to prevent inadvertent infringements by  routine legislation.3 But the ultimate legislator can never limit his own powers by law, because he can always abrogate any law he has made.4 The rule of law is therefore not a rule of the law, but a rule concerning what the law ought to be, a meta-legal doctrine or a political ideal.5 It will be effective only in so far as the legislator feels bound by it. In a democracy this means that it will not prevail unless it forms part of the moral tradition of the community, a common ideal shared and unquestioningly accepted by the majority.6\n\nIt is this fact that makes so very ominous the persistent attacks on the principle of the rule of law. The danger is all the greater because many of the applications of the rule of law are also ideals which we can hope to approach very closely but can never fully realize. If the ideal of the rule of law is a firm element of public opinion, legislation and jurisdiction will tend to approach it more and more closely. But if it is represented as an impracticable and even undesirable ideal and people cease to strive for its realization, it will rapidly disappear. Such a society will quickly relapse into a state of arbitrary tyranny. This is what has been threatening during the last two or three generations throughout the Western world.\n\nIt is equally important to remember that the rule of law restricts government  only in its coercive activities.7 These will never be the only functions of government. Even in order to enforce the law, the government requires an apparatus of personal and material resources which it must administer. And there are whole fields of governmental activity, such as foreign policy, where the problem of coercion of the citizens does not normally arise. We shall have to return to this distinction between the coercive and the non-coercive activities of government. For the moment, all that is important is that the rule of law is concerned only with the former.\n\nThe chief means of coercion at the disposal of government is punishment. Under the rule of law, government can infringe a person's protected private sphere only as punishment for breaking an announced general rule. The principle \"nullum crimen, nulla poena sine lege\"8 is therefore the most important consequence of the ideal. But clear and definite as this statement may at first seem, it raises a host of difficulties if we ask what precisely is meant by \"law.\" Certainly the principle would not be satisfied if the law merely said that whoever disobeys the orders of some official will be punished in a specified manner. Yet even in the freest countries the law often seems to provide for such acts of coercion. There probably exists no country where a person will not on certain occasions, such as when he disobeys a policeman, become liable to punishment for \"an act done to the public mischief \" or for \"disturbing the public order\" or for \"obstructing the police.\" We shall therefore not fully understand even this crucial part of the doctrine without examining the whole complex of principles which together make possible the rule of law.\n\n2. We have seen earlier that the ideal of the rule of law presupposes a very definite conception of what is meant by law and that not every enactment  of the legislative authority is a law in this sense.9 In current practice, everything is called \"law\" which has been resolved in the appropriate manner by a legislative authority. But of these laws in the formal sense of the word,10 only some\u2014today usually only a very small proportion\u2014are substantive (or \"material\") laws regulating the relations between private persons or between  such persons and the state. The great majority of the so-called laws are rather instructions issued by the state to its servants concerning the manner in which they are to direct the apparatus of government and the means which are at their disposal. Today it is everywhere the task of the same legislature to direct the use of these means and to lay down the rules which the ordinary citizen must observe. This, though the established practice, is not a necessary state of affairs. I cannot help wondering whether it might not be desirable to prevent the two types of decisions from being confused11 by entrusting the task of laying down general rules and the task of issuing orders to the administration to distinct representative bodies and by subjecting their decisions to independent judicial review so that neither will overstep its bounds. Though we may wish both kinds of decisions to be controlled democratically, this need not mean that they should be in the hands of the same assembly.12\n\nThe present arrangements help to obscure the fact that, though government has to administer means which have been put at its disposal (including the services of all those whom it has hired to carry out its instructions), this does not mean that it should similarly administer the efforts of private citizens. What distinguishes a free from an unfree society is that in the former each individual has a recognized private sphere clearly distinct from the public sphere, and the private individual cannot be ordered about but is expected to obey only the rules which are equally applicable to all. It used to be the boast of free men that, so long as they kept within the bounds of the known law, there was no need to ask anybody's permission or to obey anybody's orders. It is doubtful whether any of us can make this claim today.\n\nThe general, abstract rules, which are laws in the substantive sense, are, as we have seen, essentially long-term measures, referring to yet unknown cases and containing no references to particular persons, places, or objects. Such laws must always be prospective, never retrospective, in their effect. That this should be so is a principle, almost universally accepted but not always put into legal form; it is a good example of those meta-legal rules which must be observed if the rule of law is to remain effective.\n\n3. The second chief attribute which must be required of true laws is that they be known and certain.13 The importance which the certainty of the law has for the smooth and efficient running of a free society can hardly be exaggerated. There is probably no single factor which has contributed more to the prosperity of the West than the relative certainty of the law which has prevailed here.14 This is not altered by the fact that complete certainty of the law is an ideal which we must try to approach but which we can never perfectly attain. It has become the fashion to belittle the extent to which such certainty has in fact been achieved, and there are understandable reasons why lawyers, concerned mainly with litigation, are apt to do so. They have normally  to deal with cases in which the outcome is uncertain. But the degree of the certainty of the law must be judged by the disputes which do not lead to litigation because the outcome is practically certain as soon as the legal position is examined. It is the cases that never come before the courts, not those that do, that are the measure of the certainty of the law. The modern tendency to exaggerate this uncertainty is part of the campaign against the rule of law, which we shall examine later.15\n\nThe essential point is that the decisions of the courts can be predicted, not that all the rules which determine them can be stated in words. To insist that the actions of the courts be in accordance with pre-existing rules is not to insist that all these rules be explicit, that they be written down beforehand in so many words. To insist on the latter would, indeed, be to strive for an unattainable ideal. There are \"rules\" which can never be put into explicit form. Many of these will be recognizable only because they lead to consistent and predictable decisions and will be known to those whom they guide as, at most, manifestations of a \"sense of justice.\"16 Psychologically, legal reasoning does not, of course, consist of explicit syllogisms, and the major premises will often not be explicit.17 Many of the general principles on which the conclusions depend will be only implicit in the body of formulated law and will have to be discovered by the courts. This, however, is not a peculiarity of legal reasoning. Probably all generalizations that we can formulate depend on still higher generalizations which we do not explicitly know but which nevertheless govern the working of our minds. Though we will always try to discover those more general principles on which our decisions rest, this is probably by its nature an unending process that can never be completed.\n\n4. The third requirement of true law is equality. It is as important, but much more difficult, to define than the others. That any law should apply equally to all means more than that it should be general in the sense we have defined. A law may be perfectly general in referring only to formal characteristics of the persons involved18 and yet make different provisions for different classes  of people. Some such classification, even within the group of fully responsible citizens, is clearly inevitable. But classification in abstract terms can always be carried to the point at which, in fact, the class singled out consists only of particular known persons or even a single individual.19 It must be admitted that, in spite of many ingenious attempts to solve this problem, no entirely satisfactory criterion has been found that would always tell us what kind of classification is compatible with equality before the law. To say, as has so often been said, that the law must not make irrelevant distinctions or that it must not discriminate between persons for reasons which have no connection with the purpose of the law20 is little more than evading the issue.\n\nYet, though equality before the law may thus be one of the ideals that indicate the direction without fully determining the goal and may therefore always remain beyond our reach, it is not meaningless. We have already mentioned one important requirement that must be satisfied, namely, that those inside any group singled out acknowledge the legitimacy of the distinction as well as those outside it. As important in practice is that we ask whether we can or cannot foresee how a law will affect particular people. The ideal of equality of the law is aimed at equally improving the chances of yet unknown people but incompatible with benefiting or harming known persons in a predictable manner.\n\nIt is sometimes said that, in addition to being general and equal, the law of the rule of law must also be just. But though there can be no doubt that, in order to be effective, it must be accepted as just by most people, it is doubtful whether we possess any other formal criteria of justice than generality and equality\u2014unless, that is, we can test the law for conformity with more general rules which, though perhaps unwritten, are generally accepted, once they have been formulated. But, so far as its compatibility with a reign of freedom is concerned, we have no test for a law that confines itself to regulating the relations between different persons and does not interfere with the purely private concerns of an individual, other than its generality and equality. It is true that such \"a law may be bad and unjust; but its general and abstract formulation reduces this danger to a minimum. The protective character of the law, its very _raison d'\u00eatre_ , are to be found in its generality.\"21\n\nIf it is often not recognized that general and equal laws provide the most effective protection against infringement of individual liberty, this is due mainly to the habit of tacitly exempting the state and its agents from them and of assuming that the government has the power to grant exemptions to individuals. The ideal of the rule of law requires that the state either enforce the law upon others\u2014and that this be its only monopoly\u2014or act under the same law and therefore be limited in the same manner as any private person.22 It is this fact that all rules apply equally to all, including those who govern, which makes it improbable that any oppressive rules will be adopted.\n\n5. It would be humanly impossible to separate effectively the laying-down of new general rules and their application to particular cases unless these functions were performed by different persons or bodies. This part at least of the doctrine of the separation of powers23 must therefore be regarded as an integral part of the rule of law. Rules must not be made with particular cases in mind, nor must particular cases be decided in the light of anything but the general rule\u2014though this rule may not yet have been explicitly formulated and therefore have to be discovered. This requires independent judges who are not concerned with any temporary ends of government. The main point is that the two functions must be performed separately by two co-ordinated bodies before it can be determined whether coercion is to be used in a particular case.\n\nA much more difficult question is whether, under a strict application of the rule of law, the executive (or the administration) should be regarded as a distinct and separate power in this sense, co-ordinated on equal terms with the other two. There are, of course, areas where the administration must be free to act as it sees fit. Under the rule of law, however, this does not apply to coercive powers over the citizen. The principle of the separation of powers must not be interpreted to mean that in its dealing with the private citizen the administration is not always subject to the rules laid down by the legislature and applied by independent courts. The assertion of such a power is the very antithesis of the rule of law. Though under any workable system the administration must undoubtedly have powers which cannot be controlled by independent courts, \"Administrative Powers over Person and Property\" cannot be among them. The rule of law requires that the executive in its coercive action be bound by rules which prescribe not only when and where it may use coercion but also in what manner it may do so. The only way in which this can be ensured is to make all its actions of this kind subject to judicial review.\n\nWhether the rules by which the administration is bound should be laid down by the general legislature or whether this function may be delegated  to another body is, however, a matter of political expediency.24 This does not bear directly on the principle of the rule of law, but rather on the question of the democratic control of government. So far as the principle of the rule of law is concerned, there is no objection to delegation of legislation as such. Clearly, the delegation of the power of making rules to local legislative bodies, such as provincial assemblies or municipal councils, is unobjectionable from every point of view. Even the delegation of this power to some non-elective authority need not be contrary to the rule of law, so long as such authority is bound to announce these rules prior to their application and then can be made to adhere to them. The trouble with the widespread use of delegation in modern times is not that the power of making general rules is delegated but that administrative authorities are, in effect, given power to wield coercion without rule, as no general rules can be formulated which will unambiguously guide the exercise of such power. What is often called \"delegation of lawmaking power\" is often not delegation of the power to make rules\u2014which might be undemocratic or politically unwise\u2014but delegation of the authority to give to any decision the force of law, so that, like an act of the legislature, it must be unquestioningly accepted by the courts.\n\n6. This brings us to what in modern times has become the crucial issue, namely the legal limits of administrative discretion. Here is \"the little gap at which in time every man's liberty may go out.\"25\n\nThe discussion of this problem has been obscured by a confusion over the meaning of the term \"discretion.\" We use the word first with regard to the power of the judge to interpret the law. But authority to interpret a rule is not discretion in the sense relevant to us. The task of the judge is to discover the implications contained in the spirit of the whole system of valid rules of law or to express as a general rule, when necessary, what was not explicitly stated previously in a court of law or by the legislator. That this task of interpretation is not one in which the judge has discretion in the sense of authority to follow his own will to pursue particular concrete aims appears from the fact that his interpretation of the law can be, and as a rule is, made subject to review by a higher court. Whether or not the substance of a decision is subject to review by another such body that needs to know only the existing rules and the facts of the case is probably the best test as to whether a decision is bound  by rule or left to the discretion of the judge's authority. A particular interpretation of the law may be subject to dispute, and it may sometimes be impossible to arrive at a fully convincing conclusion; but this does not alter the fact that the dispute must be settled by an appeal to the rules and not by a simple act of will.\n\nDiscretion in a different and for our purposes equally irrelevant sense is a problem which concerns the relation between principal and agent throughout the whole hierarchy of government. At every level, from the relation between the sovereign legislature and the heads of the administrative departments down the successive steps in the bureaucratic organization, the problem arises as to what part of the authority of government as a whole should be delegated to a specific office or official. Since this assignment of particular tasks to particular authorities is decided by law, the question of what an individual agency is entitled to do, what parts of the powers of government it is allowed to exercise, is often also referred to as a problem of discretion. It is evident that not all the acts of government can be bound by fixed rules and that at every stage of the governmental hierarchy considerable discretion must be granted to the subordinate agencies. So long as the government administers its own resources, there are strong arguments for giving it as much discretion as any business management would require in similar circumstances. As Dicey has pointed out, \"in the management of its own business, properly so called, the government will be found to need that freedom of action, necessarily possessed by every private person in the management of his own personal concerns.\"26 It may well be that legislative bodies are often overzealous in limiting the discretion of the administrative agencies and unnecessarily hamper their efficiency. This may be unavoidable to some degree; and it is probably necessary that bureaucratic organizations should be bound by rule to a greater extent than business concerns, as they lack that test of efficiency which profits provide in commercial affairs.27\n\nThe problem of discretionary powers as it directly affects the rule of law is not a problem of the limitation of the powers of particular agents of government but of the limitation of the powers of the government as a whole. It is a problem of the scope of administration in general. Nobody disputes the fact that, in order to make efficient use of the means at its disposal, the government must exercise a great deal of discretion. But, to repeat, under the rule of  law the private citizen and his property are not an object of administration by the government, not a means to be used for its purposes. It is only when the administration interferes with the private sphere of the citizen that the problem of discretion becomes relevant to us; and the principle of the rule of law, in effect, means that the administrative authorities should have no discretionary powers in this respect.\n\nIn acting under the rule of law the administrative agencies will often have to exercise discretion as the judge exercises discretion in interpreting the law. This, however, is a discretionary power which can and must be controlled by the possibility of a review of the substance of the decision by an independent court. This means that the decision must be deducible from the rules of law and from those circumstances to which the law refers and which can be known to the parties concerned. The decision must not be affected by any special knowledge possessed by the government or by its momentary purposes and the particular values it attaches to different concrete aims, including the preferences it may have concerning the effects on different people.28\n\nAt this point the reader who wants to understand how liberty in the modern world may be preserved must be prepared to consider a seemingly fine point of law, the crucial importance of which is often not appreciated. While in all civilized countries there exists some provision for an appeal to courts against administrative decisions, this often refers only to the question as to whether an authority had a right to do what it did. We have already seen, however, that  if the law said that everything a certain authority did was legal, it could not be restrained by a court from doing anything. What is required under the rule of law is that a court should have the power to decide whether the law provided for a particular action that an authority has taken. In other words, in all instances where administrative action interferes with the private sphere of the individual, the courts must have the power to decide not only whether a particular action was _infra vires_ or _ultra vires_ but whether the substance of the administrative decision was such as the law demanded. It is only if this is the case that administrative discretion is precluded.\n\nThis requirement clearly does not apply to the administrative authority which tries to achieve particular results with the means at its disposal.29 It is, however, of the essence of the rule of law that the private citizen and his property should not in this sense be means at the disposal of government. Where coercion is to be used only in accordance with general rules, the justification of every particular act of coercion must derive from such a rule. To ensure this, there must be some authority which is concerned only with the rules and not with any temporary aims of government and which has the right to say not only whether another authority had the right to act as it did but whether what it did was required by the law.\n\n7. The distinction with which we are now concerned is sometimes discussed in terms of the contrast between legislation and policy. If the latter term is appropriately defined, we will indeed be able to express our main point by saying that coercion is admissible only when it conforms to general laws and not when it is a means of achieving particular objects of current policy. This manner of stating it is, however, somewhat misleading, because the term \"policy\" is also used in a wider sense, in which all legislation falls under it. In this sense legislation is the chief instrument of long-term policy, and all that is done in applying the law is to carry out a policy that has been determined in advance.\n\nA further source of confusion is the fact that within law itself the expression \"public policy\" is commonly used to describe certain pervading general principles which are often not laid down as written rules but are understood to qualify the validity of more specific rules.30 When it is said that it is the policy of the law to protect good faith, to preserve public order, or not to recognize  contracts for immoral purposes, this refers to rules, but rules which are stated in terms of some permanent end of government rather than in terms of rules of conduct. It means that, within the limits of the powers given to it, the government must so act that that end will be achieved. The reason why the term \"policy\" is used in such instances appears to be that it is felt that to specify the end to be achieved is in conflict with the conception of law as an abstract rule. Though such reasoning may explain the practice, it is clearly one which is not without danger.\n\nPolicy is rightly contrasted with legislation when it means the pursuit by government of the concrete, ever changing aims of the day. It is with the execution of policy in this sense that administration proper is largely concerned. Its task is the direction and allocation of resources put at the disposal of government in the service of the constantly changing needs of the community. All the services which the government provides for the citizen, from national defense to upkeep of roads, from sanitary safeguards to the policing of the streets,31 are necessarily of this kind. For these tasks it is allowed definite means and its own paid servants, and it will constantly have to decide on the next urgent task and the means to be used. The tendency of the professional administrators concerned with these tasks is inevitably to draw everything they can into the service of the public aims they are pursuing. It is largely as a protection of the private citizen against this tendency of an ever growing administrative machinery to engulf the private sphere that the rule of law is so important today. It means in the last resort that the agencies entrusted with such special tasks cannot wield for their purpose any sovereign powers (no _Hoheitsrechte_ , as the Germans call it) but must confine themselves to the means specially granted to them.\n\n8. Under a reign of freedom the free sphere of the individual includes all action not explicitly restricted by a general law. We have seen that it was found especially necessary to protect against infringement by authority some of the more important private rights, and also how apprehension was felt that such an explicit enumeration of some might be interpreted to mean that only they enjoyed the special protection of the constitution. These fears have proved to be only too well founded. On the whole, however, experience seems to confirm the argument that, in spite of the inevitable incompleteness of any bill of rights, such a bill affords an important protection for certain rights known to be easily endangered. Today we must be particularly aware that, as a result of technological change, which constantly creates new potential threats to individual  liberty, no list of protected rights can be regarded as exhaustive.32 In an age of radio and television, the problem of free access to information is no longer a problem of the freedom of the press. In an age when drugs or psychological techniques can be used to control a person's actions, the problem of free control over one's body is no longer a matter of protection against physical restraint. The problem of the freedom of movement takes on a new significance when foreign travel has become impossible for those to whom the authorities of their own country are not willing to issue a passport.\n\nThe problem assumes the greatest importance when we consider that we are probably only at the threshold of an age in which the technological possibilities of mind control are likely to grow rapidly and what may appear at first as innocuous or beneficial powers over the personality of the individual will be at the disposal of government. The greatest threats to human freedom probably still lie in the future. The day may not be far off when authority, by adding appropriate drugs to our water supply or by some other similar device, will be able to elate or depress, stimulate or paralyze, the minds of whole populations for its own purposes.33 If bills of rights are to remain in any way meaningful, it must be recognized early that their intention was certainly to protect the individual against all vital infringements of his liberty and that therefore they must be presumed to contain a general clause protecting against government's interference those immunities which individuals in fact have enjoyed in the past.\n\nIn the last resort these legal guaranties of certain fundamental rights are no more than part of the safeguards of individual liberty which constitutionalism provides, and they cannot give greater security against legislative infringements of liberty than the constitutions themselves. As we have seen, they can do no more than give protection against hasty and improvident action of current legislation and cannot prevent any suppression of rights by the deliberate action of the ultimate legislator. The only safeguard against this is clear awareness of the dangers on the part of public opinion. Such provisions are important mainly because they impress upon the public mind the value of  these individual rights and make them part of a political creed which the people will defend even when they do not fully understand its significance.\n\n9. We have up to this point represented those guaranties of individual freedom as if they were absolute rights which could never be infringed. In actual fact they cannot mean more than that the normal running of society is based on them and that any departure from them requires special justification. Even the most fundamental principles of a free society, however, may have to be temporarily sacrificed when, but only when, it is a question of preserving liberty in the long run, as in the case of war. Concerning the need of such emergency powers of government in such instances (and of safeguards against their abuse) there exists widespread agreement.\n\nIt is not the occasional necessity of withdrawing some of the civil liberties by a suspension of habeas corpus or the proclamation of a stage of siege that we need to consider further, but the conditions under which the particular rights of individuals or groups may occasionally be infringed in the public interest. That even such fundamental rights as freedom of speech may have to be curtailed in situations of \"clear and present danger,\" or that the government may have to exercise the right of eminent domain for the compulsory purchase of land, can hardly be disputed. But if the rule of law is to be preserved, it is necessary that such actions be confined to exceptional cases defined by rule, so that their justification does not rest on the arbitrary decision of any authority but can be reviewed by an independent court; and, second, it is necessary that the individuals affected be not harmed by the disappointment of their legitimate expectations but be fully indemnified for any damage they suffer as a result of such action.\n\nThe principle of \"no expropriation without just compensation\" has always been recognized wherever the rule of law has prevailed. It is, however, not always recognized that this is an integral and indispensable element of the principle of the supremacy of the law. Justice requires it; but what is more important is that it is our chief assurance that those necessary infringements of the private sphere will be allowed only in instances where the public gain is clearly greater than the harm done by the disappointment of normal individual expectations. The chief purpose of the requirement of full compensation is indeed to act as a curb on such infringements of the private sphere and to provide a means of ascertaining whether the particular purpose is important enough to justify an exception to the principle on which the normal working of society rests. In view of the difficulty of estimating the often intangible advantages of public action and of the notorious tendency of the expert administrator to overestimate the importance of the particular goal of the moment, it would even seem desirable that the private owner should always have the benefit of the doubt and that compensation should be fixed as high as possible without opening the door to outright abuse. This means, after all,  no more than that the public gain must clearly and substantially exceed the loss if an exception to the normal rule is to be allowed.\n\n10. We have now concluded the enumeration of the essential factors which together make up the rule of law, without considering those procedural safeguards such as habeas corpus, trial by jury, and so on, which, in the Anglo-Saxon countries, appear to most people as the chief foundations of their liberty.34 English and American readers will probably feel that I have put the cart before the horse and concentrated on minor features while leaving out what is fundamental. This has been quite deliberate.\n\nI do not wish in any way to disparage the importance of these procedural safeguards. Their value for the preservation of liberty can hardly be overstated. But while their importance is generally recognized, it is not understood that they presuppose for their effectiveness the acceptance of the rule of law as here defined and that, without it, all procedural safeguards would be valueless. True, it is probably the reverence for these procedural safeguards that has enabled the English-speaking world to preserve the medieval conception of the rule of law over men. Yet this is no proof that liberty will be preserved if the basic belief in the existence of abstract rules of law which bind all authority in their action is shaken. Judicial forms are intended to insure that decisions will be made according to rules and not according to the relative desirability of particular ends or values. All the rules of judicial procedure, all the principles intended to protect the individual and to secure impartiality of justice, presuppose that every dispute between individuals or between individuals and the state can be decided by the application of general law. They are designed to make the law prevail, but they are powerless to protect justice where the law deliberately leaves the decision to the discretion of authority. It is only where the law decides\u2014and this means only where independent courts have the last word\u2014that the procedural safeguards are safeguards of liberty.\n\nI have here concentrated on the fundamental conception of law which the traditional institutions presuppose because the belief that adherence to the external forms of judicial procedure will preserve the rule of law seems to me the greatest threat to its preservation. I do not question, but rather wish to emphasize, that the belief in the rule of law and the reverence for the forms of justice belong together and that neither will be effective without the other. But it is the first which is chiefly threatened today; and it is the illusion that it will be preserved by scrupulous observation of the forms of justice that is one  of the chief causes of this threat. \"Society is not going to be saved by importing the forms and rules of judicial procedure into places where they do not naturally belong.\"35 To use the trappings of judicial form where the essential conditions for a judicial decision are absent, or to give judges power to decide issues which cannot be decided by the application of rules, can have no effect but to destroy the respect for them even where they deserve it.\n\n* * *\n\nThe quotation at the head of the chapter is taken from John Selden's speech in the \"Proceedings in Parliament Relating to the Liberty of the Subject, 1627\u20131628,\" in Thomas Bayly Howell, _A Complete Collection of State Trials and Proceedings for High Treason and Other Crimes and Misdemeanors from the Earliest Period to the Year 1783_ (34 vols.; London: T. C. Hansard, 1816\u201328), vol. 3, p. 170. [The debate took place regarding the Proceedings on Habeas Corpus brought by Sir Thomas Darnel, Sir John Corbet, Sir Walter Earl, Sir John Heveningham, and Sir Edmund Hampton, November 1627, at the Court of King's Bench in Westminster.\u2014Ed.]\n\n1 The recent discussions of the meaning of the rule of law are very numerous, and we can list here merely some of the more significant ones: Sir Carleton Kemp Allen, _Law and Orders: An Inquiry into the Nature and Scope of Delegated Legislation and Executive Powers in England_ (London: Stevens, 1945), esp. pp. 274\u201375; Sir Ernest Barker, \"The 'Rule of Law,'\" _Political Quarterly_ , 1 (1914): 117\u201340; reprinted in Ernest Barker, _Church, State, and Study: Essays_ (London: Methuen, 1930), pp. 171\u201392; Hale Leigh Bellot, \"The Rule of Law,\" _Quarterly Review_ , 246 (1926): 346\u201365; Robin George Collingwood, _The New Leviathan; or, Man, Society, Civilization, and Barbarism_ (Oxford: Clarendon Press, 1942), chap. 39 [\"Law and Order\"], pp. 326\u201332; John Dickinson, _Administrative Justice and the Supremacy of Law in the United States_ (Cambridge, MA: Harvard University Press, 1927); Carl Joachim Friedrich, _Constitutional Government and Democracy: Theory and Practice in Europe and America_ (Boston: Little, Brown and Co., 1941); Frank Johnson Goodnow, _Politics and Administration: A Study in Government_ (New York: Macmillan, 1900); Arthur Norman Holcombe, _The Foundations of the Modern Commonwealth_ (New York: Harper and Brothers, 1923), chap. 11, pp. 436\u2013 79; Harry W. Jones, \"The Rule of Law and the Welfare State,\" _Columbia Law Review_ , 58 (1958): 143\u201356; Walter Lippman, _An Inquiry into the Principles of the Good Society_ (Boston: Little, Brown and Co., 1937); Horace Harmon Lurton, \"A Government of Law or a Government of Men,\" _North American Review_ , 193 (1911): 1\u201325; Charles Howard McIlwain, \"Government by Law,\" _Foreign Affairs: An American Quarterly Review_ , 14 (1935\u201336): 185\u201398; reprinted in his _Constitutionalism and the Changing World_ (Cambridge: Cambridge University Press, 1939), pp. 266\u201382; Franz Leopold Neumann, _The Democratic and the Authoritarian State: Essays in Political and Legal Theory_ (Glencoe, IL: The Free Press, 1957); James Roland Pennock, _Administration and the Rule of Law_ (New York: Farrer and Rinehart, 1941); Roscoe Pound, _Encyclopedia of the Social Sciences_ , vol. 13 (1934), pp. 463\u201366, S.V. \"Rule of Law,\" and \"The Rule of Law and the Modern Social Welfare State,\" _Vanderbilt Law Review_ , 7 (1953): 1\u201334; Francis Graham Wilson, _The Elements of Modern Politics: An Introduction to Political Science_ (New York: McGraw-Hill, 1936); cf. also Inns of Court Conservative and Unionist Society, _Rule of Law: A Study_ (London: Conservative Political Centre, 1955).\n\nMaxime Leroy, _La Loi: Essai sur la th\u00e9orie de l'autorit\u00e9 dans la d\u00e9mocratie_ (Paris: V. Giard and E. Bri\u00e8re, 1908); Albert Picot, _L'\u00c9tat fond\u00e9 sur le droit et le droit p\u00e9nal_ (Basel: Helbling and Lichtenhahn, 1944), pp. 201a\u20138a of the _Actes de la Soci\u00e9t\u00e9 Suisse des Juristes \/ Verhandlungen des Schweizerischen Juristenvereins \/ Atti della Societ\u00e0 svizzera dei Giuristi_ ; Marcel Waline, _L'Individualisme et le droit_ (2nd ed.; Paris: \u00c9ditions Domat Montchrestien, 1949).\n\nThe conduct of Carl Schmitt under the Hitler regime does not alter the fact that, of the modern German writings on the subject, his are still among the most learned and perceptive; see particularly his _Verfassungslehre_ (Munich: Duncker und Humblot, 1928) and _Der H\u00fcter der Verfassung_ (T\u00fcbingen: Mohr, 1931). Similarly important for the pre-Nazi state of thought are Hermann Heller, _Rechtsstaat oder Diktatur?_ (T\u00fcbingen: Mohr, 1930) and _Staatslehre_ (Leiden: A. W. Sijthoff, 1934), and Friedrich Darmst\u00e4dter, _Die Grenzen der Wirksamkeit des Rechtsstaates: Eine Untersuchung zur gegenw\u00e4rtigen Krise des liberalen Staatsgedankens_ (Heidelberg: C. Winter, 1930) and _Rechtsstaat oder Machtstaat?_ (Berlin: Rothschild, 1932). Cf. John Hamilton Hallowell, _The Decline of Liberal Ideology: With Particular Reference to German Politco-Legal Thought_ (Berkeley: University of California Press, 1943). Of the German postwar literature see particularly Franz B\u00f6hm, \"Freiheitsordnung und soziale Frage,\" in _Grundsatzfragen der Wirtschaftsordnung: Vortragszyklus_ [ _veranstaltet von der Wirtschafts-und Sozialwissenschaftlichen Fakult\u00e4t der Freien Universit\u00e4t Berlin Sommersemester 1953, Wirtschaftswissenschaftliche Abhandlungen,_ vol. 2] (Berlin: Duncker und Humblot, 1954), pp. 71\u201395; Christian-Friedrich Menger, _Der Begriff des sozialen Rechtsstaates im Bonner Grundgesetz_ (T\u00fcbingen: J. C. B. Mohr, 1953); Richard Lange, _Der Rechtsstaat als Zentralbegriff der neuesten Strafrechtsentwicklung_ (T\u00fcbingen: Mohr, 1952); Hermann Wandersleb, ed., _Recht, Staat, Wirtschaft; Schriftenreihe des Innenministers des Landes Nordrhein-Westfalen f\u00fcr staatswissenschaftliche Fortbildung_ (4 vols.; vols. 1\u20132, Stuttgart: Kohlhammer; vols. 3\u20134, D\u00fcsseldorf: L. Swann, 1949\u201353); and Ren\u00e9 Marcic, _Vom Gesetzesstaat zum Richterstaat_ (Vienna: Springer, 1957).\n\nOf special importance, mainly on the relation between democracy and the _Rechtsstaat_ , is the extensive Swiss literature in this field, largely under the influence of Fritz Fleiner and his disciple and successor, Zaccaria Giacometti. Beginning with Fleiner's _Schweizerisches Bundesstaatsrecht_ (T\u00fcbingen: Mohr, 1923; new ed. by Zaccaria Giacometti [Zurich: Polygraphischer Verlag, 1949]) and his _Institutionen des deutschen Verwaltungsrechts_ (8th ed.; T\u00fcbingen: Mohr, 1928), see Giacometti's _Die Verfassungsgerichtsbarkeit des schweizerischen Bundesgerichtes (die staatsrechtliche Beschwerde)_ (Zurich: Polygraphischer Verlag, 1933), and the volume dedicated to him under the title. _Demokratie und Rechtsstaat: Festgabe zum 60. Geburtstag von Zaccaria Giacometti_ (Zurich: Polygraphischer Verlag, 1953), especially the contribution by Werner K\u00e4gi [\"Rechtsstaat und Demokratie (Antinomie und Synthese),\" pp. 107\u201342]; Richard B\u00e4umlin, _Die rechtsstaatliche Demokratie: eine Untersuchung der gegenseitigen Beziehungen von Demokratie und Rechtsstaat_ (Zurich: Polygraphischer Verlag, 1954); Rudolph Heinrich Grossmann, _Die staats-und rechtsideologischen Grundlagen der Verfassungsgerichtsbarkeit in den Vereinigten Staaten von Amerika und in der Schweiz. Eine Studie in amerikanischem und schweizerischem Staats-und Rechtsdenken_ (dissertation; Zurich: Schulthess, 1948); Werner K\u00e4gi, _Die Verfassung als rechtliche Grundordnung des Staates: Untersuchungen \u00fcber die Entwicklungstendenzen im modernen Verfassungsrecht_ (Zurich: Polygraphischer Verlag, 1945); \"Demokratie, Gleichheit und Egalitarismus,\" in _Die Schweiz: Ein nationales Jahrbuch_ [vol. 25] (Bern: Jahrbuch Verlag, 1954), pp. 36\u201345; and esp. \"Rechtsfragen der Volksinitiative auf Partialrevision,\" in _Verhandlungen des Schweizerischen Juristenvereins_ , bk. 4 (Basel, 1956); Max Imboden, _Der Schutz vor staatlicher Willk\u00fcr_ (Z\u00fcrich: Polygraphischer Verlag, 1945); and Max Imboden, _Das Gesetz als Garantie rechtsstaatlicher Verwaltung_ (Basel: Helbing, 1954); Andreas Brunner, _Rechtsstaat gegen Totalstaat_ (dissertation; 2 vols.; Z\u00fcrich: Polygraphischer Verlag, 1948); and Juristische Fakult\u00e4ten der schweizerischen Universit\u00e4ten, ed., _Die Freiheit des B\u00fcrgers im schweizerischen Recht: Festgabe zur Hundertjahrfeier der Bundesverfassung \/ La libert\u00e9 du citoyen en droit suisse: recueil du centenaire de la constitution f\u00e9d\u00e9rale publi\u00e9 par la Facult\u00e9 de Droit des Universit\u00e9s suisses_ , by various authors (Zurich: Polygraphischer Verlag, 1948).\n\nCf. also Carel Henrik Frederik Polak, _Ordening en Rechtsstaat_ [Accepted by the Rijksuniversiteit te Leiden, 12 October 1951] (Zwolle: W. E. J. Tjeenk Willink, 1951); Lu\u00eds Legaz y Lacambra, \"El Estado de derecho,\" _Revista de administraci\u00f3n p\u00fablica_ , 2 (1951): 13\u201334 [Published in Madrid by the Instituto de Estudios Pol\u00edticos]; Felice Battaglia, \"Stato etico e Stato di diritto,\" _Rivista internazionale di filosofia di diritto_ 17 (1937): 237\u201387; and International Commission of Jurists, _Report of the International Congress of Jurists, held June 13\u201320, 1955, at Athens_ (The Hague: International Commission of Jurists, 1956).\n\n2 A clear recent statement of this basic principle of a truly liberal system occurs in Neumann, _The Democratic and the Authoritarian State_ , p. 31: \"It is the most important and perhaps the decisive demand of liberalism that interference with the rights reserved to the individual is not permitted on the basis of individual but only on the basis of general laws\"; and p. 166: \"The liberal legal tradition rests, therefore, upon a very simple statement: individual rights may be interfered with by the state only if the state can prove its claim by reference to a general law which regulates an indeterminate number of future cases; this excludes retroactive legislation and demands a separation of legislative from judicial functions.\" Cf. also the quotation in n. 12 to chapter 13, above. The seemingly slight shift in emphasis which, with the rise of legal positivism, made this doctrine ineffective comes out clearly if we compare two characteristic statements from the latter part of the last century. Adh\u00e9mar Esmein, ( _\u00c9l\u00e9ments de droit constitutionnel fran\u00e7ais et compar\u00e9_ (1896), Henry N\u00e9zard, ed. [7th ed., rev.; 2 vols.; Paris: L. Tenin, 1921], vol. 1, p. 22), sees the essence of liberty in the limitation of authority by the existence of \"des r\u00e8gles fixes, connues d'avance, qui, dans le cas donn\u00e9, _dicteront_ au souverain sa d\u00e9cision\" [\"fixed rules, known in advance, which, in any given case, _will determine_ the sovereign's decision.\"\u2014Ed.] [Italics added by Hayek.]\n\nHowever, for Georg Jellinek, _System der subjektiven \u00f6ffentlichen Rechte_ (Freiburg: J. C. B. Mohr, 1892), p. 98, \"alle Freiheit ist einfach Freiheit von gesetzwidrigem Zwange.\" [\"All freedom is nothing more than freedom from unlawful coercion.\"\u2014Ed.] In the first statement only such coercion is permissible as the law requires, in the second all coercion which the law does not forbid!\n\n3 Heinrich Stoll, \"Rechtsstaatsidee und Privatrechtslehre,\" _Jherings Jahrb\u00fccher f\u00fcr die Dogmatik des b\u00fcrgerlichen Rechts_ , 76 (1925): esp. 193\u2013204. [ _Jhering_ is an alternative spelling of _Ihering._ \u2014Ed.]\n\n4 Cf. Francis Bacon's statement: \"For a supreme and absolute power cannot conclude itself, neither can that which is in its nature revocable be made fixed\" (quoted by Charles Howard McIlwain, _The High Court of Parliament_ [New Haven: Yale University Press, 1910]. [Bacon's much-quoted statement does not appear in McIlwain's treatise, despite his devoting an extensive chapter to \"The Political History of Parliamentary Supremacy\" (pp. 336\u201393). It is, however, quoted by Dicey in his _Law of the Constitution_ , p. 62, n. 2 (Liberty Fund edition, p. 21, n. 39). The quotation originates in Bacon's _History of King Henry VII from The Works of Francis Bacon, Baron Verulam, Viscount St. Albans, and Lord High Chancellor of England_ (4 vols.; Printed by R. Gosling, 1730), vol. 3, p. 463.\u2014Ed.]\n\n5 See Georg Jellinek, _Die rechtliche Natur der Staatenvertr\u00e4ge: Ein Beitrag zur juristischen Construction des V\u00f6lkerrechts_ (Vienna: A. H\u00f6lder, 1880), pp. 2\u20133, and Hans Kelsen, _Hauptprobleme der Staatsrechtslehre entwickelt aus der Lehre vom Rechtssatze_ (T\u00fcbingen: J. C. B. Mohr, 1911), pp. 50ff; cf. Benedikt Winckler, _Soltquellensis Principiorum juris libri quinque, in quibus genuina iuris, tam Naturalis quam Positivi, principia, firmissima Iurisprudentiae fundamenta ostenduntur, ejusque summus finis ob oculos ponitur, & Divina autoritas probatur_ (Leipzig: Imprimebat Laurentius Cober, 1615 ): \"In tota jurisprudentia nihil est quod minus legaliter tractari possit quam ipsa principia.\" [\"In all jurisprudence, nothing is less able to be investigated according to the law than the first principles themselves.\" Rather than having referred to the 1615 edition, which is extremely rare, Hayek is almost certain to have consulted Winckler's essay in the reprint included in Carl von Kallenborn, _Die Vorl\u00e4ufer des Hugo Grotius_ (Abtheilung II: Kritische Ausgabe der Autoren) (Leipzig: Verlag von Gustav Mayer, 1848), pp. 45\u2013148. The quotation appears on p. 50.\u2014Ed.]\n\n6 Cf. Fritz Fleiner, _Ausgew\u00e4hlte Schriften und Reden_ (Zurich: Polygraphischer Verlag, 1941). This larger work contains Fleiner's essay _Tradition, Dogma, Entwicklung als aufbauende Kr\u00e4fte der schweizerischen Demokratie_ , pp. 288\u2013302, and L\u00e9on Duguit, _Trait\u00e9 de droit constitutionnel_ (2nd ed.; 5 vols.; Paris: E. de Boccard, 1921\u201325), vol. 1, _La r\u00e8gle de droit\u2014Le probl\u00e8me de L'\u00c9tat_ , pp. 408\u20139.\n\n7 It seems to be a misunderstanding of this point that makes Lionel Robbins (\"Freedom and Order,\" in _Economics and Public Policy_ , Arthur Smithies, ed. [Washington, DC: Brookings Institution, 1955], p.153) fear that to suggest \"a conception of government that is too limited to the execution of known laws, to the exclusion of functions of initiative and discretion that cannot without distortion be left out of the picture,\" is to oversimplify our position and expose it to ridicule.\n\n8 Cf. Stefan Glaser \"Nullum crimen sine lege,\" _Journal of Comparative Legislation and International Law_ , 3rd ser., 24 (1942): 29\u201341; Heinrich Balthassar Gerland, \"Nulla poena sine lege,\" in _Die Grundrechte und Grundpflichten der Reichsverfassung: Kommentar zum zweiten Teil der Reichsverfassung_ , Carl Nipperdey, ed. (3 vols.; Berlin: Hobbing, 1929), vol. 1, pp. 368\u201386; Jerome Hall, \"Nulla poena sine lege,\" _Yale Law Journal_ , 47 (1937\u201338): 165\u201393; L\u00e9on Julliot de La Morandi\u00e8re, _De la r\u00e8gle \"nulla poena sine lege\"_ (doctoral dissertation; Paris: L. Larose et L. Tenin, 1910); Adolf Schottl\u00e4nder, _Die geschichtliche Entwicklung des Satzes: Nulla poena sine lege_ (Strafrechtliche Abhandlungen, vol. 132; Breslau: Schletter, 1911); and Orio Giacchi, \"Precedenti canonistici del principio 'Nullum crimen sine proevia lege poenali,'\" in _Studi in onore di Francesco Scaduto_ (2 vols.; Florence: Casa editrice poligrafia universitaria del dott. C. Cya, 1936), vol. 1, pp. 433\u201349. On the position of the principle as the primary condition of the rule of law see Dicey, _Law of the Constitution_ , p. 187 [Liberty Fund edition, p. 110].\n\n9 See particularly Carl Schmitt, _Unabh\u00e4ngigkeit der Richter, Gleichheit vor dem Gesetz und Gew\u00e4hrleistung des Privateigentums nach der Weimarer Verfassung: Ein Rechtsgutachten zu den Gesetzentw\u00fcrfen \u00fcber die Verm\u00f6gensauseinandersetzung mit den fr\u00fchrer regierenden F\u00fcrstenh\u00e4usern_ (Berlin: Walter de Gruyter, 1926), and Carl Schmitt, _Verfassungslehre_ (Munich and Leipzig: Duncker und Humblot, 1928).\n\n10On this distinction see Paul Laband, _Das Staatsrecht des deutschen Reiches_ (5th ed.; 4 vols.; T\u00fcbingen: J. C. B. Mohr, 1911\u201314), vol. 2, pp. 54\u201356; Ernst Seligmann, _Der Begriff des Gesetzes im materiellen und formellen Sinne_ (Berlin: Guttentag, 1886); Albert Haenel, _Studien zum deutschen Staatsrechte_. Vol. 2: _Die organisatorische Entwicklung der deutschen Reichsverfassung. Gesetz im formellen und materiellen Sinne_ (2 vols.; Leipzig: Verlag H. Haessel, 1873\u201388); and, for a discussion of the subsequent development and current state of German teaching, Karl Zeidler, _Ma\u00dfnahmegesetz und \"klassiches\" Gesetz: Eine Kritik_ (Karlruhe: C. F. M\u00fcller, 1961); Ernest Wolfgang B\u00f6ckenf\u00f6rde, _Gesetz und gesetzgebende Gewalt: Von den Anf\u00e4ngen der deutschen Staatsrechtslehre bis zur H\u00f6he des staatsrechslichen P ositivismus_ (T\u00fcbingen: Mohr Siebeck, 1958); Hans Werner Kopp, _Inhalt und Form der Gesetze als ein Problem der Rechtstheorie; mit vergleichender Ber\u00fccksichtigung der Schweiz, Deutschlands, Frankreichs, Grossbritanniens und der USA_. (dissertation; 2 vols.; Z\u00fcrich: Polygraphischer Verlag, 1958); and especially Konrad Huber, _Rechtsgesetz und Ma\u00dfnahmegesetz: eine Studie zum rechtsstaatlichen Gesetz esbegriff_ (dissertation; Berlin: Duncker und Humblot, 1963); L\u00e9on Duguit, _Trait\u00e9 de droit constitutionnel_. Vol. 1, pp. 539\u2013 50; and Raymond Carr\u00e9 de Malberg, _La Loi: Expression de la volont\u00e9 g\u00e9n\u00e9rale: \u00e9tude sur le concept de la loi dans la constitution de 1875_ (Paris: Sirey, 1931).\n\nOf great importance in this connection is also a series of cases in American constitutional law, of which only two can be quoted here. The best-known statement is probably that by Justice [Stanley] Mathews in _Hurtado v. California_ , 110 _U.S_. 516, at 535 (1884): \"It is not every act, legislative in form, that is law. Law is something more than mere will exerted as an act of power. It must be not a special rule for a particular person or a particular case, but, in the language of Mr. Webster, in his familiar definition, 'the general law, a law which hears before it condemns, which proceeds upon inquiry, and renders judgment only after trial,' so 'that every citizen shall hold his life, liberty, property and immunities under the protection of the general rules which govern society,' and thus excluding as not due process of law, acts of attainder, bills of pains and penalties, acts of confiscation, acts reversing judgments and acts directly transferring one man's estate to another, legislative judgments and decrees, and other similar special, partial and arbitrary exertions of power under the form of legislation. Arbitrary power, enforcing its edicts to the injury of the persons and property of its subjects, is not law, whether manifested as the decree of a personal monarch or of an impersonal multitude. And the limitations imposed by our constitutional law upon the action of the governments, both State and national, are essential to the preservation of public and private rights, notwithstanding the representative character of our political institutions. The enforcement of these limitations by judicial process is the device of self-governing communities to protect the rights of individuals and minorities, as well against the power of numbers, as against the violence of public agents transcending the limits of lawful authority, even when acting in the name and wielding the force of government.\" Cf. the more recent statement in _State v. Boloff_ , 138 Or 568, at 611; 646 4 P2d 326; 7 P2d 775 (1932): \"A legislative act creates a rule for all: it is not an order or command to some individual; it is permanent, not transient. A law is universal in its application; not a sudden order to and concerning a particular person.\"\n\n11 See Walter Bagehot, _The English Constitution_ (1867) in _The Works and Life of Walter Bagehot_ , [Mrs.] Russell Barrington, ed. (London: Longman, Green, and Co., 1915), vol. 5, pp. 254\u201355: \"An immense mass, indeed, of the legislation is not, in the proper language of jurisprudence, legislation at all. A law is a general command applicable to many cases. The 'special acts' which crowd the statute book and weary Parliamentary committees are applicable to one case only. They do not lay down rules according to which railways shall be made, they enact that such and such a railway shall be made from this place to that place, and they have no bearing upon any other transaction.\" Today this tendency has gone so far that an eminent English judge has been led to ask: \"Have we not come to a time when we must find another name for statute law than Law itself ? Para-law, perhaps: or even sub-laws\" (Cyril John Radcliffe, Viscount Radcliffe of Werneth, _Law and the Democratic State: Being the Presidential Address of the Right Hon. Lord Radcliffe, President of the Holdsworth Club of the Faculty of Law in the University of Birmingham, 1954\u20131955_ [Holdsworth lecture] [Birmingham: Holdsworth Club of the University of Birmingham, 1955], p. 4. Cf. also Hermann Jahrreiss, _Freiheit and Sozialstaat_ (K\u00f6lner Universit\u00e4treden No. 17; Krefeld: Scherpe, 1957); reprinted in _Mensch und Staat: Rechtsphilosophische, staatsrechtliche und v\u00f6lkerrechtliche Grundfragen in unserer Zeit_ (Cologne: C. Heymann, 1957), p. 15: \"Wir sollten es uns einmal \u00fcberlegen, ob wir nicht hinfort unter diesem ehrw\u00fcrdigen Namen 'Gesetz' nur solche Normen setzen und Stafdrohungen nur hinter solche Normen stellen sollten, die dem Jedermann ' _das_ Gesetz' zu werden verm\u00f6gen. Sie, nur sie, seien 'Gesetze'! Alle \u00fcbrigen Regelungen\u2014die technischen Details zu solchen echten Gesetzen oder selbst\u00e4ndige Vorschriften ephemeren Charakters\u2013 sollten \u00e4u\u00dferlich abgesondert unter einem anderen Namen, als etwa 'Anordnungen' ergehen und allenfalls Sanktionen nicht strafrechtlichen Charakters vorsehen, auch wenn die Legislative sie beschlie\u00dft.\" [\"We should give serious consideration, from this point forward, whether under the honored name of 'law' we should place those norms and attach penalties only to those rules that can be universalized. They, and only they, are true laws! All other regulations\u2014the technical details that are associated with true laws or instructions of a fleeting character\u2014should be clearly separated and labeled differently, and perhaps called 'directions.' In any case, they should not be accompanied by sanctions of the character of the penal code, even were they established by the legislature.\"\u2014Ed.]\n\n12 It is interesting to speculate what the development would have been if at the time when the House of Commons successfully claimed the exclusive control over expenditure and thereby in effect the control of administration, the House of Lords had succeeded in achieving exclusive power of laying down general laws including the principles on which the private individual could be taxed. A division of competence of the two legislative chambers on this principle has never been tried but may be well worth consideration.\n\n13 See Sir Henry William Rawson Wade, \"The Concept of Legal Certainty: A Preliminary Skirmish,\" _Modern Law Review_ , 4 (1941): 183\u201399; Hermann Jahrreiss, _Berechenbarkeit und Recht_ (Leipzig: T. Weicher, 1927); Carl August Emge, _Sicherheit und Gerechtigkeit: Ihre gemeinsame metajuristische Wurzel_ (Berlin: Akademie der Wissenschaften in Kommission bei W. De Guyter, 1940); and Paul Roubier, _Th\u00e9orie g\u00e9n\u00e9rale du droit: histoire des doctrines juridiques et philosophie des valeurs sociales_ (Paris: Recueil Sirey, 1946), esp. pp. 267\u201379. See also Luis Recas\u00e9ns Siches, \"Human Life, Society, and Law: Fundamentals of the Philosophy of Law,\" in _Latin American Legal Philosophy_ , Luis Recas\u00e9ns Siches, et al., eds. (Cambridge, MA: Harvard University Press, 1948), pp. 7\u2013341.\n\n14 Cf. George Godfrey Phillips, \"Notes: The Rule of Law,\" _Journal of Comparative Legislation_ , 16 (1934): 302\u20134, and the literature there quoted. See, however, Montesquieu, _Spirit of the Laws_ , bk. 6, chap. 2, and the extensive discussion in Max Weber, _Max Weber on Law in Economy and Society_ , Max Rheinstein, ed. (Cambridge, MA: Harvard University Press, 1954); also Neumann, _The Democratic and the Authoritarian State_ , p. 40.\n\n15 It is a curious fact that the same people who stress the uncertainty of the law most often at the same time represent the prediction of judicial decisions as the sole aim of legal science. If the law were as uncertain as these authors sometimes suggest, there would exist, on their own showing, no legal science whatsoever.\n\n16 Cf. Roscoe Pound, \"Why Law Day?\" _Harvard Law School Bulletin_ , 10 (1958): 4: \"The vital, the enduring part of the law, is in principles\u2014starting points for reasoning, not in rules. Principles remain relatively constant or develop along constant lines. Rules have relatively short lives. They do not develop; they are repealed and are superseded by other rules.\"\n\n17 See Edward Hirsch Levi, _An Introduction to Legal Reasoning_ (Chicago: University of Chicago Press, 1949); reprinted from _The University of Chicago Law Review_ , 15 (1948): 501\u201374.\n\n18 Cf. Ren\u00e9 Brunet, _Le Principe d'egalit\u00e9 en droit fran\u00e7ais_ (Doctoral dissertation, Universit\u00e9 de Paris Facult\u00e9 de Droit; Paris: F. Alcan, 1910); Max Friedrich R\u00fcmelin, _Die Gleichheit von dem Gesetz: Rede gehalten bei der akademischen Preisverteilung am 6. November 1928_ (T\u00fcbingen: Mohr, 1928); Otto Mainzer, _Gleichheit vor dem Gesetz, Gerechtigkeit und Recht: Entwickelt an der Frage: welche Gewalten bindet der Gleichheitssatz in Art. 109 I RV?_ (Berlin: J. Springer, 1929); Erich Kaufmann, Hans Nawiasky, Albert Hensel, and Ottmar B\u00fchler, _Die Gleichheit vor dem Gesetz im Sinne des Art. 109 der Reichsverfassung; Der Einfluss des Steurrechts auf die Begriffsbildung des \u00f6ffentlichen Rechts_ (Ver\u00f6ffentlichungen der Vereinigung deutscher Staatsrechtslehrer, No. 33; Berlin: Walter de Gruyter, 1927); Gerhard Leibholz, _Die Gleichheit vor dem Gesetz: eine Studie auf rechtsvergleichender und rechstphilosophischer Grundlage_ (Berlin: O. Liebmann, 1925); Hans Nef, _Gleichheit und Gerechtigkeit_ (Zurich: Polygraphischer Verlag, 1941); Hans Peter Ipsen, \"Gleichheit,\" in _Die Grundrechte: Handbuch der Theorie und Praxis der Grundrechte_ , Franz Leopold Neumann, Hans Carl Nipperdey, and Ulrich Scheuner, eds. (5 vols. in 7; Berlin: Duncker und Humblot, 1954), vol. 2, pp. 111\u201398; and Eduardo Luis Llorens, _La Igualdad ante la Ley_ (Murcia: Instituto de estudios pol\u00edticos de la Universidad de Murcia, 1934).\n\n19 A good illustration from another field of how a non-discrimination rule can be evaded by provisions formulated in general terms (given by Gottfried Haberler, _The Theory of International Trade with its Applications to Commercial Policy_ [London: W. Dodge and Co., 1936], p. 339) is the German customs tariff of 1902 (still in force in 1936), which, to avoid a most-favored-nations obligation, provided for a special rate of duty for \"brown or dappled cows reared at a level of at least 300 meters above the sea and passing at least one month in every summer at a height of at least 800 meters.\"\n\n20 Cf. art. 4 of the Swiss Federal Constitution: \"Die Verschiedenheiten, die der Gesetzgeber aufstellt, m\u00fcssen sachlich begr\u00fcndet sein, d. h. auf vern\u00fcnftigen und ausschlaggebenden Erw\u00e4gungen in der Natur der Sache beruhen derart, dass der Gesetzgeber nur durch solche Unterscheidungen dem inneren Zweck der Ordnung der betreffenden Lebensverh\u00e4ltnisse.\" [The English reads: \"The distinctions that the legislator specifies must be grounded in fact; that is, they must rest on reasonable and essential aspects of the matter before him such that only by making such distinctions would the legislator do justice to the intrinsic purpose of regulating the circumstances in question.\" The quotation is not a transcription of Art. 4 of the Federal Constitution, as amended to 1 July 1953 and in place at the time _The Constitution of Liberty_ was written. Art. 4 reads: \"Alle Schweizer sind vor dem Gesetz gleich. Es gibt in der Schweiz keine Untertanenverh\u00e4ltnisse, keine Vorrechte des Ortes, der Geburt, der Familie oder Personen.\" (\"All Swiss are equal before the Law. In Switzerland there is neither subjection or privilege of locality, birth, family, or person.\") See Christopher Hughes, trans., _The Federal Constitution of Switzerland_ (Oxford: Clarendon Press, 1954), pp. 6\u20137, 173. Hayek's quotation is contained in _Die Gleich heit vor dem Gesetz_ , Erich Kaufmann, et al., eds (Berlin: Walter de Gruyter, 1927), p. 10, which itself gives a quotation from Ulrich Lampert, _Das schweizerische Bundesstaatsrecht_ (Z\u00fcrich: Orell F\u00fcssli, 1918). For a bibliographical listing of works concerning this section of the Swiss Constitution see Zaccaria Giacometti, _Schweizerisches Bundesstaatsrecht_ (Zurich: Polygraphischer Verlag, 1949), p. 401.\u2014Ed.]\n\n21 L\u00e9on Duguit, _Manuel de droit constitutionnel, th\u00e9orie g\u00e9n\u00e9rale de l'\u00e9tat, le droit de l'\u00e9tat, les libert\u00e9s publiques, organisation politique_ (3rd ed.; Paris: Fontenoing and Cie., 1918), p. 96. [\"La loi peut \u00eatre mauvaise, injuste; mais \u00e9tant formul\u00e9e par voie g\u00e9n\u00e9rale et abstraite, ce danger se trouve r\u00e9duit au minimum. Ainsi encore le caract\u00e8re protecteur de la loi, sa raison d'\u00eatre elle-m\u00eame se trouvent dans sa g\u00e9n\u00e9ralit\u00e9.\"\u2014Ed.]\n\n22 It would lead too far here to raise the question of whether the distinct attributes which Continental law attaches to \"public\" as distinct from \"private\" law are compatible with freedom under the law in the Anglo-Saxon sense. Though such a classification may be useful for some purposes, it has served to give the law which regulates the relations between the individual and the state a different character from that which regulates the relations between individuals, while it seems of the essence of the rule of law that this character ought to be the same in both fields.\n\n23 See William Searle Holdsworth's review of the 9th edition of Dicey's _Introduction to the Study of the Law of the Constitution_ , in _Law Quarterly Review_ , 55 (1939): 587\u201388, which contains one of the latest authoritative statements in England of the traditional conception of the rule of law. It deserves quotation at length, but we will reproduce only one paragraph here: \"The rule of law is as valuable a principle today as it has ever been. For it means that the Courts can see to it that the powers of officials, and official bodies of persons entrusted with government, are not exceeded and are not abused, and that the rights of citizens are determined in accordance with the law enacted and unenacted. In so far as the jurisdiction of the Courts is ousted, and officials or official bodies of persons are given a purely administrative discretion, the rule of law is abrogated. It is not abrogated if these officials or official bodies are given a judicial or quasi-judicial discretion, although the machinery through which the rule is applied is not that of the Courts.\" Cf. also Arthur Thomas Vanderbilt, _The Doctrine of the Separation of Powers and Its Present-Day Significance_ (Lincoln: University of Nebraska Press, 1953).\n\n24 See Sir Cecil Thomas Carr, _Delegated Legislation: Three Lectures_ (Cambridge: Cambridge University Press, 1921); Allen, _Law and Orders_ , esp. pp. 114\u201315; and the studies by various authors collected in _Die \u00dcbertragung rechtsetzender Gewalt im Rechtsstaat_ , Walter E. Genzer and Wolfgang Einbeck, eds. (Frankfurt: Institut zur F\u00f6rderung \u00f6ffentlicher Angelegenheiten, 1952).\n\n25 Hans Huber, \"Niedergang des Rechts und Krise des Rechtsstaates,\" in _Demokratie und Rechtsstaat: Festgabe zum 60. Geburststag von Zaccaria Giacometti_ , p. 66, writes similarly of the concept of discretion as a \"Trojan horse in constitutional, administrative law.\"\n\n26 Albert Venn Dicey, \"The Development of Administrative Law in England,\" _Law Quarterly Review_ , 31 (1915): 150. [The full quotation reads: \"When the State undertakes the management of business properly so called, and business which hitherto has been carried on by each individual citizen simply with a view to his own interest, the Government, or in the language of English law, the servants of the Crown, will be found to need that freedom of action necessarily possessed by every private person in the management of his own personal concerns.\"\u2014Ed.]\n\n27 See Ludwig von Mises, _Bureaucracy_ (New Haven: Yale University Press, 1944).\n\n28 See Ernest Freund, _Administrative Powers Over Persons and Property: A Comparative Survey_ (Chicago: University of Chicago Press, 1928), pp. 71 [Hayek is here referring to chap. 6 of Freund's book, entitled \"Administrative Discretion,\" pp. 71\u2013103\u2014Ed.]; Ralph Follen Fuchs, \"Concepts and Policies in Anglo-American Administrative Law Theory,\" _Yale Law Journal_ , 47 (1938): 538\u2013 76; Robert M. Cooper, \"Administrative Justice and the Role of Discretion,\" _Yale Law Journal_ , 47 (1938): 577\u2013602; Morris Raphael Cohen, \"Rule versus Discretion,\" _Journal of Philosophy, Psychology and Scientific Methods_ , 11(1914): 208\u201315; reprinted in Morris Raphael Cohen, _Law and the Social Order: Essays in Legal Philosophy_ (New York: Harcourt, Brace and Co., 1933), pp. 259\u201367; Fritz Morstein Marx, \"Comparative Administrative Law: A Note on Review of Discretion,\" _University of Pennsylvania Law Review_ , 87 (1938\u201339): 954\u201378; G. E. Treves, \"Administrative Discretion and Judicial Control,\" _Modern Law Review_ , 10 (1947): 276\u201391; Rudolf von Laun, _Das freie Ermessen und seine Grenzen_ (Leipzig: Deuticke, 1910); Paul Oertmann, _Die staatsb\u00fcrgerliche Freiheit und das freie Ermessen der Beh\u00f6rden_ [Vortrag gehalten in der Gehe-Stiftung zu Dresden, am 18. November 1911] (Leipzig: B. G. Teubner, 1912); Friedrich Tezner, _Das freie Ermessen der Verwaltungsbeh\u00f6rden. Kritisch-systematisch er\u00f6rtert auf Grund der \u00f6sterreichischen verwaltungsgerichtlichen Rechtsprechung_ (Leipzig and Vienna: F. Deuticke, 1924); Christian-Friedrich Menger, _System des verwaltungsrechtlichen Rechtsschutzes: eine verwaltungsgerichtliche und proze\u00dfvergleichende Studie_ (T\u00fcbingen: Mohr, 1954); and Paul Alex\u00e9ef, \"L'\u00c9tat\u2013le droit\u2013et le pouvoir discr\u00e9tionnaire des autorit\u00e9s publiques,\" _Revue internationale de la th\u00e9orie du droit_ , 3 (1928\u201329): 195\u2013219. Also Oskar Adolf Germann, \"Zur Problematik der Ermessensentschiede\" in _Festgabe zum siebzigsten Geburtstag von Erwin Ruck_ , Juristische Fakult\u00e4t der Universit\u00e4t Basel, ed. (Basel: Helbing und Lichenhahn, 1952), pp. 173\u201396; and Horst Ehmke, _\"Ermessen\" und \"unbestimmter Rechtsbegriff\" im Verwaltungsrecht_ [Recht und Staat, No . 230 \/ 231] (T\u00fcbingen: J. C. B. Mohr, 1960).\n\n29 Cf. the observation by Edgar Bodenheimer in his instructive discussion of the relation between law and administration in _Jurisprudence_ (New York: McGraw-Hill, 1940), p. 95: \"Law is mainly concerned with rights; administration is mainly concerned with results. Law is conducive to liberty and security, while administration promotes efficiency and quick decision.\"\n\n30 On this see Dennis Lloyd, Baron Lloyd of Hampstead, _Public Policy: A Comparative Study of English and French Law_ (London: University of London Press, 1953); also Hans Heinz Todsen, _Der Gesichtspunkt der Public Policy im englischen Recht unter besonderer Ber\u00fccksichtigung der Lehre von den \"contracts in restraint of trade\"_ (Hamburg: P. Evert Verlag, 1937).\n\n31What the English call \"police\"sometimes refers to the German \"Politik\"and sometimes to \"Polizei.\"The medical services police were in fact not police in the moder n sense and the police science of the ear ly nineteenth century was simply administrative science.\n\n32 Zaccaria Giacometti, _Die Freiheitsrechtskataloge als Kodifikation der Freiheit_ [Festrede des Rektors gehalten an der 122. Stiftungsfeier der Universit\u00e4t Z\u00fcrich am 29. April 1955. Jahresbericht 1954 \/ 55] (Zurich: Orell F\u00fcssli, 1955); cf. also Maurice Hauriou, _Pr\u00e9cis de droit constitutionnel_ (2nd ed.; Paris: Librairie du Recueil Sirey, 1929), p. 625; and Felice Battaglia, _Le Carte dei diritti [dalla Magna Charta alla Carta di San Francisco]_ (2nd ed.; Florence: Sansoni, 1946).\n\n33 For a none too pessimistic account of the horrors that may be in store for us see Aldous Huxley, _Brave New World: A Novel_ (London: Chatto and Windus, 1932), and _Brave New World Revisited_ (New York: Harper, 1958); and, even more alarming, because not intended as a warning but expounding a \"scientific\" ideal, Burrhus Frederic Skinner, _Walden Two_ (New York: Macmillan, 1948).\n\n34 Cf. Arthur Thomas Vanderbilt, \"The Role of Procedure in the Protection of Freedom,\" in _Conference on Freedom and the Law, May 7, 1953; Fiftieth Anniversary Celebration_ , Thurman Arnold Wesley, ed. [University of Chicago Law School Conference Series, 13] (Chicago: University of Chicago Law School, 1953), pp. 64\u201373; also Mr. Justice Frankfurter's often quoted statement: \"The history of liberty has largely been the history of observance of procedural safeguards,\" _McNabb v. United States_ 318 _U.S_. 332, at 347 (1943).\n\n35 Cyril John Radcliffe, Viscount Radcliffe of Werneth, _Holdsworth Club of Law and the Democratic State_ , p. 16; also, \"Have we not come to a time when we must find another name for statute law than Law itself ? Para-law, perhaps: or even sub-law\" (p. 4). On the situation in America see the important article by Robert Green McCloskey, \"American Political Thought and the Study of Politics,\" _American Political Science Review_ , 51 (1957), esp. the observation on p. 126 about the manifestation by American courts of \"a pervasive concern for procedural niceties coupled with broad tolerance of substantive inhibitions on freedom. . . . The American concern for procedural rights runs more deeply and steadily than the concern for substantive liberty. Indeed, so far as it goes the evidence implies that freedom in the obvious sense of liberty to think and speak and act unhindered holds no very favored place in the American hierarchy of political values.\" But there seems to be an increasing awareness of this danger, well expressed by Allen Keith-Lucas, _Decisions about People in Need: A Study of Administrative Responsiveness in Public Assistance_ (Chapel Hill: University of North Carolina Press, 1957), p. 156: \"To rely on procedure alone to produce justice is the fallacy of modern liberalism. It has made possible the legality of totalitarian regimes such as Hitler's.\"\n\n## FIFTEEN\n\nECONOMIC POLICY AND THE RULE OF LAW\n\nThe House of Representatives . . . can make no law which will not have its full operation on themselves and their friends, as well as the great mass of the society. This [circumstance] has always been deemed one of the strongest bonds by which human policy can connect the rulers and the people together. It creates between them that communion of interest, and sympathy of sentiments, of which few governments have furnished examples; but without which every government degenerates into tyranny.\n\n\u2014James Madison\n\n1. The classical argument for freedom in economic affairs rests on the tacit postulate that the rule of law should govern policy in this as in all other spheres. We cannot understand the nature of the opposition of men like Adam Smith or John Stuart Mill to government \"intervention\" unless we see it against this background. Their position was therefore often misunderstood by those who were not familiar with that basic conception; and confusion arose in England and America as soon as the conception of the rule of law ceased to be assumed by every reader. Freedom of economic activity had meant freedom under the law, not the absence of all government action. The \"interference\" or \"intervention\" of government which those writers opposed as a matter of principle therefore meant only the infringement of that private sphere which the general rules of law were intended to protect. They did not mean that government should never concern itself with any economic matters. But they did mean that there were certain kinds of governmental measures which should be precluded on principle and which could not be justified on any grounds of expediency.\n\nTo Adam Smith and his immediate successors the enforcement of the ordinary rules of common law would certainly not have appeared as government interference; nor would they ordinarily have applied this term to an alteration  of these rules or the passing of a new rule by the legislature so long as it was intended to apply equally to all people for an indefinite period of time. Though they perhaps never explicitly said so, interference meant to them the exercise of the coercive power of government which was not regular enforcement of the general law and which was designed to achieve some specific purpose.1 The important criterion was not the aim pursued, however, but the method employed. There is perhaps no aim which they would not have regarded as legitimate if it was clear that the people wanted it; but they excluded as generally inadmissible in a free society the method of specific orders and prohibitions. Only indirectly, by depriving government of some means by which alone it might be able to attain certain ends, may this principle deprive government of the power to pursue those ends.\n\nThe later economists bear a good share of the responsibility for the confusion on these matters.2 True, there are good reasons why all governmental  concern with economic matters is suspect and why, in particular, there is a strong presumption against government's actively participating in economic efforts. But these arguments are quite different from the general argument for economic freedom. They rest on the fact that the great majority of governmental measures which have been advocated in this field are, in fact, inexpedient, either because they will fail or because their costs will outweigh the advantages. This means that, so long as they are compatible with the rule of law, they cannot be rejected out of hand as government intervention but must be examined in each instance from the viewpoint of expediency. The habitual appeal to the principle of non-interference in the fight against all ill-considered or harmful measures has had the effect of blurring the fundamental distinction between the kinds of measures which are and those which are not compatible with a free system. And the opponents of free enterprise have been only too ready to help this confusion by insisting that the desirability or undesirability of a particular measure could never be a matter of principle but is always one of expediency.\n\nIn other words, it is the character rather than the volume of government activity that is important. A functioning market economy presupposes certain activities on the part of the state; there are some other such activities by which its functioning will be assisted; and it can tolerate many more, provided that they are of the kind which is compatible with a functioning market. But there are those which run counter to the very principle on which a free system rests and which must therefore be altogether excluded if such a system is to work. In consequence, a government that is comparatively inactive but does the wrong things may do much more to cripple the forces of a market economy than one that is more concerned with economic affairs but confines itself to actions which assist the spontaneous forces of the economy.\n\nIt is the purpose of this chapter to show that the rule of law provides the criterion which enables us to distinguish between those measures which are and those which are not compatible with a free system. Those that are may be examined further on the grounds of expediency. Many such measures will, of course, still be undesirable or even harmful. But those that are not must be rejected even if they provide an effective, or perhaps the only effective, means to a desirable end. We shall see that the observation of the rule of law is a necessary, but not yet a sufficient, condition for the satisfactory working of a free economy. But the important point is that all coercive action of government must be unambiguously determined by a permanent legal framework which  enables the individual to plan with a degree of confidence and which reduces human uncertainty as much as possible.\n\n2. Let us consider, first, the distinction between the coercive measures of government and those pure service activities where coercion does not enter or does so only because of the need of financing them by taxation.3 In so far as the government merely undertakes to supply services which otherwise would not be supplied at all (usually because it is not possible to confine the benefits to those prepared to pay for them), the only question which arises is whether the benefits are worth the cost. Of course, if the government claimed for itself the exclusive right to provide particular services, they would cease to be strictly non-coercive. In general, a free society demands not only that the government have the monopoly of coercion but that it have the monopoly only of coercion and that in all other respects it operate on the same terms as everybody else.\n\nA great many of the activities which governments have universally undertaken in this field and which fall within the limits described are those which facilitate the acquisition of reliable knowledge about facts of general significance.4 The most important function of this kind is the provision of a reliable and efficient monetary system. Others scarcely less important are the setting of standards of weights and measures; the providing of information gathered from surveying, land registration, statistics, etc.; and the support, if not also the organization, of some kind of education.\n\nAll these activities of government are part of its effort to provide a favorable framework for individual decisions; they supply means which individuals can use for their own purposes. Many other services of a more material kind fall into the same category. Though government must not use its power of coercion to reserve for itself activities which have nothing to do with the enforcement of the general rules of law, there is no violation of principle in its engaging in all sorts of activities on the same terms as the citizens. If in the majority of fields there is no good reason why it should do so, there are fields in which the desirability of government action can hardly be questioned.\n\nTo this latter group belong all those services which are clearly desirable  but which will not be provided by competitive enterprise because it would be either impossible or difficult to charge the individual beneficiary for them. Such are most sanitary and health services, often the construction and maintenance of roads, and many of the amenities provided by municipalities for the inhabitants of cities. Included also are the activities which Adam Smith described as \"those public works, which, though they may be in the highest degree advantageous to a great society, are, however, of such a nature, that the profit could never repay the expense to any individual or small number of individuals.\"5 And there are many other kinds of activity in which the government may legitimately wish to engage, in order perhaps to maintain secrecy in military preparations or to encourage the advancement of knowledge in certain fields.6 But though government may at any moment be best qualified to take the lead in such fields, this provides no justification for assuming that this will always be so and therefore for giving it exclusive responsibility. In most instances, moreover, it is by no means necessary that government engage in the actual management of such activities; the services in question can generally be provided, and more effectively provided, by the government's assuming some or all of the financial responsibility but leaving the conduct of the affairs to independent and in some measure competitive agencies.\n\nThere is considerable justification for the distrust with which business looks on all state enterprise. There is great difficulty in ensuring that such enterprise will be conducted on the same terms as private enterprise; and it is only if this condition is satisfied that it is not objectionable in principle. So long as government uses any of its coercive powers, and particularly its power of taxation, in order to assist its enterprises, it can always turn their position into one of actual monopoly. To prevent this, it would be necessary that any special  advantages, including subsidies which government gives to its own enterprises in any field, should also be made available to competing private agencies. There is no need to emphasize that it would be exceedingly difficult for government to satisfy these conditions and that the general presumption against state enterprise is thereby considerably strengthened. But this does not mean that all state enterprise must be excluded from a free system. Certainly it ought to be kept within narrow limits; it may become a real danger to liberty if too large a section of economic activity comes to be subject to the direct control of the state. But what is objectionable here is not state enterprise as such but state monopoly.\n\n3. Furthermore, a free system does not exclude on principle all those general regulations of economic activity which can be laid down in the form of general rules specifying conditions which everybody who engages in a certain activity must satisfy. They include, in particular, all regulations governing the techniques of production. We are not concerned here with the question of whether such regulations will be wise, which they probably will be only in exceptional cases. They will always limit the scope of experimentation and thereby obstruct what may be useful developments. They will normally raise the cost of production or, what amounts to the same thing, reduce over-all productivity. But if this effect on cost is fully taken into account and it is still thought worthwhile to incur the cost to achieve a given end, there is little more to be said about it.7 The economist will remain suspicious and hold that there is a strong presumption against such measures because their overall cost is almost always underestimated and because one disadvantage in particular\u2014namely, the prevention of new developments\u2014can never be fully taken into account. But if, for instance, the production and sale of phosphorus matches is generally prohibited for reasons of health or permitted only if certain precautions are taken, or if night work is generally prohibited, the appropriateness of such measures must be judged by comparing the over-all costs with the gain; it cannot be conclusively determined by appeal to a general principle. This is true of most of the wide field of regulations known as \"factory legislation.\"\n\nIt is often maintained today that these or similar tasks which are generally acknowledged to be proper functions of government could not be adequately performed if the administrative authorities were not given wide discretionary powers and all coercion were limited by the rule of law. There is little reason to fear this. If the law cannot always name the particular measures which the authorities may adopt in a particular situation, it can be so framed as to enable any impartial court to decide whether the measures adopted were necessary to achieve the general effect aimed at by the law. Though the variety  of circumstances in which the authorities may have to act cannot be foreseen, the manner in which they will have to act, once a certain situation has arisen, can be made predictable to a high degree. The destroying of a farmer's cattle in order to stop the spreading of a contagious disease, the tearing down of houses to prevent the spreading of a fire, the prohibition of an infected well, the requirement of protective measures in the transmission of high-tension electricity, and the enforcement of safety regulations in buildings undoubtedly demand that the authorities be given some discretion in applying general rules. But this need not be a discretion unlimited by general rules or of the kind which need to be exempt from judicial review.\n\nWe are so used to such measures being referred to as evidence of the necessity of conferring discretionary powers that it comes somewhat as a surprise that, as recently as thirty years ago, an eminent student of administrative law could still point out that \"health and safety statutes are, generally speaking, by no means conspicuous for the use of discretionary powers; on the contrary, in much of that legislation such powers are conspicuously absent. . . . Thus British factory legislation has found it possible to rely practically altogether on general rules (though to a large extent framed by administrative regulation) . . . many building codes are framed with a minimum of administrative discretion, practically all regulation being limited to requirements capable of standardization. . . . In all these cases the consideration of flexibility yielded to the higher consideration of certainty of private right, without any apparent sacrifice of public interest.\"8\n\nIn all such instances the decisions are derived from general rules and not from particular preferences which guide the government of the moment or from any opinion as to how particular people ought to be situated. The coercive powers of government still serve general and timeless purposes, not specific ends. It must not make any distinctions between different people. The discretion conferred on it is a limited discretion in that the agent is to apply the sense of a general rule. That this rule cannot be made completely unambiguous in its application is a consequence of human imperfection. The problem, nevertheless, is one of applying a rule, which is shown by the fact that an independent judge, who in no way represents the particular wishes or values of the government or of the majority of the moment, will be able to decide not only whether the authority had a right to act at all but also whether it was required by law to do exactly what it did.\n\nThe point at issue here has nothing to do with the question of whether the regulations justifying the actions of government are uniform for the whole country or whether they have been laid down by a democratically elected assembly.  There is clearly need for some regulations to be passed by local ordinances, and many of them, such as building codes, will necessarily be only in form and never in substance the product of majority decisions. The important question again concerns not the origin but the limits of the powers conferred. Regulations drawn up by the administrative authority itself but duly published in advance and strictly adhered to will be more in conformity with the rule of law than will vague discretionary powers conferred on the administrative organs by legislative action.\n\nThough there have always been pleas on the ground of administrative convenience that these strict limits should be relaxed, this is certainly not a necessary requirement for the achievement of the aims we have considered so far. It was only after the rule of law had been breached for other aims that its preservation no longer seemed to outweigh considerations of administrative efficiency.\n\n4. We must now turn to the kinds of governmental measures which the rule of law excludes in principle because they cannot be achieved by merely enforcing general rules but, of necessity, involve arbitrary discrimination between persons. The most important among them are decisions as to who is to be allowed to provide different services or commodities, at what prices or in what quantities\u2014in other words, measures designed to control the access to different trades and occupations, the terms of sale, and the amounts to be produced or sold.\n\nSo far as the entry into different occupations is concerned, our principle does not necessarily exclude the possible advisability in some instances of permitting it only to those who possess certain ascertainable qualifications. The restriction of coercion to the enforcement of general rules requires, however, that any one possessing these qualifications have an enforceable claim to such permission and that the grant of the permission depend only on his satisfying the conditions laid down as a general rule and not on any particular circumstances (such as \"local need\") which would have to be determined by the discretion of the licensing authority. Even the need for such controls could probably be rendered unnecessary in most instances by merely preventing people from pretending to qualifications which they do not possess, that is, by applying the general rules preventing fraud and deception. For this purpose the protection of certain designations or titles expressing such qualifications might well be sufficient (it is by no means evident that even in the case of doctors this would not be preferable to the requirement of a license to practice). But it is probably undeniable that in some instances, such as where the sale of poisons or firearms is involved, it is both desirable and unobjectionable that only persons satisfying certain intellectual and moral qualities should be allowed to practice such trade. So long as everybody possessing the necessary qualifications has the right to practice the occupation in question and, if necessary,  can have his claim examined and enforced by an independent court, the basic principle is satisfied.9\n\nThere are several reasons why all direct control of prices by government is irreconcilable with a functioning free system, whether the government actually fixes prices or merely lays down rules by which the permissible prices are to be determined. In the first place, it is impossible to fix prices according to long-term rules which will effectively guide production. Appropriate prices depend on circumstances which are constantly changing and must be continually adjusted to them. On the other hand, prices which are not fixed outright but determined by some rule (such as that they must be in a certain relation to cost) will not be the same for all sellers and, for this reason, will prevent the market from functioning. A still more important consideration is that, with prices different from those that would form on a free market, demand and supply will not be equal, and if the price control is to be effective, some method must be found for deciding who is to be allowed to buy or sell. This would necessarily be discretionary and must consist of _ad hoc_ decisions that discriminate between persons on essentially arbitrary grounds. As experience has amply confirmed, price controls can be made effective only by quantitative controls, by decisions on the part of authority as to how much particular persons or firms are to be allowed to buy or sell. And the exercise of all controls of quantities must, of necessity, be discretionary, determined not by rule but by the judgment of authority concerning the relative importance of particular ends.\n\nIt is thus not because the economic interests with which such measures interfere are more important than others that price and quantity controls must be altogether excluded in a free system, but because this kind of controls cannot be exercised according to rule but must in their very nature be discretionary and arbitrary. To grant such powers to authority means in effect to give it power arbitrarily to determine what is to be produced, by whom, and for whom.\n\n5. Strictly speaking, then, there are two reasons why all controls of prices and quantities are incompatible with a free system: one is that all such controls must be arbitrary, and the other is that it is impossible to exercise them in such a manner as to allow the market to function adequately. A free system can adapt itself to almost any set of data, almost any general prohibition or regulation, so long as the adjusting mechanism itself is kept functioning. And  it is mainly changes in prices that bring about the necessary adjustments. This means that, for it to function properly, it is not sufficient that the rules of law under which it operates be general rules, but their content must be such that the market will work tolerably well. The case for a free system is not that any system will work satisfactorily where coercion is confined by general rules, but that under it such rules can be given a form that will enable it to work. If there is to be an efficient adjustment of the different activities in the market, certain minimum requirements must be met; the more important of these are, as we have seen, the prevention of violence and fraud, the protection of property and the enforcement of contracts, and the recognition of equal rights of all individuals to produce in whatever quantities and sell at whatever prices they choose. Even when these basic conditions have been satisfied, the efficiency of the system will still depend on the particular content of the rules. But if they are not satisfied, government will have to achieve by direct orders what individual decisions guided by price movements will.\n\nThe relation between the character of the legal order and the functioning of the market system has received comparatively little study, and most of the work in this field has been done by men who were critical of the competitive order10 rather than by its supporters. The latter have usually been content to state the minimal requirements for the functioning of the market which we have just mentioned. A general statement of these conditions, however, raises almost as many questions as the answers it provides. How well the market will function depends on the character of the particular rules. The decision to rely on voluntary contracts as the main instrument for organizing the relations between individuals does not determine what the specific content of the law of contract ought to be; and the recognition of the right of private property does not determine what exactly should be the content of this right in order that the market mechanism will work as effectively and beneficially as possible. Though the principle of private property raises comparatively few problems so far as movable things are concerned, it does raise exceedingly difficult ones where property in land is concerned. The effect which the use of any one piece of land often has on neighboring land clearly makes it undesirable to give the owner unlimited power to use or abuse his property as he likes.\n\nBut, while it is to be regretted that economists have on the whole contributed little to the solution of these problems, there are some good reasons for this. General speculation about the character of a social order cannot produce much more than equally general statements of the principles that the  legal order must follow. The application in detail of these general principles must be left largely to experience and gradual evolution. It presupposes concern with concrete cases, which is more the province of the lawyer than of the economist. At any rate, it is probably because the task of gradually amending our legal system to make it more conducive to the smooth working of competition is such a slow process that it has had little appeal for those who seek an outlet for their creative imagination and are impatient to draw up blueprints for further development.\n\n6. There is still another point we must consider a little more closely. Since the time of Herbert Spencer11 it has become customary to discuss many aspects of our problem under the heading of \"freedom of contract.\" And for a period of time this point of view played an important role in American jurisdiction.12 There is indeed a sense in which freedom of contract is an important part of individual freedom. But the phrase also gives rise to misconceptions. In the first place, the question is not what contracts individuals will be allowed to make but rather what contracts the state will enforce. No modern state has tried to enforce all contracts, nor is it desirable that it should. Contracts for criminal or immoral purposes, gambling contracts, contracts in restraint of trade, contracts permanently binding the services of a person, or even some contracts for specific performances are not enforced.\n\nFreedom of contract, like freedom in all other fields, really means that the permissibility of a particular act depends only on general rules and not on its specific approval by authority. It means that the validity and enforcibility of a contract must depend only on those general, equal, and known rules by which all other legal rights are determined, and not on the approval of its particular content by an agency of the government. This does not exclude the possibility of the law's recognizing only those contracts which satisfy certain general conditions or of the state's laying down rules for the interpretation of contracts which will supplement the explicitly agreed terms. The existence of such recognized standard forms of contract which, so long as no contrary terms are stipulated, will be presumed to be part of the agreement often greatly facilitates private dealings.\n\nA much more difficult question is whether the law should ever provide for obligations arising out of a contract which may be contrary to the intentions of both parties, as, for example, in the case of liability for industrial accidents irrespective of negligence. But even this is probably more a question of expediency  than of principle. The enforcibility of contracts is a tool which the law provides for us, and what consequences will follow upon concluding a contract is for the law to say. So long as these consequences can be predicted from a general rule and the individual is free to use the available types of contracts for his own purposes, the essential conditions of the rule of law are satisfied.\n\n7. The range and variety of government action that is, at least in principle, reconcilable with a free system is thus considerable. The old formulae of laissez faire or non-intervention do not provide us with an adequate criterion for distinguishing between what is and what is not admissible in a free system. There is ample scope for experimentation and improvement within that permanent legal framework which makes it possible for a free society to operate most efficiently. We can probably at no point be certain that we have already found the best arrangements or institutions that will make the market economy work as beneficially as it could. It is true that after the essential conditions of a free system have been established, all further institutional improvements are bound to be slow and gradual. But the continuous growth of wealth and technological knowledge which such a system makes possible will constantly suggest new ways in which government might render services to its citizens and bring such possibilities within the range of the practicable.\n\nWhy, then, has there been such persistent pressure to do away with those limitations upon government that were erected for the protection of individual liberty? And if there is so much scope for improvement within the rule of law, why have the reformers striven so constantly to weaken and undermine it? The answer is that during the last few generations certain new aims of policy have emerged which cannot be achieved within the limits of the rule of law. A government which cannot use coercion except in the enforcement of general rules has no power to achieve particular aims that require means other than those explicitly entrusted to its care and, in particular, cannot determine the material position of particular people or enforce distributive or \"social\" justice. In order to achieve such aims, it would have to pursue a policy which is best described\u2014since the word \"planning\" is so ambiguous\u2014by the French word _dirigisme_ , that is, a policy which determines for what specific purposes particular means are to be used.\n\nThis, however, is precisely what a government bound by the rule of law cannot do. If the government is to determine how particular people ought to be situated, it must be in a position to determine also the direction of individual efforts. We need not repeat here the reasons why, if government treats different people equally, the results will be unequal, or why, if it allows people to make what use they like of the capacities and means at their disposal, the consequences for the individuals will be unpredictable. The restrictions which the rule of law imposes upon government thus preclude all those measures which would be necessary to insure that individuals will be rewarded according  to another's conception of merit or desert rather than according to the value that their services have for their fellows or, what amounts to the same thing, it precludes the pursuit of distributive, as opposed to commutative, justice. Distributive justice requires an allocation of all resources by a central authority; it requires that people be told what to do and what ends to serve. Where distributive justice is the goal, the decisions as to what the different individuals must be made to do cannot be derived from general rules but must be made in the light of the particular aims and knowledge of the planning authority. As we have seen before, when the opinion of the community decides what different people shall receive, the same authority must also decide what they shall do.\n\nThis conflict between the ideal of freedom and the desire to \"correct\" the distribution of incomes so as to make it more \"just\" is usually not clearly recognized. But those who pursue distributive justice will in practice find themselves obstructed at every move by the rule of law. They must, from the very nature of their aim, favor discriminatory and discretionary action. But, as they are usually not aware that their aim and the rule of law are in principle incompatible, they begin by circumventing or disregarding in individual cases a principle which they often would wish to see preserved in general. But the ultimate result of their efforts will necessarily be, not a modification of the existing order, but its complete abandonment and its replacement by an altogether different system\u2014the command economy.\n\nWhile it is certainly not true that such a centrally planned system would be more efficient than one based on a free market, it is true that only a centrally directed system could attempt to ensure that the different individuals would receive what someone thought they deserved on moral grounds. Within the limits set by the rule of law, a great deal can be done to make the market work more effectively and smoothly; but, within these limits, what people now regard as distributive justice can never be achieved. We shall have to examine the problems which have arisen in some of the most important fields of contemporary policy as a result of the pursuit of distributive justice. Before we do so, however, we must consider the intellectual movements which have done so much during the last two or three generations to discredit the rule of law and which, by disparaging this ideal, have seriously undermined the resistance to a revival of arbitrary government.\n\n* * *\n\nThe quotation at the head of the chapter is taken from James Madison, \"The Alleged Tendency of the Plan to Elevate the Few at the Expense of the Many Considered in Connection with Representation\" (No. 57), in Alexander Hamilton, James Madison, and John Jay, _The Federalist, or the New Constitution_ , Max Beloff, Baron Beloff, ed. (Oxford: B. Blackwell, 1948), p. 294 [Liberty Fund edition, p. 297].\n\n1 Cf. Ludwig von Mises, _Kritik des Interventionismus: Untersuchungen zur Wirtschaftspolitik und Wirtschaftsideologie der Gegenwart_ ( Jena: G. Fischer, 1929), p. 6: \"Der Eingriff ist ein von einer gesellschaftlichen Gewalt ausgehender _isolierter Befehl_ , der die Eigent\u00fcmer der Produktionsmittel und die Unternehmer zwingt, die Produktionsmittel anders zu verwenden, als sie es sonst tun w\u00fcrden.\" [\"Intervention is a _limited order_ by a social authority forcing the owners of the means of production and entrepreneurs to employ their means in a different manner than they otherwise would.\" ( _A Critique of Interventionism_ , Hans F. Sennholz, trans. [Irvington-on-Hudson, NY: Foundation for Economic Education, 1996], p. 20; Hayek's italics; the entire sentence is emphasized in the original).\u2014Ed.] See also the distinction between _produktionspolitische_ and _preispolitische Eingriffe_ elaborated later in the same work. John Stuart Mill, \"On Liberty,\" in _On Liberty and Considerations on Representative Government_ , Ronald Buchanan McCallum, ed. (Oxford: B. Blackwell, 1946), p. 85, even argues that \"the so-called doctrine of Free Trade . . . rests on grounds different from, though equally solid with, the principle of individual liberty asserted in this Essay. Restrictions on trade, or on production for purposes of trade, are indeed restraints; and all restraint, _qua_ restraint, is an evil: but the restraints in question affect only that part of conduct which society is competent to restrain, and are wrong solely because they do not really produce the results which it is desired to produce by them. As the principle of individual liberty is not involved in the doctrine of Free Trade, so neither is it in most of the questions which arise respecting the limit of that doctrine; as, for example, what amount of public control is admissible for the prevention of fraud by adulteration; how far sanitary precautions, or arrangements to protect work-people employed in dangerous occupations, should be enforced on employers.\"\n\n2 As the examination of measures of policy for their expediency is one of the chief tasks of the economists, it is not surprising that they should have lost sight of the more general criterion. John Stuart Mill, by admitting ( _On Liberty_ , p. 8) that \"there is, in fact, no recognized principle by which the propriety of government interference is customarily tested,\" had already given the impression that it was all a matter of expediency. And his contemporary, Nassau William Senior, usually regarded as much more orthodox, explicitly said so at about the same time: \"The only rational foundation of government, the only foundation of a right to govern and a correlative duty to obey, is expediency\u2014the general benefit of the community\" (quoted in Lionel Robbins, _The Theory of Economic Policy in English Classical Political Economy_ [London: Macmillan, 1952], p. 45). [Senior's comments appear in his Oxford lectures of 1847\u201352, Course 1, Lecture 6, \"The Power of Government to alter the degree in which wealth is Desirable.\" The citation appears in Marian Bowley, _Nassau Senior and Classical Economics_ (New York: Octagon Books, Inc., 1967), p. 265. \u2014Ed.] Yet both these men unquestionably took it for granted that interference with the protected sphere of the individual was permissible only where it was provided for by the general rules of law and never on mere grounds of expediency.\n\n3 The distinction is the same as that which Mill, ( _Principles_ , bk. 5, chap. 11, sec. 1, p. 942 [Liberty Fund edition, _Collected_ Works, vol. 2, p. 937]) draws between \"authoritative\" and \"unauthoritative\" government interference. It is a distinction of great importance, and the fact that all government activity has been assumed more and more to be necessarily of the \"authoritative\" character is one of the chief causes of the objectionable developments of modern times. I do not here adopt Mill's terms because it seems to me inexpedient to call his \"unauthoritative\" activities of government \"interference.\" This term is better confined to infringements of the protected private sphere, which can be done only \"authoritatively.\"\n\n4 See again the careful treatment of this in Mill, _Principles_ , bk. 5, chap. 11, sec. 3, pp. 944\u201345 [Liberty Fund edition, _Collected Works_ , vol. 2, pp. 938\u201339].\n\n5 A. Smith, _Wealth of Nations_ , bk. 5, chap. 1, pt. 2, vol. 2, p. 214 [Liberty Fund edition, vol. 2, p. 723]; cf. also the argument in favor of local, as against central, government taking charge of public works, ibid., p. 222 [Liberty Fund edition, vol. 2, p.730].\n\n6 There is, finally, the theoretically interesting, though in practice not very significant, situation in which, though certain services can be supplied by competitive private effort, either not all the cost involved or not all the benefits rendered would enter the calculations of the market and for this reason it may seem desirable to impose special charges on, or offer special grants to all who engage in those activities. These instances may perhaps be included among the measures by which government may assist the direction of private production, not by specific intervention, but by acting according to general rules.\n\nThat these cases are not of great practical significance, not because such situations may not often occur, but because it is rarely possible to ascertain the magnitude of such \"divergences between the marginal social net product and the private social net product,\" is now admitted by the author who has done more than anybody else to draw attention to them: see Arthur Cecil Pigou, \"Some Aspects of the Welfare State,\" _Diogenes_ , 7 (1954): 6: \"It must be confessed, however, that we seldom know enough to decide in what fields and to what extent the State, on account of [the gaps between private and public costs] could usefully interfere with individual freedom of choice.\"\n\n7 See Ludwig von Mises, _Kritik des Interventionismus_ , p. 6.\n\n8 Ernest Freund, _Administrative Powers Over Persons and Property: A Comparative Survey_ (Chicago: University of Chicago Press, 1928), p. 98.\n\n9 On the issue of licensing see Walter Gellhorn, _Individual Freedom and Governmental Restraints_ (Baton Rouge: Louisiana State University Press, 1956), esp. chap. 3, pp. 105\u201351). I would not have treated this matter so lightly if the final text of this chapter had not been completed before I knew this work. I believe few foreign observers and probably not many Americans are aware how far this practice has been carried in the United States in recent years\u2014so far, indeed, that it must now appear as one of the real threats to the future of American economic development.\n\n10 See particularly John Rogers Commons, _The Legal Foundations of Capitalism_ (New York: Macmillan, 1924); Walton Hale Hamilton and Douglass Adair, _The Power to Govern: The Constitution\u2014 Then and Now_ (New York: W. W. Norton, 1937); and John Maurice Clark, _Social Control of Business_ (2nd ed.; New York: Whittlesey House, McGraw-Hill, 1939); and cf. on this school, Abram Lincoln Harris, _Economics and Social Reform_ (New York: Harper, 1958).\n\n11 See especially Herbert Spencer, _Justice: Being Part IV of the Principles of Ethics_ (Authorized ed.; London: D. Appleton and Co., 1891) [Liberty Fund edition, _Principles of Ethics_ , vol. 2, pp. 19\u2013279]; and cf. Thomas Hill Green, \"Lecture on 'Liberal Legislation and Freedom of Contract,\" [1880] in _Works of T. H. Green_ (3 vols.; London: Longmans, Green, and Co., 1888), vol. 3: _Miscellanies and Memoir_ [1885\u201388], pp. 365\u201386.\n\n12 Cf. Roscoe Pound, \"Liberty of Contract,\" _Yale Law Journal_ , 18 (1908\u201309): 454\u201387.\n\n## SIXTEEN\n\nTHE DECLINE OF THE LAW\n\nThe dogma, that absolute power may, by the hypothesis of a popular origin, be as legitimate as constitutional freedom, began . . . to darken the air.\n\n\u2014Lord Acton\n\n1. Earlier in our discussion we devoted more attention than is usual to developments in Germany, partly because it was in that country that the theory, if not the practice, of the rule of law was developed furthest, and partly because it was necessary to understand the reaction against it which commenced there. As is true of so much of socialist doctrine, the legal theories which undermined the rule of law originated in Germany and spread from there to the rest of the world.\n\nThe interval between the victory of liberalism and the turn toward socialism or a kind of welfare state was shorter in Germany than elsewhere. The institutions meant to secure the rule of law had scarcely been completed before a change in opinion prevented their serving the aims for which they had been created. Political circumstances and developments which were purely intellectual combined to accelerate a development which proceeded more slowly in other countries. The fact that the unification of the country had at last been achieved by the artifice of statesmanship rather than by gradual evolution strengthened the belief that deliberate design should remodel society according to a preconceived pattern. The social and political ambitions which this situation encouraged were strongly supported by philosophical trends then current in Germany.\n\nThe demand that government should enforce not merely \"formal\" but \"substantive\" (i.e., \"distributive\" or \"social\") justice had been advanced recurrently since the French Revolution. Toward the end of the nineteenth century these ideas had already profoundly affected legal doctrine. By 1890 a  leading socialist theorist of the law could thus express what was increasingly becoming the dominant doctrine: \"By treating in a perfectly equal manner all citizens regardless of their personal qualities and economic position, and by allowing unlimited competition between them, it came about that the production of goods was increased without limit; but the poor and weak had only a small share in that output. The new economic and social legislation therefore attempts to protect the weak against the strong and to secure for them a moderate share in the good things of life. This is because today it is understood that there is no greater injustice than to treat as equal what is in fact unequal!\"1 And there was Anatole France, who scoffed at \"the majestic equality of the law that forbids the rich as well as the poor to sleep under bridges, to beg in the streets and to steal bread.\"2 This famous phrase has been repeated countless times by well-meaning but unthinking people who did not understand that they were undermining the foundations of all impartial justice.\n\n2. The ascendancy of these political views was greatly assisted by the increasing influence of various theoretical conceptions which had arisen earlier in the century and which, though in many respects strongly opposed to one another, had in common the dislike of any limitation of authority by rules of law and shared the desire to give the organized forces of government greater power to shape social relations deliberately according to some ideal of social justice. The four chief movements which operated in this direction were, in descending order of importance, legal positivism, historicism, the \"free law\" school, and the school of \"jurisprudence of interest.\" We shall only briefly consider the last three before we turn to the first, which must detain us a little longer.\n\nThe tradition which only later became known as \"jurisprudence of interest\" was a form of sociological approach somewhat similar to the \"legal realism\" of contemporary America. At least in its more radical forms it wanted to get away from the kind of logical construction which is involved in the decision of disputes by the application of strict rules of law and to replace it by a direct assessment of the particular \"interests\" at stake in the concrete case.3 The \"free law\" school was in a way a parallel movement concerned mainly with criminal law. Its objective was to free the judge as far as possible from the shackles of fixed rules and permit him to decide individual cases mainly on the basis of his \"sense of justice.\" It has often been pointed out how much the latter in particular prepared the way for the arbitrariness of the totalitarian state.4\n\nHistoricism, which must be precisely defined so that it may be sharply distinguished from the great historical schools (in jurisprudence and elsewhere)  that preceded it,5 was a school that claimed to recognize necessary laws of historical development and to be able to derive from such insight knowledge of what institutions were appropriate to the existing situation. This view led to an extreme relativism which claimed, not that we are the product of our own time and bound in a large measure by the views and ideas we have inherited, but that we can transcend those limitations and explicitly recognize how our present views are determined by circumstances and use this knowledge to remake our institutions in a manner appropriate to our time.6 Such a view would naturally lead to a rejection of all rules that cannot be rationally justified or have not been deliberately designed to achieve a specific purpose. In this respect historicism supports what we shall presently see is the main contention of legal positivism.7\n\n3. The doctrines of legal positivism have been developed in direct opposition to a tradition which, though it has for two thousand years provided the framework within which our central problems have been mainly discussed, we have not explicitly considered. This is the conception of a law of nature, which to many still offers the answer to our most important question. We have so far deliberately avoided discussing our problems with reference to this conception because the numerous schools which go under this name hold really different theories and an attempt to sort them out would require a separate book.8 But we must at least recognize here that these different schools of the law of nature have one point in common, which is that they address themselves to the same problem. What underlies the great conflict between the  defenders of natural law and the legal positivists is that, while the former recognize the existence of that problem, the latter deny that it exists at all, or at least that it has a legitimate place within the province of jurisprudence.\n\nWhat all the schools of natural law agree upon is the existence of rules which are not of the deliberate making of any lawgiver. They agree that all positive law derives its validity from some rules that have not in this sense been made by men but which can be \"found\" and that these rules provide both the criterion for the justice of positive law and the ground for men's obedience to it. Whether they seek the answer in divine inspiration or in the inherent powers of human reason, or in principles which are not themselves part of human reason but constitute non-rational factors that govern the working of the human intellect, or whether they conceive of the natural law as permanent and immutable or as variable in content, they all seek to answer a question which positivism does not recognize. For the latter, law by definition consists exclusively of deliberate commands of a human will.\n\nFor this reason, legal positivism from the very beginning could have no sympathy with and no use for those meta- legal principles which underlie the ideal of the rule of law or the _Rechtsstaat_ in the original meaning of this concept, for those principles which imply a limitation upon the power of legislation. In no other country did this positivism gain such undisputed sway in the second half of the last century as it did in Germany. It was consequently here that the ideal of the rule of law was first deprived of real content. The substantive conception of the _Rechtsstaat_ , which required that the rules of law possess definite properties, was displaced by a purely formal concept which required merely that all action of the state be authorized by the legislature. In short, a \"law\" was that which merely stated that whatever a certain authority did should be legal. The problem thus became one of mere legality.9 By the turn of the century it had become accepted doctrine that the \"individualist\" ideal of the substantive _Rechtsstaat_ was a thing of the past, \"vanquished by the creative powers of national and social ideas.\"10 Or, as an authority on administrative law described the situation shortly before the outbreak of the first World War: \"We have returned to the principles of the police state [!] to such an extent that we again recognize the idea of a _Kulturstaat_. The only difference  is in the means. On the basis of laws the modern state permits itself everything, much more than the police state did. Thus, in the course of the nineteenth century, the term _Rechtsstaat_ was given a new meaning. We understand by it a state whose whole activity takes place on the basis of laws and in legal form. On the purpose of the state and the limits of its competence the term _Rechtsstaat_ in its present-day meaning says nothing.\"11\n\nIt was, however, only after the first World War that these doctrines were given their most effective form and began to exert a great influence which extended far beyond the limits of Germany. This new formulation, known as the \"pure theory of law\" and expounded by Professor H. Kelsen,12 signaled the definite eclipse of all traditions of limited government. His teaching was avidly taken up by all those reformers who had found the traditional limitations an irritating obstacle to their ambitions and who wanted to sweep away all restrictions on the power of the majority. Kelsen himself had early observed how the \"fundamentally irretrievable liberty of the individual . . . gradually recedes into the background and the liberty of the social collective occupies the front of the stage\"13 and that this change in the conception of  freedom meant an \"emancipation of democratism from liberalism,\"14 which he evidently welcomed. The basic conception of his system is the identification of the state and the legal order. Thus the _Rechtsstaat_ becomes an extremely formal concept and an attribute of all states,15 even a despotic one.16 There are no possible limits to the power of the legislator,17 and there are no \"socalled fundamental liberties\";18 and any attempt to deny to an arbitrary despotism the character of a legal order represents \"nothing but the na\u00efvet\u00e9  and presumption of natural-law thinking.\"19 Every effort is made not only to obscure the fundamental distinction between true laws in the substantive sense of abstract, general rules and laws in the merely formal sense (including all acts of a legislature) but also to render indistinguishable from them the orders of any authority, no matter what they are, by including them all in the vague term \"norm.\"20 Even the distinction between jurisdiction and administrative acts is practically obliterated. In short, every single tenet of the traditional conception of the rule of law is represented as a metaphysical superstition.\n\nThis logically most consistent version of legal positivism illustrates the ideas which by the 1920s had come to dominate German thinking and were rapidly spreading to the rest of the world. At the end of that decade they had so completely conquered Germany that \"to be found guilty of adherence to natural law theories [was] a kind of social disgrace.\"21 The possibilities which this state of opinion created for an unlimited dictatorship were already clearly seen by acute observers at the time Hitler was trying to gain power. In 1930 a German legal scholar, in a detailed study of the effects of the \"efforts to realize the socialist State, the opposite of the _Rechtsstaat_ ,\"22 was able to point out that these \"doctrinal developments have already removed all obstacles to the disappearance of the _Rechtsstaat_ , and opened the doors to the victory of the fascist and bolshevist will of the State.\"23 The increasing concern over these developments which Hitler was finally to complete was given expression by more than one speaker at a congress of German constitutional lawyers.24 But it was too late. The antilibertarian forces had learned too well the positivist doctrine that the state must not be bound by law. In Hitler Germany and in Fascist Italy, as well as in Russia, it came to be believed that under the rule of law the state was \"unfree,\"25 a \"prisoner of the law,\"26 and that, in order to act \"justly,\" it must be released from the fetters of abstract rules.27 A \"free\" state was to be one that could treat its subjects as it pleased.\n\n4. The inseparability of personal freedom from the rule of law is shown most clearly by the absolute denial of the latter, even in theory, in the country where modern despotism has been carried furthest. The history of the development of legal theory in Russia during the early stages of communism, when the ideals of socialism were still taken seriously and the problem of the role of law in such a system was extensively discussed, is very instructive.  In their ruthless logic the arguments advanced in these discussions show the nature of the problem more clearly than does the position taken by Western socialists, who usually try to have the best of both worlds.\n\nThe Russian legal theorists deliberately continued in a direction which, they recognized, had long been established in western Europe. As one of them put it, the conception of law itself was generally disappearing, and \"the center of gravity was shifting more and more from the passing of general norms to individual decisions and instructions which regulate, assist, and co-ordinate activities of administration.\"28 Or, as another contended at the same time, \"since it is impossible to distinguish between laws and administrative regulations, this contrast is a mere fiction of bourgeois theory and practice.\"29 The best description of these developments we owe to a non-Communist Russian scholar, who observed that \"what distinguishes the Soviet system from all other despotic government is that . . . it represents an attempt to found the state on _principles_ which are the opposite of those of the rule of law . . . [and it] has evolved a _theory_ which exempts the rulers from every obligation or limitation.\"30 Or, as a Communist theorist expressed it, \"the fundamental principle of our legislation and our private law, which the bourgeois theorist will never recognize is: everything is prohibited which is not specifically permitted.\"31\n\nFinally, the Communist attacks came to be directed at the conception of law itself. In 1927 the president of the Soviet Supreme Court explained in an official handbook of private law: \"Communism means not the victory of socialist law, but the victory of socialism over any law, since with the abolition of classes with antagonistic interests, law will disappear altogether.\"32\n\nThe reasons for this stage of the development were most clearly explained by the legal theorist E. Pashukanis, whose work for a time attracted much attention both inside and outside Russia but who later fell into disgrace and disappeared.33 He wrote: \"To the administrative technical direction by subordination to a general economic plan corresponds the method of direct, technologically determined direction in the shape of programs for production and distribution. The gradual victory of this tendency means the gradual extinction of law as such.\"34 In short: \"As, in a socialist community, there was no  scope for autonomous private legal relations, but only for regulation in the interest of the community, all law was converted into administration; all fixed rules into discretion and utility.\"35\n\n5. In England developments away from the rule of law had started early but for a long time remained confined to the sphere of practice and received little theoretical attention. Though, by 1915, Dicey could observe that \"the ancient veneration for the rule of law has in England suffered during the last thirty years a marked decline,\"36 the increasingly frequent infringements of the principle attracted little notice. Even when in a 1929 book called _The New Despotism_37 appeared, in which Lord Justice Hewart pointed out how little in accord with the rule of law was the situation which had developed, it achieved a _succ\u00e8s de scandale_ but could do little to change the complacent belief that the liberties of Englishmen were safely protected by that tradition. The book was treated as a mere reactionary pamphlet, and the venom which was directed at it38 is difficult to understand a quarter of a century later, when not only liberal organs like the _Economist_39 but also socialist authors40 have come to speak of the danger in the same terms. The book did indeed lead to the appointment of an official \"Committee on Ministers' Powers\"; but its Report,41 while mildly reasserting Dicey's doctrines, tended on the whole to minimize the dangers.  Its main effect was that it made the opposition to the rule of law articulate and evoked an extensive literature which outlined an antirule-of-law doctrine which has since come to be accepted by many besides socialists.\n\nThis movement was led by a group42 of socialist lawyers and political scientists gathered around the late Professor Harold J. Laski. The attack was opened by Dr. (now Sir Ivor) Jennings in reviews of the _Report_ and the _Documents_ on which the latter was based.43 Completely accepting the newly fashionable positivist doctrine, he argued that the conception of the rule of law, in the sense in which it was used in that Report, means that \"equality before the law, the ordinary law of the land, administered by the ordinary courts . . . taken literally . . . is just nonsense.\"44 This rule of law, he contended, \"is either common to all nations or does not exist.\"45 Though he had to concede that \"the fixity and certainty of the law have been part of the English tradition for centuries,\" he did so only with evident impatience at the fact that this tradition was \"but reluctantly breaking down.\"46 For the belief shared by \"most of the members of the Committee, and most of the witnesses . . . that there was a clear distinction between the functions of a judge and the functions of an administrator,\"47 Dr. Jennings had only scorn.\n\nHe later expounded these views in a widely used textbook, in which he expressly denied that \"the rule of law and discretionary powers are contradictory\"48 or that there is any opposition \"between 'regular law' and 'administrative powers.'\"49 The principle in Dicey's sense, namely, that public authorities ought not to have wide discretionary powers, was \"a rule of action for Whigs and may be ignored by others.\"50 Though Dr. Jennings recognized that \"to a constitutional lawyer of 1870, or even 1880, it might have seemed that the British Constitution was essentially based on the individualist rule of law, and that the British State was the _Rechtsstaat_ of individualist political and  legal theory,\"51 this meant to him merely that \"the Constitution frowned on 'discretionary' powers, unless they were exercised by judges. When Dicey said that Englishmen 'are ruled by the law, and by the law alone' he meant that 'Englishmen are ruled by judges, and by judges alone.' That would have been an exaggeration, but it was good individualism.\"52 That it was a necessary consequence of the ideal of liberty under the law that only experts in the law and no other experts, and especially no administrators concerned with particular aims, should be entitled to order coercive action seems not to have occurred to the author.\n\nIt should be added that further experience appears to have led Sir Ivor to modify his views considerably. He begins and concludes a recent popular book53 with sections in praise of the rule of law and even gives a somewhat idealized picture of the degree to which it still prevails in Britain. But this change did not come before his attacks had had a wide effect. In a popular _Vocabulary of Politics_ ,54 for instance, which had appeared in the same series only a year before the book just mentioned, we find it argued that \"it is therefore odd that there should be a prevalent view that the Rule of Law is something which some people have but other people do not have, like motor cars and telephones. What does it mean, then, to be without the Rule of Law? Is it to have no laws at all?\" I fear this question correctly represents the position of most of the younger generation, grown up under the exclusive influence of positivist teaching.\n\nEqually important and influential has been the treatment of the rule of law in a widely used treatise on administrative law by another member of the same group, Professor W. A. Robson. His discussion combines a commendable zeal for regularizing the chaotic state of the control over administrative action with an interpretation of the task of administrative tribunals which, if applied, would make them entirely ineffective as a means of protecting individual liberty. He aims explicitly at accelerating the \"break-away from that Rule of Law which the late Professor A. V. Dicey regarded as an essential feature of the English constitutional system.\"55 The argument commences with an attack on \"that antique and rickety chariot,\" the \"legendary separation of powers.\"56 The whole distinction between law and policy is to him \"utterly  false,\"57 and the conception that the judge is not concerned with governmental ends but with the administration of justice a matter for ridicule. He even represents as one of the main advantages of administrative tribunals that they \"can enforce a policy unhampered by rules of law and judicial precedents. . . . Of all the characteristics of administrative law, none is more advantageous, when rightly used for the public good, than the power of the tribunal to decide the cases coming before it with the avowed object of furthering a policy of social improvement in some particular field; and of adapting their attitude towards the controversy so as to fit the needs of that policy.\"58\n\nFew other discussions of these problems show as clearly how reactionary many of the \"progressive\" ideas of our time really are! It is therefore not too surprising that such a view as Professor Robson's has rapidly found favor with the conservatives and that a recent Conservative party pamphlet on the Rule of Law echoes him in commending administrative tribunals for the fact that \"flexible and unbound by rules of law or precedent, they can be of real assistance to their Minister in carrying out his policy.\"59 This acceptance of socialist doctrine by the conservatives is perhaps the most alarming feature of the development. It has gone so far that it could be said of a conservative symposium on _Liberty in the Modern State:_60 \"So far have we travelled from the conception of the Englishman protected by the courts from the risks of oppression by the Government or its servants that no one of the contributors suggests that it would now be possible for us to go back to that nineteenth century ideal.\"61\n\nWhere these views can lead to is shown by the more indiscreet statements of some of the less-well-known members of that group of socialist lawyers. One commences an essay on _The Planned State and the Rule of Law_ by \"redefining\" the rule of law.62 It emerges from the mauling as \"whatever parliament as the supreme lawgiver makes it.\"63 This enables the author \"to assert  with confidence that the incompatibility of planning with the rule of law [first suggested by socialist authors!] is a myth sustainable only by prejudice or ignorance.\"64 Another member of the same group even finds it possible to reply to the question as to whether, if Hitler had obtained power in a constitutional manner, the rule of law would still have prevailed in Nazi Germany: \"The answer is Yes; the majority would be right: the Rule of Law would be in operation, _if_ the majority _voted_ him into power. The majority might be unwise, and it might be wicked, but the Rule of Law would prevail. For in a democracy right is what the majority makes it to be.\"65 Here we have the most fatal confusion of our time expressed in the most uncompromising terms.\n\nIt is not surprising, then, that under the influence of such conceptions there has been in Great Britain during the last two or three decades a rapid growth of very imperfectly checked powers of administrative agencies over the private life and property of the citizen.66 The new social and economic legislation has conferred ever increasing discretionary powers on those bodies and has provided only occasional and highly defective remedies in the form of a medley of tribunals of committees for appeal. In extreme instances the law has even gone so far as to give administrative agencies the power to determine \"the general principles\" whereby what amounted to expropriation could  be applied,67 the executive authority then refusing to tie itself down by any firm rules.68 Only lately, and especially after a flagrant instance of highhanded bureaucratic action was brought to the attention of the public by the persistent efforts of a wealthy and public spirited man,69 has the disquiet over these developments long felt by a few informed observers spread to wider circles and produced the first signs of a reaction, to which we shall refer later.\n\n6. It is somewhat surprising to find that in many respects developments in this direction have gone hardly less far in the United States. In fact, both the modern trends in legal theory and the conceptions of the \"expert administrator\"  without legal training have had an even greater influence here than in Great Britain; it may even be said that the British socialist lawyers we have just considered have usually found their inspiration more often in American than in British legal philosophers. The circumstances which have brought this about are little understood even in the United States and deserve to be better known.\n\nThe United States is, in fact, unique in that the stimulation received from European reform movements early crystallized into what came to be known significantly as the \"public administration movement.\" It played a role somewhat similar to that of the Fabian movement in Britain70 or of the \"socialists of the chair\" movement in Germany. With efficiency in government as its watchword, it was skilfully designed to enlist the support of the business community for basically socialist ends. The members of this movement, generally with the sympathetic support of the \"progressives,\" directed their heaviest attack against the traditional safeguards of individual liberty, such as the rule of law, constitutional restraints, judicial review, and the conception of a \"fundamental law.\" It was characteristic of these \"experts in administration\" that they were equally antagonistic to (and commonly largely ignorant of ) both law and economics.71 In their efforts to create a \"science\" of administration, they were guided by a rather na\u00efve conception of \"scientific\" procedure and showed all the contempt for tradition and principles characteristic of the extreme rationalist. It was they who did most to popularize the idea that \"liberty for liberty's sake is clearly a meaningless notion: it must be liberty to do and enjoy something. If more people are buying automobiles and taking vacations, there is more liberty.\"72\n\nIt was mainly because of their efforts that Continental European conceptions of administrative powers were introduced into the United States rather earlier than into England. Thus, as early as 1921, one of the most distinguished American students of jurisprudence could speak of \"a tendency away from courts and law and a reversion to justice without law in the form of revival of executive and even of legislative justice and reliance upon arbitrary governmental power.\"73 A few years later a standard work on administrative  law could already represent it as accepted doctrine that \"every public officer has, marked out for him by law, a certain area of 'jurisdiction.' Within the boundaries of that area he can act freely according to his own discretion, and the courts will respect his action as final and not inquire into its rightfulness. But if he oversteps those bounds, then the court will intervene. In this form, the law of court review of the acts of public officers becomes simply a branch of the law of _ultra vires_. The only question before the courts is one of jurisdiction, and the court has no control of the officer's exercise of discretion within that jurisdiction.\"74\n\nThe reaction against the tradition of stringent control of the courts over not only administrative but also legislative action had, in fact, commenced some time before the first World War. As an issue of practical politics it became important for the first time in Senator La Follette's campaign for the presidency in 1924, when he made the curbing of the power of the courts an important part of his platform.75 It is mainly because of this tradition which the Senator established that, in the United States more than elsewhere, the progressives have become the main advocates of the extension of the discretionary powers of the administrative agency. By the end of the 1930s, this characteristic of the American progressives had become so marked that even European socialists, when \"first faced with the dispute between the American liberals and the American conservatives concerning the questions of administrative law and administrative discretion,\" were inclined \"to warn them against the inherent dangers of the rise of administrative discretion, and to tell them that we [i.e., the European socialists] could vouch for the truth of the stand of the American conservative.\"76 But they were soon mollified when they discovered how greatly this attitude of the progressives facilitated the gradual and unnoticed movement of the American system toward socialism.\n\nThe conflict referred to above reached its height, of course, during the Roosevelt era, but the way had already been prepared for the developments of that time by the intellectual trends of the preceding decade. The 1920s and  early 1930s had seen a flood of antirule-of-law literature which had considerable influence on the later developments. We can mention here only two characteristic examples. One of the most active of those who led the frontal attack on the American tradition of a \"government of law and not of men\" was Professor Charles G. Haines, who not only represented the traditional ideal as an illusion77 but seriously pleaded that \"the American people should establish governments on a theory of trust in men in public affairs.\"78 To realize how completely this is in conflict with the whole conception underlying the American Constitution, one need merely remember Thomas Jefferson's statement that \"free government is founded in jealousy, not in confidence; it is jealousy and not confidence which prescribes limited constitutions, to bind those we are obliged to trust with power . . . our Constitution has accordingly fixed the limits to which, and no further, our confidence may go. . . . In questions of power, then, let no more be heard of confidence in man, but bind him down from mischief by the chains of the Constitution.\"79\n\nPerhaps even more characteristic of the intellectual tendencies of the time is a work by the late justice Jerome Frank, called _Law and the Modern Mind_ , which, when it first appeared in 1930, enjoyed a success which for the reader of today is not quite easy to understand. It constitutes a violent attack on the whole ideal of the certainty of the law, which the author ridicules as the product of \"a childish need for an authoritative father.\"80 Basing itself on psychoanalytic theory, the work supplied just the kind of justification for a contempt for the traditional ideals that a generation unwilling to accept any limitation on collective action wanted. It was the young men brought up on such ideas who became the ready instruments of the paternalistic policies of the New Deal.\n\nToward the end of the 1930s there was increasing uneasiness over these developments, which led to the appointment of a committee of investigation, the U.S. Attorney General's Committee on Administrative Procedure, whose task was similar to that of the British committee of ten years earlier. But this, too, even more than the British committee, tended in its Majority Report81 to  represent what was happening as both inevitable and harmless. The general tenor of the report is best described in the words of Dean Roscoe Pound: \"Even if quite unintended, the majority are moving in the line of administrative absolutism which is a phase of the rising absolutism throughout the world. Ideas of the disappearance of law, of a society in which there will be no law, or only one law, namely that there are no laws but only administrative orders; doctrines that there are no such things as rights and that laws are only threats of exercise of state force, rules and principles being nothing but superstition and pious wish, a teaching that separation of powers is an outmoded eighteenth century fashion of thought, that the common law doctrine of the supremacy of law had been outgrown, and expounding of a public law which is to be a 'subordinating law,' subordinating the interests of the individual to those of the public official and allowing the latter to identify one side of a controversy with the public interest and so give it a greater value and ignore the others: and finally a theory that law is whatever is done officially and so whatever is done officially is law and beyond criticism by lawyers\u2014such is the setting in which the proposals of the majority must be seen.\"82\n\n7. Fortunately, there are clear signs in many countries of a reaction against these developments of the last two generations. They are perhaps most conspicuous in the countries that have gone through the experience of totalitarian regimes and have learned the dangers of relaxing the limits on the powers of the state. Even among those socialists who not long ago had nothing but ridicule for the traditional safeguards of individual liberty, a much more respectful attitude can be observed. Few men have so frankly expressed this change of view as the distinguished dean of socialist legal philosophers, the late Gustav Radbruch, who in one of his last works said: \"Though democracy  is certainly a praiseworthy value, the _Rechtsstaat_ is like the daily bread, the water we drink and the air we breathe; and the greatest merit of democracy is that it alone is adapted to preserve the _Rechtsstaat_.\"83 That democracy does not in fact necessarily or invariably do so is only too clear from Radbruch's description of developments in Germany. It would probably be truer to say that democracy will not exist long unless it preserves the rule of law.\n\nThe advance of the principle of judicial review since the war and the revival of the interest in the theories of natural law in Germany are other symptoms of the same tendencies.84 In other Continental countries similar movements are under way. In France, G. Ripert has made a significant contribution with his study of _The Decline of Law_ , in which he rightly concludes that \"above all, we must put the blame on the jurists. It was they who for half a century undermined the conception of individual rights without being aware that they thereby delivered these rights to the omnipotence of the political state. Some of them wished to prove themselves progressive, while others believed that they were rediscovering traditional doctrine which the liberal individualism of the nineteenth century had obliterated. Scholars often show  a certain single-mindedness which prevents them from seeing the practical conclusions which others will draw from their disinterested doctrines.\"85\n\nThere has been no lack of similar warning voices86 in Great Britain, and the first outcome of the increasing apprehension has been a renewed tendency in recent legislation to restore the courts of law as the final authority in administrative disputes. Encouraging signs are also to be found in a recent report of a committee of inquiry into procedure for appeals to other than ordinary courts.87 In it the committee not only made important suggestions for eliminating the numerous anomalies and defects of the existing system but also admirably reaffirmed the basic distinction between \"what is judicial, its antithesis being what is administrative, and the notion of what is according to the rule of law, its antithesis being what is arbitrary.\" It then went on to state: \"The rule of law stands for the view that decisions should be made by known principles or laws. In general such decisions will be predictable, and the citizen will know where he is.\"88 But there still remains in Britain a \"considerable field of administration in which no special tribunal or enquiry is provided\"89 (which problem was outside the terms of reference of the committee) and where the conditions remain as unsatisfactory as ever and the citizen in effect is still at the mercy of an arbitrary administrative decision. If the process of erosion of the rule of law is to be halted, there seems to be urgent need for some independent  court to which appeal lies in all such cases, as has been proposed from several quarters.90\n\nFinally, we might mention, as an effort on an international scale, the \"Act of Athens\" adopted in June, 1955, at a congress of the International Commission of Jurists, in which the importance of the rule of law is strongly reaffirmed.91\n\nIt can hardly be said, however, that the widespread desire to revive an old tradition is accompanied by a clear awareness of what this would involve92 or that people would be prepared to uphold the principles of this tradition even when they are obstacles in the most direct and obvious route to some desired aim. These principles which not long ago seemed commonplaces hardly worth restating and which perhaps even today will seem more obvious  to the layman than to the contemporary lawyer have been so forgotten that a detailed account of both their history and their character seemed necessary. It is only on this basis that we can attempt in the next part to examine in more detail the different ways in which the various modern aspirations of economic and social policy can or cannot be achieved within the framework of a free society.\n\n* * *\n\nThe quotation at the head of the chapter is taken from Lord Acton, \"Sir Erskine May's _Democracy in Europe_ ,\" _History of Freedom_ , p. 78 [Liberty Fund edition, _Essays in the History of Liberty_ , p. 68]. The title of the chapter is borrowed from Georges Ripert, _Le D\u00e9clin du droit_ (Paris: Pichon and Durand-Auzias, 1949).\n\n1 Anton Menger, _Das b\u00fcrgerliche Recht und die besitzlosen Volksklassen_ (1890) (3rd ed.; T\u00fcbingen: H. Laupp, 1904), p. 30. [The original German reads: \"Indem man nun alle Staatsb\u00fcrger ohne R\u00fccksicht auf ihre pers\u00f6nlichen Eigenschaften und auf ihre wirtschaftliche Lage v\u00f6llig gleich behandelte und zwischen ihnen einen z\u00fcgellosen Wettbewerb zuliess, bewirkte man zwar, dass die G\u00fctererzeugung ins unendliche stieg, zugleich aber auch, dass die Armen und Schwachen an den gesteigerten G\u00fctermengen nur einen sehr geringen Anteil hatten. Daher die neue wirtschaftliche und Sozialgesetzgebung, welche bestrebt ist, den Schwachen gegen den Starken zu sch\u00fctzen und ihm an den G\u00fctern des Lebens wenigstens einen bescheidenen Anteil zu sichern. Man weiss eben heute, dass es keine gr\u00f6ssere Ungleichheit gibt, als das Ungleiche gleich zu behandeln.\"\u2014Ed.] The full consequences of this conception are worked out in that author's later book, _Neue Staatslehre_ ( Jena: Gustav Fischer, 1903). About the same time the great German criminologist, Franz Eduard von Liszt could already comment ( _Strafrechtliche Aufs\u00e4tze und Vortr\u00e4ge_. Vol. 2: _1892 bis 1904_ [2 vols.; Berlin: J. Guttentag, 1905], p. 60): \"Das heranwachsende sozialistische Geschlecht, das die gemeinsamen Interessen sch\u00e4rfer betont als seine Vorg\u00e4nger, f\u00fcr dessen Ohren das Wort 'Freiheit' einen archaistischen Klang gewonnen hat, r\u00fcttelt an diesen Grundlagen.\" [\"The coming socialist generation, which emphasizes common interests with greater force than did its predecessors and for whose ears the word 'freedom' has an archaic ring, is buffeting the foundations (of justice).\"\u2014Ed.] The infiltration of the same ideas into England is well illustrated by David George Ritchie, _Natural Rights: A Criticism of Some Political and Ethical Conceptions_ (1894) (3rd ed.; London: Allen and Unwin, 1916), p. 258: \"The claim of equality, in its widest sense, means the demand for equal opportunity\u2014the _carri\u00e8re ouverte aux talents_. The result of such equality of opportunity will clearly be the very reverse of equality of social condition, if the law allows the transmission of property from parent to child, or even the accumulation of wealth by individuals. And thus, as has often been pointed out, the effect of the nearly complete triumph of the principles of 1789\u2014the abolition of legal restrictions on free competition\u2014has been to accentuate the difference between wealth and poverty. Equality in political rights, along with great inequalities in social condition, has laid bare 'the social question'; which is no longer concealed, as it formerly was, behind the struggle for equality before the law and for equality in political rights.\"\n\n2 Anatole France, _Le Lys rouge_ (Paris: Calmann-L\u00e9vy, 1894), p. 118. [The original reads: \"La majestueuse \u00e9galit\u00e9 des lois, qui interdit au riche comme au pauvre de coucher sous les ponts, de mendier dans les rues et de voler du pain.\"\u2014Ed.]\n\n3 The tradition traces back to the later work of Rudolph von Ihering (1818\u20131882). [Von Ihering's most important works were probably _The Spirit of the Roman Laws_ (1852\u201365), originally published in German under the title _Geist des r\u00f6mischen Rechts auf den verschieden Stufen seiner Entwicklung_ (4 vols.; Leipzig: Breitkopf und H\u00e4rtel, 1852\u201365); _The Struggle for Law_ (1879), translated from the 5th German ed. of _Der Kampf um's Recht_ (Vienna: Manz, 1877); and _Law as a Means to an End_. Vol. 1: 1877; Vol. 2: 1883, which was a translation of the first volume of the 4th ed. of Ihe ring's _Der Zweck im Recht_ (Leipzig: Breitkopf und H\u00e4rtel, 1904\u201305). These works underscored Ihering's theory that self-interest was of crucial importance in shaping the law and that the process by which legal rules were maintained was self-regulating.\u2014Ed.] For the modern development see the essays collected in _The Jurisprudence of Interests: Selected Writings of Max R\u00fcmelin_ , Magdalena Schoch, ed. (Twentieth Century Legal Philosophy Series, vol. 2; Cambridge, MA: Harvard University Press, 1948).\n\n4 See, e.g., Fritz Fleiner, _Ausgew\u00e4hlte Schriften und Reden_ (Zurich: Polygraphischer Verlag, 1941), p. 438: \"Dieser Umschwung [zum totalit\u00e4ren Staat] ist vorbereitet worden durch gewisse Richtungen innerhalb der deutschen Rechtswissenschaft (z.B. die sogenannte Freirechtsschule), die geglaubt haben, dem Rechte zu dienen, indem sie die Gesetzestreue durchbrachen.\" [\"This change (this transformation toward the totalitarian state) was adumbrated by certain tendencies that marked German jurisprudence (e.g., the so-called school of free-law) that held that it was possible to serve the law by violating its integrity.\" (Interpolation Hayek's.)\u2014Ed.]\n\n5 About the character of this historicism see Menger, _Untersuchungen_ , and Sir Karl Raimund Popper, _The Poverty of Historicism_ (London: Routledge and Kegan Paul, 1957).\n\n6 Cf. my _The Counter-Revolution of Science: Studies in the Abuse of Reason_ (Glencoe, IL: Free Press, 1952), pt. 1, chap. 7 [\"The Historicism of the Scientific Approach,\" pp. 64\u201379; _Collected Works_ edition, vol. 13, pp. 126\u201341].\n\n7 On the connection between historicism and legal positivism cf. Hermann Heller, \"Bemerkungen zur staats- und rechtstheoretischen Problematik der Gegenwart,\" _Archiv f\u00fcr \u00f6_ _ff_ _entliches Recht_ , 16 (1929): 336.\n\n8 The best brief survey of the different \"natural-law\" traditions that I know of is Alessandro Passerin d'Entr\u00e8ves, _Natural Law: An Introduction to Legal Philosophy_ (Hutchinson's University Library; London: Hutchinson, 1951). [This book is the outcome of eight lectures delivered at the University of Chicago in April 1948.\u2014Ed.] It may also be briefly mentioned here that modern legal positivism derives largely from Thomas Hobbes and Ren\u00e9 Descartes, the two [The 1971 German edition reads: \"three\" and includes \"Francis Bacon.\"\u2014Ed.] men against whose rationalistic interpretation of society the evolutionary, empiricist, or \"Whig\" theology was developed, and that positivism gained its present-day ascendancy largely because of the influence of Hegel and Marx. For Marx's position, see the discussion of individual rights in the Introduction to his _Kritik der Hegelschen Rechtsphilosophie_ , in Karl Marx and Friedrich Engels, _Historisch-kritische Gesamtausgabe, Werke, Schriften, Briefe_ , David Borisovic Rjazanov [David Borisovic Gol'dendach], ed. (11 vols.; Berlin: Marx-Engels Archiv, Marx-Engels Verlag, 1927\u201332), vol. 1, pt. 1.\n\n9 Cf. Hermann Heller, _Rechtsstaat oder Diktatur?_ (T\u00fcbingen: Mohr, 1930); John Hamilton Hallowell, _The Decline of Liberalism as an Ideology, with Particular Reference to German Politico-legal Thought_ (Berkeley: University of California Press, 1943); and _The Moral Foundation of Democracy_ (Chicago: University of Chicago Press, 1954), chap. 4, \"Democracy and Liberalism,\" pp. 68\u201388, esp. p. 73.\n\n10 Richard Thoma, \"Rechtsstaatsidee und Verwaltungstrechtswissenschaft,\" in _Jahrbuch des offentlichen Rechts der Gegenwart_ , 4 (1910): 199. [The German reads: \"Diese individualistische Rechtsstaatsidee hat in der Tat ihre Rolle ausgespielt. Die sch\u00f6pferische Kr\u00e4fte der nationalen und der sozialen Ideen haben sie \u00fcberwunden. Es wird das h\u00e4ufig betont und niemand zweifelt daran.\"\u2014Ed.]\n\n11 Edmund Bernatzik, _Die Ausgestaltung des Nationalgef\u00fchls im 19. Jahrhundert. Rechtsstaat und Kulturstaat: Zwei Vortr\u00e4ge gehalten in der Vereinigung f\u00fcr staatswissenschaftliche Fortbildung in C\u014dln im April 1912_ (Hanover: Helwing, 1912), p. 56. [The German reads: \"Wir Heutigen kehren zu den Principien des Polizeistaates insofern zur\u00fcck, als wir seine Kulturstaatsidee wieden anerkennen. Der einzige Unterschied liegt in den rechlichen Mitteln. Auf Grund von Gesetzen gestattet sich der heutige Staat alles noch viel mehr als der Polizeistaat. Und so gelangte man im Laufe des 19. Jahrhunderts zu einer neuen Bedeutung des Ausdruckes ' _Rechtsstaat_.' Man verstand jetzt darunter einen Staat, dessen ganze T\u00e4tigkeit sich auf Grund von Gesetzen in rechtlichen Formen abspielt. \u00dcber den Staatszweck und die Grenzen der staatlichen Kompetenz sagt das Wort ' _Rechtsstaat_ ' in seiner heutigen Bedeutung gar nichts mehr aus und in dieser neuen Bedeutung steht daher das Wort auch nicht mehr in einem Gegensatz zum Kulturstaat.\"\u2014Ed.]. Cf. also the same author's \"Polizei und Kulturpflege,\" in _Systematische Rechtswissenschaft_ (Berlin: Teubner, 1906) [pt. 2, sec. 8 of _Die Kultur der Gegenwart. Ihre Entwicklung und ihre Ziele_ , edited by Paul Hinneberg], pp. 387\u2013426.\n\n12 The victory of legal positivism had been secured earlier, mainly through the relentless efforts of Carl Bergbohm, _Jurisprudenz und Rechtsphilosophie: kritische Abhandlungen_ (Leipzig: Duncker und Humblot, 1892), but it was in the form given to it by Hans Kelsen that it achieved a widely accepted and consistent philosophical basis. We shall here quote mainly from Kelsen's _Allgemeine Staatslehre_ (Berlin: Julius Springer, 1925), but the reader will find most of the essential ideas restated in his _General Theory of Law and State_ , Anders Wedberg, trans. (Twentieth Century Legal Philosophy Series; Cambridge, MA: Harvard University Press, 1945), which also contains a translation of an important lecture on _Die philosophischen Grundlagen der Naturrechtslehre und des Rechtspositivismus_ (Charlottenburg: Verlag Rolf Heise, 1928). [An English translation of Kelsen's _Die philosophischen Grundlagen_ appears on pp. 391\u2013446 of his _General Theory of Law and State_.\u2014Ed.]\n\n13 Hans Kelsen, _Vom Wesen und Wert der Demokratie_ (T\u00fcbingen: J. C. B. Mohr, 1920), p.10 [The German reads: \"die im Grunde genommen unrettbare Freiheit des Individuums tritt allm\u00e4hlich in den Hintergrund und die Freiheit des sozialen Kollektivums in den Vordergrund.\"\u2014Ed.]. The phrase \"im Grunde unrettbare Freiheit des Individuums\" [\"the fundamentally irretrievable freedom of the individual\"\u2014Ed.] becomes in the second edition of 1929 \"im Grunde unm\u00f6gliche Freiheit des Individuums\" [\"the in fact impossible freedom of the individual\"\u2014Ed.] ( _Von Wesen und Wert der Demokatie_ [2nd ed.; T\u00fcbingen: J. C. B. Mohr, 1929], p. 11).\n\n14 \"Losl\u00f6sung des Demokratismus von Liberalismus,\" Hans Kelsen, _Vom Wesen und Wert der Demokratie_ , p. 10.\n\n15 Hans Kelsen, _Allgemeine Staatslehre_ , p. 91. Cf. also his _Hauptprobleme der Staatsrechtslehre: entwickelt aus der Lehre vom Rechtssatze_ (2nd ed.; T\u00fcbingen: J. C. B. Mohr, 1923), p. 249, where his approach leads him consistently to assert that \"a wrong of the state must under all circumstances be a contradiction in terms.\" [\"Ein Unrecht des Staates mu\u00df unter allen Umst\u00e4nden ein Widerspruch in sich selbst sein.\"\u2014Ed.]\n\n16 Hans Kelsen, _Allgemeine Staatslehre_ , p. 335; the relevant passages read in translation: \"Entirely meaningless is the assertion that under a despotism there exists no order of law [ _Rechtsordnung_ ], [that there] the arbitrary will of the despot reigns. . . . The despotically governed state also represents some order of human behavior. This order is the order of law. To deny to it the name of an order of law is nothing but na\u00efvet\u00e9 and presumption deriving from natural-law thinking. . . . What is interpreted as arbitrary will is merely the legal possibility of the autocrat's taking on himself every decision, determining unconditionally the activities of subordinate organs and rescinding or altering at any time norms once announced, either generally or for a particular case. Such a condition is a condition of law even when it is felt to be disadvantageous. It has also its good aspects. The demand for dictatorship not uncommon in the modern _Rechtsstaat_ shows this very clearly.\" [\"Vollends sinnlos ist die Behauptung, da\u00df in der Despotie keine Rechtsordnung bestehe, sondern Willk\u00fcr des Despoten herrsche. . . . stellt doch auch der despotisch regierte Staat irgendeine Ordnung menschlichen Verhaltens dar, weil ja ohne eine solche Ordnung \u00fcberhaupt kein Staat, ja \u00fcberhaupt keine Gemeinschaft m\u00f6glich, kein Mensch als Herrscher, K\u00f6nig, F\u00fcrst qualifizierbar w\u00e4re. Diese Ordnung ist eben die Rechtsordnung. Ihr den Charakter des Rechts absprechen, ist nur eine naturrechtliche Naivit\u00e4t oder \u00dcberhebung. . . . Was als Willk\u00fcr gedeutet wird, ist nur die rechtliche M\u00f6glichkeit des Autokraten, jede Entscheidung an sich zu ziehen, die T\u00e4tigkeit der untergeordneten Organe bedingungslos zu bestimmen und einmal gesetzte Normen jederzeit mit allgemeiner oder nur besonderer Geltung aufzuheben oder abzu\u00e4ndern. Ein solcher Zustand ist ein Rechtzustand, auch wenn er als nachteilig empfunden wird. Doch hat er auch seine guten Seiten. Der im modernen _Rechtsstaat_ gar nicht seltene Ruf nach Diktatur zeigt dies ganz deutlich.\"\u2014Ed.] That this passage still represents the author's views is explicitly acknowledged by him in his essay \"Foundations of Democracy,\" _Ethics_ , no. 1, pt. 2, 66 (1955):100, n. 13; see also an earlier version of the same argument, entitled \"Democracy and Socialism,\" in _Conference of Jurisprudence and Politics_ , Scott Buchanan, ed. (Chicago: University of Chicago Law School, 1955), pp. 63\u201387.\n\n17 Hans Kelsen, _Allgemeine Staatslehre_ , p. 14.\n\n18 Hans Kelsen, _Allgemeine Staatslehre_ , pp. 154ff. [the phrase is \"die sogenannten Freiheitsrechte.\"\u2014Ed.]\n\n19 Hans Kelsen, _Allgemeine Staatslehre_ , p. 335. [The original quotation reads: \"Ihr den Charakter des Rechts absprechen, ist nur eine naturrechtliche Naivit\u00e4t oder \u00dcberhehung.\"\u2014Ed.]\n\n20 Hans Kelsen, _Allgemeine Staatslehre_ , pp. 231\u201335ff.; cf. the same author's _General Theory of Law and State_ , p. 38.\n\n21 Erich Voegelin, \"Kelsen's Pure Theory of Law,\" _Political Science Quarterly_ , 42 (1927): 269.\n\n22 Friedrich Darmst\u00e4dter, _Die Grenzen der Wirksamkeit des Rechtsstaates. Eine Untersuchung zur gegenw\u00e4rtigen Krise des liberalen Staatsgedankens_ (Heidelberg: Carl Winters, 1930), passim. (On socialism see esp. pp. 48\u201351.) Cf. Hallowell, _The Decline of Liberalism as an Ideology_ and _The Moral Foundations of Democracy_. On the further development under the Nazis see Franz Leopold Neumann, _Behemoth: The Structure and Practice of National Socialism, 1933\u20131944_ (2nd ed.; New York: Oxford University Press, 1944), and Aurel Kolnai, _The War Against the West_ (New York: Viking Press, 1938), pp. 299\u2013310.\n\n23 Darmst\u00e4dter, _Die Grenzen der Wirksamkeit des Rechtsstaates_ , p. 95. [The German reads: \"Tendenz das Wesen des Rechtsstaates im Staatsbegriff schlechthin untergehen zu lassen, . . . dem Siege und der Alleingeltung des faschistischen und bolschewistischen Staatswollens . . . derart von der Staatslehre her ein grunds\u00e4tzliches Hindernis nicht mehr engegen [stellt].\"\u2014Ed.]\n\n24 Heinrich Triepel, Comment on \"Die Reform des Wahlrechts\" ( pp. 194\u201398), and Gerhard Leibholz, \"Die Wahlrechtsreform und ihre Grundlagen\" ( pp. 159\u201390), in _Entwicklung und Reform des Beamtenrechts. Die Reform des Wahlrechts_ , Hans Gerber, ed. (Series title: Ver\u00f6ffentlichungen der Vereinigung der Deutschen Staatsrechtslehrer: Vol. 7; Berlin: Walter de Gruyter, 1932).\n\n25 Aleksandr Leonidovitch Malitzki, quoted by Boris Mirkin-Getzewitsch, _Die rechtstheoretischen Grundlagen des Sowjetstaates_ (Leipzig: Franz Deuticke, 1929), p. 117. [Translated from the French by Rita Willfort, _La th\u00e9orie g\u00e9n\u00e9rale de l'\u00e9tat sovi\u00e9tique_ (Paris: Marcel Giard, 1928).] [The quotation to which Hayek is referring reads: \"'Die Lehre vom _Rechtsstaat_ sagt die Sowjettheorie, 'ist in ihren Grundz\u00fcgen eine Doktrin vom unfreien Staat.'\" (\"The rule of law, according to Soviet theory, has as its fundamental principle the doctrine of the unfree state.\")\u2014Ed.] Cf., however, a similar discussion in Rudolph von Ihering, _Law as a Means to an End_ , Isaac Husik, trans. (Boston: Boston Book Co., 1913), pp. 314\u201315: \"Exclusive domination of the law is synonymous with the resignation on the part of society, of the free use of its hands. Society would give herself up with bound hands to rigid necessity, standing helpless in the presence of all circumstances and requirements of life which were not provided for in the law, or for which the latter was found to be inadequate. We derive from this the maxim that the State must not limit its own power of spontaneous self-activity by law any more than is absolutely necessary\u2014rather too little in this direction than too much. It is a wrong belief that the interest or the security of right and of political freedom requires the greatest possible limitation of the government by the law. This is based upon the strange notion [!] that force is an evil which must be combated to the utmost. But in reality it is a good, in which, however, as in every good, it is necessary, in order to make possible its wholesome use, to take the possibility of its abuse into the bargain.\" Cf. Otto Friedrich von Gierke, _Johannes Althusius und die Entwickling der naturrechtlichen Staatstheorien: zugleich ein Beitrag zur Geschichte der Rechtssystematik_ (Breslau: W. Koebner, 1880), p. 304, in which he remarks about the theory of the _Rechtsstaat_ put forward by Kant and Humboldt: \"Dieser _Rechtsstaat_ w\u00e4re, wenn seine Verwirklichung \u00fcberhaupt denkbar gewesen w\u00e4re, mit der vollen Unfreiheit und Ohnmacht der Staatsgewalt erkauft worden.\" [\"The _Rechtsstaat_ could have been purchased only at the cost of the total impotence and incapacity of the state, were it even conceivable.\"\u2014Ed.]\n\n26 Giacomo Perticone, \"Quelques aspects de la crise du droit public en Italie,\" _Revue internationale de la th\u00e9orie du droit \/ Internationale Zeitschrift f\u00fcr Theorie des Rechts_ , 5 (1931\u201332): 2. [The full French quotation reads: \"En se passant de toute l'\u00e9volution de la pens\u00e9e juridique, on a cru pouvoir consid\u00e9rer l'\u00c9tat de droit comme l'\u00c9tat prisonnier du droit, incapable, par cons\u00e9quent, de mouvement, de volont\u00e9, de puissance; un \u00c9tat aboulique, neutre, et ce qui s'ensuit.\" (\"During the whole of the evolution of juridical thought, one was led to the conclusion that a regime of law was one in which the State was a prisoner of the law, and as a consequence incapable of action, of will, of power, a State indecisive, emasculated, and all that which follows.\")\u2014Ed.]\n\n27 See Carl Schmitt, \"Was bedeutet der Streit um den ' _Rechtsstaat_ ,'\" _Zeitschrift f\u00fcr die gesamte Staatswissenschaft_ [ _Journal of Institutional and Theoretical Economics_ ] (T\u00fcbingen), 95 (1935): 190.\n\n28 R. Archipov, _Law in the State_ (in Russian), quoted in Boris Mirkin-Getzewitsch, _Die rechtstheoretischen Grundlagen des Sowjetstaates_ , p. 108\u20139. [The German reads: \"Ein Sowjetjurist behauptet, da\u00df 'der Schwerpunkt sich mehr und mehr von der Erlassung genereller Normen zur Setzung individueller Akte und Instruktionen verschiebe, welche die T\u00e4tigkeit der Verwaltung regeln, f\u00f6rdern und koordinieren.\"\u2014Ed.]\n\n29 Peter Ivanovitch Stuchka, _Uchenie o gosudarstve proletariata i krest'ianstva i ego konstitutsii: SSSR \/ RSFSR_ [ _Theory of the State of the Proletarians and Peasants and Its Constitution_ ] (5th ed.; Moscow: Gos. izd-vo, 1926), quoted in Boris Mirkin-Getzewitsch, _Die rechtstheoretischen Grundlagen des Sowjetstaates_ , p. 70ff. [The French translation of the Russian text (Petr Ivanovitch Stoutchka, _La th\u00e9orie de l'\u00c9tat prol\u00e9tarien et paysan et ses constitutions_ [5th ed.; Moscow, 1926], p. 194) reads: \"Dans l'impossibilit\u00e9 o\u00f9 l'on est de distinguer o\u00f9 finit la loi et o\u00f9 commence l'ordonnance administrative, cette opposition n'est qu'une pure fiction de la science et la pratique bourgeoise.\"\u2014Ed.]\n\n30 Mirkin-Getzewitsch, _Die rechtstheoretischen Grundlagen des Sowjetstaates_ , p. 107. [The German reads: \"Aber was gerade das Sowjetsystem von s\u00e4mtlichen \u00fcbrigen despotischen Staatsformen der Gegenwart und Vergangenheit unterscheidet, ist, da\u00df es nicht nur die faktische Gewalt kennt, die tats\u00e4chliche Ungesetzlichkeit, sondern da\u00df es au\u00dferdem auf einer sozialen Basis von ungeheurer Ausdehnung einen Versuch dastellt, den Staat auf Grunds\u00e4tzen zu begr\u00fcnden, die denen eines Rechtsstaates entgegengesetzt sind. . . . Der Sowjetstaat hingegen hat eine Theorie ausgearbeitet, die Herrschenden von jeder Verpflichtung, von jeder Beschr\u00e4nkung ausnimmt.\"\u2014Ed.]\n\n31 Malitzki, quoted by Mirkin-Getzewitsch, _Die rechtstheoretischen Grundlagen des Sowjetstaates_ , p. 89. [The full quotation reads: \"Hieraus folgt der fundamentale Grundsatz unserer Gesetzgebung und unseres Zivilrechtes, den die b\u00fcrgerlichen Theorien niemals anerkennen werden: Alles, was nicht speziell erlaubt worden ist, ist verboten,\" denn \"'entgegen der europ\u00e4ischen Doktrin erkl\u00e4ren wir, da\u00df Subjekt der Gewalt, Quelle des Rechtes nicht der Einzelne, sondern der Staatist.\" (\"From this follows that the fundamental principle of our legislation and our private law, which bourgeois theories will never recognize, is: everything that is not specifi cally permitted is prohibited\" because \"in contrast to European teaching, we hold that the subject of power and the source of law is not the individual but the state.\")\u2014Ed.] It has to be admitted, however, that this principle is also to be found in Aristotle, _Nicomachean Ethics_ , 1138a.1 [bk. 5, chap. 11]: \"Whatever [the law] does not bid it forbids.\"\n\n32 Quoted by Vladimir Gsovski, _Soviet Civil Law: Private Rights and Their Background Under the Soviet Regime; Comparative Survey and Translation of the Civil Code, Code of Domestic Relations, Judiciary Act, Code of Civil Procedure, Laws on Nationality, Corporations, Patents, Copyright, Collective Farms, Labor, and Related Laws_ (2 vols.; Ann Arbor: University of Michigan Law School, 1948\u201349), vol. 1, p. 170; quoted in Peter Ivanovitch Stuchka, _Entsiklopediia gosudarstva i prava_ [ _Encyclopedia of State and Law_ ] (3 vols.; Moscow: Izd-vo Kommunisticheskoi Akademii, 1925\u201327), p. 1593.\n\n33 Concerning Pashukanis's fate, Roscoe Pound observes in his _Administrative Law: Its Growth, Procedure, and Significance_ (Pittsburgh: University of Pittsburgh Press, 1942), p. 127: \"The Professor [Evgenii Bronislavovich Pashukanis] is not with us now. With the setting up of a plan by the present government in Russia, a change of doctrine was called for and he did not move fast enough in his teaching to conform to the doctrinal exigencies of the new order. If there had been law instead of only administrative orders it might have been possible for him to lose his job without losing his life.\"\n\n34 Evgenii Bronislavovich Pashukanis, _Allgemeine Rechtslehre und Marxismus: Versuch einer Kritik der juristischen Grundbegriffe_ , trans. from the 2nd Russian edition by Edith Haj\u00f3s (Vienna: Verlag f\u00fcr Literatur und Politik, 1929), p. 117. [The German reads: \"Die verwaltungstechnische Leitung durch Unterwerfung unter einen allgemeinen Wirtschaftsplan ist analog der Methode der direkten technologisch bestimmten Leitung in der Gestaltung der Programme f\u00fcr die Produktion und Verteilung. Der allm\u00e4hliche Sieg dieser Tendenz bedeutet die allm\u00e4hliche Aufgabe des Rechts als solchem.\"\u2014Ed.] An English translation of this and of a later work by Pashukanis has been published in _Soviet Legal Philosophy_ , Hugh Webster Badd, trans., Introduction by John Newbold Hazard (Cambridge, MA: Harvard University Press, 1951). For discussions, see Hans Kelsen, _The Communist Theory of Law_ (New York: Praeger, 1955); Rudolph Schlesinger, _Soviet Legal Theory: Its Social Background and Development_ (2nd ed.; London: Routledge and Kegan Paul, 1951); Lon Luvois Fuller, \"Pashukanis and Vyshinsky: A Study in the Development of Marxian Legal Theory,\" _Michigan Law Review_ , 47 (1948\u201349): 1157\u201366; and Samuel Dobrin, \"Soviet Jurisprudence and Socialism,\" _Law Quarterly Review_ , 52 (1936): 402\u201324.\n\n35 This summary of Pashukanis's argument is taken from Wolfgang Gaston Friedmann, _Law and Social Change in Contemporary Britain_ (London: Stevens and Sons, 1951), p. 154.\n\n36 Dicey, _Law of the Constitution_ (8th ed.), p. xxxviii. [Liberty Fund edition, p. lv. The 8th ed. of Dicey's book appeared in 1915.]\n\n37 Gordon Hewart, Baron Hewart, _The New Despotism_ (London: Ernest Benn Ltd., 1929).\n\n38 Characteristic of the treatment which that well justified warning received even in the United States is the following comment by Professor (now Justice) Felix Frankfurter, published in 1938: \"As late as 1929 Lord Hewart attempted to give fresh life to the moribund unrealities of Dicey by garnishing them with alarm. Unfortunately, the eloquent journalism of this book carried the imprimatur of the Lord Chief Justice. His extravagant charges demanded authoritative disposition and they received it\" (foreword to \"Current Developments in Administrative Law,\" _Yale Law Journal_ , 47 [1938]: 517). [Lord Hewart called attention to the dangers inherent in the increasingly common practice of Parliament delegating their powers to administrative tribunals, thus subverting Parliamentary government.\u2014Ed.]\n\n39 \"What is the Public Interest?\" _Economist_ , June 19, 1954, p. 952: \"The 'new despotism,' in short, is not an exaggeration, it is a reality. It is a despotism that is practised by the most conscientious, incorruptible and industrious tyrants that the world has ever seen.\"\n\n40 Richard Howard Stafford Crossman, _Socialism and the New Despotism_ [Fabian Tracts, No. 298] (London: Fabian Society, 1956).\n\n41 Committee on Ministers' Powers, _Report Presented by the Lord High Chancellor to Parliament by Command of His Majesty, April 1932_ [the Donoughmore Report], chaired from 30 October 1929 to 2 May 1931 by the Rt. Hon. The Earl of Donoughmore. Cmd. 4060 (London: His Majesty's Stationery Office, 1932); see also the Committee on Ministers' Powers, _Memoranda Submitted by Government Departments in Reply to Questionnaire of November 1929 and Minutes of Evidence_ (2 vols.; London: His Majesty's Stationery Office, 1932).\n\n42 For the description of Harold Joseph Laski, Sir Ivor Jennings, William Alexander Robson, and Herman Finer as members of the same group see William Ivor Jennings, \"Administrative Law and Administrative Jurisdiction,\" _Journal of Comparative Legislation and International Law_ , 3rd ser., 20 (1938): 103.\n\n43 Sir William Ivor Jennings, \"The Report on Ministers' Powers,\" _Public Administration_ (London), 10 (1932): 333\u201351, and Book Review [\"Official Ministers' Powers\"], 11 (1933): 109\u201314.\n\n44 Jennings, \"Report on Ministers' Powers,\" p. 342.\n\n45 Ibid., p. 343.\n\n46 Ibid., p. 345.\n\n47 Ibid., p. 345. [The quotation actually appears in Jenning's review of \"Official Ministers' Powers,\" _Public Administration_ , 11 (1933): 111.\u2014Ed.]\n\n48 Sir William Ivor Jennings, _The Law of the Constitution_ (4th ed.; London: University of London Press, 1952), p. 54.\n\n49 Ibid., p. 291.\n\n50 Ibid., p. 292.\n\n51 Ibid., p. 294.\n\n52 Ibid.\n\n53 Sir William Ivor Jennings, _The Queen's Government_ (Pelican Books; London: Penguin Books, 1954), pp. 9\u201313.\n\n54 Thomas Dewar Weldon, _The Vocabulary of Politics_ (Pelican Books; London: Penguin Books, 1953), p. 68.\n\n55 William Alexander Robson, _Justice and Administrative Law_ (3rd ed.; London: Stevens, 1951), p. xi.\n\n56 Ibid., p. 16.\n\n57 Ibid., p. 433.\n\n58 Ibid., pp. 572\u201373.\n\n59 Inns of Court, Conservative and Unionist Society, _Rule of Law: A Study_ (London: Conservative Political Centre, 1955), p. 30.\n\n60 Conservative Political Centre, _Liberty in the Modern State: Eight Oxford Lectures_ (London: Conservative Political Centre, 1957).\n\n61 _Times Literary Supplement_ , March 1, 1957 [Review of _Liberty in the Modern State: Eight Oxford Lectures_ ], p. 123. In this respect some socialists show greater concern than is noticeable in the official conservative position. Mr. R. H. S. Crossman, in the pamphlet quoted in n. 40 above ( _Socialism and the New Despotism_ , p. 19), looks forward to the next step \"to reform the Judiciary, so that it can regain the traditional function of defending individual rights against encroachment.\"\n\n62 Wolfgang Gaston Friedmann, _Law and Social Change in Contemporary Britain_ (London: Stevens and Sons, 1951), pp. 277\u2013310. One of the essays in this collection, _The Planned State and the Rule of Law_ , was published separately several years earlier (Melbourne: Melbourne University Press, 1948).\n\n63 Friedmann, _Law and Social Change in Contemporary Britain_ , p. 284.\n\n64 Ibid., p. 310. It is curious that the contention that the rule of law and socialism are incompatible, which had long been maintained by socialist authors, should have aroused so much indignation among them when it was turned against socialism. Long before I had emphasized the point in _The Road to Serfdom_ , Karl Mannheim, _Man and Society in an Age of Reconstruction: Studies in Modern Social Structure_ (London: K. Paul, Trench, Trubner and Co., 1940), p. 180, had summed up the result of a long discussion in the statement that \"recent studies in the sociology of law once more confirm that the fundamental principle of formal law by which every case must be judged according to general rational precepts, which have as few exceptions as possible and are based on logical subsumption, obtains only for the liberal-competitive phase of capitalism.\" Cf. also Franz Leopold Neumann, _The Democratic and the Authoritarian State: Essays in Political and Legal Theory_ (Glencoe, IL: The Free Press, 1957), p. 50, and Max Horkheimer, \"Bemerkung zur philosophischen Anthropologie,\" _Zeitschrift f\u00fcr Sozialforschung_ , 4 (1935): esp. 14: \"The economic basis of the significance of promises becomes less important from day to day, because to an increasing extent economic life is characterised not by contract but by command and obedience.\" [The original reads: \"Die \u00f6konomische Grundlage f\u00fcr die Bedeutung von Versprechungen wird daher schm\u00e4ler von Tag zu Tag. Denn nicht mehr der Vertrag, sondern Befehlsgewalt und Gehorsam kennzeichnen jetzt in steigendem Ma\u00df den inneren Verkehr.\"\u2014Ed.]\n\n65 Herman Finer, _The Road to Reaction_ (Boston: Little, Brown and Co., 1945), p. 60.\n\n66 Cf. Winston Spencer Churchill, \"The Conservative Case for a New Parliament,\" [Party Political Broadcast XI] _Listener_ , February 19, 1948, p. 302: \"I am told that 300 officials have the power to make new regulations, apart altogether from Parliament, carrying with them the penalty of imprisonment for crimes hitherto, unknown to the law.\" [Churchill further notes: \"A rate of war-time taxation has been maintained in a manner which has hampered and baffled enterprise and recovery in every walk of life; 700,000 more officials, all hard-working decent men and women but producing nothing themselves, have settled down upon us to administer 25,000 regulations never enforced before in time of peace\" ( p. 302).\u2014Ed.]\n\n67 Town and Country Planning Act (1947) [10 and 11 Geo. 6, chap. 51] sec. 70, subsec. (3), provides that \"regulations made under this Act with the consent of the Treasury may prescribe general principles to be followed by the Central Land Board in determining . . . whether any and if so what development charge is to be paid\" ( p. 84 of the act). It was under this provision that the Minister of Town and Country Planning was able unexpectedly to issue a regulation under which the development charges were normally \"not to be less\" than the whole additional value of the land which was due to the permission for a particular development. [The Central Land Board was established by the Town and Country Planning Act of 1947, among whose duties was to assess and levy charges on new developments of land. When planning permission was granted for any development, a charge was payable on the enhanced value of the land. In cases of default the Board could issue an order for payment, together with a penalty, enforceable as a land charge, subject to appeal to one of the county courts or the High Court. Under the bill no development was permitted to take place without consent and where permission to develop was refused, there was no right to compensation.\u2014Ed.]\n\n68 Central Land Board, _Practice Notes (First Series): Being Notes on Development Charges Under the Town and Country Planning Act, 1947_ (London: His Majesty's Stationery Office, 1949), Preface [pp. ii\u2013 iii]. It is explained there that the Notes \"are meant to describe principles and working rules in accordance with which any applicant can confidently assume his case will be dealt, unless either he shows good cause for different treatment, or the Board informs him that for special reasons the normal rules do not apply.\" It is further explained that \"a general working rule must always be variable if it does not fit a particular case\" and that the board \"have no doubt that from time to time we shall vary our policy.\" For further discussion of this measure see chap. 22, sec. 6, below.\n\n69 Cf. the official report of the Minister of Agriculture and Fisheries, _Public Inquiry Ordered by the Minister of Agriculture into the Disposal of Land at Crichel Down_ (London: History Majesty's Stationery Office, 1954) [Cmd. 9176]; and cf. also the less-known but nearly as instructive case of _Odlum v. Stratton_ (1946), before Mr. Justice Atkinson, King's Bench Division, a report of the proceedings of which has been printed by the _Wiltshire Gazette_ [ _Odlum v. Stratton: Verbatim Report of the Proceedings in the High Court of Justice, King Bench Division, before Mr. Justice Atkinson_ (Devizes, Wiltshire: Wilshire Gazette, 1946)]. [The case was the subject of comment on ministerial discretion by Lord Simon of Glaisdale in the House of Lords on 26 February 1996. Lord Simon remarked: \"( _Odlum v. Stratton_ ) was a libel action in which the professional competence of a farmer was in question. A series of reports on his competence was in the hands of the Ministry of Agriculture. The Ministry claimed that it was immune from disclosure except for two documents. Those two documents told against the plaintiff. He wanted to see the others, but those were the only two for which immunity was waived. The judge had no doubt at all, nor do I think would anyone reading a transcript have had any doubt, that the documents were divulged precisely with the objective of discrediting the plaintiff\" ( _Hansard, Lords_ , column 1265\u201366).\u2014Ed.]\n\n70 See Dwight Waldo, _The Administrative State: A Study of the Political Theory of American Public Administration_ (New York: Ronald Press Co., 1948), p. 70, n. 13; cf. also pp. 5, 15, and 40 of the same work.\n\n71 See Ibid., p. 79: \"If any person is to count for less than one in the New Order it is the Lawyer!\"\n\n72 Ibid., p. 73.\n\n73 Roscoe Pound, _The Spirit of the Common Law_ (Boston: Marshall Jones Co., 1921), p. 72; cf. also Charles Howard McIlwain, _Constitutionalism and the Changing World: Collected Papers_ (Cambridge: Cambridge University Press, 1939), p. 261: \"Slowly but surely we are drifting toward the totalitarian state, and strange to say many if not most of the idealists are either enthusiastic about it or unconcerned.\"\n\n74 John Dickinson, _Administrative Justice and the Supremacy of Law in the United States_ (Cambridge: Harvard University Press, 1927), p. 41.\n\n75 Cf. Robert Marion La Follette, _The Political Philosophy of Robert M. La Follette as Revealed in His Speeches and Writings_ , Ellen Torelle, ed. (Madison, WI: Robert M. La Follette Co., 1920), esp. pp. 179\u201381. [Art. 14 of the La Follette Progressive Republican Platform of 1920 reads: \"We denounce the alarming usurpation of legislative power, by the federal courts, as subversive of democracy, and we favor such amendments to the constitution, and thereupon, the enactment of such statutes as may be necessary, to provide for the election of all federal judges, for fixed terms not exceeding ten years, by direct vote of the people\" ( p. 419).\u2014Ed.]\n\n76 Alexander Haim Pekelis, \"Administrative Discretion and the Law of Rule,\" _Law and Social Action: Selected Essays of Alexander H. Pekelis_ , Milton Ridvas Konvitz, ed. (Ithaca, NY: Cornell University Press, 1950), p. 88; cf. Also Hans Kelsen, \"Foundations of Democracy,\" _Ethics_ , no. 1, pt. 2, 66 (1955): 77ff.\n\n77 Charles Grove Haines, _A Government of Laws or a Government of Men: Judicial or Legislative Supremacy_ (Berkeley: University of California Press, 1929), p. 37.\n\n78 Ibid., p. 18.\n\n79 Thomas Jefferson, \"Draft of the Kentucky Resolution of 1789,\" in Ethelbert Dudley Warfield, _The Kentucky Resolutions of 1798_ (2nd ed.; New York: Putman, 1894), pp. 157\u201358.\n\n80 Jerome Frank, _Law and the Modern Mind_ (New York: Brentano's, 1930), p. 21. More than a quarter of a century after the publication of this book, Thurman Wesley Arnold, in the _University of Chicago Law Review_ [\"Judge Jerome Frank,\" 24 (1957): 635], could say of it that \"more than any other it cleared the way for a new set of conceptions and ideals with respect to the relationship of the citizen to his government.\"\n\n81 Dean Acheson, chairman, _Administrative Procedure in Government Agencies: Final Report of the Committee on Administrative Procedure_ (Washington, DC: Government Printing Office, 1941). [The Federal Administrative Procedure Act of 1946 was an outgrowth of the Final Report of the Attorney General's committee on Administrative Procedure in Government Agencies, which was established in response to the immense number of administrative agencies created under the New Deal. The issues that the committee confronted were extremely contentious. So much of private conduct had been made subject to administrative regulation since 1934 and there were so few checks on the arbitrary power of administrators that many feared that the United States was on the verge of being reconstructed into a centrally planned state. To assuage them, Roosevelt requested his attorney general, Frank Murray, to strike a committee. The FAPA of 1946, which had authority over both independent agencies and those falling within the executive branch, governed the way regulations could be proposed and enacted and provided for judicial review of its decisions.\u2014Ed.]\n\n82 Roscoe Pound, \"Administrative Procedure Legislation for the 'Minority Report,'\" _American Bar Association Journal_ , 27 (1941): 678. On the present situation see Bernard Schwartz, \"Administrative Justice and Its Place in the Legal Order,\" _New York University Law Review_ , 30 (1955): 1390\u2013 1417; and Walter Gellhorn, _Individual Freedom and Governmental Restraints_ (Baton Rouge: Louisiana State University Press, 1956), esp. the remark on pp. 18\u201319, that \"some of the former upholders of the administrative process [including the author] now feel that what were mainly imaginary dangers have become real\u2014and frightening.\"\n\n83 Gustav Radbruch, _Rechtsphilosophie_ (4th ed.; Stuttgart: K. F. Koehler, 1950), p 357. [The English quotation appears in the original German as: \"Demokratie ist gewi\u00df ein preisenswertes Gut, _Rechtsstaat_ aber ist wie das t\u00e4gliche Brot, wie Wasser zum Trinken und wie Luft zum Atmen, und das Beste an der Demokratie gerade dieses, da\u00df nur sie geeignet ist, den _Rechtsstaat_ zu sichern.\" ( p. 357)\u2014Ed.] See also the significant comments in this work on the role which legal positivism has played in destroying the belief in the _Rechtsstaat_ , esp. p. 335: \"Dieses Auffassung vom Gesetz und seiner Geltung (wir nennen sie die positivistische Lehre) hat die Juristen wie das Volk wehrlos gemacht gegen noch so willk\u00fcrliche, noch so grausame, noch so verbrecherische Gesetze. Sie setzt letzten Endes das Recht der Macht gleich, nur wo die Macht ist, ist das Recht\"; and p. 352: \"Der Positivismus hat in der Tat mit seiner \u00dcberzeugung 'Gesetz ist Gesetz' den deutschen Juristenstand wehrlos gemacht gegen Gesetze willk\u00fcrlichen und verbrecherischen Inhalts. Dabei ist der Positivismus gar nicht in der Lage, aus eigener Kraft die Geltung von Gesetzen zu begr\u00fcnden. Er glaubt die Geltung eines Gesetzes schon damit erwiesen zu haben, da\u00df es die Macht besessen hat, sich durchzusetzen.\" [\"This understanding of the law and of its merits (which we call positivist theory) has made legal theorists as well as the great mass of people defenseless against laws that are arbitrary, cruel, and criminal. Ultimately this view equates law with power; that is, only where power resides is there law\" ( p. 335). \"In fact, positivism, with its claim that 'all law is law' has rendered the German legal profession defenseless against arbitrary and criminal laws. At the same time, it is impossible for positivism on its own to justify the validity of a law. For it believes that a law's validity has been proved by the power to assert itself \" ( p. 352).\u2014Ed.] It is thus not too much of an exaggeration when Emil Brunner, _Justice and the Social Order_ (New York: Harper, 1945), p. 7, maintains that \"the totalitarian state is simply and solely legal positivism in political practice.\"\n\n84 See Gottfried Dietze, \"America and Europe: Decline and Emergence of Judicial Review,\" _Virginia Law Review_ , 44 (1958): 1233\u201372, and, concerning the revival of natural law, Helmut Coing, _Grundz\u00fcge der Rechtsphilosophie_ (Berlin: Walter de Gruyter, 1950); Heinrich Mitteis, _\u00dcber das Naturrecht_ (Berlin: Akademie-Verlag, 1948); Klaus Ritter, _Zwischen Naturrecht und Rechtspositivismus: Eine rrkenntnistheoretische Auseinandersetzung mit den neueren Versuchen zur Wiederherstellung einer Rechtsmetaphysik_ (Witten-Ruhr: Luther-Verlag, 1956).\n\n85 Georges Ripert, _Le D\u00e9clin du droit_ (Paris: Pichon and Durand-Auzias, 1949), p. 192. [The French reads: \"Il nous faut tout d'abord faire le proc\u00e8s des juristes. Ce sont eux qui depuis un demi-si\u00e8cle ont affaibli la notion de droit individuel sans avoir conscience qu'ils livraient ainsi ces droits \u00e0 toute puissance de l'\u00c9tat politique. Les uns \u00e9taient d\u00e9sireux de se classer parmi les hommes de progr\u00e8s. Les autres croyaient retrouver une doctrine traditionnelle \u00e9touff\u00e9e par l'individualisme lib\u00e9ral du xixe si\u00e8cle. Les hommes de science ont souvent une certaine candeur qui ne leur permet pas d'apercevoir les cons\u00e9quences pratiques que d'autres tireront de leurs doctrines d\u00e9sint\u00e9ress\u00e9es.\"\u2014Ed.] Cf. also Paul Roubier, _Th\u00e9orie g\u00e9n\u00e9rale du droit: histoire des doctrines juridiques et philosophie des valeurs sociales_ (Paris: Recueil Sirey, 1946); Louis Auguste Paul Rougier, _La France \u00e0 la recherche d'une constitution_ (Paris: Recueil Sirey, 1952).\n\n86 See Sir Carleton Kemp Allen, _Law and Orders: An Inquiry into the Nature and Scope of Delegated Legislation and Executive Powers in England_ (London, 1945); George Williams Keeton, _The Passing of Parliament_ (London: E. Benn, 1952); Charles John Hamson, _Executive Discretion and Judicial Control: An Aspect of the Conseil d'\u00c9tat_ (London: Stevens, 1954); Cyril John Radcliffe, Viscount Radcliffe of Werneth, _Law and the Democratic State: Being the Presidential Address of the Right Hon. Lord Radcliffe, President of the Holdsworth Club of the Faculty of Law in the University of Birmingham, 1954\u20131955_ [Holdsworth lecture] (Birmingham: Holdsworth Club of the University of Birmingham, 1955); and Geoffrey Marshall, \"The Recent Development of English Administrative Law,\" _Il Politico_ , 24 (December 1959): 637\u201345.\n\n87 Committee on Administrative Tribunals and Enquiries, _Report of the Committee on Administrative Tribunals, Presented by the Lord High Chancellor to Parliament by Command of Her Majesty_ [the Franks Report], chaired by Sir Oliver Franks, Baron Franks of Headington. Cmd. 218 (London: Her Majesty's Stationery Office, 1957), p. 8, par. 37.\n\n88 Ibid., p. 6, pars.27, 29.\n\n89 Ibid., p 28, par. 120.\n\n90 See Inns of Court, Conservative and Unionist Society, _Rule of Law: A Study_ (mentioned in note 59 above), and William Alexander Robson, _Justice and Administrative Law_ (3rd ed.; London: Stevens, 1951). On similar recommendations of the Hoover Commission in the United States see the Symposium on the \"Hoover Commission and Task Force Reports on Legal Services and Procedure,\" _New York University Law Review_ , 30, no. 7 (1955): 1267\u20131417.\n\n91 The International Commission of Jurists at The Hague (now at Geneva) convened at Athens in June 1955, and adopted a resolution which solemnly declared: \"1. The State is subject to the law. 2. Governments should respect the rights of the individual under the Rule of Law and provide effective means for their enforcement. 3. Judges should be guided by the Rule of Law, protect and enforce it without fear or favor and resist any encroachments by governments or political parties on their independence as judges. 4. Lawyers of the world should preserve the independence of their profession, assert the rights of the individual under the Rule of Law and insist that every accused is afforded a fair trial.\" (See the _Report of the International Congress of Jurists, Held June 13\u201320, 1955, at Athens_ [The Hague: International Commission of Jurists, 1956], p. 9.) [These four \"resolutions\" do not, in fact, form part of the formal resolutions of the Congress but, rather, were adopted by the International Commission of Jurists as \"fundamental principles of justice . . . essential to a lasting peace throughout the world.\"\u2014Ed.]\n\nUnfortunately, since then the International Commission of Jurists (in its \"Declaration of Delhi\" of January 10, 1959) decided to introduce a \"new\" and \"dynamic\" conception of the Rule of La w which included the establishment of \"social, economic, educational and cultural conditions under which [the individual's] aspirations and dignity may be realized.\" However desirable these objectives might be, extending the notion of the Rule of Law to include these goals can only lead to making the term worthless and can only accelerate the repudiation of those constraints that the Rule of Law places on the actions of the state should these limitations stand in the way of pursuing certain social ends. Cf. \"The Declaration of Delhi,\" _Newsletter of the International Commission of Jurists_ , no. 6 (March\u2013April 1959): 1.\n\n92 It is no exaggeration when one student of jurisprudence ( Julius Stone, _The Province and Function of Law: Law as Logic, Justice, and Social Control; A Study in Jurisprudence_ [Cambridge, MA: Harvard University Press, 1950], p. 261) asserts that the restoration of the rule of law as here defined \"would strictly require the reversal of legislative measures which all democratic legislatures seem to have found essential in the last half century.\" The fact that democratic legislatures have done this does not, of course, prove that it was wise or even that it was essential to resort to this kind of measure in order to achieve what they wanted to achieve, and still less that they ought not to reverse their decisions if they recognize that they produce unforeseen and undesirable consequences.\n\n## PART III\n\nFREEDOM IN THE WELFARE STATE\n\nAbove this race of men stands an immense and tutelary power, which takes upon itself alone to secure their gratifications and to watch over their fate. That power is absolute, minute, regular, provident, and mild. It would be like the authority of a parent if, like that authority, its object was to prepare men for manhood; but it seeks, on the contrary, to keep them in perpetual childhood: it is well content that the people should rejoice, provided they think of nothing but rejoicing. For their happiness such a government willingly labors, but it chooses to be the sole agent and the only arbiter of that happiness; it provides for their security, foresees and supplies their necessities, facilitates their pleasures, manages their principal concerns, directs their industry, regulates the descent of property, and subdivides their inheritances; what remains, but to spare them all care of thinking and all the trouble of living?\n\n\u2014Alexis de Tocqueville\n\nThis quotation is taken from Tocqueville, _Democracy in America_ , vol. 2, p. 318. [The French reads: \"Au-dessus de ceux-l\u00e0 s'\u00e9l\u00e8ve un pouvoir immense et tut\u00e9laire, qui se charge seul d'assurer leur jouissance et de veiller sur leur sort. Il est absolu, d\u00e9taill\u00e9, r\u00e9gulier, pr\u00e9voyant et doux. Il ressemblerait \u00e0 la puissance paternelle si, comme elle, il avait pour objet de pr\u00e9parer les hommes \u00e0 l'\u00e2ge viril; mais il ne cherche, au contraire, qu'\u00e0 les fixer irr\u00e9vocablement dans l'enfance; il aime que les citoyens se r\u00e9jouissent, pourvu qu'ils ne songent qu'\u00e0 se r\u00e9jouir. Il travaille volontiers \u00e0 leur bonheur; mais il veut en \u00eatre l'unique agent et le seul arbitre; il pourvoit \u00e0 leur s\u00e9curit\u00e9, pr\u00e9voit et assure leurs besoins, facilite leur plaisirs, conduit leurs principales affaires, dirige leur industrie, r\u00e8gle leurs successions, devise leurs h\u00e9ritages; que ne peut-il leur \u00f4ter enti\u00e8rement le trouble de penser et la peine de vivre?\" (bk. 2, sec. 4, pt. 6; vol. 2, p. 837).\u2014Ed.] The three paragraphs which follow, or indeed the whole of chapter 6 of book 4, from which it is taken, would deserve quotation as a prologue to the following discussion.\n\n## SEVENTEEN\n\nTHE DECLINE OF SOCIALISM AND THE RISE OF THE WELFARE STATE\n\nExperience should teach us to be most on our guard to protect liberty when the Government's purposes are beneficent. Men born to freedom are naturally alert to repel invasion of their liberty by evil-minded rulers. The greatest dangers to liberty lurk in insidious encroachment by men of zeal, well meaning but without understanding.\n\n\u2014Louis Brandeis\n\n1. Efforts toward social reform, for something like a century, have been inspired mainly by the ideals of socialism\u2014during part of this period even in countries like the United States which never has had a socialist party of importance. Over the course of these hundred years socialism captured a large part of the intellectual leaders and came to be widely regarded as the ultimate goal toward which society was inevitably moving. This development reached its peak after the second World War, when Britain plunged into her socialist experiment. This seems to have marked the high tide of the socialist advance. Future historians will probably regard the period from the revolution of 1848 to about 1948 as the century of European socialism.\n\nDuring this period socialism had a fairly precise meaning and a definite program. The common aim of all socialist movements was the nationalization of the \"means of production, distribution, and exchange,\" so that all economic activity might be directed according to a comprehensive plan toward some ideal of social justice. The various socialist schools differed mainly in the political methods by which they intended to bring about the reorganization of society. Marxism and Fabianism differed in that the former was revolutionary and the latter gradualist; but their conceptions of the new society they hoped to create were basically the same. Socialism meant the common ownership of the means of production and their \"employment for use, not for profit.\"\n\nThe great change that has occurred during the last decade is that socialism in this strict sense of a particular method of achieving social justice has collapsed. It has not merely lost its intellectual appeal; it has also been abandoned by the masses so unmistakably that socialist parties everywhere are searching for a new program that will insure the active support of their followers.1 They have not abandoned their ultimate aim, their ideal of social justice. But the methods by which they had hoped to achieve this and for which the name \"socialism\" had been coined have been discredited. No doubt the name will be transferred to whatever new program the existing socialist parties will adopt. But socialism in the old definite sense is now dead in the Western world.\n\nThough such a sweeping statement will still cause some surprise, a survey of the stream of disillusionist literature from socialist sources in all countries and the discussions inside the socialist parties amply confirm it.2 To those who watch merely the developments inside a single country, the decline of socialism may still seem no more than a temporary setback, the reaction to political  defeat. But the international character and the similarity of the developments in the different countries leave no doubt that it is more than that. If, fifteen years ago, doctrinaire socialism appeared as the main danger to liberty, today it would be tilting at windmills to direct one's argument against it. Most of the arguments that were directed at socialism proper can now be heard from within the socialist movements as arguments for a change of program.\n\n2. The reasons for this change are manifold. So far as the socialist school which at one time was most influential is concerned, the example of the \"greatest social experiment\" of our time was decisive: Marxism was killed in the Western world by the example of Russia. But for a long time comparatively few intellectuals comprehended that what had happened in Russia was the necessary outcome of the systematic application of the traditional socialist program. Today, however, it is an effective argument, even within socialist circles, to ask: \"If you want one hundred per cent socialism, what's wrong with the Soviet Union?\"3 But the experience of that country has in general discredited only the Marxist brand of socialism. The widespread disillusionment with the basic methods of socialism is due to more direct experiences.\n\nThe chief factors contributing to the disillusionment were probably three: the increasing recognition that a socialist organization of production would be not more but much less productive than private enterprise; an even clearer recognition that, instead of leading to what had been conceived as greater social justice, it would mean a new arbitrary and more inescapable order of rank than ever before; and the realization that, instead of the promised greater freedom, it would mean the appearance of a new despotism.\n\nThe first to be disappointed were those labor unions which found that, when they had to deal with the state instead of a private employer, their power was greatly reduced. But the individuals also soon discovered that to be confronted everywhere by the authority of the state was no improvement upon their position in a competitive society. This happened at a time when the general rise in the standard of living of the working class (especially of the manual workers) destroyed the conception of a distinct proletarian class and, with it, the class-consciousness of the workers\u2014creating in most of Europe a situation similar to that which in the United States had always prevented the growth of an organized socialist movement.4 In the countries that had experienced a totalitarian regime there also took place a strong individualist reaction among the younger generation, who became deeply distrustful of all collective activities and suspicious of all authority.5\n\nPerhaps the most important factor in the disillusionment of socialist intellectuals has been the growing apprehension among them that socialism would mean the extinction of individual liberty. Though the contention that socialism and individual liberty were mutually exclusive had been indignantly rejected by them when advanced by an opponent,6 it made a deep impression when stated in powerful literary form by one from their own midst.7 More recently the situation has been very frankly described by one of the leading intellectuals of the British Labour Party. Mr. R. H. S. Crossman, in a pamphlet entitled _Socialism and the New Despotism_ , records how \"more and more serious-minded people are having second thoughts about what once seemed to them the obvious advantages of central planning and the extension of State ownership\";8 and he continues to explain that \"the discovery that the Labour Government's 'Socialism' meant the establishment of a number of vast, bureaucratic public corporations,\"9 of \"a vast centralised State bureaucracy [which] constitutes a grave potential threat to social democracy,\"10 had created a situation in which \"the main task of Socialists to-day is to convince the nation that its liberties are threatened by this new feudalism.\"11\n\n3. But, though the characteristic methods of collectivist socialism have few defenders left in the West, its ultimate aims have lost little of their attraction. While the socialists no longer have a clear-cut plan as to how their goals are to be achieved, they still wish to manipulate the economy so that the distribution of incomes will be made to conform to their conception of social justice.  The most important outcome of the socialist epoch, however, has been the destruction of the traditional limitations upon the powers of the state. So long as socialism aimed at a complete reorganization of society on new principles, it treated the principles of the existing system as mere encumbrances to be swept away. But now that it no longer has any distinctive principles of its own, it can only present its new ambitions without any clear picture of the means. As a result, we approach the new tasks set by the ambition of modern man as un-principled, in the original meaning of this word, as never before.\n\nWhat is significant is that, in consequence, though socialism has been generally abandoned as a goal to be deliberately striven for, it is by no means certain that we shall not still establish it, albeit unintentionally. The reformers who confine themselves to whatever methods appear to be the most effective for their particular purposes and pay no attention to what is necessary to preserve an effective market mechanism are likely to be led to impose more and more central control over economic decisions (though private property may be preserved in name) until we get that very system of central planning which few now consciously wish to see established. Furthermore, many of the old socialists have discovered that we have already drifted so far in the direction of a redistributive state that it now appears much easier to push further in that direction than to press for the somewhat discredited socialization of the means of production. They seem to have recognized that by increasing governmental control of what nominally remains private industry, they can more easily achieve that redistribution of incomes that had been the real aim of the more spectacular policy of expropriation.\n\nIt is sometimes regarded as unfair, as blind conservative prejudice, to criticize those socialist leaders who have so frankly abandoned the more obviously totalitarian forms of \"hot\" socialism, for having now turned to a \"cold\" socialism which in effect may not be very different from the former. We are in danger, however, unless we succeed in distinguishing those of the new ambitions which can be achieved in a free society from those which require for their realization the methods of totalitarian collectivism.\n\n4. Unlike socialism, the conception of the welfare state12 has no precise  meaning. The phrase is sometimes used to describe any state that \"concerns\" itself in any manner with problems other than those of the maintenance of law and order. But, though a few theorists have demanded that the activities of government should be limited to the maintenance of law and order, such a stand cannot be justified by the principle of liberty. Only the coercive measures of government need be strictly limited. We have already seen (in chap. 15) that there is undeniably a wide field for non-coercive activities of government and that there is a clear need for financing them by taxation.\n\nIndeed, no government in modern times has ever confined itself to the \"individualist minimum\" which has occasionally been described,13 nor has such confinement of governmental activity been advocated by the \"orthodox\" classical economists.14 All modern governments have made provision for the indigent, unfortunate, and disabled and have concerned themselves with questions of health and the dissemination of knowledge. There is no reason why the volume of these pure service activities should not increase with the general growth of wealth. There are common needs that can be satisfied only by collective action and which can be thus provided for without restricting individual liberty. It can hardly be denied that, as we grow richer, that minimum of sustenance which the community has always provided for those not able to look after themselves, and which can be provided outside the market, will gradually rise, or that government may, usefully and without doing any harm, assist or even lead in such endeavors. There is little reason why the government should not also play some role, or even take the initiative, in such areas as social insurance and education, or temporarily subsidize certain experimental developments. Our problem here is not so much the aims as the methods of government action.\n\nReferences are often made to those modest and innocent aims of governmental activity to show how unreasonable is any opposition to the welfare state as such. But, once the rigid position that government should not concern itself at all with such matters is abandoned\u2014a position which is defensible but has little to do with freedom\u2014the defenders of liberty commonly discover that the program of the welfare state comprises a great deal more that is represented as equally legitimate and unobjectionable. If, for instance, they admit that they have no objection to pure-food laws, this is taken to imply that they should not object to any government activity directed toward a desirable end. Those who attempt to delimit the functions of government in terms of aims rather than methods thus regularly find themselves in the position of having to oppose state action which appears to have only desirable consequences or of having to admit that they have no general rule on which to base their objections to measures which, though effective for particular purposes, would in their aggregate effect destroy a free society. Though the position that the state should have nothing to do with matters not related to the maintenance of law and order may seem logical so long as we think of the state solely as a coercive apparatus, we must recognize that, as a service agency, it may assist without harm in the achievement of desirable aims which perhaps could not be achieved otherwise. The reason why many of the new welfare activities of government are a threat to freedom, then, is that, though they are presented as mere service activities, they really constitute an exercise of the coercive powers of government and rest on its claiming exclusive rights in certain fields.\n\n5. The current situation has greatly altered the task of the defender of liberty and made it much more difficult. So long as the danger came from socialism of the frankly collectivist kind, it was possible to argue that the tenets of the socialists were simply false: that socialism would not achieve what the socialists wanted and that it would produce other consequences which they would not like. We cannot argue similarly against the welfare state, for this term does not designate a definite system. What goes under that name is a conglomerate of so many diverse and even contradictory elements that, while some of them may make a free society more attractive, others are incompatible with it or may at least constitute potential threats to its existence.\n\nWe shall see that some of the aims of the welfare state can be realized without detriment to individual liberty, though not necessarily by the methods which seem the most obvious and are therefore most popular; that others can be similarly achieved to a certain extent, though only at a cost much greater than people imagine or would be willing to bear, or only slowly and gradually as wealth increases; and that, finally, there are others\u2014and they are those particularly dear to the hearts of the socialists\u2014that cannot be realized in a society that wants to preserve personal freedom.\n\nThere are all kinds of public amenities which it may be in the interest of all members of the community to provide by common effort, such as parks  and museums, theaters and facilities for sports\u2014though there are strong reasons why they should be provided by local rather than national authorities. There is then the important issue of security, of protection against risks common to all, where government can often either reduce these risks or assist people to provide against them. Here, however, an important distinction has to be drawn between two conceptions of security: a limited security which can be achieved for all and which is, therefore, no privilege, and absolute security, which in a free society cannot be achieved for all. The first of these is security against severe physical privation, the assurance of a given minimum of sustenance for all; and the second is the assurance of a given standard of life, which is determined by comparing the standard enjoyed by a person or a group with that of others. The distinction, then, is that between the security of an equal minimum income for all and the security of a particular income that a person is thought to deserve.15 The latter is closely related to the third main ambition that inspires the welfare state: the desire to use the powers of government to insure a more even or more just distribution of goods. Insofar as this means that the coercive powers of government are to be used to insure that particular people get particular things, it requires a kind of discrimination between, and an unequal treatment of, different people which is irreconcilable with a free society. This is the kind of welfare state that aims at \"social justice\" and becomes \"primarily a redistributor of income.\"16 It is bound to lead back to socialism and its coercive and essentially arbitrary methods.\n\n6. Though _some_ of the aims of the welfare state can be achieved _only_ by methods inimical to liberty, _all_ its aims _may_ be pursued by such methods. The chief danger today is that, once an aim of government is accepted as legitimate, it is then assumed that even means contrary to the principles of freedom may be legitimately employed. The unfortunate fact is that, in the majority of fields, the most effective, certain, and speedy way of reaching a given end will seem to be to direct all available resources toward the now visible solution. To the ambitious and impatient reformer, filled with indignation at a particular evil, nothing short of the complete abolition of that evil by the quickest and most direct means will seem adequate. If every person now suffering from unemployment, ill health, or inadequate provision for his old age is at once to be relieved of his cares, nothing short of an all-comprehensive and compulsory scheme will suffice. But if, in our impatience to solve such problems  immediately, we give government exclusive and monopolistic powers, we may find that we have been shortsighted. If the quickest way to a now visible solution becomes the only permissible one and all alternative experimentation is precluded, and if what now seems the best method of satisfying a need is made the sole starting point for all future development, we may perhaps reach our present goal sooner, but we shall probably at the same time prevent the emergence of more effective alternative solutions. It is often those who are most anxious to use our existing knowledge and powers to the full that do most to impair the future growth of knowledge by the methods they use. The controlled single-channel development toward which impatience and administrative convenience have frequently inclined the reformer and which, especially in the field of social insurance, has become characteristic of the modern welfare state may well become the chief obstacle to future improvement.\n\nIf government wants not merely to facilitate the attainment of certain standards by the individuals but to make certain that everybody attains them, it can do so only by depriving individuals of any choice in the matter. Thus the welfare state becomes a household state in which a paternalistic power controls most of the income of the community and allocates it to individuals in the forms and quantities which it thinks they need or deserve.\n\nIn many fields persuasive arguments based on considerations of efficiency and economy can be advanced in favor of the state's taking sole charge of a particular service; but when the state does so, the result is usually not only that those advantages soon prove illusory but that the character of the services becomes entirely different from that which they would have had if they had been provided by competing agencies. If, instead of administering limited resources put under its control for a specific service, government uses its coercive powers to insure that men are given what some expert thinks they need; if people thus can no longer exercise any choice in some of the most important matters of their lives, such as health, employment, housing, and provision for old age, but must accept the decisions made for them by appointed authority on the basis of its evaluation of their need; if certain services become the exclusive domain of the state, and whole professions\u2014be it medicine, education, or insurance\u2014come to exist only as unitary bureaucratic hierarchies, it will no longer be competitive experimentation but solely the decisions of authority that will determine what men shall get.17\n\nThe same reasons that generally make the impatient reformer wish to organize such services in the form of government monopolies lead him also to believe that the authorities in charge should be given wide discretionary powers over the individual. If the objective were merely to improve opportunities for all by supplying certain specific services according to a rule, this could be attained on essentially business lines. But we could then never be sure that the results for all individuals would be precisely what we wanted. If each individual is to be affected in some particular way, nothing short of the individualizing, paternalistic treatment by a discretionary authority with powers of discriminating between persons will do.\n\nIt is sheer illusion to think that when certain needs of the citizen have become the exclusive concern of a single bureaucratic machine, democratic control of that machine can then effectively guard the liberty of the citizen. So far as the preservation of personal liberty is concerned, the division of labor between a legislature which merely says that this or that should be done18 and an administrative apparatus which is given exclusive power to carry out these instructions is the most dangerous arrangement possible. All experience confirms what is \"clear enough from American as well as from English experience, that the zeal of administrative agencies to achieve the immediate end they see before them leads them to see their function out of focus and to assume that constitutional limitations and guaranteed individual rights must give way before their zealous efforts to achieve what they see as a paramount purpose of government.\"19\n\nIt would scarcely be an exaggeration to say that the greatest danger to liberty today comes from the men who are most needed and most powerful in modern government, namely, the efficient expert administrators exclusively concerned with what they regard as the public good. Though theorists may still talk about the democratic control of these activities, all who have direct experience in this matter agree that (as one recent English writer put it) \"if the Minister's control . . . has become a myth, the control of Parliament is and always has been the merest fairy-tale.\"20 It is inevitable that this sort of administration  of the welfare of the people should become a self-willed and uncontrollable apparatus before which the individual is helpless, and which becomes increasingly invested with all the _mystique_ of sovereign authority\u2014the _Hoheitsverwaltung_ or _Herrschaftsstaat_ of the German tradition that used to be so unfamiliar to Anglo-Saxons that the strange term \"hegemonic\"21 had to be coined to render its meaning.\n\n7. It is not the aim of the following chapters to expound a complete program of economic policy for a free society. We shall be concerned mainly with those comparatively new aspirations whose place in a free society is still uncertain, concerning which our various positions are still floundering between extremes, and where the need for principles which will help us to sort out the good from the bad is most urgent. The problems we shall select are chiefly those which seem particularly important if we are to rescue some of the more modest and legitimate aims from the discredit which over-ambitious attempts may well bring to all actions of the welfare state.\n\nThere are many parts of government activity which are of the highest importance for the preservation of a free society but which we cannot examine satisfactorily here. First of all, we shall have to leave aside the whole complex of problems which arise from international relations\u2014not only because any serious attempt to consider these issues would unduly expand this book but also because an adequate treatment would require philosophical foundations other than those we have been able to provide. Satisfactory solutions to these problems will probably not be found as long as we have to accept as the ultimate units of international order the historically given entities known as sovereign nations. And to what groups we should entrust the various powers of government if we had the choice is far too difficult a question to answer briefly. The moral foundations for a rule of law on an international scale seem to be completely lacking still, and we should probably lose whatever advantages it brings within the nation if today we were to entrust any of the new powers of government to supra-national agencies. I will merely say that only makeshift solutions to problems of international relations seem possible so long as we have yet to learn how to limit the powers of all government effectively and how to divide these powers between the tiers of authority. It should also be said that  modern developments in national policies have made the international problems very much more difficult than they would have been in the nineteenth century.22 I wish to add here my opinion that, until the protection of individual freedom is much more firmly secured than it is now, the creation of a world state probably would be a greater danger to the future of civilization than even war.23\n\nHardly less important than the problems of international relations is that of centralization versus decentralization of governmental functions. In spite of its traditional connection with most of the problems we shall be discussing, we shall not be able to consider it systematically. While it has always been characteristic of those favoring an increase in governmental powers to support maximum concentration of these powers, those mainly concerned with individual liberty have generally advocated decentralization. There are strong reasons why action by local authorities generally offers the next best solution where private initiative cannot be relied upon to provide certain services and where some sort of collective action is therefore needed; for it has many of the advantages of private enterprise and fewer of the dangers of the coercive action of government. Competition between local authorities or between larger units within an area where there is freedom of movement provides in a large measure that opportunity for experimentation with alternative methods which will secure most of the advantages of free growth. Though the majority of individuals may never contemplate a change of residence, there will usually be enough people, especially among the young and more enterprising, to make it necessary for the local authorities to provide as good services at as reasonable costs as their competitors.24 It is usually the authoritarian planner who, in the interest of uniformity, governmental efficiency, and administrative convenience, supports the centralist tendencies and in this receives the strong support of the poorer majorities, who wish to be able to tap the resources of the wealthier regions.\n\n8. There are several other important problems of economic policy that we can mention only in passing. Nobody will deny that economic stability and the prevention of major depressions depends in part on government action.  We shall have to consider this problem under the subjects of employment and monetary policy. But a systematic survey would lead us into highly technical and controversial issues of economic theory, where the position I should have to take as the result of my specialized work in this field would be largely independent of the principles discussed in the present book.\n\nSimilarly, the subsidization of particular efforts out of funds raised by taxation, which we shall have to consider in connection with housing, agriculture, and education, raises problems of a more general nature. We cannot dismiss them simply by maintaining that no government subsidies should ever be given, since in some unquestioned fields of government activity, such as defense, it is probably often the best and least dangerous method of stimulating necessary developments and is often to be preferred to the government's taking over completely. Probably the only general principle that can be laid down with respect to subsidies is that they can never be justified in terms of the interest of the immediate beneficiary (whether it be the provider of the subsidized service or its consumer) but only in terms of the general benefits which may be enjoyed by all citizens\u2014i.e., the general welfare in the true sense. Subsidies are a legitimate tool of policy, not as a means of income redistribution, but only as a means of using the market to provide services which cannot be confined to those who individually pay for them.\n\nThe most conspicuous gap in the following survey is probably the omission of any systematic discussion of enterprise monopoly. The subject was excluded after careful consideration mainly because it seemed not to possess the importance commonly attached to it.25 For liberals antimonopoly policy has usually been the main object of their reformatory zeal. I believe I have myself in the past used the tactical argument that we cannot hope to curb the coercive powers of labor unions unless we at the same time attack enterprise monopoly. I have, however, become convinced that it would be disingenuous to represent the existing monopolies in the field of labor and those in the field of enterprise as being of the same kind. This does not mean that I share the position of some authors26 who hold that enterprise monopoly is in some respects beneficial and desirable. I still feel, as I did fifteen years ago,27 that it may be a good thing if the monopolist is treated as a sort of whipping boy of economic policy; and I recognize that, in the United States, legislation has succeeded in creating a climate of opinion unfavorable to monopoly. So far  as the enforcement of general rules (such as that of non-discrimination) can curb monopolistic powers, such action is all to the good. But what can be done effectively in this field must take the form of that gradual improvement of our law of corporations, patents, and taxation, on which little that is useful can be said briefly. I have become increasingly skeptical, however, about the beneficial character of any discretionary action of government against particular monopolies, and I am seriously alarmed at the arbitrary nature of all policy aimed at limiting the size of individual enterprises. And when policy creates a state of affairs in which, as is true of some enterprises in the United States, large firms are afraid to compete by lowering prices because this may expose them to antitrust action, it becomes an absurdity.\n\nCurrent policy fails to recognize that it is not monopoly as such, or bigness, but only obstacles to entry into an industry or trade and certain other monopolistic practices that are harmful. Monopoly is certainly undesirable, but only in the same sense in which scarcity is undesirable; in neither case does this mean that we can avoid it.28 It is one of the unpleasant facts of life that certain capacities (and also certain advantages and traditions of particular organizations) cannot be duplicated, as it is a fact that certain goods are scarce. It does not make sense to disregard this fact and to attempt to create conditions \"as if \" competition were effective. The law cannot effectively prohibit states of affairs but only kinds of action. All we can hope for is that, whenever the possibility of competition again appears, nobody will be prevented from taking advantage of it. Where monopoly rests on man-made obstacles to entry into a market, there is every case for removing them. There is also a strong case for prohibiting price discrimination so far as is possible by the application of general rules. But the record of governments in this field is so deplorable  that it is astounding that anyone should still expect that giving governments discretionary powers will do anything but increase those obstacles. It has been the experience of all countries that discretionary powers in the treatment of monopoly are soon used to distinguish between \"good\" and \"bad\" monopolies and that authority soon becomes more concerned with protecting the supposedly good than with preventing the bad. I doubt whether there are any \"good\" monopolies that deserve protection. But there will always be inevitable monopolies whose transitory and temporary character is often turned into a permanent one by the solicitude of government.\n\nBut, though very little is to be hoped for from any specific government action against enterprise monopoly, the situation is different where governments have deliberately fostered the growth of monopoly and even failed to perform the primary function of government\u2014the prevention of coercion, by granting exceptions from the general rules of law\u2014as they have been doing for a long time in the field of labor. It is unfortunate that in a democracy, after a period in which measures in favor of a particular group have been popular, the argument against privilege becomes an argument against the groups that in recent times have enjoyed the special favor of the public because they were thought to need and deserve special help. There can be no question, however, that the basic principles of the rule of law have nowhere in recent times been so generally violated and with such serious consequences as in the case of labor unions. Policy with respect to them will therefore be the first major problem that we shall consider.\n\n* * *\n\nThe quotation at the head of the chapter is taken from the dissenting opinion of Mr. Justice Louis Brandeis in _Olmstead v. United States_ 277 U.S. 438, at 479 (1927). Also see Jeremy Bentham, _Deontology; or, The Science of Morality: In Which the Harmony and Co-incidence of Duty and Self-interest, Virtue and Felicity, Prudence and Benevolence, are Explained and Exemplif ed_ (2 vols.; London: Rees, Orme, Browne, Green, and Longman, 1834), vol. 2, p. 289: \"Despotism never takes a worse shape than when it comes in the guise of benevolence.\"\n\n1 The most lively discussion of these problems is going on in Britain. See particularly Richard Howard Stafford Crossman, ed., _New Fabian Essays_ (London: Turnstile Press, 1952); and Socialist Union, _Socialism_ : _A New Statement of Principles_ (London: Lincolns-Prager, 1952). Crossman has himself written, in \"Communication: On Political Neuroses,\" _Encounter_ , 3 (May 1954): 66, that socialism today is viewed by most European socialist leaders as a \"Utopian myth . . . often remote from the realities of day to day politics.\" Also, George Douglas Howard Cole, _Is This Socialism?_ [ _New Statesman_ pamphlet] (London: New Statesman and Nation, 1954); Hugh Todd Naylor Gaitskell, _Recent Developments in British Socialist Thinking_ (London: Cooperative Union, Southern Section Education Council, 1956); Socialist Union, ed., _Twentieth Century Socialism: The Economy of Tomorrow_ (Harmondsworth, Middlesex: Penguin Books, 1956); Charles Anthony Raven Crosland, _The Future of Socialism_ (London: Jonathan Cape, 1956); Richard Howard Stafford Crossman, _Socialism and the New Despotism_ [Fabian Tracts, No. 298] (London: Fabian Society, 1956); Douglas Jay, _Socialism and the New Society_ (London: Longmans, 1962); Roy Jenkins, _The Labour Case_ (Harmondsworth, Middlesex: Penguin, 1959); Reuben Kelf-Cohen, _Nationalisation in Britain: The End of a Dogma_ (London: Macmillan, 1958); and the discussions carried on in the journals _Socialist Commentary_ and the _New Statesman_. A useful survey of these debates is Thomas Wilson, \"Changing Tendencies in Socialist Thought,\" _Lloyds Bank Review_ ( July 1956): 1\u201321. Illuminating comments on the British experiment by foreign observers are Bertrand de Jouvenel, _Probl\u00e8mes de l'Angleterre socialiste: ou, L'\u00e9chec d'une exp\u00e9rience_ (Paris: La table ronde, 1947); Clare Elmer Griffn, _Britain: A Case Study for Americans_ (Ann Arbor: University of Michigan Press, 1950); David McCord Wright, _Post-War West German and United Kingdom Recovery_ (Washington, DC: American Enterprise Association, 1957); and Johannes Messner, _Das englische Experiment des Sozialismus: auf Grund \u00f6konomischer Tatsachen und sozialistischer Selbstzeugnisse dargestellt_ (Innsbruck: Tyrolia, 1954).\n\n2 For the Continental developments see particularly Joseph Buttinger, _In the Twilight of Socialism: A History of the Revolutionary Socialists of Austria_ , Elizabeth Bruce Ashton, trans. (New York: F. A. Praeger, 1953); Karl Bednarik, _The Young Worker of Today: A New Type_ , Ren\u00e9e Tupholme, trans. (London: Faber and Faber, 1955); Fritz Klenner, _Das Unbehagen in der Demokratie: ein Beitrag zu Gegenwartsproblemen der Arbeiterbewegung_ (Vienna: Wiener Volksbuchhandlung, 1956). A similar change in attitude among American socialists is shown by Norman Thomas, _Democratic Socialism: A New Appraisal_ (New York: League for Industrial Democracy, 1953).\n\n3 Crossman, _Socialism and the New Despotism_ , p. 4. Crossman's essay contains a description of a discussion at a Fabian Summer School at Oxford in 1955.\n\n4 Crosland, _The Future of Socialism_ , and Bednarik, _The Young Worker of Today: A New Type._\n\n5 See especially Klenner, _Das Unbehagen in der Demokratie_ , pp. 66ff.\n\n6 As was made clear by the quotation from Karl Mannheim that I placed at the head of the chapter on \"Planning and the Rule of Law\" (in _The Road to Serfdom_ [Chicago: University of Chicago Press, 1944], chap. 6, pp. 72\u201387, reprinted as vol. 2 of _The Collected Works of F. A. Hayek_ , Bruce Caldwell, ed. [Chicago: University of Chicago Press, 2007], pp.112\u201323) and repeated in note 64, chap. 16, above.\n\n7 Especially George Orwell, _Nineteen Eighty-four: A Novel_ (London: Secker and Warburg, 1949); cf. also his review of _The Road to Serfdom_ , \"Grounds for Dismay,\" in the (London) _Observer_ , April 9, 1944, p. 3. [Orwell's review is more easily accessible in \"Review: _The Road to Serfdom_ by F. A. Hayek, and _The Mirror of the Past_ by Konni Zilliacus,\" in _The Collected Essays, Journalism, and Letters of George Orwell_ , Sonia Orwell and Ian Angus, eds. (4 vols.; New York: Harcourt, Brace and World, Inc., 1968), vol. 3, pp. 117\u201318.\u2014Ed.]\n\n8 Crossman, _Socialism and the New Despotism_ , p. 1.\n\n9 Ibid., p. 1.\n\n10 Ibid., p. 6.\n\n11 Ibid., p. 13. These apprehensions have clearly also affected the latest offcial statement of the British Labour party on these issues (see Labour Party [Great Britain], _Personal Freedom: Labour's Policy for the Individual and Society_ [London: Labour Party, 1956]). But, though this pamphlet deals with most of the crucial issues and shows how much the problems we have discussed have forced themselves into the foreground under a socialist regime even in a country with liberal traditions, it is, a curiously contradictory document. It not only repeats the phrase that \"freedom with gross inequalities is hardly worth having\" ( p. 7) but even expressly reasserts the basic thesis of administrative despotism that \"a minister must remain free to take different decisions in cases which are exactly similar\" ( p. 26).\n\n12 The term \"welfare state\" is comparatively new in the English language and was probably still unknown twenty-five years ago. Since the German _Wohlfahrtsstaat_ has been in use in that country for a long time and the thing it describes was first developed in Germany, the English term probably derives from the German. It deserves mention that the German term, from the beginning, was employed to describe a variant of the conception of the police state ( _Polizeistaat_ )\u2014apparently first by nineteenth-century historians to describe the more favorable aspects of eighteenth-century government. The modern conception of the welfare state was first fully developed by the German academic _Sozialpolitiker_ , or \"socialists of the chair,\" from about 1870 onward and was first put into practice by Bismarck. [The term \"socialists of the chair\" ( _Kathedersozialisten_ or _Sozialpolitiker_ ) has reference to those German professors whose sympathies were with the German Historical School, that is, those who regarded history as the sole basis upon which conclusions in the field of economics could be reached and who questioned the emphasis orthodox economics placed on the production of wealth rather than its distribution.\u2014Ed.]\n\nThe similar developments in England contemplated by the Fabians and by theorists like Arthur Cecil Pigou and Leonard Trelawney Hobhouse and put into practice by Lloyd George and Lord William Beveridge were, at least in their beginnings, strongly influenced by the German example. The acceptance of the term \"welfare state\" was assisted by the fact that the theoretical foundations that Pigou and his school had provided were known as \"welfare economics.\"\n\nBy the time F. D. Roosevelt followed in the footsteps of Bismarck and Lloyd George, the ground had been similarly well prepared in the United States, and the use made since 1937 by the Supreme Court of the \"general welfare\" clause of the Constitution naturally led to the adoption of the term \"welfare state\" already in use elsewhere. [The Supreme Court's rulings in May 1937 upholding the constitutionality of the Social Security Act by reference to the \"general welfare\" clause were _Helvering v. Davis_ (301 _U.S_. 619); _Steward Machine Company v. Davis_ (301 _U.S._ 548); and _Carmichael v. Southern Coal and Coke Co._ (201 _U.S._ 495).\u2014Ed.]\n\n13 Cf., e.g., Henry Sidgwick, _The Elements of Politics_ (London: Macmillan, 1891), chap. 4 \"Individualism and Individualistic Minimum,\" pp. 40\u201361.\n\n14 See on this particularly Lionel Robbins, _The Theory of Economic Policy in English Classical Political Economy_ (London: Macmillan, 1952).\n\n15 The preceding sentences are deliberately repeated, with only small alterations, from my book _The Road to Serfdom_ , chap. 9, pp. 119\u201320, where this subject is treated at greater length. [ _Collected Works_ edition, pp. 147\u201348.]\n\n16 Alvin Harvey Hansen, \"The Task of Promoting Economic Growth and Stability,\" address to the National Planning Association, February 26, 1956 (mimeographed). [While this address does not appear to have ever seen print, similar sentiments are contained in Hansen's _Economic Issues of the 1960s_ (New York: McGraw-Hill, 1960), passim, esp. chaps. 5 and 6, pp. 43\u201368.\u2014Ed.]\n\n17 Cf. John Stuart Mill, \"On Liberty,\" in _On Liberty and Considerations on Representative Government_ , Ronald Buchanan McCallum, ed. (Oxford: B. Blackwell, 1946), pp. 99\u2013100: \"If the roads, the railways, the banks, the insurance Offices, the great joint stock companies, the universities, and the public charities, were all of them branches of the government; if, in addition, the municipal corporations and local boards, with all that now devolves on them, became departments of the central administration; if the employees of all these different enterprises were appointed and paid by the government, and looked to the government for every rise in life; not all the freedom of the press and popular constitution of the legislature would make this or any other country free otherwise than in name. And the evil would be greater, the more effciently and scientifically the administrative machinery was constructed\u2014the more skilful the arrangements for obtaining the best qualified hands and heads with which to work it.\"\n\n18 Cf. Thomas Humphrey Marshall, _Citizenship and Social Class, and Other Essays_ (Cambridge: Cambridge University Press, 1950), p. 59: \"So we find that legislation . . . acquires more and more the character of a declaration of policy that it is hoped to put into effect some day.\"\n\n19 Roscoe Pound, \"The Rise of the Service State and Its Consequences,\" in _The Welfare State and the National Welfare: A Symposium on Some of the Threatening Tendencies of our Times_ , Sheldon Glueck, ed. (Cambridge, MA: Addison-Wesley Press, 1952), p. 226.\n\n20 Peter Wiles, \"Property and Equality,\" in _The Unservile State: Essays in Liberty and Welfare_ , George Watson, ed. (London: Allen and Unwin, 1957), p. 107. Cf. also the statement in the Inns of Court Conservative and Unionist Society, _Rule of Law: A Study_ (London: Conservative Political Centre, 1955), p. 20, and endorsed by the Franks Committee (Committee on Administrative Tribunals and Enquiries), _Report of the Committee on Administrative Tribunals and Enquiries, Presented by the Lord High Chancellor to Parliament by Command of Her Majesty_ , chaired by Sir Oliver Franks, Cmd. 218 (London: Her Majesty's Stationery Office, 1957), p. 60, par. 273, that \"whatever the theoretical validity of this argument, those of us who are Members of Parliament have no hesitation in saying that it bears little relation to reality. Parliament has neither the time nor the knowledge to supervise the Minister and call him to account for his administrative decisions.\"\n\n21 See Ludwig von Mises, _Human Action_ , pp.195\u201399 [Liberty Fund edition, vol. 4, p. 196].\n\n22 Cf. Lionel Robbins, _Economic Planning and International Order_ (London: Macmillan, 1937).\n\n23 Cf. Walter Fred Berns, \"The Case against World Government,\" _Readings in World Politics_ , Robert Allen Goldwin, ed. (3rd ed.; 3 vols.; Chicago: American Foundation for Political Education, 1955), vol. 3, pp. 62\u201375.\n\n24 Cf. George Joseph Stigler, \"The Tenable Range of Functions of Local Government (unpublished lecture, 1957) [mimeographed]. [The article which Hayek appears to have seen only in mimeographed form has since appeared in print. See Stigler, \"The Tenable Range of Functions of Local Government,\" _Federal Expenditure Policy for Economic Growth and Stability_ (Papers Submitted by Panelists Appearing Before the Subcommittee on Fiscal Policy; 35th Congress, 1st Session; Washington, DC: Joint Economic Committee, 1957), pp. 213\u2013219.\u2014Ed.]\n\n25 See the encyclopedic treatment of these problems by my friend Fritz Machlup, _The Political Economy of Monopoly: Business, Labor, and Government Policies_ (Baltimore: Johns Hopkins Press, 1952).\n\n26 See notably Joseph Alois Schumpeter, _Capitalism, Socialism, and Democracy_ (New York: Harper and Brothers, 1942), chap. 7 [\"The Process of Creative Destruction,\"], pp. 81\u201386.\n\n27 _The Road to Serfdom_ , chap. 4 [\"The Inevitability of Planning,\"], pp. 43\u201355. [ _Collected Works_ edition, pp. 91\u201399].\n\n28 Cf. Frank Hyneman Knight, \"Conflict of Values: Freedom and Justice,\" in _Goals of Economic Life_ , Alfred Dudley Ward, ed. (New York: Harper and Brothers, 1953), pp. 224\u201325: \"The public has most exaggerated ideas of the scope of monopoly as really bad and remediable, and talk of 'abolishing' it is merely ignorant or irresponsible. There is no clear line between legitimate and necessary profit and the monopoly gain that presents a problem for action. Every doctor or artist of repute has a monopoly, and monopolies are deliberately granted by law to encourage invention and other creative activities. And, finally, most monopolies work in the same ways as 'patents,' etc., and are temporary and largely balanced by losses. Moreover, by far the worst monopolist restrictions are those organized by wage earners and farmers with the connivance or direct aid of government and with public approval.\" Cf. also the earlier statement by the same author in his review \"The Meaning of Freedom,\" _Ethics_ , 52 (1941\u201342): 103: \"It is needful to state that the role of 'monopoly' in actual economic life is enormously exaggerated in the popular mind and also that a large part of the monopoly which is real, and especially the worst part, is due to the activities of government. In general (and especially in the United States under the New Deal), these have been very largely such as to promote, if not directly to create, monopoly rather than to create or to enforce the conditions of market competition. What competition actually means is simply the freedom of the individual to 'deal' with any and all other individuals and to select the best terms as judged by himself, among those offered.\"\n\n## EIGHTEEN\n\nLABOR UNIONS AND EMPLOYMENT\n\nGovernment, long hostile to other monopolies, suddenly sponsored and promoted widespread labor monopolies, which democracy cannot endure, cannot control without destroying, and perhaps cannot destroy without destroying itself.\n\n\u2014Henry C. Simons\n\n1. Public policy concerning labor unions has, in little more than a century, moved from one extreme to the other. From a state in which little the unions could do was legal if they were not prohibited altogether, we have now reached a state where they have become uniquely privileged institutions to which the general rules of law do not apply. They have become the only important instance in which governments signally fail in their prime function\u2014the prevention of coercion and violence.\n\nThis development has been greatly assisted by the fact that unions were at first able to appeal to the general principles of liberty1 and then retain the support of the liberals long after all discrimination against them had ceased and they had acquired exceptional privileges. In few other areas are progressives so little willing to consider the reasonableness of any particular measure  but generally ask only whether it is \"for or against unions\" or, as it is usually put, \"for or against labor.\"2 Yet the briefest glance at the history of the unions should suggest that the reasonable position must lie somewhere between the extremes which mark their evolution.\n\nMost people, however, have so little realization of what has happened that they still support the aspirations of the unions in the belief that they are struggling for \"freedom of association,\" when this term has in fact lost its meaning and the real issue has become the freedom of the individual to join or not to join a union. The existing confusion is due in part to the rapidity with which the character of the problem has changed; in many countries voluntary associations of workers had only just become legal when they began to use coercion to force unwilling workers into membership and to keep non-members out of employment. Most people probably still believe that a \"labor dispute\" normally means a disagreement about remuneration and the conditions of employment, while as often as not its sole cause is an attempt on the part of the unions to force unwilling workers to join.\n\nThe acquisition of privilege by the unions has nowhere been as spectacular as in Britain, where the Trade Dispute Act of 1906 conferred \"upon a trade union a freedom from civil liability for the commission of even the most heinous wrong by the union or its servant, and in short confer[red] upon every trade union a privilege and protection not possessed by any other person or body of persons, whether corporate or incorporate.\"3 Similar friendly legislation  helped the unions in the United States, where first the Clayton Act of 1914 exempted them from the antimonopoly provisions of the Sherman Act; the Norris-LaGuardia Act of 1932 \"went a long way to establish practically complete immunity of labor organizations for torts\";4 and, finally, the Supreme Court in a crucial decision sustained \"the claim of a union to the right to deny participation in the economic world to an employer.\"5 More or less the same situation had gradually come to exist in most European countries by the 1920s, \"less through explicit legislative permission than by the tacit toleration by authorities and courts.\"6 Everywhere the legalization of unions was interpreted as a legalization of their main purpose and as recognition of their right to do whatever seemed necessary to achieve this purpose\u2014namely, monopoly. More and more they came to be treated not as a group which was pursuing a legitimate selfish aim and which, like every other interest, must be kept in check by competing interests possessed of equal rights, but as a group whose aim\u2014the exhaustive and comprehensive organization of all labor\u2014 must be supported for the good of the public.7\n\nAlthough flagrant abuses of their powers by the unions have often shocked public opinion in recent times and uncritical pro-union sentiment is on the wane, the public has certainly not yet become aware that the existing legal position is fundamentally wrong and that the whole basis of our free society is gravely threatened by the powers arrogated by the unions. We shall not be concerned here with those criminal abuses of union power that have lately attracted much attention in the United States, although they are not entirely  unconnected with the privileges that unions legally enjoy. Our concern will be solely with those powers that unions today generally possess, either with the explicit permission of the law or at least with the tacit toleration of the law-enforcing authorities. Our argument will not be directed against labor unions as such; nor will it be confined to the practices that are now widely recognized as abuses. But we shall direct our attention to some of their powers which are now widely accepted as legitimate, if not as their \"sacred rights.\" The case against these is strengthened rather than weakened by the fact that unions have often shown much restraint in exercising them. It is precisely because, in the existing legal situation, unions could do infinitely more harm than they do, and because we owe it to the moderation and good sense of many union leaders, that the situation is not much worse that we cannot afford to allow the present state of affairs to continue.8\n\n2. It cannot be stressed enough that the coercion which unions have been permitted to exercise contrary to all principles of freedom under the law is primarily the coercion of fellow workers. Whatever true coercive power unions may be able to wield over employers is a consequence of this primary power of coercing other workers; the coercion of employers would lose most of its objectionable character if unions were deprived of this power to exact unwilling support. Neither the right of voluntary agreement between workers nor even their right to withhold their services in concert is in question. It should be said, however, that the latter\u2014the right to strike\u2014though a normal right, can hardly be regarded as an inalienable right. There are good reasons why in certain employments it should be part of the terms of employment that the worker should renounce this right; i.e., such employments should involve long-term obligations on the part of the workers, and any concerted attempts to break such contracts should be illegal.\n\nIt is true that any union effectively controlling all potential workers of a firm or industry can exercise almost unlimited pressure on the employer and that, particularly where a great amount of capital has been invested in specialized equipment, such a union can practically expropriate the owner and command nearly the whole return of his enterprise.9 The decisive point, however, is that this will never be in the interest of all workers\u2014except in the unlikely  case where the total gain from such action is equally shared among them, irrespective of whether they are employed or not\u2014and that, therefore, the union can achieve this only by coercing some workers against their interest to support such a concerted move.\n\nThe reason for this is that workers can raise real wages above the level that would prevail on a free market only by limiting the supply, that is, by withholding part of labor. The interest of those who will get employment at the higher wage will therefore always be opposed to the interest of those who, in consequence, will find employment only in the less highly paid jobs or who will not be employed at all.\n\nThe fact that unions will ordinarily first make the employer agree to a certain wage and then see to it that nobody will be employed for less makes little difference. Wage fixing is quite as effective a means as any other of keeping out those who could be employed only at a lower wage. The essential point is that the employer will agree to the wage only when he knows that the union has the power to keep out others.10 As a general rule, wage fixing (whether by unions or by authority) will make wages higher than they would otherwise be only if they are also higher than the wage at which all willing workers can be employed.\n\nThough unions may still often act on a contrary belief, there can now be no doubt that they cannot in the long run increase real wages for all wishing to work above the level that would establish itself in a free market\u2014though they may well push up the level of money wages, with consequences that will occupy us later. Their success in raising real wages beyond that point, if it is to be more than temporary, can benefit only a particular group at the expense of others. It will therefore serve only a sectional interest even when it obtains the support of all. This means that strictly voluntary unions, because their wage policy would not be in the interest of all workers, could not long receive the support of all. Unions that had no power to coerce outsiders would thus not be strong enough to force up wages above the level at which all seeking work could be employed, that is, the level that would establish itself in a truly free market for labor in general.\n\nBut, while the real wages of all the employed can be raised by union action only at the price of unemployment, unions in particular industries or crafts may well raise the wages of their members by forcing others to stay in less-well-paid occupations. How great a distortion of the wage structure this in fact causes is difficult to say. If one remembers, however, that some unions find it expedient to use violence in order to prevent any influx into their trade and that others are able to charge high premiums for admission (or even to  reserve jobs in the trade for children of present members), there can be little doubt that this distortion is considerable. It is important to note that such policies can be employed successfully only in relatively prosperous and highly paid occupations and that they will therefore result in the exploitation of the relatively poor by the better-off. Even though within the scope of any one union its actions may tend to reduce differences in remuneration, there can be little doubt that, so far as relative wages in major industries and trades are concerned, unions today are largely responsible for an inequality which has no function and is entirely the result of privilege.11 This means that their activities necessarily reduce the productivity of labor all around and therefore also the general level of real wages; because, if union action succeeds in reducing the number of workers in the highly paid jobs and in increasing the number of those who have to stay in the less remunerative ones, the result must be that the over-all average will be lower. It is, in fact, more than likely that, in countries where unions are very strong, the general level of real wages is lower than it would otherwise be.12 This is certainly true of most countries of Europe, where union policy is strengthened by the general use of restrictive practices of a \"make-work\" character.\n\nIf many still accept as an obvious and undeniable fact that the general wage level has risen as fast as it has done because of the efforts of the unions, they do so in spite of these unambiguous conclusions of theoretical analysis\u2014and in spite of empirical evidence to the contrary. Real wages have often risen much faster when unions were weak than when they were strong; furthermore, even the rise in particular trades or industries where labor was not organized has frequently been much faster than in highly organized and equally prosperous industries.13 The common impression to the contrary is due partly to the fact that wage gains, which are today mostly obtained in union negotiations, are for that reason regarded as obtainable only in this manner14 and even more to the fact that, as we shall presently see, union activity does in fact  bring about a continuous rise in money wages exceeding the increase in real wages. Such increase in money wages is possible without producing general unemployment only because it is regularly made ineffective by inflation\u2014 indeed, it must be if full employment is to be maintained.\n\n3. If unions have in fact achieved much less by their wage policy than is generally believed, their activities in this field are nevertheless economically very harmful and politically exceedingly dangerous. They are using their power in a manner which tends to make the market system ineffective and which, at the same time, gives them a control of the direction of economic activity that would be dangerous in the hands of government but is intolerable if exercised by a particular group. They do so through their influence on the relative wages of different groups of workers and through their constant upward pressure on the level of money wages, with its inevitable inflationary consequences.\n\nThe effect on relative wages is usually greater uniformity and rigidity of wages within any one union-controlled group and greater and non-functional differences in wages between different groups. This is accompanied by a restriction of the mobility of labor, of which the former is either an effect or a cause. We need say no more about the fact that this may benefit particular groups but can only lower the productivity and therefore the incomes of the workers in general. Nor need we stress here the fact that the greater stability of the wages of particular groups which unions may secure is likely to involve greater instability of employment. What is important is that the accidental differences in union power of the different trades and industries will produce not only gross inequalities in remuneration among the workers which have no economic justification but uneconomic disparities in the development of different industries. Socially important industries, such as building, will be greatly hampered in their development and will conspicuously fail to satisfy urgent needs simply because their character offers the unions special opportunities for coercive monopolistic practices.15 Because unions are most powerful where capital investments are heaviest, they tend to become a deterrent to investment\u2014at present probably second only to taxation. Finally, it is often union monopoly in collusion with enterprise that becomes one of the chief foundations of monopolistic control of the industry concerned.\n\nThe chief danger presented by the current development of unionism is that, by establishing effective monopolies in the supply of the different kinds of labor, the unions will prevent competition from acting as an effective regulator of the allocation of all resources. But if competition becomes ineffective as a means of such regulation, some other means will have to be adopted in  its place. The only alternative to the market, however, is direction by authority. Such direction clearly cannot be left in the hands of particular unions with sectional interests, nor can it be adequately performed by a unified organization of all labor, which would thereby become not merely the strongest power in the state but a power completely controlling the state. Unionism as it is now tends, however, to produce that very system of over-all socialist planning which few unions want and which, indeed, it is in their best interest to avoid.\n\n4. The unions cannot achieve their principal aims unless they obtain complete control of the supply of the type of labor with which they are concerned; and, since it is not in the interest of all workers to submit to such control, some of them must be induced to act against their own interest. This may be done to some extent through merely psychological and moral pressure, encouraging the erroneous belief that the unions benefit all workers. Where they succeed in creating a general feeling that every worker ought, in the interest of his class, to support union action, coercion comes to be accepted as a legitimate means of making a recalcitrant worker do his duty. Here the unions have relied on a most effective tool, namely, the myth that it is due to their efforts that the standard of living of the working class has risen as fast as it has done and that only through their continued efforts will wages continue to increase as fast as possible\u2014a myth in the assiduous cultivation of which the unions have usually been actively assisted by their opponents. A departure from such a condition can come only from a truer insight into the facts, and whether this will be achieved depends on how effectively economists do their job of enlightening public opinion.\n\nBut though this kind of moral pressure exerted by the unions may be very powerful, it would scarcely be sufficient to give them the power to do real harm. Union leaders apparently agree with the students of this aspect of unionism that much stronger forms of coercion are needed if the unions are to achieve their aims. It is the techniques of coercion that unions have developed for the purpose of making membership in effect compulsory, what they call their \"organizational activities\" (or, in the United States, \"union security\"\u2014 a curious euphemism) that give them real power. Because the power of truly voluntary unions will be restricted to what are common interests of all workers, they have come to direct their chief efforts to the forcing of dissenters to obey their will.\n\nThey could never have been successful in this without the support of a misguided public opinion and the active aid of government. Unfortunately, they have to a large extent succeeded in persuading the public that complete unionization is not only legitimate but important to public policy. To say that the workers have a right to form unions, however, is not to say that the unions have a right to exist independently of the will of the individual workers. Far  from being a public calamity, it would indeed be a highly desirable state of affairs if the workers should not feel it necessary to form unions. Yet the fact that it is a natural aim of the unions to induce all workers to join them has been so interpreted as to mean that the unions ought to be entitled to do whatever seems necessary to achieve this aim. Similarly, the fact that it is legitimate for unions to try to secure higher wages has been interpreted to mean that they must also be allowed to do whatever seems necessary to succeed in their effort. In particular, because striking has been accepted as a legitimate weapon of unions, it has come to be believed that they must be allowed to do whatever seems necessary to make a strike successful. In general, the legalization of unions has come to mean that whatever methods they regard as indispensable for their purposes are also to be treated as legal.\n\nThe present coercive powers of unions thus rest chiefly on the use of methods which would not be tolerated for any other purpose and which are opposed to the protection of the individual's private sphere. In the first place, the unions rely\u2014to a much greater extent than is commonly recognized\u2014 on the use of the picket line as an instrument of intimidation. That even so-called \"peaceful\" picketing in numbers is severely coercive and the condoning of it constitutes a privilege conceded because of its presumed legitimate aim is shown by the fact that it can be and is used by persons who themselves are not workers to force others to form a union which they will control, and that it can also be used for purely political purposes or to give vent to animosity against an unpopular person. The aura of legitimacy conferred upon it because the aims are often approved cannot alter the fact that it represents a kind of organized pressure upon individuals which in a free society no private agency should be permitted to exercise.\n\nNext to the toleration of picketing, the chief factor which enables unions to coerce individual workers is the sanction by both legislation and jurisdiction of the closed or union shop and its varieties. These constitute contracts in restraint of trade, and only their exemption from the ordinary rules of law has made them legitimate objects of the \"organizational activities\" of the unions. Legislation has frequently gone so far as to require not only that a contract concluded by the representatives of the majority of the workers of a plant or industry be available to any worker who wishes to take advantage of it, but that it apply to all employees, even if they should individually wish and be able to obtain a different combination of advantages.16 We must also regard as  inadmissible methods of coercion all secondary strikes and boycotts which are used not as an instrument of wage bargaining but solely as a means of forcing other workers to fall in with union policies.\n\nMost of these coercive tactics of the unions can be practiced, moreover, only because the law has exempted groups of workers from the ordinary responsibility of joint action, either by allowing them to avoid formal incorporation or by explicitly exempting their organizations from the general rules applying to corporate bodies. There is no need to consider separately various other aspects of contemporary union policies such as, to mention one, industry-wide or nation-wide bargaining. Their practicability rests on the practices already mentioned, and they would almost certainly disappear if the basic coercive power of the unions were removed.17\n\n5. It can hardly be denied that raising wages by the use of coercion is today the main aim of unions. Even if this were their sole aim, legal prohibition of unions would however, not be justifiable. In a free society much that is undesirable has to be tolerated if it cannot be prevented without discriminatory legislation. But the control of wages is even now not the only function of the unions; and they are undoubtedly capable of rendering services which are not only unobjectionable but definitely useful. If their only purpose were to force up wages by coercive action, they would probably disappear if deprived of coercive power. But unions have other useful functions to perform, and, though it would be contrary to all our principles even to consider the possibility of prohibiting them altogether, it is desirable to show explicitly why there is no economic ground for such action and why, as truly voluntary and non-coercive organizations, they may have important services to render. It is in fact more than probable that unions will fully develop their potential usefulness only after they have been diverted from their present antisocial aims by an effective prevention of the use of coercion.18\n\nUnions without coercive powers would probably play a useful and important role even in the process of wage determination. In the first place, there is often a choice to be made between wage increases, on the one hand, and, on the other, alternative benefits which the employer could provide at the same cost but which he can provide only if all or most of the workers are willing to accept them in preference to additional pay. There is also the fact that the relative position of the individual on the wage scale is often nearly as important to him as his absolute position. In any hierarchical organization it is important that the differentials between the remuneration for the different jobs and the rules of promotion are felt to be just by the majority.19 The most effective way of securing consent is probably to have the general scheme agreed to in collective negotiations in which all the different interests are represented. Even from the employer's point of view it would be difficult to conceive of any other way of reconciling all the different considerations that in a large organization have to be taken into account in arriving at a satisfactory wage structure. An agreed set of standard terms, available to all who wish to take advantage of them, though not excluding special arrangements in individual cases, seems to be required by the needs of large-scale organizations.\n\nThe same is true to an even greater extent of all the general problems relating to conditions of work other than individual remuneration, those problems which truly concern all employees and which, in the mutual interest of workers and employers, should be regulated in a manner that takes account of as many desires as possible. A large organization must in a great measure be governed by rules, and such rules are likely to operate most effectively if drawn up with the participation of the workers.20 Because a contract between employers and employees regulates not only relations between them but also relations between the various groups of employees, it is often expedient to  give it the character of a multilateral agreement and to provide in certain respects, as in grievance procedure, for a degree of self-government among the employees.\n\nThere is, finally, the oldest and most beneficial activity of the unions, in which as \"friendly societies\" they undertake to assist members in providing against the peculiar risks of their trade. This is a function which must in every respect be regarded as a highly desirable form of self-help, albeit one which is gradually being taken over by the welfare state. We shall leave the question open, however, as to whether any of the above arguments justify unions of a larger scale than that of the plant or corporation.\n\nAn entirely different matter, which we can mention here only in passing, is the claim of unions to participation in the conduct of business. Under the name of \"industrial democracy\" or, more recently, under that of \"co-determination,\" this has acquired considerable popularity, especially in Germany and to a lesser degree in Britain. It represents a curious recrudescence of the ideas of the syndicalist branch of nineteenth-century socialism, the least-thought-out and most impractical form of that doctrine. Though these ideas have a certain superficial appeal, they reveal inherent contradictions when examined. A plant or industry cannot be conducted in the interest of some permanent distinct body of workers if it is at the same time to serve the interests of the consumers. Moreover, effective participation in the direction of an enterprise is a full-time job, and anybody so engaged soon ceases to have the outlook and interest of an employee. It is not only from the point of view of the employers, therefore, that such a plan should be rejected; there are very good reasons why in the United States union leaders have emphatically refused to assume any responsibility in the conduct of business. For a fuller examination of this problem we must, however, refer the reader to the careful studies, now available, of all its implications.21\n\n6. Though it may be impossible to protect the individual against all union coercion so long as general opinion regards it as legitimate, most students of the subject agree that comparatively few and, as they may seem at first, minor changes in law and jurisdiction would suffice to produce far-reaching and probably decisive changes in the existing situation.22 The mere withdrawal of the special privileges either explicitly granted to the unions or arrogated by them with the toleration of the courts would seem enough to deprive them  of the more serious coercive powers which they now exercise and to channel their legitimate selfish interests so that they would be socially beneficial.\n\nThe essential requirement is that true freedom of association be assured and that coercion be treated as equally illegitimate whether employed for or against organization, by the employer or by the employees. The principle that the end does not justify the means and that the aims of the unions do not justify their exemption from the general rules of law should be strictly applied. Today this means, in the first place, that all picketing in numbers should be prohibited, since it is not only the chief and regular cause of violence but even in its most peaceful forms is a means of coercion. Next, the unions should not be permitted to keep non-members out of any employment. This means that closed-and union-shop contracts (including such varieties as the \"maintenance of membership\" and \"preferential hiring\" clauses) must be treated as contracts in restraint of trade and denied the protection of the law. They differ in no respect from the \"yellow-dog contract\" which prohibits the individual worker from joining a union and which is commonly prohibited by the law.\n\nThe invalidating of all such contracts would, by removing the chief objects of secondary strikes and boycotts, make these and similar forms of pressure largely ineffective. It would be necessary, however, also to rescind all legal provisions which make contracts concluded with the representatives of the majority of workers of a plant or industry binding on all employees and to deprive all organized groups of any right of concluding contracts binding on men who have not voluntarily delegated this authority to them.23 Finally, the responsibility for organized and concerted action in conflict with contractual obligations or the general law must be firmly placed on those in whose hands the decision lies, irrespective of the particular form of organized action adopted.\n\nIt would not be a valid objection to maintain that any legislation making certain types of contracts invalid would be contrary to the principle of freedom of contract. We have seen before (in chap. 15) that this principle can never mean that all contracts will be legally binding and enforceable. It means merely that all contracts must be judged according to the same general rules and that no authority should be given discretionary power to allow or disallow particular contracts. Among the contracts to which the law ought to deny validity are contracts in restraint of trade. Closed-and union-shop contracts fall clearly into this category. If legislation, jurisdiction, and the tolerance of executive agencies had not created privileges for the unions, the need for special legislation concerning them would probably not have arisen in  common-law countries. That there is such a need is a matter for regret, and the believer in liberty will regard any legislation of this kind with misgivings. But, once special privileges have become part of the law of the land, they can be removed only by special legislation. Though there ought to be no need for special \"right-to-work laws,\" it is difficult to deny that the situation created in the United States by legislation and by the decisions of the Supreme Court may make special legislation the only practicable way of restoring the principles of freedom.24\n\nThe specific measures which would be required in any given country to reinstate the principles of free association in the field of labor will depend on the situation created by its individual development. The situation in the United States is of special interest, for here legislation and the decisions of the Supreme Court have probably gone further than elsewhere25 in legalizing union coercion and very far in conferring discretionary and essentially irresponsible powers on administrative authority. But for further details we must refer the reader to the important study by Professor Petro on _The Labor Policy of the Free Society_ ,26 in which the reforms required are fully described.\n\nThough all the changes needed to restrain the harmful powers of the unions involve no more than that they be made to submit to the same general principles of law that apply to everybody else, there can be no doubt that the existing unions will resist them with all their power. They know that the achievement of what they at present desire depends on that very coercive power which will have to be restrained if a free society is to be preserved. Yet the situation is not hopeless. There are developments under way which sooner or later will prove to the unions that the existing state cannot last. They will find that, of the alternative courses of further development open to them, submitting to the general principle that prevents all coercion will be greatly preferable in the long run to continuing their present policy; for the latter is bound to lead to one of two unfortunate consequences.\n\n7. While labor unions cannot in the long run substantially alter the level of real wages that all workers can earn and are, in fact, more likely to lower than to raise them, the same is not true of the level of money wages. With  respect to them, the effect of union action will depend on the principles governing monetary policy. What with the doctrines that are now widely accepted and the policies accordingly expected from the monetary authorities, there can be little doubt that current union policies must lead to continuous and progressive inflation. The chief reason for this is that the dominant \"full-employment\" doctrines explicitly relieve the unions of the responsibility for any unemployment and place the duty of preserving full employment on the monetary and fiscal authorities. The only way in which the latter can prevent union policy from producing unemployment is, however, to counter through inflation whatever excessive rises in real wages unions tend to cause.\n\nIn order to understand the situation into which we have been led, it will be necessary to take a brief look at the intellectual sources of the full-employment policy of the \"Keynesian\" type. The development of Lord Keynes's theories started from the correct insight that the regular cause of extensive unemployment is real wages that are too high. The next step consisted in the proposition that a direct lowering of money wages could be brought about only by a struggle so painful and prolonged that it could not be contemplated. Hence he concluded that real wages must be lowered by the process of lowering the value of money. This is really the reasoning underlying the whole \"fullemployment\" policy, now so widely accepted.27 If labor insists on a level of money wages too high to allow of full employment, the supply of money must be so increased as to raise prices to a level where the real value of the prevailing money wages is no longer greater than the productivity of the workers seeking employment. In practice, this necessarily means that each separate union, in its attempt to overtake the value of money, will never cease to insist on further increases in money wages and that the aggregate effort of the unions will thus bring about progressive inflation.\n\nThis would follow even if individual unions did no more than prevent any reduction in the money wages of any particular group. Where unions make such wage reductions impracticable and wages have generally become, as the economists put it, \"rigid downward,\" all the changes in relative wages of the different groups made necessary by the constantly changing conditions must be brought about by raising all money wages except those of the group whose relative real wages must fall. Moreover, the general rise in money wages and the resulting increase in the cost of living will generally lead to attempts, even on the part of the latter group, to push up money wages, and several rounds of successive wage increases will be required before any readjustment of relative  wages is produced. Since the need for adjustment of relative wages occurs all the time, this process alone produces the wage-price spiral that has prevailed since the second World War, that is, since full-employment policies became generally accepted.28\n\nThe process is sometimes described as though wage increases directly produced inflation. This is not correct. If the supply of money and credit were not expanded, the wage increases would rapidly lead to unemployment. But under the influence of a doctrine that represents it as the duty of the monetary authorities to provide enough money to secure full employment at any given wage level, it is politically inevitable that each round of wage increases should lead to further inflation.29 Or it is inevitable until the rise of prices becomes sufficiently marked and prolonged to cause serious public alarm. Efforts will then be made to apply the monetary brakes. But, because by that time the economy will have become geared to the expectation of further inflation and much of the existing employment will depend on continued monetary expansion, the attempt to stop it will rapidly produce substantial unemployment. This will bring a renewed and irresistible pressure for more inflation. And, with ever bigger doses of inflation, it may be possible for quite a long time to prevent the appearance of the unemployment which the wage pressure would otherwise cause. To the public at large it will seem as if progressive inflation were the direct consequence of union wage policy rather than of an attempt to cure its consequences.\n\nThough this race between wages and inflation is likely to go on for some time, it cannot go on indefinitely without people coming to realize that it must somehow be stopped. A monetary policy that would break the coercive powers of the unions by producing extensive and protracted unemployment must be excluded, for it would be politically and socially fatal. But if we do not succeed in time in curbing union power at its source, the unions will soon be faced with a demand for measures that will be much more distasteful to the individual workers, if not the union leaders, than the submission of the unions to the rule of law: the clamor will soon be either for the fixing of wages by government or for the complete abolition of the unions.\n\n8. In the field of labor, as in any other field, the elimination of the market as a steering mechanism would necessitate the replacement of it by a system of administrative direction. In order to approach even remotely the ordering function of the market, such direction would have to co-ordinate the whole economy and therefore, in the last resort, have to come from a single central authority. And though such an authority might at first concern itself only with the allocation and remuneration of labor, its policy would necessarily lead to the transformation of the whole of society into a centrally planned and administered system, with all its economic and political consequences.\n\nIn those countries in which inflationary tendencies have operated for some time, we can observe increasingly frequent demands for an \"over-all wage policy.\" In the countries where these tendencies have been most pronounced, notably in Great Britain, it appears to have become accepted doctrine among the intellectual leaders of the Left that wages should generally be determined by a \"unified policy,\" which ultimately means that government must do the determining.30 If the market were thus irretrievably deprived of its function, there would be no efficient way of distributing labor throughout the industries, regions, and trades, other than having wages determined by authority. Step by step, through setting up an official conciliation and arbitration machinery with compulsory powers, and through the creation of wage boards, we are moving toward a situation in which wages will be determined by what must be essentially arbitrary decisions of authority.\n\nAll this is no more than the inevitable outcome of the present policies of labor unions, who are led by the desire to see wages determined by some conception of \"justice\" rather than by the forces of the market. But in no workable system could any group of people be allowed to enforce by the threat of violence what it believes it should have. And when not merely a few privileged groups but most of the important sections of labor have become effectively organized for coercive action, to allow each to act independently would not only produce the opposite of justice but result in economic chaos. When we can no longer depend on the impersonal determination of wages by the  market, the only way we can retain a viable economic system is to have them determined authoritatively by government. Such determination must be arbitrary, because there are no objective standards of justice that could be applied.31 As is true of all other prices or services, the wage rates that are compatible with an open opportunity for all to seek employment do not correspond to any assessable merit or any independent standard of justice but must depend on conditions which nobody can control.\n\nOnce government undertakes to determine the whole wage structure and is thereby forced to control employment and production, there will be a far greater destruction of the present powers of the unions than their submission to the rule of equal law would involve. Under such a system the unions will have only the choice between becoming the willing instrument of governmental policy and being incorporated into the machinery of government, on the one hand, and being totally abolished, on the other. The former alternative is more likely to be chosen, since it would enable the existing union bureaucracy to retain their position and some of their personal power. But to the workers it would mean complete subjection to the control by a corporative state. The situation in most countries leaves us no choice but to await some such outcome or to retrace our steps. The present position of the unions cannot last, for they can function only in a market economy which they are doing their best to destroy.\n\n9. The problem of labor unions constitutes both a good test of our principles and an instructive illustration of the consequences if they are infringed. Having failed in their duty of preventing private coercion, governments are now driven everywhere to exceed their proper function in order to correct the results of that failure and are thereby led into tasks which they can perform only by being as arbitrary as the unions. So long as the powers that the unions have been allowed to acquire are regarded as unassailable, there is no way to  correct the harm done by them but to give the state even greater arbitrary power of coercion. We are indeed already experiencing a pronounced decline of the rule of law in the field of labor.32 Yet all that is really needed to remedy the situation is a return to the principles of the rule of law and to their consistent application by legislative and executive authorities.\n\nThis path is still blocked, however, by the most fatuous of all fashionable arguments, namely, that \"we cannot turn the clock back.\" One cannot help wondering whether those who habitually use this clich\u00e9 are aware that it expresses the fatalistic belief that we cannot learn from our mistakes, the most abject admission that we are incapable of using our intelligence. I doubt whether anybody who takes a long-range view believes that there is another satisfactory solution which the majority would deliberately choose if they fully understood where the present developments were leading. There are some signs that farsighted union leaders are also beginning to recognize that, unless we are to resign ourselves to the progressive extinction of freedom, we must reverse that trend and resolve to restore the rule of law and that, in order to save what is valuable in their movement, they must abandon the illusions which have guided it for so long.33\n\nNothing less than a rededication of current policy to principles already abandoned will enable us to avert the threatening danger to freedom. What is required is a change in economic policy, for in the present situation the tactical decisions which will seem to be required by the short-term needs of government in successive emergencies will merely lead us further into the thicket of arbitrary controls. The cumulative effects of those palliatives which the pursuit of contradictory aims makes necessary must prove strategically fatal. As is true of all problems of economic policy, the problem of labor unions  cannot be satisfactorily solved by _ad hoc_ decisions on particular questions but only by the consistent application of a principle that is uniformly adhered to in all fields. There is only one such principle that can preserve a free society: namely, the strict prevention of all coercion except in the enforcement of general abstract rules equally applicable to all.\n\nThe quotation at the head of the chapter is taken from Henry Calvert Simons, \"Hansen on Fiscal Policy,\" _Journal of Political Economy_ , 50 (1942): 171; reprinted in _Economic Policy for a Free Society_ (Chicago: University of Chicago Press, 1948), p. 193.\n\n* * *\n\n1 Including the most \"orthodox\" political economists, who invariably supported freedom of association. See particularly the discussion in John Ramsay McCulloch, _Treatise on the Circumstances Which Determine the Rate of Wages and the Condition of the Labouring Classes_ (London: Long-man, Brown, Green, and Longmans, 1851), pp. 79\u201389, with its stress on _voluntary_ association. [McCulloch at one point notes: \"A voluntary combination among workmen is certainly in no respect injurious to any of the rights of their masters. It is a contradiction to pretend that masters have any right or title to the services of free workmen in the event of the latter not choosing to accept the price offered them for their labour. And as the existence of a combination to procure a rise in wages shows that they have not so chosen, and is proof of the want of all concord and agreement between the parties, so it is also a proof that the workmen are fairly entitled to enter into it; and that, however injurious their proceedings may be to themselves, they do not encroach on the privileges or rights of others.\"\u2014Ed.] For a comprehensive statement of the classical liberal attitude toward the legal problems involved see Ludwig Bamberger, _Die Arbeiter frage unter dem Gesichtspunkte des Vereinsrechtes_ (Stuttgart: J. G. Cotta, 1873).\n\n2 Characteristic is the description of the \"liberal\" attitude to unions in Charles Wright Mills, _The New Men of Power: America's Labor Leaders_ (New York: Harcourt, Brace, 1948), p. 21. \"In many liberal minds there seems to be an undercurrent that whispers: 'I will not criticize the unions and their leaders. There I draw the line.' This, they must feel distinguishes them from the bulk of the Republican Party and the right-wing Democrats; this keeps them leftward and socially pure.\"\n\n3 Dicey, \"Introduction,\" _Law and Opinion_ (2nd edition), pp. xlv\u2013xlvii [Liberty Fund edition, pp. 373\u201374]. He continues to say that the law \"makes a trade union a privileged body exempted from the ordinary law of the land. No such privileged body has ever before been deliberately created by an English Parliament [and that] it stimulates among workmen the fatal delusion that workmen should aim at the attainment, not of equality, but of privilege.\" Cf. also the comment on the same law, thirty years later, by Joseph Alois Schumpeter, _Capitalism, Socialism, and Democracy_ (New York: Harper and Brothers, 1942), p. 321, n. 4: \"It is difficult, at the present time, to realize how this measure must have struck people who still believed in a state and in a legal system that centered in the institution of private property. For in relaxing the law of conspiracy in respect to peaceful picketing\u2014which practically amounted to legalization of trade-union action implying the threat of force\u2014and in exempting trade-union funds from liability in action for damages _for torts_ \u2014which practically amounted to enacting that trade unions could do not wrong\u2014 this measure in fact resigned to the trade unions part of the authority of the state and granted to them a position of privilege which the formal extension of the exemption to employers' unions was powerless to affect.\" Still more recently the Lord Chief Justice of Northern Ireland said of the same act ( John Clark MacDermott, Baron MacDermott, _Protection from Power under English Law_. The Hamlyn Lectures. [London: Stevens, 1957], p. 174): \"In short, it put trade unionism in the same privileged position which the Crown enjoyed until ten years ago in respect of wrongful acts committed on its behalf.\"\n\n4 Roscoe Pound, _Legal Immunities of Labor Unions_ (Washington, DC: American Enterprise Association, 1957), p. 23, reprinted in Edward Hastings Chamberlin, et al., _Labor Unions and Public Policy_ (Washington, DC: American Enterprise Institute for Public Policy Research, 1958).\n\n5 _Hunt v. Crumboch_ 325 _U.S_. 821, at 831 (1944) (Mr. Justice Robert Jackson's dissent).\n\n6 Ludwig von Mises, _Die Gemeinwirtschaft. Untersuchungen \u00fcber den Sozialismus_ (2nd ed.; Jena: Verlag von Gustav Fischer, 1932), p. 447. [The extended German quotation reads: \"Es gen\u00fcgt festzustellen, da\u00df sie es in den letzten Jahrzehnten \u00fcberall errungen haben, weniger durch ausdr\u00fcckliche gesetzliche Zustimmung als durch stillschweigende Duldung der Beh\u00f6rden und Gerichte.\" (\"It is sufficient to say that in the last decades it has been established everywhere, less by explicit legislative sanction than by the tacit toleration of public authority and the law.\") ( _Socialism_ , Jacques Kahane, trans. [London: Jonathan Cape, 1936]).\u2014Ed.]\n\n7 Few liberal sympathizers of the trade unions would dare to express the obvious truth which a courageous woman from within the British labor movement frankly stated, namely, that \"it is in fact the business of a Union to be anti-social: the members would have a just grievance if their officials and committees ceased to put sectional interests first\" (Barbara Wootton, _Freedom under Planning_ [London: Allen and Unwin, 1945], p. 97). On the flagrant abuses of union power in the United States, which I shall not further consider here, see Sylvester Petro, _Power Unlimited: The Corruption of Union Leadership; A Report on the McClellan Committee Hearings_ (New York: Ronald Press, 1959).\n\n8 In this chapter, more than in almost any other, I shall be able to draw upon a body of opinion that is gradually forming among an increasing number of thoughtful students of these matters\u2014men who in background and interest are at least as sympathetic to the true concerns of the workers as those who in the past have been championing the privileges of the unions. See particularly William Harold Hutt, _The Theory of Collective Bargaining: A History, Analysis and Criticism of the Principal Theories Which Have Sought to Explain the Effects of Trade Unions and Employers Associations Upon the Distribution of the Product of Industry_ (London: P. S. King, 1930), and his _Economists and the Public: A Study of Competition and Opinion_ (London: Jonathan Cape, 1936); Henry Calvert Simons, \"Some Reflections on Syndicalism,\" _Journal of Political Economy_ , 52 (1944): 1\u201325, reprinted in his _Economic Policy for a Free Society_ (Chicago: University of Chicago Press, 1948), pp. 121\u201359; John Thomas Dunlop, _Wage Determination under Trade Unions_ (New York: Macmillan, 1944); Chamber of Commerce, _Economic Institute on Wage Determination and the Economics of Liberalism_ , Joseph H. Ball, moderator [Addresses delivered at an Economic Institute on Wage Determination and Economic Liberalism, held at the Chamber of Commerce, January 11, 1947] (Washington, DC: Chamber of Commerce of the United States, 1947), especially the contributions of Jacob Viner (\"The Role of Costs in a System of Economic Liberalism,\" pp. 15\u201333) and Fritz Machlup (\"Monopolistic Wage Determination as a Part of the General Problem of Monopoly,\" pp. 49\u201382); Leo Wolman, _Industry-wide Bargaining_ (Irvington-on-Hudson, NY: Foundation for Economic Education, 1948); Charles Edward Lindblom, _Unions and Capitalism_ (New Haven: Yale University Press, 1949), cf. the reviews of this book by Aaron Director (\"Book Review of _Unions and Capitalism_ ,\" _University of Chicago Law Review_ , 18 [1950]: 164\u201367), by John Thomas Dunlop (\"Review of _Unions and Capitalism_ ,\" _American Economic Review_ , 40 [1950]: 463\u201368), and by Albert Rees (\"Labor Unions and the Price System,\" _Journal of Political Economy_ , 58 [1950]: 254\u201363); David McCord Wright, ed., _The Impact of the Union: Eight Economic Theorists Evaluate the Labor Union Movement_ [Institute on the Structure of the Labor Market held at the American University, May 12\u201313, 1950] (New York: Harcourt, Brace, 1951), especially the contributions of Milton Friedman (\"Some Comments on the Significance of Labor Unions for Economic Policy,\" pp. 204\u201334) and Gottfried Haberler (\"Wage Policy, Employment, and Economic Stability,\" pp. 34\u201362); Fritz Machlup, _The Political Economy of Monopoly: Business, Labor, and Government Policies_ (Baltimore: Johns Hopkins Press, 1952); Donald Randall Richberg, _Labor Union Monopoly: A Clear and Present Danger_ (Chicago: H. Regnery Co., 1957); Sylvester Petro, _The Labor Policy of a Free Society_ (New York: Ronald Press, 1957); Benjamin Charles Roberts, _Trade Unions in a Free Society_ (London: Institute of Economic Affairs, 1959); and John Davenport's two articles, \"Labor Unions in the Free Society,\" _Fortune_ , April 1959, pp. 132\u201334, 204, 206, 211\u201312; \"Labor and the Law,\" _Fortune_ , May 1959, pp. 142\u201343, 237\u201338, 240, 242, 246.\n\nSee also Edward Hastings Chamberlin, _The Economic Analysis of Labor Union Power_ (Washington: American Enterprise Institute for Public Policy Research, 1958); Philip D. Bradley, _Involuntary Participation in Unionism_ (Washington, DC: American Enterprise Institute for Public Policy Research, 1956); and Gerard Denis Reilly, _States Rights and the Law of Labor Relations_ (Washington, DC: American Enterprise Institute for Public Policy Research, 1955). These three articles, together with Roscoe Pound, _Legal Immunities of Labor Unions_ (see n. 4, above) are reprinted in Edward Hastings Chamberlin, et al., _Labor Unions and Public Policy_ (Washington: American Enterprise Institute for Public Policy Research, 1958).\n\nOn general wage theory and the limits of the powers of the unions see also John Richard Hicks, _The Theory of Wages_ (London: Macmillan, 1932), Richard von Strigl, _Angewandte Lohntheorie: Untersuchungen \u00fcber die wirtschaftlichen Grundlagen der Sozialpolitik_ (Leipzig: F. Deuticke, 1926), and Dunlop, _Wage Determination under Trade Unions_ (cited above).\n\n9 See particularly the works by Henry Calvert Simons [\"Some Reflections on Syndicalism,\" _Journal of Political Economy_ , 52 (1944): 1\u201325, reprinted in his _Economic Policy for a Free Society_ (Chicago: University of Chicago Press, 1948), pp. 121\u201359] and William Harold Hutt [ _The Theory of Collective Bargaining: A History, Analysis, and Criticism of the Principal Theories Which Have Sought to Explain the Effects of Trade Unions and Employers Associations Upon the Distribution of the Product of Industry_ (London: P. S. King, 1930); and _Economists and the Public: A Study of Competition and Opinion_ (London: Jonathan Cape, 1936) cited in n. 8 above]. Whatever limited validity the old argument about the necessity of \"equalizing bargaining power\" by the formation of unions may ever have had, has certainly been destroyed by the modern development of the increasing size and specificity of the employers' investment, on the one hand, and the increasing mobility of labor (made possible by the automobile), on the other.\n\n10 This must be emphasized especially against the argument of Lindblom in _Unions and Capitalism_.\n\n11 Chamberlin, _The Economic Analysis of Labor Union Power_ , pp. 4\u20135, rightly stresses that \"there can be no doubt that one effect of trade union policy . . . is to diminish still further the real income of the really low income groups, including not only the low income wage receivers but also such other elements of society as 'self-employed' and small business men.\"\n\n12 Cf. Fritz Machlup in \"Monopolistic Wage Determination as a Part of the General Problem of Monopoly\" and _The Political Economy of Monopoly: Business, Labor, and Government_.\n\n13 A conspicuous example of this in recent times is the case of the notoriously unorganized domestic servants whose average annual wages (as pointed out by Milton Friedman in \"Some Comments on the Significance of Labor Unions for Economic Policy,\" David McCord Wright, ed., _The Impact of the Union: Eight Economic Theorists Evaluate the Labor Union Movement_ , p. 224) in the United States in 1947 were 2.72 times as high as they had been in 1939, while at the end of the same period the wages of the comprehensively organized steel workers had risen only to 1.98 times the initial level.\n\n14 Cf. Bradley, _Involuntary Participation in Unionism_.\n\n15 Cf. Stephen Paul Sobotka, \"Union Influence on Wages: The Construction Industry,\" _Journal of Political Economy_ , 61 (1953): 127\u201343.\n\n16 It would be difficult to exaggerate the extent to which unions prevent the experimentation with, and gradual introduction of, new arrangements that might be in the mutual interest of employers and employees. For example, it is not at all unlikely that in some industries it would be in the interest of both to agree on \"guaranteed annual wages\" if unions permitted individuals to make a sacrifice in the amount of wages in return for a greater degree of security.\n\n17 To illustrate the nature of much contemporary wage bargaining in the United States, Edward Hastings Chamberlin, in his essay _The Economic Analysis of Labor Union Power_ , pp. 40\u201341, uses an analogy which I cannot better: \"Some perspective may be had on what is involved by imagining an application of the techniques of the labor market in some other field. If A is bargaining with B over the sale of his house, and if A were given the privileges of a modern labor union, he would be able (1) to conspire with all other owners of houses not to make any alternative offers to B, using violence or the threat of violence if necessary to prevent them, (2) to deprive B himself of access to any alternative offers, (3) to surround the house of B and cut off all deliveries of food (except by parcel post), (4) to stop all movement from B's house, so that if he were for instance a doctor he could not sell his services and make a living, and (5) to institute a boycott of B's business. All of these privileges, if he were capable of carrying them out, would no doubt strengthen A's position. But they would not be regarded by anyone as part of 'bargaining'\u2014unless A were a labor union.\"\n\n18 Cf. Petro, _The Labor Policy of a Free Society_ , p. 51: \"Unions can and do serve useful purposes, and they have only barely scratched the surface of their potential utility to employees. When they really get to work on the job of serving employees instead of making such bad names for themselves as they do in coercing and abusing employees, they will have much less difficulty than they presently have in securing and keeping new members. As matters now stand, union insistence upon the closed shop amounts to an admission that unions are really not performing their functions very well.\"\n\n19 Cf. Chester Irving Barnard, \"Functions and Pathology of Status Systems in Formal Organizations,\" in _Industry and Society_ , William Foote Whyte, ed. (New York: McGraw-Hill,1946), pp. 46\u201383; reprinted in Chester Irving Barnard, _Organization and Management: Selected Papers_ (Cambridge, MA: Harvard University Press, 1949), pp. 207\u201344.\n\n20 Cf. Sumner Huber Slichter, _Trade Unions in a Free Society_ [Revision of a paper prepared for a bicentennial conference on the evolution of social institutions at Princeton University, October 8, 1946] (Cambridge, MA: Harvard University Press, 1947), p. 12, where it is argued that such rules \"introduce into industry the equivalent of civil rights, and they greatly enlarge the range of human activities which are governed by rule or law rather than by whim or caprice.\" See also Alvin Ward Gouldner, _Patterns of Industrial Bureaucracy_ (Glencoe, IL: Free Press, 1954), esp. the discussion of \"rule by rule,\" in chap. 9, \"About the Functions of Bureaucratic Rules,\" pp. 157\u201380.\n\n21 See particularly Franz B\u00f6hm, \"Das wirtschaftliche Mitbestimmungsrecht der Arbeiter im Betrieb,\" _Ordo_ , 4 (1951): 21\u2013250, and Goetz Antony Briefs, _Zwischen Kapitalismus und Syndikalismus: die Gewerk schaften am Scheideweg_ (Bern: A. Francke, 1952).\n\n22 See the essays by Jacob Viner, \"The Role of Costs in a System of Economic Liberalism\"; Gottfried Haberler, \"Wage Policy, Employment, and Economic Stability;\" Milton Friedman, \"Some Comments on the Significance of Labor Unions for Economic Policy;\" and the book by Sylvester Petro, _The Labor Policy of a Free Society._\n\n23 Such contracts binding on third parties are equally as objectionable in this field as is the forcing of price-maintenance agreements on non-signers by \"fair-trade\" laws.\n\n24 Such legislation, to be consistent with our principles, should not go beyond declaring certain contracts invalid, which is sufficient for removing all pretext for action to obtain them. It should not, as the title of the \"right-to-work laws\" may suggest, give individuals a claim to a particular job, or even (as some of the laws in force in certain American states do) confer a right to damages for having been denied a particular job, when the denial is not illegal on other grounds. The objections against such provisions are the same as those which apply to \"fair employment practices\" laws.\n\n25 See Arthur Lenhoff, \"The Problem of Compulsory Unionism in Europe,\" _American Journal of Comparative Law_ , 5 (1956): 18\u201343.\n\n26 See Sylvester Petro, _The Labor Policy of a Free Society_ , esp. pp. 235ff. and 282.\n\n27 See the articles by Gottfried Haberler, \"Creeping Inflation Resulting from Wage Increases in Excess of Productivity\" (vol. 1, pp. 137\u201346), and myself, \"Inflation Resulting from Downward Inflexibility of Wages\" (vol. 1, pp. 147\u201352), in Committee for Economic Development, _Problems of United States Economic Development_ (2 vols.; New York: Committee for Economic Development, 1958).\n\n28 Cf. Arthur Joseph Brown, _The Great Inflation, 1939\u20131951_ (London: Oxford University Press, 1955).\n\n29 See John Richard Hicks, \"Economic Foundations of Wage Policy,\" _Economic Journal_ , 65 (1955): esp. 391: \"The world we now live in is one in which the monetary system has become relatively elastic, so that it can accommodate itself to changes in wages, rather than the other way about. Instead of actual wages having to adjust themselves to an equilibrium level, monetary policy adjusts the equilibrium level of money wages so as to make it conform to the actual level. It is hardly an exaggeration to say that instead of being on a Gold Standard, we are on a Labour Standard.\" But see also the same author's later article, \"The Instability of Wages,\" _Three Banks Review_ , 31 (September 1956): 3\u201319.\n\n30 See William Henry Beveridge, _Full Employment in a Free Society_ (London: Allen and Unwin, 1944); Margaret F. W. Joseph and Nicholas Kaldor, _Economic Reconstruction after the War_ (Handbooks for discussion groups, no. 5; London: Published for the Association for Education in Citizenship by the English Universities Press, 1942); Barbara Wootton, _The Social Foundations of Wage Policy: A Study of Contemporary British Wage and Salary Structure_ (London: Allen and Unwin, 1955); and, on the present state of the discussion, Sir Daniel Thompson Jack, \"Is a Wage Policy Desirable and Practicable?\" _Economic Journal_ , 67 (1957): 585\u201390. It seems that some of the supporters of this development imagine that this wage policy will be conducted by \"labor,\" which presumably means by joint action of all unions. This seems neither a probable nor a practicable arrangement. Many groups of workers would rightly object to their relative wages being determined by a majority vote of all workers, and a government permitting such an arrangement would in effect transfer all control of economic policy to the labor unions.\n\n31 See, e.g., Barbara Wootton, _Freedom under Planning_ , p. 101: \"The continual use of terms like 'fair,' however, is quite subjective: no commonly accepted ethical pattern can be implied. The wretched arbitrator, who is charged with the duty of acting 'fairly and impartially' is thus required to show these qualities in circumstances in which they have no meaning; for there can be no such thing as fairness or impartiality except in terms of an accepted code. No one can be impartial in a vacuum. One can only umpire at cricket because there are rules, or at a boxing match so long as certain blows, like those below the belt, are forbidden. Where, therefore, as in wage determinations, there are no rules and no code, the only possible interpretation of impartiality is conservatism.\" See also Orwell de Ruyter F\u00f6nander, _Studies in Australian Law and Relations_ (Melbourne: Melbourne University Press, 1952). Also Kenneth Frederick Walker, _Industrial Relations in Australia_ (Cambridge, MA: Harvard University Press, 1956), p. 362: \"Industrial tribunals, in contrast with ordinary courts, are called upon to decide issues upon which there is not only no defined law, but not even any commonly accepted standards of fairness or justice.\" Cf. also Lady Gertrude Williams, \"The Myth of 'Fair' Wages,\" _Economic Journal_ , 66 (1956): 621\u201334.\n\n32 See Sylvester Petro, _The Labor Policy of a Free Society_ , pp. 262ff., esp. 264: \"I shall show in this chapter that the rule of law does not exist in labor relations; that there a man is _entitled_ in only exceptional cases to a day in court, no matter how unlawfully he has been harmed\"; and p. 272: \"Congress has given the NLRB [National Labor Relations Board] and its General Counsel arbitrary power to deny an injured person a hearing, Congress has closed the federal courts to persons injured by conduct forbidden under federal law. Congress did not, however, prevent unlawfully harmed persons from seeking whatever remedies they might find in state courts. That blow to the ideal that every man is entitled to his day in court was struck by the Supreme Court.\"\n\n33 The Chairman of the English Trades Union Congress, Mr. Charles Geddes, was reported in 1955 to have said: \"I do not believe that the trade union movement of Great Britain can live for very much longer on the basis of compulsion. Must people belong to us or starve, whether they like our policies or not? [Is that to be the future of the movement?] No. I believe the trade union card is an honor to be conferred, not a badge which signifies that you have got to do something whether you like it or not. We want the right to exclude people from our union if necessary and we cannot do that on the basis of 'Belong or starve.'\" [The story is reported in the _Times_ (London), May 21, 1955, p. 5, col. E, in connection with Mr. Geddes's opposition to a closed shop in the Union of Post Office Workers.\u2014Ed.]\n\n## NINETEEN\n\nSOCIAL SECURITY\n\nThe doctrine of the safety net, to catch those who fall, has been made meaningless by the doctrine of fair shares . . . for those of us who are quite able to stand.\n\n\u2014 _The Economist_\n\n1. In the Western world some provision for those threatened by the extremes of indigence or starvation due to circumstances beyond their control has long been accepted as a duty of the community.1 The local arrangements which first supplied this need became inadequate when the growth of large cities and the increased mobility of men dissolved the old neighborhood ties; and (if the responsibility of the local authorities was not to produce obstacles to movement) these services had to be organized nationally and special agencies created to provide them. What we now know as public assistance or relief, which in various forms is provided in all countries, is merely the old poor law adapted to modern conditions. The necessity of some such arrangement in an industrial society is unquestioned\u2014be it only in the interest of those who require protection against acts of desperation on the part of the needy.\n\nIt is probably inevitable that this relief should not long be confined to those who themselves have not been able to provide against such needs (the \"deserving poor,\" as they used to be called) and that the amount of relief now given in a comparatively wealthy society should be more than is absolutely necessary to keep alive and in health. We must also expect that the availability of  this assistance will induce some to neglect such provision against emergencies as they would have been able to make on their own. It seems only logical, then, that those who will have a claim to assistance in circumstances for which they could have made provision should be required to make such provision themselves. Once it becomes the recognized duty of the public to provide for the extreme needs of old age, unemployment, sickness, etc., irrespective of whether the individuals could and ought to have made provision themselves, and particularly once help is assured to such an extent that it is apt to reduce individuals' efforts, it seems an obvious corollary to compel them to insure (or otherwise provide) against those common hazards of life. The justification in this case is not that people should be coerced to do what is in their individual interest but that, by neglecting to make provision, they would become a charge to the public. Similarly, we require motorists to insure against third-party risks, not in their interest but in the interest of others who might be harmed by their action.\n\nFinally, once the state requires everybody to make provisions of a kind which only some had made before, it seems reasonable enough that the state should also assist in the development of appropriate institutions. Since it is the action of the state which makes necessary the speeding-up of developments that would otherwise have proceeded more slowly, the cost of experimenting with and developing new types of institutions may be regarded as no less the responsibility of the public than the cost of research or the dissemination of knowledge in other fields that concern the public interest. The aid given out of the public purse for this purpose should be temporary in nature, a subsidy designed to assist in the acceleration of a development made necessary by a public decision and intended only for a transitional period, terminating when the existing institution has grown and developed to meet the new demand.\n\nUp to this point the justification for the whole apparatus of \"social security\" can probably be accepted by the most consistent defenders of liberty. Though many may think it unwise to go so far, it cannot be said that this would be in conflict with the principles we have stated. Such a program as has been described would involve some coercion, but only coercion intended to forestall greater coercion of the individual in the interest of others; and the argument for it rests as much on the desire of individuals to protect themselves against the consequences of the extreme misery of their fellows as on any wish to force individuals to provide more effectively for their own needs.\n\n2. It is only when the proponents of \"social security\" go a step further that the crucial issues arise. Even at the beginning stage of \"social insurance\" in Germany in the 1880s, individuals were not merely required to make provision against those risks which, if they did not, the state would have to provide for, but were compelled to obtain this protection through a unitary organization  run by the government.2 Although the inspiration for the new type of organization came from the institutions created by the workers on their own initiative, particularly in England, and although where such institutions had also sprung up in Germany\u2014notably in the field of sickness insurance\u2014they were allowed to continue, it was decided that wherever new developments were necessary, as in the provision for old age, industrial accidents, disability, dependents, and unemployment, these should take the form of a unified organization which would be the sole provider of these services and to which all those to be protected had to belong.\n\n\"Social insurance\" thus from the beginning meant not merely compulsory insurance but compulsory membership in a unitary organization controlled by the state. The chief justification for this decision, at one time widely contested but now usually accepted as irrevocable, was the presumed greater efficiency and administrative convenience (i.e., economy) of such a unitary organization. It was often claimed that this was the only way to assure sufficient provision at a single stroke for all those in need.\n\nThere is an element of truth in this argument, but it is not conclusive. It is probably true that, at any given moment, a unified organization designed by the best experts that authority can select will be the most efficient that can be created. But it is not likely to remain so for long if it is made the only starting point for all future developments and if those initially put in charge also become the sole judges of what changes are necessary. It is an error to believe that the best or cheapest way of doing anything can, in the long run, be secured by advance design rather than by the constant re-evaluation of available resources. The principle that all sheltered monopolies become inefficient in the course of time applies here as much as elsewhere.\n\nTrue, if we want at any time to make sure that we achieve as quickly as we can all that is definitely known to be possible, the deliberate organization of all the resources to be devoted to that end is the best way. In the field of social security, to rely on the gradual evolution of suitable institutions would undoubtedly mean that some individual needs which a centralized organization would at once care for might for some time get inadequate attention. To the impatient reformer, who will be satisfied with nothing short of the immediate abolition of all avoidable evils, the creation of a single apparatus with full powers to do what can be done now appears therefore as the only appropriate method. In the long run, however, the price we have to pay for this, even in terms of the achievement in a particular field, may be very high. If we  commit ourselves to a single comprehensive organization because its immediate coverage is greater, we may well prevent the evolution of other organizations whose eventual contribution to welfare might have been greater.3\n\nIf initially it was chiefly efficiency that was stressed in support of the single compulsory organization, there were other considerations clearly also present in the minds of its advocates from the beginning. There are, in fact, two distinct, though connected, aims which a governmental organization with coercive powers can achieve but which are beyond the reach of any agency operating on business lines. A private agency can offer only specific services based on contract, that is, it can provide only for a need which will arise independently of the deliberate action of the beneficiary and which can be ascertained by objective criteria; and it can provide in this manner only for foreseeable needs. However far we extend any system of true insurance, the beneficiary will never get more than satisfaction of a contractual claim\u2014i.e., he will not get whatever he may be judged to need according to his circumstances. A monopolistic government service, on the other hand, can act on the principle of allocation according to need, irrespective of contractual claim. Only such an agency with discretionary powers will be in a position to give individuals whatever they \"ought\" to have, or make them do whatever they \"ought\" to do to achieve a uniform \"social standard.\" It will also be in a position\u2014and this is the second chief point\u2014to redistribute income among persons or groups as seems desirable. Though all insurance involves a pooling of risks, private competitive insurance can never effect a deliberate transfer of income from one previously designated group of people to another.4\n\nSuch a redistribution of income has today become the chief purpose of what is still called social \"insurance\"\u2014a misnomer even in the early days of these schemes. When in 1935 the United States introduced the scheme, the term \"insurance\" was retained\u2014by \"a stroke of promotional genius\"5\u2014 simply to make it more palatable. From the beginning, it had little to do with insurance and has since lost whatever resemblance to insurance it may ever have had. The same is now true of most of those countries which originally started with something more closely akin to insurance.\n\nThough a redistribution of incomes was never the avowed initial purpose of the apparatus of social security, it has now become the actual and admitted aim everywhere.6 No system of monopolistic compulsory insurance has  resisted this transformation into something quite different, an instrument for the compulsory redistribution of income. The ethics of such a system, in which it is not a majority of givers who determine what should be given to the unfortunate few, but a majority of takers who decide what they will take from a wealthier minority, will occupy us in the next chapter. At the moment we are concerned only with the process by which an apparatus originally meant to relieve poverty is generally being turned into a tool of egalitarian redistribution. It is as a means of socializing income, of creating a sort of household state which allocates benefits in money or in kind to those who are thought to be most deserving, that the welfare state has for many become the substitute for old-fashioned socialism. Seen as an alternative to the now discredited method of directly steering production, the technique of the welfare state, which attempts to bring about a \"just distribution\" by handing out income in such proportions and forms as it sees fit, is indeed merely a new method of pursuing the old aims of socialism. The reason why it has come to be so much more widely accepted than the older socialism is that it was at first regularly presented as though it were no more than an efficient method of providing for the specially needy. But the acceptance of this seemingly reasonable proposal for a welfare organization was then interpreted as a commitment to something very different. It was mainly through decisions that seemed to most people to concern minor technical issues, where the essential distinctions were often deliberately obscured by an assiduous and skilful propaganda, that the transformation was effected. It is essential that we become clearly aware of the line that separates a state of affairs in which the community accepts the duty of preventing destitution and of providing a minimum level of welfare from that in which it assumes the power to determine the \"just\" position of everybody and allocates to each what it thinks he deserves. Freedom is critically threatened when the government is given exclusive powers to provide certain services\u2014powers which, in order to achieve its purpose it must use for the discretionary coercion of individuals.7\n\n3. The extreme complexity and consequent incomprehensibility of the social security systems create for democracy a serious problem. It is hardly an exaggeration to say that, though the development of the immense social security apparatus has been a chief factor in the transformation of our economy, it is also the least understood. This is seen not only in the persisting beliefs8 that the individual beneficiary has a moral claim to the services, since he has paid for them, but also in the curious fact that major pieces of social security legislation are sometimes presented to the legislatures in a manner which leaves them no choice but to accept or reject them whole and which precludes any modifications by them.9 And it produces the paradox that the same majority of the people whose assumed inability to choose wisely for themselves is made the pretext for administering a large part of their income for them is in its collective capacity called upon to determine how the individual incomes are to be spent.10\n\nIt is not only the lay members of the general public, however, to whom the intricacies of social security are largely a mystery. The ordinary economist or sociologist or lawyer is today nearly as ignorant of the details of that complex and ever changing system. As a result, the expert has come to dominate in this field as in others.\n\nThe new kind of expert, whom we also find in such fields as labor, agriculture, housing, and education, is an expert in a particular institutional setup. The organizations we have created in these fields have grown so complex that it takes more or less the whole of a person's time to master them. The institutional expert is not necessarily a person who knows all that is needed to enable him to judge the value of the institution, but frequently he is the only one who understands its organization fully and who therefore is indispensable. The reasons why he has become interested in and approves of the particular institution have often little to do with any expert qualifications. But, almost invariably, this new kind of expert has one distinguishing characteristic: he is unhesitatingly in favor of the institutions on which he is expert. This is so not merely because only one who approves of the aims of the institution will have the interest and the patience to master the details, but even more because such an effort would hardly be worth the while of anybody else: the views of anybody who is not prepared to accept the principles of the existing institutions are not likely to be taken seriously and will carry no weight in the discussions determining current policy.11\n\nIt is a fact of considerable importance that, as a result of this development, in more and more fields of policy nearly all the recognized \"experts\" are, almost by definition, persons who are in favor of the principles underlying the policy. This is indeed one of the factors which tend to make so many contemporary developments self-accelerating. The politician who, in recommending some further development of current policies, claims that \"all the  experts favor it,\" is often perfectly honest, because only those who favor the development have become experts in this institutional sense, and the uncommitted economists or lawyers who oppose are not counted as experts. Once the apparatus is established, its future development will be shaped by what those who have chosen to serve it regard as its needs.12\n\n4. It is something of a paradox that the state should today advance its claims for the superiority of the exclusive single-track development by authority in a field that illustrates perhaps more clearly than any other how new institutions emerge not from design but by a gradual evolutionary process. Our modern conception of providing against risks by insurance is not the result of any one's ever having seen the need and devising a rational solution. We are so familiar with the operation of insurance that we are likely to imagine that any intelligent man, after a little reflection, would rapidly discover its principles. In fact, the way in which insurance has evolved is the most telling commentary on the presumption of those who want to confine future evolution to a single channel enforced by authority. It has been well said that \"no man ever aimed at creating marine insurance as social insurance was later created\" and that we owe our present techniques to a gradual growth in which the successive  steps due to \"the uncounted contributions of anonymous or historical individuals have in the end created a work of such perfection that in comparison with the whole all the clever conceptions due to single creative intelligences must seem very primitive.\"13\n\nAre we really so confident that we have achieved the end of all wisdom that, in order to reach more quickly certain now visible goals, we can afford to dispense with the assistance which we received in the past from unplanned development and from our gradual adaptation of old arrangements to new purposes? Significantly enough, in the two main fields which the state threatens to monopolize\u2014the provision for old age and for medical care\u2014we are witnessing the most rapid spontaneous growth of new methods wherever the state has not yet taken complete control, a variety of experiments which are almost certain to produce new answers to current needs, answers which no advance planning can contemplate.14 Is it really likely, then, that in the long run we shall be better off under state monopoly? To make the best available knowledge at any given moment the compulsory standard for all future endeavor may well be the most certain way to prevent new knowledge from emerging.\n\n5. We have seen how the practice of providing out of the public purse for  those in great want, in combination with that of compelling people to provide against these wants so that they should not become a burden on the rest, have in the end produced almost everywhere a third and different system, under which people in certain circumstances, such as sickness or old age, are provided for, irrespective of want and irrespective of whether or not they have made provisions for themselves.15 Under this system all are provided with that standard of welfare which it is thought they should enjoy, irrespective of what they can do for themselves, what personal contributions they have made, or what further contribution they are still capable of making.\n\nThe transition to this third system has generally been effected by first supplementing out of public funds what was obtained through compulsory insurance and then giving to the people as a matter of right what they have only to a small extent paid for. Making these compulsory income transfers a legal right cannot, of course, alter the fact that they can be justified only on the score of special need and that they are therefore still charity. But this character is usually disguised by giving this right to all or nearly all and simply taking out of the pockets of those who are better off a multiple of what they receive. The alleged aversion of the majority to receiving anything they know they have not earned and is given only in consideration of personal need, and their dislike of a \"means test,\" have been made the pretext of so wrapping up the whole arrangement that the individual can no longer know what he has and what he has not paid for.16 This is all part of the endeavor to persuade public opinion, through concealment, to accept a new method of income distribution, which the managers of the new machine seem from the beginning to have regarded as a merely transitional half-measure which must be developed into an apparatus expressly aimed at redistribution.17 This development can be prevented only if, from the outset, the distinction is clearly made between benefits for  which the recipient has fully paid, to which he has therefore a moral as well as a legal right, and those based on need and therefore dependent on proof of need.\n\nIn this connection we must note still another peculiarity of the unitary state machine of social security: its power to use funds raised by compulsory means to make propaganda for an extension of this compulsory system. The fundamental absurdity of a majority taxing itself in order to maintain a propaganda organization aimed at persuading the same majority to go further than it is yet willing should be obvious. Although, at least in the United States, the employment by public agencies of \"public relations\" techniques that are legitimate enough in private business has come to be widely accepted, the propriety of such agencies in a democracy spending public funds on publicity in favor of extending their activities must remain questionable. And in no other field has this become so general a phenomenon, on both a national and an international scale, as in that of social security. It amounts to nothing less than a group of specialists interested in a particular development being allowed to use public funds for the purpose of manipulating public opinion in its favor. The result is that both voters and legislators receive their information almost exclusively from those whose activities they ought to direct. It is difficult to overestimate the extent to which this factor has helped to accelerate development far beyond what the public would otherwise have allowed. Such subsidized propaganda, which is conducted by a single tax-maintained organization, can in no way be compared with competitive advertising. It confers on the organization a power over minds that is in the same class with the powers of a totalitarian state which has the monopoly of the means of supplying information.18\n\nThough in a formal sense the existing social security systems have been created by democratic decisions, one may well doubt whether the majority of the beneficiaries would really approve of them if they were fully aware of what they involved. The burden which they accept by allowing the state to divert a part of their incomes to ends of its choosing is particularly heavy in the relatively poor countries, where increase in material productivity is most urgently needed. Does anyone really believe that the average semiskilled worker in Italy is better off because 44 per cent of his employer's total outlay for his work is handed over to the state or, in concrete figures, because of the 49 cents which his employer pays for an hour of his work, he receives only 27  cents, while 22 cents are spent for him by the state?19 Or that, if the worker understood the situation and were given the choice between this and having his disposable income nearly doubled without social security, he would choose the former? Or that in France, where the figure for all workers amounts to an average of about one-third of total labor cost,20 the percentage is not more than the workers would willingly surrender for the services that the state offers in return? Or that in Germany, where about 20 per cent of the total national income is placed in the hands of the social security administration,21 this is not a compulsory diversion of a share of resources much greater than the people would expressly wish? Can it be seriously denied that most of those people would be better off if the money were handed over to them and they were free to buy their insurance from private concerns?22\n\n6. We can consider more specifically only the chief branches of social security here: the provision for old age, for permanent disablement from other causes, and for loss of the breadwinner of the family; the provision of medical and hospital care; and the protection against loss of income through unemployment.  The numerous other services that are supplied in various countries either as part of those or separately, such as maternity and children's allowances, raise distinct problems in that they are conceived as part of what is called \"population policy,\" an aspect of modern policy which we shall not consider.\n\nThe field in which most countries have committed themselves furthest and which is likely to create the most serious problems is the provision for old age and dependents (except perhaps in Great Britain, where the establishment of a free National Health Service has created problems of a similar magnitude). The problem of the aged is particularly serious, for in most parts of the Western world today it is the fault of governments that the old have been deprived of the means of support that they may have endeavored to provide for themselves. By failing to keep faith and not discharging their duty of maintaining a stable currency, governments everywhere have created a situation in which the generation going into retirement in the third quarter of our century has been robbed of a great part of what they had attempted to put aside for their retirement and in which many more people than there would otherwise have been are undeservedly facing poverty, despite their earlier efforts to avoid such a predicament. It cannot be said too often that inflation is never an unavoidable natural disaster; it is always the result of the weakness or ignorance of those in charge of monetary policy\u2014though the division of responsibility may be spread so wide that nobody is alone to blame. The authorities may have regarded whatever they tried to avert through inflation as greater evils; it is always their choice of policy, however, that brings about inflation.\n\nYet, even if we approach the problem of provision for old age, as we ought to, in full awareness of the special responsibility which governments have incurred, we can but question whether the damage done to one generation (which, in the last resort, shares the responsibility) can justify the imposition upon a nation of a permanent system under which the normal source of income above a certain age is a politically determined pension paid out of current taxation. The whole Western world is, however, tending toward this system, which is bound to produce problems that will dominate future policy to an extent yet uncomprehended by most. In our efforts to remedy one ill, we may well saddle future generations with a burden greater than they will be willing to bear, so tying their hands that, after many efforts to extricate themselves, they will probably in the end do so by an even greater breach of faith than we have committed.\n\nThe problem arises in serious form as soon as government undertakes to secure not only a minimum but an \"adequate\" provision for all the aged, regardless of the individual's need or the contributions made by him. There are two critical steps that are almost invariably taken, once the state assumes the monopoly of providing this protection: first, the protection is granted not  only to those who have through their contributions gained a claim to it, but to those who have not yet had time to do so; and, second, when the pensions are due, they are not paid out of the yield of an additional capital accumulated for the purpose and therefore out of additional income due to the efforts of the beneficiary, but are a transfer of part of the fruits of the work of those currently producing. This holds equally true whether the government nominally builds up a reserve fund and \"invests\" it in government securities (i.e., lends it to itself and in fact currently spends the money) or whether it openly covers current obligations by current taxation.23 (The conceivable, but never practiced, alternative of the government's investing the reserve funds in productive capital would rapidly produce an ever increasing governmental control of the capital of industry.) These two regular consequences of old age pensions' being provided by the state are usually also the chief reasons why this kind of organization is insisted upon.\n\nIt is easy to see how such a complete abandonment of the insurance character of the arrangement, with the recognition of the right of all over a certain age (and all the dependents or incapacitated) to an \"adequate\" income that is currently determined by the majority (of which the beneficiaries form a substantial part), must turn the whole system into a tool of politics, a play ball for vote-catching demagogues. It is vain to believe that any objective standard of justice will set a limit on the extent to which those who have reached the privileged age, even if capable of continued work, can insist on being \"adequately\" maintained by those still at work\u2014who in turn will find consolation only in the thought that at some future date, when they will be proportionally even more numerous and possess correspondingly greater voting strength, they will be in a still better position to make those at work provide for their needs.\n\nAssiduous propaganda has completely obscured the fact that this scheme of adequate pensions for all must mean that many who have at last reached the long-hoped-for time of retirement and who can retire on their savings will nevertheless be the recipients of a gratuity at the expense of those who have not yet reached it, many of whom would at once retire if they were assured of the same income,24 and that in a wealthy society not devastated by inflation it is normal that a large proportion of the retired should be more comfortably off than those still at work. How seriously public opinion has been deliberately misguided in this matter is well illustrated by the often quoted assertion (accepted by the United States Supreme Court) that in the United States in 1935, \"approximately 3 out of 4 persons 65 and older were probably dependent partly or wholly on others for support\"\u2014a statement based on statistics  which explicitly assumed that all property held by old couples was owned by the husbands and that consequently all the wives were \"dependent\"!25\n\nAn inevitable result of this situation, which has become a normal feature in other countries besides the United States, is that at the beginning of every election year there is speculation as to how much social security benefits will again be raised.26 That there is no limit to the demands that will be pressed for is most clearly shown by a recent pronouncement of the British Labour Party to the effect that a really adequate pension \"means the right to go on living in the same neighbourhood, to enjoy the same hobbies and to be able to mix with the same circle of friends.\"27 It will probably not be long before it is argued that, because the retired have more time to spend money, they must be given more than those still at work; and, with the age distribution we are approaching, there is no reason why the majority over forty should not soon attempt to make those of a lower age toil for them. It may be only at that point that the physically stronger will rebel and deprive the old of both their political rights and their legal claims to be maintained.\n\nThe British Labour document just mentioned is significant also because, besides being motivated by the desire to help the aged, it so clearly betrays the wish to make them unable to help themselves and to make them exclusively dependent on government support. An animosity toward all private pension schemes or other similar arrangements pervades it; and what is even more noteworthy is the cool assumption underlying the figures of the proposed plan that prices will double between 1960 and 1980.28 If this is the degree of inflation planned for in advance, the real outcome is indeed likely to be such that most of those who will retire at the end of the century will be dependent on the charity of the younger generation. And ultimately not morals but the fact  that the young supply the police and the army will decide the issue: concentration camps for the aged unable to maintain themselves are likely to be the fate of an old generation whose income is entirely dependent on coercing the young.\n\n7. The provision against sickness presents not only most of the problems which we have already considered but peculiar ones of its own. They result from the fact that the problem of \"need\" cannot be treated as though it were the same for all who satisfy certain objective criteria, such as age: each case of need raises problems of urgency and importance which have to be balanced against the cost of meeting it, problems which must be decided either by the individual or for him by somebody else.\n\nThere is little doubt that the growth of health insurance is a desirable development. And perhaps there is also a case for making it compulsory since many who could thus provide for themselves might otherwise become a public charge. But there are strong arguments against a single scheme of state insurance; and there seems to be an overwhelming case against a free health service for all. From what we have seen of such schemes, it is probable that their inexpediency will become evident in the countries that have adopted them, although political circumstances make it unlikely that they can ever be abandoned, now that they have been adopted. One of the strongest arguments against them is, indeed, that their introduction is the kind of politically irrevocable measure that will have to be continued, whether it proves a mistake or not.\n\nThe case for a free health service is usually based on two fundamental misconceptions. They are, first, the belief that medical needs are usually of an objectively ascertainable character and such that they can and ought to be fully met in every case without regard to economic considerations and second, that this is economically possible because an improved medical service normally results in a restoration of economic effectiveness or earning power and so pays for itself.29 Both contentions mistake the nature of the problem  involved in most decisions concerning the preservation of health and life. There is no objective standard for judging how much care and effort are required in a particular case; also, as medicine advances, it becomes more and more clear that there is no limit to the amount that might profitably be spent in order to do all that is objectively possible.30 Moreover, it is also not true that, in our individual valuation, all that might yet be done to secure health and life has an absolute priority over other needs. As in all other decisions in which we have to deal not with certainties but with probabilities and chances, we constantly take risks and decide on the basis of economic considerations whether a particular precaution is worthwhile, i.e., by balancing the risk against other needs. Even the richest man will normally not do all that medical knowledge makes possible to preserve his health, perhaps because other concerns compete for his time and energy. Somebody must always decide whether an additional effort and additional outlay of resources are called for. The real issue is whether the individual concerned is to have a say and be able, by an additional sacrifice, to get more attention or whether this decision is to be made for him by somebody else. Though we all dislike the fact that we have to balance immaterial values like health and life against material advantages and wish that the choice were unnecessary, we all do have to make the choice because of facts we cannot alter.\n\nThe conception that there is an objectively determinable standard of medical services which can and ought to be provided for all, a conception which underlies the Beveridge scheme and the whole British National Health Service, has no relation to reality.31 In a field that is undergoing as rapid  change as medicine is today, it can, at most, be the bad average standard of service that can be provided equally for all.32 But since in every progressive field what is objectively possible to provide for all depends on what has already been provided for some, the effect of making it too expensive for most to get better than average service, must, before long, be that this average will be lower than it otherwise would be.\n\nThe problems raised by a free health service are made even more difficult by the fact that the progress of medicine tends to increase its efforts not mainly toward restoring working capacity but toward the alleviation of suffering and the prolongation of life; these, of course, cannot be justified on economic but only on humanitarian grounds. Yet, while the task of combating the serious diseases which befall and disable some in manhood is a relatively limited one, the task of slowing down the chronic processes which must bring about the ultimate decay of all of us is unlimited. The latter presents a problem which can, under no conceivable condition, be solved by an unlimited provision of medical facilities and which, therefore, must continue to pre sent a painful choice between competing aims. Under a system of state medicine this choice will have to be imposed by authority upon the individuals. It may seem harsh, but it is probably in the interest of all that under a free system those with full earning capacity should often be rapidly cured of a temporary and not dangerous disablement at the expense of some neglect of the aged and mortally ill. Where systems of state medicine operate, we generally find that those who could be promptly restored to full activity have to wait for long periods because all the hospital facilities are taken up by people who will never again contribute to the needs of the rest.33\n\nThere are so many serious problems raised by the nationalization of medicine that we cannot mention even all the more important ones. But there is one the gravity of which the public has scarcely yet perceived and which is likely to be of the greatest importance. This is the inevitable transformation of doctors, who have been members of a free profession primarily responsible to their patients, into paid servants of the state, officials who are necessarily subject to instruction by authority and who must be released from the duty of secrecy so far as authority is concerned. The most dangerous aspect of the new development may well prove to be that, at a time when the increase in medical knowledge tends to confer more and more power over the minds of men to those who possess it, they should be made dependent on a unified organization under single direction and be guided by the same reasons of state that generally govern policy. A system that gives the indispensable helper of the individual, who is at the same time an agent of the state, an insight into the other's most intimate concerns and creates conditions in which he must reveal this knowledge to a superior and use it for the purposes determined by authority opens frightening prospects. The manner in which state medicine has been used in Russia as an instrument of industrial discipline34 gives us a foretaste of the uses to which such a system can be put.\n\n8. The branch of social security which seemed the most important in the period before the last war, the provision against unemployment, has become relatively unimportant in recent years. Though there can be no question that the prevention of large-scale unemployment is more important than the method of providing for the unemployed, we cannot be certain that we have permanently solved the former problem and that the latter will not again assume major importance. Nor can we be sure that the character of our provision for the unemployed will not prove to be one of the most important factors determining the extent of unemployment.\n\nWe shall again take for granted the availability of a system of public relief which provides a uniform minimum for all instances of proved need, so that no member of the community need be in want of food or shelter. The special problem raised by the unemployed is that of how and by whom any further assistance based on their normal earnings should be provided for them, if at all, and, in particular, whether this need justifies a coercive redistribution of income according to some principle of justice.\n\nThe chief argument in support of a provision in excess of the minimum  that is assured to all is that sudden and unforeseeable changes in the demand for labor occur as a result of circumstances which the worker can neither foresee nor control. There is force in this argument, so far as widespread unemployment during a major depression is concerned. But there are many other causes of unemployment. Recurrent and foreseeable unemployment occurs in most seasonal trades, and here it is clearly in the general interest either that the labor supply be so limited that the seasonal earnings will suffice to maintain the worker during the year, or that the flow of labor be maintained by periodic movements from and to other occupations. There is also the important instance in which unemployment is the direct effect of wages being too high in a particular trade, either because they have been pushed too high by union action or because of a decline in the industry concerned. In both cases the cure of unemployment demands flexibility of wages and mobility of the workers themselves; however, these are both reduced by a system which assures to all the unemployed a certain percentage of the wages they used to earn.\n\nThere is undoubtedly a case for genuine insurance against unemployment wherever practicable, insurance in which the different risks of the various trades are reflected in the premiums paid. Insofar as an industry, because of its peculiar instability, requires a reserve of unemployed most of the time, it is desirable that it induce a sufficient number to hold themselves in readiness by offering wages high enough to compensate for this particular risk. For various reasons, such a system of insurance did not seem immediately practicable in certain occupations (such as agricultural labor and domestic service), and it has been largely for this reason that state schemes for \"insurance\" were adopted,35 schemes which in fact subsidized earnings among such groups out of funds levied from contributions by other workers or by general taxation. When, however, the risk of unemployment peculiar to a particular trade is not covered out of the earnings in that trade but from outside, it means that the labor supply of such trades is subsidized to expand beyond the point which is economically desirable.\n\nThe chief significance of the comprehensive systems of unemployment compensation that have been adopted in all Western countries, however, is that they operate in a labor market dominated by the coercive action of unions and that they have been designed under strong union influence with the aim of assisting the unions in their wage policies. A system in which a worker is regarded as unable to find employment and therefore is entitled to benefit because the workers in the firm or industry in which he seeks employment are on strike necessarily becomes a major support of union wage pressure. Such a system, which relieves the unions of the responsibility for the  unemployment that their policies create and which places on the state the burden not merely of maintaining but of keeping content those who are kept out of jobs by them, can in the long run only make the employment problem more acute.36\n\nThe reasonable solution of these problems in a free society would seem to be that, while the state provides only a uniform minimum for all who are unable to maintain themselves and endeavors to reduce cyclical unemployment as much as possible by an appropriate monetary policy, any further provision required for the maintenance of the accustomed standard should be left to competitive and voluntary efforts. It is in this field that labor unions, once they have been deprived of all coercive power, can make their most beneficial contribution; indeed, they were well on the way to supplying the need when the state largely relieved them of the task.37 But a compulsory scheme of so-called unemployment insurance will always be used to \"correct\" the relative remunerations of different groups, to subsidize the unstable trades at the expense of the stable, and to support wage demands that are irreconcilable with a high level of employment. It is therefore likely in the long run to aggravate the evil it is meant to cure.\n\n9. The difficulties which social insurance systems are facing everywhere and which have become the cause of recurrent discussion of the \"crisis of social security\" are the consequence of the fact that an apparatus designed for the relief of poverty has been turned into an instrument for the redistribution of income, a redistribution supposedly based on some non-existing principle of social justice but in fact determined by _ad hoc_ decisions. It is true, of course, that even the provision of a uniform minimum for all those who cannot provide for themselves involves some redistribution of income. But there is a great deal of difference between the provision of such a minimum for all those who cannot maintain themselves on their earnings in a normally functioning market and a redistribution aiming at a \"just\" remuneration in all the more important occupations\u2014between a redistribution wherein the great majority earning their living agree to give to those unable to do so, and a redistribution wherein a majority takes from a minority because the latter has more. The former preserves the impersonal method of adjustment under which people  can choose their occupation; the latter brings us nearer and nearer to a system under which people will have to be told by authority what to do.\n\nIt seems to be the fate of all unitary, politically directed schemes for the provision of such services to be turned rapidly into instruments for determining the relative incomes of the great majority and thus for controlling economic activity generally.38 The Beveridge plan, which was not conceived by its author as an instrument of income redistribution but was promptly turned into such by the politicians, is merely the best-known instance among many. But while in a free society it is possible to provide a minimum level of welfare for all, such a society is not compatible with sharing out income according to some preconceived notion of justice. The assurance of an equal minimum for all in distress presupposes that this minimum is provided only on proof of need and that nothing which is not paid for by personal contribution is given without such proof. The wholly irrational objection to a \"means test\" for services which are supposed to be based on need has again and again led to the absurd demand that all should be assisted irrespective of need, in order that those who really need help should not feel inferior. It has produced a situation in which generally an attempt is made to assist the needy and at the same time allow them to feel that what they get is the product of their own effort or merit.39\n\nThough the traditional liberal aversion to any discretionary powers of authority may have played some role in making this development possible, it should be noted that the objection against discretionary coercion can really provide no justification for allowing any responsible person an unconditional claim to assistance and the right to be the ultimate judge of his own needs. There can be no principle of justice in a free society that confers a right to \"non-deterrent\" or \"non-discretionary\" support irrespective of proved need. If such claims have been introduced under the disguise of \"social insurance\" and through an admitted deception of the public\u2014a deception which is a source of pride to its authors40\u2014they have certainly nothing to do with the principle of equal justice under the law.\n\nThe hope is now sometimes expressed by liberals that \"the whole Welfare State apparatus must itself be regarded as a passing phenomenon,\"41 a kind of transitional phase of evolution which the general growth of wealth will soon make unnecessary. It must seem doubtful, however, whether there exists such a distinct phase of evolution in which the net effects of those monopolistic institutions are likely to be beneficial, and still more whether, once they have been created, it will ever be politically possible again to get rid of them. In poor countries the burden of the ever growing machinery is likely to slow down considerably the growth of wealth (not to mention its tendency to aggravate the problem of overpopulation) and thus to postpone indefinitely the time when it will be thought unnecessary, while in the richer countries it  will prevent the evolution of alternative institutions that could take over some of its functions.\n\nThere perhaps exists no insuperable obstacle to a gradual transformation of the sickness and unemployment allowance systems into systems of true insurance under which the individuals pay for benefits offered by competing institutions. It is much more difficult to see how it will ever be possible to abandon a system of provision for the aged under which each generation, by paying for the needs of the preceding one, acquires a similar claim to support by the next. It would almost seem as if such a system, once introduced, would have to be continued in perpetuity or allowed to collapse entirely. The introduction of such a system therefore puts a strait jacket on evolution and places on society a steadily growing burden from which it will in all probability again and again attempt to extricate itself by inflation. Neither this outlet, however, nor a deliberate default on obligation already incurred42 can provide the basis for a decent society. Before we can hope to solve these problems sensibly, democracy will have to learn that it must pay for its own follies and that it cannot draw unlimited checks on the future to solve its present problems.\n\nIt has been well said that, while we used to suffer from social evils, we now suffer from the remedies for them.43 The difference is that, while in former times the social evils were gradually disappearing with the growth of wealth, the remedies we have introduced are beginning to threaten the continuance of that growth of wealth on which all future improvement depends. Instead of the \"five giants\" which the welfare state of the Beveridge report was designed to combat, we are now raising new giants which may well prove even greater enemies of a decent way of life. Though we may have speeded up a little the conquest of want, disease, ignorance, squalor, and idleness, we may in the future do worse even in that struggle when the chief dangers will come from inflation, paralyzing taxation, coercive labor unions, an ever increasing dominance of government in education, and a social service bureaucracy with far-reaching arbitrary powers\u2014dangers from which the individual cannot escape by his own efforts and which the momentum of the overextended machinery of government is likely to increase rather than mitigate.\n\n* * *\n\nThe quotation at the head of the chapter is taken from \"Security through Squalor,\" _Economist_ (London), March 15, 1958, p. 918.\n\n1 Compare the classic explanation offered by Nassau William Senior in Lionel Robbins, _The Theory of Economic Policy in English Classical Political Economy_ (London: Macmillan, 1952), p. 140, who quotes Senior [ _Journals Kept in France and England From 1848 to 1852: With a Sketch of the Revolution of 1848_ (London: H. M. King and Co., 1871), pp. 57\u201358]: \"to guarantee _subsistence_ to all\u2014to proclaim that no man whatever his vices or even his crimes, shall die of hunger or cold \u2014is a promise that in the state of civilization of England, or of France, can be performed not merely with safety but with advantage, because the gift of subsistence may be subjected to conditions which no one will voluntarily accept.\" With respect to Germany, there is, as far as I know, not a single state in which there do not exist positive and distinct laws that no one shall starve. In all German jurisdictions of which I am aware, the municipality is required to sustain all those who cannot feed themselves.\n\n2 About Germany's, and especially Prussia's, role as a model for legislation in the area of social security and public education see Sir Ernest Barker, _The Development of Public Services in Western Europe, 1600\u20131930_ (London: Oxford University Press, 1944), pp. 69, 75, 78, 83\u201385.\n\n3 Cf. Alfred Marshall's wise statement on a universal scheme for pensions before the Royal Commission on the Aged Poor (\"Minutes of Evidence Taken Before the Royal Commission on the Aged Poor, June 5, 1893,\" _Official Papers of Alfred Marshall,_ John Maynard Keynes, ed. [London: Macmillan, for the Royal Economic Society, 1926], p. 244): \"My objections to them [universal pension schemes] are that their educational effect, though a true one, would be indirect; that they would be expensive; and that they do not contain, in themselves, the seeds of their own disappearance. I am afraid that, if started, they would tend to become perpetual. I regard all this problem of poverty as a mere passing evil in the progress of man upwards; and I should not like any institution started which did not contain in itself the causes which would make it shrivel up, as the causes of poverty itself shriveled up.\"\n\n4 Cf. Eveline Mabel Burns, \"Private and Social Insurance and the Problem of Social Security,\" _Canadian Welfare_ (February 1, 1953): 5\u201310, and (March 15, 1953): 9\u201313; reprinted in _Analysis of the Social Security System: Hearings Before a Subcommittee on the Committee on Ways and Means, House of Representatives_ (83rd Cong., 1st sess.) No. 38458 (Washington, DC: Government Printing Office, 1954), p. 1475: \"It is no longer a matter of offering each individual a choice as to how much protection he will buy at the range of premiums yielded by the calculations of the actuary. Unlike the private insurer, the government is not restricted by the fear of competition, and can safely offer differential benefits for uniform contributions, or discriminate against certain insured groups. . . . In private insurance, the purpose is to make a profit out of selling people something they want. The essential criterion governing every decision as to terms and conditions is its effect upon the continuing existence of the company. Obviously, if the company is to continue operating in a competitive world, it must offer services that people think it worth while to pay for, and run its affairs in such a way that the guarantees offered will be honoured when due. . . . In social insurance the purpose is different.\" Cf. also the same author's \"Social Insurance in Evolution,\" _American Economic Review_ , 34 (1944): 199\u2013211; and her _Social Security and Public Policy_ (New York: McGraw-Hill, 1956); and Walter Hagenbuch, _Social Economics_ (Cambridge: Cambridge University Press, 1958), p.198.\n\n5 Lewis Meriam and Karl Schlotterbeck, _The Cost and Financing of Social Security_ (Washington, DC: Brookings Institution, 1950), p. 8: \"Adoption of the term 'insurance' by the proponents of social security was a stroke of promotional genius. Thus social security has capitalized on the good will of private insurance and, through the establishment of a reserve fund, has clothed itself with an aura of financial soundness. In fact, however, the soundness of old age and survivors insurance rests not on the Social Security Reserve Fund but on the federal power to tax and to borrow.\"\n\n6 Cf. the statements of Dr. Arthur Joseph Altmeyer, United States commissioner of social security and at one time chairman of the Social Security Board [in _Analysis of the Social Security System: Hearings Before a Subcommittee on the Committee on Ways and Means, House of Representatives_ (83rd Cong., 1st sess.) No. 38458 (Washington, DC: Government Printing Office, 1954), p. 1407]: \"I am not suggesting for a moment that social security be used primarily as a method for redistributing income. That problem has to be attacked frontally and frankly through progressive taxes. . . . But I also am very much in favor of having progressive taxation cover a large part of the cost of social security benefits.\" Similarly M. Pierre Laroque, \"From Social Insurance to Social Security: Evolution in France,\" _International Labour Review_ , 57 (1948): 588: \"The French social security plan was aimed in essence at no other target than to introduce a little more justice into the distribution of the national income\"; and Gerhard Weisser, \"Soziale Sicherheit,\" in _Handw\u00f6rterbuch der Sozialwissenschaften_ , Erwin v. Beckerath, et al., eds. (Stuttgart: Gustav Fischer; T\u00fcbingen: J. C. B. Mohr, 1956), vol. 9, p. 401: \"Ein weiterer Wesenszug der Sicherungssysteme ist unter _kulturellen_ Gesichtspunkten beachtlich. Diese Systeme verwenden Teile des Volkseinkommens _zwangsweise_ zur Deckung eines bestimmten Bedarfs, der f\u00fcr _objektiv_ gegeben gehalten wird.\" [\"A further tendency of social security schemes arises when one considers it in _cultural_ terms. These schemes require, _under compulsion_ , that parts of the national income be used to underwrite the costs of a particular demand that is presented as being an _objective_ need.\"\u2014Ed.] Also Alfred M\u00fcller-Armack, \"Soziale Marktwirtschaft,\" again in the _Handw\u00f6rterbuch der Sozialwissenschaften_ , p. 391: \"Der marktwirtschaftliche Einkommensproze\u00df bietet der Sozialpolitik ein tragf\u00e4higes Fundament f\u00fcr eine staatliche Einkommensumleitung, die in Form von F\u00fcrsorgeleistungen, Renten- und Lastenausgleichszahlungen, Wohnungsbauzusch\u00fcssen, Subventionen usw. die Einkommensverteilung korrigiert.\" [\"The market economy's method of distributing income provides a solid foundation for a social policy that calls for the diversion of income at the hands of the government, which, in the form of support payments, pensions, equalization payments, allowances for the construction of housing, subsidies, and so on, changes its distribution.\"\u2014Ed.]\n\n7 Within the limited space here it is impossible to show in detail how the ambitious aims of the government social security schemes make inevitable the conferment of extensive discretionary and coercive powers on the authorities. Some of these problems are clearly shown in the interesting attempt made by A. D. Watson, _The Principles Which Should Govern the Structure and Provisions of a Scheme of Unemployment Insurance_ , to construct a scheme of private insurance achieving the same ends. On this Eveline Mabel Burns, in \"Private and Social Insurance and the Problems of Social Security\"; reprinted in _Analysis of the Social Security System_ [Hearings before a Subcommittee of the Committee on Ways and Means] House of Representatives (83rd Cong., 1st sess.) No. 38458 (Washington, DC: Government Printing Office, 1954), p. 1474, comments: \"Thus A. D. Watson, the author of what is probably the most sustained and consistent effort to relate social to private insurance, states: 'The transgression of sound insurance principles leads into the wilderness, and once in there may be no return.' Yet, in the attempt to devise the specific provisions of an unemployment insurance law, even this author finds himself forced to fall back upon the principles which run in terms of what is 'reasonable,' 'administratively feasible,' or 'practically fair.' But such words can be interpreted only in relation to some underlying purpose, some specific social environment and set of prevailing social values. The decision as to precisely what is 'reasonable' thus involves a balancing of interests and objectives.\" [Andrew Daniel Watson, at one time Chief Actuary of the Department of Insurance of the Government of Canada, was one of the world's leading authorities on the whole spectrum of social insurance legislation, including unemployment insurance, old-age insurance, and disability insurance. The quotation appears in Watson's _The Principles Which Should Govern the Structure and Provisions of a Scheme of Unemployment Insurance_ (Ottawa: The Unemployment Commission, 1948), p. 11.\u2014Ed.] This difficulty arises only if it is assumed that a scheme of private insurance must provide all that a system of government insurance could. Even with more limited objectives, private competing systems may still be preferable.\n\n8 Ample illustration of the extent to which this erroneous belief has guided policy in the United States is given in Dillard Stokes, _Social Security\u2014Fact and Fancy_ (Chicago: H. Regnery, 1956); see especially the Preface ( pp. vii\u2013x). Similar illustrations could be given for Great Britain.\n\n9 See Meriam and Schlotterbeck, _The Cost and Financing of Social Security_ , pp. 9\u201310, where it is reported of the then latest United States social security bill that it \"passed the House on October 5, 1949, under a rule that did not permit the offering of amendments from the floor or by the minority members of the Ways and Means Committee. The position taken, not without substantial merit, was that H.R. 6000 was too intricate and technical for piecemeal amendment by persons not conversant with all its complexities.\" See also Hans Achinger, _Sozialpolitik als Gesellschaftspolitik: Von der Arbeiterfrage zum Wohlfahrtsstaat_ (Hamburg: Rowohlt, 1958), p. 135: \"Dabei kommt es zu einer Geheimsprache, die es, um ein Beispiel zu nennen, neun Zehnteln der Bundestagsabgeordneten unm\u00f6glich macht, sozialpolitischen Debatten mit Verst\u00e4ndnis zu folgen.\" [\"There thus emerges a secret language which, for example, makes it impossible for nine out of ten members of parliament to understand the debates on social policy.\"\u2014Ed.]\n\n10 Cf. Ludwig von Mises, _Human Action_ (New Haven: Yale University Press, 1949), p. 613 [Liberty Fund edition, vol. 2, p. 617]: \"One may try to justify [such a system of social security] by declaring that the wage earners lack the insight and the moral strength to provide spontaneously for their own future. But then it is not easy to silence the voices of those who ask whether it is not paradoxical to entrust the nation's welfare to the decisions of voters whom the law itself considers incapable of managing their own affairs; whether it is not absurd to make those people supreme in the conduct of government who are manifestly in need of a guardian to prevent them from spending their own income foolishly. Is it reasonable to assign to wards the right to elect their guardians?\"\n\n11 An illuminating illustration of this was provided in a related field by the reception, a few years ago, of a symposium on _The Impact of the Union_ [David McCord Wright, ed., Institute on the Structure of the Labor Market held at the American University, May 12\u201313, 1950 (New York: Harcourt, Brace, 1951)], in which some of the most distinguished economists of our time had taken part [David McCord Wright, John Maurice Clark, Gottfried von Haberler, Frank Hyneman Knight, Kenneth Ewart Boulding, Edward Hastings Chamberlin, Milton Friedman, and Paul Anthony Samuelson]. Although it contained most penetrating discussions of one of our most pressing economic problems, it was treated patronizingly and condescendingly by the \"experts in labor relations.\"\n\n12 There is a further effect of the rule of the expert which deserves brief consideration. Any development which is governed by the successive decisions of a series of different experts working within the same organization is liable to be carried further because it meets with fewer real checks than it would in a competitive world. When the medical experts say that this or that is necessary and \"must\" be done, this is a datum on which the expert in administration bases his decision; and what in consequence he decides to be administratively necessary similarly becomes the datum for the lawyer in drafting the law, and so on. None of these different experts can feel that he is in a position to look at the whole and, in view of the aggregate result, to disregard any of the other experts' \"musts.\" In the past, when things were simpler and the rule was that \"the expert should be on tap but not on top,\" this was the task of the political head of the government department concerned. The complexity of the modern measures makes him almost powerless vis-\u00e0-vis the array of experts. In consequence, the resulting measures are more and more not really the result of co-ordination and mutually adjusted decisions but the product of a summation, in which one decision makes the next inevitable, although this was not foreseen by those who made the first, a process in which nobody has the power to say \"Stop!\" The resulting measures do not rest on the kind of division of labor where at each step a man is free to accept or not to accept as the basis for his decision what some other particular agency offers him. The single scheme that emerges, to which there is no alternative, is determined by the internal necessities of this process, which has little to do with any comprehension of the whole by any one mind.\n\nThere can be little doubt, indeed, that, for tasks of the magnitude of, say, the provision of medical services for a whole nation, the single comprehensive organization is not the most efficient method, even for utilizing all the knowledge already available, and still less the method most conducive to a rapid development and spreading of new knowledge. As in many other fields, the very complexity of the task requires a technique of co-ordination which does not rely on the conscious mastery and control of the parts by a directing authority but is guided by an impersonal mechanism.\n\n13 Josef Schreiegg, _Die Versicherung als geistige Sch\u00f6pfung des Wirtschaftslebens: Eine entwicklungspsychologische Untersuchung_ (Leipzig and Berlin: Teubner, 1934), pp. 59 and 58. [The original German reads: \"Es hat niemals ein Mensch das Ziel gehabt, eine Seeversicherung zu schaffen wie sp\u00e4ter eine Sozialversicherung oder Wasserleitungsversicherung\" and \"Diese Entwicklungsglieder mit ihren unz\u00e4hligen Einf\u00e4llen einzelner anonymer oder historischer Pers\u00f6nlichkeiten haben aber schlie\u00dflich zu einer Vollkommenheit des Sch\u00f6pfungswerkes gef\u00fchrt, gegen\u00fcber der kluge Einfall einer isolierten 'sch\u00f6pferischen Potenz' begreiflicherweise h\u00f6chst primitiv erscheinen mu\u00df.\"\u2014Ed.]\n\n14 On the growth of private pension schemes in Great Britain see particularly the _Report of the Committee on the Economic and Financial Problems of the Provisions for Old Age_ (Sir Thomas Phillips, chairman [London: His Majesty's Stationery Office], Cmd. 9333), and the summary of its findings in Arthur Seldon, _Pensions in a Free Society_ (London: Institute of Economic Affairs, 1957), pp. 4\u20135, where it is stated that \"in 1936, about 1,800,000 were covered in industry and commerce. By 1951 about 6,300,000 people were covered, 3,900,000 in private employment, 2,400,000 in public employment. By 1953\u201354 the total had risen to 7,100,000. It is now ( June 1957) nearing 8,500,000. This includes about 5,500,000 in private industry.\" [According to the British Department for Works and Pensions' _Statistical Summary_ of June 2005 ( p. 4), the number of recipients receiving a state pension in September 2004 was 11,500,000.\u2014Ed.]\n\nThe American developments in this field are even more striking, but the most significant fact here is the rapid development of new types of medical or health insurance (see Chester C. Nash, \"The Contribution of Life Insurance to Social Security in the United States,\" _International Labour Review_ , 72 (1955): 21\u201339, reprinted as _The Contribution of Life Insurance to Social Security in the United States_ [Geneva: International Labour Office, 1955]; and Wesley Glenn Campbell and Rita Ricardo Campbell, _Voluntary Health Insurance in the United States_ [Washington, DC: American Enterprise Association, 1960]; George Bernard de Huszar, ed. _Fundamentals of Voluntary Health Care_ [Caldwell, ID: Caxton Printers, 1962], where sections of the Campbell study are reprinted.)\n\n15 There are, unfortunately, no convenient English equivalents to the German terms describing these stages such as _F\u00fcrsorge_ , _Versicherung_ , and _Versorgung_ ; see Hans Achinger, _Soziale Sicherheit. Eine historisch-soziologische Untersuchung neuer Hilfsmethoden_ (Stuttgart: Frederick Vorwerk, 1953), p. 35, and cf. the same author's contribution to the collective volume, _Neuordnung der sozialen Leistungen: Denkschrift auf Anregung des Herrn Bundeskanzlers_ (2 vols. in 1; Cologne: Graven Verlag, 1955) [Essays by Hans Achinger, Joseph H\u00f6ffner, Hans Muthesius, Ludwig Neund\u00f6rfer], and Karl Heinrich Hansmayer, _Der Weg zum Wohlfahrtsstaat: Wandlungen der Staatst\u00e4tigkeit im Spiegel der Finanzpolitik unseres Jahrhunderts_ (Frankfurt am Main: F. Knapp, 1957).\n\n16 For numerous instances of this see Stokes, _Social Security\u2014Fact and Fancy._\n\n17 Cf. the passages quoted in n. 4, above, and, for the extent to which this aim has in fact been achieved in various countries see Alan Turner Peacock, ed., _Income Redistribution and Social Policy_ (London: Jonathan Cape, 1954). [For theoretical discussions of income redistribution see Harry Gordon Johnson, \"The Macro-Economics of Income Redistribution\" ( pp. 19\u201340), and Denstone Berry, \"Modern Welfare Analysis and the Forms of Income Redistribution\" ( pp. 41\u201351).\u2014Ed.]\n\n18 Apart from much of the publications of the International Labor Organization, the lavishly produced volume [George Roger Nelson, ed.] _Freedom and Welfare: Social Patterns in the Northern Countries of Europe_ [Sponsored by the Ministries of Social Affairs of Denmark, Finland, Iceland, Norway, and Sweden (Copenhagen: Krohns Bogtrykkeri, 1953)], is a conspicuous example of this propaganda on an international scale, the financing of which it would be interesting to inquire into.\n\n19 Bank for International Settlements, _BIS 24th Annual Report for 1953\u20131954_ [1 April 1953\u201331 March 1954] (Basel: Bank for International Settlements, 1954), p. 46. [Forty years later conditions in Italy have remained approximately the same as when these figures were generated. According to the Organization for Economic Cooperation and Development, the average effective tax rate on labor in Italy in the period 1991\u201397 was 47.3% (the comparable figure in France was 47.2% and in Germany 41.4%). See David Carey and Harry Tchilinguirian, _Average Effective Tax Rates on Capital, Labour, and Consumption_ (Economics Department Working Papers No. 258; Paris: Organization for Economic Cooperation and Development, 2000), p. 28.\u2014Ed.]\n\n20 See Laroque, \"From Social Insurance to Social Security: Evolution in France,\" p. 587, and Georges Rottier and Jean Fran\u00e7ois Albert, \"The Social Services and Income Redistribution in France,\" in _Income Redistribution and Social Policy_ , Alan T. Peacock, ed., p. 98. [According to the Organization for Economic Cooperation and Development, social security contributions in France in 2004 constituted 38% of labor costs (comparable figures for Italy and Germany were 31.8% and 34.6%), _Organization for Economic Cooperation and Development, Taxing Wages, 2003\u20132004_ (Paris: Organization for Economic Cooperation and Development, 2004), table 1.2.\u2014Ed.]\n\n21 Weisser, \"Soziale Sicherheit,\" p. 407. The corresponding percentages of the national income devoted in 1950 in the five main English-speaking countries are given by Burns, _Social Security and Public Policy_ , p. 5, as Australia 7.3, Canada 7.99, United Kingdom 11.87, New Zealand 13.18, and United States 5.53. Recent figures for European countries, given in \"Free Trade and Social Security,\" _Planning_ , 405 (1956): 142\u201355, are Germany 20.0, France 16.5, Austria 15.8, Italy 11.3, United Kingdom 11.0, and Switzerland 10.0 per cent. [The following figures show expenditures on social security as a percentage of national income for 2003: Japan 25.63%, Germany 38.77%, Sweden 44.14%, France, 39.77%, United Kingdom 26.66%, United States (1997) 20.50%. The source for these data is the Department of Research Planning and Coordination, National Institute of Population and Social Security Research, Tokyo, Japan, \"The Cost of Social Security in Japan: Fiscal Year 2004.\"\u2014Ed.]\n\n22 In Belgium, I understand, the workers and employed themselves finally put a stop to this development after, in the course of twelve years, the charge had risen from 25 to 41 per cent of wages (see Wilhelm R\u00f6pke, _Jenseits von Angebot und Nachfrage_ [Erlenbach-Zurich and Stuttgart: Rentsch, 1958], p. 295).\n\n23 See Alan Turner Peacock, _The Economics of National Insurance_ (Edinburgh: W. Hodge, 1952).\n\n24 Cf. Stokes, _Social Security\u2014Fact and Fancy_ , pp. 89ff.\n\n25 See Henry D. Allen, \"The Proper Federal Function in Security for the Aged,\" _American Economic Security_ , 10 (1953): 50. [Mr. Justice Benjamin N. Cardozo delivered the majority opinion in _Helvering v. Davis_ 301 _U.S. Reports_ 619 at 643 (1936). Quoting the Social Security Board (\"Economic Insecurity in Old Age,\" 1937, p. 15) Cardozo, on behalf of the Court, noted that \"one fifth of the aged in the United States were receiving old-age assistance, emergency relief, institutional care, employment under the works program, or some other form of aid from public or private funds; two-fifths to one-half were dependent on friends and relatives; one-eighth had some income from earnings; and possibly one-sixth had some savings or property. Approximately three out of four persons 65 or over were probably dependent wholly or partially on others for support.\" The claim is quoted in Allen's essay as an example of the propaganda circulated by the federal government of the need for broader social programs to aid retired Americans.\u2014Ed.]\n\n26 See, for example, the _Wall Street Journal_ , January 2, 1958: \"Social Security: With Elections Near, Chances Grow for New Increase in Benefits. Congress May Hike Monthly Check 5% or 10%,\" pp. 1, 13. The anticipation has proved correct.\n\n27 Labour Party (Great Britain), _National Superannuation: Labour's Policy for Security in Old Age_ (London: Published by the Labour Party, 1957), p. 30.\n\n28 Ibid., pp. 104 and 106.\n\n29 The most characteristic expression of this view will be found in the \"Beveridge Report\" (William Henry Beveridge, _Social Insurance and Allied Services_ [London: His Majesty's Stationery Office, 1942] [Cmd. 6404], secs. 426\u201339), where it is proposed that the national health service should \"ensure that for every citizen there is available whatever medical treatment he requires, in whatever forms he requires it, domiciliary or institutional, general, specialist, or consultant\" (sec. 427, p. 158), and that it should become \"a health service providing full preventive and curative treatment of every kind to every citizen without exceptions, without remuneration limit and without an economic barrier at any point to delay recourse to it\" (sec. 437, p. 162). It may be mentioned here that the annual cost of the proposed service estimated in the Beveridge Report at \u00a3170 million is now running at well over \u00a3450 million. See Brian Abel-Smith and Richard Morris Titmuss, _The Cost of the National Health Service in England and Wales_ (Cambridge: Cambridge University Press, 1956), pp. 58\u201361, and Committee of Enquiry into the Cost of the National Health Service, _Report of the Committee of Enquiry into the Cost of the National Health Service_ (January 1956) [Guillebaud Report] (London: Her Majesty's Stationery Office, 1956), [Cmd. 9663], pp. 120 and 135; cf. also Charles Anthony Raven Crosland, _The Future of Socialism_ (London: Jonthan Cape, 1956), pp. 120 and 135 [In the fiscal year ending 31 March 2006, total outlays on the National Health Service had reached \u00a391.50 billion, or 7.4 percent of GDP (United Kingdom, Office of National Statistics, _Annual Abstract of Statistics_ , \"Summary of Government Expenditure on Health, 1998\u20132006,\" table 10\u201322, p. 160.).\u2014Ed.];also John and Sylvia Jewkes, _The Genesis of the British National Heath Service_ (Oxford: Blackwell, 1961).\n\n30 Cf. Ffrangcon Roberts, _The Cost of Health_ (London: Macmillan, 1952), and Werner Bosch, _Patient, Arzt, Kasse: eine wirtschaftliche Betrachtung \u00fcber Krankheit und Gesundheit_ (Heidelberg: Quelle und Meyer, 1954); see also Ludwig von Mises, _Socialism_ (new ed.; New Haven: Yale University Press, 1951), pt. 5, chap. 34, sec. 3, pp. 476ff. [Liberty Fund edition, p. 431], and earlier German literature quoted there. [Mises refers to two works in German: Erwin Liek, _Der Arzt und seine Sendung: Gedanken eines Ketzers_ (4th ed.; Munich: J. F. Lehmanns Verlag, 1927), p. 54, and Erwin Liek, _Die Sch\u00e4den der sozialen Versicherungen und Wege zur Besserung_ (2nd ed.; Munich: J. F. Lehmanns, 1928), pp. 17 et seq.\u2014Ed.]\n\n31 See Roberts, _The Cost of Health_ , p. 129. Cf. also John Jewkes, \"The Economist and Economic Change,\" in _Economics and Public Policy_ , Arthur Smithies, ed., Brookings Lectures, 1954 (Washington, DC: Brookings Institution, 1955), p. 96: \"The important economic question [about the British National Health Service] was this: if there is a service the demand for which at zero price is almost infinitely great, if no steps are taken to increase the supply, if the cost curve is rising rapidly, if every citizen is guaranteed by law the best possible medical service, and if there is no obvious method of rationing, what will happen? I do not recall any British economist, before the event, asking these simple questions and, after the event, it is the doctors themselves and not primarily the economists, who have raised these questions.\"\n\n32 Cf. Roberts, _The Cost of Health_ , p. 116: \"Our enquiry has shown that medicine, having harnessed itself to science, has acquired the property of perpetual expansion with accelerating velocity; that it feeds upon and is in turn fed by professional ambitions and trade interests; that this process is further accentuated by its own success in that it promotes the prolongation of life in a state of medicated survival rather than cure; and that further factors making for the expansionism of medicine are the raising of the standard of living and the emotion and sentiment inseparable from the contemplation of sickness.\"\n\n33 Roberts, _The Cost of Health_ , p. 136: \"A man of eighty who sustains a fractured hip requires immediate admission to hospital and when he gets there he stays for a long time. On the other hand the person who could be cured, by a brief stay in hospital, of a minor physical defect which nevertheless impairs his working capacity may have to wait a long time.\" Dr. Roberts adds: \"This economic view of the healing art may seem callous. The charge would indeed be justified if our aim were the welfare of the State considered as a superhuman entity; and it need hardly be said that the doctor has no concern with the economic value of his patients. Our aim, however, is the welfare of the members of the State; and since our resources are insufficient to enable us to treat all disease with the efficiency which under more fortunate conditions the advance of science would make possible, we are compelled to reach a just balance between the short-term direct benefits to the individual and the long-term benefits reflected back to the individual.\"\n\n34 See Mark George Field, _Doctor and Patient in Soviet Russia_ (Cambridge, MA: Harvard University Press, 1957).\n\n35 Cf. Eveline Mabel Burns, \"Social Insurance in Evolution,\" pp. 204\u20135.\n\n36 As one of the most careful British students of these matters, John Richard Hicks, pointed out some time ago (\"The Pursuit of Economic Freedom,\" in _What We Defend: Essays in Freedom by Members of the University of Manchester_ , Ernest Fraser Jacob, ed. [London: Oxford University Press, 1942], p. 105): \"One of the reasons why we have high unemployment figures . . . is a direct consequence of our progressive social policy; our unemployment statistics are drawn up in close connection with the administration of unemployment benefit, and the right to that benefit is given very generously.\"\n\n37 See Colin Clark, _Welfare and Taxation_ (Oxford: Catholic Social Guild, 1954), p. 25.\n\n38 Cf. Barbara Wootton, \"The Labour Party and Social Services,\" _Political Quarterly_ , 24 (1953): 65\u201366: \"The future design of the social services waits upon some clearer decision as to what these services are supposed to be for. In particular, are they intended to contribute to a policy of social equality? Or are they just part of the national minimum programme enunciated in the earlier work of the Webbs\u2014measures to secure that nobody starves, or is too poor to see a doctor, or lacks a rudimentary education? It is the answers to these questions which must govern the whole future of our social services.\"\n\n39 It may be useful to recall here the classical doctrine on these matters as expressed by Edmund Burke, _Thoughts and Details on Scarcity_ , in _Works_ , vol. 7, pp. 390\u201391 [Liberty Fund edition, _Selected Works_ , vol. 3, p. 72]: \"Whenever it happens that a man can claim nothing according to the rules of commerce, and the principles of justice, he passes out of that department, and comes within the jurisdiction of mercy.\"\n\nMuch of the best critical analysis of the present tendencies in this field that I know of is contained in an essay by Walter Hagenbuch, \"The Rationale of the Social Services,\" _Lloyds Bank Review_ , n.s., 29 ( July 1953): 9\u201312; partly reproduced in the Epilogue of Walter Hagenbuch, _Social Economics_ (Cambridge: Cambridge University Press, 1958), pp. 298\u2013305, where he contends: \"Without realizing it, we may be drifting into a system in which everyone becomes permanently dependent on the State for certain basic needs and will inevitably become more and more dependent. Not only are the social services no longer self-liquidating; they are self-propagating. . . . There is surely all the difference in the world between a regime in which a few unfortunate people receive occasional and temporary benefits to tide them over their misfortune and one in which a large slice of everybody's income is continually channelled through the State. The absence of any direct links between what the individual puts in and what he takes out, the political situation that must arise when any kind of inequality of distribution is discussed, and the sheer paternalism of it all, suggest a rapid disappearance of that small stream of the national income which does not go through the social service pool, and a move towards the complete State control of all incomes. . . . We may therefore summarize the long term conflict of policy as follows: On the one hand, we may aim at a system of social services which removes poverty by making everybody poor (or everybody rich, according to how you look at it), by giving no benefits unless they are universal, and by socializing the national income. On the other hand, we may aim at a system of social services which removes poverty by raising those below the poverty line above it, by giving selective benefits to groups of people in need, adopting either a means test or the method of insurance categories, and by looking forward to the day when social services will no longer be necessary because the standard of living of even the lowest income groups is above the poverty line.\" See also the same author's \"The Welfare State and Its Finances,\" _Lloyds Bank Review_ , n.s., 49 ( July 1958): 1\u201317; Hans Willgerodt, \"Die Krisis der sozialen Sicherheit und das Lohnproblem,\" _Ordo. Jahrbuch f\u00fcr die Ordnung von Wirtschaft und Gesellschaft_ , 7 (1955): 145\u201387; Hans Achinger, _Soziale Sicherheit_ , pp. 45\u201360; Wilhelm R\u00f6pke, _Jenseits von Angebot und Nachfrage_ , chap. 4, pp. 210\u2013305; as well as Heddy Neumeister, \"Autorit\u00e4re Sozialpolitik,\" in _Ordo_ , 12 (1960 \/ 1961): 187\u2013252.\n\n40 Cf. the essay by Eveline Mabel Burns, \"Private and Social Insurance and the Problems of Social Security,\" reprinted in _Analysis of the Social Security System_ , esp. p. 1478.\n\n41 Peter Wiles, \"Property and Equality,\" in _The Unservile State: Essays in Liberty and Welfare_ , George Watson, ed. (London: Allen and Unwin, 1957), p. 100. Cf. also Elliott Dodds, \"Liberty and Welfare,\" in _The Unservile State_ , George Watson, ed., esp. p. 20: \"It has become evident that a State monopoly in Welfare has certain illiberal consequences, and our conviction is that the time has come to provide, not Welfare merely, but a varied and competitive Welfare.\"\n\n42 As against the proposals for reform in Stokes, _Social Security\u2014Fact and Fancy_ , which would amount to a repudiation of obligations already incurred, it must be said that, however great the temptation to \"wipe the slate clean\" and however great the burden already assumed may appear, this would seem to me a fatal new beginning for any attempt to create more reasonable arrangements.\n\n43 This phrase was used by Mr. Joseph Wood Krutch in an informal talk.\n\n## TWENTY\n\nTAXATION AND REDISTRIBUTION\n\nIt lies in the nature of things that the beginnings are slight, but unless great care is taken, the rates will multiply rapidly and finally will reach a point that no one could have foreseen.\n\n\u2014Franscesco Guicciardini (ca. 1538)\n\n1. In many ways I wish I could omit this chapter. Its argument is directed against beliefs so widely held that it is bound to offend many. Even those who have followed me so far and have perhaps regarded my position as on the whole reasonable are likely to think my views on taxation doctrinaire, extremist, and impractical. Many would probably be willing to restore all the freedom for which I have been pleading, provided that the injustice that they believe this would cause were corrected by appropriate measures of taxation. Redistribution by progressive taxation has come to be almost universally accepted as just. Yet it would be disingenuous to avoid discussing this issue. Moreover, to do so would mean to ignore what seems to me not only the chief source of irresponsibility of democratic action but the crucial issue on which the whole character of future society will depend. Though it may require considerable effort to free one's self of what has become a dogmatic creed in this matter, it should become evident, once the issue has been clearly stated, that it is here that, more than elsewhere, policy has moved toward arbitrariness.\n\nAfter a long period in which there was practically no questioning of the principle of progressive taxation and in which little discussion took place that was new, there has lately appeared a much more critical approach to the problem.1 There is, however, still a great need for a more searching review of the  whole subject. Unfortunately, we can attempt to present only a brief summary of our objections in this chapter.\n\nIt should be said at once that the only progression with which we shall be concerned and which we believe cannot in the long run be reconciled with free institutions is the progression of taxation as a whole, that is, the more than proportionally heavy taxation of the larger incomes when all taxes are considered together. Individual taxes, and especially the income tax, may be graduated for a good reason\u2014that is, so as to compensate for the tendency of many indirect taxes to place a proportionally heavier burden on the smaller incomes. This is the only valid argument in favor of progression. It applies, however, only to particular taxes as part of a given tax structure and cannot be extended to the tax system as a whole. We shall discuss here mainly the effects of a progressive income tax because in recent times it has been used as the main instrument for making taxation as a whole steeply progressive. The question of the appropriate mutual adjustment of the different kinds of taxes within a given system will not concern us.\n\nWe shall also not consider separately the problems which arise from the fact that, though progressive taxation is today the chief instrument of income redistribution, it is not the only method by which the latter can be achieved. It is clearly possible to bring about considerable redistribution under a system of proportional taxation. All that is necessary is to use a substantial part of the revenue to provide services which benefit mainly a particular class or to subsidize it directly. One wonders, however, to what extent the people in the lower-income brackets would be prepared to have their freely spendable income reduced by taxation in return for free services. It is also difficult to see how this method could substantially alter the differentials of the higher-income groups. It might well bring about a considerable transfer of income  from the rich as a class to the poor as a class. But it would not produce that flattening of the top of the income pyramid which is the chief effect of progressive taxation. For the comparatively well-to-do it would probably mean that, while they would all be taxed proportionately on their whole incomes, the differences in the services they receive would be negligible. It is in this class, however, that the changes in relative incomes produced by progressive taxation are most significant. Technical progress, the allocation of resources, incentives, social mobility, competition, and investment\u2014the effects of progressive taxation on all these operate mainly through its effects on this class. Whatever may happen in the future, for the present at any rate, progressive taxation is the chief means of redistributing incomes, and, without it, the scope of such a policy would be very limited.\n\n2. As is true of many similar measures, progressive taxation has assumed its present importance as a result of having been smuggled in under false pretenses. When at the time of the French Revolution and again during the socialist agitation preceding the revolutions of 1848 it was frankly advocated as a means of redistributing incomes, it was decisively rejected. \"One ought to execute the author and not the project,\" was the liberal Turgot's indignant response to some early proposals of this sort.2 When in the 1830s they came to be more widely advocated, J. R. McCulloch expressed the chief objection in the often quoted statement: \"The moment you abandon the cardinal principle of exacting from all individuals the _same proportion of their income_ or _of their property_ , you are at sea without rudder or compass, and there is no amount of injustice and folly you may not commit.\"3 In 1848 Karl Marx and Friedrich  Engels frankly proposed \"a heavy progressive or graduated income tax\" as one of the measures by which, _after_ the first stage of the revolution, \"the proletariat will use its political supremacy to wrest, by degrees, all capital from the bourgeoisie, to centralize all instruments of production in the hands of the state.\" And these measures they described as \"means of despotic inroads on the right of property, and on the conditions of bourgeois production . . . measures . . . which appear economically insufficient and untenable but which, in the course of the movement, outstrip themselves, necessitate further inroads upon the old social order, and are unavoidable as a means of entirely revolutionizing the mode of production.\"4 But the general attitude was still well summed up in A. Thiers's statement that \"proportionality is a principle, but progression is simply hateful arbitrariness,\"5 or John Stuart Mill's description of progression as \"a mild form of robbery.\"6\n\nBut after this first onslaught had been repelled, the agitation for progressive taxation reappeared in a new form. The social reformers, while generally disavowing any desire to alter the distribution of income, began to contend that the total tax burden, assumed to be determined by other considerations, should be distributed according to \"ability to pay\" in order to secure \"equality of sacrifice\" and that this would be best achieved by taxing incomes at progressive rates. Of the numerous arguments advanced in support of this, which still survive in the textbooks on public finance,7 one which looked most scientific carried the day in the end. It requires brief consideration because some still believe that it provides a kind of scientific justification of progressive taxation. Its basic conception is that of the decreasing marginal utility of successive acts of consumption. In spite of, or perhaps because of, its abstract  character, it has had great influence in making scientifically respectable8 what before had been admittedly based on arbitrary postulates.9\n\nModern developments within the field of utility analysis itself have, however, completely destroyed the foundations of this argument. It has lost its validity partly because the belief in the possibility of comparing the utilities to different persons has been generally abandoned10 and partly because it is more than doubtful whether the conception of decreasing marginal utility can legitimately be applied at all to income as a whole, i.e., whether it has meaning if we count as income all the advantages a person derives from the use of his resources. From the now generally accepted view that utility is a purely relative concept (i.e., that we can only say that a thing has greater, equal, or less utility compared with another and that it is meaningless to speak of the degree of utility of a thing by itself ), it follows that we can speak of utility (and of decreasing utility) of income only if we express utility of income in terms of some other desired good, such as leisure (or the avoidance of effort).  But if we were to follow up the implications of the contention that the utility of income in terms of effort is decreasing, we would arrive at curious conclusions. It would, in effect, mean that, as a person's income grows, the incentive in terms of additional income which would be required to induce the same marginal effort would increase. This might lead us to argue for regressive taxation, but certainly not for progressive. It is, however, scarcely worthwhile to follow this line of thought further. There can now be little doubt that the use of utility analysis in the theory of taxation was all a regrettable mistake (in which some of the most distinguished economists of the time shared) and that the sooner we can rid ourselves of the confusion it has caused, the better.\n\n3. Those who advocated progressive taxation during the latter part of the nineteenth century generally stressed that their aim was only to achieve equality of sacrifice and not a redistribution of income; also they generally held that this aim could justify only a \"moderate\" degree of progression and that its \"excessive\" use (as in fifteenth-century Florence, where rates had been pushed up to 50 per cent) was, of course, to be condemned. Though all attempts to supply an objective standard for an appropriate rate of progression failed and though no answer was offered when it was objected that, once the principle was accepted, there would be no assignable limit beyond which progression might not be carried with equal justification, the discussion moved entirely in a context of contemplated rates which made any effect on the distribution of income appear negligible. The suggestion that rates would not stay within these limits was treated as a malicious distortion of the argument, betraying a reprehensible lack of confidence in the wisdom of democratic government.\n\nIt was in Germany, then the leader in \"social reform,\" that the advocates of progressive taxation first overcame the resistance and its modern evolution began. In 1891, Prussia introduced a progressive income tax rising from 0.67 to 4 per cent. In vain did Rudolf von Geist, the venerable leader of the then recently consummated movement for the _Rechtsstaat_ , protest in the Diet that this meant the abandonment of the fundamental principle of equality before the law, \"of the most sacred principle of equality,\" which provided the only barrier against encroachment on property.11 The very smallness of the burden involved in the new schemes made ineffective any attempt to oppose it as a matter of principle.\n\nThough some other Continental countries soon followed Prussia, it took nearly twenty years for the movement to reach the great Anglo-Saxon powers. It was only in 1910 and 1913 that Great Britain and the United States adopted graduated income taxes rising to the then spectacular figures of 8\u00bc and 7 per cent, respectively. Yet within thirty years these figures had risen to 97\u00bd and 91 per cent.\n\nThus in the space of a single generation what nearly all the supporters of progressive taxation had for half a century asserted could not happen came to pass. This change in the absolute rates, of course, completely changed the character of the problem, making it different not merely in degree but in kind. All attempt to justify these rates on the basis of capacity to pay was, in consequence, soon abandoned, and the supporters reverted to the original, but long avoided, justification of progression as a means of bringing about a more just distribution of income.12 It has come to be generally accepted once more that the only ground on which a progressive scale of over-all taxation can be defended is the desirability of changing the distribution of income and that this defense cannot be based on any scientific argument but must be recognized as a frankly political postulate, that is, as an attempt to impose upon society a pattern of distribution determined by majority decision.\n\n4. An explanation of this development that is usually offered is that the great  increase in public expenditure in the last forty years could not have been met without resort to steep progression, or at least that, without it, an intolerable burden would have had to be placed on the poor and that, once the necessity of relieving the poor was admitted, some degree of progression was inevitable. On examination, however, the explanation dissolves into pure myth. Not only is the revenue derived from the high rates levied on large incomes, particularly in the highest brackets, so small compared with the total revenue as to make hardly any difference to the burden borne by the rest; but for a long time after the introduction of progression it was not the poorest who benefited from it but entirely the better-off working class and the lower strata of the middle class who provided the largest number of voters. It would probably be true, on the other hand, to say that the illusion that by means of progressive taxation the burden can be shifted substantially onto the shoulders of the wealthy has been the chief reason why taxation has increased as fast as it has done and that, under the influence of this illusion, the masses have come to accept a much heavier load than they would have done otherwise. The only major result of the policy has been the severe limitation of the incomes that could be earned by the most successful and thereby gratification of the envy of the less-well-off.\n\nHow small is the contribution of progressive tax rates ( particularly of the high punitive rates levied on the largest incomes) to total revenue may be illustrated by a few figures for the United States and for Great Britain. Concerning the former it has been stated (in 1956) that \"the entire progressive super structure produces only about 17 per cent of the total revenue derived from the individual [income] tax\"\u2014or about 8\u00bd per cent of all federal revenue\u2014and that of this, \"half . . . is taken from the taxable income brackets up through $16,000\u2013$18,000, where the tax rate reaches 50 per cent\" while \"the other half comes from the higher brackets and rates.\"13 As for Great Britain, which has an even steeper scale of progression and a greater proportional tax burden, it has been pointed out that \" _all_ surtax (on both earned and unearned incomes) only brings in about 2\u00bd per cent of all public revenue, and that if we collared every \u00a31 of income over \u00a32.000 p. a. [$5.600], we would only net an extra 1\u00bd per cent of revenue. . . . Indeed the massive contribution to income tax and surtax comes from incomes between \u00a3750 p.a. and \u00a33.000 p.a. [$2.100\u2013$8.400]\u2014i.e., just those which begin with foremen and end with managers, or begin with public servants just taking responsibility and end with those at the head of our Civil and other services.\"14\n\nGenerally speaking and in terms of the progressive character of the two tax systems as a whole, it would seem that the contribution made by progression in the two countries is between 2\u00bd and 8\u00bd per cent of total revenue, or between \u00bd and 2 per cent of gross national income. These figures clearly do not suggest that progression is the only method by which the revenue required can be obtained. It seems at least probable (though nobody can speak on this with certainty) that under progressive taxation the gain to revenue is less than the reduction of real income which it causes.\n\nIf the belief that the high rates levied on the rich make an indispensable contribution to total revenue is thus illusory, the claim that progression has served mainly to relieve the poorest classes is belied by what happened in the democracies during the greater part of the period since progression was introduced. Independent studies in the United States, Great Britain, France, and Prussia agree that, as a rule, it was those of modest income who provided the largest number of voters that were let off most lightly, while not only those who had more income but also those who had less carried a much heavier proportional burden of total taxation. The best illustration of this situation, which appears to have been fairly general until the last war, is provided by the results of a detailed study of conditions in Britain, where in 1936\u201337 the total burden of taxation on fully earned income of families with two children was 18 per cent for those with an annual income of \u00a3100 per annum, which then gradually fell to a minimum of 11 per cent at \u00a3350 and then rose again, to reach 19 per cent only at \u00a31,000.15 What these figures (and the similar data for other  countries) clearly show is not only that, once the principle of proportional taxation is abandoned, it is not necessarily those in greatest need but more likely the classes with the greatest voting strength that will profit, but also that all that was obtained by progression could undoubtedly have been obtained by taxing the masses with modest incomes as heavily as the poorest groups.\n\nIt is true, of course, that developments since the last war in Britain, and probably elsewhere, have so increased the progressive character of the income tax as to make the burden of taxation progressive throughout and that, through redistributive expenditure on subsidies and services, the income of the very lowest classes has been increased (so far as these things can be meaningfully measured: what can be shown is always only the cost and not the value of the services rendered) by as much as 22 per cent.16 But the latter development is little dependent on the present high rates of progression but is financed mainly by the contributions of the middle and upper ranges of the middle class.\n\n5. The real reason why all the assurances that progression would remain moderate have proved false and why its development has gone far beyond the most pessimistic prognostications of its opponents17 is that all arguments in support of progression can be used to justify any degree of progression. Its advocates may realize that beyond a certain point the adverse effects on the efficiency of the economic system may become so serious as to make it inexpedient to push it any further. But the argument based on the presumed justice of progression provides for no limitation, as has often been admitted by its supporters, before all incomes above a certain figure are confiscated and those below left untaxed. Unlike proportionality, progression provides no principle which tells us what the relative burden of different persons ought to be. It is no more than a rejection of proportionality in favor of a discrimination against the wealthy without any criterion for limiting the extent of this discrimination.  Because \"there is no ideal rate of progression that can be demonstrated by formula,\"18 it is only the newness of the principle that has prevented its being carried at once to punitive rates. But there is no reason why \"a little more than before\" should not always be represented as just and reasonable.\n\nIt is no slur on democracy, no ignoble distrust of its wisdom, to maintain that, once it embarks upon such a policy, it is bound to go much further than originally intended. This is not to say that \"free and representative government is a failure\"19 or that it must lead to \"a complete distrust in democratic government,\"20 but that democracy has yet to learn that, in order to be just, it must be guided in its action by general principles. What is true of individual action is equally true of collective action, except that a majority is perhaps even less likely to consider explicitly the long-term significance of its decision and therefore is even more in need of guidance by principles. Where, as in the case of progression, the so-called principle adopted is no more than an open invitation to discrimination and, what is worse, an invitation to the majority to discriminate against a minority, the pretended principle of justice must become the pretext for pure arbitrariness.\n\nWhat is required here is a rule which, while still leaving open the possibility of a majority's taxing itself to assist a minority, does not sanction a majority's imposing upon a minority whatever burden it regards as right. That a majority, merely because it is a majority, should be entitled to apply to a minority a rule which does not apply to itself is an infringement of a principle much more fundamental than democracy itself, a principle on which the justification of democracy rests. We have seen before (in chaps. 10 and 14) that if the classifications of persons which the law must employ are to result neither in privilege nor in discrimination, they must rest on distinctions which those inside the group singled out, as well as those outside it, will recognize as relevant.\n\nIt is the great merit of proportional taxation that it provides a rule which is likely to be agreed upon by those who will pay absolutely more and those who will pay absolutely less and which, once accepted, raises no problem of a separate rule applying only to a minority. Even if progressive taxation does not name the individuals to be taxed at a higher rate, it discriminates by introducing a distinction which aims at shifting the burden from those who determine the rates onto others. In no sense can a progressive scale of taxation be regarded as a general rule applicable equally to all\u2014in no sense can it be said  that a tax of 20 per cent on one person's income and a tax of 75 per cent on the larger income of another person are equal. Progression provides no criterion whatever of what is and what is not to be regarded as just. It indicates no halting point for its application, and the \"good judgment\" of the people on which its defenders are usually driven to rely as the only safeguard21 is nothing more than the current state of opinion shaped by past policy.\n\nThat the rates of progression have, in fact, risen as fast as they have done is, however, also due to a special cause which has been operating during the last forty years, namely, inflation. It is now well understood that a rise in aggregate money incomes tends to lift everybody into a higher tax bracket, even though their real income has remained the same. As a result, members of the majorities have found themselves again and again unexpectedly the victims of the discriminatory rates for which they had voted in the belief that they would not be affected.\n\nThis effect of progressive taxation is often represented as a merit, because it tends to make inflation (and deflation) in some measure self-correcting. If a budget deficit is the source of inflation, revenue will rise proportionately more than incomes and may thus close the gap; and if a budget surplus has produced deflation, the resulting fall of incomes will soon bring an even greater reduction in revenue and wipe out the surplus. It is very doubtful, however, whether, with the prevailing bias in favor of inflation, this is really an advantage. Even without this effect, budgetary needs have in the past been the main source of recurrent inflations; and it has been only the knowledge that an inflation, once started, is difficult to stop that in some measure has acted as a deterrent. With a tax system under which inflation produces a more than proportional increase in revenue through a disguised increase in taxes which requires no vote of the legislature, this device may become almost irresistibly tempting.\n\n6. It is sometimes contended that proportional taxation is as arbitrary a principle as progressive taxation and that, apart from an apparently greater mathematical neatness, it has little to recommend it. There are, however, other strong arguments in its favor besides the one we have already mentioned\u2014i.e., that it provides a uniform principle on which people paying different amounts are likely to agree. There also is still much to be said for the old argument that, since almost all economic activity benefits from the basic services of government, these services form a more or less constant ingredient of all we consume and enjoy and that, therefore, a person who commands more of the resources of society will also gain proportionately more from what the government has contributed.\n\nMore important is the observation that proportional taxation leaves the relations between the net remunerations of different kinds of work unchanged.  This is not quite the same as the old maxim, \"No tax is a good tax unless it leaves individuals in the same relative position as it finds them.\"22 It concerns the effect, not on the relations between individual incomes, but on the relations between the net remunerations for particular services performed, and it is this which is the economically relevant factor. It also does not, as might be said of the old maxim, beg the issue by simply postulating that the proportional size of the different incomes should be left unchanged.\n\nThere may be a difference of opinion as to whether the relation between two incomes remains the same when they are reduced by the same amount or in the same proportion. There can be no doubt, however, whether or not the net remunerations for two services which before taxation were equal still stand in the same relation after taxes have been deducted. And this is where the effects of progressive taxation are significantly different from those of proportional taxation. The use that will be made of particular resources depends on the net reward for services, and, if the resources are to be used efficiently, it is important that taxation leave the relative recompenses that will be received for particular services as the market determines them. Progressive taxation alters this relation substantially by making net remuneration for a particular service dependent upon the other earnings of the individual over a certain period, usually a year. If, before taxation, a surgeon gets as much for an operation as an architect for planning a house, or a salesman gets as much for selling ten cars as a photographer for taking forty portraits, the same relation will still hold if proportional taxes are deducted from their receipts. But with progressive taxation of incomes this relation may be greatly changed. Not only may services which before taxation receive the same remuneration bring very different rewards; but a man who receives a relatively large payment for a service may in the end be left with less than another who receives a smaller payment.\n\nThis means that progressive taxation necessarily offends against what is probably the only universally recognized principle of economic justice, that of \"equal pay for equal work.\" If what each of two lawyers will be allowed to retain from his fees for conducting exactly the same kind of case as the other depends on his other earnings during the year\u2014they will, in fact, often derive very different gains from similar efforts. A man who has worked very hard, or for some reason is in greater demand, may receive a much smaller reward for further effort than one who has been idle or less lucky. Indeed, the more  the consumers value a man's services, the less worthwhile will it be for him to exert himself further.\n\nThis effect on incentive, in the usual sense of the term, though important and frequently stressed, is by no means the most harmful effect of progressive taxation. Even here the objection is not so much that people may, as a result, not work as hard as they otherwise would, as it is that the change in the net remunerations for different activities will often divert their energies to activities where they are less useful than they might be. The fact that with progressive taxation the net remuneration for any service will vary with the time rate at which the earning accrues thus becomes a source not only of injustice but also of a misdirection of resources.\n\nThere is no need to dwell here on the familiar and insoluble difficulties which progressive taxation creates in all instances where effort (or outlay) and reward are not approximately coincident in time, i.e., where effort is expended in expectation of a distant and uncertain result\u2014in short, in all instances where human effort takes the form of a long and risky investment. No practicable scheme of averaging incomes can do justice to the author or inventor, the artist or actor, who reaps the rewards of perhaps decades of effort in a few years.23 Nor should it be necessary to elaborate further on the effects of steeply progressive taxation on the willingness to undertake risky capital investments. It is obvious that such taxation discriminates against those risky ventures which are worthwhile only because, in case of success, they will bring a return big enough to compensate for the great risk of total loss. It is more than likely that what truth there is in the alleged \"exhaustion of investment opportunities\" is due largely to a fiscal policy which effectively eliminates a wide range of ventures that private capital might profitably undertake.24\n\nWe must pass rapidly over these harmful effects on incentive and on investment, not because they are unimportant but because they are on the whole well enough known. We shall devote our limited space, then, to other effects which are less understood but at least equally important. Of these, one which perhaps still deserves emphasis is the frequent restriction or reduction of the  division of labor. This effect is particularly noticeable where professional work is not organized on business lines and much of the outlay that in fact would tend to increase a man's productivity is not counted as part of the cost. The tendency to \"do it yourself \" comes to produce the most absurd results when, for instance, a man who wishes to devote himself to more productive activities may have to earn in an hour twenty or even forty times as much in order to be able to pay another whose time is less valuable for an hour's services.25\n\nWe can also only briefly mention the very serious effect of progressive taxation on the supply of savings. If twenty-five years ago the argument that savings were too high and should be reduced may have had some degree of plausibility, few responsible persons today will doubt that, if we are to achieve even part of the tasks we have set ourselves, we want as high a rate of saving as people are prepared to supply. The socialist answer to those who are concerned about this effect on savings is, in fact, no longer that these savings are not needed but that they should be supplied by the community, i.e., out of funds raised from taxation. This, however, can be justified only if the long-term aim is socialism of the old kind, namely, government ownership of the means of production.\n\n7. One of the chief reasons why progressive taxation has come to be so widely accepted is that the great majority of people have come to think of an appropriate _income_ as the only legitimate and socially desirable form of reward. They think of income not as related to the value of the services rendered but as conferring what is regarded as an appropriate status in society. This is shown very clearly in the argument, frequently used in support of progressive taxation, that \"no man is worth \u00a310,000 a year, and, in our present state of poverty, with the great majority of people earning less than \u00a36 a week, only a few very exceptional men deserve to exceed \u00a32,000 a year.\"26 That this contention lacks all foundation and appeals only to emotion and  prejudice will be at once obvious when we see that what it means is that no act that any individual can perform in a year or, for that matter in an hour, can be worth more to society than \u00a310,000 ($28,000). Of course, it can and sometimes will have many times that value. There is no necessary relation between the time an action takes and the benefit that society will derive from it.\n\nThe whole attitude which regards large gains as unnecessary and socially undesirable springs from the state of mind of people who are used to selling their time for a fixed salary or fixed wages and who consequently regard a remuneration of so much per unit of time as the normal thing.27 But though this method of remuneration has become predominant in an increasing number of fields, it is appropriate only where people sell their time to be used at another's direction or at least act on behalf of and in fulfilment of the will of others. It is meaningless for men whose task is to administer resources at their own risk and responsibility and whose main aim is to increase the resources under their control out of their own earnings. For them the control of resources is a condition for practicing their vocation, just as the acquisition of certain skills or of particular knowledge is such a condition in the professions. Profits and losses are mainly a mechanism for redistributing capital among these men rather than a means of providing their current sustenance. The conception that current net receipts are normally intended for current consumption, though natural to the salaried man, is alien to the thinking of those whose aim is to build up a business. Even the conception of income itself is in their case largely an abstraction forced upon them by the income tax. It is no more than an estimate of what, in view of their expectations and plans, they can afford to spend without bringing their prospective power of expenditure below the present level. I doubt whether a society consisting mainly of \"self-employed\" individuals would ever have come to take the concept of income so much for granted as we do or would ever have thought of taxing the earnings from a certain service according to the rate at which they accrued in time.\n\nIt is questionable whether a society which will recognize no reward other than what appears to its majority as an appropriate income, and which does not regard the acquisition of a fortune in a relatively short time as a legitimate form of remuneration for certain kinds of activities, can in the long run preserve a system of private enterprise. Though there may be no difficulty in widely dispersing ownership of well-established enterprises among a large number of small owners and in having them run by managers in a position intermediate between that of an entrepreneur and that of a salaried employee, the building-up of new enterprises is still and probably always will  be done mainly by individuals controlling considerable resources. New developments, as a rule, will still have to be backed by a few persons intimately acquainted with particular opportunities; and it is certainly not to be wished that all future evolution should be dependent on the established financial and industrial corporations.\n\nClosely connected with this problem is the effect of progressive taxation on an aspect of capital formation which is different from that already discussed, namely, the place of formation. It is one of the advantages of a competitive system that successful new ventures are likely for a short time to bring very large profits and that thus the capital needed for development will be formed by the persons who have the best opportunity of using it. The large gains of the successful innovator meant in the past that, having shown the capacity for profitably employing capital in new ventures, he would soon be able to back his judgment with larger means. Much of the individual formation of new capital, since it is offset by capital losses of others, should be realistically seen as part of a continuous process of redistribution of capital among the entrepreneurs. The taxation of such profits, at more or less confiscatory rates, amounts to a heavy tax on that turnover of capital which is part of the driving force of a progressive society.\n\nThe most serious consequence, however, of the discouragement of individual capital formation where there are temporary opportunities for large profits is the restriction of competition. The system tends generally to favor corporate as against individual saving and particularly to strengthen the position of the established corporations against newcomers. It thus assists to create quasi-monopolistic situations. Because taxes today absorb the greater part of the newcomer's \"excessive\" profits, he cannot, as has been well said, \"accumulate capital; he cannot expand his own business; he will never become big business and a match for the vested interests. The old firms do not need to fear his competition: they are sheltered by the tax collector. They may with impunity indulge in routine, they may defy the wishes of the public and become conservative. It is true, the income tax prevents them, too, from accumulating new capital. But what is more important for them is that it prevents the dangerous newcomer from accumulating any capital. They are virtually privileged by the tax system. In this sense progressive taxation checks economic progress and makes for rigidity.\"28\n\nAn even more paradoxical and socially grave effect of progressive taxation is that, though intended to reduce inequality, it in fact helps to perpetuate existing inequalities and eliminates the most important compensation for that inequality which is inevitable in a free-enterprise society. It used to be the redeeming feature of such a system that the rich were not a closed group and that the successful man might in a comparatively short time acquire large resources.29 Today, however, the chances of rising into the class are probably already smaller in some countries, such as Great Britain, than they have been at any time since the beginning of the modern era. One significant effect of this is that the administration of more and more of the world's capital is coming under the control of men who, though they enjoy very large incomes and all the amenities that this secures, have never on their own account and at their personal risk controlled substantial property. Whether this is altogether a gain remains to be seen.\n\nIt is also true that the less possible it becomes for a man to acquire a new fortune, the more must the existing fortunes appear as privileges for which there is no justification. Policy is then certain to aim at taking these fortunes out of private hands, either by the slow process of heavy taxation of inheritance or by the quicker one of outright confiscation. A system based on private property and control of the means of production presupposes that such property and control can be acquired by any successful man. If this is made impossible, even the men who otherwise would have been the most eminent capitalists of the new generation are bound to become the enemies of the established rich.\n\n8. In those countries where taxation of incomes reaches very high rates, greater equality is, in effect, brought about by setting a limit to the net income that anybody can earn. (In Great Britain, during the last war, the largest net income after taxation was approximately \u00a35,000, or $14,000\u2014though this was partly tempered by the fact that capital gains were not treated as income.) We have seen that, considering the insignificant contribution which progressive taxation of the higher brackets makes to revenue, it can be justified only by the view that nobody should command a large income. But what a large income is depends on the views of the particular community and, in the last  resort, on its average wealth. The poorer a country, therefore, the lower will its permissible maximum incomes be, and the more difficult for any of its inhabitants to reach income levels that in wealthier countries are considered only moderate. Where this may lead is illustrated by a recent proposal, only narrowly defeated, of the National Planning Commission of India, according to which a ceiling of $6,300 per annum was to be fixed for all incomes (and a ceiling of $4,300 for salary incomes).30 One need only to think of the same principle being applied to the different regions of any one country, or internationally, to see its implications. These consequences certainly are a commentary on the moral basis of the belief that the majority of a particular group should be entitled to decide on the appropriate limit of incomes and on the wisdom of those who believe that in this manner they will assist the well-being of the masses. Can there be much doubt that poor countries, by preventing individuals from getting rich, will also slow down the general growth of wealth? And does not what applies to the poor countries apply equally to the rich?\n\nIn the last resort, the problem of progressive taxation is, of course, an ethical problem, and in a democracy the real problem is whether the support that the principle now receives would continue if the people fully understood how it operates. It is probable that the practice is based on ideas which most people would not approve if they were stated abstractly. That a majority should be free to impose a discriminatory tax burden on a minority; that, in consequence, equal services should be remunerated differently; and that for a whole class, merely because its incomes are not in line with those of the rest, the normal incentives should be practically made ineffective\u2014all these are principles which cannot be defended on grounds of justice. If, in addition, we consider the waste of energy and effort which progressive taxation in so many ways leads to,31 it should not be impossible to convince reasonable people of  its undesirability. Yet experience in this field shows how rapidly habit blunts the sense of justice and even elevates into a principle what in fact has no better basis than envy.\n\nIf a reasonable system of taxation is to be achieved, people must recognize as a principle that the majority which determines what the total amount of taxation should be must also bear it at the maximum rate. There can be no justified objection to the same majority deciding to grant to an economically weak minority some relief in the form of a proportionately lower taxation. The task of erecting a barrier against abuse of progression is complicated by the fact that, as we have seen, some progression in personal income taxation is probably justified as a way of compensating for the effects of indirect taxation. Is there a principle which has any prospect of being accepted and which would effectively prevent those temptations inherent in progressive taxation from getting out of hand? Personally, I do not believe that setting an upper limit which progression is not to exceed would achieve its purpose. Such a percentage figure would be as arbitrary as the principle of progression and would be as readily altered when the need for additional revenue seemed to require it.\n\nWhat is needed is a principle that will limit the maximum rate of direct taxation in some relation to the total burden of taxation. The most reasonable rule of the kind would seem to be one that fixed the maximum admissible (marginal) rate of direct taxation at that percentage of the total national income which the government takes in taxation. This would mean that if the government took 25 per cent of the national income, 25 per cent would also be the maximum rate of direct taxation of any part of individual incomes. If a national emergency made it necessary to raise this proportion, the maximum admissible rate would be raised to the same figure; and it would be correspondingly reduced when the over-all tax burden was reduced. This would still leave taxation somewhat progressive, since those paying the maximum rate on their incomes would also pay some indirect taxes which could bring their total proportional burden above the national average. Adherence to this principle would have the salutary consequence that every budget would have to be prefaced by an estimate of the share of national income which the government proposed to take as taxes. This percentage would provide the standard rate of direct taxation of incomes which, for the lower incomes, would be reduced in proportion as they were taxed indirectly. The net result would be a slight over-all progression in which, however, the marginal rate of taxation of the largest incomes could never exceed the rate at which incomes were taxed on the average by more than the amount of indirect taxation.\n\n* * *\n\nThe quotation at the head of the chapter is taken from Francesco Guicciardini, \"La decima scalata,\" in _Opere inedite_ , ed. Piero and Luigi Guicciardini (10 vols.; Florence: Barb\u00e0ra, Bianchi e comp., 1867), vol. 10, p. 377. The occasion of this observation (now translated into English as \"Two Discourses on Progressive Taxation of Land Incomes,\" _International Economic Papers_ , _No. 9_. Translations prepared for the International Economics Association [London: Macmillan, 1959], pp. 7\u201319) and the remarkable sixteenth-century discussion of progressive taxation from which it is taken deserve a brief account.\n\nIn the fifteenth century the republic of Florence, which for two hundred years had enjoyed a regime of personal freedom under the law as had not been known since ancient Athens and Rome, fell under the rule of the Medici family, who increasingly gained despotic powers by an appeal to the masses. One of the instruments they used for this purpose was progressive taxation, as Guicciardini describes elsewhere (\"Del reggimento di Firenze,\" _Opere inedite_ , vol. 2, p. 40): \"It is well known how much the nobility and the wealthy were oppressed by Cosimo and in the following time by taxation, and the reason for this, which the Medici never admitted, was that it provided a certain means of destroying in a seemingly legal manner, because they always reserved to themselves the power to knock down arbitrarily anybody they wished.\" [The Italian reads: \"\u00c8 notissimo quante nobilit\u00e0, quante ricchezze furono distrutte da Cosimo, e poi ne' tempi seguenti, colle gravezze; e questa \u00e8 stata la cagione che mai la casa de' Medici non ha consentito, che si truovi uno modo fermo, che le gravezze quasi dalla legge; perch\u00e8 hanno volunto riservarsi sempre la potest\u00e0 di battere co'modi arbitrarii chi gli pareva.\"\u2014Ed.]\n\nWhen at some time in the following century progressive taxation was again advocated, Guicciardini wrote (the date 1538, suggested by Karl Theodor von Eheberg [\"Finanzwissenschaft,\" _Handw\u00f6rterbuch der Staatswissenschaften_ , Johannes Conrad, Wilhelm Lexis, Edgar Leoning, and Ludwig Elster, eds. (3rd ed.; 8 vols.; Jena: Verlag von Gustav Fischer, 1909\u201311), vol. 4, pp. 292\u2013 315, esp. 296], is no more than a conjecture) two brilliant discourses on progressive taxation, one supporting and the second, which evidently represents his opinion, opposing it. They remained in manuscript and were published only in the nineteenth century. His basic objection is (Guicciardini, \"La Decima Scalata,\" vol. 10, p. 368) that \"the equality which we must aim at consists in this, that no citizen can oppress another, and that the citizens are all subject to the laws and the authorities, and that the voice of each who is admissible to the Council counts as much as that of any other. This is the meaning of equality in liberty, and not that all are equal in every respect.\" [\"Ma la egualit\u00e0 che si ricerca consiste in questo, che nessuno cittadino possa opprimere l'altro, che ognuno sia egualmente sottoposto alle leggi e a'Magistrati, e che la fava diognuno che \u00e8 abile a questo Consiglio, abbia tanta autorit\u00e0 l'una quanto l'altra. Cos\u00ec si intende la egualit\u00e0 nelle libert\u00e0, e non generalmente che ognuno sia pari in ogni cosa.\"\u2014Ed.]\n\nHe argues further (vol. 10, p. 372): \"It is not liberty when one part of the community is oppressed and maltreated by the rest, nor is it the end for which we have sought liberty, which was that each should with security be able to preserve his proper state.\" [\"N\u00e8 si chiama libert\u00e0, quando una parte della citt\u00e0 \u00e8 oppressata e male trattata dagli altri, n\u00e8 \u00e8 questo il fine a che furono trovate la libert\u00e0, che fu che ognuno sicuramente potessi conservare il grado suo.\"\u2014Ed.] The advocates of progressive taxation are to him (vol. 10, p. 372) \"suscitatori del popolo, dissipatori della libert\u00e0 e de buoni governi delle republiche.\" [The English reads: \"Troublemakers who squander freedom and compromise the good government of the Republic.\"\u2014Ed.] The main danger he states in the passage quoted at the head of the chapter, which may also be reproduced here in the original Italian: \"Ma a la natura delle cose, i principii cominciano piccoli, ma se l'uomo non avvertisce, moltiplicano presto a scorrono in luogo che poi nessuno \u00e8 a tempo a provvedervi.\" [The English repeats the epigraph.\u2014Ed.] Cf. on this Giuseppe Ricca-Salerno, _Storia delle dottrine finanziarie in Italia: col raffronto delle dottrine forestiere e delle istituzioni e condizioni di fatto_ (2nd ed.; Palermo: A. Reber, 1896), pp. 73\u201376; and Max Grabein, \"Beitr\u00e4ge zur Geschichte der Lehre von der Steuerprogression,\" _Finanz-archiv_ , 12 (1895): 481\u201396.\n\nSee also G\u00fcnter Schm\u00f6lders, _Progression und Regression_ [Forschungsberichte des Landes Nordrhein-Westfalen No. 624] (Cologne: Westdeutscher Verlag, 1958); Gene Erion, \"A Consideration of Some of the Attacks upon the Theory of Progressive Taxation,\" _The Southwestern Social Science Quarterly_ , 38 (March 1958): 344\u201355; Kurt Schmidt, _Die Steuerprogression_ (Basel: Kyklos-Verlag, 1960); Milton Friedman, _Capitalism and Freedom_ (Chicago: University of Chicago Press, 1962).\n\n1 Ten years ago there were only a very few economists left who opposed progressive taxation on principle, among whom Ludwig von Mises, _Human Action_ (New Haven: Yale University Press, 1949), pp. 803ff. [Liberty Fund edition, vol. 3, p. 807], and Harley Leist Lutz, _Guideposts to a Free Economy: A Series of Essays on Enterprise and Government Finance_ (New York: McGraw-Hill,1945), chap. 9 [\"Progressive Taxation,\" pp. 73\u201382] should be specially mentioned. The first of the younger generation who pointed to its dangers seems to have been David McCord Wright, _Democracy and Progress_ (New York: Macmillan, 1948), pp. 94\u2013103. The general reopening of the discussion is due mainly to the careful study of Walter J. Blum and Harry Kalven, Jr., _The Uneasy Case for Progressive Taxation_ (Chicago: University of Chicago Press, 1952), first published in the _University of Chicago Law Review_ , 19 (1952): 417\u2013520. Two earlier discussions of the problem by myself are, \"Die Ungerechtigkeit der Steuerprogression,\" [The Injustice of the Progressive Income Tax] _Schweizer Monatshefte_ , 32 (1952): 508\u201317 [later translated and published as \"The Case Against Progressive Income Taxes,\" _Freeman_ , 4 (December 28, 1953): 229\u201332] and \"Progressive Income Tax Reconsidered,\" in _On Freedom and Free Enterprise: Essays in Honor of Ludwig von Mises_ , Mary Sennholz, ed. [Presented on the Occasion of the Fiftieth Anniversary of his Doctorate, 26 February 1956] (Princeton, NJ: D. Van Nostrand Co., 1956), pp. 265\u201384. A substantial part of the latter has been incorporated in the present chapter. A recently published non-critical but highly instructive history of progressive taxation in Great Britain is Fakkri Shehab, _Progressive Taxation: A Study in the Development of the Progressive Principle in the British Income Tax_ (Oxford: Clarendon Press, 1953).\n\n2 Turgot's marginal note, \"Il faut ex\u00e9cuter 1'auteur, et non le projet, \" is reported by Friedrich von Gentz, \"\u00dcber die H\u00fclfsquellen der franz\u00f6sischen Regierung,\" _Historisches Journal_ , 3 (1799): 138. Gentz himself comments there on progressive taxation: \"Nun ist schon eine jede Abgabe, bei welcher irgend eine andere, als die reine (geometrische) Progression der Eink\u00fcnfte oder des Verm\u00f6gens zum Grunde liegt, jede, die sich auf das Prinzip einer steigenden Progression gr\u00fcndet, nicht viel besser als ein Strassenraub.\" [\"Of course every tax that is not based on a pure (geometric) progression with regard to income or wealth, every tax based on the principle of graduated progression, is not much better than highway robbery.\"\u2014Ed.] (Gentz, of course, here uses \"progression\" with regard to the absolute and not to the proportional amount of the tax.) Wilhelm Gerloff, \"Steuerwirtschaftslehre,\" in _Handbuch der Finanzwissenschaft_ (4 vols.; T\u00fcbingen: J. C. B. Mohr, 1956), vol. 2, p. 288\u201389, regarded my article as containing \"not a single new argument.\"\n\n3 [John Ramsay McCulloch], \"On the Complaints and Proposals Regarding Taxation\" [\"art. 7: 1. Scheme for a Graduated Property Tax; 2. Suggestions for the Relief of the Public Burdens\"], _Edinburgh Review_ , 57 (1833): 164. This early article was largely incorporated into the better-known expanded version in the same author's _A Treatise on the Principles and Practical Influence of Taxation and the Funding System_ (London: Printed for Longman, Brown, Green, and Longmans, 1845), p. 143. See also Robert von Mohl in the Frankfurt National Assembly (Franz Wigard, ed., _Stenographischer Bericht \u00fcber die Verhandlungen der deutschen constituirenden Nationalv ersammlung zu Frankfurt am Main_ [9 vols.; Leipzig: Druck von Breitkopf und Hartel, 1848\u201349], vol. 7, pp. 5107\u201309), who contrasts his remarks with the discussions in the French National Assembly of the same year.\n\n4 See Karl Marx, _Selected Works_ , Vladimir Viktorovich Adoratskii, ed. (2 vols.; London: Lawrence and Wishart, 1942), vol. 1, pp. 227\u201328. The quotation is from the _Communist Manifesto_. As Ludwig von Mises has pointed out, _Planning for Freedom and Other Essays and Addresses_ (South Holland, IL: Libertarian Press, 1952), p. 96 [Liberty Fund edition, p. 86], the words \"necessitate further inroads upon the old social order\" do not occur in the original version of the Communist Manifesto but were inserted by Friedrich Engels in the English translation of 1888.\n\n5 Marie Joseph Louis Adolphe Thiers, _De la propri\u00e9t\u00e9_ (Paris: Paulin, Lheureux et cie., 1848), p. 319: \"La proportionnalit\u00e9 est un principe, mais la progression n'est qu'un odieux arbitraire.\"\n\n6 John Stuart Mill, _Principles of Political Economy, with some Applications to Social Philosophy_ (1st ed.; 2 vols.; London: J. W. Parker, 1848), vol. 2, p. 353. [The first two editions of Mill's _Principles_ (1848 and 1849) read: \"It is partial [i.e., progressive] taxation that is a mild form of robbery\" (bk. 5, chap. 2, sec. 3). It is in the third edition that this passage has been altered to read: \"To tax the larger incomes at a higher percentage than the smaller is to lay a tax on industry and economy; to impose a penalty on people for having worked harder and saved more than their neighbours.\" See the Liberty Fund edition, _Collected Works_ , vol. 3, pp. 810\u201311.\u2014Ed.]\n\n7 For recent surveys of these arguments in favor of progressive taxation see Elmer D. Fagen, \"Recent and Contemporary Theories of Progressive Taxation,\" _Journal of Political Economy_ , 46 (1938): 457\u201398, and Edgard Allix, \"Die Theorie der progressiven Steuer,\" in vol. 4 of _Die Wirtschaftstheorie der Gegenwart_ (4 vols.; Vienna: Springer, 1928), pp. 246\u201362.\n\n8 I remember that my own teacher, Friedrich von Wieser, one of the founders of modern marginal utility analysis and author of the term \"marginal utility\" ( _Grenznutzen_ ), regarded it as one of his main achievements to have provided a scientific basis for just taxation. The author who had in this connection the greatest influence in the English-speaking world was Francis Ysidro Edgeworth, _Papers Relating to Political Economy_ (3 vols.; London: Published on behalf of the Royal Economic Society by Macmillan, 1925), vol. 2, pp. 234\u201370. [The relevant essays on taxation in this volume were all originally published elsewhere: \"The Subjective Element in the First Principles of Taxation,\" _Quarterly Journal of Economics_ , 14 (1910): 459\u201370 ( published in this collection as \"Minimum Sacrifice versus Equal Justice,\" [pp. 234\u201342]); \"Methods of Graduating Taxes on Income and Capital,\" _Economic Journal_ , 29 (1919):138\u201353 ( published in this collection as \"Graduation of Taxes,\" [pp. 243\u201359]); and \"Mathematical Formulae and the Royal Commission on Income Tax,\" _Economic Journal_ , 30 (1920): 398\u2013408 ( published in this collection as \"Formulae for Graduating Taxation,\" [pp. 260\u201370]).\u2014Ed.]\n\n9 As late as 1921, Sir Josiah Stamp (later Lord Stamp) could say ( _The Fundamental Principles of Taxation in the Light of Modern Developments_ [London: Macmillan, 1921], p. 40) that \"it was not until the marginal theory was thoroughly worked out on its psychological side, that progressive taxation obtained a really secure basis in principle.\" Even more recently Tibor Barna, _Redistibution of Incomes through Public Finance in 1937_ (Oxford: Clarendon Press, 1945), p. 5, could still argue that \"given the total national income, satisfaction is maximized with an equal distribution of income. This argument is based, on the one hand, on the law of diminishing marginal utility of income, and, on the other hand, on the assumption (based on the postulates of political democracy rather than economics) that persons with the same income possess the same capacity of enjoyment. In addition, the currently accepted economic doctrine denies that there is virtue in thrift (made so much easier by the existence of high incomes) _so long as there is unemployment_ , and thus the main traditional justification of inequality falls away.\"\n\n10 This conclusion can probably be regarded as firmly established in spite of the ever recurring objection that individually most of us have definite views about whether a given need of one person is greater or smaller than that of another. The fact that we have an opinion about this in no way implies that there is any objective basis for deciding who is right if people differ in their views about the relative importance of different people's needs; nor is there any evidence that they are likely to agree.\n\n11 Prussian Parliament, _Stenographische Berichte \u00fcber die Verhandlungen der durch die Allerh\u00f6chste Verordnung vom 16. Dezember 1890 einberufenen beiden H\u00e4user des Landtages: Haus der Abgeordneten_ (2 vols.; Berlin, 1891), vol. 2, p. 907: \"Die allerheiligsten politischen Grunds\u00e4tze der Gleichheit werden sich aber untreu, wenn wir an die Frage der Progressivsteuer herangehen. Da verleugnet selbst die absolute Demokratie in Hunderttausenden von Stimmen ihre Grunds\u00e4tze, wenn es sich darum handelt, den Reichen sch\u00e4rfer zu treffen.\" [\"The most sacred political principles of equality are betrayed, however, when dealing with the question of progressive taxation. Here extreme democracy contravenes these principles with hundreds of thousands of votes should the issue concern encroaching upon the wealth of the rich.\"\u2014Ed.]\n\n12 See particularly Henry Calvert Simons, _Personal Income Taxation: The Definition of Income as a Problem of Fiscal Policy_ (Chicago: University of Chicago Press, 1938), pp. 17ff. Cf. also Alan Turner Peacock, \"Welfare in the Liberal State,\" in _The Unservile State: Essays in Liberty and Welfare_ , George Watson, ed. (London: Allen and Unwin, 1957), pp. 113\u201330: \"Liberal support for such measures as progressive taxation does not rest on the utilitarian belief that an extra pound is more 'valuable' or will 'afford a greater utility' to a poor man than to a rich man. It rests on a positive dislike of gross inequality.\" [Since Hayek wrote this, substantial changes have been made in the income tax structure of most developed nations, where the top tax rates were, on average, nearly 20% lower in 2004 than they were in the 1970s. This was especially true in the United States and the United Kingdom, where the top rates were 38% and 40%, respectively. Initiated by Margaret Thatcher in Britain and Ronald Reagan in the United States, tax rates were reduced and its progressive features were sharply curtailed. In the United States, the following income tax rates on taxable income for those who filed single returns prevailed in 2007:\n\nFrom $1.00 to 7,825.00: 10%\n\nFrom 7,826.00 to 31,850.00: 15%\n\nFrom 31,851.00 to 77,100.00: 25%\n\nFrom 77,101.00 to 160,850.00: 28%\n\nFrom 160,850.00 to 349,700.00: 33%\n\nOver 349,701.00: 35%\n\nThe rates in the United Kingdom on earned income for 2005\u20132006 were:\n\nFrom \u00a31.00 to 2,230.00: 10%\n\nFrom 2,231.00 to 33,300.00: 22%\n\nOver 33,301.00: 40%\n\nIt should be kept in mind that these rates did not include social security \"contributions,\" which, in the United States and the United Kingdom, were quite substantial, nor, in the United States, did they include state or local income taxes.\u2014Ed.]\n\n13 National Association of Manufacturers, Taxation Committee, _Facing the Issue of Income Tax Discrimination: In Support of the Five-Year Plan for Income Tax Reduction_ (rev. and expanded ed.; New York: National Association of Manufacturers, 1956), p. 14.\n\n14 David Graham Hutton, \"The Dynamics of Progress,\" in _The Unservile State_ , pp. 161\u201386. This seems to be recognized now even in Labour party circles (see, for example, Charles Anthony Raven Crosland, _The Future of Socialism_ [London: Jonthan Cape, 1956], p.190).\n\n15 Cf. George Finlay Shirras and L\u00e1szl\u00f3 Rostas, _The Burden of British Taxation_ (Cambridge: Cambridge University Press, 1943), p. 56. The main results of this investigation are shown in the accompanying table. See also the earlier discussions in Committee on the National Debt and Taxation, _Report_ [chaired by Frederick Henry Smith, Baron Colwyn] (London: His Majesty's Stationery Office, 1927) [Cmd. 2800]; for the United States, Gerhard Colm and Helen Tarasov, _Who Pays the Taxes? (Allocation of Federal, State, and local taxes to consumer income brackets)_ , Temporary National Economic Committee, Monograph No. 3 (Washington, DC: Government Printing Office,1940); and John Hans Adler, \"The Fiscal System: The Distribution of Income and Public Welfare,\" in _Fiscal Policies and the American Economy_ , Kenyon Edward Poole, ed. (New York: Prentice-Hall, 1951), pp. 359\u2013409; for France, see Hubert Brochier, _Finances publiques et redistribution des revenus_ (Paris: Presse de la Fondation nationale des sciences politiques, 1950); and, for an earlier similar result for Prussia, Friedrich Julius Neumann, _Die pers\u00f6nlichen Steuern vom Einkommen: Verbunden mit Ertrags-oder mit Verm\u00f6genssteuern mit besonderer Beziehung auf w\u00fcrttembergische Verh\u00e4ltnisse_ (T\u00fcbingen: Laupp'sche Buchhandlung, 1896).\n\n16 Allan Murray Cartter, _The Redistribution of Income in Postwar Britain: A Study of the Effects of the Central Government Fiscal Program in 1948\u20131949_ (New Haven: Yale University Press, 1955), pp. 54\u201379 [chap. 11, \"The Redistribution of Income in 1948\u20131949\"], esp. p. 57); see also Alan Turner Peacock and P. R. Browning, \"The Social Services in Great Britain and the Redistribution of Income,\" in _Income Redistribution and Social Policy_ , Alan Turner Peacock, ed. (London: Jonathan Cape, 1954), pp. 139\u201377; and Richard Abel Musgrave, John Joseph Carroll, L. D. Cooke, and Lenore Frane, \"Distribution of Tax Payments by Income Groups: A Case Study for 1948,\" _National Tax Journal_ , 4 (1951): 1\u201353.\n\n17 The best-known of these pessimistic prognostications is that by William Edward Hartpole Lecky, _Democracy and Liberty_ (2 vols.; new ed.; New York: Longmans, Green, and, Co. 1899), vol. 1, p. 347 [Liberty Fund edition, vol. 1, p. 293]: \"Highly graduated taxation realises most completely the supreme danger of democracy, creating a state of things in which one class imposes on another burdens which it is not asked to share, and impels the State into vast schemes of extravagance, under the belief that the whole costs will be thrown upon others.\"\n\n18 Royal Commission on Taxation of Profits and Income, _Second Report_ (London: Her Majesty's Stationery Office, 1954) [Cmd. 9105], p. 43 [sec. 142].\n\n19 Justice [Edward Douglass] White in _Knowlton v. Moore_ 178 _U.S._ 41 at 109 (1899), quoted in Blum and Kalven, \"The Uneasy Case for Progressive Taxation,\" _University of Chicago Law Review_ , 19 (1952): 425.\n\n20 Edwin Robert Anderson Seligman, _Progressive Taxation in Theory and Practice_ (2nd ed., completely rev. and enl.; Princeton, NJ: American Economic Association Quarterly, 1908), p. 298.\n\n21 Royal Commission on Taxation of Profits and Income, _Second Report_ , p. 45 [sec. 150].\n\n22 McCulloch in [\"On the Complaints and Proposals Regarding Taxation,\"] _Edinburgh Review_ , p. 162; reprinted in _Treatise on the Principles and Practical Influence of Taxation_ , p. 141. The phrase was later used often and occurs, for example, in Francis Amasa Walker, _Political Economy_ (2nd ed., rev. and enl.; New York: H. Holt and Co., 1887), p. 491. [Walker's formulation differs slightly. He writes: \"No tax is a just tax unless it leaves individuals in the same relative condition in which it finds them.\"\u2014Ed.]\n\n23 See the detailed discussion in the Royal Commission on the Taxation of Profits and Income, _Final Report_ (London: Her Majesty's Stationery Office, 1955) [Cmd. 9474], p. 60 [secs. 186\u2013207, esp. 186]: \"It is inherent in a graduated tax that it should fall with different incidence on the uneven and the even income.\"\n\n24 It deserves notice that the same authors who were loudest in their emphasis on the alleged \"exhaustion of investment opportunities\" are now demanding that \"the effective progressivity of the income tax must be strengthened\" and emphasizing that \"the most important single other grounds confronting American politics today is the issue of progressivity of our income tax\" and seriously contend that \"we are in a situation in which the marginal _tax dollar_ can clearly yield a much higher social utility than the marginal _pay envelope_ dollar\" (Alvin Harvey Hansen, \"The Task of Promoting Economic Growth and Stability.\" Address to the National Planning Association, February 26, 1956 [mimeographed].).\n\n25 This seems to have shaken even an author so firmly convinced of the justice of progressive taxation that he wanted to apply it on an international scale (see James Edward Meade, _Planning and the Price Mechanism: The Liberal-Socialist Solution_ (London: Allen and Unwin, 1948), p. 40: \"Thus a skilled author who is taxed 19s 6d in the \u00a3 [i.e., 97\u00bd per cent] must earn \u00a3200 in order to have the money to pay \u00a35 to get some housework done. He may well decide to do the house-work himself instead of writing. Only if he is forty times more productive in writing than house-work will it be profitable for him to extend the division of labour and to exchange his writing for housework.\"\n\n26 Sir William Arthur Lewis, _The Principles of Economic Planning: A Study Prepared for the Fabian Society_ (London: D. Dobson, 1949), p. 30; the argument appears to have been used first by Leonard Trelawney Hobhouse, _Liberalism_ (Home University Library; London: Williams and Norgate, 1911), pp. 199\u2013201, who suggests that the argument for a supertax is \"a respectful doubt whether any single individual is worth to society by any means as much as some individuals obtain\" and suggests that \"when we come to an income of some \u00a35,000 a year, we approach the limit of the industrial value of the individual.\"\n\n27 Cf. Wright, _Democracy and Progress_ , p. 96: \"It must be remembered that our income-tax laws have been for the most part drawn up and enacted by people on steady salaries for the benefit of people on steady salaries.\"\n\n28 Ludwig von Mises, _Human Action_ , pp. 804\u20135 [Liberty Fund edition, vol. 3, pp. 808\u20139]. Cf. also Colin Clark, _Welfare and Taxation_ (Oxford: Catholic Social Guild, 1954), pp. 51\u201352: \"Many upholders of high taxation are sincere opponents of monopoly; but if taxation were lower and, especially, if undistributed profits were exempted from taxation, many businesses would spring up which would compete actively with the old established monopolies. As a matter of fact, the present excessive rates of taxation are one of the principal reasons for monopolies now being so strong.\" Similarly, Lionel Robbins, \"Notes on Public Finance,\" _Lloyds Bank Review_ , n.s., 38 (October 1955): 10: \"The fact that it has become so difficult to accumulate even a comparatively small fortune must have the most profound effects on the organization of business; and it is by no means clear to me that these results are in the social interest. Must not the inevitable consequence of all this be that it will become more and more difficult for innovation to develop save within the ambit of established corporate enterprise, and that more and more of what accumulation takes place will take place within the large concerns which\u2014 largely as a result of individual enterprise in the past\u2014managed to get started before the ice age descended?\"\n\n29 See Wright, _Democracy and Progress_ , pp. 94\u2013103; cf. also John Keith Butters and John Lintner, _Effects of Federal Taxes on Growing Enterprises_ (Boston: Division of Research, Harvard University Graduate School of Business Administration, 1945).\n\n30 See the report in the _New York Times_ , \"Ceiling on Income Proposed in India,\" January 8, 1956, p. 24.\n\n[New Delhi, India, January 7\u2014India's National Planning Commission has suggested putting a ceiling of the equivalent of $6,300 a year on all incomes.\n\nThat would be the most an Indian could earn from business or dividends. Salaried Indians would be limited to $4,300 a year after taxes.\n\nThe suggestions were contained in the latest memorandum on the second five-year plan. The new plan is supposed to go into operation in April and will be directed toward building up India's industrial strength.\u2014Ed.]\n\n31 Much of the expense-account waste is indirectly a consequence of progressive taxation, since, without it, it would often be in the better interest of a firm so to pay its executives as to induce them to pay their representation expenditure out of their own pockets. Much greater than is commonly understood are also the legal costs caused by progressive taxation; cf. Blum and Kalven, \"The Uneasy Case for Progressive Taxation,\" p. 431: \"It is remarkable how much of the day to day work of the lawyer in the income tax field derives from the simple fact that the tax is progressive. Perhaps the majority of his problems are either caused or aggravated by that fact.\"\n\n## TWENTY-ONE\n\nTHE MONETARY FRAMEWORK\n\nThere is no subtler, no surer means of overturning the existing basis of society than to debauch the currency. The process engages all the hidden forces of economic law on the side of destruction, and does it in a manner which not one man in a million is able to diagnose.\n\n\u2014J. M. Keynes\n\n1. The experience of the last fifty years has taught most people the importance of a stable monetary system. Compared with the preceding century, this period has been one of great monetary disturbances. Governments have assumed a much more active part in controlling money, and this has been as much a cause as a consequence of instability. It is only natural, therefore, that some people should feel it would be better if governments were deprived of their control over monetary policy. Why, it is sometimes asked, should we not rely on the spontaneous forces of the market to supply whatever is needed for a satisfactory medium of exchange as we do in most other respects?\n\nIt is important to be clear at the outset that this is not only politically impracticable today but would probably be undesirable if it were possible. Perhaps, if governments had never interfered, a kind of monetary arrangement might have evolved which would not have required deliberate control; in particular, if men had not come extensively to use credit instruments as money or close substitutes for money, we might have been able to rely on some self-regulating mechanism.1 This choice, however, is now closed to us. We know of no substantially  different alternatives to the credit institutions on which the organization of modern business has come largely to rely; and historical developments have created conditions in which the existence of these institutions makes necessary some deliberate control of the interacting money and credit systems. Moreover, other circumstances which we certainly could not hope to change by merely altering our monetary arrangements make it, for the time being, inevitable that this control should be largely exercised by governments.2\n\nThe three fundamental reasons for this state of affairs are of different degrees of generality and validity. The first refers to all money at all times and explains why changes in the relative supply of money are so much more disturbing than changes in any of the other circumstances that affect prices and production. The second refers to all monetary systems in which the supply of money is closely related to credit\u2014the kind on which all modern economic life rests. The third refers to the present volume of government expenditure and thus to a circumstance which we may hope to change eventually but which we must accept, for the time being, in all decisions about monetary policy.\n\nThe first of these facts makes money a kind of loose joint in the otherwise self-steering mechanism of the market, a loose joint that can sufficiently interfere with the adjusting mechanism to cause recurrent misdirections of production unless these effects are anticipated and deliberately counteracted. The reason for this is that money, unlike ordinary commodities, serves not by being used up but by being handed on. The consequence of this is that the effects of a change in the supply of money (or in the demand for it) do not directly lead to a new equilibrium. Monetary changes are, in a peculiar sense, \"selfreversing.\" If, for example, an addition to the stock of money is first spent on a particular commodity or service, it not merely creates a new demand which in its nature is temporary and passing, but also sets up a train of further effects which will reverse the effects of the initial increase in demand. Those who first received the money will in turn spend it on other things. Like the ripples on a pool when a pebble has been thrown into it, the increase in demand will spread itself throughout the whole economic system, at each point temporarily altering relative prices in a way which will persist as long as the quantity  of money continues to increase but which will be reversed when the increase comes to an end. Exactly the same applies if any part of the stock of money is destroyed, or even if people start holding larger or smaller amounts of cash, in relation to their receipts and outlay, than they normally do; each change of this sort will give rise to a succession of changes in demand which do not correspond to a change in the underlying real factors and which will therefore cause changes in prices and production which upset the equilibrium between demand and supply.3\n\nIf, for this reason, changes in the supply of money are particularly disturbing, the supply of money as we know it is also particularly apt to change in a harmful manner. What is important is that the rate at which money is spent should not fluctuate unduly. This means that when at any time people change their minds about how much cash they want to hold in proportion to the payments they make (or, as the economist calls it, they decide to be more or less liquid), the quantity of money should be changed correspondingly. However we define \"cash,\" people's propensity to hold part of their resources in this form is subject to considerable fluctuation both over short and over long periods, and various spontaneous developments (such as, for instance, the credit card and the travelers' check) are likely to affect it profoundly. No automatic regulation of the supply of money is likely to bring about the desirable adjustments before such changes in the demand for money or in the supply of substitutes for it have had a strong and harmful effect on prices and employment.\n\nStill worse, under all modern monetary systems, not only will the supply of money not adjust itself to such changes in demand, but it will tend to change in the opposite direction. Whenever claims for money come to serve in the place of money\u2014and it is difficult to see how this can be prevented\u2014the supply of such substitutes for money tends to be \"perversely elastic.\"4 This is a result of the simple fact that the same considerations which will make people want to hold more money will also make those who supply claims for money by lending produce fewer such claims, and vice versa. The familiar fact that, when everybody else wants to be more liquid, the banks for the same reasons will also wish to be more liquid and therefore supply less credit, is merely one instance of a general tendency inherent in most forms of credit.\n\nThese spontaneous fluctuations in the supply of money can be prevented only if somebody has the power to change deliberately the supply of some generally accepted medium of exchange in the opposite direction. This is a  function which it has generally been found necessary to entrust to a single national institution, in the past the central banks. Even countries like the United States, which long resisted the establishment of such an institution, found in the end that, if recurrent panics were to be avoided, a system which made extensive use of bank credit must rest on such a central agency which is always able to provide cash and which, through this control of the supply of cash, is able to influence the total supply of credit.\n\nThere are strong and probably still valid reasons which make it desirable that these institutions should be independent of government and its financial policy as much as possible. Here, however, we come to the third point to which we have referred\u2014a historical development which, though not strictly irrevocable, we must accept for the immediate future. A monetary policy independent of financial policy is possible so long as government expenditure constitutes a comparatively small part of all payments and so long as the government debt (and particularly its short-term debt) constitutes only a small part of all credit instruments.5 Today this condition no longer exists. In consequence, an effective monetary policy can be conducted only in co-ordination with the financial policy of government. Co-ordination in this respect, however, inevitably means that whatever nominally independent monetary au thorities still exist have in fact to adjust their policy to that of the government. The latter, whether we like it or not, thus necessarily becomes the determining factor.\n\nThis more effective control over monetary conditions by government which, it would seem, can thus be achieved is welcomed by some people. Whether we have really been placed in a better position to pursue a desirable monetary policy we shall have to consider later. For the moment the important fact is that, as long as government expenditure constitutes as large a part of the national income as it now does everywhere, we must accept the fact that government will necessarily dominate monetary policy and that the only way in which we could alter this would be to reduce government expenditure greatly.\n\n2. With government in control of monetary policy, the chief threat in this field has become inflation. Governments everywhere and at all times have been the chief cause of the depreciation of the currency. Though there have been occasional prolonged falls in the value of a metallic money, the major inflations of the past have been the result of governments' either diminishing the coin or issuing excessive quantities of paper money. It is possible that the present generation is more on its guard against those cruder ways in which currencies were destroyed when governments paid their way by issuing paper money. The same can be done nowadays, however, by subtler procedures that the public is less likely to notice.\n\nWe have seen how every one of the chief features of the welfare state which we have considered tends to encourage inflation. We have seen how wage pressures from the labor unions, combined with the current full-employment policies, work in this manner and how the heavy financial burden which governments are assuming through old age pensions are likely to lead them to repeated attempts to lighten them by reducing the value of money. We should also note here, although this may not necessarily be connected, that governments seem invariably to have resorted to inflation to lighten the burden of their fixed obligations whenever the share of national income which they took exceeded about 25 per cent.6 And we have also seen that, because under a system of progressive taxation inflation tends to increase tax revenue proportionately more than incomes, the temptation to resort to inflation becomes very great.\n\nIf it is true, however, that the institutions of the welfare state tend to favor inflation, it is even more true that it was the effects of inflation which strengthened the demand for welfare measures. This is true not only of some of those measures we have already considered but also of many others which we have yet to examine or can merely mention here, such as rent restrictions on dwellings, food subsidies, and all kinds of controls of prices and expenditures. The extent to which the effects of inflation have in recent times provided the chief arguments for an extension of government controls is too well known to need more illustration. But the extent to which, for over forty years now, developments throughout the whole world have been determined by an unprecedented inflationary trend is not sufficiently understood. It is perhaps best seen in the influence that it has had on the efforts of the generation whose working life covers that period to provide for their old age.\n\nIt will help us to see what inflation has done to the savings of the generation  now on the point of retiring if we look at the results of a little statistical inquiry.7 The aim of the inquiry was to determine what would be the present value in various countries of the accumulated savings of a person who for a period of forty-five years, from 1913 to 1958, had put aside every year the equivalent in money of the same real value and invested it at a fixed rate of interest of 4 per cent. This corresponds approximately to the return which the small saver in Western countries could have obtained from the kind of investment accessible to him, whether its actual form was a savings account, government bonds, or life insurance. We shall represent as 100 the amount that the saver would have possessed at the end of the period if the value of money had remained constant. What part of this real value would such a saver actually have had in 1958?\n\nIt seems that there is only one country in the world, namely, Switzerland, where the amount would have been as much as 70 per cent. The saver in the United States and Canada would still have been relatively well off, having been able to retain about 58 per cent. For most of the countries of the British Commonwealth and the other members of the \"sterling bloc\" the figure would have been around 50 per cent, and for Germany, in spite of the loss of all pre-1924 savings, still as much as 37 per cent. The investors in all those countries were still fortunate, however, compared with those in France or Italy, who would have retained only between 11 and 12 per cent of what the value of their savings over the period ought to have been at the beginning of 1958.8\n\nIt is usual today to dismiss the importance of this long and world-wide inflationary trend with the comment that things have always been like that and that history is largely a history of inflation. However true this may be in general, it is certainly not true of the period during which our modern economic system developed and during which wealth and incomes grew at an unprecedented rate. During the two hundred years preceding 1914, when Great Britain adhered to the gold standard, the price level, so far as it can be meaningfully measured over such a period, fluctuated around a constant level, ending up pretty well where it started and rarely changing by more than a third above or below that average level (except during the period of the Napoleonic wars, when the gold standard was abandoned).9 Similarly, in the United States, during the period 1749\u20131939 there also does not seem to have occurred a significant upward trend of prices.10 Compared with this, the rate at which prices have risen during the last quarter of a century in these and other countries represents a major change.\n\n3. Although there are a few people who deliberately advocate a continuous upward movement of prices, the chief source of the existing inflationary bias is the general belief that deflation, the opposite of inflation, is so much more to be feared that, in order to keep on the safe side, a persistent error in the direction of inflation is preferable. But, as we do not know how to keep prices completely stable and can achieve stability only by correcting any small movement in either direction, the determination to avoid deflation at any cost must result in cumulative inflation. Also, the fact that inflation and deflation will often be local or sectional phenomena which must occur necessarily as part of the mechanism redistributing the resources of the economy means that attempts to prevent any deflation affecting a major area of the economy must result in over-all inflation.\n\nIt is, however, rather doubtful whether, from a long-term point of view, deflation is really more harmful than inflation. Indeed, there is a sense in which inflation is infinitely more dangerous and needs to be more carefully  guarded against. Of the two errors, it is the one much more likely to be committed. The reason for this is that moderate inflation is generally pleasant while it proceeds, whereas deflation is immediately and acutely painful.11 There is little need to take precautions against any practice the bad effects of which will be immediately and strongly felt; but there is need for precautions wherever action which is immediately pleasant or relieves temporary difficulties involves much greater harm that will be felt only later. There is, indeed, more than a mere superficial similarity between inflation and drug-taking, a comparison which has often been made.\n\nInflation and deflation both produce their peculiar effects by causing unexpected price changes, and both are bound to disappoint expectations twice. The first time is when prices prove to be higher or lower than they were expected to be and the second when, as must sooner or later happen, these price changes come to be expected and cease to have the effect which their unforeseen occurrence had. The difference between inflation and deflation is that, with the former, the pleasant surprise comes first and the reaction later, while, with the latter, the first effect on business is depressing. The effects of both, however, are self-reversing. For a time the forces which bring about either tend to feed on themselves, and the period during which prices move faster than expected may thus be prolonged. But unless price movements continue in the same direction at an ever accelerating rate, expectations must catch up with them. As soon as this happens, the character of the effects changes.\n\nInflation at first merely produces conditions in which more people make profits and in which profits are generally larger than usual. Almost everything succeeds, there are hardly any failures. The fact that profits again and again prove to be greater than had been expected and that an unusual number of ventures turn out to be successful produces a general atmosphere favorable to risk-taking. Even those who would have been driven out of business without the windfalls caused by the unexpected general rise in prices are able to hold on and to keep their employees in the expectation that they will soon share in the general prosperity. This situation will last, however, only until people begin to expect prices to continue to rise at the same rate. Once they begin to count on prices being so many per cent higher in so many months' time, they will bid up the prices of the factors of production which determine the costs to a level corresponding to the future prices they expect. If prices then rise no more than had been expected, profits will return to normal, and the proportion of those making a profit also will fall; and since, during the period of exceptionally large profits, many have held on who would otherwise have been forced to change the direction of their efforts, a higher proportion than usual will suffer losses.\n\nThe stimulating effect of inflation will thus operate only so long as it has not been foreseen; as soon as it comes to be foreseen, only its continuation at an increased rate will maintain the same degree of prosperity. If in such a situation prices rose less than expected, the effect would be the same as that of unforeseen deflation. Even if they rose only as much as was generally expected, this would no longer provide the exceptional stimulus but would lay bare the whole backlog of adjustments that had been postponed while the temporary stimulus lasted. In order for inflation to retain its initial stimulating effect, it would have to continue at a rate always faster than expected.\n\nWe cannot consider here all the complications which make it impossible for adaptations to an expected change in prices ever to become perfect, and especially for long-term and short-term expectations to become equally adjusted; nor can we go into the different effects on current production and on investment which are so important in any full examination of industrial fluctuations. It is enough for our purpose to know that the stimulating effects of inflation must cease to operate unless its rate is progressively accelerated and that, as it proceeds, certain unfavorable consequences of the fact that complete adaptation is impossible become more and more serious. The most important of these is that the methods of accounting on which all business decisions rest make sense only so long as the value of money is tolerably stable. With prices rising at an accelerating rate, the techniques of capital and cost accounting that provide the basis for all business planning would soon lose all meaning. Real costs, profits, or income would soon cease to be ascertainable by any conventional or generally acceptable method. And, with the principles of taxation being what they are, more and more would be taken in taxes as profits that in fact should be reinvested merely to maintain capital.\n\nInflation thus can never be more than a temporary fillip, and even this beneficial effect can last only as long as somebody continues to be cheated and the expectations of some people unnecessarily disappointed. Its stimulus is due to the errors which it produces. It is particularly dangerous because the harmful aftereffects of even small doses of inflation can be staved off only by larger doses of inflation. Once it has continued for some time, even the prevention of further acceleration will create a situation in which it will be very difficult to avoid a spontaneous deflation. Once certain activities that have become extended can be maintained only by continued inflation, their simultaneous discontinuation may well produce that vicious and rightly feared process in which the decline of some incomes leads to the decline of other incomes, and so forth. From what we know, it still seems probable that we should be able to prevent serious depressions by preventing the inflations which regularly precede them, but that there is little we can do to cure them, once they have set in. The time to worry about depressions is, unfortunately, when they are furthest from the minds of most people.\n\nThe manner in which inflation operates explains why it is so difficult to resist when policy mainly concerns itself with particular situations rather than with general conditions and with short-term rather than with long-term problems. It is usually the easy way out of any temporary difficulties for both government and private business\u2014the path of least resistance and sometimes also the easiest way to help the economy get over all the obstacles that government policy has placed in its way.12 It is the inevitable result of a policy which regards all the other decisions as data to which the supply of money must be adapted so that the damage done by other measures will be as little noticed as possible. In the long run, however, such a policy makes governments the captives of their own earlier decisions, which often force them to adopt measures that they know to be harmful. It is no accident that the author whose views, perhaps mistakenly interpreted, have given more encouragement to these inflationary propensities than any other man's is also responsible for the fundamentally antiliberal aphorism, \"in the long run we are all dead.\"13 The inflationary bias of our day is largely the result of the prevalence of the short-term view, which in turn stems from the great difficulty of recognizing the more remote consequences of current measures, and from the inevitable preoccupation of practical men, and particularly politicians, with the immediate problems and the achievement of near goals.\n\nBecause inflation is psychologically and politically so much more difficult to prevent than deflation and because it is, at the same time, technically so much more easily prevented, the economist should always stress the dangers of inflation. As soon as deflation makes itself felt, there will be immediate attempts to combat it\u2014often when it is only a local and necessary process that should not be prevented. There is more danger in untimely fears of deflation than in the possibility of our not taking necessary countermeasures. While nobody is likely to mistake local or sectional prosperity for inflation, people often demand wholly inappropriate monetary countermeasures when there is a local or sectional depression.\n\nThese considerations would seem to suggest that, on balance, probably some mechanical rule which aims at what is desirable in the long run and ties  the hands of authority in its short-term decisions is likely to produce a better monetary policy than principles which give to the authorities more power and discretion and thereby make them more subject to both political pressure and their own inclination to overestimate the urgency of the circumstances of the moment. This, however, raises issues which we must approach more systematically.\n\n4. The case for \"rules versus authorities in monetary policy'' has been persuasively argued by the late Henry Simons in a well-known essay.14 The arguments advanced there in favor of strict rules are so strong that the issue is now largely on of how far it is practically possible to tie down monetary authority by appropriate rules. It may still be true that if there were full agreement as to what monetary policy ought to aim for, an independent monetary authority, fully protected against political pressure and free to decide on the means to be employed in order to achieve the ends it has been assigned, might be the best arrangement. The old arguments in favor of independent central banks still have great merit. But the fact that the responsibility for monetary policy today inevitably rests in part with agencies whose main concern is with government finance probably strengthens the case against allowing much discretion and for making decisions on monetary policy as predictable as possible.\n\nIt should perhaps be explicitly stated that the case against discretion in monetary policy is not quite the same as that against discretion in the use of the coercive powers of government. Even if the control of money is in the hands of a monopoly, its exercise does not necessarily involve coercion of private individuals.15 The argument against discretion in monetary policy rests on the view that monetary policy and its effects should be as predictable as possible. The validity of the argument depends, therefore, on whether we can devise an automatic mechanism which will make the effective supply of money change in a more predictable and less disturbing manner than will any discretionary measures likely to be adopted. The answer is not certain. No automatic mechanism is known which will make the total supply of money adapt itself exactly as we would wish, and the most we can say in favor of any mechanism (or action determined by rigid rules) is that it is doubtful whether in practice any deliberate control would do better. The reason for this doubt is partly that the conditions in which monetary authorities have to make their decisions are usually not favorable to the prevailing of long views, partly that we are not too certain what they should do in particular circumstances and  that, therefore, uncertainty about what they will do is necessarily greater when they do not act according to fixed rules.\n\nThe problem has remained acute ever since the destruction of the gold standard by the policies of the 1920s and1930s.16 It is only natural that some people should regard a return to that tried system as the only real solution. And an even larger number would probably agree today that the defects of the gold standard have been greatly exaggerated and that it is doubtful whether its abandonment was a gain. This does not mean, however, that its restoration is at present a practical proposition.\n\nIt must be remembered, in the first place, that no single country could effectively restore it by independent action. Its operation rested on its being an international standard, and if, for example, the United States today returned to gold, it would chiefly mean that United States policy would determine the value of gold and not necessarily that gold would determine the value of the dollar.\n\nSecond, and no less important, the functioning of the international gold standard rested on certain attitudes and beliefs which have probably ceased to exist. It operated largely on the basis of the general opinion that to be driven off the gold standard was a major calamity and a national disgrace. It is not likely to have much influence even as a fair-weather standard when it is known that no country is prepared to take painful measures in order to preserve it. I may be mistaken in my belief that this _mystique_ of gold has disappeared for good, but, until I see more evidence to the contrary, I do not believe that an attempt to restore the gold standard can be more than temporarily successful.17\n\nThe case for the gold standard is closely connected with the general argument in favor of an international, as against a national standard. Within the limitations we have accepted here, we cannot pursue this problem further. We will merely add that if a standard is desired which is highly automatic and can at the same time be made international, a commodity reserve standard which has been worked out in some detail appears to me still the best plan for achieving all the advantages attributed to the gold standard without its defects.18 But, though the proposals for such a standard deserve more attention than they have received, they hardly offer a practical alternative for the near future. Even if there were a chance of such a scheme being immediately adopted, there would be very little prospect of its being run as it should be, i.e., for the purpose of stabilizing only the aggregate price of the large group of commodities selected and not the prices of any of the individual commodities included.\n\n5. I certainly have no wish to weaken the case for any arrangement that will force the authorities to do the right thing. The case for such a mechanism becomes stronger as the likelihood of the monetary policy's being affected by considerations of public finance becomes greater; but it would weaken, rather than strengthen, the argument if we exaggerated what can be achieved by it. It is probably undeniable that, though we can limit discretion in this field, we never can eliminate it; in consequence, what can be done within the unavoidable range of discretion not only is very important but is likely in practice to determine even whether or not the mechanism will ever be allowed to operate.\n\nThere is one basic dilemma, which all central banks face, which makes it inevitable that their policy must involve much discretion. A central bank can exercise only an indirect and therefore limited control over all the circulating media. Its power is based chiefly on the threat of not supplying cash when it is needed. Yet at the same time it is considered to be its duty never to refuse to supply this cash at a price when needed. It is this problem, rather than the general effects of policy on prices or the value of money, that necessarily preoccupies the central banker in his day-to-day actions. It is a task which makes it necessary for the central bank constantly to forestall or counteract developments in the realm of credit, for which no simple rules can provide sufficient guidance.19\n\nThe same is nearly as true of the measures intended to affect prices and employment. They must be directed more at forestalling changes before they  occur than at correcting them after they have occurred. If a central bank always waited until rule or mechanism forced it to take action, the resulting fluctuations would be much greater than they need be. And if, within the range of its discretion, it takes measures in a direction opposite to those which mechanism or rule will later impose upon it, it will probably create a situation in which the mechanism will not long be allowed to operate. In the last resort, therefore, even where the discretion of the authority is greatly restricted, the outcome is likely to depend on what the authority does within the limits of its discretion.\n\nThis means in practice that under present conditions we have little choice but to limit monetary policy by prescribing its goals rather than its specific actions. The concrete issue today is whether it ought to keep stable some level of employment or some level of prices. Reasonably interpreted and with due allowance made for the inevitability of minor fluctuations around a given level, these two aims are not necessarily in conflict, provided that the requirements for monetary stability are given first place and the rest of economic policy is adapted to them. A conflict arises, however, if \"full employment\" is made the chief objective and this is interpreted, as it sometimes is, as that maximum of employment which can be produced by monetary means in the short run. That way lies progressive inflation.\n\nThe reasonable goal of a high and stable level of employment can probably be secured as well as we know how while aiming at the stability of some comprehensive price level. For practical purposes, it probably does not greatly matter precisely how this price level is defined, except that it should not refer exclusively to final products (for if it did, it might in times of rapid technological advance still produce a significant inflationary tendency) and that it should be based as much as possible on international rather than local prices. Such a policy, if pursued simultaneously by two or three of the major countries, should also be reconcilable with stability of exchange rates. The important point is that there will be definite known limits which the monetary authorities will not allow price movements to exceed\u2014or even to approach to the point of making drastic reversals of policy necessary.\n\n6. Though there may be some people who explicitly advocate continuous inflation, it is certainly not because the majority wants it that we are likely to get it. Few people would be willing to accept it when it is pointed out that even such a seemingly moderate increase in prices as 3 per cent per annum means that the price level will double every twenty-three and a half years and that it will nearly quadruple over the normal span of a man's working life. The danger that inflation will continue is not so much due to the strength of those who deliberately advocate it as to the weakness of the opposition. In order to prevent it, it is necessary for the public to become clearly aware of the things we can do and of the consequences of not doing them. Most competent students agree that the difficulty of preventing inflation is only political and not economic.  Yet almost no one seems to believe that the monetary authorities have the power to prevent it and will exercise it. The greatest optimism about the short-term miracles that monetary policy will achieve is accompanied by a complete fatalism about what it will produce in the long run.\n\nThere are two points which cannot be stressed enough: first, it seems certain that we shall not stop the drift toward more and more state control unless we stop the inflationary trend; and, second, any continued rise in prices is dangerous because, once we begin to rely on its stimulating effect, we shall be committed to a course that will leave us no choice but that between more inflation, on the one hand, and paying for our mistake by a recession or depression, on the other. Even a very moderate degree of inflation is dangerous because it ties the hands of those responsible for policy by creating a situation in which, every time a problem arises, a little more inflation seems the only easy way out.\n\nWe have not had space to touch on the various ways in which the efforts of individuals to protect themselves against inflation, such as sliding-scale contracts, not only tend to make the process self-accelerating but also increase the rate of inflation necessary to maintain its stimulating effect. Let us simply note, then, that inflation makes it more and more impossible for people of moderate means to provide for their old age themselves; that it discourages saving and encourages running into debt; and that, by destroying the middle class, it creates that dangerous gap between the completely propertyless and the wealthy that is so characteristic of societies which have gone through prolonged inflations and which is the source of so much tension in those societies. Perhaps even more ominous is the wider psychological effect, the spreading among the population at large of that disregard of long-range views and exclusive concern with immediate advantages which already dominate public policy.\n\nIt is no accident that inflationary policies are generally advocated by those who want more government control\u2014though, unfortunately, not by them alone. The increased dependence of the individual upon government which inflation produces and the demand for more government action to which this leads may for the socialist be an argument in its favor. Those who wish to preserve freedom should recognize, however, that inflation is probably the most important single factor in that vicious circle wherein one kind of government action makes more and more government control necessary. For this reason, all those who wish to stop the drift toward increasing government control should concentrate their efforts on monetary policy. There is perhaps nothing more disheartening than the fact that there are still so many intelligent and informed people who in most other respects will defend freedom and yet are induced by the immediate benefits of an expansionist policy to support what, in the long run, must destroy the foundations of a free society.\n\n* * *\n\nThe quotation at the head of the chapter is taken from John Maynard Keynes, _The Economic Consequences of the Peace_ (London: Macmillan, 1919), pp. 220\u201321. Keynes's observation was prompted by a similar remark attributed to Lenin to the effect that \"the best way to destroy the capitalist system was to debauch the currency.\" Cf. also Keynes's later statement in _A Tract on Monetary Reform_ (London: Macmillan, 1923), p. 40. \"The Individualistic Capitalism of to-day, precisely because it entrusts saving to the individual investor and production to the individual employer, _presumes_ a stable measuring-rod of value, and cannot be efficient\u2014perhaps can not survive\u2014without one.\" [The statement attributed to Lenin is possibly spurious. The Library of Congress has failed to uncover this or any similar statement in Lenin's writings. However, it has recently been suggested that he made the claim in an interview in 1919.\u2014Ed.]\n\n1 Cf. Ludwig von Mises, _Human Action_ , pp. 429\u201345 [Liberty Fund edition, vol. 2, pp. 432\u2013 448].\n\n2 Though I am convinced that modern credit banking as it has developed, requires some public institutions such as the central banks, I am doubtful whether it is necessary or desirable that they (or the government) should have the monopoly of the issue of all kinds of money. The state has, of course, the right to protect the name of the unit of money which it (or anybody else) issues and, if it issues \"dollars,\" to prevent anybody else from issuing tokens with the same name. And as it is its function to enforce contracts, it must be able to determine what is \"legal tender\" for the discharge of any obligation contracted. But there seems to be no reason whatever why the state should ever prohibit the use of other kinds of media of exchange, be it some commodity or money issued by another agency, domestic or foreign. One of the most effective measures for protecting the freedom of the individual might indeed be to have constitutions prohibiting all peacetime restrictions on transactions in any kind of money or the precious metals.\n\n3 The most important of these temporary and self-reversing shifts of demand which monetary changes are likely to cause are changes in the relative demand for consumers' goods and investment goods; this problem we cannot consider here without entering into all the disputed problems of business-cycle theory.\n\n4 See the more detailed discussion of these problems in my _Monetary Nationalism and International Stability_ (London: Longmans, Green, and Co., 1937).\n\n5 See Richard Sidney Sayers, _Central Banking after Bagehot_ (Oxford: Clarendon Press, 1957), pp. 85\u2013107 [chap. 7, \"The Variation of Cash Reserve Requirements,\" pp. 85\u201391, and chap. 8, \"The Determination of the Volume of Bank Deposits: England 1955\u201356,\" pp. 92\u2013107]. In light of the common assumption that \"f scal policy\" as a means of controlling the economy was unknown when liberal policy was at its peak, tw o quotations from John Mor ley, Viscount Morley ( _Life of William Ewart Gladstone_ [3 vols.; London: Macmillan, 1903]) are of interest. Morley, writing of Gladstone's f rst budget in 1853 (vol. 1, bk. 4, p. 461) observes: \"its initial boldness lay in the adoption of the unusual course of estimating the national income roughly for a period of seven years, and assuming that expenditure would remain tolerably steady for the whole period,\" and, earlier, he cites a remark made by Gladstone (vol. 2, bk. 5, p. 57): \"If you want to benef t the labouring classes and to do the maximum of good, it is not enough to operate upon the articles consumed by them; you should rather operate upon the articles that give the maximum of employment.\"\n\n6 See Colin Clark, \"Public Finance and Changes in the Value of Money,\" _Economic Journal_ , 55 (1945): 371\u201389, and compare the discussion of his thesis by Joseph A. Pechman and Thomas Mayer, \"Mr. Colin Clark on the Limits of Taxation,\" _Review of Economics and Statistics_ , 34 (1952): 232\u201342.\n\n7 The figures quoted in the text are the result of calculations made for me by Mr. Salvator V. Ferrera, whose assistance I gratefully acknowledge. They were necessarily confined to those countries for which cost-of-living index numbers were readily available for the whole of the forty-year period. I am deliberately giving in the text only round figures, because I do not believe that the results of this kind of calculation can give us more than rough indications of the orders of magnitude involved. For those who are interested I give here the results (up to one decimal place) for all the countries for which the calculation was made:\n\nThe comparatively good results these statistics show for Germany are probably due to the fact that the currency reform of 1948 was not taken into account.\n\n8 So far as France is concerned, this, of course, does not take into account the effects of the considerable further depreciation (and consequent devaluation) of the French franc in the course of 1958.\n\n9 There is no continuous index number available for the whole of this two-hundred-year period, but the approximate trend of prices can be gauged by piecing together the data given by Elizabeth Waterman Gilboy, \"The Cost of Living and Real Wages in Eighteenth Century England,\" _Review of Economics and Statistics_ , 18 (1936): 134\u201343, and Rufus Stickney Tucker, \"Real Wages of Artisans in London, 1729\u20131935,\" _Journal of the American Statistical Association_ , 31 (1936): 73\u201384. For another later study, see Phyllis Deane and William Alan Cole, _British Economic Growth, 1688\u20131959: Trends and Structure_ (Cambridge: Cambridge University Press, 1962).\n\n10 This statement is based on the index number of wholesale prices for the United States (see United States Department of Labor, Bureau of Labor Statistics, _Bureau of Labor Statistics Chart Series_ [1948] (Washington, DC: Government Printing Office, 1948), chart E-11. [The Bureau of Labor Statistics calculates that wholesale prices for all commodities (1926 = 100) was 53.5 in 1749 and 77.1 in 1939, or an annual increase in prices of approximately .1925% over the course of 190 years.\u2014Ed.]\n\n11 Cf. Wilhelm R\u00f6pke, _Welfare, Freedom and Inflation_ (London: Pall Mall Press, 1957).\n\n12 Cf. my essay, \"Full Employment, Planning, and Inflation,\" _Review of the Institute of Public Affairs_ (Melbourne, Australia), 4 (1950): 174\u201384; reprinted in _Studies in Philosophy, Politics and Economics_ (Chicago: University of Chicago Press, 1967), pp. 270\u201379; the German version of this essay, [\"Vollbesch\u00e4ftigung, Planwirtschaft und Inflation\"] appears in _Vollbesch\u00e4ftigung, Inflation und Plan-wirtschaft_ [Schweizerisches Institut f\u00fcr Auslandforschung], Albert Hunold, ed. (Erlenbach-Zurich: Eugen Rentsch Verlag, 1951), pp. 184\u201397; and Friedrich August Lutz, \"Inflationsgefahr und Konjunkturpolitik,\" _Schweizerische Zeitschrift f\u00fcr Volkswirtschaft und Statistik \/ Revue suisse d'\u00e9conomie politique_ , 93 (1957): 195\u2013205, and \"Cost-and Demand-Induced Inflation,\" _Banca Nazionale del Lavoro Quarterly Review_ , 11 (1958): 3\u201318.\n\n13 John Maynard Keynes, _A Tract on Monetary Reform_ (London: Macmillan, 1923), p. 80.\n\n14 Henry Calvert Simons's essay of that title, originally published in \"Rules versus Authorities in Monetary Policy,\" _Journal of Political Economy_ , 44 (1936): 1\u201330; reprinted in Henry Calvert Simons, _Economic Policy for a Free Society_ (Chicago: University of Chicago Press, 1948), pp. 160\u201383.\n\n15 This applies at least to the traditional instruments of monetary policy though not to such newer measures as the changes in the required reserves of the banks.\n\n16 The fatal errors begin with the British attempt after the first World War to restore the pound to its former value rather than to relink it with gold at a new parity corresponding to its reduced value. Besides the fact that this was not required by the principles of the gold standard, it was contrary to the best classical teaching. David Ricardo had explicitly said of a similar situation one hundred years earlier that he \"never should advise a government to restore a currency, which was depreciated 30 pct., to par; I should recommend, as you propose, but not in the same manner, that the currency should be fixed at the depreciated value by lowering the standard, and that no further deviations should take place\" (David Ricardo, letter to John Wheatley, September 18, 1821, _The Works and Correspondence of David Ricardo_ , Piero Sraffa, ed. (11 vols.; Cambridge: Cambridge University Press, 1951), vol. 9, p. 73 [Liberty Fund edition, vol. 9 ( _Letters 1821\u20131823_ ), p. 73].\n\n17 There is, of course, a strong case for completely freeing the trade in gold. Indeed, it would seem desirable to go considerably further in this direction; probably nothing would contribute more to international monetary stability than the different countries mutually binding themselves by treaty to place no obstacles whatever in the way of free dealing in one another's currencies. (There would probably also be a strong case for going still further and permitting their respective banks to operate freely in their territories.) But, though this would go far in the direction of restoring a stable international standard, the control of the value of this standard would still be in the hands of the authorities of the biggest countries participating in it.\n\n18 Cf. my essay on \"A Commodity Reserve Currency,\" _Economic Journal_ , 53 (1943): 176\u201384; reprinted in Friedrich August Hayek, _Individualism and Economic Order_ , pp. 209\u201319.[ _Collected Works_ edition, vol. 6, pp. 106\u201314.]\n\n19 See my essay _Monetary Nationalism and International Stability_ , (London: Longmans, Green, and Co., 1937). [ _Collected Works_ edition, vol. 6, pp. 37\u2013105.]\n\n## TWENTY-TWO\n\nHOUSING AND TOWN PLANNING\n\nIf the government simultaneously abolished housing subsidies and cut working class taxation by an amount exactly equal to the subsidies the working classes would be no worse off financially; but they would then without any doubt prefer to spend the money in other ways than on housing, and would live in overcrowded and inadequately provided houses, some because they do not know the advantages of better housing, and others because they value these too lightly in comparison with other ways of spending their money. That is the case, and the only case for housing subsidies, and it is put here in its crudest form because the matter is so often discussed in left wing literature without facing reality.\n\n\u2014William Arthur Lewis\n\n1. Civilization as we know it is inseparable from urban life. Almost all that distinguishes civilized from primitive society is intimately connected with the large agglomerations of population that we call \"cities,\" and when we speak of \"urbanity,\" \"civility,\" or \"politeness,\" we refer to the manner of life in cities. Even most of the differences between the life of the present rural population and that of primitive people are due to what the cities provide. It is also the possibility of enjoying the products of the city in the country that in advanced civilizations often makes a leisured life in the country appear the ideal of a cultured life.\n\nYet the advantages of city life, particularly the enormous increases in productivity made possible by its industry, which equips a small part of the population remaining in the country to feed all the rest, are bought at great cost. City life is not only more productive than rural life; it is also much more costly. Only those whose productivity is much increased by life in the city will reap a net advantage over and above the extra cost of this kind of life. Both the costs and the kinds of amenities which come with city life are such that the minimum income at which a decent life is possible is much higher than in the country. Life at a level of poverty which is still bearable in the country not  only is scarcely tolerable in the city but produces outward signs of squalor which are shocking to fellow men. Thus the city, which is the source of nearly all that gives civilization its value and which has provided the means for the pursuit of science and art as well as of material comfort, is at the same time responsible for the darkest blotches on this civilization.\n\nMoreover, the costs involved in large numbers living in great density not only are very high but are also to a large extent communal, i.e., they do not necessarily or automatically fall on those who cause them but may have to be borne by all. In many respects, the close contiguity of city life invalidates the assumptions underlying any simple division of property rights. In such conditions it is true only to a limited extent that whatever an owner does with his property will affect only him and nobody else. What economists call the \"neighborhood effects,\" i.e., the effects of what one does to one's property on that of others, assume major importance. The usefulness of almost any piece of property in a city will in fact depend in part on what one's immediate neighbors do and in part on the communal services without which effective use of the land by separate owners would be nearly impossible.\n\nThe general formulas of private property or freedom of contract do not therefore provide an immediate answer to the complex problems which city life raises. It is probable that, even if there had been no authority with coercive powers, the superior advantages of larger units would have led to the development of new legal institutions\u2014some division of the right of control between the holders of a superior right to determine the character of a large district to be developed and the owners of inferior rights to the use of smaller units, who, within the framework determined by the former, would be free to decide on particular issues. In many respects the functions which the organized municipal corporations are learning to exercise correspond to those of such a superior owner.\n\nIt must be admitted that, until recently, economists gave regrettably little attention to the problems of the co-ordination of all the different aspects of city development.1 Though some of them have been among the foremost critics of the evils of urban housing (some fifty years ago a satirical German  weekly could suggest that an economist be defined as a man who went around measuring workmen's dwellings, saying they were too small!) so far as the important issues of urban life are concerned, they have long followed the example of Adam Smith, who explained in his lectures that the problem of cleanliness and security, \"to wit, the proper method of carrying dirt from the streets, and the execution of justice, so far as it regards regulations for preventing crimes or the method of keeping a city guard, though useful, are too mean to be considered in a general discourse of this kind.\"2\n\nIn view of this neglect by his profession of the study of a highly important subject, an economist perhaps ought not to complain that it is in a very unsatisfactory state. Development of opinion in this field has, in fact, been led almost exclusively by men concerned with the abolition of particular evils, and the central question of how the separate efforts are to be mutually adjusted has been much neglected. Yet the problem of how the effective utilization of the knowledge and skill of the individual owners is to be reconciled with keeping their actions within limits where they will not gain at somebody else's expense is here of peculiar importance. We must not overlook the fact that the market has, on the whole, guided the evolution of cities more successfully, though imperfectly, than is commonly realized and that most of the proposals to improve upon this, not by making it work better, but by superimposing a system of central direction, show little awareness of what such a system would have to accomplish, even to equal the market in effectiveness.\n\nIndeed, when we look at the haphazard manner in which governments, with seemingly no clear conception of the forces that determined the development of cities, have generally dealt with these difficult problems, we wonder that the evils are not greater than they are. Many of the policies intended to combat particular evils have actually made them worse. And some of the more recent developments have created greater potentialities for a direct control by authority of the private life of the individual than may be seen in any other field of policy.\n\n2. We must first consider a measure which, though always introduced as a device to meet a passing emergency and never defended as a permanent arrangement, has in fact regularly become a lasting feature and in much of western Europe has probably done more to restrict freedom and prosperity than any other measure, excepting only inflation. This is rent restriction or the placing of ceilings on the rents of dwellings. Originally introduced to prevent rents from rising during the first World War, it was retained in many countries for more than forty years through major inflations, with the result  that rents were reduced to a fraction of what they would be in a free market. Thus house property was in effect expropriated. Probably more than any other measure of this kind, it worsened in the long run the evil it was meant to cure and produced a situation in which administrative authorities acquired highly arbitrary powers over the movement of men. It also contributed much toward weakening the respect for property and the sense of individual responsibility. To those who have not experienced its effects over a long period, these remarks may seem unduly strong. But whoever has seen the progressive decay of housing conditions and the effects on the general manner of life of the people of Paris, of Vienna, or even of London, will appreciate the deadly effect that this one measure can have on the whole character of an economy\u2014and even of a people.\n\nIn the first place, any fixing of rents below the market price inevitably perpetuates the housing shortage. Demand continues to exceed supply, and, if ceilings are effectively enforced (i.e., the appearance of \"premiums\" prevented), a mechanism for allocating dwelling space by authority must be established. Mobility is greatly reduced and in the course of time the distribution of people between districts and types of dwellings ceases to correspond to needs or desires. The normal rotation, in which a family during the period of full earning power of the head occupies more space than a very young or retired couple, is suspended. Since people cannot be ordered to move around, they just hold on to what they have, and the rented premises become a sort of inalienable property of the family which is handed down from generation to generation, irrespective of need. Those who have inherited a rented dwelling are often better off than they would be otherwise, but an ever increasing proportion of the population either cannot get a separate dwelling at all or can do so only by grace of official favor or by a sacrifice of capital they can ill afford or by some illegal or devious means.3\n\nAt the same time, the owner loses all interest in investing in the maintenance of buildings beyond what the law allows him to recover from the tenants  for that specific purpose. In cities like Paris, where inflation has reduced the real value of rents to a twentieth or less of what they once were, the rate at which houses are falling into an unprecedented state of decay is such that their replacement will be impracticable for decades to come.\n\nIt is not the material damage, however, that is the most important. Because of rent restriction, large sections of the population in Western countries have become subject to arbitrary decisions of authority in their daily affairs and accustomed to looking for permission and direction in the main decisions of their lives. They have come to regard it as a matter of course that the capital which pays for the roof over their heads should be provided free by somebody else and that individual economic well-being should depend on the favor of the political party in power, which often uses its control over housing to assist its supporters.\n\nWhat has done so much to undermine the respect for property and for the law and the courts is the fact that authority is constantly called upon to decide on the relative merits of needs, to allocate essential services, and to dispose of what is still nominally private property according to its judgment of the urgency of different individual needs. For example, whether \"an owner, with an invalid wife and three young children, who wishes to obtain occupation of his house [would] suffer more hardship if his request were refused than the tenant, with only one child but a bed-ridden mother-in-law, would suffer if it were granted\"4 is a problem that cannot be settled by appeal to any recognized principles of justice but only by the arbitrary intervention of authority. How great a power this sort of control over the most important decisions of one's private life confers on authority is clearly shown by a recent decision of the German Administrative Court of Appeal, which found it necessary to declare as illegal the refusal of a local government labor exchange to find work for a man living in a different area unless he first obtained from the housing authority permission to move and promise of accommodation\u2014not because neither authority was entitled to refuse his request but because their refusal involved an \"inadmissible coupling of separate interests of administration.\"5 Indeed, the co-ordination of the activities of different authorities, which the planners  so dearly want, is liable to turn what otherwise is merely arbitrariness in particular decisions into despotic power over the whole life of the individual.\n\n3. While rent restriction, even where it has been in force as far back as most people can remember, is still regarded as an emergency measure which has become politically impossible to abandon,6 efforts to reduce the cost of housing for the poorer sections of the population by public housing or building subsidies have come to be accepted as a permanent part of the welfare state. It is little understood that, unless very carefully limited in scope and method, such efforts are likely to produce results very similar to those of rent restriction.\n\nThe first point to note is that any group of people whom the government attempts to assist through a public supply of housing will benefit only if the government undertakes to supply all the new housing they will get. Provision of only part of the supply of dwellings by authority will in effect be not an addition to, but merely a replacement of, what has been provided by private building activity. Second, cheaper housing provided by government will have to be strictly limited to the class it is intended to help, and, merely to satisfy the demand at the lower rents, government will have to supply considerably more housing than that class would otherwise occupy. Third, such limitation of public housing to the poorest families will generally be practicable only if the government does not attempt to supply dwellings which are both cheaper and substantially better than they had before; otherwise the people thus assisted would be better housed than those immediately above them on the economic ladder; and pressure from the latter to be included in the scheme would become irresistible, a process which would repeat itself and progressively bring in more and more people.\n\nA consequence of this is that, as has again and again been emphasized by the housing reformers, any far-reaching change in housing conditions by public action will be achieved only if practically the whole of the housing of a city is regarded as a public service and paid for out of public funds. This means, however, not only that people in general will be forced to spend more on housing than they are willing to do, but that their personal liberty will be gravely threatened. Unless the authority succeeds in supplying as much of this better and cheaper housing as will be demanded at the rents charged,  a permanent system of allocating the available facilities by authority will be necessary\u2014that is, a system whereby authority determines how much people should spend on housing and what sort of accommodation each family or individual ought to get. It is easy to see what powers over individual life authority would possess if the obtaining of an apartment or house were generally dependent on its decision.\n\nIt should also be realized that the endeavor to make housing a public service has already in many instances become the chief obstacle to the general improvement of housing conditions, by counteracting those forces which produce a gradual lowering of the cost of building. All monopolists are notoriously uneconomical, and the bureaucratic machinery of government even more so; and the suspension of the mechanism of competition and the tendency of any centrally directed development to ossify are bound to obstruct the attainment of the desirable and technically not impossible goal\u2014a substantial and progressive reduction of the costs at which all the housing needs can be met.\n\nPublic housing (and subsidized housing) can thus, at best, be an instrument of assisting the poor, with the inevitable consequence that it will make those who take advantage of it dependent on authority to a degree that would be politically very serious if they constituted a large part of the population. Like any assistance to an unfortunate minority, such a measure is not irreconcilable with a general system of freedom. But it raises very grave problems that should be squarely faced if it is not to produce dangerous consequences.\n\n4. The greater earning power and other advantages that city life offers are to a considerable degree offset by its higher costs, which generally increase with the size of the city. Those whose productivity is greatly increased by working in the city will derive a net advantage, even though they have to pay much more for their limited dwelling space and may also have to pay for daily transportation over long distances. Others will gain a net advantage only if they do not have to spend money on travel or expensive quarters or if they do not mind living in crowded conditions so long as they have more to spend on other things. The old buildings which at most stages of the growth of a city will exist in its center, on land which is already in such great demand for other purposes that it is no longer profitable to build new dwellings on it, and which are no longer wanted by the better-off, will often provide for those of low productivity an opportunity to benefit from what the city offers at the price of very congested living. So long as they are prepared to live in them, to leave these old houses standing will often be the most profitable way of using the land. Thus, paradoxically, the poorest inhabitants of a city frequently live in districts where the value of the land is very high and the landlords draw very large incomes from what is likely to be the most dilapidated part of the city. In such a situation property of this sort continues to be available for housing  only because the old buildings, with little spent on them for repair or maintenance, are occupied at great density. If they were not available or could not be used in this manner, the opportunities for increasing their earnings by more than the additional costs of living in the city would not exist for most of the people who live there.\n\nThe existence of such slums, which in a more or less aggravated form appear during the growth of most cities, raises two sets of problems which ought to be distinguished but are commonly confused. It is unquestionably true that the presence of such unsanitary quarters, with their generally squalid and often lawless conditions, may have a deleterious effect on the rest of the city and will force the city administration or the other inhabitants to bear costs which those who come to live in the slums do not take into account. Insofar as it is true that the slum dwellers find it to their advantage to live in the center of the city only because they do not pay for all the costs caused by their decision, there is a case for altering the situation by charging the slum properties with all these costs\u2014with the probable result that they will disappear and be replaced by buildings for commercial or industrial purposes. This would clearly not assist the slum dwellers. The case for action here is not based on their interest; the problems are raised by \"neighborhood effects\" and belong to the questions of city planning, which we shall have to consider later.\n\nQuite different from this are the arguments for slum clearance based on the presumed interests or needs of slum dwellers. These pose a genuine dilemma. It is often only because people live in crowded old buildings that they are able to derive some gain from the extra earning opportunities of the city. If we want to abolish the slums, we must choose one of two alternatives: we must either prevent these people from taking advantage of what to them is part of their opportunity, by removing the cheap but squalid dwellings from where their earning opportunities lie, and effectively squeeze them out of the cities by insisting on certain minimum standards for all town dwellings;7 or we must provide them with better facilities at a price which does not cover costs and thus subsidize both their staying in the city and the movement into the city of more people of the same kind. This amounts to a stimulation of the growth of cities beyond the point where it is economically justifiable and to a deliberate creation of a class dependent on the community for the provision of what they are presumed to need. We can hardly expect this service to be provided for long without the authorities also claiming the right to decide who is and who is not to be allowed to move into a given city.\n\nAs happens in many fields, the policies pursued here aim at providing for a given number of people without taking into account the additional numbers  that will have to be provided for as a result. It is true that a part of the slum population of most cities consists of old inhabitants who know only city life and who would be even less able to earn an adequate living in rural conditions. But the more acute problem is that raised by the influx of large numbers from poorer and still predominantly rural regions, to whom the cheap accommodation in the old and decaying buildings of the city offers a foothold on the ladder that may lead to greater prosperity. They find it to their advantage to move into the city in spite of the crowded and unsanitary conditions in which they have to live. Providing them with much better quarters at an equally low cost will attract a great many more. The solution of the problem would be either to let the economic deterrents act or to control directly the influx of population; those who believe in liberty will regard the former as the lesser evil.\n\nThe housing problem is not an independent problem which can be solved in isolation: it is part of the general problem of poverty and can be solved only by a general rise in incomes. This solution, however, will be delayed if we subsidize people to move from where their productivity is still greater than the cost of living to places where it will be less, or if we prevent from moving those who believe that, by doing so, they can improve their prospects at the price of living in conditions which to us seem deplorable.\n\nThere is no space here to consider all the other municipal measures which, though designed to relieve the needs of a given population, really tend to subsidize the growth of giant cities beyond the economically justifiable point. Most of the policies concerning public utility rates which are immediately aimed at relieving congestion and furthering the growth of the outlying districts by providing services below costs only make matters worse in the long run. What has been said of current housing policies in England is equally true about most other countries: \"We have drifted into a practice of encouraging financially, out of taxes collected from the whole nation, the maintenance of over-grown and over-concentrated urban fabrics and, in the case of large cities still growing, the continuance of fundamentally uneconomic growth.\"8\n\n5. A different set of problems is raised by the fact that in the close contiguity of city living the price mechanism reflects only imperfectly the benefit or harm to others that a property owner may cause by his actions. Unlike the situation which generally prevails with mobile property, where the advantages or disadvantages arising from its use are usually confined to those who control it, the use made of a piece of land often necessarily affects the usefulness of neighboring pieces. Under the conditions of city life this applies to the actions of private owners and even more to the use made of communally owned land, such as that used for streets and the public amenities which are  so essential to city life. In order that the market may bring about an efficient co-ordination of individual endeavors, both the individual owners and the authorities controlling communal property should be so placed as to enable them to take into account at least the more important effects of their actions on other property. Only when the value of the property of individuals as well as of the city authorities reflects all the effects of the use they make of it, will the price mechanism function as it should. Without special arrangements, this condition will exist only to a limited degree. The value of any piece of property will be affected by the manner in which the neighbors use theirs and even more by the services provided and the regulations enforced by the authorities; and unless the various decisions take these effects into account, there is little likelihood that total benefits will exceed total costs.9\n\nBut though the price mechanism is an imperfect guide for the use of urban land, it is still an indispensable guide if development is to be left to private initiative and if all the knowledge and foresight dispersed among many men is to be used. There is a strong case for taking whatever practical measures can be found to cause the mechanism to operate more efficiently by making owners take into consideration all the possible effects of their decisions. The framework of rules within which the decisions of the private owner are likely to agree with the public interest will therefore in this case have to be more detailed and more adjusted to particular local circumstances than is necessary with other kinds of property. Such \"town planning,\" which operates largely through its effects on the market and through the establishing of general conditions to which all developments of a district or neighborhood must conform but which, within these conditions, leaves the decisions to the individual owner, is part of the effort to make the market mechanism more effective.\n\nThere is a very different type of control, however, which is also practiced under the name of \"town planning.\" Unlike the other, this is motivated by the desire to dispense with the price mechanism and to replace it by central direction. Much of the town planning that is in fact carried out, particularly by architects and engineers who have never understood the role that prices play in co-ordinating individual activities,10 is of this kind. Even where it is not aimed at tying future developments to a preconceived plan which prescribes  the use of every piece of land, it tends to lead to this by making the market mechanism increasingly inoperative.\n\nThe issue is therefore not whether one ought or ought not to be for town planning but whether the measures to be used are to supplement and assist the market or to suspend it and put central direction in its place. The practical problems which policy raises here are of great complexity, and no perfect solution is to be expected. The beneficial character of any measures will show itself in contributing to a desirable development, the details of which, however, will be largely unpredictable.\n\nThe main practical difficulties arise from the fact that most measures of town planning will enhance the value of some individual properties and reduce that of others. If they are to be beneficial, the sum of the gains must exceed the sum of the losses. If an effective offsetting is to be achieved, it is necessary that both gains and losses due to a measure accrue to the planning authority, who must be able to accept the responsibility of charging the individual owners for the increase in the value of their property (even if the measures causing it have been taken against the will of some of the owners) and of compensating those whose property has suffered. This can be achieved without conferring on authority arbitrary and uncontrollable powers by giving it only the right of expropriation at fair market value. This is generally sufficient to enable the authority both to capture any increments in value that its actions will cause and to buy out those who oppose the measure because it reduces the value of their property. In practice, the authority will normally not have to buy, but, backed by its power of compulsory purchase, it will be able to negotiate an agreed charge or compensation with the owner. So long as expropriation at market value is its only coercive power, all legitimate interests will be protected. It will be a somewhat imperfect instrument, of course, since in such circumstances \"market value\" is not an unambiguous magnitude and opinions about what is a fair market value may vary widely. The important point, however, is that such disputes can be decided in the last resort by independent courts and need not be left to the discretion of the planning authority.\n\nThe dangers come largely from the desire of many planners to be released from the necessity of counting all the costs of their schemes. They often plead that if they are made to compensate at market value, the cost of carrying out some improvements becomes prohibitive. Wherever this is the case, it means, however, that the proposed plan should not be carried out. Nothing ought to be treated with more suspicion than arguments used by town planners to justify expropriation below fair market value, arguments regularly based on the false contention that they can thereby reduce the social costs of the scheme. All that such a scheme amounts to is that certain costs will not be taken into account: the planners make it appear advantageous simply by placing some of the costs on the shoulders of private persons and then disregarding them.\n\nMost of what is valid in the argument for town planning is, in effect, an argument for making the planning unit for some purposes larger than the usual size of individually owned property. Some of the aims of planning could be achieved by a division of the contents of the property rights in such a way that certain decisions would rest with the holder of the superior right, i.e., with some corporation representing the whole district or region and possessing powers to assess benefits and charges to individual subowners. Estate development in which the developer retains some permanent control over the use of the individual plots offers at least one alternative to the exercise of such control by political authority. There is also the advantage that the larger planning unit will still be one of many and that it will be restrained in the exercise of its powers by the necessity of competing with other similar units.\n\nTo some extent, of course, even competition between municipalities or other political subdivisions will have a similar restraining effect. Town planners, however, frequently demand town planning on a regional or even national scale. It is true that there will always be some factors in planning which only the larger units can consider. But it is still more true that, as the area of unified planning is extended, particular knowledge of local circumstances will, of necessity, be less effectively used. Nation-wide planning means that, instead of the unit of competition becoming larger, competition will be eliminated altogether. This is certainly not a desirable solution. There is probably no perfect answer to the real difficulties which the complexity of the problem creates. But only a method which operates mainly through the inducements and data offered to the private owner and which leaves him free in the use of a particular piece of land is likely to produce satisfactory results, since no other method will make as full use of the dispersed knowledge of the prospects and possibilities of development as the market does.\n\nThere still exist some organized groups who contend that all these difficulties could be solved by the adoption of the \"single-tax\" plan, that is, by transferring the ownership of all land to the community and merely leasing it at rents determined by the market to private developers. This scheme for the socialization of land is, in its logic, probably the most seductive and plausible of all socialist schemes. If the factual assumptions on which it is based were correct, i.e., if it were possible to distinguish clearly between the value of \"the permanent and indestructible powers of the soil,\" on the one hand, and, on the other, the value due to the two different kinds of improvement\u2014that due to communal efforts and that due to the efforts of the individual owner\u2014the argument for its adoption would be very strong. Almost all the difficulties we have mentioned, however, stem from the fact that no such distinction can be drawn with any degree of certainty. In order to give the necessary scope for private development of any one piece of land, the leases that would have to be granted at fixed rents would have to be for such long periods (they would  also have to be made freely transferable) as to become little different from private property, and all the problems of individual property would reappear. Though we might often wish that things were as simple as the single-tax program assumes, we will find in it no solution to any of the problems with which we are concerned.\n\n6. The administrative despotism to which town planners are inclined to subject the whole economy is well illustrated by the drastic provisions of the British Town and Country Planning Act of 1947.11 Though they had to be repealed after a few years, they have not lacked admirers elsewhere and have been held up as an example to be imitated in the United States.12 They provided for nothing less than the complete expropriation of all gains by the owner of urban property from any major change in the use made of his land\u2014and a gain was defined as any increase in the value of the land over what it would be if a change in its use were altogether prohibited, which might, of course, be zero.13 The compensation for this confiscation of all development rights was to be a share in a lump sum set aside for that purpose.\n\nThe conception underlying the scheme was that people should be free to sell and buy land only at a price based on the assumption that the particular piece of land would be permanently devoted to its present use: any gain made from changing its use was to go to the planning authority as the price for the  permission to make the change, while any loss caused by a fall in the value of the land in its present use would affect only the owner. In instances where a piece of land had ceased to bring any return in its present use, the \"development charges,\" as the levy was called, would therefore have amounted to the full value of the land in any new use to which it could be put.\n\nAs the authority created to administer these provisions of the law was thus given complete control of all changes in the use of land outside agriculture, it was in effect given a monopoly in deciding the use of any land in Britain for new industrial or commercial uses and complete authority to employ this power to exercise effective control of all such developments. This is a power which, by its nature, cannot be limited by rules, and the Central Land Board entrusted with it made it clear from the beginning that it did not mean to limit itself by any self-imposed rules to which it would consistently adhere. The _Practice Notes_ it issued at the beginning of its activities stated this with a frankness that has rarely been equaled. They explicitly reserved the right to deviate from its announced working rules whenever \"for special reasons the normal rules do not apply\" and \"from time to time to vary [its] policy\" and to treat the \"general working rule [as] variable if it does not fit a particular case.\"14\n\nIt is not surprising that these features of the act were found unworkable and had to be repealed after seven years and before any of the compensations for the \"nationalization of the development value\" of all land had been paid. What remains is a situation in which all development of land is by permission of the planning authority, which permission, however, is presumed to be obtainable if the development is not contrary to an announced over-all plan. The individual owner thus again has an interest in putting his land to better use. The whole experiment might be regarded as a curious episode and an illustration of the follies of ill-considered legislation, if it were not in fact the logical outcome of conceptions which are widely held. All endeavors to suspend the market mechanism in land and to replace it by central direction must lead to some such system of control that gives authority complete power over all development. The abortive British experiment has not attracted wider attention because, while the law was in force, the mechanism which its administration required never came into full operation. The law and the apparatus required to administer it were so complex that nobody except the unfortunate few who got caught in its meshes ever came to understand what it was all about.\n\n7. Similar to the problems of general town planning in many respects are those of building regulations. Though they do not raise important questions  of principle, they must be briefly considered. There are two reasons why some regulation of buildings permitted in cities is unquestionably desirable. The first is the now familiar consideration of the harm that may be done to others by the erection of buildings which constitute fire or health hazards; in modern conditions the people to be considered include the neighbors and all the users of a building who are not occupants but customers or clients of occupants and who need some assurance (or at least some means of ascertaining) that the building they enter is safe. The second is that, in the case of building, the enforcement of certain standards is perhaps the only effective way of preventing fraud and deception on the part of the builder: the standards laid down in building codes serve as a means of interpreting building contracts and insure that what are commonly understood to be appropriate materials and techniques will in fact be used unless the contract explicitly specifies otherwise.\n\nThough the desirability of such regulations can hardly be disputed, there are few fields in which government regulations offer the same opportunity for abuse or have in fact been used so much to impose harmful or wholly irrational restrictions on development and so often help to strengthen the quasi-monopolistic positions of local producers. Wherever such regulations go beyond the requirement of minimum standards, and particularly where they tend to make what at a given time and place is the standard method the only permitted method, they can become serious obstructions to desirable economic developments. By preventing experimentation with new methods and by supporting local monopolies of enterprise and labor, they are often partly to blame for the high building costs and are largely responsible for housing shortages and overcrowding. This is particularly true where regulations not merely require that the buildings satisfy certain conditions or tests but prescribe particular techniques to be employed. It should be especially emphasized that \"performance codes\" of the former kind impose less restrictions on spontaneous developments than \"specification codes\" and are therefore to be preferred. The latter may at first seem to agree more with our principles because they confer less discretion on authority; the discretion which \"performance codes\" confer is, however, not of the objectionable kind. Whether or not a given technique satisfies criteria of performance laid down in a rule can be ascertained by independent experts, and any dispute, if it arises, can be decided by a court.\n\nAnother issue of some importance and difficulty is whether building regulations should be laid down by local or by central authorities. It is perhaps true that local regulations will be more liable to be abused under the influence of local monopolies and are also in other respects more likely to be obstructive. There are probably strong arguments in favor of a carefully thought-out national standard or pattern which local authorities can adopt with whatever modifications seem appropriate to them. In general, however, it seems  probable that if the codes are determined locally, the competition between local authorities will bring about a more rapid elimination of obstructive and unreasonable restrictions than would be possible if the codes were uniformly laid down by law for a whole country or large region.\n\n8. Problems of the kind raised by town planning are likely to assume great importance in the future in connection with the location of industries on a national scale. The subject is beginning to occupy the attention of the planners more and more, and it is in this area that we now encounter most often the contention that the results of free competition are irrational and harmful.\n\nHow much is there in this alleged irrationality of the actual location of industry and the supposed possibility of improving upon it by central planning? It is, of course, true that, had developments been correctly foreseen, many decisions about the location of plants would have been different and that in this sense what has happened in the past appears in retrospect as unwise. This does not mean, however, that, with the knowledge which was then available, a different decision could have been expected or that the results would have been more satisfactory if developments had been under the control of a national authority. Though we again have to deal here with a problem wherein the price mechanism operates only imperfectly and does not take into account many things we would wish to see taken into account, it is more than doubtful whether a central planner could guide developments as successfully as the market does. It is remarkable how much the market does accomplish in this respect by making individuals take into account those facts which they do not know directly but which are merely reflected in the prices. The best-known critical examination of these problems has indeed led August L\u00f6sch to conclude that \"the most important result of this book is probably the demonstration of the surprising extent to which the free forces operate favorably.\" He then goes on to say that the market \"respects all human wishes, sight unseen, whether these are wholesome or unwholesome\" and that \"the free market mechanism works much more to the common good than is generally supposed, though with certain exceptions.\"15\n\n* * *\n\nThe quotation at the head of the chapter is taken from Sir William Arthur Lewis, _The Principles of Economic Planning: A Study Prepared for the Fabian Society_ (London: D. Dobson, 1949), p. 32.\n\n1 A valuable attempt to remedy this position has recently been made in Ralph Turvey, _Economics of Real Property: An Analysis of Property Values and Patterns of Use_ (London: Allen and Unwin Ltd., 1957). Of earlier works the discussions of local taxation by Edwin Cannan, _History of Local Rates in England, in Relation to the Proper Distribution of the Burden of Taxation_ (2nd ed., much enl.; London: P. S. King and Son, 1912), and his \"Memorandum,\" in Royal Commission on Local Taxation, _Memoranda Chiefly Relating to the Classification and Incidence of Imperial and Local Taxes_ [Alexander Hugh Bruce Balfour, Baron Balfour, Chairman] (London: Her Majesty's Stationery Office, 1899) [Cmd. 9528], pp. 160\u201375, are still among the most helpful on the crucial issues. [See also Cannan's _Answers to the Questions Submitted to Him by the Royal Commission on Local Taxation_ (London, 1898).\u2014Ed.] See also Jane Jacobs, _The Death and Life of the Great American Cities_ (New York: Random House, 1961).\n\n2 Adam Smith, _Lectures on Justice, Police, Revenue, and Arms: Delivered in the University of Glasgow_ (delivered in 1763\u201364) Edwin Cannan, ed. (Oxford: Clarendon Press, 1896), p. 154 [Liberty Fund edition, _Lectures on Jurisprudence_ , p. 486].\n\n3 Cf. Milton Friedman and George Joseph Stigler, _Roofs or Ceilings? The Current Housing Problem_ (New York: Foundation for Economic Education, 1946); Bertrand de Jouvenel, _No Vacancies_ (New York: Foundation for Economic Education, 1948); Sir Roy Forbes Harrod, _Are These Hardships Necessary?_ (London: Rupert Hart-Davis, 1947); Frank Walter Paish, \"The Economics of Rent Restriction,\" _Lloyds Bank Review_ , n.s., 14 (April 1950): 1\u201317, reprinted in Frank Walter Paish, _The Post-War Financial Problem, and Other Essays_ (London: Macmillan, 1950), pp. 74\u201393; Wilhelm R\u00f6pke, _Wohnungszwangswirtschaft\u2014ein europ\u00e4isches Problem_ (D\u00fcsseldorf: Deutsche Wohnung-swirtschaft, 1951); Alfred Amonn, \"Normalisierung der Wohnungswirtschaft in grunds\u00e4tzlicher Sicht,\" S _chweizer Monatshefte_ , 33 ( June 1953): 129\u2013138; and my own earlier essays, _Das Mieterschutzproblem: national\u00f6konomische Betrachtungen_ [The Rent-Control Problem: Political-Economic Considerations] (Vienna: Steyrerm\u00fchl Verlag, 1929) and \"Wirkungen der Mietzinsbeschr\u00e4nkungen\" [The Repercussions of Rent Restrictions], _Schriften des Vereins f\u00fcr Sozialpolitik_ , 182 (Leipzig: Duncker und Humblot, 1930), pp. 253\u201370.\n\n4 The illustration is given by Frank Walter Paish in his essay, \"The Economics of Rent Restriction,\" p. 4; reprinted in _The Post-War Financial Problem, and Other Essays_ , pp. 77\u201378.\n\n5 Ernst Forsthoff, _Lehrbuch des Verwaltungsrechts_. Vol. 1: _Allgemeiner Teil_ (Munich: C. H. Beck, 1950), p. 222. [The citation to which Hayek refers reads: \"Mit anderen Worten: die Verwaltungsbeh\u00f6rde darf die Erledigung ihrer Obliegenheiten nicht mit den Interessen oder An spr\u00fcchen anderer Beh\u00f6rden oder mit der Erledigung anderer Verwaltungszwecke verkuppeln, sofern nicht eine Verbindung in der Sache selbst gegeben ist.\" (\"In other words, the administrative authority may not couple the execution of its responsibilities with the interests or demands of other authorities or with the pursuit of other administrative goals, unless such a coupling follows from the matter itself.\")\u2014Ed.]\n\n6 Only recently have determined, systematic efforts been made in both Great Britain and Germany to abolish the whole system of rent controls. Even in the United States they still exist in New York City. [In 1969 New York City enacted a rent stabilization law to replace the older rent control law. As rent controlled apartments in New York City become vacant, they normally become subject to rent stabilization, which limits the rate of rent increases and stipulates the grounds on which a landlord may evict a tenant, including the manner of eviction. Allowable rent increases are determined by a Rent Guidelines Board. In 1993, high-rent units were decontrolled.\u2014Ed.]\n\n7 This possibility has not infrequently been used in various parts of the world to drive out unpopular racial minorities.\n\n8 Sir Frederick Osborn, \"How Subsidies Distort Housing Development,\" _Lloyds Bank Review_ , n.s., 36 (April 1955): 36.\n\n9 On these problems see Ralph Turvey, _Economics of Real Property_ , and Allison Dunham, \"City Planning: An Analysis of the Content of the Master Plan,\" _Journal of Law and Economics_ , 1 (1958): 170\u201386.\n\n10 The extent to which the movement for town planning, under the leadership of such men as Frederick Law Olmsted, Patrick Geddes, and Lewis Mumford, has developed into a sort of anti-economics would make an interesting study. [Frederick Law Olmsted, Jr. (1870\u20131957), the son of America's greatest landscape architect and a founder of the American town planning movement; Patrick Geddes (1854\u20131932), Scottish biologist and an outspoken adherent of urban planning; Lewis Mumford (1895\u20131990), social critic who developed a theory that urban sprawl, the undirected and uncontrolled growth of cities, was responsible for most modern social ills.\u2014Ed.]\n\n11 It should perhaps be said, in exculpation of the British economists, that it would hardly have been possible for these absurdities ever to have become law if the decisive stage of the preparation of the legislation had not taken place at a time when the economists were almost entirely occupied with the war effort, and when the town planners had the time and a free field to put through their conception of a better postwar world. It is hardly an exaggeration to say that, at the time the act was passed, scarcely anybody in Parliament understood its implications and that probably nobody at all foresaw that the responsible minister would use the powers given to him to decree a complete confiscation of the development gain. See on the act Sir Arnold Plant, \"Land Planning and the Economic Functions of Ownership,\" _The Journal\u2014Chartered Auctioneers and Estate Agents Institute_ , 29 (1949): 284\u2013305 [While the title of the volume in which this article appears is as shown, this varies slightly from those of subsequent numbers, which carry the title _The Journal of the Chartered Auctioneers' and Estate Agents' Institute_.\u2014Ed.]; and, in addition to Ralph Turvey, _Economics of Real Property_ , see his article, \"Development Charges and the Compensation-Betterment Problem,\" _Economic Journal_ , 63 (1953): 299\u2013317, and my article \"A Levy on Increasing Effciency,\" _Financial Times_ (London), pt. 1 (April 26, 1949), \"The Economics of Development Charges,\" p. 4; pt. 2 (April 27, 1949), \"Detrimental Effects of Development Charges,\" p. 4; and pt. 3 (April 28, 1949), \"Too Little Evidence of Planning,\" p. 4.\n\n12 Charles Monroe Haar, _Land Planning Law in a Free Society: A Study of the British Town and Country Planning Act_ (Cambridge, MA: Harvard University Press, 1951); cf. my \"Review of C. M. Haar's _Land Planning Law in a Free Society_ ,\" _University of Chicago Law Review_ , 19 (1952): 620\u201326; reprinted as an Appendix under the title \"The Economics of Development Charges,\" in _Studies in Philosophy, Politics and Economics_ (Chicago: University of Chicago Press, 1967), pp. 331\u201338.\n\n13 Strictly speaking, this act was implemented by the responsible minister who had been authorized to fix the development charges at some percentage of the development gain and chose to fix them at 100 per cent.\n\n14 Central Land Board, _Practice Notes (First Series): Being Notes on Development Charges Under the Town and Country Planning Act, 1947_ (London: His Majesty's Stationery Office, 1949), Preface, pp. ii\u2013iii.\n\n15 August L\u00f6sch, _The Economics of Location_ , William Henry Woglom, trans. (New Haven: Yale University Press, 1954), pp. 343\u201344.\n\n## TWENTY-THREE\n\nAGRICULTURE AND NATURAL RESOURCES\n\nMy opinion is against an overdoing of any sort of administration, and more especially against this most momentous of all meddling on the part of authority; the meddling with the subsistence of the people.\n\n\u2014Edmund Burke\n\n1. The increase in the urban and industrial population which always accompanies the growth of wealth and civilization has in the modern Western world brought about a decrease not only in the proportion but in the absolute numbers of the agricultural population. Technological advance has so increased the productivity of human effort in the production of food that fewer men than ever before can supply the needs of a larger population. But, though an increase in population causes a proportional increase in the demand for food, as the population increase slows down and further advance mainly takes the form of a growth of income per head, less and less of this additional income is spent on an increased consumption of food. People may still be induced to spend more on food if preferred kinds are offered, but, after a certain point, per capita consumption of the cereal staples ceases to increase and may actually decrease. This increase in productivity combined with an inelastic demand means that if those engaged in agriculture are to maintain their average income (let alone keep up with the general increase in incomes), their number will have to decrease.\n\nIf such a redistribution of manpower between agriculture and other occupations takes place, there is no reason why in the long run those remaining in agriculture should not derive as much benefit from economic advance as the rest. But as long as the agricultural population is relatively too large, the change, while it proceeds, is bound to operate to their disadvantage. Spontaneous movements out of agriculture will be induced only if incomes in agriculture are reduced relative to those in urban occupations. The greater the reluctance of the farmers or peasants to shift to other occupations, the greater  the differences in incomes will be during the transitional period. Particularly when the change continues over several generations, the differences will be kept small only if the movements are relatively fast.\n\nPolicy, however, has everywhere delayed this adjustment, with the result that the problem has steadily grown in magnitude. The part of the population which has been kept in agriculture by deliberate acts of policy has grown so large that equalizing productivity between the agricultural and the industrial population would in many cases require a shift of numbers which seems altogether impracticable within any limited period of time.1\n\nThis policy has been pursued for a variety of reasons. In the European countries in which industrialization proceeded rapidly, the policy initially resulted from some vague notion about a \"proper balance\" between industry and agriculture, where \"balance\" meant little more than the maintenance of the traditional proportion between the two. In the countries which, as a consequence of their industrialization, tended to become dependent on imported food, those arguments were supported by the strategic consideration of self-sufficiency in wartime. Also it was often believed that the necessity of a transfer of population was a non-recurring one and that the problem could therefore be eased by spreading the process over a longer period. But the dominant consideration which almost everywhere led governments to interfere with it was the assurance of an \"adequate income\" to the people engaged in agriculture at the moment.\n\nThe support which the policy received from the general public was often due to the impression that the whole of the agricultural population, rather than only the less productive sections of it, was unable to earn a reasonable income. This belief was founded on the fact that the prices of agricultural products tended to fall much lower before the necessary readjustments were effected than they would have to do permanently. But it is also only this pressure of prices, which not only produces the necessary reduction in the agricultural population but leads to the adoption of the new in agricultural techniques, that will lower cost and make the survival of the suitable units possible.\n\nThe elimination of the marginal land and farms, which will reduce average costs and, by reducing supply, stop and perhaps even partly reverse the fall in product prices, is only part of the necessary readjustment. Equally important for restoring the prosperity of agriculture are the changes in its internal structure which will be induced by the changes in the relative prices of its different products. The policies pursued to assist agriculture in its difficulties, however, usually prevent those very adjustments that would make it profitable.\n\nWe can give here only one significant instance of this. As has already been said, once the general rise in incomes has exceeded a certain level, people are not likely to increase their expenditure on food unless they are offered preferred kinds. In the Western world this means mainly a substitution of high-protein foods, such as meat and dairy products, for cereals and other starchy foods. This process would be assisted if agriculture were led to produce more of these desired products at reduced relative costs. This would be brought about if the cereals were allowed to fall in price until it became profitable to use them as feed for cattle and thus indirectly produce the food that the consumers want. Such a development would prevent the total consumption of grain from shrinking as much as it would otherwise and, at the same time, decrease the costs of meat, etc. It is usually made impossible, however, by a policy of maintaining the prices of cereals at such a level that human consumption will not absorb the supply and they cannot be profitably put to other uses.\n\nThis example must suffice here as an illustration of the various ways in which the policies pursued have prevented agriculture from adapting itself to the changed conditions. With proper adaptation, a smaller number of producers (but still larger than would otherwise succeed) could increase their productivity so as to share in the general growth of prosperity. It is true, of course, that part of the trouble of agriculture is that both the character of its processes and that of the producers tend to make it peculiarly sluggish in its adaptation to change. But the remedy clearly cannot lie in making it still more resistant to adaptation. This, however, is what most of the important measures of control adopted by governments, and particularly all measures of price control, do.\n\n2. It should hardly be necessary to repeat that in the long run price controls serve no desirable purpose and that, even for a limited period, they can be made effective only if combined with direct controls of production. If they are to benefit the producers, they must be supplemented in one way or another by decisions of authority as to who is to produce, how much, and what. Since the intention is to enable the people now tilling the land to stay there and to earn an income which satisfies them, and since consumers are not willing to spend enough on food to maintain them at that level, authority must resort to forcible transfer of income. How far this is likely to be carried is best shown  by the example of Great Britain, where it is expected that the total financial assistance to agriculture will soon reach \"something like two-thirds of the aggregate net income of agriculture.\"2\n\nTwo things should be especially noted about this development. One is that in most countries the process of taking agriculture out of the market mechanism and subjecting it to increasing government direction began before the same was done in industry and that it was usually carried out with the support, or even on the initiative, of the conservatives, who have shown themselves little averse to socialistic measures if they serve ends of which they approve. The second is that the tendency was perhaps even stronger in countries where the agricultural population constituted a comparatively small part of the total but, because of a peculiar political position, was given privileges which no similar group had yet attained and which could be granted to all in no sort of system. There are few developments which give one so much cause for doubt concerning the ability of democratic government to act rationally or to pursue any intelligent designs, once it throws principles to the wind and undertakes to assure the status of particular groups. We have reached a state of affairs in agriculture where almost everywhere the more thoughtful specialists no longer ask what would be a rational policy to pursue but only which of the courses that seem politically feasible would do the least harm.\n\nIn a book such as this we can pay no attention, however, to the political necessities which the existing state of opinion imposes upon current decisions. We must confine ourselves to showing that agricultural policy has been dominated in most Western countries by conceptions which not only are self-defeating but, if generally applied, would lead to a totalitarian control of all economic activity. We cannot apply the principles of socialism for the benefit of one group only; if we do, we cannot expect to resist the demand of other groups to have their incomes similarly determined by authority according to supposed principles of justice.\n\nThe best illustration of the consequences of such policies is probably the situation which has arisen in the United States after twenty years of effort to apply the conception of \"parity.\"3 The attempt to assure to the agricultural  producers prices that stand in a fixed relation to the prices of industrial products must lead to a suspension of the forces which would bring about the necessary restriction of agricultural production to those producers operating at the lowest costs and to those products which can still be profitably produced. It is undeniable that, if these forces are to operate, the growth of incomes in agriculture during the period of transition will lag behind that of the rest of the population. But nothing we can do, short of stopping the progress of technology and wealth, will avoid the necessity of these adaptations; and the attempt to mitigate its effects by compulsory transfers of income from the urban to the agricultural population must, by delaying it, produce an ever greater backlog of postponed adaptations and so increase the difficulty of the problem.\n\nThe results of this policy in the United States\u2014the ever mounting accumulation of surplus stocks, the existence of which has become a new threat to the stability not only of American but of world agriculture, the fundamentally arbitrary and yet ineffective and irrational allocation of acreages, and so on\u2014 are too well known to need description. Few people will deny that the main problem has become that of how policy can extricate itself from the situation it has produced and that American agriculture would be in a healthier state if the government had never meddled with prices and quantities and methods of production.\n\n3. Though the irrationality and absurdity of modern agricultural policy is perhaps most easily seen in the United States, we must turn to other countries if we are to become aware of the full extent to which such policies, systematically pursued, are liable to impose restrictions on the farmer (whose \"sturdy independence\" is at the same time often referred to as an argument for maintaining him at public expense) and turn him into the most regimented and supervised of all producers.\n\nThis development has probably gone furthest in Great Britain, where a degree of supervision and control of most farming activities has been established that is not equaled this side of the iron curtain. Perhaps it is inevitable that, once farming is conducted largely at public expense, certain standards should also be enforced, and even that the penalty for what the authorities regard as bad farming should be that the offender is driven from his own  property. It is, however, a curious illusion to expect that farming will more effectively adapt itself to changing conditions if methods of cultivation are made subject to the control of a committee of neighbors and if what the majority or some superior authority regards as good farming is made the standard method universally enforced. Such restrictions may be the best way of preserving the kind of farming which we know and which many people (most of whom, one suspects, live in the city) wish to see preserved for sentimental reasons; but they can result only in the agricultural population's becoming more and more dependent.\n\nIn fact, the remarkable solicitude which the public shows in England for the fate of farming is probably due more to aesthetic than to economic considerations. The same is true to an even greater degree of the concern shown by the public in countries like Austria or Switzerland for the preservation of the mountain peasants. In all these instances a heavy burden is accepted because of the fear that the familiar face of the countryside would be changed by the disappearance of the present farming techniques and that the farmer or peasant, if he were not specially protected, would disappear altogether. It is this apprehension which causes people to be alarmed over any reduction in the agricultural population and to conjure up in their minds a picture of completely deserted villages or valleys as soon as some homesteads are abandoned.\n\nIt is, however, this very \"conservation\" which is the archenemy of a viable agriculture. It is hardly ever true that all farmers or peasants are equally threatened by any development. There are as great gaps between prosperity and poverty among farmers working under similar conditions as exist in any other occupation.4 As in all other fields, if there is to be a continuous adaptation to changing circumstances in agriculture, it is essential that the example of those individuals who are successful because they have discovered the appropriate response to a change be followed by the rest. This always means that certain types will disappear. In agriculture in particular, it means that the farmer or peasant, if he is to succeed, must progressively become a businessman\u2014 a necessary process that many people deplore and want to prevent. But the alternative for the agricultural population would be to become more and more a sort of appendage to a national park, quaint folk preserved to people the scenery, and deliberately prevented from making the mental and technological adjustments that would enable them to be self-supporting.\n\nSuch attempts to preserve particular members of the agricultural population by sheltering them against the necessity of changing strong traditions and habits must turn them into permanent wards of government, pensioners  living off the rest of the population, and lastingly dependent for their livelihood on political decisions. It would certainly be the lesser evil if some remote homesteads disappeared and in some places pastures or even forests replaced what in different conditions had been arable land. Indeed, we should be showing more respect for the dignity of man if we allowed certain ways of life to disappear altogether instead of preserving them as specimens of a past age.\n\n4. The contention that there is in agriculture no case for control of prices or production or for any kind of over-all planning, and that most of the measures of this sort have been both economically unwise and a threat to individual liberty, does not mean that there are not genuine and important problems of agricultural policy, or that government has no important functions to perform in this field. But here, as elsewhere, these tasks involve, on the one hand, the gradual improvement of the legal institutions which will make the market function more effectively and induce the individual to take fuller account of the effects of his actions and, on the other, those true service activities in which government as the agent of the people provides certain facilities, mainly in the form of information, which, at least in certain stages of development, is not likely to be provided in any other way, though here, too, government should never arrogate to itself exclusive rights but rather facilitate the growth of voluntary efforts which may in time take over these functions.\n\nTo the first category belong all those problems which in agriculture no less than in urban affairs arise from the neighborhood effects and from the more far-reaching consequences which the use of a particular piece of land may have for the rest of the community.5 Some of these problems we shall have to consider a little later in connection with the general problem of the conservation of natural resources. There are also, however, specifically agricultural problems with regard to which our legal framework and particularly the law concerning ownership and tenure could be improved. Many of the more serious defects in the working of the price mechanism can be remedied only by the evolution of appropriate units of enterprise under single control, and sometimes perhaps only by appropriate groups collaborating for certain purposes. How far such an evolution of appropriate forms of organization will go will depend largely on the character of the land law, including the possibilities that it provides, under the necessary safeguards, for compulsory expropriation. There can be little question that the consolidation of dispersed holdings inherited in Europe from the Middle Ages or the enclosures of the commons in England were necessary legislative measures to make improvements by individual  efforts possible. And it is at least conceivable, though the actual experience with \"land reforms\" gives little ground for confidence, that in certain circumstances changes in the land law may assist the breakup of latifundia which have become uneconomical but are kept in existence by certain features of the existing law. While there is room for such gradual improvement in the legal framework, the greater the freedom of experimentation allowed in the existing arrangements, the greater will be the likelihood that the changes will be made in the right direction.\n\nThere is also much scope for government action of a service character, especially in the form of spreading information. One of the real difficulties of agriculture in a dynamic society is that the very character of an agricultural population makes it likely that it will be less in touch with the advances and changes in knowledge than others. Where this means, as it often does with a peasantry adhering to traditional methods of cultivation, that most individuals do not even know that there is useful knowledge available and worth paying for, it will often be an advantageous investment for the community to bear some of the costs of spreading such knowledge. We all have an interest in our fellow citizens' being put in a position to choose wisely, and if some have not yet awakened to the possibilities which technological developments offer, a comparatively small outlay may often be sufficient to induce the individuals to take advantage of new opportunities and thence to advance further on their own initiative. Again the government should not become the sole dispenser of knowledge, with the power of deciding what the individual should and should not know. It is also possible that too much activity on the part of government will do harm by preventing the growth of more effective forms of voluntary effort. At any rate, there can be no objection of principle against such services being rendered by government; and the question as to which of these services will be worth while and to what extent they should be carried is one of expediency and raises no further fundamental issues.\n\n5. Though we cannot attempt here to consider seriously the peculiar problems of \"underdeveloped countries,\"6 we cannot leave the subject of agriculture without commenting briefly on the paradoxical fact that, while the old countries involve themselves in the most absurd complexities to prevent a shrinkage of their agricultural population, the new countries seem even more anxious to speed up the growth of the industrial population by artificial  means.7 Much of this endeavor on the latter's part seems to be based on a rather na\u00efve fallacy of the _post hoc ergo propter hoc_ variety: because historically the growth of wealth has regularly been accompanied by rapid industrialization, it is assumed that industrialization will bring about a more rapid growth of wealth. This involves a clear confusion of an intermediate effect with a cause. It is true that, as productivity per head increases as a result of the investment of more capital in tools, and even more as a result of investment in knowledge and skill, more and more of the additional output will be wanted in the form of industrial products. It is also true that a substantial increase in the production of food in those countries will require an increased supply of tools. But neither of these considerations alters the fact that if large-scale industrialization is to be the most rapid way of increasing average income, there must be an agricultural surplus available so that an industrial population can be fed.8 If unlimited amounts of capital were available and if the mere availability of sufficient capital could speedily change the knowledge and attitudes of an agricultural population, it might be sensible for such countries to impose a planned reconstruction of their economies on the model of the most advanced capitalist countries. This, however, is clearly not within the range of actual possibilities. It would seem, indeed, that if such countries as India and China are to effect a rapid rise in the standard of living, only a small portion of such capital as becomes available should be devoted to the creation of elaborate industrial equipment and perhaps none of it to the kind of highly automatized, \"capital-intensive\" plants that are characteristic of countries where the value of labor is very high, and that these countries should aim at spreading such capital as widely and thinly as possible among those uses that will directly increase the production of food.\n\nThe essentially unpredictable developments that may be produced by the application of advanced technological knowledge to economies extremely poor in capital are more likely to be speeded up if opportunity for free development is provided than if a pattern is imposed which is borrowed from societies in which the proportion between capital and labor is altogether different  from what it will be in the newer economies in the foreseeable future. However strong a case there may exist in such countries for the government's taking the initiative in providing examples and spending freely on spreading knowledge and education, it seems to me that the case against over-all planning and direction of all economic activity is even stronger there than in more advanced countries. I say this on both economic and cultural grounds. Only free growth is likely to enable such countries to develop a viable civilization of their own, capable of making a distinct contribution to the needs of mankind.\n\n6. Most sensible people in the West are aware that the problem of agricultural policy now is to extricate governments from a system of controls in which they have become entangled and to restore the working of the market. But in the related field of the exploitation of natural resources, prevalent opinion still is that the peculiar situation existing here requires governments to undertake far-reaching controls. This view is particularly strong in the United States, where the \"conservation movement\" has to a great extent been the source of the agitation for economic planning and has contributed much to the indigenous ideology of the radical economic reformers.9 Few arguments have been used so widely and effectively to persuade the public of the \"wastefulness of competition\" and the desirability of a central direction of important economic activities as the alleged squandering of natural resources by private enterprise.\n\nThere are several reasons why, in a new country that was rapidly settled by immigrants bringing with them an advanced technology, the problem of resource conservation should become more acute than it ever did in Europe. While there the evolution had been gradual and some sort of equilibrium had established itself long before ( partly, no doubt, because exploitation had done its worst at an early stage, as in the deforestation and consequent erosion of much of the southern slopes of the Alps), the rapid occupation in America of enormous tracts of virgin lands raised problems of a different order of magnitude. That the changes involved in bringing the whole of a continent for the first time under cultivation in the course of a single century should have caused upsets in the balance of nature which in retrospect seem regrettable need not surprise us.10 Most of those who complain about what has happened, however, are being wise after the event, and there is little reason to believe that, with  the knowledge available at the time, even the most intelligent governmental policy could have prevented those effects which are now most deplored.\n\nIt is not to be denied that there has been real waste; it must be emphasized, however, that the most important instance of this\u2014the depletion of the forests\u2014was largely due to the fact that they did _not_ become private property but were retained as public land and given over to private exploitation on terms which gave the exploiters no incentive for conservation. It is true that, with some kinds of natural resources, property arrangements that are generally adequate will not secure an efficient use and that special provisions of the law may be desirable with regard to them. Different kinds of natural resources raise separate problems in this respect which we must consider in turn.\n\nWith some natural resources, such as deposits of minerals, their exploitation necessarily means that they are gradually used up, while others can be made to bring a continuous return for an indefinite period.11 The usual complaint of the conservationists is that the former\u2014the \"stock resources\"\u2014are used up too rapidly, while the latter\u2014the \"flow resources\"\u2014are not used so as to give as high a permanent return as they would be capable of. These contentions are based partly on the belief that the private exploiter does not take a long enough view or does not have as much foreknowledge of future developments as the government and partly, as we shall see, on a simple fallacy which invalidates a great part of the usual conservationist argument.\n\nThere arises also in this connection the problem of the neighborhood effects, which may in certain instances lead to wasteful methods of exploitation unless the units of property are of such size that at least all the more important effects of any one owner's actions are reflected in the value of his own property. This problem arises in particular in connection with the various types of \"fugitive resources,\" such as game, fish, water, oil, or natural gas (and perhaps rain, too, in the near future) which we can appropriate only by using them up and which no individual exploiter will have an interest in conserving, since what he does not take will be taken by others. They give rise to situations in which either private property cannot exist (as with deep-sea fisheries and most other forms of wild-life resources), and we have, in consequence, to find some substitute arrangement, or where private property will lead to rational use only if the scope of unified control is made coextensive with the range within which the same resource can be tapped, as with a pool of oil. It is undeniable that where for such technological reasons we cannot have exclusive control of particular resources by individual owners, we must resort to alternative forms of regulation.\n\nIn a sense, of course, most consumption of irreplaceable resources rests on an act of faith. We are generally confident that, by the time the resource is exhausted, something new will have been discovered which will either satisfy the same need or at least compensate us for what we no longer have, so that we are, on the whole, as well off as before. We are constantly using up resources on the basis of the mere probability that our knowledge of available resources will increase indefinitely\u2014and this knowledge does increase in part because we are using up what is available at such a fast rate. Indeed, if we are to make full use of the available resources, we must act on the assumption that it will continue to increase, even if some of our particular expectations are bound to be disappointed. Industrial development would have been greatly retarded if sixty or eighty years ago the warning of the conservationists about the threatening exhaustion of the supply of coal had been heeded; and the internal combustion engine would never have revolutionized transport if its use had been limited to the then known supplies of oil (during the first few decades of the era of the automobile and the airplane the known resources of oil at the current rate of use would have been exhausted in ten years). Though it is important that on all these matters the opinion of the experts about the physical facts should be heard, the result in most instances would have been very detrimental if they had had the power to enforce their views on policy.\n\n7. The chief arguments that have persuaded people of the necessity of central direction of the conservation of natural resources are that the community has a greater interest in and a greater foreknowledge of the future than the individuals and that the preservation of particular resources raises problems different from those of the provision for the future in general.\n\nThe implications of the contention that the community has a greater interest in providing for the future than do individuals go far beyond the problems of the conservation of natural resources. The contention is not merely that certain future needs, such as security or defense, can be provided for only by the community as a whole. It is also that the community should generally devote a larger proportion of its resources to provision for the future than will result from the separate decisions of the individuals. Or, as it is often put, future needs should be valued more highly (or discounted at a lower rate of interest) by the community than is done by individuals. If valid, this contention would indeed justify central planning of most economic activity. There is, however, nothing to support this but the arbitrary judgment of those who maintain it.\n\nThere is no more justification in a free society for relieving the individuals of the responsibility for the future than there is for claiming that past generations ought to have made more provision for us than they did. The contention is made no more conclusive by the often used fallacious argument that, because government can borrow at cheaper rates, it is in a better position to  take care of future needs. It is fallacious because the advantage which governments have in this respect rests solely on the fact that the risk of failure in its investments is not borne by them but by the taxpayer; in fact, the risk is no less, so far as judgment of the worthwhileness of the particular investment is concerned. But, since governments that can recoup themselves by taxation if the investment does not bring the expected return usually count only the interest they actually pay as costs of the capital they are using, the argument operates in fact against, rather than in favor of, government investment.\n\nThe claim that the government possesses superior knowledge raises a more complex problem. It cannot be denied that there are some facts concerning probable future developments which the government is more likely to know than most of the individual owners of natural resources. Many of the more recent achievements of science illustrate this. There will always exist, however, an even greater store of knowledge of special circumstances that ought to be taken into account in decisions about specific resources which only the individual owners will possess and which can never be concentrated within a single authority. Thus, if it is true that the government is likely to know some facts known to few others, it is equally true that the government will be necessarily ignorant of an even greater number of relevant facts known to some others. We can bring together all the knowledge that is relevant to particular problems only by dispersing downward the generic knowledge available to the government, not by centralizing all the special knowledge possessed by individuals. There is probably no instance where authority can possess superior knowledge of all the facts that ought to influence a specific decision; and, while it is possible to communicate to the owners of particular resources the more general considerations that they ought to take into account, it is not possible for authority to learn all the different facts known to the individuals.\n\nThis appears perhaps most clearly where the problem concerns the rate at which stock resources, such as mineral deposits, ought to be used up. An intelligent decision presupposes a rational estimate of the future course of prices of the materials in question, and this in turn depends on forecasts of future technological and economic developments which the small individual owner is usually not in a position to make intelligently. This does not mean, however, that the market will not induce individual owners to act as if they took these considerations explicitly into account, or that such decisions should not be left to them who alone know many of the circumstances which determine the present usefulness of a particular deposit. Though they may know little about probable future developments, they will be influenced in their decisions by the knowledge of others who make it their concern to estimate such probabilities and who will be prepared to offer for the resources prices determined by these estimates. If the owner can get a higher return by selling to those who want to conserve than by exploiting the particular resource himself, he will do so.  There will normally exist a potential sale price of the resource which will reflect opinion about all the factors likely to affect its future value, and a decision based on the comparison of its value as a salable asset with what it would bring if exploited now will probably take into account more of all the relevant knowledge than could any decision of a central authority.\n\nIt has often been demonstrated that, in the case of rare natural resources, exploitation by a monopoly is likely to extend their use over a longer period and that this is perhaps the only instance where such monopolies are likely to be formed and to persist in a free economy.12 I cannot go all the way with those who use this as an argument in favor of such monopolies, because I am not persuaded that the greater degree of conservation which a monopoly would practice is desirable from a social point of view. But for those who want more conservation because they believe that the market habitually underestimates future needs, the monopolies that are likely to develop spontaneously in such instances provide the answer.\n\n8. Much of the argument for conservation, however, rests simply on an unreasoned prejudice. Its proponents take for granted that there is something particularly desirable about the flow of services that a given resource can provide at any one time and that this rate of output should be permanently maintained. Though they recognize that this is impossible with regard to stock resources, they consider it a calamity if the rate of return of flow resources is diminished below the level at which it is physically possible to maintain it. This position is often taken with regard to both the fertility of the soil in general and the stock of game, fish, etc.\n\nTo bring out the crucial point most strongly, we shall consider here the most conspicuous instance of this prejudice, where most people are inclined to accept uncritically the fallacy of much of the conservationist argument. It is the belief that the natural fertility of the soil should in all circumstances be preserved and that what is branded as \"soil mining\" should in all circumstances be avoided. It can be easily shown that as a general proposition, this is unsound and that the level at which fertility ought to be maintained has little to do with the initial condition of a given piece of land. In fact, \"soil mining\" may in certain circumstances be as much in the long-range interest of a community as the using up of any stock resource.\n\nA tract of land is often built up by cumulative deposits of organic substance to a level of fertility which, once the land is brought under cultivation, can be maintained only at costs in excess of the returns. As in certain circumstances it will be desirable to build up the fertility of a piece of land by artificially enriching it to a level at which what is annually put in will be repaid by the  increase of the product, so in certain other circumstances it will be desirable to allow the fertility to decline to the level at which investments will still pay. In some instances this may even mean that it is uneconomical to aim at permanent cultivation and that, after the accumulated natural fertility has been exhausted, the land ought to be abandoned, because in the given geographic or climatic conditions it cannot with advantage be permanently cultivated.\n\nTo use up a free gift of nature once and for all is in such instances no more wasteful or reprehensible than a similar exploitation of a stock resource. There may, of course, be other effects, known or probable, which a lasting change in the character of a tract of land may have and which ought to be taken into account: for example, as a result of temporary cultivation it may lose properties or potentialities that it possessed before and which could have been utilized for some other purpose. But this is a separate problem, one which does not concern us. We are concerned solely with examining the belief that, wherever possible, the flow of services from any natural resource should be kept at the highest level attainable. This may be accidentally valid in a particular instance, but never because of considerations which concern the attributes of a given piece of land or some other resource.\n\nSuch resources share with most of the capital of society the property of being exhaustible, and if we want to maintain or increase our income, we must be able to replace each resource that is being used up with a new one that will make at least an equal contribution to future income. This does not mean, however, that it should be preserved in kind or replaced by another of the same kind, or even that the total stock of natural resources should be kept intact. From a social as well as from an individual point of view, any natural resource represents just one item of our total endowment of exhaustible resources, and our problem is not to preserve this stock in any particular form, but always to maintain it in a form that will make the most desirable contribution to total income. The existence of a particular natural resource merely means that, while it lasts, its temporary contribution to our income will help us to create new ones which will similarly assist us in the future. This normally will not mean that we should replace any one resource with one of the same kind. One of the considerations which we shall have to keep in mind is that if one kind of resource becomes scarcer, the products depending on it will also be more scarce in the future. The foreseeable rise in the prices of products consequent upon the growing scarcity of a natural resource will indeed be one of the factors determining the amount of investment that will go to preserving this kind of resource.13\n\nPerhaps the best way of concisely stating the chief point is to say that all resource conservation constitutes investment and should be judged by precisely the same criteria as all other investment.14 There is nothing in the preservation of natural resources as such which makes it a more desirable object of investment than man-made equipment or human capacities; and, so long as society anticipates the exhaustion of particular resources and channels its investment in such a manner that its aggregate income is made as great as the funds available for investment can make it, there is no further economic case for preserving any one kind of resource. To extend investment in the conservation of a particular natural resource to a point where the return is lower than the capital it uses would bring elsewhere would reduce future income below what it would otherwise be. As has been well said, \"the conservationist who urges us 'to make greater provision for the future' is in fact urging a _lesser_ provision for posterity.\"15\n\n9. While most of the arguments advanced in favor of governmental control of private activity in the interest of conservation of natural resources are thus invalid and while there is little in them beyond an argument for providing more information and knowledge, the situation is different where the aim is the provision of amenities of or opportunities for recreation, or the preservation of natural beauty or of historical sites or places of scientific interest, etc. The kinds of services that such amenities render to the public at large, which often enable the individual beneficiary to derive advantages for which he cannot be charged a price, and the size of the tracts of land usually required make this an appropriate field for collective effort.\n\nThe case for natural parks, nature reservations, etc., is exactly of the same sort as that for similar amenities which municipalities provide on a smaller scale. There is much to be said for their being provided as far as possible by voluntary organizations, such as the National Trust in Great Britain, rather than through the compulsory powers of government. But there can be no objection to the government's providing such amenities where it happens to be the owner of the land in question or, indeed, where it has to acquire it out of funds raised by taxation or perhaps even by compulsory purchase, so long as the community approves this, in full awareness of the cost, and realizes that this is one aim competing with others and not a unique objective overriding all other needs. If the taxpayer knows the full extent of the bill he will have to foot and has the last word in the decision, there is nothing further to be said about these problems in general terms.\n\n* * *\n\nThe quotation at the head of the chapter is the concluding sentence of Edmund Burke, _Thoughts and Details upon Scarcity_ (1795), in _Works_ , vol. 7, p. 419 [Liberty Fund edition, _Selected Works_ , vol. 3, p. 92].\n\n1 See Eric Mervyn Ojala, _Agriculture and Economic Progress_ (London: Oxford University Press, 1952); Kenneth Ewart Boulding, \"Economic Analysis and Agricultural Policy,\" _Canadian Journal of Economic and Political Science_ , 13 (1947): 436\u201346, reprinted in _Contemporary Readings in Agricultural Economics_ , Harold Graham Halcrow, ed. (New York: Prentice-Hall, 1955), pp. 195\u2013220; Theodore William Schultz, _Agriculture in an Unstable Economy_ (New York: McGraw-Hill, 1945); Jean Fourastie, _Le grand espoir du XX_ _e_ _si\u00e8cle: progr\u00e8s technique, progr\u00e8s \u00e9conomique, progr\u00e8s sociale_ (Paris: Presses universitaires de France, 1949); Heinrich Niehaus, _Leitbilder der Wirtschafts- und Agrarpolitik in der modernen Gesellschaft_ (Stuttgart: H. Seewald, 1957); and Heinrich Niehaus and Hermann Priebe, _Agrarpolitik in der sozialen Marktwirtschaft. Wortlaut der Vortr\u00e4ge und Diskussion auf der F\u00fcnften Arbeitstagung der Aktionsgemeinschaft Soziale Marktwirtschaft am 13. M\u00e4rz 1956 in Bad Godesberg_ (Ludwigsburg: M. Hoch, 1956). Also William H. Peterson, _The Great Farm Problem_ (Chicago: Henry Regnery, 1959).\n\n2 Sir Ralph Enfield, \"How Much Agriculture?\" _Lloyds Bank Review_ , n.s., 32 (April 1954): 30. [In 2001, total support to agriculture in the United States amounted to 64% of the sector's contribution to GDP, while the figure for the European Union was 66% and for Japan 127%! (See the World Trade Organization, _Annual Report, 2003_ [Geneva: World Trade Organization, 2003]).\u2014Ed.].\n\n3 It perhaps deserves mention, since this is little known, that in this field, too, the inspiration for the control measures seems to have come from Germany. Cf. the account in Arthur Meier Schlesinger, Jr., _The Age of Roosevelt: The Crisis of the Old Order, 1919\u20131933_ (Boston: Houghton Mifflin, 1957), p. 110: \"In the late twenties Beardsley Ruml of the Laura Spelman Rockefeller Foundation, impressed by a program of agricultural control he observed in operation in Germany, asked John Black, now at Harvard, to investigate its adaptability to the American farm problem. In 1929 Black worked out the details of what he christened the voluntary domestic allotment plan.\" [Beardsley Ruml was at one point Treasurer of R. H. Macy, the department store, and went on to become Chairman of the Federal Reserve Bank of New York and a close advisor to President Franklin D. Roosevelt. Among his many contributions to the modern state was the idea of withholding income taxes from workers' paychecks. John Black, an advisor in farm policy in the Roosevelt administration, later joined the Harvard faculty, where he became one of the most persistent voices supporting more extensive government. Black's proteg\u00e9, John Kenneth Galbraith, was brought into the department largely at Black's urging.\u2014Ed.]\n\n4 Cf. Hilde Weber, _Die Landwirtschaft in der volkswirtschaftlichen Entwicklung: eine Betrachtung \u00fcber Besch\u00e4ftigung und Einkommen_ [inaugural dissertation, Universit\u00e4t Bonn; Sonderheft No. 161] (Hamburg: P. Parey, 1955).\n\n5 On the extent to which \"soil conservation\" has often served merely as a pretext for economic controls see Charles Meyer Hardin, _The Politics of Agriculture: Soil Conservation and the Struggle for Power in Rural America_ (Glencoe, IL: Free Press, 1952), and Frederic Benham, _Economic Aid to Under-developed Countries_ (Oxford: Oxford University Press, 1959).\n\n6 On the problems of undeveloped countries and assistance to their economic development see particularly Peter Tam\u00e1s Bauer, _Economic Analysis and Policy in Underdeveloped Countries_ (Cambridge: Cambridge University Press, 1957); Sally Herbert Frankel, _The Economic Impact on Under-developed Societies: Essays on International Investment and Social Change_ (Oxford: Blackwell, 1953); Frederic Benham, \" Reflexiones sobre los pa\u00edses insuficientemente desarrollados,\" _El Trimestre econ\u00f3mico_ , 19 (1952): 45\u201357; and Milton Friedman, \"Foreign Economic Aid, Means and Objectives,\" _Yale Review_ , n.s., 47 (1958): 500\u2013516.\n\n7 This has its complement in the fact, first pointed out, I believe, by Frank Walter Paish, that today the wealthy countries regularly overpay their farmers while the poor countries generally underpay them.\n\n8 The important and well-established fact of the necessity of the development of an agricultural surplus before rapid industrialization can bring a growth of wealth is particularly well brought out by Kenneth Ewart Boulding, \"Economic Analysis and Agricultural Policy,\" esp. p. 440, reprinted in _Contemporary Readings in Agricultural Economics_ , esp. p. 197: \"The so called 'industrial revolution' was not created by a few rather unimportant technical changes in the textile industry; it was the direct child of the agricultural revolution based on turnips, clover, four-course rotation, and livestock improvement which developed in the first half of the eighteenth century. It is the turnip, not the spinning jenny, which is the father of industrial society.\" See Samuel Pfrimmer Hays, _Conservation and the Gospel of Effciency: The Progressive Conservation Movement, 1890\u20131920_ [Harvard Historical Monographs, No. 40] (Cambridge, MA: Harvard University Press, 1960).\n\n9 It is significant that, as has been pointed out by Anthony Scott, _National Resources: The Economics of Conservation_ (Toronto: University of Toronto Press, 1955), p. 37, \"the whole school of land economics (and its cousin, institutional economics)\" largely traces back to this concern of Americans.\n\n10 Cf. Paul Bigelow Sears, \"Science and National Resources,\" _American Scientist_ , 44 (1956): 331\u2013 46, and \"The Processes of Environmental Change by Man,\" in _Man's Role in Changing the Face of the Earth_ , William Leroy Thomas, Jr., ed. [International Symposium on Man's Role in Changing the Face of the Earth, Princeton, NJ, 1955] (Chicago: University of Chicago Press, 1956), pp. 471\u201384.\n\n11 See mainly Scott, _National Resources_ ; Scott Gordon, \"Economics and the Conservation Question,\" _Journal of Law and Economics_ , 1 (1958): 110\u201321; and Siegfried von Ciriacy-Wantrup, _Resource Conservation: Economics and Policies_ (Berkeley: University of California Press, 1952).\n\n12 Cf. Ludwig von Mises, _Socialism_ , p. 392 [Liberty Fund edition, p. 350], and Scott, _National Resources_ , pp. 82\u201385.\n\n13 Cf. my _The Pure Theory of Capital_ (London: Macmillan, 1941), chap. 7, pp. 85\u201394, esp. p. 88n [reprinted as vol. 12 of _The Collected Works of F. A. Hayek_ , Lawrence H. White, ed. (Chicago: University of Chicago Press, 2007), pp. 100\u2013107, esp. p. 104, n. 2].\n\n14 See Scott, _National Resources_ , p. 8.\n\n15 Ibid., p. 97.\n\n## TWENTY-FOUR\n\nEDUCATION AND RESEARCH\n\nA general State education is a mere contrivance for moulding people to be exactly like one another: and as the mould in which it casts them is that which pleases the predominant power in the government, whether this be a monarch, a priesthood, an aristocracy, or the majority of the existing generation; in proportion as it is efficient and successful, it establishes a despotism over the mind, leading by natural tendency to one over the body.\n\n\u2014John Stuart Mill\n\n1. Knowledge is perhaps the chief good that can be had at a price, but those who do not already possess it often cannot recognize its usefulness. More important still, access to the sources of knowledge necessary for the working of modern society presupposes the command of certain techniques\u2014above all, that of reading\u2014which people must acquire before they can judge well for themselves what will be useful to them. Though our case for freedom rests to a great extent on the contention that competition is one of the most powerful instruments for the dissemination of knowledge and that it will usually demonstrate the value of knowledge to those who do not possess it, there is no doubt that the utilization of knowledge can be greatly increased by deliberate  efforts. Ignorance is one of the chief reasons why men's endeavors are often not channeled so that they are most useful to their fellows; and there are various reasons why it may be in the interest of the whole community that knowledge be brought to people who have little incentive to seek it or to make some sacrifice to acquire it. These reasons are particularly compelling in the case of children, but some of the arguments apply no less to adults.\n\nWith regard to children the important fact is, of course, that they are not responsible individuals to whom the argument for freedom fully applies. Though it is generally in the best interest of children that their bodily and mental welfare be left in the care of their parents or guardians, this does not mean that parents should have unrestricted liberty to treat their children as they like. The other members of the community have a genuine stake in the welfare of the children. The case for requiring parents or guardians to provide for those under their care a certain minimum of education is clearly very strong.1\n\nIn contemporary society, the case for compulsory education up to a certain minimum standard is twofold. There is the general argument that all of us will be exposed to fewer risks and will receive more benefits from our fellows if they share with us certain basic knowledge and beliefs. And in a country with democratic institutions there is the further important consideration that democracy is not likely to work, except on the smallest local scale, with a partly illiterate people.2\n\nIt is important to recognize that general education is not solely, and perhaps not even mainly, a matter of communicating knowledge. There is a need for certain common standards of values, and, though too great emphasis on this need may lead to very illiberal consequences, peaceful common existence would be clearly impossible without any such standards. If in long-settled communities with a predominantly indigenous population, this is not likely to be a serious problem, there are instances, such as the United States during the period of large immigration, where it may well be one. That the United States would not have become such an effective \"melting pot\" and would probably have faced extremely difficult problems if it had not been for a deliberate policy of \"Americanization\" through the public school system seems fairly certain.\n\nThe fact that all education must be and ought to be guided by definite values is, however, also the source of real dangers in any system of public education. One has to admit that in this respect most nineteenth-century liberals were guided by a na\u00efve overconfidence in what mere communication of knowledge could achieve. In their rationalistic liberalism they often presented the case for general education as though the dispersion of knowledge would solve all major problems and as though it were necessary only to convey to the masses that little extra knowledge which the educated already possessed in order that this \"conquest of ignorance\" should initiate a new era. There is not much reason to believe that, if at any one time the best knowledge which some possess were made available to all, the result would be a much better society. Knowledge and ignorance are very relative concepts, and there is little evidence that the difference in knowledge which at any one time exists between the more and the less educated of a society can have such a decisive influence on its character.\n\n2. If we accept the general argument for compulsory education, there remain these chief problems: How is this education to be provided? How much of it is to be provided for all? How are those who are to be given more to be selected and at whose expense? It is probably a necessary consequence of the adoption of compulsory education that for those families to whom the cost would be a severe burden it should be defrayed out of public funds. There is still the question, however, how much education should be provided at public expense and in what manner it should be provided. It is true that, historically, compulsory education was usually preceded by the governments' increasing opportunities by providing state schools. The earliest experiments with making education compulsory, those in Prussia at the beginning of the eighteenth century, were in fact confined to those districts where the govern  ment had provided schools. There can be little doubt that in this manner the process of making education general was greatly facilitated. Imposing general education on a people largely unfamiliar with its institutions and advantages would indeed be difficult. This does not mean, however, that compulsory education or even government-financed general education today requires the educational institutions to be run by the government.\n\nIt is a curious fact that one of the first effective systems under which compulsory education was combined with the provision of most educational institutions by the government was created by one of the great advocates of individual liberty, Wilhelm von Humboldt, only fifteen years after he had argued that public education was harmful because it prevented variety in accomplishments and unnecessary because in a free nation there would be no lack of educational institutions. \"Education,\" he had said, \"seems to me to lie wholly beyond the limits within which political agency should be properly confined.\"3 It was the plight of Prussia during the Napoleonic wars and the needs of national defense that made him abandon his earlier position. The desire for \"the development of the individual personalities in their greatest variety\" which had inspired his earlier work became secondary when desire for a strong organized state led him to devote much of his later life to the building of a system of state education that became a model for the rest of the world. It can scarcely be denied that the general level of education which Prussia thus  attained was one of the chief causes of her rapid economic rise and later that of all Germany. One may well ask, however, whether this success was not bought at too high a price. The role played by Prussia during the succeeding generations may make one doubt whether the much lauded Prussian schoolmaster was an unmixed blessing for the world, or even for Prussia.\n\nThe very magnitude of the power over men's minds that a highly centralized and government-dominated system of education places in the hands of the authorities ought to make one hesitate before accepting it too readily. Up to a point, the arguments that justify compulsory education also require that government should prescribe some of the content of this education. As we have already mentioned, there may be circumstances in which the case for authority's providing a common cultural background for all citizens becomes very strong. Yet we must remember that it is the provision of education by government which creates such problems as that of the segregation of Negroes in the United States\u2014difficult problems of ethnic or religious minorities which are bound to arise where government takes control of the chief instruments of transmitting culture. In multinational states the problem of who is to control the school system tends to become the chief source of friction between nationalities. To one who has seen this happen in countries like the old Austria-Hungary, there is much force in the argument that it may be better even that some children should go without formal education than that they should be killed in fighting over who is to control that education.4\n\nEven in ethnically homogeneous states, however, there are strong arguments against entrusting to government that degree of control of the contents of education which it will possess if it directly manages most of the schools that are accessible to the great masses. Even if education were a science which provided us with the best of methods of achieving certain goals, we could hardly wish the latest methods to be applied universally and to the complete exclusion of others\u2014still less that the aims should be uniform. Very few of the problems of education, however, are scientific questions in the sense that they can be decided by any objective tests. They are mostly either outright questions of value, or at least the kind of questions concerning which the only ground for trusting the judgment of some people rather than that of others is that the former have shown more good sense in other respects. Indeed, the very possibility that, with a system of government education, all elementary education may come to be dominated by the theories of a particular group who genuinely believe that they have scientific answers to those prob  lems (as has happened to a large extent in the United States during the last thirty years) should be sufficient to warn us of the risks involved in subjecting the whole educational system to central direction.\n\n3. In fact, the more highly one rates the power that education can have over men's minds, the more convinced one should be of the danger of placing this power in the hands of any single authority. But even if one does not rate its power to do good as highly as did some of the rationalistic liberals of the nineteenth century, however, the mere recognition of this power should lead us to conclusions almost the opposite of theirs. And if, at present, one of the reasons why there should be the greatest variety of educational opportunities is that we really know so little about what different educational techniques may achieve, the argument for variety would be even stronger if we knew more about the methods of producing certain types of results\u2014as we soon may.\n\nIn the field of education perhaps more than in any other, the greatest dangers to freedom are likely to come from the development of psychological techniques which may soon give us far greater power than we ever had to shape men's minds deliberately. But knowledge of what we can make of human beings if we can control the essential conditions of their development, though it will offer a frightful temptation, does not necessarily mean that we shall by its use improve upon the human being who has been allowed to develop freely. It is by no means clear that it would be a gain if we could produce the human types that it was generally thought we needed. It is not at all unlikely that the great problem in this field will soon be that of preventing the use of powers which we do possess and which may present a strong temptation to all those who regard a controlled result as invariably superior to an uncontrolled one. Indeed, we may soon find that the solution has to lie in government ceasing to be the chief dispenser of education and becoming the impartial protector of the individual against all uses of such newly found powers.\n\nNot only is the case against the management of schools by government now stronger than ever, but most of the reasons which in the past could have been advanced in its favor have disappeared. Whatever may have been true then, there can be little doubt that today, with the traditions and institutions of universal education firmly established and with modern transportation solving most of the difficulties of distance, it is no longer necessary that education be not only financed but also provided by government.\n\nAs has been shown by Professor Milton Friedman,5 it would now be entirely practicable to defray the costs of general education out of the public purse  without maintaining government schools, by giving the parents vouchers covering the cost of education of each child which they could hand over to schools of their choice. It may still be desirable that government directly provide schools in a few isolated communities where the number of children is too small (and the average cost of education therefore too high) for privately run schools. But with respect to the great majority of the population, it would undoubtedly be possible to leave the organization and management of education entirely to private efforts, with the government providing merely the basic finance and ensuring a minimum standard for all schools where the vouchers could be spent. Another great advantage of this plan is that parents would no longer be faced with the alternative of having to accept whatever education the government provides or of paying the entire cost of a different and slightly more expensive education themselves; and if they should choose a school out of the common run, they would be required to pay only the additional cost.\n\n4. A more difficult problem is how much education is to be provided at public expense and for whom such education is to be provided beyond the minimum assured to all. It can hardly be doubted that the number of those whose contribution to the common needs will be increased by education extended beyond a certain stage sufficiently to justify the cost will always be only a small proportion of the total population. Also, it is probably undeniable that we have no certain methods of ascertaining beforehand who among the young people will derive the greatest benefit from an advanced education. Moreover, whatever we do, it seems inevitable that many of those who get an advanced education will later enjoy material advantages over their fellows only because someone else felt it worthwhile to invest more in their education, and not because of any greater natural capacity or greater effort on their part.\n\nWe shall not stop to consider how much education is to be provided for all or how long all children should be required to attend school. The answer must depend in part on particular circumstances, such as the general wealth of the community, the character of its economy, and perhaps even climatic conditions affecting the age of adolescence. In wealthier communities the problem usually is no longer one of what schooling will increase economic efficiency but rather one of how to occupy children, until they are allowed to earn a living, in a manner that will later assist them in better using their leisure.\n\nThe really important issue is that of the manner in which those whose education is to be prolonged beyond the general minimum are to be selected. The costs of a prolonged education, in terms of material resources and still more of human ones, are so considerable even for a rich country that the desire to give a large fraction of the population an advanced education will always in some degree conflict with the desire to prolong the education for all. It also seems probable that a society that wishes to get a maximum economic return  from a limited expenditure on education should concentrate on the higher education of a comparatively small elite,6 which today would mean increasing that part of the population getting the most advanced type of education rather than prolonging education for large numbers. Yet, with government education, this would not seem practicable in a democracy, nor would it be desirable that authority should determine who is to get such an education.\n\nAs in all other fields, the case for subsidization of higher education (and of research) must rest not on the benefit it confers on the recipient but on the resulting advantages for the community at large. There is, therefore, little case for subsidizing any kind of vocational training, where the greater proficiency acquired will be reflected in greater earning power, which will constitute a fairly adequate measure of the desirability of investing in training of this kind. Much of the increased earnings in occupations requiring such training will be merely a return on the capital invested in it. The best solution would seem to be that those in whom such investment would appear to promise the largest return should be enabled to borrow the capital and later repay it out of their increased earnings, though such an arrangement would meet with considerable practical difficulties.7\n\nThe situation is somewhat different, however, where the costs of a higher education are not likely to result in a corresponding increase in the price at which the services of the better-trained man can be sold to other individuals (as is the case in the professions of medicine, the law, engineering, and so on) but where the aim is the further dispersion and increase in knowledge throughout the community at large. The benefits that a community receives from its scientists and scholars cannot be measured by the price at which these men can sell particular services, since much of their contribution becomes freely available to all. There is therefore a strong case for assisting at least some of those who show promise and inclination for the pursuit of such studies.\n\nIt is a different matter, however, to assume that all who are intellectually capable of acquiring a higher education have a claim to it. That it is in the general interest to enable all the specially intelligent to become learned is by no means evident or that all of them would materially profit by such an advanced education, or even that such an education should be restricted to those who have an unquestionable capacity for it and be made the normal or perhaps the exclusive path to higher positions. As has been pointed out recently, a much sharper division between classes might come to exist, and the less fortunate might become seriously neglected, if all the more intelli  gent were deliberately and successfully brought into the wealthy group and it became not only a general presumption but a universal fact that the relatively poor were less intelligent. There is also another problem which has assumed serious proportions in some European countries and which we ought to keep in mind, and this is the problem of having more intellectuals than we can profitably employ. There are few greater dangers to political stability than the existence of an intellectual proletariat who find no outlet for their learning.\n\nThe general problem we are faced with in all higher education, then, is this: by some method, certain young people must be selected, at an age when one cannot know with any certainty who will profit most, to be given an education that will enable them to earn a higher income than the rest; and to justify the investment, they must be selected so that, on the whole, they will be qualified to earn a higher income. Finally, we have to accept the fact that, since as a rule somebody else will have to pay for the education, those who benefit from it will thus be enjoying an \"unearned\" advantage.\n\n5. In recent times the difficulties of this problem have been greatly increased and a reasonable solution made almost impossible by the increasing use of government education as an instrument for egalitarian aims. Though a case can be made for assuring opportunities for an advanced education as far as possible to those most likely to profit from them, the control of government over education has in large measure been used to equalize the prospects of all, which is something very different. Though egalitarians usually protest against the imputation that their goal is any sort of mechanical equality which would deprive some people of advantages which cannot be provided for all, there is in education a clear indication that such is the tendency. This egalitarian stand is usually not so explicitly argued as in R. H. Tawney's _Equality_ , in which influential tract the author contends that it would be unjust \"to spend less liberally on the education of the slow than on that of the intelligent.\"8 But to some extent the two conflicting desires of equalizing opportunity and of adjusting opportunity to capacity (which, as we know, has little to do with merit in any moral sense) have become everywhere confused.\n\nIt should be admitted that, so far as education at public expense is concerned, the argument for equal treatment of all is strong. When it is combined, however, with an argument against permitting any special advantages to the more fortunate ones, it means in effect that all must be given what any child gets and that none should have what cannot be provided for all. Consistently pursued, it would mean that no more must be spent on the education of any child than can be spent on the education of every child. If this were the necessary consequence of public education, it would constitute a strong argu  ment against government's concerning itself with education beyond the elementary level, which can indeed be given to all, and for leaving all advanced education in private hands.\n\nAt any rate, the fact that certain advantages must be limited to some does not mean that a single authority should have exclusive power to decide to whom they should go. It is not likely that such power in the hands of authority would in the long run really advance education or that it would create social conditions that would be felt to be more satisfactory or just than they would otherwise have been. On the first point it should be clear that no single authority should have the monopoly of judging how valuable a particular kind of education is and how much should be invested in more education or in which of the different kinds of education. There is not\u2014and cannot be in a free society\u2014a single standard by which we can decide on the relative importance of different aims or the relative desirability of different methods. Perhaps in no other field is the continued availability of alternative ways as important as in that of education, where the task is to prepare young people for an ever changing world.\n\nSo far as justice is concerned, we should be clear that those who in the general interest most \"deserve\" an advanced education are not necessarily those who by effort and sacrifice have earned the greatest subjective merit. Natural capacity and inborn aptitude are as much \"unfair advantages\" as accidents of environment, and to confine the advantages of higher education to those that we can confidently foresee profiting most from them will necessarily increase rather than decrease the discrepancy between economic status and subjective merit.\n\nThe desire to eliminate the effects of accident, which lies at the root of the demand for \"social justice,\" can be satisfied in the field of education, as elsewhere, only by eliminating all those opportunities which are not subject to deliberate control. But the growth of civilization rests largely on the individuals' making the best use of whatever accidents they encounter, of the essentially unpredictable advantages that one kind of knowledge will in new circumstances confer on one individual over others.\n\nHowever commendable may be the motives of those who fervently desire that, in the interest of justice, all should be made to start with the same chances, theirs is an ideal that is literally impossible to realize. Furthermore, any pretense that it has been achieved or even closely approached can only make matters worse for the less successful. Though there is every case for removing whatever special obstacles existing institutions may put in the way of some, it is neither possible nor desirable to make all start with the same chances, since this can be achieved only by depriving some of possibilities that cannot be provided for all. While we wish everybody's opportunities to be as great as possible, we should certainly decrease those of most if we were  to prevent them from being any greater than those of the least fortunate. To say that all who live at the same time in any given country should start at the same place is no more reconcilable with a developing civilization than to say that this kind of equality should be assured to people living at different times or at different places.\n\nIt may be in the interest of the community that some who show exceptional capacities for scholarly or scientific pursuits should be given an opportunity to follow them irrespective of family means. But this does not confer a right on anyone to such opportunity; nor does it mean that only those whose exceptional capacities can be ascertained ought to have the opportunity or that nobody should have it unless it can be assured to all who can pass the same objective tests.\n\nNot all the qualities which enable one to make special contributions are ascertainable by examinations or tests, and it is more important that at least some of those who possess such qualities have an opportunity than that it be given to all who satisfy the same requirements. A passionate desire for knowledge or an unusual combination of interests may be more important than the more visible gifts or any testable capacities; and a background of general knowledge and interests or a high esteem for knowledge produced by family environment often contributes more to achievement than natural capacity. That there are some people who enjoy the advantages of a favorable home atmosphere is an asset to society which egalitarian policies can destroy but which cannot be utilized without the appearance of unmerited inequalities. And since a desire for knowledge is a bent that is likely to be transmitted through the family, there is a strong case for enabling parents who greatly care for education to secure it for their children by a material sacrifice, even if on other grounds these children may appear less deserving than others who will not get it.9\n\n6. The insistence that education should be given only to those of proved capacity produces a situation in which the whole population is graded according to some objective test and in which one set of opinions as to what kind of person qualifies for the benefits of an advanced education prevails throughout. This means an official ranking of people into a hierarchy, with the certified  genius on top and the certified moron at the bottom, a hierarchy made much worse by the fact that it is presumed to express \"merit\" and will determine access to the opportunities in which value can show itself. Where exclusive reliance on a system of government education is intended to serve \"social justice,\" a single view of what constitutes an advanced education\u2014and then of the capacities which qualify for it\u2014will apply throughout, and the fact that somebody has received an advanced education will be presumed to indicate that he had \"deserved\" it.\n\nIn education, as in other fields, the admitted fact that the public has an interest in assisting some must not be taken to mean that only those who are judged by some agreed view to deserve assistance out of public funds should be allowed access to an advanced education, or that nobody should be allowed to assist specific individuals on other grounds. There is probably much to be said for some members of each of the different groups of the population being given a chance, even if the best from some groups seem less qualified than members of other groups who do not get it. For this reason, different local, religious, occupational, or ethnic groups should be able to assist some of the young members, so that those who receive a higher education will represent their respective group somewhat in proportion to the esteem in which the latter hold education.\n\nIt must at least seem doubtful that a society in which educational opportunities were universally awarded according to presumed capacity would be more tolerable for the unsuccessful ones than one in which accidents of birth admittedly played a great role. In Britain, where the postwar reform of education has gone a long way toward establishing a system based on presumed capacity, the consequences already cause concern. A recent study of social mobility suggests that it now \"will be the grammar schools which will furnish the new _elite_ , an elite apparently much less assailable because it is selected for 'measured intelligence.' The selection process will tend to reinforce the prestige of occupations already high in social status and to divide the population into streams which many may come to regard, indeed already regard, as distinct as sheep and goats. Not to have been to a grammar school will be a more serious disqualification than in the past, when social inequality in the educational system was known to exist. And the feeling of resentment may become more rather than less acute just because the individual concerned realizes that there is some validity in the selection process which has kept him out of grammar school. In this respect apparent justice may be more difficult to bear than injustice.\"10 Or, as another British writer has observed more generally, \"it is  one unexpected result of the Welfare State that it should make the social pattern not less rigid but more so.\"11\n\nLet us by all means endeavor to increase opportunities for all. But we ought to do so in the full knowledge that to increase opportunities for all is likely to favor those better able to take advantage of them and may often at first increase inequalities. Where the demand for \"equality of opportunity\" leads to attempts to eliminate such \"unfair advantages,\" it is only likely to do harm. All human differences, whether they are differences in natural gifts or in opportunities, create unfair advantages. But, since the chief contribution of any individual is to make the best use of the accidents he encounters, success must to a great extent be a matter of chance.\n\n7. On the highest level the dissemination of knowledge by instruction becomes inseparable from the advance of knowledge by research. The introduction to those problems which are on the boundaries of knowledge can be given only by men whose main occupation is research. During the nineteenth century the universities, particularly those on the European Continent, in fact developed into institutions which, at their best, provided education as a by-product of research and where the student acquired knowledge by working as an apprentice to the creative scientist or scholar. Since then, because of the increased amount of knowledge that must be mastered before the boundaries of knowledge are reached, and because of the increasing numbers receiving a university education without any intention of ever reaching that stage, the character of the universities has greatly changed. The greater part of what is still called \"university work\" is today in character and substance merely a continuation of school instruction. Only the \"graduate\" or \"postgraduate\" schools\u2014in fact, only the best of these\u2014are still mainly devoted to the kind of work that characterized the Continental universities of the last century.\n\nThere is no reason to think, however, that we are not as much in need of the more advanced type of work. It is still this kind of work on which the general level of the intellectual life of a country chiefly depends. And while in the experimental sciences research institutes in which the young scientists serve  their apprenticeship are in some measure fulfilling this need, there is danger that in some fields of scholarship the democratic broadening of education may be detrimental to the pursuit of that original work that keeps knowledge alive.\n\nThere is probably less cause for concern about the supposedly inadequate number of university-trained specialists that are currently being produced in the Western world12 than about the inadequate output of men of really top quality. And though, at least in the United States, and to an increasing extent also elsewhere, the responsibility for this rests mainly with the inadequate preparation by the schools and with the utilitarian bias of institutions concerned primarily with conferring professional qualifications, we must not overlook the democratic preference for providing better material opportunities for large numbers over the advancement of knowledge, which will always be the work of the relatively few and which indeed has the strongest claim for public support.\n\nThe reason why it still seems probable that institutions like the old universities, devoted to research and teaching at the boundaries of knowledge, will continue to remain the chief sources of new knowledge is that only such institutions can offer that freedom in the choice of problems and those contacts between representatives of the different disciplines that provide the best conditions for the conception and pursuit of new ideas. However greatly progress in a known direction may be accelerated by the deliberate organization of work aiming at some known goal, the decisive and unforeseeable steps in the general advance usually occur not in the pursuit of specific ends but in the exploitation of those opportunities which the accidental combination of particular knowledge and gifts and special circumstances and contacts have placed in the way of some individual. Though the specialized research institution may be the most efficient for all tasks that are of an \"applied\" character, such institutional research is always in some measure directed research, the aim of which is determined by the specialized equipment, the particular team assembled, and the concrete purpose to which the institution is dedicated. But in \"fundamental\" research on the outskirts of knowledge there are often no fixed subjects or fields, and the decisive advances will frequently be due to the disregard of the conventional division of disciplines.\n\n8. The problem of supporting the advance of knowledge in the most effective manner is therefore closely connected with the issue of \"academic freedom.\" The conceptions for which this term stands were developed in the countries of the European Continent, where the universities were generally state  institutions; thus they were directed almost entirely against political interference with the work of these institutions.13 The real issue, however, is a much wider one. There would be nearly as strong a case against any unitary planning and direction of all research by a senate composed of the most highly reputed scientists and scholars as there is against such direction by more extraneous authorities. Though it is natural that the individual scientist should most resent interference with his choice or pursuit of problems when it is motivated by what to him seem irrelevant considerations, it might be still less harmful if there were a multiplicity of such institutions, each subject to different outside pressures, than if they were all under the unified control of one single conception of what at a given moment was in the best scientific interest.\n\nAcademic freedom cannot mean, of course, that every scientist should do what seems most desirable to him. Nor does it mean self-government of science as a whole. It means rather that there should be as many independent centers of work as possible, in which at least those men who have proved their capacity to advance knowledge and their devotion to their task can themselves determine the problems on which they are to spend their energies and where they can expound the conclusions they have reached, whether or not these conclusions are palatable to their employer or the public at large.14\n\nIn practice, this means that those men who have already proved themselves in the eyes of their peers, and who, for this reason, have been given senior positions in which they can determine both their own work and that of their juniors, should be given security of tenure. This is a privilege conferred for reason similar to those which have made it desirable to make the position of judges secure, and it is conferred not in the interest of the individual but because it is rightly believed that persons in such positions will, on the whole, serve the public interest best if they are protected against pressure from outside opinion. It is of course not an unlimited privilege, and it means merely that, once it is granted, it cannot be withdrawn except for reasons specifically provided for in the original appointment.\n\nThere is no reason why these terms should not be altered for new appointments as we gain new experience, though such new conditions cannot apply to those who already possess what in the United States is called \"tenure.\" For example, recent experience seems to suggest that the terms of appointment should specify that the occupant of such a position forfeits the privilege if he  knowingly joins or supports any movement that is opposed to the very principles on which this privilege rests. Tolerance should not include the advocacy of intolerance. On this ground I feel that a Communist should not be given \"tenure,\" though, once he has been given it without such explicit limitations, it would have to be respected like any other similar appointment.\n\nAll this applies, however, only to the special privilege of \"tenure.\" Apart from these considerations pertinent to tenure, there exists little justification for anyone claiming as a matter of right the freedom to do or teach what he likes or, on the other hand, for any hard-and-fast rule stating that anyone holding a particular opinion should be universally excluded. Though an institution aiming at high standards will soon discover that it can attract first-class talent only if it grants even its youngest members a wide choice of pursuits and opinions, no one has the right to be employed by an institution irrespective of the interests and views he holds.\n\n9. The need for protecting institutions of learning against the cruder kind of interference by political or economic interests is so well recognized today that there is not much danger of its being successfully exercised in reputable institutions. There is still need for watchfulness, especially in the social sciences, where the pressure is often exercised in the name of highly idealistic and widely approved aims. Pressure against an unpopular view is more harmful than opposition to a popular one. It should certainly be a warning to us that even Thomas Jefferson argued that in the field of government the principles taught and the texts to be followed in the University of Virginia should be prescribed by authority, because the next professor might be \"one of the school of quondam federalism\"!15\n\nToday the danger lies, however, not so much in obvious outside interference as in the increased control which the growing financial needs of research give to those who hold the purse strings. It constitutes a real threat to the interests of scientific advance because the ideal of a unified and centralized direction of all scientific efforts which it might be made to serve is shared by some of the scientists themselves. Although the first great attack which, in the name of  planning of science and under strong Marxist influence, was launched in the 1930s has been successfully repelled,16 and the discussions to which it gave rise have created a greater awareness of the importance of freedom in this field, it seems probable that the attempts to \"organize\" scientific effort and to direct it to particular goals will reappear in new forms.\n\nThe conspicuous successes which the Russians have achieved in certain fields and which are the cause of the renewed interest in the deliberate organization of scientific effort should not have surprised us and should give us no reason for altering our opinion about the importance of freedom. That any one goal, or any limited number of objectives, which are already known to be achievable, are likely to be reached sooner if they are given priority in a central allocation of all resources cannot be disputed. This is the reason why a totalitarian organization is indeed likely to be more effective in a short war\u2014 and why such a government is so dangerous to the others when it is in a position to choose the most favorable moment for war. But this does not mean that the advance of knowledge in general is likely to be faster if all efforts are directed to what now seem the most important goals or that, in the long run, the nation that has more deliberately organized its efforts will be the stronger.17\n\nAnother factor that has contributed to the belief in the superiority of directed research is the somewhat exaggerated conception of the extent to which modern industry owes its progress to the organized teamwork of the great industrial laboratories. In fact, as has been shown recently in some detail,18 a much greater proportion than is generally believed even of the chief technological advances of recent times has come from individual efforts, often from men pursuing an amateur interest or who were led to their problems by accident. And what appears to be true of the more applied fields is certainly even more true of basic research, where the important advances are, by their nature, more difficult to foresee. In this field there may indeed be danger in the current emphasis on teamwork and co-operation, and it may well be the greater individualism of the European (which is partly owing to his being less used to and therefore less dependent on ample material support) which still seems to give him some advantage over the American scientist in the most original sphere of fundamental research.\n\nThere is perhaps no more important application of our main theses than that the advance of knowledge is likely to be fastest where scientific pursuits are not determined by some unified conception of their social utility, and where each proved man can devote himself to the tasks in which he sees the best chance of making a contribution. Where, as is increasingly the case in all the experimental fields, this opportunity can no longer be given by assuring to every qualified student the possibility of deciding how to use his own time, but where large material means are required for most kinds of work, the prospects of advance would be most favorable if, instead of the control of funds being in the hands of a single authority proceeding according to a unitary plan, there were a multiplicity of independent sources so that even the unorthodox thinker would have a chance of finding a sympathetic ear.\n\nThough we still have much to learn about the best manner of managing independent funds devoted to the support of research and though it may not be certain whether the influence of the very large foundations (with their inevitable dependence on majority opinion and consequent tendency to accentuate the swings of scientific fashion) has always been as beneficial as it might have been, there can be little doubt that the multiplicity of private endowments interested in limited fields is one of the most promising features of the American situation. But though present tax laws may have temporarily increased the flow of such funds, we should also remember that the same laws make the accumulation of new fortunes more difficult, and that to that extent these sources are likely to dry up in the future. As elsewhere, the preservation of freedom in the spheres of the mind and of the spirit will depend, in the long run, on the dispersal of the control of the material means and on the continued existence of individuals who are in a position to devote large funds to purposes which seem important to them.\n\n10. Nowhere is freedom more important than where our ignorance is greatest\u2014at the boundaries of knowledge, in other words, where nobody can predict what lies a step ahead. Though freedom has been threatened even there, it is still the field where we can count on most men rallying to its defense when they recognize the threat. If in this book we have been concerned mainly with freedom in other fields, it is because we so often forget today that intellectual freedom rests on a much wider foundation of freedom and cannot exist without it. But the ultimate aim of freedom is the enlargement of those capacities in which man surpasses his ancestors and to which each generation must endeavor to add its share\u2014its share in the growth of knowledge and the gradual advance of moral and aesthetic beliefs, where no superior must be allowed to enforce one set of views of what is right or good and where only further experience can decide what should prevail.\n\nIt is wherever man reaches beyond his present self, where the new emerges and assessment lies in the future, that liberty ultimately shows its value. The  problems of education and research have thus brought us back to the leading theme of this book, from where the consequences of freedom and restriction are more remote and less visible to where they most directly affect the ultimate values. And we cannot think of better words to conclude than those of Wilhelm von Humboldt which a hundred years ago John Stuart Mill put in front of his essay _On Liberty:_ \"The grand, leading principle, towards which every argument hitherto unfolded in these pages directly converges, is the absolute and essential importance of human development in its richest diversity.\"19\n\n* * *\n\nThe quotation at the head of the chapter is taken from John Stuart Mill, \"On Liberty,\" in _On Liberty and Considerations on Representative Government_ , Ronald Buchanan McCallum, ed. (Oxford: B. Blackwell, 1946), p. 95. Cf. also Bertrand Russell, commenting on the same problem ninety-five years later in his lecture, \"John Stuart Mill,\" _Proceedings of the British Academy_ , 41 (1955): 57: \"State education, in the countries which adopt [Johann Gottfried Fichte's] principles, produces, so far as it is successful, a herd of ignorant fanatics, ready at the word of command to engage in war or persecution as may be required of them. So great is this evil that the world would be a better place (at any rate, in my opinion) if State education had never been inaugurated.\" [At the heart of Fichte's philosophical system is a passion for a system of universal education that will liberate all men from their instincts to a life based on reason.\u2014Ed.] Also consider the directives issued by one of Napoleon's ministers (quoted in Pieter Geyl, _The Revolt of the Netherlands, 1555\u20131609_ [London: Williams and Norgate Ltd., 1932], p. 140): \"Education must impart the same knowledge and the same principles to all individuals living in the same society, so that they will make, as a whole, one body, informed with one and the same understanding, and working for the common good, on the basis of uniformity of views and desires.\"\n\n1 Cf. Mill, \"On Liberty,\" pp. 94\u201395: \"It is in the case of children that misapplied notions of liberty are a real obstacle to the fulfilment by the State of its duties. One would almost think that a man's children were supposed to be literally, and not metaphorically, a part of himself, so jealous is opinion of the smallest interference of law with his absolute and exclusive control over them; more jealous than of almost any interference with his own freedom of action; so much less do the generality of mankind value liberty than power. Consider, for example the case of education. Is it not almost a self-evident axiom, that the State should require and compel the education, up to a certain standard, of every human being who is born its citizen? . . . If the government would make up its mind to require for every child a good education, it might save itself the trouble of providing one. It might leave to parents to obtain the education where and how they pleased, and content itself with helping to pay the school fees of the poorer classes of children, and defraying the entire school expenses of those who have no one else to pay for them. The objections which are argued with reason against State education do not apply to the enforcement of education by the State, but to the State's taking upon itself to direct that education; which is a totally different thing.\"\n\n2 Historically, the needs of universal military service were probably much more decisive in leading most governments to make education compulsory than the needs of universal suffrage. It is also significant that (according to Max Pohlenz, _Griechische Freiheit: Wesen und Werden eines Le bensideals_ [Heidelberg: Quelle und Meyer, 1955], p. 42): \"es unter den B\u00fcrgern des alten Athen keine Analphabeten gegeben haben soll, obw ohl dessen 'freie Demokratie jede Einmischung in das Privatle-ben [vermied]. Es gab k einen Schulzwang und keine staatlichen Schulen.'\" [\"There were allegedly no illiterate persons among the citizens of ancient Athens, though their 'free democracy also avoided all interference with private life. There was no compulsory education nor state schools.'\"\u2014Ed.]\n\n3 Wilhelm von Humboldt, _\u00dcber die Grenzen der Wirksamkeit des Staates_ (Nuremberg: Verlag Hans Carl, 1946) (written in 1792, but first completely published in Breslau in 1851 under the title _Ideen zu einem Versuch, die Gr\u00e4nzen der Wirksamkeit des Staats zu bestimmen_ ), chap. 6, summary at the beginning and the concluding sentence. [The English quotation can be found in the standard English edition, _The Sphere and Duties of Government_ , Joseph Coulthard, Jr., trans. (London: John Chapman, 1854), p. 71 (Liberty Fund edition, p. 52). The sentence Hayek here quotes is indeed the concluding sentence of chapter 6, which in German reads: \"\u00d6ffentliche Erziehung scheint mir daher ganz au\u00dferhalb der Schranken zu liegen, in welchen der Staat seine Wirksamkeit halten mu\u00df\" ( _\u00dcber die Grenzen_ , p. 85). The summary reads: \"Having seen in a preceding chapter that it is not only a justifiable but necessary end of Government to provide for the mutual security of the citizens, it here becomes our duty to enter on a more profound and explicit investigation into the nature of such a solicitude, and the means through which it acts. For it does not seem enough merely to commit the care for security to the political power as a general and unconditional duty, but it further becomes us to define the especial limits of its activity in this respect or, at least, should this general definition be difficult, or wholly impossible, to exhibit the reasons for that impossibility, and discover the characteristics by which these limits may, in given cases, be recognized\" ( p. 62; Liberty Fund edition, p.46). ( \"Eine tiefere und ausf\u00fchrlichere Pr\u00fcfung erfordert die Sorgfalt des Staats f\u00fcr die innere Sicherheit der B\u00fcrger unter einander, zu der ich mich jetzt wende. Denn es scheint mir nicht hinl\u00e4nglich, demselben blo\u00df allgemein die Erhaltung derselben zur Pflicht zu machen, sondern ich halte es vielmehr f\u00fcr notwendig, die besondern Grenzen dabei zu bestimmen oder wenn dies allgemein nicht m\u00f6glich sein sollte, wenigstens die Gr\u00fcnde dieser Unm\u00f6glichkeit auseinanderzusetzen und die Merkmale anzugeben, an welchen sie in gegebenen F\u00e4llen zu erkennen sein m\u00f6chten.\" The quotation falls on p. 77 of the 1946 German edition.)\u2014Ed.]\n\n4 Cf. Ludwig von Mises, _Nation, Staat und Wirtschaft. Beitr\u00e4ge zur Politik und Geschichte der Zeit_ (Vienna and Leipzig: Manzscher Verlag, 1919). [This work was translated into English as _Nation, State, and Economy: Contributions to the Politics and History of Our Time_ , Leland B. Yeager, trans. (New York: New York University Press, 1983). A Liberty Fund edition was released in 2006.\u2014Ed.]\n\n5 Milton Friedman, \"The Role of Government Education,\" in _Economics and the Public Interest_ , Robert Alexander Solo, ed. (New Brunswick, NJ: Rutgers University Press, 1955), pp. 123\u201344, and Friedman, _Capitalism and Freedom_ (Chicago: University of Chicago Press, 1962).\n\n6 Cf. George Joseph Stigler, \"The Economic Theory of Education\" [in an unpublished essay]. [This brief essay, twelve pages in typescript, was written in 1957. It has never appeared in print.\u2014Ed.]\n\n7 See the interesting proposals suggested by Milton Friedman in \"The Role of Government Education,\" which deserve careful study, though one may feel doubt about their practicability.\n\n8 Richard Henry Tawney, _Equality_ , Halley Stewart Lectures, 1929 (London: Allen and Unwin, 1931), p. 52.\n\n9 A problem which is not taken care of in present conditions is that presented by the occasional young person in whom a passionate desire for knowledge appears without any recognizable special gifts in the standard subjects of instruction. Such a desire ought to count for much more than it does, and the opportunity of working through college does not really solve the problem on a higher level. It has always seemed to me that there is a strong case for institutions which fulfill the functions that the monasteries fulfilled in the past, where those who cared enough could, at the price of renouncing many of the comforts and pleasures of life, earn the opportunity of devoting all the formative period of their development to the pursuit of knowledge.\n\n10 David Victor Glass, \"Introduction,\" in the volume edited by him and entitled _Social Mobility in Britain_ (London: Routledge and Kegan Paul, 1954), pp. 25\u201326; see also the review of this work by Adam Curle, \"The Scale of Prestige: Review of D. V. Glass, _Social Mobility in Britain_ ,\" in _The New Statesman and Nation_ , n.s., 48 (August 14, 1954): 190, col. 2, where it is suggested that \"the educational dilemma is that the desire to produce a more 'open' society may simply end in one which, while flexible so far as individuals are concerned, is just as rigidly stratified on an I.Q. basis as it was once by birth.\" Cf. also Michael Young, _The Rise of the Meritocracy, 1870\u20132033: An Essay on Education and Equality_ (London: Thames and Hudson, 1958).\n\n11 Sir Charles Percy Snow, quoted in _Time,_ May 27, 1957, p. 106. [The quotation originates in a letter by Snow to the (London) _Sunday Times_ of January 8, 1956. The original reads: \"it is an unexpected result of the Welfare State that in this sense it should make the social pattern not less rigid but much more so.\" The quotation as Hayek has it is an exact transcription of the _Time_ article.\u2014Ed.]\n\n12 David Mordecai Blank and George Joseph Stigler, _The Demand and Supply of Scientific Personnel_ (New York: National Bureau of Economic Research, 1957).\n\n13 It is significant that in England, where the universities were endowed corporations, each consisting of a large number of self-governing bodies, academic freedom has never become a serious issue in the manner in which it did where universities were government institutions.\n\n14 Cf. Michael Polanyi, _The Logic of Liberty: Reflections and Rejoinders_ (London: Routledge and Kegan Paul, 1951), p. 33: \"Academic freedom consists in the right to choose one's own problems for investigation, to conduct research free from any outside control, and to teach one's subject in the light of one's own opinion.\"\n\n15 Thomas Jefferson [to Joseph Carrington Cabell], February 3, 1825, in _The Writings of Thomas Je_ _ff_ _erson: Being his Autobiography, Correspondence, Reports, Messages, Addresses, and Other Writings, O_ _ffi_ _cial and Private_ , Henry Augustine Washington, ed. [Published by the Order of the Joint Committee of Congress on the Library, from the original manuscripts, deposited in the Department of State] (9 vols.; New York: J. C. Riker, 1853\u201354), vol. 7, p. 397. It should be said that Jefferson's opposition to academic freedom was quite consistent with his general position on such matters, which, in the manner of most doctrinaire democrats, made him equally oppose the independence of judges. See also his report to the President and Directors of the Literary Fund, October 5, 1824, in _Early History of the University of Virginia, as Contained in the Letters of Thomas Jefferson and Joseph C. Cabell_ , Nathaniel Francis Cabell, ed.(Richmond, VA:J.W.Randolph, 1856), p. 482.See also _Letters and Other Writings of James Madison, Fourth president of the United States, Published by Order of Congress_ (4 vols.; Philadelphia: J. B. Lippincott and Co., 1865), vol. 3, pp. 481\u201383.\n\n16 Cf. John Randal Baker, _Science and the Planned State_ (London: Allen and Unwin, 1945).\n\n17 This is not the place to enter into a discussion of the Russian educational system. But it may be briefly mentioned that its chief differences from the American system have little to do with the different social order and that, in fact, the Russians are merely following a Continental European tradition. In the critical aspects the achievements of the German or French or Scandinavian schools would repay study as much as the Russian ones.\n\n18 See John Jewkes, David Sawers, and Richard Stillerman, _The Sources of Invention_ (London: Macmillan, 1958), esp. pp. 197\u2013222.\n\n19 Wilhelm von Humboldt, _The Sphere and Duties of Government,_ Joseph Coulthard, Jr., trans. (London: John Chapman, 1854), p. 65 [Liberty Fund edition, p. 48]. [The German reads: \"Nach dem ganzen vorigen R\u00e4sonnement kommt schlechterdings alles auf Ausbildung des Menschen in der h\u00f6chsten Mannigfaltigkeit an.\" ( _\u00dcber die Grenzen der Wrksamkeit des Staates_ [Nuremberg: Verlag Hans Carl, 1946], p. 80)\u2014Ed.] John Stuart Mill took the quotation from this translation.\n\n## POSTSCRIPT\n\nWHY I AM NOT A CONSERVATIVE\n\nAt all times sincere friends of freedom have been rare, and its triumphs have been due to minorities, that have prevailed by associating themselves with auxiliaries whose objects often differed from their own; and this association, which is always dangerous, has sometimes been disastrous, by giving to opponents just grounds of opposition.\n\n\u2014Lord Acton\n\nThe quotation at the head of the Postscript is taken from Acton, _History of Freedom_ , p. 1 [Liberty Fund edition, _Essays in the History of Liberty_ , p. 1].\n\n## WHY I AM NOT A CONSERVATIVE\n\n1. At a time when most movements that are thought to be progressive advocate further encroachments on individual liberty,1 those who cherish freedom are likely to expend their energies in opposition. In this they find themselves much of the time on the same side as those who habitually resist change. In matters of current politics today they generally have little choice but to support the conservative parties. But, though the position I have tried to define is also often described as \"conservative,\" it is very different from that to which this name has been traditionally attached. There is danger in the confused condition which brings the defenders of liberty and the true conservatives together in common opposition to developments which threaten their different ideals equally. It is therefore important to distinguish clearly the position taken here from that which has long been known\u2014perhaps more appropriately\u2014 as conservatism.\n\nConservatism proper is a legitimate, probably necessary, and certainly widespread attitude of opposition to drastic change. It has, since the French Revolution, for a century and a half played an important role in European politics. Until the rise of socialism its opposite was liberalism. There is nothing corresponding to this conflict in the history of the United States, because what in Europe was called \"liberalism\" was here the common tradition on which the American polity had been built: thus the defender of the American tradition was a liberal in the European sense.2 This already existing confusion was made worse by the recent attempt to transplant to America the European type of conservatism, which, being alien to the American tradition, has acquired a somewhat odd character. And some time before this, American radicals  and socialists began calling themselves \"liberals.\" I will nevertheless continue for the moment to describe as liberal the position which I hold and which I believe differs as much from true conservatism as from socialism. Let me say at once, however, that I do so with increasing misgivings, and I shall later have to consider what would be the appropriate name for the party of liberty. The reason for this is not only that the term \"liberal\" in the United States is the cause of constant misunderstandings today, but also that in Europe the predominant type of rationalistic liberalism has long been one of the pacemakers of socialism.\n\nLet me now state what seems to me the decisive objection to any conservatism which deserves to be called such. It is that by its very nature it cannot offer an alternative to the direction in which we are moving. It may succeed by its resistance to current tendencies in slowing down undesirable developments, but, since it does not indicate another direction, it cannot prevent their continuance. It has, for this reason, invariably been the fate of conservatism to be dragged along a path not of its own choosing. The tug of war between conservatives and progressives can only affect the speed, not the direction, of contemporary developments. But, though there is need for a \"brake on the vehicle of progress,\"3 I personally cannot be content with simply helping to apply the brake. What the liberal must ask, first of all, is not how fast or how far we should move, but where we should move. In fact, he differs much more from the collectivist radical of today than does the conservative. While the last generally holds merely a mild and moderate version of the prejudices of his time, the liberal today must more positively oppose some of the basic conceptions which most conservatives share with the socialists.\n\n2. The picture generally given of the relative position of the three parties does more to obscure than to elucidate their true relations. They are usually represented as different positions on a line, with the socialists on the left, the conservatives on the right, and the liberals somewhere in the middle. Nothing could be more misleading. If we want a diagram, it would be more appropriate to arrange them in a triangle with the conservatives occupying one corner, with the socialists pulling toward the second and the liberals toward the third. But, as the socialists have for a long time been able to pull harder, the conservatives have tended to follow the socialist rather than the liberal direction and have adopted at appropriate intervals of time those ideas made respectable by radical propaganda. It has been regularly the conservatives who have compromised with socialism and stolen its thunder. Advocates of the Middle Way4 with no goal of their own, conservatives have been guided by the belief that the truth must lie somewhere between the extremes\u2014with the result that they have shifted their position every time a more extreme movement appeared on either wing.\n\nThe position which can be rightly described as conservative at any time depends, therefore, on the direction of existing tendencies. Since the development during the last decades has been generally in a socialist direction, it may seem that both conservatives and liberals have been mainly intent on retarding that movement. But the main point about liberalism is that it wants to go elsewhere, not to stand still. Though today the contrary impression may sometimes be caused by the fact that there was a time when liberalism was more widely accepted and some of its objectives closer to being achieved, it has never been a backward-looking doctrine. There has never been a time when liberal ideals were fully realized and when liberalism did not look forward to further improvement of institutions. Liberalism is not averse to evolution and change; and where spontaneous change has been smothered by government control, it wants a great deal of change of policy. So far as much of current governmental action is concerned, there is in the present world very little reason for the liberal to wish to preserve things as they are. It would seem to the liberal, indeed, that what is most urgently needed in most parts of the world is a thorough sweeping-away of the obstacles to free growth.\n\nThis difference between liberalism and conservatism must not be obscured by the fact that in the United States it is still possible to defend individual liberty by defending long-established institutions. To the liberal they are valuable not mainly because they are long established or because they are American but because they correspond to the ideals which he cherishes.\n\n3. Before I consider the main points on which the liberal attitude is sharply opposed to the conservative one, I ought to stress that there is much that the liberal might with advantage have learned from the work of some conservative thinkers. To their loving and reverential study of the value of grown institutions we owe (at least outside the field of economics) some profound insights  which are real contributions to our understanding of a free society. However reactionary in politics such figures as Coleridge, Bonald, De Maistre, Justus M\u00f6ser, or Donoso Cort\u00e8s may have been, they did show an understanding of the meaning of spontaneously grown institutions such as language, law, morals, and conventions that anticipated modern scientific approaches and from which the liberals might have profited. But the admiration of the conservatives for free growth generally applies only to the past. They typically lack the courage to welcome the same undesigned change from which new tools of human endeavors will emerge.\n\nThis brings me to the first point on which the conservative and the liberal dispositions differ radically. As has often been acknowledged by conservative writers, one of the fundamental traits of the conservative attitude is a fear of change, a timid distrust of the new as such,5 while the liberal position is based on courage and confidence, on a preparedness to let change run its course even if we cannot predict where it will lead. There would not be much to object to if the conservatives merely disliked too rapid change in institutions and public policy; here the case for caution and slow process is indeed strong. But the conservatives are inclined to use the powers of government to prevent change or to limit its rate to whatever appeals to the more timid mind. In looking forward, they lack the faith in the spontaneous forces of adjustment which makes the liberal accept changes without apprehension, even though he does not know how the necessary adaptations will be brought about. It is, indeed, part of the liberal attitude to assume that, especially in the economic field, the self-regulating forces of the market will somehow bring about the required adjustments to new conditions, although no one can foretell how they will do this in a particular instance. There is perhaps no single factor contributing so much to people's frequent reluctance to let the market work as their inability to conceive how some necessary balance, between demand and supply, between exports and imports, or the like, will be brought about without deliberate control. The conservative feels safe and content only if he is assured that some higher wisdom watches and supervises change, only if he knows that some authority is charged with keeping the change \"orderly.\"\n\nThis fear of trusting uncontrolled social forces is closely related to two other characteristics of conservatism: its fondness for authority and its lack of understanding of economic forces. Since it distrusts both abstract theories and general principles,6 it neither understands those spontaneous forces on which  a policy of freedom relies nor possesses a basis for formulating principles of policy. Order appears to the conservatives as the result of the continuous attention of authority, which, for this purpose, must be allowed to do what is required by the particular circumstances and not be tied to rigid rule. A commitment to principles presupposes an understanding of the general forces by which the efforts of society are co-ordinated, but it is such a theory of society and especially of the economic mechanism that conservatism conspicuously lacks. So unproductive has conservatism been in producing a general conception of how a social order is maintained that its modern votaries, in trying to construct a theoretical foundation, invariably find themselves appealing almost exclusively to authors who regarded themselves as liberal. Macaulay, Tocqueville, Lord Acton, and Lecky certainly considered themselves liberals, and with justice; and even Edmund Burke remained an Old Whig to the end and would have shuddered at the thought of being regarded as a Tory.\n\nLet me return, however, to the main point, which is the characteristic complacency of the conservative toward the action of established authority and his prime concern that this authority be not weakened rather than that its power be kept within bounds. This is difficult to reconcile with the preservation of liberty. In general, it can probably be said that the conservative does not object to coercion or arbitrary power so long as it is used for what he regards as the right purposes. He believes that if government is in the hands of decent men, it ought not to be too much restricted by rigid rules. Since he is essentially opportunist and lacks principles, his main hope must be that the wise and the good will rule\u2014not merely by example, as we all must wish, but by authority given to them and enforced by them.7 Like the socialist, he is less concerned with the problem of how the powers of government should be limited than with that of who wields them; and, like the socialist, he regards himself as entitled to force the value he holds on other people.\n\nWhen I say that the conservative lacks principles, I do not mean to suggest that he lacks moral conviction. The typical conservative is indeed usually a  man of very strong moral convictions. What I mean is that he has no political principles which enable him to work with people whose moral values differ from his own for a political order in which both can obey their convictions. It is the recognition of such principles that permits the coexistence of different sets of values that makes it possible to build a peaceful society with a minimum of force. The acceptance of such principles means that we agree to tolerate much that we dislike. There are many values of the conservative which appeal to me more than those of the socialists; yet for a liberal the importance he personally attaches to specific goals is no sufficient justification for forcing others to serve them. I have little doubt that some of my conservative friends will be shocked by what they will regard as \"concessions\" to modern views that I have made in Part III of this book. But, though I may dislike some of the measures concerned as much as they do and might vote against them, I know of no general principles to which I could appeal to persuade those of a different view that those measures are not permissible in the general kind of society which we both desire. To live and work successfully with others requires more than faithfulness to one's concrete aims. It requires an intellectual commitment to a type of order in which, even on issues which to one are fundamental, others are allowed to pursue different ends.\n\nIt is for this reason that to the liberal neither moral nor religious ideals are proper objects of coercion, while both conservatives and socialists recognize no such limits. I sometimes feel that the most conspicuous attribute of liberalism that distinguishes it as much from conservatism as from socialism is the view that moral beliefs concerning matters of conduct which do not directly interfere with the protected sphere of other persons do not justify coercion. This may also explain why it seems to be so much easier for the repentant socialist to find a new spiritual home in the conservative fold than in the liberal.\n\nIn the last resort, the conservative position rests on the belief that in any society there are recognizably superior persons whose inherited standards and values and position ought to he protected and who should have a greater influence on public affairs than others. The liberal, of course, does not deny that there are some superior people\u2014he is not an egalitarian\u2014but he denies that anyone has authority to decide who these superior people are. While the conservative inclines to defend a particular established hierarchy and wishes authority to protect the status of those whom he values, the liberal feels that no respect for established values can justify the resort to privilege or monopoly or any other coercive power of the state in order to shelter such people against the forces of economic change. Though he is fully aware of the important role that cultural and intellectual elites have played in the evolution of civilization, he also believes that these elites have to prove themselves by their capacity to maintain their position under the same rules that apply to all others.\n\nClosely connected with this is the usual attitude of the conservative to  democracy. I have made it clear earlier that I do not regard majority rule as an end but merely as a means, or perhaps even as the least evil of those forms of government from which we have to choose. But I believe that the conservatives deceive themselves when they blame the evils of our time on democracy. The chief evil is unlimited government, and nobody is qualified to wield unlimited power.8 The powers which modern democracy possesses would be even more intolerable in the hands of some small elite.\n\nAdmittedly, it was only when power came into the hands of the majority that further limitation of the power of government was thought unnecessary. In this sense democracy and unlimited government are connected. But it is not democracy but unlimited government that is objectionable, and I do not see why the people should not learn to limit the scope of majority rule as well as that of any other form of government. At any rate, the advantages of democracy as a method of peaceful change and of political education seem to be so great compared with those of any other system that I can have no sympathy with the anti-democratic strain of conservatism. It is not who governs but what government is entitled to do that seems to me the essential problem.\n\nThat the conservative opposition to too much government control is not a matter of principle but is concerned with the particular aims of government is clearly shown in the economic sphere. Conservatives usually oppose collectivist and directivist measures in the industrial field, and here the liberal will often find allies in them. But at the same time conservatives are usually protectionists and have frequently supported socialist measures in agriculture. Indeed, though the restrictions which exist today in industry and commerce are mainly the result of socialist views, the equally important restrictions in agriculture were usually introduced by conservatives at an even earlier date. And in their efforts to discredit free enterprise many conservative leaders have vied with the socialists.9\n\n4. I have already referred to the differences between conservatism and liberalism in the purely intellectual field, but I must return to them because the characteristic conservative attitude here not only is a serious weakness of conservatism but tends to harm any cause which allies itself with it. Con  servatives feel instinctively that it is new ideas more than anything else that cause change. But, from its point of view rightly, conservatism fears new ideas because it has no distinctive principles of its own to oppose to them; and, by its distrust of theory and its lack of imagination concerning anything except that which experience has already proved, it deprives itself of the weapons needed in the struggle of ideas. Unlike liberalism with its fundamental belief in the long-range power of ideas, conservatism is bound by the stock of ideas inherited at a given time. And since it does not really believe in the power of argument, its last resort is generally a claim to superior wisdom, based on some self-arrogated superior quality.\n\nThis difference shows itself most clearly in the different attitudes of the two traditions to the advance of knowledge. Though the liberal certainly does not regard all change as progress, he does regard the advance of knowledge as one of the chief aims of human effort and expects from it the gradual solution of such problems and difficulties as we can hope to solve. Without preferring the new merely because it is new, the liberal is aware that it is of the essence of human achievement that it produces something new; and he is prepared to come to terms with new knowledge, whether he likes its immediate effects or not.\n\nPersonally, I find that the most objectionable feature of the conservative attitude is its propensity to reject well-substantiated new knowledge because it dislikes some of the consequences which seem to follow from it\u2014or, to put it bluntly, its obscurantism. I will not deny that scientists as much as others are given to fads and fashions and that we have much reason to be cautious in accepting the conclusions that they draw from their latest theories. But the reasons for our reluctance must themselves be rational and must be kept separate from our regret that the new theories upset our cherished beliefs. I can have little patience with those who oppose, for instance, the theory of evolution or what are called \"mechanistic\" explanations of the phenomena of life simply because of certain moral consequences which at first seem to follow from these theories, and still less with those who regard it as irreverent or impious to ask certain questions at all. By refusing to face the facts, the conservative only weakens his own position. Frequently the conclusions which rationalist presumption draws from new scientific insights do not at all follow from them. But only by actively taking part in the elaboration of the consequences of new discoveries do we learn whether or not they fit into our world picture and, if so, how. Should our moral beliefs really prove to be dependent on factual assumptions shown to be incorrect, it would be hardly moral to defend them by refusing to acknowledge facts.\n\nConnected with the conservative distrust of the new and the strange is its hostility to internationalism and its proneness to a strident nationalism. Here is another source of its weakness in the struggle of ideas. It cannot alter the  fact that the ideas which are changing our civilization respect no boundaries. But refusal to acquaint one's self with new ideas merely deprives one of the power of effectively countering them when necessary. The growth of ideas is an international process, and only those who fully take part in the discussion will be able to exercise a significant influence. It is no real argument to say that an idea is un-American, un-British, or un-German, nor is a mistaken or vicious ideal better for having been conceived by one of our compatriots.\n\nA great deal more might be said about the close connection between conservatism and nationalism, but I shall not dwell on this point because it may be felt that my personal position makes me unable to sympathize with any form of nationalism. I will merely add that it is this nationalistic bias which frequently provides the bridge from conservatism to collectivism: to think in terms of \"our\" industry or resource is only a short step away from demanding that these national assets be directed in the national interest. But in this respect the Continental liberalism which derives from the French Revolution is little better than conservatism. I need hardly say that nationalism of this sort is something very different from patriotism and that an aversion to nationalism is fully compatible with a deep attachment to national traditions. But the fact that I prefer and feel reverence for some of the traditions of my society need not be the cause of hostility to what is strange and different.\n\nOnly at first does it seem paradoxical that the anti-internationalism of the conservative is so frequently associated with imperialism. But the more a person dislikes the strange and thinks his own ways superior, the more he tends to regard it as his mission to \"civilize\" others10\u2014not by the voluntary and unhampered intercourse which the liberal favors, but by bringing them the blessings of efficient government. It is significant that here again we frequently find the conservatives joining hands with the socialists against the liberals\u2014 not only in England, where the Webbs and their Fabians were outspoken imperialists, or in Germany, where state socialism and colonial expansionism went together and found the support of the same group of \"socialists of the chair,\" but also in the United States, where even at the time of the first Roosevelt it could be observed: \"the Jingo and the Social Reformer have gotten together and have formed a political party, which threatened to capture the Government and use it for their program of Caesaristic paternalism, a danger which appears now to have been averted only by the other parties having themselves adopted this programme in a somewhat milder degree and form.\"11\n\n5. There is one respect, however, in which there is justification for saying that the liberal occupies a position midway between the socialist and the conservative: he is as far from the crude rationalism of the socialist, who wants to reconstruct all social institutions according to a pattern prescribed by his individual reason, as from the mysticism to which the conservative so frequently has to resort. What I have described as the liberal position shares with conservatism a distrust of reason to the extent that the liberal is very much aware that we do not know all the answers and that he is not sure that the answers he has are certainly the right ones or even that we can find all the answers. He also does not disdain to seek assistance from whatever non-rational institutions or habits have proved their worth. The liberal differs from the conservative in his willingness to face this ignorance and to admit how little we know, without claiming the authority of supernatural sources of knowledge where his reason fails him. It has to be admitted that in some respects the liberal is fundamentally a skeptic12\u2014but it seems to require a certain degree of diffidence to let others seek their happiness in their own fashion and to adhere consistently to that tolerance which is an essential characteristic of liberalism.\n\nThere is no reason why this need mean an absence of religious belief on the part of the liberal. Unlike the rationalism of the French Revolution, true liberalism has no quarrel with religion, and I can only deplore the militant and essentially illiberal antireligionism which animated so much of nineteenth-century Continental liberalism. That this is not essential to liberalism is clearly shown by its English ancestors, the Old Whigs, who, if anything, were much too closely allied with a particular religious belief. What distinguishes the liberal from the conservative here is that, however profound his own spiritual beliefs, he will never regard himself as entitled to impose them on others and that for him the spiritual and the temporal are different spheres which ought not to be confused.\n\n6. What I have said should suffice to explain why I do not regard myself as a conservative. Many people will feel, however, that the position which emerges is hardly what they used to call \"liberal.\" I must, therefore, now face the question of whether this name is today the appropriate name for the party of liberty. I have already indicated that, though I have all my life described  myself as a liberal, I have done so more recently with increasing misgivings\u2014 not only because in the United States this term constantly gives rise to misunderstanding, but also because I have become more and more aware of the great gulf that exists between my position and the rationalistic Continental liberalism or even the English liberalism of the utilitarians.\n\nIf liberalism still meant what it meant to an English historian who in 1827 could speak of the revolution of 1688 as \"the triumph of those principles which, in the language of the present day, are denominated liberal or constitutional\"13 or if one could still, with Lord Acton, speak of Burke, Macaulay, and Gladstone as the three greatest liberals, or if one could still, with Harold Laski, regard Tocqueville and Lord Acton as \"the essential liberals of the nineteenth century,\"14 I should indeed be only too proud to describe myself by that name. But, much as I am tempted to call their liberalism true liberalism, I must recognize that the majority of Continental liberals stood for ideas to which these men were strongly opposed, and that they were led more by a desire to impose upon the world a preconceived rational pattern than to provide opportunity for free growth. The same is largely true of what has called itself Liberalism in England at least since the time of Lloyd George.\n\nIt is thus necessary to recognize that what I have called \"liberalism\" has little to do with any political movement that goes under that name today. It is also questionable whether the historical associations which that name carries today are conducive to the success of any movement. Whether in these circumstances one ought to make an effort to rescue the term from what one  feels is its misuse is a question on which opinions may well differ. I myself feel more and more that to use it without long explanations causes too much confusion and that as a label it has become more of a ballast than a source of strength.\n\nIn the United States, where it has become almost impossible to use \"liberal\" in the sense in which I have used it, the term \"libertarian\" has been used instead. It may be the answer; but for my part I find it singularly unattractive. For my taste it carries too much the flavor of a manufactured term and of a substitute. What I should want is a word which describes the party of life, the party that favors free growth and spontaneous evolution. But I have racked my brain unsuccessfully to find a descriptive term which commends itself.\n\n7. We should remember, however, that when the ideals which I have been trying to restate first began to spread through the Western world, the party which represented them had a generally recognized name. It was the ideals of the English Whigs that inspired what later came to be known as the liberal movement in the whole of Europe15 and that provided the conceptions that the American colonists carried with them and which guided them in their struggle for independence and in the establishment of their constitution.16 Indeed, until the character of this tradition was altered by the accretions due to the French Revolution, with its totalitarian democracy and social  ist leanings, \"Whig'' was the name by which the party of liberty was generally known.\n\nThe name died in the country of its birth partly because for a time the principles for which it stood were no longer distinctive of a particular party, and partly because the men who bore the name did not remain true to those principles. The Whig parties of the nineteenth century, in both Britain and the United States, finally brought discredit to the name among the radicals. But it is still true that, since liberalism took the place of Whiggism only after the movement for liberty had absorbed the crude and militant rationalism of the French Revolution, and since our task must largely be to free that tradition from the overrationalistic, nationalistic, and socialistic influences which have intruded into it, Whiggism is historically the correct name for the ideas in which I believe. The more I learn about the evolution of ideas, the more I have become aware that I am simply an unrepentant Old Whig\u2014with the stress on the \"old.\"\n\nTo confess one's self an Old Whig does not mean, of course, that one wants to go back to where we were at the end of the seventeenth century. It has been one of the purposes of this book to show that the doctrines then first stated continued to grow and develop until about seventy or eighty years ago, even though they were no longer the chief aim of a distinct party. We have since learned much that should enable us to restate them in a more satisfactory and effective form. But, though they require restatement in the light of our present knowledge, the basic principles are still those of the Old Whigs. True, the later history of the party that bore that name has made some historians doubt where there was a distinct body of Whig principles; but I can but agree with Lord Acton that, though some of \"the patriarchs of the doctrine were the most infamous of men, the notion of a higher law above municipal codes, with which Whiggism began, is the supreme achievement of Englishmen and their bequest to the nation\"a\u2014and, we may add, to the world. It is the doctrine which is at the basis of the common tradition of the Anglo-Saxon countries. It is the doctrine from which Continental liberalism took what is valuable in it. It is the doctrine on which the American system of government is based. In its pure form it is represented in the United States, not by the radicalism of  Jefferson, nor by the conservatism of Hamilton or even of John Adams, but by the ideas of James Madison, the \"father of the Constitution.\"18\n\nI do not know whether to revive that old name is practical politics. That to the mass of people, both in the Anglo-Saxon world and elsewhere, it is today probably a term without definite associations is perhaps more an advantage than a drawback. To those familiar with the history of ideas it is probably the only name that quite expresses what the tradition means. That, both for the genuine conservative and still more for the many socialists turned conservative, Whiggism is the name for their pet aversion shows a sound instinct on their part. It has been the name for the only set of ideals that has consistently opposed all arbitrary power.\n\n8. It may well be asked whether the name really matters so much. In a country like the United States, which on the whole still has free institutions and where, therefore, the defense of the existing is often a defense of freedom, it might not make so much difference if the defenders of freedom call themselves conservatives, although even here the association with the conservatives by disposition will often be embarrassing. Even when men approve of the same arrangements, it must be asked whether they approve of them because they exist or because they are desirable in themselves. The common resistance to the collectivist tide should not be allowed to obscure the fact that the belief in integral freedom is based on an essentially forward-looking attitude and not on any nostalgic longing for the past or a romantic admiration for what has been.\n\nThe need for a clear distinction is absolutely imperative, however, where, as is true in many parts of Europe, the conservatives have already accepted a large part of the collectivist creed\u2014a creed that has governed policy for so long that many of its institutions have come to be accepted as a matter of course and have become a source of pride to \"conservative\" parties who created them.19 Here the believer in freedom cannot but conflict with the conservative and take an essentially radical position, directed against popular  prejudices, entrenched positions, and firmly established privileges. Follies and abuses are no better for having long been established principles of policy.\n\nThough _quieta non movere_ may at times be a wise maxim for the statesman, it cannot satisfy the political philosopher. He may wish policy to proceed gingerly and not before public opinion is prepared to support it, but he cannot accept arrangements merely because current opinion sanctions them. In a world where the chief need is once more, as it was at the beginning of the nineteenth century, to free the process of spontaneous growth from the obstacles and encumbrances that human folly has erected, his hopes must rest on persuading and gaining the support of those who by disposition are \"progressives,\" those who, though they may now be seeking change in the wrong direction, are at least willing to examine critically the existing and to change it wherever necessary.\n\nI hope I have not misled the reader by occasionally speaking of \"party\" when I was thinking of groups of men defending a set of intellectual and moral principles. Party politics of any one country has not been the concern of this book. The question of how the principles I have tried to reconstruct by piecing together the broken fragments of a tradition can be translated into a program with mass appeal, the political philosopher must leave to \"that insidious and crafty animal, vulgarly called a statesman or politician, whose councils are directed by the momentary fluctuations of affairs.\"20 The task of the political philosopher can only be to influence public opinion, not to organize people for action. He will do so effectively only if he is not concerned with what is now politically possible but consistently defends the \"general principles which are always the same.\"21 In this sense I doubt whether there can be such a thing as a conservative political philosophy. Conservatism may often be a useful practical maxim, but it does not give us any guiding principles which can influence long-range developments.\n\n* * *\n\n1 This has now been true for over a century, and as early as 1855 John Stuart Mill in a letter to Harriet Taylor, Rome, 15 January 1855 (Friedrich August Hayek, _John Stuart Mill and Harriet Taylor: Their Correspondence and Subsequent Marriage_ [London: Routledge and Kegan Pau1, 1951], p. 216) could say that \"almost all the projects of social reformers of these days are really _liberticide_.\"\n\n2 Bernard Crick, \"The Strange Quest for an American Conservatism,\" _Review of Politics_ , 17 (1955): 365, says rightly that \"the normal American who calls himself 'a conservative' is, in fact, a liberal.\" It would appear that the reluctance of these conservatives to call themselves by the more appropriate name dates only from its abuse during the New Deal era.\n\n3 The expression is that of Robin George Collingwood, _The New Leviathan; or, Man, Society, Civilization and Barbarism_ (Oxford: Clarendon Press, 1942), p. 209.\n\n4 Cf. the characteristic choice of this title for the programmatic book by British Prime Minister Harold Macmillan, _The Middle Way: A Study of the Problem of Economic and Social Progress in a Free and Democratic Society_ (London: Macmillan, 1938). [Harold Macmillan became prime minister and leader of the Conservative Party on the resignation of Anthony Eden in January 1957. He remained in that office until he himself resigned in October 1963, when he was replaced as prime minister by his foreign secretary, Alec Douglas-Home.\u2014Ed.] Unfortunately, for the most part this is the social doctrine adopted by the Roman Catholic Church and certain conscientious Ger-man social democrats, who were able to cite one of the nation' s leading Catholic social philosophers , Oswald von Nell-Breuning, in their recent publication, Sozialdemokratische Partei Deutschlands, _Godesberger Programm: Zur Situation nach Mater Magistr a_ (Bonn: Sozialdemokratische Partei, 1962), p. 25: \"Soweit auf sozialem und \u00f6k onomischem Gebiet Differenzen in der Christlichen Soziallehre bestehen, sind sie auf jeden Fall geringer als die Differenzen zwischen Neoliberalismus und christlicher Soziallehre.\" [\"As far as there are differences within Christian social teaching in social and economic matters, these are in any case smaller than the differences between neoliberalism and Christian social teaching.\"\u2014Ed.]\n\n5 Cf. Hugh Richard Heathcote, Lord Cecil, _Conservatism_ (Home University Library; London: Williams and Norgate, 1912), p. 9: \"Natural Conservatism . . . is a disposition averse from change; and it springs partly from a distrust of the unknown.\"\n\n6 Cf. the revealing self-description of a conservative in Sir Keith Grahame Feiling, _Sketches in Nineteenth Century Biography_ (London: Longmans Green, and Co.,1930), p. 174: \"Taken in bulk, the Right have a horror of ideas, for is not the practical man, in Disraeli's words, 'one who practises the blunders of his predecessors'? For long tracts of their history they have indiscriminately resisted improvement, and in claiming to reverence their ancestors often reduce opinion to aged individual prejudice. Their position becomes safer, but more complex, when we add that this Right wing is incessantly overtaking the Left; that it lives by repeated inoculation of Liberal ideas, and thus suffers from a never-perfected state of compromise.\"\n\n7 I trust I shall be forgiven for repeating here the words in which on an earlier occasion I stated an important point: \"The main merit of the individualism which [Adam Smith] and his contemporaries advocated is that it is a system under which bad men can do least harm. It is a social system which does not depend for its functioning on our finding good men for running it, or on all men becoming better than they now are, but which makes use of men in all their given variety and complexity, sometimes good and sometimes bad, sometimes intelligent and more often stupid\" ( _Individualism and Economic Order_ [Chicago: University of Chicago Press, 1948], pp. 11\u201312). [ _Collected Works_ edition, vol. 13, p. 57.]\n\n8 Cf. Lord Acton, _Letters of Lord Acton to Mary, Daughter of the Right Hon. W. E. Gladstone_ , Herbert Woodfield Paul, ed. (2nd ed.; London: Macmillan, 1913), p. 73: \"The danger is not that a particular class is unfit to govern. Every class is unfit to govern. The law of liberty tends to abolish the reign of race over race, of faith over faith, of class over class.\"\n\n9 John Richard Hicks has rightly spoken in this connection of the \"caricature drawn alike by the young Disraeli, by Marx and by Goebbels\" (Hicks, \"The Pursuit of Economic Freedom,\" in _What We Defend: Essays in Freedom by Members of the University of Manchester_ , Ernest Fraser Jacob, ed. [London: Oxford University Press, 1942], p. 96). On the role of the conservatives in this connection see also my \"Introduction: History and Politics,\" _Capitalism and the Historians_ (Chicago: University of Chicago Press, 1954), pp. 19ff. [ _Collected Works_ edition, vol. 3, pp. 56\u201372.]\n\n10 Cf. John Stuart Mill, \"On Liberty,\" in _On Liberty and Considerations on Representative Government_ , Ronald Buchanan McCallum, ed. (Oxford: B. Blackwell, 1946), p. 83: \"I am not aware that any community has a right to force another to be civilized.\"\n\n11 John William Burgess, _The Reconciliation of Government with Liberty_ (New York: Charles Scribner's Sons, 1915), p. 380.\n\n12 Cf. Learned Hand, \"The Spirit of Liberty\" [Address delivered at the \"I Am an American Day,\" in Central Park, New York City, on May 21, 1944], in _The Spirit of Liberty: Papers and Addresses of Learned Hand_ , Irving Dillard, ed. (New York: Alfred A. Knopf, 1952), p. 190: \"The spirit of liberty is the spirit which is not too sure that it is right.\" See also Oliver Cromwell's often quoted statement in his _Letter to the General Assembly of the Church of Scotland_ [ _A letter sent to the General Assembly of the kirke of Scotland (August 3, 1650) by Oliver Cromwell Lord General of the army of the Common-wealth of England now in Scotland_ ] (London: Printed for Hanna Allen, 1650), p. 4: \"I beseech you, in the bowels of Christ, think it possible you may be mistaken.\" It is significant that this should probably be the best-remembered saying of the only \"dictator\" in British history!\n\n13 Henry Hallam, _The Constitutional History of England, Henry VII to George II_ (1827) (Everyman edition; 3 vols.; London: J. M. Dent and Sons, 1930), vol. 3, p. 90. It is often suggested that the term \"liberal\" derives from the early nineteenth-century Spanish party of the _liberales_. I am more inclined to believe that it derives from the use of the term by Adam Smith in such passages as _Wealth of Nations_ , vol. 2, p. 41 [Liberty Fund edition, vol. 1, p. 538]: \"the liberal system of free exportation and free importation\"; and vol. 2, p. 216 [Liberty Fund edition, vol. 2, p. 664]: \"allowing every man to pursue his own interest his own way, upon the liberal plan of equality, liberty, and justice.\"\n\n14 Lord Acton, in _Letters to Mary Gladstone_ , p. 44. [In a letter dated December 27, 1880, Acton writes: \"I do think that, of the three greatest Liberals, Burke is equally good in speaking and writing; Macaulay better in writing, and Mr. Gladstone better in speaking.\"\u2014Ed.] Cf. also his judgment of Tocqueville in _Lectures on the French Revolution_ (London: Macmillan, 1910), p. 357 [Liberty Fund edition, p. 308]: \"Tocqueville was a Liberal of the purest breed\u2014a Liberal and nothing else, deeply suspicious of democracy and its kindred, equality, centralisation, and utilitarianism.\" Similarly in \"Noticeable Books: Tocqueville's Souvenirs,\" in _The Nineteenth Century_ , 33 (1893): 885. The statement by Harold Joseph Laski occurs in \"Alexis de Tocqueville,\" in _The Social and Political Ideas of Some Representative Thinkers of the Victorian Age: A Series of Lectures delivered at King's College, University of London, During the Session 1931\u20131932_ , Fossey John Cobb Hearnshaw, ed. (London: G. G. Harrap and Co., 1933), p. 100, where he says that \"a case of unanswerable power could, I think, be made out for the view that he [Tocqueville] and Lord Acton were the essential liberals of the nineteenth century.\"\n\n15 As early as the beginning of the eighteenth century, an English observer could remark that he \"scarce ever knew a foreigner settled in England, whether of Dutch, German, French, Italian, or Turkish growth, but became a Whig in a little time after his mixing with us\" (quoted by George Herbert Guttridge, _English Whiggism and the American Revolution_ [Berkeley: University of California Press, 1942], p. 3). [The \"English observer\" is Francis Atterbury, _English Advice to the Freeholders of England_ (London, 1714), p. 24.\u2014Ed.]\n\n16 In the United States the nineteenth-century use of the term \"Whig\" has unfortunately obliterated the memory of the fact that in the eighteenth it stood for the principles which guided the revolution, gained independence, and shaped the Constitution. It was in Whig societies that the young James Madison and John Adams developed their political ideals (cf. Edward McNall Burns, _James Madison: Philosopher of the Constitution_ [New Brunswick, NJ: Rutgers University Press, 1938], p. 4); it was Whig principles which, as Jefferson tells us, guided all the lawyers who constituted such a strong majority among the signers of the Declaration of Independence and among the members of the Constitutional Convention (see Thomas Jefferson, _Writings of Thomas Je_ _ff_ _erson_ , Andrew Adgate Lipscomb and Albert Ellery Bergh, eds. [20 vols.; Washington, DC: Issued under the auspices of the Thomas Jefferson Memorial Association of the United States, 1903\u20134], vol. 16, p. 156). The profession of Whig principles was carried to such a point that even Washington's soldiers were clad in the traditional \"blue and buff\" colors of the Whigs, which they shared with the Foxites in the British Parliament and which was preserved down to our own days on the covers of the _Edinburgh Review_. If a socialist generation has made Whiggism its favorite target, this is all the more reason for the opponents of socialism to vindicate the name. It is today the only name which correctly describes the beliefs of the Gladstonian liberals, of the men of the generation of Maitland, Acton, Bryce, Pollock, Sidgwick, and Leslie Stephen, the last generation for whom liberty rather than equality or democracy was the main goal.\n\n17 Lord Acton, \"The Rise of the Whig,\" _Lectures on Modern History_ , John Neville Figgis and Reginald Vere Laurence, eds. (London: Macmillan, 1906), pp. 217\u201318 [Liberty Fund edition, _Essays in the History of Liberty_ , p. 107]. (I have slightly rearranged Acton's clauses to reproduce briefly the sense of his statement). [The original phrasing reads: \"Burke's address to the colonists is the logical outcome of the principles of liberty and the notion of a higher law above municipal codes and constitutions, with which Whiggism began. It is the supreme achievement of Englishmen and their bequest to the nation; but the patriarchs of the doctrine were the most infamous of men.\"\u2014Ed.]\n\n18 Cf. Saul Kussiel Padover, ed., \"Introduction: Madison as a Political Thinker,\" _The Complete Madison: His Basic Writings_ (New York: Harper, 1953), p. 10: \"In modern terminology, Madison would be labeled a middle-of-the-road liberal and Jefferson a radical.\" This is true and important, though we must remember what Edwin Samuel Corwin (\"James Madison: Layman, Publicist, and Exegete,\" _New York University Law Review_ , 27 [1952]: 285) has called Madison's later \"surrender to the overweening influence of Jefferson.\"\n\n19 Cf. the British Conservative party's statement of policy, Conservative and Unionist Central Office, _The Right Road for Britain: The Conservative Party's Statement of Policy_ (London: Conservative and Unionist Central Office, 1949), pp. 41\u201342, which claims, with considerable justification, that \"this new conception [of the social services] was developed [by] the Coalition Government with a majority of Conservative Ministers and the full approval of the Conservative majority in the House of Commons. . . . [We] set out the principle for the schemes of pensions, sickness and unemployment benefit, industrial injuries benefit and a national health scheme.\"\n\n20 Adam Smith, _Wealth of Nations_ , vol. 1, p. 432 [Liberty Fund edition, vol. 1, p. 468].\n\n21 Ibid.\n\n## ANALYTICAL TABLE OF CONTENTS\n\nIntroduction\n\n_PART I. The Value of Freedom_\n\nOne Liberty and Liberties\n\n* * *\n\n1 Freedom as absence of coercion\n\n2 Contrasted with political liberty\n\n3 Contrasted with \"inner liberty\"\n\n4 Contrasted with freedom as power\n\n5 These concepts are incommensurable\n\n6 Freedom and slavery\n\n7 Freedom, coercion, and the law\n\nTwo The Creative Powers of a Free Civilization\n\n* * *\n\n1 Civilization and the growth of knowledge\n\n2 The different uses of experience\n\n3 Transmission and communication of experience\n\n4 The case for freedom based on recognition of ignorance\n\n5 Freedom as opportunity for the unknown few\n\n6 Freedom of thought and freedom of action\n\n7 Freedom and changes in values\n\n8 Organization and competition\n\n9 Rationalism and the limitations of reason\n\nThree The Common Sense of Progress\n\n* * *\n\n1 The disillusionment about progress\n\n2 Progress and improvement\n\n3 Progress and inequality\n\n4 Experiments in living\n\n5 International aspects\n\n6 Redistribution and the speed of progress\n\n7 Material progress and the other values\n\n8 Civilization depends on continued progress\n\nFour Freedom, Reason, and Tradition\n\n* * *\n\n1 The two traditions of freedom\n\n2 The evolutionary conception\n\n3 The growth of order\n\n4 The conflicting assumptions of the two approaches\n\n5 Custom and tradition\n\n6 The rule of morals\n\n7 Superstitions about superstition\n\n8 The moral and the \"social\"\n\n9 Freedom as a moral principle\n\n10 The role of reason\n\nFive Responsibility and Freedom\n\n* * *\n\n1 The inseparability of freedom and responsibility\n\n2 Both discredited by doubt about \"freedom of the will\"\n\n3 The function of assigning responsibility\n\n4 The responsible person\n\n5 The pursuit of individual aims\n\n6 Responsibility for one's own welfare\n\n7 Education for freedom\n\n8 The range of responsibility\n\nSix Equality, Value, and Merit\n\n* * *\n\n1 Equal treatment and making equal\n\n2 The importance of individual differences\n\n3 Nature and nurture\n\n4 The family and inheritance\n\n5 Equality of opportunity\n\n6 The conflict between merit and value\n\n7 Principles of remuneration and freedom of choice\n\n8 The consequences of distribution according to merit\n\n9 Freedom and distributive justice\n\n10 Claims based on membership of particular community\n\nSeven Majority Rule\n\n* * *\n\n1 Liberalism and democracy\n\n2 Democracy a means, not an end\n\n3 Popular sovereignty\n\n4 The justification of democracy\n\n5 The process of the formation of opinion\n\n6 The need for principles and the danger of drift\n\n7 The rule of ideas\n\n8 The duties of the political philosopher\n\n9 The conditions of preserving democracy\n\nEight Employment and Independence\n\n* * *\n\n1 The increase in the employed part of the population\n\n2 The conditions of freedom of the employed\n\n3 The moral standards of the employed\n\n4 The effect of legislation determined by them\n\n5 Freedom impossible in a unified employed hierarchy\n\n6 The importance of the man of independent means\n\n7 The ethos of the wealthy\n\n8 Leadership in non-material values\n\n_PART II. Freedom and the Law_\n\nNine Coercion and the State\n\n* * *\n\n1 The meaning of coercion\n\n2 Coercion and power\n\n3 Coercion and monopoly\n\n4 Degrees of coercion\n\n5 Coercion and the assured free sphere\n\n6 Property and the protection against coercion\n\n7 Enforcement of general rules minimizes coercion\n\n8 Unavoidable coercion\n\n9 The justification of coercion\n\n10 Coercion and moral pressure\n\nTen Law, Commands, and Order\n\n* * *\n\n1 The demarcation of individual spheres by abstract rules\n\n2 The distinction between commands and laws\n\n3 Specific or concrete and general or abstract rules\n\n4 Arbitrariness, privilege, and discrimination\n\n5 Law and freedom\n\n6 The division of knowledge in acting under rules\n\n7 Order without commands\n\nEleven The Origins of the Rule of Law\n\n* * *\n\n1 Modern liberty begins in seventeenth-century England\n\n2 The sources of the ideal in ancient Athens\n\n3 And in the Roman Republic\n\n4 The English fight against privilege\n\n5 The codification of the Whig doctrine\n\n6 Developments during the eighteenth century\n\n7 Restatements by Hume, Blackstone, and Paley\n\n8 The end of the British evolution\n\nTwelve The American Contribution: Constitutionalism\n\n* * *\n\n1 The Founding Fathers and the British tradition\n\n2 Constitution as a device to limit government\n\n3 A constitution of liberty\n\n4 The state constitutions and the bills of rights\n\n5 The discovery of federalism: dividing power is limiting power\n\n6 The evolution of judicial review\n\n7 The curious story of due process\n\n8 The great crisis of 1937\n\n9 The influence of the American model\n\nThirteen Liberalism and Administration: The _Rechtsstaat_\n\n* * *\n\n1 The revulsion against absolutism\n\n2 The abortive efforts of the French Revolution\n\n3 Postrevolutionary liberalism in France\n\n4 The sources of the German _Rechtsstaat_ tradition 294\n\n5 Prussian antecedents\n\n6 The _Rechtsstaat_ as the ideal of the liberal movement 299\n\n7 The problem of administrative courts\n\n8 The misunderstanding of the Continental tradition in England\n\nFourteen The Safeguards of Individual Liberty\n\n* * *\n\n1 The rule of law a meta-legal doctrine\n\n2 The attributes of true law\n\n3 Certainty of the law\n\n4 Generality and equality\n\n5 The separation of powers\n\n6 The limits of administrative discretion\n\n7 Legislation and policy\n\n8 Fundamental rights and civil liberties\n\n9 Conditions of interference in the individual sphere\n\n10 Procedural safeguards\n\nFifteen Economic Policy and the Rule of Law\n\n* * *\n\n1 Individual freedom precludes certain methods of government\n\n2 The scope of legitimate government activities\n\n3 The scope of administrative action\n\n4 Measures excluded in principle\n\n5 The contents of private law\n\n6 \"Freedom of contract\"\n\n7 The rule of law and distributive justice\n\nSixteen The Decline of the Law\n\n* * *\n\n1 The German origins of the reaction\n\n2 Schools opposed to traditional limitations\n\n3 Legal positivism\n\n4 The fate of law under communism\n\n5 The socialist lawyers of England\n\n6 Development in the United States\n\n7 Signs of a revival of the law\n\n_PART III. Freedom in the Welfare State_\n\nSeventeen The Decline of Socialism and the Rise of the Welfare State\n\n* * *\n\n1 The end of the century of socialism\n\n2 The reasons for its decline\n\n3 The lasting effects of the socialist epoch\n\n4 The meaning of the welfare state\n\n5 The changed tasks of the defenders of liberty\n\n6 The inherent expansionism of the administrative state\n\n7 Discussion to be confined to internal policy\n\n8 Monopoly and other minor problems\n\nEighteen Labor Unions and Employment\n\n* * *\n\n1 Freedom of association\n\n2 Union coercion and wages\n\n3 Limits of union power over wages\n\n4 Methods of union coercion\n\n5 Legitimate functions of unions\n\n6 Constraining coercion\n\n7 The role of monetary policy\n\n8 The long-run prospects\n\n9 The choice before us\n\nNineteen Social Security\n\n* * *\n\n1 Public relief and compulsory insurance\n\n2 The tendencies of recent developments\n\n3 Democracy and the expert\n\n4 Growth versus design\n\n5 The expansionism of the social security apparatus\n\n6 Provisions for old age\n\n7 Health insurance versus free medicine\n\n8 Unemployment\n\n9 The crisis of social security\n\nTwenty Taxation and Redistribution\n\n* * *\n\n1 The central issue of redistribution\n\n2 The rise of progressive taxation\n\n3 The change in its justification\n\n4 Not required by financial necessity\n\n5 Progression and democracy\n\n6 Proportionality versus progressiveness\n\n7 An appropriate income as the sole permissible reward?\n\n8 The moral issue and the principles of political action\n\nTwenty-one The Monetary Framework\n\n* * *\n\n1 Money and government\n\n2 Inflation and the welfare state\n\n3 Inflation and deflation\n\n4 The illusions of inflation\n\n5 Rule versus discretion in monetary policy\n\n6 The goals of monetary policy\n\nTwenty-two Housing and Town Planning\n\n* * *\n\n1 The problems of urban life\n\n2 Rent restriction\n\n3 Public housing\n\n4 The economics of slums\n\n5 Town planning and property rights\n\n6 The control of land use\n\n7 Building regulations\n\n8 Control of the location of industry\n\nTwenty-three Agriculture and Natural Resources\n\n* * *\n\n1 Agriculture and industrial progress\n\n2 Price control and \"parity\"\n\n3 Wards of government\n\n4 Government and the dispersion of knowledge\n\n5 Agriculture in undeveloped countries\n\n6 Conservation of natural resources\n\n7 Collective and individual foresight\n\n8 Particular resources and general progress\n\n9 Amenities and the preservation of natural life\n\nTwenty-four Education and Research\n\n* * *\n\n1 The rights of children\n\n2 Education and the state\n\n3 Government management and government finance\n\n4 Education and equality\n\n5 Problems of higher education\n\n6 A new order of rank?\n\n7 The universities and research\n\n8 Academic freedom\n\n9 The finance and organization of research\n\n10 \"The development of the human individual in its richest diversity\"\n\n_POSTSCRIPT_\n\nPostscript: Why I Am Not a Conservative\n\n* * *\n\n1 Conservatism offers no alternative goal\n\n2 The triangular relation of the parties\n\n3 The basic difference between conservatism and liberalism\n\n4 The weakness of conservatism\n\n5 Rationalism, antirationalism, and irrationalism\n\n6 The name of the party of liberty\n\n7 A new appeal to the Old Whigs\n\n8 Principles and practical possibilities\n\n## CITATION INDEX\n\nAbbott, Thomas Kingsmill, 296n18\n\nAbel-Smith, Brian, \u2013422n29\n\nAcheson, Dean, \u2013362n81\n\nAchille-L\u00e9on-Victor (Duc de Broglie), 293n12\n\nAchinger, Hans, 411n9, \u2013428n39\n\nActon, Harry Burrows, 125n40\n\nActon, John Emerich Edward Dalberg (Lord Acton), , 69n28, \u2013 116n23, \u2013, \u2013202n4, 207n10, 234n4, 240n18, 259n92, , 262n5, 263n7, 263n9, \u2013, , , , 525n8, , 529n14, , 531n17\n\nAdams, John, \u2013202n4, 244n33\n\nAdams, Randolph Greenfield, 263n8\n\nAdams, Samuel, \u2013264n10\n\nAdkins, Arthur William Hope, 157n13\n\nAdler, John Hans, \u2013440n15\n\nAdler, Mortimer Jerome, \u201358n2\n\nAlbert, Jean Fran\u00e7ois, 417n20\n\nAlbertini, Rudolf von, \u2013233n1\n\nAlchian, Armen Albert, 159n15\n\nAlcmaeon, 239n16\n\nAlex\u00e9ef, Paul, 294n14, 322n28\n\nAllen, Carleton Kemp, \u2013258n86, 307n40, \u2013310n1, 364n86\n\nAllen, Henry D., 420n25\n\nAllix, Edgard, 434n7\n\nAltmeyer, Arthur Joseph, \u2013410n6\n\nAmonn, Alfred, 469n3\n\nAnsch\u00fctz, Gerhard, 301n30\n\nAntiphon, 113n19\n\nAquinas, Thomas, 157n13\n\nArchipov, R., 351n28\n\nAristotle, 58n3, 61n10, 136n6, \u2013 157nn12\u2013, \u2013238nn9\u2013, \u2013 , \u2013243n28\n\nArnold, Thurman Wesley, 361n80\n\nAronson, M. J., 215n\n\nAtterbury, Francis, 530n15\n\nAubrey, John, 237n9\n\nAuden, Wystan Hugh, 52n7\n\nAustin, John, 218n7, 260n93\n\nBacon, Francis, 218n7, 237n9, 311n4\n\nBagehot, Walter, , 314n11\n\nB\u00e4hr, Otto, 301n32\n\nBailey, Samuel, 95n9\n\nBaker, John Randal, 514n16\n\nBaldwin, Alice Mary, 263n9\n\nBamburger, Ludwig, 384n1\n\nBarker, Ernest, 59n5, 240n18, 258n90, \u2013310n1, 407n2\n\nBarna, Tibor, 435n9\n\nBarnard, Chester Irving, 188n4, 395n19\n\nBarnett, Homer Garner, 79n8\n\nBarry, Norman, \u2013\n\nBarth\u00e9lemy, Joseph Hippolyte Jean Baptiste, 290n6\n\nBartley, William W., III, 134n2\n\nBattaglia, Felice, \u2013310n1, 325n32\n\nBauchet, Pierre, 19n48\n\nBauer, Peter Tam\u00e1s, 489n6\n\nB\u00e4umlin, Richard, \u2013310n1\n\nBay, Christian, 52n7, 58n3, 124n39\n\nBecker, Carl Lotus, 53n9\n\nBellot, Hale Leigh, \u2013310n1\n\nBenedict, Ruth Fulton, 59n5, \u2013168n2\n\nBenham, Frederic, 488n5, 489n6\n\nBenn, Stanley Isaac, 61n9, 216n3\n\nBentham, Jeremy, 58n3, , \u2013 179n15, 369n\n\nBerg, G\u00fcnther Heinrich von, , \u2013 288n, \u2013300n26\n\nBergbohm, Carl, 347n12\n\nBerlin, Isaiah, \u201370n29\n\nBernatzik, Edmund, 346n11\n\nBerns, Walter, 213n21, 380n23\n\nBerry, Denstone, 415n16\n\nBeudant, Charles, 216n2\n\nBeveridge, William Henry (Baron Beveridge), 190n8, 401n30, \u2013422n29\n\nBieri, Ernst, \u2013188n3\n\nBlackstone, William, , 258nn87\u2013\n\nBlank, David Mordecai,\n\nBlum, Walter J., \u2013432n1, 449n31\n\nB\u00f6ckenf\u00f6rde, Ernest Wolfgang, 313n10\n\nBodenheimer, Edgar, 323n29\n\nBodin, Jean, \u2013300n26\n\nB\u00f6hm, Franz, \u2013310n1, 396n21\n\nBoisguillebert, Pierre de, 109n3\n\nBonesana, Cesare, \u2013289nl\n\nBosch, Werner, 422n30\n\nBoswell, James, \u201377n6\n\nBoulding, Kenneth Ewart, 102n15, 157n13, 175n13, 412n11, 483n1, 490n8\n\nBowley, Marian, \u2013331n2\n\nBracton, Henry, , 199n, \u2013237n8\n\nBradley, Philip D., \u2013388n8, 390n13\n\nBrandeis, Louis,\n\nBriefs, Goetz Antony, 396n21\n\nBroad, Charles Dunbar, \u2013136n4\n\nBrochier, Hubert, \u2013440n15\n\nBromley, John Selwyn, \u2013233n1\n\nBronfenbrenner, Martin, 69n28\n\nBrown, Arthur Joseph, 400n28\n\nBrowning, P. R., 440n16\n\nBrunet, Ren\u00e9, \u2013317n18\n\nBrunner, Andreas, \u2013310n1\n\nBrunner, Emil, 231n28, 363n83\n\nBuckland, William Warwick, 244n32\n\nB\u00fchler, Ottmar, \u2013317n18\n\nBurckhardt, Jacob, \u2013, \u2013, \u2013202n4, 213n21\n\nBurgess, John William, 527n11\n\nBurke, Edmund, , 52n7, \u2013115n22, \u2013, 123n38, 125n40, , \u2013 , \u2013202n4, \u2013236n6, , \u2013262n2, 262n4, \u2013428n39,\n\nBurnet, Gilbert, 110n9, \u2013252n61\n\nBurns, Eveline Mabel, \u2013409n4, 417n21, 425n35, 428n40\n\nBurns, Robert,\n\nBury, John Bagnell, 92n1, 94n6\n\nBusolt, Georg, \u2013241n19, 241n21\n\nButler, Joseph, 122n35\n\nButterfield, Herbert, 122n36, \u2013237n8\n\nButters, John Keith, 448n29\n\nButtinger, Joseph, 370n2\n\nCaldwell, Bruce, , , 6n10\n\nCampbell, Charles Arthur, \u2013136n4\n\nCampbell, Rita Ricardo, 414n14\n\nCampbell, Wesley Glenn, 414n14\n\nCannan, Edwin, 467n1\n\nCarey, David, 417n19\n\nCaritat, Jean-Antoine-Nicholas de (Marquis de Condorcet), 290n6\n\nCarlyle, Alexander James, 234n4\n\nCarlyle, Robert Warrand, 234n4\n\nCarlyle, Thomas, 256n83\n\nCarr, Cecil Thomas, 320n24\n\nCarroll, John Joseph, 440n16\n\nCarter, John Corrie, 215n\n\nCartter, Allan Murray, 440n16\n\nCarver, Thomas Nixon, 141n9\n\nCato, , 113n19\n\nChamberlain, Joseph, 172n7\n\nChamberlin, Edward Hastings, 386n4, \u2013388n8, 390n11, 394n17, 412n11\n\nChristern, Hermann, \u2013300n26\n\nChristoph, Johann, \u2013288n\n\nChurchill, Winston Spencer, 357n66\n\nCicero, M. Tullius, , 118n19, \u2013 245nn34\u2013\n\nCiriacy-Wantrup, Siegfried von, 492n11\n\nClark, Colin, 426n37, \u2013448n28, 455n6\n\nClark, George Norman, \u2013233n1\n\nClark, John, 105n17\n\nClark, John Maurice, 72n33, 208n14, 338n10, 412n11\n\nClarke, William, \u2013249n48\n\nCohen, Morris Raphael, 322n28\n\nCoing, Helmut, 363n84\n\nCoke, Edward, , \u2013115n22, 245n38, 247n44, , \u2013250n54\n\nCole, George Douglas Howard, 370n1\n\nCole, William Alan, 457n9\n\nColeridge, Samuel Taylor, 51n6, \u2013 179n15\n\nCollier, David S., 111n13\n\nCollingwood, Robin George, 163n18, \u2013310n1, 520n3\n\nCommager, Henry Steele, 166n1\n\nCommons, John Rogers, 67n23, \u2013 206n6, 338n10\n\nComte, Auguste, 121n33\n\nCondorcet. _See_ Caritat, Jean-Antoine-Nicholas de (Marquis de Condorcet)\n\nConrad, Hermann, \u2013295n15\n\nConstant, Benjamin, , 238n11\n\nCooke, Colin Arthur, 119n28\n\nCooke, L. D., 440n16\n\nCooley, Thomas McIntyre, 281n58\n\nCooper, Robert M., 322n28\n\nCorwin, Edward S., \u2013, 66n19, \u2013236n6, 245n36, \u2013265n12, \u2013275n37, 277n44, 278n46, \u2013 281nn54\u2013, 281n57, 282nn60\u2013, 532n18\n\nCoulanges, Numa Denis Fustel de, 238n11\n\nCranston, Maurice William, \u201358n2, 61n9, 247n43, 252n62\n\nCrick, Bernard, 519n2\n\nCroce, Benedetto, \u2013168n2\n\nCromwell, Oliver, , 250n57, 528n12\n\nCropsey, Joseph, 69n28, 119n28\n\nCrosland, Charles Anthony Raven, 145n11, 155n10, 162n17, 213n20, 370n1, 371n4, \u2013422n29, 438n14\n\nCrosskey, William Winslow, 278n47, 279n49\n\nCrossman, Richard Howard Stafford, 353n40, 356n61, 370n1, 371n3, , 372nn8\u2013\n\nCurl, Adam, \u2013510n10\n\nDarmst\u00e4dter, Friedrich, 295n17, \u2013 310n1, 349n22\u2013\n\nDarwin, Charles, 194n11\n\nDavenport, John, \u2013388n8\n\nDavies, Godfrey, \u2013249n48\n\nDavies, John, \u2013115n22\n\nDeane, Phyllis, 457n9\n\nDefoe, Daniel, 254n77\n\ndel Vecchio, Giorgio, 157n13\n\nDemosthenes, 242n27\n\nDenning, Alfred, \u2013255n78\n\nDescartes, Ren\u00e9, , \u2013114n20,\n\nDewey, John, 67nn24\u2013\n\nDicey, Albert Venn, , \u201392n, 130n51, 175n14, 179n16, 236n7, 258n90, \u2013 , 305n35, \u2013, 306nn37\u2013, 311n4, 312n8, 319n23, 321n26, , 353n36, 353n38, \u2013, \u2013386n3\n\nDickinson, H. D., \u2013, 5n8\n\nDickinson, John, 273n35, \u2013310n1, 360n74\n\nDickinson, Lowes, 83n14\n\nDiehl, Ernst, \u2013240n17, \u2013241n19\n\nDietze, Gottfried, 278n46, 363n84\n\nD\u00edez del Corral, Luis, 293n12\n\nDirector, Aaron, 87n17, \u2013388n8\n\nDisraeli, Benjamin, \u2013523n6\n\nDobrin, Samuel, 352n34\n\nDostoevsky, Fyodor, 134n1\n\nDrucker, Peter Ferdinand, \u201366n18\n\nDuguit, L\u00e9on, 226n18, 294n14, 313n10, 318n21, 331n6\n\nDunbar, Leslie Wallace, 277n44\n\nDunham, Allison, 475n9\n\nDunlop, John Thomas, \u2013388n8\n\nEbenstein, Alan, 6n13\n\nEhmke, Horst, 322n28\n\nEhrenberg, Victor, \u2013241n19, 241n21\n\nEinaudi, Luigi Roberto, 246n40\n\nEllwein, Thomas, \u2013300n26\n\nEnfield, Ralph, 485n2\n\nEngels, Friedrich, 345n8, \u2013, 434n4\n\nE\u00f6tv\u00f6s, J\u00f3zsef, \u2013110n6\n\nErion, Gene, \u2013431n\n\nEsmein, Adh\u00e9mar, 246n42, 310n2\n\nEvans-Pritchard, Edward Evan, 229n24\n\nEyck, Erich, \u2013297n20\n\nFagen, Elmer, 434n7\n\nFaguet, Emile, 108n1\n\nFalter, Gustav, \u2013297n20\n\nFarrer, Austin Marsden, \u2013136n4\n\nFeiling, Keith Grahame, \u2013523n6\n\nFenet, P. Antoine, 291n9\n\nFerguson, Adam, , 5n7, 74n1, \u201392n, 93n3, 112n14, 217n5\n\nFerrero, Guglielmo, 269n20\n\nFeuerbach, Paul Johann Anselm, 298n23\n\nFichte, Johann Gottlieb, \u2013297n20, 498n\n\nField, Mark George, 424n34\n\nFiggis, John Neville, \u2013233n1, \u2013 235nn4\u2013, 253n70\n\nFiner, Herman, \u20134n4, 172n8, 357n65\n\nFink, Zera Silver, 237n9\n\nFinley, John, Jr., \u2013238n10\n\nFleiner, Fritz, 304n34, \u2013310n1, 331n6, 344n3\n\nFlorio, John, 239n13\n\nF\u00f6nander, Orwell de Ruyter, 402n31\n\nFoot, Philippa, 136n6\n\nForbes, Duncan, \u201392n, 110n8\n\nForsthoff, Ernst, 304n34, 470n5\n\nFosdick, Dorothy, 67n24\n\nFourastie, Jean, 483n1\n\nFrance, Anatole, ,\n\nFrane, Lenore, 440n16\n\nFrank, Jerome, , 361n80\n\nFrank, Tenney, 246n40\n\nFrankel, Sally Herbert, 489n6\n\nFrankfurter, Felix, 353n38\n\nFrazer, James George, \u2013116n23\n\nFreeman, Kathleen, \u2013240n17\n\nFreund, Ernest, 322n28, 335n8\n\nFreund, Paul Abraham, 207n8\n\nFriedman, Milton, 194n12, \u2013388n8, 390n13, 396n22, 412n11, \u2013431n, 469n3, 489n6, 503n5, 505n7\n\nFriedmann, Wolfgang Gaston, 352n35, \u2013357nn62\u2013\n\nFriedrich, Carl Joachim, \u2013310n1\n\nFromm, Erich, 134n1\n\nFuchs, Ralph Follen, 322n28\n\nFuller, Lon Luvois, 352n34\n\nGabor, Andr\u00e9, 68n27\n\nGabor, Denis, 68n27\n\nGaitskell, Hugh Todd Naylor, 370n1\n\nGalbraith, John Kenneth, 98n12\n\nGallie, Walter Bryce, 156n12\n\nGaragorri, Paulino, \u2013168n2\n\nGardiner, Samuel Rawson, 249n52, 250n57\n\nGarve, Christian, \u2013297n20\n\nGeist, Rudolf von,\n\nGellhorn, Walter, 337n9, 362n82\n\nGentz, Friedrich von, 433n2\n\nGerland, Heinrich Balthassar, 312n8\n\nGermann, Oskar Adolf, 322n28\n\nGeyl, Pieter, \u2013233n1, 498n\n\nGiacchi, Orio, 312n8\n\nGiacometti, Zaccaria, \u2013268n17, \u2013 310n1, 317n20, 325n32\n\nGibbon, Edward,\n\nGierke, Otto Friedrich von, \u2013234n3, \u2013300n26, 350n25\n\nGilboy, Elizabeth Waterman, 457n9\n\nGilby, Thomas, 234n4\n\nGladstone, William Ewart, 277n45\n\nGlaser, Stefan, \u2013255n78, 312n8\n\nGlass, David Victor, \u2013510n10\n\nGneist, Rudolf von, , 301n33, 305n35\n\nGoethe, Johann Wolfgang von, \u201392n\n\nGomperz, Heinrich, \u2013136n4\n\nGooch, George Peabody, \u2013249n48\n\nGoodnow, Frank Johnson, \u2013310n1\n\nGordon, Scott, 492n11\n\nGough, John Wiedhofft, 235n5, 243n30, 249n51, 250n57, 252n63\n\nGrabein, Max, \u2013431n\n\nGraveson, Ronald Harry, 222n11\n\nGray, John, \u2013, 22n55\n\nGreen, Thomas Hill, 60n7, \u201370n29,\n\nGreene, Donald Johnson, 255n79\n\nGrene, Marjorie Glicksman, 134n1\n\nGriffin, Clare Elmer, 370n1\n\nGroethuysen, Bernard, 108n2\n\nGrossman, Rudolph Heinrich, \u2013310n1\n\nGrotius, Hugo, 157n13\n\nGruchy, Allen Garfield, \u201366n18\n\nGsovski, Vladimir, 352n32\n\nGuarini, Giovanni Battista, \u2013233n1\n\nGuicciardini, Francesco, , \u2013431n\n\nGuttridge, George Herbert, 262n4, 530n15\n\nHaar, Charles Monroe, 478n12\n\nHaberler, Gottfried, 396n22, 399n27, 412n11\n\nHaenel, Albert, 313n10\n\nHaensel, Werner, 295n17\n\nHagenbuch, Walter, \u2013428n39\n\nHaines, Charles G.,\n\nHaldane, John Burdon Sanderson, 149n3\n\nHale, Mathew, 110n9, , \u2013115n22, 252n63\n\nHale, Robert Lee, \u201368n26, \u2013206n6\n\nHal\u00e9vy, Elie, 109n4\n\nHall, Jerome, 312n8\n\nHall, Robert Lowe, 213n21\n\nHallam, Henry, 529n13\n\nHaller, William, \u2013249n48\n\nHallowell, John Hamilton, \u2013310n1, 346n9, 349n22\n\nHamilton, Alexander, , 264n11, \u2013 , 276n41, 278n48\n\nHamowy, Ronald, 11n28\n\nHamson, Charles John, 293n13, 364n86\n\nHand, Learned, \u2013173n9, 528n12\n\nHansen, Alvin Harvey, 376n16, 444n24\n\nHardin, Charles Meyer, 488n5\n\nHare, Richard Merwyn, \u2013136n4\n\nHarrington, James, , 243n30,\n\nHarris, Abram Lincoln, 338n10\n\nHarrod, Roy Forbes, 469n3\n\nHart, Herbert Lionel Adolphus, \u2013 136n4\n\nHartwell, R. M.,\n\nHashagen, Justus, 270n23\n\nHaskell, Henry Joseph, 246n40\n\nHauriou, Maurice, 325n32\n\nHavelock, Eric Alfred, \u2013238n10\n\nHayek, F. A., \u20134nn2\u2013, 5n6, 5n8, \u20137nn12\u2013, 8nn17\u2013, 9n21, 11nn26\u2013, 12nn32\u2013, \u201315nn36\u2013 , \u201318n42, 76n6, \u201386nn15\u2013, 87n18, 94n7, 98n12, 110n8, 120n32, 127n46, 130n50, 134n2, 136n5, 141n9, \u2013183n19, 194n12, 217n5, 227n19, 232n, 275n39, 345n6, 357n64, 372n6, 376n15, 381n27, 399n27, 453n4, 460n12, 463nn18\u2013, 469n3, 478nn11\u2013, 496n13, 523n7, 525n9\n\nHays, Samuel Pfrimmer, 490n8\n\nHeathcote, Hugh Richard (Lord Cecil), 522n5\n\nHeller, Hermann, \u2013310n1, 345n7, 346n9\n\nHeman, Carl Friedrich, 136n6\n\nHenne, Rudolph, 233n2\n\nHennings, August Adolph Friedrich von, 233n2\n\nHensel, Albert, \u2013317n18\n\nHermens, Ferdinand Aloys,\n\nHerodotus, \u2013202n4, , 241n20\n\nHewart, Gordon (Baron Hewart), \u2013 , 353n38\n\nHicks, John Richard, \u2013388n8, 400n29, 426n36, 525n8\n\nHintz, Otto, 299n25\n\nHobart, R. E., 136n6\n\nHobbes, Thomas, 66n21, \u2013136n4, 237n9, 238n11,\n\nHobhouse, Leonard Trelawny, 445n26\n\nHoebel, Edward Adamson, 72n33, 171n6\n\nH\u00f6hn, Karl, \u2013240n17\n\nHolcombe, Arthur Norman, \u2013310n1\n\nHoldsworth, William, \u2013115n22, 252n63, 253n70, 254n74, 255n82, 260n96, 319n23\n\nHolland, Philemond, 244n33\n\nHollar, Wenceslas, 166n\n\nHolmes, Oliver Wendell, Jr., ,\n\nHook, Sydney, \u2013136n4\n\nHooker, Richard, , 234n4\n\nHorkheimer, Max, 357n64\n\nHowell, Thomas Bayly, 255n80, 308n\n\nHuber, Hans, 320n25\n\nHuber, Konrad, 313n10\n\nHughes, Christopher, 317n20\n\nHuizinga, Johan, 146n12, \u2013239n12\n\nHumboldt, Wilhelm von, , , 297n21, , 501n3,\n\nHume, David, \u2013, 54n10, \u201392n, 100n13, 107n, \u2013116n23, , 119n28, , 125n40, 136nn5\u2013, \u2013 158nn13\u2013, , 215n, , 227n19, 244n34, 256n83, \u2013258nn85\u2013, 264n11, 267n15, \u2013269nn18\u2013, 269n22, \u2013272n32\n\nHumphreys, Robert Arthur, 263n9, 265n13, 274n36\n\nHuszar, George Bernard de, 414n14\n\nHutt, William Harold, \u2013388nn8\u2013\n\nHutton, David Graham, 438n14\n\nHuxley, Aldous,\n\nHyneman, Charles S., 19n48\n\nHyperides, 241n24\n\nIhering, Rudolph von, 72n33, 344n3, 350n25\n\nImboden, Max, \u2013310n1\n\nIngram, John Kells, 121n33\n\nIpsen, Hans Peter, \u2013317n18\n\nIsokrates, 241n22\n\nJack, Daniel Thompson, 401n30\n\nJackson, Robert Houghwout, 279n51, 281n58\n\nJacobs, Jane, 467n1\n\nJahrreiss, Hermann, 127n46, 229n25, 314n11, 315n13\n\nJames, William, 63n14\n\nJames I (king of England), 229n23\n\nJay, Douglas, 370n1\n\nJay, John, , 264n11, 273n35\n\nJefferson, Thomas, , 172n7, 513n15, 530n16\n\nJeffrey, Francis, 112n15\n\nJellinek, Georg, 58n3, 238n11, 270n23, 310n2, 311n5\n\nJenkins, Roy, 370n1\n\nJenks, Edward, 235n5\n\nJennings, William Ivor, \u2013355nn42\u2013\n\nJewkes, John, \u2013422n29, \u2013423n31, 514n18\n\nJewkes, Sylvia, \u2013422n29\n\nJhering, Rudolph von. _See_ Ihering, Rudolph von\n\nJohnson, Harry Gordon, 415n16\n\nJohnson, Samuel, \u201377n6, \u2013, 255n79\n\nJones, Arnold Hugh Martin, \u2013238n10, 242n27\n\nJones, Harry W., \u2013310n1\n\nJones, John Walter, 70n31, 226n18, \u2013 241n19, 263n9, \u2013265n12\n\nJones, William, \u2013118n26, \u2013 241n19\n\nJoseph, Margaret F. W., 401n30\n\nJouvenel, Bertrand de, , 65n17, 159n15, 196n15, 201n3, 370n1, 469n3\n\nJudson, Margaret Atwood, 250n57\n\nJunius, \u2013258n86\n\nK\u00e4gi, Werner, \u2013168n2, \u2013310n1\n\nKaldor, Nicholas, 401n30\n\nKallen, Horace Mever, 149n3\n\nKallenborn, Carl von, 311n5\n\nKalven, Harry, Jr., \u2013432n1, 449n31\n\nKant, Immanuel, 119n28, \u2013202n4, 208n14, \u2013288n, \u2013296nn17\u2013\n\nKaufmann, Erich, \u2013317n18\n\nKeeton, George Williams, 364n86\n\nKeith-Lucas, Allen, 328n35\n\nKeller, Robert von, \u2013234n3\n\nKelsen, Hans, 62n11, 109n5, \u2013 136n4, \u2013168n2, 311n5, , \u2013 349nn12\u2013, 352n34, 360n76\n\nKern, Fritz, 235n5\n\nKeynes, John Maynard, , 121n33, \u2013 179n15, \u2013191n9, , 451n, 460n13\n\nKlenner, Fritz, 370n2, 371n5\n\nKline, Bennett E., \u201381n10, 83n14\n\nKnight, Frank Hyneman, 58n3, 60n8, 69n28, \u201372n32, 151n7, 159n15, 200n1, 206n7, 382n28, 412n11\n\nKolnai, Aurel, 349n22\n\nKopp, Hans Werner, 313n10\n\nKoser, Reinhold, 295n16\n\nKossman, Ernest Heinrich, \u2013233n1\n\nKroeber, Alfred Louis, 94n5\n\nKrutch, Joseph Wood, 429n43\n\nKuhn, Manfred, 50n5\n\nLabadi\u00e9, Jean, 110n8\n\nLaband, Paul, 313n10\n\nLaboulaye, \u00c9douard, 297n21\n\nLa Follette, Robert Marion, 360n75\n\nLahaye, Louis-Marie de (Vicomte de Cormenin), 293n12\n\nLaird, John, 244n34\n\nLa Morandi\u00e8re, L\u00e9on Julliot de, 312n8\n\nLampert, Ulrich, 317n20\n\nLange, Richard, \u2013310n1\n\nLanglois, Charles Victor, 235n5\n\nLangmuir, Irving, 94n8\n\nLaroque, M. Pierre, \u2013410n6, 417n20\n\nLarsen, Jakob Aall Ottesen, \u2013241n19\n\nLasker, Eduard, 301n31\n\nLaski, Harold Joseph, 62n11, 238n11,\n\nLaslett, Peter, 252n62\n\nLaun, Rudolf von, 322n28\n\nLavoie, Don C.,\n\nLawson, George, 250n58\n\nLecky, William Edward Hartpole, , 440n17,\n\nLee, Arthur, 262n4\n\nLegaz y Lacambra, Lu\u00eds, \u2013310n1\n\nLeibholz, Gerhard, 148n1, \u2013317n18, 349n24\n\nLenhoff, Arthur, 398n25\n\nLeoni, Bruno, , 113n19\n\nLerner, Abba Ptachya, 82n13\n\nLerner, Max, \u2013168n2, 286n68\n\nLeroy, Maxime, \u2013310n1\n\nLetwin, William Lewis, 247n44\n\nLevi, Edward Hirsch, 316n17\n\nLewis, George Cornewall, 222n10\n\nLewis, William Arthur, 80n9, 445n26,\n\nLieber, Francis, , \u2013110n6\n\nLiek, Erwin, 422n30\n\nLilburne, John, 250n58, \u2013268n17\n\nLincoln, Abraham,\n\nLindblom, Charles Edward, \u2013388n8, 389n10\n\nLindsay, A. D., 296n18\n\nLintner, John, 448n29\n\nLippmann, Walter, \u2013183n19, \u2013 310n1\n\nLipsius, Justus Hermann, 242n27\n\nLipson, Leslie,\n\nLiszt, Franz Eduard von, 343n1\n\nLively, Jack, 293n12\n\nLivy, \u201363n13, , 239n14, 244n33\n\nLlorens, Eduardo Luis, \u2013317n18\n\nLloyd, Dennis (Baron Lloyd of Hamp stead), 323n30\n\nLocke, John, , \u2013136n4, , \u2013 , \u2013253nn62\u2013, 253nn71\u2013\n\nLoening, Edgar, 299n25\n\nLolme, Jean Louis de, , \u201363n13, \u2013258n86\n\nL\u00f6sch, August,\n\nLouis Philippe (king of France),\n\nLovejoy, Arthur Osborne, 116n25\n\nLowell, Abbott Lawrence, \u2013168n2, \u2013295n15, 306n36\n\nLoyola, Ignatius, 63n14\n\nL\u00fcbtow, Ulrich von, \u2013244n31, 246n40\n\nLurton, Horace Harmon, \u2013310n1\n\nLutz, Friedrich August, 460n12\n\nLutz, Harley Leist, \u2013432n1\n\nMabbott, John David, \u2013136n4, 228n21\n\nMacaulay, Thomas Babington, 174n11, 254n76,\n\nMacbeath, Alexander, \u2013116n23\n\nMacDermott, John Clark (Baron MacDermott), 201n3, \u2013386n3\n\nMacGregor, David Hutchison, 119n30\n\nMachiavelli, Niccol\u00f2, 120n32\n\nMachlup, Fritz, 381n25, \u2013388n8, 390n12\n\nMacIver, Robert Morrison, 200n1\n\nMacKay, D. M., \u2013136n4\n\nMackintosh, James, \u2013116n23\n\nMaclean, A. H., 250n58\n\nMacmillan, Harold, \u2013521n4\n\nMadison, James, , 123n38, \u2013202n4, 264n11, 273n34, 276n41, 278n48,\n\nMaggs, Douglas Blound, \u2013236n6\n\nMaine, Henry Sumner, 207n9, 207n11, 222n11\n\nMainzer, Otto, \u2013317n18\n\nMaitland, Frederic William, , \u201381n10, 112n15, 131n51, 170n5, 199n, \u2013234n3,\n\nMalberg, Raymond Carr\u00e9 de, 313n10\n\nMalinowski, Bronislaw, 69n28, 72n33, \u2013116n23, 207n12\n\nMalitzki, Aleksandr Leonidovitch, 350n25, \u2013352n31\n\nMangoldt, Hermann von, 281n56\n\nMannheim, Karl, 357n64, 372n6\n\nMarchet, Gustav, \u2013300n26\n\nMarcic, Ren\u00e9, \u2013310n1\n\nMarshall, Alfred, 181n18, 408n3\n\nMarshall, Geoffrey, 364n86\n\nMarshall, John, , 245n38, 276n42\n\nMarshall, Thomas Humphrey, 378n18\n\nMarsilius of Padua, 234n4\n\nMartin, Alfred Wilhelm von, \u2013288n\n\nMartin, Norman H., \u201381n10, 83n14\n\nMarx, Fritz Morstein, 322n28\n\nMarx, Karl, 345n8, \u2013, 434n4\n\nMason, George, \u2013272n32\n\nMason, Lowell, 189n6\n\nMathews, Stanley, 313n10\n\nMaximus, Valerius, \u201363n13\n\nMay, Thomas Erskine, 108n1\n\nMayer, Thomas, 455n6\n\nMayo, H. B., 19n48\n\nMcAdam, Edward Lippincott, 255n79\n\nMcCallum, Ronald Buchanan, \u2013 122n34\n\nMcCloskey, Henry John, 228n21\n\nMcCloskey, Robert Green, 328n35\n\nMcCulloch, John Ramsay, 384n1, 433n3, 443n22\n\nMcGovern, William Montgomery, 111n13\n\nMcIlwain, Charles Howard, \u2013, 228n22, 234nn4\u2013, 249n50, 254n77, \u2013262n2, \u2013265n12, 278n46, 286n67, 294n14, \u2013310n1, 311n4, 359n73\n\nMcKechnie, William Sharp, \u2013237n8\n\nMcLaughlin, Andrew Cunningham, 262n3, 266n14, 278n48\n\nMcNair, Arnold Duncan, 244n32\n\nMeade, James Edward, 445n25\n\nMedawar, Peter Brian, 149n3\n\nMeek, Ronald, \u201392n\n\nMeinecke, Friedrich, 228n22, 256n84\n\nMenger, Carl, 111n13, \u2013116n23, 130n51, 171n6, 216n3, 296n19, 345n5\n\nMenger, Christian-Friedrich, \u2013310n1, 322n28\n\nMenter, Anton, 343n1\n\nMeriam, Lewis, 409n5, 411n9\n\nMerriam, Charles E., \u20134n4\n\nMessner, Johannes, 370n1\n\nMetzger, Wilhelm, \u2013297n20\n\nMichelet, Jules,\n\nMi\u00e8ge, Guy, 233n2, \u2013252n61\n\nMignet, Fran\u00e7ois-Auguste-Marie-Alexis, \u2013290n3\n\nMill, James, \u2013110n6\n\nMill, John Stewart, , 92n2, 96n10, , 156n12, 172n7, 174n11, \u2013179n15, , 212n18, 213n14, 228n20, \u2013 332nn1\u2013, \u2013378n17, , 434n6, , 499n1, , 519n1, 527n10\n\nMills, Charles Wright, 184n1, 385n2\n\nMilton, John, , 142n10, \u2013, \u2013 202n4, 250n58\n\nMims, Edwin, Jr., 62n12, 166n1, 261n1\n\nMirkin-Getzewitsch, Boris, 350n25, \u2013 352nn29\u2013\n\nMises, Ludwig von, , , 60n7, \u2013 113n17, \u2013173n9, 189n5, 194n12, 209n15, 228n22, 321n27, 330n1, 334n7, 386n6, \u2013412n10, 422n30, \u2013432n1, 434n4, \u2013448n28, 451n1, 495n12, 502n4\n\nMitteis, Heinrich, 363n84\n\nMohl, Robert von, \u2013300n26, 433n3\n\nMommsen, Theodor, \u2013244n31\n\nMontague, Francis Charles, 48n3\n\nMontesquieu, , \u201363n13, 119n28, \u2013202n4, 233n2, 245nn37\u2013, 255n82, 315n14\n\nMorley, Felix, ,\n\nMorley, John (Viscount Morley), 454n5\n\nMossner, Ernest Cambpell, 256n83\n\nMotley, John Lothrop, \u2013233n1\n\nMounier, Jean-Joseph, \u2013291nn7\u2013\n\nM\u00fcller, Adam Heinrich, \u2013300n26\n\nM\u00fcller-Armack, Alfred, \u2013410n6\n\nMullett, Charles Frederic, 263nn8\u2013\n\nMumford, Lewis, 111n13\n\nMusgrave, Richard Abel, 440n16\n\nNash, Chester C., 414n14\n\nNaumann, Friedrich, \u201364n15, 110n7\n\nNawiasky, Hans, \u2013317n18\n\nNeckel, Gustav, 59n5\n\nNef, Hans, \u2013289n1, \u2013317n18\n\nNef, John Ulric, 247n43\n\nNell-Breuning, Oswald von, \u2013521n4\n\nNeratius, 113n19\n\nNeumann, Franz Leopold, 58n3, 166n1, 224n15, 286n67, 305n35, \u2013 310nn1\u2013, 315n14, 349n22, 357n64\n\nNeumann, Friedrich Julius, \u2013440n15\n\nNeumeister, Heddy, \u2013428n39\n\nNicolas of Cusa, 234n4\n\nNiehaus, Heinrich, 483n1\n\nNowell-Smith, Patrick Horace, \u2013 136n4\n\nOakeshott, Michael, 108n2, 273n35\n\nOertel, Friedrich, 246n40\n\nOertmann, Paul, 322n28\n\nOfstad, Harald, \u201358n2\n\nOjala, Eric Mervyn, 483n1\n\nOlivecrona, Karl, 218n7\n\nOliver, Henry R., Jr., 10n23\n\nOllier, Fran\u00e7ois, 114n21\n\nOrelli, Konrad von, \u2013168n2\n\nOrtega y Gasset, Jos\u00e9, \u2013168n2, , 215n, \u2013220n8\n\nOrwell, George, 372n7\n\nOsborn, Frederick, 474n8\n\nOsborn, Henry Fairchild, 116n24\n\nPadover, Saul Kussiel, 532n18\n\nPaish, Frank Walter, \u2013470nn3\u2013\n\nPaley, William, \u2013\n\nPap, Arthur, \u2013136n4\n\nPareto, Vilfredo, 121n33\n\nPashukanis, Evgenii Bronislavovich, \u2013 353nn34\u2013\n\nPasteur, Louis, 81n12\n\nPeacock, Alan Turner, 415n16, 419n23, 437n12, 440n16\n\nPease, Theodore Calvin, \u2013249n48\n\nPechman, Joseph A., 455n6\n\nPekelis, Alexander Haim, 360n76\n\nPennock, James Roland, \u2013310n1\n\nPericles, ,\n\nPerry, Ralph Barton, \u201368n26\n\nPersky, Joseph, 121n33\n\nPerticone, Giacomo, 350n26\n\nPeters, Richard Stanley, 61n9, 216n3\n\nPeterson, William H., 483n1\n\nPetro, Sylvester, \u2013388nn7\u2013, \u2013 395n18, 396n22, , 403n32\n\nPetty-Fitzmaurice, William, \u2013258n86\n\nPfizer, Paul Achatius, \u2013301nn26\u2013\n\nPhilbrook, Clarence, 181n17\n\nPhillips, George Godrey, 315n14\n\nPhillips, Henry Bayard,\n\nPhilo of Alexandria, \u2013238n10\n\nPicot, Albert, \u2013310n1\n\nPigou, Arthur Cecil, 333n6\n\nPincoffs, Edmund L., 19n48\n\nPlamenatz, John Petrov, 152n8\n\nPlanitz, Hans, \u2013234n3\n\nPlant, Arnold, 478n11\n\nPlato, 93n3, 150n6, , 241n23\n\nPlucknett, Thomas, 235n5\n\nPocock, J. G. A., \u2013115n22, 252n63\n\nPohlenz, Max, 59n6, \u2013241n19, \u2013 500n2\n\nPolak, Carel Henrik, \u2013310n1\n\nPolanyi, Michael, , \u201375n2, 76n4, 95n9, \u2013, 199n, , 512n14\n\nPollock, Frederick, \u2013234n3\n\nPopper, Karl Raimund, \u201378n7, 87n19, 94n7, 114n21, 132n54, 134n2, \u2013 173n9, 220n9, 345n5\n\nPortalis, Jean, 291n9\n\nPound, Roscoe, \u2013, \u2013310n1, 316n16, , 352n33, 359n73, , 378n19, 386n4, \u2013388n8\n\nPratt, Charles (Lord Camden), , 255n80\n\nPrice, Richard, 120n32, , 260nn94\u2013\n\nPriebe, Hermann, 483n1\n\nPriene, \u2013238n10\n\nPringsheim, Fritz, 246n41\n\nRadbruch, Gustav, 156n12, \u2013168n2, 304n34, \u2013, 363n83\n\nRadcliffe, Cyril John (Viscount Radcliffe of Werneth), 314n11, 328n35, 364n86\n\nRamsey, Frank Plumpton, 76n5\n\nRapin-Thoyras, Paul de, 233n2\n\nRashdall, Hastings, 83n14\n\nRasmussen, Steen Eiler, 195n14\n\nRawley, William, 243n29\n\nRawls, John, 228n21\n\nRay, Jean, \u2013291nn7\u2013\n\nRees, Albert, \u2013388n8\n\nRehfeldt, Bernhard, \u2013236n6\n\nReilly, Gerard Denis, \u2013388n8\n\nRenan, Ernest, \u2013122n34\n\nRetz, Jean Fran\u00e7ois Paul de Gondi de, \u201392n\n\nRheinstein, Max, \u2013236n6\n\nRicardo, David, , 462n16\n\nRicca-Salerno, Giuseppe, \u2013431n\n\nRichberg, Donald Randall, \u2013388n8\n\nRiesman, David, 147n13\n\nRipert, Georges, 342n, \u2013, 364n85\n\nRitchie, David George, 343n1\n\nRitter, Gerhard, 109n4, \u2013168n2, 201n3\n\nRitter, Klaus, 363n84\n\nRobbins, Caroline, 250n58\n\nRobbins, Lionel (Lord Robbins), , 119n30, 225n17, 312n7, \u2013331n2, 374n14, 380n22, 405n1, \u2013448n28\n\nRoberts, Benjamin Charles, \u2013388n8\n\nRoberts, Ffrangcon, \u2013424nn30\u2013\n\nRobinson, Joan, 57n1\n\nRobson, William Alexander, 365n90\n\nR\u00f6pke, Wilhelm, \u2013428n39, 458n11, 469n3\n\nRoscher, Wilhelm, 93n3\n\nRosin, Franz, \u2013300n26\n\nRossi, Peter, \u2013168n2\n\nRossiter, Clinton, \u201370n29, 262n6\n\nRostas, L\u00e1szl\u00f3, \u2013440n15\n\nRostovtzeff, Michael Ivanovitch, 246nn39\u2013\n\nRothschild, Walther, 250n57\n\nRotteck, Carl von, \u2013202n4\n\nRottier, Georges, 417n20\n\nRoubier, Paul, 315n13, 364n85\n\nRougier, Louis Auguste Paul, 364n85\n\nRousseau, Jean-Jacques, 172n7, 245n37, \u2013, \u2013289nn1\u2013\n\nRowland, Kate Mason, \u2013272n32\n\nR\u00fcegg, Walter, 244n34\n\nRuggiero, Guido de, 108n1, 293n12\n\nR\u00fcmelin, Max Friedrich, \u2013317n18\n\nRussell, Bertrand, 66n21, 212n19, 498n\n\nRutherford, Samuel, 250n56\n\nRyle, Gilbert, 76n4\n\nSabine, George Holland, 244n34\n\nSadler, John, 250n58\n\nSalander, Gustav Adolf, 270n23\n\nSalmond, John William, \u2013216n1\n\nSalomon, Max, 157n13\n\nSantillana, Giorgio de, \u201378n7\n\nSapir, Edward, 125n42, 217n6\n\nSavigny, Friedrich Carl von, \u2013216n1\n\nSawers, David, 514n18\n\nSayers, Richard Sidney, 454n5\n\nSchabel, Franz, \u2013168n2\n\nScharpff, Franz Anton von, 234n4\n\nSchiller, Ferdinand Canning Scott, \u201378n7\n\nSchiller, Friedrich, 142n10, \u2013297n20\n\nSchindler, Dietrich, \u2013132n53\n\nSchlesinger, Arthur Meier, Jr., \u2013486n3\n\nSchlesinger, Rudolph, 352n34\n\nSchlick, Moritz, \u2013136n4\n\nSchlotterbeck, Karl, 409n5, 411n9\n\nSchmidt, Richard Karl, 270n23\n\nSchmitt, Carl, \u2013168n2, \u2013300n26, \u2013310n1, 313n9, 350n27\n\nSchm\u00f6lders, G\u00fcnter, \u2013431n\n\nSchnabel, Franz, \u2013300n26\n\nSchoeck, Helmut, , 181n17\n\nSchottl\u00e4nder, Adolf, 312n8\n\nSchrader, Otto, 59n5\n\nSchreiegg, Josef, 414n13\n\nSchultz, Theodore William, 483n1\n\nSchumpeter, Joseph Alois, 170n5, 190n7, 381n26, \u2013386n3\n\nSchwartz, Bernard, 293n13, 362n82\n\nScott, Anthony, 491n9, 492n11, 495n12, 497nn14\u2013\n\nScott, Samuel Parsons, 200n2, \u2013244n31\n\nSears, Paul Bigelow, 491n10\n\nSelden, John, 228n22,\n\nSeldon, Arthur, \u201313n35,\n\nSeligman, Edwin Robert Anderson, 441n20\n\nSeligmann, Ernst, 313n10\n\nSenior, Nassau William, \u2013331n2\n\nShaw, George Bernard, \u2013134n1\n\nShehab, Fakkri, \u2013432n1\n\nShenfield, Arthur, 21n53\n\nShirras, George Finlay, \u2013440n15\n\nSiches, Luis Recas\u00e9ns, 315n13\n\nSidgwick, Henry, 48n2, 374n13\n\nSidney, Algernon, , , 244n33, \u2013252n61\n\nSieghart, Marguerite A., 293n13, 306n39\n\nSilone, Ignazio, 213n21\n\nSimon, Yves, 136n6\n\nSimons, Henry C., , \u2013388nn8\u2013, 437n12,\n\nSkidelsky, Robert, 3n3\n\nSkinner, Burrus Frederic, 126n45, 135n3,\n\nSlichter, Sumner Huber, 395n20\n\nSmart, John Jamieson Carswell, 228n21\n\nSmiles, Samuel, 184n\n\nSmith, Adam, \u2013, , , 119n28, 157n13, 225n16, \u2013, \u2013258n86, 333n5, 468n2, 529n13, 533nn20\u2013\n\nSmith, Stanley Barney, 244n34\n\nSnow, Charles Percy, 510n11\n\nSobotka, Stephen Paul, 391n15\n\nSocrates, , 93n3\n\nSolon, \u2013240n17\n\nSombart, Werner, 67n25\n\nSpencer, Herbert, , \u201378n7,\n\nSrbik, Heinrich Ritter von, \u2013168n2\n\nStahl, Friedrich Julius, \u2013300n26\n\nStamp, Josiah (Lord Stamp), 435n9\n\nStearns, Harold, 213n21\n\nStein, Freiherr vom, \u2013300n26\n\nStephen, James Fitzjames, , \u2013173n9\n\nStern, Alfred, 290n6\n\nStewart, Dugald, \u2013118n26\n\nStigler, George Joseph, 380n24, 469n3, 505n6,\n\nStillerman, Richard, 514n18\n\nStokes, Dillard, 411n8, 415n16, 419n24, 429n42\n\nStoll, Heinrich, 311n3\n\nStone, Julius, 365n92\n\nStory, Joseph, 277n43, \u2013300n26\n\nStourzh, Gerald, \u2013272n32\n\nStrabo, \u2013238n10\n\nStrigle, Richard von, \u2013388n8\n\nStubbs, William, 234n4\n\nStuchka, Peter Ivanovitch, 351n29, 352n32\n\nSufrin, Sidney C., 19n49\n\nSumner, William Graham, \u201381n10, 153n9, 156n12\n\nTalmon, Jacob Lieb, 109n4, , 111nn12\u2013, 113n18, 114n21\n\nTarde, Gabriele de, 97n11\n\nTawney, Richard Henry, 149n2, 152n8,\n\nTchilinguirian, Harry, 417n19\n\nTezner, Friedrich, 322n28\n\nThiers, Marie Joseph Louis Adolphe, , 434n5\n\nThiry, Paul Henri (Baron d'Holbach), 126n45\n\nThoma, Richard, 346n10\n\nThomas, Norman, 370n2\n\nThompson, Dorothy, \u2013284n64\n\nThucydides, \u2013238n10\u2013, , 241n22\n\nTh\u00fcr, Josef, \u2013168n2\n\nTitmuss, Richard Morris, \u2013422n29\n\nTocqueville, Alexis de, , 107n, 108n1, 127n47, 142n10, , 174n12, 280n52, 292n11, ,\n\nTodsen, Hans Heinz, 323n30\n\nToulmin, Stephen Edelston, 228n21\n\nTrevelyan, George Macaulay, 254n74\n\nTreves, G. E., 322n28\n\nTriepel, Heinrich, 349n24\n\nTrotsky, Leon,\n\nTucker, Josiah, 118n27\n\nTucker, Rufus Stickney, 457n9\n\nTurgot, Anne Robert Jacques, \u201392n,\n\nTurvey, Ralph, 467n1, 475n9, 478n11\n\nUrmson, James Opie, 228n21\n\nVanderbilt, Arthur Thomas, 327n34\n\nVattel, Emer de, , \u2013264n10\n\nVaughan, Charles Edwyn, 253n70\n\nVico, Giambattista, \u201392n\n\nViner, Jacob, \u2013388n8, 396n22\n\nVinnius, Arnoldus, 244n34\n\nVinogradoff, Paul, 240n18\n\nVirgil, 244n34\n\nVoegelin, Erich, 349n21\n\nVoet, Johannes, 244n34\n\nVoltaire, 65n17, 245n37\n\nVossler, Otto, 111n11, 235n5, \u2013262n2, 270n23\n\nWaas, Adolf, 59n5\n\nWaddington, Conrad Hal, 135n3\n\nWade, Henry William Rawson, 315n13\n\nWagner, Donald Owen, 247n44\n\nWaldo, Dwight, \u201368n26, 359nn70\u2013\n\nWaline, Marcel, \u2013310n1\n\nWalker, Francis Amasa, 443n22\n\nWalker, Kenneth Frederick, 402n31\n\nWallich, Henry Christopher, \u2013103n16\n\nWandersleb, Hermann, \u2013310n1\n\nWarfield, Ethelbert Dudley, 361n79\n\nWatkins, Frederick Mundell, 49n4\n\nWatkins, J. W. N.,\n\nWatson, Andrew Daniel, \u2013411n7\n\nWeaver, Warren, \u201378n7\n\nWeber, Hilde, 487n4\n\nWeber, Max, 72n33, 220n9, 231n27, 315n14\n\nWebster, Daniel, 280n52\n\nWebster, William Clarence, 270nn24\u2013, 271n28, 272n33\n\nWeiss, Egon, 242n27\n\nWeisser, Gerhard, \u2013410n6, 417n21\n\nWelcker, Karl Teodor, \u2013300n26\n\nWeldon, Thomas Dewar, 355n54\n\nWestermann, William Linn, , \u2013238n10\n\nWheeler, John Archibald,\n\nWhite, Edward Douglass, 441n19\n\nWhite, Leslie Alvin, 75n3\n\nWhitehead, Alfred North, 53n8,\n\nWiese, Leopold von, \u2013168n2\n\nWieser, Friedrich von, 201n3, 269n22, 434n7\n\nWilcox, Bertram Francis, \u2013206n6\n\nWiles, Peter, \u2013379n20, 428n41\n\nWilkes, John, \u2013258n86\n\nWillgerodt, Hans, \u2013428n39\n\nWilliams, Roger John, 149nn3\u2013\n\nWilson, Francis Graham, \u2013310n1\n\nWilson, James, 276n41, 278n48\n\nWilson, Thomas, 370n1\n\nWilson, Woodrow, \u201366n18\n\nWinckler, Benedikt, 311n5\n\nWirszubski, Chaim, \u2013244n31\n\nWolfe, Don Marion, \u2013249n48\n\nWolin, Sheldon Sanford, 131n52, 234n4\n\nWollheim, Richard, 218n7\n\nWoodhouse, Arthur Sutherland Pigott, \u2013249n48, 249n53, 250nn55\u2013, \u2013268n17\n\nWoodhouse, William John, \u2013240n17\n\nWoolf, Cecil Nathan Sidney, 234n4\n\nWootton, Barbara, \u20134n4, 65n17, 386n7, \u2013402nn30\u2013, 427n38\n\nWordsworth, William, 54n11\n\nWormuth, F. D., , 229n23, \u2013251nn57\u2013\n\nWright, David McCord, 370n1, \u2013 388n8, 412n11, \u2013432n1, 446n27, 448n29\n\nYoung, Michael Dunlop, 162n17, \u2013 510n10\n\nZagorin, Perez, \u2013249n48\n\nZeidler, Karl, 313n10\n\nZeno, \u2013238n10\n\nZiegler, Heinz Otto, 166n1\n\nZimmern, Alfred Eckhard, 240n18\n\nZschimmer, Eberhard, 67n25\n\n## SUBJECT INDEX\n\nabsolutism: administrative, ; in Continental Europe, \u2013; law and,\n\nabstraction: manifestations of, 217n5; rules governing society and, \u2013\n\nAbuse of Reason project,\n\naccidents: innovation and, \u2013, , ; social justice and,\n\nAct of Athens, , 365n91\n\nAct of Settlement of 1701 (England),\n\nActon, John Emerich Edward Dalberg (Lord Acton), , 529n14\n\nAdams, John, 270n26, 530n16\n\nAdams, Samuel, , \u2013264n10\n\nadaptation: growth of knowledge and, \u2013; origins of institutions and, ; progress and, ; rules and,\n\nadministration: arbitrariness and, \u2013 362n81, ; versus business, , 321n26; in Continental Europe, , \u2013295n15, \u2013; delegation of power and, , 353n38; despotism and, 372n11, ; discretion and, \u2013 , 319n23, \u2013321nn25\u2013, \u2013 , , 372n11; employed majority and, \u2013; English, , \u2013 315n12; excessively powerful, \u2013, \u2013358nn66\u2013, \u2013362nn81\u2013; housing and, \u2013; judiciary and, \u2013, \u2013, \u2013, \u2013 ; labor unions and, ; versus law, 323n29, \u2013, 352n33; legal limitation of, , ; legislation and, \u2013 ; versus market, ; overdoing by, ; public administration movement and, ; public and private spheres and, , ; of social insurance, \u2013; tribunals and, \u2013, , ; welfare state and, \u2013; zealousness of, . _See also_ executive branch of government\n\n_Agenda for a Free Society_ (Seldon),\n\nagriculture: aesthetics and, \u2013; costs and prices and, \u2013, , ; in developing world, \u2013; farmers' earnings and, 490n7; government control and, \u2013; government subsidy of, \u2013, 485n2; industry and, , \u2013, 490n8; information provision and, ; land reform and, ; parity and, \u2013; politics and, ; population shifts and, \u2013; protectionism and, ; soil conservation and, 488n5, \u2013; surplus produce of,\n\n_All Things Considered,_ \u2013\n\naltruism, responsibility and, \u2013\n\nAmerica. _See_ United States\n\nAmerican Revolution: British constitution and, \u2013; English ideals and, \u2013262n2; French Revolution and, ; higher law and, ; liberalism and, ; Whigs and, 530n16\n\nanarchy, 69n28\n\nantirationalism: British tradition and, \u201392n; Christian tradition and, ; David Hume and, , 131n52; evolutionary tradition and, ; faith and, 131n51; versus irrationalism or mysticism,\n\nAquinas, Thomas, 234n4\n\narbitrariness: in absence of rule of law, ; administrative, \u2013362n81, ; antimonopoly policy and, ; arrest and, , ; coercion and, \u2013, , ; common good and, \u2013 288n; conservatives and, ; democracy and, \u2013, \u2013; despotism and, 348n16; European rule of law and, \u2013; freedom versus, 60n7, , ; French Revolution and, \u2013290n3; housing policy and, , \u2013; inequality and, ; judiciary and, 258n87; king and Parliament and, 250n56, ; labor unions and, \u2013; law and, , , 222n10, 231n28, , \u2013252n61, 252n63, 313n10; legislature and, ; magistrates and, ; meanings of, ; Parliament (English) and, ; prevention of, \u2013; progressive taxation and, , ; _Rechtsstaat_ and, , \u2013 ; regulatory agencies and, ; rule of law and, , ; slavery and, ; socialism and, ; taxation and, ; totalitarianism and, ; town planning and, ; tyranny and, , ; U.S. Constitution and, , ; wages and, ; Whigs versus,\n\naristocracy, intellectual class and, \u2013 194n10\n\nAristotle, , 243n29\n\nArmy Debates (England), 249n53\n\nAthens, ancient: individual liberty in, \u2013, \u2013238n10; isonomy and, \u2013241n19, 242n27; rationalism and, ; resistance to tyranny in, \u2013 241n19. _See also_ Greece, ancient\n\nAustin, John, \u2013236n6\n\nAustria, , , 298n24\n\nAustria-Hungary,\n\nAustrianism, 3n3\n\nauthoritarianism, , , \u2013\n\nBacon, Francis, 65n17, 236n7, 243n29, 345n8\n\nBagehot, Walter,\n\nbanks and banking: discretion and, \u2013 ; international, 462n17; monetary policy and, ,\n\nBartolus, 234n4\n\nBelgium, 417n22, 456n7\n\nbelief: clarity of, ; rationalism and, \u2013; voluntary conformity and,\n\nBelli\u00e8vre, Pomponne de, \u201392n\n\nBentham, Jeremy, \u2013122n34, \u2013\n\nBenthamites, , , 260n93\n\nBernard, Francis, 262n3\n\nBeveridge, William, \u2013374n12\n\nBeveridge Report, \u2013422n29, ,\n\nbills of attainder, 254n76\n\nBismarck, \u2013374n12\n\nBlack, John, \u2013486n3\n\nBodin, Jean, 269n21\n\nB\u00f6hm, Franz, 10n23\n\nBonald,\n\nbourgeois society, , 231n27\n\nBrandes, Ernest, \u2013300n26\n\nBritish Commonwealth states,\n\nbureaucracy. _See_ administration\n\nBurke, Edmund: British tradition of liberty and, , ; as great liberal, , 529n14; human character and, 51n6; liberal society and, ; on rights and political institutions, ; as Whig, 131n5, , 531n17\n\nBurns, Eveline Mabel, \u2013411n7\n\nbusiness: versus bureaucracy, , 321n26; versus state enterprise, \u2013\n\nCairo lectures, \u2013, ,\n\nCalvin's case, \u2013115n22, 245n38\n\nCanada,\n\nCantillon, Richard, 116n24\n\ncapital: formation of, ; Western economic dominance and,\n\ncapitalism: inheritance and, ; predictability and, 231n27\n\nCardozo, Benjamin N., 420n25\n\nCartesian school, , ,\n\nCary, Lucius (Viscount Falkland), 173n10\n\nCase of Monopolies (England),\n\ncertainty. _See_ uncertainty and certainty\n\ncharacter, human, 51n6\n\nCharles I (England), 173n10, , 250n56, 262n4\n\nChristianity: antirationalism and, ; freedom in, \u2013116n23; neoliberalism and, \u2013521n4\n\nChristian V (Denmark), 297n2\n\nCicero, \u2013, 244n34\n\ncivilization: accidents and, , 81n11; growth of knowledge and, \u2013; human design in creation of, \u2013, \u201375nn1\u2013; knowledge versus ignorance and, \u2013; progress and, \u2013; urban life and,\n\ncivil liberties, \u201359n4, 59n5\n\ncivil rights, \u2013396n20\n\nCivil War (English),\n\nClayton Act of 1914 (United States),\n\ncoercion: access to resources and, \u2013 , ; administrative, , ; authority to order, ; bad effects of, \u2013; choice and, \u2013, 200nn1\u2013; conservatives and, ; definition of, , \u2013, \u2013, \u2013 , ; degrees of, \u2013, ; democracy and, ; destructive, ; economic affairs and, \u2013206n6, , \u2013; education policy and, , , \u2013500nn1\u2013, \u2013, ; employment and, \u2013, , \u2013 , ; ends and means and, \u2013 , 208n14; envy and, ; equal application of rules and, ; equality and, \u2013; experts and administrators and, \u2013, ; fraud and deception and, ; versus freedom, , 58n3, 60n8, ; freedom as absence of, , , ; fundamental principles and, ; labor unions and, \u2013, \u2013, \u2013, \u2013 395n18, , 403n33, ; liberalism versus democracy and, ; liberty as absence of, 64n16, , ; liberty as power and, 67n24; limited government and, ; limits on, ; majority will and, , ; manumission and, ; markets and, ; minimization of, , ; monetary policy and, , 461n15; monopoly and, \u2013, 208n14, , \u2013; morality and, \u2013, 213n21; versus ordinary difficulties, \u2013; political power and, \u2013; prevention of, \u2013, \u2013, , ; private sphere and, \u2013, \u2013, 212n19, \u2013; punishment as, ; resistance to, ; responsibility and, ; versus restraint, , 66n20; rule of law and, \u2013, \u2013 , 310n2, 312n7, , \u2013; separation of powers and, ; social justice and, ; social security and, \u2013, \u2013411nn6\u2013, , \u2013 ; versus spontaneous order, ; state monopoly of, \u2013, \u201372nn32\u2013, 208n14; taxation and, \u2013; threatened versus actual, \u2013; threat of force and, , \u2013, \u2013; town planning and, ; unpredictable, ; versus voluntary conformity, ; welfare state and, ,\n\nCoke, Edward, 236n7, , 274n36\n\nColeridge, Samuel Taylor,\n\ncommand economy,\n\ncommands: versus laws and rules, \u2013 , , 260n93; of laws versus men, 244n33\n\n_Commentaries on the Laws of England_ (Black-stone),\n\ncommon good: arbitrariness and, \u2013 288n; law and, \u2013291nn7\u2013; public amenities and, \u2013; taxation and,\n\ncommon law: British tradition of liberty and, ; Case of Monopolies (England) and, ; creation of, \u2013115n22; economic affairs and, ; flexibility of, ; judiciary and, \u2013255n78; knowledge and, \u2013, 127n47; liberty and, , 247n44; origins of law and, \u2013236n6; Roman law and, ; state necessity and, 255n80\n\nCommons, John Rogers,\n\ncommunism: academic tenure and, ; fall of, ; inheritance and, ; law and, ; origins of, \u2013\n\ncompetition, organization and, \u2013\n\nCondillac, Etienne Bonnot de, 111n10\n\nCondorcet (French philosopher),\n\nConrad II (Holy Roman Emperor), 234n4\n\nconscription,\n\nconservation movement, \u2013, 491n9\n\nconservatives and conservatism: advance of knowledge and, ; agricultural policy and, ; authoritarianism and, \u2013; coercion and morality and, 213n21; collectivism and, , ; definition of, \u2013; democracy and, \u2013; of European peasants, ; fear of change and, \u2013, \u2013 523n6, \u2013, ; freedom and, \u2013; versus Hayek's position, , \u2013; imperialism and, ; inertia and, ; lack of principle and, \u2013; liberalism and, , \u2013 , 519n2; middle way and, \u2013; morality and, \u2013; mysticism of, ; nationalism of, \u2013; as politically nonviable, ; rule of law and, , 356n61; socialism and, \u201370n29, , , \u2013, , ; U.S. Supreme Court and, ; Whigs and,\n\nconsistency, reason and, 129n48\n\nConstant, Benjamin, , \u2013, 111n10, 293n12\n\nConstantine,\n\nConstitutio Libertatis,\n\n_Constitution of Liberty, The_ (Hayek): central premise of, ; collection of essays on, \u2013; editions of, , , ; general principles underlying, \u2013; Hayek's broader project and, \u2013; Hayek's notes and, \u2013; legacy of, ; organization of, \u2013, ; origins of, \u2013, \u2013; publication of, \u2013; public reception of, \u2013; reading of with hindsight, ; reputation of, ; as response to H. D. Dickinson, ; reviews of, ; translations of, , \u2013\n\nconstitutions: Bills of Rights and, \u2013 , 270n23, 271n28; fixed, \u2013 264n10; fundamental principles and, \u2013118n26, \u2013, , \u2013 272n32; government by, 255n81; hierarchy of laws and rules and, , \u2013 268n17, \u2013; judicial review and, \u2013, 278n46; limited, 264n11; limited government and, \u2013, \u2013; long-term versus immediate objectives and, ; making of, \u2013116n23, \u2013118n26; necessity of, 119n28, \u2013; need for enforcement of, \u2013, 273n34; rationalism and, ; reason and, 273n35; _Rechtsstaat_ and, \u2013; representative government and, \u2013; safeguarding of, \u2013; of U.S. states, 281n56. _See also specific countries;_ U.S. Constitution\n\ncontracts: breaking of, ; versus command, 357n64; enforceability of, \u2013 ; labor unions and, 297n23, \u2013 ; private sphere and, , ; status and,\n\nCorbet, John, 308n\n\nCorn Bill Debate, 255n79\n\nCort\u00e8s, Donoso,\n\n_Counter-Revolution of Science, The_ (Hayek), , 85n15\n\nCromwell, Oliver, \u201392n, 528n12\n\nCulpeper, John, 173n10\n\nculture: conscious awareness of, 125n42; differentiation in social conditions and, 83n14; luxury versus necessity and, , 98n12; stagnation and, ; transmission of,\n\ncurrency: commodities and, ; debasement of, , 456n8; elasticity of, \u2013 ; issuance of, 452n2; market and, \u2013, 453n3; stability of, 451n, 462n17\n\nCushing, Luther Stearns, 215n\n\ncustom. _See_ tradition and custom\n\nCzechoslovakia, end of communism in,\n\nDahlmann, Frederich C., \u2013300n26\n\n_Darcy v. Allein,_ 247n44\n\nDarnel, Thomas, 308n\n\nDarwin, Charles, , 116n24, 124n39,\n\n_Davenant v. Hurdis,_ 247n44\n\nDavenport, John, , \u201318n42\n\nDeclaration of Independence (U.S.), 530n16\n\nDeclaration of Parliament Assembled at Westminster,\n\nDeclaration of the Rights of Man and of the Citizen, 212n18, \u2013, , 290n5,\n\nde Maistre, Joseph,\n\ndemocracy: advantages of, ; in ancient Greece, \u2013268n17; arbitrariness and, \u2013; versus authoritarianism, ; conservatives and, \u2013; definition of, \u2013, \u2013239n12; demagoguery and, ; democratic spirit and, ; direct, \u2013268n17; dogmatic, , \u2013168nn1\u2013, \u2013, ; elective despotism and, 274n36; enlargement of government and, 172n7; experts and administrators in, , \u2013183n19; homogeneity and, ; isonomy versus, \u2013; law and, ; versus liberalism, \u2013, \u2013168n2, \u2013, ; liberty and, , ; limits of, \u2013; limit to government and, , ; peaceful change and, , \u2013173n9; progressive taxation and, 440n17, ; proportional representation and, ; rule of law and, , ; social security and, \u2013; Tocqueville on, , 174n12; understanding of public affairs and, , 174n11; unlimited government and, ; voting and, . _See also_ suffrage\n\n_Democracy in America_ (Tocqueville),\n\nDenmark, 297n2, 456n7\n\nd'Entr\u00e8ves, Alessandro Passerin, 345n8\n\nDescartes, Ren\u00e9, 345n8\n\ndesign: coordination of human activities and, ; creation of civilization and, \u2013, \u201375nn1\u2013; versus evolution, \u2013; freedom and, ; French Revolution and, ; legislation and, ; order in social life and, ; origins of institutions and, , , ; rules governing society and, ; social arrangements and, \u2013, ; versus social process, ; U.S. Constitution and, , 273n35\n\ndespotism: administrative, 372n11, ; arbitrariness and, 348n16; benevolence and, 369n; education and, ; elective, 274n36; enlightened, ; new, \u2013, 353nn38\u2013, ; paternalism and, ; _Rechtsstaat_ and, \u2013 ; socialism and, ; Soviet system and, ; as type of government, \u2013 300n26\n\ndeterminism, scientific, , \u2013, 134n2, \u2013137nn6\u2013. _See also_ science\n\ndeveloping world, \u2013, \u2013\n\nDewey, John,\n\nDickinson, H. D., \u2013\n\ndisabilities, people with, \u2013\n\ndiscretion: administrative, \u2013, 319n23, \u2013321nn25\u2013, \u2013 , \u2013, , 372n11, ; banks and, \u2013; building regulations and, ; judicial, \u2013243n28, 319n23, \u2013; legal limits of, \u2013 ; meaning of, \u2013; monetary policy and, \u2013, ; rule of law and, ; social insurance and, , , ; as Trojan horse, 320n25\n\ndiscrimination: equality before the law and, ; government regulation and, \u2013; majority rule and, ; monopoly and, ; versus privilege, \u2013; progressive taxation and, \u2013, , ; trade policy and, 317n19; welfare state and,\n\nDisraeli, Benjamin, 525n9\n\ndistribution of income: agriculture and, , \u2013; fairness and, ; social insurance and, \u2013, \u2013410n6, , \u2013; socialist goals and, \u2013  ; social services and, \u2013428n39; taxation and, \u2013, 435n9, \u2013 , ; welfare state and,\n\ndistributive justice. _See_ distribution of income; redistribution of wealth\n\nDouglas-Home, Alec, \u2013521n4\n\n_Dred Scott v. Sanford,_ 279n51\n\nDub\u010dek, Alexander,\n\ndue process, , 282n60,\n\nDutch Republic, \u2013233n1\n\nEarl, Walter, 308n\n\nEastern Europe, \u2013\n\nEast Germany,\n\neconomic growth: origins of concept of, 121n33; rapid, ; versus redistribution of wealth, \u2013103n16, ; as unplannable,\n\neconomic man, \u2013\n\neconomic policy: principles of in free society, \u2013; welfare and, \u2013374n12\n\n\"Economics and Knowledge\" (Hayek), \u201377n6\n\n_Economics of Socialism, The_ (Dickinson), 5n8\n\n_Economist_ (magazine), , 353n39\n\neconomists and economics: confusion on government intervention and, \u2013 ; history and, \u2013374n12; on legal order, \u2013; orthodox, ; public policy and, \u2013331n2; special knowledge of, ; town planning and, ; urban development and, \u2013\n\neconomy: command, ; government intervention and, \u2013; government regulation and, \u2013; prevention of depressions and, \u2013; stabilization of, \u2013; wage equilibrium and, 400n29; wage-price spiral and, \u2013. _See also_ currency; inflation; trade policy\n\nEden, Anthony, \u2013521n4\n\nEdgeworth, Francis Ysidro, 435n8\n\n_Edingburgh Review,_ \u2013118n26, 530n16\n\neducation: academic freedom and, \u2013, 512nn13\u2013; Americanization and, ; compulsory, , \u2013 500nn1\u2013, \u2013, ; despotism and, ; equality and, , \u2013; excessive, \u2013; government provision and, ; higher, \u2013; merit selection and, \u2013, \u2013510n10; public versus private, \u2013, 501n3; purpose of, ; research and, \u2013 , 512n14; science and, \u2013; segregation in, ; tenure and, \u2013 ; understanding of public affairs and, , 174n11; uniformity and, 498n; vouchers for, \u2013\n\nEgypt, 456n7. _See also_ Cairo lectures\n\n_Elements of Politics, The_ (Sidgwick), 48n2\n\nEliot, John, 236n7\n\nElizabeth I (Queen of England), \u2013\n\nemotion, appeals to, ,\n\nempiricism: legal positivism and, 345n8; origins of institutions and, 112n15; theories of liberty and, ,\n\nemployment: coercion and, \u2013, ; decline of self-employment and, , 184n1, \u2013188n3, \u2013; freedom and, \u2013; full-employment policies and, \u2013, ; giftedness and, \u2013; in government, \u2013 , 189n6, \u2013378n17, ; government regulation and, \u2013, 337n9; independence and, \u2013; initiative and inventiveness and, \u2013; intellectual class and, \u2013, \u2013 194nn10\u2013; labor unions and, \u2013 , , ; monopoly of, ; pleasant versus unpleasant, 160n16; prices and, \u2013; propertylessness and, ; remuneration and, ; as right, ; security and, \u2013188nn2\u2013; by the state, , , ; unemployment and, \u2013, \u2013, 426n36; value and, ; voting and,\n\nEncyclopedists,\n\nends and means: coercion and, \u2013, 208n14; democracy and, , 170n5, ; experts and administrators and, ; labor unions and, ; law and,  ; success and failure and, ; values and, \u2013; welfare state and, \u2013\n\nEngland. _See_ Great Britain\n\nEnlightenment, French, ,\n\nEnlightenment Project, 22n55\n\nentitlement: earnings versus inheritance and, 151n7; employment and,\n\nenvironment, human inequality and, \u2013\n\nenvy: coercion and, ; inequality of opportunity and, , 162n17; inequality of outcome and, ; necessity of wealthy class and, ; as odious, ; taxation and, ,\n\nequality: in ancient Rome, ; on basis of group membership, \u2013; changes in ideal of, \u2013; coercion and, \u2013; commutative justice and, 157n13; education policy and, , \u2013; English versus French liberalism and, 108n1; envy and, ; equal pay for equal work and, \u2013; equal protection and, , 282n60; as injustice, ; isonomy and, \u2013239n12, 239n15; law and, \u2013, 242n27, , \u2013289n1, \u2013, 317n20; of legal treatment, ; liberty and, 67n24, \u2013, \u2013 431n; as luxury, ; material versus factual, , , 150n6; of opportunity, , 343n1, ; progressive taxation and, , , 436n11, ; social services and, 427n38; superiority and, ; unfreedom and, 148n1. _See also_ inequality\n\n_Equality_ (Tawney),\n\nethics: rule of law and, , 296n18; taxation and, \u2013\n\n_Ethics_ ( journal),\n\nEucken, Walter, 10n23\n\nEurope: academic freedom in, \u2013; agricultural policy in, \u2013; constitutional law in, 294n14; education in, 514n17; employed majority and, \u2013; Justinian code in, ; labor unions in, ; legal codes in, , 297n2; legislation in, \u2013236n6; new freedom and, \u201366n18; origins of law and, \u2013; revolution of 1848 in, , \u2013; socialist period in, ; standard of living and, ; strong administration in, 236n7, 359n73\n\nEuropean Union, 485n2\n\nevolution: British versus French traditions of liberty and, ; economic man and, \u2013; human variety and, ; laws of, ; legal positivism and, 345n8; origins of institutions and, , ; progress and, \u2013, 94n4; versus rationalism, \u2013; rules and, ; social and biological, \u2013; value of freedom and,\n\nexecutive branch of government: Franklin D. Roosevelt and, , \u2013284n64; judicial review and, ; U.S. Supreme Court and, . _See also_ administration; separation of powers\n\nexpectations, disappointment of,\n\nexperts and expertise: coercion by, ; democracy and, ; limitations of, ; locating information and, \u2013, \u201377n6; natural resources and, ; public administration movement and, \u2013; self-accelerating policy and, \u2013, 413n12; social security and,\n\nexpropriation, compensation and, \u2013\n\nFabianism, , \u2013374n12,\n\nfascism, \u2013, , 350n25\n\nfederalism, \u2013\n\nFederal Republic of Germany. _See_ Germany\n\nFenwick, John, 254n76\n\nFerguson, Adam, , ,\n\nFerrera, Salvator V., 456n7\n\nFitzgerald, Edward, 254n76\n\nFlorence. _See_ Italy\n\nforeign policy, political philosophy and, \u2013\n\nformations: general laws and, ; versus institutions, 85n15\n\n_Fortune_ (magazine),\n\nFoxites, 530n16\n\nFrance: administration in, , 293n12, ; antirationalism and, 131n51; conception of free society in, ; constitutional law and, 294n14; Declaration of the Rights of Man and, 212n18; depreciation of currency of, 456n8; French Code and, 226n18; inflation in, ; July Monarchy in, ; liberal tradition in, , 65n17; Napoleonic regime in, \u2013; rationalist tradition in, , ; separation of powers in, ; social insurance in, \u2013410n6, , 417nn20\u2013; theories of liberty and, \u2013, 108n1, \u2013110nn6\u2013. _See also_ French Revolution\n\nfranchise. _See_ suffrage\n\nFrankfurter, Felix, 327n34\n\nFranks, Oliver, \u2013379n20\n\nFrederick II (Prussia), , 295n16, , 297n2\n\nFrederick the Great. _See_ Frederick II (Prussia)\n\nfreedom: as absence of coercion, , ; as absence of restraint, 68n27; versus absolute power, ; of action, ; altruism versus selfishness and, \u2013 ; in antiquity, \u2013116n23, \u2013 ; of association, ; attributes of legal systems and, ; benefits to majority of, \u2013; bound service and religious vows and, ; certainty of, ; in Christianity, \u2013116n23; coercion and, 58n3, , 60n8; collective, \u2013; conservatism and, \u2013; of contract, \u2013, ; creation of law and, ; criteria for, ; definition of, , \u2013, \u201358nn1\u2013, 59n5, \u201361nn7\u2013, \u2013238n10; delayed recognition of, \u2013, 107n; dignity of the individual and, \u2013; dispersal of knowledge and, ; economic, \u201366n18, , ; employment and, \u2013, \u2013; expansionism and, ; fear of, \u2013; free sphere and, \u2013; guarantees of, ; ignorance and, , , ; individual initiative and, ; inequality and, 148n1, 372n11; injury to others and, , 212n18; inner, \u2013, \u201370n29, ; intellectual foundations for, 110n8; judgment and, ; labor unions and, ; law and, \u2013, 245n37; versus liberty, 61n10; manumission and, \u2013; metaphysical, \u2013; morality and, \u2013, 142n10; national, \u2013; new, \u201366n18, ; as omnipotence, ; personal, ; philosophy of, ; political, \u2013, 62n11, \u201363n13; positive versus negative, 67n25; power and, , 65n17, 67n25, 69n28; predictability and, ; as primary human need, 53n8; property and, 62n11; range of choice and, ; rarity of friends of, ; rationalists as enemies of, ; reduction of coercion and, ; restatement of ideal of, , 48n2; restrictions on legislative activity and, ; rule of law and, 11n28, , , \u201363n13, 254n74, ; rules of conduct and, \u2013; science and, , 135n3; security and, 67n24, 87n19; slavery and, 59n6, , \u2013; socialism and, 343n1; social security and, ; social versus physical circumstances and, , 60n8; societal versus individual, \u2013; subjective, \u2013; tradition and, ; uncertainty and, ; as undesigned, ; values and, 52n7; from want, 57n1; welfare state and, , \u2013378n17; of will, , \u2013, \u2013136n4, \u2013138nn7\u2013, 142n10, \u2013, 222n10; youth and, . _See also_ unfreedom\n\n_Freedom and the Law_ (Leoni), 12n31\n\n\"Freedom and the Rule of Law in F. A. Hayek\" (Hamowy), 11n28\n\n\"free law\" school, , 344n4\n\nfree-trade doctrines, 82n13\n\nFrench Revolution: administrative power and, ; American Revolution and, ; arbitrariness and, \u2013290n3; centralization and, 292n11; English tradition of liberty and, , \u2013 291nn7\u2013; enthusiasts for, ; ideology of, \u2013236n6; liberalism and, , \u2013; liberty and, \u2013 291nn7\u2013, \u2013; rationalism and, ; rule of law and, \u2013, 292n10; socialism and, ; social justice and, \u2013; spread of ideas after, \u2013295n15; taxation and, \u2013 ; as universally welcomed, \u2013 ; Whigs and,\n\nFreund, Ernest, 283n62\n\nfundamental principles: arbitrariness and, 258n87; constitutions and, \u2013 118n26, \u2013, , \u2013272n32; judicial review and, ; U.S. Supreme Court and, ; Virginia Declaration of Rights and, \u2013272n32\n\nGalbraith, John Kenneth, \u2013486n3\n\nGaliani, Ferdinando, \u201392n\n\nGeddes, Charles, 403n33\n\nGeddes, Patrick, 475n10\n\ngeneral good. _See_ common good\n\n_General Theory of Employment, Interest, and Money, The_ (Keynes),\n\nGeorge, Lloyd, \u2013374n12\n\nGermany: administration in, \u2013; administrative courts in, \u2013, ; agricultural policy in, \u2013486n3; antirationalism and, 131n51; Basic Law in, 267n16; Bills of Rights literature and, 270n23; constitutional law and, 294n14, \u2013300n26; constitutional monarchy and, ; education and, ; English example and, \u2013 300n26, , 301n31; German Historical School and, \u2013374n12; housing policy in, , 470n5; imperialism and, ; inflation in, ; legal positivism in, \u2013, \u2013, 363n83; liberal and national movements and, \u201364n15; liberal tradition in, ; morality in literature of, 142n10; mystique of sovereign authority and, ; natural law and, , ; Nazism in, , ; progressive taxation in, ; rent control in, 471n6; rule of law in, \u2013, , \u2013, ; social insurance in, \u2013, 407n2, , 417nn20\u2013; socialists of the chair movement and, , \u2013374n12; trade policy and, 317n19; welfare state in, \u2013374n12. See also _Rechtsstaat_\n\nGladstone, W. E., 454n5, , 529n14, 530n16\n\nGlorious Revolution (England), , \u2013 , 252n62, 262n4\n\nGneist, Rudolf von, 294n15,\n\nGod, law and, \u2013,\n\nGodwin, William,\n\nGoebbels, Joseph, 525n9\n\ngold standard, \u2013, 462nn16\u2013\n\nGoodrich, Pierre F., \u201381n10\n\ngovernment: activities of, \u2013, 333n6, \u2013, \u2013; aim of, ; benevolent, ; centralization and decentralization of, ; by consent of the governed, ; interest versus opinion and, , 166n; of laws not of men, , 243n30, , 270n26, 271n30, 279n50, , ; legislator as gardener and, \u2013132n53; monopoly on coercion and, \u2013, \u201372nn32\u2013; nonmarket goods and, ; science of, \u2013115n22; superior knowledge of, ; unlimited, ; of will versus government of law, . _See also_ government intervention; limited government\n\ngovernment intervention: admissible, ; agriculture and, \u2013; authoritative versus unauthoritative, 332n3; character versus volume of, ; coercive versus noncoercive, ; differences in treatment and, \u2013; economic activity and, \u2013, 330n1; equality and, , ; expediency and,  \u2013331n2, ; French theory of liberty and, ; income and, \u2013; inflationary policies and, ; inhibition of progress and, ; labor unions and, \u2013; manner versus extent of, \u2013; market regulation and, ; money supply and, ; monopolistic power and, \u2013; morality and, 213n21; natural bias toward, 130n51; prevention of coercion and, \u2013; price and quantity controls and, \u2013 ; private property and, \u2013; in private sphere, \u2013331n2; rationalism and, ; removal of discontent and, ; social justice and, \u2013; to stabilize economy, \u2013; unemployment and, \u2013; wages and, \u2013. _See also specific areas of intervention;_ government; limited government\n\nGreat Britain: academic freedom in, 512n13; Act of Settlement of 1701 in, ; administrative arbitrariness in, ; administrative law in, ; agricultural policy in, \u2013, \u2013 ; Army Debates in, 249n53; bills of attainder in, 254n76; Civil War and Restoration in, \u2013, \u2013 ; classical influence on, \u2013; committee on administrative procedure in, ; common law in, , 247n44; conception of law in, \u2013 236n6, , \u2013237n8; constitution and, 255n82, 262n5\u2013, \u2013, \u2013 264n10, 277n45, \u2013300n26; Corn Bill Debate in, 255n79; Cromwell's rule in, 528n12; economic leveling in, ; education reform in, \u2013; English tradition of law and, ; evolution of political institutions and, \u2013118n26, \u2013; Fabian movement in, ; factory legislation in, ; freedom in, \u2013; Glorious Revolution in, , \u2013, 252n62, 262n4; housing policy in, ; imperialism and, ; inflation in, 456n7, ; isonomy and, \u2013, \u2013241n19; king and Parliament in, \u2013, \u2013, 254n77, , 262n4; Labour Party in, 372n11; laissez faire economics and, ; legal philosophy in, ; liberalism in, ; liberty and, \u2013, 108n1, \u2013110nn6\u2013, , 119n28, \u2013, 233n2, ; monetary policy in, 462n16; monopoly in, , 247n43, ; National Health Service in, , \u2013422n29, , \u2013423n31; Petition of Grievances in, \u2013; prerogative courts in, ; progressive taxation in, \u2013, 437n12, \u2013 440n15, ; rent control in, 471n6; rule of law in, \u2013, 305n35, \u2013 ; social insurance in, , 408n3, 414n14, 417n21, , , 532n19; socialism and, 343n1, ; Star Chamber in, , , \u2013258n86; strong administration in, 236n7; Town and Country Planning Act in, 358n67, \u2013, 478n11, 478n13; Trade Dispute Act of 1906 in, ; wage policy in, ; welfare socialism and, ; Whigs and, , 530n15, , 531n17; Wilkes case in,\n\nGreat Depression,\n\nGreece, ancient: direct democracy in, \u2013268n17; education in, \u2013 500n2; hostility toward, 237n9; individual liberty in, 209n16, \u2013, \u2013 239n10\u2013; isonomy and, \u2013, \u2013239n12, \u2013241n19; legislation in, \u2013236n6; limited constitution and, 264n11; moral worth in, 142n10; rationalism in, 113n19. _See also_ Athens, ancient; Sparta, ancient\n\nGreece, modern, 456n7\n\nGuggenheim Foundation,\n\nGuizot, Fran\u00e7ois, 293n12\n\nHamilton, Alexander,\n\nHampton, Edmund, 308n\n\nHayek, F. A., \u2013; Abuse of Reason project and, ; ad hominem attacks  on, \u20134n4; apology over errors by, ; autobiographical writings of, , ; background of, \u2013; hostility to work of, \u2013; influence of, \u2013; Nobel Prize and, \u2013; as Old Whig, , ; political philosophy of, ; reputation of, \u2013; rigidity of writings of, 3n3\n\nHazlitt, Henry,\n\nhealth care: Beveridge Report and, \u2013 422n29; government provision and, , , 414n13, \u2013, \u2013 422n29; in Great Britain, ; insurance and, 414n14, ; medical decisions and, ; productivity of recipients of, , \u2013424n33; prolongation of life and, , 423n32; self-accelerating policy and, 413n12; as social security, ; standard of service and, \u2013\n\nHegel, Georg Wilhelm Friedrich, 345n8\n\nHeveningham, John, 308n\n\nhistorians,\n\nhistoricism, \u2013\n\nhistory, economics and, \u2013374n12\n\n_History of England_ (Hume),\n\nHitler, Adolf, \u20134n4, 328n35, \u2013\n\nHobbes, Thomas: critique of Edward Coke by, \u2013115n22; law as command and, 218n7; legal positivism and, 345n8; Mathew Hale's critique of, , 252n63; rationalism and, ; sovereignty and,\n\nHobhouse, LeonardTrelawny, \u2013 374n12\n\nhomosexuality, 212n19\n\nHook, Sidney,\n\nhousing: bureaucratic allocation of, \u2013 ; overcrowding in, , ; price mechanisms and, \u2013; productivity and, ; public, , ; rent control and, \u2013, 471n6; shortage of, ; subsidies for, , ; urbanization and, \u2013\n\nhuman nature: good and bad people and, , 120n32; morality and, ; rationalism versus evolutionism and, ; self-love and, ; variety of,\n\nHumboldt, Wilhelm von, \u2013118n26\n\nHume, David: antirationalism and, , 131n52; British tradition of liberty and, ; Enlightenment and, , 131n52; history of England and, ; influence of, ; liberal society and, ; political affiliation of, 256n83; reason and morality and, , 124n39; Roman influences on, 244n34\n\nHungary,\n\n_Hurtado v. California,_ 313n10\n\nHutton, James, 116n24\n\nIceland, 297n2\n\nideas, influence of, \u2013, \u2013 179nn15\u2013\n\nignorance: culture and, ; freedom and, , , ; institutionalization of, \u201381n10; versus knowledge, \u2013; political theory and, ; range of, \u2013; slavery to passions and, ; tolerance and, ; versus uncertainty, \u201381n10\n\nimmigration,\n\nimperialism, , 527n10\n\nincome. _See_ distribution of income\n\nindependence: civil society and, , \u2013191n9; versus employment, \u2013; intellectual class and, \u2013 194n10; liberty and, \u2013252n61; as privilege,\n\nIndia, , 449n30\n\nindividual initiative: in developing world, ; judgment and, 225n16\n\nindividualism, , 523n7\n\n\"Individualism: True and False\" (Hayek),\n\nindividual responsibility. _See_ responsibility, individual\n\nindividual rights: versus rights of majority, \u201366n18; state coercion and,\n\nindustry: agriculture and, , \u2013, 490n8; natural resources and, ; town planning and,\n\ninequality: democratic spirit and, ; economic growth and, \u2013103n16; envy and, ; family background and, \u2013, 153n9; freedom and, 148n1, 372n11; of income, \u2013; increases and decreases in, ; inheritance and, 151n7, , \u2013; justification of, 435n9; merit selection in education and, \u2013, \u2013510n10; nature versus nurture and, \u2013, 149n5, \u2013; of opportunity, , 162n17; of outcome, \u2013; progressive taxation and, ; progress and, \u2013; rich and poor countries and, \u2013, 102n15. _See also_ equality\n\ninflation: continuous, , ; versus deflation, \u2013, ; full employment and, ; in history, , 457n9; housing costs and, ; monetary policy and, , ; prevention of, \u2013; progressive taxation and, ; retirement savings and, , , \u2013, 456n7; social security and, ; stimulating effect of, ; wages and, \u2013\n\ninheritance: of liberties, \u2013252n61; mischief and, 227n19; philanthropy and, ; reformers and, 190n8; rise of modern capitalism and,\n\ninnovation: accidents and, \u2013, ; aptitudes and opportunities and, \u2013, 79n8; process of, 80n9\n\nInstitute of Economic Affairs (London),\n\n_Institutes of the Laws of England_ (Coke),\n\ninstitutions: conflicts of interest and, , , 120n31; versus formations, 85n15; good and bad people and, , 120n32; growth of knowledge and, \u2013, ; life, liberty, and property and, \u2013; origins of, , \u2013113n17, \u2013; rationale of, 113n19; survival of, \u2013116n23; tradition and custom and,\n\ninsurance: design versus evolution and, \u2013; mandatory, , ; medical, 414n14, \u2013; private versus social, \u2013411n7. _See also_ social security\n\nintellectual class: absence of in U.S., \u2013; asceticism of, 208n13; employment and, \u2013, \u2013 194nn10\u2013\n\ninterest, jurisprudence of,\n\nInternational Commission of Jurists, , 365n91\n\ninternational relations, limited government and, \u2013\n\ninvisible hand: human laws and, 119n28; reason and, 119n28\n\nIreland, 456n7\n\nIslam,\n\nisonomy: ancient Greece and, , \u2013, \u2013239n12, 242n27; versus democracy, \u2013; early use of word, 239n16, ; equality and, 239n15; songs celebrating, \u2013241n19\n\nItaly: early taxation in, \u2013431n, ; fascism in, , ; inflation in, ; isonomy and, \u2013; origins of liberty and, \u2013233n1; social insurance in, \u2013, 417nn19\u2013\n\nJames I (England),\n\nJapan, 417n21, 485n2\n\nJefferson, Thomas: academic freedom and, , 513n15; on free government, ; French Revolution and, ; French tradition of liberty and, 111n11; as idol of doctrinaire democrats, 274n36; influence of, 532n18; political philosophy and, 263n8; radicalism of, \u2013, 532n18\n\nJellinek, Georg, 226n18\n\nJennings, Ivor, \u2013\n\nJesuits, , 63n14\n\nJoseph II (Austria), 298n24\n\nJosephine Code (Austria), 298n24\n\njudicial review: administration and, , \u2013, ; _Marbury v. Madison_ and, 279n50; New Deal agencies and, \u2013 362n81; postwar advancement of, ; rule of law and, ; U.S. Constitution  and, \u2013275n37, \u2013, 278n46, \u2013279nn48\u2013,\n\njudiciary: administration and, \u2013 , \u2013, , \u2013; appellate power of, \u2013; certainty of law and, , 316n15; common law and, \u2013255n78; decline of law and, \u2013; discretion and, \u2013243n28, 258n87, 319n23, \u2013; excessively powerful, , 360n75; \"free law\" school and, , 344n4; general principles and, ; independence of, , , 254n74; judicial abstinence and, 279n51; procedural safeguards and, \u2013; reform of, 356n61; rights of minorities and, 313n10; role of, , 245n38; Roman law and, 245nn37\u2013 ; separation of powers and, 258n87; taxation and, \u2013. _See also_ judicial review; separation of powers\n\nJudiciary Act of 1801 (U.S.), 278n48\n\njurisprudence of interest,\n\njustice: commutative, \u2013157nn12\u2013 ; distributive, \u2013157nn12\u2013, 160n16, ; formal, 220n9; income and, 163n18; law and, ; proportional remuneration and, 160n16; public and private interest and, 227n19; taxation and, , 443n22. _See also_ social justice\n\nJustinian code, ,\n\nKant, Immanuel, , \u2013\n\nKeynes, John Maynard, , \u2013191n9\n\nKeynesianism, 3n3,\n\nknowledge: accumulation and transmission of, \u2013; adaptation and, \u2013; aptitudes and opportunities and, \u2013; versus belief, \u2013; common good and, \u2013, 127n47; components of, ; desire for, 508n9; dispersal of, \u2013, ; education and, \u2013, ; employment and giftedness and, ; existing versus new, ; freedom and, ; government's superiority in, ; growth of civilization and, \u2013; Hayek's writings on, \u201377n6; ignorance and, \u2013, ; as individual not collective, \u2013; inequality and, \u2013; law and, , \u2013; liberal and conservative attitudes toward, ; limitations of, \u2013; morality and, ; organization and, ; of possibilities, ; research and, \u2013; of rules governing society, , 217n6; science and, \u2013, 77n7; standard of living and, \u2013; unconscious processes and, \u2013; as unplannable, , \u2013; utopian constructions and, ; Western economic dominance and, \u2013, 100n13; wisdom and,\n\n_Labor Policy of the Free Society, The_ (Petro),\n\nlabor unions: abolition of, ; civil rights and, \u2013396n20; coercion and, \u2013 , , \u2013, \u2013395n18, , 403n33, ; conduct of business and, ; evolution of, \u2013; monopoly and, , \u2013; negotiations and, \u2013; power of, \u2013, 388n9, , , \u2013; privileges of, \u2013, \u2013386n3, \u2013388n8, 393n16, \u2013; public opinion and, \u2013; public policy and, \u2013, \u2013; right-to-work laws and, , 398n24; services rendered by, , \u2013395n18; socialism and, , ; strikes and boycotts and, , \u2013, , ; tribunals and, 402n31; types of contracts and, 297n23, \u2013; unemployment and, \u2013; violence and, \u2013; voluntary association and, 384n1, ; wages and, \u2013, 390n11, 390n13, 393n16, \u2013, 394n17, \u2013, \u2013, 401n30, \u2013,\n\nLa Follette, Robert Marion, ,\n\nlaissez faire economics: admissible intervention and, ; presumption in favor of, 130n51; rationalism and, \u2013\n\nLaski, Harold J.,\n\nLaura Spelman Rockefeller Foundation, \u2013486n3\n\nlaw: administrative, , 293n12, ; versus administration, 323n29, \u2013 , 352n33; ancient world and, \u2013 , \u2013115n22, \u2013; Anglo-Saxon versus Continental, 318n22; arbitrariness and, , 222n10, \u2013252n61, 252n63; attributes of, \u2013, \u2013; bad people and, 120n32; boundaries between individuals and, \u2013, \u2013216n1; Cicero and, \u2013; classical view of, ; coerciveness of, , \u201313n35; versus command, \u2013, 218n7, , 260n93; command of a sovereign and, \u2013236n6; common good and, \u2013 291nn7\u2013; communism and, ; constitutional, 294n14; creation of, \u2013 , \u2013236n6; criminal, \u2013289n1; from custom to, \u2013; decline of, \u2013; versus decrees of tyrants, 12n31; definition of, ; democratic machinery of government and, ; discretion and, \u2013243n28, , \u2013, 256n83, \u2013258n86; doctrine of the rule of, \u2013; equality and, , , , \u2013, 242n27, , \u2013289n1, \u2013, 317n20; equal protection and, , 282n60; ethics and, , 296n18; European legal codes and, , 297n2; formal versus abstract, 220n9; free, ; freedom and, \u201363n13, , \u2013, 245n37; future rather than past transactions and, , \u2013250n54; general will and, ; of God, ; government of, ; Hayek's far-reaching claims for, \u2013, \u201313n35; hierarchy of, \u2013; higher, 245n36; ideas and, 179n16; impartiality of, , ; inconvenience and, \u2013, 256n83, 285n65; as infraction of liberty, ; infringement of rights by, , 267n16; instrumental, ; Islamic, ; isonomy and, ; judiciary and, \u2013 243n28, , 245nn37\u2013; justice and, ; Justinian code and, ; as king, 241n24, ; legal positivism and, \u2013236n6; legal realism and, ; legislation and, \u2013236n6, ; liberty and, \u2013122n34, , 245n37, , \u2013289n1; majority rule and, ; making of, \u2013116n23; in material meaning, \u2013; modern theory of, ; morality and, \u2013216n1, 296n18; _nulla poena sine lege_ and, , \u2013255n78; oppression and, ; order and, ; origins of, , 215n, 226n18; permission and prohibition and, \u2013352n31; power of, ; precedent and, \u2013; predictability and, \u2013, \u2013, , 231n28; principles and, 316n16; public policy and, , \u2013; public versus private, 318n22; pure theory of, \u2013 ; of reason, ; versus regulation by government agencies, ; religion and, ; retroactive, , 271n31, , ; rights and, \u2013288n; versus rules, \u2013, , 290n6; self-interest and, 344n3; source of, \u2013 352n31; spontaneously generated order and, ; state coercion and, ; state organization and, 171n6; of state versus nature, ; statutes and, 328n35; sufficiency for a free society and, ; supremacy of, , ; tradition and, ; universal, ; welfare of the people and, ; will and, 142n10. _See also_ natural law; _Rechtsstaat_ ; rule of law\n\n_Law and the Modern Mind_ (Frank), , 361n80\n\nLaws of the Twelve Tables,\n\n_Lectures Introductory to the Study of the Law of the Constitution_ (Dicey), \u2013\n\nlegal positivism. _See_ positivism, legal\n\nlegislation and legislative power: abitrariness and, ; administration and, \u2013; in American colonies, ; constitutional limitation of, \u2013 265n12,  , , , \u2013, \u2013 , , \u2013295n15; delegation of power and, , 353n38; discovery of, \u2013236n6, ; factory, , ; formation of constitution and, \u2013 268n17, 278n48; general principles and, \u2013, \u2013268n17; government regulation and, \u2013; higher law and, ; limitations on discretion and, ; permission and prohibition and, \u2013352n31; as protector of freedom, ; public policy and, \u2013, 378n18; reasonableness standard and, , 283n62, ; retroactive, 310n2; versus rule of law, \u2013 , \u2013314nn10\u2013; separation of powers and, 258n87; social insurance and, , 411n9; special versus general acts and, , \u2013282n55; universal applicability of, \u2013. _See also_ separation of powers\n\nleisure, criticism of, \u2013\n\nLenin, V. I., 451n\n\nleveling, economic, ,\n\n_Lex, Rex_ (Rutherford), , 250n56\n\nliberals and liberalism: advance of knowledge and, ; American Revolution and, ; British versus Continental, , ; Cicero and, ; coercion and, , 213n21; commutative justice and, 156n12; confidence of, ; conservatism and, , \u2013, 519n2, ; conservative versus radical, 111n13; definition of, , \u2013 , 519n2, \u2013, 529n13; versus democracy, \u2013, \u2013168n2, \u2013, ; discretion and, ; elites and, ; English versus French, 108n1, 109n3; fiscal policy and, 454n5; French Revolution and, ; ideal versus pragmatic, 111n13; ideas and, \u2013 , \u2013179n15; labor unions and, 385n2, 386n7; limitation of administrative activity and, ; limits on rights and, 310n2; longevity of welfare state and, \u2013; merit and rewards and, 156n12; monopoly and, ; morality and, ; need for full account of, ; permission and prohibition and, 225n17; pressing task of, 48n3; procedure and, 328n35; progressive taxation and, 437n12; rationalism and, \u2013 122n34, ; _Rechtsstaat_ and, , \u2013 ; religion and, ; rule of law and, , ; skepticism of, ; socialism and, 109n5, , , ; as system of principles, ; versus totalitarianism, , 166n1; tradition and, 125n40; as transition, \u2013301n27; utilitarianism and, ; Whiggism and, 258n90, , \u2013\n\nliberty: as absence of coercion, 64n16, , ; as absence of restraint, \u2013, 66n19, 69n28, ; administrative zeal and, ; ancient Greece and, , 209n16, \u2013, \u2013239n10\u2013; ancients' understanding of, \u2013; benevolent government and, ; children and, 499n1; to choose, 64n16; civil discord and, ; common law and, , 247n44; constitution of, \u2013 237n8; decentralization of government and, ; decline and revival of philosophy of, \u2013; definition of, , \u2013, 57n1, 59n5, \u2013, , , \u2013206n6; democracy and, , ; as desirable, \u2013; versus domination, 525n8; economic, ; English conception of, , \u2013291nn7\u2013; English influence on U.S. and, \u2013; English versus French theories of, \u2013 , 108n1, ; equality and, 67n24, \u2013, \u2013431n; extinguishing of, ; versus freedom, 61n10; free trade and, 330n1; French Revolution and, \u2013291nn7\u2013, \u2013; government employment and, 189n6; human development and, \u2013; independence and, \u2013252n61; of individual versus social collective, ; inner, 64n16, ; intellectual, ; labor unions and, ; law and, , \u2013  122n34, 245n37, , \u2013289n1; legal guarantees of, ; liberties and, \u2013; limits of, \u2013216n1; limits on rights and, 310n2; majority power and, ; meaning of, \u2013; in medieval Europe, ; morality and, 123n38; mundane concerns and, ; name for party of, \u2013; Napoleonic regime and, \u2013; natural, \u2013, 119n28; organization and, \u2013; origins of, \u2013, \u2013233nn1\u2013; political and other forms of, , , ; positive versus negative concept of, \u2013, \u201370n29; as power, \u2013, 67n24, , 69n28; of the press, \u2013 ; principle versus expediency and, \u2013; procedural safeguards and, \u2013328nn34\u2013, 328n35; progressives and, , 519n1; rationalism and, ; reasons for importance of, ; responsibility and, \u2013, \u2013134n1, \u2013 ; satisfaction of wishes and, ; science of, , 216n2; as self-proving, \u2013252n61; separation of powers and, \u2013258n86; silent revolution against, \u2013; skepticism and, 528n12; slow product of, 107n; socialism and, ; subsidized housing and, \u2013 ; suppression of in name of liberty, \u2013; theories of, ; in United States, ; universality of tradition of, \u2013; unpredictability and, ; value of, \u2013; various types of, , 58n3; virtue and, 52n7, 123n38; war and, ; wealth and, \u2013, \u201368n26; welfare state and, , \u2013; will and, 136n5. _See also_ civil liberties\n\nLiberty Fund, editions cited, \u2013\n\n_Liberty in the Modern State_ (Conservative Political Centre),\n\nlimited government: in Anglo-Saxon world, ; constitutionalism and, ; critiques of Hayek's argument on, ; Daniel Webster on, 280n52; eclipse of traditions of, ; versus fascism, 350n25; federalism and, ; individualist minimum and, ; international relations and, \u2013; law enforcement and, ; liberalism and, , \u2013168n2, \u2013; origins of, ; pressure to do away with, ; rule of law and, \u2013; socialists and socialism and, \u2013, ; versus totalitarianism, ; U.S. Constitution and, \u2013, \u2013, . _See also_ government intervention\n\nlinguistics, \u2013118n26, \u2013241n19\n\nliterature: German, 142n10; Russian, 134n1\n\nLivy, ,\n\nLocke, John, , , \u2013\n\n_Logic of Scientific Discovery, The_ (Popper), 134n2\n\nLos Angeles Pamphleteers, 156n12\n\nluxury: equality as, ; necessities and, \u2013\n\nMacaulay, Thomas Babington, , 259n92, , 529n14\n\nMachlup, Fritz, , 6n13\n\nMadison, James: on Bill of Rights, 276n41; as father of Constitution, ; on goals of government, \u2013 275nn36\u2013; as liberal, 532n18; as moderate, 532n18; on rights and powers of government, ; Whigs and, 530n16\n\nMagna Carta: American colonies and, , 262n6; as constitution of liberty, \u2013237n8; Edward Coke's interpretation of, ; influence of, 234n4, , \u2013258n86; misinterpretation of, \u2013237n8; as unalterable, 250n57\n\nmajority: action by, \u2013; advancement of opinion and, ; authority and, , \u2013; discrimination and, ; employment and, \u2013; imposition of will of, ; individual freedom as benefit to, ; liberty as impossible for, 83n14; limits to rule of, , ; luxuries and, \u2013; moral rules and, ;  opinion and, , , ; principled action and, \u2013; right of against the individual, \u201366n18; rule of law and, ; self-limitation of, \u2013; taxation and, , ; temporary, , ; unlimited power of, ; wage determination and, 401n30; wisdom and, . _See also_ democracy\n\nMalthus, Thomas, 116n24\n\nMandeville, Bernard, 110n8, \u2013116n23\n\n_Marbury v. Madison,_ , 279nn50\u2013\n\nmarket: versus administration, ; agriculture and, ; ballot of, , ; government services and, 333n6; legal order and, ; limits of the market and, ; money supply and, \u2013, 453n3; natural resources and, \u2013 ; nonmarket goods and, \u2013; price and quantity controls and, \u2013 ; property values and, ; wages and, \u2013\n\nMarshall, John,\n\nMarx, Karl, , 345n8, 525n9\n\nMarxists and Marxism, \u2013, , ,\n\nMason, George, 278n48\n\nMaximus, Valerius, \u201363n13\n\nMayflower compact,\n\nMcCulloch, J. R.,\n\nmeans and ends. _See_ ends and means\n\nMedici family, taxation and, \u2013431n\n\nmemory, writing and, 93n3\n\nMerchant Tailors' Case, 247n44\n\nmerit: assessment of, \u2013; distributive justice and, 157n13; earned and unearned income and, 164n19; education and, \u2013, \u2013510n10; meaning of, 158n14; morality and, 142n10; opportunity and, \u2013; remuneration and, 160n16, ; rewards and, \u2013, \u2013157nn12\u2013 ; value and, \u2013, 157n13, \u2013; wealth and, \u2013, \u2013 157nn12\u2013,\n\nMiddle Ages, origins of law and, \u2013 , 235n5\n\nmilitary service, compulsory, , \u2013 500n2\n\nMill, John Stuart: economic man and, , 121n33; government intervention and, ; influences on, 297n21; rationalism versus custom and, \u2013 122n34; on tolerance, ; travels of,\n\nMilton, John,\n\nmind control,\n\nminority: accumulation of wealth by, ; dissenting, ; innovators as, ; judicial protection of, 313n10; opinion of, \u2013; peaceful change and, \u2013 173n9; slum clearance and, 473n7; taxation and,\n\nMises, Ludwig von, , , 379n21\n\nmonarchy, 295n16\n\nmonetary policy: coercion and, , 461n15; discretion and, \u2013; gold standard and, \u2013, 462nn16\u2013; government debt and expenditure and, , 454n5; inflation and, , , ; limitations on discretion and, ; monopoly and, 461n15; necessity of, \u2013; rules versus authorities in, \u2013\n\nmonopoly: abolition of, 382n28; in Bills of Rights, , 271n29; Case of Monopolies (England) and, , ; coercion and, , , 208n14, , \u2013; discrimination and, ; education and, ; government intervention and, \u2013, 382n28, ; health care and, ; housing and, ; on issuance of currency, 452n2; labor unions and, \u2013; in labor versus enterprise, , ; liberal opposition to, ; monetary policy and, 461n15; natural resources and, ; obstacles to entry and, \u2013; patents and, 382n28; on provision of nonmarket goods, ; social insurance and, , \u2013, , \u2013 ; state enterprise and, \u2013; taxation and, \u2013448n28; welfare and, 428n41\n\nMontesquieu: as Anglo-maniac, 111n10; English tradition of law and, ; French tradition of liberty and, \u2013; as interpreter of English institutions, ; on political liberty, \u201359n4\n\nMont P\u00e8lerin Society,\n\nmorality: altruism and, \u2013; coercion and, \u2013, 213n21; conservatives and, \u2013; of employment, ; income and, 164n19; knowledge and, ; law and, \u2013216n1; legislation and, ; moral merit and, 142n10; in private sphere, ; rationalism and, \u2013; reason and, , 124n39; rule of law and, 296n18; rules of conduct and, ; social security and, ; values and, , ; voluntary conformity and, , 123n38\n\nMorris, Gouverneur, 278n48\n\nM\u00f6ser, Justi, \u2013300n26\n\nM\u00f6ser, Justus, \u2013300n26,\n\nMumford, Lewis, 475n10\n\nMurray, Frank, \u2013362n81\n\nNapoleonic codes,\n\nNapoleonic regime,\n\nNapoleonic wars,\n\nNasica, Scipio, \u201363n13\n\nnationalism: collectivism and, ; of conservatives, \u2013; despotism and, ; distribution on the basis of nation and, \u2013; liberal movements and,\n\nnationalization,\n\nNational Labor Relations Board (U.S.), 403n32\n\nNational Recovery Administration Act (U.S.),\n\nNational Socialism, \u2013,\n\nnation states: incapacity of, \u2013 350nn25\u2013; law and, 171n6; rights of majority within, ; stability and change and, \u2013116n23; wrongs of, 348n15\n\nnatural law: declining appeal of, ; despotism and, 348n16; in Germany, ; legal positivism and, \u2013, ; legislation and, ; property and, , 227n19; traditions of, 345n8; Whiggism and,\n\nnatural resources: depletion of, , , ; fugitive, ; government control and, , \u2013, ; investment in conservation and, ; irreplaceable, ; market and, \u2013; monopoly and, ; stock versus flow, , \u2013\n\nNazism. _See_ National Socialism\n\nNew Deal, \u2013, \u2013362n81\n\n_New Despotism, The_ (Hewart), \u2013, 353n38\n\n_New Individualist Review_ (University of Chicago student publication),\n\n_Newsweek_ (magazine),\n\n_New York Times Book Review,_\n\nNew Zealand, 456n7\n\nNicolas of Cusa, 234n4\n\nNLRB. _See_ National Labor Relations Board (U.S.)\n\nNobel Prize, \u2013\n\nnonrational action, growth of knowledge and, \u2013\n\nNorris-LaGuardia Act of 1932 (United States),\n\nNorway, 297n2, 456n7\n\nOld Testament, social justice and,\n\nOlmsted, Frederick Law, 475n10\n\n_On Liberty_ (Mill),\n\nopinion: advancement of, \u2013, \u2013 , ; experts and, ; formation of, , \u2013, 174n11; freedom of, 53n8; government by, ; labor unions and, \u2013; philanthropy and, ; politics and, ; power and, ; principled action and, \u2013; safeguarding of rights and, \u2013; social reform and, ; social security and, ; statements of ideal and, ; state power and, 175n13; students and, 181n18; U.S. Supreme Court, \u2013\n\nopportunity: equality and, , 343n1;  freedom and, ; innovation and, \u2013, 79n8; merit and, \u2013\n\noppression: coercion and, , ; law and,\n\norder: cooperation and, ; definition of, ; fear of change and, \u2013; obedience and, \u2013; polycentric, ; in social life, \u2013\n\n_Ordo_ (German journal), , 10n23\n\norganization: competition and, \u2013; corporations and, ; Germany and, 110n7; knowledge and, ; liberty and, \u2013; superiority and subordination and, 229n23; theories of liberty and, , \u2013110nn6\u2013\n\n_Osborn v. Bank of United States,_ 245n38\n\nOtis, James, , \u2013264n10\n\nPaine, Thomas,\n\nPaley, William, ,\n\nParliament (English): American colonies and, , , \u2013264n10; king and, \u2013, \u2013, 254n77, ; popular will and, 263n9; power of,\n\nPashukanis, Evgenii Bronislavovich, , 352n33\n\npatents, monopoly and, 382n28\n\npaternalism, administrative discretion and,\n\npeace: democratic change and, , \u2013 173n9; limits to majority rule and, ; rapid progress and, \u2013\n\nperfection, striving for, 54n10\n\nPeru, 456n7\n\nPetition of Grievances (England), \u2013\n\nphilanthropy, \u2013\n\nPhillips, H. B.,\n\nPhysiocrats,\n\nPigou, Arthur Cecil, \u2013374n12\n\nPitt, William,\n\nplanned economy, progress in,\n\n_Planned State and the Rule of Law_ (Friedmann), \u2013\n\nplanning. _See_ public policy\n\nPlant, Arnold, 227n19\n\nPlato,\n\npolice, in England and Germany, 324n31\n\npolice state, welfare state and, \u2013374n12\n\n_Political Ideal of the Rule of Law, The_ (Hayek), , ,\n\npolitics, as art of the possible, \u2013\n\npositivism, legal: fascism and, \u2013; historicism and, ; natural law and, \u2013, ; origins of, 345n8, 347n12; rule of law and, \u2013, , , , 363n83; social justice and, ; sovereignty and, ; totalitarianism and, 363n83\n\npoverty: redistribution of wealth and, 96n10; as relative, ; urban, \u2013\n\npower: absolute, ; arbitrariness, 313n10; to coerce, ; economic versus political, ; enumeration of, 276n41; as evil, \u2013, \u2013202n4; as freedom, 67n25, 69n28; legitimacy of, ; liberty and, \u2013, 67n24, , 69n28, \u2013252n61; limitation of, , \u2013; as not self-concluding, 311n4; possession of as freedom, , 65n17; public opinion and, 175n13; subject of, \u2013352n31\n\npragmatism, socialism and, 130n51\n\npredictability: capitalism and, 231n27; government regulation and, ; Magna Carta and, 250n57; as main function of law, , 231n28; in social life, , 229nn24\u2013\n\nPrice, Richard,\n\nprice system: employment and, \u2013; necessity of, \u2013; wages and, \u2013\n\nPriestley, Joseph,\n\nprivate enterprise. _See_ business\n\nprivate property. _See_ property\n\nprivate sphere: delimitation of, , , , ; freedom and, , ; government encroachment on, , \u2013 ; government intervention and, \u2013331n2; necessary infringements of, ; protection from coercion and, \u2013, \u2013\n\nprivilege: civil, \u2013245n35; control of production and, ; versus discrimination, \u2013\n\nproduction, control of, ,\n\nprofit, legitimate versus excess, 164n19\n\nprogress: civilization and, \u2013; economic leveling and, ; enjoyment of, \u2013; goals and, \u2013; inequality and, \u2013; lack of control and, \u2013; meaning of, 94n4; opinion formation and, \u2013; rapid, , ; resistance to change and, ; as self-making, 94n5; skepticism about, ; theories of, 94n6; uncertainty and, , \u201392n, , ; wealth and, , \u2013\n\nprogressives and progressivism: administrative discretion and, ; change and, ; imperialism and, ; labor unions and, \u2013; liberty and, , 519n1; public administration movement and, ; as reactionary, ; socialism and,\n\nProhibition, 213n21,\n\nproletariat,\n\nproperty: building regulations and, \u2013 ; civil society and, \u2013191n9; commutative justice and, 157n13; contracts and, ; employment and, ; inviolability of, 226n18; laws of nature and, , 227n19; legal order and, ; natural resources and, ; neighborhood effects and, , , , ; planning in Great Britain and, 358nn67\u2013; prevention of coercion and, \u2013; private, , 207n9; private sphere and, ; protection of, ; responsibility and, ; restrictions on abatement of, \u2013245n35; rights and, ; several, , 207n9; socialization of, \u2013; town planning and, \u2013; in U.S. Constitution, . _See also_ housing\n\nPrussia: administration in, \u2013295n15, 305n35; civil code of 1751 in, ; education in, \u2013; as police state, ; progressive taxation in, ; rule of law and, \u2013, \u2013; separation of powers in, ; social insurance in, 407n2\n\npsychology, 126n45\n\npublic policy: economists and economics, \u2013331n2; employed majority and, \u2013; equal laws and, \u2013; expediency and, \u2013331n2; ideas and, 179n16, \u2013; implications of Hayek's thought for, ; labor unions and, \u2013, \u2013; law and, , \u2013; legislation and, \u2013 , 378n18; rules and, \u2013; self-accelerating, \u2013, 413n12; social justice and, ; town planning and, \u2013, 475n10, . _See also specific policy areas_\n\npublic relief: fair shares and, ; guaranteed subsistence and, \u2013, 405n1; insurance and, ; means testing and, ; as self-propagating, \u2013428n39; state monopoly and, 428n41; unemployment and, \u2013. _See also_ social security\n\nPuritans, 274n36\n\nradicalism, wealth and, 194n12\n\nrationalism: ancient world and, , 113n19; versus belief, \u2013; Christianity and, ; consistency and, 129n48; constitutionalism and, ; control and predictability and, ; definition of, 108n2; versus evolutionism, \u2013; founding of, ; French Revolution and, ; French tradition of liberty and, 65n17, ; laissez faire economics and, \u2013; liberty and, ; morality and, , 124n39, \u2013; origins of institutions and, , \u2013113n17, ; theories of liberty and, , ; tradition and custom and, , ; U.S. Constitution and, ; utilitarianism and, \u2013; Whigs and, . _See also_ antirationalism\n\nReagan, Ronald, 437n12\n\nrealism, legal,\n\nreason: constitutions and, 273n35; liberal and conservative distrust of, ; limits to power of, ; limits to uses of, \u2013. _See also_ rationalism\n\n_Rechtsstaat:_ abandonment of, ; battles over, ; development of, \u2013, \u2013300n26; disappearance of, ; English rule of law and, 305n35, \u2013 ; formalization of, \u2013; German liberal tradition and, ; from ideal to practice and, \u2013; justicialism and, , ; legal positivism and, \u2013; origins of, \u2013; praise for, ; Prussian government and, ; significance of, \u2013; versus socialism, ; state incapacity and, 350n25; taxation and, ; theory of, \u2013300n26; as word, \u2013300n26, 305n35. _See also_ rule of law\n\nredistribution of income. _See_ distribution of income\n\nredistribution of wealth: on basis of group membership, \u2013; command economy and, ; distributive justice and, 156n12; versus economic growth, \u2013103n16, ; extreme poverty and, 96n10; long-term versus short-term view and, \u2013; poor countries and, ; versus progress, \u2013; rule of law and, ; substantive versus formal justice and, \u2013 ; taxation and, . _See also_ distribution of income\n\nregulatory agencies, versus law,\n\nRehberg, August Wilhelm, \u2013300n26\n\nrelief. _See_ public relief\n\nreligion: for conservatives, socialists, and liberals, , ; enforced conformity and, ; law and, . _See also_ Christianity\n\nRenaissance, origins of liberty and, \u2013 233n1\n\nresponsibility, individual: altruism and, \u2013; assignment of, \u2013, \u2013; versus collective responsibility, 146n12, ; consequences and, , ; decline of, \u2013; dread of, \u2013134n1; education and, ; employment and giftedness and, \u2013 ; freedom of the will and, ; liberty and, \u2013; natural resources and, \u2013; necessity of, ; scientific determinism and, , 136n6; success and,\n\nRestoration (England), , \u2013\n\nrestraint: absence of, , 66n19, 68n27, 69n28; versus coercion, , 66n20; rules of government and,\n\nrights: bills of, \u2013, 270n23, \u2013 , 276n41, , ; to education, ; employment and, ; enumeration of, ; federalism and, \u2013, 274n36; _garantisme_ system and, 293n12; German approach to, \u2013236n6; law and, , 267n16, \u2013288n, 310n2; in nineteenth-century Prussia, \u2013 295n15; private sphere and, \u2013, \u2013; as procedural, \u2013; property, ; rule of law and, 287n; social justice and, \u2013; U.S. Constitution and, \u2013; to work, , 398n24\n\nRights of Man,\n\nright to vote. _See_ suffrage\n\nrisk: employment and, , ; natural disasters and, ; value and, \u2013\n\n_Road to Reaction, The_ (Finer), \u20134n4\n\n_Road to Serfdom, The_ (Hayek), \u2013, \u20134n4, \u2013,\n\nRobespierre, , 114n21\n\nRobson, William Alexander, \u2013\n\nromanticism, English liberalism and, 110n7\n\nRome, ancient: constitution of, ; economy in, ; hostility toward, 237n9; individual liberty in, ; legal tradition of, \u2013, \u2013245nn33\u2013 ; legislation in, \u2013236n6; limited constitution and, 264n11\n\nRoosevelt, Franklin D.: administrative agencies under, \u2013362n81; agricultural policy and, \u2013486n3; Court  Packing Bill of, \u2013, 285n66; executive power and, \u2013, \u2013 284n64; on freedom from want, 57n1; movement toward socialism and, ; welfare state and, \u2013374n12\n\nRoosevelt, Theodore,\n\nRousseau, Jean-Jacques, , , , \u2013\n\nrule of law: Act of Athens and, , 365n91; administrative courts and, ; arbitrariness and, \u2013, , ; attacks on principle of, ; attributes of legal systems and, ; case law and, ; coercion and, \u2013, 312n7, , \u2013; common law and, ; consolidation of, \u2013, 254n74; constitutional monarchy and, ; criteria for, , 11n28, ; critiques of Hayek's argument on, \u2013; Declaration of the Rights of Man and of the Citizen and, , ; decline of in France, 292n10; delegation and, ; democracy and, ; denial of in Russia, \u2013; discretion and, \u2013, ; economic affairs and, , \u2013; English movement away from, \u2013, 353n38; equality before the law and, ; exceptions to, ; expropriation with compensation and, \u2013; freedom and, , 254n74, ; freedom of contract and, ; general, ; general principles and, ; general will and, ; Germany and, , 305n35, ; government regulation and, ; ideal of, \u2013, ; international variations of, \u2013; isonomy and, ; judicial review and, ; justice and, ; labor unions and, \u2013396n20, , , 403n32; legal positivism and, \u2013 , , \u2013, 363n83; versus legislation, \u2013, \u2013314nn10\u2013 ; liberal movement and, ; liberty and, 293n12; meaning of _rule_ and, 244n33; morality and, 296n18; Nazism and, ; \"nulla poena sine lege\" and, 162n17, , ; opposition to, , ; procedural safeguards and, \u2013 ; Prussian contribution to, \u2013 ; _Rechtsstaat_ and, \u2013, 305n35, ; reformers' efforts to weaken, ; restoration of, , 365n92; rights and, 287n; separation of powers and, ; socialism and, , \u2013, 357n64; social justice and, \u2013; sources for Hayek's treatment of, ; Soviet theory and, 350n25; state incapacity and, 350n26; theory versus practice and, ; usefulness of, . _See also_ law\n\nrules: abstract, ; versus commands, \u2013; compulsion of custom and, \u2013, \u2013220n8; delineation of spheres and, \u2013; destructive, \u2013; functions of, \u2013; as government or dominion, 244n33; hierarchy of, , ; isonomy versus, ; versus laws and rules, \u2013, ; moral, ; as necessary for freedom, \u2013; as necessary for society, \u2013; personal, ; public policy and, \u2013; regulation of economic activity and, \u2013; self-regulation and, 344n3; voluntary conformity to, \u2013\n\nRuml, Beardsley, \u2013486n3\n\nRussia: early stages of communism in, \u2013; education in, 514n17; lawbreakers in literature of, 142n10; Marxism and, ; science in, ; state medicine in, ; welfare socialism and,\n\nsafety net. _See_ public policy\n\nSaint-Just, , 114n21\n\nSamuelson, Paul Anthony, 412n11\n\nSavigny, Friedrich Carl von, 215n\n\nScholastic philosophy, 142n10\n\n_Schweizer Monatshefte_ ( periodical),\n\nscience: academic freedom and, , \u2013; false hypotheses and, 125n41; freedom and, , 135n3;  funding for, ; human behavior and, 126n45; ignorance and, \u2013, 77n7; intellectual liberty and, ; knowledge and, \u2013, 77n7; liberty and, \u2013 , , 216n2; versus political philosophy, ; research and, \u2013, \u2013; uncertainty and, \u2013, 94n8; unpredictability and, ; utility and, 95n9\n\nscientific determinism. _See_ determinism, scientific\n\nscientism, \u2013\n\n_Second Treatise on Civil Government_ (Locke), \u2013, 252n62\n\nsecurity, freedom and, 67n24, 87n19\n\nsegregation, public education and,\n\nSenior, N. W., , 405n1\n\nseparation of powers: in American colonial constitutions, 271n30; appearance of concept of, 250n58; in Bills of Rights, ; in British constitution, , 255n82; civil society and, \u2013 252n61; Declaration of the Rights of Man and of the Citizen and, 290n5; English retreat from rule of law and, ; in France, ; French Revolution and, ; judicial review and, 278n48; party differences and, ; political philosophers on, \u2013258n86; popularization of, 255n82; prevention of coercion and, ; in Prussia, ; reaffirmation of, ; rule of law and, ; as safeguard against abuse of authority, ; safeguarding of, \u2013 ; special versus general acts and, ; U.S. Constitution and, 274n36; welfare state and, , \u2013379n20; William Paley on, \u2013\n\nSherman Act (United States),\n\nSiey\u00e8s, Abb\u00e9,\n\nSlaughterhouse Cases (U.S.), 282n60\n\nslaves and slavery: in ancient Greece, \u2013238n10; benefits of to master, 120n31; definition of, \u2013238n10; emancipation and, 282n60; government without consent and, ; manumission and, \u2013; property ownership and, , 70n31; recognition of freedom and, 59n6; voluntary,\n\nSmith, Adam: antirationalism and, ; British tradition of liberty and, ; government intervention and, \u2013 ; individualism and, 523n7; influence of, ; liberal society and, ; wrongful accusation of, 111n13\n\nsocial contract: French Revolution and, ; limits of majority rule and, ; origins of institutions and,\n\nsocial control,\n\nsocial democracy, , 109n5,\n\nsocial good. _See_ common good\n\nsocial growth, adaptive evolution and, , \u2013116n23\n\nsocial insurance. _See_ social security\n\n_Socialism and the New Despotism_ (Crossman),\n\nsocialists and socialism: agriculture and, ; American inspiration for, ; in ancient Rome, ; arbitrariness and, ; coercion and morality and, 213n21; communism and, \u2013; conservatives and, \u201370n29, , 356n61, , \u2013, ; conservatives and conservatism, ; continued appeal of, \u2013; disillusionment with, \u2013; distributive justice and, 156n12; employment and, ; equality of legal treatment and, ; freedom and, , 343n1; French Revolution and, , \u2013; hot versus cold, ; imperialism and, ; international trends in, \u2013; labor unions and, ; liberalism and, 109n5, , ; liberty and, ; limited government and, \u2013, ; morality and, ; nationalization and, ; in nineteenth-century Europe, , ; postwar enthusiasm for, \u2013; pragmatism and, 130n51; progressivism and, ; rationalism of, ; versus _Rechtsstaat,_ ; revolution of 1848 and, ; rule of law and, , \u2013  , 357n64; social justice and, \u2013 , ; standard of living and, ; syndicalist, ; taxation and savings and, ; utilitarianism and, 48n2; welfare socialism and, \u2013; Whigs and, . _See also_ National Socialism\n\nsocial justice: accidents and, ; collective rewards and punishments and, \u2013; as disturbing to Hayek, ; employed majority and, ; envy and, ; movements in support of, \u2013 ; nationalization and, ; rule of law and, \u2013; socialism and, \u2013 ; substantive versus formal, \u2013 ; wages and, \u2013; welfare state and,\n\nsocial mobility, taxation and,\n\nsocial organization, lack of conscious planning and, , 5n7,\n\nsocial process, versus design,\n\nsocial security: abolition of, 429n42; administrative apparatus for, \u2013; ambitious aims of, \u2013411n7; beneficiaries' irresponsibility and, , \u2013 412n10; branches of, \u2013; coercion and, \u2013, \u2013411nn6\u2013, \u2013; complexity of, \u2013, 411n9; dependence of the aged and, \u2013, 420n25; discretion and, ; European social insurance and, \u2013 , 407n2; funding for, \u2013410n6; means testing and, ; minimal versus adequate provision and, \u2013, \u2013, 427n38; origins of in U.S., \u2013, 409n5; politics and, \u2013 , 420nn25\u2013; private provision of, \u2013409n4, , ; propaganda and, , 416n18; public relations and, \u2013; public's support for, \u2013 ; redistribution of income and, \u2013; as self-propagating, \u2013 428n39; stages of, , 415n15; taxation and, \u2013, 437n12; universal coverage and, \u2013. _See also_ public relief\n\nSolidarity party (Poland),\n\nSolon, \u2013,\n\nSouth Africa, 456n7\n\n_Southern Economic Journal,_\n\nsovereignty: democracy and, \u2013168n2; Hobbesian conception of, ; John Locke and, ; limitations on, ; original concept of, ; of Parliament, ; popular,\n\nSoviet Union, , 350n25, ,\n\nSparta, ancient, \u2013, \u2013114nn21\u2013. _See also_ Greece, ancient _Sphere and Duties of Government, The_ (Humboldt),\n\nstandard of living, \u2013, ,\n\nstate enterprise, versus business, \u2013\n\n_State v. Bolo_ _ff_ _,_ 313n10\n\nstatus, contract and,\n\nStoic philosophers, \u2013238n10\n\nsuccess and failure, ends and means and,\n\nsuffrage: freedom and, 62n11, \u201363n13, ; limits of democracy and, \u2013; for propertyless, ; slaves and, ; tyranny and, ; universal, \u2013500n2; women's, \u2013170n4\n\nSweden, 297n2, 417n21, 456n7\n\nSwitzerland, \u2013170n4, 317n20,\n\nTacitus,\n\ntaxation: appropriate income and, \u2013 , 445n26; attorneys and, 449n31; capital formation and, ; coercion and, \u2013, \u2013; common good and, ; degree of progression in, \u2013, 437n12, 440n17, ; division of labor and, , 445n25; ethics and, \u2013; in Europe, \u2013 ; expenditures and, \u2013315n12; expense-account waste and, 449n31; graduated versus geometric progression and, 433n2; incentives and, ; inequality and, ; inflation and, ; justice and, , \u2013, 443n22; marginal utility and, \u2013, 435nn8\u2013, 437n12, 444n24; maximum  rates of, ; Medici family and, \u2013431n; monopolies and, \u2013 448n28; opposition to, \u2013; origins of progressive form of, \u2013; productivity and, ; proportional versus progressive, , 433n2, \u2013 ; public expenditure and, ; punitive rates and, , , \u2013; rates of, 437n12; redistribution and, ; rising incomes and, ; as robbery, ; salaried drafters of, 446n27; savings and, ; scientific basis for, \u2013, 435n8, ; scientific research and, ; single-tax plans and, \u2013; social security and, \u2013, 437n12; state and local, 437n12; uneven income and, 444n23; voting strength and, \u2013 ; withholding from paychecks and, \u2013486n3\n\nTaylor, Harriet, , 519n1\n\ntechnology, protected rights and, \u2013\n\nThatcher, Margaret, 437n12\n\ntheocracy, \u2013300n26\n\nThompson, George, 278n48\n\nthought, operations of,\n\n_Times Literary Supplement,_\n\nTocqueville, Alexis de: as Anglo-maniac, 111n10; British versus French traditions of liberty and, ; on bureaucracy, 293n12; French liberal thought and, ; French Revolution and, ; as liberal, , 529n14\n\ntolerance, ignorance and,\n\nTories (England), , 255n81\n\ntotalitarianism: American drift toward, 359n73; arbitrariness and, ; economic claims of, ; \"free law\" school and, 344n4; French Revolution and, ; French tradition of liberty and, ; hot and cold socialism and, ; individualist reaction to, ; legal positivism and, 363n83; versus liberalism, , 166n1; versus limited government, ; morality and, ; public relations and, ; social democracy and, , 109n5, ; war and,\n\nTrade Dispute Act of 1906 (Great Britain),\n\ntrade policy: fair trade and, 297n23; free trade and, 330n1\n\ntradition and custom, \u2013, \u2013,\n\ntruth, retention of, , 47n1\n\nTucker, Josiah, ,\n\nTurgot, Anne Robert Jacques, 111n10,\n\ntyranny: in absence of rule of law, ; isonomy versus, ; liberation movements and, ; models of resistance to, \u2013241n19; voting and,\n\nuncertainty and certainty: benefits of freedom and, ; coercion and, ; in daily work, ; economic decisions and, ; free-trade doctrines and, 82n13; versus ignorance, \u201381n10; of law, , 316n15; open society and, 87n19; progress and, , \u201392n, ; rationalism and, ; scientific discovery and, \u2013, 94n8; of success, ; true laws and, \u2013; unfreedom and, 199n\n\nunfreedom: appreciation of freedom and, ; benefits from freedoms of others and, ; commands and, 12n35; equality and, 148n1; meaning of freedom and, , ; rule of law and, 11n28; slavery and, 59n6; slavery to passions and, ; suffrage and, 62n11; sympathy for lawbreakers and, 134n1; uncertainty and, 199n; youth and, . _See also_ freedom\n\nunions. _See_ labor unions\n\nUnited Kingdom. _See_ Great Britain\n\nUnited States: administrative excess in, \u2013; administrative law in, ; agricultural policy in, \u2013, \u2013 486nn2\u2013; antimonopoly policy in, \u2013; Clayton Act of 1914 in, ; colonies and Parliament and, , \u2013; conservation movement in, \u2013, 491n9; constitutional law in,  313n10; education in, , , , 514n17; English conception of liberty and, \u2013, 262n3; Glorious Revolution (England) and, 262n4; gold standard in, ; health care and, 414n14; imperialism and, ; inflation in, , , 457n10; John Locke and, ; Judiciary Act of 1801 (U.S.) and, 278n48; labor unions in, \u2013, , 398n24; legal thought in, 280n52; liberal as term in, , ; liberals in, ; movement toward socialism in, \u2013, ; Norris-LaGuardia Act of 1932 in, ; political philosophy in, ; progressive taxation in, , 437n12, ; rent control in, 471n6; segregation in, ; Sherman Act in, ; social equality in, ; socialism in, ; social security in, , 417n21; urban planning in, ; welfare socialism and, ; Whigs and, , 530n16, \u2013. _See also_ American Revolution; U.S. Constitution\n\nUniversity of Chicago Press, , , \u2013\n\nunpredictability. _See_ uncertainty and certainty\n\nurbanization: civilization and, \u2013; costs of, \u2013, ; economists' attention to, \u2013; price mechanisms and, \u2013; responsibility and, ; slum clearance and, , 473n7; town planning and, \u2013, 475n10,\n\nU.S. Constitution: admiration for, , 277n45; Articles of Confederation and, , \u2013; Bill of Rights in, \u2013 , 276n41; British constitution and, \u2013; colonial constitutions and, \u2013, 270n26, \u2013272nn28\u2013 ; commerce clause in, 282n61; due process in, , 282n60; as experiment, ; federalism and, \u2013; Fifth Amendment of, ; Fourteenth Amendment of, \u2013, 282nn60\u2013 ; judicial review and, \u2013275n37, \u2013, 278n46, 278n48, ; limited government and, ; Ninth Amendment of, , 277n44, ; popular will and, 285n65; prohibition in, 213n21; rule of law and, ; separation of powers in, ; tradition and historical accident in, ; Whigs and, 530n16\n\n\"Use of Knowledge in Society\" (Hayek), \u201377n6\n\nU.S. Supreme Court: constitutional law and, \u2013; executive power and, \u2013; Franklin D. Roosevelt and, \u2013, 285n66; judicial abstinence and, 279n51; judicial review and, , , ; labor unions and, , , 403n32; overstepping by, ; on retroactive law, ; social security and, , 420n25; traditional role of, \u2013 , 285n65; welfare state and, \u2013 374n12. _See also specific cases_\n\nutilitarianism, 48n2, , ,\n\nutility: fitness and, 124n39; law and, ; marginal, \u2013, 435nn8\u2013, 437n12, 444n24; regulation of property and, 227n19; science and, 95n9\n\nutopianism,\n\nvalue: merit and, \u2013, 157n13, \u2013 ; risk and, \u2013\n\nvalues: altruism and, \u2013; ends and means and, \u2013; morals and,\n\n_Vanzant v. Waddell,_ \u2013282n55\n\nVico, Giambattista, \u201392n\n\nViner, Jacob,\n\nviolence: coercion and, , \u2013, \u2013; labor unions and, , \u2013\n\nVirginia Declaration of Rights, \u2013 272n32\n\nvirtue, 123n38, 142n10\n\n_Vocabulary of Politics, The_ (Weldon),\n\nvote, right to. _See_ suffrage\n\nwages: equilibrium in, 400n29; government intervention and, \u2013; inflation and, \u2013; labor unions and, \u2013 ,  390n11, 390n13, \u2013394nn16\u2013 , \u2013, \u2013, 401n30, \u2013 ; unemployment and,\n\n_Wall Street Journal,_ \u201318n42\n\nwar: liberty and, ; totalitarianism and,\n\nWashington, George, , 262n5\n\nwaste, \u2013, 195n13\n\nWatkins, J. W. N., \u201313n35\n\nwealth: complaints about, , 93n3; idle rich and, , , 195n14; leisure and, \u2013; liberty and, \u2013, \u201368n26; merit and, \u2013, \u2013 157nn12\u2013, ; necessity of wealthy class and, ; philanthropy and, \u2013 ; progress and, , \u2013; radicalism and, 194n12; taxation and, ; waste and, . _See also_ redistribution of wealth\n\nWebb, Sidney and Beatrice, 190n8, 427n38,\n\nWebster, Daniel, 280n52\n\nwelfare state: Alexis de Tocqueville on, ; coercion and, , ; education and, , 510n11; extent of, \u2013 ; freedom and, \u2013378n17; housing subsidies and, ; inflation and, ; liberty and, , \u2013; longevity of, \u2013; in nineteenth-century Europe, , ; origins of, \u2013 374n12; police state and, \u2013374n12; private versus public costs and, 333n6; as term, \u2013, \u2013374n12. _See also_ public relief; social security\n\nWentworth, Peter, 236n7\n\nWestern civilization: inequality and, ; leadership role of, \u2013; rapid progress and, ; world's borrowing from, \u2013; world's envy of, \u2013, 105n17\n\nWhigs: American colonies and, , 262nn4\u2013; arbitrary power and, ; discretionary power and, ; Edmund Burke as, 131n51, ; evolution of political institutions and, \u2013118n26; Great Britain and, , 530n15, , 531n17; Hayek as, , ; isonomy and, \u2013241n19; legal positivism and, 345n8; liberalism and, 174n11, 258n90, ; natural law and, ; political defeat of, ; _Rechtsstaat_ and, \u2013300n26; religion and, ; Thomas Aquinas as, 234n4; in United States, 530n16, \u2013\n\nWick, Warner,\n\nWilkes case,\n\nwill: freedom of, , \u2013, \u2013 136n4, \u2013138nn7\u2013, 142n10, \u2013 , 222n10; general, ; government by, 255n81; government of, ; law and, 142n10; liberty and, 136n5; of majority, ; popular, 263n9; strength of,\n\nwisdom, majority and,\n\nWorld War I,\n\nwriting, memory and, 93n3\n\nyouth, , \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Eric Eldred and Chetan K Jain.\n\n\n\n\n\n\n\n\n\n\nThe Crescent Moon\n\n\nBy Rabindranath Tagore\n\nTranslated from the original Bengali by the author\n\nwith eight illustrations in colour\n\nLondon and New York: Macmillan and Company, 1913\n\n\nTO T. STURGE MOORE\n\n\n[Frontispiece: From a drawing by Nandalall Bose--see cbeach.jpg]\n\n\n\n\n  CONTENTS\n\n\n  THE HOME\n  ON THE SEASHORE\n  THE SOURCE\n  BABY'S WAY\n  THE UNHEEDED PAGEANT\n  SLEEP-STEALER\n  THE BEGINNING\n  BABY'S WORLD\n  WHEN AND WHY\n  DEFAMATION\n  THE JUDGE\n  PLAYTHINGS\n  THE ASTRONOMER\n  CLOUDS AND WAVES\n  THE CHAMPA FLOWER\n  FAIRYLAND\n  THE LAND OF THE EXILE\n  THE RAINY DAY\n  PAPER BOATS\n  THE SAILOR\n  THE FURTHER BANK\n  THE FLOWER-SCHOOL\n  THE MERCHANT\n  SYMPATHY\n  VOCATION\n  SUPERIOR\n  THE LITTLE BIG MAN\n  TWELVE O'CLOCK\n  AUTHORSHIP\n  THE WICKED POSTMAN\n  THE HERO\n  THE END\n  THE RECALL\n  THE FIRST JASMINES\n  THE BANYAN TREE\n  BENEDICTION\n  THE GIFT\n  MY SONG\n  THE CHILD-ANGEL\n  THE LAST BARGAIN\n\n\n\n\n  LIST OF COLOURED ILLUSTRATIONS\n\n\n  FRONTISPIECE\n  THE HOME\n  THE BEGINNING\n  FAIRYLAND\n  PAPER BOATS\n  THE MERCHANT\n  THE HERO\n  BENEDICTION\n\n\n\n\n  INDEX OF THE FIRST LINES\n\n\n  Ah, these jasmines\n  Ah, who was it coloured that little frock\n  Bless this little heart\n  Child, how happy you are sitting in the dust\n  Come and hire me\n  Day by day I float my paper boats\n  I am small because I am a little child\n  If baby only wanted to, he could fly\n  If I were only a little puppy\n  If people came to know where my king's palace is\n  I long to go over there\n  Imagine, mother\n  I only said, \"When in the evening\"\n  I paced alone\n  It is time for me to go, mother\n  I want to give you something, my child\n  I wish I could take a quiet corner\n  Mother, I do want to leave off my lessons\n  Mother, let us imagine we are travelling\n  Mother, the folk who live up in the clouds\n  Mother, the light has grown grey\n  Mother, your baby is silly\n  On the seashore of endless worlds\n  O you shaggy-headed banyan tree\n  Say of him what you please\n  Sullen clouds are gathering\n  Supposing I became a _champa_ flower\n  The boat of the boatman Madhu\n  The night was dark when we went away\n  The sleep that flits on baby's eyes\n  They clamour and fight\n  This song of mine\n  When I bring you coloured toys\n  When storm clouds\n  When the gong sounds ten\n  Where have I come from\n  Who stole sleep from baby's eyes\n  Why are those tears in your eyes, my child\n  Why do you sit there on the floor\n  You say that father writes a lot of books\n\n\n\n\n\n\n[Illustration: The Home--from a drawing by Nandalall Bose--see\nchome.jpg]\n\n\nTHE HOME\n\n\nI paced alone on the road across the field while the sunset was\nhiding its last gold like a miser.\n\nThe daylight sank deeper and deeper into the darkness, and the\nwidowed land, whose harvest had been reaped, lay silent.\n\nSuddenly a boy's shrill voice rose into the sky.  He traversed\nthe dark unseen, leaving the track of his song across the hush of\nthe evening.\n\nHis village home lay there at the end of the waste land, beyond\nthe sugar-cane field, hidden among the shadows of the banana and\nthe slender areca palm, the cocoa-nut and the dark green\njack-fruit trees.\n\nI stopped for a moment in my lonely way under the starlight, and\nsaw spread before me the darkened earth surrounding with her arms\ncountless homes furnished with cradles and beds, mothers' hearts\nand evening lamps, and young lives glad with a gladness that\nknows nothing of its value for the world.\n\n\n\n\nON THE SEASHORE\n\n\nOn the seashore of endless worlds children meet.\n\nThe infinite sky is motionless overhead and the restless water is\nboisterous.  On the seashore of endless worlds the children meet\nwith shouts and dances.\n\nThey build their houses with sand, and they play with empty\nshells.  With withered leaves they weave their boats and\nsmilingly float them on the vast deep.  Children have their play\non the seashore of worlds.\n\nThey know not how to swim, they know not how to cast nets.\nPearl-fishers dive for pearls, merchants sail in their ships,\nwhile children gather pebbles and scatter them again.  They seek\nnot for hidden treasures, they know not how to cast nets.\n\nThe sea surges up with laughter, and pale gleams the smile of the\nsea-beach.  Death-dealing waves sing meaningless ballads to the\nchildren, even like a mother while rocking her baby's cradle.\nThe sea plays with children, and pale gleams the smile of the\nsea-beach.\n\nOn the seashore of endless worlds children meet.  Tempest roams\nin the pathless sky, ships are wrecked in the trackless water,\ndeath is abroad and children play.  On the seashore of endless\nworlds is the great meeting of children.\n\n\n\n\nTHE SOURCE\n\n\nThe sleep that flits on baby's eyes--does anybody know from where\nit comes?  Yes, there is a rumour that it has its dwelling where,\nin the fairy village among shadows of the forest dimly lit with\nglow-worms, there hang two shy buds of enchantment.  From there\nit comes to kiss baby's eyes.\n\nThe smile that flickers on baby's lips when he sleeps--does\nanybody know where it was born?  Yes, there is a rumour that a\nyoung pale beam of a crescent moon touched the edge of a\nvanishing autumn cloud, and there the smile was first born in the\ndream of a dew-washed morning--the smile that flickers on baby's\nlips when he sleeps.\n\nThe sweet, soft freshness that blooms on baby's limbs--does\nanybody know where it was hidden so long?  Yes, when the mother\nwas a young girl it lay pervading her heart in tender and silent\nmystery of love--the sweet, soft freshness that has bloomed on\nbaby's limbs.\n\n\n\n\nBABY'S WAY\n\n\nIf baby only wanted to, he could fly up to heaven this moment.\n\nIt is not for nothing that he does not leave us.\n\nHe loves to rest his head on mother's bosom, and cannot ever bear\nto lose sight of her.\n\nBaby knows all manner of wise words, though few on earth can\nunderstand their meaning.\n\nIt is not for nothing that he never wants to speak.\n\nThe one thing he wants is to learn mother's words from mother's\nlips.  That is why he looks so innocent.\n\nBaby had a heap of gold and pearls, yet he came like a beggar on\nto this earth.\n\nIt is not for nothing he came in such a disguise.\n\nThis dear little naked mendicant pretends to be utterly helpless,\nso that he may beg for mother's wealth of love.\n\nBaby was so free from every tie in the land of the tiny crescent\nmoon.\n\nIt was not for nothing he gave up his freedom.\n\nHe knows that there is room for endless joy in mother's little\ncorner of a heart, and it is sweeter far than liberty to be\ncaught and pressed in her dear arms.\n\nBaby never knew how to cry.  He dwelt in the land of perfect\nbliss.\n\nIt is not for nothing he has chosen to shed tears.\n\nThough with the smile of his dear face he draws mother's yearning\nheart to him, yet his little cries over tiny troubles weave the\ndouble bond of pity and love.\n\n\n\n\nTHE UNHEEDED PAGEANT\n\n\nAh, who was it coloured that little frock, my child, and covered\nyour sweet limbs with that little red tunic?\n\nYou have come out in the morning to play in the courtyard,\ntottering and tumbling as you run.\n\nBut who was it coloured that little frock, my child?\n\nWhat is it makes you laugh, my little life-bud?\n\nMother smiles at you standing on the threshold.\n\nShe claps her hands and her bracelets jingle, and you dance with\nyour bamboo stick in your hand like a tiny little shepherd.\n\nBut what is it makes you laugh, my little life-bud?\n\nO beggar, what do you beg for, clinging to your mother's neck\nwith both your hands?\n\nO greedy heart, shall I pluck the world like a fruit from the sky\nto place it on your little rosy palm?\n\nO beggar, what are you begging for?\n\nThe wind carries away in glee the tinkling of your anklet bells.\n\nThe sun smiles and watches your toilet.  The sky watches over you\nwhen you sleep in your mother's arms, and the morning comes\ntiptoe to your bed and kisses your eyes.\n\nThe wind carries away in glee the tinkling of your anklet bells.\n\nThe fairy mistress of dreams is coming towards you, flying\nthrough the twilight sky.\n\nThe world-mother keeps her seat by you in your mother's heart.\n\nHe who plays his music to the stars is standing at your window\nwith his flute.\n\nAnd the fairy mistress of dreams is coming towards you, flying\nthrough the twilight sky.\n\n\n\n\nSLEEP-STEALER\n\n\nWho stole sleep from baby's eyes?  I must know.\n\nClasping her pitcher to her waist mother went to fetch water from\nthe village near by.\n\nIt was noon.  The children's playtime was over; the ducks in the\npond were silent.\n\nThe shepherd boy lay asleep under the shadow of the _banyan_\ntree.\n\nThe crane stood grave and still in the swamp near the mango\ngrove.\n\nIn the meanwhile the Sleep-stealer came and, snatching sleep from\nbaby's eyes, flew away.\n\nWhen mother came back she found baby travelling the room over on\nall fours.\n\nWho stole sleep from our baby's eyes?  I must know.  I must find\nher and chain her up.\n\nI must look into that dark cave, where, through boulders and\nscowling stones, trickles a tiny stream.\n\nI must search in the drowsy shade of the _bakula_ grove,\nwhere pigeons coo in their corner, and fairies' anklets tinkle in\nthe stillness of starry nights.\n\nIn the evening I will peep into the whispering silence of the\nbamboo forest, where fireflies squander their light, and will ask\nevery creature I meet, \"Can anybody tell me where the\nSleep-stealer lives?\"\n\nWho stole sleep from baby's eyes?  I must know.\n\nShouldn't I give her a good lesson if I could only catch her!\n\nI would raid her nest and see where she hoards all her stolen\nsleep.\n\nI would plunder it all, and carry it home.\n\nI would bind her two wings securely, set her on the bank of the\nriver, and then let her play at fishing with a reed among the\nrushes and water-lilies.\n\nWhen the marketing is over in the evening, and the village\nchildren sit in their mothers' laps, then the night birds will\nmockingly din her ears with:\n\n\"Whose sleep will you steal now?\"\n\n\n\n\n[Illustration: From a drawing by Asit Kumar Haldar--see\ncbegin.jpg]\n\n\nTHE BEGINNING\n\n\n\"Where have I come from, where did you pick me up?\"  the baby\nasked its mother.\n\nShe answered half crying, half laughing, and clasping the baby to\nher breast,-- \"You were hidden in my heart as its desire, my\ndarling.\n\nYou were in the dolls of my childhood's games; and when with clay\nI made the image of my god every morning, I made and unmade you\nthen.\n\nYou were enshrined with our household deity, in his worship I\nworshipped you.\n\nIn all my hopes and my loves, in my life, in the life of my\nmother you have lived.\n\nIn the lap of the deathless Spirit who rules our home you have\nbeen nursed for ages.\n\nWhen in girlhood my heart was opening its petals, you hovered as\na fragrance about it.\n\nYour tender softness bloomed in my youthful limbs, like a glow in\nthe sky before the sunrise.\n\nHeaven's first darling, twin-born with the morning light, you\nhave floated down the stream of the world's life, and at last you\nhave stranded on my heart.\n\nAs I gaze on your face, mystery overwhelms me; you who belong to\nall have become mine.\n\nFor fear of losing you I hold you tight to my breast.  What magic\nhas snared the world's treasure in these slender arms of mine?\"\n\n\n\n\nBABY'S WORLD\n\n\nI wish I could take a quiet corner in the heart of my baby's very\nown world.\n\nI know it has stars that talk to him, and a sky that stoops down\nto his face to amuse him with its silly clouds and rainbows.\n\nThose who make believe to be dumb, and look as if they never\ncould move, come creeping to his window with their stories and\nwith trays crowded with bright toys.\n\nI wish I could travel by the road that crosses baby's mind, and\nout beyond all bounds;\n\nWhere messengers run errands for no cause between the kingdoms of\nkings of no history;\n\nWhere Reason makes kites of her laws and flies them, and Truth\nsets Fact free from its fetters.\n\n\n\n\nWHEN AND WHY\n\n\nWhen I bring you coloured toys, my child, I understand why there\nis such a play of colours on clouds, on water, and why flowers\nare painted in tints--when I give coloured toys to you, my child.\n\nWhen I sing to make you dance, I truly know why there is music in\nleaves, and why waves send their chorus of voices to the heart of\nthe listening earth--when I sing to make you dance.\n\nWhen I bring sweet things to your greedy hands, I know why there\nis honey in the cup of the flower, and why fruits are secretly\nfilled with sweet juice--when I bring sweet things to your greedy\nhands.\n\nWhen I kiss your face to make you smile, my darling, I surely\nunderstand what pleasure streams from the sky in morning light,\nand what delight the summer breeze brings to my body--when I kiss\nyou to make you smile.\n\n\n\n\nDEFAMATION\n\n\nWhy are those tears in your eyes, my child?\n\nHow horrid of them to be always scolding you for nothing?\n\nYou have stained your fingers and face with ink while writing--is\nthat why they call you dirty?\n\nO, fie!  Would they dare to call the full moon dirty because it\nhas smudged its face with ink?\n\nFor every little trifle they blame you, my child.  They are ready\nto find fault for nothing.\n\nYou tore your clothes while playing--is that why they call you\nuntidy?\n\nO, fie!  What would they call an autumn morning that smiles\nthrough its ragged clouds?\n\nTake no heed of what they say to you, my child.\n\nTake no heed of what they say to you, my child.\n\nThey make a long list of your misdeeds.  Everybody knows how you\nlove sweet things--is that why they call you greedy?\n\nO, fie!  What then would they call us who love you?\n\n\n\n\nTHE JUDGE\n\n\nSay of him what you please, but I know my child's failings.\n\nI do not love him because he is good, but because he is my little\nchild.\n\nHow should you know how dear he can be when you try to weigh his\nmerits against his faults?\n\nWhen I must punish him he becomes all the more a part of my\nbeing.\n\nWhen I cause his tears to come my heart weeps with him.\n\nI alone have a right to blame and punish, for he only may\nchastise who loves.\n\n\n\n\nPLAYTHINGS\n\n\nChild, how happy you are sitting in the dust, playing with a\nbroken twig all the morning.\n\nI smile at your play with that little bit of a broken twig.\n\nI am busy with my accounts, adding up figures by the hour.\n\nPerhaps you glance at me and think, \"What a stupid game to spoil\nyour morning with!\"\n\nChild, I have forgotten the art of being absorbed in sticks and\nmud-pies.\n\nI seek out costly playthings, and gather lumps of gold and\nsilver.\n\nWith whatever you find you create your glad games, I spend both\nmy time and my strength over things I never can obtain.\n\nIn my frail canoe I struggle to cross the sea of desire, and\nforget that I too am playing a game.\n\n\n\n\nTHE ASTRONOMER\n\n\nI only said, \"When in the evening the round full moon gets\nentangled among the branches of that _Kadam_ tree, couldn't\nsomebody catch it?\"\n\nBut dada [_elder brother_] laughed at me and said, \"Baby,\nyou are the silliest child I have ever known.  The moon is ever\nso far from us, how could anybody catch it?\"\n\nI said, \"Dada how foolish you are!  When mother looks out of her\nwindow and smiles down at us playing, would you call her far\naway?\"\n\nStill said, \"You are a stupid child!  But, baby, where could you\nfind a net big enough to catch the moon with?\"\n\nI said, \"Surely you could catch it with your hands.\"\n\nBut dada laughed and said, \"You are the silliest child I have\nknown.  If it came nearer, you would see how big the moon is.\"\n\nI said, \"Dada, what nonsense they teach at your school!  When\nmother bends her face down to kiss us does her face look very\nbig?\"\n\nBut still dada says, \"You are a stupid child.\"\n\n\n\n\nCLOUDS AND WAVES\n\n\nMother, the folk who live up in the clouds call out to me--\n\n\"We play from the time we wake till the day ends.\n\nWe play with the golden dawn, we play with the silver moon.\n\nI ask, \"But, how am I to get up to you?\"  They answer, \"Come to\nthe edge of the earth, lift up your hands to the sky, and you\nwill be taken up into the clouds.\"\n\n\"My mother is waiting for me at home,\" I say.  \"How can I leave\nher and come?\"\n\nThen they smile and float away.\n\nBut I know a nicer game than that, mother.\n\nI shall be the cloud and you the moon.\n\nI shall cover you with both my hands, and our house-top will be\nthe blue sky.\n\nThe folk who live in the waves call out to me--\n\n\"We sing from morning till night; on and on we travel and know\nnot where we pass.\"\n\nI ask, \"But, how am I to join you?\"  They tell me, \"Come to the\nedge of the shore and stand with your eyes tight shut, and you\nwill be carried out upon the waves.\"\n\nI say, \"My mother always wants me at home in the evening--how can\nI leave her and go?\"\n\nThen they smile, dance and pass by.\n\nBut I know a better game than that.\n\nI will be the waves and you will be a strange shore.\n\nI shall roll on and on and on, and break upon your lap with\nlaughter.\n\nAnd no one in the world will know where we both are.\n\n\n\n\nTHE CHAMPA FLOWER\n\n\nSupposing I became a _champa_ flower, just for fun, and\ngrew on a branch high up that tree, and shook in the wind with\nlaughter and danced upon the newly budded leaves, would you know\nme, mother?\n\nYou would call, \"Baby, where are you?\"  and I should laugh to\nmyself and keep quite quiet.\n\nI should slyly open my petals and watch you at your work.\n\nWhen after your bath, with wet hair spread on your shoulders, you\nwalked through the shadow of the _champa_ tree to the\nlittle court where you say your prayers, you would notice the\nscent of the flower, but not know that it came from me.\n\nWhen after the midday meal you sat at the window reading\n_Ramayana_, and the tree's shadow fell over your hair and\nyour lap, I should fling my wee little shadow on to the page of\nyour book, just where you were reading.\n\nBut would you guess that it was the tiny shadow of your little\nchild?\n\nWhen in the evening you went to the cow-shed with the lighted\nlamp in your hand, I should suddenly drop on to the earth again\nand be your own baby once more, and beg you to tell me a story.\n\n\"Where have you been, you naughty child?\"\n\n\"I won't tell you, mother.\"  That's what you and I would say\nthen.\n\n\n\n\n[Illustration: From a drawing by Abanindranath Tagore--see\ncfairy.jpg]\n\n\nFAIRYLAND\n\n\nIf people came to know where my king's palace is, it would vanish\ninto the air.\n\nThe walls are of white silver and the roof of shining gold.\n\nThe queen lives in a palace with seven courtyards, and she wears\na jewel that cost all the wealth of seven kingdoms.\n\nBut let me tell you, mother, in a whisper, where my king's palace\nis.\n\nIt is at the corner of our terrace where the pot of the\n_tulsi_ plant stands.\n\nThe princess lies sleeping on the far-away shore of the seven\nimpassable seas.\n\nThere is none in the world who can find her but myself.\n\nShe has bracelets on her arms and pearl drops in her ears; her\nhair sweeps down upon the floor.\n\nShe will wake when I touch her with my magic wand, and jewels\nwill fall from her lips when she smiles.\n\nBut let me whisper in your ear, mother; she is there in the\ncorner of our terrace where the pot of the _tulsi_ plant\nstands.\n\nWhen it is time for you to go to the river for your bath, step up\nto that terrace on the roof.\n\nI sit in the corner where the shadows of the walls meet together.\n\nOnly puss is allowed to come with me, for she knows where the\nbarber in the story lives.\n\nBut let me whisper, mother, in your ear where the barber in the\nstory lives.\n\nIt is at the corner of the terrace where the pot of the\n_tulsi_ plant stands.\n\n\n\n\nTHE LAND OF THE EXILE\n\n\nMother, the light has grown grey in the sky; I do not know what\nthe time is.\n\nThere is no fun in my play, so I have come to you.  It is\nSaturday, our holiday.\n\nLeave off your work, mother; sit here by the window and tell me\nwhere the desert of Tepantar in the fairy tale is?\n\nThe shadow of the rains has covered the day from end to end.\n\nThe fierce lightning is scratching the sky with its nails.\n\nWhen the clouds rumble and it thunders, I love to be afraid in my\nheart and cling to you.\n\nWhen the heavy rain patters for hours on the bamboo leaves, and\nour windows shake and rattle at the gusts of wind, I like to sit\nalone in the room, mother, with you, and hear you talk about the\ndesert of Tepantar in the fairy tale.\n\nWhere is it, mother, on the shore of what sea, at the foot of\nwhat hills, in the kingdom of what king?\n\nThere are no hedges there to mark the fields, no footpath across\nit by which the villagers reach their village in the evening, or\nthe woman who gathers dry sticks in the forest can bring her load\nto the market.  With patches of yellow grass in the sand and only\none tree where the pair of wise old birds have their nest, lies\nthe desert of Tepantar.\n\nI can imagine how, on just such a cloudy day, the young son of\nthe king is riding alone on a grey horse through the desert, in\nsearch of the princess who lies imprisoned in the giant's palace\nacross that unknown water.\n\nWhen the haze of the rain comes down in the distant sky, and\nlightning starts up like a sudden fit of pain, does he remember\nhis unhappy mother, abandoned by the king, sweeping the cow-stall\nand wiping her eyes, while he rides through the desert of\nTepantar in the fairy tale?\n\nSee, mother, it is almost dark before the day is over, and there\nare no travellers yonder on the village road.\n\nThe shepherd boy has gone home early from the pasture, and men\nhave left their fields to sit on mats under the eaves of their\nhuts, watching the scowling clouds.\n\nMother, I have left all my books on the shelf--do not ask me to\ndo my lessons now.\n\nWhen I grow up and am big like my father, I shall learn all that\nmust be learnt.\n\nBut just for to-day, tell me, mother, where the desert of\nTepantar in the fairy tale is?\n\n\n\n\nTHE RAINY DAY\n\n\nSullen clouds are gathering fast over the black fringe of the\nforest.\n\nO child, do not go out!\n\nThe palm trees in a row by the lake are smiting their heads\nagainst the dismal sky; the crows with their draggled wings are\nsilent on the tamarind branches, and the eastern bank of the\nriver is haunted by a deepening gloom.\n\nOur cow is lowing loud, tied at the fence.\n\nO child, wait here till I bring her into the stall.\n\nMen have crowded into the flooded field to catch the fishes as\nthey escape from the overflowing ponds; the rain water is running\nin rills through the narrow lanes like a laughing boy who has run\naway from his mother to tease her.\n\nListen, someone is shouting for the boatman at the ford.\n\nO child, the daylight is dim, and the crossing at the ferry is\nclosed.\n\nThe sky seems to ride fast upon the madly-rushing rain; the water\nin the river is loud and impatient; women have hastened home\nearly from the Ganges with their filled pitchers.\n\nThe evening lamps must be made ready.\n\nO child, do not go out!\n\nThe road to the market is desolate, the lane to the river is\nslippery.  The wind is roaring and struggling among the bamboo\nbranches like a wild beast tangled in a net.\n\n\n\n\n[Illustration: From a drawing by Surendranath Ganguli--see\ncboat.jpg]\n\n\nPAPER BOATS\n\n\nDay by day I float my paper boats one by one down the running\nstream.\n\nIn big black letters I write my name on them and the name of the\nvillage where I live.\n\nI hope that someone in some strange land will find them and know\nwho I am.\n\nI load my little boats with _shiuli_ flowers from our\ngarden, and hope that these blooms of the dawn will be carried\nsafely to land in the night.\n\nI launch my paper boats and look up into the sky and see the\nlittle clouds setting their white bulging sails.\n\nI know not what playmate of mine in the sky sends them down the\nair to race with my boats!\n\nWhen night comes I bury my face in my arms and dream that my\npaper boats float on and on under the midnight stars.\n\nThe fairies of sleep are sailing in them, and the lading is their\nbaskets full of dreams.\n\n\n\n\nTHE SAILOR\n\n\nThe boat of the boatman Madhu is moored at the wharf of Rajgunj.\n\nIt is uselessly laden with jute, and has been lying there idle\nfor ever so long.\n\nIf he would only lend me his boat, I should man her with a\nhundred oars, and hoist sails, five or six or seven.\n\nI should never steer her to stupid markets.  I should sail the\nseven seas and the thirteen rivers of fairyland.\n\nBut, mother, you won't weep for me in a corner.\n\nI am not going into the forest like Ramachandra to come back only\nafter fourteen years.\n\nI shall become the prince of the story, and fill my boat with\nwhatever I like.\n\nI shall take my friend Ashu with me.  We shall sail merrily\nacross the seven seas and the thirteen rivers of fairyland.\n\nWe shall set sail in the early morning light.\n\nWhen at noontide you are bathing at the pond, we shall be in the\nland of a strange king.\n\nWe shall pass the ford of Tirpurni, and leave behind us the\ndesert of Tepantar.\n\nWhen we come back it will be getting dark, and I shall tell you\nof all that we have seen.\n\nI shall cross the seven seas and the thirteen rivers of\nfairyland.\n\n\n\n\nTHE FURTHER BANK\n\n\nI long to go over there to the further bank of the river,\n\nWhere those boats are tied to the bamboo poles in a line;\n\nWhere men cross over in their boats in the morning with ploughs\non their shoulders to till their far-away fields;\n\nWhere the cowherds make their lowing cattle swim across to the\nriverside pasture;\n\nWhence they all come back home in the evening, leaving the\njackals to howl in the island overgrown with weeds,\n\nMother, if you don't mind, I should like to become the boatman of\nthe ferry when I am grown up.\n\nThey say there are strange pools hidden behind that high bank,\n\nWhere flocks of wild ducks come when the rains are over, and\nthick reeds grow round the margins where waterbirds lay their\neggs;\n\nWhere snipes with their dancing tails stamp their tiny footprints\nupon the clean soft mud;\n\nWhere in the evening the tall grasses crested with white flowers\ninvite the moonbeam to float upon their waves.\n\nMother, if you don't mind, I should like to become the boatman of\nthe ferryboat when I am grown up.\n\nI shall cross and cross back from bank to bank, and all the boys\nand girls of the village will wonder at me while they are\nbathing.\n\nWhen the sun climbs the mid sky and morning wears on to noon, I\nshall come running to you, saying, \"Mother, I am hungry!\"\n\nWhen the day is done and the shadows cower under the trees, I\nshall come back in the dusk.\n\nI shall never go away from you into the town to work like father.\n\nMother, if you don't mind, I should like to become the boatman of\nthe ferryboat when I am grown up.\n\n\n\n\nTHE FLOWER-SCHOOL\n\n\nWhen storm clouds rumble in the sky and June showers come down,\n\nThe moist east wind comes marching over the heath to blow its\nbagpipes among the bamboos.\n\nThen crowds of flowers come out of a sudden, from nobody knows\nwhere, and dance upon the grass in wild glee.\n\nMother, I really think the flowers go to school underground.\n\nThey do their lessons with doors shut, and if they want to come\nout to play before it is time, their master makes them stand in a\ncorner.\n\nWhen the rains come they have their holidays.\n\nBranches clash together in the forest, and the leaves rustle in\nthe wild wind, the thunder-clouds clap their giant hands and the\nflower children rush out in dresses of pink and yellow and white.\n\nDo you know, mother, their home is in the sky, where the stars\nare.\n\nHaven't you seen how eager they are to get there?  Don't you know\nwhy they are in such a hurry?\n\nOf course, I can guess to whom they raise their arms: they have\ntheir mother as I have my own.\n\n\n\n\n[Illustration: From a drawing by Asit Kumar Haldar--see\ncmerchant.jpg]\n\n\nTHE MERCHANT\n\n\nImagine, mother, that you are to stay at home and I am to travel\ninto strange lands.\n\nImagine that my boat is ready at the landing fully laden.\n\nNow think well, mother, before you say what I shall bring for you\nwhen I come back.\n\nMother, do you want heaps and heaps of gold?\n\nThere, by the banks of golden streams, fields are full of golden\nharvest.\n\nAnd in the shade of the forest path the golden _champa_\nflowers drop on the ground.\n\nI will gather them all for you in many hundred baskets.\nMother, do you want pearls big as the raindrops of autumn?\n\nI shall cross to the pearl island shore.  There in the early\nmorning light pearls tremble on the meadow flowers, pearls drop\non the grass, and pearls are scattered on the sand in spray by\nthe wild sea-waves.\n\nMy brother shall have a pair of horses with wings to fly among\nthe clouds.\n\nFor father I shall bring a magic pen that, without his knowing,\nwill write of itself.\n\nFor you, mother, I must have the casket and jewel that cost seven\nkings their kingdoms.\n\n\n\n\nSYMPATHY\n\n\nIf I were only a little puppy, not your baby, mother dear, would\nyou say \"No\" to me if I tried to eat from your dish?\n\nWould you drive me off, saying to me, \"Get away, you naughty\nlittle puppy?\"\n\nThen go, mother, go!  I will never come to you when you call me,\nand never let you feed me any more.\n\nIf I were only a little green parrot, and not your baby, mother\ndear, would you keep me chained lest I should fly away?\n\nWould you shake your finger at me and say, \"What an ungrateful\nwretch of a bird!  It is gnawing at its chain day and night?\"\n\nThen, go, mother, go!  I will run away into the woods; I will\nnever let you take me in your arms again.\n\n\n\n\nVOCATION\n\n\nWhen the gong sounds ten in the morning and I walk to school by\nour lane,\n\nEvery day I meet the hawker crying, \"Bangles, crystal bangles!\"\n\nThere is nothing to hurry him on, there is no road he must take,\nno place he must go to, no time when he must come home.\n\nI wish I were a hawker, spending my day in the road, crying,\n\"Bangles, crystal bangles!\"\n\nWhen at four in the afternoon I come back from the school,\n\nI can see through the gate of that house the gardener digging the\nground.\n\nHe does what he likes with his spade, he soils his clothes with\ndust, nobody takes him to task if he gets baked in the sun or\ngets wet.\n\nI wish I were a gardener digging away at the garden with nobody\nto stop me from digging.\n\nJust as it gets dark in the evening and my mother sends me to\nbed,\n\nI can see through my open window the watchman walking up and\ndown.\n\nThe lane is dark and lonely, and the street-lamp stands like a\ngiant with one red eye in its head.\n\nThe watchman swings his lantern and walks with his shadow at his\nside, and never once goes to bed in his life.\n\nI wish I were a watchman walking the streets all night, chasing\nthe shadows with my lantern.\n\n\n\n\nSUPERIOR\n\n\nMother, your baby is silly!  She is so absurdly childish!\n\nShe does not know the difference between the lights in the\nstreets and the stars.\n\nWhen we play at eating with pebbles, she thinks they are real\nfood, and tries to put them into her mouth.\n\nWhen I open a book before her and ask her to learn her a, b, c,\nshe tears the leaves with her hands and roars for joy at nothing;\nthis is your baby's way of doing her lesson.\n\nWhen I shake my head at her in anger and scold her and call her\nnaughty, she laughs and thinks it great fun.\n\nEverybody knows that father is away, but if in play I call aloud\n\"Father,\" she looks about her in excitement and thinks that\nfather is near.\n\nWhen I hold my class with the donkeys that our washerman brings\nto carry away the clothes and I warn her that I am the\nschoolmaster, she will scream for no reason and call me dada.\n[_elder brother_ ]\n\nYour baby wants to catch the moon.  She is so funny; she calls\nGanesh Ganush.  [_Ganesh, a common name in India, also that of\nthe god with the elephant's head._]\n\nMother, your baby is silly, she is so absurdly childish!\n\n\n\n\nTHE LITTLE BIG MAN\n\n\nI am small because I am a little child.  I shall be big when I am\nas old as my father is.\n\nMy teacher will come and say, \"It is late, bring your slate and\nyour books.\"\n\nI shall tell him, \"Do you not know I am as big as father?  And I\nmust not have lessons any more.\"\n\nMy master will wonder and say, \"He can leave his books if he\nlikes, for he is grown up.\"\n\nI shall dress myself and walk to the fair where the crowd is\nthick.\n\nMy uncle will come rushing up to me and say, \"You will get lost,\nmy boy; let me carry you.\"\n\nI shall answer, \"Can't you see, uncle, I am as big as father.  I\nmust go to the fair alone.\"\n\nUncle will say, \"Yes, he can go wherever he likes, for he is\ngrown up.\"\n\nMother will come from her bath when I am giving money to my\nnurse, for I shall know how to open the box with my key.\n\nMother will say, \"What are you about, naughty child?\"\n\nI shall tell her, \"Mother, don't you know, I am as big as father,\nand I must give silver to my nurse.\"\n\nMother will say to herself, \"He can give money to whom he likes,\nfor he is grown up.\"\n\nIn the holiday time in October father will come home and,\nthinking that I am still a baby, will bring for me from the town\nlittle shoes and small silken frocks.\n\nI shall say, \"Father, give them to my dada [_elder\nbrother_], for I am as big as you are.\"\n\nFather will think and say, \"He can buy his own clothes if he\nlikes, for he is grown up.\"\n\n\n\n\nTWELVE O'CLOCK\n\n\nMother, I do want to leave off my lessons now.  I have been at my\nbook all the morning.\n\nYou say it is only twelve o'clock.  Suppose it isn't any later;\ncan't you ever think it is afternoon when it is only twelve\no'clock?\n\nI can easily imagine now that the sun has reached the edge of\nthat rice-field, and the old fisher-woman is gathering herbs for\nher supper by the side of the pond.\n\nI can just shut my eyes and think that the shadows are growing\ndarker under the _madar_ tree, and the water in the pond\nlooks shiny black.\n\nIf twelve o'clock can come in the night, why can't the night come\nwhen it is twelve o'clock?\n\n\n\n\nAUTHORSHIP\n\n\nYou say that father writes a lot of books, but what he writes I\ndon't understand.\n\nHe was reading to you all the evening, but could you really make\nout what he meant?\n\nWhat nice stories, mother, you can tell us!  Why can't father\nwrite like that, I wonder?\n\nDid he never hear from his own mother stories of giants and\nfairies and princesses?\n\nHas he forgotten them all?\n\nOften when he gets late for his bath you have to go and call him\nan hundred times.\n\nYou wait and keep his dishes warm for him, but he goes on writing\nand forgets.\n\nFather always plays at making books.\n\nIf ever I go to play in father's room, you come and call me,\n\"what a naughty child!\"\n\nIf I make the slightest noise, you say, \"Don't you see that\nfather's at his work?\"\n\nWhat's the fun of always writing and writing?\n\nWhen I take up father's pen or pencil and write upon his book\njust as he does,--a, b, c, d, e, f, g, h, i,--why do you get\ncross with me, then, mother?\n\nYou never say a word when father writes.\n\nWhen my father wastes such heaps of paper, mother, you don't seem\nto mind at all.\n\nBut if I take only one sheet to make a boat with, you say,\n\"Child, how troublesome you are!\"\n\nWhat do you think of father's spoiling sheets and sheets of paper\nwith black marks all over on both sides?\n\n\n\n\nTHE WICKED POSTMAN\n\n\nWhy do you sit there on the floor so quiet and silent, tell me,\nmother dear?\n\nThe rain is coming in through the open window, making you all\nwet, and you don't mind it.\n\nDo you hear the gong striking four?  It is time for my brother to\ncome home from school.\n\nWhat has happened to you that you look so strange?\n\nHaven't you got a letter from father to-day?\n\nI saw the postman bringing letters in his bag for almost\neverybody in the town.\n\nOnly, father's letters he keeps to read himself.  I am sure the\npostman is a wicked man.\n\nBut don't be unhappy about that, mother dear.\n\nTo-morrow is market day in the next village.  You ask your maid\nto buy some pens and papers.\n\nI myself will write all father's letters; you will not find a\nsingle mistake.\n\nI shall write from A right up to K.\n\nBut, mother, why do you smile?\n\nYou don't believe that I can write as nicely as father does!\n\nBut I shall rule my paper carefully, and write all the letters\nbeautifully big.\n\nWhen I finish my writing, do you think I shall be so foolish as\nfather and drop it into the horrid postman's bag?\n\nI shall bring it to you myself without waiting, and letter by\nletter help you to read my writing.\n\nI know the postman does not like to give you the really nice\nletters.\n\n\n\n\n[Illustration: From a drawing by Nandalall Bose--see chero.jpg]\n\n\nTHE HERO\n\n\nMother, let us imagine we are travelling, and passing through a\nstrange and dangerous country.\n\nYou are riding in a palanquin and I am trotting by you on a red\nhorse.\n\nIt is evening and the sun goes down.  The waste of\n_Joradighi_ lies wan and grey before us.  The land is\ndesolate and barren.\n\nYou are frightened and thinking--\"I know not where we have come\nto.\"\n\nI say to you, \"Mother, do not be afraid.\"\n\nThe meadow is prickly with spiky grass, and through it runs a\nnarrow broken path.\n\nThere are no cattle to be seen in the wide field; they have gone\nto their village stalls.\n\nIt grows dark and dim on the land and sky, and we cannot tell\nwhere we are going.\n\nSuddenly you call me and ask me in a whisper, \"What light is that\nnear the bank?\"\n\nJust then there bursts out a fearful yell, and figures come\nrunning towards us.\n\nYou sit crouched in your palanquin and repeat the names of the\ngods in prayer.\n\nThe bearers, shaking in terror, hide themselves in the thorny\nbush.\n\nI shout to you, \"Don't be afraid, mother.  I am here.\"\n\nWith long sticks in their hands and hair all wild about their\nheads, they come nearer and nearer.\n\nI shout, \"Have a care!  you villains!  One step more and you are\ndead men.\"\n\nThey give another terrible yell and rush forward.\n\nYou clutch my hand and say, \"Dear boy, for heaven's sake, keep\naway from them.\"\n\nI say, \"Mother, just you watch me.\"\n\nThen I spur my horse for a wild gallop, and my sword and buckler\nclash against each other.\n\nThe fight becomes so fearful, mother, that it would give you a\ncold shudder could you see it from your palanquin.\n\nMany of them fly, and a great number are cut to pieces.\n\nI know you are thinking, sitting all by yourself, that your boy\nmust be dead by this time.\n\nBut I come to you all stained with blood, and say, \"Mother, the\nfight is over now.\"\n\nYou come out and kiss me, pressing me to your heart, and you say\nto yourself,\n\n\"I don't know what I should do if I hadn't my boy to escort me.\"\n\nA thousand useless things happen day after day, and why couldn't\nsuch a thing come true by chance?\n\nIt would be like a story in a book.\n\nMy brother would say, \"Is it possible?  I always thought he was\nso delicate!\"\n\nOur village people would all say in amazement, \"Was it not lucky\nthat the boy was with his mother?\"\n\n\n\n\nTHE END\n\n\nIt is time for me to go, mother; I am going.\n\nWhen in the paling darkness of the lonely dawn you stretch out\nyour arms for your baby in the bed, I shall say, \"Baby is not\nthere!\"--mother, I am going.\n\nI shall become a delicate draught of air and caress you; and I\nshall be ripples in the water when you bathe, and kiss you and\nkiss you again.\n\nIn the gusty night when the rain patters on the leaves you will\nhear my whisper in your bed, and my laughter will flash with the\nlightning through the open window into your room.\n\nIf you lie awake, thinking of your baby till late into the night,\nI shall sing to you from the stars, \"Sleep mother, sleep.\"\n\nOn the straying moonbeams I shall steal over your bed, and lie\nupon your bosom while you sleep.\n\nI shall become a dream, and through the little opening of your\neyelids I shall slip into the depths of your sleep; and when you\nwake up and look round startled, like a twinkling firefly I shall\nflit out into the darkness.\n\nWhen, on the great festival of _puja_, the neighbours'\nchildren come and play about the house, I shall melt into the\nmusic of the flute and throb in your heart all day.\n\nDear auntie will come with _puja_-presents and will ask,\n\"Where is our baby, sister?  Mother, you will tell her softly,\n\"He is in the pupils of my eyes, he is in my body and in my\nsoul.\"\n\n\n\n\nTHE RECALL\n\n\nThe night was dark when she went away, and they slept.\n\nThe night is dark now, and I call for her, \"Come back, my\ndarling; the world is asleep; and no one would know, if you came\nfor a moment while stars are gazing at stars.\"\n\nShe went away when the trees were in bud and the spring was\nyoung.\n\nNow the flowers are in high bloom and I call, \"Come back, my\ndarling.  The children gather and scatter flowers in reckless\nsport.  And if you come and take one little blossom no one will\nmiss it.\"\n\nThose that used to play are playing still, so spendthrift is\nlife.\n\nI listen to their chatter and call, \"Come back, my darling, for\nmother's heart is full to the brim with love, and if you come to\nsnatch only one little kiss from her no one will grudge it.\"\n\n\n\n\nTHE FIRST JASMINES\n\n\nAh, these jasmines, these white jasmines!\n\nI seem to remember the first day when I filled my hands with\nthese jasmines, these white jasmines.\n\nI have loved the sunlight, the sky and the green earth;\n\nI have heard the liquid murmur of the river through the darkness\nof midnight;\n\nAutumn sunsets have come to me at the bend of a road in the\nlonely waste, like a bride raising her veil to accept her lover.\n\nYet my memory is still sweet with the first white jasmines that I\nheld in my hand when I was a child.\n\nMany a glad day has come in my life, and I have laughed with\nmerrymakers on festival nights.\n\nOn grey mornings of rain I have crooned many an idle song.\n\nI have worn round my neck the evening wreath of _bakulas_\nwoven by the hand of love.\n\nYet my heart is sweet with the memory of the first fresh jasmines\nthat filled my hands when I was a child.\n\n\n\n\nTHE BANYAN TREE\n\n\nO you shaggy-headed banyan tree standing on the bank of the pond,\nhave you forgotten the little child, like the birds that have\nnested in your branches and left you?\n\nDo you not remember how he sat at the window and wondered at the\ntangle of your roots that plunged underground?\n\nThe women would come to fill their jars in the pond, and your\nhuge black shadow would wriggle on the water like sleep\nstruggling to wake up.\n\nSunlight danced on the ripples like restless tiny shuttles\nweaving golden tapestry.\n\nTwo ducks swam by the weedy margin above their shadows, and the\nchild would sit still and think.\n\nHe longed to be the wind and blow through your rustling branches,\nto be your shadow and lengthen with the day on the water, to be a\nbird and perch on your top-most twig, and to float like those\nducks among the weeds and shadows.\n\n\n\n\n[Illustration: From a drawing by Surendranath Ganguli--see\ncbene.jpg]\n\n\nBENEDICTION\n\n\nBless this little heart, this white soul that has won the kiss of\nheaven for our earth.\n\nHe loves the light of the sun, he loves the sight of his mother's\nface.\n\nHe has not learned to despise the dust, and to hanker after gold.\n\nClasp him to your heart and bless him.\n\nHe has come into this land of an hundred cross-roads.\n\nI know not how he chose you from the crowd, came to your door,\nand grasped your hand to ask his way.\n\nHe will follow you, laughing and talking, and not a doubt in his\nheart.\n\nKeep his trust, lead him straight and bless him.\n\nLay your hand on his head, and pray that though the waves\nunderneath grow threatening, yet the breath from above may come\nand fill his sails and waft him to the haven of peace.\n\nForget him not in your hurry, let him come to your heart and\nbless him.\n\n\n\n\nTHE GIFT\n\n\nI want to give you something, my child, for we are drifting in\nthe stream of the world.\n\nOur lives will be carried apart, and our love forgotten.\n\nBut I am not so foolish as to hope that I could buy your heart\nwith my gifts.\n\nYoung is your life, your path long, and you drink the love we\nbring you at one draught and turn and run away from us.\n\nYou have your play and your playmates.  What harm is there if you\nhave no time or thought for us.\n\nWe, indeed, have leisure enough in old age to count the days that\nare past, to cherish in our hearts what our hands have lost for\never.\n\nThe river runs swift with a song, breaking through all barriers.\nBut the mountain stays and remembers, and follows her with his\nlove.\n\n\n\n\nMY SONG\n\n\nThis song of mine will wind its music around you, my child, like\nthe fond arms of love.\n\nThis song of mine will touch your forehead like a kiss of\nblessing.\n\nWhen you are alone it will sit by your side and whisper in your\near, when you are in the crowd it will fence you about with\naloofness.\n\nMy song will be like a pair of wings to your dreams, it will\ntransport your heart to the verge of the unknown.\n\nIt will be like the faithful star overhead when dark night is\nover your road.\n\nMy song will sit in the pupils of your eyes, and will carry your\nsight into the heart of things.\n\nAnd when my voice is silent in death, my song will speak in your\nliving heart.\n\n\n\n\nTHE CHILD-ANGEL\n\n\nThey clamour and fight, they doubt and despair, they know no end\nto their wranglings.\n\nLet your life come amongst them like a flame of light, my child,\nunflickering and pure, and delight them into silence.\n\nThey are cruel in their greed and their envy, their words are\nlike hidden knives thirsting for blood.\n\nGo and stand amidst their scowling hearts, my child, and let your\ngentle eyes fall upon them like the forgiving peace of the\nevening over the strife of the day.\n\nLet them see your face, my child, and thus know the meaning of\nall things; let them love you and thus love each other.\n\nCome and take your seat in the bosom of the limitless, my child.\nAt sunrise open and raise your heart like a blossoming flower,\nand at sunset bend your head and in silence complete the worship\nof the day.\n\n\n\n\nTHE LAST BARGAIN\n\n\n\"Come and hire me,\" I cried, while in the morning I was walking\non the stone-paved road.\n\nSword in hand, the King came in his chariot.\n\nHe held my hand and said, \"I will hire you with my power.\"\n\nBut his power counted for nought, and he went away in his\nchariot.\n\nIn the heat of the midday the houses stood with shut doors.\n\nI wandered along the crooked lane.\n\nAn old man came out with his bag of gold.\n\nHe pondered and said, \"I will hire you with my money.\"\n\nHe weighed his coins one by one, but I turned away.\n\nIt was evening.  The garden hedge was all aflower.\n\nThe fair maid came out and said, \"I will hire you with a smile.\"\n\nHer smile paled and melted into tears, and she went back alone\ninto the dark.\n\nThe sun glistened on the sand, and the sea waves broke waywardly.\n\nA child sat playing with shells.\n\nHe raised his head and seemed to know me, and said, \"I hire you\nwith nothing.\"\n\nFrom thenceforward that bargain struck in child's play made me a\nfree man.\n\n\n\n\nTHE END\n\n\n\n\n\n\n\n\n\n\nEnd of Project Gutenberg's The Crescent Moon, by Rabindranath Tagore\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nAlso by Bruce Pollock\n\nIn Their Own Words: Lyrics and Lyricists\n\nThe Face of Rock & Roll: Images of a Generation\n\nPlaying for a Change\n\nMe, Minsky, & Max\n\nIt's Onlly Rock and Roll\n\nThe Disco Handbook\n\nWhen Rock was Young\n\nWhen the Music Mattered: Rock in the 1960s\n\nHipper Than Our Kids: A Rock & Roll Journal of the Baby Boom Generation\n\nRock Song Index: The 7500 Most Important Songs of the Rock Era, 1944-2000\n\nWorking Musicians: Defining Moments from the Road, the Studio, and the Stage(E-book)\n\nBy the Time We Got to Woodstock: The Great Rokck 'n' Roll Revolution of 1969\n\nWhen the Music Mattered: Portraits from the 1960s (e-book reissue)\n\nIt's Only Rock and Roll (E-book reissue)\n\nCopyright \u00a9 2011 by Bruce Pollock\n\nAll rights reserved. No part of this book may be reproduced in any form, without written permission, except by a newspaper or magazine reviewer who wishes to quote brief passages in connection with a review.\n\nPublished in 2011 by Backbeat Books\n\nAn Imprint of Hal Leonard Corporation\n\n7777 West Bluemound Road\n\nMilwaukee, WI 53213\n\nTrade Book Division Editorial Offices\n\n33 Plymouth St., Montclair, NJ 07042\n\nPhotos courtesy of Photofest\n\nBook design by Michael Kellner\n\nLibrary of Congress Cataloging-in-Publication Data\n\nPollock, Bruce.\n\nIf you like the Beatles: here are over 200 bands, films, records and other oddities that you will love \/ Bruce Pollock.\n\np. cm.\n\nISBN 9781617130700\n\n1. Beatles. 2. Rock musicians\u2013England. 3. Rock music\u2013History and criticism. I. Title.\n\nML421.B4P65 2011\n\n782.42166092'2--dc23\n\n2011028882\n\nwww.backbeatbooks.com\nCONTENTS\n\nINTRODUCTION: BEFORE THERE WERE THE BEATLES...\n\n**1. CROSSING OVER**\n\n**2. THE GREAT AMERICAN SONGBOOK**\n\n**3. ROCKABILLY**\n\n**4. R &B**\n\n**5. THE SONGWRITERS**\n\n**6. GEORGE MARTIN**\n\n**7. FOLK ROCK**\n\n**8. ACID ROCK**\n\n**9. THE AMERICAN BEATLES**\n\n**10. APPLE RECORDS**\n\n**11. THE ANTI-BEATLES**\n\n**12. THE AMERICAN ROOTS REVIVAL**\n\n**13. THE BRITISH INVASIONS**\n\n**14. HERE TODAY**\n\nSUGGESTED LISTENING\n\nThe Beatles, 1964. (\u00a9 CBS.)\nIntroduction\n\nBEFORE THERE WERE THE BEATLES\n\nAn untold number of hardy souls were prompted to work on their vocal chops and dance moves after seeing Elvis Presley's hip-swiveling debut on The Ed Sullivan Show on September 9, 1956.\n\nBuddy Holly's Ed Sullivan stint in December 1957, however, failed to produce any sort of onslaught of jangly guitar bands with a sensitive songwriter at the helm to challenge the prevailing dominance of the Presley rock 'n' roll model: charismatic lead singer flanked by faceless backup players.\n\nBut within a few months of the Beatles' three consecutive appearances on The Ed Sullivan Show, on February 9, 16, and 23, 1964, ninety percent of the aspiring musicians in America were buying matching suits, ties, and instruments and growing their hair long.\n\nBy 1960, the fevered run of wildcat rockabilly and salacious R&B singles that defined the 1950s had given way to malleable girl groups, post-doo-wop Italian crooners, and identical R&B acts with identical time steps. For all-American white collegiate types anxious to be sophisticated, the fraternity house harmonies of folk music gently tweaked the system, from the Kingston Trio and the Brothers Four to the Highwaymen and the Chad Mitchell Trio.\n\nIn Philadelphia, Dick Clark was grooming the first graduating class of teen idols\u2014Fabian, Dion, Paul Anka, Bobby Rydell, Freddy Cannon, Neil Sedaka, Frankie Avalon, Brenda Lee, Connie Francis\u2014for lounge careers in Las Vegas or at the Copacabana. In Detroit, Berry Gordy was looking to make black music respectable by emulating the song factories of New York, Los Angeles, and Nashville, which clung to their franchises in the sometimes thrilling, sometimes tired Tin Pan Alley tradition.\n\nBuddy Holly died in a plane crash in February 1959; Eddie Cochran, in a car crash in April 1960. In 1961 Gene Vincent, Jerry Lee Lewis, Chuck Berry, Little Richard, and the Everly Brothers were either already in or heading toward creative limbo; Elvis was busy making movies; even Wanda Jackson had cleaned up her act. Bob Dylan blew in from Minnesota, a secret rock 'n' roll fan in the guise of a weathered beatnik poet. But Dylan was an acquired taste for a small downscale market.\n\nWith no one else to be enamored of, the spreading rock 'n' roll audience became, during the 1960\u201363 period, enamored of itself. In the image of its new president and his New Frontier, this crowd was old enough to reinvent its antiquated inner Elvis in a new kind of downtown nightlife gyration called the twist. For two years this dull thud dominated the radio dial and the aspirations of local bands.\n\nThrough 1963 the idea of a rock band not tied to the mindless rhythm of the twist or the equally mindless (albeit seductive) rhythm of surf music seemed to exist only in Portland, Oregon, where the instrumental group the Wailers, of \"Tall Cool One\" fame, still occupied a legendary place in the area's rock pantheon.\n\nPaul Revere and the Raiders sent the instrumental track \"Like Long Hair,\" to the Top 40 in 1961, but Revere got drafted and the momentum faded. By 1963 he was back, vying with another local band, the Kingsmen, for the next crack at a 1956 Richard Berry favorite called \"Louie Louie\" that was kicking up dust on the circuit. Revere got the regional hit, the Kingsmen's version went national. But both paid a price. Signed by Columbia Records' Mitch Miller, who was noted for his antipathy towards rock 'n' roll, Revere and the Raiders were aced by the Kingsmen when Miller failed to promote the Raiders' single.\n\nThe Kingsmen may have won the battle, but they lost the war when lead singer Jack Ely quit after \"Louie Louie\" became a hit to form a new band. The Kingsmen's bass player, who owned the group name, drew the wrath of fans when he lip-synched to Ely's voice at gigs. Paul Revere and the Raiders wouldn't reach the Top 40 again until the end of 1965. No wonder no other major American label wanted to sign anything that smelled like a real rock band.\n\nThe most successful new American bands of the 1962\u201364 period, the Four Seasons (Vee-Jay) and the Beach Boys (Capitol), were essentially throwbacks to earlier, softer eras. The Four Seasons were already warhorses; post-doo-wop Italian soul veterans, with Frankie Valli in the Frankie Lymon falsetto role. The Beach Boys were the Four Freshmen on surfboards, with Brian Wilson still in the creative closet, under the thumb of his father.\n\nPresident Kennedy was more popular than any rock star. Comedians like Lenny Bruce, Mort Sahl, and Shelley Berman were more relevant. Dylan was in Greenwich Village now, starting to tap into the outcast rage rock 'n' roll had once channeled. Dylan walked off The Ed Sullivan Show when Sullivan wouldn't let him sing a song about the John Birch Society. You wouldn't find Dylan on Hootenanny, the folk music TV show that blacklisted Pete Seeger. On the other hand, Joey Dee and the Starliters played for the president's wife, showing just how entrenched in the establishment this harmless rock 'n' roll had become.\n\nEngland wasn't on the typical rock fan's radar then, but a few discerning folks were in on the secret. England was where old rock 'n' roll still mattered. Gene Vincent settled there. Eddie Cochran died there. Buddy Holly had a string of posthumous hits there. In 1964 Paul Simon spent his first divorce from Art Garfunkel busking there. In 1966 the Everly Brothers recorded an album called Two Yanks in England there. Amid folk, blues, and classic rock 'n' roll, the Beatles soldiered on, relentlessly playing seven shows a night, the most eclectic bar band in the world. They could segu\u00e9 from era to era as easily as they went from genre to genre. In the meantime, their original songs were starting to get some notice. Once \"Please Please Me\" hit Number One on the UK singles chart in February 1963, they would notch eleven more in a row (not counting the Twist and Shout EP).\n\nThe Beatles. Not John Lennon and the Beatles nor Paul McCartney and the Beatles, as rock 'n' roll history would have put it (John and Paul were both too competitive, especially with each other, to ever let that happen.). Instead, they were a self-contained rock 'n' roll band with three singers and two (sometimes three) great songwriters; even better than the Crickets. But despite the mania the Beatles inspired in England, American music honchos hedged their bets; the first Beatles singles of 1964 in the U.S. were on five different labels.\n\nBefore any of these groups inspired by the Beatles on The Ed Sullivan Show could find their way to a recording studio, bands from England schooled on American rock 'n' roll were all over the map. Even Bob Dylan took notice, especially that week in April 1964 when the Beatles had all five of the Top Five records. Or that day in September when the Animals took Dylan's arrangement (really Dave Van Ronk's) of \"The House of the Rising Sun\" to Number One. Later, when the Rolling Stones appeared, half of those aspiring musicians traded in their suits, ties, and smiles for grungy T-shirts and jeans, thus creating the image of the sullen rock band that's been in place ever since, from the Stooges to Nirvana to the White Stripes, at the same time casting the Beatles as hopelessly clean-cut sellouts to American capitalism. But the Beatles had lived the Stones' image for years. Before Brian Epstein spiffed them up for mass consumption they were as sullen as the Replacements.\n\nAnd Mick Jagger was an economics major.\n\nWhen the Beatles retired from live performing after about sixty American concert dates, they also suffered in comparison with the Stones\u2014who have since been on the road for forty years\u2014especially when you consider that those sixty concerts were marginal affairs, limited by the logistical nightmares their fame had created. On the other hand, if the live experience of seeing the Beatles ceased to exist after their journeyman stint in Hamburg, those first three Ed Sullivan Show appearances were enough to establish a rock band template long missing from American shores. They were enough to hoist the various Beatle singles moldering in radio's slush pile into the Top Five. And they were enough to open the ports at Ellis Island to the whole of singing England.\n\nFrom there on, with uncanny regularity, whenever a lull in the forward thrust of jangly guitar-based rock 'n' roll set upon the music industry, a Beatlesque solution would emerge to right the ship. Considering the wealth of music that informed the Beatles pre-fame set list, the extraordinary afterlife of the approximately 200 recorded originals they produced and the wealth of music they inspired with their songs, and with their attitude, there is a super-rich catalog of material to hum and sing and contemplate in these pages... if you like the Beatles.\n\nBilly Fury, 1960.\n1\n\nCROSSING OVER\n\nLandlocked to the conservative programming choices of the BBC in the late 1950s and early 1960s, the Beatles and other British musicians were sustained in their formative years by pirate radio and the import market to provide them with their nightly rush of American singles and roots music. By 1962 the Beatles, partial to stateside rock 'n' roll and the early stalwarts of R&B, became one of the most successful and sought-after cover bands on the continent, prized as much for their versatility as their energy, as apt to toss in a show tune or a traditional evergreen as to showcase the latest Brill Building gem.\n\nTheir status was established even before they were introduced to the musical and sartorial preferences of their future manager, record store owner and budding rock journalist Brian Epstein, who saw \"the future of rock and roll\" one afternoon in 1961 (predating Jon Landau's sighting of Bruce Springsteen by a good dozen years), when he heard the Beatles at Liverpool's Cavern Club. Theirs was a repertoire that certainly impressed their eventual producer George Martin, whose limitless good taste ran from show music and skiffle to the cinematic scores of Johnny Dankworth and the wacked-out comedy of Peter Sellers and Spike Milligan (The Goon Show), and Peter Cook and Dudley Moore (Beyond the Fringe).\n\nOther considerations aside, perhaps the most significant early influence on the Beatles, individually, collectively, or at least subconsciously, was the major crush they seemed to have, like almost every other British schoolboy of the time, on Rita Tushingham. Tushingham was a young actress who played Jo in the 1961 movie version of A Taste of Honey, an adaptation of a play written in 1958 by Shelagh Delaney and set in Saltford, a mere forty-five minutes from Liverpool. Prematurely frumpy, Tushingham was the poor man's Mandy Rice-Davies; the anti-Jean Shrimpton, whose career arguably peaked with her performance as the ugly duckling seventeen-year-old shop girl in the poignant and profound film adaptation.\n\nAs popular as they had become on their side of the pond, the Beatles, in their heart of hearts, were still products of the same dead-end streets and thwarted desires as the characters in the play and subsequent movie version of A Taste of Honey, looking for a way out. (John's father had been a sailor who, more or less, never returned.) You could see it in their choices of mates, many of whom eerily reflected the sad-eyed Jo. There was the drab sixteen-year-old hairdresser\/groupie Maureen Cox, who Ringo started dating in 1962 when he was twenty-two. He would marry her three years later. In that same year of 1962, John married the marginally better-looking and more educated Cynthia Powell after getting her pregnant. He had met her in art school in 1957 when he was seventeen and she, eighteen. In 1964 the twenty-year-old George fell for the slightly buck-toothed shampoo girl and aspiring nineteen-year-old model Pattie Boyd. They married early in 1966. Only Paul had his eyes on a glamorous prize signally beyond his middle class reach, the seventeen-year-old doctor's daughter Jane Asher, who he met in 1963 when he was twenty-one. She would introduce him to high society and high culture, playing the part of artist muse (as Suze Rotolo would for Bob Dylan). Although they became engaged in 1967, they never married.\n\nPaul's sentimental and nostalgic streak would have made him a sucker for A Taste of Honey and its evocative occasional theme, which was written by Bobby Scott for the 1960 Broadway production of the play starring Joan Plowright as Jo and Billy Dee Williams as her departed lover. While music was an essential emotional element in the play (characters literally danced onto the stage), provided by the Apex Jazz Trio (Johnny Walback, cornet; Barry Wright, guitar; and Christopher Capon, double bass), the Scott tune eventually became more famous than the show, especially after the movie version won the composer a Grammy for best instrumental recording.\n\nAlthough Martin Denny's recording of \"A Taste of Honey\" was the original version, Herb Alpert, a consultant on the film, had the biggest recording of it a few years later. There were also significant covers floating around England by British musicians, including pianist Victor Feldman and clarinetist Acker Bilk. After some appropriately bittersweet lyrics were added by sometime-actor Ric Marlow, a host of vocal versions commenced. One of the first was by the Beatles, who performed it live on the BBC in October 1962 and included it on their first EP, with Paul singing lead and double tracked in the chorus.\n\nUnfortunately, the Beatles' version (as well as one by Lenny Welch, which actually beat them to the marketplace by a month) had significantly different lyrics than subsequent versions by artists as disparate as Tony Bennett, Morgana King, and the Temptations. (In its history, the song has amassed over 200 covers.) Those who copied the Beatles' rendition skipped the poetry and left up for question whether or not the deliverer of the honey-flavored kiss returned to his first love as promised. Other artists were truer to the lyrics as well as the premise of the play. The movie definitively (if depressingly) clarified that the lover never came back and poor Jo went to her grave waiting for another one. According to Marlow, the Beatles felt the original version was \"too English.\" \"They asked permission to change it. As long as they paid me and gave no one else credit, I agreed,\" said Marlow.\n\nNevertheless, the original version's more realistic outcome was no doubt preferred by Tushingham fanatics like Stephen Morrissey, founder of the Smiths, who featured a scene from the movie in the video accompanying the group's first 45, \"Hand in Glove,\" quoting from the script in his lyrics as well. McCartney, being just short of a true believer, borrowed a line from the script to serve as the title of his tune \"Your Mother Should Know.\" But this was still a far cry from Morrissey's devotion; the play's author, Shelagh Delaney, is pictured on the record sleeve for the Smiths' single \"Girlfriend in a Coma.\"\n\nBut the more comfortable narrative fit the Beatles' grand scheme (which was also Brian Epstein's) far better than the kitchen sink ethos of Delaney and the hedging rewrite. As opposed to the raw emotions and sloppy stage habits displayed by homegrown British rockers such as Billy Fury, Johnny Kidd, and Tommy Steele, the Beatles, under the tutelage of Epstein and Martin, dressed in suits and ties and were known to perform abject show tunes like \"'Til There Was You\" and traditional folk songs like \"My Bonnie.\" (Researchers have thus far failed to dig up any version of \"The Big Ship Sails on the Alley Alley Oh,\" the haunting final number from A Taste of Honey.)\n\nAll this may have brought them closer to their ultimate goal: the leaving of Liverpool and making it in America, but it also aligned them more squarely with the polished pop of the perennial superstar-in-waiting, Cliff Richard, and his backup band the Shadows.\n\nWhile the Beatles appreciated Richard's relentless productivity (and the fact that he and the Shadows performed \"Voice in the Wilderness\" in Expresso Bongo, the excellent 1961 art house flick about Cliff's career), they had to be unnerved by his continued failure to dent the U.S. marketplace. By the end of 1962 he'd had nineteen Top 10 hits in the UK and none in the states, failing to capitalize on the Top 30 showing of \"Living Doll\" in 1959. They had no use for his backup band, either, although the Shadows, led by Hank Marvin, were quite popular under their own name, accounting for six Top 10 singles between 1960 and 1962, among them \"Apache\" and \"Wonderful Land,\" none of which crossed over. That their influence was hard to escape is revealed by the fact that the first original Beatles song to be professionally recorded was a Shadows-like instrumental written by John and George in 1961 that was eventually titled \"Cry for a Shadow\" and released as the B side of \"Why.\" It was the only time these two ever collaborated.\n\nIronically, after the Beatles' unsuccessful audition for Mike Smith at Decca Records, Brian Epstein paid Tony Meehan, former drummer for the Shadows, to produce the same fifteen-song Decca audition tape that wound up in front of George Martin a few months later. But there was no way the Beatles would continuing dressing up and dumbing down for Brian and George if they were destined to follow the same limited path as Cliff Richard.\n\nThings surely would have been different had they not been turned down by the coolest producer on the planet, Joe Meek, whose production of the appropriately spacey \"Telstar\" by his in-house band the Tornadoes (1962) was the second British single to top the American charts (coming just weeks after Acker Bilk's smooth instrumental \"Stranger on the Shore\"). As flaky and mercurial as Phil Spector, Meek was anything but meek in his dealings with the outside world, especially the musicians who fell under the sway of his power trip. But like Spector, he was forgiven his eventually lethal quirks (such as murdering his landlady before turning the gun on himself) because of his other-worldly talent. (Meek is known for accusing Spector of ripping him off.) Meek was the opposite of George Martin in every conceivable way except in his musical genius. He could segue from the skifflesque folk music of Lonnie Donegan (as the engineer on Donegan's \"Cumberland Gap\") to the outrageous antics of Screaming Lord Sutch, with room in his middle brain left over for tributes to Eddie Cochran and his ultimate idol, Buddy Holly. One of his acts, Emile Ford and the Checkmates, was the first black group to top the UK charts with the 1911 tune \"What Do You Want to Make Those Eyes at Me For\" in 1959.\n\nR&B and skiffle credibility notwithstanding, it is doubtful that strong personalities such as Lennon and McCartney would have soldiered well in Meek's program for world domination. After all, aside from \"Telstar,\" does anyone outside of England remember \"Johnny Remember Me\" by John Leyton, \"Be Mine\" by Lance Fortune, \"Angela Jones\" by Michael Cox (not to be confused with \"Amanda Jones\" by the Rolling Stones), \"Just Like Eddie\" by Heinz, \"Tribute to Buddy Holly\" by Mike Berry, or the entire oeuvre of the brassy belter Glenda Collins (\"Baby It Hurts\")? Even the Tornadoes' follow-up to \"Telstar,\" \"Ridin' the Wind,\" did miserably in the U.S., failing to climb past Number 62.\n\nFew British singles were fortunate enough to cross over in those pre-Beatles days, especially into the Top 10. Lonnie Donegan, the King of Skiffle, did it twice: once in 1956 with \"Rock Island Line\" and again in 1961 with his biggest hit, \"Does Your Chewing Gum Lose Its Flavor (On the Bedpost Overnight).\" Quirky crooner Frank Ifield (an eventual Beatles label mate) struck gold in 1962 with his yodeling version of the Jimmy Dorsey hit \"I Remember You,\" but received diminishing chart returns on three other remakes. Before their 1964 onslaught, the Beatles hit the invisible wall three times in 1963, with \"Please Please Me,\" \"From Me to You,\" and \"She Loves You.\" American radio programmers were only slightly more favorable to Del Shannon's version of \"From Me to You,\" which Shannon heard while playing a concert with the fellows that year. Although the record only peaked as high as Number 77, it was an important milestone nonetheless; the first Lennon and McCartney-penned tune to reach those shores. A couple of years later, Peter (the brother of Paul's girlfriend Jane Asher) and Gordon would take the Shannon composition \"I Go to Pieces\" into the American Top 10. By then the whole landscape had changed permanently.\n\nAfter the Beatles hit it big, England broke wide open, becoming a market and a scene, a proving ground and a hotbed, supplying the American charts and audiences with a constant flow of quality material. For many years it seemed as if anyone with a British accent could get a record deal in the U.S., complete with a forty-city tour and several hit singles; anybody, that is, except for an act deemed hopelessly pre-Beatles, a list that was headed by Cliff Richard. (After almost twenty years in the business and several failed U.S. tours, Richard finally crashed the American Top 10 in 1976 with \"Devil Woman,\" but by then it was too late to have made much of a difference.)\n\nBilly Fury, England's second biggest male rock star, never came close, although he charted consistently in the UK from 1959 (\"Maybe Tomorrow\") to 1965 (\"In Thoughts of You\"). Fury's place in pre-Beatles history was secured in 1960 when he recorded the seminal British rock album The Sound of Fury, produced by Jack Good for Decca Records, and again in 2003 when a statue of him was erected at the Albert Dock in Liverpool. Intentionally meant to emulate the 1950s Memphis sound of Elvis Presley at Sun Records, the album featured ten tracks written by Fury and performed in a fury of immediacy by his studio band, which included the top British rock guitarist of the day, the smoking Joe Brown, doing his best impression of Scotty Moore. The leadoff track \"That's Love\" hit the Top 20. In tribute to Fury, (who had offered to hire the Silver Beatles, as they were then called, as a backup band, but was turned down when Lennon refused to fire Stu Sutcliffe), the Beatles were known to perform one of his American rockabilly favorites, Eddie Fontaine's \"Nothin' Shakin.'\" They also covered (courtesy of George Harrison) Joe Brown's arrangement of \"The Sheik of Araby\" as well as \"A Picture of You,\" one of Brown's biggest hits. In 1974 Fury had a part in the well-regarded British movie That'll Be the Day, based on the life of John Lennon. Joe Brown sustained a significant career in show business for fifty years, remaining close to George Harrison and eventually securing an MBE of his own.\n\nTommy Steele, whose UK chart run peaked in 1957 with a Number One cover of the Guy Mitchell hit \"Singing the Blues,\" was effectively finished as a recording artist by 1960, although he did manage to cross over to the U.S. in 1965 by landing a role in the Broadway musical Half a Sixpence. In the show, he played the role of Arthur Kipps, a simple orphan who inherits and then loses a fortune. After playing in over 500 performances, Steele easily segued into the 1967 movie version of the show, having already had experience as an old movie hand in the UK. (He played himself in 1957's The Tommy Steele Story.)\n\nOther rockers of the era didn't fare nearly as well, including original Beatles mentor Tony Sheridan, for whom they eventually landed a gig as backup band in Hamburg, Germany. Sheridan had given them their first taste of recording (as the Beat Brothers) on the single \"My Bonnie\" (produced by Bert Kaempfert), which reached the Top 5 in Germany in 1961 (and the Top 30 in the U.S. three years later), surely upping their profile back home in Liverpool in the days just prior to Brian Epstein becoming their manager. Sheridan also had a significant stint working with noted English-born, American-raised crackpot Vince Taylor and his band, the Playboys. Oddly enough, both Taylor and Sheridan wound up being swallowed up by religious cults before their careers were through, with Sheridan taking the name Swami Probhu Sharan.\n\nNot to be outdone in the crackpot department, David \"Screaming Lord\" Sutch claimed he was an Earl and ran for Parliament several times after his exemplary career in show biz tanked. But this was not before he graduated from the Joe Meek stable (\"'Til the Following Night\") into the clutches of a pre-Led Zeppelin Jimmy Page, with whom he collaborated on upwards of a half-dozen numbers, the best being \"Flashing Lights.\" Page produced Sutch's 1970 album Lord Sutch and Heavy Friends, which included sidemen Jeff Beck, John Bonham, Nicky Hopkins, and Noel Redding. A screamer on the order of Screamin' Jay Hawkins, who was known for tunes like \"All Black and Hairy,\" \"Jack the Ripper,\" and \"Murder in the Graveyard,\" Sutch had an act based more on costumery than musicality, making him a logical precursor to Alice Cooper.\n\nPerhaps England's hardest rocker was Frederick Heath, who came to fame as Johnny Kidd in the heavy guitar trio Johnny Kidd and the Pirates. After breaking in with \"Please Don't Touch\" in 1959, Heath achieved immortality with his next hit, the Gene Vincent-inspired \"Shakin' All Over,\" which reached the top of the British charts in 1960 and influenced a generation of rockers (if not the Beatles, then definitely their cohorts the Who, who covered the tune, as did the Guess Who, the Canadian band that launched their U.S. career with it in 1965). Kidd's biggest hit, \"I'll Never Get Over You,\" came in 1963, the year before the flood of Merseybeat bands that started in the wake of the Beatles' success wiped out the market for such gritty sounds and sentiments. Three years later, Kidd died in a car crash.\n\nThe least explainable non-crossover to cross paths with the Beatles in the early sixties was Helen Shapiro, whose chart career peaked when she was seventeen after a magnificent three-year run, during which time she accounted for two huge international best sellers: \"You Don't Know\" (which never made the U.S. charts) and 1961's \"Walkin' Back to Happiness\" (which spent a grand total of one week at the bottom of the U.S. charts). The Beatles took Shapiro on tour, performed with her on TV, and wrote a song especially for her titled \"Misery.\" But Helen's label and management (and probably Helen herself) agreed that the downcast lyrics did not fit her profile; hers was a much more empowering message than Jo's in A Taste of Honey. Thus, history was dropped into the lap of Kenny Lynch, whose recording of \"Misery\" made him the first artist to cover a Lennon and McCartney tune; not that it did him much good in the long (or the short) run. It fell to Cilla Black, former Cavern Club hat check girl and Brian Epstein prot\u00e9g\u00e9, to take the femme rock mantle from Helen Shapiro and wear it throughout the sixties, becoming the second most popular act in England after the Beatles.\n\nBut it would be unfair to cast Helen Shapiro as the Rita Tushingham of rock 'n' roll, although her inability to transcend the boundaries of her British existence is just as touching. She was never as waifish as lovely Rita anyway; never a shop girl like Jo. She was more on the order of a suburban pom-pom girl (sounding like a 35-year-old martini-swilling diva), more on the order of Lesley Gore-meets-Shirley Bassey. And she did quite well staying put, fashioning an outstanding career during which she crossed over (physically, if not saleswise) with 1963's LP Helen in Nashville.\n\nFats Waller, circa 1940s.\n2\n\nTHE GREAT AMERICAN SONGBOOK\n\nPaul McCartney's father was a musician who led a jazz band in the 1920s. John Lennon's mother taught John how to play the banjo and introduced him to Elvis Presley records. The mother of original drummer Pete Best ran a nightclub called the Casbah in the basement of their family home in Liverpool. None of the Beatles were casual music fans, even growing up. The music that coursed through their houses, mostly on the radio, but also perhaps on the Black and White Minstrel Show on TV spanned almost a century of songs. John Lennon's first composition \"Hello Little Girl,\" written in 1958, has as one of its influences a song John's mother used to hum around the house when he was growing up called \"Scatter-Brain,\" which was a big hit in 1939 for Frankie Masters, a moderate success for Benny Goodman (featuring Louise Tobin on vocal), and a favorite of jazz chanteuse Mildred Bailey. Although the two songs share little in common except a jaundiced view of women and their effect on men, it was this grounding in the Great American Songbook that would educate and separate the various Beatles from their more provincial peers on the Liverpool and, later, Hamburg rock scenes.\n\nStarting out in musical life as a skiffle band the Quarrymen, the earliest incarnation of the Beatles, would only naturally feature in their repertoire a great old-time tune like \"Puttin' on the Style,\" which enjoyed a 1957 run on the charts courtesy of skiffle king Lonnie Donegan, the Pete Seeger of the British folk revival. Donegan was partial to American folk songs by Leadbelly, the Carter Family, and Woody Guthrie. The song, which was written in 1875, harkens back to the early days of country music. Vernon Dalhart, the light opera singer who became country music's first star, recorded it in 1925.\n\nOf the myriad hits Vernon Dalhart (one of many aliases) accounted for on 184 different record labels during his career, \"Puttin' on the Style\" stands out as one of his more upbeat efforts. Although he established his musical identity by recording morbid ballads like \"The Prisoner's Song,\" \"The Wreck of the Old 97,\" and \"The Death of Floyd Collins,\" here Dalhart dallies ostensibly in the realms of an old-time fiddling dance tune. Yet what's interesting about this is that out of all of the old-time fiddling dance tunes to choose from (not to mention the deathly serious Woody Guthrie and Leadbelly epics made available through Lonnie Donegan), on their very first day of singing together, both Lennon and McCartney were drawn to a lyric that good-naturedly critiqued the silly affectations of \"the young folks.\" Even at this early stage of their partnership, the Lennon worldview was at play and Paul was focusing on the dance, with Lennon smiling as he inserted the knife.\n\nLater on, while working as a backup band for Tony Sheridan, the individual Beatles, (including drummer Pete Best) may have had less of a say in their choice of material for their first professional recording session, produced by Bert Kaempfert, which resulted in the single \"My Bonnie.\" This Scottish folk tune had been around forever and was constantly requested by the German sailors who flocked to the Top Ten Club to see Tony Sheridan. (It has been suggested that the Bonnie in question is Charles Edward Stuart, aka \"Bonnie Prince Charlie.\") The song's lone pop moment prior to this occurred in 1938, when a hit recording of it was made by the Scottish-born singer Ella Logan, more famous for her starring role in Broadway's Finian's Rainbow. But it's probable that if the other Beatles had any influence at all on their mercurial leader, they might have directed him to an arrangement by guitar virtuoso Duane Eddy, aka \"The King of Twang.\" Eddy's rocking \"Bonnie Came Back\" was the B side of his 1959 UK single \"Movin' n' Groovin'\" as well as a crowd favorite at local soccer matches. It peaked on the U.S. charts at Number 26 early in 1960 (with the spelling of the flip side changed to \"Moovin' n' Groovin'\") and was included on the album $1,000,000 Worth of Twang, which came out at the end of the year, containing such eventual UK smashes as \"Because They're Young\" (Number 2) and \"Forty Miles of Bad Road\" (Number 11). Between 1959 and 1961 Eddy was all over the British charts, placing five songs in the Top 10. Whether it was George Harrison or Tony Sheridan who played the great guitar solo in the middle of the tune, it was no one but Duane Eddy who was being channeled. (On the Eddy track a nifty drum solo, probably played by Jimmy Troxel, replaced the guitar break.) \"My Bonnie,\" as credited to Tony Sheridan and the Beat Brothers, sold over 100,000 copies in Germany in 1962 and hit the Top 5.\n\nThe Beatles got to perform four other songs with Sheridan at the Kaempfert sessions: \"When the Saints Go Marching In\" (which wound up as the B side of \"My Bonnie,\" titled \"The Saints\"), \"Nobody's Child\" (a Hank Snow country hit from 1949), a knock-off of Chicago blues man Jimmy Reed's \"Take Out Some Insurance on Me Baby,\" and the Sheridan composition \"Why.\" As a bonus, they were allowed time to record two tracks under their own name, selecting the Lennon-Harrison original instrumental \"Cry for a Shadow\" and the evergreen \"Ain't She Sweet.\" \"Cry for a Shadow\" was the B side of \"Why\" when it made the U.S. charts in 1964. \"Ain't She Sweet\" hit the Top 20 three months later, with \"Nobody's Child\" on the flip side.\n\nThe Beatles were big fans of the Weavers and Fats Domino, both of whom covered \"When the Saints Go Marching In,\" and so they had to be thrilled to trot out their best Gene Vincent performance on \"Ain't She Sweet.\" Long one of their idols, the gimpy Vincent had an odd predilection for dredging up ancient material from the heyday of Tin Pan Alley. During his checkered career he took on such moldy chestnuts as \"By the Light of the Silvery Moon\" (1910) as well as standards of a more recent vintage, including \"Up a Lazy River\" (1931), \"Summertime\" (1935), and \"Pistol Packin' Mama\" (1943). \"Ain't She Sweet\" (1927) came from Vincent's 1956 album, Bluejean Bop, which also contained \"Peg o' My Heart\" (1913) and \"That Old Gang of Mine\" (1923). The Beatles were possibly listening when Vincent sang \"Over the Rainbow,\" the soaring Judy Garland Oscar winner from The Wizard of Oz (1939) on his 1959 album, Sounds Like Gene Vincent, and immediately started including it in their live shows.\n\nThe Vincent pedigree alone was sometimes enough for a tune to make it into the Beatles' late night songbook. But on their ill-fated audition at Decca on January 1, 1962, the inclusion of four non-Vincent-endorsed standards has long troubled diehards, who blame this lapse on their manager, the square's square, Brian Epstein. Paul McCartney, for one, has gone to great pains to justify the band's genuine appreciation of such material, a prime example being \"'Till There Was You,\" the key ballad from the Meredith Willson musical The Music Man. And while it's pretty hard to imagine any of the Beatles having the same kind of affinity for the show they may have had for A Taste of Honey, the claim that McCartney was introduced to it on a Peggy Lee album (played for him by a relative) rings only slightly truer. Nevertheless, McCartney certainly didn't shy away from singing it at as many important venues and in front of as many important people as he could, from the Queen of England to Ed Sullivan himself (they even included it on their first world tour, which kicked off in Copenhagen).\n\nIn later years, Paul put his money where his mouth was, going so far as to have MPL Communications, his publishing company, purchase the entire Meredith Willson catalog (joining works by Buddy Holly), which included \"76 Trombones,\" \"It's Beginning to Look a Lot Like Christmas,\" and \"Banners and Bonnets\" (the theme of the Salvation Army).\n\nSimilarly, it is unlikely that McCartney or any of the other Beatles heard \"September in the Rain\" on a Peggy Lee album, although Lee did perform it on radio in the 1940s, backed by her husband Dave Barbour's band. The biggest hit version of the song was Guy Lombardo's in 1937. More recently, R&B temptress Dinah Washington brought it back to the Top 10 in 1959. But the most likely source for the Beatles learning this tune is, in fact, Epstein, who was on record as having loved Washington's version. Moreover, it was also a favorite of his prot\u00e9g\u00e9, Cilla Black, who sang it on The Ed Sullivan Show in 1965.\n\nThe Beatles were also fans of the Coasters, the R&B novelty act that got much of their material from the stellar Brill Building duo, Jerry Leiber and Mike Stoller. As striving professionals in the songwriting trade, the ever upwardly mobile team of Lennon and McCartney found the careers of Leiber and Stoller greatly appealing. The Coasters were fond of the occasional standard themselves, having done an entire album of them in 1960, which included \"Moonlight in Vermont,\" \"Autumn Leaves,\" and \"Satin Doll.\" Also released that year (although not on their album of standards) was the Latin makeout song \"Besame Mucho,\" released as a single with \"Besame Mucho, Part 2\" on the B side. The Beatles were so enamored with this track that they led off their 1962 audition for EMI Records with their own version. The only thing that prevented the Beatles from owning the franchise on the cover was the sterling electric surf version released later that year by former Shadows bass player Jet Harris, which launched his solo career.\n\nThe fourth and final tune from the audition was nearer and dearer to their skiffle roots, \"The Sheik of Araby,\" a slinky number published in 1921 that was apt to be found in the repertoires of American jug bands of the 1960s as well as the 1920s. The song also found favor among New Orleans jazz musicians, Fats Domino, and, in a strange bit of transmogrification, Cilla Black. Written to capitalize on Rudolph Valentino's silent classic The Sheik, it was sung at the Decca audition by George Harrison in response to a popular version by hot guitarist Joe Brown and the Bruvvers. It was eventually put out as a bootleg in England, paired with \"September in the Rain\" on the B side.\n\nIt wasn't only McCartney who professed a fondness for the golden oldies of Tin Pan Alley. When asked about the genesis of their early single \"Please Please Me,\" John Lennon made a point of mentioning Bing Crosby's \"Please\" as a reference for its lyrical dexterity. This may have been less a case of him showing off his encyclopedic memory for random snatches of the music of his childhood (or, since \"Please\" dates back to 1932, his mother's teen years) than the more mundane probability that he'd recently heard yodeling balladeer Frank Ifield's version of the song, which was released as a single around that time. The Beatles also performed Ifield's signature number \"I Remember You,\" which crossed over to the Top 10 in America just before the Beatles' arrival. Before that, it was a hit for Jimmy Dorsey in 1942, and prior to that, for Ada Jones in 1909. To return the favor, Ifield inadvertently punched the Beatles' ticket to America as well as his own. Chicago indie Vee-Jay Records, which had been known for R&B and gospel records, became interested in the pop field and signed a deal with EMI to license Ifield's music in the U.S. After the success of Ifield's \"I Remember You,\" EMI attorney Paul Marshall recommended the Beatles' \"Please Please Me\" as EMI's next pop offering. The song wound up making the Top 40 in Chicago in 1963, the Beatles' first appearance on any American chart. Despite recording ancient standards during their early sessions, the Beatles exhibited a snarky attitude toward non-rock 'n' roll material. On a December 1963 edition of the popular British TV variety show Morecambe and Wise, they joined the hosts in a rollicking finale of \"On Moonlight Bay,\" a number written in 1912 and updated more recently by Doris Day (the mortal opposite of Rita Tushingham). When they recorded \"Three Coins in the Fountain\" (made popular in the U.S. by the Four Aces) in 1963, they may have been thinking less of the 1954 Frank Sinatra UK chart topper than The Goon Show lampooning of it called \"Three Goons in the Fountain.\" It's uncertain as to whether they performed either version of the song during one of two stints at the Three Coins club on Fountain Street in Manchester.\n\nOn The Complete Beatles Home Recordings of 1963, the Fab Four indulged in \"Tammy,\" the Oscar-nominated song from the 1957 teen romance flick Tammy and the Bachelor, starring Debbie Reynolds as Tammy Tyree, Leslie Nielsen (later of Police Story fame) as the bachelor, and Walter Brennan as Grandpa. The film was hardly on the level of the gritty reality demonstrated by A Taste of Honey, and neither the movie nor the song by Debbie Reynolds could have been anywhere near and dear to the inmost hearts of even the least discriminating member of the group. Despite this, the Reynolds hit inspired a cover version by Duane Eddy, which appeared on his 1961 album Girls, Girls, Girls. Folkster Trini Lopez may have heard the Beatles sing it during the twenty-day day gig he shared with them at Paris' Olympia Theater in 1964, just before they left for America, since it shows up on his 1965 LP The Love Album. Debbie Reynolds' wispy rendition did have its charm, hitting Number Two in the UK in 1957, not the last case of a song far outliving the movie in which it made its debut.\n\nGiven the general pandemonium of their concert tours, with a set list pretty much fixed in stone from opening night onward, the best place to have heard the Beatles showcasing the breadth of their taste in the Great American Songbook would have been at a German nightclub in the early sixties (after hoisting a few cold ones). As with many a working band that came before them, the Beatles' sheer length and number of sets played in one night forced the lads to consult the grainier reaches of their memory banks at least once or twice before the evening's work was through. Thus, if you caught them on a good night, you might have heard them segu\u00e9 from giant to giant: from an Elvis B side to a Fats Waller novelty; from a Prohibition era shaker to a bow to the end of World War I, courtesy of the father of the electric guitar, Wisconsin's Les Paul.\n\n\"The World Is Waiting for the Sunrise\" was written in 1919, one of many songs created at a time when an uplifting tone suggested that a new era of high spirits was about to commence.\n\nAlthough technically not a true blue page in the Great American Songbook (since its authors, Eugene Lockhart and Ernest Seitz, hailed from Toronto), the song entered its epilog in 1922 through a hit recording by midwesterner Isham Jones and his dance band. In 1951 it made the Top 10 again, with a version by Les Paul, who made it a showcase for his vocalist-wife Mary Ford, featuring double-tracked vocals and his own incredibly slithery fretwork on his custom-made Gibson. Thereafter, it was picked up by facile guitar players like Chet Atkins and Carl Perkins, and in England, the Beatles' favorite, Joe Brown. On the 1985 TV special Carl Perkins & Friends, George Harrison duets with Carl on the tune, although it's clear that George is only there to get a close-up view of his idol in action. Ringo joined the festivities, reuniting with George on stage for the first time since the seventies.\n\nConsidered historically comparable to the \"Swinging Sixties,\" the \"Roaring Twenties\" was a flush post-war period of parties and prosperity, hampered only by Prohibition. The drug of choice in the twenties was alcohol (in the sixties it was grass and acid), with even casual imbibers viewed as outlaws in the eyes of the feds. If the Beatles were the leading edge of the rock revolution that marked the decadent youth culture in the sixties, jazz was the musical vehicle that was transporting the youth of the twenties into the ditch of moral decay. Sexual liberation abounded in both decades, with long-haired earth mother types sans brassieres freely cohabiting with their equally long-haired hippie consorts. In the twenties, cigarette-smoking flappers in clinging skirts stoked the lusts of otherwise buttoned-up young gentlemen. In both decades, dance was the key outlet for this pent-up sexual tension.\n\nIn the sixties, topless go-go dancers like San Francisco's Carol Doda burst forth from the confines of the repressed fifties to flaunt their wares in public. Doda's performance was a natural (or perhaps, silicone-enhanced) outgrowth of the twist, which earned its place in the decade's freedom-oriented ethos with an emphasis on each partner's solo gymnastic abilities as opposed to the rigid couples-only approach of the fox trot and the Lindy. In the late teens and early twenties the shimmy was the equivalent of the twist, representing a then-new low in commercial debauchery, almost the equal of jazz itself.\n\n\"I Wish I Could Shimmy Like My Sister Kate,\" (or \"Sister Kate\") was written in 1919 by jazz violinist A.J. Piron, either to celebrate the life of a murdered madam or as the progenitor of a sexy new dance craze. The song extolled the non-virtuous virtues of the symbolic Kate, who excelled at doing the latest dance sensation to sweep the nation. There are several contenders for the first racy lady to bring the shimmy to a mass audience. Much-married Polish actress Gilda Gray is known to have shimmied in front of New York City sophisticates at the Winter Garden in 1919, the same year she brought the dance to Hollywood in The Virtuous Vamp. In 1922 she did it on Broadway in Ziegfeld Follies of 1922. Bee Palmer, an occasional singer with the New Orleans Rhythm Kings, probably did the shimmy to the group's \"Shim-Me-Sha-Wobble,\" a number that combined the era's most popular dances: the shimmy, the shim sham, and the wobble. Palmer appeared as a chorus girl in Ziegfeld Follies of 1918. More notorious than either of those two was Mae West, whose image is on the sheet music edition of \"Everybody Shimmies Now,\" which she sang in 1919 with Sophie Tucker. Seven years later, West wrote and starred in Sex, an experimental theater piece that played for a year until it was busted for obscenity. As a result, Mae was sentenced to a week in the slammer for corrupting the morals of the country's youth. She allegedly served her time wearing silk panties under her prison garb. Years later, Madonna (to say nothing of Lindsay Lohan) would take note. Although Madonna has yet to be jailed for any of her offenses, she did perpetrate a coffee table book of nude photographs of herself and others called Sex, which was published just before her album Erotica was released.\n\nLong a hotbed of commercialized erotica, New Orleans welcomed Kate into the repertoires of some of its finest jazz musicians of the twenties, including Sidney Bechet, Henry \"Red\" Allen, Kid Ory, and King Oliver. It may have been publicly introduced in 1923 by blues singer Anna Jones, accompanied on stride piano by Fats Waller. In 1960, the Olympics, a California R&B group, recorded it as \"Shimmy Like Kate,\" and during the Great American Folk Scare of the 1960s, it was again revamped by consummate folk picker Dave Van Ronk, the irrepressible Jim Kweskin sans the Jug Band, and Harvard bluesman Tom Rush.\n\nFats Waller, one of the ultimate showmen in musical history, outlasted the 1920s as well as the 1930s. With his impressive chops, a smile wide enough to reach the cheap seats, and notably droll lyrics, he was a worthy precursor of Louis Jordan and the jump blues that would jump start rock 'n' roll by the end of the 1940s. From Waller's massive repertoire, the Beatles chose the good-natured but sly \"Your Feet's Too Big\" (1939), which aptly showcased their superior taste as well as their equally superior fashion sense. (A few years later, Waller updated the premise in \"Your Socks Don't Match.\")\n\nWidely covered in the U.S. and the UK (even Helen Shapiro had a crack at it), \"Your Feet's Too Big,\" composed by Fred Fisher (\"Peg o' My Heart\"), was also a perfect fit for the repertoire of the Ink Spots, another important rock 'n' roll forerunner, whose mellow harmonies set the stage for doo-wop. Another famous song from the Fisher catalogue recorded by the Ink Spots was the non-charting \"That's When Your Heartaches Begin,\" recorded in 1941. The song eventually found its way into the early repertoire of former gospel\/country singer Elvis Presley, who recorded it in 1953 at the Sun studios in Memphis. The Beatles likely heard it when the Presley camp released it as the B side of his tenth million seller \"All Shook Up,\" which topped the UK charts for seven weeks in 1957 while also hitting the R&B charts. But Elvis, despite being a walking Great American Songbook himself, gave the Beatles much more than a sneer and a ducktail to look at when they began contemplating their career in earnest in the late fifties and early sixties.\n\nEddie Cochran, circa late 1950s.\n3\n\nROCKABILLY\n\nIn 1955 and 1956, rockabilly and rock 'n' roll were one and the same. Their moves, grooves, attitudes, outfits, and sounds were synonymous: loud, brash, self-destructive, anti-establishment, and a pie-in-the-face (if not a fist-in-the-face) to tradition. But tradition has been traditionally a worthy opponent against the forces of change (or at least the forces of youth). There is a myth that explains how the Beatles got their name from the 1953 renegade biker film The Wild One (starring Marlon Brando) in which a rival gang led by Lee Marvin is called the Beetles. (But the Beatles could not have seen the film since it was banned in the UK until 1967.) The point, however, is well taken. Youth is ever resourceful in its age-old battle against authority; black market discs prevailed in the underground, why not black market movies? The 1956 film Blackboard Jungle, however, was not banned in the UK, although city fathers and mothers probably wished it had been, after it opened in England, with Bill Haley's \"Rock Around the Clock\" blaring its incendiary call to arms. The song featured an iconic guitar solo by twenty-one-dollar-a-day session man Danny Cedrone, who first played it on Haley's 1952 recording of \"Rock the Joint.\" In Britain, Teddy Boys danced in the aisles during film showings and rioted in the streets afterwards. Rockabilly had arrived.\n\nBy 1960, as sappy teen idol Bobby Rydell hit the Top 20 in Britain with a lame song called \"Wild One,\" rock 'n' roll and rockabilly had long since started to separate, with one adding aftershave, a skinny tie, Brylcreem, a string section, a deferential attitude toward adults, and a taste for standards, and the other defiantly defending the Alamo of its lonely preferences where they could be surrounded and eliminated, one by one.\n\nAfter several years of failed singles, brothers Johnny and Dorsey Burnette, founders of the Rock 'n Roll Trio, had a falling out and broke up their band. As a solo artist, Johnny sweetened his sound, cleaned up his act, and for his troubles was rewarded with \"You're Sixteen,\" which went to Number 4 in England in 1961. Three years later, his fishing boat was rammed by a cabin cruiser in California and Johnny drowned. He was 30. Meticulous in their choice of live material, the Beatles covered Johnny's \"Lonesome Tears in My Eyes,\" a Rock 'n Roll Trio tune from 1957 co-written by Dorsey, guitarist Paul Burlison, and Al Mortimer. In 1969 John and Paul borrowed the song's lead guitar break and used it on \"The Ballad of John and Yoko.\" Once he was set free from Beatle obligations, the 34-year old Ringo Starr paid tribute to Johnny in 1974 with an unseemly version of \"You're Sixteen,\" which topped the charts in America and made number 4 in the UK.\n\nGene Vincent survived several accidents in his short lifetime, including one in England in 1960 that killed rockabilly idol Eddie Cochran, and a motorcycle spill in Korea while serving in the Navy, which left him with a limp. Cochran was twenty-two and was finishing up a tour of England and preparing to return to Los Angeles where he was to be married to songwriter Sharon Sheeley (who was also injured in the wreck along with Vincent). After influencing a host of British rockers including Tommy Steele and Billy Fury, Vincent settled in England in the sixties. In 1971 he died at age thirty-six after tripping at his mother's house and rupturing a bleeding ulcer. Vincent was known for his taste for standards; five of the ten cuts on his first album were evergreens, making it acceptable for the Beatles to stretch in that direction. They would record Vincent's Top 10 hit \"Be Bop a Lula\" as well as its follow-up, \"Bluejean Bop.\" A John Lennon favorite, \"Be Bop a Lula\" was the first track on Lennon's 1975 album Rock and Roll and one of the songs he was playing on the day he was introduced to Paul McCartney, who, in turn, showed him Eddie Cochran's \"Twenty Flight Rock,\" which Paul would set down on his 1988 album Back in the USSR. The next year, Paul recorded \"Be Bop a Lula\" for his acoustic album Unplugged.\n\nRoy Orbison, who was a big influence on the Beatles' second single \"Please Please Me,\" toured the UK with the group in 1963 just prior to his world turning black. Possessing a plaintive, operatic upper register, Orbison was a melancholy figure who married his teenage bride Claudette Frady in 1957. Although they divorced in 1964, they remarried the following year. In 1966 Claudette died instantly after her motorcycle was hit by a truck. Two years later, while he was out on the road, Orbison's Tennessee home burned down with two of his children inside. Haunted, Orbison made it to fifty-two, succumbing to a heart attack shortly after re-conquering the charts with a new hit single (\"You Got It\") and rejuvenating his career as a member of the super group the Traveling Wilburys, along with Tom Petty, Jeff Lynne, Bob Dylan, and George Harrison.\n\nIn 1959 twenty-two year old Buddy Holly died in a plane crash along with the Big Bopper and Ritchie Valens. It was a tragedy; it was the lifestyle; but it also became part of the legend. In 1954 thirty-three-year-old guitarist Danny Cedrone fell down a flight of stairs while possibly in the throes of a heart attack and died of a broken neck, just ten days after recording \"Rock Around the Clock\" with Bill Haley. Eight months later, Blackboard Jungle was released, which would have made Cedrone a hero. In 1956 Carl Perkins, riding high on the success of \"Blue Suede Shoes,\" was on his way to appear on The Perry Como Show when he nearly drowned in a car crash that critically injured his brother Jay. (Jay would die from a brain tumor two years later.) Although alcohol wasn't involved most of the time, rockabilly cats feasted on their hard-drinking image; it was just a consequence of life on the road.\n\nAlthough his career was derailed by the timing of the accident, Perkins survived and eventually made a tour of England in 1964. He backed off a bit from rockabilly, joined Johnny Cash's band, and became a forerunner of the rougher, tougher \"outlaw\" country music movement. Perkins lasted till the ripe old age of sixty-five, becoming a rockabilly guitar\/songwriting guru for George Harrison as well as Paul McCartney. Ringo Starr's vocal spotlight on the album Beatles for Sale (Beatles '65 in the U.S.) was \"Honey Don't,\" the B side of Perkins' \"Blue Suede Shoes.\" (When \"Honey Don't\" first entered the Beatles' repertoire in 1962, it was sung by John Lennon.) Ringo also recorded Perkins' \"Sure to Fall\" and \"Honey Don't\" on subsequent solo albums. In 1969 John sang \"Blue Suede Shoes\" with the Plastic Ono Band when he was moonlighting on the Beatles in Toronto. In all, the Beatles covered ten songs associated with Carl, second only to the twelve they covered by Chuck Berry; besides \"Blue Suede Shoes\" and \"Honey Don't,\" the others were \"Matchbox,\" \"Everybody's Trying to Be My Baby,\" \"Sure to Fall,\" \"Sawdust Dance Floor,\" \"Tennessee,\" \"Glad All Over,\" \"Lend Me Your Comb,\" and \"Your True Love.\"\n\nThey only covered one song by Eddie Fontaine, the rousing \"Nothin' Shakin'\" from 1958, his only hit. In 1965, after appearing in several movies including The Girl Can't Help It with Jayne Mansfield and Little Richard, Fontaine was tried and convicted of hiring someone to murder his wife. Born Brian Holden, Vince Taylor went to Hollywood High, moved to London, then Paris, strewing rockabilly identities like false credit cards. At one time Vince Taylor and the Playboys contained Tony Sheridan in their ranks. It is debatable whether Taylor influenced Sheridan and how much Sheridan influenced the Beatles when they were his backing band, but all were on the scene together, drinking from the same tap. In Paris at the end of 1963, Taylor cut a single with \"Memphis, Tennesee\" on one side and \"A Shot of Rhythm and Blues\" on the other, two tracks the Beatles favored.\n\nSoon after Ritchie Valens went down in an Iowa cornfield, Tex-Mex rocker Chan Romero was cloning him much in the same way Bobby Vee and Tommy Roe were cloning Buddy Holly. Romero's manager (who may have been Valens' mother) even went so far as to approach Valens' label with his demos. Out of this ploy came \"Hippy Hippy Shake\" in 1962, a single that found its way overseas, where it was adopted by a bunch of bands, most notably the Beatles. It was then co-opted by another Liverpool group, the Swinging Blue Jeans, who had a transatlantic hit with it in 1964, breaking into the Top 5 in England. As their fame ascended and they were asked to judge an episode of the popular UK TV show Jukebox Jury on December 7, 1963, among the records the four Beatles were asked to critique was that very same version of \"Hippy Hippy Shake.\" They eventually gave it four thumbs up, but with slightly grudging commentary. \"I like the Bill Harry version,\" said John. \"We used to do it,\" George acknowledged. \"Good, but not as good as Chan Romero,\" said Ringo. \"Nobody remembers Chan Romero,\" added Paul.\n\nWhen last seen, Romero was singing Christian music. The Swinging Blue Jeans (or at least singer Ray Ennis) spent the next fifty years dining out on that song on the oldies circuit.\n\nThe Beatles modeled their harmonies on records by the Everly Brothers, and although they covered three of the Everlys' biggest hits in their early sets (\"Bye Bye Love,\" \"Wake Up Little Susie,\" and \"Cathy's Clown\"), it was the more obscure \"So How Come (No One Loves Me)\" that they performed live on the BBC, with the rarely heard harmonizing team of George and John. They found this rather lackluster track (with its historically inaccurate message) on the 1960 album A Date with the Everly Brothers, bypassing more obvious material like the Little Richard rocker \"Lucille\" and the monster rock ballad \"Love Hurts.\"\n\nIn a couple of years, those sentiments would, ironically, be more than apt for the Everly Brothers themselves. After fourteen Top 10 records in the U.S. between 1957 and 1962, they couldn't buy another hit in America for 30 years. (They survived somewhat longer in England, with the U.S. stiff, \"The Price of Love,\" their last Top 10 item, coming in 1965.) The Everlys hit the UK Top 40 a few times through the end of 1967 with \"I'll Never Get over You,\" \"Love Is Strange,\" and \"It's My Time.\" In 1973, Two Yanks in England reflected their conversion to all things British, with a cover of Manfred Mann's \"Pretty Flamingo\" and seven tracks written by L. Ransford, a pseudonym for the Hollies (Graham Nash, Allan Clarke, and Tony Hicks), which included the standout \"So Lonely.\"\n\nBy then, both brothers were addicted to speed and Don had suffered a nervous breakdown. It all fell apart at a 1973 concert at California's Knott's Berry Farm when Phil smashed his guitar during a set before stalking off the stage. This was no tribute to Pete Townshend or Jimi Hendrix; it was the beginning of a ten-year period when Don and Phil were no longer on speaking terms with each other.\n\nBut the musicians who loved them, especially those in England, wouldn't let the legendary brothers fade into dysfunctional obscurity. In 1984 a reunion concert at the Royal Albert Hall was miraculously pulled off, breaking the ice at last. The album EB 84 resulted, with the leadoff track \"On the Wings of a Nightingale,\" a chart single written by Paul McCartney, some kind of a nightingale indeed.\n\nBy these fast and loose standards, Elvis Presley's ride from fame to fortune has to be considered nearly exemplary, if not virtually domesticated. Long since revealed in his many biographies as a simple good old boy trapped in a Graceland not entirely of his own making, Elvis was nonetheless extremely sensitive, insecure, and competitive about his place in the rock 'n' roll scheme of things.\n\nWhile Carl Perkins lay in a near coma in a New York City hospital, Scotty Moore, Bill Black, and D. J. Fontana, in town for a gig with Elvis, came to call, bearing Elvis's good wishes. But no Elvis. Weeks later he sent a telegram wishing Perkins a speedy recovery. Did he harbor a secret grudge that Carl's version of \"Blue Suede Shoes\" had outpaced his own on the charts? Who knows? But is it a coincidence that Elvis never recorded another Carl Perkins song?\n\nWhen the Beatles finally met the King in the summer of 1965, they were both ensconced in their respective huge Los Angeles mansions, like two pop cultural ships in the night, one bound for glory, the other headed for a reef in the middle of the North Atlantic. It had to have been a tense, stilted afternoon. Whereas a year earlier, Bob Dylan had famously turned them on to pot, Elvis apparently didn't offer them so much as a peanut butter and banana sandwich. On the other hand, the lads were probably already stoned when they arrived and Elvis may have been loaded on a cocktail of barbiturates. The Memphis Mafia was there, all of Elvis's childhood pals. The color TV was on with the sound off; Muddy Waters was on the stereo. The Beatles played pool with Elvis's bodyguards. Elvis played some bass and the fellows eventually jammed and talked gear. Priscilla stopped by to curtsy like a proper hausfrau. Elvis had met her when he was twenty-five and she, fourteen. They didn't marry until 1967 after she turned twenty-one. On their way out the door, the Beatles were given souvenir holsters.\n\nThis was the shell of a husk that Presley had become, a brooding recluse reduced to making movies for the money and then sitting at home making fun of them on TV with the sound off. He had suffered through a drought of twenty-two straight singles without having a Number One hit. It had been three-and-a-half years since \"Good Luck Charm\" in 1962.\n\nThere are many quotes about the astounding effect Elvis had on people upon hearing his records for the very first time. For most, it was like a door or a window opening, the earth itself opening up to a new day, a new order. Suddenly, blind men could see; frigid women got all tingly; people went out and bought guitars; sideburns started appearing underneath the ears of otherwise ordinary clods in corduroys. Elvis turned people on to the previously hidden exotic worlds of R&B and country music, which often appeared on opposite sides of his 45s. His first single in 1954 featured \"That's All Right,\" a song from the repertoire of blues man Arthur \"Big Boy\" Crudup, with the flip side, \"Blue Moon of Kentucky,\" coming from bluegrass legend Bill Monroe. (McCartney paid homage to \"Blue Moon of Kentucky\" on his 1991 acoustic album Unplugged.) Presley's last Sun single repeated the pattern, with Little Junior Parker's scorching \"Mystery Train\" paired with \"I Forgot to Remember to Forget,\" a country rocker co-written by Charlie Feathers. The latter song was performed live by the Beatles on the BBC. Elvis' fourth Sun single, \"Baby Let's Play House,\" was an early favorite of the Quarrymen. Paul McCartney stated that \"In Spite of All the Danger,\" the first original song recorded by the Quarrymen, was influenced by Elvis. On his acoustic album, Paul played tribute to Elvis's second Sun single, R&B great Roy Brown's \"Good Rockin' Tonight.\" \"Hound Dog\" found its way into John Lennon's 1972 Madison Square Garden concert performance. As late as 1999, McCartney was still channeling Presley, recording \"Party,\" from the 1957 movie Loving You, on his album Run Devil Run. The track became a big hit in the UK that year, whereas Wanda Jackson, rockabilly's only certified wild woman for a painfully brief time, had a U.S. hit with the song in 1960. Carl Perkins's version of \"Blue Suede Shoes\" also succumbed to Elvis mania, with Elvis's version breaking the Top 10 in England and Carl's nowhere to be found. British kids had a slightly different Elvis experience than American teens, as \"Paralyzed,\" \"King Creole,\" \"A Mess of Blues,\" \"Girl of My Best Friend,\" and especially the Bert Kaempfert-penned \"Wooden Heart\" enjoyed hit status in the UK, while in the U.S. they either stalled at the bottom of the charts or were not released as singles at all.\n\nIn 1965 you could forgive Elvis if in the midst of that dry spell all he really wanted to do was kick back with his pals and live off his legend and his royalties. His manager, Colonel Tom Parker, would of course not allow this. Like Brian Epstein, the Colonel had his gaze fixed firmly on the horizon. When confronted with Elvis going through the motions on \"Kiss Me Quick,\" which they reviewed on their guest-starring stint on Juke Box Jury in December 1963, the Beatles had to be honest. \"The last few years Elvis has been going down the nick,\" said Ringo. \"I like his voice,\" said Paul. \"What I don't like are his songs.\" \"Elvis is great,\" echoed George. \"His songs are rubbish.\"\n\nPerhaps word of their critiques had reached Elvis by the time they had their summit a year-and-a-half later.\n\nThe irony is that Elvis was only dogging it, laying low, and playing possum until this Beatles thing died out. He'd withstood folk rock, hadn't he? Surf music? He preferred Hawaiian numbers, gospel music, Christmas songs, and tunes from his own movies, such as \"Do the Clam,\" \"Puppet on a String,\" and \"Your Time Hasn't Come Yet Baby.\" By 1969 his ship turned around and headed back to Memphis, while it was the Beatles who were heading for divorce court. Starting with his 1968 TV special, Elvis found his voice, soaring on \"If I Can Dream\" and \"Memories.\" He also found the top of the charts in 1969 when \"Suspicious Minds\" hit Number One, one month after the whole youth culture united under the peace signs of Woodstock. Feeling particularly confident at this time, he even recorded a couple of Beatles tunes for the first time, cutting \"Hey Jude\" in Memphis (1969) and \"Yesterday\" at a Las Vegas concert (1970). But his memory was apparently a lot stronger than his confidence. He didn't release \"Hey Jude\" until 1971, after the Beatles had safely broken up, a kind of belated get-well card to the lads.\n\nLittle Richard, circa 1950s.\n4\n\nR&B\n\nAs opposed to their encounter with Elvis, the Beatles' first meeting with their other main early idol, Little Richard, was a lot more productive. Having heard him on record in the 1950s, they joined him for two concert dates in England in October 1962. Forced back into rock 'n' roll by a hungry audience after an extended tour in the ministry, Richard shared a bill with the Beatles at Hamburg's Star Club during the first two weeks in November. He treated the boys to steaks, read from the Bible, introduced them to organist Billy Preston, and taught Paul how to sing his trademarked \"woo,\" which he had learned from gospel singer Marion Williams, certainly a turning point in their history. A chalice had been passed.\n\nIf the Hamburg years were formative, the Little Richard chapter was climactic for the band. As perhaps the rock world's most incendiary performer, Little Richard influenced other incendiary types including James Brown and James Osterberg (Iggy Pop). He inspired one of the Beatles' signature live numbers, a mashup of Leiber and Stoller's \"Kansas City\" and \"Hey-Hey-Hey-Hey!\" (the B side of \"Good Golly, Miss Molly\"), which the lads learned after they heard Little Richard do it onstage in England. Two years later they brought it out of mothballs to open a last-minute show in Kansas City, bringing down the half-filled house (Municipal Stadium, home of the Kansas City A's, whose owner, Charles O. Finley, had booked the Beatles for the unprecedented sum of $150,000). \"Long Tall Sally,\" the first of four Top 10 hits, was a song about an ugly girl in Richard's New Orleans neighborhood, covered by the Beatles on their second U.S. album. The song's original B side, \"Slippin' and Slidin,' was recorded by John Lennon on his 1975 album Rock and Roll, along with a medley of \"Rip It Up\" and \"Ready Teddy,\" the last of Richard's 1956 hits. As performed by the Beatles, \"Rip It Up\" was part of a medley containing \"Blue Suede Shoes\" and Big Joe Turner's \"Shake, Rattle and Roll.\" They also performed \"Ooh My Soul,\" Richard's last Top 40 hit, from 1958.\n\nAt the end of 1958, Little Richard got his calling from above and abandoned rock 'n' roll for seminary school. With Elvis in the army, Buddy Holly in the ground, Chuck Berry fending off his impending incareration, Jerry Lee Lewis on the moral blacklist for marrying his underage cousin, Alan Freed on the payola hot seat, and everyone else having sold out to Dick Clark, rock 'n' roll in 1959 was on life support in the U.S., alive and well apparently only in England. Back in New Orleans, mystical melting pot of all things rhythmic, they hadn't yet gotten the word. Bumps Blackwell, Little Richard's producer at Specialty Records, was already grooming his replacement in the form of edgy bad boy Larry Williams. Williams's raucous style was particularly appealing to the Beatles, who recorded both sides of his 1958 single \"Dizzy Miss Lizzy\" (which found its way onto the Help album in 1965) and \"Slow Down\" (which they released as a single in 1964 and included on Something New). The album also contained Williams's semi-autobiographical \"Bad Boy\" as well as the \"Kansas City \/ Hey-Hey-Hey-Hey!\" medley. His first hit \"Short Fat Fannie\" and the album track \"Peaches and Cream\" were also Beatle favorites.\n\nLarry Williams was ultimately unable to withstand the curse of 1959. By the end of the year he had been dropped from his label, after being collared by police for dealing drugs. He spent 1960 in jail. Although they were sometimes rivals, Williams and Little Richard wound up working together, with Williams producing The Explosive Little Richard on OKeh in 1967, an album that was later dismissed by Richard as being \"too Motown.\" Despite the Beatles' continued devotion to his music during their solo careers (John's \"Bony Maronie\" and Paul's \"She Said Yeah\"), Williams' downward spiral continued. In the seventies, Williams was fighting with Richard over a drug debt when Williams pulled a gun. In 1980 Williams was found dead in L.A. of gunshot wounds\u2014possibly self-inflicted, possibly not\n\nThe kingpin of all things New Orleans was Fats Domino, whose first crossover hit single \"Ain't That a Shame\" (in 1955 after a dozen R&B chart hits) made it into the early repertoire of the Quarrymen. It was the first song Julia Lennon taught her son John to play on the banjo. Domino's sixty-sixth and last chart single was a cover of the Beatles' \"Lady Madonna,\" which spent two weeks on the charts in 1968. In 1975, Lennon recorded Domino's \"Ain't That a Shame\" for his Rock and Roll album.\n\nThe Beatles continued to mine one of rock's most fertile proving grounds when they picked up on Lee Dorsey's \"Ya Ya,\" which was produced by Allen Toussaint. The lads recorded it with Tony Sheridan, and John Lennon would also include it on his Rock and Roll album.\n\nFrom there, they diligently made their way through the rest of the history of R&B. \"Some Other Guy\" came from the New York City doo-wop streets of Chantels producer Richard Barrett. From Cincinnati came the Stereos' \"I Really Love You,\" a record cherished by George Harrison until he released it as a single in 1981. Although the Beatles recorded the racy back alley blues of Little Willie John's \"Leave My Kitten Alone,\" they preferred the white cover version by Johnny Preston. They sampled crooner Roy Hamilton by way of Elvis's version of \"I'm Gonna Sit Right Down and Cry (Over You).\" They were surely hip to early Ray Charles, favoring his 1950s R&B hits including \"I Got a Woman\" (1954), \"Hallelujah I Love Her So\" (1955), \"Get on the Right Track Baby\" (1958), and \"What'd I Say\" (1959). But they reserved a special place in their hearts and their repertoire for \"Mister Moonlight,\" the B side of the 1961 single the eponymous \"Dr. Feelgood\" by the Georgia-born barrelhouse piano player Willie Perryman (also known as Piano Red).\n\nIn the evolution of black music and rock 'n' roll history, Hank Ballard's \"The Twist\" has been considered both a highlight and a lowlight, depending on who you talk to. The Beatles were in at the very start of the twist craze. Ironically, the song and dance that it inspired served to indelibly mark the Beatles' own transitional point in their career. Although Chubby Checker's version of \"The Twist\" reached Number One in the U.S. in 1960, Checker didn't even crack the UK's Top 100 favorite artists that year. In 1960 the fading old school still reigned, typified by Freddy Cannon, Bobby Rydell, Neil Sedaka, Brenda Lee, and Connie Francis, none of whom were at the moment twisting. Only later would Francis and Rydell try to cash in on the craze; Connie with an aimless album called Do the Twist with Connie Francis and Bobby collaborating with label mate Chubby Checker on Bobby Rydell \/ Chubby Checker.\n\nBy then it was almost 1962 and the whole Peppermint Lounge thing happened. The twist had made it to the cover of Life magazine, endorsed by Jackie O and Murray the K. It was the first baby boomer dance; a revolutionary sidestep and a blast of white noise for the organ-guitar-and-bass trio Joey Dee and the Starliters, who had a major success with \"Peppermint Twist\u2013Part 1.\" Dee's hit was written by Henry Glover, whose \"Teardrops on Your Letter\" was the A side of Hank Ballard's original version of \"The Twist.\" Although Ballard never quarreled with the royalties he earned as the song's sole writer, he never achieved Checker's world-renowned status with the song. Checker's version of \"The Twist\" pulled off a rare feat\u2014topping the charts twice; once in 1960, and the second time in 1962 when it even made the Top 10 in England. \"People forget,\" Hank said, \"when Chubby got hot, we got hot, too.\"\n\nHe was referring to his group the Midnighters, famous for salacious R&B hits of the mid-fifties such as \"Work with Me Annie\" and \"Annie Had a Baby.\" Ballard's Midnighters had been on the circuit for years, but in 1960 they made the Top 10 with the singles \"Finger Poppin' Time\" and \"Let's Go, Let's Go, Let's Go.\"\n\nBallard's quote could have referred to any act that adopted or co-opted the twist's insidious beat, a beat that would take over the American charts for the next two years, effectively washing the slate clean of all the well-scrubbed American Bandstand crooners stuck on the old-school moves of the Lindy and the stroll. The atmosphere was ripe for something new and fresh to emerge after the dancers had worn themselves out. At first this only meant Chubby Checker and Hank Ballard. Ballard had released the first version of \"The Twist\" a couple of weeks before Chubby lapped him, courtesy of his national exposure on Dick Clark's Saturday night TV show. Despite the success of Checker's cover, Hank's version still was able to crack the Top 30. In late 1960 Ballard released \"The Hoochi Coochi Coo,\" which made it to Number 23. Six more dance titles hit the charts in 1961, including \"The Float,\" \"The Continental Walk,\" \"Let's Go Again (Where We Went Last Night),\" and \"Keep on Dancing.\" His last charting record, in early 1962, asked the rhetorical question, \"Do You Know How to Twist?\"\n\nA spokesman, if not an ambassador for the American music industry's practice of grinding a good thing into (gold) dust, Checker's plush 1961 started early when \"The Hucklebuck\" followed \"The Twist\" in October 1960. Then, in quick succession came \"Pony Time,\" \"Dance the Mess Around,\" \"Let's Twist Again,\" \"The Fly,\" the re-release of \"The Twist,\" and \"Twistin' U.S.A.,\" interrupted only by a Checker-Bobby Rydell duet on \"Jingle Bell Rock.\" In 1962 Checker resumed where he had left off with \"Slow Twistin'\" (with Dee Dee Sharp), \"La Paloma Twist,\" \"Dancin' Party,\" \"Limbo Rock,\" and \"Popeye the Hitchhiker,\" interrupted only by the re-entry on the charts of \"Jingle Bell Rock.\" But Chubby wasn't through. In 1963 he released \"Let's Limbo Some More\" and \"Twist It Up\" before switching gears from the twist to Bahamian folk songs.\n\nBy this time even Frank Sinatra had gotten in on the act, recording \"Ev'rybody's Twistin'\" while Count Basie came up with \"The Basie Twist.\" Top songs in 1962 included \"Mashed Potato Time\" by Dee Dee Sharp, \"The Wah Watusi\" by the Orlons, \"Dear Lady Twist\" by Gary (U.S.) Bonds, \"Twistin' the Night Away\" by Sam Cooke, \"Percolator (Twist)\" by Billy Joe and the Checkmates, and \"Soul Twist\" by King Curtis.\n\nIt had become a national dance craze; the forerunner of disco. Nowhere was the twist craze more rampant than in New York City, where Joey Dee and the Starliters became the house band to royalty at the newly-hip Peppermint Lounge on Forty-fifth Street near Broadway. Dee was a bit less hungry than some of his adversaries in the marketplace. After blazing his way to the top with the classic \"Peppermint Twist\u2013Part 1,\" he followed it up with \"Hey Let's Twist,\" but didn't return to the Top 10 until he covered the Isley Brothers' \"Shout\u2013Part 1\" the following month. Perhaps he was too busy making the movie Hey Let's Twist, which co-starred Alan Freed's favorite blonde bombshell Jo-Ann Campbell, but Dee wouldn't make it back to the Top 40 for another year when he cashed in with \"Hot Pastrami with Mashed Potatoes\u2013Part 1.\" (Dee's songs had a tendency to stretch out. If singles had had four sides, he could have filled them all.)\n\nOriginally recorded by the Phil Spector-produced Top Notes, \"Shout\" made the Top 50 in 1959 with a joyous, frenetic version by the Isley Brothers, arguably kick-starting the whole twisting trend. When the Beatles began performing \"Shout\" in Germany, they couldn't have known the tectonic shift about to commence, in music as well as of their generation. By the time they covered the Isleys' belated follow-up \"Twist and Shout\" in 1962, they might have had a clue. The Isleys' \"Twist and Shout\" hit the Top 20 in 1962, but the success of the Beatles' cover version resulted in its predecessor becoming all but forgotten.\n\nWhen the Beatles' \"Twist and Shout\" was released in America, it went all the way to Number 2, as part of the group's 1964 chart onslaught. (In England, it even became the title of a 45 EP.) The record became a token, a memento, a throwback, and a fond farewell to an era irrevocably over. The era of the twist had been washed away by the tides, to be replaced for a while by the Beatles and only the Beatles, then by any band with a British accent, and finally by creative singers, songwriters, and bands on both sides of the Atlantic, leading the way to an unprecedented renaissance of rock 'n' roll music.\n\nIn Detroit, the renaissance had been going on since 1960, when Berry Gordy left the prize fighting ring to go after an even bigger prize. While Sun Records' Sam Phillips was famous for saying that he was looking for a white boy who could replicate the black sound, Gordy was polishing up a black sound and gearing it for a white audience. The result was Motown Records. Scouring the neighborhoods, high schools, filling stations, and his own back office for talent, Gordy's mission was to define the decade in black music. For his house band, he hired the best session men he knew, including Robert White on guitar, Bennie Benjamin on drums, and James Jamerson on bass, the aggregate becoming known as the Funk Brothers, a name ascribed by Benjamin. (Gordy detested the word \"funk.\") Motown's headquarters, which housed the recording studio, became known as Hitsville U.S.A. Gordy put his writers in cubicles and had them compete for the choicest assignments. He defined his goal quite clearly with the title of the label's first Top 40 hit: \"Money (That's What I Want)\" by Barrett Strong. The record was released in 1959 and peaked at Number 23 in February 1960.\n\nThe Beatles were quick to pick up on the record even though it wasn't a hit in England; at the time, Motown was merely a drain on Berry Gordy's limited bank account, but that would change in short order. Like most music freaks, the Beatles were avid record collectors, and since manager Brian Epstein owned Liverpool's popular NEMS record shop, it enabled them to sample new sides to their heart's content. The obvious message of \"Money\" appealed to all of them, but especially to John Lennon, easily the hungriest of the four. Very soon after hearing it, the Beatles added the song to their act. In 1963 they recorded it, using the song as the rousing closer to their second British album, With the Beatles.\n\nBarrett Strong's Motown career as a songwriter flourished as he collaborated mainly with Norman Whitfield on a string of massive hits including \"I Heard It Through the Grapevine\" for Gladys Knight and the Pips and then Marvin Gaye, and \"Cloud Nine,\" \"Ball of Confusion,\" and \"Papa Was a Rolling Stone\" for the Temptations. By that time the sixties were over and Motown had become a franchise equal to that of the Beatles.\n\nPerhaps the Beatles' favorite Motown artist was Smokey Robinson, as much a consummate singer, musician, producer, arranger, and songwriter as Paul or John. They loved the sound of his group the Miracles, whose recording of \"Shop Around,\" co-written by Berry Gordy, became Motown's first Top 10 hit in December 1960. Their next big hit was \"You've Really Got a Hold on Me,\" released in 1962. By that time Smokey had the chops and the confidence of the boss to write the whole song himself. He was already handling that chore for Mary Wells, writing two of her three 1962 hits, \"Two Lovers\" and \"The One Who Really Loves You,\" while co-writing the third, \"You Beat Me to the Punch\" with guitarist Ronnie White. The Beatles adored Wells, Motown's first solo female hit maker, and took her on a tour of Europe in 1964. \"You Really Got a Hold on Me\" (with \"You've\" changed to \"You\") was included on the Beatles' second album. Ignoring all of the Miracles' hits and possibly with a hint of snobbery, John Lennon named \"I've Been Good to You\" (the B side of 1962's \"What's So Good About Good-by\") as the Beatles' favorite Miracles track.\n\nBesides Mary Wells, Brenda Holloway was another Motown lady who toured with the Beatles, landing a prestigious opening slot during their 1965 U.S. tour. Holloway launched her Motown career in 1964 with the Top 20 hit \"Every Little Bit Hurts.\" She is probably better known as the lead singer on the original 1967 version of \"You've Made Me So Very Happy,\" which she co-wrote with her sister Patrice, Frank Wilson, and Berry Gordy. Although it was her last charting single, it became the first for the reconfigured jazz-rock group Blood, Sweat and Tears, whose version reached the top of the charts in 1969. In 1961 the feisty, streetwise Marvelettes provided Motown with its first Number One hit, \"Please Mr. Postman,\" which the Beatles covered on their second album.\n\nThe Beatles had a soft spot for girl groups. The girl group phenomenon was a mini-trend transcending white and black artists, ranging from doo-wop to rock. It was largely a producer's medium, with Phil Spector at the reins of the Crystals and the Ronettes, Luther Dixon firmly in control of the Shirelles, Richard Barrett calling the signals for the Chantels, and Shadow Morton the wizard behind the Shangri-Las. Ronettes lead singer Veronica Bennett, who later married Spector (becoming Ronnie Spector), claimed that Phil's jealousy of the Beatles prevented her from joining the Ronettes when they toured the U.S. with the Beatles in 1966 U.S. tour. She did, however, tour with the Rolling Stones (the Stones opened for the Ronettes) and in 1971 recorded a single that flopped for Apple Records.\n\nAlthough the Beatles were fans of all of the above, the songs they chose to cover came from the catalogs of some of the lesser-known girl groups, including \"Chains\" by the Cookies (Little Eva's backup band), \"Keep Your Hands Off My Baby\" by Little Eva herself, and \"Devil in Her Heart\" (changed from \"Devil in His Heart\") by a completely obscure girl group from Detroit called the Donays, whose records on the tiny Brent label George Harrison had discovered. Bypassed even by Motown, the Donays' record earned an improbable annuity for its author Richard Drapkin after the Beatles covered it.\n\nEver the iconoclast even in the girl-group realm, John claimed allegiance not to Rosie and the Originals' classic \"Angel Baby,\" but to the song's B side, \"Give Me Love,\" featuring a vocal by Bluford D. Wade, a San Diego R&B singer who was recruited for the session. Similarly, when the Beatles covered songs by the Shirelles, instead of the obvious choice \"Baby It's You,\" they selected \"Boys,\" the B side of the Shirelles' first Number One single \"Will You Love Me Tomorrow.\" On their recording of the song (with a vocal by Ringo, his first on a Beatles disc) the song's female-oriented lyrics were ignored.\n\nThe girl group sound was a favorite of all of the Beatles, but one particular song had particularly daunting repercussions on George Harrison. In 1971 a lawsuit was filed charging Harrison with plagiarism for lifting the melody of the Chiffons' 1962 recording of \"He's So Fine,\" written by Ronnie Mack, and using it in his 1970 Number One record \"My Sweet Lord.\" In 1976 Harrison was found guilty of \"subconsciously\" copying the Chiffons' song. Damages weren't assessed until 1981, but by that time, Harrison's former manager Allen Klein had arranged to purchase the rights to \"He's So Fine\" from its publisher, Bright Tunes. (Ultimately, Harrison was able to acquire the song himself.) In 1975, the Chiffons, in the spirit of good fun, recorded their own version of \"My Sweet Lord.\" Harrison, in turn, recorded \"This Song,\" which satirically poked fun at the entire incident.\n\nDuring his first visit to the U.S. in 1963 (visiting his sister Louise in Benton, Illinois), George stumbled onto an album by James Ray on the tiny Caprice label that included the recent hit \"If You Gotta Make a Fool of Somebody.\" Ray was a homeless street singer who was living on a Washington, D.C. rooftop when he was discovered by songwriter Rudy Clark. Ray's story intrigued George, who brought the LP back to the Beatles. \"If You Gotta Make a Fool of Somebody\" made it into the group's repertoire, inspiring another British group, Freddie and the Dreamers, to record it. Their version made it into the Top 10 in the UK. In 1987, George recorded another song from Ray's album, \"Got My Mind Set on You\" (originally titled \"I've Got My Mind Set on You\"), which became his third Number One single. But this came too late for the diminutive Ray, who had succumbed to a drug overdose many years before.\n\nJohn Lennon was probably responsible for introducing the band to the music of country soul pioneer and part-time bus driver Arthur Alexander. The Alabama native's first hit \"You Better Move On\" launched Rick Hall's Fame Studios in Muscle Shoals, which paved the way for soul music, a black alternative to Motown during the 1960s, led by stalwarts such as Otis Redding, Aretha Franklin, Wilson Pickett, and Joe Tex. The Rolling Stones nabbed \"You Better Move On\" and put it on their album December's Children (And Everybody's). The Beatles settled for the B side, \"A Shot of Rhythm and Blues,\" which became a staple of their live act along with both sides of Alexander's next single, \"Where Have You Been (All My Life)\" (written by the team of Barry Mann and Cynthia Weil) and \"Soldier of Love.\" They recorded his third single, \"Anna (Go to Him),\" which was issued on their U.S. debut album Introducing the Beatles on Vee-Jay.\n\nLike James Ray, Alexander also died before his reputation caught up with him, of a heart attack at age 53. Even so, his place in history is secure. In addition to the Beatles and the Stones, two of the other Four Horsemen of twentieth century rock also favored his work. Bob Dylan covered Alexander's very first single \"Sally Sue Brown\" on his 1988 album Down in the Groove, and Elvis Presley had a big hit with Dennis Linde's \"Burning Love\" in 1972, which Alexander had recorded the previous year.\n\nBuddy Holly, 1958. (\u00a9 BBC.)\n5\n\nTHE SONGWRITERS\n\nThe package called the Beatles needed one more piece before it could be presented to the music industry as something more than just another talented cover band: songwriting. In those days, and for the previous hundred years of mainstream pop music, there was more of a separation of church and state between songwriters and performers than would later be expected, if not tolerated. The Beatles' success as songwriters effected a tidal change in the nature of the process, relegating the pure songwriter to the sidelines and creating a requirement for songs to come from within a band itself. Before the Beatles, rock 'n' roll bands wrote the occasional song but mainly covered the hits of the day, whether that day occurred last week or twenty years ago. The Beatles had already distinguished themselves from the local pack by their exquisite taste in covering obscure B sides. But with the dawning of the sixties, it was time to take a closer look at what made good songs tick, and more importantly, who wrote them.\n\nAs collectors and connoisseurs, the Beatles were definitely aware of the tiny type printed underneath the title on a 45, which indicated the writer credits. Paul McCartney was probably aware of the publishing credits as well. So it comes as no surprise that the group's favorite songwriter of the era was Chuck Berry. What made Berry's songs different from those by Little Richard was that they were complete, well-crafted songs, able to exist apart from the performance, whether it was an onstage riot act or studio accident. A Little Richard song is inexorably linked with a Little Richard performance; it is hard to think of one without the other. Richard's songs were little more than vehicles for his vocal pyrotechnics, more often than not, of the screaming variety. But Chuck Berry's gift was so unique that dozens of his songs wound up becoming statements of a universal rock 'n' roll coming of age, fit to be sung by anyone wanting to touch that experience. The Beatles, of course, had dabbled in writing songs, with Paul in particular being fairly prolific. But these were, for the most part, perfunctory ditties that became as transitory as grade school crushes. Berry was the one who first put the idea of a song as a work of art into their heads. Surely it had been there already. The Great American Songbook was filled with the efforts of the Gershwins, Berlins, Porters, and Kerns of their parents' generation. But those were mainly theater writers who wrote for characters on the stage or screen. In Berry's world, the stage was a lot closer to the street, and the theater was all of teenage America. This was a vision the Beatles were eager to absorb.\n\nComing from a background in the blues, rooted in its structure, and a storyteller by nature, Berry understood the three-cornered triangle of rock 'n' roll's immediate antecedents\u2014R&B, country music, and pop\u2014better than anyone except Elvis Presley. And since Berry was a writer as well as a performer (and a mean guitarist), he was arguably at least as influential a figure (if not moreso) than Presley. His breakthrough 1955 hit \"Maybellene\" was based on a 1938 recording of \"Ida Red\" by western swing's Bob Wills. The song was a traditional fiddle tune that was covered a year later by Grand Ole Opry star Roy Acuff. Originally re-naming it \"Ida May,\" Berry was instructed by label owner and producer Leonard Chess to contemporize the title. Berry's pianist Johnnie Johnson credits Chess with suggesting the name \"Maybellene,\" after spying a mascara box on the floor of the studio. A former hairdresser himself, Berry was no particular fan of the song, but saw an opening and was not about to run away from it. \"Maybellene\" told of a hot chick and a hot car, two of the most enduring truisms in the teenage experience. Songwriting co-credit was initially given to Alan Freed and Russ Fratto, who helped promote the record, but their dubious contributions were eventually deleted from record labels. Although the Beatles never covered \"Maybellene,\" Elvis Presley performed it on the popular Louisiana Hayride radio and television programs. It was also covered by rocker Gene Vincent, pop singer Tommy Sands, up-and-coming country star Marty Robbins (whose version was misspelled \"Maybelline\"), and wacky English rockabilly icon Vince Taylor.\n\nThe Beatles preferred \"Memphis, Tennessee,\" a story of a divorced man who misses his four-year-old daughter, which they used at their ill-fated Decca Records audition in 1962. They especially liked Berry's songs that celebrated rock 'n' roll, such as \"Roll Over, Beethoven,\" \"Rock and Roll Music,\" and \"Johnny B. Goode,\" the latter concerning a country boy similar to Berry himself who could play a guitar \"just like ringing a bell.\" \"Johnny B. Goode\" was covered by one of the Beatles' favorite local bands, Joe Brown and the Bruvvers, as well as by Elvis, Buddy Holly, and Jerry Lee Lewis. They would share \"Carol\" with the Rolling Stones and \"Too Much Monkey Business\" with the Kinks (who also used \"Beautiful Delilah\" to kick off their debut album). With its frenetic wordplay and cynical worldview, \"Too Much Monkey Business\" was a song that set the stage for Bob Dylan's \"Subterranean Homesick Blues\" as well as Elvis Costello's \"Pump It Up\" and REM's \"It's the End of the World as We Know It (and I Feel Fine).\"\n\nRock 'n' roll icon Jerry Lee Lewis favored Berry's \"No Particular Place to Go.\" His partner in crime, Elvis Presley, did a bravura job on \"Promised Land.\" In England, the Animals went for \"Around and Around\" and \"Memphis, Tennessee\" on their debut album. The Stones did \"Around and Around\" on 12x5 and \"I'm Talking About You\" and \"You Can't Catch Me\" on December's Children. Rolling Stone bassist Bill Wyman included \"You Never Can Tell\" on his 2004 live album and DVD Let the Good Times Roll. John Lennon put \"You Can't Catch Me\" and \"Sweet Little Sixteen\" on his 1975 album Rock and Roll and the Yardbirds took on \"Let It Rock.\"\n\nNot surprisingly, the only American rock 'n' roll female to show a true affinity for the work of Chuck Berry was Wanda Jackson, who toured with Elvis during the late 1950s. True to her wild and wicked but perhaps partly manufactured persona, Wanda recorded \"Brown Eyed Handsome Man\" for her 1961 album Right or Wrong. A recording of the song by Buddy Holly reached the Top 5 in the UK three years after the singer's death in a plane crash.\n\nHolly had been a fixture on the British charts since \"That'll Be the Day\" hit Number One in 1957. An American original from Lubbock, Texas, Holly based the song on a catchphrase from the John Wayne movie The Seachers. After his death in 1959, Holly remained popular in England where eighteen sides made the charts, none of which came close to becoming a hit in the U.S. Included among these are some of his best, albeit overlooked classics, including \"Peggy Sue Got Married,\" \"Heartbeat,\" \"True Love Ways,\" \"Listen to Me,\" \"Wishing,\" and \"Love's Made a Fool of You.\"\n\nThe Beatles were aware of Holly from the very beginning. As the Quarrymen, their first amateur record in 1958 consisted of a version of \"That'll Be the Day\" backed with a McCartney-Harrison original titled \"In Spite of All the Danger.\" Holly's acerbic attitude and horn-rimmed glasses appealed to John, and his uncanny melodic sense to Paul. If Chuck Berry taught them wordplay and storytelling, Holly taught them how to write a wrenching love song with a lonesome edge and a rockabilly twang, a sort of blending of Roy Orbison and the Everly Brothers. Plus, he had his own band, the Crickets. It was a package the lads found irresistible. They named themselves the Beatles in honor of Holly's group, with tunes like \"Everyday,\" \"It's So Easy,\" and \"Maybe Baby\" becoming staples in their nightly regimen. When they auditioned for Decca, one of the songs they performed was Holly's \"Cryin', Waitin,' Hopin',\" the B side of \"Peggy Sue Got Married,\" one of his finest posthumous singles.\n\nOnly one Buddy Holly song appeared on a Beatles album, the haunting \"Words of Love,\" which had flopped as a single for Buddy in 1957. It appears on Beatles for Sale, which was released in the UK in December 1964. (The Beatles also recorded the Holly B side \"Mailman, Bring Me No More Blues\" during the infamous 1969 Get Back sessions.) Over in England, Peter and Gordon were paying attention. In 1965 they released Holly's \"True Love Ways\" as a single, which made the Top 20.\n\nAnother Liverpool band, the Searchers, beat them both to the gate when they recorded Holly's \"Listen to Me\" in 1963 for their debut album Sugar and Spice. Fab friend and touring companion Jackie De Shannon, a pretty good writer herself, covered \"Maybe Baby\" early in her career. Another of the Beatles' favorite opening acts was Tommy Roe, who was virtually a Holly clone.\n\nBobby Vee also enjoyed a nice career based on the Holly sound. In the aftermath of Holly's death on February 3, 1959, the fifteen-year-old Vee and his band the Shadows filled in for the singer at Holly's next scheduled tour stop in Moorhead, Minnesota. This led to a recording contract with Soma Records and the release of Vee's first single, \"Suzie Baby.\" In 1962 Vee recorded Bobby Vee Meets the Crickets with surviving members of Holly's band, on which he covered \"Peggy Sue\" and \"Well... All Right.\"\n\nWanda Jackson put her own version of \"It Doesn't Matter Anymore\" on her 1959 album Rockin' with Wanda. This song, Holly's last U.S. hit, was written by Paul Anka, entering the charts three weeks after Holly's plane went down. \"Well... All Right\" and eleven other Holly tunes appear on Skeeter Davis Sings Buddy Holly (RCA Victor, 1967) on which Holly's Clear Lake bus mate, Waylon Jennings, plays guitar. In 2002, Connie Francis recorded Connie Sings Buddy for the UK Pye label.\n\nEnglish acts copied Holly less but appreciated him more. The Rolling Stones nabbed \"Not Fade Away,\" the B side of \"Oh Boy,\" for their first single; Freddie and the Dreamers put \"It Doesn't Matter Anymore\" and \"Early in the Morning\" on their first album; Johnny Kidd was a fan of \"Oh Boy\"; Billy Fury covered \"Maybe Baby\"; Herman's Hermit's performed \"Heartbeat\" on tour and put it on their second album; and the Hollies (who denied naming themselves after Holly) did an album of his songs as well. Cliff Richard and Peter and Gordon came late to the party, in some sense covering the Beatles covering Holly. Hank Marvin, Richard's guitarist in the Shadows, did an instrumental set of Holly cover tunes, and John Lennon put \"Peggy Sue\" on his Rock and Roll album.\n\nBut Paul McCartney had the last laugh on everyone. He bought Buddy Holly's song catalog.\n\nWhile the Beatles respected not only performers who could write but writers like Holly who performed in their own bands, they did not scorn songwriters who punched a time clock, worked in cubicles, wrote on deadline, or went home on the subway to their husband or wife after work. They were hip to the Brill Building and everything it stood for.\n\nThe Brill Building was where many of the best songs in popular music history had been written; opening its doors at 1619 Broadway in 1931. When Tin Pan Alley relocated from Manhattan's mid-twenties to Broadway's forties, the Brill Building, located at Forty-ninth and Broadway, became the friendly confines into which generations of publishing companies came to occupy one- and two-man offices, living on top of each other, tenement style, much like early songwriters in the ghettos of the Lower East Side at the turn of the twentieth century. Amid its rattle and roll, neophytes and journeymen pitched their demos, often starting at the top floor and working their way down until they got a sale or hit the street. As the big band era gave way to R&B, early rock 'n' roll was foreshadowed by the likes of Jesse Stone, Charlie Singleton and Rosemary McCoy, Rudy Tombs, and others. Alan Freed had an office in the building. Elvis Presley had his publishing company here. Relocating to the Brill Building from Los Angeles were Jerry Leiber and Mike Stoller, who had \"Hound Dog,\" \"Jailhouse Rock,\" and the entire Coasters catalog in their back pockets. Former blues singer Doc Pomus, who wrote \"Lonely Avenue\" for Ray Charles, teamed up with Mort Shuman to write for the Drifters, Dion and the Belmonts, Jimmy Clanton, and Bobby Rydell.\n\n\"Once we wrote two songs for Bobby Rydell,\" Doc Pomus told me. \"His A&R man rejected one of them. Now, there was a rock and roll show and we went to it and Jimmy Clanton was working there. I told Jimmy, 'We wrote a song for you.' The song was originally titled 'Go Bobby Go,' so we changed it to 'Go Jimmy Go.' Somewhere there's a tape of Bobby Rydell singing 'Go Bobby Go.'\n\n\"I don't even remember writing 'Teenager in Love,' Doc said. \"It was an assignment. We had a song that we'd already written called 'Teenager in Love' and Dion liked the lyric to it and wanted us to change the melody. So we changed the melody.\n\n\"I met John Lennon at a BMI dinner; in fact we spent the whole dinner together. One of the biggest kicks I had was when Lennon told me that one of the first songs the Beatles ever did was 'Lonely Avenue.' And he was telling me that originally all they wanted to do was reach a point, like Morty and myself or like Carole King and Gerry Goffin, where they could make enough money to survive writing songs.\"\n\nLocated a few blocks up Broadway, Don Kirshner's Aldon Music empire was considered synonymous with the Brill Building sound, housing some of the top songwriting teams in the business, among them Gerry Goffin and Carole King, whose first Number One was \"Will You Love Me Tomorrow\" by the Shirelles. Goffin himself was a devoted Chuck Berry fan. \"I think Chuck Berry wrote the best lyrics to describe what it was like in teenage America in those days,\" Gerry said. \"I think his was a more accurate picture than mine. I didn't realize how good his lyrics were until I got a job and had to write them every day.\"\n\nIronically, it was the success of the Beatles and their ingrown songwriting prowess that paved the way for a complete re-ordering of the hit making process. \"First it was just sort of pop lyrics,\" said Goffin. \"Then all of a sudden poetry got involved.\"\n\nBut it would be misleading to state that the coming of the Beatles immediately wiped out the writing community at the Brill Building. If anything, the Beatles and their contemporaries who followed their every move shined a light on New York City writers, much as they had re-introduced Chuck Berry to the rock and roll cognoscenti, along with Carl Perkins, Arthur Alexander, and Gene Vincent. Led by the Beatles' example, England became a vast new marketplace for tailor-made pop songs. The Beatles appreciated Goffin and King's slightly off-kilter, underappreciated sides, including \"Chains\" by the Cookies, which barely scraped the Top 20, and \"Don't Ever Change,\" a Top 5 UK hit for the Crickets that never charted in the U.S. At their Decca audition they sang Bobby Vee's \"Take Good Care of My Baby\" and gave a shout to Goffin and King's babysitter Little Eva when they added \"Keep Your Hands Off My Baby\" to their act as well.\n\nIn 1964, at the height of Beatlemania, Herman's Hermits had a hit with Goffin and King's \"I'm Into Something Good,\" which was originally recorded by Earl-Jean, a sister of one of the Cookies. Dusty Springfield broke through with \"Wishin' and Hopin,'\" written by Burt Bacharach and Hal David; the magisterial Bacharach also wrote \"Baby It's You\" for the Shirelles with David's brother Mack and Shirelles producer Luther Dixon; Manfred Mann hit the Top 5 with the Jeff Barry \/ Ellie Greenwich tune \"Do Wah Diddy Diddy,\" and the Animals went for the Barry Mann \/ Cynthia Weil classic \"We Gotta Get Out of This Place.\" Herman's Hermits revived \"Silhouettes,\" a Bob Crewe \/ Frank Slay tune much beloved by John Lennon that was once a hit for the Rays. Lennon said it influenced the writing of \"No Reply.\" Peter and Gordon plucked Phil Spector's \"To Know You Is to Love You\" (made famous by the Teddy Bears as \"To Know Him Is to Love Him\") from a Beatles set list for their 1965 version, as they had done with Buddy Holly's \"True Love Ways\" a few months earlier.\n\nThe Searchers were especially partial to the Brill Building oeuvre. They broke through in England with \"Sweets for My Sweet,\" a Pomus and Shuman hit for the Drifters. A year later they covered another one, \"I Count the Tears.\" They did the Barry and Greenwich rocker \"Da Doo Ron Ron,\" originally a hit for the Crystals. They took on the Bacharach and David tune \"This Empty Place\" right after Dionne Warwick was done with it. Their first hit was \"Needles and Pins\" in 1964, written by Sonny Bono and Jack Nitzsche, part of Phil Spector's Gold Star Studio team and introduced by Jackie De Shannon a year before. The Searchers returned the favor, bringing De Shannon's \"When You Walk in the Room\" to the Top 40 and later recording her \"Can't Help Forgiving You.\" Their biggest hit was Leiber and Stoller's \"Love Potion Number Nine,\" originally a hit for the Clovers. They came back with two more Leiber and Stoller numbers: \"Some Other Guy,\" a longtime Beatles favorite, and \"Tricky Dicky,\" in addition to \"Stand by Me,\" written by Leiber and Stoller with Ben E. King. Marianne Faithfull chose Jackie De Shannon's \"Come and Stay with Me\" as a follow-up to \"As Tears Go By,\" penned by Mick Jagger and Keith Richards.\n\nBy 1964, the team of Leiber and Stoller, who virtually invented the rock 'n' roll sound the Beatles were weaned on, had just about run its course. They had by now been reduced to writing tepid movie songs for Elvis Presley like \"Bossa Nova Baby\" from Fun in Acapulco (1963). Covers of their \"Spanish Harlem\" and \"Lucky Lips\" by Cliff Richard (1962) weren't making it either. A posthumous take on \"(You're So Square) Baby I Don't Care\" featuring Buddy Holly flopped in the U.S. The Coasters, who the team had nurtured since they were the Robins in 1955, hadn't had a hit since 1961's \"Little Egypt\" (later covered by Elvis).\n\nThe Beatles had used \"Searchin'\" and \"Three Cool Cats,\" (the B side of \"Charlie Brown\") at their Decca audition. They performed \"Young Blood\" on the BBC, \"Some Other Guy\" (originally a hit for Richard Barrett) at the Cavern Club, and \"Ruby Baby\" (the Drifters hit) with Tony Sheridan. They brought down the house with Little Richard's version of \"Kansas City,\" especially when they played that Midwestern city. But by 1965 the Beatles were well into their own songs; in many minds, Lennon and McCartney had by now replaced Leiber and Stoller as the preeminent rock 'n' roll songwriting team in the business.\n\nIn 1965 Leiber and Stoller produced \"Go Now\" by Bessie Banks. (The Moody Blues would cover it later that year, launching their own career.) \"Down Home Girl,\" a flop for Alvin Robinson, was covered by the Rolling Stones. (They had been running their own label, Red Bird Records, home to the ultimate New York City girl group the Shangri-Las, but in 1966 they sold their interest in it.) A symbol of their ennui came in 1969 when Peggy Lee's \"Is That All There Is\" became their last major hit, by which time Lennon and McCartney's own careers as a team was sputtering as well.\n\nRoger Moore as James Bond, 1973. (\u00a9 MGM\/UA.)\n6\n\nGEORGE MARTIN\n\nWhen the Beatles first met George Martin, no one could have been a more unlikely candidate to produce their records. At 36, he was nearly twice their ages. From his patrician upbringing to his choice of neckties, the Parlophone Records producer and talent scout had a style and stature that was diametrically opposed to the young Liverpudlian upstarts. But Martin and the erstwhile super group (especially John Lennon) clicked, partly because of a shared passion for the lunacy of the BBC's The Goon Show.\n\nMartin's involvement with the Goons wasn't just as a fan and a devotee; he'd actually produced records for the Goons, including their classic album The Bridge on the River Wye, which spoofed the Academy Award-winning movie The Bridge on the River Kwai. Martin was a friend of Spike Milligan, the show's creative voice, who he met through Peter Sellers (whose early Parlophone albums Martin also produced). The breakout single from Sellers' second album was a parody of Lonnie Donegan's recording of \"Puttin' on the Style,\" a nineteenth-century pop song that was in the early repertoire of the skiffle-mad Quarrymen. Later on, Sellers did one of the first covers of \"A Hard Day's Night,\" performed in the style of Sir Laurence Olivier as Richard III.\n\nThe half-hour wacky weekly Goon Show had been a staple in sophisticated British households since its immaculate conception in 1951 and was a forerunner to an entire new wave of British comedy, inspiring Peter Cook and Dudley Moore to go Beyond the Fringe. (Martin produced their album as well in 1961.)\n\nThe affliction with the Goons immediately endeared Martin to John Lennon, who was just entering adolescence at the time the Goons took to the airwaves, and whose view of reality was inalterably shaped by their mind-expanding programs. The myriad sounds rattling around inside Spike Milligan's mind were accomplished through the insidious use of ingenious sound effects, engineered through a mastery of echo, reverb, multiple edits, and playing with recording tape speeds, all of which would become hallmarks of the Beatles' studio repertoire, and especially, John Lennon's mad, unspooling, and acid-soaked creative vision. To be a Goon Show freak turned out to an absolute requirement for producing the Beatles, as the group grew and changed, and as their studio techniques (and minds) expanded.\n\nEver since he came to Parlophone, George Martin had been cutting his teeth on comedy records. His earliest credits include Peter Ustinov's 1955 LP Mock Mozart, which involved fooling around with tape effects and overdubs. In addition to Beyond the Fringe, Martin recorded David Frost's satirical TV show That Was the Week That Was. He also directed shows at Peter Cook's trailblazing nightclub, the Establishment, where Australian jet-setter Dame Edna Everage (Barry Humphries) was known to perform alongside visiting U.S. personages like Lenny Bruce. Martin worked with satirists Michael Flanders and Donald Swann, who took on any and all subjects for their comedy songs (much as the droll, Harvard-educated Tom Lehrer did in the U.S.), producing their albums At the Drop of a Hat and its follow up, At the Drop of Another Hat. A rabid fan of the cinema, Martin worked with actress-singer Joan Sims, a regular in twenty-four Carry On films, and produced her two best-known singles: \"Hurry Up Gran \/ Oh Not Again Ken\" and \"Spring Song \/ Men.\"\n\nMartin's appreciation for comedy and studio wizardry may have found a kindred spirit in John Lennon, but his classical training (on piano and oboe) and fondness for film scores made an impression on the more tradition-minded Paul McCartney. As a fan of Johnny Dankworth's 1960 score for Saturday Night and Sunday Morning (another \"kitchen sink\" drama \u00e0 la A Taste of Honey), Martin produced Dankworth's \"Experiments with Mice,\" jazzy variations on \"Three Blind Mice\" that hit the Top 10 in the UK in 1956. In 1961 Dankworth wrote the theme for the popular British TV spy series The Avengers. That same year Martin produced Dankworth's first Numer One record, a version of the jazz standard \"You're Driving Me Crazy,\" performed by a nine-piece trad band called the Temperance Seven, which featured vocalist Paul Macdowell. The Dankworth-Martin association continued in 1976, when Martin produced the album Born on a Friday for Dankworth's wife, jazz singer Cleo Laine.\n\nMartin was equally adept at international folk music, producing the 1955 hit \"The Bluebell Polka,\" by the prolific Scottish accordion player Jimmy Shand. Australian troubadour Rolf Harris had two Top 10 hits produced by Martin: \"Sun Arise\" and \"Tie Me Kangaroo Down, Sport,\" the latter song crossing over to Number 3 in the U.S. in 1963. Martin's talents also extended to skiffle music, a British-born melding of jazz and traditional folk music. In 1957 he produced the Top 10 hit \"Don't You Rock Me Daddy-O\" for the Vipers, the predominant skiffle group of the era. (Years later, Paul McCartney would record one of their tunes, \"No Other Baby,\" on his revival album, Run Devil Run.) At one time, the Vipers included Jet Harris and Tony Meehan, who along with guitarist Hank Marvin would achieve fame as Cliff Richard's band the Shadows, which would eventually move out of his shadow and go out on their own.\n\nThe Shadows' first hit was a cover of a 1954 instrumental written by Bert Weedon called \"Apache\" that had been recorded by another Martin artist named Jerry Lordan. Lordan's version failed to kick up much action, but in 1960 the Shadows' cover spent five weeks at Number One in the UK. In America, Danish guitarist Jorgen Ingmann's version reached Number Two the following year. (Lordan wrote another Number One record for the Shadows, \"Wonderful Land.\") When Harris and Meehan struck out on their own, they did so with Lordan's surf-sounding instrumental \"Diamonds,\" which topped the UK charts for three weeks in 1963. The Beatles would also use \"Diamonds\" extensively in their live act.\n\nThe one blank on George Martin's resume was rock 'n' roll. He passed on Tommy Steele, which cost him a potential seven Top 10 records, preferring his backup band, the Vipers. To correct this mistake he signed Jim Dale, who quit the business to become a comic and then an actor after having one hit, \"Be My Girl.\" Jerry Lordan became more successful as a writer. Martin's big signing of 1961, Shane Fenton and the Fentones, went bust after their lead singer died just after the group sent their audition tape to the BBC; in natural fashion, the BBC was oblivious to this and asked them to audition in person. The surviving group members decided to honor Fenton (at the request of his grieving mom) by not breaking up and recruited the band's roadie, Bernard Jewry, to assume the lead role at the audition. The audition was a success and in short order they were recording for EMI (which released a Best of the Fentones album in 2003). After their single \"I'm a Moody Guy\" hit the Top 20 in 1962, further singles like \"Walk Away\" and \"It's All Over Now\" failed to sustain their momentum. At the beginning of the next decade, Jewry (aka Shane Fenton II) became Alvin Stardust and achieved considerably more fame.\n\nGeorge Martin's track record showed that he was not just eclectic; he was an authority on a variety of musical styles in addition to being a schooled musician; hardly the empty corporate suit perceived by the gang of scruffy punks from Liverpool. What appealed to all of the Beatles initially was the simple fact that he was willing to sign them. As they were surprised to find out, Martin could also recognize a hit single when he heard one; however, this was not the case with the Beatles' first release. Martin favored a tune called \"How Do You Do It?\" a song that was written by Mitch Murray, whose collaborative efforts with Peter Callander would result in a string of hits for Merseybeat artists including Freddie and the Dreamers, Gerry and the Pacemakers, Cliff Richard and the Shadows, and Georgie Fame. The Beatles, on the other hand, were unanimously in favor of using their own song \"Love Me Do.\"\n\nAs predisposed as the Beatles may have been to cooperate with this sophisticated authority figure, they were still cocky enough to trust their own instincts on matters musical. Brian Epstein was permitted to dress them in suits and ties and he also prevailed upon them to quit drinking and popping pills on stage. But when it came to issuing their first single, they drew the line. In addition to objecting to their choice of material, Martin found Ringo Starr wanting as a drummer, and had his own man Andy White in the wings, ready to sit in. In this case, the band relented. Although the first pressing of the single version of \"Love Me Do\" featured Ringo on drums, later editions, including the album version, included White on drums and Ringo on tambourine.\n\nWhen it came time to record their second single a few weeks later, order was restored to the universe with the Orbison-Everly-Holly-inspired \"Please Please Me,\" featuring John Lennon playing harmonica in a style he had picked up from Delbert McClinton (\"Hey, Baby!\"). Beginning with this record, Ringo played drums from then on. Martin had little to do but congratulate the team on what he accurately predicted would be their first Number One hit, a prediction that was born out on two UK music charts, although the record didn't reach the U.S. listings until the onslaught of Beatlemania in 1964.\n\nBut Martin didn't just sit in the dugout like the manager of the New York Yankees as the Beatles coasted to twenty-seven world championships. He still made most of the musical decisions, remaining in the middle of the scrum, and demonstrating his amazing versatility even after the Beatles started flexing their studio muscles and questioning every rule in the book. Martin had to, at one moment, keep up with a frantic John Lennon on Chuck Berry's \"Rock and Roll\" and convince Paul McCartney to employ a string quartet on \"Yesterday\" at another. On \"Yellow Submarine,\" Martin was in his full Goon Show glory, with bells, whistles, chains, and tin cans galore in the foreground while John and Paul played Peter Sellers and Spike Milligan in the background. On \"Tomorrow Never Knows,\" Martin and his brilliant engineer Geoff Emerick somehow managed to satisfy Lennon's request for the sound of a thousand Tibetan monks without going over budget. On \"In My Life\" he even pulled a baroque harpsichord sound out of his arsenal by speeding up a piano (playing it himself). For \"Eleanor Rigby\" he flexed his scoring chops, inspired by Bernard Hermann's Psycho to arrange an accompaniment for string octet. For another session, he brought in a French horn (\"For No One\"), and when McCartney heard David Mason play piccolo trumpet on a Royal Philharmonic Orchestra broadcast of Bach's \"Brandenburg Concerto No. 2,\" Martin, who was friends with Mason, brought him in to overdub a solo on \"Penny Lane.\" When John Lennon wanted specific sounds for \"Strawberry Fields Forever\" and \"I Am the Walrus,\" Martin, without a second thought for his own safety, jumped inside of Lennon's acid-expanded mind, found what was rattling around in there, and made an arrangement of it.\n\nAfter the Beatles broke up, Martin was preparing a score for the James Bond film Live and Let Die when an opportunity arose to write the opening title theme. Martin immediately called Paul McCartney, just as he had done with theme songs for other Bond films when he hired two other previous clients, Shirley Bassey (\"Goldfinger\") and Matt Monro (\"From Russia with Love\").\n\nBrian Epstein saw to it that \"How Do You Do It?\" found a home with another of his clients, Gerry and the Pacemakers, whose version, also under George Martin's supervision, spent three weeks at Number One in 1963. Epstein kept Martin busy with his other main acts, Billy J. Kramer and the Dakotas and ex-hat check girl Cilla Black. Compared to the enormous, riveting, soap operatic changes of everyday life with the Beatles, however, producing these other acts was probably as interesting as sleepwalking to Martin. His association with middle-of-the-road crooner Matt Monro had to be equally humdrum. But Martin probably realized something of a career wet dream when in 1964 he got the chance to work with British singing legend Alma Cogan, producing her song \"It's You.\" Although it wasn't a hit, Martin produced a pair of Beatle cover songs for her to record: \"Eight Days a Week,\" one of her finest performances, and \"Help!\" which was on the B side. In return, Cogan befriended the Beatles and invited them to her lavish parties, giving them a taste of what lay in wait on the other side of a few more hit singles.\n\nBy the mid-sixties, drugs and intramural tensions were becoming more of an influence on the Beatles than Martin. But his leaving EMI was due to their failure to financially acknowledge his success with the Beatles. He then opened his own AIR studios. He still worked with his main clients on landmark achievements like the endlessly creative Sgt. Pepper's Lonely Hearts Club Band, but now he had another, slightly more prosperous revenue stream. The first act he signed there was a white R&B contingent called the Action, who were routinely mistaken for black due to their covers of tunes by the Temptations (\"Since I Lost My Baby\"), the Marvelettes (\"I'll Keep Holding On\"), and Bob and Earl (\"Harlem Shuffle\"). Martin next produced the duo of David and Jonathan (Roger Greenaway and Roger Cook), who would go on to write such classics as \"Long Cool Woman (In a Black Dress)\" for the Hollies and \"I'd Like to Teach the World to Sing (In Perfect Harmony)\" for the New Seekers. Under their own alias, the best they could come up with was the 1966 period piece \"Lovers of the World Unite,\" which went to Number 7 in the UK. Ironically, their biggest hit was a cover of Lennon and McCartney's \"Michelle,\" which made the Top 20 in the U.S.\n\nIn 1967 Martin's affinity for novelties asserted itself again with his production of the Ivor Cutler Trio's album Ludo, which featured such Goon-like titles like \"Good Morning! How Are You? Shut Up!\" and \"A Suck of My Thumb.\" A favorite of John Lennon's, Ivor Cutler was aptly cast as Mr. Bloodvessel in the Beatles television film Magical Mystery Tour. During the protracted squabbles interrupted by the occasional bursts of genius that characterized \"The White Album\" and Abbey Road, Martin couldn't escape the Beatles connection. For a time, the Scaffold, a trio that included Paul McCartney's brother Mike (using the pseudonym Mike McGear), was the house band at the Establishment, a London nightclub co-founded by Peter Cook. (The Bonzo Dog Doo-Dah Band performed there as well.) The Scaffold had their first hit \"Thank U Very Much\" in 1968, followed by \"Lily the Pink\" (an adaptation of the folk song \"The Ballad of Lydia Pinkham\"), which went to Number One, aided and abetted by Jack Bruce on bass, and Elton John and Graham Nash on backing vocals. Between 1969 and 1971 Martin produced records by Edwards Hand, a Beatlesque group consisting of Rod Edwards and Roger Hand. Three albums resulted, including the eponymous Edwards Hand, which was followed by Stranded and Rainshine, achieving little more than a vision of Beatles lite.\n\nAfter shepherding Ringo Starr's Sentimental Journey, a solo album of pop standards, to the marketplace in 1970, Martin continued along his eclectic way, with excursions into roots rock (Seatrain's 2nd Album and Marblehead Messenger), new age (The Paul Winter Consort's Icarus), classical vocal harmonies (The King's Singers, whose repertoire ranged from eighteenth-century oratorios and Christmas songs to Flanders and Swann and Lennon-McCartney covers), to the most classical of jazz rock (Apocalypse by John McLaughlin and the Mahavishnu Orchestra, featuring Jean-Luc Ponty on violin and the London Symphony Orchestra, conducted by Michael Tilson Thomas).\n\nIn 1974 Martin took over producing records by America, a London-based trio of U.S. Air Force brats that had their first hit in 1971 with the Neil Young-influenced \"A Horse with No Name.\" The song went to Number 3 in England, and a year later, it topped the U.S. charts. Three albums later, the group, consisting of Dewey Bunnell, Gerry Beckley, and Dan Peek, decided to go in a different direction and hired Martin, who added strings and brass to their next four albums: Holiday, Hearts, Hideaway, and Harbor. The result was a bunch of U.S. chart successes that included such Beatle-influenced hits as \"Tin Man,\" \"Sister Golden Hair,\" \"Daisy Jane,\" and \"Today's the Day.\" Ironically, none of these songs cracked the UK Top 40, illustrating perhaps the record-buying audience's desire to move on from the Beatlesque to something a little more reflective of those angry times. By 1977 the punk rock sound of the Damned, the Sex Pistols, and the Clash would be more to their liking.\n\nPunk was not a sound that appealed to George Martin. He took refuge in the other dominant genre of the 1970s, arena-based heavy metal guitar-driven pyrotechnics. As tasteful as he was, he turned for his guitar fix to a master of the form, Jeff Beck. Originally with the seminal British heavy metal band the Yardbirds, which at one time also featured Jimmy Page and Eric Clapton, Beck had been fronting his own group since 1968's bluesy Truth (featuring Rod Stewart on lead vocals). With Blow by Blow (1974) and Wired (1975), he burnished his reputation as an instrumentalist by expanding into the realm of fusion. Standout tracks from Blow by Blow include Stevie Wonder's \"Cause We've Ended as Lovers\" and keyboardist Max Middleton's \"Freeway Jam.\" Martin revived his arranging chops with stellar work on \"Scatterbrain\" (not the 1939 big band standard preferred by John Lennon's mom) and \"Diamond Dust.\" And guess who suggested the Lennon and McCartney cover of \"She's a Woman\"? For Wired, Beck brought Jan Hammer and Narada Michael Walden in from the Mahavishnu Orchestra; the Charles Mingus classic \"Goodbye Pork Pie Hat\" would become a permanent part of Beck's repertoire.\n\nMartin worked with three songwriters in the mid-to-late seventies, only one of whom broke through to performing artist glory. The group American Flyer was a quartet consisting of \u00e9migr\u00e9s from other rock groups including Craig Fuller (Pure Prairie League), Steve Katz (Blood, Sweat and Tears), Doug Yule (Velvet Underground), and Eric Kaz (Blues Magoos). Kaz's signal claim to fame was the torch song \"Love Has No Pride,\" which he co-wrote with Libby Titus. It was recorded by Bonnie Raitt (1972), Linda Ronstadt (1973), and Tracy Nelson (1974), but failed to transform American Flyer into anything more than a platform for future demos.\n\nJimmy Webb (\"MacArthur Park,\" \"Wichita Lineman,\" \"Galveston,\" \"By the Time I Get to Phoenix,\" and many others) was eagerly looking forward to following writers like Randy Newman and Leonard Cohen into the singer-songwriter niche, but an ill-conceived collection of his early demos released in 1968 was trashed by critics, thwarting his efforts. After an absence of three years, Webb emerged with his sixth album in 1977, titled El Mirage, which was produced by George Martin. With various superstars dropping in to lend support, Webb put together as fine a track list as ever, but still failed to cross over. A song from that album, \"The Highwayman,\" became a huge country hit for the super group of Willie Nelson, Waylon Jennings, Johnny Cash, and Kris Kristofferson, calling themselves the Highwaymen. (The name reflected their constant touring and outlaw spirit.) Webb's \"The Moon's a Harsh Mistress\" (named for a science fiction novel by Robert Heinlein) is one of his greatest songs; an especially haunting version was recorded by Linda Ronstadt. And any opportunity to hear his venerable gem \"P. F. Sloan\" again is a moment worth savoring.\n\nNeil Sedaka started a successful singing career in the late 1950s under the tutelage of Brill Building baron (once removed to 1650 Broadway) Don Kirshner. But in 1963, after a four-year run on the Top 10 (about average for teen idols in those days), he was considered washed up at the age of 23. Undaunted, he continued to record, write, and tour the UK and Australia until 1971 when his top ten hit of 1959, \"Oh! Carol\" (written about Carole King, whose Tapestry LP was becoming a worldwide sensation) was reissued in the UK and he subsequently moved his family: lock, stock, and housekeeper Mary, to London. The albums Emergence and Solitaire, released in 1971 and 1972, respectively, started his comeback. In 1974, Sedaka's Back was released on Elton John's startup Rocket label (soon to be home to Kiki Dee). The album produced a Number One hit (\"Laughter in the Rain\") as well as the original version of the Sedaka-Greenfield composition \"Love Will Keep Us Together,\" which hit Number One in 1975 as recorded by the Captain and Tennille. George Martin produced Sedaka's 1977 album A Song, which was recorded for Elektra. Although it was a classy affair, it yielded only one chart single, \"Amarillo.\"\n\nIn the early 1980s Martin moved restlessly, recording hard rock (No Place to Run by UFO), arena rock (All Shook Up by Cheap Trick), pop-rock (Time Exposure by Little River Band), and new wave experimentalism (Quartet by Ultravox), at the same time reuniting with Paul McCartney on the landmark Grammy-nominated CD Tug of War. Begun toward the end of 1980, production on the album was interrupted for two months after the murder of John Lennon in New York City. Tug of War, which could have been a metaphor for both McCartney and Martin's Beatles experiences, features \"Here Today,\" Paul's tribute to John. Ringo Starr sat in on drums on the second track \"Take It Away,\" which also featured Martin playing piano. The Beatles' longtime hero Carl Perkins was featured on \"Get It,\" but the highlight of the album was undoubtedly Paul's duet with Stevie Wonder on \"Ebony and Ivory,\" a song that topped the charts both in the U.S. and the UK.\n\nAnother indelible moment in a career studded with them occurred in 1997 when George Martin was brought in to produce an update to \"Candle in the Wind,\" Elton John's response to the death of Diana, Princess of Wales in an automobile accident, with Bernie Taupin recrafting his original 1973 lyrics. Martin added a flutist and a string quartet to the arrangement, resulting in a record that became the best-selling single in UK history, with some thirty-three million copies sold worldwide, much of it during the record's first week of release. Although \"Candle in the Wind\" was originally written about tortured pop cultural icon Marilyn Monroe, it could easily have been attributed to the Beatles, George Martin's early discovery, another act whose flame burned briefly but brightly and whose legend will live forever.\n\nBob Dylan, circa 1960s. (\u00a9 CBS, INC.)\n7\n\nFOLK ROCK\n\nOn August 28, 1964, the Beatles had the first of five meetings with Bob Dylan. The initial encounter took place at a suite in New York's Delmonico Hotel where the Beatles were staying. The Dylan summit was a good deal more casual than their first awkward encounter with Elvis Presley would be a year later. Dylan had misheard the lyrics to \"I Want to Hold Your Hand\" (thinking the phrase \"I can't hide\" was \"I get high\"), so when he casually passed John Lennon a joint, it was because he assumed that the Beatles were already into pot. Lennon quickly passed it to Ringo who defied druggy etiquette by smoking the entire joint instead of passing it on. It turned out that although they might have sampled a smoke or two in Hamburg, the Beatles preferred the old fashioned stuff: liquor and pills. Dylan took care of that lapse of hipness immediately, but it wouldn't be the first time in musical history that a slight misinterpretation resulted in the world changing.\n\n\"I think what initially attracted me to the Beatles was the fact that they were using folk changes,\" said Roger McGuinn, founder of the Byrds, in an interview I conducted. \"I'd been listening to Bob Gibson, who used some pretty slick chord progressions for a folk singer. He used a lot of passing chords like the Beatles ended up doing. So when I heard the Beatles. I went, 'Oh, I love those changes.' I loved their harmonies of the fourths and fifths, which were also folk music kind of harmonies, like sea shanties. I think because they had been a skiffle band and because they came from where they did, that music was just in the air there. They were doing their version of the fifties rock 'n' roll\/rockabilly sound and the folk thing combined. That's what it was. When I got to meet them, I found out they didn't know they were doing that. They didn't know how to fingerpick and they didn't play banjos or mandolins or anything. They weren't coming from where I was coming from at all, which I'd given them credit for. I thought they knew all that stuff and were just being real slick about it. But it was just kind of an accident. It was a great accident.\"\n\nFolk rock was already being officially christened in the world outside the locked door of the Delmonico hotel room by the Animals, who had just sent \"House of the Rising Sun\" to the top of the UK chart and were in the process of doing the same in the U.S. with an arrangement cribbed from Dylan's first album. (Dylan, in turn, had borrowed it from Dave Van Ronk, the erstwhile \"Mayor of MacDougal Street,\" who sometimes let Bob crash on his couch.) Roger McGuinn may have caught a whiff of a sea shanty when he heard \"Baby's in Black\" later in the year, but Lennon had already recorded what most people think of as his first Dylan-influenced song, \"I'm a Loser,\" two weeks prior to the meeting at the Delmonico. The Beatles had been aware of Dylan ever since 1963, when George Harrison brought a copy of the Freewheelin' album to Paris for them all to sample. \"I'm a Loser\" was Dylanesque only in comparison to the less word-oriented songs Lennon and McCartney were accustomed to writing. Although the Beatles had already begun the process of reinventing popular music structures, they were still hungering for something more substantial, both musically as well as lyrically. As the leading edge and voice of his generation, Dylan gave the Beatles exactly what they were looking for.\n\nPaul McCartney was so intrigued by how he behaved while on weed that he instructed his road manager to write down everything he said, just in case he should decode the secret of the universe in a folk song. About a month later, he had already started pushing his limits, both vocally and instrumentally, on songs such as \"She's a Woman,\" possibly urged by the widened perceptions he gained on pot but no doubt motivated by the desire to capture a piece of Dylan's college-age audience, an older demographic than the teenyboppers that started Beatlemania in the U.S. that February. John Lennon was a little more internally motivated; it was he who was responsible for slipping the muted phrase \"turn me on when I get lonely\" into the lyrics of \"She's a Woman.\"\n\nDylan had been similarly changed by the Beatles, for reasons relating more to McCartney's mindset than Lennon's. If marijuana unblocked mental barriers in the Beatles' creative process, Dylan, whose drug-fueled free associations were already running rampant, also gained insights about the real world after his encounter with the group. He'd already assimilated this information soon after hearing \"I Want to Hold Your Hand\" on a Colorado radio station. Dylan's realizations about the Beatles' success were confirmed when Billboard's April 4, 1964 issue showed the unprecedented occurrence of five Beatles singles occupying all of the top slots on its best selling singles chart as well as eleven of the Top 100, not including the tribute songs \"We Love You Beatles\" (based on the Conrad Birdie anthem from Bye Bye Birdie) by the Caravelles and \"A Letter to the Beatles\" by the Four Preps. All told, the Beatles placed twenty singles on the Top 100 in 1964 along with six other Lennon and McCartney tunes recorded by Billy J. Kramer and the Dakotas and Cilla Black (who were also managed by Brian Epstein). Although Dylan had been accepted by folk purists, he saw the wider fame and fortune available to him via a singles-buying populace consisting mainly of teenage girls. To Dylan, switching from folk to rock appeared to be an attractive and very doable proposition.\n\nDylan's latest release, Another Side of Bob Dylan, contained several indications of his already waning desire to continue his role as protest music's folk champion, among them \"It Ain't Me Babe,\" \"All I Really Want to Do,\" and the non-protest protest song \"Chimes of Freedom.\" After his confab with the Fab Four in August, Dylan considered the relative lack of commercial success of Another Side and moved swiftly toward embracing electric instruments. A sudden leap to rock 'n' roll was not feasible for Dylan, who didn't want to alienate the hip intelligentsia that made up his fan base, but to the diehard folk elitist he began showing a compromise in his style on his next effort, Bringing It All Back Home. Beginning in January 1965, Dylan appropriated the updated Chicago blues style of Greenwich Village crony John Hammond Jr., the son of the Columbia Records mogul who first recorded him. Dylan also utilized the sterling session men from Hammond's seminal So Many Roads album, including Robbie Robertson, Garth Hudson, and Levon Helm, soon to become members of Dylan's backup group the Band. They were joined by guitarist Michael Bloomfield of the Paul Butterfield Blues Band, who was soon to become infamous as Dylan's whirling dervish electric guitar accomplice at the 1965 Newport Folk Festival.\n\nDylan's producer Tom Wilson had already begun the conversion process of overdubbing rock arrangements on the previously acoustic Wednesday Morning, 3 A.M. by the earnest New York City folkies Simon and Garfunkel. The album juxtaposed folk chestnuts like \"You Can Tell the World\" alongside the Dylan anthem \"The Times They Are A-Changin'\" and Paul Simon's early gems \"Bleecker Street\" and \"He Was My Brother,\" as well as an acoustic version of \"The Sounds of Silence.\" This latter song, when retrofitted with a rock 'n' roll accompaniment a year later, went on to achieve legendary status as a Number One single.\n\nBy the end of 1964, the Beatles were adding blues changes and feedback (\"I Feel Fine\") to their Dylan influence. They were no doubt affected by the burgeoning blues rock scene in England, which included the Animals, the Yardbirds, and the Rolling Stones. Although Paul McCartney added some folk flourishes to \"I'll Follow the Sun,\" the remainder of the Beatles' year was devoted to covers of tunes by their fifties' heroes: Little Richard, Carl Perkins, Buddy Holly, and Chuck Berry.\n\nBerry was also an influence on Dylan, whose 1965 electrified blues rocker \"Subterranean Homesick Blues\" was based on Berry's \"Too Much Monkey Business,\" from the 1957 LP After School Session. (Chuck released St. Louis to Liverpool, his second studio album since getting out of jail, in late 1964.)\n\nThe song would become Dylan's first Top 40 single as an artist, making its debut on the charts in April 1965. Dylan's famous deadpan cue card-flipping pantomime of the song, featuring cameo appearances by poet Allen Ginsberg and Dylan's road manager Bob Neuwirth, served as an introduction to D. A. Pennebaker's 1967 documentary Don't Look Back, which tracked Dylan's 1965 tour of the UK.\n\nWhile Dylan was moving inexorably toward rock, the Beatles were back in the studio in February, recording their first all-acoustic number \"You've Got to Hide Your Love Away.\" The song, sung by John Lennon, was the first to show the influence Dylan had on the group. Lennon sang the song in the movie Help!, whose soundtrack album, released in August, reached the Top 10 in the U.S. Despite this obvious nod to the folk movement, the Help! soundtrack also featured \"Ticket to Ride,\" which Lennon would later proclaim as \"the first heavy metal song.\"\n\nNevertheless, with the two reigning generals of the incipient rock revolution coming together to exploit the choicest elements of their previously separate creative orientations (to say nothing of the implicit unification of their warring music audiences), the stylistic merger of the Beatles and Bob Dylan brought about a landmark period in rock 'n' roll history, as 1965 became the biggest year for thrilling singles since the arrival of Elvis Presley in 1956.\n\nEvery act turned on by either Dylan or the Beatles was forced by the new realities of the marketplace to up their game. The year started with the Beatles' \"I Feel Fine\" at the top of the charts with \"She's a Woman\" close behind at Number 5. Phil Spector had achieved a career peak with his production of the Righteous Brothers' \"You've Lost That Lovin' Feelin,'\" which headed to Number One in February. The Kinks followed up their entry into heavy metal, \"You Really Got Me,\" with \"All Day and All of the Night.\" The Rolling Stones released \"Heart of Stone,\" followed by \"The Last Time,\" with \"Play with Fire\" as its B side. The Beatles followed up \"She's a Woman\" with \"Ticket to Ride.\" The Animals followed up their version of John Lee Hooker's \"Boom Boom\" with \"Don't Let Me Be Misunderstood.\" At the end of February, the Moody Blues arrived with Leiber and Stoller's \"Go Now,\" while the Who made a cameo appearance in March with \"I Can't Explain.\" In April, Bob Dylan officially entered the radio fray with \"Subterranean Homesick Blues,\" followed by his spiritual successors Donovan (\"Catch the Wind\"), the Byrds (\"Mr. Tambourine Man\"), and Them with Van Morrison (\"Gloria\" and \"Here Comes the Night\"). In June the Rolling Stones delivered their first masterpiece, \"Satisfaction.\" In July came \"I Got You Babe\" by Sonny & Cher (along with Cher's solo cover of Dylan's \"All I Really Want to Do),\" \"California Girls\" by the Beach Boys, \"You Were on My Mind\" by We Five, \"Heart Full of Soul\" by the Yardbirds, and Dylan's monumental \"Like a Rolling Stone,\" capping a scintillating month. In August \"Help!\" arrived, the Turtles covered Dylan's \"It Ain't Me Babe,\" the Animals made an epic out of \"We've Gotta Get Out of This Place,\" Donovan borrowed freely from Dylan in \"Colours,\" Barry McGuire sang P. F. Sloan's desperate \"Eve of Destruction,\" and the Lovin' Spoonful countered with the sprightly \"Do You Believe in Magic.\" In September both Donovan and Glen Campbell covered Buffy Sainte-Marie's anti-war lament \"Universal Soldier,\" while Joan Baez covered protest poet Phil Ochs's \"There but for Fortune,\" and Paul McCartney went totally MOR with \"Yesterday.\"\n\nBob Dylan was back in October with the rancorous \"Positively 4th Street.\" The Stones returned with the druggy \"Get Off My Cloud.\" During the rest of October, the British group the Silkie claimed one-hit-wonder status with a cover of \"You've Got to Hide Your Love Away,\" the Byrds turned the Pete Seeger prayer \"Turn! Turn! Turn!\" into a monster hit, and the Yardbirds updated Bo Diddley's \"I'm a Man.\" November gave us the setting for a future Bruce Springsteen oratorio, \"It's My Life,\" by the Animals, \"I Can Never Go Home Anymore\" by New York City's finest star-crossed girl group, the Shangri-Las, \"Don't Think Twice\" by the Wonder Who (aka the Four Seasons), \"Let's Get Together\" (the Dino Valenti-penned anthem in its first single incarnation) by the We Five, the pepped-up arrangement of \"The Sounds of Silence\" by Simon and Garfunkel, the Hollies' first Top 40 hit \"Look Through Any Window,\" and a new one by the Spoonful, \"You Didn't Have to Be So Nice.\" December was a light month, with only the Kinks' \"A Well Respected Man\" and the Beatles' powerful tandem of \"We Can Work It Out\" and \"Day Tripper\" fulfilling the needs of rock 'n' roll's rabid new upper demo.\n\nMotown enjoyed a good year in 1965 as well, with the Miracles, Supremes, Temptations, Four Tops, and Marvin Gaye scoring with machine-like efficiency. On his way to perfecting funk, James Brown achieved the first two Top 10 hits of his career with \"Papa's Got a Brand New Bag\" and \"I Got You (I Feel Good).\" His prime tormentor Curtis Mayfield wrote his career-defining mellow anthem \"People Get Ready\" for his group, the Impressions.\n\nAs much delight as this sustained run of great singles was producing for the delirious AM radio listening audience, a more essential shift in music business priorities was even then in the wind, led by the unrestrained productivity of the two entities at the top. In March, Dylan's Bringing It All Back Home arrived, complete with howling epics like \"Gates of Eden,\" \"It's All Over Now, Baby Blue\" and the mind-blowing eleven-minute \"It's Alright Ma (I'm Only Bleeding),\" whose signature line, in which Dylan painted a portrait of a naked President Lyndon Johnson, certainly sang better than it would have looked had MTV been around at the time. The album also contained the full version of \"Mr. Tambourine Man\" (only one verse had been used by the Byrds in their Number One translation), the fierce Newport '65 opener \"Maggie's Farm,\" and other cryptic gems like \"She Belongs to Me,\" which Rick Nelson (nee Ricky) used to inaugurate his grown-up period in 1969.\n\nDylan's effort was topped a mere five months later in August with Highway 61 Revisited. With this album, the game Dylan had entered just a few short months before was inalterably changed. Michael Bloomfield assumed the lead guitar position later occupied by Robbie Robertson, while Al Kooper (of the Blues Project, by way of the Royal Teens) assumed his perch at the Hammond organ for Dylan's historic recording of \"Like a Rolling Stone.\" This scabrously exhilarating and supremely danceable song simply blew the roof off the established music business. With Dylan singing it at the Newport Folk Festival on the day of its official release, the record alienated whatever remained of Dylan's purist crowd, replacing them with a new nationwide audience of rock hipsters who would now regard albums like works of art and rock stars as poets. (Occasionally, as in the cases of Ed Sanders, owner of the Peace Eye Bookstore, and beat poet\/Fugs co-founder Tuli Kupferberg, this worked in reverse.) The album also featured Dylan landmarks of unrestrained hostility and cinematic angst, including \"Ballad of a Thin Man,\" \"Just Like Tom Thumb's Blues,\" \"It Takes a Lot to Laugh, It Takes a Train to Cry,\" and the mind-boggling postcard-from-the-hanging \"Desolation Row.\"\n\nOn a two-week tour of North America in August, the Beatles faced a frenzied welter of screaming audiences (New York City, Toronto, Atlanta, Houston, Chicago, Bloomington, Minneapolis, Portland, San Diego, Los Angeles, and San Francisco), all the while watching as \"Like a Rolling Stone\" became as inescapable in 1965 as \"I Want to Hold Your Hand\" was in 1964. Inexorably, it moved its way up the charts, from Number 26 to Number 16 to Number 5, finally peaking at Number 2 in the first week of September. Just as chilling were its implications. No longer would the 45 rpm single be the sole measuring stick of rock 'n' roll success. Or even the major one. No longer would screaming teenyboppers be the most desired payoff. The Beatles were becoming a juvenile embarrassment. If they truly wanted to remain competitive (and they did, or at least John and Paul did), they would have to start regarding the 33 1\/3 long-playing album as their primary canvas.\n\nIn the grip of this creative quantum leap, they started working feverishly on Rubber Soul in October, and had it out on the street less than two months later, complete with the double-sided non-album single \"We Can Work It Out\" backed with \"Day Tripper.\" John's most obvious nod to Dylan was yet to come: \"Norwegian Wood,\" a cryptic ode to sexual revenge and\/or getting stoned. Rubber Soul also contained John's most heartfelt lyric to date, the autobiographical travelog \"In My Life,\" later covered by the folk world's headmistress Judy Collins. While \"The Word\" continued the Beatles' long association with what Dylan once described as \"just a four-letter word,\" \"Nowhere Man\" found John embracing an Age of Aquarius flower-power sensibility, as he pondered an image of his suffering self, exhibiting symptoms of the same disease of irrelevance as the exiled fellow in the jumpsuit eating a peanut butter and banana sandwich who they'd met less than two months before.\n\nAlthough \"We Can Work It Out\" was the Beatles fastest selling single since \"I Want to Hold Your Hand,\" their pitched death battle love affair with Bob Dylan was not settled by one mere knockout blow. The two heavyweights were to meet again in New York (1965) and London (April 1966) after Dylan returned from Nashville with the completed Blonde on Blonde. By now, the Beatles were starting work on what would become Revolver. Hearing Dylan's new UK chart entry \"Rainy Day Women #12 and 35,\" Paul immediately sat down and wrote \"Yellow Submarine.\"\n\nShortly thereafter, Dylan was filmed with John Lennon in a limo by D. A. Pennebaker. In their brief conversation, the sparring was palpable. As opposed to a folk rock lightweight like Donovan, who Dylan dispensed with a half-hearted jab in Don't Look Back, Lennon was an equal, as adept at the empty-eyed put-on as the sardonic master from Minnesota. Had either of them been able to drop their guard, they might have had much to share about fame and its consequences, creativity and its limits, marijuana verses LSD verses alcohol, or abstention. Dylan was still routinely getting booed off stages as a traitor to the cause; the Beatles would soon discover how a slip of the tongue about Jesus could result in a lot of lost business in the American South's Bible Belt. It was in the studio where each found their haven and their respite. In Nashville, studio rates were low. At Abbey Road, they were non-existent.\n\nBut sooner or later, reality had to intrude. When faced with gargantuan Blonde on Blonde epics like \"Sad Eyed Lady of the Lowlands,\" or even mini-epics like \"Visions of Johanna\" and \"Memphis Blues Again,\" the cunning \"Leopard-skin Pill-box Hat,\" the perceptive \"Just Like a Woman,\" and the tumultuous \"4th Time Around\" (a song perhaps pointed at John Lennon), Lennon, for one, had to marvel and laugh. He'd played around with words himself, usually in his books, but this was ridiculous. Lennon and McCartney had their own concerns, no less profound, but expressed them a bit more simply. McCartney painted a portrait of urban loneliness in \"Eleanor Rigby\" at least as stark and probably a good deal more empathetic than \"Visions of Johanna.\" Harrison's \"Taxman\" might not have been as subtly cutting as \"Leopard-skin Pill-box Hat,\" but \"Doctor Robert\" (which was left off the U.S. edition of Revolver) was. And with \"She Said, She Said,\" \"Tomorrow Never Knows,\" and \"Rain\" (the B side of the non-album single \"Paperback Writer\"), the Beatles proved conclusively that, late to the party or not, no one could write about the drug experience better than they could.\n\nBut Revolver wouldn't come out in America until August. Blonde on Blonde started reaping praise in June. At the same time, on both sides of the Atlantic, competition was rampant. Paul Simon, previously an unknown busker in London, was now being hailed as a folk rock icon with The Sounds of Silence, with most of its songs taken from a solo album he'd recorded in England in 1965. The Paul Simon Song Book had those tunes in their original form, including \"I Am a Rock,\" \"Kathy's Song,\" \"Leaves That Are Green,\" and \"He Was My Brother.\" The Byrds, Dylan's favorite cover band, came out with Turn! Turn! Turn! featuring some of their own Beatlesque material, such as \"Set You Free This Time\" and \"It Won't Be Wrong.\" Their 1966 singles \"Eight Miles High,\" \"5D (Fifth Dimension),\" and \"Mr. Spaceman\" rivaled the Beatles' in charting the unknown regions of inner space. (\"She Said, She Said\" evolved from a meeting of the two groups, during which they and actor Peter Fonda all sampled LSD.) In New York, the Lovin' Spoonful's John Sebastian did some formidable work whose chief distinguishing feature was its relentlessly upbeat nature. Paul liked \"Daydream\" enough to base Revolver's \"Good Day Sunshine\" on its free-floating vibe. In Los Angeles, the Beach Boys, formerly mindless though harmonious devotees of surf music (as the Beatles were devoted followers of R&B and rockabilly), were finding their voice through Brian Wilson. In Pet Sounds, released in May, Wilson unveiled a soaring vision of youth's sweet dreams, complicated by drugs and reality, with \"God Only Knows,\" \"Wouldn't It Be Nice,\" \"Caroline, No,\" and especially \"I Know There's an Answer\" and \"I Just Wasn't Made for These Times.\" Perilously close to his own terrain, McCartney knew, in hearing this album, that the mission of Revolver would be to not only top Pet Sounds, but to totally blow Wilson out of the water. (At this, he unfortunately succeeded all too well.)\n\nIn England more R&B-oriented bands were now using the Beatles' example to start writing more and more of their own material. The Rolling Stones'Aftermath, released in April, was their first LP consisting solely of Jagger-Richards compositions, including some nasty tracks like \"Mother's Little Helper,\" \"Stupid Girl,\" and \"Under My Thumb.\" The Yardbirds, now featuring Jeff Beck and Jimmy Page, wrote all their own tracks for Roger the Engineer, which was released in the summer. (Only \"Over, Under, Sideways, Down\" became a hit.) The Who, however, had become a real threat. Led by the ambitious Pete Townshend, their first three singles, \"I Can't Explain,\" \"Anyway, Anyhow, Anywhere,\" and especially \"My Generation\" were some of the strongest of 1965.\n\n\"I wrote the first two songs while I was living in complete squalor, getting stoned every night and listening to Jimmy Reed,\" Townshend told me. \"I was influenced a lot by the Stones. I liked a lot of the Beatles stuff but I was never influenced that much by it. A lot of it was melodic in a way, although it sounded great when they did it; when you tried to find out what it was that made it tick and react to it musically, it was very sort of Italian lovesongs sort of stuff, like 'Yesterday.' How can you be influenced by that?\"\n\nApparently Paul McCartney was listening very carefully to Townshend as the Beatles finished up Revolver, and after a desultory tour of America in August, set to work on their next project toward the end of 1966. \"The Beatles really liked the mini-opera on A Quick One,\" said Townshend. \"Paul McCartney was saying that they were in the recording studio and were doing something similar and that was affecting what they were doing.\"\n\nBy this time, as if living out his own closing lines to onetime soul mate Suze Rotolo in the searing \"Ballad in Plain D,\" Bob Dylan had tested the \"chains of the skyway\" and found them unforgiving. A July 1966 motorcycle crash in Woodstock took him suddenly and mysteriously out of the running for king of psychedelic free association tone poetry, leaving the field wide open for John Lennon. When he was next heard from outside of Woodstock the world had changed again, due in no small part to the success of the project the Beatles were working on in late 1966 and early 1967: Sgt. Pepper.\n\nFrank Zappa, circa 1971.\n8\n\nACID ROCK\n\nIn some circles, July 29, 1966 is probably thought of as The Day the Music Died... Again.\n\nIt is not known whether the Beatles sent any of their emissaries to visit Bob Dylan in the hospital after his motorcycle accident early in August (like Elvis sending his band to visit the laid-up Carl Perkins) or whether Bob Dylan was even in the hospital at all, but the result of Dylan's poor driving skills caused a huge shift in the musical priorities of at least one, if not both of the leading players in the sixties rock revolution. Folk rock took a sharp turn to the right as Dylan emerged at the end of 1967 with the sparse, cryptic John Wesley Harding. The pastoral vibes of the Big Pink hideaway in Woodstock, where he recuperated through endless jam sessions with the Band, turned into The Basement Tapes and effectively took the air out of his prose balloon, deflating the entire scene and its attendant megalomaniacal prophesies in the bargain. With records by the Byrds, Gram Parsons, and the Band's first solo album, country rock espoused a back-to-Big-Pink ethos far removed from the street-fighting politics that dominated the fall of Aquarius.\n\nThe Beatles had their own problems, which they attempted to patch up with a press conference in Chicago preceding their first American concert of 1966, in which John Lennon tearfully proclaimed that neither he nor the rest of the band was bigger than Jesus. Nobody believed him, but it didn't matter. Once they completed the last date of the tour in San Francisco, they announced they were done with touring forever. From now on Abbey Road would be their Big Pink, where they could noodle around to their heart's content (and on EMI's dime).\n\nIn 1966 San Francisco folk rock merged with LSD to form acid rock. A heady mixture of rock, folk, blues, and self indulgence, the genre seemed, at times, to be a test of the limits an audience could be induced to listen to Jerry Garcia's guitar solos. As it turned out, these limits were virtually unlimited. Garcia and his former jug band were now the Grateful Dead, a kind of house band for Ken Kesey's Merry Pranksters and their Trips Festivals, mind-expanding environmental art showcases where LSD was freely available. One of the side benefits of LSD, it turned out, was the ability to not only tolerate, but to get deeply into extended guitar solos, whether perpetrated by Garcia or any of the dozens of B. B. King-influenced players around town, in bands called Big Brother and the Holding Company, the Jefferson Airplane, the Steve Miller Band, Mother Earth, Moby Grape, and Country Joe and the Fish. In 1966, largely because of these bands, San Francisco was not only attracting every disaffected young pot smoker on the continent, hoping to find free love on Haight Street, but also every junior A&R man with a checkbook.\n\nOn one of his independent visits to San Francisco in 1967, Paul took note of Kesey's Day-Glo painted bus, in which the Merry Pranksters traveled from place to place spreading their anarchic message of \"turn on, tune in, drop out\" as preached by Dr. Timothy Leary, the radical Stanford professor, and decided to create the film Magical Mystery Tour. By then the Beatles were well into this philosophy, although as practiced by McCartney (who did not admit to taking LSD until 1967), dropping out didn't involve giving up music. In fact, by turning on and tuning in, he was producing some of his greatest stuff ever, as evidenced by the just completed Sgt. Pepper's Lonely Hearts Club Band, which he was quick to impress upon Brian Wilson, whose home he visited in Los Angeles, playing him a snatch of \"She's Leaving Home.\" Wilson, whose Pet Sounds had inspired McCartney, was working on Smile at the time, a potential magnum opus of Gershwinesque epics like \"Surf's Up\" and \"Cabinessence\" along with quirky gems like \"Vegetables\" that might have kept him in the same breath as the Beatles. But the pressure was too much. He never finished his dream album, releasing instead the tepid Smiley Smile around the same time as Sgt. Pepper, with only \"Heroes and Villains\" and \"Good Vibrations\" to redeem it.\n\nIn Los Angeles, Frank Zappa, leader of the symphonic and satirical Mothers of Invention, generally dismissed the San Francisco scene as so much high school posturing, preferring Edgar Varese to B. B. King. He also preferred L.A.'s authenticity, of the sort that could only be found in the exclusive Laurel Canyon neighborhood where Joni Mitchell was shacked up with David Crosby and\/or Graham Nash, Carole King was jamming with James Taylor, and everyone famous lived in a house once owned by someone even more famous. (The house where Zappa lived with his wife and guests such as the GTOs, Alice Cooper, and various other freeloaders and sidemen, and where he created the Mothers' debut double album Freak Out, had previously belonged to cowboy star Tom Mix, whose horse was reportedly buried beneath the basement.)\n\nFurthering the mystical drift of the times and the Beatles' music, George Harrison spent the first few weeks after the 1966 tour ended in India on sabbatical, taking a master class with renowned sitar player Ravi Shankar, whose music he was introduced to in 1965 by the Byrds' Roger McGuinn. Intrigued by the sitar ever since playing around with one on the set of Help! George had used it on several tunes: \"Norwegian Wood,\" \"Love You To,\" and \"Tomorrow Never Knows.\" Upon returning to London, George was eager to insinuate its droning presence into a new song. It wasn't just the droning presence of the sitar, but the droning Indian philosophy that he'd been absorbing as well. This opportunity would come up in March 1967, three months into the recording of Sgt. Pepper, when Harrison invited members of the Asian Music Center to join him in the studio to play indigenous Indian instruments like the dilruba, swarmandal, tambura, and tabla on \"Within You, Without You,\" about as cogent a recap of his current anti-materialist mode of thinking as he'd yet written.\n\nBy then, John and Paul were deeply into what Paul had already earmarked as their version of Freak Out, Frank Zappa's multi-leveled, often cacophonous statement of life in Los Angeles. Released in June 1966, Freak Out came complete with riot police, unrestrained paranoia, random snatches of street corner doo-wop, the occasional audience-baiting snarl, and two mind-blowing re-creations of an acid trip (or at least songs that made much more sense if you were on acid), \"Help, I'm a Rock\" and \"The Return of the Son of Monster Magnet.\" Naturally, Paul's vision for what turned out to be Sgt. Pepper was a bit more muted, whimsical, and nostalgic, originally devised as a kind of sentimental return to the Beatles' respective childhood haunts, with \"Penny Lane\" and \"Strawberry Fields Forever\" two of the first three tracks of the concept completed (the first was \"When I'm Sixty-Four.). Although these were great tracks, they were a far cry from a freak out, which even Paul realized when he summoned the group into a session following the completion of \"Penny Lane.\" As a favor to a friend in the avant-garde circles in which he was then running (courtesy of girlfriend Jane Asher's brother Peter), Paul had agreed to contribute a piece to a \"happening\" called \"The Carnival of Light.\" The ensuing fourteen-minute mash of mumbles, screams, odd notes, and echoes was claimed by Paul to be his answer to the experimental music of Karlheinz Stockhausen or John Cage. But a thumb in the eye of Frank Zappa seems more likely to have been the case.\n\nThis was all rendered moot by a contractual anomaly of the British music business, whereby when \"Penny Lane\" and \"Strawberry Fields Forever\" were first released as singles, their use on the album in progress was precluded, scotching the nostalgia premise from the get-go.\n\nWhatever the Beatles' original plans were for the sequencing of Sgt. Pepper, these were rendered moot by a development in the communications business in America, leading late in 1966 to the opening up of the FM band on the radio dial, previously a province of dead air and classical music. Banned from duplicating the programming of their AM equivalents, radio station owners were forced to somehow fill the time. Into this breach rode the intrepid record companies with their newly-signed roster of bands that were under the influence of Dylan, the Beatles, and LSD (not necessarily in that order), bands that were just then disdaining the outmoded three-minute single in favor of the nine-minute tone poem or the five-minute guitar solo (often in the same song). By early 1967 \"free form\" FM radio had popped up in college towns across the country, commandeering larger markets like New York (WOR and WNEW), San Francisco (KMPX), and Los Angeles (KMET). Sgt. Pepper's release in June 1967 came just in time to serve as the unofficial soundtrack for what the San Francisco Chamber of Commerce proclaimed was \"The Summer of Love,\" with enough free-form FM outlets in place to blast the soundtrack nationwide (three times in a row in its entirety if the DJ felt like it). AM stations generally stuck with the 1967 singles \"Penny Lane\" \/ \"Strawberry Fields Forever,\" \"All You Need Is Love\" \/ \"Baby You're a Rich Man,\" and \"Hello Goodbye\" \/ \"I Am the Walrus,\" all of which appeared on the Magical Mystery Tour soundtrack album, although FM jocks and fans could have experienced Sgt. Pepper as it was originally intended had any of them been privy to McCartney's abandoned plans.\n\nIn the meantime, as George had gone to India as if in preparation for the masterpiece-in-waiting, John Lennon experienced his own life-changing epiphany. On November 9, 1966, he was attending the opening of an art exhibit at the Indica Gallery in London titled Unfinished Paintings and Objects when he met the woman who was to eventually influence the remainder of his career and life: Yoko Ono.\n\nUnlike George and his studies of Indian philosophy and the sitar, Yoko had little or no impact on Sgt. Pepper. At first she saw John merely as a potential buyer of her unfinished paintings and objects. But pretty soon she would insinuate herself into his life, influencing the rest of his Beatles output in a manner far more openly and completely than George's Indian instruments or philosophy ever could. For a while it was neck and neck as to who would have the greater sway over the remainder of the Beatles' time as a functioning rock band.\n\nFor most fans, to even contemplate the notion of a music scene without the Beatles in it was a prospect more depressing than dealing with the absent Dylan, who spent the rest of 1966 and all of 1967 on the sidelines. In the meantime, marijuana had become pass\u00e9 in the face of the psychic challenges offered by LSD, resulting in the decline of the brainy if undanceable folk rock in favor of the all-day, all-over, out-of-body experience of psychedelia, personified by Jerry Garcia's smile, Jimi Hendrix's use of the phasing effect (borrowed from the Beatles' \"Tomorrow Never Knows\"), and the warped poetry of Lewis Carroll. Carroll had become a cult figure among San Francisco hippies after the Jefferson Airplane immortalized his children's book Alice in Wonderland with their Top 10 hit \"White Rabbit.\" About this time, the Incredible String Band, a particular favorite of John's, recorded \"The Mad Hatter's Song\" for their second album, the hugely influential The 5000 Spirits or the Layers of the Onion, a Middle Eastern-sounding opus that also included \"Chinese White,\" \"The Hedgehog's Song,\" \"My Name Is Death,\" and the haunting \"First Girl I Loved.\" Second to none in his devotion to Carroll (and LSD), John set to work on \"I Am the Walrus\" in September 1967, the title inspired by the Carroll poem The Walrus and the Carpenter. Still mourning the recent death of Beatles manager Brian Epstein and seething about what Dylan had been able to \"get away with\" during his prime, Lennon was in rare bilious form, exorcising and entertaining his demons while prodding George Martin to reach new heights in the art of production. It was about as close to madness as a single was apt to get (apart from Napoleon XIV's 1966 hit \"They're Coming to Take Me Away, Ha-Haa!\" which was all about insanity).\n\nHad John ventured any closer to the abyss, he might have followed in the path of the unfortunate Syd Barrett, founder of Pink Floyd, and author of their first two singles, \"Arnold Layne\" and the follow up, \"See Emily Play,\" a big UK hit in the summer of 1967. Recording their first album Piper at the Gates of Dawn at Abbey Road under the auspices of former Beatles engineer Norman Smith, Pink Floyd used a lot of the studio tricks that the Beatles pioneered. They were introduced to the Beatles while Sgt. Pepper was being completed. Not much was said, but a couple of weeks later, on April 29, Pink Floyd was the headline attraction at a major psychedelic event of music, drugs, and poetry at the Alexandra Palace called \"The 14-Hour Technicolour Dream,\" which also featured Savoy Brown, the Pretty Things, the Soft Machine, and the Crazy World of Arthur Brown. Pink Floyd apparently took the stage at just about the fourteenth hour, or sometime in the early morning of the following day. LSD had been declared illegal, so many audience members were stoned on STP, a new and much stronger variant. One attendee was Syd Barrett, whose lead guitar playing that morning was particularly inspired on tunes like \"Astronomy Domine\" and \"Interstellar Overdrive.\" If Pink Floyd was anointed at that event as heirs to the psychedelic future, it may have been one of the last great performances their lead guitarist and main songwriter would ever give. For much of the next year, until he was formally kicked out of the band, his mind and life would be dominated by hallucinogens, leading to his eventual hospitalization and an early retirement.\n\nThe ever-competitive Lennon was there to witness the festivities, also in a similar state of risk, as evidenced by his recent spaced-out trilogy of psychedelic gems: \"Strawberry Fields Forever,\" \"A Day in the Life,\" and \"Lucy in the Sky with Diamonds.\" Five months later, \"I Am the Walrus\" continued the trend, making it painfully clear that none of the other Beatles had enough influence over the headstrong Lennon to stop him, even if they had wanted to do so.\n\nAlso on the bill that night was multi-media artist Yoko Ono, whose contribution, called \"Cut Piece,\" was modeled on some of her earlier groundbreaking performances. Audience members were summoned to the stage, handed a pair of scissors, and instructed to snip the clothes off a model perched on a stepladder. (On other occasions, Yoko generously sacrificed her own wardrobe for her art.) Attending the concert with John and Paul was John Dunbar, owner of the Indica Gallery, who'd hosted the exhibit where John first met Yoko, which would have given John easy access to her that night. But apparently the two gentlemen, suffering from a bit of psychedelic claustrophobia, slipped out for a smoke and the meeting didn't happen. But it was just another example, beyond music, beyond drugs, of Yoko's karmic inevitability.\n\nMeanwhile, free from the rigors of touring and the admonishments of Brian Epstein, the Beatles lounged in the studio, working on Magical Mystery Tour, an album and television film devised by McCartney mainly to give the band something to do in the wake of Epstein's passing. Early in the process, a recent signing in the EMI stable caught their fancy: the Bonzo Dog Doo-Dah Band. Previously, George Martin had produced the trad jazz group the New Temperance Seven. In addition to playing parodies of their music, Bonzo wits Neil Innes and Viv Stanshall indulged in Frank Zappian satire. They were quickly enlisted to play a song from their first album Gorilla in Magical Mystery Tour, the Presleyesque \"Death Cab for Cutie,\" which would become the name of a popular indie band some thirty years later. In 1968 Paul McCartney co-produced \"I'm the Urban Spaceman,\" a Top five UK hit from the album The Doughnut in Granny's Greenhouse (named for the U.S. hit single), which also contained \"We Are Normal,\" \"Can a Blue Man Sing the Whites,\" and \"Trouser Press,\" the latter becoming the name of a popular indie music magazine five years later. Innes would eventually ally himself with the Monty Python comedy troupe and then joined Paul McCartney's brother Mike in the band Grimms before arriving at his ultimate destiny as a Beatle parodist in the Rutles along with Pythonite Eric Idle. Their crowning achievement was a 1976 Saturday Night Live skit that was turned into a full-fledged TV movie in 1978 called All You Need Is Cash, a fab faux documentary complete with Beatlesque soundtrack album. Although George and John were big fans of the parodies, Paul McCartney was lukewarm at best. ATV Music, the Beatles' then-publisher, was also not amused, suing Innes for copyright infringement. The fact that the settlement put the Lennon-McCartney brand alongside Innes's on songs like \"Ouch!\" \"Piggy in the Middle,\" \"Cheese and Onions,\" and \"Hold My Hand\" is a satirical statement in itself.\n\nThe Beatles had already been roundly spoofed by Zappa in the Mothers of Invention's third album We're Only in It for the Money, released in the spring of 1968, which contained such all-purpose anti-Summer of Love sentiments as \"Who Needs the Peace Corps?\" \"Flower Punk,\" and \"Concentration Moon.\"\n\nBy then the Beatles were safely ensconced in the protective aura of the Maharishi Mahesh Yogi in Rishikesh, India, where they learned how to meditate and wean themselves off LSD (but not marijuana). Ringo left the compound after ten days due to his food allergies. Paul left a short time later, perhaps in a last ditch attempt to save his relationship with Jane Asher. (He failed.) George and John stayed the longest, decamping only after being convinced by a member of their entourage that the Maharishi had been making improper sexual advances to one of the women campers. During their stay, the Beatles were uncommonly productive, stockpiling dozens of songs that would fill out their remaining three albums, including John's \"Sexy Sadie,\" a fierce indictment of the humbled guru.\n\nIf Lennon was miffed about leaving, George Harrison was still pretty mellow, having spent January in Bombay, where he scored the film Wonderwall, a voyeuristic tale in which French singer Jane Birkin played a character named Penny Lane. Out of this experience came the basis for one of his best songs, \"The Inner Light,\" recorded just before returning to India to huddle with the Maharishi. George's score to Wonderwall would be the first release on the Beatles' new boutique record label, Apple.\n\nBut the good vibes didn't last more than a month. It was then that John Lennon, obviously back on LSD, spoke to the Apple executive board for the first time, amending his August 1966 announcement in Chicago by stating that he alone was Jesus Christ. Nobody dared contradict him. A day later, with his wife Cynthia out of town, John finally consummated his relationship with Yoko, but not before recording an entire album of semi-atmospheric trance music with her titled Unfinished Music No. 1: Two Virgins. Two weeks later, when sessions started on what was to become \"The White Album\" at the end of May, the bad vibes got ugly when John violated the sanctuary of the studio by bringing Yoko along. Even worse, he insisted on making his political views the subject of the Beatles next single, the waffling \"Revolution.\" Ostensibly a \"make-love-not-war\" treatise, \"Revolution\" was not meant to be part of the countercultural drift into violence that marked 1968 and 1969. In its first rendition, which ended up on \"The White Album,\" Lennon seemed to want it both ways, counting himself both \"out\" and \"in\" such a revolution. At that contentious first session, the song devolved into an extended jam, which John (and the by now indispensible Yoko) turned into a semi-atmospheric bit of uneasy listening they called \"Revolution 9,\" John's answer to Paul's answer to Frank Zappa's experimental music. Later, John re-recorded an up-tempo version that was used on the B side of \"Hey Jude.\" In this version, he revealed his true feelings (he wanted \"out\"). He would later make his way back into the middle of things, but the Beatles would be all but broken up by then.\n\nAfter this inauspicious start, the sojourn to Rishikesh took on even more significance, as the individual Beatles had plenty of material at their disposal to work on, rather than having to count on their usual studio chemistry to provide inspiration. This cosmic windfall proved to be a good thing, given that Paul was breaking up with Jane and hooking up with Linda Eastman, John was shacking up with Yoko while divorcing Cynthia, and George was becoming infatuated with a new love, the Moog synthesizer, which he used to promising effect on \"Here Comes the Sun,\" \"Because,\" and his 1969 solo album Electronic Sound, released in May, 1969.\n\nIn the end, if it was love that undid them, it was not the kind they usually sang about; it was John's obsessive love for drugs, sex, and Yoko, and Paul's for control, work, and the Beatles.\n\nUnder Yoko's influence, John's final contribution to \"The White Album\" was the poignant love song to his mother \"Julia,\" a symbolic passing of the torch to his new muse. The album, simply titled The Beatles, was released in October 1968, a month before the election of Richard M. Nixon effectively signaled the beginning of the end of the countercultural revolution that John Lennon would belatedly join. The remainder of the decade would find him tethered to Yoko in various experiments in art, political theater, and questionable (if undeniably personal) music: Unfinished Music No. 1: Two Virgins (November 1968), Unfinished Music No. 2: Life with the Lions (May 1969), and Wedding Album (November 1969). One month later, the Plastic Ono Band's Live Peace in Toronto 1969 came out, featuring \"Give Peace a Chance,\" \"Cold Turkey\" (John's admission of his heroin addiction), and Yoko's inimitable \"Don't Worry Kyoko (Mummy's Only Looking for Her Hand in the Snow).\"\n\nWhile preparing individual solo albums in January 1969, the other Beatles attempted and failed to record another album (Let It Be), filmed a rooftop concert (which was interrupted by the police), brought in the shady and divisive Allen Klein to handle their business affairs (over Paul's objections), and in February, finally started the more fruitful sessions that would result in Abbey Road. In March, John and Paul married their new paramours. John and Yoko started using heroin in June and got into a car accident in July. Yoko then brought a bed into the studio for the rest of the deteriorating Abbey Road sessions, which wouldn't wrap until August.\n\nOutside the doors of Abbey Road, the competition was heating up. At the end of 1968 the Rolling Stones issued the highly regarded Beggar's Banquet. In December 1969 they would follow it up with the unfortunately titled Let It Bleed, which seemed like a Mick Jagger joke aimed at the Beatles' abandoned Let It Be album (Jagger denied it). A day after the album's release, the humor turned grisly when the Stones presided over the Hell's Angels stabbing death of a member of the audience at the Altamont Speedway Free Festival. Forbidding tracks like \"Gimme Shelter\" and \"Midnight Rambler\" aptly summed up the blood-soaked year of the album's inception, which also included the Charles Manson killing spree in the Hollywood Hills. (The scenario unfortunately involved several Beatles songs, most famously \"Helter Skelter,\" the Beatles' attempt to outdo Pete Townshend's thunderous rock sound with the Who.)\n\nTownshend, whose mini-opera \"A Quick One While He's Away\" influenced Paul McCartney during the sessions for Sgt. Pepper, had moved forcefully into maxi-opera mode, creating Tommy in May 1969. As the hit \"Pinball Wizard\" raced up the British Top 10, McCartney was perhaps inspired to create the famous medley that dominated Side 2 of Abbey Road. Townshend and McCartney, however, never admitted to being influenced by each other. Instead, they credited a pair of 1967 singles by Keith Hopkins (aka Keith West), the relatively obscure \"Excerpt from a Teenage Opera (Grocer Jack)\" and its follow up \"Sam,\" both coming from an unproduced rock opera. West went on to join the psychedelic band Tomorrow, which released its only eponymous album in 1968.\n\nIn defiance of the year's prevailing foul mood, Bob Dylan continued his drift from the imposed requirements of his audience by perpetrating an innocuous country album, Nashville Skyline, which was released in April 1969. Despite the close proximity of his home to the Woodstock Music and Art Fair, which took place in August, Dylan left town well before the hordes descended. Unlike Lennon, he realized that the moment for such flowery sentiments had long since passed. As if to confirm this theory, Elvis Presley, playing on the momentum of his 1968 TV special, triumphantly returned to Las Vegas at the end of August. \"Suspicious Minds,\" his first Number One single since 1962, was released in September, the same month Abbey Road came out. Once again, the two ships were moving in opposite directions, in a reversal of their positions when they first met in 1965.\n\nIn 1970 John Lennon recorded \"Instant Karma\" with his old idol, irascible producer Phil Spector. So impressed was John by Spector's quick turnaround that he immediately invited him to salvage the previously abandoned Let It Be project. Spector, who was at loose ends since playing a drug dealer in Easy Rider, was more than happy to tackle the job. His arrangement of \"The Long and Winding Road,\" done without Paul's knowledge or approval, proved to be the final indignity. Although the Beatles had unofficially broken up after finishing Abbey Road, Paul made it official after hearing Spector's work.\n\nLet It Be was released in May 1970, with \"The Long and Winding Road\" ironically serving as the band's twentieth and last American Number One hit. One month later, Self Portrait, Dylan's half-assed pastiche of assorted covers and live performances was released. It was one of the worst-reviewed albums of his career (and deservedly so), and seemed to signal the end of his career as well. But Dylan would have the last laugh in 1975, when the first of several artistic and commercial comebacks occurred with the release of Blood on the Tracks, followed in short order by the official release of The Basement Tapes and 1976's Desire. The rest of the 1970s would provide him with four more hit albums. By this time, Elvis Presley was dead and buried and the Beatles were no more than a memory of collective brilliance marred by their all too human individual achievements.\n\nThe Byrds.\n9\n\nTHE AMERICAN BEATLES\n\nSo alarming and unprecedented was the Beatles' 1964 takeover of the American Top 40 singles chart that it took a moment for the arbiters of culture to adjust to it. But their output was so perfectly pitched to the moment, artistically polished, and prolific that it was difficult if not impossible to find any American rock band capable of copying it. At first, all radio programmers and their hungry audiences could do was let in a slew of the Beatles' Merseybeat brethren to handle the overflow, including Gerry and the Pacemakers, the Dave Clark Five, Herman's Hermits, the Seekers, and Billy J. Kramer and the Dakotas, not to mention the Rolling Stones, the Kinks, the Yardbirds, and the Who.\n\nPrior to the Beatles' arrival on the U.S. charts in 1964, the sound of a real live rock band on Top 40 radio or anywhere else was a rare commodity. Only in 1959 did rock bands make any sort of a dent, but these were instrumental groups such as Johnny and the Hurricanes, the Wailers, the Fireballs, Rock-a-Teens, the Virtues, the Megatrons, the Wild-Cats, and the Frantics. In 1961, Paul Revere and the Raiders came on like another instrumental group with \"Like Long Hair.\"\n\nFrom the end of 1961 through 1963, New York City's Joey Dee and the Starliters had nothing else on their mind but extolling the virtues of the twist. On the other coast, the surf craze began in late 1961, popularized by guitarist Dick Dale (\"Let's Go Trippin'\") and the Beach Boys (\"Surfin\"). This was also primarily an instrumental genre, with groups like the Surfaris, the Marketts, the Pyramids, and the Rivingtons offering very little in the way of lyric substance. The Rivingtons' \"Papa-Oom-Mow-Mow\" (1962) was a virtual doo-wop send up. The lyrics to \"Wipe Out\" by the Surfaris (1963) consisted of the title phrase maniacally cackled by the group's manager. Similarly single-minded, the Beach Boys followed up \"Surfin\" with \"Surfin' Safari,\" \"Surfin' U.S.A.,\" and \"Surfer Girl,\" which took them to the end of 1963, at which point Brian Wilson began to emerge with \"Be True to Your School\" and the lonely, wounded \"In My Room.\" This introspective phase didn't last, however, as \"Fun, Fun, Fun,\" \"I Get Around,\" and \"Dance, Dance, Dance\" dominated 1964, interrupted only by the plaintive \"When I Grow Up (To Be a Man)\" and the contemplative B side \"Don't Worry Baby.\" By this time, surf music (and its complementary sidetrack, hot rod music) had invaded inland ports as various as Minneapolis, Minnesota (\"Surfin' Bird\" by the Trashmen), Nashville, Tennessee (\"G.T.O.\" by Ronny and the Daytonas), South Bend, Indiana (\"California Sun\" by the Rivieras) and Boulder, Colorado (\"Baja\" by the Astronauts).\n\nAside from the fluke success of \"Louie Louie\" by the Kingsmen in 1963, vocal groups prevailed, including a second wave of post-doo-wop, Italian white soul acts, including Randy and the Rainbows, the Duprees, the Belmonts (sans Dion, who was now accompanied by the Del Satins), the Excellents, Vito and the Salutations, the Capris, and the Four Seasons. In addition, girl-group sounds were waiting in the wings by the Ronettes, the Shirelles, the Chantels, the Angels, the Exciters, the Jaynetts, and the Shangri-Las. Motown was already happening with its stable of lite R&B, with an occasional genius track by Smokey Robinson, the Temptations, the Four Tops, or Marvin Gaye thrown in. The teen idols of Dick Clark's American Bandstand, including Frankie Avalon, Bobby Rydell, Freddy Cannon, and Fabian were holding on to their 1950s moments. The big news at the time was the folk music revival, brought on by the enlightened politics of J.F.K., with Peter, Paul and Mary (covering Dylan), the New Christy Minstrels (covering Pete Seeger), the Chad Mitchell Trio (with future Byrds founder McGuinn backing them up on guitar), and the Big Three (with Cass Elliot, later of the Mamas and the Papas), plus holdovers from the previous folk music revival: the Kingston Trio, the Brothers Four, and the Highwaymen.\n\nIn the entire history of rock 'n' roll to that point, going back to its rockabilly beginnings circa 1956, only the Crickets could qualify as a self-contained four-man unit, led by a brilliant songwriter (Buddy Holly) and his in-house backup band. Admittedly, Gene Vincent had his Blue Caps, and Elvis had D. J., Scotty, and Bill, but those guys weren't looked upon as co-equals with the lead singer; they were studio assemblages, touring necessities. Even the Crickets evolved into Buddy Holly and the Crickets (to some extent for contractual reasons, but also because that was the tradition; even the Rock 'n Roll Trio morphed into Johnny Burnette and the Rock 'n Roll Trio). In any case, Gene Vincent hadn't had a hit since 1957, Buddy Holly had been gone since 1959, and Elvis Presley was leaning more toward moviemaking than rock, mired in a fourteen-song losing streak without a Number One record. (During 1962 and 1963 he completed movies nine through fourteen.) American kids were too busy doing the twist and watching themselves being celebrated doing the twist to worry about picking up an instrument or writing a song.\n\nInto this void stepped the Beatles, who were first of all from England, where Gene Vincent had settled in 1963; where Buddy Holly and the Crickets had had five Top 20 hits since 1960; and where five of the Everly Brothers' ten U.S. flops in 1962 and 1963 made the UK Top 40. The Beatles had been raised on Gene, Buddy, Don and Phil, and Elvis; they devoured every obscure R&B B side they could find in Brian Epstein's record store. They were enamored of Motown. They dug Arthur Alexander. They virtually cloned the Brill Building's two-minute hit formula. Plus, they had their own look: a little bit preppy and a little bit scruffy. In fact, they were a little bit too good for rock 'n' roll, but not so arch as the cardigan-clad folk singers. They were bright, witty, and possessed an undeniable chemistry. When they shook their shaggy heads in unison, they became the first headbangers. And there were four of them. For girls that meant four separate love objects. For guys that meant four separate rock 'n' role identities to choose from.\n\nSoon after the Beatles' first appearance on The Ed Sullivan Show in February 1964, Roger McGuinn (then known as Jim), was walking down a street in Greenwich Village, his hair creeping out past his collar, carrying an electric guitar. As he walked, he could hear club owners whispering to each other as he passed by, \"What we need is four of him.\"\n\nWith so few rock bands to choose from, a bunch had to be created pretty much on the spot. Luckily, the Beatles' TV appearance prodded a significant number of wide-eyed kids to pick up electric guitars, and an even more significant number of somber solo folkies to abandon their principles and go for the gusto and the gold in groups. By 1965, McGuinn and his new group were at the top of the chart with Dylan's \"Mr. Tambourine Man,\" and these same taste makers were calling the Byrds \"The American Beatles.\"\n\nBut this was not entirely accurate. There could never be an American Beatles, certainly not after 1964. The Beatles had done too many things too perfectly, nailing the time and the place, the energy and the magic, and the look and the sound to allow for such a happenstance. The Byrds did have some nice Beatlesque harmonies and a decent songwriting bent, but they were much more allied with Dylan and the nascent folk rock sound, which the Beatles dabbled in but certainly not exclusively. The problem and the beauty of the Beatles was that they dabbled in everything and the only thing they treasured was a good song, of which they wrote many more than they knew what to do with.\n\nThe Beatles wrote so many good songs that within a year or two they changed the shape of the Top 40, to say nothing of the record and radio business. Prior to their arrival, dating back to 1957, the number of singles entered onto the Top 100 went steadily up, from 483 in 1957 to 681 in 1961. During the heyday of the twist, the numbers dwindled a bit, but in 1964, the year of the Beatles arrival, 714 singles made the Top 100, an almost ten per cent increase over the previous year. The numbers soared in the next two years, peaking at an all-time high of 743 in 1966. There was then a steady decline until 1974, when only 492 singles made the charts, a level not seen since 1957. This phenomenon can be partly credited to the Beatles' shift in 1967 to Sgt. Pepper mode, as the album became a work of art and the single was reduced to an afterthought. This attitude, like most of the Beatles' attitudes, quickly permeated the thinking of the rest of the American music community as well as the record labels, who were now more concerned with the album chart rather than the singles chart.\n\nIf there could never be any one American Beatles, at least their sound could be broken up into more easily clonable pieces and dispersed to the clamoring throngs of Beatle-inspired bands that started showing up for record company auditions as early as 1965, ready to take their place on the radio in 1966. And thus, seven new rock group-oriented Top 40 styles evolved to co-exist with the recently-minted folk rock of 1965: sunshine pop, garage rock (the high school dropout's answer to 1965's frat rock), acid rock, psychedelic pop, psychedelic rock, baroque rock, and power pop. Many of these labels were fluid, with the same bands and songs being identified for each one. Many of the bands disowned such labels as soon as they found out about them. They were only playing what was in their hearts or (sadly) issuing what the record company told them was the hit. And from 1965 through the end of the sixties and into the seventies they produced something that sounded like the Beatles. This might have frustrated if not broken up a lot of young bands, but nevertheless, it provided an amazing amount of enjoyment to connoisseurs of the 45 in its waning heyday.\n\nOne of the next bands to be dubbed with the American Beatles jinx was the Remains, a group from Boston that was more like a garage band version of the Rolling Stones. Although they signed with Epic Records and once toured with the Beatles in 1966, Barry Tashian's band never had a charting single or album, despite being championed by influential rock critic Jon Landau. Their first and last singles, \"Why Do I Cry\" and \"Don't Look Back,\" were collected on the garage band compendium Nuggets: Original Artyfacts from the First Psychedelic Era, curated by noted rock 'n' roll geek and Patti Smith guitarist Lenny Kaye.\n\nOpenly coveting the same jinx was the Monkees, a blatantly contrived media takeover designed to capture the least discriminating portion of the Beatles' audience. The Monkees owed a debt to the format of the 1950s TV show The Adventures of Ozzie and Harriet, in which teen idol Ricky Nelson entertained the masses with a new song every week. It also drew inspiration from the Beatles' own A Hard Day's Night, which featured the lads romping through the strawberry fields of superstardom without raising a sweat. The Monkees were Don Kirshner's cash cow, a way to capture the image without all the hard work, and secondarily, to give his stable of former Brill Building songwriters (now relocated to L.A.) an easy outlet for their material.\n\nIronically, the Monkees' records succeeded far better than their TV show. The exceptionally well-crafted hit \"Last Train to Clarksville\" was provided by staff songwriter-producers Tommy Boyce and Bobby Hart, kicking off the concept in high style in September 1966. This was followed in December by Neil Diamond's \"I'm a Believer\" and \"A Little Bit Me, a Little Bit You\" the following March, Goffin and King's \"Pleasant Valley Sunday\" in July, Kingston Trio stand-in John Stewart's \"Daydream Believer\" in November, and Boyce and Hart's \"Valleri,\" their last Top 10 hit, in March 1968. (The latter featured one of the all-time great Top 40 guitar parts, played by studio ace Louie Shelton.) Boyce and Hart produced and wrote most of the tunes for the group's first album, The Monkees, which spent an amazing thirteen weeks at Number One, displacing Revolver. Since Sgt. Pepper wouldn't arrive for another nine months, this gave the team plenty of time to release More of the Monkees, which came out in February 1967 and sat at the top spot for eighteen weeks. Their third album Headquarters spent only one week at the top before Sgt. Pepper arrived to restore order to the universe. But after Sgt. Pepper peaked, the Monkees returned to the top spot with Pisces, Aquarius, Capricorn & Jones, Ltd, remaining there until the end of 1967. In 1968 the show was cancelled and so were the Monkees, but not before giving the world the epic psychedelic movie Head, which critic Pauline Kael called \"some potsmoker's dream,\" a fact that by itself should have guaranteed the Monkees a decent post-TV career.\n\nThe only thing wrong with this equation was that the Monkees weren't really a rock band, or any kind of band at all. Or were they?\n\n\"I've heard that Mick said later that we weren't the Monkees any more than Lorne Greene was a Cartwright,\" their bass player Peter Tork told me. \"At the same time, we were the Monkees. It was a unique phenomenon, to be a member of a group that wasn't a group and yet was a group. If we'd been a group, we would have fought to be a group, or we would have broken up as a group. But we were a project, a TV show, a record-making machine. Being a member of a synthetic group, I suffered from the criticisms\u2014those no talent schmucks from the street\u2014while in the meantime, I wasn't able to make the music I thought needed to be made. From the producers you'd run into a lot of 'You guys are not the Lovin' Spoonful, so shut up.'\"\n\nIf Mick Dolenz and Davy Jones left a lot to be desired musically, Peter Tork and Mike Nesmith were proven folk musicians. Like Dylan, Tork left college in Minnesota to play the basket houses of Greenwich Village before beating out Stephen Stills for the TV role of a lifetime. \"When the Monkees made their pilot,\" he said, \"the four of us got on stage and we were supposed to be doing a dance set. Mike had his guitar. I had my bass. Mick knew two beats on the drums. During breaks in the filming we asked the stage crew to fire up the amps and, never having played together before on the same stage, we knocked out a song and the audience liked us. Some people from Capitol Records who heard us said they would have signed us even if we hadn't had the TV show.\"\n\nIf true, this would have been either a testament to the Monkees' native ability, or an example of the desperation sweeping the music business in 1966, even at the Beatles' own label, to keep up with the demand for product. Once signed, however, recording presented its own challenges. \"Davy played nothing but tambourine,\" said Tork, \"so he had his part down after the second take. Mickey never did believe he was a decent drummer. Mike wanted to produce his own records. He wanted total control. I was the only one who believed in the group per se, and so there I was, all by myself, wanting a group, with nobody to be a group with.\"\n\nPart of the evolution\/revolution wrought by the Beatles was the opportunity they created for rock and pop rock bands to not only co-exist but to dominate the industry. This may have resulted in some great music being heard on the radio, but most of these bands had never paid their dues in an established underground setting, like the Beatles did in Hamburg. They also rarely had devoted management and a sympathetic producer, like the Beatles had in Brian Epstein and George Martin. Many were immediately thrust into the deep waters of sudden fame, unfamiliar studios and producers, unchartered tour dates and unruly fan bases, internecine skirmishes among members, and the pressure to produce a follow-up single to the miracle first hit. All they knew about how to be a band was that the Beatles (to say nothing of the Monkees) made it look simple.\n\nEven when a band's manager and producer turned out to be the father of the keyboard player who was the co-author of the band's first hit, it was no guarantee of longevity, or even another Top 10 hit. Such was the case with the Left Banke, whose ineffable \"Walk Away Renee\" was co-written by group member Michael Brown. Brown's manager\/producer was his father Harry Lookofsky, a classical violinist whose influence gave rise to the song's sound and subsequent labeling as \"baroque rock,\" noted for the inclusion of instruments such as flute and harpsichord. After writing their second single, the Top 20 hit, \"Pretty Ballerina,\" Brown started exhibiting some profoundly anti-social behavior. Although both hits had been inspired by his unrequited love for his bass player's girlfriend, it was simply not acceptable for him to replace all the other members of the band with studio musicians for the next single, which was derailed by a lawsuit on behalf of the offended players.\n\nAfter Brown quit the band, the Left Banke's career quickly dried up, leaving a lot of frustrated fans of baroque rock in their wake. But it was even more frustrating for fans of Michael Brown. His next move was as an unofficial part of the group Montage, producing and writing most of their self-titled 1969 album, which led off with \"I Shall Call Her Mary,\" a track written about his new muse, Mary Weiss of the Shangri-Las. The album also included a cover of the Left Banke's third single \"Desiree.\" That year Brown reunited with the Left Banke's lead singer Steve Martin on a song called \"Myrah,\" which failed to chart. In 1971 Brown sang two songs in the movie Hot Parts (backed by the Left Banke), \"Love Songs in the Night\" and \"Two by Two.\" With Ian Lloyd singing lead, Brown was involved with the first Stories album, but left before their breakthrough hit of 1973, a Number One cover of Hot Chocolate's \"Brother Louie,\" which appeared on their second album About Us. The mercurial Brown then moved on to the Beckies, whose only album came out in 1976.\n\nIn 1966, folk rock was still the prevailing sound on the charts, with groups shaped by the coffee house circuit making the transition to a new generation of rock theaters and clubs (Ciro's, the Troubadour, and the Trip in L.A.; the Matrix and the Fillmore in San Francisco; the Cafe Au Go Go, Electric Circus, and Fillmore East in New York City; and the Grande Ballroom in Detroit, among others). The movement was led by Dylan (\"Rainy Day Women #12 & 35,\" \"I Want You\"), along with groups such as the Byrds (\"Eight Miles High,\" \"It Won't Be Wrong\"), the Lovin' Spoonful (\"Daydream,\" \"Summer in the City\"), and Simon & Garfunkel (\"The Sounds of Silence,\" \"I Am a Rock,\" \"A Hazy Shade of Winter\"). Harmony-based vocal groups like the Mamas and the Papas (\"California Dreamin',\" \"Monday Monday\") joined the party, along with the Association (\"Along Comes Mary\"), the Outsiders (\"Time Won't Let Me\"), the Grass Roots (\"Where Were You When I Needed You\"), and the Cyrkle (\"Red Rubber Ball\").\n\nManaged by Brian Epstein, the Cyrkle also toured with the Beatles in 1966, but, as with the Remains (and the Cryan' Shames, out of Chicago) they didn't last for more than a year beyond their first hit, with two founding members cutting out for the less competitive realms of jingle writing. The Mamas and the Papas received a lot of ink as hippie scene makers; especially when John Phillips helped put together the famed Monterey Pop Festival and Mama Cass acted as Hollywood's ultimate hostess to the superstars. Still, the jangly San Francisco band the Grass Roots actually had more Top 40 hits.\n\nMost of the other San Francisco bands drifted toward psychedelia, aided by the easy availability of LSD. By 1966, the Beau Brummels, one of the scene's pioneering British-sounding folk rock bands, was already on the commercial decline, having peaked in 1965 with \"Laugh Laugh\" and \"Just a Little,\" both produced by Sylvester Stewart (aka Sly Stone) for the tiny Autumn label. Before going out on his own with Sly and the Family Stone, Stewart also produced \"Dance with Me,\" the first single for the Mojo Men in 1965. Their next single, \"Sit Down, I Think I Love You,\" written by Stephen Stills, made the Top 40 in 1967. The Beau Brummels dropped out of the singles market in 1966 after covering Bob Dylan's \"One Too Many Mornings.\" The Sopwith Camel put out \"Hello Hello\" at the end of 1966, but were into an East Coast Lovin' Spoonful vibe, soon to be hopelessly out of fashion when the acid rock triumvirate of the Jefferson Airplane (\"Somebody to Love,\" \"White Rabbit\"), Moby Grape (\"Omaha\"), and Country Joe and the Fish (\"Not So Sweet Martha Lorraine\") hit the charts in 1967, followed by Janis Joplin and Big Brother and the Holding Company (\"Piece of My Heart\") in 1968. The Grateful Dead, anchors of the scene, were too busy playing free concerts for Ken Kesey and the Merry Pranksters to worry about singles until \"Uncle John's Band\" de-virginized them in 1970. Singles were the province of garage rock bands like the Shadows of Knight, whose glorious \"Gloria,\" written by Van Morrison, would be covered by the garage band universe. In 1965, the Leaves (featuring Dino Valenti) released \"Hey Joe, Where You Gonna Go?\" whose title was shortened to \"Hey Joe\" the following year, when it was recorded by Jimi Hendrix.\n\nAcid rock, aka psychedelic rock or psychedelic pop, may have really started in 1966 in Austin, Texas, with Roky Erickson and his band, the Thirteenth Floor Elevators (\"You're Gonna Miss Me\"). This group toured with Grace Slick and the Great Society before Slick joined the Jefferson Airplane. (Janis Joplin considered joining the band before she left for San Francisco to sing with Big Brother and the Holding Company.) But sustained use of LSD and a marijuana possession charge got them into trouble. In 1969 Erickson chose to be committed to a mental hospital rather than face jail time. Easter Everywhere (1967) is generally considered their best album.\n\nAlso falling victim to drugs and musical indecision was Love, Arthur Lee's Byrds-influenced Beatlesque psychedelic rock band that lasted from 1966 to 1968. Their top single was \"7 and 7 Is\" (1966), but they were better known for two fine albums: Da Capo (1967) and Forever Changes (1968). Like the newly studio-bound Beatles, Love disdained touring, but the band quickly disintegrated after Bryan McLean, Lee's guitarist and collaborator, left due to a heroin addiction.\n\nOther than Love, Los Angeles was sunshine city on the charts of 1967, with groups like Harpers Bizarre (\"Come to the Sunshine\"), the Turtles (\"Happy Together\"), the Yellow Balloon (\"Yellow Balloon\"), the Sunshine Company (\"Happy\"), the Peanut Butter Conspiracy (\"It's a Happening Thing\"), and the Merry-Go-Round (\"Live\"), mining the vibes put forth by the Beatles' 1967 singles output. Emitt Rhodes, founder of the Merry-Go-Round, would mine those vibes even further after he broke up the band to embark on a solo career, producing the very McCartneyesque album Emitt Rhodes in 1970.\n\nOut of the dark side of the sunshine experience (sunshine was also the name of a type of acid) came the garage rock sound of the Seeds (\"Pushin' Too Hard\") and the psychedelic rock of the Strawberry Alarm Clock (\"Incense and Peppermints\") and Sagittarius (\"My World Fell Down\"), the latter group a studio creation of Gary Usher. The folk rock super group the Buffalo Springfield (\"For What It's Worth\") outgrew its humble Top 40 beginnings to produce future members of Poco and Crosby, Stills, Nash and Young. Harpers Bizarre, whose members included future Van Halen producer Ted Templeman, had one lone Top 20 hit: a cover of Paul Simon's \"The 59th Street Bridge Song (Feelin' Groovy).\" They were also fond of Van Dyke Parks' much-admired \"High Coin.\" The Turtles had calmed down some since launching their career with a Dylan cover (\"It Ain't Me Babe\"), but they were still cool enough to release Harry Nilsson's \"The Story of Rock and Roll\" as a single.\n\nBy 1968, even the Beatles, safely ensconced at Abbey Road, were feeling those dark vibes, with the scary B side \"I Am the Walrus\" setting the tone at the end of 1967. The next year would find them unusually somber and philosophical, releasing \"Lady Madonna\" backed with \"The Inner Light,\" followed by \"Hey Jude\" with the politically charged \"Revolution\" on the flip side. Their final three singles of the decade didn't get much brighter, all of them alluding to desperate times inside the castle walls, starting with Paul's plea to John to \"Get Back,\" continuing with John's obstinate answer in \"The Ballad of John and Yoko,\" and followed by the jarring use of the phrase \"over me\" by Lennon in \"Come Together.\" Its B side, George Harrison's \"Something,\" signaled George's imminent departure as well, like the once dependent spouse who suddenly starts to earn a living.\n\nIn 1970 the Beatles' two defining eulogies were released, \"Let It Be\" and \"The Long and Winding Road,\" which were preceded by John's initial Plastic Ono Band efforts, the less-than-revolutionary political rallying cry \"Give Peace a Chance,\" the harrowing \"Cold Turkey,\" and the zeitgeist-defining \"Instant Karma.\" By the end of the year, when George added \"My Sweet Lord,\" the Beatles were over, Simon and Garfunkel were over, the Buffalo Springfield were over, the Monkees were long since over, the Lovin' Spoonful were over, the Mamas and the Papas were over, and the Byrds' career had ended with \"Jesus Is Just Alright\" (after being shot in the back in the somber \"Ballad of Easy Rider,\" co-written by Bob Dylan). The Turtles' Top 40 career ended in 1970 with a re-release of \"Eve of Destruction\" (from their first album in 1965), which spent two weeks at number 100.\n\nAfter abandoning the Beatlesque Nazz (\"Hello It's Me\") in 1969 and the Beatlesque Runt (\"We Gotta Get You a Woman\") in 1970, Todd Rundgren went solo with the Beatlesque LP Something \/ Anything? (\"I Saw the Light\") in 1972. After three Top 20 hits in two years (\"Woodstock,\" \"Teach Your Children,\" and \"Ohio\"), Crosby, Stills and Nash (and sometimes Young) broke apart into separate solo identities. The Spoonful's John Sebastian went solo. The Mothers of Invention's Frank Zappa went solo. Janis Joplin left Big Brother and the Holding Company to go solo. Alex Chilton left the Box Tops in search of the lost chord. Among the last of the American Beatles were Raspberries, arriving on the Cleveland scene in 1972 when the only Beatle clone around was the expatriate trio America (\"A Horse with No Name,\" \"I Need You,\" \"Ventura Highway\"). America was formed in London, and after self-producing their first three albums, handed the reins to George Martin for several more hits (\"Tin Man,\" \"Lonely People,\" \"Sister Golden Hair,\" \"Daisy Jane\"). With no such connections and no such luck, Eric Carmen gave the world a thrilling blast of Beatlemania with \"Go All the Way\" in 1972 before lapsing into diminishing returns. Two years later, and still a year away from embarking on his own solo career, Carmen's parting shot with Raspberries was the prescient \"Overnight Sensation (Hit Record),\" which made the Top 20. But by then it was too late and Carmen went on to a Beatlesque solo career.\n\nA much sadder case of wasted time and potential was that of Alex Chilton and Big Star. Emerging in Memphis with the Box Tops when he was a teenager in 1967, Chilton had already helped create iconic hits including \"The Letter,\" \"Cry Like a Baby,\" \"I Met Her in Church,\" and \"Soul Deep.\" Like the Beatles, Chilton was looking to expand beyond the teenybopper stage. After a meeting with the Byrds' Roger McGuinn in New York City, he returned to Memphis and formed Big Star with new songwriting partner Chris Bell. The result was #1 Record, released in the spring of 1972, and containing such perfect Beatlesque pop rock songs as \"When My Baby's Beside Me,\" \"Don't Lie to Me,\" \"Thirteen\" (based on seeing the Beatles), and \"In the Street,\" which was later covered by Beatle fanatics Cheap Trick as the theme for the TV series That 70s Show. Due to label ineptitude and internal distribution problems, the album tanked. Incensed and depressed, Bell quit the band. Radio City, released early in 1974, had two of Chilton's finest songs, \"Back of a Car\" and \"September Gurls,\" but it, too, was heard by almost no one except record critics, who got their records for free anyway. Third \/ Sister Lovers, completed at the end of 1974, consisted of moody epics like \"Holocaust,\" but was considered too intricate and emotional for radio; the marketplace having by now gone way beyond anything even remotely resembling the American Beatles sound.\n\nIn its place were the original Beatles, now into their solo careers. McCartney issued eleven albums in the 1970s, including two solo efforts and nine with his group Wings. Lennon issued eight, four solo and four with the Plastic Ono Band. Harrison and Starr also issued eight albums apiece. This seemed sufficient enough to mollify the stranded, wounded hordes of Beatle fans, a small percentage of whom came out for Tom O'Horgan's off-Broadway show, Sgt. Pepper's Lonely Hearts Club Band on the Road, which played at the Beacon Theatre in New York City in 1974 and 1975. At this point, an obscure band called Klaatu attempted the previously unheard-of feat of becoming, for a brief, shimmering moment in 1976, the Canadian Beatles. But Klaatu's self-titled album (ironically recorded for Capitol) was revealed to be nothing but a slightly Beatlesque hype, although that notion alone propelled the Toronto group's two releases into the album Top 40, with the single \"Sub Rosa Subway\" making the Top 100 and \"Calling Occupants of Interplanetary Craft\" gaining a Top 40 cover by the Carpenters.\n\nThis distant whiff of Beatlemania was enough, however, to inspire Steve Leber and David Krebs, managers of Aerosmith, to fabricate an entire Broadway production called Beatlemania in 1977, a winningly clonelike spectacle that ran for over a thousand performances, surviving through 1979 before spawning several traveling companies, one of which even played London. The production eventually merged in the early eighties with a somewhat similar tribute concept called Rain that started in California and has continued to run ever since. Although Beatlemania: The Album was received as warmly by nostalgia buffs as the show, the original cast of Joe (Pecorino), Mitch (Weissman), Les (Fradkin), and Justin (McNeill) hardly had the same historical resonance as John, Paul, George, and Ringo. The 1981 film version, Beatlemania: The Movie proved to be a stiff.\n\nBut nowhere near as stiff as the film musical version of Sgt. Pepper's Lonely Hearts Club Band that appeared in 1978, for which George Martin, as its musical director and producer of the soundtrack album, must be forever hanging his head in shame. Based on Tom O'Horgan's modest-by-comparison off-Broadway show, the film starred the Bee Gees: brothers Barry, Robin, and Maurice Gibb. Fresh from their disco reinvention in Saturday Night Fever, the brothers did a complete career 180 by regressing to their Beatle clone roots. The soundtrack featured twenty-one tracks from Sgt Pepper and Abbey Road, with the Bee Gees appearing on ten titles, including \"Getting Better,\" \"Good Morning, Good Morning,\" \"She's Leaving Home,\" \"Golden Slumbers \/ Carry That Weight\" (with Peter Frampton), and \"Because\" (with Alice Cooper). Robin Gibb had a hit with \"Oh! Darling,\" Earth, Wind and Fire scored with \"Got to Get You into My Life,\" and Aerosmith did a credible job on \"Come Together.\" In the spirit of re-creating the famous Pepper album cover, which had already been re-created to death (most notably by Frank Zappa), producer Robert Stigwood persuaded a huge number of otherwise unemployed C- and D-list celebrities to participate in the film. Among them, only Donovan, Peter Noone of Herman's Hermits, disgruntled Apple artist Jackie Lomax, and maybe Del Shannon (\"From Me to You\") had anything more than the most tenuous ties to the Beatles. Carole Channing? Chita Rivera? Connie Stevens? Gwen Verdon? Sha-Na-Na? Jim Dandy of Black Oak Arkansas? Leif Garrett? The saddest part of the whole fiasco was that it was supposed to be taken seriously.\n\nMeanwhile, with the American psyche once again primed for Beatlemania, Rockford, Illinois' Cheap Trick was the first group to take advantage of it in 1977 with genuine hard rock fervor (\"I Want You to Want Me,\" \"Surrender\") and Rick Nielsen's antic presence, contributing twisted lyrics and playing a series of wonderful guitars. Detroit-based the Romantics (\"What I Like About You,\" \"Talking in Your Sleep\"), and Northern California's Tommy Tutone (\"867-5309 \/ Jenny\") mined the skinny tie look and perfect pop predilections. Todd Rundgren was even more overt with his new outfit Utopia, producing 1979's homage to the Fab Four Deface the Music, a masterful tribute to all the myriad facets of the Beatles' musical development. Coming a little late to the party was Dick Clark, who helmed the fictional documentary The Birth of the Beatles (1979), which contained upwards of twenty early tracks, including two versions of \"I Saw Her Standing There\" (one with Ringo on drums and one with Pete Best), \"Blue Suede Shoes,\" \"Lawdy Miss Clawdy\" (performed by Rory Storm), and a lot of Little Richard and Chuck Berry covers, culminating with \"P.S. I Love You\" and \"Love Me Do.\"\n\nWith all this cool, crass, overt, and latent Beatlemania in the air, it would seem the 1980s would be ripe for a full-scale return to the kind of jangly, melodic songs pioneered and inspired by the Beatles. To some degree this happened. But not for reasons anyone anticipated.\n\nJames Taylor, circa 1970s.\n10\n\nAPPLE RECORDS\n\nPerhaps the most obvious initial showcase for both the music that influenced the Beatles and the music they influenced (apart from their individual and group efforts) is the output from Apple Records (1968 to 1973) and its even shorter-lived Zapple subsidiary (a cheeky nod to Frank Zappa's Bizarre Label), which only released two records, John and Yoko's Unfinished Music No. 2: Life with the Lions and George's Electronic Music. But bizarre is the only word to describe the random assortment of signings that took place at the wildly mismanaged label in the waning years of the 1960s. The only element more bizarre was the daily workings of the label itself, a perfect expression of the generally demented atmosphere of unlimited money, drugs, na\u00efvet\u00e9, and entitlement that permeated the Beatles' camp.\n\nApple served partly as a vanity press for all the Beatles' personal indulgences, including John's tape recorded indiscretions with Yoko, George's early efforts on the Moog, and Ringo's nostalgic fling with pop standards. In view of this, it was surprising that Apple had as much success as it did. Keeping things in the family, the Beatles named Peter Asher (Jane's brother, of Peter and Gordon fame), as head of A&R, a position he would keep even after Paul and Jane broke up. One of his first signings was an unknown folk rocker from Boston named James Taylor, who was then living in England. Taylor's self-titled album, released in December 1968, contained several memorable tunes, including \"Sunshine Sunshine,\" \"Rainy Day Man,\" and \"Carolina in My Mind,\" the latter featuring McCartney on bass and an uncredited George Harrison singing backup. George got an extra special boost from the album, using Taylor's \"Something in the Way She Moves\" as the first line of his breakout song \"Something.\" Fitting in perfectly with the prevailing zeitgeist, Taylor was hospitalized for drug treatments and sent back to America at the time of his sole Apple release, and was thus unable to promote it. That his career at Apple lasted for only the one album was a costly music industry anomaly; an example of either Apple's complete lack of business smarts or their advanced sense of themselves as a loose, artist-friendly collective, with contracts being a part of the older order (i.e., pass\u00e9). In either case, once on Warner Bros., Taylor immediately zoomed to stardom as a prime mover in the new, mellow singer-songwriter era along with Brill Building graduate Carole King.\n\nInundated by tapes from every hopeful on the continent, Apple signed artists chiefly based on the suggestions of friends. The first and only Number One single that didn't have a Beatle attached to it was by Welsh folk singer Mary Hopkin, who was recommended to Paul McCartney by svelte supermodel Twiggy, who happened to catch her on the TV talent competition Opportunity Knocks. Produced by Paul, the song that was selected might have come straight from the ancient songbag of the Quarrymen. \"Those Were the Days\" was based on an old Russian folk tune that had been translated and updated by Gene Raskin. Although it had been a staple in the repertoire of the folk group the Limeliters since 1962, it's possible that McCartney heard the song after reverting to his old Beatles habit of flipping through singles, finding it on the B side of an Engelbert Humperdinck hit, \"The Last Waltz,\" which had held the top spot in the UK for five weeks in 1967. The Beatles' pedigree at the time was so strong that Hopkin's version easily withstood a cover by Sandie Shaw that came out a month or so later, which failed to chart in either the UK or the U.S. (It did, however, make the Top 40 in Australia.) Hopkins' version hit Number One in the UK and Number 2 in the U.S. Her follow-up was a Lennon and McCartney tune that the Beatles never recorded called \"Goodbye,\" which peaked at Number 2. Her last hit, \"Knock Knock Who's There,\" came in second in the Eurovision Song Contest in 1970.\n\nThe Apple signings were basically the province and playground of the mercurial tastes of Paul and George, with John and Ringo represented to a lesser extent. The first four Apple singles were released simultaneously. Two were hits: the Beatles' \"Hey Jude\" and Hopkin's \"Those Were the Days.\" The other two were unmitigated flops: \"Sour Milk Sea\" by Jackie Lomax (a tune George Harrison abandoned at \"The White Album\" sessions) and \"Thingumybob,\" a brass band version of Paul McCartney's theme for the TV show of the same name, played by the well-known Black Dyke Mills Band, with \"Yellow Submarine\" on the B side.\n\n\"Sour Milk Sea,\" which George wrote in Rishikesh, reflected George's familiar Indian philosophy but the title might have been one of the most unappealing ever given a single. The record didn't exactly jump-start the already stalled career of Lomax, a transplanted British soul singer who had been a friend of the extended Beatle family since Brian Epstein brought him back from America in 1967. After Epstein died, Harrison stepped in as his surrogate musical guru. One would think that with George producing and a band consisting of Eric Clapton on guitar, McCartney on drums, Ringo on bass, and Nicky Hopkins on keyboards, the record could have transcended its title on credibility alone. But it failed to chart, and neither did the revealingly titled album Is This What You Want?, released in the spring of 1969. Lomax was soon back in America, picking up the pieces of what remained of his career.\n\nThe label's next discovery, the Iveys, fared a good deal better, although not at first. The band, which featured songwriters Tom Evans and Pete Ham, was brought to Apple by the Beatles' loyal but underpaid road manager and go-fer Mal Evans (no relation to Tom). The Beatlesque \"Maybe Tomorrow\" was the Iveys' first release, but since it only charted well in three countries, Allen Klein, who had just been brought in to unravel the mess at Apple, decided to release their album Maybe Tomorrow, only in West Germany, Japan, and Italy (although it had gone to Number One in Holland). A year later, the band, renamed Badfinger, recorded McCartney's \"Come and Get It,\" which became a huge transatlantic hit, spurred by its appearance in the Peter Sellers\/Ringo Starr movie The Magic Christian along with two other Iveys songs, \"Rock of Ages\" and \"Carry on Till Tomorrow.\" Badfinger's first album capitalized on the success of \"Come and Get It\" without it being the actual soundtrack. Essentially, Magic Christian Music was a re-release of the quickly deleted Iveys album that included \"Maybe Tomorrow.\" Their second album, No Dice, contained another worldwide hit, \"No Matter What\" and a future pop standard \"Without You,\" which quirky singer\/songwriter and future Lennon drinking buddy Harry Nilsson brought to the top of the charts in 1972, the same year Badfinger's hit \"Day After Day\" peaked at number four, \"Without You\" was also covered by Mariah Carey in 1994. Presaging the fate of nearly every other Apple artist (aside from the four ex-Beatles), Badfinger ended badly, with both Pete Ham and Tom Evans committing suicide (Ham in 1975 and Evans in 1983). In 1976, original mentor Mal Evans was shot to death by police, who thought he was brandishing a loaded rifle during a domestic dispute.\n\nIn general, 1969 was as bad a year at Apple as it was for the imploding Beatles. Singles by non-entities called Trash (\"Road to Nowhere,\" followed up by a cover of \"Golden Slumbers \/ Carry That Weight\") and Brute Force (\"King of Fuh\") competed with an obvious Harrison signing, the Radha Krishna Temple (\"Hare Krishna Mantra\"), and future UK superstars Hot Chocolate (\"Give Peace a Chance\") for the dubious honor of most disappointing record. The woefully out-of-place Modern Jazz Quartet released their second of two mediocre albums, Space, which followed Under the Jasmine Tree to the cut-out bins. Little Richard and Ray Charles keyboard player Billy Preston broke the trend with \"That's the Way God Planned It,\" which fell short of making the UK Top 10. Aside from Abbey Road, the rest of Apple's time was taken up with assorted Beatle solo projects, including John and Yoko's Wedding Album, the Plastic Ono Band's Live Peace in Toronto 1969, Ringo's Sentimental Journey, and in April 1970, Paul McCartney's debut solo album McCartney, delivered just prior to the release of the Beatles' last album, Let It Be.\n\nProduced by George Harrison, \"That's the Way God Planned It\" could have been a knockout, considering the superstar backup band Harrison assembled behind the always stellar Preston, which included Eric Clapton on guitar, Keith Richards on bass, Ginger Baker on drums, and Doris Troy singing backup. But neither the title tune nor its follow up, \"Everything's Alright,\" gave it any momentum. Preston stuck around, however, for a second album, Encouraging Words, which may have been one of his best. On this record, George gave Preston the honor of recording two songs he was planning to use on his first solo album, \"All Things Must Pass\" and a song Harrison would take to the top by the end of the year, \"My Sweet Lord.\" Preston also collaborated with Harrison on \"Sing One for the Lord,\" and just to be on the politically safe side, he covered Lennon and McCartney's \"I've Got a Feeling\" in amazing fashion. When this also flopped, Preston escaped to A&M, where his career would soar.\n\nThanks to the Preston connection and a burgeoning career in England as a backup singer, Doris Troy landed the mixed blessing of having an Apple release as well. Known for co-writing and singing \"Just One Look\" in 1963 (covered by Linda Ronstadt, the Hollies, Anne Murray, Lulu, and Harry Nilsson, among others), Troy and her eponymous Apple album also benefitted from the Beatles' good name by being able to assemble stellar sidemen, among them Peter Frampton, Stephen Stills, Eric Clapton, Leon Russell, and Delaney and Bonnie. (The latter's 1969 Apple album, Accept No Substitute, was cancelled, to their subsequent delight, and released on Elektra, where it made the U.S. album charts.) On her self-titled album, Doris Troy showed exemplary taste in covers (Stephen Stills' \"Special Care\" and Joe South's \"Games People Play\"), but the single releases of \"Ain't That Cute\" and \"Jacob's Ladder\" (the latter with \"Get Back\" as its B side) went nowhere.\n\nIn 1971, Apple moved full scale into the movie soundtrack business with three initial releases. Raga was a straight documentary based on the life of sitar master Ravi Shankar. Produced by Ringo, Cometogether was a vanity project for director Saul Swimmer, who had previously lensed the Beatle documentary Let It Be and would go on to greater glory with his filming of George's The Concert for Bangladesh. The soundtrack album for Cometogether included the hit single \"I Can Sing a Rainbow \/ Love Is Blue\" by the Dells, but nothing by the Youngbloods, whose music was in the movie. The blockbuster of the three was the cult classic El Topo (the Mole). One of the original \"midnight movies,\" predating The Rocky Horror Picture Show by four years, this surreal film influenced a number of important tastemakers of the era, including David Lynch, John Lennon, and Bob Dylan. Lynch would later apply his twisted cinematic sense to television's Twin Peaks. Dylan would go on to star in the similar Pat Garrett and Billy the Kid (from whence came the brilliant \"Knockin' on Heaven's Door\"). John Lennon would commission British classical pianist and arranger John Barham, who worked with George on Wonderwall Music and All Things Must Pass, to re-record El Topo's score as written by the film's director and star, Alejandro Jodorowsky. The film played in art houses for years, principally for fans of the acid western genre, which included (in addition to Pat Garrett and Billy the Kid) The Shooting (1966), Easy Rider (1969), Two Lane Blacktop with James Taylor (1971), and Greaser's Palace (1972).\n\nApple's other foray into the classical realm resulted in two albums by British composer John Tavener, including his first important piece, The Whale, based on the biblical story of Jonah, which was premiered at the London Sinfonietta in 1968 and released on Apple in 1970. Celtic Requiem, featuring soprano June Barton, followed the next year.\n\nOne of the highlights of 1971, in a strictly countercultural sense, was \"God Save Us\" (originally \"God Save Oz\"), written and demoed by John Lennon (co-authored by Yoko) with the single release performed by Bill Elliott and the Elastic Oz Band. The song was written to protest the obscenity trial of the editors of the British underground magazine Oz for their May 1970 Schoolkidz Oz issue. In an example of the repressive times following the fall of Aquarius, the editors were found guilty (much like the Chicago Seven), but were acquitted on appeal. Yoko issued the original demo in The John Lennon Anthology, released in 1998.\n\nOn a musical basis, the year belonged to two female singers of totally opposite styles and sensibilities, who endured a totally different experience at Apple. Ronnie Spector was a longtime Beatle favorite and one of the greatest voices in rock 'n' roll history. The imprisoned wife of eccentric producer Phil Spector, the former Ronette recorded one single for the label, George's \"Try Some, Buy Some,\" which was co-produced by Harrison and Phil, who was now the new head of A&R at Apple after completing the infamous Let It Be remix. The record, however, found Ronnie Spector tragically miscast in mood and message. George later put the song on his album Living Spirits in the Material World, using the same arrangement. David Bowie covered it on 2003's Reality. (John Lennon was especially partial to the song's B side, \"Tandoori Chicken.\")\n\nAfter failing at Apple, Ronnie met a similar fate in the years after her divorce from Phil, as she attempted to make a comeback. Her magnificent take on Billy Joel's \"Say Goodbye to Hollywood,\" with Bruce Springsteen and the E Street Band, backed with Miami Steve Van Zandt's \"Baby Please Don't Go,\" failed to chart in 1977. In 1980 Siren was produced by girl group sound aficionado Genya Ravan (Goldie and the Gingerbreads and Ten Wheel Drive), which featured the Ramones' \"Here Today, Gone Tomorrow.\" Ronnie's 1986 cameo with Eddie Money on \"Take Me Home Tonight\" led to 1987's Unfinished Business (living up to its title); the Desmond Child track \"Love on a Rooftop\" became famous primarily for inspiring Child to write similar future hits for Bon Jovi. Joey Ramone would be back in 1999 to produce She Talks to Rainbows, an album (released in Australia only) famous for Ronnie's legendary version of \"Don't Worry Baby,\" a song originally written for her by Brian Wilson.\n\nThe other female voice belonged, of course, to Yoko Ono, although here the word \"voice\" fails to do justice to the range of sounds and feelings conveyed by the singer, in turn ethereal and hellish, impossible to listen to, and impossible to resist. Her first album, the experimental, atmospheric, and just plain weird Yoko Ono \/ Plastic Ono Band, is the soundtrack to a ritual sacrifice. Released at the end of 1970, it featured Ornette Coleman on the second side's opening track, \"AOS.\" Two other tracks became Lennon B sides, the scathing \"Why\" (which backed \"Mother\") and \"Touch Me\" (which backed \"Power to the People\"). \"Greenfield Morning (I Pushed an Empty Baby Carriage All Over the City)\" was an early leap into noise rock.\n\nOn her 1971 double album Fly, Yoko's focus exploded with the sixteen-minute \"Mind Train,\" which dominated the first side, and the twenty-three-minute \"Fly\" dominating the second. \"Don't Worry Kyoko (Mummy's Only Looking for Her Hand in the Snow\"), the B side of \"Cold Turkey,\" is here in all its frenzied glory. The keen-like \"Mind Holes\" is the soundtrack to a Japanese acid western, but \"Midsummer New York\" is a relatively straight-ahead rocker. The standout track is the grim and haunting \"Mrs. Lennon,\" which would lend its aura to Alex Chilton's \"Holocaust,\" as performed by Big Star on Sister Lovers from 1974.\n\nThe third album, Approximately Infinite Universe, backed by the New York City rock band Elephant's Memory, came out early in 1973, with twenty-four discrete tracks, including the poignant \"Song for John\" (the feminist's answer to Helen Reddy), \"What a Mess\" (which equates abortion to masturbation), \"Death of Samantha\" (in which Yoko sounds eerily like Kate Bush), and the R&B-flavored \"I Felt Like Smashing My Face in a Clear Glass Window\" (in which Yoko confronts her split with John, which would last until 1975). The subject dominates her final Apple album Feeling the Space, in which she vents with spleen and gusto on such tracks as \"Run, Run, Run,\" \"Angry Young Woman,\" and \"Woman Power.\" Combined, her four albums spent a total of nine weeks on the Billboard chart, but during those weeks, women with exotic sensibilities as varied as Lene Lovich, Sinead O'Connor, and Bjork were obviously listening.\n\nElephant's Memory got to make their own album for Apple in 1972, while Ringo's favorite, Chris Hodge, only released a couple of singles: \"We're on Our Way\" and \"Goodbye Sweet Lorraine.\" The Sundown Playboys, a Cajun band from Louisiana, had one single, \"Saturday Night Special,\" which came in through the slush pile. Although Elephant's Memory had achieved a bit of prior fame for placing two songs in the film Midnight Cowboy (\"Jungle Gym at the Zoo\" and \"Old Man Willow\"), their self-titled Apple album did little to sustain their momentum. They can also be heard backing John Lennon on half of Some Time in New York City. On the other half, John jams with his favorite Beatles antagonists, Frank Zappa's Mothers of Invention.\n\nOne of the last Apple signings was that of two brothers from New Jersey, Lon and Derrek van Eaton. Their album Brothers, released in September 1972, got lost in the shuffle of the breakups and lawsuits that split the band and the label apart. Unfortunately, The Pope Smokes Dope, released in April 1972 by David Peel and the Lower East Side, did see the light of day. Reflecting John Lennon's belated attempt to join the radical element at the butt end of the Revolution, Peel's singsong street polemics (already well established in Have a Marijuana and The American Revolution) fit all too well into Lennon's limited but still antic sense of political theater. With an offensively simplistic presentation, the always exhibitionistic Peel's songs rarely went beyond their obvious titles. Thus, potentially revealing subjects like \"I'm a Runaway,\" \"Chicago Conspiracy,\" \"Ballad of New York City \/ John Lennon and Yoko Ono,\" \"Bob Dylan,\" and the \"Okie from Muskogee\" parody \"Hippie from New York City\" merely revel in their idiocy, and in so doing, ironically paint a portrait of an era and a revolution headed for a crack-up.\n\nAs with the Beatles, it was a crack-up John Lennon seemed to be perversely looking forward to.\n\nThe Sex Pistols.\n11\n\nTHE ANTI-BEATLES\n\nIn his induction speech for Jimmy Page at the Rock and Roll Hall of Fame ceremonies in New York City in 1994, U2 guitarist Dave Evans, otherwise known as the Edge, admitted, \"We spent a lot of years in the garage trying to sound like anyone other than Jimmy Page.\"\n\nBy the 1980s, young musicians in fledgling rock bands had already had more than enough of Page's signature riffs in Led Zeppelin. Although they never released a single in the UK and had only one Top 10 hit in the U.S. (\"Whole Lotta Love\"), they'd been the preeminent rock band since their arrival in 1969. Their arena-sized sound and badass image had been imprinted upon a generation of headbanging, hotel room trashing, heavy metal bands throughout the world, even without the help of MTV. Over the course of ten years, while their 1972 epic \"Stairway to Heaven\" had long since replaced \"Louie Louie\" and \"Johnny B. Goode\" as the essential rite of lead guitar passage, other extravaganzas like \"Dazed and Confused,\" \"All My Love,\" \"Good Times, Bad Times,\" \"Heartbreaker,\" \"The Lemon Song,\" \"Going to California,\" \"Ramble On,\" and \"Kashmir\" from their nine regular albums made them impossible to avoid, if not abjectly imitate. (The combined sales of these albums were over eighty-nine times platinum.) Thus, to make any sort of mark as an original band with a guitar player in it required a total deconstruction if not demolition of Page: the man, the player, and the myth.\n\nBut this is how eras change in rock music, when the young finally declare their independence from their older brother's rock 'n' roll by setting fire to the legends and dancing amid the cinders.\n\nWhich is not to say Jimmy Page had this in mind in 1969 when he gladly did the two-step on the Beatles' grave, supplanting them in the U.S. and England as the top band in the land. The emergence of Led Zeppelin from the ruins of the always underappreciated Yardbirds must have delighted not just Page, but his compatriots on the British blues scene, who were rendered obsolete the minute the Beatles internationalized the Merseybeat craze. More likely, he was just taking the next evolutionary step following Clapton's Cream and Townshend's Who, mixing the power of the blues with the super power of the Marshall stack to impress the boys, and with Robert Plant as lead singer throwing in some crotch-grabbing teenybop moves, impressing the young girls as well. The fact they ratcheted up the Beatles' Middle Eastern Maharishi yearnings with references to demented mystic Alistair Crowley was only icing on the cake. The fact that Ozzy Osbourne emerged like a behemoth from the moors of Birmingham a year later in Black Sabbath, upping the heavy metal ante in Black Sabbath (\"N.I.B,\" \"The Wizard,\" \"Black Sabbath\"), Paranoid (\"Paranoid,\" \"Iron Man,\" \"War Pigs\") and Master of Reality (\"Children of the Grave\") past antisocial and into the realms of the psychotic was just the cherry on top.\n\nBut the anti-Beatles reaction had been planted in the populace way before that. At first, all Bob Dylan fans were de facto anti-Beatles, although Dylan himself claimed to be an early convert to their thrilling innocence. But the more rabid folkies who eventually disowned Dylan for his move to electricity only saw the Beatles as plastic pop. Frank Zappa lampooned them in We're Only in It for the Money, soon after Sgt. Pepper, but Zappa lampooned everything, and the album was more a commentary on flower power than the Beatles. But the fact that the Beatles aligned themselves so closely with the prevailing hippie culture of acid, love beads, and meditation made them ripe to take the hit for all the excesses of their era and to effectively take the era down with them when they disintegrated.\n\nIf Led Zeppelin was noisier than the Beatles ever hoped to be, it was the Velvet Underground that really stood in opposition to everything the Beatles represented. \"Let's say we were a little bit sarcastic about the love thing,\" said founding member Lou Reed. \"They thought acid was going to solve everything. You take acid and you solve the problems of the universe. And we just said, 'Bullshit. You people are fucked. That's not the way it is and you're kidding yourself.'\"\n\nFar from thinking heroin was the answer, Lou Reed nonetheless wrote anthems for a drug subculture that was anything but loving and peaceful on their first album The Velvet Underground and Nico, released in 1967, which included early nods to the decadent decade that was to follow. \"Heroin,\" \"I'm Waiting for the Man,\" and \"Black Angel's Death Song\" were leavened only by bittersweet morning-after hangover laments like \"Femme Fatale,\" \"Sunday Morning,\" and \"I'll Be Your Mirror.\" In Nico they found a voice as ghostly and forbidding as Janis was earthy and Grace was cool. That same year, Nico put out Chelsea Girls, the desultory title track coming from an experimental film by Andy Warhol about the legendary bohemian enclave, the Chelsea Hotel. The album also contained probably the first version of Jackson Browne's classic lament \"These Days,\" the rare Dylan treasure \"I'll Keep It with Mine,\" and Tim Hardin's searching \"Eulogy to Lenny Bruce,\" a kind of bookend to \"European Son: To Delmore Schwartz\" from the Velvets' first album. References to black comedy and dead poets (Schwartz died in 1966) aligned the band much closer to the Beats than the Beatles (or the hippies). Like Reed, Jack Kerouac came to despise the love generation (for co-opting his sacred message in his ultimate manifesto, On the Road, for their own heedless purposes).\n\nAs much as the Rolling Stones had been historically seen as the only true anti-Beatles group since their inception, this was really more of a ploy to keep their separate publicists busy. Maybe they didn't vacation together on the same yachts, but theirs was basically a healthy competition, highlighted by the Stones' 1969 album Let It Bleed, an accidentally witty takeoff on the Beatles' Let It Be, which it beat to the marketplace by six months. While Let It Be and Abbey Road were remote affairs, connected less to the end of an era than to the end of the Beatles, Let It Bleed, released a day before the tragic death at the Altamont concert, consciously or not documented the violence of a bloody year on the streets and campuses and even at the outdoor rock festivals of the world. \"You Can't Always Get What You Want,\" the B side of the non-album \"Honky Tonk Women,\" stands in stark contrast to the Beatles' much more na\u00efve \"All You Need Is Love.\"\n\nThe Stones' emergence, marked by 1968's Beggar's Banquet (\"Stray Cat Blues,\" \"Salt of the Earth,\" \"Sympathy for the Devil\") as men of, by, and for the people also contrasts with the Beatles of this era, who were far from the people, either convening with their guru in India or arguing with each other in the recording studio. When both bands finally decided to confront the bloody politics of that defining year, it was the Stones who beat the Beatles to the American singles charts by a week with \"Street Fighting Man,\" putting it on the A side, while the Beatles relegated the tamer sounding \"Revolution\" to the B side. And even without the Beatles to kick around, the Stones found plenty of motivation in their newly found songwriting chops, with \"Brown Sugar,\" \"Wild Horses,\" \"Bitch,\" \"Sister Morphine,\" and \"Dead Flowers\" (from 1971's Sticky Fingers) moving them squarely into Velvet Underground musical and lyrical territory.\n\nBy the end of 1970, the Velvet Underground was seriously imploding, Beatles style, with co-founder John Cale leaving after 1968's White Light \/ White Heat (\"Sister Ray,\" \"I Heard Her Call My Name\") and Lou Reed leaving in 1970 after writing two of the band's most enduring tracks, \"Sweet Jane\" and \"Rock and Roll,\" for the ironically titled Loaded (as in \"loaded\" with hits). Like many in the ranks of the deposed folk rock era he ridiculed, Reed started his solo career in 1972 with Lou Reed, produced by Richard Robinson, who'd previously worked with Flamin' Groovies on Flamingo. Later that year, Transformer came out, produced by David Bowie and Mick Ronson, with the fluke hit single \"Walk on the Wild Side,\" a veritable 1968 flashback.\n\nAny certified 1968 flashback would also have to include two anti-Beatle bands from Detroit, both signed by Elektra Records on the same day. The MC5, managed by John Sinclair, radical leader of the White Panthers, went in for blatant political diatribes on Kick out the Jams like \"Motor City Is Burning\" and the hit single \"Kick out the Jams,\" which they performed, expletives included, on the lawn of the International Amphitheatre in Chicago, where the 1968 Democratic National Convention was being staged. Thus, they were too topical to last and too political to appeal to anyone but Weathermen sympathizers. The Psychedelic Stooges, fronted by James Osterberg (aka Iggy Pop) were a different case. Sensing the shift in public temperament away from blissful optimism, they dropped Psychedelic from their name, leaving a timeless mix of rock 'n' roll dupes of the counterculture and the Three Stooges. Blasting into 1969 with anthems to cosmic ennui like \"1969,\" \"No Fun,\" and \"I Wanna Be Your Dog\" (from The Stooges), they completely undid the Beatle ethos of harmony (both inner and outer, to say nothing of musical). Disdaining the intellectual pretensions of a Lou Reed and thumbing their noses at the virtuosic aura of a Jimmy Page, the ragged, hungry, shameless Stooges established for the ages the nihilistic anything-goes punk ethos that would dominate the hearts of many rock 'n' roll musicians and fans from then on. The Sex Pistols, avatars of England's punk renaissance of 1976-77, covered \"No Fun\" on their first album. In 1970 Fun House followed (\"Loose,\" \"TV Eye,\" \"L.A. Blues,\" \"Dirt\"). Plagued by low sales, their classic Raw Power (\"Search and Destroy\") from 1973 was their final effort before Iggy Pop went solo in 1977 with The Idiot; a prodigal son returning in the midst of the punk renaissance.\n\nWhile the Stooges were too short-lived and commercially limited to do any immediate damage, by far the most visible and virulent anti-Beatle musician on the planet in 1969 was someone with a unique insider's knowledge of the whole schtick from the very beginning: John Lennon.\n\nIf it wasn't the worldly if not otherworldly Japanese artist Yoko Ono who turned John against his childhood pals (as many women in a romantic relationship will do), it was inevitable that either John or Paul would outgrow the claustrophobic hell on Earth they'd created with the Beatles. As competitive as they always were, especially with each other, it was only a question of who would quit first. Unlike the TV show Bewitched, where you could simply cast another Darren in the place of the departed Darren, it would have been impossible for the Beatles to exist without either of their two creative driving forces.\n\nAnd so, while John was doing everything he could to antagonize his former blood brothers by instituting Yoko as a virtual member of group, by introducing the fellows to the crass accountant Allen Klein, by inviting the mercurial Phil Spector into the mix, and by constantly threatening to make the Plastic Ono Band his full-time gig, it was actually Paul, pushed to the brink once too often, who made his exit public first, infuriating John.\n\nWhile still with the Beatles, John used the Plastic Ono Band as a vehicle for his ragged performance of mainly rock 'n' roll oldies at the Toronto Peace Festival. Live Peace in Toronto 1969 did feature Yoko's vocal pyrotechnics on \"Don't Worry Kyoko\" and John's admission of heroin use in \"Cold Turkey.\" But by the time of his next album, John Lennon\/Plastic Ono Band, the Beatles had broken up, and John found it more of a receptacle for his myriad confessions (\"Mother,\" \"My Mummy's Dead,\" \"God,\" \"Love,\" Working Class Hero\"). Sufficiently cleansed, he then directed his anger toward his former partner in Imagine, released in 1971. Hidden behind the utopian sentiments of the title song were the lethal mortar blasts of \"How Do You Sleep,\" featuring George Harrison on slide guitar. This song was apparently written in answer to Paul's own Lennon-bashing \"Too Many People,\" the B side of the 1971 hit \"Uncle Albert \/ Admiral Halsey,\" also found on Ram. True to form, Ringo's take on the breakup was a bit more accommodating in \"Early 1970,\" the B side of the 1971 hit \"It Don't Come Easy.\" Reserving all his B sides for Yoko, Lennon added scathing attacks to his album Imagine; \"Crippled Inside,\" \"Gimme Some Truth,\" and \"Jealous Guy\" further decimated the Beatle myth of godlike perfection. Some Time in New York City, released in 1972, smashed in the headstone, with MC5-like political rants about John Sinclair, Angela Davis, Attica, the troubles in Northern Ireland, and the plight of women, dark but relevant stuff the Beatles never would have touched. A few years later, British bands like the Sex Pistols (\"God Save the Queen\") and the Clash (\"London Calling\") would make provocative political statements de rigueur in the nascent punk scene of 1976.\n\nBefore he stepped aside in 1974 to allow the punks space to further mock the Beatles, John Lennon would first have to leave Yoko to get some space of his own. Mitigated a bit by Yoko's stage-managing of the affair from New York City, Lennon's year-and-a-half drunken sojourn to L.A. at least attracted some interesting musical bedfellows, among them Harry Nilsson, who'd been befriended by the Beatles as far back in 1967 when they declared the Pepperesque Pandemonium Shadow Show their favorite album of the year. The album featured \"Cuddly Toy\" (recorded by the Monkees), \"Ten Little Indians\" (a single by the Yardbirds), and a cogent rendition of \"She's Leaving Home,\" the Pepper track that caused Brian Wilson such heartache. Naturally enough, in between trashing hotels and getting thrown out of nightclubs, Lennon produced Nilsson's 1974 mishmash of an album Pussy Cats, which included covers of songs by Dylan, Jimmy Cliff, the Drifters, and Bill Haley and His Comets.\n\nSufficiently chastened, Lennon was summoned home by Yoko in 1975 to father a child and get himself back together. But before he semi-retired from the scene, he tacitly endorsed the theatrics of former folkie David Bowie, now the crown prince of a prominent anti-Beatles movement called glam, by collaborating with him on the 1975 single \"Fame,\" which became Bowie's first U.S. Number One hit.\n\nWith a dash of makeup, a touch of glitter, a bit of lipstick, and teased hair to match the Shangri-Las in their prime, a number of early 1970s bands took the \"are you a boy or are you a girl?\" accusations of the mid-sixties to flamboyant lengths in order to satirize them. Even Mick Jagger adopted this look for a while. The New York Dolls tried to get away with this on the streets of New York in 1973, but luckily they were based in Greenwich Village and had the streetwise chops to back up their outrageous performances. Songs like \"Trash,\" \"Personality Crisis,\" and \"Looking for a Kiss\" (from their first album New York Dolls, produced by Todd Rundgren) confirmed this. When they got Shadow Morton to produce their second album, those who knew his prior relationship to the sacred Shangri-Las were moved to the point of grateful tears. To them, Too Much Too Soon didn't disappoint (\"Human Being,\" \"Stranded in the Jungle,\" \"There's Gonna Be a Showdown\"). The trouble was, nobody else bought it. Soon, guitarist Johnny Thunders left the band and formed the Heartbreakers with punk scene pioneer Richard Hell (who had departed from Television) and former Dolls drummer Jerry Nolan. Formed in 1975, the band took two years to land a record deal, which happened only after Malcolm McLaren, the mad mastermind behind the Sex Pistols and a big Dolls fan, got them a gig touring with England's reigning punk royals: the Sex Pistols, the Clash, and the Damned. Unfortunately, the tour ended before it could even start, but the Heartbreakers' record L.A.M.F. came out on the Track label, featuring \"Chinese Rocks\" (written by Dee Dee Ramone and Richard Hell) and \"Born Too Loose.\" From the Dolls, lead singer David Johansen went on to a career as a popular entertainer (usually dressed in men's clothes). In 1991, Johnny Thunders, a long-time heroin addict, died under mysterious circumstances.\n\nBy the time the Heartbreakers made their album, glam had already given way to punk as the best anti-Beatles response going. American bands like Alice Cooper and Kiss distilled the camp satire for mass consumption as if it were a rock 'n' roll musical (e.g., the tribal travesty Hair). Former Zappa signee Cooper went heavy on the ghoulish eye shadow and featured ritual beheadings as part of his act. Kiss dressed up as fully realized comic book characters, breathing fire and spitting blood. But only David Bowie went so far as to become his own alter ego in his personal rock opera, in the role of a lifetime as Ziggy Stardust (modeled on Iggy Pop and Lou Reed), who first appeared in 1972's The Rise and Fall of Ziggy Stardust and the Spiders from Mars (\"Starman,\" \"Suffragette City\"). In 1972, Bowie provided glam with its defining anthem, \"All the Young Dudes,\" a Top 10 hit in England that he wrote for Mick Ronson's band Mott the Hoople. It appeared on their fifth album All the Young Dudes. Ronson and Bowie produced Lou Reed's 1972 album Transformer (\"Walk on the Wild Side,\" \"Vicious\") on which Lou appears to be wearing eyeliner on the cover.\n\nIn America, glam turned into arena rock relatively quickly, as working class heterosexuality reclaimed the music. But in effete Merry Olde England the genre reigned until punk obliterated it in the hearts of the musically obsessed; for everyone else, it remained extremely popular. In fact, some of the original glam acts went on to become the biggest hit makers of the decade. Gary Glitter, for instance had twelve Top 10 hits in the 1970s. Marc Bolan's T-Rex had eleven. The Sweet had ten. Bigger than all of them, Slade had thirteen, but they were really closer to the heavy metal offshoot known as sludge than glam. Not to be outdone, David Bowie had twelve hits (adding eleven more in the 1980s, including \"Under Pressure,\" a duet with Queen). Queen wound up with eighteen Top 10 hits, but only eight of them came in the 1970s. The quintessentially well-groomed Roxy Music (featuring Bryan Ferry) had ten Top 10 hits, but four of them came in the 1980s (Ferry had four Top 10 solo hits and one more in the eighties), by which time glam had morphed into early dance music and electronica.\n\nDance music and electronica were the last things on the mind of the four non-brothers who formed Ramones in 1974, their name itself tweaking Paul McCartney's stage name in the Silver Beatles (\"Paul Ramon,\" a name he also used as a sideman on two 1969 Steve Miller tracks, \"My Dark Hour\" and \"Celebration Song\"). If their look was mock-Beatles, their songs and their whole attitude was a mockery of the fat cat FM radio \/ arena sound of long guitar and drum solos and introspective lyrics brought on by too much money and too much LSD. It was more like a leap backward to the rockabilly rage of Gene Vincent. If Ramones were the anti-Beatles, by their time of their first album in 1976, they were the anti-Led Zeppelin as well. As a concept it was a home run, even if Ramones, featuring \"Blitzkrieg Bop,\" \"Beat on the Brat,\" \"Judy Is a Punk,\" and \"Now I Wanna Sniff Some Glue,\" was more like the commercial equivalent of a bloop single.\n\nNot surprisingly, Ramones were welcomed by the rising, raging punk movement in England, where they opened for Flamin' Groovies and were befriended by the Clash and the Sex Pistols. Subsequent seventies albums: Leave Home (\"Gimme Gimme Shock Treatment,\" \"Sheena Is a Punk Rocker,\" \"Pinhead\"), Rocket to Russia (\"Rockaway Beach,\" \"Teenage Lobotomy\"), Road to Ruin (\"I Wanna Be Sedated\"), and the Phil Spector-produced End of the Century (\"Rock 'n' Roll High School,\" \"Do You Remember Rock 'n' Roll Radio?\" \"Chinese Rocks,\") invariably did better in England than the U.S., with their remake of \"Baby, I Love You,\" Spector's Ronettes classic, hitting the Top 10. Even 1985's wonderful anti-Reagan song, \"Bonzo Goes to Bitburg,\" hit the Top 100 there.\n\n\"Chinese Rocks\" first appeared on the album by the Heartbreakers and was covered by the Sex Pistols' notorious bassist Sid Vicious on his 1979 live solo album Sid Sings, miserably recorded in 1978 and released after he committed suicide while awaiting a murder rap. Vicious's work was a classic deconstruction of several eras of music as well his own work with the Sex Pistols. In addition, Vicious also took on Las Vegas (Frank Sinatra's \"My Way,\" written by Paul Anka), the Brill Building (the Monkees' (\"I'm Not Your) Stepping Stone,\" by Boyce and Hart), rockabilly (Eddie Cochran's \"Something Else\"), glam (\"Born to Lose\" by the Heartbreakers), and classic punk (\"I Wanna Be Your Dog\" and \"Search and Destroy\" by the Stooges).\n\nBy the time of Sid's death, the Sex Pistols' mission had been accomplished. With anti-anthems like \"God Save the Queen\" (their answer to the Beatles' impish \"Her Majesty\"), \"Anarchy in the UK,\" \"Holidays in the Sun,\" and \"Pretty Vacant\" (from 1977's Never Mind the Bollocks Here's the Sex Pistols), they'd said it all and spat it all in the face of the bloated music industry, to say nothing of Britain's decaying working-class culture. If the Damned had gotten there first, with their 1976 single \"New Rose\" presaging a new anti-studio, anti-virtuoso feeling in the air, the Sex Pistols brought it all crashing and burning to the surface within a few short, scathing, well-publicized years.\n\nThe Clash were just as uncompromising, recalling in their elemental rage no less a prophet of protest than Phil Ochs, who committed suicide only a year before their first album, The Clash, re-established the d\u00e9class\u00e9 form to its prior relevance, with incendiary tracks like \"London's Burning,\" \"White Riot,\" and the reggae-inspired \"Police and Thieves.\" Give 'Em Enough Rope followed with \"Stay Free\" and \"All the Young Punks,\" their answer to David Bowie and the false front of glam. 1979's London Calling provided their credo, \"Death or Glory.\"\n\nFollowing the lead of Ramones, the Clash, and the Sex Pistols, punk rock stormed the barricades of Europe and America, giving rise to venom-fueled music unsurpassed in any era, as well as a return to the funky ethos of the independent 45. As early as 1974 on the Lower East Side of New York, Patti Smith, a rock 'n' roll poet with the voice of a Shangri-La, gave us \"Piss Factory.\" Richard Hell's old band Television recorded \"Little Johnny Jewel.\" In 1976, Hell and his new band the Voidoids released \"Blank Generation.\" In Cleveland, Pere Ubu released \"Final Solution\" and then \"Thirty Seconds over Tokyo.\" In Akron, Devo released \"Monogloid.\" From Cleveland, the Dead Boys, produced by Genya Ravan, gave us \"Sonic Reducer,\" from Young and Loud and Snotty in 1977. Punk finally reached California in 1978 with \"Wasted\" by Black Flag and \"Adult Books\" by X (with \"We're Desperate\" on the B side). In 1979, the Dead Kennedys released \"California \u00dcber Alles.\"\n\nEngland, however, was still storm central. Iggy Pop went to London to record The Idiot (\"China Girl,\" \"Sister Midnight\") in 1977 and Lust for Life (\"The Passenger\") in 1978, produced by David Bowie, to whom many in the punk community claimed he'd sold his soul, while others said it presaged the post-punk era of electronica. \"Orgasm Addict\" by the Buzzcocks was a key 1977 single (followed by their era-defining \"Ever Fallen in Love\"). Their sometime touring partners Joy Division released the well-regarded Unknown Pleasures in 1979, preceded by the non-album single \"Transmission.\" Then there was the thrilling agony of \"Oh Bondage, Up Yours,\" by X-Ray Spex. As opposed to all the others, X-Ray Spex was led by a girl, teenager Poly Styrene. In 1978, Susan Dallion took a page out of the Yoko Ono playbook, moving over from the avant-garde art scene to become Siouxsie of Siouxsie and the Banshees. On their first album The Scream, the banshee label was particularly apt on their cover of \"Helter Skelter.\" In the U.S. the album included their only UK Top 10 hit, the brilliant \"Hong Kong Garden.\" 1978 was the year Kate Bush was discovered by Pink Floyd's David Gilmour, sounding a bit like Yoko Ono on her debut single \"Wuthering Heights,\" from the musically bountiful The Kick Inside (\"Man with the Child in His Eyes,\" \"Them Heavy People\"), which spent four weeks at Number One on the UK charts. The Slits, fronted by three females who appear clad mainly in mud on the album cover, toured with the Clash and were seen in The Punk Rock Movie before their first album Cut (\"Typical Girls\") arrived in 1979. More of a merger of Yoko and Nico, transplanted Detroit girl Lena Lovich made her debut with \"Lucky Number,\" originally the B side of their Tommy James and the Shondells cover of \"I Think We're Alone Now.\"\n\nTwo of the most gifted artists of the punk period quickly shed that sometimes pejorative classification for the more civilized new wave label. The Police (Sting, Andy Summers, and Stewart Copeland) brought a reggae-lite touch to songs like \"So Lonely,\" \"Can't Stand Losing You,\" and \"Roxanne,\" from 1978's Outlandos d'Amour. They had no trouble parlaying their virtuosity into eight Top 20 hits in the U.S. and ten Top 10 hits in the UK, including \"Invisible Sun\" (concerning the troubles in northern Ireland). Sting was even more successful as a solo artist, producing ten Top 20 hits on his own by the end of the century. Elvis Costello was the other prolific product of this era, arriving almost fully formed in 1977 with My Aim Is True (\"Alison,\" \"Less Than Zero,\" \"Watching the Detectives\"). With his Buddy Holly horn rims and a second coming of Presley swagger, Costello spewed songwriting angst and wordplay equal to Dylan in his heyday or John Lennon in his anti-Beatles prime, topping his debut effort with This Year's Model, in which he took on Bob Dylan doing Chuck Berry in \"Pump It Up.\" Thereafter, he explored genres including country, R&B, chamber music, and classic pop, using the same mordant facility. All that was left to complete the circle was to write \"Veronica\" (1989) with Paul McCartney, which resulted in his biggest U.S. hit.\n\nBy the end of the decade, Ramones were already caught in their own limited formula, although that didn't prevent them from releasing fourteen albums, concluding with Adios Amigos in 1995. The Sex Pistols were gone. The Clash were releasing double and triple albums. They made the American Top 40 in 1980 with \"Train in Vain (Stand by Me),\" which caused everyone to doubt the band's remaining authenticity. Picking up the chalice, the Gang of Four released \"Damaged Goods\" in 1978, with \"(Love Like) Anthrax\" on the B side, containing the line \"I feel like a beetle (Beatle?) on its back.\" Turning punk's depressive outlook into a genre in itself, the Cure released the controversial \"Killing an Arab\" in 1978. Bauhaus followed with the nine-minute epic \"Bela Lugosi's Dead.\"\n\nOn May 18, 1980, the emotionally volatile fans of the genre received a kick in the teeth when Joy Division's driving force Ian Curtis committed suicide just before the band's first American tour. Their new song \"Love Will Tear Us Apart\" thus became his epitaph. One of the strange side effects of this suicide was that it prevented Martin Harnett, who produced both Joy Divison albums, from feeling up to taking on a new assignment. Thus, the U2 project fell into the hands of a young recording engineer named Steve Lillywhite, who'd recently played the glockenspiel on Siouxsie and the Banshees' hit single \"Hong Kong Gardens,\" which was also hailed in the press for its guitar parts. He did the same on \"I Will Follow,\" the leadoff track from U2's debut album Boy, which was released October 20, 1980. The song was an immediate underground hit. At last they'd found their Jimmy Page alternative sound!\n\nBut their euphoria was to be short lived, when on the evening of December 8 they learned, along with the rest of the worldwide musician and music fan community, that John Lennon had been shot in the back by a crazed lunatic outside of the Dakota Apartments in New York City. John's comeback single \"(Just Like) Starting Over,\" would hit the top of the American charts three weeks later, informing the public of his new personal and musical maturity.\n\nClearly, a reappraisal was in the wind.\n\nR.E.M., early 1980s.\n12\n\nTHE AMERICAN ROOTS REVIVAL\n\nAmerican rock 'n' roll fans, especially those of a certain age, had been mourning the loss of Elvis Presley since August 16, 1977, when their musical icon was found dead in his bathroom. After his comeback with \"Suspicious Minds\" in 1969, he'd had twenty more Top 40 singles, including some of the greatest performances of his career, including \"Rubberneckin'\" (from his last movie, A Change of Habit), \"Burnin' Love\" (his last Top 10 hit), James Taylor's \"Steamroller Blues,\" and Chuck Berry's \"Promised Land.\" In addition, he ended a thirteen-year absence from the country Top 10 in 1970 with \"There Goes My Everything.\" Between 1973 and 1975 he put four more songs into the upper reaches of the country charts: \"I've Got a Thing About You, Baby,\" \"Help Me,\" \"It's Midnight,\" and \"Hurt.\" In 1976, \"Moody Blue\" became his first country Number One since 1957. His next Number One wouldn't come until four days after he died, when \"Way Down\" was awarded the top spot for a week. A few months later, \"My Way\" was his first posthumous country Top 10. In 1978, \"Unchained Melody\" and \"Are You Sincere\" joined it. (The always tasteful Sid Vicious released his own version of \"My Way\" in 1978.) In 1979, Elvis's ghostly hold on the country charts continued with \"There's a Honky Tonk Angel (Who Will Take Me Back In).\" And in 1981, \"Guitar Man\" became his last country Number One, released a few weeks after John Lennon was murdered, peaking a week after Yoko Ono's \"Walking on Thin Ice\" began its ten-week run on the American singles chart.\n\nThus conjoined in sorrow, fans of two overlapping eras of memories and magic came together to stanch the drift of the anti-Elvis\/anti-Beatles forces in the world, to say nothing of the music industry. This would mean a return to the roots, epitomized by the early rock 'n' roll of Elvis and Gene Vincent, and harmony-based vocals attached to jangly guitars, epitomized by Buddy Holly, the Everly Brothers, the Beatles, Bob Dylan, and the Byrds. As an example of how outre the notion of the jangly guitar in a Top 40 single setting had become by 1977, when Gainesville, Florida's Tom Petty and the Heartbreakers released \"American Girl,\" it was classified as alternative rock. And when transplanted Miami native Debbie Harry, enamored of the Shangri-Las, formed Blondie on the Lower East Side of Manhattan and released \"X Offender\" (produced by ex-Brill building hand and Angels producer Richard Gottehrer), it was called punk rock.\n\nOutsized decibels reigned in America, whether at the disco, the converted hockey rink, or in the lungs of the angry punks within shouting distance of those two fat cat palaces. The best a former Beatles fan could hope for was the occasional melodic single from Cheap Trick. But their first few albums were produced by Tom Werman, who also produced Aerosmith, so such moments were usually buried under a professional gloss of studio noise. Early in 1980 they put out the EP Found All the Parts, containing their version of \"Day Tripper.\" When they snagged Beatles producer George Martin for their next full length album All Shook Up, hopes were raised among the faithful. But the album was a commercial flop.\n\n\"My Sharona,\" by the perfectly packaged Los Angeles group the Knack, was a similar flashback to the power pop era, but the power petered out after \"Good Girls Don't\" and \"Baby Talks Dirty\" revealed them more as precursors to eighties heavy metal hair bands like Ratt and Poison.\n\nCloser to the target was Shoes, a Zion, Illinois garage band (actually a living room band), that put out a brilliant independent power pop single in 1977 called \"Tomorrow Night.\" It was re-released on their 1979 major label debut Present Tense. But further albums like Tongue Twister and Boomerang failed to capitalize on its underground success.\n\nThe first authentic taste of the changing zeitgeist was provided by Robert Gordon, a member of the punk rock group Tuff Darts, who appeared in the independent film Unmade Beds in 1976, which also featured Blondie's Debbie Harry. Gordon soon came to the attention of Blondie's producer Richard Gottehrer, who produced Gordon's first three albums, the first two of which featured rockabilly guitar legend Link Wray, with nods to giants of the forgotten form including \"Summertime Blues\" and \"Twenty Flight Rock\" (Eddie Cochran), \"Boppin' the Blues\" (Carl Perkins), \"The Way I Walk\" (Jack Scott), \"A Red Cadillac and a Black Moustache\" (Warren Smith), and \"Red Hot\" (Billy \"The Kid\" Emerson). Released right after the King's passing in 1977, \"Red Hot\" made the charts.\n\nIn 1980, the Blasters reaffirmed Gordon's basic sentiments with the classic \"American Music\" from the album of the same name, the first effort in the prolific output of the Alvin Brothers (Phil and Dave). Dave Alvin's 1987 solo album Romeo's Escape featured \"The Fourth of July,\" which remains a classic of the genre. The Latin roots rock band Los Lobos broke through in 1984 with How Will the Wolf Survive?, more than fulfilling the promise of their first independent release in 1978, Just Another Band from East L.A.\n\nRobert Gordon's next major impact on the new roots revival moved him closer to the jangly guitar camp with the 1981 album Are You Gonna Be the One, on which he covered three tunes by newcomer Marshall Crenshaw: \"She's Not Mine Anymore,\" \"But, But,\" and \"Someday, Someway.\" Crenshaw's version of the latter song cracked the Top 40. But Crenshaw wasn't exactly a newcomer; in the late '70s he played John Lennon in the traveling Beatles-clone extravaganza Beatlemania. During the early part of his subsequent solo career, which started in 1982, he continued to play that up as well as down, until his quirky songwriting chops took over with tunes like \"Cynical Girl,\" \"Brand New Lover,\" \"Whenever You're on My Mind\" (covered by Ronnie Spector), and \"You're My Favorite Waste of Time\" (covered by Bette Midler). In 1987 he played Buddy Holly in the movie La Bamba, about the life of fallen rockabilly idol Ritchie Valens.\n\nSan Francisco's Flamin' Groovies had been waiting their whole career for this kind of music to return, having been dubbed hopelessly retro ever since their 1969 debut album. After losing Roy Loney, one of their founding members, they moved to England, where they had the good fortune to meet fellow rockabilly fanatic Dave Edmunds, who had almost single-handedly kept the genre alive with albums like 1975's Subtle As a Flying Mallet, which contained covers of early rock classics like \"Maybe,\" \"Da Doo Ron Ron,\" \"Baby, I Love You,\" and \"A Shot of Rhythm and Blues.\" In 1976 Edmunds produced the Groovies' Shake Some Action, which contained covers of tunes by the Beatles (\"Misery\"), Chuck Berry (\"Don't You Lie to Me\"), and Larry Williams (\"She Said Yeah\"). In 1978, The Flamin' Groovies Now arrived, with covers of the Beatles (\"There's a Place\"), the Stones (\"Paint It, Black,\" \"Blue Turns to Grey\"), and the Byrds (\"I'll Feel a Whole Lot Better\"). Jumpin' in the Night, issued in 1979, went even further, with covers of songs by Bob Dylan (\"Absolutely Sweet Marie\"), the Beatles (\"Please Please Me\"), and the Byrds (\"5D,\" \"Lady Friend,\" and \"It Won't Be Wrong\"). By this time Edmunds had returned to his own recording career, releasing Get It (1977), Trax on Wax 4 (1978), and Repeat When Necessary (1979). With Nick Lowe, Edmunds formed Rockpile, which released Seconds of Pleasure (1980), sold with a bonus EP called Dave Edmunds and Nick Lowe Sing the Everly Brothers. This contained their versions of \"Take a Message to Mary,\" \"Crying in the Rain,\" \"When Will I Be Loved,\" and \"Poor Jenny.\" In 1984, Edmunds took things one step further when he produced the Everly Brothers' comeback album EB '84, which contained the hit single \"On the Wings of a Nightingale,\" written by Paul McCartney.\n\nTaking a similar route as Flamin' Groovies, from the U.S. to Dave Edmunds' UK doorstep, were Stray Cats, whose name was taken from the 1974 rock flick Stardust. When they arrived in London in 1980, they caught the attention of every important band in town. Their first single \"Runaway Boys,\" produced by Edmunds, hit the UK Top 10 in December, shortly after John Lennon was shot. Two months later they recorded \"Rock This Town,\" which wasn't released in the U.S. until the fall of 1982. It finally reached the Top 10 in December. Neither their eponymous first album nor its follow-up Gonna Ball were released in the U.S., but the album that did eventually come out, Built for Speed, combined tracks from each, including \"Rock This Town,\" \"Stray Cat Strut,\" and the Eddie Cochran cover \"Jeanie, Jeanie, Jeanie.\"\n\nThe Athens, Georgia-based B-52s mixed the hairstyles and attitudes of early sixties girl groups with a sensibility of the approaching electronic dance era. In 1978, their monster track \"Rock Lobster\" hit the UK Top 40, and along with \"Cool,\" a 1979 single by four University of Georgia art students calling themselves Pylon, put Athens on the rock 'n' roll map. One of America's great college towns, Athens not only spawned a music scene but an entire sub-genre known as college rock. Not since the heyday of the baby boom elitism known as folk rock in the mid-sixties did a sound and feel take over a hungry sophisticated demographic, which had been enraged by the working class high school dropout mentality currently besieging their music. No group epitomized this genre better than R.E.M.\n\nLearning their instruments at frat parties and rathskellers as they went along, Michael Stipe, Peter Buck, Mike Mills, and Bill Berry would soon become names almost as recognizable as John, Paul, George, and Ringo. Their first homegrown single \"Radio Free Europe,\" was recorded at Mitch Easter's Drive-In Studios, which was literally that: a drive-in studio housed in Easter's family garage in Winston-Salem, North Carolina, a mere five-hour drive from Athens and an hour-and-a-half as the crow flies from Chapel Hill, another famous college town.\n\nEaster had some credibility to back up his modest digs. His band the Sneakers had released In the Red in 1978, which aptly summed up the spirit of the coming anti-Zeppelin, anti-punk, and anti-Disco revolution. In 1972, Jonathan Richman recorded an homage to the Velvet Underground's \"Sister Ray\" called \"Roadrunner,\" a glistening ode to being alone in a car in New England on a Saturday night. This thrilling single didn't see the light of day until the belated but timely release of The Modern Lovers in 1976. Although the Modern Lovers were influenced by the Velvet Underground, Jonathan Richman was the mortal opposite of Lou Reed, a gentle soul trapped in the body of a compulsive performer. While \"Roadrunner\" has gained a number of punchy rock performances over the years (the Sex Pistols and Joan Jett and the Blackhearts among them), Richman pretty much disowned them, striking out in a different direction with a different band the same year the album came out, and one that was a lot closer to the late seventies' return to Americana.\n\nTwo of the Sneakers, Chris Stamey and Will Rigby, went on to form the dB's; in 1981 they released Stands for Decibels, with \"Black and White\" as the first single. Next came Repercussion in 1982, which included the single \"Amplifier.\" Around the same time, Mitch Easter formed Let's Active, gaining a label deal by virtue of a perfect demo EP in 1983 called Afoot. That same year, he produced R.E.M.'s first album Murmur with Don Dixon, another local Chapel Hill hero, who had actually formed the band Arrogance in his University of North Carolina dorm room. Arrogance released a number of albums locally and had two shots at national recognition on major labels: Rumors in 1976 and Suddenly in 1980, but were unable to get over. He went on to produce albums by such stalwarts of the genre as Marshall Crenshaw, Richard Barone, Guadalcanal Diary, Tommy Keene, the Smithereens, Matthew Sweet, Marti Jones, and Beat Rodeo.\n\nIncluded in Murmur, the defining track that launched a thousand college bands was the re-recorded version of \"Radio Free Europe,\" which made the singles charts. The band and Don Dixon claim they still preferred the original 1981 version on law student Jonny Hibbert's Hib-Tone label, the first of four releases, which also included singles by the Throbs, Three Hits, and Harold Kelling (lead guitarist of the quirky artsy Atlanta-based group the Hampton Grease Band) before Hibbert closed up shop. Prior to their first full album, R.E.M. issued an EP called Chronic Town, from which came the shadowy but compelling video for \"Wolves, Lower,\" which may have been seen by the band's few intrepid friends on 120 Minutes, an alternative rock program on fledgling cable channel MTV.\n\nNeither rap nor metal nor British new wave, jangly roots bands rarely showed up anywhere else on that rigidly programmed operation in the early eighties. But that suited their underground bent just fine. All around the country, the back-to-rock-and-roll movement was taking root. In Dayton, Ohio, Robert Pollard gave up the bar band circuit for his own twisted path, which centered around a string of limited run independent releases, one of which found its way into the record collection of R.E.M., immediately upping his profile immensely. In 1994, Pollard and friends finally quit their day jobs after Bee Thousand (Scat, 93) established them on the jangly college circuit.\n\nIn New Jersey, the Smithereens released their first two EPs, Girls About Town (1980) and Beauty and Sadness (1983), the latter particularly evoking the Beatles (if the Beatles, that is, had grown up listening to the Beatles, the Velvet Underground, and Ozzy Osbourne). Also from Jersey came the Feelies, which put the city of Hoboken on the musical map with Crazy Rhythm (1980), just in time to influence R.E.M. A highlight track was their cover of the Beatles' \"Everybody's Got Something to Hide (Except Me and My Monkey).\" Influenced by the Smithereens and the Feelies, Fountains of Wayne broke out of New Jersey as well, by way of Massachusetts a decade later, led by clever songwriter Adam Schlesinger, whose quirky output (\"Stacy's Mom\"), established the anti-Springsteen, anti-Bon Jovi sound. By this time the quirky\/clever niche sometimes associated with the Beatles had been honed to a verbally manic pitch by the Brooklyn, New York-based duo They Might Be Giants (\"Ana Ng\"), the New Hope, Pennsylvania duo Ween (\"Push Th' Little Daisies\"), and the Toronto, Ontario duo of Ed Robertson and Steve Page, lead vocalists of Barenaked Ladies (\"Be My Yoko Ono\").\n\nThree hours south of Toronto and a few years earlier in Jamestown, New York, near Buffalo, the members of the newly-christened 10,000 Maniacs added intense and dreamy seventeen-year-old lead singer Natalie Merchant to the lineup in 1982 and released the EP Human Conflict Number Five. Secrets of the I Ching followed in 1983, which included the popular track \"My Mother the War,\" which also found its way onto 1985's The Wishing Chair. Their breakout album In My Tribe came in 1987, produced by Paul McCartney's almost-brother-in-law Peter Asher. Also in the area listening hard were Goo Goo Dolls, who followed the Replacements' model from punky sloppiness to Beatlesque balladry, culminating in their big 1995 hit \"Name.\" Down in New York City, the experimental no wave band Sonic Youth, went in the opposite direction, putting out their 1982 debut album, the self-titled Sonic Youth. Their sound was relatively mellow and rootsy when compared to the atonal explorations and jangly noise that would follow, among them the epics \"Teenage Riot\" and \"Silver Rocket,\" from the 1988 masterwork Daydream Nation.\n\nIn Milwaukee, Wisconsin, a stone's throw from Madison, home to the University of Wisconsin, and perhaps the most legendary of all college towns (especially during the late sixties), the Violent Femmes put out their own eponymous mixture of folk and punk anthems, an album containing their most famous tracks, many of which have gone on to a pop cultural life way beyond the scope of the band's original impact. \"Blister in the Sun\" was featured on an episode of the classic teen angst TV series My So-Called Life, in the semi-classic John Cusack-as-hit-man movie Grosse Pointe Blank, the 2007 coming-of-age film Adventureland, and referenced in John Green's coming-of-age novel Paper Towns. (It also appears in the movie Rocket Science along with \"Kiss Off.\") \"Add It Up\" is played by Ethan Hawke in the 1994 generation-co-defining movie Reality Bites. \"Good Feeling\" appears on an episode of the popular TV show How I Met Your Mother as Marshall and Lily's \"song.\"\n\nFurther north on Interstate 35 in Minneapolis, just 200 miles south of Bob Dylan's birthplace in Hibbing, the Replacements carved out a space in the American rock dream of do-it-yourself glory following the new wave-influenced Suburbs to the local TwinTone label (which released Suburbs' first EP in 1978). Influenced by Dylan, Alex Chilton, and Kiss, the Replacements' first three releases, the 1981 album Sorry Ma, Forgot to Take out the Trash, the 1982 EP Stink, and the 1983 album Hootenanny represented their sloppy-drunk, let-it-all-hang-out-on-stage-Beatles-in-Hamburg phase. By the time Let It Be came out in 1984, Paul Westerberg had emerged as a songwriter who was not afraid to reveal his passionate jangly guitar side, in songs like \"Sixteen Blue,\" \"Unsatisfied,\" and \"I Will Dare,\" which features a stellar rockabilly guitar solo by R.E.M.'s Peter Buck. Soul Asylum followed in their footsteps at TwinTone, along with the Jayhawks.\n\nAs usual, California brought the psychedelic influence to the party. While San Francisco had never truly recovered from the fall of the Summer of Love, down in L.A., a whole scene, dubbed \"the Paisley Underground\" by Michael Quercio (a member of one of its core bands, the Three O'Clock) emerged in 1981 as a love-and-peace-and-jangly-guitar alternative to bands like the Dead Kennedys, who were themselves an in-your-face punk rock alternative to the Beatles. The Three O'Clock's 1983 album Sixteen Tambourines featured the college hit \"Jet Fighter.\" They put out five albums and called it a day after 1988's Vermillion.\n\nThe biggest band in the area, however, wasn't part of that scene. The Go-Gos were more like reincarnated surfer babes than flower children, an all-girl band with one foot in the pro and the other in the anti-Beatles camp, aligned with both the punk and power pop sounds along with a healthy commercial overlay of British new wave. Their first two singles, \"Our Lips Are Sealed\" (1981) and a re-recorded \"We Got the Beat\" (1982) were big hits, with the latter peaking at number two. The album Beauty and the Beat was number one for six weeks.\n\nAs opposed to the punky pop pretensions of the Go-Gos, the bands in the Paisley Underground were resolutely true to their name. The Velvet Underground-influenced Dream Syndicate was formed at the University of California Davis by Steve Wynn, whose seven-and-a-half-minute opus \"The Days of Wine and Roses\" (the title track of their first full-length album in 1983) was his high water mark. By this time Wynn had his own Down There label, on which he released the 1982 self-titled Green on Red EP. In 1985, Wynn and Green on Red vocalist Dan Stuart put out The Lost Weekend under the names Danny and Dusty, with great tracks like \"Miracle Mile,\" \"Song for the Dreamers,\" and a cover of Bob Dylan's \"Knockin' on Heaven's Door.\" The Long Ryders, featuring Sid Griffin on guitar and autoharp, harkened back on their 1983 release 10-5-60 to the Byrds and the Flying Burrito Brothers, and forward to the alt-country sound of Uncle Tupelo, Wilco, Son Volt, and the Jayhawks.\n\nEasily the most successful of these bands was the all-girl Bangles, a foursome consisting of Susanna Hoffs and Vicki Peterson on guitars, Debbi Peterson on drums, and Annette Zilinskas on bass. Their first EP Bangles was released in 1983 along with the single \"The Real World\" (ten years prior to the debut of the MTV program of the same name) before the label folded and Zilinskas quit. She was replaced by Michael Steele, previously of the all-girl band the Runaways. Bangles' next release was All Over the Place (1984), which showed their Beatles influences on covers of songs by Emitt Rhodes (\"Live\") and Katrina and the Waves (\"Going Down to Liverpool\"). Different Light, released in 1986, further added to their tasteful credibility with songs written by Prince (\"Manic Monday\"), Alex Chilton (\"September Gurls\"), Jules Shear (\"If She Knew What She Wants\") and Akron, Ohio scene maker Liam Sternberg (\"Walk Like an Egyptian\"). \"Manic Monday\" hit Number Two on the charts in 1986 while \"Walk Like an Egyptian\" reached Number One.\n\nThe rootsy heartland sound was just about the only competition on the singles chart in the early eighties for the anti-rootsy sound coming from England. Previously, the mood of music fans was too dismally anti-Beatles to sustain such a movement after the 1975 success of \"Born to Run\" by New Jersey's premier heartland rocker Bruce Springsteen. But the Boss roared into a much more favorable climate in 1984 with an unprecedented run of seven Top 10 singles from his album Born in the USA. The way had already been greased by jangly old Tom Petty and the Heartbreakers, who hit with \"Don't Do Me Like That\" (1979), \"Refugee\" (1980), \"The Waiting\" (1981), and \"You Got Lucky\" (1982). Along with Springsteen came John Cougar Mellencamp, Indiana's hippest performer since James Dean, with I Need a Lover\" (1979), \"Ain't Even Done with the Night\" (1981), \"Hurts So Good\" (1982), and \"Pink Houses\" (1983). Huey Lewis and the News, a bluesy bar band from San Francisco, hit it big in 1984 with \"Heart and Soul,\" \"I Want a New Drug,\" \"The Heart of Rock and Roll,\" and \"If This Is It,\" with all making the Top 10. Original retro rocker John Fogerty of Creedence Clearwater Revival had a successful comeback album in 1985 called Centerfield with \"The Old Man Down the Road\" hitting the Top 10. Even Alex Chilton came back in 1985 (after a five year album layoff), releasing the soulfully rootsy EP Feudalist Tarts, which contained \"Thank You John,\" his tribute to John Lennon.\n\nTwo female singers who couldn't have been more different also emerged in 1984 to help define their gender for the remainder of the decade. Brooklyn's Cyndi Lauper evoked girl group history in her tormented voice on the album She's So Unusual, released in 1983, which transformed indie rock obscurities like Robert Hazard's \"Girls Just Want to Have Fun\" and Tom Gray's \"Money Changes Everything\" into empowering anthems. The album produced four hit records, including \"Time After Time,\" which went to Number One. Lauper's quirky persona led her to an erratically eclectic career after that, including TV appearances on Mad About You, The Simpsons, As the World Turns, and The Celebrity Apprentice and in a Broadway production of The Threepenny Opera. A good friend of Yoko Ono's, Lauper was part of Lennon-Ono's Peace Choir along with Adam Ant, Bruce Hornsby, Sebastian Bach, Lenny Kravitz, Randy Newman, Iggy Pop, Bonnie Raitt, and Little Richard (who officiated at her wedding the same year). She performed Sean Lennon's rewrite of \"Give Peace a Chance\" as a response to the 1991 Gulf War.\n\nThe debut of Bay City, Michigan's future diva Madonna Ciccone, was only slightly less sensational, with three Top 20 hits in 1984: \"Holiday,\" \"Borderline,\" and \"Lucky Star.\" But Madonna would remain the more consistent hit maker and female role model, adapting to the shifting sonic trends of the next two decades with the relentless smarts and intuition of a corporate barracuda.\n\nThis all-American logjam on the Top 40 not only led to a long-awaited renaissance of listening pleasure for fans of singles on the radio, but produced the unexpected side effect of fueling one of the last great underground rock scenes of the decade (if not the century) in and around the Roxbury, Massachusetts-based Fort Apache Studios. Barred from airspace by a new vinyl ceiling clogged with radio-friendly superstars (seven singles from one album eliminated a bunch of slots that might otherwise have gone to three or four different bands), they were thus free to thrash out their fledgling styles, lineups, and romantic entanglements leading to classic songs without undue interference from the bean counters. While this resulted in some endearing artifacts left behind for historians and local scene junkies to treasure, it also made for careers long on potential and short on beans: witness early scene makers the Lemonheads' prophetically-titled first EP Laughing All the Way to the Cleaners, released in 1986. Lead singer Evan Dando was among those who would fail the loudest, as pop star and potential new boyfriend, with 1992's \"It's a Shame About Ray\" being his surviving legacy.\n\nBy then, attention was being paid to the scene by virtue of 'Til Tuesday's 1985 Top 10 hit \"Voices Carry,\" inspired by lead singer and songwriter Aimee Mann's tortured love life. Mann would eventually end her relationship with the band and resume a tortured career as a solo artist with Whatever in 1993, which featured songs called \"I Should've Known,\" \"Put Me on Top,\" \"I Could Hurt You Now,\" and \"I've Had It.\" Her next album was called I'm with Stupid.\n\nIn a neighborhood that was dangerous at night and iffy by day, the Fort Apache studios were right next door to a deserted ghetto schoolyard, looking out on a row of empty lots filled with abandoned cars; in other words, it was the perfect setting for uncompromising rock 'n' roll. Among the early discoveries of studio manager Gary Smith was Throwing Muses, a Rhode Island band featuring Kristen Hersh and her stepsister Tanya Donelly. Smith suggested the band move to Boston, which it did. At the Rat in Boston, Smith took an interest in a band opening for Throwing Muses called the Pixies (the brainchild of University of Massachusetts students Black Francis and Joey Santiago), which he would later produce. Throwing Muses self-titled debut came out in 1986 on the British 4AD label featuring nine songs by Hersh and one by Donelly. Their second album House Tornado was produced by Smith and engineered by Paul Kolderie at Fort Apache. This album featured nine songs by Hersh and two by Donelly. By this time Smith had delivered the legendary The Purple Tape, the Pixies' original demo tape, which was turned into the Come on Pilgrim EP. Surfer Rosa followed (\"Bone Machine\") and with Doolittle, they had arrived (\"Debaser,\" \"Here Comes Your Man,\" \"Monkey Gone to Heaven\"). At this point, original bass player Kim Deal started flirting with a side project called the Breeders, along with the disaffected Tanya Donelly. Their first release Pod, came out in 1990 on 4AD (which was also the Pixies' label), featuring eleven songs written or co-written by Kim Deal, one Deal-Donelly collaboration, and a cover of the Beatles' \"Happiness Is a Warm Gun.\"\n\nSmith produced Earwig (1989) and Sunburn (1990), the second and third albums by the Blake Babies, a band Juliana Hatfield started at the Berklee College of Music with John Strohm. Smith also produced Hatfield's first solo album, Hey Babe in 1992, on which she was joined by her once and future, on-again off-again love interest Evan Dando on backup vocals and guitar. Hatfield's commercial high point occurred when her track \"Spin the Bottle\" was used in the 1994 film Reality Bites, in which Dando made a cameo appearance.\n\nOne of the greatest single tracks to emerge from Fort Apache was \"Freak Scene\" by Dinosaur Jr. in 1988, which led off the band's third album Bug, a heartfelt summation of the indie rock ethos, both through its evocative, painful lyrics and its evocative, painful guitar part, a direct descendant of Neil Young's work on \"Cinammon Girl.\" After completing this album, bass player Lou Barlow was drummed from the band, which left him plenty of time to focus on his side project Sebadoh, whose debut album, The Freed Man, containing 31 tracks, was released in 1989. His magnum opus, \"Gimme Indie Rock,\" released in 1991, eventually led the band to sign with Sub Pop, whose first release, issued in 1992, was Smash Your Head on the Punk Rock. Barlow soon stumbled into a soundtrack assignment for the film Kids, from which \"The Natural One\" emerged as a fluke Top 30 single in 1995 under the name the Folk Implosion.\n\nAs Lou Barlow recalled, \"The Natural One\" was recorded at Fort Apache. Summing up the indie rock experience almost as eloquently as his partner J. Mascis had, Barlow went on to explain, \"I knew it would be okay, because it was Fort Apache. I knew I could walk in there cold and it would just work out. We recorded the song in six hours. How beautiful is that? For me, it was the creative thrill of it and also flying in the face of any kind of doing what you're told. Maybe even at that time at Fort Apache there was a certain feeling of the right way to do things. The right mike to put on this. The right guitar to play this part. But I always really disliked those notions.\n\n\"Nevertheless, to wind up with a hit single was totally weird,\" he said. \"Once we had the success we obviously completely complicated the situation. I hate the feeling that when I walk into a situation, someone's expecting me to do something brilliant. When it comes to 'here we are following up our major label hit,' that sucks.\"\n\nUltimately, it would be the Seattle-based Sub Pop label that would define indie rock beginning in 1987, with single, EP, and album releases by a virtual who's who of the scuzzy, seat-of-the-pants genre. Starting with Green River's first EP Dry as a Bone, they had a franchise on the anti-Paisley sound of grunge with Soundgarden's first two EPs, Screaming Life (1987) and Fopp (1988), the 1988 combined Sonic Youth \/ Mudhoney single on which Sonic Youth covered Mudhoney's \"Touch Me, I'm Sick\" on one side and Mudhoney covered Sonic Youth's \"Halloween\" on the other. Mudhoney's first two albums, Mudhoney (1989) and Every Good Boy Deserves Fudge (1991) led to Nirvana's 1989 debut Bleach. The Sub Pop Record Club offered an array of singles to delight fans of the form, including \"The Wagon\" by Dinosaur Jr. (1990), Hole's second single, \"Dicknail\" (1991), Modest Mouse's first single \"Broke\" (1996), the White Stripes' Captain Beefheart trilogy, \"Of the Special Things to Do,\" \"China Pig,\" and \"Ashtray Heart\" (2000), \"I Will Be Grateful for This Day, I Will Be Grateful for Each Day to Come\" by Bright Eyes (2001), and Iron and Wine's \"Call Your Boys\" (2002). In 2001, they released the first of three albums by the Shins, Oh, Inverted World, featuring the generational mood-defining \"New Slang,\" which emerged in the soundtrack to the generational mood defining movie Garden State. That soundtrack also included Iron and Wine's cover of \"Such Great Heights\" from the 2003 album Give Up by the Postal Service, Death Cab for Cutie founder Ben Gibbard's side project.\n\nThe previous generation's mood-defining movie (sharing the marquee with Reality Bites) was 1997's Good Will Hunting, which featured six depressive songs by Portland, Oregon's anti-rock star Elliott Smith, including the haunting \"Angeles\" and the Oscar-nominated \"Miss Misery.\" The film takes place in South Boston, within hailing distance of the original Roxbury location of Fort Apache Studios, which had long since relocated to the slightly more upscale city of Cambridge.\n\nCompleting a twenty-year cosmic circle, alt-country pioneer Steve Earle arrived by way of Nashville with Guitar Town in 1986, the first \"New Dylan\" since Steve Forbert in 1979. Deeper underground, R. Stevie Moore continued sending urgent recorded missives from his living room, as he had been doing since the late sixties. Indigo Girls emerged in 1987 from Atlanta's Emery University with Strange Fire, the first \"New Simon and Garfunkel\" since the originals last reunited with \"My Little Town\" in 1975. In 1990 Paisley Underground veteran Jason Falkner of the Three o'Clock formed Jellyfish and released Bellybutton, the first (and only) \"New Badfinger,\" with two Badfinger covers and Paul McCartney's \"Jet,\" which was available only in the UK. In 1991, Neutral Milk Hotel, the next R.E.M., emerged from Athens, Georgia, led by Jeff Mangum. Their album was released on Elephant 6, a label run by an outcast fraternity of musicians and bands that would include Apples in Stereo, Of Montreal, and the Olivia Tremor Control, as well as others that would break through later in the decade. Neutral Milk Hotel released In the Aeroplane Over the Sea in 1998 and broke up the following year before their work could be deemed an underground masterpiece (eventually with significant overground sales figures to back it up). In 1993, Gin Blossoms, out of Tempe, Arizona, channeled the great jangly sound of the Grass Roots with the hits \"Hey Jealousy\" and \"Found out About You.\" In Los Angeles a year later, Rivers Cuomo channeled Buddy Holly in Weezer's hit single and accompanying evocative video \"Buddy Holly.\" (Like his father Bob, Jakob Dylan looked to the Byrds for his rock 'n' roll entry point with the Wallflowers' \"6th Avenue Heartache\").\n\nIn 1991, Tanya Donelly formed Belly, with the album Star arriving in 1993, containing fourteen tracks written solely by her (with one co-write), including the hit single, \"Feed the Tree.\"\n\nBut George Harrison almost had them all beat. In 1988, he performed the miraculous by creating the Traveling Wilburys, a super group to rival the Beatles, joined by Bob Dylan, Roy Orbison, Tom Petty, and Jeff Lynne. Their first single \"Handle with Care\" was a smash. There was talk of a tour, but Roy Orbison died of a heart attack before the second album, the perversely titled The Traveling Wilburys, Volume Three could be produced, and the Wilburys never traveled.\n\nThey might've been big.\n\nOasis, circa 1994.\n13\n\nTHE BRITISH INVASIONS\n\nIn a way, the entire decade of the eighties could have been called \"The Revenge of George Harrison.\"\n\nKnown as the quiet Beatle, the mysterious Beatles, the brooding Beatle, the sad Beatle, the spiritual Beatle, the sanctimonious Beatle, the hard-luck Beatle, and to John and Paul, the Baby Brother, George struck a particularly sympathetic note when he lost his wife to Eric Clapton in 1974. But it's safe to say that none of those sympathetic notes can be found on his landmark 1969 solo album Electronic Sound, on which George encounters the Moog synthesizer, changing music forever.\n\nAs he unconsciously did a year later with \"My Sweet Lord,\" George co-opted an entire tune on this album from something he'd heard before. \"No Time or Space,\" which consumed all of Side Two, was derived from a demonstration lesson given to him by his original Moog guru Bernie Krause, who was credited as an \"assistant\" on the album but not given any credit at all on the CD reissue. (For an additional cosmic joke, the CD reversed the two song titles on the original LP.)\n\nParalleling the course taken by Ronnie Mack, author of \"He's So Fine,\" (the song \"My Sweet Lord\" was based on), Krause took his former student to court. But lost in the legal shuffle was George's embracing of the synthesizer, which first showed up on four songs from Abbey Road, \"Because,\" \"Here Comes the Sun,\" \"I Want You (She's so Heavy),\" and \"Maxwell's Silver Hammer.\" Like all of the Beatles' various sonic inventions, the synthesizer soon became an essential thread in the fabric not only of psychedelic music, but progressive rock, fusion, funk, disco, and the British Invasion of the 1980s.\n\nKrause and his partner Paul Beaver introduced the first Moog to the hipsterati at the Monterey International Pop Festival in 1967. Shortly after, Krause hooked one up for Ray Manzarek of the Doors to use on the title track for Strange Days. Mick Jagger also fiddled around with a Moog on the Stones' Their Satanic Majesties Request. Beaver contributed \"Synthesthesia\" to the soundtrack of the Peter Fonda \/ Jack Nicholson pre-Easy Rider psychedelic epic The Trip and also played on \"Goin' Back,\" \"Natural Harmony,\" and \"Space Odyssey\" from The Notorious Byrd Brothers. His Moog work shows up on \"Star Collector\" and \"Daily Nightly\" from the Monkees' Pisces, Aquarius, Capricorn & Jones Ltd. Krause also appeared on the Electric Flag's debut album and Simon and Garfunkel's Bookends (\"Save the Life of My Child\"). After showing George the ropes, Krause was hired to work his magic on Jackie Lomax's 1969 Apple debut Is This What You Want?\n\nIn 1968, composer Walter Carlos (still four years away from becoming Wendy) showcased the Moog in the platinum-selling Switched-on Bach, influencing long-haired rock musicians like Keith Emerson and Rick Wakeman to add it into their repertoire, forming the progressive basis for ELP and Yes. Jan Hammer and John McLaughlin incorporated it into the Mahavishnu Orchestra's trailblazing Inner Mounting Flame, giving rise to the much scoffed at fusion genre. In 1969 The Minotaur by Dick Hyman and His Electric Eclectics brought a taste of space age pop to the mix. Gershon Kingsley's 1969 album Music to Moog By featured covers of \"Nowhere Man\" and \"Paperback Writer.\" In 1972, Stan Free, Kingsley's partner in the First Moog Quartet, covered Kingsley's \"Popcorn\" under the name Hot Butter, which became a huge international hit. From 1972 to 1976, Stevie Wonder took up the chalice in five albums, from Music of My Mind through Songs in the Key of Life. In 1974 the future of electronic music virtually got its template from Ralf H\u00fctter and Florian Schneider, the masterminds behind Kraftwerk, whose Autobahn was a blockbuster, with an edited version of the title track becoming a hit single in the U.S. as well as in England. Bernie Worrell brought the synth to Parliament's 1977 hit \"Flashlight,\" and Giorgio Morodor translated it to the disco crowd in Donna Summer's megahits \"Love to Love You Baby\" and \"I Feel Love.\"\n\nIn the heyday of punk, it was time for the Moog to be reconsidered. In Akron, Ohio, the geek rock band Devo put out Q: Are We Not Men? A: We Are Devo! with the anti-anthemic \"Jocko Homo\" and a unique approach to \"(I Can't Get No) Satisfaction.\" But it wasn't until 1980's Freedom of Choice (\"Whip It\") that synthesizers became a prominent part of their act. In England, Gary Numan and Tubeway Army put out the album Replicas early in 1979, with the hit UK single \"Are Friends Electric?\" Numan followed with the solo album The Pleasure Principle and his only U.S. hit, \"Cars.\" Both were dominated by the mini-Moog, as were Numan's five other Top 10 UK singles from 1979 to 1982: \"Complex,\" \"We Are Glass,\" \"I Die: You Die,\" \"She's Got Claws,\" and \"We Take Mystery (to Bed).\" The sound of the latest invasion had arrived.\n\nThis was far different from the sound of the first British Invasion, which was, of course, much more directly Merseyesque. Owing to the lack of American rock bands ready to fill the void, Liverpool and its Mersey River environs became a viable scene in 1964 and the model for future record label gold rushes into San Francisco; Akron, Ohio; Athens, Georgia; and Seattle, Washington. Groups managed by Brian Epstein made out like cheap high school sluts on the coattails of the Beatles. The first of these, Gerry and the Pacemakers, gladly recorded \"How Do You Do It,\" the song the Beatles turned down during their initial meeting with George Martin, spending three weeks at Number One in England in 1963 and one week at Number 9 in the U.S. a year later. Their next tune (also a Number One) was \"I Like It,\" lifted from the repertoire of the Dave Clark Five. Their place in Merseybeat history was secured forever with the nostalgic \"Ferry Cross the Mersey\" from the movie of the same name. Billy J. Kramer and the Dakotas benefitted from the Brian connection by snagging four future Top 10 singles written by Lennon and McCartney: \"Do You Want to Know a Secret\" (which was a hit in the UK only), \"Bad to Me,\" \"I'll Keep You Satisfied,\" and \"From a Window.\" Cilla Black, Brian's only female client, launched her career with Lennon and McCartney's \"Love of the Loved.\" She went on to become a huge star in England with eleven Top 10 hits, including \"Anyone Who Had a Heart\" and \"You're My World,\" which was her biggest hit in the U.S., peaking at Number 26. The jangly Mersey Sound's first American iteration came via the Cyrkle (also managed by Epstein), which had a couple of singles written for them, respectively, by Paul Simon and the Seekers' Bruce Woodley: \"Red Rubber Ball\" (a 1966 hit, originally a flop for the Seekers) and \"I Wish You Could Be Here\" (a 1967 flop).\n\nSince England itself would only rank twenty-eighth in size among American states (after Arkansas and before Alabama, two slots larger than New York), the invasion quickly spread to the UK's furthest reaches, encompassing the sound of groups that had been influenced by the concurrent blues revival. When including the Beatles, the acts of the initial British Invasion accounted for twenty percent of the year's Top 40 records (60 out of 302 singles). These included lite-rock Merseyites (Herman's Hermits and the Dave Clark Five), pop rock entities (the Honeycombs and the Bachelors), folkie groups (the Hollies, the Searchers, the Zombies, Peter and Gordon, and Chad and Jeremy), and pop chanteuses (Dusty Springfield, Petula Clark, and Sandie Shaw). These lighter weight groups stood alongside harder rocking bands like the Rolling Stones, the Animals, the Kinks, the Nashville Teens, and Manfred Mann. Not to be forgotten was Top 40 mainstay Cliff Richard, who'd first visited the chart shores in 1959, and Beatles label mate and quasi-yodeler Frank Ifield, who peaked in 1963 with \"I Remember You.\" Further down the charts, debut singles could be found by such stellar British artists as the Swinging Bluejeans, with the local Liverpool favorite \"Hippy Hippy Shake,\" plucky belter Lulu, the folkie Overlanders, Brian Poole and the Tremeloes (the group Decca signed after turning down the Beatles), and Scott English, an American singer-songwriter based (fittingly enough) in London. (English later wrote a song called \"Brandy\" that Clive Davis would retitle and Barry Manilow would turn into the hit \"Mandy.\")\n\nIn 1965 British acts did even better, accounting for twenty-five percent of the American Top 40, with the Who, the Yardbirds, Donovan, the Troggs, Them (from Belfast, featuring Van Morrison), the Spencer Davis Group (featuring Stevie Winwood), Tom Jones (from Wales), the Walker Brothers, Freddie and the Dreamers, Wayne Fontana and the Mindbenders, Roy Head, Jonathan King, the Fortunes, the Moody Blues, Shirley Bassey, the Silkie, Georgie Fame, Matt Monro, Ian Whitcomb, and future Monkee Davy Jones joining the party.\n\nBy this time enough American bands had come into existence to restore a bit of parity, although the extremely Beatlesque Bee Gees would soon be on their way from Australia, Rod Stewart would come along with Jeff Beck Group, and Eric Clapton's new power trio Cream, featuring Ginger Baker on drums and Jack Bruce on bass, bringing the two groups an advanced level of virtuosity to the mix.\n\nWhile the first British Invasion may have been a two-to-four-year frenzy of a scene being squeezed to its last drop, the presence of British singers and bands on the American charts was no passing fad. It's impossible to rewrite history, but had the Beatles not crossed over; had they thrived solely in England as did Mud or the Status Quo, it is quite possible that the market for English sounds in America after 1964 would have petered out. Instead, starting with 1964, there wouldn't be a year where at least one British single didn't top the American charts until 1996! Only nine songs made Number One that year. (The tenth biggest song of the year was by Welsh lass Donna Lewis.) The next year more than made up for this when Elton John's worldwide smash remake of \"Candle in the Wind\" (to mourn the passing of Princess Diana) became the top song of the year and the Spice Girls (and their requisite Spicemania) became a Beatlesque fad.\n\nThe impact of the Beatles-led invasion cannot be understated. Especially on the FM dial. This is where Led Zeppelin reigned for a decade as the demigods of heavy metal along with Black Sabbath, Deep Purple, and Motorhead, until they were edged off the mountaintop by a second wave of British heavy metal, including Judas Priest, Iron Maiden, and Def Leppard. FM was the delivery system that propelled Pink Floyd's The Dark Side of the Moon to a record-setting three generations on the album charts. Here were stations that welcomed in all its eight-minute song glory the genre known as progressive rock, featuring Yes, Genesis, Emerson, Lake and Palmer, King Crimson, Jethro Tull, the Alan Parsons Project, and Queen. It is where the hard rock of Brit bands like Bad Company, Free, UFO, Peter Frampton, Ritchie Blackmore's Rainbow, and Paul McCartney's Wings were considered as American as Mom's Apple Pie (an American rock band known primarily for a salacious album cover). Like white acts mistaken for black on the doo-wop circuit of the fifties, it's possible the few American rock bands that infiltrated the FM playlists of the 1970s were initially thought of as being British, although in actuality they came from New England (Aerosmith), New York (Blue Oyster Cult and Mountain), and the Midwest (Styx and Grand Funk Railroad). Some, like Fleetwood Mac and Foreigner, were actually half-British. The Jimi Hendrix Experience was formed, fed, and nurtured in England, only to be shipped back to the states as a new entity in much the same way the Beatles remodeled rock 'n' roll.\n\nEven as anti-Beatlemania threatened to obliterate their jangly legend in the mid-seventies (mainly courtesy of punk rock, a mostly British rage-infused genre), there was always at least one band left standing to keep the flame a flicker. In the U.S., the Knack and Cheap Trick were dubbed power pop for their Beatlesque leanings. In the UK, Jeff Lynne made it his self-professed mission to top \"Strawberry Fields Forever\" after joining the Move in 1970 and changing his group's name to the Electric Light Orchestra. ELO's second UK hit was a cover of Chuck Berry's \"Roll Over Beethoven\" (which the Beatles also covered), which almost cracked the U.S. Top 40. They went on to twelve Top 10 hits in the UK from 1972 to 1979 (five in the U.S.), nearly keeping the candle glowing all by their lonesome.\n\nBut ELO was not alone in keeping the candle glowing, as they were joined in their belief in the jangly essence of Beatlemania in 1973 by the obscure newspaper Liverpool Echo, which was also the name of a band featuring Martin Briley. (Briley would re-emerge in 1983 with the Top 40 hit \"The Salt in My Tears.\") Much more successful was 10cc, a group rife with first invasion roots. They were formed in 1972 by Eric Stewart and Graham Gouldman, who had been in the Mindbenders. Gouldman had written hits for Herman's Hermits and the Hollies. Kevin Godley and Lol Creme, in the tradition of John Lennon, were art students with a caustic edge. Although they were rejected by Apple, they would have been perfect in the label's declining days. Their songs were a mixture of high art and low comedy; an underground sensibility with a commercial foundation. The studio they worked out of and co-owned was named Strawberry Studios after \"Strawberry Fields Forever.\" As a group they were behind the ghosting of several hits for New York's bubblegum impresarios Kazenetz & Katz, providing music, lyrics, voices, and arrangements for the Ohio Express and Crazy Elephant. But their biggest breakthrough was the transatlantic smash \"I'm Not in Love,\" which in some circles is considered the best pop record of all time. More importantly, it earned them a big record company advance. Godley and Creme left in 1976 to pursue other options, among them the new field of music video direction, which was about to become an art form in the eighties. The two directed some of the best of the early MTV crop, including George Harrison's \"When We Was Fab,\" Duran Duran's \"Girls on Film,\" and the Peter Gabriel \/ Kate Bush duet \"Don't Give Up.\" In 1992, 10cc reunited for an album called Meanwhile, which featured \"Don't Break the Promises,\" co-authored by Stewart, Gouldman, and a promising young fellow named Paul McCartney.\n\nBy this time \"Rollermania\" was upon the land (in full Tartan regalia), the biggest teen craze in England since the Beatles for a new generation of thirteen year old girls too young to have experienced the real thing. The Bay City Rollers were from Edinburgh, Scotland, just 200 miles up the road from Liverpool. In their early albums they covered the Ronettes' \"Be My Baby\" and the Four Seasons' \"Bye, Bye, Baby,\" the latter becoming the Number One song in the UK in 1975. To echo the reasoning of Capitol Records in the early sixties, this was just enough to convince Arista Records' Clive Davis to take a chance on them in America. In October 1975, Davis had the band play their 1973 flop \"Saturday Night\" on a live feed for television's Saturday Night Live with Howard Cosell, an ill-fated attempt by the garrulous sportswriter to morph into Ed Sullivan. The song's appearance had its Sullivanian effect nonetheless, reaching the top of the U.S. charts during the first week of January 1976. The Bay City Rollers would have six more Top 40 hits in the U.S. in the next couple of years, before lapsing, as the Beatles had before them, into drug revelations and withering lawsuits.\n\nWith so much Beatlemania in the air, naturally someone had to emerge to correct it. This was the mission of All You Need Is Cash, a 1978 TV \"mockumentary\" on the life and times of the Rutles, produced by Monty Python's Eric Idle and Saturday Night Live's Lorne Michaels. After bombing in the U.S. on NBC, it achieved due recognition by the more discerning British TV audience a week later. An outgrowth of an episode of the Idle-Innes 1975\u201376 series Rutland Weekend Television (Britain's version of the droll Canadian sketch show SCTV), the Rutles lovingly spoofed every aspect of the Beatles legend, and included fourteen songs written by Innes, which were featured on the soundtrack album. (Six more were added to the CD reissue.) The cameos that graced All You Need Is Cash represented a highlight reel of British and American comedic talent, including Monty Python's Michael Palin and original Saturday Night Live cast members John Belushi, Dan Aykroyd, Gilda Radner, and Bill Murray. SNL writer\/performers Al Franken and Tom Davis were also on hand, as well as frequent host Paul Simon. Mick Jagger and then wife Bianca didn't want to miss out on the fun, while fellow Stone Ronnie Wood played a Hell's Angel. Noted Liverpool performance poet Roger McGough, (who'd been in the Scaffold with Paul McCartney's brother Mike McGear) played himself. Of all the Beatles, the one who appreciated the joke most was, the droll Beatle himself, George Harrison, who played a TV newscaster presiding over the ransacking of the station, much as all four Beatles presided over the public demise of Apple Records.\n\nBut in the next decade, an even better revenge was in store for Harrison.\n\nBy then, the whimsically progressive Beatlesque group Supertramp had broken through with Breakfast in America (\"The Logical Song\"). The new songwriting team of Chris Difford and Glenn Tilbrook from Squeeze was being hailed as the best since Lennon and McCartney (or at least since John and Taupin), with albums like 1979's Cool for Cats (\"Cool for Cats,\" \"Up the Junction\"), 1980's Argybargy (\"Pulling Mussels [from the Shell],\" \"Another Nail in My Heart,\" If I Didn't Love You\"), 1981's East Side Story (\"Tempted\"), and 1982's Sweets from a Stranger (\"Black Coffee in Bed\"), after which they broke up. Originally a progressive punk band, the Soft Boys (A Can of Bees) eventually produced one genuinely Lennonesque artist, Robyn Hitchcock, who covered John Lennon's \"Cold Turkey\" on that album and would go on to underground fame with tracks like \"Madonna of the Wasps\" (1989) and \"So You Think You're in Love\" (1991). But by far the most Beatlesque new band of the era was XTC, led by the obsessive, stage-stricken Andy Partridge. Releasing their first full album, the punk-flavored White Music in 1978, XTC gradually evolved so that by 1986 they had become a full-service, lush, psychedelic-era Beatles-art-rock-throwback with Skylarking, featuring Partridge doing John Lennon on the surprise hit single \"Dear God.\" (originally the B side of Colin Moulding's exceptionally Beatlesque \"Grass.\") \"Summer's Cauldron\" and \"Earn Enough for Us\" are other standouts from the album, which was produced by noted Beatles connoisseur Todd Rundgren.\n\nNineteen-eighty-six was the peak year for the biggest British invasion of America since the mid-sixties, with British acts accounting for a whopping thirty-four percent of the year's Top 40 (68 out of 211 singles), including nine Number One songs. Even the Beatles returned to the charts in 1986 with a reissue of \"Twist and Shout,\" which lapsed at Number 23.\n\nAlthough British acts were a consistent feature on American playlists during the 1970s, the latest invasion began as a trickle soon after John Lennon was shot on December 8, 1980. Latent Beatle fans around the world reacted to his death, which occurred one month after his much-anticipated comeback single, \"(Just Like) Starting Over,\" was released. In England the song had already slipped to Number 21 only to leap to Number One on December 20. One week after that, it was Number One in America. In 1981, \"Woman,\" another single from the album, replaced it on the U.S. Top 10, joined a few months later by George Harrison's tribute, \"All Those Years Ago.\" Ringo's anguished \"Wrack My Brain,\" also written by George, made the Top 40.\n\nIn 1982, \"Ebony and Ivory,\" Paul's semi-detached duet with Stevie Wonder from Tug of War, topped the chart. Buried in the album was Paul's poignant tribute to his former partner, \"Here Today.\"\n\nBut the real clue to the future was to be found in several other hits. Beginning with Gary Numan's 1980 song \"Cars,\" synth-based songs started to cross the Atlantic, the choicest among them being \"Don't You Want Me\" by Human League, \"I Ran\" by A Flock of Seagulls, \"Tainted Love\" by Soft Cell, and \"Stand or Fall\" by the Fixx. Aided by video exposure over MTV, George's long-envisioned synth revolution was upon us. In 1983, the British presence on the Top 40 doubled. While McCartney put a pair of singles with his business partner Michael Jackson on the top of the charts (\"Say Say Say\" reaching Number One; \"The Girl Is Mine\" making it to Number Two), American music fans not especially attuned to activities across the pond were introduced to a slew of acts that would define the sound of modern rock for the rest of the decade; among them Eurythmics, Culture Club, Duran Duran, Thomas Dolby, the Pretenders, Spandau Ballet, Madness, Kajagoogoo, Naked Eyes, Adam Ant, Wham, Ultravox, Tears for Fears, Heaven 17, Haysi Fantaysi, Bananarama, ABC, the Thompson Twins, the Psychedelic Furs, Dexy's Midnight Runners, and just for good measure, U2.\n\nNot a bad year.\n\nThe singles results for 1984 were even more impressive, with England taking care of thirty percent of the Top 40, mainly with acts that debuted in 1983, which were joined by Frankie Goes to Hollywood and the Style Council. John Lennon's \"Nobody Told Me,\" from the posthumous Milk and Honey album, hit the Top Five. In the UK, Julian Lennon had a Top 10 hit with \"Too Late for Goodbyes,\" which peaked in America at the beginning of 1985, a few months after \"Valotte.\" The new year maintained the same level of dominance for British acts, with Depeche Mode and Bronski Beat added to the populace along with headier creations from the Smiths, Prefab Sprout, and the Housemartins (which spun off into the Beautiful South in 1990). In 1986 Paul McCartney had his last Top 10 hit of the century, the theme from the movie Spies Like Us. But it was also a banner year for Ozzy Osbourne, making his solo debut with \"Shot in the Dark,\" and John and Andy Taylor, taking a break from Duran Duran (John with \"I Do What I Do,\" the theme for the movie 9 1\/2 Weeks, and Andy with \"Take It Easy\"). Mike Rutherford (formerly of Genesis) returned with Mike and the Mechanics. Steve Hackett (also from Genesis) put together GTR with Steve Howe (from Yes). The Art of Noise aligned with both Duane Eddy (to recreate \"Peter Gunn\") and the computer-generated Max Headroom (to create \"Paranoimia\"). The year's most auspicious debuts were by the Pet Shop Boys (\"West End Girls\") and Simply Red (\"Holding Back the Years\"). The year's least auspicious debut was by the Cure, whose \"In Between Days (Without You)\" spent one week at Number 99. But it was a start.\n\nBy 1987, the Invasion was on the wane, although two of the most Beatlesque groups were established, New Zealand's Crowded House (\"Don't Dream It's Over\" and \"Something So Strong\") and Karl Wallinger's World Party (\"Ship of Fools\"). George Harrison also had a good year, with his cover of James Ray's \"Got My Mind Set on You\" becoming his first Number One since 1973. Feeling the tickle of competition, Paul McCartney started collaborating with the closest person to John Lennon he could find, Elvis Costello. Their maiden effort was \"Back on My Feet,\" which was used as the B side of \"Once Upon a Long Ago,\" Paul's last Top 10 hit in the UK (although unreleased in the U.S.). Elvis wound up co-writing four songs for Paul's 1989 album Flowers in the Dirt: \"You Want Her Too,\" \"That Day Is Done,\" \"Don't Be Careless Love,\" and \"My Brave Face\" (Paul's last appearance on the Top 40). To return the favor, McCartney co-wrote two songs for Costello's 1989 album Spike, \"Pads, Paws and Claws\" and \"Veronica,\" the latter becoming Elvis's biggest U.S. hit.\n\nGeorge had an even better year in 1988. The video for his nostalgic \"When We Was Fab,\" directed by Godley and Creme, got six nominations at the MTV Video Music Awards and featured a Hall of Fame lineup of cameos, including Ringo Starr, Elton John, Paul Simon, co-writer Jeff Lynne, Gary Wright, drummer-for-the-stars Ray Cooper, ex-Beatles publicist Derek Taylor, and ex-Beatles road manager Neil Aspinall. Later that year, George formed the Traveling Wilburys, a super group to end all super groups, with two of his idols (Roy Orbison and Bob Dylan), his main acolyte (Jeff Lynne), and Dylan's main acolyte (Tom Petty). Getting together over dinner in Malibu to record a B side for Harrison's single \"This Is Love,\" the guys came up with \"Handle with Care,\" which was not only strong enough to be released as a single on its own, but it inspired them to hang out long enough to complete a pseudonymous album over nine days in May. Traveling Wilburys, Vol. 1 went to Number 3 on the U.S. album chart and sold three million copies.\n\nTake that, Derek & the Dominos.\n\nThe last important mini-British Invasion took place from 1995 through 1997, which featured Beatles-influenced Brit-pop bands (the Stone Roses, Suede, Take That, Radiohead, Elastica, Pulp, Supergrass, Blur, Everything but the Girl, and Oasis), precious dream pop purveyors (Belle and Sebastian, the Sundays, the Cranberries, the Corrs, the Las, the Boo Radleys, and Mazzy Star), harder rockers (Bush, Sponge and P. J. Harvey), electronica stalwarts (the Chemical Brothers, Prodigy, and White Town), acid jazz (Jamiroquai's \"Virtual Insanity\"), and the anarchic world beat of Chumbawamba (\"Tubthumping\").\n\nThis set the stage for the Spice Girls: Victoria, Melanie B., Emma, Melanie C., and Geri, Britain's answer to New England's New Kids on the Block, who had six Top 10 hits in 1989. This led to another girl group explosion to rival that of the late 1950s and early sixties, including Boyzone, Eden's Crush, Dream, B'Witched, All Saints, SWV, TLC, Destiny's Child (Beyonce's proving ground) and culminating with the manufacture of the Pussycat Dolls. This was then answered by a boy band revival of the New Kids sound featuring Hanson, the Backstreet Boys, 'N Sync, Five, LFO, O-Town, 98 Degrees, solo comebacks by old New Kids Jordan Knight and Joey McIntyre, and culminating with the cloning of the Jonas Brothers.\n\nThe final crowning touch in all this end-of-the-century synthmania-turned Beatlemania was the unearthed and unfinished John Lennon demos donated by Yoko Ono to the Fab Three to help them promote their upcoming Anthology project, consisting of a documentary, three double albums and a book. Released in 1996, \"Free as a Bird\" and \"Real Love\" were the last official \"Beatles\" recordings and the first since \"The Long and Winding Road\" in 1970. They were both quite successful, as was the Anthology project.\n\nBut not quite as successful as Oasis, which broke from the pack in 1995 with their definitively Beatlesque second album (What's the Story) Morning Glory, which became the third best-selling album in UK history, behind a greatest hits collection by Queen and the one and only Sgt. Pepper's Lonely Hearts Club Band. Among its choice singles (\"Don't Look Back in Anger,\" \"Some Might Say,\" \"Champagne Supernova,\" \"Roll with It\") is a track called \"Wonderwall,\" an obvious tribute to George Harrison's Wonderwall Music from 1968.\n\nTake that, Big Brothers.\n\nThe Beatles, circa 1969.\n14\n\nHERE TODAY\n\nSince the death of George Harrison in 2001, the Beatles' lore, legend, and legacy has been carried forward (as it probably should be), by Paul McCartney, who has released six albums in the new millennium, including two live collections and a classical work. During the same period, his first, last, and only remaining competition for generational supremacy, Bob Dylan, has put out seven, including two weird movie soundtrack compilations (Masked and Anonymous and I'm Not There) and a Christmas album. The Beatles' catalog has withstood a few soundtrack compilations as well, including I Am Sam (2002), Across the Universe (2007), and Nowhere Boy (2010).\n\nI Am Sam starred Sean Penn, Dakota Fanning, and Michelle Pfeiffer in the sentimental story of a developmentally-challenged man trying to raise his young daughter Lucy by himself. The soundtrack, however, is a challenging mix of Beatle covers by an all-star contemporary cast, including Aimee Mann and Michael Penn (\"The Two of Us\"), Sarah McLachlan (\"Blackbird\"), the Wallflowers (\"I'm Looking through You\"), Sheryl Crow (\"Mother Nature's Son\"), the Black Crowes (\"Lucy in the Sky with Diamonds\"), Eddie Vedder (\"You've Got to Hide Your Love Away\"), Ben Folds (\"Golden Slumbers\"), Ben Harper (\"Strawberry Fields Forever\"), Paul Westerberg (\"Nowhere Man\"), Nick Cave (\"Let It Be\"), and Rufus Wainwright (\"Across the Universe\").\n\n\"Across the Universe\" lends its name to a movie in which various Beatles songs conspire to tell the story of the psychedelic sixties, which is in part the story of the Beatles. Here, most of the Beatle covers are done by the actors, although not necessarily to their best advantage. The performances by Bono on \"I Am the Walrus\" and \"Lucy in the Sky with Diamonds\" are not among his choicest moments either. Joe Cocker comes off best, in a self-parodying version of \"Come Together.\"\n\nNowhere Boy is the story of a young John Lennon, focusing on the early rock 'n' roll that informed his coming of age in the late fifties. As a sampler of the period, it's a good one, including \"Twenty Flight Rock\" (Eddie Cochran), \"Hound Dog\" (Big Mama Thornton), \"Shake, Rattle and Roll\" (Elvis Presley), \"Wild One\" (Jerry Lee Lewis), \"Be Bop a Lula\" (Gene Vincent), and \"Hard Headed Woman\" (Wanda Jackson). Disc One concludes with John Lennon singing \"Mother.\" Disc Two includes \"Roll Over Beethoven\" (Chuck Berry), \"Peggy Sue\" (Buddy Holly), \"Rip It Up\" (Little Richard), \"Baby, Let's Play House (Elvis Presley), \"Rock Around the Clock\" (Bill Haley and His Comets), \"Money\" (Barrett Strong), \"Ain't That a Shame\" (Fats Domino), and the find of the package, \"Brand New Cadillac\" by Vince Taylor and His Playboys.\n\nPerhaps the weirdest-grandest-tribute-exploitation of Beatles material was the 2006 show simply called Love, conceived and presented by the acrobatic dance troupe Cirque du Soleil at the Mirage Hotel in Las Vegas. Blessed by George Harrison in 2000, it took three years to gain the other necessary signatures from Paul, Ringo, and the surviving widows of John and George. Musically directed by George Martin and his son Giles, the biographical storyline samples and re-orchestrates 130 Beatle songs into twenty-eight discrete pieces in a free floating biographical collage. (In 2006 noted experimental choreographer Twyla Tharp had a similar crack at Dylan's catalog in the Broadway flop The Times They Are-a Changin'.)\n\nFor fans of a less elaborate yet more musically satisfying Beatles homage, the all-star tribute band the Fab Faux, featuring Will Lee and Jimmy Vivino, is recommended. Their always creative and virtuosic approach to the material has gained them the reputation as being the best of the approximately infinite number of working Beatles tribute bands.\n\nAlthough there hasn't been another official British Invasion on the order of the one led by Oasis in the decade of the single digits, England's biggest contribution recently seems to be in the realm of soulful chanteuse Lisa Stansfield, the first white female ever to top the British R&B charts (with \"All Around the World\" in 1990). Since then, Beth Orton has broken through on the coffee house circuit, Dido has charmed the dancehalls, Joss Stone has displayed her massive blues chops, the late Amy Winehouse took home a boatload of Grammys, and biracial belter Leona Lewis won the X Factor (the British version of American Idol) and translated it into American success). They were followed by Lily Allen, Adele, Duffy, and Florence and the Machine.\n\nMale solo artists and bands in the UK have fared much worse, with the most popular among them failing to cross over with any sort of impact, as with Ireland's Westlife, which hit Number One in the UK with its first seven singles, only one of which made the U.S. charts. The top UK male artist of the nineties, Take That veteran Robbie Williams, is Cliff Richard all over again; big in England, and an afterthought in the U.S. James Blunt had an international best-seller with \"You're Beautiful\" in 2006, but this seems so far to have been a one-shot. Alternative acts like Badly Drawn Boy, Travis, and Arctic Monkeys haven't traveled especially well. Arctic Monkeys' last three albums have been hits, but they haven't charted a U.S. single since \"I Bet You Look Good on the Dancefloor\" in 2005. Scotland's Franz Ferdinand has done better, with two hit albums and several U.S. alternative hits, including \"No You Girls\" in 2009, but their biggest mainstream single was \"Take Me Out,\" which peaked at Number 66 in 2003. Folk rocker David Gray has had a smattering of U.S. album success, with the Top 60 reissue of \"Babylon\" (2000) being his biggest hit. The only new British act to have achieved fame in the U.S. on a Beatlesque scale is London's Coldplay, which broke through in 2000 with \"Yellow,\" following it up with hits like \"Speed of Sound\" and \"Viva La Vida.\"\n\nOnce again, as it was before the Beatles arrived, the American music scene is dominated by teen idols and heavily R&B and dance-oriented singles. Once again, pure rock bands are an endangered species on radio and TV in the era of Pro Tools; so much so that when the Strokes arrived in 2000 (\"Last Nite\") they were hailed as the saviors of a dying genre, much as Arcade Fire (\"The Suburbs\") assumed the mantle in 2010. In the meantime, bands ranging across the spectrum from heavy metal to pop rock still catch a moment or two in the ear or on the internet to keep rock fans dreaming of a better day, among them the White Stripes, Kings of Leon, Maroon 5, Linkin Park, Incubus, the Killers, Velvet Revolver, the post-punk Green Day, the ageless Foo Fighters, the spit-and-polished Nickelback, and the Air Supply-incarnate the Fray.\n\nFor those in search of Beatle footprints, they still abound, several genres wide and three generations deep, wherever the sound of psychedelic rock, sunshine pop, or baroque pop rock can be heard, whether in the form of Denver's Apples in Stereo (\"Strawberryfire\"), Dallas' Polyphonic Spree (\"Sunshine\"), Athens' of Montreal (\"Every Day Feels Like Sunday\"), Brooklyn's MGMT (\"Love Always Remains\"), or New York City's Vampire Weekend (\"A-Punk\").\n\nFor those in search of the real thing, 2010 brought about a historic merger of the two most famous corporations bearing the Apple name, when the Beatles finally allowed their catalog to be downloaded by visitors to the legendary song dispensary site on the internet.\n\nIf you like the Beatles, you probably already have most of these songs permanently downloaded into your brain. But now you can mix them with an earful of alternatives. And as far as alternatives, there are enough for Strawberry Fields to last forever.\nSUGGESTED LISTENING\n\n100 Covers\n\nWhile few artists can claim anywhere near 100 covers of their material, the Beatles have well into the thousands of their approximately 200 original songs. Of these covers, I have selected 100 that seem a particularly apt match of performer and song.\n\nPeter Sellers|  _Can't Buy Me Love_  \n---|---  \nPeggy Lee|  _A Hard Day's Night_  \nAlma Cogan|  _Eight Days a Week_  \nElla Fitzgerald|  _Savoy Truffle_  \nFrank Sinatra|  _Something_  \nNancy Sinatra|  _Run for Your Life_  \nMae West|  _Day Tripper_  \nCliff Richard|  _Things We Said Today_  \nJoe Brown|  _Here Comes the Sun_  \nSandie Shaw|  _Love Me Do_  \nThe Swinging Bluejeans|  _She Loves You_  \nThe Tremeloes|  _Good Day Sunshine_  \nThe Crickets|  _I Want to Hold Your Hand_  \nSonny Curtis|  _P.S. I Love You_  \nElvis Presley|  _Hey Jude_  \nThe Miracles|  _And I Love Her_  \nFats Domino|  _Lady Madonna_  \nJohn Mayall|  _No Reply_  \nThe Who|  _I Saw Her Standing_ There  \nDeep Purple|  _We Can Work It Out_  \nOzzy Osbourne|  _In My Life_  \nWar|  _A Day in the Life_  \nJoe Cocker|  _She Came in Through the Bathroom Window_  \nBob Dylan|  _Nowhere Man_  \nJoan Baez|  _Eleanor Rigby_  \nRichie Havens|  _Rocky Raccoon_  \nThe Beach Boys|  _You've Got to Hide Your Love Away_  \nUnderground Sunshine|  _Birthday_  \nFree Design|  _Michelle_  \nArt Garfunkel|  _I Will_  \nThe Amboy Dukes|  _I Feel Fine_  \nThe Grateful Dead|  _Lucy in the Sky with Diamonds_  \nJimi Hendrix and Jim Morrison|  _Tomorrow Never Knows_  \nMC5|  _Come Together_  \nAretha Franklin|  _Let It Be_  \nOtis Redding|  _Drive My Car_  \nThe Hollies|  _If I Needed Someone_  \nThe Bee Gees|  _Please Please Me_  \nMarianne Faithfull|  _I'm a Loser_  \nJames Taylor|  _Yesterday_  \nBilly Preston|  _Blackbird_  \nTodd Rundgren|  _While My Guitar Gently Weeps_  \nDavid Bowie|  _This Boy_  \nTiny Tim|  _Girl_  \nJeff Beck|  _She's a Woman_  \nHarry Nilsson|  _You Can't Do That_  \nThe Carpenters|  _Ticket to Ride_  \n10cc|  _Paperback Writer_  \nJohn Denver|  _Mother Nature's Son_  \nElton John|  _Get Back_  \nBilly Joel|  _I'll Cry Instead_  \nNils Lofgren|  _Any Time at All_  \nBlood, Sweat & Tears|  _Got to Get You into My Life_  \nCheap Trick|  _Magical Mystery Tour_  \nJeff Lynne|  _With a Little Help from My Friends_ \/ _Nowhere Man_  \nSupertramp|  _Two of Us_  \nElvis Costello|  _All You Need Is Love_  \nThe Damned|  _Help!_  \nDead Kennedys|  _Back in the USSR_  \nBilly Bragg|  _Revolution_  \nPaul Weller|  _Don't Let Me Down_  \nSiouxsie & the Banshees|  _Dear Prudence_  \nLydia Lunch|  _Why Don't We Do It in the Road_  \nPatti Smith|  _Within You Without_ You  \nRosanne Cash|  _I Don't Want to Spoil the Party_  \nEmmylou Harris|  _Here, There and Everywhere_  \nThe Dillards|  _I've Just Seen a Face_  \nSteve Earle|  _I'm Looking Through You_  \nRickie Lee Jones|  _For No One_  \nBryan Ferry|  _You Won't See Me_  \nRobin Gibb|  _Oh! Darling_  \nXTC|  _Strawberry Fields Forever_  \nFlamin' Groovies|  _Misery_  \nThe Residents|  _Revolution 9_  \nThe Feelies|  _Everybody's Got Something to Hide (Except Me and My Monkey)_  \nTanya Donelly | _Long Long Long_  \nThe Pixies|  _Wild Honey Pie_  \nHusker Du|  _Helter Skelter_  \nMatthew Sweet and Susanna Hoffs | _And Your Bird Can Sing_  \nU2|  _Rain_  \nSuede|  _Across the Universe_  \nStevie Ray Vaughan|  _Taxman_  \nBjork|  _Fool on the Hill_  \nOasis|  _I Am the Walrus_  \nWorld Party|  _Happiness Is a Warm Gun_  \nCornershop|  _Norwegian Wood_  \nThe Corrs|  _The Long and Winding Road_  \nTori Amos|  _She's Leaving Home_  \nSmash Mouth|  _Getting Better_  \nAmy Winehouse|  _All My Loving_  \nMichelle Shocked|  _Lovely Rita_  \nThey Might Be Giants|  _Yellow Submarine_  \nRancid|  _Ob-La-Di, Ob-La-Da_  \nShe and Him|  _I Should Have Known Better_  \nElliott Smith|  _Because_  \nSufjan Stevens|  _What Goes On_  \nMaroon 5|  _If I Fell_  \nJonas Brothers|  _Hello Goodbye_  \nJulian Lennon|  _When I'm 64_  \nSean Lennon|  _Julia_\n\nEarly Covers\n\nIn their early days, Lennon and McCartney parceled out a few of their slightly substandard songs to other local acts, who usually did quite well with them. Chart numbers are for the UK.\n\n_Love of the Loved_|  Cilla Black| 1963 #35  \n---|---|---  \n_I Wanna Be Your Man_ | The Rolling Stones| 1963 #12 B side U.S.  \n_I'm in Love_|  The Foremost| 1963 #9  \n_Hello Little Girl_|  The Foremost| 1963 #17  \n_Tip of My tongue_|  Tommy Quickly| 1963\u2014  \n_I'll Be on My Way_|  Billy J. Kramer and the Dakotas| 1963 B side  \n_Do You Want to Know a Secret_|  Billy J. Kramer and the Dakotas| 1963 #1 B side U.S.  \n_Bad to Me_|  Billy J. Kramer and the Dakotas| 1963 #1  \n\n#9 U.S.  \n_I'll Keep You Satisfied_|  Billy J. Kramer and the Dakotas| 1963 #5  \n\n#30 U.S.  \n_From a Window_|  Billy J. Kramer and the Dakotas| 1964 #10  \n\n#23 U.S.  \n_It's for You_|  Cilla Black| 1964 #7   \n\n#79 U.S.  \n_A World Without Love_|  Peter and Gordon| 1964 #1   \n\n#1 U.S.  \n_Like Dreamers Do_ | Applejacks| 1964 #20  \n_Nobody I Know_|  Peter and Gordon| 1964 #10   \n\n#12 U.S.  \n_I Don't Want to See You Again_|  Peter and Gordon| 1964\u2014#16 U.S.  \n_One and One Is Two_|  The Strangers with Mike Shannon| 1964\u2014  \n_That Means a Lot_|  P. J. Proby| 1965 #30  \n_If You've Got Trouble_|  The Bristols| 1965\u2014  \n_You've Got to Hide Your Love Away_|  The Silkie| 1965 #28  \n\n#10 U.S.  \n_Woman_|  Peter and Gordon | 1966 #28   \n\n#14 U.S.  \n_Step Inside Love_|  Cilla Black| 1967 #8  \n_Cat Call_ ( _Catswalk_ )| Chris Barber | 1967\u2014  \n_Goodbye_|  Mary Hopkin| 1969 #2   \n\n#13 U.S.  \n_It's Four You_|  The Beatnix| (Raven, 1998) An import album containing most of these tracks, as sung by a Swedish Beatle tribute band\n\nChapter by Chapter\n\nPre-Beatles\n\nThe Beach Boys|  _Surfin' Safari_ (Capitol, 1962)  \n---|---  \n| _Surfin' U.S.A._ (Capitol, 1963)  \n| _Surfer Girl_ (Capitol, 1963)  \n| _Little Deuce Coupe_ (Capitol, 1963)  \nThe Four Seasons|  _Sherry and 11 Others_ (Vee Jay, 1962)  \n| _Big Girls Don't Cry and Twelve Others_ (Vee Jay, 1963)  \n| _Ain't That a Shame and 11 Others_ (Vee Jay, 1963)  \n| _Golden Hits of the 4 Seasons_ (Vee Jay, 1963)  \nThe Kingsmen|  _The Kingsmen in Person_ (Wand, 1964)  \nDick Dale and His Del-tones|  _Surfer's Choice_ (Deltone, 1963)  \n| _Checkered Flag_ (Capitol, 1963)  \nThe Surfaris|  _Wipe Out_ (Dot, 1963)  \n| _The Surfaris Play Wipe Out and Others_ (Dot, 1963)  \nTrashmen|  _Surfin' Bird_ (Garrett, 1964)\n\nCrossing Over\n\nCliff Richard|  _It's All in the Game_ (Epic, 1964)  \n---|---  \nThe Shadows|  _The Shadows_ (EMI, 1961)  \n| _Out of the Shadows_ (EMI, 1962)  \nJoe Meek|  _The Legendary Joe Meek_ (Castle Pulse, 2005)  \nTornadoes|  _The Original Telstar_ (London, 1963)  \nScreaming Lord Sutch|  _Lord Sutch and Heavy Friends_ (Wounded Bird, 1970)  \n| _Hands of Jack the Ripper_ (Wounded Bird, 1972)  \nLonnie Donegan|  _More Tops with Lonnie_ (Pye, 1961)  \n| _Golden Age of Lonnie Donegan_ (Golden Guinea, 1962)  \nFrank Ifield|  _Jolly What! The Beatles and Frank Ifield_ (Vee Jay, 1964)  \nJoe Brown and the Bruvvers|  _The Piccadilly\/Pye Anthology_ (Piccadilly, 1994)  \nTommy Steele|  _Half a Sixpence_ (RCA Victor, 1965)  \nBilly Fury|  _The Sound of Fury_ (Decca, 1960)  \nVince Taylor and the Playboys|  _Le Rock C'est Ca!_ (Universal, 2003)  \nTony Sheridan| The Beatles with  \n| _Tony Sheridan and Guests_ (MGM, 1964)  \nJohnny Kidd and the Pirates|  _I'll Never Get Over You_ (HMV, 1963)  \n| _The Best of Johnny Kidd and the Pirates_ (EMI, 2008)  \nHelen Shapiro|  _Helen in Nashville_ (See For Miles, 1963)  \nCilla Black|  _Is It Love?_ (Capitol, 1965)  \n| _Cilla_ (Parlophone, 1965)  \n| _Cilla Sings a Rainbow_ (Parlophone, 1966)\n\nThe Great American Songbook\n\nMeredith Willson|  _The Music Man (Original Cast)_ (Capitol, 1958)  \n---|---  \n| _The Music Man (Soundtrack)_ (Warner Bros., 1962)  \nPeggy Lee|  _Latin \u00e1 La Lee_ (Capitol, 1960)  \nDinah Washington | _September in the Rain_ (Mercury, 1963)  \nDoris Day|  _On Moonlight Bay_ (Columbia, 1951)  \nFrank Sinatra|  _This Is Sinatra_ (Capitol, 1956)  \nFats Waller|  _The Very Best of Fats Waller_ (Collector's Choice, 2000)  \nLes Paul|  _The Hit Makers_ (Capitol, 1953)  \nMae West|  _Way Out West_ (Tower, 1966)  \nThe Ink Spots|  _The Anthology_ (MCA, 1998)\n\nRockabilly\n\nJohnny Burnette and the Rock 'n Roll Trio|  _Johnny Burnette and the Rock 'n Roll Trio_ (Coral, 1956)  \n---|---  \nEddie Cochran|  _Eddie Cochran_ (Liberty, 1960)  \n| _Cherished Memories_ (Liberty, 1962)  \n| _Memorial Album_ (Liberty, 1962)  \n| _Singing to My Baby_ (Liberty, 1962)  \nGene Vincent|  _Bluejean Bop_ (Capitol, 1956)  \n| Gene Vincent and His Blue Caps(Capitol, 1957)  \n| _Gene Vincent Rocks! And the Blue Caps Roll_ (Capitol, 1958)  \nRoy Orbison|  _Crying_ (Monument, 1962)  \n| _Roy Orbison's Greatest Hits_ (Monument, 1962)  \n| _In Dreams_ (Monument, 1963)  \nCarl Perkins|  _Dance Album of Carl Perkins_ (Sun, 1958)  \n| _Whole Lotta Shakin'_ (Columbia, 1958)  \nEddie Fontaine|  _Rock with Me_ (Charly, 1988)  \nChan Romero|  _Hippy Hippy Shake_ (Del-Fi, 1995)  \nThe Everly Brothers|  _The Everly Brothers_ (Cadence, 1958)  \n| _It's Everly Time_ (Warner Bros., 1960)  \n| _The Fabulous Style of the Everly Brothers_ (Warner Bros., 1960)  \n| _A Date with the Everly Brothers_ (Warner Bros., 1960)  \n| _The Golden Hits of the Everly Brothers_ (Warner Bros., 1960)  \nElvis Presley|  _Elvis Presley_ (RCA Victor, 1956)  \n| _Elvis_ (RCA Victor, 1956)  \n| _Love Me Tender_ (RCA Victor, 1956)  \n| _Lovin' You_ (RCA Victor, 1957)  \n| _Jailhouse Rock_ (RCA Victor, 1957)  \n| _King Creole_ (RCA Victor, 1958)  \n| _For LP Fans Only_ (RCA Victor, 1959)  \n| _A Date with Elvis_ (RCA Victor, 1959)  \n| _Elvis Is Back_ (RCA Victor, 1960)  \n| _G. I. Blues_ (RCA Victor, 1960)  \n| _Flaming Star_ (RCA Victor, 1960)  \n| _Something for Everybody_ (RCA Victor, 1961)  \n| _Wild in the Country_ (RCA Victor, 1961)  \n| _Blue Hawaii_ (RCA Victor, 1961)  \n| _Pot Luck_ (RCA Victor, 1962)  \n| _Follow That Dream_ (RCA Victor, 1962)  \n| _Kid Galahad_ (RCA Victor, 1962)  \n| _Girls! Girls! Girls!_ (RCA Victor, 1962)  \n| _It Happened at the World's Fair_ (RCA Victor, 1962)  \n| _Fun in Acapulco_ (RCA Victor, 1963)  \nWanda Jackson|  _Rockin' with Wanda_ (Capitol, 1960)\n\nRhythm & Blues\n\nLittle Richard|  _Here's Little Richard_ (Specialty, 1957)  \n---|---  \nLarry Williams|  _Here's Larry Williams_ (Specialty, 1959)  \nFats Domino|  _Fats Is Back_ (Reprise, 1968)  \nThe Shirelles|  _Greatest Hits_ (Scepter, 1963)  \nThe Chiffons|  _Sweet Talkin' Guy_ (Laurie, 1960)  \nLittle Willie John|  _Fever_ (King, 1956)  \nLittle Eva|  _Llllloco-Motion_ (Dimension, 1962)  \nRay Charles|  _Ray Charles_ (Atlantic, 1958)  \n| _The Great Ray Charles_ (Atlantic, 1959)  \n| _What'd I Say_ (Atlantic, 1959)  \n| _The Genius of Ray Charles_ (Atlantic, 1962)  \nPiano Red|  _The American Roots of the British Invasion_ (Varese Sarabande, 2002)  \nRoy Lee Johnson|  _Roy Lee Johnson and the Villagers_ (Stax, 1973)  \nJoey Dee & the Starliters|  _Doin' the Twist, Recorded Live at the Peppermint Lounge_ (Roulette, 1961)  \n| _Hey, Let's Twist!_ (Roulette, 1962)  \n| _Back at the Peppermint Lounge \/ Twistin' with Joey Dee and His Starliters_ (Roulette, 1962)  \nThe Isley Brothers|  _Twist and Shout_ (Wand, 1962)  \nThe Miracles|  _The Fabulous Miracles_ (Tamla, 1963)  \n| _The Miracles on Stage_ (Tamla, 63)  \n| _Doin' Mickey's Monkey_ (Tamla, 1963)  \nMary Wells|  _Two Lovers and Other Great Hits_ (Motown, 1963)  \n| _Love Songs to the Beatles_ (20th Century Fox, 1965)  \nThe Marvelettes|  _Greatest Hits_ (Tamla, 1966)  \nJames Ray|  _James Ray_ (Caprice, 1961)  \nArthur Alexander|  _You Better Move On_ (Dot, 1962)\n\nThe Songwriters\n\nChuck Berry|  _After School Session_ (Chess, 1958)  \n---|---  \n| _One Dozen Berrys_ (Chess, 1959)  \n| _Berry on Top_ (Chess, 1960)  \n| _Chuck Berry on Stage_ (Chess, 1963)  \n| _Newest Juke Box Hits_ (Chess, 1964)  \n| _St. Louis to Liverpool_ (Chess, 1964)  \nThe Crickets|  _The Chirping Crickets_ (Coral, 1958)  \nBuddy Holly|  _The Complete Buddy Holly_ (MCA, 1969)  \nJackie De Shannon|  _Jackie De Shannon_ (Collector's Choice, 1963)  \n| _Don't Turn Your Back on Me_ (Liberty, 1964)  \nJerry Leiber and Mike Stoller|  _The Leiber and Stoller Story, Vols. 1, 2, 3_ (Ace, 2004-07)  \nPhil Spector|  _Greatest Hits_ (Warner Bros., 1977)  \n| _Back to Mono_ (ABKCO, 1991)  \nThe Ronettes|  _Presenting the Fabulous Ronettes Featuring Veronica_ (Philles, 1964)  \nRonnie Spector|  _Siren_ (Polish, 1980)  \n| _Unfinished Business_ (Columbia, 1987)  \nThe Coasters|  _The Coasters_ (Atco, 1956)  \n| _Greatest Hits_ (Atco, 1958)  \n| _One by One_ (Atco, 1960)  \n| _Coast Along_ (Atco, 1961)  \nDoc Pomus and Mort Shuman  \nGerry Goffin and Carole King  \nBarry Mann and Cynthia Weil|  _The Brill Building Sound_ (Era, 93)  \nCarole King|  _Pearls: Songs of Goffin and King_ (Scarface, 1980)  \nBurt Bacharach and Hal David|  _Reach Out_ (A&M, 1967)  \n| _Make It Easy on Yourself_ (A&M, 1969)  \nJeff Barry and Ellie Greenwich|  _Let It Be Written, Let It Be Sung_ (Verve, 1973)\n\nGeorge Martin\n\nMonty Python|  _Matching Tie and Handkerchief_ (Arista, 1973)  \n---|---  \nThe Firesign Theater|  _Don't Crush That Dwarf, Hand Me the Pliers_ (Columbia, 1970)  \nFlanders and Swann|  _At the Drop of a Hat_ (EMI, 1960)  \n| _At the Drop of Another Hat_ (EMI, 1964)  \nJohnny Dankworth|  _The Best of Johnny Dankworth_ (Redial, 1999)  \nCleo Laine|  _Born on a Friday_ (RCA Victor, 1976)  \nJimmy Shand|  _Bluebell Polka_ (Castle, 2007)  \nRolf Harris|  _Tie Me Kangaroo Down, Sport & Sun Arise_ (Epic, 1963)  \nThe Vipers|  _The Coffee Bar Session_ (Parlophone, 1957)  \nMatt Monro|  _My Kind of Girl_ (Warwick, 1961)  \nThe Action|  _Rolled Gold_ (Reaction, 2002)  \nDavid and Jonathan|  _Michelle_ (Capitol, 1966)  \nIvor Cutler|  _Who Tore Your Trousers_ (Decca, 1961)  \n| _Ludo_ (Rev-Ola, 1967)  \nThe Scaffold|  _Lily the Pink_ (Parlophone, 1969)  \nEdwards Hand|  _Edwards Hand_ (Lightning Tree Eire, 1969)  \n| _Stranded_ (Lightning Tree Eire, 1970)  \n| _Rainshine_ (Regal Zonophone, 1971)  \nMahavishnu Orchestra|  _Apocalypse_ (Columbia, 1974)  \nAmerica|  _Holiday_ (Warner Bros., 1974)  \n| _Hearts_ (Warner Bros., 1975)  \n| _Hideaway_ (Warner Bros.,1976)  \n| _Harbor_ (Warner Bros., 1977)  \nJeff Beck|  _Blow by Blow_ (Epic, 1976)  \n| _Wired_ (Epic, 1977)  \nAmerican Flyer|  _American Flyer_ (United Artists, 1976)  \nJimmy Webb|  _El Mirage_ (Collector's Choice, 1977)  \nNeil Sedaka|  _A Song_ (Elektra, 1977)  \nPaul McCartney|  _Tug of War_ (Columbia, 1982)  \nVarious|  _Produced by George Martin_ (Capitol, 2001)\n\nFolk Rock\n\nThe Animals|  _The Animals_ (MGM, 1964)  \n---|---  \nBob Dylan|  _Another Side of Bob Dylan_ (Columbia, 1965)  \n| _Highway 61 Revisited_ (Columbia, 1965)  \n| _Blonde on Blonde_ (Columbia, 1966)  \nThe Brothers Four|  _A Beatles Songbook_ (Columbia, 1966)  \nThe Byrds|  _Mr. Tambourine Man_ (Columbia, 1965)  \n| _Turn! Turn! Turn!_ (Columbia, 1966)  \n| _Fifth Dimension_ (Columbia, 1966)  \nSimon and Garfunkel|  _Wednesday Morning, 3 AM_ (Columbia, 1966)  \n| _Sounds of Silence_ (Columbia, 1966)  \nThe Association|  _And Then... Along Comes the Association_ (Warner Bros., 1966)  \nDonovan|  _Sunshine Superman_ (Epic, 1966)  \nThe Lovin' Spoonful|  _Daydream_ (Kama Sutra, 1966)  \n| _Hums of the Lovin' Spoonful_ (Kama Sutra, 1966)\n\nAcid Rock\n\nThe Beach Boys|  _Pet Sounds_ (Capitol, 1966)  \n---|---  \nThe Steve Miller Band|  _Your Saving Grace_ (Capitol, 1969)  \nFrank Zappa and the Mothers of Invention|  _Freak Out_ (Verve, 1967)  \n| _We're Only in It for the Money_ (Verve, 1968)  \nRavi Shankar|  _Live at Monterey_ (EMI\/Angel, 1998)  \nThe Incredible|   \nString Band|  _The 5000 Spirits or Layers of the Onion_ (Elektra, 1967)  \nPink Floyd|  _The Piper at the Gates of Dawn_ (EMI, 1967)  \nThe Bonzo Dog Doo-Dah Band|  _Gorilla_ (Liberty, 1967)  \nTomorrow|  _Tomorrow_ (EMI, 1968)  \nTraffic|  _Mr. Fantasy_ (Island, 1967)\n\nThe American Beatles\n\nThe Remains|  _The Remains_ (Epic, 66)  \n---|---  \nThe Monkees|  _The Monkees_ (Colgems, 1966)  \n| _More of the Monkees_ (Colgems, 1967)  \n| _Headquarters_ (Colgems, 1967)  \n| _Pisces, Aquarius, Capricorn & Jones Ltd_ (Colgems, 1967)  \nThe Left Banke|  _Walk Away Renee \/ Pretty Ballerina_ (Smash, 1967)  \nLove|  _Forever Changes_ (Elektra, 1967)  \nThe Merry-Go-Round|  _The Merry-Go-Round_ (A&M, 1967)  \nHarry Nilsson|  _Pandemonium Shadow Show_ (RCA, 1968)  \nMontage|  _Montage_ (Laurie, 1969)  \nStories|  _Stories_ (Kama Sutra, 1972)  \nThe Beckies|  _The Beckies_ (Sire, 1976)  \nThe Grass Roots|  _Let's Live for Today_ (Dunhill, 1967)  \nThe Beau Brummels|  _Introducing the Beau Brummels_ (Autumn, 1965)  \n13th Floor Elevators|  _The Psychedelic Sounds of the 13th Floor Elevators_ (Sunspots, 66)  \nEmitt Rhodes|  _Emitt Rhodes_ (Dunhill, 1970)  \nTodd Rundgren|  _Something \/ Anything_ (Bearsville, 1972)  \nRaspberries|  _Raspberries_ (Capitol, 1972)  \n| _Fresh_ (Capitol, 1973)  \n| _Side Three_ (Capitol, 1973)  \n| _Starting Over_ (Capitol, 1974)  \nEric Carmen|  _Eric Carmen_ (Arista, 1975)  \nBig Star|  _#1 Record_ (Ardent, 1972)  \nAlex Chilton|  _Like Flies on Sherbert_ (Peabody, 1979)  \nCheap Trick|  _In Color_ (Epic, 1977)  \nThe Knack|  _Get the Knack_ (Capitol, 1979)  \n| _But the Little Girls Understand_ (Capitol, 1980)  \nThe Romantics|  _Romantics_ (Nemperor, 1980)  \nTommy Tutone|  _Tommy Tutone 2_ (Columbia, 1981)  \nUtopia|  _Deface the Music_ (Bearsville, 1980)  \nBeatlemania|  _Original Cast Recording_ (Pair, 1977)  \nVarious|  _Sgt. Pepper's Lonely Hearts Club Band (Soundtrack)_ (RSO, 1978)\n\nApple Records\n\nJames Taylor|  _James Taylor_ (Apple, 1969)  \n---|---  \nMary Hopkin|  _Postcard_ (Apple, 1969)  \nJackie Lomax|  _Is This What You Want?_ (Apple, 69)  \nBadfinger|  _Magic Christian Music_ (Apple, 1970)  \n| _No Dice_ (Apple, 1970)  \n| _Straight Up_ (Apple, 1971)  \nBilly Preston|  _That's the Way God Planned It_ (Apple, 69)  \nYoko Ono|  _Yoko Ono \/ Plastic Ono Band_ (Apple, 1971)  \n| _Fly_ (Apple, 1971)  \n| _Approximately Infinite Universe_ (Apple, 1973)  \n| _Feeling the Space_ (Apple, 1973)  \nElephant's Memory|  _Elephant's Memory_ (Apple, 72)  \nDavid Peel|  _The Pope Smokes Dope_ (Apple, 73)\n\nThe Anti-Beatles\n\nLed Zeppelin|  _Led Zeppelin_ (Atlantic, 1969)  \n---|---  \nThe Velvet Underground|  _The Velvet Underground and Nico_ (Verve, 1967)  \nThe Stooges|  _The Stooges_ (Elektra, 1969)  \nJohn Lennon|  _John Lennon \/ Plastic Ono Band_ (Apple, 1970)  \nHarry Nilsson|  _Pussy Cats_ (RCA, 1974)  \nDavid Bowie|  _The Rise and Fall of Ziggy Stardust and the Spiders from Mars_ (RCA, 1972)  \nNew York Dolls|  _New York Dolls_ (Mercury, 1973)  \nLou Reed|  _Transformer_ (RCA, 1972)  \nGary Glitter|  _Glitter_ (Bell, 1972)  \nRamones|  _Ramones_ (Sire, 1976)  \nThe Rutles|  _The Rutles_ (Warner Bros., 1978)  \nDevo|  _Q: Are We Not Men? A: We Are Devo!_ (Warner Bros., 1978)  \nX-Ray Spex|  _Germ Free Adolescents_ (Virgin, 1978)  \nSiouxsie and the Banshees|  _The Scream_ (Polydor, 1978)  \nLydia Lunch|  _Stateless_ (Stiff, 1979)  \nElvis Costello|  _Spike_ (Warner Bros., 1989)\n\nThe American Roots Revival\n\nElvis Presley|  _From Elvis in Memphis_ (RCA, 1969)  \n---|---  \nTom Petty and the Heartbreakers|  _Tom Petty and the Heartbreakers_ (Shelter, 1977)  \nShoes|  _Present Tense_ (Elektra, 1979)  \n| _Tongue Twister_ (Elektra, 1981)  \nRobert Gordon|  _Robert Gordon with Link Wray_ (Private Stock, 1977)  \n| _Fresh Fish Special_ (Private Stock, 1978)  \nMarshall Crenshaw|  _Marshall Crenshaw_ (Warner Bros., 1982)  \nFlamin' Groovies|  _Shake Some Action_ (Sire, 1976)  \nDave Edmunds|  _Repeat When Necessary_ (Swan Song, 1979)  \nRockpile|  _Seconds of Pleasure_ (Columbia, 1980)  \nThe Everly Brothers|  _EV84_ (Mercury, 1984)  \n| _Born Yesterday_ (Mercury, 1986)  \nStray Cats|  _Built for Speed_ (EMI-America, 1982)  \nR.E.M.|  _Murmur_ (I.R.S., 1983)  \n| _Reckoning_ (I.R.S., 1984)  \nGuided by Voices|  _Bee Thousand_ (Scat, 1994)  \nJonathan Richman|  _Modern Lovers_ (Beserkley, 1976)  \n| _Jonathan Richman & the Modern Lovers_ (Beserkley, 1977)  \n| _Rock & Roll with the Modern Lovers_ (Beserkley, 1977)  \nThe Smithereens|  _Meet the Smithereens_ (Koch, 2007)  \n| _B-Sides the Beatles_ (Koch, 2009)  \nThe Feelies|  _Only Life_ (A&M, 1988)  \nFountains of Wayne|  _Welcome Interstate Managers_ (S-Curve, 2003)  \nThey Might Be Giants|  _Lincoln_ (Bar\/None, 1988)  \nBarenaked Ladies|  _Gordon_ (Reprise, 1992)  \nWeen|  _The Mollusk_ (Elektra, 1997)  \nGoo Goo Dolls|  _A Boy Named Goo_ (Warner Bros., 1995)  \n10,000 Maniacs|  _In My Tribe_ (Elektra, 1987)  \nThe Replacements|  _Let It Be_ (Twin Tone, 1984)  \nThe Three O'Clock|  _Arrive Without Traveling_ (I.R.S., 1985)  \nBangles|  _All Over the Place_ (Columbia, 1984)  \nThe Lemonheads|  _It's a Shame About Ray_ (Atlantic, 1992)  \n'Til Tuesday|  _Voices Carry_ (Epic, 1985)  \nSteve Earle|  _Guitar Town_ (MCA, 1986)  \nR. Stevie Moore|  _Meet the R. Stevie Moore_ (Cherry Red, 2008)  \nIndigo Girls|  _Indigo Girls_ (Epic, 1989)  \nJellyfish|  _Bellybutton_ (Charisma, 1990)  \nNeutral Milk Hotel|  _In the Aeroplane Over the Sea_ (British Domino, 1998)  \nGin Blossoms|  _New Miserable Experience_ (A&M, 1993)  \nWeezer|  _Weezer_ (DGC, 1994)  \nThe Wallflowers|  _Bringing Down the Horse_ (Interscope, 1996)  \nBelly|  _Star_ (Sire, 1993)  \nTraveling Wilburys|  _Volume One_ (Wilbury, 1988)  \n| _Volume Three_ (Wilbury, 1990)  \nFraternal Order of the All | _Greetings from Planet Love_ (Dome, 1998)  \nModest Mouse|  _This Is a Long Drive for Someone with Nothing to Think About_ (Up, 1996)  \nThe Shins|  _Oh, Inverted World_ (Sub Pop, 2001)  \n| _Chutes Too Narrow_ (Sub Pop, 2003)  \n| _Wincing the Night Away_ (Sub Pop, 2007)  \nThe Postal Service|  _Give Up_ (Sub Pop, 2004)\n\nThe British Invasions\n\nBeaver and Krause|  _In a Wild Sanctuary_ (Warner Bros., 1970 )  \n---|---  \nWalter Carlos|  _Switched-On Bach_ (Columbia, 1969)  \nDick Hyman|  _Moog: The Electric Eclectics of Dick Hyman_ (Command, 1969)  \nGershon Kingsley|  _Music to Moog By_ (Dagored, 1969)  \nHot Butter|  _Popcorn_ (Musicor, 1972)  \nGary Numan|  _The Pleasure Principle_ (Beggar's Banquet, 1979)  \nGerry and the Pacemakers|  _Don't Let the Sun Catch You Crying_ (Laurie, 1964)  \n| _Gerry and the Pacemakers' Second Album_ (Laurie, 1964)   \n| _Ferry Cross the Mersey_ (United Artists, 1965)  \nThe Dave Clark Five|  _Glad All Over_ (Epic, 1964)  \n| _The Dave Clark Five Return_ (Epic, 1964)  \n| _American Tour_ (Epic, 1964)  \n| _Coast to Coast_ (Epic, 1965)  \n| _Weekend in London_ (Epic, 1965)  \n| _Having a Wild Weekend_ (Epic, 1965)  \n| _I Like It Like That_ (Epic, 1965)  \nHerman's Hermits|  _Introducing Herman's Hermits_ (MGM, 1965)  \n| _Herman's Hermits On Tour_ (MGM, 1965)  \n| _The Best of Herman's Hermits_ (MGM, 1965)  \nThe Seekers|  _The Seekers_ (Marvel, 1965)  \n| _The New Seekers_ (Capitol, 1965)  \n| _A World of Our Own_ (Capitol, 1965)  \nThe Kinks|  _You Really Got Me_ (Reprise, 1964)  \n| _Kinks-Size_ (Reprise, 1965)  \n| _Kinda Kinks_ (Reprise, 1965)  \n| _Kinks Kinkdom_ (Reprise, 1965)  \n| _The Kink Kontroversy_ (Reprise, 1966)  \n| _Kinks Greatest Hits_ (Reprise, 1966)  \n| _Face to Face_ (Reprise, 1967)  \n| _The Live Kinks_ (Reprise, 1967)  \n| _Something Else by the Kinks_ (Reprise, 1968)  \n| _Arthur (Or the Decline and Fall of the British Empire)_ (Reprise, 1969)  \nThe Rolling Stones|  _England's Newest Hitmakers_ (London, 1964)  \n| _12 x 5_ (London, 1964)  \n| _The Rolling Stones, Now!_ (London, 1965)  \n| _Out of Our Heads_ (London, 1965)  \n| _December's Children (and Everybody's)_ (London, 1965)  \n| _Big Hits (High Tide and Green Grass)_ (London, 1966)  \n| _After-Math_ (London, 1966)  \n| _Got Live if You Want It_ (London, 1966)  \n| _Between the Buttons_ (London, 1967)  \n| _Flowers_ (London, 1967)  \n| _Their Satanic Majesties Request_ (London, 1967)  \n| _Beggars Banquet_ (London, 1968)  \n| _Through the Past, Darkly (Big Hits, Vol. 2)_ (London, 1969)  \n| _Let It Bleed_ (London, 1969)  \nBilly J. Kramer and the Dakotas|  _Little Children_ (Imperial, 1965)  \nThe Yardbirds|  _For Your Love_ (Epic, 1965)  \n| _Having a Rave Up_ (Epic, 1965)  \n| _Over Under Sideways Down_ (Epic, 1966)  \n| _Yardbirds' Greatest Hits_ (Epic, 1967)  \n| _Little Games_ (Epic, 1967)  \nThe Hollies|  _Hear! Here!_ (Imperial, 1966)  \n| _Bus Stop_ (Imperial, 1966)  \n| _Stop! Stop! Stop!_ (Imperial, 1967)  \n| _The Hollies' Greatest Hits_ (Imperial, 1967)  \n| _Evolution_ (Epic, 1967)  \nThe Who|  _The Who Sell Out_ (Decca, 1968)  \nThe Jeff Beck Group|  _Truth_ (Sony Music Distributor, 1968)  \n| _Beck-ola_ (Sony Music Distributor, 1969)  \nThe Zombies|  _Odyssey and Oracle_ (Date, 1969)  \nThe Troggs|  _Wild Thing_ (Fontana, 196)  \nRoy Head|  _Treat Me Right_ (Scepter, 1965)  \nGeorgie Fame|  _Yeh Yeh_ (Imperial, 1965)  \nTraffic|  _Traffic_ (Unitied Artists, 1968)  \nThe Bee Gees|  _Bee Gees' First_ (Atco, 1967)  \n| _Horizontal_ (Atco, 1968)  \n| _Idea_ (Atco, 1968)  \n| _Odessa_ (Atco, 1969)  \nThe Small Faces|  _There Are but Four Small Faces_ (Immediate, 1966)  \nRod Stewart|  _The Rod Stewart Album_ (Mercury, 1969)  \n| _Gasoline Alley_ (Mercury, 1969)  \nElectric Light Orchestra|  _Eldorado_ (United Artists, 1974)  \n10cc|  _Sheet Music_ (UK, 1974)  \n| _The Original Soundtrack_ (Mercury, 1975)  \n| _100 cc_ (UK, 1975)  \nLiverpool Echo|  _Liverpool Echo_ (Rev-Ola, 2005)  \nBay City Rollers|  _Bay City Rollers_ (Arista, 1975)  \n| _Rock n' Roll Love Letter_ (Arista, 1976)  \n| _Dedication_ (Arista, 1976)  \n| _It's a Game_ (Arista, 1977)  \nSupertramp|  _Crime of the Century_ (A&M, 1974)  \n| _Crisis? What Crisis?_ (A&M, 1975)  \n| _Even in the Quietest Moments_ (A&M, 1977)  \n| _Breakfast in America_ (A&M, 1978)  \nSqueeze|  _Argybargy_ (A&M, 1980)  \n| _East Side Story_ (A&M, 1981)  \n| _Sweets from a Stranger_ (A&M, 1982)  \n| _Singles: 45's and Under_ (A&M, 1983)  \nXTC|  _Skylarking_ (Geffen, 1986)  \nBananarama|  _Deep Sea Skiving_ (London, 1983)  \n| _Bananarama_ (London, 1984)  \n| _True Confessions_ (London, 1986)  \n| _Wow!_ (London, 1987)  \nPrefab Sprout|  _Two Wheels Good_ (Epic, 1985)  \nCrowded House|  _Crowded House_ (Capitol, 1986)  \n| _Temple of Low Men_ (Capitol, 1988)  \nWorld Party|  _Private Revolution_ (Chrysalis, 1986)  \n| _Goodbye Jumbo_ (Chrysalis, 1991  \n| _Thank You World_ (Chrysalis, 1993)  \nThe La's|  _The La's_ (Go! Discs, 1990)  \nBoo Radleys|  _Ichabod and I_ (Action, 1990)  \nStone Roses|  _Stone Roses_ (Silvertone, 1990)  \nSuede|  _Suede_ (Nude\/Columbia, 1993)  \nBelle and Sebastian|  _Tigermilk_ (Jeepster, 1999)  \nCornershop|  _When I Was Born for the 7th Time_ (Luaka Bop, 1998)  \nOasis|  _(What's the Story) Morning Glory_ (Creation, 1995)  \nSpice Girls|  _Spice_ (Virgin, 1997)  \n| _Spiceworld_ (Virgin, 1997)  \nNew Kids on the Block|  _Hangin' Tough_ (Columbia, 1988)  \nHanson|  _Middle of Nowhere_ (Mercury, 1997)  \nBackstreet Boys|  _Backstreet Boys_ (Jive, 1997)  \n'N Sync|  _'N Sync_ (RCA, 1998)  \nThe Jonas Brothers|  _It's About Time_ (Columbia, 2006)\n\nHere Today\n\nVarious|  _I Am Sam (Soundtrack)_ (V2, 2002)  \n---|---  \nVarious|  _Across the Universe (Soundtrack)_ (Interscope, 2002)  \nVarious|  _Nowhere Boy (Soundtrack)_ (Sony, 2010)  \nThe Beatles \/ Cirque du Soleil|  _Love (The Beatles Album)_ (Apple\/Capitol, 2006)  \nCircue du Soleil|  _All Together Now (DVD)_ (Apple Corp., 2006)  \nPaul McCartney|  _Memory Almost Full_ (Universal, 2007)  \nBob Dylan|  _Together Through Life_ (Columbia, 2009)  \nWestlife|  _Westlife_ (Arista, 2000)  \nRobbie Williams|  _The Ego Has Landed_ (Capitol, 1999)  \nColdplay|  _Parachutes_ (Netwerk, 2000)  \n| _A Rush of Blood to the Head_ (Capitol, 2002)  \nJames Blunt|  _Back to Bedlam_ (Custard, 2005)  \nDavid Gray|  _White Ladder_ (ATO, 2000)  \nTravis|  _The Man Who_ (Independente, 2000)  \nFranz Ferdinand|  _Franz Ferdinand_ (Domino, 2004)  \nArctic Monkeys|  _Whatever People Say I Am, That's What I'm Not_ (Domino, 2006)  \nApples in Stereo|  _Her Wallpaper Reverie_ (SpinArt, 1999)  \nStrokes NYC|  _Is This Is_ (RCA, 2001)  \n| _Rooms on Fire_ (RCA, 2004)  \nPolyphonic Spree|  _Together We Are Heavy_ (Hollywood, 2004)  \nOf Montreal|  _The Sunlandic Twins_ (Polyvinyl, 2005)  \nVampire Weekend|  _Vampire Weekend_ (XL, 2008)  \nMGMT|  _Time to Pretend (EP)_ (Cantara, 2009)  \nArcade Fire|  _The Suburbs_ (Merge, 2010)\n\n_The Beatles Day by Day: The Originals_ (Mischief, 09)\n\nIf you only want to listen to one album to understand where the Beatles came from, this is the one to get, even if it has four sides and eighty-one tracks. Conceived to replicate the jam session the Beatles engaged in prior to tackling the Let It Be album, with all the original artists, this is about the best cross section of American rock and roots music around (with a ringer from England tossed in here and there out of patriotic duty). It's also the best indication, if any more were needed, of the Beatles' incredible knowledge of the history of music prior to their arrival in it.\n\nAs far as what happened after their arrival... I could write a book.\n\nDISC ONE\n\n_Shake, Rattle & Roll_| Big Joe Turner  \n---|---  \n_Cannonball_|  Duane Eddy  \n_Short Fat Fannie_|  Larry Williams  \n_Johnny B. Goode_|  Chuck Berry  \n_At the Hop_|  Danny and the Juniors  \n_Rip It Up_|  Little Richard  \n_Take These Chains_|  Larry Williams  \n_All Shook Up_|  Elvis Presley  \n_Whole Lotta Shakin' Goin' On_|  Jerry Lee Lewis  \n_Diggin' My Potatoes_|  Lonnie Donegan  \n_Knee Deep in the Blues_|  Guy Mitchell  \n_Mailman, Bring Me No More Blues_|  Buddy Holly  \n_Crazy Feet_|  Fred Astaire  \n_Blue Suede Shoes_|  Carl Perkins  \n_Maybellene_|  Chuck Berry  \n_Lawdy Miss Clawdy_|  Lloyd Price  \n_Kansas City_|  Little Richard  \n_Take This Hammer_|  Leadbelly  \n_That's All Right_|  Elvis Presley  \n_Move It_|  Cliff Richard\n\nDISC TWO\n\n_Bye Bye Love_|  The Everly Brothers  \n---|---  \n_Bo Diddley_|  Bo Diddley  \n_Be Bop a Lula_|  Gene Vincent  \n_Lucille_|  Little Richard  \n_Midnight Special_|  Leadbelly  \n_Sweet Little Sixteen_|  Chuck Berry  \n_Hallelujah, I Love Her So_|  Ray Charles  \n_Shake, Rattle and Roll_|  Big Joe Turner  \n_The Peanut Vendor_|  Stan Kenton  \n_When Irish Eyes Are Smiling_|  Bing Crosby  \n_That'll Be the Day_|  Buddy Holly  \n_Fools Like Me_|  Jerry Lee Lewis  \n_It's Only Make Believe_|  Conway Twitty  \n_Slippin' and Slidin'_|  Little Richard  \n_Queen of the Hop_|  Bobby Darin  \n( _You're So Square_ ) _Baby, I Don't Care_|  Elvis Presley  \n_School Day_|  Chuck Berry  \n_High Noon_ ( _Do Not Forsake Me_ )| Tex Ritter  \n_When the Saints Go Marching In_|  Louis Armstrong\n\nDISC THREE\n\n_Singing the Blues_|  Guy Mitchell  \n---|---  \n_You Are My Sunshine_|  Jimmie Davis  \n_Tiger Rag_|  Original Dixieland Jazz _Band_  \n_High School Confidential_|  Jerry Lee Lewis  \n_Little Demon_|  Screamin' Jay Hawkins  \n_Don't Be Cruel_|  Elvis Presley  \n_Your True Love_|  Carl Perkins  \n_Jenny Jenny_|  Little Richard  \n_Low Down Blues_|  Larry Williams  \n_Ramrod_|  Duane Eddy  \n_Rock Island Line_|  Lonnie Donegan  \n_Right String Baby_ ( _But the Wrong Yo Yo_ )| Carl Perkins  \n_Rock and Roll Music_|  Chuck Berry  \n_Not Fade Away_|  Buddy Holly  \n_Catch a Falling Star_|  Perry Como  \n_I Got Stung_|  Elvis Presley  \n_St. Louis Blues_|  Louis Armstrong  \n_Send Me Some Lovin'_|  Little Richard  \n_Cocaine Blues_|  Reverend Gary Davis  \n_You Can't Catch Me_|  Chuck Berry  \n_Leaning on a Lamppost_|  George Formby\n\nDISC FOUR\n\n_Honky Tonk, Parts 1 & 2_| Bill Doggett  \n---|---  \n_Lotta Lovin'_|  Gene Vincent  \n_Miss Ann_|  Little Richard  \n_True Love_|  Elvis Presley  \n_Great Balls of Fire_|  Jerry Lee Lewis  \n_Smoke Stack Lightning_|  Howlin' Wolf  \n_Brown Eyed Handsome Man_|  Chuck Berry  \n_Gone, Gone, Gone_|  Carl Perkins  \n_My Baby Left Me_|  Elvis Presley  \n_Sure to Fall_|  Carl Perkins  \n_Midnight Special_|  Lonnie Donegan  \n_Maybe Baby_|  Buddy Holly  \n_Mack the Knife_|  Louis Armstrong  \n_Long Tall Sally_|  Little Richard  \n_Honey Hush_|  Big Joe Turner  \n_Friendship_|  Judy Garland and Johnny Mercer  \n_Milkcow Blues Boogie_|  Elvis Presley  \n_Tiger Rag_|  Duke Ellington  \n_Thirty Days_|  Chuck Berry  \n_Tea for Two_|  Tommy Dorsey  \n_Adagio for Strings_|  NBC Symphony Orchestra\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nSON of MAN\n\nA BIOGRAPHY  \nof JESUS\n\nHENRY H HARRIS\nFor Lauren, my wife and best friend  \nGod has given me many gifts\u2014you are the most precious\nCONTENTS\n\nPrologue\n\nChapter 1: A Voice from the Wilderness\n\nChapter 2: Inception\n\nChapter 3: The First Passover\n\nChapter 4: The God of All Clans\n\nChapter 5: Galilee of the Gentiles\n\nChapter 6: Authority\n\nChapter 7: Lord of the Sabbath\n\nChapter 8: The Divine Conspiracy\n\nChapter 9: Wisdom's Children\n\nChapter 10: Dark Sayings from of Old\n\nChapter 11: Identity\n\nChapter 12: Homecoming\n\nChapter 13: The Bread of Life\n\nChapter 14: Tradition\n\nChapter 15: The Turning Point\n\nChapter 16: A Kingdom of Children\n\nChapter 17: Light of the World\n\nChapter 18: Pride and Prejudice\n\nChapter 19: The Gathering Storm\n\nChapter 20: Kingdom in the Midst\n\nChapter 21: Deathless\n\nChapter 22: The Passover Plot\n\nChapter 23: On a Donkey's Colt\n\nChapter 24: Confrontation\n\nChapter 25: A Second Coming\n\nChapter 26: The Lord's Supper\n\nChapter 27: Betrayed\n\nChapter 28: Condemned\n\nChapter 29: Crucified\n\nChapter 30: Raised\n\nEpilogue\n\nAppendix: A Biblical Chronology of Jesus's Ministry\n\nAbout This Book\n\nAbout the Author\nPROLOGUE\n\n\"We did not follow cleverly devised myths when we made known to you the power and coming of our Lord Jesus Christ\u2014we were eyewitnesses of his majesty.\"\n\n\u20142 Peter 1:16\n1\n\nA VOICE FROM THE WILDERNESS\n\nSpring\u2013Autumn, AD 29\n\nA voice cries out: \"Clear a path in the wilderness. Make a straight highway in the desert for our God. Let every valley be filled and every mountain and hill be made flat. Let the crooked places be made straight and the rough places smooth. Then the glory of the Lord will appear for all to see. The Lord himself has promised this.\"\n\n\u2014Isaiah 40:3\u20135\n\nA RELIGIOUS STUPOR rests upon the land. Seven centuries have passed since the prophet Isaiah predicted a voice would call from the wilderness to prepare Israel for the Lord's arrival.\n\nMost people believe the old prophecies shouldn't be taken literally, but some still look with prayerful anticipation toward the desolate lands west of the Salt Sea, where the prophets said a new exodus will begin. \"I will lead her into the wilderness and speak tenderly to her there,\" the Lord told Hosea. \"I will make the valley of trouble into a door of hope.\"\n\nThis is a tremendous promise, but hundreds of generations have come and gone, and only the names of the oppressors have changed. \"How long, oh Lord?\" is the question on the lips of many as the poor become poorer and the weak become helpless.\n\nRecently there have been reports from the Jordan River of a Nazarite hermit who is reviving interest in the old predictions and announcing the arrival of the Messiah. \"Repent,\" John bar Zechariah demands, \"for the kingdom of heaven is at hand.\"\n\nLike the prophets before him, his counsel is blunt: prepare to meet your God\u2014He will be here soon. To the poor, who are convinced that only the Lord can deliver them from their hopeless situation, John's news is encouraging. To others, his words sound more like a warning. Maybe even a threat.\n\nJohn's appearance is as startling as his message. He's in his early thirties, but years in the desert have weathered him. His dark skin is cracked and leathery, his beard is long and matted, and his hair is uncut and cascades down his back in thick coils. His body, lean from a diet of honey and locusts, is clothed in a coarse camel-hair cloak that is cinched at his waist with a leather tie.\n\nJohn confines his ministry to the Jordan River Valley, but he is known in Jerusalem. His father was a priest in the Abijah division, and everyone assumed that one day John would take his father's place, but the son of Zechariah has a different destiny. Before John was born, the angel Gabriel proclaimed that he would be filled with the Holy Spirit from the womb and, like only Samson and Samuel before him, a Nazarite for life.\n\nJohn proclaims the coming of the Messiah up and down the two hundred miles of switchbacking river that is compressed into a sixty-five-mile stretch between Lake Gennesaret and the Salt Sea.\n\nMost who come to the Jordan are from Judea, and they travel the road the Romans built on the Ascent of Adumim. It is paved, but it is also steep and dangerous. While Jerusalem sits comfortably 2,500 feet above sea level, the Jordan Valley, just twenty miles away, is 1,500 feet below sea level and 4,000 feet below Jerusalem. To complicate matters, the remoteness of the Jericho road makes it appealing to thieves and robbers.\n\nRumors about the man known as \"the Baptist\" spring up in every town and village in Israel. Some believe he's the Messiah. Others note that John dresses like Elijah and has a similar message. He even ministers in the same location where the ancient prophet mounted a fiery chariot bound for heaven. Has Elijah returned?\n\nMany go to the Jordan to judge for themselves. Unlike the lukewarm homilies they normally hear, the Baptist's messages are convicting. John points them to God's Word, and when they confess their sins, he baptizes them and exhorts them to prepare for the coming of the one who will change everything.\n\nJohn's baptism, like his appearance and message, is unorthodox. The rite itself is familiar. Every house of status has its own mikveh for ceremonial immersions, and Jewish communities throughout the land provide public pools. The Pharisees, Sadducees, and Essenes regularly immerse themselves to ensure their cleanliness before God, but John's baptism is administered one time only because it signifies something different. His baptism isn't another purification ritual\u2014it's a public declaration by those being baptized that they will live obediently in anticipation of the Messiah. It is more like a proselyte baptism, but John is immersing repentant Israelites, not Gentiles seeking to convert to Judaism.\n\nAs the months pass, more and more people make pilgrimages to the Jordan, hoping to be made right with God, hoping that somehow life will be better after they hear John's message. Not everyone who comes, however, desires to be made right with God. Religious leaders descend from Jerusalem to challenge John's authority to administer rituals they are convinced are the prerogative of priests, not locust eaters. They stand warily on the fringes of the crowd, hoping to either get the Baptist into their camp or to discredit him.\n\nJohn sees the men in the elegant robes but ignores them. He has been sent to proclaim God's salvation to the people the elite look down upon and dismiss as am ha'aretz\u2014\"people of the land\"\u2014the uneducated working poor who make up 90 percent of the population.\n\nThe religionists are convinced that the commoners are too rustic to keep the hundreds of laws and traditions necessary to remain blameless, yet it is the people of the land who are responding to John's invitation. Daily the banks of the river are lined with expectant faces, some traveling from as far away as Upper Galilee and Gaulanitis. Children play at the water's edge while their parents sit under the trees and listen to John explain what faithfulness looks like in everyday life.\n\nThis, too, was prophesied. Before John's birth the angel Gabriel declared, \"He will turn many of the children of Israel to the Lord their God, and he will go before him in the spirit and power of Elijah to turn the hearts of parents to the children and the disobedient to the wisdom of the just\u2014to make ready for the Lord a people prepared.\"\n\nAs the months pass, the angel's prophecy is fulfilled.\n\nOne day when John is preaching near Bethany-by-the-Jordan, a large number of priests and teachers of the Law are in the audience. John turns to them, pauses long enough to be sure he has their attention, and says, \"You children of vipers, who warned you to flee from the coming wrath?\"\n\nA popular legend describes how, during birthing, baby vipers will chew their way out of their mother's womb, killing her in the process. John accuses the religious leaders of doing the same to Israel. He knows what's in their hearts and will not pretend they are the high and holy men they want others to think they are.\n\n\"Produce the kind of fruit that demonstrates you have changed,\" the Baptist demands.\n\nKnowing they rely on their heritage as a divinely chosen people to give them right standing with God, John warns, \"Don't smugly think to yourself you can say, 'We have Abraham as our father,' for I tell you that God is able to raise up children for Abraham from these rocks. And even now the ax is lying at the root of the trees. Every tree that does not bear good fruit is cut down and thrown into the fire.\"\n\nThe Pharisees and Sadducees smolder and turn away, but the crowd overhears John's warning and is frightened. If the pious are doomed, what hope is there for the simple people? \"What should we do then?\" they ask.\n\nJohn encourages them to live generously. \"Let the one who has two tunics share with the one having none, and the one who has food should do the same.\"\n\nHearing John's instruction to live by faith, a tax collector then asks, \"Teacher, what about us? What should we do?\"\n\nJohn won't condemn their occupation, but he knows they are allowed to keep any extra monies they can extort. \"Collect nothing more than the amount you have been commanded to collect,\" he tells them.\n\nEmboldened by John's gracious answer, another despised group asks the same question. \"What about us\u2014what should we do?\" one of the Levite soldiers asks.\n\n\"Don't take money from anyone by force or accuse anyone falsely,\" John says, \"and be content with your wages.\"\n\nHe speaks with such authority that some wonder aloud if he is the long-awaited Christ. The Messiah, the people of Israel are convinced, will be the answer to all their troubles. He will single-handedly crush their enemies, restore righteousness to the nation, and make Jerusalem the seat from which the world will be governed for all eternity.\n\nThe Baptist shakes his head at their speculations. \"I baptize you with water to show that you have repented,\" John tells them, \"but the one coming after me is stronger than I\u2014I am not worthy to bend down and untie the strap of his sandal. I baptize you with water, but he will baptize you with the Holy Spirit and with fire. His winnowing fork is in his hand to clear his threshing floor, gathering the wheat into his barn but burning the chaff with unquenchable fire.\"\n\nThe crowd is familiar with the winnowing process, and the metaphor is disturbing. After threshing, the wheat (including husks, straw, and plant debris) is tossed high in the air with a shovel-like wooden fork. The heavy wheat kernel falls to the floor while the remaining debris\u2014the chaff\u2014is blown aside by the wind. And because Israel has few fuel resources, the chaff is often used for kindling. John says that people are like wheat. Some are grain and some are chaff. The grain will be blessed, but the chaff will be thrown into hell.\n\nThe priests and scholars scowl and walk away. They would like to lure the renegade prophet to Jerusalem to answer to the high priest, but John will not be diverted from his mission. He remains at the river, proclaiming the Messiah's coming and warning of impending judgment.\n\nONE DAY AS John is preaching and baptizing, a man quietly slips into the crowd and makes his way to the water's edge. He is clothed in Galilean homespun and appears to be in his early thirties. His height and weight are average\u2014he's slightly over five feet tall and weighs about 120 pounds. His muscular build and rough hands give the impression of a workman, perhaps a tekton who builds with wood and stone. He's not a handsome man, but his smile is warm and his eyes sparkle with the inquisitiveness of a child. His hair is cut short and gently curls around his ears. Those who bother to look see a Jewish everyman from the northlands.\n\nJesus comes meekly and without fanfare, one of many seeking to be baptized, but as he walks toward the water, the Holy Spirit stirs in John. The Baptist wonders if he is in the presence of the Anointed One, but he's confused\u2014if Jesus is the Messiah, why would he seek baptism? Why would he want to be identified with sinners?\n\nWhen Jesus draws near, John pleads his inadequacy: \"I need to be baptized by you, and yet you come to me?\"\n\n\"Permit it at this time,\" Jesus says, \"for in this way it is fitting for us to fulfill all righteousness.\"\n\nJohn does not understand Jesus's reasoning, but when he's told it is fitting to do all things that are God's will, he immediately immerses the man. As Jesus rises from the water, he is praying. He opens his eyes and sees the Holy Spirit descending.\n\nThe Baptist also glimpses the Spirit, and all his questions are answered. Long before this day God had told him, \"He on whom you see the Spirit descend and remain, this is he who baptizes with the Holy Spirit.\"\n\nAs John stands in the river allowing the significance of what has happened to wash over him, he hears a voice from heaven say to Jesus, \"You are My son, the beloved\u2014with you I am well pleased.\"\n\nThe words are from Psalm 2 and Isaiah 42, in which David describes the coronation of the Messiah and Isaiah identifies a righteous servant of God who will be beaten beyond recognition and killed. The Voice designates Jesus as both the servant and the king.\n\nJesus, his public anointing complete, immediately receives his first assignment from the Holy Spirit. With the Jordan's water still dripping from his tunic, he walks west into the wilderness. His first appointment will be with his oldest and most deadly adversary.\n2\n\nINCEPTION\n\nAutumn\u2013Winter, AD 29\n\nThe old has passed away; behold, the new has come.\n\n\u20142 Corinthians 5:17\n\nTHE BROAD DESERTS and barren chalk peaks of the Judean wilderness are part of the legendary midbar\u2014the dry lands east and south of Jerusalem where many of Israel's most painful memories are stored. The Torah records how God freed His people from Egypt and brought them to the land He promised to Abraham. Within weeks they were at the border, but the Israelites refused to enter. Their fear of the country's inhabitants was greater than their trust in the Lord. All but two of that faithless generation died in the desert, and forty years later their children followed Joshua and Caleb across the Jordan to claim their inheritance.\n\nIsaiah prophesied that one day a deliverer would come from these dry lands to lead God's people in a new exodus, but until then the midbar would remain a feared place avoided by all but outlaws and ascetics. Moses, who lived in these wastelands for eighty years, described them as \"a vast and terrifying wilderness with fiery serpents and scorpions and thirsty ground where there was no water.\" Other Scriptures describe the midbar as horbah (dry, desolate, ruined); tsiyyah (land of drought); jeshimon (desolation); and tohu (a chaotic void).\n\nHere among the lions and leopards and snakes and scorpions, Jesus is to fast for forty days and contend with a powerful angel who has been carefully molding history into his own image for thousands of years. The angel has never met a mortal he could not enlist, deceive, or overpower, and if it were not for divine interference, the entire human race would by now be his slaves.\n\nOver the years he has been known by many names. Once, as a leader in the special class of guardian angels known as cherubim, he was Helel ben Shachar (Shining Star, Son of the Dawn), but that was long ago. These days he only appears as an angel of light when it serves his purposes, and with Jesus there's no need for pretense.\n\nJesus fasts and prays for several weeks. The Devil is content to simply watch and wait. A month passes, and still Satan takes no action. This moment has been a long time coming, and he wants to savor it.\n\nAfter forty days Jesus is starved and near exhaustion. The beguiler approaches meekly with concern in his eyes. Feigning worry about Jesus's health, he suggests, \"Since you are the Son of God, tell these stones to become bread.\"\n\nThe recommendation sounds harmless, even compassionate. The man is obviously starving\u2014why not use the resources at his command to help himself? But the suggestion is not as innocent as it appears. Jesus is being tempted to question God's provision. Satan is suggesting that Jesus will be better off if he takes care of himself rather than waiting for God to get around to it.\n\n\"It is written,\" Jesus says, \"'Man does not live on bread alone, but on every word that comes from the mouth of God.'\"\n\nWhen the Devil realizes Jesus will not act independently of the Father, he supernaturally transports him to the top of the temple in Jerusalem and whispers in his ear, \"Since you are the Son of God, throw yourself down from here, for it is written, 'He will command His angels to protect you,' and 'they will lift you in their hands so that you never hit your foot against a stone.'\"\n\nJesus looks at the valley floor 450 feet below and reflects on the psalm the demon quoted from. The ninety-first psalm describes one who trusts in the Lord. Such a person is to be fearless. \"When he calls to Me,\" God says, \"I will answer him. I will be with him in trouble\u2014I will rescue him and honor him.\"\n\nThe Devil hopes to persuade Jesus to test this promise. Surely his Father will not allow him to be hurt. Why not let go and let God confirm He is who He says He is?\n\n\"Again it is written,\" Jesus reminds the tempter, \"'You shall not put the Lord your God to the test.'\"\n\nFor a second time the account of the wilderness journey is called in as evidence. When God miraculously provided food for the Israelites, they weren't satisfied. Even though they had seen His deliverance with their own eyes and had tasted it daily with their own lips, they still doubted God's goodness and demanded, \"Is the Lord among us or not?\"\n\nJesus refuses to jump from the heights of the temple because he knows that same cynical challenge will be his if he forces God to rescue him. He chooses to trust without testing.\n\nAfter his first two attempts fail, the Devil takes Jesus to a high mountain and shows him a vision of all the kingdoms of the world\u2014the splendor of imperial Rome, the grandeur of ancient Greece, the beauty of the Mediterranean isles. \"All these I will give you,\" Satan promises, \"if you will bow down and worship me.\"\n\nIt's an offer of vast political power. Satan suggests that as king of the world, Jesus can eliminate injustice and poverty. Wars will cease. Laws can be made and enforced to ensure everyone is fed and well taken care of. And, as an added bonus, Jesus can have all of this now. No need to suffer to bring it about, and certainly no need for a cross. The world's problems can be solved politically if only the right leader will take the reins.\n\nThis is an incredibly appealing proposal, one Satan has been buying souls with for centuries, but Jesus has no interest in the Devil's bargain. While he can appreciate the positive changes a godly government would bring, Jesus also knows it would be a shallow, short-term fix. Saving people from hunger and injustice is not what they most need. What they most need is to be reconciled with God\u2014something the painless solution offered by the evil one can never provide.\n\n\"Begone, Satan!\" Jesus commands, \"For it is written, 'You shall worship the Lord your God and Him only shall you serve.'\"\n\nWhen Jesus quotes the same command he is being tempted to break, the Devil realizes his first face-off with the Messiah has been a failure, but he knows there will be other opportunities. He retreats into the shadows as angels descend from heaven to care for Jesus.\n\nAT THE JORDAN, six weeks have passed. John continues to proclaim the Messiah's arrival, and the nation's religious rulers grow increasingly alarmed with the prophet's popularity, especially among the poor. The leaders have seen the way the military responds when the Pax Romana is violated, and they can't afford to have a rogue prophet creating unrest. The high priest sends a delegation to gather evidence they can use to silence the Baptist.\n\nWhen the priests and Pharisees reach John's camp, they ask probing questions about his identity. First and foremost they want to know if he claims to be the Christ.\n\n\"I am not the Messiah,\" John says.\n\n\"If you're not the Messiah, then who are you? Are you Elijah?\"\n\n\"No, I am not.\"\n\n\"Then who? Are you the Prophet that Moses spoke of?\"\n\n\"No.\"\n\nThe Baptist answers their questions but offers no further explanation, and eventually they tire of guessing. \"Who are you?\" they demand. \"Give us an answer to take back to those who sent us from Jerusalem. What do you have to say about yourself?\"\n\nJohn replies with the words of the prophet Isaiah: \"I am the voice of one shouting in the wilderness, 'Make straight the way for the Lord.'\"\n\nHis answer only flusters them more. \"If you're not the Messiah or Elijah or the Prophet, then why do you baptize people?\"\n\nNow their words betray their real concern\u2014John is performing unsanctioned religious rituals.\n\nJohn offers an enigmatic answer: \"I baptize with water, but among you stands one you do not know. He is the one who comes after me, the thongs of whose sandals I'm not worthy to untie. This was he of whom I said, 'He who comes after me has surpassed me because he was before me.'\"\n\nWhat does that mean? the priests and Pharisees ask themselves. Frustrated with the prophet's unwillingness to give them a simple answer to a simple question, they return to Jerusalem.\n\nThey are, however, comforted by one thought: John is too outspoken for his own good. Especially promising is the Baptist's recent decision to confront Herod Antipas about his affair with his sister-in-law, the wife of Herod Philip. A quiet divorce had been arranged, but John is publicly rebuking Antipas and announcing God's opposition to the union. When Jerusalem's leaders reflect on this, some conclude that the prophet may one day go too far, and their problem will resolve itself.\n\nJohn the Baptist is completely unconcerned about repercussions. He knows his part of the story is nearing its end. This was confirmed two months ago when the sky opened and he heard the voice of God. But where has the Messiah disappeared to and where is he now?\n\nThen one day as John is teaching, Jesus walks by. John motions toward him and says, \"Behold, the Lamb of God who takes away the sin of the world! This is the one I meant when I said, 'A man who comes after me has surpassed me because he was before me.' I myself did not know him, but the reason I came baptizing with water was that he might be revealed to Israel. I saw the Spirit come down from heaven as a dove and remain on him. I would not have known him, except that the One who sent me to baptize with water told me, 'The man on whom you see the Spirit come down and remain is he who will baptize with the Holy Spirit.'\"\n\nJohn pauses, looks into the eyes of his disciples, and says, \"I have seen and I testify that this is the Chosen One of God.\"\n\nFor months the Baptist has been proclaiming the imminent arrival of an unknown Messiah. The Christ now has a face and a name.\n\nThe next afternoon John sees Jesus walking by and again says, \"Behold, the Lamb of God!\"\n\nTwo of his disciples, Andrew bar Jonah and John bar Zebedee, get up and begin walking behind Jesus, trusting the Baptist's testimony that the humbly dressed and unassuming man they are following is the Messiah.\n\nWhen Jesus notices they are behind him, he turns and asks, \"What do you want?\"\n\nThe young men indicate their desire to become disciples with a simple question: \"Rabbi, where are you staying?\"\n\nJesus smiles and says, \"Come, and you will see.\" Andrew and John follow Jesus to his lodgings and spend the remainder of the day with him.\n\nBy the time they leave, Andrew is convinced John the Baptist is right about Jesus. He runs and finds his brother, Simon, and exuberantly proclaims, \"We've found the Messiah!\"\n\nSimon is fascinated by Andrew's story and even more impressed when he meets Jesus. He still has questions about how an ordinary-looking man from Nazareth could be the Messiah, but Simon can sense there is something profoundly different about this teacher.\n\nToward the end of their conversation, Jesus surprises Simon by giving him an Aramaic nickname: \"You are Simon bar Jonah\u2014you will be called Kephas.\"\n\nSimon wonders why he will be known as \"Rock\" (which in Greek is Petros\u2014Peter). He certainly doesn't feel like one, but the next day when Jesus announces he is leaving for Galilee, the new Rock and his companions follow. For Simon Peter, Andrew, and John, Capernaum is home. They are fishermen on the north shore of the harp-shaped Lake Gennesaret, recently renamed Lake Tiberias by Herod Antipas in honor of the emperor.\n\nAs the four men make their way toward the north shore, they meet Phillip, a friend from Simon and Andrew's childhood home of Bethsaida. He's a quiet man who, like many, wrestles with his faith, but he is greatly encouraged when he meets Jesus.\n\nHe is soon convinced that Jesus is the Messiah, and later, when he sees his friend Nathanael bar Tolmai sitting under a fig tree, he shares the good news with him: \"We've found the one Moses wrote about in the Law and the Prophets. He's Jesus bar Joseph from Nazareth.\"\n\nAt the mention of Jesus's hometown, Nathanael becomes skeptical. He is from Cana, nine miles north of Nazareth, and has little positive to say about the village. \"Can anything good come out of Nazareth?\" he asks.\n\nPhilip laughs and says, \"Come and see!\"\n\nNathanael has his doubts, but his friend's enthusiasm is infectious, and soon they're looking for the new rabbi.\n\nJesus sees Nathanael approaching and says, \"Behold, here is truly an Israelite in whom there is no deceit!\"\n\nJesus contrasts Nathanael with his forefather Jacob, who regularly deceived people. Nathanael is startled. He has never met this man. \"How do you know me?\" he asks.\n\n\"I saw you while you were still under the fig tree before Philip called you.\"\n\nJesus says it casually, and his observation sounds innocuous to Philip and the others, but his words stun Nathanael. There is no way Jesus could have known that he was under the fig tree. Nathanael now sees why his friend became a follower. \"Rabbi,\" he exclaims, \"you are the Son of God\u2014 you are the King of Israel!\"\n\n\"You believe because I told you I saw you under the fig tree,\" Jesus says. Then he turns to those who are following and adds, \"You will see greater things than that. I tell you the truth, you will see heaven open and the angels of God ascending and descending on the Son of Man.\"\n\nJesus's words shock his new admirers. He is claiming to be the connection between heaven and earth, God and man. Jacob had named the place where he saw the vision of the ladder Bethel, \"house of God.\" Jesus says God's house is no longer a place. Bethel has become flesh and dwells among them.\n\nTHE FIVE ARE still thinking about Jesus's words two days later when they travel to Nathanael's hometown of Cana for the wedding of a family friend. When they arrive they find Jesus's mother already there, helping coordinate the reception.\n\nIn small villages, weddings are the biggest local events of the year. The entire town turns out for the festivities. Traditionally, the ceremony begins with an evening processional: the groom, accompanied by his friends, walks to the home of the bride, takes her arm in his, and leads the wedding party back to his home for a reception that lasts for several days.\n\nThe wedding procession unfolds as planned, and the guests are noisily seated at tables filled with fine food and flagons of wine. All are enjoying themselves and the celebration is in full stride, when one of the servants realizes that the wine supply is inadequate.\n\nJesus's mother learns of the problem and knows that in a small town like Cana this will not only shame the couple, it will permanently tarnish their reputation. Mary finds Jesus and urgently whispers, \"They have no wine.\"\n\nMary hopes her son will understand the gravity of the situation, but instead of responding sympathetically, Jesus mildly chastises her. \"Woman, what does this have to do with me and you? My hour has not yet come.\"\n\nJesus gently lets his mother know that now his life must follow a different timetable. Mary doesn't understand her son's words but knows it is best to simply trust him. She turns to the servants and says, \"Do whatever he tells you.\"\n\nJesus considers the circumstances and decides this could be a good teaching opportunity. Looking around, he sees six stone water jars that earlier were used by the wedding guests for rites of purification. Each jar holds two to three measures (sixteen to twenty-seven gallons), but the vessels have served their purpose and are now empty.\n\nJesus tells the servants, \"Fill the jars with water.\"\n\nThe attendants think this is an unusual request, but they go to the well, and soon the receptacles are filled to the brim. Jesus tells one of the men, \"Now draw some out and take it to the overseer of the feast.\"\n\nThe servant initially balks at the instructions. He fears what his busy boss will do when given a goblet of wash water to sample, but after a moment he dutifully delivers the drink and awaits the overseer's response. The attendant is surprised when his master does not ask the obvious question, and even more surprised to see that the cup he brought is now filled with red wine.\n\nThe banquet coordinator, a man well experienced at dealing with the complexities involved in supplying food and drink for multiday festivities, casually takes a sip of the wine. His eyebrows immediately shoot up. He looks to the attendant for an explanation, but the servant bows his head and says nothing, afraid to tell his master the wine had moments before been well water.\n\nThe wedding organizer, savoring the richness in his cup, loudly honors the bridegroom. \"Everyone sets out the good wine first, and then, when people have become drunk, he sets out the cheaper wine. But you have kept the good wine until now.\"\n\nThe guests shout their approval, and the party continues. Only a handful of people see Jesus's first sign, but everyone at the wedding enjoys his gift. The prophet Amos predicted the Messiah's coming would be heralded by an abundance of wine in the land\u2014\"New wine will drip from the mountains and flow from all the hills\"\u2014and on this day a small village in Galilee has gotten a foretaste of that new wine and learned about a God who saves the best for last.\n\nWhen the wedding celebration ends, Jesus invites his mother and brothers to travel to the lake with him and his new followers. The spring morning is pleasant, the road slopes gently downhill, and they make the sixteen-mile walk to Capernaum easily in a day.\n\nAndrew, Simon, and John reunite with their families and tell them how the Baptist led them to Jesus and what happened in Cana. No one believes the wild story about water becoming wine, but the young fishermen know what they saw and tasted.\n\nThe teacher from Nazareth intrigues the men, and none of them are ready to return to their nets. Besides, it's time for Passover and the Feast of Unleavened Bread\u2014eight of the most sacred days on the Jewish calendar. Jesus has announced he will be attending the festival, and his five followers can't wait to see what will happen. If water was turned into wine for an unimportant party in an out-of-the-way place like Cana, what might happen at one of the most important holidays of the year with the entire nation watching?\n3\n\nTHE FIRST PASSOVER\n\nSpring\u2013Summer, AD 30\n\nBehold, I am sending my messenger to clear a path before me. Then the Lord whom you seek will suddenly come to his temple\u2014even the messenger of the covenant in whom you delight\u2014behold, he is coming, says the Lord of hosts. But who can endure the day of his coming, and who can stand when he appears? For he is like a refiner's fire and like a launderer's soap.\n\n\u2014Malachi 3:1\u20132\n\nTHE MONTH OF Nisan has arrived. The cool rainy season has passed, and wildflowers blanket the land. The rose of Sharon mentioned in the Song of Solomon covers the sandy coastal plains, and the hills are speckled with brightly colored anemones, irises, and daffodils. Solomon described it as the season when \"the flowers appear on the earth, the time of singing has come, and the voice of the turtledove is heard in our land.\"\n\nNature's time of renewal and rebirth also signals the start of the festival season. Hag ha Pesah, celebrated on Nisan 14, is followed by the weeklong Hag ha Matzot. Both commemorate the night Israel was liberated from slavery by the blood of a lamb.\n\nThe Law no longer requires the physical presence of every adult Jewish male, but all devout Jews want to be in Jerusalem for the Passover. Ships from faraway lands bring thousands into the ports of Caesarea, Joppa, Ptolemais, and Dora. Caravans of boisterous pilgrims singing the Psalms of Ascent throng the roads to the capital.\n\nBy the second week of Nisan, Jerusalem is overflowing, and a large tent city has appeared on the plain outside the northern gates. Thousands of sheep and cattle dot the hillsides. Jerusalem, normally a city of 30,000, swells to a population of more than 150,000 during the feasts, creating major problems for those seeking food and lodging.\n\nPreparation begins several weeks before the festival\u2014roads to the city are repaired, bridges are reinforced, tombs near walkways are whitewashed so that pilgrims will not accidentally defile themselves, homes are scoured and scrubbed, food and drink reserves are set aside, and everything possible is done to beautify Jerusalem and the surrounding valleys for the eight-day commemoration.\n\nFor young men who travel light, Jerusalem is less than a four-day walk from Capernaum. Like most Jews coming from the north, Jesus and his new followers take the road along the Jordan. They arrive in Jericho in three days, and early the next morning they make the strenuous seven-hour climb up the Ascent of Adumim. The grade is steep, but the men are looking forward to the festival, and the hours pass quickly. By noon, the red hills are clothed in green. Soon after passing Bethany, they crest the Mount of Olives and get their first view of the holy city lying three hundred feet below them.\n\nIn the afternoon sun, Jerusalem, built of the same tawny limestone it rests upon, has the appearance of a rumpled old lion skin draped over four hills. The nearest hill, just across the Kidron Valley, is Mount Moriah. Purchased centuries ago by King David to be the site of a house dedicated to God, the original temple has been destroyed, but in its place is the ambitious building project of Herod the Great. What Ezra once described as an \"open square before the house of God\" is now twice that size\u2014a massive thirty-seven-acre plaza enclosed by tall colonnaded porches.\n\nWithin an hour Jesus and his companions are standing on the temple mount in the Court of the Gentiles. Tables have been set up under one of the stoas to accommodate the money changers, who convert currency to the only coinage acceptable for the yearly temple tax\u2014Tyrean shekels and half-shekels, chosen for their purity of silver. Each year the money changers are allowed to open for business a month before Passover in towns throughout Israel, and twenty days before the feast they are permitted to trade on the temple grounds.\n\nNearby are the stalls of the animal merchants. Stacked around the columns are cages filled with doves and pigeons. Livestock pens spill out into the plaza. The stench of fresh manure mingles with the aroma of frankincense and myrrh wafting down from the inner courts. Those in need of a sacrifice move from pen to pen in search of value, while the vendors extol the quality of their livestock.\n\nIn the past, the animal merchants were located under the trees east of town in the Kidron Valley. Viewed from one perspective, allowing the merchants and money changers into the temple precincts provides a convenient service for those attending the festival. Bringing an animal a long distance is difficult, and if it is deemed blemished, the expense and effort has been wasted. Thus it has become more pragmatic, even for those who live in Jerusalem, to buy the preapproved animals sold at the temple. Over the years, this has come to be seen as a mutually beneficial exchange. God gets an appropriate sacrifice, and the merchants make a nice profit.\n\nJesus watches the money changers weigh coins and haggle over everything from the weight and purity of the coins being exchanged to the amount of their fee (a surcharge of 4 to 8 percent is customary, the exact amount depending on the skills of the negotiators).\n\nAs Jesus takes in the sights, smells, and sounds around him, he reflects on the original purpose of the temple. The house of God was to be set apart for worship, but instead of the sound of prayer, there is the fearful bleating of sheep. Instead of reverent conversations about the greatness of God, there is haggling over mammon.\n\nIn his Father's house.\n\nJesus's new followers notice he is upset but fail to understand why. There is nothing out of the ordinary about this Passover. They are surprised when he walks over to one of the animal stalls, picks up a handful of cords, and twists them into a whip. They are even more surprised to see him take his newly formed scourge into the bazaar and begin overturning tables and opening livestock pens.\n\nAs he drives the animals from the stoa, he turns to the sellers of doves and gestures toward their cages. \"Take these away!\" he commands. \"Do not turn my Father's house into a market!\"\n\nThe Roman troops stationed in the Antonia Fortress overlooking the temple grounds notice the disturbance but assume it's just another religious squabble between rival Jewish sects. Since only a small area of the immense people-filled plaza is being disrupted, the soldiers ignore it.\n\nIn the courtyard, the money changers are scrabbling across the pavement and arguing violently over the gold and silver strewn on the ground. The merchants look helplessly at the temple police and wonder why the troublemaker isn't being arrested. Yet the policemen, all Levites, hear Jesus calling the temple \"my Father's house\" and realize this is no ordinary agitator. They turn to the Sadducee priests, who provide oversight for the market.\n\nThe priests scowl at Jesus but do not give the order to arrest him. This is not the first time they have been opposed by those who believe the temple should be set apart solely for the worship of God. They have no interest in that debate. Rather than addressing Jesus's accusation, they challenge his authority to interfere with God's ordained leadership. \"What sign do you show us for doing these things?\" they demand.\n\nIronically, they ask for exactly what Jesus has just given them, so he gives them an even more obscure sign. \"Destroy this temple,\" Jesus says, \"and in three days I will raise it up.\"\n\nThe Jewish leaders scoff. \"It has taken forty-six years to build this temple, and will you raise it up in three days?\"\n\nJesus answers by turning and walking away.\n\nThe temple officials fume as they watch Jesus disappear into the crowd. The Galilean has attacked the financial machinery of the religious festival structure, and few threats are taken more seriously. The temple provides oversight and funding for every aspect of Jewish culture. Politics, religion, and law are virtually inseparable in the Jewish world, and Jesus has come to the center of that world during one of the biggest festivals and made a loud and symbolic protest. Property has been damaged, money has been scattered, and animals have been turned out into the streets. What Jesus has done isn't merely disrespectful\u2014it's a major offense against the entire system.\n\nWithin hours the seventy members of the Great Sanhedrin gather to discuss how to deal with the false prophet. The ruling council is a diverse meld of aristocratic Sadducee priests, influential city elders, and ultraorthodox Pharisees, many of whom are lawyers and scholars known as scribes. Though severely limited in power by Rome, the Sanhedrin exerts tremendous influence over religious Jews, especially in Jerusalem.\n\nAfter a heated discussion, all the councilmen are agreed that Jesus had no authority to do what he did, and all are agreed that his boast about being able to rebuild the temple in three days is absurd, but they are divided about the action he took with the money changers and animal merchants. The Sadducees reject Jesus's charge that the temple is being defiled; they are confident the bazaar is a blessing to both themselves and the people who buy there. Some of the Pharisees, however, are sympathetic. They, too, must pass the marketplace on their way to the temple. They, too, hear the frightened cries of the animals and the haggling over money. They, too, smell the manure. While none are willing to voice support for Jesus, some in the Sanhedrin secretly admire what the young man has done.\n\nTHE SEVEN-DAY FEAST of Unleavened Bread follows the Day of Passover. Jesus spends the week in the temple plaza, preaching to large crowds. Many praise him, but Jesus knows their flattery is insincere. Scribes and priests also regularly listen in; some come solely to gather evidence against him, but others come to hear what he has to say about God.\n\nAmong the curious is a scribe named Nicodemus. He is intrigued by Jesus's message and wants to hear more, but the temple mount is not the place for a high-ranking member of the Sanhedrin to have a conversation with an out-of-favor itinerant preacher. Still, there is something about Jesus that fascinates Nicodemus, and one evening he arranges a private meeting.\n\nWanting Jesus to know he is a potential friend, Nicodemus says, \"Rabbi, we know you are a teacher come from God, for no one can do these signs that you do unless God is with him.\"\n\nJesus looks into the learned Pharisee's eyes and says, \"Truly, truly I tell you, unless one is born again from above, he is not able to see the kingdom of God.\"\n\nNicodemus is confused; obviously Jesus is serious, but he speaks in riddles and demands the impossible. \"How can a man be born when he is old?\" Nicodemus asks. \"Can he enter a second time into his mother's womb and be born?\"\n\nJesus replies with an illustration from the Scriptures: \"Truly, truly I tell you, unless one is born of water and Spirit, he cannot enter the kingdom of God. That which is born of the flesh is flesh, and that which is born of the Spirit is spirit. Do not be amazed that I said to you, 'You must be born again from above.' The wind blows where it will, and its sound you hear, but you do not know where it comes from or where it goes. So is everyone who is born of the Spirit.\"\n\nNicodemus now begins to understand. The prophets have long used water\u2014so precious to those who live in a dry land\u2014to represent the Holy Spirit. The wind is also a common metaphor for the Spirit, but Nicodemus can't make sense of how it all works in everyday life. \"How can these things be?\" he asks.\n\nJesus shakes his head and says, \"You're the teacher of Israel, and you don't know these things?\"\n\nNicodemus, stung by Jesus's observation, falls silent.\n\nJesus, knowing his words will eventually find a home in Nicodemus's heart, broadens his admonition to include the Pharisee's fellow Sanhedrists. \"Truly, truly I tell you, that which we know, we speak, and that which we've seen, we bear witness to, but you don't receive our testimony. If you don't believe the earthly things I have told you, how will you believe if I tell you the heavenly things? No one has ascended into heaven except the one who descended from heaven, the Son of Man. And as Moses lifted up the serpent in the wilderness, so must the Son of Man be lifted up, so that whoever believes in him may have eternal life.\"\n\nNicodemus nods in partial understanding. Jesus is speaking of the brass serpent that saved those who looked to it for deliverance. But who is this son of man he speaks of, and how will he be raised up like the serpent on the pole? By the time their conversation is over and Nicodemus is walking home, his whole world has shifted.\n\nA FEW DAYS later, when the Feast of Unleavened Bread is over, Jesus begins teaching in the many small villages that are suburbs to Jerusalem. Crowds gather everywhere he goes. Those who profess faith in Jesus are pointed to his disciples, who baptize them.\n\nIt's an amazing time for the people of Israel, for while Jesus is preaching in the hill towns of Judea, John the Baptist is teaching and baptizing at the springs near the town of Salim. Yet when Jesus begins attracting larger crowds, some of John's disciples grow resentful. The Baptist once was the premier prophet in the land. Now this carpenter from Nazareth, a mere candidate for baptism a few months ago, is eclipsing the master.\n\nThe more the Baptist's disciples think about the situation, the more indignant they become. One day they approach John and say, \"Rabbi, the one who was with you on the other side of the Jordan, the one you talked about\u2014behold, he is baptizing, and everyone is coming to him!\"\n\nJohn understands their bitterness. They remember when Jesus was a nobody from nowhere. They were surprised when John called Jesus the Lamb of God and encouraged them to give him their allegiance. A few Galilean fishermen had responded, but most chose to stay with the Baptist. They preferred a prophet to a lamb.\n\nJohn knows that those who complain have never really understood his role. \"A person cannot receive anything unless it has been given him from heaven,\" John tells them. \"You yourselves can testify that I said, 'I am not the Messiah, but I have been sent ahead of him.' The one who has the bride is the bridegroom, but the friend of the bridegroom who stands and hears him rejoices greatly at the bridegroom's voice. Therefore this joy of mine has been made full.\"\n\nThe Baptist looks into the eyes of his disciples to see if his words are being received. He is delivering the same message he has been preaching for months. He is not here to draw attention to himself\u2014he's here to announce the arrival of the Messiah.\n\nThe one they are jealous of is the very one for whom he came to prepare the way. John hopes the analogy helps them better understand his part in God's plan. The friend of the groom assures the wedding gets off to the right start\u2014an important but limited role. John knows what his disciples want to hear, and his next words don't come easily. \"Jesus must increase,\" he tells them, \"but I must decrease.\"\n\nJohn's followers are shocked to hear how resigned their master is to his dwindling popularity, but soon they have a greater concern. The Baptist's preaching has created powerful enemies, and one of the deadliest is Herodias, granddaughter of Herod the Great, and niece and new wife of Herod Antipas. She had once been married to his half-brother, Herod Phillip, but after a torrid affair with Antipas in Rome, she divorced Phillip and moved into his brother's palace in Galilee.\n\nJohn is publicly denouncing the union. He will not pretend it is anything less than legitimized adultery.\n\nHerodias is furious and wants to kill the brash seer, but her husband will not consent. Herod Antipas also wants to put John to death, but he fears public reaction. The people of the land have embraced the Baptist as a prophet from God, and silencing him must be done in a way that does not disturb the Pax Romana.\n\nAntipas decides his best plan of attack is to eliminate the Baptist gradually. He sends his soldiers to the Jordan. John's disciples look on in horror as their master is arrested and unceremoniously taken south to a dungeon in Machaerus. Once little more than a rugged military outpost, the isolated peak is now the easternmost of Herod's renovated fortresses.\n\nWord of John's imprisonment soon spreads throughout the land. The multitudes wonder what Jesus will do about John's arrest. Many are surprised when Jesus makes no strong pronouncements and delivers no ultimatums. Instead, it appears he wants to run from the conflict. He tells his disciples it is time they went back to Galilee.\n\nJesus knows there will be a day of reckoning, but that confrontation must wait. The Holy Spirit has prompted him to go north, but not by the route Jews prefer. Jesus is to take the road through Samaria.\n4\n\nTHE GOD OF ALL CLANS\n\nAutumn, AD 30\n\n\"At that time,\" declares the Lord, \"I will be the God of all the clans of Israel, and they shall be my people. . . . Again you shall plant vineyards on the mountains of Samaria; the planters shall plant and shall enjoy the fruit.\"\n\n\u2014Jeremiah 31:1, 5\n\nNINE HUNDRED YEARS before Jesus's birth, the hill country of Samaria was the heart of the northern kingdom of Israel. The once illustrious Way of the Patriarchs, the primary north-south highway in Israel, bisects the region. Father Abraham had travelled it, and near Shechem had been promised by God, \"To your seed I will give this land.\" His grandson Jacob had not only travelled the road, he had owned the land. Then Jacob, renamed Israel, had given the land to his son Joseph.\n\nThose were glorious days, but they are long past. Today the name of this region is spoken as a curse, if it is spoken at all. Pious Jews avoid even passing through it. Still, the highway through Samaria is the shortest route between Galilee and Jerusalem, and Jews often take the road despite their racial and religious bigotry.\n\nIt is a prejudice centuries in the making. After Solomon's death, the nation was divided. Jerusalem became the capital of the southern kingdom of Judah, and the city of Samaria became the capital of the northern kingdom of Israel. Neither was faithful to God, but the evil done in the northern kingdom by rulers like Ahab and Jezebel led Judah to sever all ties with Israel.\n\nA few generations later, Assyria conquered the northern kingdom. Following their general conquest policy, the Assyrians deported all but the poorest Israelites. Their homes were given to refugees from other defeated territories: people from Babylon, Cuthah, Avva, Hamath, and Sepharvaim. The new residents brought their religious practices with them. A particularly grim ritual was central to the religion of the people of Sepharvaim. They worshipped a Babylonian sun god named Adrammelech and his consort, Anammelech\u2014goddess of the moon. This couple, like their companion god, Molech, required child sacrifice. For a time, the land once home to Jacob and Joseph became home to dark lords who demanded babies as burnt offerings.\n\nThe Israelites who remained in the land ignored God's prohibition against marrying outside the faith, and within a few generations they were assimilated. These new people became known as Samaritans. In deference to the god of the land, they adopted the trappings of Judaism, and when Ezra and the Jews were allowed to return from exile to rebuild the temple, a number of Samaritans volunteered to help. When the Jews rejected their offer, the Samaritans built a temple on Mount Gerizim and discarded the writings of every forefather who had been sympathetic toward Jerusalem. The words of Samuel, David, and Solomon were stricken from their Scriptures, as were the prophets and chronicles of the kings. Ultimately only the Torah survived\u2014Genesis, Exodus, Leviticus, Deuteronomy, and Numbers became the Samaritan Bible.\n\nFor centuries the hatred between the Jews and Samaritans simmered, occasionally erupting in bloodshed. When the Maccabean revolt freed Judea from Greek rule, the Jewish leader John Hyrcanus destroyed the city of Shechem and burned the temple on Mount Gerizim to the ground. This did not stop the Samaritans from worshipping on Gerizim, nor did it stop them from retaliating whenever the opportunity provided itself.\n\nOne Passover when Jesus was a boy, Samaritan grave robbers crept around the temple mount and littered it with human bones, rendering the area unusable for worship until ceremonially cleansed. These days the Pax Romana has reduced the violence, but the Jews and Samaritans still despise one another.\n\nJesus, who has been in Judea since the Passover, is prompted by the Holy Spirit to return to Galilee by way of Samaria. As he walks north, the road falls gently, and the mountains give way to rolling hills. By late morning, the cool air Jesus and his five companions enjoyed at dawn has vanished. They enter the region of Samaria tired, hungry, and thirsty.\n\nAt noon they arrive at Jacob's well\u2014a major crossroad. To the west lie the plains of Sharon and the Mediterranean; to the north the road descends into Galilee. A short distance away is the village of Sychar. Peter, Andrew, John, Phillip, and Nathaniel go into town to buy food. Jesus decides to remain at the well and enjoy the lull of midday.\n\nHis solitude is interrupted by a lone woman with a water pot. Jesus immediately knows she is no ordinary woman. Otherwise, she would have come in the cool of the day with her neighbors. But the woman walking toward Jesus is not like the other women in town.\n\nShe notices a young Jewish man sitting on the well's limestone curb. She is surprised when he doesn't stand and walk away\u2014this is no place for a man and woman to be alone. She approaches cautiously, expecting the man to ignore her, but when she draws near he says, \"Give me a drink.\"\n\nThe woman jerks to a stop, and her eyes narrow. \"How is it that you, a Jew, ask for a drink from me, a Samaritan woman?\"\n\nJesus understands her confusion but refuses to be drawn into her world. He smiles and says, \"If you knew the gift of God, and who it is that is saying to you, 'Give me a drink,' you would have asked him, and he would have given you living water.\"\n\nThe woman is skeptical. Living water\u2014water that comes directly from God via rains and natural springs\u2014is plentiful in Samaria, but she knows every water source near the village. And even if he did know of a secret spring, how could he give her a drink from it? \"Sir,\" she points out, \"you have nothing to draw water with, and the well is deep. Where do you get that living water? Are you greater than our father Jacob? He gave us the well and drank from it himself, as did his sons and his livestock.\"\n\nJesus nods and says, \"Everyone who drinks this water will thirst again, but whoever drinks the water that I will give him will never thirst again\u2014the water that I will give him will become in him a spring of water gushing up to eternal life.\"\n\nThe woman shakes her head and laughs. \"Sir, give me this water, so that I will not be thirsty or have to keep coming here to draw water.\"\n\n\"Go,\" Jesus says, \"call your husband and come here.\"\n\nThe woman blushes, and her gaze falls to the ground. Her reply is just above a whisper. \"I have no husband.\"\n\nShe says as little as possible, hoping the Jewish stranger will change the subject. He doesn't. \"You're right in saying, 'I have no husband,'\" Jesus says, \"for you have had five husbands, and the one you have now is not your husband. What you have said is true.\"\n\nHis words are delivered gently and prefaced with praise for her honesty, but they remind her of all the promises made and broken, all the lies and deceit, all the anger and bitterness. And how, after five dead ends, she has given up on the covenant of marriage.\n\nThe woman looks into Jesus's eyes, expecting to find condemnation. She sees none. A spark of hope is kindled, but she is uncomfortable talking about her past. Too many memories. She politely attempts to change the subject. \"Sir, I can see that you are a prophet. Our ancestors worshiped on this mountain, but you say that people ought to worship in Jerusalem.\"\n\nJesus refuses to be drawn into the ancient conflict. For him, it is irrelevant. As he had pointed out to the Jews in Jerusalem, the issue is not where to worship God, but how to worship Him. \"Believe me, woman,\" Jesus says, \"the time is coming when you will worship the Father neither on this mountain nor in Jerusalem. You Samaritans worship what you do not know. We worship what we know, for salvation is from the Jews. But the time is coming\u2014indeed it has arrived\u2014when the true worshipers will worship the Father in spirit and truth, for they are the kind of worshipers the Father is seeking. God is spirit, and those who worship Him must worship in spirit and truth.\"\n\nThe woman struggles to understand. The young Jew is speaking of the new age to be ushered in by the Prophet. God had said to Moses, \"I will raise up for them a prophet like you from among their brothers, and I will put my words in his mouth, and he shall speak to them all that I command him.\" Moses then announced to God's people, \"The Lord your God will raise up for you a prophet like me from among you, from your brothers\u2014it is to him you shall listen.\"\n\nThe woman is familiar with the promise and longs for the arrival of the Prophet-like-Moses. He is the Messiah (Anointed) to the Jews and the Taheb (Restorer) to the Samaritans. The woman's Messiah is not a political or military leader like David but a teacher of righteousness like Moses. \"I know that Messiah is coming,\" she says. \"When he comes, he will tell us all things.\"\n\nLike Nicodemus, the woman \"knows\" about the Messiah but can't recognize him when he is standing before her. \"I am he,\" Jesus tells her. \"The one speaking to you.\"\n\nThe conversation is interrupted by the men returning from town. As the disciples walk toward the well, they are shocked to see their rabbi talking cordially with a woman. History is full of stories about what can come from meetings between men and women at wells. Besides, rabbis are to have little or nothing to do with Samaritan women, all of whom are deemed unclean. \"Menstruants from the cradle\" is a coarse but popular description.\n\nThe woman does not speak or make eye contact with the men. She is lost in her thoughts. Had she met the Taheb? Is it possible the mighty Restorer would take the form of this gentle young man? But why would the Messiah, lord of the entire universe, be concerned about an unimportant person like her? Surely the Prophet had better ways to spend his day. None of it made sense. And yet, the man had told her secrets only God knew.\n\nShe is confused about Jesus's identity, but this is news she cannot keep to herself. Leaving her water pot at the well, she runs into Sychar and tells everyone she meets about the Jewish stranger. \"Come,\" she says, \"see a man who told me everything I ever did! Can this be the Messiah?\"\n\nThe townspeople are wary but willing to investigate. A large delegation heads toward Jacob's well.\n\nAt the well, the disciples are sitting in the shade drinking cool water and enjoying the bread they bought in the village. One of the men turns to Jesus and says, \"Rabbi, eat something.\"\n\nImmersed in prayer and gazing into Samaria, Jesus says, \"I have food to eat that you do not know about.\"\n\nThe men do not understand what he means and ask one another who might have given him food. Jesus interrupts their speculations: \"My food is to do the will of the One who sent me and to accomplish His work. Don't you say, 'Four months more and then comes the harvest'? Look, I tell you, lift up your eyes and see that the fields are white for harvest. Even now the reaper receives wages and gathers a crop for eternal life, so that the sower and the reaper may rejoice together. For in this the saying is true, 'One sows and another reaps.' I sent you to reap what you have not worked for. Others have toiled, and you have reaped the benefit of their labor.\"\n\nPeter, Andrew, John, Philip, and Nathaniel are familiar with the proverb about sowing and reaping, but Jesus has turned it upside down. The saying is derived from the common farming practice of hiring planters in the spring and harvesters in the fall. Those who plant do not see the fruit of their labors. Jesus says things are different in the kingdom of God. Here both sower and reaper experience joy because they share in the same blessing. There is no need to wait four months to see the fruit, Jesus says, because this is a radically different kind of farming. Kingdom agriculture is described in the Book of Amos: \"'Behold, the days are coming,' declares the Lord, 'when the plowman will overtake the reaper and the treader of grapes him who sows the seed; the mountains will drip sweet wine, and all the hills will flow with it.'\"\n\nJesus, like Amos, speaks of the time when Israel will be so prosperous that blessings will overlap one another. And Jesus says that time has already begun. It is, in his words, the time \"to do the will of the One who sent me and to accomplish His work.\" And any doubt about what Jesus is speaking of disappears when the disciples see a large group of white-clad Samaritans walking toward them through the fields. Jesus greets them with a warm smile and begins teaching them.\n\nJesus's companions are amazed at the openness of the Samaritans to the Word of God, and they begin to understand what Jesus meant about the fields being ready to harvest immediately after being sown. Jesus spends two days proclaiming the good news to the Samaritans, and many put their faith in him, convinced that the long-awaited Taheb has come to his people in Samaria.\n5\n\nGALILEE OF THE GENTILES\n\nAutumn\u2013Winter, AD 30\n\nThere will be a time in the future when Galilee of the Gentiles, which lies along the road that runs between the Jordan and the sea, will be filled with glory. The people who walk in darkness will see a great light. For those who live in a land of deep darkness, a light will shine.\n\n\u2014Isaiah 9:1\u20132, NLT\n\nGALILEE IS THE northern border of Israel. In the days of Joshua, the region was known as Galil Haggoyim because it was geographically galil (encircled) by goyim (Gentiles). Nine hundred years ago, King Solomon was indebted to King Hiram of Tyre and offered him twenty cities in Galilee as payment. Hiram surveyed the land and told Solomon he deemed it worthless.\n\nMany in Judea today would agree. They believe the foreign influences in Galilee produce Jews who are either liberals or revolutionaries. The Romans, who have quelled several disturbances in the region over the decades, are also suspicious of Galileans.\n\nIn truth, most of the people of Galil Haggoyim are faithful to God. They study the Scriptures and teach them to their children. They keep the Sabbath. They attend the festivals in Jerusalem. They live according to the Law of Moses.\n\nTales of Jesus's exploits in Judea reach the hills of Galilee long before he arrives. He travels slowly through the towns west of the lake preaching a simple message: \"The time has been fulfilled! The kingdom of God is near\u2014repent and trust in the good news.\"\n\nJesus eventually arrives in Cana. A large crowd gathers and he begins teaching. A government official who works for Herod Antipas pushes through the throng. His voice quivers as he pleads with Jesus to return with him to Capernaum and heal his dying child.\n\nThe crowd listens carefully to the request, but Jesus knows they are more interested in his miracles than his message. He looks into the faces surrounding him and says, \"Unless you see signs and wonders, you will not believe.\"\n\nThe Herodian official is desperate. \"Sir,\" he begs, \"come down before my child dies.\"\n\nJesus looks into the father's fear-filled eyes and says, \"Go\u2014your son lives.\"\n\nThe man is stunned by Jesus's confident pronouncement. His child is twenty miles away. Healings from that distance are unheard of, but the official believes that somehow Jesus has cured his son. He immediately sets off for home and walks until darkness forces him to stop. He continues at first light, and when he nears Capernaum, his servants greet him with good news\u2014his son has made a miraculous recovery. The man asks a few questions and determines that the boy's health improved at the exact time Jesus pronounced him well.\n\nEveryone in Capernaum knows about the healing of the Herodian official's son by the time Jesus arrives a few days later. The townspeople are excited to learn that the visiting teacher will be preaching at the synagogue, but Jesus has not come to visit\u2014he plans to make Capernaum his home.\n\nBuilt on a rich alluvial plain six hundred feet below sea level, Capernaum is a small but busy city on the north shore of Lake Gennesaret. Her thousand or so residents enjoy warm winters and long summers. Farmers plant and harvest a wide variety of crops ten months of the year, and as good as the farming is, the fishing is even better. Two miles west of town is the small cove of Tabgha. Warm mineral springs enter the lake here, creating plankton that draws large schools of musht (tilapia) in the cooler months.\n\nCapernaum also benefits from its strategic location near the eastern frontier of Herod Antipas's tetrarchy (his half-brother, Herod Philip, rules the region east of the Jordan). Antipas has set up customs stations to tax everything that crosses the border and comes in from the lake. Taxes are also levied on merchants transporting goods on the ancient highway that skirts the northwestern edge of town. In olden days, it was called the Way of the Sea. Nowadays known by its Latin name, the Via Maris is one of the three major trade routes in Israel. It is in constant use by merchants from the west and south traveling across the Mediterranean plain to and from Damascus and the east.\n\nThere is a Roman garrison stationed east of Capernaum. The centurion who commands the troops built a synagogue for the city, and his patronage insures a cordial relationship with the city elders. He and his men live in well-appointed quarters that are large enough to include a caldarium, tepidarium, and frigidarium\u2014a full Roman bath.\n\nCapernaum's location and diverse population of farmers, fishermen, craftsmen, government workers, merchants, and soldiers creates an environment that is wholly Jewish, but with an international ambience.\n\nNot only is Capernaum uniquely situated on the water and a major trade route\u2014it is also visually striking. The city's black basalt construction stands in bold contrast to the blue lake and green hills surrounding it. An eight-foot-wide esplanade runs along the top of the seawall for 2,500 feet; several long piers extend from the walkway to accommodate the many boats that come and go daily. Shops line the waterfront. Broad boulevards radiate north from the lake into neighborhoods of rustic multifamily dwellings made of unhewn lava stone.\n\nThe synagogue, built from the same black volcanic rock, stands just behind the shops. Nearby is the large home of Simon Peter and Andrew. The brothers hail from Bethsaida, three miles to the east, but they relocated their families to Capernaum. Like the other residences in town, their home is what the Romans call an insula\u2014a walled compound of small rooms clustered around an open courtyard. A narrow exterior stairway leads to the roof\u2014additional living and working space in this climate where winters are mild and summers long and hot.\n\nOne morning while Jesus is walking by the lake, he sees two of his sometimes-disciples, Simon Peter and Andrew, fishing from the shore with cast nets. Twenty feet in diameter and weighted with lead, the large net must be thrown in a way that allows it to spread completely on the surface. A cast net is extremely effective, but only in the hands of a skilled fisherman.\n\nJesus walks over to where they are fishing, and the three men exchange greetings. \"Come,\" Jesus tells them, \"follow me, and I will make you fishers of people.\"\n\nSimon Peter and Andrew are intrigued by Jesus's words. The prophet Jeremiah had spoken about fishers of people, but the context was God's judgment. The Lord promised to \"send for many fishers\" to catch those who \"have polluted My land with the carcasses of their detestable idols, and have filled My inheritance with their abominations.\"\n\nSimon and Andrew don't fully understand the invitation, but they leave their nets and follow Jesus. They continue down the lakeshore and come upon two more sometimes-disciples. James and John are near the shore in a boat with their father and the fishermen they employ. Jesus gives the sons of Zebedee the same cryptic invitation he gave the sons of Jonah. They immediately stop preparing their nets and make their way to shore, but, like Simon and Andrew, James and John only partially understand Jesus's invitation. How they will become fishers of people is not clear to them.\n\nTHE SUN RISES on the seventh day, and the town is quiet. The docks, normally filled with fish and fishermen, stand empty except for a few birds looking for breakfast. The townspeople are in their homes preparing for what is sure to be an exciting day. The new rabbi is preaching this morning.\n\nThe synagogue fills to overflowing. The worship service unfolds as usual\u2014the congregation sings psalms and offers prayers, they recite the Shema, and there is a reading from the Law and a reading from the Prophets. Jesus then walks to the front, sits, and begins to teach. The audience usually hears messages about the traditions of the elders, but the new teacher has a different approach\u2014the traditions are not even mentioned. Instead, he teaches directly from Scripture.\n\nThe people sit spellbound as Jesus proclaims the good news of the kingdom. But then, the message is interrupted by a loud shriek from someone in the audience. A murmur arises when the congregation discovers the frightening noise is coming from one of their neighbors.\n\nThe man's body is coiled and tense; his breathing is labored. \"Let us alone!\" He spits out the words. \"What do you want with us, Jesus? Nazarene! Have you come to destroy us? I know who you are\u2014the Holy One of God!\"\n\nThe crowd gasps and steps back. Jesus is unmoved. He knows it is not the man who is taunting him but an ancient spiritual being who, until now, has been able to work in and through the man undetected. The demon's words are a threat\u2014it says that if Jesus destroys it, the man in its possession will also die.\n\n\"Silence!\" Jesus commands. \"Come out from him!\"\n\nThe man falls to the ground, writhing in convulsions. The congregation watches, speechless, as the loud cries diminish and the tremors subside. The exorcist walks over and helps the man to his feet. Those who see the expulsion shake their heads in disbelief. They have heard reports of elaborate rituals and incantations that can free a possessed soul, but here is one who can dispatch powerful demons with just a word. And Jesus not only banished the demon\u2014he restored the man to health. \"What is this?\" they say to one another. \"A new teaching\u2014with authority! Even the unclean spirits obey his command.\"\n\nJesus walks out of the synagogue and up the street to Simon Peter's home. When he turns in to the courtyard, he is told that Simon's mother-in-law has a dangerously high fever. Jesus enters her room, touches her hand, and offers a word of rebuke to the fever that is similar to the word of rebuke he spoke to the evil spirit in the synagogue. The fever, like the demon, flees. The healing is so complete that she immediately gets up and, after profusely thanking Jesus, begins preparing lunch for them.\n\nHer recovery, like the exorcism, becomes widely known. When the Sabbath ends at sundown and people can resume their normal activities, a crowd gathers at Peter's home. All evening the compound's two courtyards are filled with townspeople seeking healing, and Jesus heals all who come to him.\n\nThe new rabbi's fame spreads quickly. He becomes Capernaum's biggest celebrity. Locals brag to visitors about encounters they've had with Jesus, but few know him. Everyone loves to see him perform miracles, but most people ignore his calls to repentance. The crowds grow daily, and soon Jesus is in constant demand.\n\nTo prepare himself for the long days of ministry, Jesus often slips into the hills before dawn to pray. One morning he is late returning, and the throng demands that Peter, Andrew, James, and John retrieve him. The men dutifully search for Jesus and find him sitting quietly in prayer. \"Everyone is looking for you!\" they exclaim.\n\nHis disciples do not understand that this is one of the reasons Jesus sought solitude in the first place. \"Let's go on to the neighboring towns so I can preach there also,\" Jesus tells them, \"for that is why I came.\"\n\nThe men wonder why Jesus would want to leave Capernaum when things are going so well. When a group of townsfolk who followed the disciples arrive and plead with Jesus to return to Capernaum, he declines. \"I must preach the good news of the kingdom of God to the other towns as well, for I was sent for this purpose.\"\n\nJESUS LEAVES CAPERNAUM and takes his message into the towns and villages of Galilee. Demons are banished. Diseases are cured. The lame walk, the blind see, and the deaf hear.\n\nWhen Jesus returns a few weeks later, Capernaum is filled with visitors hoping to see the new teacher. The crowds become so large that many are unable to get close enough to hear Jesus speak. One morning, when he is teaching by the lake, Jesus sees two boats sitting idle while their owners\u2014Simon, Andrew, James, and John\u2014clean their nets after a night of fishing. To take advantage of the natural amphitheater created by the water and surrounding hillsides, Jesus asks Simon Peter to row his boat a short distance offshore. The boat is large\u2014more than twenty feet long and designed for a crew of four\u2014but Peter skillfully maneuvers it to a place where the acoustics will allow the multitude to hear the message.\n\nJesus encourages his audience to change the way they think and to trust in God rather than in their own cleverness. The people are impressed by Jesus's teaching, but they can't imagine how they would survive if they depended completely on God. They don't disagree with Jesus's explanation of the Scriptures, but the new rabbi is too extreme in his application. How easy it is for a young single man to promote radical discipleship, but what do you say to your family when the crops fail or you catch no fish? Do you ask them to eat empty promises?\n\nWords will never answer those kinds of questions, and Jesus doesn't try. Instead, he turns to Simon Peter and says, \"Launch out into the deep and let down your nets for a catch.\"\n\nTo a professional fisherman, the request is absurd. Everyone knows fishing in the deep is done at night when the fish feed. During the day the fish in the deep hide under rocks, safe from nets. And even if there were fish in the deep, in the daylight they could see the layered net and easily avoid it. Simon has fished this lake his entire life. There is simply no value in going into the deep during the day with this kind of net. A cast net would work, but only in the shallows. A dragnet would work in the deep if there were enough men to pull it in, but the only net on board is the trammel.\n\nThe expert fisherman is blunt. \"Master, we've worked hard all night and haven't caught anything.\"\n\nSimon pauses, hoping for sympathy. Jesus does not reply, and Simon continues, \"But at your word, I will let the nets down.\"\n\nSimon rows toward the deep. James and John are surprised to see their partner going out again, and even more surprised when he beckons them to follow. John and James look at each other. They are being invited to go fishing in the wrong part of the lake at the wrong time of day with the wrong kind of net. The sons of Zebedee shake their heads, but they launch their boat and follow.\n\nThe lake bottom drops steeply. Simon is soon laying his net in deep water. James and John are surprised to see their partner's boat begin to list as the net is being set. The trammel net Simon Peter is extending is actually a compound of three nets; fish easily swim through a coarsely woven net, then into a fine mesh net, and then into another coarse net, forming a pocket. When the fish try to escape, they become hopelessly entangled in the three nets.\n\nIt is a wonderfully effective method for catching fish, but the net now trails behind them for hundreds of feet, and loud groans from the boat's timbers tell Simon they are in real danger\u2014the fish can only be disentangled by hand, a slow process, and his boat will soon be pulled under by the weight gathering in the net.\n\nHe frantically motions to James and John. They quickly row over and begin pulling fish into their boat, but the net is so full that both boats nearly sink. By the time the net is out of the water, the boats are overflowing with fish and just above the waterline.\n\nThe object lesson has a profound effect on the fishermen, and particularly Simon Peter. He falls at Jesus's feet. \"Go away from me, Lord, for I am a sinful man!\"\n\nCenturies earlier when God called Isaiah to be His spokesperson, Isaiah confessed he was an unclean man with unclean lips. Nevertheless, his response to God's call was, \"Here I am Lord\u2014send me.\" Jesus looks into Simon Peter's heart and sees the same willingness and zeal he saw in Isaiah. \"Do not be afraid,\" Jesus tells him. \"From now on you will be catching people.\"\n\nPeter, Andrew, James, and John abandon their fishing careers and become full-time apprentices of Jesus. The profit from the two boatloads of fish will be tremendous, and the monies will allow the men to provide for their families while they are away. Father Zebedee will oversee the fishing business. From this day forward, Peter, Andrew, John, and James have a new vocation: fishing for souls.\n6\n\nAUTHORITY\n\nWinter, AD 30\n\n\"Anyone who chooses to do the will of God will know whether my teaching is from God or whether I am speaking on my own authority.\"\n\n\u2014John 7:17\n\nTHE FIRST YEAR of Jesus's ministry draws to a close. He has spent most of it in the two hundred small villages that dot the hills of Galilee. One afternoon while walking between towns, Jesus and his four apprentices are accosted by a leper.\n\nPeter, Andrew, James, and John fall back in fright. No malady is more feared by the Jews. Because leprosy is highly contagious, it is painful not only physically but socially as well. The Torah demands that anyone who is afflicted with the disease be \"set outside the camp.\" A leper must \"wear torn clothes, the hair of his head disheveled, and he shall cover his mouth and cry out, 'Unclean, unclean.'\"\n\nThese are only a few of the restrictions placed upon lepers. Contracting leprosy means rapidly declining health, constant pain, no family, no friends, no home, and exclusion from the community. Leprosy is essentially a living death sentence. Anyone who touches or comes near a leper becomes ceremonially unclean and must go through the same seven-day purification rites prescribed for those who come in contact with a corpse.\n\nLepers are treated as the living dead because people assume that leprosy is divine retribution. The story of God's punishment of Miriam is well known and often cited. People don't talk about being healed of leprosy; they speak of being cleansed, in the same manner one is cleansed of sin.\n\nThe man groveling at Jesus's feet pleads, \"Lord, if you are willing, you can make me clean.\"\n\nThe leper believes Jesus has the power to make him whole, but he questions Jesus's willingness. Like his fellow Jews, the leper is more inclined to expect God's judgment than His mercy.\n\nJesus stoops to the ground and gently places his hands upon the ulcerated body prostrated before him. \"I am willing,\" Jesus says. \"Be cleansed.\"\n\nThe leper's eyes grow wide as he watches the oozing white sores that cover his body dry up and vanish.\n\nThe disciples are awed and confused. They believe only God can cure leprosy\u2014but Jesus had not said, \"God is willing.\" He said, \"I am willing.\"\n\nJesus, who knows the heart of everyone he meets, knows the man he healed is ready to tell the world about his miraculous restoration. Jesus gives him a stern directive: \"See that you say nothing to anyone, but go, show yourself to the priest, and make the offering that Moses commanded, as proof to them.\"\n\nThe man is told to follow the two-stage procedure outlined in the Torah. The ceremonial cleansing begins with a thorough physical examination by a priest. If there is no sign of the disease, two birds are presented as an offering to God. The person being cleansed sacrifices one of the birds. The living bird is then dipped in the blood of one slain. The priest sprinkles blood on the supplicant seven times and releases the bird that was washed in the blood.\n\nThe process is completed eight days later at the temple when the priest presents a guilt offering, sin offering, burnt offering, and grain offering on behalf of the supplicant. The four atonement rituals are rich in imagery that Jesus wants the man to grasp. The sacrifices have never been about the animals\u2014they represent the person making the offering.\n\nJesus demands full obedience to the Mosaic Law for two reasons. The man will learn and experience the deeper significance of the rituals, and others will see God at work\u2014particularly the priest who examines him.\n\nThe man healed of leprosy nods his head in agreement and departs, but he disobeys Jesus's instructions to tell only the priest. He shares his story with anyone who will listen. The multitudes coming to hear Jesus grow so large that he is forced to bypass the towns and preach in remote areas by the lake and on hillsides.\n\nWHEN THE RAINY season arrives, Jesus goes home to Capernaum. Word of his return spreads quickly. Especially interested in the news is a group of men with a paralyzed friend. Doctors have been unable to help, but the men believe Jesus can heal him. They put their friend on a stretcher and walk toward the residence where Jesus is teaching, but when they draw near, the men realize they've underestimated Jesus's popularity. The insula is overflowing, and the crowd is spilling into the street. There's not enough room in the doorway for even a child to enter.\n\nThe men are discouraged but determined. They look around for an alternate entry and notice an exterior stairway to the rooftop.\n\nRoofs in Capernaum are flat and supported by large wooden beams that span unmortared lava stone walls. The beams are interlaced with smaller poles and thatched with straw and reeds. Several inches of moist clay follow. The clay bakes in the sun and is compacted with a roof roller. The deck becomes an additional living and working space\u2014a second story in the open air.\n\nThe men look from the congested street to the empty rooftop and have an idea. They find some rope and carry their friend up the stairway.\n\nJesus and a few of his disciples are in a room filled with townspeople and a large group of religious scholars who have come from throughout Judea and Galilee to investigate Jesus and condemn him as a false prophet. Jesus knows why the scribes are there, but he graciously responds to all their questions.\n\nThe discussion is interrupted when small pieces of dirt begin falling from the ceiling. Soon large clods of clay are raining down. The conversation ceases, and everyone stares at the rift widening above them. Hands are seen frantically removing thatch. An anguished face appears in the opening. He sees Jesus, and the men tear even harder at the sod. The reason for the demolition soon becomes clear\u2014a stretcher is slowly lowered through the breach. The room grows quiet as a man whose body is twisted by paralysis comes to rest at the feet of Jesus.\n\nJesus smiles as he looks into the eyes of the paralytic and then up into the faces of his friends. The lawyers from Jerusalem approach faith as knowledge to be gained; the paralytic and his friends understand faith to be an active demonstration of trust.\n\nBut the men's faith doesn't prepare them for what happens next. Jesus says to the paralytic, \"Be encouraged, child. Your sins are forgiven.\"\n\nJesus does not reply to the request for physical healing. Instead, he responds to the man's most urgent need. Unforgiven sin is far worse than the inability to walk, and Jesus knows the deeper truth\u2014all suffering is rooted in humanity's separation from God.\n\nThe scribes are furious. \"Why is this man speaking this way?\" they murmur. \"He is blaspheming. Who can forgive sins but God alone?\"\n\nThe religious scholars believe Jesus is guilty of blaspheming God, an offense punishable by death under Mosaic Law. To blaspheme someone is to slander them. The lawyers think Jesus has slandered God by circumventing the only authorized method of forgiveness\u2014securing God's pardon requires a priest and an appropriate sacrifice at the temple.\n\n\"Why do you harbor evil thoughts in your hearts?\" Jesus asks. \"Which is easier\u2014to say, 'Your sins are forgiven,' or to say, 'Stand up and walk'? But so that you may know that the Son of Man has authority on earth to forgive sins . . . .\" Jesus pauses and turns to the paralytic. \"I say to you, rise. Take your mat and go home.\"\n\nThe man stands, picks up his mat, and walks out the door. Fear grips the room. One of the townspeople says, \"We've never seen anything like this!\" Another nods in agreement. \"We've seen amazing things today.\"\n\nEveryone in the audience is awestruck, but not everyone is praising God. Some are angry. They cannot understand how a blasphemer could perform such a miracle. The scribes are convinced that not even the Messiah will be able to forgive sins\u2014only God can do that. Who is this son of man the Galilean claims has the authority to forgive sins?\n\nJESUS DOES NOT stay to answer their questions. He makes his way out of the room, through the courtyard, and into the street. He walks south a few blocks to the international highway that skirts the town. A customs station is located here to collect taxes from fishermen coming in from the lake and merchants transporting goods on the Via Maris. The lucrative confluence of borders, waterways, and an international trade route requires several customs agents.\n\nTax collecting, introduced three centuries earlier by Ptolemy II, is part of a complicated system that ultimately demands nearly thirty percent of a working person's income.\n\nThere are two direct taxes: a land tax that is paid in grain at harvest and a poll tax that requires a regular census. These levies go directly to Rome, but the taxpayers tolerate them. People understand that building and maintaining roads, aqueducts, and a social infrastructure requires money.\n\nThere are also indirect taxes that vary by region. In Jerusalem, there are house levies and sales taxes. In Galilee, there are transportation tolls and fishing tariffs. These indirect taxes provide numerous opportunities for graft because corruption and bribery are built into the system. At the top is an emperor who wants peace in his empire and an uninterrupted supply of grain and tribute. He awards oversight of the tax collection to savvy individuals who convince him they can keep both the peace and the money coming in.\n\nOnce upon a time in Israel that person was the Edomite-turned-Jew\u2014Herod. Before the client-king died, he obtained Caesar's permission to divide his kingdom among three of his sons. That was thirty years ago. His son Antipas has maintained the peace in Galilee and Perea. His son Philip has kept Gaulanitis, Trachonitis, Batanea, and Paneas quiet and prosperous. Herod's eldest son, Archelaus, was given the prized territories of Judea, Samaria, and Idumea. He was unable to maintain the Pax Romana, and in the tenth year of his reign he was stripped of his tetrarchy and exiled to Gaul.\n\nSince then, governance of southern Israel has passed to the proconsul of Syria, but he and his troops are in Damascus, far to the north. To insure the Pax Romana in the south, a large military force is garrisoned on the coast at Caesarea Maritima. The troops are under the authority of a prefect\u2014a civilian appointed by the emperor from the lower order of Roman aristocracy. Judea has been a notoriously difficult area to govern and has had five prefects in twenty years. The latest is a no-nonsense Italian of the Equestrian order named Pontius Pilate.\n\nThe Romans are aware of Jewish prejudice against Gentiles and never personally engage in the collection of taxes. In Judea, that responsibility is awarded to affluent members of the Sanhedrin, who then subcontract smaller areas to the highest bidders. The process is similar in the regions governed by Herod Antipas and Herod Phillip. Massive wealth is generated by leasing toll stations to affluent tax farmers.\n\nThe tax farmers hire subordinates to do the actual collecting. The customs agent is the lowest rung of the tax system, and thus its public face. Agents always work in the region they were recruited from; familiarity with the terrain and people assures a more comprehensive taxation.\n\nThe amount levied on merchants depends on the avarice of the taxman and the charm of the merchant. To the public, stopping people in the road, rifling through their belongings, and demanding money appears to be little more than government-sanctioned robbery. The fact that several tax offices are located along the same road, each demanding a different toll for the same goods, adds to the perception.\n\nMoreover, because customs agents are required to come in contact with Gentiles and work on the Sabbath, they are considered unclean. Their status in the community is comparable to that of a thief or a money changer. Because a tax collector is unclean, he is excommunicated from the synagogue. Because he has no credibility in the community, he is barred from being a witness in court. He may be wealthy and have a home in the heart of the city, but culturally he is outside the camp.\n\nThe job demands a high price socially, but it pays well, and there are always people willing to trade a good reputation for a good-paying job. Matthew Levi, sitting at his station in the Via Maris tax office, is just such a man.\n\nLately, however, Matthew has been rethinking not only his career choice but his entire life. He had been in the audience one day when Jesus preached about the kingdom of heaven. When Jesus said that God's grace extends even to outsiders, Matthew had felt something shift in his spirit, and he turned back to the God he had abandoned. Now Matthew comes to work each day wondering if the Lord might have any use for someone who traded his birthright for a bowl of porridge.\n\nMatthew sees Jesus approaching and assumes he is on his way to the lake, but Jesus doesn't take his usual route. He walks up to the customs booth and tells the tax collector, \"Follow me.\"\n\nThese are words an outsider would never expect to hear from a rabbi. Matthew is not even sure he heard the teacher correctly, but the look in Jesus's eyes tells him the invitation is sincere.\n\nMatthew has a choice. He can remain a tax collector and serve God as best he can within the limitations of his job. He knows the teacher would not fault him for that. Like John the Baptist, Jesus has no problem with customs agents becoming his apprentices. What's important, Jesus says, is to be just and exercise integrity, whatever your job.\n\nMatthew Levi, however, is struggling with something more significant than his livelihood\u2014he's wrestling with his identity. He knows Jesus is offering him a new life, but the new life will require him to abandon his old one. For Matthew, it is no longer a difficult choice. He immediately resigns his position and becomes an apprentice of Jesus.\n\nSoon the entire city is talking about Jesus inviting a tax collector to be his disciple. The religious leaders are convinced that being clean requires one to disassociate himself from sinners. Jesus disagrees. He believes the Scriptures describe a God who is so identified with sinners that the Messiah himself will be \"numbered with the transgressors.\"\n\nThe upcoming feast in Jerusalem will provide an opportunity for that numbering to begin in earnest.\n7\n\nLORD OF THE SABBATH\n\nSpring, AD 31\n\n\"I will raise up for them a prophet like you from among their fellow Israelites, and I will put My words in his mouth. He will tell them everything I command him. Whoever refuses to listen to the words that prophet speaks in My name will answer to Me.\"\n\n\u2014Deuteronomy 18:18\u201319\n\nTHE ANNUAL FEASTS that commemorate God's mighty acts are central to life in Israel. They are more than celebrations of God's grace; they are affirmations of the covenant He made with His people. When the people act out the prescribed liturgies of the various festivals, they are not simply remembering the event\u2014in a sense, they are participating in it, declaring anew their identity as God's chosen people.\n\nAll the holy days are special, and some, like Passover and Tabernacles, are extra special, but none have the honor of being Israel's holiest day. That day does not come annually but weekly. Sabbath will forever be the most sacred day on the Jewish calendar. A hedge of laws and traditions protect it, and woe be to anyone who violates even the least of them.\n\nJesus is in Jerusalem for one of the feasts and is walking north of the temple plaza near the twin pools known singularly as the Pool of Bethesda. The religious elite never come here\u2014too many unclean sinners\u2014but the pools are popular with the sick and afflicted. Many believe that when the water stirs, it has the power to heal. This belief is rarely spoken of in public because the religious authorities condemn it, but the many infirm who gather daily near the waters cling to the superstition in hope of a cure.\n\nJesus winds his way through the crowd to the covered colonnades surrounding the pools. Hundreds of ailing people lie in the shade of the porticos and watch for any movement in the water. Among the sufferers is a dour-faced old man who has been disabled for thirty-eight years. Jesus walks over to where he is lying, stoops down so he can address the man face-to-face, and asks, \"Do you want to be healed?\"\n\nThe invalid's eyes narrow. Perhaps a polite reply will make the naive young man go away. \"Sir,\" he explains, \"I have no one to put me into the pool when the water is stirred up, and while I am going, another steps down ahead of me.\"\n\nThe hopeless man's answer doesn't surprise Jesus. Thirty-eight years of misery have taught him to expect little and trust no one. These days, the best he can do is position himself by the water, where he knows he will be unable to help himself.\n\nThe invalid's heart is as emaciated as his body. He has no faith in Jesus or anyone else. Jesus looks the shriveled man in the eye. \"Get up,\" he commands. \"Pick up your mat and walk.\"\n\nAs the words are spoken, the paralytic's legs, long atrophied into helpless twigs, begin to swell as new muscle and sinew forms. Energy courses through his body, and somehow, after thirty-eight years, he is able to stand. Those who are nearby gasp in disbelief.\n\nThe healed man is dumbfounded. He has spent almost four decades lying in the dust. He rolls up his straw mat and walks away from the pool. His first steps are awkward and unsure, but soon he is striding toward the stairs leading to the temple mount. He only goes a short distance before someone shouts, \"It's the Sabbath, and it's not lawful for you to carry your mat.\"\n\nLike many of the Sabbath traditions, the restriction concerning moving something \"from one domain to another\" began as an attempt to honor the Sabbath but devolved into legalism. Many feel the Sabbath laws are too restrictive, but no one is willing to challenge the traditions.\n\nThe man immediately stops and drops his bedroll. He had been so distracted that he had not realized the healer had commanded him to do something unlawful. The last thing the man wants is trouble with the authorities, and he knows how he can deflect their attention. \"The man who healed me\u2014he told me, 'Pick up your mat and walk.'\"\n\nThis news shifts the interrogator's interest to the one guilty of the more grievous sin of enticing others to break the Sabbath. \"Who is the man who told you, 'Pick up your mat and walk'?\"\n\nThe old man confesses that he doesn't know; there had been shouting and confusion, and the physician slipped away unnoticed. The leaders are frustrated. Without a way to identify the healer, they can do nothing but warn the healed man and release him.\n\nThe former invalid makes his way to the temple mount and mingles with the thousands of pilgrims there for the feast. Surrounded by strangers, he immerses himself in the celebration.\n\nSeemingly from nowhere Jesus walks up and says, \"Behold\u2014you are well. Stop sinning so that nothing worse happens to you.\"\n\nThe old man is stung by the implication. The healer somehow knows his long-kept secret\u2014all his years of misery were self-inflicted, caused by sinful choices he alone had made. The \"worse\" he's being warned of is God's condemnation. The man becomes indignant and thinks this is quite a pronouncement from someone who himself is guilty of breaking God's laws. He excuses himself and hurries away to inform the authorities that he has learned the identity of the miracle-worker.\n\nThe religious leaders immediately recognize the name. The scribes find Jesus teaching in the plaza and begin persecuting him for violating the Sabbath.\n\n\"My Father is always working,\" Jesus tells them, \"and I too am working.\"\n\nThe temple authorities bristle. The Galilean's presumption in calling God his father is shocking enough, but by claiming to work alongside God, he makes himself equal with God. The charge of undermining the Sabbath has now been eclipsed by a graver allegation: the charge of blasphemy.\n\nInstead of tactfully meeting their objections, Jesus makes an even more alarming claim: \"Truly, truly, I tell you, the Son can do nothing on his own, but only what he sees the Father doing\u2014whatever the Father does, the Son also does. The Father loves the Son and shows him everything He is doing, and He will show him greater works than these, so that you will be amazed. For just as the Father raises the dead and gives them life, so also the Son gives life to whomever he wishes. The Father judges no one, but has given all judgment to the Son, so that all may honor the Son just as they honor the Father. Whoever does not honor the Son does not honor the Father who sent him.\n\n\"Truly, truly, I tell you, whoever hears my word and believes Him who sent me has eternal life and does not come into judgment but has passed from death into life.\n\n\"Truly, truly, I tell you that a time is coming\u2014and is now here\u2014when the dead will hear the voice of the Son of God, and those who hear will live. For just as the Father has life in Himself, so He has granted the Son also to have life in himself. And He has given him authority to execute judgment because he is the Son of Man. Do not be amazed at this, for an hour is coming when all who are in their graves will hear his voice and come out\u2014those who have done good to a resurrection of life, and those who have done evil to a resurrection of judgment.\n\n\"I can do nothing on my own. As I hear, I judge, and my judgment is just, because I do not seek my own will but the will of Him who sent me.\"\n\nJesus pauses and gazes into the surprised faces surrounding him. No prophet has ever made claims like these, and the audience believes that not even the Messiah will have the ability to do the works Jesus says he will do. They understand Jesus to be speaking of having the kind of authority only God has.\n\nMost in the audience reject his words because self-testimony is not considered acceptable. Jesus introduces a second witness: \"If I testify on my own behalf, my testimony is not valid. There is another who testifies on my behalf, and I know that the testimony He gives about me is true.\n\n\"You sent messengers to John, and he testified to the truth. Not that I receive testimony from a human source, but I say these things so that you may be saved. John was a burning and shining lamp, and for a while you were willing to rejoice in his light, but my testimony is weightier than John's, for the works that the Father has given me to accomplish\u2014the very works that I am doing\u2014testify on my behalf that the Father has sent me. And the Father who sent me has Himself testified on my behalf.\"\n\nThe religious leaders are not being asked to take a blind leap of faith; they are being offered verifiable physical evidence\u2014the testimony of John the Baptist and Jesus's miraculous works. Both speak clearly to his identity. Before their eyes he is doing things only God can do.\n\nSadly, most in the audience are deaf and blind. Jesus delivers a damning indictment: \"You've never heard God's voice or seen His form, and you do not have His word abiding in you because you do not believe the one whom the Father sent. You search the Scriptures because you think that in them you have eternal life, and it is they that testify about me, yet you are unwilling to come to me to have life.\"\n\nJesus identifies the fatal mistake the religious authorities make\u2014they worship the Scriptures rather than the one the Scriptures point to.\n\n\"I do not accept glory from people,\" Jesus tells them, \"but I know you. I know that you do not have the love of God within you. I have come in my Father's name, and you do not accept me. If another comes in his own name, you will accept him. How can you believe when you accept glory from one another and do not seek the glory that comes from the only God?\n\n\"Do not think that I will accuse you before the Father. There is one who accuses you\u2014Moses, in whom you have set your hope\u2014if you believed Moses, you would believe me because he wrote about me. But if you do not believe his writings, how will you believe my words?\"\n\nJesus leaves his question hanging in the air. No one responds, but there is little left to say. Jesus's claims are declarations of deity that leave his audience only two options: they can accept his claims, in which case he is to be worshipped and obeyed, or they can reject them, in which case he is to be branded a false prophet or dismissed as a madman.\n\nTHE FESTIVAL ENDS and Jesus returns to Galilee, but his words echo in the temple plaza for weeks. The Jewish leaders condemn him among themselves but say little about him in public. Jesus's popularity precludes any overt action, so they send emissaries to shadow him and collect evidence they can use to discredit him with the multitudes. They are convinced there is only one way to deal with the renegade prophet, and from this day forward the Jewish authorities plot to kill Jesus.\n\nOne Sabbath day in the late spring, Jesus and his disciples are walking on a path that separates two grain fields (thousands of acres of barley and wheat are planted in Galilee). Hungry, they pluck a few kernels, gently rub them in their hands to remove the chaff, and eat them. Pharisees who have been spying on them see the infraction and complain to Jesus, \"Look, your disciples are doing what is not lawful to do on the Sabbath!\"\n\nThe Pharisees are not accusing the disciples of stealing grain; Moses had made provision for travelers and the poor to glean from the fields. The accusation is that they worked on the Sabbath. In fact, according to the widely accepted \"forty minus one\" laws of what constitutes work on the Sabbath, the disciples were guilty of multiple violations: by plucking the grain they were reaping; by rubbing it between their fingers they were threshing; by blowing away the chaff they were winnowing.\n\nJesus has a different understanding of the Sabbath. Reasoning from the Scriptures, he asks the religious experts, \"Haven't you read what David did when he and his companions were hungry, how he went into the house of God and ate the bread of the Presence, which it was not lawful for him or his companions to eat, but only for the priests?\"\n\nJesus knows the Pharisees do not think David or the high priest sinned when they ate the showbread from the tabernacle. They are right about that, but they are mistaken about why there was no sin involved. They believe David was exempted from the Law because of a life-threatening situation. Jesus wants them to see the real reason. The high priest did not give them the bread because he had pity on starving men. This is clear from the account in the writings of Samuel: the priest first asked David if he and his men were ceremonially unclean. If that had been the case, the priest would not have given the holy bread to them no matter how hungry they were. David got the bread for one reason\u2014he was acknowledged as an ambassador of the Lord on a royal mission.\n\nThe story of David isn't offered as an excuse but as a precedent. The religionists refuse to see any further than their traditions, and Jesus provides another illustration. \"Or haven't you read in the Law how on the Sabbath the priests in the temple desecrate the Sabbath and are guiltless? But I tell you that something greater than the temple is here.\"\n\nJesus says that if priests can work on the Sabbath without incurring God's disapproval, so can his disciples. The religious leaders are infuriated, and Jesus rebukes them with the words of the prophet Hosea. \"If you had known what this means, 'I want mercy, not sacrifice,' you would not have condemned the guiltless.\"\n\nKey to Jesus's declaration is the biblical concept of hesed (translated \"mercy\" in Hosea). In the broadest sense, hesed is an active, positive concern for the good of others. Hosea identifies the religious leaders' problem\u2014they are more interested in appearing righteous than in being righteous. Their attitude is shown not only in their lack of compassion but also in their inability to grasp the simplest truths of God's covenant.\n\nCentral to the covenant is God's hesed\u2014His \"forgiving loving-kindness\" toward his people. If God had not adopted this merciful attitude, His covenant with Israel would have been broken by their first sin. By establishing the way of sacrifice, God provided a way for the relationship to continue.\n\nThe problem, Jesus says, is that hesed is not a part of these Pharisees' lives. They do not know God's forgiving loving-kindness and can only live within the rigid confines of the law. They are content to sacrifice animals and use Sabbath laws as boundary markers to point out who is in God's favor and who is out.\n\nJesus wants them to understand that God designed the Sabbath to be a servant, not a taskmaster. \"The Sabbath was made for man, not man for the Sabbath,\" Jesus says. \"So the Son of Man is Lord even of the Sabbath.\"\n\nThe Sabbath, according to the Scriptures, belongs to God alone. To claim authority over the Sabbath is akin to claiming equality with God. The religious leaders sense that Jesus is blaspheming, but they are frustrated\u2014the Galilean cleverly couches his words and uses ambiguous phrases like \"son of man.\" He's a sinner, but a wily one.\n\nON ANOTHER SABBATH, Jesus is teaching in the synagogue, and a man is there whose right hand is withered. The traditions of the elders decree that a physician can only work on the Sabbath when life is threatened; healing a deformed hand could easily be deferred. The Pharisees want to bait Jesus into violating the Sabbath traditions. One of the Pharisees motions toward the man with the paralyzed hand and asks Jesus, \"Is it lawful to heal on the Sabbath?\"\n\nJesus looks into their hard faces and smiles. \"Which one of you who has a sheep, if it falls into a pit on the Sabbath, will not take hold of it and lift it out? How much more is a human being worth than a sheep? So it is lawful to do good on the Sabbath.\"\n\nThis application goes far beyond the question of what is lawful. Jesus reverses the prohibition and makes it a positive law: whatever is \"good\" to do is permissible.\n\nThe scribes find this a completely unacceptable notion. It leaves too many decisions in the hands of the people. People are like sheep\u2014how would they survive without the Law and the traditions to shepherd them? The religionists are unfamiliar with a God who allows that much freedom; they confer with one another and glare at Jesus.\n\nJesus turns to the man with the withered hand and says, \"Come and stand here.\"\n\nThe disabled man is startled; this is his worst nightmare. It's bad enough that every time he comes to worship and lifts his hands in prayer his fellow congregants can see his gnarled fist and speculate about what sin caused it, but now he is being called to the center of the synagogue to be put on display. The temptation to run away is strong, but the man courageously steps forward. There is something in Jesus's words that gives him hope.\n\nWith the man at his side, Jesus turns to the religious experts and poses a twofold question. \"I ask you, is it lawful on the Sabbath to do good or to do harm? To save life or to destroy it?\"\n\nThe first part of his question (whether it is lawful to do good or to do harm) refers to healing the man; the second part of the question (whether it is lawful on the Sabbath to save life or destroy it) is Jesus's way of telling the authorities that he knows they are plotting his death even on the Sabbath.\n\nJesus wants them to understand that simply professing concern for those who are hurting is not enough\u2014human need must be addressed whenever and wherever it is found. Religion isn't about boundary-keeping; it's about showing God's love in tangible ways. The litmus test for all theology and morality is its response to the weakest and most vulnerable members of the community. Therefore, in Jesus's mind, it is not only permissible to heal on the Sabbath, it is the right and godly thing to do.\n\nThe scribes and Pharisees strongly disagree but find themselves in a dilemma because of the way Jesus has reframed the Sabbath issue. They say nothing, hoping Jesus will act.\n\nJesus, deeply distressed by their stubbornness, says to the man, \"Stretch out your hand.\"\n\nAgain the man feels vulnerable. What he most fears is now before him, and a choice must be made\u2014he can refuse and spare himself the humiliation, or he can risk embarrassment and obey. He chooses faith and offers his twisted fist to Jesus. He is immediately restored to health.\n\nThe congregation erupts in praise, but the religious leaders are enraged. The prophet from Galilee has crossed a boundary and must pay, in blood. They spend the afternoon plotting with members of the Herodian sect to kill Jesus. The Pharisees and Herodians\u2014so named because of their deference to the royal family\u2014are normally adversaries. The Pharisees view them as little more than starstruck sycophants, but the Herodians have the ear of the region's ruler and may prove valuable.\n\nAware of the plot to kill him, Jesus and his disciples leave Capernaum and make camp on the shore of Lake Gennesaret. While safe from the scribes and Pharisees here, Jesus is not safe from the multitudes. Initially, his audience is primarily Galileans, but later, people from all over Syria come to the lake to hear him preach and see him heal. And not just Jews.\n\nAs he begins his second year of public ministry, Jesus is incredibly popular, but he knows there's trouble on the horizon. His claims to be Lord of the Sabbath and greater than the temple are unforgivable sins in the eyes of the authorities. As the opposition mounts, Jesus continues to draw large crowds, but most come out of curiosity. Jesus decides the time has come to choose a smaller group for specialized training. His plan is simple but fraught with a dozen reasons to fail.\n8\n\nTHE DIVINE CONSPIRACY\n\nSpring, AD 31\n\n\"Everyone then who hears these words of mine and puts them into practice will be like a wise man who built his house on the rock.\"\n\n\u2014Matthew 7:24\n\nTHE RAINS CEASE, and spring comes to Galilee. It has been a year since Jesus announced his presence in Jerusalem by symbolically cleansing the temple. Since then he has violated numerous traditions and incurred the wrath of both sacred and secular authorities. Recent death threats in Capernaum have forced him to leave town. He and a large number of men and women are encamped in the hills above Tabgha, two miles west of Capernaum.\n\nJesus knows his time on earth will be brief. For his mission to succeed, he must train others to carry on his work. One evening he goes into the hills alone and spends the night in prayer. When morning comes, he returns to camp and chooses twelve men from among his apprentices to receive special instruction.\n\nThe number is parabolic. God called twelve tribes to be with Him and become His envoys to the nations; Jesus calls twelve men to be with him and become his envoys to the world. Apostles\u2014sent-forth ones\u2014is the name Jesus gives the group. They will be eyewitnesses to everything he does and have unlimited access to him.\n\nThere are two pairs of brothers in the twelve: Simon and Andrew bar Jonah, and their volatile fishing partners, James and John bar Zebedee. Jesus also calls Philip of Bethsaida, Nathanael bar Tolmai (Bartholomew) of Cana, Matthew Levi the tax collector, Thomas (also known as Didymus; both names mean \"twin\"), James bar Alphaeus, Simon the Cananaean (\"zealous one\"), and Judas Thaddaeus.\n\nThese eleven apostles are from Galilee, and all are am ha'aretz\u2014uneducated people of the land. The twelfth apostle is the one non-Galilean. His name is Judas bar Simon. He is also known as Iscariot because he is from the town of Kerioth in southern Judea.\n\nJesus leads the newly appointed apostles down from the hills. The crowds who have been milling around the lakeshore see Jesus and begin walking up to meet him.\n\nJesus stops at a level place and allows the multitude to gather around him and the apostles. Some of the men and women are followers of Jesus, but most in the crowd are still evaluating the new rabbi. Jesus sits on the grass and explains what it means to follow him and live in the kingdom of heaven.\n\n\"Blessed are the poor in spirit, for theirs is the kingdom of heaven.\n\n\"Blessed are those who mourn, for they will be comforted.\n\n\"Blessed are the meek, for they will inherit the earth.\n\n\"Blessed are those who hunger and thirst for righteousness, for they will be filled.\n\n\"Blessed are those who show mercy, for they will be shown mercy.\n\n\"Blessed are the pure in heart, for they will see God.\n\n\"Blessed are the peacemakers, for they will be called God's children.\n\n\"Blessed are those who are persecuted because of righteousness, for theirs is the kingdom of heaven.\n\n\"Blessed are you when others hate you and exclude you and insult you and persecute you and make all sorts of false accusations against you because of the Son of Man. Rejoice and leap for joy! Great is your reward in heaven, for that is how their ancestors persecuted the prophets who came before you.\n\n\"But woe to you who are rich, for you are receiving your consolation in full.\n\n\"Woe to you who are well-fed now, for you will hunger.\n\n\"Woe to you who are laughing now, for you will mourn and weep.\n\n\"Woe to you when all people speak well of you, for that is how their ancestors treated the false prophets.\"\n\nThe people on the hillside are confused. Jesus's blessings and woes are contrary to what they have been taught all their lives. They believe wealth and power are signs of God's approval, but Jesus paints a radically different picture. He condemns those who trust in themselves and echoes God's declaration written in the scroll of Isaiah: \"I dwell in the high and holy place, and also with those who are contrite and lowly in spirit.\"\n\nJesus looks into the faces of his apprentices and says, \"You are the salt of the earth, but if the salt becomes flavorless, how can it be made salty again? It is no longer good for anything except to be thrown out and trampled under people's feet.\n\n\"You are the light of the world. A city set on a hill cannot be hidden. Nor do people light a lamp and put it under a basket\u2014they put it on the lampstand so that it gives light to everyone in the house. In the same way, let your light shine before others so that they can see your good works and give glory to your Father in heaven.\n\n\"Do not suppose that I came to abolish the Law or the Prophets; I did not come to abolish them but to fulfill them. I tell you truly, until heaven and earth pass away, not the smallest letter or slightest stroke of the pen will pass away from the Law until everything is accomplished. Therefore, whoever sets aside one of the smallest of these commandments and teaches others to disregard them will be called the least in the kingdom of heaven, but whoever does them and teaches them will be called great in the kingdom of heaven. For I tell you that unless your righteousness goes far beyond that of the scribes and Pharisees, you will never enter the kingdom of heaven.\"\n\nA puzzled look appears on the faces of the apostles. How could anyone be more devout than the religious elite?\n\nThe men are confused because they misunderstand the focus of the Scriptures. Like the scribes and Pharisees, they think righteousness is about external purity and appropriate behavior. Jesus says that righteousness is more internal than external. God is interested in matters of the heart\u2014the attitudes that create the bad behaviors.\n\nJesus offers six examples of how this internal righteousness manifests itself.\n\n\"You have heard that it was said to those of old, 'You shall not murder,' and that whoever commits murder will be liable to judgment. But I tell you that everyone who is angry with his brother or sister without cause will be liable to judgment, whoever calls his brother or sister 'worthless' will be liable to a court trial, and whoever calls them 'fool' will be liable to the fires of hell.\n\n\"So if you are offering your gift at the altar and there remember that your brother or sister has something against you, leave your gift there in front of the altar\u2014go first and be reconciled with your brother or sister\u2014then come offer your gift.\n\n\"Reconcile quickly with your opponent in a lawsuit\u2014while the two of you are still on the way to court\u2014otherwise your accuser will hand you over to the judge, and the judge to the guard, and you will be thrown in prison. I tell you truly, you won't get out until you have repaid the last penny.\n\n\"You have heard that it was said, 'Do not commit adultery,' but I tell you that every man who looks at a woman who is not his wife and wants to have sex with her has already committed adultery with her in his heart. If your right eye causes you to stumble, tear it out and throw it away\u2014you're better off losing one part of your body than having your whole body thrown into hell. And if your right hand causes you to stumble, cut it off and throw it away\u2014you're better off losing one part of your body than having your whole body go to hell.\n\n\"It was said, 'Whoever divorces his wife must give her a certificate of divorce,' but I tell you that everyone who divorces his wife for any reason except sexual unfaithfulness makes her the victim of adultery, and anyone who marries a divorced woman himself commits adultery.\n\n\"Again, you have heard that it was said to people long ago, 'Do not swear falsely,' and 'Carry out the oaths you made to the Lord,' but I tell you, do not swear at all\u2014neither by heaven, for it is God's throne, nor by the earth, for it is His footstool, nor by Jerusalem, for it is the city of the great king. And do not swear by your own head, for you cannot make a single hair white or black. Let what you say be simply 'Yes' or 'No.' Anything more than that comes from evil.\n\n\"You have heard that it was said, 'An eye for an eye and a tooth for a tooth,' but I tell you, do not resist an evil person. Instead, if anyone slaps you on the right cheek, turn the other cheek to him as well. And if anyone wants to sue you and take your tunic, give him your cloak as well. And if anyone conscripts you as his porter for one mile, go with him two miles. Give to the one who begs from you, and do not turn away from the one who wants to borrow from you. If anyone takes what belongs to you, do not demand it back.\n\n\"You have heard that it was said, 'Love your neighbor and hate your enemy,' but I tell you who are listening: love your enemies. Do good to those who hate you, bless those who curse you, and pray for those who mistreat you, so that you may show yourselves to be true children of your Father who is in heaven, for He makes His sun rise on both the evil and the good and sends rain on both the just and the unjust. If you love those who love you, what reward will you get? Don't even tax collectors do as much? And if you welcome only those of your own circle, what more are you doing than others? Don't even Gentiles do as much? Therefore, be perfect, as your heavenly Father is perfect.\"\n\nThe people on the hillside shake their heads in amazement. They are accustomed to hearing homilies that explain the letter of the Law. Jesus directs his followers to the spirit of the ancient Scriptures\u2014to look beneath the surface and determine the source of the behavior. Addressing attitudes like anger and lust will eliminate murder and adultery and a host of other evils.\n\nJesus then describes what righteousness looks like when applied to the three cardinal disciplines of Judaism: almsgiving, prayer, and fasting.\n\n\"Be careful not to practice your righteousness in front of other people in order to be noticed by them, otherwise you have no reward from your Father, who is in heaven. So when you give to the needy, do not announce it with trumpets as the hypocrites do in the synagogues and the streets so that they will be esteemed by others. I tell you truly, they have received their reward. But when you give to the needy, don't let your left hand know what your right hand is doing, so that your almsgiving is in secret. Then your Father, who sees what is done in secret, will reward you.\n\n\"And when you pray, don't be like the hypocrites, for they love to pray standing in the synagogues and on the street corners so that people will notice them. I tell you truly, they have received their reward. But when you pray, go into your most private room, shut the door, and pray to your Father, who is in secret. Then your Father, who sees what is done in secret, will repay you.\n\n\"And when you pray, don't babble on like the Gentiles do\u2014they think they will be heard because of their many words. Don't be like them, for your Father knows what you need before you ask Him. This, then, is how you should pray: 'Our Father, who is in heaven, may Your name be held in reverence. May Your kingdom come. May Your will be done on earth as it is in heaven. Give us today the bread we need for the coming day. And forgive us our debts, as we also have forgiven our debtors. And let us not be brought into temptation but rescue us from the evil one.'\n\n\"For if you forgive others for their offenses, your heavenly Father will forgive you as well, but if you do not forgive others, neither will your Father forgive your offenses.\n\n\"And when you fast, don't put on a gloomy face like the hypocrites do, for they neglect their appearance so that everyone will notice they are fasting. I tell you truly, they have received their reward. But as for you, when you fast, anoint your head and wash your face so that others can't see that you are fasting, but only your Father, who is in secret, can. Then your Father, who sees what is done in secret, will repay you.\"\n\nThe apostles nod and whisper to one another. Much of what Jesus says makes sense. God's children should reflect His character in their everyday activities.\n\nJesus then addresses two of people's greatest challenges: wealth and worry.\n\n\"Do not store up treasures for yourselves on earth, where moth and rust destroy and where thieves break in and steal. Instead, store up treasures for yourselves in heaven, where neither moth nor rust can ruin them, and no thieves can break in and steal them. For where your treasure is, there your heart will also be.\n\n\"The eye is the lamp of the body, so if your eye is healthy, your whole body will be full of light, but if your eye is bad, your whole body will be full of darkness. So if the light that is in you is darkness, how deep that darkness is!\n\n\"No one can be the slave of two masters\u2014either he will hate the one and love the other or he will be devoted to one and despise the other. You cannot serve both God and mammon.\n\n\"That's why I tell you to not worry about the food and drink you need to live or about the clothes you need for your body. Isn't life more than food and the body more than clothing? Take a good look at the wild birds\u2014they neither sow nor reap nor gather into barns, and yet your heavenly Father feeds them. Are you not worth more than birds? And who among you can add a single hour to his life by worrying?\n\n\"And why worry about clothing? Learn a lesson from the wildflowers and how they grow\u2014they neither toil nor spin, yet I tell you that not even Solomon in all his splendor was dressed like one of these. If that's how God clothes the grass of the field, which is here today and thrown into the oven tomorrow, will he not do much more for you, O you of little faith?\n\n\"So don't be anxious, saying, 'What are we to eat?' or 'What are we to drink?' or 'What are we to wear?'\u2014all these things the Gentiles strive for\u2014your heavenly Father already knows that you need them. Seek first the kingdom of God and His righteousness, and all these things will be given to you as well. So don't worry about tomorrow; tomorrow will bring its own worries. Today's troubles are enough for today.\"\n\nJesus pauses to allow the audience to feel the weight of his instruction. He has taught about the disciple's relationship with God. Now he turns to an equally important subject\u2014how his apprentices are to treat others.\n\n\"Be merciful, just as your Father is merciful. Judge not, and you will not be judged. Condemn not, and you will not be condemned. Forgive, and you will be forgiven. Give, and it will be given to you. Good measure\u2014pressed down, shaken together, running over\u2014will be put into your lap. You will be judged by the same standard by which you judge others, and the measure you give will be the measure you get back.\n\n\"Can the blind lead the blind? Will they not both fall into a pit? An apprentice is not above his teacher, but when he's fully trained, he will be like his teacher.\n\n\"Why do you see the speck in your brother's eye but fail to notice the log in your own eye? Or how can you say to your brother, 'Brother, let me take the speck out of your eye,' when all the time there is a log in your own eye? You hypocrite\u2014first take the log out of your own eye, then you will see clearly enough to remove the speck out of your brother's eye.\n\n\"Don't give what is holy to dogs, and don't throw your pearls in front of pigs, lest they trample them underfoot and then turn and tear you to pieces.\n\n\"Ask, and it will be given to you; seek, and you will find; knock, and the door will be opened to you. For everyone who asks, receives, and everyone who seeks, finds, and to everyone who knocks, the door will be opened. Who is there among you who, if his son asks for a loaf of bread, will give him a stone? Or if the child asks for a fish, will he give him a snake? So if you, bad as you are, know how to give good gifts to your children, how much more will your Father in heaven give good things to those who ask Him!\n\n\"So whatever you would like others to do to you, do also to them, for this sums up the Law and the Prophets.\"\n\nThe people on the hillside can only shake their heads. Even the apostles are struggling to understand how they could survive if they adopted the lifestyle Jesus is describing. Jesus knows that most in the audience think his instructions are unrealistic, but he insists that there are only two paths.\n\n\"Enter through the narrow gate,\" Jesus says. \"For wide is the gate and broad is the road that leads to destruction, and there are many who take it. But narrow is the gate and hard is the road that leads to life, and there are few who find it.\n\n\"Watch out for false prophets who come to you in sheep's clothing but inwardly are savage wolves. You will recognize them by their fruits. Every tree is known by the fruit it bears. Grapes aren't gathered from thorn bushes or figs from thistles, are they? In the same way, every healthy tree produces good fruit, but the diseased tree produces bad fruit. A healthy tree can't produce bad fruit, and a diseased tree can't produce good fruit. Every tree that does not bear good fruit is cut down and thrown into the fire. Thus you will recognize them by their fruits. The good person from the good treasure of his heart produces good, and the evil person from his evil treasure produces evil, for out of the abundance of the heart the mouth speaks.\n\n\"Why do you call me 'Lord, Lord,' and not do what I say? Not everyone who says to me, 'Lord! Lord!' will enter the kingdom of heaven but only the person who does the will of my Father who is in heaven. On that day, many will say to me, 'Lord! Lord! Did we not prophesy in your name, and cast out demons in your name, and do many mighty works in your name?' Then will I declare to them, 'I never knew you\u2014depart from me, you evildoers.'\n\n\"Everyone who comes to me and hears my words and puts them into practice will be like a wise man building a house, who dug deep and laid the foundation on solid rock. The rain fell and the rivers flooded and the winds blew and beat against that house but could not shake it because it was built upon the rock. But everyone who hears my words and does not put them into practice will be like a foolish man who built his house on the sand. The rain fell and the rivers flooded and the winds blew and beat against that house, and immediately it collapsed, and the ruin of that house was great.\"\n\nWhen Jesus stands to announce he is done, some on the hillside shake their heads and dismiss his words as fanciful and idealistic, but others try to imagine what life would be like in the kingdom Jesus describes.\n9\n\nWISDOM'S CHILDREN\n\nSpring\u2013Autumn, AD 31\n\nDoes not wisdom cry out? Does not understanding raise her voice? . . . . Blessed is the person who listens to me, watching daily at my gates, waiting beside my doors. For whoever finds me finds life and obtains favor from the Lord, but whoever fails to find me injures himself\u2014all who hate me love death.\n\n\u2014Proverbs 8:1, 34\u201336\n\nJESUS HAS CHOSEN twelve men and given them an overview of the apprentice life. Their training will begin immediately. Jesus and his new apostles come down from the hills above Tabgha and walk east toward Capernaum. When they arrive, they are met by a group of civic leaders who have come on behalf of one of the city's patrons, a Gentile centurion. His servant is paralyzed and in agony. The city elders ask Jesus to intercede. \"He is worthy to have you do this for him, for he loves our nation, and it was he who built our synagogue.\"\n\nCenturions are \"commanders-of-a-hundred\"\u2014normally eighty infantrymen plus support personnel. Men are promoted to the rank because of their prowess on the battlefield, not because of their spiritual sensitivity and benevolence, but Jesus senses there is more to this centurion.\n\nStill, this is a request from Jews, and Jews are rarely interested in helping Gentiles, so Jesus gently challenges them. \"Shall I come and heal him?\" His question is rhetorical and aimed at their prejudice.\n\nJesus, his disciples, and the city leaders walk toward the centurion's villa east of town. A small crowd of townspeople follows. Before they reach the military leader's home, they are met by another group of emissaries. The centurion has had second thoughts about the appropriateness of his initial request. Through his friends, the centurion says to Jesus, \"Lord, do not trouble yourself, for I am not worthy to have you come under my roof. Neither do I now consider myself worthy to come to you.\"\n\nThe military officer understands authority and recognizes it in Jesus. \"Just give the command, and my servant will be healed,\" he says. \"For I, too, am a man under authority, with soldiers under me. And I say to one, 'Go,' and he goes, and to another, 'Come,' and he comes, and to my servant, 'Do this,' and he does it.\"\n\nJesus, moved by the centurion's poverty of spirit, turns to the crowd and says, \"I tell you truly, I have not found anyone in Israel with faith like this. And I tell you that many will come from the east and west and recline at the table with Abraham, Isaac, and Jacob in the kingdom of heaven, while the sons of the kingdom will be thrown into the outer darkness\u2014there will be weeping and gnashing of teeth.\"\n\nThe people begin murmuring. All are familiar with the messianic banquet Jesus is referring to. According to Isaiah, those at the celebration will enjoy \"a feast of rich food, a feast of well-aged wine.\" Everyone in Israel longs to recline at that table, for an invitation to it means God's eternal blessing. Some of the teacher's words are comforting\u2014Abraham, Isaac, and Jacob will have honored status at the head of the table\u2014but surrounding them, Jesus said, would be Gentiles. The crowd, convinced that only Jews will be invited to the messianic banquet, scowl and shake their heads.\n\nJesus turns to the centurion's emissaries. \"Go,\" he tells them. \"As you believed, so let it be done for you.\"\n\nThe officer's friends thank Jesus and rush back to the villa, but when they arrive they find the servant healthy and talking with the centurion. When this news gets back to the crowd, their anger turns to awe and confusion. The evening ends without incident, but Jesus's words have enraged the townspeople and invigorated those who plot his death.\n\nJesus knows he is no longer safe in Capernaum, and early one morning he leaves town and travels west into the hills. His disciples and a crowd of the curious follow. Late in the afternoon, Jesus reaches the outskirts of Nain. A funeral procession is coming out of the city. In front is a bier bearing the body of a young man who had died earlier that day.\n\nThe two crowds meet near the city gate. Jesus looks beyond the bier to the boy's widowed mother, who is sobbing inconsolably. Her deepest fear has come to pass\u2014death has robbed her of her entire family. Except for the mercies of God, she is alone in the world and helpless.\n\n\"Do not weep,\" Jesus tells the woman. He walks over to the wooden plank holding the body and touches it. Both crowds come to an immediate halt. The people who are following Jesus wonder why he's interfering with a funeral, and the mourners from Nain wonder why the procession has been interrupted.\n\nBut no one expects what happens next. Jesus speaks directly to the body on the bier. \"Young man, I say to you, arise.\"\n\nAn audible gasp escapes the crowd as the once-dead man sits up and begins speaking. Those carrying the bier fall to the ground in fear. They had anointed his cold body for burial hours earlier\u2014now he's alive and talking.\n\nJesus walks the young man over to his speechless mother, and the two crowds erupt in praise. \"A great prophet has risen among us!\" says one. \"God has visited His people!\" says another.\n\nThe revivification is seen by many, and soon everyone in Judea has heard about the prophet who, like Elijah, raised a widow's son from the dead. There are similarities between the two revivifications, but even more striking is the difference. Elijah prostrated himself before the Lord with prayers and supplication, and after a time God responded by reviving the dead man. With Jesus, the process was quite different\u2014Jesus spoke directly to the corpse, and as the words were spoken, death became life. All who witnessed the miracle, and especially the new apostles, are wondering the same thing: Who is this man who can heal even the dead?\n\nEVENTUALLY, THE STORIES about Jesus reach far down the Jordan Valley and into the bowels of Machaerus, Herod's desert fortress east of the Salt Sea, where John the Baptist is imprisoned. His months in the lightless dungeon have darkened his understanding. Where there had once been clarity, there is now doubt.\n\nSome of the reports about Jesus are encouraging, but it appears to John that Jesus is content to simply roam the hills of Galilee as a healer and itinerant preacher. Where is the one who will come with his winnowing fork to sift and separate? Where is the one who will lay his ax to the root of the trees?\n\nJohn, no longer certain, sends two of his disciples to Jesus with an urgent question. They immediately set out for Galilee, and a few days later they relay John's question to Jesus: \"Are you the one who is coming, or are we to look for someone else?\"\n\nJesus reflects on the question and the doubts that motivated it. John has been a faithful servant. He has done everything God has asked and has been rewarded with imprisonment in a rat-infested hole. John's confusion is understandable, but Jesus doesn't give him a direct answer. Instead, John is asked to consider how the Scriptures describe the work of Messiah. \"Go and tell John what you have seen and heard,\" Jesus tells the Baptist's disciples. \"The blind receive their sight, the lame walk, lepers are cleansed, the deaf hear, the dead are raised, and the poor have good news preached to them. And blessed is the one who does not stumble because of me.\"\n\nReferencing the prophecies of Isaiah, Jesus offers implicit proof of his identity\u2014proof John's disciples have been eyewitnesses to. But to John and those with ears to hear, Jesus has a more subtle answer: the passages he alluded to that speak of the good news also speak of God's judgment. Jesus would have John see that while he may not be meeting the Baptist's expectations, he is even now in the process of sifting and separating. John is gently admonished not to allow his expectations to trip him up.\n\nJohn's disciples thank Jesus and begin the long trek south to Machaerus. As they walk away, Jesus turns to the crowd and says, \"What did you go out into the wilderness to see? A reed wavering in the wind? What, then, did you go out to see? A man dressed in soft clothing? Behold, those who are beautifully clothed and live in luxury are in palaces. What, then, did you go out to see? A prophet? Yes, I tell you, and more than a prophet. This is the one about whom it is written, 'Behold, I send My messenger ahead of you, who will prepare your way before you.'\n\n\"I tell you, among those born of women, no one is greater than John, yet the one who is least in the kingdom of heaven is greater than he. From the days of John the Baptist until now the kingdom of heaven is subjected to violence, and the violent plunder it. For all the prophets and the Law prophesied until John, and if you are willing to accept it, he is the Elijah destined to come. Whoever has ears, let them hear.\"\n\nJesus is reminding the crowd of John's true identity. Until John, there had not been a prophet in Israel for four hundred years, only the bat-kol, the \"daughter of a voice,\" but that all changed when John came in from the wilderness. God had sent them an Elijah-like prophet who was not only greater than Elijah, he was greater than all the prophets. John is greater not because he is more faithful than those before him but because of his role in ushering in the kingdom of heaven.\n\nNevertheless, a major transition is underway\u2014the covenant God gave Moses is being fulfilled and will soon be superseded by the New Covenant. Meanwhile, the kingdom of heaven is being established on earth, and the difference between this kingdom and all others is so vast that even the most insignificant person in the new era will be greater than anyone in the old.\n\nGod told Jeremiah, \"I will put My law in their minds and write it on their hearts. I will be their God, and they will be My people. No longer will a man teach his neighbor, or a man his brother, saying, 'Know the Lord,' because they will all know Me, from the least of them to the greatest, for I will forgive their wickedness and remember their sins no more.\"\n\nThe prophecy is being fulfilled. The New Covenant is coming in, but it must be ratified by blood, and Jesus knows that John will not live to see the gifts and revelations of that soon-coming day.\n\nMuch of John's confusion is due to his expectation that when the Messiah comes, all opposition will be swept away. Instead, as Jesus affirms, the introduction of God's kingdom is being met with fierce resistance\u2014John is in prison; the nation's leaders are condemning Jesus; and most of the people, while enthusiastic about the miracles, resist his teaching.\n\nJesus identifies the source of the resistance. \"To whom, then, shall I compare the people of this generation?\" he asks. \"What are they like? They are like children sitting in the marketplace and calling to one another, 'We played the flute for you, and you did not dance; we sang a dirge, and you did not weep.' For John the Baptist came eating no bread and drinking no wine, and you say, 'He has a demon.' The Son of Man has come eating and drinking, and you say, 'Look! A glutton and a drunkard, a friend of tax collectors and sinners!' Yet wisdom is justified by all her children.\"\n\nJesus describes his generation as people who want to play religious games but are never pleased with God's response. Because John does not dance to their tunes, he is slandered. Because Jesus will not respond to their laments, he is condemned. Jesus concludes that, despite the evidence, many will find excuses to reject him. As always, hard hearts give birth to invincible ignorance.\n\n\"Woe to you, Chorazin!\" Jesus warns. \"Woe to you, Bethsaida! For if the miracles that were done in you had been done in Tyre and Sidon, they would have repented long ago in sackcloth and ashes. And indeed I tell you, it will be more tolerable on the Day of Judgment for Tyre and Sidon than for you. And you, Capernaum\u2014will you really be 'exalted to heaven'? No, rather, 'You will be brought down to Hades,' for if the miracles done in you had been done in Sodom, it would have remained until this day. And indeed I tell you all, it will be more tolerable on the Day of Judgment for the land of Sodom than for you.\"\n\nWoe is a word that was often on the lips of the ancient prophets. It means \"how greatly you will suffer!\" Jesus condemns Capernaum, Chorazin, and Bethsaida because with greater privilege comes greater responsibility. Having seen Jesus do miracle after miracle in their presence, their responsibility is to turn humbly from their sins. Instead, they demand their agenda and condemn both John and Jesus.\n\nJesus assures them they will bear full responsibility for their sin. His bluntness is not born of anger, and his words are more lament than condemnation. Like those who made similar prophecies before him, his strong language is an attempt to awaken souls.\n\nMany in the audience slumber in self-righteousness, but some are being roused. Jesus's next words are a prayer of thanks for those who are being awakened. Jesus opens his arms, lifts his eyes, and says, \"I praise You, Father, Lord of heaven and earth, that You have hidden these things from the wise and learned and revealed them to little children; yes, Father, for such was Your gracious will.\"\n\nJesus pauses and surveys his audience. He wants no one to misunderstand his next words. \"All things have been entrusted to me by my Father, and no one recognizes the Son except the Father\u2014nor does anyone recognize the Father except the Son and anyone to whom the Son chooses to reveal Him.\"\n\nThe people once again begin murmuring. Jesus knows they are struggling with the exclusivity of his claim, and he assures them that God doesn't want any of them to perish. \"Come to me,\" Jesus says, \"all who are weary and heavy laden, and I will give you rest. Take my yoke upon you and learn from me, for I am meek and lowly in heart, and you will find rest for your souls, for my yoke is easy, and my burden is light.\"\n\nJesus promises them that God will embrace anyone who turns to him. His invitation stirs deep emotions in the listeners. All can identify with feelings of being tired and worn out. All struggle under various yokes, and particularly the \"yoke of the Law,\" as it is called by the rabbis.\n\nJesus offers a different kind of yoke. He says that those who are willing to wear his yoke will learn the true meaning of rest. There will still be tragedy and heartache and death and disease, but his apprentices will be able to see the larger picture of how God works all things toward His good purposes in the lives of those who love Him.\n\nSome in the audience accept Jesus's invitation, but most shake their heads and turn away.\n\nNAIN, LIKE ALL towns in the region, has a few prominent citizens who belong to the party of the Pharisees. One is a man named Simon; he has listened intently to Jesus's claims and wants to hear more from the teacher who sounds like a blasphemer but can raise the dead. He organizes a banquet in his home, where he can question Jesus in a more controlled environment.\n\nSimon's hastily offered invitations are quickly accepted. All are curious about the miracle-worker, and by the time the teacher and his apprentices arrive, Simon's home is filled with guests.\n\nJesus enters in through the doorway and makes eye contact with the host, but the Pharisee turns away without acknowledging him. A few minutes pass. Jesus and his disciples are not offered even the common courtesies all hosts are expected to provide. At a minimum, a guest can expect a greeting and a kiss from the host (either on the cheek, or, if the guest is a rabbi or distinguished person, on the hand), and some water and olive oil, which acts as a soap to remove the road dust. Servants would be standing by to wash the feet of valued guests.\n\nJesus and his disciples are ignored. It is an obvious slight, and everyone in the room notices immediately. The other guests whisper to one another and look nervously at the young rabbi standing near the door. When Jesus becomes aware that he is being dishonored, he says nothing and meekly takes his place at the table, where others are already reclining.\n\nHis gentle response is lost on no one in the room. There are about two dozen dinner guests, but there are also a number of the village poor sitting quietly against the back walls in the hope that at the end of the meal they will receive some of the leftovers. In this way, the host can demonstrate his generosity and prove his \"door is open to God\" because it is open to the poor.\n\nOne of the poor is a young woman who quietly watches Jesus being publicly disrespected. She knows a lot about how that feels\u2014for her it's a daily occurrence. She has made many bad choices in her life, and everyone in town knows it. She understands her status, but what had Jesus done to deserve this? All he says and does is good and kind. Earlier she had heard his invitation to enter his rest and had accepted. Since that day, her life has made a dramatic turnaround. She no longer does the things that destroyed her integrity.\n\nWhen she heard Jesus would be at the banquet, she saw an opportunity to express her appreciation. She brought an alabaster bottle of perfume to anoint him with, but her plan was undone by the unexpected behavior of the host. Jesus immediately reclined at the table, and only his feet are still reachable. But since they haven't been washed and she has no water or towel, she can't anoint even his feet.\n\nDeep sobs well up in her chest; she closes her eyes and bites her lip to hold back the tears, but she is soon is weeping uncontrollably. No longer able to maintain her anonymity, she rises from the floor, walks over to Jesus, and collapses at his feet. There are no words that can express her love. There are no words that can communicate the hurt she is feeling for the way Jesus is being disrespected. She can only grasp his feet and weep.\n\nThe dinner party comes to a complete halt as all eyes turn to Jesus. They are curious about how he will respond to this obvious act of impropriety. Minutes pass and Jesus says nothing, apparently content to let the woman remain sobbing at his feet.\n\nEventually, the woman realizes her profuse tears have provided a way for her to honor her savior. She washes his feet with her tears and bathes them in kisses of gratitude. She has nothing to dry with, so she releases her long hair to serve as a towel. After several minutes of drying and more kisses, she breaks open her alabaster bottle, and the room is filled with a pungent sweetness. She pours out the perfume and begins to kiss and anoint the feet of Jesus.\n\nThe host is appalled as he watches the slow drama unfold. According to religious tradition, a woman who lets down her hair in public is guilty of sexual immorality (for, as the Scriptures note, a woman's hair is enticing). How can Jesus claim to be a holy man and allow this immoral woman to kiss and fondle his feet? Why doesn't he push this obvious sinner away? It is said that he is a prophet, but how can he be a prophet when he can't even identify a sinner? These questions and more are in the mind of the host.\n\nSimon is an important man in this small town and knows everyone. He has seen the woman plying her trade in the village for years, but he has also seen the change in her since she began following Jesus. He knows she has walked away from her former life, yet he and his cronies have, thus far, refused to acknowledge her change of heart. Until they do, she will remain a pariah in her community.\n\nJesus wants to give Simon an opportunity to reconsider, but the Pharisee has already made up his mind. In his judgment, both Jesus and the woman are sinners, and Jesus is a fraud for claiming to be a prophet. Jesus knows that Simon is silently condemning him, and he answers the Pharisee's thoughts.\n\n\"Simon, I have something to say to you.\"\n\nThe brooding host is startled; Jesus used a phrase that often means \"get ready to hear something you are not going to want to hear.\" But Simon no longer has much interest in what the self-styled prophet has to say and replies perfunctorily, \"Say it, Teacher.\"\n\nJesus tells Simon a story he knows the wealthy man will be able to relate to: \"A certain moneylender had two debtors. One owed five hundred denarii and the other fifty. When neither could repay him, he graciously forgave the debt of both.\" Jesus then follows up the short story with a question: \"So which of them will love him more?\"\n\nThis is a scenario a wealthy man like Simon knows well. His entire culture is built on a foundation of patronal ethics in which the generosity of a benefactor would help and thus obligate the receiver to return the favor in some way, even if it is only by being loyal to the patron.\n\nSimon knows and uses this system well; in fact, this very banquet would obligate his guests. Because Jesus stays within the patronal system Simon is familiar with, the answer is obvious. \"I suppose the one to whom he forgave more,\" Simon concedes.\n\n\"You have judged rightly,\" Jesus says. He is hoping to transform Simon's thinking about the whole concept of debt (which, in Aramaic, is also the word for \"sin\").\n\nWhat Jesus is suggesting is revolutionary\u2014if everyone were to adopt his philosophy of forgiveness, all the rules of interpersonal relationships would change. Discrimination and class privilege would disappear because everyone would be on the same level.\n\nJesus does not press this point to its logical conclusion. Instead, he keeps the conversation within the framework of the patronal system Simon embraces. In order to emphasize his point, Jesus turns his back to the Pharisee and faces the woman\u2014but he continues to speak to Simon.\n\n\"Do you see this woman?\" he asks. His question causes her to look up. When their eyes meet, she blushes and quickly bows her head.\n\nJesus continues to look lovingly at her while he rebukes the host. \"I entered your house! You gave me no water for my feet, but she has wet my feet with her tears and wiped them with her hair. You gave me no kiss, but from the time I came in she has not ceased to kiss my feet. You did not anoint my head with oil, but she has anointed my feet with perfume. Therefore, I tell you, her sins\u2014which are many\u2014are forgiven, because she loved much. But he who is forgiven little, loves little.\"\n\nThe Pharisee bristles, but before he can speak, Jesus says to the woman, \"Your sins are forgiven.\"\n\nThe table, quiet during the confrontation, now erupts in conversation. The guests ask one another, \"Who is this, who even forgives sins?\" It is incomprehensible that any man, even the Messiah, could offer to do that which is the prerogative of God alone.\n\nJesus knows their thoughts but ignores them. His primary concern is the poor and broken woman lying before him. He looks into her eyes and speaks a truth she has known since the day she repented and accepted his invitation: \"Your faith has saved you.\"\n\nJesus doesn't excuse or minimize her sins, but, at the same time, he does not condemn her. Her outward actions make it clear that she has experienced an inward change of heart, and the only word Jesus has for her is a pronouncement of God's shalom. \"Go in peace,\" he tells his new apprentice.\n\nJesus, his work in Nain complete, once again takes to the roads of Galilee. No village is too small to minister in, and many weeks are spent in the hills\u2014Jesus working, his disciples watching. Then word comes from the south: John the Baptist is dead. The details are sketchy, but it is reported that he was beheaded at one of Herod's lavish parties.\n\nSome say Herod was the mastermind, but others suggest the murder was orchestrated by his wife, Herodias, who passionately hated the prophet. It is rumored that she and her daughter devised a scheme that resulted in John's head on a platter, but Jesus knows who is behind the murder, and that one is not flesh and blood.\n\nThe one who would prepare the way for Messiah is now dead, but Elijah has accomplished his mission. The way has been prepared. While his enemies celebrate and plot their next move, Jesus withdraws and awaits instructions from the Father.\n10\n\nDARK SAYINGS FROM OF OLD\n\nAutumn\u2013Winter, AD 31\n\nGive ear, O my people, to my teaching; incline your ears to the words of my mouth! I will open my mouth in a parable\u2014I will pour forth dark sayings from of old . . . . I will utter what has been hidden since the foundation of the world.\n\n\u2014Psalm 78:1\u20132; Matthew 13:35\n\nMONTHS PASS AS Jesus unhurriedly makes his way through the many villages in the hills west of Lake Gennesaret. Though he spends little time in Sepphoris and Tiberias, the region's two largest cities, his name is often heard in their streets, and his teachings have infiltrated even the palace. Joanna, the wealthy wife of Chuza, Herod's business manager, has become a devoted follower of Jesus. She slips away whenever he is teaching near Tiberias, and often gives money to the apostles to help meet the needs of the rabbi and those he has called to walk with him. Her husband is one of the most powerful men in the kingdom, and he shares the concerns of his master about the growing popularity of the prophet who is preaching to large crowds, sometimes just a short distance from the royal residence.\n\nNews about Jesus also regularly reaches the ruling council of the Jews in Jerusalem. As his fame grows, the Sanhedrin fears that the hysteria being stirred up in the north might spread to the entire nation and draw unwanted attention from Rome. They recall earlier uprisings that began in Galilee but created hardship for all Israel. After several evenings of long and heated discussion in the Hall of Hewn Stones, a plan is devised\u2014a delegation of experts in the Law will make the three-day journey to the north shore of Lake Gennesaret. They are confident they can discredit the troublemaker and diffuse the potential crisis.\n\nWithin a week the legal team is in Capernaum. The lawyers meet with the local Pharisees and are soon current on the activities of their adversary. They learn Jesus has recently returned from the hills and is staying in the home of two fishermen. Every morning when Andrew and Simon Peter open their doors to the public, the religious scholars are waiting at the courtyard gate.\n\nReports about Jesus have also regularly made their way to his mother, brothers, and sisters in Nazareth. They learn how recent events in nearby Nain have swollen the crowds to the point that Jesus often no longer stops to eat. It is said he teaches and heals from early morning until late in the evening, then retreats alone into the hills for all-night prayer vigils. The more his family hears about his activities, the more concerned they become. Perhaps his popularity has gone to his head, or maybe he just doesn't know how to say no, but it appears to the family that the eldest son has lost his way and will do nothing to help himself.\n\nAfter a long and difficult discussion, the family decides to intervene. His sisters will remain at home while his mother and brothers make the day-long journey to the lake to bring Jesus back to Nazareth\u2014separate him from the hangers-on by force, if necessary.\n\nEarly one morning they set out for Capernaum. That morning finds Jesus doing what he does every day, all day\u2014teaching about the kingdom and healing those who come to him. The scribes from Jerusalem sit in a far corner of the room observing. Morning becomes afternoon, and still the ailing come. Jesus turns none away.\n\nOne of the saddest cases that day is a man who is so helpless he must be carried in. He is a blind mute and would incite everyone's deepest pity\u2014except he is also demon-possessed. He cannot speak, but he occasionally growls and makes menacing guttural sounds. The lawyers from Jerusalem whisper to one another, curious to see what the healer will do.\n\nJesus speaks a word. The man's sight is restored, and he shouts praises to God.\n\nThe crowd is amazed\u2014who can restore sight and speech with just a word? This is the kind of mighty work the Scriptures say the Messiah will do. But how could this meek itinerant preacher and healer who spends all his time in insignificant Galilee be the Anointed One? Nothing is clear, but one in the room sees the Messianic possibility and tentatively says, \"Surely this man is not the Son of David . . . . is he?\"\n\nThe scribes and Pharisees from Jerusalem stir uncomfortably. This is not the conclusion they want the people to come to, but they must respond carefully. There have been hundreds of witnesses to dozens of miracles, and Jesus's authority over illness and demons cannot be denied. But while they can't deny he has power, they can challenge the source of his power. They unleash the plan crafted in the Hall of Hewn Stones. One of the lawyers points his finger accusingly at Jesus. \"He is possessed by Beelzebul!\"\n\nAnother nods in agreement and adds, \"It is only by the power of Beelzebul, the ruler of the demons, that this man casts out demons.\"\n\nBeelzebul, one of Satan's many nicknames, means \"lord of the dwelling.\" They are accusing Jesus of being a sorcerer who heals and exorcises with powers he has received from the Devil.\n\nAmazed at their hardness of heart, Jesus asks, \"How can Satan cast out Satan? Every kingdom that is divided against itself is laid waste, and no city or household that is divided against itself will be able to survive. And if Satan casts out Satan, he is divided against himself. So then how could his kingdom survive? His end has come.\"\n\nJesus's reasoning is simple: demons are Satan's faithful servants. Why would the Devil give one of his servants the authority to destroy his other servants? Satan would be undermining his own kingdom.\n\nThe religious experts struggle to understand. They define the kingdom of heaven as a future event and the kingdom of Satan as an abstract concept\u2014Jesus identifies both kingdoms as current realities that his miracles testify to. Jesus is suggesting that there is only one reasonable conclusion to draw from a work that frees a soul from demonic possession while restoring both sight and speech, but the Pharisees are not listening.\n\nJesus sharpens his point and makes it personal. \"You say that I cast out demons by Beelzebul, but if it is by the power of Beelzebul that I cast out the demons, by whose power do your sons cast them out? Therefore, it is they who will be your judges. But if it is by the power of the Spirit of God that I cast out the demons, then the kingdom of God has come upon you.\"\n\nJesus is warning the religious scholars that if they claim his exorcisms are the results of sorcery, they will be forced to condemn all other Jewish exorcists as sorcerers as well\u2014but if others are exorcising demons by the power of God, so is he.\n\nJesus recounts an ancient parable to help the lawyers grasp the profound truth he has demonstrated. \"When a strong man, fully armed, guards his own dwelling, his possessions are secure. How can someone enter a strong man's house and steal his goods unless he first binds the strong man? But when one stronger than the strong man attacks and overcomes him, he takes the armor in which the man trusted and divides his spoils.\"\n\nJesus's words recall God's promise to personally rescue His people. The Lord told Isaiah, \"Those who wait for Me shall not be put to shame. Can the prey be taken from the mighty? Can the captives of a tyrant be rescued? Even the captives of the mighty will be taken, and the prey of the tyrant will be rescued, for I will contend with those who contend with you, and I will save your children.\"\n\nJesus tells the parable of the strong man to demonstrate that not only is he not in collusion with the Devil, he is in an all-out war against him. The blind-mute demoniac had been one of the strong man's possessions. By healing him, Jesus had demonstrated to the \"lord of the dwelling\" who the true Lord is\u2014he had entered Satan's kingdom, bound him, and repossessed a soul.\n\nThere is a choice before everyone in the room that afternoon, and the teacher's next words make it plain there is no neutral ground: \"Whoever is not with me,\" Jesus says, \"is against me, and whoever does not gather with me scatters.\n\n\"I tell you truly, people may be forgiven every other sin and blasphemy, but blasphemy against the Spirit will not be forgiven. And if anyone speaks a word against the Son of Man, they will be forgiven, but whoever speaks against the Holy Spirit will not be forgiven, either in this age or in the age to come, but is guilty of an eternal sin.\"\n\nThe lawyers sent by the Sanhedrin perceive Jesus is attempting to reverse their charge of blasphemy by accusing them of what Moses called sinning \"with a high hand\" (the image is of raising one's fist at God in deliberate defiance)\u2014a sin the Jews believe is unforgivable. Jesus says God will forgive every sin but one\u2014attributing the work of God to Satan\u2014what Isaiah described as \"calling evil good and good evil.\"\n\nJesus challenges them to decide who he really is. \"Either make the tree good and its fruit good, or make the tree bad and its fruit bad, for the tree is known by its fruit.\" Jesus demands they make up their minds about him\u2014either he is an evil person doing evil works, or he is a good person doing good works.\n\nGod is not the author of evil, and Satan isn't interested in doing good. Accusing Jesus of doing good by Satan's power is equivalent to attributing the work of the Holy Spirit to the Devil. This is a lethal mistake that cannot be forgiven because it cuts one off from the only means of salvation.\n\nAnd there is no chance that Jesus has misjudged the important men from Jerusalem. It is by their own words they stand condemned. \"You brood of vipers!\" Jesus says. \"How can you, being evil, say anything good? For out of the overflow of the heart the mouth speaks. The good person out of his good treasure brings forth good, and the evil person out of his evil treasure brings forth evil. I tell you, on the Day of Judgment people will give account for every empty word they speak\u2014for by your words you will be justified, and by your words you will be condemned.\"\n\nJesus would have them see how words spoken in haste are often more revealing because they come straight from the heart. His warning falls on deaf ears. \"Teacher,\" the lawyers say, \"we want to see a sign from you.\"\n\nJesus is not interested in satisfying the whims of evil people who have already proven they will reject whatever sign they are given. \"It's a wicked and adulterous generation that demands a sign,\" Jesus tells them. \"But no sign will be given to it except the sign of the prophet Jonah\u2014for just as Jonah was three days and three nights in the belly of the sea monster, so the Son of Man will be three days and three nights in the heart of the earth.\n\n\"The men of Ninevah will rise up at the judgment with this generation and condemn it, for they repented in response to Jonah's preaching, and behold\u2014something greater than Jonah is here.\n\n\"The queen of the south will be raised up at the judgment with this generation and condemn it, for she came from the ends of the earth to hear the wisdom of Solomon, and behold\u2014something greater than Solomon is here.\"\n\nThe religious scholars are not sure what Jesus is claiming, but he is clearly not honoring their request. He has chosen to respond with a counterchallenge. The idol-worshipping Queen of Sheba and the pagan people of Ninevah repented when God's messenger spoke to them, Jesus says, but the Jews remain obdurate despite hearing a much greater testimony.\n\nHe tells the scribes a darkly humorous story to underline his warning. \"When the unclean spirit has been cast out of a person, it wanders through waterless places seeking rest, but finds none. Then it says, 'I will return to my house from which I was expelled.' And when it comes, it finds the house empty, swept, and put in order. Then it goes and recruits seven other spirits more wicked than itself, and together they enter and dwell there, and the final state of that person is worse than the first. And so it will be with this evil generation.\"\n\nA Jewish superstition declares that demons like to dwell in desolate, arid places, but Jesus disagrees. He says demons much prefer a human home to one in the desert, and if the person they have been expelled from is not filled with obedience to another Spirit, the demon will return with a vengeance and make the person's situation much worse than before his halfhearted resolution.\n\nOne of the women in the room, moved by Jesus's words, says, \"Blessed is the womb that bore you, and the breasts at which you nursed!\"\n\nJesus shakes his head and says, \"Blessed, rather, are those who hear the Word of God and obey it.\"\n\nSomeone comes to the door and announces the teacher's family has arrived from Nazareth and wishes to speak with him. Jesus knows his mother and brothers plan to force him to come home. His family's intentions are good, but their conclusions are similar to those of his enemies. Jesus looks around the room and asks, \"Who is my mother, and who are my brothers?\"\n\nHe pauses a few moments to allow the question to sink in, then motions toward the disciples seated around him and says, \"Behold, here are my mother and my brothers! My mother and my brothers are those who hear the Word of God and do it. Anyone who does the will of my Father in heaven is my brother and sister and mother.\"\n\nWhen his words are relayed to his family in the courtyard, they are hurt and confused, but say nothing. Jesus knows that the source of their confusion is their inability to grasp the nature of the kingdom of heaven. He wants to teach about the kingdom in depth, but he wants to share the message with a larger audience.\n\nJESUS INVITES HIS listeners to the lake. They are joined by a crowd of townspeople. Hundreds press around Jesus at the water's edge. He steps into a small fishing boat and is rowed a short distance from the shore to take advantage of a natural amphitheater. He then begins teaching them about the kingdom of heaven in parables\u2014stories that are \"thrown alongside\" a truth to make it more understandable.\n\n\"Listen,\" Jesus tells those gathered around him. \"Once a sower went out to sow his seed. As he sowed, some fell along the path and was trampled underfoot, and the birds of the air came and devoured it. Some fell on rocky ground, where they did not have much soil. Immediately they sprang up because they had no depth of soil, and when the sun rose they were scorched because they had no moisture, and because they had no root they withered away. Some fell among thornbushes, and the thorns grew up and choked them, and they yielded no grain. And some fell into good soil and produced grain, growing up and increasing and yielding thirty-fold and sixty-fold and a hundredfold. He who has ears to hear, let him hear.\"\n\nThe audience is intrigued by the parable but has no idea what it means. At first it sounds like the story of a wasteful farmer who sows indiscriminately (three of the four areas he plants bear no crop), but the story has a surprising ending\u2014the good ground is so fertile that it not only compensates for the farmer's profligate sowing, it also rewards him with an exceptional crop. A normal yield is less than tenfold. This farmer has at least tripled that. But who, the audience wonders, is the farmer? And what do the various soils point to?\n\nJesus sees their quizzical looks but doesn't explain the parable. Instead, he throws out another story. \"The kingdom of God is like a man who scatters seed on the ground,\" Jesus says. \"Night and day, whether he is asleep or awake, the seed still grows, but he does not know how it grows. On its own, the earth produces grain: first the blade, then the ear, then the full grain in the ear. But when the grain is ripe, he immediately sends forth the sickle because the harvest has come.\"\n\nThe images of the sickle and harvest are common references to the Day of the Lord. Most people in Israel expect the kingdom of God to come upon them like an earthquake. They believe the Messiah will one day dramatically appear to liberate his people. Jesus's parable warns against that kind of thinking and suggests the kingdom will come quietly.\n\nGod's kingdom will be established, but its growth will be mysterious. \"The kingdom of heaven can be compared to a man who sowed good seed in his field,\" Jesus says. \"But while his men were sleeping, his enemy came and sowed weeds among the wheat and went away. When the plants sprouted and produced a crop, then the weeds appeared also, and the servants of the master of the house came to him and said, 'Master, didn't you sow good seed in your field? How then does it have weeds?'\n\n\"'It is an enemy who has done this,' the master replied.\n\n\"So the servants said to him, 'Then do you want us to go and pull out the weeds?'\n\n\"But the master said, 'No, lest in gathering the weeds you root up the wheat along with them. Let both grow together until the harvest, and at harvesttime I will tell the reapers, 'First gather the weeds and tie them in bundles to be burned, but gather the wheat into my barn.'\"\n\nJesus pauses and looks into the confused faces surrounding him. \"With what can we compare the kingdom of God?\" he asks. \"What parable shall we use for it? It's like a grain of mustard seed that a man took and sowed in his field. It's the smallest of all seeds, yet, when it is sown, it springs up and grows larger than all the garden plants. It becomes a tree and puts out large branches so that the birds of the air come and roost in its shade.\n\n\"The kingdom of heaven is like leaven that a woman took and hid in three measures of flour until all the dough had risen.\"\n\n\"The kingdom of heaven is like treasure hidden in a field, which a man found and hid. Then in his joy he goes and sells all that he has and buys that field.\n\n\"Again, the kingdom of heaven is like a merchant in search of fine pearls, and when he had found one pearl of exceedingly great value, he went and sold all he had and bought it.\n\n\"Again, the kingdom of heaven is like a dragnet that was thrown into the sea and caught fish of every kind. When it was full, the fishermen pulled it ashore, sat down, and sorted the good into containers, but threw away those that were worthless. So it will be at the end of the age\u2014the angels will come and separate the wicked from the righteous, and they will throw the wicked into the fiery furnace, where there will be weeping and gnashing of teeth.\"\n\nJesus shares parable after parable about the mysterious way the kingdom grows, but he explains none of them. When Jesus is finished and the crowd disperses, he and his followers share a meal and rest by the lakeside. The conversation turns to the teaching earlier that afternoon. One of the disciples asks, \"Why do you speak to them in parables?\"\n\nJesus pauses long enough to ensure he has his apprentices' full attention, then says, \"Because to you it has been given to know the mysteries of the kingdom of heaven, but to them it has not been given. The secret of the kingdom of God has been given to you, but for those outside everything comes in parables, so that 'they may be ever seeing but never perceiving, and ever hearing but never understanding. Otherwise, they might turn and be forgiven.'\n\n\"In them the prophecy of Isaiah is fulfilled that says, 'You will be ever hearing but never understanding\u2014you will be ever seeing but never perceiving, for this people's heart has become calloused. They can barely hear with their ears, and they have closed their eyes. Otherwise, they would see with their eyes, hear with their ears, understand with their heart, and turn so that I might heal them.'\n\n\"But blessed are your eyes, because they see, and your ears, because they hear. I tell you truly that many prophets and righteous people longed to see what you see and did not see it, and to hear what you hear and did not hear it.\"\n\nJesus says that some will hear God's invitation and turn to Him, but others, no matter how often they hear the truth, will refuse to repent and be healed.\n\nThe disciples smile and nod in agreement, but they are confused about the meaning of the stories. They ask Jesus to explain the parable of the sower.\n\n\"Do you not understand this parable?\" Jesus asks. \"How, then, will you understand any of the parables? Hear then the meaning of the parable of the sower. The seed is the Word of God. The sower sows the Word. Some people are like seed along the path where the Word is sown. When anyone hears the message of the kingdom and does not understand it, the evil one comes and immediately snatches away the Word from their heart so that they may not believe and be saved.\n\n\"The ones on the rocky ground are those who hear the Word and immediately receive it with joy, but they are rootless and don't last very long\u2014when trouble or persecution arises on account of the Word, they immediately fall away.\n\n\"The ones sown among thorns are those who hear the Word, but the worries of this world and the false allure of wealth and the desires for other things come in and choke the Word, and it proves unfruitful.\n\n\"But those sown in the good soil are those who understand and accept the Word. These are those who, when they hear the Word, hold it fast in an honest and good heart and bear fruit with patient endurance. These do indeed produce a crop\u2014thirty, sixty, or even one hundred times what was sown.\"\n\nJesus's apprentices begin to understand\u2014there is one message but many results. The soils represent the different responses people have to the message of the kingdom, and the disciples are already familiar with all of them. Often they meet people who hear God's Word but give little thought to it. Often they see people who are initially filled with joy but walk away at the first sign of adversity. Often they talk with those who are distracted by wealth and its attendant worries. As for the good soil, the apostles need look no further than their own experiences to understand the fruitfulness Jesus speaks of.\n\nWhy does the parable sound so simple when the master explains it?\n\nEmboldened by Jesus's willingness to explain the parable of the sower, they ask about another confusing image. \"Explain to us the parable of the weeds of the field.\"\n\n\"The one who sows the good seed,\" Jesus says, \"is the Son of Man. The field is the world, and the good seed are the children of the kingdom. The weeds are the children of the evil one, and the enemy who sowed them is the Devil. The harvest is the end of the age, and the reapers are angels.\n\n\"Just as the weeds are gathered and burned in the fire, so will it be at the end of the age. The Son of Man will send his angels, and they will gather out of his kingdom all the stumbling blocks and all lawbreakers and throw them into the fiery furnace, where there will be weeping and gnashing of teeth. Then the righteous will shine like the sun in the kingdom of their Father. He who has ears, let him hear.\"\n\nThe disciples nod thoughtfully as Jesus explains the parable. Ultimately there are but two paths\u2014each person is walking either toward the light or into the darkness. One day all those who reject God will be judged. Until then, Jesus's followers will grow up alongside the children of the evil one.\n\nThis is why it is imperative his apprentices respond to God's Word. \"No one lights a lamp and then covers it with a bowl or puts it under a bed,\" Jesus tells them. \"A lamp is put on a stand, so that those who enter may see the light. For nothing is hidden that will not be revealed, nor is anything concealed that will not be known and come to light. If anyone has ears to hear, let him hear.\"\n\n\"Pay close attention to what you hear\u2014the measure you give will be the measure you get, and still more will be added. For to the one who has, more will be given, and from the one who has not, even what he thinks he has will be taken away.\"\n\nJesus slowly looks around the circle and into the heart of each disciple. He knows a fuller understanding of the parables will require time and prayerful reflection, but his followers must at least have a basic grasp of each parable's meaning, so Jesus asks, \"Have you understood all these things?\"\n\nThe apprentices realize there is still much to learn, but regarding today's lessons, they think they understand and collectively respond, \"Yes.\"\n\nJesus nods and challenges them to teach others what they have learned from him. \"Therefore every scribe who has become a disciple of the kingdom of heaven is like a master of a house who brings out of his treasure chest things new and old.\"\n11\n\nIDENTITY\n\nAutumn\u2013Winter, AD 31\n\nAnd they will give him the name God-with-Us.\n\n\u2014Isaiah 7:14\n\nJESUS HAS SPENT the afternoon teaching parables about the kingdom of God. After the crowds leave, Jesus and the apostles share a meal, and he tells them they will spend the night traveling to the Decapolis. This will be a new experience for the apprentices; until now they have only ministered in the Jewish villages west of the lake. The Decapolis, named in honor of the region's ten most important cities, is on the eastern shore of the lake and predominantly Gentile.\n\nSoon the men are heading southeast in one of the fishermen's boats. The evening is mild, the lake is flat, and the trip promises to be uneventful. As the boat pulls away from the shore, Jesus, exhausted after a long day, makes his way to the stern and is soon sleeping soundly.\n\nThe trip is pleasant until darkness falls and the boat is far from shore. Then the wind begins to build. The professional fishermen look up nervously as clouds fill the sky. Located in a basin nearly seven hundred feet below sea level, the lake is usually calm, but occasionally cool winds roll over the mountains and clash with the warm air on the lake's surface, and fierce squalls can arise quickly. The fishermen see the weather changing and are reminded why the lake is often called the \"sea\" of Galilee\u2014within a matter of minutes the gentle breeze they enjoyed while leaving the western shore has been transformed into a low whistling howl that is spawning six-foot swells.\n\nAs the boat dips in and out of deep troughs of black water, it begins taking on water faster than the men can bail it out. The apostles become convinced they are about to drown. They turn to Jesus but find him still asleep in the stern.\n\n\"Master, Master,\" one of the disciples cries out, \"we're perishing!\"\n\nAnother shouts, \"Save us, Lord!\"\n\nA third asks the question brewing in all their minds, \"Teacher, don't you care that we're perishing?\"\n\nTheir clamor awakens Jesus. He sees the fear in their faces and shouts above the tempest, \"Why are you afraid, O you of little faith?\"\n\nThe disciples look at one another in exasperation but say nothing. Isn't it obvious why they are afraid?\n\nJesus stands and addresses the storm: \"Quiet!\" he commands. \"Be still!\"\n\nAs the words are spoken, the winds cease, and the surface becomes calm. The change is so instantaneous and so complete that the apostles become more afraid of Jesus than they were of the storm. They turn to one another and ask, \"What sort of man is this that he commands even the winds and the sea, and they obey him?\" And behind that question is a deeper one\u2014who but God has that kind of authority?\n\nIn the morning, they land near the Roman settlement of Gergesa. The apostles are thankful to reach shore safely, but they are soon wondering why they came\u2014as they step out of the boat they see two naked, obviously deranged, Gentiles running toward them. They have a reckless and violent look about them, and the disciples conclude they are about to be attacked by madmen, but when they look to their master for guidance, they see Jesus is calmly awaiting the men's arrival.\n\nThe two were once respected members of the community, but nowadays are feared and avoided by all. Family and former friends are convinced the two are under the control of demons. Attempts to incarcerate and restrain them have been futile\u2014they are possessed by a superhuman strength that can break through even iron chains.\n\nLately, the men have taken up residence in a graveyard north of town. Burial caves have been cut into the stone hillside that are large enough to hold several ossuaries and still provide shelter to anyone desperate enough to need it. The men no longer terrorize the town, but the demons who possess the men exact their price. They menace all who pass their way, and when there are no hapless travelers to intimidate, the demons turn on the two bodies they possess, tormenting them into cutting themselves with sharp rocks.\n\nThe dark forces amassed in the two men running down the hill are like none the disciples have encountered. The twelve brace themselves for an assault. Instead, the two wild-eyed men fall writhing at Jesus's feet.\n\nThe apostles still aren't clear about the identity of the one they follow, but the demons know whose presence they are in, and tremble.\n\nJesus knows his adversaries are not the two men but a multitude of fallen angels who indwell them. He addresses them as one. \"Come out of the man, you unclean spirit!\"\n\nThe demons beg him to leave them alone. \"What have you to do with us, Jesus, Son of the Most High God? I adjure you in God's name, do not torment us. Have you come here to torment us before the time?\"\n\nThe demons fear Jesus because they know he not only has the power to evict them from the two men, if he chooses, he can banish them to eternal torture in a bottomless abyss. Seeking another option, any option, the demons see a large herd of swine on a nearby slope. \"If you're going to cast us out,\" they plead, \"send us into the herd of pigs.\"\n\nIt is a curious request, but one Jesus determines he can use to demonstrate what is taking place here\u2014two more souls are being repossessed from the strong man. Jesus looks into the eyes of the men and addresses the spirits within. \"What is your name?\"\n\nThe demons, still trying to avoid destruction, answer evasively, offering a number rather than a name. \"My name is Legion, for we are many.\"\n\nThe disciples now better understand the confrontation they are witnessing. \"Legion\" is a military number approximating six thousand.\n\n\"Go!\" Jesus tells the demons.\n\nAs he speaks, the two men fall to the ground and become still, but bedlam breaks out on the hillside\u2014loud, tortured squeals and snorts erupt from the herd. The pigs begin jostling, biting, and kicking one another. Then, as one, the possessed swine stampede. The pigs hurtle headlong down the steep embankment and into the lake, where they all drown.\n\nEveryone watching is stunned. They walk to the embankment and stare into the roiling waters. The swine herders are at first mystified, but soon begin piecing together what happened. They see Jesus kneeling beside the two demoniacs, talking quietly with them and treating their wounds, and the herdsmen realize they are in the presence of a powerful exorcist. Terrified of Jesus's abilities and afraid they will somehow be blamed for what happened, the pig herders scatter, some to the towns and others to nearby farms. They tell everyone they meet about the strange occurrence at the lake.\n\nThose who hear the account find the story too incredible to be true. How could an exorcism and the mishap with the swine be connected? Hundreds travel to the lake to see for themselves. When they arrive, they are surprised to find the two men who were once the bane of the territory now calmly sitting at Jesus's feet with his other disciples. The two are clean, clothed, and acting peaceably.\n\nThe fearful herdsmen again recount the story. The Gerasenes listen thoughtfully and walk to the embankment to view the carcasses in the water. They determine the story of the herdsmen is true. Then they, too, become afraid. What kind of man has such authority? What might the powerful Jewish sorcerer do next?\n\nThe region's residents don't want to find out. They ask Jesus to get back in his boat and return to wherever he came from. Some are distraught about the financial loss from the destruction of the herd. Others are more concerned about the intentions of the Jewish magician\u2014unlike their own gods, this fellow is assertive and unpredictable and does not appear open to negotiation.\n\nJesus nods and tells the Gerasenes he will respect their wishes. He directs his disciples to ready the boat; they will leave immediately. The two men who have been restored do not want to be separated from Jesus and beg him to take them with him, but their new master has a different mission for them. \"Go home to your friends, and tell them how much God has done for you, and how He has had mercy on you.\"\n\nThe men are disappointed that they will not be able to accompany Jesus, but they are excited about the opportunity to proclaim God's goodness. As Jesus enters the boat, he knows he will not return to Gergesa, but he also knows the region will not be without a witness. History will record little of it, but many souls will come to know Jesus through the testimony of two men who were once home to a legion of demons.\n\nSOON AFTER JESUS returns to Capernaum, Matthew Levi hosts a banquet to introduce his old tax collector friends to his new master. Dinner invitations are rare for customs agents. No one in respectable society will socialize with them. Their only friends are others like themselves, and they are intrigued by the notion of a holy man who will share a meal with tax collectors.\n\nOther townspeople are annoyed. They can't understand why Jesus is openly defying the sacred traditions by eating with sinners. They view tax collectors as worse than even lepers\u2014the leper, at least, does not choose his condition.\n\nThe evening arrives. Tax collectors from all over the region make their way to the party. A wonderful meal is served, and Matthew's guests find Jesus fascinating to talk with. He's humorous and full of intriguing stories, and by the time the celebration ends, the customs agents have a completely different perspective. According to Jesus, God is loving toward all and willing to accept anyone\u2014even tax collectors.\n\nJesus's gentle treatment of society's outcasts kindles deep feelings in the hearts of the tax collectors but infuriates the local Pharisees. They are tempted to confront Jesus with his sin, but experience has taught them that directly challenging the Nazarene doesn't end well. They approach the apostles and ask, \"Why does your teacher eat with tax collectors and sinners?\"\n\nThe disciples, who have wondered the same thing at times, take the question to their master. Jesus recognizes the significance of the query; in a shame-based culture, a public complaint like this is a challenge. The Pharisees want to shame and censor Jesus, and they will succeed if he can't provide an acceptable answer to their charge.\n\nJesus offers a simple proverb. \"Those who are well have no need of a physician, but those who are sick do.\" Jesus points out that a spiritual teacher who avoids sinners would be as foolish as a doctor who avoids those who are ill.\n\nJesus then turns the tables on the religionists by shaming them with the words of the prophet Hosea. He tells the Pharisees, \"Go and learn what this means: 'I desire mercy, and not sacrifice.' For I came to call not the righteous but sinners to repentance.\"\n\nThe issue, Jesus explains, is much bigger than observing boundary markers like table fellowship and remaining ceremonially clean. Relationships always take precedence over ritual\u2014beginning first and foremost with one's relationship with God.\n\nIn Hosea's day, God's people were more attentive to religious rites than to the needs of others. Those ancient people focused on ceremonial pomp and purity but ignored the heart of worship: God's hesed, the forgiving loving-kindness his people are to both enjoy and reflect. Mere outward conformity to God's Law and ceremonial correctness will never do, Jesus says, and any righteousness that places ritual above mercy is no righteousness at all.\n\nThe disciples report Jesus's words to the Pharisees. The religious teachers scoff and walk away, vowing to one day silence the quick-witted false prophet who somehow seems to come up with one clever answer after another.\n\nJesus is not surprised by the Pharisees' unwillingness to acknowledge their sin. Because they have no godly sorrow, they see no need for repentance, and the way to their salvation remains hidden.\n\nSeveral days pass, and gossip about Matthew's party continues to circulate. Much of the talk is about Jesus's religious laxity. His casual approach to the fences erected to protect God's Law is causing deep concern among the pious. Some disciples of John the Baptist come with a question about one of Judaism's most prominent boundary markers. \"Why do we and the Pharisees fast, but your disciples do not fast?\"\n\nFasting, along with prayer and almsgiving, is one of the three pillars of Judaism and a widely respected worship practice. The Mosaic Law requires the nation to fast one day a year on Yom Kippur, but fasting is popular in every season and is deemed appropriate whenever one is dissatisfied with life. Sometimes one fasts in response to a loss, sometimes to signify repentance, and at other times to plead for God's intercession, but fasting is always to be an act of worship.\n\nThe Pharisees zealously fast every Monday and Thursday. John's disciples also fast frequently, as do the Essenes, Sadducees, and others who are serious about their religion. In their minds, it is inconceivable that a rabbi would not teach his disciples to fast. If Jesus wants to be taken seriously, he must honor the sacred traditions.\n\nJesus answers their question by identifying why people fast in the first place\u2014fasting signifies that one is dissatisfied with something in themselves or their world. Jesus then explains why he and his disciples do not fast often. \"Surely the wedding guests are not to mourn while the bridegroom is with them, are they?\"\n\nHis question mines one of the richest veins of Jewish life. There are appropriate seasons for mourning and dissatisfaction with the status quo, but wedding celebrations are times for feasting, not fasting. The wedding imagery makes sense to those listening, but according to the Scriptures, the bridegroom is not a man. The groom is God.\n\nAs the people reflect on the wedding image, Jesus says, \"But the days will come when the bridegroom is taken away from them\u2014then they will fast.\"\n\nThe audience doesn't understand why the groom would be expelled from his own wedding. Jesus offers two parables and a proverb that may help.\n\n\"No one tears a piece of cloth from a new garment and patches it on an old garment,\" Jesus says. \"The new piece will pull away from the old and a worse tear will be made, and the patch from the new will not match the old.\n\n\"And no one puts new wine into old wineskins. If he does, the new wine will burst the skins, spilling the wine and destroying the skins. No, new wine must be put into fresh wineskins.\n\n\"And no one after drinking old wine desires new wine, for he says, 'The old is good.'\"\n\nJesus's observations about patches and wine are simplistic when applied to tailoring and winemaking but profound when applied in the realm of the spirit. Jesus has given them a timeless principle\u2014the new cannot be put on or into the old without disastrous results. The religious establishment is attempting to shame Jesus into keeping the traditions, but he knows the expanding fermentation of the gospel will require a much larger and more adaptable container than Judaism.\n\nJesus is not criticizing the Old Covenant; he is simply saying that the Mosaic Law, like an intact old wineskin, has admirably served its purpose\u2014it has prepared the way for the gospel\u2014but the new must have a fresh container.\n\nMany in the audience shake their heads and turn away. If Jesus wants to be accepted, he needs to patch his teachings onto those of the venerable elders. The new way is unfamiliar and ignores the sacred tenets and traditions. Why abandon those? The old is good.\n\nJUST THEN, THE door bursts open and a man collapses in tears at Jesus's feet. His name is Jairus, and he is known to Jesus and everyone in the room. He organizes worship at the synagogue and is one of Capernaum's most prominent citizens. His twelve-year-old child is gravely ill, and physicians have been unable to cure her. \"My little daughter is at the point of death,\" Jairus sobs. \"Come and lay your hands on her so that she may be made well and live.\"\n\nJesus immediately stands and walks with Jairus out the door and into a large crowd that is gathered in the street. Jesus finds it difficult to make his way through the jostling throng; everyone wants to be near him. One in particular is a woman who has a gynecological bleeding disorder that has rendered her ceremonially unclean. She has suffered for twelve years and has spent her entire life savings seeking a cure, but to no avail. She has no hope in physicians, but she believes Jesus can heal her\u2014if only she can get close enough.\n\nShe sees Jesus come out of a house with another man and walk away. Her heart quickens. She feels she cannot wait another day and begins forcing her way through the hundreds of people between her and the healer. Getting close to him is difficult, but she will not be denied. Ducking, squeezing, and crawling, she comes within a few feet of Jesus, but he, too, is pushing through the crowd. It appears he is on an urgent mission.\n\nThe woman sees her opportunity slipping away and desperately reasons that perhaps this holy man of God, like the altar of the Tabernacle, can make holy anything that touches him. Stretching out her hand, she can reach but the hem of his garment. Instantly she feels a change within and knows she has been healed.\n\nJesus stops, turns, and scans the crowd. \"Who touched my garments?\" he asks.\n\nSimon Peter shrugs and says, \"You see the crowd pressing around you, and yet you say, 'Who touched me?'\"\n\n\"Someone touched me,\" Jesus explains, \"for I know that power has gone out from me.\"\n\nThe healer continues to stand in the middle of the street surveying the crowd and asking who touched him. He asks not because he doesn't know who touched him but so that the woman might feel the full impact of her restoration.\n\nThe woman he healed is rejoicing that her life-destroying condition has disappeared, but she is also afraid. The healer somehow knows it was her, and she is about to be exposed. Her fear grows, and soon she is trembling uncontrollably. She throws herself at Jesus's feet and confesses. She tells everyone why she touched his garment and how she was healed.\n\nThe crowd begins murmuring. The woman has admitted her uncleanness. She should not even be here. The woman steels herself to receive the healer's wrath, but rather than reprimanding her, Jesus says, \"Daughter, your faith has made you well\u2014go in peace and be healed of your affliction.\"\n\nThe woman's heart leaps within her, but her joy is dampened when word comes from Jairus's home that Jesus is too late. \"Your daughter is dead,\" Jairus is told. \"Why trouble the teacher any further?\"\n\nJairus's chin falls to his chest. He nods sadly and turns to thank Jesus for his willingness to come, but the healer tells Jairus to ignore the news. \"Do not fear,\" Jesus says. \"Only believe, and she will be well.\"\n\nThe synagogue leader is confused but does not question Jesus. They soon arrive at his home. Dozens are gathered. Loud wails and moans can be heard from within. When Jairus's wife sees them enter the courtyard, she runs to her husband and falls into his arms, sobbing.\n\nJesus instructs everyone to remain outside except the child's parents and three of the apostles. They enter the house and find singers from the mourner's guild weeping and chanting dirges. Several flutists provide a haunting accompaniment.\n\n\"Why are you making a commotion and weeping?\" Jesus asks. Then he turns to the child's mother and comforts her. \"Do not weep, for she is not dead, only sleeping.\"\n\nWhen the professional mourners hear Jesus's pronouncement, they laugh derisively. They've been around death long enough to know it when they see it. The girl isn't sleeping. She's dead.\n\nJesus tells the mourners to leave and walks into the room where the body lies. He goes directly to the child's bed, takes her small cold hand in his, and says, \"Little girl, I say to you, arise.\"\n\nImmediately the child sits up and looks into the eyes of the kind-faced stranger who is holding her hand. Her parents gasp in disbelief and clutch each other tightly. The girl sees her parents' tears but doesn't understand why they are crying. Jesus tells the child's mother to prepare her some food and charges both parents to keep what happened a secret.\n\nHis work accomplished, Jesus walks home with a large throng following. In the crowd are two blind men who are convinced Jesus is the Messiah. As they stumble along behind him, they shout, \"Have mercy on us, Son of David.\"\n\nJesus hears their cries, but he has already performed two very public miracles and doesn't want to stir the crowd up further, so he walks home without acknowledging them. The men, however, are persistent. They know the Messiah will be able to heal blindness, and their faith in Jesus is so strong that they follow him through the front door of his home.\n\nOnce inside and out of sight of the crowd, Jesus asks, \"Do you believe that I have the power to do this?\"\n\nThe men reply in unison, \"Yes, Lord.\"\n\nJesus lays his hands on their eyes and says, \"In response to your faith, be it done for you.\"\n\nAs the word is spoken, their sight returns. Jesus sternly warns them, \"See that no one knows about it.\"\n\nJesus walks back into the street. There are still dozens of people milling around, wondering if anything else will happen. A few men walk up the street carrying a mute who is demon possessed. Jesus dispatches the evil spirit and restores the man's ability to speak. Those who see the exorcism proclaim, \"Never was anything like this seen in Israel.\"\n\nA Pharisee in the crowd attempts to dampen their amazement. \"He casts out demons by the prince of demons.\"\n\nAgain Jesus hears his enemies' favorite allegation. Jesus ignores the taunt. His words and deeds are sufficient testimony to his identity\u2014let those who have eyes see.\n12\n\nHOMECOMING\n\nWinter, AD 31\u201332\n\n\"He's just a carpenter\u2014Mary's boy. We've known him since he was a kid. We know his brothers, James, Justus, Jude, and Simon, and his sisters. Who does he think he is?\"\n\n\u2014Mark 6:3 The Message\n\nOF THE MORE than two-hundred towns in Galilee, none are more special to Jesus than the one he grew up in, yet in two years of public ministry he has not taught in his hometown. As Jesus and his disciples enter Nazareth, he knows the challenges that await him.\n\nWhen it's announced that he will be speaking on the coming Sabbath, all his former neighbors make plans to attend. They have heard much about his exploits, and many have known Jesus since he was a child, but the rumors circulating about him do not sound at all like the gentle boy they once knew.\n\nSabbath arrives. When the call to worship is given, the building is filled to overflowing. The services open, as always, with prayers of confession led by one of the elders. After prayer, the revered Shema (hear!) is recited by the congregation in unison. It opens with six powerful words that go to the heart of their faith: Shema Yisrael, Adonai Elohainu, Adoni Echad. HEAR ISRAEL\u2014the LORD our GOD, the LORD is ONE. The Shema is followed by more prayer.\n\nAs the congregation prays, the synagogue attendant prepares for the next part of the liturgy: he brings a scroll containing a portion of the Torah to the table in front. One of Jesus's old neighbors comes forward, reads from the Law, and returns the scroll to the attendant, who puts it back in its place and removes a scroll with writings from one of the prophets.\n\nJesus moves to the reading table and is given a scroll containing the prophecies of Isaiah. He unfurls the sheepskin and reads a familiar passage. \"The Spirit of the Lord is upon me because he has anointed me to preach good news to the poor. He has sent me to proclaim release to the captives and recovery of sight to the blind, to set free those who are oppressed, to proclaim the year of the Lord's favor . . . .\"\n\nJesus stops reading in the middle of Isaiah's prophecy and carefully rolls up the scroll. All eyes are fixed on the young rabbi as he returns the writings to the attendant and sits down to teach. Jesus surveys the faces of people who have known him for almost thirty years. He smiles and says, \"Today this Scripture has been fulfilled in your hearing.\"\n\nJesus then begins teaching them what Isaiah was referring to in the passage he read, and as he reveals the meaning of the Scriptures, the men and women turn to one another in astonishment. Where had he learned these deep truths? And he presents them so confidently! Isn't this Joseph's son?\n\nHis neighbors are impressed with his ability to teach, but Jesus knows they do not believe his claim that the Day of the Lord has begun. Isaiah's words about the Messiah's work should have prompted everyone in the synagogue to consider the implications of Jesus's claim to be the prophecy's fulfillment. The promised \"year of the Lord's favor\" would also be a \"year of Jubilee,\" when all debts were to be forgiven and all slaves freed. Moses and the prophets had spoken of such a year, and Jesus says it is now here\u2014he has brought it.\n\nThe people of Nazareth shake their heads in disbelief. How could this meek young man possess that kind of power? The townsfolk are more interested in why Jesus has not done the kinds of miracles in Nazareth he has done in other places\u2014if those stories are even true. The people will need more proof. They want an unmistakable sign from heaven.\n\nKnowing their thoughts, Jesus says, \"Surely you will quote me that proverb, 'Physician, heal yourself'\u2014everything we have heard you did at Capernaum, do here in your hometown as well.\"\n\nJesus is aware of the fears and prejudices that motivate their beliefs. He responds not with a sign but with a rebuke and a history lesson. \"I tell you truly, no prophet is accepted in his homeland.\"\n\nThe congregation winces at the reminder. The persecution of prophets in Israel is well documented, and Jesus offers an embarrassing example. \"In truth I tell you that there were many widows in Israel in Elijah's day when the heavens were shut for three years and six months, and a severe famine came over all the land\u2014yet Elijah was sent to none of them, but only to a widow in Zarephath, in the land of Sidon.\"\n\nJesus has pulled a story from some of Israel's darkest days. Back then the nation was divided. The southern kingdom was the land of Judea. The northern kingdom, which included Samaria and Galilee, retained the name Israel, and was ruled by an evil king named Ahab and his queen, Jezebel. They led Israel to trade the God of their fathers for a storm god named Baal. This created a deep rift not only between them and the people of Judea but also between them and God. His response to their idolatry was a drought and famine that lasted three and a half years. Daily the people prayed for the storm god to bring rain, but not a drop came, not even the morning dew. Thousands died in Israel, but God's prophet was sent only to a Gentile woman in Phoenicia.\n\nThe people in the synagogue know their history and are insulted by the suggestion that they are like their forefathers. Jesus responds with an even more offensive example. \"There were many lepers in Israel in the time of Elisha the prophet,\" Jesus reminds them, \"and none of them were cleansed\u2014only Naaman the Syrian was.\"\n\nJesus mentions an event he knows will infuriate every Jew in the synagogue. Leprosy and other skin diseases are understood to be outward manifestations of inward corruption, so, to a Jew, to be cleansed of leprosy is equivalent to being cleansed of sin. In Elijah's day, many Jews suffered from incurable skin diseases. God healed none of them. Instead, He cleansed a Gentile\u2014and not just any Gentile. He healed the general of the Syrian army that was at war with Israel.\n\nThere is no mistaking Jesus's message: the people of Nazareth are not entitled to special treatment just because he grew up there. God is not a respecter of persons and is more interested in where a person's heart is than where he lives. Jesus wants his old neighbors to see it is not God's will that they seek blessings only for those who look and think and act like them. He would have them understand that God is more than a regional Savior. As Isaiah said long ago, the Messiah would come not just to the Jews but to every nation. Particularly galling to the Nazarenes is Jesus's insistence that, as in the cases of the Phoenician widow and the Syrian general, receiving God's blessings is a matter of faith, not ancestry.\n\nAs the implications of Jesus's words sink in, the synagogue erupts in protest; the angry villagers resent being told they are worse off than Gentile women and lepers. There is shouting and shoving as the townspeople force Jesus out of the building and toward a rocky cliff on the outskirts of town. Many are ready to execute judgment on the infuriating prophet immediately, but in the commotion and confusion Jesus slips away. Where he has gone, they do not know, but his words will be long remembered.\n\nUNDAUNTED BY HIS rejection in Nazareth, Jesus continues to travel from town to town in Galilee. He works from early morning until evening, but the number of those in need continues to grow. His compassion for the harassed and helpless of Israel is boundless, but physically he has an obvious limitation\u2014he can only be in one place at a time. Jesus, however, has known all along that his mission is not his alone.\n\n\"The harvest is plentiful,\" he tells those who follow him, \"but the workers are few. Therefore ask the Lord of the harvest to send out workers into His harvest.\"\n\nThe harvest, Jesus says, is the work of the Creator. Ultimately, it is God who draws souls to Himself. Nevertheless, those who follow Jesus are to pray for more workers because the Lord has chosen to give His people key roles in His work. Some will plant, others will water, and God will give the increase.\n\nThe apostles are divided into six pairs and told to prepare for a missionary journey. Jesus then gives them specific instructions about where to go and what to do. \"Do not go among the Gentiles and enter no town of the Samaritans, but go instead to the lost sheep of the house of Israel. As you go, proclaim this message: 'The kingdom of heaven has arrived.' Heal the sick, raise the dead, cleanse the lepers, cast out demons.\"\n\nTo the apostles, the mission Jesus describes sounds like the one they have been on for two years, with one significant difference\u2014until now, their role has been to watch and learn from their master's example. Now it will be their hands God uses to heal, their commands that awaken the dead and banish demons, and their preaching that God uses to save souls.\n\nThe apostles struggle to understand the breadth of the commission, and Jesus reminds them that they will be offering only what they themselves were given. \"Freely you received,\" Jesus says. \"Freely give.\"\n\nAccording to Jesus, God has a different strategy for their sustenance, and His provision is so generous, there will be no need to bring money or supplies. The clothes on their back and the sandals on their feet will be sufficient. The staff they leave with will bring them home. \"Take nothing for your journey,\" Jesus tells them. \"Don't take gold or silver or copper in your money belts. Don't take bread or bag or sandals or staff or a second tunic, for the worker is worthy of his keep.\"\n\nTraveling light will allow them to respond quickly to the Spirit's leading. More importantly, his apprentices will learn to rely upon God and one another for every necessity. Jesus explains, \"Whatever town or village you enter, make inquiries as to who in it is worthy, and stay with them until you leave. As you enter the house, wish peace upon it, and if that household is worthy of it, let your peace come upon it. But if it is not worthy, let your peace return to you. And if anyone will not welcome you and listen to your message, as you go out of that house or town, shake off the dust from your feet as a testimony against them. I tell you truly, on the Day of Judgment it will be more bearable for the land of Sodom and Gomorrah than for that town.\n\n\"Behold, I am sending you out as sheep in the midst of wolves, so be as wise as serpents and as innocent as doves. Be cautious of people, for they will hand you over to the courts and flog you in their synagogues, and you will be brought before governors and kings because of me, as a witness to them and the Gentiles. When they hand you over, don't worry about what to say or how to say it. What you are to say will be given to you at that time, for it is not you who speak but the Spirit of your Father speaking through you.\n\n\"A brother will hand over his brother to be killed, and a father his child. Children will rise against their parents and have them put to death. You will be hated by all because of my name, but it is the person who remains faithful to the end who will be saved. When they persecute you in one town, flee to the next, for I tell you truly, you will not have gone through all the towns of Israel before the Son of Man comes.\n\n\"A disciple is not above his teacher, nor a servant above his master. It is enough for the disciple to become like his teacher, and the servant like his master. If they have called the master of the house Beelzebul, how much more the members of his household?\n\n\"So have no fear of them, for nothing is covered that will not be uncovered, or hidden that will not be known. What I tell you in the dark, you are to speak in the light, and what you hear whispered, you are to proclaim from the rooftops.\n\n\"And do not fear those who kill the body but cannot kill the soul. Instead, fear the One who can destroy both soul and body in hell.\"\n\nJesus's knowledge of eternity creates an urgency in his invitation. He wants the twelve to know what is at stake. Hell is not an abstract doctrine or children's tale; it is an eternal shop of horrors.\n\nBut Jesus says there is good news\u2014because of the Father's love and provision for His creation, there is no need for anyone to go to perdition. \"Are not two sparrows sold for a penny?\" Jesus asks. \"Yet not one of them will fall to the ground without your Father's knowledge. Indeed, even the hairs of your head are all numbered. So fear not\u2014you are worth more than many sparrows!\"\n\nBut Jesus wants the twelve to know that every soul they encounter has a choice to make. \"Everyone who acknowledges me before others, I also will acknowledge before my Father who is in heaven,\" Jesus says. \"But whoever denies me before others, I also will deny before my Father who is in heaven.\n\n\"Do not think that I have come to bring peace on the earth\u2014I have not come to bring peace but a sword. Indeed, I have come to set a man against his father, and a daughter against her mother, and a daughter-in-law against her mother-in-law\u2014a person's enemies will be those of his own household. Whoever loves father or mother more than me is not worthy of me, and whoever loves son or daughter more than me is not worthy of me. And whoever does not take his cross and follow me is not worthy of me. Whoever finds his life will lose it, and whoever loses his life for my sake will find it.\n\n\"Whoever welcomes you, welcomes me, and whoever welcomes me, welcomes the One who sent me. Anyone who welcomes a prophet because he is a prophet will receive a prophet's reward, and anyone who welcomes a righteous person because he is a righteous person will receive a righteous person's reward. And whoever gives even a cup of cold water to one of these little ones because he is a disciple, I tell you truly, he will by no means lose his reward.\"\n\nWhen Jesus concludes his teaching, the apostles know he has told them truths that reach far beyond the current mission, but they have been given more than enough instruction about where to go and what to do on this assignment.\n\nAs Jesus and his followers move to the next town, the six teams scatter in different directions. For weeks they travel throughout the region telling people about Jesus and healing those who are ill and oppressed.\n\nThe exploits of Jesus and the apostles are often discussed in the palace of Herod Antipas. Tiberias is the largest port on the lake, and the docks resound with stories of miracles happening all over Galilee. Everyone is interested in news about Jesus, and none more than Antipas. Most alarming to Herod is the persistent rumor that John the Baptist has returned from the dead.\n13\n\nTHE BREAD OF LIFE\n\nSpring, AD 32\n\n\"Oh, for some meat! We remember the fish we used to eat for free in Egypt, and the cucumbers, melons, leeks, onions, and garlic. But now our appetite has dried up. All we ever see is this manna!\"\n\n\u2014Numbers 11:4\u20136\n\nTHE SPRING RAINS make the hills green and the roads muddy, yet nothing can dampen the success of the twelve who were sent into the countryside to preach and heal. From all over the land come stories of miracles wrought by the apostles. The blind see, the deaf hear, the lame walk, lepers are healed, and the demon-possessed are set free. There are even far-fetched tales of people being raised from the dead.\n\nThis last rumor is particularly disturbing to the region's ruler. Herod Antipas is not a Jew and is more superstitious than religious, but he believes in the resurrection of the dead and is haunted by his decision to kill John the Baptist. Antipas knows John was a prophet and fears God will punish him, but the tetrarch fears Rome more than he fears God. He is concerned about reports that Jesus is announcing the arrival of a new kingdom. Kingdoms require kings, and Galilee has long been a birthplace of revolutionaries. If Rome interprets Jesus's preaching as politically motivated, there could be serious consequences.\n\nJesus knows that Antipas is paranoid, and he's seen how the ruler responds to perceived threats. With the return of the apostles from their mission, the hundreds who once followed Jesus from town to town now number in the thousands. Further, the large crowds are so demanding that he and his apostles often miss meals and lose sleep. The apostles don't complain, but Jesus can see the price they are paying and decides that now would be a good time to retreat for some much-needed rest. One morning he gathers the twelve and says, \"Come away by yourselves to a desolate place and rest awhile.\"\n\nThe crowd is dismissed, and the thirteen get in a boat and begin rowing north along the shore. The crowd, however, refuses to disperse. They can see the boat Jesus is in and simply walk along the shore following it. When the boat reaches the north shore, Jesus instructs those rowing to head east toward the Golan Mountains. Located just east of the Jordan in the tetrarchy of Phillip, Gaulanitis is a wilderness area ideal for solitude and rest, but by the time they arrive there is a crowd the size of a small city. In addition to those who followed the boat around the lake, a large multitude of locals arrive.\n\nIt becomes clear to Jesus that his plan to rest must wait; he instructs the rowers to make for shore. More than fifteen thousand people await them. Jesus is hungry and exhausted, but his compassion compels him to teach and heal all afternoon. When evening approaches, the twelve, sensitive to the size of the multitude and fearing a crisis is developing, come to Jesus and say, \"This is a desolate place, and the hour is now late. Send the crowd away into the surrounding countryside and villages to find lodging and buy themselves something to eat.\"\n\nGiven the remote location, the request is reasonable, but rather than alleviating the crisis, Jesus says \"They don't need to go away\u2014you give them something to eat.\"\n\nSeeing the stunned look on the apostles' faces and wanting to stretch their faith, Jesus turns to Phillip and casually asks, \"Where can we buy bread so that these people may eat?\"\n\nPhillip, who grew up in nearby Bethsaida, is flummoxed by the question. Surely Jesus knows that buying bread would be impossible at this late hour. Besides, even if food were available, where would they get the money to buy it? Phillip throws up his hands in exasperation and says, \"Two hundred denarii worth of bread would not be enough for each of them to get even a little.\"\n\nJesus nods and asks, \"How many loaves do you have? Go and see.\"\n\nSoon Andrew returns and reports, \"There is a boy here who has five barley loaves and two fish, but what are they for so many?\"\n\n\"Bring them here to me,\" Jesus replies. He then tells the apostles to organize the crowd to facilitate serving. \"Have them sit down in groups of about fifty each.\"\n\nThe apostles make their way through the multitude, and soon all are seated in groups on the lush green carpet provided by the spring rains. Jesus then says a prayer of thanksgiving to the Father and gives the bread and fish to the apostles to distribute. The five barley loaves are the size of a saucer and finger-thick, and the two fish are hand-size. When divided into twelve baskets, the provision looks impossibly inadequate, but the twelve do as they are told. They are shocked as person after person reaches into the baskets and retrieves bread and fish. Thousands are not only fed but filled.\n\nWhen everyone has eaten all they want, Jesus tells the apostles, \"Gather up the leftover fragments so that nothing may be lost.\"\n\nWhen they collect the leftover bread and fish, they bring it to Jesus. The twelve large baskets are now filled with food. The disciples are reminded of two passages in the Scriptures. One tells the story of how Elisha miraculously provided food to fill a hundred men from twenty small barley loaves, but even more striking is the parallel with the accounts in the Torah and the Psalms that recall the time when God \"gave them bread from heaven in abundance.\"\n\nThose who were miraculously fed by Jesus see the connection and proclaim, \"This is indeed the prophet who is to come into the world!\" They are alluding to \"the prophet like Moses\" foretold in Deuteronomy who would lead God's people to freedom in a new exodus.\n\nJesus knows that populist and revolutionary sentiments burn passionately in the crowd, and he perceives that if he were to reject their offer, they could easily start a rebellion, publicly declare him their guerrilla king, and thus conscript him. There are already in the crowd five thousand men ready to swear allegiance to the cause, and if they recruit in Galilee, birthplace of the passionate Zealot movement, they could easily raise a large army.\n\nHowever, Jesus has no interest in such a kingship; it would defeat his mission. He tells the apostles to get in the boat and go west toward Bethsaida until he can free himself from the crowd. He assures them he will join them later, but they must leave immediately. The apostles hastily board the boat and row away. Jesus turns again to the multitude. There is nothing he can say or do to change their mind, so he doesn't try. He determines that his time would be better spent in prayer, and there is much he needs to talk with his Father about.\n\nJesus quietly walks toward the mountains. Most in the crowd do not see him climb into the Golan Heights and disappear into the night.\n\nAS THE APOSTLES row across the lake, they discuss the amazing day and wonder aloud why Jesus sent them away so hurriedly. They are soon off the coast of Bethsaida, where Jesus had instructed them to wait. When Jesus doesn't arrive, the fishermen see the weather deteriorating and decide to continue toward Capernaum.\n\nThey row back across the lake, but their progress is slowed by a fierce west wind. They labor at the oars for hours with little progress. The headwind grows into a tempest. The howling gale and rough seas remind the apostles of a similarly frightening storm they were in earlier, but with one significant difference\u2014Jesus isn't in the boat with them this time.\n\nTheir master, however, has not abandoned them. From the heights where he is praying, Jesus sees the storm develop and knows the apostles are far from shore, but he also knows his apprentices are in the caring hands of his Father. He senses no urgency and remains in prayer until the dead of night.\n\nA few hours before dawn, Jesus walks down from the mountain and onto the water. The apostles, more than three miles ahead and far from shore, continue to struggle, hope ebbing in the rising gale. They suspect the waves have tossed them off course and strain to find their bearings. Peering into the predawn darkness, they wonder if their eyes are playing tricks on them\u2014they see a man-shaped apparition walking on the water.\n\nThey shudder. All have heard tales of maritime assaults by night specters and the spirits of men drowned at sea. One cries out, \"It's a ghost!\"\n\nThe apparition answers with a familiar voice, \"Take courage. I am here. Do not be afraid.\"\n\nThe disciples are terrified. How can this be? Men cannot walk on waves. Simon Peter impetuously responds, \"Lord, if it's you, command me to come to you on the water.\"\n\nTo his surprise, Jesus says, \"Come.\"\n\nPeter steps out of the boat and onto the water. When he does not sink into the sea, he begins walking tentatively toward his master. All is well until he takes his eyes off Jesus, and his faith turns to fear. As he sinks under the waves, Simon Peter cries, \"Lord, save me!\"\n\nJesus reaches out his hand, takes hold of him, and says, \"You of little faith, why did you doubt?\"\n\nAs Jesus speaks the words, Peter rises and is soon standing beside Jesus on the surface of the lake. Hand in hand, they walk through the tempest to the boat. When they step inside, the wind immediately ceases, and the water becomes calm. The men are astounded and fall to their knees, confessing, \"Truly you are the Son of God.\"\n\nTheir words profess his deity, but Jesus knows they only have a partial understanding of the one they worship. None of them have grasped the message bound in the feeding of the multitude. They are in awe, but it is an awe born of misunderstanding. Knowing they will later comprehend the meaning of what they have seen, Jesus says nothing.\n\nThe shoreline appears, and they see they are approaching the plain of Gennesaret a few miles southwest of Capernaum. As they moor the boat, they are recognized by several on the shore. Jesus is a renowned healer, and soon many who are ill are headed toward the lake. Jesus heals all who come to him, then returns to Capernaum.\n\nOn the other side of the lake are thousands of people who wake up hungry and find Jesus gone. They know he had come to the region by boat, but they had also seen his disciples leave without him the night before. The crowd knows that Jesus lives in Capernaum; they assume he slipped away in the darkness and walked home.\n\nMost return to their homes, but hundreds of people head toward Capernaum. They find Jesus at the synagogue. They want more of what they received the day before but do not know how to ask. Seeking to start a conversation, one of the men says, \"Rabbi, when did you come here?\"\n\nJesus nods and says, \"Truly, truly, I tell you, you are seeking me not because you saw signs but because you ate your fill of the loaves. Work not for the food that perishes but for the food that sticks with you and nourishes eternal life, which the Son of Man will give to you. For on him God the Father has set His seal.\"\n\nThe crowd is stung by Jesus's knowledge of their true intent but encouraged by his discussion of works they might do in order to obtain God's favor. They ask, \"What must we do to do the works God requires?\"\n\nThey want a precise definition of how one is made right with God, but Jesus knows they are steeped in a tradition that says they must earn their salvation by good works. He answers enigmatically, \"This is the work of God\u2014that you believe in the one He has sent.\"\n\nThe audience is stunned. Jesus's answer runs contrary to everything they have been taught. Just believe in him? What about the Law of Moses and the sacred traditions? By what authority does he make such pronouncements?\n\nOne in the crowd challenges Jesus. \"Then what sign will you perform that we may see it and believe you? What will you do? Our fathers ate the manna in the wilderness; as it is written, 'He gave them bread from heaven to eat.'\"\n\nThe irony of their request is not lost on Jesus. Many in the crowd had seen (and were filled by) just such a sign the day before. Moreover, they completely miss the point of the wilderness account they reference. Jesus says, \"Truly, truly, I tell you, it was not Moses who gave you the bread from heaven but my Father who gives you the true bread from heaven. For the bread of God is he who comes down from heaven and gives life to the world.\"\n\nThe audience is intrigued but lacks understanding. \"Sir,\" they say, \"give us this bread always.\"\n\nJesus smiles and says, \"I am the bread of life. Whoever comes to me will never hunger, and whoever believes in me will never thirst. But as I told you, you have seen me and still you do not believe. Everyone the Father gives me will come to me, and whoever comes to me I will never cast out, for I have come down from heaven not to do my own will but the will of the One who sent me. And this is the will of the One who sent me: that I lose none of all that He has given me but raise it up on the last day. This is the will of my Father: that everyone who looks on the Son and believes in him will have eternal life, and I will raise him up on the last day.\"\n\nWhen the audience realizes that Jesus is claiming to have come from heaven, they begin grumbling, \"Is this not Jesus, the son of Joseph, whose father and mother we know? How can he now say, 'I have come down from heaven'?\"\n\nJesus hears the murmuring and says, \"Do not grumble among yourselves. No one can come to me unless the Father who sent me draws him, and I will raise him up on the last day. It is written in the Prophets, 'And they will all be taught by God.' Everyone who has heard from the Father and has learned comes to me\u2014not that anyone has seen the Father except the one who is from God; only he has seen the Father.\n\n\"Truly, truly, I tell you, whoever believes has eternal life. I am the bread of life. Your fathers ate the manna in the wilderness, yet they died. This is the bread that comes down from heaven, so that one may eat of it and not die. I am the living bread that came down from heaven. Anyone who eats this bread will live forever, and the bread that I will give for the life of the world is my flesh.\"\n\nThe murmuring grows louder as the people quarrel about the meaning of Jesus's words. It's obvious that he's not talking about cannibalism, but if his words are figurative, what do they represent? For most, the metaphor is impenetrable\u2014hardness of heart prevents them from recognizing that Jesus is comparing feeding to faith.\n\nJesus, however, does not explain the metaphor. Instead, he intensifies it. \"Truly, truly, I tell you, unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. Whoever eats my flesh and drinks my blood has eternal life, and I will raise him up on the last day, for my flesh is true food, and my blood is true drink. Whoever eats my flesh and drinks my blood abides in me, and I in him. Just as the living Father sent me and I live because of the Father, so the one who feeds on me will live because of me. This is the bread that came down from heaven; it is not like the bread your fathers ate. They later died. But whoever eats this bread will live forever.\"\n\nThe grumbling grows louder. Someone in the audience says, \"This is a harsh message,\" and many nod their heads in agreement. They find the teacher's words not only obscure but offensive.\n\n\"Does this cause you to stumble?\" Jesus asks. \"Then what if you see the Son of Man ascend to where he was before? It is the Spirit who gives life; the flesh accomplishes nothing. The words that I have spoken to you are spirit and are life, yet there are some of you who do not believe. This is why I told you that no one can come to me unless it is given to him by the Father.\"\n\nWhen Jesus concludes his message, some of his disciples know they have heard deep truth that must be carefully unpacked, but many shake their heads in disgust and walk away. There is no need to follow this teacher any longer\u2014he makes outlandish claims and refuses to substantiate them. He promises food that can eternally satisfy, yet their stomachs still growl. He asserts he is greater than Moses but rejects those who would crown him king. He says the only work that will meet God's requirements is to believe in him, yet he offers no corroborating proof from heaven. At best, this fellow is filled with empty words.\n\nWithin a few days, it becomes obvious that a dramatic shift has taken place. The multitudes who once followed Jesus are now nowhere to be found. No longer is he celebrated in Capernaum, or even wanted. Jesus sees the effect this is having on the twelve. He gathers them together and asks, \"Do you want to go away as well?\"\n\nSimon Peter answers, \"Lord, to whom shall we go? You have the words of eternal life, and we have come to believe and know that you are the Holy One of God.\"\n\nJesus knows his apprentices well and senses a note of superiority in the response. Peter is pretentiously claiming that the apostles are not like those who left. Jesus wants to remind them it was he who chose them, not the other way around. Further, there is nothing superior about them or their knowledge. His next words are carefully chosen to wake them up.\n\n\"Did I not choose you, the twelve?\" he asks. \"Yet one of you is a devil.\"\n14\n\nTRADITION\n\nSpring\u2013Summer, AD 32\n\nDepart, depart, go out from there\u2014touch no unclean thing . . . . purify yourselves.\n\n\u2014Isaiah 52:11\n\nTWO HUNDRED YEARS before the birth of Jesus, Israel was being assimilated into a Greek culture that had already enveloped most of the world. Hellenist philosophy, art, architecture, and language eclipsed even the Romans, who had defeated them. No country was beyond their influence, including Israel. Attitudes and practices that had long distinguished the Jews from other nationalities were becoming blurred, and Israel would have been Hellenized like the lands around them except for the resistance of a small group of faithful Jews known as the Hassidim\u2014the Pious Ones.\n\nThe Hassidim were dedicated to preserving the distinctive characteristics of Judaism and aligned themselves with the priest, Mattathias, who started a revolt against the rule of the vicious anti-Semitic king of Syria, Antiochus IV. When the old priest died, his sons continued the fight, and Israel was eventually liberated.\n\nThe victory of the Maccabees ushered in eighty years of independence. The Hassidim restored temple worship and reinstituted the Law of Moses. Politically, the nation was led by the Hasmoneans, a dynasty of priest-kings named in honor of Mattathias's great-grandfather.\n\nThe Hassidim and the Hasmoneans were allies until the powerful influence of Hellenization again enticed the nation's rulers to promote Greek culture and values. By this time, however, there was theological and political dissension within the Hassidim.\n\nSome advocated complete physical separation from Gentiles. They identified themselves as Essenes and withdrew from society to live in their own neighborhoods in Jerusalem and other large cities. Some of the more zealous Essenes took vows of celibacy and established Qumran, an ascetic monastic community in the wilderness near the Salt Sea.\n\nOthers within the Hassidim preferred a less radical approach. They, too, insisted on separation but chose to work within the system to effect the changes they sought. This faction, known as the Pharisees, was willing, to a degree, to do business and interact with the non-devout. This brought condemnation from the Essenes, who labeled them \"seekers after smooth things,\" but the Pharisees weren't lax\u2014they simply took a different route to holiness. To insulate themselves from the lure of Hellenization, they established rules and traditions.\n\nTheir focus was ritual purity; thus, many of their injunctions addressed how to remain ceremonially clean while surrounded by Gentiles and Jews who did not obey the Torah. They built a fence of oral laws around the written Law to protect it from possible infringements. These oral laws became known as the traditions of the elders. The desire to protect the Torah from being corrupted was admirable, but pharisaical zeal often took their precautions to absurd lengths.\n\nNevertheless, throughout Jesus's lifetime the Pharisees have been the largest and most respected of the five major Jewish sects. Pharisaic practices are deeply entrenched throughout Israel and viewed by the majority of pious Jews as nonnegotiable. To violate the traditions of the elders is to rebel against God\u2014and here the Pharisees come into direct conflict with Jesus.\n\nIn the two years Jesus has been in public ministry, he has violated the oral laws hundreds of times. The Sanhedrin decides it is time for a reckoning. A group of scribes and Pharisees are sent from Jerusalem to rebuke Jesus for his lack of respect for the traditions.\n\nShortly after they arrive in Capernaum, the lawyers observe the apostles ignoring the oral laws concerning ritual hand washing. They ask Jesus, \"Why do your disciples violate the tradition of the elders? They eat with impure hands.\"\n\nJesus shakes his head and says, \"Why do you violate the commandment of God for the sake of your tradition? You hypocrites! Isaiah prophesied rightly about you when he said, 'These people honor Me with their lips, but their heart is far from Me; in vain do they worship Me\u2014their teachings are but rules taught by men.'\n\n\"You have skillfully set aside the commandment of God in order to observe your tradition. For God said, 'Honor your father and your mother,' and, 'Whoever speaks evil of his father or mother must surely die.' But you say that if someone tells his father or his mother, 'Any help you might have expected from me is corban,' then you no longer permit him to honor his parents. Thus you cancel out the Word of God by the tradition you pass down\u2014and you do many similar things.\"\n\nThe Pharisees wince when they hear the word corban. Jesus is accusing them of ignoring a law of primary importance\u2014the fifth commandment\u2014in order to keep one of secondary importance: the law of honoring vows. Leviticus and Numbers define corban as a sacrificial offering to God. When someone designates a portion of their property as corban, it is no longer theirs; it belongs to God. In practice, that means it belongs to the temple treasury. What makes the tradition so attractive is how it is enforced\u2014the Pharisees allow the donor to defer giving the gift as long as he chooses. Thus a selfish person can simultaneously protect his property, shirk his responsibility to honor his parents, and appear to be righteous.\n\nJesus says this tradition not only circumvents the Law, it nullifies it.\n\nThere is no response from the Pharisees; they are not interested in discussing the merits of the traditions. Besides, Jesus has not answered their question\u2014why does he refuse to teach his disciples to cleanse themselves of impurity?\n\nJesus hasn't forgotten their question. It is of great importance. Purity is one of the major tenets of the faith, but the Pharisees have completely misunderstood why God instituted purity laws. They believe the purpose of the laws is to protect God's people from outside evils.\n\n\"Hear me, all of you, and understand,\" Jesus says. \"It isn't what goes into the mouth that makes a person unclean but what comes out of the mouth\u2014this is what makes a person unclean.\"\n\nThe Pharisees scoff and walk away. With the confrontation over and the day spent, Jesus and his disciples head home, but the afternoon's heated discussion still occupies their thoughts. The twelve fear that Jesus has crossed a dangerous line with the nation's spiritual leaders. One of the men timidly asks, \"Do you know that the Pharisees were offended by what you said?\"\n\nJesus nods and says, \"Every plant that my heavenly Father has not planted will be uprooted. Leave them\u2014they are blind guides, and if the blind lead the blind, both will fall into a pit.\"\n\nWhen Simon Peter realizes that Jesus is adamant about the irrelevancy of the Pharisees, he changes the subject. He has strong convictions about ritual purity and is struggling with what Jesus said about what defiles a person. \"Explain the parable to us,\" he asks.\n\nJesus looks into the faces of the disciples and says, \"Are you still without understanding? Don't you see that nothing that enters a person from the outside can make him unclean? For whatever goes into his mouth doesn't go to his heart but into his stomach and is expelled. But what comes out of the mouth comes from the heart, and these are the things that make a person unclean. For from within, out of people's hearts, come evil thoughts, sexual immorality, theft, murder, adultery, covetousness, wickedness, deceit, lewdness, envy, slander, pride, foolishness, and false testimony. It is these things that make a person unclean, but to eat with unwashed hands does not defile anyone.\"\n\nJesus says the source of uncleanness is the heart, and it begins with evil thoughts. The traditions of the elders focus on behavior, but Jesus says people must first change the way they think\u2014they must repent. Until there is genuine change in the way a person understands who he is in relation to God and others, any attempt to draw close to God is futile.\n\nFOR TWO YEARS, Jesus has walked the hills of Galilee proclaiming the good news. It's time to take the gospel outside the boundaries of Israel. Jesus announces they will leave immediately for Syria.\n\nThey walk northwest toward the coastal city of Tyre, twenty-five miles away. Located in a bustling port on the Mediterranean, Tyre is the southernmost city in Syria and is world-renowned for three things: the purple dye they make from the shells of murex snails, the purity of their silver coins (Tyrian silver is the only coin deemed acceptable for Jerusalem's annual temple tax), and their worship of the Greek hero Heracles, whom the Romans call Hercules.\n\nThe city is crowded and busy, but the hills surrounding it are quiet and isolated. It is here Jesus wants to spend time training his apprentices, but even in this remote country his fame precedes him. One woman, a lifelong resident of Tyre, is convinced Jesus has come in answer to her prayers. When she hears Jesus has entered the region, she walks toward the border to find him. Later that day she sees a dozen or so Jewish men in the distance and knows she has found Jesus.\n\n\"Have mercy on me, O Lord, Son of David,\" she cries out. \"My daughter is severely oppressed by a demon. Have mercy on me, O Lord, Son of David!\"\n\nThe twelve assume their master will quiet the Canaanite woman, but Jesus continues walking as if he hears nothing. The woman's pleas grow louder and more insistent. \"Have mercy on me, O Lord, Son of David!\"\n\nJesus knows his disciples see no value in the woman. His recent lesson about what God defines as unclean fell on deaf ears. This Gentile woman is shouting her acknowledgement of Jesus as the Messiah, but the apostles only see Jesus ignoring the plea of an obnoxious idolator. As he should. Godly rabbis are to talk with neither women nor Gentiles.\n\n\"Send her away,\" they tell Jesus. \"For she is crying out after us.\"\n\nWhen Jesus hears the request to get rid of her, he stops and waits for the woman to catch up with them. When she draws near, Jesus turns to his disciples and says, \"I was sent only to the lost sheep of the house of Israel.\"\n\nHis answer appears to affirm his followers' prejudice. Their racism is deeply ingrained, and Jesus knows it will take more than a verbal reprimand to change their thinking. He wants to provide them with a living parable and stretch the woman's faith at the same time.\n\nThe woman falls at Jesus's feet with a simple but fervent plea: \"Lord, help me!\"\n\n\"Let the children be fed first,\" Jesus replies, \"for it is not right to take the children's bread and throw it to the dogs.\"\n\nThe disciples wince when they hear their prejudices verbalized. Like most Jews, they are convinced Gentiles are dogs (the analogy is so common it has become proverbial), but hearing the proverb applied to a humble woman who is pleading for her daughter's life is unsettling.\n\nThe woman humbly acknowledges her status and responds with a proverb of her own. \"Yes, Lord,\" she says. \"Yet even the dogs under the table eat the children's crumbs.\"\n\nJesus is delighted with her answer, and the apostles are amazed at the wit and wisdom demonstrated by someone they had judged to be little better than a wild dog. \"O woman,\" Jesus exclaims, \"great is your faith! Let it be done for you as you desire\u2014the demon has left your daughter.\"\n\nThe mother rejoices and runs home. The disciples are embarrassed by the lesson and realize there is much they need to learn.\n\nJESUS SPENDS SEVERAL weeks in Syria teaching the apostles. He travels north toward Sidon, then turns eastward. Some still come to Jesus for healing, but the isolated locations afford him the time he needs to train his disciples. They walk east until they come to the arid heights of the Golan Mountains in the Decapolis. They were here weeks earlier, but this time there are no crowds following from Galilee, and the people they meet are mostly Gentiles.\n\nThey, too, have heard of Jesus. Crowds gather, hoping he will do for them what he has done among the Jews. Jesus does not disappoint. The lame walk, the blind see, demons are displaced, and the Gentiles praise the God of Israel.\n\nAmong those brought for healing is a man who is both deaf and mute. Jesus ordinarily heals such people with just a word, but this time he chooses a more visible demonstration: he touches the deaf man's ears, then applies some of his saliva to the mute's tongue. Lifting his eyes, Jesus commands, \"Be opened!\" and instantly the man is made whole.\n\nThe crowd is overjoyed. They know how severe the man's condition was and recognize how much power resides in the healer. \"He has done all things well,\" they say to one another. \"He even makes the deaf hear and the mute speak.\"\n\nThe number of Gentiles coming to Jesus astonishes his disciples. What he did among the children of Israel he now does among the Gentiles. After Jesus has been in the region for several days, the multitude swells to nearly ten thousand men, women, and children. They have little in the way of provisions but refuse to leave because Jesus is among them.\n\nJesus calls his disciples together and says, \"I have compassion on the multitude. They've already stayed here with me for three days, and they have nothing to eat. If I send them home hungry, they will collapse on the way, and some of them have come a long distance.\"\n\nThe apostles are caught off guard by Jesus's statement and ask, \"Where could we get enough bread in such a desolate place to feed so great a crowd?\"\n\n\"How many loaves do you have?\" Jesus asks.\n\n\"Seven, and a few small fish.\"\n\nJesus asks the twelve to seat the crowd on the ground and have them prepare to eat. He then takes the bread, gives thanks to the Father, and divides the seven loaves into seven large baskets, each with a portion of the fish. As his disciples share the food, they realize there is enough for everyone to eat their fill. Jesus has again miraculously multiplied the meager resources of his followers.\n\nWhen the people finish eating, the disciples gather the remainder and fill the seven baskets. Jesus tells them to pack the baskets into their boat. They will be traveling back west to Magdala. As they row away, hundreds of Gentiles wave to them from the shore. The apostles reflect on the surprising new direction Jesus has taken and wonder how they will be received when they return to Galilee.\n\nThey come to shore a few miles north of Tiberias in the fishing village of Magdala. News of Jesus's arrival travels quickly. He has many enemies in the area and is soon accosted by a group of Pharisees and Sadducees. Hoping to discredit him, they demand he show them a sign from heaven.\n\nJesus offers them a lesson from the weather instead. \"When it's evening you say, 'It will be fair weather, for the sky is red,' and in the morning you say, 'It will be stormy today because the sky is red and threatening.' You know how to interpret the appearance of the sky, but you can't interpret the signs of the times. Why does this generation seek a sign? I tell you truly, no sign will be given to this generation except the sign of Jonah.\"\n\nJesus offers no further explanation, and the religionists are left to ponder the meaning of the sign of Jonah. Jesus has said all he has to say to Galilee's religious leaders. He directs the apostles back into the boat, and they row north toward Bethsaida.\n\nNear the end of the crossing, the twelve realize that in their haste to leave Magdala they forgot to bring food for the trip. Jesus overhears them discussing the lack of bread and says, \"Watch out\u2014beware of the leaven of the Pharisees and Sadducees and that of Herod.\"\n\nJesus is reminding them of the confrontation just hours earlier, but the disciples can think only of their growing hunger. They conclude that Jesus's comment about leaven is his way of chastising them for forgetting to bring food.\n\nJesus shakes his head and says, \"You of little faith, why are you talking about having no bread? Do you still not perceive or understand? Have your hearts become hardened? Having eyes, do you not see? Having ears, do you not hear? And do you not remember? When I broke the five loaves for the five thousand, how many basketfuls of pieces did you pick up?\"\n\n\"Twelve.\"\n\n\"And when I broke the seven loaves for the four thousand, how many basketfuls of pieces did you pick up?\"\n\n\"Seven.\"\n\nJesus nods and says, \"How can you still not understand that I wasn't talking to you about loaves of bread? But be on your guard against the yeast of the Pharisees and Sadducees.\"\n\nThe teacher looks into their eyes and is encouraged. The twelve are beginning to understand.\n\nWhen they reach the shore, they moor the boat and walk toward Bethsaida. Shortly after they enter the town, a blind man is brought to Jesus. The healer takes him by the hand and leads him outside the village. The twelve watch Jesus spit in the man's eyes and then rub the spittle in with his hands. \"Do you see anything?\" Jesus asks.\n\nThe man blinks a few times and squints. \"I see people,\" he says, \"but they look like trees, walking.\"\n\nThe one touch by Jesus has only partially restored his sight. Jesus again places his hands on the man's eyes, and his sight is fully restored. Jesus then tells the man to go home without broadcasting his healing.\n\nAs the man walks away, Jesus looks into the faces of his followers to see if they have understood the lesson. Like the man healed by Jesus, the apostles must move from blindness to full sight\u2014they must move from no understanding of their master's identity to full comprehension.\n\nAnd this, as Jesus intimated in the lesson, will take more than one touch. Ultimately, it will require an empty tomb.\n15\n\nTHE TURNING POINT\n\nSummer, AD 32\n\n\"Who do you say I am?\"\n\n\u2014Matthew 16:15\n\nALMOST THREE YEARS have passed since Jesus was anointed by John in the Jordan. The large crowds that once followed him have dwindled. His diminishing acclaim is due, in part, to the efforts of the nation's religious leaders to discredit him, but more damaging to Jesus's reputation is the message he brings. His teaching goes against the grain of some of Judaism's most honored beliefs and practices.\n\nJesus is undeterred. He expected his gospel to be widely misunderstood and knew there would be fierce opposition. But while there are no surprises for Jesus, there is the realization that his ministry in Galilee has come to an end. In a few months, it will be Passover; Jesus must prepare his apprentices for what will happen to him in Jerusalem and what awaits them afterward.\n\nFrom Bethsaida, they walk twenty-five miles into Traconitis, the northern boundary of ancient Israel at the base of Mount Hermon. Once home to the tribe of Dan, the area has long been populated by Gentiles.\n\nThe region's tetrarch, Herod Phillip, recently renamed the primary city Caesarea Phillipi as a tribute to Caesar and himself, but for many generations the city has been known as Paneas, in honor of its most popular deity: the half-man, half-goat god Pan.\n\nSeventeen hundred feet above the lake and overlooking the north end of the Jordan River Valley, the region's appeal to Jesus is its isolation; the sparsely populated area will allow him time to teach uninterrupted.\n\nAfter a hike up the mountainside, the disciples stop for the evening. After dinner, Jesus asks the twelve, \"Who do people say that I am?\"\n\nThe apostles reply, \"Some say John the Baptist, others say Elijah, and others say Jeremiah or one of the prophets of old has risen.\"\n\n\"But who do you say I am?\"\n\nThe men look at one another, hesitant to speak. Finally, Simon Peter confesses what most of them believe: \"You are the Christ\u2014the Son of the living God.\"\n\nJesus nods and says, \"Blessed are you, Simon bar Jonah, for flesh and blood has not revealed this to you but my Father who is in heaven has.\"\n\nPeter is not the first of the twelve to call Jesus the Messiah, and recently all the disciples acknowledged him as the Christ when he came to them walking on the water. But something important is being missed.\n\nThe apostles, like the crowds, only understand the coming of the Messiah in a political context\u2014a \"Son of David\" who will liberate Israel from her oppressors and restore the prosperity she enjoyed during the reign of the first David. Peter and the other disciples cannot comprehend a Christ who will suffer and die. This is simply not his role.\n\nJesus's apprentices do not grasp the nature of his mission, but they have reached a watershed in their understanding\u2014they know he is the Messiah. Now they must learn how the Son of David and the Suffering Servant are the same man.\n\nThis will take time. The belief in a militaristic Messiah is deeply entrenched in the Jewish mind-set, and Jesus knows that changing this way of thinking will not happen in isolation. There must be a community. There must be participation. Only by partnering with God and others on the path will they come to understanding.\n\nJesus responds to Peter's declaration with one of his own. \"And I tell you, you are Peter, and on this rock I will build my church, and the gates of Hades shall not prevail against it. I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall have been bound in heaven, and whatever you loose on earth shall have been loosed in heaven.\"\n\nJesus says that Simon Peter\u2014Rocky\u2014will become a foundation stone in God's house, the church.\n\nThe apostles understand little of Jesus's teaching, but the concepts are familiar. Ekklesia (church) is the word the Septuagint translators chose to replace the Hebrew word qahal\u2014the community of God's covenant people during the exodus. The apostles have been appointed to lead a faithful remnant in a new qahal, and Jesus promises that even the gates of Hades (a common Semitic metaphor for death) will not prevail against his church. In Christ, God's community will be eternally unstoppable.\n\nEven more surprising to the apostles is the role they are to play in the new qahal. Peter will be given \"the keys to the kingdom.\" Keys are a familiar symbol of authority. They open and close doors. They control who has access and who does not.\n\nLater Jesus will confer this role on the entire church, but Peter will be the kingdom's chief steward and lead administrative officer. He will be like the one God told the prophet Isaiah about. \"I will place on his shoulder the key to the house of David. What he opens, none shall shut\u2014what he shuts, none shall open. I will fasten him like a peg in a secure place.\"\n\nEssential to Peter's role as chief steward is mindfulness of whose kingdom he holds the keys to. First and foremost, stewards are servants. All the authority they wield is delegated. Stewards do not determine who is let in or put out; they simply obey their master's decrees. As Jesus said, that which the church binds or looses on earth must be that which has already been bound or loosed by God. Peter and the apostles are to function as ambassadors of the kingdom of heaven.\n\nThe twelve hear Jesus for the first time affirm he is the long-awaited Messiah, but he tells them that for now this information must be kept secret. He charges them to not use the title in public, partly because it will stir up notions of nationalism, but also because the disciples are still attached to false ideas about the Messiah.\n\nThe people of Israel have been confused about the Messiah for centuries. In ancient Israel, a person appointed to a special task was sometimes anointed with oil as a symbol of God's blessing. They would be deemed a messiah\u2014an \"anointed one.\" Over the course of many generations, the term came to identify a particular person\u2014a coming king who would free God's people and establish a kingdom of righteousness. Particularly influential in the development of this concept were the Psalms of Solomon, a collection of extra-biblical writings that prophesied the Messiah would restore David's kingdom to its former glory. The only image the disciples have of the Messiah is one of a conquering liberator.\n\nJesus tells them, \"It is necessary for the Son of Man to suffer much and be rejected by the elders and chief priests and scribes and be killed and on the third day be raised.\"\n\nThe twelve are dumbfounded. Jesus believes his death is somehow necessary? The rabbi is obviously confused about the arc of the Messiah.\n\nSimon Peter takes Jesus by the arm and walks him away from the circle. \"God forbid, Lord!\" he whispers. \"This shall never happen to you.\"\n\nMoments earlier Peter had spoken for God; now he speaks for the enemy. Jesus confronts the source of the suggestion. \"Get behind me, Satan!\" he tells Peter. \"You are a stumbling block to me, for you are not setting your mind on the things of God but on the things of man.\"\n\nPeter's declaration promises the same political shortcut the Devil proposed in the wilderness, and Jesus knows that so long as the rock upon whom the church is to be built holds that belief, he will be a stumbling stone. Thus the admonition to \"get behind\" and assume the appropriate position of an apprentice.\n\n\"If anyone wishes to come behind me,\" Jesus says, \"let him deny himself, take up his cross daily, and follow me\u2014for whoever wishes to save his life will lose it, but whoever loses his life for my sake will save it.\"\n\nJesus wants his followers to understand that discipleship is more than a decision\u2014it is a way of life. It's about choosing obedience to God even when faithfulness leads into harm's way. Jesus says it's about following him down a path that ends at a cross.\n\n\"What does it profit a person to gain the whole world only to lose or forfeit his soul?\" Jesus asks. \"For what can a person give in exchange for his soul?\"\n\nJesus admonishes his followers to take the long view: today's decisions will have eternal consequences.\n\n\"Whoever is ashamed of me and of my words in this adulterous and sinful generation, the Son of Man will also be ashamed of when he comes in the glory of the Father with the holy angels. Then he will repay each person according to what that person has done.\"\n\nJesus speaks of a day of reckoning when all accounts will be settled. Then he adds enigmatically, \"But I tell you truly, there are some standing here who will not taste death before they see the kingdom of God come with power.\"\n\nJESUS KNOWS HIS revelation has not been understood, and he spends the next six days alone with the apostles, reinforcing his teaching about what awaits him in Jerusalem. Late one afternoon, they make camp at the base of a mountain, and Jesus invites Peter, James, and John to climb into the heights with him for prayer. They near the summit as darkness falls and begin praying, but the apostles are exhausted after a long day and fall asleep.\n\nThe sound of voices awakens them. They lift their heads and rub their eyes to be sure they are not dreaming\u2014the Jesus they know has been transformed. His face is shining, and his clothing is radiating light.\n\nAs their heads clear, the disciples realize their master is talking with two men. From the conversation, they learn they are in the presence of Moses and Elijah, two of Israel's preeminent prophets. Moses foretold the coming of a final prophet who would lead God's people to a new kingdom, and it was prophesied that before that day came, Elijah would return and \"turn the hearts of fathers to their children and the hearts of children to their fathers.\" This restoration of relationships was understood to be the harbinger of the restoration of all things.\n\nThe conversation is concluding, but the disciples hear enough to know that Jesus, Moses, and Elijah are discussing the suffering Jesus will experience in Jerusalem. The topic frightens the three apostles, and none are willing to ask for more information, but when the two prophets prepare to leave, Peter wants to extend the meeting.\n\nNot knowing what to say, he suggests the disciples do something to commemorate the occasion. \"Rabbi,\" he says, \"it is good that we're here. Let us make three tents, one for you and one for Moses and one for Elijah.\"\n\nPeter's intention is good; he wants to honor three of the most important people in Israel's history, but he is misinterpreting what he sees\u2014this is not a conversation between three prophets.\n\nAs Peter speaks, an unnaturally radiant cloud envelopes the mountaintop. From the cloud comes a voice. \"This is My beloved Son, My chosen one. With him I am well pleased\u2014listen to him.\"\n\nAt Jesus's baptism, the Father had quoted Psalm 2 and the prophet Isaiah to identify the Son as both the Messiah and the Suffering Servant. The Father repeats those two references and adds \"listen to him.\" These three words point to a third title for Jesus. Peter, James, and John are seeing the fulfillment of Moses's prophecy in Deuteronomy: \"The Lord your God will raise up for you a prophet like me from among you, from your brothers\u2014you must listen to him.\"\n\nThe disciples understand the significance of the prediction. To listen to God is to comprehend what He is saying, but listening doesn't end there. Ultimately, to listen to God is to obey Him. Hence God's admonition following Moses's prophecy: \"I will put My words in his mouth, and he will tell them everything I command him. Whoever refuses to listen to My words that he shall speak in My name will answer to Me.\"\n\nPeter, James, and John are familiar with the Scriptures that surround the command and realize that God has transferred all of Moses's authority to Jesus. And more, for the one spoken of by Moses is the final prophet who will usher in the consummation of the age. When the three hear this pronouncement, they tremble and fall facedown on the ground.\n\nJesus walks over to his terrified disciples and touches them. \"Rise,\" he says, \"and have no fear.\"\n\nWhen they lift their heads, they see they are alone with Jesus. Moses, Elijah, and the luminous cloud have all vanished. The Jesus they know has returned.\n\nMorning breaks, and they walk down the mountain to rejoin the others. On the way, Jesus says, \"Don't tell anyone what you have seen until the Son of Man has been raised from the dead.\"\n\nPeter, James, and John readily agree, but they have no idea what Jesus is talking about and are afraid to ask. As they walk, the three exchange confused whispers about the mysterious reference to being raised from the dead and wonder if Jesus has told them another parable.\n\nParticularly troubling to the disciples is their inability to reconcile Jesus's words with what they have been taught all their life about the chronology of the eschaton. The prophet Malachi closed the canon of Scripture with God's promise to \"send Elijah the prophet before the great and awesome Day of the Lord.\" Jesus had just spoken with Elijah. Is what they saw on the mountain a fulfillment of that prophecy?\n\nFurther, it was commonly taught that Elijah would \"restore all things\" before the Messiah arrives, but how could the Messiah be killed in an environment in which justice and righteousness have been restored? Jesus's reasoning doesn't make sense.\n\nThe disciples ask, \"Why do the scribes say that first Elijah must come?\"\n\nJesus responds with a rhetorical question that frames the disciples' dilemma. \"Elijah does indeed come first to restore all things,\" he tells them. \"Yet how is it written concerning the Son of Man that he should suffer many things and be treated with contempt? But I tell you that Elijah has already come, and they did not recognize him but did to him whatever they wanted. In the same way, the Son of Man is destined to suffer at their hands.\"\n\nJesus acknowledges Elijah's role in the restoration of righteousness, but he reminds them that the Scriptures also speak of a righteous sufferer who will come before Earth's ultimate restoration. Isaiah called him \"the Servant of the Lord\" and predicted he would be despised and rejected by men, a man of sorrows who would bear the sins of many\u2014a man who would ultimately be \"pierced for our transgressions and crushed for our iniquities.\"\n\nJesus is asking his followers to reconsider Isaiah's prophecy, and particularly what it says about the necessity of the Servant's death\u2014even though the Servant will be innocent of any wrongdoing, he will be \"numbered with the transgressors,\" and it will be \"the will of the Lord to crush him.\"\n\nIt is essential that Jesus's apprentices grasp this truth. All they can think about is the victorious reign of the Messiah, but there can be no exaltation of the Messiah until there has been a humiliation of the Servant. The two are inseparably linked, for it is the Servant's sacrificial death that makes the Messiah's reign possible in the first place.\n\nJesus is saying that the disciples' interpretation of the Scriptures is right, but their history is wrong. Elijah has already fulfilled his mission. He was rewarded with rejection and martyrdom\u2014the same destiny Jesus expects for himself.\n\nThe three nod knowingly; Jesus is speaking about John the Baptist. That much they understand.\n\nJESUS, PETER, JAMES, and John reach the foot of the mountain and find the other apostles embroiled in a dispute with a group of religious scholars.\n\nWhen the crowd sees Jesus approaching, they run to greet him, but Jesus continues walking toward the nine, who are being accosted by the scribes.\n\n\"What are you arguing about with them?\" Jesus asks.\n\nBefore anyone can answer, a man steps out of the crowd, kneels before Jesus, and says, \"Lord, have mercy on my son, my only child, for he is possessed by a spirit that makes him mute. He has seizures and suffers terribly. Whenever the spirit seizes him, he screams and is thrown to the ground in convulsions. He foams at the mouth and grinds his teeth and becomes rigid. It scarcely ever leaves and is destroying him. I brought him to your disciples and asked them to cast it out, but they could not.\"\n\nJesus realizes he has walked in on a failed exorcism attempt and says, \"You faithless and perverted generation, how long must I stay with you and put up with you?\"\n\nJesus pauses to allow his question to sink in, and then turns to the boy's father. \"Bring your son to me.\"\n\nWhen the boy is brought to Jesus, the demon within the child recognizes the Christ and attempts to intimidate him. The spirit throws the boy to the ground. He begins convulsing and rolling in the dirt, foam bubbling from his mouth.\n\nJesus watches the behavior for a few moments, then asks the boy's father, \"How long has he been like this?\"\n\n\"From childhood,\" the man replies. \"And it has often cast him into fire and water, to destroy him.\"\n\nThe father pauses to consider his next words. Earlier he had boldly asked Jesus's apprentices to exorcise the demon, and they were unable. Why would their master fare any better? The boy seems possessed by an inexorable force. Still, any assistance would be appreciated.\n\n\"If you can do anything,\" the father pleads, \"have compassion on us and help us.\"\n\nThe father's faith is tentative. Once hopeful of a complete cure, he will now settle for whatever help he can get. Jesus knows the man's faith is small and wants to stretch it. He uses the man's own words to do it.\n\n\"'If you can'?\" Jesus replies. \"All things are possible for the one who believes.\"\n\nImmediately the man cries out, \"I believe\u2014help my unbelief!\"\n\nJesus looks at the child writhing on the ground but speaks to the malevolent spirit within. \"Mute and deaf spirit, I command you: come out of him and never enter him again.\"\n\nAs the words are spoken, an unearthly shriek comes from deep within the boy. He shudders violently and goes into convulsions. Then the screaming and shaking stops. The boy's body lies limp and still in the dust. To the onlookers, it appears the exorcism has gone terribly wrong.\n\nWord spreads through the crowd: \"The boy is dead.\"\n\nThe exorcist responds by taking the boy by the hand, lifting him up, and walking him over to his father. The crowd is astonished and shouts praises. Jesus bids them farewell and walks south.\n\nThe apostles who failed to exorcise the demon are thankful for the way things turned out but bothered by their lack of success. \"Why were we not able to cast it out?\" they ask.\n\n\"Because of your little faith,\" Jesus says. \"For I tell you truly, if you have faith the size of a mustard seed, you will say to this mountain, 'Move from here to there,' and it will move. Nothing will be impossible for you. But this kind cannot be cast out by anything but prayer.\"\n\nJesus is telling the twelve that their faith is virtually nonexistent. Worse, what little faith they do have is misdirected. They had been given the authority to heal, but had treated the gift like it was a magical ability they could apply at will.\n\nJesus points out that an apprentice's power lies not in incantations but in his relationship with God. God does not demand exceptional faith\u2014even a minuscule amount is sufficient to move a mountain\u2014but at the heart of \"mustard seed\" faith is a willingness to obey God. Having a title, even one as lofty as apostle, is of no value if one is not regularly receiving God's guidance and implementing it.\n\nAnd this is precisely the challenge Jesus faces. His disciples believe their problems are sins and bad habits, but the source is much deeper\u2014they are unwilling to trust God's Word enough to live by its principles.\n\nJesus knows there is much work to be done in the hearts of his apprentices, but he also knows his hour is rapidly approaching.\n\nIn a few short months, he will be dead.\n16\n\nA KINGDOM OF CHILDREN\n\nSummer\u2013Autumn, AD 32\n\n\"Unless you turn and become like children, you will never get into the kingdom of heaven.\"\n\n\u2014Matthew 18:3\n\nJESUS AND THE apostles return to Galilee after several weeks in the region of Caesarea Phillipi. In the past, he has drawn large crowds. Now he bypasses the towns and villages. There are no miraculous healings, no exorcisms, no sermons\u2014his teaching is limited to the twelve and focused on what lies ahead. Jesus must prepare his followers for a future they cannot begin to imagine.\n\n\"Let these words sink into your ears,\" Jesus tells them. \"The Son of Man is destined to be betrayed into the hands of people who will kill him, and on the third day he will be raised.\"\n\nThe words are plain, and the disciples now understand that what Jesus told them earlier was not a parable; he is prophesying his death. What they can't understand is how this could be part of God's plan. And who except God is mighty enough to deliver the Messiah to death? And what does a resurrection have to do with any of this? The twelve have no answers but are afraid to ask.\n\nJesus and his apprentices walk south toward the lake and eventually arrive in Capernaum. The town is quiet, and the men inconspicuously make their way to the home of Peter and Andrew.\n\nThey do not, however, escape the notice of the townspeople. The Levites who collect the temple tax have no record of Jesus paying, so when they see Peter in the street one day, they ask, \"Does your teacher not pay the tax?\"\n\nThey are not asking about a civil tax but one that, since the days of the Maccabees, has been annually requested of every Jewish male between the ages of twenty and fifty. Its purpose is honorable\u2014the maintenance and beautification of the temple\u2014but the tax is controversial.\n\nWhile most Jews pay the levy, two small but vocal minorities do not. The party of the Sadducees reject it as a modern invention of the Pharisees, and the Essenes, in keeping with the Scriptures that mention the tax, instruct their followers to pay it only once in their lifetime.\n\nThe temple tax is sourced in the Exodus account of a half-shekel required of every male member of the community when the census was taken. The Scriptures describe each Israelite's offering as \"a ransom for his life\" to be used for the upkeep of the Lord's tabernacle in the wilderness, \"that it may bring the people of Israel to remembrance before the Lord.\"\n\nThe levy is a wellspring of Jewish pride and is viewed by most as a patriotic duty. The religious representatives would be content with a \"no\" to their question\u2014a refusal could be useful in undermining the prophet's popularity. But Peter answers yes. Of course Jesus will pay the tax.\n\nPeter's reply comes quickly, but as he walks away he is less sure of his answer, and when he returns home, he is startled when Jesus says, \"What do you think, Simon? The kings of the earth\u2014from whom do they levy duties and tax? From their own sons or from others?\"\n\nThe apostle is at a loss for words. The master is obviously addressing what happened earlier, but how could he know about that conversation?\n\n\"From others,\" Simon stammers.\n\nJesus smiles and says, \"Well, then, the sons are exempt. But so that we don't cause them to stumble, go down to the lake and cast a hook. Take the first fish that comes up; when you open its mouth, you will find a coin. Take that and give it to them for you and me.\"\n\nJesus is convinced that he and the children of the King are free from obligation, but he's happy to surrender personal privileges when they become stumbling blocks. The apostles' attitudes are different. Their inclination is to think of themselves first. Arguments often break out among the twelve about who is the greatest apostle. They may be an elite group, but all of them are insecure and competitive.\n\nOne day after a particularly heated quarrel, the apostles approach Jesus meekly and ask, \"Who is the greatest in the kingdom of heaven?\"\n\nThe question sounds innocent, but Jesus knows what motivates it and asks, \"What were you arguing about on the way?\"\n\nThe men say nothing and stare at the ground. Jesus sits down and asks his apprentices to gather around him. \"If anyone wishes to be first,\" Jesus tells them, \"he must be last of all and servant of all.\"\n\nJesus calls a young child to come to him. Embracing the child, he addresses his followers. \"I tell you truly, unless you turn and become like children, you will never get into the kingdom of heaven. Anyone who will take the lowly position of this child is the greatest in the kingdom of heaven.\n\n\"Whoever receives a little child like this in my name receives me, and whoever receives me, receives not just me but the One who sent me. For whoever is the least among you is the greatest.\"\n\nThe apostles are concerned about their place in the heavenly pecking order. Jesus wants them to strive for greatness, but he warns them that God's definition of greatness is different from theirs\u2014in God's economy, the ones awarded the highest positions are those who serve others and give no thought to rank and privilege.\n\nJesus's recommendation to seek the status of a child is hard for the twelve to receive. A child has no status\u2014all he is and all he has comes from his parents. Jesus says life in the kingdom is the same way.\n\nThe apostles hear little of Jesus's instruction; their thoughts are focused on how they can gain stature in the church. \"Teacher,\" John says, \"we saw someone casting out demons in your name, and we tried to stop him because he's not following us.\"\n\nJesus shakes his head and says, \"Do not stop him, for no one who does a miracle in my name will be able soon afterward to speak evil of me. Whoever is not against us is for us. For I tell you truly, whoever gives you a cup of water to drink because you belong to Christ will by no means lose his reward.\n\n\"But whoever causes one of these little ones who believe in me to stumble, it would be better for him to have a heavy millstone hung around his neck and to be drowned in the depths of the sea. Woe to the world because of stumbling blocks, for stumbling blocks are bound to come, but woe to the one through whom the stumbling block comes!\"\n\nJesus then admonishes those who allow their behavior to trip them up. \"If your hand causes you to stumble,\" Jesus says, \"cut it off and throw it away. It is better for you to enter life crippled than with two hands to go to Gehenna, to the unquenchable fire. And if your foot causes you to stumble, cut it off. It is better for you to enter life lame than with two feet to be thrown into Gehenna. And if your eye causes you to stumble, tear it out. It is better for you to enter the kingdom of God one-eyed than to have two eyes and be thrown into Gehenna, 'where their worm does not die, and the fire is not quenched.'\"\n\nThe word pictures are as memorable as they are gruesome. Jesus quotes Isaiah's prophecy about Gehenna, the final abode of people who rebel against God.\n\nGehenna (a compound word meaning valley of Hinnom) was originally the southern border of the city of Jerusalem. The prophet Jeremiah identified it as a pagan holy place where children were sacrificed to the god Molech. King Josiah ended that by desecrating the valley and turning it into a refuse dump. It became the destination for things that had no value. Human excrement, animal carcasses, and other rubbish were dumped there. The fires in the valley constantly burned, and over the years the site became symbolic of the place of divine punishment. \"Gehenna\" became a place of endless torment with no hope of relief. Gehenna represents the eternal hell of all who refuse to accept God's invitation to life.\n\nJesus solemnly warns the twelve to stop trivializing and misinterpreting what it means to be an apprentice. It's about obedience and servanthood, not status and privilege.\n\n\"For everyone,\" Jesus tells them, \"will be salted with fire.\"\n\nThe apostles are not sure what Jesus is referring to. Salt and fire are widely used metaphors, but the only place they are used together is in sacrificial worship\u2014salt is a symbol of the covenant and is applied to every sacrifice before it is consumed by the fire.\n\nThe prophet Malachi had said the Messiah would be \"like a refiner's fire\" and would \"purify the sons of Levi and refine them like gold and silver.\" John the Baptist, drawing on Malachi's words, had prophesied that the Messiah would baptize with the Holy Spirit and with fire.\n\nIn the Scriptures, fire often refers to trials that God uses to make His people fit for the kingdom. Believers, however, are far from passive victims in the process. \"Salt is good,\" Jesus tells his apprentices, \"but if the salt has lost its saltiness, how can you make it salty again? Have salt in yourselves, and be at peace with one another.\"\n\nSalt means different things to different people. To the homemaker it is a preservative; to the cook it is a flavor enhancer; to those before the altar it speaks of purity; to the rabbis, salt symbolizes wisdom. But there is no need for the disciples to choose between these options\u2014all are relevant to the work Jesus is calling them to.\n\nJesus's admonition to be at peace with one another reminds the apostles how the conversation began\u2014they were squabbling about status. \"See to it that you do not despise one of these little ones,\" he tells the future leaders of his church. \"For I tell you that their angels in heaven continually see the face of my Father who is in heaven.\"\n\nThe message is clear\u2014the least matter much to God, and He is always watching over them.\n\n\"What do you think?\" Jesus asks. \"If someone has a hundred sheep and one of them wanders away, won't he leave the ninety-nine on the hillsides and go in search of the one that went astray? And if he manages to find it, I tell you truly that he rejoices over it more than over the ninety-nine that never went astray. In the same way, it is not the will of your Father in heaven that one of these little ones should perish.\"\n\nThe parable is carefully chosen. Jesus knows the day will come when his church will have to resolve her conflicts without his physical presence, and the parable of the one and ninety-nine is taught to prepare his people to help one another when that day arrives. Everyone falls out of step with God at times, and there must be a path for reconciliation.\n\nJesus teaches his disciples four progressive steps to take when a fellow church member wanders away from God.\n\n\"If your brother sins,\" Jesus says, \"go and tell him his fault, between you and him alone. If he listens to you, you have gained your brother. But if he does not listen, take one or two others along with you, so that 'every charge may be established by the evidence of two or three witnesses.' If he refuses to listen to them, tell it to the church. And if he refuses to listen even to the church, let him be to you as a Gentile and a tax collector.\"\n\nJesus's teaching echoes the command from the Torah to \"reason frankly with your neighbor so you will not share in his guilt.\" If one disciple sees another disciple wandering from the truth, the first course of action is to, in a spirit of gentleness, personally and privately speak with the wanderer. If the one-on-one conversation results in repentance and reconciliation, that is to be the end of it.\n\nBut if the person refuses to listen, Jesus says to follow God's instruction in Deuteronomy\u2014a second meeting takes place in the presence of objective witnesses. Perhaps the additional counselors can help both parties become clear about God's will. And, again, if there is repentance, the matter is to be closed.\n\nBut if there is still hardness of heart, a third step is to be taken\u2014the entire church is informed. Hopefully a multitude of witnesses will persuade the wandering soul to return, but if there is still no repentance, one last drastic step is to be taken\u2014the community of faith is to dissociate herself from the person. The one who refuses to turn back to God is to be treated like \"a Gentile and a tax collector.\" In the Torah, this was called being \"cut off\" from the assembly.\n\nIn Christ's community of faith, believers are to teach, encourage, and admonish one another, but this is impossible when a person refuses to listen to Jesus. He is to be excluded from the church.\n\nBut even this final step is not about retribution\u2014it speaks to rehabilitation and reconciliation. No animosity is to be shown toward the one set outside; he is to be loved and prayed for. The hope of the church is that the Holy Spirit will use the severance to convict the wanderer's heart and lead him to repent and return to the fold.\n\nThe church, however, is to take an active role in the process. \"I tell you truly,\" Jesus says, \"whatever you bind on earth shall have been bound in heaven, and whatever you loose on earth shall have been loosed in heaven.\"\n\nThe apostles have heard these words before; Jesus had used them in Caesarea Phillipi when he appointed Simon Peter to be the chief steward of the kingdom. Simon still stands as the first among equals, but now Jesus expands the mandate to the rest of the disciples. The keys to the kingdom will be entrusted to the church.\n\nJesus then identifies the source of his people's power. \"Moreover,\" Jesus says, \"I tell you that if two of you on earth agree about anything you ask for, it will be done for you by my Father in heaven. For where two or three are gathered in my name, I am there in the midst of them.\"\n\nIn Judaism, a minimum of ten adult males is required for corporate worship to be valid. Jesus says that in his church, two men or women in agreement is a quorum when they gather under his authority.\n\nThe lesson has been about corporate forgiveness, and the disciples understand the process, but they are unclear about personal offenses. Many rabbis, citing the prophet Amos, say that God forgives three times but no more, and His people should do the same.\n\nSimon Peter asks, \"Lord, how many times will my brother sin against me, and I still forgive him? As many as seven times?\"\n\nPeter wants to frame the question in a way that sounds gracious, so he doubles the number in Amos and adds one more for good measure. Seven is the number of completeness. Peter feels he is being more than generous and is shocked when Jesus replies, \"I tell you, not just seven times but as many as seventy-seven times.\"\n\nJesus draws on a story from Genesis and contrasts the limitless vengeance threatened by Lamech with the limitless forgiveness God offers. Jesus wants his followers to understand that forgiveness isn't about keeping a tally of wrongs done\u2014forgiveness is an attitude that reflects the heart of God.\n\n\"The kingdom of heaven,\" Jesus says, \"may be compared to a king who wanted to settle accounts with his servants. When he began to settle up, one was brought to him who owed him ten thousand talents. The man could not pay the debt, so his master ordered him to be sold, together with his wife and children and all that he had, so that payment could be made.\n\n\"The servant fell at his feet, imploring him, 'Be patient with me, and I will repay you everything.' And out of pity for him, the master of that servant released him and forgave him the debt.\n\n\"But when that servant went out, he found one of his fellow servants who owed him a hundred denarii. Grabbing him by the throat, he began to choke him, saying, 'Pay me back what you owe me!'\n\n\"His fellow servant fell down and pleaded with him, 'Be patient with me, and I will repay you.' But he refused and threw him in prison until he could repay the debt.\n\n\"His fellow servants saw what had taken place. They were horrified and reported to their master what they had seen. Then his master summoned him and said to him, 'You wicked servant! All that debt I forgave you because you pleaded with me. Wasn't it your duty to show mercy on your fellow servant, just as I showed mercy on you?' And his master was furious and delivered him to the torturers until he could pay all his debt.\"\n\nJesus pauses to be sure his apprentices have understood the story, and then adds, \"My heavenly Father will treat every one of you the same way unless you forgive your brother from your heart.\"\n\nThe parable is shocking. In an honor-and-shame-based society such as Israel, revenge is not only acceptable, it is lauded. An eye for an eye is justice. Jesus doesn't disagree but suggests there is an alternative: mercy.\n\nThe monetary amounts mentioned in the parable tell the tale. The second debtor owed a hundred denarii\u2014a substantial amount given the average day laborer was paid one denarii. At that rate, it would take about four months to repay the debt. The first debtor, on the other hand, owed ten thousand talents. One talent was the equivalent of six thousand denarii; to repay ten thousand talents would require two hundred thousand years\u2014far too great a debt to ever hope to repay\u2014and this is Jesus's point. God has forgiven each of his children an incalculable debt, and they are obliged to pass that mercy on.\n\nThe story reminds the disciples of what Jesus had taught about forgiveness in the sermon on the mountainside. Jesus had looked each of them in the eye and said, \"If you forgive others for their offenses, your heavenly Father will forgive you as well, but if you do not forgive others, neither will your Father forgive your offenses.\"\n\nForgiveness, Jesus teaches, is reciprocal. When a person experiences God's forgiveness, his eyes are opened, and he can see the immeasurable debt he has been released from. His repentance will bring perspective. He will see how minor the sins committed against him are when compared to the sins he has committed against God. His nature will change\u2014the forgiven will become forgiving.\n\nBut those who do not forgive demonstrate they have not changed. Despite God's offer of full forgiveness, they refuse to turn to Him. Instead, like the man in the parable, they demand what is owed them. And that is exactly what they will receive.\n\nWHEN AUTUMN ARRIVES, the nation's attention turns to Sukkoth, the Feast of Booths, the country's most popular festival. One day when Jesus is visiting his family, his brothers encourage him to attend the celebration. \"Leave here and go to Judea,\" they tell him, \"so that your disciples there may also see the works you are doing. For no one does anything in secret if he seeks to be well known. If you're doing these things, show yourself to the world!\"\n\nHis brothers' proposal is reasonable; the huge festival crowd could translate into thousands of followers, but Jesus is not interested in pandering to the multitudes. \"My time has not yet come,\" he tells his brothers, \"but your time is always here. The world cannot hate you, but it hates me, because I testify about it that its works are evil. You go up to the feast. I am not going up to this feast because my time has not yet fully come.\"\n\nJesus tells his brothers that anytime is right for them to travel to the festival; they are causing no trouble, and no one is seeking to harm them. Jesus, on the other hand, has been marked for death. He has intentionally missed this year's entire festival season so far. He will not go to Jerusalem until God instructs him to do so.\n\nThose instructions come just as the celebration is getting started. The Holy Spirit prompts Jesus to go to Sukkoth, but Jerusalem is a three-day walk, and he will not arrive until the festivities are well underway.\n\nThe shortest route to the capital is the ridge road through Samaria. Jesus sends a couple of disciples ahead to a Samaritan village to arrange for lodging, but when the villagers learn they are on their way to Jerusalem, they refuse them hospitality.\n\nWhen the two return with the news, James and John are furious. \"Lord!\" they shout. \"Do you want us to call fire down from heaven to consume them?\" The two brothers are inspired by Elijah, who had called fire down on two companies of Samaritan soldiers not far from this very village. Was there not biblical precedence to destroy these Christ rejecters?\n\nJesus shakes his head and rebukes the aptly nicknamed Sons of Thunder. The villagers weren't rejecting the Messiah\u2014they were rejecting a group of Jews on their way to a religious festival in Jerusalem. Jesus reminds them that there are other villages they can stay in.\n\nAs they continue walking, they are joined by a teacher of the Law who tells Jesus, \"I will follow you wherever you go.\"\n\nFew religious scholars have recognized Jesus's true identity, and he would gladly have the scribe as a disciple, but he knows the man's understanding of what it means to follow him is shallow.\n\n\"The foxes have holes and birds of the air have roosts,\" he tells the lawyer, \"but the Son of Man has nowhere to lay his head.\"\n\nThe man is being encouraged to count the cost\u2014following Jesus will lead to a cross.\n\nJesus continues walking south. Along the way, he meets a man and says, \"Follow me.\"\n\nThe offer is intriguing, but the man is not willing to risk the condemnation of his family and community. \"Lord,\" he replies, \"first let me go and bury my father.\"\n\nThe man's father is not dead or dying. Otherwise, the man would be home attending to his family. His request is motivated by the traditional duty of a son to live at home to care for his mother and father until their death.\n\nThe man assumes that Jesus will not tell his disciples to violate the expectations of family and friends, but this is exactly what Jesus demands. \"Leave the dead to bury their own dead,\" he tells the man, \"but as for you, go and proclaim the kingdom of God.\"\n\nAs Jesus continues walking toward Jerusalem, he has a third encounter with a man interested in becoming a disciple. \"I will follow you, Lord,\" the man promises, \"but first let me take leave of those at my home.\"\n\nThe man expresses a desire to follow Jesus, but his request is insincere. The man is not asking to be allowed time to go home and say good-bye to his family. He's asking if he may \"take leave\" of his parents. To take leave is not to inform them of his decision but to ask for permission to leave.\n\nJesus knows the man is more interested in pleasing his parents than in following him and says, \"No one who puts his hand to the plow and looks back is fit for the kingdom of God.\"\n\nJesus chooses the image of a plowman to stress the primacy of discipleship. A Palestinian plow is very light, and the plowman must perform several operations at the same time. With his left hand, he keeps the tool upright and lifts it over rocks while applying whatever pressure is necessary to make the appropriate furrow depth. In his right hand is a six-foot goad tipped with an iron spike to keep the oxen in line. Then, with both hands full, he must constantly look ahead to keep the furrow straight. It's a job that demands total concentration. Jesus says apprenticeship to him is like that.\n\nJesus wants those who are interested in following him to know that apprenticeship is not a task to be added to other interests. Discipleship will not only disrupt life's normal rhythms, it will redefine them.\n17\n\nLIGHT OF THE WORLD\n\nOctober, AD 32\n\nIn him was life, and that life was the light of all people. The light shines in the darkness, and the darkness has not mastered it.\n\n\u2014John 1:4\u20135\n\nISRAEL'S YEARLY RELIGIOUS calendar revolves around a cycle of sevens. In the Torah, God commands seven festivals to be held in the first seven months, and each is to be an expression of the nation's most sacred day: the Sabbath. Three of the feasts are considered so important that attendance is obligatory for all Jewish men, yet, as this year's festival season draws to a close with the Feast of Sukkoth, Jesus has attended none of them.\n\nSukkoth, the Feast of Booths, is held six months after Passover and is the nation's most popular festival. Like the other feasts, it combines thanksgiving for God's provision with a retelling of sacred history. Sukkoth was prescribed by God to remind His people of the lessons they learned in the wilderness and to recall the time when God dwelt in a temporary shelter alongside them.\n\nIn the Torah it is written, \"All native Israelites shall dwell in booths so that your descendants may know that I made the people of Israel dwell in booths when I brought them out of the land of Egypt\u2014I am the Lord your God.\"\n\nIn obedience to God's Word, dwellings made of green boughs are constructed throughout the land. Temporary shelters surround the city walls and sit in courtyards and atop many roofs, and for eight days the people of Israel live outdoors in the leafy booths and recite the lessons learned during forty years in the wilderness.\n\nMuch time is spent recalling the teachings of Moses, but equal time is spent celebrating God's goodness. It's autumn, and the second harvest is in. Grapes are becoming new wine; olives have been pressed to yield their oil; and larders are filled with ripe figs, dates, and pomegranates.\n\nIt is a time of abundance, and normally the people are in high spirits, but this year a pall lies over the feast. Jesus is being talked about all over town, and opinions vary. Some say he is a good man, and others are convinced he is a deceiver, but all the discussions take place privately because no one wants to be identified as a follower of Jesus. They have been warned of the consequences.\n\nWhen Jesus misses the first days of the festival, the Jewish leaders assume he will not attend, and his arrival in the middle of the week is unexpected. They find Jesus on the temple mount preaching to a large crowd. Those in the audience are surprised by his grasp of the Scriptures and his ability to apply them. Several ask, \"How does this man have such learning when he has never been taught?\"\n\nJesus overhears the question. \"My teaching is not my own,\" he tells the crowd. \"It comes from the One who sent me.\" Like other rabbis, Jesus claims his authority comes from another, but unlike other rabbis, Jesus says his authority comes directly from God.\n\n\"Anyone who chooses to do the will of God will know whether my teaching is from God or whether I am speaking on my own authority.\"\n\nAccording to Jesus, recognizing truth doesn't require theological training. Truth is self-authenticating\u2014but only when the one evaluating it is committed to doing God's will. Knowing the truth requires trusting God enough to act on what He says. Then one will be able to discern whether Jesus's words are from God.\n\nBut this goes to the heart of his opponents' problem\u2014they do not seek to know God's will. They are more interested in impressing one another.\n\n\"The one who speaks on his own authority seeks his own glory,\" Jesus tells them, \"but the one who seeks the glory of the One who sent him is true, and in him there is no unrighteousness.\"\n\nThe Jews are proud of their heritage and how God entrusted them with His Law, but Jesus wants them to realize that there is a big difference between receiving the Law and living it. \"Hasn't Moses given you the Law?\" he asks the crowd. \"Yet none of you keeps the Law. Why are you looking for an opportunity to kill me?\"\n\nThe indictment shocks his audience. One shouts, \"You have a demon!\"\n\nAnother asks, \"Who is trying to kill you?\"\n\nSome in the crowd are pilgrims from faraway lands, but most live locally and know exactly what the teacher is talking about. When Jesus was in Jerusalem last year, he incurred the wrath of the Sanhedrin by healing a man on the Sabbath and then claiming God's authority to do so. The desire of the Sanhedrin to silence Jesus is a secret to no one in town.\n\nJesus knows they are still upset about last year's Sabbath healing. He looks into the angry faces surrounding him and says, \"I did one work, and you are all amazed. This is why Moses gave you circumcision\u2014not that it's from Moses, but from the patriarchs\u2014and you circumcise a man on the Sabbath. If a man can receive circumcision on the Sabbath so that the law of Moses may not be broken, are you angry with me because on the Sabbath I made a man's whole body well? Stop judging by mere appearances\u2014judge with right judgment.\"\n\nMany in the crowd nod in agreement; it is commonly accepted that some of God's commandments override others. Jesus uses a classic example\u2014circumcision. The rite preceded Mosaic Law and was never negated by it. According to the rabbis, the rite makes the boy being circumcised \"complete.\" Thus circumcision has always taken precedence over Sabbath labor restrictions. Jesus's argument is simple\u2014if circumcision, which \"completes\" one body part, takes precedence over Sabbath law, then how much more so an act that brings wholeness to the entire body?\n\nThe crowd begins to wonder. How could this be the man the religious leaders want to silence? What he says makes perfect sense. The people whisper questions to one another. Several who live in Jerusalem ask, \"Isn't this the man they're trying to kill? And here he is, speaking publicly, and they say nothing to him!\"\n\nHas the Sanhedrin had a change of heart? Some ask, \"Can it be the authorities really know that this is the Messiah?\"\n\nOthers are skeptical and point out, \"But we know where this man comes from, and when the Messiah appears, no one will know where he comes from.\"\n\nThe rabbis taught that while the birthplace of the Messiah had been revealed by Micah, the rest of his identity would be shrouded in mystery until he revealed himself. Most in the crowd are convinced there is nothing mysterious about the carpenter from Nazareth.\n\n\"You know me, and you know where I am from?\" Jesus asks. \"I have not come of my own accord, but the One who sent me is true\u2014the One you do not know. I know Him, for I am from Him, and He sent me.\"\n\nSome in the crowd are ready to make a citizen's arrest. Jesus's words are not only insulting, they are blasphemous. But others in the audience hear Jesus's confession differently and say, \"The Messiah, whenever he comes\u2014will he do more signs than this man has done?\"\n\nThe Pharisees become alarmed when they hear people voicing faith in Jesus and rush to the nearby Hall of Hewn Stones. They report the mood of the audience to the Sanhedrin, and the temple guard is immediately dispatched to arrest Jesus. Within a few minutes the policemen are in the crowd and pushing their way to the front to arrest the teacher, when they hear Jesus say, \"I will be with you a little longer, and then I go to the One who sent me. You will seek me but you will not find me, and where I am you cannot come.\"\n\nJesus's words halt the soldiers, and they become part of the questioning crowd. \"Where does this man intend to go that we will not find him?\" they ask.\n\nOne in the throng scoffs, \"Surely he isn't planning to go to the Diaspora and teach the Greeks?\"\n\nAnother is confused and asks, \"What does he mean by saying, 'You will seek me but you will not find me, and where I am you cannot come'?\"\n\nJesus, aware of the Sanhedrin's plot to arrest him, does not stay to answer questions. While the crowd is debating the meaning of his message, Jesus quietly walks away and is not seen at the temple until the last day of the feast.\n\nMEANWHILE, THE DAILY rituals of the holy week constantly remind the celebrants of how God took care of them for forty years in the wilderness. Two rites in particular capture the heart of the festival.\n\nThe most important is the water rite, designed to recall how God saved Israel by bringing water from the rock. It has been a sacred part of the feast for more than two hundred years. Every morning of the festival week, the crowds participate in an elaborate ritual. A holy procession walks south from the temple, singing and making music as they travel the quarter mile to the Pool of Siloam. One of the priests fills a golden pitcher with living water from the Gihon Spring, which feeds the pool. Thousands of pilgrims line the parade route and shout praises as the procession returns to the temple. The water is brought to the altar, and is poured out in conjunction with the morning drink offering while a Levitical choir sings the Hallel (Psalms 113\u2013118). When the choir reaches Psalm 118, the pilgrims join in the celebration by holding up their etrogs (citron fruit) and waving their lulavs (fans made from branches of willow, myrtle, and palm) and shouting, \"Give thanks to the Lord!\"\n\nOn the seventh day of the feast, the entire ritual is repeated seven times. The altar is saturated with water and prayers for rain in the coming year. More importantly, there are prayers for an outpouring of the Holy Spirit. Seven times the people sing the passage in Psalm 118 that affirms, \"The stone that the builders rejected has become the cornerstone.\" Seven times they profess, \"Blessed is he who comes in the name of the Lord!\" When the closing verse is sung, the temple walls reverberate with the declaration, \"Give thanks to the Lord, for He is good\u2014His love endures forever!\"\n\nAs the final words of the ceremony are spoken, the people in the temple plaza lift their lulavs and shout praises for several minutes.\n\nAs their cries of thanksgiving die, another voice is heard. Jesus has reappeared at the temple. \"If anyone thirsts,\" he cries out, \"let him come to me and drink. Whoever believes in me, as the Scripture says, 'Rivers of living water will flow from within him.'\"\n\nThe apostles have heard this offer before; Jesus gave a woman in Samaria a similar invitation.\n\nSome who hear Jesus's words are impressed. Several in the crowd turn to their neighbor and say, \"This really is the Prophet.\"\n\nOthers say, \"This is the Messiah!\"\n\nBut some shake their heads and say, \"Is the Messiah to come from Galilee? Doesn't the Scripture say that the Messiah will be a descendant of David and will be born in Bethlehem, the village where David lived?\"\n\nWhen the Levites return to the Sanhedrin empty-handed, the council demands an explanation. \"Why didn't you bring him?\"\n\nThe guards have but one defense. \"No one ever spoke like this man!\"\n\nAnnoyed with the excuse and angry that Jesus is not under arrest, the leaders shout, \"Have you also been deceived? Have any of the authorities or the Pharisees believed in him? But this crowd that does not know the Law\u2014they are accursed!\"\n\nThe Levites know better than to respond, but one of the Sanhedrin's own brings up a procedural issue. Nicodemus, one of the council's most illustrious teachers, says, \"Surely our law doesn't condemn a man without first giving him a hearing and learning what he is doing?\"\n\nThe question infuriates the council. Technically, Nicodemus is right, but now is not the time for subtleties of the law\u2014the foundations of the faith are being challenged. \"Are you from Galilee as well?\" they snarl. \"Search the Scriptures, and you will see that no prophet arises from Galilee.\"\n\nNicodemus knows as well as they do that important prophets have come from Galilee, but correcting them would serve no good purpose, so he falls silent. His admonition, however, slows their deliberations, and they are unable to agree on a plan before they are forced to suspend the meeting in order to prepare for the closing ceremonies of Sukkoth.\n\nThe sun sets, and the city grows dark for the first time in six nights. The evenings during the festival were devoted to a ritual that offered another reminder from the wilderness\u2014the pillar of fire that ultimately led Israel to the Promised Land.\n\nThe rite takes place in the Court of Women. In the courtyard are four pillars. Each rises seventy five feet (fifty cubits), and during the feast they become lamp stands. Each pillar is crowned with four golden bowls that are kept filled with olive oil. Beginning the second evening of the feast, these massive menorahs are lit nightly and are bright enough to illuminate every courtyard in town. Under the glow of the lamps, God's people dance and sing hymns late into the evening.\n\nOne of the most prominent features of the nightly ceremony takes place on the fifteen broad steps that go down into the Court of Women. A Levite choir and orchestra are assembled on the top step. As they descend each step, they sing one of the fifteen Psalms of Ascent (120\u2013134), singing the final psalm in the courtyard surrounded by worshipers.\n\nFor the past six evenings the candelabra have been lit, and the celebration has gone far into the night, but tonight the menorahs stand unattended. Sunset has ushered in the eighth day of the feast.\n\nAs night falls, Jesus is preaching at the treasury adjoining the now-dark Court of Women. \"I am the light of the world,\" he proclaims. \"Whoever follows me will not walk in darkness but will have the light of life.\"\n\nSeveral Pharisees hear Jesus's claim. They ignore the invitation and focus on a technicality. \"You are testifying on your own behalf,\" they challenge. \"Your testimony is not valid.\"\n\nThe Pharisees see Jesus's claim as self-testimony. But because they are only interested in discrediting him, they misunderstand what he is saying and make the same allegations they made last year when he healed a man on the Sabbath. Back then Jesus had provided several witnesses to corroborate his claim, but his opponents had refused to hear them.\n\nKnowing their hearts, Jesus answers, \"Even if I testify on my own behalf, my testimony is valid because I know where I came from and where I'm going, but you don't know where I came from or where I'm going. You judge according to human standards\u2014I judge no one. Yet even if I do judge, my judgment is valid, for it is not I alone who judge, but I and the Father who sent me. And even in your Law it is written that the testimony of two people is valid. I am one witness, and the One who sent me is the other\u2014the Father.\"\n\nThe religious leaders confer. They had asked for corroborating testimony, and Jesus is claiming an invisible father as his second witness. \"Where is your father?\" they demand.\n\nThe Pharisees want to cross-examine the alleged witness, but Jesus tells them, \"You know neither me nor my Father. If you knew me, you would also know my Father.\"\n\nJesus realizes that no amount of discussion will convince the religionists, so he repeats something he said earlier, but this time with a much sharper point. \"I am going away, and you will search for me,\" he tells them, \"and you will die in your sin. Where I am going, you cannot come.\"\n\nThe verdict is harsh, but his audience is not listening. One smirks and says, \"Will he kill himself, since he says, 'Where I am going, you cannot come'?\"\n\nJesus looks into their cold flat eyes and says, \"You are from below; I am from above. You are of this world; I am not of this world. I told you that you would die in your sins, for unless you believe that I am, you will die in your sins.\"\n\nJesus explains the cause of the misunderstanding\u2014they are from two different worlds. The Pharisees are from a world where humans have the final word and are the ultimate authority. Jesus is from a kingdom that is submitted and obedient to God, and he admonishes them to turn around or face eternal damnation.\n\nBut the Pharisees hear none of that. What they hear is Jesus saying, \"Unless you believe that I am . . . .\" but not finishing his statement. \"I am\" what? The teacher makes no sense.\n\n\"Who are you?\" they ask.\n\nJesus smiles and says, \"Just what I have been telling you from the beginning.\"\n\nHe is offering the Pharisees a simple answer to their question\u2014his words reveal his identity. \"I have much to say and to judge concerning you,\" Jesus continues. \"But the One who sent me is true, and what I have heard from Him I declare to the world.\"\n\nFor some in the audience, there is the beginning of comprehension, but most are hopelessly confused. Jesus knows it cannot be otherwise and says, \"When you have lifted up the Son of Man, then you will know that I am, and that I do nothing on my own authority, but I speak just as the Father taught me. And the One who sent me is with me. He has not left me alone, for I always do the things that are pleasing to Him.\"\n\nAs Jesus says these things, many in the crowd nod in agreement and profess faith in him. Jesus knows there are many kinds of faith, and he says to those who profess belief, \"If you dwell in my word, you are truly my disciples, and you will know the truth, and the truth will set you free.\"\n\nMany in the audience are deeply offended at Jesus's implication. \"We're descendants of Abraham and have never been enslaved to anyone!\" they exclaim. \"How can you say, 'You will be set free'?\"\n\nJesus knows the crowd's faith is fickle and says, \"Truly, truly, I tell you, everyone who practices sin is a slave to sin. The slave doesn't remain in the house forever. The son remains forever. So if the son sets you free, you will be free indeed. I know that you are descendants of Abraham\u2014yet you seek to kill me because my word has no place in you. I speak of what I have seen with my Father, and you, too, do what you have heard from your father.\"\n\nThe people protest, \"Abraham is our father.\"\n\nJesus shakes his head and says, \"If you were children of Abraham, you would be doing the works Abraham did, but now you seek to kill me, a man who has told you the truth that I heard from God. This is not what Abraham did. You are doing the works of your father.\"\n\n\"We were not born of fornication!\" they shout. \"We have one Father\u2014God!\"\n\n\"If God were your Father,\" Jesus says, \"you would love me, because I came from God. I did not come on my own\u2014He sent me. Why do you not understand my message? It is because you are not able to hear my word. You are children of your father, the Devil, and you want to carry out your father's desires. He was a murderer from the beginning and does not stand in the truth because there is no truth in him. When he lies, he speaks his native language, for he is a liar and the father of lies. But because I speak the truth, you do not believe me. Who among you can prove me guilty of sin? If I'm telling the truth, why don't you believe me? Whoever is of God hears the words of God. You don't hear them because you are not of God.\"\n\nJesus claims the crowd's unbelief stems not from their lack of comprehension but from their spiritual paternity. Like their father, they are unwilling to trust and obey God. That would require submission, and they despise anyone telling them what to do.\n\n\"Did we not rightly say that you are a Samaritan and demon-possessed?\" they scoff.\n\nJesus replies softly, \"I do not have a demon, but I honor my Father, and you dishonor me. Yet I do not seek my own glory\u2014there is One who seeks and judges. Truly, truly, I tell you, whoever keeps my word will never see death.\"\n\nThis pronouncement infuriates the crowd even more. Surely these are the words of a deceiver or a lunatic. \"Now we know that you are demon-possessed!\" they exclaim. \"Abraham died, as did the prophets, yet you say, 'If anyone keeps my word, he will never taste death.' Are you greater than our father Abraham? He died, and so did the prophets. Who do you think you are?\"\n\nJesus responds to their angry accusations by gently explaining, \"If I glorify myself, my glory is nothing. My Father, whom you claim as your God, is the One who glorifies me. You do not know Him, but I know Him. If I were to say that I do not know Him, I would be a liar like you, but I do know Him, and I keep His word. Your father Abraham looked forward to the time when he would see my day. He saw it and was glad.\"\n\nThis comment brings guffaws from the Pharisees. \"You're not yet fifty years old, and you've seen Abraham?\"\n\nThe suggestion is preposterous. Who does this fellow think he is?\n\nJesus smiles and says, \"Truly, truly, I tell you, before Abraham was, I Am.\"\n\nThere is a moment of silence as the listeners process what Jesus has claimed, then everyone is shouting at once. The prophet is declaring the Divine Name is his own.\n\nThe Jews have no legal authority to execute lawbreakers, but the mob is so infuriated they are willing to risk Roman reprisal. There are construction projects on the temple mount, and vigilantes start gathering stones to kill the false prophet, but when they look up, Jesus is gone.\n\nA full report is made to the Sanhedrin, and even those who sympathize with Jesus wonder what caused him to make such outrageous statements.\n18\n\nPRIDE AND PREJUDICE\n\nOctober\u2013December, AD 32\n\n\"Whoever listens to you, listens to me, and whoever rejects you, rejects me, and whoever rejects me, rejects the One who sent me.\"\n\n\u2014Luke 10:16\n\nTHE FEAST OF Sukkoth is over, and the pilgrims have returned to their homes, but Jesus remains in Jerusalem. He wants to provide the nation's capital every opportunity to hear the gospel, and he knows that despite threats from the Sanhedrin to excommunicate anyone sympathetic to his message, there are souls in Jerusalem ready to turn to God.\n\nOne day as he and the apostles walk down the street, they pass a blind beggar. It is the Sabbath, and the twelve are reflecting upon how God works in the world. One gestures toward the beggar and callously asks, \"Rabbi, who sinned\u2014this man or his parents\u2014that he was born blind?\"\n\nThe apostles believe that sin and suffering are always integrally linked. The rabbis teach there is no death without sin (for God said, \"The soul who sins will die\"), and there is no suffering without guilt (for God said, \"If they violate My statutes and do not keep My commandments, I will punish their transgression\"). The apostles have been taught since childhood that certain maladies and deformities are divine punishment for particular sins. In their minds, there is no question about the reason for the man's blindness. They are simply curious about the details.\n\nJesus can only shake his head at the question. His apprentices sound like Job's comforters. Yes, the Scriptures teach infirmities are sometimes the result of personal sin, and certainly there are children born with disorders caused by the destructive lifestyle of a parent, but this does not mean all afflictions are punishments for sin. There is another possibility.\n\n\"Neither this man nor his parents sinned,\" Jesus says, \"but this happened so that the works of God might be displayed in him. We must work the works of the One who sent me while it is day. Night is coming when no one can work. As long as I am in the world, I am the light of the world.\"\n\nThe disciples ponder Jesus's words as they watch him squat close to the ground, spit in the dirt, and stir it with his finger to make mud. He takes a small amount and smears it on the man's eyelids. \"Go\u2014wash in the pool of Siloam,\" Jesus instructs him.\n\nThe blind man does not understand, but he obeys. He slowly makes his way down the hill to the pool and rinses the mud from his eyes. He blinks, then squints in the bright afternoon sun\u2014the man born blind can now see.\n\nWhen the healed man returns home, his neighbors are shocked. \"Isn't this the man who used to sit and beg?\" one asks.\n\nSome nod and say, \"Yes, it's him.\"\n\nOthers can't believe it's the same man. \"No,\" they say, \"it's not him, but he looks like him.\"\n\n\"I am the man!\" the beggar insists.\n\nAn unconvinced neighbor asks, \"Then how were your eyes opened?\"\n\nThe one born blind replies, \"The man called Jesus made mud and put it on my eyes and told me, 'Go to Siloam and wash.' So I went and washed and received my sight.\"\n\nThis sounds impossible. There must be more to the story. \"Where is he?\" the neighbors ask.\n\nThe man can only shrug and say, \"I don't know.\"\n\nHis neighbors are at a loss, but one suggests that since this is obviously a supernatural incident, perhaps the religious experts can explain what has happened. They bring the beggar to the synagogue, and the healed man recounts how he received his sight as simply as he can. \"He put mud on my eyes, and I washed, and I see.\"\n\nThe explanation draws raised eyebrows and an immediate response from the Pharisees who are judging his case. \"This man is not from God\u2014he doesn't keep the Sabbath.\" The accusation stems from Jesus's violation of the traditions of the elders by \"working\" on the Sabbath to make the mud and apply it to the man's eyes.\n\nThe Pharisees' hatred will not allow them to believe that the man is telling the truth. In the entire history of the world there has never been a person with the ability to heal congenital blindness. Further, the ability to cure blindness is to be one of the signature abilities of the coming Messiah, which prompts some who are listening to ask, \"How can a sinner perform such miraculous signs?\"\n\nAn argument breaks out, but when it becomes evident that neither side is gaining ground, the Pharisees turn again to the beggar. \"What do you say about him?\" they ask. \"It was your eyes he opened.\"\n\nThe restored one immediately answers, \"He's a prophet!\"\n\nThe Pharisees scoff at the notion. They have heard enough to realize they will not get reliable testimony, but at least they can expose the beggar as a liar. Since he claims he was congenitally blind, they need only question his parents.\n\nHis mother and father are located and brought before the judges. \"Is this your son, who you say was born blind? How is it that he now sees?\"\n\nThe healed man's parents are overjoyed that their son can see, but they also fear the Pharisees. \"We know that he's our son and that he was born blind,\" they admit, \"but we don't know how he can see or who opened his eyes. Ask him\u2014he's old enough to speak for himself.\"\n\nThe Pharisees shake their heads. More useless testimony. They again summon the man who claims he was healed and admonish him to recant. \"Give glory to God!\" they demand. \"We know this man is a sinner.\"\n\nBut the one who was made well refuses to condemn Jesus. \"Whether he is a sinner or not, I don't know. One thing I do know\u2014I was blind, but now I see.\"\n\nHis inquisitors' eyes narrow. \"What did he do to you?\" they ask. \"How did he open your eyes?\" They seek specific violations they can use to arrest the prophet.\n\nBut the restored man has heard enough. He has given his testimony. Emboldened by his increasing comprehension of what has happened, he wryly suggests, \"I've told you already, and you didn't listen. Why do you want to hear it again? You don't want to become his disciples, too, do you?\"\n\nThe Pharisees turn crimson; one shakes his finger in the beggar's face and rails, \"You are this man's disciple\u2014we are disciples of Moses. We know that God spoke to Moses, but as for this fellow, we don't even know where he comes from!\"\n\nThe formally blind man shakes his head in mock wonder. \"Now that's amazing!\" he says. \"You don't know where he comes from, and yet he opened my eyes. We know that God does not listen to sinners, but He does listen to those who worship Him and do His will. No one has ever heard of opening the eyes of a man born blind. If this man were not from God, he could do nothing.\"\n\nThe enlightened beggar's answer is not theologically airtight, but close enough to send the Pharisees into a rage. \"You were born steeped in sin!\" they curse. \"And you are teaching us?\"\n\nThe religious leaders have heard enough. The man is immediately thrown out of the synagogue and told not to return.\n\nNews of the excommunication travels quickly. Jesus hears what happened and goes in search of the man. When he finds him, he asks, \"Do you believe in the Son of Man?\"\n\nThe reference is cryptic, and the man does not understand. \"And who is he, sir? Tell me, so that I may believe in him.\"\n\nJesus looks into eyes once blind and says, \"You have seen him\u2014he is the one who is speaking to you.\"\n\nWhen the man hears this, he falls to his knees and says, \"Lord, I believe.\"\n\n\"For judgment I came into this world,\" Jesus tells him, \"so that the blind may see and those who see may become blind.\"\n\nSome Pharisees overhear Jesus's words. Convinced he could not possibly be talking about them, they ask, \"Are we also blind?\"\n\n\"If you were blind,\" Jesus replies, \"you would have no sin. But since you say, 'We see,' your sin remains.\n\n\"Truly, truly, I tell you, he who does not enter the sheepfold through the door but climbs in by another way\u2014that one is a thief and a robber. But the one who enters through the door is the shepherd of the sheep. To him the doorkeeper opens, and the sheep hear his voice, and he calls his own sheep by name and leads them out. When he has brought out all his own, he goes ahead of them, and the sheep follow him because they know his voice. But a stranger they will not follow\u2014they will flee from him because they do not recognize the voice of strangers.\"\n\nSheep and shepherds are well known to those listening. Thieves, wolves, and lions roam the countryside, and protective enclosures like sheepfolds are essential. There are different kinds of sheep pens; Jesus describes the more permanent kind often found in villages. Open to the sky and made of stone, its walls rise six feet or more and are covered with thorns to further dissuade predators. A heavy wooden door bars the sheepfold's single entrance. Access is controlled by a gatekeeper who is on duty throughout the night.\n\nCommunities can bring hundreds of sheep into the fold without concern about the flocks becoming mixed because every sheep knows the distinctive call of its shepherd and will follow him alone. When the shepherd is ready to move, he simply calls to his own and begins walking. His flock will always follow.\n\nThe audience is familiar with shepherds and sheep, but they can't make sense of the word picture. They do not understand that Jesus is responding to the ban on Sabbath healing. Jesus is alluding to Ezekiel's pronouncements when he was commanded to \"prophesy against the shepherds of Israel, prophesy and say to them, 'Thus says the Lord God: woe to the shepherds of Israel who only take care of themselves! Should not shepherds take care of the flock?'\"\n\nJesus wants the nation's current leaders to see their spiritual kinship to those who ruled in Ezekiel's day. God's condemnation of those shepherds was searing. \"The weak you have not strengthened, the sick you have not healed, the broken you have not bound up, the scattered you have not brought back, the lost you have not sought, but with force and harshness you have dominated them.\"\n\nThe religionists can't see themselves as spiritual descendants of those evil shepherds, so Jesus expands the metaphor. \"Truly, truly, I tell you, I am the door for the sheep. All who came before me are thieves and robbers, but the sheep did not listen to them. I am the door. Whoever enters through me will be saved and will go in and out and find pasture.\"\n\nJesus is still using the shepherding image, but by referring to himself as the door, he indicates he has shifted the imagery from the village sheepfolds to the makeshift folds erected in the wilderness. When a shepherd is forced to keep his sheep in open country, the pen may be as simple as a circle of thorny brambles piled high. The entrance is left open, and the shepherd himself becomes the door by sleeping in the portal.\n\nJesus again picks up the thread in Ezekiel. \"The thief comes only to steal and kill and destroy,\" he warns. \"I came that they may have life, and have it abundantly. I am the good shepherd. The good shepherd lays down his life for the sheep.\"\n\nJesus says that all the messianic pretenders who came before him had false motives. \"The hired hand, who is not the shepherd and doesn't own the sheep, sees the wolf coming and abandons the sheep and flees, and the wolf attacks the flock and scatters the sheep. The man runs away because he is a hired hand and cares nothing for the sheep. I am the good shepherd. I know my own, and my own know me, just as the Father knows me and I know the Father, and I lay down my life for the sheep.\"\n\nSome in the audience nod in agreement, but many have faces of stone. Jesus wants Israel's leaders to see that God's mission goes far beyond the petty traditions they busy themselves with. \"I have other sheep that are not of this fold,\" he tells them. \"I must bring them also, and they will listen to my voice, and they will become one flock with one shepherd.\"\n\nThe prophet Jeremiah spoke of a day when the Gentiles would \"no longer stubbornly follow their own evil heart\" but would turn to the Lord. And in the scrolls of Isaiah, God declared that the Servant of the Lord would not only restore Israel, he would be \"a light of the nations so that My salvation may reach to the end of the earth.\" God promised that one day \"My house shall be called a house of prayer for all peoples.\"\n\nJesus is announcing the arrival of that day.\n\nThe broadening of God's mission to encompass the Gentiles will bring profound changes to God's ancient people, and Jesus knows many in Israel will oppose those changes to the death, but he will not shrink from his mission.\n\n\"The Father loves me because I lay down my life so that I might take it up again. No one takes it from me, but I lay it down voluntarily. I have authority to lay it down, and I have authority to take it up again. This charge I have received from my Father.\"\n\nTo some, the claims are absurd. \"He has a demon and is insane,\" they conclude. \"Why listen to him?\"\n\nBut others are not so sure. \"These are not the words of one who is possessed by a demon,\" they say. \"Can a demon open the eyes of the blind?\"\n\nOpinions are divided, but Jesus knows what the ultimate verdict will be, and he knows it will not be long in coming.\n\nWINTER IS APPROACHING, and Jesus wants to bring his message to the people of Judea one more time before the rains disrupt travel. He chooses seventy-two apprentices from his followers and divides them into thirty-six pairs.\n\n\"The harvest is plentiful,\" he tells them, \"but the workers are few. Pray, therefore, to the Lord of the harvest to send out workers into His harvest. Go! Behold, I send you out as lambs in the midst of wolves. Carry neither moneybag nor knapsack nor sandals, and greet no one on the road. Whatever house you enter, first say, 'Peace to this house!' And if a child of peace is there, your peace will rest on that person, but if not, it will return to you. And remain in the same house, eating and drinking what they provide, for the worker deserves to be paid. Don't move around from house to house, and whenever you enter a town and they receive you, eat what is set before you. Heal the sick who are there and tell them, 'The kingdom of God is come upon you.'\"\n\nThe image Jesus paints of being lambs surrounded by wolves is disturbing, but he tells his apprentices they need not fear; they need only pray. Prayer, Jesus assures them, is their most powerful resource. By staying in constant communication with the Father, they will be taken care of every step of the way. There is no need to encumber themselves with supplies, or even money\u2014their heavenly Father will meet their needs.\n\nThey are told that when they enter a town, they are to seek a person of peace\u2014someone who will receive them as God's emissaries. The disciples are to stay with this person and minister to the village, healing the sick and crushing all demonic opposition.\n\nJesus knows, however, that some communities will be hostile to his message. \"But whenever you enter a town and they do not receive you,\" Jesus tells the seventy-two, \"go into its streets and say, 'Even the dust of your town that clings to our feet we wipe off in protest against you. Nevertheless know this\u2014the kingdom of God has come.'\"\n\nThere will be no neutral territory on this mission. Jesus's message will be either received or rejected. When it is spurned, the disciples are to perform a prophetic act of judgment. They are to announce the arrival of the kingdom and symbolically declare the village unclean by wiping the town's dust from their feet.\n\nThey are then to move on to the next town and allow God to deal with those who reject His message. \"I tell you,\" Jesus proclaims, \"on that day it will be better for Sodom than for that town.\n\n\"Woe to you, Chorazin! Woe to you, Bethsaida! For if the mighty works done in you had been done in Tyre and Sidon, they would have repented long ago, sitting in sackcloth and ashes. It will be better at the judgment for Tyre and Sidon than for you. And you, Capernaum, will you be exalted to heaven? No, you will be brought down into Hades.\"\n\nThe disciples are surprised at the harsh assessment of the towns they have spent so much time in, but Jesus wants them to see beneath the veneer of piety. Ultimately it all comes down to one thing. \"Whoever listens to you,\" Jesus says, \"listens to me, and whoever rejects you, rejects me, and whoever rejects me, rejects the One who sent me.\"\n\nHis instructions complete, Jesus blesses the seventy-two and sends them forth.\n\nWhen they return a few weeks later, they bring news of the mission's success. \"Lord,\" they exclaim, \"even the demons submit to us in your name!\"\n\nJesus rejoices with his apprentices, and as they celebrate he tells them, \"I saw Satan fall from heaven like a flash of lightning. Behold, I have given you authority to trample on serpents and scorpions and over all the power of the enemy\u2014nothing will hurt you. But don't rejoice that the spirits submit to you; rejoice that your names are written in heaven.\"\n\nJesus lifts his eyes and says, \"I thank You, Father, Lord of heaven and earth, that You have hidden these things from the wise and the learned and have revealed them to little children. Yes, Father, it pleased You to do it this way.\"\n\nJesus's prayer is contrary to the teachings of the rabbis. Conventional wisdom says only the educated elite can know the secrets of God, but Jesus says the elite can't understand God's truth\u2014only little kids can.\n\n\"All things have been handed over to me by my Father,\" Jesus announces. \"No one truly knows the Son except the Father, and no one truly knows the Father except the Son and those to whom the Son chooses to reveal Him.\"\n\nA murmur spreads through the crowd. How can the teacher claim that he alone knows God, and that he alone gets to decide whom to reveal God to?\n\nJesus offers no evidence to support his claim. Instead, he turns to his disciples and says, \"Blessed are the eyes that see what you see, for I tell you that many prophets and kings desired to see the things you see, and did not see them, and to hear the things you hear, and did not hear them.\"\n\nThe disciples are interested in learning more about this secret knowledge, but Jesus is interrupted by a religious scholar who stands and says, \"Teacher, what must I do to inherit eternal life?\"\n\nThe question sounds harmless, but Jesus knows it is a test. There is a question behind this question, and the man is there to challenge him. Jesus deflects the test by sending the scribe to the Scriptures. \"What is written in the Law?\" Jesus asks. \"How do you read it?\"\n\nThe scholar responds by quoting two key passages from the Torah. \"'You shall love the Lord your God with all your heart and with all your soul and with all your strength and with all your mind,' and 'love your neighbor as yourself.'\"\n\n\"You have answered well,\" Jesus says. \"Do this, and you will live.\"\n\nThe scribe has answered his own query with two excellent texts, but the passages describe one who has devoted body, mind, and soul to loving his neighbor. This troubles the religionist\u2014he knows he is unwilling to love those who are not like him. Seeking to justify his narrow criteria for who qualifies as a neighbor in the text he cited, the scholar asks Jesus, \"And who is my neighbor?\"\n\nJesus knows the lawyer's real question and answers with a parable. \"A man was going down from Jerusalem to Jericho, and he fell into the hands of robbers who stripped him and beat him and went away, leaving him half dead. Now, a priest happened to be going down the same road, and when he saw the man, he passed by on the other side. Likewise a Levite, when he came to the place and saw him, passed by on the other side. But a Samaritan who was traveling that way came upon the man, and when he saw him, he had compassion and went to him. He poured oil and wine on the man's wounds and bandaged him. Then he put him on his own animal and brought him to an inn and took care of him. The next day he took out two denarii and gave them to the innkeeper, saying, 'Take care of him, and when I return, I'll reimburse you for any extra expense you may have.'\"\n\nJesus pauses to let the story to sink in, then asks, \"Which of these do you think became a neighbor to the man who fell into the hands of robbers?\"\n\nJesus's question places the emphasis on what makes someone a neighbor. And as the parable ultimately reveals, it is not race or social standing or even living in the same neighborhood that makes someone a neighbor. A person becomes a neighbor by compassionately caring for those God brings into his life.\n\nThe scribe grudgingly recognizes this, but is loathe to admit there is such a thing as a good Samaritan. He can only bring himself to say, \"The one who showed him mercy.\"\n\nJesus nods and says, \"You go and do likewise.\"\n\nThe lawyer is left to ponder his opportunity. Jesus leaves to visit friends in Bethany, a two-mile walk from Jerusalem. Within an hour, he and a few of his disciples arrive in the village and are met by their friends Martha and Mary.\n\nJesus and his companions are welcomed into their home and provided pillows to rest upon while a meal is prepared. The men recline on the cushions, and Jesus begins teaching. Martha goes to the kitchen, but Mary places herself at Jesus's feet. The disciples are surprised\u2014women never do this\u2014but since Jesus says nothing, they say nothing.\n\nIn the kitchen, Martha labors alone and fumes. Her sister is completely inconsiderate. Jesus surely sees this, yet he allows Mary to sit idly while her responsible sibling does all the work. Soon Martha is out of the kitchen and striding across the room. She glares at Mary, then complains to Jesus. \"Lord, do you not care that my sister has left me to serve alone? Tell her to help me.\"\n\nJesus does care, but Mary is not his current concern. He smiles and says, \"Martha, Martha, you are worried and troubled by many things\u2014there is need of only one thing. Mary has chosen the better part, and it will not be taken away from her.\"\n\nMartha drops her eyes and swallows hard, but she knows Jesus is right. More importantly, she knows he loves her and only wants God's best for her. She returns to the kitchen and soon a wonderful meal is enjoyed by all. It is late by the time the visitors stand to offer their thanks for the gracious hospitality. Good-byes are exchanged, and the company departs for Jerusalem.\n\nWINTER AND THE month of Kislev arrive. Jesus plans to remain in the capital until Hanukkah, which begins on the twenty-fifth of the month. He spends many days teaching in and around Jerusalem and many hours in prayer. One day, when Jesus is returning from an extended time of prayer, one of the disciples says, \"Lord, teach us to pray, as John taught his disciples.\"\n\n\"When you pray,\" Jesus tells them, \"say 'Father, hallowed be Your name. Your kingdom come. Give us each day our daily bread and forgive us our sins, for we forgive everyone who is indebted to us. And let us not be brought into temptation.'\"\n\nThe disciples have heard a version of this prayer before. It is the kind of prayer that reflects the heart of one who has been forgiven and indelibly changed.\n\nMeditating on the prayer will help his apprentices understand what they should pray for, but Jesus also wants them to understand how to pray, and he tells them a parable. \"Can you imagine one of you has a friend, and you go to him at midnight and say, 'Friend, lend me three loaves; a friend of mine has arrived on a journey, and I have nothing to set before him,' and the one inside answers, 'Don't bother me! The door is already locked, and my children are with me in bed, and I can't get up and give you anything'?\"\n\nThe question is long and twisted, but Jesus's humor is not lost on the disciples. He is asking them to imagine an impossible scenario\u2014a neighbor refusing to help provide hospitality when an unexpected guest arrives in town. Hospitality is so deeply ingrained in the culture and the fabric of village life that something like this would simply never happen.\n\nJesus says, \"I tell you, even if he will not get up and give the neighbor anything because he is his friend, yet because of the neighbor's bold shamelessness he will get up and give him whatever he needs. So I tell you\u2014ask and it will be given to you, seek and you will find, knock and the door will be opened to you. For everyone who asks receives, and the one who seeks finds, and to the one who knocks, the door will be opened.\"\n\nNow the point of the parable becomes plain\u2014if the petitioner can go boldly and without shame to request help from a neighbor who may not even like him, how much more boldly can he petition the God who loves him? The key, according to Jesus, is to ask, seek, and knock\u2014pray, pursue God, and be persistent.\n\n\"Which father among you whose child asks for a fish will give a snake instead of a fish? Or if the child asks for an egg will give a scorpion? If you, then, who are evil, know how to give good gifts to your children, how much more will your heavenly Father give the Holy Spirit to those who ask Him!\"\n\nJesus's point is simple: loving parents would never give their child something harmful, and if this is true of fallible parents, how much more so of the heavenly Father. And if earthly parents give \"good gifts,\" how does one describe the gift the Father will give?\n\nThe analogy makes sense to the apostles, but they do not understand Jesus's reference to the gift of the Holy Spirit. The men are intrigued by the possibilities, but no one asks Jesus to explain, and they will not fully understand this gift until Pentecost.\n19\n\nTHE GATHERING STORM\n\nDecember, AD 32\n\n\"Do you think I have come to bring peace to the earth?\"\n\n\u2014Luke 12:51\n\nHANUKKAH APPROACHES, AND Jerusalem prepares for an influx of pilgrims. The winter rains slow preparations but do little to dampen people's spirits. Homes all over town are being readied for the popular eight-day celebration. But while families prepare festive candles to brighten the feast, the most powerful men in the nation sit in dark halls and plot the death of Jesus.\n\nHis recent comments likening Israel's religious leaders to the evil shepherds in Ezekiel's day cannot be allowed to stand, but the council must proceed cautiously\u2014Jesus is popular with the rabble. They decide their best approach is to isolate Jesus from the crowds. Not coincidentally, Jesus begins getting invitations to private banquets hosted by the city's elite.\n\nJesus knows why the doors of Jerusalem's leading citizens are opening to him, and he welcomes the opportunity to discuss the issues that separate them. At a lavish luncheon in the home of a prominent Pharisee, Jesus exchanges greetings with the host, then casually walks past the ceremonial washbasins to an open place at the foot of the table and reclines to eat.\n\nThere is a murmur around the table. While ritual hand washing before meals is not commanded in Scripture, it is an important tradition and a common practice. The guests stir restlessly and are unsure how to respond to the slight.\n\nJesus looks to the head of the table and says, \"Now, you Pharisees clean the outside of the cup and dish, but inside you are full of extortion and wickedness. Fools! Didn't the One who made the outside make the inside also? So give for alms those things that are within, and behold, everything will be clean for you. But woe to you Pharisees, for you tithe mint and rue and herbs of all kinds, but you bypass justice and the love of God. These were the things you ought to have done, without neglecting the others.\"\n\nThe Pharisees are surprised at the bold direction the conversation is taking. Their concern was a violation of the hand-washing code, but Jesus's mention of dishes expands the scope of the discussion to include all ritual cleansing.\n\nHis complaint is simple: the Pharisees ascribe importance to trivial matters while ignoring what is essential. Jesus says that if they will simply pay attention to building integrity in their hearts and demonstrating the Father's forgiving loving-kindness to others, they will become cleaner than any ritual can make them.\n\nJesus looks into the obdurate faces surrounding him for any hint of repentance. Seeing none, he continues. \"Woe to you Pharisees, for you love the seat of honor in the synagogues and to be greeted with respect in the marketplaces. Woe to you, for you are like unmarked graves that people walk over without knowing it.\"\n\nThe graveyard analogy infuriates those around the table. Walking on a grave, even unknowingly, makes a person ceremonially unclean. According to Jesus, not only are the Pharisees themselves unclean, they are contaminating those they teach.\n\nSome at the table are unwilling to receive the rebuke. A religious scholar says, \"Teacher, when you say these things you insult us also.\"\n\nThe lawyer assumes that the charges leveled against the Pharisees do not apply to scribes, but in Jesus's eyes there is little difference between the two.\n\n\"Woe to you lawyers also!\" Jesus declares. \"For you weigh the people down with burdens they can scarcely carry, and you will not lift a finger to help them carry the load. Woe to you, for you build the tombs of the prophets whom your fathers killed. Thus, you are witnesses, and you agree with the deeds of your fathers: they did the killing; you do the building.\n\n\"Because of this, the Wisdom of God said, 'I will send them prophets and apostles. Some they will kill and others they will persecute.' As a result, this generation will be held responsible for the blood of all the prophets shed from the foundation of the world\u2014from the blood of Abel to the blood of Zechariah, who perished between the altar and the sanctuary. Yes, I tell you, it will be charged against this generation.\"\n\nThe allegation is clear. The scribes, like the Pharisees, preach trivia. Further, the spirit that motivated the ancient Israelites to reject and kill God's prophets is alive and well in the hearts of those gathered around the table. They build elaborate monuments to the prophets, but they ignore their teachings. Worse, they block others from hearing the message.\n\n\"Woe to you lawyers!\" Jesus says. \"For you have taken away the key of knowledge\u2014you did not enter yourselves, and you hindered those who were entering.\"\n\nThe wide-ranging condemnation of the religious system creates chaos at the table. Jesus stands and walks out the door, but several scribes and Pharisees follow him into the street, peppering him with malicious questions. They receive no response.\n\nJESUS CONTINUES TO preach daily to large crowds on the temple mount. One day as he is teaching, a man in the audience calls out to him, \"Teacher, tell my brother to divide the family inheritance with me.\"\n\nThe request is not unusual; rabbis are often called upon to resolve such disputes because inheritance laws are in the Scriptures, and rabbis are tasked with interpreting and applying Scripture. But the man's question betrays his true intent. He is not seeking justice; he simply wants the teacher to take his side against his brother.\n\nJesus shakes his head and says, \"Man, who made me a judge or arbitrator over you?\"\n\n\"Watch out!\" Jesus warns the crowd. \"Be on your guard against all forms of greed, for a man's life does not consist in the abundance of his possessions.\"\n\nJesus shares a parable to help them remember the warning. \"The land of a certain rich man was very productive, and he thought to himself, 'What shall I do? I have no place to store my crops.' Then he said, 'This is what I'll do\u2014I'll tear down my barns and build larger ones, and there I will store all my grain and my goods. And I'll say to my soul, \"Soul, you have stored up enough good things to last you for many years. Relax! Eat, drink, be merry.\"' But God said to him, 'Fool! This night your soul is required of you. And the things you've prepared, whose will they be?'\"\n\nJesus pauses a few moments to let the story seep in, and says, \"This is how it will be for anyone who accumulates treasure for himself and is not rich toward God.\"\n\nJesus is not condemning wealth; he's making an observation about how easily it can become a god. \"Be on guard. Keep awake,\" he warns. \"Let your loins be girded and keep your lamps burning.\"\n\nJesus knows how easy it is to be lulled into a false sense of security and urges his apprentices to be watchful\u2014they are to have their loins girded and their lamps burning. Both images picture readiness. When men wanted to be ready to move quickly, they would gird their loins by lifting their long garments and binding them around their hips.\n\nGod had told His people during the first exodus to gird their loins and be ready to move quickly. Jesus repeats the admonition for those who will follow him in the second.\n\n\"Be like servants who were put in charge,\" Jesus says. \"Each with their work and watching for their master to return home from the wedding banquet, so that when he comes and knocks, they can immediately open the door for him. Blessed are those servants whom the master finds keeping watch when he comes. I tell you truly, he will gird himself for service and have them recline at the table, and he will come and serve them. Therefore be alert, for you do not know when the master of the house will come\u2014in the evening or at midnight or when the rooster crows or in the morning\u2014lest he come suddenly and find you asleep.\n\n\"And what I say to you, I say to everyone. Watch! But know this\u2014if the master of the house had known what hour the thief was coming, he would have stayed awake and not let his house be broken into. Therefore, you also must be ready, for the Son of Man is coming at an unexpected hour.\"\n\nThe message to followers of Jesus is simple: be ready at all times for anything. There is no way of knowing or predicting the future. The disciples understand the basic message, but they are unclear about whom Jesus is warning. Peter asks, \"Lord, are you telling this parable to us or to everyone?\"\n\nJesus smiles and says, \"Who then is the faithful and wise steward, whom the master will put in charge of his household, to give them their food allowance at the proper time? Blessed is that servant if his master finds him doing this when he returns! I tell you truly, the master will put him in charge of all he owns. But if that servant says in his heart, 'My master is gone for a long time,' and he begins to beat the menservants and maids, and to eat and drink and get drunk, the master of that wicked servant will come on a day when he does not expect him and at an hour he does not know, and will cut him in pieces and assign him a place with the unfaithful. And the servant who knows his master's will but doesn't get ready or carry out his master's wishes will be beaten with many blows. But the one who does not know and does things that deserve punishment will be beaten with few blows. From everyone to whom much has been given, much will be required, and from the one to whom much has been entrusted, even more will be demanded.\"\n\nThe answer to Simon Peter's question is clear: the parable is for everyone. All will be held accountable for the gifts and abilities given them. But Jesus wants his disciples to understand that this judgment does not take place only in the future; it is happening even now.\n\n\"I came to cast fire on the earth,\" Jesus tells them, \"and how I wish it were already kindled! I have a baptism to be baptized with, and how great is my distress until it is completed! Do you think I have come to bring peace to the earth? No, I tell you, but rather division. For from now on in one house there will be five divided, three against two and two against three. Father will be divided against son and son against father, mother against daughter and daughter against mother, mother-in-law against her daughter-in-law and daughter-in-law against mother-in-law.\"\n\nBorrowing and strengthening the words of the prophet Micah, Jesus emphasizes that the judgment has already begun\u2014it begins the moment a decision is made about Jesus.\n\n\"When you see a cloud rising in the west,\" Jesus observes, \"you immediately say, 'It's going to rain,' and it does. And when you see the south wind blowing, you say, 'There will be scorching heat,' and there is. Hypocrites! You know how to interpret the appearance of earth and sky\u2014why do you not know how to interpret the present time?\n\n\"And why do you not judge for yourselves what is right? As you go with your accuser before the magistrate, make every effort on the way to settle with him, lest he drag you to the judge, and the judge hands you over to the officer, and the officer throws you in prison. I tell you, you won't get out until you have paid the very last lepton.\"\n\nThe disciples have heard Jesus use this image before, but then he was speaking about relationships with others. Now he speaks about a relationship with God. The \"debts\" in the parable are sins. Jesus's counsel is simple: seek reconciliation with the One you have sinned against as soon as possible, while you are \"on the way\" to Judgment Day. Otherwise, the entire debt must be worked off, and not even eternity can afford that much time.\n\nThe talk about judgment makes the crowd uncomfortable. They want to feel better about their standing with God, so when someone mentions a group of Galileans who were recently executed by Pilate, others in the audience join in. They assume Jesus will be sympathetic with the plight of the worshippers who were unjustly killed, but Jesus redirects the conversation back to their own culpability.\n\n\"Do you think that these Galileans were worse sinners than all the other Galileans because they suffered these things?\" Jesus asks. \"No, I tell you, but unless you repent, you will all likewise perish. Or those eighteen who were killed when the tower in Siloam fell and killed them\u2014do you think that they were worse offenders than all others living in Jerusalem? No, I tell you, but unless you repent, you will all likewise perish.\"\n\nThe crowd begins grumbling. They have been taught to take comfort in the fact that bad things only happen to bad people.\n\nJesus, mindful of the nation's indifference to God's mercy, offers a warning veiled in a parable. \"A man had a fig tree planted in his vineyard,\" Jesus says, \"and he came looking for fruit on it and found none. So he said to the vinedresser, 'Look, for three years now I've come looking for fruit on this fig tree, and still I find none. Cut it down. Why waste the soil?' But the vinedresser replied, 'Master, leave it alone for one more year until I dig around it and put fertilizer on it. Then if it bears fruit next year, well and good; but if not, you can cut it down.'\"\n\nThe parable offers hope, but insists Israel's destiny is in her hands. The Owner of the tree is patient\u2014He will withhold judgment for another season, but not forever.\n\nOne Sabbath Jesus is preaching in the synagogue, and a woman who suffers from a severe back condition is present. Her affliction has fused her spine into a perpetual arch, and for eighteen years her only view of the world has been from knee height. Jesus invites her to come to him. All eyes follow the woman as she laboriously shuffles to the front of the synagogue.\n\nWhen she reaches Jesus, he lays his hands on her back and says, \"Woman, you are freed from your infirmity.\" As he speaks, the woman feels her spine strengthen, and for the first time in nearly two decades she stands eye-to-eye with her neighbors.\n\nShe begins babbling praises to God, but an angry voice silences her. \"There are six days on which work should be done!\" the voice booms. \"Come on those days and be healed\u2014not on the Sabbath day!\"\n\nThe invective comes from the ruler of the synagogue. He is convinced Jesus has violated the Sabbath, and his rebuke is a public challenge. By affirming the traditions of the elders, he asserts his authority to interpret the Scriptures and condemns the healer's lawlessness.\n\nThe traditions are widely respected, and several in the congregation nod in agreement, but Jesus shakes his head and says, \"Hypocrites! Does not each of you on the Sabbath untie his ox or take his donkey from the manger and lead it out to give it water? And ought not this woman\u2014a daughter of Abraham whom Satan bound for eighteen years\u2014be set free from this bondage on the Sabbath day?\"\n\nThe congregation erupts in praise. The red-faced synagogue ruler can only stare at the ground.\n\nA FEW DAYS later, Hanukkah begins. It is not one of the feasts decreed in the Torah but a festival commemorating the rededication of the temple in the days of the Maccabees. One hundred and fifty years ago the Jews ousted the Syrian King, Antiochus Epiphanes, after he defiled the temple by sacrificing pigs on the altar and erecting a statue of Zeus in the Holy of Holies. Each year on the twenty-fifth day of Kislev, the eight-day festival commemorates Israel's short-lived religious and national independence.\n\nBecause Hanukkah does not demand a trip to the temple and the feast can be observed in homes, Jerusalem is only a fraction of the size it had been a few weeks earlier when the temporary shelters of Sukkoth blanketed the hills. Nevertheless, thousands are in town to celebrate what is affectionately known as the Festival of Lights.\n\nOn one of the feast days, Jesus is teaching in Solomon's Colonnade, a covered porch that stretches two hundred yards along the eastern side of the temple, so named because it is believed the foundation of the first temple lies beneath it.\n\nJesus's message hasn't changed in three years, and his opponents are becoming increasingly frustrated with his unwillingness to identify himself as the Christ. \"How long will you keep us in suspense?\" they grumble. \"If you're the Messiah, tell us plainly.\"\n\nJesus shrugs and says, \"I did tell you, and you don't believe. The works I do in my Father's name speak for me, but you don't believe because you are not my sheep. My sheep hear my voice, and I know them, and they follow me. I give them eternal life, and they will never perish, and no one will wrest them out of my hand. My Father, who has given them to me, is greater than all, and no one can wrest them out of the Father's hand. I and the Father are one.\"\n\nWhen the audience hears Jesus claim to be one with God, there is a loud murmur, and some of the men stoop to the ground for rocks.\n\n\"I have shown you many good works from the Father,\" Jesus shouts above the din. \"For which of these works are you going to stone me?\"\n\nA man holding a large rock responds angrily, \"It's not for a good work that we're going to stone you, but for blasphemy, because you, a mere man, make yourself out to be God.\"\n\nJesus nods and says, \"Is it not written in your Law, 'I said, \"you are gods\"?' If He called them gods to whom the Word of God came\u2014and Scripture cannot be annulled\u2014what about the one whom the Father set apart and sent into the world? Why do you accuse me of blasphemy because I said, 'I am God's son'? If I'm not doing the works of my Father, then don't believe me. But if I do them, believe the works even if you don't believe me, so that you may know and understand that the Father is in me, and I am in the Father.\"\n\nThe crowd knows the Torah identifies Israel as \"God's firstborn,\" and they have all read how Moses \"became a god\" to Pharaoh.\n\nJesus cites Psalm 82, which calls humans \"gods.\" The passage is understood by many rabbis to identify Israel at the time the Law was given, and the Jews are comfortable being called God's children, yet they are ready to stone Jesus for claiming to be God's son.\n\nJesus determines it is time to leave Judea. He walks east from Jerusalem on the Jericho road. When he comes to the Jordan, he turns north and follows the river to Batanea, the isolated region where he began his ministry three years earlier. His plan is to use the remaining few months before Passover to prepare his disciples for his death, but despite the remote location, thousands find him.\n\nJesus teaches daily in the towns and along the river. He encourages people to put their faith in God rather than rely on their Jewish heritage. One day someone asks, \"Lord, will those who are saved be few?\"\n\nThe teacher smiles; the one who asks the question has been listening. \"Strive to enter through the narrow door,\" Jesus says, \"for many, I tell you, will try to enter and will not be able to. Once the master of the house has gotten up and shut the door, you will stand outside and knock on the door, saying, 'Lord, open to us.' And he will answer you, 'I do not know where you come from.' Then you will say, 'We ate and drank with you, and you taught in our streets.' But he will reply, 'I tell you, I do not know where you come from. Depart from me, all you workers of unrighteousness!' There will be weeping and gnashing of teeth when you see Abraham and Isaac and Jacob and all the prophets in the kingdom of God, but you will be thrown out. And people will come from east and west and from north and south and recline at the table in the kingdom of God. And behold, the last shall become first, and the first shall become last.\"\n\nJesus never answers the question about how many will be saved. Instead, he explains how salvation works. He likens it to a door that opens only from the inside. Once God opens the way to salvation, anyone can go in, yet the opening is narrow and requires one to turn in order to enter. But Jesus says the invitation will one day expire, and the narrow door will become a closed door. Today is the day to turn and enter.\n\nThe crowd is not surprised to hear the plea for an immediate response; prophets have long stressed urgency. What startles them is Jesus's revelation that Gentiles will have prominent places in the kingdom while many children of Abraham will be refused admittance.\n\nThe notion that heathens will become God's chosen people is incomprehensible to the audience, yet Jesus is teaching directly from the Scriptures. Isaiah spoke of an eschatological banquet in Jerusalem that would be held not just for the Jews but \"for all peoples.\"\n\nThe Isaiah passage is familiar to Jesus's audience, but their deeply rooted racism has blinded them. Some rabbis even teach that the \"invitation\" Gentiles will receive to the banquet will be an invitation to be judged by the Messiah. Other rabbis are more lenient, allowing that Gentiles may attend if they become Jewish proselytes, but Jesus has distorted the teaching beyond recognition. There is no way that God would invite the children of Cain to the feast but leave Abraham's children locked outside.\n\nAs the crowd grapples with Jesus's pronouncement, a group of Pharisees approaches him and says, \"Get away from here\u2014Herod wants to kill you.\"\n\nJesus nods and says, \"Go tell that fox, 'Behold, I cast out demons and perform cures today and tomorrow, and on the third day I complete my task. In any case, today, tomorrow, and the next day I must be on my way\u2014it is impossible for a prophet to be killed outside of Jerusalem.'\"\n20\n\nKINGDOM IN THE MIDST\n\nDecember, AD 32\u2013February, AD 33\n\n\"Behold, the kingdom of God is in the midst of you.\"\n\n\u2014Luke 17:21\n\nTHE RAINY SEASON is waning. Snow still covers the higher elevations, but the Jordan Valley where Jesus is wintering is subtropical and experiencing torrential downpours. Between storms Jesus and his disciples travel between the villages along the river. There is little public ministry; most of their time is devoted to training.\n\nJesus continues to receive requests to attend banquets hosted by religious leaders hoping to discredit him. One Sabbath he accepts an invitation to lunch at the home of an influential Pharisee, and as soon as he enters he knows a trap has been laid.\n\nThe trap is in the form of a guest who was invited under false pretenses. The man's bloated limbs make it obvious that he is afflicted with dropsy. Jesus knows the man is out of place at this luncheon because it is widely believed that this condition is a punishment for sin. Yet, here he sits in the home of a Pharisee, awkwardly trying to fit in.\n\nA large number of scribes and Pharisees are reclined around the table enjoying casual conversation and ignoring the dropsical man.\n\nJesus asks, \"Is it lawful to heal on the Sabbath or not?\"\n\nEveryone at the table has a ready answer, but all remain silent. Jesus knows why they won't respond and refuses to subject the man with edema to their evil games. He heals the man and sends him home.\n\nJesus then turns to his tablemates and asks, \"If one of you has a son or an ox that falls into a well on a Sabbath day, will you not rush to pull him out?\"\n\nAgain the Pharisees do not respond, but this time for a different reason\u2014they have no way of answering without incriminating themselves. But Jesus isn't interested in winning a debate; he wants them to see how their desire for status is blocking their entrance into the kingdom of God.\n\n\"When you're invited by someone to a wedding banquet,\" he tells his fellow guests, \"don't sit in a place of honor, for someone more distinguished than you may have been invited by your host. Then the host, who invited both of you, may come to you and say, 'Give this person your place.' Then in humiliation you will head to the lowest place. Instead, when you're invited, go and sit in the lowest place, so that when your host comes he may say to you, 'Friend, move up higher.' Then you will be honored in the presence of all who sit at the table with you. For everyone who exalts himself will be humbled, and the one who humbles himself will be exalted.\"\n\nJesus's counsel is based on the familiar but often ignored wisdom of Solomon. In a culture that values personal honor above all else, being forced to move to a lower seat with everyone watching is an excruciating experience, but Jesus wants them to understand that there is no need to seek the higher places\u2014their identity is not defined by where they sit at the table.\n\nHe then turns to the host with similar counsel: \"When you give a luncheon or dinner, don't invite your friends or your brothers or your relatives or your rich neighbors, lest they also invite you in return and you are repaid. But when you give a feast, invite the poor, the crippled, the lame, and the blind\u2014you will be blessed because they cannot repay you, for you will be repaid at the resurrection of the righteous.\"\n\nBy the time Jesus is finished, his tablemates are shaking their heads in disbelief. If a system like this were to be implemented, the very fabric of society would unravel. Like the rest of the Roman Empire, Jewish culture operates according to the quid pro quo ethics of reciprocity. Gifts are never given freely; unwritten laws dictate that when someone gives something to another or helps them in any way, the recipient is obligated to return the favor. If there cannot be reciprocity in kind, which is often the case when the gift is from a wealthy person to one who is impoverished, the recipient is morally bound to be loyal and of service to his patron. This web of obligation weaves the empire together. Without the hierarchical order the patronage system brings, society could not function.\n\nJesus disagrees. He views the gift-and-obligation system as harmful, and he is provided an opportunity to explain why when one of the guests responds to his remark about the resurrection of the righteous.\n\n\"Blessed is the one who will eat bread in the kingdom of God!\" a man exclaims.\n\nThe declaration is made in the hope of drawing a comment from the teacher about the highly anticipated messianic banquet. Jesus accommodates the request with a parable.\n\n\"A man prepared a great banquet and invited many guests,\" Jesus says. \"When the time for the banquet came, he sent his servant to tell those who had been invited, 'Come, for everything is now ready.'\n\n\"But they all began to make excuses. The first said to him, 'I've bought a piece of cropland, and I must go see it. Please accept my regrets.' Another said, 'I've bought five yoke of oxen, and I'm on my way to try them out. Please accept my regrets.' Another said, 'I've married, so I can't come.'\n\n\"The servant returned and reported these things to his master. The master of the house became angry and told his servant, 'Go at once into the streets and lanes of the city and bring in the poor and crippled and blind and lame.'\n\n\"After the servant had done this, he said, 'Master, what you commanded has been done, and there is still room.'\n\n\"So the master told the servant, 'Go out into the highways and hedges and urge people to come in so that my house may be filled.'\"\n\nThe parable makes it plain that there is a broad conspiracy to insult the host of the banquet. All the guests had earlier confirmed they would attend, but on the day of the feast, when the table is overflowing with food, and the host sends word to his guests that the celebration is beginning, one by one they refuse to come. Three refusals are alluded to, but these are only representative. All those invited\u2014enough people to fill a banquet hall\u2014renege on their promise.\n\nOne asks to be excused because he has bought some farmland sight unseen, and he is off to see his purchase. The alibi is absurd\u2014no one would buy a piece of cropland until they had inspected every inch of it and knew everything from the soil's arability to how much sun the field would receive in the winter.\n\nAnother sends his regrets because he's on his way to try out five pair of oxen he recently purchased. This is another preposterous excuse. Given the cost of these prized farm animals and the many considerations involved in evaluating them, a farmer would not even bid on a yoke of oxen until conducting a careful physical examination and seeing the pair work together.\n\nAnother guest doesn't even bother to ask to be excused; he simply informs the host he will not attend because he recently married and will be busy with his wife. This is another blatant fabrication; the banquet is not being held during the man's honeymoon, otherwise he would never have accepted the invitation in the first place. His intent is to dishonor the host. And so it goes with all who had originally accepted the invitation.\n\nThe host is angry, but the food has been prepared, and the banquet will go forward. The servant is sent out again with invitations, and when there is still room at the table, he is sent a third time. The master will force no one to attend, but he wants a full banquet hall and will go to great lengths to persuade people to come to his feast.\n\nThe scribes and Pharisees are confounded by how the parable ends. It is widely held that a person afflicted with a disease like dropsy is accursed and will not have a seat at the Lord's table or a place in the kingdom. Jesus says that not only will the blind and maimed have a place at the table, for some it will be a seat of honor.\n\nThe pious Pharisees will not accept the idea that they are the outsiders described in the story, so Jesus personalizes the parable. He looks into the eyes of those reclined around the table and says, \"I tell you, none of those men who were invited will taste my banquet.\"\n\nThe banquet Jesus refers to is the messianic banquet\u2014a metaphor for eternal life. The scribes and Pharisees perceive they are being likened to those who dishonor the Host and reject His salvation. They're right about that, but Jesus is not condemning them; he's inviting them to change. Jesus is urging them to realize that it is not bodily defects or diseases that bar one from heaven but an unwillingness to participate in the life God offers. Everyone who accepts the Host's invitation will be at the banquet.\n\nWhen it becomes evident to Jesus that the influential Pharisee and his guests are not interested in accepting his invitation, Jesus thanks the man for his hospitality and leaves. He and his followers walk to the next town. Jesus spends his days teaching his apprentices about what it means to be his disciple. Using the hyperbole common in Jewish preaching, he is blunt about the cost: \"Whoever comes to me and does not hate father and mother and wife and children and brothers and sisters, yes, and even life itself, cannot be my disciple. Whoever does not carry his cross and come after me cannot be my disciple.\"\n\nJesus uses two powerful images to explain the priorities of those who follow him, and both portray apprenticeship as allegiance. The startling analogies are offered to help the crowd realize that the preeminence of God is at the heart of Jesus's gospel.\n\nThe call to hate one's self is not a call to self-loathing; Jesus has taught his followers to love others as they love themselves. Nor is the command to hate family to be taken literally. The audience understands the hyperbole\u2014Jesus is saying one's love for God should be so deep that, by comparison, the love of anything else is hate.\n\nThe ultimatum about cross bearing is even more daunting. The only time a cross is borne is when a convicted criminal is carrying it to the place of his crucifixion. Jesus says his followers will live like one who is condemned and on his way to execution\u2014a person obviously not interested in amassing wealth or status.\n\nAll the talk about crucifixion and hate makes it clear that discipleship has a price. Jesus offers two metaphors to encourage potential apprentices to count the cost.\n\n\"Suppose one of you wants to build a tower,\" Jesus says. \"Will he not first sit down and estimate the cost to see if he has enough money to complete it? Otherwise, when he has laid a foundation and isn't able to finish, all who see it will ridicule him, saying, 'This fellow began to build and was not able to finish.'\n\n\"Or suppose a king is going out to engage another king in war. Will he not first sit down and consider whether he is strong enough with ten thousand men to oppose the one who comes against him with twenty thousand? And if he can't, he will send an emissary to discuss the terms of peace while the enemy is still a long way off.\n\n\"In the same way, any of you who does not renounce all that he has cannot be my apprentice. Salt is good, but if the salt has lost its taste, how can its saltiness be restored? Flavorless salt is fit neither for the soil nor the manure pile\u2014it is thrown away. He who has ears to hear, let him hear.\"\n\nThe Pharisees who are in the audience frown and shake their heads. Why would they want to follow Jesus? They whisper to one another, \"This fellow welcomes sinners and eats with them.\"\n\nJesus knows what motivates their complaint and tells three parables to describe how far this kind of grumbling is from the heart of God.\n\n\"Suppose one of you has a hundred sheep and loses one of them. Won't he leave the ninety-nine in the open country and go after the one that is lost until he finds it? And when he finds it, he joyfully puts it on his shoulders and goes home. He calls together his friends and neighbors and tells them, 'Rejoice with me\u2014I've found my lost sheep!'\"\n\nJesus pauses to allow the scribes and Pharisees to absorb the story, and then says, \"I tell you that, in the same way, there will be more joy in heaven over one sinner who repents than over ninety-nine righteous persons who have no need of repentance.\n\n\"Or suppose a woman has ten silver coins and loses one of them. Won't she light a lamp and sweep the house and carefully search until she finds it? And when she finds it, she calls her friends and neighbors and says, 'Rejoice with me\u2014I've found my lost coin!' In the same way, I tell you, there is joy in the presence of the angels of God over one sinner who repents.\"\n\nThe scribes and Pharisees look at one another and shrug. Why are they being told these stories?\n\nJesus shares a third parable that will make his point clear. \"There was a man who had two sons. The younger son said to his father, 'Father, give me the portion of property that is coming to me.' So the father divided his estate between them. A few days later, the younger son gathered all he had and traveled to a distant land, and there he squandered his wealth in wild living.\n\n\"When he had spent everything, a severe famine arose in that country and he began to be in need, so he hired himself out to one of the citizens of that region, who sent him to his farm to feed pigs. He longed to fill himself with the pods the pigs ate, but no one gave him anything.\n\n\"He came to his senses and said, 'How many of my father's hired hands have more than enough bread, but here I am dying of hunger! I will get up and go to my father and say, \"Father, I have sinned against heaven and before you. I am no longer worthy to be called your son. Treat me like one of your hired hands.\"'\n\n\"So he set off and went to his father. But while he was still a long way off, his father saw him and was filled with compassion and ran and embraced him and kissed him. Then the son said to him, 'Father, I have sinned against heaven and before you; I am no longer worthy to be called your son.'\n\n\"But the father said to his servants, 'Quick! Bring the finest robe and put it on him. Give him a ring for his hand and sandals for his feet. And get the fattened calf and kill it. Let's have a feast and celebrate, for my son was dead and now he lives; he was lost, but now is found!' And they began to celebrate.\"\n\nThe scribes and Pharisees assume the lesson is over. Jesus has told three parables, and all make the same point\u2014God cares inordinately for the lost. But Jesus is not done with the parable. There is another message, this one directed toward those who grumble about the way he welcomes sinners. Jesus continues the story by announcing the arrival of the prodigal's brother.\n\n\"Now his older son was in the field, and when he returned and drew near to the house, he heard music and dancing. He called one of the servants and asked what was going on, and the servant replied, 'Your brother is back, and your father has killed the fattened calf because he has returned safe and sound.'\n\n\"The older brother became angry and refused to go in. His father came out and pleaded with him, but he said to his father, 'Listen! These many years I've slaved for you, and I never disobeyed your orders, yet you've never given me even a young goat so that I might celebrate with my friends. But when this son of yours came back after squandering your money on prostitutes, you killed the fattened calf for him!'\n\n\"And his father said to him, 'My son, you are always with me, and all that is mine is yours. But it was fitting to celebrate and be glad, for your brother was dead and is now alive\u2014he was lost and has been found.'\"\n\nBy the time Jesus completes the parable, the frowns on the faces of the Pharisees and scribes have turned to scowls. They walk away cursing Jesus under their breath and brood on how they can silence him.\n\nPASSOVER IS ONLY a few weeks away, and Jesus still has much to teach his fledgling church. Soon these men and women will be the sole stewards of the Christian faith, and they cannot naively wander into the future.\n\nOne day Jesus tells them a strange parable he hopes will spur a more active faith. \"There was a rich man who had an estate manager, and charges were brought to him that this man was squandering his possessions. So he summoned the estate manager and said, 'What is this I hear about you? Turn in a complete account of your handling of my property, for you can no longer be manager.'\n\n\"The estate manager said to himself, 'What shall I do now? My master is taking the management position away from me. I'm not strong enough to dig, and I'm ashamed to beg. I've decided what I can do so that when I'm dismissed from my position people will welcome me into their homes.'\n\n\"The manager invited each person who owed money to his master to come in and discuss their debt. He asked the first, 'How much do you owe my master?'\n\n\"The man said, 'Eight hundred gallons of olive oil.'\n\n\"The estate manager said to him, 'Take your bill, sit down, and quickly write four hundred.'\n\n\"Then he asked another, 'And how much do you owe?'\n\n\"'A thousand bushels of wheat.'\n\n\"The estate manager said to him, 'Take your bill and write eight hundred.'\n\n\"The master commended the dishonest manager because he acted shrewdly.\"\n\nThe parable ends with a surprising twist; the conniving steward is praised for his cleverness and foresight. Then Jesus makes the application: \"For the children of this world are more shrewd in dealing with their own kind than are the children of light. I tell you, use worldly wealth to make friends for yourselves, so that when it's gone, you will be welcomed into the eternal dwellings.\"\n\nJesus loves to populate his parables with colorful characters, and the estate manager is a memorable one. He is clearly a scoundrel and is identified as dishonest from the outset, yet Jesus says this child of the world has a lesson for the children of light\u2014he used the wealth he had oversight of to make friends with those who would help him in his time of need. Jesus says his followers should do likewise. They should use their money to make friends who will help when they are in need. But the \"friends\" Jesus mentions are not other people\u2014these are friends who can \"receive you into the eternal dwellings.\" These friends are God and His angels.\n\nJesus would have his audience understand that, in a sense, everyone is like the manager in the story. Each person is given someone else's stuff to take care of. Those who are God's friends trust Him to take care of them in all circumstances. They will invest God's resources in God's causes. Good stewardship, Jesus says, is about integrity, and is thus intimately related to devotion.\n\n\"Whoever is faithful in little is also faithful in much,\" Jesus says, \"and whoever is dishonest in little is also dishonest in much. So if you haven't been trustworthy in handling unrighteous mammon, who will entrust to you the true riches? And if you haven't been trustworthy with someone else's property, who will give you property of your own?\n\n\"No servant can serve two masters, for either he will hate the one and love the other, or he will be devoted to the one and despise the other. You cannot serve God and Mammon.\"\n\nSome Pharisees are listening in, and when they hear Jesus disparage the love of money, they ridicule him.\n\nJesus turns to them and says, \"You're the ones who justify yourselves in the sight of others, but God knows your hearts\u2014that which is prized by men is an abomination in the sight of God.\n\n\"There was a rich man who was clothed in purple and fine linen and who ate sumptuously every day. At his gate lay a poor man named Lazarus, covered with sores, who longed to be fed with what fell from the rich man's table. Even the dogs would come and lick his sores.\n\n\"The time came when the poor man died and was carried by the angels to be with Abraham. The rich man also died and was buried. In Hades, where he was being tormented, he lifted up his eyes and saw Abraham in the far distance and Lazarus at his side. He called out, 'Father Abraham, have mercy on me and send Lazarus to dip the tip of his finger in water and cool my tongue, for I am in agony in these flames.'\n\n\"But Abraham said, 'Child, remember that during your lifetime you received your good things, while Lazarus received bad things, but now he is comforted here, and you are in anguish. And besides all this, between us and you a great chasm has been fixed, so that those who might wish to pass from here to you cannot do so, and none may cross from there to us.'\n\n\"The rich man said, 'Then, Father Abraham, I beg you to send him to my father's house, for I have five brothers. Let him warn them, so they don't end up in this place of torment.'\n\n\"But Abraham said, 'They have Moses and the prophets\u2014let them hear them.'\n\n\"And the rich man said, 'No, Father Abraham! If someone goes to them from the dead, they will repent.'\n\n\"But Abraham said to him, 'If they won't hear Moses and the prophets, neither will they be convinced even if someone rises from the dead.'\"\n\nThe parable is not directed toward the rich but toward those who make wealth their god and turn their backs on the beggar at their gate. The Pharisees, frustrated with the false prophet who only speaks in parables, shake their heads in disgust and walk away.\n\nTHE WINTER RAINS subside, and the hills blossom with new life. Jesus continues to train the apostles for the intense persecution ahead. To underline how important faithfulness will be, he one day says to the twelve, \"Who among you who has a servant plowing or tending sheep says to him when he has come in from the field, 'Come immediately and take your place at the table'? Will he not rather say to him, 'Prepare supper for me\u2014get yourself ready and serve me while I eat and drink, and afterward you will eat and drink'? Does he thank the servant because he did what was commanded?\"\n\nJesus pauses, then says, \"So you also, when you have done all that you were commanded, say, 'We are worthless servants\u2014we have only done what we were obligated to do.'\"\n\nJesus uses the concept of servitude to describe the ideal attitude of his apprentices\u2014when they do all they are commanded to do and complete their assigned tasks, they understand that they have only done their duty. Slaves do not seek status\u2014they seek to do the will of their master.\n\nOne day as Jesus and his followers enter a village, he is met by ten lepers who cry out, \"Jesus, Master! Have mercy on us!\"\n\nJesus walks over to the ten, greets them, and simply says, \"Go and show yourselves to the priests.\"\n\nAn official examination by a priest is required in order for a leper to be reinstated into society, but this, of course, is always done after the healing, not before. To obey Jesus will require faith, but the ten want to be healed, so they walk away together in search of a priest. As they are walking to the synagogue, their leprosy vanishes. As the sores dry up and disappear, the ten become aware of what has happened. Nine of the healed lepers continue toward the synagogue. One runs back and falls on his face at Jesus's feet.\n\nThe disciples are shocked\u2014the only one who expressed gratitude was a Samaritan.\n\n\"Were not ten cleansed?\" Jesus asks. \"Where are the other nine? Has no one returned to give glory to God except this foreigner?\"\n\nThe healer helps the man to his feet. \"Get up and go your way. Your faith has saved you.\"\n\nThis is a powerful lesson for the disciples. God's grace goes out to all, but many never receive its ultimate benefit. Faith is more than accepting God's kindness\u2014it is turning back to the Lord.\n\nThe lesson is lost on the Pharisees. They want to know when God's kingdom will come. They are convinced that spectacular signs will herald the Messiah's establishment of the kingdom, but Jesus warns the sign seekers they are looking in the wrong direction.\n\n\"The kingdom of God is not coming in ways that can be observed,\" Jesus tells them, \"nor will they say, 'Look, here it is!' or 'There it is!' for behold, the kingdom of God is in the midst of you.\"\n\nThe religionists seek a sign in the heavens, yet God's greatest sign is standing before them. If they want to see and experience the kingdom of heaven, there is no need to look for signs announcing its address. It is here now\u2014present in the person and promises of Jesus.\n\nThe religious leaders see none of this and sneer at Jesus's asinine claim that the kingdom is in their midst. Everyone knows that the kingdom of God will not appear on earth until the Messiah comes.\n21\n\nDEATHLESS\n\nFebruary\u2013March, AD 33\n\n\"Do not be surprised at this\u2014the time is coming when the dead will hear his voice and come out of their graves.\"\n\n\u2014John 5:28\u201329\n\nSPRING HAS COME to Israel. The winter rains have left the grass tall and green, and the hills are speckled with wildflowers. The trees blossom, leaf, and bear tiny fruit. The winter barley is beginning to ripen, and the wheat will soon follow. The grain harvest is but a few weeks away, around the time of Passover.\n\nJesus is in Galilee, near the border of Samaria, preparing his followers for what awaits him in Jerusalem. There is much to be done and little time, but before Jesus can continue the apostles' training, he must address their fears. He tells them a story about an unrighteous judge.\n\n\"There was a judge in a certain city who neither feared God nor respected people, and there was a widow in that city who kept coming to him and saying, 'Grant me justice against my adversary.'\n\n\"For a while the judge refused, but later he said to himself, 'Though I neither fear God nor respect people, yet because this widow keeps bothering me, I will grant her justice so that she will not wear me out by her continual coming.'\"\n\nJesus pauses to let his apprentices reflect on the story, and then adds, \"Hear what the unrighteous judge says. And will not God grant justice to His chosen ones who cry out to Him day and night? Will He delay long in helping them? I tell you, He will vindicate them speedily. Nevertheless, when the Son of Man comes, will he find faith on earth?\"\n\nThe disciples draw two lessons from the parable. First, if an unjust judge can be persuaded to administer justice for someone he doesn't care about, how much more can God be trusted to vindicate those He dearly loves?\n\nThe second lesson is intrinsically related\u2014those who follow Jesus must learn to trust God's timing. Jesus encourages his apprentices to emulate the widow's perseverance in prayer. There will be trials and hardships and times when they will wonder if God is even listening. Jesus says that when those days inevitably come, continue to cry out to God to grant justice\u2014God will bring vindication in due season. Meanwhile, pray and persevere and know that God has good reasons for allowing things to unfold as they do. He will not be late in fulfilling His promise.\n\nThere are some in the audience who can't imagine this teaching is addressed to them. After all, they are not like the rabble. Jesus challenges them to look in the mirror.\n\n\"Two men went up to the temple to pray,\" Jesus says, \"one a Pharisee and the other a tax collector. The Pharisee, standing by himself, prayed, 'God, I thank you that I am not like other people\u2014extortioners, rogues, adulterers, or even like this tax collector. I fast twice a week; I give a tenth of all my income.' But the tax collector, standing far off, would not even lift his eyes to heaven, but beat his breast, saying, 'O God, be merciful to me, a sinner!'\n\nJesus pauses, then says, \"I tell you, this man returned home justified, rather than the other. For everyone who exalts himself will be humbled, but the one who humbles himself will be exalted.\"\n\nJesus says a reversal is coming. The self-righteous need to rethink their inclination to believe they are better than others. God is not interested in relative righteousness.\n\nTHE TRAINING OF the twelve continues. They travel east of the Jordan to a more sparsely populated region, but Jesus cannot escape the tests and traps laid by the nation's religious leaders. One day a group of Pharisees asks, \"Is it lawful for a man to divorce his wife for any reason he wants?\"\n\nFew subjects are more volatile than a man's rights regarding divorce, and Israel's religious scholars are not in agreement concerning what the Scriptures teach about ending a marriage. To determine under what circumstances divorce is permissible, they invariably go to the passage in Deuteronomy and point out that Moses said a husband can end the marriage if he finds \"any matter of indecency\" in his wife.\n\nThe scribes debate how to apply this passage. The rabbinic school of Shammai focuses on the word indecency and defines it very narrowly; little outside of sexual immorality is viable grounds for divorce. The Hillel school gives more weight to the any matter clause and broadens the definition of what constitutes indecency\u2014in effect, Hillel's interpretation gives the husband the right to divorce his wife for almost any reason. Not surprisingly, this is the favored interpretation among men. Women have few marriage rights, and even those are often ignored.\n\n\"What did Moses command you?\" Jesus asks.\n\n\"Moses allowed a man to write a certificate of divorce and send her away,\" they reply.\n\nJesus shakes his head. He had asked them what Moses commanded; they responded by telling him what Moses permitted. To find what Moses commanded, they must go to the first book he wrote, which is where Jesus takes them.\n\n\"Have you not read that from the beginning that the Creator 'made them male and female,' and said, 'For this reason a man will leave his father and mother and be joined to his wife, and the two will become one flesh'? As a result, they are no longer two, but one flesh. Therefore, what God has joined together, let no one separate.\"\n\nJesus goes back to the days before humanity's fall to remind them of God's original purpose for marriage, but the Pharisees protest. \"So why did Moses give the commandment to 'give her a divorce certificate and send her away'?\"\n\nThe counterquestion makes it obvious that the religionists are not listening. Jesus says, \"Moses, in response to your hardness of heart, permitted you to divorce your wives, but it was not this way at the beginning.\"\n\nJesus points out that Moses provided a concession, not a commandment. He would have them see that marriage is more than a legal bond. Genesis says the man and woman become one flesh. A blood relationship is formed that is only to be dissolved by death. If the Pharisees want to know the divine perspective on divorce, it is readily available in the writings of the prophet Malachi. But the hard glares of the Pharisees tell Jesus that these men are not interested in hearing what God has to say.\n\nLater that evening the disciples ask for a fuller explanation, and Jesus devotes significant time to teaching them about God's will concerning marriage. He stresses that the union must never be devalued, because the marriage covenant is not based on what is comfortable and convenient\u2014it is sourced in God's very purpose for creation. Marriage reflects the Lord's covenant relationship with His people. Thus, it is to be held inviolate.\n\n\"Whoever divorces his wife, except for sexual immorality, and marries another, commits adultery against her,\" Jesus says. \"And if she divorces her husband and marries another, she commits adultery. And he who marries a woman divorced from her husband commits adultery.\"\n\nThe twelve know Jesus's narrow view on divorce and are not surprised by his blunt statements, but they are again reminded how much this interpretation restricts a man's options. One jokingly remarks, \"If that's the way it is for a man with his wife, it's better not to marry.\"\n\nThe disciples chuckle at the humorous notion\u2014no man would choose celibacy over even the worst wife. Not that it's an option; marriage and fatherhood are considered religious obligations in Israel, but Jesus uses the humorous comment to make a serious point about celibacy.\n\n\"Not everyone can accept this saying,\" Jesus tells his apprentices, \"but only those who have been given the ability, for there are eunuchs who were born that way from their mother's womb, and there are eunuchs who have been made eunuchs by men, and there are eunuchs who have made themselves eunuchs for the sake of the kingdom of heaven. Whoever is able to accept this, let him accept it.\"\n\nJesus's teaching about celibacy goes against the traditions of the elders. In Judaism, marriage is a duty. The celibate life is never recommended, yet Jesus says God gives some the gift of celibacy in order to further kingdom goals. His metaphor of castration is shocking but assures the lesson will be remembered. There is also a word of encouragement for those who are physically impotent or have been castrated\u2014while others may forbid them to participate at the temple, they are welcome in the kingdom of God.\n\nJESUS DAILY OFFERS God's salvation to the people of the Jordan River Valley, and his message of grace and mercy is popular with the poor and downtrodden. Children are especially attracted to the gentle teacher and often sit at his feet listening to his stories. Parents also continuously bring infants and toddlers to be blessed and healed. In a land where six out of ten children never reach their sixteenth birthday, Jesus has looked into the fear-filled eyes of many mothers and fathers, and he never tires of blessing and caressing children.\n\nThe disciples, however, grow increasingly concerned about the noisy little distractions running around the camp. The teacher is an important man. He has better things to do than bless babies and play with kids. One day when the children are particularly exuberant, a disciple begins rebuking one of the parents.\n\nJesus overhears his frustrated apprentice and interrupts him. \"Permit the children to come to me\u2014do not hinder them, for to such as these belongs the kingdom of God. I tell you truly, anyone who will not receive the kingdom of God like a child will never enter it.\"\n\nJesus says that to receive the kingdom, one's attitude toward God must be like that of a toddler toward her parents. It is a relationship of innate trust. Those who behave as autonomous \"adults\" toward God will not be able to enter because the kingdom is made up of the childlike. While his followers ponder the lesson on discipleship, Jesus takes several children into his arms and blesses them.\n\nLater, when Jesus is preparing to leave, a well-dressed young man steps from the crowd, kneels before him, and asks, \"Good teacher, what must I do to inherit eternal life?\"\n\nThe man is sincere, but his use of flattery indicates he has learned nothing from Jesus's message. Moments earlier he heard Jesus teach that eternal life is a gift to be received\u2014now he asks Jesus how he can earn it. The wealthy man is trapped in the same status-seeking system his peers are\u2014a compliment requires one in return. The rich man honored Jesus with a respectful title; he expects a proper response.\n\nBut Jesus will not participate. \"Why do you call me good?\" Jesus asks. \"No one is good except God alone.\"\n\nJesus is willing to discuss eternal life, but he will first establish the criteria. The man views himself and others like him as generally good, but Jesus wants him to know that relative goodness is not under discussion when it comes to eternal life. The good is God alone. It is obvious to Jesus that the rich man's spirit is troubled. Otherwise, he would be content to obey the Torah.\n\n\"You know the commandments,\" Jesus reminds him. \"'Do not murder,' 'Do not commit adultery,' 'Do not steal,' 'Do not bear false witness,' 'Do not defraud,' 'Honor your father and mother,' and, 'You shall love your neighbor as yourself.'\"\n\nJesus provides a representative list that includes five of the ten commandments. He does not mention the first four (which deal with one's relationship with God), or the tenth, which concerns covetousness. Instead, he focuses the wealthy ruler's attention on the commandments that deal with personal relationships.\n\nThe young man hastily answers, \"Teacher, all these I have kept from my youth. Where do I still fall short?\"\n\nThe man is convinced that he has faithfully observed God's commandments, but he is concerned that this won't be enough to qualify him for eternal life.\n\nJesus knows what is haunting the ruler and keeping him in denial and pushing him to inquire what more he can do.\n\n\"You lack one thing,\" Jesus says. \"If you want to be complete, go, sell all that you have and distribute the money to the poor, and you will have treasure in heaven. Then come follow me.\"\n\nThe audience is stunned by Jesus's counsel, and the young man is appalled. Jesus has exposed what the man has been struggling with for years\u2014the tenth commandment against covetousness. But this is only part of it, because by violating the tenth commandment, he also violates the first. God views covetousness as worship of the god Mammon.\n\nGiven the young man's heart, Jesus's prescription is that he renounce his status and become like a little child. He is to donate his wealth to the poor and become part of Jesus's entourage.\n\nThe young man's thoughts are jumbled. The teacher suggests something that, in theory, could be done, but the remedy is imprudent. What would happen to the family name? If he were to give everything away to join this ragtag band, his family would become the laughingstock of the community. No, the man decides, the counsel is foolhardy and must be ignored. He walks away in despair.\n\nJesus is saddened by the young man's response. As he watches him leave, he says, \"How difficult it will be for those who have wealth to enter the kingdom of God!\"\n\nThe apostles are amazed. They have been taught their entire life that earthly prosperity signifies heaven's approval.\n\nJesus sees the astonishment in their faces and says, \"Children, how difficult it is to enter the kingdom of God. It's easier for a camel to go through the eye of a needle than for a rich person to enter the kingdom of God.\"\n\nThe apostles shake their heads in disbelief. \"Then who can be saved?\"\n\nThe proverb Jesus shared with them says it is impossible to earn eternal life, even for the affluent. The disciples cannot fathom how this could be. Is it not the wealthy who fund God's projects? Without the beneficence of the rich, who would provide alms for the poor? Who would build and endow the synagogues and schools? Who would fund the orphanages?\n\nJesus knows the source of his apprentices' confusion. His declaration that eternal life is impossible to earn has undermined their understanding of salvation. His followers ascribe much of the good in society to the patronage system. If the wealthy who do so much good in the world won't be saved, who will?\n\nThis is the conclusion Jesus hoped they would reach, and he assures them, \"For humans this is impossible, but not for God\u2014all things are possible with God.\"\n\nThe disciples appreciate the encouraging words but are still concerned. \"Look,\" Simon Peter says, \"we've left everything and followed you. What then will there be for us?\"\n\n\"I tell you truly,\" Jesus assures them, \"at the renewal of all things, when the Son of Man sits on his glorious throne, you who have followed me will also sit on twelve thrones, judging the twelve tribes of Israel. And everyone who has left house or wife or brothers or sisters or mother or father or children or lands for my sake and for the sake of the kingdom of God will receive a hundred times as much in this age\u2014houses and brothers and sisters and mothers and children and lands, with persecutions\u2014and in the coming age, eternal life. But many who are first will be last, and the last first.\"\n\nJesus alludes to Daniel's prophecy to describe the enthronement of the Son of Man at the end of days. On that day there will be great rewards for those who abandoned status-seeking to follow Jesus. In addition to eternal life and manifold wealth, they will be awarded positions of leadership. But the rewards are not limited to the twelve\u2014Jesus says everyone who makes the kingdom their top priority will be compensated a hundredfold in both this world and the next.\n\nJesus's praise is sincere, but it doesn't apply to everything Simon Peter said. The apostle's words were only partly right, for Peter is trapped in the same status consciousness as the young ruler who walked away\u2014there is a sense of entitlement and a desire for preeminence. Jesus shares a parable to help his followers see things from God's perspective.\n\n\"The kingdom of heaven is like a landowner who went out early in the morning to hire laborers for his vineyard. After agreeing with the laborers on a denarius for the day, he sent them into his vineyard. When he went out about nine o'clock he saw others standing idle in the marketplace and said to them, 'You also go into the vineyard, and I will pay you whatever is right.' And off they went. Going out again about noon and at three in the afternoon, he did the same. At about five o'clock, he went out and found others standing around and said to them, 'Why do you stand here idle all day?' 'Because no one has hired us,' they replied. He said to them, 'Off you go, too, into the vineyard.'\n\n\"And when evening came, the owner of the vineyard said to his foreman, 'Call the laborers and pay them their wages, beginning with the last to the first.'\n\n\"When those hired about five o'clock came, each of them received a denarius, so when those hired first came, they assumed they would receive more, but each of them also received a denarius. On receiving it, they began to grumble against the landowner, saying, 'Those men who were hired last worked only one hour, and you've made them equal with us who have endured a whole day's work in the scorching heat.'\n\n\"'Friend,' the landowner replied to one of them, 'I haven't cheated you. Didn't you agree with me on one denarius? Take your pay and go. I want to give the one who was hired last the same as I gave to you. Am I not allowed to do what I wish with what belongs to me? Or are you envious because I am generous?'\n\n\"So the last will be first, and the first last.\"\n\nThe disciples are bewildered by the parable. They can't detach themselves from their culture's meritocracy, so what Jesus describes sounds unfair. Rewards should be proportional to work done.\n\nJesus disagrees. He says the kingdom of heaven does not operate like earthly realms, and some who come into the kingdom late will surpass some who came in early.\n\nThe implications for the disciples' entitlement notions are profound. Jesus says that no one is superior to anyone else in God's kingdom. This is as true for individuals as it is for nations. Distinctions like Jew and Gentile and male and female and master and slave do not exist in the kingdom. A spirit of unity transcends those descriptors. There are different roles, but everyone shares an equal and incomprehensibly glorious treasure\u2014eternal life and the opportunity to participate in God's endless adventures.\n\nThe teaching about God's unbounded generosity has multiple applications, and the apprentices want to hear more about the kingdom of heaven, but further explanation will have to wait. A messenger arrives from Bethany with urgent news. Jesus's dear friend, Lazarus, brother of Martha and Mary, is near death.\n\nThe plea from the sisters is terse. \"Lord, the one you love is ill.\"\n\nJesus nods and says, \"This illness will not end in death. It is for the glory of God, so that the Son of God may be glorified through it.\"\n\nJesus knows he cannot get to Bethany before Lazarus's death, so he awaits the Father's instructions. Two days later Lazarus dies, and Jesus says, \"Let us go to Judea again.\"\n\nWorried looks cloud the faces of his followers. Bethany is a suburb of Jerusalem. \"Rabbi, a short while ago the Jews were seeking to stone you, and you're going back there again?\"\n\nOnly a few weeks have passed since the debacle at the Feast of Dedication, and the venomous threats of the Jerusalemites still haunt the disciples. To return now would be unwise.\n\n\"Are there not twelve hours in the day?\" Jesus asks. \"If anyone walks in the day, he does not stumble because he sees the light of this world. But if anyone walks around at night, he stumbles because the light is not in him.\"\n\nJesus then tells them the reason for the trip. \"Our friend Lazarus has fallen asleep, but I go to wake him.\"\n\nThe disciples don't understand that Jesus is speaking figuratively, and they use the news as an excuse not to go. \"Lord, if he has fallen asleep, he will be all right.\"\n\nJesus shakes his head and says, \"Lazarus has just died, and I am glad for your sake I was not there\u2014so that you may believe. But let us go to him.\"\n\nThe twelve nod gloomily and begin to pack. They are convinced no good can come of this. Thomas, like the rest of the apostles, believes that returning to Bethany is a mistake, but this afternoon he is ready to follow Jesus anywhere. \"Let us go as well,\" Thomas says, \"so that we may die with him.\"\n\nThe apostle's courage is admirable but misguided.\n\nWhen Jesus and his followers arrive in Bethany, the town is enduring a spring cleaning in anticipation of the many thousands who will pass nearby on their way to the temple for the Passover celebration. It has been four days since Lazarus died, and the fourth day of grieving is particularly sad. The rabbis teach that the spirit of one who has died lingers around the body for three days looking for a way to reenter, but after that, the body's physical deterioration will block the spirit from ever returning. Thus, traditionally, there are three days of weeping. Then, on the fourth day, when all hope of recovery is abandoned, heavy mourning begins.\n\nOn the afternoon of the fourth day, Martha gets word that Jesus has arrived and is at the outskirts of the village. She runs to him and says, \"Lord, if you had been here, my brother would not have died. But even now I know that whatever you ask from God, God will give you.\"\n\nMartha is confident that if Jesus had been there in time he would have healed her brother, and she wants him to know she has not lost her faith in him. Jesus smiles and says, \"Your brother will rise again.\"\n\nMartha bites her lip; she has heard these words dozens of times over the past few days. Like most Jews, she believes that one day God's people will be restored to bodily life, but on this first day of heavy mourning, those words are cold comfort.\n\n\"I know that he will rise again in the resurrection on the last day.\"\n\nJesus looks into Martha's eyes and says, \"I am the resurrection and the life. Whoever believes in me will live even though he dies, and whoever lives and believes in me will never die. Do you believe this?\"\n\nMartha returns his gaze and says, \"Yes, Lord. I believe that you are the Christ, the Son of God, who has come into the world.\"\n\nHer faith is personal, and her confidence is complete\u2014Jesus is the Messiah, and his words have given her hope that will sustain her despite the loss of her brother.\n\nJesus knows his promise was not fully heard, but Martha will hear it soon enough. She returns home to tell her sister about her conversation with Jesus. When Mary hears that Jesus requested she come see him, she quickly rises and runs in the direction Martha points, tears streaming down her cheeks. Many neighbors and friends see her hasty departure and assume she is going to the tomb. They follow to console her.\n\nWhen she reaches Jesus, she falls at his feet and sobs, \"Lord, if you had been here, my brother would not have died.\"\n\n\"Where have you laid him?\" Jesus asks.\n\nMary wipes the tears from her cheeks and says, \"Lord, come and see.\"\n\nAs they walk to the tomb, Jesus weeps. Some say, \"See how he loved Lazarus!\"\n\nBut others are confused and ask, \"Could not he who opened the eyes of the blind man also have prevented this man from dying?\"\n\nWhen they arrive at the tomb, instead of mourning, Jesus gives an abrupt command. \"Roll away the stone!\"\n\nMartha is horrified. \"Lord,\" she whispers, \"by this time, there will be an odor\u2014he has been dead four days.\"\n\nJesus turns and solemnly reminds her, \"Did I not tell you that if you believed you would see the glory of God?\"\n\nMartha bows her head and steps back, her mind racing to understand Jesus's words.\n\nThe stone is large, but several men are able to roll it aside. The spectators grow quiet; many shake their heads in disbelief and brace themselves for the stench.\n\nJesus lifts his eyes toward the heavens, and his voice rings over the graveyard. \"Father, thank You for hearing me. I know that You always hear me, but I said this for the benefit of the crowd standing here, so that they may believe that You sent me.\"\n\nJesus then turns to the open tomb and commands, \"Lazarus, come out!\"\n\nA low murmur arises; now all are shaking their heads. The man has been dead four days. His spirit is in Hades, and his body is in corruption. Even a fool should know that Lazarus is beyond hope.\n\nBut just then a tiny, almost imperceptible sound escapes the tomb. A hush falls over the gathering. Did they imagine it? No. There it is again, a shuffling sound, and it's growing louder.\n\nThe crowd gasps when a shadow appears in the doorway. A man staggers from the tomb wrapped head to foot in burial linen. He steps outside and halts, his face still covered by the kerchief placed there on the day of his death. For several long moments, there is a stunned silence. Then Jesus turns to the two sisters, smiles, and says, \"Unbind him and let him go.\"\n22\n\nTHE PASSOVER PLOT\n\nMarch, AD 33\n\n\"What are we going to do? . . . . If we let him go on like this, everyone will believe in him, and the Romans will come and take away both our place and our nation!\"\n\n\u2014John 11:47\u201348\n\nTHE FIRST MAJOR festival of the year is less than two weeks away, and Jerusalem is preparing for crowds that will increase the population fivefold. The lights burn late in the Hall of Hewn Stones, but the Sanhedrin's deliberations have nothing to do with the upcoming holy days. Israel's ruling council is consumed with a report just in from Bethany\u2014Jesus is back and has raised a man who had been dead four days. There appears to be no end to the powers of the Galilean sorcerer. Every step they have taken to silence him has failed.\n\n\"What are we going to do?\" a frustrated councilman asks. \"This man is performing many miraculous signs. If we let him go on like this, everyone will believe in him, and the Romans will come and take away both our place and our nation!\"\n\nThe complaint reveals the Sanhedrin's true concern\u2014their status and security. Their eye is toward Rome. They fear a messianic movement forming behind Jesus that could cost Israel both their quasi-autonomy and temple privileges. It has happened before, and the religious rulers dread a recurrence.\n\nA loud voice pierces the confusion; the men turn and look into the reddened face of the high priest, Joseph Caiaphas. Appointed by the Roman prefect, Valerius Gratus, twelve years earlier, he now has a strong bond with Gratus's successor, Pontius Pilate. Caiaphas is rude and ill-tempered, but he is also influential, and his words have weight.\n\n\"You know nothing at all!\" he tells the Sanhedrin. \"You don't realize that it is better for you that one man die for the people than the whole nation perish.\"\n\nThe councilmen, impressed, nod and whisper to one another. The high priest has spoken God's truth. By the end of deliberations that evening no official action has been taken, but a decision has been made. All that remains is arranging the Nazarene's demise in a way that will create the least disturbance.\n\nJesus is in Bethany when he learns of the Sanhedrin's unofficial verdict. He knows he will confront Caiaphas, but now is not the time. He tells his followers to pack. They will be leaving immediately.\n\nJesus and his disciples walk northeast to the village of Ephraim. Here on the frontier of the wilderness Jesus prays and prepares for the maelstrom ahead.\n\nAfter resting a few days in Ephraim, he leads his followers to the Jordan, and they turn south toward Jericho. The road is already crowded with pilgrims. Many are Galileans who are going to the city early in order to be ritually purified before the Passover.\n\nAnticipation of what might happen in Jerusalem is higher than usual this year. It has been widely reported that Jesus publicly blasphemed God during Hanukkah and came near to being stoned on the temple mount. Many wonder if the Nazarene will show himself again so soon.\n\nJesus is aware of the speculation swirling around him as he joins the throngs headed to the festival. The revelers are boisterous and in good spirits. They laugh and sing the Psalms of Ascent as they walk. The apostles, on the other hand, are somber and say little. The teacher has given them much to think about, most of it disturbing, and with every step toward the capital, their sense of foreboding grows.\n\nJesus compounds their apprehension later that morning when he calls the twelve aside and says, \"Look, we're going up to Jerusalem, and all the predictions of the prophets about the Son of Man will be fulfilled. He will be betrayed to the chief priests and scribes, and they will condemn him to death and deliver him over to the Gentiles to be mocked and spit on and flogged and crucified and killed, and after three days he will rise.\"\n\nThe apostles shake their heads and look at one another. The one they have identified as the Messiah is saying he must die. The twelve assume that Jesus is telling them some kind of parable. He can't be speaking literally.\n\nLater that morning when they stop to rest, the two sons of Zebedee approach Jesus with their mother at their side. They have brought her because older women can make requests that would be inappropriate coming from anyone else. When she steps forward, Jesus smiles and says, \"What do you wish?\"\n\nShe gestures toward James and John and says, \"Declare that these two sons of mine are to sit one on your right and one on your left in your kingdom.\"\n\nJesus looks at the two brothers who instigated the request. James and John are religiously studying their sandals.\n\n\"You don't know what you are asking,\" he tells them. \"Can you drink the cup that I am about to drink or to be baptized with the baptism I am about to be baptized with?\"\n\nThe two look up with hope in their eyes and answer in unison, \"We can!\"\n\nThe brothers have no idea that Jesus is speaking about being immersed into death, and Jesus knows that when he undergoes that brutal baptism next week, his disciples will be hiding in fear. But he also knows that faith is a process, and a day will come when they will be ready to suffer and die with him.\n\n\"You will indeed drink my cup and be baptized with the baptism I am baptized with,\" Jesus says. \"But to sit on my right and left is not mine to grant. These places belong to those for whom they have been prepared by my Father.\"\n\nThe sons of Zebedee nod and quietly retreat. When the other apostles hear what happened, they become angry. They, too, have an interest in status. Jesus again calls the twelve aside for instruction, this time on a well-worn topic.\n\n\"You know that the rulers of the Gentiles lord it over them, and their great men impose their authority on them\u2014it is not to be like that among you. Whoever among you who wants to become great is to be your servant, and whoever would be first is to be the slave of all, just as the Son of Man came not to be served but to serve and to give his life as a ransom for many.\"\n\nThe apprentices struggle with Jesus's summary of the fifty-third chapter of Isaiah, a passage he has adopted as his own. The prophecy speaks of a righteous servant who will be \"an offering for sin\" and die as a ransom for his people, but what does that have to do with the glorious destiny of the Messiah?\n\nTHEY COME TO the ancient oasis of Jericho. The old city on the hill whose walls fell long ago now lies in ruins. Jesus walks past the rubble and south toward the new city in the valley. New Jericho sprang up as a suburb around an opulent residence built by the Hasmonean dynasty. The home was later expanded into three palaces by Herod the Great to serve as his winter residence. No expense was spared to build the extravagant escape. In addition to ornate Roman pools and bath houses, Herod built a gymnasium, a theater, and a track for horse-and-chariot races, all surrounded by lush tropical gardens.\n\nAfter Herod's death, this portion of his kingdom was given to his son, Archelaus. He developed the winter palace even further, enveloping it in a large grove of palms, but more than twenty years have passed since Archelaus was deposed for incompetence and exiled to Gaul. The palaces are now under Roman jurisdiction and are used less frequently, but the city has continued to grow because of its strategic location. With Passover a week away, the town is filled with people on their way to Jerusalem.\n\nJesus enters Jericho, and a growing number of people recognize him. Soon he is at the center of a noisy crowd walking down the road.\n\nA blind beggar who is simply known as bar Timaeus (son of Timaeus) is sitting on the side of the road with a another blind mendicant. Bartimaeus and his companion hear the commotion and ask what is happening. When they are told Jesus of Nazareth is passing by, they begin shouting, \"Lord, have mercy on us! Son of David!\" Again and again they plead, \"Jesus, have mercy on us! Son of David!\" When the crowd commands them to be silent, the two men cry out even louder, \"Lord, have mercy on us! Son of David!\"\n\nJesus stops and requests the two be brought to him. Several run and tell the beggars, \"Take heart! Get up. He's calling you!\"\n\nWhen Bartimaeus hears the invitation, he struggles to his feet, throws off his cloak, and walks on wobbly legs toward what sounds like the center of the crowd, his partner close behind. Soon the two blind men are standing before Jesus.\n\n\"What do you want me to do for you?\" Jesus asks.\n\n\"Lord, we want our eyes to be opened\u2014we want to see!\"\n\nJesus smiles and touches the men's eyes saying, \"Receive your sight; your faith has saved you.\"\n\nThe two men instantly regain their sight and begin praising God.\n\nA few minutes later, the band of disciples is once again moving toward Jerusalem, their number now increased by two. All are in a celebratory mood, Jesus's prophecy of impending crucifixion a distant memory.\n\nAs they walk toward the outskirts of town, Jesus stops under a sycamore tree and looks up at a small man perched in the branches. Jesus smiles and extends his arm. \"Zacchaeus, hurry and come down, for I must stay at your house today.\"\n\nJesus is addressing one of the wealthiest and most despised men in Jericho. Zacchaeus is small in stature, but he has great power and vast estates. He is the region's chief tax collector and has a dark reputation. Zacchaeus hurries down from his perch and says he would be honored to have Jesus in his home.\n\nMany in the crowd grumble, \"He has gone to be the guest of a sinful man.\"\n\nThe tax magnate has no illusions about his past; he knows he has leveraged his position to extort money, and he is ready to make restitution. \"Behold, half of my possessions, Lord, I will give to the poor, and if I have defrauded anyone of anything, I am going to pay back four times as much.\"\n\nJesus smiles and says, \"Today salvation has come to this house, for he, too, is a son of Abraham. For the Son of Man came to seek and to save the lost.\"\n\nWhen Jesus says, \"today salvation has come,\" the crowd's thoughts turn to one of their favorite subjects\u2014God's timetable for the coming of the Messiah. The gift of sight given the two beggars heightens expectations. Healing the blind is a signature ability of the Christ. Many believe the long-awaited kingdom of God is about to appear. They have been taught since childhood that the Messiah will appear dramatically, and what better time for the unveiling of the Messiah than Passover?\n\nJesus tells them a parable to help them understand that the kingdom will unfold in two stages and not all at once as they expect.\n\n\"A man of noble birth traveled to a faraway country to receive for himself a kingdom and then return. Summoning ten of his servants, he gave them ten minas and said to them, 'Do business with these until I come back.'\n\n\"But his citizens hated him and sent a delegation after him, saying, 'We don't want this man to rule over us.'\n\n\"When he received his kingdom and returned, he ordered the servants to whom he had given the money to be summoned so that he might find out what they had gained by doing business.\n\n\"The first came before him, saying, 'Lord, your mina has made ten minas more.'\n\n\"And he said to him, 'Well done, good servant! Because you have been faithful with the little I entrusted to you, you shall have authority over ten cities.'\n\n\"And the second came, saying, 'Lord, your mina has made five minas.'\n\n\"And he said to him, 'And you are to be over five cities.'\n\n\"Then another came, saying, 'Lord, here is your mina. I have kept it stored in a handkerchief because I was afraid of you\u2014you are a severe man. You take what you did not deposit and reap what you did not sow.'\n\n\"The nobleman said to him, 'I will condemn you from your own mouth, evil servant! You knew, did you, that I was a severe man, taking what I did not deposit and reaping what I did not sow? Why then did you not put my money in the bank? Then when I returned I could have collected it with interest.'\n\n\"And he said to those who stood by, 'Take the mina from him and give it to the one who has the ten minas.'\n\n\"And they said to him, 'Lord, he has ten minas!'\n\n\"The king replied, 'I tell you that to everyone who has, more will be given, but from the one who has not, even what he has will be taken away. But as for these enemies of mine who did not want me to reign over them\u2014bring them here and slaughter them before me.'\"\n\nWhen Jesus finishes the story, his audience begins whispering questions to one another. The parable is complex, and they struggle to decipher the different characters and how any of it relates to the messianic kingdom.\n\nIn essence, it's the story of a nobleman who entrusts a portion of his estate to his servants while he goes away to secure a vassal kingship. On his return there is a reckoning. The amount the servants were given was small (a mina was equivalent to about four months wages for a laborer). The assignment was a test of their faithfulness.\n\nAs the story unfolds, it becomes evident that the third servant does not know his master. To justify his inaction, he misrepresents the king as harsh and impossible to please, but the monarch's treatment of the other servants proves that accusation false. Justice can sometimes be interpreted as sternness, but in truth, this lord is generous and merciful. The wicked servant cannot see this. As the eighteenth psalm says: to those who are merciful, blameless, and pure, God shows Himself merciful, blameless, and pure. But to the crooked, God appears to be twisted.\n\nThe evil servant could have simply put the money in the bank and the king would have been satisfied, but it turns out the servant is no different from the country's citizens who refused to be ruled, and his end will be the same. Ultimately, all who oppose the king will get their wish\u2014they will be placed outside his kingdom.\n\nJesus finishes his teaching and then spends the rest of the evening visiting with Zacchaeus. He rises early the next morning to make the arduous sixteen-mile ascent to the Mount of Olives. The cool spring air is bracing, the sun is at their back, and the mountain passes are not yet crowded. They arrive in Bethany by early afternoon and make their way to the home of Lazarus, Martha, and Mary.\n\nSunset ushers in the Sabbath, a simple meal with friends, and rest. Jesus is told a banquet is being held in his honor the following evening. The dinner will be hosted by Martha and Mary in the home of Simon the Leper.\n\nAS THE SUN sets on Sabbath the next day, many in the village gather at the home of Simon to fete Jesus. While the guests are talking, Mary inconspicuously walks behind Jesus carrying a large alabaster flask. With a twist of her wrist, she snaps off the flask's slender neck. The room is filled with the exotic aroma of Indian nard, the fragrance of choice for the well-to-do. All eyes are drawn to the woman with the flask. They watch her lavishly pour the rare liquid over Jesus's head. Jesus remains still as the costly oil runs through his hair, onto his shoulders, and down his back. The dinner guests watch speechlessly as Mary then moves to his feet and anoints them with the remaining nard. The guests grow even more alarmed when she shamelessly lets down her long hair and begins to wipe Jesus's feet.\n\nA murmur springs up around the table, but Mary is oblivious to the stares and whispers. Minutes pass, and the apostles have time to think about the cost of the oil Mary used to anoint Jesus. They exchange looks and whisper, \"Why such waste?\"\n\nJudas, who carries the money box, verbalizes the thoughts of the others. \"Why wasn't this perfume sold for three hundred denarii and given to the poor?\"\n\nJesus, rather than exposing Judas's motive for asking the question, honors Mary's devotion. \"Leave her alone. Why do you trouble the woman? She has done a good thing for me. You will always have the poor with you, and you can help them whenever you want, but you will not always have me. She has done what she could. She has anointed my body ahead of time to prepare it for burial, and I tell you truly, wherever this good news is proclaimed in the whole world, what this woman has done will be told in memory of her.\"\n\nThe disciples and dinner guests are surprised at Jesus's response. The value of the perfume was significant, the amount one would have to work an entire year to earn. How could pouring it on someone in a moment of spontaneous extravagance ever be acceptable? A respectable young man was recently told he must give all his wealth to the poor, yet this woman, it seems, is praised for her wastefulness.\n\nJesus, however, did not honor Mary because she flattered him. He honored her to make a statement. His apprentices underestimate the hatred of the religious leaders. The recent miracle in Bethany and resultant popularity of Lazarus has galvanized the Sanhedrin. They believe their worst fears are about to be realized\u2014a sorcerer with the power to raise the dead is going to bring Israel to ruin.\n\nThe necessity of killing Jesus is a given, but now it appears more drastic action is required. The chief priests meet and decide that Lazarus must also die. He is a walking testimony to the power of the Nazarene. So long as Lazarus is alive, people will be favorably inclined toward Jesus. But the nation's religious leaders have a plan, and soon everyone will know the truth about the Galilean.\n23\n\nON A DONKEY'S COLT\n\nMarch, AD 33\n\nShout in triumph, O people of Jerusalem! Behold, your king is coming to you.\n\n\u2014Zechariah 9:9\n\nTHE TEMPLE IN Jerusalem is the highest point in the Holy City, but to the east, across the Kidron Valley, is a round ridge that was a place of worship long before the temple was built. Samuel, the last of the judges, worshiped on this mountain, as did King David. In Ezekiel's day, when the glory of the Lord departed the temple, it moved to this ridge. Zechariah prophesied the Messiah would reveal himself to the world from this mountain. Everyone in Israel is aware of the significance of the Mount of Olives, and it is not by coincidence that Jesus makes the mount his final earthly residence.\n\nHe is staying with his friends Lazarus, Martha, and Mary in the village of Bethany on the mount's eastern slope. For several days, Jesus has been within two miles of Jerusalem, but he has not yet entered the city. He is awaiting a specific day on the Jewish calendar: the tenth of Nisan\u2014the day each year when the lambs to be sacrificed on Passover are brought into Jerusalem.\n\nWhen the day arrives, Jesus tells his disciples they will be going to the city that afternoon for the presentation of the lambs. Walking west from Bethany, they are soon at the summit and approaching the town of Bethphage. Jesus tells two of his apprentices, \"Go into the village ahead. Upon entering, you will find a donkey tied there with her colt. Untie them and bring them to me. If anyone says anything to you, you are to say, 'The Lord needs them,' and he will release them at once.\"\n\nThe two hurry into the village and find the mother and foal where Jesus said they would be. When the disciples begin untying the donkeys, the animals' owner walks over and questions them.\n\nThe two men smile and repeat Jesus's words. The donkeys' owner nods and allows them to take the animals.\n\nThe two donkeys are brought to Jesus, and he tells his followers he will ride the foal into the city. His disciples do not understand why their master is so particular about how he enters the city, but they prepare the animals, and soon Jesus is riding the colt down the mountainside.\n\nThe road is filled with people coming from Jericho. Many are Galileans, and they recognize Jesus as he rides above the throng. Spiritual fervor is high, and some believe Jesus will be crowned king of the Jews in Jerusalem.\n\nJesus would have them recognize him as a king, but his choice of mount should alert them to the kind of king he is. The donkey symbolizes two things to the Jews: humility and peace. In olden days, a king went into battle astride a warhorse, but he rode a donkey when he came in peace.\n\nLong ago the prophet Zechariah predicted Zion's king would be gentle and come riding into Jerusalem on a donkey's colt, but he also prophesied that the future king's realm would extend \"to the ends of the earth.\"\n\nThe implications of Zechariah's prophecy are lost on the crowd, but they sense something extraordinary is happening and want to honor Jesus. Several remove their cloaks and spread them across the road. Others throw down palm branches and shout, \"Hosanna!\" (Give salvation now!).\n\nThe procession nears the city gates, and they are met by another large crowd coming out of Jerusalem. They, too, are waving palm branches and crying out, \"Hosanna! Blessed is he who comes in the name of the Lord, even the king of Israel!\"\n\nThe two groups merge, shouting, \"Hosanna to the Son of David!\" and \"Blessed is the coming kingdom of our father, David!\"\n\nThe noisy crowd draws the attention of the priests who oversee the shops on the western slope of the Mount of Olives, the traditional marketplace for animal merchants. Because it is Nisan 10, the day the sacrificial animals must be presented, the valley is filled with bleating lambs and worshippers seeking an acceptable substitute.\n\nThe priests and Pharisees hear the commotion and see the multitudes turning to Jesus. They shake their heads and say, \"It's useless. Look\u2014the world has gone after him.\"\n\nThe religionists had planned to keep quiet and bide their time, but when the crowd becomes effusive in their praise, they fear momentum is building to proclaim Jesus as Messiah. One of the Pharisees says, \"Teacher, rebuke your disciples.\"\n\nJesus smiles and says, \"I tell you, if they were silent, the stones would start shouting.\"\n\nThe procession nears the massive city walls. The hosannas continue, but Jesus grows melancholy and tears stream down his face. \"If only you had recognized on this day the things that lead to peace!\" he laments. \"But now it is hidden from your eyes. Indeed, the days will come upon you when your enemies will set up a barricade around you and surround you and hem you in on every side. They will dash you and your children within you to the ground, and they will not leave within you one stone upon another, because you did not recognize the time of your visitation.\"\n\nJesus is not prophesying a disaster to come in the distant future\u2014in less than forty years, Jerusalem will lie in ruins.\n\nJesus enters the city, and the crowds around him merge with the thousands of visitors filling the streets. His entry goes unnoticed at the busy gate, and the significance of his arrival atop the colt is understood by no one. Jesus dismounts the foal and walks up the temple steps. His apprentices follow tentatively. Death threats forced them to leave the city in haste a few months ago, and nothing has changed.\n\nThe disciples are relieved when the journey ends anticlimactically. Jesus walks around the plaza, surveys the activities, and leaves. He and his followers are soon out of the city and walking back up the hill to spend a quiet evening with Lazarus and his sisters.\n\nTHE NEXT MORNING Jesus rises early. After a time of prayer, he gathers the apostles and walks toward Jerusalem. As they near the summit of the Mount of Olives, Jesus sees a fig tree in full foliage a short distance from the road. He knows the unripe figs won't be palatable this time of year, but he's hungry and walks over to pluck a few of the paggim to eat on his way to Jerusalem. He is surprised to find the tree has no fruit but realizes the tree's deceptive barrenness can serve as a parable.\n\nHe says to the tree, \"May no one ever eat fruit from you again.\"\n\nFig trees are symbols of peace and prosperity and have long been associated with God's people. The prophets used the fig tree to report Israel's standing with God\u2014Hosea, Joel, and Micah all pleaded with the people to repent of their spiritual barrenness, but they refused time and again, leading God to pronounce through the prophet Jeremiah, \"There are no grapes on the vine, nor figs on the fig tree.\"\n\nFor Jesus, the words of the prophets are not a condemnation of Israel's people but an indictment against her leaders. They honor God with their words, but their hearts are far from Him.\n\nWithin an hour, Jesus is in the city and on the temple mount; he goes directly to the marketplace that has been set up beneath the Royal Stoa in the Court of the Gentiles. At the first Passover of his public ministry, Jesus lodged a loud, symbolic protest. He has not disrupted the temple's commerce since then because he knows the next protest will set in motion events that will put him on a cross, but the waiting is past.\n\nThe Passover is only two days away, and thousands are gathered in the temple plaza. Jesus strides through the marketplace. Tables are tossed, sheep pens are opened, and coins and animals scatter in all directions. Several of the merchants and money changers saw this happen three years ago and do not wait for Jesus to get to them; they fold up shop and melt into the crowd.\n\nJesus turns to the priests who are overseeing the commerce and says, \"Is it not written, 'My house shall be called a house of prayer for all the nations'? But you have made it a den of thieves!\"\n\nWhen Jesus describes them as thieves, he is not accusing them of stealing from others\u2014it is God they rob. The Lord's temple is to be a place of worship and prayer, not a bazaar and not a bank.\n\nJesus's rebuke is an invitation to repent, but the leaders aren't listening. The crowd at the temple, however, is taking in every word. His miracles amaze the audience, as does his knowledge of the Scriptures. The children who are present run through the crowd laughing and singing, \"Hosanna to the Son of David!\"\n\nThe chief priests and scribes are watching everything unfold, and when they hear the children singing messianic praises, their faces redden. They glare at Jesus and say, \"Do you hear what the children are saying?\"\n\nJesus smiles and says, \"Yes, and have you never read, 'Out of the mouth of infants and nursing babies you have prepared praise'?\"\n\nThe eighth psalm says God's glory is expressed in the cooing of babies; Jesus says this makes the children's praise appropriate. But the religious leaders cannot make the connection\u2014the psalm Jesus quoted from is a psalm of praise to God.\n\nIn the crowd that morning is a group of Gentiles. They were born and raised in a culture that has many gods but few morals, and the monotheistic worship and ethics of Judaism are appealing. The Jews refer to this kind of Gentile as a God-fearer. They are often referred to as \"half-proselytes\" by the devout because while they appreciate much that Judaism offers, they are unwilling to be circumcised and submit to the dietary laws and other restrictions.\n\nThey have heard Jesus teach and would like to meet him but are afraid to approach. They bring their request to one of his apostles. \"Sir,\" they say to Philip, \"we wish to see Jesus.\"\n\nThis creates a dilemma for Philip; he has heard the master say his mission is to the lost sheep of the house of Israel, and he is not sure if bringing the Gentiles to Jesus would be appropriate. He goes to his fellow apostle, Andrew, for counsel. Andrew, too, knows about Jesus's mission to the Jews, but he has also heard Jesus say he will turn away no one who comes to him. For Andrew, it's an easy decision because he loves introducing people to Jesus.\n\nPhilip and Andrew bring the Gentiles' entreaty to their master, but Jesus hears more than an appeal in their words.\n\n\"The hour has come for the Son of Man to be glorified,\" Jesus replies. \"Truly, truly, I tell you, unless a grain of wheat falls into the ground and dies, it remains alone, but if it dies, it bears much fruit.\"\n\nJesus uses the planting metaphor to explain why he is about to be glorified in a most unexpected way. Those listening struggle to understand. Rather than explain the metaphor, Jesus applies it to anyone interested in becoming his apprentice.\n\n\"Whoever loves his life will lose it,\" Jesus says. \"Yet whoever hates his life in this world will keep it for eternal life. If anyone serves me, let him follow me, and where I am, there my servant will be as well. If anyone serves me, the Father will honor him.\"\n\nThe promise of God's favor is welcomed, but the crowd has also heard Jesus speak of a cross. The teacher wants them to know their apprenticeship will ultimately result in blessings; meanwhile, there will be crosses.\n\nJesus's own cross will soon be a physical reality, and as the time draws near, the horror of what lies ahead is on his mind. \"Now is my soul troubled,\" he confesses. \"And what shall I say? Father, save me from this hour! But for this purpose I have come to this hour. Father\u2014glorify Your name!\"\n\nThe heavens begin to rumble. Some believe it is thunder and others think they have heard an angel, but it is the voice of the Father answering Jesus's prayer that His name be glorified.\n\n\"I have glorified it,\" the Voice proclaims, \"and I will glorify it again.\"\n\nJesus realizes that those who do not know God cannot recognize his voice, but they should at least recognize the preternatural rumbling as God's affirmation of Jesus's prayer.\n\n\"This voice has come not for my benefit but for yours,\" Jesus tells the crowd. \"Now is the judgment of this world; now the ruler of this world will be cast out. And I, when I am lifted up from the earth, will draw all people to myself.\"\n\nJesus says two things are about to happen simultaneously\u2014he will draw all people to himself, and Satan will be dealt a death blow. Both will be accomplished, Jesus says, when he is \"lifted up.\"\n\nThis term confuses the crowd; they understand it to be a metaphor for death on a cross, and this does not fit their perception of the Messiah. \"We heard from the Law that the Christ remains forever,\" a man says. \"So how can you say the Son of Man must be lifted up? Who is this Son of Man?\"\n\nThey want to hear more about the identity of the Son of Man. He can't be the Christ, because the Christ will not die, so what kind of being is this Son of Man?\n\nJesus does not answer their question. Instead, he says, \"The light is among you for a little while longer. Walk while you have the light, so that darkness will not overtake you. The one who walks in the darkness does not know where he is going. While you have the light, believe in the light, so that you may become children of light.\"\n\nThose who turn toward the light, Jesus says, will inherit the identity of the light. His message convinces some he is the Messiah, but they say nothing; they have leadership positions they would lose if they confessed their faith.\n\nJesus invites them to move from the shadows. \"Whoever believes in me, believes not in me but in the One who sent me. And whoever sees me sees the One who sent me. I have come into the world as light, so that whoever believes in me will not remain in darkness.\n\n\"If anyone hears my words and does not keep them, I do not condemn him, for I did not come to judge the world but to save the world. Whoever rejects me and does not receive my words has a judge: the word I have spoken will be his judge on the last day. For I have not spoken on my own authority, but the Father who sent me has Himself given me a commandment\u2014what to say and what to speak. And I know that His commandment is eternal life. Whatever I say, therefore, is just what the Father has told me to say.\"\n\nJesus says he is God's Word because he only speaks what the Father tells him to speak.\n\nThe crowd hears his claim, but most shake their heads and walk away.\n\nThe hour is late, and it has been a long day; Jesus and the apostles retire to Bethany.\n\nTHE NEXT MORNING they rise early and walk back down the Mount of Olives. They pass the fig tree Jesus had cursed the day before, and the apostles are shocked to see the large tree has withered. \"Rabbi, look!\" Peter says. \"The fig tree you cursed has withered.\"\n\nJesus looks into the faces of his apprentices and says, \"Have faith in God. I tell you truly, if you have faith and do not doubt, you will not only do what has been done to the fig tree, but even if you say to this mountain, 'Be taken up and thrown into the sea,' and you do not doubt in your heart but believe what you say will come to pass, it will be done for you. Therefore I tell you, whatever you ask in prayer, believe that you have received it, and it will be yours. And when you stand praying, forgive anything you have against anyone so that your Father in heaven will also forgive you your failings.\"\n\nThe disciples ponder Jesus's words as they walk to the city and climb the steps to the temple mount. The hour is early, but thousands of pilgrims are milling about the plaza.\n\nJesus goes into one of the stoas in the Court of the Gentiles and begins teaching. A short while later he is interrupted by the sounds of angry voices. He stands and sees several men dragging a half-dressed woman. She is thrown to the pavement before him. Jesus glances at the sobbing woman who lies shivering at his feet, and then looks into the eyes of the men who brought her, and waits.\n\n\"Teacher,\" the Pharisees say, \"this woman has been caught in the act of adultery. The Law of Moses commands us to stone such women. So what do you say?\"\n\nJesus shakes his head as they set their trap. He knows the only way they could have caught this woman in the physical act was by lying in wait. Instead of answering their question, Jesus bends to the ground and begins writing in the dust. The scribes and Pharisees press for an answer, but Jesus ignores them and continues to write on the ground.\n\nAfter a few minutes, Jesus stands and addresses the woman's accusers. He agrees that she should be executed\u2014so long as it is done according to the Scriptures.\n\n\"Let him who is without sin among you be the first to throw a stone at her.\"\n\nJesus then bends down and resumes writing in the dust. The prerequisite he calls for is not an attempt to make it impossible to execute her; it is the Law of Moses. When the lawgiver outlined the steps necessary to convict someone of an offense, a key stipulation was that those who make the accusation have no culpability in the offense. Jesus is not saying the first stone must be thrown by a sinless person; he is reminding them that the Law of Moses says the accusers cannot be guilty of twisting the truth in order to get a conviction.\n\nThe words have a profound effect on the woman's accusers. No one speaks, and no stones are picked up; the men all know they entrapped the woman. The Holy Spirit convinces each of his guilt, and one by one, starting with the eldest, the men who brought the charges slither into the crowd and disappear.\n\nWhen her accusers are gone, Jesus asks the adulteress to stand. \"Woman, where are they?\" he asks. \"Has no one condemned you?\"\n\nThe woman for the first time lifts her eyes. She examines the faces surrounding her and says, \"No one, sir.\"\n\n\"Neither do I condemn you,\" Jesus says. \"Go, and from now on, sin no more.\"\n\nThe woman has not sought forgiveness or shown any sign of repentance. Nevertheless, like many before her, she is a recipient of God's grace\u2014she is given an opportunity to change, no strings attached. What she does with that opportunity is up to her.\n24\n\nCONFRONTATION\n\nMarch, AD 33\n\n\"Woe to you, scribes and Pharisees. Hypocrites! You shut the kingdom of heaven in people's faces\u2014you won't go in yourselves, and you won't let anyone else in either. . . . You snakes, you brood of vipers, how can you escape being condemned to hell?\"\n\n\u2014Matthew 23:13, 33\n\nTHE DIFFERENT SECTS that make up the Sanhedrin squabble about even the smallest matters, but the priests, Pharisees, and elders are of one mind about Jesus. He must be silenced. A plan is devised, and a cadre of scholars and priests is assembled. The quasi-official delegation finds Jesus in the plaza teaching.\n\n\"By what authority are you doing these things?\" they demand. \"And who gave you this authority?\"\n\n\"I will ask you one question,\" Jesus says. \"Answer me, and I will tell you by what authority I do these things. The baptism of John\u2014where did it come from? Was it from heaven or from humans? Answer me.\"\n\nJesus is using the standard rabbinic technique of answering a question with a counterquestion that, if answered correctly, will provide the answer to the original question. If the inquisitors can identify the source of John's authority, they will have identified the source of Jesus's authority as well.\n\nThe priests and scholars, however, are not interested in determining truth; they have come to destroy a threat. They huddle and discuss how to respond, but can't agree. \"If we say, 'From heaven,' he will say, 'Then why didn't you believe him?' But if we say, 'From man,' all the people will stone us, for they are convinced John was a prophet.\"\n\nThe delegation from the Sanhedrin can't come up with a safe answer, so they feign ignorance. \"We do not know,\" they say.\n\n\"Then neither will I tell you by what authority I do these things,\" Jesus says. \"But what do you think? A man had two sons. He went to the first and said, 'Son, go and work in the vineyard today.' The son answered, 'I don't want to,' but later he changed his mind and went. The man went to his other son and said the same thing. 'I will, sir,' he answered, but he didn't go.\"\n\nJesus pauses, then asks, \"Which of the two obeyed his father?\"\n\nThe faces of the priests and scholars grow tight. \"The first,\" they answer warily.\n\nJesus nods and says, \"I tell you truly, the tax collectors and the prostitutes will go into the kingdom of God before you. For John came and showed you the way of righteousness, and you didn't believe him, but the tax collectors and the prostitutes believed him. And even when you saw this, you refused to change your minds and believe him.\"\n\nThe lesson about the identity of the Baptist has come full circle. John had spent many months in Judea, and the chief priests and prominent citizens Jesus is addressing are very familiar with John's message.\n\n\"Hear another parable,\" Jesus says. \"There was a landowner who planted a vineyard. He put a fence around it, dug a winepress in it, and built a watchtower. He then leased it to tenant farmers and traveled away. At harvesttime he sent a servant to the tenants to get some of the fruit of the vineyard, but the farmers beat him and sent him away empty-handed. He sent them another servant; they beat him on the head and treated him shamefully and sent him away empty-handed. He sent a third servant. They wounded and threw this one out also. He sent another, and him they killed. And so it went with many others\u2014some they beat and some they killed. Then the owner of the vineyard said, 'What shall I do? I will send my beloved son. Perhaps they will respect him.' But when the tenants saw the son, they said to one another, 'This is the heir. Come, let us kill him, and the inheritance will be ours.' And they took him and threw him out of the vineyard and killed him.\"\n\nJesus ends the parable abruptly and asks, \"When the owner of the vineyard comes, what will he do to those farmers?\"\n\nThe priests and scholars confer briefly and answer, \"He will bring those evildoers to an evil end and lease the vineyard to other farmers who will give him his share of the fruit at harvesttime.\"\n\nJesus nods and says, \"He will come and destroy those farmers and give the vineyard to others.\"\n\nThe crowd listening to the story is shocked by the behavior of the murderous tenants and several exclaim, \"May this never be!\"\n\nJesus looks into the faces surrounding him. Within forty-eight hours these same people will not only witness the death of the son, most in the crowd will vindicate the evil tenants who kill him.\n\nThe parable is the history of Israel in story form and is sourced in Isaiah's song of the unfruitful vineyard. Centuries earlier, God had prompted the prophet to write lyrics lamenting Israel's barrenness. \"My beloved had a vineyard on a very fertile hillside. He dug it and cleared it of stones and planted it with choice vines. He built a watchtower and dug out a winepress. He looked for it to yield grapes, but it yielded wild grapes. And now, O inhabitants of Jerusalem and men of Judah, judge between me and my vineyard\u2014what more was there to do for my vineyard that I have not done?\"\n\nThe promise Israel once possessed is likened to a vineyard that God richly provided for. It would be expected that a vineyard so well taken care of would yield much fruit, but the nation and her leaders have a long history of murdering the messengers of the vineyard owner.\n\nJesus looks the lawyers in the eye and challenges them. \"Have you never read in the Scriptures, 'The stone that the builders rejected has become the cornerstone; this was the Lord's doing, and it is wonderful to see'? Therefore I tell you, the kingdom of God will be taken away from you and given to a people producing its fruits.\"\n\nJesus quotes the famous Hallel psalm but gives it a different application. When Psalm 118 was written, the rejected stone represented Israel. Jesus applies the image to himself. He wants them to realize that a change is taking place in the vineyard. The wicked tenants\u2014the corrupt priesthood and elders who lead the nation\u2014will be removed, but Jesus does not say they will be replaced by another group of leaders. He says the kingdom will be given to a nation identified by Daniel as \"the people of the saints of the Most High.\" This new nation will be neither Jew nor Gentile but a community of all who follow Jesus. These will produce godly fruit and become the new vineyard.\n\nAll this, says the psalmist, is the Lord's doing.\n\nJesus shares another reference to the stone, this time from the prophet Isaiah. \"The one who falls on this stone will be broken to pieces, and when it falls on anyone, it will crush him.\"\n\nThe Jewish leaders bristle but say nothing, and Jesus tells them another story. \"The kingdom of heaven is like a king who prepared a wedding banquet for his son. He sent his servants to summon those who had been invited to the banquet, but they refused to come. So he sent other servants and said, 'Tell those who have been invited, \"Look, I have the meal ready. My oxen and fattened calves have been butchered, and everything is ready. Come to the wedding banquet.'\" But they made light of it and went their way, one to his farm, another to his business, while the rest of them seized the king's servants and abused and killed them.\n\n\"The king was furious. He sent out his troops and destroyed those murderers and burned their city. Then he said to his servants, 'The wedding banquet is ready, but those who were invited were not worthy. Go out to the street corners and invite everyone you find to the wedding.' So the servants went out into the streets and brought in everyone they could find, good and bad alike, and the wedding hall was filled with guests. But when the king came in to inspect the guests, he noticed a man who was not wearing wedding clothes and said to him, 'Friend, how did you get in here without wedding clothes?' The man was speechless. Then the king told the servants, 'Bind him hand and foot and throw him into the outer darkness, where there will be weeping and gnashing of teeth.' For many are called but few are chosen.\"\n\nThe scribes, chief priests, and elders realize they have been bested by the quick-witted Galilean and withdraw.\n\nWHEN WORD OF the defeat gets back to the ruling council, some Pharisees offer an ingenious plan to ensnare Jesus, but they are unwilling to initiate it. They still sting from their last attempt to trap him, so they enlist several men skilled in rhetoric to present the question as their own.\n\nThe sophists find Jesus in the temple plaza and approach obsequiously. \"Teacher, we know that you have integrity and teach the way of God accurately, and you do not care about anyone's opinion, for you are not swayed by status. Tell us, then, what you think\u2014is it lawful to pay the poll tax to Caesar or not?\"\n\nThe hirelings are confident they can damage the reputation of Jesus regardless of his answer. If he says the yearly tax to Rome should be paid, he can be discredited with the Jews, and if he says God's people are under no obligation to pay, he can be turned over to Pilate on the charge of sedition.\n\nJesus ignores the flattery and says, \"Why are you trying to trap me? Hypocrites! Show me the coin that is used for paying the poll tax.\"\n\nWhen one of the elders reaches into his pocket and pulls out a denarius, Jesus asks the crowd, \"Whose likeness and inscription is this?\"\n\nMany in the audience shout, \"Caesar's.\"\n\nJesus smiles and says, \"Then render to Caesar the things that are Caesar's, and to God the things that are God's.\"\n\nThe wordsmiths who brought the question concede and withdraw.\n\nTHE SADDUCEES HEAR of the Pharisees' failure and are impressed with Jesus's political savvy. They will take a different approach; they will expose the lay preacher's theological naivet\u00e9. The priests find Jesus in the plaza and have their servants make a path for them through the crowd. Soon they have pushed and shoved their way to the front.\n\nOne of the Sadducees smirks and says, \"Teacher, Moses said, 'If a man dies having no children, his brother is to marry the widow and raise up offspring for his brother.' Now there were seven brothers. The first married and died, and since he had no offspring, he left his wife to his brother. The same thing happened with the second and third, down to the seventh. Then last of all, the woman died. So when they rise again, whose wife will she be? All seven were married to her.\"\n\nThe hypothetical dilemma they propose is based on the Sadducees' belief that there will not be a bodily resurrection; they are convinced their question uncovers the absurdity of the doctrine.\n\nJesus shakes his head and says, \"You are wrong, because you know neither the Scriptures nor the power of God. The people of this age marry and are given in marriage, but in the age to come, those considered worthy of being raised from the dead will neither marry nor be given in marriage. Indeed, they can no longer die, because they are like angels and are children of God, being children of the resurrection.\n\n\"And as for the dead being raised, have you not read in the book of Moses, in the story about the bush, how Moses himself showed that the dead rise\u2014for God spoke to him, saying, 'I am the God of Abraham, and the God of Isaac, and the God of Jacob'? He is not the God of the dead but of the living, for all live to him. You are very wrong.\"\n\nThe priests are infuriated but can say nothing to refute Jesus's point. His simple grammar lesson has negated their entire argument.\n\nOthers from the Sanhedrin see the exchange and grudgingly admit, \"Teacher, you have spoken well.\"\n\nWHEN THE PRIESTS' failure becomes known, the Pharisees again take the initiative. Their ruse with the coin didn't work; perhaps a more theological approach will fare better. A small group of lawyers approach Jesus. A learned scholar presents the test question. \"Teacher,\" the scribe asks respectfully, \"what is the most important commandment in the Law?\"\n\nThe question issues from the rabbinic belief that the Law of Moses contains six hundred and thirteen commandments, and while all are important, some are more significant than others. The rabbis debate which commandments carry more weight and how these can be prioritized. The religious scholar is hoping Jesus will cite a commandment he can refute with a weightier one.\n\nJesus, however, goes to the essence of the Law. \"The most important one is, 'Hear, O Israel: The Lord our God, the Lord is one. And you shall love the Lord your God with all your heart and with all your soul and with all your mind and with all your strength.' This is the great and first commandment, and the second is like it: 'You shall love your neighbor as yourself.' All the Law and the writings of the prophets hang on these two commandments. No other commandment is greater than these.\"\n\nThe scribe shakes his head in appreciation and amazement. \"Well said, Teacher. You are right in saying He is one, and there is no other besides Him. To love Him with all your heart, and with all your understanding, and with all your strength, and to love your neighbor as yourself is more important than all the burnt offerings and sacrifices.\"\n\nJesus smiles and says, \"You are not far from the kingdom of God.\"\n\nThe lawyer blushes and looks to the ground, his spirit in turmoil.\n\nHis peers from the Sanhedrin can't rescue him. They, too, are frustrated.\n\nJesus knows they are at their collective wit's end and wants to encourage them to move beyond their superficial understanding of the Messiah. \"Why do the scribes say that the Messiah is the Son of David?\" Jesus asks. \"What do you think about the Messiah? Whose son is he?\"\n\n\"The son of David,\" they reply warily.\n\nJesus nods and asks, \"Then how is it that David, inspired by the Spirit, calls him Lord? For David himself says in the Book of Psalms, 'The Lord said to my Lord, \"Sit at my right hand until I make your enemies your footstool.\"' David himself calls him Lord, so how is the Messiah his son?\"\n\nJesus is not challenging the fact that the Messiah would be a descendant of David; he is explaining the unique relationship between the two. The term Son of David is accurate in one sense. Jesus is David's physical progeny. But ultimately it is an inadequate title. The Jews have yet to understand that the Messiah is not only David's descendant\u2014he is also David's Lord.\n\nThe religious scholars can't explain why David would address his son as Lord, and Jesus turns to the audience. \"Beware of the scribes and the Pharisees who sit on Moses's seat,\" he warns. \"Do and observe whatever they tell you, but don't do as they do, because they preach but don't practice. They tie up heavy, cumbersome burdens and put them on other people's shoulders, but they are not willing to lift a finger to move them. Everything they do is for show\u2014they make their phylacteries broad and their fringes long, and they like to walk around in long robes. They love the place of honor at banquets and the best seats in the synagogues and greetings in the marketplaces and being called rabbi by others. They devour widows' houses and then make a show of saying long prayers. They will receive harsher condemnation.\"\n\nThe warning is blunt: honor the position the religious leaders hold as spokespersons for God and honor the teachings of Moses they expound, but be discerning and do not imitate their lifestyle\u2014the teachers have no integrity and don't practice what they preach.\n\nMany in the crowd nod in agreement. Jesus's assessment of the religious leaders is harsh but accurate. The people have seen how the rabbis relish celebrity status.\n\nJesus admonishes his followers to avoid honorific titles. \"You are not to be called 'Rabbi,' for you have one teacher, and you are all classmates. And call no one on earth 'Father,' for you have one Father, who is in heaven. Nor are you to be called 'Master,' for you have one master\u2014the Messiah. The greatest among you is to be your servant. Whoever lifts himself up will be brought low, and whoever lowers himself will be lifted up.\"\n\nJesus is addressing a challenge he knows his church will battle throughout history\u2014status seeking\u2014the desire to be \"lifted up.\" Jesus mentions three honorific titles of his day\u2014rabbi, father, and master\u2014to make a point about all titles: they are part of the world's system and have no value in God's kingdom.\n\nJesus turns again to Israel's religious rulers and tells them how God sees them. \"Woe to you, scribes and Pharisees. Hypocrites! You shut the door to the kingdom of heaven in people's faces. You won't go in yourselves, and you won't let anyone else in either.\n\n\"Woe to you, scribes and Pharisees. Hypocrites! You travel across land and sea to recruit a single proselyte, and then you make the new convert into twice the child of hell you yourselves are.\n\n\"Woe to you, blind guides! You say, 'If anyone swears by the temple, it means nothing, but if anyone swears by the gold in the temple, he is bound by his oath.' You blind fools! Which is greater\u2014the gold or the temple that makes the gold sacred? You also say, 'If anyone swears by the altar, it means nothing, but if anyone swears by the offering that is on the altar, he is bound by his oath.' How blind you are! Which is greater\u2014the offering, or the altar that makes the offering sacred? So whoever swears by the altar swears by it and by everything placed on it, and whoever swears by the temple swears by it and by the One who dwells in it. And whoever swears by heaven swears by the throne of God and by the One who sits on it.\n\n\"Woe to you, scribes and Pharisees. Hypocrites! You tithe your mint and dill and cumin, but you have neglected the weightier matters of the law: justice and mercy and faithfulness. These you ought to have done while not neglecting the others. You blind guides\u2014you strain out a gnat but swallow a camel!\n\n\"Woe to you, scribes and Pharisees. Hypocrites! You clean the outside of the cup and the plate, but inside they are full of greed and self-indulgence. Blind Pharisee! First clean the inside of the cup so that its outside may be clean as well.\n\n\"Woe to you, scribes and Pharisees. Hypocrites! You are like whitewashed tombs that appear beautiful from the outside, but inside they are full of the bones of the dead and all kinds of uncleanness. It's like that with you\u2014outwardly you give people the impression that you're righteous, but inside you are full of hypocrisy and lawlessness.\n\n\"Woe to you, scribes and Pharisees. Hypocrites! You build the tombs of the prophets and decorate the graves of the righteous, and you say, 'If we had lived in the time of our ancestors, we wouldn't have taken part with them in shedding the blood of the prophets.' Thus you testify against yourselves that you are the descendants of those who murdered the prophets. Fill up, then, the measure of your ancestors!\n\n\"You snakes, you brood of vipers, how can you escape being condemned to hell? Look, that is why I send you prophets and wise men and scribes. Some you will kill and crucify, and some you will flog in your synagogues and persecute in one town after another, so that you may bear responsibility for all the righteous blood that is being shed on earth, from the blood of righteous Abel to the blood of Zechariah the son of Barachiah, whom you murdered between the sanctuary and the altar. I tell you truly, all these things will come upon this generation.\"\n\nThe audience is accustomed to hearing scathing pronouncements; in the Scriptures the prophets often rebuked those who did evil. The crowd is not shocked by the strong invective, but they are surprised by the breadth of the charges. Jesus describes a system that is hopelessly opposed to God and tells the religious leaders to \"fill up the measure\" of their ancestors\u2014to go ahead and complete what their forefathers started.\n\nThe conversation is painful; Jesus loves these people, even those who oppose him, and he laments the future they have chosen for themselves. \"Jerusalem, Jerusalem, who kills the prophets and stones those who are sent to her, how often have I wanted to gather your children together as a hen gathers her chicks under her wings, and you were not willing. Behold, your house is left to you desolate. For I tell you, you will not see me again until you say, 'Blessed is he who comes in the name of the Lord.'\"\n\nJesus says God's judgment on the temple is irrevocable\u2014God has abandoned it for another. But those who are listening hear a note of hope for Israel. In the psalm that Jesus quoted, God expresses time and again a \"steadfast love that endures forever\" for His people.\n\nJudgment is coming, Jesus warns, but it is not too late to turn. The priests and people still have time to say with the psalmist, \"Save us, we pray, O Lord! Blessed is he who comes in the name of the Lord!\"\n\nWhen Jesus sees no remorse in the faces of the religious leaders, he walks across the broad plaza and climbs the steps to the Court of the Women, where the temple treasury is located. In the courtyard stand thirteen shofar-shaped chests that collect the people's freewill offerings. Beyond are the rooms where designated offerings are accepted. The courtyard is filled with people bringing money to the temple. Jesus sits on a bench across from the collection trumpets. A number of wealthy men parade by, their gifts announced with pomp and gravity by the priests overseeing the treasury.\n\nInto the bustling throng walks a destitute widow. She slowly makes her way to one of the receptacles and drops in two lepta, the smallest coins in circulation. Sixty-four lepta equal one denarius, which is the average daily wage, so the woman's offering is equivalent to fifteen minutes of work, as small a sum as the priests have seen all day.\n\nJesus has a different perspective. He turns to his disciples and says, \"I tell you truly, this poor widow put more into the treasury than all the others. They all contributed out of their surplus, but she, out of her poverty, has put in everything she had\u2014all she had to live on.\"\n\nJesus views the widow's mites as a greater contribution to the treasury than the large sums being cast into the collection boxes because God takes the motivation of the giver into account when assessing an offering's value. And this woman's gift, Jesus says, is priceless.\n25\n\nA SECOND COMING\n\nApril, AD 33\n\nOn that day his feet shall stand on the Mount of Olives to the east of Jerusalem, and the Mount of Olives will be split in two.\n\n\u2014Zechariah 14:4\n\nFORTY MONTHS HAVE passed since Jesus was baptized in the Jordan. Few are convinced he is the Messiah; he does the works foretold of the Anointed One, but his words are those of the Suffering Servant. Even his most devoted followers do not understand the man they call Lord.\n\nThe nation's religious leaders, on the other hand, know all they need to know about Jesus\u2014he is a threat to the nation. If he is proclaimed Messiah by the large crowds during the Passover, there will be riots in Jerusalem and harsh repercussions from Rome.\n\nThe priests and Pharisees have no need to worry. Jesus has preached his last message on the temple mount. He gathers the twelve and walks toward the Eastern Gate. As he and the apostles cross the courtyard, one of the men gestures toward the temple and says, \"Look, Teacher, what impressive stones and what magnificent buildings!\"\n\nJesus stops and glances up\u2014the afternoon sun reflecting off the white marble walls and gold adornments gives the temple the appearance of a mountain afire.\n\n\"Do you see these great buildings?\" Jesus asks. \"I tell you truly, the days will come when not one stone will be left upon another\u2014it will all be demolished.\"\n\nThe disciples are shocked by the prediction and remain silent as they walk from the temple down into the Kidron Valley and up the Jericho Road.\n\nAs they near the summit of the Mount of Olives, Jesus leads the twelve off the road. They walk a short distance, and he invites the apostles to sit and rest a few minutes. From this height they can see the entire temple plaza, and because the entrance to the Holy Place faces east, they can look directly into the darkness that cloaks the Holy of Holies.\n\nAs the twelve look at the temple, they reflect on what Jesus said earlier. They cannot imagine life without the temple and assume its destruction will initiate the revealing of the Messiah. Peter, Andrew, James, and John walk over to where Jesus is resting and sit down beside him. \"Teacher, tell us, when will these things happen, and what will be the sign of your coming and the completion of the age?\"\n\nJesus sees the confusion in their hearts and says, \"Watch out. Don't let anyone mislead you, for many will come in my name saying, 'I am the Messiah!' and 'The time is near!' They will lead many astray\u2014do not follow them.\n\n\"And when you hear of wars and rumors of war and revolutions, do not be alarmed, for these things must take place first, but the end will not follow immediately. For nation will rise against nation and kingdom against kingdom, and there will be famines and plagues and earthquakes in various places, and there will be terrors and great signs from heaven.\"\n\nJesus speaks of natural disasters, social turmoil, and political upheavals that will take a long time to germinate, develop, and come to fruition. He tells his apprentices that there will be a span of years before the destruction of the temple, and another span of years before the consummation of the age. Key events must happen before each can take place\u2014events the faithful must be prepared for and participate in.\n\nJesus explains: \"Before all this occurs, they will seize you and persecute you, but these are just the beginning of the birth pains. Be on your guard, for you will be handed over to the local councils, and you will be beaten in synagogues and prisons, and you will be brought before governors and kings\u2014all on account of my name.\n\n\"This will turn out to be an opportunity to tell them about me, so when they arrest you and bring you to trial, settle it in your minds not to worry or prepare your defense in advance. Just say whatever is given you at that time, for it is not you who speaks but the Holy Spirit. I will give you words and a wisdom that none of your adversaries will be able to withstand or contradict.\n\n\"Then they will deliver you to be persecuted and put to death, and you will be hated by all nations because of my name. At that time many will be ensnared and betray one another and hate one another, and many false prophets will appear and deceive many. You will be betrayed even by parents and relatives and friends, and some of you they will put to death. But not a hair of your head will perish.\n\n\"Because of rampant lawlessness, the love of many will grow cold, but the person who remains faithful to the end will be saved. And this good news of the kingdom must be proclaimed all over the world as a testimony to all the nations\u2014then the end will come.\"\n\nJesus explains the role his followers play in a much larger story\u2014the redemption of all creation. Jesus warns them that the way will not be trouble-free. Persecution lies ahead, and the mission, which until now has been limited to Palestine, must be expanded to the entire world before the end will come.\n\nThe apostles cannot grasp the scope of the assignment. While the twelve have never traveled beyond their homeland, they know the earth is vast and filled with people of many tribes and languages. With a mission this large, there is no way to know how long it will take, but they know the end will not happen immediately, and it will not come gently.\n\nTo prepare his followers for the future, Jesus speaks of two occurrences: the destruction of the temple and the consummation of the age. He wants them to see the first event as a pattern and portent of the second.\n\n\"When you see Jerusalem surrounded by armies,\" Jesus says, \"then know that its desolation has come near. When you see what Daniel the prophet spoke about\u2014the abomination of desolation standing in the holy place\u2014then those in Judea must flee to the mountains. Those who are in the city must leave, and those in the country must not return to the city. The one on the housetop must not go down into the house to pack, and the one who is in the field must not return even to get his cloak, for these are days of vengeance, to fulfill all that is written.\n\n\"How dreadful it will be for pregnant women and nursing mothers in those days, for there will be great distress upon the land and wrath upon this people. They will fall by the edge of the sword and be taken away as captives into all nations, and Jerusalem will be trampled underfoot by the Gentiles until the times of the Gentiles are fulfilled.\n\n\"Pray that you won't have to escape in winter or on the Sabbath, for in those days there will be tribulation greater than at any time since the world began. There will never be anything like it again, and if those days had not been cut short, no one at all would be saved, but for the sake of God's chosen ones, those days will be cut short.\n\n\"Then if anyone tells you, 'Look, here is the Messiah!' or 'Look, there he is!' do not believe it,\" Jesus says. \"For false messiahs and false prophets will appear and perform great signs and wonders to deceive, if possible, even the elect.\n\n\"Behold\u2014I have forewarned you. So, if they tell you, 'Look, he is in the wilderness,' don't go out there. If they say, 'Look, he is in the inner rooms,' don't believe it. For as lightning flashes across the sky from east to west, so will be the coming of the Son of Man.\"\n\nJesus's words bewilder the apostles. He quotes the prophet Daniel's prediction about the abomination of desolation and tells his disciples to see this desecration of the temple as a sign that it is time to flee the city. Jerusalem is about to experience divine judgment. Both the city and the temple will be destroyed.\n\nJesus wants his followers to see the near-term destruction of Jerusalem as an omen. His reference to the coming of the Son of Man is a fulfillment of Daniel's prophecy and refers to the end of the age, the time when God will set up \"the kingdom that will never be destroyed.\"\n\nIt is to the establishment of this kingdom Jesus now turns his attention: \"But immediately after the tribulation of those days there will be signs in sun and moon and stars\u2014the sun will be darkened, and the moon will not give its light, and the stars will fall from heaven\u2014and here on earth the nations will be in turmoil, perplexed by the roaring seas and waves. People will faint from fear and foreboding of what is coming upon the world, for the powers of the heavens will be shaken.\n\n\"Then will appear in heaven the sign of the Son of Man, and all the tribes of the earth will mourn as they see the Son of Man coming on the clouds of heaven with power and great glory. He will send out his angels with a loud trumpet blast, and they will gather his chosen people from the four winds\u2014from the ends of the earth to the ends of heaven.\n\n\"Now, when these things begin to take place, straighten up and raise your heads because your redemption is drawing near.\"\n\nThe apostles listen with increasing amazement. Jesus predicts a day when evil will run rampant and many Christ-followers will fall away. The prophecies from the scrolls of Isaiah, Ezekiel, and Joel are familiar, but the disciples do not know if Jesus is speaking literally or figuratively.\n\nThe prophets used these images figuratively to point to political and spiritual disturbances, and God's wrath being visited on those who oppose him, but Jesus also speaks of disturbances in the physical heavens. He says his arrival will be like lightning, that he will come on the clouds, and that amid trumpet blasts he will summon the redeemed of all the ages to join him in the air as he descends to the earth.\n\nThis is not the description of a future the apostles have ever heard or even imagined. Yet Jesus assures them that the cataclysmic events he predicted will come to pass.\n\n\"Now learn this lesson from the fig tree and all trees,\" Jesus says. \"When their tender shoots come out and the leaves appear, you know that summer is near. In the same way, when you see these things taking place, know that the kingdom of God is near\u2014right at the door. I tell you truly, this generation will not pass away until all these things happen. Heaven and earth will pass away, but my words will not pass away.\"\n\nJesus says that all the travails must take place before the end, but it is not clear to the apostles whether he is talking about the end of the temple, the end of the world, or both.\n\n\"But concerning that day and hour,\" Jesus says, \"no one knows, not even the angels of heaven, nor the Son, but only the Father. Be on guard so that your hearts are not weighed down with debauchery and drunkenness and the worries of this life. Otherwise, that day will come upon you suddenly, like a trap, for it will come upon all who dwell on the face of the earth. Be always on the watch, and pray that you might have the strength to escape all that is about to happen and to stand before the Son of Man.\"\n\nJesus says God's people must be watchful and faithful; his coming will find many unprepared. \"Just as it was in the days of Noah, so will it be at the coming of the Son of Man. The people knew nothing about what would happen. They were eating and drinking and marrying and being given in marriage\u2014until the day Noah entered the ark and the flood came and destroyed them all. That's the way the coming of the Son of Man will be.\n\n\"It was the same in the days of Lot. People were eating and drinking, buying and selling, planting and building\u2014but on the day Lot left Sodom, fire and sulfur rained from heaven and destroyed them all. So will it be on the day when the Son of Man is revealed.\n\n\"On that day, anyone on the housetop who has belongings in the house must not go down to get them, and anyone who is in the field must not return home. Remember Lot's wife.\n\n\"I tell you, in that night there will be two in one bed. One will be taken, the other left. Then two men will be in the field; one will be taken and one left. Two women will be grinding at the mill; one will be taken and the other left. So stay awake, for you don't know what day your Lord is coming.\"\n\nThe apostles are confused; they want to participate in the conversation but can think of only one question: \"Where, Lord?\"\n\nOut of curiosity, they ask where this judgment will take place, but in Jesus's mind the question has no relevance. He knows that when the time comes, the scavengers will reveal the location. \"Where the carcass is,\" Jesus says, \"there the vultures will gather.\"\n\nAccording to Jesus it will be a time of great joy for those who are ready, but a day of untold horrors for those who are not. Jesus tells three parables to emphasize the importance of being prepared.\n\n\"At that time the kingdom of heaven will be like ten bridesmaids who took their lamps and went to meet the bridegroom. Five were foolish and five were wise. When the foolish girls took their lamps, they took no oil with them, but the wise bridesmaids took flasks of oil with their lamps.\n\n\"When the bridegroom was delayed, they all became drowsy and fell asleep, but at midnight there was a shout: 'Look! Here comes the bridegroom! Go out to meet him.'\n\n\"Then all the bridesmaids woke up and got their lamps ready, and the foolish said to the wise, 'Give us some of your oil. Our lamps are going out.'\n\n\"But the wise answered, 'No. We don't have enough for all of us. Go to the store and buy some for yourselves.'\n\n\"But while they went to buy it, the bridegroom came, and those who were ready went with him to the wedding celebration and the door was shut.\n\n\"Later the other bridesmaids came saying, 'Lord, lord, open up for us!'\n\n\"But he answered, 'I tell you truly, I do not know you.'\"\n\nJesus pauses, looks into the faces of his followers, and says, \"So stay awake, for you know neither the day nor the hour.\"\n\nJesus is teaching about personal accountability, and like many of his stories, it has a twist at the end. At a normal wedding, the doors would not be locked, and tardy bridesmaids would not be refused entrance by the groom. But this is not simply a story about a wedding\u2014it is an explanation of the consummation of the age.\n\nFurther, the parable implies the bridegroom's return will be delayed. In the interim, people are to ready themselves for his arrival. The foolish bridesmaids did not take their relationship with the bridegroom seriously, and despite being afforded every opportunity, they chose to ignore the one they claimed to be waiting for.\n\nJesus says that many offer God only lip service; they claim to be followers, but they will not repent and prepare, and spiritual readiness is not something that can be borrowed. All souls must prepare to meet their God.\n\nJesus says, \"It will be like a man going on a journey who summoned his servants and entrusted his possessions to them. To one he gave five talents, to another two, to another one\u2014to each according to his ability. Then he went away. The servant who had received five talents went at once and traded with them and made five more. In the same way, the servant who was given two made another two. But the one who received one talent went off, dug a hole in the ground, and buried his master's money.\n\n\"A long time later the master of those servants returned and settled accounts with them. The one who had received five talents came bringing five talents more, saying, 'Master, you entrusted me with five talents. Look, I gained another five talents.'\n\n\"His master said to him, 'Well done, good and faithful servant. You were trustworthy with a few things; I will put you in charge of many things. Enter into the joy of your master.'\n\n\"And the one who had the two talents also came forward, saying, 'Master, you entrusted me with two talents. Look, I gained another two talents.'\n\n\"His master said to him, 'Well done, good and faithful servant. You were trustworthy with a few things; I will put you in charge of many things. Enter into the joy of your master.'\n\n\"Then the servant who had received the one talent approached saying, 'Master, I knew you to be a hard man, reaping where you did not sow and gathering where you did not scatter seed, so I was afraid, and I went and hid your talent in the ground. Look, here is what belongs to you!'\n\n\"But his master answered him, 'You wicked and slothful servant! So you knew that I reap where I have not sown and gather where I have not scattered seed? Then you should have invested my money with the bankers; then at my coming I would have gotten it back with interest.\n\n\"'So take the talent from him and give it to the one who has the ten talents, for to everyone who has, more will be given, and he will have an abundance. But from the one who has not, even what he has will be taken away. And throw the worthless servant into the outer darkness. There will be weeping and gnashing of teeth.'\"\n\nThe disciples are familiar with this parable; Jesus has told it before in a slightly different form. Jesus wants his followers to realize that everything they possess ultimately belongs to God: wealth, health, intelligence, abilities, even life itself. Each person is given a portion of God's abundance, and one day there will be a reckoning. Those who make excuses and do nothing with their God-given gifts demonstrate they have no interest in living in God's kingdom.\n\nJesus tells his apprentices a third story to underline the urgency of responding to his invitation. \"When the Son of Man comes in his glory, and all the angels with him, he will sit on his glorious throne. All the nations will be assembled in front of him, and he will separate people one from another as a shepherd separates the sheep from the goats.\n\n\"He will put the sheep on his right and the goats on his left. Then the King will say to those on his right, 'Come, you who are blessed by my Father, inherit the kingdom prepared for you from the foundation of the world. For I was hungry and you gave me something to eat; I was thirsty and you gave me something to drink; I was a stranger and you made me your guest; I needed clothes and you clothed me; I was sick and you visited me; I was in prison and you came to me.'\n\n\"Then the righteous will answer him, saying, 'Lord, when did we see you hungry and feed you, or thirsty and give you a drink? And when did we see you as a stranger and invite you in, or needing clothes and clothed you? And when did we see you sick or in prison and go to visit you?'\n\n\"And the King will reply, 'I tell you truly, whatever you did for one of the least of these brothers and sisters of mine, you did for me.'\n\n\"Then he will say to those on his left, 'Depart from me, accursed ones, into the eternal fire prepared for the Devil and his angels. For I was hungry and you gave me nothing to eat; I was thirsty and you gave me nothing to drink; I was a stranger and you did not make me your guest; I needed clothes and you did not clothe me; I was sick and in prison and you did not visit me.'\n\n\"Then they, too, will answer, 'Lord, when did we see you hungry or thirsty or a stranger or needing clothes or sick or in prison and not help you?'\n\n\"Then he will answer them, 'I tell you truly, whatever you did not do for one of the least of these, you did not do for me.'\n\n\"And these people will go away into eternal punishment, but the righteous into eternal life.\"\n\nThe apostles struggle to understand the point of the story. Jesus describes a final judgment when there will be only two kinds of people present\u2014those who received God's messengers and those who rejected them. Both groups will be surprised by how closely Jesus identifies with his disciples. According to Jesus, eternal destinies will be determined on the basis of how a person treats his followers. This is not a new teaching. Jesus is reaffirming a lesson he taught when the twelve and seventy-two were sent out: to reject the messenger is to reject the message, and to reject the message is to reject the One who sent it.\n\nHis apprentices are still trying to grasp Jesus's pronouncements when he says, \"As you know, in two days it will be the Passover, and the Son of Man will be handed over to be crucified.\"\n\nThis prediction confuses his disciples even more. Jesus says he will soon be dead, yet he seems perfectly content with how things are unfolding. Why? How can the Messiah's death accomplish anything?\n\nWHILE THE APOSTLES sit on the Mount of Olives listening to the prophecy, Caiaphas is at his home in Jerusalem consulting with a group of priests and elders about how to silence the Galilean. Some advocate an immediate arrest; others fear the crowds and counsel prudence. \"Not during the festival,\" they say. \"The people may riot.\"\n\nWhat the ruling council needs is an insider who can provide them with information they can use to arrest Jesus away from the crowds.\n\nLater that evening their prayers are answered. When Judas Iscariot appears at the high priest's door, the religious leaders can hardly believe their good fortune.\n\nThe apostle gets right to the point. \"How much will you give me if I hand Jesus over to you?\"\n\nThey briefly haggle price, but the betrayer's demands are modest; he simply wants to be done with Jesus. They settle on thirty pieces of silver, the value of a slave in Moses's day.\n\nJudas instructs the priests to have the temple guard ready to move quickly. He then goes out into the night and back up the Mount of Olives to join Jesus and his friends. As he climbs the hill, he contemplates his decision to betray the man he pledged allegiance to and wonders how it will all end.\n26\n\nTHE LORD'S SUPPER\n\nApril 2, AD 33\n\n\"I will bring you out . . . . I will deliver you . . . . I will redeem you . . . . I will take you as My own.\"\n\n\u2014Exodus 6:6\u20137\n\nTHURSDAY MORNING FINDS Jesus and the twelve in Bethany. Last night's teaching went late, and the apostles are looking forward to a day of rest. Because the Jewish day is reckoned from sunset to sunset, this evening is technically the start of Hag ha Pesah, but the significant activities of the Passover won't begin until tomorrow, when the sacrificial lambs are slaughtered.\n\nThe Torah dictates that the lambs are to be sacrificed in the last hours of Nisan 14. The lamb is then flayed and roasted whole. The other foods are prepared while the lamb cooks, and the meal is eaten that evening after the date changes to Nisan 15, the first day of Hag ha Matzot, the festival of Unleavened Bread.\n\nThe apostles assume they will be going to Jerusalem early tomorrow to make arrangements for the Seder, but shortly after sundown Jesus tells Peter and John, \"Go and prepare for us to eat the Passover.\"\n\nThe request mystifies the two apostles. The Seder could be prepared this evening, but where would they get the lamb, the centerpiece of the entire meal? It will be impossible to offer a sacrifice at the temple until tomorrow afternoon. Why does the master want to celebrate the Passover a day early and without the lamb?\n\nPeter and John have questions, but all they ask is, \"Where will you have us prepare it?\"\n\n\"As you enter the city,\" Jesus replies, \"a man carrying a jar of water will meet you. Follow him into the house he enters and say to the master of the house, 'The Teacher says to tell you, \"My time is at hand. Where is the guest room where I may eat the Passover with my disciples?\"' He will show you a large, furnished upper room; prepare it there.\"\n\nPeter and John again exchange inquiring glances. Apparently the master has everything arranged except the food preparation. The two men are soon on their way to Jerusalem. They enter the city gates and see an unusual sight\u2014a man carrying a water jar. The two apostles follow the man into a home. They repeat Jesus's words to the homeowner and are immediately taken to a spacious upper room that is fully furnished for a banquet.\n\nThe apostles thank the man for his kindness and walk to the market to purchase fruits, vegetables, wine, bread, salt, and herbs. They take the food back to the upper room and prepare the meal. The bitter herbs are chopped, the wine is mixed with the right amount of water, the fruits for the charoseth are stewed, the bread is wrapped in a napkin, dipping sauces are made, and the table is set.\n\nIt is well after dark when Jesus and the other apostles enter Jerusalem and make their way to the upper room.\n\nAt formal banquets, dinner guests do not sit in chairs at tables; they recline on pillows around a low, U-shaped banquet table, propping themselves on their left elbow. This leaves only one hand free to eat with, so meals traditionally consist of bread, dipping sauces, and bite-size finger foods.\n\nJesus sits at the head of the table, but no other seats are assigned. The apostles may sit where they choose. There is room for one on either side of Jesus. John bar Zebedee takes one of those places, Judas Iscariot the other.\n\nWhen everyone is settled in, Jesus says, \"I have eagerly desired to eat this Passover with you before I suffer. For I tell you, I will not eat this meal again until it finds fulfillment in the kingdom of God.\"\n\nThe twelve shift uncomfortably and look at one another. No one wants to ask Jesus to explain the cryptic comment, and Jesus is content to begin the Seder by pouring the first of the four cups of wine to be shared during the meal. He offers a prayer of thanksgiving for God's provision and passes the cup to the apostles saying, \"Take this and divide it among yourselves, for I tell you that I will not drink again of the fruit of the vine until the kingdom of God comes.\"\n\nThe apostles share the cup and ponder Jesus's ominous words, but say nothing.\n\nJesus stands and removes his outer garments. The twelve have no idea why their master is disrobing. They watch incredulously as he strips to his loincloth, wraps a long towel around his waist, pours a basin of water, and begins washing the apostles' feet.\n\nThe men are red-faced and speechless. Jesus is performing one of the most demeaning jobs imaginable. When he comes to Simon Peter, the exasperated apostle says, \"Lord, are you going to wash my feet?\"\n\nJesus nods and says, \"You don't understand now what I'm doing, but after these things you will understand.\"\n\nThere is much that will not become clear until later, and Jesus counsels patience. Simon isn't listening. \"No,\" he says. \"You will never wash my feet.\"\n\nAll Peter can see is the social inappropriateness of the act. Jesus should be doing works that befit his royal status, not performing the menial chores of a slave. But the chief apostle is misinterpreting his master's actions.\n\nJesus looks into Peter's eyes and says, \"Unless I wash you, you have no share with me.\"\n\nPeter is chastened but continues to misunderstand. \"Lord,\" he says, \"then wash not only my feet but my hands and my head as well!\"\n\n\"A person who has had a bath is completely clean,\" Jesus replies. \"He only needs to wash his feet.\"\n\nJesus looks around the table and adds, \"And you are clean, but not every one of you.\"\n\nJudas tenses, but Jesus says no more. Jesus finishes his work, removes the towel, dresses, and resumes his place at the table. He looks into twelve bewildered faces and says, \"Do you understand what I've done for you? You call me 'Teacher' and 'Lord,' and you are right, for this is what I am. So if I, your Lord and Teacher, have washed your feet, you also ought to wash one another's feet. I have given you a pattern\u2014what I've done, you do. Truly, truly I tell you, a servant is not greater than his master, nor is a messenger greater than the one who sent him. Now that you know these things, you will be blessed if you do them.\n\n\"I am not speaking about all of you; I know the ones I have chosen. But this is to fulfill the Scripture, 'The one who eats my bread has lifted his heel against me.'\n\n\"I am telling you this now, before it happens, so that when it does happen you will believe that I am he. Truly, truly I tell you, whoever receives the one I send receives me, and whoever receives me receives the One who sent me.\"\n\nThe twelve look at one another; eleven of the apostles are convinced there has been a mistake, but Jesus is unswerving. \"Truly, truly I tell you, one of you will betray me\u2014one who is eating with me. The Son of Man goes as it is written of him, but woe to the one who betrays him! It would have been better for that man if he had not been born.\"\n\nEleven of the men have no idea who Jesus is referring to and are plunged into sorrow. They begin asking one another who would do such a thing. Several turn to Jesus, eyes wide, voices hushed. \"Is it I, Lord?\"\n\nJesus won't identify his betrayer, and Simon Peter grows frustrated; he motions to John to ask for more information.\n\nJohn turns to Jesus and whispers, \"Lord, who is it?\"\n\nJesus picks up a piece of matzoh and says, \"It's the one I'll give this morsel of bread to when I have dipped it.\"\n\nHe leans forward, sops the bread in the sauce, and gives it to the apostle on his right. Judas Iscariot is writhing inside but maintains a calm demeanor. He smiles and accepts the morsel, but feels obligated to say something, so he asks the question everyone else has asked, \"Is it I, Rabbi?\"\n\nJesus looks into the eyes of his betrayer and nods. \"You have said so. What you are going to do, do quickly.\"\n\nWithout a word, Judas gets up and leaves the room. The eleven overhear Jesus's instructions and assume that because Judas is in charge of the money box he is being sent to purchase supplies or give alms; at Passover the temple courts are kept open all night and are filled with helpless people seeking aid.\n\nJesus says nothing about Judas's sudden departure, but he knows where Judas is going because Judas didn't leave the room alone\u2014his benefactor went with him.\n\nJesus became aware of the presence of Satan as soon as he entered the room. The evil angel has been waiting a long time for this night. Jesus may have bettered him in the wilderness, but tonight things will be different. The arch-demon has carefully laid a trap that is sure to end with Jesus dead. And when Judas takes the bread and squanders his last chance to repent, Satan gleefully enters the apostle. And it is to both Judas and his benefactor Jesus issues the command: \"What you are going to do, do quickly.\"\n\nWITH THE TWO now unwittingly on their way to fulfill God's plan, Jesus turns to the eleven. He picks up a new loaf of unleavened bread, prays for the Father's blessing, and breaks it. Each disciple is given a piece. \"Take, eat,\" Jesus says. \"This is my body, which is given for you. Do this in remembrance of me.\"\n\nThe eleven eat the bread and reflect on the new significance Jesus is giving it. They have been instructed to eat his broken body. The disciples know he is using figurative speech, but even so, using such terms to describe faith is shocking.\n\nJesus, however, did not use the image to offend them; he used it to provide an analogy. Just as the Jews eat unleavened bread to remind them of their deliverance from Egypt, followers of Jesus are to associate his death with their deliverance from sin.\n\nThe meal continues, but when they come to the third traditional cup, the cup of blessing, Jesus again gives the rite a new significance. He blesses the wine and says, \"Take this, and divide it among yourselves. This cup is the new covenant in my blood that is poured out for many for the forgiveness of sins. Do this, as often as you drink it, in remembrance of me.\"\n\nAnother shocking image. Drinking blood of any kind is forbidden. But Jesus chose this image carefully. His allusion is to a passage in Exodus. \"Moses took the blood and threw it on the people and said, 'Behold the blood of the covenant that the Lord has made with you in accordance with all these words.'\"\n\nJesus is reminding the eleven that the ancient covenant God made with His people had to be ratified with blood\u2014the same is true of the new covenant spoken of by Jeremiah. \"The days are coming, declares the Lord, when I will make a new covenant with the house of Israel and the house of Judah, not like the covenant I made with their fathers on the day I took them by the hand to bring them out of the land of Egypt\u2014My covenant that they broke, though I was their husband, declares the Lord. This is the covenant I will make with the house of Israel after those days, declares the Lord: I will put my law within them, and I will write it on their hearts. And I will be their God, and they shall be My people.\"\n\nEveryone in Israel looks forward to the inauguration of the new covenant. Jesus links the shedding of his blood to its fulfillment.\n\nJesus also alludes to the prophet Isaiah's words about the Servant of the Lord who \"poured out his soul to death and was numbered with the transgressors, yet he bore the sin of many and makes intercession for the transgressors.\"\n\nAs the eleven reflect on Jesus's teaching, they realize he is bringing together all the images he has used during his ministry. In his few statements over the bread and wine, he has referred to the Son of Man, the Messiah, the Son of David, and the Suffering Servant\u2014yet he used none of those titles.\n\nJesus knows that one day his followers will understand the significance of what he has said and done at the table, but today the eleven are only partially listening. As Jesus was explaining the new covenant and its blessings, the apostles were coveting rewards of another kind and squabbling about who will be the greatest in the kingdom.\n\n\"The kings of the Gentiles lord it over them, and those who exercise authority over them call themselves benefactors,\" Jesus says. \"But you are not to be that way. Rather, let the greatest among you become like the youngest, and the leader like one who serves. For who is greater\u2014the one who reclines at the table or the one who serves? Is it not the one who reclines at the table? But I am among you as one who serves.\"\n\nThe subject Jesus addresses is not a new one; they earlier saw a powerful demonstration of these words when their master picked up the basin and towel. But none of it seems right to the apostles; they all grew up in a world where striving to become great is a good and God-honoring pursuit. The world defines greatness in terms of power and authority, but Jesus says that service is the hallmark of greatness. He seeks leaders who will give and share and serve\u2014not those who want to control others. Jesus's counsel is to follow his example.\n\nThe apostles still have a lot of room for improvement, but Jesus knows they have remained constant and have teachable hearts. He wants them to know that God has noticed.\n\n\"You are those who have stayed with me in my trials,\" Jesus says, \"and I assign to you the authority my Father assigned to me, so that you might eat and drink at my table in my kingdom and sit on thrones judging the twelve tribes of Israel.\n\n\"Now is the Son of Man glorified, and God is glorified in him. If God is glorified in the Son, God will also glorify the Son in Himself, and He will glorify him very soon.\"\n\nThe glorification Jesus speaks of is his death and resurrection. The glory of both Father and Son will be revealed at the cross, and Jesus must prepare his followers for a future without his physical presence. He looks into the faces of his followers and says, \"Dear children, I am with you only a little longer. I say to you what I said to the Jewish authorities: 'Where I am going, you cannot come.'\n\n\"A new commandment I give to you\u2014love one another. Just as I have loved you, you are to love one another. By this everyone will know that you are my apprentices\u2014if you have love for one another.\"\n\nJesus is teaching them how to live in his absence, but Peter is struggling with the comment about leaving. \"Lord, where are you going?\"\n\n\"Where I'm going now, you can't follow,\" Jesus says. \"But you will follow me later.\"\n\nPeter doesn't fully understand what Jesus is talking about, but he realizes it has something to do with death. \"Lord, why can't I follow you now? I will lay down my life for you.\"\n\nJesus looks into Simon's eyes. \"Will you lay down your life for me?\"\n\nThe room becomes still. Jesus does not want his next words to be misunderstood. \"Tonight you will all fall away because of me, for it is written, 'I will strike the shepherd, and the sheep of the flock will be scattered.' But after I have been raised, I will go ahead of you into Galilee.\"\n\nSimon Peter says what they are all thinking. \"Even if everyone else falls away because of you, I never will.\"\n\n\"Simon, Simon,\" Jesus says, \"Listen! Satan demanded to sift all of you like wheat, but I have prayed for you\u2014that your faith may not fail. And when you have turned again, strengthen your brothers.\"\n\nPeter disagrees. \"Lord, I'm ready to follow you both to prison and to death!\"\n\nPeter's intentions are good, but he overestimates himself and underestimates the forces aligned against them.\n\nJesus makes a grim prediction. \"I tell you, Peter, before the rooster crows twice today, you will deny three times that you know me.\"\n\nPeter grimaces and shakes his head. This is preposterous. \"Even if I must die with you,\" he promises, \"I will not deny you!\"\n\nThe others agree, and for the next few minutes the air is filled with oaths of loyalty and vows of faithfulness.\n\nJesus does not pursue the point; he continues giving final instructions to prepare the eleven for the changes that are about to take place. He asks, \"When I sent you out without moneybag, knapsack, or sandals, did you lack anything?\"\n\n\"Nothing,\" the apostles reply.\n\nThe men are being asked to recall how they were provided for when the twelve and later the seventy-two were sent out and told to rely on the \"people of peace\" they would meet along the way to provide food and shelter. This worked well when the gospel message was new and people were willing to listen, but for the most part people were unwilling to change, and the cumulative response of the nation was rejection. A new strategy must be implemented.\n\n\"But now,\" Jesus says, \"let the one who has a moneybag take it, and likewise a knapsack. And let the one who has no sword sell his cloak and buy one. For I tell you that this Scripture must be fulfilled in me: 'And he was numbered with the transgressors.' And indeed, what is written about me is being fulfilled.\"\n\nJesus uses the words of Isaiah to explain why the changes must happen: the people of Israel are even now numbering him with the transgressors\u2014they are determining he is a false prophet.\n\nJesus uses the metaphor of a sword to teach his apprentices that they need to prepare spiritually and mentally for resistance. The apostles think Jesus is speaking literally and do a weapons inventory.\n\n\"Look, Lord, here are two swords.\"\n\nJesus sighs and says, \"That's enough.\" The eleven are seriously misinterpreting his teaching, but all is not lost. There are other resources. God knew his new community of faith would need more guidance than Jesus would be able to provide in his time on earth. It is to this important topic Jesus now turns.\n\n\"Let not your hearts be troubled,\" Jesus says. \"Believe in God; believe also in me. In my Father's house are many dwelling places. If this were not so, would I have told you that I am going to prepare a place for you? And if I go and prepare a place for you, I will come again and take you with me, so that where I am, you will also be. And you know the way to where I am going.\"\n\nThe eleven do not understand that it is on the cross that Jesus will secure a place for them in heaven, and Thomas says, \"Lord, we don't know where you're going. How can we know the way?\"\n\nThomas wants a map, but Jesus says, \"I am the way and the truth and the life\u2014no one comes to the Father except through me. If you know me, you will know my Father as well, and you do know Him and have seen Him.\"\n\nThe apostle Philip, like the others, does not yet realize what he knows and wants supernatural confirmation. \"Lord,\" he says, \"show us the Father, and we'll be satisfied.\"\n\nPhilip wants visible proof. He wants to see God with his own eyes. \"Have I been with you all this time, Philip, and you still don't know me? Whoever has seen me has seen the Father. How can you say, 'Show us the Father'? Do you not believe that I am in the Father and the Father is in me? The words that I speak to you I do not speak on my own authority, but the Father who dwells in me\u2014He does His works through me. Believe me when I say that I am in the Father, and the Father is in me; otherwise, have faith in me because of the works themselves.\n\n\"Truly, truly I tell you, the one who has faith in me will do the same works that I am doing, and he will do even greater works than these because I am going to the Father. I will do whatever you ask in my name so that the Son may bring glory to the Father.\"\n\nJesus is explaining that before the new covenant is initiated, he will be restricted physically, but after his death and resurrection he will be able to work through all his disciples simultaneously. The important lesson for his followers is that fruitfulness depends on a vibrant prayer life. Requests offered in Jesus's name will be answered\u2014not because there is magic in prayer but because praying in Jesus's name means praying in alignment with God's will.\n\n\"If you love me, you will keep my commandments. And I will ask the Father, and He will give you another Paraclete who will be with you forever. He is the Spirit of truth, whom the world cannot receive because it neither sees Him nor knows Him. You know Him, for He dwells with you and will be in you.\n\n\"I will not leave you orphaned\u2014I'm coming to you. In a little while, the world will no longer see me, but you will see me. Because I live, you also will live. On that day, you will know that I am in my Father, and you are in me, and I am in you. Those who accept my commandments and keep them are the ones who love me, and those who love me will be loved by my Father. I, too, will love them and reveal myself to them.\"\n\nJesus describes the work he has been doing as that of a paraclete. Broadly defined, a paraclete is \"one who is called alongside.\" The term is used in court to refer to legal counselors and advocates, but this is just one application. It is also commonly used to refer to someone who comes alongside another to strengthen and encourage.\n\nJesus has been his apprentices' counselor and strengthener and encourager for more than three years. Tonight all that will change.\n\nJesus knows that at the cross the eleven will feel abandoned, and he assures them that despite the way things look, his death will not orphan them. Jesus promises not only to send another paraclete who will never leave them, soon he will return also. People who are wrapped up in the world won't see him, he says, but his disciples will.\n\nThis prompts a question from Thaddaeus. \"Lord, why are you going to reveal yourself only to us and not to the world?\"\n\nThaddaeus does not understand that those who reject God are deaf and blind. Worldly people would not see God even if he were to appear before them in the flesh.\n\nJesus smiles and says, \"Those who love me will keep my word, and my Father will love them, and we will go to them and make our home with them. Anyone who does not love me will not keep my words, and the word you hear is not mine\u2014it's from the Father who sent me.\n\n\"I've told you these things while I am still with you, but the Paraclete\u2014the Holy Spirit, whom the Father will send in my name\u2014will teach you all things and remind you of all the things I told you.\n\n\"Peace I leave with you; my peace I give to you. I do not give you the kind of peace the world gives. Don't let your hearts be troubled and don't be afraid. You heard me say, 'I am going away, and I will come back to you.' If you loved me, you would rejoice that I am going to the Father, for the Father is greater than I. Now I've told you this before it happens, so that when it comes to pass you will believe.\n\n\"I will not be talking with you much longer like this, because the ruler of this world is coming. He has no power over me, but I do as the Father has commanded me so that the world will know that I love the Father.\"\n\nJesus pauses and looks into the faces of the apostles. There is more to say, but the hour is late. The lesson can continue as they walk. \"Come,\" Jesus says. \"Let's be going.\"\n\nThe men stand and prepare to leave, but there is one more element in the Seder\u2014they must sing the portion of Psalm 118 that completes the Hallel. As they walk out of the upper room, the walls reverberate with words that have never had more meaning than they have tonight.\n\nI will praise You\u2014You answered my prayer and have become my salvation.\n\nThe stone the builders rejected has become the chief cornerstone.\n\nThis is the Lord's doing, and it is amazing to see.\n\nThis is the day that the Lord has made;\n\nLet us rejoice and be glad in it.\n\nSave us, we pray, O Lord! O Lord, we pray, give us success!\n\nBlessed is the one who comes in the name of the Lord!\n\nWe bless you from the house of the Lord.\n\nThe Lord is God, and He has given us light.\n\nBind the festival sacrifice with cords to the horns of the altar!\n\nYou are my God and I praise You. You are my God and I exalt You.\n\nO give thanks to the Lord, for He is good\u2014His steadfast love endures forever!\n27\n\nBETRAYED\n\nApril 2\u20133, AD 33\n\n\"The one I kiss is the one you want.\"\n\n\u2014Matthew 26:48\n\nTHE HOUR IS late when Jesus and the eleven make their way down the stairway from the upper room. The streets are still busy, but it has been a long day, and most are hurriedly making their way to their lodgings. Jesus and eleven apostles are walking to a garden on the Mount of Olives, where he likes to pray.\n\nThe men travel the back streets, and Jesus teaches as they walk. During the Seder, he had spoken of a mutual indwelling. He said that he would be in his apprentices, and his apprentices would be in him. Jesus expands and deepens the lesson by reinterpreting a well-known metaphor.\n\n\"I am the true vine, and my Father is the gardener. He removes every branch in me that does not bear fruit, and every branch that does bear fruit He prunes so that it may bear more fruit.\n\n\"You are already clean because of the message I have spoken to you. Remain in me, and I will remain in you. Just as a branch can't bear fruit unless it remains in the vine, neither can you bear fruit unless you remain in me.\n\n\"I am the vine; you are the branches. Those who remain in me and I in them will bear much fruit, for apart from me you can do nothing. Anyone who does not remain in me is like a branch that is thrown away and withers. These branches are picked up and thrown into the fire and burned.\n\n\"If you remain in me and my words remain in you, you may ask for whatever you wish, and it will be done for you. By this my Father is glorified: that you bear much fruit and so prove yourselves to be my apprentices.\"\n\nThe apostles partially understand. They are familiar with God being portrayed as a gardener, but in the Scriptures the vine represents Israel and is often described as unfruitful. Time and again the nation was warned that if they remained unfaithful, the divine gardener would one day replant the vineyard.\n\nJesus says the wild vine that produced sour grapes has been replaced with a new vine, and those who are organically connected to the new vine will yield fruit that glorifies God.\n\nJesus explains the union. \"As the Father has loved me, so have I loved you. Remain in my love. If you keep my commandments, you will remain in my love, just as I have kept my Father's commandments and remain in His love. I've told you these things so that my joy may be in you, and that your joy may be complete.\n\n\"This is my commandment, that you love one another as I have loved you. Greater love has no one than this: that he lay down his life for his friends. You are my friends if you do what I command. I no longer call you servants, because the servant doesn't know what his master is doing. Instead, I've called you friends because I've told you everything that I heard from my Father.\n\n\"You did not choose me, but I chose you and appointed you to go and bear fruit\u2014fruit that will last\u2014so that the Father will give you whatever you ask for in my name. I am giving you these commands so that you will love one another.\"\n\nJesus describes a community that will be identified by their allegiance to God and their love for one another, but he warns his disciples that living this way will put them on a collision course with the spiritual rebels who populate the world.\n\n\"If the world hates you, know that it hated me before it hated you. If you belonged to the world, the world would love you as its own, but you don't belong to the world. I chose you out of the world, so the world hates you.\n\n\"Remember what I told you: a servant is not greater than his master. If they persecuted me, they will also persecute you. If they obeyed my teaching, they will obey yours also.\n\n\"But they will do all these things to you because of my name, for they do not know the One who sent me. If I had not come and spoken to them, they would have no sin, but now they have no pretense for their sin.\n\n\"Whoever hates me also hates my Father. If I had not done among them the works that no one else has done, they would not be guilty of sin, but now they have seen and hated both me and my Father. But this happened so that the word written in their Law might be fulfilled: 'They hated me without a cause.'\n\n\"But when the Paraclete comes, whom I will send to you from the Father\u2014the Spirit of truth who comes from the Father\u2014He will testify about me. And you also must testify, for you have been with me from the beginning.\"\n\nJesus prescribes a twofold approach toward a world violently opposed to his message: the witness of the Spirit and the testimony of his disciples. Jesus tells them that God will make sure the gospel is spread, but His messengers must prepare for persecution, or they will see their faith swept away.\n\nUntil now, Jesus has taken the brunt of the abuse, but he knows that after he's gone the wrath now directed at him will be turned on his apprentices.\n\n\"I have told you these things to keep you from stumbling. They will expel you from the synagogues; in fact, the time will come when whoever kills you will think he is serving God. They will do these things because they know neither the Father nor me. I've told you these things so that when they happen, you will remember I told you about them.\n\n\"I didn't tell you these things earlier because I was with you, but now I'm going to the One who sent me, yet none of you asks me, 'Where are you going?' Instead, you grieve because of what I told you. But I tell you the truth\u2014it's better for you that I go away, because if I don't go away, the Paraclete won't come to you. But if I go, I will send Him to you. When He comes, He will convict the world of sin and righteousness and judgment. Of sin, because they do not believe in me; of righteousness, because I go to the Father, and you will see me no longer; of judgment, because the ruler of this world has been condemned.\"\n\nThe apostles hear little after Jesus announces that he is leaving them, and what they do hear makes little sense. How could life be better without Jesus to guide them?\n\nJesus knows his disciples' understanding of these things will have to wait. \"I still have many things to tell you,\" Jesus says, \"but you cannot bear them now. When the Spirit of truth comes, He will guide you in all the truth, for He will not speak on His own. He will speak only what He hears, and He will tell you the things that are to come.\n\n\"He will bring me glory because He will tell you whatever He receives from me. All that belongs to the Father is mine\u2014that's why I said the Spirit will tell you whatever He receives from me.\n\n\"In a little while you won't see me anymore, but a little while after that you will see me.\"\n\nJesus is speaking of his death and resurrection. The apostles don't make the connection. They turn to one another and whisper, \"What's this he's telling us? 'In a little while you won't see me, but a little while after that you will see me' and 'because I go to the Father'? What does he mean by 'a little while'? We don't know what he's talking about.\"\n\nJesus overhears them and asks, \"Are you trying to find out from one another what I meant when I said, 'In a little while you won't see me anymore, but a little while after that you will see me'? Truly, truly I tell you, you will weep and mourn, but the world will rejoice. You will grieve, but your grief will turn into joy.\n\n\"A woman who is giving birth is in anguish because her time has come, but when she has given birth to the baby, she forgets her anguish because of her joy\u2014a child has been born into the world.\n\n\"So also you have grief now, but I will see you again, and your hearts will rejoice, and no one will take your joy away from you. In that day you will ask nothing of me. Truly, truly I tell you, whatever you ask of the Father in my name, He will give you. So far you have asked for nothing in my name. Ask, and you will receive so that your joy may be full.\n\n\"I have said these things to you in figures of speech. The time will come when I will no longer speak to you figuratively but will tell you about the Father in plain language. In that day you will ask in my name\u2014and I am not saying that I will ask the Father on your behalf, for the Father himself loves you because you have loved me and have believed that I came from God. I came from the Father and have come into the world; now I am leaving the world and going back to the Father.\"\n\nWhen Jesus finishes his teaching, the eleven nod enthusiastically and say, \"Now you are speaking in plain terms and not using figurative speech! Now we can see that you know all things, and there is no need to question you. Now we believe that you came from God.\"\n\nJesus can only shake his head. The apostles' zeal is admirable but misinformed. They believe the day Jesus spoke of earlier has already come. They don't realize that these things cannot happen until after the resurrection.\n\n\"Now you believe?\" Jesus asks.\n\nNo one answers. The tone of Jesus's voice suggests that he questions their understanding, and his next words confirm it. \"Listen,\" Jesus says, \"a time is coming\u2014indeed it has already come\u2014when you will be scattered, each to his own, and you will leave me alone. Yet I am not alone, for the Father is with me. I have told you these things so that in me you may have peace. In the world you will have anguish, but take heart\u2014I have overcome the world.\"\n\nThe apostles don't understand why Jesus is proclaiming victory but ask no questions. Jesus and the eleven walk out the city gate, cross the trickling Brook Kidron, and begin climbing the Mount of Olives. It is near midnight, but the paschal moon illuminates the entire valley.\n\nTHEY WALK A short distance up the hillside, and Jesus stops and begins to pray. \"Father, the hour has come. Glorify Your Son so that the Son may glorify You, just as You have given him authority over all people so that he may give eternal life to all those You have given him. And this is eternal life, that they know You, the only true God, and the one You have sent\u2014Jesus Christ. I brought glory to You on earth by accomplishing the work You gave me to do. So now, Father, bring me into the glory we shared before the world existed.\n\n\"I have made You known to the people You gave me from the world. They were Yours, and You gave them to me, and they have kept Your Word. Now they know that everything You have given me comes from You because I gave them the words You gave me. They accepted Your words and have come to know with certainty that I came from You, and they believe that You sent me.\n\n\"I ask on their behalf. I am not praying for the world but for the ones You have given me, for they are Yours. All that is mine is Yours, and all that is Yours is mine, and my glory is shown through them. I am coming to You. I will remain in the world no longer, but they are still in the world. Holy Father, protect them by the power of Your name\u2014the name You gave me\u2014so that they may be one, as we are one.\n\n\"While I was with them, I protected them in Your name that You have given me. I guarded them, and not one was lost except the son of perdition, in order for the Scripture to be fulfilled. Now I am coming to You, and I say these things while I am still in the world so that my apprentices may be filled with my joy.\n\n\"I have given them Your word, and the world has hated them because they are not of the world, just as I am not of the world. I do not ask that You take them out of the world but that You protect them from the evil one. They are not of the world, just as I am not of the world.\n\n\"Make them holy in the truth\u2014Your word is truth. Just as You sent me into the world, I have sent them into the world. And for their sake I consecrate myself so that they also may be made holy in truth.\n\n\"I pray not only for these disciples but also for the ones who will believe in me through their message, so that all of them might be one. Just as You, Father, are in me, and I am in You, may they also be in us, so that the world may believe that You have sent me. I have given them the glory You gave me, so they may be one just as we are one\u2014I in them and You in me. May they become completely one, so that the world may know that You sent me and have loved them just as You have loved me.\n\n\"Father, I want those You have given me to be with me where I am, so they can see my glory, the glory that You have given me because You loved me before the foundation of the world.\n\n\"Righteous Father, even though the world does not know You, I know You, and these disciples know that You have sent me. I have made You known to them, and I will continue to make You known so that Your love for me may be in them, and I will be in them.\"\n\nJESUS ENDS HIS prayer and begins walking up the Mount of Olives. The men make their way slowly up the hill. There are hundreds of pilgrims camped in the shadows of the ancient trees.\n\nThey eventually come to a large walled orchard and garden. Named Gethsemane because of the oil press in it, the garden is familiar to the disciples. They have been here frequently with Jesus, and the eleven assume they will be spending the night here and not returning to Bethany.\n\nWhen they enter the garden gate, Jesus invites Andrew, Phillip, Bartholomew, Matthew, Thomas, James, Simon, and Thaddaeus to find a place to rest, and asks Peter, James, and John to accompany him further for prayer.\n\nIt is obvious to the three that Jesus is troubled. They walk a short distance, and he turns to them. \"My soul is extremely sorrowful\u2014to the point of death. Stay here and keep watch with me.\"\n\nPeter, James, and John are concerned about their master's anguish, but they are emotionally and physically exhausted. When Jesus asks them to pray for God to strengthen and prepare them for the ordeal ahead, the three agree, find a comfortable place to sit under a tree, and are soon sleeping soundly.\n\nJesus, his energy sapped and his steps leaden, walks a short distance and falls to his knees. \"Abba, Father, everything is possible for You. Remove this cup from me. Nevertheless, I want Your will to be done, not mine.\"\n\nJesus knows the cup of God's wrath is full, and he has no death wish. As he prays, he is visited by an angel; there is no message of deliverance, only encouragement.\n\nEventually, he rises and goes to check on the disciples.\n\n\"Simon,\" he asks, \"are you sleeping?\"\n\nThe three disciples wake up to find Jesus kneeling beside them. \"You couldn't stay awake with me for a single hour? Watch and pray that you will not give in to temptation. The spirit is willing, but the flesh is weak.\"\n\nThe apostles rub their eyes and try to wake up as Jesus walks away, but soon they drift back to sleep. Jesus continues praying. He has known this day was coming, but now that it's here, he struggles to comprehend the enormity of what is before him. Yet his resolve is unshakable.\n\n\"My Father, if this cup can't be taken away unless I drink it, may Your will be done.\"\n\nAfter a few minutes, Jesus notices Peter, James, and John are asleep. He wakes them and exhorts them to watch and pray with him. The disciples are embarrassed and vow to themselves to stay awake, but when Jesus goes away a third time, they again fall asleep.\n\nJesus resumes praying, but soon his attention is drawn to the city. The Father has answered his prayer. A crowd carrying torches and weapons tumbles out the eastern gate, down into the valley, and straight up the Mount of Olives toward Gethsemane.\n\nHe returns to Peter, James, and John. They have slumbered through their opportunity to be strengthened. \"Sleep on and rest!\" he says.\n\nThe apostles stir and open their eyes.\n\n\"Look!\" Jesus says. \"The time has come\u2014the Son of Man is betrayed into the hands of sinners. Get up. Let's go. Look, my betrayer has arrived.\"\n\nThe urgency in Jesus's voice startles the apostles to consciousness. They look around groggily, not comprehending. Their eyes grow wide when the see a large group of men with swords and clubs walking up the hill toward them.\n\nPeter, James, and John scramble to their feet. The rabbi is already striding toward the gate. The apostles follow. In the torchlight, they see more than two hundred men approaching, an unusual alliance of priests, Pharisees, elders, scribes, temple police, and Roman soldiers. The eleven are terrified when they see the angry mob, but horrified to see a trusted friend leading them.\n\nThe apostles brace for an attack, but the throng stops a short distance away, and Judas Iscariot meekly approaches alone. Jesus steps forward; he knows the signal Judas prearranged with the leaders, and as his former apprentice walks toward him, Jesus says, \"Judas, are you betraying the Son of Man with a kiss? Friend, do what you came to do.\"\n\nJudas smiles nervously and says, \"Greetings, Rabbi!\" He embraces Jesus and affectionately kisses him on the cheek.\n\nAs Judas pulls away, Jesus is looking directly into his eyes, but rather than acknowledging the kiss, he turns to the mob and asks, \"Who are you looking for?\"\n\n\"Jesus of Nazareth,\" a voice calls out.\n\n\"I am he,\" Jesus says.\n\nAs the words are spoken, a great fear strikes the men, and they fall back, shaken\u2014by what, they are not sure.\n\nJesus asks again, \"Who are you looking for?\"\n\nThe men regain their composure and answer, \"Jesus of Nazareth.\"\n\n\"I told you that I am he.\" Then gesturing toward the eleven, he adds, \"So if you're looking for me, let these others go their way.\"\n\nThe arresting officers have no interest in the eleven so long as they do not get in the way. Swords drawn, the soldiers move toward Jesus. Near the front is Malchus, one of the high priest's most trusted slaves, sent to ensure that the arrest goes as planned and Jesus is brought directly to his home.\n\nThe apostles realize their master is about to be taken prisoner and ask, \"Lord, shall we strike with the sword?\"\n\nBefore Jesus can answer, Simon Peter draws a sword and slashes wildly. Malchus sees the steel coming and moves quickly; the blade misses his skull but slices off his right ear. The man curses as he clutches his head to staunch the bleeding.\n\n\"No more of this!\" Jesus commands.\n\nHe walks over to the high priest's slave and touches the wound. It is instantly healed. Malchus, speechless, wonders where this man they are arresting got the power to restore a severed ear.\n\nJesus turns to Peter. \"Put your sword back in its place, for all who draw the sword will die by the sword. Don't you realize that I could call upon my Father, and He would immediately send me more than twelve legions of angels? But how then would the Scriptures be fulfilled that say it must happen this way? Shall I not drink the cup the Father has given me?\"\n\nPeter, numb and confused, drops his sword. Jesus turns to the leaders of the alliance. \"Have you come with swords and clubs to arrest me as if I were a bandit? When I was with you day after day teaching in the temple courts, you didn't arrest me. But this is your time\u2014the time when darkness reigns.\"\n\nWhen the apostles hear Jesus proclaiming that his arrest is part of God's plan, it sounds like foolishness. The story of the Messiah can't end this way. Why is the master meekly giving himself up? There are no answers. Convinced that all is lost, the eleven flee into the night.\n\nJesus, now alone before the mob, does not resist being bound and pushed down the hill.\n\nThe priests and leaders who have been frustrated in their attempts to silence Jesus are jubilant but eager to move quickly; dawn is only a few hours from now, and there is much to be done. The hard part, however, is behind them. The wily Nazarene is finally under their control, and if all goes as planned, he will be dead by nightfall.\n28\n\nCONDEMNED\n\nApril 3, AD 33\n\n\"It is better for you that one man die for the people than the whole nation perish.\"\n\n\u2014Joseph Caiaphas, High Priest, AD 18\u201336\n\nA LARGE GROUP of city elders, lawyers, priests, and Roman soldiers escort the prisoner through the eastern gate. They climb the steep stairway to the Court of the Gentiles and walk across the temple plaza.\n\nTheir destination is the Hasmonean Palace on the hill west of the temple. It is too early to formally try Jesus before the Sanhedrin\u2014the law demands all trials be held during the day. Still, there is much that can be accomplished before dawn, and if all goes well, that trial will be a mere formality.\n\nThe soldiers escort the accused to the home of the high priest and transfer authority to the temple police. Then they and most of the crowd hastily depart to get some sleep. Tomorrow will be a busy day.\n\nIn the shadows watching the transfer is Simon Peter and another disciple who knows the high priest. The other disciple slips into the small crowd escorting Jesus into the palace compound. A few minutes later he reappears, says a few words to the servant girl watching the gate, and Peter is granted entrance.\n\nPeter walks into a large courtyard filled with dozens of slaves, servants, and policemen. It is a cold Nisan night, and the men and women are huddled around charcoal fires awaiting orders. He joins one of the groups to listen for news about Jesus, but soon senses he is being watched. The gatekeeper is staring at him. She's trying to remember where she has seen him before. When the chief apostle looks at her, she says, \"You, too, were with that Nazarene, Jesus. Aren't you one of the man's disciples?\"\n\nPeter frowns. \"I don't know what you're talking about\u2014I don't know him, woman.\"\n\nThe apostle's voice is confident, but the servant girl's question unnerves Peter, and he walks away from the fire. In the distance a cock crows.\n\nJESUS IS NEARBY in a large ornately furnished room in the heart of the palace. He is bound in thick ropes and standing before a well-dressed elderly man. Annas, son of Seth, was appointed high priest of Israel by Quirinius when Jesus was a child, and the old man would still hold the office if not for the political machinations of the Romans. According to the Torah, the appointment to high priest is for life, but Pilate's predecessor, Valarius Gratus, found Annas uncooperative and ousted him after nine years. Little changed, however, because the office remained in the family. Currently his devoted son-in-law, Joseph Caiaphas, is high priest, and Annas is grooming five more sons to take over after him.\n\nJesus has been brought to Annas for questioning and to allow Caiaphas time to gather enough councilmen to form a quorum. The savvy old politician has heard much about the Galilean and hopes to learn some of the sorcerer's secrets. Two things are of particular interest: the size of Jesus's forces and the secret teachings he is dispensing.\n\n\"I have spoken openly to the world,\" Jesus tells his inquisitor. \"I have always taught in synagogues and in the temple, where all the Jews come together, and I have said nothing in secret. Why do you ask me? Ask those who have heard me\u2014they know what I said.\"\n\nNo one in the room is accustomed to hearing Annas challenged. One of the minor officials who is standing near Jesus slaps him hard in the face. \"Is that how you answer the high priest?\"\n\nJesus looks the man in the eye and says, \"If what I said is wrong, point out the wrong; but if what I said is right, why did you hit me?\"\n\nThe wizened politico has seen enough fanatics in his day to know he will get nothing from this one. He orders the Levites to take Jesus to his son-in-law for final disposition.\n\nOUTSIDE IN THE courtyard the temperature has dropped to near freezing. The soldiers, servants, and slaves draw closer to the charcoal fires and one another. Peter again walks near one of the fires. The servant girl who questioned him earlier is also at the fire and still suspicious. She points at Peter and says, \"This man is one of them.\"\n\nAnother woman agrees. \"This man was with Jesus of Nazareth.\"\n\nThe men take a closer look at the stranger. One finally nods and says, \"You also are one of them.\"\n\nPeter senses he is about to be exposed and swears, \"Man, I am not!\"\n\nThrowing his arms up in disgust, Peter stomps away from the fire. But some are not convinced; they detect a Galilean accent. They watch the stranger go to the forecourt and stand for a long time near the gate. When Peter eventually walks over to another charcoal pit, several suspicious Jerusalemites follow.\n\nThey squeeze into the circle and stare across the coals at the outsider. Peter barely notices them. He is looking at Jesus. His master, bound and surrounded by temple police, has been brought into the courtyard. He looks unharmed; it appears they are moving him to a different part of the palace.\n\nPeter's attention is brought back to the fire pit by familiar voices. \"Surely you too are one of them\u2014your accent betrays you\u2014you are a Galilean.\"\n\nAnother man standing at the pit squints, straining to see the newcomer's features in the coal's faint light. He had been in Gethsemane when his kinsman, Malchus, had his ear cut off and miraculously restored. He, too, thinks the stranger looks familiar. \"Didn't I see you in the garden with him?\"\n\nFear grips Peter, and all he can think to do is save himself. \"Man, I don't know what you are talking about\u2014I do not know the man.\"\n\nAs Simon Peter denies knowing Jesus for the third time, he hears a rooster crow in the distance. His eyes grow wide. He turns and sees Jesus looking at him. The awful prophecy\u2014the one that could never happen\u2014has come to pass.\n\nThe apostle stands by the coal fire, his mind moving slowly, erratically. Was it just a few hours ago he cursed Judas for betraying the Master?\n\nPeter stumbles away from the fire, out of the courtyard, and into the street, weeping inconsolably.\n\nINSIDE THE PALACE, Caiaphas has not yet gathered a quorum, but there are sufficient councilmen to hold an inquiry and establish the facts of the case.\n\nJesus is brought into a large crowded room. The law demands the testimony of reliable witnesses in order to establish guilt, but Caiaphas can't find two whose stories agree.\n\nWitness after witness appears, but their testimony is inconsistent. Several allude to comments Jesus made about the temple three years earlier. One says, \"This man said, 'I am able to destroy the temple of God and rebuild it in three days.\"\n\nAnother testifies, \"We heard him say, 'I will destroy this temple made with hands, and in three days I will build another made without hands.'\"\n\nThe allegations pile up, but the high priest knows that none will survive formal court. Frustrated, he stands and glares at Jesus. \"Have you no answer to these men's testimony against you?\"\n\nJesus does not reply.\n\nCaiaphas realizes that he must find a way to force Jesus to incriminate himself. He uses his authority as high priest to put Jesus under oath. \"I adjure you by the living God\u2014tell us if you are the Messiah, the Son of God.\"\n\n\"You say that I am,\" Jesus says.\n\nHe then quotes David and Daniel and seals his fate. \"Indeed I tell you, from now on you will see the Son of Man seated at the right hand of power and coming on the clouds of heaven.\"\n\nThe informal hearing disintegrates into a sea of angry voices. Caiaphas grabs the collar of his robe and rips it, signifying a guilty verdict, and shakes his finger at Jesus. \"He has blasphemed! What further need do we have for witnesses? You've heard his blasphemy\u2014what is your judgment?\"\n\nThe response is immediate. \"He deserves death!\"\n\nJesus has claimed to be God's co-regent. The inquiry is over; Caiaphas has more than enough evidence to convict when the Sanhedrin convenes. As he walks from the room, he tells the guards to prepare the condemned man for transport.\n\nThe temple police tie the prisoner's hands and blindfold him. The Galilean has cost them a night's sleep, and they are ready to return the favor. One of the men spits in Jesus's face and slams a fist to his head. \"Prophesy to us, Messiah\u2014who was it that hit you?\"\n\nThe other guards laugh and join in the taunting and beating. It stops only when daylight comes and they are forced to transfer the prisoner to the council chambers. A quorum of the Sanhedrin is present, and they listen to a summary of the high priest's findings. Following the report, they turn to the accused. \"If you're the Messiah, tell us.\"\n\nJesus shakes his head and says, \"If I tell you, you will not believe, and if I question you, you will not answer. But from now on the Son of Man shall be seated at the right hand of the power of God.\"\n\n\"Are you, then, the Son of God?\" they ask.\n\n\"You say that I am.\"\n\nThis is enough evidence for the Sanhedrin. \"What further testimony do we need? We've heard it ourselves from his own mouth.\"\n\nThe council members prepare an indictment they are confident will get Jesus convicted of sedition, but the charges must be formulated quickly; Pilate only listens to court cases early in the morning.\n\nLISTENING NEARBY IS the betrayer. Remorse sets in, and Judas realizes he made a mistake. The chief priests and elders he had originally negotiated with are standing outside the door of the temple. Judas walks up to them, his cupped hands filled with silver. \"I have sinned by betraying innocent blood.\"\n\nThe men laugh. \"What's that to us? That's your problem.\"\n\nThe ex-apostle, humiliated and out of ideas, hurls the coins to the temple floor and flees, his thoughts dark and confused. Iscariot heads toward the outskirts of town and contemplates his options. He can see only one. He finds a rope and a tree and hangs himself.\n\nIN THE CITY, Jesus is being transferred to the prefect's headquarters. Caiaphas assumes that since Pilate had authorized his soldiers the night before to assist in the arrest, the governor will simply endorse their decision to execute him. The indictment must be changed, of course; blasphemy is not punishable by death under Roman law. Political charges must be substituted, but the Jewish leaders are confident they can use the prisoner's own words to prove he is a revolutionary and a threat to Rome.\n\nA large delegation of chief priests accompanied by scribes and elders lead Jesus to the governor. Many of Pilate's troops are barracked in the Antonio Fortress on the temple mount, but he and his wife prefer the comforts of the old palace built in the upper city by Herod the Great.\n\nWhen the religious leaders arrive at the governor's residence, they turn Jesus over to the Roman guard and request Pilate come outside to hear their case. Today is the first full day of the festival, and the priests are scheduled to oversee important ceremonies. If they enter the home of a Gentile, they will become ceremonially unclean and unable to participate in the Passover.\n\nPilate has lived long enough among the Jews to understand some of their peculiar customs. He normally accommodates them to avoid conflict, but when he walks out onto his porch to meet the Sanhedrin that morning, he knows they are being deceitful. The Jews aren't about to hand over one of their own simply to help Rome. There is something they aren't telling him.\n\n\"What accusation do you bring against this man?\"\n\nThe question catches the leaders off guard. They were expecting a cursory endorsement, not a reopening of the trial. Their reply is surly. \"If this man were not doing evil, we would not have delivered him over to you.\"\n\nPilate grimaces\u2014it's far too early in the morning to put up with such insolence. \"Take him yourselves and judge him by your own law.\"\n\nThe leaders look quickly at their sandals and are abruptly reminded how much they need the governor's cooperation. Sufficiently humbled, they reply, \"It is not lawful for us to put anyone to death.\"\n\nWhen Pilate does not respond, they bring an indictment against Jesus containing three cleverly devised charges. \"We found this one perverting our nation and forbidding us to pay tribute to Caesar and saying that he himself is an anointed one\u2014a king.\"\n\nPilate can see their charges are motivated by envy, but he is interested in the third accusation and goes inside to interrogate Jesus. When he sees the prisoner, he wonders why there is such animosity toward him. The man standing bound before him looks anything but royal or powerful. His peasant garb is torn and grimy. His face is swollen and cut. His hair is tangled and caked with blood. Pathetic, certainly, but not a threat.\n\n\"Are you the king of the Jews?\"\n\n\"You are saying so,\" Jesus replies. \"Do you say this of your own accord, or did others talk to you about me?\"\n\nPilate is astounded. \"Am I a Jew? Your own nation and the chief priests handed you over to me!\" Pilate glares at the accused. \"What have you done?\"\n\n\"My kingdom,\" Jesus says, \"is not of this world. If my kingdom were of this world, my servants would have fought so that I would not have been handed over to the Jews\u2014but my kingdom is not from here.\"\n\n\"So you are a king?\"\n\n\"You say that I am a king. For this purpose I was born and for this purpose I have come into the world\u2014to testify to the truth. Everyone who is of the truth listens to my voice.\"\n\nPilate shakes his head and mutters, \"What is truth?\" This is the most unorthodox defense he has ever been offered. \"Don't you hear how many charges they are bringing against you? Have you no answer?\"\n\nJesus offers none. Pilate turns and paces the floor, thinking. He knows little about the accused, but it is clear the charges have no merit. The prefect walks back out onto the porch and announces, \"I don't find this man guilty of anything.\"\n\nThe religious leaders throw up their hands and protest, \"He stirs up the people, teaching throughout all Judea\u2014from Galilee, where he began, even to this place.\"\n\nWhen Pilate hears Jesus is a Galilean, he grows quiet. Perhaps there is a way he can avoid condemning the man after all. Herod Antipas is also in Jerusalem for the Passover. Since Jesus is from the province Herod governs, perhaps Herod will accept jurisdiction. The governor instructs his soldiers to take the prisoner to the tetrarch and hopes he has seen the last of Jesus.\n\nANTIPAS IS EXCITED when he learns that Jesus is coming. For years he has heard rumors about the Nazarene wonder-worker. Perhaps Jesus can be persuaded to perform a miracle for his lord.\n\nHerod's spirit deflates when the prisoner is brought into the room. Can this gaunt and bleeding little man be the one he has heard so much about? How very average. How very unimpressive.\n\nAccompanying Jesus are representatives from the Sanhedrin who repeat the charges they made before Pilate. The tetrarch listens carefully to the indictment, then turns to hear Jesus's defense.\n\nThe prisoner says nothing. More charges are brought. Herod pleads with Jesus to answer, but every accusation is met with silence.\n\nAfter numerous attempts, Antipas grows bored and frustrated. It is clear to him that the charges have no merit, but he has no interest in getting involved in a messy political situation\u2014what he wants is a demonstration from a famous magician. When it becomes clear that the accused will not cooperate, Herod decides he must make his own entertainment, so he begins taunting the raggedy mute. Soon everyone in the room is playing along. When he tires of the game, Herod drapes a beautiful cloak around the harmless fool who would be king and returns him to the governor.\n\nMOST OF THE city is now awake, and many are already worshipping in the temple plaza. The large group of dignitaries escorting a prisoner to the governor's residence draws a crowd, and when Pilate walks out onto his porch to meet the delegation from the Sanhedrin, he sees a number of townspeople have joined them.\n\nHe addresses the high priests. \"You brought this man to me claiming he is misleading the people. I have examined him here in your presence, and I find no basis for any of your accusations against him. Neither did Herod, for he sent him back to us. Look, he has done nothing deserving death, so I will punish and release him.\"\n\nAs Pilate pronounces the verdict, the chief priests stiffen, then huddle together to confer. A few minutes later they begin whispering into the crowd. A murmur fills the courtyard. People are discussing the governor's custom of releasing one Jewish prisoner at Passover, and Jesus isn't the only famous man under arrest. He's not even the only famous Jesus in custody. The Romans are holding a man named Jesus Barabbas, who many consider to be a hero because he advocates armed rebellion and has himself killed for the cause.\n\nPilate overhears the conversations in the crowd and is amazed. The nation's spiritual leaders are so filled with envy and hatred, they are encouraging the crowd to condemn a harmless rabbi and free a known robber and murderer. The governor reflects on the irony and laughs. He despises the Jews and decides to have some fun at the expense of the Sanhedrin. He throws his arms open to the crowd and asks, \"Which of the two do you want me to release for you: Jesus Barabbas, or Jesus who is called Christ?\"\n\nPilate looks toward the religious leaders and sneers. \"Do you want me to release for you the king of the Jews?\"\n\nThe priests, scribes and elders scowl and again begin whispering among the crowd. As Pilate waits, an aide brings him a message from his wife. It is a warning: \"Have nothing to do with that righteous man, for I have suffered much because of him today in a dream.\"\n\nPilate nods grimly. He has no desire to participate, but sees no way to remove himself. Hopefully the people of Jerusalem will vindicate the Galilean. He looks over the sea of faces and repeats his offer. \"Which of the two do you want me to release for you?\"\n\nThis time there is no hesitancy. \"Barabbas!\" the crowd shouts. \"Away with this fellow, and release to us Barabbas.\"\n\nPilate sighs and asks, \"Then what shall I do with the one you call the king of the Jews?\"\n\nThe people have requested grace for a terrorist; surely they will seek mercy for the gentle teacher as well.\n\nThe people cry out, \"Let him be crucified!\"\n\nPilate sees the crowd turning into a mob, and he blunders by trying to reason with them. \"Why, what evil has he done?\" Pilate says. \"I have found no crime in him deserving death. I will therefore punish and release him.\"\n\nLoud shouts drown Pilate out; the mob is gripped by bloodlust. \"Let him be crucified!\" they demand.\n\nPilate stands and walks back inside. The morning is passing, the mob is growing, and the situation is deteriorating. To appease the crowd, the governor orders his men to take the prisoner to an adjacent courtyard and punish him. The soldiers drag Jesus out, strip him naked, and throw him facefirst against a rough-hewn olivewood post. His bound wrists are tied high to the pillar, exposing his back and sides.\n\nOne of the soldiers picks up a scourge and walks slowly toward the condemned. He is skilled with the whip and knows just how much pain to inflict to accomplish the desired effect.\n\nThe Romans have three levels of whippings: the fustigatio, the least severe and usually prescribed for minor offenses, the flagellatio, for more serious offenses, and the verberatio, a life-threatening scourging normally reserved for those on their way to execution.\n\nPilate sentences Jesus to the fustigatio, but even this mildest of the beatings extracts a terrible price. The whip is made of multiple leather thongs plaited around found objects\u2014pieces of jagged metal and bone shards and lead weights and fishhooks. In skilled hands, it is an instrument capable of delivering anything from a warning to a death sentence.\n\nWhen the soldiers are satisfied the prisoner has received a worthy fustigatio, they release him from the post; it is time for the games to begin, and this prisoner will be the perfect pawn for a popular soldier's game. After all, what could be more fun than playing the game of king with a man who thinks he is a king?\n\nProps are needed. One of the soldiers fashions a circlet that mimics the one worn by Caesar but is made of thorns. His mates roar with laughter as he forces the crude crown on Jesus's head. Another finds a spare tunic to serve as his royal robe. They encircle Jesus and begin slapping and pushing and taunting him. \"Hail, king of the Jews!\"\n\nWhen the soldiers are finished, they inform Pilate. The governor evaluates the work his men have done on Jesus and is confident the crowd will be appeased.\n\nPilate walks back outside and the people grow quiet. He gestures toward the doorway and says, \"Look, I am bringing him out to you so that you may know that I find him not guilty.\" Jesus is ushered to the porch wearing his new crown and wrapped in his mock regal robe. A loud murmur ripples through the crowd, and Pilate assumes they have seen the foolishness of their indictment. He points toward the impotent prisoner and says, \"Behold the man!\"\n\nBut Pilate has underestimated the Sanhedrin's hatred. When the chief priests see Jesus, they begin shouting, \"Crucify him! Crucify him!\" Soon the chant is picked up by the mob, and all are clamoring, \"Crucify him!\"\n\nPilate tells the soldiers to take the accused back inside. He waits for the din to subside, but for several minutes the crowd continues to shout, \"Crucify him! Crucify him!\"\n\nPilate turns to the priests who incited the riot. \"Take him yourselves and crucify him\u2014I find no fault in him.\"\n\nFor a third time the governor has pronounced Jesus not guilty, but the Sanhedrin refuses to accept the verdict. One of the councilmen reminds Pilate, \"We have a law, and according to that law he ought to die because he claimed to be the Son of God.\"\n\nThe charge of blasphemy is being made, but all Pilate hears is that Jesus professes to be a son of God, and to Pilate that conjures images from his own religious beliefs. He is convinced there are divine men who walk the earth\u2014mortals endowed by the gods with supernatural abilities.\n\nPilate recalls Jesus mentioning he had a kingdom in another world, and fear grips him\u2014the last thing he wants to do is answer to the gods for harming one of their chosen ones. He quickly walks inside and within moments is in Jesus's face. \"Where are you from?\" he whispers.\n\nJesus looks into Pilate's eyes but says nothing.\n\nThe uncomfortable silence frustrates the prefect. \"You refuse to speak to me?\"\n\nJesus says nothing, and Pilate walks away even more afraid. After reflecting for a few moments, he turns back toward Jesus. His eyes narrow and he asks, \"Do you not realize that I have authority to release you and authority to crucify you?\"\n\nJesus nods and says, \"You would have no authority over me at all unless it had been given you from above. Therefore he who handed me over to you has the greater sin.\"\n\nPilate has no idea what Jesus is talking about, but the answer disturbs him; he is now convinced he wants nothing to do with any of this. He walks back onto the porch and pleads with the mob to change their mind. All his requests are met with angry howls. The confrontation climaxes when a politically savvy council member delivers a veiled threat. \"If you release this man, you are not Caesar's friend. Everyone who makes himself a king opposes Caesar.\"\n\nThis is a prospect Pilate cannot dismiss. He knows Tiberius Caesar is by nature a suspicious man, and a report from a distant land that the prefect is allowing someone to call himself a king will not be ignored. The emperor wants peaceful territories that produce crops to fill Rome's larder, not hotbeds of rebellion, and the governor knows that if he can't maintain the Pax Romana, he will be deposed.\n\nPilate realizes that justice in this case may have to give way to political expediency, but he desperately wants to avoid that if possible. At the same time, he loathes the Jews and loves to disturb them. He brings the accused out again and taunts, \"Behold your king!\"\n\nOutraged, the religious leaders cry, \"Away with him! Away with him! Crucify him!\"\n\nPilate can't resist goading them. \"Shall I crucify your king?\"\n\nOne of the chief priests answers angrily, \"We have no king but Caesar.\"\n\nPilate is tempted to laugh at the thought of Jewish allegiance to the emperor, but he knows he is trapped in an impossible situation. He ceremoniously walks over to a basin and begins washing his hands, imitating the Jewish rite of purification. \"I am innocent of this man's blood,\" he tells the crowd. \"See to it yourselves.\"\n\nThe mob roars in approval, \"His blood be on us and on our children!\"\n\nPilate marvels. In the name of God these people request the release of a known insurrectionist and murderer, yet they demand the life of an innocent man accused of the same crime. The exasperated governor instructs the guards to free Barabbas and begin the crucifixion process with the Galilean. He will join two other criminals condemned to die today.\n\nThe Roman troops are disgruntled by the addition; they know this third man is popular, and that means crowd control and more work. There is one consolation though: they will have another turn at the whip with him. The first part of the crucifixion process is a dreadful scourging known as the verberatio. When done properly, the criminal is flayed to the bone but remains alive enough to experience the agony of the cross and a lingering death.\n\nBlow after blow falls on Jesus's already lacerated body. When the scourging stops, the soldier is dripping with sweat, and Jesus's back is an indefinable red mass. His face, torn and bleeding before the verberatio began, has taken numerous blows; his visage is now so marred that he barely looks human.\n\nThey lift the prisoner to his feet. Jesus attempts to stand, but his legs buckle. The soldiers catch him and prop him against the post to assess the damage they have inflicted. They determine he will survive the scourging and is good for a little more sport. They drape a red tunic over Jesus's shoulders and place a wooden staff in his hands to serve as his scepter. The soldiers then close in around him and begin pushing him around the circle, slapping him and spitting in his face. Several snatch the staff from his hand and bring it down upon his head; others kneel in homage and salute with a mock ave Caesar: \"Hail, king of the Jews!\"\n\nThe abuse continues until Jesus's weakened condition forces them to stop. They don't want to have to carry the laughingstock king to the cross. They strip him of the blood-soaked tunic and dress him in his own clothes, but for fun they leave the king his crown.\n29\n\nCRUCIFIED\n\nNisan 14, 3793\n\nChrist redeemed us from the curse of the law by becoming a curse on our behalf, for it is written, \"Cursed is everyone who is hung on a tree.\"\n\n\u2014Galatians 3:13\n\nA CROWD HAS gathered in the street outside the old Herodian palace that serves as the Praetorium when Pilate is in Jerusalem. It is rumored that the controversial teacher from Galilee has been arrested for sedition and blasphemy. The details are few, but some say he is being tried by the Roman governor this morning.\n\nWhen the onlookers see Jesus being dragged into the street by a four-man execution squad, several gasp. The gentle teacher who preached daily at the temple has been scourged so severely that he is barely recognizable. Many shake their heads as the soldiers lead the condemned man to a large crossbeam, place one end of it on his shoulder, and push him into the street.\n\nJesus staggers a few steps and falls on his face. The soldiers curse and lift him to his feet. He stumbles forward and makes it as far as the city gate before collapsing again. The guards kick him in the side and order him to stand. Jesus tries to rise but slumps under the weight.\n\nThe soldiers start grumbling; at this pace they will be all morning with this wretched assignment. They look into the crowd and conscript a young man to carry the patibulum. Simon, a native of Cyrene who is in town to celebrate the Passover, dutifully shoulders the beam and follows.\n\nAn audience trails the procession. Some are curious pilgrims, some are enemies, some are sympathizers, and some are family. A small but vocal minority are women from Jerusalem who are lamenting the Galilean's fate.\n\nJesus stops and turns to the women. \"Daughters of Jerusalem, do not weep for me\u2014weep for yourselves and your children. For behold, the days are coming when they will say, 'Blessed are the barren and the wombs that never bore and the breasts that never nursed!' Then they will cry out to the mountains, 'Fall on us,' and to the hills, 'Cover us.'\"\n\nDaughters of Jerusalem is a phrase used in the Scriptures to refer to the entire population of the city. Jesus says their mourning is misdirected; they need to be concerned about themselves and their families. A judgment will overtake this generation that will be so terrifying that the residents of Jerusalem will plead for creation itself to end their misery. Jesus wants Israel to realize something far worse than a crucifixion can come upon a nation.\n\nJesus asks an enigmatic question. \"If they do these things when the wood is green, what will happen when it is dry?\"\n\nHe is asking them to consider what they are seeing. If God will allow a fiery judgment to fall on one who is full of life, what will be the fate of a nation whose faith is dead?\n\nThe soldiers prod Jesus to continue walking, and they are soon at their destination: a hill north of the city that the locals call Golgotha because of its eerie resemblance to a skull. The Romans like the site. In addition to being conveniently close to the Praetorium, it is on a well-traveled road and thus is an effective venue to demonstrate what happens to those who challenge the emperor.\n\nCrucifixion is an indescribably painful and humiliating way to die, but the Jews have a further incentive to abhor the cross: the Torah says that anyone who is hanged on a tree is accursed by God.\n\nThe priests who follow the procession know this passage well and are gloating. Once Jesus is crucified, the people of Israel will be scandalized by the suggestion that a gallows bird is the Messiah.\n\nWhen the execution detail reaches Golgotha, the other condemned men are already hanging, but there is an empty stake between them reserved for the day's most infamous criminal.\n\nSimon of Cyrene is relieved of the beam and dismissed. One of the soldiers pours a cup of wine mixed with myrrh and puts it to Jesus's lips to torment him. He tastes the bitter liquid, winces, and turns away.\n\nThe men laugh, strip off his clothing, and throw him to the ground. One holds his arm to the patibulum while another drives a six-inch iron spike through the base of his palm and into the olive wood. His other arm is stretched out, and the process is repeated.\n\nTwo of the soldiers lift the beam to their shoulders, and then the other two come from beneath with forked poles and boost the spar high so that passersby can see where following this rebel will take them.\n\nAs he is lifted up, Jesus's body pitches forward; he groans as the nails take the weight of his body. His legs are placed together and turned to the side with knees slightly bent. Another long spike is hammered just under the ankles of both heels and deep into the tree.\n\nHe is now crucified and enduring one of the most exquisite tortures ever devised.\n\nJesus looks into faces filled with hatred and prays, \"Father, forgive them, for they know not what they do.\"\n\nOne of the soldiers props a ladder to the cross, climbs it, and nails a titula with three inscriptions to the top of the upright. The charge of sedition is written in Aramaic, Greek, and Latin so that all will know his offense: \"Jesus of Nazareth, the King of the Jews.\"\n\nWhen the priests see the title, they are furious. They demand the soldiers remove the sign, and when they are told it is there by order of the governor, they rush down the hill and inform the high priest. Caiaphas immediately goes to the Praetorium to request the sign be changed.\n\n\"Do not write 'The King of the Jews,'\" he tells Pilate, \"but rather 'This man said I am King of the Jews.'\"\n\nPilate hears the complaint but is unsympathetic. The danger of being reported to Caesar is now past, and the governor is not about to miss an opportunity to goad the men who threatened him. \"What I have written,\" he tells Caiaphas, \"I have written.\"\n\nThe high priest is livid but can only turn and walk away.\n\nAt the execution site, the four soldiers are bored and begin dividing up the meager spoils they are inheriting from the condemned. The items are few and of little value, but the men still quarrel over who gets what. When they come to Jesus's seamless tunic, they decide to gamble for it. Things are quiet on the hill, and the men are soon lost in their dice game.\n\nOblivious to the soldiers is a small group of Jesus's family and friends who have moved close to the cross. All but one are women from Galilee who have been a key part of Jesus's ministry since the beginning.\n\nThey have given money and supplies; they have cooked and cleaned; they have run errands and drawn water; they have ministered to women and children in crisis, but there is much they have been unable to do because of their gender. Nevertheless, the women do what they can\u2014they faithfully follow Jesus and minister to him whenever they see a need. And they are committed to doing this regardless of what happens.\n\nJesus looks down from the cross. His vision is blurred, but he recognizes the familiar faces of his loved ones. He is particularly interested in one woman who is present\u2014his mother.\n\nMary is biting her lip and weeping silently. She is now a widow in her fifties, and Jesus wants to assure her that she will want for nothing. He looks into his mother's anxious eyes, then to John, then back to Mary.\n\nHe smiles weakly and says, \"Woman, behold, your son!\"\n\nThen he looks at his apprentice. \"Behold, your mother!\"\n\nOthers standing near the cross hear Jesus speaking and join in, but they want to take the conversation in a different direction.\n\n\"You who would destroy the temple and rebuild it in three days, save yourself! If you're the Son of God, come down from the cross.\"\n\nSeveral onlookers begin laughing, and the group from the Sanhedrin wants to provoke the crowd. \"He saved others,\" one says, \"but he can't save himself.\"\n\nAnother crows, \"Let him save himself if he's the Messiah of God, His Chosen One!\"\n\nAnother man snickers and says, \"Let this Messiah, the king of Israel, come down now from the cross so that we may see and believe in him. He trusts in God; let God deliver him now\u2014if He desires him. For he said, 'I am the Son of God.'\"\n\nThis brings guffaws and more derision. One of the soldiers offers Jesus some posca, the vinegar wine the soldiers are allowed to drink while on duty, and says, \"If you're the king of the Jews, save yourself!\"\n\nThe ridicule becomes so widespread that even the two insurrectionists hanging on either side of Jesus join in. Eventually, one of them begins thinking about what he is doing and becomes silent. He has heard of Jesus, but until now has given Jesus's teachings little consideration.\n\nThe man on the other cross, however, continues to mock. \"Are you not the Messiah? Save yourself\u2014and us!\"\n\nThis hypocrisy is too much for the criminal now repenting of his earlier remarks, and he rebukes the scoffer. \"Do you not fear God? Are you not under the same condemnation? And indeed we are punished justly\u2014we're getting what we deserve for our deeds. But this man has done nothing wrong.\"\n\nHe then pauses and looks to the man in the middle. He knows he doesn't deserve to ask, but he musters the courage to say, \"Jesus, remember me when you come into your kingdom.\"\n\nJesus, himself near death, looks into the fearful man's eyes and assures him, \"I tell you truly, today you will be with me in Paradise.\"\n\nWhile Jesus is speaking to his new disciple, the people on Golgotha notice something strange happening. It is noon, yet the sun is growing dim. Within minutes, the entire land is immersed in an eerie twilight.\n\nThe preternatural darkness is unnerving. Several onlookers determine it is a bad omen and quickly walk down the hill to seek refuge within the city walls. Those who spent the morning mocking Jesus are now quiet, hoping the gloom will lift.\n\nIt doesn't. Three hours pass, and still a pall hangs over the land. The ominous silence is broken by a loud cry from the cross. \"My God, my God, why have you forsaken me?\"\n\nSeveral bystanders say, \"Listen, he's calling Elijah.\"\n\nThey have misunderstood Jesus. They hear his cry to Eli (the Aramaic word for God) and think he is a calling Elijah. Many believe that God sometimes sends the prophet to rescue the righteous from suffering.\n\nThey think they hear Jesus say something about being forsaken and strain to hear him talk more about who has abandoned him, but when he finally speaks, all he says is, \"I thirst.\"\n\nOne of the women runs to the bucket of posca, dips a sponge in it, and puts it to Jesus's lips. Others try to stop her. \"Wait!\" they say, \"let's see whether Elijah will come to save him.\"\n\nBut the ancient prophet does not appear. Instead, Jesus moistens his lips with the vinegar and cries out, \"It is finished! Father, into Your hands I commit my spirit.\"\n\nWith these words, Jesus bows his head and dies.\n\nAs Jesus releases his last breath, a violent earthquake rocks Jerusalem.\n\nAt the temple, it is time for the afternoon sacrifice and the slaughter of the lambs for the Passover. The priests at the altar are jolted by the quake and hear a loud ripping sound coming from inside the Holy Place. They run to investigate the damage but stop in the doorway and stare in disbelief into the Holy of Holies. The thick curtain that keeps the most sacred space in perpetual darkness has been torn from top to bottom\u2014a tear not unlike the way a judge rips his robe to pronounce guilt.\n\nLight now streams into the Holy of Holies. The Levites, trembling, run to find the high priest, but travel in the city is difficult. The streets have sustained extensive damage; foundation stones have been moved, and numerous roofs have collapsed. In the Kidron Valley, where there are a large number of tombs, the quake is so strong that many of the graves lie open, the stones once covering them in pieces on the ground.\n\nThe only good news many in Jerusalem can see is that the darkness has receded, and the sun is once again shining.\n\nThe Roman centurion overseeing the execution detail is awestruck. He shakes his head and says, \"Certainly this man was innocent\u2014truly he was the Son of God!\"\n\nIt is now midafternoon and only a few hours remain before sunset and the beginning of the Sabbath. The Jews petition Pilate to hasten the death of the men on the crosses; the living corpses will defile the paschal Sabbath about to begin.\n\nThe governor, familiar with Jewish idiosyncrasies, has no objections and orders a crurifragium\u2014with their legs broken, the condemned can't push themselves up to free their diaphragm, and they will quickly asphyxiate.\n\nA stout iron club is used to crush the shinbones of the first two, but when the executioners come to the man in the middle, they find he has already expired. To confirm he is dead, one of the soldiers thrusts his lance between the malefactor's ribs, piercing his heart. As he withdraws the blade, blood and water pour from the wound. The centurion supervising the detail officially pronounces Jesus dead and walks back into town to report the task is complete.\n\nThe centurion is not the only one on his way to the governor's residence. Soon after Jesus dies, Pilate receives an unexpected visit from Joseph of Arimathea. The wealthy councilman is requesting permission to give Jesus an honorable burial. Joseph knows his petition could be misconstrued and have him branded a rebel sympathizer, but he will not stand idly by while Jesus's body is thrown into a criminal's grave.\n\nUntil now, few have known Joseph's sympathies, but he is not Jesus's only friend in the Sanhedrin. Nicodemus, the Pharisee who visited Jesus three years earlier, is also now ready to identify himself as a disciple. While Joseph is petitioning Pilate for the body, Nicodemus is hastily gathering the supplies they will need. He has little time, but much money, and doors open to him.\n\nAt the Praetorium, Pilate is considering Joseph's request. Ordinarily the governor grants appeals by family and friends for a criminal's body. Except in one circumstance\u2014when the crime is sedition. Roman policy throughout the empire is to leave the bodies of rebels and revolutionaries hanging on the cross to rot and become carrion, and when their bones are picked clean, they are tossed on the refuse heap to be burned.\n\nIn Israel, however, the execution process has been altered to placate Jewish convictions about ritual cleanness; a body hanging on the cross during the Sabbath will defile the land.\n\nAfter Pilate consults the centurion and confirms Jesus's death, Joseph's petition is granted. The governor has no qualms about releasing the body to the prominent Sanhedrist; he knows the indictment against Jesus was false. Pilate watches the old man hurry away and laughs; this is sure to infuriate Joseph's peers.\n\nON GOLGOTHA, THE soldiers are packing their gear; they're ready for a hot meal and more posca. When Joseph informs them of the governor's authorization, the soldiers are barely interested. They gesture toward the body and continue working.\n\nJoseph and his servants carefully take Jesus down from the cross and to a freshly hewn tomb in a nearby garden. Joseph had purchased the grave for himself and his family; now it will be Jesus's final resting place.\n\nThe sepulcher, like others nearby, is cut into a sheer rock face, a remnant of the quarry that was here centuries earlier. The entrance is small, perhaps three feet in height, and leads into an antechamber tall enough to stand in and prepare the body. Because family tombs must be entered numerous times over the years, a large boulder is placed at the entrance to dissuade hungry animals.\n\nFor the tombs of the wealthy there is a more secure seal: A track several inches deep is notched in the ground at the entrance. In the slot sits a four-foot stone disk that serves as the door. The track is cut on an incline, making the door easy to close but difficult to open; several men will be needed to roll the heavy stone back up the slope once it is sealed.\n\nJoseph instructs his servants to place Jesus's body on the stone slab in the antechamber. Nicodemus soon arrives with the necessary supplies. In accordance with Jewish custom, the body is carefully washed and then wrapped in clean linen.\n\nIn the tomb, the aroma is intense. Nicodemus, wanting to honor his Lord in death in a way he did not in life, has purchased more than sixty-five pounds of powdered myrrh and aloes\u2014thirty thousand denarii worth of spices\u2014an amount fit to bury a king.\n\nWatching from a distance are two disciples, Mary Magdalene and Mary, wife of Clopas. The two women have not let Jesus out of their sight since he was thrown into the street at the Praetorium, but they do not speak to the two wealthy strangers. The women recognize them only as members of the council that condemned their Lord to death.\n\nIt will be Sabbath within an hour, and the two Marys know there is little they can do for Jesus until the Sabbath has passed, but there is one thing\u2014they can purchase and mix the spices they will use to anoint his body. The two women are grieving as they leave the tomb, but they are comforted to know where their master's body will be when they return Sunday morning.\n\nThe sun sets and the paschal Sabbath begins. Because the Sabbath this year falls on the first day of the festival of Unleavened Bread, it is a Great Sabbath. Ordinarily this would be a time of heightened celebration, but many in the city are thinking about today's events. First there was the strange darkness that enveloped the land from noon until three. Then came the earthquake. Then tonight, when the full moon rose, it was blood red. Some attribute the events to coincidence. Others suspect they are signs from heaven and wonder what God is telling them.\n\nSunrise finds the temple plaza filled with thousands of people, but not everyone is worshipping. Dozens of Jesus's followers are huddled in small groups, grieving and confused and struggling to be thankful.\n\nThe nation's religious leaders are also not worshipping. They are discussing a prediction Jesus made that he would rise from the dead. The Sadducees who dominate the Sanhedrin find the notion of resurrection laughable, but they are not naive. What is to stop the deceiver's disciples from vandalizing the grave site and spreading lies? Priests and Pharisees are dispatched to the governor's residence to request that a Roman security detail be assigned to the tomb.\n\nThe emissaries go to Pilate and say, \"Sir, we remember that when that impostor was still alive he said, 'After three days I will be raised.' Therefore order the tomb be made secure until the third day so that his disciples can't come and steal him and then tell the people, 'He has been raised from the dead.' That last deception would be worse than the first.\"\n\nPilate smiles and wonders what kind of dread this man instilled in them that they fear him even in death. But the governor has no interest in wasting good men on a fool's errand.\n\n\"You have a guard of soldiers. Go make it as secure as you can.\"\n\nThis is not the answer the leaders want, but at least they will be able to seal and stand guard at the tomb. The priests and Pharisees thank the governor and depart. Within an hour the grave site is swarming with temple police. The Levites roll away the stone, confirm the body is in the sepulcher, and roll the stone over the door. To assure no one breaks in undetected, the tomb is sealed, and a guard is set.\n\nWith the body of the impostor now secure, the priests and Pharisees decide they can finally relax. Tomorrow will be the third day. The Nazarene will be proven a liar and left to rot in his grave, and they can get on with the business of serving God.\n30\n\nRAISED\n\nApril\u2013May, AD 33\n\n\"I will put enmity between you and the woman, and between your offspring and her offspring; he will bruise your head, and you will bruise his heel.\"\n\n\u2014Genesis 3:15\n\nIN THE PREDAWN darkness, a group of Galilean women quietly dress and make their way toward the Gennath Gate. Mary Magdalene, Joanna, Mary the mother of James, Salome, and several others are bringing aromatic oils to anoint Jesus's body. They saw the abundance of spices brought to the tomb by the wealthy Sanhedrists who buried him, but these women have faithfully served Jesus since the early days in Capernaum, and they want to anoint their master with fragrances they have lovingly mixed with their own hands.\n\nThe women know nothing about the guard posted at the sepulcher; their concern is the large rock that blocks the entrance. They wonder aloud, \"Who will roll away the stone for us from the door of the tomb?\"\n\nAt the grave site, the Levites are nearing the end of their watch. The night has been uneventful, and the men are looking forward to some hot food and a bed.\n\nJust before dawn, the earth begins to quake, and a brilliant light appears in the garden. The guards stand motionless and gape\u2014the brightness is a person.\n\nThe luminous presence effortlessly rolls away the stone and perches atop it. The Levites guarding the tomb are no-nonsense professionals, but they have no defense against angels. Their weapons fall to the ground, and they faint from fear.\n\nWhen the earthquake strikes, the women are nearing the city gate. Mary Magdalene senses something is amiss and bolts toward the garden. When she reaches the open sepulcher, she abruptly stops. She sees the guards lying unconscious near the entrance and assumes someone has stolen the body. Peter and the apostles must be alerted. Mary Magdalene flees the garden, running as fast as she can.\n\nA few minutes later the other women arrive. Seeing the stone rolled away and the men on the ground, they assume the worst but decide to assess the damage inside the tomb before reporting the theft. Joanna, Salome, Mary, and the other women enter and are amazed to find that while the body is missing, the grave clothes and a fortune in precious spices rest undisturbed on the table. It appears the body has simply vanished.\n\nAs they stand looking at the empty grave clothes, they are joined by two angels. The disciples fall to their knees with their faces to the ground.\n\n\"Do not be afraid,\" one of the messengers says. \"You seek Jesus of Nazareth, who was crucified. Why do you seek the living among the dead? He is not here but has risen! Remember how he told you while he was still in Galilee, 'The Son of Man must be delivered into the hands of sinful men and be crucified and on the third day rise.'\"\n\nThe women slowly nod their heads as they recall the Master's predictions. Everything has happened just as he foretold.\n\n\"Come.\" The angel points toward the table of linen and spices. \"See the place where they laid him. Then go quickly and tell his disciples and Peter that he has been raised from the dead and is going ahead of you to Galilee\u2014there you will see him just as he told you.\"\n\nThe angel becomes very still and says, \"This is my message to you.\" Then both angels disappear.\n\nThe women immediately set out to find the apostles, but as they are leaving the garden they are met by the last person they would expect to meet.\n\nJesus walks up to them and says, \"Greetings!\"\n\nThe women fall to the ground and clutch his feet. They want to touch him to be sure he is really there.\n\n\"Do not be afraid,\" Jesus says. \"Go and tell my brothers to go to Galilee\u2014that's where they will see me.\"\n\nAT THE SEPULCHER, the guards are reviving. They stand on wobbly legs and stagger toward the city to report the empty tomb. When they tell the priests they were assaulted by an angel, a hasty meeting of the Sanhedrin is called.\n\nThe leaders aren't interested in discussing the implications of a divine visitation\u2014their goal is to control the story. Several options are discussed, and ultimately they decide to bribe the witnesses.\n\nThe Levites are called before the ruling council. The men are fearful\u2014if it is determined that they have lied, they will be executed for dereliction of duty. The policemen are pleasantly confused when a priest begins handing out leather pouches. They open the bags, look at one another, and smile.\n\nOne of the priests explains what they must do to earn the money: \"Tell people, 'His disciples came by night and stole him away while we were asleep.' And if this comes to the governor's ears, we will satisfy him and keep you out of trouble.\"\n\nThe Levites pocket the bribe and vow to keep the truth to themselves.\n\nIN ANOTHER PART of town, Mary Magdalene has found Peter and John.\n\n\"They've taken the Lord out of the tomb, and we don't know where they put him!\"\n\nThe two apostles are quickly out the door and running toward the grave site. Mary Magdalene follows, weeping disconsolately.\n\nJohn arrives at the sepulcher first but hesitates to enter. He bends down and peers into the shadows. He can see the grave clothes, but no body.\n\nPeter arrives and immediately enters; John follows cautiously. The men are not ready for what they find. The linen cloths and spices lie untouched, but the napkin that was placed over Jesus's face has been neatly folded and is sitting to the side.\n\nAs the men leave the tomb, they are struggling to understand what has happened, but in John a change is taking place. He knows Jesus has somehow done what he promised. Lost in thought and considering an impossibility, the men say little to one another as they walk back to town.\n\nMary Magdalene sees the apostles leave. Sobbing, she walks to the tomb and stoops to look inside. Sitting on the table are two angels in bright raiment. \"Woman,\" they ask, \"why are you weeping?\"\n\nMary Magdalene repeats what she told Peter and John. \"They've taken my Lord away, and I don't know where they put him!\"\n\nMary hears a noise and turns to see a man standing behind her. He asks her the same question the angels asked.\n\n\"Woman, why are you weeping? Whom are you seeking?\"\n\nMary Magdalene doesn't recognize the man and assumes he is the caretaker. \"Sir,\" she pleads, \"if you've carried him away, tell me where you've put him\u2014I will take him!\"\n\nEmbarrassed by her boldness, she blushes and looks to the ground.\n\nThe stranger gently says, \"Mary.\"\n\nShe looks up and cries, \"Teacher!\" and falls to his feet.\n\n\"Do not cling to me,\" Jesus tells her, \"for I have not yet ascended to the Father. But go to my brothers and tell them, 'I am ascending to my Father and your Father, to my God and your God.'\"\n\nMary Magdalene, her strength renewed, rises and runs from the garden. When she arrives at the home where the apostles are gathered, she discovers her friends have also seen the angels and spoken to Jesus. Mary tells everyone what happened at the tomb and gives them the message Jesus gave her.\n\nPeter and John listen carefully but say nothing; they are still trying to make sense of what they saw in the tomb. The other apostles aren't as ambivalent; they believe the women are suffering from grief-induced delirium. Jesus is dead.\n\nThere is disagreement among the men and women who have been following Jesus. Some are hopeful, but others are discouraged and convinced the cause is lost. Later that day, two disciples decide to go home to the village of Emmaus. As they walk, they get into a heated debate about what happened to Jesus.\n\nSoon they are overtaken by another pilgrim who is leaving Jerusalem. They assume he's headed home early from the festival as they are.\n\nThe man comes alongside them and asks, \"What were you discussing with each other as you walked?\"\n\nThe two stop and turn to the stranger. One of them, a man named Cleopas, is astounded.\n\n\"Are you the only visitor to Jerusalem who doesn't know the things that have happened there these last few days?\"\n\n\"What kind of things?\" the stranger asks.\n\n\"Things concerning Jesus of Nazareth, who was a prophet mighty in deed and word before God and all the people, and how our chief priests and leaders handed him over to be condemned to death and crucified him. We were hoping that he was the one to redeem Israel, but it has already been three days since all this happened.\n\n\"Then some women in our group amazed us. They were at the tomb early this morning, and when they didn't find his body there, they came back and told us they had seen a vision of angels who said that he was alive. Some of those who were with us went to the tomb and found it just as the women had said, but they didn't see Jesus.\"\n\nThe stranger looks into the eyes of Cleopas and his companion and gently replies, \"Oh how foolish you are, and how slow of heart to believe all that the prophets spoke! Was it not necessary for the Messiah to suffer these things and enter into his glory?\"\n\nWhen the men do not answer, the stranger launches into a long explanation of how everything that had happened over the past three days had been foretold in the Scriptures.\n\nAs the mysterious traveler explains the necessity of the cross, the gloom in the disciples lifts. By the time they reach Emmaus, Cleopas and his companion are renewed and wanting to hear more from the peculiar teacher.\n\n\"Stay with us,\" they plead, \"evening is near, and the day is already spent.\"\n\nThe stranger agrees, and they go to one of the men's homes for dinner. A simple meal is prepared, and the visitor is asked to offer thanks. He takes a loaf of bread, prays, then breaks it and gives some to Cleopas and his friend. As the disciples take the bread, they realize they are dining with Jesus. Their mouths drop open, and the men look at each other in disbelief, but when they turn back to Jesus, he is not there.\n\nWhen the men recover from the shock, they ask each other, \"Did not our hearts burn within us as he talked to us on the road and opened up the Scriptures to us?\"\n\nThere is no question in their minds\u2014Jesus has risen. They jump up from the table and return to Jerusalem. Night has fallen by the time Cleopas and his companion reach the city.\n\nTen of the eleven apostles are there when they arrive. The two disciples describe their encounter with Jesus on the Emmaus road and discover they are not the only ones who have seen him. One of the apostles says, \"The Lord has risen indeed, and has appeared to Simon!\"\n\nThe men become excited as they discuss what all this might mean, and they don't notice that their circle has grown by one. The new arrival says, \"Peace be with you!\"\n\nThe men are startled and step back, unsure how the one before them got past the locked door. They assume they are in the presence of a ghost.\n\n\"Why are you frightened?\" the visitant asks. \"Why do doubts arise in your hearts? Look at my hands and my feet\u2014see that it's really me. Touch me and see for yourselves. A spirit doesn't have flesh and bones as you see I have.\"\n\nJesus then offers his hands for their inspection and shows them his pierced side. The men are overcome by astonishment and can't believe they are speaking with Jesus.\n\nJesus knows they are having trouble believing their eyes and asks, \"Do you have anything here to eat?\"\n\nThe men nod and offer him a piece of broiled fish. As Jesus eats, the apostles marvel. This is too good to be true.\n\nJesus smiles and says, \"Peace be with you. As the Father has sent me, even so I am sending you.\"\n\nHe then takes a large breath and exhales, saying, \"Receive the Holy Spirit. If you forgive anyone's sins, they are forgiven; if you do not forgive them, they are not forgiven.\"\n\nLike the washing of the apostles' feet at Passover, this is an acted parable. Soon the Holy Spirit will indwell and empower the Lord's church to preach the good news, and the new community will be used to both comfort the broken and confront the proud.\n\nWhen Jesus concludes his instructions, he leaves as mysteriously as he arrived.\n\nThe ten sit in silence trying to process what happened. By the time the apostle Thomas returns, they are convinced they did not imagine Jesus's visit.\n\n\"We've seen the Lord!\" they tell Thomas.\n\nThomas frowns and shakes his head. \"Unless I see the nail marks in his hands and put my finger where the nails were and put my hand into his side, I will never believe.\"\n\nThomas is convinced that Jesus is dead, and the ten say no more.\n\nThe apostles remain in Jerusalem another week until the festival ends. Most of their time is spent behind locked doors awaiting arrest as coconspirators, but the time passes uneventfully.\n\nOn the first day of the week, the apostles are preparing to go home to Galilee. Jesus again appears in their midst and says, \"Peace be with you.\"\n\nHe walks over to Thomas, stretches forth his palms, and says, \"Put your finger here and look at my hands. Put your hand into the wound in my side\u2014stop doubting and believe!\"\n\nBut Thomas no longer wants to probe the wounds. He falls to his knees and confesses, \"My Lord and my God!\"\n\n\"You believe because you've seen me,\" Jesus says. \"Blessed are those who haven't seen, yet have come to believe.\"\n\nWith these words, Jesus vanishes from their sight.\n\nTHE DISCIPLES ARE soon on their way to Galilee. When they arrive in Capernaum, they are unsure how to proceed. They were told to go home and wait, and they have. Now what?\n\nA week passes with no sign of Jesus. Simon Peter announces, \"I'm going fishing.\"\n\nSix of the apostles join him. The men spend all night on the lake. Time and again they pull in empty nets. By sunrise they have had enough and head in.\n\nThey see a stranger standing on the shore.\n\n\"Young men,\" he calls, \"have you no fish?\"\n\nThe apostles are too tired to engage in conversation, and simply answer no.\n\n\"Cast the net on the right side of the boat,\" the stranger tells them. \"You will find some.\"\n\nIt's been a long and futile night, but they humor the stranger and halfheartedly cast the net one more time. When they begin to pull it in, the weight tells them the net is full. All hands begin feverishly pulling in the catch, but as they haul in the fish, the apostle John realizes this is all too familiar. When he looks again at the man on shore, he recognizes him. He nudges Peter and says, \"It's the Lord!\"\n\nPeter turns and recognizes Jesus. He leaps into the water and begins swimming. The other disciples pull in the overflowing net and row in. When they land, they find Jesus has built a small charcoal fire and is cooking fish and heating bread.\n\n\"Bring some of the fish you've just caught,\" he tells them.\n\nPeter retrieves several fish, and Jesus adds them to those already on the coals. When the fish are cooked, Jesus invites the apostles to have breakfast with him. The disciples walk over timidly. They know it is Jesus who is serving them, but they don't understand why, and none have the courage to ask.\n\nThey take the food and eat in silence. When they're finished, Jesus turns to Peter. He gestures toward the other disciples and asks, \"Simon bar Jonah, do you love me more than these?\"\n\nPeter is not expecting the question. He has often boasted of his faithfulness in the past, but this morning he is content to appeal to Jesus's knowledge.\n\n\"Yes, Lord,\" he stammers, \"you know that I love you.\"\n\n\"Feed my lambs,\" Jesus responds.\n\nA few minutes later Jesus repeats his question. \"Simon bar Jonah, do you love me?\"\n\nPeter, deeply grieved, again relies on Jesus's knowledge. \"Yes, Lord. You know that I love you.\"\n\n\"Tend my sheep.\"\n\nJesus then asks a third time, \"Simon bar Jonah, do you love me?\"\n\nPeter doesn't know what to say and can only repeat himself. \"Lord, you know everything. You know that I love you!\"\n\nJesus nods and says, \"Feed my sheep. Truly, truly I tell you, when you were young, you dressed yourself and went wherever you wanted, but when you grow old, you'll stretch out your hands, and another will dress you and take you where you don't want to go.\"\n\nAs Jesus turns and walks toward the lakeshore, Peter follows. His mind is racing. Three times Peter stood at a charcoal fire and denied Jesus. Now at another fire he has thrice been challenged with a similar question.\n\nThe good news for Simon Peter is that he is right about Jesus knowing the condition of his heart. Jesus knows Peter has repented and is ready to serve God whenever and wherever God wants.\n\nJesus smiles at his apprentice and says, \"Follow me.\"\n\nPeter is relieved that he will be allowed to remain in the fold despite his apostasy, but he's concerned about Jesus's prediction that when he is old he will \"stretch out his hands\"\u2014a common metaphor for crucifixion. Peter accepts Jesus's prediction without questioning the implications, but when he notices his friend John walking close behind them, he asks, \"Lord, what about him?\"\n\nJesus stops and looks into Peter's eyes. \"If I want him to remain alive until I return, what is that to you? You follow me!\"\n\nJesus tells the eleven to organize a meeting of those who believe in him. He will be leaving soon and has some parting instructions for his apprentices.\n\nThe apostles spread the message, and several days later more than five hundred people gather on a hillside in Galilee. Some are convinced Jesus has risen from the dead, and they worship him. Others are skeptical.\n\nJesus looks over the multitude and proclaims, \"All authority in heaven and on earth has been given to me. Therefore, go and make apprentices of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always, to the end of the age.\"\n\nThe commission is simple. The apostles are to matheteusate\u2014make apprentices. They are to train any who are willing to walk in the Way. They are to pass on what they were taught by Jesus to faithful men and women who will then proclaim it to future generations. They are to baptize people in water, but more importantly, they are to immerse them in an environment where they can experience life with the Father, Son, and Spirit.\n\nWhen Jesus finishes his message, he instructs the eleven to return to Jerusalem and wait.\n\nWITHIN A WEEK the apostles are in the city, but their time with Jesus in Galilee has changed them. They no longer cower in fear behind locked doors. Daily they are in the temple plaza worshipping and praising God.\n\nJesus visits the eleven often during the next few weeks. When the festival of Pentecost (held fifty days after Passover) draws near, Jesus reminds them, \"This is what I told you while I was still with you\u2014everything written about me in the Law of Moses and the Prophets and the Psalms must be fulfilled. It was written that the Messiah would suffer and rise from the dead on the third day, and that repentance and forgiveness of sins are to be proclaimed in his name to all nations, beginning in Jerusalem.\n\n\"You are witnesses of these things. Behold, I am sending the promise of my Father upon you, but stay in the city until you are clothed with power from on high, for John baptized with water, but you will be baptized with the Holy Spirit not many days from now.\"\n\nJesus concludes his teaching and invites the apostles to walk with him. The route is familiar. They walk out of the city into the Kidron Valley and take the road east. They walk past the Garden of Gethsemane and up the Mount of Olives. When they reach the summit, Jesus stops and turns toward the temple. The colorful Babylonian tapestries that adorn the entrance are slowly undulating in the breeze.\n\nThe disciples can sense something dramatic is about to happen, and they assume Jesus is finally going to assert his messianic authority. All their lives they have been taught from the Scriptures that the pouring out of the Spirit will accompany the reign of the Messiah and will reestablish Israel's wealth and power.\n\n\"Lord,\" the eleven ask, \"has the time come to restore the kingdom to Israel?\"\n\nJesus shakes his head. \"That is not for you to know. The Father alone has the authority to set those dates and times. But you will receive power when the Holy Spirit comes upon you, and you will be my witnesses in Jerusalem and throughout Judea and Samaria and to the ends of the earth.\"\n\nJesus then lifts his hands and begins pronouncing blessings on them. As he speaks, he begins to rise. The apostles watch, dumbstruck, as Jesus drifts upward and is enveloped by a cloud.\n\nAs the men stand gazing into the heavens, they are joined by two angels who say, \"Men of Galilee, why are you standing here staring into the sky? Jesus has been taken from you into heaven, but one day he will return from heaven in the same way you saw him go.\"\nEPILOGUE\n\n\"Now Jesus did many other signs in the presence of the disciples which are not recorded in this book. Were every one of them to be recorded, I suppose that the world itself could not contain the books that would be written, but these are written so that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in His name.\"\n\n\u2014John 20:30\u201331; 21:25\nAPPENDIX\n\nA BIBLICAL CHRONOLOGY OF JESUS'S MINISTRY\n\nChapter 1: A Voice from the Wilderness\n\n1) John the Baptist and Scripture: Matthew 3:1\u20136; Mark 1:2\u20136; Luke 3:1\u20136; Luke 1:5\u201319; Isaiah 40:3\u20135; Hosea 2:14\u201315; Malachi 3:1\u20133; 4:5\u20136\n\n2) John's preaching of repentance: Matthew 3:7\u201310; Luke 3:7\u20139\n\n3) John's reply to questioners: Luke 3:10\u201314\n\n4) John's messianic preaching: Matthew 3:11\u201312; Mark 1:7\u20138; Luke 3:15\u201318\n\n5) Baptism of Jesus: Matthew 3:13\u201317; Mark 1:9\u201311; Luke 3:21\u201322; Psalm 2:7; Isaiah 42:1\n\nChapter 2: Inception\n\n1) The temptation: Matthew 4:1\u201311; Mark 1:12\u201313; Luke 4:1\u201313; Exodus 17:1\u20137; Deuteronomy 6:13, 16; Psalm 91:9\u201315; Ezekiel 28:12\u201319; Isaiah 14:12\u201315\n\n2) John the Baptist's testimony: John 1:15, 19\u201334; Isaiah 40:3\n\n3) Jesus's first contact with five of the twelve: John 1:35\u201351; Genesis 27:34\u201336; 28:10\u201313\n\n4) The first sign\u2014water to wine: John 2:1\u201312; Amos 9:13\n\nChapter 3: The First Passover\n\n1) First temple cleansing: John 2:13\u201325; Ezra 10:9; Zechariah 14:20\u201321; Malachi 3:1\u20133; Song of Solomon 2:12\n\n2) The discourse with Nicodemus: John 3:1\u201321; Ezekiel 36:25\u201326; Numbers 21:4\u20139\n\n3) John's testimony about Christ: John 3:22\u201336\n\n4) John imprisoned at Machaerus: Matthew 14:3\u20135; Mark 6:17\u201318; Luke 3:19\u201320\n\n5) Jesus decides to leave Judea: Matthew 4:12; Mark 1:14a; John 4:1\u20133\n\nChapter 4: The God of All Clans\n\n1) The woman from Samaria: John 4:4\u201343; Amos 9:13; Deuteronomy 18:15\u201319; Jeremiah 31:1, 5\n\nChapter 5: Galilee of the Gentiles\n\n1) Jesus returns to Galilee: Matthew 4:17; Mark 1:14b\u201315; Luke 4:14\u201315; John 4:45\n\n2) The healing of a royal official's son from a distance: John 4:46\u201354\n\n3) The calling of the first four disciples: Matthew 4:18\u201322; Mark 1:16\u201320; Jeremiah 16:16\u201318\n\n4) Sabbath exorcism at the synagogue in Capernaum: Mark 1:21\u201328; Luke 4:31\u201337\n\n5) The healing of Peter's mother-in-law: Matthew 8:14\u201315; Mark 1:29\u201331; Luke 4:38\u201339\n\n6) The sick healed in the evening: Matthew 8:16\u201317; Mark 1:32\u201334; Luke 4:40\u201341\n\n7) Jesus leaves Capernaum: Mark 1:35\u201338; Luke 4:42\u201343\n\n8) First tour of Galilee: Matthew 4:23; Mark 1:39; Luke 4:44\n\n9) A supernatural catch: Luke 5:1\u201311; Isaiah 6:5, 8\n\nChapter 6: Authority\n\n1) The cleansing of a leper: Matthew 8:2\u20134; Mark 1:40\u201345; Luke 5:12\u201315; Leviticus 13:45\u201346; 14:1\u201320; Numbers 12:1\u201310\n\n2) The healing of the paralytic lowered through the roof: Matthew 9:2\u20138; Mark 2:1\u201312; Luke 5:17\u201326\n\n3) The calling of Matthew Levi: Matthew 9:9; Mark 2:13\u201314; Luke 5:27\u201328; Isaiah 53:2\u201312\n\nChapter 7: Lord of the Sabbath\n\n1) Sabbath healing at the pool of Bethesda: John 5:1\u201318\n\n2) Jesus's defense of the Sabbath healing: John 5:19\u201347\n\n3) Sabbath grain-picking dispute: Matthew 12:1\u20138; Mark 2:33\u201328; Luke 6:1\u20135; 1 Samuel 21:1\u20136; Exodus 25:30; Leviticus 24:5\u20139; Numbers 28:9\u201310; Hosea 6:6\n\n4) Synagogue Sabbath healing of a man with a withered hand: Matthew 12:9\u201315a; Mark 3:1\u20136; Luke 6:6\u201311\n\n5) Jesus heals multitudes by the lake: Matthew 12:15b\u201316; Mark 3:7\u201312; Luke 6:17\u201319\n\nChapter 8: The Divine Conspiracy\n\n1) The choosing of the twelve: Matthew 10:2\u20134; Mark 3:13\u201319; Luke 6:12\u201316\n\n2) Setting of the Sermon on the Mount\/Plain: Matthew 5:1\u20132; Luke 6:17\u201320a\n\n3) Blessings and woes: Matthew 5:3\u201312; Luke 6:20b\u201326; Isaiah 57:15\n\n4) Salt of the earth, light of the world: Matthew 5:13\u201316\n\n5) Fulfilling the Law of Moses: Matthew 5:17\u201348; Luke 6:27\u201336\n\n6) True and false religion: Matthew 6:1\u201318\n\n7) Wealth and worry: Matthew 6:19\u201334\n\n8) How to treat others: Matthew 7:1\u201312; Luke 6:37\u201342\n\n9) The two ways: Matthew 7:13\u201327; Luke 6:43\u201349\n\nChapter 9: Wisdom's Children\n\n1) The healing of the centurion's servant: Matthew 8:5\u201313; Luke 7:1\u201310; Isaiah 25:6\n\n2) The raising of the widow's son in Nain: Luke 7:11\u201317; 1 Kings 17:17\u201323\n\n3) John the Baptist's question and Jesus's answer: Matthew 11:2\u20136; Luke 7:18\u201323; Isaiah 35:4\u20136\n\n4) Jesus's witness about John: Matthew 11:7\u201319; Luke 7:24\u201335; Malachi 3:1; Jeremiah 31:31\u201334\n\n5) Woes pronounced on Galilean cities: Matthew 11:20\u201324\n\n6) Thanksgiving and invitation: Matthew 11:25\u201330\n\n7) A sinful woman anoints Jesus at a Pharisee's table: Luke 7:36\u201350; Song of Songs 6:5\n\n8) Death of John the Baptist: Matthew 14:6\u201313a; Mark 6:21\u201329\n\nChapter 10: Dark Sayings from of Old\n\n1) Jesus's family seeks to protect him: Mark 3:20\u201321\n\n2) Collusion with Beelzebul accusation: Matthew 12:22\u201330; Mark 3:22\u201327; Luke 11:14\u201323; Isaiah 49:23\u201325\n\n3) The sin against the Holy Spirit: Matthew 12:31\u201337: Mark 3:28\u201330; Numbers 15:29\u201331; Isaiah 5:20\n\n4) The sign of Jonah: Matthew 12:38\u201342; Luke 11:29\u201332; Jonah 1:17; 3:4\u201310; 2 Chronicles 9:1\n\n5) The return of the evil spirit: Matthew 12:43\u201345; Luke 11:24\u201328\n\n6) Jesus defines his true family: Matthew 12:46, 48\u201350; Mark 3:31\u201335; Luke 8:19\u201321\n\n7) Parable of the sower: Matthew 13:1\u20139; Mark 4:1\u20139; Luke 8:4\u20138\n\n8) Parable of the seed growing secretly: Mark 4:26\u201329\n\n9) Parable of the wheat and the weeds: Matthew 13:24\u201330\n\n10) Parable of the mustard seed: Matthew 13:31\u201332; Mark 4:30\u201332\n\n11) Parable of the leaven: Matthew 13:33\n\n12) Two parables: the hidden treasure and the pearl: Matthew 13:44\u201346\n\n13) Parable of the net: Matthew 13:47\u201350\n\n14) The reason Jesus speaks in parables: Matthew 13:10\u201317; Mark 4:10\u201312; Luke 8:9\u201310; Isaiah 6:9\u201310\n\n15) Explanation of the parable of the sower: Matthew 13:18\u201323; Mark 4:13\u201320; Luke 8:11\u201315\n\n16) Explanation of the parable of the wheat and the weeds: Matthew 13:36\u201343\n\n17) Revelation as light to be heeded: Mark 4:21\u201325; Luke 8:16\u201318\n\n18) Kingdom scribes and treasures old and new: Matthew 13:51\u201352\n\nChapter 11: Identity\n\n1) Jesus stills a storm: Matthew 8:23\u201327; Mark 4:35\u201341; Luke 8:22\u201325\n\n2) The Gadarene demoniacs: Matthew 8:28\u201334; Mark 5:1\u201320; Luke 8:26\u201339\n\n3) Levi's banquet: Matthew 9:10\u201313: Mark 2:15\u201317; Luke 5:29\u201332; Hosea 6:6\n\n4) A question about fasting: Matthew 9:14\u201317: Mark 2:18\u201322; Luke 5:33\u201339; Isaiah 54:5\u20136; 62:4\u20135; Hosea 2:16\u201320\n\n5) Jairus's daughter and the woman with a hemorrhage Matthew 9:18\u201326; Mark 5:21\u201343; Luke 8:40\u201356\n\n6) Two blind men healed: Matthew 9:27\u201331\n\n7) The mute demoniac: Matthew 9:32\u201334\n\nChapter 12: Homecoming\n\n1) Rejected in Nazareth: Matthew 13:54\u201358; Mark 6:1\u20136a; Luke 4:16\u201330; Deuteronomy 6:4\u20139; Isaiah 61:1\u20132; 1 Kings 17:1\u201324; 18:1; 2 Kings 5:1\u201314\n\n2) The commissioning of the twelve: Matthew 9:35\u201338; 10:1, 5\u201342; Mark 6:6b\u201313; Luke 9:1\u20136\n\nChapter 13: The Bread of Life\n\n1) Opinions About Jesus: Matthew 14:1\u20132; Mark 6:14\u201316; Luke 9:7\u201310\n\n2) Return of the twelve: Mark 6:30\u201331; Luke 9:10a\n\n3) Feeding of the five thousand: Matthew 14:13\u201321; Mark 6:32\u201344; Luke 9:10b\u201317; John 6:1\u201315; 2 Kings 4:42\u201344; Exodus 16:4; Nehemiah 9:15; Deuteronomy 18:15\u201319; Isaiah 54:13; Jeremiah 31:33\u201334\n\n4) Jesus walks on water: Matthew 14:22\u201333; Mark 6:45\u201352; John 6:16\u201321\n\n5) Healings at Gennesaret: Matthew 14:34\u201336; Mark 6:53\u201356\n\n6) The Bread of Life discourse: John 6:22\u201359\n\n7) Reaction to the discourse: John 6:60\u201371\n\nChapter 14: Tradition\n\n1) Conflicts about the traditions of the elders: Matthew 15:1\u201320; Mark 7:1\u201323; Isaiah 29:13; Deuteronomy 23:21\u201323; Numbers 30:1\u201316; Exodus 20:12; 21:17\n\n2) The Syrophoenician woman: Matthew 15:21\u201328; Mark 7:24\u201330\n\n3) Heals a deaf mute and many others: Matthew 15:29\u201331; Mark 7:31\u201337\n\n4) Feeding of the 4,000: Matthew 15:32\u201339a; Mark 8:1\u20139\n\n5) The Pharisees seek a sign: Matthew 15:39b\u201316:4; Mark 8:10\u201313\n\n6) The leaven of the Pharisees: Matthew 16:5\u201312; Mark 8:14\u201321\n\n7) A blind man is healed in Bethsaida: Mark 8:22\u201326\n\nChapter 15: The Turning Point\n\n1) Peter's confession: Matthew 16:13\u201320; Mark 8:27\u201330; Luke 9:18\u201321\n\n2) Jesus foretells his death and resurrection: Matthew 16:21\u201328; Mark 8:31\u20139:1; Luke 9:22\u201327\n\n3) The transfiguration: Matthew 17:1\u20138; Mark 9:2\u20138; Luke 9:28\u201336a; Psalm 2:7; Isaiah 42:1; Deuteronomy 18:15\u201319\n\n4) The coming of Elijah: Matthew 17:9\u201313; Mark 9:9\u201313; Luke 9:36b; Isaiah 53:2\u201312; Malachi 4:5\u20136\n\n5) Jesus heals a demon\u2013possessed boy: Matthew 17:14\u201320; Mark 9:14\u201329; Luke 9:37\u201343a\n\nChapter 16: A Kingdom of Children\n\n1) Jesus again foretells his death: Matthew 17:22\u201323; Mark 9:30\u201332; Luke 9:43b\u201345\n\n2) Payment of the temple tax: Matthew 17:24\u201327; Exodus 30:12\u201316; Nehemiah 10:32\u201333\n\n3) True greatness: Matthew 18:1\u20135; Mark 9:33\u201337; Luke 9:46\u201348\n\n4) The outsider exorcist: Mark 9:38\u201341; Luke 9:49\u201350\n\n5) Warnings about temptations: Matthew 18:7\u20139; Mark 9:42\u201350; 2 Kings 23:10; Isaiah 66:24; Jeremiah 7:31\u201332; Malachi 3:1\u20133\n\n6) Parable of the lost sheep: Matthew 18:10\u201314\n\n7) On reproving a fellow disciple: Matthew 18:15\u201318; Leviticus 19:17; Deuteronomy 19:15\n\n8) Two or three gathered in my name: Matthew 18:19\u201320\n\n9) On reconciliation: Matthew 18:21\u201322; Amos 1:3\u201313; Genesis 4:23\u201324\n\n10) Parable of the unforgiving servant: Matthew 18:23\u201335\n\n11) Jesus's unbelieving brothers: John 7:1\u20139\n\n12) Decision to go to Jerusalem secretly: Luke 9:51; John 7:10\n\n13) Jesus is rejected by the Samaritans: Luke 9:52\u201356; 2 Kings 1:1\u201317\n\n14) On Following Jesus: Matthew 8:19\u201322; Luke 9:57\u201362\n\nChapter 17: Light of the World\n\n1) Jesus at Sukkoth\u2014mixed reactions: John 7:11\u201352; Leviticus 23:33\u201343; Numbers 29:12\u201339; Deuteronomy 16:13\u201317; Nehemiah 8:13\u201318; Psalm 78:16; 118:1\u201329; Isaiah 58:11\n\n2) Jesus claims to be the light of the world, the truth, and Abraham's superior: John 8:12\u201359\n\nChapter 18: Pride and Prejudice\n\n1) Jesus heals a man born blind: John 9:1\u201341\n\n2) Jesus the Good Shepherd with a new flock: John 10:1\u201321; Ezekiel 34:2\u201316; Jeremiah 3:17; Isaiah 49:6; 56:7\n\n3) The mission of the seventy-two: Luke 10:1\u201316\n\n4) The return of the seventy-two: Luke 10:17\u201320\n\n5) Jesus's thanksgiving to the Father: Luke 10:21\u201324\n\n6) The lawyer's question and the parable of the good Samaritan: Luke 10:25\u201337\n\n7) Mary and Martha: Luke 10:38\u201342\n\n8) The disciple's prayer: Luke 11:1\u20134\n\n9) Parable of the bold friend at midnight: Luke 11:1\u20134\n\n10) Encouragement to pray: Luke 11:9\u201313\n\nChapter 19: The Gathering Storm\n\n1) Jesus rebukes the scribes and Pharisees: Luke 11:37\u201354\n\n2) Warning against greed: Luke 12:13\u201315\n\n3) Parable of the rich fool: Luke 12:16\u201321\n\n4) Faithfulness in the midst of watchfulness: Luke 12:35\u201348; Mark 13:33\u201337\n\n5) Jesus declares his mission is to bring division: Luke 12:49\u201353; Micah 7:6\n\n6) Interpreting the times: Luke 12:54\u201356\n\n7) Agreement with one's accuser: Luke 12:57\u201359\n\n8) A question about sin and the parable of the unfruitful fig tree in the vineyard: Luke 13:1\u20139\n\n9) Sabbath healing of a crippled woman: Luke 13:10\u201317\n\n10) Jesus declares his unity with the Father at Hanukkah: John 10:22\u201339; Exodus 4:22; 7:1; Psalm 82:6\u20137\n\n11) Jesus moves to Batanea: John 10:40\u201342\n\n12) Exclusion from the kingdom: Luke 13:22\u201330; Isaiah 25:6\u20138\n\n13) A warning to Herod Antipas: Luke 13:31\u201333\n\nChapter 20: Kingdom in the Midst\n\n1) Jesus heals man with dropsy: Luke 14:1\u20136\n\n2) Teaching on humility: Luke 14:7\u201314; Proverbs 25:6\u20137\n\n3) Parable of the great banquet: Luke 14:15\u201324\n\n4) Conditions of discipleship: Luke 14:25\u201333\n\n5) Parable of salt: Luke 14:34\u201335\n\n6) Parable of the lost sheep: Luke 15:1\u20137\n\n7) Parable of the lost coin: Luke 15:8\u201310\n\n8) Parable of the forgiving father: Luke 15:11\u201332\n\n9) Parable of the unjust steward: Luke 16:1\u20139\n\n10) On faithfulness in what is least: Luke 16:10\u201312\n\n11) On serving two masters: Luke 16:13\n\n12) Jesus rebukes the Pharisees: Luke 16:14\u201315\n\n13) Parable of the rich man and Lazarus: Luke 16:19\u201331\n\n14) Parable of the dutiful servant: Luke 17:7\u201310\n\n15) The cleansing of ten lepers: Luke 17:11\u201319\n\n16) The kingdom is in the midst of you: Luke 17:20\u201321\n\nChapter 21: Deathless\n\n1) Parable of the unjust judge: Luke 18:1\u20138\n\n2) Parable of the Pharisee and the tax collector: Luke 18:9\u201314\n\n3) Departure to Judea: Matthew 19:1\u20132; Mark 10:1\n\n4) On divorce and celibacy: Matthew 19:3\u201312; Mark 10:2\u201312; Genesis 1:27; 2:24; Deuteronomy 24:1; Malachi 2:16\n\n5) Jesus blesses the children: Matthew 19:13\u201315; Mark 10:13\u201316\n\n6) The rich young ruler: Matthew 19:16\u201322; Mark 10:17\u201322; Luke 18:18\u201323\n\n7) On riches and the rewards of discipleship: Matthew 19:23\u201330; Mark 10:23\u201331; Luke 18:24\u201330; Daniel 7:13\u201314, 27\n\n8) Parable of the laborers in the vineyard: Matthew 20:1\u201316\n\n9) The raising of Lazarus: John 11:1\u201344\n\nChapter 22: The Passover Plot\n\n1) The Jewish reaction to the raising of Lazarus: John 11:45\u201354\n\n2) Jesus again foretells his death and resurrection: Matthew 20:17\u201319; Mark 10:32\u201334; Luke 18:31\u201334\n\n3) The request for precedence by the sons of Zebedee: Matthew 20:20\u201328; Mark 10:35\u201345; Luke 22:24\u201327; Isaiah 53:1\u201312\n\n4) Healing of the blind men at Jericho (Bartimaeus): Matthew 20:29\u201334; Mark 10:46\u201352; Luke 18:35\u201343\n\n5) Zacchaeus the tax farmer: Luke 19:1\u201310\n\n6) Parable of the ten minas: Luke 19:11\u201328\n\n7) The hostility of the Sanhedrin: John 11:55\u201357\n\n8) The anointing in Bethany: Matthew 26:6\u201313; Mark 14:3\u20139; John 12:1\u201311\n\nChapter 23: On a Donkey's Colt\n\n1) The triumphal entry into Jerusalem: Matthew 21:1\u20139; Mark 11:1\u201310; Luke 19:29\u201344; John 12:12\u201319; Exodus 12:3\u20136; 2 Samuel 15:30\u201332; Ezekiel 11:22\u201323; Zechariah 9:9; 14:3\u20134\n\n2) Jesus curses a fruitless fig tree: Matthew 21:18\u201319; Mark 11:12\u201314; Jeremiah 8:13\n\n3) Second temple cleansing: Matthew 21:10\u201317; Mark 11:15\u201319; Luke 19:45\u201348; Isaiah 56:7; Jeremiah 7:11; Psalm 8:2\n\n4) Jesus explains the significance of his death: John 12:20\u201350\n\n5) Explanation of the withered fig tree: Matthew 21:20\u201322; Mark 11:20\u201325; Jeremiah 8:13\n\n6) The woman caught in adultery: John 7:53\u20148:11; Deuteronomy 17:7\n\nChapter 24: Confrontation\n\n1) A question about authority: Matthew 21:23\u201327; Mark 11:27\u201333; Luke 20:1\u20138\n\n2) Parable of the two sons: Matthew 21:28\u201332\n\n3) Parable of the wicked tenants: Matthew 21:33\u201346; Mark 12:1\u201312; Luke 20:9\u201319; Isaiah 5:1\u20137; 8:14\u201315; Psalm 118:22\u201323\n\n4) Parable of the great supper: Matthew 22:1\u201314\n\n5) On paying tribute to Caesar: Matthew 22:15\u201322; Mark 12:13\u201317; Luke 20:20\u201326\n\n6) A question about the resurrection: Matthew 22:23\u201333; Mark 12:18\u201327; Luke 20:27\u201340; Exodus 3:1\u20136\n\n7) A question about the greatest commandment: Matthew 22:34\u201340; Mark 12:28\u201334\n\n8) A question about the Messiah: Matthew 22:41\u201346; Mark 12:35\u201337a; Luke 20:41\u201344\n\n9) Jesus publicly denounces the religious leaders: Matthew 23:1\u201336; Mark 12:37b\u201340; Luke 20:45\u201347\n\n10) Jesus's lament over Jerusalem: Matthew 23:37\u201339\n\n11) The widow's mite: Mark 12:41\u201344; Luke 21:1\u20134\n\nChapter 25: A Second Coming\n\n1) Prediction of the destruction of the temple: Matthew 24:1\u20132; Mark 13:1\u20132; Luke 21:5\u20136\n\n2) Events before the end: Matthew 24:3\u20138; Mark 13:3\u20138; Luke 21:7\u201311\n\n3) Persecutions foretold: Matthew 24:9\u201314; Mark 13:9\u201313; Luke 21:12\u201319\n\n4) The Desolation: Matthew 24:15\u201322; Mark 13:14\u201320; Luke 21:20\u201324; Daniel 2:44\u201345; 9:27; Isaiah 13:10; 34:4; Joel 2:31; Ezekiel 10:3, 18; 11:23\n\n5) False christs and false prophets: Matthew 24:23\u201327; Mark 13:21\u201323; Luke 17:23\u201324\n\n6) The Second Coming: Matthew 24:29\u201331; Mark 13:24\u201327; Luke 21:25\u201328\n\n7) Parable of the fig tree: Matthew 24:32\u201336; Mark 13:28\u201332; Luke 21:29\u201333\n\n8) Watchfulness enjoined: Luke 21:34\u201336\n\n9) Be ready, for the Day will come suddenly: Matthew 24:37\u201342; Luke 17:26\u201337\n\n10) Parable of the Ten Virgins: Matthew 25:1\u201313\n\n11) Parable of the talents: Matthew 25:14\u201330\n\n12) The sheep and the goats: Matthew 25:31\u201346\n\n13) Plot to kill Jesus: Matthew 26:1\u20135; Mark 14:1\u20132; Luke 22:1\u20132\n\n14) Judas's betrayal: Matthew 26:14\u201316; Mark 14:10\u201311; Luke 22:3\u20136; Exodus 21:32; Zechariah 11:7\u201313\n\nChapter 26: The Lord's Supper\n\n1) Preparation for the Passover: Matthew 26:17\u201319; Mark 14:12\u201316; Luke 22:7\u201313\n\n2) The last supper: Matthew 26:20; Mark 14:17; Luke 22:14\u201317; John 13:1\u20132\n\n3) Jesus washes the disciples feet: John 13:3\u201320; Psalm 41:9\n\n4) Jesus foretells his betrayal: Matthew 26:21\u201325; Mark 14:18\u201321; Luke 22:21\u201323; John 13:21\u201330\n\n5) The Lord's Supper: Matthew 26:26\u201329; Mark 14:22\u201325; Luke 22:15\u201320; 1 Corinthians 11:24\u201325; Exodus 24:8; Jeremiah 31:31\u201334; Isaiah 53:12\n\n6) The new way of greatness: Luke 22:24\u201330\n\n7) A new commandment: John 13:31\u201335\n\n8) Peter's denial predicted: Matthew 26:31\u201335; Mark 14:26\u201331; Luke 22:31\u201334; John 13:36\u201338; Zechariah 13:7\n\n9) The two swords: Luke 22:35\u201338\n\n10) The way, the truth, the life: John 14:1\u201314\n\n11) The promise of the Paraclete: John 14:15\u201326\n\n12) Peace pronounced: John 14:27\u201331\n\n13) Meal conclusion and departure: Matthew 26:30; Mark 14:26; Luke 22:39; Psalm 118:21\u201329\n\nChapter 27: Betrayed\n\n1) The true vine: John 15:1\u20138; Isaiah 5:1\u20137\n\n2) Love, obedience, abiding: John 15:9\u201317\n\n3) The world's hatred: John 15:18\u201325; John 13:16; Matthew 10:24; Psalm 35:19; 69:4\n\n4) The witness of the Paraclete: John 15:26\u201327\n\n5) On persecutions: John 16:1\u20134\n\n6) The work of the Paraclete: John 16:5\u201315\n\n7) Sorrow turned to joy: John 16:16\u201322; Isaiah 66:14\n\n8) Access to the Father: John 16:23\u201328\n\n9) Tribulation is coming, but Jesus has overcome the world: John 16:29\u201333\n\n10) Jesus's intercessory prayer: John 17:1\u201326\n\n11) Gethsemane: Matthew 26:36\u201346; Mark 14:32\u201342; Luke 22:39\u201346; John 18:1\n\n12) Jesus is arrested: Matthew 26:47\u201356; Mark 14:43\u201352; Luke 22:47\u201353; John 18:2\u201312\n\nChapter 28: Condemned\n\n1) Jesus examined by Annas and Peter's first denial: Matthew 26:69\u201370; Mark 14:66\u201368; Luke 22:56\u201357; John 18:12\u201324\n\n2) Peter's second denial: Matthew 26:71\u201372; Mark 14:69\u201370a; Luke 22:58; John 18:25\n\n3) Peter's third denial: Matthew 26:73\u201375; Mark 14:70b\u201372; Luke 22:59\u201362; John 18:26\u201327\n\n4) Examined by Caiaphas: Matthew 26:57\u201368; Mark 14:53\u201365; Luke 22:63\u201365; Psalm 110:1; Daniel 7:13\n\n5) Formally condemned by the Sanhedrin and delivered to Pilate: Matthew 27:1\u20132; Mark 15:1; Luke 22:66\u201423:1\n\n6) Judas commits suicide: Matthew 27:3\u201310; Jeremiah 19:1\u201313; Zechariah 11:12\u201313\n\n7) First examination by Pilate: Matthew 27:11\u201314; Mark 15:2\u20135; Luke 23:2\u20137; John 18:28\u201338\n\n8) Examination by Herod: Luke 23:8\u201312\n\n9) Pilate declares Jesus innocent: Luke 23:13\u201316\n\n10) The trial and the choice\u2014Jesus or Barabbas: Matthew 27:15\u201323; Mark 15:6\u201314; Luke 23:18\u201323; John 18:39\u201419:15\n\n11) Pilate delivers Jesus to be crucified: Matthew 27:24\u201326; Mark 15:15\n\n12) Jesus mocked by the soldiers: Matthew 27:28\u201331; Mark 15:17\u201320; John 19:2\u20133 (see 10 for context)\n\nChapter 29: Crucified\n\n1) Journey to Golgotha: Matthew 27:32; Mark 15:21; Luke 23:26\u201332; John 19:16b\u201317; Galatians 3:13; Deuteronomy 21:22\u201323\n\n2) The crucifixion: Matthew 27:33\u201337; Mark 15:22\u201326; Luke 23:33\u201334; John 19:18\u201327\n\n3) Jesus is mocked on the cross: Matthew 27:38\u201343; Mark 15:27\u201332a; Luke 23:35\u201338\n\n4) The two thieves: Matthew 27:44; Mark 15:32b; Luke 23:39\u201343\n\n5) Death of Jesus: Matthew 27:45\u201356; Mark 15:33\u201341; Luke 23:44\u201349; John 19:28\u201337; Psalm 22:1\n\n6) The burial of Jesus: Matthew 27:57\u201361; Mark 15:42\u201347; Luke 23:50\u201356; John 19:38\u201342\n\n7) The guard at the tomb: Matthew 27:62-66\n\nChapter 30: Raised\n\n1) The women come to the tomb (Mary runs ahead, then quickly leaves): Matthew 28:1; Mark 16:1\u20133; Luke 24:1; John 20:1\u20132\n\n2) The other women arrive and are met by two angels: Matthew 28:2\u20137; Mark 16:4\u20137; Luke 24:2\u20138\n\n3) Jesus appears to the women after they leave the garden: Matthew 28:8\u201310; Mark 16:8\n\n4) The report of the guard: Matthew 28:11\u201315\n\n5) Peter and John go to the tomb: John 20:3\u201310; Luke 24:12\n\n6) Mary returns after Peter and John leave: Luke 24:9\u201311; John 20:11\u201318\n\n7) The Emmaus road appearance: Luke 24:13\u201335\n\n8) Jesus appears to the apostles (without Thomas): Luke 24:36\u201343; John 20:19\u201323\n\n9) Jesus appears to the apostles (with Thomas): John 20:24\u201329\n\n10) Further appearances over forty days: John 21:1\u201323; Acts 1:3; 1 Corinthians 15:5\u20136\n\n11) The great commission: Matthew 28:16\u201320\n\n12) Parting instructions and ascension: Luke 24:44\u201353; Acts 1:4\u201311\nABOUT THIS BOOK\n\nSon of Man harmonizes the four Gospels to tell the story of Jesus's ministry years. No fictionalized elements or dialog have been added. All events and quotations are taken from the texts of Matthew, Mark, Luke, and John. All of Jesus's recorded words and actions are included, and duplicate accounts have been harmonized.\n\nSeveral widely respected harmonies were consulted to establish the chronology of events: Synopsis of the Four Gospels by Kurt Aland (Editor), A Harmony of the Gospels: New American Standard Edition by Robert L. Thomas and Stanley N. Gundry, A Harmony of the Gospels: The New International Version by Orville E. Daniel, and Jesus according to Scripture: Restoring the Portrait from the Gospels by Darrell L. Bock.\nABOUT THE AUTHOR\n\nHENRY H HARRIS knows Jesus. He has been Henry's constant companion for more than fifty years.\n\nIn addition to being a pastor and a church planter, Henry is an award-winning author who has published more than 500 articles about life in Christ. For thirty years he has taught the in-Christ lifestyle to both large and small groups in churches, conferences, and retreats, de-mystifying what it means to be an apprentice to Christ, clarifying the discipleship process, and helping people understand how we can become more like Jesus.\n\nHenry has been happily married to his wife, Lauren, since 1976. They live in San Juan Bautista, California.\nSON OF MAN: A BIOGRAPHY OF JESUS\n\nCopyright \u00a9 2015 by Henry H Harris\n\nAll rights reserved. No part of this book may be reproduced in any form without written permission from the publisher except for the use of brief quotations in a book review.\n\nPublished 2015 by inChrist Press\n\nISBN 978-0-9963933-2-4\n\nEdited by Susanne Lakin and Pennie Embry\n\neBook design by Maureen Cutajar\n\nCover design by Dane Low\n\nCover image by Rembrandt (Head of Christ, circa 1656)\n\nAll Scripture quotations in the text are the author's translations, paraphrases, and harmonies of the Gospels of Matthew, Mark, Luke, and John.\n\nThe epigraph in Chapter 5 is taken from the HOLY BIBLE, NEW LIVING TRANSLATION (NLT), Copyright\u00a9 1996, 2004, 2007 by Tyndale House Foundation. Used by permission of Tyndale House Publishers, Inc., Carol Stream, Illinois 60188. All rights reserved. Used by permission.\n\nThe epigraph in Chapter 12 is taken from THE MESSAGE: THE BIBLE IN CONTEMPORARY ENGLISH, copyright\u00a91993, 1994, 1995, 1996, 2000, 2001, 2002. Used by permission of NavPress Publishing Group\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \n#### The Eagle of Toledo\n\nThis book is dedicated to the memory of Mar\u00eda Jos\u00e9 Ant\u00f3n Jornet, and to Mar.\n\n#### Contents\n\nAuthor's Note\n\nAcknowledgements\n\nPrologue: The Reference Point\n\n1. 'Nunca Ha Sido Ni\u00f1o' ('He Never Was A Child')\n\n2. The Melon Thief\n\n3. Cycling's Most Famous Ice Cream\n\n4. The Gypsy Rider\n\n5. Pointing the Finger\n\n6. Loro\u00f1o v Bahamontes \u2013 A Knife in the Table\n\n7. The Dangers of 'Go-Fast'\n\n8. 1959 \u2013 'That Man From Toledo'\n\n9. 'The Bullfighter on Two Wheels'\n\n10. Goodbye to the Vuelta\n\n11. Exile\n\n12. A Pirate With a Pair of Pliers\n\n13. The Master and the Apprentice\n\n14. Into the Bushes\n\n15. La Vuelta a Toledo\n\nPalmares\n\nBibliography\n\nIndex\n\nCopyright\n\n#### Author's Note\n\nFor the sake of clarity for non-cycling fans and also because that is what the people of Catalonia call it, I have used _Volta a Catalunya_ or _Volta_ to describe the region's premier stage race (also known, outside Spain, as the Tour of Catalunya or Tour of Catalonia). For similar reasons I have used _Vuelta al Pa\u00eds Vasco_ rather than the Tour of the Basque Country, _Giro d'Italia_ or _Giro_ rather than the Tour of Italy, and the _Vuelta_ or the _Vuelta a Espa\u00f1a_ for the Tour of Spain. Given its overriding importance, the Tour de France is sometimes referred to simply as the Tour. (As Bahamontes likes to say: ' _Le Tour is le Tour_.')\n\nOn a completely different note, this biography contains extracts from a number of interviews with Federico Mart\u00edn Bahamontes but it is not an authorised version of his life. As Bahamontes put it: 'You can do it, provided you leave me in peace.' And, by and large, I did.\n\nAlasdair Fotheringham\n\nMarch 2012\n\n#### Acknowledgements\n\nFirst and foremost, my greatest thanks goes to Naomi, for making this book both possible and worth doing.\n\nSteve Farrand has been a rock of support throughout, and Jacinto Vidarte has provided everything from photographs of Bahamontes to a much-appreciated dry sense of humour about the whole project. To these two in particular, thanks.\n\nI am also very grateful to my mother, Alison Harding, for her support, knowledge and advice on reading and writing books, thinking about books and getting them published; to my brother William, for sound advice about the same, useful phone numbers, specific comments and general encouragement (not to mention getting me interested in cycling in the first place); to my father, Alex, for his interest and comments, as well as sharing the odd invaluable memory about 1950s and 1960s cycling; and to all three of them for helping to keep me motivated. Thank you.\n\nThanks also to Spanish cycling historian Javier Bodegas, for his support, comments, information and interest; to Philippe Bouvet of _L'Equipe_ for phone numbers, advice, and infectious enthusiasm; to Josu Garai for allowing me to take over _MARCA's_ photocopy machine for several hours; to Alex Hoskins, for being the ideal co-pilot around a large part of Spain and France in search of elderly former bike riders and the right kind of Patxaran; to Richard Moore for much-needed encouragement when the book was beginning to take shape; and at the other end of affairs, to Sam Harrison of Aurum and Martin Smith for their hard work and skilful editing when the book was being hammered into its final form. Also to (in alphabetical order) Geoff Brown, Alberto Contador, Pedro Delgado, Raul Esgueva, Hugh Gladstone, Rupert Guinness, John Herety, Miguel Indurain, Graham Jones, Alain Laiseka, Unai Larrea, Margarita Lobo, Edward Pickering, Dave Prichard, David Randall, Jorge Quintana, Phil Sheehan, Francois Thomazeau, Benito Urraburu, Antonio Valdivia Molina and Graham Watson. Finally to my agent, Mark Stanton of Jenny Brown Associates, both for his astute handling of all the boring bits of getting a book published as well as keeping the nerves of a first-time author from becoming overly jangled.\nPrologue\n\n#### The Reference Point\n\nBefore Alberto Contador made his mark on stage racing in the late 2000s, his hometown of Pinto, outside Madrid, was best known for being the geographical centre of Spain. Following his rise to fame, every time that Contador stood on the town hall balcony, waving at the cheering crowds after taking his latest Grand Tour, the people of the humdrum dormitory town must have felt they were at the centre of the sporting world, too.\n\nDuring one of these celebrations, someone tried to gatecrash the party. As Contador left the balcony and walked down the stairs to a reception, the bell at the main doors locked to keep out intruders, began to ring furiously. At the same time a voice could be heard shouting: 'Let me in! Let me in! It's Bahamontes!' Sure enough, it was. Smartly dressed in his usual dark suit and tie, his shock of wavy, unruly hair as neatly combed as it could be, Federico Mart\u00edn Bahamontes was outside bestowing embraces on friends and fans alike, signing autographs and grinning from ear to ear. After Bahamontes was admitted, Contador duly obliged by having his photograph taken for the press alongside the man known as the Eagle of Toledo. It was a symbolic moment: Spain's first winner of the Tour de France, and arguably the greatest climber the sport has known, and his latest heir (albeit a somewhat controversial one given Contador's positive test for the banned performance-enhancing drug clenbuterol).\n\nIn his home country, Bahamontes is considered to be the man who paved the way for future generations of Spanish cyclists who dreamed of taking the sport's most prestigious trophy. Unlike similarly iconic figures in other countries, and double Tour winner Fausto Coppi in Italy and Britain's 1965 World Champion Tom Simpson spring to mind, Bahamontes is anything but historically remote. Time and again he pops up, giving away prizes at race finishes, making statements to the press, and in the case of top riders like Contador or 2008 Tour de France winner Carlos Sastre, giving them reams of sound, often longwinded, advice from the other end of a mobile telephone. Even though that advice may be unsolicited, Bahamontes seems boundlessly confident that as the most important original source of inspiration for Spanish professional cycling, his presence and comments will always be appreciated. And is that so surprising given that his Tour de France win had such an enormous impact in post-Civil War Spain that it extended far beyond sport?\n\nBahamontes' Tour win in 1959 was the biggest international sporting breakthrough for Spain in a year when the country was, at long last, making advances socially and economically as well. The most important of these was undoubtedly the definitive end of the international isolation which had followed the victory of General Franco's forces over a democratically elected Republican government in the 1936\u201339 Civil War, and his regime's support in the first part of World War Two for Nazi Germany and Fascist Italy while technically neutral. Both Hitler and Mussolini had backed the General's coup and it was nearly two decades before the United States and the other Allies were fully willing to welcome Spain back into their midst.\n\nOn a sporting level, as arguably the country's biggest individual success at any point since Franco took power, Bahamontes' victory in the Tour 'proved' to the man on the street that Spain was returning to full membership of the international community, and marked an important watershed. Bahamontes' success coincided with the end of two devastating decades of economic recession, so bad they are still referred to as 'the years of hunger'. On 21 July, 1959, with Bahamontes' Tour glory less than a week old, the Government introduced their 'Stabilisation Plan', a series of radical reforms that heralded the modernisation of the Spanish economy.\n\nBahamontes, then, was not just opening a new chapter in Spanish sport, he was a surefire sign that the country's worst years were over, and that at home and abroad times were changing for the better. It was therefore small wonder that Franco's regime provided no less than fourteen military bands for the victory celebrations in Toledo. That Bahamontes' triumph in Paris came on 18 July, the anniversary of the start of Franco's uprising against the Republic, was an even happier coincidence for the country's rulers, not least because it reinforced that hallowed date in the Spanish calendar.\n\nEven better, winning the Tour was viewed as beating the despised French at their own game. France was resented by Franco for being the sporadic political ally of the defeated Republic and the closest example of democratic prosperity, as well as the alleged source of the liberal, subversive and modernist values that had corrupted his idealised view of traditional, authoritarian Roman Catholic Spain. Such a strong political interpretation of sporting achievement may strike twenty-first century readers as odd; however it continued to infect Spanish cycling right up until the 1990s when Miguel Indurain's string of victories in the Tour was seen as proof positive that Spain had finally made it into mainstream Europe. Indurain's runaway victory in the 1992 Tour, which visited all the European Union member states who had signed that year's Maastricht Treaty \u2013 including Spain \u2013 was the icing on the cake.\n\nBack in 1959, Bahamontes' Tour win represented a massive breakthrough. And like any breakthrough it would also shape the future. Victory in the greatest multi-day race of all left Spain's cyclists obsessed with stage racing. Even today, one-day Classics and track racing, two other major facets of the sport, barely receive the media attention or sponsorship they deserve. 'The mountains [are] the basic truth of cycling for Spaniards,' wrote Carlos Arribas, longstanding sports correspondent for _El Pa\u00eds_ , as recently as January 2012, 'and the cyclist [is] the only being capable of using the strength of his legs to challenge the force of gravity and fly.'\n\nAt the same time, being Spain's first Tour winner has left Bahamontes with a sporting sinecure for life. It is a label he has never ceased to exploit, even now when it is looking somewhat timeworn. Indeed, when Bahamontes receives journalists it is rarely at his home in Toledo; he prefers that they come to the headquarters of his _pe\u00f1a,_ his fan club. It is apparently irrelevant that this fan club, housed in a shabby warehouse on the outskirts of Toledo, is semi-defunct, the red lettering on the sign outside fading fast, and its website closed down. Bahamontes appears confident that the visitor will go away with a full appreciation of his importance. During interviews Bahamontes always sits behind an immense mahogany desk beneath a vast framed photograph of himself, besuited and smiling out from some time in the 1960s or 1970s. The life-size picture takes up almost the entire wall space and towers above the journalist. And if a reminder was needed of Bahamontes' nickname, you do not have to look far: to his left an alabaster eagle stares balefully and blankly out at the world from the top of a metre-high column. The dusty shelves in the room are crammed with trophies and photograph albums which are dedicated to his entire career and carefully labelled in Bahamontes' spidery scrawl.\n\nAs he talks, non-stop, giving answers that rarely last less than five minutes, postcards showing him in his pomp will be passed unrequested across the table, as well as posters of his _palmares_ , his achievements. The highlights are obviously his one Tour de France victory and the six King of the Mountains titles in the race, the latter a record that stood for nearly half a century. But there is more, much more. As Bahamontes never tires of pointing out, he took the King of the Mountains title in every stage race he competed in; even if this is not strictly true, his achievements are staggering nonetheless. Just one example: for fifty years no other professional crossed more of the Tour's highest category mountain passes in first place.\n\nSomewhere in this process of racking up victories and streaking away in the mountains, Bahamontes developed into cycling's most emblematic climber: the figure who, the moment the road steepened, would automatically power through the peloton, legs hammering away; the ultimate specialist in stretching the bunch beyond the limits of pain on the cols to the point where they would have to give up and let him go. 'Ask any Frenchman who the greatest climber is? And you know who'll they say . . . me,' Bahamontes once said. 'Probably the greatest climber ever,' concurred Sam Abt, the revered cycling correspondent of the _New York Times_ , a comment reinforced by the fact it was made during Abt's obituary of Charly Gaul, himself a major contender for the accolade.\n\nOne reason Bahamontes is undeniably right is that while other two-wheeled kings of the mountains \u2013 Pantani, Robert Millar, Van Impe \u2013 might have more impressive results to their name, none could match the Eagle for his impetuous, impassioned, hell-bent style of attack. Even in the comparatively anarchic racing of the 1950s, Bahamontes was praised as a throwback to a golden age of cycling when the sport was (supposedly) a spontaneous, non-stop _fiesta_ of uncalculating, hot-blooded attempts to drop the rest of the field. 'Whenever there were mountains, you'd know that Bahamontes would attack,' British sprinter Barry Hoban once said. 'Always. And you wouldn't see him again until the next morning.' Brian Robinson, Britain's first Tour stage winner, adds: 'He would pound away on those little gears, get a gap, turn around and look at us to see if anybody was able to follow. Normally, nobody could. Then he'd do it all over again. And then he'd be gone.'\n\nWhat frightened Bahamontes' rivals was that there was always an element of sheer enjoyment in his attacks, a slightly zany tendency to attack for the hell of it. He was, as arch-rival Raymond Poulidor says, a _provocateur._ 'He was a pure climber, able to stay off the front on the climbs even when he wasn't in form. I don't really think he was really too bothered about whether a climb was at the start or the finish of a stage. All he wanted was to be at the top first.'\n\nBahamontes legends abound. He was so sure of his racing power, they said, that he once attacked in the Alps then stopped at the top of a col to eat an ice-cream while waiting for the pack. Or, so it was rumoured, he would tell his team-mates they could all abandon and head for the team hotel at the foot of the last climb because he could handle the ride to the summit finish alone. His sense of honour could be so touchy that, allegedly, he once got off his bike during a race and attacked a spectator with a bike-pump for insulting him. His temper was so fierce, they said, that he abandoned a Vuelta a Espa\u00f1a after officials refused to re-admit a disqualified team-mate. He rode so deliberately slowly that day that he was booed and whistled along the route, and he finished more than an hour behind the rest of the field.\n\nHowever, for all he might be surrounded by stories and anecdotes, Bahamontes was also a solitary figure, and remains so. Once asked if he had ridden for a team who supported him, Bahamontes snapped back: 'Never. I was always alone.' Throughout his career he never had a full-time manager, no personal mechanic, no long-standing faithful _soigneur_ : none of the typical trappings top professional cyclists build up over the years. Given to fits of child-like petulance, Bahamontes often accused fellow riders and race directors of betraying his trust and abandoning him to his fate. Even the Spanish state, he claims, was prepared to sacrifice his chances of victory in the Vuelta a Espa\u00f1a for the sake of its political interests. Barring a handful of allies, such as team worker Julio San Emeterio, Bahamontes was widely regarded as self-centred and self-isolating, convinced of his own sporting genius but rarely capable of appreciating efforts made to help him.\n\n'Federico?' former team-mate, and Tour of Spain winner, Bernardo Ruiz once said to me. 'He was never grateful to anybody for anything.' The Tour's quasi-official newspaper _L'Equipe_ called him: 'A sombre, melancholy figure like the El Greco figures from his native region of Castille,' adding bitingly: 'Bahamontes was too whimsical, too Bohemian, too capricious to win the Tour more than once.' Others are less polite. '[As a racer] he was what we could call \"madness personified,\"' says Josu Loro\u00f1o, a former cycling journalist and son of Bahamontes' deadliest rival, Jes\u00fas Loro\u00f1o. Josu elsewhere claims: 'It's always been about \"me, me, me\" . . . Yes, he [Bahamontes] was great, but he never appreciated what others could do.'\n\nHowever, this image of Bahamontes the eccentric individualist was far from being completely unintentional: while convinced that the world, by and large, was out to get him, this self-imposed isolation also contained ingredients of a craftily designed publicity stunt that allowed him to pursue his real objectives under a smokescreen. As Bahamontes never tires of pointing out, his 1959 Tour win came about partly because the other favourites, notably the French, thought his only interest was in the King of the Mountains jersey. By the time they realised their mistake, the Eagle of Toledo had flown.\n\nBahamontes still cannot resist the opportunity to add to the myth. During interviews, he will bring down from the shelves a copy of the book he had specially produced for the half-century celebrations of that success. Though a kind enough gesture, you cannot help noticing there are two or three dozen identical volumes still sitting on the bookshelf awaiting future admirers. In an adjacent room, the dust-enshrouded personality cult continues with another vast photograph of Bahamontes. This time, close to the end of his career, he is in full flight on a mountain climb: typically, his head and body jut forward, arms locked in an 'L' position, fingers splayed wide over the handlebars and legs beating out a relentless rhythm as he seeks one last win.\n\nEven after retiring following his last Tour in 1965, which he abandoned little more than a week in because his team allegedly refused to pay him, the almost manic energy that had driven him throughout his career remained intact. For years he ran amateur and professional teams. In the off-season, Julio Jim\u00e9nez, who won the King of the Mountains title three times in the 1960s, would join him in cruising around Spain in Bahamontes' gleaming Mercedes, milking a series of invitations as guests of honour at criteriums, winter club meets and inaugurations. Indeed, Bahamontes' restless energy seems to increase rather than diminish. When I met him early one morning in the spring of 1993, he walked so fast through the streets of Toledo, shouting at anybody who looked remotely sleepy that they should 'wake up, it's day-time', it was almost impossible to keep up.\n\nVisiting him never fails to produce dramatic moments. I once spent half an hour waiting in his Mercedes as he went into the local police station to nag them to fix a drain cover that was sticking up in the road. He was so keen to get into the building he forgot to put the handbrake on and the car, parked on a roundabout with me in it, rolled ten yards downhill and collided with a traffic bollard. He got back in and drove away as if nothing had happened.\n\nYet behind this nervous bundle of energy, the regrets and barely concealed bitterness remain. Bahamontes firmly believes he was the best climber cycling has seen; all that prevented him from winning more Tours was the lack of a decent team, a surfeit of loyal team-mates and the absence of summit finishes, which was fatal for a poor descender like him. However, the truth is far more complex. Bahamontes' Tour victory in 1959 was at least partly assisted by an internecine power struggle between France's top riders, who secretly preferred a Spaniard to win rather than one of their home rivals. And too often his impulsive nature, his sometimes excessive ambition and his bizarre abandons, left him overly isolated, even in the context of the cut-throat world of 1950s cycling.\n\n'If Bahamontes had been different, he'd have won a whole lot more,' reasons Jim\u00e9nez. The problem as Jim\u00e9nez saw it?: 'He [Bahamontes] wanted everything.' However, Bahamontes' single-mindedness should not detract from his successes, nor what he represents. It is fair to say that he is both one of the last great surviving figures from an era Spain wishes to forget, and one of the few living legends from a time for which cycling fans remain relentlessly nostalgic.\n\nIn Spain, quite apart from the political exploitation of his victories, Bahamontes' achievements became part of popular culture across the generations. 'He was a god in a country that, after the Civil War, was desperate for myths,' said Angel Arroyo, second in the 1983 Tour, in an interview. Older Spaniards remember clustering around the radio in the village bar to listen to his exploits in the 1959 Tour, the first live broadcast of a sporting event in Spain. Decades later, the Eagle of Toledo would still form part of children's street games. 'We'd cut out pictures of cyclists from magazines and put them inside the caps of Coca-Cola bottles with a piece of glass on top to keep them stuck in there,' recalls 1988 Tour de France winner Pedro Delgado. 'If you were lucky, you'd get Bahamontes. Then in your street \u2013 which usually wasn't paved \u2013 you'd make a circuit out of the sand with curves and hills and you'd flick the cap around it with your finger. The first one across the line won. Sometimes you'd secretly file down the edges to make the lid go faster. I suppose you'd call that our equivalent of doping!'\n\nCuriously enough, this game of _chapas_ is first mentioned in connection with Bahamontes in the sports daily _MARCA_ on 19 July, 1959, the day after he won the Tour. It claimed that children across Spain have 'changed from playing at footballers to playing with lids of pop bottles with their favourites glued to them . . . the lucky ones get Bahamontes, of course'. More than a decade later, as Delgado points out, Spanish children were still doing it.\n\n'That game was how I first heard of Bahamontes. In the 1970s there weren't many televisions in Spain, it was just what your parents had told you about him when the Tour came round. When Luis Oca\u00f1a won the Tour in 1973, everybody in the neighbourhood would pile down to watch the TV in the local bar and say, \"It's like Bahamontes all over again\". It made you think, \"Well, if Oca\u00f1a was like this, then what must Bahamontes have done?\"'\n\nAs tarmac and television spread across Spain, games like _chapas_ have long since disappeared from the lives of Spanish children, but Bahamontes remains the key reference point for his country's cycling. Others would follow, but none has had the dramatic potency of the lone climber, soaring high in the mountains, oblivious to the rest of the field and at the centre of his own universe.\n\n'I talk a lot to him, or rather he talks a lot to me, but I've always liked to listen to him,' says Carlos Sastre, who grew up in \u00c1vila, close to Toledo. 'When I was a junior, I was on the point of signing for his team, and he would go on to me about what I should and shouldn't do, saying I should sign with him because I'd be treated as if I was going to race for Real Madrid. I didn't sign, I wanted to race in the same team as Jos\u00e9 Mar\u00eda Jim\u00e9nez [Sastre's now late brother-in-law, another top climber] but I can remember him buzzing around in the races he'd organised, pointing out which riders were good. He's very direct, he calls a spade a spade and if he thinks you're a bastard, he'll tell you. He's given me loads of advice. He'd ring me up in a Vuelta and tell me, \"Don't trust him, he's a bastard\", or \"do that\". And I'd say to myself, \"If Fede's said that, it has to be for a good reason\".'\n\n'He is synonymous with cycling at its highest level,' says Contador. 'Bahamontes' personality is unique, not just because he opened up Spanish cycling to the world but because he has an innovative, inexhaustible way of seeing life in general.'\n\nDelgado believes: 'Bahamontes was the father of the sport and above all you'd think, my goodness, if there was such a big difference even in my time between what we had in Spain and abroad in terms of material and training techniques, how much bigger must the difference have been back then? It was cycling at its most epic, its most legendary. Another world.'\n\nAs with any legend, though, contradictions abound. Yet perhaps this is not so surprising. Getting accurate reports of what has happened in a bike race can be complicated, even now. There may be up to two hundred starters and television tends to focus on the leading few. This fact is something all riders, not just Bahamontes, cheerfully exploit to ensure they come out looking the best. As one top Spanish cycling journalist put it to me: 'It used to be impossible to be sure of what really happened before the live TV coverage came on, which was only when there were a couple of hours left to race. If the newspaper's sports desk asked you for an on-the-spot mid-stage report of what was going on, you'd just dive into the nearest public phonebox and make it up.'\n\nMore than half a century ago, that potential for being 'economical with the truth' was far greater. Before cycling was televised, and with fifty years separating the riders from the races, their versions of events tend to be even more disparate. At the same time, the clouds in the memories of men almost all in their ninth decade grow steadily thicker. As for Bahamontes, he has only had one full-scale biography published about him. That was back in 1969 and it has long been out of print. All the more reason, then, to dust down tales of those exploits, and to do so now before the hammering on the town hall doors falls silent for good.\nChapter One\n\n#### 'Nunca Ha Sido Ni\u00f1o' ('He Never Was A Child')\n\nOn the outside wall of the church of Val de Santo Domingo, the village in the province of Toledo where Federico Bahamontes was born on 9 July, 1928, is a half-metre high plaque that reads 'Fallen for God and Spain'. It is embossed with Spanish Fascist Party symbols and the name 'Jos\u00e9 Antonio Primo de Rivera', the party's leader until he was executed in the first months of the Civil War. Beneath are listed the fourteen men from the village who died for General Franco's Nationalist Spain during the bitter three-year conflict. For the Republicans of Val de Santo Domingo, there is still no similar memorial. As was the case until Franco's death in 1975, even now, more than three decades later, the opponents to his rule have no tangible commemoration here.\n\nIf the divisions that split Spain for forty years remain unchanged on Val de Santo Domingo's church wall, time has stood still in other ways in Bahamontes' home village: the plaque on the central square, too, has not changed its name from Plaza del General\u00edsimo. However, this is not as uncommon as it seems. In many of the pueblos of deepest Spain, not just Val de Santo Domingo, remnants of Franco's Spain linger on in street names, plaques on church walls. A few even have his honorific 'El Caudillo' [The Leader] added to the village name itself. And hidden from sight there are far grimmer reminders: the bodies of one hundred and forty thousand Republicans murdered in mass executions are still buried in unmarked graves along roads and in ravines across the country.\n\nThe names and memorial still on display in Val de Santo Domingo are also apt reminders that without Franco and the Civil War, the Eagle of Toledo might never have achieved his status as one of Spanish cycling's greatest legends. Curiously, though, public recognition of Bahamontes' links with the village are almost non-existent. There is one street named after him, but nothing on the signs at the village entrances to indicate that this is his birthplace. Certainly there is nothing as sophisticated as a museum, as there are in many famous bike riders' home towns. Moreover, the house where Bahamontes was born, situated on a slight rise just outside the town and opposite a huge restaurant built principally to feed workers from a nearby iron foundry, was knocked down a few years ago. Demolishing what might have become a site of pilgrimage for cycling fans was hardly a profitable move. The humble house made way instead for an unintended monument to the country's new economic malaise: one of Spain's myriad phantom industrial estates, where the tarmac, street lights and traffic signals are all in place, but the recession took hold before any factories were actually built.\n\nIn fact, Val de Santo Domingo is like hundreds of anonymous agricultural villages in the remoter regions of Spain. Its centre is a huddle of single-storey, century-old, terraced houses on poorly maintained roads around the main square and church. Farther out there are more recently-built, garish, double-storey residences, each with its own concreted garden space, tiny swimming pool and garage. Beyond that are a clutch of factories, none of which look particularly active, and in the background, the vast, windswept, uncultivated, unfenced and unchanging fields of central Castille.\n\nVal de Santo Domingo's deserted streets, even at 11 o'clock on a weekday summer's morning, give the place a decidedly spooky feel. But near a hermitage church overlooking the village at least one inhabitant is visible: an old man, Pablo Rodr\u00edguez, is walking his dog. Rodr\u00edguez vividly recalls Bahamontes coming back from the 1959 Tour and visiting Val de Santo Domingo for a celebration dinner at his school. He still has the photograph taken of all the schoolchildren, himself included, then aged eight, with Bahamontes in the middle. 'He used to come here a lot after he retired,' Rodr\u00edguez recalls, 'to visit a friend of his, Marcelino, who ran the bar on the corner next to the main road. But when Marcelino died, that was it. I've barely seen him since, except on the telly.'\n\nIndeed, just the day before, Bahamontes had popped up on the nation's screens when he appeared on the podium of the 2011 _Vuelta a Espa\u00f1a_ in nearby Talavera, helping France's Sylvain Chavanel don his red jersey as race leader. But in Val de Santo Domingo nowadays Bahamontes is barely a memory. His family left for Toledo when he was eight, and Rodr\u00edguez claims there are hardly any of Bahamontes' relatives left in the village: 'Just a cousin or two'. That is only true in the living sense: the graves in the local churchyard are jammed together, just a few centimetres apart, and perhaps one in every four of those buried there has Mart\u00edn, the surname of Bahamontes' father, either for their first or second name.\n\nRodr\u00edguez followed Bahamontes' career closely, and like so many of his fans he feels that despite winning so much Bahamontes' achievements are a fraction of what they could have been. 'It was Coppi who really guided him,' Rodr\u00edguez says in a convinced tone of voice, referring to the all-too-brief period when Bahamontes rode for the Italian champion's team. 'Without him, he wouldn't have won that Tour in 1959. Bahamontes should have won more though. But with the sort of head he has, could you really have expected it?'\n\nToo far from Toledo to become one of the typical Spanish dormitory towns with their avenues of high-rise housing, the church tower is still the highest landmark in Val de Santo Domingo, just as it was in 1928. Behind the locked church doors the font where Bahamontes was baptised remains in place, and so too does the parish register of births, baptisms, marriages and deaths. Perhaps appropriately for a rider whose career was often shrouded in conflicting accounts, myths and half-truths, there is even some uncertainty about Bahamontes' name: in the register of baptisms he is called Alejandro. Equally strangely, though his two surnames are Mart\u00edn Bahamontes, he is not known by his first, as is customary in Spain, but by his second. 'My uncle Federico was younger than my father, Juli\u00e1n, but Federico was the head of the family,' Bahamontes explains. 'And after the baptism he said, \"This one's going to be called after me\". So I was. As for Bahamontes, I used my mother's name because there were lots of Mart\u00edns out there.'\n\nAfter Federico was born, his parents had three more children, all girls, each born a year apart from 1929 to 1931. It was not a large family for the time, but thankfully in those days a road-mender's job came with a rent-free house next to the section of highway for which he was responsible. In early 1936 a position came up for a foreman on a large estate just outside Toledo owned by a local aristocrat, the Duke of Montoya. Juli\u00e1n's parents had run a farm in his youth and his agricultural knowledge was enough to land him the more profitable job.\n\nInitially the move was a real step up for Bahamontes' family, even if his father still earned only 3.5 pesetas a month (around 48 euros today). The family's income was supplemented by eggs, milk, meat and vegetables from the farm, and rabbits would appear on the kitchen table some mornings thanks to Juli\u00e1n's poaching. As for Federico, he was able to attend a charity school in Toledo run by nuns. Not surprisingly for someone as restless as Bahamontes, formal education was not something he remembers fondly. 'We spent hours sitting at desks trying to join up fancy-looking letters in blue-lined exercise books, and that was pretty much it,' Bahamontes recalls. 'It was boring as hell.'\n\nTedious as learning copperplate handwriting might have been, life on the Montoya estate was good for Bahamontes and his family. They had food on their plates, a roof over their heads and his father's job looked secure. However, in the space of one summer morning, a few months after they had arrived, this new-found stability was to shatter permanently.\n\nAt 7 a.m. on 21 July, 1936, an infantry captain named Vela-Hidalgo stood in the vast Alc\u00e1zar fortress in Toledo, reading out an official declaration of war. If that was not startling enough, this declaration was especially momentous because it was being made against the Spanish Republic, an institution the Army was sworn to protect. Doubtless, Vela-Hidalgo drew on the same justifications for this armed rebellion as one of its main ringleaders, General Franco, had used shortly after the uprising started four days earlier. Franco claimed that Spain was in mortal danger because 'revolutionary hordes, backed by foreigners, were destroying our monuments and paralysing the nation with their strikes'. To make matters worse, Franco argued, the Constitution had all but disintegrated and anarchists were stalking the country. If the rebel general was to be believed, Spain was hell-bent on self-destruction and in such dramatic circumstances it was up to the Armed Forces, as the country's moral guardians, to restore order. Seizing power in the process was merely a necessary evil \u2013 or a cynical and brutal abuse of military force, depending on how you looked at it.\n\nMilitary rebellions were far from uncommon in Spain, with thirty-seven attempted uprisings between 1814 and 1874 alone, twelve of them successful. Spain's only previous Republic, in the 1870s, had been brought down by the Army's intervention, and the still fledgling Second Republic (1931\u201339) had succeeded the military dictatorship (1923\u201330) of Primo de Rivera. As recently as 1931, there had been another attempt by the Army to overthrow the Republic, led by Franco's colleague General Sanjurjo.\n\nAt the time Spain was one of the poorest and least industrialised countries in Western Europe. Her established political classes were largely more interested in maintaining their own status than in improving the lot of the country's underprivileged workers. During the first five years of the Republic \u2013 Spain's first full-scale democracy \u2013 there had been some limited improvements for the working class, particularly in the education and status of women. But the country had also suffered a steady increase in political murders and general strikes as well as several attempts at all-out revolution, the most significant erupting in Asturias in October 1934.\n\nMeanwhile, the bulk of the Spanish Army's vast surplus of officers \u2013 at the start of the century there was one general for every hundred rank-and-file soldiers \u2013 did little but conspire for the restoration of traditional, authoritarian government. The working class, for its part, was increasingly frustrated by the Republic's failure to implement the urgently-needed land, education and legal reforms that would iron out some of the tremendous social inequalities in the country, and help reduce levels of unemployment running at almost twenty per cent.\n\nAdding to this simmering political tension were the Nationalistic movements in Spain's peripheral regions: Galicia, the Basque Country and Catalonia. Culturally very different to the rest of the peninsula, these areas were strongly Separatist and large parts of their populations dreamed of splitting away from what they viewed as a corrupt and inept central government. In the wealthy Basque Country and Catalonia, the Separatists also possessed the financial clout to make their case.\n\nThe social and political turmoil came to a head when a loose alliance of left-leaning political parties, the Popular Front, ousted the right-wing Government in the February 1936 general election. For the right, losing power democratically was like an open invitation to begin their latest military-led plot to return to government by force. As early as February, Franco even tried to convince the caretaker Prime Minister, Manuel Portela Valladares, not to allow the Popular Front to take power. At the opposite end of the political spectrum, hundreds of left-wing prisoners were freed even before a general amnesty was decreed. The incoming Popular Front, meanwhile, was disappointed to find that Spain's economic problems were even worse than in 1931. Sabotage of the peseta's value by right-wing industrialists had produced a radical drop in foreign investment, and the right-wing press churned out reports claiming the country was completely ungovernable, often fabricating political crimes as a means to justify the imminent uprising. Indeed, street violence, even if much of it was deliberately provoked by the Fascist-inspired Falange, was steadily growing. By 1936, even Spanish parliamentary deputies started attending debates armed; in April, a bomb was thrown, allegedly by a member of Spain's militarised Civil Guard police force, at President Manuel Aza\u00f1a during the Republic's anniversary celebrations. More shooting broke out at the civil guard's funeral.\n\nOn the streets strikers and demonstrators were regularly attacked by right-wing gunmen in hit-and-run raids. In Madrid in June, for example, the Falange machine-gunned pickets during a seventy thousand-strong building strike. Others would regularly drive through working-class districts, shooting indiscriminately. At the same time, land raids on the giant _latifundia_ estates in the south by local socialists and anarchists intent on farming them as cooperative communes free from money, religion and marriage, were becoming more commonplace.\n\nThe right-wing coalition opposed to the Republic (the Fascists, to their enemies) included the Monarchists, and the majority of right-wing parties including Spain's Falange. Later this alliance would became known, albeit in an amalgamated form, as _El Movimiento Nacional_ , or _FET de la JONS_ , the only political organisation permitted under Franco's dictatorship. The driving forces, as usual, were the upper echelons of the Armed Forces, with the support of the majority within the Spanish Catholic Church.\n\nThe final straw for the Right \u2013 even though the uprising was already well planned by this stage \u2013 was the retaliatory murder of a leading right-wing politician, Calvo Sotelo, on 13 July, 1936. Four days later, the _coup d'etat_ began; first among the garrisons in Spanish Morocco, where the Republic was weakest, and then, on 18 July, by the Army in mainland Spain.\n\nBroadly speaking, the Nationalists' common goals were an authoritarian state, the defence of the interest of the industrialists and landowners, the continuing domination of Spain by the Catholic Church and strongly centralised government. The Republicans, however, fought for a wide-ranging and diametrically opposed set of goals, from anarchist or socialist revolution through to independence for the Basque Country and Catalonia, to a simple desire to resist the continuing rise of the extreme Right across Western Europe.\n\nAlongside the anti-interventionist stance of the democratic governments of Britain, France and the United States, and the support for the Nationalists provided by Hitler and Mussolini, the diversity of political aims among the Republicans was one of the main causes of their eventual defeat. In July 1936, in cities like Seville and Oviedo where the authorities refused to arm the workers quickly enough, the military rebellion triumphed. However, in others, like Madrid and Barcelona, Army barracks were quickly surrounded and the uprising crushed within a few days. Indeed, the coup met with far more resistance than Franco and his co-conspirators had anticipated. A mixture of trade union militiamen, a handful of regular army units, Basque and Catalan Separatists, and foreign volunteers known as the International Brigades, rose to the defence of the troubled Republic. The result was the three-year Civil War that devastated Spain.\n\nThe Toledo that the Bahamontes family moved to in early 1936 was little more than a large town of around thirty thousand inhabitants. However, both militarily and historically it was far more significant. Singled out by the Roman historian Livy as being a well-fortified town as early as 192 BC, it was later the Visigoths' capital of central Spain. Following the Moorish conquest of most of Spain in the eighth century it was viewed as one of Europe's most culturally advanced and ethnically tolerant cities. Through much of the Middle Ages, Arab, Jewish and native Spanish communities lived side by side, and its market places, craftsmen and university thrived on what was an unusual example of successful integration. Today the architecture in the older parts of Toledo, particularly its labyrinthine Jewish quarter, is a vivid reminder of its bedrock of different cultures. However, the city's tolerant ethos came to an abrupt end in 1480 when, during their process of uniting Spain for the first time, the 'Catholic Kings' Fernando II of Arag\u00f3n and Isabella I of Castille announced that the Jewish residents of Toledo were forbidden to live in Christian areas. Far worse was to come in 1492. While across Spain the Arabs were forced to convert to Christianity, the Jews were expelled en masse.\n\nSixteenth-century Castille was Spain's industrial and commercial powerhouse, and while Toledo was only the country's capital for a brief period, it was the spiritual centre of Spain's Roman Catholicism for much longer. It had always been famous for its high quality metalwork, and as Spain's foreign policies grew increasingly belligerent following the country's unification in 1492, so too did the demands for the weaponry made in Toledo. 'Toledo blades' became world famous, and a symbol of the city: even today, winners of bike race stages that finish in Toledo will often receive a large commemorative sword.\n\nEven though Toledo lost its political stature in 1561 when Felipe II adopted Madrid as his capital, the town remained of key military importance. At the time of the Civil War, the Alc\u00e1zar fortress housed Spain's main military academy and Toledo's proximity to Madrid made it an ideal forward position for an assault on the Republican capital. Furthermore, as one of the largest military buildings in Spain, dominating the skyline of old Toledo, the Alc\u00e1zar attained huge symbolic value. Perhaps unsurprisingly, then, as soon as the Civil War started, Toledo took a real hammering. Though by no means a major battle, the attempt by the Republicans to gain control of the Alc\u00e1zar was bitterly fought, and its successful defence by Franco's supporters was raised to near-mystical levels on the Nationalist side.\n\nThe alleged execution of the 16-year-old son of the Alc\u00e1zar commander after he refused to surrender (other reports say the boy was killed in reprisal for an air-raid) was seized on with zeal by Nationalist propagandists. Moreover, Franco's decision to divert his march on Madrid towards Toledo to ensure the Alc\u00e1zar did not fall into Republican hands is widely believed to have provided the definitive consolidation of his hold on the Nationalist leadership.\n\nAll this came at a high price to the city, though. By the time Franco visited the Alc\u00e1zar on 29 September after his supporters had held out for two months huge sections of Toledo held by the Republicans had been destroyed, both by artillery fire and inaccurate air-raids. Photographs from the siege's aftermath show the fortress looking like a sandcastle swept over by a succession of high tides. Towers lean at crazy angles, walls have caved in and foundations laid bare by vast craters. Little more than a quarter of the building above ground remained standing. Outside in the city itself, contemporary photographs show little more than a mass of tangled house beams, piles of bricks and gutted buildings. Destruction of this magnitude was far from uncommon in the Spanish Civil War, and the numbers of refugees ran into the millions. Among them were Bahamontes and his family.\n\nA veteran combatant of the Spanish war in Cuba in the 1890s, Bahamontes' father was too old to be recruited by either side. However, he still came within a whisker of becoming one of the Civil War's earliest victims. As news broke that a Republican column of troops and militia was on its way, Franco's forces retreated to the Alc\u00e1zar in Toledo and many civilians were enlisted to accompany the General's men to act as reinforcements, or more accurately cannon fodder. Juli\u00e1n was one of them. Realising his chances of survival were low \u2013 as it turned out, only half of the Alc\u00e1zar's one thousand defenders made it through the siege uninjured \u2013 he managed to escape undetected. That was far from being the end of his troubles, though. The Duke of Montoya's estate was the ideal location for the Republic's artillery in their siege of the Alc\u00e1zar. When Juli\u00e1n returned home, it was to find the whole area had been taken over by militiamen who, Bahamontes says scornfully, 'had come down from Madrid by taxi. And they called themselves Communists!'\n\nCommunists or not, Juli\u00e1n was ordered to open up the estate's storehouses so they could steal cooking oil. When he refused, they threatened to shoot him. Juli\u00e1n took to his heels for a second time that day and managed to shake off the ensuing manhunt by hiding in a doorway. By the time he reached the family house later that night, spent shell cases lay scattered across the surrounding farmland and the Republic's battalion of 102-millimetre guns was stationed in the middle of it, blazing away at the fortress. In a matter of hours, Toledo had gone up in flames and the whole area had become a war zone. Under the circumstances, Juli\u00e1n, his wife Victoria and their family had only one option: to join the long line of refugees making their way towards Madrid and relative safety.\n\nThe family reached the Spanish capital in the back of a cart in late July 1936. Once there they settled in a vast refugee camp set up in the grounds of Madrid University. 'We were living like gypsies, under canvas,' Bahamontes recalls. 'There were people there in their thousands.'\n\nFor the first few months of the war, after the Army's attempt to take the city had failed, Madrid remained the seat of the Republican Government. By late September the city was under siege from Franco's best forces at which point the Republican Government decamped to Valencia. But the combination of armed workers' militias, a few regular army units and later the International Brigades, saved the city for the Republic. As the initial turmoil subsided, and despite air-raids and trenches across many of the main streets, Madrid regained something akin to pre-war normality. In the centre, restaurants and expensive shops reopened, even though the front-line was only a few kilometres away. Indeed, Gran Via, one of the city's main thoroughfares, was known as 'Howitzer Alley' because of the number of shells that fell there. While refugees in cities like Barcelona were often installed in the homes of middle-class families who had fled, in Madrid they were largely left to fend for themselves. The Bahamontes family was no exception.\n\nAs soon as possible Juli\u00e1n moved his family again, away from the refugee camp, to stay with his sister in a flat in a cul-de-sac near El Retiro Park in central Madrid. Juli\u00e1n went to register their new address \u2013 'he had no choice if he wanted a ration card for the family,' as Bahamontes recalls \u2013 and this time he was promptly roped into the Republican Army. Juli\u00e1n's age meant he could only form part of the reserve and he was detailed to work in supplies. His experience with animals led to him being put in charge of a mule team hauling provisions from a vast depot at Atocha railway station on Madrid's south side up to the front at Brunete, a few kilometres outside the city. It was dangerous work, but it had some advantages. As Bahamontes points out, Juli\u00e1n could steal some of the Army's provisions.\n\nFederico, meanwhile, was also busy contributing to his family's upkeep by pilfering firewood from trees cut down in El Retiro Park. As the park had been closed, Bahamontes remembers that the only way to get the wood out was to perch between the iron spikes of the railings. It was a tight squeeze for an eight-year-old, but just about manageable. 'Once I'd got there, I'd throw a rope with a hook on it into the park and drag the fallen branches towards me,' Bahamontes says. 'Then we'd heave the branches over the railings, drag them home, and use the rope again to haul them up to my aunt's flat for fuel for cooking.'\n\nAs the air raids increased, Federico, his mother and sisters were sent to stay with relatives in the village of Villarubia de Santiago, to the south of Madrid. 'We had an aunt with an olive tree plantation, that was enough of a reason to go,' Bahamontes explains. With an increasing lack of food in Madrid, the capital certainly held little attraction. Toledo, now in Franco's hands, was unreachable, so the Bahamontes family spent the rest of the conflict in a village in one of the last Republican zones to surrender.\n\nWhen the war ended with Franco's victory on 1 April, 1939, the family were joined by Juli\u00e1n and remained in Villarubia de Santiago for another two years. There was nowhere else for them to go: the Duke of Montoya's estate had been swept away by the Civil War, and as a soldier on the losing side Juli\u00e1n was not entitled to any kind of pension. The only job he could find was breaking rocks as a road-mender's mate. Federico, now aged eleven, was expected to help him. 'His job was to split them up with a pick, and then I would break them up into smaller pieces,' Bahamontes recalls. 'That was it.'\n\nFor two years, Bahamontes' childhood consisted of little more than moving from one section of road to another and breaking stones. The monotony of the work can only be imagined. But as Bahamontes puts it, there was no other option. That said, since professional cyclists were dubbed 'the convicts of the road' by the media at the time, there is a slightly ironic edge to how he spent his early youth.\n\nIn early 1941 the Bahamontes family decided to return to Toledo. They could hardly have chosen a worse time. When Bahamontes started winning top international cycle races he quickly became a sporting hero for Franco's regime. It is ironic, then, that Bahamontes' youth was so distorted by the consequences of the war that placed Franco in power: mass poverty, an economy in ruins, famine, widespread ill-health and disease.\n\nNor did things improve quickly for the Spanish after the Civil War. There are a number of reasons for this: poor economic management by Franco's government; the seemingly endemic corruption in the one-party, military dictatorship he created; and, finally, the conflict in the rest of Europe and the Allies' partial blockade of the country. Even the end of World War Two had little positive effect on Spain's economy. A report by _The Daily Telegraph_ 's Spanish correspondent in the summer of 1946 from the city of Cordoba in the southern region of Andalusia leaves no doubt about the levels of poverty. 'All the usual revolting signs of famine are there: children with hideously swollen stomachs, fragile limbs and wizened, emaciated faces, women like human scarecrows with enormous eyes who are unable to move as their joints are swollen,' he wrote.\n\nHispanist Gerald Brenan, who travelled through the country in 1949, was equally graphic. 'The widespread corruption causes shame and dismay,' Brenan wrote in his account of his travels, _The Face of Spain._ 'The system of government controls is the despair of businessmen, whilst the severe inflation has reduced the middle and lower classes to great straits and the agricultural labourers to starvation. The feeling given out by Spain today is that of a country whose road to \u2013 I do not say prosperity \u2013 but simply any humanly tolerable condition is blocked.'\n\nAs for Toledo itself in the 1940s, Brenan describes it as 'a strange, dark, almost ominous city . . . built on a bare rocky hill in a loop of the Tagus \u2013 a fortress if there ever was one \u2013 it has, through the greater part of its history, been [a] citadel. What a rabbit warren its streets and houses and churches make! Like Fez it reeks of the Middle Ages: like Lhasa, of monks. Yet the thing that most impressed me on this occasion was the proximity of the bare, rocky hills beyond the river gorge, walking in narrow, crooked lanes . . . that harsh, waterless sierra, with its iron-coloured boulders looks as if it rose from the end of the street. Toledo, one says to oneself \u2013 though it is not quite true \u2013 is a fortress built in a desert.'\n\nBut like the rest of Spain, Toledo had almost nothing left to defend. If many Spanish middle-class families, according to Brenan, could only afford one full meal a day if they wanted to buy decent clothes, for ordinary working-classes families like Bahamontes', life became a desperate hand-to-mouth struggle against starvation. While the cost of living was five times higher in the 1950s than it had been before the war, it was only in 1954 that average incomes in Spain returned to pre-Civil War levels.\n\n'Federico never was a child,' his mother said on the eve of his greatest victory, the 1959 Tour de France. 'It just wasn't possible.' Like so many Spaniards under Franco, Bahamontes had to grow up far too fast for any notion of childhood.\n\nThe Bridge of Saint Martin, or _Puente de San Mart\u00edn_ to give it its name in Spanish, has always been one of Toledo's most popular attractions for travellers. That is partly because you cannot miss it. Built in the fourteenth century across the River Tagus to supplement the older, narrower _Puente de Alc\u00e1ntara_ , _San Mart\u00edn_ boasts five giant arches, the largest with a massive forty-metre span. At each end of the bridge there are two more gigantic archways set into towers; each tower is heavily fortified and bristles with cannon. It is the first man-made landmark anyone sees when they approach the city's outskirts, and it would have been as unforgettable and unmistakable a sight for visitors in the Middle Ages as San Francisco's Golden Gate is for globetrotters today.\n\nSix centuries later, as a child growing up in Toledo in the 1940s, Federico Mart\u00edn Bahamontes looked on the _Puente de San Martin_ rather more irreverently. True to his reputation for keeping an eye on the main chance, and particularly in pursuit of food in those days, the bridge was an ideal location for the odd spot of robbery. 'We'd climb up one of the towers just before a vegetable lorry went underneath,' Bahamontes recalls with an impish grin. 'When the lorry got there, it would have to slow because the archway's so narrow. And then we'd drop on top of the load and fill up a bag with beetroot. Or we'd slit open a sack with sugar or beets in it, fill up a bag, and steal those instead.' This kind of derring-do may sound like the stuff of Errol Flynn movies, but for Bahamontes and his teenage accomplices, stealing vegetables was no pastime: it was part of the daily fight to stave off hunger. 'Orange peel, stale bread, vine shoots, rotten fruit and cats \u2013 I ate them all in my childhood,' Bahamontes recalls. 'If I caught a cat, I'd cut off its paws and head, and skin it. Then my mother would gut it, fill up the cavities with salt, pepper and vegetables, and pop it in the oven. They tasted delicious. We called them \"baby goats\".'\n\nWhile the family could afford few scruples about where they got their food from, they were equally practical about what they wore. 'My mother would make us clothes out of hand-me-downs, given to her by people who took pity on us. She'd get old shoes, and remake them with canvas undersoles so we had something to wear on our feet. It didn't get so bad we had to beg, but we were always in debt. Even if we managed to pay our bills, we were never out of the woods. Always _atascados_ [bogged down].'\n\nBahamontes had one very dangerous way of making money: selling live ammunition dug out of old Civil War trenches around Toledo for scrap. 'It could have blown our heads off, but you don't think about that when you're starving.'\n\nThough Bahamontes says he never saw anybody die of hunger in or after the war, it was not uncommon. A report from the American embassy in Madrid estimated that infant mortality rates in Spain's poorest areas in the early 1940s ran at fifty per cent or worse. Living so close to the edge for so long, and working from such a young age \u2013 'I started at eleven and never stopped' \u2013 have clearly left their mark. Five decades on he can still recall exactly how much he was paid for one of his earliest jobs as a market worker.\n\nThroughout his life Bahamontes has shown what some regard as an obsession with money and financial security. His longstanding 1960s rival and friend Raymond Poulidor recalls:. 'He came up to me in a criterium and said, \"You remember that place where such and such happened\" \u2013 he was capable of protesting about something that had happened two years before, too \u2013 \"I bought you a stamp that day. You owe me for a stamp\". He had it all written down in a little book. To be honest, it made me laugh.' Back in the 1940s, though, the struggle for even a stamp's worth of money was a far more serious matter. As Bahamontes says, 'When you talk about times like that, it sounds like a kid's battle. But you wait until it happens to you.'\n\nApart from road-mending and robbing lorries, Bahamontes was regularly taken out into the fields by his father to help cut wheat or hay. When the harvest was over, however, father and son had to look for whatever work they could find for the winter. 'We came back in 1941 and in Toledo it was the most desperate year of a desperate era,' Bahamontes recalls. 'There was barely enough food to be had for anybody. My mother would get a lump of bread and some garlic, boil it up in some water, and we'd have that for supper. You know those [tv survival] programmes where they eat cockroaches on desert islands? That was nothing compared to '40 and '41'\n\nThe family of six lived in a three-roomed flat, with no electricity, no heating and no running water. Fuel was rarely available; in any case, as 'the coal carts had to come all the way from Madrid and the roads were so bad they would only get through once every three or four days'. Gainful employment was difficult to find, and Bahamontes is convinced his father's personality often worked against him. As Bahamontes puts it: 'He was shy, too shy, a nothing, a zero. If it hadn't been for my mother, we'd have all died in a ditch.'\n\nTo describe a parent like this, particularly when Bahamontes says he got on well with his father, sounds unbelievably harsh. But in his defence, the family bordered on the brink of starvation for most of his childhood and the majority of his waking hours were spent in search of food. Under the circumstances, shyness simply meant you stayed hungry longer. Fortunately Bahamontes had a similar temperament to his mother, and he was tenacious and versatile with it. By hanging around Toledo market long enough, he was finally hired to unload a cart, receiving fifteen centimos per one hundred and fifty kilo load. This was a ridiculously low sum, around twenty-five pence in modern currency, but it was money nonetheless.\n\nSlowly but surely Bahamontes became a regular hired hand at the market. 'You'd meet one person, make friends with them, and then you'd get to know another,' is how he explains it. 'It was all about contacts. And not wanting to be hungry.' Starting at six in the morning, he would begin his day by unloading lorries, then move on to picking out spoiled fruit and vegetables from the loads. He even had a nickname: _El Lechuga_ \u2013 the Lettuce-boy. This second job in the market was much more to his liking, because he quickly opened up a sideline with private auctions of any fruit rejected as unsellable. 'I'd gather it all up and sell it out the front. It wasn't too hard; people were desperate,' he recalled. 'One peseta fifty centimos [\u00a32 today], the box. Then after a while, I got enough together to buy a cart and I'd go around five different fruit shops selling produce to them direct.'\n\nBahamontes' business instincts were getting sharper and sharper, something that would prove invaluable later in his life. But with work in the market over by 10 a.m., by mid-morning he would find himself at a loose end again. So in 1946, when he was just eighteen, he decided to move into other, more illicit, business. To do that, though, he needed a more sophisticated mode of transport.\n\nBahamontes' first bike cost him one hundred and fifty pesetas (\u00a3250 in modern money). For an occasional market labourer, it was a small fortune. 'It was second-hand, too, and had no brakes or tyres either, which was a real worry because if you wanted to buy new ones you had to return the previous set,' Bahamontes recalls. However, he was convinced it was worth the investment. His plan was to ride from village to village, picking up illegal loads of bread, beans and flour, and selling them on in Toledo. Throughout Spain, villages had got off far more lightly in the Civil War, hiding stores of food from the authorities. With rationing still in force until the 1950s, the black market flourished. Typically, Bahamontes has no problem recalling how much money he made. 'The villagers would make a profit of at least two pesetas [\u00a33] a bike load. If they'd sold it wholesale, they'd have made far less. Also, I could sell it cheaper than the official rate. So we were all on to a good thing.'\n\nBahamontes was not the only Spanish cycling star who started riding a bike for illegal profit. Carmelo Morales, a top Spanish rider of the 1950s, started riding a bike aged eleven to deliver fish his family had 'forgotten' to pay the municipal tax on. He earned his nickname of _El Jabal\u00ed_ after he dodged the Civil Guard by diving into the bushes and wriggling away through the undergrowth so fast they told villagers that Morales was just as difficult to catch as a wild boar.\n\nBernardo Ruiz, the winner of the Vuelta a Espa\u00f1a in 1948, also admitted that he 'only began riding a bike to shift goods around'. When asked what kind of things he would transport, he gave an embarrassed smile, and said: 'Well, black market goods.' Bahamontes, on the other hand, positively delights in revealing his 'criminal' past. As he puts it: 'It's all part of the legend.'\n\nBahamontes' parents were initially opposed to him buying the bike. But when they realised it had been purchased with the sole aim of making money on the sly, they gave their approval. Of course, black marketeering had its risks. The Civil Guard, the semi-militarised police force who handled all rural crime, watched the roads and anyone caught in possession of illegal goods faced a prison sentence. Bahamontes, though, had bought a racing bike, complete with drop handlebars, with good reason: to ensure he could make a fast getaway. Like his fellow conspirators from Toledo, he had other tactics, too. 'We deliberately timed it so we'd go to the villages when it was hottest. That way we'd be on the roads when the police were asleep under a tree. And we'd warn each other if our paths crossed \u2013 you know, just like cars flash their headlights nowadays when there's a speed trap coming \u2013 so you'd know if you had to change roads. We knew all the shortcuts and we knew their schedules. It worked.'\n\nBahamontes had a few close shaves, but riding the back lanes on his bike gave him a considerable advantage over the Civil Guard, who were either on foot or horseback. 'And I reckoned,' Bahamontes says with slightly disturbing nonchalance, 'that they would never shoot at a black marketeer.' Fortunately, he never had to find out. He managed to steer clear of the Civil Guard for nearly two years, though he paid a high price for his nearest miss. 'One of my friends gave me a tip-off they [the Civil Guard] were coming up the road I wanted to use and I had to hide under a bridge. I managed to get there just in time, but I was standing in stagnant water for so long I got bitten by a mosquito and ended up with typhoid.' Typhoid is not actually transmitted by mosquitoes, but whatever led to Bahamontes' illness, and whatever the illness was, the effects were highly unpleasant. 'I ended up so ill I couldn't leave the house for two months. I had bouts of high fever that were so severe they thought I might die. When the fever came down I was so hungry I would break open the padlock my mother had put on the larder door to steal whatever I could get. I was as thin as an anorexic. I must have weighed about forty-five kilos [a little over seven stones] at the worst point. I was so weak I couldn't walk. Two of my sisters would carry me between them around the town so I could get some fresh air. All my hair fell out, too. Before it had been straight, and when it grew again after I was cured it came out much curlier.' He was given a doctor's certificate, which stated that he had chronic chest problems and recommended he avoid all types of physically demanding sports. As Bahamontes would delight in telling journalists in the years to come, that obviously included cycling. But, of course, he continued riding. After all, blind obedience to authority has never been in his nature.\n\nSo what led Bahamontes to move from black marketeer to bike rider? The simple answer is money, and the talent to make it.\n\nBut if the driving force was economic, it was pure chance that first pushed Bahamontes to enter a race. On Wednesday, 18 July, 1947, as they would do every year on the anniversary of the start of the Civil War until his death nearly three decades later, General Franco's supporters celebrated his victory and rise to power. The country would grind to a halt. But not everybody spent the so-called _Fiesta Nacional_ in a public show of loyalty to the ruling regime. After his 1959 Tour win, the still severely censored Spanish media would claim that Bahamontes had made enough money from his market work to buy the bike on which he rode his first race a dozen years before. But, in fact, Bahamontes had spent that morning riding the thirty-one kilometres from Toledo to the town of Torres to pick up an illegal shipment of bread, flour and beans and bring it back in bags slung over his handlebars. For the black marketeers, 18 July was business as usual. The only difference for Bahamontes was that because the street market where he had his legal job was closed for the celebrations, he could make an earlier start.\n\nBy midday he was back in Toledo, his illicit consignment disposed of, and his bike pointing homewards. But a chance meeting with two other teenaged black marketeers took Bahamontes' life in a different direction altogether. His partners-in-crime were competing in a bike race, organised as part of the _Fiesta Nacional_ in the nearby village of Minasalbas. When they invited Bahamontes to join them, he jumped at the chance.\n\nAt this point it is worth recalling that Bahamontes had never taken part in a competition of any kind before. His one attempt to play a sport, in his case soccer, had ended when his father crept into his room in the middle of the night and cut his football to ribbons. He wanted to be sure his son had no distractions from work. However, when Bahamontes reached the start of his first race, he came up against an even bigger obstacle than his father: Spanish bureaucracy. It transpired he needed a licence to take part. Luck remained on his side. After a quick word with a government official, the required document was obtained and filled in. Even today in Spain, this kind of simple bureaucratic process can take weeks. That day, for Bahamontes at least, it took no time at all; perhaps because the race was sponsored by the Ministry of Education itself and was part of the official celebrations of Franco's victory, they felt the more riders they got to the start-line the better.\n\nNow Bahamontes was the owner of his first race licence and had become a _bone fide_ amateur bike rider. Unfortunately he had come completely unprepared. His work bike was a racer, true enough, but he had no sports clothes. Instead, as he was to recall thousands of times, he rode his first race in his usual trousers and a borrowed baseball shirt with reinforced shoulder pads. As for food, a lemon and a banana were his only sustenance; by the end of the event, he was so hungry he wolfed down the lot, banana skin and lemon peel included.\n\nThe race, around forty-five kilometres long, started well enough. As soon as the flag dropped, not knowing the textbook tactic was to conserve strength until the end, he made a move that was to become 'vintage Bahamontes'. As a handwritten note says on the first page in his privately published account of the 1959 Tour de France, 'my tactic from the beginning was attack, attack and attack again'. It would have caught the rest by surprise. Unfortunately there was a mechanical problem: Bahamontes' bike chain kept coming off. And not for the last time he was caught on a descent. Not all was lost, though. After sitting in the bunch for the rest of the race, Bahamontes amazingly finished second. Encouraged by this, he decided to come back the following week for more.\n\nBahamontes' next race was sixty kilometres long, and he still wore the same basketball shirt and trousers. At least he had sorted out the problem with his chain. On the downside, the opposition was far stronger. Of the twenty-five riders taking part, the youngest was Bahamontes' age, the oldest twenty-two or twenty-three. Fortunately Bahamontes had a secret weapon: 'Team Fede', a lorry-load of supporters. With his three younger sisters on board, as well as their respective boyfriends, the lorry drove along behind the race. Bahamontes could not see them, but knowing they were there was a major boost to his morale. The flag dropped. Almost immediately, he put his head down, stomped on the pedals and opened a gap. It was only his second race, but he already had a trademark move. By the halfway point, Bahamontes was still out in front, alone and pedalling his way slowly but steadily across the arid semi-desert which surrounds Toledo. His only company in the miles of featureless flatlands was a one-man police motorbike escort. A long way behind, a low dust cloud indicated the main pack of chasers. It had become a one-bike race. At one point, the outrider dropped back level with Bahamontes and looked across at him. 'Hey,' he shouted, 'aren't you bored out here by yourself?' Bahamontes looked back. 'Don't worry,' he replied, 'I'm doing just fine.'\nChapter Two\n\n#### The Melon Thief\n\nAfter victory in only his second race, Bahamontes endured another summer of backbreaking, poorly paid agricultural work with his father before deciding to start racing as a full-time amateur. The logic of the choice was almost purely financial. 'I could make more money out of winning one race than my father would make out of working in an entire harvest,' Bahamontes recalls.\n\nHowever, piecing together his progress in the early years of his career is anything but simple. The economic dificulties that faced races he took part in as an amateur are largely responsible: frequently events already curtailed for three years by the Civil War in the 1930s would appear and disappear again. This endemic problem would even affect the country's flagship professional event, the Vuelta a Espa\u00f1a. Moreover, the erratic newspaper reporting in a country governed by a military dictatorship, and in the throes of its worst economic depression in a century, also contributed. 'The Spanish press makes a curious study,' writer Gerald Brenan observed in 1949. 'The first thing one notices is there is scarcely any news given in the Madrid papers about Spain. One is not told, for example, that the factories in Barcelona are only working two days a week. The foreigner casting his eye over the press might well suppose that nothing happens in the Peninsula except football matches, religious ceremonies and bullfights.' The arbitrary nature of the reporting meant that the Volta a Catalunya stage race, for example, was enormously popular in its own region and received huge amounts of column inches in the local newspapers. Beyond the Catalan frontiers, however, despite being Spain's biggest race between 1951 and 1954, it barely made an impact. In 1953 _ABC_ , the leading Madrid-based newspaper of the time, published exactly four paragraphs, all results, on the race's final stage.\n\nThe racing conditions themselves reflected Spain's extreme poverty and almost non-existent infrastructure. An early Government-funded documentary of the 1946 Vuelta a Espa\u00f1a, designed to promote Spain in the best possible light and shown in Spanish cinemas, cannot avoid revealing much of the country's true condition. For nearly a month, and on a course lasting three thousand eight hundred kilometres, the forty-six-strong peloton raced the length and breadth of Spain: from Madrid, south through Extremadura to Seville, then north and east to San Sebastian, via Barcelona, and finally back, through Santander and Valladolid, to Madrid again. Because it covered so much terrain, the documentary is probably one of the most complete visual 'snapshots' of Spain since the Civil War ended seven years before. The scars of the conflict are still clearly visible. Not just in the black-clad widows, many of them young women, who form dark knots in the crowds lining the route. The country itself is barely back on its feet: none of the roads outside the cities are tarmacked; the surfaces often just a tangle of ruts and badly spaced cobblestones. The consequences of that disrepair are also evident: every two minutes or so an injured rider is shown slumped on the side of the road as orderlies bandage him up.\n\nAnd not even a propaganda film can hide the occasional glimpse of bombed-out buildings in the background as the riders pedal by. The lack of infrastructure extends to the race itself where the black, high-sided support cars look pre-War in appearance \u2013 pre-World War One, that is. Handkerchiefs are used as flags, there are no banners except at the finish line, no barriers or advertising on or off the riders' dark clothing. The heavy-looking racing bikes have no numbers; instead they are pinned to the riders' backs. Certainly there is nothing like a winner's podium, no cups or bouquets, at least until the twenty-nine riders who complete the Vuelta reach the final stage finish at an athletics stadium in Madrid.\n\nThe poverty is unmissable, too. The wiriness of the spectators in the large crowds is visible even at a distance. In many cases their threadbare Sunday-best clothes hang off them, and a fair proportion of the women and children are barefoot, particularly outside the cities. One tall male spectator is wrapped in nothing but a blanket. Most telling of all is the difference in physical condition of the public and the cyclists: even after three weeks' hard racing the riders, traditionally scrawny alongside 'normal' people, look comparatively healthy and well-fed.\n\nCars, too, are a rarity. In one shot the few vehicles visible are outnumbered by horses and carts. Bikes, though, abound; many have been converted for carrying heavy loads. At one point three lads, very probably black marketeers to judge by the large, unmarked boxes and bags behind their saddles, look worriedly at the camera for a second before haring off at top speed. However, the biggest difference between these images of Spain and contemporary footage of, say, Great Britain, is the almost total absence of features you would take for granted in a twentieth-century European landscape: telegraph poles and wires, pavements, streetlights, parked cars, tarmac. There are barely any shops, few blocks of flats, no billboards, no parks. Remove the travelling circus of the Vuelta and you could be in the Spain of the 1840s: the sense of grinding economic misery is palpable.\n\nIn an interview in 1962, Spain's most successful racer of the 1940s, Juli\u00e1n Berrendero, recalled: 'We raced as if we had been abandoned by everybody. We had no help. We had bikes with just three gears rather than the ten you normally have now.' Asked how tough it was to race, he answered: 'If I suffered half now of what I had to put up with in my best years, I would be a millionaire. I put up with everything. In the 1937 Tour I fell and they gave me six stitches in my knee. The wound burst open, but I kept racing. They stitched me up, I went to bed, started again, it burst open again, and so on and so on until I completed the race. These days nobody could stand that sort of torture \u2013 nobody.'\n\nBorn in the village of San Agustin de Guadalix near Madrid, Berrendero was toughened by his harsh youth which included working long hours at a clay pigeon shoot for two pesetas (six euros) a day. But Berrendero's recollection of his racing days show that life in the peloton made significant demands for very little reward. The races, he confirms, were run with minimal infrastructure, but the riders got on with it. 'We were much more resistant to everything \u2013 cold, heat, rain. I don't know if it was because we were so desperate for the money, but there were no wages, no start money, and if you abandoned you had to make your own way home. When we raced the Vuelta a Espa\u00f1a there were only half-a-dozen support cars and two or three journalists to keep the whole country informed. Now there are dozens of them.' Berrendero, amazingly, seemed to have a certain nostalgia for the savage racing conditions of two decades earlier. 'These days the trade teams have hugely improved things financially for the riders. Cycling has lost nothing as a spectacle, but it's lost the grandeur of those superhuman efforts. There are far more people involved, but their love of cycling has diminished.'\n\nIf the Vuelta a Espa\u00f1a was the country's premier bike race, what must the provincial races of Bahamontes' early years have been like? With the country's transport and communication infrastructure so poor, and such long distances between the major cities, the bike racing 'scene' in the late 1940s and early 1950s was a series of chronically disconnected hubs of activity. Some, like Catalonia, were very dynamic; others, such as Andalusia, only sporadically so. What they all had in common was scant knowledge about what was going on elsewhere in the cycling world. It was something Bahamontes exploited to the full, whether consciously or not.\n\nRacing across the country, Bahamontes pops up under different names: first as Mart\u00edn, secondly Mart\u00edn Bahamontes, and thirdly, and most frequently, 'Bahamonde'. 'Bahamontes' only starts to appear in the newspapers with any degree of consistency in late 1953. Ask him these days about the name-changing and Bahamontes just grins. But he had reason to be happy with all of them. He should have used Mart\u00edn, his father's surname, but preferred his mother's. General Franco's second surname was Bahamonde, and for an ambitious young man like Bahamontes, letting the public think he shared a family name with the _General\u00edsimo_ was presumably appealing, if only because it caught the attention of the press. Most importantly, though, this ambiguity kept rivals in the dark about whether he was competing in a race until he appeared at the starting line. By then, of course, it was too late. Their concern was understandable since Bahamontes had discovered a formula that guaranteed him success: forming a combine with another rider, also from Toledo.\n\n'His name was Ladislau Soria and we were both strong riders. He didn't climb well but was good on the flat, and with me it was the other way round,' Bahamontes recalls. 'His brother, who was a telegraph operator at [nearby] Aranjuez railway station, organised our trips. He was our manager, but he didn't charge us any money. He'd ring up the villages and towns where he knew the annual _fiesta_ was coming up and he'd ask, \"Are you going to put on a bike race as part of the celebrations?\" If the answer was yes, then off we'd head. Nine times out of ten, we'd win.'\n\nWith cars scarce and the roads in terrible condition, there was only one way Bahamontes and Soria could travel between races: as hobos on the railways. 'We'd go in the guards van of goods trains so we wouldn't have to pay. [Soria's] mother would cook us up some breaded pork chops and omelettes, and we'd put them in a wicker basket for later.' With his transport, racing programme and sustenance all laid on by his colleague, Bahamontes was on to a winner. He did make a contribution, though, albeit a minor one: 'When I chipped in with a bit of bread that was us sorted for food.'\n\nSleepless nights and spectacularly long bike rides, often in the dark, were part and parcel of the dynamic duo's campaigns. 'There was the time we went from Toledo to Motilla del Palancar [three hundred kilometres] in one night. We travelled from Toledo to Aranjuez [60 kilometres] by bike, Aranjuez to Cinco Casas by train, then rode from Cinco Casas to Tomelloso [twenty kilometres] in the middle of the night in the pitch dark. Then it was back on the train again and we got to Motilla at seven in the morning. We ate a couple of bananas, drank a litre of milk, slept on four chairs pulled together for an hour, then went out to race.'\n\nThis 'hobo' image is not as surprising as it sounds. In many ways the Spanish cyclists of the late 1940s and early 1950s had more in common with other 'rootless' lifestyles of the time \u2013 like tinkers, itinerant musicians and theatre companies \u2013 than they did with professional sportsmen. The first time Jes\u00fas Loro\u00f1o, one of Bahamontes' main rivals, ran into the Eagle of Toledo was in 1953 when they were both en route to a bike race. 'He was lying under a tree, sucking on a lump of fat, and looking like a tramp in scruffy workaday clothes.' Loro\u00f1o's description was confirmed by another leading rider, Miguel Poblet. 'I first came across Bahamontes when I did my Military Service, and I competed in the military national championships in Toledo, riding for Catalonia. Suddenly riding along behind me I noticed this really scrawny kid wearing rag-tag clothes. He overtook us even though he had nothing to do with the race! Admittedly we had our rifles and bandoliers, but he still overtook us. And when I next saw Bahamontes, in a Volta a Catalunya, he told me it had been him.'\n\nHitching rides on goods trains meant there was little certainty about when the pair would get back to Toledo, or whether their supplies would last them. If money was tight the only solution was to steal fruit and vegetable crops from fields. Bahamontes recalls one nightmare ride from a race in Burgos to Toledo [400 kilometres] in which he and Soria ate nothing but purloined, half-ripe, watermelons. Bahamontes' partnership with Soria was a makeshift system but it was highly effective. 'We stitched them up time and again. I won one amateur Tour of Andalusia by over an hour, as well as four stages and a hill-climb to Sierra Nevada, then did the same all over again in the Tour of Cadiz. And I always finished alone. It got to the point that riders wouldn't want to start if they knew we were racing.'\n\nThis lucrative collaboration was interrupted in July 1949 when Bahamontes was summoned for National Service. Billeted at Canillejas, close to Madrid, the sudden loss of the freedom to travel more or less where he wanted, coupled with life in an army barracks after two years of making money hand over fist, came as a shock to Bahamontes. 'For eighteen months my life consisted of trying to eat as many black beans and lentils as I could,' he recalls. 'That's what I wanted to eat, anyway. But for some reason the saucepans in the barracks never seemed to hold anything bar water. We were always hungry. I'd ride over to Toledo to see my girlfriend [Fermina, who Bahamontes had met in the local market where she worked] and she gave me parcels of sausages and chocolate.'\n\nIn fact, financially, his life was picking up. He and his family had built their own house, principally out of rubble from bombed-out Civil War buildings. By 1950, aged just twenty-two, Bahamontes had opened a shop in Toledo, a one-roomed 'hole in the wall' where he rented out bikes. As he was doing his military service at the time, a former amateur rider who would become a life-long business associate, Faustino Su\u00e1rez, took over as manager. Fermina manned the cash register, as she would do in a series of bike shops Bahamontes was to own in Toledo. 'She was there on the front-line for her whole life,' Bahamontes says. 'Wherever there's money, the women are always there. They're the ones who look after it the best.'\n\nBahamontes was breaking through in other areas as well, taking the King of the Mountains title in the amateur Tour of \u00c1vila in 1948. The earliest reference to him in a national daily newspaper, albeit under one of his aliases, came in _ABC_ 's sporadic coverage of the same race on 26 May, 1949. After three stages, the first two having gone unreported, one 'Bahamonde de Toledo' was lying fifth, a minute and a half down. The final stage, on 27 May, was won by Bahamontes, though most _ABC_ readers may not have noticed: the story was tucked away in the bottom right-hand corner of page seventeen. Nonetheless, Bahamontes' win was there. A bold eight-line series of results records that at the end of a stage one hundred and nine kilometres long, which ran from Arevalo to \u00c1vila, 'Bahamonde' outstripped the pack in his last chance for victory. Soria, his combine companion, was fifth.\n\nEven if the media coverage was patchy, the prize money offered was still irresistible for even the least talented rider, and Bahamontes was far from that. Bernardo Ruiz recounts that when he first travelled to the Volta a Catalunya race in 1945, aged twenty-one, he had three hundred and fifty pesetas [\u00a3240 in modern money] in his back pocket and had never been outside the region of Valencia in his life. After winning the Volta a Catalunya, he came back home with several times that amount. Compared with Bahamontes' two peseta mark-up on each bike load from his black market trade in Toledo, there is no question which was the more profitable profession; it also explains why Bahamontes stuck at cycling so determinedly.\n\nFrom the earliest days his preference for a spectacular racing style over an effective one shone through. 'He always had to win alone,' recalled Luis L\u00f3pez Nicolas, an amateur rider at the time and later a journalist with _El Alc\u00e1zar_ newspaper. 'Sitting in and waiting for a sprint was too comfortable an option for him. He was a nervous rider who launched attack after attack, most of them inconsequential.' Bahamontes bought a lighter bike with his winnings: it weighed just twelve kilos rather than twenty, which increased his chances of victory. But he had a propensity to be involved in crashes, even if they were not all of his own making. According to _El Alc\u00e1zar_ one of the most spectacular was in the 1948 amateur national championships when a spectator crossing the road brought down the peloton. Bahamontes was in the thick of the ensuing pile-up, but still ran fifth. An animal was responsible for Bahamontes' next big crash, which took place when he was leading the 1948 Tour of \u00c1vila: during a descent a pig ran into the road and right into the pack. 'As the crash happened just after a corner, more and more riders blasted round the bend at 60 k.p.h. and slammed into the mass of bodies,' _El Alc\u00e1zar_ reported. 'One, Isidro L\u00f3pez de Salamanca, had his head split open like a watermelon and had to be taken to hospital. Fortunately he survived. The skin was ripped off the whole of Bahamontes left side \u2013 arm, leg, thigh and face \u2013 but fortunately he was all right apart from that.' Bahamontes made a supreme effort to try to catch an Asturian rider named Su\u00e1rez, who was lying second overall and who had dodged the crash. In an eighty-kilometre pursuit, Bahamontes managed to overtake several other riders, but lack of food, coupled with exhaustion and the after-effects of the crash forced him to abandon. Nevertheless it was an indication of the innate tenacity which would pay him dividends in the future.\n\nBahamontes' narrowest escape in his amateur years, though, did not happen while he was racing, but when he was taking one of his late-night rides between events with Soria. After finishing first and third in a race near Toledo, the two had to make it to Cuenca, two hundred kilometres further east, for another event the following day. But there were no train connections. Having ridden more than one hundred and fifty kilometres in pitch darkness, Bahamontes hit a stone in the road and went over his handlebars. Bleeding heavily from two major wounds, one in the arm, the other in his shoulder, he was unconscious for half an hour. 'Soria was really nervous, he thought I'd killed myself,' Bahamontes told _El Alc\u00e1zar._ The two arrived in Cuenca at 5 a.m., woke up four hours later and once again finished first and third, even though Bahamontes could only grip his handlebars with one hand. Sometimes Bahamontes' mishaps were caused deliberately by his rivals. At one race in Toledo, there were so many tin-tacks on the road he punctured eight times. But having sent his father home to get all the spare tyres he could find, Bahamontes still finished.\n\n'In 1948 he raced in San Sebasti\u00e1n for the first time, at the Ministry of Education's national champonships, and that was when he first saw the sea,' L\u00f3pez Nicolas recalls. 'Third overall, and second in the King of the Mountains competition, that was also where he told me he first felt really confident about his chances on the climbs.'\n\nSporadically, but steadily, more results for Bahamontes and his various aliases began to appear in the press: second in the opening stage of the Tour of \u00c1vila 1950; second in a hilly stage of the Tour of Madrid in 1951. He also took two overall victories in a stage race: at the Tour of \u00c1vila in 1951 and at the Tour of Cadiz the same year, rounding off his success in Cadiz with another stage win on the final day. Bahamontes also claims he won the Spanish Amateur National Championships in 1949. However, according to Spanish Federation records, what would have marked a breakthrough victory never actually happened: instead the title went to Eduardo Gadea.\n\nBut winning races was not the only source of income to be found in cycling: Bahamontes recalls he threw the following year's Amateur National Championships to 'Luciano Montero, while [Juan] Campillo came third. I did it because I needed the money'. If Bahamontes had reached a position where he was selling a race as important as the Spanish Nationals, this was ample proof that he had mastered the game and was ready to compete at a higher level.\n\nHis first opportunity to take on the professionals came in the week-long Tour of Asturias in 1953. It was one of the toughest professional races on the Spanish calendar and was frequently decided in the daunting Picos de Europa mountain range. The initial challenge for Bahamontes, though, was actually getting to the start. 'It was a five hundred-kilometre journey from Toledo and I did it all by bike,' he remembers. 'That sounds bad, but the long trip north had given me a great opportunity to hone my form and find a good race rhythm. So when I got to Asturias I was more than ready to go.' Nonetheless, he faced strong opposition. Barring those participating in the Giro d'Italia, which ran concurrently, all Spain's top professionals were present, plus some powerful national teams from Italy and Belgium. None of this deterred Bahamontes. Riding as an _independiente_ , or semi-professional, unsponsored, unaffiliated rider, he won the opening stage and the first professional King of the Mountains jersey of his career. He finished a more than respectable twenty-first overall out of the sixty-nine starters.\n\nThe stage win was no fluke, either. In a 120-kilometre run from Mieres to Luarca, incorporating three classified climbs, the 24-year-old took off halfway through. Sixty kilometres and two cols later, he was still alone at the head of the field. Just as in his amateur days, Bahamontes was determined to win in his trademark solo style. Also typical was his plummet in the overall classification on the next stage, held on the afternoon of the same day.\n\nFollowing his triumph in Asturias, Bahamontes competed in more races in Madrid. In one of them, the one-day Vuelta a los Puertos, the first signs emerged of his decade-long conflict with Jes\u00fas Loro\u00f1o, Spain's other top contemporary rider. Bahamontes and Loro\u00f1o broke away on the Navacerrada pass outside Madrid, on the steeper side known as 'The Seven Hairpins'. However, after arguing about who was going to set the pace on the front, the duo were reeled in. It was a minor incident, but it set the tone for a rivalry that was to divide Spanish cycling fans into _Bahamontistas_ and _Loro\u00f1istas._ Now all this success meant it was time for Bahamontes to make the move to the richest area of Spain and the only region where bike racing was truly flourishing: Catalonia.\n\nFor any self-respecting professional in Spain in the 1950s, a spell in Barcelona was inevitable. As the area least affected by the economic depression, Catalonia was 'where it was all happening', as Bahamontes put it. He explained: 'Catalonia had more money than the rest of Spain combined and it was much more advanced in all sorts of ways. Just take sport: in Madrid there was only one football stadium and one bullring, while Barcelona didn't only have that, it even had its own velodrome. [The singer-actresses] Lola Flores, Carmen Sevilla and [the boxer] Fred Galiana . . . all us big stars went there, artists and athletes alike. You could say my sporting roots are in Barcelona, and it was where I finally retired, too, with my last race [in 1965] in the Montjuic park.'\n\nBarcelona was also where Bahamontes found the first of the two key backers of his career: Santiago Mostajo, a businessman and former rider who twice won the Spanish national championships. Mostajo was a hugely influential figure in Barcelona's cycling scene at the time, and not just as a race organiser and team director. He was also one of those shadowy figures so often found behind the scenes in cycling and without whom the sport would grind to a halt: an affluent fan who was happy to take a few riders under his wing, help finance their racing and provide everything from transport to moral support, all in exchange for a little reflected glory. Mostajo's set-up involved four or five riders who were, as Bahamontes puts it, paid 'under the counter'. Indeed, Bahamontes suspected Mostajo was paying them using money that officials from Barcelona's town hall had given him to build a second velodrome. 'The velodrome was going to be an open-air one, but it never happened,' says Bahamontes. 'Basically he had us for nothing.'\n\nOthers, who are more generous to Mostajo, point out that the businessman put up riders in his own house, rent-free, with meals and laundry thrown in. And the second velodrome _was_ built \u2013 it just took a very long time (until 1961) to happen. 'Mostajo just wanted to be part of the scene,' comments 1955 Tour of Spain runner-up Antonio Jim\u00e9nez Quiles, who stayed in Mostajo's house as an amateur rider a few years after Bahamontes. 'He was a former rider himself and his son, also called Santiago, was a racer, but he wasn't very good. OK, he had a pretty dodgy used-tyre business and there was that stuff with the velodrome, but he never kept the prize money. Nobody else was doing what he did for cyclists in Barcelona. He just loved the sport.'\n\nMostajo directed Bahamontes' first professional squad, a small, eight-man team sponsored by Balanzas Berkel, a company who manufactured scales and weighing machines. Whatever the source, by the standards of his fellow Spaniards Bahamontes was now earning a reasonable wage of around five hundred pesetas a month. But the situation was not to his liking and his association with Mostajo did not last long. However, before the two parted company, the older man gave Bahamontes' career a significant boost. Despite his prot\u00e9g\u00e9's professional status, Mostajo helped Bahamontes into Europe's most prestigious training camp for amateurs, the _Campo Simplex._ The money may not have been good, but as Jim\u00e9nez Quiles points out, the logistical and psychological support Mostajo provided for Bahamontes was crucial. And the closeness between them comes across clearly in one of the few photographs to survive from Bahamontes' early career. Taken after his biggest victory of 1953, the King of the Mountains jersey in the Volta a Catalunya, it shows the finish line of the race and there, waiting to fling his arms around the young rider, is Mostajo. Both men, the trainer and his star pupil, display huge grins.\n\nIt was symptomatic of the economic gulf that separated Catalonia from the rest of Spain that the Volta a Catalunya, the region's top stage race, was for a while Spain's biggest cycling event. Nine days' long, as top Catalan sports newspaper _El Mundo Deportivo_ put it in its 1953 preview of the race, 'the Volta a Catalunya is our own little Tour de France'. Without the competition of the Vuelta a Espa\u00f1a, which did not run between 1950 and 1955 due to financial problems, the Volta enjoyed massive media coverage in Catalonia. At the time, _El Mundo Deportivo_ published six or eight pages each day, and three or four of those would be dedicated exclusively to the Volta. And all written, its journalists were keen to point out, on typewriters provided free by Olivetti.\n\nEven today, one of road-racing's greatest attractions is that it is free to watch, and in the grip of an economic recession the Catalan public turned out for the two-wheeled entertainment in their thousands. That was not all. Villages would sponsor prizes, known as _primes_ , for the first rider through the main square; climbs would be jammed with spectators; and during the race newspapers would be plastered with advertisements for cycling-related products. At the end of the Volta a Catalunya, fourteen different sponsors were listed in the newspaper's reports as backing various prizes, from Cinzano for the King of the Mountains to Olivetti for the sprint competitions (as well as the journalists' typewriters).\n\nIf Asturias had seen Bahamontes fighting against a depleted field, that was not the case in the Volta a Catalunya of 1953, where nine four-man Spanish regional squads took part, as well as teams from Belgium, Germany and Holland. Only the French were absent after three of their single five man-squad failed to turn up. In controversial circumstances, the victor of the opening stage in Montjuic park was all-rounder Miguel Poblet, the previous year's winner and, in 1956, Spain's first leader of the Tour de France. Another rider, Miguel Bover, was actually first across the line, but he was penalised for accepting a wheel from a spectator. The rules were unclear at the time, stating that a rider could not receive 'a bike' from a spectator, and Bover's team claimed that a wheel was 'not a whole bike, but a part of one'. Unsurprisingly, the race officials disagreed, and Bover was demoted.\n\nThough sprinting was never his strong point, Bahamontes had been active in a break and that afternoon, in the stage's second section to Gerona, he showed again he was undaunted by the calibre of the opposition. Midway through the stage, after an earlier break of eight had disintegrated when a massive rut in the road wrecked several of their bikes, the Italians Giancarlo Grosso and Donato Zampini broke away. The bulk of the chasing pack eased back and let the pair go, but Bahamontes and the young Belgian rider Marcel Janssens managed to bridge the gap. As the riders reached Gerona, the quartet were a mere twenty seconds ahead, and the peloton appeared confident they could pull them back on the four laps of the city centre that followed. However, to their surprise, it emerged that times for the overall classification were taken at the start of the first lap, not at the finish line, meaning Grosso could claim the lead and Bahamontes a prestigious result as first Spaniard. To make matters even more confusing, the four lapped the pack and sprinted for the stage victory in the middle of the main field. Afterwards the top Spaniards were furious, but as the lead home rider in his biggest race to date, Bahamontes was ecstatic. 'Bahamonde [sic] is the big news of the race,' commented _El Mundo Deportivo_ , who brushed aside the protests about the rules with the testy comment 'although it would have been better if things had been explained to the pack before the race reached Gerona'.\n\nThings reverted to normal the following day on stage three to Granollers when Francisco Alomar, of Spain, took over as leader. Grosso had unfortunately drunk some water from a public fountain and suffered a severely upset stomach, losing nearly forty minutes. Bahamontes, meanwhile, remained third overall and fourth in the King of the Mountains competition. However, the next day, stage four, he made his first appearance on the front page of a Spanish newspaper. The photograph shows him in a plain jersey, with inner tubes strapped across his back, and staring piercingly at the camera. He was f\u00eated because he had performed above expectations on an exceptionally mountainous stage which followed the main road from Barcelona to Puigcerda and the French border before darting westwards to Andorra. Bahamontes did more than pull through, crossing the massive Collada de Tosas pass in the Pyrenean foothills in sixth place. He was then ninth into Andorra after a 200-kilometre trek through some of the wildest uplands of Catalunya.\n\nVictory went to Salvador Botella, another Mostajo prot\u00e9g\u00e9. Botella had made a lone attack that enabled him to gain a four-minute forty-second lead on Alomar. But it was the performance of the little-known Bahamontes that everybody was talking about. He was 'one of the Spanish revelations of the race', claimed _El Mundo Deportivo'_ s headline. His slide from third to seventh mattered little, the report insisting that by riding so well as a rookie professional 'even such defeats carry the whiff of victory'.\n\nThe next two stages developed into a phony war. Despite his racing against the clock being theoretically weak, Bahamontes lost just one place overall, dropping to eighth, in the fifty-four-kilometre midday individual time-trial that, incredibly by modern standards, was jammed between a forty-three-kilometre mass-start stage from Andorra in the morning, and a sixty-kilometre stage down to Lerida in the afternoon. Bahamontes continued to move up the mountains classification after he broke away briefly on stage seven to lead across the Collado de Lilla climb. However, it was on stage eight the next day that he really shone.\n\nBahamontes' achievements on the 251-kilometre ride from Tarragona on the coast to the Pyrenean town of Berga all but confirmed in a single day's racing that he was capable of becoming Spain's greatest stage racer. There were some spectacular twists and turns in the overall battle. Against all expectations, Botella cracked completely, leaving the way open for Zampini to gain a seven-minute advantage and the lead. Once again, though, Bahamontes stole the show, with a 146-kilometre solo breakaway through the streets of Barcelona, culminating in a lone ascent of the capital's best-known climb, the Tibidalbo. What Mostajo must have thought when his prot\u00e9g\u00e9, sporting the bright pink jersey of the King of the Mountains leader, arrived in Barcelona a full five minutes ahead of the pack can only be imagined. The media were certainly impressed. 'We need a few riders like Bahamontes,' _El Mundo Deportivo_ 's cycling correspondent wrote, 'who climb well, perform brilliantly on the flat and whose courage distinguishes them above all others. We need this type of rider, the kind who know how to fling themselves into the battle, with the sole intention of provoking a war. All the other teams' well-studied strategies went up in smoke thanks to one lone man's courage.' It was a story that would repeat itself time and time again.\n\nAfter a break of nearly an hour for lunch \u2013 a tradition that would soon die out \u2013 the riders set off from the summit of the Tibidalbo, separated by time gaps based on when they had reached the top of the climb. Bahamontes knew he had to make the most of his four-minute advantage, and after scooping up prime after prime, it was only at the foot of La Trona, the third and final climb of the day, that he was finally reeled in. Exhausted by such a long early break, he trailed across the line twenty-fifth, fourteen minutes down on Zampini, the new leader. In a lesson perhaps too well learnt by Bahamontes, the impact of such an impressive breakaway far outlasted the moment he was caught by the pack. As _El Mundo Deportivo_ pointed out: 'The crowds had heard about what he had achieved in the morning and applauded him even though he was dropped.'\n\nNot for the last time, Bahamontes' sanity was called into question. The newspaper described his breakaway as 'the adventure of a madman', concluding: 'It would be no bad thing if cycling were full of madmen like him.' Nor was such 'lunacy' unprofitable: Bahamontes earned thousands of pesetas from his breakaway. In the town of Manresa alone, the fans offered their own 'prime' of five hundred pesetas (\u00a3600 in modern money) for the first rider to pass through the centre; that was as much as he earned from Mostajo in a month. Moreover, Bahamontes' maverick style was praised in the press as 'harking back to the golden age of our sport' when legend had it that impetuous lone attacks, rather than cold-blooded conservatism, were the order of the day.\n\nBahamontes had shown he could compete with the established riders of the time. Former national champion Francisco Masip, Jos\u00e9 Serra, who finished third in the 1950 Tour of Spain, and Antonio Gelabert, the 1950 Volta winner, were all in the race. However, they all preferred to play a waiting game than risk burning themselves out in breakaways. The younger generation, like Bahamontes and Botella, exploited their caution to the full. Botella finally regained the overall lead for good after Zampini cracked on the final day, conveniently, the skeptical might say, in the middle of nowhere.\n\nIf eclipsing the big names was a major coup for Bahamontes, so too was the way his triumph was presented to the public by the press as a throwback to a long-lost era. _El Mundo Deportivo_ even went so far as to compare Bahamontes to Vicente Trueba \u2013 the 'Flea of Torrelavega' who won the Tour de France's first King of the Mountains jersey in 1933 \u2013 as well as other intrepid climbers from Spanish cycling history, like Federico Ezquerra and Julian Berrendero. Ezquerra, who was born in 1909, was the second Spaniard to win a Tour de France stage but is more widely remembered for his spectacular solo assault on the infamous Galibier climb in 1934. Berrendero, born three years later, won the Vuelta a Espa\u00f1a and the Volta a Catalunya twice, as well as the King of the Mountains jersey in the Tour and Vuelta a Espa\u00f1a. For Bahamontes to be compared with those two was high praise indeed, and suggested a country eager to recapture past glories. 'Bahamontes proves our heritage of climbers has not disappeared,' the paper said.\n\nJust as today's cycling fans regard the 1950s and 1960s as a golden age of dashing heroics, so sixty years ago there was a sense that the sport had become excessively calculating. Bahamontes, therefore, was seen as a new hero who would break the mould, gain iconic status and increase newspaper sales in the process. Everybody, in short, was a winner.\nChapter Three\n\n#### Cycling's Most Famous Ice Cream\n\nLong before the days of budget flights, Spain had already become an important training destination for Britain's amateur cyclists. However, back in the early 1950s, while very few, if any, riders from the United Kingdom could say they had ridden alongside Federico Mart\u00edn Bahamontes, one, at least, could claim to have shared jugs of red wine with the future Tour winner across a dinner table in Cannes. That rider was Morecambe-born Ian Brown. He was selected in 1954 to take part in the Simplex training camp, sponsored by a components company for top amateurs from around Europe, which was where he met Bahamontes.\n\nIn charge of the camp was the former French cycling great Charles Pelissier, who had won eight stages in a single Tour, in 1930, a record equalled only by two other riders. He had also been joint leader of the race in 1931, the year Brown was born. 'Pelissier had us ride alternate days hard and easier,' Brown recalls. 'The more straightforward ones were approximately 100-kilometre rides along the coast and the _corniches_ in a big loop. The harder day was around one hundred and sixty kilometres, riding up through the Alps, then at a reasonably hard pace down to Cannes for lunch. The ride back from Cannes was supposedly an easy semi-individual ride along the coast road to the camp headquarters at Monte Carlo. Somehow Bahamontes and I always ended up at the same table for lunch in this really nice restaurant in Cannes. He didn't speak English, and I didn't speak Spanish and his French was even worse than my schoolboy French. Despite this we always ended up together, purposely the last two riders out of the restaurant. There'd always be this _pichet_ of wine on the table so we'd down that.\n\n'But it wasn't the wine. Really we'd stayed there so we'd be the last out and back on our bikes, which we were as soon as the last of the other guys had gone. The aim was to catch and pass all the others on the way back to Monte Carlo. We were jumping on to the back of trucks, scooters, motorbikes and any other vehicles we could tuck in behind, doing \"bit and bit\" [taking turns to lead]. It was real fun, way better than riding on the wet slushy roads in England. I must admit the red wine we were allowed to have with lunch helped make it more fun, too. And we'd always catch everyone before we'd get to Monte Carlos. It was hilarious.'\n\nIt is somehow sobering to think of Bahamontes and his British friend, unable to communicate, but forming a loose, slightly inebriated alliance to score a few harmless points off the other amateurs. Up to that point, Brown and Bahamontes' careers had been curiously similar: neither had travelled abroad before, and both were still finding their feet outside their own country. British and Spanish cycling had been isolated from the rest of continental Europe, though for radically different reasons. For much of the 1940s athletes from Franco's Spain had been subjected to an extensive international boycott; Britain's sporting isolation was more cultural: before the pioneering Brian Robinson, in the 1950s, amateurs rarely ventured across the English Channel. 'Federico was always very open, very polite, a fun rider,' Brown says. 'But I also got the impression he was very glad to be out of Spain, what with Franco and the political repression.'\n\nBrown, one of the five-man 'Northern' team in the 1953 Tour of Britain, was the top rider in his category in the race for several days 'until I went on a breakaway', he remembers wryly, 'and blew to bits'. Nonetheless, his high-profile ride secured Brown his invite to the month-long camp in Monte Carlo. 'A train ticket turned up in the post in late January 1954, and away I went,' he says. 'It was wonderful to get down south and see yellow mimosa and orange blossom for the first time in my life. Then in Monte Carlo there was the palace across the bay from our hotel. We weren't allowed to dry our kit on the balcony, though \u2013 the royals might have seen it.' The Spaniard with whom Brown would make friends, meanwhile, was also thanking his lucky stars he was there: technically an _independiente_ , or freelance professional cyclist, Mostajo had wangled Bahamontes into the amateurs' camp thanks to his contacts within the parent company. For both, Simplex represented a hugely important rung on the ladder towards a full professional career. Agents and teams from across Europe would be keeping an eye on them and snap up those judged to be the most promising and talented.\n\nBrown recalls: 'During the last few days all the riders, including me and Fede, were taken to a clinic and given medical tests to see if we really had it in us physically. I had no idea what the tests were for. Being an out-of-the-way northerner, I did not even know there were such things as sports doctors. The next day we were all individually interviewed by Pelissier, who told us who among us were going to be super champions. Unfortunately I was not one of them, but [in Pelissier's words] \"that Spanish rider who you drink a lot of wine with at lunch in Cannes, and race back to Monte Carlo with, he is\".'\n\nNot only was Pelissier right about Bahamontes, but Brown says he was spot on about other riders, too. '[He said that] the Italian [Guido Messina] who was the oldest rider there, an ex-soccer player, he was going to be a super champion, too \u2013 and he was world pursuit champion that year. The Belgian rider [Jef] Planckaert, he was going to be a super champion, and in fact he made second in the Tour de France [1962]. And there was a Swiss rider, too, but unfortunately he was killed in a race going through an unlit tunnel.' Last but not least, Brown recalls Pelissier told him: 'The Luxembourg rider you ride with, he will be a super champion, too.' That was Charly Gaul, winner of the Tour de France in 1958 and known as 'the Angel of the Mountains'. Unlike Gaul, though, who became one of Bahamontes' most formidable rivals, Brown's path only crossed once more with the Eagle of Toledo following their slightly tanked-up training rides along the Cote D'Azur: in the 1955 Vuelta a Espa\u00f1a.\n\nAs the first Spaniard to attend a _Campo Simplex_ , as Bahamontes called it, his presence represented a breakthrough for his country's cycling at international level. Even though his federation rather shortsightedly threatened to revoke Bahamontes' professional racing licence for attending a camp meant for amateurs, the fact that a former rider as prestigious as Pelissier had selected Bahamontes was a major step forward. So, too, was Bahamontes' first international victory. It came just before the camp finished when his Balanzas-Berkel squad took part in the early-season Mont Agel hill-climb on the France-Monaco border. Held on the last day of February, it had the standard format for these events of a short, flattish opening segment whose sole purpose was to warm up the riders' legs, followed by the main challenge: a lengthy grind up to a mountain-top summit, in the case of Mont Agel, 3,700 feet above sea level. In other words, it was a race designed for thoroughbred climbers like Bahamontes, who would win seventeen of this type in his career. For his first, after pulling out of the pack three kilometres from the top, Bahamontes overtook French favourite Peruggi less than one hundred metres from the finish.\n\nThe victory did not escape the notice of Franco's regime; Bahamontes' team made sure of that. When _El Mundo Deportivo_ attempted to contact the team's director, Santiago Mostajo, by telephone at his Barcelona home to discuss the win, his daughter told them her father was already out at the Post Office sending a telegram about the success to General Moscard\u00f3, the Nationalist hero of the Alc\u00e1zar siege, and by 1954 president of the Spanish Olympic Committee and the Government's sports minister. Equally remarkable was _El Mundo Deportivo'_ s exploitation of the win to symbolise the renewal of Spain's moral and spiritual welfare. That a newspaper should try to forge such a far-fetched link between the two concepts is hard to conceive \u2013 until it is remembered that at the time the Spanish media was used by Franco's regime as a vehicle for its propaganda.\n\n_El Mundo Deportivo'_ s front-page report of Bahamontes' Mont Agel victory in its edition of 1 March, 1954, proved to be a precursor to the same kind of elegiac, nationalistic theories of innate Spanish sporting and human talent that were rife in 1959 after his Tour de France win. The article starts off with a fairly predictable description of Bahamontes as a 'Quijote of Toledo', a comparison to one of the most traditional Spanish literary ideals of the warrior-knight from Castille. But then citing the theories of Adolf Hitler, of all people, on the relationship between humanity and history, followed by a paean to the natural greatness and civilising influence of the Iberian macho, was an indication of how Nazi ideology was still politically acceptable in 1950s Spain. Piling on this pseudo-grandiose significance on one small victory in France illustrates how desperate the Spanish authorities were to exploit any kind of international success by a local athlete. Headlined 'Bahamontes wins the European criterium of cycling', the report then claimed that: 'History repeats itself, the [new] generations are created with the good that they inherited. When this sacred truth disappears, society will be chaos and primitive brutality will return. As Hitler wrote in his memoirs, \"He who is indifferent to history is a man with no ears, no face, of course that man can live, but what does his life have?\"' After comparing Bahamontes again to Quijote, and 'giving a warm welcome to this young man from Toledo', the writer then claims that 'while many novels and films drag us towards the conventional and morbid, it's encouraging to find that sport has sparked the re-forging of an ancient lineage. What is good and uplifting for national pride reaches deeper places . . . and Bahamontes, a Spaniard, has spoken in one of the Alps' highest summits, giving a new example that the Hispanic race has not extinguished and that its desire to exist and triumph is reborn with new motivation, just as one day it was a civilising light in two worlds . . .'\n\nThough the article can best be described as tepid Fascist propaganda wrapped up in a sports report, it demonstrated that right from the start Bahamontes' career was being exploited for political reasons. The role of sport, in any case, had become clear just a few years after the Civil War ended. By 1945 the running of sport in Spain was put into the hands of a specifically created delegation of the _Movimiento Nacional,_ the only political party permitted by Franco's regime, containing both Fascists and Monarchists. A law introduced in 1945 had specified the sports delegation was to be 'a service provided by _El Movimiento_ , and it is responsible for the direction and promotion of sport in all its aspects and modalities . . . it is the supreme authority'. The head of sport, or 'chief sports delegate', would also be named by Franco himself. The sports delegation's statutes rammed home the message about its importance with article 42 announcing that 'all the members of National [sports] Federations and the heads of clubs or sporting societies must have proven adhesion to the _Movimiento'_. In other words, on paper at least, only the party faithful would be allowed to have an officially approved career in Spanish sport. And there could be no doubt, either, as to who would take the credit for their success.\n\nIf nothing else, Bahamontes' victory showed that post-Civil War Spain was beginning to have an impact abroad in professional cycling. It had taken its time. The Spanish national team's first post-war participation in the Tour de France of 1949 ended with all six riders abandoning in less than a week. The riders were poorly equipped and given no expenses, though the Spanish Federation had promised seventy-five kilos of sugar in lieu of cash payment for their participation. But as Lucy Fallon and Adrian Bell point out in their book on the Vuelta a Espa\u00f1a, _Viva la Vuelta!,_ the sugar disappeared into the black market before the riders saw it. That the sports director provided by the Federation was completely deaf was hardly a morale boost. Though the riders were criticised for lacking moral fibre when they pulled out \u2013 they were labelled 'dwarves of the road' by the Spanish press \u2013 their exits were not down to poor physical condition. Rather, they lacked decent logistical support, which in turn was due to a dearth in funding. When one Spanish rider, Dalmacio Langarica (Bahamontes' future team director), had a mechanical problem on stage five, the team car was in such poor condition it took nearly forty minutes to reach him. With the whole team instructed to wait for their leader, they failed to finish inside the time limit.\n\nAfter turning down Tour director Jacques Goddet's invitation to return in 1950 to avoid repeating the embarrassment, the Spaniards were back in 1951 and did far better than expected. Bernardo Ruiz took two mountain stages, one in the Massif Central after a day-long breakaway, the second in the Alps, and finished ninth overall. If that was not impressive enough, in 1952 Ruiz became the first Spanish rider to make it into the top three of the Tour de France. However, he was never in a position to beat the winner, Fausto Coppi. The following year, the Basque climber Jes\u00fas Loro\u00f1o clinched the King of the Mountains title by taking off alone on the Aubisque pass in the Pyrenees and winning at Cauterets, six minutes ahead of his closest pursuer, on the way to finishing fifth overall.\n\nIf the Spanish were beginning to make their mark in the Tour for the first time in twenty years, the country itself was slowly beginning to emerge from the economic consequences of the Civil War. By 1953 the Franco regime was starting to receive the huge loans it desperately needed after allowing the United States to build military bases there as part of their Cold War strategy against the Soviet Union. Another five years of recession followed, but at least by 1954 average Spanish incomes had finally regained pre-Civil War levels. Franco, though, showed no signs of ceding power, and constantly played up the alleged Communist threat to Spain, both for the benefit of the United States and internal rivals. 'Every newspaper gives the impression that war is imminent and that the whole of Europe down to the Pyrenees will be overrun by Russian armies,' Gerald Brenan commented. 'This anti-Red ballyhoo helps keep together the two parties on which the regime depends, the Monarchists and Falangists. But for the widespread fear of Communism . . . Franco would have left long ago.' Even if the country's economic position was no longer critical, the dictatorship was there to stay.\n\nThough the 13.4-kilometre pass of the Col de la Romeyere is far from being the hardest Alpine climb the crowd gathered on Monday, 26 July, 1954, for the seventeenth stage of the Tour de France, feeling a sense of mounting expectation. On this very hot day, the question foremost in their minds is, naturally, who will be first across the top? However, if any of them are expecting it to be Frenchman Louison Bobet, the 1953 Tour winner, who is lying second in the King of the Mountains competition, they are in for a big disappointment. Instead, the first rider across the summit, leading a break of four, is Spanish. And given that Federico Bahamontes has already made his mark in the Pyrenees, and has been on the attack in every mountain stage so far, there is no mistaking him, either. His shorts are rolled up high, and his head is raised and staring fixedly into the middle distance. Bahamontes is already well-known as the young leader of the King of the Mountains competition, even though there is not yet a distinctive jersey for the category. However, there is confusion in the crowd when Bahamontes crosses the summit first, claims the points and then immediately rides his bike off the road towards a metal cart selling ice cream. He gets off and without even so much as offering to pay, grabs a cornet. Bahamontes is mobbed by the crowd as he consumes his 'bonus prize', some trying to rip off his race number, and even his clothes, if the victim is to be believed. However, the story of Bahamontes licking away at an ice cream at the top of the Romeyere is guaranteed a permanent niche in the history of cycling.\n\nBike riders stopping for a snack in the middle of a race is not as uncommon an occurrence as might be thought. In one Vuelta a Espa\u00f1a in the early 2000s, the entire peloton apparently pulled over at a road-side hot-dog stall for a bite to eat. Yet there are key differences: the Vuelta a Espa\u00f1a's 'hot-dog moment' was early on an endlessly flat, tedious stage somewhere in the bleak, empty moorlands of northern Spain; it was not at the top of the first Alpine climb in the Tour de France of 1954. To stop for an ice cream while at the head of the biggest bike race in the world seems bizarre and utterly subversive; it is as if the rider is almost wilfully disconnecting himself from his sport. Small wonder that if there is one story that cycling fans know about Bahamontes, it is that he was 'the Spanish climber who ate that ice cream on top of a mountain during the Tour'. Or as Bahamontes put it to me two decades ago in an interview for the magazine _Cycle Sport_ : 'I'm never going to hear the last of that ruddy ice cream.'\n\nHis explanation for what happened is very straightforward: 'I only stopped because two of my spokes were broken and I had to wait for assistance. I had got away with three guys, one of them a Belgian. The Belgian's team-car came up to him to tell him not to collaborate because that was only going to favour me. When his car came past me, it struck a stone, which bounced up and broke my spokes. The Romeyere was a shortish climb but very tough, with some very steep sections. When I got to the top, with my spokes broken, I was nervous and really angry. There was no sign of [Spanish team director Juli\u00e1n] Berrendero. So I stopped. The summit was packed, just like every summit of the Tour. But there were two ice cream carts. I picked up a cone from one of them and put in a scoop of vanilla ice cream. At the time, I was more angry with Berrendero than anything else.'\n\nThe impromptu ice cream stop was not the only unusual incident on top of the Romeyere, as Bahamontes reveals. 'As it was a good breakaway we had fourteen minutes' advantage \u2013 not like they call breaks these days, a proper one \u2013 but I didn't know that. I had to wait so long \u2013 the bunch had split apart and my director was stuck in race \"traffic\" \u2013 I actually got bored after eating the ice cream and went down to a stream a little way away. I got some water into a _bidon_ and when the bunch came past me I threw the water all over them.' Dousing the peloton in water earned Bahamontes a fine. However, it was the ice cream incident that made it into sporting history.\n\nBahamontes knew he would be riding in his first Tour as early as the summer of 1953, almost a year before. 'Even then, [team director] Juli\u00e1n Berrendero told me to get ready for the Tour, and when I asked him what my objective would be when the time came to take part, he told me, \"To try and win it\",' Bahamontes recalls. Today, making overall victory a target for a rookie Tour de France rider a year in advance would be ridiculous. However, it is an indication of how fast Bahamontes was progressing. After his King of the Mountains jersey in the Volta a Catalunya in 1953, big results began to pour in. In mid-October Bahamontes won an astonishing five stages in the Tour of Malaga, and by the spring and summer of 1954 he had grown so successful he sold another victory, in exchange for a contract for the following season. Bahamontes claims that in the one-day Classic, the Vuelta a los Puertos, he let a rival from the La Solera team win. 'I had been first over two of the climbs that day, los Leones and another one, but I was interested in going to Barcelona because that was where the money was. The man in charge of La Solera said to me during the race, \"If you let him [Bahamontes' rival] win, you've got a contract in Catalonia for next year\", and I said, \"That's a deal\".' The deal-maker kept his word: after racing in 1954 for three Barcelona-based teams \u2013 Circulo Barcelonista-Yaste, Balanzas-Berkel and Splendid \u2013 in 1955 Bahamontes had a La Solera jersey on his back.\n\nAfter his years as a black marketeer, swift negotiating came naturally to Bahamontes. As Mostajo's influence faded in late 1954, Bahamontes insists he became much more of a one-man show, just like his early days. 'It's true I received advice from another Catalan businessman, Evaristo Murtra \u2013 and I called him \" _padrino_ [godfather]\". He sponsored me sometimes, like when I had my bike shop in Toledo. But basically I fought every battle by myself. I did the deals by myself, and if I won it was all by myself as well. Always. I work alone better than anybody else can.' The Spanish expression he uses is _me muevo solo,_ or 'I move alone', which implies the physical sense of riding without back-up. It is a phrase that neatly sums up his career and even his life. And in the mountains of France, and for the first time on a worldwide stage, Bahamontes was going to prove that he could 'move alone' better than any other bike rider.\n\nPredictably, even though he had been told he would be racing in the 1954 Tour so early, Bahamontes' final selection was not without a minor drama. After winning the King of the Mountains in the Tour of Asturias for a second year, as well as in the Bicicleta Eibarresa in the Basque Country, Bahamontes claims that Berrendero came up to him and without further ado confirmed his promise with the words: 'You, you're going to the Tour de France.' Somewhat implausibly, Bahamontes recollects his initial reaction was to refuse to go because he had no suitcase and did not speak French. However, after demanding to be allowed to telephone his parents to discuss his dilemma, Bahamontes quickly changed his mind, raising suspicions that his initial refusal was merely attention-seeking. Given this definitive invitation to ride had only come in the third week of June, and the Tour started on 8 July, Bahamontes did little specific training for the race. In any case, his role was what the Spanish call 'a free electron', meaning he had no responsibility to his team leaders, and had complete freedom to concentrate on his own speciality, the mountains. So with eleven other Spanish riders, one of them a reserve, he flew from Barcelona to Amsterdam for the start of the Tour: Bahamontes' twelve-year relationship with the race that defined his career was about to begin.\n\nThe French sports daily, _L'Equipe_ , rated the Spanish line-up in 1954 as one of the strongest because it combined relative youth with experience. It says a great deal about the speed with which Bahamontes was progressing that despite being the youngest team member at nearly twenty-six, his presence was taken for granted by the Spanish media. 'Of the ten riders, there are no possible doubts about four of them: [1948 Vuelta winner] Ruiz, [former national champion Francisco] Masip, [1953 Volta a Catalunya winner Salvador] Botella and Bahamontes,' argued El Mundo Deportivo before criticising the inclusion of others who had not performed well enough to merit selection. 'I don't think we can win,' Berrendero said, 'but we'll do our best to get as high up the overall classification and I have high hopes for the mountain stages.' Berrendero even attempted to place Ruiz, Bahamontes and Manolo Rodr\u00edguez, who finished second in the 1950 Vuelta a Espa\u00f1a, on an equal footing as co-leaders. However, the fast-track promotion of Bahamontes and Rodr\u00edguez caused a near-mutiny within the team, and as a result Ruiz was declared sole leader. Bahamontes had to seek consolation in the huge headline that appeared in _L'Equipe_ just before the start declaring him as the 'centre-forward' of the Spanish team. Besides, he knew he had permission from Berrendero to race as freely as he wanted. Crucially for Bahamontes, Jes\u00fas Loro\u00f1o, King of the Mountains the previous year, was missing through injury, while all-rounder Miguel Poblet had pulled out at his own request because of poor form.\n\nThere were many other top names missing from the Tour, too, including all the Italians. Their federation refused to enter following arguments with the Tour organisers when they tried to incorporate commercial publicity on their riders' clothing. Their decision not to start was made easier when Fausto Coppi, the 1949 and 1952 Tour winner, had had a freak training accident in June, fracturing his skull after a wheel came off a lorry in front of him. Only the French, whose 'A' team was led by 1953 winner Louison Bobet, looked as if they would provide serious opposition.\n\nThe Spanish were certainly protective of their riders before the Grand Depart in Amsterdam, the first time the race had started outside France in its history. The build-up featured a spectacular pre-race launch in central Amsterdam, with the presentation of six former winners, including the race's first champion in 1903, Maurice Garin, by now eighty-three, and Odiel Defraye, who won in 1912. Berrendero banned his squad from eating cherries at the presentation dinner for fear they would catch dysentery, though they were permitted to eat melons, oranges and bananas. However, Berrendero's request that the cooks at their hotel use 'the delicious local Dutch butter', as he called it, to fry the riders' food fell on deaf ears; to his annoyance, the Dutch chefs had already opted for olive oil, which they thought the Spaniards would prefer. Bahamontes seemed unconcerned about all the fuss to the point that he was already playing up to his role as the loose cannon in the team. 'Tell the fans \"hello\" from me,' he instructed one journalist, 'and that they should forgive me if I do something \"mad\" in the race.'\n\nFor the first half of the Tour, as it slowly wended its way southwards, Bahamontes gave no indication that he had any acts of insanity planned. However, no sooner did the Tour reach the Pyrenees, on stage eleven, than he went on the rampage. It was a painfully early morning start at 8.15 bearing in mind the riders would get up three hours earlier to cram their bodies with the massive food intake required. Bahamontes was wide awake, though, and tried to forge the early break of the day along with the top French sprinter Andr\u00e9 Darrigade and three others. That move fizzled out, but on the descent of the Soulor pass Bahamontes was in the thick of the action again, forming part of an eight-strong breakaway group which included Charly Gaul. It was on the Aubisque, the first classified climb in the Tour's King of the Mountains competition that year, that Bahamontes finally shed a Tour pack for good for the first time. Initially chased by a Belgian, Richard Van Genechten, to the delight of the large numbers of Spanish fans who had crossed the nearby border, Bahamontes went clear. However, after reaching the summit of the Aubisque in first position, and with a forty-five-second advantage to boot, disaster struck. On a notoriously tricky descent made worse by fog, Bahamontes crashed and eventually struggled into the finish at Pau in thirty-second place, seven minutes down. Nonetheless, the Spanish press was jubilant. 'Bahamontes has confirmed that the great hopes we had in him were justified, although undoubtedly he did not take enough risks on the descent,' trumpeted _ABC_ the next day, Its caveat was perhaps harsh considering how many riders have crashed on that descent in the Tour's history. But it set the general tone for future press coverage: Bahamontes' descending skills would be called into question time and again.\n\nIn the Pyrenees the next day Bahamontes was back on the attack. Indeed, he came as close as he would in the whole of the 1954 Tour to a stage win: he lost by half a wheel in Luchon to the French 'discovery' of the race, Gilbert Bauvin, from a break of three. Before that, though, Bahamontes produced a typical 'yo-yoing' performance in which he pulled ahead on several occasions before falling back. He managed to cross in first place two of the Tour's best-known Pyrenean cols, the Tourmalet and the Peyresourde, with a third place on the Aspin in between. After crossing the summit of the Peyresourde with a one hundred-metre advantage over the Frenchman Jean Mallejac, yet again on the descent he was caught by the chasers, Mallejac and Bauvin. But though Bauvin proved to be the fastest at the finish in Luchon, Bahamontes had established a six-point advantage over Bobet in the King of the Mountains competition.\n\nIn a strategy set to repeat itself over the years, Bahamontes continued to rack up points in the King of the Mountains competition day after day by making an early attack then letting himself be swept up by the main pack, and then attacking again to gain more points. That was the tactic he used on stage fourteen from Toulouse to Millau, and again on stage fifteen, putting in a first-hour move with Frenchmen Rafael Geminiani and Robert Varnajo and taking second place on the Causse de Sauveterre climb. Finally on stage sixteen, on the Col de Pertuis and Col de la Republique, Bahamontes was again in the break, though he was not the first over the climb. His slightly less aggressive attitude as the race wore on led the Spanish press to speculate whether he was losing the form that had led him to shine in the Pyrenees. That was far from being the case.\n\nOn the first Alpine climb, la Romeyere, Bahamontes was on the move once more. This particular attack, though, thanks to what happened at the summit, would have a far greater long-term impact on cycling fans than he could have imagined. Ice cream breaks notwithstanding, what was particularly impressive was how Bahamontes battled throughout the Alps at the same consistently high level as he had during the Pyrenees, even if there were the same radical swings in performance within each stage. On stage eighteen, after scooping the points on the first two 'easy' climbs of the Laffrey and Bayard, where he outpowered Bobet, Bahamontes fell adrift on the Izoard. However, despite starting the climb in thirtieth place, he rode so hard that by the summit only Bobet managed to stay ahead of him, and that by a margin of just fifty-five seconds. Come the finish in Brian\u00e7on, in any case, Bahamontes was in an unassailable position in the mountains classification with thirty-three points more than Bobet, who had all but won the Tour that day. Yet Bahamontes wanted more.\n\nThe final big day in the Alps included an early assault of the infamous Galibier, the highest climb that year, with a special prime of one hundred thousand francs for the first rider to reach the summit. Though the 1955 Vuelta a Espa\u00f1a winner Jean Dotto had taken off alone, Bahamontes refused to let him go clear, overtaking the Frenchman and reaching the summit alone. Only a broken pedal on the descent prevented him from clinching his first Tour stage.\n\nEven so, Bahamontes' achievements in his first Tour were enormous. 'A couple of years ago, this cyclist was just wearing alpargatas [in those days, Spain's cheap alternative to leather shoes],' pointed out _ABC._ 'This has been amazing progress.' Not only did Bahamontes \u2013 dubbed, for the first time that year 'The Eagle of Toledo' by one admiring French journalist \u2013 take the King of the Mountains title by a huge margin, he was first over the Tour's two legendary climbs, the Tourmalet and Galibier, as well as the Aubisque. Of the biggest climbs that year, only the Izoard escaped him, and only Bobet, the overall winner, could beat him there. In fact, on every Tour stage with a classified climb in it, Bahamontes had increased his points total, and of the twenty-five classified climbs he had been first over twelve of them. Overexaggerating wildly, Bahamontes claimed at the finish in Paris that the mountains of France were 'nothing' in comparison with the steep slopes of the city of Toledo. 'We've got real climbs there, they are more difficult than any of the ones that they talk about here,' he said. 'All this talk about \"cols\", but none of them are as steep as Doctor Mara\u00f1on Street in Toledo. The roads here in France are too good, too well-surfaced, they're like riding across a dancehall. If the roads weren't so smooth they'd never have dropped me on the descents and on the flat.'\n\nBahamontes ended the race with ninety-five points, nearly double those of Bobet. Such a startling breakthrough came with a hidden price, though. Just as when he took the King of the Mountains crown in the 1953 Volta a Catalunya, such fast-earned success meant Bahamontes risked becoming too focused on one objective so early in his career. By ignoring the greater prize of the overall title so often, he would be accused of deliberately losing time at certain points in his races, and as such wasting his talent.\n\nAs Bahamontes says, it was not until 1959 that he seriously thought about winning the Tour itself. Yet earlier on in his career, fighting for the general classification was not an unrealistic prospect. In the final time-trial of the 1954 Tour, for example, Bahamontes was the best-placed Spaniard, even beating the more experienced Ruiz by twenty-four seconds. That achievement was buried in all the excitement about Bahamontes' climbing prowess. However, with the benefit of hindsight, what more could he have done overall if he had decided to work on improving his time-trialling? Or if there had been more summit finishes than just the handful in the whole of the 1950s?\n\nMuch was made of Bahamontes' poor descending, and it was a major chink in his armour. But as was so often the case with Bahamontes he deliberately exploited this defect. 'I knew everybody thought I was lousy at going downhill,' he reveals, 'so I decided to make the most of that impression. Every time the race went down a mountain, while others were concentrating on going as fast as possible, I was concentrating on recovering as best I could and eating as much as I could. Every morning I'd get out a stage map, and mark the points where I should eat with a cross before I got caught. Then when it came to the next climb I'd get caught, but I'd have eaten some more food, my batteries were recharged, and I was ready to go on the attack again. But that wasn't something I was going to reveal to the press in case my rivals got wind of my tactics. That would have been like in that story [about a besieged fortress] where they [the defenders] send out a message, \"Send us bullets, we're running out!\"'\n\nIf Bahamontes' descending tactics brought mixed reactions, it was not thought he could improve much in one other key area, that of weight. One of the most frequently used strategies for improving performance in cycling is for a rider to slim down, but in Bahamontes' case that was not practical. In 1954, he finished the Tour weighing sixty-six kilos, just one less than when he started. 'He is so thin,' _ABC_ reported, 'and his cheekbones stick out so much that he is in serious danger of cutting himself each time he shaves.'\n\nBack home, Bahamontes was showered with praise, with his first Tour de France title far outshining Ruiz's well-deserved, but hardly spectacular, eighteenth place overall. 'Bahamontes, a real discovery for the world of cycling, has confirmed beyond any doubt in this, his first stage race outside Spain, what we always knew about him,' claimed a report in _ABC_ before the race had even reached Paris. A photograph of Bahamontes filled the whole front page of the paper. 'All that climbing is much, much easier for him than it is for the vast majority of cyclists,' the article continued, 'and never before has a King of the Mountains racked up such an impressive quantity of triumphs: [first on] three first category climbs \u2013 at the time, the top ranking for climbs in terms of difficulty \u2013 [first on] five second categories climbs and [first on] as many third categories as he cared to take. The name of Bahamontes, familiar to hundreds of millions of Europeans, generates more excitement and enthusiasm than any other Spanish bike rider. Hundreds of fans charge up to him, greet him, open their arms to him . . . \" _Vive L'Espagne_ \", one Frenchman, who wishes he was Spanish, shouts.'\n\nIt is not surprising, then, that even a fortnight after the event, Bahamontes' return to Spain was marked by a huge reception at Barcelona airport which lasted an entire afternoon. When he arrived in Toledo at the end of August, despite a low-key performance at the World Championships where he abandoned, an estimated twenty-five thousand people filled the Plaza del General\u00edsimo to cheer him. Met earlier by the mayor of Toledo with his car in Madrid, thousands of banner-waving Bahamontes supporters had crammed into open lorries so they could follow the local boy's journey home. Some waited up to fifteen kilometres away from the city so they could be guaranteed the best spot in the makeshift procession. Once in Toledo, rockets were fired and bunting rained down on Bahamontes. Speeches were made in his honour by the city's top brass before Bahamontes laid bouquets at the feet of the statue of the Virgin Mary in the local cathedral. The only upset came after Bahamontes was invited to kick off a local football match between Toledo and a side from Madrid. The Toledo club president gave him a bejewelled brooch, and Bahamontes promptly lost it. After a frantic search, a ten-year-old girl finally found it and, rather than making off with it, handed it over to Bahamontes, who duly re-pinned it to his chest. Nothing, it seemed, could go wrong for Spain's newest sports star that day. Bahamontes freely admits now: 'That year was the most emotional I ever got about the Tour de France, even more so than in 1959. The reaction in Spain really affected me.'\n\nHowever, his success that year can not be allowed to cloud the implications of the ice cream incident. Not for the first time in his career, Bahamontes had shown a tendency to over-react to the most minor of events. It is not stretching a point to say that briefly Bahamontes had all but opted out of the race because of two broken spokes. He was leading comfortably on the climb but he still ended up losing eleven minutes by the finish. Such excessive sensitivity to even the smallest of misfortunes, coupled with a fragile ego, is not infrequent in top professional bike riders. Indeed, one team director I talked to likened one of his top riders to an expensive make of sports car: 'It goes wonderfully, but the smallest mechanical problem and the whole thing gets shot to hell.' However, with Bahamontes that extreme sensitivity seemed to be even closer to the surface than usual. What probably aggravated this particular character trait was that Bahamontes knew that even if he did fall wide of the mark on occasions, and even if he playacted, his talent was so great he could still win prizes beyond the wildest dreams of a former teenaged black marketeer and half-starved street urchin.\n\nIn fact Bahamontes seemed to pick up prizes even when he did not want them. In the 1950s the final stage is far from being the glorified criterium [exhibition race] it is now, in which only the sprinters are racing flat out. In 1954, when Bahamontes punctured late on, and lost a place to finish twenty-fifth overall, he was given the prize for 'Most Unfortunate Rider'. Then there were post-Tour criteriums, where the top riders would make more than they did in the race itself. That was certainly true for Bahamontes after Daniel Dousset, France's top agent for criterium contracts, took charge of his contracts. Just as many riders do now, Bahamontes prioritised criteriums over returning home straightaway. While he made five hundred thousand francs [\u00a39,000 in modern money] from his first King of the Mountains title, much of it was distributed to his team-mates. Where Bahamontes could really cash in was in the post-Tour events, and he calculated he made seven hundred thousand francs [\u00a311,000] in 1954 alone. 'I made more money than I would have done in five years as a professional,' Bahamontes pointed out. 'A top pro in Spain would have made about five thousand pesetas a month [\u00a31,300] while I [as a rookie pro] was making one thousand pesetas [\u00a3270] a month.'\n\nThe kudos from the King of the Mountains win was so good, in fact, that when he crashed badly at a criterium in Belleville, France, and the Spanish press got wind of it, Bahamontes had Dousset to ring up the organisers of the forthcoming Barcelona criterium to reassure them that even with a badly injured knee and elbow he was still coming. It seemed not even Dousset could stop Bahamontes from inside wheeler-dealing, though. A newspaper later reported Bahamontes had 'sub-contracted' his place in several French criteriums in 1954 to another Spaniard, Francisco Alomar, so that Bahamontes could race in the Volta a Catalunya, where, of course, he would be paid by the organisers as well.\n\nAs a further side-effect of his King of the Mountains success, Bahamontes came into closer contact with some of General Franco's most fervent supporters. One of these came at a formal reception in honour of his victory with Doctor Enrique Pla y Daniel, the Archbishop of Toledo and the first highly-placed figure of the Spanish Catholic church to support Franco's uprising. Then, straight after he had returned from his criterium campaign, Bahamontes met General Moscard\u00f3, the Nationalist hero of the Alc\u00e1zar siege. In peacetime, Moscard\u00f3 had become the chief of Franco's personal _corps de garde_ and as the sports delegate in Franco's government, the most powerful figure in sport in the country. According to media reports, their conversation rather improbably centred on how Bahamontes could improve in his track racing. Yet if Bahamontes was being sounded out by Moscard\u00f3 for his potential as a sporting idol by the ruling regime, as seems more likely, there was no immediate external sign of it. That summer the Government-controlled media at least preached caution about Spain's new star. 'Bahamontes is the only Spaniard who can win the Tour de France,' _ABC_ wrote, 'but he can mess it up, too. Moving fast [in sport] is a risky process, and Bahamontes is already receiving many laurels for him to sleep on. He has done nothing more than start.'\n\nAt least one other rider was blunt about Bahamontes' potential, and his potential limitations. As Tour winner Louison Bobet put it after the race: 'I have never seen such a great climber. Bahamontes can achieve a lot, if he stops clowning around.'\nChapter Four\n\n#### The Gypsy Rider\n\nIt is a cold, misty, early spring morning in Orihuela, an elegant, slightly battered-looking spa town in southeast Spain, bisected both by the wide, shallow waters of the River Segura and lines of six-foot-high palm trees. Close to the centre is the town's casino, a gracefully appointed nineteenth-century building with spacious, high-ceilinged rooms, wall-length windows and well-varnished wooden floors. The silence is broken only by the clattering feet of a stray, bow-tied waiter. That morning Bernardo Ruiz, the 1948 Vuelta a Espa\u00f1a winner, and I are almost alone. Yet, despite his age, and the absence of other customers, to judge from the furtive way Ruiz keeps looking around to check we are not being overheard we could be plotting a bank robbery. The explanation for his behaviour finally arrives when Ruiz places his coffee down with the cautiously calculated movements of an eighty-something-year-old, then leans forward and all but hisses: 'Does Federico know you're doing a book on him? And does he know you're talking to me?' More than half a century has passed, but Ruiz is apparently still worried sick about the consequences of what he can tell a journalist about bike racing and Bahamontes in the 1950s. Old habits, it seems, die hard. For Federico Bahamontes' generation, the so-called law of _omerta_ that has reigned for so long in cycling \u2013 about drugs, race-fixing, secret alliances and open treacheries, not to mention their real opinion of fellow riders \u2013 remains almost unbreakable.\n\nRuiz, a six-foot, muscular man, is a conundrum. He spills the beans on a lot of issues that cast Bahamontes in a poor light. At the same time, however, he still refuses to divulge a lot of what he knows about his old rival. He almost seems upset that the unwritten code of honour obliges him to stay silent about a fellow professional for whom (and this is being polite) he has scant sympathy. Along with Jes\u00fas Loro\u00f1o, and occasionally Miguel Poblet, Ruiz was one of Bahamontes' biggest rivals in Spain. However, that rivalry is not the main reason for his lack of appreciation for the Eagle of Toledo. What appears to be mutual dislike, fuelled by accusations of betrayal and mistrust throughout their careers, is reinforced because Ruiz cannot comprehend what made Bahamontes tick as a person. Yet they would frequently find themselves thrown together in the same team, most notably in the national squad for major Tours. Indeed, for a year, in 1960, Ruiz was Bahamontes' sports director in the trade team Faema. To judge by our conversation, it was not a happy twelve-month period for either of them.\n\nInterestingly, Bahamontes has no such problem with Ruiz or the rest of his contemporaries because he thinks they are too far beneath him. He has such a low opinion of them that he refuses point-blank to discuss them, once dismissing them all with the throwaway comment: 'How can I call them rivals, when none of them could beat me?' Ruiz, on the other hand, is cagey, and unhappy that he feels obliged to be so. The conversation is broken by silences, moments when he shakes his head as his voice trails off while muttering: 'The things I could tell you, the things I could say . . .' There is no getting past him, though, when he does not want to talk. Even when I promise that the book will be published first in English, he responds: 'Yes, but there are [Spanish] journalists who will read it.'\n\nEven so, of the few Spanish riders still alive who witnessed Bahamontes' career close up, Ruiz's reminiscences are telling: without giving too much away, he depicts his former rival as a rider whose excessive individualism made him an isolated figure, incapable of asking for or receiving assistance at crucial times. When I ask him if Bahamontes was someone who remembered favours done for him in the past, Ruiz stiffens visibly and finally booms out: 'Fede? He's never been grateful to anybody in his entire life.' As Ruiz sees it, 'He should have won [the 1957 Vuelta], but he didn't know how to position himself in the bunch to stop them wrecking his chances, and that's . . . why we always beat him.' And Ruiz later adds. 'He was a magnificent climber but he did not know how to use his own strength; he was out of control. One day he'd beat you, then the next he'd do nothing. [Personally], I couldn't work out what was going on inside his head.'\n\nRuiz had plenty of opportunities to try to find out. According to Spanish Cycling Federation figures, there were just twenty-three fully-fledged Spanish professionals in 1954, and across the three categories immediately below them ( _Primera aspirante\/Segunda aspir ante\/Primera independiente_ ) there were only another thirty-eight. It was a small world, where the same old faces appeared time and again in races; one where friendships formed rapidly and rivalries could harden just as quickly. In the case of Ruiz and Bahamontes, things went wrong on a personal level almost from day one. Ruiz admits he was initially at fault, but Bahamontes' ability to take everything to extremes turned a minor row into a major one. 'I first met him in 1954, in the Tour of Asturias,' Ruiz remembers. 'He was fighting to beat Rik Van Looy in the overall, and I don't know why but I said something stupid. We had raced over [the climb of] Pajares the previous day and he'd done well. Then in the next day's time-trial, which started with a descent, he started off just before me. As we waited there on the line to start, I jokingly said to him, \"Be careful, you'd better get out of the way on the downhill or I'll mount you from behind like you were a horse\", basically meaning, \"Either you get out of my way or I'm going to ride over the top of you\", because I knew he was a bad descender. It was a bit foggy and when I finally saw him, about forty or fifty metres ahead of me on the descent, I shouted out and, maybe because he took fright at my shouting, he took a curve badly and ended up going off the road. At the finish he told everybody I'd pushed him off the road, which was a total lie. I hadn't touched him, but when I won the race overall I got booed by the crowd all because of his big mouth. It's funny now, but back then that was what would happen . . . Bahamontes was a great climber, but he would lose his self-control easily and end up paying the price. He was always a bit naive, too. In the Vuelta a Espa\u00f1a when I was his sports director and he abandoned [1960], he went off the rails completely, again.'\n\nThere were other reasons why they fell out. With such a small group of top professionals in Spain, it was not too hard for races to be fixed. However, as Ruiz explains, Bahamontes was reluctant to play the game, as he showed in the Volta a Catalunya in 1953 with his dramatic lone attack that upset the plans of his more conservative superiors. When deals were struck, Ruiz says, Bahamontes could all too often renege on them: he was too much of a loner and perhaps too distrustful to maintain 'honour among thieves'. 'Believe me, the number of things we did so that Fede could win races . . .' Ruiz says almost mournfully. 'But because of his character or whatever, he couldn't adapt. If something is agreed upon, then you have to stick to it. Within the teams, we'd agree before the race, \"I'm going to do this, I'm going to do that, I'll let you sit on my wheel because if we break away you'll work for me afterwards, and so on\". But once the start whistle blew, what can I say? Everything we'd agreed beforehand went down the pan.'\n\nIt is ironic that, according to Ruiz, one of the biggest problems Bahamontes caused his fellow professionals was not that he refused to buy and sell races, but that he was too taken with winning for them to trust him to keep his word. Indeed, Bahamontes remains contradictory on the subject today, sometimes admitting to having sold races, at other times denying it. 'With time, Federico wasn't so wild,' says Antonio Jim\u00e9nez Quiles, who finished second in the Vuelta a Espa\u00f1a in 1955, aged just nineteen. 'At first he thought he was the King of Kings. Later, he got taken down a peg or two and he became much more calculating. But he was always a really individual guy, always separate from the rest.'\n\nAntonio Jim\u00e9nez Quiles and Bahamontes became acquainted in 1955 when the Vuelta a Espa\u00f1a returned after a five-year absence. However, at a time of severe political repression, prison camps and economic misery, that year's Vuelta a Espa\u00f1a was hardly a showpiece event. 'It was bloody miserable,' says Ian Brown, who rode for the Great Britain team. 'There were all these guys standing around with guns and helmets, nobody looked happy. When we flew into Spain, we did some training beforehand and went past a massive concrete building, maybe half a kilometre long, and there was a long line of guys all chained together. I asked a Spanish rider we were training with, \"What's that?\", and he said, \"You didn't see anything, it's political\". We started off in Bilbao and went down towards Barcelona, and the roads were terrible all the way. I got four punctures in one day.'\n\nAfter his success in the Tour de France the year before, Bahamontes was in Spain's national 'A' team. But according to Jim\u00e9nez Quiles and Ruiz, the idea of riders working for each other in the national selection was risible. 'There was never a friendly atmosphere,' Ruiz recalls. 'We were all fighting each other every day . . . and everyone would have problems, to a greater or lesser extent, with all the rest.' Jim\u00e9nez Quiles says: 'All the teams were the same \u2013 a bunch of guys flung together, and that was as far as it got. You ate together, you slept together, then you got on your bike and it was a case of \"last one to the finish is a poof\". It wasn't like these days where you have a single leader and some cohesion.' Jim\u00e9nez Quiles recalls that he was the only Andalusian rider at the 1955 Vuelta in a mixed team of Catalans and riders from Aragon. 'All the Catalans would sit in one table in their fine track suits and with their nice food, and I'd be all by myself at another,' he says. 'During the race, I got no help from them whatsoever. I ended up paying a guy from another team to give me his bike if I ended up crashing.' Another point of assistance was Bahamontes himself, but at a price. According to Jim\u00e9nez Quiles, throughout his career Bahamontes 'had an entire shop in his suitcase. He would sell gears, spokes, rims, inner tubes, the lot. He'd get them from France and sell them over here to the rest of us. You could get two types of inner tubes back then, the ones that were fresh out of the factory, and which would lose their tread after you'd ridden about two hundred kilometres. Then there were the ones which had been kept hung up for a year and dried out properly, and they had really good treads. Bahamontes would charge you a one hundred per cent mark-up on those. He would only take cash, no credit.' Somewhat uncomfortably to modern ears, Quiles adds: 'He was a right gypsy.' And he was not the only one to give Bahamontes that name: gypsy ( _el_ _Gitano_ _)_ was the nickname by which the entire peloton eventually came to know Bahamontes. Moreover, Bahamontes does not appear to object to it. Miguel Poblet, another of Bahamontes' rivals in the 1950s, and today someone who wears a toup\u00e9, says: 'When I ring Bahamontes up, I'll say, \"Hey, gypsy, how's it going?\" and he'll answer back, \"Hey, wiggy, you ok?\"' Given the degree of animosity in Spain towards that ethnic group, then and now, the nickname seems sadly apt for a loner like Bahamontes.\n\nWhile Jim\u00e9nez Quiles was isolated within his second-string regional team, yet still managed to finish runner-up to France's Jean Dotto, Bahamontes had a similarly miserable time in Spain's 'A' team, and ended up twenty-first. Though a breakaway on stage one with Loro\u00f1o looked promising, things began to go wrong for Bahamontes almost as soon as the Vuelta started. On stage five to Lerida, Bahamontes' left knee starting to give him grief. Calcium injections were little help, not surprisingly since their only effect is long-term strengthening of the bones. However, in those days the sport's medics were often little more than quacks, and riders would be given the most bizarre substances. By the rest day in Barcelona, there were rumours that Bahamontes would not only be unable to finish, but that the Tour de France was out of the question, too. However, Bahamontes' Barcelona-based professional team, La Solera-Cacaolat, came to the rescue by hiring a doctor, Joaquim Cabot, specifically to treat their star rider. Cabot confirmed that Bahamontes' knee injury was due to a twisted tendon, but was unable to identify the cause. Bahamontes initially claimed it was due to being kicked by a horse in his youth. Nonetheless, Cabot said he could continue racing. On the stage from Barcelona to Tortosa, Bahamontes was back in action, trying to get in several breaks. It was not the best of ideas. On the stage to Cuenca, Bahamontes was already in trouble, and when he got off the bike to greet his mother as the race reached Madrid, reporters noted he was limping again.\n\nMeanwhile, the 'A' team was falling apart as a squad with Loro\u00f1o bellowing for the benefit of the journalists: 'Team-mates? Where are my team-mates?' as he crossed the finish line at Lerida, after he had apparently been abandoned to his fate when he was dropped. Asked by a reporter in Barcelona why there were problems inside the team, Loro\u00f1o answered testily: 'Because we all want to win and only one of us can.' According to Chico P\u00e9rez's _Historia de la Vuelta,_ Loro\u00f1o added elsewhere: 'We've declared war all right, but it's a war between us team-mates.' Bahamontes was quoted by the newspaper _El Correo Catalan_ , as saying: 'We're all fed up with Loro\u00f1o and with what he says.' So much for team spirit.\n\nThe disunity in the top team contrasted with the success of the regional Spanish racers, one of whom, Rene Marigil, even took the yellow jersey for a couple of stages before Dotto moved into the lead. By the time the race reached Madrid Jim\u00e9nez Quiles was in the top three overall, and Loro\u00f1o was reduced to cheap criticism. He claimed that the reason for the top riders' relatively poor performance was because the 'B' [regional] teams are all watching the 'A' team like hawks, before hastily rectifying that with: 'I mean, everybody is watching us.' With three stages left, the press held out little hope for the best-placed Spaniards, with _ABC_ asking: 'What kind of support can Marigil count on? The truth is, very little. A lack of agreement between the riders of the \"A\" team has had fatal consequences. With a few exceptions, each rider has fought the race as an individual, not as a team. The fact that Loro\u00f1o was in yellow meant that Bahamontes did not make too big an effort and Poblet \u2013 after racing through his region \u2013 decided to pull out altogether. The second-level [regionals] have given our top men a lesson. The \"A\" team had the chance to win stages and make money and instead they've all attacked at the wrong time or without a clear aim.'\n\nIt was not only the riders in the Vuelta who had problems. The race co-director, Luis Bergareche, was badly injured when he fell out of a fourth-floor window in his hotel, plunging through a canopy and landing on a taxi bonnet. 'A' team trainer Juli\u00e1n Berrendero, meanwhile, received a public warning from the race organisers for 'talking to the press and not to his team'. Spanish hopes of a last-minute turnaround in fortunes sank even further when Marigil pulled out injured. Not even treatment with hydrocaine, a cocaine-based medication, could help him.\n\nAs for Bahamontes, he had tried to get over Leones, the climb in that year's Vuelta closest to Toledo, at the head of the field but was beaten to the punch by an Italian rider. Then, on the stage to Valladolid, to add to his misery he rode for the last hour on a spectator's bike after lending his own to Loro\u00f1o, who had punctured twice. It was a rare moment of fraternity between two riders normally at daggers drawn. To cap it all, later on the same stage Bahamontes fell and hurt his elbow and shoulder. His knee was also playing up again, increasingly seriously and when it was announced with three days of the Vuelta a Espa\u00f1a to go that Bahamontes would not be doing the Tour de France, nobody was particularly surprised.\n\nIt has almost become a party trick. Bahamontes grabs your hand and puts it on the side of his knee, saying: 'Feel that?' And what you detect beneath his trousers is a lumpy protrusion roughly the size of a twenty pence piece. That was how he reacted in 1993 when I first interviewed him and asked him why there had been such a lean period in the Tour de France between his first King of the Mountains title in 1954 and the second in 1958. When I asked the same question in 2008, more because I suspected I would get the same response than because I was interested in the answer, out shot his hand again to provide the same explanation. 'That injury cost me a ride at the Tour in 1955 and I thought it would cost me my career,' Bahamontes says. 'It was so bad I could hardly walk, let alone ride my bike.' Bahamontes traced the root of the problem, a partly dried-out meniscus (the disks of cartilage that cushion the knee) to a badly-aligned shoe plate. It was just a few millimetres out of place, but that was enough for it to act as the metaphorical snowball that, eventually, provokes an avalanche. 'It altered the position of my foot, which in turn meant the angle of my leg was out of kilter every time I pedalled,' he says. 'And that caused some tissue in the knee joint to rub together. The injury ended up leaving me completely lame. I tried walking with Fermina half-supporting me and after fifty metres it was so painful I collapsed. I couldn't get on my bike, and as I couldn't get on my bike, I couldn't race. The thought of an operation really scared me. That and not being able to race.'\n\nSuch insecurity about a career-threatening injury is only to be expected, particularly as Bahamontes could not forget the abject poverty of his childhood and beyond. If Bahamontes wanted to see the consequences of being out of a job, he only had to look around at the pinched faces and ragged clothing that predominated in Spain then. Bahamontes was on the horns of a simple dilemma: racing was his ticket to financial security, but racing hard with an injured knee would only aggravate the problem.\n\nIn fact, things had started going sour for Bahamontes almost as soon as he stepped off the podium in Paris. After injuring his hip in a post-Tour criterium, then abandoning during the World Championship following a lacklustre performance, Bahamontes pulled out of the Volta a Catalunya with stomach problems. He blamed the water he drank from a fountain in the town of Matar\u00f3, but there were rumours he had overdone it on the post-Tour celebrations and was under-prepared.\n\nThe next setback was financial, and much closer to home: it involved Toledo's main cycling club and their star rider. According to a lengthy letter sent by the club to the newspaper _ABC_ , Bahamontes and fellow Spanish professional, Mariano Corrales, had agreed to take part in a criterium during the city's week-long fiesta in October. But, the letter went on, just a few hours before the race was due to take place Bahamontes and Corrales turned up in the club's offices, demanding 'start money' ['appearance money' in cycling jargon], which was refused. The organisers wrote that since the race was funded by the town council, they had reminded Bahamontes it was being run on a shoestring. They even claimed that Bahamontes had promised to race for free. Bahamontes and Corrales refused to take part a second time, and the start was delayed by twenty minutes before the race finally got under way without them. However, when the two turned up at the finish and demanded payment a third time, the organisers reacted by explaining over the loudspeaker system why the local hero was not in the peloton. Both riders were duly booed and wolf-whistled by the crowd in stark contrast to the ecstatic welcome Bahamontes had received just a few months before. Bahamontes says he has no recollection of any of this happening, but whatever the truth he did not come out of the affair well. 'Bahamontes has lost popularity and was whistled at by people from his own city,' a semi-humourous column in _ABC_ reported in early December, 'the problem being they have yet to see their hero turn a single pedal stroke. They have only read about the Aubisque in the newspapers.'\n\nThe next mini-scandal was physical, rather than financial, and blew up when Bahamontes allegedly lashed out at Miguel Poblet after the first stage of an early-season race, the Tour of Andalusia. As Poblet recalls it: 'We were in a break together and the bunch was on the point of catching us with about three kilometres to go, and I said to Federico, \"I'm not going to work with you because we're going to be caught and I'll have to sprint\". Well I won, and he got second or third, just a little behind me, and he was pissed off because I hadn't worked with him in the last kilometre. So he hit me with his bike pump. That's his nerves for you. He was very nervous, very temperamental, but it didn't go any further. The next day we didn't even remember what had happened.' The Catalan laughs away the incident now, but there was at least one newspaper report that the injury was so bad Poblet had to abandon and could not train for several days. Poblet's sports director told _ABC'_ s cycling correspondent that his rider had been kicked in the stomach by Bahamontes, 'in front of everybody, and I believe him'. Bahamontes, the sports director continued, was annoyed because Poblet had chased him down three or four times close to the finish and Bahamontes' attempts to block his path by zig-zagging across the road failed. Bahamontes' recollection of events is very different: 'He said I kicked him in the nuts but I never touched him. If I had kicked him there, how would he have been able to ride his bike?'\n\nYet amid the controversies, awards for Bahamontes' 1954 Tour success continued to roll in. The most important was the Baron de Guell Cup from the Ministry of Sports, one of biggest prizes in Spain for sporting achievements. The Spanish Cycling Federation named him their 'rider of the year'. In late October, aged just twenty-seven, his first biography was published, costing 1.5 pesetas [fifty pence] a copy.\n\nThe racing season started well enough for Bahamontes. He began it by beating future Vuelta winner Jean Dotto's record for the hill climb held on the infamously steep Mont Faron, just outside Toulouse, in February. He clocked the fastest time at both intermediate checkpoints, finishing the quickest by fifteen seconds. Over time he would prove that when it came to single-climb races he was all but invincible. 'It wasn't as tough as they said,' Bahamontes remarked afterwards, 'but now I have to win a race on the flat because if I don't I'll end up being considered a climbing specialist.' Dotto stated emphatically: 'There is nothing to do against Bahamontes in the mountains. He is simply the best of all us.' As if to confirm Dotto's words, Bahamontes then won the GP Monte Carlo despite not starting the race until the peloton was already out of sight. He had gone for a walk along the seafront and had not realised the race had begun.\n\nWith the knee injury threatening his embryonic career, Bahamontes sought emergency medical treatment after the Vuelta a Espa\u00f1a in 1955. 'According to the doctor I saw in Madrid, L\u00f3pez Quiles, I had to have the knee treated, obviously,' he says. 'My knee swelled up and they drained the wound every few days. But thank goodness he said I didn't need an operation. Just a lot of rest. However, Bahamontes did not follow the medical advice. Instead after the Vuelta, he embarked on a lengthy campaign to prove his form ahead of the Tour de France, recording an impressive run of wins, including the Tour of Asturias and the Bicicleta Eibarresa stage races, and the Vuelta a los Puertos one-day event. It all seemed to be going well, and when the Federation doctor carried out a medical examination immediately before the Tour, the newspapers were able to report that 'several X-rays' indicated the knee was completely cured and there was no bone damage whatsoever. On 24 June, _ABC_ reported that Bahamontes had confirmed again that he was 'ready to take part in the Tour'. With a week to go he was named on the official list of participants. However, after placing an unremarkable nineteenth in the National Championships in Barcelona on 29 June, Bahamontes and his team-mates were given another medical. This time the result was different and Bahamontes was substituted by Antonio Gelabert, the new national champion. To describe this as a bombshell is no exaggeration. Bahamontes had won the Tour of Asturias less than a fortnight before, but the Federation's official report could not have been more damning. According to _ABC_ , a Federation communique stated: 'Having given all the riders their check-up, we have confirmed [their physical fitness] with the exception of Federico Mart\u00edn Bahamontes, who is suffering subjective pains in his left knee, and we therefore believe his participation in the Tour is not recommendable, given that possibly he will not perform well.' Calling Bahamontes' injury 'subjective' was like a green light to the press for some ferocious criticism of the Eagle of Toledo and they went at it with gusto. 'Our trump card of the Tour, Bahamontes, has gone up in smoke,' said _El Mundo Deportivo_ before displaying the blackest of humour and claiming: 'We hope that our explanation why this has happened \u2013 our man from Toledo has commited suicide \u2013 goes up in smoke, too, and that he gets well soon and that the phase he is going through is as brief as possible. If the official reason \u2013 \"doctor's orders\" \u2013 sounds odd given his statements of a few days earlier, we don't want to go into the real causes. My opinion is that Bahamontes, the real, great Bahamontes whose racing [was so spectacular it] caused Europe to grind to a halt last year, has been a victim of the \"frivolous\" Bahamontes. He saw money, contracts and laurels, and instead of taking care of himself and his future, he has let himself be worshipped, forgetting that riding a bike is such a tough sport.'\n\nHaving thrown Bahamontes to the wolves, the Federation then attempted belatedly to try and repair the damage to Bahamontes' image. 'His knee is suffering from malaise,' Federation president Alejandro del Caz claimed, 'and that's not what he wants, and neither do we.' With the Tour de France lost and his reputation for unreliability considerably reinforced, Bahamontes' knee finally received the rest it needed. When he returned in early August he was forced to pull out of the Vuelta a Galicia stage race, which caused a minor controversy because his team had no idea why he had abandoned, and the organisers insisted his team-mates had to quit as well. But Bahamontes bounced back in September when he took two stages of the Volta a Catalunya. By October it was business as usual as he planned the season ahead. It was typical Bahamontes. Just as he could swing unpredictably from feeling upbeat to exuding total gloom, on the few occasions he was injured seriously in his career he appeared equally capable of regaining his race condition in record time.\n\nHis knee, though, still clicks when he walks. And ever since 1955 he has been grabbing visitors' hands and getting them to feel the twenty-pence-sized hole that marks his old war wound.\nChapter Five\n\n#### Pointing the Finger\n\nIf Federico Mart\u00edn Bahamontes' career was put on hold by injuries during the summer of 1955, the following year it was on an upward swing again and he had chances of victory in all three major Tours. Yet he failed in all of them. Each time Bahamontes pointed the finger at something or somebody else being responsible, but never at himself. Then as now, self-criticism is something the Eagle of Toledo rarely indulges in.\n\nIt has been argued that 1956 represented Bahamontes' best chance of winning the Vuelta a Espa\u00f1a. What the race highlighted again was that Bahamontes was no team player. According to Bernardo Ruiz, as early as 1955 his team-mates had discovered that Bahamontes was too much of an individualist when it came to making deals, even when he would have benefited from them.\n\n'Bahamontes was a gentle person,' says Brian Robinson, the Briton who finished eighth in that year's Vuelta, 'except when he was shouting at his team-mates or he got his hackles up with his team director. He could be very fiery then. When they [the Spanish team] did start arguing they would go on shouting for ever. They couldn't agree on anything. Just getting them to agree a menu for dinner would have been difficult enough.'\n\nRobinson is the British cyclist who knew Bahamontes and the Spanish the best. He first came across Bahamontes in 1956 on the Vuelta, and a year later they were nominally team-mates in the St. Raphael-Geminiani squad, though Bahamontes barely raced for them. Their careers would run parallel until Robinson retired in 1963. 'Bahamontes was a friendly, bubbly guy, I don't think I ever saw him down,' recalls Robinson, now in his early eighties. 'We'd talk in a kind of French, though his was a bit of a patois and he mangled Spanish and French together. If he had a puncture, a _crevaison_ , he'd yell out, \" _Je clavais! Je clavais_!\"'\n\nRobinson also travelled around Spain with Bahamontes, which gave him an insight into how widely appreciated he was on home soil. 'I once did a week's criteriums with him, Anquetil, Darrigade and various other local Spaniards. First we were riding down in Benidorm \u2013 which was just a fishing quay in those days and we spent the whole evening riding over cobbles \u2013 and then all across Andalusia, mostly on what seemed like football fields. There were good crowds and Fede was hugely popular. We stayed in the best hotels and got good money, paid on the nail, too. They couldn't muck around with the big names. Particularly Fede. They even did special flamenco dancing in these dives every evening, with the words in the songs all about Bahamontes.'\n\nHowever, if Robinson had no objections to riding alongside Bahamontes in exhibition races he would try and steer clear of him when it came to serious road-racing. 'We'd be friendly enough and if there was a gap in the bunch you'd shove him in it, no problem. He'd remember favours, and in criteriums it was all about putting on a show with as little effort as possible. You all pitched in. When it came to breaks, too, he'd work. At least I don't remember him ever \"swinging a leg\". But the problem was that the Spanish were more erratic than other nations in the peloton, tossing their bikes about, because they were more impulsive. They weren't as steady as others. [Miguel] Poblet was good because he had track experience, and [Jes\u00fas] Loro\u00f1o was solid enough; otherwise you always had to be careful if you were riding behind a Spaniard. If that Spaniard was Bahamontes, \"who was always switching about\", you had to be doubly careful.'\n\nEven though he could not speak any Spanish, Robinson had no problem appreciating their different personalities. He got on so well with Poblet, in fact, that he assisted him to his first Milan-San Remo win in 1957 by leading him out despite being on a different team. 'The team manager had said we should help him, it was something to do with us having the same bike company,' Robinson recalls. Robinson still finished third, at the time the best result in a major Classic by a Briton since the nineteenth century. 'Loro\u00f1o was calm as anything, and Poblet could get upset,' Robinson says. 'But Fede was definitely the most excitable of them all, he couldn't keep still. He always had very fixed ideas, too. Once he got an idea, you couldn't change his mind.'\n\nTo the rest of the cycling community, the Spanish stood out for reasons other than their poor bike-handling and constant arguing, not least their collective lack of money. Unofficially the Spanish were rated fourth in the world behind the Italians, French and Belgians, which was an amazing achievement considering it was just five years since they had returned to the international scene. But their finances placed them a lot further down the list. 'The Spanish were always a bit ramshackle as a national squad,' says Robinson. 'Normally a team would have its own bikes, but the Spanish would turn up and take the yellow Tour bikes the organisers would provide if you couldn't afford your own.' Despite their 'low-budget' look, Robinson knew Bahamontes was deadly in the mountains, and had to be respected. 'He used to go' \u2013 and Robinson makes a highly expressive sound to describe the noise of Bahamontes' pedals turning at high speed \u2013 ' _tsch-tsch-tsch-tsch_ and get a hundred metres, have a little look back at us, stay there, then go _tsch.-tsch-tsch-tsch_ again, really pedalling away on the low gears, get another hundred metres. Then he'd be gone. He'd just ride away. He wouldn't have won the prize for elegance in his riding. He had that funny position, more of a sitting position, with his arms looking so stiff holding the bars close up. But if he came to a race you'd know that he could drop me, Geminiani, Anquetil, the good boys if you like, on the climbs. He had the edge on all of us. When he put his mind to it.'\n\nSpain's ten-man 'A' team for the Vuelta a Espa\u00f1a in 1956 was designed to put behind them the misfortunes of the previous season. That was the theory. In practice, the squabbling began before a pedal was turned in anger. Loro\u00f1o was furious that Luis Puig, the team's sports director, had decided Salvador Botella was to fight for the overall while he and Bahamontes would only go for the King of the Mountains competition. 'Botella has no right to be the boss,' Loro\u00f1o stated before the race left Bilbao. 'I've just won the Bicicleta Eibarresa [stage race], yet that doesn't seem to interest my sports director. I can't stand that. I'm the strongest and I should be the boss.'\n\nRobinson was there with fellow Britons Tony Hoar and Ian Steel thanks to an invitation from manager Phillipe Louviot to ride under a flag of convenience for the Swiss team. It was not a luxurious experience for any of the riders with little support from the organisers or teams. After each stage the peloton would grab their suitcases from the army jeeps which were acting as support vehicles, balancing their bags on the handlebars, and head off to low-budget _pensiones_ anything up to twenty kilometres away. 'The race was a bit of a shambles,' Robinson recalls, 'and the living conditions were so bad in Spain generally that I came back with dysentery. I was passing blood and spent six weeks off the bike. I don't remember ever getting any prize money, either. I remember you could get a made-to-measure suit of English Worcester completed for you in a night for a pound. So the guy who made it couldn't have earned much. And the food was really poor. It was pre-refrigeration days and all we got was eggs or a scrawny chicken that had been running around in the wilderness, no red meat and no fish unless you were by the sea. And I can't remember once seeing butter.'\n\nEven in 1959, when he next rode the Vuelta, Robinson says it was much like the infamous _Monty Python_ sketch where the cafe menu offered nothing but spam. 'They kept on giving us eggs and more eggs, and that got to some people. There was one hotel run by nuns and we were all seated round the dinner table when they came out with a great platter of eggs, and [Raphael] Geminiani suddenly shouted out, \"I'm sick of bloody _huevos!_ \" and threw the whole thing, bang!, against the wall.' It was not easy to look for alternatives, either. 'There'd be soldiers at every crossroads, and when we tried to snaffle some grapes once from a field, one of them started loading up his gun, putting the cartridges in, and making like he'd shoot us. We got out of there in a hurry.' Nutrition was such an issue in the 1956 Vuelta that the French team pulled out with food poisoning after claiming the lunchtime sandwiches provided were going rotten. The organisers responded by saying that the official responsible had given the team the wrong batch by mistake when he grabbed them from the back seat of the car. Assuming they were not lying, it is unclear for whom the dodgy _bocadillos_ had been intended. But if the British and French were unhappy, nobody knew about the lack of food in Spain better than the Spanish themselves. 'The Vuelta a Espa\u00f1a and the Tour de France were light years apart,' says Ruiz, 'right down to the food. I used to say to my team-mates when we did the Tour that even if we were at the back of the peloton it was better to be in France than in Spain, if only because we'd get good stuff to eat. You'd go into the hotel dining room after each Tour stage and you could have what you wanted \u2013 meat, jam, butter, bread. Here in Spain, we had nothing.'\n\nIf the food was dire, Robinson says the roads were equally poor. 'You'd see these women with baskets of stones on their heads, trying to repair them. But they had no proper surface. The only roads with macadam on them were on the coasts and that was very poor quality. When it snowed on one stage, they just cancelled it straightaway and made us go on trains, sitting on wooden slatted seats just like in the utility trains we'd had back home. With no toilets, the sanitary gutters ran through the middle of the roads, too, and it wasn't pleasant riding through all that.' Punctures and crashes were frequent on what were little more than cart-tracks. As a result, solid back-up from the team or even a hand from a friendly rival was crucial. However, Bahamontes' lack of negotiating skills and preference for going it alone meant, just when he needed them most, he was guaranteed neither.\n\nThe high point of Bahamontes' 1956 Vuelta came when he was just eight seconds behind leader Angelo Conterno of Italy after stage fourteen. With only three days' racing left he should have been poised for his first major Tour win. In fact, he soon found himself battling merely to survive, and battling with his team-mates, too. The first backwards step came with a major crash on stage fifteen, caused by the ruts in an unsurfaced road, which left Bahamontes with thigh and knee injuries as well as a broken set of handlebars. Loro\u00f1o then opened up a minute's gap on the main pack by the finish in Bilbao and put extra pressure on Bahamontes' second place. The next morning, as rain fell in Bilbao and mist closed in, a heavily anesthetised Bahamontes set off for Vitoria on stage sixteen. A puncture early on did not cause him any undue problems, but another on the climb of La Herrera most certainly did. Though Ruiz and three other Spaniards dropped back to assist him, by the time they regained the peloton following a furious pursuit Conterno and the rest of the the Italian team had raced away. At the summit of La Herrera, Loro\u00f1o and Conterno had established an advantage of ninety seconds over Bahamontes; by the finish in Vitoria, after two further punctures, it had stretched to more than three minutes. Bahamontes had dropped to fourth overall and was nearly four minutes back. With one stage left, and in the space of two days, Bahamontes had almost slid out of contention.\n\nSome claim that Bahamontes' injuries were the main reason he was so suddenly out of the running. But a key Spanish team-mate has told me that Bahamontes lost the Vuelta a Espa\u00f1a that year because of an internal conflict: he refused to go along with plans to break a pact to keep Conterno in yellow. This version of events might sound implausible, but there are three facts in the team-mate's favour. Firstly, though Bahamontes was three and a half minutes down overall, with a mountain stage still to come, the gap was not unbridgeable. Secondly, though injured on stage fifteen, Bahamontes had still managed to drop the entire field on the first climb of Urkiola, so he was in good form. Thirdly, and most significantly, after stage sixteen had finished in Vitoria a rumour started up in the team hotels that Conterno, the race leader, had fallen seriously ill with bronchitis. So maybe there was still something to play for. The questions remained, though: was Bahamontes interested in playing and did he know how to?\n\nAccording to the team-mate, with a first Vuelta a Espa\u00f1a victory at stake, that evening word came that the Italians had requested that Conterno be allowed to keep his lead on the final day and other reports say they were willing to make cash payments to those who agreed. It has been claimed members of the Spanish team promised that they would not challenge the Italian. The Belgians, who were to provide Conterno with invaluable assistance the next day, certainly agreed. However, there were wheels within wheels. Despite Loro\u00f1o being the best-placed Spanish rider, in second spot, the would-be pact-breakers within the Spanish team wanted Bahamontes to go on the attack, and wreck all the previous arrangements to keep Conterno in yellow. 'In Vitoria all the teams had made agreements about who would win the stage, stay second, or keep the general classification as it stood,' the team member told me. Bahamontes, he claims, 'initially agreed to break the pact, but as he'd actually got quite a good placing overall he changed his mind and that annoyed a lot of people.'\n\nBahamontes version of events is far less convoluted: he attempted to attack on the following stage, but failed to get away because Conterno first held onto his shorts, and then because Conterno received 'external assistance.' Ultimately, the row that took place between Bahamontes and the rest of the Spanish team in Vitoria lacked any relevance since Conterno was helped on the last stage by the Belgians who pushed him up and over the Sollube, the last climb of the race. Conterno was given a thirty-second penalty for the obvious cheating, but still won the race from Loro\u00f1o by thirteen seconds. 'Conterno received all kind of assistance,' Loro\u00f1o claimed later. 'He rode up the Sollube without even realising he was going uphill.' And Bahamontes? He had offered Loro\u00f1o some limited support on the Sollube, but it was not the devastating solo move that others had been hoping he would make. If Bahamontes' foibles had provoked antipathy among his team-mates before, he was now in danger of becoming permanently isolated.\n\nTo modern sensibilities, such overt wheeler-dealing may seem shocking. But in a Spain suffering its worst economic recession of the century the pressure to make a fast buck was enormous. Though the money earned from cycling was often comparable to top football players in Spain, there was a key difference betweeen the two sports: no rider was guaranteed a salary from his team sponsor. Until the very end of the decade Spanish team sponsors rarely paid their riders anything except perhaps a few bonuses for victories, making prize money the only certain income. Local clubs would sometimes provide their professional riders with a basic subsidy which Bahamontes says was around one thousand pesetas [\u00a3300 in today's money] a month in the early 1950s. Others might come under the wing of directors like Santiago Mostajo, who in 1954 was paying his riders five hundred pesetas [\u00a3120] a month, though that was not enough to live on.\n\n'It was simple: whenever I went to a race in Bilbao, say,' explains Ruiz, 'I'd have to win, otherwise I wouldn't eat dinner that night. Then I'd go to Mallorca, on the other side of Spain, and it would be the same \u2013 first or second, or no food. I'd make damn sure I won, too, because if you didn't you wouldn't get a hotel to sleep in that night, either. You'd fight and fight with the riders to win, then fight and fight with the organisers to get them to pay for your hotel.' It is therefore not surprising that the top riders were so frequently at odds when their whole livelihoods depended on winning.\n\n'Given how much the train cost, you'd ride from Barcelona up to the Basque Country, seven or eight hundred kilometres at a time, ten or twelve of you in a line, praying the police wouldn't stop you because you'd be fined for riding in a group [which was considered dangerous to other traffic],' says Ruiz. 'If it rained you'd sleep wherever it caught you. You'd have a masseur lined up in Madrid or whatever, stop there, get a massage and a bit of kip, then go on.' Under such circumstances it seems logical that riders would work together to ensure certain results were fixed, if only to ensure a minimum income. But Bahamontes did not want to pitch in with the rest of his contemporaries. Instead he 'moved alone', as he liked to say. For that, at least initially, he paid a high price.\n\nIf Bahamontes was proving to be cycling's equivalent of Greta Garbo, it is not an unusual trait among climbers. In more modern times Scotland's Robert Millar also cut a lonely path in the peloton, while Charly Gaul became a notorious recluse after he retired, living alone in a wood with his shotguns and his dog, the crampons he invariably wore on his shoes the only reminder of his past. As a racer, though, Gaul was the complete opposite to Bahamontes in many ways: while Bahamontes was most effective in warm weather, Gaul seemed to come alive whenever the temperature dropped and the heavens opened. It was his ability to blast away in the coldest, most miserable of conditions that enabled Gaul to win both the 1956 Giro and the 1958 Tour. He is most famous in France for the attack in a rainstorm across the Chartreuse mountains that regained fourteen minutes on race leader Geminiani. Only the national team system, which put the rider from little Luxembourg at a distinct disadvantage, probably prevented him from winning more.\n\nThere could be no doubting Gaul's genius at breaking away whenever the roads steepened. Geminiani called the former abattoir worker 'a murderous climber'. French writer Antonie Blondin, perhaps the greatest of cycling journalists, described him as 'Mozart on two wheels'. 'When we raced we were irreconcilable,' Bahamontes told me when Gaul died in 2005. 'Going uphill even he admitted I was better, but when it rained he was impossible to beat.' But for all his affinity to Gaul, Bahamontes was kept at a distance like everybody else. 'Charly was friendly enough,' recalls Robinson, 'but you could never get close to him, never really get to know him.' And where Bahamontes was outspoken and fiery, Gaul was gloomy and withdrawn. 'He gives the impression that an evil deity has forced him into a cursed profession,' commented one writer.\n\nTeam-mates had the same problem with trying to get to know Bahamontes, which meant he received scant sympathy when things went wrong as they so often did in 1950s stage racing. It was a risky policy on Bahamontes' part. As his knee injury proved the year before, nobody could be sure they would not need a helping hand. In Bahamontes' case, his physiognomy meant he had to depend on team-mates. According to Ruiz: 'Bahamontes always had one or two days where he'd have a sudden physical crisis, and depending on the terrain he'd either be able to get over that crisis or not.' Known as 'the bonk' in British cycling circles, this sudden loss of energy is due to a rapid depletion of glycogen stores in the muscles. In Spanish it is called _una crisis_ or, more poetically, _una visita del hombre del mazo_ , a visit by the man with a mace. 'On the stage to Vitoria [in the 1956 Vuelta] he started to claim his brakes weren't working properly and he had punctures,' says Ruiz. 'That was why he had to stop on La Herrera, not for punctures. I had to stay with him to help him get there, using whatever means necessary, but I'm not going to tell you how.' Another member of Bahamontes' team is not so reticent on how this was probably achieved. 'He'd jam his arm between your elbow and you'd drag him alongside you,' says Jim\u00e9nez Quiles.\n\nLike so many climbers, Bahamontes lacked the technical skills for weaving his way through the bunch and relied on a 'sherpa' to guide him. He would also need help regaining time lost on the downhill sections. 'He and Charly Gaul had this thing in their legs so that whenever we got to the mountains it was \"Goodbye boys, we're off now\",' says Robinson. 'But Bahamontes was a bad descender, even worse than Gaul. And they'd both trail around at the back quite a lot. I can't remember Bahamontes doing a turn on the front. That's not the way to race and that limited the damage both could do.'\n\nEighty-three riders started stage eighteen of the Giro d'Italia in 1956; forty-two finished. One photograph sums up what they had to endure that day. It shows the previous year's winner, the Italian Fiorenzo Magni, halfway up one of the stage's five major cols. At that point it is snowing lightly. But in weather conditions still rated as the most extreme in the race's history, Magni will also have to battle through hailstorms, torrential rain and blizzards for nine hours and two hundred and forty-two kilometres. 'This day surpassed anything seen before in terms of pain, suffering and difficulty,' wrote the former Tour organiser Jacques Goddet. With behaviour that seems to border on the lunatic, Magni \u2013 who has been riding with a broken collarbone since stage twelve \u2013 has the end of one length of inner tube from his tyres clenched between his teeth and the other wrapped around his handlebars. He is trying to use the miniscule additional leverage he receives by pulling with the inner tube both to keep the front half of his bike as high out of the snow as possible and reduce the pain in his shoulder to a minimum.\n\nFor Magni to have reached the point where he is reduced to such painful measures just to keep going epitomises the extremes to which the participants were pushed on this stage. The exit of Ni\u00f1o de Filippis, the provisional leader, was one of the most memorable: with his fingers glued around the handlebars with cold, and having ridden himself to a standstill, he and his bike simply keeled over in unison. Magni, however, made it through. He finished third, just twelve minutes down on stage winner Charly Gaul, who also moved into the race lead. Bahamontes would have moved into top spot overall had he stayed with Magni. But around fifty kilometres from the finish, and the final ascent to the Bondone, Bahamontes abandoned. He then had to make his way through snow and freezing fog to the nearest farmhouse to seek shelter. 'It was impossible to race. There had been landslides and stones as big as a cupboard were all over the road,' Bahamontes recalls. 'Charly Gaul ended up partially deformed by the cold and I had frostbite in my hands and feet. I couldn't use them properly for a month.'\n\nAs winds reached speeds of 70 k.p.h., riders plunged their hands into bowls of warm water supplied by nearby inhabitants or downed glass after glass of brandy to try and regain some energy. Gaul was reduced to descending at a snail's pace because his fingers could not pull on the brakes. When he reached the finish he was trembling so violently his clothes had to be cut from him. 'I'd never known such bad weather. It was really frightening,' Bahamontes continues. 'We just stopped wherever we could, two here, six there, wherever you were you abandoned.' Bahamontes asserts that 'nobody got to the top on a bike', which is untrue of Gaul. However, his claims that a large number of the 'finishers' that day reached the line by car, has been corroborated. 'Me and [fellow Spaniard] Jes\u00fas Galdeano sneaked into somebody's house and put on some pajamas we saw lying about,' he says. 'Finally we were picked up by a lorry. We all quit. Anybody who says the opposite is lying because then they came by the hotels the next day asking who wanted to finish the Giro. But I hadn't reached the finish line so I wasn't starting.'\n\nWhile he now says it was the extreme cold that caused him to abandon, in a profile of Bahamontes' career cycling historian Javier Bodegas claims that was not the case. Rather, Bahamontes roundly blamed his sports director at the Girardengo-ICEP team for his race's premature end, saying he had sold out to Gaul and had deliberately driven ahead of Bahamontes in his team car on the ascent of the Bondone. There were also reports that when Bahamontes saw what had happened to De Filippis, falling like a deadweight on to the road, he was so frightened he stopped instantly. Whatever really happened, Bahamontes would surely have led the Giro had he finished the stage, and then gone on to become the first Spanish winner. He had started just six minutes down on the leader and ten minutes ahead of Gaul. Instead, Gaul, his face wrinkled with cold, his hands and feet blue, gained a seven-minute advantage on his nearest pursuer, Alessandro Fantini. 'Charly has the skin of a hippo,' remarked Geminiani; _El Mundo Deportivo_ called the stage 'a hecatomb', slaughter on a large scale.\n\nGaul's epic ride also brought him the record for overturning the largest overall disadvantage in the Giro to take the leader's jersey. It is a feat that, more than half a century on, no one has matched. Meanwhile, Spain's first winner of the Giro was not Federico Bahamontes in 1956, but Miguel Indurain in 1992.\n\n'There is always the Tour,' _El Mundo Deportivo'_ s editorial reflected after Bahamontes' exit from the Giro. But it was not as straightforward as that. Initially Bahamontes did not want to go to France; once there, he did not want the Tour to finish. 'If the 1956 Tour had lasted another week I'd have won it,' Bahamontes now claims. 'I lost nearly fifty-six minutes in the first two stages and every day I was pulling back time to the point where I finished fourth.'\n\nBefore the race, Bahamontes had suffered from sinusitis and stomach problems. 'My stomach swelled up,' he says, 'and it was so bad I had to cut my shorts open at the waist. That was when I started to use braces regularly. Fortunately the Tour doctor [Pierre Dumas] understood me well and gave me pills with carbon to get rid of all the extra gas and that sorted me out. In fact every day I was getting better and better and everybody was waiting to see if I would win.'\n\nRobinson has one word to describe the 1956 Tour, won by outsider Roger Walkowiak, and that is 'odd'. He says: 'Each day was like a blank sheet of paper. You had no idea what was going to happen. Breaks would go, but nobody really controlled anything.' The absence of Louison Bobet, winner of the three previous Tours, following a painful operation on a testicle, left a vacuum that no one was able to fill. In its post-race analysis, _L'Equipe_ commented: 'The Tour had a new appearance this year. No Bobet, because of his operation, and no [Jean] Robic because of an accident. Charly Gaul should have been the number one favourite, but he became overly involved in a fruitless combat with Stan Ockers for the overall and Federico Mart\u00edn Bahamontes for the King of the Mountains competition. Overall, the Tour lacked panache, particularly given \"Walko\" did not win a single stage.'\n\nBahamontes recalls: 'I was in the same breaks as Walkowiak, but as he was ahead of me in the overall I couldn't win. Still, which other rider apart from me could have been fifty minutes down and end up fourth, at ten minutes overall? I'm sure the Tour de France has never seen such a huge comeback. Everybody thought I could win.' However, the race was simply not long enough for Bahamontes.\n\nThings also did not go his way in the King of the Mountains competition. His tactic of attacking on the mountains, and then waiting for the pack, fell foul of a Tour that ran out of control and where breakaways were coming from unlikely scenarios. Bahamontes' strategy depended on a stage developing predictably, and the 1956 Tour was anything but that. _L'Equipe_ went so far as to describe it as 'hair-raising'.\n\nThough in poor physical condition early on, Bahamontes started to improve on stage four to Rouen, where he took third place in a short individual race against the clock, thereby proving that his time-trialling was better than people thought. However, as he points out: 'Even though the differences were not huge at that point overall, certainly not to start worrying about losing the race, we then lost a lot of morale in the next stage, when five of the Spanish crashed.' To make matters worse, on stage seven to Angers, a thirty-one-man attack, including Walkowiak, gained eighteen minutes on all the favourites. Another significant move on stage ten added nearly a quarter of an hour on the bunch. Bahamontes, significantly, was not part of either of them.\n\nAfter the first stage in the Pyrenees, Bahamontes found himself lying thirty-fifth overall, nearly fifty minutes behind race leader Andr\u00e9 Darrigade. Just the sight of Darrigade, a sprinter, donning yellow after a day that should have been reserved for the climbers confirmed Robinson's assessment that this Tour de France was indeed 'odd'. On paper the time gaps for riders like Bahamontes seemed insurmountable. Or as Bahamontes once rather poetically put it: 'They were as big as the distances between continents, and we had to row our boats day and night to try and arrive with enough time to conquer them.' However, that was only on paper. On the road Bahamontes pulled back four minutes on the second stage in the Pyrenees. Then he regained seventeen minutes in a five-rider break on a flat stage between Toulouse and Montpellier. 'My boat,' as he put it, 'could see land in sight on the continent ahead.'\n\nSuddenly back in the top twenty at a mere twenty-one minutes nineteen seconds behind new leader Jean Adriaenssens of Belgium, Bahamontes progress had not gone unnoticed. The prestigious _Miroir-Sprint_ magazine published an article entitled: 'And if Bahamontes were to win?' He pulled back another seven minutes on stage sixteen, another flattish day, and moved into thirteenth place overall. Andr\u00e9 Leducq, a Tour winner twice in the early 1930s and a team director in the 1950s, was quoted as saying: 'If he climbs as well as he did two years ago, Toledo may light up again . . . he's only eighteen minutes down and it wouldn't take a miracle for him to pull them back.' Bahamontes put it even more graphically. 'As I waited in the team hotel I couldn't stop ticking the minutes off for the next day's start in the Alps. I was sure that Fermina would light a candle for me when she heard it was getting so close . . . because what were eighteen minutes to a climber of my class?' Bahamontes, usually one of the most pragmatic individuals in the world, even went so far as to say: 'I will buy her a garage . . . she will no longer have to worry about the future. She is everything to me.'\n\nBut for all the emotionally hyped-up, gung-ho image Bahamontes presents of himself, it was not quite as straightforward as that. It was only logical that the closer he got to the yellow jersey, the nicer the things the press said about him, the less likely the rest of the pack were to let him go. Yet another massive break on a stage through the Alps to Turin enabled him to shave seven more minutes off the race leader's advantage, and move into ninth overall. At the same time, Walkowiak was proving a much trickier customer to shake off than everyone had anticipated. 'We never gambled on him being so dangerous, and we never raced together against him,' said Bahamontes. 'Still there was one big day in the Alps to go [the two hundred and fifty kilometres from Turin to Grenoble] and for sure Walkowiak wasn't sleeping easy in his bed the night before.'\n\nThe next day, Bahamontes attacked from the gun, taking Gaul with him. But he was predictably dropped on the descent of the mammoth Alpine pass, the Croix de Fer. Bahamontes finished in a three-man chasing group behind Gaul and moved up to sixth. However, it looked as though his poor downhill racing would cost him at least a place on the podium. It was during this stage that one of the most persistent of the Bahamontes myths emerged: that he threw his bike down a ravine in frustration at not being able to chase down Gaul. Bahamontes, however, denies its veracity. 'If I'd been able to stay with Gaul, everything would have changed,' he says. 'I'd have made up at least seven minutes on Walkowiak, but once again my descending skills let me down. Instead, Walkowiak got his moment of glory by making it into two breakaways on days when the big names forgot about him. He will always be remembered as the least deserving winner of the Tour de France.'\n\nThough 'winning _a la_ Walkowiak' has now entered cycling terminology as meaning 'to win from a flukey breakaway attack', not everybody agreed with Bahamontes' cold assessment. 'Little Walkowiak has beaten the biggest stars, so that makes him a \"big star\", too,' _L'Equipe_ argued. 'Maybe there were no great names in this race, but it was still a great Tour.'\n\nOne reason Gaul was not overly prepared to form an alliance with Bahamontes in the Alps had nothing to do with the overall classification. While Bahamontes had not made the King of the Mountains jersey a specific objective, with three days left there was just one point separating the pair at the top of the ranking. As a result, on the second last stage with classified climbs, when Gaul and Bahamontes _did_ get away together ahead of Walkowiak, there was anything but mutual collaboration. However, Gaul's refusal to do more than mark Bahamontes, and grab some more mountain points in the process, ended with the Spaniard gaining only another ninety seconds on Walkowiak overall. 'The King of the Mountains was decided on the eve of the race's arrival in Paris,' Bahamontes says, 'and Gaul formed an alliance with other riders so that I would not score any points.' In other words Bahamontes was outwitted. He ended up fourth in the overall classification and second in the King of the Mountains. But he was bullish enough to swear blind to anyone who would listen in the Parc des Princes, where the Tour finished, that in 1957 he would win the race.\n\nBahamontes' season petered out after that, so that 1956 became the only year of his career that Bahamontes finished with no victories at all. He dodged the Spanish Nationals because he was 'resting up for the World Championships', then abandoned them, claiming that the course did not suit him, though not before courting controversy by refusing to travel on the Spanish Federation's specially chartered flight. Finally, he again bypassed the Volta a Catalunya in favour of more profitable criteriums in France.\n\nThe highpoint of his year, in fact, had nothing to do with bikes. On 3 November Bahamontes married his longstanding girlfriend, Fermina Aguilar Sanchez. Like her new husband, Fermina had been working in the market in Toledo when they met. One of Federico's main charms for her, she informed the press, was that he 'always behaved like a real gentleman'. For example, she said: 'When we go out riding our bikes, he never overtakes me, not even on a climb.' A glance at the witness list \u2013 the Mayor of Toledo, the head of the Spanish Cycling Federation Alejandro del Caz, as well as his close business associate Evaristo Murtra \u2013 and the location (Toledo Cathedral) made it clear that socially Bahamontes was an established figure. In sporting terms, though, with three successive near-misses in the Grand Tours, his position was far less stable. Rather than mark a high point in the career of a young professional, 1956 established a pattern of unpredictability, isolation and blame games in Bahamontes' performances that was to dog the whole of his career.\nChapter Six\n\n#### Loro\u00f1o v Bahamontes \u2013 A Knife in the Table\n\nFederico Mart\u00edn Bahamontes seems to have fallen out with a large number of members of the Spanish peloton at one time or another. The most spectacular and longest-running dispute was with Jes\u00fas Loro\u00f1o. It turned into an eight-year duel which tapped directly into some of the most important divisions within Spanish society of the time and is still referred to as the most intense rivalry between two individual athletes in the history of sport in the country.\n\n'Bahamontes versus Loro\u00f1o is something that is hard to understand now, but at the time it was so serious it split Spain down the middle,' Josu Loro\u00f1o, son of the father, confirms between mouthfuls of croissant one rainy spring Sunday morning in a bar near Bilbao. 'It got to the point where there were families I knew who wouldn't talk to one another because they were either _Bahamontistas_ or _Loro\u00f1istas._ But I truly believe that just as Loro\u00f1o wouldn't have been so good without Bahamontes, so Bahamontes wouldn't have been so good without Loro\u00f1o. They were rivals, but they needed each other, too.'\n\nJes\u00fas Loro\u00f1o stopped racing in 1962 and died almost two decades ago, but he remains a star of the sport in the Basque Country, the heartland of cycling in Spain. If Loro\u00f1o is less famous internationally than Bahamontes or Miguel Poblet it is because he rarely raced abroad; when he did, though, he usually won as he showed in the Tour de France in 1953. Born in 1925 into a farming family in the village of Lattabetzu, close to Bilbao, Loro\u00f1o's first job at eleven was digging trenches in the 'Iron Belt', a line of fortifications built to defend the city in the first year of the Civil War. Five of his eight brothers and sisters were imprisoned in labour camps after the Basque Country fell to the Nationalists in July 1937, but Loro\u00f1o earned his living as a woodcutter. He became one of the region's top amateur cyclists, despite only being able to train at night (his mother used to threaten to take his woodcutter's axe to his bike in case he fell ill from sweating so heavily while out in the dark, cold Basque winters). Nonetheless, like so many Spaniards in the 1940s he thought his best option was to emigrate, in his case to Chile with one of his brothers. Before he could head to South America, however, Loro\u00f1o had to fulfil his military service. As it happened, his commanding officer was a cycling fan who not only encouraged Loro\u00f1o to train, but persuaded him to enter his first major race, the Subida a Naranco hill-climb in Asturias. Loro\u00f1o won, netting the considerable sum of seven thousand pesetas (\u00a33,500), and beat all the established domestic stars of the time. As a result he opted to stay in Spain and start a professional career in which he won the Vuelta a Espa\u00f1a and Volta a Catalunya as well as a King of the Mountains title in the Tour de France. He became Bahamontes' deadliest rival and even when they had both retired the mutual antagonism never really died down.\n\n'Fede was and sometimes is like a child,' Loro\u00f1o told one of Spain's top cycling reporters, Josu Garai, shortly before he died in 1994. 'I was stronger than him time-trialling, and he never beat me, not on the flat, not on the climbs, not even in hill-climbs which suited him perfectly because you started the climb \"cold\". Well, he only beat me once, and that was in a sprint I messed up. But he refused to recognise my triumphs.'\n\nIn the public's eyes, particularly in a cycling heartland like the Basque Country, Loro\u00f1o's most important success was winning the toughest mountain stage en route to the King of the Mountains jersey in the 1953 Tour. 'My father told me his King of the Mountains jersey had a huge impact,' says Josu, once a cycling journalist, and now a councillor for the Basque Nationalist Party in Getxo, a dormitory town of Bilbao. 'When he came back from France that year the celebrations were far bigger than when he won the Vuelta. The reason was that for the Basques and for Spain, France in the 1950s was like another planet. Even twenty years later when I was growing up in the 1970s you went across the border to buy things that you just couldn't get in Spain. As late as the 1980s, when I was a journalist covering the Tour de France I remember the telecommunications in France were light years ahead. So just imagine when some kid like my father goes over the border in the 1950s and beats the French at their own game.'\n\nHowever, Loro\u00f1o was unable to defend his King of the Moutains title in 1954. And who should take it but Bahamontes. 'My father couldn't go because he was laid up in hospital for three months after crashing in the GP Eibar when he collided with a motorbike coming the wrong way. Jacques Goddet [the Tour director] had called my father his favourite [to win the Tour outright] that year, and that was his biggest regret: not being able to go to the Tour that year, far more so than in 1959.' Josu says the antagonism between Bahamontes and his father did not get serious until the 1955 Vuelta. But the fact Bahamontes had identical aims for the 1954 Tour as Loro\u00f1o had the year before, and succeeded in claiming the King of the Mountains jersey while its previous owner lay in a hospital bed, all added greatly to the grist of the pair's rivalry.\n\nSport apart, there was also a major social divide between the two. For the Basques, Loro\u00f1o was a well-built, no-nonsense, working-class hero from Bilbao's industrial and agricultural hinterland, who was taking on the more eccentric, erratic wisp of a climber from some dusty city down south. Certainly their personalities could not have been more different. Loro\u00f1o was dour, pragmatic, straightforward and a direct kind of talker. Bahamontes, on the other hand, was far more Quixotic. Or as _El Pa\u00eds_ described him at the age of eighty: 'Affable, unique, contradictory, ultra-individualist and daring . . . as agile as an adolescent and as restless as a hyperactive child.' The biggest difference, though, was political. Loro\u00f1o's family were diehard Basque Separatists, whereas Bahamontes, from authoritarian, central Spain, was so popular with Franco he even met him personally at least once. As a result cycling's two leading figures came to symbolise one of the most damaging, deep-rooted divisions within Franco's Spain: the Basque Country versus Madrid.\n\nIf the 'Basque question' remains an important issue in Spanish politics even today, it was one of Franco's biggest headaches during his forty-year dictatorship. Of the eleven states of emergency Franco had to declare in Spain, six were in the Basque regions of Vizcaya and Guipuizcoa. Until a definitive ceasefire was declared in October 2011, the fifty-year drive by Basque terrorist group E.T.A. for an independent Euskal Herria was one of western Europe's most deeply ingrained and bloodiest conflicts. E.T.A.'s existence has historical links to the repression that followed the Civil War when the Basques were on the losing side. Despite being one of the most religious areas of Spain, which should in theory have placed them alongside Franco in the war, the political strength of the Basque separatist movement was such that two of the three Basque Country regions, Guipuizcoa and Vizcaya remained loyal to the Republic. Though they were cut off from the rest of Republican Spain from the earliest days of the war, the Basques held out for almost a year in Vizcaya. They pinned down a large number of Franco's troops and eased the military pressure on the main Republican strongholds. If their spirited resistance during and after the Civil War was not enough to make the Basque Country unpopular with the ruling regime and its supporters, the differences between Spain's most northerly region and the rest of the peninsula were far more extensive than whether the Basque _ikurri\u00f1a_ flag replaced the Spanish tricolor over the town halls of San Sebastian and Bilbao.\n\nThose differences certainly outweigh any similarities. Take economics, for example: for decades many of Spain's traditional 'heavy' industries like mining, ironworks or shipbuilding were concentrated in the Basque region. That meant the Basques were far better off than the agricultural centre and south. Half a century ago, John Hooper said in his book _The Spaniards:_ 'To visit the Basque Country is to take a trip in time back to the industrial North of Europe earlier this century. Its belching smokestacks, grimy buildings and dogged, pasty-faced inhabitants are the very stuff of Lowry paintings.' That was a far cry from the austere Castillian fortress towns like Toledo and the rugged, dry sierras of Teruel and Extremadura or the semi-deserts of Andalusia.\n\nBut the Basque Country had cultural and legal differences as well as its own language, Euskera. Deep into the twentieth century the Basques retained their traditional right to veto Spanish law and numerous other economic and social privileges. These _fueros_ created envy and resentment between the Basque Country and the rest of the country and Franco banned them as soon as he gained power.\n\nSport, meanwhile, has always been vastly more popular in the Basque Country than in the rest of Spain. But the political implications of sport have always been close to the surface in the north. For reasons no one can quite fathom that has been particularly true of cycling. It was not just a question of one team versus another or one individual versus another: entire races could fall foul of the political tensions. After Basque separatist violence caused the Vuelta a Espa\u00f1a's final stages to be all but cancelled in 1978, Spain's biggest race did not return to the region until 2011. The fears that the race might be subject to terrorist attacks was no exaggeration: in 2005 E.T.A. threatened to attack it with rocket-launchers.\n\nIt was, therefore, only natural that the Loro\u00f1o versus Bahamontes rivalry intermingled with the Basque Country versus Madrid conflict. For central Spain the fact Bahamontes came from the town of Toledo just outside Madrid, combined with their ingrained dislike of the Basques, automatically made him their champion. Bahamontes might have been a shade temperamental at times, but he was clearly capable of taking on Loro\u00f1o and the Basques and that was largely all that mattered. Josu Loro\u00f1o is convinced that Spain's backward economic circumstances, as well as the political situation, fanned the flames of the rivalry. 'They fought each other during a crucial period for Spain \u2013 the post-Civil War era, a time of extreme [political] repression. Society's only safety valve was sport. It was the opium of the people.'\n\nOther countries had damaging rivalries between their top cyclists, particularly France, and in 1958 and 1959 Bahamontes would take full advantage of that. However, as Josu points out, none of those rivalries had the breadth of Bahamontes versus Loro\u00f1o, which had social, political, economic and sporting conflict rolled into one. Loro\u00f1o's son believes the closest example would be that between Coppi and Bartali which divided Italy in the 1940s and early 1950s. But, in comparison, Loro\u00f1o versus Bahamontes was wilder and less stage-managed than the elegant face-off between their Italian equivalents. It is hard, for instance, to imagine the genteel Bartali slamming his dinner knife into the table in front of Coppi if the two of them had raced in the same team. That is apparently what Loro\u00f1o did as a 'conversation opener' when he was once having dinner with Charly Gaul and Bahamontes during a Giro d'Italia. The 'knife in the table' anecdote, more reminiscent of a bad Western than cycling, has never been confirmed. But that was the whole point of Bahamontes versus Loro\u00f1o: it was a conflict with its own laws of engagement that went beyond everyday reality. Whatever outrageous stories the press made up about them, no one could prove that they were not true. 'Nowadays when you can see the whole stage of the Tour de France on television their rivalry couldn't have got so big,' Josu Loro\u00f1o comments. 'Back then, though, when the only way people could get information was by going to the bar or shop that had the only radio in their street or village, it was far easier for journalists to make things up \u2013 not necessarily lie, but to imagine stuff, fantasise, exaggerate. Actually, the press and the fans were behind quite a few of their so-called disputes. Neither of them complained, though.'\n\nQuite apart from the mutual dislike there was also fear in the case of Bahamontes. Josu claims that Bahamontes was sometimes so terrified of his father during their racing careers that he would hide in his hotel room to avoid him. As for Jes\u00fas Loro\u00f1o, he considered Bahamontes to be 'mentally weak . . . sitting in his room saying, [and Josu puts on a fake falsetto] \"Loro\u00f1o wants to hit me, I'm not coming out, Loro\u00f1o wants to hit me\".' Yet the two were not always on such bad terms. Josu Loro\u00f1o first encountered Bahamontes at the restaurant in Bilbao which his father opened after retiring, and where Bahamontes would sometimes drop in for a meal if he was in town. There were even times, he claims, when Bahamontes spent a few days as Loro\u00f1o's house guest. However, media-wise, any examples of friendliness between Loro\u00f1o and Bahamontes were brushed under the carpet for the simple reason that the rivalry was a newspaper reporter's dream.\n\nAnd Bahamontes and Loro\u00f1o fulfilled their roles to perfection: they cheerfully traded insults, refused to work together on the national team, and accused each other of betrayal, cheating and foul play. When it exploded it was pure dynamite. Physical aggression among cycling fans has traditionally been all but non-existent. Bahamontes versus Loro\u00f1o, though, was the exception that proved the rule. When Bahamontes was selected for the Tour and Loro\u00f1o was dropped, his fans threw stones at the national trainer's shop in Bilbao, spat at his wife and sent hate mail. On at least one occasion they tried to beat him up. In turn, contemporary photographs show Bahamontes' supporters parading in the streets wearing shirts with the words 'R.I.P. Loro\u00f1o' stitched in black ribbons on the front. Josu recalls: 'With time, it got very complicated and more divisive, not less. Families would be split down the middle because Loro\u00f1o had \"rebel\" supporters in the rest of Spain and Bahamontes had some in the Basque Country. For example, there was a well-known sports writer, Francisco G. Ubieta, who wrote for the _La Gaceta del Norte_ newspaper here in the north. He was a sympathiser of the Franco regime, but at the same time he was a real Loro\u00f1o fan.'\n\nHowever, over time, as happens with most conflicts, the original reasons for their falling-out became forgotten and the rivalry became self-perpetuating. But did it ever, to Josu's knowledge, get seriously violent between the two? 'Well,' he replies '[they say] there was the time that they hit each other with bike pumps, and what I have heard is that once he grabbed Bahamontes by the throat. That's what they say, anyway . . . and then there was the time Bahamontes refused to come out of the room because he was scared my father would hit him. Some kind of row, I imagine.'\n\nWhat really separated Loro\u00f1o and Bahamontes was that they could not have been more different as people and were therefore almost doomed to rub each other up the wrong way. Bahamontes refuses to discuss Loro\u00f1o. 'What's the point, if nobody's heard of him in France or Italy?' Loro\u00f1o was equally contemptuous of his rival. 'Bernardo Ruiz has told him more than once, \"Loro\u00f1o beat you fairly\",' Loro\u00f1o said in 1994, 'and it's about time Bahamontes recognised it.'\n\n'They had very different personalities,' Josu Loro\u00f1o says. 'One of them was what we would call madness personified, the other would always play it straight down the line. Perhaps my father was a bit too stiff-necked at times, but in any case he was the complete opposite to Bahamontes. My father was the kind of guy who when he said something was \"A\" it was \"A\" even if he was wrong.' Loro\u00f1o adds later 'He went so far as to say Bahamontes was weak \u2013 mentally weak \u2013 that he had no character.' And Josu says Bahamontes had a second problem with Loro\u00f1o: 'Bahamontes suffered from Loro\u00f1itis: when he saw my father he used to shit himself. My father dominated his mind.'\n\nNot only was 1957 the year that Bahamontes' conflict with Loro\u00f1o reached its peak, he was also in serious trouble with his latest team. Bahamontes had quit Faema, Spain's most prestigious squad, for whom he had barely raced in 1956, for a much smaller one, Mobylette. He had been convinced that he was the victim of a persecution campaign at Faema, only to find exactly the same problems with Mobylette. 'I left Faema because I was supposed to be leader and the mechanic was puncturing every wheel I had.' But when he moved to Mobylette he alleges that Loro\u00f1o only won the Volta a Catalunya 'because they paid off everybody else, including my team. The director I had was bought off by Miguel Torrello, the Faema boss,' he says, 'and then when I punctured five or six times on one stage, I can remember the Faema trainer laughing about what had happened.'\n\nIt would be easy to write Bahamontes off as being paranoid, but there were incidents that suggested he was the victim of a loosely organised conspiracy. The Volta a Catalunya of 1955 was a case in point. On stage six, Jim\u00e9nez Quiles was in a break with Loro\u00f1o and Bahamontes, and he says the Basque rider did everything to convince him to work against Bahamontes. 'I got into a move with the two of them with about twenty kilometres to go and they started making long, long drives on the front, much bigger than the ones they do nowadays,' says Jim\u00e9nez Quiles. 'It looked like they were trying to wipe each other out. Finally I got away ahead of both of them and my director came up to me on a descent and told me to stop because Loro\u00f1o had dropped Bahamontes. When Loro\u00f1o caught up he persuaded me that I should work with him and let him win, purely because Bahamontes was behind, and he wanted to beat him so badly, and so I did. Jes\u00fas gave me the trophies and all the prizes from that stage \u2013 the trophies are over there in that cabinet if you want to look at them \u2013 because he said they belonged to me anyway. All he cared about was beating Bahamontes.'\n\nThings were no better in the national team. 'Fede had more enemies in the national team for the Vuelta a Espa\u00f1a in 1957 than out of it,' Bernardo Ruiz recalls. 'The line-up consisted of all my team-mates who I had recruited to get on the squad. Then they kicked me out to make way for Bahamontes because I was apparently past my sell-by date at thirty-three. So there he was, alone in a team and surrounded by all my team-mates.' At this point the simplest solution would have been for Bahamontes to get out his wallet and buy himself some allies. But Bahamontes' refusal to do so, perhaps because a man who had endured such poverty in his early years might be reluctant to pay for loyalty, let him down dearly. 'The entire team was effectively my Faema squad and Bahamontes wouldn't spend anything to gain the team's sympathy. He was a magnificent rider, but that was a very serious defect,' Ruiz says, particularly since it was considered almost normal at the time to buy and sell races.\n\nBy 1957, Bahamontes' enemies had two natural magnets in Ruiz and Loro\u00f1o. Just like the bulk of the country, the Spanish peloton divided into _Loro\u00f1istas_ and _Bahamontistas._ Barring a very few key Bahamontes _domestiques_ , like Julio San Emeterio, and a handful of notable 'neutrals' such as Miguel Poblet, Loro\u00f1o had far more supporters in the bunch. These included Ruiz and his young ally Salvador Botella, already the winner of the Volta a Catalunya back in 1953. One catalyst in the Loro\u00f1o-Bahamontes rivalry was Spain's team trainer, Luis Puig. An avuncular type given to bland, meaningless statements of goodwill that proved ideal for a political career that eventually led to him becoming president of the U.C.I., cycling's governing body, Puig was totally incapable of handling his two top riders. Instead, Puig relied excessively on Ruiz, who, unbeknown to him, was in the Loro\u00f1o camp, and then promptly washed his hands of any major conflict. It took the Spanish Federation two years, and a huge scandal in the Tour de France, to realise that with Loro\u00f1o and Bahamontes in the line-up, someone willing to bang heads together would be far more appropriate for the job.\n\nMeanwhile, the Loro\u00f1o-Bahamontes conflict continued unabated to the point where their _domestiques_ lined up and took pot shots at their leaders as well, even 30 years after the event. 'Bahamontes is the kind of bloke that if somebody he dislikes goes on an attack he goes after him even if it's flat and he [Bahamontes] is chasing all by himself,' Jes\u00fas Galdeano, one of Loro\u00f1o's most loyal supporters, told _El Mundo_ in the 1980s. 'He doesn't believe in his own possibilities. He wins in the mountains and gets half an hour, then he loses half an hour on the flat and blames everybody else.' Poblet, a rival but also a friend of Bahamontes, says: 'Fede wasn't a bad guy. He didn't kill anybody. He was always a happy sort, but he behaved differently to most other people. His personality was different. He was wilful, too: when he wanted to do something he'd do it whether it was right or wrong. I can remember we were in the same team, with Charly Gaul, in a Tour of Italy [1958] and there was a stage finish with a ten-kilometre climb just before, then a descent, then a few kilometres of flat. And I said to him, \"Don't attack before the climb, wait until there are a couple of kilometres to go, I'll watch your back, and then if I can, I'll win the sprint from the group behind\". Then what does he do? He attacks right at the start of the climb, gets a minute or a minute and a half, then gets caught on the descent by Anquetil and the rest. Fortunately I won the sprint, but he finished dead last when he could have won. Loro\u00f1o was much straighter, at least with me. But Bahamontes would tell you one thing and then the next day he'd go and do another.' Jim\u00e9nez Quiles adds: 'In any case, what did he expect? Those days if you were in front and you saw a team-mate attack, you'd know it was up to you to go after him.'\n\nIronically, Bahamontes' 1957 season got off to a flying start when he outsprinted Loro\u00f1o for victory in a Tour of the Levante stage \u2013 the one Loro\u00f1o said he messed up \u2013 and then took his second win in three years in the Mont Faron hill-climb in France. The Vuelta a Espa\u00f1a started equally well, too, despite Loro\u00f1o gaining a couple of minutes' advantage from a break on stage one. Forty-eight hours later Bahamontes bounced back with a vengeance to claim his first stage win and the leader's jersey after a solo attack in the mountains of Asturias. Loro\u00f1o, on the other hand, was at nearly thirteen minutes, tenth overall and itching to counter-attack for all that he and Bahamontes were in the same squad. Ruiz, meanwhile, exiled to a regional squad but with his usual Faema team-mates filling out the Spanish 'A' team, was watching and waiting in the wings. Knowing that Bahamontes hated the cold, Loro\u00f1o took advantage of a snowstorm en route to Le\u00f3n on stage four to move ahead. However, with the weather conditions becoming worse on the Pajares pass the organisers decided to suspend the stage. Loro\u00f1o was so furious at his misfortune that he continued pedalling uphill and had to be forcibly removed from his bike.\n\nBahamontes lost the lead briefly in Madrid to another team-mate, Botella, a close ally of Ruiz's. This was the first time that Bahamontes was indisputably let down by his team, and in particular Puig who failed to inform him of the sizable gap that Botella and other breakaways had gained. When Bahamontes found out, it was only thanks to the race organisers, not his team. He counter-attacked almost instantly, though not before having a shouting match with Puig for failing to tell him earlier. Just to complicate matters, when Bahamontes punctured the next day on the Navacerrada climb on a stage around the sierras of Madrid, Loro\u00f1o voluntarily gave him a spare wheel. Though it was in both their interests that Botella did not gain too much time, Loro\u00f1o later regretted this, telling the _La Gaceta del Norte_ newspaper: 'I was the idiot who saved Bahamontes in Navacerrada.'\n\nBahamontes was determined not to let the race slip through his fingers, and bounced back brilliantly two days later when he asked for and received support from the French in a break on the way to Cuenca. By allowing Roger Walkowiak and his compatriot Gilbert Bauvin to take first and second places, Bahamontes moved back into yellow. Loro\u00f1o, meanwhile, had dropped to eleventh and was nearly sixteen minutes behind. Puig confirmed to the press that Bahamontes would be the team leader from now on while Loro\u00f1o went for the King of the Mountains. All went well on the long drop down from the tablelands of Cuenca to the coast at Valencia, but the next day on the coast road to Tortosa, Bahamontes was to discover just how friendless he was.\n\n'I whispered in Loro\u00f1o's ear, \"Come with me\",' is how Ruiz explains the start of the mass attack in which Bahamontes lost twenty-one minutes and the yellow jersey to his arch-rival. Poetically appealing though Ruiz's recollection of events is, the truth is slightly more mundane. Ruiz and two other riders powered up the coast road in the first hour, and when they had opened a twenty-second gap Loro\u00f1o came across with four others. Then the race was on. But there was no doubt who the move was directed against. Another 1950s Spanish professional, Luis Ota\u00f1o, recalls Loro\u00f1o telling him: 'During the break Ruiz was yelling, \"Go, go, Jes\u00fas, Bahamontes is dropped and you're going to win the Vuelta. You're better than him, fuck it, go, ride!\"' Moreover, it was rumoured that Bahamontes' team-mates participated in the conspiracy. Some reports claimed that rather than help their stranded leader, they went as far as holding on to his shorts to prevent him counter-attacking. Certainly he received little support. 'With Loro\u00f1o ahead, and with all my Faema team-mates and friends in the national squad behind, nobody would work for Bahamontes,' Ruiz recounts with a hint of smugness. 'Bahamontes was the leader, he could have won that Vuelta, he should have won it, but he didn't know how to win over his team-mates.' Evaristo Murtra, Bahamontes' 'godfather', who was following the race, confirmed: 'He had no support whatsoever. Nobody wanted to help him, and the last anybody saw of Puig was when the break had gained two minutes, and he shot past in his jeep indicating with his hands to the riders that they should take it easy and not work to bring back Loro\u00f1o's break.'\n\nRuiz says: 'So we reached Tortosa miles ahead, [Italian Bruno] Tognacini gets the stage, Loro\u00f1o got first overall, I moved into second and Tognacini third, and that's when the scandals started. Even the Federation had to intervene.' It was a brutal, barefaced betrayal, but even fifty-five years on, Ruiz has no regrets. 'Fede says that we robbed him of that Vuelta. But he was vulnerable, he let himself lose it. And the ones who knew how to race, well, just like in this Vuelta, we beat him.' As for Loro\u00f1o, he claimed to be surprised at his success, but equally pitiless. 'I never thought I'd get the yellow jersey,' he says. 'From two-thirds of the way through the stage it was pretty clear what was happening. But if Bahamontes couldn't do anything then it was because he couldn't.'\n\nSomewhere in Bahamontes' office, amid all the rows of photograph albums and collections of newspaper cuttings, lurks a piece of paper that allegedly explains why he did not win the Vuelta in 1957. Given that at one point Bahamontes had an advantage of nearly sixteen minutes over Loro\u00f1o it seems almost more difficult for Bahamontes to have lost the race than to have won it. However, that was what happened.\n\nBahamontes' hand thrums on one knee and a ferocious glare comes over his face as he gives his explanation for snatching defeat from the jaws of victory. He says he was 'robbed' by a government telegram, received while in Valencia, instructing him to throw the race in favour of Loro\u00f1o. In the middle of a military dictatorship there was no ignoring an official communication. That the rider who benefited happened to be his sworn enemy only added insult to injury. According to Bahamontes the instructions came from 'above', which is how Spaniards refer to the higher echelons of power. At the very least, Bahamontes says, it was the Minister of Sport Jos\u00e9 Antonio Elola who put his name to the telegram.\n\nWhoever decreed that Bahamontes should lose, if, indeed, such a decree was made at all, the logic behind the decision would have been purely political: Loro\u00f1o was Basque, the Vuelta was Basquerun, and Euskadi was the most troublesome region Franco had to deal with. Rather than let someone from the Madrid area win, Bahamontes believes that Franco's henchmen let the Basques get one over the centre of Spain to satisfy local pride. In some ways it was like the Roman emperors who kept the noisiest parts of the populace happy with 'bread and circuses'.\n\nIt is an extraordinary claim to make, but Bahamontes says he still has a telegram from the Ministry of Sport to prove it. He has certainly kept others, like the telegram from Toledo town hall promising him some prime building land in the city after the Tour victory in 1959, which he is always quick to locate in a photograph album for anyone curious enough to want to see it. Yet when asked if it is possible to see the document that proves he lost the Vuelta as a result of political manoeuvring, he says 'sure, sure', half-gets out of his seat, looks around and then slumps down again, as if he cannot be bothered. Or maybe he knows that the telegram cannot be found.\n\nEven so, he is still adamant about his side of the story. 'It was all because of a telegram from Jos\u00e9 Antonio Elola telling me that even though I was leader Loro\u00f1o had to win,' Bahamontes says now. 'I'm convinced Puig had a part to play in that, too, because Elola was staying in Puig's chalet at the time as a guest. In any case, the telegram said I had to settle for second and the King of the Mountains, and let Loro\u00f1o win because he was Basque and the race was Basque. I had to take it or leave it.' When interviewed a month after the Vuelta Bahamontes claimed: 'I didn't move because Loro\u00f1o was ahead.'\n\nThe press was equally confused by what happened on the stage to Tortosa. 'I've been told Bahamontes was under orders not to [counter-] attack,' commented _El Mundo Deportivo_ 's correspondent, 'and I admit it may be true. I can't understand what happened, though, and why a rider who fought so hard to get the jersey in Madrid was so prepared to sit back and let it go out of his reach so calmly. Isn't there one of Aesop's fables where somebody spends years and years putting together a fortune, then calmly throws it all down a mineshaft? That's what Bahamontes did yesterday with his yellow jersey.'\n\nHe was not the only reporter who suspected a possible government connection to the race's outcome. In the cycling gossip section of _La Regi\u00f3n_ , an Oviedo newspaper, an anonymous journalist going only by the name of 'Ricardo' wrote a one-line reference to the Tortosa stage, saying: 'Is this sport? Let us salute our \"comrade\" Elola for this case.' The correspondent of the pro-Loro\u00f1o _La Gaceta del Norte_ took a completely different point of view, running a story headlined: 'Puig-Bahamontes conspiracy against Loro\u00f1o'. But if there was such a plot then it is fair to say Puig and Bahamontes resoundingly lost it. _L'Equipe_ seems to have had a clearer idea of what Loro\u00f1o's attack represented. They ran a one-word headline for their Vuelta coverage the next day: 'Heresy'.\n\nRegardless of whether he messed up the race of his own volition or was annoyed because he had been told to throw it, newspaper reports claimed that Bahamontes spent most of that night's meal in the team hotel in Tortosa flinging insults at all and sundry. Finally, Loro\u00f1o grabbed Bahamontes and demanded to know what his problem was. At that point Bahamontes apparently fled to his room, asking that his food be sent up and refusing to come down while Loro\u00f1o was around.\n\nThough one journalist claimed that 'the Valencia-Tortosa stage was the execution wall for Bahamontes, and Loro\u00f1o and Luis Puig were the ones firing the rifles', in fact things were not much better between team director and Loro\u00f1o. While no friend of Bahamontes, and after failing to ensure the support of his team-mates, it later emerged Puig had sat even more firmly on the fence by trying to stop Loro\u00f1o from attacking, too. 'Puig kept blocking the road with his car, telling Loro\u00f1o he had to sit up,' Josu Loro\u00f1o recalls, 'but my father would dodge past, saying he was having none of it. Puig even shouted at him to stop at one point, and my father yelled back, \"Not even if the Civil Guard come and get me!\" And in those days, that was quite something to say about the _Guardia Civil_.'\n\nThe following day's stage between Barcelona and Zaragoza went calmly enough until ten kilometres to go when the Loro\u00f1o-Bahamontes war broke out again. Bahamontes went on the attack and Loro\u00f1o, who had been waiting for the move, immediately chased him down. Still riding a few metres ahead of the bunch, the two then began arguing with each other at the tops of their voices. Their row enabled a number of top foreign riders to take advantage and go clear. Indeed, by Zaragoza the leaders had gained a minute's lead on the Bahamontes-Loro\u00f1o group. Neither seemed to care. Loro\u00f1o then continued to claim to all and sundry that Bahamontes had been attacking him 'non-stop' since Barcelona, and 'that was how he paid me back for the help I gave him on the stage to Madrid'. The minor matter of his attack on Bahamontes while he was leading the race at Valencia was, strangely enough, never mentioned by Loro\u00f1o.\n\nTension levels in the Spanish 'A' team reached the point that after a crucial eighty-one-kilometre time-trial from Zaragoza to Huesca, the Spanish Cycling Federation decided they had to intervene. Their warning that both riders would be excluded from the race unless the insults stopped, coincidentally, arrived by telegram. A truce was officially agreed and toasts were drunk to mutual loyalty for the press's benefit. Loro\u00f1o and Bahamontes even shook hands publicly, and the media dubbed it 'the Huesca Pact'. Nobody was fooled, though, except perhaps the Federation; behind the scenes the war continued unabated.\n\nLater, the organising newspaper _El Correo Espa\u00f1ol_ ran a story in which it claimed Puig had taken the two riders into his hotel room, showed them the Federation telegram and told them in no uncertain terms to stop fighting otherwise they would both be excluded. He also apparently told Bahamontes, _sotto voce_ , that if Loro\u00f1o did not win the Vuelta, 'there was no way either he or Bahamontes would be allowed inside Bilbao [where the race finished]'. To add to the intrigue, it later emerged that Bahamontes tried to abandon the race the same night, and was only persuaded not to by his 'godfather', the Catalan businessman Evaristo Murtra.\n\nBahamontes rode flat out in the time-trial the following day, using a twenty-eight-spoke front wheel from a track bike, despite them being prohibited for road-racing. He was given a one-minute penalty which dropped him from second, six seconds behind Loro\u00f1o to third in the time-trial. But he had the minor satisfaction of ousting Ruiz from second place overall. 'They told me those wheels were banned, but then why were the French team all using them?' Bahamontes complained to me more than fifty years later. Loro\u00f1o bitterly criticised Bahamontes at the time-trial finish, saying: 'It's really annoyed me that he uses a wheel that he knows is illegal, even telling the race officials about it, because he'll get more benefit from using it than whatever he was going to be penalised. I'd have rather died on the road than lose the yellow jersey to him that way.'\n\nFor the last three days, with nothing to lose, Bahamontes attacked as and when he liked. However, given Loro\u00f1o and Ruiz were now firm allies, he was effectively fighting against two teams not one. Furthermore, Ruiz was happy to tip off Loro\u00f1o if Bahamontes made a move. 'Bahamontes was officially going for mountains points whenever he attacked, but there was one time when he got a one hundred-metre advantage on a climb,' Ruiz revealed to me. 'I told Loro\u00f1o, \"Don't trust him, he only needs to be twenty metres ahead, not leading by minutes\". And so Bahamontes was chased down. They kept on insulting each other all the time, though. It was incredible. Like something out of a film.'\n\nBahamontes rode brilliantly in the final mountain stages, taking out his rage at losing the overall title by attaining his first Vuelta King of the Mountains title with ease. Not that Loro\u00f1o or Ruiz were prepared to let him get too much of an advantage. On the stage to Bayonne Loro\u00f1o chased down every charge by Bahamontes, 'who attacked and attacked and attacked', according to _El Mundo Deportivo_ , 'turning around to see each time what Loro\u00f1o could do'. The report continued: 'But Loro\u00f1o went from strength to strength, hammering on the pedals every which way in order to be totally sure he was gaining the maximum distance possible. When he brought back Bahamontes to fifty metres, Bahamontes sat up, looking at him with a mixture of hatred and admiration. Loro\u00f1o kept on baring his teeth as if to show that even if he ran out of physical strength he'd defend himself by biting.'\n\nAgain, Bahamontes' apparent change of mind over the instructions to lose the race so late on raises yet more doubts about the existence of a telegram that nobody, barring Bahamontes, has apparently seen. However, it is difficult to conceive another explanation. Why should a rider who was in total control of a race suddenly allow a rival he loathed to gain such a huge margin and throw it all away, even if then he changed his mind again and tried to get it back? Ultimately, there are four possible explanations for Bahamontes' behaviour on the Valencia-Tortosa stage: extreme carelessness; a sudden attack of indifference; lack of team support; or external orders, perhaps from Elola. Whichever reason is the right one, fifty years on it is pretty clear that Bahamontes' bitterness endures. Furthermore, given he cannot turn the clock back he has no choice but to belittle the entire event. 'The Vuelta?' Bahamontes says. 'In those days it was just a bit of a joke.'\n\n'That telegram?' says Josu Loro\u00f1o. 'If he's got it, then he should produce it, which he never has. The day he shows somebody that telegram, that's an end to any arguments.' Until he does, Loro\u00f1o has his own interpretation. 'It's basically sour grapes. He's pissed, too, that he never won the Volta, the Vuelta or the Bicicleta Vasca. He's got a magnificent _palmares_ , but in that area he feels sore.' Loro\u00f1o argues that from a political point of view Bahamontes' claims are illogical. 'Given what my father was, from a Basque Nationalist family with two brothers in exile and a third doing time in a hard-labour camp in Cadiz because of the Civil War, the ruling regime would have far prefered Bahamontes to have won. Nobody can deny Bahamontes' achievements which are huge. But at the same time, he shouldn't deny what other riders achieved. It's all \"me, me, me\" and the rest are a heap of shit.'\nChapter Seven\n\n#### The Dangers of 'Go-Fast'\n\n7 July, 1957, Eastern France: In many ways it is the least significant of acts. In a quiet country lane, a bike rider draws up on a roadside, dismounts, takes off his watch and his shoes, and sits down on the ground. Yet how can it be insignificant? Federico Mart\u00edn Bahamontes, the race's big favourite, is abandoning the Tour de France. Instantly a huddle of journalists, spectators, race officials and team personnel crowd around him. They offer him water \u2013 it was already 36 degrees when the stage started in Besan\u00e7on a couple of hours back in one of the worst heatwaves of the 1950s \u2013 but Bahamontes refuses it. 'It's over, I'm not going on,' he insists.\n\nNot everybody is convinced: a Spanish team mechanic attempts to put his race shoes back on his feet, but Bahamontes grabs them from him and flings them into a ditch. The same mechanic then attempts to pick Bahamontes up and physically put him on the bike. 'Fuck, I said I'm not going on,' Bahamontes yells at him. He leans so heavily on the mechanic that he has to let him fall back down on to the ground.\n\nWhen his sports director Luis Puig gets out of his car and demands an explanation, Bahamontes says nothing but points at a swelling in his left arm. It has been caused, it will later emerge, by an injection given to him by Puig himself. Bahamontes cannot steer his bike, he claims, though that has not stopped him from taking part in the first break of the day.\n\nGrainy photographs of the scene show a glassy-eyed Bahamontes drawn up into a semi-fetal position, lying at the centre of a crowd which is steadily growing in size and variety. Two of his Spanish team-mates, Carmelo Morales and Antonio Ferraz, see his bike flung on the ground, stop, and plead with him to get back on into the race. 'Do it for the sake of your mother, Fede!' Ferraz shouts at Bahamontes over the heads of the crowd. Back from somewhere in the middle of the knot of people comes the muffled but increasingly determined answer: 'No.' 'For Fermina!' 'No!' 'For Spain!' 'No!' Then Ferraz plays his last card, and theoretically the best: 'For Franco!' Again Bahamontes spits out the answer: 'No!' He then picks himself up and gets in the race's 'broom wagon', the vehicle for riders who have abandoned.\n\nMeanwhile, Puig's initial sympathy has been replaced by anger. He refuses to believe that the injection is the real cause of the abandon and he attempts to provoke Bahamontes by claming he is not 'man enough' to continue. 'Go on then, quit,' Puig sneers. 'But when you're getting changed out of your clothes and putting on your trousers, take a look down at your chest and you'll see you've grown tits!' Bahamontes just ignores him and the broom wagon heads off: a spectacularly undignified exit for a Tour favourite, and arguably one of the most melodramatic and farcical in the race's history.\n\nLong-term, Bahamontes' departure will force Spanish cycling's top brass to finally face up to the country's chronic lack of unity and carry out a much-needed overhaul of the Federation. Above all it will produce a change of national team director, from Luis Puig to Dalmacio Langarica, from a director with whom Bahamontes was incapable of collaborating and who oversaw his most humiliating abandon of any bike race, to one whom he described as 'the best I ever had'. In fact, one of the most important seeds of Bahamontes' success in the 1959 Tour, the crowning moment of his career, is sown on a day when success is probably the last thing on his mind.\n\nBahamontes was interviewed a few days later in a hairdresser's shop in Toledo by the newspaper _Informaciones._ 'With the \"closed\" sign on the door and the guy in the chair, where better place to have a chat?' the reporter shrewdly observed in his article. During the conversation Bahamontes claimed that he had no choice but to abandon. '[Race director Jacques] Goddet told me that even if I reached the finish outside the time limit, I'd be allowed to continue,' he said. 'And I wanted to continue but my arm was far too swollen and painful. I had fever, sinusitis and laryngitis as well. I had to ride with my arm on my _musette_ [food bag] for the first eighty-five kilometres, the only part of the stage where things were calm. I didn't abandon for good reasons or bad reasons, I abandoned because I had no other choice.'\n\nNot everybody believed him. 'He was playing chicken,' Bernardo Ruiz told the Mayor of Barcelona, Juan Antonio Samaranch, a few days later. 'I've had those injections and they hurt a bit, but not that much. In any case, Fede is stronger with one arm than most of us are with two.'\n\nThe injection Puig had administered the night before was variously reported as containing lime, calcium, vitamins B and C or a combination of all four. The problem was not the mixture of medicines themselves, outlandish as it sounds. Rather, according to Bahamontes, Puig messed up the injection, jamming the needle in twice, which caused severe swelling and pain in his entire arm. In consequence, he could not control his bike, so had to stop.\n\nBut quite why Puig had given Bahamontes the injection in the first place was harder to establish. According to Puig, Bahamontes' morale was low and he needed a boost. And as he cheerfully, and very unwisely, told journalists: 'It's perfectly natural for me to give the riders injections.' The U.C.I. did not see it quite like especially since in France only doctors were allowed to inject patients. Nor were the Tour de France organisers impressed; even if the police cast a blind eye to some laws being infringed by the Tour, the only doctor allowed to administer injections was the official medic, Pierre Dumas. As a result of Puig's bragging, on the same day that Bahamontes abandoned Goddet told the Spanish Federation to make sure Puig 'did not return to the Tour next year, because he is unwelcome'. Politely worded as Goddet's message was it was still a bombshell: being unable to direct the national team in the biggest race in the sport meant Puig's career as national coach was over, and the French knew it. And Bahamontes? 'Federico Bahamontes has fallen from grace,' Goddet later wrote in his usual poetical vein in _L'Equipe_ , 'just at the point when he was the sole favourite of the race, and his departure leaves the way open to new challengers. It's perfectly logical that Federico suffered as a result of the injection his sports director gave him in his left arm. But what's not so comprehensible is that the Spanish champion has not tried harder to overcome his problem and to give it time to pass.' He was not the only person who thought that.\n\nThe scandal did not fade into the background immediately. The Spanish Federation banned Bahamontes and, separately, Miguel Poblet from racing abroad for a month, both for abandoning without due reason. The aim was to stop them making money from post-Tour criteriums. In Poblet's case this seems particularly harsh. Sixth in the Giro, and therefore with reasonable hopes of a strong ride in the Tour, the race was going to have a stage finish in his home town of Barcelona and he had been keen to win there. However, he was forced to abandon on stage four because of a boil 'the size of a golf ball in the part that most hurts a cyclist. Whenever I sit down, I see stars'. Bahamontes was also required to re-state his explanations for abandoning before a medical tribunal in Madrid in the autumn and ordered to undergo a health check before he was allowed to race again. 'I had to go to Madrid to prove I hadn't taken anything [illegal],' he said. 'Puig was a doctor and before the Alps, he injected me with Binerva and Redoxon [the brand names for vitamins B and C] in my arm. I was in agony. [Team-mate and future Vuelta winner Antonio] Su\u00e1rez told me that I shouldn't give up, but I refused to go down a climb steering with only one arm. By then the damage had been done, anyway. It fucked up my Tour completely.'\n\nThe word that was never mentioned throughout the controversy was, of course, doping. Though lists of banned substances existed at that time, very rarely was any concerted action taken to fight drugs. However, allusions to the problem are plain: Tour organisers made dark hints in their official report of the affair about 'practices of which we don't approve'. Then there were the unstated aims of the medical commission who investigated Bahamontes on his return. As far as I have been able to establish from a cross-section of professional cyclists and team doctors, calcium has no performance-enhancing benefits whatsoever. Nor does lime. Either Bahamontes did not know that and fell for some of the quackery prevalent in cycling at the time, or he did not know what was in the syringe but assumed that if Puig was administering it, it could only do him good. Either way, Puig had crossed a red line given that officially the only provider of legal medicines for the riders \u2013 be it injections or otherwise \u2013 was the Tour's doctor, Pierre Dumas. Brian Robinson, the British rider, recalls: 'Dumas would tell us which vitamins to buy. He told me the American products were the best. I'd bring it over with me from England and he'd inject me with it. He used to come by every couple of days. I remember I went to the local chemists in Yorkshire and asked for vitamin B and C in five hundred milligram ampoules, and they said, \"What do you want with that?\" When I told them, they said, \"Bloody hell, it's what we give horses!\"'\n\nUnlike Robinson, many riders wanted more than Dumas would give them. Far more. Les Woodland's _Yellow Jersey Guide to the Tour de France_ records that in cycling half a century ago: 'There were _soigneurs, fakirs_ , who came from the six-day [criterium] races. Their value was in the contents of their case. Riders took anything they were given, even bee stings and toad extract.' This was not a new development. As one of the toughest endurance sports, cycling had been infected with doping as early as the 1890s. It was particularly prevalent at track events where strychnine and arsenic were among the substances used. In 1896, the death of Bordeaux-Paris winner Arthur Linton was suspected, though never proved satisfactorily, to have been the first in sport caused by drug use. And that was only the beginning. As William Fotheringham's _Cyclopedia_ notes, in 1924 professionals Henri and Francis Pelissier once gave a graphic description of their doping to leading Tour journalist Albert Londres. Opening a pillbox, Henri showed Londres 'cocaine for the eyes . . . chloroform for the gums, and do you want to see the pills? We ride on dynamite. When the mud is washed off us, we are white as sheets, we are drained by diarrhoea, we dance jigs in our bedroom instead of sleeping'. Just to add to the grotesque horror of the interview, which was entitled 'Convicts of the Road' when published, Francis claimed that six of his ten toenails had fallen out, the strong implication being that it was due to doping.\n\nBy the 1930s drug-taking was so widespread that historian Benjo Maso records that the riders' contracts in the Tour stated that the costs of 'stimulants, tonics and doping' had to be paid for by the competitors. And two decades later Bahamontes' old training partner Ian Brown was to discover that, while it was possible to complete a race without dope, the decision not to take drugs could destroy what had looked a promising career. Brown had been disappointed not to be named as one of the future stars at the end of the Simplex camp. 'As I said to [Charles] Pelissier, I didn't want to go back to England. In fact I was a bit desperate to get out of the normal life of Britain,' he says.\n\nBrown's professional career started almost by chance, though, thanks to Simplex generously putting him up for a week in a hotel outside Brussels' Gare du Nord 'because Pelissier said the only racing on at that time of year, if I wanted to do it, was in Belgium. So I had to go there'. Its only initial drawback was that it was in the middle of the city's red-light district. 'There were all these sexy women in the shop windows, which was a bit of a surprise; we didn't have anything like that in Morecambe,' Brown recalls. 'But there was a race the following Wednesday, and I got third and a load of money when I hadn't even realised where the finish was and that I'd crossed the line. And I thought, \"Holy crikey, this is easy!\"'\n\nHowever, while some of his discoveries about life and professional racing abroad were appealing, others were not and led him to retire early. Brown makes it very clear when interviewed why he packed up prematurely: drugs, principally amphetamines, 'though there was other stuff going around. What, I don't know, because I never took it'. Brown has a collective term for all the substances, in any case: 'go-fast'. The evidence of Brown and other British riders indicates that there was little drug-taking in cycling on their side of the Channel. However, in mainland Europe the opposite was true, and Brown decided, with decidedly bleak logic, to opt out. 'I knew a lot of riders who later died by taking too much \"go-fast\", guys in their twenties or thirties, and I wasn't taking that stuff,' he says. 'Somebody who'd been Belgian pro champion twice, about ten years older than me, once offered me something called _la bombe atomique_ to win races. The next spring when I came back to the cafeteria where I was staying, he'd died. I'd see them head off the side of the road in the middle of races to get injections. And in Belgian races, with about two hundred riders, it was hard and fast to start with, so with about fifty kilometres to go they'd all take their pills. They called it \"parachute time\" because all the pills were in little bits of paper and when the bits of paper came flying back through the bunch they looked like parachutes.' Brown's career fizzled out in 1958, he says, 'because I knew if I'd gone back to try and win races the following season, I'd be asked to take stuff and I wasn't doing that'.\n\nBahamontes, on the other hand, maintains that it was still possible to win without doping. Not only does he state categorically that he did not use drugs, he claims there was little evidence of it, and that he only once saw 'somebody put something in a flask'. At most, he says, his success was caffeine-induced; in the 1959 Tour he drank two cups of coffee before destroying the field in an uphill time-trial at Puy de D\u00f4me. His comments could sound like a deliberate under-exaggeration given the extent of drug-taking that went on in the 1950s when there were no anti-doping controls in racing. In addition to Brown, it is easy to find people who disagree that the sport was relatively clean. When Josu Loro\u00f1o, the son of Bahamontes' arch-rival Jes\u00fas Loro\u00f1o, was asked what he had heard about drug-taking in that era, he answered, without batting an eyelid: 'There were a lot of amphetamines going around.'\n\nHowever, Bahamontes decries the idea that he personally could have taken drugs, saying: 'Your health takes priority over everything. If you don't take care of it, it's like not taking care of your wife or your car. Wherever you go, make sure your car has good tyres and good petrol, and make sure your wife has money so she's got nice clothes and good food. Because if you don't, then \u2013 pardon my being so direct \u2013 somebody else is going to fuck your wife and somebody else is going to steal your car. I never took anything, never. I saw a _soigneur_ put something in a _bidon_ , once, and that was it.'\n\nBut Bahamontes also refers to many of his former rivals 'being in the grave already'. As he puts it: 'You can be alive and not have much money, or have all the glory and be dead these thirty years past.' Bahamontes does not state it outright, but the implication is there: he got through the major Tours without drugs; others didn't and paid the ultimate price.\n\nRobinson partly supports Bahamontes' claims that it was possible to compete and win major races like the Tour without drugs. 'I could go along with that: that it was due to his athletic ability. I know I got over those hills myself [without drugs],' says Robinson. 'Anquetil . . . said you can't ride the Tour on water, but I proved you can . . . with a little bit of vitamin C, a bit of cola and a bit of this or that. Nothing drastic. On the other hand, I did the job as it was supposed to be done. I didn't bugger about at night. That's why [Tom] Simpson' \u2013 Britain's greatest professional who died of a mixture of amphetamine abuse and dehydration on the Mont Ventoux in 1967 \u2013 'and I split. He went to Belgium to further his career in that line, which was sad because he was the one [rider] who didn't need it.'\n\nRobinson's view that it was possible to race at a high level even if you were clean is corroborated by Brown, who says: 'You could ride just as well without [drugs] if you were crafty. I was better at stage races than I was at one-day races, because if you weren't taking \"go-fast\" then you'd be hanging for grim death in the first two days, but in the third you would be all right.' It should be noted that Brown's experience is based largely on week-long stage races, in which he did very well without being outstanding. However, the demands of week-long races are nothing compared to a three-week Grand Tour. Brown only did one of those, the Vuelta a Espa\u00f1a in 1955, from which he abandoned. Bahamontes rode up to three major Tours in one year, and one season finished them all. Furthermore, while Bahamontes had numerous major highpoints in his career, Brown says simply and bleakly: 'It was just make money and eat and sleep well. I never had one [a highpoint], there weren't any at all.'\n\nWhile officially turning a blind eye, the Tour de France organisers were far from unaware what their riders were up to. Dumas himself spoke of 'medicine from the heart of Africa . . . healers laying on hands or giving out irradiating balms, feet plunged into unbelievable mixtures which could lead to eczema, so-called magnetised diets and everything else you could imagine. In 1953 and 1954 (Tours de France) it was all magic, medicine and sorcery. After that, they started reading _Vidal_ [the French medicine directory].' At best, the organisers carried out damage limitation when the doping got too blatant, as was the case in the 1955 Tour after the abandon of Jean Mallejac and Charly Gaul falling ill. Mallejac lost consciousness while ascending the Mont Ventoux and pulled out, almost certainly after using a 'bad batch' of drugs. Gaul was allegedly close to following suit, according to media reports of the time, because of the same batch. However, Gaul's director claimed in _L'Equipe_ that the rider had been victim of an attempted poisoning by being 'forced to dope'. Given the scandal surrounding Mallejac's abandon, the Tour issued a _communiqu\u00e9_ urging directors 'to watch how their riders take care of themselves' and advising that they should 'stop them using certain products that have not been prescribed by a doctor'. But that was as far as it went.\n\nSeven years later, as Bahamontes neared the end of his career, very little had changed. In the 1962 Tour, Gastone Nencini, the 1960 winner, and several other riders pulled out after eating what they claimed was rotten fish from the same hotel. The Tour organisers published a _communiqu\u00e9_ apparently proving their excuse was a lie, and warning grimly that 'the physical limits which athletes of various disciplines seek to attain are beyond all medical and psychological control'. However, they then suggested the best solution to a situation which they were incapable of solving was that Dumas should do a doctor's round of the riders' rooms each evening. The implication was that even if they could not stop them taking drugs, they would at least know what they were.\n\nIn that context, for Puig to be injecting his riders with calcium and other substances is hardly strange. However, to admit it in public was too blatant for the cycling establishment to swallow. Puig later stated that he 'only gave Bahamontes the injection because we couldn't find the race doctor in the team hotel'. Since there was no reason why Dumas should have been in the hotel, this excuse was ignored by the Tour authorities: they wanted Puig out. It is quite possible, of course, that Puig was only providing a badly-administered placebo to Bahamontes. As Robinson says: 'There was so much talk about what it does. \"You got to take this, you've got to take that\". A lot of it [the benefits of doping] is purely in your head.'\n\nAfter what was a bizarre abandon considering that by all accounts Bahamontes' injuries did not justify it, conspiracy theories abounded. One of the more popular was that Bahamontes was faced with a re-run of his Vuelta 1957 duel against Loro\u00f1o and psychologically could not take it. There were definitely parallels. Again Bahamontes had moved into a dominating position early in the Tour, surviving a wasp-sting, the blazing heat and the dreaded cobbled farm-tracks on the stage to Roubaix, to become sole leader of the Spanish team. No sooner had he done so, though, than Loro\u00f1o managed to counter-attack, not once but twice in two mountain stages in the Vosges, and move ahead in the overall. This was eerily reminiscent of the Vuelta stage from Valencia to Tortosa. For Bahamontes such a situation must have been impossible to accept, particularly as since his Vuelta defeat he had slowly been regaining a strong position in Spanish cycling. Now he was faced with that all-too-familiar scenario.\n\nThe season had started so well. In May Bahamontes had produced a dominating performance in the Tour of Asturias, his talismanic race. He won the first, the most mountainous stage, and led the race from start to finish before scooping the King of the Mountains jersey. On top of that the pace set by his Mobylette team was so intense that thirty-four of the seventy-five-strong peloton had abandoned by the finish. By then Bahamontes was showering his team-mates with stage wins. The only disagreeable incident for Mobylette during the race was when one of Bahamontes' team-mates, Rene Marigil, attempted to beat up an Italian rival for trying to get in a break by riding in the slipstream of the Italian team's car. Marigil was allowed to continue racing despite the controversy that followed, and even won a mountain stage. As a morale-boost Bahamontes could not have asked for more. Incidentally, his days of riding by bike to races like Asturias had long gone. As Bernard Ruiz says: 'The top riders of the time, like me and Bahamontes, earned more money than football players.' So, as one of Spain's top earning athletes, Bahamontes now travelled almost everywhere by aeroplane, car or train.\n\nBahamontes, though, was far less successful in his one venture abroad between the Vuelta and Tour, abandoning the GP Midi-Libre, then a one-day race, with cramp after a spectacular early mountain attack. Though he also pulled out after two stages of the Bicicleta Vasca, his last Tour warm-up race, it was suggested that his form was so good there was no point in him 'over-cooking it'. The only downside from Bahamontes' point of view was that the Spanish team's line-up was full of unfriendly faces like Ruiz and Loro\u00f1o. Bahamontes chose to make Ruiz, rather than Loro\u00f1o, the subject of his complaints. With hindsight it was an unwise move, though it revealed the depth of the disunity within the squad. Bahamontes said at the time: 'The whole line-up is wrong, but in particular the presence of Bernardo Ruiz, whom as everybody knows has been my public enemy number one for the last seven years. He envies me and I don't know why, but if he comes to France then I know I won't be able to expect anything from him at all. I suppose he might be useful as a _domestique_ , just like Loro\u00f1o. [Actually] if he comes, I refuse to be leader. He can be leader. I'll go as a _domestique_.' What particularly worried Bahamontes was that Ruiz and Puig were close, both being from the Levante area of Spain, and that Ruiz's far stronger character and veteran status would give him greater authority than the team manager. 'Ruiz is thirty-three years old and at that age should go for lesser objectives,' Bahamontes said. 'Everybody knows that if he goes to the Tour, he'll be the team manager because that's what he's always been. They ask him everything, right down to what the plans are for the stage and what time they ought to have dinner each night. I don't think he should be selected just because he won the [Vuelta] stage from Valencia to Tortosa.' Puig adopted his usual pose of sticking his head in the sand, claiming: 'Of course there will be unanimity. The race is so hard, Ruiz and Bahamontes are bound to get on.' In a sentence which neatly summed up Puig's inability to sort out internal team conflicts, he added: 'We'll agree on everything beforehand, just like we did in the Vuelta a Espa\u00f1a in Bilbao.' Again it was Bahamontes who paid the price.\n\nThe Tour de France itself started brilliantly for Bahamontes. He was in two breaks on the first three stages and propelled himself into to fourth place overall. Significantly, at the same time Loro\u00f1o was most definitely on the back foot. After losing a little time on stage one \u2013 'the Belgians wouldn't let me move even a metre ahead' \u2013 Bahamontes bounced back the next day by making it into a move of nineteen that gained more than three minutes on 1956 winner Roger Walkowiak, and nearly five on France's rising star Jacques Anquetil. Twenty-four hours later, a timely shove by team-mate Su\u00e1rez propelled Bahamontes into another big break and he shot back up the overall classification to fourth, the best-placed of the favourites. The race was affected by a heatwave, which lasted ten days, and riders all but stampeded to take on water every time the race went past a drinking fountain. It was so debilitating that Charly Gaul abandoned on stage two because he could not stand the high temperatures. More importantly for Bahamontes was the incident that cost Loro\u00f1o his status as co-team leader. On the same day as Gaul's exit Loro\u00f1o decided to dismount and stand under a garden hose to cool down. He was enjoying his cold shower so much that by the time he came to his senses, more than quarter of an hour had passed, the race was miles ahead and he was fortunate to finish inside the time limit.\n\nStage four to Roubaix, however, marked the beginning of a relentless series of setbacks for Bahamontes. In one day alone he was stung by a wasp, hit hard in the eye by a bottle [flung unintentionally by another rider during the struggle to be first at a drinking fountain] and suffered a severe bout of sunstroke. The heat affected everyone. Of the one hundred and sixteen starters, twenty-eight abandoned in the first four days, while reporters' throats dried up so badly while dictating their copy by telephone they were reduced to swigging cough mixture to keep talking. At one point it looked as if Bahamontes was going to join the list of abandonments: he began weaving all over the road, dropping back from group to group. As he did so, he gave the Spanish correspondents covering the Tour more and more cause for alarm. '\"Sunstroke!\" Su\u00e1rez yelled at us when we drove alongside him and asked him what was wrong with Bahamontes,' recorded _El Mundo Deportivo's_ correspondent. The news from Dumas, the Tour doctor, was even more unsettling. 'Bahamontes had a kind of fit because of the heat,' he told the Spanish journalists, 'and he got off his bike, half-fainting. He recovered fairly quickly, though. When I put him back on it he was off like a rocket.' So much so that he regained eight minutes on the pack in three kilometres, albeit with some team support.\n\nPhotographs of Bahamontes at Roubaix after the stage show a miserable-looking figure; the swelling above his right eye caused by the wasp-sting was clearly visible. He was criticised by _El Mundo Deportivo_ 's man for failing to combat the sunstroke by putting lettuce leaves on his neck and nape 'as most riders do in extreme heat'. However, while Bahamontes had not lost time he was looking in trouble. Stage five, a longish grind from Roubaix to Charleroi in Belgium over some of the region's best-known cobbled climbs including the infamous Mur de Grammont, dealt Bahamontes a severe blow. Lightly-built climbers like Bahamontes are notoriously vulnerable to this kind of terrain because they cannot ride over the hills in low gears like the heavier Classics specialists. So while Bahamontes had survived the _pav\u00e9s_ of Roubaix, the Belgian murs took a punishing toll. The bunch split completely after an attack masterminded by Anquetil and riders reached the finish line in dribs and drabs. Bereft of team support, not even the encouragement provided by a solid wall of Belgian fans along the stage's one hundred and seventy kilometres could help Bahamontes. He slid from fourth to fourteenth overall. Loro\u00f1o saw his opportunity and took it with both hands. By sneaking into breaks in two stages in the Vosges mountains, Loro\u00f1o regained twenty-seven minutes on the main favourites, climbing thirty-eight places to seventh overall. Suddenly Bahamontes' role as sole team leader was severely undermined. The parallels between the Tour de France and the Vuelta a Espa\u00f1a were evident, right down to Bahamontes' bizarrely passive attitude when his arch-rival went up the road. According to the Spanish team mechanic, Bahamontes' usual tactic when racing alongside Loro\u00f1o was to 'hide at the back of the bunch. Then if there was a break and the bunch split behind chasing, Bahamontes would sneak into the forward half without anybody noticing and he could claim he was protecting Loro\u00f1o's position as well as his own'. That was the usual scenario. However, one Spanish newspaper correspondent noted the change: 'When I saw Baha halfway through the second Vosges stage with the Loro\u00f1o break long gone, he was right at the rear of the main bunch, and I yelled, \"Always at the back!\" His only response was to smile.'\n\nAs Bahamontes slid from fourteenth to sixteenth, and Loro\u00f1o moved into the top ten, it was little wonder that his morale was sinking and that Puig should provide him with medical assistance. However, again with hindsight, it was not a wise move. 'Maybe a little lime spilled out of the syringe, but that's not why he abandoned,' was Puig's later interpretation of events. '[In fact] he wasn't ready to face up to the challenges of the Tourmalet [tackled the day after Bahamontes abandoned]. He was scared of failing in the mountains and he didn't have a strong enough sense of vocation.'\n\nBahamontes' business advisor, Evaristo Murtra, partly went along with Puig but like others he pointed out that the Federation and manager had failed to live up to their responsibilities by not designating Bahamontes as sole leader from the start of the Tour. 'Bahamontes' morale has suffered after seeing how much freedom Loro\u00f1o and Ruiz had,' Murtra told the press in a reference to Puig allowing Loro\u00f1o to get into two breaks in the Vosges without being reprimanded. 'Combined with the heat, the injection and his constantly being watched by the French, it all got too much for him.' Murtra had only turned up because he wanted to see his rider's attacks on the Tourmalet in person. Instead, both he and Bahamontes returned to Spain empty-handed, Murtra by aeroplane, Bahamontes by train, in a third-class compartment, which was all that the Federation were willing to pay for their fallen star.\n\nIf Loro\u00f1o's breakaways had provided a replay of Bahamontes' worst sporting nightmares, the knock-on effects of his abandon within the Federation were considerable. Not all of them were bad, though. The entire medical commission \u2013 three different doctors \u2013 resigned. This gave the Federation the opportunity to bring in a fresh generation of young medics who were not so inclined to accept at face value the quackery and outlandish cures that were so popular in cycling, as well as post-Civil War Spain. Even more importantly, Puig was gradually phased out and Dalmacio Langarica was brought in as coach for the 1958 Tour.\n\nFor Bahamontes, the arrival of Langarica was a huge turning point. He describes Langarica simply as 'the best coach in the Spanish team that I had, because he knew how to take decisions'. That was despite Langarica being a friend, former team-mate and regular room-mate of Loro\u00f1o's when he was a professional. Crucially, Langarica was far tougher than Puig. From the word go, Langarica told both Bahamontes and Loro\u00f1o in no uncertain terms that he was not going to take any nonsense from either of them. He also showed, albeit indirectly, which rider he favoured when he politely but firmly condemned Puig for allowing Loro\u00f1o to attack in the Vosges in 1957 when Bahamontes was already in a commanding position overall. 'Bahamontes is Spain's classiest rider, the greatest the country has ever known. While Loro\u00f1o did what he had to, it [the overall strategy of letting Loro\u00f1o attack] was throwing stones against our own roof,' Langarica was quoted as saying in an interview in August 1957. 'In that context, Bahamontes' abandon made perfect sense.'\n\nBahamontes will have welcomed this support even if it was clear Langarica did not believe a word of his explanation about the injection and attributed his abandon purely to his rivalry with Loro\u00f1o. Furthermore, unlike Puig, Langarica showed a keen understanding of Bahamontes' personality. 'Fede is like a child,' he used to say. 'If he doesn't like the people in charge of him, he'll lead you a merry dance. But if you know how to handle him, he'll come quietly enough.'\n\nOn 9 July, 1958, Bahamontes turned thirty. He could not have asked for a better birthday present: his first stage win in the Tour de France. The build-up to the victory on a short, sharp one hundred and twenty-kilometre stage through the Pyrenees from Pau to Luchon, was surprisingly straightforward. With the Aspin and the Peyresourde cols as the two main challenges of the day, Bahamontes needed no telling that it was a comparatively short distance from the summit of the Peyresourde down to Luchon, and if he gained enough time on the climb he could be in control. 'We got [team-mate Fernando] Manzaneque into a break with three French riders and then in the village of Sainte-Marie-de-Campan [at the foot of the Aspin] I hit the pedals hard twice, more than just at cruising speed, and left them all behind,' Bahamontes remembers. 'I overtook the rest of the guys ahead on the Aspin, and my advantage was one and a half minutes by the summit. The descent to Arrau was thirteen kilometres long and really awful, the hardest in the Pyrenees, but I actually gained thirty seconds, which ended up being fundamental. Then on the Peyresourde the race blew apart because Gaul and Geminiani attacked and I gave it even more on the climb, knowing that somebody had lit the fuse behind. Finally I got two minutes and a bit on Gaul and two and a half minutes on Bobet and Geminiani, and even though there were fifteen kilometres downhill left, I was determined not to get caught. I finished alone with about two minutes on the field. I'd promised my wife to do something dramatic and give Toledo a triumph to remember, and that's what happened.'\n\nHow come it looked so easy? There were a number of key differences between Bahamontes in 1957 and Bahamontes in 1958, not least that he was in exceptionally good form and morale. He finished sixth in the Vuelta after another bust-up with Loro\u00f1o, which cost them both the race. That was the last race where he would be directed by Luis Puig. 'Just as well he'd never direct me in the Tour again,' Bahamontes commented. Bahamontes nonetheless took a significant consolation prize in the King of the Mountains competition. Then he followed up by scoring his first stage win in the Giro. Bahamontes was part of a powerful Faema squad that year and had decided to use the Giro as Tour preparation. On stage four, with a summit finish in the ski station of Superga, Bahamontes inflicted stinging defeats on Charly Gaul and Loro\u00f1o, leading them home by twenty-seven and thirty-seven seconds respectively. Not huge, but enough to convince him that his decision to race the Giro had been the right one.\n\nAnother major difference was that on the 1958 Tour there was no Jes\u00fas Loro\u00f1o. After racing flat-out in both the Vuelta and Giro, Loro\u00f1o decided that the Tour would be asking too much of himself. His absence meant Bahamontes was finally the undisputed leader in a major Tour. In addition Bahamontes' relationship with his team-mates was apparently improving, albeit intermittently. The reported group cheating at the 1958 Spanish National Championships, categorically denied by Bahamontes, but given lavish eyewitness coverage in _El Mundo Deportivo_ , would seem to hint at this, too. According to _History of the Spanish National Championships_ by Javier Bodegas and Juan Dorronsoro, the 100-kilometre time-trial was completely fixed. In theory it was an individual event, but Bahamontes allegedly joined forces with two other Faema riders, first Salvador Botella, who waited for Bahamontes after a timely puncture, then Francisco Moreno, and the three rode as a unit over the second half of the course. To make matters worse, Bodegas and Dorronsoro claim, the police failed to stop traffic in either direction, allowing riders to benefit from assistance provided by motorbikes. Indeed, Bahamontes' illegal pacer crossed the finishing line just ahead of him. As a result of the alleged race-fixing, the favourite Loro\u00f1o had no chance despite building a fifty-second lead at halfway. He finally finished third behind Bahamontes and Botella.\n\nYet more significant than Loro\u00f1o's absence from the Tour was the change in the man behind the steering wheel in the Spanish national team. The replacement of Puig by Langarica meant Bahamontes could count on a degree of team loyalty, not least because it effectively ended Ruiz's role as the power behind the director's throne. However, Bahamontes respected Langarica for more than that: he also liked his team tactics and was willing to use them. 'Langarica dropped the previous director's strategies,' Bahamontes said in an interview at the time. 'He moved with the times. He was practical. Previously, we attacked where we could, but on the day I won we attacked at exactly the right moment to be sure I got my first of seven Tour stages. It was a daring move, but it was efficent, too.'\n\nCrucially, the Spanish took a large leaf out of the French team's book with Langarica drawing on the lessons learnt from Bahamontes' defeat the previous year. In 1957, after his superb start to the Tour, the French had refused to let Bahamontes go up the road under any circumstances, whereas Loro\u00f1o had been given a considerably longer leash. The tactic was to deliberately favour the less dangerous rider. This time round the boot was on the other foot and the switch in strategy was particularly effective in the Alps where Bahamontes gained a second stage win over the Vars and Izoard passes. Realising the squabbling French were willing to give Bahamontes some room for manoeuvre, Langarica encouraged Bahamontes to come up with a specific plan to win the stage. Initially doubtful, 'because plans are one thing and reality another', Bahamontes finally went along with Langarica since he believed the director had already got him his first Tour stage win in the Pyrenees. 'The night before we talked about the stage and Langarica said, \"You have to make the most of the war between the French and the way they're dealing with Gaul. They're all going to be watching each other, and you have to take advantage of that\",' Bahamontes recalled. Sure enough, after the Vars climb had whittled down the main pack to just a dozen, Bahamontes broke away on the Izoard, crossing the summit with a fifty-seven-second advantage on his closest pursuer, the Italian Nino Catalano. When he reached Brian\u00e7on to claim his second stage win, his advantage had only dropped to fifty seconds. It indicated that his poor descending was no longer automitically the handicap it had been.\n\nIn the past Bahamontes had used the tactics of eating on the descents, so he could attack with a full stomach on the next, and making early breaks to rack up points in the King of the Mountains competitions before easing back after a couple of cols. However, these strategies limited him, almost deliberately so. If the other contenders for the overall prizes did not 're-fuel' on the descents it was at least partly because they knew they risked losing time. In 1958, though, Bahamontes partly changed tack, and it is no coincidence that at the same time his descending radically improved. Bahamontes puts this down to greater experience, but it is clear from the way he won those two stages that his previous unwritten, unspoken agreement with the other favourites that he would stay away to the finish after making an early attack no longer applied. Rather than just focus on the King of the Mountains title, for the first time in the Tour Bahamontes was starting to think bigger. Now he was able to trust his sports director to come up with a plan that did not weaken his status compared with other rivals inside the Spanish camp. The reward was not only two stage wins but the King of the Mountains title as well. _L'Equipe_ still criticised him, though: 'In any other rider, what Bahamontes did would have produced admiration and respect. Not many other riders can get two big mountain stage wins, easily get the overall in the King of the Mountains title and finish eighth overall after being forty minutes down on the race leader prior to the Pyrenees. Only an athlete like Bahamontes can pull off an achievement of this calibre. But it loses its edge when we consider what a rider of his class is actually capable of doing. His tactics are extremely simplistic, and worthy of condemnation, consisting of playing dead in the first few days of the race so he loses any importance to the main favourites.'\n\nThe judgment is unduly harsh, but it contains some grains of truth. Bahamontes had been ill in the first part of the race, suffering from suspected appendicitis, but even so he had under-exploited his talent, and that needed to change. Not that Langarica could resolve all of Bahamontes' defects: he still had an almost pathological inability to race in cold weather. The day after he won in Grenoble, as the rain teemed down Bahamontes was dropped by Gaul on the Col de Luitel, the second of five Alpine cols. On a stage with nearly one hundred kilometres of climbing Bahamontes ended up losing five seconds short of half an hour on Gaul and dropped to tenth overall. He pulled back a couple of places overall in the race's final seventy-four-kilometre time-trial at Besancon to reach Paris in eighth place. Once again it showed that even if he was nowhere near as good as specialists like Jacques Anquetil or Roger Rivi\u00e8re, at the very least he could hold his own. And the following year in the Tour de France, that ability was to prove crucial.\nChapter Eight\n\n#### 1959 \u2013 'That Man From Toledo'\n\nLuis Ota\u00f1o has barely opened the door of his fourth floor flat before he comes out with a joke about the rider he helped guide to victory in the Tour de France, back in 1959. And it is not a polite one. 'I bet you anything in the world, Federico's asked you for money to do his biography, hasn't he?' he says. Before I have time to deny it, he yells through to his wife in the kitchen, 'Hey, I told you, Bahamontes asked him for money!' Chuckling to himself, Ota\u00f1o leads me through to a sunlit living room. We are high above the town centre traffic in Renteria, just outside San Sebastian. Through the windows the green foothills of the Spanish Pyrenees are visible in the distance. Just beyond them, but out of sight, lie the roads and climbs of France where more than fifty years ago Ota\u00f1o rode as part of the Spanish team who had Bahamontes at its head.\n\nCareer-wise, the Pyrenees mark a dividing line for Ota\u00f1o. On home soil he recorded numerous top-five finishes in the Vuelta, including second overall behind Raymond Poulidor in 1964. However, once on the other side of the mountains it was a different story: Ota\u00f1o rode in eight Tours de France and each time it was purely as a _domestique_ , a behind-the-scenes worker for whoever was Spanish team leader. That included Federico Mart\u00edn Bahamontes, of course. 'But not Fede's personal _domestique_ , eh,' Ota\u00f1o says with a grin. 'That was [Juan] Campillo. Bahamontes used to have him wash his socks and clean his shoes, he did.' The bald, sturdy man with a piercing stare lets rip a cackle. 'Me, I used to argue with Fede all the time, asking him, \"Why are you doing this?\" and \"Why are you doing that?\" even though I was younger than him. And he would mutter, \"Anquetil this\" or \"Anquetil that\". If Anquetil hadn't been there or if there had been more summit finishes, I'm sure Fede would have won a lot more.'\n\nOf Bahamontes' eleven team-mates on that 1959 Tour, Ota\u00f1o is one of three still living. So what was Bahamontes like to work for? 'Odd,' Ota\u00f1o says. 'On and off the bike. For example, he used to insist on us not eating any tomatoes for dinner because he said they were bad for our health, and we'd ignore him. When Loro\u00f1o used to drink a bottle of wine a night, that would piss him off even more because even a little wine used to make Baha dizzy. If he didn't like something then he tried to make sure none of us had it.' Not that they paid him much attention. Nor did the top French riders when he addressed them. 'Mostly, they'd laugh at his French. He'd say, \" _J'ai . . . gran fusil_ , _j'ai . . . gran fusil ['I have the top gun']\",'_ says Ota\u00f1o.\n\nCommunication was not at a premium in any case. Unlike today the countdown to racing did not start with anything remotely approaching a meeting to discuss strategy. Instead, Ota\u00f1o says: 'We'd talk a little bit among ourselves, decide who would help who. But that was as far as it went. There was no real plan, no tactics. At most, \"Wait for him if he punctures\". We'd mostly all wait for Fede when he did puncture, even Bernardo Ruiz in 1958. It'd always be pretty dramatic, though, lots of people shouting, \"Fede's punctured! Fede's punctured!\" In fact, everything was a drama with Fede.'\n\nDespite their antipathy, Ota\u00f1o agrees with Bahamontes on one key point: that incoming team director Dalmacio Langarica played a crucial role in pulling them together, and that he did not beat around the bush when it came to letting the riders know who was boss, either. 'Langarica was very tough, but good. He knew what he was doing,' says Ota\u00f1o. 'I remember when he came into the Tour de France hotel he brought a big walking stick with him. He put it down the middle of the dining table where we were sitting as if to say, \"Don't mess with me\". Only [Jes\u00fas] Manzaneque and [Antonio] Su\u00e1rez didn't pay much attention to him. Su\u00e1rez was always banging on about how he'd won the Vuelta and the Nationals, and they both wanted their own chance to go for the overall.' In the event, Manzaneque finished fourteenth in Paris, and Su\u00e1rez abandoned.\n\nDespite two 'renegades', support from nine team-mates was not bad compared with what Bahamontes had been used to on previous Grand Tours. However, some of that number are convinced that Bahamontes only succeeded because of them. Speaking in an interview in 2008, Jos\u00e9 Gomez del Moral said: 'If it wasn't for me, Bahamontes wouldn't have won. There was as much about him that was cack-handed as there was pure talent. He should have won four or five Tours and he only won one. The rest of the Spaniards finished outside the time limit and I was the one who guided him through the Dolomites.' Since the Dolomites did not feature on the 1959 Tour itinerary, or any other for that matter, it is difficult to know how seriously to take Gomez del Moral's comments. That said, his lack of sympathy towards Bahamontes is notably similar to Ota\u00f1o and his description of how he first fell out with Bahamontes sounds plausible enough. 'I first came across him in a race in Ja\u00e9n [in Andalusia], a hard one through a lot of hills that was one hundred and fifty kilometres long. It was called the Turkey Race because it took place at Christmas and because apart from a cash prize you got a turkey. My father heard Bahamontes boast he was going to give us a feather each from the turkey. Finally, though, I won.' By 1959, however, it was Bahamontes who ruled the roost.\n\nAs a rider twenty kilos heavier than Bahamontes, and no climber, Ota\u00f1o's main job was to lead the pack on the flat and watch out for breaks. Bahamontes required a lot of 'nannying', particularly when it came to being shepherded through the bunch. Ota\u00f1o reveals his subversive sense of humour again when he grudgingly admits that Bahamontes was not the worst strategist. 'He wasn't bad. He knew how to calculate, even if he calculated wrongly most of the time. On the stage to Aurillac, for example, he told us, \"Work until such-and-such a point, and then I'll go for it\". And he did.'\n\nWith his severely limited capability in the mountains, Ota\u00f1o's direct contact with Bahamontes was mainly in the early stages, as well as providing support when the climber was in trouble. He had a particularly inglorious personal mission: pushing his leader along on the rainy stages when Bahamontes was having one of his off-days. One of those came in 1958 on the stage into Aix-les-Bains, won by Charly Gaul, when Bahamontes lost thirty minutes. 'It was nearly dark, thunder and stormy, almost nobody around and Bahamontes was crying, grabbing any food he could get, and Langarica kept on yelling at me, \"Help him! Help him!\"' The pair finally finished twenty-nine minutes fifty-five seconds down. 'If I hadn't stayed with him, I'd have been second,' Ota\u00f1o says with no indication of resentment, just what he considers to be an indisputable fact. 'He would say, \"I'm the best, I'm the best\", but we pushed him just the same, his hand holding on to your leg. And he would fret while we did it, saying, \"What if somebody's attacking now?\" Us _domestiques_ did nothing more than push the big names, you see?'\n\nA solid ride in the Vuelta, when he finished eighth overall, guaranteed Ota\u00f1o a slot in the 1959 Tour team. However, nothing he says suggests he had any enthusiasm for Bahamontes' quest that year. This is more surprising than it sounds. As Charly Wegelius, Britain's top mountain _domestique_ in the last decade, has pointed out: 'Even if you don't like your leader, the fact that you're all riding together with a common objective ends up binding you. You get emotionally involved. It's inevitable.' Not so, it seems, for Ota\u00f1o. He went through the motions all right, but he seems devoid of any passion for the Tour in general, let alone for Spain's first win in it. Part of this lack of motivation could be because there was no guaranteed money from the Federation for taking part, not even expenses. 'I once got called up to Madrid to go over my taxes, and they said, \"You've raced the Tour, so you've earned x\". And I said, \"No, I've not earned a penny\". The same went for the World Championships. The kit the Federation gave you was rubbish, right down to the trousers [for wearing in the evenings]. We got nothing, absolutely nothing, not even a bike. We had to pay for all our equipment, right down to the inner tubes of our wheels. All we got was what we could make in prizes. I often thought somebody must have run off with the money from the travel expenses we were supposed to get.'\n\nAs if that was not enough to lower morale, the Spanish also had the most ramshackle equipment of all the top teams. 'In 1958, my bike had a range of gears from fourteen to twenty-two,' says Ota\u00f1o. 'I didn't have any higher gears. For time-trials, climbs, the lot. Try getting over the Alps with a bike like that. Bahamontes might have had slightly better gearing, maybe a twenty-four, but not much. Still, that bike was an improvement on the equipment Bahamontes used to have. I can remember when he first raced up here in Euskadi, he didn't have enough money to buy any race shorts; he'd wear a pair of those baggy dungarees like mechanics used to have.'\n\nIt was not just their clothing and equipment that were dire. With two mechanics, two _soigneurs_ , a sports director and two jeeps for twelve riders, the back-up team was stretched to the limits, although any accidents that the consequent fatigue inevitably caused were seen as amusing rather than annoying. Indeed, when a Spanish mechanic fell asleep at the wheel during the 1959 Tour, and fell out of the vehicle, his misfortune was greeted with hoots of laughter. No one seemed to be worried if he was injured.\n\nIf Ota\u00f1o and the other Spanish riders were not one hundred per cent concentrated on the racing, it was hard to blame them considering they were almost economic refugees from a half-starved Spain. Ota\u00f1o says: 'The best thing about the Tour was the food. In the Vuelta all we'd get was meat so tough it was black. It looked like they'd taken a horse and cart and cut the whole thing to pieces, cart included. To follow would be boiled rice pudding in paper, and the paper would be soaked in the liquid from the rice so you'd eat that, too. In France, you'd get some amazing food, some of it even came wrapped up . . . chicken, all sorts.'\n\nHowever, despite the better 'fuel' on offer in France, Ota\u00f1o did not make it to Paris. He had to abandon on stage thirteen. 'It was on the stage to Aurillac,' he says. 'It was baking hot, the tarmac was all twisted in the heat, and I punctured close to a control point. There wasn't anybody there, but I found a bag lying on the ground with an inner tube in it, so I used that. Then I punctured again, and [French rival Jean] Robic gave me another inner tube, and I changed it again. By that point Bahamontes was twenty minutes ahead and there were forty kilometres left to race. I said to myself, \"What's the point of killing myself to finish if I'm going to be outside the time limit?\", so I quit on the spot. Langarica wasn't too happy. He said they would have let me back into the race if I'd finished the stage. But there were four of us Spaniards outside the limit that day, so I don't know what would have happened.'\n\nThough not there to see it in person Ota\u00f1o was not surprised Bahamontes won the Tour that year. 'To win in 1959 he needed Langarica, that was for sure. Without Langarica he wouldn't have done it. But we could see it coming. He was in good shape. He's got a great _palmares._ What surprises me, though, is that he didn't win more.' It was not easy for Ota\u00f1o to discover why. Even though they were team-mates for two years in the trade team Margnat in the early 1960s, Bahamontes tended to keep his own counsel. 'There were stages like one in Brian\u00e7on in 1962 when we asked him why he hadn't won instead of some Belgian non-climber [Emile Daems], and he muttered something about Anquetil. Very odd, because Baha could have won it.' The professional relationship between the pair ended as soon as Ota\u00f1o packed his suitcase in the Margnat team hotel one autumn evening in 1963 and headed for another squad. On a personal level, it remains non-existent.\n\n'I read he said that if he'd had the support that Pedro Delgado or Miguel Indurain had, he'd have won several Tours,' Ota\u00f1o says now. 'All I can say is that some days us _domestiques_ would be going well, on others we'd be out the back with our tongues sticking out. We'd hold on for as long as possible; we'd try to be there. But we did what we could, whatever the circumstances. And he never said thank you.' In anybody else that lack of gratitude might be considered rude. But given Bahamontes' constant reference to how he 'moved alone', as well as his tendency to criticise team-mates for being disloyal or useless, it is hardly that surprising. What is strange is that Bahamontes did have some solid, loyal _domestiques_ , like Julio San Emeterio and Juan Campillo; indeed, from his days at Margnat, Andr\u00e9 Darrigade still prides himself on the amount of work he did for Bahamontes. True, nobody in the 1950s would have gone to the extremes of Britain's twenty-first century star Mark Cavendish, whose first words after a victory are almost always to thank his team-mates. But Bahamontes' relationships, even by 1959 when he no longer had to worry about Loro\u00f1o or Ruiz, never seem to have been fully functional. That did not even change under his greatest manager, Fausto Coppi, whose dealings with his _domestiques_ appear to have been extremely good. 'Compared with say Eddy Merckx, Coppi's _domestiques_ were more menial,' says William Fotheringham, who wrote the biography _Fallen Angel_ about the Italian champion. 'Coppi was more austere and certainly more capricious, even getting them to look after his daughter while he was with his mistress. I certainly can't imagine Coppi playing cards with them and larking about like Merckx did. They also never got to race for themselves in big races, unlike Merckx's. It was more feudal. That said, Coppi clearly appreciated them and they were devoted to him.' That is not something that can be said of Bahamontes.\n\nPerhaps part of the problem was Bahamontes' character. Climbers tend to be reclusive. Charly Gaul certainly was; so was Britain's best climber, Robert Millar. 'I think Millar's default mode was suspicion,' says his biographer, Richard Moore. 'He distrusted people, which made him an awkward leader. Yet at Peugeot he had great support from [team-mates] Allan Peiper and Sean Yates. Ronan Pensec was fond of him as well as Pascal Simon, though Millar felt the French generally resented him. He wasn't really suited to leading a team; he couldn't rally the troops.' However, neither Millar nor Gaul laid into team-mates with as much relentless gusto as Bahamontes did throughout his career.\n\nAnother reason _domestiques_ often have such poor opinions of their leaders, and vice versa, is connected to the claustrophobic lifestyle they endure during a major Tour. For nearly a month there is no escape from the other members of the team: they eat, sleep, room and ride together. The constant travelling means there is no opportunity to sneak off to the cinema or for a quick coffee as there would be in, say, a major football tournament. The only valid comparison in sport for such lengthy, intimate and unwanted contact with another person of the same sex is probably ocean racing. That, in turn, has parallels to life in prison. Both are intense bonding-experiences while counting the days until you are back on _terra firma_ , or on the right side of the walls. In cycling, once the final stage is over and the celebrations start, those harsh memories tend to fade very fast and the benefits of bonding remain. However, during the two hours I talk to Ota\u00f1o there is no sense whatsoever that he is proud to have been part of the first Spanish team to win the Tour de France. His attitude is partly because of the contradiction that underpins professional cycling: it is the only team sport where an individual wins. So no matter how many kilometres of pushing their leader they do, how often they clean his shoes, how many hours of the day they spend together and how many rice-paper containers they eat, the rest of the squad receives almost no recognition in return. If they had won football's World Cup, they would barely be able to get out of the front door without somebody offering to buy them a drink or do their shopping.\n\nThis degree of neglect strikes home even harder when Ota\u00f1o's wife comes in halfway through the interview, camera in hand and takes a picture of me talking to her husband. There could hardly be a more graphic illustration of how completely forgotten Ota\u00f1o's contribution to one of a handful of sporting highpoints in an impoverished, war-torn, politically repressed country, has been. So on the day, fifty-two years later, when a stray foreign journalist turns up to talk about the 1959 Tour, Ota\u00f1o and his wife want a photograph for their album, if only to be sure it really happened. Strange as it sounds, my only regret about interviewing Ota\u00f1o is that it mattered to him and his wife that much.\n\nThe start of Federico Bahamontes' _annus mirabilis_ was predictably dogged by the latest in his disputes with Jes\u00fas Loro\u00f1o. To his credit, Bahamontes had spent most of the previous six months avoiding his arch-rival. During the winter he had deliberately chosen not to re-sign for Faema, where the pair would have remained team-mates in name at least, opting instead to race for Fausto Coppi's Tricolfilina squad. Bahamontes had prepared the ground carefully for his move by inviting Coppi to go greyhound hunting with him in Toledo during cycling's closed season. Coppi, partly drawn by curiosity, and partly by his interest in signing Bahamontes, was keen to accept. After a morning's hunting, a midday meal of typical Madrid stew and some joking over how to drink wine out of a wineskin that invariably ends up with wine all over the novice's face, Coppi 'popped the question'. Coppi's logic was simple: as 1959 was to be his last season he would be racing more in spirit than anything else and he needed a star rider. Bahamontes' attempts to bluff and make his future employer nervous by pretending he had not heard the question, failed completely. 'You heard me perfectly,' Coppi told him. But even Bahamontes was not expecting what Coppi told him next: that he could win the Tour.\n\nBahamontes has since claimed that this was the turning point for which he had been waiting. Not so much for joining the team, who turned out to be far weaker than he would have liked, or the half a million pesetas a year wages [\u00a385,000 in modern money] which he never fully received, but because of the faith an international star (and double Tour winner and a rider considered a pioneer in so many areas) was prepared to show in him.\n\nThe deal was agreed there and then and Bahamontes' sadly brief spell as a team-mate of Coppi's began. Given that Coppi would only live for another year, and that 1959 was Bahamontes' most successful, what would have happened if Bahamontes and Coppi had collaborated together for longer? But the little time they had together they used to the full.\n\nThere were several reasons why Coppi worked so well with Bahamontes, says Raphael Geminiani. For instance, the older man realised the degree to which the younger rider's real potential had been underexploited. 'Coppi had his team, and his sponsor, though not a lot of money, and he knew that Bahamontes was a climber, and a climber in the Tour always plays a pivotal role,' explains Geminiani, 'So he told Bahamontes to forget the King of the Mountains and stick with Anquetil and Rivi\u00e8re. He told him to switch strategy and he gained his trust. It's as simple as that. He transformed him.'\n\nBut it was not just mind-games that Coppi was so good at. As Geminiani points out, the other techniques for racing which the Italian used as a rider, and briefly as a team director, were way ahead of his times. 'Coppi operated with a level of authority about all sorts of things that nobody else had,' says Geminiani. 'He was really advanced in terms of race strategy, training, equipment, nutrition. He was a pioneer in a lot of things. He knew how to direct a team when director and he inspired me as a manager. To give an example, Coppi's principles in the evening were to let the riders have their bath and massage, talk to them afterwards, but never discuss the next day's racing until after their meal. Then out the strategy would come: this, that, that, this, that . . . like a game of chess. Not before and certainly not five minutes before the start. That way, the message really sinks in.' With an impulsive rider, that was not always straightforward. But as Langarica observed, Bahamontes had a child-like nature, meaning that if treated the right way he would usually come round to your point of view.\n\nHowever, allying with Coppi would not help on a day-to-day basis in the event itself where riders again raced in national teams. In the last few weeks before the Tour the question of how Loro\u00f1o and Bahamontes could possibly race under the same flag intensified. Bahamontes, with his usual direct style, had made it quite clear what his position was. 'It's him or me,' he told Langarica in private conversations. On the record he was only slightly more polite. 'We haven't spoken in over a year and I won't eat at the same table as him,' Bahamontes told the Spanish press. 'But if he's leader, I'll play my part. He'll have my help, even though I don't actually rate him as a rider.' Loro\u00f1o was even more uncompromising. He insisted he would not support any other Spanish rider under any circumstances, and 'particularly not that man from Toledo'. To complicate matters further, another Spaniard, Antonio Su\u00e1rez, had won the Vuelta a Espa\u00f1a in April. He also claimed that he could be a good leader of the Spanish squad in France. Langarica finally decided to go for Bahamontes and dropped Loro\u00f1o, a choice 'that man from Toledo' unsurprisingly thought was the right one. 'I could win the Tour and Loro\u00f1o couldn't,' Bahamontes reasoned later. 'And Loro\u00f1o hadn't won a single race in the whole of 1959, either.'\n\nHowever, the situation got completely out of hand when Langarica called all the riders to an emergency meeting in the Spanish Federation's offices to explain his decision. At the same time a list of the Tour riders was published outside the office building. 'In those days,' Bahamontes recalls, 'the line-up was as important in Spain as whoever got to play in the football World Cup.' Loro\u00f1o refused point blank to accept his exclusion and the conversation grew more and more heated. Then when he heard Su\u00e1rez had been admitted to the Tour team, talking gave way to shouting. Loro\u00f1o stormed out, slamming the door behind him. 'A few minutes later a gentleman whom I had never seen before came in asking for me,' Langarica recalled in an interview with _Informaciones._ 'He said to me, \"You can't direct a Spanish team because you are an undesirable Commie Separatist\".' Provoked by what was arguably one of the most insulting of insults in 1950s Spain, Langarica lashed out at the fan, who responded by thumping him back. 'I must confess,' Langarica said rather laconically later, 'that I lacked enough internal calm and I reacted violently.' The fighting only stopped when one of Langarica's fingers was broken after he swung a particularly enthusiastic punch. Then, as Langarica was making his way downstairs to leave the building, the scuffle broke out again.\n\nThe Loro\u00f1o supporter \u2013 rumoured to be a journalist from _La Gaceta del Norte_ \u2013 was finally ejected, but the press were waiting outside and the news spread fast. An overly frank telephone interview Langarica gave during the row to a 'friendly' journalist turned out to have been taped and the resulting conversation published verbatim. It ensured that the latest Bahamontes-Loro\u00f1o row remained in the headlines for several days. The revenge of Loro\u00f1o's fans did not end there, either: a few days later, while on the Tour, Langarica learnt that his bike shop in Bilbao had been attacked and its front window broken. Throughout the race he received anonymous hate-mail. Loro\u00f1o made a last-minute unsuccessful appeal to Langarica to ride in France, saying he would accept team orders. But he recognised how badly things had got out of hand and belatedly acknowledged the decision to leave him behind. His supporters' behaviour earned him a fine, though, as well as a two-month suspension from racing.\n\nBahamontes, meanwhile, told Langarica that he would have no regrets about excluding Loro\u00f1o. 'I told him I'd show him what I could do and I'd make [1958 winner] Charly Gaul suffer in the process!'\n\nLeaving Loro\u00f1o at home united the Spanish team. In a Tour which French cycling historians believe their country lost because of internal divisions that was no small matter. Bahamontes, though, was still as insecure as ever. No sooner had he seen off Loro\u00f1o than he began to worry about Su\u00e1rez's loyalty instead. But even he acknowledged the significance of Loro\u00f1o's absence. 'The atmosphere in the team will be better because of the absence of one of our team-mates,' Bahamontes rather smugly told _L'Equipe._ 'Our team is more cohesive than last year.'\n\nBahamontes had few other pre-Tour requests apart from asking for all the Spanish newspapers to be delivered to him twice a week. 'I want to be sure which of the journalists are my friends,' he said to the correspondent of Spanish sports daily _MARCA._ It was a warning he clearly intended to go around the press room. However, Bahamontes knew that he could count on one loyal supporter all the way around France. Jacques Daud\u00e9 was a French restaurant owner who became a fan after the ice-cream incident on the Romeyere in 1954. Daud\u00e9 followed Bahamontes throughout the 1959 Tour, paying his own way and organising sweepstakes on a Bahamontes' victory among the journalists. Bahamontes would always stop immediately after each stage to talk to Daud\u00e9. In return Daud\u00e9 would invariably produce a bottle of Perrier, a comb and some eau-de-cologne for Bahamontes to perform his post-race _toilette_ while he held on to the Spaniard's bike frame.\n\nWith Loro\u00f1o gone and Ruiz missing, Bahamontes was the only member of the Spanish squad with real Tour experience. Of his eleven team-mates who boarded a flight to France for the race start at Mulhouse, Su\u00e1rez was now the only possible threat to his authority. How much of a challenge Suar\u00e9z posed is hard to estimate. He was four years younger than Bahamontes and had his Vuelta win to reinforce his credibility, whereas Bahamontes had made another controversial abandon in his home race. Following a blistering start, including a spectacular stage win in Granada, Bahamontes had missed some crucial moves in the second week. After falling out with Fausto Coppi, Bahamontes competed in the Vuelta 'on loan' for the K.A.S. team. Coppi was particularly unhappy when he finally made it into a break only for Bahamontes to promptly chase it down. Bahamontes eventually pulled out on stage eleven, allegedly in protest after five of his key rivals from Faema were re-admitted despite failing to make the cut-off time. However, according to _Viva La Vuelta_ , the history of the race by Lucy Fallon and Adrian Bell, when asked for an explanation Bahamontes spat out: 'Because it's easier to go by car.' Bahamontes later alleged another reason: that he abandoned as the result of an anthrax infection, a disease usually affecting those working with animals. Yet more reports from the time suggest he had appendicitis.\n\nFurthermore, in June Bahamontes lost the Spanish national championships, a time-trial round Madrid, to Su\u00e1rez. Conversely, he came within a whisker of winning the Tour de Suisse that month despite a painfully poor start. Indeed, he commented later: 'I remember some woman in a bikini pushing me up a climb. I was creeping.' But he then picked up his form so fast that he took two stages, the King of the Mountains and had received 'an offer to buy the race off me'. This he refused, perhaps unwisely given he did not win. Yet despite those results Bahamontes still felt the need to assert his leadership once and for all. As usual with Bahamontes, there was only one way he knew to do that: attack. But this time he did things slightly differently: rather than wait for the mountains, Bahamontes went for it on the flat.\n\nThe Tour's first stage from Mulhouse to Metz was held in a downpour. It was the kind of weather that would normally have had Bahamontes grumbling at the back of the peloton. Instead he went on the rampage. Forty kilometres from the finish he suddenly lunged out of the pack. Two more riders, Rolf Graf of Switzerland and Frenchman Michel Vermeulin went with him. Fourteen others finally bridged across, and in a breakneck finish on a circuit through Metz they opened a gap of more than a minute. It was a sign of how fast they were going that the average speed for the stage was above forty-two kilometres an hour, which was very high for the era. Andr\u00e9 Darrigade took the stage, but the rest of Bahamontes' main rivals for the yellow jersey were nowhere. Best of all for 'that man from Toledo', nor was Su\u00e1rez. It was an audacious move and one which meant any attempt by Su\u00e1rez to claim the Spanish leadership had, at least partly, been stomped on. 'Su\u00e1rez had tried to break away earlier on several occasions, but it did not work out,' _MARCA_ 's correspondent wrote that evening. 'His problem is he doesn't like the rain.' Neither did Bahamontes, or so his rivals thought, but he had stolen a march on them of one minute twenty-nine seconds. 'It's very important I've done this because it shows my team-mates I'm not just here to win the King of the Mountains prize,' Bahamontes told reporters afterwards. 'This time I'm after the overall as well.'\n\nThe extent of Bahamontes' fear of Su\u00e1rez was shown later by his actions on stage eleven from Bagneres de Bigorre to St. Gaudens when Bahamontes attacked early on with Charly Gaul. The stage was structured so that the two big climbs of the day, the Aspin and Peyresourde, were crossed a long way from the finish, yet Bahamontes could not resist the opportunity. Bahamontes knew he and Gaul had no real chance of staying away, but he persuaded his unwilling accomplice to continue their near-suicidal attack. Bahamontes recalled: 'I had to shout at him all the way through a village to convince him.' The reason for Bahamontes' enthusiasm was that he had learnt Su\u00e1rez had been dropped from the bunch.\n\nBut if stage one had given Bahamontes the upper hand in his battle to ensure he was sole Spanish leader of the 1959 Tour de France, then stage two from Metz to Namur showed Su\u00e1rez was not willing to lie down and take his punishment so easily. Close to the finish he launched several attacks. His multiple moves failed to work out, though, and Bahamontes made a counter-charge at the finish on the steep uphill climb to the fortress of Namur. Only Gaul could follow Bahamontes. Just when it seemed they would be fighting for the yellow jersey, a policeman stationed at the entrance to one of the fortress's numerous tunnels misdirected the pair. By the time Bahamontes and Gaul had found their way out of the tunnel and on to the proper route again they had been caught by the peloton and they all reached the finish together. Su\u00e1rez complained bitterly afterwards that Bahamontes had failed to let him 'ride my own race, one that does no one any harm, particularly as all the favourites were asleep'. Bahamontes responded: 'Even if it had worked out, all we'd have got would have been a stage win.' He was clearly sticking to his line that the idea of Su\u00e1rez fighting for the overall classification was inconceivable.\n\nIf the Spanish were merely bickering, the French were riven by serious in-fighting. Sprinter Darrigade, the race leader, and the French squad's co-leader Jacques Anquetil were now refusing to talk to each other. Anquetil had apparently not taken kindly to Darrigade working in the first day's break which had directly helped Bahamontes. It looked like a minor tiff, but in fact it was just the beginning of the French team's troubles. The underlying problem was that all of their top contenders were in the same squad. It contained Anquetil, three-times Tour winner Louison Bobet, 1958 runner-up Raphael Geminiani and the up-and-coming Roger Rivi\u00e8re. Team trainer Marcel Bidot had tried to avoid conflicts before the start with the so-called 'Poigny pact' between all four under which they agreed to work for the common goal of a French win. 'It was risky having so many big names in the same team,' Bahamontes recognises. 'But there was more to it than that. Anquetil and Rivi\u00e8re were both time-triallists and that intensified their rivalry. They disliked each other off the bike, too. As for Bobet and Geminiani, they were at the end of their careers and this Tour represented their last chance to go out on top. Finally, when you're only in the same team for one race like the Tour, money becomes a lot more important, too.'\n\nThings had gone awry between the 'Big Four' in the previous year's Tour. Geminiani partly lost the race in the Alps because Rivi\u00e8re and Anquetil had been unwilling to help him. When Geminiani reached the finish line he realised his best chance of winning the Tour had gone up in smoke and he accused Anquetil and Rivi\u00e8re of being 'more treacherous than Judas'. That hardly augured well for the 1959 race. However, in addition to such a strong senior squad, there were three other French teams, the so-called 'regionals', who might lend a hand. Indeed, forty per cent of the riders in the Tour that year were French. For Bahamontes, his best hope was that there were too many big names in one single squad for power to be shared successfully. As Bahamontes put it with his usual directness: 'Which director was going to be brave enough to argue with any or all of them, and even more so when the Tour was the one race that they all rode for the same team?'\n\nThe Spanish team was almost unrecognisable at the end of stage three. Just like the rest of the peloton, they were covered in mud from head to foot, having ridden more than half the course through an interminable series of treacherous, waterlogged country lanes in northern France. By the time they reached the finish at the Roubaix velodrome, their nerves were shattered and their bodies jangled by the huge cobblestones that pockmark that part of the country. _MARCA_ rather predictably dubbed the stage 'Hell'. The Spaniards suffered twelve punctures between them and Rene Marigil damaged his bike so badly he had to change it three times. Su\u00e1rez had to change his twice. Remarkably, Bahamontes neither punctured nor crashed. Others were not so fortunate. While most of the Spaniards, including Bahamontes, had avoided the one big pile-up of the day, 1947 winner Jean Robic fractured one of his fingers. He struggled on but never fully recovered and was finally eliminated for falling too far behind on the stage to Chalone-sur-Sa\u00f4ne, just two days before the finish.\n\nWhile the Spaniards were never experts in racing over cobbles and had enough trouble staying upright, the battle for the overall continued and the classification had changed considerably. Fortunately for Bahamontes, though, they were not important in the long-term. The biggest change had come after an opportunist ten-man move, not containing any top favourites, had broken away early on the Grammont 'Wall' in Belgium, which had caused Bahamontes such grief in 1957. Since none of the riders was considered to be any real threat, the break was allowed to gain more than eleven minutes on the main peloton. The French were particularly happy to see it go, gambling that their one representative in it, Robert Cazala, would be fastest on home soil in the velodrome. The gamble paid off. Cazala, who was only in his second year as a professional, not only won the stage but found himself leading the biggest bike race in the world. He would stay in the yellow jersey for a further six days, the highpoint of his career.\n\nFor the next week the race wound its way first westwards to the Atlantic, then southwards to the foot of the Pyrenees at Bayonne. Bahamontes concentrated on staying out of trouble. 'There is a midweek time-trial, which though it's short, is where I would like to make my presence known,' Bahamontes told reporters early on in the Tour. 'But essentially I'm riding with the same kind of strategy that Gaul did in 1958: lay low and wait for the mountains.' Cunningly, he even opted to shadow Gaul, who had a much stronger team to back him up. Though not particularly great mountain riders, they were experts at sheltering their climber against attacks by the French and Belgian squads. Where one climber could shelter, Bahamontes reasoned, so could two.\n\nIf Bahamontes' spirits were rising 'the further away we get from those _pav\u00e9s_ and the horrible weather in the north', the race itself suffered a major tragedy en route when a five-year-old child was hit and killed by a vehicle from the Tour's publicity caravan on the stage to Rennes.\n\nThe time-trial on stage six was won by Roger Rivi\u00e8re, though it had little long-term significance. However, during the race against the clock Bahamontes felt he had been the victim of yet another French 'conspiracy', and he made sure the world knew it, too. This time Bahamontes' opportunity to accuse his rivals revolved around the starting order for the riders, something usually decided by their position in the overall classification. Bahamontes, being twelve places better placed than Jacques Anquetil, should have started the time-trial later than the Frenchman. Instead he found at the start-line that Anquetil was starting immediately after him, not before. The disadvantage for Bahamontes was that Anquetil would be able to use him as a 'moving target' and adjust his pace accordingly. On top of that, given Anquetil's position as one of the world's top time-triallists, Bahamontes was sure to be overtaken. Being 'caught' is always a huge blow to any rider's morale and even more so for a rider with a fragile ego like Bahamontes. His vigorous protests were ignored and he had to make the best of a bad job. When he was caught by Anquetil, Bahamontes simply fell in behind and adapted his pace to the Frenchman's, limiting the time deficit to two minutes. He even sprinted for the line, beating Anquetil by a wheel's breadth. The gesture was not received well by the race organisers, who promptly fined him one thousand francs. Apart from the financial penalty Bahamontes had no other reason to complain. He had finished tenth on the stage, the best-place Spaniard, which helped reinforce his position over Su\u00e1rez. In addition, while Rivi\u00e8re had gained three minutes on Bahamontes, none of the other favourites had taken more than two. For someone who rode only reasonably well in time-trials, as a damage-limitation exercise it was a significant triumph. Even better, the Tour now headed southwards towards the hotter weather. 'I function on solar energy,' Bahamontes liked to say, 'and with any luck my rivals will melt on the roads of the Pyrenees.'\n\nAs it turned out, he did not have to wait that long for two opponents to fall by the wayside. On stage nine from Bordeaux to Bayonne temperatures soared into the high thirties. After one hundred and fifty kilometres of racing through the airless pine forests of south-west France, the peloton was ready to crumble under the slightest pressure. The honour of wrecking the peloton fell to the Belgian team, who had been given a roasting by their director at Bordeaux for failing to ride aggressively enough. So it was predictable that it should be one of their number, Marcel Janssens, who led the attack. Janssens's move quickly split the bunch and twenty-two riders forged ahead, including most of the favourites. However, both Raphael Geminiani and Su\u00e1rez missed it, as did Cazala, the race leader since Roubaix. Su\u00e1rez finally lost six minutes, while Geminiani finished twelve minutes down, thereby relinquishing all hope of victory in Paris. France's 'Big Four' had become the 'Big Three'. Eddy Pauwels, a young, inexperienced Belgian rider, took over in the yellow jersey, but with the Pyrenees looming he had little hope of remaining there for long. Bahamontes, meanwhile, was poised to strike.\n\nSeventeen kilometres long, the Tourmalet is the highest and most formidable mountain pass of the Pyrenees. Its nickname in French is _l'incontournable,_ the unavoidable one. 'We gave it that name because it's the only way through that part of the Pyrenees,' Phillippe Bouvet, _L'Equipe_ 's head journalist, told the British magazine _Cycle Sport_ a few years back, 'and because it's been a part of the Tour for so long all of its riders must face it at one point or another.' In 1949, the Tourmalet was the starting point of what remains arguably the most spectacular comeback in Tour history. Fausto Coppi was thirty minutes behind the leaders when the race reached the climb, but his attack there was so stunning he went on to win the race by an incredible ten minutes. Now, ten years later, the riders would take on the biggest single challenge of the Tour on the race's first day in the mountains when they often had problems adapting physically from multiple stages on the flat.\n\nStrange, then, that when eleven riders moved ahead in the 1959 race just before the Tourmalet, the French team failed to react. Surely they would want to control one of the most important stages of the race and win it? Instead, the breakaway group opened a gap of eleven minutes by the foot of the mountain. Behind, the phony war of the Tour's first ten days continued. Finally, after an hour of waiting for the French to increase the pace and start reeling in the break, Bahamontes could contain himself no longer. Eleven kilometres from the summit, on a series of broad hairpins just before the village of Bar\u00e8ges, he charged up the road. It was the first big mountain attack of the Tour and it looked very good indeed. Gaul, Rivi\u00e8re and Anquetil followed at first, but with another blast on the pedals, Bahamontes and Gaul were clear. This could have been a race-winning move, the sort that has earned riders countless victories in the Tour. But then, to the bewilderment of the crowds of Spanish fans lining the route, Bahamontes eased back, inviting Gaul to move ahead and keep the pace high. Gaul did so, but noticing Bahamontes' reticence he did not go flat out either. By the summit their advantage over the first chasing group of Rivi\u00e8re and Fernando Manzaneque was two minutes eight seconds: a good margin but by no means decisive. From the summit of the Tourmalet there was another twenty-eight kilometres to the finish at Bagn\u00e8res-de-Bigorre. But rather than widen the gap Bahamontes and Gaul crossed the line with an advantage of just one minute twenty-two seconds over Anquetil, Rivi\u00e8re and Bobet.\n\nThe big question after the stage was why Bahamontes and Gaul had failed to put more daylight between themselves and the French when they had the chance. Even the traditionally loyal _MARCA_ ran a headline the next day: 'Disappointing Performance By Top Climbers on the Tourmalet', which was a none-too-veiled criticism of the Spaniard's failure to drive home his advantage. Bahamontes' explanation was that he had wanted to prove to the French who was the strongest man in the mountains, but with Gaul on his wheel he had not wanted to risk an all-out attack. 'I'm biding my time,' was all he would say. 'I'll go for it when it suits me.' With the benefit of hindsight, Bahamontes' strategy was probably the right one, even if it inevitably provoked a sense of anti-climax. If he and Gaul had opened up a bigger margin on the rest of the field with two weeks' racing still to go they would have had an uphill task defending their lead all the way to the Alps and beyond. The French would almost certainly have become more united against the two clear enemies, and for Bahamontes with an inexperienced Tour team to back him that was too great a risk. At least short-term, then, Bahamontes needed Gaul as an ally, not as a rival for yellow.\n\nBahamontes certainly seemed fairly relaxed. In his daily column for the local Toledo newspaper _El Alc\u00e1zar_ , he barely discussed the race during the Pyrenees. Instead he thanked the 'Frutos Ramos' wholesalers of Toledo for sending a box of fresh Spanish fruit every day for the team. He also claimed, in his latest conspiracy theory, that the Italians were getting preferential treatment because 'they've posted a supporter on every corner of the Pyrenean climbs with bottles of water. I hope the race officials do something about this so we're racing on a level playing field'. But there was no comment on why after being so gung-ho in the flatter stages, he was suddenly no longer as aggressive as he could have been in the mountains.\n\nHowever, there were good reasons for Bahamontes' patient policy. The French squad still dominated the race after one day's racing in the Pyrenees, but their ferocious discussions about who was the leader continued apace. The longer Bahamontes sat back, it seemed, the worse the disagreements grew. 'Our boat is still sailing, but it's leaking from all sides,' was French trainer Marcel Bidot's graphic description of their disunity. 'We have to work harder together and control the breaks, because that's exactly what we didn't do today.' Bidot had formulated the famous 'Poigny pact' between France's top four riders. But rather than work in their favour, the pact became an excuse for the quartet to watch each other like hawks for the slightest sign of weakness as well as failing to agree basic tactics like chasing down breaks.\n\nWith Bahamontes not willing to declare open war, and the French too divided to do so this lack of aggression led to some bizarre consequences. The second stage through the Pyrenees was won by Darrigade, a sprinter, not a climber. Even stranger, the Tour left the mountains with the overall classification led by Michel Vermeulin, a rider for the lowly Paris-Nord'Est team whose full-time job was as a telegraph operator in Paris. Vermeulin was part of the eleven-man break over the Tourmalet, but he could have had little idea of what effect the move would have or that he would end up wearing the yellow jersey. Anquetil and Rivi\u00e8re could only look on enviously from the sidelines. 'The Tour is being taken over by greenhorns,' was Bahamontes' acerbic analysis. 'Something's going to have to happen.'\n\nBahamontes' words proved prophetic. On stage thirteen's trek through the hills of central France the race was finally blown apart. The French were mainly responsible, but it was a day in which the cracks in their team became chasms. For two hundred and nineteen kilometres, on a day of blazing heat and rugged terrain, the Poigny agreement disintegrated as the French relentlessly chased down each another's attacks. Anquetil remained nominally in contention, but France's chances of winning the Tour had suffered a serious setback. Meanwhile, in the muddle of the mutual self-destruction, eight riders failed to finish within the official time limit and were eliminated from the race. The two riders who gained most were Bahamontes and Henry Anglade, the French national champion who bizarrely rode for the regional Centre-Midi team rather than the country's top squad. The rider who suffered most was the winner of the previous year's Tour, Charly Gaul. He had been forced to stop at one point on the Col de Monsalvy climb to beg a farmer for some water and eventually crossed the line more than twenty minutes down on Bahamontes. As he put it: 'I was so exhausted I thought I would end up crawling all the way there in my underpants. I could hardly see, there was so much sweat pouring down my face.' The heat was so extreme that even at 10.30 in the morning, before the stage started, Tour officials distributed free sunglasses and sun cream to the riders. 'We Spanish didn't wear them, but we put them in our pockets,' Bahamontes recalled later, 'just in case.' Gaul, though, was in need of far greater help.\n\nGiven how badly the stage ended for the French, it was ironic that it was their decision to attack which caused them to suffer such a stinging defeat. After fifty kilometres, the French 'A' squad had a team worker, Rene Privat, and Darrigade up the road in a seven-man break. By the first feeding station at Rodez, when the leading group had only a two-minute advantage, Bahamontes attacked. The move had been planned well in advance. Bahamontes had even ordered the Spanish team's car driver to wait with his food bag on the opposite side of the road and wear a red jersey so he would be sure to see him. Nonetheless, with one hundred and forty kilometres left, it looked an insanely bold move. Despite the riders already suffering badly in the heat, Anquetil responded immediately by jumping on Bahamontes' wheel. Rivi\u00e8re and Bobet, happy to see one of their internal rivals take part in such a suicidal long-distance attack, opted to stay in the bunch. Four other riders, among them Anglade, reigning world champion Ercole Baldini and his team-mate Roberto Falaschi of Italy, also followed Bahamontes and Anquetil up the road.\n\nFor thirty kilometres, the six-man chasing group hovered somewhere in no-man's land between the early break of seven and the bunch. Finally, on a small third category climb, after Falaschi had driven so hard he cracked completely, the two front moves merged. When Radio Tour reported that the two leading breaks had become a twelve-man group, panic overtook the peloton. After sitting so smugly behind, Bobet and Rivi\u00e8re quickly realised that rather than overplaying their hand, Anquetil and Anglade now held the trump cards. Bobet lost his nerve and broke away from the bunch in an attempt to bridge the gap alone. But Anquetil, on hearing the news that Bobet was chasing, decided to increase his pace at the front of the leading group as well. The situation, then, was that one French rider from the 'A' team was chasing while his team-mate was riding as hard as possible ahead. The much-vaunted French _entente cordiale_ had suddenly come apart at the seams. Quickly realising that this was too good an opportunity to miss, Bahamontes started to collaborate with Anquetil. Behind yet more French riders came to Rivi\u00e8re's assistance, while ahead Anglade, Bahamontes and Anquetil had formed what proved to be a highly effective working alliance.\n\nIn the blazing heat the leading break shrank to nine. Privat was so exhausted he later collapsed on the line and needed to be taken to hospital. However, Bahamontes, Anquetil and Anglade were in no mood to throw in the towel despite the depleting numbers. On the second category climb of Montsalvy, sixty-five kilometres from the finish, Bobet's lone chase between bunch and break came to an end when he cracked completely. He was overtaken by the Rivi\u00e8re-led group and ended up twenty minutes down, all hope of adding another victory to his three previous Tour wins totally destroyed. He was not the only rider to suffer so badly. Gaul reeled off the road on the Montsalvy and dismounted in a desperate search for water. He, too, lost all chance of victory. The bunch began to crumble as Rivi\u00e8re tried to raise the pace even more. But he had run out of _domestiques_ to support his counter-attack and only the Belgian Jos Hoevenaers offered some feeble collaboration. The rest were exhausted. Whatever orders Bidot had given to Rivi\u00e8re behind or Anquetil in front, they were ignored. The speed in the break ahead was so great that press cars and officials had serious problems overtaking them on the Midi's narrow, winding roads. When they finally did, close to the finish at Aurillac Velodrome, it was to find the leading break had now shrunk to just four: Britain's Brian Robinson, Anglade, Anquetil and Bahamontes. 'Robinson went for it [in the closing metres] for the stage win, and I followed him, but I was too impetuous,' Bahamontes recalls. 'We reached the velodrome together, but then Anglade went high up the side of the track and he used his speed from descending again to overtake me to win. Then I saw Anquetil pass me, too.' Bahamontes took third, Robinson fourth.\n\nBehind the four stage leaders, it was carnage. Hoevenaers, who took over the yellow jersey, led in a group containing Rivi\u00e8re at nearly four minutes, while Gaul and Bobet crossed the line a massive twenty minutes forty seconds down. To add insult to injury Gaul had another thirty seconds tagged on to his time for accepting pushes from spectators. Bahamontes, meanwhile, had shot up the overall classification from ninth to fifth and was just over seven minutes behind the leader; twenty-four hours earlier he had been more than fourteen minutes behind.\n\n'We've wiped out Gaul and Bobet, and Rivi\u00e8re is almost out of the running, too. It has been an excellent day,' Bahamontes reflected. It was made even sweeter by news that Su\u00e1rez had been forced to abandon. As the Tour reached its crucial phase Bahamontes found himself in an ideal position overall. Barring the inexperienced Anglade, he had become the best-placed climber, and still to come was a mountain time-trial followed by three major stages in the Alps. Suddenly Bahamontes had moved from being just another potential contender to the rider most likely to win the Tour.\n\nSince a Spaniard had never won the race before, the country was understandably abuzz with the news of Bahamontes' success. For the first time in his career, Bahamontes felt certain he could live up to their expectations. 'The war has started and it is the most ferocious combat on two wheels of recent times,' reported the newspaper _Informaciones._ 'Bahamontes is the favourite of this year's Tour.' _El Alc\u00e1zar_ even put a photograph of Bahamontes on its front page for the first time during the 1959 Tour with the headline underneath: 'Congratulations, Fede . . . Fede could become the real star of Spanish cycling.' The Toledo publication added: 'Even Jacques Goddet came up to our newspaper's correspondent at Aurillac, pointed at Bahamontes and said, \"There you have the winner of this year's race\".' Bahamontes would later recall: 'At Aurillac, the time and the day had finally arrived that I was sure that I could win.' However, at the time he did not let the opportunity pass to criticise his team-mates and the rest of the break, saying yet again: 'Nobody would help me.' In his later years he claimed he was fully focused on his future Tour de France win. 'The prince was about to become a king and the eyes of an entire country were waiting for my coronation.' It would not be a long wait, either.\n\nAlmost every account of the 1959 Tour, at least outside Spain, has perpetuated the widespread belief that the race that year was all but decided between the agents for the criteriums, the closest that some top riders of the era, including Bahamontes, had to a manager. The theory is straightforward: Rivi\u00e8re, Anquetil and Bahamontes had the same agent, Daniel Dousset, while Anglade was with Roger Piel. Therefore, it was in Anquetil and Rivi\u00e8re's interests that if they could not win, then another Dousset 'client' should, namely Bahamontes. 'The 1959 Tour was the war between the managers,' says Raymond Poulidor. 'Dousset was the well-established figure \u2013 I used to call him Al Capone \u2013 and Piel the breakaway competition. There wasn't any difference in what they charged: both got ten per cent. But Dousset was only interested in big riders, and Piel was happy to go with the smaller ones while they were still on the way up. In 1959, [that was] Anglade. By the time Dousset got to me, though, I was with Piel; it was too late.' Not everybody agrees that Piel and Dousset made such a big difference to the race. Raphael Geminiani, himself a wheeler-dealer of considerable skill, and unlike Poulidor, part of the 1959 Tour peloton, is convinced Dousset's role was minor to the outcome. But the myth has persisted. Three years later, in 1962, when a satirical 'A\u2013Z of cycling' pamphlet did the rounds of the Tour de France press room, 'A' was for [Rudy] Altig \u2013 described as 'a treacherous German in a Spanish comedy' in reference to his 'robbing' team-mate Anquetil of the 1962 Vuelta victory \u2013 'D' for Dousset: 'When he arrives, the Tour is over and everybody can go home.'\n\nWhile the theory that Dousset helped shape the outcome of the Tour probably contains some grains of truth, it is impossible to prove. Either way it is certainly far too simplistic to say: 'Bahamontes won the Tour thanks to Anquetil and Rivi\u00e8re.' As Robert Millar once said about the Vuelta a Espa\u00f1a he lost to Pedro Delgado because of a Spanish combine in 1985: 'Other riders can't make you win, but they can make sure you lose.' That is almost certainly what happened to Anglade in 1959. His compatriots' tactics wrecked his race. Not that he would definitely have won, but what they wrecked was his _chance_ to win. And there is a big difference between the two. A more pertinent question is whether Anquetil and Rivi\u00e8re combining against Anglade, with perhaps Dousset as 'puppet-master', devalues Bahamontes greatest win? The answer has to be no, not least because if that is the case then numerous other victories in major Tours during the 1950s would also have to be discredited.\n\nEven if we rewind just one year to 1958, the French national team's refusal to help Geminiani, who, like Anglade, was in a regional team, arguably cost their fellow Frenchman the Tour, too. But does anybody claim Charly Gaul's victory was a gift, even if he benefited from these divisions? 'Bahamontes was strong enough to be there in the breaks and there on the attack,' points out _L'Equipe_ 's veteran cycling journalist Philippe Bouvet. 'Whatever the French got up to, nobody can take that away.' Anglade himself has refused to condemn the combine that lost him his chance of winning the Tour; half a century ago such skulduggery was far more part and parcel of racing than it is today. Besides, for Bahamontes to win he had to be in the right place at the right time. As Raphael Geminiani pointed out to me: 'It's not like Anquetil and Rivi\u00e8re were called up on the phone by Dousset during the stage. They decided to do what they did, not Dousset. He wasn't in the French national team car giving orders.'\n\nTwo days after Bahamontes' knock-out performance in Aurillac, the Tour's next big challenge was a 12.5-kilometre uphill time-trial at the Puy de D\u00f4me, an extinct volcano near Clermont Ferrand. The road up the Puy is well-surfaced and not particularly narrow, and appears deceptively easy at first; it is the second part that causes the damage to riders. Like a helterskelter at a fairground its steepest section winds remorselessly round and round the edge of the Puy to the summit. On the right, painted on the cliffside, are the names of former winners: on the left, a low stone wall, and then a precipice. Though not as long and demanding as the single hardest climb in France, Mont Ventoux, the Puy's gradients are far more challenging. And, like Ventoux, there are no breaks, either, no false flats to catch your breath; it is just one long exercise in pain. The climb had only been used on one previous occasion, during the 1952 Tour, and its appearance then had an encouraging historical precedent for Bahamontes. Seven years before, his trade team boss Fausto Coppi had blasted off to victory there, dropping Dutchman Jan Nolten in the last three hundred metres. However, there was one crucial difference: by the time the Tour reached the Puy de D\u00f4me stage in 1952, Coppi already had the race sewn up; in 1959, Bahamontes still had to prove he could win it.\n\nOne of Bahamontes' weapons on his most important Tour de France stage win was an artificial stimulant, albeit a legal one. 'I didn't use to drink coffee in those days, but as I was heading towards the start of the stage I was feeling nervous. I knew it was going to be a very important day,' Bahamontes recalls. 'So when I saw a publicity vehicle from the Tour I stopped and grabbed a coffee. An expresso. Then, because it didn't look like much in the cup, I thought, \"Hey, have another one, at least that way you'll notice you swallowed something\".'\n\nApart from the caffeine coursing through his veins, Bahamontes had other advantages on his side as the Puy de D\u00f4me loomed, the most important being that standing-start mountain time-trials were his speciality. Indeed, he had won the early-spring hill-climb to Arrate on several occasions. This was only the second time a mountain time-trial had been held during the Tour and Bahamontes had finished second to Gaul in the first one on Mont Ventoux the previous year. He was, therefore determined to make the most of it. In his favour, for the rider who 'functioned on solar energy', was the heatwave France was still enduring. Bahamontes says Langarica told him it was the hottest summer in France for a century. By way of a morale boost the team director cunningly pointed out that Gaul would be suffering more than most in the searing heat as a result. Additionally, for once the Spanish were not to be outdone technically, either. A special combination of gears was put on Bahamontes bike \u2013 rear cogs of fourteen, sixteen, nineteen, twenty-one, twenty-two and twenty-three with a forty-four and fifty-one double chainring on the front \u2013 were designed to cope with the opening section on the flat. He also had special lightweight silk 'tub' tyres, which were sometimes risky in wet weather but not in the baking heat. Not huge advances by any means, but given the usual ramshackle nature of the team they made a difference.\n\nThere were other, smaller issues that were tilting things towards Bahamontes. Rather than driving the Spanish team car, Langarica opted to open the car's sun-roof and lean out of the window to give his star rider orders and encouragement. For Bahamontes, with his love of dramatic gestures, having his director so close was an important plus. Then, though Bahamontes did not know it, Gaul was not his only key rival in trouble. Clermont-Ferrand is an important railway junction and Rivi\u00e8re, who would be starting two minutes before, had been kept up all night by the noise from trains in the marshalling yard close to his hotel.\n\nMost importantly of all, perhaps, the mountain time-trial was a section where Bahamontes needed no external assistance or team-mates. On the Puy de D\u00f4me he was able to do what he always said he did best and 'move alone': and moving alone, he soared.\n\nEven before Bahamontes reached the steepest uphill section it was clear he was going to have an exceptionally good day. After four and a half kilometres of rolling terrain he was clocking the fastest time and was already fourteen seconds ahead of Gaul, fifty-one seconds up on Anquetil, and more than a minute in front of Rivi\u00e8re. With his shorts rolled up extra high so he felt more comfortable, and sitting in his classic climbing position, hands slumped slightly over the handlebars, Bahamontes said before the start he was 'super-nervous and ready to eat the road'. Clearly the coffee had already had quite an effect. Before halfway he had Rivi\u00e8re in his sights and two-thirds of the way up he overtook the Frenchman. Needless to say Bahamontes was the fastest through every checkpoint. By the time he reached the summit the damage he had inflicted on the field was colossal: Gaul had beaten him by thirty-one seconds on the more difficult Ventoux climb in 1958, but over such a short distance he was the closest at one minute twenty-six seconds. Anglade was at three minutes and the two great French time-trial specialists, Rivi\u00e8re and Anquetil, were even further back. This was not a defeat, it was a rout. While Jos Hoevenaers clung on to the yellow jersey, Bahamontes was just four seconds behind, and with the Alps to come his path to overall victory looked increasingly free of obstacles.\n\nNow only Anglade, thirty-nine seconds behind Bahamontes, could be considered a viable threat. Anquetil, the strongest of France's 'Big Four', was now more than five minutes back, and Rivi\u00e8re more than seven. Bahamontes revealed that Rivi\u00e8re's relatively strong performance in the middle section had been the spur to do even more damage, exactly at the point where the climb was steepest and hurt his rivals the most. 'I'd decided not to go too hard,' Bahamontes said, 'because I didn't think the gaps would be that big in such a short a distance. I thought I'd catch Rivi\u00e8re quite quickly, though, and I was surprised that I didn't so I started going flat out to try and get past him. It was only afterwards that I realised he had got fourth and I discovered I'd done a far better time than I'd thought possible.' Bahamontes' time was so good, in fact, that he was the only rider to average more than 20 k.p.h.\n\nThe French took their defeat in radically different ways. Anquetil made the most of Rivi\u00e8re being overtaken by Bahamontes to take a sideswipe at his rival and tell reporters: 'That'll teach Rivi\u00e8re to boast.' As for Jean Robic, who finished fifty-second, nearly ten minutes back, he gave the most laconic of answers when asked about his experiences. 'It went brilliantly,' he said, 'I actually had to brake quite often to try and slow down a bit.'\n\nAmazingly for such a short distance nine riders finished outside the time limit. However, the race officials applied a rule that allowed them arbitrarily to increase the percentage of the winner's time within which riders had to finish. As a result, only four were eliminated. One, though, was Bahamontes' team-mate Aniceto Utset, the 1956 Volta winner. 'That was our only worry after the Puy de D\u00f4me,' Bahamontes said. 'I was running out of team-mates. By that point, I only had six left with three big Alpine stages to come.' However, two of the other three were powerful French _domestiques_ and that could only strengthen his hand.\n\nThe loss of Utset had reduced the Spanish team from twelve starters to just seven. Under the circumstances it was important to look for allies. Bahamontes did not have to look far. Charly Gaul had been performing so badly there were rumours that he had actually left the race, and the 1958 winner was keen to show his face at the front.\n\nStage seventeen to Grenoble, where Gaul and Bahamontes launched their joint attacks, did not look overly promising. Of its two second category climbs, la Romeyere, where Bahamontes had eaten his ice cream back in 1954, was more than sixty kilometres from the finish. Assuming their rivals were organised then in theory they would make mincemeat of the two climbers on the long, flat run-in to Grenoble. However, that was the point: Bahamontes' main rivals, the French, _weren't_ organised. They had shown that again the day before. Rivi\u00e8re and Anquetil had planned a joint ambush of Bahamontes on the lumpy run through the Massif Central from Clermont-Ferrand to Grenoble. Rivi\u00e8re was so convinced of his chances he had even requested a special gear, a thirteen-cog sprocket, so he could blast away that much quicker. However, when Rivi\u00e8re was revving up for the move on a descent where they knew Bahamontes would be most vulnerable, Jean Grazyck, a French _domestique_ , came rushing through the peloton with a message. He said: 'Jacques says to tell you he's not coming. His legs hurt.' And so the attack was aborted. Even more telling was that Anquetil did not bother to talk in person to Rivi\u00e8re: with internal communication so limited it hardly suggested they were really prepared to collaborate against Bahamontes.\n\nThen again, perhaps they did not need to worry, given that Bahamontes' first attack on the stage from Saint Etienne to Grenoble the next day bordered on the suicidal. Desperate to take the yellow jersey, and ignoring Langarica's orders for once, Bahamontes rocketed off the front on the Col de Gran Bois with more than one hundred and ninety kilometres left to go. He was joined by Gaul and a Frenchman from Anglade's Centre-Midi squad, Francois Huot, but their lead was only thirty-five seconds at the summit. This was too small and too soon in the day. The trio eased back and were absorbed by the bunch. On the Romeyere, though longer and with some brief, steeper sections, Bahamontes went for it again. Taking advantage of a crash that split the bunch, he attacked 'and this time I wasn't going to stop for any ruddy ice cream at the top'.\n\nInitially alone, Bahamontes blasted up the steady but largely shallow climb with such speed that by the summit he had a sixty-seven-second advantage on Gaul, and was nearly three minutes clear of the bunch. While there was no stop for a cornet, food was still on Bahamontes' mind, an indication that he recognised that the sixty kilometres to Grenoble would be no easy task. 'He crossed the line at the summit of the Romeyere yelling, \"Bananas! Bananas!\" at us,' _MARCA_ reported. 'We weren't sure what was happening, but if he had the strength to shout for food, we knew he had the strength to pedal too.' In what is surely one of the more surreal moments in Tour de France history, _MARCA_ 's correspondent and his colleagues then embarked on a 'banana hunt' to find Bahamontes the food he had requested. The sight of a bunch of sweaty, middle-aged foreign journalists, dressed primarily in singlets and shorts, bursting into local houses and asking for bananas is worthy of French farce at its best. But their mission proved unsuccessful. 'We coudn't find any bananas, not one, all the way to Grenoble,' _MARCA_ reported mournfully, 'and we checked in a lot of houses.'\n\nCrucially, rather than plough on alone, Bahamontes soon realised that Gaul was 'the best company I could have. He needed the stage win, I needed the lead. He was miles back overall and no threat to me. It all made sense'. Equally importantly, Gaul was able to guide Bahamontes on the descent. By the foot of the Romeyere the duo's advantage had stretched to four minutes. Then, even on the flat valley floor that stretches for nearly fifty kilometres through Voreppe and Saint-Egreve en route to Grenoble the two climbers managed to hold off an entire pack. While the conspiracy theorists point to this failure to reel them in as another sign of French indifference to Bahamontes winning the Tour, media reports at the time suggest that on this occasion the French were united in trying to close down the breakaways. It was Rivi\u00e8re who had co-led the chase early in the stage and Rivi\u00e8re again who worked hardest to keep Bahamontes and Gaul at bay towards Grenoble. On neither occasion did the chase work. But rather than their failure to catch the two escapees, there is a more important question: why did Rivi\u00e8re, and to a lesser extent Anquetil, try to keep Bahamontes out of the yellow jersey and maintain Anglade's chances on one day, yet once Bahamontes had the lead why did they then collaborate to keep him there and wreck Anglade's Tour the next? Assuming that this was a sophisticated anti-Anglade conspiracy, the answer could be that on the road to Grenoble Rivi\u00e8re and Anquetil's plan was to make a symbolic contribution to Anglade's chances rather than a real one. That they only pulled back twenty-seven seconds of the four-minute deficit over forty-five kilometres lends strength to this theory. From the French team's point of view, the advantage of keeping Anglade in contention was that the roles of Rivi\u00e8re and Anquetil as 'king-makers' would remain intact, roles which if Bahamontes became a clear Tour winner at this juncture, would be irrelevant. They needed to maintain the illusion of an Anglade-Bahamontes duel for as long as possible before their cards were turned up on the table.\n\nIn any case, the results still stand and they cannot fail to impress. Whatever the reasons for the failure to chase down Gaul and Bahamontes, they cannot detract from stage seventeen of the Tour de France 1959 being one of the few occasions when two lightweight climbers managed to break and stay away from the pack over a forty-five-kilometre stretch of flat terrain. It was a monumental achievement. On top of that, while Bahamontes showed strategic sense in waiting for Gaul and striking the bargain with him, he would have needed to ride to the limits of his physical strength to maintain that deal in the two-hour battle against the peloton.\n\nWhen they reached the Grenoble velodrome Bahamontes had to respect the unspoken deal with Gaul and let him win the sprint. In practical terms he had no choice: he had a punctured front wheel. 'The crowd whistled and booed when I let Charly get the stage,' he recalls. 'It was only when the race commentator held up the front wheel between his finger and thumb and showed them I had punctured that they started to applaud me, too. Letting him win was the right thing to do. I had got the yellow jersey and increased my lead in the King of the Mountains competition. That was enough.' Bahamontes was not the first Spaniard to lead the Tour (Miguel Poblet had worn the yellow jersey for a day in 1955 in Dieppe), but with a four-minute fifty-one-second advantage over Anglade, he was certainly the first from Spain with a real chance of winning it. 'We just thought he was going for the mountain points,' Anquetil told _Informaciones_ at the finish. 'If he isn't just interested in the King of the Mountains, then the Tour is his.'\n\nAs usual, Bahamontes opted for a melodramatic description to sum up his achievements. 'Grenoble was the city where three hundred years before Bayard the \"Knight With No Fear\" had died, I showed I had no fear, either,' he wrote in his privately published account of the 1959 Tour. It was not strictly true as Bayard was born close to Grenoble, but died in Italy, though Bahamontes' point was clear. 'The Alps were still to come,' he went on, 'but for me the race was done and dusted because I was clearly superior in all areas. The only possible problem was the lack of a strong team.'\n\nIn his daily column in _El Alc\u00e1zar_ , Bahamontes said: 'Getting the yellow jersey was not as hard as I expected. In my terrain, the mountains, nobody bothers me. Anquetil is my closest enemy and he is at nine minutes, Rivi\u00e8re and Baldini at over eleven, and Gaul at over twenty.' Pointedly, he did not mention Anglade. However, as he rode around the velodrome on a lap of honour, clad in yellow and soaking up the applause, Bahamontes could be forgiven for not wanting to think about the next day's racing. Just getting this far was impressive enough.\n\nThe eighteenth stage from the Lautaret valley in France to Val de Aosta in Italy started and finished with the same rider in yellow. And Bahamontes managed to maintain an almost identical advantage overall. However, what happened during those two hundred and forty-three kilometres caused one of the biggest scandals in French sport as an unholy, though unspoken alliance between Bahamontes, Anquetil and Poulidor all but ensured the Spaniard, not Anglade, won the Tour.\n\nThe mammoth Alpine stage across the Galibier, Iseran and Petit St. Bernard passes, which included sixty-nine kilometres of climbing, should have suited Bahamontes. Instead, he twice came within a whisker of losing the lead. The French, logically enough, attacked Bahamontes on his two weakest points, the descents and in wet weather, and on both occasions their strategy worked perfectly. But the scandal arose because they then failed to follow through with their initial success: it seemed to be enough for them to have Bahamontes in trouble, not out for the count. The first attack came on the descent of the Iseran, where Geminiani took off on what Bahamontes later called a 'suicide mission', and took with him Anglade, Anquetil and eight of the top eleven overall. Neither Gaul nor Bahamontes were present, and Anglade upped the pace as best he could to haul back some of his five-minute deficit. However, neither Anquetil nor Rivi\u00e8re offered their help. After five kilometres of climbing on the Petit St. Bernard Bahamontes and Gaul had managed to rejoin the main group, thus resolving what Langarica later called 'one of the most dangerous moments of the entire race'.\n\n'They could have had me up against the ropes,' Bahamontes recalled, 'but the fact that Anglade was in a regional team was impossible for them to handle psychologically.' However, if Anquetil and Rivi\u00e8re deserve at least part of the blame, the bulk of the responsibility has to rest on the shoulders of Marcel Bidot, the French team trainer. Bidot had no money to make from criteriums and no deals with Daniel Dousset, just his duty to ensure a French victory. 'Rather than organise the break with Anglade, for some reason Bidot stayed driving alongside me,' Bahamontes recalls, 'to be sure, so he said later, there were no \"dodgy tactics\". Under what were very critical circumstances, given he gave up on what were his real duties, Bidot was a real ally for me.'\n\nThe second crunch moment came forty-five kilometres from home on the descent of Petit St. Bernard. Bahamontes had no problems keeping up with the rest of the pack on the climb, but when the heavens opened on the descent he was in trouble when he punctured. The only good news was that he was not the only favourite to suffer: so, too, did Anquetil, Rivi\u00e8re and Geminiani; one Italian, Michele Gismondi, punctured six times. Indeed, just six riders in the front group, among them Bahamontes and Anglade, did not have to dismount at one point or another in the freezing rain. One by one, the field slowly picked their way down the treacherously wet single-track dirt road. If racing was the last thing on their mind it would have been hard to blame them. That, though, was not the case.\n\nAnglade took advantage of the drop in pace to go clear and steadily increased his lead on the pack. When Gaul and several others joined him, Bahamontes was again in trouble. With only Gomez del Moral to help him, Bahamontes' fear that lack of team support could be his undoing looked more than justified. But again Anquetil and Rivi\u00e8re were willing to give him their support, this time directly as they hammered away at the front of the group to chase down Anglade. In the end, the gap between Anglade's group and Bahamontes was halved to forty-seven seconds. But that was nothing compared to the damage done to French morale. 'At the finish, all that anybody could talk about was the way Anquetil and Rivi\u00e8re had betrayed their country,' Bahamontes said. 'On France's national holiday [14 July] as well. What a great way to commemorate it.' To increase the host nation's humiliation, Louison Bobet abandoned the Tour in the most melodramatic of fashions on the summit of the Iseran. He never returned.\n\n'I had to make a great effort on this stage,' Bahamontes said in _El Alc\u00e1zar_ , 'but I'm sure my enemies have worked even harder. Some people were thinking I would attack, but when you're in yellow you can't take any risks. The shame is that it's not sunny otherwise I would annihilate all my rivals.'\n\nWhile the Spanish press chose to ignore how Anglade had been stabbed in his back \u2013 'an uneventful day' was _MARCA_ 's anodyne description of the stage \u2013 the French were forced to admit that beating Bahamontes was now almost impossible. 'It's not complicated,' claimed Rivi\u00e8re. 'Bahamontes has won the Tour.' Langarica, the Spanish team director, did not agree, claiming that Bahamontes' advantage was insufficient for him to be sure of victory. 'If Bahamontes doesn't attack on the last mountain stage, he could lose the lead in the final time-trial,' Langarica warned. But he then explained, presumably tongue in cheek, that Bahamontes had only lost time on the final descent of the Aosta stage 'because he had more to lose than the rest, so had to be more cautious'. Had it not been for the French disunity, though, Bahamontes might have lost the race completely.\n\nAcross Spain, and particularly in Toledo, the news that Bahamontes was just days away from winning the Tour de France left few people unaffected. 'When the news broke that Fede had taken the yellow jersey, you could hear the cheers right the way across Toledo,' reported _MARCA._ 'Shop windows across the city were filled with posters congratulating Fede on his victory in the Tour. And the demand for yellow ribbons to pin to lapels was so great that the haberdashers ran out of yellow cloth. One photographer's shop displayed an enormous photo of Fede with a yellow ribbon pinned to his chest. And in Bahamontes' bike shop it was as packed as Charmartin [one of Madrid's two main railway stations] on a cup final day.'\n\nCongratulatory telegrams galore were sent to the team hotels, though Fermina Bahamontes reported that in her one telephone conversation with her husband, the connection had been so poor they were unable to hear each other. However, the couple would soon be reunited as she was heading for Paris in a relative's Seat car, along with the president of the Spanish Cycling Federation, Alejandro del Caz. All that was needed now, of course, was for Bahamontes to win.\n\nIf the first Alpine stage of the 1959 Tour was dominated by thunderstorms and French self-destruction, the race's final day in the mountains, from Aosta to Annecy, was a damp squib. Once again, the weather was the main reason. The Col du Gran St. Bernard, the first climb of the day and the most difficult, was shrouded in fog and the riders refused point-blank to attack if it meant risking their necks on the descent. The only excitement came when Gaul started yelling at Bahamontes after a Spaniard, Carmelo Morales, attacked close to the summit. 'His only objective was to stop my rivals gaining any points in the King of the Mountains competition,' Bahamontes primly explained a little later. On the descent, with cars crawling at ten kilometres an hour through the thick fog, the whole peloton regrouped. 'The descent was lethal,' said Bahamontes. 'I had some broken spokes, but I didn't stop for fear of being attacked. Everybody was waiting for a chance to attack me.'\n\nThe Forclaz, far shorter, much steeper, but fog-free, did a lot more damage. An attack by Switzerland's Rolf Graf and regional French rider Gerard Saint shredded the bunch to twenty-two with Bahamontes' support reduced to just Morales. It was at this point that Bahamontes, having played it conservatively all day, took the one option still open to him and attacked. Yet again Bahamontes showed that even if the civil war between the French had all but guaranteed him the Tour, nobody deserved the win more. Anglade attempted to follow, but was dropped, thereby proving that for all the French rivalry, in their first and only full-on mountain duel as the two top challengers, Bahamontes beat Anglade hands down. By the summit, with Gaul by his side again, Bahamontes had a two-minute five-second advantage on the field. For once the French rallied round the common cause of helping Anglade, but they only seemed to do so when it was clear he was going to lose. On the descent Anglade's bad luck continued with a puncture. By Annecy, though Graf won the stage Bahamontes had added another sixty-three seconds to his overall advantage. With his recently-arrived trade team boss Fausto Coppi nodding sagely by his side at the finish, Bahamontes pointed out that: 'The French tried it on the first day in the Alps, but in the second they had to admit they were beaten.'\n\nThe sixty-nine-kilometre time-trial round Dijon remained, but even if it all went disastrously wrong at least Bahamontes now had an unassailable lead in the King of the Mountains classification. Whatever the final result, he would go down in cycling history as the first rider to take the mountains' title three times. 'I had one bottle of champagne ready to celebrate that particular achievement,' Bahamontes said, 'but I had the feeling I might need a few more.' Anglade, now five minutes forty seconds adrift, was more forthright about his chances. 'What will it take to beat Bahamontes now?' he said to reporters. 'Somewhere between here and Paris I'd need to cut his legs off. That might do it.'\n\nThe time-trial was more confirmation of what had been clear since Annecy: that Bahamontes was going to win the Tour de France. The citizens of Toledo continued to dream up more exotic ways of celebrating Bahamontes' victory with the unofficial prize going to the shopkeeper who put a bike in his window with a live eagle perched on top. Bahamontes, meanwhile, prepared for the most important time-trial of his career with a rather more mundane twenty-kilometre training ride through the Bourgogne vineyards. And then he was off. As Bahamontes preferred, Langarica avoided any vocal encouragement from the following car, simply providing him with advice and updates on the number of kilometres covered. At the time, with no radio communication between riders and the team cars, there was no way for Bahamontes could know how his rivals were doing except from the timekeepers' boards at fixed points on the side of the road. There was a brief moment of panic when Bahamontes suddenly slipped to three minutes behind Anglade after fifty kilometres, more than sixty per cent of his overall advantage. However, that was as bad as it got: Bahamontes opened up the throttle and finished ninety-nine seconds down on Anglade, the Tour safely in the bag.\n\nBahamontes explained later that he preferred only to go all-out at the end of the time-trial 'because if you go hard earlier and start thinking you might be about to crack, that could make you feel even more nervous'. Langarica said: 'He's lost more or less what he should have, but we didn't want to go too hard because the race isn't finished yet.'\n\nAs still tends to happen in cycling whenever there is an exceptional performance, certain elements of the press, in this case the Italians, tried to breathe life into some insinuations, albeit minor ones, that Bahamontes was guilty of doping. Even after the time-trial, amid all the euphoria, Langarica felt obliged to respond to these accusations. However, given how clumsy his comments were, he might have been better off saying nothing at all. 'For the record Bahamontes didn't \"take\" anything,' Langarica said, 'although I'm sure there were other riders who did today.' Now, though, barring one final, marathon stage to Paris in which the leader was traditionally not attacked, the Tour was effectively over.\n\nWhile Anglade remained in second place overall, the battle for third was far tighter. Finally, despite Rivi\u00e8re taking a second time-trial victory, Anquetil moved on to the podium, displacing Francois Mahe, whose one moment of Tour glory remained leading the race way back in 1953.\n\nThe media attention was now all on Bahamontes. Yet while the camera flashes popped, the journalists scribbled away in their notebooks and the praise poured in, there were no tears or emotional tributes to those who had helped him along the way. Instead Bahamontes, despite having sealed the biggest prize cycling could offer, kept his eyes firmly on the financial benefits. One of the first questions he was asked was what he would do when he completed the Tour. Bahamontes' disarmingly frank response was: 'Talk to Dousset to see what kind of criterium money I can get. Anquetil, Rivi\u00e8re, Bobet and Baldini race for a quarter of a million francs each, and I don't want any less.'\n\nHis first message to the good folk of Toledo was equally pragmatic: 'Tell the Mayor if he's willing to give me that land he promised me for my house [if I won] then that would be wonderful.'\nChapter Nine\n\n#### 'The Bullfighter on Two Wheels'\n\nFrom the Nationalist Government's point of view, Federico Mart\u00edn Bahamontes could not have chosen a better day to win Spain's first Tour de France. The final chapter of the country's greatest individual sporting achievement of the Franco era was written on 18 July, the day the General\u00edsimo had begun the so-called 'glorious uprising' back in 1936. Fast forward twenty-three years: the newspapers' front pages across Spain carried the usual anniversary photograph of General Franco, complete with thanks to the man who had apparently singlehandedly made it possible for generations of Spaniards to be brought up in an atmosphere of, and to use his words quoted in _MARCA_ , 'peacefulness, hard work and renewal'. However, in 1959, in addition to the images of Franco there were huge photographs of Bahamontes pedalling away during the last three hundred and thirty-one kilometres towards Paris and victory.\n\nThe treatment of the two subjects by a conservative, traditional newspaper like _ABC_ was typical: on its cover it carried a full-size picture of Franco, and inside a lengthy editorial insisting that the Spanish remain 'on guard and alert' against the 'Soviet virus' of socialism and anarchism. 'It has been twenty-three years since the uprising and we can still find new and legitimate reasons to justify it,' _ABC_ trumpeted. 'Look how there is barely the slightest difference between the last will and testament of [Russian emperor] Peter the Great and the Communist International Congress' plans for expansion. But we remain alert and [maintain] the same spirit . . . of 1936, [fighting] against those who attempted to subjugate our country to the rule of the hammer and sickle.'\n\nThe editorial's aim was clearly to remind Spaniards why Franco was so good for them politically. But after a series of strategically-placed advertisements came some of the best evidence about why Franco had been so good for Spanish sport, too: a picture of a triumphant Bahamontes in Paris. The subliminal message was depressingly simple: just as Franco was writing a glorious page in Spanish political history, now Bahamontes was in the process of doing the same in sport, approved and overseen by the ruling regime, of course. No doubt about it: this was a victory signed, stamped, sealed and sent out from the Nationalist headquarters in Madrid in triplicate on the tissue-like multi-coloured folios so beloved of twentieth-century bureaucratic Spain. What else could be possible, given that the Spaniard had managed to win on such a hallowed date as 18 July?\n\nThe telegram sent by the governor of Vizcaya province to Bahamontes and Dalmacio Langarica, care of the Spanish embassy in Paris, shows just how neatly Nationalist and sporting pride blended together thanks to the '18 July factor'. It read: 'Your triumph embodies our Spanish pride and our admiration, enthusiasm and praise goes to the hero of the Vuelta [sic] in a victory for our fatherland that is, above all, [placed] in the indelible context of such a glorious date.' The tone of the telegram suggests that the sender might be congratulating one of Franco's generals on winning a battle against the Communists, rather than the winner of a bike race. Celebrations for Bahamontes' win became so absorbed by the anniversary of the Civil War that it effectively became a political hijacking. Bahamontes quickly discovered this for himself. When he arrived at the Spanish embassy in Paris during the evening of 18 July for the _de rigueur_ round of congratulations, it was to find that they were already in the midst of a party to commemorate Franco's uprising. Bahamontes' success simply became an excuse for an even bigger and more patriotic fiesta.\n\n'Bahamontes' win made an already beautiful date even more beautiful,' exclaimed the editorial in _MARCA,_ the most important sports daily in Spain outside Catalonia. It then went on to outline how victory in such a non-elitist sport like cycling could transform all areas of Franco's supposedly ideal society. 'Bahamontes' triumph pushes the pedals of the priest as he goes to say mass in a village, of the worker as he goes to his building site, of the road-mender as he goes to work on his stretch of road, of the message boy as he rushes through the streets of Madrid at midnight to the newspaper printer's with this article. Football is too specialised in comparison with cycling. From today onwards, all the bicycles in Spain will be lighter, happier, as if the wind of Bahamontes' triumph were pushing them onwards.'\n\nLooked at with fifty years of hindsight, such claims seem corny at best, sinister at worst: a crass line of Fascist propaganda, idealising the working class and reinforcing the illusion of a harmonious pyramid of a society topped off by a benevolent, beloved dictator. Yet it was difficult to find anything else in the Spanish press. The Madrid daily _Informaciones_ , for example, not known for its hard-line pro-Franco stance, had an almost identical presentation of Bahamontes' victory. It started with a speech by Franco and an editorial celebrating the anniversary of Franco's uprising and Spain's successful onward march to economic stability. At the bottom of the front page were photographs of Bahamontes. 'Spain is not rich but it is on the way towards becoming so . . . in twenty years we have done more than in the last hundred,' the editorial claimed. 'With Franco we advance and conquer new international credibility in a world in which Red Imperialism barely had any problems until we, the Spanish, took up arms against them. And now we work in the common task of fighting Communism, whose only real defeat has been at the hands of Franco.' Given Bahamontes' defeat of the French was just underneath these claims it would have been difficult not to associate such political 'victories' with his success. This association continued inside the newspaper. The first report in _Informaciones_ was dedicated to Franco's inauguration of the Ministry of Housing headquarters in Madrid and the handing over of 20,931 low-rent, state-owned houses to needy families. Then came a double-page special describing in lurid detail the 'Communist Plot' of July 1936, which Franco's rebellion foiled. And after that it was straight on to the Tour de France, and another feather in Franco's cap.\n\nCuriously enough, Toledo's 'local' newspaper, _El Alc\u00e1zar_ , was one of the rare cases where Bahamontes' victory was afforded a low degree of prominence. But that was down to its early press deadline which meant it could not include pictures from Paris. Instead _El Alc\u00e1zar_ opened with details of a study showing that Spaniards were on average two centimetres taller than a decade before, and continued with a report of military manoeuvres in Castille, complete with photographs of Franco peering at a model landscape and a spectacular fake nuclear explosion, giant mushroom cloud included. A feature on page four reported on the 'crisis in British pubs'. The article claimed in shocked tones: 'The ones in London date from Queen Victoria's time and nowadays have stacks of greasy sandwiches in display cabinets on the bars.' Only now did _El Alc\u00e1zar_ describe the opening section of the final stage into Paris. However, it made up for lost time in the edition of 20 July (no newspapers were published on 19 July) when the front page consisted of a large portrait of Bahamontes. Inside there was a special pull-out that included a first-person piece, 'Fede \u2013 how I won the Tour', plus all the details of the celebrations in Toledo, and the first segment of a week-long mini-biography entitled 'This is the Eagle of Toledo'.\n\nBahamontes' Tour victory was the only sport shown on Spanish television on 18 July, 1959. Not surprisingly, since the footage would have been flown in from France, the fifteen-minute programme of race coverage was shown late at night on the country's one channel. However, the number of people who had access to a television set in 1950s Spain was extremely small, and radio together with the cinema newsreels were a far more important means of communication. Typically, villagers in rural Spain would gather around the _pueblo_ 's one radio to listen to major events. Indeed, it acted as a social 'glue'. But the broadcasts were no freer of the shackles of Franco's censorship than any other type of media and his regime was able to exploit shamelessly the success of Bahamontes and other leading sporting stars, most notably the football team Real Madrid. By way of example, in 1955 when Real Madrid won the shortlived Latin Cup tournament, Franco gave the entire team the Imperial Medal of Yoke and Arrows, the symbols of the Fascist Party, as part of his 18 July celebrations.\n\nHowever, the most significant method by which Bahamontes could be pushed forward as the poster boy of Franco's regime was through the _noticias-documentales,_ widely known in Spain as _no-dos._ These were eleven-minute newsreels, produced from 1943 onwards, which were shown before feature films in Spanish cinemas. They were the only newsreels permitted and were an obligatory part of all schedules across the country. 'They were the main access to any kind of information on current affairs for a large percentage of Spain's population,' says Margarita Lobo, an expert in _no-dos_ at Spain's Ministry of Culture. 'Given the level of illiteracy in Spain at the time, more than the press, they were the most important form of internal propaganda. People would go to the cinema which, appealingly for an impoverished population, was cheap and warm, for up to seven hours at a stretch, from three to ten, and watch the double bill of films twice over. This meant watching the _no-dos_ twice over, too. The _no-dos_ were very anodyne and always had the same structure. They would start off with a news item, some kind of inauguration of a building by Franco, say, then be followed by a catastrophe \u2013 a bridge collapsing, maybe, or a flood. But this catastrophe would always be in a foreign country, to remind the public that in Spain nothing so awful happened. Then there would be the Spanish news section, more often than not featuring Franco, and finally the \"miscellaneous\" section, which would include sport.'\n\nLobo says that Bahamontes would have been obvious, prime-time, _no-do_ material in 1959. 'Up until 1950, internationally Spain didn't \"exist\" because of the boycotts. From the 1950s onwards Spain slowly started to create links with the outside world again, particularly with the United States. As a result, any event [such as winning the Tour] that could be used to bring us closer together was given huge importance. If the _no-dos_ exploited any kind of success so intensely, it was because they had very few other options. In Bahamontes' time there were barely any other top-level athletes \u2013 perhaps a horse rider or two as well as the gymnast Joaquin Blume [winner at the European Championships in 1957, but whose career was cut short in its prime when he was killed in a plane crash in 1959]. On top of that, if a person did not have a high level of sympathy for the regime then they wouldn't be given as much importance.' That is why Jes\u00fas Loro\u00f1o, for example, rarely appeared in the _no-dos:_ nor was Loro\u00f1o's King of the Mountains title in the 1953 Tour covered, yet Bahamontes crossing the Tourmalet in first place in 1954 was given its own special report.\n\nOn 27 July 1959, Bahamontes' victory opened the sports section with paeans of patriotic praise ringing out. 'The most famous race in the world sees a brilliant performance by our Spanish hero, taking our first ever triumph,' the commentator crowed. He then listed Bahamontes achievements (King of the Mountains, a stage win, fourth in the points prize and so on) before finally describing him as a 'bullfighter on two wheels'. A month later, on his return to Spain after the criterium _tourn\u00e9e_ ('Triple the number of engagements we expected,' according to his agent, Daniel Dousset), a homage in the comparatively minor city Matar\u00f3, just outside Barcelona, was given headline billing. The film showed Bahamontes receiving rapturous applause in a rather dark outdoor velodrome, Bahamontes receiving a gold watch in recognition of his success, Bahamontes wobbling rather nervously round the track into the gloom. 'He has done twenty-eight criteriums in thirty days,' says the commentator in awed tones, 'and travelled twelve thousand kilometres.' Its position on the _no-dos_ news agenda is designed to give the public the impression that in Franco's Spain such successes can almost be taken for granted. But only successes: after the disastrous 1960 tour for the Spanish, the only sports the July _no-dos_ covered were a horse race in Hamburg, car racing in Reims and a regatta in Kiel. It was as if the Tour had been cancelled.\n\nHowever, the bottom line of the authorities' desire to appropriate Bahamontes' triumph and make it a universal success was down to the fact they had absolutely nothing to lose by doing so. Coming from Castille, Bahamontes was the perfect rider on whom to base the regime's unspoken aspirations. Unlike Loro\u00f1o or Catalan Miguel Poblet, there was no chance of a region with separatist goals appropriating the victory. Even so, thanks to Langarica's role in Bahamontes' success, fans in the largely anti-Franco Basque Country could join in the fun and enjoy a rare excuse for pro-Spain celebrations. That was equally true in Asturias, one of the staunchest Republican areas in the Civil War but also a cycling heartland. Bahamontes' regular success in their local Tour, with two overall victories and two wins in the King of the Mountains competition, had brought him huge popularity. In the final days before he reached Paris local newspapers had recalled repeatedly that Bahamontes' first major triumph had been in the Tour of Asturias in 1953. Six years later fans poured into the newspaper offices in Oviedo, seeking news from France. As a result, the offices had to be barricaded and loudspeakers installed to broadcast news of the race to those outside.\n\nBahamontes implicitly recognises that his victory was appropriated by Franco's regime, recalling that when he met Franco, 'the Caudillo [Franco] told me I had planted the Spanish flag at a height it had never reached before.' However he also insists that 'it was a great moment for all the Spanish Communists in France who'd been there since the [Civil] War. They could all come out of hiding and say, \"We're proud to be Spanish again\"'. As Bahamontes sees it, his victory briefly managed to close the old Spanish Civil War wounds \u2013 but politically at least, Franco's propaganda machine, always interested in keeping those wounds open, saw his Tour win very differently, and the winner's personal opinion would barely have counted.\n\nUnsurprisingly, though, Toledo could not be matched for 'Bahamania'. From the moment he took the lead in Grenoble the streets and shop windows had been plastered in yellow ribbons and the local council put up a giant neon sign over the town hall saying 'Bahamontes King of Toledo'. When Radio Toledo confirmed at 4.40 on the afternoon of 18 July that he had won the Tour outright, the city went Bahamontes crazy. For more than a week the daily post was delivered in vans adorned with yellow ribbons, the market where Bahamontes had worked was similarly decked out and the city's bars re-dubbed the standard 'ca\u00f1a' measure of beer a 'leader' or a 'yellow jersey'. Some doctors' prescription forms even had the words 'Bahamontes will win the Tour' stamped across the bottom. 'All the rockets in Toledo's one pyrotechnics shop have sold out and to judge from the noise here it often sounds like they were all used to celebrate Bahamontes' win at the same time,' said _Informaciones._ 'Every car I see with a Toledo number plate still has yellow ribbons on it,' wrote _El Alc\u00e1zar_ 's columnist German L\u00f3pez a few days later, 'and the streets of Madrid are still full of people with Bahamontes flags.' A newsagent in Toledo added: 'All my newspapers are sold, two hours before their usual time. I don't think anybody, as an individual, has ever managed to sell so many newspapers as Bahamontes just because of what he's managed to do.'\n\nJust like the most famous Spanish bullfighters, it was only a question of time before Bahamontes had a ' _pasodoble_ ' (the traditional folksong and dance) dedicated to him. Entitled with predictable unoriginality, 'Bahamontes King of the Tour', and like many _pasodobles_ eminently forgettable, the song starts by claiming that 'on his steel steed, he rides towards Paris, what honour and glory will await him there . . .' Dire though it may have been, it was an indication of how deeply Bahamontes' success had struck home in Spain.\n\nThe unofficial prize for the most diehard Bahamontes fan, though, must go to Se\u00f1or Talavera, president of his fan-club in Toledo. Talavera purchased a live eagle and planned to fly with it to Paris so he could release the bird in the Parc des Princes when Bahamontes climbed on to the podium. However, last-minute visa problems meant both he and the eagle remained in Spain.\n\nEven if an eagle from Toledo failed to make it to the Parc de Princes velodrome in time for the finish of the Tour, the Eagle of Toledo himself, resplendent in his yellow jersey, was most definitely present. Yet even as he received the final bouquets of flowers, and Fermina broke down in tears on the stadium steps, Bahamontes insists that the scale of his had not sunk in. 'I don't think it will do, to be honest, until I cross the Spanish frontier,' he said. 'I do know that I won this race more with my head than with my legs.' For all Bahamontes' climbing skills his greatest win did not come just because of his ability to get up mountains faster than anybody else, it was also down to the tactics worked out with Langarica. As Tour director Jacques Goddet pointed out: '[Strategically] he has changed considerably. He is not the rider I used to know. The reputation he has of being a dreamer is no longer valid. I'm sure he's had to keep a handle on his temper and impetuosity at times, too.' Goddet also recognised that Bahamontes had outwitted his rivals by anticipating them, claiming the lead before they expected. 'He moved into yellow a day early, and for that reason on the last day in the Alps he barely had to push himself. If there had been any real battles in the Alps, in any case, he was so strong he would have been at a real advantage. If he had done what he did this year in previous Tours, I'm sure he would have won the race before.'\n\nBahamontes acknowledged the huge differences Langarica had made in planning his strategies. 'I could never have won with Puig,' he said after the time-trial in Puy de D\u00f4me. 'Langarica was much more intelligent and Puig had never ridden a bike.' Langarica was also more practical. Every evening, without fail, Langarica would go to Bahamontes' room with a copy of the overall classification and the stage route for the next day. It might take hours, but only after they had hammered out an agreement on the strategy to be employed would Langarica get up and leave.\n\n'The French team was tied up in knots by their friendships,' argued _Informaciones._ 'Bidot went for compromise, even if he knew this fantastic \"cocktail\" of riders would not yield any particular result. Langarica, on the other hand, dispensed with Loro\u00f1o. And it would be useful to ask all those critics [of that decision] what they think of a result when a Spaniard tops the Tour de France for the first time. The years of [directing the Spanish team by] Luis Puig, incapable of taking the same kind of responsibility that Langarica did, make it clear what the consequences would have been.' Langarica told _Informaciones_ : 'I never managed to convince Federico he could win until this year. I could not accept the war that went on inside the Spanish team in 1958. Then, after one man from Vizcaya [Loro\u00f1o], advised by another man from Vizcaya, said he would not take part . . . a real Spaniard arose, Bahamontes, labelled as being mad and with all that past history of failures, abandons and betrayals. He called me to tell me he would come with me to the Tour, and that's where it all started.' Team-mate Carmelo Morales said simply: 'He would have won more often had he been directed better, sooner, in his career.'\n\nThe influence of Fausto Coppi on Bahamontes, for many the most crucial factor in his victory, was only highlighted in the Italian press. The man who Bahamontes said had inspired him to win during the course of a long conversation over the meal in Talavera the previous winter, had been convinced his star rider would take the Tour as soon as the day after the Puy de D\u00f4me. 'My only fear,' Coppi told Italian journalists, 'was that Bahamontes would do something stupid, because of the strange ideas that sometimes run through his mind.'\n\nBahamontes told _El Alc\u00e1zar:_ 'I had a huge sense of responsibility before the race started. Hadn't I been forced to abandon the Vuelta a Espa\u00f1a? I knew that everybody would expect more of me, and I expected more of myself. That said, I was a bit hurt by the way the French newspapers only put me as sixth favourite before the start rather than making me an indisputable top favourite. But, in fact, not racing the Giro and being forced to abandon the Vuelta made all the difference. I was a lot stronger and in the mood for a fight. By the time I had won the Puy de D\u00f4me time-trial, the race was practically won. Gaul even came up and congratulated me then, saying, \"This year it's yours. Don't let it slip out of your hands\".'\n\nThe French, predictably, were hammered mercilessly by their national press and fans, who whistled and booed Jacques Anquetil as he stood on the podium in Paris in third place. Apart from pointing out that Bahamontes had defeated four former Tour winners in Jean Robic, Louison Bobet, Charly Gaul and Anquetil, _L'Equipe_ added that he had defeated 'French egotism'. The disputes rumbled on, with the blame being placed on the overly conservative approach of Bobet, who was conveniently absent in Paris after his abandon. Roger Rivi\u00e8re, meanwhile, directed his ire at another of his other team-mates. 'If I had known Jacques Anquetil a bit better I'd have won the Tour,' he said. 'And without Bobet in our line-up, either Jacques or I would have beaten Bahamontes.' He said the French had expected the winner to self-destruct, and that they had underrated Anglade, even though he was the French national champion. 'We watched everybody we were supposed to \u2013 Gaul, Ercole Baldini \u2013 but we forgot about Anglade and Bahamontes. The only times we thought about Bahamontes we remembered his eccentricities, which generally end up wrecking his chances. So we let Bahamontes seem to lord it over the rest of us and then we expected Charly Gaul to wipe him out. But instead of that Gaul did nothing and Bahamontes, whom nobody rated, ended up taking the yellow all the way to Paris.'\n\nThe French were not the only ones to underestimate Bahamontes. 'We all thought that Bahamontes was just going for the mountain points, and look what happened,' said Baldini, the reigning world champion, and 1958 Giro winner. _Informaciones_ summed it up: 'In sporting terms, the Tour has been mediocre. Some of the stars were not up to scratch and others feared to lose what they had. Only Bahamontes has really given it any life.'\n\nAndr\u00e9 Darrigade shakes his head as he tells me: 'That year it was a disaster.' The Frenchman's green jersey in the race, plus his stunning victory in the World Championships, effectively saved his country's season. He was one of the few riders who willingly sacrificed his chances for his team-mates. 'He was the perfect _domestique_ and our default winner,' Bidot said. 'If he had raced more for himself and he hadn't been so unlucky, he'd have won ten more stages.' Darrigade is unfailingly polite and does not make a single disparaging comment about Bahamontes. He seems curiously innocent, too, and fifty years on he still finds it difficult to believe that two of his country's greatest stars could destroy the chances of a third (Anglade) simply because their egos would not let him triumph.\n\nDarrigade watched the greatest team France had fielded in a decade disintegrate in front of him; the only words to describe what he seems to feel are barely disguised despair. 'It was really, really bad. Bobet and Anquetil: that [their relationship] was terrible, and Rivi\u00e8re and Anquetil, that was even worse. Anglade could have won the Tour when he broke away, but Anquetil and Rivi\u00e8re agreed to bring Federico back up to Anglade because they preferred Federico to win. On top of that, Geminiani and Coppi were close friends and don't forget that Coppi was Bahamontes' manager. The rivalry was a problem. Each one preferred Bahamontes to win rather than the rest.'\n\nFor Spanish cyclists, Bahamontes' success ended their role in the Tour as a sideshow confined to the mountains. 'Climbers of the calibre of Vicente Trueba and [Juli\u00e1n] Berrendero were known here as the Fleas of the Pyrenees,' recalled leading French journalist Rene Dunan of the winners of the King of the Mountains titles in 1933 and 1936 respectively. 'They only appeared when the Tour approached the mountains near their country, and their victories were applauded all the more warmly because they were clearly without any long-term significance. The Spanish were like marionettes who did their \"number\" on the stage and then left. They created the legend of the Spanish climber, but they've always been considered eccentrics.' Bahamontes might initially have appeared to be similar to Trueba and Berrendero, but his win brought Spanish cycling up to a completely different level.\n\nOn a wider plain, the Tour de France was just one of many examples of progress being made by Spain in 1959. That winter United States President Eisenhower paid his first visit, thereby symbolising the end of the country's political isolation. Severo Ochoa's Nobel Prize for Medicine confirmed that scientifically Spain was no longer a backwater. Economically, too, Spain broke out of the recession that had gripped the country since the Civil War. In the future, 1958 was to become the last of the so-called ' _A\u00f1os de Hambre_ ' [Years of Hunger], and 1959 the first of the ' _A\u00f1os del Milagro_ ', the years of Spain's economic miracle. And this was not just Nationalist propaganda: in 1959 a 'Stablisation Plan' was implemented, backed by the International Monetary Fund, and the peseta devalued; Spain's economic revival began with a radical increase in foreign investment. On the day of Bahamontes' victory, there was also a high-level political meeting in Brussels to decide whether Spain had advanced enough to join the Organisation for European Economic Cooperation, the O.E.C.E. or O.E.C.D. as it was later known. A week later Spain was in.\n\nThe biggest development, though, was in tourism. The ending of the requirement for Western Europeans to obtain entry visas that year might not seem an earth-shattering change, but it was to have a huge effect on the travel industry. It opened up the country to package holidays in which Spain would do a roaring trade during the 1960s and 1970s. At the same time Spain began its transition to a far more urban society; the number of agricultural workers fell by half between 1960 and 1976 as they sought better living standards in the towns and cities.\n\nBahamontes appeared in his last _no-do_ as an athlete in 1964 when they recorded an homage to him in Madrid. The newsreel amply reflected the developments and giant strides the country had made in just five years. The film opened with news of a trade centre in Bilbao, then introduced a new long-distance bus 'complete with a bar, a hostess and headphones, just like the best aeroplanes'. A longer feature showed university students going to help out in the country, pitchforking hay and teaching classes of grimy-nailed farmers' wives, 'wiping out the last traces of illiteracy from the country', according to the documentary. If such activities, vehicles or buildings were unthinkable even ten years before, so too was the location of Bahamontes's homage: Madrid's Sports Palace. As he and several other top riders wheeled around a specially-built artificial track on special track bikes, waving to the fans, behind them were advertisements galore: for rum, cigarettes, a festival and even _El Cortefiel_ , Spain's first major chain of department stores. Compared to a decade earlier, the opulence and comfort on offer in Spanish society was manifest.\n\nIn 1959, though, the fruits of change lay ahead. From the regime's point of view Bahamontes' Tour de France victory was just the sort of image-changing international success the country needed. Apart from an indisputable display of modernity, it was also proof positive that even for the poorest, least developed sectors of Spanish society there was some light at the end of the tunnel after decades of misery and civil conflict. 'Bahamontes represents the transition of a person from the \"Spain of hunger\" to the \"Spain of development\",' historian Manuel Espin said in an interview published on 21 June, 2009, in the newspaper _El Mundo_. 'He took the path from that hard, battered Spain to a nation that started to open up to the world.'\n\nHowever, no matter what a trailblazer Bahamontes was for cycling in his country, the fact remained that when the Eagle of Toledo finally got his claws into the Tour he was not the young, fresh-faced Anquetil of 1957, or Eddy Merckx in 1969, set to dominate for nearly a decade. Bahamontes was thirty-one. As such, Bahamontes was therefore no pioneer: rather, after years of trying to get the pieces of the jigsaw the right way round, in 1959 the puzzle had finally fallen into place.\n\nAnd in 1960, while Spain made giant leaps forward in social and economic areas, for Bahamontes it was back to square one.\nChapter Ten\n\n#### Goodbye to the Vuelta\n\nFederico Mart\u00edn Bahamontes basked in the glory of the greatest triumph of his career for the rest of the year. However, within months of the start of 1960 it all went sour. He was about to become embroiled in the greatest sporting scandal of his life.\n\nBahamontes' season had gone askew almost from the start of 1960. Initially it looked as though he was en route to making a fortune by building on his success in the Tour de France. The first setback came when Fausto Coppi's team collapsed when the Italian died in January from malaria. Bahamontes bounced back quickly by signing for Faema, his former enemies, allegedly for the astronomical sum of eight hundred thousand pesetas [\u00a3450,000 in modern money] a year. He was not deterred by the presence of Bernardo Ruiz, his old rival, as sports director or having Antonio Su\u00e1rez, who had attempted to usurp him as top Spanish contender in the Tour the previous year, as a team-mate.\n\nIt was just as much of a rollercoaster once he was on the bike. Following a crash on 13 March in the Tour of Levante, Bahamontes spent thirty-six days with his fractured femur in plaster. He returned, after just twenty-five days' training, to win the Arrate hill-climb. Then came the Vuelta a Espa\u00f1a. It was a race Bahamontes was on the point of not starting because he was ill and had fallen out with his team. Later, on 17 May, he would tell _MARCA_ that he had not been able to 'move his guts' since before the start of the race, hence his unwillingness to take part.\n\nBut that was only the beginning of his problems. As Chico P\u00e9rez writes in _The Vuelta 1935\u201385:_ 'The great Federico was constantly polemical and in only ten days went from the \"genius\" of Spanish cycling to its \" _enfant terrible_ \".' The Vuelta had barely begun in Gijon when Su\u00e1rez crashed in the opening team time-trial and suffered major bruising and huge scrapes all over his body (the dreaded road-rash as cyclists call it). Bahamontes also went down in the crash, ripping off half his right thumb-nail. Without Bahamontes wishing it, Su\u00e1rez's accident had conveniently sidelined his main opposition from inside the team. Then, on stage five from Orense to Zamora, Bahamontes underlined his suddenly improving form and position in the squad by embarking on a long-distance attack. To call it long-distance is actually an under-exaggeration: he was off the front alone for two hundred and forty-three kilometres, the longest single breakaway of his career, and the longest in the history of the Vuelta at that time. But since he was the reigning Tour winner his move was impetuous in the extreme: the peloton were never going to let a figure of his stature establish such a commanding lead so early in the race. Bahamontes was caught thirty-two kilometres from the finish, first by Frenchman Antoine Abat\u00e9 and then by the bunch. _El Mundo Deportivo'_ s headline the next day, 'A genius or a madman?' summed up the general consensus. Likening his 'stupid, useless effort' to the adventures of Don Quixote, the writer's only explanation was 'that like every big star, he has ended up having a psychological dependence on getting into the newspapers'.\n\nBahamontes' strange attack contributed to a widespread feeling that something was not quite right in the Vuelta. The underlying unease continued even as Bahamontes attempted another dramatic breakaway the next day on the stage from Zamora to Madrid in which he was accompanied by Faema team-mate Fernando Manzaneque. This time the gamble paid off: the move gained almost four minutes on the peloton, and even if the stage win went to an earlier breakaway, the leader's jersey still ended up on the shoulders of Manzaneque. Bahamontes, though, was still twelve minutes down despite those two gung-ho performances. Conscious that he could not lose any more time he fought back hard when Charly Gaul made his one and only major attack of the race on stage seven, a mountainous trek through the sierras of Madrid. Gaul jumped twenty-one places to fourth overall and ahead of Bahamontes by seventy-four seconds. Despite complaining of an upset stomach, Bahamontes ended the stage with a tenuous lead in the King of the Mountains competition.\n\nThe narrow gap between Gaul and Bahamontes briefly raised expectations of a duel between the two mountain kings: however, that particular battle was overshadowed by a more significant turn of events the following day when the entire race started to go off the rails. The stage from Madrid to Zaragoza was an extremely long run, one of five exceeding two hundred and fifty kilometres and seven in a row that were more than two hundred kilometres in length. The riders requested another feeding station mid-stage, and when the organisers refused they went on the first of five major go-slows which turned the Vuelta into a farce. As a result of the first protest the race reached Zaragoza at nearly half past nine at night, two and a half hours later than scheduled. The thirty thousand spectators whistled and booed as the riders inched their way around the finishing circuit. But worse was to come. The near-unanimous apathy continued for another four days. There were isolated incidents of picket-breaking: an eleven-rider break of non-favourites gained a forty-minute advantage over the bunch on the stage into Barcelona; a ten-man break claimed a nineteen-minute advantage on the next stage into Barbastro and, after a rest-day, a four-man break finished twenty-four minutes ahead into Logro\u00f1o. Just a handful of young Belgian riders seemed interested in racing, and they dominated the overall. The rest of the reduced peloton of forty-seven was effectively on strike, and the race had become, as Chico P\u00e9rez put it, 'a total disaster'. Race organiser Luis Bergareche begged the directors to get their riders to do more than pedal mechanically at ridiculously low speeds, but nobody paid him any heed.\n\nBahamontes and Gaul were now more than fifty-one minutes down on Belgian leader Armand Desmet. 'No \"Bahamontes miracle\" today', _El Mundo Deportivo_ sardonically reported. However, the next stage from Logro\u00f1o broke the pattern of the previous five days. A breakaway by Bahamontes, all but from the gun, managed to stick this time. Two hundred kilometres later at the finish in San Sebastian he was still ahead of the field with a three-minute twenty-two-second advantage over his closest pursuer. Such a spectacular move could have saved the race's image. However, after four days of trundling across Spain the general mood of disillusion among the press and fans was so great that day-long breakaways like Bahamontes' had a decidedly hollow ring to them. If the rest of the field were so apathetic, what did it matter what one rider achieved? Had Bahamontes opened up a bigger gap on the stage \u2013 ten minutes, say, or even twenty \u2013 it might have resurrected a sporting corpse. But the organisation shot themselves in the foot. For some reason nobody had bothered to give Bahamontes details of the time difference he had over his rivals as they should have done. Instead, Bahamontes pedalled away for two hundred kilometres with no idea of the effect he was having until he punctured close to the finish. By then, though, it was too late. His stage win strengthened his lead in the King of the Mountains competition and in the race's 'most combative rider' classification. However, since he only moved from twenty-eighth to twenty-first overall this was barely noticed. Bahamontes' answers to the press at the finish \u2013 even if they were serious \u2013 did little to alter the view that the race had become meaningless. When asked what his objectives had been with such a long attack, without batting an eyelid, he replied: 'To get over the mountains first and finish half an hour ahead of the field. Why not? That's what has happened every day up until now.' When Su\u00e1rez staged an eight-hour breakaway to win the 'Queen Stage' to Vitoria the following day it gained the same lukewarm reaction from observers. There were even rumours that the race was so discredited there might not be a Vuelta in 1961.\n\nThen, in the early hours of the morning after the Vitoria stage, the news began to circulate that Bahamontes was threatening to stage a 'go-slow' because the organisers were refusing to readmit Julio San Emeterio, Bahamontes' prized _domestique_ after he had failed to finish inside the time limit. If carried out, Bahamontes' departure would be a major blow to the race, especially as it came just hours after Gaul had quit stage fourteen for no clear reason other than he could not be bothered to continue. The organisers could ill afford to lose another star. Yet they were faced with a dilemma: if Bahamontes' demands were met, and San Emeterio reinstated, then two other teams, Licor 43 and K.A.S., said they would abandon en masse. With the race already reduced to thirty-four riders the organisers risked either losing Bahamontes or another twelve riders, five of whom were in the top ten. According to Bernardo Ruiz, now Faema director, the reason the teams decided to put the Vuelta organisers in such a position was because they had learnt of Bahamontes' strategy to save his Vuelta and knew that San Emeterio was crucial to it. Bahamontes had deliberately told San Emeterio to take it easy, even if he finished outside the time limit, because the next day they would make a joint attack. Unfortunately for Bahamontes, San Emeterio blabbed about this cunning plan when he finished the stage: small wonder the rival teams did not want to see him start, particularly when they discovered the organisers had initially acceded to Bahamontes' demands.\n\n'I directed Bahamontes that year and he was a man out of control,' Ruiz told me. 'Julio San Emeterio finished outside the time limit in Vitoria and said afterwards that he had taken it easy because he wanted to win the stage to Santander. At first the organisers readmitted him without anyone knowing, but then the other teams got wind of it, protested and they kicked him out again. I went to talk to the director of the Vuelta and I said, \"Look, does it really matter to you [if San Emeterio is readmitted or not]? I've got Fede sitting outside your office and he's a shade annoyed about all this. And Bergareche said to me, \"Don't get me annoyed about this either. I can't do it because of what San Emeterio has said about winning in Santander\".' Ruiz went back to Bahamontes and told him that San Emeterio was out of the race. 'So I said to Federico, \"It's not going to happen\", and his response was, \"Right then, I'm going to finish outside the time limit to see if they exclude me, too\". And that's what he did. On the next stage he went deliberately slowly and of course they threw him out. I said to him, \"Who the hell do you think you are?\"'\n\nEven half a century afterwards Ruiz's anger towards Bahamontes is still palpable to the extent that he talks in the present, as if the controversy had happened yesterday and not in 1960. 'What do you want me to do, [should I] kill him? [During the stage of the go-slow] I give him a couple of riders to bring him up to the front of the bunch and he stops, he tells them to go on, because \"I'm going to finish outside the time limit\".' And then Ruiz repeats: 'What do you want me to do, kill him?'\n\nThe confrontation with Bahamontes, and the general mood of discontent among the riders, was not all the Vuelta had to face that grim morning in Vitoria. On the same day one newspaper ran a story in which Lucor 43 rider Miguel Pacheco claimed he had been 'exiled' from Faema, the team with whom he had initially signed that year, because of his good relationship with Bahamontes. 'I could not imagine my friendship would end up with me being barely considered for races [by Faema],' he said. 'I have been declared an outlaw.' Pacheco claimed that as Bahamontes' decision to join Faema 'was not seen favourably, they took it out on me because I was his friend'. He even said that attempts had been made to bribe Licor 43 so that he would not take part in the Vuelta, and that Faema had formed an alliance with the Belgians to try and squeeze him out of third place overall. Quite apart from highlighting Bahamontes' difficult position in the team, Pacheco's allegations were effectively confirmed on the final stage to San Sebastian when he punctured and the Faema-Belgian alliance tried to eliminate him. Pacheco's claims were the latest bombshell for a Vuelta already cowering under the table, and they did not go down well with the fans lining the roadside and reading their morning newspapers as they waited for the race. After Gaul's abandon Bahamontes remained the main target for the public's dissatisfaction. And given his duel with Jes\u00fas Loro\u00f1o, Bahamontes' popularity was precarious anyway with fans from the Basque Country where the race was due to pass. While news of the Pacheco story circulated at the start in Vitoria, Bahamontes was in the middle of a blazing row with Bergareche about San Emeterio. As Bahamontes pointed out, in the 1959 Vuelta a similar situation had occurred when Roger Rivi\u00e8re and five members of the Faema team had been allowed back in despite finishing outside the time limit. At the time Bahamontes' protests had gone unheard and he had pulled out of the race. A year later his protests on the start-line were just as fruitless. San Emeterio was also present, ready to start, but Bergareche refused to succumb to what he described as 'blackmail' and threatened to take the case to the Federation. Bahamontes took his appeal to the international _commissaire_ for the race, but it did no good: San Emeterio was out and he stayed out.\n\nThe stage kicked off with spectators shouting insults at Bahamontes and these increased when they realised that he was about to stage what looked like a deliberate go-slow. The newspaper reports about Pacheco only added to the tension. Around the eighty-kilometre mark, just outside Bilbao near the town of Sollube, Bahamontes could no longer handle the constant jeering and insults and dismounted, bicycle pump in hand.\n\n'He assaulted a group of fans both verbally and physically,' the official report into the incident said later. At that point a pro-Bahamontes supporter intervened in an attempt to calm everybody down. It did not succeed. According to newspaper reports, as well as Antonio Jim\u00e9nez Quiles's account of events, Bahamontes' team-mate Herrero Berrendero dismounted as well. In the midst of the argy-bargy and shouting he apparently accidently struck the Bahamontes supporter, thinking he was about to attack his leader with a bottle. 'I didn't hit anyone,' Bahamontes told me. 'There's a photograph of me with my arm raised and a pump _about_ to strike someone, but I didn't manage to do it. It was Herrero Berrendero and he got the wrong person.' At this point Luis Bergareche intervened and told Bahamontes that he had to go on riding, and reportedly adding: 'If you don't, I will take you to the U.C.I. and I assure you that you will remember me for the rest of your days.' Finally everybody calmed down and the race continued.\n\nIf Bahamontes had intended to cause chaos and delays he certainly succeeded. The fact that thirteen other riders joined him, albeit protesting about something else, was another indication that the Vuelta was coming apart at the seams. 'Half the time we were starving because they wouldn't let us take on extra food,' Jim\u00e9nez Quiles, who joined the protest, recalls. 'It got to the point where we'd end up stopping, get off our bikes and break into people's houses to rob bread. So that's why I was in that group with Bahamontes that day.'\n\nKeen to see the outcome of the fourteen-rider 'strike', around one hundred and fifty vehicles containing journalists and race followers formed a massive convoy grinding along at snails' pace behind the mini-peloton. But if they could not see much of the race they had no need to worry about getting bored. As the press cars rolled through one village Bergareche and Faema representative Miguel Torell\u00f3 could be seen standing on the pavement arguing at the tops of their voices. Then, at the entrance to the town of Castro Urdiales the fans' ill-feeling towards Faema and its director Bernardo Ruiz was clearly expressed in a huge poster reading: 'Bernardo: this is a country of _hidalgos_ : no traitors here'. Bergareche also ran the gauntlet of spectator protests, which had developed into death threats. After the controversial abandon of another Vuelta rider, P\u00e9rez Frances, a poster greeted the race director at the entrance to one town in northern Spain: 'Bergareche: P\u00e9rez Frances was born here'. That might have been inoffensive in itself except that the message appeared to be written in blood. Another sign went up in San Emeterio's village, which the Vuelta passed through the day after Bahamontes' exit: 'Bergareche, San Emeterio will be your cemetery'. If anything was designed to make Bergareche feel less favourably towards Bahamontes and the riders in general it was messages like these.\n\nFifty kilometres from the finish in Santander four of the fourteen realised that they were almost certainly facing expulsion and darted away from the Bahamontes group. They finished between thirty-five and forty-five minutes down on the winner, the Belgian Arthur DeCabooter, twenty minutes or more behind the rest of the field, but more importantly just inside the cut. There was no reprieve for the rest. Like Bahamontes, they were summarily thrown off the race. According to the official inquiry into the stage, as well as some well-publicised accusations from Faema sports director Miguel Torrello, Bahamontes was the ringleader of the rebellion: he had allegedly urged the others to remain with him and even refused any support from his team-mates who had briefly dropped back to try and guide him back to the main peloton. Instead, the report concluded, Bahamontes had continued his two-wheeled protest. Combined with the incident involving the bicycle pump and the fan, there were a multitude of reasons why Bahamontes could not be allowed to continue. Quite apart from anything else the organisers had race regulations of the most mundane kind to back them up: by the finish the Bahamontes group was more than fifty minutes slower than DeCabooter and they were outside the time limit. End of story.\n\nBergareche told the newspaper _El Diario Montan\u00e9s:_ 'With people like this it is impossible to organise a Vuelta. My only regret is that I tried to do so.' Bahamontes responded in kind: 'With these people [organising it], I'm not racing the Vuelta again.' His defence to the press, which was contradicted by everybody from the organisers to eye-witnesses, was not that he was annoyed about San Emeterio, but that he 'did not feel well and could not go any faster. Do you want me to die on the roadside just because the race is going to Bilbao?'\n\nThe organisers were not the only ones upset by the scandal. Antonio Anglade, Faema's chief executive, said he was 'obliged to consider' withdrawing sponsorship of the team, as well as expressing 'displeasure at what has happened'. He added: 'These events are far beneath the category of this race and the dignity of this sport.' _El Mundo Deportivo_ summed up the prevailing mood: 'Bahamontes has not known how to digest the huge number of homages and tributes he gained after his phenomenal win last year and he believes he can wreck the essence of this sport. There is a path to take for every rider who does not consider himself a demigod: either withdraw from the race or withdraw from his profession.'\n\nThe Vuelta itself was in major trouble. Quite apart from the fact that only around a quarter of the starters made it to the finish, the winner Frans De Mulder was a Belgian of very little standing whatsoever. His winning margin of fifteen minutes from another team-mate, Armand Desmet, confirmed the abysmally low level of competition in the race.\n\nIf Bahamontes' reputation was damaged, the long-term financial consequences were minimal. Though the Vuelta's official communiqu\u00e9 confirmed he had been excluded from the race, the Federation's investigative committee fined him the risible sum of twenty thousand pesetas and declared the case closed. If Herrero Berrendero was involved in the fracas with spectators, as Jim\u00e9nez Quiles claims, that was not even mentioned. There have been suggestions that Bahamontes was too important a figure, after his Tour de France win, for the Federation to mete out the level of suspension he probably deserved. But the Tour was less than a month away and for the defending champion not to go to France was unthinkable.\n\nYet there was a price to pay for letting him walk away from the Vuelta of 1960 all but unscathed: Bahamontes had been allowed to put one over on his country's cycling authorities. Giving one rider that much unspoken power was not the wisest strategy, particularly in view of the way he had already behaved in the Vuelta. 'Everything that happens here is a consequence of his personal interpretation [of events] or a consequence of his absence,' _El Mundo Deportivo_ complained the day after his expulsion. In a lengthy interview with _Alc\u00e1zar_ , Bahamontes denied that he had even got off his bike and that, contrary to what the photograph published in _ABC_ suggested, he had not planned to hit a spectator on the roadside. Somehow Bahamontes managed to draw his old enemy Jes\u00fas Loro\u00f1o into the conflict, claiming that the person he was gunning for was 'an individual in Loro\u00f1o's team car who had been laying into me for three or four days'. Bahamontes also claimed that Faema director Torrello had only warned two of the riders in the go-slow protest that if they did not reach the finish within time the jury would show no mercy and they would not be allowed to start the next day, and he was not one of them. 'I am ill,' he said, 'and that's why I was unable to race properly. I should be able to cure myself. If they ask me to do the Tour, I'll go, but that doesn't depend on me.'\n\nWhoever was right or wrong, the failure of those in authority to punish Bahamontes more stringently was an admission of their dependency on him for results. However, if the short-term consequences were minimal or even beneficial to Bahamontes, in the longer-term they had a considerable knock-on effect: the events of May 1960 saw a definitive rupture between the rider and Spanish cycling. Except for a final, low-key appearance in the 1965 Vuelta, Bahamontes never returned to his country's biggest race; from 1961 he spent the last four and a half years of his career in foreign squads.\n\nThe clearest indication of what the Bahamontes affair meant in the wider political arena came from the reaction of Franco's regime: they simply blanked him of their _no-dos_. He only made fleeting appearances in the big-screen documentaries during 1963 and 1964; as Margarita Lobo, the _no-do_ expert, pointed out, anyone who failed to deliver the goods disappeared from view. In the newspapers, never again would Bahamontes be hailed as the social and sporting example to follow in the unqualified way he was in 1959. Instead the old labels of the 'unpredictable genius' were revived to describe him, and they stuck for the rest of his career. 'Physically Bahamontes is gifted for racing, but mentally he is not at the same level,' claimed _MARCA_ in 1963. Later that summer, after an equally controversial Tour, _ABC_ described him as having 'justified fame as a champion, but at the same time [also] for being over the top and abnormal'. The authorities were no more sympathetic. 'Any problem can affect his nervous system,' Federation president Alejandro Del Caz said. 'He is like a child and has to be treated as such.'\n\nAs for the Vuelta, it never reached such a low point again, not even in 1978 when protests by Basque separatists forced the last stage to be cancelled. As _MARCA's_ special correspondent wrote: 'The 1960 Vuelta is over: may God forgive it.'\n\nIf Bahamontes turned around both his season and his career after a disastrous Vuelta in 1959 by his performance in the Tour, there was to be no repeat in 1960. Quite the opposite, in fact. 'The Eagle of Toledo has become a corn-fed chicken and corn is all he has in his head,' Tour director Jacques Goddet said after Bahamontes pulled out on the second stage in France. 'He has lost his honour as a cyclist and has shown himself to be unworthy of the attention he received last year from his compatriots who were so proud to have a Tour winner in their midst. [The Eagle] has let himself be stripped of his plumage without any desire to put up a fight.'\n\nIndeed, Bahamontes was a shadow of the 1959 champion. After losing time early on during the second stage across northern Belgium to Malo les Bains in Dunkirk due to a mechanical problem, Bahamontes began drifting out of the back again in the final hour of racing the same day. 'I couldn't respond when the peloton accelerated,' he recalled later. 'My guts hurt too much.' The constant bashing along the cobbled roads of north-west Flanders that day probably made any pain even worse. Eight riders from the fourteen-man Spanish team stayed with him for support, but it was no good. Forty-five kilometres from the finish Bahamontes slowed to a standstill, wrapped himself in a blanket and sat down by the side of the road. Then, for the second time in four years, the last his team-mates saw of him was a gaunt, dejected figure, slumped in the back of the race's 'broom wagon'. His compatriot's noisy attempts to persuade him to continue had failed, though this time there were no appeals to his patriotic duty or loyalty to his wife. Those minor details apart, history really was repeating itself.\n\nIt is not difficult to understand why Goddet was so furious: Bahamontes' abandon so early left the Tour director without the leading favourite for no clear reason other than a lack of motivation; for a former Tour winner to act like that was nothing short of treachery. Physically, for all he claimed his stomach hurt, Bahamontes did not seem to be in bad shape. Unlike 1957 there were no wasp stings, no reports of sunstroke or suspect injections. Nor did anybody see this one coming: Bahamontes had finished with the main bunch in the first section of the opening stage, between Lille and Brussels. Though he lost nearly three minutes to Anglade and Britain's Tom Simpson, so too had Rivi\u00e8re, for example. And while he lost nearly four and a half minutes to Rivi\u00e8re in a twenty-seven-kilometre time-trial on the race's second section in the afternoon, that was only slightly more than he had lost to the Frenchman in the final time-trial at Dijon during the 1959 Tour.\n\nAgain, with the benefit of hindsight there was perhaps one indication of what was to come: Bahamontes' anger at team-mate Fernando Manzaneque because he had misunderstood a race official's instructions and as a result the Spanish team leader had to start the time-trial earlier than he wanted. Instead, Bahamontes had to roll only a few minutes after eating his lunch, the time when his digestive system apparently hurt the most and he found it harder to pedal. But the team, it seemed, had no real idea that anything was wrong right up until he decided to pull out. 'If the doctors don't know what's wrong with him, how am I expected to know?' Team trainer Juli\u00e1n Berrendero fumed over the phone to a journalist that night. 'He's sitting in his [hotel] room refusing to eat anything but pills.' Team-mate Rene Marigil added: 'It's incomprehensible that he abandons like this. If he was so ill he shouldn't have asked the rest of us to lose time as well. All of us, barring three Spanish riders, stayed back with him.'\n\nIf everyone was at a loss for an explanation it was hardly unfamiliar territory. This was Bahamontes' fourth controversial abandon in just over three years and his enemies were quick to take advantage. 'It is just like 1957 all over again,' his old director Luis Puig said, 'except there is less of a show than there was back then. But the substance of the matter is the same.'\n\nOn one point at least Puig was wrong. The scandal in Spain surrounding Bahamontes' latest premature exit was far larger than three years before: Bahamontes was the country's first Tour winner after all. His victory then was supposed to be representative of the new, dynamic, prosperous Spain. Instead, the grainy agency photographs which appeared in the next day's newspapers showed an ill-looking rider in Spanish team kit, clearly ignoring the cameras and jammed into the back seat of what looked like an Army jeep. It gave Bahamontes the appearance of a war refugee trying to clutch on to the last shreds of personal dignity as his world collapses around him; certainly not that of a top athlete or the poster-boy for Spain's international success.\n\nNeither did Bahamontes' behaviour in France on the way home do anything to deflate the sense of national disquiet. After catching a train in Dunkirk to Paris he stopped off at his friend Jacques Daud\u00e9's restaurant. There an immaculately suited Bahamontes was photographed drinking some light soup. It was the only thing, he claimed, that his stomach could handle at the time. However, he looks perfectly healthy in the picture which was exactly what he did not need if he wanted the Spanish authorities to feel better disposed towards him. The photograph is odd for another reason: the presence of Swiss rider Willy Trepp, the only other occupant of the broom wagon when Bahamontes abandoned. For some bizarre reason he stayed with the Spaniard the next day, too, all the way to Daud\u00e9's restaurant. It contributed to the sense that something was out of place, but nobody knew quite what. This time, though, the Federation was not going to let Bahamontes off the hook so lightly.\n\nNo sooner had Bahamontes set foot in Spain than he was summoned to the Federation by its president, Alejandro Del Caz. Bahamontes turned up for the appointment half an hour late, but even he was not able to avoid the formal, full-scale inquiry into why he had abandoned. As usual with Bahamontes, the reasons for pulling out never quite added up. One of the more surreal stories behind his Tour-start-and-instant-abandon was the case of the poor translation service. Del Caz had apparently tried to get in touch with Bahamontes in France five days before the race began. Bahamontes had arrived early to see a doctor to try and sort himself out, though he later claimed he was taking part in criteriums. Del Caz called to tell him he had to return to Madrid for the obligatory pre-Tour medical check-up. However, Bahamontes was unreachable, and Del Caz was sidetracked into talking to a Tour de France official who apparently assured him that Bahamontes was well enough to start the Tour. It emerged later that Del Caz had little idea what the official was telling him. The conversation had been in French and by the time the official translator was roped in, Del Caz had already got it into his head that he was being told Bahamontes was in good shape. By the time the opposite was discovered to be true Bahamontes was out of the Tour and on his way home. This could have reflected badly on Del Caz, of course, except Bahamontes was equally insistent that things had been going fine. How could he possibly have wanted to start a race, Bahamontes said, if he knew he was going to have to abandon so soon? Bahamontes pointed out he had even paid twenty thousand francs (around \u00a310,000 today) to the French doctor who saw him before the Tour to ensure all was in order. (There are unconfirmed reports this was Dumas, the Tour's own medic.) Even after the opening time-trial, Bahamontes was still telling journalists that his stomach pains had disappeared.\n\nHowever, there were wheels within wheels: much later Bahamontes claimed that he had been under massive pressure from the Federation to start the Tour, which he had not wanted to do. Others, he argued, should have read the signs correctly and noticed, for example, that he had finished nine minutes down on the leaders in the Tour des Pyrenees, a warm-up race immediately before the Tour. 'I shouldn't have started,' Bahamontes said. 'I felt really bad, as bad as in the Vuelta. But I wanted to please everybody.' Neither the last-minute trip to Paris to try and cure a stomach upset that had plagued him since the Vuelta, nor the few days' rest in Melun at Jacques Daud\u00e9's country house, did much to solve the problem. The key point, though, was that there was too much at stake for Bahamontes _not_ to start. The Spanish authorities were so keen to have Bahamontes in the Tour line-up that they were prepared to bend their own rules over the obligatory check-up and assume a French official was telling them what they wanted to hear about his condition. While Bahamontes knew he was ill he felt obliged to start. Berrendero, who had the least to lose over the whole affair, thought it was nothing to do with illness or cramps, 'just that our _vedette_ [diva] had a real attack of nerves. It was completely out of the ordinary.'\n\nMeanwhile, as the Spanish newspapers ran full-page photographs of Bahamontes' abandon, it seemed that everybody had an opinion, which they were invited to express. One magazine _,_ _Informaci\u00f3n_ , even placed advertisements in the daily newspapers asking readers 'from Madrid, Toledo and Spain', to send in their reactions for a special edition about the issue.\n\nOver the years another possible contributory element to his untimely exit has emerged. Following disagreements a year earlier, it seems the night before the start in Lille an unwilling Bahamontes was forced to sign a deal with his team-mates over how prize money would be shared out. Could the answer to the Bahamontes question have been financial? Bahamontes has denied that he had been paid a prime by the Tour organisers to start that year, so there was no initial slice of the cake for the others to have. Luis Ota\u00f1o, for one, said that he received his correct share from the 1959 race. However, if disputes over money did affect Bahamontes, it would not be the last time financial disagreements tipped the balance when it came to him pulling out of races of whatever size. It is perhaps worth noting, too, that when Del Caz gave Bahamontes a public roasting in the Federation's offices, one of the things he berated Bahamontes for was putting financial interests above representing Spain.\n\nWhat is most striking about how the Federation dealt with Bahamontes is their total lack of faith in what he told them. From the word go they insisted on seeing the reports from the French medic who had examined Bahamontes before the race to compare them with their own doctors' analyses. At one point this was not going to be just a physical check-up, either: Del Caz wanted Bahamontes to see a psychiatrist. Initially the Federation president had defended Bahamontes, but he ended up roundly criticising him for failing to communicate with the Federation in the run-up to the Tour. As the scandal grew, any sympathy for Bahamontes evaporated completely.\n\nHowever, unlike 1957 when it was relatively straightforward for the Federation to give Bahamontes the green light again, this time their investigation was exhaustive. He was poked and prodded by nine different doctors, from specialists in the heart and lungs, ear, nose and throat, the digestive tracts, the stomach, as well as an ophthalmologist, the Federation's head of medical services, a family surgeon and another medic with unspecified skills. They found four specific physical problems: breathing, chronic throat and sinus infections and dyspepsia, which produced large quantities of wind. None of these were exactly life-threatening even if combined, as Bahamontes claimed, with a major dental trouble. Bahamontes later revealed that he was suffering from caries in four canine teeth and had to have them removed in the aftermath of the Tour. Strangely, given how thoroughly he was investigated, this operation was not mentioned anywhere in the official report. However, when the problems were combined they were enough for the Federation to 'declare him useless', as _ABC_ rather cruelly put it, and take away his racing licence. That effectively barred him from racing in any of the profitable post-Tour criteriums had there been any invitations in any case. Typically, Bahamontes was so annoyed his licence being withdrawn that he initially refused to apply to renew it. It was thanks to his wife that his career did not finish there and then. 'Finally it was Fermina who did it,' Bahamontes recalls. 'She and [Bahamontes business associate] Evaristo Murtra, went and got the licence sorted out again behind my back. Evaristo told me later he knew that was the only way it would get resolved \u2013 if Fermina did it.'\n\nThe Federation turned a blind eye when he made his return to racing on 31 August, four weeks earlier than he was supposed to. However, his performance in the two-day GP Priego de Cordoba was a minor triumph. Though it was hardly a major race, Bahamontes took both stages, the King of the Mountains classification and the overall. Suddenly it looked as though a strong end to the season was in prospect. The looks were deceptive. In the Spanish regional championships, his final race of the year, Bahamontes made another abandon described in the press as 'incomprehensible and unjustified'. Bahamontes' three-man Castille squad were the reigning champions in the team time-trial event held in front of a huge home crowd in Madrid's Casa del Campo park. However, after the first four of eight laps the Castillians were losing time and were ninety seconds down on the time set by the Catalans. They were still just about on course for a respectable result until first Fernando Manzaneque and then Bahamontes lost contact with team-mate Su\u00e1rez for reasons none of them were prepared to explain. In Manzaneque's case, the local media claimed it was a lack of general fitness. Indeed, he collapsed on the side of the road and needed medical assistance. Su\u00e1rez and Bahamontes raced on without Manzaneque, and with a minimum of two riders required to finish could still have defended their title. But rather than work together the two longstanding rivals began racing on opposite sides of the road. Su\u00e1rez finally dropped Bahamontes and continued alone. Bahamontes then abandoned and cunningly left Su\u00e1rez and Manzaneque to face the outraged fans at the finish by flinging his bike into the back of a car and driving off. His high-speed getaway probably neatly summed up his feelings about 1960: he could not get away from it quickly enough.\n\nWhy Bahamontes should have cracked so badly in the two most important events of the year was probably down to a lack of external guidance. The two biggest factors missing from his life in 1960 were his directors: Fausto Coppi died and Dalmacio Langarica was unavailable to manage the national squad in the Tour. He had resigned on 21 May, 1960, telling the press it was because of expanding business in the bike shop he owned. That did not stop him returning to directing later on, though, with the legendary K.A.S. squad. Bahamontes has always maintained that these two individuals were the keys to his racing success, albeit in different ways: Coppi had inspired him to win the Tour; Langarica had shown him how to do it. Crucially, Bahamontes respected them both in a way he never did Luis Puig or Bernardo Ruiz.\n\nIn the 1960 Vuelta with Faema, it is arguable that Bahamontes' volatile personality, combined with his increased sense of self-importance after the Tour win, was a fatal combination. His ego would not let him accept that one of his main support riders could be eliminated from the race. Having Ruiz, his old enemy, as director can only have heightened the tension inside the team. As far as the Tour was concerned, it seems Bahamontes underestimated his own problems.\n\nHowever, with his worst season following hard on the heels of his most outstanding one, not even Bahamontes could avoid the key question: how on earth was he going to put his career back on track?\nChapter Eleven\n\n#### Exile\n\nAndr\u00e9 Darrigade is a gentle, softly-spoken man who is keen to please. As he talks he springs around the room, dragging out boxes and bulging envelopes containing mountains of photographs from the time when he was a friend and team-mate of Federico Mart\u00edn Bahamontes. His office in the well-appointed bungalow in one of the more chic areas of Biarritz is crammed from floor to ceiling with trophies, medals and pictures of his victories. And they are not from the smallest of races. The winner of a staggering twenty-two stages from fourteen starts in the Tour de France, as well as being the 1958 world champion, Darrigade was one of France's greatest cyclists. A sprinter by profession, he won the opening stage of the Tour five times out of a possible six: not even Mark Cavendish, the 2011 world champion and unquestionably the fastest rider of modern times, has matched that yet.\n\nEven half a century on, and with limited contact between the two, Darrigade still feels a great deal of affection for Bahamontes, who was a respected colleague in the French Margnat squad for four years from 1962 to 1965. Darrigade's presence was a key reason why Bahamontes stayed so long with the team and did not keep switching at the end of each season as he had done before. For the first time in his professional career he felt sufficiently appreciated to want to stay. With a team-mate like Darrigade that is hardly surprising. Bahamontes once told me that Darrigade was his 'only friend among the foreigners in the peloton'. Darrigade recalls: 'We liked him, we really did. He wouldn't shut up, but he talked to everybody. He was a good rider. He could be cunning strategically and he wasn't the type who'd go out and party on stage races. In fact, I can remember him once ordering the entire team to get to bed.'\n\nThe universally popular Darrigade was a top sprinter who could climb well. But he was also a superb _domestique_ to the extent that he was sometimes criticised by directors for sacrificing his own chances to help the team's overall contenders. It was rare to find that degree of selflessness in the cut-throat world of cycling in the 1950s and 1960s. 'There was never a problem with Fede as a leader,' Darrigade told me. 'I'd do my work for the team and then I'd sprint, too. I did that for Anquetil, for Bobet and I did it for Federico. If he was ahead in the mountains, I'd try to slow down the rest [by chasing down counter-attacks], including Anquetil.'\n\nAs a heftily-built, tall sprinter, Darrigade's principle role within the team was to take care of Bahamontes ahead of the mountains, guiding him through the pack. By positioning himself just ahead of Bahamontes he could ensure that his leader had plenty of elbow room and would not get blocked in. 'It was difficult for Federico,' Darrigade says. 'Without me he'd have had to try to find enough space by himself. That's why he lost so often with the Spanish because they wouldn't pay him much attention. He was particularly vulnerable in side-winds and in echelons and he'd invariably lose time there in the north, in Belgium. So when he got to the mountain stages he'd be out of touch. I could warn him when problems were coming up. I would look at the maps and warn him what was going on, right down to looking at the width of the road. It was the kind of stuff I'd done with Anquetil countless times.'\n\nLike Miguel Poblet, Darrigade says that Bahamontes post-1962 was a different rider than in his younger days. 'Federico was willing to learn new tactics, like when he went training for an hour in 1964 before the start of a big mountain stage in Andorra, and he did the same again in Luchon the next day. Even before the stage started, he'd ridden over the Peyresourde pass to warm up. And as I knew Anquetil and Poulidor really well, I could tell how they were feeling and pass that information on to Federico, too.' Both Darrigade and Bahamontes point out that their new director at Margnat, former French professional Raoul Remy, was another catalyst in Bahamontes' renewal. Unlike Fausto Coppi, Remy was not a pioneer in terms of technology. But he did not have to be. Some of the changes Remy introduced were very straightforward, and quickly integrated, but they produced major results. For instance, in 1963, he suggested changing the gearing on Bahamontes' time-trial bike, so that he used a fifty-three-inch chain-ring rather than a fifty-two; it enabled Bahamontes to improve significantly when riding against the clock. Other innovations, such as bringing Darrigade and Jean Grazyck into the team, took longer to engineer but were equally beneficial.\n\nDarrigade repeatedly expresses his incomprehension that Bahamontes had received such poor support from the Spanish in the past. He partly blames Bahamontes, though, saying: 'It was in his character.' Things changed at Margnat after the Tour returned to trade teams rather than national squads in 1962. 'He was our leader,' says Darrigade. 'We knew he could win the Tour, so he got a team that worked with him. _C'est normal, non?_ ' However, as Darrigade emphasises, by now Bahamontes was thirty-four and within sight of retirement age for a professional. 'The tragedy was that it all happened too late.'\n\nBahamontes' career can be characterised as a series of brilliant flashes interspersed with sudden, dramatic falls from grace. However, after the turmoil of 1960 he went through a comparatively calm patch for two years which led to some of his most consistent and impressive results. The most significant development during 1961 and 1962 was that Bahamontes discovered that even at his age, and after a two-year absence from the race, he still had it in him to go for the overall classification in the Tour. Equally importantly, he also found that he was far better off in a French trade team than in any Spanish squad at the time. Neither discovery was straightforward: Bahamontes missed the 1961 Tour because his trade team, the Italians V.O.V., did not have the money to enter. In his one Grand Tour that year, the Giro d'Italia, he abandoned, citing injuries, though it later transpired the team had been unable to pay its riders for the entire season. If leaving a team because of money problems was not new to Bahamontes \u2013 he had run up against similar problems when he raced for Tricolfilina and would again, he claimed, in his final year with Margnat \u2013 he did not simply turn his back on the foreign option, as he might have done before. Rather than return home to ride for, say, Faema, Bahamontes decided to stay abroad in 1962 and join the French team. 'I was respected more outside Spain than in it,' he explains. He was also able to stipulate some basic conditions in this contract: he would not ride in the Classics as he had been forced to do with V.O.V; a definite ride in the Tour de France; plus the signing of a loyal _domestique_ , Juan Campillo.\n\nIn a sense the decision to stay abroad was the culmination of an ongoing process that began with Dalmacio Langarica and the Tour in 1958. Bahamontes had realised then that with the right tactics, and the right man in the driving seat as sports director, he could do more than just win the King of the Mountains prize. But after changing his attitude towards the Tour even more after Coppi's insistence that he could win it, 1960 proved Bahamontes was as good at forgetting advice as he was at taking it on board. However, in 1961 he lurched back towards taking a more objective, less impulsive approach to racing. After the 1960 debacle it seemed he might be changing for good on a personal level. 'Federico was very different in his later years, very different,' recalls Miguel Poblet. 'When he started racing, and we were sharing rooms, he was a right tearaway. You couldn't teach him anything. I remember at some point, in some race, we started being given these clear plastic cards, and I'd rip out the page of the route book and put it inside them. Bahamontes was lying on his bed, and he says, \"What's that? Can you do me one please?\" So I did. And when the race was over I ask him how using the card had gone. And he said, \"What card? It's still in my jersey. I don't understand it at all!\" That was Bahamontes in his early years. Later on, the more he raced, he got more sophisticated. He was willing to learn. There was a transformation over three or four years. It made sense because either you changed or you'd have to be a numbskull and Bahamontes was never a numbskull. He just liked to say what he thought.' One thing did not change, though, according to Poblet: 'He was very disorganised in the hotel room. His suitcase was so chaotic you'd have to walk round it on tiptoe and get the _soigneur_ to come and clear it up. But he was growing up all the same. With age he got a whole lot sharper.'\n\nBahamontes' various analyses of the advantages of being with Margnat vary wildly: while he admits in his account of the 1959 Tour that he had a better structured season with the French team, and that Remy was a great director, on other occasions when asked if he ever had a good team, he responds categorically: 'Never.' He elucidates: 'In the companies [teams] I joined, if there were [good riders], they didn't pay them, so they weren't there long. In a sporting sense it was always a disaster. If I'd ridden with [the legendary Spanish team of the 1960s] K.A.S., I'd have won more than one [Tour].' It is certainly true that in 1962 Margnat did not have a strong squad, underlined by the fact that they had only three finishers in the Tour, one of them Bahamontes. To their credit Margnat brought in some top reinforcements in 1963 as Bahamontes had requested.\n\nAt the same time Bahamontes recognises that going to Margnat solved a dilemma: it meant he did not risk offending either of the two top Spanish teams, K.A.S. and Ferrys. 'Margnat was the team who paid the best,' he says. 'They all turned up in Toledo \u2013 K.A.S., Ferrys and Margnat \u2013 with offers of half a kilo [five hundred thousand pesetas \u2013 \u00a3210,000 in today's money] each. I had a letter and a verbal offer from Raoul Remy [at Margnat] and I said to myself, \"If I race with Ferrys, then K.A.S. will race against me all season; if I race with K.A.S., then Ferrys will be against me\". So it was neither K.A.S. nor Ferrys; I was off to France. I took Juan Campillo with me [in 1964] and Antonio Blanco [in 1964]. I went for 475,000 pesetas a year, [\u00a3200,000] which was less than I'd get in a Spanish team. But the difference was they even paid my social security.' On top of that, he points out, as a 'foreigner', he could charge start money for races in Spain, which was apparently impossible if he was racing with a local squad.\n\nOne race Bahamontes did not ride again was the Vuelta. After the events of 1960 that probably suited both sides perfectly. Since his Tour win he had a lot to lose in smaller races he would be expected to win, and little to gain. In France, though, he could concentrate on the Tour with a squad who would support him. After a comparatively lacklustre 1961, Bahamontes' return to the Tour in 1962 followed an impressive first half of the season. Just as in 1961, Bahamontes hit the ground running. Whereas twelve months earlier his successes petered out after he abandoned the Giro when his old femur injury played up, in 1962, the good results stretched into July and beyond.\n\nBahamontes' return to form started with memorable victories in the Mont Faron time-trial and road race. 'I told my team-mates that when we got to Toulon [at the foot of the climb] they could go to the hotel and I'd handle the last part myself.' It was no idle boast. Just as the year before he won two hill-climbs back-to-back near Nice. But this time he followed them up with wins in May and June, on the Arrate hill-climb for a fifth time, then in a stage in the Tour of Romandie, widely used as the warm-up race for the Giro d'Italia. Even more encouragingly he was fourth and King of the Mountains in the week-long Dauphin\u00e9 Lib\u00e9r\u00e9, France's third biggest stage race after the Tour and Paris-Nice.\n\n'He was in great shape in the Dauphin\u00e9 Lib\u00e9r\u00e9,' confirmed fellow Spaniard Jos\u00e9 Bern\u00e1rdez. 'If he was brave enough to descent fast then he'd be unbeatable. But I don't think he's a match for Anquetil and Van Looy's strong teams.' _MARCA_ was wary about Bahamontes' Tour chances, saying that he was 'in great shape, but we'll have to give it a couple of stages to see what his real condition is like'.\n\n_MARCA's_ caution proved to be wise. On the opening stage from Nancy to Spa, a front group of twenty-three containing all the race favourites barring Bahamontes and Gaul, gained a massive margin. While Gaul lost two minutes, Bahamontes finished seven minutes forty-two seconds down with just Campillo supporting him. It was not quite game over but it was definitely not the return to the Tour for which Bahamontes had been looking.\n\nMost of the media attention centred on the post-stage bust-up between Belgium's sprint legend Rik Van Looy and stage winner Rudy Altig, the Vuelta champion. But the Spanish journalists were waiting for Bahamontes, who informed them he had been caught out. 'We rode well but the bunch split. It was hot and we were unlucky,' was Bahamontes' facile explanation. This time, though, Bahamontes did not just abandon and go home. He simply continued to lose chunks of time here, there and everywhere: ninety seconds to Jacques Anquetil on stage two's time-trial; six minutes on the first section of stage seven to Henry Anglade and Tom Simpson; and three minutes to Anquetil in the afternoon time-trial the same day. The totals were adding up and Bahamontes had little to say to the press apart from bland clich\u00e9s: 'The important thing is to stay close to the front,' he said after one stage; 'The Spanish are doing well, I'll try to do something in the mountains,' after another. Perhaps more notably, he recognised that Luis Ota\u00f1o and Antonio Su\u00e1rez 'are the ones defending our country the best'.\n\nThere were parallels to his disastrous performance in the first part of the 1956 race when he found himself even further behind the frontrunners. But unlike six years before this Tour was following a standard pattern rather than being what Brian Robinson called 'a blank sheet every day where you never knew what was going to happen'; it was going to be harder for Bahamontes to regain time. There were occasional attacks which raised hopes and when he chased down a break at the finish of the stage into San Malo it earned him his first mention of the race in _L'Equipe._ There were also brief flashes of the old self-confidence. As Raymond Poulidor recalls, Bahamontes would be telling riders when and where on the course he would eliminate them. In 1962 it was not just the more impressionable Tour rookies like Poulidor who could receive their marching orders from the Eagle of Toledo, it was big names like Rik Van Looy, too. 'Van Looy was attacking every single stage with his famous Red Guard. We got to the foot of the Pyrenees racing at 45 k.p.h. all the way from the start,' Poulidor told me. 'Then on the morning of the first Pyrenean stage we're all there at the signing-on, me, Van Looy and Bahamontes . . . and Bahamontes goes up to Van Looy and says, \"You, this evening, you're on the train out of here. It's me that's got the keys to the race now\". And, sure enough, that day Bahamontes was on the attack in the Pyrenees and that evening Van Looy was on the train, just like Bahamontes had told him.' And that was Bahamontes' level of confidence on an average Tour.\n\nMore than half the Tour had passed when the race finally headed into the Pyrenees and Bahamontes duly took off. He went over the Tourmalet, Aspin and Peyresourde in first place. It was as spectacular a performance as any of his earlier mountain rides and earned him the front cover of _MARCA_ with the headline: 'The Eagle Takes Flight Again.' However, while it racked him up a healthy twenty-three-point lead in the King of the Mountains competition, in terms of the overall it was all but meaningless. Caught by Rolf Wolfshohl on the Tourmalet's descent, and again on the Aspin, Bahamontes was duly reined in sixty kilometres from the top of the Peyresourde. Game over. 'When you see that [Robert] Cazala, a sprinter who got over middling-size hills in reasonable shape, won the stage, that just about sums it all up,' Bahamontes fumed. 'I couldn't have won. To go on would have been crazy.' Next up, though, was an 18.5-kilometre mountain time-trial from Luchon to Superbagn\u00e8res, one of Bahamontes specialities and he duly delivered. But though he won the stage by eighty-five seconds over race leader Josef Planckaert, and gained seven minutes on the Belgian overall, he was still nearly twelve minutes behind the yellow jersey and remained pessimistic. 'I can't win, it's impossible,' he said. 'I've taken ninety seconds on Anquetil, but he'll regain six minutes on me in the time-trials.' As was so often the case in Bahamontes' mind, others were responsible for the situation. 'My team isn't sufficently strong for me to pull this off,' he claimed, 'though I'm ready to say that I prefer it to the one that was with me in 1959.' Bahamontes may have been critical of the Margnat team but they were not as weak as he made out: Ota\u00f1o, for example, was lying fifth overall after Superbagn\u00e8res.\n\nUnhappily, the Alps lost their relevance for Bahamontes in terms of the overall before he even reached them. Like Ota\u00f1o, he lost a huge amount of time the day after his stage win and trailed into Carcassonne more than thirteen minutes down on all the other favourites. Having praised him for his classy performance during the two days in the Pyrenees, _MARCA_ now highlighted his erratic nature, saying: 'He is a formidable rider \u2013 he gets formidable wins, then formidable defeats.' This tongue-in-cheek attitude was reinforced when they dragged up an old anecdote in which Bahamontes had apparently said he had suffered from a non-existent illness called _limaquitis_. 'Remember when he caught _limaquitis_? Nobody knew what it was because it was only an illness for him, and only he knew what that illness was.' For once Bahamontes blamed himself for a momentary lack of attention: he was at the back of the bunch when a sudden acceleration wrenched it into two. It left him with only the King of the Mountains jersey and another stage win as possible goals over the eight remaining days before Paris.\n\nBahamontes succeeded with the former, but not the latter. He blasted away alone in the Alps on the Tour's first assault of the Restefond Bonnette, Europe's highest pass at 2,802 metres above sea level. On the Izoard he was twenty seconds ahead of the main favourites. However, he was easily caught on the drop into Brian\u00e7on and had to settle for sealing his fourth King of the Mountains jersey. Once again that competition had been a one-man show: Bahamontes had a colossal sixty-point advantage over Italy's Imerio Massignan, who had won the title in his absence in 1961. At the finish in the Parc des Princes the French press dubbed the only three Margnat riders to complete the course (Ota\u00f1o, Campillo and Bahamontes) 'The Three Musketeers'. They had all finished inside the top thirty which led journalists to wonder what would have happened had the race been ridden with national teams.\n\nBahamontes' popularity remained high in Spain as one journalist from _El Mundo Deportivo_ testified when he came down into the lobby of the team hotel after Superbagn\u00e8res to find it overflowing with telegrams congratulating Bahamontes on his win. His old French friend, Jacques Daud\u00e9, was equally delighted by Bahamontes' return to winning ways. Just as in 1959, Daud\u00e9 funded his own trip around the Tour so he could be there at each finish with a bottle of water for his favourite rider. He had also become Bahamontes' minder, forcing journalists, and particularly Spanish ones, to ask him permission to talk to Bahamontes. 'It's easier if you're a foreigner,' one huffy Spanish journalist wrote.\n\nThe Tour route in 1962 did not particularly favour Bahamontes as he constantly pointed out. It contained four time-trials, albeit one of them uphill and one of them a team time-trial, making one hundred and eleven kilometres of racing against the clock. Made to measure, in other words, for Anquetil. Many of the finishes were also placed a long way from the top of the last mountain pass: it occurred after both the Tourmalet and the Izoard. Bahamontes was undoubtedly not helped by a lack of team support, not least because the last of the seven Frenchmen who started had abandoned in the Alps. But that would have been more significant if Bahamontes had been in contention, and that, as he admits, was at least partly his own fault.\n\nSo the 1962 Tour was a classic case of whether the glass was half-full or half-empty. Bahamontes felt the pros outweighed the cons. 'The important thing about 1962 was that it put me back on the map, even for those people who had claimed that I was on the way down,' he says. That in itself, given he was now thirty-three, was no mean achievement. So, too, was Bahamontes finally finding a trade team that suited him \u2013 and vice versa.\nChapter Twelve\n\n#### A Pirate With a Pair of Pliers\n\nMy ears hurt. Raphael Geminiani has just yelled, 'Who? Where? Who says Anquetil tried to buy Bahamontes?' at the top of his voice. For good measure he has slammed his fist on the kitchen table. Even at eighty-six, ' _Le Grand Fusil_ ' \u2013 Mr Top Gun is probably the closest translation \u2013 remains a force to be reckoned with. Interviewing Geminiani is not for the fainthearted. At six feet tall, he is a mountainous man with huge hands and tree-trunk thighs to match his girth. His laugh is like a long, dry drain and he has a leery, shark-like grin. If Geminiani had a previous life, it probably had something to do with bottles of rum, high seas and buried treasure. Instead he played the role of pirate on dry land. He had a near-legendary ability for wheeler-dealing and was one of the great managers, representing everybody from Louison Bobet to Anquetil, Lucien Aimar to Stephen Roche, and even briefly Eddy Merckx. Geminiani was and remains cycling's ultimate buccaneer, a champion of the sport's non-conformists. He was expelled from France's 'A' squad to a regional team in 1958 for being a firebrand. But only a raging storm in the mountains of Chartreuse, and a climber of the talent of Charly Gaul, prevented 'Gem' from embarrassing France's top team by winning the yellow jersey outright.\n\nGeminiani was equally fearless when it came to taking on race organisers and cycling's governing body, the U.C.I. Indeed, he staged a near one-man battle to have the names of trade sponsors included on team jerseys, something that is now taken for granted. In fact, in one race as a team director he instructed his riders to take off their jackets to reveal their logo-encrusted jerseys only seconds before a stage was due to begin to beat the ban on sponsorship. Just as fascinating is Geminiani's penchant for ridiculing the authorities. Typical is the story of the pet donkey he received from a fan at the 1958 Tour start and named after the French national trainer, Marcel Bidot, in revenge for his non-selection to France's 'A' team for that year's race. Parading it around the streets of Brussels where the race began, Geminiani then presented it as a gift to Bidot before as many photographers as he could assemble at the World's Fair, held in the Belgian capital that year. It was the kind of publicity stunt he took a positive delight in.\n\nThat, though, was more than half a century ago. These days, as Geminiani potters around his village near Clermont-Ferrand in central France, his giant frame squeezed into a tiny car, cigarette jammed into one corner of his mouth, he looks almost inoffensive. But this is the man who had one of cycling's stormiest and most intensely-lived careers.\n\nRight now, he is furious that anyone could suggest Anquetil wanted to pay off Federico Mart\u00edn Bahamontes. He yells again at the top of his voice: 'Where? Who?' In that mythical former life Geminiani would probably have been jabbing a sword in my chest. A neighbour has dropped in with a home-grown lettuce the size of a large cannonball for Geminiani's lunch; he watches this verbal dust-up across the kitchen table with all the impassivity of a countryman who has seen these kind of fireworks many times before. At this point I cop out. I do not have the nerve to tell Geminiani that the allegation comes from Bahamontes himself, who describes how Geminiani rang him one night during the Tour and offered to buy the race from him on Anquetil's behalf.\n\nIn any case, I have given Geminiani the right of reply. When he finally calms down he insists: 'I only ever had one rider in my team, a Belgian, who once sold a Paris-Roubaix, and when I threw him out and Anquetil said I'd been tough on him, you know what I said? \"My sponsor doesn't pay riders to make their rivals win\". But that's typical, they [the press] never say it's the strongest who wins, it's always a cheat, a doper or someone who's been bought. You say that Bahamontes helped Anquetil? I've got two sons and four daughters, and my honour, and I'm telling you [on all of them]: no. Never did anybody help Anquetil win anything. Whatever.' I leave it at that. After all, this is a man who allegedly swung a bicycle pump so hard during a Giro d'Italia he knocked five teeth out of a fan's mouth. His biography is titled _400 Fits of Bad Temper and Rifle Shots_. I have no desire to become the victim of the four hundred and first. As he reaches for his fifth cigarette in about an hour, and pours another black coffee, we continue.\n\nGeminiani was one of the first French riders to get to know Bahamontes well, and that as far back as 1956. So he was able to witness first hand the growth in the Spaniard's popularity in France. He also knew his weak points. 'We did the whole of the _tournee_ [post-race series of criteriums] of the Tour de France in 1956 in my car. It was me who drew up the route, me who chose the hotels and me who \u2013 he-he-he \u2013 chose the restaurants,' claims Geminiani between loud sucks on his cigarette. 'Daniel [Dousset] had told me to look after Bahamontes because he had no idea about France.' In the criteriums, Geminiani recalls, Bahamontes proved to be just the sort of ally he wanted. 'I remember we went to a crit in Chateaulin, the best paid in France. We got to the start and there was the Bobet mafia and the Anquetil mafia. And I said to Gerard Saint, \"Gerard, you do the first five laps and get all the primes\". Then, \"Federico, it's you [to another five laps] and afterwards it's me\". It worked. The mafia only got to win the fifty-franc primes and we got all the hundred thousand primes. And that evening in the distribution of the primes at Chateaulin town hall, I remember saying to the two mafias, \"All right, lads? Those fifty-franc primes feel good?\"' He hoots with laughter at the memory.\n\n'With time, we grew to trust each other. I taught him some French. He didn't speak much, but when he did people liked it.' Just as Andr\u00e9 Darrigade had suggested, Geminiani says that Bahamontes was enormously popular, 'even more so here than in Spain. They liked him here, he was _spectaculaire._ Everybody knew about that ice cream incident and they loved him for it. Maybe he only ate it because he'd needed a wheel change, but it caught their attention. But he wasn't so big in Spain because he'd never won the Vuelta. Whatever.'\n\nThe massive round of applause Bahamontes received in 1962 at the Tour presentation in Nancy goes some way to indicating how much he was appreciated in France as the leader of a French team. And there is some more intriguing confirmation of his fame in the most unlikely of places: the film _Amelie._ The heroine finds a box of toys left behind in a flat and upon returning them to their owner, a middle-aged man, he is reminded of watching Bahamontes win the 1959 Tour. Bahamontes concurs with the verdicts of Geminiani and Darrigade about how he is perceived in France. He once told me: 'Go anywhere there and ask a Frenchman who the best climber in the world has ever been? Bahamontes.' Geminiani, being Geminiani, could see Bahamontes' popularity in France was something he could exploit. When he sought alliances to defeat Bahamontes, he went to the Spanish riders. 'Given the situation it was easy to work with them against Bahamontes,' he says. 'It wasn't just that he was the man to beat. The aim of the game for the Spanish was to beat the man who was popular in France. But I can't say anything bad about him, he was somebody I liked.'\n\nFor all that Bahamontes was popular in France, it was no longer something he could take for granted on the other side of the Pyrenees. By early 1962 Bahamontes was no longer the number one sports star, let alone cycling star, in Spain. Two years of failing to make an impact in Grand Tours, and the rise of new stars like Angelino Soler and Jos\u00e9 P\u00e9rez Franc\u00e9s, contributed to a drop in interest in the man from Toledo. Furthermore, Bahamontes raced for a foreign team and never took part in the Vuelta. By the time his old rival Jes\u00fas Loro\u00f1o retired at the end of 1962 the two rarely raced against each other any more: Bahamontes competed mainly abroad and Loro\u00f1o seldom ventured out of Spain. Besides, the external factors of an inward-looking, isolated and economically-backward Spain that had helped drive their conflict had begun to fade. Not that Bahamontes was complaining. Racing for a French team automatically made the Tour the number one target of his season, something which in the Italian squad V.O.V., with its huge interest in the Giro and the home market, had by no means been guaranteed. Bahamontes was happy to be where he was, too: in 1963, for the first time in his career, he opted to stay with the same team for more than one season. Margnat were delighted with his success in the King of the Mountains in 1962, but their signing of two double winners of the Tour's points jersey, Jean Grazyck and Darrigade, showed they had bigger ambitions for Bahamontes than the previous summer. Indeed, ensuring he did not lose time on the flat could only mean one thing: Margnat wanted their star rider to have a crack at the overall title.\n\nIf news of Bahamontes' switch of strategy towards regaining the Tour overall had filtered out of Margnat's team before the race start, most French fans would not have been displeased. However, among French riders his new-found ambition drew a mixed reaction. 'We used to detest Bahamontes, but only because he was so good,' Raymond Poulidor says. 'Whatever the stage he would always be positioned a long way forwards on his bike, pedalling furiously. He'd attack at the foot of the first col and we wouldn't see him until the evening of the same stage. We didn't want to let him get out of our sight, but we didn't have any choice. He used the \"coffee-grinder\" style of pedalling: he would use a big gear to get away for his first attack, but then he'd spin a small gear really fast and keep up the same speed to create the really big time gap. If you tried to go with him, it was like committing suicide.'\n\nEven in 1963, when he turned thirty-five, Bahamontes made a great start to the season, gaining resentment and respect in equal measure from those who were less prepared. Poulidor recalls: 'It was always the same. Every year. I remember the Mont Agel hill-climb race early in my first year as a pro in 1960. Nobody had done any training \u2013 I'd maybe ridden three or four hundred kilometres before it \u2013 and we just had a pump on our bikes to carry as little weight as possible. Bahamontes, though, had a pump, a bottle and spare wheels on his. But who was the first at the top? Bahamontes.'\n\nIn 1963, then, Bahamontes came to the Tour de France with far more ambitious plans than 'just' another King of the Mountains jersey. 'Raoul Remy [Margnat's director] told me that there was no point in going for it again, people would take that for granted,' Bahamontes recalls. 'On top of that, the route was far more mountainous than usual. We had to go for the overall.' Anquetil won the Tour despite the organisers doing their best to level the playing field by cutting more than thirty kilometres from his strongest suit of time-trialling, reducing it from one hundred and eleven kilometres in 1962 to seventy-seven in 1963. However, what nobody expected was, as Tour director Jacques Goddet put it: 'The one rival who would be at his level was our old acquaintance, Federico Bahamontes.'\n\nBahamontes' crowning moment of glory came in the ski resort of Isere. A day after taking an Alpine stage win in a lone break, for the first time in four years he captured his first yellow jersey since 1959. True, his overall advantage over Anquetil was almost risibly small: just three seconds. Yet given he celebrated his thirty-fifth birthday the day he took the lead in the Tour for the second time in his career, it was a landmark achievement.\n\nEverything had fallen into place in 1963 starting with morale-boosting King of the Mountains wins in both the Midi Libre and Dauphin\u00e9 Lib\u00e9r\u00e9 warm-up races in June. Significantly for his team, on the first stage of the Tour Bahamontes had managed to get into the earliest possible break alongside Britain's Alan Ramsbottom and Belgian Eddy Pauwels and gained an eighty-eight-second advantage over Anquetil. For a squad like Margnat's this was crucial: just as today, much store was set in a team by a contender's ride in the Tour's prologue. Bahamontes gaining time _outside_ the mountains so early in 1963 showed his troops that their often erratic leader was on form and keen to do battle. Small wonder Margnat rallied around him, and he had some powerful team-mates to back him up. 'Bahamontes had a great team for the flat in 1963,' Poulidor recalls. 'Darrigade and Grazyck were two key players. His only weakness, curiously enough, was in the mountains: not many riders to support him there.' _MARCA_ waxed lyrical about this early success: 'He has turned up with the same kind of strength he used to have as a junior. Bahamontes is \"Baha\" again, flying over the Tour.' However, they added a sting in the tail, noting that in previous years, 'Anquetil has shown he is the best'.\n\nBahamontes' early break meant that even after an unremarkable seventh place for Margnat in the team time-trial on stage two he still remained fourth overall. He continued to go from strength to strength, staying out of trouble on the _pav\u00e9s_ to Roubaix. Then he lost just ninety-eight seconds to Anquetil in the first individual time-trial on a technically difficult course which contained sixty dangerous curves in 24.5 kilometres. 'I'm in great shape, but I don't drop them like I did before,' Bahamontes said, 'but at least I'm not losing time on the flat, either.'\n\nAnquetil tried his best to ambush Bahamontes before the mountains, attacking with Raymond Poulidor, Henry Anglade and Jean Stablinski on the long stage from Limoges to Bordeaux. However, Margnat combined with the Basque squad K.A.S. to pull back the move, prompting Bahamontes to tell journalists: 'Finally I am in a good team. Other times I raced the Tour as an individual. What do you think Van Looy or Anquetil could do without team-mates?' The difference between 1963 and previous years was that sometimes at least Bahamontes alternated his criticisms of his team-mates with dollops of praise. Margnat must have been doing something right.\n\nHowever, while Bahamontes and his team rode far above expectations on the flat, things did not go to plan in the Pyrenees. All his attacks were neutralised, and though he took a solid hold on the King of the Mountains competition with some early breaks Margnat's only stage win came thanks to Darrigade, who secured the twentieth of his twenty-two career Tour victories. It was earlier in that stage that Bahamontes made a serious error. After attacking on one mountain section very early on, and despite having team-mate Claude Mattio with him for support in atrocious weather conditions, he then opted to sit up at a point where he could have inflicted serious damage on his rivals. 'He missed out because he listened to his director's instructions to wait for the peloton,' _MARCA_ claimed later. 'If he had not done that it would have been Bahamontes on the highest step of the Tour de France podium that year.' Others agreed and also blamed the team. 'Fede lost the race in the Pyrenees,' Rik Van Looy said later. 'He couldn't handle the rain. If he had been in my team he and no other would have won the Tour for the simple reason I would have thought for him. That isn't what those who directed him did.'\n\nThe Alps were a different story. After Jacques Anquetil gained a thirty-second time bonus on Bahamontes at Aurillac, Bahamontes fired back the next day en route to Grenoble when he launched a devastatingly effective, short-range attack over the Col de Porte. Just nineteen kilometres from the finish Bahamontes blasted away alone on the first category climb. After reaching the summit with a two-minute advantage, for once he managed to maintain the gap all the way to Grenoble. It was the city where he had taken the yellow jersey four years before; now he was just twenty-four hours away from recapturing it in Isere. 'It was one of the most brilliant operations the Spaniard has ever carried out,' _MARCA_ claimed, before warning: 'But it did not get us the result we wanted. Unless Bahamontes takes five minutes in any one of these stages any earlier moves will have been in vain.'\n\nTaking the yellow jersey was almost a formality. On the Iseran pass the leader Gilbert Desmet of Belgium cracked completely, but with no real reaction from the main contenders. Desmet slid backwards, the rest trundled upwards. Bahamontes and the rest of the top names grouped together between the three-metre high snowdrift walls at the summit with only one brief attack from Poulidor to liven up proceedings. Bahamontes sardonically observed later that it was the only one Poulidor made in the entire race. However, Poulidor was not uppermost in Bahamontes' mind as they came down from the Iseran. With Desmet out of the running, Bahamontes was now ahead of Anquetil. One major Alpine stage and one long time-trial remained; the duel for the 1963 Tour, Bahamontes observed, was about to begin.\n\nOne of cycling's hidden attractions is that a lot of what remains memorable in an event is not what happens in the race but what happens around it. In the 1963 Tour it was the off-race controversy surrounding the crucial last Alpine stage that has come to dominate fans' memories, even though the key incident lasted all of twenty seconds. At the foot of the Forclaz, the last key climb of the race, Anquetil raised his hand to denote his bike needed repairing. Raphael Geminiani sped forward and cut the brake cable on Anquetil's bike with a pair of pliers when the officials, even the one in his car, were either not paying attention or could not see it. As a result of the 'broken cable' Geminiani was able to claim Anquetil should have a replacement bike, and quickly provided him with a lighter model. As a result of those twenty seconds, the three weeks and three thousand or more kilometres of the 1963 Tour were delivered into Anquetil's hands. Because as Geminiani puts it: 'If it hadn't been for the Forclaz, Bahamontes would have won the Tour.'\n\nSitting at his kitchen table now, Geminiani says: 'Bahamontes was the big rival we defeated that day and Poulidor lost seven minutes, but it was the organisers who made a mistake.' To say that Geminiani relishes discussing the Forclaz, and the tale of Anquetil's cable, is no exaggeration. He gives a lengthy, no-holds-barred explanation of what happened. He freely admits he cut the cable, but insists the fault was not his: it was the organisers' for failing to spot it.\n\n'That year, before the race, it was said that on that stage we'd go over the Grand St. Bernard pass, go down to Martigny and then take the A road over the Forclaz. Not a hard climb. But there was a landslide [he laughs heartily at the memory] and that blocked the Forclaz. So . . . I was lucky enough that someone told me beforehand we would have to go over the pass on the old sheep track that we'd gone over in 1948 when everybody went up it on foot. I looked at the pieces of paper the organiser had given us, and there was no change on it . . . but I knew that we'd have to be careful and change bikes, which was prohibited. So I prepared another bike with a twenty-six-tooth sprocket [which was far more suitable for an untarmacked climb]. I told Anquetil what we'd do, but obviously we'd need some kind of mechanical problem.' He chuckles again. 'So I made the problem happen: I cut his cable. I told the official watching that Jacques' rear cable was broken and that we'd need a replacement bike. And that is how, when Bahamontes attacked on the Forclaz, he dropped everybody, but not Anquetil. Not on his light bike.'\n\nThat was not the only thing the officials failed to notice that day, Geminiani points out. He outlines a second, less well-known, game-changing piece of cheating. 'When we got to the other side, I gave Anquetil his old bike back, but with his time-trial gearing instead of his usual mountain gearing to do the last, flatter, thirty kilometres to Chamonix.' Bahamontes had tried his utmost to shake off the Frenchman, but he had been outwitted. When the pair reached Chamonix, Anquetil took advantage of the latest switch of bikes to outsprint Bahamontes and win the stage. The time bonus of a minute \u2013 Bahomontes received a thirty-second bonus for second place \u2013 meant Anquetil landed the yellow jersey by a comfortable margin of twenty-seven seconds.\n\nWith only Anquetil's strong suit of a time-trial to come, Bahamontes' dream of a second Tour had evaporated. For all _MARCA_ claimed that he lost the Tour in the Pyrenees, and rebuked him for attacking earlier in the same day and wasting energy, the Forclaz had been his great opportunity to regain that time and turn the tables again. But thanks to Geminiani's sleight-of-hand that was rendered impossible. 'Bahamontes' defeat by Anquetil shows he had real talent,' Geminiani recognises. 'At Chamonix, it took a really strong Anquetil to beat him. And, like I said, if it hadn't been for the Forclaz, he'd have won the Tour.'\n\nWhen Geminiani reached the finish he faced a storm of protests about cutting the cable. But he tackled them head-on.\n\n'Remy protested for Bahamontes, [manager Antonin] Magne for Poulidor and I was surrounded by officials and journalists, and so on,' he remembers. 'Magne said, \"You cut the cable!\" And I said, \"Did you see it? That's slander, be careful, eh?\" So I said to the race official, \"Can you tell this gentleman what happened?\" And he said, \"I saw there was a mechanical problem.\" And I said, \"Do I have the right to change a bike if there's a mechanical?\" \u2013 \"Yes\" \u2013 \"Well that's what I've done.\" And Remy said, \"Well I could have done that, too!\" and I said, \"Well, did you?\" \u2013 \"No\" \u2013 \"In that case, your rights to do so are denied\".' He howls with laughter at the memory.\n\n'Me, though, I'd done my job, respecting the rules,' Geminiani insists to me, tongue clearly in cheek. Then he chortles at having beaten officialdom and the opposition in one fell swoop. Not to mention winning the Tour.\n\nApart from his specific accusation that Geminiani tried to buy the race from him, Bahamontes has very mixed feelings about the 1963 Tour. Keenly aware that he could have won the race, a torrent of resentment runs just beneath the surface when he discusses it. 'Geminiani was really scared of what I could do that year,' he says. 'When I attacked over the Grand St. Bernard, he got everybody he could to chase after me. I'd gone on the attack, waited to see if anybody came, nobody did, and [Margnat director] Raoul Remy came up and said, \"Where are you off to?\" and I said I just wanted to get to the top to settle the King of the Mountains competition. And then when I got to the summit of the Grand St. Bernard I had five minutes. And if you're at the top of a climb with five minutes' advantage, what do you do?' So on Bahamontes went, despite being weighed down by a large potato, kept in his back pocket to try to prevent an over-large _maillot jaune_ from ballooning out. 'All the teams were chasing like crazy, the Belgians and the Dutch and Gem's team, because they knew I could win the Tour. Then there was only the Forclaz left and Geminiani was there with Anquetil. Geminiani gets a kick these days out of telling people how just at the point where it got toughest he changed Anquetil's bike and pushed him up the climb round the steepest corners. On the climb itself Poulidor got dropped when I attacked and there was just Anquetil left. He sat on my wheel and he was suffering badly because I went up on the left-hand side so he'd be riding into the wind as well. But Geminiani had changed his bike, cutting the cables with some pliers, and then he pushed Anquetil round each bend. On the descent Anquetil attacked, I didn't have good brakes and my foot came out of the pedal on a bend and he got fifty metres on me.'\n\nAfter Forclaz, on a bike with gears suited for a sprint, wrapping up the race was easy for Anquetil. 'There must only have been three kilometres left when Geminiani told Anquetil to ride like it was a time-trial, but I was ready for that attack,' Bahamontes recalls. 'Then a motorbike belonging to the race organisation came up \u2013 it had no right to be there \u2013 and helped Anquetil attack again. And that wasn't the only time that happened by a long shot.'\n\nAt other moments, though, Bahamontes was capable of recognising that Anquetil rode the 1963 Tour 'like a great champion. He was well organised, he had no need of Poulidor'. Yet almost in the same breath, in an interview with _L'Equipe_ at the finish in Paris, Bahamontes once again accused Raymond Poulidor of 'working for Anquetil on the Grand Saint-Bernard. Without that, I would have got six minutes. I'm not mad. I saw them talking together two days before that'.\n\n_MARCA_ said in their post-race analysis: 'He is the best Spanish rider in history and he deserved the triumph had it not been for those errors he made. At thirty-five, to go on being a great figure in his sport at the same level as the winner has its merit.' The Franco regime woke up to the fact that Bahamontes was back on top of his game, though it was noticeable that they took advantage of Jos\u00e9 P\u00e9rez Franc\u00e9s's third place behind Bahamontes, as well as Angelino Soler's sixth place overall, before singling out Bahamontes only briefly. It was a long way from the ecstatic reception and political exploitation that Bahamontes' Tour win had received four years earlier. 'I congratulated him and all the Spanish who have made the race Spanish again,' commented Jos\u00e9 Antonio Elola, who remained in control of sport. 'Even in different teams, you [the Spanish] cannot deny that your style of racing and sporting strategies have united you above brand names and tactics.'\n\nThe most intriguing analysis of Bahamontes' near-miss came from Damian Pla, P\u00e9rez Franc\u00e9s's director, who claimed that had it not been for Anquetil, 'Bahamontes would have lost twenty minutes on the flat', and with it all chance of a podium finish. 'Anquetil ended up blocking the race [by being so strong], he made everybody race more conservatively,' he said. That sounds like sour grapes because Bahamontes finished ahead of P\u00e9rez Franc\u00e9s. Besides, barring that crucial error of not opening up a gap in the Pyrenees, he hardly raced conservatively: as early as the first stage Bahamontes was taking time from Anquetil. As for Bahamontes' claims that Anquetil had been helped to win the Tour, the issue was skilfully fudged by _L'Equipe._ The newspaper shrewdly pointed out: 'Federico accuses, comments and regrets with the same degree of sincerity.' However, the writer then dodged the issue of Anquetil by focusing on another rider altogether: Poulidor, who finished eighth. 'His arguments lack any foundation . . .' _L'Equipe_ blithely added. 'Poulidor was always behind him, he was never able to bring back this devil of a man who flew towards the summits.' Anquetil did, however, even if occasionally he needed a pair of pliers and a shrewd, piratical manager to be sure.\nChapter Thirteen\n\n#### The Master and the Apprentice\n\nIn 2011, Luxembourg's Andy Schleck jumped away from the pack on the Alpine stage over the Izoard pass and up the mythical Galibier climb and rode the sixty-five kilometres to the finish alone. His closest pursuer was two minutes seven seconds adrift and it took him within fifteen seconds of the yellow jersey. Ultimately he finished second overall in that year's Tour de France.\n\nThe move was hailed as a throwback to the glorious days of long-distance mountain attacking. It was almost Bahamontes-esque in its execution. Indeed, it was almost a carbon-copy of another devastating break by Federico Mart\u00edn Bahamontes back in 1964. Bahamontes attacked at the foot of the first of four Pyrenean cols with just one other rider, Julio Jim\u00e9nez, for company. After riding over the Peyresourde, Aspin, and Tourmalet together he dropped Jim\u00e9nez on the Aubisque and reached the summit alone. Forty-five kilometres later he crossed the line in Pau nearly two minutes before anyone else. As a result Bahamontes moved to within thirty-five seconds of the yellow jersey in second place. He finished third overall in Paris.\n\nThe similarities between the two moves, forty-seven years apart, are clear. Bahamontes' spectacularly long breakaway provided perhaps his greatest stage win of all in the Tour. However, outside Spain his two hundred-kilometre dash for glory is still considered merely a sideshow in the drama of the 1964 Tour. For proof of that lack of interest you could do worse than look at _L'Equipe'_ s mammoth 1,500-page history of the Tour: Bahamontes' two wins in 1964 \u2013 in Brian\u00e7on and Pau \u2013 do not even earn a mention. In fact, the only reference to the Eagle of Toledo is in one (wildly inaccurate) headline: 'Poulidor climbs better than Bahamontes'. His lack of recognition that year is down to the fact that he was considered the third man in what is still viewed as an epic two-way duel between Jacques Anquetil and Raymond Poulidor. Often cited as one of the five most exciting Tours in living memory, it was the year of Anquetil's near-collapse coming out of Andorra, of Poulidor's clown-like mechanical problems near Toulouse and of his stunning fightback to Luchon. Above all, it is the year of the Anquetil-Poulidor _t\u00eate-a-t\u00eate_ duel, shoulders straining against one another, on the slopes of the Puy de D\u00f4me. Bahamontes actually finished ahead of both of them on the Puy, but he was never considered to be a serious challenger. In the words of Raphael Geminiani, Anquetil's director: 'Bahamontes was all spectacle. He wasn't a threat.'\n\nYet when Bahamontes rode alone across the summit of the Aubisque he had an advantage of six minutes on the pack and the lead in the Tour de France was within his grasp. Bahamontes remains convinced that he could have won the Tour that year. Geminiani, as a biased observer, might not agree, but an astute neutral like Andr\u00e9 Darrigade says: 'He's not wrong; he could have.' Darrigade explains: 'When Bahamontes attacked that day in the Pyrenees Anquetil used the Pelforth team against him. There was a sort of conspiracy, or manipulation of events, against him.' And then, of course, there was Julio Jim\u00e9nez.\n\nRemembering how difficult Bahamontes' relationship was with his predecessor as Tour's King of the Mountains, Jes\u00fas Loro\u00f1o, it is perhaps reasonable to suppose that Bahamontes' relationship with the man who succeeded him was never going to be a marriage made in heaven. As Darrigade points out, though, it is ironic that on that stage from Luchon to Pau in 1964 Jim\u00e9nez was the man who held the key to a second Tour victory for Bahamontes. But Bahamontes failed to reach out and claim the crown. 'If he'd been a bit craftier,' Darrigade observes, 'he'd have said to Jim\u00e9nez, \"You take the stage win, and I'll go for the Tour\". If the two had ridden together and worked together all the way to the finish Bahamontes could have done it. He could have won the Tour.'\n\nTrue, with Jim\u00e9nez riding for K.A.S. and Bahamontes for Margnat, the alliance of Spain's two great climbers on stage sixteen was never going to be more than a temporary fusion of mutual interests. Of course, both wanted to win the King of the Mountains title, and both wanted to finish as high in the overall classification as possible. But under normal circumstances an understanding between the pair was far from unprecedented in cycling. Putting as much daylight between themselves and their rivals as possible was to their mutual advantage and far more likely if they combined forces. After that the two could divide the spoils, Bahamontes going for the Tour and Jim\u00e9nez for the stage and the King of the Mountains title. The problem was that while Bahamontes wanted the overall, he did not want to relinquish the King of the Mountains jersey. As a result, his best chance of winning the race that year disappeared on the long road to Pau.\n\nBahamontes made his move after three kilometres. 'I attacked early to see if I could get rid of the _domestiques_ ,' he claims, 'and force the leaders to come across. When I heard Julio Jim\u00e9nez was trying to bridge across my director, Remy, told me to wait. I let him move ahead on the summits of the three climbs that followed. But when he had taken them all and moved ahead of me on the mountains classification he then refused to work any more. At the top of the Aubisque I had six minutes thirteen seconds on the bunch, more than double the advantage I'd had at the foot, but that was when the suffering started. I had all that way to go to Pau, but I was alone, nobody was there to work with me and that's what I needed, above all, when I had a group with Anquetil, Poulidor [and nine other riders] chasing behind. With a head-wind in the last ten kilometres the ride to the finish seemed to be interminable, and behind they rapidly reduced my advantage. But I still had nearly two minutes on them at the finish.'\n\nWith Spain able to watch the race live on television for the first time that year, what proved to be Bahamontes' last full Tour, and his last stage win, turned into a memorable dessert to serve for his supporters back home. 'It was,' Bahamontes once said, 'one of the most meritorious days of my career, but my hard work did not allow me to reap the full rewards. The distance from the summit of the Aubisque to the finish in Pau was so far that it allowed the chasing group, unable to beat me individually, to beat me as a unit.' Overall, though, Bahamontes had made serious inroads, moving to just thirty-five seconds behind leader Georges Groussard, while Anquetil was fifty-one seconds back and Poulidor at a minute.\n\nThough Bahamontes took a trouncing in the time-trial that followed the next day, losing four minutes in forty-two kilometres, there was still an outside chance of regaining time a few days later on the Puy de D\u00f4me, the race's final summit finish. Instead, Bahamontes was unable to follow Julio Jim\u00e9nez's attack late on, admitting that he miscalculated, and his second place at the finish line simply allowed Bahamontes to consolidate his third place overall. A podium place was no mean achievement for a thirty-six year old and his two mountain stage wins were confirmation that he was still mounting an aggressive campaign, and not just riding conservatively.\n\nOn occasions Bahamontes has attempted to explain away his defeat by saying that he wanted to finish third in the 1964 because he had already finished first, second and fourth overall. At other moments, though, more realistic feelings take over. '[In 1964] I took my sixth and final overall classification in the King of the Mountains,' he once wrote. 'But above all I felt that I had missed a great opportunity.' The real shame was that in all the hullabaloo surrounding Poulidor and Anquetil hardly anyone outside Spain noticed.\n\nBahamontes stood on the finish line in Brian\u00e7on, a week into the race, and pronounced: 'I have prepared myself exclusively to try and win the Tour de France.' At that point, after the first major mountain stage of 1964, he was clearly the best placed of the favourites to do so. There was only one rider ahead of Bahamontes after Brian\u00e7on \u2013 Georges Groussard \u2013 a reliable but not exceptionally talented _domestique_. The Frenchman had played his cards perfectly by getting into a break in the Vosges mountains and being allowed to go clear by the big names. Two days later he was into a lead he would hold for nine stages. For once, though, Bahamontes had managed to come through the first week of the Tour without losing too much time. Compared to Anquetil, the key reference point, he was around two minutes down, most of it lost in a twenty-kilometre team time-trial on stage three. However, it was one thing for a veteran like Bahamontes to stay in contention, another to move ahead. But Bahamontes, being Bahamontes, took the first opportunity he could. His declaration of war on the Tour's eighth stage was anything but subtle: he let his troops soften up the opposition in the first one hundred and seventy kilometres, before going clear on the second category Telegraphe. He widened the gap yet further on the most challenging Alpine climb of them all, the Galibier, then maintained his advantage on the long drop down the valley to Brian\u00e7on. Second home was Poulidor, at ninety-two seconds, with Anquetil finishing a further seventeen seconds back. Overall Bahamontes was behind Groussard, but he was thirty-two seconds up on Poulidor, with Anquetil eighth nearly two minutes back. Asked if he dared attack the quadruple Tour winner, Bahamontes answered defiantly: 'The aim is to try. Both Anquetil and his _domestiques_ want to stop me; they'll have to suffer.' However, Bahamontes' onslaught was not quite as beneficial as it could have been: in hindsight, a later, punchier attack on the Galibier, even if it did not rack him up as many King of the Mountains points, might have been more productive in terms of the overall.\n\nThe next day Bahamontes' attack over the La Bonette-Restefond climb enabled him to crown what was (and remains) the highest tarmacked mountain pass in France alone and ahead of the field for the second time in three years. It was a magnificent achievement, and Bahamontes wheeltracks \u2013 in the Tour at least \u2013 remained unfollowed for nearly thirty years until Scot Robert Millar became the next rider in the race to cross the summit of the pass, in 1993. Though not exceptionally steep La Bonette Restefond is daunting for its exceptional length \u2013 twenty-three kilometres \u2013 and its fifteen hundred metres of climbing. The altitude is another intimidating factor since anything above two thousand metres, like La Bonette-Restefond's summit, causes oxygen debt. The appalling road surfaces also pose problems. In 1964 it was only a gravel descent and in 1993 I can remember driving along poorly tarmacked roads, barely a car's width across and with no barriers separating us from a vast, impossibly deep ravine.\n\nMillar told British magazine _Cycling Weekly_ at the time that he had been 'honoured' to be the first rider to reach La Bonette-Restefond's summit since Bahamontes, and in a curious echo of Bahamontes' rider four decades earlier, Scot and Spaniard employed identical tactics on the descent. Just as Bahamontes had done in the foggy drop off La Bonette-Restefond in 1962, I can recall Millar saying he used the race motorbike's brakelights to guide him down as fast as possible, too. When the motorbike braked, so did Millar.\n\nHowever, there were more parallels \u2013 none of the attacks worked out. Just as Millar's lone break in 1993 ended with him being caught on the flat roads that followed La Bonette-Restefond, so both of Bahamontes' breaks proved fruitless. This was no surprise the second time round in 1964 as the stage ended in Monaco, 130 kilometres further on. To add insult to injury, Anquetil outsprinted Tom Simpson at the finish for a minute's bonus: so for all his exploits, including taking the lead in the King of the Mountains competition, Bahamontes ended up losing time on his main rival.\n\nFor Bahamontes the big problem remained the same as it had been in every Tour since 1954: the summits were too far from the finishes for him to benefit. As a descender whose ability ranged from poor to mediocre, it was a real disadvantage. In 1964 that handicap became evident on the Bonette-Restefond stage; five days later, on the road from Andorra to Toulouse, when the whole field had Anquetil up against the ropes, the message was really rammed home.\n\nFor Bahamontes the planning for the Andorra stage started twenty-four hours earlier when he learnt that Anquetil was spending the rest day at a barbecue. 'They told me, \"Anquetil's having a right party in the house up there, he's drinking champagne\",' Bahamontes recalls. 'So I said to myself, \"Tomorrow all that champagne's going to be coming out of his ears\", and I got hold of [team-mate Jos\u00e9] Segu and I said, \"Tomorrow we'll have breakfast three hours earlier than usual, and then ride the twenty-five or thirty kilometres to the start.\"' The objective was to be fully 'warmed up' as soon as the race started the first ascent of the day, the twenty-kilometre Envalira pass out of Andorra which began almost before the stage had left the start town of La Vella. 'When we started, we were all fired up and \u2013 boom! \u2013 we went off \u2013 boom! \u2013 on the attack. After two kilometres they came up and told us that Anquetil had abandoned, and then the whole of his team had had to drop back to help him get over.'\n\nBahamontes was far from being the only one with the idea of putting Anquetil to the test on the Envalira, even though it emerged that his abandon was no more than a rumour. The photograph of Anquetil at the party stuffing himself with lamb from a barbecue, and Geminiani pouring him a drink, rather than embarking on the usual rest day's light training, had appeared in that morning's newspapers. Geminiani is adamant that it was just a mock-up for the press. Mock-up or not it was certainly unwise: Henry Anglade and the Pelforth team, defending Georges Groussard's yellow jersey, along with Raymond Poulidor, all went training before the start with the same intention. The mass attack they then staged was arguably the only way to sink so strong a rider as Anquetil, and the Frenchman was four minutes adrift by the summit of the Envalira, albeit partly as a result of his alleged indigestion.\n\nNot for the last time, with three mountain stages ahead, and only two minutes down, Bahamontes seemed to be in the driving seat. But then Bahamontes and Poulidor, who led over the summit in dense fog, made two strategic mistakes. The first was not to wait for the first chase group, all of whom would have been more than happy to collaborate to eliminate Anquetil from the running. Secondly, they underestimated Anquetil's self-esteem: he pulled out all the stops on the descent, taking risks in the fog that neither Bahamontes nor Poulidor had been prepared to, and regained his position in the peloton. 'The fog saved him,' Poulidor would later say. 'The cars had their lights switched on because of the fog and that was a reference point [for Anquetil on the descent]. Without that fog, we would have waited for the group behind and he would never have seen us again.'\n\n'Anquetil was kaput,' recalls Geminiani. 'They had all attacked and he'd been dropped one hundred metres after the start.' But his star rider's problems had nothing to do with the previous day's lamb or wine, he said. 'The problem was that some astrologer had predicted he'd die on stage thirteen and it had been published in _France Soir_ , the biggest newspaper around at the time. All the _Poulidoristes_ had sent him copies of the newspaper, and Anquetil was like, \"What do I do? Go home?\"' Anquetil was so worried that Geminiani even got his wife to visit to persuade him to calm down and not pull out. Anquetil spent most of the rest day in bed. Geminiani says: 'I got really pissed off with him lying there in bed and dragged him off to that barbecue. But the next morning he didn't want to get on his bike. I made him do it. One hundred metres after the start he was dropped. He was so far back he was the last rider over the Envalira. Then, at the summit, the fog was so thick it seemed like the end of the world. And I was so pissed off I went up to Jacques and yelled at him, \"Hey! If you're going to die like that astrologer says, die at the head of the race and not in front of the broom wagon!\"'\n\nAnd that, by and large, was that. What had started as a mass mutiny against the winner of the three previous Tours fizzled out. Anquetil pulled off a stunning descent, first catching the Groussard group and then Bahamontes.\n\nIt is difficult to disagree with Bahamontes when he says the Tour was weighted in Anquetil's favour. It was not just the route or the lack of summit finishes, it also helped Anquetil that the organisers split the individual time-trials into three different stages. Each had a one-minute time bonus and they were all won by the Frenchman. Additionally, the Tour's decision to revert to trade teams made it impossible for Bahamontes to exploit any disunity between the major stars in the French team like he had in the 'bad old days' when the Tour was run with national squads. With television cameras now filming the race it was far harder to work against theoretical allies or pull a fast one on your team-mates without anybody noticing. All Bahamontes could do was play Anquetil at his own game and try and form alliances with rivals to outmanoeuvre him. And who better, in theory, than Julio Jim\u00e9nez?\n\nJulio Jim\u00e9nez settles back in the armchair in the first-floor flat he calls home in \u00c1vila. Now well into his eighth decade, and nearly five decades after one of the defining moments of the 1964 Tour, when asked what his problem was that day with Bahamontes, Jim\u00e9nez's answer is 'he [Bahamontes] wanted everything.' This is the Julio Jim\u00e9nez so memorably described by the late Geoffrey Nicholson in _The Great Bike Race_ as a 'small, bird-like figure . . . he was also balding, grey-faced and never looked particularly well, but on the first steep slope he would prance away as if he had springs in his calves'. Jim\u00e9nez, who weighed fifty-seven kilos (eight stone thirteen pounds) when in form, has thickened with age. However, he still zooms about his living room at speed and with a bounce in his stride that would do credit to Zebedee in _The Magic Roundabout_. He constantly picks up and puts down pieces of paper and photographs, his head darting from side to side and talking nineteen to the dozen. Clearly none of the nervous energy that saw him secure three King of the Mountains titles in the Tour and Vuelta, and finish second in the Tour in 1967, has been lost. In his own slightly skittish way Jim\u00e9nez is as keen as Bahamontes to ensure the visitor cannot miss the scale of his success. For one thing, he uses the pennants that he received for each of his five Tour mountain stage wins as rather painful head-rests (they are studded with brass knobs) for his sofa. And if the hat-trick of stags' heads he received as best climber in the Tour's Alpine stages between 1965\u201367 were lined up a little lower in his hallway, a tall guest might be tempted to hang his coat on them.\n\nDelve into Jim\u00e9nez's personal and sporting history and there are many other similarities with Bahamontes, right down to them both wearing grimy, non-regulation sports jerseys in their first races: Bahamontes in his basketball shirt complete with shoulderpads, Jim\u00e9nez in 'an \u00c1vila football shirt that had been washed so many times you could barely read the letters'. And the parallels continue: both come from working-class backgrounds in central Spain. Though not as poor as the Bahamontes family, Jim\u00e9nez's parents could not afford any luxuries like bicycles, either. In fact, Jim\u00e9nez only acquired his first bike because his mother worked as a maid for an army general who had several. Jim\u00e9nez spent so much time staring at them, he recalls, that the general felt sorry for him and ended up buying Jim\u00e9nez a tricycle. Both Bahamontes and Jim\u00e9nez started their professional careers late: Bahamontes at almost twenty-five, Jim\u00e9nez at twenty-seven. Finally, they both spent large parts of their early twenties making their own way across Spain to races. Just a few years after Bahamontes, Jim\u00e9nez would travel up to five hundred and fifty kilometres to Asturias and the Basque Country for his first big races. But in a sign of Spain's slowly improving economy he did so on a battered Vespa, not a racing bike like Bahamontes.\n\nJim\u00e9nez was nicknamed the 'Watchmaker of \u00c1vila' because that was his original trade. He also cut his cycling teeth in Barcelona, staying in Santiago Mostajo's house. He received his first professional contract in 1961 with a small Catalan squad. Three years later K.A.S. gave him the chance he so desperately wanted: to ride the Tour. Jim\u00e9nez's first stage win came with an attack on the approach roads to the Envalira pass into Andorra in 1964. Like Bahamontes, he was a poor descender, but he still retained enough of an advantage on the pack at the Envalira summit to make it down to the finish to victory. It was on stage fourteen to Pau that Bahamontes and Jim\u00e9nez, the old master and the young apprentice, found themselves together at the head of the field for the first and last time. There were four mountain passes ahead of them. As radios across Spain transmitted reports of the developing breakaway, crowds gathered in bars and the nation held its collective breath. With two such adept climbers at the head of the field who knew what would happen?\n\nBack on the road in the Pyrenees, however, there was a problem. And it was not an easy one to resolve. 'The thing was we were a bit pissed off with each other,' Jim\u00e9nez says now, 'It wasn't like we were working together at first. We would each make long accelerations on either side of the road until his director came up and told him to work with me. At that point, the idea was, \"I would be the first across each climb\" \u2013 which Jim\u00e9nez was \u2013 \"and he would go for the [yellow] jersey\". We did a couple more climbs together. The break was going all right, [even if] Fede was sprinting for the points at the top of the climbs.' That, of course, went against their agreement. 'Then on the last climb I had a tough moment and I got dropped,' says Jim\u00e9nez. Strategically, if Bahamontes really believed he could challenge for the overall it would have made more sense for him to wait. However, Jim\u00e9nez claims that he did not have the courage to ask Bahamontes to do so, and Bahamontes never suggested it. 'To tell you the truth, I'd done absolutely everything I could to help him in that break and I was shattered. He even told me, earlier on, not to go so hard because there was a long way left to go. So that was it. We exchanged a couple of words and he went on. He was my idol; it was my first Tour. How was I going to say to him, \"Go a bit easier because on the other side of the Aubisque there's sixty kilometres to go and you might need me there?\"'\n\nAs Jim\u00e9nez hints, that kind of distance without any further climbs, is an extremely long way for any lone rider to fend off a group of chasing rivals. But Bahamontes and Jim\u00e9nez could not agree on a tactic that would benefit both of them. 'He wanted everything: to go over the top alone and win the stage and he forced the pace deliberately [to drop me] because you can tell when someone does that,' Jim\u00e9nez says. 'So I went up at my own pace, and I said to myself, \"If I manage to catch him, I'm going to attack him and he's going to pay for what he did\".'\n\nBy the time he was halfway down the other side of the Aubisque Jim\u00e9nez was under no illusions that he would catch Bahamontes again. He stopped and bought himself a Coca-Cola and sandwich in a bar. These days that would probably get him fired from his team; back then it was seen as perfectly normal. While he knocked back his Coke, his director told him to hurry up because a large chasing group was about to go through. Sure enough, about five minutes later the likes of Poulidor, Groussard and Anquetil came hammering down the road. Jim\u00e9nez gulped down his quick snack and latched on to the back. He insists, perhaps a little too much, that he did not assist in the ongoing pursuit of Bahamontes, something that Bahamontes claims both Jim\u00e9nez and K.A.S. team-mate Francisco Gabica had done. 'I got on their wheel and I barely moved from the back, I swear, though I didn't take one turn on the group. I swear it,' says Jim\u00e9nez. 'But in any case they were flying and when we got to the square in the middle of Pau he just had a couple of minutes. However, if he'd waited for me we might have gone up [the climb] a minute slower, but I think on the flat we'd have held them off for longer, and he'd have been closer to getting the yellow jersey.' Instead, Bahamontes' determination to retain the King of the Mountains jersey had betrayed him and ultimately the greater prize eluded him.\n\nThe tale has been told and retold so often that the reality of what was actually said between Jim\u00e9nez and Bahamontes is irretrievably buried under a blanket of half-truths. But Bahamontes' desire to take both the King of the Mountains title and fight for the overall, rather than sacrifice the first, was not the only problem. When asked if Bahamontes could be trusted in 'gentlemen's agreements', whilst Bahamontes insists he respected all deals with Jim\u00e9nez, Jim\u00e9nez answers straightaway. 'Not in my case,' he says. That, it emerges, was the reason for the lack of collaboration at the start of the stage which lost them both time: Jim\u00e9nez claims he had already been too badly burned in at least one previous breakaway with Bahamontes. 'I don't remember where, but there was some mountain stage in the Tour that year [almost certainly stage nine to Monaco where Bahamontes took the lead in the King of the Mountains], and where he'd told me the day before, \"Look, if I go on the attack, don't follow me because it could ruin the break, and I'm going for the lead\". And I thought I'd respect that because he was a god, an idol, in Spain and I was a piece of shit in comparison. However, when I talked to my director, Dalmacio Langarica [Bahamontes' director in 1959], he said, \"Don't trust him\". But I did. And, sure enough, Fede attacked and got away. But then his team-mates started fighting me for the points that remained on the third category climbs in the King of the Mountains. When I asked him later what was happening with his team-mates I realised that he wanted the King of the Mountains competition, too, and that he didn't care whether he finished second or third in the overall. The Mountains was the one he was after. That would be fair enough, but it was the way that he did it. That wears you out and pisses you off, and you end up feeling like you're stupid.'\n\nJim\u00e9nez is not the only one to claim he had this sort of problem with Bahamontes. Poulidor, too, has recollections of Bahamontes \u2013 who says he cannot remember the event \u2013 'pulling a fast one' on him the first time they raced against each other. 'After Bahamontes won the Tour in 1959, I was an \"independent\" [non-sponsored] rider and we both got in a break in the Bol d'Or de Monedieres-Chaumeil. I attacked alone on the climb and the only one who could get across was Bahamontes. He said he wouldn't go for the King of the Mountains prize at the top, which surprised me because I knew that was almost more important for him than winning. But I fell for it, we worked together, and then he outsprinted me at the summit. Then he told me as we rode towards the finish that he wasn't going to work any longer, but I didn't fall for it a second time. And even though we had two minutes' advantage at twenty kilometres we were both caught and finished fourth. To tell the truth, he gave me a very poor first impression of him.' Fast forward five years to the 1964 Tour and Jim\u00e9nez felt the same as Poulidor. But this time Bahamontes arguably paid a heavier price.\n\nFast forward another thirty-five years and Jim\u00e9nez is still resentful that Bahamontes will not give him the recognition he feels he deserves. He recalls when he and Bahamontes went to visit the Vuelta and Bahamontes was handing out postcards with his _palmares_. When somebody told Jim\u00e9nez that Bahamontes had left him one and he went to pick it up, he saw it read, 'To my friend and _domestique_ , [signed] Federico Mart\u00edn Bahamontes'. As Jim\u00e9nez points out, quite apart from being a put-down, he could never have been Bahamontes' _domestique_ , given the two never rode in the same team!\n\nOnce started Jim\u00e9nez cannot stop coming out with examples of Bahamontes' unwillingness to acknowledge his merits. There is little to be gained in listing them all, but several illustrate the point about how Bahamontes' pride did not permit him to admit that riders he claimed 'could never beat me', had in fact done so. At times it seems Bahamontes is trying to re-write history. 'There was the Subida [hill-climb] a Urkiola, where I beat him, and Bahamontes said he'd never ridden it,' says Jim\u00e9nez. 'I had to get someone to download some classifications off the Internet and show them to him; that was the only way he would accept it. I said to him, \"Look, you didn't even finish second behind me, you got third!\"' Then there are the calculated putdowns. Jim\u00e9nez scoffs at Bahamontes' insistence that he 'let' him win on the Puy de D\u00f4me in 1964: 'And I joked to him maybe I'd have let him win, if he'd come up with a ruddy great cheque. I wasn't anybody then and needed the money' Another time Bahamontes was quoted as saying that his only rival in the mountains was Charly Gaul. 'And only when it rained.' Jim\u00e9nez was not even mentioned.\n\nAt the root is the age-old rage of the apprentice who knows he is outstripping the master but who never feels he gets the recognition he deserves, and the master who feels his time is passing but is determined to go out anything but quietly. Dignified it is not, but there is no denying it is part of human nature. Jim\u00e9nez says: 'The most reasonable thing for a star to say [on approaching retirement] is that \"the young riders are breathing down my neck\". And that \"my successor was the Watchmaker of \u00c1vila who won three Kings of the Mountains and should have won a fourth\".' That is how numerous other stars have behaved, Jim\u00e9nez says. Bahamontes, though, was always a case apart. Ultimately, these futile spats boil down to the fact that neither rider was willing to let the other gain predominance in the public eye if it meant they risked losing out. That, perhaps, was why Bahamontes was not prepared to let Jim\u00e9nez share in his ride to glory in Pau; he wanted to prove he remained the boss even if it meant losing the Tour.\n\nNor was this rivalry straightforward, either. While Jim\u00e9nez admits that every Spanish climber was naturally keen to beat Bahamontes, because outdoing the star of the sport automatically brought added recognition, there were outbreaks of friendship and camaraderie. The two have travelled together to countless events around Spain. As recently as 2010 they went to the Vuelta presentation in Benidorm 'in a benzene-powered Audi with the race director all the way from the Paseo de la Castellana [in central Madrid]'. Yet amid the bonhomie, fast cars and backslapping, the resentment and mistrust lingered. In 1970, with Bahamontes long since retired and Jim\u00e9nez desperate for a team as his career faded, he approached his old rival's professional team, La Casera, and asked for a ride. Jim\u00e9nez did not have the nerve, or perhaps he had too much pride, to ask Bahamontes directly. In any case the answer came back immediately through an intermediary: No. 'I suppose that was because he thought they would forget about me quicker,' Jim\u00e9nez says. And echoing Josu Loro\u00f1o's words about Bahamontes, he says: 'It was all me, me, me . . . only he existed.'\n\nIn fairness to Bahamontes it is possible that Bahamontes rejected Jim\u00e9nez purely because he did not think he was worth having on his team; he was at the tail-end of his career, after all. However, if it was the case that Jim\u00e9nez paid the price for Bahamontes' unwillingness to share the glory in 1964, that reticence seems to have backfired completely. Others think it would have made no difference. When I interviewed Geminiani he claimed he did not believe that Bahamontes could have challenged in the 1964 Tour. 'He and Jim\u00e9nez were welcome to put on all the shows they wanted to, but we knew how much time he had and we based our game around that. Apart from on the Forclaz [in 1963] he was never Anquetil's rival.' However, the way Anquetil forged alliances with Pelforth to reel in Bahamontes, both in the Alps and Pyrenees, suggest the opposite. Darrigade also feels that Bahamontes could have won the Tour. He might not have been a major threat like Poulidor, but threat he was, hence their frantic chase on the descent of the Aubisque. You can not imagine, either, that Anquetil would have forgotten Bahamontes' victory in 1959 or how close he came to winning in 1963.\n\nThat day in the Pyrenees the mistrust between the two top Spaniards was arguably what cost Bahamontes the chance of winning the Tour. The King of the Mountains had been 'his' for too long for him to give it up so easily. Most likely Bahamontes preferred to hedge his bets, thinking that even if he worked with Jim\u00e9nez there was no guarantee that he would win overall. But if the two pair had worked together in 1964 the history of the Tour might read very differently indeed.\nChapter Fourteen\n\n#### Into the Bushes\n\nIt took Federico Mart\u00edn Bahamontes five months from the moment retiring from cycling first crossed his mind to his final, irrevocable decision. The process, during 1965, was in keeping with much of his career: occasionally farcical in appearance, often controversial in substance and nearly always driven, beneath the surface, by hard economics. The first and biggest sign that Bahamontes, now thirty-seven, was nearing the end of the road came in the 1965 Tour. After finishing second in 1963 and third in 1964, he had regained the status of favourite, particularly in the absence of Jacques Anquetil \u2013 allegedly because the Frenchman felt he had reached his sell-by date and did not want to risk being beaten. However, Bahamontes abandoned on stage ten, blaming knee injuries at the time, but later suggesting his main sponsors, Margnat, had not been paying him. 'The whole company was going to the dogs,' Bahamontes says, 'and they didn't pay anybody \u2013 neither me, nor Ota\u00f1o, nor Darrigade.' Andr\u00e9 Darrigade has denied this was the case and that he was paid regularly by Margnat during his time there. When I passed this information on to Bahamontes he looked, to say the least, thunderstruck. Bahamontes says he had a major argument about the lack of money with his sports director, Raoul Remy, in the morning before the start of stage nine between Dax and Bagneres. When Remy dodged the issue and suggested he talk to the company director, Bahamontes rode off in a huff. He finished second last, forty minutes down.\n\nYet more fruitless discussions followed that evening in the team hotel. The next day on another Pyrenean stage from Bagneres de Bigore to Ax-les-Thermes, Bahamontes went on the attack just before the Portet-d'Aspet climb 'to show them what they would be missing'. However, if anyone expected him to reach the summit alone and at the head of the field as had been the script of so many Tours, they would to be sorely disappointed. Instead Bahamontes opened a gap, then when out of sight bizarrely rode off the road and into some bushes. 'I wanted to give them all a surprise; it was a joke,' he tells me thirty-six years later, still smiling to himself at his antics. 'They wouldn't be able to find me, the bushes were quite high. Of course, when they did find me all hell broke loose.'\n\nFor someone famed for his eccentricity and individualism it is somehow appropriate that the final image of Bahamontes on two wheels in the Tour de France is arguably the most surreal of his entire career: he is riding a road-bike, but he is not actually riding it on a road. Taken from behind, in the photograph Bahamontes can be seen heading over a broad grassy verge, well past a motorbike parked on the side of the road, and pedalling towards a dense bank of long grass with a line of trees beyond. A couple of seconds later and he would have disappeared completely. With the end of his career beckoning, the picture gives the impression that roads and civilisation no longer interest Bahamontes; complete with bike, team kit and bottle of water he is off to find himself in the jungle like the lead character in Jos\u00e9ph Conrad's novel _Heart of Darkness_. In fact, Bahamontes' Kurtz moment quickly passed. The next photographs taken that day show the cyclist putting on a tracksuit beside a team-car, talking with his director, and ready to abandon. It was his first step on the road to retirement and arguably the most important.\n\nHis Margnat team-mate Andr\u00e9 Darrigade recalls: 'When he left the Tour, it was a big disappointment. He was the team leader, after all; we'd all worked for him. Nobody could see it coming.' Bahamontes said at the time that he could not continue because of the knee injury, and 'you can't race the Tour on one leg'. Given he was in a breakaway when he abandoned that excuse is unconvincing. It was, though, consistent with previous controversial abandons. His antics were also highly reminiscent of the child-like behaviour he had shown back in the days when he staged go-slows in the Vuelta or pulled out of the Tour after two stages. Perhaps it was another indication that after years of self-control in the Margnat squad, Bahamontes was reverting to his 'true' self.\n\nCoincidentally, the first rider across the Portet-D'Aspet that day was Julio Jim\u00e9nez, who had won the previous day's stage after a breakaway. 'I heard Bahamontes had been dropped and turned on the heat,' Jim\u00e9nez says. 'To be honest I wanted to pay him back for 1964.' That was not lost on the press either, and Jim\u00e9nez was again hailed widely as Bahamontes' successor. Extensive interviews appeared with both riders with the Spanish magazine _Actualidad Espa\u00f1ola_ , on 15 July, the day after the Tour finished. Tellingly, Jim\u00e9nez, who won the King of the Mountains jersey that year as well as stages in the Alps and Pyrenees, has a full-page photograph on the cover and receives four pages inside; for Bahamontes, there are just two pages. There was no doubt who was the coming man. The interviews revealed that the rivalry between the two was now a full-scale power struggle for the top spot in Spanish cycling: Jim\u00e9nez was determined to see Bahamontes gone, while Bahamontes was adamant that he would fight back. Underneath the headline, 'The Eaglet takes over from the Eagle', and resplendent in his Spanish national champion's jersey, Jim\u00e9nez claims at first: 'I can't believe that he [Bahamontes] is finished, I'd prefer to think he's not gone . . . he deserves a dignified exit'. But, perhaps rather too quickly, he reflects that Bahamontes' retirement is, in fact, 'due to happen soon', adding: 'Time waits for no man. Your natural abilities fade and if you don't abandon in due time you're bound to sink.'\n\nThen Jim\u00e9nez really sticks the knife in, suggesting that Bahamontes had failed to realise he was past his sell-by date. 'Curiously enough, he'd told me that morning that he was going better than ever in the flat, and that observation made me doubt that he was,' said Jim\u00e9nez. 'He's a climber and I'm a climber. Whenever I notice I'm going better on flat stages I know that my climbing form is going to be worse. Strange but true.' Sensing blood, the interviewer asks: 'Would you give Federico any advice?' Sure enough, Jim\u00e9nez recommends that 'he should already have retired. Nothing better than an honourable retirement at the height of his powers. A minute later [on stage nine] and he'd have finished outside the time limit.' Leaving nothing to the reader's imagination, he added: 'This is the rider who forced so many others to finish outside the time limit when he won his stages and he was about to suffer the same fate. How humiliating for Federico!' Showing a keen knowledge of what drove Bahamontes on, Jim\u00e9nez segued into the financial reasons for a retirement and argued that Fede's 'earning days' on the bike were long past. 'The best thing he could do now is stay at home. He's got a business going and he's not going to lack for anything. It's understandable that somebody is unwilling to renounce their best source of income when it seems like easy money. But sooner or later there aren't any rich [sponsors] left, and he shouldn't wait until they throw him out \u2013 not somebody as prestigious as Federico.'\n\nAs a declaration of how prepared Jim\u00e9nez, already thirty and no spring chicken, was to step into Bahamontes shoes the _Actualidad_ interview could hardly have been clearer. However, such a violent attack by one cycling star on another was also perhaps an indication of how unsure Jim\u00e9nez was that Bahamontes was really going to retire. It was true that Bahamontes had ridden the Vuelta in 1965 for the first time in five years and finished tenth overall without any of his usual fireworks. 'It was,' as _Actualidad_ put it, 'as if Bahamontes had abandoned the race, even though he was still there.' Bahamontes' disinterest in the Vuelta was such that Poulidor recalls he won a mountain time-trial, the Spaniard's speciality: 'I overtook six or seven riders, who had all started before me at minute intervals. The last one I caught was Bahamontes'.\n\nBut Jim\u00e9nez was right to be worried. In other races that year Bahamontes had shown there was still life in the old dog yet. If you ignored the Vuelta and the Tour, Bahamontes had not had a disastrous season at all: not only had he taken second place in the Arrate hill-climb he had captured an overall win in the Circuit du Provencal ahead of Jan Janssens, a future Tour de France winner, and Britain's Tom Simpson. Could Bahamontes really be finished? Not if his defiant first answer in the _Actualidad_ interview was to be believed: 'I'm not dead, though. Me, dead? Leave off it, please. The deader people take me for the more alive I am.' However, the longer the interview went on, the more Bahamontes played it hot and cold, hinting at one point he could retire, but at another that his racing days were still not over. Presumably still holding out for payment from Margnat, Bahamontes blamed his Tour abandon on his old meniscus injury from 1955. However, when asked about the future, in different answers he was either unusually non-committal or adamant that his Tour days were finished. Something, the journalist concluded in the article, was brewing in Bahamontes' mind, but it was not clear what. 'I'm thirty-seven, not thirty-eight as some people say, and though I've lost some of my top-end speed, with time I've gained a lot of experience,' he said. 'I know I'm due to perform less well with age but I don't muck about as much as I used to.' That comment was barely credible given his recent premature exit from the Tour. 'I had and I have class,' Bahamontes nonetheless insisted. 'If I didn't I wouldn't be able to propose doing what I've got planned.' 'Such as?' asked the journalist. 'Well . . . things,' Bahamontes responded clumsily. 'Things that some people don't believe and which I don't care about if they do.' But he did reveal: 'This was my last Tour. You can be sure of that', adding by way of a rather trite explanation that: 'It's very hard.' As for racing the Vuelta, Bahamontes argued: 'That would depend on the sponsor.'\n\nHowever, when asked directly if he would retire, Bahamontes seemed to leave his options open. 'It has to come some time. That's life. I don't race with the same idealism or willingness to sacrifice myself to the sport as when I started.' Such an off beat, mixed response showed it was not going to be easy for Bahamontes to leave the sport behind. There again, he had not ruled it out. By September, retirement had loomed a little closer when his sporting pride suffered a huge blow, thanks to the Federation.\n\nOddly enough, Bahamontes' beef with the Spanish Federation this time was not down to disagreements over Grand Tours, his strongest suit, but an event in which he had never shone: the World Championships. Indeed, his best result in the World Championships was seventeenth place in 1961. Long after the World's road race, held that year in San Sebastian on 5 September, Bahamontes claimed that his dream had been to end his career in front of a home crowd and wearing the rainbow jersey of the world champion. Whether he would actually have retired had he fulfilled this highly unrealistic ambition is hard to say. But the humiliation Bahamontes suffered in the process of having to fork out money to watch a race he had desperately wanted to take part in was sufficient to tip him even closer to the edge. 'They didn't just not select me, they made me pay for my ticket to get into the circuit,' Bahamontes fumes. 'I'd ridden the Tour of Luxembourg to get into shape, and to get to San Sebastian in time I abandoned on the last stage thinking that I was in the Worlds. And then they didn't choose me. But on top of making me pay, when they [the organisers] realised I was in the stands they came up and asked me if I would cut the ribbon at the start. I said, \"Don't come bothering me, I've paid for my ticket, I'm just a spectator \u2013 for one thing it's like you don't know me, and for another you come looking for me specially\".' Bahamontes lays the blame on Gabriel Saura, the national coach, 'for being so envious of me. I had wanted to retire as world champion, but Saura was so jealous he made me pay for it, and I've never forgiven him for it, in all my life. I was ready to take part. Did he really not think I was one of the top eight riders in Spain?'\n\nBahamontes' assumption that he was one of the top eight riders in Spain, and therefore due automatic selection for the World Championships, was far from Saura's view. Bahamontes had already, without any reason offered, decided not to take part in the National Championships, which did not go down well. As Saura explained to _MARCA_ : 'I can only say that Bahamontes has not raced anywhere he could have shown that he has recovered from his abandon in the French race, which he said was due to injury.' Abandoning the Tour of Luxembourg did not, contrary to Bahamontes' opinion, help his case.\n\nOn 8 September, following the row with the Federation, Bahamontes announced pointblank that he would not race any more. He then apparently called up the Volta a Catalunya organiser, the U.D. Sants cycling club, to tell them just two days before he was due to race there that he was retiring from the sport. 'I'm demoralised, and it's the right thing to do. I've thought about it, and when you have no morale, it's no good,' he told the Catalan newspaper _El Noticiero Universal_. He raced on, nonetheless, taking part in the Subida a Naranco hill-climb in Asturias where he finished eighth. Given the mixed signals, some like Margnat director Raul Remy did not believe he was going to retire. 'I consider Bahamontes to be a good rider, and with enough experience to continue for another year,' Remy said, 'though he should choose his races more carefully and pay particular attention to how he races the Tour.' The organisers of the Subida a Arrate hill-climb attempted to persuade Bahamontes to continue at least until their event the following spring. Even as late as 1 October Bahamontes was undecided, saying: 'It'll be me who decides whether I retire or not. I belong to a commercial firm [trade team] and the rest is rubbish. I'm still keen, I've still got fans, and I'm still keen to fight.'\n\nStill wavering wildly over what to do, the final decision was not taken until he had ridden what was traditionally the race that brought down the curtain on the Spanish season, the Subida de Montjuic in Barcelona. Even more appropriately for Bahamontes' last race, Montjuic was his signature event, a hill-climb. It was held in two sections with a mass start and an uphill time-trial. Bahamontes did not win either. He was beaten into second place in both races by Raymond Poulidor. However, in warm sunshine, and watched by huge crowds, he received enormous rounds of applause in the city where his professional career had effectively begun. 'We saw Bahamontes, risen again from the ashes, and climbing formidably well,' claimed _El Mundo Deportivo'_ s correspondent, before adding, a shade pretentiously, that Bahamontes 'received an effusive welcome from the fans who seemed inspired by enthusiasm [we would see] in the olden days . . . the public came here to savour the finest essence of cycling'. And for their main photograph to illustrate the event, what better than Bahamontes powering up a climb between a wall of fans one last time?\n\nBahamontes says he had barely finished in Barcelona when his longstanding Catalan business associate, Evaristo Murtra, told him in no uncertain terms that his career was over and that he should concentrate on his business in Toledo. 'He came up to me and said, \"Right, now you are going to Toledo to sell bikes and motorbikes in your shop with your wife\". I disagreed, saying, \"Now that they pay me double, I'm going to stop?\" Murtra replied, \"It's not worth it. What's best for you are motorbikes and bikes and that's the future. That's where the easy money is.\"' With that alluring prospect in mind Bahamontes could have announced he had ended his career there and then. However, there were rumours that he would continue racing at least until the Six Days of Madrid track race, an exhibition event he had won the previous year. Here again versions vary of what actually happened. While Bahamontes insists to this day that he had decided to stop after Montjuic, the Madrid organisers told _MARCA_ they were convinced he was going to be taking part in their event, because 'he wants to end his career in front of a home crowd, in Madrid'. However, it was claimed that a disagreement over the appearance money Bahamontes would be paid ruined what would have been a more spectacular send-off, especially as results are often pre-arranged in such exhibition events: he would have had a reasonable chance of going out with his arms aloft in victory one last time.\n\n'He asked for two hundred thousand pesetas [\u00a336,000],' the organisers told _MARCA_ , 'which is what we paid him the previous year. But we offered him what [Spanish track start] Guillermo Timoner was going to get, and he didn't want to do the selection test for the event, anyway.' Afterwards, the organisers were adamant that 'his absence wasn't noticed'.\n\nWhatever the truth of these claims, Bahamontes did not take part. He was moving on. An ex-cyclist now, he quickly adopted a new role as expert commentator on Spanish cycling. That winter he took part in a round-table conference in Bilbao with his old rival Jes\u00fas Loro\u00f1o, which predictably turned into a rose-tinted look back at their careers and how much better racing had been in the past. While Loro\u00f1o said he believed Spanish cycling 'needed a rivalry like me and Federico had', Bahamontes claimed: 'If you take a look at my _palmares_ there is no rider in Spain who's done better than me this year. [Top Spanish riders] Patxi Gabica, [Valentin] Uriona and Jos\u00e9 P\u00e9rez Frances could have got better results,' he claimed. In fact, Gabica and Uriona had both won two stages in the Volta a Catalunya, while P\u00e9rez Franc\u00e9s had won a stage of the Tour and finished sixth overall. 'But they just sit behind other riders' wheels [in the pack]. They're cold-blooded and never attack.' Since Bahamontes had just opted to retire because it made more sense economically this was at best a case of the pot calling the kettle black.\n\nThere is a certain irony that, rather than back out because he felt he had nothing more to give to the sport or that his time was up, Spain's top athlete of the 1950s retired using exactly the same kind of hardheaded financial logic with which he started riding a bike in the first place. In that sense, at least, Bahamontes had come full circle.\nChapter Fifteen\n\n#### La Vuelta A Toledo\n\n_When you're going well, you don't suffer, you enjoy it. Nothing can go wrong because you can see that it's not you that's suffering, it's the rest of the field. Every turn of your pedal feels like a fresh round of applause a singer receives._\n\n**Federico Mart\u00edn Bahamontes**\n\nAugust may be the hottest month of the year in Spain, but it still feels pretty chilly at 7 a.m. as dawn breaks over the broad, busy avenue in Toledo where Federico Mart\u00edn Bahamontes' fan club, his _Pe\u00f1a,_ is based. A shadowy group of men in their forties and upwards traipse back and forth between the building and a cluster of half a dozen vehicles parked nearby. As the sun rises the stickers on the hefty four-wheel drive cars become slowly legible. Across the top of the windscreens each one reads ' _Vuelta a Toledo 2011_ '. The chain gang in shorts and T-shirts heaving boxes and folders out of the _Pe\u00f1a_ are the race's back-up staff, and they are putting the finishing touches to that day's final stage of one of the biggest under-23 races in central Spain.\n\nThe reason for the painfully early start is simple: the race starts rolling at 9.30 a.m. and is finished by midday, well before the real heat kicks in. 'Fede? The boss doesn't come til later,' one says of those carrying boxes containing the trophies for the presentation. Sure enough, just before 8 a.m., Bahamontes arrives in person, bustling around and barking orders. The convoy of six vehicles moves off soon afterwards towards the start, which is held in the most uninspiring of settings: slap bang in the middle of a semi-deserted industrial estate on the outskirts of town. On one side there is a line of wholesale dealers closed for August, on the other a huge metal fence and a severely under-used petrol station the size of a football stadium. It looks like the ideal location for a particularly gruesome gangland murder. However, the police motorbike planted in the centre of the road, blue light whirling, is only there to deter traffic. At one end a race steward in bright yellow safety vest guides team cars and race vehicles to their parking spots. In the middle Bahamontes blasts away on his whistle, stomping up and down the lines of cars to make sure they park absolutely where he wants them. The race's centre of activity is just in front of a shop selling knock-down furniture. Three girls stand behind a long trestle table and serve coffee and snacks to the handful of spectators, a trio of blue-shirted officials from the U.C.I. and the small army of organisers and hangers-on.\n\nWhat the start lacks in scenery, though, it makes up for in good vibrations. The twelve-strong motorised corps of Civil Guards wander around munching croissants, chattering away cheerfully and, in the way Spanish policemen do, half-hiding the morning's first cigarettes in cupped hands behind their backs. Riders weave their way to the signing-on table just outside the shop, then dart back to their team cars, one hand on the handlebars, the other clutching a cup of coffee. Nobody seems stressed: for the organisers and police 9 August marks the last leg of their work, and though the classification is still undecided the participants know they will also be heading home soon. At one point Bahamontes waves over the race leader, a tall amiable Frenchman from the G.S.C. Blagnac club and demands to know: _'Tu aller gagner?' 'Je pense que oui,'_ replies the younger man with a broad smile before quickly pedalling away. Bahamontes, meanwhile, looks a shade smug about being able to speak the rider's language sufficiently well to hold the brief conversation. There is more of a communication problem, though, when the race doctor forms a brief huddle with the event's one injured rider, a Japanese amateur, whose knee and arm are a mass of injuries and seeping blood from a crash the previous day. The doctor bellows in Spanish at the rider's French director: 'Tell him he can take the bandages off in a couple of days.' The director looks blank, the Japanese rider even blanker. 'Two days,' the doctor shouts again before making strange handsigns.\n\nThings, though, could not be friendlier. There are no restrictions, V.I.P. areas or team buses to deter the public from getting close to the riders. Though the location for the start is hardly ideal, the spectacular finish at the Plaza del Zocodover in the heart of Toledo's old town more than makes up for that. The riders and team staff look pleased with the warm weather and free food as they ogle the gaggle of publicity girls, barely out of their teens and teetering in ridiculously white high-heels. To judge from the slogan plastered on one girl's T-shirt \u2013 ITV Oca\u00f1a \u2013 a government M.O.T. vehicle centre is a sponsor.\n\nWith no houses in sight there are few children present. One who is must be about two years old and he is clutching his grandmother's arm. As they pass Bahamontes the boy receives a quick lesson in cycling history. 'That man is called Bahamontes and he won a big bike race called the Tour de France,' Grandma says. The two-year-old looks far more interested in the huge, empty filling station opposite than a grizzled man more than eighty years his senior, but it is a nice gesture all the same.\n\nLike any sports event, part of this race's appeal is its predictability. Here, a PR vehicle consisting of a large man with a small microphone, a booming voice and a couple of loudspeakers strapped to the roof, draws up and begins informing everybody of what they already know. 'This is the Vuelta a Toledo, the fourth and final stage, and it's about to start.' Finally, after much to-ing and fro-ing and more pointless bellowing to corral the riders, Bahamontes dons the most incongruous headgear imaginable: a sailing cap. It clashes dreadfully with his sharply-ironed trousers, sensible brown shoes and immaculate white top and makes him look like a bank manager who has wandered into a landlocked marina. But as his way of indicating the race is about to start, it certainly stands out.\n\nAnd the race starts with a bang, too. Running in a long southwesterly loop out and back to Toledo, the pack shatters almost immediately on two short but painfully steep climbs on the southern, rural side of the River Tagus. At the top of each one the bunch is in a line and gasping for breath, edging over the summit in groups of five or six. The race leader is just behind the first group of riders, looking far less happy than when he spoke to Bahamontes. On the roller-coaster roads which follow, past vast cattle ranches and foul-smelling pig farms, there is little chance of regrouping.\n\nThe twenty or so members of the race organisation act like sheepdogs around a worried, highly mobile flock. They warn the riders of upcoming dangers by waving flags or arms; they honk their horns at roadside fans to warn them the race is approaching; they power ahead to check that any obstacles have been cleared, then remove all the evidence of signs, barricades and bottles that showed Toledo's biggest bike race has just passed through. It is a non-stop operation but it proceeds smoothly. Since Bahamontes has been in charge for all the race's forty-six years, that is perhaps not so surprising. As for Bahamontes himself, he is a co-pilot in a two-seat, open-top sportscar, preceded by the two front-runners from the race's police escort. While a U.C.I. official runs the internal workings of the race, keeping an eye on the head of the pack, Bahamontes is responsible for ensuring any last-minute hitches are cleared. It is also impossible not to notice that as the spearhead of the race he soaks up the maximum attention possible. La Vuelta a Toledo is, in a sense, the Vuelta de Bahamontes, which is appropriate as Vuelta also means 'return' or 'comeback' in Spanish, as well as 'tour'. If on one level it is a race, on another it represents Bahamontes going back to his roots on a circuit of the villages and towns where he raced as an amateur. That said, despite being Toledo's greatest sports star Bahamontes is more than happy to knuckle down. He pulls down barriers, whizzes across from one side of the road to another at junctions with red tape to discourage oncoming vehicles, and yells and waves should any official vehicle be too close to the pack for his liking.\n\n'Fede's never been one to work in a team and we're all used to that,' comments Basilio L\u00f3pez, a close friend and head of sports at a local town hall, who drives a car for race guests and V.I.P.s in his holiday time. 'He's just too hands-on to delegate responsibilities. I keep on telling him that he should just ease back and let others take over. But he's just too keen to get involved. It's the philosophy of the workers who were born under Franco: you don't stop working till you drop.' Basilio has been part of the organisation for the last decade and reveals that the heart of the tour is the Bahamontes' _pe\u00f1a_ itself. While barely functioning now as a supporters' club, its main role is to run four or five bike races. In a town with no cycling clubs like Toledo that is hugely important. As the effects of the recession continue to affect Spain the number of teams has dropped though, by around a third. The chances of another rider following the wheeltracks of a former winner like modern-day Spanish Classics star Juan Antonio Flecha and making the big time are increasingly limited.\n\nHowever, even if Bahamontes spends a large part of his year chasing up money from the towns and villages who host the tour \u2013 he has the list of debtors kept permanently in his inside jacket pocket \u2013 the commercial sponsors remain solid. Wurth and the M.O.T. centre at nearby Oca\u00f1a have been loyal for years, and if Coca-Cola's sponsorship of the signing-on table at the start is hardly the most generous of efforts by the biggest multi-national present, at least there are free cans of fizzy pop to drink as a result. There are national backers as well for a bulging portfolio of different prizes within the race: RENFE, the Spanish state railway, and the Spanish branch of Shimano, the bike components manufacturer, back two different sprint competitions. A local transport company has forked out for the King of the Mountains prize and a Toledo-based builder specialising in swimming pools funds the points jersey competition.\n\nAs usual in bike races the smaller the town and the further it is from the big cities the greater the interest. When the race passes through Galvez, a town where Bahamontes used to go to pick up bags of corn and wheat as a black marketeer more than sixty years ago, it seems that every front doorstep has a little child squatting on it, with grandma sitting on a canvas chair just outside and elder brothers and sisters lined along the pavement. On the outskirts of Toledo, though, barely anybody stops to look.\n\nWhile returning to Toledo ahead of the race Basilio stops to give a co-worker a lift: it is the _pe\u00f1a's_ treasurer, Jos\u00e9 Ignacio. The _pe\u00f1a_ is still going, he reveals, but it is crumbling at the edges. Almost all its members are in their sixties or older and the days are long gone when its numbers ran into the thousands and they had separate sections for hunting and skeet shooting. Now the number of card-carrying _pe\u00f1istas_ is down to just one hundred. Ignacio remembers the day of glory in 1959 when Toledo was awash with yellow ribbons and the bars served 'yellow jerseys' and 'leaders' rather than glasses of beer. 'I'd joined the _pe\u00f1a_ a few years before,' he recalls, 'but when Fede won I was on my honeymoon. Still, anywhere I'd go, the minute I told them I was part of his _pe\u00f1a_ , it was drinks on the house for me,' he cackles. 'That's the spirit of Spain that we've now lost,' Basilio reflects. 'The old, real Spain.'\n\nBahamontes might have fallen out with the Spanish Cycling Federation after the events of 1965 but as a recently retired rider he had no intention of leaving the world of cycling. He bought his bike shop premises outright from its former owner after apparently helping to speed up the process by threatening to apply pressure in Toledo's town hall after the owner had been embroiled in a scandal involving a priest. Almost immediately he started directing the first in a series of amateur and professional teams. Initially encouraged by his success with junior teams, and using his fan-club headquarters as the logistics base for all these teams, the 'La Casera-Pe\u00f1a Bahamontes' ran as a ten-rider amateur team in 1968, and then with a squad of twenty-three in 1969. It became a professional team in 1970 and quickly moved into the middle ranks of the Spanish cycling hierachy. With the help of experienced co-director Miguel Moreno, La Casera took a stage victory in the Giro d'Italia in 1970 through Miguel Mari Lasa as well as the sprints competition in the Vuelta a Espa\u00f1a. Joaquim Galera even won Bahamontes' old favourite, the Arrate hill-climb in the Basque Country.\n\nWith Lasa gone, though, 1971 and 1972 were leaner years. But the team bounced back in 1973 when Pedro Torres took the team's first (and only) stage win in the Tour de France as well as the King of the Mountains competition. (Jos\u00e9 Luis Abilleira had pulled off the same 'double' in the Vuelta where La Casera also won the team prize.) Then in 1974 Abilleira landed the King of the Mountains jersey in the Vuelta a Espa\u00f1a for the second year running. However, with major arguments breaking out behind the scenes between Bahamontes and his riders that proved to be La Casera-Bahamontes' last season as a professional squad.\n\nFrom Bahamontes' point of view, La Casera-Bahamontes was a victim of its own success. He became increasingly disappointed and frustrated with his riders, who, he claims, grew more and more demanding with each triumph. 'We built the team up category by category, from juvenile to junior to amateur to professional and we signed the best riders from all the other teams,' Bahamontes recalls. 'We had an advantage in that we had a big backer [La Casera] and the owner told me to do the maths and we turned pro. We had some good riders, and we were the first team out there to buy a bus for the riders to travel in to the starts. But after racing Milan-San Remo one year the riders wanted to go back by plane rather than in the bus, and I told them they couldn't, not after all the money we'd made the owner pay for it. If they didn't like the bus they could ride back home on their bikes.' That, says Bahamontes, was the beginning of the end. 'Then they started to make more and more demands [across the board]: they wanted to be Napoleon without having won the war. I had teams [amateur and profesional] for nineteen years. But [the professional team didn't fold] because of what it cost, the problem was what the cyclists demanded \u2013 and what they couldn't have.'\n\nBahamontes did not just run a shop and a bike team. Almost as soon as he retired his _pe\u00f1a_ took over the role of Toledo's main cycling club, organising the prestigious four-day Vuelta a Toledo and other stage races. Another long-term project was a bike museum, located in the unlikely setting of an industrial tools factory, forty minutes' drive from Toledo. It contains around twenty frames from Bahamontes' career, as well as bikes donated by everybody who was anybody in Spanish cycling, from Miguel Indurain to Oscar Freire, the three-times world champion. It even includes the wooden-rimmed models Bahamontes used during his days as a black-marketeer. In addition, there are heavy woollen jerseys, posters, photographs and other paraphernalia from his past. Had Bahamontes' plans to transfer the museum to Toledo's city centre succeeded it would have been more appealing to normal tourists. But for bike fans the trip to the Wurth factory is well worthwhile.\n\nBahamontes had no qualms about milking the celebrity circuit during his retirement. For years he would travel the length and breadth of Spain, and occasionally France, in a gleaming Mercedes to attend everything from velodrome openings and cycling club prize-givings to annual get-togethers of former star riders. There is still no stopping him. In 2011 he twice presented the prizes at the finishes of different stages of the Vuelta, in Talavera and El Escorial, and was there again in Madrid, springing on to the podium to hand over the King of the Mountains' jersey to David Moncouti\u00e9, of France. He is always willing to turn up for cycling celebrations or ceremonies, sometimes uninvited, as Alberto Contador once discovered. He is a regular interviewee in various forms of media on the state of Spanish cycling, though he is none too polite about modern cycling in general. Indeed, he said in an interview a few years ago: 'It's no longer the same since it became pure business. It's got a lot softer, too compassionate. Breakaways in my time weren't like the ones [1988 Tour winner Pedro] Delgado used to do, in the last five hundred metres and with a big team. These days if somebody punctures the rest of the field is supposed to wait. Back when I was riding you'd use that to get an extra couple of minutes. Sport's no longer got that edge to it. It's like if you cut someone's eyebrow in boxing, should you stop trying to hit it? Of course not.'\n\nTo many observers, Bahamontes seems incapable of slowing down to a more settled lifestyle. It has got to the point where it is jocularly suggested that his multifarious activities would wear out someone twenty years his junior. 'Bahamontes or how to freeze time', ran the headline in Spanish sports daily _AS_ during the 2011 Vuelta. Underneath was a photograph of Bahamontes at the race taking pictures with the Agfa Optima III camera he has had since 1960. 'Federico was a popular hero in a Spain that was just opening up to the world,' the article said, 'and a media phenomenon, perhaps the first of [Spanish] sport . . . [he was] over the top, provocative, extravagant, impulsive and vain. A genius of the bicycle.'\n\nWhen asked the secret of his longevity Bahamontes always has two answers: the first, he claims, is that he did not take dope, with the tacit implication that others of his generation did and paid a high price. The second is work. 'If you come tomorrow, on Saturday and Sunday, you'll see me doing the same as I did today,' he told _El Pa\u00eds_ in June 2011 'I started work at eleven, flinging stones into a truck [when he and his father were road-menders]; now I work from the morning until two or three when I stop for lunch. And half an hour later I'm back in there again. I get up at seven [in the morning] and I don't stop.' In a sense it is a continuation of his life as a professional rider. He has always been 'the last to go to bed and the first to get up'. At weekends he occupies himself on his small area of farmland, checking and tending crops, but his weekday routine centres on office work at the _pe\u00f1a_. Much of his time is spent chasing up unpaid bills from recession-hit town halls who have hosted the starts or finishes of the Vuelta a Toledo. In one of our most recent interviews, he showed me the foot-high pile of paperwork detailing money due. Periodically he will drive off to hound the defaulters in person. It seems the final chapter of Spain's leading sportsman from the first half of the Franco regime could be that of an unofficial, ageing debt-collector.\n\nHis lifestyle will probably remain frenetic. It is in his genes. He told _El Pa\u00eds_ : 'My maternal grandfather had the same character. He was a small guy, but dynamic. My mother always used to say \" _alto y flojo, como el hinojo\"_ ['tall and weak, just like fennel'] because when they [people] are taller, they can't reach down so easily and they can't work so hard.' And how did he feel about reaching eighty-three? 'You can't turn back the clock and be in the thick of the action all the time.' But if he did not work, he admits: 'I would be sunk.'\n\nHe is still married to Fermina and has frequent visitors to his _pe\u00f1a_ , but there are still flashes of his old solitary self when he was the Eagle of Toledo flying through the mountains of France, Italy and Spain. When I asked him once if he had any good friends, either in sport or out of it, his answer was: 'Just one: Evaristo Murtra.' But that was all. And in interviews he gives the strong impression that his relationship with Murtra was founded on business rather than a personal friendship. However, that sense of isolation does not seem to bother him; it just sums up his life-philosophy As he puts it: ' _Solo me muevo mejor que nadie_ ' ('alone, I move better than anybody else'). And given what he achieved flying solo, can that really be doubted?\n\nThe _palmares_ cannot fail to impress: six King of the Mountains titles and Spain's first victory in the Tour de France; two podium finishes in the same race when aged over thirty-five; seven Tour stage wins plus notable victories in every other major stage race of his time. The big unanswered question in Bahamontes' career, though, is whether he could have done even more? Or was he riding so far above his usual level in 1959 that further victories in the Tour were never really on the cards? And was he the world's best climber of all time? There will never be definitive answers. Yes, Richard Virenque may have one more King of the the Mountains title, and Lucien Van Impe more mountain-top stage wins and as many mountain titles as Bahamontes. But Virenque did not win the Tour and Van Impe never came close to Bahamontes' record of crossing Tour mountain passes in first place fifty-three times. And without Bahamontes, so the story goes, there might not have been a Van Impe at all: the Spaniard provided the inspiration for the Belgian to get on a bike in the first place. Indeed, it is said that Bahamontes used his influence to get Van Impe his first professional contract after seeing him win the King of the Mountains title in the Tour de L'Avenir, France's top race for promising amateurs. When I asked cycling's all-time top rider Eddy Merckx what he thought of Bahamontes' place in cycling history, he said: 'There are different eras, different people and it's very hard to compare. But he was certainly one of a very few, one of the very, very greatest.'\n\nThe question remains, though, whether a rider who could climb so well that others, like Raymond Poulidor, say that to attempt to follow him was to 'like committing suicide', might have acheived more than near-blinding, but erratic, flashes of brilliance. So why didn't he? Quite apart from Geminiani's skulduggery in 1963, his failure to collaborate with Julio Jim\u00e9nez in 1964, or even that elusive telegram from the Franco Government during the 1957 Vuelta, the single biggest factor in his failure to shine even more brightly must be poor management. On a good day, he could be the best climber there has been. However, lack of consistency was his biggest downfall: to gain it a volatile, erratic personality like Bahamontes needed a firm hand directing him; too rarely did he have one.\n\nFrom Geminiani to a man I met near a graveyard in Val de Santo Domingo, the common consensus is that Coppi was the best manager he had, certainly in terms of his motivational skills. Dalmacio Langarica was the one Bahamontes valued most for his practical abilities, but Coppi was the one who inspired him to greatness and the man with his finger on the pulse when it came to technological advances in the sport too.\n\nThe tragedy, though, is that even if the Tricolfilina-Coppi team's financial difficulties had not existed, Coppi's untimely death in early 1960 leaves unanswered the question of what Bahamontes' career might have been with the Italian by his side for longer. And Langarica, his other guiding light, was only with Bahamontes for one Tour. For that reason 1959 became a watershed for Bahamontes. When he had the natural strength of a young man he did not have the team or director he needed; then when he did have the team and the directors, particularly post-1959 with Raoul Remy, he no longer had the natural power of a younger man. Remy in any case, made some strategic errors such as when he ordered Bahamontes to sit up during his Pyrenean attack in 1963. It was only in 1959, when Bahamontes was thirty-one, with Coppi overseeing him and Langarica's faultless guidance of the team in the Tour that the twin elements of brute strength and sound strategy fully blended. When that happened, nothing could stop the Eagle of Toledo.\n\nOne indication of just how perfectly the planets lined up for Bahamontes that year is that he remains one of only six riders to win the Tour's overall classification and the King of the Mountains in the same race. It is surely no coincidence that Coppi, Bahamontes' mentor, took both titles in 1949 and 1952. But since 1959 only Eddy Merckx (in 1969 and 1970) and Carlos Sastre, by default in 2008 after Austrian Bernard Kohl was banned for doping, have managed the difficult double.\n\nIt was inevitable that such a huge triumph for Spain was exploited so ruthlessly by General Franco's regime. Starved of international recognition, and in constant need to justify its existence to the Spanish population as memories of the Civil War faded, sport provided both a pressure valve on social unrest for the Franco dictatorship and as a propaganda vehicle. As a Castillian taking what was arguably Spain's biggest individual athletic triumph since the Civil War, Bahamontes' Tour de France win was too good an opportunity to miss. As such, Bahamontes acts as a bridge between the two key periods of the Franco era: a reminder of the harsh, post-Civil War recession and international isolation of the earliest years, and the transition to the economic well-being of the 1960s and 1970s.\n\nIn terms of sport, at least, Bahamontes is one of the last, and certainly the most important, link between the two eras. But even today in Spain he is rarely cited as such. One of the rare high-profile analyses of Bahamontes' significance for Spanish society came in a keynote speech made by _MARCA's_ former editor, Eduardo Inda, to celebrate the seventieth anniversary of the newspaper in 2011. In it he singled out Bahamontes as the one person who 'gained our affection because he confirmed that the \"botched-up\" Spain was giving way, little by little, to the Spain of excellence. His Tour and his six Kings of the Mountains prizes carried us to seventh heaven'. Inda went on to describe what the success of Bahamontes and other athletes meant to the people of Spain at the time. 'We feel grateful, emotionally, to the most veteran members [of our sport] because you made our long night of dictatorshp more bearable. With sport, the suffering was less painful, even knowing that the former regime used sport generally . . . as a sleeping pill which would make us forget the anxiety we had for freedom and the harsh economic and social post-war reality . . . they gave colour to a black-and-white Spain, to that Spain we have recorded in our minds as if it were an interminable _no-do_.' As Inda sees it, Bahamontes and his co-stars at the time played a beneficial role even while they were being exploited by the regime to help justify its existence.\n\nThen there is his continuing role as the father figure for Spanish cycling, as Pedro Delgado calls him, and in particular for their most valued facet of the sport, stage racing. 'I'm from a much later generation,' says Miguel Indurain, Spain's five-times Tour de France winner and for many his country's top all-time athlete. 'I grew up watching Merckx and Hinault, but we all knew how Bahamontes was the one who staged the break-through for us in the Tour. In the mountains, he was unmatchable, even if consistency never was his strong point.'\n\nPerhaps thinking of Jes\u00fas Loro\u00f1o and Vicente Treuba, Indurain says: 'There were riders before him who were important, but Bahamontes was the Tour pioneer. He was an impulsive, very fiery rider \u2013 and that's still the case now. He just keeps going, he never stops. But he was hugely important, particularly when you consider that back then the Tour de France was as big as [football's] World Cup here in Spain. Not like now.'\n\nThe pioneering aspect of Bahamontes is one of his most appealing legacies. Perhaps one of the hardest aspects for fans to accept is how big a part money had to play in his career. Yet he was far from being the only big name to view cycling as a business as much as a sport. In William Fotheringham's biography of Tom Simpson, he writes of the British star: 'His contemporaries all noted his financial hunger.' And according to Paul Howard's biography, one potential reason why Jacques Anquetil failed to go for a sixth Tour de France was that it did not make financial sense. 'For him [Anquetil], the notion of a palmares was absurd if it didn't add value commercially,' top French cycling journalist Philippe Brunel tells Howard.\n\nApart from the precarious nature of his profession, as Carlos Sastre points out, this was an era in Spain when the struggle for survival was not just a question of being better or worse off. Like a large percentage of the population, including many of its greatest cyclists, Bahamontes lived through an era in which the spectre of starvation was never far away. It is hardly a surprise that whatever 'life-philosophy' he and other Spanish professionals learnt then was applied so aggressively to their racing. If this explains some of the sport's less savoury aspects, then arguably the same fear of financial ruin also helped drive Bahamontes and his fellow professionals along the path to greatness. 'You mustn't forget he was living in a time when nutrition as we know it didn't exist. It was a question of survival, pure and simple,' Sastre believes. 'That made them fight hard for sustenance and that kind of fighting experience was what made them such tough cyclists. Now we have all we want to eat and we just think about stupid things like buying cars. In the jungle, you're looking for food, not for the best car or the nicest house, just to survive.'\n\nRaphael Geminiani concurs with Sastre's analysis, but goes further. 'He was more worried about making money than anything else,' he says. 'And that's why he didn't do his job so well, because he wasn't sufficently dedicated to his profession.' However, if money was the reason which drove Bahamontes to racing, and which spurred him on in his amateur years, ultimately it may have been what let him down, too. Right from the start the richer Bahamontes became, the more King of the Mountains titles he amassed, the less willing he seems to have been to opt for the riskier goal of going for the overall titles. There were exceptions, like the 1959 and 1963 Tours, but by then it seems the overall had become increasingly more difficult for Bahamontes to avoid. Matters reached a head in 1964: had he reached a deal with Julio Jim\u00e9nez that had worked, such as conceding the King of the Mountains title, Bahamontes could have found the ally he needed on the long road to Pau. The margin between himself and Anquetil going into the time-trial that followed would have been smaller. But Bahamontes could not give up his 'surefire guarantee' of the King of the Mountains and beneath that lies a fascinating contradiction: the less likely his attacks were to succeed in winning the biggest prize, the greater his reputation as cycling's best 'pure' climber, and as the dreamer who never gave up. Or, to use a very Castillian metaphor, the eternal tilter at windmills.\n\nAll great sports stars have their critics and there are those who believe that rather than Bahamontes' career going largely unrewarded, in 1959 he simply got lucky and was in the right place at the right time when the French started to fall out among themselves. Geminiani, who was riding in the French 'A' team that year as one of the 'Big Four', does not agree. 'Bahamontes was not the climber he's been taken for,' Geminiani says. 'He was more spectacular than efficient. But then just when Bahamontes was getting on in years, just when Charly Gaul was at the height of his powers, just when he has all of the top French riders \u2013 Anglade, Rivi\u00e8re, Anquetil \u2013 lined up against him, Bahamontes goes off and wins the Tour. It's true they say that he was able to take advantage of the war between the agents, but that's not the case. Anquetil did not want Rivi\u00e8re to win \u2013 true. Rivi\u00e8re did not want Anquetil to win \u2013 true. But that Anquetil and Rivi\u00e8re ganged up against Anglade \u2013 not true. They stayed out of it. It wasn't their fight. All we saw that year was an Anglade-Bahamontes duel. And Bahamontes deserved to win it.'\n\nAnother damaging factor for Bahamontes' career was his erratic, sometimes eccentric, personality. But at least in 1959 Coppi and Langarica were there to resolve those problems. The knock-on effect was noticed by the newspapers. 'He is as hyper-nervous as Coppi, but now his brain controls those nerves,' argued _El Alc\u00e1zar_ after his Tour win. 'That extreme nervousness is what all great champions have . . . Coppi, aware he was close to losing a Tour [in 1952], once told a mechanic to get the right wheel as soon as possible or he'd shoot his head off. That kind of temperament is a two-edged weapon. If used rightly then it's a huge advantage. Bahamontes' issues make him seem what he is not \u2013 vain, proud and anti-social \u2013 when in reality he is simply childlike.'\n\nIn his role as a latter-day commentator Bahamontes is wont to say: 'People don't enjoy cycling as much as they used to because there aren't any more breakaways like mine.' Note the use of the word 'breakaways' rather than 'wins'. Therein may lie the nub of why Bahamontes remains such an inspiring figure: because his concept of racing borders on the artistic, or as _L'Equipe_ once said of him, on the 'Bohemian'. It is almost an anti-sporting way of racing a sport. Climbing up mountains at the head of the pack just because they are there, not because of what you can gain, underlines the fact that sport is ultimately an exercise in futility, albeit a fascinating one. That futility can even have a subversive edge as when Bahamontes stopped for his ice cream on top of the Romeyere. The effect was unintentional, but to the world it looked as if he was walking away deliberately from his chance of glory. And given his self-destructive racing style in the years that followed, the tantrums and the multiple abandons, that is why the image of Bahamontes opting for an ice cream, rather than continuing, retains so much power. Even if he did not mean it, it looked like 'typical Bahamontes'.\n\nBahamontes lingers in the memory precisely because of his wayward eccentricity and love of seemingly pointless, impetuous, attacking. Unfulfilled potential is far more romantic than a series of brilliant results, as Bahamontes, talking to reporters after his fruitless break across the Bonette-Restefond in the 1964 Tour, was keenly aware. As he put it: 'This will have made the bars in Toledo a lot of money this afternoon.' However, there was always an eminently practical bent to it all, too: after blasting across the mountain pass at the head of the Tour, Bahamontes was also the same man who stuffed his suitcase full of spare bike parts to sell back home in Spain. He was the one whose first thoughts when he talked to reporters after winning the Tour in 1959 was how much money he could make out of it. And that is perhaps his greatest achievement, that Bahamontes was driven by a desire for financial gain, but simultaneously maintained the rather far-fetched but appealing image of the eccentric climber forging his own solitary, almost dream-like path far from the vices of civilisation.\n\nTo race just for the money is straightforward enough, and so it is to race idiosyncratically. To do both, simultaneously, for twelve years as a professional is a tough call indeed. It is something only a handful of athletes have managed to pull off as convincingly as the Eagle of Toledo.\n\nIf the grimy, run-down setting for the start of the Vuelta a Toledo was a pretty faithful reflection of life in the suburbs of postindustrial Spain, the finish could hardly be more 'teepical Spanish', as the natives themselves say when the country's caricatures are so brazenly displayed. It is staged in the central Zocodover square at the top of a sharp, cobbled rise into old Toledo. The feel that this is the touristy part of town is reinforced by the woman in a tight flamenco dress, and even tighter hairbun, standing on top of the podium and belting out traditional folksongs. The crowd of around one hundred is mostly foreign, too. Eventually an MC from the race organisation takes over. Not even his inane patter \u2013 'they're coming very fast, the bunch is all over the place, they're coming very fast, what's going to happen?' \u2013 causes the crowd to thin. All that matters is that they are enjoying themselves: it is a fine, sunny day and they are about to witness a free spectacle. For bike races in the summer that is the way it has always been.\n\nMeanwhile, at the finish, which stretches down one cobbled side of triangular square in the shadow of the town hall, Bahamontes is once again in full flight. Striding up and down between the barriers and under the finishing gantry, it is Bahamontes who waves through what little race traffic there is before the peloton itself turns up. He is constantly peering at the descent to see if the riders are on their way. He bustles, he fusses, he is never still for more than ten seconds. The characteristic energy he exudes is irritating and fascinating in equal measure. When the riders finally appear the barriers are thick with the public. Bahamontes disappears from view as a blizzard of glittering bike frames and brightly coloured, tensed bodies, roar past. One, a shovel-jawed, burly teenaged Spaniard in a gaudy red, yellow and blue jersey, bellows with delight as he scoots across the line for both the stage win and, it later emerges, overall victory.\n\nNo sooner has the stage finished than Bahamontes is back in action. Beneath the curtain behind the podium his brown leather shoes can be seen darting around at high speed. The announcer spends a long time requesting riders, most leaning on team car bonnets and comparing stories before the long trip home, to come up for their prizes. Finally, perhaps half an hour after the race is finished, the ceremony starts. 'A big hand for Teresa, the godmother of the Vuelta a Toledo,' the MC says as another publicity girl teeters across the stage, clutching a bag of cakes and sweets provided by local manufacturer Miguela\u00f1ez instead of bouquets or a trophy. The Frenchman who was leading the race overnight, looks gloomy at getting beaten but still receives the prize for best under-23 rider, courtesy of the M.O.T. centre. He cheers up appreciably when he receives a couple of kisses from the PR girls. Then, when a couple of slightly inebriated friends in the crowd start bellowing his name and pointing at the sky, he positively beams.\n\nEach time the winner of a different trophy comes out, Bahamontes again dashes out from the wings, fidgeting behind the publicity girls, pulling at their sleeves to ensure they stand correctly on the podium for the photographs. There is one cringeworthy moment when he all but manhandles a small child, clearly the son of a sponsor and allowed on to the podium as a treat, out of the way of the prizewinner. 'Can't he just enjoy it?' says one member of the public, a bike rider himself judging by his shaven legs. 'He's so nervous about everything.' There are a few moments when Bahamontes is recognised: 'Fede, you're the best', one man shouts, and another asks for a photograph with the 1959 Tour winner. But that is it: his time, it seems, has gone. Or perhaps not. When the riders have all had their pictures taken, raised their arms and brandished their trophies, Bahamontes leaps on to the stage again clutching handfuls of postcards bearing his picture and a list of his palmares which he flings gleefully into the crowd. Is it a pure ego trip? There is only one answer to that.\n\nHowever, in a country where the sport is drying up in the worst recession since the 1950s, his bike race endures. In fact, the Vuelta a Toledo is the only Spanish race of its level that has never been cancelled, not once in its forty-six-year history. And it is a world away from the Ministry of Education race held nearby that offered Bahamontes his start back in 1947. But the connection between the two persists in the shape of a man still driven onwards by the same manic energy that enabled him to rule the Tour de France mountains for more than a decade.\n\nAnd the legend, albeit increasingly distorted, lives on as well. 'Bahamontes?' the taxi driver says as I head away. 'Not many people know who he is now, but he was the greatest in his time.'\n\nThese are perhaps the final images we should retain of the Eagle of Toledo, the ones from the Vuelta a Toledo in August 2011: Bahamontes, arms waving in the air, whistle blowing, as he guides his race through the province where it all began; Bahamontes, determined to maintain his place in the public's consciousness and act as an inspiration for the younger generations of riders even if it all comes down to an eighty-three-year-old flinging pieces of paper into the crowd; Bahamontes, standing at the top of the climb to the Zocodover square, alone again naturally, but waiting to see who will be the fastest rider to power up the hill and join him.\n\nBahamontes being Bahamontes, he might even get an ice cream.\n\n#### Palmares\n\n**Federico Mart\u00edn Bahamontes**\n\nPlace of birth: Val de Santo Domingo, Spain\n\nDate of birth: 9 July, 1928\n\nTurned professional: 1953\n\nRetired: 1965\n\n**1953 TEAM: BALANZAS BERKEL\/SPLENDID**\n\n**Wins**\n\nTour of Asturias: stage 1 and King of the Mountains (K.O.M.)\n\nCircuito del Sardinero: stage 1\n\nTour of Malaga: overall\n\nVolta a Catalunya: K.O.M\n\n**Selected places**\n\nVolta a Catalunya: 8th\n\nTour of Asturias: 21st\n\n**1954 TEAM: CIRCULO BARCELONISTA-YASTA\/SPLENDID-D'ALESSANDRO**\n\n**Wins**\n\nMt. Agel hill-climb\n\nTour of Mallorca: stage 3 (first sector)\n\nTour of Asturias: K.O.M.\n\nTour de France: K.O.M.\n\nBicicleta Eibarresa: K.O.M.\n\n**Selected places**\n\nTour de France: 25th; stage 18, 5th\n\nBarcelona\u2013Palamos: 2nd\n\nBicicleta Eibarresa: 2nd\n\nTour of Mallorca: 3rd\n\nTrofeo Masferrer: 5th\n\nSpanish Regional Championships: 7th\n\n**1955 TEAM: PE\u00d1A LA SOLERA**\n\n**Wins**\n\nTour of Asturias: overall, K.O.M. and stage 6\n\nBicicleta Eibarresa: stage 2\n\nVolta a Catalunya: stages 7 (2nd sector) and 11\n\nMont Faron hill-climb\n\nCl\u00e1sica de los Puertos\n\nGP Monte Carlo\n\n**Selected places**\n\nVuelta a Espa\u00f1a: 21st; stage 2, 2nd\n\n**1956 TEAM: FAEMA\/GIRARDENGO-ICEP**\n\n**Wins**\n\nNone\n\n**Selected places**\n\nMont Faron: 2nd\n\nTour de France: 4th; K.O.M. 2nd\n\nVuelta a Espa\u00f1a: 4th: K.O.M. 2nd)\n\nGiro d'Italia: stage 7, 2nd; stage 16, 2nd\n\n**1957** **TEAM** **:** **MOBYLETTE** **GAC** **\/** **ST** **-** **RAPHAEL** **-** **GEMINIANI**\n\n**Wins**\n\nVuelta a Espa\u00f1a: 2nd\n\nTour of Levante: stage 3\n\nTour of Asturias: overall, stage 1 and K.O.M.\n\nMount Faron hill-climb\n\nGP Jerez\n\nGP San Lucar de Barrameda\n\nGP Toledo\n\n**Selected places**\n\nVuelta: stage 3 and K.O.M.\n\nGP Europe: 3rd\n\nTour de France: stage 3, 4th\n\n**1958 TEAM: FAEMA**\n\n**Wins**\n\nTour de France: stage 14, stage 20 and K.O.M.\n\nVuelta: K.O.M.\n\nGiro d'Italia: stage 4\n\nSpanish National Championships\n\nArrate hill-climb\n\n**Selected places**\n\nVuelta a Espa\u00f1a: 6th\n\nTour de France: 8th\n\nGiro d'Italia: 17th\n\n**1959 TEAM: TRICOLFILINA-COPPI\/CONDOR\/KAS**\n\n**Wins**\n\nTour de France: overall, stage 15 and K.O.M.\n\nVuelta a Espa\u00f1a: stage 4\n\nSpanish Mountains Championship\n\nArrate hill-climb\n\nAntequera\u2013Cabra\u2013Antequera: stage 1\n\nTour de Suisse: stages 3 (b) and 5 and K.O.M.\n\nTour of Levante: K.O.M.\n\nTour of Cerde\u00f1a: K.O.M.\n\n14 criteriums\n\n**Selected places**\n\nTour de France: stage 17, 2nd; stage 13, 3rd; stage 19, 4th\n\nVuelta a Espa\u00f1a: stage 1 (1st sector), 4th; stage 1 (2nd sector), 3rd\n\nMont Faron hill-climb: 2nd\n\nSpanish Regional Championship\n\nSpanish National Championships: 2nd\n\n**1960 TEAM: FAEMA**\n\n**Wins**\n\nVuelta a Espa\u00f1a: stage 13\n\nGP Priego: two stages 1 and 2, overall and K.O.M.\n\nTour of Asturias: K.O.M.\n\nArrate hill-climb\n\n3 criteriums\n\n**1961 TEAM: V.O.V.**\n\n**Wins**\n\nArrate hill-climb\n\nMonaco\u2013Mont Agel hill-climb\n\nNice\u2013Mont Agel hill-climb\n\nTour of Cerde\u00f1a: stage 4\n\n**Selected places**\n\nWorld Championships: 17th\n\nSpanish Regional Championships: 2nd\n\n**1962 TEAM: MARGNAT-PALOMA**\n\n**Wins**\n\nTour de France: stage 13 and K.O.M.\n\nArrate hill-climb\n\nMont Faron hill-climb\n\nMont Faron time-trial\n\nTour of Romandie: 7th\n\nDauphin\u00e9 Lib\u00e9r\u00e9: K.O.M.\n\n4 criteriums\n\n**Selected places**\n\nTour de France: 14th; stage 19, 3rd\n\nDauphin\u00e9 Lib\u00e9r\u00e9: 4th\n\nTour of Romandie: stage 3\n\n**1963 TEAM: MARGNAT-PALOMA**\n\n**Wins**\n\nMont Faron time-trial\n\nTour de France: stage 15 and K.O.M.\n\nMidi Libre: K.O.M.\n\nDauphin\u00e9 Lib\u00e9r\u00e9: 4th\n\n**Selected places**\n\nTour de France: 2nd\n\nTour of Romandie: 2nd\n\nDauphin\u00e9 Lib\u00e9r\u00e9: K.O.M.\n\n**1964 TEAM: MARGNAT-PALOMA**\n\n**Wins**\n\nTour de France: stages 8 and 16 and K.O.M.\n\nNaranco hill-climb\n\nMont Faron hill-climb\n\nMont Faron time-trial\n\nMidi Libre: stage 5\n\nSix Days of Madrid (with Rik Van Steenbergen)\n\n**Selected places**\n\nTour de France: 3rd\n\nDauphin\u00e9 Lib\u00e9r\u00e9: 6th\n\n**1965 TEAM: MARGNAT-PALOMA**\n\n**Wins**\n\nCircuit du Proven\u00e7al: overall\n\n**Selected places**\n\nVuelta a Espa\u00f1a: 10th; stage 4, 5th; stage 16, 6th\n\nMontjuic hill-climb: 2nd\n\nUrkiola hill-climb: 8th\n\nNaranco hill-climb: 8th\n\n#### Bibliography\n\nA NOTE FROM THE AUTHOR ON SOURCES\n\nAmong the many people willing to give up their time to be interviewed, for which many thanks, first and foremost is, of course, Federico Mart\u00edn Bahamontes himself.\n\nMany thanks, too, to the following willing interviewees (in alphabetical order) for their time and invaluable insights:\n\nIan Brown (8 March, 2011)\n\nAndr\u00e9 Darrigade (7 April, 2011)\n\nRaphael Geminiani (11 July, 2011)\n\nMiguel Indurain (22 March, 2012)\n\nAntonio Jim\u00e9nez Quiles (15 April, 2011)\n\nJulio Jim\u00e9nez (13 September, 2011)\n\nJosu Loro\u00f1o (9 April, 2011)\n\nLuis Ota\u00f1o (8 April, 2011)\n\nMiguel Poblet (21 March, 2011)\n\nRaymond Poulidor (1 July, 2011)\n\nBrian Robinson (29 March, 2011)\n\nBernardo Ruiz (18 January, 2011)\n\nCarlos Sastre (15 January, 2012)\n\nBOOKS\n\n_(Published in English)_\n\nBeevor, Antony \u2013 _The Spanish Civil War,_ Penguin, 1982.\n\nBorkenau, Franz \u2013 _The Spanish Cockpit_ , Faber & Faber, 1937.\n\nBrenan, Gerald \u2013 _The Spanish Labyrinth_ , Cambridge University Press, 1950.\n\nBrenan, Gerald \u2013 _The Face of Spain,_ The Turnstile Press, 1950.\n\nFallon, Lucy & Bell, Adrian \u2013 _Viva La Vuelta_ , Mousehold, 2005.\n\nFotheringham, William \u2013 _Put Me Back On My Bike: In Search of Tom Simpson_ , Yellow Jersey Press, 2002.\n\nFotheringham, William \u2013 _Roule Britannia: A History of Britons in the Tour de France_ , Yellow Jersey Press, 2009.\n\nFotheringham, William \u2013 _Fallen Angel: the Passion of Fausto Coppi,_ Yellow Jersey Press, 2009.\n\nHooper, John \u2013 _The Spaniards_ , Viking Press, 1986.\n\nHoward, Paul \u2013 _Sex, Lies and Handlebar Tape: The Remarkable Life of Jacques Anquetil,_ Mainstream, 2009.\n\nMcGann, Bill & Carol \u2013 _The Story of the Tour de France_ , _Volume 1_ , Dog Ear Publishing, 2006.\n\nMoore, Richard \u2013 _In Search of Robert Millar_ , Harper Collins, 2007.\n\nThomas, Hugh \u2013 T _he Spanish Civil War,_ Simon & Schuster, 1994.\n\n_(Published in Spanish)_\n\nBodegas, Javier & Dorronsoro, Juan \u2013 _Con Ficha de la Espa\u00f1ola 1960 \u2013 2003_ , Uriziar Edizioak, 2003.\n\nBodegas, Javier & Dorronsoro, Juan \u2013 _Historia de la Bicicleta Eibarresa\/Euskal Bizikleta_ , Dorleta, 2000.\n\nBodegas, Javier & Dorronsoro, Juan \u2013 _Historia del Campeonato de Espa\u00f1a,_ Urizar, 2003.\n\nGarai, Josu \u2013 _Ciclismo del Norte,_ Recoletos, 1994.\n\nMart\u00edn Bahamontes, Federico \u2013 _La Sombra del Aguila,_ Wurth, 2009.\n\nP\u00e9rez, Chico & Guerra, Adrian \u2013 _Vuelta Ciclista a Espa\u00f1a 1935\u20131985,_ Caja Postal, 1986.\n\nPreston, Paul \u2013 _Francisco Franco, Caudillo de Espa\u00f1a,_ Grijalbo, 1998.\n\n_(Published in French)_\n\nVarious \u2013 _Tour de France 100 Ans_ , L'Equipe, 2002.\n\nPAMPHLETS\n\n_(Published in Spanish)_\n\nBodegas, Javier \u2013 _Bahamontes,_ Dorleta, 2000.\n\nBodegas, Javier \u2013 _Jes\u00fas Loro\u00f1o,_ Dorleta, 2000.\n\nNEWSPAPERS AND MAGAZINES\n\n_(Published in English)_\n\n_Cycle Sport_\n\n_Velo News_\n\n_The Independent_\n\n_The Guardian_\n\n_(Published in Spanish)_\n\n_MARCA_\n\n_El Pa\u00eds_\n\n_El Mundo_\n\n_ABC_\n\n_El Mundo Deportivo_\n\n_AS_\n\n_Deia_\n\n_La Vanguardia_\n\n_Urtekaria_\n\n_El Alc\u00e1zar_\n\n_Informaciones_\n\n_(Published in French)_\n\n_L'Equipe_\n\n_Le Dauphin\u00e9 Lib\u00e9r\u00e9_\n\n#### Index\n\nAbat\u00e9, Antoine ref1\n\n_ABC_ ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14\n\nAbilleira, Jos\u00e9 Luis ref1\n\nAbt, Sam ref1\n\n_Actualidad Espa\u00f1ola_ ref1, ref2, ref3\n\nAdriaenssens, Jean ref1\n\nAimar, Lucien ref1\n\nAlc\u00e1zar fortress, Toledo ref1, ref2\n\nAlomar, Francisco ref1, ref2, ref3\n\nAltig, Rudy ref1, ref2\n\nAmateur Championships, Spain:\n\n1948 ref1, ref2\n\n1949 ref1\n\n1950 ref1\n\nAndalusia, Tour of ref1, ref2\n\nAnglade, Antonio ref1\n\nAnglade, Henry ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18\n\nAnquetil, Jacques ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33, ref34, ref35, ref36, ref37, ref38, ref39, ref40, ref41, ref42, ref43, ref44, ref45, ref46, ref47, ref48, ref49, ref50, ref51, ref52, ref53, ref54, ref55, ref56\n\nArrate hill-climb ref1, ref2\n\n1960 ref1\n\n1962 ref1\n\n1965 ref1\n\n1966 ref1\n\n1970 ref1\n\n_AS_ ref1\n\nAsturias, Tour of:\n\n1953 ref1, ref2, ref3\n\n1954 ref1, ref2\n\n1955 ref1\n\n1957 ref1\n\n\u00c1vila, Tour of:\n\n1948 ref1, ref2\n\n1950 ref1\n\n1951 ref1\n\nAza\u00f1a, Manuel ref1\n\nBahamontes, Federico:\n\n1953 season ref1\n\n1954 season ref1\n\n1955 season ref1\n\n1956 season ref1\n\n1957 season ref1\n\n1958 season ref1\n\n1959 season ref1\n\n1960 season ref1\n\n1961, 1962 seasons ref1, ref2\n\n1963 season ref1\n\n1964 season ref1\n\n1965 season ref1\n\namateur racing ref1\n\nattacking style ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33, ref34, ref35, ref36, ref37\n\nattacks a spectator ref1, ref2\n\nattacks other cyclists ref1\n\nbike shop ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9\n\nbiography ref1, ref2\n\nbirth ref1, ref2\n\ncelebrity circuit ref1\n\ncheating ref1, ref2, ref3, ref4\n\nchildhood ref1\n\nclimber ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29\n\ncrashes ref1, ref2, ref3, ref4, ref5, ref6\n\ndescents ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13\n\ndoping ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\neccentric nature ref1, ref2, ref3, ref4, ref5, ref6\n\nego, fragile ref1, ref2\n\nerratic nature ref1, ref2, ref3, ref4\n\nfame ref1, ref2, ref3, ref4, ref5\n\nfans ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33\n\nfinances ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33, ref34, ref35, ref36, ref37, ref38, ref39, ref40, ref41, ref42, ref43, ref44, ref45, ref46\n\nfirst bike ref1\n\nfirst professional race ref1\n\nfirst race ref1\n\ngypsy ref1\n\nhobo image ref1\n\nice cream in Tour de France, eats ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\nignores greater prizes for King of the Mountains title ref1\n\nillness ref1, ref2, ref3, ref4, ref5\n\ninjections ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\ninjuries ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18\n\nKing of the Mountains ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33, ref34, ref35, ref36, ref37, ref38, ref39, ref40, ref41, ref42, ref43, ref44, ref45, ref46, ref47, ref48, ref49, ref50, ref51, ref52, ref53, ref54, ref55, ref56, ref57, ref58, ref59, ref60, ref61, ref62, ref63, ref64, ref65, ref66, ref67, ref68\n\nlife after retirement ref1\n\nloner ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20\n\nmadness ref1, ref2, ref3, ref4, ref5, ref6\n\nmanic energy ref1, ref2, ref3\n\nmarries ref1\n\nnames ref1, ref2\n\nNational Service ref1\n\nnicknames ref1, ref2\n\nposition on bike ref1, ref2, ref3\n\nrecord ref1, ref2\n\nretires ref1, ref2, ref3, ref4\n\nrivalries _see under individual rival name_\n\nsells races ref1, ref2\n\nteam director ref1\n\ntemperament ref1, ref2, ref3, ref4, ref5, ref6\n\nToledo cycling club and ref1\n\nweather and ref1, ref2, ref3, ref4, ref5, ref6, ref7\n\nyounger riders and ref1, ref2, ref3 _see also under individual rider name_\n\n_see also under individual event name_\n\nBahamontes, Fermina (wife\/girlfriend) ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12\n\nBahamontes, Juli\u00e1n (father) ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\nBahamontes, Victoria (mother) ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10\n\nBalanzas Berkel team ref1, ref2, ref3, ref4\n\nBaldini, Ercole ref1, ref2, ref3, ref4\n\nBaron de Guell Cup ref1\n\nBartali, Gino ref1\n\nBasque Country ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18\n\nBasque separatist movement ref1, ref2, ref3\n\nBauvin, Gilbert ref1, ref2, ref3\n\nBergareche, Luis ref1, ref2, ref3, ref4, ref5, ref6\n\nBern\u00e1rdez, Jos\u00e9 ref1\n\nBerrendero, Herrero ref1, ref2\n\nBerrendero, Juli\u00e1n ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9\n\nBicicleta Eibarresa:\n\n1954 ref1\n\n1955 ref1\n\n1956 ref1\n\nBicicleta Vasca ref1\n\n1957 ref1\n\nBidot, Marcel ref1, ref2, ref3, ref4, ref5, ref6, ref7\n\nBlanco, Antonio ref1\n\nBlondin, Antonie ref1\n\nBobet, Louison ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17\n\nBodegas, Javier ref1, ref2\n\nBotella, Salvador ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9\n\nBouvet, Philippe ref1, ref2\n\nBover, Miguel ref1\n\nBrenan, Gerald ref1, ref2, ref3\n\nBrown, Ian ref1, ref2, ref3, ref4, ref5\n\nBrunel, Philippe ref1\n\nCabot, Joaquim ref1\n\nCadiz, Tour of ref1\n\n1951 ref1\n\nCampillo, Juan ref1, ref2, ref3, ref4, ref5, ref6, ref7\n\nCampo Simplex ref1, ref2, ref3, ref4\n\nCatalano, Nino ref1\n\nCatalonia ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9\n\nCavendish, Mark ref1, ref2\n\nCazala, Robert ref1, ref2, ref3\n\nCentre-Midi team ref1, ref2\n\nChavanel, Sylvain ref1\n\nCircuit du Proven\u00e7al, 1965 ref1\n\nCivil War, Spanish (1936\u201339) ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22\n\nContador, Alberto ref1, ref2, ref3, ref4\n\nConterno, Angelo ref1, ref2\n\nCoppi, Fausto ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21\n\nCorrales, Mariano ref1, ref2\n\n_Cycle Sport_ ref1, ref2\n\n_Cycling Weekly_ ref1\n\nDaems, Emile ref1\n\nDarrigade, Andr\u00e9 ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18\n\nDaud\u00e9, Jacques ref1, ref2, ref3, ref4\n\nDauphin\u00e9 Lib\u00e9r\u00e9:\n\n1962 ref1\n\n1963 ref1\n\nDe Mulder, Frans ref1\n\nDeCabooter, Arthur ref1, ref2\n\nDefraye, Odiel ref1\n\nDel Caz, Alejandro ref1, ref2, ref3, ref4, ref5, ref6\n\nDelgado, Pedro ref1, ref2, ref3, ref4, ref5, ref6\n\nDesmet, Armand ref1, ref2\n\nDesmet, Gilbert ref1\n\ndomestiques ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13\n\nDotto, Jean ref1, ref2, ref3, ref4\n\nDousset, Daniel ref1, ref2, ref3, ref4, ref5, ref6, ref7\n\nDumas, Pierre ref1, ref2, ref3, ref4, ref5, ref6, ref7\n\nDunan, Rene ref1\n\nE.T.A. ref1, ref2\n\nEibar, GP ref1\n\n_El Alc_ \u00e1 _zar_ ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11\n\n_El Correo Catalanref1_\n\n_El Correo Espa\u00f1ol_ ref1\n\n_El Diario Montan\u00e9s_ ref1\n\n_El Movimiento Nacional_ ( _FET de la JONS_ ) ref1, ref2\n\n_El Mundo Deportivo_ ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21\n\n_El Noticiero Universal_ ref1\n\n_El Pa\u00eds_ ref1, ref2, ref3\n\nElola, Jos\u00e9 Antonio ref1, ref2, ref3, ref4\n\nEmeterio, Julio San ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\nEspin, Manuel ref1\n\nEzquerra, Federico ref1\n\nFaema ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20\n\nFalange ref1, ref2\n\nFalaschi, Roberto ref1\n\n_Fallen Angel_ (Fotheringham) ref1\n\nFantini, Alessandro ref1\n\nfascism ref1, ref2, ref3, ref4, ref5, ref6, ref7\n\nFerraz, Antonio ref1\n\nFerrys team ref1\n\nFilippis, Ni\u00f1o de ref1, ref2\n\nFlecha, Juan Antonio ref1\n\nFranc\u00e9s, Jos\u00e9 P\u00e9rez ref1, ref2, ref3, ref4, ref5\n\nFranco, General ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33, ref34, ref35, ref36, ref37, ref38\n\nFreire, Oscar ref1\n\nGabica, Francisco ref1\n\nGabica, Patxi ref1\n\nGadea, Eduardo ref1\n\nGaldeano, Jes\u00fas ref1, ref2\n\nGalera, Joaquim ref1\n\nGarai, Josu ref1\n\nGarin, Maurice ref1\n\nGaul, Charly ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33, ref34, ref35, ref36, ref37, ref38, ref39, ref40, ref41, ref42, ref43, ref44, ref45, ref46, ref47, ref48, ref49\n\nGelabert, Antonio ref1, ref2\n\nGeminiani, Rafael ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23\n\nGirardengo-ICEP team ref1, ref2\n\nGiro d'Italia ref1, ref2, ref3\n\n1953 ref1\n\n1956 ref1\n\n1957 ref1\n\n1958 ref1, ref2\n\n1961 ref1, ref2\n\n1970 ref1\n\nGismondi, Michele ref1\n\nGoddet, Jacques ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\nGraf, Rolf ref1, ref2\n\nGrazyck, Jean ref1, ref2, ref3, ref4\n\nGrosso, Giancarlo ref1\n\nGroussard, Georges ref1, ref2, ref3, ref4, ref5\n\nHerria, Euskal ref1\n\nHoevenaers, Jos ref1, ref2\n\n'Huesca Pact, the' ref1\n\nHuot, Francois ref1\n\nInda, Eduardo ref1\n\nIndurain, Miguel ref1, ref2, ref3, ref4, ref5\n\n_Informaci\u00f3n_ ref1\n\n_Informaciones_ ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\nInternational Brigades ref1, ref2\n\nItaly, tour of, 1958 ref1\n\nJanssens, Marcel ref1, ref2, ref3\n\nJim\u00e9nez, Julio ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12\n\nK.A.S. ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\nKohl, Bernard ref1\n\n_L'Equipe_ ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17\n\nLa Casera team ref1, ref2\n\n_La Gaceta del Norte_ ref1, ref2, ref3, ref4\n\n_La Regi\u00f3n_ ref1\n\nLa Solera team ref1, ref2, ref3\n\nLangarica, Dalmacio ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28\n\nLasa, Miguel Mari ref1\n\nLeducq, Andr\u00e9 ref1\n\nLevante, Tour of:\n\n1957 ref1\n\n1960 ref1\n\nLicor 43 team ref1, ref2\n\nLinton, Arthur ref1\n\nLobo, Margarita ref1, ref2\n\nLondres, Albert ref1\n\nL\u00f3pez, Basilio ref1, ref2\n\nL\u00f3pez, German ref1\n\nLoro\u00f1o, Jes\u00fas ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33, ref34, ref35, ref36, ref37, ref38, ref39\n\nLoro\u00f1o, Josu ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\nMadrid, Tour of, 1951 ref1\n\nMagne, Antonin ref1\n\nMagni, Fiorenzo ref1, ref2\n\nMahe, Francois ref1\n\nMalaga, Tour of, 1953 ref1\n\nMallejac, Jean ref1, ref2\n\nManzaneque, Fernando ref1, ref2, ref3, ref4, ref5\n\nManzaneque, Jes\u00fas ref1\n\n_MARCA_ ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22\n\nMargnat team ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17\n\nMarigil, Rene ref1, ref2, ref3, ref4, ref5\n\nMasip, Francisco ref1, ref2\n\nMaso, Benjo ref1\n\nMassignan, Imerio ref1\n\nMattio, Claude ref1\n\nMerckx, Eddy ref1, ref2, ref3, ref4, ref5, ref6\n\nMidi Libre, GP:\n\n1957 ref1\n\n1963 ref1\n\nMilan-San Remo ref1\n\n1957 ref1\n\nMillar, Robert ref1, ref2, ref3, ref4, ref5\n\n_Miroir-Sprint_ ref1\n\nMobylette ref1, ref2, ref3\n\nMoncouti\u00e9, David ref1\n\nMont Agel hill-climb:\n\n1954 ref1\n\n1960 ref1\n\nMonte Carlo, GP ref1\n\nMoore, Richard ref1\n\nMoral, Jos\u00e9 Gomez del ref1, ref2\n\nMorales, Carmelo ref1, ref2, ref3, ref4\n\nMoreno, Francisco ref1\n\nMoreno, Miguel ref1\n\nMoscard\u00f3, General ref1, ref2\n\nMostajo, Santiago ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\nMont Faron hill-climb:\n\n1955 ref1\n\n1957 ref1\n\n1962 ref1\n\nMurtra, Evaristo ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\nNicolas, Luis L\u00f3pez ref1, ref2\n\nNolten, Jan ref1\n\n_noticias-docummentales_ ( _no-dos_ ) ref1, ref2, ref3, ref4\n\nOchoa, Severo ref1\n\nOta\u00f1o, Luis ref1, ref2, ref3, ref4, ref5, ref6\n\nPacheco, Miguel ref1\n\nPauwels, Eddy ref1, ref2\n\nPeiper, Allan ref1\n\nPelforth team ref1, ref2, ref3\n\nPelissier, Charles ref1, ref2, ref3, ref4\n\nPelissier, Francis ref1\n\nPelissier, Henri ref1\n\nPensec, Ronan ref1\n\nPeugeot ref1\n\nPiel, Roger ref1, ref2\n\nPlanckaert, Josef 'Jef' ref1, ref2\n\nPoblet, Miguel ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16\n\n'Poigny pact' ref1, ref2, ref3\n\nPoulidor, Raymond ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26\n\nPriego de Cordoba, GP ref1\n\nPrivat, Rene ref1\n\nPuig, Luis ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22\n\nPyrenees, Tour des, 1960 ref1\n\nQuiles, Antonio Jim\u00e9nez ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10\n\nQuiles, L\u00f3pez ref1\n\nRamsbottom, Alan ref1\n\nRemy, Raoul ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9\n\nRivera, Jos\u00e9 Antonio Primo de ref1, ref2\n\nRivi\u00e8re, Roger ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25\n\nRobic, Jean ref1, ref2, ref3, ref4, ref5\n\nRobinson, Brian ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13\n\nRoche, Stephen ref1\n\nRodr\u00edguez, Manolo ref1\n\nRodr\u00edguez, Pablo ref1\n\nRomandie, Tour of, 1962 ref1\n\nRuiz, Bernardo ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33\n\nSaint, Gerard ref1, ref2\n\nSalamanca, Isidro L\u00f3pez de ref1\n\nSamaranch, Juan Antonio ref1\n\nSastre, Carlos ref1, ref2, ref3, ref4\n\nSaura, Gabriel ref1\n\nSchleck, Andy ref1\n\nSegu, Jos\u00e9 ref1\n\nSerra, Jos\u00e9 ref1, ref2\n\nSimon, Pascal ref1\n\nSimpson, Tom ref1, ref2, ref3, ref4, ref5, ref6, ref7\n\nSix Days of Madrid, 1965 ref1\n\nSoler, Angelino ref1, ref2\n\nSoria, Ladislau ref1, ref2, ref3\n\nSotelo, Calvo ref1\n\nSpain:\n\nCivil War (1936) ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19\n\neconomy\/poverty ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19\n\nfascism ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\nrole of sport in\/use of Bahamontes victories ref1, ref2, ref3, ref4, ref5\n\ntourism ref1\n\nSpanish Cycling Federation ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28\n\nSpanish Fascist Party ref1, ref2, ref3, ref4\n\nSpanish National Championships ref1, ref2, ref3\n\n1955 ref1\n\n1956 ref1\n\n1958 ref1, ref2\n\n_Spanish National Championships_ (Bodegas\/Dorronsoro) ref1\n\nSplendid team ref1, ref2\n\nSt. Raphael-Geminiani team ref1, ref2\n\nStablinski, Jean ref1\n\nSteel, Ian ref1\n\nSu\u00e1rez, Antonio ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18\n\nSu\u00e1rez, Faustino ref1\n\nSubida a Naranco hill-climb, ref1, ref2\n\n1965 ref1\n\nSubida de Montjuic, 1965 ref1\n\nSuisse, Tour de, 1958 ref1\n\nTimoner, Guillermo ref1\n\nTognacini, Bruno ref1\n\nToledo ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33, ref34\n\nTorell\u00f3, Miguel ref1, ref2, ref3, ref4\n\nTorres, Pedro ref1\n\nTour de France:\n\n1903 ref1\n\n1930 ref1\n\n1931 ref1\n\n1933 ref1, ref2\n\n1936 ref1\n\n1937 ref1\n\n1949 ref1, ref2, ref3, ref4\n\n1952 ref1, ref2, ref3, ref4, ref5\n\n1953 ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\n1954 ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9\n\n1955 ref1, ref2, ref3, ref4\n\n1956 ref1, ref2, ref3, ref4\n\n1957 ref1, ref2, ref3, ref4, ref5, ref6\n\n1958 ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13\n\n1959 ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22, ref23, ref24, ref25, ref26, ref27, ref28, ref29, ref30, ref31, ref32, ref33, ref34, ref35, ref36, ref37, ref38\n\n1960 ref1, ref2, ref3\n\n1961 ref1\n\n1962 ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10\n\n1963 ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\n1964 ref1, ref2, ref3, ref4\n\n1965 ref1, ref2\n\n1967 ref1, ref2\n\n1973 ref1, ref2\n\n1983 ref1\n\n1988 ref1, ref2\n\n1992 ref1\n\n2008 ref1\n\n2011 ref1\n\npost-Tour criteriums ref1, ref2, ref3\n\nTrepp, Willy ref1\n\nTricolfilina ref1, ref2, ref3, ref4\n\nTrueba, Vicente ref1, ref2, ref3\n\nU.C.I. ref1, ref2, ref3, ref4, ref5, ref6, ref7\n\nUbieta, Francisco G. ref1\n\nUrdiales, Castro ref1\n\nUriona, Valentin ref1\n\nUtset, Aniceto ref1\n\nV.O.V. ref1, ref2, ref3\n\nValladares, Manuel Portela ref1\n\nVan Genechten, Richard ref1\n\nVan Impe, Lucien ref1, ref2\n\nVan Looy, Rik ref1, ref2, ref3, ref4, ref5\n\nVarnajo, Robert ref1\n\nVermeulin, Michel ref1, ref2\n\nVirenque, Richard ref1\n\n_Viva la Vuelta!_ (Bell\/Fallon) ref1, ref2\n\nVolta a Catalunya ref1, ref2, ref3, ref4\n\n1945 ref1\n\n1950 ref1\n\n1953 ref1, ref2, ref3, ref4, ref5, ref6\n\n1954 ref1\n\n1955 ref1, ref2, ref3\n\n1957 ref1\n\n1965 ref1\n\n_Vuelta 1935-85, The_ (P\u00e9rez) ref1, ref2\n\nVuelta a Espa\u00f1a ref1, ref2, ref3, ref4, ref5\n\n1946 ref1\n\n1948 ref1, ref2\n\n1950 ref1, ref2\n\n1953 ref1\n\n1955 ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8\n\n1956 ref1, ref2\n\n1957 ref1, ref2, ref3, ref4\n\n1958 ref1, ref2\n\n1959 ref1, ref2\n\n1960 ref1, ref2, ref3\n\n1962 ref1\n\n1964 ref1\n\n1965 ref1, ref2\n\n1978 ref1\n\n2011 ref1, ref2\n\nVuelta a Galicia, 1955 ref1\n\nVuelta a los Puertos:\n\n1953 ref1, ref2\n\n1955 ref1\n\nVuelta a Toledo ref1, ref2, ref3, ref4\n\nWalkowiak, Roger ref1, ref2, ref3, ref4, ref5\n\nWegelius, Charly ref1\n\nWolfshohl, Rolf ref1\n\nWorld Championships ref1\n\n1954 ref1\n\n1955 ref1\n\n1956 ref1\n\n1958 ref1\n\n1959 ref1\n\n1961 ref1\n\n1965 ref1, ref2\n\nYates, Sean ref1\n\nZampini, Donato ref1, ref2, ref3, ref4\n\n#### Copyright\n\nFirst published in 2012  \nby Aurum Press Ltd, 7 Greenland Street, London NW1 0ND\n\nThis eBook edition first published in 2012\n\nAll rights reserved  \n\u00a9 Alasdair Fotheringham, 2012\n\nThe right of Alasdair Fotheringham to be identified as author of this work has been asserted in accordance with Section 77 of the Copyright, Designs and Patents Act 1988\n\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 publishers' rights, and those responsible may be liable in law accordingly.\n\neBook conversion by CPI Group\n\nISBN 978\u20131\u201378131\u2013037-3\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n### Sobre _La Era de Huesos_\n\n\u00abUna oscura, aguerrida y maravillosa novela de fantas\u00eda.\u00bb\n\n__\n\n_The_ _Observer_\n\n\u00ab _La Era de Huesos_ nos transporta a un submundo del futuro, donde la clarividencia existe. Un mundo de unas dimensiones impresionantes, repleto de fantas\u00eda, distop\u00eda e intriga.\u00bb __\n\n__\n\n_Vogue_\n\n\u00abEl ritmo de _La Era de Huesos_ raramente se ralentiza, y la fuerte e ingeniosa Paige resulta una hero\u00edna memorable.\u00bb __\n\n_BookPage_\n\n__\n\n\u00abMe atrap\u00f3 irremediablemente... Samantha Shannon es una autora joven con un gran futuro por delante, de todo menos dist\u00f3pico.\u00bb __\n\n_Stylist_\n\n__\n\n\u00abUna fantas\u00eda oscura y exquisitamente representada. Est\u00e1 descrita de una forma \u00fanica. _La Era de Huesos_ es una lectura obligada.\u00bb\n\nKAMI GARC\u00cdA, autora de _Hermosas criaturas_\n\n__\n\n\u00abSamantha Shannon tiene un talento enorme e ingenioso, y una imaginaci\u00f3n desbordante. Ha empezado a trabajar a todo gas.\u00bb\n\nSUSAN HILL, autora de _La mujer de negro_\n\n\u00abShannon escribe tan bien que consigue mantener el inter\u00e9s desde el principio... Un mundo de ficci\u00f3n original y entretenido... La verdad, el final me tuvo absolutamente en vilo.\u00bb\n\n__\n\n_The Telegraph_\n\n\u00abMarca la llegada de una escritora brit\u00e1nica extraordinariamente talentosa, preparada para desafiar el monopolio de las listas de m\u00e1s vendidos internacionales de las series _Crep\u00fasculo_ de Stephenie Meyer y _Los juegos del hambre_ de Suzanne Collins.\u00bb\n\n_The British Fantasy Society_\n\n\u00abNo encuentro palabras para describir la maravillosa y confiada escritura de Samantha Shannon, sobre todo en un debut literario... La historia que m\u00e1s me ha atrapado en el \u00faltimo a\u00f1o, y la primera novela m\u00e1s absorbente y convincente que he le\u00eddo desde _Jonathan Strange y el se\u00f1or Norrell._ La recomiendo fervientemente.\u00bb\n\n__\n\n_Forbidden Planet International Blog_\n\n__\n\n\u00abEn _La Era de Huesos_ encontramos los adolescentes y la distop\u00eda de _Los juegos del hambre_ , sumados a una historia de amor al estilo de _Crep\u00fasculo_. Eso s\u00ed, con una mayor calidad literaria.\u00bb\n\n_SFX Magazine_\n\n__\n\n\u00abCon ecos de _Los juegos del hambre_ y de la trilog\u00eda _Cr\u00f3nicas del_ _mago negro_ de Trudi Canavan, este es un excelente debut literario que atrapar\u00e1 al lector de principio a fin, y lo dejar\u00e1 con ganas de m\u00e1s.\u00bb\n\n_Irish Examiner_\n\n__\n\n\u00ab _La Era de Huesos_ surge del inter\u00e9s de Shannon por obras como _El cuento de la criada_ y _La naranja mec\u00e1nica_ , pero tambi\u00e9n de su fascinaci\u00f3n por la poes\u00eda de John Donne.\u00bb\n\n_The_ _Sunday Times_\n\n_Para los so\u00f1adores_\n\nPor encima de esta tierra y de la raza humana que la puebla, existe un reino de esp\u00edritus, un mundo invisible, pero que nos rodea y est\u00e1 en todas partes.\n\nCHARLOTTE BRONT\u00cb, _Jane Eyre_\n\nMe gusta pensar que al principio \u00e9ramos m\u00e1s. No muchos, supongo. Pero s\u00ed m\u00e1s que ahora.\n\nSomos la minor\u00eda que el mundo no acepta. No nos acepta fuera del \u00e1mbito de la fantas\u00eda, que est\u00e1 en la lista negra. Somos como los dem\u00e1s. A veces actuamos como los dem\u00e1s. En muchos aspectos, somos como otro cualquiera. Estamos por todas partes, en cualquier calle. Llevamos una vida que podr\u00edais considerar normal, siempre que no os fijarais demasiado.\n\nNo todos nosotros sabemos lo que somos. Algunos mueren sin llegar a saberlo. Algunos lo sabemos, y nunca nos descubren. Pero estamos aqu\u00ed.\n\nCreedme.\n\nDesde los ocho a\u00f1os, hab\u00eda vivido en esa parte de Londres que se llamaba Islington. Iba a un colegio privado para chicas, y a los diecis\u00e9is me puse a trabajar. Eso fue en el a\u00f1o 2056. AS 127, seg\u00fan el calendario de Scion. Se esperaba de los j\u00f3venes que empezaran a ganarse la vida donde pudieran, y normalmente era detr\u00e1s de alg\u00fan tipo de mostrador. Hab\u00eda mucha oferta de empleo en el sector servicios. Mi padre cre\u00eda que yo llevar\u00eda una vida sencilla; que era inteligente pero poco ambiciosa, que me contentar\u00eda con cualquier trabajo que la vida me ofreciera.\n\nMi padre se equivocaba, como siempre.\n\nDesde los diecis\u00e9is a\u00f1os hab\u00eda trabajado en el mundo del hampa de Scion Londres (SciLo, como lo llam\u00e1bamos en las calles). Trataba con implacables bandas de videntes, todas dispuestas a derribarse unas a otras para sobrevivir. Esas bandas formaban parte de un sindicato que abarcaba la ciudadela entera, dirigido por el Subse\u00f1or. Empujados hacia los bordes de la sociedad, nos ve\u00edamos obligados a delinquir para prosperar. Y por eso nos odiaban a\u00fan m\u00e1s. Hac\u00edamos reales las historias que contaban de nosotros.\n\nYo ten\u00eda mi sitio en aquel caos. Era una dama, la protegida de un mimetocapo. Mi jefe, Jaxon Hall, era el mimetocapo responsable del sector I-4. \u00c9ramos seis los que trabaj\u00e1bamos directamente para \u00e9l. Nos llam\u00e1bamos los Siete Sellos.\n\nNo pod\u00eda cont\u00e1rselo a mi padre. \u00c9l cre\u00eda que trabajaba de dependienta en un bar de ox\u00edgeno, un empleo mal pagado pero legal. Era una mentira f\u00e1cil. Si hubiera tratado de explicarle por qu\u00e9 me pasaba el d\u00eda con delincuentes, no lo habr\u00eda entendido. Mi padre no sab\u00eda que yo me parec\u00eda m\u00e1s a ellos que a \u00e9l.\n\nTen\u00eda diecinueve a\u00f1os el d\u00eda que mi vida cambi\u00f3. Por entonces mi nombre ya sonaba en las calles. Tras una semana especialmente dura en el mercado negro, ten\u00eda previsto pasar el fin de semana con mi padre. Jax no entend\u00eda por qu\u00e9 necesitaba un poco de tiempo libre (para \u00e9l, no hab\u00eda nada digno de nosotros fuera del sindicato), pero \u00e9l no ten\u00eda una familia, y yo s\u00ed. O no ten\u00eda una familia viva. Y a pesar de que mi padre y yo nunca hab\u00edamos estado muy unidos, sent\u00eda que no deb\u00edamos perder el contacto. Una cena de vez en cuando, alguna que otra llamada de tel\u00e9fono, un regalo por Novembertide. El \u00fanico problema era su lista interminable de preguntas. \u00bfD\u00f3nde trabajaba? \u00bfQui\u00e9nes eran mis amigos? \u00bfD\u00f3nde viv\u00eda? Yo no pod\u00eda contestar. La verdad era peligrosa. Si se hubiera enterado de a qu\u00e9 me dedicaba, es posible que \u00e9l mismo me hubiera mandado a la colina de la Torre. Quiz\u00e1 deber\u00eda haberle contado la verdad. Quiz\u00e1 eso lo habr\u00eda matado. Fuera como fuese, no me arrepent\u00eda de haber entrado en el sindicato. Mi trabajo era deshonesto, pero estaba bien pagado. Y como siempre dec\u00eda Jax, era mejor ser un forajido que un fiambre.\n\nEse d\u00eda llov\u00eda. El \u00faltimo d\u00eda que fui a trabajar.\n\nUn equipo de soporte vital manten\u00eda mis constantes. Parec\u00eda muerta, y en cierto modo lo estaba: mi esp\u00edritu se hab\u00eda separado parcialmente de mi cuerpo. Era un delito por el que habr\u00edan podido condenarme a la horca.\n\nHe dicho que trabajaba en el sindicato. Dejadme que lo aclare: era una especie de hacker mental. M\u00e1s que leer otras mentes, era una especie de radar de mentes, en sinton\u00eda con lo que pasaba en el \u00e9ter. Percib\u00eda los matices de los onirosajes, y la presencia de esp\u00edritus solitarios. Cosas que estaban fuera de m\u00ed. Cosas que los videntes normales no pod\u00edan percibir.\n\nJax me utilizaba como herramienta de vigilancia. Mi trabajo consist\u00eda en seguir la pista de cualquier actividad et\u00e9rea en su secci\u00f3n. A menudo me hac\u00eda vigilar a otros videntes, para averiguar si ocultaban algo. Al principio, solo me ped\u00eda que observara a personas que estaban en la misma habitaci\u00f3n (personas a las que yo pod\u00eda ver, o\u00edr y tocar), pero pronto se dio cuenta de que yo pod\u00eda ir m\u00e1s all\u00e1. Pod\u00eda percibir cosas que suced\u00edan en otro sitio: un vidente que pasaba por la calle, una reuni\u00f3n de esp\u00edritus en Covent Garden. Mientras tuviera soporte vital, pod\u00eda captar el \u00e9ter en un radio de dos kil\u00f3metros alrededor de Seven Dials. As\u00ed que si Jaxon necesitaba que alguien cotilleara lo que estaba pasando en el I-4, pod\u00edas apostar cualquier cosa a que me llamar\u00eda a m\u00ed. Dec\u00eda que yo ten\u00eda potencial para ir a\u00fan m\u00e1s lejos, pero Nick no quer\u00eda que lo intentara. No sab\u00edamos qu\u00e9 consecuencias podr\u00eda acarrearme.\n\nToda forma de clarividencia estaba prohibida, por supuesto, pero aquella con la que se pod\u00eda ganar dinero era directamente pecado. Ten\u00edan un t\u00e9rmino especial para designarlo: mimetodelincuencia. Comunicaci\u00f3n con el mundo de los esp\u00edritus, con la intenci\u00f3n expresa de obtener beneficios econ\u00f3micos. El sindicato se basaba en la mimetodelincuencia.\n\nLa clarividencia pagada en efectivo estaba muy extendida entre quienes no lograban entrar en ninguna banda. Nosotros lo llam\u00e1bamos limosnear; Scion lo llamaba traici\u00f3n. El m\u00e9todo oficial de ejecuci\u00f3n de quienes comet\u00edan esos delitos era la asfixia por nitr\u00f3geno, comercializada bajo la marca NiteKind. Todav\u00eda recuerdo los titulares: \u00abCastigo sin dolor: el \u00faltimo milagro de Scion\u00bb. Dec\u00edan que era como quedarse dormido, como tomarse una pastilla. Todav\u00eda hab\u00eda ejecuciones p\u00fablicas en la horca, y alg\u00fan que otro caso de tortura por alta traici\u00f3n.\n\nYo comet\u00eda alta traici\u00f3n por el simple hecho de respirar.\n\nPero volvamos a ese d\u00eda. Jaxon me hab\u00eda conectado al equipo de soporte vital y me hab\u00eda enviado a hacer un reconocimiento del sector. Yo llevaba tiempo cercando una mente que rondaba por all\u00ed, un visitante frecuente del sector 4. Hab\u00eda hecho todo lo posible para ver sus recuerdos, pero siempre hab\u00eda sucedido algo que me lo hab\u00eda impedido. Aquel onirosaje no se parec\u00eda a nada que yo hubiera visto hasta aquel momento. Incluso Jax estaba perplejo. Por las diferentes capas de mecanismos de defensa, habr\u00eda jurado que su due\u00f1o ten\u00eda miles de a\u00f1os de edad, pero no pod\u00eda ser eso. Era algo diferente.\n\nJax era muy desconfiado. Lo que correspond\u00eda en esos casos era que si un nuevo clarividente llegaba a su sector se anunciara a \u00e9l en un plazo de cuarenta y ocho horas. Jax dec\u00eda que deb\u00eda de haber otra banda implicada, pero ninguna de las del I-4 ten\u00eda experiencia suficiente para obstaculizar mis reconocimientos. Ninguna sab\u00eda lo que yo pod\u00eda hacer. No era Didion Waite, que dirig\u00eda la segunda banda m\u00e1s grande de la zona. No eran los limosneros muertos de hambre que frecuentaban Dials. No eran los mimetocapos territoriales especializados en hurto et\u00e9reo. Aquello era otra cosa.\n\nCientos de mentes pasaban a mi lado lanzando destellos plateados en la oscuridad. Iban deprisa por las calles, como sus due\u00f1os. Yo no reconoc\u00eda a esas personas. No pod\u00eda ver sus caras; solo vislumbraba los bordes de sus mentes.\n\nHab\u00eda salido de Dials. Mi percepci\u00f3n estaba m\u00e1s al norte, aunque no pod\u00eda precisar d\u00f3nde. Segu\u00ed aquella sensaci\u00f3n de peligro que tan bien conoc\u00eda. La mente del desconocido estaba cerca. Me llev\u00f3 por el \u00e9ter correteando como una luci\u00e9rnaga, sorteando otras mentes. Se mov\u00eda deprisa, como si me hubiera notado. Como si intentara huir.\n\nNo deb\u00eda perseguir esa luz. No sab\u00eda ad\u00f3nde me llevar\u00eda, y ya me hab\u00eda alejado demasiado de Seven Dials.\n\n\u00abJaxon te ha dicho que lo encuentres.\u00bb Era un pensamiento lejano. \u00abSe va a enfadar.\u00bb Segu\u00ed adelante, a una velocidad mucho mayor de la que jam\u00e1s hab\u00eda alcanzado con el cuerpo. Luch\u00e9 contra las limitaciones de mi f\u00edsico. Ya pod\u00eda distinguir la mente de aquel solitario. No era plateada, como las otras; no, la suya era oscura y fr\u00eda, una mente de hielo y piedra. Corr\u00ed hacia ella. Estaba tan cerca... No pod\u00eda dejarla escapar.\n\nEntonces el \u00e9ter tembl\u00f3 a mi alrededor y, de repente, el intruso desapareci\u00f3. Su mente volv\u00eda a estar fuera de mi alcance.\n\nAlguien me zarande\u00f3.\n\nMi cord\u00f3n arg\u00e9nteo (la conexi\u00f3n entre mi cuerpo y mi esp\u00edritu) era extremadamente sensible. Era lo que me permit\u00eda percibir onirosajes a distancia. Tambi\u00e9n pod\u00eda devolverme a mi piel. Cuando abr\u00ed los ojos, Dani estaba ilumin\u00e1ndome la cara con una linterna de bolsillo. \u00abLas pupilas reaccionan \u2014dijo para s\u00ed\u2014. Estupendo.\u00bb\n\nDanica. El genio del grupo, con una inteligencia que solo Jax superaba. Era tres a\u00f1os mayor que yo y ten\u00eda todo el encanto y la sensibilidad de un golpe a traici\u00f3n. Cuando la contrataron, Nick la clasific\u00f3 como soci\u00f3pata. Jax dijo que era parte de su personalidad.\n\n\u2014Despierta, So\u00f1adora. \u2014Me dio un cachete\u2014. Bienvenida al mundo de la carne.\n\nLa bofetada me doli\u00f3: era buena se\u00f1al, aunque desagradable. Levant\u00e9 una mano para quitarme la mascarilla de ox\u00edgeno.\n\nLa guarida fue cobrando definici\u00f3n. La vivienda de Jax era un almac\u00e9n secreto de contrabando, lleno de pel\u00edculas, m\u00fasica y libros prohibidos, todo amontonado en estantes y recubierto por una gruesa capa de polvo. Hab\u00eda una colecci\u00f3n de fasc\u00edculos de terror, de esos que pod\u00edas conseguir en Covent Garden los fines de semana, y un mont\u00f3n de panfletos grapados. Era el \u00fanico sitio del mundo donde yo pod\u00eda leer, ver y hacer lo que se me antojara.\n\n\u2014No deber\u00edas despertarme as\u00ed \u2014dije. Dani conoc\u00eda las normas\u2014. \u00bfCu\u00e1nto rato he estado all\u00ed?\n\n\u2014\u00bfD\u00f3nde?\n\n\u2014\u00bfD\u00f3nde crees?\n\nDani chasc\u00f3 los dedos.\n\n\u2014Ya, claro. En el \u00e9ter. Perdona. No lo he contado.\n\nImprobable: siempre me cronometraba.\n\nMir\u00e9 el reloj Nixie azul del equipo. Lo hab\u00eda fabricado Dani, y lo llamaba Sistema de Auxilio Mortal, o SAM. Monitorizaba y controlaba mis funciones vitales cuando percib\u00eda el \u00e9ter a distancia. Vi las cifras y me dio un vuelco el coraz\u00f3n.\n\n\u2014Cincuenta y siete minutos. \u2014Me frot\u00e9 las sienes\u2014. \u00bfMe has tenido una hora en el \u00e9ter?\n\n\u2014Puede ser.\n\n\u2014\u00bfUna hora entera?\n\n\u2014Las \u00f3rdenes son las \u00f3rdenes. Jax dijo que quer\u00eda que descifraras esa mente misteriosa antes del anochecer. \u00bfHas podido?\n\n\u2014Lo he intentado.\n\n\u2014Eso quiere decir que no has podido. Te has quedado sin bonificaci\u00f3n. \u2014Se bebi\u00f3 el caf\u00e9 de un trago\u2014. Todav\u00eda no puedo creer que perdieras a Anne Naylor.\n\nEra inevitable que sacara aquello a colaci\u00f3n. Unos d\u00edas antes me hab\u00edan enviado a la sala de subastas a reclamar un esp\u00edritu que le correspond\u00eda a Jax: Anne Naylor, la famosa fantasma de Farringdon. Hab\u00edan ofrecido m\u00e1s que yo.\n\n\u2014Jam\u00e1s habr\u00edamos conseguido a Naylor \u2014dije\u2014. Didion estaba muy pendiente del martillo, despu\u00e9s de lo que pas\u00f3 la \u00faltima vez.\n\n\u2014Si t\u00fa lo dices. De todas formas, no s\u00e9 qu\u00e9 habr\u00eda hecho Jax con una duende. \u2014Dani me mir\u00f3\u2014. Dice que te ha dado el fin de semana libre. \u00bfPor qu\u00e9?\n\n\u2014Razones psicol\u00f3gicas.\n\n\u2014\u00bfQu\u00e9 significa eso?\n\n\u2014Significa que tus aparatos y t\u00fa me est\u00e1is volviendo loca.\n\nMe tir\u00f3 el vaso vac\u00edo.\n\n\u2014Cuido de ti, golfa. Mis aparatos no pueden funcionar solos. Podr\u00eda largarme a comer y dejar que se te secara esa birria de cerebro que tienes.\n\n\u2014Hoy se me podr\u00eda haber secado.\n\n\u2014Qu\u00e9 pena me das. Ya sabes c\u00f3mo funciona esto: Jax da las \u00f3rdenes, nosotros las cumplimos, nos pagan. Si no te gusta, vete a trabajar para Hector.\n\n_Touch\u00e9_.\n\nDani me dio mis botas gastadas de piel. Me las puse.\n\n\u2014\u00bfD\u00f3nde est\u00e1n todos?\n\n\u2014Eliza duerme. Ha tenido un episodio.\n\nSolo dec\u00edamos \u00abepisodio\u00bb cuando alguno de nosotros ten\u00eda un encuentro casi fatal, que en el caso de Eliza era una posesi\u00f3n no solicitada. Mir\u00e9 hacia la puerta de su taller.\n\n\u2014\u00bfEst\u00e1 bien?\n\n\u2014Lo estar\u00e1 cuando haya dormido un poco.\n\n\u2014Supongo que Nick la habr\u00e1 visto.\n\n\u2014Lo he llamado. Est\u00e1 con Jax en Chat's. Me ha dicho que te acompa\u00f1ar\u00e1 a casa de tu padre a las cinco y media.\n\nChateline's era uno de los \u00fanicos sitios a los que pod\u00edamos ir a comer, un bar restaurante con clase en Neal's Yard. El due\u00f1o hab\u00eda hecho un trato con nosotros: le d\u00e1bamos buenas propinas, y \u00e9l no les dec\u00eda a los centinelas lo que \u00e9ramos. La propina costaba m\u00e1s que la comida, pero val\u00eda la pena si quer\u00edas salir una noche.\n\n\u2014Pues llega tarde \u2014dije\u2014. Algo debe de haberlo retenido.\n\nDani cogi\u00f3 el tel\u00e9fono.\n\n\u2014No, no te molestes. \u2014Me recog\u00ed el pelo debajo de la gorra\u2014. No quiero interrumpirlos.\n\n\u2014No puedes ir en tren.\n\n\u2014S\u00ed puedo.\n\n\u2014Vas a palmar.\n\n\u2014No me pasar\u00e1 nada. Hace semanas que no vigilan la l\u00ednea. \u2014Me levant\u00e9\u2014. \u00bfDesayunamos juntas el lunes?\n\n\u2014Puede ser. Le debo unas horas extras a la bestia. \u2014Mir\u00f3 la hora\u2014. Ser\u00e1 mejor que te vayas. Son casi las seis.\n\nTen\u00eda raz\u00f3n. Ten\u00eda menos de diez minutos para llegar a la estaci\u00f3n. Cog\u00ed mi chaqueta y corr\u00ed hacia la puerta, saludando con un r\u00e1pido \u00abHola, Pieter\u00bb al fantasma que estaba en el rinc\u00f3n; a modo de respuesta, \u00e9l emiti\u00f3 un resplandor tenue, aburrido. No vi el destello, pero lo sent\u00ed. Pieter volv\u00eda a estar deprimido. A veces, estar muerto lo aflig\u00eda.\n\nTen\u00edamos una forma establecida de trabajar con los esp\u00edritus, al menos en nuestro sector. Con Pieter, por ejemplo, uno de nuestros fantasmas asesores (una musa, t\u00e9cnicamente): Eliza dejaba que la poseyera durante unas tres horas diarias, y dedicaba ese tiempo a pintar una obra maestra. Cuando terminaba, yo iba corriendo a Covent Garden y vend\u00eda el cuadro a alg\u00fan coleccionista de arte incauto. Pero Pieter era muy temperamental. A veces pasaban meses sin que tuvi\u00e9ramos ninguna obra nueva.\n\nEn una guarida como la nuestra no hab\u00eda cabida para la \u00e9tica. Suele pasar cuando obligas a una minor\u00eda a moverse en la clandestinidad. Suele pasar cuando el mundo es cruel. No hab\u00eda m\u00e1s remedio que seguir adelante. Sobrevivir como fuera, sacar un poco de pasta. Prosperar a la sombra del Arconte de Westminster.\n\nLa base de mi trabajo (de mi vida) estaba en Seven Dials. Concretamente, seg\u00fan el exclusivo sistema de divisi\u00f3n urbana de Scion, en el sector 4 de la cohorte I, o I-4. El sector estaba construido alrededor de una columna que se erig\u00eda en un cruce cerca del mercado negro de Covent Garden. En esa columna hab\u00eda seis relojes de sol.\n\nCada sector ten\u00eda su propio mimetocapo, hombre o mujer. Juntos formaban la Asamblea Antinatural, que en teor\u00eda gobernaba el sindicato, aunque cada uno hac\u00eda lo que le parec\u00eda en su sector. Dials se encontraba en el centro de la cohorte, donde el sindicato ten\u00eda m\u00e1s fuerza. Por eso lo hab\u00eda elegido Jax. Por eso segu\u00edamos all\u00ed. Nick era el \u00fanico que ten\u00eda su propia casa, m\u00e1s al norte, en Marylebone; solo la utiliz\u00e1bamos en caso de emergencia. En los tres a\u00f1os que yo llevaba trabajando para Jaxon, esto solo hab\u00eda ocurrido una vez, el d\u00eda que la Divisi\u00f3n de Vigilancia Nocturna (DVN) hab\u00eda hecho una redada en Dials con el fin de detectar alg\u00fan rastro de clarividencia. Un recadista nos avis\u00f3 con unas dos horas de antelaci\u00f3n. Conseguimos desaparecer en la mitad de ese tiempo.\n\nEra una t\u00edpica noche de marzo, fr\u00eda y lluviosa. Percib\u00eda esp\u00edritus. Dials hab\u00eda sido una barriada en la \u00e9poca pre-Scion, y todav\u00eda hab\u00eda gran cantidad de almas desconsoladas que deambulaban alrededor de la columna a la espera de nuevos encargos. Convoqu\u00e9 a una bandada de esp\u00edritus; un poco de protecci\u00f3n nunca ven\u00eda mal.\n\nScion era el no va m\u00e1s en seguridad amaur\u00f3tica. Toda referencia a la otra vida estaba prohibida. Frank Weaver nos consideraba antinaturales, y como muchos Grandes Inquisidores antes que \u00e9l, hab\u00eda ense\u00f1ado al resto de Londres a detestarnos. A menos que fuera imprescindible, solo sal\u00edamos en horas seguras. Es decir, cuando dorm\u00eda la DVN y tomaba el relevo la Divisi\u00f3n de Vigilancia Diurna (DVD). Los agentes de la DVD no eran videntes. No les estaba permitido emplear la misma brutalidad que a sus hom\u00f3logos nocturnos. Al menos, no en p\u00fablico.\n\nLos agentes de la DVN eran diferentes. Clarividentes uniformados. Obligados a servir durante treinta a\u00f1os antes de someterse a la eutanasia. Un pacto diab\u00f3lico, seg\u00fan algunos, pero que les proporcionaba una garant\u00eda de treinta a\u00f1os de vida desahogada. La mayor\u00eda de los videntes no ten\u00edan tanta suerte.\n\nLondres hab\u00eda acumulado tanta muerte en su pasado que era dif\u00edcil encontrar un sitio donde no hubiera esp\u00edritus. Formaban una red de seguridad. Aun as\u00ed, ten\u00edas que confiar en que los que consiguieras fueran buenos. Si utilizabas un fantasma d\u00e9bil, tal vez solo lograra aturdir a tu agresor durante unos segundos. Los mejores eran los esp\u00edritus que hab\u00edan tenido una vida violenta. Por eso ciertos esp\u00edritus se pagaban tan bien en el mercado negro. Por Jack el Destripador se habr\u00edan pagado millones si alguien hubiera conseguido encontrarlo. Hab\u00eda quienes todav\u00eda aseguraban que el Destripador era Eduardo VII: el pr\u00edncipe ca\u00eddo, el Rey Sangriento. Scion afirmaba que \u00e9l hab\u00eda sido el primer clarividente, pero yo nunca me lo hab\u00eda cre\u00eddo. Prefer\u00eda pensar que siempre hab\u00edamos existido.\n\nFuera oscurec\u00eda. El cielo estaba te\u00f1ido del dorado del atardecer, y la luna era una fina sonrisa blanca. Debajo se alzaba la ciudadela. The Two Brewers, el bar de ox\u00edgeno de la otra acera, estaba abarrotado de amaur\u00f3ticos. Gente normal. Los videntes dec\u00edamos que estaban aquejados de amaurosis, del mismo modo que ellos dec\u00edan que nosotros lo est\u00e1bamos de clarividencia. A veces los llamaban \u00abcarro\u00f1os\u00bb.\n\nNunca me hab\u00eda gustado esa palabra, su referencia a la putrefacci\u00f3n. Me parec\u00eda hip\u00f3crita llamarlos as\u00ed, dado que \u00e9ramos nosotros quienes convers\u00e1bamos con los muertos.\n\nMe abroch\u00e9 la chaqueta y me tap\u00e9 los ojos con la visera de la gorra. Cabeza agachada, ojos abiertos. Esa era la ley que yo acataba, y no las de Scion.\n\n\u2014Se\u00f1ora, le leo la buenaventura por un chel\u00edn. \u00a1Solo un chel\u00edn, se\u00f1ora! El mejor or\u00e1culo de Londres, se\u00f1ora, se lo prometo. \u00bfTiene algo para un pobre limosnero?\n\nLa voz pertenec\u00eda a un hombre delgado, acurrucado bajo una chaqueta tambi\u00e9n delgada. Hac\u00eda tiempo que no ve\u00eda a ning\u00fan limosnero. No abundaban en el centro de la cohorte, donde la mayor\u00eda de los videntes pertenec\u00edan al sindicato. Le\u00ed su aura. No era un or\u00e1culo, sino un adivino; y un adivino muy est\u00fapido, pues los mimetocapos despreciaban a los mendigos. Me dirig\u00ed hacia \u00e9l.\n\n\u2014\u00bfQu\u00e9 demonios haces? \u2014Lo agarr\u00e9 por el cuello de la chaqueta\u2014. \u00bfTe has vuelto loco?\n\n\u2014Por favor, se\u00f1orita. Estoy muerto de hambre \u2014dijo \u00e9l, con voz ronca a causa de la deshidrataci\u00f3n. Ten\u00eda los t\u00edpicos tics faciales de los adictos al ox\u00edgeno\u2014. No me queda ni un chavo. No se lo diga al Vinculador, se\u00f1orita. Solo quer\u00eda...\n\n\u2014Pues l\u00e1rgate de aqu\u00ed. \u2014Le puse unos billetes en la mano\u2014. No me importa ad\u00f3nde vayas, pero vete de la calle. C\u00f3mprate una dosis. Y si ma\u00f1ana tienes que limosnear, hazlo en la cohorte VI, no aqu\u00ed. \u00bfMe has entendido?\n\n\u2014Gracias, se\u00f1orita.\n\nRecogi\u00f3 sus escasas posesiones, entre las que hab\u00eda una bola de cristal, o de un material m\u00e1s barato que el cristal. Lo vi salir corriendo en direcci\u00f3n al Soho. Pobre hombre. Si malgastaba ese dinero en un bar de ox\u00edgeno, al poco tiempo volver\u00eda a estar en las calles. Muchos lo hac\u00edan: se conectaban a una c\u00e1nula y aspiraban aire aromatizado durante horas. Era la \u00fanica droga recreativa que pod\u00eda obtenerse en la ciudadela. Fuera lo que fuese lo que hiciera ese limosnador, estaba desesperado. Quiz\u00e1 lo hubieran echado del sindicato, o lo hubiera rechazado su familia. No pensaba pregunt\u00e1rselo.\n\nNadie preguntaba.\n\nNormalmente, la estaci\u00f3n I-4B estaba llena. A los amaur\u00f3ticos no les importaba viajar en tren. No ten\u00edan auras que los delataran. La mayor\u00eda de los videntes evit\u00e1bamos el transporte p\u00fablico, pero a veces los trenes eran m\u00e1s seguros que las calles. La DVN no ten\u00eda suficientes agentes para cubrir toda la ciudadela. Los controles al azar en los trenes eran poco frecuentes.\n\nHab\u00eda seis sectores en cada una de las seis cohortes. Si quer\u00edas salir de tu sector, sobre todo por la noche, necesitabas un permiso de viaje y una buena dosis de suerte. Los metrovigilantes se desplegaban al anochecer. Eran una subdivisi\u00f3n de la DVN, formada por videntes a quienes les garantizaban poder llevar una vida est\u00e1ndar. Serv\u00edan al Estado para seguir vivos.\n\nYo nunca me hab\u00eda planteado trabajar para Scion. A veces los videntes eran crueles unos con otros (y, hasta cierto punto, yo entend\u00eda a los que se volv\u00edan contra sus semejantes), pero aun as\u00ed sent\u00eda cierta afinidad con ellos. Habr\u00eda sido incapaz de detener a uno; sin embargo, cuando llevaba dos semanas trabajando a destajo y a Jax se le olvidaba pagarme, estaba tentada de hacerlo.\n\nEscane\u00e9 mi pase cuando solo me quedaban dos minutos. Una vez que hube pasado las barreras, solt\u00e9 a mi bandada. A los esp\u00edritus no les gustaba que los llevaras demasiado lejos de sus guaridas y, si los obligaba, no me ayudar\u00edan.\n\nMe dol\u00eda la cabeza. El medicamento que Dani me hab\u00eda administrado por v\u00eda intravenosa estaba dejando de hacer efecto. Una hora en el \u00e9ter... Desde luego, Jaxon estaba forzando mis l\u00edmites.\n\nEn el and\u00e9n, un Nixie luminoso verde mostraba los horarios de los trenes; por lo dem\u00e1s, hab\u00eda poca luz. La voz pregrabada de Scarlett Burnish se o\u00eda por los altavoces:\n\n\u00abEste tren para en todas las estaciones del sector 4 de la cohorte I, direcci\u00f3n norte. Por favor, tengan preparados sus pases para la inspecci\u00f3n. Est\u00e9n atentos a los boletines ofrecidos por las pantallas de seguridad. Gracias y buenas noches\u00bb.\n\nPara m\u00ed no era una buena noche en absoluto. No hab\u00eda comido nada desde el amanecer. Jax solo me dejaba parar para comer si estaba de muy buen humor, lo que ocurr\u00eda muy de vez en cuando.\n\nApareci\u00f3 otro mensaje en las pantallas de seguridad: TDR: TECNOLOG\u00cdA DE DETECCI\u00d3N RADIEST\u00c9SICA. Los otros pasajeros no se dieron cuenta. Pon\u00edan esos anuncios continuamente:\n\n\u00abEn una ciudadela tan poblada como Londres, es normal pensar que podr\u00eda estar usted viajando junto a un individuo antinatural. \u2014En la pantalla apareci\u00f3 una pantomima en la que cada silueta representaba a un ciudadano. Una se volvi\u00f3 roja\u2014. La SciOECI est\u00e1 poniendo a prueba el escudo TDR en la terminal de Paddington, as\u00ed como en el Arconte. Para 2061, esperamos tener instalado el escudo TDR en el ochenta por ciento de las estaciones del centro de la cohorte, lo que nos permitir\u00e1 reducir la presencia de agentes de polic\u00eda antinaturales en el metro. Para m\u00e1s informaci\u00f3n, dir\u00edjanse a Paddington, o pregunten a un agente de la DVD\u00bb.\n\nAparecieron otros anuncios, pero yo me qued\u00e9 pensando en ese. El TDR era la mayor amenaza para la poblaci\u00f3n vidente de la ciudadela. Seg\u00fan Scion, pod\u00eda detectar el aura a una distancia de seis metros. Si sus planes no sufr\u00edan un gran retraso, en 2061 no podr\u00edamos pisar la calle. Era t\u00edpico de los mimetocapos: a ninguno se le hab\u00eda ocurrido una soluci\u00f3n. Se limitaban a seguir peleando entre ellos. Y a pelear sobre sus peleas.\n\nLas auras vibraban en la calle, por encima de m\u00ed. Yo era una especie de diapas\u00f3n que zumbaba con su energ\u00eda. Para distraerme, saqu\u00e9 mi pase, que llevaba mi fotograf\u00eda, nombre, direcci\u00f3n, huellas dactilares, lugar de nacimiento y profesi\u00f3n. \u00abSe\u00f1orita Paige E. Mahoney, residente nacionalizada del I-5. Nacida en Irlanda en 2040. Trasladada a Londres en 2048 en circunstancias especiales. Empleada de un bar de ox\u00edgeno del I-4, de ah\u00ed el permiso de viaje. Rubia. Ojos grises. Metro setenta y cinco. Sin rasgos distintivos salvo los labios oscuros, seguramente a causa del h\u00e1bito de fumar.\u00bb\n\nYo no hab\u00eda fumado en la vida.\n\nUna mano h\u00fameda me agarr\u00f3 por la mu\u00f1eca. Me sobresalt\u00e9.\n\n\u2014Me debes una disculpa.\n\nMir\u00e9, desafiante, a un hombre de cabello oscuro con bomb\u00edn y un sucio fular blanco. Deber\u00eda haberlo reconocido por su hedor: Hector de Haymarket, uno de nuestros rivales menos higi\u00e9nicos. Siempre ol\u00eda a cloaca. Por desgracia, tambi\u00e9n era el Subse\u00f1or, el jefazo del sindicato. A su territorio lo llamaban Devil's Acre, como la barriada de la \u00e9poca victoriana.\n\n\u2014Ganamos la partida. Con todas las de la ley. \u2014Apart\u00e9 el brazo\u2014. \u00bfNo tienes nada que hacer, Hector? Lavarte los dientes, por ejemplo.\n\n\u2014Y t\u00fa podr\u00edas jugar limpio, tramposa. Y aprender a ser m\u00e1s respetuosa con tu Subse\u00f1or.\n\n\u2014No soy ninguna tramposa.\n\n\u2014Yo creo que s\u00ed. \u2014Hablaba en voz baja\u2014. Por muchos aires que se d\u00e9 ese capo vuestro, los siete sois unos mentirosos y unos estafadores de mierda. Dicen que eres la m\u00e1s astuta del mercado negro, mi querida So\u00f1adora. Pero desaparecer\u00e1s. \u2014Me acarici\u00f3 la mejilla con un dedo\u2014. Todos acaban desapareciendo.\n\n\u2014T\u00fa tambi\u00e9n desaparecer\u00e1s.\n\n\u2014Ya lo veremos. Y pronto. \u2014Las siguientes palabras que pronunci\u00f3 me las susurr\u00f3 al o\u00eddo\u2014: Que tengas un buen viaje y llegues sana y salva a casa, golfilla.\n\nSe esfum\u00f3 por el t\u00fanel de salida.\n\nTen\u00eda que andarme con mucho cuidado cuando Hector estaba cerca. Como Subse\u00f1or no ten\u00eda ning\u00fan poder real sobre los otros mimetocapos (su \u00fanica tarea era convocar reuniones), pero ten\u00eda muchos seguidores. Estaba picado desde que mi banda hab\u00eda vencido a sus lacayos jugando al tarocchi, dos d\u00edas antes de la subasta de Naylor. A los hombres de Hector no les gustaba perder. Y Jaxon, que siempre los provocaba, no ayudaba mucho. La mayor\u00eda de los de mi banda hab\u00edan evitado que los pusieran en la lista negra, sobre todo manteni\u00e9ndose al margen; pero Jax y yo \u00e9ramos demasiado insolentes. La So\u00f1adora P\u00e1lida (as\u00ed era como me llamaban en las calles) estaba en su lista de sentenciados. El d\u00eda en que me acorralaran, podr\u00eda darme por muerta.\n\nEl tren lleg\u00f3 con un minuto de retraso. Ocup\u00e9 un asiento vac\u00edo. Solo hab\u00eda otra persona en el vag\u00f3n: un hombre que iba leyendo _El descendiente_. Era vidente, un m\u00e9dium. Me puse en tensi\u00f3n. Jax ten\u00eda enemigos, y muchos videntes sab\u00edan que yo era su dama. Tambi\u00e9n sab\u00edan que vend\u00eda cuadros que no pod\u00eda haber pintado el verdadero Pieter Claesz.\n\nSaqu\u00e9 mi tableta de datos y seleccion\u00e9 mi novela autorizada favorita. Sin una bandada que me protegiera, la \u00fanica medida de seguridad que pod\u00eda adoptar era parecer tan normal y amaur\u00f3tica como fuera posible.\n\nMientras pasaba las p\u00e1ginas, vigilaba al otro pasajero con el rabillo del ojo. Yo sab\u00eda que \u00e9l me ten\u00eda en su radar, pero ninguno de los dos dijo nada. Dado que no me hab\u00eda agarrado por el cuello y no me hab\u00eda pegado hasta dejarme inconsciente, deduje que no deb\u00eda de ser un aficionado al arte al que hubieran embaucado recientemente.\n\nMe arriesgu\u00e9 a echar un vistazo a su ejemplar de _El descendiente_ , el \u00fanico peri\u00f3dico que segu\u00eda public\u00e1ndose en papel. El papel se prestaba demasiado a usos inadecuados; con las tabletas de datos, en cambio, solo pod\u00edamos bajarnos los pocos medios aprobados por el censor. Vi las t\u00edpicas noticias. Dos j\u00f3venes ahorcados por alta traici\u00f3n, un centro comercial sospechoso clausurado en el sector 3. Tambi\u00e9n hab\u00eda un art\u00edculo largo en el que se rechazaba la idea \u00abantinatural\u00bb de que Gran Breta\u00f1a estaba pol\u00edticamente aislada. El periodista llamaba a Scion \u00abun imperio en fase embrionaria\u00bb. Llevaban diciendo eso desde que yo ten\u00eda uso de raz\u00f3n. Si Scion todav\u00eda estaba en fase embrionaria, os aseguro que yo no quer\u00eda estar por all\u00ed cuando saliera del \u00fatero.\n\nHab\u00edan transcurrido casi dos siglos desde que se hab\u00eda instaurado Scion en respuesta a la amenaza de una \u00abepidemia de clarividencia\u00bb percibida por el imperio. La fecha oficial era 1901, a\u00f1o en que se atribuyeron cinco asesinatos espantosos a Eduardo VII. Aseguraban que el Rey Sangriento hab\u00eda abierto una puerta que ya no podr\u00eda volver a cerrarse; que hab\u00eda tra\u00eddo al mundo la plaga de la clarividencia; y que sus seguidores estaban por todas partes, reproduci\u00e9ndose y matando, obteniendo su poder de una fuente de una maldad terrible.\n\nA continuaci\u00f3n lleg\u00f3 Scion, una rep\u00fablica construida con el fin de erradicar la enfermedad. A lo largo de los cincuenta a\u00f1os siguientes se hab\u00eda convertido en una m\u00e1quina de perseguir a videntes, donde todas las pol\u00edticas importantes giraban en torno a los antinaturales. Los asesinatos siempre los comet\u00edan los antinaturales. La violencia aleatoria, los robos, las violaciones, los incendios... todo suced\u00eda por culpa de los antinaturales. Con el tiempo, el sindicato de videntes se hab\u00eda desarrollado en la ciudadela, hab\u00eda formado un hampa organizada, y hab\u00eda ofrecido refugio a los clarividentes. Desde entonces, Scion se esforzaba a\u00fan m\u00e1s para erradicarnos.\n\nCuando hubieran instalado el TDR, el sindicato se vendr\u00eda abajo y Scion lo ver\u00eda todo. Ten\u00edamos dos a\u00f1os para ponerle remedio, pero con Hector como Subse\u00f1or, yo no abrigaba muchas esperanzas. De momento, su mandato solo nos hab\u00eda aportado corrupci\u00f3n.\n\nEl tren pas\u00f3 sin incidentes por tres estaciones. Acababa de terminar un cap\u00edtulo cuando se apagaron las luces y el tren se detuvo. Me di cuenta de lo que estaba pasando una mil\u00e9sima de segundo antes que el otro pasajero, que se enderez\u00f3 en el asiento.\n\n\u2014Van a registrar el tren.\n\nIntent\u00e9 decir algo, confirmar sus temores, pero mi lengua parec\u00eda un trapo doblado.\n\nApagu\u00e9 la tableta. Se abri\u00f3 una puerta en la pared del t\u00fanel. El Nixie del vag\u00f3n anunci\u00f3 una ALERTA DE SEGURIDAD. Sab\u00eda qu\u00e9 pasar\u00eda a continuaci\u00f3n: aparecer\u00edan dos metrovigilantes. Siempre hab\u00eda un jefe, generalmente un m\u00e9dium. Nunca hab\u00eda presenciado uno de esos controles, pero sab\u00eda que muy pocos videntes se libraban de ellos.\n\nEl coraz\u00f3n me martilleaba en el pecho. Mir\u00e9 al otro pasajero tratando de calibrar su reacci\u00f3n. Era m\u00e9dium, aunque no especialmente poderoso. Lo sab\u00eda, aunque no pudiera explicar c\u00f3mo; mis antenas lo detectaban, sencillamente.\n\n\u2014Tenemos que salir de este tren. \u2014Se levant\u00f3\u2014. \u00bfQu\u00e9 eres, guapa? \u00bfUn or\u00e1culo?\n\nNo le contest\u00e9.\n\n\u2014S\u00e9 que eres vidente. \u2014Tir\u00f3 de la manija de la puerta\u2014. Venga, tesoro, no te quedes ah\u00ed sentada. Tiene que haber alguna forma de salir de aqu\u00ed. \u2014Se sec\u00f3 el sudor de la frente con la manga\u2014. Ten\u00eda que haber un control precisamente hoy, precisamente el d\u00eda...\n\nNo me mov\u00ed. No hab\u00eda forma de escapar. Las ventanas estaban selladas; las puertas, cerradas con seguro. Y se nos hab\u00eda agotado el tiempo. Los haces de las linternas iluminaron el vag\u00f3n.\n\nMe qued\u00e9 muy quieta. Metrovigilantes. Deb\u00edan de haber detectado a cierto n\u00famero de videntes en el vag\u00f3n, o no habr\u00edan apagado las luces. Yo sab\u00eda que pod\u00edan ver nuestras auras, pero querr\u00edan averiguar qu\u00e9 clase de videntes \u00e9ramos exactamente.\n\nEntraron en el vag\u00f3n: un invocador y un m\u00e9dium. El tren segu\u00eda movi\u00e9ndose, pero no hab\u00edan vuelto a encenderse las luces. Se dirigieron al hombre primero.\n\n\u2014\u00bfNombre?\n\nEl hombre se enderez\u00f3.\n\n\u2014Linwood.\n\n\u2014\u00bfMotivo del viaje?\n\n\u2014Vengo de visitar a mi hija.\n\n\u2014De visitar a tu hija. \u00bfSeguro que no vas a una sesi\u00f3n de espiritismo, m\u00e9dium?\n\nAquellos dos quer\u00edan pelea.\n\n\u2014Tengo los certificados del hospital. Mi hija est\u00e1 muy enferma \u2014replic\u00f3 Linwood\u2014. Tengo permiso para visitarla todas las semanas.\n\n\u2014Si vuelves a abrir la boca, se te van a acabar los permisos. \u2014Se volvi\u00f3 hacia m\u00ed y me grit\u00f3\u2014: \u00a1T\u00fa! \u00bfD\u00f3nde est\u00e1 tu pase?\n\nLo saqu\u00e9 del bolsillo.\n\n\u2014\u00bfY el permiso de viaje?\n\nSe lo di; hizo una pausa para leerlo.\n\n\u2014Trabajas en el sector 4.\n\n\u2014S\u00ed.\n\n\u2014\u00bfQui\u00e9n ha expedido este permiso?\n\n\u2014Bill Bunbury, mi supervisor.\n\n\u2014Ya. Pero necesito ver algo m\u00e1s. \u2014Me ilumin\u00f3 los ojos con la linterna\u2014. No te muevas.\n\nAguant\u00e9 sin parpadear.\n\n\u2014No tiene visi\u00f3n espiritista \u2014coment\u00f3\u2014. Debes de ser un or\u00e1culo. Hac\u00eda tiempo que no ve\u00eda ninguno.\n\n\u2014Yo no hab\u00eda visto un or\u00e1culo con tetas desde los a\u00f1os cuarenta \u2014observ\u00f3 el otro metrovigilante\u2014. Esto les va a encantar.\n\nSu superior sonri\u00f3. Ten\u00eda un coloboma en cada ojo, una se\u00f1al de visi\u00f3n espiritista permanente.\n\n\u2014Est\u00e1s a punto de hacerme muy rico, jovencita \u2014me dijo\u2014. Deja que vuelva a examinarte los ojos.\n\n\u2014No soy un or\u00e1culo \u2014dije.\n\n\u2014Claro que no. Cierra el pico y abre esos ojitos.\n\nCasi todos los videntes me tomaban por un or\u00e1culo. Un error f\u00e1cil. Las auras eran similares; de hecho, eran del mismo color.\n\nEl vigilante me separ\u00f3 los p\u00e1rpados del ojo izquierdo con los dedos y me examin\u00f3 las pupilas con el fino haz de la linterna buscando el coloboma. Linwood corri\u00f3 hacia la puerta abierta y les lanz\u00f3 un esp\u00edritu (su \u00e1ngel guardi\u00e1n) a los metrovigilantes. El \u00e1ngel se estrell\u00f3 contra el vigilante de refuerzo e hizo un revoltijo con sus sentidos.\n\nPero el primer metrovigilante era muy r\u00e1pido. Antes de que los dem\u00e1s pudi\u00e9ramos movernos, hab\u00eda hecho aparecer una bandada de duendes.\n\n\u2014No te muevas, m\u00e9dium.\n\nLinwood lo mir\u00f3 fijamente. Era un hombre de escasa estatura, de unos cuarenta a\u00f1os, delgado pero nervudo, con cabello casta\u00f1o oscuro y las sienes encanecidas. Yo no pod\u00eda ver a los duendes (no pod\u00eda ver pr\u00e1cticamente nada, porque me deslumbraba la linterna), pero me estaban debilitando tanto que no pod\u00eda moverme. Cont\u00e9 hasta tres. Nunca hab\u00eda visto a nadie controlar a un duende, y mucho menos a tres. Not\u00e9 un sudor fr\u00edo en la nuca.\n\nCuando el \u00e1ngel gir\u00f3 sobre s\u00ed mismo para volver a atacar, los duendes empezaron a describir c\u00edrculos alrededor del metrovigilante.\n\n\u2014Ven con nosotros por las buenas, m\u00e9dium \u2014dijo este\u2014, y convenceremos a nuestros jefes para que no te torturen.\n\n\u2014Hagan lo que tengan que hacer, caballeros. \u2014Linwood levant\u00f3 una mano\u2014. Con los \u00e1ngeles a mi lado, no le tengo miedo a ning\u00fan hombre.\n\n\u2014Eso dicen todos, se\u00f1or Linwood. Pero cuando ven la Torre, ya no se acuerdan.\n\nLinwood lanz\u00f3 su \u00e1ngel hacia el fondo del vag\u00f3n. No vi la colisi\u00f3n, pero esta sacudi\u00f3 violentamente todos mis sentidos. Me obligu\u00e9 a levantarme. La presencia de los tres duendes estaba minando mis fuerzas; Linwood ten\u00eda mucha labia, pero era evidente que a \u00e9l tambi\u00e9n le afectaba y que estaba haciendo todo lo posible para fortalecer a su \u00e1ngel. Mientras el invocador controlaba a los duendes, el segundo metrovigilante recitaba el treno: una serie de palabras que compel\u00edan a los esp\u00edritus a morir por completo, envi\u00e1ndolos lejos del alcance de los videntes. El \u00e1ngel tembl\u00f3. Para hacerlo desaparecer habr\u00edan necesitado saber su nombre completo, pero mientras uno de los dos siguiera recitando, el \u00e1ngel ser\u00eda demasiado d\u00e9bil para proteger a su hu\u00e9sped.\n\nLa sangre me lat\u00eda en las sienes. Ten\u00eda la garganta cerrada, los dedos entumecidos. Si no hac\u00eda nada, nos detendr\u00edan a los dos. Me vi en la Torre, sometida a torturas, condenada a la horca...\n\nNo, no estaba dispuesta a morir ese d\u00eda.\n\nCuando los duendes se cernieron sobre Linwood, le sucedi\u00f3 algo a mi visi\u00f3n. Me centr\u00e9 en los metrovigilantes. Sus mentes vibraban junto a la m\u00eda, dos aros pulsantes de energ\u00eda. O\u00ed el golpe de mi cuerpo al caer al suelo.\n\nSolo pretend\u00eda desorientarlos, ganar tiempo para huir. Contaba con el factor sorpresa: me hab\u00edan infravalorado, porque los or\u00e1culos necesitaban una bandada para representar un verdadero peligro.\n\nYo no.\n\nMe invadi\u00f3 una oleada negra de miedo. Mi esp\u00edritu se separ\u00f3 de mi cuerpo y se introdujo en el del primer metrovigilante. Antes de darme cuenta de lo que estaba haciendo, me hab\u00eda estrellado en su onirosaje. No contra \u00e9l, sino dentro de \u00e9l, a trav\u00e9s de \u00e9l. Lanc\u00e9 su esp\u00edritu al \u00e9ter, dejando su cuerpo vac\u00edo. Antes de que su compinche pudiera reaccionar, le hab\u00eda sucedido lo mismo.\n\nMi esp\u00edritu regres\u00f3 a mi cuerpo. Not\u00e9 un fuerte dolor detr\u00e1s de los ojos. Jam\u00e1s hab\u00eda sentido un dolor semejante; era como si me clavaran pu\u00f1ales en el cr\u00e1neo, como si me ardiera el cerebro; estaba tan caliente que no pod\u00eda moverme ni pensar. Era vagamente consciente del suelo pringoso del vag\u00f3n contra mi mejilla. Fuera lo que fuese lo que acababa de hacer, no ten\u00eda ninguna prisa por repetirlo.\n\nEl tren se balance\u00f3. Deb\u00eda de estar llegando a la siguiente estaci\u00f3n. Me incorpor\u00e9 apoy\u00e1ndome en los codos, y me temblaron los m\u00fasculos por el esfuerzo.\n\n\u2014\u00bfSe\u00f1or Linwood?\n\nNo me contest\u00f3. Me arrastr\u00e9 hasta donde estaba tumbado. Cuando el tren pas\u00f3 junto a una luz de servicio, vi su cara.\n\nEstaba muerto. Los duendes le hab\u00edan extra\u00eddo el esp\u00edritu. Su pase estaba en el suelo. William Linwood, cuarenta y tres a\u00f1os. Dos hijos, una con fibrosis qu\u00edstica. Casado. Empleado de banca. M\u00e9dium.\n\n\u00bfSab\u00edan su mujer y sus hijos que ten\u00eda una vida secreta? \u00bfO eran amaur\u00f3ticos y no sab\u00edan nada?\n\nTen\u00eda que recitar el treno, o Linwood quedar\u00eda atrapado en aquel vag\u00f3n para siempre.\n\n\u2014William Linwood \u2014dije\u2014, vete al \u00e9ter. Est\u00e1 todo arreglado. Todas las deudas est\u00e1n saldadas. Ya no tienes que morar entre los vivos.\n\nEl esp\u00edritu de Linwood flotaba cerca de su cuerpo. El \u00e9ter produjo un susurro cuando \u00e9l y su \u00e1ngel desaparecieron.\n\nEntonces se encendieron las luces, y se me cort\u00f3 la respiraci\u00f3n: hab\u00eda dos cuerpos m\u00e1s en el suelo.\n\nMe agarr\u00e9 a una barra y me levant\u00e9. Ten\u00eda la palma sudorosa y apenas pod\u00eda sujetarme. El primer metrovigilante estaba a solo unos palmos, muerto; todav\u00eda ten\u00eda la expresi\u00f3n de sorpresa en la cara.\n\nLo hab\u00eda matado. Hab\u00eda matado a un metrovigilante.\n\nSu compa\u00f1ero no hab\u00eda tenido tanta suerte. Estaba tendido boca arriba, con los ojos fijos en el techo; le resbalaba un hilillo de saliva por la barbilla. Cuando me acerqu\u00e9 a \u00e9l, se sacudi\u00f3 un poco. Not\u00e9 un escalofr\u00edo, y el sabor de la bilis abras\u00e1ndome la garganta. No hab\u00eda empujado su esp\u00edritu lo suficientemente lejos, y se hab\u00eda quedado flotando en las partes m\u00e1s oscuras de su mente: las partes secretas, silenciosas, donde no pod\u00eda habitar ning\u00fan esp\u00edritu. Hab\u00eda enloquecido. No: yo lo hab\u00eda hecho enloquecer.\n\nApret\u00e9 las mand\u00edbulas. No pod\u00eda dejarlo as\u00ed; ni siquiera un metrovigilante merec\u00eda semejante destino. Le puse las manos, fr\u00edas, sobre los hombros y me arm\u00e9 de valor para practicarle la eutanasia. El metrovigilante dio un gru\u00f1ido y susurr\u00f3:\n\n\u2014M\u00e1tame.\n\nTen\u00eda que hacerlo. Se lo deb\u00eda.\n\nPero no pod\u00eda. No pod\u00eda matarlo.\n\nCuando el tren entr\u00f3 en la estaci\u00f3n I-5C, me coloqu\u00e9 junto a la puerta. Subieron otros pasajeros y vieron los cad\u00e1veres, pero ya era demasiado tarde para atraparme. Yo ya hab\u00eda subido a la calle y me hab\u00eda calado la gorra para ocultar mi cara.\n\nEntr\u00e9 sin hacer ruido en el piso y colgu\u00e9 mi chaqueta. Vic, el vigilante de seguridad que trabajaba a jornada completa en el complejo Golden Crescent, estaba haciendo su ronda cuando me col\u00e9 por el portal, y no vio la palidez extrema de mi cara ni mis manos temblorosas cuando saqu\u00e9 la tarjeta para abrir la puerta.\n\nMi padre estaba en el sal\u00f3n. Vi sus pies, enfundados en unas zapatillas, apoyados en la otomana. Estaba viendo ScionVista, el canal de noticias que cubr\u00eda todas las ciudadelas de Scion, y en la pantalla Scarlett Burnish anunciaba que acababan de cerrar el metro que atravesaba la cohorte I.\n\nCada vez que o\u00eda esa voz, me estremec\u00eda. A sus veinticinco a\u00f1os, Burnish era la Gran Anecdotista m\u00e1s joven de la historia: la ayudante del Gran Inquisidor, encargada de transmitir su voz y su inteligencia a Scion. La gente la llamaba \u00abla puta de Weaver\u00bb, quiz\u00e1 por celos. Ten\u00eda la piel clara y unos labios enormes, y llevaba un grueso perfilador de ojos rojo a juego con su pelo, recogido en un elegante mo\u00f1o trenzado. Sus vestidos de cuello alto siempre me hac\u00edan pensar en la horca.\n\n\u00abEn el \u00e1mbito internacional, el Gran Inquisidor de la Rep\u00fablica Francesa, Beno\u00eet M\u00e9nard, visitar\u00e1 al Inquisidor Weaver con motivo de las fiestas de Novembertide de este a\u00f1o. A falta de ocho meses, el Arconte ya est\u00e1 haciendo los preparativos de lo que promete ser una visita francamente estimulante.\u00bb\n\n\u2014\u00bfPaige?\n\nMe quit\u00e9 la gorra.\n\n\u2014Hola.\n\n\u2014Ven a sentarte conmigo.\n\n\u2014Voy enseguida.\n\nMe fui derecha al cuarto de ba\u00f1o. Estaba empapada de sudor.\n\nHab\u00eda matado a una persona. La hab\u00eda asesinado. Jax siempre me hab\u00eda cre\u00eddo capaz de cometer un asesinato incruento, pero yo nunca le hab\u00eda cre\u00eddo. Ahora era una asesina. Y peor a\u00fan: hab\u00eda dejado pistas: un superviviente. Adem\u00e1s hab\u00eda perdido mi tableta, que ten\u00eda mis huellas dactilares por todas partes. No se contentar\u00edan con someterme a NiteKind; eso habr\u00eda sido demasiado f\u00e1cil. Me esperaban la tortura y la horca, sin ninguna duda.\n\nNada m\u00e1s entrar en el cuarto de ba\u00f1o, vomit\u00e9 en el v\u00e1ter. Cuando lo hube sacado todo menos los \u00f3rganos, temblaba tan violentamente que apenas me ten\u00eda en pie. Me quit\u00e9 la ropa y me met\u00ed, tambaleante, en la ducha. El agua caliente me aporre\u00f3 la piel.\n\nEsa vez hab\u00eda ido demasiado lejos. Por primera vez en la vida, no me hab\u00eda limitado a tocar otros onirosajes, sino que los hab\u00eda invadido.\n\nA Jaxon le iba a encantar.\n\nSe me cerraban los ojos. Ve\u00eda desarrollarse una y otra vez la escena del tren. No me hab\u00eda propuesto matarlos, solo quer\u00eda darles un empuj\u00f3n, lo suficiente para provocarles una migra\u00f1a, quiz\u00e1 hacer que les sangrara la nariz. Causar una distracci\u00f3n para ganar tiempo.\n\nPero el p\u00e1nico se hab\u00eda apoderado de m\u00ed. El miedo a que me encontraran. El miedo a convertirme en otra v\u00edctima an\u00f3nima de Scion.\n\nPens\u00e9 en Linwood. Los videntes nunca se proteg\u00edan unos a otros, a menos que pertenecieran a la misma banda, y sin embargo su muerte me produc\u00eda un gran cargo de conciencia. Me llev\u00e9 las rodillas hasta la barbilla y me sujet\u00e9 la dolorida cabeza con ambas manos. Si hubiera sido m\u00e1s r\u00e1pida... Hab\u00edan muerto dos personas, y otra hab\u00eda enloquecido; y a menos que tuviera mucha suerte, yo ser\u00eda la siguiente.\n\nMe acurruqu\u00e9 en el rinc\u00f3n de la ducha, con las rodillas pegadas al pecho. No pod\u00eda quedarme escondida all\u00ed toda la vida. Al final siempre te encontraban.\n\nTen\u00eda que pensar. Scion ten\u00eda un procedimiento de contenci\u00f3n para esas situaciones. Despu\u00e9s de despejar la estaci\u00f3n y detener a los posibles testigos, llamar\u00edan a un narco (un experto en drogas et\u00e9reas) y administrar\u00edan \u00e1ster azul. Eso restituir\u00eda temporalmente los recuerdos de mis v\u00edctimas, y los har\u00eda visibles. Cuando hubieran grabado las partes relevantes, le practicar\u00edan la eutanasia al metrovigilante que hab\u00eda sobrevivido y llevar\u00edan su cad\u00e1ver a la morgue del II-6. A continuaci\u00f3n revisar\u00edan sus recuerdos y buscar\u00edan entre ellos la cara de su verdugo. Y entonces me encontrar\u00edan.\n\nLas detenciones no siempre ten\u00edan lugar por la noche. A veces te atrapaban de d\u00eda, cuando sal\u00edas a la calle. Una linterna alumbr\u00e1ndote los ojos, una aguja en el cuello, y estabas perdido. Nadie informaba de tu desaparici\u00f3n.\n\nDe momento no pod\u00eda pensar en el futuro. Un intenso dolor volvi\u00f3 a sacudirme la cabeza y me devolvi\u00f3 al presente.\n\nRepas\u00e9 mis opciones. Pod\u00eda volver a Dials y esconderme un tiempo en nuestro cubil, pero si los centinelas ya hab\u00edan salido a buscarme, los conducir\u00eda hasta Jax. Adem\u00e1s, no pod\u00eda volver al sector 4, porque hab\u00edan cerrado las estaciones. Me costar\u00eda encontrar un taxi pirata, y por la noche se reforzaban los sistemas de seguridad.\n\nPod\u00eda quedarme en casa de alguna amiga, pero las pocas amigas que ten\u00eda fuera de Dials eran amaur\u00f3ticas, chicas del colegio con las que ten\u00eda poco contacto. Si les dec\u00eda que me persegu\u00eda la polic\u00eda secreta porque hab\u00eda matado a una persona con mi esp\u00edritu, me tomar\u00edan por loca. Y seguramente me denunciar\u00edan.\n\nMe puse una bata vieja; fui descalza hasta la cocina y puse un cazo de leche a calentar. Era lo que hac\u00eda siempre cuando volv\u00eda a casa; no me conven\u00eda alterar la rutina. Mi padre hab\u00eda dejado fuera mi taza favorita, una grande con la inscripci\u00f3n: \u00abNo hay nada como un caf\u00e9 por la ma\u00f1ana\u00bb. Nunca me hab\u00eda entusiasmado el ox\u00edgeno aromatizado, o Floxy\u00ae, la alternativa de Scion al alcohol. El caf\u00e9 era m\u00e1s o menos legal. Todav\u00eda estaban investigando si la cafe\u00edna pod\u00eda desencadenar la clarividencia o no. Pero \u00abNo hay nada como el ox\u00edgeno aromatizado por la ma\u00f1ana\u00bb no habr\u00eda tenido tanto gancho, claro.\n\nUtilizar mi esp\u00edritu hab\u00eda tenido consecuencias en mi cabeza. Apenas pod\u00eda mantener los ojos abiertos. Mientras vert\u00eda la leche, mir\u00e9 por la ventana. Mi padre ten\u00eda un gusto impecable en lo relativo al dise\u00f1o de interiores. Tambi\u00e9n ayudaba que tuviera dinero suficiente para pagar una vivienda de alta seguridad en el exclusivo Barbican Estate. El apartamento era amplio y luminoso. Los pasillos ol\u00edan a popurr\u00ed y a ropa limpia. Hab\u00eda grandes ventanas cuadradas en todas las habitaciones. La m\u00e1s grande estaba en el sal\u00f3n, un ventanal enorme que cubr\u00eda toda la pared orientada a poniente, junto a las elaboradas cristaleras por las que se acced\u00eda al balc\u00f3n. De ni\u00f1a sol\u00eda contemplar la puesta de sol desde esa ventana.\n\nFuera, la ciudadela segu\u00eda con su ritmo vertiginoso. Por encima de nuestro complejo se alzaban los tres edificios de arquitectura brutalista de Barbican Estate donde viv\u00edan los funcionarios de Scion. En lo alto de la Torre Lauderdale estaba la pantalla de transmisi\u00f3n del I-5. Desde esa pantalla se proyectaban todas las ejecuciones p\u00fablicas los domingos por la ma\u00f1ana. En ese momento mostraba la insignia est\u00e1tica del sistema de Scion (un s\u00edmbolo rojo semejante a un ancla) y una sola palabra escrita con letras negras, SCION, todo sobre un as\u00e9ptico fondo blanco. Luego estaba aquel espantoso eslogan: EL LUGAR M\u00c1S SEGURO. Deber\u00eda haber sido \u00abEl lugar m\u00e1s inseguro\u00bb. Al menos para nosotros.\n\nMientras me beb\u00eda la leche a sorbitos, observ\u00e9 aquel s\u00edmbolo y me cagu\u00e9 en \u00e9l. Luego lav\u00e9 la taza, llen\u00e9 un vaso de agua y me fui a mi habitaci\u00f3n. Ten\u00eda que llamar a Jaxon.\n\nMi padre me intercept\u00f3 en el pasillo.\n\n\u2014Espera, Paige.\n\nMe par\u00e9.\n\nMi padre, irland\u00e9s de nacimiento, con una mata de cabello pelirrojo, trabajaba en la agencia de investigaciones cient\u00edficas de Scion. Cuando no estaba en el trabajo, estaba garabateando f\u00f3rmulas en su tableta y hablando extasiado sobre bioqu\u00edmica cl\u00ednica, una de sus dos licenciaturas. No nos parec\u00edamos en nada.\n\n\u2014Hola. Perdona que llegue tan tarde. He hecho unas horas extras.\n\n\u2014No tienes por qu\u00e9 disculparte. \u2014Me hizo se\u00f1as para que entrara en el sal\u00f3n\u2014. Deja que te prepare algo de comer. Te veo paliducha.\n\n\u2014Estoy bien, solo un poco cansada.\n\n\u2014Mira, hoy he estado leyendo un art\u00edculo sobre el circuito de ox\u00edgeno. Ha habido un caso horrible en el IV-2. Personal mal pagado, ox\u00edgeno sucio, clientes que sufren ataques epil\u00e9pticos... Muy desagradable.\n\n\u2014Pues los bares del centro est\u00e1n bien. Los clientes exigen calidad. \u2014Mi padre empez\u00f3 a poner la mesa\u2014. \u00bfQu\u00e9 tal t\u00fa en el trabajo?\n\n\u2014Bien. \u2014Levant\u00f3 la cabeza y me mir\u00f3\u2014. Paige, ese trabajo tuyo en el bar...\n\n\u2014\u00bfQu\u00e9 pasa?\n\nUna hija trabajando en las esferas m\u00e1s bajas de la ciudadela: no pod\u00eda haber nada m\u00e1s embarazoso para un hombre de su posici\u00f3n. Qu\u00e9 inc\u00f3modo deb\u00eda de sentirse cuando sus colegas le preguntaban por sus hijos, suponiendo que ser\u00edan m\u00e9dicos o abogados. C\u00f3mo deb\u00edan de cuchichear cuando se enteraban de que yo trabajaba en una barra de bar, y no en un bufete de abogados. Minti\u00e9ndole le hac\u00eda un favor. Mi padre jam\u00e1s habr\u00eda podido asimilar la verdad: que yo era una antinatural, una delincuente.\n\nY una asesina. Solo de pensarlo me daban n\u00e1useas.\n\n\u2014Ya s\u00e9 que yo no soy nadie para decirlo, pero creo que deber\u00edas plantearte volver a solicitar una plaza en la universidad. Ese empleo tuyo es un callej\u00f3n sin salida. Mal pagado, sin porvenir. En cambio, la universidad...\n\n\u2014No. \u2014Lo dije m\u00e1s alto de lo que era mi intenci\u00f3n\u2014. Me gusta mi trabajo. Lo escog\u00ed yo.\n\nTodav\u00eda me acordaba del d\u00eda en que la directora del colegio me hab\u00eda entregado las notas finales. \u00abLamento que no hayas solicitado plaza en la universidad, Paige \u2014me hab\u00eda dicho\u2014, pero quiz\u00e1 sea lo mejor. Has faltado demasiado a clase, y eso no se considera correcto en una joven distinguida. \u2014Me hab\u00eda entregado una carpeta delgada, con el emblema del colegio grabado en las tapas de piel\u2014. Aqu\u00ed tienes una carta de recomendaci\u00f3n de tus tutoras. Destacan tus aptitudes para la educaci\u00f3n f\u00edsica, el franc\u00e9s y la historia de Scion.\u00bb\n\nNo me importaba. Siempre hab\u00eda odiado el colegio: el uniforme, el dogma. Marcharme de all\u00ed hab\u00eda sido lo mejor de mis a\u00f1os de formaci\u00f3n.\n\n\u2014Yo podr\u00eda hacer algo \u2014dijo mi padre. Le habr\u00eda encantado tener una hija instruida\u2014. Podr\u00edas volver a solicitar el ingreso.\n\n\u2014En Scion el nepotismo no funciona \u2014dije\u2014. Deber\u00edas saberlo.\n\n\u2014A m\u00ed no me dejaron elegir, Paige. \u2014Le tembl\u00f3 un m\u00fasculo de la mejilla\u2014. No tuve ese lujo.\n\nNo me apetec\u00eda mantener esa conversaci\u00f3n. No quer\u00eda pensar en lo que mi padre hab\u00eda dejado atr\u00e1s.\n\n\u2014\u00bfTodav\u00eda vives con tu novio? \u2014me pregunt\u00f3.\n\nLa mentira del novio hab\u00eda sido un error. Desde que me lo invent\u00e9, mi padre siempre hab\u00eda querido conocerlo.\n\n\u2014He roto con \u00e9l \u2014dije\u2014. No nos iba bien. Pero no pasa nada. Suzette tiene sitio en su apartamento. Te acuerdas de ella, \u00bfno?\n\n\u2014\u00bfSuzy, tu amiga del colegio?\n\n\u2014S\u00ed.\n\nMientras hablaba, not\u00e9 una fuerte punzada de dolor en un lado de la cabeza. No pod\u00eda esperar a que mi padre preparara la cena. Ten\u00eda que llamar a Jaxon, contarle lo que hab\u00eda pasado. Ya.\n\n\u2014Mira, me duele un poco la cabeza \u2014dije\u2014. \u00bfTe importa que me acueste pronto?\n\nMi padre vino a mi lado y me tom\u00f3 la barbilla con una mano.\n\n\u2014Siempre te duele la cabeza. Est\u00e1s demasiado cansada. \u2014Me pas\u00f3 el pulgar por la cara, por las ojeras\u2014. Est\u00e1n dando un documental muy interesante. \u00bfNo te apetece verlo? Puedes tumbarte en el sof\u00e1.\n\n\u2014Ma\u00f1ana, a lo mejor. \u2014Le apart\u00e9 la mano con suavidad\u2014. \u00bfTienes alg\u00fan analg\u00e9sico?\n\nAl cabo de un momento, mi padre asinti\u00f3 con la cabeza.\n\n\u2014En el cuarto de ba\u00f1o. Ma\u00f1ana por la ma\u00f1ana preparar\u00e9 un desayuno completo al estilo de Ulster, \u00bfte parece? Quiero que me cuentes muchas cosas, _seillean_.\n\nLo mir\u00e9 fijamente. No me hab\u00eda preparado el desayuno desde que yo ten\u00eda doce a\u00f1os, ni me hab\u00eda llamado con ese apodo desde que viv\u00edamos en Irlanda. De eso hac\u00eda diez a\u00f1os. En otra vida.\n\n\u2014\u00bfPaige?\n\n\u2014Vale \u2014dije\u2014. Hasta ma\u00f1ana.\n\nMe fui a mi habitaci\u00f3n. Mi padre no dijo nada m\u00e1s. Dej\u00f3 la puerta entreabierta, como hac\u00eda siempre que yo estaba en casa. Nunca hab\u00eda sabido c\u00f3mo tratarme.\n\nLa habitaci\u00f3n de invitados estaba tan caldeada como siempre. Mi antiguo dormitorio. Me hab\u00eda mudado a Dials nada m\u00e1s terminar el colegio, pero mi padre nunca hab\u00eda alojado a ning\u00fan inquilino, porque no lo necesitaba. Oficialmente, yo todav\u00eda viv\u00eda all\u00ed. Era mejor dejarlo as\u00ed en el registro. Abr\u00ed la puerta del balc\u00f3n que discurr\u00eda entre mi cuarto y la cocina. Mi piel hab\u00eda pasado de estar fr\u00eda a arder, y notaba una extra\u00f1a sensaci\u00f3n de tirantez en los ojos, como si llevara horas mirando fijamente una fuente de luz. Lo \u00fanico que ve\u00eda era la cara de mi v\u00edctima, y la vacuidad (la locura) reflejada en la del que hab\u00eda dejado con vida.\n\nHab\u00eda infligido esos da\u00f1os en cuesti\u00f3n de segundos. Mi esp\u00edritu no solo era un explorador, sino que era un arma. Jaxon llevaba tiempo esperando que sucediera algo as\u00ed.\n\nCog\u00ed el tel\u00e9fono y llam\u00e9 a la habitaci\u00f3n de Jaxon en el refugio. Jaxon contest\u00f3 inmediatamente:\n\n\u2014\u00a1Vaya, vaya! Cre\u00eda que me hab\u00edas abandonado todo el fin de semana. \u00bfD\u00f3nde est\u00e1 el incendio, coraz\u00f3n? \u00bfTe has pensado mejor lo de las vacaciones? No las necesitas, \u00bfverdad? Ya te lo dec\u00eda yo. No puedo prescindir de mi andarina dos d\u00edas enteros. S\u00e9 buena, querida. Estupendo. Me alegro de que est\u00e9s de acuerdo. Por cierto, \u00bfhas conseguido a Jane Rochford? Te traspasar\u00e9 unos cuantos billetes de mil m\u00e1s por si los necesitas. Pero, por favor, no me digas que ese desgraciado estirado de Didion nos ha robado a Anne Naylor y a...\n\n\u2014He matado a una persona.\n\nSilencio.\n\n\u2014\u00bfA qui\u00e9n? \u2014dijo Jax con voz rara.\n\n\u2014Una pareja de metrovigilantes intentaba detener a un m\u00e9dium y...\n\n\u2014\u00bfLos has matado?\n\n\u2014He matado a uno.\n\nJax inspir\u00f3 bruscamente.\n\n\u2014\u00bfY el otro?\n\n\u2014Lo he mandado a su zona hadal.\n\n\u2014Un momento. \u00bfLo has hecho con tu...? \u2014Como no contest\u00e9, se ech\u00f3 a re\u00edr. Le o\u00ed dar palmadas contra el tablero de la mesa\u2014. Por fin. \u00a1Por fin! \u00a1Paige, peque\u00f1a taumaturga, lo has hecho! \u00bfVes como no puedes perder el tiempo con esas sesiones de espiritismo? Y ese tipo, el metrovigilante, \u00bfse ha quedado convertido en vegetal?\n\n\u2014S\u00ed \u2014contest\u00e9\u2014. \u00bfEstoy despedida?\n\n\u2014\u00bfDespedida? \u00a1Por el _zeitgeist_ , tesoro, claro que no! Llevo a\u00f1os esperando que saques provecho de tu talento. Te has abierto como la flor de ambros\u00eda que eres, mi adorable prodigio. \u2014Me lo imagin\u00e9 dando una calada a su puro para celebrarlo\u2014. Vaya, vaya, mi onir\u00e1mbula ha entrado por fin en otro onirosaje. Y solo ha tardado tres a\u00f1os. Y dime, \u00bfhas podido salvar al vidente?\n\n\u2014No.\n\n\u2014\u00bfNo?\n\n\u2014Ten\u00edan tres duendes.\n\n\u2014Venga ya. Ning\u00fan m\u00e9dium podr\u00eda controlar a tres duendes.\n\n\u2014Pues ese m\u00e9dium s\u00ed pod\u00eda. Me confundi\u00f3 con un or\u00e1culo.\n\nJax ri\u00f3 por lo bajo.\n\n\u2014Aficionados.\n\nMir\u00e9 por la ventana, hacia la torre. Hab\u00eda aparecido otro mensaje: LES INFORMAMOS DE QUE HAY RETRASOS INESPERADOS EN EL METRO.\n\n\u2014Han cerrado el metro \u2014dije\u2014. Ahora me est\u00e1n buscando.\n\n\u2014No te dejes llevar por el p\u00e1nico, Paige. Es poco decoroso.\n\n\u2014Bueno, espero que tengas un plan. Han bloqueado toda la red. Y necesito largarme de aqu\u00ed.\n\n\u2014No te preocupes por eso. Aunque intenten extraerle los recuerdos y lo consigan, el cerebro de ese metrovigilante no es m\u00e1s que pur\u00e9 de patata. \u00bfEst\u00e1s segura de que lo empujaste hasta su zona hadal?\n\n\u2014S\u00ed.\n\n\u2014Entonces tardar\u00e1n como m\u00ednimo doce horas en extraerle los recuerdos. Me sorprende que ese pobre desgraciado siga vivo.\n\n\u2014\u00bfQu\u00e9 me est\u00e1s diciendo?\n\n\u2014Te estoy diciendo que ser\u00e1 mejor que esperes y no te des de cabeza con una persecuci\u00f3n. Est\u00e1s m\u00e1s segura con tu papa\u00edto de Scion que aqu\u00ed.\n\n\u2014Tienen esta direcci\u00f3n. No puedo quedarme sentada esperando a que vengan a detenerme.\n\n\u2014No te van a detener, cari\u00f1o m\u00edo. Hazme caso. Qu\u00e9date en casa, duerme un poco, y ma\u00f1ana por la ma\u00f1ana enviar\u00e9 a Nick con el coche. \u00bfQu\u00e9 te parece?\n\n\u2014No me gusta.\n\n\u2014No tiene por qu\u00e9 gustarte. T\u00fa duerme para estar guapa y fresca. Aunque no te hace ninguna falta \u2014a\u00f1adi\u00f3\u2014. Por cierto, \u00bfpuedes hacerme un favor? P\u00e1sate por Grub Street ma\u00f1ana y recoge esas eleg\u00edas de Donne que tiene Minty, \u00bfquieres? No puedo creer que haya vuelto su esp\u00edritu, es completamente...\n\nLe colgu\u00e9.\n\nJax era un capullo. Un genio, s\u00ed, pero tambi\u00e9n un adulador, un agarrado, un insensible y un capullo, como todos los mimetocapos. Pero \u00bfa qui\u00e9n pod\u00eda acudir yo? Con un don como el m\u00edo, era m\u00e1s vulnerable si estaba sola. Jax era el menor de dos males.\n\nNo pude evitar sonre\u00edr al pensar eso. Que Jaxon Hall fuera el menor de dos males dec\u00eda mucho de c\u00f3mo estaba el mundo.\n\nNo pod\u00eda dormir. Ten\u00eda que prepararme. Hab\u00eda una minipistola en un caj\u00f3n, escondida bajo un mont\u00f3n de ropa. Tambi\u00e9n hab\u00eda una primera edici\u00f3n de uno de los panfletos de Jaxon, _Sobre los m\u00e9ritos de la antinaturalidad_ ; __ en \u00e9l enumeraba __ los distintos tipos de vidente, seg\u00fan sus investigaciones. Mi ejemplar estaba lleno de anotaciones hechas por \u00e9l: ideas nuevas, n\u00fameros de contacto de videntes. Cargu\u00e9 la pistola y saqu\u00e9 una mochila de debajo de la cama. Mi mochila de emergencia, guardada all\u00ed dos a\u00f1os atr\u00e1s, lista para el d\u00eda que tuviera que huir. Met\u00ed el panfleto en el bolsillo delantero. No pod\u00eda arriesgarme a que lo encontraran en casa de mi padre.\n\nMe tumb\u00e9 en la cama, sin quitarme la ropa, con una mano sobre la pistola. O\u00ed truenos a lo lejos, en la oscuridad.\n\nDeb\u00ed de quedarme dormida. Cuando despert\u00e9, not\u00e9 algo raro. El \u00e9ter estaba demasiado abierto. Hab\u00eda videntes en el edificio, en la escalera. No era la se\u00f1ora Heron, la anciana del piso de arriba, que utilizaba un andador y siempre sub\u00eda en ascensor. Eran las botas de una unidad de asalto.\n\nHab\u00edan venido a buscarme. Por fin hab\u00edan venido.\n\nMe levant\u00e9 de inmediato. Me puse una chaqueta encima de la camisa, los zapatos y los mitones. Me temblaban las manos. Para eso era para lo que Nick me hab\u00eda entrenado: para correr como un animal salvaje. Pod\u00eda llegar a la estaci\u00f3n si me lo propon\u00eda, pero esa carrera pondr\u00eda a prueba mi resistencia. Tendr\u00eda que encontrar y parar un taxi para llegar al sector 4. Los taxis piratas paraban a cualquiera con tal de sacarse unas monedas, aunque fueras un vidente fugitivo.\n\nMe colgu\u00e9 la mochila, met\u00ed la pistola en un bolsillo de la chaqueta y abr\u00ed la puerta del balc\u00f3n, que el viento hab\u00eda cerrado. La lluvia me aporre\u00f3 la ropa. Cruc\u00e9 el balc\u00f3n, me sub\u00ed al antepecho de la ventana de la cocina, me agarr\u00e9 al borde del tejado y me sub\u00ed a \u00e9l de un fuerte tir\u00f3n. Cuando llegaron al apartamento, yo ya hab\u00eda empezado a correr.\n\n\u00a1Pum! Echaron la puerta abajo sin llamar, sin avisar. Al cabo de un momento el sonido de un disparo raj\u00f3 la noche. Me obligu\u00e9 a seguir corriendo. No pod\u00eda volver. Nunca mataban a amaur\u00f3ticos sin motivo, y menos a\u00fan a empleados de Scion. Seguramente deb\u00eda de haber sido solo un disparo tranquilizador, para controlar a mi padre mientras me deten\u00edan. Para abatirme a m\u00ed iban a necesitar algo mucho m\u00e1s fuerte.\n\nLa urbanizaci\u00f3n estaba tranquila. Mir\u00e9 por encima del borde del tejado, para inspeccionarla. No vi al vigilante de seguridad, que deb\u00eda de estar otra vez haciendo la ronda. No tard\u00e9 mucho en encontrar el furg\u00f3n policial en el aparcamiento: una furgoneta con las ventanas tapadas y los faros encendidos. Si alguien se hubiera molestado en mirar, habr\u00eda visto el s\u00edmbolo de Scion en las puertas traseras.\n\nSalv\u00e9 una brecha y trep\u00e9 a una cornisa peligrosamente resbaladiza. Mis zapatos y mis guantes se agarraban bastante bien, pero tendr\u00eda que vigilar d\u00f3nde pon\u00eda los pies. Pegu\u00e9 la espalda a la pared y avanc\u00e9 despacio hacia una escalerilla de emergencia; la lluvia me pegaba el pelo a la cara. Trep\u00e9 hasta el balc\u00f3n de hierro forjado del siguiente piso y, una vez all\u00ed, forc\u00e9 una ventana peque\u00f1a. Atraves\u00e9 el apartamento vac\u00edo a toda velocidad, baj\u00e9 los tres tramos de escalones y sal\u00ed por el portal del edificio. Necesitaba salir a la calle, perderme por un callej\u00f3n oscuro.\n\nLuces rojas. La DVN estaba aparcada justo delante, bloqueando mi ruta de huida. Di media vuelta, cerr\u00e9 la puerta de golpe y activ\u00e9 el cierre de seguridad. Con manos temblorosas, cog\u00ed un hacha contra incendios de una vitrina, romp\u00ed la ventana de la planta baja y sal\u00ed por ella a un peque\u00f1o patio. Me cort\u00e9 en los brazos con el cristal. Volv\u00eda a estar bajo la lluvia, trepando por ca\u00f1os de desag\u00fce y alf\u00e9izares, agarr\u00e1ndome por los pelos, hasta que llegu\u00e9 al tejado.\n\nCuando los vi se me cort\u00f3 la respiraci\u00f3n. El exterior del edificio estaba infestado de hombres con camisa roja y chaqueta negra. Varias linternas me enfocaron, me apuntaron a los ojos. Era la primera vez que ve\u00eda ese uniforme en Londres. \u00bfEran de Scion?\n\n\u2014No te muevas.\n\nEl que estaba m\u00e1s cerca avanz\u00f3 hacia m\u00ed. Llevaba una pistola en la mano; usaba guantes. Percib\u00ed un aura muy intensa y me apart\u00e9. El l\u00edder de los soldados era un m\u00e9dium extremadamente poderoso. La luz de las linternas revel\u00f3 una cara demacrada, unos ojos peque\u00f1os y una boca grande de labios finos.\n\n\u2014\u00a1No corras, Paige! \u2014me grit\u00f3 desde el otro extremo del tejado\u2014. \u00bfPor qu\u00e9 no sales de debajo de la lluvia?\n\nHice un r\u00e1pido barrido del entorno. El edificio de al lado era un bloque de oficinas abandonado. Tendr\u00eda que dar un salto enorme, porque deb\u00eda de haber unos seis metros de separaci\u00f3n, y debajo hab\u00eda una calle concurrida. Nunca hab\u00eda intentado saltar tanta distancia, pero, a menos que quisiera atacar al m\u00e9dium y abandonar mi cuerpo, tendr\u00eda que intentarlo.\n\n\u2014Mejor que no \u2014dije, y ech\u00e9 a correr.\n\nLos soldados dieron un grito de alarma. Salt\u00e9 a una parte m\u00e1s baja del tejado. El m\u00e9dium me persigui\u00f3. O\u00eda sus pasos por el tejado, pegados a los m\u00edos; estaba entrenado para aquellas persecuciones. No pod\u00eda permitirme parar ni un instante. Era ligera y lo bastante delgada para colarme entre barrotes y por debajo de las vallas, pero mi perseguidor tambi\u00e9n. Cuando dispar\u00e9 con la pistola por encima del hombro, \u00e9l se agach\u00f3 sin detenerse. El viento arrastr\u00f3 su risa, de modo que no supe calcular a qu\u00e9 distancia estaba.\n\nVolv\u00ed a guardarme la pistola en la chaqueta. No ten\u00eda sentido disparar; no pod\u00eda acertar. Flexion\u00e9 los dedos y me prepar\u00e9 para agarrar un canal\u00f3n. Me ard\u00edan los m\u00fasculos y ten\u00eda los pulmones a punto de estallar. Un dolor en el tobillo me alert\u00f3 de una lesi\u00f3n, pero ten\u00eda que continuar. Pelear o huir. Correr o morir.\n\nEl m\u00e9dium salt\u00f3 por encima de la cornisa, r\u00e1pido y fluido como el agua. La adrenalina corr\u00eda por mis venas. Mis piernas se mov\u00edan con fuerza y la lluvia me golpeaba en los ojos. Salt\u00e9 por encima de tubos flexibles y conductos de ventilaci\u00f3n. A medida que ganaba velocidad, trataba de dirigir mi sexto sentido hacia el m\u00e9dium. Ten\u00eda una mente poderosa, tan r\u00e1pida como su cuerpo. No consegu\u00eda inmovilizarla, ni siquiera definirla. No pod\u00eda hacer nada para disuadirlo.\n\nLa adrenalina iba paliando el dolor de mi tobillo. De pronto me encontr\u00e9 ante un abismo de quince pisos. Al otro lado de la brecha hab\u00eda un canal\u00f3n y, m\u00e1s all\u00e1, una escalera de incendios. Si lograba bajar por ella, podr\u00eda desaparecer en las arterias palpitantes del sector 5. Podr\u00eda huir. S\u00ed, lo conseguir\u00eda. O\u00eda la voz de Nick alent\u00e1ndome: \u00abLas rodillas hacia el pecho. Los ojos en el punto donde esperas aterrizar\u00bb. Ahora o nunca. Me di impulso hacia arriba y me lanc\u00e9 por encima del precipicio.\n\nMi cuerpo choc\u00f3 contra una pared s\u00f3lida de ladrillo. Se me parti\u00f3 el labio, pero no perd\u00ed el conocimiento. Mis dedos se agarraron al canal\u00f3n. Mis pies golpearon la pared. Emple\u00e9 la poca fuerza que me quedaba para impulsarme hacia arriba, y el borde del canal\u00f3n se me clav\u00f3 en las manos. Se me cay\u00f3 una moneda de la chaqueta, y se precipit\u00f3 hacia la calle oscura.\n\nMi victoria dur\u00f3 poco. Cuando consegu\u00ed subirme al borde del tejado, con las palmas de las manos en carne viva, un dolor atroz me recorri\u00f3 la espalda. Estuve a punto de soltarme, pero segu\u00ed agarrada al tejado con una mano. Estir\u00e9 el cuello para mirar hacia atr\u00e1s, jadeando. Ten\u00eda un dardo largo y estrecho clavado en la parte baja de la espalda.\n\nFlux. Ten\u00edan flux.\n\nEl f\u00e1rmaco se extendi\u00f3 r\u00e1pidamente por mis venas. Al cabo de seis segundos todo mi torrente sangu\u00edneo estaba comprometido. Pens\u00e9 dos cosas: la primera, que Jax me matar\u00eda; y la segunda, que no importaba, porque de todas formas iba a morir. Me solt\u00e9 del tejado.\n\nNada.\n\nDur\u00f3 una eternidad. No recordaba cu\u00e1ndo hab\u00eda empezado, y no ve\u00eda el fin.\n\nRecordaba movimiento, un rugido ronco, que me hab\u00edan atado a una superficie dura. Luego una aguja, y el dolor apoder\u00e1ndose de m\u00ed.\n\nLa realidad estaba deformada. Me hallaba cerca de una vela, pero la llama crec\u00eda hasta alcanzar las dimensiones de un infierno. Estaba atrapada en un horno. El sudor me goteaba por los poros como la cera. Estaba hecha de fuego. Ard\u00eda. Me sal\u00edan ampollas y me chamuscaba; luego me congelaba, anhelaba acercarme a una fuente de calor, cre\u00eda que iba a morir de fr\u00edo. No hab\u00eda t\u00e9rmino medio, solo un dolor infinito, ilimitado.\n\nEl AUP Fluxion 14 era el resultado de un proyecto de colaboraci\u00f3n entre los departamentos m\u00e9dico y militar de Scion. Produc\u00eda un efecto paralizante llamado fantasmagor\u00eda, que los videntes resentidos llamaban \u00abpeste cerebral\u00bb: una v\u00edvida serie de alucinaciones, causada por distorsiones del onirosaje humano. Intent\u00e9 superar una visi\u00f3n tras otra, chillando cada vez que el dolor se hac\u00eda demasiado intenso para soportarlo en silencio. Si tuviera que definir el infierno, dir\u00eda que es eso: un chute de flux.\n\nHac\u00eda arcadas tratando en vano de expulsar el veneno de mi cuerpo; el pelo se me adher\u00eda a la cara, mojada de l\u00e1grimas y sudor. Lo \u00fanico que quer\u00eda era que terminara todo. Algo ten\u00eda que librarme de aquella pesadilla, ya fuera el sue\u00f1o, la inconsciencia o la muerte.\n\n\u2014Bueno, tesoro. No queremos que te mueras todav\u00eda. Hoy ya hemos perdido a tres.\n\nUnos dedos fr\u00edos me acariciaron la frente. Arque\u00e9 la espalda, me apart\u00e9. Si no quer\u00edan que muriera, \u00bfpor qu\u00e9 me hac\u00edan aquello?\n\nPasaban flores secas roz\u00e1ndome los ojos. La habitaci\u00f3n se enroll\u00f3 formando una h\u00e9lice, daba vueltas y vueltas hasta que no supe d\u00f3nde estaba arriba y d\u00f3nde abajo. Mord\u00ed una almohada para ahogar mis gritos. Not\u00e9 sabor a sangre y deduje que hab\u00eda mordido otra cosa: mi labio, mi lengua, mi mejilla; no sab\u00eda qu\u00e9.\n\nEl flux no sal\u00eda de tu cuerpo as\u00ed como as\u00ed. Por muchas veces que vomitaras u orinaras, segu\u00eda circulando, transportado por tu sangre, reproducido por tus propias c\u00e9lulas, hasta que lograbas inyectarte el ant\u00eddoto. Intent\u00e9 suplicar, pero no consegu\u00eda articular palabra. Me invad\u00eda una oleada tras otra de dolor, hasta que me convenc\u00ed de que iba a morir.\n\nEntonces percib\u00ed otra voz.\n\n\u2014Basta. A esta la necesitamos viva. Trae el ant\u00eddoto, o me encargar\u00e9 de que te administren el doble de la dosis que le han administrado a ella.\n\n\u00a1El ant\u00eddoto! Quiz\u00e1 sobreviviera. Intent\u00e9 ver algo m\u00e1s all\u00e1 del velo ondulado de las visiones, pero solo consegu\u00ed distinguir la llama de la vela.\n\nEstaban tardando demasiado. \u00bfD\u00f3nde estaba mi ant\u00eddoto? No importaba mucho. Quer\u00eda dormir, sumirme en el m\u00e1s largo de los sue\u00f1os.\n\n\u2014Su\u00e9lteme \u2014dije\u2014. D\u00e9jeme salir.\n\n\u2014Est\u00e1 hablando. Trae agua.\n\nEl fr\u00edo borde de un vaso choc\u00f3 contra mis dientes. Beb\u00ed a grandes tragos, sedienta. Levant\u00e9 la cabeza e intent\u00e9 ver la cara de mi salvador.\n\n\u2014Por favor \u2014dije.\n\nUnos ojos me miraron. Estallaron en llamas.\n\nY de pronto ces\u00f3 la pesadilla. Me sum\u00ed en un sue\u00f1o negro y profundo.\n\nCuando despert\u00e9, me qued\u00e9 muy quieta.\n\nTodav\u00eda ten\u00eda sensibilidad suficiente para hacerme una idea de d\u00f3nde estaba: tumbada boca abajo en un colch\u00f3n duro. Ten\u00eda la garganta abrasada. Era un dolor tan intenso que me oblig\u00f3 a recobrar el sentido, aunque solo fuera para buscar agua. Entonces me di cuenta de que estaba desnuda, y me sobresalt\u00e9.\n\nMe tumb\u00e9 sobre un costado y apoy\u00e9 el peso del cuerpo en un codo y una cadera. Ten\u00eda restos de v\u00f3mito en las comisuras de la boca. Cuando pude enfocar, busqu\u00e9 el \u00e9ter. Hab\u00eda otros videntes conmigo en aquella c\u00e1rcel.\n\nMis ojos tardaron un rato en adaptarse a la penumbra. Estaba en una cama individual, con s\u00e1banas fr\u00edas y h\u00famedas. A mi derecha hab\u00eda una ventana con barrotes, sin cristal. El suelo y las paredes eran de piedra. Una fuerte corriente de aire me hizo estremecer. Echaba nubes de vaho por la boca al respirar. Me tap\u00e9 con la s\u00e1bana hasta los hombros. \u00bfQui\u00e9n demonios me hab\u00eda quitado la ropa?\n\nEn un rinc\u00f3n hab\u00eda una puerta entreabierta. Vi una luz. Me levant\u00e9 y evalu\u00e9 mis fuerzas. Tras asegurarme de que no iba a caerme, fui hacia la luz. Lo que encontr\u00e9 fue un lavabo rudimentario. La luz proven\u00eda de una sola vela. Hab\u00eda un v\u00e1ter viejo y un grifo oxidado, colocado a bastante altura en la pared. El grifo estaba fr\u00edo al tacto. Gir\u00e9 una v\u00e1lvula que hab\u00eda cerca y me sepult\u00f3 un diluvio de agua helada. Intent\u00e9 girar la v\u00e1lvula hacia el otro lado, pero el agua se resisti\u00f3 a calentarse m\u00e1s de medio grado. Decid\u00ed ir moj\u00e1ndome las extremidades una a una, coloc\u00e1ndolas bajo aquella cosa que no merec\u00eda llamarse ducha. No hab\u00eda toallas, as\u00ed que me sequ\u00e9 con las s\u00e1banas de la cama, sin dejar de envolverme con una. Intent\u00e9 abrir la puerta de la habitaci\u00f3n, pero estaba cerrada con llave.\n\nMe picaba todo el cuerpo. No ten\u00eda ni idea de d\u00f3nde estaba, ni por qu\u00e9 estaba all\u00ed, ni qu\u00e9 pensaban hacerme mis captores. Nadie sab\u00eda qu\u00e9 les pasaba a los detenidos, porque nunca hab\u00eda vuelto ninguno.\n\nMe sent\u00e9 en la cama y respir\u00e9 hondo varias veces. Todav\u00eda estaba d\u00e9bil tras tantas horas de fantasmagor\u00eda, y no necesitaba un espejo para saber que parec\u00eda, m\u00e1s que nunca, un cad\u00e1ver.\n\nNo temblaba solo de fr\u00edo. Estaba desnuda y sola en una habitaci\u00f3n oscura, con barrotes en la ventana y sin rastro de una ruta de huida. Deb\u00edan de haberme llevado a la Torre. Y me hab\u00edan quitado la mochila y el panfleto. Me acurruqu\u00e9 contra el pilar de la cama e hice todo lo posible por conservar el calor corporal mientras el coraz\u00f3n me lat\u00eda con fuerza. Ten\u00eda un nudo en la dolorida garganta.\n\n\u00bfLe har\u00edan algo a mi padre? \u00c9l era valioso (buena materia prima), pero \u00bflo perdonar\u00edan por haber albergado a una vidente? Eso era encubrimiento de traici\u00f3n. Pero mi padre era importante. No pod\u00edan perderlo.\n\nPerd\u00ed la noci\u00f3n del tiempo durante un rato. Me sum\u00ed en un sue\u00f1o irregular. Por fin la puerta se abri\u00f3 de golpe, y despert\u00e9 bruscamente.\n\n\u2014Lev\u00e1ntate.\n\nUna l\u00e1mpara de queroseno entr\u00f3, oscilante, en la habitaci\u00f3n. La sujetaba una mujer. Ten\u00eda la piel morena y lustrosa, y una estructura \u00f3sea elegante; era m\u00e1s alta que yo. El cabello, largo y rizado, era negro, como el vestido de talle alto, cuyas mangas le llegaban hasta las puntas de los dedos enguantados. Era imposible discernir su edad: podr\u00eda haber tenido veinticinco a\u00f1os, o cuarenta. Me ce\u00f1\u00ed la s\u00e1bana alrededor del cuerpo mientras la observaba.\n\nMe llamaron la atenci\u00f3n tres cosas de aquella mujer. La primera, que ten\u00eda los ojos amarillos. No era ese tono ambarino que, con seg\u00fan qu\u00e9 luz, pod\u00edas llamar amarillo. Eran realmente amarillos, un poco verdosos, y resplandec\u00edan.\n\nLa segunda, su aura. Era vidente, pero nunca hab\u00eda visto un aura como la suya. No habr\u00eda sabido explicar por qu\u00e9 era tan extra\u00f1a, pero mis sentidos no encajaban bien con ella.\n\nY la tercera (y la que me hel\u00f3 la sangre en las venas), su onirosaje. Era id\u00e9ntico al que hab\u00eda percibido en el I-4, ese que no hab\u00edamos sabido identificar. El del desconocido. Mi instinto me impulsaba a atacarla, pero sab\u00eda que no podr\u00eda abrir una brecha en un onirosaje como aquel, y mucho menos en el estado en que me encontraba.\n\n\u2014\u00bfEstamos en la Torre? \u2014pregunt\u00e9 con voz ronca.\n\nLa mujer ignor\u00f3 mi pregunta. Me acerc\u00f3 la l\u00e1mpara a la cara y me escudri\u00f1\u00f3 los ojos. Me pregunt\u00e9 si todav\u00eda estar\u00eda sufriendo peste cerebral.\n\n\u2014T\u00f3mate esto \u2014dijo.\n\nMir\u00e9 las dos p\u00edldoras que ten\u00eda en la mano.\n\n\u2014T\u00f3matelas.\n\n\u2014No \u2014dije.\n\nMe peg\u00f3. Not\u00e9 sabor a sangre. Quer\u00eda devolverle el golpe, pelear, pero estaba tan d\u00e9bil que apenas pude levantar la mano. Me tom\u00e9 las p\u00edldoras con dificultad, por culpa de la herida del labio.\n\n\u2014T\u00e1pate \u2014dijo mi captora\u2014. Si vuelves a desobedecerme, me asegurar\u00e9 de que no salgas de esta habitaci\u00f3n. Al menos, no con la piel sobre los huesos.\n\nMe tir\u00f3 un fardo de ropa.\n\n\u2014Rec\u00f3gela.\n\nNo quer\u00eda que me volviera a pegar. Esa vez no lo aguantar\u00eda. Apret\u00e9 las mand\u00edbulas y recog\u00ed la ropa del suelo.\n\n\u2014P\u00f3ntela.\n\nMir\u00e9 la ropa; me goteaba sangre del labio, y apareci\u00f3 una mancha en el blus\u00f3n blanco que ten\u00eda en las manos. Ten\u00eda las mangas largas y el escote cuadrado. Tambi\u00e9n hab\u00eda un faj\u00edn negro, a juego con los pantalones, los calcetines y las botas, ropa interior negra y un chaleco negro con una peque\u00f1a ancla blanca bordada. El s\u00edmbolo de Scion. Me vest\u00ed con movimientos r\u00edgidos, obligando a mis fr\u00edas extremidades a moverse. Cuando termin\u00e9, la mujer se volvi\u00f3 hacia la puerta.\n\n\u2014S\u00edgueme. No hables con nadie.\n\nFuera de la habitaci\u00f3n hac\u00eda un fr\u00edo tremendo, y la ra\u00edda alfombra no contribu\u00eda a suavizar la temperatura. En su d\u00eda deb\u00eda de haber sido roja, pero estaba deste\u00f1ida y manchada de v\u00f3mito. Mi gu\u00eda me llev\u00f3 por un laberinto de pasillos de piedra, con ventanitas con barrotes y antorchas encendidas. Aquello parec\u00eda demasiado luminoso, demasiado crudo, tras la luz fr\u00eda y azulada de las farolas de Londres.\n\n\u00bfSer\u00eda un castillo? No sab\u00eda de nadie en dos mil kil\u00f3metros a la redonda de Londres que tuviera un castillo (no hab\u00edamos tenido monarcas desde la reina Victoria). Quiz\u00e1 fuera una de aquellas viejas c\u00e1rceles de Categor\u00eda D. A menos que fuera la Torre.\n\nMe aventur\u00e9 a echar un vistazo fuera. Era de noche, pero distingu\u00ed un patio iluminado por varios faroles. No sab\u00eda cu\u00e1nto tiempo hab\u00eda estado bajo los efectos del flux. \u00bfMe habr\u00eda visto esa mujer mientras sufr\u00eda? \u00bfRecib\u00eda \u00f3rdenes de la DVN, o recib\u00edan ellos \u00f3rdenes suyas? Quiz\u00e1 trabajara para el Arconte, pero me extra\u00f1aba que emplearan a una vidente. Y pod\u00eda ser otras cosas, pero desde luego era vidente.\n\nLa mujer se par\u00f3 ante una puerta. Empujaron a un ni\u00f1o al pasillo. Era un cr\u00edo muy flaco, con cara de rata, con una mata de pelo rubio rojizo y todos los s\u00edntomas de la intoxicaci\u00f3n por flux: ojos vidriosos, cara p\u00e1lida, labios azules. La mujer lo mir\u00f3 de arriba abajo.\n\n\u2014\u00bfNombre?\n\n\u2014Carl \u2014contest\u00f3 \u00e9l con voz ronca.\n\n\u2014\u00bfC\u00f3mo dices?\n\n\u2014Carl. \u2014Se notaba que estaba desesperado de dolor.\n\n\u2014Muy bien, te felicito por haber sobrevivido al Fluxion 14, Carl. \u2014Por su tono de voz no parec\u00eda que lo celebrara en absoluto\u2014. Quiz\u00e1 no vuelvas a disfrutar de unas horas de sue\u00f1o hasta dentro de un tiempo.\n\nCarl y yo nos miramos. Yo sab\u00eda que deb\u00eda de ofrecer un aspecto tan lamentable como \u00e9l. Seguimos recorriendo pasillos y fuimos recogiendo a m\u00e1s videntes cautivos. Sus auras eran potentes y distintivas; fui adivinando qu\u00e9 era cada uno. Un profeta. Una quirom\u00e1ntica (o palmista) con el cabello corto, te\u00f1ido de azul el\u00e9ctrico. Un tase\u00f3grafo. Un or\u00e1culo con la cabeza rapada. Una chica morena y delgada, de labios finos, seguramente una suspirante, que por lo visto ten\u00eda un brazo roto. Ninguno aparentaba m\u00e1s de veinte a\u00f1os, ni menos de quince. Todos estaban p\u00e1lidos y enfermizos a causa del flux. Al final \u00e9ramos diez. La mujer se volvi\u00f3 hacia su peque\u00f1o reba\u00f1o de monstruos.\n\n\u2014Me llamo Pleione Sualocin \u2014dijo\u2014. Ser\u00e9 vuestra gu\u00eda en vuestro primer d\u00eda en Sheol I. Esta noche asistir\u00e9is al serm\u00f3n de bienvenida. Hay una serie de normas sencillas que deb\u00e9is cumplir. No pod\u00e9is mirar a ning\u00fan refa\u00edta a los ojos. Deb\u00e9is mirar al suelo, donde corresponde, a menos que os inviten a hacer otra cosa.\n\nSin apartar la mirada de sus pies, la palmista levant\u00f3 una mano.\n\n\u2014\u00bfQu\u00e9 es un refa\u00edta?\n\n\u2014Pronto lo averiguar\u00e9is. \u2014Pleione hizo una pausa\u2014. Y otra norma: no deb\u00e9is hablar a menos que un refa\u00edta se dirija a vosotros. \u00bfTen\u00e9is alguna duda?\n\n\u2014S\u00ed. \u2014Era el tase\u00f3grafo. No miraba al suelo\u2014. \u00bfD\u00f3nde estamos?\n\n\u2014Lo sabr\u00e9is enseguida.\n\n\u2014\u00bfCon qu\u00e9 derecho nos tratan as\u00ed? Yo ni siquiera estaba limosneando. No he transgredido la ley. \u00a1Demuestren que tengo aura! Pienso volver a la ciudad, y usted no es nadie para...\n\nSe detuvo. Le salieron sendas gotas de sangre de los ojos. Hizo un d\u00e9bil ruido y se derrumb\u00f3.\n\nLa palmista dio un grito.\n\nPleione mir\u00f3 al tase\u00f3grafo. Cuando volvi\u00f3 a mirarnos a los dem\u00e1s, ten\u00eda los ojos azules como una llamarada de gas. Desvi\u00e9 la mirada.\n\n\u2014\u00bfAlguna pregunta m\u00e1s?\n\nLa palmista se tap\u00f3 la boca con una mano.\n\nNos condujeron a una habitaci\u00f3n peque\u00f1a con las paredes y el suelo h\u00famedos, oscura como una cripta. Pleione nos encerr\u00f3 bajo llave y se march\u00f3.\n\nAl principio nadie se atrevi\u00f3 a hablar. La palmista sollozaba, al borde de la histeria. Los otros todav\u00eda estaban demasiado d\u00e9biles como para hablar. Yo me apart\u00e9 y me sent\u00e9 en un rinc\u00f3n. Ten\u00eda la piel de gallina bajo las mangas del blus\u00f3n.\n\n\u2014\u00bfTodav\u00eda estamos en la Torre? \u2014pregunt\u00f3 un augur\u2014. Parece la Torre.\n\n\u2014C\u00e1llate \u2014repuso alguien\u2014. C\u00e1llate ya.\n\nAlguien empez\u00f3 a rezarle al _zeitgeist_ , nada menos. Como si fuera a servir de algo. Apoy\u00e9 la barbilla en las rodillas. No quer\u00eda saber qu\u00e9 nos har\u00edan. No sab\u00eda cu\u00e1nto resistir\u00eda si me aplicaban el submarino. Hab\u00eda o\u00eddo hablar a mi padre del submarino, dec\u00eda que solo te dejaban respirar unos segundos. Dec\u00eda que no era tortura, sino terapia.\n\nUn profeta se sent\u00f3 a mi lado. Era calvo y de hombros anchos. No lo ve\u00eda muy bien en aquella penumbra, pero distingu\u00ed sus grandes ojos, oscuros e intensos. Me tendi\u00f3 una mano.\n\n\u2014Me llamo Julian.\n\nNo parec\u00eda asustado, m\u00e1s bien tranquilo.\n\n\u2014Paige \u2014dije yo. Era mejor no usar apellidos. Carraspe\u00e9 y dije\u2014: \u00bfDe qu\u00e9 cohorte eres?\n\n\u2014Del IV-6.\n\n\u2014Yo, del I-4.\n\n\u2014Ese es el territorio del Vinculador Blanco, \u00bfno? \u2014Asent\u00ed\u2014. \u00bfDe qu\u00e9 parte?\n\n\u2014Soho \u2014respond\u00ed. Si dec\u00eda que estaba en Dials, \u00e9l deducir\u00eda que pertenec\u00eda al c\u00edrculo de Jaxon.\n\n\u2014Te envidio. Me habr\u00eda encantado vivir en el centro.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014All\u00ed el sindicato tiene mucha fuerza. En mi sector nunca pasa gran cosa. \u2014Hablaba en voz baja\u2014. \u00bfLes has dado motivos para detenerte?\n\n\u2014Mat\u00e9 a un metrovigilante. \u2014Me dol\u00eda la garganta\u2014. \u00bfY t\u00fa?\n\n\u2014Tuve una discusi\u00f3n sin importancia con un centinela. Resumiendo: el centinela ya no est\u00e1 con nosotros.\n\n\u2014Pero t\u00fa eres profeta.\n\nLa mayor\u00eda de los videntes miraban con desprecio a los profetas, una clase de adivinos. Como todos los adivinos, se comunicaban con los esp\u00edritus a trav\u00e9s de objetos; en el caso de los profetas, cualquier cosa reflectante. Jax odiaba con toda su alma a los adivinos (\u00abFarsantes, tesoro, ll\u00e1malos farsantes\u00bb). Y a los augures, ahora que lo pienso.\n\nJulian debi\u00f3 de leerme el pensamiento.\n\n\u2014No crees que un profeta sea capaz de matar a nadie.\n\n\u2014No con esp\u00edritus. No podr\u00edas controlar a una bandada lo bastante grande.\n\n\u2014Entiendes de videntes. \u2014Se frot\u00f3 los brazos\u2014. Tienes raz\u00f3n. Le dispar\u00e9. Pero eso no les impidi\u00f3 detenerme.\n\nNo le contest\u00e9. Del techo goteaba un agua helada que me ca\u00eda en el pelo y me resbalaba por la nariz. La mayor\u00eda de los otros prisioneros estaban callados. Un chico se mec\u00eda adelante y atr\u00e1s, en cuclillas.\n\n\u2014Tienes un aura rara. \u2014Julian me mir\u00f3\u2014. No s\u00e9 qu\u00e9 eres. Dir\u00eda que un or\u00e1culo, pero...\n\n\u2014Pero \u00bfqu\u00e9?\n\n\u2014Hace tiempo que no oigo hablar de ninguna mujer or\u00e1culo. Y no creo que seas una sibila.\n\n\u2014Soy acutum\u00e1ntica.\n\n\u2014\u00bfQu\u00e9 hiciste? \u00bfLe clavaste una aguja a alguien?\n\n\u2014Algo as\u00ed.\n\nFuera se oy\u00f3 un estruendo, seguido de un grito espantoso. Todos dejamos de hablar.\n\n\u2014Eso ha sido un berserker \u2014dijo una voz masculina cargada de temor\u2014. No ir\u00e1n a meter a un berserker aqu\u00ed, \u00bfverdad?\n\n\u2014Los berserkers no existen \u2014asever\u00e9.\n\n\u2014\u00bfNo has le\u00eddo _Sobre los m\u00e9ritos_?\n\n\u2014S\u00ed. Pero solo es un tipo hipot\u00e9tico.\n\nNo pareci\u00f3 que mi afirmaci\u00f3n lo aliviara mucho. Acordarme del panfleto hizo que se me helara a\u00fan m\u00e1s la sangre. Pod\u00eda estar en cualquier sitio, en manos de cualquiera: una primera edici\u00f3n del panfleto m\u00e1s sedicioso de la ciudadela, lleno de anotaciones y detalles de contactos. Era imposible que yo estuviera en posesi\u00f3n de una cosa as\u00ed sin conocer a su autor.\n\n\u2014Van a volver a torturarnos. \u2014La suspirante sosten\u00eda el brazo roto contra el pecho\u2014. Quieren algo. Si no, no nos habr\u00edan dejado salir.\n\n\u2014Salir \u00bfde d\u00f3nde? \u2014pregunt\u00e9.\n\n\u2014De la Torre, idiota. Donde todos hemos pasado los dos \u00faltimos a\u00f1os.\n\n\u2014\u00bfDos a\u00f1os? \u2014Se oy\u00f3 una risa medio hist\u00e9rica que proven\u00eda de un rinc\u00f3n\u2014. Querr\u00e1s decir nueve. Nueve a\u00f1os.\n\nOtra risotada, y luego una risita.\n\nNueve a\u00f1os. Que nosotros supi\u00e9ramos, a los detenidos se les ofrec\u00edan dos opciones: entrar en la DVN o ser ejecutados. No hab\u00eda ninguna necesidad de almacenar gente.\n\n\u2014\u00bfPor qu\u00e9 nueve? \u2014pregunt\u00e9.\n\nNo hubo respuesta desde el rinc\u00f3n. Al cabo de un minuto Julian dijo:\n\n\u2014\u00bfAlguien m\u00e1s se ha preguntado por qu\u00e9 no estamos muertos?\n\n\u2014A todos los dem\u00e1s los mataron. \u2014Era una voz nueva\u2014. Yo pas\u00e9 meses all\u00ed. A los otros videntes de mi ala los ahorcaron a todos. \u2014Una pausa\u2014. Nos han escogido por algo.\n\n\u2014SciOECI \u2014susurr\u00f3 alguien\u2014. Vamos a ser conejillos de Indias, \u00bfverdad? Los m\u00e9dicos nos quieren diseccionar.\n\n\u2014Esto no es SciOECI \u2014dije yo.\n\nHubo un largo silencio, solo interrumpido por las amargas l\u00e1grimas de la palmista. Por lo visto no pod\u00eda controlarse. Al final Carl se dirigi\u00f3 a la suspirante:\n\n\u2014Dices que deben de querer algo, susu. \u00bfQu\u00e9 crees que podr\u00edan querer?\n\n\u2014Cualquier cosa. Nuestra visi\u00f3n.\n\n\u2014No pueden quitarnos nuestra visi\u00f3n \u2014dije.\n\n\u2014Por favor. T\u00fa ni siquiera tienes visi\u00f3n. No quieren a videntes discapacitados.\n\nContuve el impulso de romperle el otro brazo.\n\n\u2014\u00bfQu\u00e9 le ha hecho al tase\u00f3grafo? \u2014La palmista estaba temblando\u2014. Sus ojos... \u00a1Lo ha hecho sin moverse siquiera!\n\n\u2014Pues yo ten\u00eda claro que nos iban a matar \u2014dijo Carl, como si no entendiera por qu\u00e9 los dem\u00e1s est\u00e1bamos tan preocupados. No ten\u00eda la voz tan ronca\u2014. Yo aceptar\u00eda cualquier cosa que no fuera la horca, \u00bfy vosotros?\n\n\u2014Pues puede que nos cuelguen \u2014dije, y se qued\u00f3 callado.\n\nOtro chico, tan p\u00e1lido que parec\u00eda que el flux le hubiera consumido toda la sangre de las venas, estaba empezando a hiperventilar. Ten\u00eda pecas en la nariz. Hasta ese momento no me hab\u00eda fijado en \u00e9l; no ten\u00eda ni rastro de aura.\n\n\u2014\u00bfD\u00f3nde estamos? \u2014pregunt\u00f3. Apenas pod\u00eda articular palabra\u2014. \u00bfQui\u00e9n... qui\u00e9nes sois?\n\nJulian lo mir\u00f3.\n\n\u2014T\u00fa eres amaur\u00f3tico \u2014dijo\u2014. \u00bfPor qu\u00e9 te han tra\u00eddo aqu\u00ed?\n\n\u2014\u00bfAmaur\u00f3tico?\n\n\u2014Debe de ser un error. \u2014El or\u00e1culo parec\u00eda aburrido\u2014. Lo matar\u00e1n de todas formas. Mala suerte, chico.\n\nDaba la impresi\u00f3n de que el chico iba a desmayarse. Se puso en pie de un salto y tir\u00f3 de los barrotes.\n\n\u2014\u00a1No s\u00e9 qu\u00e9 hago aqu\u00ed! \u00a1Quiero irme a mi casa! \u00a1No soy antinatural! \u2014Estaba a punto de llorar\u2014. \u00a1Lo siento! \u00a1Siento lo de la piedra!\n\nLe tap\u00e9 la boca con una mano.\n\n\u2014Basta. \u2014Algunos de los otros lo insultaron\u2014. \u00bfQuieres que te deje frito a ti tambi\u00e9n?\n\nEstaba temblando. Deb\u00eda de tener unos quince a\u00f1os, pero era muy inmaduro. Me acord\u00e9 de otros tiempos, cuando yo tambi\u00e9n era una ni\u00f1a asustada y sola.\n\n\u2014\u00bfC\u00f3mo te llamas? \u2014le pregunt\u00e9 tratando de que mi voz sonara amable.\n\n\u2014Seb. Seb Pearce. \u2014Se cruz\u00f3 de brazos, tratando de retraerse\u2014. \u00bfSois todos... antinaturales?\n\n\u2014S\u00ed, y haremos cosas antinaturales con tus \u00f3rganos internos si no cierras el pico \u2014le espet\u00f3 una voz.\n\nSeb se estremeci\u00f3.\n\n\u2014No le hagas caso \u2014le dije\u2014. Me llamo Paige. Este es Julian.\n\nJulian se limit\u00f3 a asentir con la cabeza. Por lo visto, me correspond\u00eda a m\u00ed charlar con el amaur\u00f3tico.\n\n\u2014\u00bfDe d\u00f3nde eres, Seb?\n\n\u2014De la cohorte III.\n\n\u2014El anillo \u2014observ\u00f3 Julian\u2014. Muy bonito.\n\nSeb desvi\u00f3 la mirada. Le temblaban los labios de fr\u00edo. Seguro que cre\u00eda que lo descuartizar\u00edamos y nos ba\u00f1ar\u00edamos en su sangre en un frenes\u00ed ocultista. Yo hab\u00eda ido a una escuela de secundaria del anillo, que era como llam\u00e1bamos en la calle a la cohorte III.\n\n\u2014Cu\u00e9ntanos qu\u00e9 pas\u00f3 \u2014dije.\n\nSeb mir\u00f3 a los dem\u00e1s. No pod\u00eda reprocharle su temor. Desde que ten\u00eda uso de raz\u00f3n le hab\u00edan explicado que los clarividentes eran el origen de todos los males del mundo, y ahora estaba en la c\u00e1rcel con ellos.\n\n\u2014Un chico de sexto me meti\u00f3 contrabando en la cartera \u2014dijo. Seguramente una piedra de adivinaci\u00f3n, el _numen_ m\u00e1s frecuente en el mercado negro\u2014. El director me vio intentando devolv\u00e9rselo en clase. Pens\u00f3 que me lo hab\u00eda dado uno de esos mendigos. Llamaron a los centinelas del colegio para que me registraran.\n\nUn chico de Scion, sin duda. Si su colegio ten\u00eda sus propios centinelas, deb\u00eda de provenir de una familia tremendamente rica.\n\n\u2014Tard\u00e9 horas en convencerlos de que me hab\u00edan tendido una trampa. Tom\u00e9 un atajo para volver a mi casa. \u2014Seb trag\u00f3 saliva\u2014. Hab\u00eda dos hombres vestidos de rojo en la esquina. Intent\u00e9 pasar de largo, pero me oyeron. Llevaban m\u00e1scaras. No s\u00e9 por qu\u00e9, pero ech\u00e9 a correr. Ten\u00eda miedo. Entonces o\u00ed un disparo y... creo que me desmay\u00e9. Y despert\u00e9 vomitando.\n\nNo sab\u00eda qu\u00e9 efecto tendr\u00eda el flux en los amaur\u00f3ticos. Lo l\u00f3gico era que aparecieran s\u00edntomas f\u00edsicos (v\u00f3mitos, sed, terror inexplicable), pero no la fantasmagor\u00eda.\n\n\u2014Qu\u00e9 horror \u2014dije\u2014. Seguro que todo esto es solo una equivocaci\u00f3n terrible.\n\nEstaba segura. Un chico amaur\u00f3tico de buena familia como Seb no pintaba nada all\u00ed.\n\nSeb parec\u00eda un poco m\u00e1s animado.\n\n\u2014\u00bfMe dejar\u00e1n volver a casa?\n\n\u2014No \u2014dijo Julian.\n\nAguc\u00e9 el o\u00eddo. Pasos. Pleione hab\u00eda vuelto. Abri\u00f3 la puerta, agarr\u00f3 al primer prisionero que encontr\u00f3 y lo levant\u00f3 con una sola mano.\n\n\u2014Seguidme. Recordad las normas.\n\nSalimos del edificio por una puerta de doble batiente; la suspirante guiaba a la palmista. El aire, g\u00e9lido, nos cortaba cada cent\u00edmetro de piel descubierta. Me sobresalt\u00e9 cuando llegamos a la horca (quiz\u00e1 aquello s\u00ed fuera la Torre), pero Pleione pas\u00f3 de largo. No ten\u00eda ni idea de qu\u00e9 le hab\u00eda hecho al tase\u00f3grafo, ni de qu\u00e9 hab\u00eda sido aquel grito, pero no pensaba preguntarlo. Cabeza agachada, ojos abiertos. Esa iba a ser mi norma all\u00ed.\n\nNos gui\u00f3 por calles desiertas, iluminadas por l\u00e1mparas de gas y mojadas tras una noche de lluvia intensa. Julian se coloc\u00f3 a mi lado. A medida que and\u00e1bamos, los edificios eran cada vez m\u00e1s grandes, pero no eran rascacielos, ni mucho menos. No hab\u00eda armazones met\u00e1licos, ni luz el\u00e9ctrica. Aquellos edificios eran antiguos y extra\u00f1os, construidos en una \u00e9poca en que dominaba otra est\u00e9tica. Fachadas de piedra, puertas de madera y ventanas con cristales emplomados te\u00f1idos de rojo intenso y violeta. Cuando doblamos la \u00faltima esquina, se present\u00f3 ante nosotros una imagen que nunca podr\u00e9 olvidar.\n\nLa calle que se extend\u00eda delante era extra\u00f1amente ancha. No se ve\u00eda ni un coche: solo una hilera larga y sinuosa de viviendas destartaladas que la recorr\u00eda de punta a punta. Las paredes de contrachapado sosten\u00edan los techos de chapa de zinc. A ambos lados de esa peque\u00f1a poblaci\u00f3n se alzaban otros edificios m\u00e1s altos. Ten\u00edan puertas de madera maciza, ventanas altas y almenas, como los castillos de la \u00e9poca victoriana. Me recordaron tanto a la Torre que tuve que mirar hacia otro lado.\n\nA escasa distancia de la chabola m\u00e1s cercana hab\u00eda un grupo de figuras esbeltas en un escenario al aire libre. Estaban rodeados de velas que iluminaban sus caras enmascaradas. Bajo las tablas sonaba un viol\u00edn. M\u00fasica de videntes, una m\u00fasica que solo los suspirantes pod\u00edan interpretar. Una nutrida audiencia los contemplaba desde abajo. Todos los espectadores llevaban un blus\u00f3n rojo y un chaleco negro.\n\nLas figuras empezaron a bailar, como si hubieran estado esperando nuestra llegada. Eran todos clarividentes; de hecho, all\u00ed todos eran clarividentes: los bailarines, los espectadores, todos. Jam\u00e1s hab\u00eda visto a tantos videntes en el mismo sitio, compartiendo apaciblemente el espacio. Deb\u00eda de haber un centenar de observadores alrededor del escenario.\n\nAquello no era ninguna reuni\u00f3n secreta en un t\u00fanel subterr\u00e1neo. Aquello no era el brutal sindicato de Hector. Aquello era diferente. Cuando Seb me dio la mano, no lo apart\u00e9 de m\u00ed.\n\nEl espect\u00e1culo se prolong\u00f3 unos minutos. No todos los espectadores prestaban atenci\u00f3n. Algunos hablaban entre ellos mientras que otros abucheaban a los actores. Estaba segura de haber o\u00eddo a alguien decir \u00abcobardes\u00bb. Despu\u00e9s del baile, una chica con mallas negras se subi\u00f3 a una plataforma m\u00e1s alta. Llevaba el cabello, casta\u00f1o oscuro, recogido en un mo\u00f1o, y una m\u00e1scara dorada con los extremos con forma de alas. Se qued\u00f3 all\u00ed de pie un momento, quieta como una estatua; de pronto salt\u00f3 de la plataforma y agarr\u00f3 dos largas cortinas rojas que hab\u00edan soltado desde el telar del escenario. Enrosc\u00f3 los brazos y las piernas alrededor de las cortinas y, tras trepar hasta una altura de seis metros, adopt\u00f3 una pose. Recibi\u00f3 algunos aplausos del p\u00fablico.\n\nTodav\u00eda estaba muy confusa por efecto del flux. \u00bfQu\u00e9 era aquello, una especie de secta de videntes? Cosas m\u00e1s raras hab\u00eda o\u00eddo. Hice un esfuerzo y examin\u00e9 la calle. De algo s\u00ed estaba segura: aquello no era SciLo. No hab\u00eda nada que delatara la presencia de Scion. Grandes edificios antiguos, espect\u00e1culos p\u00fablicos, l\u00e1mparas de gas y una calle adoquinada: era como si hubi\u00e9ramos retrocedido en el tiempo.\n\nDe pronto supe exactamente d\u00f3nde estaba.\n\nTodos hab\u00edamos o\u00eddo hablar de la ciudad perdida de Oxford. Formaba parte del programa de estudios de Scion. Un incendio hab\u00eda destruido la universidad en oto\u00f1o de 1859. Lo que quedaba estaba clasificado como Sector Restringido de Clase A. Estaba terminantemente prohibido acceder all\u00ed, por temor a una indefinible contaminaci\u00f3n. Scion la hab\u00eda borrado de los mapas. Yo hab\u00eda le\u00eddo en los archivos de Jaxon que un periodista intr\u00e9pido de _El Pendenciero_ hab\u00eda intentado entrar all\u00ed en 2036 y amenazado con publicar un art\u00edculo, pero unos francotiradores hicieron salir su coche de la carretera y nunca m\u00e1s se supo de \u00e9l. _El_ _Pendenciero_ , un peri\u00f3dico sensacionalista, tambi\u00e9n desapareci\u00f3. Hab\u00eda intentado demasiadas veces revelar los secretos de Scion.\n\nPleione se dio la vuelta hacia nosotros. La oscuridad nos imped\u00eda verle bien la cara, pero sus ojos segu\u00edan ardiendo.\n\n\u2014Es indecoroso quedarse mirando \u2014dijo\u2014. No deb\u00e9is llegar tarde al serm\u00f3n.\n\nPero aquel baile segu\u00eda atrayendo nuestras miradas. La seguimos, pero no consigui\u00f3 que apart\u00e1ramos la vista.\n\nDesfilamos detr\u00e1s de Pleione hasta llegar ante una gran verja de hierro forjado. Dos hombres abrieron la verja; ambos se parec\u00edan a nuestra gu\u00eda: los mismos ojos, la misma piel satinada, las mismas auras. Pleione pas\u00f3 majestuosa a su lado.\n\nSeb ten\u00eda muy mal color. Le cog\u00ed la mano y no se la solt\u00e9 mientras recorr\u00edamos los jardines del edificio. Aquel amaur\u00f3tico no ten\u00eda por qu\u00e9 importarme, pero parec\u00eda demasiado vulnerable para que lo dejara solo. La palmista estaba llorando. Solo el or\u00e1culo, que iba frot\u00e1ndose los nudillos, parec\u00eda no tener miedo. Mientras and\u00e1bamos, se nos unieron varios grupos m\u00e1s de reci\u00e9n llegados, todos vestidos de blanco. La mayor\u00eda parec\u00edan asustados, si bien unos pocos se mostraban entusiasmados. Nuestro grupo iba api\u00f1\u00e1ndose a medida que llegaban m\u00e1s prisioneros.\n\nNos estaban arreando como si fu\u00e9ramos ganado.\n\nEntramos en una sala alargada de techos altos. Unos estantes de color verde oliva llenos de preciosos libros antiguos cubr\u00edan las paredes hasta el techo. En una de las paredes hab\u00eda once vidrieras. La decoraci\u00f3n era cl\u00e1sica, con suelos de m\u00e1rmol con dibujos geom\u00e9tricos. Los prisioneros nos empuj\u00e1bamos unos a otros formando filas. Yo me qued\u00e9 entre Julian y Seb, con los sentidos en alerta m\u00e1xima. Julian tambi\u00e9n estaba en tensi\u00f3n. Miraba uno a uno a los cautivos, evalu\u00e1ndolos. Formaban un verdadero crisol: una muestra representativa de los distintos tipos de videntes, desde augures y adivinos hasta m\u00e9diums y sensores.\n\nPleione nos hab\u00eda dejado. Ahora estaba subida a un estrado, lo mismo que otros ocho seres que deduje que tambi\u00e9n deb\u00edan de ser refa\u00edtas. Mi sexto sentido tembl\u00f3.\n\nUna vez que estuvimos todos reunidos, un silencio sepulcral se apoder\u00f3 de la sala. Una mujer dio un paso adelante y empez\u00f3 a hablar.\n\n\u2014Bienvenidos a Sheol I.\n\nLa oradora med\u00eda casi dos metros. Sus facciones eran perfectamente sim\u00e9tricas: nariz recta y larga, p\u00f3mulos prominentes, ojos hundidos. La luz de las velas se reflejaba en su pelo y en su lustrosa piel. Vest\u00eda de negro, como los otros, pero las mangas y los costados de su t\u00fanica, acuchillados, dejaban entrever un forro dorado.\n\n\u2014Soy Nashira Sargas. \u2014Ten\u00eda una voz fr\u00eda, grave\u2014. Soy la soberana de sangre de la raza de los refa\u00edtas.\n\n\u2014\u00bfEs una broma? \u2014susurr\u00f3 alguien.\n\n\u2014\u00a1Chisss! \u2014susurr\u00f3 otro.\n\n\u2014En primer lugar, debo disculparme por el angustioso inicio de vuestra estancia aqu\u00ed, sobre todo si hab\u00e9is pasado un tiempo alojados en la Torre. La gran mayor\u00eda de los clarividentes tienen la impresi\u00f3n de que van a ser ejecutados cuando son llamados a nuestro lado. Utilizamos Fluxion 14 para asegurarnos de que el traslado a Sheol sea seguro y sencillo. Una vez sedados, se os lleva en tren a unas instalaciones de detenci\u00f3n, donde sois vigilados. Se os han confiscado la ropa y los objetos personales.\n\nMientras escuchaba examin\u00e9 a la mujer asom\u00e1ndome al \u00e9ter. Su aura no se parec\u00eda a nada que yo hubiera percibido hasta entonces. Me habr\u00eda gustado verla. Era como si hubiese cogido varios tipos de aura diferentes y hubiera forjado con ellos un extra\u00f1o campo energ\u00e9tico.\n\nPero hab\u00eda algo m\u00e1s. Una frialdad ins\u00f3lita. La mayor\u00eda de las auras emit\u00edan una se\u00f1al tenue y c\u00e1lida, como cuando pasas por delante de un calentador; la suya, en cambio, me produc\u00eda escalofr\u00edos.\n\n\u2014Comprendo que os haya sorprendido ver esta ciudad. Quiz\u00e1 la conozc\u00e1is por el nombre de Oxford. Vuestro gobierno neg\u00f3 su existencia hace dos siglos, mucho antes de que nacierais. Supuestamente se puso en cuarentena despu\u00e9s de un incendio, pero es mentira. La cerraron para que nosotros, los refa\u00edtas, pudi\u00e9ramos venir a vivir aqu\u00ed.\n\n\u00bbLlegamos hace dos siglos, en 1859. Vuestro mundo hab\u00eda alcanzado lo que llamamos el \"umbral et\u00e9reo\". \u2014Hizo una pausa y evalu\u00f3 nuestra reacci\u00f3n\u2014. La mayor\u00eda de vosotros sois clarividentes. Sab\u00e9is que vivimos rodeados de esp\u00edritus, demasiado cobardes o testarudos para enfrentarse a la muerte definitiva en el coraz\u00f3n del \u00e9ter. Pod\u00e9is comunicaros con ellos, y ellos pueden protegeros y guiaros. Pero esa conexi\u00f3n tiene un precio. Cuando el mundo corp\u00f3reo experimenta una superpoblaci\u00f3n de esp\u00edritus errantes, estos producen profundas fisuras en el \u00e9ter. Y cuando esas fisuras se ensanchan demasiado, el umbral et\u00e9reo se rompe.\n\n\u00bbCuando la Tierra rompi\u00f3 su umbral, qued\u00f3 expuesta a una dimensi\u00f3n m\u00e1s elevada, el Inframundo, donde residimos nosotros. Ahora hemos venido aqu\u00ed. \u2014Nashira dirigi\u00f3 la mirada a mi fila de prisioneros\u2014. Vosotros, los humanos, hab\u00e9is cometido muchos errores. Hab\u00e9is llenado vuestra f\u00e9rtil tierra de cad\u00e1veres, la hab\u00e9is cargado de esp\u00edritus errantes. Ahora pertenece a los refa\u00edtas.\n\nMir\u00e9 a Julian y vi mi miedo reflejado en sus ojos. Esa mujer ten\u00eda que estar loca.\n\nSe produjo un silencio en la sala. Nashira Sargas ten\u00eda toda nuestra atenci\u00f3n.\n\n\u2014Los miembros de mi pueblo, los refa\u00edtas, somos todos clarividentes. Entre nosotros no hay amaur\u00f3ticos. Desde que se produjo la escisi\u00f3n entre nuestros mundos, nos hemos visto obligados a compartir el Inframundo con una raza par\u00e1sita, los emim. Son seres salvajes, bestiales, a los que les gusta la carne humana. Si no fuera por nosotros, habr\u00edan traspasado el umbral. Habr\u00edan venido a destruiros.\n\nLoca. Estaba loca.\n\n\u2014Fuisteis todos detenidos por humanos que trabajan para nosotros. Se llaman \u00abcasacas rojas\u00bb. \u2014Nashira se\u00f1al\u00f3 a una fila de hombres y mujeres, todos vestidos de rojo, que estaban al fondo de la biblioteca\u2014. Desde nuestra llegada nos hemos hecho cargo de muchos clarividentes humanos. A cambio de protecci\u00f3n, os entrenamos para destruir a los emim (y, as\u00ed, proteger a la poblaci\u00f3n de \u00abnaturales\u00bb) formando parte de un batall\u00f3n penal. Esta ciudad act\u00faa como baliza para esos seres, los aleja del resto del mundo corp\u00f3reo. Cuando abren una brecha en sus muros, llamamos a los casacas rojas para que los destruyan. Esas brechas se anuncian mediante una sirena, pues exponen a la poblaci\u00f3n a un grave riesgo de mutilaci\u00f3n.\n\n\u00abY tambi\u00e9n hay un grave riesgo de que todo esto est\u00e9 pasando solo en mi cabeza\u00bb, me dije.\n\n\u2014Os ofrecemos este destino como alternativa a lo que os ofrece Scion: la pena de muerte en la horca o mediante asfixia. O, como algunos de vosotros ya hab\u00e9is experimentado, una larga y oscura condena en la Torre.\n\nEn la fila que ten\u00eda detr\u00e1s, una chica empez\u00f3 a gimotear. Las personas que ten\u00eda a los lados la hicieron callar.\n\n\u2014No tenemos que trabajar juntos, por supuesto. \u2014Nashira empez\u00f3 a pasearse a lo largo de la primera fila\u2014. Cuando vinimos a este mundo, comprobamos que era vulnerable. Solo un porcentaje m\u00ednimo de vosotros sois clarividentes, y a\u00fan sois menos los que ten\u00e9is habilidades m\u00ednimamente \u00fatiles. Habr\u00edamos podido dejar que los emim os atacaran. Habr\u00eda estado justificado, debido a lo que vosotros le hab\u00e9is hecho a este mundo.\n\nSeb me estaba aplastando la mano. Not\u00e9 un d\u00e9bil zumbido en los o\u00eddos. Aquello era rid\u00edculo. Una broma de mal gusto. O eso, o peste cerebral. S\u00ed, deb\u00eda de ser peste cerebral. Scion intentaba hacernos pensar que nos hab\u00edamos vuelto locos. Y quiz\u00e1 fuera verdad.\n\n\u2014Pero tuvimos clemencia. Nos apiadamos de vosotros. Negociamos con nuestros gobernantes, y creamos esta peque\u00f1a isla. Nos cedieron esta ciudad, que nosotros llamamos Sheol I, y cada diez a\u00f1os nos enviaban a cierto n\u00famero de clarividentes. Nuestra fuente principal era y sigue siendo Londres. Fue esa capital la que trabaj\u00f3 siete d\u00e9cadas para desarrollar el sistema de seguridad de Scion. Scion ha aumentado considerablemente la posibilidad de reconocer, reubicar y rehabilitar a los clarividentes en una nueva sociedad, lejos de los llamados amaur\u00f3ticos. A cambio de ese servicio, nosotros nos hemos comprometido a no destruir vuestro mundo. En lugar de destruirlo, nos hemos propuesto controlarlo.\n\nNo estaba segura de haber entendido lo que estaba diciendo, pero una cosa estaba clara: si dec\u00eda la verdad, Scion no era m\u00e1s que un gobierno t\u00edtere. Subordinado. Y nos hab\u00eda vendido.\n\nLa verdad es que no me sorprendi\u00f3.\n\nLa chica de la fila de detr\u00e1s no pudo soportarlo m\u00e1s. Dio un grito ahogado y ech\u00f3 a correr hacia la puerta.\n\nLe dispararon. No pudo hacer nada.\n\nHubo gritos por todas partes. Y sangre. Seb me clav\u00f3 las u\u00f1as en la mano. En medio del caos, un refa\u00edta se adelant\u00f3 y grit\u00f3:\n\n\u2014\u00a1Silencio!\n\nEl ruido ces\u00f3 de inmediato.\n\nBajo la cabeza de la chica se form\u00f3 un charco de sangre. Ten\u00eda los ojos abiertos. En su cara persist\u00eda una expresi\u00f3n de angustia, de terror.\n\nEl asesino era humano y vest\u00eda de rojo. Enfund\u00f3 su rev\u00f3lver y junt\u00f3 las manos detr\u00e1s de la espalda. Dos compa\u00f1eras suyas cogieron el cad\u00e1ver por los brazos y lo arrastraron hasta el exterior.\n\n\u2014Siempre hay un casaca amarilla \u2014dijo una de ellas, lo bastante alto para que lo oy\u00e9ramos todos.\n\nEl suelo de m\u00e1rmol estaba manchado. Nashira nos mir\u00f3 con gesto imperturbable.\n\n\u2014Si alguien m\u00e1s quiere huir, este es el momento de hacerlo. Pod\u00e9is estar seguros de que hay sitio en la tumba.\n\nNadie se movi\u00f3.\n\nSe produjo un silencio, y me aventur\u00e9 a echar un vistazo al estrado. Uno de los refa\u00edtas me miraba.\n\nDeb\u00eda de llevar rato examin\u00e1ndome, esperando a que yo lo viera, buscando un atisbo de disconformidad; su mirada se uni\u00f3 a la m\u00eda. Su piel, de un dorado oscuro semejante al de la miel, hac\u00eda resaltar sus ojos, amarillos y de p\u00e1rpados gruesos. Era el m\u00e1s alto de los cinco varones, con cabello casta\u00f1o oscuro y \u00e1spero; vest\u00eda un traje negro bordado. Lo envolv\u00eda un aura extra\u00f1a, difuminada, ensombrecida por las otras presentes en la sala. Era la cosa m\u00e1s hermosa y terrible que yo jam\u00e1s hab\u00eda contemplado.\n\nNot\u00e9 un doloroso espasmo en las tripas. Baj\u00e9 la vista al suelo. \u00bfSer\u00edan capaces de dispararme solo por mirar?\n\nNashira segu\u00eda hablando y pase\u00e1ndose entre las filas.\n\n\u2014Los clarividentes hab\u00e9is desarrollado un gran poder a lo largo de los a\u00f1os. Est\u00e1is acostumbrados a sobrevivir. El simple hecho de que est\u00e9is aqu\u00ed, de que hay\u00e1is evitado que os capturen hasta ahora, es un testimonio de vuestra capacidad de adaptaci\u00f3n colectiva. Vuestros dones han resultado valios\u00edsimos para mantener a raya a los emim. Esa es la raz\u00f3n por la que, a lo largo de diez a\u00f1os, recogemos a cuantos podemos de vosotros, y os mantenemos en la Torre esperando vuestra transici\u00f3n desde Scion. Llamamos a esas levas decenales Eras de Huesos. Esta es la Era de Huesos XX.\n\n\u00bbEn su debido momento recibir\u00e9is vuestro n\u00famero de identificaci\u00f3n. Los que sois clarividentes ser\u00e9is asignados a un guardi\u00e1n refa\u00edta. \u2014Se\u00f1al\u00f3 a sus compa\u00f1eros\u2014. Vuestro guardi\u00e1n es vuestro amo en todo. \u00c9l o ella evaluar\u00e1 vuestras habilidades y calcular\u00e1 vuestro valor. Si alguno de vosotros muestra cobard\u00eda, recibir\u00e1 el blus\u00f3n amarillo: el de los cobardes. Los que sois amaur\u00f3ticos, es decir, los pocos de vosotros que no entend\u00e9is ni una palabra de lo que estoy diciendo \u2014a\u00f1adi\u00f3\u2014, trabajar\u00e9is en nuestras residencias. Para servirnos.\n\nMe pareci\u00f3 que Seb hab\u00eda dejado de respirar.\n\n\u2014Si no aprob\u00e1is el primer examen, o si os gan\u00e1is el blus\u00f3n amarillo dos veces, os pondremos al cuidado del Capataz, que os formar\u00e1 para ser actores. Los actores son los responsables de nuestro entretenimiento, y del entretenimiento de nuestros empleados.\n\nReflexion\u00e9 sobre las dos opciones: monstruo de feria o servicio militar obligatorio. Me temblaron los labios, y la mano que ten\u00eda libre form\u00f3 un pu\u00f1o. Hab\u00eda imaginado muchos motivos para que detuvieran a los videntes, pero ninguno como aquel.\n\nTr\u00e1fico de humanos. No, tr\u00e1fico de videntes. Scion nos hab\u00eda convertido en esclavos.\n\nO\u00ed llorar a varias personas; otros se hab\u00edan quedado entre absortos y horrorizados. Nashira no pareci\u00f3 notarlo. Ni siquiera hab\u00eda pesta\u00f1eado cuando hab\u00eda muerto la chica. No se hab\u00eda inmutado.\n\n\u2014Los refa\u00edtas no olvidamos. Los que os adapt\u00e9is a este sistema ser\u00e9is recompensados. Los que no, castigados. Nosotros no queremos que eso suceda, pero si nos falt\u00e1is al respeto, sufrir\u00e9is. A partir de ahora, esto ser\u00e1 vuestra vida.\n\nSeb se desmay\u00f3. Julian y yo lo mantuvimos en pie entre los dos, pero era un peso muerto.\n\nLos nueve refa\u00edtas bajaron del estrado. Mantuve la cabeza agachada.\n\n\u2014Estos refa\u00edtas han ofrecido sus servicios como guardianes \u2014nos inform\u00f3 Nashira\u2014. Ahora decidir\u00e1n a qui\u00e9nes de vosotros quieren llevarse.\n\nDe los nueve, siete empezaron a pasearse por la sala, entre las hileras de prisioneros. El \u00faltimo (ese al que yo hab\u00eda mirado) se qued\u00f3 junto a Nashira. No me atrev\u00ed a mirar a Julian, pero dije en voz baja:\n\n\u2014Esto no puede ser verdad.\n\n\u2014M\u00edralos \u2014repuso \u00e9l sin apenas mover los labios. Lo o\u00ed porque est\u00e1bamos muy juntos, uno a cada lado de Seb\u2014. No son humanos. Son de alg\u00fan otro sitio.\n\n\u2014\u00bfTe refieres a eso que ha llamado Inframundo? \u2014Cerr\u00e9 la boca porque pas\u00f3 un refa\u00edta a mi lado, y luego continu\u00e9\u2014: La \u00fanica otra dimensi\u00f3n que existe es el \u00e9ter. No hay nada m\u00e1s.\n\n\u2014El \u00e9ter existe junto al mundo de la carne: alrededor de nosotros, y no fuera de nosotros. Esto es diferente.\n\nDentro de m\u00ed borbote\u00f3 una risa desquiciada.\n\n\u2014Scion se ha vuelto majara.\n\nJulian no dijo nada. Al otro lado de la sala una refa\u00edta cogi\u00f3 a Carl por el codo.\n\n\u2014XX-59-1 \u2014dijo\u2014, te reclamo.\n\nCarl trag\u00f3 saliva; lo condujeron al estrado, pero mantuvo una actitud valerosa. Una vez que lo hubieron dejado all\u00ed, los refa\u00edtas siguieron dando vueltas, como ladronzuelos escogiendo a una v\u00edctima adinerada.\n\nMe pregunt\u00e9 qu\u00e9 criterio aplicar\u00edan para elegirnos. \u00bfSer\u00eda malo para Carl que lo hubieran escogido tan pronto?\n\nPasaban los minutos. Las filas iban menguando. La suspirante, que ahora se llamaba XX-59-2, se march\u00f3 con Carl. El or\u00e1culo se fue con Pleione; no parec\u00eda que le interesara mucho aquel tr\u00e1mite. Un refa\u00edta de rostro cruel arrastr\u00f3 a la palmista hasta el estrado. La chica empez\u00f3 a llorar, suplicando en vano. Luego se llevaron a Julian. XX-59-26. Me lanz\u00f3 una mirada, asinti\u00f3 con la cabeza y acompa\u00f1\u00f3 a su nuevo guardi\u00e1n al estrado.\n\nCambiaron otros doce nombres por n\u00fameros. Llegaron al 38. Solo qued\u00e1bamos ocho: los seis amaur\u00f3ticos, un cantor y yo. Alguien tendr\u00eda que escogerme. Varios refa\u00edtas me hab\u00edan examinado, fij\u00e1ndose mucho en mi cuerpo y mis ojos, pero ninguno me hab\u00eda reclamado. \u00bfQu\u00e9 pasar\u00eda si ninguno me escog\u00eda?\n\nEl cantor, un chico menudo con el pelo recogido en trencitas cosidas, se march\u00f3 con Pleione: 39. La \u00fanica vidente que quedaba era yo.\n\nLos refa\u00edtas miraron a Nashira, quien a su vez nos mir\u00f3 a los que qued\u00e1bamos. Se me tens\u00f3 la columna vertebral.\n\nEntonces el que me hab\u00eda estado mirando dio unos pasos adelante. No dijo nada, pero se acerc\u00f3 a Nashira y lade\u00f3 la cabeza hacia m\u00ed. Nashira me mir\u00f3; levant\u00f3 una mano y me hizo se\u00f1as con un largo dedo. Llevaba guantes negros, igual que Pleione. Todos llevaban guantes negros.\n\nSeb segu\u00eda inconsciente. Intent\u00e9 deslizarlo hasta el suelo, pero se aferr\u00f3 a m\u00ed. Al percatarse del apuro en que me encontraba, otro amaur\u00f3tico me lo quit\u00f3 de los brazos.\n\nTodas las miradas estaban fijas en m\u00ed cuando camin\u00e9 por el suelo de m\u00e1rmol y me par\u00e9 ante aquella pareja. De cerca, Nashira parec\u00eda mucho m\u00e1s alta, y el otro refa\u00edta me sacaba casi dos palmos.\n\n\u2014\u00bfTu nombre?\n\n\u2014Paige Mahoney.\n\n\u2014\u00bfDe d\u00f3nde eres?\n\n\u2014Cohorte I.\n\n\u2014Pero tienes otro origen.\n\nDeb\u00edan de haber visto mis documentos.\n\n\u2014De Irlanda \u2014contest\u00e9.\n\nUn leve temblor recorri\u00f3 toda la sala.\n\n\u2014\u00bfDe Scion Belfast?\n\n\u2014No, de la Irlanda libre.\n\nAlguien dio un grito ahogado de asombro.\n\n\u2014Ya veo. Un esp\u00edritu libre. \u2014Sus ojos parec\u00edan bioluminiscentes\u2014. Tu aura nos tiene intrigados. Dime, \u00bfqu\u00e9 eres?\n\n\u2014Soy cifradora \u2014respond\u00ed.\n\nMe mantuve impert\u00e9rrita bajo su atenta mirada.\n\n\u2014Tengo buenas noticias para ti, Paige Mahoney. \u2014Nashira le puso una mano en el brazo a su acompa\u00f1ante\u2014. Le has llamado la atenci\u00f3n al consorte de sangre: Arcturus, Custodio de los Mesarthim. Ha decidido ser tu guardi\u00e1n.\n\nLos refa\u00edtas se miraron. No dijeron nada, pero me pareci\u00f3 que sus auras se tensaban.\n\n\u2014No es habitual que se interese por un humano \u2014continu\u00f3 Nashira en voz baja, como si me confiara un secreto muy bien guardado\u2014. Puedes considerarte muy afortunada.\n\nYo no me sent\u00eda nada afortunada. Estaba horrorizada.\n\nEl consorte de sangre se inclin\u00f3 hasta ponerse a mi nivel. Tuvo que inclinarse mucho. No desvi\u00e9 la mirada.\n\n\u2014XX-59-40. \u2014Ten\u00eda una voz grave y suave\u2014. Te reclamo.\n\nAs\u00ed que ese tipo iba a ser mi amo. Lo mir\u00e9 a los ojos, pese a saber que no deb\u00eda. Quer\u00eda saber qu\u00e9 cara ten\u00eda mi enemigo.\n\nYa hab\u00edan escogido a todos los videntes. Nashira elev\u00f3 la voz y se dirigi\u00f3 a los seis amaur\u00f3ticos.\n\n\u2014Vosotros seis esperar\u00e9is aqu\u00ed. Enviaremos una escolta que os llevar\u00e1 al cuartel. Los dem\u00e1s ir\u00e9is con vuestros guardianes a las residencias. Buena suerte a todos, y recordad: aqu\u00ed sois libres de tomar las decisiones que quer\u00e1is. Solo espero que tom\u00e9is las correctas.\n\nDicho esto, se dio la vuelta y ech\u00f3 a andar. La siguieron dos casacas rojas. Yo me qued\u00e9 all\u00ed plantada con mi guardi\u00e1n, petrificada.\n\nArcturus fue hacia la puerta y me hizo una se\u00f1a con la mano para que lo siguiera. Como no obedec\u00ed al instante, se par\u00f3 y se qued\u00f3 esperando.\n\nTodos me miraban. Me daba vueltas la cabeza. Lo vi todo rojo, y luego blanco.\n\nLo segu\u00ed.\n\nLa primera luz del alba acariciaba las torres de los edificios. Los videntes salieron detr\u00e1s de sus guardianes, en grupos de tres o cuatro. Yo era la \u00fanica que ten\u00eda un guardi\u00e1n para m\u00ed sola.\n\nArcturus se puso a mi lado. Demasiado cerca. Se me agarrot\u00f3 la espalda.\n\n\u2014Debes saber que aqu\u00ed dormimos durante el d\u00eda.\n\nNo dije nada.\n\n\u2014Tambi\u00e9n debes saber que no tengo por costumbre aceptar inquilinos. \u2014Qu\u00e9 forma tan bonita de llamar a los prisioneros\u2014. Si apruebas tus ex\u00e1menes, vivir\u00e1s conmigo de forma permanente. Si no, me ver\u00e9 obligado a desalojarte. Y aqu\u00ed las calles no son nada acogedoras.\n\nSegu\u00ed callada. Ya sab\u00eda que las calles no eran acogedoras. No pod\u00edan ser mucho peores que las de Londres.\n\n\u2014No eres muda \u2014dijo\u2014. Habla.\n\n\u2014No sab\u00eda que pudiera hablar sin permiso.\n\n\u2014Te concedo ese privilegio.\n\n\u2014No tengo nada que decir.\n\nArcturus me escudri\u00f1\u00f3 con la mirada. Sus ojos conten\u00edan un calor mortecino.\n\n\u2014Estamos instalados en la residencia de Magdalen. \u2014Se coloc\u00f3 de espaldas al amanecer\u2014. Supongo que tienes fuerzas suficientes para andar, \u00bfno, chica?\n\n\u2014S\u00ed, puedo andar \u2014contest\u00e9.\n\n\u2014Estupendo.\n\nNos pusimos en marcha. Salimos del edificio; en la calle, el siniestro espect\u00e1culo hab\u00eda terminado. Vi a la contorsionista cerca del escenario, guardando sus sedas en una bolsa. Nuestras miradas se encontraron un momento, y luego ella mir\u00f3 hacia otro lado. Ten\u00eda el aura delicada de una cartom\u00e1ntica. Y los cardenales de una prisionera.\n\nMagdalen era un edificio espl\u00e9ndido. Era de otra \u00e9poca, de otro mundo. Ten\u00eda una capilla, campanarios y altas ventanas de cristal tras las que brillaba la intensa luz de las antorchas. Mientras nos acerc\u00e1bamos y entr\u00e1bamos por una puerta peque\u00f1a, sonaron cinco campanadas. Un chico con blus\u00f3n rojo nos salud\u00f3 con una reverencia cuando entramos en un soportal. Segu\u00ed a Arcturus al interior en penumbra. Subimos por una escalera de caracol de piedra y nos detuvimos ante una puerta maciza, que \u00e9l abri\u00f3 con una llave peque\u00f1a de bronce.\n\n\u2014Por aqu\u00ed \u2014me dijo\u2014. Este ser\u00e1 tu nuevo hogar. La Torre del Fundador.\n\nMe asom\u00e9 a mi c\u00e1rcel.\n\nTras la puerta hab\u00eda una gran c\u00e1mara rectangular. Los muebles eran francamente opulentos. Las paredes, blancas, estaban desprovistas de adornos; lo \u00fanico que hab\u00eda colgado en ellas era un emblema con un dibujo en blanco y negro coronado con tres flores. Un tablero de ajedrez con los escaques oblicuos. A ambos lados de las ventanas, que daban a unos patios, colgaban unas gruesas cortinas rojas. Hab\u00eda dos butacas ante una espl\u00e9ndida chimenea donde ard\u00eda la le\u00f1a, y en un rinc\u00f3n, un div\u00e1n rojo con un mont\u00f3n de cojines de seda encima. A su lado hab\u00eda un reloj de p\u00e9ndulo. En un gram\u00f3fono colocado sobre un escritorio de madera oscura sonaba \u00abGloomy Sunday\u00bb, y junto a la gran cama con dosel hab\u00eda una mesilla de noche elegante. Una gruesa alfombra estampada cubr\u00eda el suelo.\n\nArcturus cerr\u00f3 la puerta con una llave que a continuaci\u00f3n se guard\u00f3.\n\n\u2014No s\u00e9 mucho acerca de los humanos. Quiz\u00e1 tengas que recordarme cu\u00e1les son tus necesidades. \u2014Dio unos golpecitos en el escritorio con un dedo\u2014. Aqu\u00ed hay unas sustancias medicinales. Tienes que tomarte una pastilla de cada todas las noches.\n\nNo dije nada, pero le ech\u00e9 una ojeada a su onirosaje. Antiguo y extra\u00f1o, endurecido por el tiempo. Un farol m\u00e1gico en el \u00e9ter.\n\nAquel intruso del I-4 era, sin duda, uno de ellos.\n\nNot\u00e9 que sus ojos le\u00edan m\u00e1s all\u00e1 de mi cara. Estudiaban mi aura; estaba tratando de averiguar qu\u00e9 tipo de carga se hab\u00eda echado encima. O qu\u00e9 tesoro escondido hab\u00eda descubierto. Ese pensamiento aviv\u00f3 mi odio.\n\n\u2014M\u00edrame.\n\nEra una orden. Levant\u00e9 la barbilla y lo mir\u00e9. Por nada del mundo iba a dejar que viera el miedo que me inspiraba.\n\n\u2014No tienes visi\u00f3n espiritista \u2014observ\u00f3\u2014. Eso ser\u00e1 un inconveniente aqu\u00ed. A menos que tengas alguna forma de compensarlo, por supuesto. Quiz\u00e1 un sexto sentido m\u00e1s potente.\n\nNo repliqu\u00e9. Siempre hab\u00eda so\u00f1ado con tener aunque solo fuera una visi\u00f3n espiritista parcial, pero segu\u00eda siendo ciega espiritista. No ve\u00eda las lucecitas del \u00e9ter; solo las notaba. Sin embargo Jaxon nunca lo hab\u00eda considerado una debilidad.\n\n\u2014\u00bfTienes alguna pregunta?\n\nExaminaba cada cent\u00edmetro de mi cara con sus implacables ojos.\n\n\u2014\u00bfD\u00f3nde dormir\u00e9?\n\n\u2014Har\u00e9 que te preparen una habitaci\u00f3n. De momento dormir\u00e1s aqu\u00ed. \u2014Se\u00f1al\u00f3 el div\u00e1n\u2014. \u00bfAlgo m\u00e1s?\n\n\u2014No.\n\n\u2014Hoy tengo que salir. Durante mi ausencia puedes familiarizarte con la ciudad. Volver\u00e1s todos los d\u00edas antes del amanecer. Si suena la sirena, volver\u00e1s inmediatamente a esta habitaci\u00f3n. Si robas o tocas algo, lo sabr\u00e9.\n\n\u2014S\u00ed, se\u00f1or.\n\nEl \u00abse\u00f1or\u00bb se me escap\u00f3.\n\n\u2014T\u00f3mate esto. \u2014Me tendi\u00f3 una pastilla\u2014. T\u00f3mate otra ma\u00f1ana por la noche, junto con las otras.\n\nNo la cog\u00ed. Arcturus llen\u00f3 un vaso de agua de una jarra, sin mirarme. Me acerc\u00f3 el vaso y el comprimido. Me humedec\u00ed los labios.\n\n\u2014\u00bfQu\u00e9 pasa si no me la tomo?\n\nHubo un largo silencio.\n\n\u2014Era una orden \u2014dijo por fin\u2014. No te lo he pedido. Te lo he ordenado.\n\nEl coraz\u00f3n me palpitaba. Hice rodar el comprimido entre los dedos. Era de color verde gris\u00e1ceo. Me lo tragu\u00e9. Ten\u00eda un sabor amargo.\n\nArcturus cogi\u00f3 el vaso.\n\n\u2014Otra cosa. \u2014Arcturus me agarr\u00f3 por la nuca con la mano que ten\u00eda libre, y me oblig\u00f3 a mirarlo. Not\u00e9 un escalofr\u00edo\u2014. Te dirigir\u00e1s a m\u00ed \u00fanicamente por mi t\u00edtulo solemne: Custodio. \u00bfEntendido?\n\n\u2014S\u00ed.\n\nTuve que esforzarme para decirlo. \u00c9l me mir\u00f3 a los ojos, grab\u00e1ndome su mensaje a fuego, y entonces afloj\u00f3 la mano.\n\n\u2014Cuando regrese, iniciaremos tu entrenamiento. \u2014Se dirigi\u00f3 hacia la puerta\u2014. Que duermas bien.\n\nNo puede evitarlo: solt\u00e9 una d\u00e9bil y amarga risotada.\n\nArcturus volvi\u00f3 un poco la cabeza. Vi que sus ojos se vaciaban. No dijo nada m\u00e1s y se march\u00f3. La llave gir\u00f3 en la cerradura.\n\nLos destellos rojizos del sol entraban por la ventana. La luz me sac\u00f3 de un sue\u00f1o profundo. Ten\u00eda mal sabor de boca. Por un instante cre\u00ed estar en mi dormitorio del I-5, lejos de Jax, lejos del trabajo.\n\nEntonces lo record\u00e9. Las Eras de Huesos. Los refa\u00edtas. Un disparo y un cad\u00e1ver.\n\nNo, no me encontraba en el I-5.\n\nLos cojines estaban en el suelo; los hab\u00eda tirado mientras dorm\u00eda. Me incorpor\u00e9 y mir\u00e9 alrededor; me frot\u00e9 la agarrotada nuca. Me dol\u00eda la parte baja de la espalda, y me martilleaba la cabeza. Una de mis \u00abresacas\u00bb, como las llamaba Nick. De Arcturus, el Custodio, no hab\u00eda ni rastro.\n\nEl gram\u00f3fono segu\u00eda lament\u00e1ndose. Reconoc\u00ed la \u00abDanse Macabre\u00bb de Saint-Sa\u00ebns inmediatamente, y eso me alarm\u00f3: Jax la escuchaba cuando estaba de muy mal humor, generalmente tom\u00e1ndose una copa de vino a\u00f1ejo. Siempre me hab\u00eda puesto los pelos de punta. Apagu\u00e9 el gram\u00f3fono, descorr\u00ed las cortinas y ech\u00e9 un vistazo al patio orientado a levante. Hab\u00eda un guardia refa\u00edta apostado junto a una puerta gigantesca de roble, de doble hoja.\n\nHab\u00edan dejado un uniforme limpio encima de la cama. Encontr\u00e9 una nota manuscrita sobre la almohada, escrita en tinta negra:\n\nEspera a que suene la campana.\n\nPens\u00e9 en lo que nos hab\u00edan dicho en el serm\u00f3n. Nadie hab\u00eda mencionado ninguna campana. Arrugu\u00e9 la nota y la tir\u00e9 a la chimenea, donde otros trozos de papel esperaban a que los quemaran.\n\nPas\u00e9 unos minutos registrando la habitaci\u00f3n, revisando cada rinc\u00f3n. Las ventanas no ten\u00edan barrotes, pero no pod\u00edan abrirse. En las paredes no encontr\u00e9 junturas ocultas ni paneles deslizantes. Hab\u00eda otras dos puertas, una de ellas oculta tras unas gruesas cortinas rojas y cerrada con llave. La otra conduc\u00eda a un gran cuarto de ba\u00f1o. Como no vi ning\u00fan interruptor, me llev\u00e9 una l\u00e1mpara de aceite y entr\u00e9. La ba\u00f1era era del mismo m\u00e1rmol negro que el suelo de la biblioteca, y la rodeaban unas cortinas transparentes. Un espejo dorado ocupaba casi toda la pared; me acerqu\u00e9, pues sent\u00eda curiosidad por saber si la mutilaci\u00f3n que hab\u00eda sufrido mi vida se reflejaba en mi cara.\n\nNo, no se reflejaba. Con excepci\u00f3n del corte en el labio, estaba igual que antes de que me capturaran. Me sent\u00e9 en la oscuridad y me puse a pensar.\n\nLos refa\u00edtas hab\u00edan cerrado su trato en 1859, exactamente dos siglos atr\u00e1s. Si no recordaba mal lo que hab\u00eda aprendido en el colegio, eso correspond\u00eda al mandato de lord Palmerston. Mucho antes del fin de la monarqu\u00eda, en 1901, cuando una nueva Rep\u00fablica de Inglaterra asumi\u00f3 el poder y declar\u00f3 la guerra a la antinaturalidad. Esa rep\u00fablica hab\u00eda llevado al pa\u00eds por casi tres d\u00e9cadas de adoctrinamiento y propaganda, hasta que en 1929 pas\u00f3 a llamarse Scion. Fue entonces cuando eligieron al Primer Inquisidor, y Londres se convirti\u00f3 en la primera ciudadela de Scion. Todo eso me hac\u00eda pensar que, de alguna forma, la llegada de los refa\u00edtas hab\u00eda desencadenado el advenimiento de Scion. Todas esas sandeces sobre la antinaturalidad solo eran un cuento para saciar a esos seres llegados de qui\u00e9n sab\u00eda d\u00f3nde.\n\nInspir\u00e9 hondo. Ten\u00eda que haber alguna otra explicaci\u00f3n. No sab\u00eda c\u00f3mo, pero lo entender\u00eda. De momento mi prioridad consist\u00eda en salir de all\u00ed. Hasta que lo consiguiera, buscar\u00eda respuestas en ese lugar. No pod\u00eda largarme sin m\u00e1s, ahora que sab\u00eda ad\u00f3nde enviaban a los videntes. No pod\u00eda olvidar cuanto hab\u00eda visto y o\u00eddo.\n\nPrimero buscar\u00eda a Seb. Su condici\u00f3n de amaur\u00f3tico y su ignorancia hac\u00edan que estuviera m\u00e1s asustado que los dem\u00e1s. Solo era un ni\u00f1o: no se merec\u00eda eso. Cuando lo hubiera encontrado, buscar\u00eda a Julian y a los otros detenidos de la Era de Huesos XX. Quer\u00eda saber m\u00e1s cosas sobre los emim, y hasta que volviera mi guardi\u00e1n, ellos eran mi \u00fanica fuente de informaci\u00f3n.\n\nSon\u00f3 la campana de la torre de mi edificio, y al cabo de un momento son\u00f3 otra a lo lejos. \u00abEspera a que suene la campana.\u00bb Deb\u00eda de haber toque de queda.\n\nPuse la l\u00e1mpara en el borde de la ba\u00f1era. Mientras me echaba agua fr\u00eda en la cara, consider\u00e9 mis opciones. De momento lo mejor que pod\u00eda hacer era seguirles la corriente a los refa\u00edtas. Si sobreviv\u00eda el tiempo suficiente, intentar\u00eda ponerme en contacto con Jax. Jax vendr\u00eda a buscarme; nunca dejaba atr\u00e1s a un vidente. Al menos, a ning\u00fan vidente que trabajara para \u00e9l. M\u00e1s de una vez le hab\u00eda visto dejar morir a limosneros.\n\nEn la habitaci\u00f3n cada vez estaba m\u00e1s oscuro. Abr\u00ed el caj\u00f3n del medio del escritorio. Dentro hab\u00eda tres bl\u00edsters de pastillas. No quer\u00eda tom\u00e1rmelas, pero supon\u00eda que el Custodio las contar\u00eda para asegurarse de que lo hab\u00eda hecho. A menos que yo las tirara.\n\nSaqu\u00e9 una de cada bl\u00edster. Roja, blanca y verde. Ninguna estaba etiquetada.\n\nLa ciudad estaba llena de no humanos, llena de cosas que yo todav\u00eda no entend\u00eda. Quiz\u00e1 esas pastillas sirvieran para protegerme de algo: toxinas, radiaci\u00f3n... Esa contaminaci\u00f3n sobre la que nos hab\u00eda advertido Scion. Quiz\u00e1 no fuera mentira. Quiz\u00e1 deb\u00eda tom\u00e1rmelas. Tarde o temprano tendr\u00eda que hacerlo, cuando \u00e9l regresara.\n\nPero todav\u00eda no hab\u00eda llegado. De momento no pod\u00eda verme. Tir\u00e9 las tres pastillas por el desag\u00fce del lavamanos. Que se tomara \u00e9l sus medicinas y se atragantara con ellas.\n\nCuando intent\u00e9 abrir la puerta, comprob\u00e9 que no estaba cerrada con llave. Baj\u00e9 los escalones de piedra y volv\u00ed a los soportales. La residencia era enorme. Al llegar a la puerta de la calle, vi que una chica huesuda con la nariz rosada y el cabello de un rubio sucio hab\u00eda sustituido al chico de rojo. Al acercarme a ella, levant\u00f3 la vista.\n\n\u2014Hola \u2014me salud\u00f3\u2014. T\u00fa debes de ser nueva.\n\n\u2014S\u00ed.\n\n\u2014Bueno, has iniciado tu viaje en un sitio estupendo. Bienvenida a Magdalen, la mejor residencia de Sheol I. Soy XIX-49-33, portera de noche. \u00bfEn qu\u00e9 puedo ayudarte?\n\n\u2014Quiero salir.\n\n\u2014\u00bfTienes permiso?\n\n\u2014No lo s\u00e9.\n\nY tampoco me importaba.\n\n\u2014Muy bien. Voy a comprobarlo. \u2014Su sonrisa estaba volvi\u00e9ndose tensa\u2014. \u00bfPuedes decirme tu n\u00famero?\n\n\u2014XX-59-40.\n\nLa chica consult\u00f3 su registro. Cuando encontr\u00f3 la p\u00e1gina, me mir\u00f3 con los ojos muy abiertos.\n\n\u2014T\u00fa eres a la que ha hospedado el Custodio.\n\nBueno, si quer\u00eda llamarlo \u00abhospedar\u00bb...\n\n\u2014Nunca hab\u00eda hospedado a ning\u00fan humano \u2014continu\u00f3\u2014. En Magdalen muy pocos lo hacen. Generalmente solo hay refas, con unos pocos asistentes humanos. Tienes mucha suerte de alojarte con \u00e9l, \u00bflo sabes?\n\n\u2014Eso me han dicho \u2014repuse\u2014. Si no te importa, tengo unas cuantas preguntas sobre este sitio.\n\n\u2014Adelante.\n\n\u2014\u00bfD\u00f3nde puedo conseguir comida?\n\n\u2014El Custodio ha dejado instrucciones sobre eso. \u2014Me puso en la mano un pu\u00f1ado de agujas despuntadas, anillos de hojalata baratos y dedales\u2014. Toma. Son _numa_. Los bufones siempre los necesitan. Puedes cambiarlos por comida en los puestos de ah\u00ed fuera (hay una especie de asentamiento ocupa), aunque no es muy buena. Yo esperar\u00eda a que tu guardi\u00e1n te diera de comer.\n\n\u2014\u00bfHay probabilidades de que lo haga?\n\n\u2014Puede ser.\n\nBueno, eso ya estaba aclarado.\n\n\u2014\u00bfD\u00f3nde est\u00e1 ese asentamiento? \u2014pregunt\u00e9.\n\n\u2014En el Broad. Tuerce a la derecha al salir de Magdalen, y luego gira por la primera a la izquierda. Ya lo ver\u00e1s. \u2014Pas\u00f3 la hoja de su registro\u2014. Recuerda, no debes sentarte en zonas p\u00fablicas sin permiso, ni entrar en ninguna residencia. Tampoco debes llevar nada que no sea el uniforme. Ah, y sobre todo, debes volver aqu\u00ed antes del amanecer.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Ver\u00e1s, los refas duermen de d\u00eda. Supongo que es m\u00e1s f\u00e1cil ver a los esp\u00edritus despu\u00e9s de la puesta de sol.\n\n\u2014Y eso hace que el entrenamiento sea m\u00e1s f\u00e1cil.\n\n\u2014Exacto.\n\nNo me gustaba nada esa chica.\n\n\u2014\u00bfT\u00fa tienes un guardi\u00e1n?\n\n\u2014S\u00ed. Pero ahora est\u00e1 fuera.\n\n\u2014\u00bfD\u00f3nde?\n\n\u2014No lo s\u00e9. Pero estoy segura de que ha ido a hacer algo importante.\n\n\u2014Entiendo. Gracias.\n\n\u2014De nada. Que pases una buena noche. Y recuerda \u2014a\u00f1adi\u00f3\u2014, no cruces el puente.\n\nMenudo lavado de cerebro. Sonre\u00ed y segu\u00ed adelante.\n\nAl salir de la residencia empec\u00e9 a echar vaho por la boca, y me pregunt\u00e9 d\u00f3nde me hab\u00eda metido. El Custodio. Susurraban su nombre como si fuera una oraci\u00f3n, una promesa. \u00bfPor qu\u00e9 \u00e9l era diferente de los otros? \u00bfQu\u00e9 significaba \u00abconsorte de sangre\u00bb? Me promet\u00ed que m\u00e1s tarde intentar\u00eda enterarme. De momento necesitaba comer. Luego buscar\u00eda a Seb. Por lo menos ten\u00eda un sitio donde dormir cuando volviera; tal vez \u00e9l no hubiera tenido tanta suerte.\n\nSe hab\u00eda formado una fina niebla. No parec\u00eda que hubiera electricidad en la ciudad. A mi izquierda hab\u00eda un puente de piedra cuyos lados estaban iluminados con l\u00e1mparas de gas. Ese deb\u00eda de ser el puente que me hab\u00edan prohibido cruzar. Una hilera de guardias vestidos de rojo bloqueaba el camino entre la ciudad y el mundo exterior. Al ver que me quedaba quieta, los diez me apuntaron con sus armas. Armas de Scion, militares. Con las diez miras clavadas en mi espalda, me puse en marcha, dispuesta a explorar la peque\u00f1a ciudad.\n\nLa calle discurr\u00eda junto a los jardines de Magdalen, separada de la residencia por un alto muro. Pas\u00e9 por delante de tres s\u00f3lidas puertas de madera, todas custodiadas por un humano ataviado con el blus\u00f3n rojo. El muro estaba rematado con p\u00faas de hierro. Agach\u00e9 la cabeza y segu\u00ed las indicaciones de 33. La siguiente calle estaba tan desierta como la primera, y no hab\u00eda l\u00e1mparas de gas que me alumbraran el camino. Cuando sal\u00ed de la oscuridad, con las manos cortadas por el fr\u00edo, vi que me encontraba en algo parecido al centro de la ciudad. A mi izquierda se alzaban dos grandes edificios. El que estaba m\u00e1s cerca ten\u00eda columnas y un front\u00f3n decorado, como el Gran Museo de la cohorte I. Pas\u00e9 de largo y llegu\u00e9 al Broad. En los escalones de las puertas y los alf\u00e9izares de las ventanas hab\u00eda velitas encendidas. Los sonidos de la vida humana invad\u00edan la noche.\n\nA lo largo del centro de la calle hab\u00edan montado tenderetes desvencijados y puestos de comida iluminados con faroles sucios. Eran raqu\u00edticos, l\u00fagubres. A ambos lados hab\u00eda sendas hileras de casuchas rudimentarias, chozas y tiendas hechas de chapa de zinc ondulada, contrachapado y pl\u00e1stico: toda una barriada de chabolas en pleno centro de la ciudad.\n\nY la sirena. Un modelo mec\u00e1nico viejo con un solo cuerno enorme. Nada que ver con los sofisticados equipos electr\u00f3nicos de los puestos de avanzada de la DVN, dise\u00f1ados para su uso en caso de emergencia nacional. Confi\u00e9 en no o\u00edr nunca el sonido que sal\u00eda de sus rotores. Lo \u00faltimo que necesitaba era que me persiguiera una m\u00e1quina asesina.\n\nEl olor a carne asada me arrastr\u00f3 hacia la barriada. Me dol\u00eda el est\u00f3mago de hambre. Entr\u00e9 en un t\u00fanel oscuro y estrecho y me dej\u00e9 guiar por el olfato. Las chabolas parec\u00edan conectadas por una serie de t\u00faneles de contrachapado, remendados con trozos de tela y metal. Ten\u00edan pocas ventanas, y estaban iluminadas con velas y l\u00e1mparas de queroseno. Yo era la \u00fanica persona que vest\u00eda blus\u00f3n blanco. Toda aquella gente llevaba ropa mugrienta. Los colores no mejoraban el aspecto de su cutis cetrino ni de sus ojos, apagados e inyectados en sangre. No vi a nadie que pareciera sano. Esos deb\u00edan de ser los actores humanos que no hab\u00edan aprobado los ex\u00e1menes y hab\u00edan sido condenados a entretener a los refa\u00edtas durante el resto de su vida, y seguramente tambi\u00e9n en su otra vida. La mayor\u00eda eran adivinos o augures, las dos clases de videntes m\u00e1s frecuentes. Unos cuantos se fijaron en m\u00ed, pero pasaron de largo. Me dio la impresi\u00f3n de que no quer\u00edan entretenerse mucho mir\u00e1ndome.\n\nEl origen del olor era una gran sala cuadrada con un agujero en el techo de chapa de zinc para dejar salir el humo y el vaho. Me sent\u00e9 en un rinc\u00f3n oscuro y trat\u00e9 de no llamar la atenci\u00f3n. Estaban sirviendo la carne en lonchas finas como una oblea; todav\u00eda estaba rosada y tierna en el centro. Los actores repart\u00edan los platos de carne con verduras y sacaban crema de unas soperas de plata. Se peleaban por la comida, se la met\u00edan en la boca a toda prisa y se chupaban los dedos para recoger el jugo caliente. Antes de que pudiera preguntar, un vidente me puso un plato en las manos. Estaba esquel\u00e9tico e iba vestido con poco m\u00e1s que unos harapos. Llevaba unas gafas con los cristales gruesos muy rayados.\n\n\u2014\u00bfMayfield todav\u00eda est\u00e1 en el Starch? \u2014me pregunt\u00f3.\n\n\u2014\u00bfMayfield?\n\n\u2014S\u00ed, Abel Mayfield. \u2014Repiti\u00f3 el nombre tratando de vocalizar mejor\u2014. \u00bfTodav\u00eda est\u00e1 en el Arconte? \u00bfTodav\u00eda es Gran Inquisidor?\n\n\u2014Mayfield lleva a\u00f1os muerto.\n\n\u2014\u00bfQui\u00e9n hay ahora?\n\n\u2014Frank Weaver.\n\n\u2014Ah, vale. No tendr\u00e1s un ejemplar de _El descendiente_ , \u00bfverdad?\n\n\u2014Me lo han confiscado todo. \u2014Mir\u00e9 alrededor buscando un sitio donde sentarme\u2014. \u00bfDe verdad cre\u00edas que Mayfield todav\u00eda era Inquisidor?\n\nEra imposible no conocer la identidad del Inquisidor. Exceptuando a Scarlett Burnish, Weaver era el alma de Scion.\n\n\u2014Oye, no te pongas as\u00ed. \u00bfC\u00f3mo iba a saberlo? Solo nos llegan noticias una vez cada diez a\u00f1os. \u2014Me agarr\u00f3 por el brazo y me llev\u00f3 hasta un rinc\u00f3n\u2014. Dime, \u00bfvolvi\u00f3 a publicarse _El pendenciero_?\n\n\u2014No. \u2014Intent\u00e9 soltarme, pero el chico no me dej\u00f3.\n\n\u2014\u00bfSigue Sinatra en la lista negra?\n\n\u2014S\u00ed.\n\n\u2014Qu\u00e9 pena. \u00bfY el Fleapit? \u00bfLo encontraron?\n\n\u2014Cyril, acaba de llegar. Creo que necesita comer un poco.\n\nAlguien se hab\u00eda percatado del aprieto en que me hallaba. Cyril se volvi\u00f3 hacia una joven que lo miraba con los brazos cruzados y la barbilla levantada.\n\n\u2014Eres una cascarrabias de mierda, Rymore. \u00bfHa vuelto a salirte el diez de espadas?\n\n\u2014S\u00ed, cuando pensaba en ti.\n\nCyril la mir\u00f3 con odio, cogi\u00f3 su plato y se larg\u00f3. Intent\u00e9 agarrarlo por la camisa, pero era \u00e1gil como un carterista. La chica sacudi\u00f3 la cabeza. Ten\u00eda una cara de facciones peque\u00f1as, muy expresiva, enmarcada por una mata de rizos negros. Los labios, pintados de rojo, destacaban como una herida reciente contra su piel.\n\n\u2014Te dieron el serm\u00f3n anoche, hermanita. \u2014Ten\u00eda una forma peculiar de pronunciar las erres\u2014. Tu est\u00f3mago no habr\u00eda soportado esa clase de comida.\n\n\u2014Com\u00ed ayer por la ma\u00f1ana \u2014repliqu\u00e9.\n\nNo supe si re\u00edrme de que aquella cr\u00eda tan peque\u00f1aja me llamara \u00abhermanita\u00bb.\n\n\u2014Es el flux, cr\u00e9eme. Te ha jodido el cerebro. \u2014Mir\u00f3 alrededor y a\u00f1adi\u00f3\u2014: R\u00e1pido. Ven conmigo.\n\n\u2014\u00bfAd\u00f3nde?\n\n\u2014Tengo un refugio. All\u00ed podremos hablar.\n\nNo me convenc\u00eda la idea de seguir a una desconocida, pero necesitaba hablar con alguien. La segu\u00ed.\n\nMi gu\u00eda parec\u00eda conocer a todo el mundo. Por el camino entrechoc\u00f3 las manos con varias personas, sin dejar de vigilarme para ver si todav\u00eda la segu\u00eda. La ropa que llevaba parec\u00eda en mejor estado que la de los otros actores: una blusa fina con mangas acampanadas y unos pantalones que le quedaban cortos. Deb\u00eda de estar congelada. Descorri\u00f3 una cortina deshilachada.\n\n\u2014R\u00e1pido \u2014volvi\u00f3 a decir\u2014. O nos ver\u00e1n.\n\nDetr\u00e1s de la cortina estaba en penumbra, pero un hornillo de parafina manten\u00eda las sombras a raya. Me sent\u00e9. Un mont\u00f3n de s\u00e1banas sucias y un coj\u00edn formaban una cama rudimentaria.\n\n\u2014\u00bfSiempre recoges a los perros callejeros?\n\n\u2014A veces. S\u00e9 por lo que se pasa cuando se llega aqu\u00ed. \u2014La chica se sent\u00f3 junto al hornillo\u2014. Bienvenida a la Familia.\n\n\u2014\u00bfFormo parte de una familia?\n\n\u2014Ahora s\u00ed, hermana. Y no se trata de una secta ni nada parecido, si es eso lo que est\u00e1s pensando. Es una familia formada para protegerse. \u2014Se puso a manipular el hornillo\u2014. Me imagino que vienes del sindicato.\n\n\u2014Es posible.\n\n\u2014Yo no. En el centro no necesitaban a gente como yo. \u2014Una d\u00e9bil sonrisa acarici\u00f3 sus labios\u2014. Llegu\u00e9 aqu\u00ed en la Era de Huesos anterior.\n\n\u2014\u00bfCu\u00e1nto tiempo hace de eso?\n\n\u2014Diez a\u00f1os. Yo ten\u00eda trece. \u2014Extendi\u00f3 una mano encallecida. Dud\u00e9 un instante, y luego se la estrech\u00e9\u2014. Liss Rymore \u2014dijo.\n\n\u2014Paige.\n\n\u2014\u00bfXX-59-40?\n\n\u2014S\u00ed.\n\nLiss detect\u00f3 mi desagrado.\n\n\u2014Lo siento \u2014se disculp\u00f3\u2014. Es la fuerza de la costumbre. O quiz\u00e1 que me han lavado el cerebro.\n\n\u2014T\u00fa \u00bfqu\u00e9 n\u00famero tienes?\n\n\u2014XIX-49-1.\n\n\u2014\u00bfC\u00f3mo es que sabes mi n\u00famero?\n\nVerti\u00f3 un poco de alcohol de quemar en el hornillo.\n\n\u2014En una ciudad tan peque\u00f1a como esta, las noticias vuelan. No nos llega informaci\u00f3n del exterior. No les gusta que nos enteremos de lo que pasa ah\u00ed fuera, en el mundo libre. Si es que se puede llamar libre a Scion. \u2014Prendi\u00f3 una llama azul\u2014. Todo el mundo habla de tu n\u00famero.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014\u00bfNo te has enterado? Arcturus Mesarthim nunca hab\u00eda alojado a un humano en su residencia. De hecho, jam\u00e1s se ha interesado por los humanos. Siento decirlo, pero eso ha sido un notici\u00f3n. Es lo que pasa cuando no puedes leer los peri\u00f3dicos sensacionalistas.\n\n\u2014\u00bfSabes por qu\u00e9 me eligi\u00f3 a m\u00ed?\n\n\u2014Supongo que Nashira se ha encaprichado de ti. \u00c9l es el consorte de sangre, su prometido. Nosotros procuramos no acercarnos a \u00e9l. Aunque la verdad es que nunca sale de esa torre. \u2014Puso una cacerola sobre el hornillo\u2014. Antes de hablar, d\u00e9jame prepararte algo de comer. Lo siento, hace a\u00f1os que los bufones no comemos en mesas.\n\n\u2014\u00bfLos bufones?\n\n\u2014Es como los casacas llaman a los actores. No les caemos muy bien.\n\nCalent\u00f3 un poco de caldo y lo verti\u00f3 en un cuenco. Le ofrec\u00ed unos cuantos anillos, pero ella neg\u00f3 con la cabeza.\n\n\u2014Invita la casa.\n\nDi un sorbo de caldo. Era inodoro y transparente, y sab\u00eda a rayos, pero estaba caliente. Liss me observ\u00f3 mientras yo dejaba el cuenco limpio.\n\n\u2014Toma. \u2014Me pas\u00f3 un pedazo de pan rancio\u2014. _Skilly_ y _toke_. Ya te acostumbrar\u00e1s. La mayor\u00eda de los guardianes se olvidan de que necesitamos comer con regularidad.\n\n\u2014All\u00ed hab\u00eda carne. \u2014Se\u00f1al\u00e9 la sala central.\n\n\u2014Eso solo es para celebrar la Era XX. Este _skilly_ lo he preparado yo antes con los jugos. \u2014Se sirvi\u00f3 un cuenco\u2014. Contamos con que los carro\u00f1os no nos dejen morir de hambre. Esta porquer\u00eda viene de las cocinas \u2014dijo se\u00f1alando el hornillo y la cacerola\u2014. Se supone que solo pueden cocinar para los casacas rojas, pero, cuando pueden, roban algo de comida y nos la dan. Sin embargo, se muestran menos dispuestos a ayudarnos desde que pillaron a una de las chicas.\n\n\u2014\u00bfQu\u00e9 pas\u00f3?\n\n\u2014A la carro\u00f1a le dieron una paliza. Al vidente al que estaba alimentando lo castigaron con cuatro d\u00edas de privaci\u00f3n de sue\u00f1o. Cuando lo soltaron, estaba delirando.\n\nPrivaci\u00f3n de sue\u00f1o. Era un castigo nuevo. Las mentes videntes funcionaban en dos niveles: vida y muerte. Era agotador. Mantener despierto a un vidente durante cuatro d\u00edas equival\u00eda a hacerle enloquecer.\n\n\u2014\u00bfQui\u00e9n trae la comida a la ciudad?\n\n\u2014Ni idea. Quiz\u00e1 la traigan en tren. Hay una l\u00ednea que une Londres con Sheol I. Nadie sabe d\u00f3nde est\u00e1n las entradas del t\u00fanel, evidentemente. \u2014Acerc\u00f3 m\u00e1s los pies al hornillo\u2014. \u00bfCu\u00e1nto cre\u00edste que te hab\u00eda durado la peste cerebral?\n\n\u2014Una eternidad.\n\n\u2014Pues dur\u00f3 cinco d\u00edas. Dejan a los novatos en el infierno cinco d\u00edas antes de administrarles el ant\u00eddoto.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Para que se enteren lo antes posible de cu\u00e1l es su sitio. Aqu\u00ed no eres m\u00e1s que un n\u00famero, a menos que te ganes los colores. \u2014Liss se sirvi\u00f3 un cuenco de caldo\u2014. As\u00ed que est\u00e1s en Magdalen.\n\n\u2014S\u00ed.\n\n\u2014Supongo que estar\u00e1s harta de o\u00edrlo, pero puedes considerarte afortunada. Magdalen es uno de los sitios m\u00e1s seguros para los humanos.\n\n\u2014\u00bfCu\u00e1ntos hay?\n\n\u2014\u00bfCu\u00e1ntos humanos?\n\n\u2014No, cu\u00e1ntas residencias.\n\n\u2014Ah, vale. Bueno, cada residencia es un peque\u00f1o distrito. Hay siete para humanos: Balliol, Corpus, Exeter, Merton, Oriel, Queens y Trinity. Nashira vive en la Residencia del Suzerano, que es donde recibiste el serm\u00f3n. Luego est\u00e1n la Casa, un poco m\u00e1s al sur; el Castillo (la penitenciar\u00eda); y este vertedero, el Poblado. La calle se llama el Broad. La otra calle paralela es el Paseo Magdalen.\n\n\u2014Y m\u00e1s all\u00e1, \u00bfqu\u00e9 hay?\n\n\u2014Campos desiertos. Nosotros lo llamamos la Tierra de Nadie. Est\u00e1 sembrada de minas y trampas.\n\n\u2014\u00bfNadie ha intentado nunca cruzarla?\n\n\u2014S\u00ed.\n\nSe le tensaron los hombros. Di otro sorbo de _skilly_.\n\n\u2014\u00bfQu\u00e9 tal en la Torre?\n\nLa mir\u00e9.\n\n\u2014A m\u00ed no me llevaron a la Torre.\n\n\u2014Naciste con buena estrella, \u00bfno? \u2014Frunc\u00ed el entrecejo, y Liss sacudi\u00f3 la cabeza\u2014. Reclutan a los videntes de cada Era de Huesos durante diez a\u00f1os. Algunos se pasan una d\u00e9cada en la Torre antes de que los destinen aqu\u00ed.\n\n\u2014\u00bfEs una broma?\n\nEso explicar\u00eda lo de aquel pobre diablo que llevaba all\u00ed nueve a\u00f1os.\n\n\u2014No. Son muy astutos en lo referente a mantenernos sumisos. Conocen nuestros puntos d\u00e9biles, saben c\u00f3mo destrozarnos. Diez a\u00f1os en la Torre acaban con cualquiera.\n\n\u2014\u00bfQu\u00e9 son?\n\n\u2014Ni idea. Solo s\u00e9 que no son humanos. \u2014Moj\u00f3 un poco de pan en el _skilly_ \u2014. Se comportan como dioses. Les gusta que los tratemos como si lo fueran.\n\n\u2014Y nosotros somos sus adoradores.\n\n\u2014No solo eso. Les debemos la vida. No nos dejan olvidar que nos est\u00e1n protegiendo de los zumbadores, y que la esclavitud es \u00abpor nuestro bien\u00bb. Dicen que es mejor ser esclavos que estar muertos. O fuera, victimizados por el Inquisidor.\n\n\u2014\u00bfQu\u00e9 son los zumbadores?\n\n\u2014Los emim. Nosotros los llamamos as\u00ed.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Siempre los hemos llamado as\u00ed. Creo que se lo inventaron los casacas rojas. Son ellos los que tienen que combatirlos.\n\n\u2014\u00bfLo hacen muy a menudo?\n\n\u2014Depende de la \u00e9poca del a\u00f1o. Atacan mucho m\u00e1s en invierno. Debes estar atenta a la sirena. Un solo toque sirve para llamar a los casacas rojas. Si el tono empieza a cambiar, m\u00e9tete dentro. Significa que vienen.\n\n\u2014Sigo sin entender qu\u00e9 son. \u2014Part\u00ed un trozo de pan\u2014. \u00bfSe parecen a los refa\u00edtas?\n\n\u2014He o\u00eddo muchas historias. A los casacas rojas les gusta asustarnos. \u2014Las llamas del hornillo proyectaban sombras danzarinas en su cara\u2014. Dicen que los emim pueden adoptar diferentes formas. Puedes morir por el simple hecho de estar cerca de ellos. Hay quien asegura que pueden arrancarte el esp\u00edritu del cuerpo. Otros los llaman \u00abgigantes pudrientes\u00bb; a saber lo que significa eso. Otros dicen que son huesos andantes que necesitan piel para cubrirse. No s\u00e9 qu\u00e9 hay de cierto en todo eso, pero lo que s\u00ed es cierto es que comen carne humana. Son adictos a la carne humana. No te sorprendas si ves por aqu\u00ed a gente a la que le falta alguna extremidad.\n\nSus palabras deber\u00edan haberme asqueado, pero me dejaron atontada. No parec\u00eda real. Liss estir\u00f3 un brazo y movi\u00f3 la cortina para que no pudieran vernos desde fuera. Me fij\u00e9 en un mont\u00f3n de seda de colores.\n\n\u2014Eres la contorsionista \u2014dije.\n\n\u2014\u00bfTe ha gustado mi actuaci\u00f3n?\n\n\u2014S\u00ed, mucho.\n\n\u2014As\u00ed es como me gano la vida aqu\u00ed. Por suerte, aprend\u00ed deprisa. Antes actuaba en los teatros callejeros. \u2014Se relami\u00f3 para limpiarse los labios\u2014. Te vi pasar anoche con Pleione. Tu aura dio mucho que hablar.\n\nNo dije nada. Hablar de mi aura pod\u00eda ser peligroso, sobre todo con una chica a la que acababa de conocer.\n\nLiss se qued\u00f3 escrut\u00e1ndome.\n\n\u2014\u00bfTienes visi\u00f3n espiritista?\n\n\u2014No.\n\nEra verdad.\n\n\u2014\u00bfPor qu\u00e9 te detuvieron?\n\n\u2014Mat\u00e9 a un metrovigilante.\n\nCierto.\n\n\u2014\u00bfC\u00f3mo?\n\n\u2014Con un pu\u00f1al \u2014respond\u00ed\u2014. En un momento de exaltaci\u00f3n.\n\nFalso.\n\nLiss se qued\u00f3 mir\u00e1ndome largo rato. Ella ten\u00eda visi\u00f3n permanente, un rasgo t\u00edpico de los adivinos. Ve\u00eda mi aura, roja, con la misma claridad con que ve\u00eda mi cara. Si investigaba un poco, descubrir\u00eda a qu\u00e9 categor\u00eda correspond\u00eda yo.\n\n\u2014No te creo. \u2014Tamborile\u00f3 con los dedos en el suelo\u2014. T\u00fa nunca has derramado tanta sangre.\n\nEra buena, para ser adivina.\n\n\u2014No eres un or\u00e1culo \u2014expuso, como si hablara sola\u2014. He visto a muchos or\u00e1culos. Eres demasiado tranquila para ser una furia, y desde luego no eres m\u00e9dium. De modo que tienes que ser... \u2014el reconocimiento se reflej\u00f3 en sus ojos\u2014 una onir\u00e1mbula. \u2014Volvi\u00f3 a mirarme\u2014. \u00bfCorrecto?\n\nLe sostuve la mirada. Liss, en cuclillas, se ech\u00f3 un poco hacia atr\u00e1s.\n\n\u2014Bueno, ahora lo entiendo.\n\n\u2014\u00bfQu\u00e9 entiendes?\n\n\u2014Por qu\u00e9 Arcturus te ha elegido. Nashira nunca ha encontrado a ning\u00fan andar\u00edn, y est\u00e1 deseando dar con uno. Quiere asegurarse de que est\u00e1s bien protegida. Si eres la humana de Arcturus, nadie se atrever\u00e1 a tocarte. Si Nashira cree que existe la posibilidad, por remota que sea, de que seas una andarina, ir\u00e1 por ti.\n\n\u2014\u00bfY eso?\n\n\u2014Esto no te va a gustar.\n\nA esas alturas, dudaba mucho que algo pudiera sorprenderme.\n\n\u2014Nashira tiene un don \u2014dijo Liss\u2014. \u00bfTe fijaste en qu\u00e9 aura tan rara desprend\u00eda? \u2014Asent\u00ed\u2014. No tiene una sola capacidad. Puede recorrer diversos caminos por el \u00e9ter.\n\n\u2014Eso es imposible. Todos tenemos un solo don.\n\n\u2014Mira, olv\u00eddate de la realidad. Sheol I tiene sus propias reglas. Ac\u00e9ptalo ya, y todo ser\u00e1 mucho m\u00e1s f\u00e1cil. \u2014Acerc\u00f3 las maltrechas rodillas a la barbilla\u2014. Nashira tiene cinco \u00e1ngeles guardianes. No s\u00e9 c\u00f3mo lo hace, pero consigue que se queden con ella.\n\n\u2014\u00bfEs vinculadora?\n\n\u2014No lo sabemos. Debe de haberlo sido en alg\u00fan momento, pero su aura se ha corrompido.\n\n\u2014\u00bfQu\u00e9 la ha corrompido?\n\n\u2014Los \u00e1ngeles. \u2014Frunc\u00ed el entrecejo, y ella suspir\u00f3\u2014. Esto solo es una teor\u00eda. Creemos que puede utilizar los dones que ten\u00edan los \u00e1ngeles cuando viv\u00edan.\n\n\u2014Eso no pueden hacerlo ni los vinculadores.\n\n\u2014Exacto. \u2014Me mir\u00f3 a los ojos\u2014. Si quieres un consejo, mant\u00e9n la cabeza agachada. No dejes que se note lo que eres. Si Nashira se entera de que eres una andarina, est\u00e1s perdida.\n\nMantuve una expresi\u00f3n neutra. Mis tres a\u00f1os en el sindicato me hab\u00edan habituado al peligro, pero ese sitio era diferente. Tendr\u00eda que aprender a esquivar nuevas amenazas.\n\n\u2014\u00bfC\u00f3mo puedo evitar que lo averig\u00fce?\n\n\u2014Ser\u00e1 dif\u00edcil. Te examinar\u00e1n para que reveles tu don. Eso es lo que significan los blusones. Rosa despu\u00e9s del primer examen, rojo despu\u00e9s del segundo.\n\n\u2014Pero t\u00fa suspendiste los ex\u00e1menes.\n\n\u2014S\u00ed, por suerte. Ahora respondo ante el Capataz.\n\n\u2014\u00bfQui\u00e9n era tu guardi\u00e1n?\n\nLiss dirigi\u00f3 la mirada hacia el hornillo.\n\n\u2014Gomeisa Sargas.\n\n\u2014\u00bfQui\u00e9n es?\n\n\u2014El otro soberano de sangre. Siempre hay dos, un var\u00f3n y una f\u00e9mina.\n\n\u2014Pero Arcturus est\u00e1...\n\n\u2014Comprometido con Nashira, s\u00ed. Pero \u00e9l no es de \u00abla sangre\u00bb \u2014dijo con un deje de asco\u2014. Solo los miembros de la familia Sargas pueden acceder a la corona. Los soberanos de sangre no pueden formar pareja: eso ser\u00eda incestuoso. Arcturus es de otra familia.\n\n\u2014Entonces es el pr\u00edncipe consorte.\n\n\u2014Es el consorte de sangre. Es lo mismo. \u00bfM\u00e1s _skilly_?\n\n\u2014No, gracias. \u2014Liss meti\u00f3 el cuenco en un barre\u00f1o lleno de agua grasienta\u2014. \u00bfQu\u00e9 hiciste para suspender los ex\u00e1menes?\n\n\u2014Me mantuve humana. \u2014Compuso una sonrisita\u2014. Los refas no son humanos. Por mucho que se parezcan a nosotros, no son como nosotros. No tienen nada aqu\u00ed. \u2014Se dio unos golpecitos en el pecho\u2014. Si quieren que trabajemos con ellos, tienen que librarse de nuestras almas.\n\n\u2014\u00bfC\u00f3mo?\n\nAntes de que Liss pudiera responder, descorrieron la cortina. Un refa\u00edta muy delgado apareci\u00f3 en el umbral.\n\n\u2014T\u00fa \u2014le gru\u00f1\u00f3 a Liss, y ella se llev\u00f3 las manos a la cabeza\u2014. Lev\u00e1ntate. V\u00edstete. Holgazana de mierda. \u00bfY con una invitada? \u00bfAcaso eres una reina?\n\nLiss se levant\u00f3. Toda su fuerza hab\u00eda desaparecido, y de pronto parec\u00eda peque\u00f1a y fr\u00e1gil. Le temblaba la mano izquierda.\n\n\u2014Lo siento, Suhail \u2014dijo\u2014. 40 es nueva aqu\u00ed. Quer\u00eda explicarle las normas de Sheol I.\n\n\u201440 ya deber\u00eda conocer las normas de Sheol I.\n\n\u2014Perd\u00f3name.\n\nEl refa\u00edta levant\u00f3 una mano enguantada como si fuera a pegarle.\n\n\u2014S\u00fabete a las sedas.\n\n\u2014Cre\u00eda que esta noche no ten\u00eda que actuar. \u2014Retrocedi\u00f3 hacia el rinc\u00f3n de la chabola\u2014. \u00bfHas hablado con el Capataz?\n\nMir\u00e9 con detenimiento al interrogador. Era alto y ten\u00eda los ojos dorados, como los otros refa\u00edtas, pero no percib\u00ed aquella mirada extraviada que hab\u00eda apreciado en sus semejantes. Cada arruga de su cara estaba cargada de odio.\n\n\u2014No necesito hablar con el Capataz, miserable marioneta. 15 sigue indispuesto. Los casacas rojas esperan que su idiota favorita lo sustituya. \u2014Estir\u00f3 los labios mostrando los dientes\u2014. A menos que quieras reunirte con \u00e9l en la penitenciar\u00eda, tienes que actuar dentro de diez minutos.\n\nLiss se retrajo. Meti\u00f3 los hombros hacia el pecho y mir\u00f3 hacia otro lado.\n\n\u2014Entiendo \u2014dijo.\n\n\u2014As\u00ed me gusta, esclava.\n\nAl salir, el refa\u00edta arranc\u00f3 la cortina. Ayud\u00e9 a Liss a recogerla del suelo. Temblaba ostensiblemente.\n\n\u2014\u00bfQui\u00e9n era?\n\n\u2014Suhail Chertan. El Capataz siempre est\u00e1 un poco tenso bajo todo ese maquillaje teatral. Responde ante Suhail si nosotros hacemos algo mal. \u2014Se dio unos toquecitos en los ojos con la manga\u2014. 15 es Jordan, el chico al que sometieron a privaci\u00f3n de sue\u00f1o. Es el otro contorsionista.\n\nLe quit\u00e9 la cortina de las manos y vi que ten\u00eda el blus\u00f3n manchado de sangre.\n\n\u2014\u00bfTe has cortado?\n\n\u2014No es nada.\n\n\u2014Algo te ha pasado.\n\nLa sangre siempre significaba algo.\n\n\u2014Estoy bien. \u2014Se frot\u00f3 la cara, y le aparecieron unas manchas rojas bajo los ojos\u2014. Solo me ha cogido un poco de brillo.\n\n\u2014\u00bfC\u00f3mo dices?\n\n\u2014Se ha alimentado de m\u00ed.\n\nEstaba segura de que no la hab\u00eda o\u00eddo bien.\n\n\u2014\u00bfQue se ha alimentado?\n\nLiss sonri\u00f3.\n\n\u2014\u00bfSe les olvid\u00f3 mencionar que los refas se alimentan de aura? \u00bfPor qu\u00e9 ser\u00e1 que siempre se les olvida?\n\nTen\u00eda la cara manchada de sangre. Se me hizo un nudo en el est\u00f3mago.\n\n\u2014Eso es imposible. El aura no sustenta la vida \u2014dije\u2014. Sustenta la videncia. No...\n\n\u2014Sustenta su vida.\n\n\u2014Pero... eso significar\u00eda que son algo m\u00e1s que clarividentes. Significar\u00eda que son la encarnaci\u00f3n del \u00e9ter.\n\n\u2014Puede que lo sean. \u2014Liss se ech\u00f3 una manta ra\u00edda sobre los hombros\u2014. Para eso estamos aqu\u00ed los bufones. Nosotros solo somos m\u00e1quinas de aura. Forraje para refas. De vosotros, los casacas, no se alimentan. Ese es vuestro privilegio. \u2014Mir\u00f3 el hornillo\u2014. A menos que suspend\u00e1is los ex\u00e1menes.\n\nMe qued\u00e9 un momento callada. No pod\u00eda concebir que los refa\u00edtas se alimentaran de aura. El aura era una conexi\u00f3n con el \u00e9ter, \u00fanica para cada vidente. No me explicaba c\u00f3mo pod\u00edan utilizarla para sobrevivir.\n\nPero esa informaci\u00f3n arrojaba luz sobre Sheol I. Por eso acog\u00edan a los videntes. Por eso no liquidaban a los actores si no pod\u00edan combatir a los emim. No los quer\u00edan solo para que bailaran, como es l\u00f3gico. Eso eran distracciones necias, para que no se aburrieran con tanto poder. No solo \u00e9ramos sus esclavos, sino tambi\u00e9n su fuente de alimento. Por eso pag\u00e1bamos nosotros por los errores humanos, y no los amaur\u00f3ticos.\n\nY pensar que solo unos d\u00edas atr\u00e1s estaba en Londres, viviendo mi vida en Seven Dials sin saber que exist\u00eda esa colonia.\n\n\u2014Alguien tiene que pararlos \u2014dije\u2014. Esto es una locura.\n\n\u2014Llevan doscientos a\u00f1os aqu\u00ed. \u00bfNo crees que alguien los habr\u00eda parado ya?\n\nMe volv\u00ed. Me martilleaba la cabeza.\n\n\u2014Lo siento. \u2014Liss me mir\u00f3\u2014. No quiero asustarte, pero llevo diez a\u00f1os aqu\u00ed. He visto luchar a muchos, a gente que quer\u00eda volver a su antigua vida, y todos acabaron muertos. Al final dejas de intentarlo.\n\n\u2014\u00bfEres profeta?\n\nSab\u00eda que no lo era, pero quer\u00eda ver si me ment\u00eda.\n\n\u2014Lectora. \u2014Era un t\u00e9rmino anticuado para denominar a los cartom\u00e1nticos, un t\u00e9rmino del argot callejero de una d\u00e9cada atr\u00e1s\u2014. Lo supieron la primera vez que le\u00ed las cartas.\n\n\u2014\u00bfQu\u00e9 viste?\n\nAl principio cre\u00ed que no me hab\u00eda o\u00eddo. Entonces fue hasta el fondo de la chabola y se arrodill\u00f3 junto a una cajita de madera. Sac\u00f3 una baraja de cartas de tarot atadas con una cinta roja y me dio una. El Loco.\n\n\u2014Siempre supe que estaba destinada a que me tocara la china \u2014dijo\u2014. Y ten\u00eda raz\u00f3n.\n\n\u2014\u00bfPuedes ech\u00e1rmelas?\n\n\u2014En otro momento. Ahora tienes que irte. \u2014Liss sac\u00f3 una pastilla de colofonia del arc\u00f3n\u2014. Ven a verme otro d\u00eda, hermana. No puedo protegerte, pero llevo una d\u00e9cada aqu\u00ed. Quiz\u00e1 pueda impedir que te mates. \u2014Me sonri\u00f3 cansada\u2014. Bienvenida a Sheol I.\n\nLiss me indic\u00f3 c\u00f3mo llegar a la Casa Amaur\u00f3tica, que era adonde el Guardi\u00e1n Gris (Graffias Sheratan, el refa encargado de controlar al peque\u00f1o grupo de trabajadores amaur\u00f3ticos) hab\u00eda llevado a Seb; y me dio un poco de carne y pan para que se los diera.\n\n\u2014Que no te vea Graffias \u2014me advirti\u00f3.\n\nHab\u00eda aprendido mucho en solo cuarenta minutos. La revelaci\u00f3n m\u00e1s inquietante era que me hallaba en el radar de Nashira; no me entusiasmaba la perspectiva de convertirme en su esp\u00edritu esclavo para la eternidad. Siempre me hab\u00eda dado un miedo tremendo no acabar en el centro del \u00e9ter, el lugar donde mueren todas las cosas. Me horrorizaba la idea de ser un esp\u00edritu inquieto, un poco de munici\u00f3n con la que los videntes podr\u00edan comerciar y a la que podr\u00edan maltratar. Con todo, eso nunca me hab\u00eda impedido llamar a bandadas de esp\u00edritus para que me protegieran, ni pujar en nombre de Jax por la enojad\u00edsima Anne Naylor, que era una cr\u00eda cuando la asesinaron.\n\nAdem\u00e1s, la advertencia de Liss me pon\u00eda muy nerviosa. \u00abAl final dejar\u00e1s de intentarlo.\u00bb Se equivocaba.\n\nLa Casa Amaur\u00f3tica quedaba fuera de la zona donde se erig\u00edan las residencias. Tuve que pasar por varias calles abandonadas para llegar hasta all\u00ed. Hab\u00eda visto mapas de la ciudad en un ejemplar antiguo de _El_ _Pendenciero_ (otro objeto de inter\u00e9s que Jax le hab\u00eda quitado a Didion Waite) y ten\u00eda una idea aproximada de d\u00f3nde estaban la mayor\u00eda de sus monumentos. Me dirig\u00ed hacia el norte por la calle principal. Vi a unos casacas rojas apostados delante de ciertos edificios, pero solo me miraron de pasada. Deb\u00eda de haber alg\u00fan tipo de barrera que nos imped\u00eda escapar, adem\u00e1s de las minas de la Tierra de Nadie. \u00bfCu\u00e1ntos videntes habr\u00edan muerto intentando atravesarla?\n\nAl cabo de pocos minutos encontr\u00e9 el edificio. Era discreto y austero, con una peque\u00f1a media luna de hierro sobre la cancela. Lo que hubiera escrito all\u00ed con anterioridad hab\u00eda sido sustituido por las palabras CASA AMAUR\u00d3TICA. Debajo hab\u00eda una frase en lat\u00edn: DOMUS STULTORUM. No quer\u00eda saber qu\u00e9 significaba. Me asom\u00e9 entre los barrotes y mi mirada se encontr\u00f3 con la de un guardia refa\u00edta. Ten\u00eda el cabello rizado, casta\u00f1o oscuro, y lo llevaba suelto sobre los hombros. El labio inferior, carnoso, le confer\u00eda un aire enfurru\u00f1ado. Deb\u00eda de ser Graffias.\n\n\u2014Espero que tengas un buen motivo para estar cerca de la Casa Amaur\u00f3tica \u2014dijo con una voz cargada de desd\u00e9n.\n\nNo se me ocurri\u00f3 ninguno. La proximidad de ese ser me helaba la sangre.\n\n\u2014No \u2014contest\u00e9\u2014. Pero tengo esto.\n\nLe ense\u00f1\u00e9 mis _numa_ : anillos, dedales, agujas. Graffias me mir\u00f3 con tanto odio, tanto asco, que me estremec\u00ed. Casi prefer\u00eda aquella otra mirada, cruel.\n\n\u2014Yo no acepto sobornos. Ni necesito esas despreciables alhajas humanas para acceder al \u00e9ter.\n\nMe guard\u00e9 las despreciables alhajas humanas en el bolsillo. Qu\u00e9 idea tan est\u00fapida. Claro que no necesitaban esas chorradas. Eran la moneda de los mendigos.\n\n\u2014Lo siento \u2014dije.\n\n\u2014Vuelve a tu residencia, casaca blanca, o llamar\u00e9 a tu guardi\u00e1n para que venga a castigarte.\n\nSac\u00f3 una bandada de esp\u00edritus. Me di la vuelta, me alej\u00e9 de la verja hasta quedar fuera de su campo de visi\u00f3n y no gir\u00e9 la cabeza. Cuando estaba a punto de echar a correr hacia Magdalen, o\u00ed una vocecilla por encima de mi cabeza.\n\n\u2014\u00a1Espera, Paige!\n\nVi asomar una mano entre los barrotes de una ventana del segundo piso. Afloj\u00e9 los hombros, aliviada. Era Seb.\n\n\u2014\u00bfEst\u00e1s bien?\n\n\u2014No. Por favor, Paige. S\u00e1came de aqu\u00ed. Por favor \u2014dijo con voz entrecortada\u2014. Tengo que salir de aqu\u00ed. Siento... Siento mucho haberte llamado antinatural. Lo siento...\n\nVolv\u00ed la cabeza para comprobar que no hubiera nadie mir\u00e1ndome. Trep\u00e9 por la fachada del edificio, met\u00ed una mano bajo mi blus\u00f3n y le di el paquete de comida a Seb.\n\n\u2014Te sacar\u00e9 de aqu\u00ed. \u2014Le apret\u00e9 la mano helada desde el otro lado de los barrotes\u2014. Har\u00e9 todo lo que pueda para sacarte, pero tienes que darme tiempo.\n\n\u2014Me matar\u00e1n. \u2014Desenvolvi\u00f3 el paquete con dedos temblorosos\u2014. Me morir\u00e9 antes de que puedas liberarme.\n\n\u2014\u00bfQu\u00e9 te hacen?\n\n\u2014Me hacen fregar el suelo hasta que me sangran las manos, y luego tengo que seleccionar cristales rotos, y escoger los trozos m\u00e1s puros para sus ornamentos. \u2014Me fij\u00e9 en sus manos, llenas de cortes profundos y sucios\u2014. Ma\u00f1ana tengo que empezar a trabajar en las residencias.\n\n\u2014\u00bfDe qu\u00e9 clase de trabajo se trata?\n\n\u2014Todav\u00eda no lo s\u00e9. No quiero saberlo. Creen que soy un... que soy uno de los vuestros. \u2014Ten\u00eda la voz tomada\u2014. \u00bfPara qu\u00e9 me quieren?\n\n\u2014No lo s\u00e9. \u2014Me fij\u00e9 en su ojo derecho, hinchado e inyectado en sangre\u2014. \u00bfQu\u00e9 te ha pasado en el ojo?\n\n\u2014Uno me peg\u00f3. Yo no hab\u00eda hecho nada, Paige, te lo prometo. Dijo que era escoria humana. Dijo que...\n\nAgach\u00f3 la cabeza y le temblaron los labios. Era su primer d\u00eda y ya lo hab\u00edan utilizado de saco de boxeo. \u00bfC\u00f3mo iba a sobrevivir una semana, un mes? \u00bfO una d\u00e9cada, como Liss?\n\n\u2014C\u00f3mete eso. \u2014Le apret\u00e9 las manos alrededor del paquete\u2014. Ma\u00f1ana, intenta ir a Magdalen.\n\n\u2014\u00bfT\u00fa vives all\u00ed?\n\n\u2014S\u00ed. Seguramente mi guardi\u00e1n no estar\u00e1. Podr\u00e1s darte un ba\u00f1o y comer algo. \u00bfDe acuerdo?\n\nSeb asinti\u00f3 con la cabeza. Parec\u00eda que deliraba; seguro que sufr\u00eda una conmoci\u00f3n cerebral. Necesitaba ir a un hospital, que lo viera un m\u00e9dico. Pero all\u00ed no hab\u00eda m\u00e9dicos. Seb no le importaba a nadie.\n\nAquella noche yo no pod\u00eda hacer nada m\u00e1s por \u00e9l. Le di un apret\u00f3n en el brazo, me solt\u00e9 de la ventana, ca\u00ed de pie y me dirig\u00ed hacia el centro de la ciudad.\n\nLlegu\u00e9 a la residencia al alba. El portero de d\u00eda, vestido de rojo, me entreg\u00f3 una llave de la c\u00e1mara del Custodio.\n\n\u2014D\u00e9jala encima de su mesa \u2014dijo\u2014. Ni se te ocurra qued\u00e1rtela.\n\nNo le contest\u00e9. Sub\u00ed por la oscura escalera evitando a los dos guardias. Me horrorizaba ver brillar sus ojos en los pasillos; parec\u00edan reflectores naturales en la oscuridad. Y se supon\u00eda que esa era una residencia segura: no quer\u00eda imaginarme c\u00f3mo deb\u00edan de ser las otras.\n\nSonaron las campanas de la torre: era la se\u00f1al para que los humanos volvieran a sus prisiones. Ya en la c\u00e1mara, cerr\u00e9 la puerta con llave y dej\u00e9 la llave encima de la mesa. No hab\u00eda ni rastro del Custodio. Encontr\u00e9 una caja de cerillas en un caj\u00f3n y las us\u00e9 para encender unas velas. En ese mismo caj\u00f3n hab\u00eda tres pares id\u00e9nticos de guantes de piel negros, y un gran anillo de plata con una piedra roja engarzada.\n\nEn una pared hab\u00eda una vitrina de madera oscura de palisandro. Abr\u00ed las puertas de cristal, y mi sexto sentido me dio una punzada. Dentro hab\u00eda una colecci\u00f3n de instrumentos; reconoc\u00ed algunos porque los hab\u00eda visto en el mercado negro. Algunos eran _numa_. La mayor\u00eda eran solo baratijas: una tabla de escritura espiritista, un poco de tiza, una pizarra de espiritismo... la clase de art\u00edculos in\u00fatiles que los amaur\u00f3ticos asociaban con la clarividencia. Otros, como la bola de cristal, los utilizaban los profetas para predecir el futuro. Yo no era adivina; ninguno de aquellos objetos me serv\u00eda para nada. Como Graffias, no necesitaba objetos para tocar el \u00e9ter.\n\nLo que necesitaba era un equipo de soporte vital. Hasta que encontrara alg\u00fan aparato de ox\u00edgeno, tendr\u00eda que vigilar la frecuencia con que me desprend\u00eda de mi esp\u00edritu. As\u00ed era como ampliaba mi percepci\u00f3n del \u00e9ter: pod\u00eda desplazar mi esp\u00edritu de su ubicaci\u00f3n natural y llevarlo hasta los l\u00edmites de mi onirosaje. El problema era que si me demoraba demasiado, mi reflejo respiratorio se deten\u00eda.\n\nMe llam\u00f3 la atenci\u00f3n un estuche peque\u00f1o, rectangular, con una estilizada flor de madera grabada en la tapa. Ocho p\u00e9talos. Solt\u00e9 el cierre y lo abr\u00ed. Dentro hab\u00eda cuatro viales que conten\u00edan un l\u00edquido viscoso de color rojo muy oscuro, casi negro. Cerr\u00e9 el estuche. No quer\u00eda saber m\u00e1s.\n\nNotaba un dolor sordo en un ojo. No vi ropa de dormir; no s\u00e9 por qu\u00e9 esperaba encontrarla. Al Custodio no le importaba lo que yo llevara puesto ni si dorm\u00eda bien. Lo \u00fanico que le interesaba era que respirara.\n\nMe quit\u00e9 las botas y me tumb\u00e9 en el div\u00e1n. En la habitaci\u00f3n hac\u00eda un fr\u00edo tremendo con la chimenea apagada, pero no me atrev\u00ed a tocar las s\u00e1banas de la cama del Custodio. Apoy\u00e9 la mejilla en un gran coj\u00edn de terciopelo.\n\nEl flux me hab\u00eda dejado d\u00e9bil y cansada. Mientras rondaba por los umbrales del sue\u00f1o, mi esp\u00edritu entraba y sal\u00eda del \u00e9ter. Iba rozando onirosajes, captando oleadas de recuerdos. El dolor y la sangre eran denominadores comunes. En esa residencia hab\u00eda otros refas, pero sus mentes eran absolutamente impenetrables. Los humanos estaban m\u00e1s abiertos, pues el miedo debilitaba sus defensas. Sus onirosajes desprend\u00edan una luz intensa, molesta, que delataba aflicci\u00f3n. Al final me qued\u00e9 dormida.\n\nMe despertaron los crujidos del suelo de madera. Abr\u00ed los ojos y vi entrar al Custodio por la puerta. La \u00fanica luz, aparte de las dos velas que quedaban encendidas, era la que desprend\u00edan sus ojos. Cruz\u00f3 la habitaci\u00f3n y vino hasta mi rinc\u00f3n. Me hice la dormida. Al cabo de un rato que me pareci\u00f3 eterno, se march\u00f3. Ya no caminaba con tanto cuidado, y por el ritmo de sus pasos me di cuenta de que cojeaba. Cerr\u00f3 la puerta del cuarto de ba\u00f1o.\n\n\u00bfQu\u00e9 pod\u00eda haber capaz de lastimar a un ser como un refa\u00edta?\n\nTard\u00f3 unos minutos en salir; el coraz\u00f3n me golpeaba tan fuerte en el pecho que pod\u00eda contar cada uno de sus latidos. Cuando vi que giraba el picaporte, volv\u00ed a taparme la cabeza con los brazos. El Custodio sali\u00f3 completamente desnudo. Cerr\u00e9 los ojos.\n\nSegu\u00ed fingiendo que dorm\u00eda mientras \u00e9l iba hacia la cama; por el camino tir\u00f3 al suelo una bola de cristal. Unas ondas oscilaron por el \u00e9ter. Corri\u00f3 las cortinas del dosel de la cama y desapareci\u00f3 de mi vista. Hasta que su mente se seren\u00f3, no abr\u00ed los ojos y me incorpor\u00e9. No detect\u00e9 ning\u00fan movimiento.\n\nDescalza, me acerqu\u00e9 a la cama y met\u00ed los dedos entre las cortinas, separ\u00e1ndolas lo suficiente para ver al Custodio. Estaba tumbado de lado, tapado con las s\u00e1banas, y su piel brillaba en la penumbra. El cabello, enmara\u00f1ado, le cubr\u00eda la cara. Mientras lo observaba, una luz tenue brill\u00f3 bajo las s\u00e1banas, cerca de donde \u00e9l ten\u00eda el brazo derecho.\n\nMe acerqu\u00e9 a su onirosaje. Algo hab\u00eda cambiado. No lo entend\u00eda muy bien, pero percib\u00eda que hab\u00eda algo que no estaba como deber\u00eda. Todos los onirosajes ten\u00edan una especie de luz invisible: un resplandor interior, imperceptible para los sentidos amaur\u00f3ticos. La luz vital del Custodio se estaba apagando.\n\nPermanec\u00eda quieto como un muerto. Mir\u00e9 las s\u00e1banas y vi que estaban salpicadas de un l\u00edquido amarillo verdoso, ligeramente luminoso. Ten\u00eda un d\u00e9bil olor met\u00e1lico. Not\u00e9 como si tiraran de mi sexto sentido, como si estuviera inhalando el \u00e9ter. Retir\u00e9 la pesada ropa de cama.\n\nEl Custodio ten\u00eda una mordedura que supuraba en el lado interior del brazo. Tragu\u00e9 saliva. Vi las marcas de unos dientes, la piel toscamente desgarrada. La herida rezumaba gotas de luz. Sangre.\n\nEra su sangre.\n\nDeb\u00eda de haber dicho a los otros refa\u00edtas ad\u00f3nde iba. Los otros deb\u00edan de saber que hab\u00eda ido a hacer algo peligroso. Si se mor\u00eda, no encontrar\u00edan ninguna prueba que me incriminara.\n\nEntonces record\u00e9 lo que me hab\u00eda dicho Liss en la choza. \u00abLos refas no son humanos. Por mucho que se parezcan a nosotros, no son como nosotros.\u00bb __\n\nComo si fuera a importarles que no hubiera pruebas. Pod\u00edan inventarse las pruebas. Pod\u00edan decir lo que quisieran. Si el Custodio mor\u00eda en su cama, podr\u00edan decir que yo lo hab\u00eda asfixiado. As\u00ed Nashira tendr\u00eda una excusa para matarme pronto.\n\nQuiz\u00e1 deb\u00eda hacerlo. Esa era mi gran oportunidad para librarme de \u00e9l. Ya hab\u00eda matado una vez; pod\u00eda volver a hacerlo.\n\nTen\u00eda tres opciones: quedarme all\u00ed sentada vi\u00e9ndolo morir, matarlo o tratar de impedir que muriera. Prefer\u00eda verlo morir, pero intu\u00eda que ser\u00eda mejor salvarlo. En Magdalen estaba relativamente a salvo. Lo \u00faltimo que quer\u00eda hacer, de momento, era mudarme.\n\n\u00c9l todav\u00eda no me hab\u00eda hecho da\u00f1o, pero seguramente me lo har\u00eda. Para poseerme tendr\u00eda que subyugarme, torturarme, obligarme a obedecerlo por todos los medios que fueran necesarios. Si lo mataba ahora, quiz\u00e1 me salvara. Estir\u00e9 un brazo hacia una almohada. Pod\u00eda hacerlo, pod\u00eda asfixiarlo. \u00abS\u00ed, vamos, m\u00e1talo.\u00bb Dobl\u00e9 los dedos, agarr\u00e9 la tela de algod\u00f3n. \u00ab\u00a1M\u00e1talo!\u00bb\n\nNo, no pod\u00eda. Se despertar\u00eda. Se despertar\u00eda y me partir\u00eda el cuello. Y aunque no me lo partiera, no podr\u00eda escapar. Los vigilantes que hab\u00eda fuera me mandar\u00edan a la horca por asesinato.\n\nTen\u00eda que salvarlo.\n\nAlgo me dec\u00eda que no deb\u00eda tocar las s\u00e1banas. No me fiaba de ese l\u00edquido. El resplandor parec\u00eda radiactivo, y no pod\u00eda olvidar las advertencias de Scion respecto a la contaminaci\u00f3n. Fui al caj\u00f3n y saqu\u00e9 un par de guantes. Eran enormes, confeccionados para manos de refa\u00edta. Con ellos puestos, mis dedos perd\u00edan toda la destreza. Desgarr\u00e9 una de las s\u00e1banas m\u00e1s limpias. Eran muy finas, no abrigaban nada. Cuando hube hecho unas cuantas tiras largas, me las llev\u00e9 al cuarto de ba\u00f1o y las empap\u00e9 en agua caliente. Quiz\u00e1 no funcionara, pero quiz\u00e1 s\u00ed: tal vez despertara y pudiera pedir ayuda a los otros refa\u00edtas. Si ten\u00eda suerte, claro.\n\nVolv\u00ed a la habitaci\u00f3n y me seren\u00e9. El Custodio estaba absolutamente inm\u00f3vil, y el fr\u00edo de su piel me atravesaba los guantes. A todas luces parec\u00eda muerto. Ten\u00eda la piel gris\u00e1cea. Escurr\u00ed la s\u00e1bana y me puse a trabajar en la herida. Al principio lo hac\u00eda con cuidado, pero \u00e9l no se mov\u00eda, y comprend\u00ed que no iba a despertar.\n\nFuera, detr\u00e1s de las ventanas, la luz empezaba a cambiar. Escurr\u00ed la s\u00e1bana sobre la herida y limpi\u00e9 la sangre y la arenilla incrustada en la carne. Al cabo de lo que me parecieron varias horas, hab\u00eda conseguido algo. Ve\u00eda subir y bajar el pecho del Custodio al comp\u00e1s de su respiraci\u00f3n, sus d\u00e9biles degluciones. Con otro trozo de s\u00e1bana hice un parche que coloqu\u00e9 sobre la herida, y le sujet\u00e9 el ap\u00f3sito improvisado con el faj\u00edn de mi blus\u00f3n. Luego volv\u00ed a taparlo con la ropa de cama. Ahora ya era asunto suyo sobrevivir o no.\n\nDespert\u00e9 unas horas m\u00e1s tarde.\n\nDeduje, por el silencio reinante, que no hab\u00eda nadie en la habitaci\u00f3n. La cama estaba hecha. Hab\u00edan puesto s\u00e1banas limpias. Las cortinas estaban atadas con tiras de tela bordada, y la p\u00e1lida luz de la luna ba\u00f1aba las paredes.\n\nEl Custodio se hab\u00eda ido.\n\nPor los cristales de las ventanas resbalaban gotas de condensaci\u00f3n. Me sent\u00e9 junto al fuego. Era imposible que me hubiera imaginado toda aquella escena, a menos que todav\u00eda estuviera sufriendo los efectos del flux. Pero hab\u00eda tomado el ant\u00eddoto. Ya ten\u00eda la sangre limpia. Eso significaba que el Custodio, por el motivo que fuera, hab\u00eda vuelto a marcharse.\n\nEncima de la cama hab\u00eda un uniforme nuevo, junto con otra nota escrita con la misma letra. Rezaba, sencillamente:\n\nMa\u00f1ana.\n\nAs\u00ed que no se hab\u00eda muerto mientras dorm\u00eda. Y mi entrenamiento se hab\u00eda retrasado un d\u00eda m\u00e1s.\n\nLos guantes hab\u00edan desaparecido; deb\u00eda de hab\u00e9rselos llevado \u00e9l. Entr\u00e9 en el cuarto de ba\u00f1o y me lav\u00e9 las manos con agua caliente. Me puse el uniforme, saqu\u00e9 las tres pastillas de los bl\u00edsters y las tir\u00e9 por el desag\u00fce del lavamanos. Quer\u00eda seguir recabando informaci\u00f3n. No me importaba lo que dijera Liss: no pod\u00edamos aceptar la situaci\u00f3n sin m\u00e1s. No me importaba si los refas llevaban all\u00ed doscientos a\u00f1os o dos millones: no pensaba permitir que abusaran de mi clarividencia. Yo no era su soldado, ni Liss era su merienda.\n\nLa portera de noche anot\u00f3 mi salida de la residencia en el registro. Fui al Poblado y me compr\u00e9 un cuenco de gachas. Sab\u00edan a lo que parec\u00edan, cemento, pero me obligu\u00e9 a com\u00e9rmelas. La actriz me dijo en voz baja que Suhail rondaba por all\u00ed; no pod\u00eda sentarme a comer. Le pregunt\u00e9 si sab\u00eda d\u00f3nde pod\u00eda encontrar a Julian, y se lo describ\u00ed lo mejor que pude. Me indic\u00f3 que mirara en las residencias centrales; me dio sus nombres, me dijo d\u00f3nde estaba cada una y luego sigui\u00f3 ocup\u00e1ndose de su hornillo de parafina.\n\nMe qued\u00e9 de pie en un rinc\u00f3n oscuro. Mientras com\u00eda, observaba a la gente que pululaba a mi alrededor. Todos ten\u00edan la misma mirada apagada. El colorido de su ropa resultaba casi insultante, como los graffiti en una l\u00e1pida.\n\n\u2014Dan ganas de vomitar, \u00bfverdad?\n\nLevant\u00e9 la cabeza. Era la suspirante a la que hab\u00eda visto la primera noche. Llevaba un vendaje sucio en un brazo. Parec\u00eda aburrida, y se sent\u00f3 a mi lado.\n\n\u2014Me llamo Tilda.\n\n\u2014Yo soy Paige.\n\n\u2014Ya lo s\u00e9. Dicen que acabaste en Magdalen. \u2014En una mano ten\u00eda un cigarrillo liado de cualquier manera de cuyo extremo sal\u00eda un humo denso que ol\u00eda a especias y perfume. Reconoc\u00ed el aroma del \u00e1ster morado\u2014. Toma.\n\n\u2014No, gracias.\n\n\u2014No seas tonta, solo es un poco de _regal_. Es mejor que el _tincto_.\n\nEl _tincto_ , o l\u00e1udano, era el vicio preferido de los amaur\u00f3ticos dispuestos a correr el riesgo de alterar su estado mental. No a todos les gustaba el Floxy.\n\nDe vez en cuando deten\u00edan a alg\u00fan amaur\u00f3tico sospechoso de antinaturalidad, y luego la DVN descubr\u00eda que se hab\u00eda intoxicado con l\u00e1udano. A los videntes no nos hac\u00eda gran cosa; no era lo bastante fuerte para alterar nuestros onirosajes. Tilda deb\u00eda de consumirlo por hacer algo.\n\n\u2014\u00bfDe d\u00f3nde lo has sacado? \u2014le pregunt\u00e9, pues dudaba de que los refas permitieran el consumo de drogas et\u00e9reas.\n\n\u2014Hay un narco que lo vende a pondos. Dice que est\u00e1 aqu\u00ed desde la Era XVI.\n\n\u2014\u00bfCuarenta a\u00f1os?\n\n\u2014Desde que ten\u00eda veintiuno. Antes he estado hablando con \u00e9l. Parece simp\u00e1tico. \u2014Volvi\u00f3 a ofrecerme el porro\u2014. \u00bfSeguro que no quieres una calada?\n\n\u2014No, paso. \u2014Me qued\u00e9 mir\u00e1ndola fumar. Tilda ten\u00eda toda la pinta de los adictos al \u00e1ster, o cortesanos, como ellos mismos se hac\u00edan llamar; solo una yonqui habr\u00eda dicho \u00abpondos\u00bb en lugar de \u00ablibras\u00bb. Quiz\u00e1 pudiera ayudarme\u2014. \u00bfC\u00f3mo es que no est\u00e1s entren\u00e1ndote?\n\n\u2014Mi guardi\u00e1n se ha marchado a no s\u00e9 d\u00f3nde. \u00bfY t\u00fa?\n\n\u2014Por lo mismo. \u00bfQui\u00e9n es tu guardi\u00e1n?\n\n\u2014Terebell Sheratan. Parece una cabrona, pero todav\u00eda no ha intentado vapulearme.\n\n\u2014Ya. \u2014Sigui\u00f3 fumando\u2014. \u00bfSabes qu\u00e9 son esas pastillas que nos dan?\n\nTilda asinti\u00f3.\n\n\u2014La blanca es un anticonceptivo normal y corriente. Es raro que no la hayas reconocido.\n\n\u2014\u00bfUn anticonceptivo? \u00bfPara qu\u00e9?\n\n\u2014Para que no nos reproduzcamos, evidentemente. Y para que no menstruemos. A ver, \u00bfqui\u00e9n querr\u00eda parir en un sitio como este?\n\nTen\u00eda raz\u00f3n.\n\n\u2014\u00bfY la roja?\n\n\u2014Es un suplemento de hierro.\n\n\u2014\u00bfY la verde?\n\n\u2014\u00bfC\u00f3mo?\n\n\u2014La tercera pastilla.\n\n\u2014No hay tercera pastilla.\n\n\u2014Es un comprimido \u2014insist\u00ed\u2014. De color verde gris\u00e1ceo. Tiene un sabor amargo.\n\nTilda neg\u00f3 con la cabeza.\n\n\u2014Lo siento, ni idea. Si me traes una, le echar\u00e9 un vistazo.\n\nSe me contrajo el est\u00f3mago.\n\n\u2014Vale \u2014dije. Tilda se dispon\u00eda a dar otra calada, pero la interrump\u00ed\u2014: T\u00fa estabas con Carl, \u00bfno? En el serm\u00f3n.\n\n\u2014Yo no me relaciono con ese renegado. \u2014Arque\u00e9 una ceja; Tilda exhal\u00f3 un humo de color lila\u2014. \u00bfNo te has enterado? Es un traidor. Sorprendi\u00f3 a una carro\u00f1a pas\u00e1ndole comida a Ivy, esa palmista del pelo azul, y se chiv\u00f3 a su guardi\u00e1n. Tendr\u00edas que haber visto lo que le hicieron.\n\n\u2014Cu\u00e9ntame.\n\n\u2014Le dieron una paliza. Le afeitaron la cabeza. No quiero acordarme. \u2014Le temblaba un poco la mano\u2014. Si eso es lo que hay que hacer para sobrevivir aqu\u00ed, prefiero que me manden al \u00e9ter. Me ir\u00e9 sin rechistar.\n\nSe produjo un silencio. Tilda tir\u00f3 su porro de \u00e1ster.\n\n\u2014\u00bfSabes en qu\u00e9 residencia est\u00e1 Julian? \u2014pregunt\u00e9 al cabo de un rato\u2014. 26.\n\n\u2014\u00bfEl chico calvo? Creo que en Trinity. Puedes echar un vistazo por la verja que hay en la parte de atr\u00e1s. Los novatos se entrenan all\u00ed, en el jard\u00edn. Pero procura que no te vean.\n\nLa dej\u00e9 li\u00e1ndose otro porro.\n\nEl \u00e1ster era muy apreciado. Seguramente era la planta de la que m\u00e1s se abusaba en las calles. La adicci\u00f3n estaba muy extendida en sitios como Jacob's Island. Las flores pod\u00edan ser de color blanco, azul, rosa o morado, y cada una ten\u00eda un efecto diferente en el onirosaje. Eliza fue adicta al \u00e1ster blanco durante a\u00f1os; me lo hab\u00eda contado. Comparado con el azul, que recuperaba los recuerdos, el \u00e1ster blanco produc\u00eda un efecto que llam\u00e1bamos \u00abencalado\u00bb, o p\u00e9rdida parcial de memoria. Durante un tiempo, Eliza hab\u00eda olvidado hasta su apellido. Despu\u00e9s se enganch\u00f3 al morado, pues dec\u00eda que la ayudaba a pintar. Me hab\u00eda hecho jurar que jam\u00e1s probar\u00eda ninguna droga et\u00e9rea, y yo no ten\u00eda ning\u00fan motivo para incumplir mi promesa.\n\nMe preocupaba saber que a m\u00ed me daban una pastilla m\u00e1s que a los otros prisioneros. A menos que lo raro fuera que a Tilda solo le dieran dos. Tendr\u00eda que pregunt\u00e1rselo a alguien m\u00e1s.\n\nLa residencia de Trinity estaba vigilada por el lado que daba a la calle. Borde\u00e9 el barrio de chabolas y me gui\u00e9 por mi limitado conocimiento de la ciudad para averiguar d\u00f3nde estaba la parte trasera de la residencia. Acab\u00e9 ante la empalizada que cercaba sus extensos jardines. Tilda ten\u00eda raz\u00f3n: hab\u00eda un grupo de casacas blancas en el jard\u00edn, dirigidos por una refa. Julian se encontraba entre ellos. Empujaban a unos esp\u00edritus por el aire con unos bastones rebordeados, alumbr\u00e1ndose con unas l\u00e1mparas de gas que emit\u00edan una luz verdosa. Al principio cre\u00ed que eran _numa_ , objetos por los que pod\u00eda circular el \u00e9ter de los que los adivinos obten\u00edan su poder; pero nunca hab\u00eda visto que se utilizaran objetos para controlar a los esp\u00edritus.\n\nDej\u00e9 que actuara mi sexto sentido. Los onirosajes de los humanos estaban agrupados en el \u00e9ter, y la refa funcionaba como una especie de eje que atra\u00eda a los humanos como insectos hacia un farol.\n\nLa refa eligi\u00f3 ese momento para cargar contra Julian. Enarbol\u00f3 su bast\u00f3n y le lanz\u00f3 un esp\u00edritu furioso. \u00c9l cay\u00f3 de espaldas al suelo, aturdido.\n\n\u2014En pie, 26. \u2014Julian no se movi\u00f3\u2014. Lev\u00e1ntate.\n\nNo pod\u00eda levantarse. Y no me extra\u00f1\u00f3: un esp\u00edritu furioso acababa de golpearle en la cara. Ning\u00fan vidente habr\u00eda podido levantarse despu\u00e9s de un golpe as\u00ed.\n\nSu guardiana le propin\u00f3 una fuerte patada en un lado de la cabeza. Los otros casacas blancas retrocedieron tambale\u00e1ndose, como si a continuaci\u00f3n la refa fuera a pegarles a ellos; sin embargo, se limit\u00f3 a mirarlos con frialdad antes de darse la vuelta y echar a andar hacia la residencia, con la t\u00fanica negra ondeando. Los humanos se miraron y la siguieron. Ninguno se qued\u00f3 a ayudar a Julian, que estaba tendido en el suelo, en posici\u00f3n fetal. Intent\u00e9 abrir la verja, pero estaba cerrada con una gruesa cadena.\n\n\u2014\u00a1Julian! \u2014lo llam\u00e9.\n\nHizo una mueca de dolor y alz\u00f3 la cabeza. Al verme, se levant\u00f3 y fue hacia la verja. Ten\u00eda la cara cubierta de sudor. Detr\u00e1s de \u00e9l los faroles se apagaron.\n\n\u2014En el fondo me adora \u2014dijo esbozando una sonrisa\u2014. Soy su alumno aventajado.\n\n\u2014\u00bfQu\u00e9 clase de esp\u00edritu era eso?\n\n\u2014Un fantasma viejo. \u2014Se frot\u00f3 los ojos\u2014. Lo siento, todav\u00eda veo cosas.\n\n\u2014\u00bfQu\u00e9 ves?\n\n\u2014Caballos. Libros. Fuego.\n\nEl fantasma le hab\u00eda dejado una impresi\u00f3n de su muerte. Era una de las facetas desagradables del combate contra esp\u00edritus.\n\n\u2014\u00bfQui\u00e9n era esa refa? \u2014pregunt\u00e9.\n\n\u2014Se llama Aludra Chertan. No entiendo por qu\u00e9 se ha ofrecido voluntaria para ser guardiana. Nos odia.\n\n\u2014Todos nos odian. \u2014Ote\u00e9 la extensi\u00f3n de c\u00e9sped. Aludra no hab\u00eda vuelto\u2014. \u00bfPuedes salir?\n\n\u2014Puedo intentarlo. \u2014Se llev\u00f3 una mano a la cabeza e hizo una mueca\u2014. \u00bfTu guardi\u00e1n ya se ha cebado contigo?\n\n\u2014Casi no lo he visto. \u2014Prefer\u00ed no mencionar lo que hab\u00eda pasado la noche anterior, por precauci\u00f3n.\n\n\u2014Ayer Aludra se ceb\u00f3 con Felix. Cuando recobr\u00f3 el conocimiento, no pod\u00eda parar de temblar. Y aun as\u00ed le oblig\u00f3 a entrenar.\n\n\u2014\u00bfC\u00f3mo le fue?\n\n\u2014Estaba aterrorizado. Durante dos horas no pudo percibir el \u00e9ter.\n\n\u2014Hacerle eso a un vidente es una locura. \u2014Gir\u00e9 la cabeza para ver si hab\u00eda alg\u00fan vigilante\u2014. Yo no dejar\u00e9 que se ceben conmigo.\n\n\u2014Quiz\u00e1 no puedas evitarlo. \u2014Desenganch\u00f3 un farol de la verja\u2014. Tu guardi\u00e1n es muy famoso. \u00bfDices que casi no lo has visto?\n\n\u2014Sale mucho.\n\n\u2014\u00bfAd\u00f3nde va? \u00bfQu\u00e9 hace?\n\n\u2014No tengo ni idea.\n\nJulian se qued\u00f3 mir\u00e1ndome. A esa distancia vi que ten\u00eda visi\u00f3n espiritista permanente, como Liss. Los videntes con visi\u00f3n parcial pod\u00edan activar y desactivar su don, pero Julian ve\u00eda peque\u00f1as hebras de energ\u00eda continuamente.\n\n\u2014A ver si puedo salir \u2014dijo\u2014. No he comido nada desde ayer por la ma\u00f1ana. O por la noche. Yo qu\u00e9 s\u00e9.\n\n\u2014\u00bfTe dar\u00e1n permiso?\n\n\u2014Puedo pedirlo.\n\nLo vi alejarse hacia la residencia, y se me ocurri\u00f3 pensar que tal vez no volviera a salir.\n\nLo esper\u00e9 cerca del Poblado. Estaba a punto de desistir cuando vi el destello de un blus\u00f3n blanco. Julian sali\u00f3 por un peque\u00f1o portal, tap\u00e1ndose la cara con una mano. Le hice se\u00f1as.\n\n\u2014\u00bfQu\u00e9 ha pasado?\n\n\u2014Lo que era inevitable. \u2014Parec\u00eda congestionado\u2014. Me ha dicho que pod\u00eda comer, pero que no podr\u00eda oler la comida, ni apreciar su sabor.\n\nSe quit\u00f3 la mano de la cara. Aspir\u00e9 bruscamente entre los dientes. Le resbalaba sangre por la barbilla, le estaban saliendo cardenales bajo los ojos y ten\u00eda la nariz roja e hinchada.\n\n\u2014Necesitas hielo. \u2014Lo empuj\u00e9 detr\u00e1s de una pared de contrachapado\u2014. Vamos. Los actores tendr\u00e1n algo para curarte.\n\n\u2014Estoy bien. No creo que la tenga rota. \u2014Se toc\u00f3 el puente de la nariz\u2014. Tenemos que hablar.\n\n\u2014Podemos hablar mientras comemos.\n\nMientras atravesaba el Poblado con Julian, iba buscando alguna arma. Cualquier cosa habr\u00eda servido: una horquilla afilada, un fragmento de cristal o metal. Pero no vi nada. Si los actores estaban desarmados, no tendr\u00edan forma de defenderse si los emim entraban en la ciudad. Los refas y los casacas rojas eran su \u00fanica protecci\u00f3n.\n\nLlegamos a la choza de la comida y obligu\u00e9 a Julian a beberse un cuenco de _skilly_ y a comer un poco de _toke_ ; luego di los _numa_ que me quedaban a un adivino a cambio de un paquete de acetaminof\u00e9n robado. No quiso decirme a qui\u00e9n se lo hab\u00eda robado, ni c\u00f3mo, y se perdi\u00f3 entre el gent\u00edo en cuanto tuvo las agujas en la mano. Deb\u00eda de ser un acutum\u00e1ntico aut\u00e9ntico. Llev\u00e9 a Julian a un rinc\u00f3n oscuro.\n\n\u2014T\u00f3mate esto \u2014le dije\u2014. Y que nadie te vea.\n\nJulian no dijo nada. Se meti\u00f3 dos pastillas en la boca y se las trag\u00f3. Encontr\u00e9 un trapo y un poco de agua en una chabola vac\u00eda. Julian se limpi\u00f3 la sangre.\n\n\u2014Bueno \u2014dijo con voz un poco pastosa\u2014, \u00bfqu\u00e9 sabemos sobre los emim?\n\n\u2014Yo, nada.\n\n\u2014He estado investigando sobre c\u00f3mo funciona este sitio. \u00bfTe interesa?\n\n\u2014Claro que me interesa.\n\n\u2014Los casacas blancas realizan el entrenamiento b\u00e1sico durante unos d\u00edas. Combate con esp\u00edritus, sobre todo: tienes que demostrar que puedes formar bandadas, y cosas as\u00ed. Luego haces el primer examen. Entonces es cuando tienes que verificar tu don.\n\n\u2014\u00bfVerificarlo?\n\n\u2014Demostrar que es \u00fatil. Los adivinos tienen que hacer una predicci\u00f3n. Los m\u00e9diums tienen que incitar a una posesi\u00f3n. Ya te imaginas.\n\n\u2014\u00bfQu\u00e9 es lo que consideran \u00fatil?\n\n\u2014Tienes que hacer algo para demostrar tu lealtad. Estuve hablando de eso con el portero de Trinity. No quiso contarme gran cosa, pero dijo que su predicci\u00f3n hab\u00eda facilitado que llevaran a alguien a Sheol I. Tienes que mostrarles lo que quieren ver, aunque con eso pongas a otro humano en peligro.\n\nSe me hizo un nudo en la garganta.\n\n\u2014\u00bfY el segundo examen?\n\n\u2014Est\u00e1 relacionado con los emim. Supongo que si sobrevives pasas a ser un casaca roja.\n\nRecorr\u00ed la choza con la mirada. Entre los actores hab\u00eda un par de blusones amarillos.\n\n\u2014Mira \u2014dijo Julian en voz baja\u2014. Esa del rinc\u00f3n. F\u00edjate en sus dedos.\n\nMir\u00e9 hacia donde me indicaba. Una joven se tomaba el _skilly_ mientras hablaba con un hombre de aspecto enfermizo. Le faltaban tres dedos de una mano. Recorr\u00ed la habitaci\u00f3n con la mirada y vi otras lesiones: manos amputadas, cicatrices de mordeduras y ara\u00f1azos en brazos y piernas.\n\n\u2014Se ve que les gusta la carne humana \u2014coment\u00e9; Liss no me hab\u00eda mentido.\n\n\u2014Eso parece. \u2014Julian me ofreci\u00f3 su cuenco\u2014. \u00bfQuieres acab\u00e1rtelo?\n\n\u2014No, gracias.\n\nNos quedamos un rato callados. No volv\u00ed a mirar, pero no pod\u00eda parar de pensar en las lesiones que hab\u00edan sufrido aquellas personas. Las hab\u00edan ro\u00eddo como si fueran huesos de pollo, y luego las hab\u00edan tirado a la basura. Estaban en peligro continuo en ese tugurio miserable y desprotegido.\n\nNo quer\u00eda que los refa\u00edtas supieran lo que yo era, pero para aprobar el primer examen tendr\u00eda que demostr\u00e1rselo.\n\n\u00bfQuer\u00eda aprobar los ex\u00e1menes? Me pas\u00e9 los dedos por el pelo, pensativa. Tendr\u00eda que esperar a que regresara el Custodio y enterarme de qu\u00e9 quer\u00eda que hiciera. Me di cuenta de que el Custodio controlaba mi destino.\n\nLlevaba unos minutos observando a los actores cuando vi una cara conocida: Carl. Se produjo un silencio. Los actores le abrieron paso y agacharon las cabezas. Estir\u00e9 el cuello y vi qu\u00e9 era lo que estaban mirando: su blus\u00f3n rosa. \u00bfQu\u00e9 hac\u00eda Carl en el Poblado?\n\n\u2014Tilda me ha dicho que Carl hab\u00eda aprobado su primer examen \u2014le dije a Julian\u2014. \u00bfQu\u00e9 crees que ha tenido que hacer? \u00bfSolo delatar a Ivy?\n\n\u2014Es adivino. Seguramente solo tuvo que encontrar a su difunta t\u00eda en una taza de t\u00e9 \u2014dijo.\n\n\u2014Eso lo hacen los augures. \u00bfT\u00fa no eres adivino?\n\n\u2014Yo nunca he afirmado que sea adivino. \u2014Esboz\u00f3 una sonrisa\u2014. No eres la \u00fanica que tiene un aura enga\u00f1osa.\n\nEso me dio que pensar. Se consideraba a los adivinos la clase m\u00e1s baja de clarividentes; al menos eran los m\u00e1s frecuentes. Quiz\u00e1 Julian lo considerara insultante. O quiz\u00e1 yo no fuera tan buena identificando a videntes como afirmaba Jax.\n\nJax. Me pregunt\u00e9 qu\u00e9 estar\u00eda haciendo. Si estar\u00eda preocupado por m\u00ed o no. Seguro que estaba preocupado: yo era su onir\u00e1mbula, su dama. No sab\u00eda qu\u00e9 har\u00eda para encontrarme. Quiz\u00e1 a Dani o a Nick se les ocurriera algo. Ellos ten\u00edan empleos en Scion. En alg\u00fan sitio ten\u00eda que haber una base de datos de prisioneros que el Arconte manten\u00eda en secreto.\n\n\u2014Intentan sobornarlo. \u2014Julian estaba mirando a dos actores; le ofrec\u00edan _numa_ a Carl y hablaban con \u00e9l\u2014. Deben de creer que ahora tiene poder sobre los refas.\n\nLo parec\u00eda. Carl los rechaz\u00f3 y ellos se apartaron.\n\n\u2014Julian \u2014dije\u2014, \u00bfa ti cu\u00e1ntas pastillas te dan?\n\n\u2014Una.\n\n\u2014\u00bfC\u00f3mo es?\n\n\u2014Redonda y roja. Creo que es hierro. \u2014Se tom\u00f3 el _skilly_ \u2014. \u00bfPor qu\u00e9? \u00bfCu\u00e1ntas te dan a ti?\n\nClaro. Scion fabricaba una inyecci\u00f3n para la contracepci\u00f3n masculina, pero no ten\u00eda sentido esterilizar a ambos sexos. Carl me ahorr\u00f3 tener que contestar esa pregunta.\n\n\u2014Y entonces mir\u00e9 dentro de la piedra \u2014le estaba diciendo a un casaca blanca, bajo la atenta mirada de varios bufones\u2014, y decid\u00ed hacer una predicci\u00f3n acorde con sus deseos. Resulta que est\u00e1 ansiosa por encontrar a un tal Vinculador Blanco y, por supuesto, nada m\u00e1s ver su cara supe exactamente d\u00f3nde estaba. Por lo visto es el capo del I-4.\n\nSent\u00ed un fr\u00edo mortal. Ese era Jaxon.\n\n\u2014Paige \u2014dijo Julian.\n\n\u2014Estoy bien. Esp\u00e9rame aqu\u00ed, solo ser\u00e1 un momento.\n\nMe levant\u00e9 sin pensar y fui derecha hacia Carl. Lo agarr\u00e9 por el blus\u00f3n y lo arrastr\u00e9 hasta un rinc\u00f3n. \u00c9l se qued\u00f3 perplejo.\n\n\u2014\u00bfQu\u00e9 viste? \u2014le susurr\u00e9.\n\nCarl me miraba con gesto de sorpresa, como si me hubiera crecido otra cabeza.\n\n\u2014\u00bfQu\u00e9?\n\n\u2014\u00bfQu\u00e9 le dijiste del Vinculador Blanco, Carl?\n\n\u2014Me llamo XX-59-1.\n\n\u2014No me importa. Dime qu\u00e9 viste.\n\n\u2014Eso no es asunto tuyo. \u2014Le ech\u00f3 un vistazo a mi blus\u00f3n blanco\u2014. Se ve que no has progresado tan deprisa como todos esperaban. \u00bfQu\u00e9 ha pasado? \u00bfHas decepcionado a tu guardi\u00e1n especial?\n\nAcerqu\u00e9 la cara a la suya. A esa escasa distancia Carl a\u00fan se parec\u00eda m\u00e1s a una rata.\n\n\u2014Esto no es ning\u00fan juego, Carl \u2014dije en voz baja\u2014. Y no me gustan los renegados. Dime qu\u00e9 viste.\n\nLos faroles que ten\u00edamos m\u00e1s cerca parpadearon. Nadie pareci\u00f3 notarlo (los actores ya no nos prestaban atenci\u00f3n), pero Carl s\u00ed. Vi un destello de temor en sus ojos.\n\n\u2014No vi exactamente d\u00f3nde estaba \u2014admiti\u00f3\u2014, pero vi un reloj de sol.\n\n\u2014\u00bfSe lo dijiste?\n\n\u2014S\u00ed.\n\n\u2014\u00bfQu\u00e9 quiere del Vinculador?\n\nLe agarr\u00e9 m\u00e1s fuerte del blus\u00f3n.\n\n\u2014No lo s\u00e9. Yo solo hice lo que me ordenaba. \u2014Se apart\u00f3 de m\u00ed\u2014. \u00bfPor qu\u00e9 me haces tantas preguntas?\n\nMe rug\u00eda la sangre en los o\u00eddos.\n\n\u2014Por nada. \u2014Le solt\u00e9 el blus\u00f3n\u2014. Lo siento. Es que estoy nerviosa por los ex\u00e1menes.\n\nCarl se relaj\u00f3, halagado.\n\n\u2014Es l\u00f3gico. Estoy seguro de que pronto conseguir\u00e1s tu siguiente color.\n\n\u2014Y despu\u00e9s, \u00bfqu\u00e9 pasa?\n\n\u2014\u00bfDespu\u00e9s del rosa? \u00a1Entramos en el batall\u00f3n, por supuesto! Estoy deseando echarles el guante a los cabrones de los zumbadores. Dentro de nada conseguir\u00e9 el blus\u00f3n rojo.\n\nYa lo ten\u00edan cautivado. Ya era un soldado, un asesino en potencia. Compuse una sonrisa forzada y me march\u00e9.\n\nCarl ten\u00eda motivos para estar orgulloso. Era un buen profeta. Hab\u00eda utilizado a Nashira para enfocar a un sujeto, para verlo en la superficie reluciente del _numen_ de su elecci\u00f3n. Ese era el don de los adivinos, y tambi\u00e9n de algunos augures. Pod\u00edan encajar su don en los deseos de otra persona, el solicitante, para leerles el futuro. Los cartom\u00e1nticos y los palmistas lo hac\u00edan continuamente. Y dijera lo que dijese Jaxon, muchas veces resultaba \u00fatil. El \u00e9ter era como Scionet: una red de onirosajes, cada uno de los cuales conten\u00eda informaci\u00f3n a la que se pod\u00eda acceder pulsando un bot\u00f3n. El solicitante proporcionaba una especie de motor de b\u00fasqueda, un medio para ver con los ojos de los esp\u00edritus que vagaban a la deriva.\n\nCarl hab\u00eda encontrado a una solicitante perfecta en Nashira. No solo hab\u00eda visto a Jax, sino que tambi\u00e9n hab\u00eda visto una pista acerca de su paradero: uno de los seis relojes de sol de la columna de Seven Dials.\n\nTen\u00eda que prevenir a Jax, y pronto. Ignoraba qu\u00e9 pod\u00eda querer Nashira de \u00e9l, pero no pensaba permitir que lo llevara a Sheol I.\n\nJulian me sigui\u00f3 afuera.\n\n\u2014Paige. \u2014Me sujet\u00f3 por la manga\u2014. \u00bfQu\u00e9 te ha dicho?\n\n\u2014Nada.\n\n\u2014Est\u00e1s muy p\u00e1lida.\n\n\u2014Estoy bien. \u2014Cuando vi el pan que ten\u00eda en la mano me acord\u00e9 de Seb\u2014. \u00bfVas a comerte eso?\n\n\u2014No. \u00bfLo quieres?\n\n\u2014No es para m\u00ed. Es para Seb.\n\n\u2014\u00bfYa lo has encontrado?\n\n\u2014S\u00ed, en la Casa Amaur\u00f3tica.\n\n\u2014Vale. Tiene gracia: en Londres encierran a los videntes y aqu\u00ed, a los amaur\u00f3ticos.\n\n\u2014Quiz\u00e1 para ellos tenga sentido. \u2014Me guard\u00e9 el pan en la manga\u2014. Nos vemos ma\u00f1ana al anochecer, \u00bfvale?\n\n\u2014Vale. \u2014Hizo una pausa\u2014. Si puedo salir, claro.\n\nLa Casa Amaur\u00f3tica estaba a oscuras cuando llegu\u00e9. Hasta las l\u00e1mparas de fuera estaban apagadas. Prefer\u00ed no intentar convencer a Graffias para que me dejara pasar, y trep\u00e9 directamente por una bajante.\n\n\u2014\u00bfSeb?\n\nNo hab\u00eda luz en la habitaci\u00f3n. Ol\u00eda a aire viciado. Seb no me contest\u00f3.\n\nMe agarr\u00e9 a los barrotes y me puse en cuclillas en el alf\u00e9izar.\n\n\u2014Seb \u2014insist\u00ed\u2014. \u00bfEst\u00e1s ah\u00ed?\n\nNo, no estaba. No hab\u00eda ning\u00fan onirosaje en esa habitaci\u00f3n. Los amaur\u00f3ticos tambi\u00e9n ten\u00edan onirosajes, aunque sin color. No detect\u00e9 matices emocionales, ni actividad espiritual. Seb se hab\u00eda esfumado.\n\nQuiz\u00e1 se lo hubieran llevado a trabajar a alguna residencia. Quiz\u00e1 volviera m\u00e1s tarde.\n\nO quiz\u00e1 aquello fuera una trampa.\n\nMe saqu\u00e9 el pan de la manga, lo pas\u00e9 entre los barrotes y baj\u00e9 por el ca\u00f1o. No me sent\u00ed a salvo hasta que volv\u00ed a tener los pies en el suelo.\n\nPero esa sensaci\u00f3n no dur\u00f3 mucho. Nada m\u00e1s volverme hacia el centro de la ciudad, not\u00e9 que me agarraban con fuerza del brazo. Unos ojos abrasadores se clavaron en los m\u00edos.\n\nEstaba quieto como una estatua. Llevaba una camisa negra de cuello alto, con ribete dorado. Las mangas le cubr\u00edan el brazo que yo le hab\u00eda vendado.\n\nMe mir\u00f3 con gesto inexpresivo. Me humedec\u00ed los labios y trat\u00e9 de pensar alguna excusa.\n\n\u2014Bueno \u2014dijo, y se me acerc\u00f3 m\u00e1s\u2014. Vendas heridas y alimentas a los esclavos amaur\u00f3ticos. Qu\u00e9 curioso.\n\nLa repugnancia que sent\u00ed me hizo dar una sacudida con el brazo, y \u00e9l no me lo impidi\u00f3. Yo pod\u00eda pelear si no estaba acorralada, pero entonces vi a los otros. Cuatro refas, dos varones y dos f\u00e9minas. Los cuatro ten\u00edan ese onirosaje acorazado. Adopt\u00e9 una postura defensiva, y ellos se rieron de m\u00ed.\n\n\u2014No seas necia, 40.\n\n\u2014Lo \u00fanico que queremos es hablar contigo.\n\n\u2014Pues hablad \u2014dije con una voz que no parec\u00eda la m\u00eda.\n\nEl Custodio segu\u00eda mir\u00e1ndome fijamente. Bajo la luz de una l\u00e1mpara de gas cercana, sus ojos ard\u00edan con un nuevo color. \u00c9l no se hab\u00eda re\u00eddo como hab\u00edan hecho los dem\u00e1s.\n\nMe sent\u00ed rodeada, como un animal acorralado. Intentar salir de esa situaci\u00f3n no habr\u00eda sido solamente est\u00fapido, sino suicida.\n\n\u2014Ir\u00e9 \u2014dije, y el Custodio asinti\u00f3 con la cabeza.\n\n\u2014Terebell \u2014dijo\u2014, ve a ver a la soberana de sangre. Dile que tenemos a XX-59-40 bajo custodia.\n\n\u00bfBajo custodia? Mir\u00e9 a la refa\u00edta. Deb\u00eda de ser la guardiana de Tilda y de Carl, Terebell Sheratan. Me volvi\u00f3 a mirar con ojos firmes y amarillos. Su cabello, negro y reluciente, rodeaba su cara como una capucha.\n\n\u2014S\u00ed, consorte de sangre \u2014dijo.\n\nSe coloc\u00f3 en cabeza de la escolta. Yo mantuve la vista en el suelo.\n\n\u2014Vamos \u2014dijo el Custodio\u2014. La soberana de sangre nos espera.\n\nNos dirigimos al centro de la ciudad. Los guardias se quedaron atr\u00e1s, a una distancia respetuosa del Custodio. Me fij\u00e9 en que sus ojos s\u00ed ten\u00edan un color distinto: eran naranja. Se dio cuenta de que lo estaba mirando.\n\n\u2014Si tienes alguna pregunta \u2014dijo\u2014, puedes hacerla.\n\n\u2014\u00bfAd\u00f3nde vamos?\n\n\u2014A que hagas tu primer examen. \u00bfAlgo m\u00e1s?\n\n\u2014\u00bfQui\u00e9n te mordi\u00f3?\n\nMantuvo la mirada al frente. Al cabo de un momento dijo:\n\n\u2014Te retiro el permiso para hablar.\n\nEstuve a punto de hacerme una herida en la lengua. Capullo. Me hab\u00eda pasado horas limpi\u00e1ndole las heridas. Habr\u00eda podido matarlo. Deb\u00ed matarlo.\n\nEl Custodio conoc\u00eda muy bien la ciudad. Recorrimos varias calles hasta llegar a la parte trasera de otra residencia, la misma donde nos hab\u00edan dado el serm\u00f3n. Fuera hab\u00eda una placa que rezaba: RESIDENCIA DEL SUZERANO. Los guardias saludaron con la cabeza cuando el Custodio pas\u00f3 ante ellos, y se llevaron un pu\u00f1o al pecho. \u00c9l no se molest\u00f3 en saludarlos. Las puertas se cerraron detr\u00e1s de nosotros. El sonido met\u00e1lico de los cerrojos hizo que se me tensaran los m\u00fasculos. Pase\u00e9 la mirada por las paredes, buscando hasta en el \u00faltimo recoveco algo a lo que pudiera agarrarme con las manos y los pies. Las fachadas de los edificios estaban recubiertas de enredaderas, arom\u00e1ticas madreselvas, hiedras y glicinas, pero solo hasta una altura de unos palmos por encima del suelo. Despu\u00e9s estaban las ventanas. Recorrimos un sendero de color arena que bordeaba un \u00f3valo de hierba en el que se alzaba una sola farola con los cristales rojos.\n\nAl final del sendero hab\u00eda una puerta. El Custodio no me mir\u00f3, pero se detuvo.\n\n\u2014No digas nada de las heridas \u2014me dijo en voz tan baja que apenas le o\u00ed\u2014, o tendr\u00e1s motivos para lamentar haberme salvado la vida.\n\nLe hizo se\u00f1as a su escolta. Dos refas se colocaron a sendos lados de la puerta; el otro, un var\u00f3n de cabello rizado y mirada deslumbrante, se coloc\u00f3 a mi otro lado. Flanqueada por los guardias, traspuse la puerta y entr\u00e9 en el fresco interior del edificio.\n\nMe hallaba en una habitaci\u00f3n estrecha y recargada, con paredes de piedra de color marfil. Por la pared de la izquierda se desparramaban manchas de colores c\u00e1lidos gracias a la luz de la luna refractada por las vidrieras de colores. Distingu\u00ed cinco placas conmemorativas, pero no tuve tiempo de pararme a leerlas: me estaban conduciendo hacia un arco iluminado. Subimos tres pelda\u00f1os de m\u00e1rmol negro, y entonces el Custodio se arrodill\u00f3 y agach\u00f3 la cabeza. El guardia me mir\u00f3 fijamente, y yo imit\u00e9 al Custodio.\n\n\u2014Arcturus.\n\nUna mano enguantada le levant\u00f3 la barbilla. Me arriesgu\u00e9 a mirar.\n\nHab\u00eda aparecido Nashira. Esa noche llevaba un vestido negro que la cubr\u00eda desde el cuello y que ondulaba como el agua bajo la luz de las velas. Bes\u00f3 al Custodio en la frente, y \u00e9l le puso una mano sobre el vientre.\n\n\u2014Veo que has tra\u00eddo a nuestro peque\u00f1o prodigio \u2014dijo Nashira con la mirada puesta en m\u00ed\u2014. Buenas noches, XX-40.\n\nMe mir\u00f3 de arriba abajo, y tuve la sensaci\u00f3n de que intentaba leer mi aura. Levant\u00e9 unas barreras preventivas. El Custodio no se movi\u00f3. No le ve\u00eda la cara.\n\nDetr\u00e1s de la pareja hab\u00eda una hilera de refa\u00edtas, todos con capa y capucha. Sus auras llenaban la capilla y apenas le dejaban espacio a la m\u00eda. Era la \u00fanica humana presente all\u00ed.\n\n\u2014Supongo que sabes por qu\u00e9 est\u00e1s aqu\u00ed \u2014dijo Nashira.\n\nMantuve la boca cerrada. Sab\u00eda que me hab\u00eda buscado un problema llev\u00e1ndole comida a Seb, pero pod\u00eda haberme buscado un problema por much\u00edsimas cosas m\u00e1s: vendar al Custodio, escabullirme, ser humana. Lo m\u00e1s probable era que Carl hubiera informado de mi inter\u00e9s por la visi\u00f3n de Nashira. O quiz\u00e1 supieran lo que yo era.\n\n\u2014La hemos encontrado delante de la Casa Amaur\u00f3tica \u2014declar\u00f3 el guardia. Era el vivo retrato de Pleione; la forma de sus ojos era id\u00e9ntica\u2014. Escabull\u00e9ndose en la oscuridad como una rata de alcantarilla.\n\n\u2014Gracias, Alsafi. \u2014Nashira me mir\u00f3, pero no me invit\u00f3 a levantarme\u2014. Tengo entendido que le has llevado comida a uno de los empleados amaur\u00f3ticos, 40. \u00bfHay alguna raz\u00f3n que lo justifique?\n\n\u2014Que lo est\u00e1is matando de hambre y maltrat\u00e1ndolo como si fuera un animal. Necesita un m\u00e9dico, un hospital.\n\nMi voz reson\u00f3 por la oscura capilla. Los refa\u00edtas encapuchados guardaban silencio.\n\n\u2014Siento mucho que pienses as\u00ed \u2014dijo Nashira\u2014, pero a los ojos de los refa\u00edtas, los ojos que ahora son los responsables de tu pa\u00eds, los humanos y las bestias se encuentran al mismo nivel. Nosotros no tenemos m\u00e9dicos para las bestias.\n\nSent\u00ed que palidec\u00eda de ira, pero me mord\u00ed la lengua. Si replicaba, solo conseguir\u00eda que mataran a Seb.\n\nNashira se dio la vuelta. El Custodio se levant\u00f3 y yo lo imit\u00e9.\n\n\u2014Quiz\u00e1 recuerdes, 40, que en el serm\u00f3n os explicamos que nos gusta examinar a los humanos que recogemos durante las Eras de Huesos. Ver\u00e1s, enviamos a nuestros casacas rojas a buscar humanos con aura, pero no siempre podemos identificar las habilidades de esas auras. Confieso que en el pasado cometimos algunos errores. A veces, un caso que parec\u00eda prometedor resulta menos emocionante que un cartom\u00e1ntico vagabundo. Pero no me cabe duda de que t\u00fa ser\u00e1s mucho m\u00e1s interesante. Tu aura te precede. \u2014Me hizo se\u00f1as para que me acercara y a\u00f1adi\u00f3\u2014: Ven, mu\u00e9stranos tus talentos.\n\nEl Custodio y Alsafi se separaron de m\u00ed. Nashira y yo nos quedamos cara a cara, solas.\n\nSe me tensaron los m\u00fasculos. \u00bfQu\u00e9 quer\u00edan, que peleara con ella? Yo no habr\u00eda podido vencerla. Sus \u00e1ngeles y ella destrozar\u00edan mi onirosaje. Los notaba girando alrededor de Nashira, preparados para defender a su hu\u00e9sped.\n\nPero entonces record\u00e9 lo que me hab\u00eda contado Liss: que Nashira quer\u00eda un onir\u00e1mbulo. Intent\u00e9 pensar. Quiz\u00e1 yo pudiera hacer algo que ella no tuviera poder para impedir, alguna ventaja que pudiera utilizar contra ella.\n\nPens\u00e9 en lo que hab\u00eda pasado en el tren. Sin un onir\u00e1mbulo o un or\u00e1culo en su s\u00e9quito, Nashira no pod\u00eda alterar el \u00e9ter. Y a menos que hubiera consumido el esp\u00edritu de un ilegible, yo podr\u00eda soltar mi esp\u00edritu en su mente.\n\nPodr\u00eda matarla.\n\nMi plan A se vino abajo cuando volvi\u00f3 Alsafi. Llevaba en brazos un cuerpo fr\u00e1gil, un cuerpo con una bolsa negra cubri\u00e9ndole la cabeza. Sentaron al prisionero en una silla y lo esposaron a ella. Se me quedaron los dedos entumecidos. \u00bfSer\u00eda uno de los otros? \u00bfHabr\u00edan encontrado Dials, habr\u00edan encontrado a mi banda?\n\nPero no percib\u00eda ning\u00fan aura: ten\u00eda que tratarse de un amaur\u00f3tico. Pens\u00e9 en mi padre y me dio un mareo, pero aquel cuerpo era demasiado peque\u00f1o, demasiado delgado.\n\n\u2014Creo que os conoc\u00e9is \u2014dijo.\n\nLe quitaron la bolsa de la cabeza. Me qued\u00e9 helada.\n\nSeb. Se lo hab\u00edan llevado. Ten\u00eda los ojos tan hinchados que parec\u00edan dos ciruelas; le colgaban mechones de pelo ensangrentado alrededor de la cara, y ten\u00eda los labios partidos y manchados de sangre. El resto de su rostro estaba recubierto de una costra de sangre seca. Ya hab\u00eda visto otras palizas graves, cuando las v\u00edctimas de Hector ven\u00edan arrastr\u00e1ndose hasta Seven Dials a pedirle ayuda a Nick, pero nunca nada como aquello. Nunca hab\u00eda visto a una v\u00edctima tan joven.\n\nEl guardia le arre\u00f3 otro porrazo en la mejilla. Seb estaba casi inconsciente, pero consigui\u00f3 levantar un poco la cabeza y mirarme.\n\n\u2014Paige.\n\nSu voz quebrada hizo que se me agolparan las l\u00e1grimas en los ojos. Me volv\u00ed hacia Nashira.\n\n\u2014\u00bfQu\u00e9 le hab\u00e9is hecho?\n\n\u2014Nada. Se lo vas a hacer t\u00fa.\n\n\u2014\u00bfQu\u00e9?\n\n\u2014Ya va siendo hora de que te ganes tu pr\u00f3ximo blus\u00f3n, XX-40.\n\n\u2014\u00bfDe qu\u00e9 demonios est\u00e1s hablando?\n\nAlsafi me peg\u00f3 tan fuerte en la cabeza que casi me derrib\u00f3. Me agarr\u00f3 del pelo y me dio la vuelta para que pudiera mirarlo.\n\n\u2014No uses estas palabras en presencia de la soberana de sangre. Cuida tu lenguaje o te coser\u00e9 la boca.\n\n\u2014Paciencia, Alsafi. Deja que se enfade. \u2014Nashira levant\u00f3 una mano\u2014. Al fin y al cabo, en el tren estaba enfadada.\n\nMe zumbaban los o\u00eddos. Aparecieron dos caras en mi memoria. Dos cuerpos en el suelo del vag\u00f3n. Uno, muerto; el otro, loco. Mis v\u00edctimas. Mis presas.\n\nEse era mi examen. Para ganarme el nuevo blus\u00f3n, ten\u00eda que matar a un amaur\u00f3tico.\n\nTen\u00eda que matar a Seb.\n\nNashira deb\u00eda de haber adivinado lo que yo era. Deb\u00eda de haber adivinado que mi esp\u00edritu pod\u00eda abandonar mi cuerpo, su ubicaci\u00f3n natural. Que era capaz de matar sin derramamiento de sangre. Quer\u00eda ver c\u00f3mo lo hac\u00eda. Quer\u00eda verme bailar. Quer\u00eda saber si val\u00eda la pena robarme ese don.\n\n\u2014No \u2014dije.\n\nNashira no se inmut\u00f3.\n\n\u2014\u00bfNo? \u2014Como no dije nada, continu\u00f3\u2014: La negativa est\u00e1 descartada. Obedecer\u00e1s o nos veremos obligados a deshacernos de ti. Seguro que el Gran Inquisidor se alegrar\u00e1 de corregir tu insolencia.\n\n\u2014Pues m\u00e1tame \u2014repliqu\u00e9\u2014. \u00bfA qu\u00e9 esperas?\n\nNinguno de los trece jueces dijo nada. Nashira tampoco. Se qued\u00f3 mir\u00e1ndome fijamente. Tratando de averiguar si me estaba marcando un farol.\n\nAlsafi no se anduvo con rodeos. Me agarr\u00f3 por la mu\u00f1eca y me arrastr\u00f3 hasta la silla. Me retorc\u00ed y patale\u00e9 mientras \u00e9l me rodeaba el cuello con un musculoso brazo.\n\n\u2014Hazlo \u2014me gru\u00f1\u00f3 al o\u00eddo\u2014, o te aplastar\u00e9 las costillas y te ahogar\u00e1s en tu propia sangre. \u2014Me sacudi\u00f3 tan fuerte que me tembl\u00f3 la vista\u2014. Mata al chico. Ahora mismo.\n\n\u2014No.\n\n\u2014Obedece.\n\n\u2014No.\n\nAlsafi me apret\u00f3 m\u00e1s fuerte. Le clav\u00e9 las u\u00f1as en la manga. Le ara\u00f1\u00e9 un costado, y encontr\u00e9 el cuchillo que llevaba al cinto. Era peque\u00f1o, del tama\u00f1o de un abrecartas, pero servir\u00eda. Bast\u00f3 una pu\u00f1alada somera para hacer que me soltara. Me sub\u00ed a un banco; todav\u00eda ten\u00eda el cuchillo en la mano.\n\n\u2014No te me acerques \u2014le advert\u00ed.\n\nNashira ri\u00f3 y los jueces la imitaron. Al fin y al cabo, para ellos yo solo era otra clase de int\u00e9rprete. Otra humana endeble con la cabeza llena de confeti y fuegos artificiales.\n\nPero el Custodio no ri\u00f3. No apartaba la vista de mi cara. Lo apunt\u00e9 con el cuchillo.\n\nNashira vino hacia m\u00ed.\n\n\u2014Impresionante \u2014coment\u00f3\u2014. Me gustas, XX-40. Tienes temple.\n\nMe temblaba la mano.\n\nAlsafi se mir\u00f3 el corte que le hab\u00eda hecho en el brazo. Su piel rezumaba un fluido luminoso. Cuando mir\u00e9 el cuchillo, vi que la hoja estaba recubierta de esa sustancia.\n\nSeb lloraba. As\u00ed el pu\u00f1al con m\u00e1s fuerza, pero ten\u00eda las manos sudadas. No pod\u00eda atacar a todos aquellos refa\u00edtas con un abrecartas. Adem\u00e1s, no se me daba nada bien el combate con armas blancas, y no habr\u00eda podido lanzar un cuchillo con precisi\u00f3n.\n\nExceptuando a los cinco \u00e1ngeles que rodeaban a Nashira, no hab\u00eda esp\u00edritus con los que hacer una bandada. Tendr\u00eda que acercarme mucho m\u00e1s para soltar a Seb. Y luego tendr\u00eda que encontrar la forma de salir los dos de all\u00ed con vida.\n\n\u2014Arcturus, Aludra: desarmadla \u2014orden\u00f3 Nashira\u2014. Sin usar esp\u00edritus.\n\nUno de los jueces se quit\u00f3 la capucha. Era una mujer.\n\n\u2014Ser\u00e1 un placer \u2014dijo.\n\nLa examin\u00e9. Era la guardiana de Julian: un ser taimado, con cabello rubio liso y ojos felinos. El Custodio se qued\u00f3 detr\u00e1s de ella. Med\u00ed sus auras.\n\nAludra era una salvaje. Pod\u00eda parecer civilizada, pero percib\u00ed que estaba conteni\u00e9ndose para no babear. Estaba deseando pelear, excitada por la debilidad de Seb, y sedienta de mi aura. El Custodio era m\u00e1s oscuro, m\u00e1s fr\u00edo, y sus intenciones eran m\u00e1s misteriosas, pero eso lo hac\u00eda m\u00e1s letal. Si no consegu\u00eda leer su aura, no podr\u00eda predecir sus movimientos.\n\nDe pronto se me ocurri\u00f3 una cosa. La sangre del Custodio me hab\u00eda hecho sentir m\u00e1s cerca del \u00e9ter. Quiz\u00e1 volviera a funcionar. Inhal\u00e9, sosteniendo el cuchillo cerca de mi cara. El aroma fr\u00edo despert\u00f3 mis sentidos. El \u00e9ter me envolvi\u00f3 como un agua helada y me sumergi\u00f3. Di una sacudida con la mu\u00f1eca y lanc\u00e9 el cuchillo hacia la cara de Aludra, apunt\u00e1ndole entre los ojos. Ella lo esquiv\u00f3 por los pelos. Mi punter\u00eda hab\u00eda mejorado mucho.\n\nAludra agarr\u00f3 un pesado candelabro y se abalanz\u00f3 sobre m\u00ed.\n\n\u2014Ven aqu\u00ed, ni\u00f1a \u2014dijo\u2014. Baila conmigo.\n\nMe ech\u00e9 hacia atr\u00e1s. Con el cr\u00e1neo destrozado no iba a servirle de nada a Seb.\n\nAludra me embisti\u00f3. Su misi\u00f3n era derribarme y cebarse con lo que quedara de m\u00ed. Si mis sentidos no se hubieran agudizado, seguramente lo habr\u00eda conseguido. Rod\u00e9 sobre m\u00ed misma para evitarla y, en lugar de aplastarme, el candelabro se estrell\u00f3 contra la cabeza de una estatua. Me puse r\u00e1pidamente en pie, salt\u00e9 por encima del altar y corr\u00ed por la capilla, pasando al lado de los refas encapuchados que se encontraban en los bancos.\n\nAludra recuper\u00f3 su arma. O\u00ed silbar el aire cuando la lanz\u00f3 hacia el fondo de la capilla. El candelabro pas\u00f3 por encima de la cabeza de Seb, que grit\u00f3 mi nombre.\n\nFui hacia la puerta, que estaba abierta; pero un guardia la cerr\u00f3 desde fuera, y me encerr\u00f3 en la capilla con mi p\u00fablico. Como no tuve tiempo de reducir la velocidad, me empotr\u00e9 contra la puerta. El impacto me cort\u00f3 la respiraci\u00f3n, y perd\u00ed el equilibrio. Me golpe\u00e9 la cabeza contra el m\u00e1rmol. Al cabo de una mil\u00e9sima de segundo, el candelabro se estrell\u00f3 contra la puerta. Apenas me dio a tiempo a moverme antes de que cayera al suelo, justo donde hac\u00eda un instante estaban mis piernas. El ruido reson\u00f3 por la capilla como una campanada.\n\nNotaba un dolor sordo en la parte trasera del cr\u00e1neo, pero no hab\u00eda tiempo para descansar. Aludra me hab\u00eda alcanzado. Me rode\u00f3 el cuello con sus dedos enguantados y me apret\u00f3 la garganta con los pulgares. Me ahogaba. Mis ojos se llenaron de sangre y dej\u00e9 de ver. Aludra me estaba robando el aura. Sus ojos se iluminaron con una luz roja y abrasadora.\n\n\u2014Basta, Aludra.\n\nNo pareci\u00f3 que lo hubiera o\u00eddo. Not\u00e9 un sabor met\u00e1lico.\n\nEl cuchillo estaba en el suelo, a mi lado. Estir\u00e9 los dedos hacia \u00e9l, pero Aludra me agarr\u00f3 la mu\u00f1eca.\n\n\u2014Ahora me toca a m\u00ed.\n\nSolo ten\u00eda una opci\u00f3n. Cuando Aludra me acerc\u00f3 el cuchillo a la mejilla, empuj\u00e9 mi esp\u00edritu hacia el \u00e9ter.\n\nEn forma de esp\u00edritu ve\u00eda con otros ojos, en otro plano. All\u00ed s\u00ed ten\u00eda visi\u00f3n espiritista. El \u00e9ter era un vac\u00edo silencioso, tachonado de esferas semejantes a estrellas; cada esfera era un onirosaje. Aludra estaba f\u00edsicamente cerca de m\u00ed; su \u00abesfera\u00bb, por lo tanto, no estaba muy lejos. Habr\u00eda sido suicida intentar entrar en su mente (era muy antigua, muy poderosa), pero sus ansias de aura hab\u00edan debilitado sus defensas. \u00abAhora o nunca\u00bb, me dije, y me lanc\u00e9 contra su mente.\n\nAludra no estaba en guardia y yo fui muy r\u00e1pida. Llegu\u00e9 a su medianoche antes de que ella se diera cuenta de lo que hab\u00eda pasado. Cuando se percat\u00f3, me vi expulsada con la fuerza de una bala. Volv\u00eda a estar dentro de mi cuerpo, con la vista fija en el techo de la capilla. Aludra, arrodillada, se sujetaba la cabeza con ambas manos.\n\n\u2014\u00a1Sacadla de aqu\u00ed! \u00a1Sacadla! \u2014gritaba\u2014. \u00a1Es onir\u00e1mbula!\n\nMe levant\u00e9 con dificultad, jadeando, y me lanc\u00e9 contra el Custodio, que me sujet\u00f3 por los hombros. Me hinc\u00f3 los dedos enguantados. No intentaba hacerme da\u00f1o: solo quer\u00eda sujetarme, contenerme; pero mi esp\u00edritu era como una planta carn\u00edvora y reaccionaba al peligro. Casi sin propon\u00e9rmelo, volv\u00ed a intentar el mismo ataque.\n\nEsa vez ni siquiera llegu\u00e9 a tocar el \u00e9ter. No pod\u00eda moverme.\n\nEl Custodio. Era \u00e9l. Esta vez era \u00e9l quien me estaba robando la energ\u00eda, sorbi\u00e9ndome el aura. Me sent\u00ed atra\u00edda por \u00e9l como una flor por la luz del sol; y, conmocionada, no pude hacer nada.\n\nEntonces par\u00f3. Fue como si se partiera un cable que nos un\u00eda. \u00c9l ten\u00eda los ojos de un rojo intenso como la sangre.\n\nLo mir\u00e9 fijamente. El Custodio dio un paso atr\u00e1s y mir\u00f3 a Nashira. Se produjo un silencio. Entonces los refa\u00edtas encapuchados se levantaron y aplaudieron.\n\nMe qued\u00e9 sentada en el suelo, aturdida.\n\nNashira se arrodill\u00f3 a mi lado y me puso una mano enguantada en la cabeza.\n\n\u2014Preciosa. Mi peque\u00f1a onir\u00e1mbula.\n\nEstaba perdida: Nashira lo sab\u00eda.\n\nSe levant\u00f3 y se volvi\u00f3 hacia Seb, que observaba la escena, aterrorizado. Con un ojo entreabierto sigui\u00f3 la trayectoria de Nashira, que se coloc\u00f3 detr\u00e1s de su silla.\n\n\u2014Gracias por tus servicios. Te estamos muy agradecidos. \u2014Le puso las manos a ambos lados de la cabeza\u2014. Adi\u00f3s.\n\n\u2014\u00a1No, por favor! \u00a1No quiero morir! \u00a1Paige!\n\nNashira le gir\u00f3 bruscamente la cabeza hacia un lado. Seb abri\u00f3 m\u00e1s los ojos, y de sus labios escap\u00f3 un grito ahogado.\n\nLo hab\u00eda matado.\n\n\u2014\u00a1No! \u2014exclam\u00e9. No pod\u00eda creer lo que acababa de ver. Me qued\u00e9 mir\u00e1ndola\u2014. Has... has...\n\n\u2014Demasiado tarde. \u2014Nashira solt\u00f3 la cabeza de Seb, y esta cay\u00f3 hacia un lado\u2014. Podr\u00edas haberlo hecho t\u00fa, 40. Sin dolor. Solo ten\u00edas que hacer lo que te he pedido.\n\nFue su sonrisa lo que me impuls\u00f3. Porque sonre\u00eda. Me lanc\u00e9 contra ella con la sangre hirvi\u00e9ndome en las venas. El Custodio y Alsafi me tomaron los brazos y me sujetaron. Patale\u00e9 y me retorc\u00ed hasta que qued\u00e9 empapada de sudor.\n\n\u2014\u00a1Zorra! \u2014grit\u00e9\u2014. \u00a1Zorra! \u00a1Zorra asquerosa! \u00a1Seb ni siquiera era vidente!\n\n\u2014Tienes raz\u00f3n, no lo era. Pero los esp\u00edritus amaur\u00f3ticos son los mejores sirvientes, \u00bfno te parece?\n\nAlsafi estuvo a punto de dislocarme un hombro. Le clav\u00e9 las u\u00f1as en el brazo al Custodio. En el brazo malo, el que yo le hab\u00eda curado. \u00c9l tens\u00f3 los m\u00fasculos; no me import\u00f3.\n\n\u2014Os matar\u00e9 \u2014dije, dirigi\u00e9ndome a todos. Casi no pod\u00eda respirar, pero lo dije\u2014. Os matar\u00e9. Juro que os matar\u00e9.\n\n\u2014No hace falta que nos jures nada, 40. Deja que nosotros juremos por ti.\n\nAlsafi me tir\u00f3 al suelo y me golpe\u00e9 la cabeza contra el duro m\u00e1rmol. Perd\u00ed moment\u00e1neamente la visi\u00f3n. Intent\u00e9 levantarme, pero algo me inmovilizaba. Una rodilla sobre mi espalda. Estir\u00e9 los dedos por el suelo. Entonces not\u00e9 un dolor atroz en el hombro, el dolor m\u00e1s terrible que jam\u00e1s hab\u00eda sentido. Ardiente, demasiado ardiente. Ol\u00ed a carne quemada. No pude evitarlo: grit\u00e9.\n\n\u2014Juramos tu lealtad eterna a los refa\u00edtas. \u2014Nashira no desviaba la mirada de m\u00ed\u2014. La juramos con la marca de fuego. XX-59-40, est\u00e1s unida para siempre al Custodio de los Mesarthim. Desde ahora renuncias a tu verdadero nombre para el resto de tus d\u00edas. Tu vida nos pertenece.\n\nEra mi piel lo que ol\u00eda a quemado. Solo pod\u00eda pensar en el dolor que sent\u00eda.\n\nYa estaba. Hab\u00edan matado a Seb, y ahora iban a matarme a m\u00ed. Vi el destello de una aguja.\n\nTen\u00eda demasiado flux en la sangre.\n\nCorr\u00ed describiendo c\u00edrculos por mi onirosaje. El flux lo hab\u00eda deformado, hab\u00eda roto las formas y los colores.\n\nO\u00eda los latidos de mi coraz\u00f3n, el aire abras\u00e1ndome al pasarme por la garganta y la nariz.\n\n\u00abMe est\u00e1n matando.\u00bb Eso pens\u00e9 mientras luchaba contra mi mente, mientras la ve\u00eda desmenuzarse como la le\u00f1a en un horno. Ya estaba. Nashira hab\u00eda descubierto qu\u00e9 era. Me hab\u00eda envenenado, y ahora me estaba muriendo. No pod\u00eda durar mucho; al fin y al cabo, los cad\u00e1veres no pod\u00edan conservar el onirosaje. Entonces ese pensamiento se deshilach\u00f3 y pas\u00f3, y me qued\u00e9 rondando por las partes m\u00e1s oscuras de mi conciencia.\n\nEntonces la encontr\u00e9. Mi zona soleada, donde habitaba la belleza. La seguridad. El calor. Corr\u00ed hacia ella, pero era como correr por la arena h\u00fameda. Unas nubes oscuras se me adher\u00edan y tiraban de m\u00ed hacia las nubes y las sombras. Forceje\u00e9 para defenderme del flux, me retorc\u00ed y patale\u00e9 para soltarme de \u00e9l, y ca\u00ed rodando hacia la luz, hacia el prado de flores.\n\nTodos ten\u00edamos un onirosaje, un hermoso espejismo dentro de nuestra mente. En sue\u00f1os, hasta los amaur\u00f3ticos ve\u00edan su zona soleada, aunque no con mucha claridad. Los videntes pod\u00edan ver el interior de su propia mente, pod\u00edan vivir all\u00ed hasta morir de hambre. Mi zona soleada era un prado de flores rojas, un campo que ondulaba y cambiaba seg\u00fan mi estado de \u00e1nimo. Ve\u00eda fragmentos del mundo que me rodeaba, notaba el movimiento de la tierra cuando vomitaba el escaso contenido de mi est\u00f3mago. Pero dentro de mi mente estaba tranquila, mientras el flux causaba estragos a mi alrededor. Me tumb\u00e9 entre las flores y esper\u00e9 a que llegara el final.\n\nVolv\u00eda a estar en la habitaci\u00f3n de Magdalen. El gram\u00f3fono desgranaba una melod\u00eda, otro de los temas prohibidos favoritos de Jaxon, \u00abDid You Ever See a Dream Walking?\u00bb. Estaba tumbada boca abajo en el div\u00e1n, desnuda de cintura para arriba. Me hab\u00edan recogido el pelo en un mo\u00f1o.\n\nMe llev\u00e9 una mano a la cara. Piel. Piel fr\u00eda, pegajosa. Estaba viva. Dolorida, s\u00ed, pero viva. No me hab\u00edan matado.\n\nEl dolor me imped\u00eda permanecer quieta. Intent\u00e9 incorporarme, pero el peso de mi cabeza solo me dej\u00f3 levantarme unos cent\u00edmetros. Notaba una laceraci\u00f3n abrasadora detr\u00e1s del hombro derecho. Otro dolor, punzante, en la ingle me indicaba el lugar donde me hab\u00edan inyectado el f\u00e1rmaco; pero esa vez los da\u00f1os eran m\u00e1s profundos.\n\nEl flux era de los pocos f\u00e1rmacos que funcionaban mejor inyectados en las arterias que en las venas. Ten\u00eda el muslo caliente e hinchado. Mi pecho sub\u00eda y bajaba al ritmo de mi respiraci\u00f3n. Estaba ardiendo de fiebre. El refa que me hab\u00eda hecho aquello hab\u00eda sido muy cruel adem\u00e1s de muy torpe. Recordaba vagamente haber visto a Suhail sonri\u00e9ndome con lascivia antes de que se apagaran las luces.\n\nQuiz\u00e1 hubieran intentado matarme. Quiz\u00e1 estuviera muri\u00e9ndome.\n\nVolv\u00ed la cabeza hacia un lado. Hab\u00edan encendido la chimenea.\n\nY hab\u00eda alguien en la habitaci\u00f3n: mi guardi\u00e1n.\n\nEstaba sentado en su butaca contemplando las llamas. Lo mir\u00e9 con odio desde el div\u00e1n y volv\u00ed a notar sus manos sujet\u00e1ndome, impidi\u00e9ndome salvar a Seb. \u00bfSe sent\u00eda culpable por aquel asesinato gratuito? \u00bfLe importaban algo los indefensos esclavos de la Casa Amaur\u00f3tica? Me pregunt\u00e9 si habr\u00eda algo que le importara. Hasta sus interacciones con Nashira parec\u00edan mec\u00e1nicas. \u00bfHabr\u00eda algo que lo hiciera vibrar?\n\nEl Custodio debi\u00f3 de advertir que lo estaba mirando porque se levant\u00f3. Me qued\u00e9 muy quieta; me dol\u00edan tantas partes del cuerpo que no me atrev\u00eda a moverme. El Custodio se arrodill\u00f3 a mi lado. Levant\u00f3 una mano, y me retraje. Pos\u00f3 el dorso de los dedos sobre mi ardiente mejilla. Volv\u00eda a tener los ojos de un dorado neutro.\n\nLa fiebre me hab\u00eda resecado la garganta.\n\n\u2014Su esp\u00edritu \u2014alcanc\u00e9 a decir. Hablar era un suplicio\u2014. \u00bfSe solt\u00f3?\n\n\u2014No.\n\nTuve que emplear todas mis fuerzas para enmascarar mi dolor. Si nadie hab\u00eda recitado el treno, Seb quedar\u00eda atrapado. Todav\u00eda estaba asustado y solo; y lo peor de todo: todav\u00eda estaba prisionero.\n\n\u2014\u00bfPor qu\u00e9 no me ha matado? \u2014Las palabras me irritaron la garganta\u2014. \u00bfPor qu\u00e9 no ha acabado definitivamente conmigo?\n\nEl Custodio no me contest\u00f3. Tras examinarme el hombro, cogi\u00f3 un c\u00e1liz de la mesilla de noche. Estaba lleno hasta el borde de un l\u00edquido oscuro. Mir\u00e9 a mi guardi\u00e1n; \u00e9l me acerc\u00f3 el c\u00e1liz a los labios, sujet\u00e1ndome la parte de atr\u00e1s de la cabeza con una mano. Intent\u00e9 apartarme. El Custodio dej\u00f3 escapar un d\u00e9bil gru\u00f1ido.\n\n\u2014Esto reducir\u00e1 la hinchaz\u00f3n de la pierna \u2014dijo\u2014. Bebe.\n\nGir\u00e9 la cabeza. El Custodio me apart\u00f3 la copa de los labios.\n\n\u2014\u00bfNo quieres curarte?\n\nLo mir\u00e9 fijamente.\n\nDeb\u00eda de haber sido un accidente que sobreviviera. No hab\u00eda ninguna raz\u00f3n para que no me hubieran matado.\n\n\u2014Te han marcado \u2014dijo\u2014. Tienes que dejar que te cure la herida durante unos d\u00edas, o se te infectar\u00e1.\n\nGir\u00e9 la cabeza para mirarme el hombro, tap\u00e1ndome los pechos con las s\u00e1banas.\n\n\u2014\u00bfMarcado? \u00bfCon qu\u00e9? \u2014Resegu\u00ed la herida por la piel, tirante, y me temblaron los dedos. _XX-59-40_. \u00a1No!\u2014. Cerdo asqueroso. Te matar\u00e9. Cuando est\u00e9s dormido...\n\nMe dol\u00eda demasiado la garganta. Par\u00e9 de hablar; respiraba entrecortadamente. El Custodio escudri\u00f1\u00f3 mi cara, como si intentara descifrar un texto escrito en un idioma extranjero.\n\nNo era idiota. \u00bfPor qu\u00e9 me miraba as\u00ed? Me hab\u00edan marcado como a un animal. O como algo peor. Con un n\u00famero.\n\nSolo se o\u00edan mis jadeos. El Custodio apoy\u00f3 una mano enguantada en mi rodilla. Apart\u00e9 la pierna, y la descarga de dolor me lleg\u00f3 hasta los dedos del pie.\n\n\u2014No me toques.\n\n\u2014La marca dejar\u00e1 de dolerte con el tiempo \u2014dijo \u00e9l\u2014, pero la arteria femoral es otra cosa.\n\nDesliz\u00f3 la mano y me apart\u00f3 las s\u00e1banas de la pierna. Cuando me vi el muslo, cre\u00ed que iba a volver a desmayarme. Estaba hinchad\u00edsimo y cubierto de cardenales que se extend\u00edan casi hasta la rodilla. La zona alrededor de la ingle estaba negra e inyectada en sangre. El Custodio me aplic\u00f3 una liger\u00edsima presi\u00f3n en la pierna, apenas suficiente para accionar un gatillo de precisi\u00f3n. Vomit\u00e9.\n\n\u2014Esta herida no se curar\u00e1 sola. Las heridas causadas por el flux no sanan sin un segundo ant\u00eddoto m\u00e1s fuerte.\n\nCre\u00ed que me morir\u00eda si ejerc\u00eda una pizca m\u00e1s de presi\u00f3n.\n\n\u2014Vete al infierno \u2014dije con un hilo de voz.\n\n\u2014El infierno no existe. Solo existe el \u00e9ter.\n\nApret\u00e9 los dientes y tembl\u00e9 del esfuerzo por no sollozar. El Custodio apart\u00f3 la mano de mi pierna y se dio la vuelta.\n\nNo sabr\u00eda decir cu\u00e1nto rato permanec\u00ed all\u00ed tumbada, d\u00e9bil y delirante. Solo pod\u00eda pensar en c\u00f3mo deb\u00eda de estar disfrutando \u00e9l con aquello, viendo que volv\u00edamos a representar cada uno el papel que nos correspond\u00eda. Esa vez era \u00e9l quien ten\u00eda poder sobre m\u00ed, poder para verme sufrir y sudar. Y esa vez era \u00e9l quien ten\u00eda el remedio.\n\nEmpezaba a clarear. El reloj avanzaba. El Custodio estaba sentado en su butaca e iba echando le\u00f1a al fuego. No ten\u00eda ni idea de a qu\u00e9 esperaba. Si lo que pretend\u00eda era que yo cambiara de opini\u00f3n con respecto al remedio, iba a pasarse mucho rato all\u00ed. Tal vez solo le hubieran ordenado vigilarme, asegurarse de que no me suicidara. No voy a decir que no me lo planteara. El dolor era insoportable. Ten\u00eda la pierna r\u00edgida, y se me contra\u00eda espasm\u00f3dicamente. La piel, hinchada, estaba tensa y brillante, como una ampolla a punto de reventar.\n\nTranscurr\u00edan las horas, y el Custodio iba de un sitio a otro: la ventana, la butaca, el cuarto de ba\u00f1o, el escritorio, la butaca otra vez. Como si yo no estuviera all\u00ed. En una ocasi\u00f3n sali\u00f3 de la habitaci\u00f3n y volvi\u00f3 con un poco de pan caliente, pero lo rechac\u00e9. Quer\u00eda que pensara que estaba en huelga de hambre. Quer\u00eda recuperar mi poder. Quer\u00eda hacerle sentir tan insignificante como me hab\u00eda sentido yo.\n\nEl dolor del muslo no remit\u00eda, sino todo lo contrario. Me apret\u00e9 la piel oscurecida. Segu\u00ed apretando m\u00e1s y m\u00e1s, hasta que vi destellos luminosos. Ten\u00eda esperanzas de que eso me hiciera perder el conocimiento, porque as\u00ed dispondr\u00eda de unas horas de alivio; pero lo \u00fanico que consegu\u00ed fue volver a vomitar. El Custodio me mir\u00f3 mientras yo arrojaba una bilis \u00e1cida en un barre\u00f1o. Ten\u00eda la mirada inexpresiva. Estaba esperando a que yo cediera, a que le suplicara.\n\nMir\u00e9 el barre\u00f1o; lo ve\u00eda todo borroso, pero comprob\u00e9 que estaba empezando a vomitar gruesos co\u00e1gulos de sangre. Dej\u00e9 caer la cabeza sobre los cojines.\n\nDeb\u00ed de perder el conocimiento. Cuando despert\u00e9, volv\u00eda a oscurecer. Julian deb\u00eda de estar pregunt\u00e1ndose d\u00f3nde me hab\u00eda metido, suponiendo que hubiera podido salir de su residencia, lo cual era muy dudoso. Mi cerebro pod\u00eda concentrarse en esas cosas porque todo el dolor hab\u00eda desaparecido, inexplicablemente.\n\nIgual que toda la sensibilidad en la pierna.\n\nEl miedo se apoder\u00f3 de m\u00ed. Intent\u00e9 mover los dedos de los pies, girar el tobillo, pero no consegu\u00ed nada.\n\nEl Custodio volv\u00eda a estar a mi lado.\n\n\u2014Tal vez deber\u00eda mencionar \u2014dijo\u2014 que, si no tratamos la infecci\u00f3n, es muy probable que pierdas la pierna. O la vida.\n\nLe habr\u00eda escupido, pero los v\u00f3mitos me hab\u00edan deshidratado. Sacud\u00ed la cabeza. Estaba perdiendo la visi\u00f3n.\n\n\u2014No seas necia. \u2014Me agarr\u00f3 la cabeza y me oblig\u00f3 a mirarlo\u2014. Necesitas las piernas.\n\nMe ten\u00eda acorralada. Ten\u00eda raz\u00f3n: no pod\u00eda perder la pierna. Necesitaba poder correr. Esa vez, cuando me sujet\u00f3 la cabeza con una mano, abr\u00ed la boca y beb\u00ed del c\u00e1liz. El l\u00edquido sab\u00eda muy mal, a tierra y a metal. El Custodio asinti\u00f3 con la cabeza.\n\n\u2014Muy bien.\n\nLe lanc\u00e9 una mirada que quer\u00eda ser de odio, pero el cosquilleo que me recorri\u00f3 la pierna la suaviz\u00f3. Me beb\u00ed hasta la \u00faltima gota de aquel l\u00edquido repugnante, y me limpi\u00e9 los labios con la mano.\n\nEl Custodio volvi\u00f3 a levantar las s\u00e1banas. Mi muslo ya estaba recuperando sus dimensiones normales.\n\n\u2014Ahora estamos en paz \u2014dije. Me ard\u00eda la garganta\u2014. Ya est\u00e1. Yo te cur\u00e9 y t\u00fa me has curado a m\u00ed.\n\n\u2014T\u00fa nunca me has curado.\n\n\u2014\u00bfC\u00f3mo? \u2014dije, titubeante.\n\n\u2014No he sufrido ninguna herida.\n\n\u2014Pero \u00bfno te acuerdas?\n\n\u2014Eso no pas\u00f3.\n\nNo dud\u00e9 ni un instante de que lo que hab\u00eda pasado fuera real. El Custodio todav\u00eda llevaba los brazos cubiertos, de modo que no pod\u00eda mostrarle las pruebas; pero hab\u00eda sucedido. Por mucho que \u00e9l lo negara.\n\n\u2014Entonces deb\u00ed de equivocarme \u2014dije.\n\nEl Custodio no desvi\u00f3 la mirada de mis ojos. Me miraba con inter\u00e9s. Un inter\u00e9s fr\u00edo, desapasionado.\n\n\u2014S\u00ed \u2014dijo\u2014. Te equivocaste.\n\nY esa fue mi advertencia.\n\nSon\u00f3 la campana de la torre. El Custodio mir\u00f3 por la ventana.\n\n\u2014Puedes irte. No est\u00e1s en condiciones de empezar a entrenarte esta noche, pero deber\u00edas buscar algo de comer. \u2014Se\u00f1al\u00f3 la urna que estaba sobre la repisa de la chimenea\u2014. Ah\u00ed dentro tienes m\u00e1s _numa_. Coge todo lo que necesites.\n\n\u2014No tengo ropa.\n\n\u2014Eso es porque te corresponde un nuevo uniforme. \u2014Levant\u00f3 un blus\u00f3n rosa\u2014. Felicidades, Paige. Te han ascendido.\n\nEsa fue la primera vez que me llam\u00f3 por mi nombre.\n\nTen\u00eda que largarme de all\u00ed: eso fue lo primero que pens\u00e9 cuando sal\u00ed al intenso fr\u00edo de la noche. Sheol I estaba igual que antes, como si Seb nunca hubiera recorrido sus calles; yo, en cambio, hab\u00eda cambiado. En lugar del blus\u00f3n blanco, llevaba otro de color rosa claro. El ancla bordada en mi chaleco nuevo era del mismo rosa horrible. Estaba manchada.\n\nNo pod\u00eda hacer el siguiente examen. De ninguna manera. Si en el primero hab\u00edan matado a un ni\u00f1o, \u00bfqu\u00e9 ser\u00edan capaces de hacerme en el segundo? \u00bfCu\u00e1nta sangre habr\u00eda que derramar para que me convirtiera en casaca roja? Deb\u00eda irme. Ten\u00eda que haber alguna forma de huir, aunque para ello hubiera que sortear las minas. Cualquier cosa ser\u00eda mejor que esa pesadilla.\n\nMientras me abr\u00eda camino por el Poblado, con la pierna derecha d\u00e9bil y pesada, un fr\u00edo extra\u00f1o se extendi\u00f3 por mis tripas. Cada vez que un actor me miraba, mudaba la expresi\u00f3n. Su gesto se volv\u00eda inexpresivo, o agachaba la cabeza. Mi blus\u00f3n era una advertencia: \u00abSoy una renegada, una traidora. Al\u00e9jate de m\u00ed. Soy una asesina\u00bb.\n\nNo, no era ninguna asesina. A Seb lo hab\u00eda matado Nashira, no yo; pero eso no lo sab\u00edan los actores. Ellos despreciaban a cualquiera que no fuera un casaca blanca. Deber\u00eda haberme quedado en Magdalen aquella noche; pero entonces habr\u00eda tenido que estar con el Custodio, y no soportaba pasar ni un minuto m\u00e1s en su compa\u00f1\u00eda. Recorr\u00ed, cojeando, aquellos pasadizos claustrof\u00f3bicos. Necesitaba encontrar a Liss. Ella pod\u00eda ayudarme a salir de aquella pesadilla. Ten\u00eda que haber una forma.\n\n\u2014\u00bfPaige?\n\nMe par\u00e9; me temblaba la pierna. Ten\u00eda que hacer un esfuerzo agotador para andar. Liss hab\u00eda asomado la cabeza. Le ech\u00f3 un vistazo a mi blus\u00f3n rosa y se puso tensa.\n\n\u2014Hola, Liss.\n\n\u2014Aprobaste.\n\nSu cara se ensombreci\u00f3.\n\n\u2014S\u00ed \u2014afirm\u00e9\u2014, pero...\n\n\u2014\u00bfA qui\u00e9n ayudaste a apresar?\n\n\u2014A nadie. \u2014Me mir\u00f3 con gesto de incredulidad, y comprend\u00ed que ten\u00eda que cont\u00e1rselo\u2014. Intentaron obligarme a matar a... Seb. El amaur\u00f3tico. \u2014Agach\u00e9 la cabeza\u2014. Y ahora \u00e9l est\u00e1 muerto.\n\nLiss dio un respingo.\n\n\u2014Vale \u2014dijo\u2014. Pues hasta luego.\n\n\u2014Liss, esc\u00fachame, por favor. T\u00fa no...\n\nCorri\u00f3 la cortina de su puerta de un tir\u00f3n y me dej\u00f3 con la palabra en la boca. Exhausta, resbal\u00e9 hasta el suelo con la espalda pegada a la pared. No era uno de ellos.\n\n\u00abSeb.\u00bb Dije su nombre mentalmente, tratando de hacer salir a su esp\u00edritu de donde lo hubieran escondido, pero no sali\u00f3 nada del \u00e9ter. Ni siquiera una leve punzada. Ni siquiera logr\u00e9 nada a\u00f1adiendo su apellido; seguramente me faltaba alg\u00fan nombre. Una vez muerto, el ni\u00f1o que tanto hab\u00eda confiado en m\u00ed, que tan seguro estaba de que yo lo salvar\u00eda, era un desconocido para m\u00ed.\n\nMe dio la impresi\u00f3n de que la cortina me miraba con odio. Liss deb\u00eda de considerarme pura escoria. Cerr\u00e9 los ojos y trat\u00e9 de ignorar el dolor sordo del muslo. Quiz\u00e1 encontrara a otro casaca rosa con quien intercambiar informaci\u00f3n, pero me resist\u00eda a intentarlo. No pod\u00eda confiar en ellos. La mayor\u00eda eran asesinos. La mayor\u00eda hab\u00edan delatado a alguien. Si quer\u00eda hablar con alguien que no fuera un renegado, tendr\u00eda que demostrarle a Liss que pod\u00eda confiar en m\u00ed. Me levant\u00e9 haciendo un esfuerzo que me dej\u00f3 empapada de sudor y me dirig\u00ed a la choza de la comida. Tal vez encontrara a Julian all\u00ed. Seguramente \u00e9l tampoco querr\u00eda hablar conmigo, pero quiz\u00e1 me diera una oportunidad.\n\nMe llam\u00f3 la atenci\u00f3n una luz. Un hornillo. Un grupo de actores fumaban en un cobertizo diminuto, tumbados sobre los costados, agitando de vez en cuando una mano como si quisieran atrapar algo. \u00c1ster, otra vez. Tilda estaba entre ellos, con la cabeza apoyada en un coj\u00edn, el blanco blus\u00f3n sucio y arrugado como un pa\u00f1uelo de papel usado. Busqu\u00e9 a tientas en mi chaleco el comprimido verde que me hab\u00eda llevado. Me arrodill\u00e9 junto a Tilda procurando no lastimarme la pierna.\n\n\u2014Tilda.\n\nAbri\u00f3 un poco los ojos.\n\n\u2014\u00bfQu\u00e9 pasa?\n\n\u2014He tra\u00eddo la pastilla.\n\n\u2014Un momento. Todav\u00eda estoy reinando. Dame un minuto, cielo. O quiz\u00e1 dos. O cinco. \u2014Se puso boca abajo; una risa silenciosa la sacudi\u00f3\u2014. El onirosaje se ha vuelto morado. \u00bfEres real?\n\nEsper\u00e9 a que se agotara el efecto del \u00e1ster. Tilda se pas\u00f3 un minuto entero riendo; estaba roja como un tomate. Notaba el desenfreno de su aura, c\u00f3mo se sacud\u00eda y cambiaba por efecto de la droga. Los otros videntes no daban se\u00f1ales de querer despertar. Con manos temblorosas, Tilda se frot\u00f3 la cara y asinti\u00f3.\n\n\u2014Vale, ya estoy destronada. \u00bfD\u00f3nde est\u00e1 esa pastilla?\n\nSe la di. Tilda la examin\u00f3 desde todos los \u00e1ngulos. Pas\u00f3 un dedo por su superficie, analizando su textura. La parti\u00f3 por la mitad. Aplast\u00f3 una mitad entre los dedos. Oli\u00f3 el residuo, lo prob\u00f3.\n\n\u2014Tu guardiana ha vuelto a salir \u2014dije.\n\n\u2014Sale mucho. \u2014Me devolvi\u00f3 los restos de la pastilla\u2014. Es herbal, pero no sabr\u00eda decirte qu\u00e9 hierba lleva.\n\n\u2014\u00bfConoces a alguien que pueda dec\u00edrmelo?\n\n\u2014Podr\u00edas probar en la casa de empe\u00f1os. A lo mejor el tipo que me vendi\u00f3 el \u00e1ster sabe dec\u00edrtelo. La contrase\u00f1a es _specchio_.\n\n\u2014Ir\u00e9 a verlo. \u2014Me levant\u00e9\u2014. Te dejo con tu \u00e1ster.\n\n\u2014Gracias. Hasta luego.\n\nSe desplom\u00f3 sobre el coj\u00edn. Me pregunt\u00e9 qu\u00e9 les har\u00eda Suhail si los descubr\u00eda.\n\nTard\u00e9 un rato en encontrar la casa de empe\u00f1os. En el Poblado hab\u00eda muchas habitaciones, la mayor\u00eda ocupadas por grupos de dos o tres personas. Se pasaban el d\u00eda en aquellas peque\u00f1as chozas, apretujados alrededor de un hornillo de parafina, y dorm\u00edan sobre s\u00e1banas que apestaban a humedad y orines. Com\u00edan lo que encontraban. Si no encontraban nada, pasaban hambre. Permanec\u00edan juntos por dos motivos: porque no hab\u00eda suficiente espacio, y por el fr\u00edo intenso que hac\u00eda en la ciudad. No hab\u00eda instalaciones higi\u00e9nicas, ni m\u00e1s medicinas que las que pudieran obtener robando. Era un sitio adonde se iba a morir.\n\nLa casa de empe\u00f1os estaba escondida detr\u00e1s de una serie de gruesas cortinas. Ten\u00edas que saber d\u00f3nde buscar; yo la encontr\u00e9 despu\u00e9s de interrogar a una bufona. La chica parec\u00eda reacia a revelarme la informaci\u00f3n, y me previno sobre los sobornos y los precios elevados, pero se\u00f1al\u00f3 en la direcci\u00f3n correcta.\n\nVigilando la tienda estaba el cantor al que hab\u00eda visto el d\u00eda del serm\u00f3n. Se hallaba sentado en un coj\u00edn, jugando con unos dados. Ya no llevaba el blus\u00f3n blanco; deb\u00eda de haber suspendido el examen. \u00bfPara qu\u00e9 pod\u00edan querer los refa\u00edtas a un cantor?\n\n\u2014Hola \u2014lo salud\u00e9.\n\n\u2014Hola \u2014me contest\u00f3 con una voz dulce y pura; la voz de un cantor, sin duda.\n\n\u2014\u00bfPuedo hablar con el prestamista?\n\n\u2014\u00bfContrase\u00f1a?\n\n\u2014 _Specchio_.\n\nEl chico se levant\u00f3. Ten\u00eda los p\u00e1rpados del ojo derecho hinchados debido a una infecci\u00f3n. Descorri\u00f3 las cortinas y entr\u00e9.\n\nLas casas de empe\u00f1os de Londres eran establecimientos por lo general peque\u00f1os y sin licencia de las peores zonas de la cohorte central. Tambi\u00e9n hab\u00eda muchas en Chapel, en el II-6. Aquella no era diferente. El prestamista hab\u00eda montado su negocio en una especie de tienda de campa\u00f1a hecha con pa\u00f1uelos parecidos a los que usaba Liss en sus actuaciones. La mitad del espacio, iluminado por una sola l\u00e1mpara de parafina, se hab\u00eda convertido en un palacio de espejos. El prestamista, sentado en un sill\u00f3n de piel maltrecho, contemplaba las superficies de cristal moteado. Era un hombre de cabello cano con demasiada barriga para ser actor. Los espejos delataban su especialidad: la catoptromancia.\n\nCuando entr\u00e9, se puso el mon\u00f3culo en un ojo y examin\u00f3 mi reflejo. Ten\u00eda los ojos empa\u00f1ados de los profetas expertos.\n\n\u2014Creo que no te hab\u00eda visto nunca. Ni en mis espejos ni en mi tienda.\n\n\u2014Era de Huesos XX \u2014dije.\n\n\u2014Entiendo. \u00bfDe qui\u00e9n eres?\n\n\u2014De Arcturus Mesarthim.\n\nEstaba harta de ese nombre; de o\u00edrlo, de pronunciarlo.\n\n\u2014Vaya, vaya. \u2014Se dio unas palmadas en la panza\u2014. De modo que eres su inquilina.\n\n\u2014\u00bfC\u00f3mo te llamas?\n\n\u2014XVI-19-16.\n\n\u2014Me refiero a tu verdadero nombre.\n\n\u2014Ya no me acuerdo, pero los actores me llaman Duckett. Por si prefieres usar los nombres reales.\n\n\u2014Pues s\u00ed.\n\nMe agach\u00e9 para echar un vistazo a sus existencias. Casi todos los art\u00edculos eran _numa_ : espejos de mano rajados, botellas de cristal, cuencos y tazas, perlas, bolsas llenas de huesos de animales, cartas y piedras de adivinaci\u00f3n. Luego estaban las plantas. \u00c1ster, brezo blanco, salvia, tomillo y otras plantas de quemar. Tambi\u00e9n hab\u00eda art\u00edculos m\u00e1s pr\u00e1cticos, esenciales para la supervivencia. Examin\u00e9 el mont\u00f3n: s\u00e1banas, cojines mullidos, cerillas, unas pinzas, alcohol de uso t\u00f3pico, aspirina y oxitetraciclina, latas de Sterno, un cuentagotas de \u00e1cido fus\u00eddico, vendas y desinfectantes. Cog\u00ed una vieja caja de yesca.\n\n\u2014\u00bfDe d\u00f3nde has sacado todo esto?\n\n\u2014De aqu\u00ed y de all\u00e1.\n\n\u2014Supongo que los refas no saben que lo tienes.\n\nEsboz\u00f3 una sonrisa.\n\n\u2014Dime, \u00bfc\u00f3mo funciona esta tienda ilegal?\n\n\u2014Bueno, supongamos que eres espatulom\u00e1ntica y necesitas huesos para practicar la clarividencia. Si te confiscan los huesos, tienes que buscar m\u00e1s. \u2014Se\u00f1al\u00f3 una bolsa con la etiqueta \u00abRata com\u00fan\u00bb\u2014. Yo te encargo una tarea. Puedo pedirte que me traigas determinado art\u00edculo, o que entregues un mensaje por m\u00ed; cuanto m\u00e1s valioso sea el objeto que necesites, m\u00e1s peligrosa ser\u00e1 la tarea. Si consigues realizarla, yo te doy los huesos. Si lo que quieres es un pr\u00e9stamo por un tiempo limitado, tienes que entregarme cierto n\u00famero de _numa_ , que yo te devuelvo cuando t\u00fa me restituyes el objeto. Es un sistema sencillo, pero eficaz.\n\nNo parec\u00eda una casa de empe\u00f1os convencional, que prestaba dinero a cambio de art\u00edculos empe\u00f1ados.\n\n\u2014\u00bfCu\u00e1nto cobras por dar informaci\u00f3n?\n\n\u2014Eso depende de la informaci\u00f3n que busques.\n\n\u2014\u00bfQu\u00e9 es esto? \u2014pregunt\u00e9 ense\u00f1\u00e1ndole la pastilla.\n\nSe qued\u00f3 mir\u00e1ndola un momento; entonces se quit\u00f3 el mon\u00f3culo y la cogi\u00f3. Le temblaban los dedos, gruesos.\n\n\u2014Por esto \u2014dijo\u2014 podr\u00eda darte cualquier cosa que quisieras de la tienda. Gratis.\n\nArrugu\u00e9 el entrecejo.\n\n\u2014\u00bfQuieres qued\u00e1rtelo?\n\n\u2014S\u00ed, claro. Es muy valioso. \u2014Se puso la mitad en la palma de la mano\u2014. \u00bfDe d\u00f3nde lo has sacado?\n\n\u2014La informaci\u00f3n tiene un precio, Duckett.\n\n\u2014Si me traes m\u00e1s, nunca te cobrar\u00e9 nada. Ll\u00e9vate lo que quieras. Un art\u00edculo por p\u00edldora.\n\n\u2014Si no me dices qu\u00e9 es, no hay trato.\n\n\u2014Dos art\u00edculos.\n\n\u2014No.\n\n\u2014La informaci\u00f3n es peligrosa. No se le puede poner precio. \u2014Acerc\u00f3 el comprimido a la l\u00e1mpara de parafina\u2014. Lo que puedo decirte es que es una p\u00edldora herbal, y que es inofensiva. \u00bfTe basta con eso?\n\nDos art\u00edculos por cada pastilla. All\u00ed hab\u00eda art\u00edculos que pod\u00edan salvar vidas en el Poblado.\n\n\u2014Tres \u2014dije\u2014, y trato hecho.\n\n\u2014Excelente. Sabes hacer negocios. \u2014Junt\u00f3 las yemas de los dedos y a\u00f1adi\u00f3\u2014: \u00bfQu\u00e9 m\u00e1s sabes hacer?\n\n\u2014Soy acutum\u00e1ntica.\n\nEra la mentira a la que sol\u00eda recurrir. Una especie de test de competencia. Me gustaba ver si me cre\u00edan o no. Duckett ri\u00f3 entre dientes.\n\n\u2014No eres adivina. Si yo tuviera visi\u00f3n, dir\u00eda que est\u00e1s en el otro extremo del espectro. Tienes un aura caliente. Como las brasas. \u2014Tamborile\u00f3 con los dedos en un espejo\u2014. A\u00fan resultar\u00e1 que la de este a\u00f1o ha sido otra era interesante.\n\nMe puse en tensi\u00f3n.\n\n\u2014\u00bfC\u00f3mo dices?\n\n\u2014Nada, nada. Hablaba solo. Cuando llevas cuarenta a\u00f1os aqu\u00ed, es lo mejor que puedes hacer para no volverte loco. \u2014Sus labios esbozaron una sonrisa\u2014. Dime, \u00bfqu\u00e9 te parece el Custodio?\n\nVolv\u00ed a dejar la caja de yesca encima de la mesa.\n\n\u2014Cre\u00eda que era obvio \u2014dije.\n\n\u2014En absoluto. Aqu\u00ed hay gran variedad de opiniones. \u2014Duckett pas\u00f3 el pulgar por la lente de su mon\u00f3culo\u2014. Muchos opinan que el consorte de sangre es el m\u00e1s atractivo de los refa\u00edtas.\n\n\u2014Quiz\u00e1 t\u00fa pienses as\u00ed. Yo lo encuentro repugnante. \u2014Le sostuve la mirada\u2014. Me llevo mis cosas.\n\nDuckett se recost\u00f3 en la silla. Cog\u00ed una lata de Sterno, unas aspirinas y el \u00e1cido fus\u00eddico.\n\n\u2014Ha sido un placer hacer negocios contigo \u2014dijo\u2014. \u00bfSe\u00f1orita...?\n\n\u2014Mahoney. Paige Mahoney. \u2014Le di la espalda\u2014. Por si prefieres utilizar los nombres reales.\n\nSal\u00ed de aquel cubil con la mirada de Duckett clavada en la espalda.\n\nMe sent\u00eda interrogada. Estaba segura de no haber contestado nada que no debiera. Hab\u00eda dicho exactamente lo que pensaba del Custodio. No ten\u00eda ni idea de por qu\u00e9 Duckett quer\u00eda o\u00edrme decir otra cosa.\n\nAl salir le tir\u00e9 el \u00e1cido fus\u00eddico al cantor. \u00c9l me mir\u00f3 con la cabeza ladeada.\n\n\u2014Para el ojo \u2014le dije.\n\nParpade\u00f3, y yo segu\u00ed caminando.\n\nCuando llegu\u00e9 a la choza que buscaba, llam\u00e9 golpeando la pared con los nudillos.\n\n\u2014\u00bfLiss? \u2014No me contestaron, y volv\u00ed a llamar\u2014. Soy yo, Liss. Paige.\n\nDescorrieron la cortina. Liss sujetaba un farolillo.\n\n\u2014D\u00e9jame en paz \u2014dijo con una voz cargada de resentimiento\u2014. Por favor. Yo no hablo ni con rosas ni con rojos. Lo siento: no hablo. Tendr\u00e1s que buscar a otros casacas, \u00bfvale?\n\n\u2014Yo no mat\u00e9 a Seb. \u2014Le ofrec\u00ed el Sterno y las aspirinas\u2014. Mira, Duckett me ha dado esto. \u00bfPodemos hablar un momento?\n\nMir\u00f3 mis art\u00edculos y luego me mir\u00f3 a la cara. Arrug\u00f3 la frente y frunci\u00f3 los labios.\n\n\u2014Bueno \u2014concedi\u00f3\u2014, ser\u00e1 mejor que entres.\n\nNo llor\u00e9 cuando le cont\u00e9 lo de mi examen. No pude. Jax odiaba las l\u00e1grimas. (\u00abEres una chica dura de los bajos fondos, querida. Comp\u00f3rtate como tal, hazme el favor.\u00bb) Incluso all\u00ed, donde \u00e9l jam\u00e1s podr\u00eda verme, sent\u00eda que observaba cada uno de mis movimientos. Sin embargo, recordar el cuello roto de Seb me produc\u00eda n\u00e1useas. No pod\u00eda olvidar la expresi\u00f3n de sorpresa en sus ojos, ni c\u00f3mo hab\u00eda gritado mi nombre. Cuando termin\u00e9 de contarle la historia, me sent\u00e9 y estir\u00e9 la pierna r\u00edgida ante m\u00ed.\n\nLiss me acerc\u00f3 un vaso humeante.\n\n\u2014B\u00e9bete esto. Si quieres protegerte de Nashira, necesitas conservar tu fortaleza. \u2014Se recost\u00f3\u2014. Ahora ya sabe qu\u00e9 eres.\n\nDi un sorbo. El l\u00edquido sab\u00eda a menta.\n\nMe escoc\u00edan los ojos y todav\u00eda me dol\u00eda la garganta, pero no iba a llorar por Seb. Habr\u00eda sido irrespetuoso llorar estando Liss sentada a mi lado. Ella ten\u00eda la cara hinchada, moratones en el cuello y el hombro dislocado, y sin embargo hab\u00eda antepuesto mi bienestar al suyo. \u00abAhora formas parte de la Familia, hermana\u00bb, me hab\u00eda dicho. Con una sola mano, me aplic\u00f3 un emplasto caliente en la marca del hombro. Ya no notaba aquel ardor tan intenso, pero Liss me asegur\u00f3 que la quemadura me dejar\u00eda cicatriz. De eso se trataba: de recordarme, todos los d\u00edas, a qui\u00e9n pertenec\u00eda.\n\nJulian dorm\u00eda bajo una s\u00e1bana deste\u00f1ida. Su guardiana se hab\u00eda marchado con su familia, los Chertan. Antes de que se quedara dormido, yo le hab\u00eda dado una aspirina. Su nariz ten\u00eda mejor aspecto. Como yo no hab\u00eda aparecido al amanecer, hab\u00eda ido all\u00ed a buscarme, y Liss lo hab\u00eda convencido para que se quedara. Juntos hab\u00edan arreglado la choza lo mejor que hab\u00edan podido, pero, aun as\u00ed, parec\u00eda una nevera. Liss me hab\u00eda invitado, de todas formas, a quedarme a pasar la noche, y yo estaba decidida a aceptar su invitaci\u00f3n. Necesitaba alejarme de Magdalen.\n\nLiss abri\u00f3 la lata de Sterno con un abrelatas viejo.\n\n\u2014Gracias por tra\u00e9rmelo. Hac\u00eda mucho tiempo que no ve\u00eda calor enlatado. \u2014Sac\u00f3 una cerilla y prendi\u00f3 el alcohol gelatinoso. Brot\u00f3 una llama limpia y azul\u2014. \u00bfTe lo ha dado Duckett?\n\n\u2014S\u00ed, pero se lo he pagado.\n\n\u2014\u00bfCon qu\u00e9?\n\n\u2014Con una de mis pastillas.\n\nLiss arque\u00f3 una ceja.\n\n\u2014\u00bfPara qu\u00e9 la quiere?\n\n\u2014La quiere porque me dan una pastilla que no le dan a nadie m\u00e1s. No tengo ni idea de qu\u00e9 es.\n\n\u2014Si puedes utilizarlas para sobornar a Duckett, vale la pena que las guardes. Las tareas que pone son peligrosas. Hace ir a la gente a las residencias a robar para \u00e9l. Y muchas veces los descubren.\n\nHizo una mueca de dolor y se toc\u00f3 el hombro. Le quit\u00e9 la lata de Sterno de la mano y la puse entre nosotras dos.\n\n\u2014Ha sido Gomeisa, \u00bfverdad? \u2014dije.\n\n\u2014A veces se aburre con las cartas. Y no siempre le gusta lo que le ense\u00f1an. \u2014Se tumb\u00f3 boca arriba y se puso la almohada bajo la nuca\u2014. No importa. No lo veo mucho. La mayor parte del tiempo creo que ni siquiera est\u00e1 en la ciudad.\n\n\u2014\u00bfEras su \u00fanica humana?\n\n\u2014S\u00ed. Por eso me odia. Estaba exactamente en la misma situaci\u00f3n que t\u00fa, elegida por un refa que jam\u00e1s hab\u00eda reclamado a un humano. Cre\u00eda que yo ten\u00eda potencial; cre\u00eda que pod\u00eda ser una de las mejores arrancahuesos de Sheol I.\n\n\u2014\u00bfArrancahuesos?\n\n\u2014As\u00ed es como llamamos a los casacas rojas. Gomeisa cre\u00eda que yo me ganar\u00eda ese color. Pero lo decepcion\u00e9.\n\n\u00bbMe pidi\u00f3 que le echara las cartas a un buf\u00f3n. Sospechaban que era un traidor, que hab\u00eda intentado huir. Yo sab\u00eda que era verdad. Las cartas lo habr\u00edan delatado. Me negu\u00e9 a ech\u00e1rselas.\n\n\u2014Yo tambi\u00e9n me negu\u00e9 a obedecer, pero ella descubri\u00f3 lo que soy. \u2014Me frot\u00e9 las sienes\u2014. Y ahora Seb est\u00e1 muerto.\n\n\u2014Aqu\u00ed mueren muchos amaur\u00f3ticos. La habr\u00eda palmado hicieras lo que hicieses. \u2014Volvi\u00f3 a incorporarse\u2014. Va, comamos algo.\n\nAbri\u00f3 el arc\u00f3n de madera. Me qued\u00e9 mirando lo que hab\u00eda dentro: un paquete de caf\u00e9 soluble, latas de jud\u00edas y cuatro huevos.\n\n\u2014\u00bfDe d\u00f3nde has sacado eso?\n\n\u2014Lo encontr\u00e9.\n\n\u2014\u00bfD\u00f3nde?\n\n\u2014Un amaur\u00f3tico lo escondi\u00f3 cerca de su residencia. Son las sobras de las provisiones para la Era de Huesos. \u2014Liss cogi\u00f3 una olla de hierro y la llen\u00f3 de agua de una botella\u2014. Ya ver\u00e1s. Comeremos como reinas. \u2014Puso la olla encima del Sterno\u2014. \u00bfC\u00f3mo te encuentras, Jules?\n\nNuestras voces deb\u00edan de haberlo despertado. Julian apart\u00f3 la s\u00e1bana y se sent\u00f3 con las piernas cruzadas.\n\n\u2014Mejor. \u2014Se apret\u00f3 un poco la nariz\u2014. Gracias por las medicinas, Paige.\n\n\u2014De nada. \u00bfCu\u00e1ndo te examinas?\n\n\u2014Ni idea. Se supone que Aludra tiene que ense\u00f1arnos a sublimar, pero se pasa la mayor parte del tiempo maltrat\u00e1ndonos.\n\n\u2014\u00bfA sublimar?\n\n\u2014A convertir objetos normales en _numa_. \u00bfTe acuerdas de los bastones que est\u00e1bamos utilizando la otra noche, cuando viniste a verme? Estaban sublimados. Puede utilizarlos cualquiera, no solo los adivinos.\n\n\u2014Y \u00bfqu\u00e9 hacen?\n\n\u2014Ejercen cierta influencia sobre los esp\u00edritus m\u00e1s cercanos, pero no pueden utilizarse para ver el \u00e9ter.\n\n\u2014Entonces, en realidad no son _numa_.\n\n\u2014Pero son peligrosos \u2014terci\u00f3 Liss\u2014. Los carro\u00f1os pueden usarlos. Lo \u00faltimo que nos falta es un arma et\u00e9rea que Scion pueda utilizar.\n\nJulian sacudi\u00f3 la cabeza.\n\n\u2014Scion jam\u00e1s usar\u00eda _numa_. Les repugna la clarividencia.\n\n\u2014Pero los refa\u00edtas no.\n\n\u2014Dudo mucho que les gusten los refas \u2014dije\u2014. Son clarividentes. Lo que pasa es que no tienen m\u00e1s remedio que obedecer porque tienen a los emim a la vuelta de la esquina.\n\nEl agua ya herv\u00eda. Liss la verti\u00f3 en tres vasos de pl\u00e1stico y a\u00f1adi\u00f3 el caf\u00e9. Llevaba d\u00edas sin oler el caf\u00e9, o semanas. \u00bfCu\u00e1nto tiempo llevaba all\u00ed?\n\n\u2014Toma. \u2014Liss me dio una taza a m\u00ed y otra a Julian\u2014. \u00bfD\u00f3nde te tiene Aludra, Jules?\n\n\u2014En una habitaci\u00f3n sin luz. Debe de ser una antigua bodega. Dormimos en el suelo. Felix tiene claustrofobia y Ella a\u00f1ora a su familia. Se pasan el d\u00eda llorando, no me dejan dormir.\n\n\u2014Haz que te echen. Aqu\u00ed la vida es dura, pero no tanto como cuando tienes un guardi\u00e1n. Solo nos comen si estamos en el sitio inadecuado en el momento inadecuado. \u2014Liss bebi\u00f3 de su vaso\u2014. Hay gente que no lo soporta. Ten\u00eda una amiga que se vino aqu\u00ed conmigo, pero suplic\u00f3 a su guardi\u00e1n que le diera otra oportunidad. Ahora es arrancahuesos.\n\nNos bebimos el caf\u00e9 en silencio. Liss hirvi\u00f3 los huevos y nos los comimos a palo seco.\n\n\u2014Estaba pensando... \u2014dijo Julian\u2014. \u00bfLos refas pueden volver al sitio de donde salieron?\n\n\u2014Supongo \u2014dijo Liss.\n\n\u2014Es que no entiendo por qu\u00e9 se quedan aqu\u00ed. Porque no siempre han estado aqu\u00ed. \u00bfD\u00f3nde consegu\u00edan auras antes de encontrarnos a nosotros?\n\n\u2014Quiz\u00e1 tenga que ver con los zumbadores \u2014dije\u2014. Nashira dijo que eran una raza parasitaria, \u00bfno?\n\nJulian asinti\u00f3.\n\n\u2014\u00bfCrees que los zumbadores les quitaron algo?\n\n\u2014\u00bfLa cordura?\n\nJulian ri\u00f3.\n\n\u2014S\u00ed. A lo mejor eran buena gente hasta que los zumbadores se la chuparon toda.\n\nLiss no ri\u00f3.\n\n\u2014Podr\u00eda ser el umbral et\u00e9reo \u2014dije\u2014. Nashira mencion\u00f3 que aparecieron cuando se rompi\u00f3.\n\n\u2014Creo que nunca lo sabremos. \u2014Liss parec\u00eda tensa\u2014. Dudo mucho que vayan a difundirlo.\n\n\u2014\u00bfPor qu\u00e9 no? Si ellos son tan poderosos y nosotros tan d\u00e9biles, \u00bfqu\u00e9 necesidad hay de mantener el secreto?\n\n\u2014El conocimiento es poder \u2014dijo Julian\u2014. Ellos lo tienen, y nosotros no.\n\n\u2014Te equivocas, hermano. El conocimiento es peligroso. \u2014Liss acerc\u00f3 las rodillas a la barbilla. Era lo mismo que hab\u00eda insinuado Duckett\u2014. Una vez que sabes algo, no puedes librarte de ello. Tienes que llevarlo contigo. Siempre.\n\nJulian y yo nos miramos. Liss llevaba mucho tiempo all\u00ed; tal vez fuera mejor que sigui\u00e9ramos sus consejos. O no. Tal vez sus consejos acabaran mat\u00e1ndonos.\n\n\u2014Liss \u2014dije\u2014, \u00bfnunca te planteas rebelarte contra ellos?\n\n\u2014Todos los d\u00edas.\n\n\u2014Pero no lo haces.\n\n\u2014Me planteo sacarle los ojos a Suhail con mis propias manos. \u2014Lo dijo apretando los dientes\u2014. Me planteo acribillar a balazos a Nashira. Me planteo degollar a Gomeisa. Pero s\u00e9 que ellos me matar\u00edan antes a m\u00ed, y por eso no lo hago.\n\n\u2014Pero, si piensas as\u00ed, te quedar\u00e1s atrapada aqu\u00ed para siempre \u2014replic\u00f3 Julian con suavidad\u2014. \u00bfEs eso lo que quieres?\n\n\u2014Claro que no. Quiero volver a mi casa, aunque no sepa muy bien qu\u00e9 significa eso. \u2014Liss volvi\u00f3 la cabeza\u2014. Ya s\u00e9 lo que pensar\u00e9is de m\u00ed. Que no tengo agallas.\n\n\u2014Liss \u2014dije\u2014, no hemos querido decir...\n\n\u2014S\u00ed. No os lo reprocho. Pero ya que os interesa tanto saber, os voy a decir una cosa. Hubo una rebeli\u00f3n durante la Era XVIII, en 2039. Toda la poblaci\u00f3n humana de Sheol I se rebel\u00f3 contra los refas. \u2014El dolor que se reflejaba en sus ojos la envejec\u00eda\u2014. Murieron todos: amaur\u00f3ticos, videntes, todos. Como no hab\u00eda casacas rojas para combatirlos, los emim entraron y los mataron a todos. Y los refas no hicieron nada por impedirlo.\n\nMir\u00e9 a Julian, que no apartaba la vista de Liss.\n\n\u2014Dijeron que era lo que merec\u00edan. Por su desobediencia. Fue lo primero que nos dijeron cuando llegamos. \u2014Se pasaba las cartas de una mano a otra\u2014. S\u00e9 que los dos sois combativos, pero no quiero veros morir aqu\u00ed. No de esa forma.\n\nSus palabras me hicieron callar. Julian se pas\u00f3 una mano por la cabeza y se qued\u00f3 contemplando la llama del hornillo.\n\nNo volvimos a hablar de la rebeli\u00f3n. Nos comimos las jud\u00edas y reba\u00f1amos las latas hasta dejarlas limpias. Liss ten\u00eda la baraja sobre el regazo. Al cabo de un momento Julian carraspe\u00f3.\n\n\u2014\u00bfD\u00f3nde viv\u00edas, Liss? Antes de venir aqu\u00ed.\n\n\u2014En Cradlehall. Est\u00e1 cerca de Inverness.\n\n\u2014\u00bfC\u00f3mo es Scion por all\u00ed?\n\n\u2014Igual que aqu\u00ed. Todas las grandes ciudades se rigen por el mismo sistema, solo que con menos fuerzas de seguridad que Londres. Est\u00e1n sometidas a la legislaci\u00f3n inquisitorial, como la ciudadela.\n\n\u2014\u00bfPor qu\u00e9 viniste al sur? \u2014pregunt\u00e9\u2014. Estoy convencida de que las Tierras Altas eran m\u00e1s seguras para los videntes.\n\n\u2014\u00bfPor qu\u00e9 va la gente a SciLo? Por trabajo. Por dinero. Necesitamos comer, igual que los amaur\u00f3ticos. \u2014Liss se ech\u00f3 una s\u00e1bana sobre los hombros\u2014. A mis padres les daba miedo vivir en el centro de Inverness. All\u00ed los videntes no est\u00e1n organizados; no hay nada parecido al sindicato. Mi padre crey\u00f3 que deb\u00edamos probar suerte en la ciudadela. Invertimos todos nuestros ahorros en trasladarnos a Londres. Hablamos con algunos capos, pero ninguno necesitaba adivinos. Cuando se nos acab\u00f3 el dinero, ten\u00edamos que limosnear para conseguir un catre donde pasar la noche.\n\n\u2014Y os descubrieron.\n\n\u2014Mi padre enferm\u00f3, y no pod\u00eda salir a la calle. Ten\u00eda m\u00e1s de sesenta a\u00f1os, y en las calles cog\u00eda todo tipo de virus. Lo sustitu\u00ed en el sitio donde sol\u00eda colocarse. Se me acerc\u00f3 una mujer y me pidi\u00f3 que le echara las cartas. \u2014Pas\u00f3 el pulgar por el borde de la baraja\u2014. Yo ten\u00eda nueve a\u00f1os. No me di cuenta de que era de la DVN.\n\n\u2014\u00bfCu\u00e1nto tiempo pasaste en la Torre? \u2014le pregunt\u00f3 Julian.\n\n\u2014Cuatro a\u00f1os. Me hicieron el submarino varias veces, quer\u00edan que les dijera d\u00f3nde estaban mis padres. Yo les dec\u00eda que no lo sab\u00eda.\n\nContarnos su historia no la ayudar\u00eda a sentirse mejor. Cambi\u00e9 de tema:\n\n\u2014\u00bfY t\u00fa, Julian? \u00bfD\u00f3nde viv\u00edas?\n\n\u2014En Morden. En el IV-6.\n\n\u2014Es el sector m\u00e1s peque\u00f1o, \u00bfverdad?\n\n\u2014S\u00ed, por eso el sindicato no se preocupa mucho por \u00e9l. Ten\u00eda una banda peque\u00f1a, pero no nos dedic\u00e1bamos a la mimetodelincuencia. Solo hac\u00edamos sesiones de espiritismo de vez en cuando.\n\nSent\u00ed una punzada de nostalgia. Echaba de menos a mi banda.\n\nJulian no tard\u00f3 en rendirse al agotamiento. El Sterno estaba qued\u00e1ndose sin combustible. Liss esper\u00f3 hasta que el fuego se apag\u00f3 del todo. Me hice la dormida, pero no pod\u00eda parar de pensar en la Era XVIII. Deb\u00eda de haber muerto mucha gente. Sus familias no debieron de saber nada m\u00e1s de ellos. No debi\u00f3 de haber juicios, ni apelaciones. Semejante injusticia me produc\u00eda n\u00e1useas. No me extra\u00f1aba que a Liss le diera miedo pelear.\n\nY entonces son\u00f3 la sirena.\n\nJulian se despert\u00f3 de golpe. El sonido fue aumentando de intensidad progresivamente, hasta convertirse en un bramido estridente. Mi cuerpo reaccion\u00f3 de inmediato: un cosquilleo en las piernas, el coraz\u00f3n desbocado.\n\nO\u00edmos pasos por los pasadizos. Julian descorri\u00f3 la cortina que hac\u00eda las veces de puerta. Pasaron corriendo tres casacas rojas; uno llevaba una potente linterna. Liss abri\u00f3 los ojos, pero permaneci\u00f3 inm\u00f3vil.\n\n\u2014Tienen pu\u00f1ales \u2014observ\u00f3 Julian.\n\nLiss se acurruc\u00f3 en un rinc\u00f3n de la choza. Recogi\u00f3 su baraja de cartas, se abraz\u00f3 las rodillas con el brazo sano y agach\u00f3 la cabeza.\n\n\u2014Ten\u00e9is que iros \u2014dijo\u2014. \u00a1Corred!\n\n\u2014Ven con nosotros \u2014dije\u2014. Cu\u00e9late en alguna residencia. Aqu\u00ed no est\u00e1s...\n\n\u2014\u00bfQu\u00e9 quer\u00e9is, recibir una paliza de Aludra? \u00bfO del Custodio? \u2014Nos mir\u00f3 con severidad\u2014. Llevo diez a\u00f1os haciendo esto. Largaos de aqu\u00ed.\n\nJulian y yo nos miramos. Ya lleg\u00e1bamos tarde. No sab\u00eda qu\u00e9 me har\u00eda el Custodio, pero ambos sab\u00edamos lo violenta que era Aludra Chertan. Esta vez quiz\u00e1 lo matara. Salimos de la choza y echamos a correr.\n\nCuando llegu\u00e9 a la residencia todav\u00eda aullaban las sirenas. XIX-49-33 no me abri\u00f3 la puerta hasta despu\u00e9s de llevar un buen rato llamando y gritando mi n\u00famero por encima del estruendo. Tras comprobar que era una humana, tir\u00f3 de m\u00ed y cerr\u00f3 r\u00e1pidamente la puerta, y jur\u00f3 que la pr\u00f3xima vez no me dejar\u00eda entrar si era tan lenta para obedecer las \u00f3rdenes m\u00e1s b\u00e1sicas. La dej\u00e9 echando los cerrojos, muy agitada y con dedos temblorosos.\n\nLas sirenas cesaron cuando llegu\u00e9 al patio. Esa vez los emim no hab\u00edan logrado entrar en la ciudad. Me recog\u00ed el pelo con las manos y trat\u00e9 de recobrar el aliento. Al cabo de un minuto me obligu\u00e9 a mirar hacia el umbral y la escalera de caracol. Ten\u00eda que hacerlo. Esper\u00e9 un momento m\u00e1s para serenarme, y entonces sub\u00ed a la torre, su torre. Se me pon\u00eda la piel de gallina solo de pensar en dormir en la misma habitaci\u00f3n que \u00e9l; en compartir su espacio, su calor, el aire que respiraba.\n\nCuando llegu\u00e9, la llave estaba en la puerta. La hice girar y entr\u00e9 con sigilo.\n\nNo con suficiente sigilo. En cuanto traspuse el umbral, mi guardi\u00e1n se levant\u00f3. Echaba fuego por los ojos.\n\n\u2014\u00bfD\u00f3nde estabas?\n\n\u2014Fuera \u2014contest\u00e9 levantando una endeble barrera mental.\n\n\u2014Ten\u00edas que volver aqu\u00ed si sonaba la sirena.\n\n\u2014Cre\u00ed que te refer\u00edas a Magdalen, y no a esta habitaci\u00f3n en concreto. Deber\u00edas ser m\u00e1s claro.\n\nO\u00ed la insolencia de mi propia voz. Los ojos del Custodio se oscurecieron, y sus labios formaron una l\u00ednea recta y apretada.\n\n\u2014Te dirigir\u00e1s a m\u00ed con el respeto debido \u2014dijo\u2014, o no te permitir\u00e9 salir de esta habitaci\u00f3n.\n\n\u2014No has hecho nada para ganarte mi respeto.\n\nLe sostuve la mirada, y \u00e9l a m\u00ed. Como no me mov\u00ed ni desvi\u00e9 la mirada, \u00e9l pas\u00f3 a mi lado y cerr\u00f3 de un portazo. No me inmut\u00e9.\n\n\u2014Si oyes la sirena \u2014dijo\u2014, deja lo que est\u00e9s haciendo y vuelves inmediatamente a esta habitaci\u00f3n. \u00bfEntiendes?\n\nMe qued\u00e9 mir\u00e1ndolo sin contestar. El Custodio se agach\u00f3 hasta que su cara qued\u00f3 al nivel de la m\u00eda.\n\n\u2014\u00bfTengo que repet\u00edrtelo?\n\n\u2014Preferir\u00eda que no \u2014respond\u00ed.\n\nEstaba segura de que me iba a pegar. Nadie pod\u00eda hablarle as\u00ed a un refa. Pero lo \u00fanico que hizo fue erguirse cuan alto era.\n\n\u2014Ma\u00f1ana iniciaremos tu entrenamiento \u2014anunci\u00f3\u2014. Espero que est\u00e9s preparada cuando suene la campana.\n\n\u2014\u00bfQu\u00e9 entrenamiento?\n\n\u2014Para tu siguiente casaca.\n\n\u2014No la quiero.\n\n\u2014Entonces tendr\u00e1s que hacerte actriz. Tendr\u00e1s que pasarte el resto de la vida siendo objeto de las burlas y los insultos de los casacas rojas. \u2014Me mir\u00f3 de arriba abajo\u2014. \u00bfQuieres ser una bufona? \u00bfUna payasa?\n\n\u2014No.\n\n\u2014En ese caso ser\u00e1 mejor que me obedezcas.\n\nSe me contrajo la garganta. Pese a lo que llegaba a odiar a aquel ser, ten\u00eda motivos para temerlo. Record\u00e9 su expresi\u00f3n de crueldad en la capilla, cuando se hab\u00eda plantado ante m\u00ed y me hab\u00eda absorbido el aura. Para los videntes, el aura era tan vital como la sangre o el agua. Sin aura, sufrir\u00eda un choque espiritual y acabar\u00eda muerta o loca, deambulando sin conexi\u00f3n alguna con el \u00e9ter.\n\nFue hasta las cortinas y las descorri\u00f3; la portezuela que hab\u00eda detr\u00e1s estaba entreabierta.\n\n\u2014Los amaur\u00f3ticos han vaciado el piso de arriba para ti. A menos que te ordene otra cosa, tienes que quedarte all\u00ed todo el tiempo. \u2014Hizo una pausa\u2014. Tambi\u00e9n debes saber que est\u00e1 prohibido que tengamos cualquier contacto f\u00edsico directo, excepto durante el entrenamiento. Ni siquiera con guantes.\n\n\u2014Entonces, si te viera entrar herido en esta habitaci\u00f3n \u2014dije\u2014, \u00bftendr\u00eda que dejarte morir?\n\n\u2014S\u00ed.\n\n\u00abMentiroso.\u00bb No pude controlarme y le solt\u00e9:\n\n\u2014Esa es una orden que obedecer\u00e9 con mucho gusto.\n\nEl Custodio se qued\u00f3 mir\u00e1ndome, impasible. Casi me enfureci\u00f3 ver lo poco que le afectaba mi falta de respeto. Ten\u00eda que hacer algo que lo hiciera saltar. Se limit\u00f3 a meter la mano en el caj\u00f3n y a sacar mis pastillas.\n\n\u2014T\u00f3matelas.\n\nSab\u00eda que no ten\u00eda sentido discutir. Las cog\u00ed.\n\n\u2014B\u00e9bete esto \u2014a\u00f1adi\u00f3, y me dio un vaso\u2014. Ve a tus habitaciones. Conviene que ma\u00f1ana est\u00e9s descansada.\n\nCerr\u00e9 la mano derecha hasta formar un pu\u00f1o. Estaba harta de sus \u00f3rdenes. Deb\u00ed dejar que se desangrara. \u00bfPor qu\u00e9 demonios le hab\u00eda vendado la herida? \u00bfQu\u00e9 clase de delincuente era yo, curando a mis enemigos? Jax se habr\u00eda muerto de risa si me hubiera visto. \u00abTesoro \u2014me habr\u00eda dicho\u2014, no tienes lo que hay que tener.\u00bb Y quiz\u00e1 tuviera raz\u00f3n. No lo ten\u00eda, todav\u00eda.\n\nPas\u00e9 al lado del Custodio evitando rozarlo siquiera. Antes de entrar en el oscuro pasillo, lo vi mirarme. Cerr\u00f3 la puerta con llave detr\u00e1s de m\u00ed.\n\nOtra escalera de caracol me condujo al piso superior de la torre. Ech\u00e9 una ojeada a mi nueva morada: una habitaci\u00f3n espaciosa y vac\u00eda. Me record\u00f3 a la penitenciar\u00eda, con el suelo h\u00famedo y las ventanas con barrotes. La l\u00e1mpara de parafina que ard\u00eda en la repisa de la ventana proporcionaba una luz y un calor escasos. Junto a la l\u00e1mpara estaba la cama, con barandilla y un colch\u00f3n lleno de bultos. Las s\u00e1banas eran penosas comparadas con los exquisitos mantos de terciopelo de la cama con dosel del Custodio; de hecho, toda la habitaci\u00f3n ol\u00eda a inferioridad humana; pero cualquier cosa era mejor que compartir habitaci\u00f3n con mi guardi\u00e1n.\n\nRevis\u00e9 cada rinc\u00f3n y cada recoveco de la habitaci\u00f3n, como hab\u00eda hecho con la del piso de abajo. No hab\u00eda salida, por supuesto, pero s\u00ed cuarto de ba\u00f1o. Dentro hab\u00eda un v\u00e1ter, un lavamanos y algunos productos de higiene.\n\nPens\u00e9 en Julian, que deb\u00eda de estar en su s\u00f3tano oscuro, y en Liss, temblando en su choza. Liss no ten\u00eda cama. No ten\u00eda nada. Mi habitaci\u00f3n no era bonita, pero era mucho m\u00e1s c\u00e1lida y limpia que cualquier vivienda del Poblado. Y m\u00e1s segura. Ten\u00eda paredes de piedra que me proteg\u00edan de los emim. Liss solo contaba con cortinas deshilachadas.\n\nComo no me hab\u00edan dado ropa de dormir, me qued\u00e9 en ropa interior. No hab\u00eda espejo, pero sab\u00eda que estaba adelgazando. El estr\u00e9s, la intoxicaci\u00f3n con flux y la falta de alimentos nutritivos empezaban a pasarme factura. Baj\u00e9 la llama de la l\u00e1mpara y me met\u00ed entre las s\u00e1banas.\n\nNo me sent\u00eda cansada, pero al poco rato estaba dormitando. Y pensando. Pensaba en el pasado, en aquella serie de d\u00edas extra\u00f1os que me hab\u00edan conducido hasta all\u00ed. Record\u00e9 el d\u00eda que hab\u00eda conocido a Nick. Fue \u00e9l quien nos puso a Jaxon y a m\u00ed en contacto. Nick, el hombre que me hab\u00eda salvado la vida. Cuando yo ten\u00eda nueve a\u00f1os, poco despu\u00e9s de llegar a Inglaterra, mi padre y yo salimos de Londres y fuimos al sur en uno de sus \u00abviajes de negocios\u00bb. Hab\u00eda tenido que anotar nuestros nombres en una lista de espera para que nos permitieran salir de la ciudadela. Tras meses de espera, recibimos por fin el permiso para visitar a Giselle, una vieja amiga de mi padre. Giselle viv\u00eda al final de una cuesta empedrada, en una casa de color rosa con un tejado que sobresal\u00eda por encima de las ventanas. El terreno circundante me record\u00f3 a Irlanda: una belleza abierta y suntuosa, una naturaleza virgen y agreste; eso que Scion hab\u00eda destruido. Al anochecer, cuando mi padre no me ve\u00eda, yo trepaba al tejado y me acurrucaba contra la alta chimenea de ladrillo. Desde all\u00ed contemplaba el cielo y los frondosos bosques de las colinas, y recordaba a mi primo Finn y a los otros fantasmas de Irlanda, y echaba tanto de menos a mis abuelos que me dol\u00eda el coraz\u00f3n. Nunca hab\u00eda entendido por qu\u00e9 no hab\u00edan venido con nosotros.\n\nPero lo que yo quer\u00eda era ver el mar, el prodigioso mar, el camino reluciente que se extend\u00eda hasta las tierras libres. Era al otro lado del mar donde Irlanda me esperaba para llevarme a casa, a la pradera cenicienta, al \u00e1rbol partido de la canci\u00f3n de los rebeldes. Mi padre me prometi\u00f3 que ir\u00edamos a verlo, pero estaba demasiado ocupado con Giselle. Siempre se quedaban hablando hasta muy entrada la noche.\n\nYo era demasiado peque\u00f1a para entender qu\u00e9 significaba realmente vivir en aquel pueblo. Los videntes quiz\u00e1 corrieran peligro en la ciudadela, pero no pod\u00edan huir a aquellos idilios campestres. Lejos del Arconte, los amaur\u00f3ticos de pueblo se pon\u00edan muy nerviosos. Las sospechas de antinaturalidad eran una constante en aquellas comunidades tan cerradas. Ten\u00edan la costumbre de vigilarse unos a otros, atentos por si descubr\u00edan una bola de cristal o una piedra de adivinaci\u00f3n, dispuestos a llamar al puesto de avanzada de Scion m\u00e1s cercano, o a tomarse la justicia por su mano. Un aut\u00e9ntico clarividente no habr\u00eda durado ni un d\u00eda all\u00ed. Y aunque hubiera durado, no hab\u00eda trabajo. Hab\u00eda que cultivar la tierra, pero no hac\u00edan falta muchas manos porque ten\u00edan m\u00e1quinas para ocuparse de eso. Los videntes solo pod\u00edan ganarse bien la vida en la ciudadela.\n\nNo me gustaba alejarme de la casa, y menos a\u00fan sin mi padre. La gente hablaba demasiado, miraba demasiado, y Giselle les hablaba y los miraba sin tapujos. Era una mujer severa, delgada y de facciones duras, con un anillo en cada dedo y unas venas largas, como cuerdas, muy marcadas en los brazos y el cuello. No me ca\u00eda bien. Pero un d\u00eda, desde el tejado, ote\u00e9 un remanso de paz: un prado de amapolas, un charco rojo bajo el cielo de hierro.\n\nTodos los d\u00edas, cuando mi padre cre\u00eda que yo estaba arriba jugando, iba a aquel campo y me pasaba horas leyendo con mi nueva tableta de datos, mientras las amapolas cabeceaban a mi alrededor. Fue en ese campo donde tuve mi primer encuentro real con el mundo de los esp\u00edritus. Con el \u00e9ter. Yo todav\u00eda no sab\u00eda que era clarividente. La antinaturalidad todav\u00eda era un cuento para una ni\u00f1a de nueve a\u00f1os, un coco sin facciones definidas. Todav\u00eda ten\u00eda que entender aquel sitio. Solo sab\u00eda lo que me hab\u00eda contado Finn: que a la gente mala del otro lado del mar no les gustaban las ni\u00f1as como yo. Ya no estaba a salvo.\n\nAquel d\u00eda descubr\u00ed lo que Finn hab\u00eda querido decir. Cuando entr\u00e9 en el prado, percib\u00ed la enojada presencia de una mujer. No la vi, pero la sent\u00ed. La sent\u00ed en las amapolas y en el viento. La sent\u00ed en la tierra y en el aire. Estir\u00e9 un brazo con la esperanza de entender qu\u00e9 era.\n\nY de pronto me vi en el suelo. Sangrando. Fue mi primer encuentro con un duende, un esp\u00edritu furioso que pod\u00eda entrar en el mundo corp\u00f3reo.\n\nAl poco apareci\u00f3 mi salvador. Un hombre joven, alto y fuerte, con cabello muy rubio y un rostro amable. Me pregunt\u00f3 c\u00f3mo me llamaba. Le contest\u00e9 balbuceando. Cuando vio el brazo herido me envolvi\u00f3 en su abrigo y me llev\u00f3 a su coche. Llevaba la palabra \u00abScionmed\u00bb bordada en la camisa. Vi que sacaba una aguja y sent\u00ed p\u00e1nico. \u00abMe llamo Nick \u2014dijo\u2014. Est\u00e1s a salvo, Paige.\u00bb\n\nLa aguja me atraves\u00f3 la piel. Me doli\u00f3, pero no llor\u00e9. Poco a poco todo se oscureci\u00f3.\n\nY so\u00f1\u00e9. So\u00f1\u00e9 con amapolas que brotaban con dificultad del polvo. Nunca hab\u00eda so\u00f1ado en color y, en cambio, ahora lo \u00fanico que ve\u00eda eran las flores rojas y el sol del atardecer. Me proteg\u00edan, desprendi\u00e9ndose de sus p\u00e9talos y cubriendo mi afiebrado cuerpo. Despert\u00e9 en una cama con s\u00e1banas blancas. Llevaba el brazo vendado. El dolor hab\u00eda desaparecido.\n\nEl hombre rubio estaba a mi lado. Recuerdo su sonrisa; no era m\u00e1s que un esbozo, pero me hizo sonre\u00edr. Parec\u00eda un pr\u00edncipe.\n\n\u2014Hola, Paige \u2014dijo. Le pregunt\u00e9 d\u00f3nde estaba\u2014. Est\u00e1s en un hospital. Yo soy tu m\u00e9dico.\n\n\u2014Pareces demasiado joven para ser m\u00e9dico \u2014dije. \u00abY no das suficiente miedo\u00bb, pens\u00e9\u2014. \u00bfCu\u00e1ntos a\u00f1os tienes?\n\n\u2014Dieciocho. Todav\u00eda estoy estudiando.\n\n\u2014No me habr\u00e1s hecho una chapuza al coserme el brazo, \u00bfverdad?\n\n\u2014Te he cosido la herida lo mejor que he podido \u2014dijo riendo\u2014. Ya me dir\u00e1s qu\u00e9 te parece.\n\nMe coment\u00f3 que hab\u00eda avisado a mi padre, y que ya estaba en camino. Le dije que estaba mareada; contest\u00f3 que era normal, y que tendr\u00eda que descansar para que se me pasara. Todav\u00eda no pod\u00eda comer, pero \u00e9l me conseguir\u00eda algo bueno para cenar. Se qued\u00f3 el resto del d\u00eda conmigo, y solo me dej\u00f3 para ir a buscar unos bocadillos y un zumo de manzana a la cafeter\u00eda del hospital. Mi padre me hab\u00eda ense\u00f1ado que no deb\u00eda hablar con desconocidos y, sin embargo, no le ten\u00eda miedo a aquel chico tan amable y educado.\n\nEl doctor Nicklas Nyg\u00e5rd, al que hab\u00edan trasladado de la ciudadela Scion Estocolmo, me mantuvo con vida aquella noche. Me ayud\u00f3 a superar el choque que me hab\u00eda producido convertirme en clarividente. De no ser por \u00e9l quiz\u00e1 no habr\u00eda podido soportarlo.\n\nMi padre me llev\u00f3 a casa unos d\u00edas m\u00e1s tarde. Conoc\u00eda a Nick porque hab\u00edan coincidido en un congreso de medicina. Nick estaba haciendo pr\u00e1cticas en el pueblo antes de ocupar una plaza fija en la SciOECI. Nunca me dijo qu\u00e9 hac\u00eda en el campo de amapolas. Mientras mi padre me esperaba en el coche, Nick se arrodill\u00f3 delante de m\u00ed y me tom\u00f3 las manos. Recuerdo que pens\u00e9 que era guap\u00edsimo, y que sus cejas formaban un arco perfecto sobre sus preciosos ojos verde invierno.\n\n\u2014Paige \u2014me dijo en voz baja\u2014, esc\u00fachame. Lo que voy a decirte es muy importante. Le he dicho a tu padre que te atac\u00f3 un perro.\n\n\u2014Pero fue una mujer.\n\n\u2014S\u00ed, pero esa mujer era invisible, _s\u00f6tnos_. Hay adultos que no saben nada de las cosas invisibles.\n\n\u2014Pero t\u00fa s\u00ed \u2014dije, segura de su sabidur\u00eda.\n\n\u2014Yo s\u00ed. Pero no quiero que los otros adultos se r\u00edan de m\u00ed, y por eso no se lo cuento. \u2014Me acarici\u00f3 la mejilla\u2014. No debes hablarle a nadie de ella, Paige. Nunca. Ser\u00e1 nuestro secreto. \u00bfMe lo prometes?\n\nDije que s\u00ed con la cabeza. Habr\u00eda podido prometerle cualquier cosa, pues me hab\u00eda salvado la vida. Me qued\u00e9 mir\u00e1ndolo por la ventanilla cuando mi padre me meti\u00f3 en el coche para volver a la ciudadela. Nick levant\u00f3 una mano y me dijo adi\u00f3s. Segu\u00ed mir\u00e1ndolo hasta que doblamos una esquina.\n\nTodav\u00eda ten\u00eda cicatrices del ataque. Formaban un racimo en el centro de la palma de mi mano izquierda. El esp\u00edritu me hab\u00eda hecho otros cortes en el brazo, hasta el codo, pero los de la mano fueron los que me dejaron cicatriz.\n\nCumpl\u00ed mi promesa. Durante siete a\u00f1os jam\u00e1s dije ni una palabra de lo ocurrido. Guard\u00e9 el secreto de Nick en mi coraz\u00f3n, como una flor nocturna, y \u00fanicamente pensaba en \u00e9l cuando estaba sola. Nick sab\u00eda la verdad. Nick ten\u00eda la clave. Durante todo ese tiempo me pregunt\u00e9 qu\u00e9 habr\u00eda sido de Nick, y si se habr\u00eda acordado alguna vez de aquella ni\u00f1a irlandesa a la que sac\u00f3 del campo de amapolas. Y tras siete largos a\u00f1os tuve mi recompensa: Nick volvi\u00f3 a encontrarme.\n\nOjal\u00e1 pudiera encontrarme ahora.\n\nNo se o\u00eda nada en el piso de abajo. A medida que avanzaban las horas, aguzaba el o\u00eddo por si o\u00eda pasos, o la d\u00e9bil melod\u00eda del gram\u00f3fono, pero lo \u00fanico que o\u00eda era aquel silencio impenetrable.\n\nPas\u00e9 el resto del d\u00eda sumida en un sue\u00f1o irregular. Ten\u00eda fiebre, provocada por los restos del \u00faltimo ataque con flux. De vez en cuando me despertaba sobresaltada, acosada por im\u00e1genes del pasado. \u00bfLlevaba antes otra ropa que no fueran esos blusones, esas botas? \u00bfHab\u00eda conocido un mundo donde no hab\u00eda esp\u00edritus ni muertos errantes? \u00bfUn mundo sin emim, sin refa\u00edtas?\n\nMe despertaron unos golpes en la puerta. Apenas tuve tiempo de taparme con la s\u00e1bana cuando el Custodio entr\u00f3 en la habitaci\u00f3n.\n\n\u2014Pronto sonar\u00e1 la campana. \u2014Dej\u00f3 un uniforme limpio a los pies de la cama\u2014. V\u00edstete.\n\nMe qued\u00e9 mir\u00e1ndolo en silencio. \u00c9l me sostuvo la mirada un momento antes de salir y cerrar la puerta. No pod\u00eda hacer nada. Me levant\u00e9, me recog\u00ed el pelo en un mo\u00f1o y me lav\u00e9 con agua helada. Me puse el uniforme y me abroch\u00e9 el chaleco hasta la barbilla. Mi pierna, por lo visto, ya estaba curada.\n\nEncontr\u00e9 al Custodio hojeando una novela en su habitaci\u00f3n. Era un ejemplar de _Frankenstein_ con las tapas sucias de polvo. Scion no permit\u00eda esa clase de literatura fant\u00e1stica. Nada donde hubiera monstruos o fantasmas. Nada que hiciera referencia a la antinaturalidad. Not\u00e9 un cosquilleo en los dedos, ansiosos por agarrar ese libro y pasar las p\u00e1ginas. Lo hab\u00eda visto en la estanter\u00eda de Jaxon, pero nunca hab\u00eda encontrado tiempo para leerlo. El Custodio dej\u00f3 el libro a un lado y se levant\u00f3.\n\n\u2014\u00bfEst\u00e1s preparada?\n\n\u2014S\u00ed.\n\n\u2014Muy bien. \u2014Hizo una pausa y a\u00f1adi\u00f3\u2014: Dime, Paige, \u00bfc\u00f3mo es tu onirosaje?\n\nMe pill\u00f3 por sorpresa. Entre los videntes se consideraba de mala educaci\u00f3n hacer una pregunta tan directa.\n\n\u2014Un campo de flores rojas.\n\n\u2014\u00bfQu\u00e9 flores?\n\n\u2014Amapolas.\n\nNo dijo nada. Cogi\u00f3 sus guantes, se los puso y salimos de la habitaci\u00f3n. Todav\u00eda no hab\u00eda sonado la campanada nocturna, pero el portero nos dej\u00f3 pasar sin hacer preguntas. Nadie le hac\u00eda preguntas a Arcturus Mesarthim.\n\nLlevaba tiempo sin ver la luz del d\u00eda. El sol empezaba a ponerse, y suavizaba los bordes de los edificios. Sheol reluc\u00eda en medio de la neblina. Yo cre\u00eda que \u00edbamos a entrenar en un sitio cubierto, pero el Custodio me llev\u00f3 hacia el norte, m\u00e1s all\u00e1 de la Casa Amaur\u00f3tica, por territorio desconocido.\n\nLos edificios de las afueras de la ciudad estaban abandonados. Estaban en ruinas, con los cristales de las ventanas rotos; algunas paredes y techos parec\u00edan quemados. Quiz\u00e1 fuera cierto que hab\u00eda habido un gran incendio. Pasamos una calle de casas apretujadas unas contra otras. Era un pueblo fantasma. All\u00ed no hab\u00eda ni rastro de seres vivos. Percib\u00ed la presencia de esp\u00edritus, esp\u00edritus resentidos que quer\u00edan recuperar las casas que hab\u00edan perdido. Algunos eran duendes, bastante d\u00e9biles. Yo no me fiaba, pero el Custodio no parec\u00eda asustado. Ninguno se le acerc\u00f3.\n\nLlegamos al final de la ciudad. Echaba nubes de vaho por la boca al respirar. Ante m\u00ed se extend\u00eda una pradera hasta donde alcanzaba la vista. La hierba llevaba tiempo seca, y el suelo brillaba, cubierto de escarcha. Era raro, porque est\u00e1bamos a principios de primavera. Hab\u00edan cercado la pradera con una valla de unos diez metros de alto, coronada con alambre de espino. Al otro lado de la valla hab\u00eda \u00e1rboles, tambi\u00e9n recubiertos de una fina capa de escarcha. Crec\u00edan alrededor de la pradera y me imped\u00edan ver lo que hab\u00eda m\u00e1s all\u00e1. Un letrero oxidado rezaba: PUERTO PRADERA. SOLO ENTRENAMIENTO. AUTORIZADO EL USO DE PODERES MORT\u00cdFEROS. Junto a la verja estaba el poder mort\u00edfero mismo: un refa var\u00f3n.\n\nLlevaba el rubio cabello recogido en una coleta. A su lado hab\u00eda una figura sucia y delgada con la cabeza afeitada: Ivy, la palmista. Vest\u00eda un blus\u00f3n amarillo, el distintivo de los cobardes; un desgarr\u00f3n en el cuello le dejaba un hombro huesudo expuesto al fr\u00edo. Le vi la marca: XX-59-24. Custodio avanz\u00f3, y yo lo segu\u00ed. Al vernos, el guardi\u00e1n de Ivy hizo una reverencia.\n\n\u2014Hete aqu\u00ed a la concubina real \u2014dijo\u2014. \u00bfQu\u00e9 te trae por Puerto Pradera?\n\nAl principio cre\u00ed que me hablaba a m\u00ed. Nunca hab\u00eda o\u00eddo a los refas hablarse con semejante repugnancia. Entonces me di cuenta de que el otro guardi\u00e1n miraba con fijeza y odio al Custodio.\n\n\u2014He venido a entrenar a mi humana. \u2014El Custodio miraba hacia la pradera\u2014. Abre la puerta, Thuban.\n\n\u2014Un poco de paciencia, concubina. \u00bfVa armada?\n\nSe refer\u00eda a m\u00ed. A la humana.\n\n\u2014No \u2014contest\u00f3 el Custodio\u2014. No va armada.\n\n\u2014\u00bfN\u00famero?\n\n\u2014XX-59-40.\n\n\u2014\u00bfEdad?\n\nEl Custodio me mir\u00f3.\n\n\u2014Diecinueve \u2014contest\u00e9.\n\n\u2014\u00bfTiene visi\u00f3n espiritista?\n\n\u2014Esas preguntas son irrelevantes, Thuban. No me gusta que me traten como a un cr\u00edo, y mucho menos que lo haga un cr\u00edo.\n\nThuban se limit\u00f3 a mirarlo. Calcul\u00e9 que deb\u00eda de tener m\u00e1s de veinticinco a\u00f1os; no era ning\u00fan cr\u00edo, desde luego. Ni el rostro de Thuban ni el del Custodio revelaban enojo, pero bastaba con o\u00edrlos.\n\n\u2014Tienes tres horas hasta que Pleione traiga a su reba\u00f1o. \u2014Empuj\u00f3 la verja y la abri\u00f3\u2014. Si 40 intenta huir, le disparar\u00e1n en el acto.\n\n\u2014Y a ti, si vuelves a faltarles el respeto a tus mayores, te desterrar\u00e1n en el acto.\n\n\u2014La soberana de sangre no lo permitir\u00eda.\n\n\u2014No tiene por qu\u00e9 enterarse. Un accidente as\u00ed no es demasiado dif\u00edcil de ocultar. \u2014El Custodio descollaba sobre \u00e9l\u2014. No me asusta tu apellido, Sargas. Soy el consorte de sangre, y pienso ejercer el poder que conlleva mi posici\u00f3n. \u00bfMe he explicado bien, Thuban?\n\nThuban lo mir\u00f3 desde abajo con un ardor azulado en los ojos.\n\n\u2014S\u00ed \u2014respondi\u00f3 en voz baja\u2014, consorte \u00abde sangre\u00bb.\n\nEl Custodio pas\u00f3 a su lado. Yo no supe c\u00f3mo interpretar aquella conversaci\u00f3n, pero fue un gustazo presenciar que un Sargas recib\u00eda una tunda verbal. Traspuse la verja detr\u00e1s del Custodio, y Thuban le arre\u00f3 una bofetada a Ivy. La chica gir\u00f3 la cabeza. No ten\u00eda l\u00e1grimas en los ojos, pero su cara estaba p\u00e1lida e hinchada, y se la ve\u00eda m\u00e1s delgada que antes. Ten\u00eda los brazos manchados de sangre y suciedad. Deduje que la manten\u00edan encerrada sobre sus propios excrementos. Record\u00e9 que Seb tambi\u00e9n me hab\u00eda mirado as\u00ed, como si toda la esperanza del mundo se hubiera desmoronado.\n\nEstaba decidida a sacarle partido a aquella sesi\u00f3n de entrenamiento. Quer\u00eda hacerlo por Seb, por Ivy y por todos los que vendr\u00edan despu\u00e9s.\n\nPuerto Pradera era inmenso. El Custodio caminaba por \u00e9l a grandes zancadas, tan largas que me costaba seguirlo. Yo avanzaba con dificultad, y al mismo tiempo intentaba calcular las dimensiones de la pradera. Era dif\u00edcil, porque la luz disminu\u00eda; pero alcanc\u00e9 a distinguir, a ambos lados, unas vallas de alambre fino entretejido y recubierto de hielo que divid\u00edan el terreno en grandes ruedos. La parte superior de los postes estaba curvada; algunos ten\u00edan unos gruesos soportes de los que colgaban faroles. Una torre de vigilancia se erig\u00eda en el lado de poniente, y dentro pude ver la silueta de un humano o un refa.\n\nPas\u00e9 al lado de una charca poco profunda. La superficie, helada, era lisa como un espejo, perfecta para emplearse como medio de adivinaci\u00f3n. Pens\u00e9 que todo en aquella pradera era perfecto para el combate espiritista. El suelo era s\u00f3lido; el aire, di\u00e1fano y fresco. Y hab\u00eda esp\u00edritus. Los notaba alrededor de m\u00ed, por todas partes. Me pregunt\u00e9 qu\u00e9 clase de alambrada era la que cercaba la pradera. \u00bfHabr\u00edan ideado la manera de retener a los esp\u00edritus?\n\nNo. A veces los esp\u00edritus pod\u00edan alterar el mundo de la carne, pero no estaban sometidos a restricciones f\u00edsicas. Solo los vinculadores pod\u00edan retenerlos. Su orden (el quinto orden) pod\u00eda forzar los l\u00edmites entre el mundo de la carne y el \u00e9ter.\n\n\u2014Las vallas no est\u00e1n electrificadas \u2014dijo el Custodio, que me hab\u00eda visto observ\u00e1ndolas\u2014, sino cargadas con energ\u00eda et\u00e9rea.\n\n\u2014\u00bfC\u00f3mo puede ser?\n\n\u2014Bater\u00edas et\u00e9reas. Una fusi\u00f3n de pericia refa\u00edta y humana, aplicada por primera vez en 2045. Vuestros cient\u00edficos trabajan en la tecnolog\u00eda h\u00edbrida desde principios del siglo XX. Nosotros solo sustituimos la energ\u00eda qu\u00edmica de una bater\u00eda por un duende cautivo, un esp\u00edritu que puede interactuar con el mundo corp\u00f3reo. As\u00ed se crea un campo de repulsi\u00f3n.\n\n\u2014Pero los duendes pueden huir de sus v\u00ednculos \u2014razon\u00e9\u2014. \u00bfC\u00f3mo los captur\u00e1is?\n\n\u2014Utilizamos un duende dispuesto a colaborar, por supuesto.\n\nMe qued\u00e9 perpleja. La palabra \u00abduende\u00bb y la locuci\u00f3n \u00abdispuesto a colaborar\u00bb eran tan opuestas como \u00abguerra\u00bb y \u00abpaz\u00bb.\n\n\u2014Nuestro asesoramiento tambi\u00e9n condujo a la invenci\u00f3n del Fluxion 14 y la Tecnolog\u00eda de Detecci\u00f3n Radiest\u00e9sica \u2014prosigui\u00f3\u2014. Esta \u00faltima todav\u00eda est\u00e1 en fase experimental. Seg\u00fan los \u00faltimos informes que hemos recibido, Scion casi ha conseguido perfeccionarla.\n\nApret\u00e9 un pu\u00f1o. Claro, los refa\u00edtas eran los responsables de la TDR. Dani nunca hab\u00eda entendido de d\u00f3nde la hab\u00edan sacado.\n\nAl cabo de un rato el Custodio se detuvo. Hab\u00edamos llegado a un \u00f3valo de hormig\u00f3n de tres metros de ancho. Cerca se encendi\u00f3 una l\u00e1mpara de gas.\n\n\u2014Empecemos \u2014dijo el Custodio.\n\nEsper\u00e9.\n\nSin previo aviso, el Custodio hizo como si fuera a darme un pu\u00f1etazo en la cara. Lo esquiv\u00e9.\n\nCuando fue a darme con el otro pu\u00f1o, desvi\u00e9 el golpe con un brazo.\n\n\u2014Otra vez.\n\nCada vez me atacaba m\u00e1s deprisa, oblig\u00e1ndome a defenderme desde todos los \u00e1ngulos. Par\u00e9 todos los golpes con las manos abiertas.\n\n\u2014Aprendiste a pelear en las calles.\n\n\u2014Puede ser \u2014dije.\n\n\u2014Otra vez. Intenta detenerme.\n\nEsta vez dirigi\u00f3 ambas manos hacia mi escote, como si fuera a agarrarme por el cuello. Un ratero hab\u00eda intentado eso conmigo una vez. Torc\u00ed el torso hacia la izquierda y llev\u00e9 el brazo derecho en la misma direcci\u00f3n, alej\u00e1ndole las manos de mi cuello. Not\u00e9 la fuerza de esas manos; pero el Custodio me solt\u00f3. Levant\u00e9 un codo hacia su mejilla, un movimiento con el que hab\u00eda conseguido derribar al ratero. El Custodio me estaba dejando ganar.\n\n\u2014Excelente \u2014dijo. Dio unos pasos hacia atr\u00e1s\u2014. Pocos humanos llegan aqu\u00ed preparados para formar parte de un batall\u00f3n penal. Les llevas ventaja a la mayor\u00eda, pero con un emite no podr\u00e1s utilizar esas t\u00e1cticas tan simples. Tu gran baza es tu capacidad para alterar el \u00e9ter.\n\nVi un destello plateado. El Custodio ten\u00eda una daga en la mano. Se me tensaron los m\u00fasculos.\n\n\u2014Por lo que he podido comprobar, tu don se activa con el peligro. \u2014Me apunt\u00f3 en el pecho con la daga. La punta casi me tocaba\u2014. Demu\u00e9stramelo.\n\n\u2014No s\u00e9 c\u00f3mo.\n\n\u2014Ya veo.\n\nCon una sacudida de la mu\u00f1eca, subi\u00f3 la daga hasta mi cuello. Not\u00e9 una descarga de adrenalina. El Custodio se inclin\u00f3 hacia m\u00ed.\n\n\u2014Esta daga se ha utilizado para derramar sangre humana \u2014dijo en voz muy baja\u2014. Sangre como la de tu amigo Sebastian.\n\nMe puse a temblar.\n\n\u2014Y quiere m\u00e1s sangre. \u2014La hoja de la daga se desliz\u00f3 por mi cuello\u2014. Nunca ha probado la sangre de un So\u00f1ador.\n\n\u2014No me das miedo. \u2014El temblor de mi voz me delataba\u2014. No me toques.\n\nPero me toc\u00f3. La hoja de la daga ascendi\u00f3 por mi cuello y mi barbilla y me roz\u00f3 los labios. Levant\u00e9 los pu\u00f1os y le apart\u00e9 la mano. El Custodio solt\u00f3 la daga, me asi\u00f3 las mu\u00f1ecas con una mano y me las sujet\u00f3 contra el suelo de hormig\u00f3n. Ten\u00eda una fuerza asombrosa: yo no pod\u00eda mover ni un solo m\u00fasculo.\n\n\u2014Me pregunto... \u2014dijo mientras me levantaba la barbilla con la punta de la daga\u2014. Si te corto el cuello, \u00bfcu\u00e1nto tardar\u00e1s en morir?\n\n\u2014No te atrever\u00e1s \u2014dije, desafiante.\n\n\u2014Ah, \u00bfno?\n\nIntent\u00e9 darle con la rodilla en la entrepierna, pero \u00e9l me agarr\u00f3 el muslo y me baj\u00f3 la pierna. Todav\u00eda notaba debilidad en ella; no le cost\u00f3 mucho. Estaba consiguiendo hacerme parecer fr\u00e1gil. Logr\u00e9 soltar una mano, pero \u00e9l me retorci\u00f3 el brazo detr\u00e1s de la espalda. No tanto como para hacerme da\u00f1o, pero s\u00ed lo suficiente para inmovilizarme.\n\n\u2014As\u00ed llevas las de perder \u2014me dijo al o\u00eddo\u2014. Emplea tu fuerza.\n\n\u00bfAcaso aquel ser no ten\u00eda ninguna debilidad? Pens\u00e9 en todos los puntos vulnerables de los humanos: ojos, ri\u00f1ones, plexo solar, nariz, entrepierna... Nada de todo eso estaba a mi alcance. Iba a tener que soltarme y correr. Ech\u00e9 el peso del cuerpo hacia atr\u00e1s, entre las piernas del Custodio, y con un solo movimiento me puse en pie. Aprovech\u00e9 el instante que tard\u00f3 \u00e9l en levantarse para echar a correr por la pradera. Si me quer\u00eda, tendr\u00eda que venir a buscarme.\n\nEra imposible huir. El Custodio me estaba alcanzando. Record\u00e9 mis sesiones de entrenamiento con Nick y cambi\u00e9 de direcci\u00f3n. Segu\u00ed corriendo, en la oscuridad, alej\u00e1ndome de la torre de vigilancia. En una valla como aquella ten\u00eda que haber alg\u00fan punto d\u00e9bil, un hueco en la alambrada por el que pudiera colarme. Luego tendr\u00eda que ocuparme de Thuban. Pero ten\u00eda mi esp\u00edritu. Pod\u00eda hacerlo. S\u00ed, pod\u00eda.\n\nPese a tener una agudeza visual excelente, a veces pod\u00eda ser incre\u00edblemente cegata. Al cabo de un minuto me hab\u00eda perdido. Lejos del \u00f3valo de hormig\u00f3n y las l\u00e1mparas de gas, iba dando traspi\u00e9s por la extensa pradera. Y el Custodio estaba all\u00ed, persigui\u00e9ndome. Corr\u00ed hacia una de las l\u00e1mparas. Mi sexto sentido se estremeci\u00f3 a medida que me acercaba a la valla. Para cuando llegu\u00e9 a unos dos metros, ten\u00eda n\u00e1useas y notaba las extremidades flojas y torpes.\n\nPero ten\u00eda que intentarlo. Agarr\u00e9 el alambre helado.\n\nNo puedo describir con precisi\u00f3n la sensaci\u00f3n que se apoder\u00f3 de mi cuerpo. Lo vi todo negro; luego, todo blanco; y, por \u00faltimo, rojo. Se me puso la piel de gallina. Un centenar de recuerdos pasaron ante mis ojos, recuerdos de un grito en un campo de amapolas; y tambi\u00e9n otros, nuevos: los recuerdos del duende. Lo hab\u00edan asesinado. Un estr\u00e9pito ensordecedor sacudi\u00f3 todo mi cuerpo. Mi est\u00f3mago dio una sacudida tremenda. Ca\u00ed al suelo y vomit\u00e9.\n\nDeb\u00ed de quedarme all\u00ed un minuto, atormentada por im\u00e1genes de sangre sobre una alfombra de color crema. Le hab\u00edan disparado con una escopeta. Le hab\u00eda explotado el cr\u00e1neo, roci\u00e1ndolo todo de masa encef\u00e1lica y fragmentos de hueso. Me zumbaban los o\u00eddos. Cuando recobr\u00e9 el conocimiento, me fallaba la coordinaci\u00f3n. Me arrastr\u00e9 por el suelo, parpadeando para ahuyentar aquellas visiones sangrientas. Ten\u00eda una quemadura blancuzca en la palma de la mano. La marca de un duende.\n\nAlgo pas\u00f3 roz\u00e1ndome la oreja. Mir\u00e9 hacia arriba y vi otra torre de vigilancia y al vigilante que hab\u00eda dentro.\n\nEra un dardo de flux.\n\nEl vigilante me dispar\u00f3 un segundo dardo. Me puse en pie como pude y ech\u00e9 a correr hacia el este; pero no tard\u00e9 en llegar ante otra torre, y los disparos me hicieron virar hacia el sur. Cuando vi el \u00f3valo comprend\u00ed que me dirig\u00eda de nuevo hacia el Custodio.\n\nEl siguiente dardo me dio en el hombro. El dolor fue instant\u00e1neo e insoportable. Me arranqu\u00e9 el dardo. Me sangr\u00f3 la herida, y de pronto sent\u00ed n\u00e1useas y me desorient\u00e9. Actu\u00e9 lo bastante deprisa para detener el efecto del f\u00e1rmaco (tardaba unos cinco segundos en autoinyectarse), pero el mensaje era claro: \u00abVuelve al \u00f3valo, o te dispararemos\u00bb. El Custodio estaba esper\u00e1ndome.\n\n\u2014Me alegro de que hayas vuelto.\n\nMe quit\u00e9 el sudor de la frente con el dorso de la mano.\n\n\u2014Veo que no me est\u00e1 permitido correr.\n\n\u2014No. A menos que quieras que te d\u00e9 el blus\u00f3n amarillo que reservamos para los cobardes.\n\nMe lanc\u00e9 contra \u00e9l, cegada por la ira, y le hinqu\u00e9 el hombro en el abdomen. Dada su estatura, no pas\u00f3 nada. Se limit\u00f3 a agarrarme por el blus\u00f3n y a apartarme. Di contra el suelo con el mismo hombro.\n\n\u2014No puedes pelear conmigo con las manos. \u2014Empez\u00f3 a pasearse por el borde del \u00f3valo\u2014. Ni huir de un emite. Eres una onir\u00e1mbula. Tienes el poder de vivir y morir a tu antojo. Saquea mi onirosaje. \u00a1Hazme enloquecer!\n\nUna parte de m\u00ed se desprendi\u00f3. Mi esp\u00edritu sali\u00f3 volando y recorri\u00f3 el espacio que me separaba del Custodio. Raj\u00f3 el anillo exterior de su mente, como un cuchillo cortando una seda tensa. Atraves\u00f3 la parte m\u00e1s oscura de su onirosaje, forzando barreras asombrosamente poderosas, y apunt\u00f3 hacia la lejana mancha de luz de su zona soleada, pero no fue tan f\u00e1cil como lo hab\u00eda sido en el tren. El centro de su onirosaje estaba muy lejos, y mi esp\u00edritu ya estaba siendo expulsado. Como una goma el\u00e1stica tensada en exceso, me vi lanzada de nuevo a mi mente. El peso de mi propio esp\u00edritu me derrib\u00f3. Me golpe\u00e9 la cabeza contra el suelo.\n\nPoco a poco volv\u00ed a ver las l\u00e1mparas de gas. Me incorpor\u00e9 apoy\u00e1ndome en los codos; me dol\u00edan las sienes. El Custodio segu\u00eda de pie. No hab\u00eda conseguido arrodillarlo, como a Aludra, pero hab\u00eda alterado ligeramente su percepci\u00f3n. Se pas\u00f3 una mano por la cara y sacudi\u00f3 la cabeza.\n\n\u2014Bien \u2014dijo\u2014. Muy bien.\n\nMe levant\u00e9. Me temblaban las piernas.\n\n\u2014Intentas cabrearme \u2014dije\u2014. \u00bfPor qu\u00e9?\n\n\u2014Porque por lo visto funciona. \u2014Se\u00f1al\u00f3 la daga\u2014. Hag\u00e1moslo otra vez.\n\nLo mir\u00e9 mientras trataba de recobrar el aliento.\n\n\u2014\u00bfOtra vez?\n\n\u2014Puedes hacerlo mejor. Apenas has rozado mis defensas. Quiero que hagas mella en ellas.\n\n\u2014No puedo hacerlo otra vez. \u2014Ve\u00eda puntos negros\u2014. Esto no funciona as\u00ed.\n\n\u2014\u00bfPor qu\u00e9 no?\n\n\u2014Porque dejo de respirar.\n\n\u2014\u00bfNo sabes bucear?\n\n\u2014\u00bfQu\u00e9?\n\n\u2014El humano medio puede contener la respiraci\u00f3n al menos durante treinta segundos sin sufrir da\u00f1os irreversibles. Es tiempo m\u00e1s que suficiente para atacar otra mente y volver a tu cuerpo.\n\nNunca me lo hab\u00eda planteado as\u00ed. Nick siempre se hab\u00eda asegurado de que tuviera soporte vital para percibir el \u00e9ter desde lejos.\n\n\u2014Tienes que pensar en tu esp\u00edritu como un m\u00fasculo que sale de su posici\u00f3n natural \u2014dijo el Custodio\u2014. Cuanto m\u00e1s lo usas, m\u00e1s fuerte y m\u00e1s r\u00e1pido se vuelve, y mejor soporta tu cuerpo las repercusiones. Podr\u00e1s saltar r\u00e1pidamente de un onirosaje a otro antes de que tu cuerpo d\u00e9 contra el suelo.\n\n\u2014T\u00fa no lo entiendes \u2014dije.\n\n\u2014Ni t\u00fa. Sospecho que el incidente del tren fue la primera vez que entraste en otro onirosaje. \u2014No movi\u00f3 la daga\u2014. Pasea por el m\u00edo. Te desaf\u00edo.\n\nEscudri\u00f1\u00e9 su cara. Estaba invit\u00e1ndome a entrar en su mente, a poner en peligro su cordura.\n\n\u2014En realidad no te importa. Solo me est\u00e1s entrenando \u2014dije. Empezamos a movernos en c\u00edrculo\u2014. Nashira te pidi\u00f3 que me escogieras. S\u00e9 lo que quiere.\n\n\u2014No. Te escog\u00ed yo. Reclam\u00e9 tu instrucci\u00f3n. Y lo \u00faltimo que quiero \u2014dijo avanzando hacia m\u00ed\u2014 es que me averg\u00fcences con tu incompetencia. \u2014Su mirada era dura como la piedra\u2014. Vuelve a atacarme. Y esta vez hazlo bien.\n\n\u2014No. \u2014Lo pondr\u00eda en evidencia. Le har\u00eda pasar verg\u00fcenza. Que se avergonzara de m\u00ed tanto como mi padre\u2014. No pienso matarme solo para que Nashira te conceda una estrella de oro.\n\n\u2014Quieres hacerme da\u00f1o \u2014replic\u00f3, en voz m\u00e1s baja\u2014. Est\u00e1s resentida conmigo. Me odias. \u2014Levant\u00f3 la daga\u2014. Destr\u00fayeme.\n\nAl principio no hice nada. Entonces me acord\u00e9 de las horas que hab\u00eda pasado cur\u00e1ndole el brazo, y de c\u00f3mo me hab\u00eda amenazado. Record\u00e9 que se hab\u00eda apartado y hab\u00eda visto morir a Seb. Volv\u00ed a lanzarle mi esp\u00edritu.\n\nEn el rato que pasamos en la pradera, apenas logr\u00e9 fracturar su onirosaje. No pod\u00eda ir m\u00e1s all\u00e1 de su zona hadal, ni siquiera cuando \u00e9l retiraba casi todas sus defensas, porque su mente era demasiado poderosa. No paraba de provocarme. Me dec\u00eda que era d\u00e9bil, que era pat\u00e9tica, que era una verg\u00fcenza para los clarividentes. Que no le extra\u00f1aba que los humanos solo sirvi\u00e9ramos para ser esclavos. \u00bfQuer\u00eda vivir en una jaula, como un animal? \u00c9l estaba dispuesto a complacerme. En un primer momento las provocaciones surtieron efecto, pero a medida que avanzaba la noche, sus insultos iban enfureci\u00e9ndome menos. Al final eran solo frustrantes; no bastaban para obligar a mi esp\u00edritu a salir.\n\nEntonces el Custodio lanz\u00f3 la daga. Apunt\u00f3 lejos de m\u00ed, pero verla volar bast\u00f3 para que mi esp\u00edritu se soltara. Cada vez que lo hac\u00eda, me ca\u00eda al suelo. Si se me escapaba un pie fuera del \u00f3valo, me lanzaban un dardo de flux. No tard\u00e9 en aprender a prever el silbido del dardo y a agacharme antes de que la aguja diera en el blanco.\n\nConsegu\u00ed salir cinco o seis veces de mi cuerpo. Sent\u00eda como si me abrieran la cabeza cada vez. Al final no pod\u00eda m\u00e1s. Ve\u00eda doble, y ten\u00eda una fuerte migra\u00f1a localizada sobre el ojo izquierdo. Me dobl\u00e9 por la cintura, casi sin aliento. \u00abNo muestres debilidad. No muestres debilidad _.\u00bb_ Se me doblaban las rodillas.\n\nEl Custodio se arrodill\u00f3 delante de m\u00ed y me abraz\u00f3 por la cintura. Intent\u00e9 apartarlo, pero ten\u00eda los brazos flojos como cuerdas.\n\n\u2014Basta \u2014dijo\u2014. No sigas resisti\u00e9ndote.\n\nMe cogi\u00f3 en brazos. Yo nunca hab\u00eda saltado repentinamente, en una sucesi\u00f3n tan r\u00e1pida, y no sab\u00eda si mi cerebro lo soportar\u00eda. Me dol\u00eda el fondo de los ojos. No pod\u00eda mirar el farol.\n\n\u2014Lo has hecho bien. \u2014El Custodio me mir\u00f3\u2014. Pero puedes hacerlo mucho mejor.\n\nNo pude contestarle.\n\n\u2014\u00bfPaige?\n\n\u2014Estoy bien \u2014repuse arrastrando las palabras.\n\nEl Custodio me crey\u00f3. Ech\u00f3 a andar hacia la verja, todav\u00eda conmigo en brazos.\n\nAl cabo de un rato volvi\u00f3 a dejarme en el suelo. Fuimos en silencio hasta la entrada, donde Thuban hab\u00eda abandonado su puesto. Ivy estaba sentada con la espalda apoyada en la valla; ten\u00eda la cabeza entre las manos y le temblaban los hombros. Cuando nos acercamos a la verja, se levant\u00f3 y descorri\u00f3 el cerrojo. El Custodio la mir\u00f3 de pasada.\n\n\u2014Gracias, Ivy.\n\nElla lo mir\u00f3. Ten\u00eda l\u00e1grimas en los ojos. \u00bfCu\u00e1nto tiempo hac\u00eda que nadie la llamaba por su verdadero nombre?\n\nFuimos hacia la ciudad fantasma; el Custodio guardaba silencio. Yo estaba medio dormida. Si hubiera estado con Nick, \u00e9l me habr\u00eda obligado a acostarme y a descansar varias horas, y me habr\u00eda rega\u00f1ado.\n\nHasta que pasamos por delante de la Casa Amaur\u00f3tica, el Custodio no volvi\u00f3 a dirigirme la palabra.\n\n\u2014\u00bfIntentas a menudo sentir el \u00e9ter desde lejos?\n\n\u2014Eso no es asunto tuyo \u2014respond\u00ed.\n\n\u2014En tus ojos hay muerte. Muerte y hielo. \u2014Se volvi\u00f3 y me mir\u00f3\u2014. Es extra\u00f1o, porque tu ira los hace arder.\n\nLo mir\u00e9 y dije:\n\n\u2014Tus ojos tambi\u00e9n cambian.\n\n\u2014\u00bfA qu\u00e9 crees que se debe eso?\n\n\u2014No lo s\u00e9. No s\u00e9 nada de ti.\n\n\u2014Cierto. \u2014El Custodio me mir\u00f3 de arriba abajo\u2014. Ens\u00e9\u00f1ame la mano.\n\nTras un momento de vacilaci\u00f3n, le mostr\u00e9 la mano derecha. La quemadura ten\u00eda un aspecto horrible, un lustre nacarado. El Custodio se sac\u00f3 del bolsillo un vial lleno de l\u00edquido, verti\u00f3 unas gotas sobre su dedo enguantado y me lo aplic\u00f3 en la cicatriz. Vi la herida desaparecer sin dejar rastro. Retir\u00e9 la mano.\n\n\u2014\u00bfC\u00f3mo lo has hecho?\n\n\u2014Se llama amaranto. \u2014Se guard\u00f3 el vial y me mir\u00f3\u2014. Dime, Paige, \u00bfle tienes miedo al \u00e9ter?\n\n\u2014No. \u2014Notaba un cosquilleo en la palma de la mano.\n\n\u2014\u00bfPor qu\u00e9 no?\n\nEra mentira. El \u00e9ter me daba miedo. Cuando forzaba demasiado mi sexto sentido, me arriesgaba a morir, o al menos a padecer alguna lesi\u00f3n cerebral. Jax me hab\u00eda advertido desde el principio de que si trabajaba para \u00e9l, probablemente mi esperanza de vida se reducir\u00eda en unos treinta a\u00f1os, tal vez m\u00e1s. Todo depend\u00eda de la suerte.\n\n\u2014Porque el \u00e9ter es perfecto \u2014dije\u2014. No hay guerras. No hay muerte, porque all\u00ed todo est\u00e1 muerto ya. Y no hay sonido. Solo silencio. Y seguridad.\n\n\u2014En el \u00e9ter no hay nada seguro. Y no est\u00e1 exento de guerras y muerte.\n\nObserv\u00e9 su perfil mientras \u00e9l contemplaba el negro cielo. No expulsaba vaho al respirar, como yo. Pero por un instante, un momento brev\u00edsimo, vislumbr\u00e9 algo humano en su cara. Una expresi\u00f3n pensativa, casi amarga. Entonces el Custodio se volvi\u00f3 hacia m\u00ed, y aquello que me hab\u00eda parecido ver desapareci\u00f3.\n\nPasaba algo raro en el Poblado. Un grupo de casacas rojas que estaban en cuclillas en la calle adoquinada hablaban en voz baja, observados por bufones silenciosos. Mir\u00e9 al Custodio y trat\u00e9 de discernir si aquello le preocupaba. Si le preocupaba, no se le notaba. Fue hacia el grupo, y la mayor\u00eda de los bufones se metieron en sus chozas.\n\n\u2014\u00bfQu\u00e9 pasa?\n\nUn casaca roja levant\u00f3 la vista, vio qui\u00e9n hab\u00eda hablado y se apresur\u00f3 a agachar la cabeza. Ten\u00eda el blus\u00f3n manchado de barro.\n\n\u2014Est\u00e1bamos en el bosque \u2014dijo con voz ronca\u2014. Nos perdimos. Los emim...\n\nInstintivamente, el Custodio se llev\u00f3 una mano al antebrazo.\n\nLos casacas rojas estaban alrededor de un chico de unos diecis\u00e9is a\u00f1os. Le faltaba la mano derecha e iba manchado de sangre. Apret\u00e9 las mand\u00edbulas. Le hab\u00edan arrancado la mano de cuajo, como si se la hubiera pillado en una m\u00e1quina. El Custodio analiz\u00f3 la escena sin revelar ni una pizca de emoci\u00f3n.\n\n\u2014Dices que os hab\u00e9is perdido \u2014dijo\u2014. \u00bfQu\u00e9 guardi\u00e1n iba con vosotros?\n\n\u2014El heredero de sangre.\n\nEl Custodio dirigi\u00f3 la mirada hacia la calle.\n\n\u2014Deber\u00eda haberlo imaginado \u2014dijo.\n\nSe qued\u00f3 all\u00ed plantado, y yo clav\u00e9 los ojos en su espalda. El casaca roja temblaba de pies a cabeza y ten\u00eda la cara cubierta de sudor. Si alguien no le vendaba el mu\u00f1\u00f3n, se iba a morir. Al menos hab\u00eda que echarle una manta por encima.\n\n\u2014Llev\u00e1dselo a Oriel. \u2014El Custodio se apart\u00f3 del grupo\u2014. Terebell se ocupar\u00e1 de \u00e9l. Los dem\u00e1s, volved a vuestras residencias. Los amaur\u00f3ticos os curar\u00e1n las heridas.\n\nEscudri\u00f1\u00e9 sus duras facciones en busca de alg\u00fan indicio de calidez, pero no encontr\u00e9 nada. No le importaba. Yo no sab\u00eda por qu\u00e9 segu\u00eda mir\u00e1ndolo.\n\nLos casacas rojas levantaron a su amigo y se encaminaron hacia un callej\u00f3n; por el camino fueron dejando un reguero de sangre.\n\n\u2014Necesita ir a un hospital \u2014me obligu\u00e9 a decir\u2014. No tienes idea de c\u00f3mo...\n\n\u2014Se ocupar\u00e1n de \u00e9l.\n\nEntonces se qued\u00f3 callado, y su mirada se endureci\u00f3. Deduje que eso significaba que con mi comentario me hab\u00eda pasado de la raya.\n\nPero empezaba a preguntarme d\u00f3nde estaba trazada la raya. El Custodio nunca me pegaba. Me dejaba dormir. Cuando est\u00e1bamos solos, me llamaba por mi nombre verdadero. Hasta me hab\u00eda dejado atacar su mente, se hab\u00eda expuesto a mi esp\u00edritu, un esp\u00edritu capaz de abocarlo a la locura. No entend\u00eda por qu\u00e9 corr\u00eda ese riesgo. Incluso Nick recelaba de mi don. (\u00abLl\u00e1malo respeto sano, _s\u00f6tnos_.\u00bb)\n\nFuimos hacia la residencia, y por el camino me solt\u00e9 el mo\u00f1o. Estuve a punto de volver a abandonar mi cuerpo cuando unas manos me asieron la melena de rizos h\u00famedos.\n\n\u2014\u00a1XX-40! Es un placer volver e verte \u2014dijo una voz con un deje ir\u00f3nico, muy aguda para ser masculina\u2014. Te felicito, Custodio. Con el blus\u00f3n est\u00e1 a\u00fan m\u00e1s hermosa.\n\nMe volv\u00ed y mir\u00e9 al hombre que ten\u00eda detr\u00e1s. Tuve que controlarme para no retroceder.\n\nEra el m\u00e9dium que me hab\u00eda perseguido por los tejados del I-5, pero esa noche no iba armado con una pistola de flux. Llevaba un extra\u00f1o uniforme con los colores de Scion. Hasta su cara hac\u00eda juego con \u00e9l: boca roja, cejas negras, la tez espolvoreada con \u00f3xido de zinc. Deb\u00eda de estar rozando la cuarentena, y llevaba un grueso l\u00e1tigo de cuero en la mano. Habr\u00eda jurado que el l\u00e1tigo estaba manchado de sangre. Deb\u00eda de ser el Capataz, el encargado de mantener a raya a los bufones. Detr\u00e1s de \u00e9l estaba el or\u00e1culo de la primera noche. Me mir\u00f3 con unos ojos desconcertantes: uno, oscuro y penetrante; el otro, de color avellana claro. Su blus\u00f3n era del mismo color que el m\u00edo.\n\nEl Custodio los mir\u00f3.\n\n\u2014\u00bfQu\u00e9 quieres, Capataz?\n\n\u2014Perdona que te importune. Solo quer\u00eda volver a ver a la So\u00f1adora. He seguido sus progresos con gran inter\u00e9s.\n\n\u2014No es actriz. Sus progresos no tienen ning\u00fan inter\u00e9s.\n\n\u2014Desde luego. Pero da gusto mirarla. \u2014Me lanz\u00f3 una sonrisa\u2014. Perm\u00edteme darte la bienvenida a Sheol I. Soy Beltrame, el Capataz. Espero que mi dardo de flux no te haya dejado cicatriz.\n\nReaccion\u00e9. No pude evitarlo.\n\n\u2014Si me entero de que le hiciste da\u00f1o a mi padre...\n\n\u2014No te he dado permiso para hablar, XX-40.\n\nEl Custodio me mir\u00f3 fijamente. El Capataz ri\u00f3 y me dio unas palmaditas en la mejilla. Me apart\u00e9 de \u00e9l.\n\n\u2014Tranquila. A tu padre no le ha pasado nada. \u2014Se hizo una se\u00f1al sobre el pecho\u2014. Palabra de honor.\n\nDeber\u00eda haber sentido alivio, pero su descaro me enfurec\u00eda. El Custodio mir\u00f3 al acompa\u00f1ante del Capataz.\n\n\u2014\u00bfQui\u00e9n es? \u2014pregunt\u00f3.\n\n\u2014XX-59-12. \u2014El Capataz le puso una mano en el hombro\u2014. Es un sirviente muy leal de Pleione. Ha progresado mucho en sus estudios estas \u00faltimas semanas.\n\n\u2014Ya veo. \u2014El Custodio lo examin\u00f3 con la mirada, evaluando su aura\u2014. \u00bfEres un or\u00e1culo?\n\n\u2014S\u00ed, Custodio. \u201412 agach\u00f3 la cabeza.\n\n\u2014La soberana de sangre debe de estar muy complacida con tus progresos. No hab\u00edamos tenido ning\u00fan or\u00e1culo desde la Era de Huesos XVI.\n\n\u2014Espero encontrarme muy pronto entre los que est\u00e1n a su servicio, Custodio. \u2014Ten\u00eda un ligero acento del norte.\n\n\u2014No lo dudes, 12. Creo que lo har\u00e1s muy bien contra tu emite. 12 est\u00e1 a punto de hacer su segundo examen \u2014explic\u00f3 el Capataz\u2014. Precisamente volv\u00edamos a Merton para reunirnos con el resto de su batall\u00f3n. Pleione y Alsafi lo dirigir\u00e1n.\n\n\u2014\u00bfSaben los Sualocin lo del casaca roja herido? \u2014pregunt\u00f3 el Custodio.\n\n\u2014S\u00ed. Buscan al emite que lo mordi\u00f3.\n\nEl Custodio mud\u00f3 ligeramente la expresi\u00f3n.\n\n\u2014Te deseo mucha suerte en tu empe\u00f1o, 12 \u2014dijo, y 12 volvi\u00f3 a agachar la cabeza.\n\n\u2014Pero tengo otro motivo para interrumpirte, antes de marcharnos \u2014a\u00f1adi\u00f3 el Capataz\u2014. Estoy aqu\u00ed para invitar a la onir\u00e1mbula. Si me lo permites.\n\nEl Custodio se volvi\u00f3 y lo mir\u00f3. El Capataz interpret\u00f3 su silencio como un permiso para continuar.\n\n\u2014Estamos organizando una celebraci\u00f3n muy especial en honor de esta Era de Huesos, XX-40. La vig\u00e9sima. \u2014Abri\u00f3 un brazo hacia el Poblado\u2014. Nuestros mejores actores. Un fest\u00edn para los sentidos. Una saturnal de m\u00fasica y baile para hacer alarde de todos nuestros chicos y chicas.\n\n\u2014Te refieres al Bicentenario \u2014dijo el Custodio.\n\nEra la primera vez que yo o\u00eda esa palabra.\n\n\u2014Exactamente. \u2014El Capataz sonri\u00f3\u2014. La ceremonia durante la cual se firmar\u00e1 el Gran Tratado Territorial.\n\nAquello no sonaba nada bien. De pronto me ceg\u00f3 una visi\u00f3n.\n\nNick, como todo or\u00e1culo, pod\u00eda enviar im\u00e1genes mudas a trav\u00e9s del \u00e9ter. Las llamaba _khresmoi_ , una palabra griega. Yo nunca hab\u00eda podido pronunciarla, y por eso las llamaba \u00abinstant\u00e1neas\u00bb. 12 tambi\u00e9n ten\u00eda ese don. Vi un reloj cuyas dos agujas se\u00f1alaban las doce y, a continuaci\u00f3n, cuatro columnas y una escalinata. Al cabo de un momento parpade\u00e9, y las im\u00e1genes desaparecieron. Cuando abr\u00ed los ojos, vi que 12 me miraba.\n\nFue solo un segundo.\n\n\u2014Estoy al corriente del Tratado \u2014iba diciendo el Custodio\u2014. Ve al grano, Capataz. 40 est\u00e1 agotada.\n\nEl Capataz no se inmut\u00f3 ante la brusquedad del Custodio. Deb\u00eda de estar acostumbrado a que lo trataran con desprecio. Se limit\u00f3 a esbozar una blanda sonrisa.\n\n\u2014Me gustar\u00eda invitar a 40 a actuar con nosotros el d\u00eda del Bicentenario. Me impresionaron su fuerza y su agilidad la noche que la captur\u00e9. Es un gran placer para m\u00ed invitarla a ser nuestra actriz principal, junto con XIX-49-1 y con XIX-49-8.\n\nIba a rechazar la invitaci\u00f3n, en un tono con el que me habr\u00eda ganado un castigo severo, cuando el Custodio dijo:\n\n\u2014Soy su guardi\u00e1n y lo proh\u00edbo. \u2014Lo mir\u00e9\u2014. 40 no es actriz, y a menos que suspenda sus ex\u00e1menes antes del Bicentenario, sigue estando bajo mi custodia. \u2014El Custodio mir\u00f3 a los ojos al Capataz y a\u00f1adi\u00f3\u2014: 40 es onir\u00e1mbula. Es la onir\u00e1mbula que te encargaron traer a esta colonia. No permitir\u00e9 que la exhiban ante los emisarios de Scion como si fuera una vulgar profeta. Esa es una funci\u00f3n propia de tus humanos, no de los m\u00edos.\n\nEl Capataz ya no sonre\u00eda.\n\n\u2014Muy bien. \u2014Agach\u00f3 la cabeza y dej\u00f3 de mirarme\u2014. Vamos, 12. Tu desaf\u00edo te espera.\n\n12 me mir\u00f3 con disimulo y enarc\u00f3 una ceja con gesto interrogante. Asent\u00ed con la cabeza. 12 se dio la vuelta y sigui\u00f3 al Capataz hacia el Poblado, caminando sin prisas. No parec\u00eda asustarle eso a lo que iba a enfrentarse.\n\nEl Custodio me mir\u00f3 fijamente.\n\n\u2014\u00bfConoces al or\u00e1culo?\n\n\u2014No.\n\n\u2014No te quitaba los ojos de encima.\n\n\u2014Perd\u00f3name, amo \u2014repliqu\u00e9\u2014, pero \u00bftengo prohibido hablar con los otros humanos?\n\nNo dej\u00f3 de mirarme. Me pregunt\u00e9 si los refas sab\u00edan captar el sarcasmo.\n\n\u2014No \u2014contest\u00f3\u2014. Tienes permiso.\n\nPas\u00f3 a mi lado sin decir nada m\u00e1s.\n\nNo dorm\u00ed bien. Me dol\u00eda mucho la cabeza; el dolor se concentraba en mi sien izquierda. Tumbada entre las s\u00e1banas, vi consumirse la vela.\n\nEl Custodio no me hab\u00eda enviado directamente a mi habitaci\u00f3n. Me hab\u00eda ofrecido un poco de comida y agua, que yo hab\u00eda aceptado porque estaba al borde de la deshidrataci\u00f3n. Luego se hab\u00eda sentado junto al fuego y se hab\u00eda quedado observando las llamas. Yo hab\u00eda tardado unos diez minutos en preguntarle si pod\u00eda retirarme ya, y \u00e9l me hab\u00eda contestado con un \u00abs\u00ed\u00bb cortante.\n\nArriba hac\u00eda fr\u00edo. El cristal de las ventanas era muy delgado y dejaba pasar el aire. Me envolv\u00ed con las finas s\u00e1banas, temblando. Al cabo de un rato me qued\u00e9 dormida. Las palabras del Custodio resonaban en mis o\u00eddos: que en mis ojos hab\u00eda muerte y hielo. De vez en cuando volv\u00edan a asaltarme las im\u00e1genes de XX-12, grabadas todav\u00eda en mi onirosaje. No era la primera vez que ve\u00eda im\u00e1genes oraculares. Una vez Nick me hab\u00eda mostrado una instant\u00e1nea en la que me ca\u00eda de un tejado bajo y me romp\u00eda el tobillo, y a la semana pas\u00f3 exactamente eso. Nunca volv\u00ed a dudar de sus predicciones meteorol\u00f3gicas.\n\nXX-12 me hab\u00eda pedido que me reuniera con \u00e9l a las doce. No ve\u00eda ninguna raz\u00f3n para no acudir a la cita.\n\nCuando despert\u00e9, estaban dando las once. Me lav\u00e9, me vest\u00ed y baj\u00e9 a la c\u00e1mara del Custodio. No se o\u00eda nada. Las cortinas estaban descorridas y dejaban entrar la luz de la luna. Por primera vez desde hac\u00eda varios d\u00edas, encontr\u00e9 una de sus notas encima del escritorio.\n\nAverigua lo que puedas sobre los emim.\n\nMe recorri\u00f3 un escalofr\u00edo. Si ten\u00eda que investigar sobre los zumbadores, quer\u00eda decir que estaba destinada a enfrentarme a ellos. Tambi\u00e9n significaba que ten\u00eda libertad para ver a 12. De alguna forma estar\u00eda obedeciendo \u00f3rdenes. 12 acababa de realizar su segundo examen. Me pregunt\u00e9 qu\u00e9 habr\u00eda visto durante la noche. Por fin iba a obtener datos fiables sobre los emim, suponiendo, claro estaba, que no se hubieran comido a 12.\n\nMinutos antes de medianoche baj\u00e9 la escalera y cerr\u00e9 la puerta detr\u00e1s de m\u00ed. Hab\u00eda llegado la hora de hacer los deberes.\n\nLa portera nocturna no me salud\u00f3 al verme. Cuando le ped\u00ed m\u00e1s _numa_ , me los dio, pero sigui\u00f3 con expresi\u00f3n altiva. Todav\u00eda estaba dolida por el incidente de la sirena.\n\nFuera hac\u00eda fr\u00edo, y la llovizna enturbiaba la atm\u00f3sfera. Fui al Poblado y me compr\u00e9 el desayuno: un poco de _skilly_ en un vaso de pl\u00e1stico. Lo compr\u00e9 con un pu\u00f1ado de agujas y anillos. Me obligu\u00e9 a dar un par de sorbos y me dirig\u00ed hacia el edificio que los bufones llamaban Hawksmoor, el gran centinela de piedra que custodiaba la biblioteca y su patio.\n\n12 me esperaba detr\u00e1s de una columna. Llevaba un blus\u00f3n rojo, nuevo, y ten\u00eda un corte en la mejilla. Cuando vio mi taza de _skilly_ , enarc\u00f3 una ceja.\n\n\u2014\u00bfTe comes eso?\n\nDi un sorbo.\n\n\u2014\u00bfPor qu\u00e9? \u00bfQu\u00e9 comes t\u00fa?\n\n\u2014Lo que me da mi guardiana.\n\n\u2014No todos somos arrancahuesos. Por cierto: felicidades.\n\nMe tendi\u00f3 la mano y yo se la estrech\u00e9.\n\n\u2014Me llamo David.\n\n\u2014Yo, Paige.\n\n\u2014Paige. \u2014Me taladr\u00f3 con uno de sus oscuros ojos; el otro parec\u00eda menos enfocado\u2014. Si no tienes nada mejor que hacer con tu tiempo, me gustar\u00eda llevarte a dar un paseo.\n\n\u2014Cualquiera dir\u00eda que soy un perro.\n\nRi\u00f3 sin mover los labios.\n\n\u2014Por aqu\u00ed \u2014dijo\u2014. Si alguien pregunta, te llevo a que te interroguen por un incidente.\n\nBajamos juntos por una calle estrecha, hacia la Residencia del Suzerano. David era unos cinco cent\u00edmetros m\u00e1s alto que yo; ten\u00eda los brazos largos y el torso grueso. No estaba muerto de hambre como los bufones.\n\n\u2014Un poco arriesgado, \u00bfno? \u2014le pregunt\u00e9.\n\n\u2014\u00bfQu\u00e9?\n\n\u2014Hablar conmigo. Ahora eres un casaca roja.\n\nSonri\u00f3.\n\n\u2014No cre\u00ed que fueras una presa tan f\u00e1cil. Ya has ca\u00eddo en su trampa, \u00bfverdad?\n\n\u2014\u00bfQu\u00e9 quieres decir?\n\n\u2014Segregaci\u00f3n, 40. Ves que soy un casaca roja y piensas que no deber\u00eda hablar contigo. \u00bfEso te ha ense\u00f1ado tu guardi\u00e1n?\n\n\u2014No. Pero funciona as\u00ed.\n\n\u2014\u00bfLo ves? Para eso existe este sitio: para lavarnos el cerebro. Para hacernos sentir inferiores. \u00bfPor qu\u00e9 crees que dejan a la gente en la Torre durante a\u00f1os? \u2014Como no contestaba, sacudi\u00f3 la cabeza\u2014. Venga, 40. El submarino, el aislamiento, los d\u00edas sin comida. Despu\u00e9s de eso, este sitio parece un rinc\u00f3n del para\u00edso. \u2014Ten\u00eda parte de raz\u00f3n\u2014. Tendr\u00edas que o\u00edr al Capataz. Cree que los refas deber\u00edan gobernarnos, que deber\u00edan ser nuestra nueva monarqu\u00eda.\n\n\u2014\u00bfPor qu\u00e9 va a pensar eso?\n\n\u2014Porque le han lavado el cerebro.\n\n\u2014\u00bfCu\u00e1nto tiempo lleva aqu\u00ed?\n\n\u2014Que yo sepa, solo desde la Era de Huesos XIX, pero es fiel como un perro. Lleva tiempo tratando de sacar a buenos videntes del sindicato.\n\n\u2014Entonces es un captador.\n\n\u2014S\u00ed, pero no es muy bueno. Nashira quiere sustituirlo. Quiere a alguien capaz de percibir el \u00e9ter a un nivel m\u00e1s elevado.\n\nIba a preguntarle algo m\u00e1s, pero me contuve. Distingu\u00ed, a trav\u00e9s de la neblina gris, un edificio circular con una gran c\u00fapula. Se alzaba en una plaza desierta, grande y pesado, enfrente de la Residencia del Suzerano. Detr\u00e1s de las ventanas se ve\u00eda una luz ambarina.\n\n\u2014\u00bfQu\u00e9 es eso? \u2014pregunt\u00e9 se\u00f1al\u00e1ndolo.\n\n\u2014Los bufones lo llaman la Sala. He intentado averiguar para qu\u00e9 es, pero por lo visto a nadie le gusta hablar de ella. A los humanos no nos dejan entrar.\n\nSigui\u00f3 adelante sin mirarla siquiera. Apret\u00e9 el paso para alcanzarlo.\n\n\u2014Dec\u00edas que trataba de sacar a videntes del sindicato \u2014dije intentando retomar el hilo de nuestra conversaci\u00f3n\u2014. \u00bfPor qu\u00e9?\n\n\u2014No hagas demasiadas preguntas, 40.\n\n\u2014Cre\u00eda que ese era el motivo de esta cita.\n\n\u2014Podr\u00eda serlo. Tambi\u00e9n podr\u00eda ser que sencillamente me ca\u00edste bien. Ya hemos llegado.\n\nNuestro destino era una iglesia antigua. En su d\u00eda deb\u00eda de haber sido magn\u00edfica, pero se estaba derrumbando. Las ventanas no ten\u00edan cristales, la torre hab\u00eda quedado reducida a un esqueleto, y unos tablones cerraban el acceso al porche, orientado al sur. Arque\u00e9 una ceja.\n\n\u2014\u00bfSeguro que sabes lo que haces?\n\n\u2014No es la primera vez. Adem\u00e1s \u2014dijo col\u00e1ndose por debajo de un list\u00f3n\u2014, seg\u00fan el Capataz est\u00e1s acostumbrada a las estructuras inseguras. \u2014Mir\u00f3 m\u00e1s all\u00e1 de mi hombro\u2014. R\u00e1pido. El Guardi\u00e1n Gris.\n\nMe col\u00e9 entre los tablones justo a tiempo: Graffias pas\u00f3 por delante de la entrada guiando a tres amaur\u00f3ticos desnutridos. Segu\u00ed a David por la iglesia. Gran parte del techo se hab\u00eda derrumbado y cubr\u00eda el suelo de la capilla. Las vigas de madera y el hormig\u00f3n hab\u00edan destrozado los bancos, y hab\u00eda cristales por el suelo. Me abr\u00ed paso entre los escombros.\n\n\u2014\u00bfQu\u00e9 ha pasado aqu\u00ed?\n\nDavid no me contest\u00f3.\n\n\u2014Hay que subir ciento veinticuatro escalones \u2014dijo\u2014. \u00bfTe animas?\n\nNo esper\u00f3 a que le contestara. Lo segu\u00ed hacia la escalera. Estaba acostumbrada a trepar; hab\u00eda trepado por cientos de edificios en la cohorte I. Casi todos los escalones estaban todav\u00eda intactos, y no tardamos mucho en llegar al final de la escalera. Un fuerte viento me sacudi\u00f3 el pelo y me lo apart\u00f3 de la cara. Ol\u00eda mucho a humo. David apoy\u00f3 los brazos en una balaustrada de piedra.\n\n\u2014Me gusta este sitio. \u2014Se sac\u00f3 un porro de la manga y lo encendi\u00f3 con una cerilla\u2014. Una atalaya.\n\nNos hall\u00e1bamos en un balc\u00f3n, justo debajo de la torre semiderruida. Faltaba parte de la balaustrada, y otro letrero advert\u00eda de la peligrosidad de la estructura. Alc\u00e9 la vista hacia las estrellas.\n\n\u2014Has aprobado tu segundo examen \u2014dije\u2014. Si quieres que hablemos, cu\u00e9ntame algo de los emim.\n\nDavid exhal\u00f3 el humo con los ojos cerrados. Ten\u00eda los dedos manchados.\n\n\u2014\u00bfQu\u00e9 quieres saber, exactamente?\n\n\u2014Quiero saber qu\u00e9 son.\n\n\u2014No tengo ni idea.\n\n\u2014Debes de haber visto a alguno.\n\n\u2014No muy bien. El bosque est\u00e1 oscuro. S\u00e9 que parec\u00eda humano (al menos ten\u00eda cabeza, brazos y piernas), pero se mov\u00eda como un animal. Y apestaba a cloaca. Y sonaba como una cloaca.\n\n\u2014\u00bfSonaba como una cloaca?\n\n\u2014A moscas, 40. Bzzzz.\n\n\u00abLos zumbadores.\u00bb\n\n\u2014\u00bfY su aura? \u2014insist\u00ed\u2014. \u00bfTen\u00eda aura?\n\n\u2014No, o al menos no la vi. Hac\u00eda que pareciera que el \u00e9ter se estaba derrumbando \u2014continu\u00f3\u2014. Como si hubiera un agujero negro alrededor de su onirosaje.\n\nNo, no me hac\u00eda ninguna gracia enfrentarme a una cosa as\u00ed. Mir\u00e9 hacia abajo, hacia la ciudad.\n\n\u2014\u00bfLo mataste?\n\n\u2014Lo intent\u00e9. \u2014Se fij\u00f3 en mi expresi\u00f3n y le dio unos golpecitos al porro\u2014. Nos pusieron a unos cuantos all\u00ed, todos casacas rosa. Dos grupos. Nos acompa\u00f1aban dos casacas rojas, 30 y 25. Nos dieron un pu\u00f1al a cada uno y nos dijeron que rastre\u00e1ramos al zumbador como pudi\u00e9ramos. 30 nos dijo que los pu\u00f1ales solo serv\u00edan para que nos sinti\u00e9ramos mejor. La mejor forma de localizar a aquella cosa era utilizar el \u00e9ter.\n\n\u00bbUno de los casacas rosa era rabdomante, as\u00ed que hicimos unos cuantos _sortes_ con ramitas. 30 nos dio una botella que conten\u00eda sangre de un tipo al que le hab\u00edan arrancado la mano, al que podr\u00edamos usar como solicitante. Mojamos las ramitas en la sangre, y el rabdomante las lanz\u00f3. Se\u00f1alaron hacia el oeste. Seguimos lanzando _sortes_ y cambiando de direcci\u00f3n. Como es l\u00f3gico, el zumbador tambi\u00e9n se mov\u00eda, de modo que no llegamos a ninguna parte. 21 propuso que lo hici\u00e9ramos venir adonde est\u00e1bamos nosotros. Encendimos una hoguera e hicimos una sesi\u00f3n para invocar a los esp\u00edritus del bosque.\n\n\u2014\u00bfHay muchos?\n\n\u2014S\u00ed. Todos los idiotas que intentaron huir por el campo de minas, o eso dicen los casacas rojas.\n\nDisimul\u00e9 un estremecimiento.\n\n\u2014Esperamos unos minutos all\u00ed sentados. Los esp\u00edritus desaparecieron. O\u00edmos ruidos. Empezaron a llegar moscas del bosque y a sub\u00edrsenos por los brazos. Entonces vimos salir una cosa gigantesca e hinchada de la nada. Al cabo de dos segundos el emite le hab\u00eda agarrado el pelo a 19 con la boca; casi le lleva tambi\u00e9n la piel \u2014a\u00f1adi\u00f3\u2014. 19 se puso a gritar, y eso confundi\u00f3 a la cosa. Le arranc\u00f3 un mech\u00f3n de pelo y fue a atacar a 1.\n\n\u2014\u00bfCarl?\n\n\u2014No s\u00e9 sus nombres. 1 se puso a chillar como un cochinillo e intent\u00f3 clavarle el pu\u00f1al. No consigui\u00f3 nada. \u2014Ech\u00f3 una ojeada al extremo de su porro\u2014. El fuego se estaba apagando, pero yo todav\u00eda lo ve\u00eda. Intent\u00e9 atacarlo con una imagen. Pens\u00e9 en luz blanca e intent\u00e9 introducirla en el onirosaje del zumbador para cegarlo. De pronto sent\u00ed como si me atropellaran la mente, y como si hubiera un charco de aceite en el \u00e9ter. Estaba todo oscuro y muerto. Todos los esp\u00edritus de la zona procuraban alejarse de aquel desastre. 20 y 14 echaron a correr. 30 les grit\u00f3 que eran unos amarillos, pero estaban demasiado asustados para volver. 10 lanz\u00f3 un pu\u00f1al y le dio a 5. 5 cay\u00f3 al suelo. El zumbador se le ech\u00f3 encima. El fuego se apag\u00f3, y qued\u00f3 todo completamente oscuro. 5 grit\u00f3 pidiendo ayuda.\n\n\u00bbYa nadie ve\u00eda nada. Us\u00e9 el \u00e9ter para saber d\u00f3nde estaba aquella cosa. Se estaba comiendo a 5, que ya estaba muerto. Agarr\u00e9 a la cosa por el cuello y lo separ\u00e9 de 5. Se me qued\u00f3 su piel mojada y muerta pegada a las manos. Se volvi\u00f3 hacia m\u00ed. Vi unos ojos blancos en la oscuridad que me miraban fijamente. De pronto sal\u00ed volando por los aires, sangrando como un cerdo degollado.\n\nSe baj\u00f3 el cuello del blus\u00f3n y se destap\u00f3 un vendaje. Debajo ten\u00eda cuatro boquetes. Alrededor de las heridas la piel estaba gris y surcada de capilares rotos.\n\n\u2014Parecen heridas de duende \u2014coment\u00e9.\n\n\u2014No lo s\u00e9. \u2014Volvi\u00f3 a taparse las heridas con la venda\u2014. No pod\u00eda moverme. La cosa ven\u00eda hacia m\u00ed, manch\u00e1ndome de sangre. 10 estaba intentando ayudar a 5, pero entonces se levant\u00f3. Ten\u00eda un \u00e1ngel guardi\u00e1n, el \u00fanico esp\u00edritu que no se hab\u00eda largado. Se lo tir\u00f3 al zumbador. Lanc\u00e9 otra imagen hacia su onirosaje. La cosa grit\u00f3, muy fuerte. Empez\u00f3 a alejarse reptando. Hac\u00eda un ruido espantoso, y arrastraba un pedazo de 5. 21 le hab\u00eda prendido fuego a una rama y se la lanz\u00f3 al zumbador. Ol\u00ed a carne quemada. Entonces me desmay\u00e9. Despert\u00e9 en Oriel, cubierto de vendajes.\n\n\u2014Y os dieron a todos el blus\u00f3n rojo.\n\n\u2014A 20 y a 14, no. A ellos les dieron el amarillo. Y tuvieron que recoger los restos de 5.\n\nNos quedamos unos minutos en silencio. Yo no pod\u00eda dejar de pensar en 5, al que se hab\u00edan comido vivo en el bosque. No sab\u00eda su verdadero nombre, pero confiaba en que alguien hubiera recitado el treno. Qu\u00e9 muerte tan horrible.\n\nDirig\u00ed la mirada a lo lejos y distingu\u00ed un punto de luz; desde all\u00ed parec\u00eda poco m\u00e1s que la llama de una vela.\n\n\u2014\u00bfQu\u00e9 es eso?\n\n\u2014Una hoguera.\n\n\u2014\u00bfQu\u00e9 est\u00e1n quemando?\n\n\u2014Cad\u00e1veres de zumbadores. O cad\u00e1veres humanos. Depende de qui\u00e9n gane. \u2014Tir\u00f3 el porro\u2014. Creo que utilizan los huesos para no s\u00e9 qu\u00e9 adivinaciones.\n\nMientras \u00e9l dec\u00eda eso, me pasaron cenizas ante los ojos. Atrap\u00e9 una pizca con un dedo. Los augures conectaban con el \u00e9ter a trav\u00e9s del mundo natural: el cuerpo humano, animales y plantas, los elementos. Constitu\u00edan uno de los \u00f3rdenes m\u00e1s bajos, seg\u00fan Jaxon.\n\n\u2014A lo mejor los atrae el fuego \u2014dije\u2014. Dijeron que esta ciudad era una baliza.\n\n\u2014Es una baliza et\u00e9rea, 40. Est\u00e1 llena de videntes, esp\u00edritus y refas. Piensa en c\u00f3mo funciona el \u00e9ter.\n\n\u2014\u00bfC\u00f3mo es que sabes tanto del \u00e9ter? \u2014Me volv\u00ed y lo mir\u00e9\u2014. No eres del sindicato. \u00bfQui\u00e9n eres?\n\n\u2014Un cifrador, igual que t\u00fa.\n\nMe qued\u00e9 callada, rechinando los dientes.\n\n\u2014Tienes m\u00e1s preguntas \u2014me dijo tras un breve silencio\u2014. \u00bfSeguro que quieres hac\u00e9rmelas?\n\n\u2014No empieces.\n\n\u2014Que no empiece \u00bfa qu\u00e9?\n\n\u2014A decirme lo que quiero saber. Quiero respuestas. \u2014Lo dije atropelladamente\u2014. Quiero saberlo todo sobre el sitio donde se supone que voy a pasar el resto de mi vida. \u00bfNo lo entiendes?\n\nAcodados en la balaustrada, contempl\u00e1bamos la Sala. Por temor a que se derrumbara, procur\u00e9 no apoyarme demasiado en la piedra.\n\n\u2014En fin \u2014a\u00f1ad\u00ed\u2014, \u00bfpuedo hacerte esas preguntas?\n\n\u2014Esto no es ning\u00fan juego de sal\u00f3n, 40. No he venido aqu\u00ed a jugar a las adivinanzas. Te he tra\u00eddo para ver si de verdad eres una onir\u00e1mbula.\n\n\u2014Lo soy, de carne y hueso.\n\n\u2014No siempre, por lo que yo s\u00e9. A veces sales de tu cuerpo. \u2014Me mir\u00f3 de arriba abajo\u2014. Te sacaron de la cohorte central. Del sanctasanct\u00f3rum del sindicato. Debiste de ser poco cuidadosa.\n\n\u2014Poco cuidadosa, no. Poco afortunada. \u2014Lo mir\u00e9 a los ojos\u2014. \u00bfPor qu\u00e9 les preocupa tanto el sindicato?\n\n\u2014Porque se queda con los mejores videntes. Esconde a todos los vinculadores, los onir\u00e1mbulos y los or\u00e1culos. A los representantes de los \u00f3rdenes m\u00e1s elevados, a los que Nashira querr\u00eda tener en su colonia. Por eso les preocupa el sindicato, 40. Por eso van a aprobar esa nueva ley.\n\n\u2014\u00bfQu\u00e9 dice esa ley?\n\n\u2014Nashira lleva tiempo intentando hacerse con buenos videntes, pero est\u00e1n todos protegidos por las bandas. Hasta que encuentren la forma de disolver a los capos de Londres, no tienen m\u00e1s remedio que buscar en otros lugares. Esa ley promete que en el plazo de dos a\u00f1os se establecer\u00e1 Sheol II, y Scion Par\u00eds ser\u00e1 su ciudadela de siega. \u2014Sigui\u00f3 con un dedo las heridas que ten\u00eda en el pecho\u2014. Y \u00bfqui\u00e9n va a detenerlos, si los emim nos matar\u00e1n si lo intentamos?\n\nMe recorri\u00f3 un fr\u00edo extra\u00f1o.\n\nDe modo que Nashira consideraba que el sindicato era una amenaza; eso era nuevo para m\u00ed. Para m\u00ed, los mimetocapos eran una pandilla de ego\u00edstas, aprovechados e interesados; al menos as\u00ed eran los de la cohorte central. La Asamblea Antinatural llevaba a\u00f1os sin reunirse; hab\u00edan dejado que los mimetocapos hicieran lo que quisieran en sus respectivas zonas, pues Hector estaba demasiado ocupado con el juego y la prostituci\u00f3n para ocuparse de ellos. Y, sin embargo lejos de all\u00ed, en Sheol I, la soberana de sangre de los refa\u00edtas tem\u00eda a aquella chusma desmandada.\n\n\u2014Ahora eres uno de sus fieles seguidores \u2014dije mirando su blus\u00f3n rojo\u2014. \u00bfVas a ayudarlos?\n\n\u2014No soy fiel, 40. Solo lo aparento. \u00bfAlguna vez has visto sangrar a un refa?\n\nNo supe qu\u00e9 contestar.\n\n\u2014Su sangre se llama ectoplasma. Es la m\u00e1xima obsesi\u00f3n de Duckett. Los refas son algo as\u00ed como el \u00e9ter materializado. Su sangre es \u00e9ter licuado. Si ves el ectoplasma, ves el \u00e9ter. Si lo bebes; te conviertes en \u00e9ter. Como ellos.\n\n\u2014Pero eso significa que los amaur\u00f3ticos podr\u00edan llegar a utilizar el \u00e9ter, \u00bfno? Bastar\u00eda con que tocaran un poco de ectoplasma.\n\n\u2014Correcto. Para los carro\u00f1os, en teor\u00eda, el ecto actuar\u00eda como una especie de suced\u00e1neo del aura. A corto plazo, por supuesto. Los efectos secundarios solo duran unos quince minutos. Sin embargo, si aplic\u00e1ramos la ciencia y resolvi\u00e9ramos un par de problemas, apuesto algo a que podr\u00edamos vender p\u00edldoras de clarividencia instant\u00e1nea dentro de pocos a\u00f1os. \u2014Se qued\u00f3 contemplando la ciudad\u2014. Ese d\u00eda llegar\u00e1, 40. Seremos nosotros los que experimentaremos con esos cabrones, y no al rev\u00e9s.\n\nLos refas hab\u00edan cometido un grave error d\u00e1ndole el blus\u00f3n rojo a David. Era evidente que los odiaba.\n\n\u2014Puedes hacerme una pregunta m\u00e1s \u2014dijo.\n\n\u2014Vale. \u2014Hice una pausa, y entonces me acord\u00e9 de Liss\u2014. \u00bfQu\u00e9 sabes de la Era de Huesos XVIII?\n\n\u2014No sab\u00eda si me lo preguntar\u00edas. \u2014Apart\u00f3 otro list\u00f3n, revelando una ventana rota\u2014. Ven, te lo ense\u00f1ar\u00e9.\n\nLo segu\u00ed.\n\nEn esa habitaci\u00f3n hab\u00eda esp\u00edritus. Me habr\u00eda gustado poder ver cu\u00e1ntos; calcul\u00e9 que ocho o nueve. El aire estaba enrarecido, impregnado del olor empalagoso de las flores marchitas. En un rinc\u00f3n hab\u00eda una especie de altar. Un \u00f3valo de metal burdamente cortado, rodeado de humildes ofrendas: cabos de vela, varillas de incienso rotas, un ramito seco de tomillo, etiquetas con nombres. En medio hab\u00eda un ramito de ran\u00fanculos y azucenas. Las que ol\u00edan eran las azucenas: eran flores frescas. David se sac\u00f3 una linterna del bolsillo.\n\n\u2014Aqu\u00ed tienes las ruinas de la esperanza.\n\nMe acerqu\u00e9 m\u00e1s. En el metal hab\u00eda grabadas unas palabras.\n\nPOR LOS CA\u00cdDOS\n\n28 DE NOVIEMBRE DE 2039\n\n\u2014A\u00f1o 2039 \u2014dije\u2014. La Era de Huesos XVIII.\n\nUn a\u00f1o antes de mi nacimiento.\n\n\u2014Hubo una rebeli\u00f3n el d\u00eda de Novembertide. \u2014David segu\u00eda iluminando el altar con la linterna\u2014. Un grupo de refas se alz\u00f3 contra los Sargas. La mayor\u00eda de los humanos los apoyaban. Intentaron matar a Nashira y evacuar a los humanos a Londres.\n\n\u2014\u00bfQu\u00e9 refas?\n\n\u2014Nadie lo sabe.\n\n\u2014\u00bfQu\u00e9 pas\u00f3?\n\n\u2014Un humano los traicion\u00f3. XVIII-39-7. Una d\u00e9bil filtraci\u00f3n en la base, y todo el montaje se vino abajo. Nashira tortur\u00f3 a los refas implicados. Los marc\u00f3. A los humanos los mataron los emim. A todos. Corre el rumor de que solo hubo dos supervivientes, aparte de Duckett: el traidor y la ni\u00f1a.\n\n\u2014\u00bfLa ni\u00f1a?\n\n\u2014Duckett me lo cont\u00f3 todo. \u00c9l se libr\u00f3 porque era demasiado cobarde para rebelarse. Les suplic\u00f3 de rodillas que le permitieran vivir. Me cont\u00f3 que aquel a\u00f1o hab\u00edan tra\u00eddo aqu\u00ed a una ni\u00f1a de unos cuatro o cinco a\u00f1os. XVIII-39-0.\n\n\u2014\u00bfPara qu\u00e9 traer\u00edan a una ni\u00f1a tan peque\u00f1a?\u2014. Se me hizo un nudo en el est\u00f3mago\u2014. Los ni\u00f1os no pueden combatir a los zumbadores.\n\n\u2014Ni idea. Duckett cree que quer\u00edan ver si sobrevivir\u00eda.\n\n\u2014\u00bfC\u00f3mo iba a sobrevivir? Un cr\u00edo de cuatro a\u00f1os no durar\u00eda ni dos d\u00edas en semejante pocilga.\n\n\u2014Exacto.\n\nSe me empezaron a revolver las tripas.\n\n\u2014Muri\u00f3, \u00bfno?\n\n\u2014Duckett jura que no encontraron su cad\u00e1ver. \u00c9l se encarg\u00f3 de recoger los cad\u00e1veres \u2014dijo David\u2014. Formaba parte del trato a cambio de su supervivencia. Dice que nunca encontr\u00f3 a la ni\u00f1a, pero esto lo contradice.\n\nAlumbr\u00f3 una de las ofrendas: un osito de peluche sucio, con ojos de bot\u00f3n. Llevaba una nota colgada del cuello. La acerqu\u00e9 a la linterna de David.\n\nXVIII-39-0\n\nNinguna vida se vive en vano\n\nSe produjo un silencio, y al cabo de un momento lo interrumpi\u00f3 un repique lejano. Volv\u00ed a dejar el oso entre las flores.\n\n\u2014\u00bfQui\u00e9n hizo todo esto? \u2014Me dol\u00eda la garganta al hablar\u2014. \u00bfQui\u00e9n puso este altar aqu\u00ed?\n\n\u2014Los bufones. Y los marcados. Los misteriosos refas que se sublevaron contra Nashira.\n\n\u2014\u00bfTodav\u00eda viven?\n\n\u2014Nadie lo sabe. Pero yo me juego algo a que no. \u00bfPor qu\u00e9 los dejar\u00eda Nashira libres en la ciudad, sabiendo que la hab\u00edan traicionado?\n\nMe temblaban los dedos. Los ocult\u00e9 bajo las mangas de mi blus\u00f3n.\n\n\u2014Ya he visto suficiente \u2014dije.\n\nDavid me acompa\u00f1\u00f3 a Magdalen. Todav\u00eda faltaban unas horas para el amanecer, pero no quer\u00eda ver a nadie m\u00e1s. Esa noche no.\n\nCuando apareci\u00f3 la torre de la residencia, me volv\u00ed hacia David.\n\n\u2014No s\u00e9 por qu\u00e9 has hablado conmigo \u2014dije\u2014, pero gracias.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Por ense\u00f1arme el altar.\n\n\u2014De nada \u2014dijo con gesto sombr\u00edo\u2014. Te concedo una pregunta m\u00e1s. Con la condici\u00f3n de que pueda contestarla en menos de un minuto.\n\nReflexion\u00e9. Todav\u00eda ten\u00eda muchas preguntas por hacer, pero hab\u00eda una que llevaba incordi\u00e1ndome desde hac\u00eda d\u00edas.\n\n\u2014\u00bfPor qu\u00e9 las llaman \u00abEras de Huesos\u00bb?\n\nDavid sonri\u00f3.\n\n\u2014\u00abEra\u00bb es un sin\u00f3nimo de \u00absiega\u00bb. Cada \u00abEra de Huesos\u00bb es una campa\u00f1a de diez a\u00f1os durante los cuales los refa\u00edtas, tal como pactaron con Scion, \u00abrecolectan\u00bb humanos. Para ellos solo somos eso: huesos. Cad\u00e1veres. Y por eso a los casacas rojas nos llaman \u00abarrancahuesos\u00bb: porque ayudamos a traerlos a la colonia.\n\nMe qued\u00e9 helada. Hab\u00eda una parte de m\u00ed que se hab\u00eda planteado quedarse all\u00ed. Ahora solo quer\u00eda marcharme.\n\n\u2014\u00bfC\u00f3mo sabes todo eso? \u2014pregunt\u00e9\u2014. Dudo mucho que te lo hayan contado los refas.\n\n\u2014Lo siento, no m\u00e1s preguntas. He hablado demasiado.\n\n\u2014Podr\u00edas estar mintiendo.\n\n\u2014No miento.\n\n\u2014Podr\u00eda delatarte a los refas. \u2014Me mantuve firme\u2014. Podr\u00eda contarles todo lo que sabes.\n\n\u2014S\u00ed, pero entonces tendr\u00edas que contarles tambi\u00e9n todo lo que t\u00fa sabes. \u2014Me sonri\u00f3, y comprend\u00ed que hab\u00eda perdido\u2014. Puedes quedar en deuda conmigo por la informaci\u00f3n que te he dado, o bien puedes saldar tu deuda ahora mismo.\n\n\u2014\u00bfC\u00f3mo?\n\nA modo de respuesta me acarici\u00f3 la cara. Con la otra mano me asi\u00f3 por la cintura. Me puse en tensi\u00f3n.\n\n\u2014No, eso no \u2014dije.\n\n\u2014\u00bfPor qu\u00e9 no? \u2014Empez\u00f3 a acariciarme la cintura; su cara estaba cada vez m\u00e1s cerca de la m\u00eda\u2014. \u00bfTe tomas la pastilla?\n\n\u2014\u00bfAs\u00ed es como quieres que te pague? \u2014Lo apart\u00e9 de un empuj\u00f3n\u2014. Vete al infierno, casaca roja.\n\nDavid me miraba fijamente.\n\n\u2014Hazme un favor \u2014dijo\u2014. Busca esto en Merton. A ver si descubres algo. Eres m\u00e1s lista de lo que yo cre\u00eda. \u2014Me puso un sobre en la mano\u2014. Que tengas dulces sue\u00f1os, 40.\n\nSe alej\u00f3 y yo me qued\u00e9 all\u00ed un momento, r\u00edgida y fr\u00eda; entonces me apoy\u00e9 en la pared. No deber\u00eda haber ido con \u00e9l all\u00ed. Sab\u00eda que no era recomendable caminar con desconocidos por calles oscuras. \u00bfD\u00f3nde estaba mi intuici\u00f3n?\n\nHab\u00eda descubierto demasiadas cosas en una sola noche. Liss nunca hab\u00eda mencionado que los refas hab\u00edan participado en el levantamiento de la Era de Huesos XVIII. Quiz\u00e1 no lo supiera.\n\n\u00abLos marcados.\u00bb Deb\u00eda buscarlos, buscar a los que nos hab\u00edan ayudado. O quiz\u00e1 debiera agachar la cabeza y vivir mi nueva vida; eso era lo m\u00e1s sensato, lo m\u00e1s f\u00e1cil.\n\nA\u00f1oraba a Nick. A\u00f1oraba a Jax. A\u00f1oraba mi antigua vida. S\u00ed, antes era una delincuente, pero viv\u00eda rodeada de amigos. Hab\u00eda decidido vivir con ellos. Mi estatus de dama de un mimetocapo me hab\u00eda protegido de gente como David. Nadie se hab\u00eda atrevido a tocarme nunca en mi propio territorio.\n\nPero aquel no era mi territorio. All\u00ed no ten\u00eda poder. Por primera vez quer\u00eda volver a la protecci\u00f3n que ofrec\u00edan los muros de piedra de Magdalen. Ansiaba la protecci\u00f3n que me garantizaba la presencia del Custodio, aunque la odiara. Me met\u00ed el sobre en el bolsillo y me dirig\u00ed hacia la puerta.\n\nCuando llegu\u00e9 a la Torre del Fundador, esperaba encontrar una habitaci\u00f3n vac\u00eda. Pero lo que encontr\u00e9 fue sangre. Sangre de refa.\n\nLa habitaci\u00f3n estaba patas arriba. Hab\u00eda cristales rotos, instrumentos rotos, una cortina desenganchada de la barra y manchas relucientes de color amarillo verdoso en las losas del suelo y en la alfombra. Entr\u00e9 pisando los cristales. La vela del escritorio estaba apagada, igual que las l\u00e1mparas de parafina. Hac\u00eda un fr\u00edo tremendo. Sent\u00eda el \u00e9ter por todas partes. Me puse en guardia, preparada para lanzarle mi esp\u00edritu a un posible agresor.\n\nLas cortinas de la cama estaban echadas. Detr\u00e1s hab\u00eda otro onirosaje. \u00abRefa\u00bb, pens\u00e9.\n\nFui hacia la cama. Me par\u00e9 a escasa distancia de las cortinas y trat\u00e9 de pensar racionalmente en lo que estaba a punto de hacer. Sab\u00eda que all\u00ed detr\u00e1s estaba el Custodio, pero no sab\u00eda en qu\u00e9 estado. Quiz\u00e1 estuviera herido, o dormido, o muerto. No estaba convencida de querer saberlo.\n\nIntent\u00e9 serenarme. Flexion\u00e9 los dedos antes de asir con ellos la gruesa tela. Apart\u00e9 la cortina.\n\nEstaba desplomado en la cama, quieto como un cad\u00e1ver. Me sub\u00ed a la cama y lo zarande\u00e9.\n\n\u2014\u00bfCustodio?\n\nNada.\n\nMe sent\u00e9 en la cama. El Custodio hab\u00eda hecho hincapi\u00e9 en que no deb\u00eda tocarlo, ni ayudarlo si suced\u00eda algo as\u00ed; pero esta vez los da\u00f1os parec\u00edan mucho m\u00e1s graves. Ten\u00eda la camisa empapada. Intent\u00e9 darle la vuelta, pero pesaba como un muerto. Estaba comprobando si respiraba cuando estir\u00f3 una mano y me agarr\u00f3 la mu\u00f1eca.\n\n\u2014T\u00fa \u2014dijo con voz \u00e1spera\u2014. \u00bfQu\u00e9 haces aqu\u00ed?\n\n\u2014Estaba...\n\n\u2014\u00bfQui\u00e9n te ha visto entrar?\n\nMe qued\u00e9 inm\u00f3vil.\n\n\u2014La portera de noche.\n\n\u2014\u00bfAlguien m\u00e1s?\n\n\u2014No.\n\nEl Custodio se incorpor\u00f3 apoy\u00e1ndose en un codo. Se llev\u00f3 una mano al hombro. Todav\u00eda llevaba puestos los guantes.\n\n\u2014Ya que est\u00e1s aqu\u00ed \u2014dijo\u2014, qu\u00e9date para verme morir. Seguro que te gustar\u00e1.\n\nTemblaba de pies a cabeza. Intent\u00e9 pensar en algo malvado que decirle, pero me sali\u00f3 algo muy diferente:\n\n\u2014\u00bfQu\u00e9 te ha pasado?\n\nNo me contest\u00f3. Estir\u00e9 un brazo y fui a tocarle la cabeza. Me apret\u00f3 m\u00e1s la mu\u00f1eca.\n\n\u2014Tienes que dejar que las heridas respiren \u2014dije.\n\n\u2014Ya lo s\u00e9.\n\n\u2014Pues hazlo.\n\n\u2014No me digas lo que tengo que hacer. Quiz\u00e1 me est\u00e9 muriendo, pero no estoy sometido a tus \u00f3rdenes. Eres t\u00fa la que est\u00e1 sometida a las m\u00edas.\n\n\u2014\u00bfCu\u00e1les son tus \u00f3rdenes?\n\n\u2014Que me dejes morir en paz.\n\nPero a la orden le faltaba fuerza. Le quit\u00e9 la mano del hombro. Ten\u00eda una mordedura.\n\n\u00abZumbador.\u00bb\n\nLe ardieron los ojos, como si alguna sustancia qu\u00edmica vol\u00e1til hubiera reaccionado en su interior. Por un instante cre\u00ed que iba a matarme. Mi esp\u00edritu presion\u00f3 contra los umbrales de mi mente, ansioso por atacar.\n\nEntonces sus dedos me aflojaron la mu\u00f1eca. Escudri\u00f1\u00e9 su rostro.\n\n\u2014Tr\u00e1eme agua \u2014dijo con un hilo de voz\u2014. Y... sal. Busca en la vitrina.\n\nNo ten\u00eda m\u00e1s remedio que obedecer. Abr\u00ed la vitrina; el Custodio me sigui\u00f3 con la mirada. Cog\u00ed un salero de madera, un cuenco dorado y una jarra de agua, junto con unos pa\u00f1os de hilo. El Custodio se desabroch\u00f3 los lazos de la parte superior de la camisa. Ten\u00eda el pecho empapado de sudor.\n\n\u2014En el caj\u00f3n hay unos guantes \u2014dijo apuntando con la barbilla al escritorio\u2014. P\u00f3ntelos.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Haz lo que te digo.\n\nApret\u00e9 las mand\u00edbulas, pero obedec\u00ed.\n\nEn el caj\u00f3n, junto a los guantes, estaba su daga de mango negro, enfundada y limpia. Al verla me detuve un instante; le di la espalda al Custodio y cog\u00ed los guantes. Ni siquiera dejar\u00eda mis huellas dactilares. Con el pulgar, saqu\u00e9 la daga de la funda.\n\n\u2014Yo en tu lugar no lo intentar\u00eda.\n\nMe par\u00e9 en seco.\n\n\u2014El acero no mata a los refa\u00edtas \u2014dijo con voz d\u00e9bil\u2014. Si me clavaras esa daga en el coraz\u00f3n, no dejar\u00eda de latir.\n\nEl silencio se espes\u00f3.\n\n\u2014No te creo \u2014dije\u2014. Podr\u00eda vaciarte las entra\u00f1as. Est\u00e1s demasiado d\u00e9bil para huir.\n\n\u2014Si quieres correr el riesgo, adelante. Pero hazte esta pregunta: \u00bfpor qu\u00e9 dejamos que los casacas rojas lleven armas? Si vuestras armas pudieran matarnos, \u00bfpor qu\u00e9 \u00edbamos a ser tan idiotas como para armar a nuestros prisioneros? \u2014Sus ojos se clavaban en mi espalda\u2014. Muchos lo han intentado. Y ya no est\u00e1n aqu\u00ed.\n\nUn fr\u00edo cosquilleo me recorri\u00f3 el brazo. Dej\u00e9 la daga en el caj\u00f3n.\n\n\u2014No veo por qu\u00e9 deber\u00eda ayudarte \u2014dije\u2014. La \u00faltima vez ni siquiera me lo agradeciste.\n\n\u2014Olvidar\u00e9 que ibas a matarme.\n\nEl reloj de p\u00e9ndulo segu\u00eda el ritmo de mi pulso. Mir\u00e9 por fin hacia atr\u00e1s. El Custodio me miraba; la luz de sus ojos se estaba extinguiendo. Cruc\u00e9 despacio la habitaci\u00f3n y dej\u00e9 las cosas en la mesilla de noche.\n\n\u2014\u00bfQui\u00e9n te ha hecho eso? \u2014pregunt\u00e9.\n\n\u2014Ya lo sabes. \u2014Apoy\u00f3 la espalda en el cabecero; ten\u00eda las mand\u00edbulas r\u00edgidas\u2014. Has estado investigando.\n\n\u2014Un emite.\n\n\u2014S\u00ed.\n\nSu confirmaci\u00f3n me hel\u00f3 la sangre. En silencio, mezcl\u00e9 la sal y el agua en el cuenco. El Custodio me observaba. Moj\u00e9 y retorc\u00ed un pa\u00f1o y me inclin\u00e9 sobre su hombro derecho. La visi\u00f3n de la herida y el olor que desprend\u00eda me echaron atr\u00e1s.\n\n\u2014Esto est\u00e1 gangrenado \u2014dije.\n\nLa herida supuraba y ten\u00eda un color gris\u00e1ceo. Alrededor, la piel estaba ardiendo. Calcul\u00e9 que la temperatura del Custodio deb\u00eda de ser alrededor del doble de la de los humanos; notaba su calor a trav\u00e9s de los guantes. Alrededor de la mordedura, los tejidos empezaban a pudrirse. Lo que necesitaba era un antipir\u00e9tico. No ten\u00eda quinina, que era lo que sol\u00eda utilizar Nick para bajarnos la fiebre. Era f\u00e1cil conseguirla en los bares de ox\u00edgeno (la utilizaban para producir fluorescencias), pero dudaba mucho que la encontrara all\u00ed. Tendr\u00eda que apa\u00f1\u00e1rmelas con aquella soluci\u00f3n salina y un poco de suerte.\n\nEstruj\u00e9 el pa\u00f1o y moj\u00e9 la herida. Los m\u00fasculos del brazo del Custodio se endurecieron y se le marcaron los tendones de la mano.\n\n\u2014Lo siento \u2014dije, y enseguida me arrepent\u00ed, pues \u00e9l no se hab\u00eda disculpado cuando hab\u00eda visto que me marcaban, ni cuando hab\u00eda visto morir a Seb; \u00e9l no sent\u00eda nada.\n\n\u2014Habla \u2014me dijo.\n\nLo mir\u00e9.\n\n\u2014\u00bfQu\u00e9?\n\n\u2014Me duele. Un poco de distracci\u00f3n me vendr\u00e1 bien.\n\n\u2014Como si te interesara lo que yo pueda contarte \u2014dije sin pensar.\n\n\u2014Me interesa \u2014replic\u00f3. Estaba muy sereno, teniendo en cuenta el estado en que se encontraba\u2014. Me gustar\u00eda saber algo m\u00e1s de la persona con la que comparto habitaciones. S\u00e9 que eres una asesina \u2014me puse en tensi\u00f3n\u2014, pero debes de ser algo m\u00e1s que eso. Si no es as\u00ed, me equivoqu\u00e9 al reclamarte.\n\n\u2014Yo no te ped\u00ed que me reclamaras.\n\n\u2014Pero lo hice.\n\nSegu\u00ed lavando la herida. No ve\u00eda por qu\u00e9 ten\u00eda que ser cuidadosa con \u00e9l, as\u00ed que apretaba un poco m\u00e1s de la cuenta.\n\n\u2014Nac\u00ed en Irlanda \u2014dije\u2014. En un pueblo llamado Clonmel. Mi madre era inglesa. Huy\u00f3 de Scion.\n\nAsinti\u00f3 d\u00e9bilmente con la cabeza.\n\n\u2014Viv\u00eda con mi padre y mis abuelos en el Golden Vale \u2014continu\u00e9\u2014, la regi\u00f3n lechera del sur. Era muy bonito. Nada que ver con las ciudadelas de Scion. \u2014Retorc\u00ed el pa\u00f1o y volv\u00ed a empaparlo\u2014. Pero entonces Abel Mayfield dio rienda suelta a su codicia. Quer\u00eda Dubl\u00edn. Estallaron las revueltas de Molly. La masacre de Mayfield.\n\n\u2014Mayfield \u2014dijo el Custodio mirando hacia la ventana\u2014. S\u00ed, me acuerdo de \u00e9l. Un personaje desagradable.\n\n\u2014\u00bfLo conoc\u00edas?\n\n\u2014He conocido a todos los gobernantes de Scion desde 1859.\n\n\u2014Pero si... Eso significa que tienes como m\u00ednimo doscientos a\u00f1os.\n\n\u2014S\u00ed.\n\nIntent\u00e9 no flaquear.\n\n\u2014Cre\u00edamos que est\u00e1bamos a salvo \u2014prosegu\u00ed\u2014, pero al final la violencia lleg\u00f3 al sur, y tuvimos que marcharnos.\n\n\u2014\u00bfQu\u00e9 le pas\u00f3 a tu madre? \u2014El Custodio me miraba a los ojos\u2014. \u00bfSe qued\u00f3 all\u00ed?\n\n\u2014Muri\u00f3. Desprendimiento de placenta. \u2014Me ech\u00e9 hacia atr\u00e1s\u2014. \u00bfD\u00f3nde hay otra mordedura?\n\nEl Custodio se abri\u00f3 la camisa. Ten\u00eda otra herida en el pecho. No habr\u00eda sabido decir si se la hab\u00edan hecho unos dientes, unas garras u otra cosa. Empec\u00e9 a aplicarle agua con el pa\u00f1o, y \u00e9l tens\u00f3 los m\u00fasculos.\n\n\u2014Sigue \u2014dijo.\n\nPor lo visto no me encontraba aburrida.\n\n\u2014Cuando yo ten\u00eda ocho a\u00f1os nos fuimos a Londres.\n\n\u2014\u00bfVoluntariamente?\n\n\u2014No. A mi padre lo reclut\u00f3 la SciOECI ese a\u00f1o. \u2014Interpret\u00e9, por su silencio, que no conoc\u00eda esa abreviatura\u2014. \u00abScion: Organizaci\u00f3n Especial para la Ciencia y la Investigaci\u00f3n.\u00bb\n\n\u2014Conozco las siglas. \u00bfPor qu\u00e9 lo reclutaron?\n\n\u2014Era pat\u00f3logo forense. Trabajaba mucho para la polic\u00eda irlandesa. Scion le pidi\u00f3 que buscara una explicaci\u00f3n cient\u00edfica de por qu\u00e9 la gente se volv\u00eda clarividente, y de por qu\u00e9 los esp\u00edritus persisten despu\u00e9s de la muerte. \u2014Me di cuenta de que hablaba con tono cortante\u2014. Mi padre cree que la clarividencia es una enfermedad. Cree que tiene cura.\n\n\u2014Entonces es que no percibe tu clarividencia.\n\n\u2014Es amaur\u00f3tico. \u00bfC\u00f3mo iba a percibirla?\n\nTras una pausa, el Custodio me pregunt\u00f3:\n\n\u2014\u00bfTienes ese don desde tu nacimiento?\n\n\u2014No del todo. Desde muy peque\u00f1a percib\u00eda auras y esp\u00edritus. Y un d\u00eda me atac\u00f3 una duende. \u2014Me ech\u00e9 hacia atr\u00e1s para secarme el sudor de la frente\u2014. \u00bfCu\u00e1nto tiempo te queda?\n\n\u2014No estoy seguro. La sal retrasa lo inevitable, pero no mucho tiempo \u2014dijo con indiferencia\u2014. \u00bfCu\u00e1ndo desarrollaste la capacidad de desplazar tu esp\u00edritu?\n\nHablar me estaba tranquilizando. Decid\u00ed ser sincera, aunque solo fuera porque seguramente \u00e9l ya lo sab\u00eda todo de m\u00ed. Nashira sab\u00eda que yo era irlandesa; deb\u00edan de tener todo tipo de expedientes. Quiz\u00e1 el Custodio me estuviera poniendo a prueba, comprobando si le ment\u00eda o no.\n\n\u2014Despu\u00e9s de que me atacara la duende, empec\u00e9 a tener un sue\u00f1o recurrente. O eso cre\u00eda yo: que era un sue\u00f1o. \u2014Vert\u00ed un poco m\u00e1s de agua sobre la herida\u2014. So\u00f1aba con un prado de flores. Cuanto m\u00e1s me adentraba en el prado, m\u00e1s oscuro se volv\u00eda. Cada noche iba un poco m\u00e1s lejos, hasta que un d\u00eda llegu\u00e9 al borde y salt\u00e9. Me precipit\u00e9 por el \u00e9ter, fuera de mi cuerpo. Me despert\u00e9 en una ambulancia. Mi padre me dijo que me hab\u00eda levantado son\u00e1mbula, hab\u00eda ido al sal\u00f3n y hab\u00eda dejado de respirar. Dijeron que deb\u00eda de haber entrado en coma.\n\n\u2014Pero sobreviviste.\n\n\u2014S\u00ed. Y no sufr\u00ed da\u00f1os cerebrales. La hipoxia cerebral es uno de los peligros de mi... afecci\u00f3n \u2014a\u00f1ad\u00ed. No me gustaba hablarle de m\u00ed, pero supon\u00eda que era mejor que lo supiera. Si me obligaba a entrar en el \u00e9ter y a quedarme all\u00ed demasiado tiempo sin soporte vital, mi cerebro pod\u00eda sufrir lesiones irreversibles\u2014. Tuve suerte.\n\nEl Custodio observaba mientras yo segu\u00eda lav\u00e1ndole la herida.\n\n\u2014Eso me har\u00eda pensar que, por precauci\u00f3n, no entras en el \u00e9ter muy a menudo \u2014especul\u00f3\u2014; sin embargo, pareces familiarizada con \u00e9l.\n\n\u2014Instinto. \u2014Desvi\u00e9 la mirada\u2014. Si no tomas alg\u00fan medicamento, no te bajar\u00e1 la fiebre.\n\nNo era del todo falso. Mi don era instintivo, pero lo que no pensaba contarle era que me hab\u00eda educado y entrenado un mimetocapo que me manten\u00eda conectada a un equipo de soporte vital.\n\n\u2014Esa duende \u00bfte dej\u00f3 alguna cicatriz?\n\nMe quit\u00e9 un guante y le mostr\u00e9 la mano izquierda. Dej\u00e9 que me examinara las marcas. No era frecuente que un vidente en ciernes se expusiera de forma tan violenta al \u00e9ter.\n\n\u2014Supongo que ya hab\u00eda una brecha en m\u00ed, algo que dej\u00f3 entrar el \u00e9ter \u2014dije\u2014. La duende se limit\u00f3 a... desollarme.\n\n\u2014\u00bfEs as\u00ed como lo ves? \u00bfUna invasi\u00f3n del \u00e9ter?\n\n\u2014\u00bfC\u00f3mo quieres que lo vea?\n\n\u2014No har\u00e9 comentarios sobre mi propia opini\u00f3n. Pero muchos clarividentes explican que invaden el \u00e9ter, y no al rev\u00e9s. Lo ven como si molestaran a los muertos. \u2014No esper\u00f3 a que yo replicara\u2014. He visto eso otras veces. Los ni\u00f1os son vulnerables a los cambios repentinos de su clarividencia. Si se exponen al \u00e9ter antes de que su aura se haya desarrollado del todo, esta puede volverse inestable.\n\nRetir\u00e9 la mano.\n\n\u2014Yo no soy inestable.\n\n\u2014Pero tu don s\u00ed lo es.\n\nNo pod\u00eda discutir. Ya hab\u00eda matado con mi esp\u00edritu; eso era una prueba indudable de inestabilidad.\n\n\u2014Lo que tengo en las heridas es un tipo de necrosis \u2014dijo el Custodio\u2014, pero solo afecta a los refa\u00edtas. El cuerpo humano tiene medios propios para combatirla. \u2014Hizo una pausa; esper\u00e9\u2014. La necrosis de los refa\u00edtas se puede curar con sangre humana. Si el torrente sangu\u00edneo no est\u00e1 afectado, los humanos pueden sobrevivir a una mordedura. \u2014Me se\u00f1al\u00f3 la mu\u00f1eca\u2014. Medio litro de tu sangre me salvar\u00eda la vida.\n\nSe me hizo un nudo en la garganta.\n\n\u2014\u00bfPretendes beberte mi sangre?\n\n\u2014S\u00ed.\n\n\u2014\u00bfQu\u00e9 eres? \u00bfUn vampiro?\n\n\u2014Cre\u00eda que los habitantes de Scion no le\u00edan sobre vampiros.\n\nMe puse en tensi\u00f3n. Menudo fallo. Solo un capo del sindicato pod\u00eda tener acceso a literatura relacionada con los vampiros o con cualquier otro ser sobrenatural. En mi caso era una obra espantosa, _Los vampiros de Vauxhall_ , escrita por un m\u00e9dium an\u00f3nimo de Grub Street. Para compensar la escasez de obras literarias de Scion, se inventaba toda clase de historias a partir de leyendas folcl\u00f3ricas del otro mundo. Sus cuentos ten\u00edan t\u00edtulos como _T\u00e9 con el tase\u00f3grafo_ y _El fiasco de las hadas_. El mismo autor hab\u00eda escrito unos cuantos libros potables sobre videntes, como _Los misterios de la isla de Jacob_. Me arrepent\u00ed de no haberlos le\u00eddo.\n\nEl Custodio debi\u00f3 de interpretar mi silencio como un s\u00edntoma de inquietud.\n\n\u2014No, no soy vampiro, ni ning\u00fan otro ser sobre los que puedas haber le\u00eddo \u2014dijo\u2014. No me alimento de carne ni de sangre. No me produce ninguna satisfacci\u00f3n pedirte este favor. Pero me estoy muriendo, y resulta que tu sangre (en esta ocasi\u00f3n, dado el car\u00e1cter de mis heridas) puede curarme.\n\n\u2014No parece que te est\u00e9s muriendo.\n\n\u2014Cr\u00e9eme, es la verdad.\n\nNo quer\u00eda saber c\u00f3mo hab\u00edan descubierto que la sangre humana pod\u00eda combatir aquella infecci\u00f3n. Ni siquiera sab\u00eda si era verdad.\n\n\u2014\u00bfPor qu\u00e9 iba a confiar en ti? \u2014pregunt\u00e9.\n\n\u2014Porque te salv\u00e9 de la humillaci\u00f3n de tener que actuar en la troupe de imb\u00e9ciles del Capataz. Por poner solo un ejemplo.\n\n\u2014\u00bfY si necesito dos?\n\n\u2014Te deber\u00e9 un favor.\n\n\u2014\u00bfCualquier favor?\n\n\u2014Cualquier cosa menos tu libertad.\n\nMi libertad: las palabras que acababan de morir en mis labios. El Custodio se hab\u00eda anticipado a mi petici\u00f3n. Deb\u00ed saber que era pedir demasiado; sin embargo, quiz\u00e1 fuera \u00fatil que el Custodio me debiera un favor.\n\nCog\u00ed un trozo de cristal del suelo, un fragmento de vial, y me cort\u00e9 la mu\u00f1eca. Le ofrec\u00ed mi sangre; \u00e9l entrecerr\u00f3 los ojos.\n\n\u2014T\u00f3mala \u2014dije\u2014. Antes de que me lo piense mejor.\n\nEl Custodio me mir\u00f3 fijamente, interpretando la expresi\u00f3n de mi cara. Entonces me asi\u00f3 la mu\u00f1eca y se la acerc\u00f3 a la boca.\n\nSu lengua me roz\u00f3 la herida. Not\u00e9 una leve presi\u00f3n cuando sus labios se cerraron sobre ella, mientras me apretaba el brazo para hacer que saliera m\u00e1s sangre. Le lat\u00eda el cuello mientras beb\u00eda a un ritmo constante. No hubo arrebato ni frenes\u00ed. El Custodio enfocaba aquello como un procedimiento m\u00e9dico, cl\u00ednico y distante, ni m\u00e1s ni menos.\n\nCuando me solt\u00f3 la mu\u00f1eca, me ech\u00e9 hacia atr\u00e1s demasiado deprisa. El Custodio me ayud\u00f3 a recostarme sobre las almohadas.\n\n\u2014Despacio \u2014dijo.\n\nFue al cuarto de ba\u00f1o; hab\u00eda recuperado las fuerzas. Cuando volvi\u00f3, llevaba un vaso de agua fr\u00eda. Me puso un brazo detr\u00e1s de la espalda y me ayud\u00f3 a incorporarme. Beb\u00ed. El agua estaba endulzada.\n\n\u2014\u00bfEst\u00e1 Nashira al corriente de todo esto? \u2014pregunt\u00e9, y el rostro del Custodio se ensombreci\u00f3.\n\n\u2014Quiz\u00e1 te interrogue sobre mis ausencias. Y sobre mis heridas \u2014dijo.\n\n\u2014As\u00ed que no sabe nada.\n\nNo me contest\u00f3. Me recost\u00f3 en los gruesos cojines de terciopelo, asegur\u00e1ndose de que ten\u00eda la cabeza bien apoyada. Se me estaban pasando las n\u00e1useas, pero todav\u00eda me sangraba la herida de la mu\u00f1eca. Al verlo, el Custodio cogi\u00f3 un rollo de gasa de la mesilla de noche. Mi gasa. Reconoc\u00ed la goma con que la hab\u00eda sujetado. Deb\u00eda de haberla sacado de mi mochila. Al verla en sus manos, me dio un escalofr\u00edo. Me acord\u00e9 del panfleto desaparecido. \u00bfLo tendr\u00eda \u00e9l? Me agarr\u00f3 la mu\u00f1eca. Sus manos, enormes, me vendaron la herida con cuidado. Supuse que era su forma de darme las gracias. Cuando la sangre dej\u00f3 de traspasar la gasa, me sujet\u00f3 el vendaje con un alfiler y me apoy\u00f3 el brazo sobre el pecho. Yo no dejaba de mirarlo.\n\n\u2014Creo que hemos quedado en tablas \u2014dijo\u2014. Tienes una habilidad especial para sorprenderme en situaciones delicadas. Lo l\u00f3gico ser\u00eda que te complacieras con mis momentos de debilidad, y sin embargo me ofreces tu sangre. Me limpias las heridas. \u00bfPor qu\u00e9 lo haces?\n\n\u2014Quiz\u00e1 necesite un favor. Y no me gusta ver morir a otros. No soy como t\u00fa.\n\n\u2014Juzgas con excesiva ligereza.\n\n\u2014Viste que ella lo mataba. \u2014Deber\u00eda haber temido pronunciar esas palabras, pero ya no me importaba\u2014. Te quedaste mirando. Deb\u00edas de saber lo que ella iba a hacer.\n\nEl Custodio permaneci\u00f3 impasible. Desvi\u00e9 la mirada.\n\n\u2014A lo mejor soy un sepulcro blanqueado \u2014dijo.\n\n\u2014\u00bfUn qu\u00e9?\n\n\u2014Un hip\u00f3crita. Me gusta esa expresi\u00f3n. Ya s\u00e9 que me detestas, pero cumplo mi palabra. \u00bfY t\u00fa? \u00bfCumples la tuya?\n\n\u2014\u00bfAd\u00f3nde quieres llegar?\n\n\u2014Lo que ha sucedido esta noche no debe salir de esta habitaci\u00f3n. Quiero saber si guardar\u00e1s el secreto.\n\n\u2014\u00bfPor qu\u00e9 iba a hacerlo?\n\n\u2014Porque revelarlo no te ayudar\u00e1.\n\n\u2014Podr\u00eda liberarme de ti.\n\nMe pareci\u00f3 que mudaba la expresi\u00f3n.\n\n\u2014S\u00ed, te liberar\u00eda de m\u00ed \u2014dijo\u2014, pero tu vida no mejorar\u00eda. Tal vez te echaran a la calle; o, si no, tal vez te asignaran otro guardi\u00e1n, y no todos son tan liberales como yo. Lo que correspond\u00eda era que te matara a palos por algunas de las cosas que me has dicho estos \u00faltimos d\u00edas. Pero yo s\u00e9 lo que vales. No como otros.\n\nFui a replicar, pero no llegu\u00e9 a hacerlo. Era verdad: no quer\u00eda tener otro guardi\u00e1n, sobre todo si todos eran como Thuban.\n\n\u2014As\u00ed que quieres que guarde tu secreto. \u2014Me frot\u00e9 la mu\u00f1eca\u2014. Y \u00bfqu\u00e9 me ofreces a cambio?\n\n\u2014Intentar\u00e9 protegerte. Aqu\u00ed podr\u00edas morir por infinitas causas, y t\u00fa no pones mucho de tu parte para evitarlas.\n\n\u2014Tarde o temprano tendr\u00e9 que morir. Ya s\u00e9 qu\u00e9 quiere Nashira de m\u00ed. No puedes protegerme.\n\n\u2014Cierto, tal vez no pueda protegerte eternamente. Pero supongo que querr\u00e1s sobrevivir a los ex\u00e1menes.\n\n\u2014\u00bfPara qu\u00e9?\n\n\u2014Puedes demostrarle lo fuerte que eres. T\u00fa no eres una casaca amarilla. Puedes aprender a luchar.\n\n\u2014No quiero luchar.\n\n\u2014S\u00ed quieres. Eres luchadora por naturaleza.\n\nEl reloj del rinc\u00f3n dio la hora.\n\nNo me parec\u00eda correcto tener a un refa como aliado. Por otra parte, eso aumentar\u00eda significativamente mis posibilidades de seguir con vida. El Custodio pod\u00eda ayudarme a conseguir provisiones, a sobrevivir el tiempo suficiente para huir de all\u00ed.\n\n\u2014Muy bien \u2014dije\u2014. No se lo contar\u00e9 a nadie. Pero me debes un favor. \u2014Levant\u00e9 la mu\u00f1eca\u2014. Por la sangre.\n\nAcababa de pronunciar esas palabras cuando la puerta se abri\u00f3 de golpe. Una refa irrumpi\u00f3 en la habitaci\u00f3n: Pleione Sualocin. Primero mir\u00f3 c\u00f3mo estaba la habitaci\u00f3n; luego me mir\u00f3 a m\u00ed y, por \u00faltimo, al Custodio. Sin decir palabra, le lanz\u00f3 un tubo de cristal. El Custodio lo atrap\u00f3 con una mano. Lo mir\u00e9.\n\nSangre. Sangre humana. Llevaba una peque\u00f1a etiqueta gris triangular. Y un n\u00famero: AXIV. Amaur\u00f3tico 14.\n\n\u00abSeb.\u00bb\n\nMir\u00e9 al Custodio; \u00e9l baj\u00f3 la cabeza, como si acab\u00e1ramos de compartir un peque\u00f1o secreto. Me invadi\u00f3 una repugnancia visceral. Me levant\u00e9, d\u00e9bil todav\u00eda por la p\u00e9rdida de sangre, y sub\u00ed atropelladamente la escalera que llevaba a mi celda.\n\nLa primera vez que vi a Nick Nyg\u00e5rd ten\u00eda nueve a\u00f1os. La segunda, ten\u00eda diecis\u00e9is.\n\nEst\u00e1bamos en el trimestre de verano de 2056, y las alumnas de und\u00e9cimo curso de mi colegio del III-5 hab\u00edamos iniciado el per\u00edodo m\u00e1s importante de nuestras vidas. Pod\u00edamos seguir estudiando dos a\u00f1os m\u00e1s y hacer los dos cursos de preparaci\u00f3n para la universidad, o marcharnos y buscar trabajo. En un intento de convertir a las indecisas, la directora hab\u00eda organizado una serie de conferencias de personajes inspiradores: agentes de la DVD, anecdotistas de los medios de comunicaci\u00f3n e, incluso, un pol\u00edtico del Arconte, el ministro de Migraci\u00f3n. Aquel d\u00eda iban a hablarnos de las ciencias m\u00e9dicas. Doscientas alumnas nos api\u00f1\u00e1bamos en la sala de conferencias, con nuestro uniforme negro, nuestros lazos rojos y nuestras blusas blancas. La se\u00f1orita Briskin, la profesora de qu\u00edmica, subi\u00f3 a la tarima.\n\n\u2014Buenos d\u00edas, ni\u00f1as. Me alegro de veros tan contentas y motivadas. Muchas de vosotras hab\u00e9is expresado vuestro inter\u00e9s por la investigaci\u00f3n cient\u00edfica como carrera \u2014no era mi caso\u2014, as\u00ed que esta podr\u00eda ser una de las conferencias m\u00e1s inspiradoras. \u2014Hubo algunos aplausos\u2014. Nuestro orador de hoy ya tiene una carrera sumamente emocionante. \u2014Yo no estaba tan convencida\u2014. Lo transfirieron de la Universidad de Scion Estocolmo en 2046, termin\u00f3 sus estudios en Londres, y trabaja para la SciOECI, el mayor complejo de investigaci\u00f3n de la cohorte central. Es un gran honor para nosotras contar con su presencia. \u2014En las primeras filas se apreci\u00f3 cierto revuelo\u2014. Por favor, recibamos con un fuerte aplauso a nuestro orador, el doctor Nicklas Nyg\u00e5rd.\n\nLevant\u00e9 la cabeza de golpe. Era \u00e9l. Nick.\n\nNo hab\u00eda cambiado nada. Era tal como yo lo recordaba: alto, de facciones suaves, guapo. Todav\u00eda joven, aunque en sus ojos se adivinaba la carga de una vida adulta. Vest\u00eda traje negro y corbata roja, como todos los funcionarios de Scion. Llevaba el cabello peinado hacia atr\u00e1s con gomina, un estilo de moda en Estocolmo. Sonri\u00f3, y las monitoras se enderezaron en sus asientos.\n\n\u2014Buenos d\u00edas, se\u00f1oritas.\n\n\u2014Buenos d\u00edas, doctor Nyg\u00e5rd.\n\n\u2014En primer lugar, gracias por invitarme. \u2014Baraj\u00f3 sus hojas con las mismas manos que me hab\u00edan suturado el brazo herido cuando yo ten\u00eda nueve a\u00f1os. Me mir\u00f3 fijamente y sonri\u00f3. Se me aceler\u00f3 el coraz\u00f3n\u2014. Espero que esta charla resulte instructiva, pero no me ofender\u00e9 si os qued\u00e1is dormidas.\n\nRisas. Los funcionarios no sol\u00edan ser tan chistosos. Yo no pod\u00eda quitarle los ojos de encima. Siete a\u00f1os pregunt\u00e1ndome d\u00f3nde podr\u00eda estar, y de pronto \u00e9l ven\u00eda a mi colegio. Una imagen extra\u00edda de mi memoria. Nos habl\u00f3 de sus investigaciones sobre las causas de la antinaturalidad, de sus experiencias como estudiante en dos ciudadelas de Scion. Brome\u00f3 y anim\u00f3 al p\u00fablico a participar, haciendo preguntas y contestando las que le hac\u00edan las alumnas. Hasta hizo sonre\u00edr a la directora. Cuando son\u00f3 el timbre, sal\u00ed antes que nadie de la sala y me dirig\u00ed al pasillo.\n\nTen\u00eda que encontrarlo. Llevaba siete a\u00f1os intentando entender qu\u00e9 hab\u00eda pasado en el prado de amapolas. All\u00ed no hab\u00eda ning\u00fan perro. \u00c9l era el \u00fanico que pod\u00eda explicarme qu\u00e9 era lo que me hab\u00eda dejado aquellas cicatrices fr\u00edas en la mano. El \u00fanico que pod\u00eda darme respuestas.\n\nEnfil\u00e9 el pasillo abri\u00e9ndome paso entre un grupo de alumnas parlanchinas de octavo curso. All\u00ed estaba, ante la puerta de la sala de profesoras, estrech\u00e1ndole la mano a la directora. Al verme se le ilumin\u00f3 la mirada.\n\n\u2014Hola \u2014dijo.\n\n\u2014Doctor Nyg\u00e5rd... \u2014Apenas pod\u00eda articular palabra\u2014. Su discurso ha sido... muy inspirador.\n\n\u2014Gracias. \u2014Volvi\u00f3 a sonre\u00edr, y me taladr\u00f3 con la mirada. Lo sab\u00eda. Se acordaba\u2014. \u00bfC\u00f3mo te llamas?\n\nS\u00ed, lo sab\u00eda. Not\u00e9 un cosquilleo en las palmas de las manos.\n\n\u2014Es Paige Mahoney \u2014se adelant\u00f3 la directora poniendo \u00e9nfasis en mi apellido. Mi apellido irland\u00e9s. Me mir\u00f3 de arriba abajo fij\u00e1ndose en mi lazo mal hecho y en mi blazer desabrochado\u2014. Ser\u00e1 mejor que te vayas a clase, Paige. La se\u00f1orita Anville est\u00e1 muy disgustada por tus ausencias.\n\nSe me encendieron las mejillas.\n\n\u2014Estoy seguro de que la se\u00f1orita Anville podr\u00e1 prescindir de Paige unos minutos \u2014dijo Nick, y compuso una sonrisa irresistible\u2014. Me encantar\u00eda charlar un poco con ella.\n\n\u2014Es usted muy amable, doctor Nyg\u00e5rd, pero \u00faltimamente Paige ha tenido que quedarse con frecuencia en la enfermer\u00eda. Va un poco retrasada y necesita asistir a todas sus clases. \u2014Se volvi\u00f3 hacia \u00e9l y, bajando la voz, a\u00f1adi\u00f3\u2014: Es irlandesa. Estos irlandeses tienen sus propias ideas respecto a la gesti\u00f3n del tiempo y el trabajo.\n\nDe pronto se me nubl\u00f3 la vista. Not\u00e9 una fuerte presi\u00f3n en el cr\u00e1neo, como si estuviera a punto de explotarme. A la directora le sali\u00f3 un hilillo de sangre de la nariz.\n\n\u2014Est\u00e1 sangrando, se\u00f1orita \u2014dije.\n\n\u2014\u00bfC\u00f3mo? \u2014Mir\u00f3 hacia abajo, y la sangre le manch\u00f3 la blusa\u2014. \u00a1Oh! \u00a1Qu\u00e9 desastre! \u2014Se tap\u00f3 la nariz\u2014. No te quedes ah\u00ed con la boca abierta, Paige. Ve a buscarme un pa\u00f1uelo.\n\nIba a estallarme la cabeza. Ante mis ojos se extendi\u00f3 una red gris que restring\u00eda mi visi\u00f3n. Nick me mir\u00f3 fijamente mientras le daba un paquete de pa\u00f1uelos de papel a la directora.\n\n\u2014Quiz\u00e1 deber\u00eda sentarse, directora. \u2014Le puso una mano en la espalda\u2014. Enseguida volver\u00e9 con usted.\n\nEn cuanto la directora se march\u00f3, Nick me mir\u00f3 y dijo:\n\n\u2014\u00bfHa pasado esto otras veces? \u00bfAlguien m\u00e1s ha sangrado por la nariz estando t\u00fa cerca?\n\nLo dijo en voz baja. Al cabo de un momento asent\u00ed con la cabeza.\n\n\u2014\u00bfAlguien se ha fijado?\n\n\u2014Nunca me han llamado antinatural. Todav\u00eda. \u2014Busqu\u00e9 su mirada\u2014. \u00bfSabes por qu\u00e9 pasa esto?\n\nGir\u00f3 la cabeza; luego dijo:\n\n\u2014Es posible.\n\n\u2014Expl\u00edcamelo. Por favor.\n\n\u2014\u00bfDoctor Nyg\u00e5rd? \u2014La se\u00f1orita Briskin asom\u00f3 la cabeza por la puerta de la sala de profesores\u2014. El consejo escolar quiere hablar con usted.\n\n\u2014Voy enseguida.\n\nLa se\u00f1orita Briskin se march\u00f3, y Nick me dijo al o\u00eddo:\n\n\u2014Volver\u00e9 dentro de unos d\u00edas. No te matricules en la universidad, Paige. Todav\u00eda no. Conf\u00eda en m\u00ed.\n\nMe dio un apret\u00f3n en la mano. Y de pronto desapareci\u00f3, tan aprisa como hab\u00eda venido. Me qued\u00e9 con mis libros apretados contra el pecho, donde el coraz\u00f3n me lat\u00eda desbocado; ten\u00eda las mejillas ardiendo y las manos sudadas. No hab\u00eda dejado de pensar en Nick ni un solo d\u00eda, y ahora \u00e9l hab\u00eda regresado. Me seren\u00e9 y fui a mi aula; todav\u00eda me costaba ver y pensar. Nick se acordaba de mi nombre. Sab\u00eda que yo era aquella ni\u00f1a peque\u00f1a a la que hab\u00eda salvado.\n\nDudaba mucho que volviera. Yo no pod\u00eda ser tan importante para \u00e9l, sobre todo ahora que \u00e9l hab\u00eda triunfado en el mundo. Pero dos d\u00edas m\u00e1s tarde lo encontr\u00e9 esper\u00e1ndome en la puerta del colegio. Esa ma\u00f1ana hab\u00eda pasado una cosa muy extra\u00f1a: hab\u00eda so\u00f1ado despierta con un coche plateado. La imagen me hab\u00eda asaltado cuando estaba en clase de franc\u00e9s, y me hab\u00eda producido n\u00e1useas. Ahora ese mismo coche estaba all\u00ed fuera, y Nick estaba sentado al volante. Llevaba gafas de sol. Fui como son\u00e1mbula hasta la portezuela del coche, alej\u00e1ndome de las otras ni\u00f1as. Nick asom\u00f3 la cabeza por la ventana.\n\n\u2014Hola, Paige.\n\n\u2014No cre\u00ed que volvieras \u2014dije.\n\n\u2014Por lo que le pas\u00f3 a la directora.\n\n\u2014S\u00ed.\n\n\u2014He venido precisamente por eso. \u2014Se baj\u00f3 un poco las gafas para que pudiera verle los ojos. Parec\u00eda cansado\u2014. Si quieres saber m\u00e1s, puedo explic\u00e1rtelo, pero no aqu\u00ed. \u00bfQuieres venir conmigo?\n\nGir\u00e9 un momento la cabeza. Las otras alumnas no nos estaban mirando.\n\n\u2014Vale \u2014dije.\n\n\u2014Gracias.\n\nNick me llev\u00f3 lejos del colegio. Mientras circul\u00e1bamos hacia la cohorte central, me miraba de vez en cuando. Yo guardaba silencio. Me mir\u00e9 en el espejo retrovisor y vi que estaba colorada. Me mor\u00eda de ganas de hablar con \u00e9l, pero era incapaz de articular una sola frase coherente. Al cabo de unos minutos Nick dijo:\n\n\u2014\u00bfLe contaste a tu padre lo que hab\u00eda pasado en el prado?\n\n\u2014No.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Porque t\u00fa me dijiste que no se lo contara a nadie.\n\n\u2014Muy bien. \u2014Apret\u00f3 m\u00e1s el volante\u2014. Voy a contarte un mont\u00f3n de cosas que no entender\u00e1s, Paige. Ya no eres la que eras antes de aquel d\u00eda, y debes saber por qu\u00e9.\n\nMantuve la mirada al frente. No hac\u00eda falta que Nick me lo dijera. Yo ya sab\u00eda que era diferente mucho antes del incidente del prado de las amapolas; desde muy peque\u00f1a ten\u00eda una sensibilidad especial. A veces notaba un ligero temblor cuando alguien pasaba a mi lado, como si mis dedos hubieran rozado un cable con corriente. Pero a partir de aquel d\u00eda las cosas hab\u00edan cambiado. Ahora no solo sent\u00eda a la gente, sino que pod\u00eda hacerle da\u00f1o. Pod\u00eda hacer que sangrara, que le doliera la cabeza y que se le empa\u00f1aran los ojos. A veces me quedaba dormida en clase, y al cabo de un rato despertaba empapada de un sudor fr\u00edo. La enfermera del colegio me conoc\u00eda mejor que a ninguna otra alumna.\n\nAlgo estaba emergiendo de mi interior, empujando para salir al mundo. Al final, el mundo lo ver\u00eda.\n\n\u2014Yo puedo ayudarte a controlarlo \u2014dijo\u2014. Puedo protegerte.\n\nYa me hab\u00eda protegido una vez.\n\n\u2014\u00bfTodav\u00eda puedo confiar en ti?\n\nEscudri\u00f1\u00e9 su cara, esa cara que no hab\u00eda olvidado. Nick me mir\u00f3 fijamente.\n\n\u2014Siempre \u2014respondi\u00f3.\n\nFuimos a un bar de trabajadores de Silk Street y pedimos caf\u00e9. Era la primera vez que lo probaba y, aunque no dije nada, me pareci\u00f3 que sab\u00eda a barro. Hablamos un rato de mi vida. Le habl\u00e9 del colegio, del trabajo de mi padre; pero no era eso por lo que est\u00e1bamos all\u00ed, y ambos lo sab\u00edamos.\n\n\u2014Paige, habr\u00e1s o\u00eddo hablar de la antinaturalidad \u2014dijo\u2014. No quiero asustarte, pero muestras signos de ella.\n\nSe me hizo un nudo en la garganta. Nick no trabajaba para Scion.\n\n\u2014No te preocupes. \u2014Puso una mano sobre la m\u00eda, y me sent\u00ed reconfortada\u2014. No voy a delatarte. Voy a ayudarte.\n\n\u2014\u00bfC\u00f3mo?\n\n\u2014Me gustar\u00eda que vinieras a hablar con un amigo m\u00edo.\n\n\u2014\u00bfQui\u00e9n es?\n\n\u2014Una persona en la que conf\u00edo. Una persona que est\u00e1 interesada en ti.\n\n\u2014\u00bfEs...?\n\n\u2014S\u00ed. Yo tambi\u00e9n lo soy. \u2014Me apret\u00f3 la mano\u2014. Antes has tenido una enso\u00f1aci\u00f3n. Has visto mi coche. \u2014Me qued\u00e9 mir\u00e1ndolo, perpleja\u2014. Ese es mi don, Paige. Puedo enviar im\u00e1genes. Puedo hacer que los dem\u00e1s vean cosas.\n\n\u2014Ir\u00e9... \u2014Ten\u00eda la boca seca\u2014. Ir\u00e9 a hablar con \u00e9l.\n\nLe dej\u00e9 recado a la secretaria de mi padre de que volver\u00eda tarde a casa. Nick me llev\u00f3 en su coche a un peque\u00f1o restaurante franc\u00e9s de Vauxhall. All\u00ed nos esperaba un hombre alto y delgado de cerca de cuarenta a\u00f1os. Su mirada, chispeante, revelaba inteligencia y una especie de agitaci\u00f3n nerviosa. Ten\u00eda la tez blanca como la cera y una densa mata de pelo casta\u00f1o oscuro, y los labios p\u00e1lidos y ligeramente fruncidos. Los p\u00f3mulos eran muy marcados. Llevaba un fular amarillo y un chaleco negro bordado, con reloj de bolsillo.\n\n\u2014T\u00fa debes de ser Paige \u2014dijo con una voz grave, con un deje de jocosidad\u2014. Yo soy Jaxon Hall.\n\nMe tendi\u00f3 una mano huesuda, y se la estrech\u00e9.\n\n\u2014Hola \u2014dije.\n\nMe dio un apret\u00f3n firme y formal. Me sent\u00e9. Nick se sent\u00f3 a mi lado.\n\nCuando vino el camarero, Jaxon Hall no pidi\u00f3 nada de comer, sino solo un vaso de _mecks_ , un vino sin alcohol. Era una bebida muy cara, lo que indicaba que ten\u00eda gustos sofisticados.\n\n\u2014Quiero hacerte una propuesta, se\u00f1orita Mahoney. \u2014Jaxon Hall dio un gran sorbo de _mecks_ \u2014. El doctor Nyg\u00e5rd vino a verme ayer y me inform\u00f3 de que puedes infligir ciertas... anomal\u00edas m\u00e9dicas a otras personas. \u00bfCorrecto?\n\nMir\u00e9 a Nick.\n\n\u2014Tranquila \u2014me dijo, y sonri\u00f3\u2014. No es de Scion.\n\n\u2014No me insultes, por favor. \u2014Jaxon dio otro sorbo de _mecks_ \u2014. Estoy m\u00e1s lejos del Arconte que la cuna de la tumba. Aunque esos dos estados tampoco est\u00e1n muy alejados, pero t\u00fa ya me entiendes.\n\nNo estaba segura de entenderlo. Desde luego, no se comportaba como un funcionario de Scion.\n\n\u2014\u00bfSe refiere a las hemorragias nasales? \u2014pregunt\u00e9.\n\n\u2014S\u00ed, exactamente. Fascinante. \u2014Ten\u00eda las manos entrelazadas encima de la mesa\u2014. \u00bfAlgo m\u00e1s?\n\n\u2014Dolores de cabeza. A veces, migra\u00f1as.\n\n\u2014Y \u00bfc\u00f3mo te sientes cuando eso pasa?\n\n\u2014Cansada. Mareada.\n\n\u2014Entiendo. \u2014Recorri\u00f3 mi rostro con la mirada. Ten\u00eda unos ojos fr\u00edos y escrutadores; me dio la impresi\u00f3n de que ve\u00eda m\u00e1s all\u00e1 de m\u00ed\u2014. \u00bfQu\u00e9 edad tienes?\n\n\u2014Diecis\u00e9is \u2014contest\u00e9.\n\n\u2014Estar\u00e1s a punto de terminar los estudios. A menos \u2014a\u00f1adi\u00f3\u2014 que te propongan ir a la universidad.\n\n\u2014No lo creo.\n\n\u2014Excelente. Pero los j\u00f3venes aspir\u00e1is a encontrar trabajo en la ciudadela. \u2014Tamborile\u00f3 con los dedos en la mesa\u2014. Me gustar\u00eda ofrecerte un empleo para toda la vida.\n\nFrunc\u00ed el entrecejo.\n\n\u2014\u00bfQu\u00e9 clase de empleo?\n\n\u2014Un empleo bien pagado. Un empleo que te proteger\u00e1. \u2014Jaxon escudri\u00f1\u00f3 mi cara\u2014. \u00bfSabes qu\u00e9 es la clarividencia?\n\n\u00abClarividencia.\u00bb La palabra prohibida. Ech\u00e9 un vistazo alrededor, pero nadie nos miraba. Ni nos escuchaba, o eso me pareci\u00f3.\n\n\u2014Es lo mismo que... antinaturalidad \u2014dije.\n\nJaxon esboz\u00f3 una sonrisa.\n\n\u2014As\u00ed la llama el Arconte. Pero \u00bfsabes qu\u00e9 significa esa palabra?\n\n\u2014Es... una especie de percepci\u00f3n extrasensorial. Saber cosas que est\u00e1n ocultas.\n\n\u2014Y \u00bfd\u00f3nde est\u00e1n escondidas?\n\nVacil\u00e9 un momento.\n\n\u2014\u00bfEn el subconsciente?\n\n\u2014S\u00ed, a veces. Y otras veces \u2014apag\u00f3 de un soplido la vela del centro de la mesa\u2014 en el \u00e9ter.\n\nMe qued\u00e9 mirando el humo, atra\u00edda por \u00e9l. Me recorri\u00f3 un escalofr\u00edo.\n\n\u2014\u00bfQu\u00e9 es el \u00e9ter?\n\n\u2014El infinito. Provenimos de \u00e9l, vivimos dentro de \u00e9l, y al morir volvemos a \u00e9l. Pero no todos nosotros estamos dispuestos a despedirnos del mundo f\u00edsico.\n\n\u2014Jax \u2014dijo Nick en voz baja\u2014, esto iba a ser una introducci\u00f3n, no una conferencia en toda regla. Paige tiene diecis\u00e9is a\u00f1os.\n\n\u2014Quiero saberlo \u2014dije.\n\n\u2014Paige...\n\n\u2014Por favor \u2014insist\u00ed; necesitaba saberlo.\n\nSu expresi\u00f3n se suaviz\u00f3. Se recost\u00f3 en el asiento y bebi\u00f3 un poco de agua.\n\n\u2014Como quieras.\n\nJaxon, que nos miraba con las cejas enarcadas, frunci\u00f3 los labios antes de continuar.\n\n\u2014El \u00e9ter es un plano de existencia superior \u2014dijo\u2014. Existe paralelamente al plano corporal. Los clarividentes, las personas como nosotros, tienen la capacidad de recurrir al \u00e9ter.\n\nEstaba sentada en un restaurante con dos antinaturales.\n\n\u2014\u00bfC\u00f3mo? \u2014pregunt\u00e9.\n\n\u2014Bueno, hay infinitas formas de hacerlo. Llevo quince a\u00f1os tratando de clasificarlas.\n\n\u2014Pero \u00bfqu\u00e9 significa \u00abrecurrir al \u00e9ter\u00bb?\n\nHacer preguntas sobre la clarividencia era una transgresi\u00f3n sumamente emocionante.\n\n\u2014Significa que puedes comunicarte con los esp\u00edritus \u2014aclar\u00f3 Nick\u2014. Con los difuntos. Las diferentes clases de vidente pueden hacerlo de diferentes maneras.\n\n\u2014Entonces \u00bfel \u00e9ter es como la otra vida?\n\n\u2014Es... un purgatorio, por decirlo as\u00ed \u2014dijo Jaxon.\n\n\u2014S\u00ed, es como la otra vida \u2014a\u00f1adi\u00f3 Nick.\n\n\u2014Tendr\u00e1s que disculpar al doctor Nyg\u00e5rd. Solo intenta ser delicado. \u2014Jaxon bebi\u00f3 m\u00e1s _mecks_ \u2014. Por desgracia, la muerte no es delicada. Me gustar\u00eda instruirte sobre el verdadero significado de la clarividencia, muy alejado del enfoque tristemente sesgado de esa condici\u00f3n que hace Scion. Es un milagro, no una perversi\u00f3n. Tienes que entenderlo, querida, o apagar\u00e1n ese adorable resplandor.\n\nSe quedaron ambos callados cuando el camarero me trajo la ensalada. Volv\u00ed a mirar a Jax.\n\n\u2014Cu\u00e9ntame m\u00e1s cosas.\n\nJaxon sonri\u00f3.\n\n\u2014El \u00e9ter es el \u00aborigen\u00bb del que en ocasiones Scion se atreve a hablar \u2014dijo\u2014. El reino de los muertos sin reposo. La fuente a la que presuntamente tuvo acceso su Rey Sangriento durante una sesi\u00f3n espiritista, lo que le hizo cometer cinco asesinatos espantosos y desencadenar una epidemia de clarividencia que se extendi\u00f3 por todo el mundo. Todo eso son sandeces, por supuesto. El \u00e9ter no es m\u00e1s que el plano espiritual, y los clarividentes son las personas con capacidad para acceder a \u00e9l. No hubo ninguna epidemia. Siempre hemos existido. Algunos somos buenos; otros son malos, si es que existe la maldad. Pero, seamos lo que seamos, no somos una enfermedad.\n\n\u2014Entonces Scion minti\u00f3.\n\n\u2014S\u00ed. Acost\u00fambrate a esa idea. \u2014Jaxon encendi\u00f3 un puro\u2014. Eduardo bien pudo ser Jack el Destripador, pero dudo mucho que fuera clarividente. Era demasiado torpe.\n\n\u2014No tenemos ni idea de por qu\u00e9 lo atribuyeron todo a la clarividencia \u2014intervino Nick\u2014. Es un misterio que solo entiende el Arconte.\n\n\u2014\u00bfC\u00f3mo funciona?\n\nTen\u00eda la piel de gallina. Cab\u00eda la posibilidad de que yo fuera clarividente. Cab\u00eda la posibilidad de que fuera uno de ellos.\n\n\u2014No todos los esp\u00edritus entran pac\u00edficamente en el coraz\u00f3n del \u00e9ter, donde creemos que las personas hallan una especie de muerte definitiva \u2014prosigui\u00f3 Jaxon. Se notaba que estaba disfrutando con aquella disertaci\u00f3n\u2014. Algunos se entretienen, y vagan entre el plano corp\u00f3reo y el plano espiritual. Mientras se hallan en ese estado, los llamamos vagabundos. Conservan su personalidad, y con la mayor\u00eda se puede establecer contacto. Solo tienen cierto grado de libertad, y normalmente no tienen inconveniente en ayudar a los videntes.\n\n\u2014Est\u00e1s hablando de personas reales. Personas muertas \u2014dije\u2014. \u00bfMueves las cuerdas y ellos bailan?\n\n\u2014Correcto.\n\n\u2014Pero \u00bfpor qu\u00e9 lo hacen?\n\n\u2014Porque de ese modo pueden quedarse cerca de sus seres queridos. \u2014Dio un resoplido, como si no pudiera entenderlo\u2014. O cerca de personas a las que quieren acosar. Sacrifican su libre albedr\u00edo por una especie de inmortalidad.\n\nEmpec\u00e9 a comerme la ensalada. Era como masticar un pedazo de algod\u00f3n mojado.\n\n\u2014No comienzan como esp\u00edritus, por supuesto. \u2014Jaxon me dio unos toques en el dorso de la mano\u2014. T\u00fa tienes un cuerpo f\u00edsico. Puedes caminar en el plano corp\u00f3reo. Pero tambi\u00e9n tienes una conexi\u00f3n \u00edntima con el \u00e9ter. Nosotros lo llamamos onirosaje. El paisaje de la mente humana.\n\n\u2014Un momento, un momento. Hablas todo el rato en plural. \u00bfQui\u00e9nes son ese plural, concretamente? \u00bfLos clarividentes?\n\n\u2014S\u00ed. Es una comunidad efervescente. \u2014Nick me sonri\u00f3\u2014. Pero muy discreta.\n\n\u2014Se puede identificar a los videntes por el aura. As\u00ed fue como te reconoci\u00f3 Nick \u2014continu\u00f3 Jaxon. Mi creciente inter\u00e9s lo estaba animando\u2014. Mira, todos tenemos un onirosaje. Una ilusi\u00f3n de seguridad, una especie de _locus_ _amoenus_. Lo entiendes, \u00bfno? \u2014Yo no estaba muy segura de entenderlo\u2014. Los videntes tienen onirosajes de colores. Los dem\u00e1s los tienen en blanco y negro. Ven su onirosaje cuando sue\u00f1an. Por tanto, los amaur\u00f3ticos sue\u00f1an en blanco y negro. Los videntes, en cambio...\n\n\u2014\u00bfSue\u00f1an en color?\n\n\u2014Los videntes no sue\u00f1an, querida m\u00eda. O no sue\u00f1an como los amaur\u00f3ticos. Ese placer fr\u00edvolo solo lo tienen ellos. Pero el color del onirosaje de un clarividente atraviesa su forma corp\u00f3rea y crea un aura. Los diferentes tipos de vidente tienden a tener auras parecidas. Ya aprender\u00e1s a clasificarlos.\n\n\u2014\u00bfYo puedo ver las auras?\n\nJax y Nick se miraron. Nick se llev\u00f3 las manos hacia la cara y se quit\u00f3 dos finas lentes de contacto de los ojos. Me dio un escalofr\u00edo.\n\n\u2014M\u00edrame los ojos, Paige.\n\nNo hizo falta que me lo dijera dos veces. Recordaba aquellos ojos como si hubiera sido ayer. Aquel verde gris\u00e1ceo exquisito, aquellas finas l\u00edneas que irradiaban de sus iris. En lo que no me hab\u00eda fijado era en el peque\u00f1o defecto, con forma de ojo de cerradura, que ten\u00eda en la pupila derecha.\n\n\u2014Algunos videntes tienen una especie de tercer ojo. \u2014Se recost\u00f3 en el asiento\u2014. Pueden ver auras, y tambi\u00e9n pueden ver a los vagabundos. Puedes tener visi\u00f3n parcial, como yo, con un solo coloboma, o visi\u00f3n completa, como Jax.\n\nJaxon me mir\u00f3 y abri\u00f3 mucho los ojos. Ten\u00eda aquel defecto en los dos ojos.\n\n\u2014Yo no tengo eso \u2014dije\u2014. \u00bfSignifica que soy clarividente, pero que no tengo un tercer ojo?\n\n\u2014La carencia de visi\u00f3n es muy frecuente en los \u00f3rdenes m\u00e1s elevados. Tu don no requiere que veas esp\u00edritus. \u2014Jaxon me mir\u00f3 complacido\u2014. T\u00fa sientes las auras y a los vagabundos, pero no los percibes visualmente.\n\n\u2014En realidad, eso no es ning\u00fan inconveniente \u2014aport\u00f3 Nick\u2014. Sin la ayuda adicional de la visi\u00f3n, tu sexto sentido estar\u00e1 mucho m\u00e1s afinado.\n\nPese a que el restaurante estaba caldeado, el fr\u00edo se estaba extendiendo por todo mi cuerpo. Mir\u00e9 a los dos hombres, aquellas dos caras tan distintas.\n\n\u2014Y yo \u00bfqu\u00e9 tipo de clarividente soy?\n\n\u2014Eso es lo que queremos averiguar. A lo largo de los a\u00f1os he clasificado siete \u00f3rdenes de clarividencia. Creo que t\u00fa, querida m\u00eda, perteneces al orden m\u00e1s elevado, lo que te convierte en una de las clarividentes m\u00e1s singulares del mundo moderno. Y como creo que no estoy equivocado... \u2014sac\u00f3 una carpeta de su lujosa cartera de piel\u2014, me gustar\u00eda que firmaras un contrato de trabajo.\n\nMe mir\u00f3 a los ojos.\n\n\u2014Podr\u00eda anotar una variedad infinita de n\u00fameros en este cheque, Paige. \u00bfCu\u00e1nto har\u00e1 falta para que te quedes conmigo?\n\nEl coraz\u00f3n me martilleaba en el pecho.\n\n\u2014Una copa, para empezar.\n\nJaxon se ech\u00f3 para atr\u00e1s.\n\n\u2014Nick, p\u00eddele un _mecks_ a esta se\u00f1orita. Se queda con nosotros.\n\nEl Custodio y yo pasamos varias noches sin hablarnos. Tampoco reanudamos mis entrenamientos. Todas las noches me marchaba nada m\u00e1s sonar la campana, y al pasar a su lado ni siquiera lo miraba. \u00c9l s\u00ed me miraba, pero nunca me imped\u00eda salir. A veces yo deseaba que lo hiciera, porque as\u00ed habr\u00eda podido desahogar mi rabia.\n\nUna noche intent\u00e9 ir a ver a Liss. Estaba lloviendo, y quer\u00eda calentarme junto a su hornillo. Pero no pod\u00eda ir, despu\u00e9s de lo que hab\u00eda pasado con el Custodio. Tras haber ayudado otra vez al enemigo, no me habr\u00eda atrevido a mirar a Liss a los ojos.\n\nPronto encontr\u00e9 un nuevo refugio, un lugar donde pod\u00eda estar a solas: el soportal que cubr\u00eda los escalones de la entrada de Hawksmoor. En su d\u00eda deb\u00eda de haber sido una construcci\u00f3n majestuosa, pero ahora esa misma grandeza le confer\u00eda un aire tr\u00e1gico: fr\u00eda, triste y con las esquinas erosionadas, se dir\u00eda que aguardaba el regreso de una \u00e9poca que tal vez no volviera nunca. Ese rinc\u00f3n se convirti\u00f3 en mi refugio. Iba all\u00ed todas las noches y, si no hab\u00eda arrancahuesos de guardia, me colaba en la biblioteca abandonada y me llevaba un mont\u00f3n de libros al soportal. Hab\u00eda tantas novelas prohibidas que empec\u00e9 a preguntarme si ser\u00eda all\u00ed adonde Scion las enviaba todas. Jax habr\u00eda vendido su alma para hacerse con ellas. Si hubiera tenido un alma que vender.\n\nHab\u00edan pasado cuatro noches desde la sangr\u00eda. Yo segu\u00eda sin entender por qu\u00e9 hab\u00eda ayudado al Custodio. \u00bfA qu\u00e9 estaba jugando conmigo? Pensar que se hab\u00eda bebido mi sangre me produc\u00eda n\u00e1useas. No quer\u00eda ni acordarme de lo que hab\u00eda hecho.\n\nLlov\u00eda mucho; hab\u00eda decidido quedarme dentro, en la biblioteca, y hab\u00eda dejado una ventana entreabierta. Si ven\u00edan a buscarme, los oir\u00eda. No dejar\u00eda que me pillaran desprevenida, como en el I-5. Hab\u00eda encontrado un libro titulado _Otra vuelta de tuerca_ escondido entre los estantes. Me tumb\u00e9 boca abajo bajo un escritorio y encend\u00ed una lamparita de aceite.\n\nEn el Broad reinaba el silencio. La mayor\u00eda de los bufones estaban empezando a ensayar para la celebraci\u00f3n del Bicentenario. Se hab\u00eda extendido el rumor de que el Gran Inquisidor en persona iba a asistir al acto. Ten\u00edan que conseguir impresionarlo con nuestra nueva forma de vida, o quiz\u00e1 no permitiera que se prolongara aquel \u00abacuerdo especial\u00bb. Aunque no ten\u00eda mucha alternativa. Con todo, ten\u00edamos que demostrarle que \u00e9ramos \u00fatiles, aunque solo fuera para entretener. Que val\u00edamos un poco m\u00e1s de lo que costar\u00eda administrarnos NiteKind.\n\nSaqu\u00e9 el sobre que me hab\u00eda dado David. Dentro hab\u00eda una hoja amarillenta arrancada de un cuaderno, con un fragmento de texto. La examin\u00e9. Parec\u00eda como si se le hubiera ca\u00eddo una vela encima: las esquinas estaban duras, impregnadas de cera, y en el medio hab\u00eda un gran agujero producido por una quemadura. En una esquina de la hoja hab\u00eda un boceto emborronado; parec\u00eda una cara, pero estaba deformada y descolorida. Solo pude descifrar unas pocas palabras.\n\n\u2013 \u2013 refa\u00edtas son \u2013 \u2013 de doble \u2013 \u2013 En el \u2013 \u2013 llamados \u2013 \u2013 confinado \u2013 \u2013 pueden \u2013 \u2013 per\u00edodos de tiempo ilimitados, pero \u2013 \u2013 nueva forma, cuando \u2013 \u2013 hambre \u2013 \u2013 incontrolable e \u2013 \u2013 energ\u00eda alrededor de la presunta \u2013 \u2013 flor roja, la \u2013 \u2013 \u00fanico m\u00e9todo \u2013 \u2013 naturaleza del \u2013 \u2013 y solo entonces pueden \u2013 \u2013\n\nIntent\u00e9 hilvanar otra vez las palabras, encontrar alg\u00fan patr\u00f3n. No era muy dif\u00edcil conectar los fragmentos sobre el hambre y la energ\u00eda, pero no se me ocurr\u00eda qu\u00e9 pod\u00eda significar la flor roja.\n\nEl sobre conten\u00eda otra cosa: un daguerrotipo desva\u00eddo. La fecha \u00ab1842\u00bb estaba garabateada en una esquina. Lo contempl\u00e9 largo rato, pero solo consegu\u00ed distinguir unas manchas blancas sobre negro. Me guard\u00e9 el sobre en el blus\u00f3n y mordisque\u00e9 un poco de _toke_ rancio. Cuando se me cans\u00f3 la vista, apagu\u00e9 la l\u00e1mpara y me acurruqu\u00e9 en posici\u00f3n fetal.\n\nMi cabeza era una mara\u00f1a de cabos sueltos. El Custodio y sus heridas. Pleione llev\u00e1ndole la sangre de Seb. David y su inter\u00e9s por mi bienestar. Y Nashira, con sus ojos que todo lo ve\u00edan.\n\nMe obligu\u00e9 a pensar solo en el Custodio. Todav\u00eda me enfurec\u00eda cuando pensaba en la sangre de Seb, embotellada y etiquetada, lista para su consumo. Confiaba en que se la hubieran extra\u00eddo cuando todav\u00eda estaba vivo, y no de su cad\u00e1ver. Luego estaba Pleione. Ella le hab\u00eda llevado la sangre al Custodio; deb\u00eda de saber que iba a sufrir una necrosis, o como m\u00ednimo que pod\u00eda sufrirla. Deb\u00eda de haber previsto la necesidad de llevarle sangre humana antes de que fuera demasiado tarde. Como Pleione se hab\u00eda retrasado, el Custodio hab\u00eda tenido que beber mi sangre. Fuera lo que fuese lo que estuviera haciendo, lo hac\u00eda con la complicidad de Pleione.\n\nEl Custodio guardaba un secreto. Y yo tambi\u00e9n. Yo estaba ocultando mi conexi\u00f3n con los bajos fondos, eso que a Nashira tanto le interesaba. Si \u00e9l aceptaba mi silencio, yo tambi\u00e9n aceptar\u00eda el suyo.\n\nMe toqu\u00e9 el brazo vendado. Esa herida, que segu\u00eda sin cicatrizar, era para m\u00ed tan repugnante como la marca. Si me dejaba una cicatriz, jam\u00e1s olvidar\u00eda la pena y el miedo que hab\u00eda sentido cuando me la hab\u00eda hecho. Un miedo muy parecido al que hab\u00eda sentido la primera vez que me hab\u00eda enfrentado al mundo de los esp\u00edritus. Miedo de ser lo que era. De lo que pod\u00eda llegar a ser.\n\nDeb\u00ed de quedarme dormida. Un fuerte dolor en la mejilla me devolvi\u00f3 a la realidad.\n\n\u2014\u00a1Paige!\n\nLiss me estaba zarandeando. Abr\u00ed los ojos, hinchados y enrojecidos.\n\n\u2014Paige, \u00bfqu\u00e9 demonios haces aqu\u00ed? Ya ha amanecido. Los arrancahuesos te est\u00e1n buscando.\n\nLevant\u00e9 la cabeza, adormilada.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Porque el Custodio se lo ha ordenado. Ten\u00edas que estar en Magdalen hace una hora.\n\nLiss ten\u00eda raz\u00f3n: una luz dorada estaba ti\u00f1endo el cielo. Liss me ayud\u00f3 a levantarme.\n\n\u2014Tienes suerte de que no te hayan encontrado aqu\u00ed. Est\u00e1 prohibido.\n\n\u2014\u00bfC\u00f3mo me has encontrado?\n\n\u2014Antes yo tambi\u00e9n ven\u00eda a este sitio. \u2014Me sujet\u00f3 por los hombros y me mir\u00f3 a los ojos\u2014. Tienes que suplicarle al Custodio que te perdone. Si le suplicas, quiz\u00e1 no te castigue.\n\nCasi me ech\u00e9 a re\u00edr.\n\n\u2014\u00bfSuplicarle?\n\n\u2014Es la \u00fanica forma.\n\n\u2014No pienso suplicarle nada.\n\n\u2014Te pegar\u00e1.\n\n\u2014No me importa. Tendr\u00e1n que llevarme ante \u00e9l.\n\nMir\u00e9 por la ventana.\n\n\u2014\u00bfTendr\u00edas problemas si me encontraran en tu chabola?\n\n\u2014Es mejor eso a que te encuentren aqu\u00ed. \u2014Me agarr\u00f3 la mu\u00f1eca\u2014. Vamos. No tardar\u00e1n en buscar por aqu\u00ed.\n\nEscond\u00ed la l\u00e1mpara y el libro bajo una estanter\u00eda de una patada. Bajamos la escalinata a toda prisa y salimos al exterior. La atm\u00f3sfera ol\u00eda a lluvia inminente.\n\nLiss me hizo esperar hasta que hubo comprobado que no hab\u00eda nadie cerca. Cruzamos el patio, pasamos por el h\u00famedo soportal y salimos al Broad. El sol brillaba por encima de los edificios. Liss separ\u00f3 dos paneles de contrachapado que estaban sueltos y nos colamos en el Poblado. Me gui\u00f3 entre corrillos de actores. Sus objetos personales, rescatados de la basura, estaban esparcidos por los pasadizos, como si hubieran registrado las chozas. Vi a un chico al que le sangraban los ojos apoyado en una pared. Todos susurraban al vernos pasar.\n\nMe met\u00ed en la choza. Julian esperaba all\u00ed sujetando un cuenco de _skilly_ que apoyaba sobre una rodilla. Levant\u00f3 la cabeza y dijo:\n\n\u2014Buenos d\u00edas.\n\nMe sent\u00e9.\n\n\u2014\u00bfTe alegras de verme?\n\n\u2014Supongo. \u2014Me sonri\u00f3\u2014. Aunque solo sea para acordarme de que necesito urgentemente un despertador.\n\n\u2014\u00bfNo deber\u00edas estar en tu residencia?\n\n\u2014Estaba a punto de irme; pero, ahora que has venido, tengo la sensaci\u00f3n de que iba a perderme la fiesta.\n\n\u2014\u00a1Est\u00e1is locos! \u2014nos reprendi\u00f3 Liss\u2014. Aqu\u00ed se toman muy en serio el toque de queda, Jules. Os va a caer una buena a los dos.\n\nMe pas\u00e9 la mano por el pelo h\u00famedo.\n\n\u2014\u00bfCu\u00e1nto tardar\u00e1n en encontrarnos?\n\n\u2014No mucho. Pronto volver\u00e1n a revisar las habitaciones. \u2014Se sent\u00f3\u2014. \u00bfPor qu\u00e9 no os vais?\n\nTen\u00eda todos los m\u00fasculos del cuerpo en tensi\u00f3n.\n\n\u2014No pasa nada, Liss \u2014dije\u2014. Me arriesgar\u00e9.\n\n\u2014Los arrancahuesos son implacables. No te escuchar\u00e1n. Y te lo advierto, el Custodio te matar\u00e1 si...\n\n\u2014No me importa.\n\nLiss apoy\u00f3 la cabeza en una mano. Mir\u00e9 a Julian. Ya no llevaba el traje de novato, sino un blus\u00f3n rosa.\n\n\u2014\u00bfQu\u00e9 has tenido que hacer?\n\n\u2014Nashira me pregunt\u00f3 qu\u00e9 era \u2014respondi\u00f3\u2014. Le dije que era palmista, pero evidentemente no supe leerle las manos. Hizo entrar en la habitaci\u00f3n a una amaur\u00f3tica y la hizo atar a una silla. Me acord\u00e9 de Seb y le pregunt\u00e9 a Nashira si me dejaba usar agua para hacer la predicci\u00f3n.\n\n\u2014\u00bfEres hidrom\u00e1ntico?\n\n\u2014No, pero no quiero que Nashira sepa qu\u00e9 soy. Fue lo primero que se me ocurri\u00f3. \u2014Se frot\u00f3 la cabeza\u2014. Llen\u00f3 un cuenco dorado y me orden\u00f3 buscar a una tal Antoinette Carter.\n\nArrugu\u00e9 el entrecejo. Antoinette Carter hab\u00eda sido una celebridad en Irlanda a principios de los a\u00f1os cuarenta. Recordaba que era una mujer delgada de mediana edad, fr\u00e1gil y enigm\u00e1tica. Ten\u00eda un programa de televisi\u00f3n, _Las verdades de Toni_ , que se emit\u00eda todos los jueves por la noche. Le tocaba las manos a la gente y aseguraba ver su futuro, que predec\u00eda con voz grave y comedida. Cancelaron el programa despu\u00e9s de la Incursi\u00f3n de 2046, cuando Scion invadi\u00f3 Irlanda, y Carter pas\u00f3 a la clandestinidad. Todav\u00eda publicaba un panfleto ilegal, _Stingy Jack_ , que denunciaba las atrocidades de Scion.\n\nPor razones que nosotros desconoc\u00edamos, Jaxon le hab\u00eda pedido a un falsante llamado Leon (un experto en enviar mensajes fuera de Scion) que estableciera contacto con ella. Yo nunca supe cu\u00e1l hab\u00eda sido el resultado. Leon era un buen falsante, pero llevaba tiempo esquivar los sistemas de seguridad de Scion.\n\n\u2014Es una fugitiva \u2014dije\u2014. Viv\u00eda en Irlanda.\n\n\u2014Pues ya no est\u00e1 en Irlanda.\n\n\u2014\u00bfQu\u00e9 viste? \u2014No me gust\u00f3 la expresi\u00f3n de su cara\u2014. \u00bfQu\u00e9 le dijiste?\n\n\u2014No te va a gustar. \u2014Dio un suspiro antes de continuar\u2014. Le dije que hab\u00eda visto unos relojes de sol. Record\u00e9 que Carl los hab\u00eda mencionado, y cre\u00ed que parecer\u00eda veros\u00edmil que yo los hubiera visto tambi\u00e9n.\n\nDesvi\u00e9 la mirada. Nashira andaba buscando a Jaxon. Tarde o temprano descubrir\u00eda d\u00f3nde estaban esos relojes de sol.\n\n\u2014Lo siento. Me dar\u00eda con la cabeza contra la pared. \u2014Julian se frot\u00f3 la frente\u2014. \u00bfPor qu\u00e9 son tan importantes esos relojes de sol?\n\n\u2014No puedo dec\u00edrtelo. Lo siento. Pero pase lo que pase... \u2014mir\u00e9 hacia la entrada de la choza\u2014 Nashira no debe volver a o\u00edr hablar de esos relojes de sol. Podr\u00eda poner en peligro a unos amigos m\u00edos.\n\nLiss se ech\u00f3 una manta por encima de los hombros.\n\n\u2014Paige \u2014dijo\u2014, me parece que tus amigos han intentado ponerse en contacto contigo.\n\n\u2014\u00bfQu\u00e9 quieres decir?\n\n\u2014Gomeisa me llev\u00f3 al Castillo. \u2014Su rostro se tens\u00f3\u2014. Estaba en mi celda, barajando las cartas para hacerle la lectura, y de pronto me sent\u00ed atra\u00edda hacia el Colgado. Cog\u00ed la carta, y estaba invertida. Vi el \u00e9ter. La cara de un hombre. Me record\u00f3 a la nieve.\n\n\u00abNick.\u00bb Los adivinos siempre dec\u00edan eso de Nick cuando lo ve\u00edan: que era como la nieve.\n\n\u2014\u00bfQu\u00e9 te envi\u00f3?\n\n\u2014Una imagen de un tel\u00e9fono. Creo que intenta averiguar d\u00f3nde est\u00e1s.\n\nUn tel\u00e9fono. Claro: \u00e9l no sab\u00eda d\u00f3nde estaba. La banda no sab\u00eda que Scion me hab\u00eda apresado, aunque a esas alturas ya deb\u00edan de sospechar algo. Nick quer\u00eda que lo llamara, quer\u00eda o\u00edrme decir que estaba bien. Deb\u00eda de haberle llevado d\u00edas encontrar el camino adecuado por el \u00e9ter. Si volv\u00eda a intentarlo mediante una sesi\u00f3n de espiritismo, quiz\u00e1 lograra enviarme un mensaje. No entend\u00eda por qu\u00e9 se lo hab\u00eda enviado a Liss. \u00c9l conoc\u00eda mi aura; en teor\u00eda habr\u00eda sido mucho m\u00e1s f\u00e1cil encontrarla. Quiz\u00e1 fueran las pastillas, o alg\u00fan tipo de interferencia provocada por los refas; pero no importaba.\n\nHab\u00eda intentado ponerse en contacto conmigo. No iba a abandonar.\n\nLa voz de Julian me sac\u00f3 de mi ensimismamiento:\n\n\u2014\u00bfDe verdad conoces a otros saltadores, Paige? \u2014Lo mir\u00e9, y \u00e9l se encogi\u00f3 de hombros\u2014. Cre\u00eda que el s\u00e9ptimo orden era el m\u00e1s raro.\n\n\u00abSaltadores\u00bb: una palabra cargada de connotaciones. Un orden de videntes, como los adivinos y los augures. Era la categor\u00eda a la que yo pertenec\u00eda: los videntes que pod\u00edan alterar o entrar en el \u00e9ter. Jax hab\u00eda desencadenado la gran separaci\u00f3n de los videntes en los a\u00f1os treinta, cuando ten\u00eda aproximadamente mi edad. Todo hab\u00eda empezado con _Sobre los m\u00e9ritos de la antinaturalidad_ , que se hab\u00eda extendido como una plaga por los bajos fondos de la videncia. En esa obra hab\u00eda identificado siete \u00f3rdenes de clarividentes: adivinos, augures, m\u00e9diums, sensores, furias, guardianes y saltadores. Afirmaba que los tres \u00faltimos eran muy superiores a los otros. Era una forma novedosa de abordar la clarividencia, que hasta entonces nadie hab\u00eda categorizado; pero los \u00f3rdenes inferiores no se lo hab\u00edan tomado bien. Las guerras de bandas que ocasion\u00f3 hab\u00edan durado dos cruentos a\u00f1os. Al final los editores de Jax hab\u00edan retirado el panfleto, pero las rencillas persist\u00edan.\n\n\u2014S\u00ed \u2014dije\u2014. Solo a uno. Es un or\u00e1culo.\n\n\u2014Pues debes de tener un puesto muy alto en el sindicato.\n\n\u2014S\u00ed, bastante alto.\n\nLiss me sirvi\u00f3 un cuenco de _skilly_. Si ten\u00eda alguna opini\u00f3n formada sobre el panfleto, no la expres\u00f3.\n\n\u2014Jules \u2014dijo\u2014, \u00bfte importa que hable un momento a solas con Paige?\n\n\u2014Claro que no. Me quedar\u00e9 fuera vigilando por si vienen los rojos.\n\nSali\u00f3 de la choza. Liss se qued\u00f3 mirando el hornillo.\n\n\u2014\u00bfQu\u00e9 pasa? \u2014pregunt\u00e9.\n\nSe ci\u00f1\u00f3 la manta.\n\n\u2014Estoy preocupada por ti, Paige.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Tengo un mal presentimiento sobre esa celebraci\u00f3n, el Bicentenario. Ya s\u00e9 que no soy un or\u00e1culo, pero veo cosas. \u2014Sac\u00f3 la baraja\u2014. \u00bfMe dejas que te eche las cartas? Hay ciertas personas a las que siento la necesidad de hacerles una predicci\u00f3n.\n\nVacil\u00e9. Yo solo hab\u00eda usado las cartas para jugar al tarocchi.\n\n\u2014Como quieras.\n\n\u2014Gracias. \u2014Puso la baraja entre las dos\u2014. \u00bfAlguna vez te han le\u00eddo los signos? \u00bfUn adivino o un augur?\n\n\u2014No, nunca.\n\nMe hab\u00edan preguntado muchas veces si quer\u00eda que me hicieran una predicci\u00f3n, pero nunca hab\u00eda estado convencida de que fuera buena idea asomarme al futuro. A veces Nick me daba pistas, pero yo no sol\u00eda dejarle entrar en detalles.\n\n\u2014Vale. Dame la mano.\n\nLe tend\u00ed la mano derecha, y Liss me la cogi\u00f3. Su rostro adopt\u00f3 una expresi\u00f3n de concentraci\u00f3n intensa. Sac\u00f3 siete cartas de la baraja y las puso boca abajo en el suelo.\n\n\u2014Utilizo la extensi\u00f3n el\u00edptica. Leo tu aura, y luego escojo siete cartas y las interpreto. No todos los lectores interpretamos igual cada una de las cartas, as\u00ed que no te enfades si oyes algo que no te gusta. \u2014Me solt\u00f3 la mano\u2014. La primera indicar\u00e1 tu pasado. Me mostrar\u00e1 algunos de tus recuerdos.\n\n\u2014\u00bfVes los recuerdos?\n\nLiss esboz\u00f3 una sonrisa. Eso era algo de lo que todav\u00eda se enorgullec\u00eda.\n\n\u2014Los lectores podemos utilizar objetos, pero en realidad no encajamos en ninguna categor\u00eda. Hasta _Sobre los m\u00e9ritos_ lo reconoc\u00eda. Yo lo considero algo positivo.\n\nDio la vuelta a la primera carta.\n\n\u2014Cinco de copas \u2014dijo. Cerr\u00f3 los ojos\u2014. Perdiste algo cuando eras muy peque\u00f1a. Hay un hombre de pelo rojizo. Son sus copas las que se derraman.\n\n\u2014Mi padre \u2014dije.\n\n\u2014S\u00ed. Est\u00e1s de pie a su lado, hablando con \u00e9l. No te contesta. Mira fijamente un retrato. \u2014Sin abrir los ojos, Liss gir\u00f3 la siguiente carta. Estaba del rev\u00e9s\u2014. Esto es el presente \u2014prosigui\u00f3\u2014. El rey de bastos, invertido. \u2014Frunci\u00f3 los labios\u2014. Te domina. No puedes escapar de su control.\n\n\u2014\u00bfEl Custodio?\n\n\u2014No lo creo. Pero tiene poder. Espera demasiado de ti. Le tienes miedo.\n\n\u00abJaxon.\u00bb\n\n\u2014La siguiente representa el futuro. \u2014Liss gir\u00f3 la carta y aspir\u00f3 entre los dientes\u2014. El Diablo. Esta carta representa impotencia, restricci\u00f3n, temor; pero te lo impones t\u00fa misma. El Diablo representa a alguien, pero no puedo verle la cara. Por mucho poder que esa persona tenga sobre ti, conseguir\u00e1s librarte de ella. Intentar\u00e1 hacerte creer que est\u00e1s unida a ella para siempre, pero no ser\u00e1 as\u00ed, aunque te lo parezca.\n\n\u2014\u00bfTe refieres a una pareja? \u2014Notaba una opresi\u00f3n en el pecho\u2014. \u00bfUn novio? \u00bfO es el Custodio?\n\n\u2014Podr\u00eda ser. No lo s\u00e9. \u2014Forz\u00f3 una sonrisa\u2014. No te preocupes. La siguiente carta te indicar\u00e1 qu\u00e9 tienes que hacer cuando llegue el momento.\n\nMir\u00e9 la cuarta carta.\n\n\u2014\u00bfLos Amantes?\n\n\u2014S\u00ed. \u2014Hab\u00eda bajado la voz\u2014. No veo gran cosa. Hay tensi\u00f3n entre el esp\u00edritu y la carne. Demasiada tensi\u00f3n. \u2014Sus dedos se desplazaron hacia la siguiente carta\u2014. Influencias externas.\n\nNo sab\u00eda si quer\u00eda continuar. Hasta el momento, Liss solo hab\u00eda dicho una cosa positiva, e incluso esa iba a resultar dolorosa. No esperaba que salieran Los Amantes, desde luego.\n\n\u2014La Muerte, invertida. La Muerte es una carta que les sale a menudo a los videntes. Suele aparecer en la posici\u00f3n del pasado o el presente. Pero aqu\u00ed, invertida... No estoy segura. \u2014Los ojos le temblaron bajo los p\u00e1rpados\u2014. A partir de aqu\u00ed, mi visi\u00f3n se vuelve confusa. Las cosas no est\u00e1n nada claras. S\u00e9 que el mundo cambiar\u00e1 a tu alrededor, y que har\u00e1s todo lo que puedas para resistirte. La muerte actuar\u00e1 de diferentes maneras. Si retrasas el cambio, prolongar\u00e1s tu sufrimiento.\n\n\u00bbLa sexta carta. Tus esperanzas y tus miedos. \u2014La cogi\u00f3 y la acarici\u00f3 con el pulgar\u2014. Ocho de espadas.\n\nLa carta ten\u00eda dibujada a una mujer encerrada en un c\u00edrculo de espadas que apuntaban hacia abajo. Llevaba los ojos vendados. Liss estaba sudando y le brillaba la piel.\n\n\u2014Te veo. Tienes miedo. \u2014Le temblaba la voz\u2014. Veo tu cara. No puedes moverte en ninguna direcci\u00f3n. Puedes quedarte quieta, atrapada, o sentir el dolor de las espadas.\n\nDeb\u00edan de ser las cartas m\u00e1s negativas que Liss hab\u00eda visto jam\u00e1s. Yo no ten\u00eda ningunas ganas de ver la \u00faltima.\n\n\u2014Y el resultado final. \u2014Cogi\u00f3 la \u00faltima carta\u2014. La conclusi\u00f3n de todas las anteriores.\n\nCerr\u00e9 los ojos. El \u00e9ter tembl\u00f3.\n\nNo llegu\u00e9 a ver la carta. Tres personas irrumpieron en la choza, y Liss se sobresalt\u00f3. Los arrancahuesos me hab\u00edan encontrado.\n\n\u2014\u00a1Vaya, vaya! Creo que hemos encontrado a la fugitiva. Y a su c\u00f3mplice. \u2014Uno de ellos agarr\u00f3 a Liss por la mu\u00f1eca y la levant\u00f3 de un tir\u00f3n\u2014. \u00bfQu\u00e9, ech\u00e1ndole las cartas a tu invitada?\n\n\u2014Solo estaba...\n\n\u2014Solo estabas usando el \u00e9ter. En privado \u2014dijo una voz femenina, desde\u00f1osa\u2014. \u00bfAcaso no sabes que solo puedes echarle las cartas a tu guardi\u00e1n, 1?\n\nMe levant\u00e9.\n\n\u2014Creo que es a m\u00ed a quien busc\u00e1is.\n\nSe volvieron los tres hacia m\u00ed. La chica era un poco mayor que yo; ten\u00eda el cabello largo y desgre\u00f1ado, y una frente prominente.\n\nLos dos chicos se parec\u00edan tanto que ten\u00edan que ser hermanos.\n\n\u2014Es verdad. Es a ti. \u2014El m\u00e1s alto de los dos apart\u00f3 a Liss de un empuj\u00f3n\u2014. \u00bfVas a venir por las buenas, 40?\n\n\u2014Depende de ad\u00f3nde quer\u00e1is llevarme \u2014contest\u00e9.\n\n\u2014A Magdalen, desgraciada. Ya ha amanecido.\n\n\u2014Ir\u00e9 yo sola.\n\n\u2014Te escoltaremos \u2014dijo la chica mir\u00e1ndome con profundo desprecio\u2014. Son las \u00f3rdenes. Has violado las normas.\n\n\u2014\u00bfVais a imped\u00edrmelo?\n\nLiss sacudi\u00f3 la cabeza, pero no le hice caso. Mir\u00e9 fijamente a la chica, que ten\u00eda las mand\u00edbulas muy apretadas.\n\n\u2014Adelante, 16.\n\n16 era el m\u00e1s bajo de los dos chicos, pero era muy corpulento. Me agarr\u00f3 por la mu\u00f1eca. Torc\u00ed r\u00e1pidamente el brazo hacia la derecha, y se le abri\u00f3 la mano. Le hinqu\u00e9 el pu\u00f1o entre las clav\u00edculas y lo empuj\u00e9 hacia su hermano.\n\n\u2014He dicho que ir\u00e9 yo sola.\n\n16 se llev\u00f3 las manos al cuello. El otro chico se abalanz\u00f3 sobre m\u00ed. Esquiv\u00e9 su brazo, levant\u00e9 una pierna y le propin\u00e9 una patada en el est\u00f3mago que le cort\u00f3 la respiraci\u00f3n. La chica me pill\u00f3 desprevenida: me agarr\u00f3 un mech\u00f3n de pelo y tir\u00f3 de \u00e9l. Me golpe\u00e9 la cabeza contra la pared de chapa met\u00e1lica. 16 se ech\u00f3 a re\u00edr entre resuellos mientras su hermano me inmovilizaba contra el suelo.\n\n\u2014Me parece que tienes que aprender a ser m\u00e1s respetuosa \u2014dijo y, jadeando, me tap\u00f3 la boca con una mano\u2014. Seguro que a tu guardi\u00e1n no le importar\u00e1 que te ense\u00f1e una lecci\u00f3n. Adem\u00e1s, nunca est\u00e1 por aqu\u00ed.\n\nEmpez\u00f3 a manosearme el pecho. Me hab\u00eda tomado por una presa f\u00e1cil, una chica indefensa. No sab\u00eda que estaba ante una dama. Le pegu\u00e9 con la frente en toda la nariz. El chico maldijo en voz alta. La chica me agarr\u00f3 los brazos. Le mord\u00ed la mu\u00f1eca, y chill\u00f3.\n\n\u2014\u00a1Zorra de mierda!\n\n\u2014\u00a1Su\u00e9ltala, Kathryn! \u2014Liss la sujet\u00f3 por el blus\u00f3n y la separ\u00f3 de m\u00ed\u2014. \u00bfQu\u00e9 te ha pasado? \u00bfTe has vuelto tan cruel como Kraz?\n\n\u2014He madurado. No quiero ser como t\u00fa y vivir rodeada de mierda \u2014le espet\u00f3 Kathryn\u2014. Eres pat\u00e9tica. Una bufona repugnante y pat\u00e9tica.\n\nMi agresor sangraba abundantemente por la nariz, pero no pensaba rendirse. Me ca\u00edan gotas de su sangre en la cara. Me tir\u00f3 del blus\u00f3n y rompi\u00f3 una costura. Le empuj\u00e9 el pecho; mi esp\u00edritu estaba a punto de estallar. El impulso de atacar era tan intenso que se me empa\u00f1aron los ojos.\n\nY entonces apareci\u00f3 Julian. Ten\u00eda un derrame en un ojo y un corte reciente en la mejilla. Deb\u00edan de haberle pegado antes de entrar en la choza. Agarr\u00f3 al chico por el cuello con un brazo.\n\n\u2014\u00bfEs as\u00ed como os pon\u00e9is calientes los arrancahuesos? \u2014Nunca lo hab\u00eda visto tan furioso\u2014. \u00bfSolo os gusta si ellas se resisten?\n\n\u2014Est\u00e1s muerto, 26 \u2014dijo mi agresor con voz estrangulada\u2014. Espera a que tu guardiana se entere de esto.\n\n\u2014Cu\u00e9ntaselo. A ver si te atreves.\n\nMe baj\u00e9 el blus\u00f3n con manos temblorosas. El casaca roja levant\u00f3 los brazos para protegerse. Julian le asest\u00f3 un gancho brutal en la mand\u00edbula. La sangre le salpic\u00f3 el blus\u00f3n al chico y lo dej\u00f3 lleno de manchas oscuras. Le salt\u00f3 un trozo de diente de la boca.\n\nKathryn arremeti\u00f3 a golpes contra Liss; le dio en la cara con el dorso de la mano, y Liss dej\u00f3 escapar un grito. Ese grito me sobresalt\u00f3. Era el grito de Seb, solo que esta vez todav\u00eda no era demasiado tarde. Me levant\u00e9 del suelo con la intenci\u00f3n de derribar a Kathryn, pero 16 me agarr\u00f3 por la cintura. Era m\u00e9dium, pero no estaba utilizando esp\u00edritus. Quer\u00eda ver sangre.\n\n\u2014\u00a1Suhail! \u2014grit\u00f3.\n\nEl alboroto hab\u00eda atra\u00eddo a un grupo de bufones. Entre ellos tambi\u00e9n hab\u00eda un casaca blanca. Lo reconoc\u00ed: era el chico de las trencitas cosidas, el cantor.\n\n\u2014\u00a1Ve a buscar a Suhail, in\u00fatil! \u2014le grit\u00f3 Kathryn. Ten\u00eda a Liss sujeta por el pelo\u2014. \u00a1Corre!\n\nEl chico no se movi\u00f3. Ten\u00eda unos ojos grandes y oscuros, con largas pesta\u00f1as. Ya no estaban infectados. Lo mir\u00e9 y sacud\u00ed la cabeza.\n\n\u2014No \u2014dijo.\n\n\u2014\u00a1Traidor! \u2014le grit\u00f3 16.\n\nAlgunos actores huyeron al o\u00edr esa palabra. Empuj\u00e9 a 16, y empec\u00e9 a sudar bajo el blus\u00f3n. Ve\u00eda un resplandor en los bordes de mi visi\u00f3n.\n\nEl hornillo. Vi las llamas ascendiendo por los tablones.\n\nLiss logr\u00f3 soltarse de Kathryn y apart\u00f3 de un empuj\u00f3n a 16. Julian lo sujet\u00f3 y lo arrastr\u00f3 lejos de nosotras.\n\nUna nube de humo empezaba a llenar la choza. Liss se puso a recoger sus cartas, intentando reunir la baraja. Kathryn le empuj\u00f3 la cabeza hacia abajo y la inmoviliz\u00f3. Liss dio un grito apagado.\n\n\u2014Eh, mira \u2014dijo Kathryn mostr\u00e1ndome una carta\u2014. Creo que esta es para ti, XX-40.\n\nEn la carta hab\u00eda dibujado un hombre tendido boca abajo, con diez espadas clavadas.\n\nLiss intent\u00f3 quitarle la carta.\n\n\u2014\u00a1No! Esa no era la...\n\n\u2014\u00a1Cierra el pico, asquerosa! \u2014Kathryn la inmoviliz\u00f3. Forceje\u00e9 con 16, pero me estaba haciendo una llave de cabeza\u2014. \u00bfTe quejas de lo dura que es tu vida? \u00bfCrees que es muy duro bailar para ellos mientras nosotros estamos ah\u00ed fuera arriesg\u00e1ndonos a que los zumbadores nos coman vivos?\n\n\u2014No ten\u00edas por qu\u00e9 volver, Kathy...\n\n\u2014\u00a1C\u00e1llate! \u2014Kathryn le golpe\u00f3 la cabeza contra el suelo. Estaba demasiado furiosa para preocuparse por el fuego\u2014. Todas las noches voy al bosque y veo que le arrancan los brazos a la gente, solo para que los emim no vengan aqu\u00ed y os deg\u00fcellen a todos. Y todo para que t\u00fa puedas seguir jugando a las cartas y haciendo virguer\u00edas con tus cintas. No quiero volver a ser como t\u00fa, \u00bfme oyes? \u00a1Los refas han visto algo m\u00e1s importante en m\u00ed!\n\nJulian se llev\u00f3 a 16 afuera. Intent\u00e9 recoger las cartas, pero Kathryn se me adelant\u00f3.\n\n\u2014Buena idea, 40 \u2014dijo, furiosa\u2014. D\u00e9mosle una lecci\u00f3n a esa asquerosa casaca amarilla.\n\nLanz\u00f3 toda la baraja a las llamas.\n\nLas consecuencias fueron inmediatas. Liss dio un grito espantoso, desgarrador. Jam\u00e1s hab\u00eda o\u00eddo a ning\u00fan humano producir un sonido parecido. Se me pusieron los pelos de punta. Las cartas ardieron como hojas secas. Lizz intent\u00f3 rescatar una, pero le sujet\u00e9 la mu\u00f1eca.\n\n\u2014\u00a1Es demasiado tarde, Liss!\n\nNo me hizo caso. Meti\u00f3 una mano en el fuego, gritando una y otra vez \u00abNo, no\u00bb con voz ahogada.\n\nSin m\u00e1s combustible que la parafina derramada, el fuego no tard\u00f3 en apagarse. Liss se qued\u00f3 arrodillada, con las manos enrojecidas y brillantes, mirando fijamente los restos chamuscados. Ten\u00eda la tez gris\u00e1cea y los labios amoratados. Sollozaba desconsoladamente mientras se mec\u00eda adelante y atr\u00e1s. La abrac\u00e9 y me qued\u00e9 mirando el fuego como atontada. El cuerpo menudo de Liss se estremec\u00eda.\n\nSin sus cartas, Liss ya no podr\u00eda conectar con el \u00e9ter. Tendr\u00eda que ser muy fuerte para sobrevivir al shock.\n\nKathryn me agarr\u00f3 por el hombro.\n\n\u2014Si hubieras venido con nosotros, esto no habr\u00eda pasado. \u2014Se limpi\u00f3 la sangre de la nariz\u2014. Lev\u00e1ntate.\n\nMir\u00e9 a Kathryn y lanc\u00e9 una pizca de mi esp\u00edritu contra su mente. Ella se encogi\u00f3, apart\u00e1ndose de m\u00ed.\n\n\u2014No te me acerques \u2014dije.\n\nEl humo me escoc\u00eda los ojos, pero no desvi\u00e9 la mirada. Kathryn intent\u00f3 re\u00edr, pero empez\u00f3 a sangrarle la nariz.\n\n\u2014Eres un monstruo. \u00bfQu\u00e9 eres, una especie de furia?\n\n\u2014Las furias no pueden afectar el \u00e9ter.\n\nPar\u00f3 de re\u00edr.\n\nSe oy\u00f3 un grito ahogado al otro lado de la cortina, y Suhail irrumpi\u00f3 en la choza apartando a empujones a los aterrorizados actores. Mir\u00f3 alrededor: el humo, el desorden. Kathryn se arrodill\u00f3 y agach\u00f3 la cabeza.\n\nMe qued\u00e9 quieta. Suhail me agarr\u00f3 por el pelo y acerc\u00f3 mi cara a la suya.\n\n\u2014Vas a morir \u2014dijo\u2014. Hoy.\n\nSe le pusieron los ojos rojos.\n\nEntonces comprend\u00ed que lo dec\u00eda en serio.\n\nEl portero de d\u00eda se qued\u00f3 mirando cuando Suhail pas\u00f3 tirando de mi mu\u00f1eca. Me dol\u00eda la garganta y ten\u00eda sangre en las mejillas. Me arrastr\u00f3 por la escalera y llam\u00f3 a la puerta del Custodio.\n\n\u2014\u00a1Arcturus!\n\nO\u00ed un repique de campanas a lo lejos. Liss hab\u00eda dicho que el Custodio me matar\u00eda por llegar pasado el amanecer. \u00bfQu\u00e9 me har\u00eda por resistirme a que me detuvieran?\n\nSe abri\u00f3 la puerta, y la imponente silueta del Custodio se destac\u00f3 contra la penumbra de la habitaci\u00f3n. Sus ojos eran dos agujeritos de luz. Me qued\u00e9 paralizada. Me hab\u00edan comido el aura, y eso me hab\u00eda provocado una especie de ataque epil\u00e9ptico. No sent\u00eda el \u00e9ter. Nada. Si mi guardi\u00e1n intentaba matarme en ese momento, yo no podr\u00eda hacer nada para evitarlo.\n\n\u2014Ya la hemos encontrado. \u2014Suhail me dio un empuj\u00f3n\u2014. Estaba escondida en el Poblado. Ha intentado provocar un incendio.\n\nEl Custodio nos mir\u00f3 alternadamente. Los indicios saltaban a la vista: los ojos de Suhail, mis mejillas manchadas de sangre.\n\n\u2014Te has cebado en ella \u2014dijo.\n\n\u2014Tengo derecho a cebarme en los humanos.\n\n\u2014En esta no. Te has excedido. A la soberana de sangre no le gustar\u00e1 tu falta de comedimiento.\n\nNo pod\u00eda verle la cara a Suhail, pero supuse que deb\u00eda de estar sonriendo con desd\u00e9n.\n\nSe produjo un silencio, y tos\u00ed: una tos seca, \u00e1spera. Temblaba de pies a cabeza. El Custodio se fij\u00f3 en el desgarr\u00f3n de mi blus\u00f3n.\n\n\u2014\u00bfQui\u00e9n ha hecho eso?\n\nPermanec\u00ed callada. El Custodio se agach\u00f3 para ponerse a mi nivel.\n\n\u2014\u00bfQui\u00e9n ha sido? \u2014Su voz me produjo un estremecimiento en el pecho\u2014. \u00bfUn casaca roja?\n\nAsent\u00ed con un lev\u00edsimo movimiento de cabeza. El Custodio mir\u00f3 a Suhail.\n\n\u2014\u00bfPermites a los casacas rojas violar a humanos durante tu guardia?\n\n\u2014No me importa qu\u00e9 m\u00e9todos empleen.\n\n\u2014No queremos que se reproduzcan, Suhail. No tenemos tiempo ni medios para encargarnos de un embarazo.\n\n\u2014Las pastillas los esterilizan. Adem\u00e1s, de sus fornicaciones se encarga el Capataz.\n\n\u2014Har\u00e1s lo que te ordene.\n\n\u2014Sin duda. \u2014Suhail me mir\u00f3 con aquellos espeluznantes ojos rojos\u2014. Pero volvamos a lo que nos ocupa. P\u00eddele perd\u00f3n a tu amo, 40.\n\n\u2014No \u2014dije.\n\nMe dio un bofet\u00f3n. Me tambale\u00e9 hacia un lado y di contra la pared. Ve\u00eda manchas de colores.\n\n\u2014\u00a1P\u00eddele perd\u00f3n a tu amo, XX-59-40!\n\n\u2014Tendr\u00e1s que golpearme mucho m\u00e1s fuerte.\n\nAlz\u00f3 una mano, dispuesto a satisfacerme; pero antes de que pudiera pegarme, el Custodio le sujet\u00f3 el brazo.\n\n\u2014Ya me encargar\u00e9 de ella en privado \u2014dijo\u2014. No te corresponde a ti castigarla. Despierta al Capataz y ocupaos del alboroto. No quiero que estos sucesos alteren las horas de sol.\n\nSe miraron fijamente. Suhail solt\u00f3 un d\u00e9bil gru\u00f1ido, se dio la vuelta y sali\u00f3 por la puerta. El Custodio lo vio marchar. Al cabo de un momento me cogi\u00f3 por el hombro y me hizo entrar en la habitaci\u00f3n.\n\nTodo estaba como siempre: las cortinas corridas, el fuego en la chimenea. En el gram\u00f3fono sonaba \u00abMr. Sandman\u00bb. La cama, tan c\u00f3moda y caliente. Me habr\u00eda gustado tumbarme, pero ten\u00eda que permanecer de pie. El Custodio cerr\u00f3 la puerta con llave y se sent\u00f3 en su butaca. Esper\u00e9; todav\u00eda estaba mareada del golpe que hab\u00eda recibido.\n\n\u2014Ven aqu\u00ed.\n\nNo tuve m\u00e1s remedio que obedecer. El Custodio me mir\u00f3; incluso sentado era casi tan alto como yo. Ten\u00eda los ojos tenues y transparentes, como el licor de chartreuse.\n\n\u2014\u00bfTe atrae la muerte, Paige? \u2014No le contest\u00e9\u2014. No me importa lo que pienses de m\u00ed, pero en esta ciudad hay ciertas normas que debes obedecer. Una de ellas es el toque de queda.\n\nSegu\u00ed sin decir nada. No pensaba darle la satisfacci\u00f3n de asustarme.\n\n\u2014\u00bfC\u00f3mo era ese casaca roja?\n\n\u2014Moreno. Unos veinte a\u00f1os. \u2014Mi voz sonaba ronca\u2014. Hab\u00eda otro chico que se le parec\u00eda, 16. Y una chica, Kathryn.\n\nMientras hablaba, un fuerte espasmo me sacudi\u00f3 el est\u00f3mago. Chivarse a un refa era vergonzoso. Entonces record\u00e9 la cara de Liss y su dolor, y mi determinaci\u00f3n se vio fortalecida.\n\n\u2014Ya s\u00e9 qui\u00e9nes son. \u2014El Custodio dirigi\u00f3 la mirada hacia el fuego\u2014. Son hermanos; m\u00e9diums los dos. XIX-49-16 y 17. Est\u00e1n aqu\u00ed desde que eran bastante m\u00e1s j\u00f3venes que t\u00fa. \u2014Entrelaz\u00f3 las manos\u2014. Me asegurar\u00e9 de que nunca m\u00e1s se les permita hacerte da\u00f1o.\n\nDeber\u00eda haberle dado las gracias, pero no lo hice.\n\n\u2014Si\u00e9ntate \u2014continu\u00f3\u2014. Tu aura se regenerar\u00e1.\n\nMe dej\u00e9 caer en la otra butaca. Empezaban a dolerme las costillas y las piernas. El Custodio me observaba.\n\n\u2014\u00bfTienes sed?\n\n\u2014No.\n\n\u2014\u00bfHambre?\n\n\u2014No.\n\n\u2014Debes de tener hambre. Esas gachas que preparan los actores hacen m\u00e1s mal que bien.\n\n\u2014No tengo hambre.\n\nEra mentira. El _skilly_ no era mucho m\u00e1s que agua, y mi est\u00f3mago anhelaba algo sustancioso y caliente.\n\n\u2014Es una l\u00e1stima. \u2014El Custodio se\u00f1al\u00f3 la mesilla de noche\u2014. Te hab\u00eda preparado un poco de comida.\n\nLa hab\u00eda visto nada m\u00e1s entrar. Hab\u00eda dado por hecho que la bandeja era para \u00e9l, pero entonces record\u00e9 de qu\u00e9 se alimentaba. Claro que no era para \u00e9l.\n\nComo no me mov\u00eda, el Custodio me puso la bandeja y unos pesados cubiertos de plata en el regazo. Mir\u00e9 la comida y sent\u00ed mareo. Huevos pasados por agua, partidos por la mitad de modo que se derramaba la yema, dorada y caliente. Un plato de cristal de cebada perlada, con pi\u00f1ones y unas gruesas jud\u00edas negras que brillaban como gotas de \u00f3nix. Una pera mondada, empapada en co\u00f1ac. Un racimo de orondas uvas negras. Pan integral con mantequilla.\n\n\u2014C\u00f3metelo.\n\nApret\u00e9 los pu\u00f1os.\n\n\u2014Tienes que comer, Paige.\n\nEstaba decidida a desobedecerle, a tirarle la bandeja por encima; pero estaba mareada, ten\u00eda la boca seca y lo \u00fanico que quer\u00eda era comerme aquella maldita comida. Cog\u00ed la cuchara y tom\u00e9 un poco de cebada. Las alubias estaban calientes; los pi\u00f1ones, dulces y prietos. Sent\u00ed un alivio intenso, y mi dolor de est\u00f3mago empez\u00f3 a disminuir.\n\nEl Custodio volvi\u00f3 a su asiento. Me observ\u00f3 en silencio mientras yo com\u00eda. Notaba el peso de su mirada, penetrante y abrasadora. Cuando hube terminado, dej\u00e9 la bandeja en el suelo. Todav\u00eda notaba el calor del co\u00f1ac en la lengua.\n\n\u2014Gracias \u2014dije.\n\nHabr\u00eda preferido no decirlo, pero ten\u00eda que decir algo. El Custodio tamborile\u00f3 con los dedos en el brazo de la butaca.\n\n\u2014Ma\u00f1ana por la noche quiero retomar tu entrenamiento \u2014dijo\u2014. \u00bfTienes alguna objeci\u00f3n?\n\n\u2014No tengo alternativa.\n\n\u2014\u00bfY si la tuvieras?\n\n\u2014No la tengo \u2014insist\u00ed\u2014, as\u00ed que no importa.\n\n\u2014Hablo hipot\u00e9ticamente. Si pudieras elegir, si pudieras controlar tu destino, \u00bfseguir\u00edas entrenando conmigo, o preferir\u00edas presentarte al siguiente examen sin m\u00e1s preparaci\u00f3n?\n\nMis labios iban a contestar con dureza, pero me mord\u00ed la lengua.\n\n\u2014No lo s\u00e9 \u2014dije.\n\nEl Custodio ech\u00f3 m\u00e1s le\u00f1a al fuego.\n\n\u2014Debes de encontrarte ante un dilema. Tu c\u00f3digo moral dice \u00abno\u00bb, pero tu instinto de supervivencia dice \u00abs\u00ed\u00bb.\n\n\u2014Ya s\u00e9 pelear. Soy m\u00e1s fuerte de lo que parece.\n\n\u2014S\u00ed, es verdad. Tu huida del Capataz da testimonio de tu fuerza. Y tu don es una gran baza, desde luego: ni siquiera los refa\u00edtas esperar\u00edan que un segundo esp\u00edritu invadiera su onirosaje. Tienes el factor sorpresa a tu favor. \u2014El fuego se reflejaba en sus ojos\u2014. Pero primero tienes que vencer tus l\u00edmites. Hay una raz\u00f3n por la que te cuesta tanto abandonar tu cuerpo: todos tus movimientos est\u00e1n excesivamente controlados. Tus m\u00fasculos est\u00e1n constantemente en tensi\u00f3n, preparados para hacerte correr; es como si percibieras peligro hasta en el aire que respiras. Resulta doloroso verlo; es peor que ver dar caza a un ciervo. Al menos un ciervo puede correr hacia su manada. \u2014Se inclin\u00f3 hacia delante\u2014. \u00bfD\u00f3nde est\u00e1 tu manada, Paige Mahoney?\n\nNo supe qu\u00e9 responder. Entend\u00eda a qu\u00e9 se refer\u00eda, pero mi manada, mi reba\u00f1o, eran Jax y el resto de la banda. Y no pod\u00eda decir ni una sola palabra de su existencia.\n\n\u2014No tengo ninguna \u2014dije\u2014. Soy un lobo solitario.\n\nNo se dej\u00f3 enga\u00f1ar.\n\n\u2014\u00bfQui\u00e9n te ha ense\u00f1ado a trepar por los edificios? \u00bfQui\u00e9n te ha ense\u00f1ado a disparar un arma? \u00bfQui\u00e9n te ayud\u00f3 a adentrarte en el \u00e9ter, a desplazar tu esp\u00edritu de su ubicaci\u00f3n natural?\n\n\u2014Aprend\u00ed yo sola.\n\n\u2014Mientes.\n\nMeti\u00f3 una mano bajo la butaca. Me puse en tensi\u00f3n. Mi mochila de emergencia. Una de las correas colgaba de un hilo.\n\n\u2014Podr\u00edas haber muerto la noche que huiste del Capataz. Si no pereciste esa noche fue solo porque tu mochila se enganch\u00f3 con una cuerda de tender cuando perdiste el conocimiento y evit\u00f3 que cayeras. Cuando me enter\u00e9 de lo ocurrido, sent\u00ed curiosidad.\n\nAbri\u00f3 la cremallera de la mochila. Apret\u00e9 las mand\u00edbulas. Lo que hab\u00eda all\u00ed dentro era m\u00edo, no suyo.\n\n\u2014Quinina \u2014dijo el Custodio hurgando en el interior\u2014. Adrenalina, dexanfetamina y cafe\u00edna. Medicamentos b\u00e1sicos. Somn\u00edferos. Hasta un arma de fuego. \u2014Sac\u00f3 mi pistola\u2014. Esa noche ibas muy bien preparada, Paige. No como los otros.\n\nMe estremec\u00ed. Ni rastro del panfleto. O lo hab\u00eda escondido en alg\u00fan sitio, o hab\u00eda ido a parar a otras manos.\n\n\u2014Seg\u00fan tu documento de identidad, eres oxista, camarera de un bar de ox\u00edgeno. Por lo que me ha contado el Capataz de las ciudadelas de Scion, el salario en esos bares es bajo. Eso me lleva a pensar que estos art\u00edculos no los compraste t\u00fa. \u2014Hizo una pausa\u2014. \u00bfQui\u00e9n los compr\u00f3?\n\n\u2014Eso no es asunto tuyo.\n\n\u2014\u00bfSe los robaste a tu padre?\n\n\u2014No pienso decirte nada m\u00e1s. Mi vida antes de llegar aqu\u00ed no te pertenece.\n\nEl Custodio cavil\u00f3 un momento antes de volver a mirarme a los ojos.\n\n\u2014Tienes raz\u00f3n \u2014dijo\u2014, pero ahora tu vida s\u00ed me pertenece.\n\nHinqu\u00e9 las u\u00f1as en la butaca.\n\n\u2014Si est\u00e1s abierta a plantearte el concepto general de la supervivencia, empezaremos a entrenar de nuevo ma\u00f1ana. Pero tu instrucci\u00f3n incluir\u00e1 un aspecto nuevo. \u2014Apunt\u00f3 a mi butaca con la barbilla y a\u00f1adi\u00f3\u2014: Todas las noches te sentar\u00e1s ah\u00ed y hablar\u00e1s conmigo al menos durante una hora.\n\n\u2014Prefiero morir \u2014dije sin pensarlo.\n\n\u2014Bueno, puedes morir si quieres. Tengo entendido que si fumas suficiente \u00e1ster morado, quedar\u00e1s atrapada en tu onirosaje y tu cuerpo se marchitar\u00e1 por falta de agua. \u2014Se\u00f1al\u00f3 la puerta\u2014. Vete, si quieres. Mu\u00e9rete. No vuelvas a mirarme. No veo ninguna raz\u00f3n para prolongar tu sufrimiento.\n\n\u2014\u00bfNo se enfadar\u00e1 la soberana de sangre?\n\n\u2014Es posible que s\u00ed.\n\n\u2014\u00bfTe importa?\n\n\u2014Nashira es mi prometida, no mi guardiana. Ella no influye en c\u00f3mo trato a los humanos que tengo a mi cargo.\n\n\u2014Y \u00bfc\u00f3mo piensas tratarme?\n\n\u2014Como mi alumna, no como mi esclava.\n\nGir\u00e9 la cabeza y apret\u00e9 las mand\u00edbulas. No quer\u00eda ser su alumna. No quer\u00eda volverme como \u00e9l, jugar en su terreno, atacar a mis semejantes.\n\nNotaba un d\u00e9bil escozor de los sentidos: estaba empezando a sentir de nuevo el \u00e9ter.\n\n\u2014Si me tratas como a una alumna \u2014dije\u2014, yo quiero tratarte como a un mentor, y no como a un amo.\n\n\u2014Me parece un trato justo. Pero a los mentores hay que mostrarles respeto. Espero eso de ti. Y tambi\u00e9n espero que todas las noches te quedes conmigo una hora haci\u00e9ndome compa\u00f1\u00eda.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Tienes potencial para ir del \u00e9ter al mundo corp\u00f3reo a tu antojo \u2014respondi\u00f3\u2014. Pero si no aprendes a quedarte quieta, incluso ante la presencia de tus enemigos, no te servir\u00e1 de mucho. Y no vivir\u00e1s mucho tiempo.\n\n\u2014Y t\u00fa no quieres que muera.\n\n\u2014No. Creo que ser\u00eda un desperdicio terrible de una vida singular. Tienes un gran potencial, pero necesitas un mentor.\n\nSus palabras me produjeron retortijones. Yo ya ten\u00eda un mentor. Jaxon Hall era mi mentor.\n\n\u2014Me gustar\u00eda consultarlo con la almohada \u2014dije.\n\n\u2014Por supuesto. \u2014Se levant\u00f3, y repar\u00e9 una vez m\u00e1s en lo alto que era. Yo ni siquiera le llegaba por los hombros\u2014. Ten en cuenta que s\u00ed puedes decidir. Pero te recomiendo, como mentor, que pienses en quienes te dieron esto. \u2014Con una sacudida de la mu\u00f1eca, me lanz\u00f3 la mochila\u2014. \u00bfQuerr\u00edan ellos que murieras en vano, o preferir\u00edan verte pelear?\n\nEl granizo golpeaba el tejado de la torre. Me frot\u00e9 las manos ante la l\u00e1mpara de parafina; ten\u00eda los labios y los dedos entumecidos de fr\u00edo.\n\nTen\u00eda que valorar la oferta del Custodio. No quer\u00eda trabajar con \u00e9l, pero necesitaba aprender a sobrevivir en aquel lugar, al menos hasta que encontrara la forma de volver a Londres. De volver con Nick, con Jax. De volver a escapar de los _centis_ , a dedicarme a la mimetodelincuencia; a estafar a Didion Waite y a birlarle esp\u00edritus, a mofarme de Hector y de sus chicos. Eso era lo que quer\u00eda. Aprender m\u00e1s cosas sobre mi don quiz\u00e1 me ayudara a salir de all\u00ed.\n\nJaxon siempre hab\u00eda dicho que ser onir\u00e1mbulo significaba algo m\u00e1s que tener un sexto sentido agudizado. Yo ten\u00eda potencial para pasearme por cualquier sitio, incluso por otros onirosajes. Lo hab\u00eda demostrado al matar a aquellos dos metrovigilantes. El Custodio quiz\u00e1 pudiera ense\u00f1arme m\u00e1s cosas; pero yo me resist\u00eda a que fuera mi maestro. \u00c9l y yo \u00e9ramos enemigos naturales; no ten\u00eda sentido fingir lo contrario. Y, sin embargo, \u00e9l me hab\u00eda observado mucho: c\u00f3mo me conten\u00eda, mi tensi\u00f3n, mi vigilancia. Jax siempre me dec\u00eda que ten\u00eda que soltarme, dejarme llevar, pero eso no significaba que pudiera confiar en el hombre que me ten\u00eda encerrada en una habitaci\u00f3n oscura y fr\u00eda.\n\nVaci\u00e9 la mochila alumbr\u00e1ndome con la d\u00e9bil luz de la l\u00e1mpara. La mayor\u00eda de mis objetos personales segu\u00edan all\u00ed: las jeringuillas (vac\u00edas), el material, la pistola (sin munici\u00f3n, por supuesto). El tel\u00e9fono me lo hab\u00edan confiscado. Solo faltaba otra cosa: _Sobre los m\u00e9ritos de la antinaturalidad_.\n\nNot\u00e9 un hormigueo por todo el cuerpo. Si el Custodio se lo hubiera ense\u00f1ado a Nashira, ella me habr\u00eda llamado para interrogarme. Los refa\u00edtas ya deb\u00edan de conocer el panfleto, pero no hab\u00edan visto mi ejemplar.\n\nMe tumb\u00e9 en el colch\u00f3n procurando no lastimarme m\u00e1s las heridas y me tap\u00e9 hasta la barbilla. Los muelles rotos se me clavaban en los omoplatos. Hab\u00eda recibido tres golpes en la cabeza en tres minutos, y estaba cansada. Mir\u00e9 el mundo exterior a trav\u00e9s de los barrotes, con la esperanza de hallar all\u00ed la respuesta, pero no encontr\u00e9 nada. Solo el ineludible anochecer.\n\nCuando se puso el sol, son\u00f3 la campanada nocturna. Ya me hab\u00eda familiarizado con ese sonido; era como la alarma de un despertador. Para cuando me hube vestido, hab\u00eda tomado una decisi\u00f3n dif\u00edcil. Intentar\u00eda volver a entrenar con \u00e9l, si lo soportaba. Eso significar\u00eda someterme a una hora de conversaci\u00f3n, pero cre\u00eda poder aguantarlo. Cre\u00eda poder llenar esa hora de mentiras.\n\nEl Custodio me esperaba junto a la puerta. Me mir\u00f3 de arriba abajo.\n\n\u2014\u00bfYa has tomado una decisi\u00f3n?\n\n\u2014S\u00ed \u2014respond\u00ed manteniendo las distancias\u2014. Entrenar\u00e9 contigo, pero solo si admites no ser mi amo.\n\n\u2014Eres m\u00e1s sensata de lo que cre\u00eda. \u2014Me dio una chaqueta negra con bandas rosa en las mangas\u2014. Ponte esto. Lo necesitar\u00e1s para tu siguiente examen.\n\nMe puse la chaqueta y me la abroch\u00e9. El forro era grueso y abrigado. El Custodio me tendi\u00f3 una mano. En la palma ten\u00eda las tres pastillas. No las cog\u00ed.\n\n\u2014\u00bfPara qu\u00e9 sirve la verde?\n\n\u2014Eso no es asunto tuyo.\n\n\u2014Quiero saber para qu\u00e9 es. Nadie m\u00e1s la toma.\n\n\u2014Porque t\u00fa eres diferente. \u2014No retir\u00f3 la mano\u2014. Ya s\u00e9 que no te tomas las pastillas. No tengo ning\u00fan inconveniente en hacer que te las tomes por la fuerza.\n\n\u2014Me encantar\u00eda ver c\u00f3mo lo intentas.\n\nEscudri\u00f1\u00f3 mi cara, y se me pusieron los pelos de punta.\n\n\u2014Preferir\u00eda no tener que recurrir a eso \u2014dijo.\n\nYo sab\u00eda que era una batalla perdida. Pod\u00e9is llamarlo instinto criminal. Era como volver a estar en el mercado negro, peleando con Didion por Anne Naylor. Hab\u00eda cosas en las que el Custodio estaba dispuesto a transigir, pero esa no era una de ellas. Me dije que ya le llevar\u00eda a Duckett la pastilla verde del d\u00eda siguiente.\n\nMe tragu\u00e9 los comprimidos con un vaso de agua. El Custodio me cogi\u00f3 la barbilla con una mano enguantada.\n\n\u2014Hay una raz\u00f3n.\n\nGir\u00e9 la cabeza. \u00c9l me mir\u00f3 un momento, y a continuaci\u00f3n abri\u00f3 la puerta. Lo segu\u00ed por la escalera de caracol hasta los soportales. Unas estatuas de piedra grotescas montaban guardia en el patio. Hab\u00eda bajado la temperatura, y una fina capa de escarcha lo cubr\u00eda todo. Me abrac\u00e9 el torso para conservar el calor. El Custodio me gui\u00f3 al exterior, pero no salimos a la calle, sino que tomamos la direcci\u00f3n opuesta; pasamos por una verja de hierro forjado y cruzamos una pasarela sobre un r\u00edo verde azulado. El reflejo intenso de la luna brillaba en la superficie del agua. Hab\u00eda parado de granizar, y el suelo estaba cubierto de hielo.\n\nMientras recorr\u00edamos un sendero de tierra, el Custodio se enroll\u00f3 una manga de la camisa. La herida de la primera vez le supuraba. Estaba empezando a cicatrizar, pero todav\u00eda no estaba curada.\n\n\u2014\u00bfSon venenosos? \u2014pregunt\u00e9\u2014. Los zumbadores.\n\n\u2014Los emim son portadores de una infecci\u00f3n llamada semimpulso, que si no se trata produce locura e incluso la muerte. Comen cualquier tipo de carne, fresca o podrida.\n\nVi que, mientras habl\u00e1bamos, la herida empezaba a curarse.\n\n\u2014\u00bfC\u00f3mo lo haces? \u2014pregunt\u00e9, vencida por la curiosidad\u2014. Est\u00e1 cicatrizando.\n\n\u2014Utilizo tu aura.\n\nDi un respingo.\n\n\u2014\u00bfC\u00f3mo dices?\n\n\u2014Ya debes de saber que los refa\u00edtas nos alimentamos de aura. Para m\u00ed es m\u00e1s f\u00e1cil alimentarme cuando mi hu\u00e9sped no sabe que lo estoy haciendo.\n\n\u2014\u00bfHas comido de mi aura?\n\n\u2014S\u00ed. \u2014Me mir\u00f3 y a\u00f1adi\u00f3\u2014: Pareces enfadada.\n\n\u2014Mi aura no te pertenece. \u2014Me apart\u00e9 de \u00e9l, asqueada\u2014. Ya me has arrebatado la libertad. No tienes ning\u00fan derecho a quitarme el aura.\n\n\u2014No te he quitado la suficiente para perjudicar tu don. Me cebo en los humanos a peque\u00f1as dosis, dejando tiempo para la regeneraci\u00f3n. Otros son menos considerados. Y cr\u00e9eme \u2014agreg\u00f3 mientras volv\u00eda a bajarse la manga\u2014, no te har\u00eda ninguna gracia que contrajera semimpulso en tu presencia.\n\nLo mir\u00e9, y \u00e9l no se resisti\u00f3 a mi examen.\n\n\u2014Tus ojos. \u2014Me fij\u00e9 en sus ojos, a la vez hechizada y repelida\u2014. Por eso cambian.\n\nNo lo neg\u00f3. Ya no ten\u00eda los ojos amarillos, sino de un rojo oscuro con un resplandor tenue. El color de mi aura.\n\n\u2014No era mi intenci\u00f3n ofenderte \u2014dijo\u2014, pero as\u00ed es como debe ser.\n\n\u2014\u00bfPor qu\u00e9? \u00bfPorque t\u00fa lo dices?\n\nSigui\u00f3 andando y no me contest\u00f3. Lo segu\u00ed. Me pon\u00eda enferma pensar que mi guardi\u00e1n pod\u00eda alimentarse de m\u00ed.\n\nAl cabo de unos minutos el Custodio se detuvo. Nos hall\u00e1bamos rodeados de una neblina azulada. Me sub\u00ed el cuello de la chaqueta.\n\n\u2014Lo notas \u2014dijo el Custodio\u2014. El fr\u00edo. \u00bfAlguna vez te has preguntado por qu\u00e9 aqu\u00ed hay escarcha a principios de primavera?\n\n\u2014Estamos en Inglaterra. Hace fr\u00edo.\n\n\u2014No tanto como aqu\u00ed. F\u00edjate bien. \u2014Me cogi\u00f3 una mano y me quit\u00f3 el guante. Me ardieron los dedos de fr\u00edo\u2014. Por aqu\u00ed cerca hay un punto fr\u00edo.\n\nVolv\u00ed a ponerme el guante.\n\n\u2014\u00bfUn punto fr\u00edo?\n\n\u2014S\u00ed. Se forman cuando un esp\u00edritu ha morado mucho tiempo en un determinado sitio, creando una abertura entre el \u00e9ter y el mundo corp\u00f3reo. \u00bfNunca te has fijado en el fr\u00edo que hace cuando hay esp\u00edritus cerca?\n\n\u2014Supongo que s\u00ed.\n\nEra verdad que los esp\u00edritus me produc\u00edan sensaci\u00f3n de fr\u00edo, pero no le hab\u00eda dado mucha importancia.\n\n\u2014En teor\u00eda, los esp\u00edritus no moran entre los dos mundos. Absorben calor para sustentarse. Sheol I est\u00e1 rodeada de puntos fr\u00edos; aqu\u00ed la actividad et\u00e9rea es mucho m\u00e1s elevada que en la ciudadela. Por eso los emim se sienten atra\u00eddos hacia nosotros m\u00e1s que hacia la poblaci\u00f3n amaur\u00f3tica de Londres. \u2014Se\u00f1al\u00f3 la franja de tierra que ten\u00edamos delante\u2014. \u00bfC\u00f3mo crees que podr\u00edas encontrar el epicentro de un punto fr\u00edo?\n\n\u2014La mayor\u00eda de los videntes ver\u00edan al esp\u00edritu \u2014dije\u2014. Tienen el tercer ojo.\n\n\u2014Pero t\u00fa no.\n\n\u2014No.\n\n\u2014Los que carecen de visi\u00f3n tambi\u00e9n pueden hacerlo. \u00bfHas o\u00eddo hablar de la rabdomancia?\n\n\u2014Tengo entendido que no sirve para nada \u2014dije. Jax me lo hab\u00eda dicho infinidad de veces\u2014. Los rabdom\u00e1nticos afirman poder encontrar el camino de regreso desde cualquier lugar. Dicen que cuando se pierden pueden lanzar _numa_ , y que los esp\u00edritus los orientan en la direcci\u00f3n correcta. Pero no funciona.\n\n\u2014Quiz\u00e1 tengas raz\u00f3n, pero no puedes decir que no sirve para nada. Todos los tipos de clarividencia sirven para algo.\n\nNot\u00e9 que me ard\u00edan las mejillas. No era verdad que creyera que los rabdom\u00e1nticos no serv\u00edan para nada, pero Jax siempre me hab\u00eda dicho que eran unos in\u00fatiles. No pod\u00edas trabajar para Jaxon Hall y no compartir sus opiniones sobre esas cuestiones.\n\n\u2014Entonces \u00bfpara qu\u00e9 es \u00fatil? \u2014pregunt\u00e9. El Custodio me mir\u00f3 y no dijo nada\u2014. Se supone que tienes que instruirme. Instr\u00fayeme.\n\n\u2014Muy bien. Si te interesa aprender... \u2014El Custodio ech\u00f3 a andar\u2014. La mayor\u00eda de los rabdom\u00e1nticos creen que cuando sus _numa_ caen, se\u00f1alan hacia su casa, hacia un tesoro escondido o hacia lo que sea que est\u00e9n buscando. Eso acaba volvi\u00e9ndolos locos. Porque a lo que apuntan sus _numa_ no es al oro, sino al epicentro del punto fr\u00edo m\u00e1s cercano. A veces recorren kil\u00f3metros, y no hallan lo que buscan. Pero s\u00ed encuentran algo: una puerta secreta. Lo que no saben es qu\u00e9 hay que hacer para abrirla.\n\nSe detuvo. Yo estaba temblando. Hac\u00eda fr\u00edo y el aire estaba enrarecido. Respir\u00e9 hondo.\n\n\u2014Los seres vivos no soportan bien los puntos fr\u00edos \u2014dijo\u2014. Toma.\n\nMe dio una petaca de plata con tap\u00f3n de rosca. Me qued\u00e9 mir\u00e1ndola.\n\n\u2014Es solo agua, Paige.\n\nBeb\u00ed. Estaba demasiado sedienta para rechazarla. El Custodio cogi\u00f3 la petaca y se la guard\u00f3. El agua me despej\u00f3 la cabeza.\n\nLa tierra ante nosotros estaba helada, dura, como si fuera pleno invierno. Me casta\u00f1eteaban los dientes. El esp\u00edritu responsable de aquel punto fr\u00edo rondaba por all\u00ed cerca. Como no se nos acercaba, el Custodio se puso en cuclillas al borde del hielo, sac\u00f3 una daga y se la acerc\u00f3 al brazo. Di un paso adelante.\n\n\u2014\u00bfQu\u00e9 haces?\n\n\u2014Voy a abrir la puerta.\n\nSe hizo un corte en la mu\u00f1eca. Las gotas de ectoplasma cayeron sobre el hielo. El punto fr\u00edo se agriet\u00f3 por el medio, y el aire se volvi\u00f3 blanco. Me vi rodeada de formas y voces. \u00ab\u00a1So\u00f1adora! \u00a1So\u00f1adora!\u00bb Me tap\u00e9 los o\u00eddos, pero segu\u00eda oy\u00e9ndolas. \u00ab\u00a1So\u00f1adora, no vayas m\u00e1s all\u00e1! \u00a1Da media vuelta!\u00bb Levant\u00e9 la cabeza, y volv\u00eda a estar rodeada de oscuridad.\n\n\u2014\u00bfPaige?\n\n\u2014\u00bfQu\u00e9 ha pasado?\n\nEstaba mareada y me dol\u00eda la cabeza.\n\n\u2014He abierto el punto fr\u00edo.\n\n\u2014Con tu sangre.\n\n\u2014S\u00ed.\n\nVi que la mu\u00f1eca hab\u00eda dejado de sangrarle. Todav\u00eda ten\u00eda los ojos rojos, pues mi aura segu\u00eda cur\u00e1ndole las heridas.\n\n\u2014As\u00ed que los puntos fr\u00edos se pueden abrir \u2014dije.\n\n\u2014T\u00fa no puedes. Yo s\u00ed.\n\n\u2014Porque los puntos fr\u00edos llevan al \u00e9ter. \u2014Hice una pausa\u2014. \u00bfPuedes utilizarlos para llegar al Inframundo?\n\n\u2014S\u00ed. As\u00ed fue como llegamos aqu\u00ed. Imagina que hay dos velos que separan el \u00e9ter y tu mundo, el mundo de la vida. Entre esos velos est\u00e1 el Inframundo, un estado intermedio entre la vida y la muerte. Cuando los rabdom\u00e1nticos encuentran un punto fr\u00edo, encuentran la manera de moverse entre ambos velos. Para entrar en mi mundo, el reino de los refa\u00edtas.\n\n\u2014\u00bfLos humanos pueden ir?\n\n\u2014Int\u00e9ntalo.\n\nLo mir\u00e9. Apunt\u00f3 con la barbilla al punto fr\u00edo, y di un paso sobre el hielo. No sucedi\u00f3 nada.\n\n\u2014La materia corp\u00f3rea no puede sobrevivir m\u00e1s all\u00e1 del velo \u2014dijo el Custodio\u2014. Tu cuerpo no puede trasponer ese umbral.\n\n\u2014\u00bfY los rabdom\u00e1nticos?\n\n\u2014Ellos tambi\u00e9n son de carne y hueso.\n\n\u2014Entonces \u00bfpor qu\u00e9 lo abres?\n\nEl sol ya se hab\u00eda puesto.\n\n\u2014Porque ha llegado la hora \u2014dijo\u2014 de que te enfrentes al Inframundo. No entrar\u00e1s, pero lo ver\u00e1s.\n\nEmpez\u00f3 a sudarme la frente. Me apart\u00e9 del hielo. Percib\u00eda esp\u00edritus por todas partes.\n\n\u2014La noche es la hora de los esp\u00edritus. \u2014El Custodio alz\u00f3 la mirada hacia la luna\u2014. Es cuando los velos son m\u00e1s finos. Los puntos fr\u00edos son como desgarrones en una tela.\n\nFij\u00e9 la vista en el punto fr\u00edo. Hab\u00eda algo all\u00ed que hac\u00eda vibrar mi esp\u00edritu.\n\n\u2014Paige, esta noche tendr\u00e1s dos tareas \u2014dijo el Custodio volvi\u00e9ndose hacia m\u00ed\u2014. Ambas pondr\u00e1n a prueba los l\u00edmites de tu cordura. \u00bfMe creer\u00e1s si te digo que te ayudar\u00e1n?\n\n\u2014Lo dudo \u2014respond\u00ed\u2014, pero adelante.\n\nEl Custodio no me revel\u00f3 ad\u00f3nde \u00edbamos. Me gui\u00f3 por otro sendero que discurr\u00eda por los jardines de Magdalen. Notaba esp\u00edritus por todas partes: en el aire, en el agua; esp\u00edritus de los difuntos que en vida hab\u00edan paseado por all\u00ed. No los o\u00eda; pero, como se hab\u00eda abierto un punto fr\u00edo, los percib\u00eda con tanta intensidad como si fueran presencias vivas.\n\nMe mantuve cerca del Custodio a mi pesar. Supon\u00eda que, si alguno de aquellos esp\u00edritus era maligno, \u00e9l podr\u00eda repelerlo m\u00e1s eficazmente que yo.\n\nLa oscuridad se intensificaba a medida que nos adentr\u00e1bamos en los jardines y nos alej\u00e1bamos de los faroles de Magdalen. El Custodio guard\u00f3 silencio cuando atravesamos una pradera h\u00fameda donde, en lugar de c\u00e9sped, hab\u00eda malas hierbas que me llegaban por las rodillas.\n\n\u2014\u00bfAd\u00f3nde vamos? \u2014pregunt\u00e9, con las botas y los calcetines empapados.\n\nEl Custodio no me contest\u00f3.\n\n\u2014Dijiste que era tu alumna, no tu esclava \u2014le record\u00e9\u2014. Quiero saber ad\u00f3nde vamos.\n\n\u2014Al interior del jard\u00edn.\n\n\u2014\u00bfPor qu\u00e9?\n\nVolvi\u00f3 a callar.\n\nCada vez hac\u00eda m\u00e1s fr\u00edo, un fr\u00edo antinatural. Tras lo que me parecieron horas, el Custodio se par\u00f3 por fin y se\u00f1al\u00f3.\n\n\u2014All\u00ed.\n\nAl principio no lo vi. Cuando mis ojos se acostumbraron, el contorno del animal apareci\u00f3 bajo la tenue luz de la luna. Era un ser de cuatro patas con pelaje sedoso. Ten\u00eda el cuello blanco como la nieve, y una cara estrecha y alargada, con ojos oscuros y un peque\u00f1o hocico negro. Me pregunt\u00e9 cu\u00e1l de los dos deb\u00eda de parecer m\u00e1s sorprendido.\n\nUna cierva. No ve\u00eda ciervos desde que viv\u00eda en Irlanda, cuando mis abuelos me llevaron a las monta\u00f1as Galtee. Me invadi\u00f3 un entusiasmo infantil.\n\n\u2014Qu\u00e9 bonita es \u2014observ\u00e9.\n\nEl Custodio fue hacia la cierva, atada a un poste.\n\n\u2014Se llama Nuala.\n\n\u2014Es un nombre irland\u00e9s.\n\n\u2014S\u00ed, el diminutivo de Fionnuala. Significa hombros blancos, u hombros claros.\n\nVolv\u00ed a fijarme y vi que ten\u00eda dos grandes manchas blancas a ambos lados del cuello.\n\n\u2014\u00bfQui\u00e9n le puso ese nombre?\n\nEn Scion era arriesgado ponerles nombres irlandeses a los ni\u00f1os o las mascotas. Pod\u00edan sospechar que simpatizabas con los alborotadores de las revueltas de Molly.\n\n\u2014Fui yo.\n\nLe solt\u00f3 el collar que llevaba alrededor del cuello. Nuala le dio un empujoncito con el morro. Cre\u00ed que echar\u00eda a correr, pero se qued\u00f3 quieta mirando al Custodio. \u00c9l le habl\u00f3 en un idioma que no supe identificar y le acarici\u00f3 el cuello; me pareci\u00f3 que ella lo escuchaba, cautivada.\n\n\u2014\u00bfQuieres darle de comer? \u2014El Custodio se sac\u00f3 una manzana roja de la manga\u2014. Le encantan las manzanas.\n\nMe la lanz\u00f3. Nuala me mir\u00f3 y movi\u00f3 el hocico.\n\n\u2014Con cuidado \u2014dijo el Custodio\u2014. Se asusta f\u00e1cilmente cuando hay un punto fr\u00edo abierto cerca.\n\nYo no quer\u00eda asustarla, pero si no le ten\u00eda miedo al Custodio, \u00bfc\u00f3mo iba a ten\u00e9rmelo a m\u00ed? Estir\u00e9 el brazo ofreci\u00e9ndole la manzana. La cierva olfate\u00f3 el fruto. El Custodio dijo algo m\u00e1s, y entonces Nuala cogi\u00f3 la manzana.\n\n\u2014Tendr\u00e1s que perdonarla. Tiene mucha hambre. \u2014Le dio unas palmaditas en el cuello y sac\u00f3 otra manzana\u2014. Tengo muy pocas oportunidades de venir a verla.\n\n\u2014Pero si vive en Magdalen.\n\n\u2014S\u00ed, pero he de tener cuidado. No est\u00e1 permitido tener animales dentro de la ciudad.\n\n\u2014Entonces \u00bfpor qu\u00e9 la tienes?\n\n\u2014Por la compa\u00f1\u00eda que me hace. Y por ti.\n\n\u2014\u00bfPor m\u00ed?\n\n\u2014Nuala te estaba esperando. \u2014Se sent\u00f3 sobre una roca plana y dej\u00f3 que Nuala fuera hacia los \u00e1rboles\u2014. Eres onir\u00e1mbula. \u00bfQu\u00e9 significa eso para ti?\n\nNo me hab\u00eda llevado all\u00ed para dar de comer a una cierva.\n\n\u2014Que estoy en sinton\u00eda con el \u00e9ter.\n\n\u2014\u00bfQu\u00e9 m\u00e1s?\n\n\u2014Puedo percibir otros onirosajes desde lejos. Y la actividad et\u00e9rea en general.\n\n\u2014Exactamente. Ese es tu don primario, lo esencial: una sensibilidad al \u00e9ter acentuada, una conciencia que no poseen la mayor\u00eda de los clarividentes. Proviene de tu cord\u00f3n arg\u00e9nteo, que es flexible. Te permite desplazar tu esp\u00edritu del centro de tu onirosaje, ensanchar tu percepci\u00f3n del mundo. Eso har\u00eda enloquecer a la mayor\u00eda de los clarividentes. Pero cuando entrenamos en la pradera, te anim\u00e9 a lanzar tu esp\u00edritu contra mi onirosaje. A atacarlo. \u2014Sus ojos ard\u00edan en la penumbra\u2014. Tienes potencial para hacer algo m\u00e1s que sencillamente percibir el \u00e9ter. Puedes alterarlo. Puedes alterar a otras personas.\n\nSegu\u00ed callada.\n\n\u2014Cuando eras m\u00e1s peque\u00f1a quiz\u00e1 pod\u00edas hacer da\u00f1o a otras personas. Quiz\u00e1 pod\u00edas ejercer presi\u00f3n sobre sus onirosajes. Quiz\u00e1 ellas advirtieran algo: les sangraba la nariz, se les nublaba la visi\u00f3n...\n\n\u2014S\u00ed.\n\nNo ten\u00eda sentido negarlo, porque \u00e9l ya lo sab\u00eda.\n\n\u2014Algo cambi\u00f3 en el tren \u2014continu\u00f3\u2014. Tu vida peligraba. Tem\u00edas que te detuvieran. Y, por primera vez en tu vida, emergi\u00f3 ese poder que tienes dentro.\n\n\u2014\u00bfC\u00f3mo te has enterado?\n\n\u2014Lleg\u00f3 un informe de que hab\u00edan matado a un metrovigilante. Sin derramamiento de sangre, sin arma alguna, sin que quedara ni una sola se\u00f1al en su cuerpo. Nashira supo al instante que aquello hab\u00eda sido obra de un onir\u00e1mbulo.\n\n\u2014Podr\u00eda haberlo hecho un duende.\n\n\u2014Los duendes siempre dejan una se\u00f1al. T\u00fa deber\u00edas saberlo.\n\nNot\u00e9 tirantez en las cicatrices de mi mano.\n\n\u2014Nashira quer\u00eda capturarte viva \u2014prosigui\u00f3 el Custodio\u2014. La DVN hace detenciones violentas y torpes, como muchos de nuestros casacas rojas. Cerca de la mitad de esas detenciones acaban en muerte. Y eso no pod\u00eda pasar contigo. A ti no pod\u00edamos perderte. Por eso Nashira envi\u00f3 al Capataz, su especialista en captura de clarividentes.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Porque quiere aprender tu secreto.\n\n\u2014No hay ning\u00fan secreto. Es lo que soy.\n\n\u2014Y tambi\u00e9n es lo que Nashira quiere ser. Le encantan los dones inusuales, entre ellos el tuyo.\n\n\u2014Pues \u00bfpor qu\u00e9 no me lo quita? Habr\u00eda podido matarme cuando mat\u00f3 a Seb. \u00bfPara qu\u00e9 esperar?\n\n\u2014Porque quiere entender el alcance de tus capacidades, pero no quiere esperar eternamente.\n\n\u2014No actuar\u00e9 para ti \u2014dije\u2014. Todav\u00eda no soy una bufona.\n\n\u2014No te he pedido que act\u00faes para m\u00ed. \u00bfQu\u00e9 necesidad hay? En la capilla vi de qu\u00e9 eres capaz. Lanzaste tu esp\u00edritu contra la mente de Aludra. Lo vi en la pradera, cuando entraste en la m\u00eda. Pero dime \u2014se inclin\u00f3 hacia m\u00ed, mir\u00e1ndome con sus ojos abrasadores\u2014, \u00bfhabr\u00edas podido poseernos a alguno de los dos?\n\nSe produjo un silencio, interrumpido solo por el ulular aflautado de un b\u00faho. Mir\u00e9 hacia arriba y vi la luna, envuelta en jirones de nube. Por un breve instante me vi transportada al despacho de Jax, la primera vez que abordamos el tema de la posesi\u00f3n.\n\n\u00abQuerida \u2014me hab\u00eda dicho\u2014, hasta ahora has sido una estrella. Qu\u00e9 digo, un luminar. Eres, sin ning\u00fan g\u00e9nero de duda, una joya, un Sello a punto de estallar; pero ahora me gustar\u00eda asignarte una nueva tarea. Una tarea que te pondr\u00e1 a prueba, pero que tambi\u00e9n te satisfar\u00e1. \u2014Me pidi\u00f3 que obligara a mi mente a entrar en la suya, que intentara controlar su cuerpo. Esa idea me hab\u00eda conmocionado. Hab\u00eda hecho un intento desganado, pero su mente era excesivamente compleja y yo no hab\u00eda podido sondearla\u2014. Bueno \u2014hab\u00eda dicho Jax dando una calada a su puro\u2014, ha valido la pena intentarlo, querida m\u00eda. Y ahora, vete. Hay cartas que repartir, y juegos a los que jugar.\u00bb\n\nQuiz\u00e1 lo habr\u00eda logrado. Si de veras hubiera querido, habr\u00eda podido apoderarme del cuerpo de Jax y apagar aquel maldito puro, pero esa capacidad era precisamente lo que me asustaba. Controlar a otro implicaba demasiada responsabilidad. Aun con la promesa de un aumento de sueldo. Deambular\u00eda por la mente de Londres, pero nunca la controlar\u00eda. Ni por todo el dinero del mundo.\n\n\u2014\u00bfPaige?\n\nSal\u00ed de mi ensimismamiento.\n\n\u2014No \u2014dije\u2014. No habr\u00eda podido poseer a Aludra. Ni poseerte a ti.\n\n\u2014\u00bfPor qu\u00e9 no?\n\n\u2014No puedo poseer a otra persona. Y mucho menos a un refa.\n\n\u2014Pero \u00bfte gustar\u00eda?\n\n\u2014No. No puedes obligarme a hacerlo.\n\n\u2014No tengo intenci\u00f3n de obligarte. Solo te estoy ofreciendo la oportunidad de \u00abampliar tus horizontes\u00bb, como dec\u00eds vosotros.\n\n\u2014Causando dolor.\n\n\u2014Si se hace bien, la posesi\u00f3n no tiene por qu\u00e9 causar dolor. No pretendo que poseas a un humano. Y menos esta noche, por supuesto.\n\n\u2014Entonces \u00bfqu\u00e9 quieres?\n\nMir\u00f3 a lo lejos, m\u00e1s all\u00e1 de la pradera, y yo lo imit\u00e9. La cierva tocaba unas flores con la pezu\u00f1a y las observaba oscilar.\n\n\u2014Nuala \u2014dije.\n\n\u2014S\u00ed.\n\nLa cierva baj\u00f3 la cabeza y olfate\u00f3 la hierba. Nunca me hab\u00eda planteado practicar la posesi\u00f3n con animales. La mente de los animales era muy diferente de la humana (menos compleja, menos consciente), pero eso tal vez lo hiciera a\u00fan m\u00e1s dif\u00edcil. Quiz\u00e1 ni siquiera fuera posible que yo encajara mi esp\u00edritu humano en el cuerpo de un animal. \u00bfPensar\u00eda como un humano cuando tuviera un onirosaje de animal? Y tambi\u00e9n me preocupaban otras cosas. \u00bfLe har\u00eda da\u00f1o a la cierva? \u00bfSe resistir\u00eda a mi infiltraci\u00f3n, o me dejar\u00eda entrar?\n\n\u2014No lo s\u00e9 \u2014dije\u2014. Es demasiado grande. A lo mejor no puedo controlarla.\n\n\u2014Buscar\u00e9 algo m\u00e1s peque\u00f1o.\n\n\u2014\u00bfQu\u00e9 quieres sacar con esto exactamente? \u2014Como no contestaba, continu\u00e9\u2014: Para estar ofreci\u00e9ndome una oportunidad, insistes mucho.\n\n\u2014Quiero que aproveches esta oportunidad. No lo niego.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Porque quiero que sobrevivas.\n\nLe sostuve la mirada un momento, tratando de descifrarlo, pero no pude. Las caras de los refa\u00edtas ten\u00edan algo que dificultaba hacer conjeturas emocionales.\n\n\u2014De acuerdo \u2014ced\u00ed\u2014. Un animal m\u00e1s peque\u00f1o. Un insecto, un roedor, tal vez un p\u00e1jaro. Algo con conciencia limitada.\n\n\u2014Muy bien.\n\nIba a darse la vuelta, pero se par\u00f3. Me mir\u00f3 y se sac\u00f3 algo del bolsillo: un colgante con una cadena fina.\n\n\u2014Ponte esto \u2014dijo.\n\n\u2014\u00bfPor qu\u00e9?\n\nSe march\u00f3 sin contestarme. Me sent\u00e9 sobre una roca peque\u00f1a, y contuve un escalofr\u00edo de emoci\u00f3n. Seguro que Jax lo habr\u00eda aprobado, aunque no ten\u00eda tan claro si Nick lo habr\u00eda hecho.\n\nMir\u00e9 el colgante. Ten\u00eda aproximadamente la longitud de mi pulgar, y estaba entretejido formando unas alas. Cuando lo acarici\u00e9 con un dedo, hubo un d\u00e9bil temblor en el \u00e9ter. Deb\u00eda de estar sublimado. Me pas\u00e9 la cadenilla por la cabeza.\n\nNuala volvi\u00f3 al cabo de un rato, cuando se cans\u00f3 de la hierba. Yo estaba acurrucada contra la roca, con las manos en los bolsillos de la chaqueta. Hac\u00eda un fr\u00edo terrible, y sal\u00edan nubes blancas de vaho de mi boca. \u00abHola\u00bb, dije. Nuala me olisque\u00f3 el pelo, como si tratara de averiguar qu\u00e9 era; luego dobl\u00f3 las patas y se acurruc\u00f3 a mi lado. Apoy\u00f3 la cabeza en mi regazo y dio una especie de bufido de satisfacci\u00f3n. Me quit\u00e9 los guantes y le acarici\u00e9 las orejas. La piel le ol\u00eda a almizcle. Notaba los latidos de su coraz\u00f3n, fuertes y constantes. Nunca hab\u00eda estado tan cerca de un animal salvaje. Intent\u00e9 imaginar qu\u00e9 deb\u00eda de sentirse siendo un cervato: sosteni\u00e9ndose sobre cuatro patas, viviendo en el bosque en estado salvaje.\n\nPero yo no viv\u00eda en estado salvaje. Hab\u00eda vivido m\u00e1s de una d\u00e9cada en una ciudadela de Scion, y hab\u00eda perdido todo mi salvajismo. Supon\u00eda que por eso me hab\u00eda juntado con Jax. Para aferrarme a los vestigios de mi antiguo yo.\n\nAl cabo de un momento decid\u00ed tantear el terreno. Cerr\u00e9 los ojos y solt\u00e9 mi esp\u00edritu. Nuala ten\u00eda un onirosaje permeable, fino y fr\u00e1gil como una pompa de jab\u00f3n. Con los a\u00f1os, los humanos acumul\u00e1bamos capas de resistencia, pero los animales no ten\u00edan esa coraza emocional. En teor\u00eda pod\u00eda controlarla. Le di un empujoncito a su onirosaje.\n\nNuala dio un resoplido de alarma. Le acarici\u00e9 las orejas para tranquilizarla. \u00abLo siento \u2014dije\u2014. No volver\u00e9 a hacerlo.\u00bb Entonces apoy\u00f3 la cabeza en mi regazo, pero estaba temblando. Ella no sab\u00eda que hab\u00eda sido yo quien le hab\u00eda hecho da\u00f1o. La acarici\u00e9 suavemente bajo la barbilla. Para cuando volvi\u00f3 el Custodio, nos hab\u00edamos quedado las dos dormidas. Me despert\u00f3 toc\u00e1ndome la mejilla. Nuala levant\u00f3 la cabeza, pero el Custodio la calm\u00f3 con una palabra y enseguida volvi\u00f3 a dormirse.\n\n\u2014Ven conmigo \u2014me dijo\u2014. Te he encontrado otro cuerpo.\n\nSe sent\u00f3 a mi lado en la roca. Me impresion\u00f3 verlo bajo la luz de la luna: su contorno perfectamente recortado, sus rasgos marcados, su piel radiante.\n\n\u2014\u00bfQu\u00e9 es? \u2014pregunt\u00e9.\n\n\u2014Mira.\n\nTen\u00eda las manos juntas y ahuecadas; las yemas de sus dedos apenas se tocaban. Mir\u00e9 y vi un insecto fr\u00e1gil: una mariposa o una polilla. Costaba distinguirlo con tan poca luz.\n\n\u2014Cuando la he encontrado estaba inactiva \u2014dijo\u2014. Todav\u00eda est\u00e1 en estado let\u00e1rgico. He pensado que as\u00ed ser\u00eda m\u00e1s f\u00e1cil.\n\nEntonces era una mariposa. Temblaba ligeramente en sus manos.\n\n\u2014Los puntos fr\u00edos asustan a los animales \u2014dijo en voz baja\u2014. Perciben que hay un conducto abierto hacia el Inframundo.\n\n\u2014\u00bfPor qu\u00e9 lo has abierto?\n\n\u2014Ahora lo ver\u00e1s. \u2014Me mir\u00f3 a los ojos\u2014. \u00bfEst\u00e1s dispuesta a intentar una posesi\u00f3n?\n\n\u2014S\u00ed, lo intentar\u00e9.\n\nSus ojos resplandecieron, ardientes como brasas.\n\n\u2014Seguramente ya lo sabes \u2014dije\u2014, pero mi cuerpo se va a desplomar cuando lo abandone. Te agradecer\u00eda que me sujetaras.\n\nTuve que obligarme a decir esas palabras. Detestaba tener que pedirle un favor, aunque fuera algo tan obvio y sencillo.\n\n\u2014Por supuesto.\n\nDesvi\u00e9 la mirada.\n\nInspir\u00e9 hondo y solt\u00e9 mi esp\u00edritu. Inmediatamente se me nublaron los sentidos y vi mi onirosaje. Ya sent\u00eda el \u00e9ter; fue fortaleci\u00e9ndose a medida que me acercaba al borde del prado de amapolas, donde estaba oscuro. El \u00e9ter estaba esper\u00e1ndome all\u00ed.\n\nSalt\u00e9.\n\nVi que mi cord\u00f3n arg\u00e9nteo se desenredaba de mi onirosaje, proporcion\u00e1ndome un camino de regreso. El onirosaje del Custodio estaba cerca. La mariposa solo era un puntito a su lado, un grano de arena junto a una canica. Me deslic\u00e9 dentro de su mente. Mi receptor no se retrajo, no dio ni la m\u00e1s leve sacudida.\n\nMe hallaba en un mundo de sue\u00f1os. Un mundo de color ba\u00f1ado en una luz ocre. La mariposa pasaba los d\u00edas aliment\u00e1ndose en las flores, cuyos vivos colores compon\u00edan todos sus recuerdos. Aromas de ambros\u00eda llegaban flotando de todas partes, lavanda, hierba y rosas. Camin\u00e9 por el onirosaje cubierto de roc\u00edo y me dirig\u00ed hacia la zona m\u00e1s luminosa. Ca\u00eda polen formando remolinos de los \u00e1rboles cargados de flores y quedaba prendido en mi pelo. Nunca me hab\u00eda sentido tan libre. No hab\u00eda resistencia, ni siquiera el m\u00e1s leve mecanismo de defensa. Era tan indoloro, tan f\u00e1cil y hermoso; sent\u00eda como si me hubieran quitado unos gruesos grilletes. Parec\u00eda todo tan natural. Eso era lo que ansiaba mi esp\u00edritu: deambular por tierras extra\u00f1as. No soportaba estar atrapado en un \u00fanico cuerpo todo el tiempo. Ten\u00eda ansias de conocer mundo.\n\nLlegu\u00e9 a la zona soleada y lo vi: una lev\u00edsima hebra de esp\u00edritu, de color rosa. Frunc\u00ed los labios y sopl\u00e9, y se desliz\u00f3 hacia las partes m\u00e1s oscuras.\n\nHab\u00eda llegado el momento de la prueba real. Si yo lo hab\u00eda entendido bien (y si Jax, que me lo hab\u00eda explicado, ten\u00eda raz\u00f3n), entrar en la zona soleada me permitir\u00eda hacerme con el control de mi nuevo cuerpo.\n\nNada m\u00e1s entrar en el c\u00edrculo, una luz intensa inund\u00f3 todo el onirosaje: una luz dorada que me envolv\u00eda e impregnaba mis ojos, mi piel y mi sangre. Me ceg\u00f3. El mundo se convirti\u00f3 en un diamante hecho a\u00f1icos, un asterisco de colores luminosos.\n\nDurante unos momentos no hubo nada. Mi cuerpo se hab\u00eda esfumado y no notaba nada. Y entonces despert\u00e9.\n\nLo primero que sent\u00ed fue p\u00e1nico. \u00bfD\u00f3nde estaban mis brazos, mis piernas? Un momento: ve\u00eda (apenas), pero todo estaba te\u00f1ido de morado, y el verde de la hierba era tan intenso que me da\u00f1aba los ojos. Un espasmo sacudi\u00f3 mis endebles extremidades. Aquello era parecido a la peste cerebral, pero mucho peor. Estaba aplastada, asfixiada, gritando sin labios ni voz. Y \u00bfqu\u00e9 eran esas cosas que ten\u00eda pegadas a los costados? Intent\u00e9 moverme, y esas cosas se estremecieron, como si estuviera agonizando.\n\nSin propon\u00e9rmelo, sal\u00ed de la mariposa y volv\u00ed a mi cuerpo. Temblaba de pies a cabeza y respiraba a boqueadas. Resbal\u00e9 de la roca y ca\u00ed al suelo a cuatro patas.\n\n\u2014\u00bfPaige?\n\nMe dieron arcadas. Se me llen\u00f3 la boca de un sabor \u00e1cido y repugnante, pero no vomit\u00e9 nada.\n\n\u2014No volver\u00e9 a hacerlo jam\u00e1s \u2014dije.\n\n\u2014\u00bfQu\u00e9 ha pasado?\n\n\u2014Nada. Era... era tan f\u00e1cil, pero de repente... \u2014Me desabroch\u00e9 la cremallera de la chaqueta; me costaba respirar\u2014. No puedo hacerlo.\n\nEl Custodio guardaba silencio. Se qued\u00f3 mirando mientras yo me enjugaba el sudor de la frente y procuraba no hiperventilar.\n\n\u2014Pues lo has hecho \u2014dijo entonces\u2014. Aunque haya sido doloroso, lo has hecho. Ha movido las alas.\n\n\u2014Cre\u00eda que me iba a morir.\n\n\u2014Pero lo has conseguido.\n\nMe apoy\u00e9 en la roca.\n\n\u2014\u00bfCu\u00e1nto he durado?\n\n\u2014Cerca de medio minuto.\n\nMejor de lo que yo esperaba, pero aun as\u00ed lamentable. Jaxon se habr\u00eda partido de risa.\n\n\u2014Siento decepcionarte \u2014dije\u2014. A lo mejor es que no soy tan buena como otros onir\u00e1mbulos.\n\nEl Custodio me mir\u00f3 con gesto serio.\n\n\u2014Eres buena \u2014dijo\u2014. Pero, si no te lo crees, no podr\u00e1s desarrollar todo tu potencial.\n\nAbri\u00f3 la mano y la mariposa ech\u00f3 a volar. Segu\u00eda viva. No la hab\u00eda matado.\n\n\u2014Est\u00e1s enfadado \u2014dije.\n\n\u2014No.\n\n\u2014Entonces \u00bfpor qu\u00e9 me miras as\u00ed?\n\n\u2014\u00bfC\u00f3mo te miro? \u2014Sus ojos se hab\u00edan enfriado.\n\n\u2014No importa.\n\nCogi\u00f3 un pu\u00f1ado de ramas menudas que estaban apoyadas contra la roca. Entrechoc\u00f3 dos piedras y encendi\u00f3 una peque\u00f1a hoguera utilizando las ramas como encendaja. Desvi\u00e9 la mirada. \u00abQue tiene la rabia.\u00bb Yo no estaba all\u00ed para hacer de titiritera con los animales.\n\n\u2014Descansaremos aqu\u00ed unas horas. \u2014El Custodio no me mir\u00f3\u2014. Necesitas dormir antes de abordar la segunda parte de tu examen.\n\n\u2014\u00bfSignifica eso que he aprobado la primera?\n\n\u2014Claro que has aprobado. Has pose\u00eddo a la mariposa. Era lo \u00fanico que te hab\u00eda pedido. \u2014Se qued\u00f3 contemplando las llamas\u2014. Nada m\u00e1s.\n\nAbri\u00f3 su mochila y despleg\u00f3 un sencillo saco de dormir negro.\n\n\u2014Toma \u2014dijo\u2014. Tengo que hacer una cosa. Aqu\u00ed estar\u00e1s a salvo, al menos un rato.\n\n\u2014\u00bfVuelves a la ciudad?\n\n\u2014S\u00ed.\n\nNo ten\u00eda m\u00e1s remedio que obedecer, aunque no me hac\u00eda ninguna gracia dormir en un sitio donde hab\u00eda tantos esp\u00edritus sueltos. Hab\u00edan aparecido m\u00e1s, y hac\u00eda m\u00e1s fr\u00edo. Me quit\u00e9 las botas y los calcetines mojados, y los puse a secar junto a las llamas; me met\u00ed en el saco y cerr\u00e9 la cremallera. No estaba suficientemente abrigada, pese a la chaqueta y el chaleco, pero era mejor que nada.\n\nEl Custodio tamborileaba con los dedos en la rodilla contemplando la oscuridad. Sus ojos eran dos brasas ardientes en estado de alerta. Me di la vuelta y alc\u00e9 la vista hacia la luna. Qu\u00e9 oscuro estaba el mundo. Qu\u00e9 oscuro y qu\u00e9 fr\u00edo.\n\n\u2014Date prisa, Pip. Vamos.\n\nMi primo Finn me tir\u00f3 m\u00e1s fuerte del brazo. Yo ten\u00eda seis a\u00f1os y est\u00e1bamos en el abarrotado centro de Dubl\u00edn, rodeados de gente que se desga\u00f1itaba.\n\n\u2014No puedo seguirte, Finn \u2014dije, pero \u00e9l no me hizo caso; era la primera vez que mi primo no me escuchaba.\n\nEra una ma\u00f1ana fr\u00eda de febrero de 2046; el sol invernal derramaba su oro blanco sobre el Liffey, y se supon\u00eda que est\u00e1bamos en el cine. Yo estaba pasando las vacaciones de mitad de trimestre en casa de mi t\u00eda Sandra. Ella le hab\u00eda encargado a Finn cuidar de m\u00ed mientras estaba en el trabajo, ya que mi primo no ten\u00eda clases. Yo quer\u00eda ir a ver una pel\u00edcula y a comer en Temple Bar, pero Finn dijo que ten\u00edamos que hacer otra cosa: ir a ver la estatua de Molly Malone. Me asegur\u00f3 que era importante. Demasiado importante para no hacerlo. Un d\u00eda muy especial. \u00abVamos a hacer historia, Pip\u00bb, me hab\u00eda asegurado apret\u00e1ndome la mano que yo llevaba enfundada en un mit\u00f3n.\n\nCuando me lo dijo, arrugu\u00e9 un poco la nariz. La historia era para el colegio. Yo adoraba a Finn (era alto, gracioso e inteligente, y cuando ten\u00eda algo de calderilla siempre me compraba caramelos), pero hab\u00eda visto a Molly centenares de veces. Y me sab\u00eda de memoria la letra de su canci\u00f3n.\n\nNos acercamos a la estatua, a cuyo alrededor se hab\u00eda congregado la gente. Mir\u00e9 entre asustada y emocionada a todas aquellas personas con la cara colorada que cantaban. Finn se puso a cantar con ellas a voz en grito, y yo me un\u00ed tambi\u00e9n, pese a no entender qu\u00e9 hac\u00edamos all\u00ed. Pens\u00e9 que deb\u00eda de ser una fiesta popular.\n\nFinn se puso a hablar con sus amigos del Trinity College sin soltarme de la mano. Iban todos vestidos de verde y agitaban grandes pancartas. Yo ya sab\u00eda leer lo suficiente para descifrar casi todas las palabras, pero hab\u00eda una que no conoc\u00eda: \u00abScion\u00bb. Estaba escrita en todas las pancartas. Las ve\u00eda en lo alto, unas en ingl\u00e9s y otras en irland\u00e9s. \u00ab\u00a1Abajo Mayfield! _\u00c9ire go br\u00e1ch!_ \u00a1Dubl\u00edn dice no!\u00bb Le tir\u00e9 de la manga a Finn.\n\n\u2014\u00bfQu\u00e9 pasa, Finn?\n\n\u2014Nada, Paige. Calla un momento. \u00ab\u00a1Fuera Scion! \u00a1Abajo Scion! \u00a1Scion fuera de Dubl\u00edn!\u00bb\n\nYa est\u00e1bamos cerca de la estatua, empujados por la multitud. Molly siempre me hab\u00eda gustado; ten\u00eda un rostro bondadoso. Sin embargo, ese d\u00eda la encontr\u00e9 diferente. Le hab\u00edan puesto una bolsa en la cabeza y una soga al cuello. Se me empa\u00f1aron los ojos.\n\n\u2014No me gusta, Finn.\n\n\u2014\u00ab\u00a1Fuera Scion! \u00a1Abajo Scion! \u00a1Scion fuera de Dubl\u00edn!\u00bb\n\n\u2014Quiero irme a casa.\n\nLa novia de Finn, Kay, me mir\u00f3 arrugando el ce\u00f1o. Siempre me hab\u00eda ca\u00eddo bien. Ten\u00eda el pelo muy bonito, de un casta\u00f1o rojizo oscuro que brillaba como el cobre, muy rizado, y los brazos muy blancos y cubiertos de pecas. Finn le hab\u00eda regalado un anillo Claddagh que ella llevaba con el coraz\u00f3n apuntando hacia el cuerpo. Ese d\u00eda iba vestida de negro, y se hab\u00eda pintado las mejillas de color verde, blanco y naranja.\n\n\u2014Podr\u00edan ponerse violentos, Finn \u2014dijo\u2014. \u00bfNo ser\u00eda mejor que la llevaras a casa? \u2014Como mi primo no le contestaba, le dio un empuj\u00f3n\u2014. \u00a1Finn!\n\n\u2014\u00bfQu\u00e9 pasa?\n\n\u2014\u00a1Lleva a Paige a casa! Cleary tiene bombas caseras en el coche, por amor de Dios...\n\n\u2014Ni hablar. No me perder\u00eda esto por nada del mundo. Si entran esos cabrones, ya no podremos sacarlos.\n\n\u2014Tiene seis a\u00f1os. No deber\u00eda estar viendo esto. \u2014Kay me dio la mano\u2014. Si no la llevas t\u00fa, lo har\u00e9 yo. Tu madre se avergonzar\u00eda de ti.\n\n\u2014No. Quiero que lo vea.\n\nSe arrodill\u00f3 delante de m\u00ed y se quit\u00f3 la gorra. Ten\u00eda el pelo alborotado. Finn se parec\u00eda a mi padre, pero ten\u00eda un semblante que denotaba franqueza y bondad, y unos ojos azules como el cielo de verano. Me puso las manos sobre los hombros.\n\n\u2014Paige Eva \u2014dijo poni\u00e9ndose muy serio\u2014, \u00bfsabes qu\u00e9 est\u00e1 pasando?\n\nDije que no con la cabeza.\n\n\u2014Van a llegar unos malos por el mar. Nos van a encerrar en nuestra ciudad y no nos dejar\u00e1n salir, y convertir\u00e1n este sitio en una ciudad-c\u00e1rcel como la suya. No nos dejar\u00e1n volver a cantar nuestras canciones, ni visitar a nadie fuera de Irlanda. Y las personas como t\u00fa, Pip... no les gustan.\n\nMir\u00e9 a Finn a los ojos y entend\u00ed lo que quer\u00eda decir. Finn siempre hab\u00eda sabido que yo ve\u00eda cosas. Yo sab\u00eda d\u00f3nde viv\u00edan todos los fantasmas de Dubl\u00edn. \u00bfMe convert\u00eda eso en una mala persona?\n\n\u2014Pero \u00bfpor qu\u00e9 lleva Molly una bolsa en la cabeza, Finn? \u2014pregunt\u00e9.\n\n\u2014Porque eso es lo que les hacen los malos a las personas que no les caen bien. Les ponen una bolsa en la cabeza y una soga al cuello.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Para matarlas. Tambi\u00e9n a las ni\u00f1as peque\u00f1as como t\u00fa.\n\nMe puse a temblar. Me dol\u00edan los ojos. Ten\u00eda un nudo en la garganta, pero no lloraba. Era valiente. Muy valiente, como Finn.\n\n\u2014\u00a1Ya los veo, Finn! \u2014exclam\u00f3 Kay.\n\n\u2014\u00ab\u00a1Fuera Scion! \u00a1Abajo Scion!\u00bb\n\nEl coraz\u00f3n me lat\u00eda muy deprisa. Finn me enjug\u00f3 las l\u00e1grimas y me puso su gorra en la cabeza.\n\n\u2014\u00ab\u00a1Scion fuera de Dubl\u00edn!\u00bb\n\n\u2014Ya vienen, Paige, y tenemos que detenerlos. \u2014Me agarr\u00f3 por los hombros\u2014. \u00bfVas a ayudarme a detenerlos?\n\nAsent\u00ed.\n\n\u2014\u00a1Dios m\u00edo, Finn, tienen tanques!\n\nY entonces mi mundo explot\u00f3. Los malos hab\u00edan apuntado sus ca\u00f1ones y hab\u00edan disparado sus dardos de fuego contra los manifestantes.\n\nMe despert\u00f3 el ruido de disparos dentro de mi cabeza.\n\nNotaba la piel fr\u00eda y h\u00fameda, pero por dentro estaba ardiendo. Aquel recuerdo me hab\u00eda abrasado todo el cuerpo. Todav\u00eda me parec\u00eda ver a Finn, su rostro transido de odio. Finn, que siempre me llamaba Pip.\n\nMe puse a patalear dentro del saco de dormir. Hab\u00edan pasado trece a\u00f1os y todav\u00eda o\u00eda disparos. Segu\u00eda viendo a Kay, con los ojos abiertos, sorprendida por la muerte. Su blusa manchada de sangre. Un disparo en el coraz\u00f3n. Eso fue lo que hizo que Finn corriera hacia los soldados y me dejara atr\u00e1s, agachada bajo la carretilla de Molly. Grit\u00e9 su nombre una y otra vez, pero Finn ya no regres\u00f3.\n\nNo volv\u00ed a verlo nunca.\n\nDespu\u00e9s de eso ya no recordaba casi nada. S\u00e9 que alguien me llev\u00f3 a casa. S\u00e9 que llor\u00e9 por Finn hasta quedarme af\u00f3nica. Y s\u00e9 que mi padre no permiti\u00f3 que mi t\u00eda Sandra volviera a verme hasta el d\u00eda del funeral. Despu\u00e9s dej\u00e9 de llorar. Las l\u00e1grimas no hac\u00edan que las personas volvieran. Me enjugu\u00e9 el sudor de la cara con la camisa. Deb\u00eda de estar todav\u00eda en los jardines de Magdalen. Me volv\u00ed hacia un lado; ten\u00eda tanto fr\u00edo que no me notaba los pies, y me ovill\u00e9 dentro del saco.\n\nEl fuego deb\u00eda de haberse apagado. Llov\u00eda, pero no me hab\u00eda mojado. Estir\u00e9 un brazo hacia arriba y mis dedos rozaron un toldo de lona, un refugio improvisado. Me puse la capucha de la chaqueta y sal\u00ed despacio de debajo de aquel toldo.\n\n\u2014\u00bfCustodio?\n\nNo hab\u00eda ni rastro de \u00e9l. Ni de la cierva. Ni de la hoguera.\n\nYa no temblaba solo de fr\u00edo. \u00bfAd\u00f3nde pod\u00eda haber ido? Me hab\u00eda dicho que iba a la ciudad, pero en realidad no hab\u00edamos salido de Sheol I. Magdalen y sus jardines formaban parte del complejo de residencias. Solo nos hab\u00edamos alejado algo m\u00e1s de un kil\u00f3metro del punto fr\u00edo, o ni siquiera eso.\n\nEl viento arreciaba. Me guarec\u00ed bajo el toldo. No hab\u00eda ninguna raz\u00f3n para que el Custodio me hubiera dejado all\u00ed sola. A lo mejor no hab\u00eda dormido tanto rato como cre\u00eda. Me puse los calcetines y las botas y mir\u00e9 dentro del saco de dormir. Para mi sorpresa, dentro hab\u00eda unas pocas provisiones: unos guantes, una jeringuilla hipod\u00e9rmica de adrenalina y una peque\u00f1a linterna de plata, junto con un sobre de papel Manila con mi nombre escrito. Reconoc\u00ed la letra de mi guardi\u00e1n y abr\u00ed el sobre.\n\nBienvenida a la Tierra de Nadie. Tu examen es sencillo: tienes que volver a Sheol I en el menor tiempo posible. No tienes comida, agua ni mapa. Utiliza tu don. Conf\u00eda en tu instinto.\n\nY hazme este favor: sobrevive a esta noche. Estoy seguro de que preferir\u00edas que no te rescataran.\n\nBuena suerte.\n\nMe qued\u00e9 con la nota en la mano un momento, y luego la romp\u00ed en pedazos.\n\nSe iba a enterar. Se iba a enterar, y bien. Intentaba asustarme, pero no iba a conseguirlo. \u00abSobrevive a esta noche.\u00bb \u00bfQu\u00e9 se supon\u00eda que significaba eso? Muy fr\u00e1gil deb\u00eda de considerarme si no me cre\u00eda capaz de vencer un poco de viento y un poco de lluvia. Si pod\u00eda apa\u00f1\u00e1rmelas en las calles s\u00f3rdidas de SciLo, pod\u00eda arregl\u00e1rmelas en un bosque oscuro. En cuanto a la comida, \u00bfpara qu\u00e9 la necesitaba? Tampoco me hab\u00eda dejado tirada en medio del desierto, \u00bfno?\n\nSaqu\u00e9 la cabeza de la improvisada tienda y vi una caja marcada con el s\u00edmbolo de ScionIdus, el brazo armado del gobierno: dos l\u00edneas en \u00e1ngulo recto formando una horca, con tres l\u00edneas m\u00e1s cortas atravesadas en la l\u00ednea vertical. Dentro hab\u00eda otra nota.\n\nTen cuidado con los dardos. Si se rompen, el \u00e1cido que contienen te provocar\u00e1 una parada card\u00edaca. En caso de emergencia, usa la bengala y enviar\u00e9 a una brigada de casacas rojas.\n\nNo vayas hacia el sur.\n\nIlumin\u00e9 el contenido de la caja con la linterna: una pistola de ca\u00f1\u00f3n largo, una pistola de bengalas, un viejo encendedor Zippo, una navaja y tres dardos de plata presurizados. En los lados llevaban impresos los s\u00edmbolos de toxicidad y corrosividad, junto con las palabras \u00ab\u00c1cido Fluorh\u00eddrico (HF)\u00bb. Una pistola tranquilizadora y un pu\u00f1ado de dardos de \u00e1cido. \u00bfNo pod\u00eda haberme devuelto mi pistola? Bueno, algo tendr\u00eda que hacer, a menos que quisiera quedarme toda la noche en aquel claro. Enroll\u00e9 el saco de dormir y lo met\u00ed en una bolsa, pero no desmont\u00e9 la tienda. Pod\u00eda servirme de indicador para asegurarme de que no corr\u00eda en c\u00edrculos.\n\nEntonces vi que hab\u00eda algo alrededor del campamento: un c\u00edrculo de diminutos cristales blancos. Me arrodill\u00e9 y los toqu\u00e9 con los dedos; saqu\u00e9 la punta de la lengua y los prob\u00e9.\n\n\u00abSal.\u00bb\n\nEl campamento estaba montado en el centro de un c\u00edrculo de sal.\n\nMe qued\u00e9 muy quieta. Entre los videntes se rumoreaba que la sal pod\u00eda repeler a los esp\u00edritus (lo llamaban alomancia), pero yo sab\u00eda que no era verdad. Desde luego, no serv\u00eda para ahuyentar a los duendes. \u00bfLa habr\u00eda puesto all\u00ed solo para asustarme?\n\nMe sub\u00ed la capucha, me abroch\u00e9 la chaqueta hasta arriba y guard\u00e9 mis escasos pertrechos. Met\u00ed los dardos y la pistola en la mochila, protegi\u00e9ndolos con el saco de dormir, y me puse la pistola de bengalas en la cinturilla. La navaja me la met\u00ed dentro de la bota, y la jeringuilla, en la chaqueta. Me puse los guantes.\n\nEstaba deseando volver y plantarme ante aquel desgraciado. Me lo imaginaba c\u00f3modamente sentado junto a la chimenea encendida, mirando el reloj, contando los minutos hasta mi regreso.\n\nSe iba a enterar. No pensaba pasar desapercibida. Yo era la So\u00f1adora P\u00e1lida, y le iba a demostrar por qu\u00e9. Iba a saber por qu\u00e9 Jax me hab\u00eda elegido: porque hab\u00eda sobrevivido, contra todo pron\u00f3stico.\n\nCerr\u00e9 los ojos e intent\u00e9 detectar actividad et\u00e9rea, pero no hab\u00eda nada. Ning\u00fan onirosaje. Estaba sola. Cuando abr\u00ed los ojos, me llam\u00f3 la atenci\u00f3n el cielo. Ten\u00eda suerte de haber despertado cuando lo hab\u00eda hecho: las estrellas estaban a punto de desaparecer detr\u00e1s de una nube, y como ya se hab\u00eda puesto el sol, no ten\u00eda otra forma de orientarme. No vi Sirio, as\u00ed que busqu\u00e9 el Cintur\u00f3n de Ori\u00f3n. Gracias a los apasionados discursos de Nick sobre astronom\u00eda sab\u00eda que si encontrabas el Cintur\u00f3n, pod\u00edas situar el norte, que estaba m\u00e1s o menos en la direcci\u00f3n opuesta. Tambi\u00e9n sab\u00eda d\u00f3nde estaba en relaci\u00f3n con Sheol I. Localic\u00e9 las tres estrellas y me volv\u00ed despacio hasta encarar mi camino. Lo que ten\u00eda ante m\u00ed era un denso tramo de bosque tupido y oscuro.\n\nEl coraz\u00f3n me lat\u00eda con fuerza. Nunca me hab\u00eda asustado la oscuridad, pero me obligar\u00eda a depender de mi sexto sentido para detectar cualquier movimiento. Seguramente se trataba de eso: de ponerme a prueba.\n\nMir\u00e9 hacia atr\u00e1s. El bosque del otro lado del claro estaba igual de oscuro. El sendero me llevar\u00eda hacia el sur, lejos de la colonia.\n\n\u00abNo vayas hacia el sur.\u00bb __\n\nSab\u00eda a qu\u00e9 jugaba el Custodio. Esperaba que yo obedeciera, como buena humana. Pero \u00bfpor qu\u00e9 iba a ir hacia el norte, si el norte me devolver\u00eda a la esclavitud y al Custodio, que era quien me hab\u00eda puesto all\u00ed? No necesitaba demostrarle nada. Me volv\u00ed hacia el Cintur\u00f3n. Ir\u00eda hacia el sur. Me marchar\u00eda de aquel lugar horrible.\n\nEl fuerte viento atravesaba las copas de los \u00e1rboles y me helaba la piel h\u00fameda. Ahora o nunca. Si me pon\u00eda a pensar qu\u00e9 pod\u00eda haber o no escondido all\u00ed dentro, cuando acabara ya no tendr\u00eda valor para moverme. Apret\u00e9 las mand\u00edbulas y entr\u00e9 en el bosque.\n\nEstaba muy negro. No ve\u00eda nada. La lluvia hab\u00eda ablandado el terreno, dej\u00e1ndolo esponjoso y empapado. Mis pies no hac\u00edan ruido alguno mientras avanzaba entre los robles, a buen paso, a veces corriendo, ayud\u00e1ndome de las manos para tantear el camino entre las ramas. Con el delgado haz de mi linterna distingu\u00eda una d\u00e9bil bruma que abrazaba los troncos de los \u00e1rboles y quedaba suspendida formando una fina manta sobre el suelo que tapaba mis botas. No hab\u00eda luz natural. Rec\u00e9 para que la linterna no se apagara. Llevaba el s\u00edmbolo de Scion; deb\u00eda de pertenecer a un lote prestado por la DVN. Eso me produc\u00eda cierto alivio, pues los art\u00edculos fabricados por Scion no sol\u00edan estropearse.\n\nDe pronto se me ocurri\u00f3 pensar que me hallaba fuera de los l\u00edmites normales de Sheol I. Aquel lugar se llamaba Tierra de Nadie por alg\u00fan motivo: porque no pertenec\u00eda a nadie. Quiz\u00e1 fuera propiedad de Scion, pero quiz\u00e1 no. No ten\u00eda ni idea de ad\u00f3nde me llevar\u00eda esa ruta, pero s\u00ed sab\u00eda que Oxford estaba al norte de Londres. Iba en la direcci\u00f3n correcta. La chaqueta y los pantalones eran lo bastante oscuros para no llamar la atenci\u00f3n si hab\u00eda alguien vigilando, y mi sexto sentido estaba m\u00e1s afinado que nunca. Pod\u00eda pasar sin ser vista al lado de cualquier vigilante refa\u00edta. Pod\u00eda trepar por una valla con la misma facilidad con que pod\u00eda colarme por debajo de ella. Y si alguien me atacaba, pod\u00eda emplear mi don. Lo sabr\u00eda con antelaci\u00f3n.\n\nPero entonces me acord\u00e9 de lo que hab\u00eda dicho Liss de aquel lugar a mi llegada: \u00abCampos desiertos. Lo llamamos la Tierra de Nadie\u00bb. Eso me habr\u00eda animado de no ser por lo que hab\u00eda dicho despu\u00e9s, cuando le pregunt\u00e9 si alguna vez alguien hab\u00eda intentado huir por la ruta sur. \u00abS\u00ed\u00bb, se hab\u00eda limitado a contestar. Solo \u00abs\u00ed\u00bb. Una confirmaci\u00f3n indudable de que aquel camino entra\u00f1aba peligros. Otros videntes lo hab\u00edan tomado y hab\u00edan muerto. Quiz\u00e1 a ellos tambi\u00e9n los hubieran puesto a prueba as\u00ed. \u00bfConsist\u00eda el examen, simplemente, en resistir la tentaci\u00f3n de huir? Pensarlo me hizo romper a sudar. Minas de tierra, bombas trampa: seguro que las ten\u00edan. Imagin\u00e9 c\u00e1maras en los \u00e1rboles que observaban cada uno de mis movimientos, esperando a que pisara una mina. Reduje un poco el paso.\n\nNo, no. Ten\u00eda que continuar. Pod\u00eda salir de all\u00ed. Ellos confiaban en que yo pensar\u00eda as\u00ed, en que adoptar\u00eda una actitud conservadora. Estuve a punto de torcer hacia el norte, pero mi determinaci\u00f3n me hizo continuar. No pude evitar imaginarme al Custodio, a David y al Capataz junto al fuego, entrechocando sus copas al verme pisar una mina. \u00abBien, caballeros, brindemos por la onir\u00e1mbula \u2014dir\u00eda el Capataz\u2014. La humana m\u00e1s idiota que jam\u00e1s hemos tra\u00eddo a Sheol I.\u00bb Y \u00bfqu\u00e9 escribir\u00edan en mi l\u00e1pida? \u00bfPAIGE MAHONEY, o solo XX-59-40? Y eso suponiendo que quedaran suficientes restos de m\u00ed que recoger y meter en una tumba, claro.\n\nMe par\u00e9 y me apoy\u00e9 en un \u00e1rbol. Era una locura. \u00bfPor qu\u00e9 me imaginaba esas cosas? El Custodio no soportaba al Capataz. Cerr\u00e9 los ojos, apret\u00e9 los p\u00e1rpados y me imagin\u00e9 a otro grupo: Jaxon, Nick y Eliza. Estaban en la ciudadela, esper\u00e1ndome, busc\u00e1ndome. Si consegu\u00eda salir de aquel bosque, podr\u00eda volver con ellos. Al cabo de un momento abr\u00ed los ojos. Y me qued\u00e9 mirando lo que hab\u00eda en el suelo.\n\nHuesos. Huesos humanos. Un esqueleto con blus\u00f3n blanco hecho jirones; le faltaban las piernas desde las rodillas. Retroced\u00ed, y estuve a punto de tropezar y caerme. Algo cruji\u00f3 bajo mi pie. Un cr\u00e1neo.\n\nJunto al esqueleto hab\u00eda una bolsa. Una mano todav\u00eda sujetaba la correa. La solt\u00e9, y produjo un ruido de huesos secos. Las moscas se atiborraban de los restos de carne: unas moscas gigantescas, con pelos negros, saciadas de tejido muerto. Echaron a volar cuando le arranqu\u00e9 la bolsa a su difunto propietario. Alumbr\u00e9 el contenido con la linterna: un poco de pan podrido y una botella vac\u00eda.\n\nSent\u00ed un sudor fr\u00edo. Ilumin\u00e9 con la linterna hacia la derecha. A escasa distancia se abr\u00eda un cr\u00e1ter entre las hojas, anegado de agua de lluvia. Hab\u00eda trozos de hueso y fragmentos de mina esparcidos por el suelo.\n\nAs\u00ed que era verdad: hab\u00eda un campo de minas.\n\nApoy\u00e9 la espalda en el tronco de un roble. No pod\u00eda atravesar un campo de minas a oscuras. Me separ\u00e9 poco a poco del \u00e1rbol y pas\u00e9 por encima del esqueleto. \u00abNo pasa nada, Paige.\u00bb Me temblaban las piernas; torc\u00ed hacia el norte y volv\u00ed sobre mis pasos. No me hab\u00eda alejado mucho del claro. Lo conseguir\u00eda. Solo me hab\u00eda apartado un poco de los huesos cuando tropec\u00e9 con una ra\u00edz y di contra el suelo. Me qued\u00e9 r\u00edgida, con todos los m\u00fasculos en tensi\u00f3n y el coraz\u00f3n lati\u00e9ndome desbocado, pero no se produjo ninguna explosi\u00f3n.\n\nMe apoy\u00e9 en los codos, busqu\u00e9 dentro de la chaqueta, saqu\u00e9 el Zippo y lo encend\u00ed con el pulgar. Brot\u00f3 una llama limpia. Una ruta hacia el \u00e9ter. Yo no era augur (el fuego no era mi aliado), pero pod\u00eda utilizarlo para realizar una sesi\u00f3n espiritista en miniatura. \u00abNecesito un gu\u00eda \u2014susurr\u00e9\u2014. Si hay alguien ah\u00ed, que venga hasta esta llama.\u00bb\n\nAl principio no pas\u00f3 nada. La llama temblaba y parpadeaba. Entonces mi sexto sentido se activ\u00f3, y de entre los \u00e1rboles sali\u00f3 un joven esp\u00edritu. Me levant\u00e9.\n\n\u2014Necesito llegar a mi campamento. \u2014Le acerqu\u00e9 el mechero\u2014. \u00bfPuedes guiarme?\n\nNo le o\u00ed decir nada, pero se desplaz\u00f3 hacia el camino por el que yo hab\u00eda llegado. Not\u00e9 que era el esp\u00edritu del casaca blanca muerto, y ech\u00e9 a correr tras \u00e9l. No hab\u00eda ninguna raz\u00f3n para que me enga\u00f1ara.\n\nNo tard\u00e9 en ver el c\u00edrculo de sal. La lluvia apag\u00f3 el encendedor, pero el esp\u00edritu no se alej\u00f3 de m\u00ed. Tard\u00e9 unos minutos en serenarme. Era duro aceptarlo, pero no ten\u00eda m\u00e1s alternativa que dirigirme hacia el norte. Comprob\u00e9 que mis cosas segu\u00edan all\u00ed y ech\u00e9 a andar de nuevo hacia los \u00e1rboles, con la linterna en una mano y el Zippo en la otra, y con el esp\u00edritu sigui\u00e9ndome de cerca.\n\nCuando llevaba una media hora andando, con el esp\u00edritu alrededor de los hombros como una soga, me par\u00e9 para comprobar que segu\u00eda teniendo el Cintur\u00f3n de Ori\u00f3n a mis espaldas. Correg\u00ed un poco el rumbo y volv\u00ed a sumergirme en la oscuridad. Me dol\u00edan los o\u00eddos y la nariz, y mi sexto sentido me provocaba estremecimientos. Apenas me notaba los dedos de los pies. Me par\u00e9 y me agarr\u00e9 las rodillas mientras respiraba hondo para calmar los nervios. Nada m\u00e1s inhalar, ol\u00ed algo. Conoc\u00eda bien ese olor: olor a muerto.\n\nEl haz de la linterna flaqueaba. El hedor a carne podrida era cada vez m\u00e1s intenso. Segu\u00ed andando un minuto m\u00e1s, y entonces encontr\u00e9 la fuente: otro cad\u00e1ver.\n\nParec\u00eda el cad\u00e1ver de un zorro. Mechones de pelo rojizo, apelmazado y manchado de sangre seca; las cuencas de los ojos repletas de gusanos. Me tap\u00e9 la nariz y la boca con la manga de la chaqueta. El hedor era espantoso.\n\nQuienquiera que hubiera hecho aquello, estaba en el bosque conmigo.\n\n\u00abNo te pares, Paige. Sigue adelante.\u00bb La linterna parpade\u00f3. Nada m\u00e1s ponerme en marcha, o\u00ed partirse una ramita.\n\n\u00bfMe lo hab\u00eda imaginado? No, claro que no. Ten\u00eda muy buen o\u00eddo. O\u00eda mi sangre palpitar en los o\u00eddos. Apoy\u00e9 la espalda en un \u00e1rbol e intent\u00e9 respirar haciendo el m\u00ednimo ruido posible.\n\nUn vigilante. Un casaca roja haciendo la ronda nocturna. Pero entonces o\u00ed unos pasos muy pesados, demasiado pesados para ser humanos. Apagu\u00e9 la linterna y me la guard\u00e9 en el bolsillo. De nada iba a servirme llevarla en la mano: si la encend\u00eda, delatar\u00eda mi posici\u00f3n.\n\nAguc\u00e9 el o\u00eddo. No ve\u00eda nada, pero o\u00ed otra pisada m\u00e1s cerca. Y luego, ruido de dientes masticando el esqueleto. Alguien hab\u00eda encontrado el zorro.\n\nO hab\u00eda vuelto por \u00e9l.\n\nProteg\u00ed la llama del mechero con la mano ahuecada. Mi coraz\u00f3n hac\u00eda cosas raras. No estaba segura de si se hab\u00eda acelerado tanto que solo percib\u00eda un zumbido, o si hab\u00eda dejado de latir. Detr\u00e1s de m\u00ed, el esp\u00edritu se estremeci\u00f3.\n\nPasaban los minutos. Esper\u00e9. Tendr\u00eda que moverme, pero estaba convencida de que hab\u00eda algo cerca.\n\nTres ruiditos guturales.\n\nSe me tensaron todos los m\u00fasculos del cuerpo. Respiraba por la nariz, manteniendo los labios fuertemente cerrados. No sab\u00eda qu\u00e9 hab\u00eda sido ese ruido, pero ten\u00eda la certeza de que no lo hab\u00eda producido ning\u00fan humano. Hab\u00eda o\u00eddo a los refas hacer ruidos raros, pero nunca un sonido tan desagradable y visceral.\n\nUna r\u00e1faga de viento apag\u00f3 la llama del mechero. Mi esp\u00edritu gu\u00eda desapareci\u00f3.\n\nEl miedo me paraliz\u00f3 durante un minuto. Entonces me acord\u00e9 de la pistola que llevaba en la mochila. Era una estupidez dispararle a aquella cosa, pero quiz\u00e1 pudiera distraerla y ganar algo de tiempo para avanzar. Me plante\u00e9 trepar a un \u00e1rbol, pero lo descart\u00e9. Los \u00e1rboles no eran mi fuerte. Era mejor que buscara otro sitio donde esconderme. Lo m\u00e1s sensato parec\u00eda buscar terreno elevado. Si consegu\u00eda llegar a un lugar seguro, podr\u00eda alumbrar a aquel ser con la linterna y ver qu\u00e9 era. Cerr\u00e9 el mechero y me lo guard\u00e9 en la mochila.\n\nYa con la pistola en la mano, me dispuse a sacar un dardo. Ten\u00eda la impresi\u00f3n de que hac\u00eda mucho ruido cada vez que me mov\u00eda: con cada exhalaci\u00f3n, con cada susurro de mi chaqueta. Por fin toqu\u00e9 el cilindro fr\u00edo y liso de un dardo con las yemas de los dedos. Sab\u00eda c\u00f3mo cargar una pistola de balas, pero tard\u00e9 unos minutos en cargar aquella arma con la que no estaba familiarizada, a oscuras, con las manos sudadas, procurando por todos los medios no hacer ruido. En cuanto lo hube conseguido, levant\u00e9 los brazos, apunt\u00e9 y dispar\u00e9.\n\nCuando el dardo se clav\u00f3, chisporrote\u00f3 como la grasa en una sart\u00e9n. El ser corri\u00f3 hacia el origen del ruido, produciendo al moverse una especie de zumbido. Moscas.\n\nNo, no era un animal.\n\nSent\u00ed n\u00e1useas. Hab\u00eda o\u00eddo decir muchas cosas sobre los emim, pero nunca me hab\u00eda imaginado que me enfrentar\u00eda a uno. Pese a lo que hab\u00eda o\u00eddo el d\u00eda del serm\u00f3n, y pese a haber visto a aquel casaca roja que hab\u00eda perdido una mano, casi hab\u00eda empezado a creer que no exist\u00edan. Hasta ese momento.\n\nTuve que esforzarme para mantenerme en pie. Me temblaban las manos y me casta\u00f1eteaban los dientes. No pod\u00eda respirar ni pensar. \u00bfOir\u00eda aquella cosa mis latidos? \u00bfOler\u00eda mi miedo? \u00bfEstaba babeando ya, intuyendo mi carne, o ten\u00eda que acercarme m\u00e1s para que me detectara?\n\nCargu\u00e9 otro dardo en la pistola. El zumbador olfate\u00f3 el sitio desde donde yo hab\u00eda disparado. Cerr\u00e9 los ojos y busqu\u00e9 el \u00e9ter.\n\nAlgo iba mal, muy mal. Todos los esp\u00edritus hab\u00edan huido, como si tuvieran miedo; pero \u00bfpor qu\u00e9 temer\u00edan los esp\u00edritus a un ser del mundo f\u00edsico? Ellos no pod\u00edan volver a morir. Fuera cual fuese la causa, no hab\u00eda nada a lo que recurrir.\n\nRepar\u00e9 en que ya no pod\u00eda o\u00edr al zumbador. Ten\u00eda las manos sudadas y me costaba sujetar la pistola. Pod\u00eda morir en cualquier momento.\n\nTodo aquello deb\u00eda de ser un montaje. Nashira nunca hab\u00eda pretendido que me ganara los colores. Lo \u00fanico que quer\u00eda era verme muerta.\n\n\u00abNo ser\u00e1 hoy \u2014me dije\u2014. No ser\u00e1 hoy, Nashira.\u00bb\n\nSal\u00ed corriendo de detr\u00e1s de los \u00e1rboles. Mis botas pisaban con fuerza; el coraz\u00f3n me martilleaba en el pecho. \u00bfD\u00f3nde se hab\u00eda metido? \u00bfMe hab\u00eda visto ya?\n\nAlgo me golpe\u00f3 entre los omoplatos. Qued\u00e9 ingr\u00e1vida un instante, suspendida en la oscuridad. Luego choqu\u00e9 contra el suelo. Se me dobl\u00f3 una mu\u00f1eca y se me parti\u00f3. Contuve un grito, pero no a tiempo. La pistola hab\u00eda desaparecido. Ya no hab\u00eda ninguna posibilidad de encontrarla. Volv\u00ed a o\u00edr a la cosa: estaba cerca de m\u00ed, la ten\u00eda encima. Con la mano ilesa, me saqu\u00e9 la navaja de la bota.\n\nMe olvid\u00e9 de mi esp\u00edritu. Clav\u00e9 la navaja en una masa blanda. Me resbal\u00f3 algo h\u00famedo por la mu\u00f1eca. \u00abZzzz.\u00bb Otra pu\u00f1alada. Y otra. \u00abZzzz. Zzzz.\u00bb Me ca\u00edan unas cosas peque\u00f1as y redondas en la cara. Parpade\u00e9 para quit\u00e1rmelas de los ojos y las escup\u00ed de la boca. Unos dedos me agarraron por el cuello, y not\u00e9 un aliento hediondo y c\u00e1lido junto a la mejilla. Otra pu\u00f1alada, y otra. \u00abZzzz.\u00bb Unos dientes entrechocaron junto a mi oreja. Volv\u00ed a clavar la navaja en la carne y la hund\u00ed con todas mis fuerzas. La hoja cort\u00f3 m\u00fasculos y cart\u00edlagos.\n\nDe pronto, el ser hab\u00eda desaparecido. Qued\u00e9 libre. Ten\u00eda las manos cubiertas hasta las mu\u00f1ecas de un l\u00edquido apestoso y espeso como jarabe. La bilis ascendi\u00f3 por mi garganta abras\u00e1ndome la boca y la nariz.\n\nLa linterna estaba tirada en el suelo, a unos tres metros. Me arrastr\u00e9 hasta ella, protegi\u00e9ndome la mu\u00f1eca contra el pecho. No era la primera vez que me la romp\u00eda: me dol\u00eda a morir. Me arrastr\u00e9 impuls\u00e1ndome con un solo brazo, con la navaja entre los dientes, empapada de sudor. El olor a cad\u00e1ver me revolv\u00eda el est\u00f3mago y me produc\u00eda violentas arcadas.\n\nAgarr\u00e9 la linterna y alumbr\u00e9 detr\u00e1s de m\u00ed. Vi sombras oscuras entre los \u00e1rboles. M\u00e1s pasos. M\u00e1s zumbadores. \u00ab\u00a1No!\u00bb\n\nMe dol\u00eda la cabeza. Se me nubl\u00f3 la visi\u00f3n. \u00abNo quiero morir.\u00bb Poseer a la mariposa me hab\u00eda debilitado m\u00e1s de lo que hab\u00eda previsto. \u00abCorre.\u00bb Met\u00ed una mano dentro de la chaqueta y saqu\u00e9 la jeringuilla: era mi \u00faltimo recurso. No quer\u00eda disparar la pistola de bengalas, no era un recurso. No quer\u00eda perder la partida.\n\nAdrenalina Autoinyectable ScionMed. Mucho m\u00e1s potente que el c\u00f3ctel de f\u00e1rmacos diluido que Jax utilizaba para mantenerme despierta. Me clav\u00e9 la aguja en el muslo a trav\u00e9s de los pantalones.\n\nNot\u00e9 un dolor intenso. Maldije en voz alta, pero no extraje la aguja. Un chorro de adrenalina se introdujo en el m\u00fasculo. La adrenalina de Scion estaba dise\u00f1ada para despertar todo tu cuerpo; no se limitaba a ayudarlo a funcionar, sino que eliminaba el dolor y te fortalec\u00eda. A los _centis_ se les suministraba constantemente. Mis m\u00fasculos ganaron flexibilidad; mis piernas ganaron fuerza. Me levant\u00e9 del suelo y ech\u00e9 a correr. La adrenalina no ten\u00eda efecto sobre mi sexto sentido, pero me ayudaba a concentrarme en el \u00e9ter.\n\nEl zumbador ten\u00eda un onirosaje oscuro y tenebroso; parec\u00eda un agujero negro en el \u00e9ter. Si trataba de entrar en \u00e9l, no llegar\u00eda muy lejos. De todas formas lo intent\u00e9, aunque sin abandonar mi cuerpo del todo.\n\nMe envolvi\u00f3 una nube negra. Mi onirosaje se oscureci\u00f3, y los m\u00e1rgenes de mi visi\u00f3n se estrecharon. Necesitaba repelerlo; con un salto brusco quiz\u00e1 lo lograra. Mi esp\u00edritu se proyect\u00f3 fuera de mi cuerpo y agriet\u00f3 los bordes del onirosaje de aquella cosa, que solt\u00f3 un grito espeluznante. Dej\u00e9 de o\u00edr sus pasos. Al mismo tiempo, un dolor atroz me oblig\u00f3 a volver a mi onirosaje. Ca\u00ed al suelo parando el golpe con las palmas de las manos. Me puse de nuevo en pie. Jadeaba.\n\nEl bosque dej\u00f3 paso a una pradera. Distingu\u00ed las torres de la Casa. La ciudad. Era la ciudad.\n\nLa adrenalina circulaba a toda velocidad por mis venas y regaba mis m\u00fasculos. Corr\u00ed hacia mi prisi\u00f3n con la mu\u00f1eca colgando a un costado, como una pecadora arrepentida. Prefer\u00eda ser prisionera que fiambre.\n\nEl zumbador grit\u00f3. Su grito reson\u00f3 por cada c\u00e9lula de mi cuerpo. Salt\u00e9 por encima de una alambrada y segu\u00ed corriendo. En lo alto de la Casa hab\u00eda una torre de vigilancia; dentro deb\u00eda de haber un casaca roja armado. Pod\u00edan detener al zumbador; pod\u00edan matarlo. El sudor me empapaba la ropa. Ya no faltaba mucho. Todav\u00eda no notaba el dolor, pero sab\u00eda que me hab\u00eda desgarrado un m\u00fasculo. Pas\u00e9 al lado de un letrero oxidado que rezaba: AUTORIZADO EL USO DE PODERES MORT\u00cdFEROS. Bien. Nunca hab\u00eda necesitado los poderes mort\u00edferos m\u00e1s que ahora. Ya ve\u00eda la torre de vigilancia. Cuando me dispon\u00eda a sacar la pistola de bengalas y gritar pidiendo ayuda, me sent\u00ed inmovilizada.\n\nUna red. Una gruesa red de alambre me cubr\u00eda por completo. \u00ab\u00a1No, no! \u00a1Matadlo!\u00bb, grit\u00e9 a pleno pulm\u00f3n. Forceje\u00e9 como un pez que ha mordido el anzuelo. \u00bfPor qu\u00e9 me hab\u00edan atrapado? \u00a1Yo no era el enemigo! \u00abClaro que lo eres\u00bb, dijo una voz dentro de mi cabeza; pero yo ya no escuchaba nada. Ten\u00eda que soltarme de la red. El zumbador se acercaba. Me destrozar\u00eda, como hab\u00eda hecho con el zorro.\n\nUn desgarr\u00f3n. Una voz pronunciando mi nombre: \u00abC\u00e1lmate, Paige, no pasa nada, ya est\u00e1s a salvo\u00bb; pero no me fiaba de esa voz. Era la voz que yo tem\u00eda. Sal\u00ed arrastr\u00e1ndome de debajo de la red e intent\u00e9 volver a correr. Entonces alguien me sujet\u00f3 y me ech\u00f3 hacia atr\u00e1s. \u00ab\u00a1Conc\u00e9ntrate, Paige! \u00a1Utiliza tu miedo!\u00bb No pod\u00eda enfocar. El miedo me volv\u00eda salvaje. El coraz\u00f3n me lat\u00eda tan deprisa que no pod\u00eda respirar. Perd\u00eda la visi\u00f3n. Ten\u00eda la boca seca. \u00bfSegu\u00eda de pie? \u00ab\u00a1A tu derecha, Paige! \u00a1At\u00e1calo!\u00bb\n\nMir\u00e9 hacia la derecha. No distingu\u00ed qu\u00e9 era, pero no era humano. Mi miedo alcanz\u00f3 su punto m\u00e1ximo. Me lanc\u00e9 hacia el \u00e9ter. Hacia nada. Y, de pronto, hacia algo.\n\nLo \u00faltimo que vi fue mi cuerpo cayendo al suelo. Pero no lo vi con mis ojos, sino con los de una cierva.\n\nEn la vida hay cosas que no olvidas jam\u00e1s. Cosas que excavan muy hondo, que anidan en la zona hadal. Dorm\u00ed como un tronco, esperando a que mi cerebro borrara el terror que hab\u00eda sentido en el bosque.\n\nDormir era lo que me salvaba, el intervalo sereno entre estar despierta y onirambulear. Jax y los dem\u00e1s nunca hab\u00edan entendido por qu\u00e9 me gustaba tanto dormir. Despu\u00e9s de pasar horas en el \u00e9ter siempre necesitaba descansar, y Nadine lo encontraba muy gracioso. \u00abEst\u00e1s loca, Mahoney \u2014me dec\u00eda\u2014. Llevas horas roncando, y ahora pretendes seguir durmiendo. Menuda jeta. Con la pasta que te pagan.\u00bb\n\nNadine Arnett, la simpat\u00eda personificada. Era el \u00fanico miembro de la banda al que no echaba de menos.\n\nCuando despert\u00e9 era de noche. Llevaba la mu\u00f1eca sujeta con una f\u00e9rula. En lo alto hab\u00eda un dosel de terciopelo. Estaba en la cama del Custodio. \u00bfQu\u00e9 hac\u00eda en su cama?\n\nMe costaba pensar. No recordaba muy bien qu\u00e9 hab\u00eda pasado. Me sent\u00eda como el d\u00eda en que Jaxon me hab\u00eda dejado probar el vino. Me mir\u00e9 la mano. La f\u00e9rula me imped\u00eda mover la mu\u00f1eca. Quer\u00eda levantarme, salir de aquella cama, pero notaba el cuerpo demasiado caliente y pesado para moverme. \u00abSedada\u00bb, pens\u00e9. Y no me import\u00f3. Nada me importaba.\n\nCuando volv\u00ed a abrir los ojos, estaba m\u00e1s alerta. O\u00ed una voz conocida. El Custodio hab\u00eda vuelto, y no estaba solo. Me arrastr\u00e9 hacia las cortinas y las separ\u00e9.\n\nLa chimenea estaba encendida. El Custodio estaba de pie de espaldas a m\u00ed, hablando en un idioma que no reconoc\u00ed. Su voz, grave y melodiosa, resonaba como la m\u00fasica en un sal\u00f3n. Enfrente ten\u00eda a Terebell Sheratan. Ella sujetaba un c\u00e1liz con una mano y se\u00f1alaba la cama una y otra vez. El Custodio negaba con la cabeza.\n\n\u00bfEn qu\u00e9 idioma hablaban?\n\nSintonic\u00e9 con los esp\u00edritus m\u00e1s cercanos, fantasmas que hab\u00edan vivido all\u00ed en otros tiempos; se dir\u00eda que danzaban al son de la conversaci\u00f3n del Custodio y Terebell. Era exactamente lo mismo que suced\u00eda cuando Nadine tocaba el piano, o cuando un cantor cantaba una balada en la calle. Los cantores (pol\u00edglotas, estrictamente hablando) entend\u00edan y sab\u00edan hablar una lengua que solo conoc\u00edan los esp\u00edritus, pero ni el Custodio ni Terebell eran cantores. Ninguno de los dos ten\u00eda aura de pol\u00edglota.\n\nJuntaron las cabezas y examinaron un objeto. Me fij\u00e9 mejor. Y me qued\u00e9 helada.\n\nEra mi tel\u00e9fono.\n\nTerebell le daba vueltas en una mano y pasaba el pulgar sobre las teclas. La bater\u00eda se hab\u00eda agotado hac\u00eda mucho tiempo.\n\nSi ten\u00edan mi tel\u00e9fono y la mochila, tambi\u00e9n deb\u00edan de tener el panfleto. \u00bfEstar\u00edan intentando ver mis n\u00fameros? Deb\u00edan de sospechar que conoc\u00eda al autor del panfleto. Si encontraban el n\u00famero de Jaxon, podr\u00edan seguirle la pista hasta Seven Dials, y de pronto la visi\u00f3n de Carl cobrar\u00eda sentido.\n\nTen\u00eda que recuperar el tel\u00e9fono.\n\nTerebell se lo guard\u00f3 en la camisa. El Custodio le dijo algo. Ella acerc\u00f3 la frente a la de \u00e9l, sali\u00f3 de la habitaci\u00f3n y cerr\u00f3 la puerta maciza. El Custodio se qued\u00f3 un momento donde estaba, mirando hacia la ventana, para luego dirigir la atenci\u00f3n a la cama. Hacia m\u00ed.\n\nApart\u00f3 las cortinas y se sent\u00f3 sobre la colcha.\n\n\u2014\u00bfC\u00f3mo te encuentras?\n\n\u2014Vete al cuerno.\n\nSus ojos ard\u00edan.\n\n\u2014Ya veo que mejor.\n\n\u2014\u00bfQu\u00e9 hace Terebell con mi tel\u00e9fono?\n\n\u2014As\u00ed Nashira no lo encontrar\u00e1. Sus casacas rojas podr\u00edan utilizarlo para extraer informaci\u00f3n sobre tus amigos del sindicato.\n\n\u2014Yo no tengo amigos en el sindicato.\n\n\u2014No me mientas, Paige.\n\n\u2014No estoy mintiendo.\n\n\u2014Otra mentira.\n\n\u2014Claro, porque t\u00fa eres siempre tan sincero. \u2014Le sostuve la mirada\u2014. Me dejaste con esa cosa. Me dejaste sola, a oscuras, con un zumbador.\n\n\u2014Sab\u00edas que aparecer\u00eda. Sab\u00edas que tendr\u00edas que enfrentarte a un emite. Adem\u00e1s, te lo advert\u00ed.\n\n\u2014\u00bfC\u00f3mo demonios me advertiste?\n\n\u2014Los puntos fr\u00edos, Paige. As\u00ed es como viajan.\n\n\u2014Entonces \u00bflo soltaste t\u00fa?\n\n\u2014No corr\u00edas peligro. Ya s\u00e9 que estabas asustada, pero necesitaba que poseyeras a esa cierva.\n\nMe clav\u00f3 la mirada. Se me sec\u00f3 la boca.\n\n\u2014\u00bfHiciste todo eso solo para que pudiera poseer a Nuala? \u2014Me humedec\u00ed los labios\u2014. Por eso abriste el punto fr\u00edo. \u2014Asinti\u00f3 con la cabeza\u2014. Soltaste al zumbador. \u2014Volvi\u00f3 a asentir\u2014. Hiciste que me asustara tanto que...\n\n\u2014S\u00ed. \u2014No se avergonzaba\u2014. Sospechaba que tu don se activaba mediante emociones fuertes: rabia, odio, tristeza... y miedo. El miedo es tu verdadero detonante. Poni\u00e9ndote en el l\u00edmite m\u00e1ximo del terror mental, te obligu\u00e9 a poseer a Nuala, haci\u00e9ndote creer que ella era el zumbador que te hab\u00eda perseguido por el bosque. Pero nunca se me habr\u00eda ocurrido ponerte en peligro.\n\n\u2014Podr\u00eda haber muerto.\n\n\u2014Tom\u00e9 ciertas precauciones. Te lo repito: en ning\u00fan momento corriste un peligro inminente.\n\n\u2014Mentira. Si crees que un c\u00edrculo de sal es una precauci\u00f3n, est\u00e1s como una cabra \u2014dije exaltada\u2014. Has debido de disfrutar vi\u00e9ndome...\n\n\u2014No, Paige. Intento ayudarte.\n\n\u2014Vete al infierno.\n\n\u2014Ya existo en cierto nivel del infierno.\n\n\u2014Pues existe en uno que no est\u00e9 cerca del m\u00edo.\n\n\u2014Lo siento, pero t\u00fa y yo hemos hecho un trato. \u2014Me sostuvo la mirada\u2014. Te espero dentro de diez minutos. Me debes una hora de conversaci\u00f3n.\n\nMe habr\u00eda gustado escupirle, pero me contuve. Sal\u00ed de la habitaci\u00f3n y sub\u00ed al piso de arriba.\n\nNo le revelar\u00eda nada m\u00e1s sobre m\u00ed misma. Ya sab\u00eda demasiado de mi vida privada, y no pod\u00eda permitir que descubriera mi relaci\u00f3n con Jax. Nashira estaba buscando a la banda. Si descubr\u00eda que era una de las principales aliadas de Jax, seguramente me obligar\u00eda a apresarlo yo misma. Decid\u00ed fingir que estaba traumatizada por el encuentro con el zumbador y que casi no pod\u00eda hablar.\n\nVolv\u00ed a o\u00edr el ruido \u00e1spero de aquella respiraci\u00f3n. Cerr\u00e9 los ojos. El recuerdo se extingui\u00f3.\n\nLlevaba una bata fina encima de la ropa, que ol\u00eda a sudor y a muerte. Fui al cuarto de ba\u00f1o y me desnud\u00e9. Me hab\u00edan dejado un uniforme rosa nuevo. Me frot\u00e9 la piel con jab\u00f3n y agua caliente. Quer\u00eda eliminar hasta el m\u00e1s leve recuerdo de aquel olor.\n\nCuando me mir\u00e9 en el espejo me di cuenta de que todav\u00eda llevaba el colgante. Me lo quit\u00e9. No me hab\u00eda servido de nada.\n\nCuando volv\u00ed a la c\u00e1mara, el Custodio estaba sentado en su butaca favorita. Me se\u00f1al\u00f3 la otra, enfrente de la suya.\n\n\u2014Por favor.\n\nMe sent\u00e9. La butaca parec\u00eda enorme.\n\n\u2014\u00bfMe has sedado?\n\n\u2014Despu\u00e9s de la posesi\u00f3n sufriste una especie de ataque epil\u00e9ptico. \u2014Se qued\u00f3 mir\u00e1ndome\u2014. \u00bfIntentaste poseer al emite?\n\n\u2014Quer\u00eda ver su onirosaje.\n\n\u2014Entiendo. \u2014Cogi\u00f3 su copa\u2014. \u00bfQuieres beber algo?\n\nEstuve tentada de pedirle algo ilegal (vino, quiz\u00e1), pero no ten\u00eda fuerzas para seguir provoc\u00e1ndolo.\n\n\u2014Caf\u00e9 \u2014dije.\n\nTir\u00f3 de una borla escarlata conectada a una campanilla anticuada.\n\n\u2014Enseguida lo traer\u00e1n \u2014dijo.\n\n\u2014\u00bfQui\u00e9n? \u00bfUn amaur\u00f3tico?\n\n\u2014S\u00ed.\n\n\u2014Los trat\u00e1is como si fueran mayordomos.\n\n\u2014Esclavos, Paige. Dej\u00e9monos de eufemismos.\n\n\u2014Pero su sangre es valiosa.\n\nDio un sorbo de la copa que ten\u00eda en la mano. Me cruc\u00e9 de brazos y esper\u00e9 a que iniciara la conversaci\u00f3n.\n\nEl gram\u00f3fono volv\u00eda a estar en marcha. Reconoc\u00ed la canci\u00f3n, \u00abI Don't Stand a Ghost of a Chance (With You)\u00bb, la versi\u00f3n de Sinatra. Figuraba en la lista negra de Scion por el simple hecho de que la palabra \u00abfantasma\u00bb aparec\u00eda en el t\u00edtulo, aunque no ten\u00eda nada que ver con los fantasmas. \u00a1Ay, c\u00f3mo echaba de menos a La Voz!\n\n\u2014\u00bfTodos los discos prohibidos acaban en tus manos? \u2014pregunt\u00e9 haciendo un esfuerzo supremo para sonar desinteresada.\n\n\u2014No, van a parar a la Casa. De vez en cuando voy all\u00ed y cojo un par para mi gram\u00f3fono.\n\n\u2014\u00bfTe gusta nuestra m\u00fasica?\n\n\u2014No toda. Sobre todo la del siglo XX. Vuestros idiomas me parecen interesantes, pero no me gustan las producciones musicales modernas.\n\n\u2014De eso tiene la culpa el censor. Si no fuera por vosotros, no lo habr\u00eda.\n\nAlz\u00f3 la copa y dijo:\n\n\u2014 _Touch\u00e9_.\n\nAl cabo de un momento no pude reprimirme y pregunt\u00e9:\n\n\u2014\u00bfQu\u00e9 es eso?\n\n\u2014Esencia de flor de amaranto, mezclada con tinto.\n\n\u2014Nunca hab\u00eda o\u00eddo hablar del amaranto.\n\n\u2014Esta variedad no crece en la Tierra. Cura la mayor\u00eda de las lesiones espirituales. Si hubieras tomado amaranto tras tu encuentro con el duende, la herida no te habr\u00eda dejado una cicatriz tan profunda. Tambi\u00e9n curar\u00eda parte de los da\u00f1os sufridos por tu cerebro si usaras tu esp\u00edritu muy a menudo sin soporte vital.\n\nVaya, vaya. Un remedio para mi cerebro. Si Jaxon se enterara de las propiedades del amaranto, no me dejar\u00eda dormir ni un minuto.\n\n\u2014\u00bfPor qu\u00e9 lo bebes?\n\n\u2014Viejas heridas. El amaranto calma el dolor.\n\nHubo un breve silencio. Me tocaba a m\u00ed hablar.\n\n\u2014Esto es tuyo \u2014dije, y le acerqu\u00e9 el colgante.\n\n\u2014Qu\u00e9datelo.\n\n\u2014No lo quiero.\n\n\u2014Insisto. Quiz\u00e1 no ahuyente a los emim, pero podr\u00eda salvarte la vida si te atacara un duende.\n\nLo dej\u00e9 en el brazo de la butaca. El Custodio lo mir\u00f3, y luego me mir\u00f3 a m\u00ed.\n\nLlamaron a la puerta. Entr\u00f3 un chico que deb\u00eda de tener mi edad; era un poco mayor que yo, como mucho. Llevaba un blus\u00f3n gris, y ten\u00eda los ojos inyectados en sangre. Sin embargo, era atractivo; parec\u00eda salido de un cuadro. Su cabello, muy rubio, enmarcaba una cara de facciones angulosas, y ten\u00eda los labios y las mejillas rosa como p\u00e9talos. Los ojos eran de un azul l\u00edquido, transparente. Me pareci\u00f3 detectar el rastro tembloroso de un aura a su alrededor.\n\n\u2014Caf\u00e9, por favor, Michael \u2014le dijo el Custodio\u2014. \u00bfLo tomas con az\u00facar, Paige?\n\n\u2014No, gracias \u2014contest\u00e9. Michael salud\u00f3 inclinando la cabeza y se march\u00f3\u2014. \u00bfQu\u00e9 es, tu esclavo personal?\n\n\u2014Michael fue un regalo de la soberana de sangre.\n\n\u2014Qu\u00e9 rom\u00e1ntico.\n\n\u2014No tanto. \u2014El Custodio mir\u00f3 hacia las ventanas\u2014. No se puede hacer nada cuando Nashira quiere algo. O a alguien.\n\n\u2014Ya me lo imagino.\n\n\u2014Ah, \u00bfs\u00ed?\n\n\u2014Ya s\u00e9 que tiene cinco \u00e1ngeles.\n\n\u2014As\u00ed es. Pero m\u00e1s que una fuerza son una debilidad. \u2014Volvi\u00f3 a beber de su copa\u2014. La soberana de sangre sufre bajo la influencia de sus supuestos \u00e1ngeles.\n\n\u2014Estoy segura de que ellos lo lamentan.\n\n\u2014La odian.\n\n\u2014\u00bfEn serio?\n\n\u2014S\u00ed. \u2014Era evidente que le divert\u00eda mi desd\u00e9n\u2014. Solo llevamos dos minutos hablando, Paige. Procura no gastar todo tu sarcasmo de golpe.\n\nMe habr\u00eda gustado matarlo, pero no pod\u00eda.\n\nEl chico volvi\u00f3 con una cafetera. Dej\u00f3 la bandeja en la mesa, con un plato abundante de casta\u00f1as asadas espolvoreadas con canela. Se me hizo la boca agua con aquel aroma dulce. Cerca del puente de Blackfriars hab\u00eda un vendedor ambulante que en invierno vend\u00eda casta\u00f1as. Esas parec\u00edan mejores, con la c\u00e1scara agrietada y un interior blanco y tierno. Tambi\u00e9n hab\u00eda fruta: trozos de pera, cerezas relucientes y manzanas rojas en tajos.\n\nMichael hizo una se\u00f1a, y el Custodio neg\u00f3 con la cabeza.\n\n\u2014Gracias, Michael. Nada m\u00e1s.\n\nEl chico volvi\u00f3 a saludar antes de salir. Me dieron ganas de gritarle; no soportaba aquella actitud sumisa.\n\n\u2014Cuando dices \u00absupuestos\u00bb \u00e1ngeles \u2014dije, oblig\u00e1ndome a serenarme\u2014, \u00bfqu\u00e9 quieres decir exactamente?\n\nEl Custodio hizo una pausa.\n\n\u2014Come, por favor.\n\nCog\u00ed una casta\u00f1a del plato; todav\u00eda estaba caliente, reci\u00e9n salida del horno. Sab\u00eda a calidez y a invierno.\n\n\u2014Estoy seguro de que sabes qu\u00e9 es un \u00e1ngel: un alma que regresa a este plano para proteger a aquel por quien dio la vida \u2014dijo\u2014. Nosotros conocemos la existencia de \u00e1ngeles y arc\u00e1ngeles, y supongo que los videntes de a pie tambi\u00e9n. \u2014Asent\u00ed\u2014. Nashira puede mandar a \u00e1ngeles de un tercer nivel.\n\n\u2014Ah, \u00bfs\u00ed?\n\n\u2014Puede atrapar a cierta clase de esp\u00edritus.\n\n\u2014Entonces es vinculadora.\n\n\u2014Es algo m\u00e1s que una simple vinculadora, Paige. Si decide matar a un clarividente, no solo puede atrapar su esp\u00edritu, sino utilizarlo. Mientras el esp\u00edritu est\u00e1 vinculado a ella, su presencia afecta al aura de Nashira. Esa deformaci\u00f3n es lo que le permite tener varios dones a la vez.\n\nSe me derram\u00f3 el caf\u00e9 en el regazo.\n\n\u2014\u00bfTiene que matarlo ella misma?\n\n\u2014S\u00ed. Los llamamos \u00ab\u00e1ngeles ca\u00eddos\u00bb. \u2014Me mir\u00f3\u2014. Y est\u00e1n condenados a quedarse para siempre con su asesino.\n\n\u2014Eres malvado. \u2014La taza cay\u00f3 junto a mis pies\u2014. \u00bfC\u00f3mo pretendes que hable contigo, que te trate como si fueras humano, cuando tu prometida puede hacer algo as\u00ed? \u00bfC\u00f3mo puedes mirarla a la cara?\n\n\u2014\u00bfAcaso he dicho que yo haya llamado alguna vez a un \u00e1ngel ca\u00eddo?\n\n\u2014Pero has matado.\n\n\u2014T\u00fa tambi\u00e9n.\n\n\u2014Eso no viene al caso.\n\nLa expresi\u00f3n del Custodio hab\u00eda cambiado: ya no hab\u00eda ni rastro de burla en ella.\n\n\u2014No s\u00e9 qu\u00e9 podr\u00e9 hacer por este mundo \u2014dijo\u2014, pero no dejar\u00e9 que sufras ning\u00fan da\u00f1o.\n\n\u2014No necesito que me protejas. Deshazte de m\u00ed. Enjar\u00e9tame a otro. Ya no quiero ser tu alumna. Quiero cambiar de guardi\u00e1n. Quiero irme con Thuban. Env\u00edame con Thuban.\n\n\u2014No sabes lo que es tener un guardi\u00e1n de los Sargas, Paige.\n\n\u2014No me digas lo que quiero. Quiero...\n\n\u2014Quieres volver a sentirte segura. \u2014Se levant\u00f3; la mesita de caf\u00e9 nos separaba\u2014. Quieres que te trate como Thuban y los otros tratan a sus humanos, porque as\u00ed sentir\u00edas que tienes todo el derecho a odiar a los refa\u00edtas. Pero, como no te hago ning\u00fan da\u00f1o, y como intento comprenderte, huyes. S\u00e9 por qu\u00e9 lo haces, claro. T\u00fa no entiendes mis motivos. Te preguntas una y otra vez por qu\u00e9 quiero ayudarte, y no llegas a ninguna conclusi\u00f3n. Pero eso no significa que no haya conclusi\u00f3n, Paige. Significa que todav\u00eda tienes que descubrirla.\n\nMe recost\u00e9 en la butaca. El caf\u00e9 caliente me hab\u00eda traspasado los pantalones. Cuando lo vio, el Custodio dijo:\n\n\u2014Voy a buscarte algo.\n\nFue hasta el armario. Yo echaba fuego por los ojos. Casi pod\u00eda o\u00edr a Jax rega\u00f1\u00e1ndome: \u00abQu\u00e9 tonta eres. M\u00edrate, est\u00e1s a punto de llorar. \u00a1Levanta la cabeza, querida! \u00bfQu\u00e9 quieres? \u00bfCompasi\u00f3n? \u00bfL\u00e1stima? No las vas a obtener de \u00e9l, como tampoco las obten\u00edas de m\u00ed. El mundo es un matadero, dama m\u00eda. Empu\u00f1a tus armas. Ens\u00e9\u00f1ame c\u00f3mo le haces sufrir\u00bb _._\n\nEl Custodio me acerc\u00f3 un blus\u00f3n negro.\n\n\u2014Espero que te sirva. Quiz\u00e1 te venga un poco grande, pero te abrigar\u00e1.\n\nAsent\u00ed. El Custodio me dio la espalda. Me puse el blus\u00f3n por la cabeza. Ten\u00eda raz\u00f3n: me llegaba por las rodillas.\n\n\u2014Ya est\u00e1 \u2014dije.\n\n\u2014\u00bfQuieres sentarte?\n\n\u2014Como si pudiera elegir.\n\n\u2014Te estoy dejando elegir.\n\n\u2014No s\u00e9 qu\u00e9 quieres que diga.\n\n\u2014Lo ideal ser\u00eda que me contaras qui\u00e9n ha sido tan cruel contigo en el pasado para hacerte pensar que no puedes fiarte de nadie. \u2014Volvi\u00f3 a su butaca\u2014. Pero s\u00e9 que eso no vas a cont\u00e1rmelo. Quieres proteger a esos amigos tuyos.\n\n\u2014No s\u00e9 de qu\u00e9 me hablas.\n\n\u2014Ya, claro.\n\nNo pude contenerme m\u00e1s.\n\n\u2014De acuerdo, tengo amigos videntes. \u00bfNo es lo normal, que los videntes tengan amigos videntes?\n\n\u2014No. El sindicato de Londres se ha fortalecido con los a\u00f1os. Nosotros capturamos sobre todo a marginados, a los que viven solos o en las calles, porque no saben controlar sus poderes. O porque sus familias los han echado de casa. Por eso muchos nos prestan sus servicios de buen grado: sus semejantes los han maltratado. Y si bien los refa\u00edtas los tratamos como a ciudadanos de segunda clase, les ofrecemos la oportunidad de permitirse un poco de \u00e9ter. Los ponemos por grupos, hacemos que vuelvan a pertenecer a una estructura social. \u2014Se\u00f1al\u00f3 la puerta\u2014. Michael era pol\u00edglota; tengo entendido que vosotros los llam\u00e1is \u00abcantores\u00bb. A sus padres les asustaba tanto su fraseolog\u00eda que intentaron exorcizarlo. Su onirosaje se derrumb\u00f3, y despu\u00e9s perdi\u00f3 el habla casi por completo.\n\nNo supe qu\u00e9 decir. Hab\u00eda o\u00eddo hablar de personas cuyo onirosaje se derrumbaba; era lo que le hab\u00eda pasado a uno de los chicos de mi banda, Zeke. As\u00ed era como te convert\u00edas en un ilegible: el onirosaje se regeneraba, pero levantaba una capa tras otra de armadura, impidiendo cualquier ataque espiritual.\n\n\u2014Los casacas rojas lo capturaron hace dos a\u00f1os. Malviv\u00eda en las calles de Southwark; era un ilegible sin dinero ni comida. Lo llevaron a la Torre y sospecharon que era antinatural, pero yo hice que lo trajeran aqu\u00ed antes de tiempo. Lo tratan como a un amaur\u00f3tico, pero todav\u00eda tiene aura. Yo le ense\u00f1\u00e9 a hablar de nuevo. Espero que alg\u00fan d\u00eda encuentre el \u00e9ter, y que pueda cantar como antes. Con las voces de los muertos.\n\n\u2014Un momento \u2014dije\u2014. \u00bfLe ense\u00f1aste t\u00fa?\n\n\u2014S\u00ed.\n\nEl silencio invad\u00eda cada rinc\u00f3n y cada grieta de la habitaci\u00f3n. El Custodio cogi\u00f3 su copa.\n\n\u2014\u00bfQui\u00e9n eres? \u2014pregunt\u00e9. Me mir\u00f3\u2014. Eres el consorte de sangre de una soberana Sargas. Trabajas para su gobierno desde 1859. Has apoyado el tr\u00e1fico de videntes, has visto desarrollarse todo un sistema alrededor de eso. Les has ayudado a divulgar mentiras, odio y temor. \u00bfPor qu\u00e9 ayudas a los humanos?\n\n\u2014Eso no puedo dec\u00edrtelo. Del mismo modo que t\u00fa no quieres revelarme qui\u00e9nes son tus amigos, yo no te revelar\u00e9 cu\u00e1les son mis motivos ocultos.\n\n\u2014\u00bfMe lo dir\u00edas si averiguaras qui\u00e9nes son mis amigos?\n\n\u2014Es posible.\n\n\u2014\u00bfLo sabe Michael?\n\n\u2014No todo. Michael me ha demostrado una gran lealtad, pero no puedo confiar plenamente en \u00e9l dado su fr\u00e1gil estado mental.\n\n\u2014\u00bfPiensas lo mismo de m\u00ed?\n\n\u2014No dispongo de suficiente informaci\u00f3n para confiar en ti, Paige. Pero eso no significa que no puedas ganarte mi confianza. De hecho \u2014a\u00f1adi\u00f3 recost\u00e1ndose en la butaca\u2014, hoy se te presentar\u00e1 una oportunidad.\n\n\u2014\u00bfQu\u00e9 quieres decir?\n\n\u2014Ya lo ver\u00e1s.\n\n\u2014A ver si lo adivino. Mataste a un adivino y le robaste su poder, y ahora crees poder predecir mi futuro.\n\n\u2014Yo no robo dones. Pero conozco muy bien a Nashira, lo suficiente para prever sus movimientos. S\u00e9 cu\u00e1ndo le gusta atacar.\n\nEl reloj de p\u00e9ndulo dio las once. El Custodio lo mir\u00f3.\n\n\u2014Ya ha transcurrido una hora \u2014dijo\u2014. Eres libre para irte. Quiz\u00e1 deber\u00edas ir a visitar a tu amiga, la cartom\u00e1ntica.\n\n\u2014Liss sufre un choque espiritual \u2014dije. El Custodio me mir\u00f3\u2014. Los casacas rojas arrojaron sus cartas al fuego. \u2014Ten\u00eda un nudo en la garganta\u2014. No he vuelto a verla desde entonces.\n\n\u00abP\u00eddele ayuda. \u2014Me debat\u00eda conmigo misma\u2014. Preg\u00fantale si puede conseguirle otras cartas. Seguro que dir\u00e1 que s\u00ed. Ayud\u00f3 a Michael.\u00bb\n\n\u2014Es una l\u00e1stima \u2014dijo\u2014. Es una actriz con gran talento.\n\nMe obligu\u00e9 a decirlo:\n\n\u2014\u00bfQuerr\u00edas ayudarla?\n\n\u2014No tengo ninguna baraja. Ella necesita su enlace con el \u00e9ter. \u2014Me mir\u00f3 a los ojos\u2014. Tambi\u00e9n necesitar\u00eda amaranto.\n\nMe qued\u00e9 quieta en el sitio y vi al Custodio coger una cajita de encima de la mesa. Parec\u00eda una caja de rap\u00e9 antigua, hecha de madreperla y trocitos de oro. En el centro de la tapa estaba la flor de ocho p\u00e9talos, la misma que hab\u00eda en la caja donde guardaba sus viales. La abri\u00f3 y sac\u00f3 una botellita de aceite de color azul.\n\n\u2014Eso es extracto de \u00e1ster \u2014dije.\n\n\u2014Muy bien.\n\n\u2014\u00bfPor qu\u00e9 lo tienes?\n\n\u2014Utilizo dosis peque\u00f1as de la flor estrella para ayudar a Michael. Le ayuda a recordar su onirosaje.\n\n\u2014\u00bfFlor estrella?\n\n\u2014As\u00ed es como los refa\u00edtas llamamos al \u00e1ster. Es una traducci\u00f3n literal de nuestra lengua: _glossolalia_ , o _gloss_.\n\n\u2014\u00bfEs la lengua que hablan los cantores?\n\n\u2014S\u00ed. La antigua lengua del \u00e9ter. Michael ya no puede hablarla, pero la entiende. Igual que los suspirantes.\n\nAs\u00ed que los cantores pod\u00edan entender a los refa\u00edtas. Interesante.\n\n\u2014\u00bfVas a darle \u00e1ster... ahora?\n\n\u2014No. Solo quer\u00eda organizar mi colecci\u00f3n de f\u00e1rmacos requisados \u2014dijo. No sab\u00eda si estaba bromeando o no. Seguramente no\u2014. Algunas, como la an\u00e9mona, pueden usarse para hacernos da\u00f1o. \u2014Sac\u00f3 una flor roja de la caja\u2014. Ciertos venenos deben mantenerse lejos de las manos humanas. \u2014Me miraba fijamente\u2014. No nos gustar\u00eda que los utilizaran, pongamos por caso, para infiltrarse en la Casa. Eso pondr\u00eda en peligro nuestros suministros m\u00e1s secretos.\n\n\u00abFlor roja. \u2014Record\u00e9 la nota de David\u2014. \u00danico m\u00e9todo.\u00bb \u00bfEl \u00fanico m\u00e9todo para matar a los refa\u00edtas?\n\n\u2014No \u2014dije\u2014. No nos gustar\u00eda.\n\nLa Colonia estaba tranquila. No hab\u00eda vuelto a ver a Liss desde que Suhail me hab\u00eda escoltado hasta Magdalen; no hab\u00eda tenido ocasi\u00f3n de ir a ver c\u00f3mo estaba y comprobar si hab\u00eda sobrevivido a la p\u00e9rdida de su baraja.\n\nLa encontr\u00e9 consciente, pero ausente. Ten\u00eda los labios p\u00e1lidos y la mirada perdida, desenfocada. Estaba en pleno choque espiritual.\n\nJulian y el actor con gafas al que hab\u00eda visto el primer d\u00eda, Cyril, se hab\u00edan impuesto la misi\u00f3n de cuidarla. Le daban de comer, le cepillaban el pelo, le curaban las quemaduras de las manos y hablaban con ella. Liss estaba all\u00ed tumbada, r\u00edgida y sudorosa, murmurando sobre el \u00e9ter. Como ya no pod\u00eda conectar con \u00e9l, su tendencia natural era abandonar su cuerpo y unirse con el \u00e9ter. Nosotros ten\u00edamos que contener esa tendencia y hacer que Liss se quedara con nosotros.\n\nFui a la casa de empe\u00f1os de Duckett y cambi\u00e9 dos pastillas por un Sterno, unas cerillas y una lata de jud\u00edas. En la tienda no hab\u00eda ninguna baraja de cartas. Las hab\u00eda confiscado todas aquella casaca roja, Kathryn, para asegurarse de que Liss segu\u00eda sufriendo. Kathryn ten\u00eda suerte de que el Custodio le hubiera prohibido acercarse a m\u00ed.\n\nCuando volv\u00ed a la choza, Julian levant\u00f3 la cabeza. Ten\u00eda los ojos enrojecidos de cansancio. En lugar del blus\u00f3n rosa llevaba una camisa hecha jirones y unos pantalones de tela.\n\n\u2014Has tardado mucho, Paige.\n\n\u2014Estaba ocupada. Ya te lo explicar\u00e9. \u2014Me arrodill\u00e9 junto a Liss\u2014. \u00bfHa comido?\n\n\u2014Ayer consegu\u00ed que comiera un poco de _skilly_ , pero lo vomit\u00f3 todo.\n\n\u2014\u00bfY las quemaduras?\n\n\u2014Mal. Necesitamos sulfadiazina de plata.\n\n\u2014Tenemos que intentar conseguir que coma algo. \u2014Le acarici\u00e9 los rizos h\u00famedos y le pellizqu\u00e9 la mejilla\u2014. \u00bfLiss?\n\nTen\u00eda los ojos abiertos, pero no me contest\u00f3. Encend\u00ed el Sterno. Cyril tamborileaba con los dedos en una rodilla.\n\n\u2014Venga, Rymore \u2014le dijo con fastidio\u2014. No puedes dejar las sedas tanto tiempo.\n\n\u2014Un poco de cari\u00f1o no le vendr\u00eda mal \u2014dijo Julian.\n\n\u2014No hay tiempo para eso. Suhail vendr\u00e1 a buscarla pronto. Tiene que actuar conmigo.\n\n\u2014\u00bfTodav\u00eda no se han enterado?\n\n\u2014Nell ha estado cubri\u00e9ndola. Se parecen bastante con el traje y la m\u00e1scara: misma estatura, mismo color de pelo. Pero Nell no es tan buena. Se cae. \u2014Cyril mir\u00f3 a Liss\u2014. Rymore nunca se cae.\n\nJulian puso las jud\u00edas a calentar. Busqu\u00e9 una cuchara e incorpor\u00e9 un poco a Liss rode\u00e1ndola con un brazo. Ella neg\u00f3 con la cabeza.\n\n\u2014No.\n\n\u2014Tienes que comer un poco, Liss \u2014le dijo Julian asi\u00e9ndole la fr\u00eda mu\u00f1eca, pero Liss no reaccion\u00f3.\n\nCuando las jud\u00edas estuvieron calientes, Julian le ech\u00f3 la cabeza hacia atr\u00e1s a Liss. Empec\u00e9 a darle de comer, pero Liss apenas pod\u00eda tragar. Las jud\u00edas le resbalaban por la barbilla. Cyril cogi\u00f3 la lata y reba\u00f1\u00f3 lo que quedaba con los dedos. Me puse en cuclillas y me qued\u00e9 mirando a Liss arrebujada en las s\u00e1banas.\n\n\u2014Esto no puede ser.\n\n\u2014Pues no podemos hacer nada. \u2014Julian apret\u00f3 un pu\u00f1o\u2014. Aunque encontr\u00e1ramos una baraja, no tenemos ninguna garant\u00eda de que vaya a funcionar. Ser\u00eda como ponerle un brazo o una pierna nuevos. Liss podr\u00eda rechazarla.\n\n\u2014Tenemos que intentarlo. \u2014Mir\u00e9 a Cyril\u2014. \u00bfNo hay m\u00e1s cartom\u00e1nticos por aqu\u00ed?\n\n\u2014Solo muertos.\n\n\u2014Aunque hubiera alguno, no podemos usar la baraja de otro \u2014dijo Julian en voz baja\u2014. Eso ser\u00eda peor que un asesinato.\n\n\u2014Entonces se la robaremos a los refas \u2014dije. Robar era mi especialidad\u2014. Voy a entrar en la Casa. All\u00ed debe de haber suministros.\n\n\u2014Morir\u00e1s \u2014dijo Cyril sin una pizca de aflicci\u00f3n.\n\n\u2014He sobrevivido a un zumbador. No me pasar\u00e1 nada.\n\nJulian me mir\u00f3 con gesto de sorpresa.\n\n\u2014\u00bfLos has visto?\n\n\u2014Viven en el bosque. El Custodio me dej\u00f3 all\u00ed con uno.\n\n\u2014\u00bfSignifica eso que has aprobado los ex\u00e1menes? \u2014La desconfianza se dibuj\u00f3 en su cara\u2014. \u00bfAhora eres una casaca roja?\n\n\u2014No lo s\u00e9. Cre\u00eda que s\u00ed, pero... \u2014Me tir\u00e9 del blus\u00f3n\u2014. Esto no es rojo.\n\n\u2014Menos mal. \u2014Hizo una pausa\u2014. \u00bfC\u00f3mo era? El zumbador.\n\n\u2014R\u00e1pido. Agresivo. No lo vi muy bien. \u2014Le mir\u00e9 la ropa\u2014. \u00bfT\u00fa todav\u00eda no has visto ninguno?\n\nEsboz\u00f3 una sonrisa.\n\n\u2014Aludra me ech\u00f3 por llegar pasado el toque de queda. Me temo que me han degradado a buf\u00f3n.\n\nCyril estaba temblando.\n\n\u2014Su mordedura es mortal \u2014susurr\u00f3\u2014. No deber\u00edas volver all\u00ed.\n\n\u2014Quiz\u00e1 no tenga alternativa \u2014dije. Cyril apoy\u00f3 la cabeza en los brazos\u2014. Jules, p\u00e1same una s\u00e1bana.\n\nLo hizo, y envolv\u00ed con ella a Liss. No dejaba de temblar. Le frot\u00e9 los helados brazos para ayudarla a entrar en calor. Ten\u00eda ampollas en los dedos.\n\n\u2014\u00bfLo dices en serio, Paige? \u2014me pregunt\u00f3 Julian\u2014. Lo de entrar en la Casa.\n\n\u2014El Custodio dice que all\u00ed hay suministros. Un almac\u00e9n secreto, cosas que nosotros no deber\u00edamos ver. Quiz\u00e1 haya sulfadiazina de plata.\n\n\u2014\u00bfNo se te ha ocurrido pensar que deben de estar vigilados? \u00bfNi que el Custodio podr\u00eda estar mintiendo?\n\n\u2014Me arriesgar\u00e9.\n\nJulian dio un suspiro y a\u00f1adi\u00f3:\n\n\u2014Supongo que no puedo imped\u00edrtelo. Y \u00bfqu\u00e9 piensas hacer si consigues entrar?\n\n\u2014Voy a robar todo lo que pueda, cualquier cosa que pueda utilizar para defenderme, y luego me ir\u00e9. Si alguien se anima, puede venir conmigo. Si no, ir\u00e9 sola. Pase lo que pase, no pienso pudrirme aqu\u00ed el resto de mi vida.\n\n\u2014No vayas \u2014dijo Cyril\u2014. Morir\u00e1s. Como murieron otros antes. Se los comieron los zumbadores. Y a ti tambi\u00e9n te comer\u00e1n.\n\n\u2014Basta, Cyril, por favor. \u2014Julian me mir\u00f3 y a\u00f1adi\u00f3\u2014: Ve a la Casa, Paige. Intentar\u00e9 reunir una tropa.\n\n\u2014\u00bfUna tropa?\n\n\u2014Venga, Paige. \u2014La llama del hornillo se reflejaba en sus ojos\u2014. No pensar\u00e1s marcharte sin pelear, \u00bfverdad?\n\n\u2014\u00bfSin pelear? \u2014dije arqueando las cejas.\n\n\u2014No pensar\u00e1s largarte y hacer como si no hubiera pasado nada. Scion lleva dos siglos haciendo esto, Paige. No va a dejarlo as\u00ed como as\u00ed. \u00bfQu\u00e9 les impedir\u00e1 volver a traerte aqu\u00ed en cuanto llegues a SciLo?\n\nJulian ten\u00eda raz\u00f3n.\n\n\u2014\u00bfQu\u00e9 sugieres que haga?\n\n\u2014Una fuga masiva. Nos vamos todos y los dejamos sin videntes de que alimentarse.\n\n\u2014Aqu\u00ed hay m\u00e1s de doscientos humanos. No podemos irnos todos, como si tal cosa. Adem\u00e1s, en el bosque hay minas de tierra. \u2014Acerqu\u00e9 las rodillas a la barbilla\u2014. Ya sabes lo que pas\u00f3 durante la Era XVIII. No quiero tener tantas muertes en mi conciencia.\n\n\u2014No las tendr\u00e1s en tu conciencia. La gente quiere irse, Paige; lo que pasa es que les falta valor. Todav\u00eda. Si conseguimos causar una distracci\u00f3n, podremos sacarlos por el bosque. \u2014Me puso una mano en el brazo\u2014. T\u00fa eres del sindicato. Eres irlandesa. \u00bfNo crees que ya va siendo hora de demostrar a los refas que no mandan ellos? \u00bfQue no pueden seguir rob\u00e1ndonos as\u00ed? \u2014Como no contestaba, me dio un apret\u00f3n\u2014. Vamos a demostrarles que, despu\u00e9s de doscientos a\u00f1os, todav\u00eda tienen algo que temer.\n\nYa no le ve\u00eda la cara. Ve\u00eda a Finn aquel d\u00eda en Dubl\u00edn, dici\u00e9ndome que deb\u00eda pelear.\n\n\u2014A lo mejor tienes raz\u00f3n \u2014conced\u00ed.\n\n\u2014Claro que la tengo. \u2014Compuso una sonrisa que mitig\u00f3 brevemente su cara de cansancio\u2014. \u00bfA cu\u00e1ntos crees que necesitamos?\n\n\u2014Empieza con los que tengan buenas razones para odiar a los refa\u00edtas. Los bufones. Los casacas amarillas. Los amaur\u00f3ticos. Ella, Felix, Ivy. Luego, los casacas blancas.\n\n\u2014\u00bfQu\u00e9 les digo?\n\n\u2014Nada, todav\u00eda. Lim\u00edtate a hacer preguntas. Averigua si estar\u00edan dispuestos a intentar una fuga.\n\nJulian mir\u00f3 a Cyril.\n\n\u2014No. \u2014Cyril sacudi\u00f3 la cabeza. Detr\u00e1s de las gafas destrozadas, el miedo hac\u00eda brillar sus ojos\u2014. Yo no. Ni hablar, hermano. Nos matar\u00e1n. Son inmortales.\n\n\u2014No son inmortales. \u2014La llama del Sterno estaba reduci\u00e9ndose\u2014. Se les puede hacer da\u00f1o. Me lo ha dicho el Custodio.\n\n\u2014Podr\u00eda estar mintiendo \u2014insisti\u00f3 Julian\u2014. Estamos hablando del prometido de Nashira. Del consorte de sangre. Su mano derecha. \u00bfPor qu\u00e9 conf\u00edas en \u00e9l?\n\n\u2014Porque creo que ya se ha rebelado contra ella antes. Creo que es uno de los marcados.\n\n\u2014De los \u00bfqu\u00e9?\n\n\u2014Un grupo de refas que iniciaron la rebeli\u00f3n de la Era XVIII. Los torturaron y los marcaron.\n\n\u2014\u00bfQui\u00e9n te ha contado eso?\n\n\u2014Un arrancahuesos, XX-12.\n\n\u2014\u00bfTe f\u00edas de un arrancahuesos?\n\n\u2014No, pero me ense\u00f1\u00f3 el altar que levantaron por las v\u00edctimas.\n\n\u2014Y crees que el Custodio es uno de esos \u00abmarcados\u00bb \u2014dijo. Asent\u00ed con la cabeza\u2014. Supongo que habr\u00e1s visto esas cicatrices.\n\n\u2014No. Me parece que las oculta.\n\n\u2014\u00bfTe parece? Con eso no basta, Paige.\n\nIba a contestar cuando entr\u00f3 alguien en la habitaci\u00f3n. Me qued\u00e9 de piedra. Era el Capataz.\n\n\u2014Vaya, vaya. \u2014Arque\u00f3 las perfiladas cejas\u2014. Se ve que entre nosotros hay un impostor. \u00bfQui\u00e9n era la que estaba con las sedas, si XIX-1 estaba aqu\u00ed?\n\nMe levant\u00e9, y Julian tambi\u00e9n.\n\n\u2014Tiene choque espiritual \u2014dije. Mir\u00e9 al Capataz a los ojos\u2014. As\u00ed no puede actuar.\n\nEl Capataz se arrodill\u00f3 junto a Liss y le toc\u00f3 la frente. Ella intent\u00f3 apartarse.\n\n\u2014\u00a1Esto es terrible! \u2014Le pas\u00f3 los dedos por el pelo\u2014. \u00a1Terrible! No puedo perder a 1. Mi peque\u00f1a 1 es especial.\n\nLiss empez\u00f3 a chillar en medio de fuertes espasmos.\n\n\u2014\u00a1Vete! \u2014dec\u00eda\u2014. \u00a1Vete!\n\nJulian agarr\u00f3 al Capataz por el hombro y le dio un fuerte empuj\u00f3n.\n\n\u2014No la toques.\n\nMe puse a su lado. Cyril, sentado en cuclillas, se mec\u00eda adelante y atr\u00e1s. Al principio el Capataz se qued\u00f3 estupefacto; luego rompi\u00f3 a re\u00edr. Se levant\u00f3 y se puso a dar palmadas de alegr\u00eda. Meti\u00f3 una mano enguantada dentro de su chaqueta.\n\n\u2014\u00bfQu\u00e9 es esto, ni\u00f1os? \u00bfUna rebeli\u00f3n? \u00bfHe dejado entrar a dos lobos hambrientos en mi reba\u00f1o?\n\nCon una sacudida de la mu\u00f1eca desenroll\u00f3 su l\u00e1tigo, una herramienta dise\u00f1ada para manejar el ganado.\n\n\u2014No voy a permitir que corromp\u00e1is a 1. Ni a ninguno de mis peque\u00f1os. \u2014Restall\u00f3 el l\u00e1tigo mir\u00e1ndome a m\u00ed\u2014. Quiz\u00e1 todav\u00eda no seas actriz, 40, pero lo ser\u00e1s. Vuelve con tu guardi\u00e1n.\n\n\u2014No.\n\n\u2014No vamos a irnos. \u2014El rostro de Julian irradiaba determinaci\u00f3n\u2014. No vamos a dejar sola a Liss.\n\nEl Capataz sacudi\u00f3 el l\u00e1tigo. Julian se tambale\u00f3 y empez\u00f3 a sangrar por una nueva herida en la mejilla.\n\n\u2014Ahora eres de los m\u00edos, chico, y ser\u00e1 mejor que lo recuerdes. \u2014Separ\u00e9 los pies y los plant\u00e9 firmemente en el suelo. El Capataz me lanz\u00f3 una sonrisa\u2014. No hay ninguna necesidad, 40. Cuidar\u00e9 bien de 1.\n\n\u2014No puedes obligarme a irme. Mi guardi\u00e1n es Arcturus. \u2014Me mantuve firme\u2014. Me encantar\u00eda ver c\u00f3mo le explicas por qu\u00e9 me has pegado.\n\n\u2014No tengo intenci\u00f3n de pegarte, andarina. Solo quiero arrearte.\n\nVolvi\u00f3 a restallar el l\u00e1tigo. Julian le lanz\u00f3 un pu\u00f1etazo y desvi\u00f3 el latigazo. Ya est\u00e1bamos otra vez, como con los arrancahuesos. Pero esa vez \u00edbamos a ganar.\n\nSent\u00ed un arrebato y me lanc\u00e9 contra el Capataz. Le asest\u00e9 un pu\u00f1etazo en la mand\u00edbula, y \u00e9l gir\u00f3 bruscamente la cara. Julian lo derrib\u00f3 con una patada detr\u00e1s de las rodillas. El Capataz dej\u00f3 de apretar el l\u00e1tigo un momento; intent\u00e9 quit\u00e1rselo, pero no pude. Dibuj\u00f3 una mueca entre una sonrisa y un gru\u00f1ido, ense\u00f1\u00e1ndome los dientes. Julian le hizo una llave de cuello. Consegu\u00ed arrancarle el l\u00e1tigo y levant\u00e9 la mano para golpearlo con \u00e9l, pero me lo quitaron. Una bota se estrell\u00f3 contra mi est\u00f3mago, y choqu\u00e9 contra la pared.\n\nSuhail. Deb\u00ed imagin\u00e1rmelo. Siempre que aparec\u00eda el Capataz, su superior no tardaba mucho en hacerlo. Igual que en las calles: el mat\u00f3n y el jefe.\n\n\u2014Pens\u00e9 que te encontrar\u00eda aqu\u00ed, mequetrefe. \u2014Me agarr\u00f3 por el pelo\u2014. Otra vez causando problemas, \u00bfverdad?\n\nLe escup\u00ed. Me peg\u00f3 tan fuerte que vi las estrellas.\n\n\u2014No me importa qui\u00e9n sea tu guardi\u00e1n, perra. La concubina no me da miedo. Si no te corto el cuello es porque la soberana de sangre me ha pedido que venga a buscarte.\n\n\u2014Seguro que le encanta o\u00edr que llamas \u00abconcubina\u00bb al Custodio, Suhail \u2014atin\u00e9 a decir\u2014. \u00bfQuieres que se lo cuente?\n\n\u2014Cu\u00e9ntale lo que quieras. La palabra de un humano significa menos que el balbuceo incoherente de un perro.\n\nMe carg\u00f3 sobre un hombro. Forceje\u00e9 y grit\u00e9, pero no quise arriesgarme a utilizar mi esp\u00edritu. El Capataz golpe\u00f3 a Julian en la cabeza con el canto de la mano y lo tir\u00f3 al suelo. Lo \u00faltimo que vi fue a Julian y a Liss, ambos a merced de un hombre del que ya no pod\u00eda defenderme.\n\nLa Residencia del Suzerano parec\u00eda mucho m\u00e1s fr\u00eda y oscura que el d\u00eda del serm\u00f3n. Estaba sola con Suhail, y seguramente tambi\u00e9n estar\u00eda sola con Nashira. Empec\u00e9 a notar peque\u00f1os espasmos que me recorr\u00edan las piernas.\n\nSuhail no me llev\u00f3 a la sala del serm\u00f3n ni a la capilla. Me arrastr\u00f3 por los pasillos y me meti\u00f3 en una habitaci\u00f3n de techos altos con ventanas en arco. La iluminaba una ara\u00f1a de luces en la que ard\u00edan velas, y una chimenea enorme. La luz proyectaba sombras m\u00f3viles en la b\u00f3veda nervada del techo.\n\nEn el centro de la habitaci\u00f3n hab\u00eda una larga mesa de comedor. A la cabecera de la mesa, sentada en una silla con tapizado rojo, estaba Nashira Sargas. Llevaba un vestido negro de cuello alto, de dise\u00f1o geom\u00e9trico, escult\u00f3rico.\n\n\u2014Buenas noches, 40.\n\nNo dije nada. Hizo una se\u00f1a con la mano.\n\n\u2014Ya puedes irte, Suhail.\n\n\u2014S\u00ed, soberana de sangre. \u2014Suhail me empuj\u00f3 hacia ella\u2014. Hasta la pr\u00f3xima \u2014me susurr\u00f3 al o\u00eddo\u2014, perra.\n\nSali\u00f3 por la puerta sin decir nada m\u00e1s. Me qued\u00e9 en la habitaci\u00f3n en penumbra, mirando a la mujer que quer\u00eda matarme.\n\n\u2014Si\u00e9ntate \u2014me orden\u00f3.\n\nIba a sentarme en la silla del extremo opuesto de la mesa, a unos cuatro metros de ella, pero Nashira se\u00f1al\u00f3 la que ten\u00eda m\u00e1s cerca, a su izquierda, en el lado m\u00e1s alejado de la chimenea. Rode\u00e9 la mesa y me sent\u00e9; el m\u00e1s leve movimiento me produc\u00eda dolor de cabeza. Suhail no se hab\u00eda comedido lo m\u00e1s m\u00ednimo al asestarme aquel \u00faltimo pu\u00f1etazo.\n\nNashira no me quitaba los ojos de encima. Unos ojos de color absenta. Me pregunt\u00e9 en qui\u00e9n se habr\u00eda cebado esa noche.\n\n\u2014Est\u00e1s sangrando.\n\nJunto a los cubiertos hab\u00eda una servilleta sujeta con un grueso aro de oro. Me limpi\u00e9 el labio, hinchado, con ella, y manch\u00e9 de sangre el lino de color marfil. Dobl\u00e9 la servilleta para ocultar la mancha y me la puse en el regazo.\n\n\u2014Supongo que debes de estar asustada \u2014dijo Nashira.\n\n\u2014No.\n\nDeber\u00eda estarlo. Lo estaba. Esa mujer lo controlaba todo. Era su nombre el que se susurraba a oscuras, sus \u00f3rdenes las que pon\u00edan fin a vidas. Sus \u00e1ngeles ca\u00eddos se deslizaban alrededor, sin alejarse mucho de su aura.\n\nEl silencio crec\u00eda. Yo no sab\u00eda si mirarla o no. Con el rabillo del ojo vi que algo reflejaba la luz del fuego: una campana de cristal que estaba en el centro mismo de la mesa. Bajo el cristal hab\u00eda una flor marchita, con los p\u00e9talos marrones y arrugados, que se sosten\u00eda mediante un fino alambre. Ignoraba de qu\u00e9 tipo de flor se trataba, pues estaba irreconocible. No se me ocurr\u00eda ninguna raz\u00f3n por la que Nashira pudiera tener una flor muerta en el centro de su mesa de comedor; pero, claro, era Nashira. Viv\u00eda rodeada de cosas muertas.\n\nRepar\u00f3 en mi inter\u00e9s.\n\n\u2014Hay cosas que est\u00e1n mejor muertas \u2014dijo\u2014. \u00bfNo te parece?\n\nYo no pod\u00eda desviar la mirada de la flor. Y no estaba segura, pero me pareci\u00f3 que mi sexto sentido temblaba.\n\n\u2014S\u00ed \u2014coincid\u00ed.\n\nNashira levant\u00f3 la vista. Hab\u00eda varias hileras de rostros de yeso sobre las ventanas, por lo menos cincuenta en cada una de las paredes m\u00e1s largas. Me sent\u00ed atra\u00edda hacia la que ten\u00eda m\u00e1s cerca y la examin\u00e9. Era una cara de mujer, con la expresi\u00f3n relajada y una sonrisa enigm\u00e1tica. La mujer parec\u00eda tranquila, como si durmiera.\n\nDe pronto sent\u00ed n\u00e1useas. Era _La desconocida del Sena_ , la famosa m\u00e1scara mortuoria francesa. Jax ten\u00eda una r\u00e9plica en su guarida. Seg\u00fan \u00e9l, aquella mujer, a la que encontraba hermosa, hab\u00eda sido la obsesi\u00f3n de los bohemios a finales del siglo XIX. Eliza le hab\u00eda obligado a taparla con una s\u00e1bana. Dec\u00eda que le pon\u00eda los pelos de punta.\n\nMir\u00e9 despacio alrededor, abarcando toda la habitaci\u00f3n. Todas las caras eran m\u00e1scaras mortuorias. Tuve que contener las arcadas. Nashira no solo coleccionaba esp\u00edritus de videntes, sino tambi\u00e9n sus caras. Me acord\u00e9 de Seb. \u00bfY si Seb estaba all\u00ed tambi\u00e9n? Baj\u00e9 la mirada, pero segu\u00eda teniendo el est\u00f3mago revuelto.\n\n\u2014Tienes mala cara \u2014observ\u00f3 Nashira.\n\n\u2014Estoy bien.\n\n\u2014Me alegro de o\u00edrlo. Ser\u00eda una l\u00e1stima que enfermaras en esta etapa crucial de tu estancia en Sheol I. \u2014Sin dejar de mirarme, pas\u00f3 un dedo enguantado por el cuchillo que ten\u00eda junto al plato\u2014. Mis casacas rojas se reunir\u00e1n con nosotras dentro de unos minutos, pero antes quer\u00eda hablar contigo en privado.\n\n\u00bbEl consorte de sangre ha ido inform\u00e1ndome de tus progresos. Me dice que ha hecho todo lo posible por sacar a la luz tu don, pero que no has conseguido la posesi\u00f3n plena de un onirosaje, ni siquiera la del de un animal. \u00bfEs eso cierto?\n\nNashira no lo sab\u00eda.\n\n\u2014S\u00ed, es cierto \u2014confirm\u00e9.\n\n\u2014Qu\u00e9 pena. Y, sin embargo, te enfrentaste a un emite y sobreviviste. Incluso heriste a ese ser. Por esa raz\u00f3n Arcturus cree que deber\u00edamos ascenderte a casaca roja.\n\nNo supe qu\u00e9 decir. El Custodio no le hab\u00eda contado lo de la mariposa. Ni lo de la cierva. Eso significaba que no quer\u00eda que Nashira descubriera mis aptitudes; sin embargo, s\u00ed quer\u00eda que me hicieran casaca roja. \u00bfA qu\u00e9 estaba jugando esta vez?\n\n\u2014Qu\u00e9 callada est\u00e1s \u2014observ\u00f3 Nashira. Ten\u00eda una mirada glacial\u2014. El d\u00eda del serm\u00f3n no te mostraste tan t\u00edmida.\n\n\u2014Me dijeron que solo deb\u00eda hablar cuando me lo ordenaran.\n\n\u2014Pues te lo ordeno.\n\nMe habr\u00eda gustado decirle que se metiera sus \u00f3rdenes donde le cupieran. Ya hab\u00eda sido insolente con el Custodio, y no me habr\u00eda importado serlo tambi\u00e9n con ella; pero Nashira todav\u00eda ten\u00eda la mano sobre el cuchillo, y su fr\u00eda mirada revelaba una falta de escr\u00fapulos total. Al final, procurando aparentar sumisi\u00f3n, dije:\n\n\u2014Me alegro de que el consorte de sangre me considere digna del blus\u00f3n rojo. Lo he hecho lo mejor que he podido en los ex\u00e1menes.\n\n\u2014No lo dudo. Pero no nos confiemos. \u2014Se recost\u00f3 en la silla\u2014. Quiero hacerte unas preguntas. Antes de tu banquete de investidura.\n\n\u2014\u00bfBanquete de investidura?\n\n\u2014S\u00ed. Enhorabuena, 40. Ya eres casaca roja. Hay que presentarte a tus nuevos colegas, todos fieles a m\u00ed. Incluso por encima de sus respectivos guardianes.\n\nMe palpitaban las sienes. Casaca roja. Arrancahuesos. Hab\u00eda llegado a los niveles m\u00e1s altos de Sheol I, el c\u00edrculo de allegados de Nashira Sargas.\n\n\u2014Quiero hablar contigo sobre Arcturus \u2014dijo Nashira mirando el fuego de la chimenea\u2014. Creo que compart\u00eds dependencias.\n\n\u2014Tengo mi propia habitaci\u00f3n. En el piso de arriba.\n\n\u2014\u00bfAlguna vez te ha pedido que salgas de ella?\n\n\u2014Solo para entrenar.\n\n\u2014\u00bfNada m\u00e1s? \u00bfQuiz\u00e1 para charlar un rato?\n\n\u2014No le interesa hablar conmigo \u2014respond\u00ed\u2014. Dudo mucho que yo pudiera decir algo de inter\u00e9s para el consorte de sangre.\n\n\u2014Tienes mucha raz\u00f3n.\n\nMe mord\u00ed la lengua. Nashira no ten\u00eda ni idea de cu\u00e1nto le interesaba al Custodio, ni de todo lo que \u00e9l me hab\u00eda ense\u00f1ado en sus propias narices.\n\n\u2014Supongo que habr\u00e1s explorado sus dependencias. \u00bfHay algo en la Torre del Fundador que te haya llamado la atenci\u00f3n? \u00bfAlgo que se salga de lo normal?\n\n\u2014Tiene unos extractos de plantas que no conozco.\n\n\u2014Flores.\n\nAsent\u00ed con la cabeza, y ella cogi\u00f3 algo de la mesa. Un broche, deslustrado por el tiempo, con forma de la flor de la caja de rap\u00e9 del Custodio.\n\n\u2014\u00bfHas visto este s\u00edmbolo en la Torre del Fundador?\n\n\u2014No.\n\n\u2014Lo dices muy segura.\n\n\u2014Estoy segura. No lo he visto nunca.\n\nMe mir\u00f3 a los ojos. Intent\u00e9 sostenerle la mirada.\n\nO\u00ed cerrarse una puerta a lo lejos. Una fila de casacas rojas entraron en la habitaci\u00f3n; los acompa\u00f1aba un refa al que no reconoc\u00ed.\n\n\u2014Bienvenidos, amigos \u2014los salud\u00f3 Nashira\u2014. Sentaos, por favor.\n\nEl refa se toc\u00f3 el pecho con un pu\u00f1o y sali\u00f3 de la habitaci\u00f3n. Escudri\u00f1\u00e9 las caras de los humanos. Veinte arrancahuesos, todos bien alimentados e impecables. Los veteranos de la Era de Huesos XIX iban a la cabeza. Kathryn se encontraba entre ellos, al igual que 16 y 17. Cerraba la fila Carl, con blus\u00f3n rojo y con el pelo peinado con raya. Me lanz\u00f3 una mirada de reproche. Seguramente nunca hab\u00eda visto a un casaca rosa sentado a la mesa de la soberana de sangre.\n\nTodos tomaron asiento. Carl se vio obligado a ocupar la \u00fanica silla libre, la que estaba enfrente de m\u00ed. David se sent\u00f3 unos cuantos asientos m\u00e1s all\u00e1. Ten\u00eda otro corte en la cabeza, cosido con unas cuantas suturas adhesivas. Con las cejas arqueadas, contemplaba las m\u00e1scaras mortuorias.\n\n\u2014Me alegro de que hay\u00e1is podido venir esta noche. Gracias a vuestros incesantes esfuerzos, esta semana no hemos sufrido ning\u00fan ataque de emim que valga la pena mencionar. \u2014Nashira fue mir\u00e1ndolos uno a uno\u2014. Dicho esto, no debemos olvidar que esos seres constituyen una amenaza constante. Su brutalidad no tiene remedio y, por culpa de la fractura del umbral, tampoco hay forma de encerrarlos en el Inframundo. Vosotros sois lo \u00fanico que se interpone entre los cazadores y sus presas.\n\nTodos asintieron. Todos se lo cre\u00edan. Bueno, David quiz\u00e1 no. \u00c9l segu\u00eda observando una m\u00e1scara, con un amago de sonrisa en los labios.\n\nMi mirada se encontr\u00f3 con la de Kathryn, sentada al otro lado de la mesa. Un cardenal enorme le ocupaba todo un lado de la cara. 16 y 17 ni siquiera me miraron. Mejor. Si me miraban, quiz\u00e1 no fuera capaz de contener el impulso de lanzarles el cuchillo de mi cubierto. Liss segu\u00eda en su choza, muri\u00e9ndose, por culpa suya.\n\n\u201422 \u2014Nashira se volvi\u00f3 hacia el arrancahuesos sentado a su derecha\u2014, \u00bfc\u00f3mo est\u00e1 11? Tengo entendido que sigue en Oriel.\n\nEl joven carraspe\u00f3.\n\n\u2014Est\u00e1 un poco mejor, soberana de sangre. No hay se\u00f1ales de infecci\u00f3n.\n\n\u2014Su valor no ha pasado inadvertido.\n\n\u2014Se sentir\u00e1 honrado de o\u00edrlo, soberana de sangre.\n\n\u00abS\u00ed, soberana de sangre. No, soberana de sangre.\u00bb A los refas les encantaba que les acariciaran el ego.\n\nNashira volvi\u00f3 a dar unas palmadas. Cuatro amaur\u00f3ticos entraron por una portezuela; cada uno llevaba una bandeja, y con ellos entr\u00f3 un aroma abrumador a hierbas. Michael era uno de ellos, pero no me mir\u00f3. Se afanaron en repartir un banquete magn\u00edfico por la mesa, alrededor de la campana de cristal. Uno nos sirvi\u00f3 vino blanco muy fr\u00edo en las copas. Se me hizo un nudo en la garganta. Las bandejas estaban rebosantes de comida. Pollo delicadamente cortado, tierno y suculento, con la piel crujiente y dorada, con relleno de salvia y cebolla; salsa de carne, espesa y con un olor dulz\u00f3n; salsa de ar\u00e1ndanos; verduras al vapor y patatas asadas; rollizas salchichas envueltas con panceta. Era un fest\u00edn digno del Inquisidor. A una se\u00f1al de Nashira, los arrancahuesos se pusieron a comer. Com\u00edan deprisa, pero sin la urgencia salvaje del hambre.\n\nMe dol\u00edan las tripas. Quer\u00eda comer. Pero entonces me acord\u00e9 de los bufones, que sobreviv\u00edan a base de grasa y pan duro en sus tugurios. All\u00ed dentro hab\u00eda tanta comida, y all\u00ed fuera, tan poca. Nashira repar\u00f3 en mis reservas.\n\n\u2014Come.\n\nEra una orden. Puse unas cuantas lonchas de pollo y un poco de verdura en mi plato. Carl se bebi\u00f3 el vino de un trago, como si fuera agua.\n\n\u2014Ten cuidado, 1 \u2014le dijo una de las chicas\u2014. No vayas a encontrarte mal otra vez.\n\nLos dem\u00e1s rieron. Carl compuso una sonrisa.\n\n\u2014Eso solo me ha pasado una vez. Cuando todav\u00eda era un casaca rosa.\n\n\u2014S\u00ed, dejad tranquilo a 1. Se ha ganado el vino. \u201422 le dio un pu\u00f1etazo amistoso en el brazo\u2014. Todav\u00eda es un novato. Adem\u00e1s, todos pasamos un mal rato con nuestro primer zumbador.\n\nHubo murmullos de aprobaci\u00f3n.\n\n\u2014Yo me desmay\u00e9 \u2014admiti\u00f3 la chica que hab\u00eda hablado antes. Una exhibici\u00f3n desinteresada de solidaridad\u2014. La primera vez que vi uno.\n\nCarl sonri\u00f3.\n\n\u2014Pero eres muy buena en combate espiritista, 6.\n\n\u2014Gracias.\n\nObserv\u00e9 en silencio sus muestras de camarader\u00eda. Era repugnante, pero no estaban actuando. A Carl no solo le gustaba ser un casaca roja; era algo m\u00e1s que eso: estaba como pez en el agua en ese extra\u00f1o nuevo mundo. Yo lo entend\u00eda, hasta cierto punto. Era lo mismo que hab\u00eda sentido yo cuando empec\u00e9 a trabajar para Jaxon. Quiz\u00e1 lo que le pasaba a Carl era que nunca hab\u00eda encontrado un sitio en el sindicato.\n\nNashira los observaba. Deb\u00eda de disfrutar con aquella farsa semanal. Humanos est\u00fapidos, adoctrinados, riendo de las duras pruebas a que los hab\u00eda sometido, completamente rendidos a ella, comiendo su comida. Qu\u00e9 poderosa deb\u00eda de sentirse. Qu\u00e9 satisfecha de s\u00ed misma.\n\n\u2014T\u00fa todav\u00eda eres rosa \u2014dijo una voz aguda que me llam\u00f3 la atenci\u00f3n\u2014. \u00bfHas peleado con alg\u00fan zumbador?\n\nLevant\u00e9 la cabeza y vi que todos me miraban.\n\n\u2014S\u00ed, anoche \u2014contest\u00e9.\n\n\u2014Es la primera vez que te veo. \u201422 arque\u00f3 las pobladas cejas\u2014. \u00bfA qu\u00e9 batall\u00f3n perteneces?\n\n\u2014No pertenezco a ning\u00fan batall\u00f3n.\n\nLa conversaci\u00f3n se estaba poniendo interesante.\n\n\u2014En alguno tienes que estar \u2014dijo otro chico\u2014. Eres una casaca roja. \u00bfQu\u00e9 otros humanos hay en tu residencia? \u00bfQui\u00e9n es tu guardi\u00e1n?\n\n\u2014Mi guardi\u00e1n solo tiene un humano. \u2014Mir\u00e9 a 22 y esboc\u00e9 una sonrisa\u2014. Quiz\u00e1 lo hayas visto por ah\u00ed. Es el consorte de sangre.\n\nEl silencio se prolong\u00f3 durante lo que me parecieron horas. Di un sorbo de vino. No estaba acostumbrada al alcohol, y not\u00e9 una extra\u00f1a sensaci\u00f3n en la lengua.\n\n\u2014Me alegro de que el consorte de sangre haya escogido a una inquilina humana tan capacitada, 40 \u2014dijo Nashira, y solt\u00f3 una risita. Su risa era desconcertante; era como o\u00edr una campana que tocaba una nota equivocada\u2014. Se enfrent\u00f3 a un zumbador ella sola, sin su guardi\u00e1n.\n\nM\u00e1s silencio. Supuse que ninguno de ellos hab\u00eda entrado nunca en el bosque sin ir acompa\u00f1ado de un refa, y mucho menos se hab\u00eda enfrentado \u00e9l solo a un zumbador. 30 aprovech\u00f3 la ocasi\u00f3n para expresar la misma duda que yo me estaba planteando:\n\n\u2014\u00bfSignifica eso que \u00e9l no lucha contra los emim, soberana de sangre?\n\n\u2014El consorte de sangre tiene prohibido enfrentarse a ellos. Como futura pareja m\u00eda, no ser\u00eda apropiado que hiciera el trabajo de los casacas rojas.\n\n\u2014Claro, soberana de sangre.\n\nEstaba segura de que Nashira me miraba a m\u00ed. Segu\u00ed comi\u00e9ndome las patatas.\n\nEl Custodio s\u00ed peleaba contra los emim; yo misma le hab\u00eda limpiado las heridas. Lo hac\u00eda pese a esa prohibici\u00f3n de la que hab\u00eda hablado Nashira, y ella no ten\u00eda ni idea, o a lo sumo lo sospechaba.\n\nDurante unos minutos solo se oy\u00f3 el tintineo de los cubiertos. Me com\u00ed las verduras con salsa de carne y segu\u00ed pensando en los enfrentamientos secretos del Custodio con los emim. Pese a no tener necesidad de poner su vida en peligro, hab\u00eda decidido ir a su encuentro y pelear contra ellos. Ten\u00eda que haber alguna explicaci\u00f3n.\n\nLos casacas rojas hablaban en voz baja. Intercambiaban informaci\u00f3n sobre sus respectivas residencias y ensalzaban la belleza de los edificios antiguos. A veces hablaban con desd\u00e9n de los bufones (\u00abSon unos cobardes, incluso los m\u00e1s simp\u00e1ticos\u00bb). Kathryn paseaba la comida por el plato y se estremec\u00eda cada vez que mencionaban el Poblado. 30 todav\u00eda estaba colorada, mientras que Carl masticaba con excesivo \u00edmpetu, alternando los bocados con tragos de su segunda copa de vino. Cuando todos los platos hubieron quedado limpios, volvieron los amaur\u00f3ticos y recogieron la mesa, en la que dejaron tres bandejas de postres. Nashira esper\u00f3 a que los casacas rojas se sirvieran antes de volver a tomar la palabra.\n\n\u2014Ahora que hab\u00e9is comido y bebido, amigos, vamos a distraernos un poco.\n\nCarl se limpi\u00f3 la melaza de los labios con la servilleta. Una troupe de bufones entr\u00f3 en la habitaci\u00f3n. Entre ellos hab\u00eda un suspirante; a un movimiento de cabeza de Nashira, se coloc\u00f3 el viol\u00edn en el hombro y empez\u00f3 a tocar una melod\u00eda suave y alegre. Los otros empezaron a realizar gr\u00e1ciles acrobacias.\n\n\u2014Centr\u00e9monos, pues \u2014dijo Nashira sin prestar la m\u00e1s m\u00ednima atenci\u00f3n a la actuaci\u00f3n\u2014. Si alguno de vosotros ha conversado alguna vez con el Capataz, quiz\u00e1 sepa lo que hace para ganarse el sustento. Es mi captador para las Eras de Huesos. Desde hace unas d\u00e9cadas, intento captar a videntes valiosos del sindicato criminal de Scion Londres. Todos lo conoc\u00e9is, sin duda; algunos de vosotros quiz\u00e1 hasta hay\u00e1is formado parte de \u00e9l.\n\n30 y 18 se removieron en los asientos. No recordaba haber visto sus caras por el sindicato, pero yo siempre hab\u00eda trabajado dentro de los l\u00edmites del sector I-4 y, solo ocasionalmente, en los I-1 y I-5. Hab\u00eda otros treinta y tres sectores de los que pod\u00edan provenir.\n\nNadie miraba a los actores. Su actuaci\u00f3n era perfecta, y a nadie le importaba.\n\n\u2014En Sheol I buscamos calidad, no solo cantidad. \u2014Nashira ignor\u00f3 las miradas ce\u00f1udas de la mitad de su audiencia\u2014. En las \u00faltimas d\u00e9cadas he observado una disminuci\u00f3n constante de la diversidad entre los clarividentes que capturamos. Los refa\u00edtas respetamos y valoramos todas vuestras habilidades, pero todav\u00eda necesitamos muchos talentos para enriquecer esta colonia. Debemos aprender unos de otros. No basta con traer cartom\u00e1nticos y palmistas.\n\n\u00bbXX-59-40 es un buen ejemplo de la clase de clarividentes que buscamos ahora. Es nuestra primera onir\u00e1mbula. Tambi\u00e9n necesitamos sibilas y berserkers, vinculadores e invocadores, y un par de or\u00e1culos m\u00e1s: cualquier g\u00e9nero de clarividente que pueda aportar m\u00e1s perspicacia a nuestras tropas.\n\nKathryn me mir\u00f3 con sus amoratados ojos. Ahora ya sab\u00eda con certeza que yo no era una furia.\n\n\u2014Creo que todos podr\u00edamos aprender mucho de 40 \u2014dijo David alzando su copa\u2014. Estoy dese\u00e1ndolo.\n\n\u2014Una actitud excelente, 12. S\u00ed, esperamos aprender mucho de 40 \u2014confirm\u00f3 Nashira, y me mir\u00f3\u2014. Esa es la raz\u00f3n por la que ma\u00f1ana voy a enviarla a realizar una misi\u00f3n externa.\n\nLos veteranos se miraron. Carl se puso colorado como la charlota de fresa.\n\n\u2014XX-59-1 tambi\u00e9n ir\u00e1. Y t\u00fa, 12 \u2014continu\u00f3 Nashira. Carl estaba euf\u00f3rico. David miraba dentro de su copa con una sonrisa en los labios\u2014. Ir\u00e9is con uno de nuestros s\u00e9niors de la Era de Huesos XIX, que vigilar\u00e1 vuestra actuaci\u00f3n. 30, supongo que puedo contar contigo para eso.\n\n30 asinti\u00f3 con la cabeza y dijo:\n\n\u2014Ser\u00e1 un honor, soberana de sangre.\n\n\u2014Estupendo.\n\nCarl estaba sentado en el borde de la silla.\n\n\u2014\u00bfEn qu\u00e9 consistir\u00e1 la misi\u00f3n, soberana de sangre?\n\n\u2014Tenemos que resolver una situaci\u00f3n delicada. Como ya saben 1 y 12, he pedido a la mayor\u00eda de los casacas blancas que hicieran predicciones del paradero de un grupo denominado los Siete Sellos. Pertenecen al sindicato de clarividentes.\n\nNo levant\u00e9 la mirada.\n\n\u2014Sabemos que los Siete Sellos poseen varios tipos de clarividentes poco comunes, entre ellos un or\u00e1culo y un vinculador. De hecho, ese tal Vinculador Blanco es el elemento clave del grupo. A partir de predicciones recientes, hemos deducido que van a reunirse en Londres pasado ma\u00f1ana. El sitio se llama Trafalgar Square, en la cohorte I, y la reuni\u00f3n se celebrar\u00e1 a la una de la madrugada.\n\nHab\u00edan acumulado una cantidad de detalles incre\u00edble. Pero, con tantos videntes haciendo predicciones a la vez, concentrando su energ\u00eda en determinado sector del \u00e9ter, no deber\u00eda haberme sorprendido. Con ello conseguir\u00edan un efecto parecido al de una sesi\u00f3n espiritista.\n\n\u2014\u00bfAlguno de vosotros sabe algo sobre los Siete Sellos? \u2014Como nadie contestaba, Nashira me mir\u00f3\u2014. 40, t\u00fa debes de haber tenido alguna relaci\u00f3n con el sindicato. De no ser as\u00ed no habr\u00edas podido permanecer escondida en Londres tanto tiempo. \u2014Me mir\u00f3 fijamente\u2014. Cu\u00e9ntame lo que sabes.\n\nCarraspe\u00e9.\n\n\u2014Las bandas son muy herm\u00e9ticas \u2014dije\u2014. He o\u00eddo alg\u00fan cotilleo, pero...\n\n\u2014\u00bfAlg\u00fan cotilleo?\n\n\u2014Rumores \u2014aclar\u00e9\u2014. Habladur\u00edas.\n\n\u2014Expl\u00edcate mejor.\n\n\u2014Todos sabemos sus nombres falsos.\n\n\u2014Y \u00bfqu\u00e9 nombres son esos?\n\n\u2014El Vinculador Blanco, la Visi\u00f3n Roja, el Diamante Negro, la So\u00f1adora P\u00e1lida, la Musa Martirizada, la Furia Encadenada y la Campana Silenciosa.\n\n\u2014Conoc\u00eda la mayor\u00eda de esos nombres. No as\u00ed el de So\u00f1adora P\u00e1lida. \u2014Me alegr\u00e9\u2014. Eso me hace pensar que hay otra onir\u00e1mbula. Qu\u00e9 coincidencia, \u00bfno? \u2014Tamborile\u00f3 con los dedos en la mesa\u2014. \u00bfSabes d\u00f3nde tienen su base?\n\nNo pod\u00eda negarlo. Nashira hab\u00eda visto mi documentaci\u00f3n.\n\n\u2014S\u00ed \u2014respond\u00ed\u2014. En el I-4. Es donde yo trabajaba.\n\n\u2014\u00bfNo es inusual que dos onir\u00e1mbulas vivan tan cerca la una de la otra? Seguro que tambi\u00e9n te habr\u00edan contratado a ti.\n\n\u2014Ellos no lo sab\u00edan. Yo procuraba no llamar la atenci\u00f3n \u2014ment\u00ed\u2014. Esa So\u00f1adora es la dama del I-4, la protegida del Vinculador. Me habr\u00eda hecho matar si se hubiera enterado de que ten\u00eda una rival. A las bandas dominantes no les gusta la competencia.\n\nEstaba segura de que Nashira estaba jugando conmigo. Nashira no era idiota. Ya deb\u00eda de haber atado cabos: el panfleto, la So\u00f1adora P\u00e1lida, los Siete Sellos trabajando en el I-4. Sab\u00eda perfectamente qui\u00e9n era yo.\n\n\u2014Si la So\u00f1adora P\u00e1lida es una onir\u00e1mbula, el Vinculador Blanco podr\u00eda estar escondiendo a una de las clarividentes m\u00e1s codiciadas de la ciudadela \u2014dijo\u2014. Raras veces se nos presenta la oportunidad de a\u00f1adir joyas tan valiosas a nuestra corona. Tu papel en esta misi\u00f3n es vital, 40. Si hay alguien capaz de reconocer a la onir\u00e1mbula de los Siete Sellos, es otra onir\u00e1mbula.\n\n\u2014S\u00ed, soberana de sangre \u2014dije con la garganta muy tensa\u2014, pero \u00bfpor qu\u00e9 van a reunirse los Siete Sellos a esa hora?\n\n\u2014Como ya he dicho, 40, se trata de una situaci\u00f3n delicada. Parece ser que unos clarividentes irlandeses est\u00e1n intentando establecer contacto con el sindicato de Londres. Su l\u00edder es una fugitiva irlandesa llamada Antoinette Carter. Los Siete Sellos van a reunirse con ella.\n\nAs\u00ed que Jax lo hab\u00eda conseguido. Me pregunt\u00e9 c\u00f3mo se las habr\u00eda ingeniado Antoinette para colarse en la ciudadela. Era casi imposible cruzar el mar de Irlanda. Otros videntes hab\u00edan intentado salir del pa\u00eds, la mayor\u00eda para dirigirse a Am\u00e9rica, pero pocos lo hab\u00edan logrado. No pod\u00edas cruzar el oc\u00e9ano en un bote. Y aunque alguien lo hubiera conseguido, Scion nunca habr\u00eda dejado que lo supi\u00e9ramos.\n\n\u2014Es fundamental que no se cree un sindicato criminal an\u00e1logo en Dubl\u00edn. De ah\u00ed nuestro inter\u00e9s por impedir que se celebre esa reuni\u00f3n. Tu misi\u00f3n consiste en capturar a Antoinette Carter. Creo que ella tambi\u00e9n es una clase poco frecuente de clarividente, y quiero averiguar qu\u00e9 poderes oculta exactamente. La segunda misi\u00f3n es capturar a los Siete Sellos. El Vinculador Blanco es el objetivo principal.\n\nJaxon. Mi capo.\n\n\u2014Te supervisar\u00e1n el consorte de sangre y su prima. Espero resultados. Si Carter consigue volver a Irlanda, os considerar\u00e9 responsables. \u2014Nashira nos mir\u00f3 uno por uno: a 30, a David, a Carl y a m\u00ed\u2014. \u00bfMe hab\u00e9is entendido?\n\n\u2014S\u00ed, soberana de sangre \u2014dijeron 30 y Carl, mientras David hac\u00eda girar el vino en su copa.\n\nYo no dije nada.\n\n\u2014Tu vida aqu\u00ed est\u00e1 a punto de dar un giro, 40. Esta misi\u00f3n te permitir\u00e1 hacer un buen uso de tu don. Espero que muestres gratitud por las largas horas que Arcturus ha dedicado a tu entrenamiento. \u2014Nashira desvi\u00f3 la mirada del fuego y me mir\u00f3 a los ojos\u2014. Tienes un gran potencial. Si no intentas sacarle el mejor partido, me encargar\u00e9 de que nunca m\u00e1s vuelvas a pisar las protegidas salas de Magdalen. Por m\u00ed puedes pudrirte en las calles, con el resto de los desgraciados.\n\nEn su mirada no hab\u00eda ni rastro de emoci\u00f3n, pero s\u00ed de hambre. A Nashira Sargas se le estaba empezando a agotar la paciencia.\n\nA los miembros quinto y sexto de nuestra banda los encontraron a principios de 2057, un a\u00f1o despu\u00e9s de mi ingreso.\n\nLlegaron durante una ola de calor tremenda. Uno de los recadistas de Jaxon inform\u00f3 de la presencia de dos clarividentes nuevos en el I-4. La pareja formaba parte de un grupo de turistas que hab\u00eda llegado para participar en el congreso de verano de la universidad que se celebraba todos los a\u00f1os con gran \u00e9xito. Tra\u00edan a centenares de turistas j\u00f3venes y entusiastas desde pa\u00edses no adeptos a Scion, a los que pensaban devolver a sus lugares de origen convertidos en abogados de las pol\u00edticas anticlarividentes. Esos programas ya hab\u00edan encontrado apoyo en algunos lugares de Estados Unidos, donde las opiniones sobre Scion llevaban d\u00e9cadas divididas. El bienintencionado recadista hab\u00eda descubierto dos auras y hab\u00eda ido corriendo a avisar a su mimetocapo, pero entonces se enter\u00f3 de que los reci\u00e9n llegados no eran residentes permanentes del I-4. No ten\u00edan ni idea de la existencia del sindicato. Quiz\u00e1 ni siquiera supieran que eran videntes.\n\nEl recadista hab\u00eda informado de que uno de los dos turistas, la chica, era, casi con toda seguridad, suspirante. Jax estaba impresionado. Los suspirantes, seg\u00fan me dijo, pertenec\u00edan al orden de los sensores; conoc\u00edan el funcionamiento del \u00e9ter, los olores, sonidos y ritmos de los esp\u00edritus. Pod\u00edan o\u00edr sus voces y sus vibraciones, y hasta utilizarlas para tocar instrumentos. \u00abUn don interesante \u2014admiti\u00f3\u2014, pero en absoluto innovador.\u00bb Los sensores eran menos comunes que los m\u00e9diums, pero no mucho. El cuarto orden de la clarividencia. Sin embargo, en la ciudadela no hab\u00eda muchos, y a Jaxon le gustaban las rarezas.\n\nLo que le interesaba era la otra mitad de la pareja. El recadista hab\u00eda hablado de un aura extraordinaria, entre el naranja y el rojo. El aura de una furia.\n\nJax llevaba a\u00f1os recorriendo las calles en busca de una furia, y aquella era la primera vez que sus esperanzas pod\u00edan cumplirse. No pod\u00eda creer que hubiera tenido tanta suerte. \u00c9l ten\u00eda una visi\u00f3n, un proyecto. Jaxon Hall no se contentaba con tener una banda; qu\u00e9 va. \u00c9l quer\u00eda un joyero, la _cr\u00e8me de la cr\u00e8me_ de la sociedad vidente. Quer\u00eda que la Asamblea Antinatural lo envidiara m\u00e1s que a ning\u00fan otro mimetocapo.\n\n\u2014Los convencer\u00e9 para que se queden \u2014hab\u00eda dicho y, se\u00f1al\u00e1ndome con el bast\u00f3n, hab\u00eda a\u00f1adido\u2014: Ya lo ver\u00e1s, dama m\u00eda.\n\n\u2014Tienen una vida en su pa\u00eds, Jax. Tienen familia. \u2014Yo no estaba nada convencida\u2014. \u00bfNo crees que necesitar\u00e1n tiempo para pens\u00e1rselo?\n\n\u2014No hay tiempo para eso, querida. Si se marchan, no volver\u00e1n. Tienen que quedarse.\n\n\u2014Est\u00e1s so\u00f1ando.\n\n\u2014No, no estoy so\u00f1ando. Pero \u00bfquieres jugarte algo? \u2014Me tendi\u00f3 una mano\u2014. Si pierdes, me haces dos encargos gratis. Y me limpias el espejo antiguo.\n\n\u2014\u00bfY si gano?\n\n\u2014Te pagar\u00e9 el doble por los mismos encargos. Y no tendr\u00e1s que limpiarme el espejo.\n\nLe estrech\u00e9 la mano.\n\nJaxon ten\u00eda un pico de oro. Yo sab\u00eda perfectamente qu\u00e9 habr\u00eda dicho mi padre de \u00e9l: \u00abEse tipo ha besado la piedra de la elocuencia\u00bb. Jaxon ten\u00eda algo que hac\u00eda que quisieras complacerlo, que quisieras ver brotar ese brillo en sus ojos. Sab\u00eda que conseguir\u00eda que la pareja se quedara. Tras localizar su hotel y pagar a un limosnero para que averiguara sus nombres, les envi\u00f3 una invitaci\u00f3n a un \u00abacto extraoficial\u00bb en una cafeter\u00eda de moda de Covent Garden. Yo misma se la entregu\u00e9 al conserje, en un sobre dirigido a la se\u00f1orita Nadine L. Arnett y el se\u00f1or Ezekiel S\u00e1enz.\n\nNos contestaron diciendo que eran hermanastros. Ambos resid\u00edan en Boston, la reluciente capital de Massachusetts. El d\u00eda de la entrevista, Jaxon nos mantuvo informados por correo electr\u00f3nico.\n\nFabuloso. Esto es fabuloso.\n\nElla es susu, sin ninguna duda. Muy elocuente. Y muy maleducada. __\n\nEl hermano me intriga. No consigo descifrar su aura. Inquietante. __\n\nNick, Eliza y yo esperamos una hora m\u00e1s, y entonces lleg\u00f3 la confirmaci\u00f3n definitiva.\n\nSe quedan. Paige, al espejo hay que darle fuerte. __\n\nFue la \u00faltima vez que apost\u00e9 contra Jaxon Hall.\n\nPasaron dos d\u00edas. Mientras Eliza hac\u00eda sitio en la guarida para los reci\u00e9n llegados, yo acompa\u00f1\u00e9 a Nick a recogerlos a Gower Street. El plan era hacerlos desaparecer del mapa, como si los hubieran secuestrado y asesinado. Dejar\u00edamos pistas: alguna prenda manchada de sangre, alg\u00fan pelo. A Scion le encantar\u00eda. Podr\u00edan utilizarlo para anunciar m\u00e1s cr\u00edmenes antinaturales; pero lo m\u00e1s importante era que no vendr\u00edan a buscar a los hermanos desaparecidos.\n\n\u2014\u00bfCre\u00e9is que Jax los ha convencido para que se queden? \u2014dije mientras camin\u00e1bamos.\n\n\u2014Ya conoces a Jax. Ser\u00eda capaz de convencerte para que saltaras desde un acantilado.\n\n\u2014Pero deben de tener familia. Y Nadine todav\u00eda est\u00e1 estudiando.\n\n\u2014Quiz\u00e1 no les haya ido bien all\u00ed, _s\u00f6tnos_. Al menos, en Scion los videntes pueden entender lo que son. All\u00ed deben de pensar que est\u00e1n locos. \u2014Se puso las gafas de sol\u2014. En ese sentido, Scion es una maravilla.\n\nEn cierto modo ten\u00eda raz\u00f3n. Fuera de Scion no exist\u00eda ninguna pol\u00edtica oficial sobre los clarividentes; no estaban reconocidos legalmente, ni ten\u00edan estatus de minor\u00eda; solo aparec\u00edan en la ficci\u00f3n. Sin embargo, ten\u00eda que ser mejor eso que ser sistem\u00e1ticamente perseguidos y ejecutados, como nosotros. Yo no acababa de entender que quisieran quedarse.\n\nEstaban esper\u00e1ndonos delante de la universidad. Nick levant\u00f3 una mano y dijo:\n\n\u2014Hola. \u00bfEres Zeke? \u2014El desconocido asinti\u00f3\u2014. Yo soy Nick.\n\n\u2014Y yo, Paige \u2014dije.\n\nMe fij\u00e9 en los ojos de Zeke, del color del t\u00e9 negro, y en su cara delgada, que denotaba nerviosismo. Deb\u00eda de tener veintitantos a\u00f1os; era flaco para su elevada estatura, con las mu\u00f1ecas estrechas y la piel acostumbrada al sol.\n\n\u2014Est\u00e1is con Jaxon Hall, \u00bfverdad?\n\nSu voz ten\u00eda un acento raro. Con la mano que ten\u00eda libre se enjug\u00f3 el sudor de la frente, y entrev\u00ed una cicatriz vertical.\n\n\u2014S\u00ed, pero no vuelvas a pronunciar su nombre. La DVD podr\u00eda estar cerca. \u2014Nick sonri\u00f3\u2014. Y supongo que t\u00fa eres Nadine \u2014a\u00f1adi\u00f3 mirando a la suspirante.\n\nNadine ten\u00eda los ojos y los rasgos nerviosos de su hermano, pero ah\u00ed se acababa el parecido. Llevaba el pelo te\u00f1ido de rojo, y parec\u00eda que se lo hubieran cortado con regla. Las ciudadelas de Scion tend\u00edan a seguir la moda y a emplear el argot de la d\u00e9cada en la que se hab\u00edan establecido; en SciLo todos llev\u00e1bamos ropa de tonos neutros, al estilo victoriano, y aquella camisa amarilla, los vaqueros y los zapatos de tac\u00f3n de aguja eran como un letrero que rezara \u00abturista\u00bb y \u00abdiferente\u00bb.\n\n\u2014S\u00ed, eso creo \u2014dijo.\n\nNick mir\u00f3 a Zeke con los ojos un poco entornados. Yo tambi\u00e9n estaba intentando clasificar su aura. Al darse cuenta, Nadine se acerc\u00f3 m\u00e1s a su hermano.\n\n\u2014\u00bfQu\u00e9 pasa?\n\n\u2014Nada. Lo siento \u2014se disculp\u00f3 Nick. Mir\u00f3 por encima de sus cabezas, hacia la universidad, y luego los mir\u00f3 primero a uno y luego al otro\u2014. Hemos de darnos prisa. Supongo que os lo hab\u00e9is pensado bien, porque una vez que hay\u00e1is salido de ese edificio, no habr\u00e1 vuelta atr\u00e1s.\n\nZeke mir\u00f3 a su hermana. Ella, con los brazos cruzados, se mir\u00f3 los zapatos.\n\n\u2014S\u00ed, estamos seguros \u2014dijo Zeke\u2014. Hemos tomado una decisi\u00f3n.\n\n\u2014Pues entonces, vamos.\n\nAl final de la calle nos apretujamos los cuatro en un pirata. Nadine hurg\u00f3 en su bolso y sac\u00f3 unos auriculares. Sin decir nada, se los puso y cerr\u00f3 los ojos. Me pareci\u00f3 que le temblaban los labios.\n\n\u2014A Monmouth Street, por favor \u2014le dijo Nick al conductor.\n\nEl taxi arranc\u00f3 lentamente. Por suerte para nosotros, los piratas no ten\u00edan licencia. Se ganaban muy bien la vida con sus clientes clarividentes.\n\nMonmouth Street era donde viv\u00eda Jax: un tr\u00edplex encima de una peque\u00f1a boutique. Yo me quedaba a menudo a dormir all\u00ed, y le dec\u00eda a mi padre que estaba en casa de alguna amiga. No era exactamente mentira. Durante meses hab\u00eda ido enter\u00e1ndome de c\u00f3mo funcionaba la sociedad clarividente: la estructura de las bandas, los nombres de sus l\u00edderes, el protocolo y la enemistad entre diferentes sectores. Ahora Jaxon pon\u00eda a prueba mi don y me ense\u00f1aba a ser uno de ellos.\n\nPocas semanas despu\u00e9s de incorporarme a mi nuevo trabajo, hab\u00eda conseguido sacar a mi esp\u00edritu de su sitio conscientemente. Y al momento hab\u00eda dejado de respirar. A Jax y a Eliza les entr\u00f3 p\u00e1nico; creyeron que me hab\u00edan matado. Nick, sin perder los nervios, como buen m\u00e9dico, me hab\u00eda reanimado inyect\u00e1ndome adrenalina directamente en el coraz\u00f3n y, aunque despu\u00e9s me doli\u00f3 el pecho durante una semana, estaba muy orgullosa. Hab\u00edamos ido los cuatro a Chateline's a celebrarlo, y Jax hab\u00eda ordenado que la pr\u00f3xima vez me pusieran soporte vital.\n\nEncajaba con aquella gente. Ellos entend\u00edan mi extra\u00f1o mundo, un mundo que yo solo estaba empezando a descubrir. En Seven Dials hab\u00edamos creado un peque\u00f1o mundo, un mundo de delincuencia y color. Ahora hab\u00eda un desconocido entre nosotros. Quiz\u00e1 dos, si Nadine acababa resultando interesante.\n\nTante\u00e9 sus onirosajes. El de Nadine era normal, pero el de Zeke... Bueno, el de Zeke era interesante. Una presencia oscura y pesada en el \u00e9ter.\n\n\u2014\u00bfDe d\u00f3nde eres, Zeke? \u2014pregunt\u00f3 Nick.\n\n\u2014Nac\u00ed en M\u00e9xico \u2014contest\u00f3\u2014, pero ahora vivo con Nadine.\n\nNo dio m\u00e1s explicaciones. Mir\u00e9 por encima del hombro y dije:\n\n\u2014\u00bfHab\u00edas estado en alguna ciudadela de Scion?\n\n\u2014No. No ten\u00eda muy claro que fuera buena idea.\n\n\u2014Pero has venido.\n\n\u2014Quer\u00edamos salir un poco. La universidad de Nadine nos hab\u00eda ofrecido plazas en el congreso. Yo sent\u00eda curiosidad por Scion. \u2014Se mir\u00f3 las manos\u2014. Me alegro de que decidi\u00e9ramos venir. Hac\u00eda a\u00f1os que nos sent\u00edamos diferentes, pero... Bueno, el se\u00f1or Hall nos ha contado por qu\u00e9.\n\nNick parec\u00eda intrigado.\n\n\u2014\u00bfCu\u00e1l es la postura oficial respecto a la clarividencia en Estados Unidos?\n\n\u2014La llaman PES, percepci\u00f3n extrasensorial. Lo \u00fanico que dicen es que bajo la ley de Scion es una enfermedad reconocida, y que el CCE la est\u00e1 investigando. No quieren adoptar ninguna postura firme al respecto. Creo que no lo har\u00e1n nunca.\n\nMe habr\u00eda gustado preguntarles sobre sus familias, pero mi instinto me aconsej\u00f3 dejarlo para m\u00e1s adelante.\n\n\u2014Jaxon est\u00e1 muy contento de que os qued\u00e9is con nosotros. \u2014Nick sonri\u00f3\u2014. Espero que os guste esto.\n\n\u2014Ya os acostumbrar\u00e9is \u2014dije yo\u2014. Yo al principio lo odiaba. Cuando me contrat\u00f3 Jaxon, todo mejor\u00f3. El sindicato se ocupar\u00e1 de vosotros.\n\n\u2014\u00bfNo eres inglesa? \u2014me pregunt\u00f3 Zeke.\n\n\u2014No. Irlandesa.\n\n\u2014Cre\u00eda que muy pocos irlandeses hab\u00edan podido huir de las revueltas de Molly.\n\n\u2014Yo, por ejemplo.\n\n\u2014Fue una tragedia. La m\u00fasica irlandesa es preciosa \u2014a\u00f1adi\u00f3\u2014. \u00bfConoces la canci\u00f3n de los alborotadores?\n\n\u2014\u00bfLa que trata sobre Molly?\n\n\u2014No, la otra. La que cantaron al final de las revueltas, cuando lloraban a los muertos.\n\n\u2014Te refieres a \u00abAn Ember Morning\u00bb.\n\n\u2014S\u00ed, eso es. \u2014Hizo una pausa, y entonces a\u00f1adi\u00f3\u2014: \u00bfPuedes cantarme un trozo? \u2014Nick y yo re\u00edmos a la vez. Zeke se puso rojo\u2014. Lo siento \u2014dijo\u2014. Es que me encantar\u00eda o\u00edrla bien cantada. Si no os importa demasiado. Antes pod\u00eda escuchar a Nadine, pero... ya no canta.\n\nNick me mir\u00f3. Una suspirante que no cantaba. A Jaxon no le iba a gustar.\n\n\u2014Paige... \u2014dijo en voz baja, y repar\u00e9 en que Zeke segu\u00eda mir\u00e1ndome, esperando mi respuesta.\n\nNo sab\u00eda si podr\u00eda cantar la canci\u00f3n. En Scion estaba prohibida la m\u00fasica irlandesa, y especialmente la m\u00fasica irlandesa revolucionaria. De peque\u00f1a, yo ten\u00eda un marcado acento irland\u00e9s, pero cuando nos trasladamos a ScionLo lo hab\u00eda abandonado por miedo al sentimiento antirland\u00e9s, cada vez m\u00e1s extendido en Scion. Con solo ocho a\u00f1os, ya notaba las miradas extra\u00f1as que me dirig\u00eda la gente cuando pronunciaba algo de una forma que ellos consideraban rara. Me pasaba horas delante del espejo, imitando a los locutores de los noticiarios, hasta que consegu\u00ed un impecable acento de colegio privado ingl\u00e9s. Todav\u00eda ca\u00eda bastante mal a mis compa\u00f1eras (me llamaron \u00abMolly Mahoney\u00bb durante a\u00f1os), pero al final, un peque\u00f1o grupo de alumnas me acept\u00f3, seguramente porque mi padre patrocinaba el baile del colegio.\n\nQuiz\u00e1 recordaba la canci\u00f3n gracias a mi primo. Mir\u00e9 por la ventana y me o\u00ed recitar:\n\n_Asomaba el mes de octubre_\n\n_y la ma\u00f1ana ard\u00eda, mi amor._\n\n_El fuego bramaba en la pradera color miel._\n\n_Ven, fantasma del valle,_\n\n_aqu\u00ed estoy, en las cenizas donde ruges._\n\n_Erin te espera para llevarte a casa._\n\n__\n\n_Vi una llama en el cielo, coraz\u00f3n,_\n\n_y la fr\u00eda ma\u00f1ana de octubre se hizo noche._\n\n_El humo invad\u00eda la pradera color miel._\n\n_Escucha, esp\u00edritu del sur,_\n\n_cerca del \u00e1rbol hendido yo te espero,_\n\n_ahora que el coraz\u00f3n de Irlanda yace, roto, junto al mar._\n\nHab\u00eda m\u00e1s estrofas, pero de pronto me par\u00e9. Me acord\u00e9 de mi abuela cant\u00e1ndole esa canci\u00f3n a Finn en su funeral, el que celebramos en secreto en el Golden Vale. Solo \u00e9ramos seis. No hab\u00eda cad\u00e1ver que enterrar. Fue all\u00ed donde mi padre anunci\u00f3 su traslado, que le obligar\u00eda a dejar a mis abuelos solos ante la ocupaci\u00f3n militar del sur por parte de Scion. Zeke estaba muy serio. Al cabo de un momento Nick me apret\u00f3 la mano.\n\nPara cuando llegamos a Monmouth Street, en el taxi hac\u00eda un calor insoportable. Le puse unos billetes en la mano al conductor. \u00c9l me devolvi\u00f3 uno.\n\n\u2014Por esa canci\u00f3n tan bonita \u2014dijo\u2014. Bendita seas, querida.\n\n\u2014Gracias.\n\nPero dej\u00e9 el billete en el asiento. No pod\u00eda aceptar dinero por un recuerdo.\n\nAyud\u00e9 a Nick a descargar las maletas. Nadine sali\u00f3 del taxi y se quit\u00f3 los auriculares. Le lanz\u00f3 una mirada fulminante al edificio. Me fij\u00e9 en su bolsa, de un dise\u00f1ador de Nueva York; tendr\u00eda que deshacerse de ella. Los productos norteamericanos se vend\u00edan como rosquillas en el Garden. Yo me hab\u00eda imaginado que llevar\u00eda un estuche de alg\u00fan instrumento musical, pero no hab\u00eda nada por el estilo. A lo mejor no era suspirante. Hab\u00eda por los menos otras tres variedades de sensores.\n\nAbr\u00ed con mi llave la puerta roja con una placa dorada que rezaba: THE LENORMAND AGENCY. Para el mundo exterior, \u00e9ramos una respetable agencia de arte. Dentro ya no \u00e9ramos tan honrados.\n\nAl final de la escalera estaba Jax, vestido con sus mejores galas: chaleco de seda, cuello postizo blanco, reloj de bolsillo reluciente y puro encendido. Llevaba una taza de caf\u00e9 de cristal en la mano. Intent\u00e9 imaginar, sin \u00e9xito, c\u00f3mo se pod\u00edan compatibilizar el puro y el caf\u00e9.\n\n\u2014Zeke, Nadine. Me alegro de volver a veros.\n\nZeke le estrech\u00f3 la mano.\n\n\u2014Y yo a usted, se\u00f1or Hall.\n\n\u2014Bienvenidos a Seven Dials. Como ya sab\u00e9is, soy el mimetocapo de este territorio. Y ahora vosotros pertenec\u00e9is a mi c\u00edrculo de \u00e9lite.\n\nJax miraba a Zeke a la cara, pero yo sab\u00eda que estaba concentrado en leerle el aura.\n\n\u2014Supongo que hab\u00e9is salido de Gower Street sin llamar la atenci\u00f3n.\n\n\u2014No nos ha visto nadie. \u2014Zeke se puso en tensi\u00f3n\u2014. \u00bfQu\u00e9 es eso de ah\u00ed? \u00bfUn esp\u00edritu?\n\nJax gir\u00f3 la cabeza.\n\n\u2014S\u00ed, es Pieter Claesz, un pintor de vanidades holand\u00e9s. Es una de nuestras musas m\u00e1s prol\u00edficas. Muri\u00f3 en 1660. Pieter, ven a saludar a nuestros nuevos amigos.\n\n\u2014Enc\u00e1rgate t\u00fa, Zeke. Yo estoy cansada. \u2014Nadine no miraba a Pieter, que hab\u00eda ignorado la orden de Jax. Nadine no ten\u00eda visi\u00f3n\u2014. Quiero una habitaci\u00f3n para m\u00ed sola. No comparto mis espacios \u2014dijo mirando con dureza a Jax\u2014. Que quede claro.\n\nEsper\u00e9 para ver c\u00f3mo reaccionaba Jax. No ten\u00eda una cara muy expresiva, pero se le inflaron las aletas de la nariz. No era buena se\u00f1al.\n\n\u2014Tendr\u00e1s lo que te den \u2014dijo.\n\nNadine se enfureci\u00f3. Previendo un enfrentamiento, Nick le puso un brazo sobre los hombros a Nadine.\n\n\u2014Claro que tendr\u00e1s tu propia habitaci\u00f3n \u2014dijo, y me mir\u00f3 con cara de fastidio por encima de la cabeza de la chica. Tendr\u00edamos que poner a Zeke en un sof\u00e1\u2014. Eliza lo est\u00e1 organizando todo. \u00bfTe apetece beber algo?\n\n\u2014S\u00ed, por favor. \u2014Mir\u00f3 a Jax arqueando las cejas\u2014. Por lo visto, todav\u00eda hay europeos que s\u00ed saben tratar a una mujer.\n\nJaxon se qued\u00f3 de piedra, como si le hubieran dado una bofetada. Nick se llev\u00f3 a Nadine hacia la _kitchenette_.\n\n\u2014Yo no soy europeo \u2014dijo Jax apretando los dientes.\n\nNo pude evitar sonre\u00edr.\n\n\u2014Me asegurar\u00e9 de que nadie te moleste \u2014dije.\n\n\u2014Gracias, Paige. \u2014Jaxon se irgui\u00f3 cuan alto era y agreg\u00f3\u2014: Ven a mi despacho, Zeke. Tenemos mucho que hablar.\n\nZeke subi\u00f3 el siguiente tramo de la escalera sin quitarle los ojos de encima a Pieter, que flotaba frente a su \u00faltimo cuadro. Antes de que yo pudiera decir nada, Jaxon me agarr\u00f3 por el brazo.\n\n\u2014Su onirosaje \u2014dijo en voz baja\u2014. \u00bfC\u00f3mo es?\n\n\u2014Oscuro \u2014dije\u2014. Y...\n\n\u2014Excelente. No digas nada m\u00e1s.\n\nSubi\u00f3 la escalera al trote, con el puro en una comisura de la boca. Me qued\u00e9 all\u00ed sola con las tres maletas y un pintor muerto; pese a que le ten\u00eda aprecio a Pieter, no era persona de muchas palabras.\n\nMir\u00e9 la hora: las once y media. Eliza volver\u00eda pasados unos minutos. Prepar\u00e9 caf\u00e9 y fui a sentarme en el sal\u00f3n, donde un lienzo de John William Waterhouse ocupaba el lugar de honor: una mujer de cabello oscuro con vestido rojo largo y suelto escudri\u00f1ando una bola de cristal. Jax hab\u00eda pagado mucho dinero a un comerciante por tres cuadros de Waterhouse que figuraban en la lista negra. Tambi\u00e9n hab\u00eda un retrato de Eduardo VII con traje de gala. Abr\u00ed la ventana y me puse a leer el nuevo panfleto en que estaba trabajando Jaxon, _Sobre las maquinaciones de los muertos itinerantes_. De momento me hab\u00eda hablado de cuatro tipos de esp\u00edritus: \u00e1ngeles guardianes, fantasmas, musas y psicopompos; todav\u00eda no hab\u00eda llegado al cap\u00edtulo de los duendes.\n\nEliza apareci\u00f3 a las doce, ensimismada como era su costumbre. Me dio un recipiente de fideos de Lisle Street.\n\n\u2014Hola. Supongo que no habr\u00e1s convencido a Pieter para que vuelva a pintar _Vanitas con viol\u00edn y bola de cristal_ , \u00bfverdad?\n\nEliza Renton, cuatro a\u00f1os mayor que yo, era la m\u00e9dium mediante trance de Jax. Su especialidad era el plagio de cuadros. Nacida en el mism\u00edsimo centro de Londres, hab\u00eda trabajado en un teatro clandestino de The Cut hasta los diecinueve a\u00f1os; entonces hab\u00eda respondido a un panfleto de Jaxon, y \u00e9l la hab\u00eda contratado. Desde entonces, era la principal fuente de ingresos de Jax. Ten\u00eda la piel clara y aceitunada, los ojos verdes y una melena rubia de tirabuzones. Nunca le faltaban admiradores (hasta los esp\u00edritus la adoraban), pero Jax nos impon\u00eda su pol\u00edtica de \u00abno implicaci\u00f3n\u00bb, y Eliza la respetaba.\n\n\u2014Todav\u00eda no. Creo que sufre un bloqueo art\u00edstico. \u2014Dej\u00e9 el panfleto a un lado\u2014. \u00bfYa conoces a los reci\u00e9n llegados?\n\n\u2014Solo a Nadine. Me ha dicho \u00abhola\u00bb, y punto. \u2014Eliza se sent\u00f3 a mi lado\u2014. \u00bfSeguro que es una susu?\n\nAbr\u00ed el recipiente de fideos humeantes.\n\n\u2014Yo no he visto ning\u00fan instrumento, pero todo es posible. \u00bfHas visto a Zeke?\n\n\u2014Me he asomado un momento al despacho. Tiene el aura naranja oscuro.\n\n\u2014Entonces es una furia.\n\n\u2014Pero no lo parece. Parece incapaz ni de darle un susto a un fantasma. \u2014Se puso los snacks de gambas encima de la rodilla\u2014. Bueno, si Pieter se pone testarudo, tengo un hueco en mi programa. \u00bfQuieres intentar salirte otra vez?\n\n\u2014No. Mientras Jax no traiga el soporte vital, no.\n\n\u2014Claro, claro. Me parece que el ventilador llegar\u00e1 el martes. No hay prisa. \u2014Me pas\u00f3 un cuaderno de bocetos y un l\u00e1piz\u2014. Hace tiempo que quer\u00eda preguntarte... \u00bfPuedes dibujar tu onirosaje?\n\n\u2014\u00bfDibujarlo? \u2014pregunt\u00e9 al tiempo que cog\u00eda el cuaderno y el l\u00e1piz.\n\n\u2014S\u00ed. No que dibujes las flores, ni nada de eso. Solo el contorno tal como se ver\u00eda a vista de p\u00e1jaro. Estamos intentando entender el trazado del onirosaje humano, pero es dif\u00edcil porque ninguno de nosotros podemos salir de nuestras zonas soleadas. Creemos que, como m\u00ednimo, hay tres zonas, pero necesitamos que t\u00fa hagas un boceto para ver si nuestras teor\u00edas coinciden. \u00bfPuedes hacerlo?\n\nMe sent\u00ed motivada, satisfecha de tener un cometido. Estaba resultando muy \u00fatil dentro del grupo.\n\n\u2014Claro \u2014dije.\n\nEliza encendi\u00f3 el televisor. Me puse a trabajar en el boceto. Dibuj\u00e9 un c\u00edrculo con un punto en el centro, rodeado de tres anillos.\n\nDel televisor sal\u00eda la m\u00fasica de fondo de ScionVista. Scarlett Burnish le\u00eda las noticias del mediod\u00eda. Eliza se\u00f1al\u00f3 la pantalla mientras se com\u00eda las galletas.\n\n\u2014\u00bfCrees que en realidad es mayor que Weaver, pero que se ha hecho tantas operaciones que ya no pueden salirle arrugas?\n\n\u2014No, sonr\u00ede demasiado. \u2014Segu\u00ed dibujando. Hab\u00eda conseguido algo que se parec\u00eda a un ojo de buey con cinco divisiones\u2014. Bueno, hemos determinado que esto \u2014dije, y di unos golpecitos en el centro del c\u00edrculo\u2014 es la zona soleada.\n\n\u2014De acuerdo. La zona soleada es donde los esp\u00edritus tienen que permanecer para conservar una mente sana. El cord\u00f3n arg\u00e9nteo es una especie de red de seguridad. Impide que los videntes salgan de esa zona.\n\n\u2014A casi todos, pero a m\u00ed no.\n\n\u2014Exactamente. Esa es tu singularidad. Digamos que la mayor\u00eda tenemos unos tres cent\u00edmetros de cord\u00f3n entre nuestro cuerpo y nuestro esp\u00edritu \u2014dijo midiendo con los dedos\u2014. T\u00fa tienes casi dos kil\u00f3metros. Puedes caminar hasta el anillo exterior de tu onirosaje, lo que significa que puedes sentir el \u00e9ter mucho m\u00e1s all\u00e1 que nosotros. Tambi\u00e9n puedes sentir otros onirosajes. Nosotros solo sentimos esp\u00edritus y auras, y no desde muy lejos. Yo, ahora, no siento a Jaxon ni a los dem\u00e1s.\n\nYo s\u00ed los sent\u00eda.\n\n\u2014Pero tengo mis l\u00edmites.\n\n\u2014Por eso hemos de tener cuidado. Todav\u00eda no conocemos tus l\u00edmites. Quiz\u00e1 puedas salir de tu cuerpo, y quiz\u00e1 no. Todav\u00eda est\u00e1 por ver.\n\nAsent\u00ed con la cabeza. Jaxon me hab\u00eda explicado su teor\u00eda sobre los onir\u00e1mbulos varias veces, pero Eliza era mucho mejor maestra.\n\n\u2014\u00bfQu\u00e9 pasar\u00eda si intentaras salir de tu zona soleada? Hipot\u00e9ticamente hablando.\n\n\u2014Ver\u00e1s, creemos que la segunda zona es donde tienen lugar las \u00abpesadillas\u00bb amaur\u00f3ticas. A veces, si est\u00e1s muy tenso o nervioso, el cord\u00f3n te deja llegar hasta tan lejos. M\u00e1s all\u00e1, empiezas a notar un fuerte tir\u00f3n hacia el centro. Si salieras m\u00e1s all\u00e1 de la zona crepuscular, empezar\u00edas a enloquecer.\n\nArque\u00e9 una ceja.\n\n\u2014Soy rarita, \u00bfverdad?\n\n\u2014No, no, Paige. No digas eso. Ninguno de nosotros somos raros. Eres un milagro. Una saltadora. \u2014Me quit\u00f3 el cuaderno de las manos\u2014. Le dir\u00e9 a Jax que le eche un vistazo a esto en cuanto haya terminado. Le encantar\u00e1. \u00bfTe quedas en casa de tu padre esta noche? \u00bfNo ten\u00edas que quedarte con \u00e9l los viernes?\n\n\u2014Tengo trabajo. Didion cree que ha encontrado a William Terriss.\n\n\u2014\u00a1Vaya, no hace falta que digas nada m\u00e1s! \u2014Se dio la vuelta y me mir\u00f3\u2014. Oye, ya sabes lo que dicen del sindicato: que una vez que entras, ya no sales nunca. \u00bfSeguro que no te has arrepentido?\n\n\u2014Qu\u00e9 va, todo lo contrario.\n\nEliza me sonri\u00f3. Fue una sonrisa extra\u00f1a, casi nost\u00e1lgica.\n\n\u2014Vale \u2014dijo\u2014. Voy arriba. Tengo que tranquilizar a Pieter.\n\nSali\u00f3 de la habitaci\u00f3n envuelta en el tintineo de sus brazaletes. Empec\u00e9 a sombrear los aros de mi dibujo, haci\u00e9ndolos cada vez m\u00e1s oscuros.\n\nSegu\u00eda trabajando unas horas m\u00e1s tarde cuando Jax baj\u00f3 del piso de arriba. No tardar\u00eda en ponerse el sol. Ten\u00eda que salir pitando a ver a Didion, pero antes quer\u00eda pasar mi dibujo al ordenador. Jax estaba como afiebrado.\n\n\u2014\u00bfPasa algo, Jax?\n\n\u2014Un ilegible \u2014dijo en voz baja\u2014. Ay, Paige, mi querid\u00edsima Paige. Nuestro querido se\u00f1or S\u00e1enz es un ilegible.\n\nNunca olvidar\u00e9 la cara del Custodio cuando me vio con el blus\u00f3n rojo. Por primera vez vi miedo en sus ojos.\n\nSolo dur\u00f3 una mil\u00e9sima de segundo. Pero lo vi, aunque solo fuera un instante: un atisbo de inseguridad, m\u00e1s d\u00e9bil que la llama de una vela. Se qued\u00f3 mir\u00e1ndome mientras yo me dirig\u00eda hacia mi habitaci\u00f3n.\n\n\u2014Paige.\n\nMe par\u00e9.\n\n\u2014\u00bfC\u00f3mo ha ido tu banquete de investidura?\n\n\u2014Ha sido instructivo. \u2014Resegu\u00ed con los dedos el ancla roja del chaleco\u2014. Ten\u00edas raz\u00f3n. Me ha hecho algunas preguntas sobre ti.\n\nHubo un breve y tenso silencio. El Custodio ten\u00eda todos los m\u00fasculos de la cara en tensi\u00f3n.\n\n\u2014Y t\u00fa las has contestado. \u2014Lo dijo con una frialdad que me sorprendi\u00f3\u2014. \u00bfQu\u00e9 le has dicho? Tengo que saberlo.\n\nNo iba a suplicarme: era demasiado orgulloso. Ten\u00eda las mand\u00edbulas apretadas, y sus labios formaban una l\u00ednea recta. Me pregunt\u00e9 qu\u00e9 estar\u00eda pas\u00e1ndole por la cabeza. A qui\u00e9n avisar, ad\u00f3nde huir. Qu\u00e9 hacer a continuaci\u00f3n.\n\n\u00bfCu\u00e1nto tiempo pod\u00eda hacerlo sufrir?\n\n\u2014Dijo una cosa que s\u00ed me llam\u00f3 la atenci\u00f3n. \u2014Me sent\u00e9 en el div\u00e1n\u2014. Que el consorte de sangre tiene prohibido pelear con los emim.\n\n\u2014S\u00ed. Estrictamente prohibido. \u2014Tamborile\u00f3 con los dedos en el brazo de la butaca\u2014. Le has contado lo de mis heridas.\n\n\u2014No, no le he contado nada.\n\nVi que mudaba la expresi\u00f3n. Al cabo de un momento se sirvi\u00f3 un poco de amaranto de la licorera.\n\n\u2014Entonces te debo la vida \u2014dijo.\n\n\u2014Bebes mucho amaranto \u2014coment\u00e9\u2014. \u00bfEs para las cicatrices?\n\nMe lanz\u00f3 una mirada.\n\n\u2014Las cicatrices.\n\n\u2014S\u00ed, las cicatrices.\n\n\u2014Tengo mis motivos para beber amaranto.\n\n\u2014\u00bfQu\u00e9 motivos?\n\n\u2014Motivos de salud. Ya te lo dije. Viejas heridas. \u2014Dej\u00f3 la copa encima de la mesa\u2014. Decidiste no decirle a Nashira que he desobedecido. Estoy intrigado: me gustar\u00eda saber por qu\u00e9.\n\n\u2014No soy ninguna traidora.\n\nNo se me hab\u00eda pasado por alto su evasiva: las cicatrices y las viejas heridas eran lo mismo.\n\n\u2014Entiendo. \u2014El Custodio dirigi\u00f3 la mirada hacia la chimenea vac\u00eda\u2014. As\u00ed que le has ocultado informaci\u00f3n a Nashira, y sin embargo te han dado el blus\u00f3n rojo.\n\n\u2014T\u00fa mismo se lo recomendaste.\n\n\u2014S\u00ed, pero no sab\u00eda si le parecer\u00eda bien. Sospecho que Nashira tiene motivos ocultos.\n\n\u2014Me han ordenado salir a cumplir una misi\u00f3n. Al exterior. Ma\u00f1ana.\n\n\u2014A la ciudadela \u2014conjetur\u00f3\u2014. Sorprendente.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Despu\u00e9s de lo que le cost\u00f3 sacarte de la ciudadela, resulta extra\u00f1o que ahora te env\u00ede all\u00ed.\n\n\u2014Quiere utilizarme de se\u00f1uelo para hacer salir a una banda de Londres, los Siete Sellos. Cree que tienen un onir\u00e1mbulo, y que yo sabr\u00e9 reconocer a otro como yo. \u2014Esper\u00e9, pero el Custodio no reaccion\u00f3. \u00bfAcaso sospechaba de m\u00ed?\u2014. Saldremos ma\u00f1ana por la noche, con tres casacas rojas y otro refa\u00edta.\n\n\u2014\u00bfQui\u00e9n?\n\n\u2014Tu prima.\n\n\u2014Ah, s\u00ed. \u2014Junt\u00f3 las yemas de los dedos\u2014. Situla Mesarthim es la mercenaria m\u00e1s leal de Nashira. T\u00fa y yo tendremos que andarnos con cuidado cuando ella est\u00e9 cerca.\n\n\u2014\u00bfSignifica eso que volver\u00e1s a tratarme como si fuera tu esclava?\n\n\u2014Ser\u00e1 temporal, pero necesario. Situla y yo no somos amigos. La han escogido a ella para que me vigile.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Transgresiones cometidas en el pasado. \u2014Vio mi expresi\u00f3n de curiosidad y agreg\u00f3\u2014: Es mejor que no sepas nada. Lo \u00fanico que necesitas saber es que yo no mato a menos que sea absolutamente necesario.\n\nTransgresiones cometidas en el pasado. Viejas heridas. Eso solo pod\u00eda significar una cosa, y ambos sab\u00edamos qu\u00e9; sin embargo, no garantizaba que pudiera confiar en el Custodio. Aunque fuera un marcado.\n\n\u2014Necesito dormir un poco \u2014dije\u2014. Hemos quedado en la residencia de Nashira ma\u00f1ana al anochecer.\n\nEl Custodio asinti\u00f3 con la cabeza sin mirarme. Recog\u00ed mis botas, fui a mi habitaci\u00f3n y lo dej\u00e9 bebi\u00e9ndose su remedio.\n\nEn lugar de dormir, me pas\u00e9 casi todo el d\u00eda pensando en los diferentes panoramas que pod\u00edan presentarse cuando lleg\u00e1ramos a Londres. El plan, seg\u00fan la reuni\u00f3n informativa posterior a la cena, consist\u00eda en esperar hasta que Carter llegara a la base de la columna de Nelson, donde se hab\u00eda dado cita con un representante de los Sellos. Los dem\u00e1s los rodear\u00edamos y los atacar\u00edamos. Por lo visto, Nashira cre\u00eda que nos plantar\u00edamos all\u00ed, disparar\u00edamos a Carter, tomar\u00edamos a unos cuantos prisioneros y volver\u00edamos tan campantes a Sheol I a tiempo para la campanada diurna.\n\nPero yo sab\u00eda que no iba a ser as\u00ed. Conoc\u00eda a Jax. Jax proteg\u00eda bien a sus inversiones. Jam\u00e1s se le ocurrir\u00eda enviar a un \u00fanico representante a reunirse con Antoinette: ir\u00eda toda la banda. Los centinelas manten\u00edan las calles vigiladas durante la noche, y dominaban el combate espiritista. Adem\u00e1s, tendr\u00edamos que lidiar con el p\u00fablico, y como en la calle habr\u00eda videntes, pod\u00edamos acabar en medio de una pelea de dimensiones considerables. Una pelea en la que yo vestir\u00eda los colores de un bando y desear\u00eda que ganara el contrario.\n\nMe di la vuelta, nerviosa. Aquella era mi ocasi\u00f3n para escapar, o al menos para hacer correr la voz. Ten\u00eda que conseguir hablar con Nick, si \u00e9l no me mataba primero. O me cegaba con sus visiones. \u00c9l era mi \u00fanica oportunidad.\n\nAl final desist\u00ed de quedarme dormida. Fui al cuarto de ba\u00f1o, me lav\u00e9 la cara y me recog\u00ed el pelo en un mo\u00f1o. Me hab\u00eda crecido bastante, hasta los hombros. La lluvia golpeaba los cristales de las ventanas. Me puse el uniforme, el blus\u00f3n rojo de los traidores y baj\u00e9 a la c\u00e1mara del Custodio. El reloj de p\u00e9ndulo marcaba casi las siete. Me sent\u00e9 junto al fuego. Cuando son\u00f3 la hora, el Custodio apareci\u00f3 en la puerta, con el cabello y la ropa empapados de lluvia.\n\n\u2014Es la hora.\n\nMe levant\u00e9 y sal\u00ed por la puerta; el Custodio la cerr\u00f3 con llave y baj\u00f3 conmigo por la escalera de piedra.\n\n\u2014Todav\u00eda no te he dado las gracias \u2014me dijo mientras recorr\u00edamos los soportales\u2014. Por tu silencio.\n\n\u2014No me las des todav\u00eda.\n\nLas calles estaban en silencio. Las piedras de granizo medio derretidas cruj\u00edan bajo las suelas de mis botas. Cuando llegamos a la residencia, dos refa\u00edtas nos escoltaron hasta la biblioteca, donde nos esperaba Nashira. El Custodio y ella representaron su saludo ritual: \u00e9l le puso una mano sobre el abdomen, y ella le acerc\u00f3 los labios a la frente. Esa vez me fij\u00e9 en m\u00e1s cosas: en la rigidez de los movimientos del Custodio, en que nunca miraba a Nashira a los ojos, en que ella le pasaba una mano por el pelo, sin mirarlo. Me recordaron a un perro y a su ama.\n\n\u2014Me alegro de que los dos hay\u00e1is podido venir esta noche \u2014dijo Nashira. Como si nosotros hubi\u00e9ramos podido elegir\u2014. 40, te presento a Situla Mesarthim.\n\nSitula era casi tan alta como el Custodio. Se notaba que eran parientes: ten\u00edan el mismo tono de cabello, casta\u00f1o ceniza, la misma piel de color miel, las mismas facciones marcadas y los ojos hundidos. Salud\u00f3 con una inclinaci\u00f3n de cabeza al Custodio, que segu\u00eda arrodillado.\n\n\u2014Hola, primo. \u2014El Custodio inclin\u00f3 la cabeza. Situla me mir\u00f3. Ten\u00eda los ojos azules\u2014. XX-59-40, esta noche me tratar\u00e1s como si fuera tu segunda guardiana. Supongo que sabes a qu\u00e9 me refiero.\n\nAsent\u00ed con la cabeza. El Custodio se levant\u00f3 y mir\u00f3 a su prometida.\n\n\u2014\u00bfD\u00f3nde est\u00e1n los otros humanos?\n\n\u2014Prepar\u00e1ndose, por supuesto. \u2014Le dio la espalda\u2014. Y t\u00fa deber\u00edas hacer lo mismo, mi fiel servidor.\n\nEl aura del Custodio se nubl\u00f3, como si se estuviera fraguando una tormenta en su onirosaje. Se dio la vuelta y fue hacia unos gruesos cortinajes de color carmes\u00ed. Una chica amaur\u00f3tica se apresur\u00f3 a seguirlo; llevaba unas prendas de ropa en las manos.\n\n\u2014Formar\u00e1s pareja con 1 \u2014me dijo Nashira\u2014. Ir\u00e9is los dos con Arcturus. Situla se llevar\u00e1 a 30 y a 12.\n\nDavid sali\u00f3 de detr\u00e1s de las cortinas; llevaba pantal\u00f3n negro, botas y un ligero chaleco antibalas. Al verlo me sobresalt\u00e9. Era id\u00e9ntico al Capataz la noche que me hab\u00edan capturado.\n\n\u2014Buenas noches, 40 \u2014me dijo.\n\nMantuve la boca cerrada. David sonri\u00f3 y sacudi\u00f3 la cabeza, como si yo le hiciera gracia. Se me acerc\u00f3 un amaur\u00f3tico.\n\n\u2014Tu ropa.\n\n\u2014Gracias.\n\nSin mirar a David, pas\u00e9 con mi ropa detr\u00e1s de las cortinas. All\u00ed hab\u00eda un vestidor improvisado. Me quit\u00e9 el uniforme y me puse el nuevo: primero, una camisa roja de manga larga, y luego el chaleco antibalas (marcado con el ancla roja, igual que el chaleco), y una chaqueta negra con un brazalete rojo en la manga. A continuaci\u00f3n me puse los guantes y los pantalones, ambos de una tela negra y flexible, y unas robustas botas de cuero. Con ese atuendo podr\u00eda correr, trepar y pelear. En la chaqueta hab\u00eda una jeringa de adrenalina y una pistola de flux. Para cazar videntes.\n\nUna vez equipada, volv\u00ed a donde estaban los otros tres humanos. Carl me sonri\u00f3.\n\n\u2014Hola, 40.\n\n\u2014Hola, Carl.\n\n\u2014\u00bfC\u00f3mo te sienta el blus\u00f3n nuevo?\n\n\u2014Es de mi talla, si es a eso a lo que te refieres.\n\n\u2014No, me refiero a c\u00f3mo te sienta ser una casaca roja.\n\nLos tres me miraban fijamente.\n\n\u2014Genial \u2014contest\u00e9 tras una pausa.\n\n\u2014S\u00ed, es genial \u2014coincidi\u00f3 Carl\u2014. Quiz\u00e1 acertaran concedi\u00e9ndote tantos privilegios.\n\n\u2014O quiz\u00e1 se equivocaran \u2014terci\u00f3 30 sac\u00e1ndose la melena de debajo de la camisa. Era m\u00e1s alta que yo, ancha de hombros y caderas\u2014. Eso lo averiguaremos en las calles.\n\nLe ech\u00e9 otro vistazo a 30. A juzgar por su aura, deb\u00eda de ser una adivina, pero poco com\u00fan. Quiz\u00e1 clerom\u00e1ntica. No era demasiado inusual. Deb\u00eda de haber ido ascendiendo sin reparar en los medios.\n\n\u2014Exacto \u2014dije.\n\n30 aspir\u00f3 con fuerza por la nariz.\n\nEl regreso del Custodio tuvo un efecto asombroso sobre la actitud de 30. Le hizo una delicada reverencia y murmur\u00f3 \u00abconsorte de sangre\u00bb. Carl, a su vez, salud\u00f3 con una inclinaci\u00f3n. Yo me qued\u00e9 quieta, con los brazos cruzados. El Custodio ech\u00f3 un vistazo a su club de fans, pero no respondi\u00f3 a ninguno de los tributos; en lugar de eso me mir\u00f3 a m\u00ed, que estaba a cierta distancia. 30 se llev\u00f3 un chasco. Pobre 30.\n\nEl cambio de vestuario hab\u00eda transformado a mi guardi\u00e1n. En lugar del traje anticuado de los refa\u00edtas, iba vestido como un habitante adinerado de Scion, el tipo de personaje al que ning\u00fan ladr\u00f3n inteligente intentar\u00eda birlar la cartera.\n\n\u2014Os llevar\u00e1n a la cohorte I en dos veh\u00edculos de recogida \u2014dijo Nashira\u2014. Despejar\u00e1n el tr\u00e1fico para dejaros pasar. Deber\u00edais estar de regreso aqu\u00ed antes de que suene la campanada diurna.\n\nLos cuatro humanos asentimos con la cabeza. El Custodio sacudi\u00f3 ligeramente los hombros y se dirigi\u00f3 hacia la puerta.\n\n\u2014XX-40, XX-1 \u2014nos llam\u00f3.\n\nCarl se sorprendi\u00f3 tanto como si Novembertide se hubiera adelantado. Corri\u00f3 tras el Custodio, guard\u00e1ndose la pistola de flux en la chaqueta por el camino. Me dispon\u00eda a seguirlo cuando Nashira me agarr\u00f3 por el brazo con una mano enguantada. Me qued\u00e9 quieta y contuve el impulso de soltarme.\n\n\u2014S\u00e9 qui\u00e9n eres \u2014dijo acerc\u00e1ndome la cara\u2014. S\u00e9 de d\u00f3nde provienes. Si no me traes a un onir\u00e1mbulo, deducir\u00e9 que no me equivocaba y que eres la So\u00f1adora P\u00e1lida. Esa comprobaci\u00f3n tendr\u00e1 consecuencias para todos nosotros. \u2014Me lanz\u00f3 una mirada que me dej\u00f3 helada; entonces me dio la espalda y fue hacia la puerta\u2014. Te deseo un buen viaje, XX-59-40.\n\nEn el puente hab\u00eda dos veh\u00edculos esperando con las luces apagadas. Antes de meternos dentro y cerrar las puertas con seguro, nos vendaron los ojos a los cuatro. Yo me sent\u00e9 con Carl; no ve\u00eda nada, pero o\u00eda el motor. Deb\u00edan de temer que averigu\u00e1ramos la ruta para salir de la colonia.\n\nUn pelot\u00f3n de centinelas se hab\u00eda desplazado a Sheol I para acompa\u00f1arnos, pero aun as\u00ed el procedimiento era complicado. La ciudad era una colonia penitenciaria, y salir de ella resultaba tan fastidioso como si fu\u00e9ramos prisioneros en libertad provisional. En una de las subestaciones exteriores de Scion nos implantaron unos chips localizadores subcut\u00e1neos por si intent\u00e1bamos huir, y nos examinaron las huellas dactilares y las auras. Me extrajeron una muestra de sangre y me dejaron un cardenal en el pliegue interno del codo. Por fin cruzamos la \u00faltima frontera y volvimos a Scion Londres. Volvimos al mundo real.\n\n\u2014Ya pod\u00e9is quitaros las vendas \u2014dijo el Custodio.\n\nMe apresur\u00e9 a quitarme la m\u00eda.\n\n\u00a1Mi ciudadela! Pas\u00e9 un dedo por el cristal de la ventana, siguiendo las luces que se reflejaban en mis ojos. El coche pas\u00f3 por delante del gigantesco centro comercial del barrio de White City, en el II-3. Nunca se me hab\u00eda ocurrido pensar que echar\u00eda de menos aquellas calles sucias de color gris plomo, pero las extra\u00f1aba, y tambi\u00e9n pujar para conseguir esp\u00edritus y jugar al tarocchi y trepar a los edificios con Nick para ver la puesta de sol. Quer\u00eda salir del coche y zambullirme en el coraz\u00f3n envenenado de Londres.\n\nCarl hab\u00eda estado intranquilo durante la primera parte del trayecto, no paraba de mover una pierna y de acariciar su pistola de flux; pero en la autopista se hab\u00eda quedado dormido. Me hab\u00eda dicho que 30 se llamaba Amelia, y que su guardi\u00e1n era un tal Elnath Sarin. Era clerom\u00e1ntica, como yo sospechaba, y se le daban especialmente bien los dados. Tard\u00e9 un rato en recordar la palabra exacta: astragalom\u00e1ntico. Estaba empezando a oxidarme. En otros tiempos Jax me examinaba a diario sobre los siete \u00f3rdenes de la clarividencia.\n\nVolv\u00ed a mirar a Carl. Llevaba el pelo sucio. Sus ojeras me revelaron que estaba tan cansado como yo; pero \u00e9l no ten\u00eda cardenales. Nuevas traiciones deb\u00edan de haberle proporcionado seguridad. Abri\u00f3 los ojos, como si hubiera notado que lo estaba observando.\n\n\u2014No intentes huir.\n\nLo dijo con un hilo de voz. Como no contest\u00e9, se desplaz\u00f3 un poco hacia m\u00ed.\n\n\u2014No lo permitir\u00e1n. \u00c9l no lo permitir\u00e1. \u2014Mir\u00f3 al Custodio a trav\u00e9s de la pantalla de vidrio\u2014. Sheol es un sitio seguro para nosotros. \u00bfPor qu\u00e9 quieres irte?\n\n\u2014Porque no es nuestro sitio.\n\n\u2014Te equivocas, es nuestro \u00fanico sitio. All\u00ed podemos ser clarividentes. No tenemos que escondernos.\n\n\u2014T\u00fa no eres idiota, Carl. Sabes perfectamente que es una c\u00e1rcel.\n\n\u2014Y \u00bfla ciudadela no lo es?\n\n\u2014No.\n\nCarl mir\u00f3 su pistola. Yo mir\u00e9 por la ventana.\n\nUna parte de m\u00ed sab\u00eda a qu\u00e9 se refer\u00eda Carl. Por supuesto que la ciudadela era una c\u00e1rcel (Scion nos ten\u00eda encerrados como animales), pero all\u00ed no ve\u00edamos que pegaban a la gente, ni que la dejaban morir tirada en la calle.\n\nApoy\u00e9 la frente en el cristal de la ventana. No, eso no era verdad. Hector lo hac\u00eda. Y Jaxon. Lo hac\u00edan todos los mimetocapos de la ciudadela. Solo recompensaban a los que eran \u00fatiles. Al resto los echaban y los dejaban pudrirse.\n\nPero para m\u00ed la banda era lo m\u00e1s parecido a una familia. En la ciudadela no ten\u00eda que hacerle reverencias a nadie. Era la dama del I-4. Ten\u00eda un nombre.\n\nNo tardamos en llegar a Marylebone. Mientras el Custodio contemplaba el territorio de la ciudadela, desconocido para \u00e9l, me pregunt\u00e9 si habr\u00eda estado en Londres alguna vez. Seguramente s\u00ed, ya que hab\u00eda conocido a inquisidores anteriores. Me daba escalofr\u00edos pensar que hab\u00eda habido refas por las calles. Que hab\u00edan estado en el Arconte. Incluso en el I-4.\n\nEl conductor era un hombre callado y corpulento; llevaba gafas de montura met\u00e1lica y traje, con pa\u00f1uelo de bolsillo y corbata de seda roja. En la oreja izquierda llevaba un Ductaphone que pitaba a cada momento. Era morboso y fascinante ver lo organizado que estaba todo. Scion ten\u00eda todas sus bases cubiertas: nadie pod\u00eda descubrir la existencia de Sheol I. Era una ciudad cerrada a cal y canto.\n\nEl Custodio le hizo se\u00f1as al conductor para que parara en una esquina. El hombre asinti\u00f3 y sali\u00f3 del coche. Cuando volvi\u00f3, llevaba una gran bolsa de papel. El Custodio me la pas\u00f3 por la trampilla.\n\n\u2014Despi\u00e9rtalo \u2014me dijo, y se\u00f1al\u00f3 a Carl, que hab\u00eda vuelto a quedarse dormido.\n\nDentro de la bolsa hab\u00eda dos cajas de cart\u00f3n calientes de Brekkabox, la tienda de comida r\u00e1pida m\u00e1s famosa de la ciudadela. Zarande\u00e9 un poco a Carl.\n\n\u2014\u00a1Despierta!\n\nCarl se sobresalt\u00f3. Abr\u00ed mi caja y encontr\u00e9 un bocadillo enrollado, una servilleta y un tarro de gachas de avena. Mir\u00e9 al Custodio por el espejo retrovisor, y \u00e9l dio una leve sacudida con la cabeza. Desvi\u00e9 la mirada.\n\nEl coche entr\u00f3 en el sector 4. Mi sector. Me sudaba el cuero cabelludo, y el sudor me produc\u00eda picor. Mi padre viv\u00eda a solo veinte minutos de all\u00ed, y nos est\u00e1bamos acercando mucho a Seven Dials. Prest\u00e9 atenci\u00f3n por si recib\u00eda algo de Nick, pero el \u00e9ter estaba en completo silencio. Varios cientos de onirosajes presionaban contra el m\u00edo y me distra\u00edan del mundo de la carne. Cuando me concentr\u00e9 en los m\u00e1s cercanos, no sent\u00ed nada inusual, ninguna nueva oleada de emoci\u00f3n. Aquella gente no ten\u00eda ni idea de la existencia de los refa\u00edtas ni de la colonia penitenciaria. No les importaba ad\u00f3nde iban los antinaturales mientras no estuvieran a la vista.\n\nNuestro coche se detuvo en el Strand, donde nos esperaba un centinela. Todos los que estaban de guardia se parec\u00edan: altos, anchos de hombros, generalmente m\u00e9diums. Al salir del coche, evit\u00e9 mirar a aquel hombre; dej\u00e9 las cajas del desayuno vac\u00edas bajo el asiento.\n\nEl Custodio, con su imponente estatura, no estaba nervioso en absoluto.\n\n\u2014Buenas noches, centinela.\n\n\u2014Buenas noches, Custodio. \u2014El centinela se toc\u00f3 la frente con tres dedos, uno en el centro y otro sobre cada ojo; a continuaci\u00f3n salud\u00f3 alzando la mano. Era la se\u00f1al oficial de su clarividencia, su tercer ojo\u2014. \u00bfPodr\u00eda confirmarme que tiene a Carl Dempsey-Brown y a Paige Mahoney bajo su custodia?\n\n\u2014S\u00ed, confirmado.\n\n\u2014\u00bfN\u00fameros de identificaci\u00f3n?\n\n\u2014XX-59-1 y 40.\n\nEl centinela tom\u00f3 nota. Me pregunt\u00e9 qu\u00e9 le habr\u00eda hecho darles la espalda a los suyos. Quiz\u00e1 un capo cruel.\n\n\u2014Vosotros dos, no olvid\u00e9is que est\u00e1is bajo custodia. Est\u00e1is aqu\u00ed para ayudar a los refa\u00edtas. Una vez cumplida vuestra misi\u00f3n, volver\u00e9is directamente a Sheol I. Si alguno de los dos intenta revelar la ubicaci\u00f3n de Sheol I, os dispararemos. Si alguno de los dos intenta establecer contacto con la poblaci\u00f3n, o con alg\u00fan miembro del sindicato, os dispararemos. Si alguno de los dos intenta hacerle da\u00f1o a vuestro guardi\u00e1n, o a un centinela, os dispararemos. \u00bfHa quedado claro?\n\nBueno, hab\u00eda dejado bastante claro que, hici\u00e9ramos lo que hici\u00e9semos, nos disparar\u00edan.\n\n\u2014Lo hemos entendido \u2014dije.\n\nPero el centinela no hab\u00eda terminado. Sac\u00f3 un tubo de plata y un par de guantes de l\u00e1tex de un compartimiento de su cintur\u00f3n. \u00abOtra aguja no, por favor\u00bb, me dije.\n\n\u2014T\u00fa primero. \u2014Me agarr\u00f3 una mu\u00f1eca\u2014. Abre la boca.\n\n\u2014\u00bfQu\u00e9?\n\n\u2014Abre-la-boca.\n\nQuise mirar al Custodio, pero su silencio me indicaba que no ten\u00eda objeciones a aquel procedimiento. Antes de que hubiera podido obedecer, el centinela me abri\u00f3 la boca a la fuerza. Le habr\u00eda pegado un mordisco al muy desgraciado. Me frot\u00f3 los labios con el extremo de pl\u00e1stico del tubo cubri\u00e9ndolos de una sustancia fr\u00eda y amarga.\n\n\u2014Cierra.\n\nNo ten\u00eda alternativa, as\u00ed que cerr\u00e9 la boca. Cuando intent\u00e9 volver a abrirla, no pude. Abr\u00ed mucho los ojos. \u00ab\u00a1Mierda!\u00bb\n\n\u2014Solo es un poco de adhesivo d\u00e9rmico. \u2014El centinela agarr\u00f3 a Carl por un brazo\u2014. Se va solo al cabo de dos o tres horas. No queremos correr riesgos, porque todos los sindis os conoc\u00e9is.\n\n\u2014Pero si yo no soy... \u2014protest\u00f3 Carl.\n\n\u2014C\u00e1llate.\n\nY por fin obligaron a Carl a cerrar el pico.\n\n\u2014XIX-49-30 no va sellada. Miradla a ella para recibir las \u00f3rdenes \u2014dijo el centinela\u2014. Por lo dem\u00e1s, ce\u00f1\u00edos a vuestros objetivos.\n\nMe empuj\u00e9 los labios con la lengua, pero no se movieron. A aquel centinela deb\u00eda de encantarle controlar a ex miembros del sindicato.\n\nTras sellarnos la boca, el centinela salud\u00f3 al Custodio y volvi\u00f3 al edificio triste y gris del que hab\u00eda salido. Fuera hab\u00eda una placa que rezaba: CIUDADELA SCION LONDRES \u2013 PUESTO DE MANDO DE LA DVN \u2013 COHORTE I SECTOR 4, y un mapa de la zona que cubr\u00eda ese puesto de mando. Alcanc\u00e9 a ver un letrero que indicaba la situaci\u00f3n del centro comercial de Covent Garden, el centro del mercado negro. Ojal\u00e1 consiguiera llegar hasta all\u00ed. Quiz\u00e1 todav\u00eda estuviera a tiempo.\n\nCarl trag\u00f3 saliva. Llev\u00e1bamos a\u00f1os viendo esas placas, y sin embargo nos sobrecog\u00edan. Mir\u00e9 al Custodio.\n\n\u2014Situla y sus humanos llegar\u00e1n a la plaza desde el lado oeste \u2014dijo\u2014. \u00bfEst\u00e1is preparados?\n\nNo s\u00e9 c\u00f3mo esperaba que contest\u00e1ramos. Carl asinti\u00f3 con la cabeza. El Custodio se sac\u00f3 dos m\u00e1scaras de la chaqueta.\n\n\u2014Tomad \u2014dijo, y nos dio una a cada uno\u2014. As\u00ed ocultar\u00e9is vuestra identidad.\n\nNo eran m\u00e1scaras normales y corrientes. Ten\u00edan unos rasgos uniformes e inexpresivos, con peque\u00f1as ranuras para los ojos y orificios para respirar bajo la nariz. Cuando me puse la m\u00eda, se me adhiri\u00f3 inmediatamente a la piel. No llamar\u00eda la atenci\u00f3n de los atareados ciudadanos de Scion, y sin embargo impedir\u00eda que la banda me reconociera. Y como ten\u00eda los labios sellados, no podr\u00eda pedir ayuda.\n\nQu\u00e9 bien lo hab\u00edan organizado todo.\n\nEl Custodio me mir\u00f3 un momento antes de ponerse la m\u00e1scara. Por los orificios para sus ojos sal\u00eda una luz estremecedora. Por primera vez me alegr\u00e9 de estar peleando en su bando.\n\nFuimos caminando hacia la columna de Nelson. Al igual que la de Seven Dials, el Monumento y la mayor parte del resto de las columnas, se iluminaba con luz roja o verde dependiendo del grado de seguridad de cada momento. En ese momento estaba verde, igual que las fuentes. Hab\u00eda una patrulla de centinelas repartida a intervalos regulares por el Strand; seguramente les hab\u00edan ordenado que nos apoyaran si era necesario. Nos miraron con cautela cuando pasamos a su lado, pero ninguno se movi\u00f3. Todos llevaban carabinas M4. La DVN no divulgaba su verdadera funci\u00f3n en la ciudad, pero todos sab\u00edamos que eran algo m\u00e1s que polic\u00edas. No te dirig\u00edas a un vigilante nocturno para quejarte de algo, como quiz\u00e1 s\u00ed hicieras con un agente de la DVD. Solo te dirig\u00edas a ellos si la situaci\u00f3n era desesperada, y nunca si eras vidente. Ni siquiera a los amaur\u00f3ticos les gustaba acerc\u00e1rseles. Al fin y al cabo, eran antinaturales.\n\nCarl, con las manos en los bolsillos, no paraba de doblar los dedos. \u00bfC\u00f3mo iba a ingeni\u00e1rmelas para salir de all\u00ed sin matar a ninguno de mi banda? Ten\u00eda que haber alguna forma de hacerles saber qui\u00e9n era. Ten\u00eda que avisarlos, porque, si no, vendr\u00edan conmigo a la colonia penitenciaria. Y no pod\u00eda permitir que Nashira los apresara.\n\nTrafalgar Square estaba iluminada con luz artificial, pero no lo suficiente para que llam\u00e1ramos la atenci\u00f3n. Situla, Amelia y David se acercaron desde el otro lado de la plaza y desaparecieron detr\u00e1s de uno de los cuatro leones de bronce que vigilaban la columna de Nelson. El Custodio se agach\u00f3 hasta que sus ojos quedaron a la altura de los m\u00edos.\n\n\u2014Carter no tardar\u00e1 \u2014dijo en voz baja\u2014. Tenemos que esperar a que establezca contacto con el Sello. No dej\u00e9is que os capturen bajo ning\u00fan concepto.\n\nCarl asinti\u00f3 con la cabeza.\n\n\u2014Cuando la zona quede despejada, la DVN nos escoltar\u00e1 de nuevo hasta el veh\u00edculo. Si los Sellos salen de los l\u00edmites de la cohorte, abortar\u00e9is la misi\u00f3n.\n\nRomp\u00ed a sudar. Seven Dials estaba en el centro de la cohorte I. Si la banda intentaba volver a la base, los seguir\u00edan hasta all\u00ed.\n\nFaltaban dos minutos para que sonara el Big Ben. El Custodio mand\u00f3 a Carl a sentarse en los escalones de la columna; era adivino y, por tanto, el menos sospechoso. Una vez que Carl se hubo instalado all\u00ed, el Custodio me llev\u00f3 m\u00e1s all\u00e1 de la fuente, hasta el pedestal de una estatua. En total hab\u00eda siete, una para cada una de las personas que hab\u00edan hecho posible la fundaci\u00f3n y el mantenimiento de Scion: Palmerston, Salisbury, Asquith, MacDonald, Zettler, Mayfield y Weaver. El s\u00e9ptimo pedestal siempre llevaba una reproducci\u00f3n del Inquisidor que gobernaba en ese momento, junto con su lema.\n\nEl Custodio se detuvo detr\u00e1s de una estatua y se qued\u00f3 escudri\u00f1ando mi m\u00e1scara.\n\n\u2014Perd\u00f3name \u2014dijo\u2014. No sab\u00eda que os iban a sellar.\n\nNo di muestras de haberle o\u00eddo. Ten\u00eda que concentrarme en respirar por la nariz.\n\n\u2014No mires todav\u00eda. Carter est\u00e1 esperando junto a la base de la columna, tal como estaba planeado.\n\nYo no quer\u00eda hacer aquello. Quer\u00eda que Antoinette se marchara de all\u00ed. Quer\u00eda irrumpir en su onirosaje, obligarla a huir.\n\nY entonces los not\u00e9.\n\nEran ellos, sin duda. Se acercaban desde diferentes direcciones. Jax deb\u00eda de haber movilizado a toda la banda, a los seis Sellos que quedaban. \u00bfReconocer\u00eda mi aura al instante, o me tomar\u00eda por otra onir\u00e1mbula de los alrededores, por mucho que eso le extra\u00f1ara?\n\n\u2014Percibo a un m\u00e9dium \u2014dijo el Custodio\u2014. Y a un suspirante.\n\nEran Eliza y Nadine. Mir\u00e9 hacia la base de la columna de Nelson. Y s\u00ed, all\u00ed estaba Antoinette.\n\nAntoinette llevaba una levita y un sombrero negro de ala ancha del que asomaban mechones de pelo pelirrojo entrecano. En lo poco que vi de su cara apreci\u00e9 arrugas que en el programa de televisi\u00f3n le hab\u00edan disimulado. Sujetaba una boquilla, en la que hab\u00eda un cigarrillo de lo que parec\u00eda \u00e1ster morado. Una desfachatez: nadie fumaba drogas et\u00e9reas en p\u00fablico.\n\nLa perspectiva de pelear contra Toni Carter bastaba para que me pusiera enferma por los nervios. En su programa sufr\u00eda ataques muy violentos antes de hacer sus predicciones; era un atractivo a\u00f1adido que hab\u00eda disparado los \u00edndices de audiencia. Me imagin\u00e9 c\u00f3mo ser\u00eda peleando. Nick pon\u00eda en duda que fuera un or\u00e1culo: los or\u00e1culos jam\u00e1s perd\u00edan el control de esa forma.\n\nNadine lleg\u00f3 primero. Llevaba un blazer de raya diplom\u00e1tica, solo parcialmente abrochado. Sin ninguna duda, ocultaba unas pistolas. Luego fueron apareciendo los dem\u00e1s, uno a uno, aunque sin que se notara que se conoc\u00edan unos a otros. Solo los un\u00edan sus auras. Cuando vi a Nick, cre\u00ed que iba a estallar; que me echar\u00eda a llorar, a re\u00edr, a cantar. Iba muy disfrazado. Era l\u00f3gico, pues ten\u00eda una carrera brillante en Scion. Llevaba peluca oscura y sombrero, y gafas de cristales oscuros. Unos palmos m\u00e1s all\u00e1, Jax daba golpecitos con su bast\u00f3n. El Custodio, a mi lado, guard\u00f3 silencio. Se le oscurecieron los ojos cuando uno de sus objetivos se acerc\u00f3 m\u00e1s a Antoinette. Hab\u00edan elegido a Eliza para adelantarse. La segu\u00eda Dani, con los labios apretados. Tambi\u00e9n iba disfrazada.\n\nEn su lugar, yo primero habr\u00eda establecido contacto con Antoinette con uno de mis \u00abempujoncitos\u00bb, para comprobar que no hubiera moros en la costa; pero Eliza no ten\u00eda esa capacidad. El \u00e9ter la dominaba a ella, y no al rev\u00e9s. Levant\u00f3 cuatro dedos de la mano derecha y tres de la izquierda y se los pas\u00f3 por el pelo, como si se lo desenredara. Antoinette capt\u00f3 el mensaje. Se acerc\u00f3 a Eliza y le tendi\u00f3 la mano. Eliza se la dio.\n\nSitula fue la primera en atacar. No me di ni cuenta y ya se le hab\u00eda subido encima a Antoinette y estaba estrangul\u00e1ndola. El Custodio fue hacia Zeke, mientras Carl lanzaba un esp\u00edritu que estaba all\u00ed cerca contra Eliza. Deb\u00eda de ser Nelson, el esp\u00edritu m\u00e1s potente de la plaza; Eliza se estrell\u00f3 contra uno de los leones y, aferrada a su propio pecho, grit\u00f3 con voz estrangulada: \u00ab\u00a1No puedo controlar los vientos y el mar, ni a m\u00ed mismo en la hora de la muerte!\u00bb. A continuaci\u00f3n sali\u00f3 Amelia, pero Nick, furioso porque hab\u00eda visto a Eliza retorci\u00e9ndose de dolor, le hizo un placaje. David sujet\u00f3 a Jax; o intent\u00f3 sujetar a Jax, porque Dani le lanz\u00f3 un pu\u00f1etazo que le hizo sangrar por la boca. En menos de diez segundos, yo era la \u00fanica que todav\u00eda no hab\u00eda salido a pelear.\n\nA m\u00ed ya me parec\u00eda bien, pero a Jaxon no.\n\nMe vio enseguida; o mejor dicho, vio a otro enemigo enmascarado. Agarr\u00f3 una bandada de seis esp\u00edritus y me la lanz\u00f3. Ten\u00eda que moverme, y deprisa: los esp\u00edritus de Trafalgar pod\u00edan plantear una amenaza grave. Le dispar\u00e9 un dardo de flux, pero apunt\u00e9 muy por encima de su cabeza. Jax se agach\u00f3 de todos modos, y la bandada se dispers\u00f3. \u00abD\u00e9jalo \u2014pens\u00e9\u2014. No me obligues a atacarte.\u00bb\n\nPero Jaxon no se rend\u00eda. Estaba furioso. Le hab\u00edamos estropeado los planes. Se lanz\u00f3 contra m\u00ed blandiendo su bast\u00f3n. Intent\u00e9 pegarle una patada en el abdomen para contenerlo, pero no le di bastante fuerte. Me agarr\u00f3 por el tobillo, flexion\u00f3 los brazos y me dio la vuelta. Sent\u00ed dolor. \u00abVenga, mu\u00e9vete.\u00bb\n\nNo fui suficientemente r\u00e1pida. Jax me propin\u00f3 una patada en el costado con su bota con puntera de acero, y ca\u00ed al suelo. Me hinc\u00f3 una rodilla en el pecho. Vi alzarse su pu\u00f1o, una mancha borrosa, y entonces algo s\u00f3lido me dio en la parte de la cara que no ten\u00eda protegida. Jax llevaba nudilleras de metal. Volvi\u00f3 a pegarme en las costillas. Not\u00e9 que se me part\u00eda algo, y un fuerte dolor. Y otra vez. Levant\u00e9 un brazo para parar un cuarto pu\u00f1etazo. Jax echaba fuego por los ojos; estaba sediento de sangre. Me iba a matar.\n\nNo ten\u00eda alternativa. Como ten\u00eda el cuerpo inmovilizado utilic\u00e9 mi esp\u00edritu. Jax no se lo esperaba. No estaba concentrado en mi aura. El golpazo contra su onirosaje lo derrib\u00f3, y se le cay\u00f3 el bast\u00f3n al suelo. Me levant\u00e9 como pude. Me dol\u00edan la cara y las costillas, y no pod\u00eda abrir el ojo derecho. Me qued\u00e9 agachada, apoyada en las rodillas, y me obligu\u00e9 a respirar por la nariz. No sab\u00eda que Jax pudiera ser tan brutal.\n\nMe llam\u00f3 la atenci\u00f3n un alarido. Cerca de una de las fuentes, Nadine hab\u00eda abandonado el combate con esp\u00edritus y hab\u00eda inmovilizado a Amelia en el suelo. Saqu\u00e9 la jeringuilla que llevaba en la chaqueta, la abr\u00ed con dedos ensangrentados y me clav\u00e9 la aguja en la mu\u00f1eca. Al cabo de unos segundos el dolor disminuy\u00f3. Segu\u00eda sin ver bien con el ojo derecho, aunque el izquierdo lo ten\u00eda intacto.\n\nLa mira roja de una escopeta vacilaba por mi pecho. Deb\u00edan de tener francotiradores en los edificios.\n\nTen\u00eda que haber alguna manera de salir de all\u00ed.\n\nCon energ\u00edas renovadas, corr\u00ed hacia la fuente, donde Amelia pataleaba en vano. Por mucho que quisiera que ganara Nadine, no pod\u00eda quedarme tan tranquila viendo morir a otro ser humano. Le hice un placaje agarr\u00e1ndola por la cintura y me lanc\u00e9 con ella hacia la fuente. El agua se ti\u00f1\u00f3 de rojo al cambiar las luces indicadoras del nivel de seguridad. Nadine emergi\u00f3 medio segundo despu\u00e9s que yo. Apretaba las mand\u00edbulas y ten\u00eda los m\u00fasculos del cuello en tensi\u00f3n. Me apart\u00e9.\n\n\u2014\u00a1Qu\u00edtate esa m\u00e1scara, zorra! \u2014me grit\u00f3. La apunt\u00e9 con mi pistola de flux.\n\nNadine empez\u00f3 a describir c\u00edrculos alrededor de m\u00ed. Se abri\u00f3 el blazer y sac\u00f3 una navaja. Siempre hab\u00eda preferido el acero a los esp\u00edritus.\n\nNotaba los latidos de mi coraz\u00f3n por todo el cuerpo, hasta en las yemas de los dedos. Nadine casi nunca fallaba con una navaja en las manos, y el chaleco antibalas solo me ofrec\u00eda una protecci\u00f3n relativa: si me golpeaba por encima del pecho, pod\u00eda darme por muerta. Y justo entonces apareci\u00f3 David. Cuando Nadine se dispon\u00eda a lanzar la navaja, \u00e9l le clav\u00f3 un dardo de flux entre los om\u00f3platos. Nadine abri\u00f3 mucho los ojos. Se tambale\u00f3, oscil\u00f3 y se dobl\u00f3 sobre el borde de la fuente. David la sac\u00f3 del agua y le asi\u00f3 la cabeza con ambas manos. Nos hab\u00edan ordenado no matar, pero, con la exaltaci\u00f3n, David parec\u00eda haberlo olvidado. \u00bfQu\u00e9 importancia pod\u00eda tener una susu?\n\nNo me par\u00e9 a pensar: lanc\u00e9 mi esp\u00edritu. Zeke jam\u00e1s me perdonar\u00eda si dejaba morir a su hermana.\n\nMe pas\u00e9. Solo necesit\u00e9 un segundo en la cabeza de David para hacerle soltar a Nadine. Un segundo m\u00e1s tarde volv\u00eda a estar en mi cuerpo y corr\u00eda hacia David. Cargu\u00e9 contra \u00e9l con todas mis fuerzas, lo golpe\u00e9 en un costado y ambos ca\u00edmos al suelo.\n\nLo vi todo negro. Acababa de poseer a David. Solo hab\u00eda sido un instante, pero le hab\u00eda movido el brazo.\n\nPor fin hab\u00eda pose\u00eddo a un humano.\n\nDavid se llev\u00f3 las manos a la cabeza. No hab\u00eda sido precisamente cuidadosa con \u00e9l. Me levant\u00e9 con dificultad y parpade\u00e9 para hacer desaparecer un aluvi\u00f3n de estrellas blancas. Antoinette y Situla se hab\u00edan esfumado.\n\nDej\u00e9 a Nadine junto a David y me apart\u00e9 corriendo de la fuente, con la ropa empapada. Me sub\u00ed a un le\u00f3n e inspeccion\u00e9 la escena. Ambos grupos se hab\u00edan abierto en abanico por la plaza. Zeke, que no era muy buen luchador, hab\u00eda abandonado sabiamente el barco al ver que el Custodio iba hacia \u00e9l; se hab\u00eda puesto el pasamonta\u00f1as y estaba intercambiando golpes con Amelia. El Custodio se fij\u00f3 en Nick, que acababa de dejar a Carl sin sentido con una bandada. Observ\u00e1ndolos, cre\u00ed que se me parar\u00eda el coraz\u00f3n: mi guardi\u00e1n y mi mejor amigo. Volv\u00ed a bajar al suelo, atemorizada. Ten\u00eda que ayudar a Nick. El Custodio pod\u00eda matarlo.\n\nEntonces apareci\u00f3 Eliza, furiosa. Me asaltaron esp\u00edritus desde todas direcciones. Los esp\u00edritus siempre se aliaban con los m\u00e9diums. Tres marineros franceses irrumpieron en mi onirosaje. Me tambale\u00e9, cegada por sus recuerdos: las olas imponentes, las explosiones de los mosquetes, las llamas que ard\u00edan en la cubierta del _Achille_ , gritos, caos... Y entonces Eliza me dio un empuj\u00f3n y me ca\u00ed. Levant\u00e9 todas mis defensas mentales para tratar de apartar de m\u00ed a los invasores.\n\nMe qued\u00e9 un momento incapacitada. Eliza intent\u00f3 inmovilizarme con las rodillas.\n\n\u2014\u00a1No os mov\u00e1is de ah\u00ed, chicos!\n\nMi onirosaje se estaba inundando. Las balas de ca\u00f1\u00f3n lo atravesaban. Ca\u00eda madera ardiendo ante mis ojos. Eliza alz\u00f3 las manos para desenmascararme.\n\n\u00ab\u00a1No, no!\u00bb No pod\u00eda dejar que me viera, o la DVN le disparar\u00eda. Con un tremendo esfuerzo, obligu\u00e9 a los esp\u00edritus a dispersarse y apart\u00e9 a Eliza de una patada. Eliza grit\u00f3 de dolor, y sent\u00ed una punzada de remordimiento. Gir\u00e9 sobre m\u00ed misma justo a tiempo para desviar el bast\u00f3n de Jax con mi pistola de flux.\n\n\u2014Vaya, vaya. Una andarina uniformada \u2014dijo en voz baja\u2014. \u00bfD\u00f3nde te encontraron? \u00bfD\u00f3nde te escond\u00edas? \u2014Se me acerc\u00f3 y escudri\u00f1\u00f3 los orificios para los ojos de la m\u00e1scara\u2014. No puedes ser mi Paige. \u2014Me apart\u00f3 el brazo con el bast\u00f3n; mis m\u00fasculos se tensaron\u2014. Entonces \u00bfqui\u00e9n eres?\n\nAntes de que yo pudiera reaccionar, Jax se vio empujado hacia atr\u00e1s por una bandada enorme, mucho mayor que la que habr\u00eda podido reunir cualquier humano. El Custodio. Me levant\u00e9 e intent\u00e9 recuperar la pistola, pero Jax bland\u00eda su bast\u00f3n a ciegas. Apart\u00e9 la cabeza hacia la izquierda instintivamente. Demasiado despacio. Not\u00e9 una quemadura en la oreja: un calor limpio e intenso. Una cuchilla. Consegu\u00ed asir la pistola, pero Jax me la quit\u00f3 de la mano con un segundo golpe. La cuchilla del extremo del bast\u00f3n me recorri\u00f3 todo el brazo; atraves\u00f3 la tela de mi chaqueta y me cort\u00f3. Un grito apagado intent\u00f3 salir por mi garganta. Not\u00e9 un fuerte dolor en el brazo.\n\n\u2014\u00a1Venga, andarina, usa tu esp\u00edritu! \u2014Jaxon me apunt\u00f3 con la cuchilla, riendo\u2014. Utiliza tu dolor. Deja atr\u00e1s las heridas.\n\nAmelia lanz\u00f3 otra bandada contra Jax. Yo la hab\u00eda salvado, y ahora ella me salvaba a m\u00ed. Nick devolvi\u00f3 el fuego, y Amelia se agazap\u00f3 detr\u00e1s de un le\u00f3n. Zeke estaba tumbado en el suelo, inm\u00f3vil. \u00abNo est\u00e9s muerto \u2014pens\u00e9\u2014. Que no te hayan matado, por favor.\u00bb\n\nVi un destello de pelo rojo. Antoinette hab\u00eda vuelto. Se le hab\u00eda volado el sombrero, y no me extra\u00f1\u00f3: estaba en una especie de trance de batalla. Se le sal\u00edan los ojos de las \u00f3rbitas, ten\u00eda las aletas de la nariz muy abiertas, y su esp\u00edritu era una llamarada que inutilizaba las farolas azules de la ciudadela, dise\u00f1adas para calmar las mentes exaltadas. Una lluvia de pu\u00f1os, piernas y esp\u00edritus cay\u00f3 sobre Situla, impidi\u00e9ndole clavarle la navaja a Antoinette. Situla le lanz\u00f3 un fantasma. Antoinette se apart\u00f3 \u00e1gilmente y lo esquiv\u00f3.\n\nY de pronto, sin avisar, sali\u00f3 disparada. El Custodio la vio echar a correr entre la gente, que gritaba.\n\n\u2014\u00a1Detenla! \u2014me grit\u00f3.\n\nSal\u00ed detr\u00e1s de Antoinette. Aquella era mi oportunidad de escapar. Un centinela me dej\u00f3 pasar al ver mi uniforme, pero le cerr\u00f3 el paso a una amaur\u00f3tica. Un hombre me agarr\u00f3 por la chaqueta (un suspirante), pero yo iba demasiado deprisa, y me solt\u00f3. Mi mente era un haz de luz pura. Antoinette iba derecha hacia el Arconte de Westminster. Tomar esa direcci\u00f3n era una locura, pero a m\u00ed no me importaban los motivos que pudiera tener, pues me estaba ofreciendo una oportunidad valios\u00edsima. Enfrente del Arconte hab\u00eda una estaci\u00f3n de metro, que siempre estaba abarrotada de metrovigilantes, pero tambi\u00e9n de ciudadanos que iban al trabajo. Si me quitaba la m\u00e1scara y la chaqueta, podr\u00eda pasar las barreras y perderme entre la multitud. Las columnas de la entrada me proteger\u00edan de la DVN, y para llegar a Green Park solo hab\u00eda una parada. Desde all\u00ed podr\u00eda llegar a Dials. Si ese plan fallaba, ir\u00eda hacia el T\u00e1mesis. Pod\u00eda tirarme al agua y nadar. Estaba dispuesta a hacer cualquier cosa para escapar.\n\nLo conseguir\u00eda. Estaba segura de que lo conseguir\u00eda.\n\nMis piernas se mov\u00edan con \u00edmpetu. El dolor del brazo era brutal, pero no pod\u00eda parar. El trance de Antoinette parec\u00eda haberle dado alas. Ning\u00fan ser humano pod\u00eda correr tanto, a menos que lo guiaran esp\u00edritus. Intent\u00e9 no perder de vista sus auras mientras serpenteaba entre hordas de gente y coches.\n\nUn taxi fren\u00f3 bruscamente delante de Antoinette. Situla y ella lo esquivaron cada una por un lado y se metieron en la masa de peatones. Yo tom\u00e9 el camino m\u00e1s corto: segu\u00ed corriendo, me sub\u00ed al cap\u00f3 del taxi y, desde all\u00ed, al techo, y baj\u00e9 resbalando por el otro lado. Antoinette pas\u00f3 como una exhalaci\u00f3n. Situla la sigui\u00f3 un par de segundos m\u00e1s tarde, esquivando a otros transe\u00fantes. Uno de ellos muri\u00f3. No pod\u00eda detenerme. Si aflojaba aunque solo fuera un momento, perder\u00eda de vista a Antoinette y Situla. Cuando ya cre\u00eda que iban a explotarme los pulmones, llegamos al final de Whitehall.\n\nSeg\u00fan el mapa, nos encontr\u00e1bamos en el centro de la ciudadela: cohorte I, sector 1. Los videntes evitaban esa zona como la peste. Alc\u00e9 la vista hacia el Arconte de Westminster; la sangre me resbalaba por los dedos. La esfera del reloj estaba iluminada con luz roja, y las agujas y los d\u00edgitos eran negros y se destacaban sobre la superficie iluminada. All\u00ed era donde danzaban los t\u00edteres de Frank Weaver. Si mi vida no hubiera estado amenazada, me habr\u00eda gustado dejar alg\u00fan graffiti exquisito en las paredes.\n\nCorr\u00ed hacia el Starch. Situla iba un poco por delante de m\u00ed. Cuando lleg\u00f3 al puente, Antoinette se volvi\u00f3 hacia su enemiga. Su piel tensa sobre los huesos parec\u00eda una fina capa de pintura, y ten\u00eda los labios fruncidos y blancos.\n\n\u2014Est\u00e1s rodeada, or\u00e1culo. \u2014Situla avanz\u00f3 hacia ella\u2014. R\u00edndete.\n\n\u2014No me llames or\u00e1culo, criatura. \u2014Antoinette alz\u00f3 una mano\u2014. Veamos si averiguas qu\u00e9 soy.\n\nLa atm\u00f3sfera se enfri\u00f3.\n\nA Situla no le impresion\u00f3 la amenaza; no ten\u00eda nada que temer de un simple ser humano. Avanz\u00f3 hacia Antoinette, pero antes de que pudiera intentar nada, se vio levantada del suelo y lanzada hacia atr\u00e1s, y estuvo a punto de caer del puente. Me sobresalt\u00e9. Un esp\u00edritu. Un quebrajador. Me asom\u00e9 al \u00e9ter para intentar identificarlo. Era semejante a un \u00e1ngel guardi\u00e1n, muy viejo y poderoso.\n\nUn arc\u00e1ngel. Un \u00e1ngel que se quedaba en la misma familia varias generaciones, incluso despu\u00e9s de morir la persona a la que hab\u00edan salvado. Eran muy dif\u00edciles de exorcizar. El treno no lo desterrar\u00eda por mucho tiempo.\n\nSitula se puso en pie.\n\n\u2014Qu\u00e9date quieta. \u2014Dio un paso m\u00e1s\u2014. Vamos a averiguar qu\u00e9 eres.\n\nAtrap\u00f3 un esp\u00edritu al vuelo, y luego otro, y otro, hasta reunir una bandada temblorosa. Antoinette manten\u00eda la mano alzada, pero se le crisp\u00f3 el rostro cuando Situla empez\u00f3 a cebarse en ella. Se le pusieron los ojos de un rojo intenso. Por un instante cre\u00ed que Antoinette iba a derrumbarse. Le sali\u00f3 una gota de sangre del ojo izquierdo. Entonces dio una sacudida con un brazo, y el arc\u00e1ngel sali\u00f3 disparado hacia Situla. La bandada fue a su encuentro; el \u00e9ter se abri\u00f3 de golpe, y ech\u00e9 a correr.\n\nLa mayor\u00eda de los centinelas ten\u00edan visi\u00f3n espiritista. La colisi\u00f3n de los esp\u00edritus los distraer\u00eda. No me ver\u00edan. No pod\u00edan verme. Ten\u00eda que volver a Dials. Corr\u00ed hacia la estaci\u00f3n I-1A.\n\nBajo mis botas el puente se estremec\u00eda con tanta energ\u00eda. No me detuve. Ve\u00eda el letrero de la estaci\u00f3n al otro lado de la calle. Me desprend\u00ed de la chaqueta y del chaleco antibalas. As\u00ed correr\u00eda m\u00e1s, y cuando me hubiera quitado la maldita m\u00e1scara, ya no parecer\u00eda una casaca roja. Ser\u00eda solo una chica con camisa roja. Mir\u00e9 los edificios y busqu\u00e9 puntos de apoyo en las fachadas. Si no consegu\u00eda entrar en la estaci\u00f3n, tendr\u00eda que salir de all\u00ed trepando. Si consegu\u00eda llegar a los tejados, estar\u00eda a salvo.\n\nDe pronto not\u00e9 otra cosa. Un dolor.\n\nNo me par\u00e9, pero de pronto me costaba m\u00e1s correr. No pod\u00eda ser una herida grave. El arc\u00e1ngel no se me hab\u00eda acercado. La que le preocupaba era Situla: ella s\u00ed supon\u00eda una amenaza. Deb\u00eda de haberme desgarrado un m\u00fasculo.\n\nNot\u00e9 un calor pegajoso bajo las costillas. Cuando mir\u00e9, vi que mi camisa roja se estaba ti\u00f1endo de otro rojo diferente, y que ten\u00eda un peque\u00f1o orificio redondo un poco m\u00e1s arriba de la cadera.\n\nMe hab\u00edan disparado, como a aquellos estudiantes irlandeses.\n\nTen\u00eda que seguir corriendo. Me di impulso hacia la calle, donde todav\u00eda hab\u00eda mucho tr\u00e1fico proveniente del Embankment. \u00abVenga, Paige, corre.\u00bb Nick podr\u00eda curarme. Solo ten\u00eda que llegar a Dials. Ya ve\u00eda la estaci\u00f3n. Volvieron a dispararme, pero no me dieron. Ten\u00eda que salir de su alcance. Me obligu\u00e9 a seguir adelante, pero el dolor iba en aumento y no pod\u00eda cargar el peso en el lado derecho. Mi carrera tambaleante se hab\u00eda convertido en cojera. Delante de la estaci\u00f3n hab\u00eda unas columnas. Si consegu\u00eda llegar hasta ellas, podr\u00eda contener la hemorragia y desaparecer.\n\nCorr\u00ed detr\u00e1s de un autob\u00fas, utiliz\u00e1ndolo para ocultarme, y llegu\u00e9 a la primera columna del otro lado de la calle. Me hab\u00eda quedado sin fuerzas.\n\nIntent\u00e9 seguir avanzando, pero not\u00e9 una fuerte punzada de dolor en la cintura. Se me doblaron las rodillas.\n\nQu\u00e9 r\u00e1pido me sobreven\u00eda la muerte, como si llevara a\u00f1os esperando ese momento. El mundo f\u00edsico se desdibuj\u00f3 hasta formar una neblina. Vi pasar destellos. Todav\u00eda o\u00eda los ruidos de la pelea, pero en el \u00e9ter, no en la calle.\n\nAdi\u00f3s, onir\u00e1mbula.\n\nNo me quedaba mucho tiempo. Pod\u00edan volver a dispararme. Me arrastr\u00e9 detr\u00e1s de una columna, donde no pudieran verme desde la estaci\u00f3n, donde los pasajeros intentaban averiguar de d\u00f3nde ven\u00eda tanto ruido. Me acurruqu\u00e9 contra la pared. La herida, peque\u00f1a, no paraba de sangrar. Me la tap\u00e9 con las manos temblorosas. La mordaza me imped\u00eda separar los labios.\n\nNo conseguir\u00eda llegar a Dials. Aunque me subiera a un tren, me detendr\u00edan al apearme. Reparar\u00edan en la sangre de mis manos. Al menos no hab\u00eda muerto en Sheol I; eso habr\u00eda sido el colmo de la desgracia. All\u00ed, en SciLo, Nashira no pod\u00eda alcanzarme.\n\nHab\u00eda alguien a mi lado, agarr\u00e1ndome un brazo. Lo ol\u00ed antes de verlo. Alcanfor.\n\n\u00abNick.\u00bb\n\nNo me reconoci\u00f3. No pod\u00eda reconocerme. Me levant\u00f3 la barbilla exponiendo mi cuello a su navaja.\n\n\u2014Maldita traidora.\n\n\u00abNick.\u00bb Me ard\u00eda la herida. Ten\u00eda la manga empapada de sangre.\n\n\u2014Mu\u00e9strame la cara \u2014dijo Nick. M\u00e1s tranquilo, apenado\u2014. No s\u00e9 qu\u00e9 eres, pero eres vidente. Saltador. Quiz\u00e1 lo recuerdes cuando veas la \u00faltima luz.\n\nMe quit\u00f3 la m\u00e1scara. Al verme, se rompi\u00f3 algo en su interior.\n\n\u2014\u00a1Paige! \u2014dijo con voz estrangulada\u2014. Oh, no, Paige. _F\u00f6rl\u00e5t mig_. \u2014Me apret\u00f3 el costado con las manos tratando de detener la hemorragia\u2014. Lo siento. Lo siento. Cre\u00ed... Jaxon me pidi\u00f3... \u2014Claro. Jaxon quer\u00eda atrapar al onir\u00e1mbulo. Me hab\u00eda disparado Nick, no Scion\u2014. Pero \u00bfqu\u00e9 te han hecho? \u2014Le temblaba la voz. Me doli\u00f3 mucho verlo tan desconsolado\u2014. Te pondr\u00e1s bien, te lo prometo. M\u00edrame, Paige. \u00a1M\u00edrame!\n\nMe costaba mucho enfocar la mirada. Me pesaban los p\u00e1rpados. Llev\u00e9 una mano hacia su camisa. Nick me sostuvo la cabeza contra el pecho.\n\n\u2014No pasa nada, cielo. \u00bfAd\u00f3nde te llevaron?\n\nNegu\u00e9 con la cabeza. Nick me acarici\u00f3 el pelo sudado; eso me tranquiliz\u00f3. Quer\u00eda quedarme all\u00ed. No quer\u00eda que me llevaran otra vez a aquel sitio.\n\n\u2014Paige, ni se te ocurra cerrar los ojos. Dime ad\u00f3nde te llevaron esos cerdos.\n\nVolv\u00ed a negar con la cabeza. No pod\u00eda dec\u00edrselo, no sin mi voz.\n\n\u2014Venga, _s\u00f6tnos_. Tienes que decirme d\u00f3nde es. Para que vuelva a encontrarte, como la otra vez. \u00bfTe acuerdas?\n\nTen\u00eda que dec\u00edrselo. Nick ten\u00eda que saberlo. No pod\u00eda morir sin decirle d\u00f3nde era. Ten\u00eda que salvar a otros, a los otros videntes de la ciudad perdida. Pero entonces vi una silueta, el contorno de un hombre.\n\nNo, de un hombre no.\n\nDe un refa\u00edta.\n\nTen\u00eda los dedos manchados de sangre. Levant\u00e9 un brazo y trac\u00e9 las tres primeras letras en la pared. Nick las ley\u00f3.\n\n\u2014Oxford \u2014dijo\u2014. \u00bfTe llevaron a Oxford?\n\nBaj\u00e9 la mano. El hombre sin rostro se acercaba en la oscuridad. Nick levant\u00f3 la cabeza.\n\n\u2014No. \u2014Se le tensaron los m\u00fasculos\u2014. Voy a llevarte a casa \u2014dijo, y empez\u00f3 a levantarme\u2014. No dejar\u00e9 que vuelvan a llevarte all\u00ed.\n\nDesenfund\u00f3 una pistola. Le rode\u00e9 el cuello con un brazo. Quer\u00eda que lo intentara, que echara a correr, que me sacara de otro campo de amapolas; pero si le dejaba hacerlo, lo matar\u00edan. Nos matar\u00edan a los dos. La sombra nos perseguir\u00eda hasta Dials. Le tir\u00e9 de la camisa, y al mismo tiempo negu\u00e9 con la cabeza, pero \u00e9l no me entend\u00eda. La sombra se cruz\u00f3 en nuestro camino. Nick asi\u00f3 fuertemente la pistola, hasta que se le pusieron los nudillos blancos, y apret\u00f3 el gatillo. Una vez, dos veces. Grit\u00e9 con los labios sellados. \u00ab\u00a1Corre, Nick!\u00bb. Pero \u00e9l no pod\u00eda o\u00edrme, no pod\u00eda saberlo. Se le cay\u00f3 la pistola de la mano, y se qued\u00f3 p\u00e1lido. Una gigantesca mano enguantada lo agarr\u00f3 por el cuello. Intent\u00e9 apartarla con las \u00faltimas fuerzas que me quedaban.\n\n\u2014Ella viene conmigo. \u2014Era el Custodio. Parec\u00eda un demonio\u2014. Huye, or\u00e1culo.\n\nLa vida se me escapaba de las manos. O\u00ed el coraz\u00f3n de Nick junto a mi oreja, not\u00e9 sus dedos trabarse detr\u00e1s de mi espalda. La luz disminu\u00eda. Hab\u00eda llegado la muerte.\n\nEl tiempo se convirti\u00f3 en una serie de momentos, con vac\u00edos intercalados. A veces hab\u00eda luces. A veces, voces. Me parec\u00eda ir en un coche, notar un balanceo.\n\nMe cortaron la camisa. Intent\u00e9 apartar aquellas manos indiscretas, pero mi cuerpo no me obedeci\u00f3. Reconoc\u00ed la densa niebla de los f\u00e1rmacos. De pronto me encontr\u00e9 arropada en la cama del Custodio, tumbada sobre el costado izquierdo. Ten\u00eda el pelo mojado. Me dol\u00eda todo el cuerpo.\n\n\u2014\u00bfPaige?\n\nUna voz que parec\u00eda submarina. Emit\u00ed un sonido d\u00e9bil, entre un sollozo y un estertor. Me ard\u00eda el pecho. Y el brazo. \u00abNick.\u00bb Estir\u00e9 una mano a tientas.\n\n\u2014R\u00e1pido, Michael. \u2014Me dieron la mano\u2014. Espera, Paige.\n\nDeb\u00ed de volver a desmayarme. Cuando despert\u00e9, me sent\u00eda pesada, blanda y deforme como un edred\u00f3n. No ten\u00eda sensibilidad en el brazo derecho. Me dol\u00eda al respirar, pero pod\u00eda abrir la boca. Respiraba agitadamente.\n\nMe apoy\u00e9 en un codo y me pas\u00e9 la lengua por los dientes. Estaban todos, y enteros.\n\nSentado en su butaca, el Custodio miraba su gram\u00f3fono. Me habr\u00eda gustado romper aquel aparato. Aquellas voces no ten\u00edan derecho a sonar tan animadas. Cuando el Custodio vio que me mov\u00eda, se levant\u00f3.\n\n\u2014Paige.\n\nSe me aceler\u00f3 el coraz\u00f3n. Me apret\u00e9 contra el cabecero de la cama y record\u00e9 sus ojos en la oscuridad.\n\n\u2014\u00bfLo has matado? \u2014Me sequ\u00e9 el sudor del labio\u2014. \u00bfHas matado al or\u00e1culo?\n\n\u2014No. Sigue vivo.\n\nCon cuidado, sin dejar de mirarme a la cara, me ayud\u00f3 a sentarme. Al moverme, not\u00e9 el tir\u00f3n de la v\u00eda intravenosa que llevaba en la mano.\n\n\u2014No veo bien.\n\nTen\u00eda la voz ronca, pero al menos pod\u00eda hablar.\n\n\u2014Tienes un hematoma periorbital.\n\n\u2014\u00bfQu\u00e9 es eso?\n\n\u2014Un ojo morado.\n\nMe pas\u00e9 un dedo por la piel fina de la parte superior de la mejilla. Jax me hab\u00eda pegado fuerte. Ten\u00eda todo el lado derecho de la cara hinchado.\n\n\u2014As\u00ed que hemos vuelto \u2014dije.\n\n\u2014Intentaste huir.\n\n\u2014Claro que intent\u00e9 huir. \u2014No pude disimular la amargura de mi voz\u2014. \u00bfCrees que quiero morirme aqu\u00ed y acosar a Nashira el resto de la eternidad? \u2014El Custodio se qued\u00f3 mir\u00e1ndome. Se me hizo un nudo en la garganta\u2014. \u00bfPor qu\u00e9 no me dejas irme a casa?\n\nUna d\u00e9bil mancha verde estaba borr\u00e1ndose de sus ojos. Deb\u00eda de haberse cebado en Eliza.\n\n\u2014Existen motivos \u2014dijo.\n\n\u2014Excusas.\n\nSe qued\u00f3 callado largo rato. Cuando volvi\u00f3 a hablar, no fue para explicarme por qu\u00e9 hab\u00eda vuelto a llevarme a aquella cloaca.\n\n\u2014Tienes una colecci\u00f3n de lesiones impresionante. \u2014Me coloc\u00f3 bien las almohadas y me ayud\u00f3 a apoyarme en ellas\u2014. Jaxon Hall es mucho m\u00e1s despiadado de lo que hab\u00edamos imaginado.\n\n\u2014Dame la lista.\n\n\u2014Un ojo morado, dos costillas rotas, un labio partido, una oreja cortada, contusiones, laceraciones en el brazo derecho, herida de bala en el torso. Es incre\u00edble que consiguieras correr hasta el puente.\n\n\u2014La adrenalina. \u2014Escudri\u00f1\u00e9 su cara y a\u00f1ad\u00ed\u2014: \u00bfTe hirieron?\n\n\u2014Solo alg\u00fan rasgu\u00f1o.\n\n\u2014Entonces solo me usaron a m\u00ed de saco de boxeo.\n\n\u2014Te enfrentaste a un grupo muy poderoso de clarividentes y sobreviviste, Paige. No hay que avergonzarse de la propia fuerza.\n\nPero yo s\u00ed me avergonzaba. Eliza me hab\u00eda inmovilizado, Nick me hab\u00eda disparado y Jax me hab\u00eda dado una paliza. Eso no era ser fuerte. El Custodio me acerc\u00f3 un vaso de agua a los labios y beb\u00ed a rega\u00f1adientes.\n\n\u2014\u00bfSabe Nashira que intent\u00e9 escapar?\n\n\u2014S\u00ed, claro.\n\n\u2014\u00bfQu\u00e9 me har\u00e1?\n\n\u2014Te han quitado el blus\u00f3n rojo. \u2014Dej\u00f3 el vaso en la mesilla de noche\u2014. Ahora eres una casaca amarilla.\n\nEl color de los cobardes. Consegu\u00ed soltar una risa amarga, pero me dolieron las costillas.\n\n\u2014Me importa un cuerno el blus\u00f3n que me asigne. Lo que quiere es matarme, con casaca roja o sin ella. \u2014Me temblaron los hombros\u2014. Ll\u00e9vame con ella. Acabemos con esto.\n\n\u2014Est\u00e1s herida y cansada, Paige. Cuando te encuentres mejor, seguramente no lo ver\u00e1s todo tan negro.\n\n\u2014\u00bfCu\u00e1ndo ser\u00e1 eso?\n\n\u2014Si quieres, ma\u00f1ana podr\u00e1s volver a tu habitaci\u00f3n.\n\nFrunc\u00ed el entrecejo, pero me dolieron todos los m\u00fasculos de la cara.\n\n\u2014\u00bfMa\u00f1ana?\n\n\u2014Antes de marcharnos de Londres, le ped\u00ed al conductor que recogiera scimorfina y antiinflamatorios del edificio de la SciOECI. Dentro de un par de d\u00edas estar\u00e1s completamente recuperada.\n\nLa scimorfina era car\u00edsima.\n\n\u2014\u00bfViste a mi padre en el SciOECI?\n\n\u2014No, yo no entr\u00e9 en el edificio. Solo unos cuantos pol\u00edticos del Arconte est\u00e1n al corriente de nuestra existencia.\n\nMir\u00f3 la v\u00eda intravenosa que llevaba en la mano. Comprob\u00f3, sin quitarse los guantes, que el esparadrapo estuviera bien pegado.\n\n\u2014\u00bfPor qu\u00e9 llevas esos guantes? \u2014dije con una chispa de rabia\u2014. \u00bfEncuentras demasiado asquerosos a los humanos?\n\n\u2014Son las normas de Nashira.\n\nMe ardieron las mejillas bajo los moretones. Deb\u00eda de haberse pasado horas cur\u00e1ndome; ten\u00eda que admitirlo, por muy mal que me cayera.\n\n\u2014\u00bfQu\u00e9 les ha pasado a los otros? \u2014pregunt\u00e9.\n\n\u20141 y 12 salieron ilesos. Situla qued\u00f3 en estado latente, pero ya se ha recuperado. \u2014Hizo una pausa\u2014. 30 ha muerto.\n\n\u2014\u00bfMuerto? \u00bfC\u00f3mo?\n\n\u2014Ahogada. La encontramos en la fuente.\n\nLa noticia me estremeci\u00f3. Amelia no me ca\u00eda especialmente bien, pero nunca hab\u00eda deseado su muerte. Me pregunt\u00e9 qu\u00e9 miembro de la banda la habr\u00eda matado.\n\n\u2014\u00bfY Carter?\n\n\u2014Logr\u00f3 huir. Un veh\u00edculo la recogi\u00f3 en el puente antes de que pudi\u00e9ramos capturarla.\n\nPor lo menos, Carter se hab\u00eda librado. No sab\u00eda cu\u00e1l era su poder, pero no quer\u00eda que Nashira se lo quitara.\n\n\u2014\u00bfY los Sellos?\n\n\u2014Huyeron. Jam\u00e1s hab\u00eda visto tan furiosa a Nashira.\n\nSent\u00ed un alivio inmenso. Estaban todos bien. La banda conoc\u00eda muy bien el I-4, todos sus rincones secretos y sus refugios; no debi\u00f3 de costarles mucho desaparecer, aunque Nadine y Zeke estuvieran heridos. Todos los videntes del sector respond\u00edan ante Jax. A ambos se los habr\u00edan llevado sus recadistas. Volv\u00ed a mirar al Custodio.\n\n\u2014Me salvaste.\n\nSus ojos se pasearon por mi rostro.\n\n\u2014S\u00ed.\n\n\u2014Si me entero de que le hiciste algo a ese or\u00e1culo...\n\n\u2014No le hice da\u00f1o. Lo dej\u00e9 marchar.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Porque sab\u00eda que era amigo tuyo. \u2014Se sent\u00f3 en el borde de la cama\u2014. Lo s\u00e9, Paige. S\u00e9 que eres el Sello que faltaba all\u00ed. Solo un necio no lo habr\u00eda descubierto.\n\nLe sostuve la mirada.\n\n\u2014\u00bfVas a dec\u00edrselo a Nashira?\n\nSe qued\u00f3 mir\u00e1ndome largo rato. Fueron los segundos m\u00e1s largos de mi vida.\n\n\u2014No \u2014contest\u00f3\u2014, pero Nashira no es tonta. Hace mucho tiempo que sospecha qui\u00e9n eres. Tarde o temprano lo sabr\u00e1.\n\nLos nervios me retorc\u00edan el est\u00f3mago. El Custodio se levant\u00f3 y camin\u00f3 hasta la chimenea.\n\n\u2014Ha surgido una complicaci\u00f3n. \u2014Fij\u00f3 la vista en las llamas\u2014. T\u00fa y yo nos hemos salvado el uno al otro de la primera muerte. Estamos en deuda, unidos por una deuda vital. Esas deudas tienen consecuencias.\n\n\u2014\u00bfUna deuda vital? \u2014Intent\u00e9 recordar; los restos de la morfina no me dejaban concentrarme\u2014. \u00bfCu\u00e1ndo te he salvado yo la vida?\n\n\u2014Tres veces. Me limpiaste las heridas, lo que me permiti\u00f3 ganar tiempo para pedir ayuda la primera noche. Me diste tu sangre, impidiendo que contrajera semimpulso. Y cuando Nashira te invit\u00f3 a su mesa, me protegiste. Si le hubieras dicho la verdad, me habr\u00edan ejecutado. He cometido muchos cr\u00edmenes de carne cuyo castigo es la muerte.\n\nNo sab\u00eda qu\u00e9 era un crimen de carne, y no lo pregunt\u00e9.\n\n\u2014Y ahora t\u00fa me has salvado la vida a m\u00ed.\n\n\u2014Te he salvado la vida varias veces.\n\n\u2014\u00bfCu\u00e1ndo?\n\n\u2014Prefiero no divulgar esa informaci\u00f3n. Pero cr\u00e9eme: me debes la vida m\u00e1s de tres veces. Eso significa que t\u00fa y yo ya no somos simplemente guardi\u00e1n y alumna, ni amo y esclava.\n\n\u2014\u00bfQu\u00e9? \u2014dije sacudiendo la cabeza.\n\nEl Custodio apoy\u00f3 un brazo en la repisa de la chimenea y se qued\u00f3 contemplando las llamas.\n\n\u2014El \u00e9ter nos ha marcado a los dos. Ha identificado nuestra tendencia a protegernos el uno al otro, y ahora estamos obligados a protegernos el uno al otro eternamente. Estamos unidos por un cord\u00f3n \u00e1ureo.\n\nMe habr\u00eda echado a re\u00edr de la solemnidad con que hablaba, pero me dio la impresi\u00f3n de que no estaba bromeando. Los refas no bromeaban.\n\n\u2014\u00bfUn cord\u00f3n \u00e1ureo?\n\n\u2014S\u00ed.\n\n\u2014\u00bfTiene algo que ver con el cord\u00f3n arg\u00e9nteo?\n\n\u2014Por supuesto. Se me hab\u00eda olvidado. Supongo que s\u00ed tiene alguna relaci\u00f3n... pero el cord\u00f3n arg\u00e9nteo es personal, \u00fanico de cada individuo, mientras que el cord\u00f3n \u00e1ureo lo forman dos esp\u00edritus.\n\n\u2014\u00bfQu\u00e9 demonios es?\n\n\u2014Ni yo lo s\u00e9 muy bien. \u2014Verti\u00f3 el oscuro contenido de un vial en su copa\u2014. Por lo que s\u00e9, el cord\u00f3n \u00e1ureo es una especie de s\u00e9ptimo sentido y se forma cuando dos esp\u00edritus se salvan el uno al otro de la primera muerte un m\u00ednimo de tres veces. \u2014Alz\u00f3 la copa y bebi\u00f3 un sorbo\u2014. A partir de ahora, t\u00fa y yo siempre sabremos del otro. Dondequiera que est\u00e9s, podr\u00e9 encontrarte a trav\u00e9s del \u00e9ter. \u2014Hizo una pausa y a\u00f1adi\u00f3\u2014: Siempre.\n\nSolo tard\u00e9 unos segundos en asimilar sus palabras.\n\n\u2014No \u2014dije\u2014. Es imposible. \u2014El Custodio dio otro sorbo de amaranto y dije, m\u00e1s alto\u2014: Demu\u00e9stralo. Demu\u00e9strame que existe ese \u00abcord\u00f3n \u00e1ureo\u00bb.\n\n\u2014Si insistes...\n\nEl Custodio dej\u00f3 la copa en la repisa de la chimenea.\n\n\u2014Imaginemos por un instante que estamos en Londres. Es de noche, y estamos en el puente. Pero esta vez es a m\u00ed a quien han disparado. Te pedir\u00e9 ayuda.\n\nEsper\u00e9.\n\n\u2014Esto no es m\u00e1s que... \u2014empec\u00e9 a decir, pero entonces not\u00e9 algo. Un d\u00e9bil zumbido en mis huesos, una m\u00ednima vibraci\u00f3n. Se me puso la piel de gallina. Dos palabras se materializaron en mi mente: \u00abpuente\u00bb, \u00abayuda\u00bb.\n\n\u2014Puente, ayuda \u2014dije en voz baja\u2014. No.\n\nNo pod\u00eda ser. Me volv\u00ed y mir\u00e9 hacia el fuego. Ahora el Custodio ten\u00eda su propia campanilla espiritual para llamarme. Al cabo de un momento mi conmoci\u00f3n se torn\u00f3 rabia. Me habr\u00eda gustado romperle todos los viales, pegarle pu\u00f1etazos en la cara; cualquier cosa menos compartir un v\u00ednculo con \u00e9l. Si el Custodio pod\u00eda rastrearme por el \u00e9ter, jam\u00e1s me librar\u00eda de \u00e9l.\n\nY yo ten\u00eda la culpa. Por salvarle la vida.\n\n\u2014Ignoro qu\u00e9 otros efectos puede tener sobre nosotros \u2014dijo el Custodio\u2014. Quiz\u00e1 puedas obtener poder de m\u00ed.\n\n\u2014No quiero tu poder. Deshazte del v\u00ednculo. R\u00f3mpelo.\n\n\u2014Para romper los v\u00ednculos del \u00e9ter hacen falta algo m\u00e1s que palabras.\n\n\u2014Has sabido llamarme con \u00e9l \u2014dije con voz tr\u00e9mula\u2014. Debes de saber c\u00f3mo romperlo.\n\n\u2014El cord\u00f3n es un enigma, Paige. No tengo ni idea.\n\n\u2014Lo has hecho a prop\u00f3sito. \u2014Me apart\u00e9 de \u00e9l, asqueada\u2014. Me has salvado la vida para crear ese cord\u00f3n, \u00bfverdad?\n\n\u2014\u00bfC\u00f3mo quieres que haya planeado una cosa as\u00ed, cuando no ten\u00eda forma de saber si t\u00fa me salvar\u00edas a m\u00ed la vida? Odias a los refa\u00edtas. \u00bfPor qu\u00e9 ibas a intentar salvar a uno?\n\nEra una buena pregunta.\n\n\u2014No puedes acusarme de ser una paranoica \u2014repliqu\u00e9.\n\nVolv\u00ed a recostarme en las almohadas y me sujet\u00e9 la cabeza con ambas manos. El Custodio vino a sentarse otra vez a mi lado, pero tuvo la sensatez de no tocarme.\n\n\u2014Paige, t\u00fa no me temes. S\u00e9 que me odias, pero no me tienes miedo. Y, sin embargo, le temes al cord\u00f3n.\n\n\u2014Eres un refa\u00edta.\n\n\u2014Y t\u00fa me juzgas por eso. Por ser el prometido de Nashira.\n\n\u2014Es sanguinaria y cruel. Y, aun as\u00ed, t\u00fa la escogiste.\n\n\u2014\u00bfEso crees?\n\n\u2014Bueno, como m\u00ednimo la aceptaste.\n\n\u2014Los Sargas escogen ellos mismos a sus parejas. Los dem\u00e1s no tenemos ese privilegio. \u2014Baj\u00f3 la voz hasta reducirla a un d\u00e9bil gru\u00f1ido\u2014. Si quieres que te diga la verdad, la detesto. Su sola presencia me produce n\u00e1useas.\n\nLo mir\u00e9 e intent\u00e9 interpretar la expresi\u00f3n de su cara. Ten\u00eda un semblante sombr\u00edo, como si estuviera arrepentido. Vio que lo observaba y mud\u00f3 la expresi\u00f3n.\n\n\u2014Entiendo \u2014dije.\n\n\u2014No, no lo entiendes. Nunca lo has entendido.\n\nMir\u00f3 hacia otro lado. Esper\u00e9. Como no se movi\u00f3, interrump\u00ed el silencio.\n\n\u2014Pues me gustar\u00eda entenderlo.\n\n\u2014No s\u00e9 si puedo confiar en ti. \u2014La luz de sus ojos se apag\u00f3\u2014. Creo que eres digna de confianza. Es evidente que eres leal a las personas a las que m\u00e1s quieres. Ser\u00eda lamentable compartir un cord\u00f3n \u00e1ureo con alguien en quien no puedo confiar, y que no conf\u00eda en m\u00ed.\n\nAs\u00ed que quer\u00eda confiar en m\u00ed. Y estaba pidi\u00e9ndome que confiara en \u00e9l; proponi\u00e9ndome un intercambio, una tregua. Pod\u00eda pedirle cualquier cosa, lo que quisiera, y \u00e9l me la conceder\u00eda.\n\n\u2014D\u00e9jame entrar en tu onirosaje \u2014dije.\n\nHe de reconocer que no mostr\u00f3 sorpresa.\n\n\u2014\u00bfQuieres ver mi onirosaje?\n\n\u2014No, no quiero solo verlo. Quiero entrar en \u00e9l. Si s\u00e9 qu\u00e9 hay en tu mente, quiz\u00e1 pueda confiar en ti.\n\nAdem\u00e1s, me intrigaba el onirosaje de los refa\u00edtas. Deb\u00eda de haber algo digno de verse detr\u00e1s de tanto blindaje.\n\n\u2014Eso requerir\u00eda un grado de confianza equivalente por mi parte. Tendr\u00eda que confiar en que no vas a vulnerar mi cordura.\n\n\u2014Exactamente.\n\nReflexion\u00f3 un momento.\n\n\u2014Est\u00e1 bien \u2014concedi\u00f3.\n\n\u2014\u00bfEn serio?\n\n\u2014Si te sientes con suficiente fuerza, s\u00ed. \u2014Se dio la vuelta y me mir\u00f3\u2014. \u00bfAfectar\u00e1 la morfina a tu don?\n\n\u2014No. \u2014Cambi\u00e9 de postura y me sent\u00e9\u2014. Tal vez te haga da\u00f1o.\n\n\u2014Me aguantar\u00e9.\n\n\u2014He matado a gente entrando en su onirosaje.\n\n\u2014Ya lo s\u00e9.\n\n\u2014Y \u00bfc\u00f3mo sabes que no voy a matarte?\n\n\u2014No lo s\u00e9. Tengo que correr el riesgo.\n\nAdopt\u00e9 un gesto inexpresivo. Aquella era mi oportunidad para destrozarlo, para aplastar su onirosaje como si aplastara una mosca contra una pared.\n\nSin embargo, sent\u00eda una gran curiosidad. En realidad nunca hab\u00eda visto otro onirosaje; solo hab\u00eda vislumbrado destellos, visiones fugaces a trav\u00e9s del \u00e9ter. Pero el jard\u00edn iridiscente de la mariposa... Quer\u00eda volver a ver eso. Quer\u00eda sumergirme. Y all\u00ed estaba el Custodio, ofreci\u00e9ndome su mente.\n\nSer\u00eda fascinante ver un onirosaje que hab\u00eda tenido miles de a\u00f1os para desarrollarse. Y tras la repentina confesi\u00f3n del Custodio acerca de Nashira, quer\u00eda saber m\u00e1s sobre su pasado. Quer\u00eda saber c\u00f3mo era Arcturus Mesarthim por dentro.\n\n\u2014De acuerdo \u2014dije.\n\nSe sent\u00f3 a mi lado. Su aura toc\u00f3 la m\u00eda y roz\u00f3 mi sexto sentido.\n\nLo mir\u00e9 a los ojos. Amarillos. A tan poca distancia pude ver que no ten\u00eda colobomas. Pero era imposible que careciera de visi\u00f3n.\n\n\u2014\u00bfCu\u00e1nto tiempo puedes quedarte? \u2014me pregunt\u00f3.\n\nLa pregunta me pill\u00f3 desprevenida.\n\n\u2014No mucho. A menos que tengas un AMBU a mano. \u2014Entrecerr\u00f3 los ojos\u2014. Es una especie de mascarilla de ox\u00edgeno. Me proporciona respiraci\u00f3n artificial cuando mi cuerpo deja de funcionar.\n\n\u2014Entiendo. Y si tienes ese aparato, \u00bfpuedes permanecer \u00aba la deriva\u00bb durante un per\u00edodo m\u00e1s prolongado?\n\n\u2014En teor\u00eda, s\u00ed. Nunca lo he probado en un onirosaje. Solo en el \u00e9ter.\n\n\u2014\u00bfPor qu\u00e9 te hacen hacer eso?\n\nLos dos sab\u00edamos perfectamente qui\u00e9nes eran \u00abellos\u00bb. El instinto me aconsejaba no decir nada, pero el Custodio ya sab\u00eda que trabajaba para Jaxon Hall.\n\n\u2014Porque el sindicato funciona as\u00ed \u2014contest\u00e9\u2014. Los capos esperan obtener una compensaci\u00f3n a cambio de la protecci\u00f3n que ofrecen.\n\nSu aura estaba cambiando.\n\n\u2014Ya entiendo. \u2014Estaba bajando sus defensas, abriendo sus puertas\u2014. Estoy preparado.\n\nMe incorpor\u00e9 un poco m\u00e1s, recost\u00e1ndome en las almohadas. Entonces cerr\u00e9 los ojos, inspir\u00e9 hondo y entr\u00e9 en mi onirosaje.\n\nEl prado de amapolas era un cuadro borroso. Todo se difuminaba, suavizado por la morfina que todav\u00eda circulaba por mi torrente sangu\u00edneo. Pas\u00e9 entre las flores, camino del \u00e9ter. Cuando llegu\u00e9 a la \u00faltima linde, empuj\u00e9 con las manos y vi que la ilusi\u00f3n de mi cuerpo se esfumaba ante mis ojos. En tu onirosaje, solo te pareces a ti mismo si tu mente te percibe as\u00ed. En cuanto desaparec\u00ed, adopt\u00e9 mi forma de esp\u00edritu. Fluida, amorfa. Un resplandor tenue y an\u00f3nimo.\n\nHab\u00eda visto el onirosaje del Custodio desde fuera en otra ocasi\u00f3n, pero aun as\u00ed me estremeci\u00f3. Parec\u00eda una canica negra, apenas perceptible en la oscuridad silenciosa del \u00e9ter. Cuando me acerqu\u00e9 a \u00e9l, una ondulaci\u00f3n recorri\u00f3 su superficie. El Custodio estaba retirando todas esas capas de blindaje que hab\u00eda ido acumulando a lo largo de siglos. Me deslic\u00e9 m\u00e1s all\u00e1 de los muros, hasta su zona hadal. Hab\u00eda llegado a ese punto durante nuestras sesiones de entrenamiento, pero solo en arranques irregulares. Ahora pod\u00eda ir m\u00e1s all\u00e1. Avanc\u00e9 por la oscuridad hacia el centro de su mente.\n\nMe pasaron cenizas junto a la cara. Al adentrarme en territorio desconocido, se me puso la imaginaria piel de gallina. En la mente del Custodio reinaba un silencio absoluto. Generalmente, los anillos externos estaban llenos de espejismos, alucinaciones correspondientes a los miedos o las congojas de la persona, pero all\u00ed no hab\u00eda nada. Solo silencio.\n\nEl Custodio me esperaba en su zona soleada, si es que podemos llamarla as\u00ed, porque parec\u00eda, m\u00e1s bien, iluminada por la luna. Estaba cubierto de cicatrices, y su piel no ten\u00eda ni pizca de color. As\u00ed era como \u00e9l se ve\u00eda a s\u00ed mismo. Me pregunt\u00e9 qu\u00e9 aspecto tendr\u00eda yo. Ahora estaba en su onirosaje, y jugaba con sus reglas. Vi que mis manos estaban igual que siempre, aunque ten\u00edan un d\u00e9bil resplandor. Mi nueva oniroforma. Pero \u00bfve\u00eda \u00e9l mi verdadera cara? Yo pod\u00eda tener cualquier expresi\u00f3n: sumisa, demente, ingenua, cruel... No ten\u00eda ni idea de qu\u00e9 pensaba \u00e9l de m\u00ed, y nunca lo averiguar\u00eda. En los onirosajes no hab\u00eda espejos. Yo nunca ver\u00eda a la Paige que \u00e9l hab\u00eda creado.\n\nLlegu\u00e9 a una \u00e1rida extensi\u00f3n de arena. No sabr\u00eda decir qu\u00e9 esperaba encontrar, pero desde luego no era aquello. El Custodio inclin\u00f3 la cabeza.\n\n\u2014Bienvenida a mi onirosaje. Perd\u00f3name por la escasa decoraci\u00f3n \u2014dijo, pase\u00e1ndose sin rumbo fijo\u2014. No suelo recibir invitados.\n\n\u2014No hay nada. \u2014Echaba vaho por la boca al respirar\u2014. Absolutamente nada \u2014dije sin exagerar.\n\n\u2014En nuestro onirosaje es donde nos sentimos m\u00e1s seguros \u2014dijo el Custodio\u2014. Quiz\u00e1 yo me sienta m\u00e1s seguro cuando no pienso en nada.\n\n\u2014Pero es que tampoco hay nada en las partes oscuras.\n\nNo contest\u00f3 nada. Me adentr\u00e9 un poco m\u00e1s en la niebla.\n\n\u2014No veo nada. Eso me sugiere que dentro de ti no hay nada. Ni pensamientos, ni conciencia. Ni miedo. \u2014Me volv\u00ed y lo mir\u00e9\u2014. \u00bfTodos los refa\u00edtas tienen el onirosaje vac\u00edo?\n\n\u2014Yo no soy onir\u00e1mbulo, Paige. Solo puedo imaginar c\u00f3mo deben de ser los onirosajes de los otros.\n\n\u2014\u00bfQu\u00e9 eres?\n\n\u2014Puedo hacer que los dem\u00e1s sue\u00f1en sus recuerdos. Puedo entretejerlos, crear una falsa ilusi\u00f3n. Veo el \u00e9ter a trav\u00e9s de la lente del onirosaje, y a trav\u00e9s de la hierba hipn\u00f3tica.\n\n\u2014Onirom\u00e1ntico. \u2014No pod\u00eda dejar de mirarlo a los ojos\u2014. Eres un traficante de sue\u00f1o.\n\nJax estaba convencido de que ten\u00edan que existir. Hab\u00eda catalogado a los onirom\u00e1nticos a\u00f1os atr\u00e1s, mucho antes de escribir _Sobre los m\u00e9ritos_ , pero nunca hab\u00eda encontrado a ninguno que demostrara su teor\u00eda: un tipo de vidente que pod\u00eda atravesar el onirosaje, recoger recuerdos y componer con ellos lo que los amaur\u00f3ticos llamaban \u00absue\u00f1os\u00bb.\n\n\u2014Me hac\u00edas so\u00f1ar. \u2014Inspir\u00e9 hondo\u2014. He estado rememorando cosas desde que llegu\u00e9 aqu\u00ed. C\u00f3mo me volv\u00ed onir\u00e1mbula, c\u00f3mo me encontr\u00f3 Jaxon. Eras t\u00fa. T\u00fa me hac\u00edas so\u00f1ar todo eso. Por eso lo sab\u00edas, \u00bfno es as\u00ed?\n\nMe sostuvo la mirada.\n\n\u2014Para eso era la tercera pastilla \u2014dijo\u2014. Conten\u00eda una hierba llamada salvia, que te hac\u00eda so\u00f1ar tus recuerdos. Es la hierba que me ayuda a tocar el \u00e9ter, mi _numen_. Flu\u00eda por tus venas. Despu\u00e9s de varias pastillas, pod\u00eda acceder a tus recuerdos a mi antojo.\n\n\u2014Me has... drogado \u2014dije con dificultad\u2014 para entrar en mi mente.\n\n\u2014S\u00ed. Mientras t\u00fa espiabas onirosajes para Jaxon Hall.\n\n\u2014No es lo mismo. Yo no me quedaba sentada junto a la chimenea observando recuerdos como si viera... una especie de pel\u00edcula. \u2014Me apart\u00e9 poco a poco de \u00e9l\u2014. Esos recuerdos son m\u00edos. Son privados. Hasta miraste... \u00a1debes de haberlo visto todo! Hasta lo que sent\u00eda por... por...\n\n\u2014Nick. Estabas enamorada de \u00e9l.\n\n\u2014C\u00e1llate. Cierra tu sucia boca.\n\nSe call\u00f3.\n\nMi oniroforma se estaba derrumbando. Antes de que pudiera salir de all\u00ed por mis propios medios, me vi arrojada de su onirosaje, como una hoja arrastrada por el viento. Cuando despert\u00e9 en mi cuerpo, le puse las manos en el pecho y lo empuj\u00e9.\n\n\u2014No te me acerques.\n\nMe dol\u00eda la cabeza. No pod\u00eda mirar al Custodio, y mucho menos estar cerca de \u00e9l. Al intentar levantarme, not\u00e9 el tir\u00f3n de la v\u00eda en la mano.\n\n\u2014Lo siento \u2014dijo.\n\nMe ard\u00edan las mejillas. Le hab\u00eda ofrecido una pizca de confianza, solo una pizca, y \u00e9l se hab\u00eda aprovechado de m\u00ed. Me hab\u00eda quitado siete a\u00f1os de memoria. Me hab\u00eda quitado a Finn. Y a Nick.\n\nSe qued\u00f3 un minuto all\u00ed. Quiz\u00e1 esperara que yo dijera algo m\u00e1s. Yo me mor\u00eda de ganas de gritarle hasta quedarme sin voz, pero no pod\u00eda hacerlo. Quer\u00eda que se marchara. Como no me mov\u00eda, corri\u00f3 las gruesas cortinas alrededor de la cama, encerr\u00e1ndome en una jaula oscura.\n\nTard\u00e9 horas en dormirme. Lo o\u00eda sentado a su mesa, escribiendo; lo \u00fanico que nos separaba eran las cortinas.\n\nMe escoc\u00edan los ojos y la nariz, y ten\u00eda la garganta dolorida. Por primera vez desde hac\u00eda muchos a\u00f1os deseaba que desapareciera todo. Quer\u00eda que todo volviera a ser normal, como cuando era peque\u00f1a, antes de que el \u00e9ter me abriera de un rasg\u00f3n.\n\nMir\u00e9 el dosel. Por mucho que a veces lo deseara, no hab\u00eda normalidad posible. Nunca la hab\u00eda habido. Los conceptos \u00abnormal\u00bb y \u00abnatural\u00bb eran las mayores mentiras que hab\u00edamos creado. Los humanos y sus peque\u00f1as mentes. Adem\u00e1s, quiz\u00e1 la normalidad no fuera conmigo.\n\nCuando el Custodio puso en marcha el gram\u00f3fono, empez\u00f3 a entrarme sue\u00f1o. No hab\u00eda pasado mucho tiempo dentro de su onirosaje, pero lo hab\u00eda hecho sin soporte vital. Me qued\u00e9 adormilada. Unas voces chisporroteantes se mezclaban entre ellas.\n\nDeb\u00ed de quedarme dormida. Cuando despert\u00e9, ya no ten\u00eda la v\u00eda en la mano. En su lugar hab\u00eda una tirita.\n\nSon\u00f3 la campanada diurna. Sheol I dorm\u00eda durante el d\u00eda, pero por lo visto yo no iba a poder dormir. Lo \u00fanico que pod\u00eda hacer era levantarme y enfrentarme a \u00e9l.\n\nLo odiaba con toda mi alma. Me daban ganas de romper el espejo, notar c\u00f3mo se resquebrajaba el cristal bajo mis nudillos. No deb\u00ed tomarme aquellas pastillas.\n\nQuiz\u00e1 s\u00ed fuera lo mismo que hac\u00eda yo. Yo tambi\u00e9n espiaba a otras personas; pero no husmeaba en su pasado. Yo solo ve\u00eda lo que ellas cre\u00edan ser, pero no lo que eran. Ve\u00eda destellos de personas: los bordes y las esquinas, el d\u00e9bil resplandor de onirosajes lejanos. No como \u00e9l. Ahora lo sab\u00eda todo de m\u00ed, todo lo que yo hab\u00eda intentado ocultar. El Custodio siempre hab\u00eda sabido que yo pertenec\u00eda a la banda de los Siete Sellos. Lo sab\u00eda desde la primera noche.\n\nPero no se lo hab\u00eda dicho a Nashira. No le hab\u00eda contado lo de la mariposa y la cierva, ni le hab\u00eda revelado mi verdadera identidad. Nashira deb\u00eda de haber descubierto que yo formaba parte del sindicato, pero no hab\u00eda sido \u00e9l quien se lo hab\u00eda dicho.\n\nDescorr\u00ed las cortinas. La luz dorada del sol entraba a raudales en la torre y hac\u00eda brillar los instrumentos y los libros. Cerca de la ventana, Michael, el chico amaur\u00f3tico, serv\u00eda un desayuno en una mesita. Me mir\u00f3 y sonri\u00f3.\n\n\u2014Hola, Michael.\n\n\u00c9l me saludo con la cabeza.\n\n\u2014\u00bfD\u00f3nde est\u00e1 el Custodio?\n\nSe\u00f1al\u00f3 la puerta.\n\n\u2014\u00bfTe ha comido la lengua el gato?\n\nSe encogi\u00f3 de hombros. Me sent\u00e9. Michael me acerc\u00f3 un plato con un mont\u00f3n de tortitas.\n\n\u2014No tengo hambre \u2014dije\u2014. No pienso aceptar su comida. Me la ofrece porque se siente culpable.\n\nMichael suspir\u00f3, me puso un tenedor en la mano y pinch\u00f3 con \u00e9l una tortita.\n\n\u2014Est\u00e1 bien, pero si lo vomito todo, t\u00fa tendr\u00e1s la culpa.\n\nMichael hizo una mueca de asco. Solo para complacerlo, espolvore\u00e9 las tortitas con az\u00facar moreno.\n\nMichael no dejaba de observarme mientras iba de aqu\u00ed para all\u00e1 por la habitaci\u00f3n, haciendo la cama y colocando bien las cortinas. Las tortitas me abrieron el apetito. Acab\u00e9 comi\u00e9ndome todo el mont\u00f3n, junto con dos cruasanes con mermelada de fresa, un cuenco de cereales, cuatro tostadas con mantequilla, un plato de huevos revueltos, una manzana roja con la carne blanca y crujiente, tres tazas de caf\u00e9 y medio litro de zumo de naranja helado. Cuando ya no pude comer m\u00e1s, Michael me entreg\u00f3 un sobre de papel Manila sellado.\n\n\u2014Conf\u00eda en \u00e9l.\n\nEra la primera vez que le o\u00eda hablar. Su voz no era m\u00e1s que un susurro.\n\n\u2014T\u00fa \u00bfconf\u00edas en \u00e9l?\n\nAsinti\u00f3 con la cabeza, recogi\u00f3 el desayuno y se march\u00f3. Y aunque era de d\u00eda, no cerr\u00f3 la puerta con llave. Romp\u00ed el sello de cera del sobre y desdobl\u00e9 la hoja de papel grueso que hab\u00eda dentro. Ten\u00eda una cenefa de oro. El mensaje dec\u00eda lo siguiente:\n\nPaige:\n\nPerd\u00f3name por ofenderte. Pero, aunque est\u00e9s enojada conmigo, debes saber que solo pretend\u00eda entenderte. No puedes culparme de tu negativa a ser entendida.\n\nMenuda disculpa. Pese a todo, segu\u00ed leyendo:\n\nTodav\u00eda es de d\u00eda. Ve a la Casa. All\u00ed encontrar\u00e1s cosas que yo no puedo proporcionarte.\n\nNo te entretengas. Si te preguntan, di a los guardias que vas a recoger un lote de \u00e1ster para m\u00ed.\n\nNo te precipites al juzgar, peque\u00f1a So\u00f1adora.\n\nArrugu\u00e9 la hoja y la lanc\u00e9 a la chimenea. Por el mero hecho de escribirla, el Custodio estaba alardeando de su nueva confianza en m\u00ed. Yo pod\u00eda llev\u00e1rsela a Nashira. Ella reconocer\u00eda su caligraf\u00eda, sin duda. Pero yo no quer\u00eda ayudar a Nashira de ninguna manera. Odiaba al Custodio por retenerme en aquel sitio, pero necesitaba entrar en la Casa.\n\nSub\u00ed al piso de arriba y me puse el uniforme nuevo: el blus\u00f3n amarillo y el chaleco gris con el ancla amarilla. Un amarillo intenso, visible desde lejos. 40 la cobarde. 40 la rajada. Por una parte me gustaba, pues significaba que hab\u00eda desobedecido las \u00f3rdenes de Nashira. Yo nunca hab\u00eda aspirado a vestir el blus\u00f3n rojo.\n\nDespacio, pensativa, volv\u00ed a la c\u00e1mara del Custodio. Segu\u00eda sin saber si quer\u00eda organizar una fuga masiva, aunque yo s\u00ed que quer\u00eda marcharme. Iba a necesitar provisiones para el viaje de regreso. Comida, agua. Armas. \u00bfNo hab\u00eda dicho el Custodio que la flor roja pod\u00eda herirlos?\n\nLa caja de rap\u00e9 estaba encima de la mesa, con la tapa abierta. Dentro hab\u00eda muestras de varias plantas: ramitos de laurel, hojas de sic\u00f3moro y roble, bayas de mu\u00e9rdago, \u00e1ster azul y blanco; y un paquete de hojas secas con la etiqueta \u00abSalvia Divinorum\u00bb. Su _numen_. Debajo, en un rinc\u00f3n de la caja, un vial lleno de un fino polvo negro azulado. La etiqueta dec\u00eda \u00abanemone Coronaria\u00bb. Quit\u00e9 el corcho y percib\u00ed un olor penetrante. El polen de la flor roja. Aquellos granitos quiz\u00e1 me mantuvieran con vida. Cerr\u00e9 el vial y me lo guard\u00e9 en el chaleco.\n\nDeb\u00eda de haber guardias apostados fuera durante el d\u00eda, pero pod\u00eda burlarlos. Sab\u00eda c\u00f3mo hacerlo. Y yo no era una vulgar casaca amarilla, por mucho que Nashira Sargas me hubiera clasificado como tal. Yo era la So\u00f1adora P\u00e1lida.\n\nYa era hora de demostr\u00e1rselo.\n\nMe invent\u00e9 el cuento de que iba a buscar \u00e1ster para mi guardi\u00e1n, y le dije al nuevo portero de d\u00eda que lo consultara con \u00e9l si ve\u00eda alg\u00fan inconveniente. No le entusiasm\u00f3 la idea: estuvo a punto de echarme a la calle de una patada cuando ley\u00f3 en el libro de registro qui\u00e9n era mi guardi\u00e1n. Ni siquiera mencion\u00f3 la mochila que llevaba a la espalda. Nadie quer\u00eda hacer enfadar a Arcturus Mesarthim.\n\nResultaba extra\u00f1o ver la ciudad a plena luz del d\u00eda. Me imaginaba que el Broad estar\u00eda vac\u00edo (no hab\u00eda ni rastro de los sonidos ni los olores de siempre), pero necesitaba hacer una cosa antes de llegar a la Casa. Recorr\u00ed los callejones del Poblado. Goteaba agua por todas las rendijas y las grietas, porque hab\u00eda pasado una tormenta. Encontr\u00e9 la choza que buscaba y apart\u00e9 la gastada cortina. Dentro estaba Julian: dorm\u00eda abrazado a Liss para mantenerla caliente. El aura de Liss se estaba apagando, como una vela a la que ya no le queda mecha. Me agach\u00e9 a su lado y vaci\u00e9 mi mochila. Escond\u00ed un paquete de comida en el pliegue del codo del brazo que Julian ten\u00eda libre, donde no podr\u00eda verlo ning\u00fan guardia que pasara por all\u00ed, y los tap\u00e9 a ambos con mantas blancas limpias. Dej\u00e9 una caja de cerillas en el arc\u00f3n.\n\nVer la miseria en que viv\u00edan me convenci\u00f3 de que estaba haciendo lo correcto. Ellos necesitaban mucho m\u00e1s de lo que yo hab\u00eda robado de la Torre del Fundador. Necesitaban lo que hab\u00eda en la Casa.\n\nEl choque espiritual era un proceso lento. Para superarlo, ten\u00edas que luchar con todas tus armas. Solo sobreviv\u00edan los m\u00e1s fuertes. Liss no hab\u00eda recobrado el conocimiento desde que hab\u00edan destruido sus cartas, exceptuando unos breves momentos de lucidez. Si no se recuperaba pronto, perder\u00eda su aura y sucumbir\u00eda a la amaurosis. Su \u00fanica esperanza era encontrar otra baraja, aunque no hab\u00eda ninguna garant\u00eda de que conectara con ella. Yo estaba decidida a registrar la Casa hasta que encontrara unas cartas.\n\nEn la calle no vi a ning\u00fan guardia, pero yo sab\u00eda que deb\u00eda de haber vig\u00edas. Por si acaso, trep\u00e9 a un edificio y busqu\u00e9 un camino por los tejados, usando cornisas y pilastras para recorrer la ciudad. Vigilaba mucho d\u00f3nde pon\u00eda los pies, pero iba muy lenta: ten\u00eda el brazo derecho completamente r\u00edgido, y el resto del cuerpo todav\u00eda muy dolorido y cubierto de cardenales.\n\nLa Casa se ve\u00eda desde casi dos kil\u00f3metros. Sus dos torres se alzaban en medio de la neblina. Cuando ya estaba cerca, baj\u00e9 a un callej\u00f3n; la distancia hasta la siguiente pared era demasiado grande para saltar. Detr\u00e1s de un muro se erig\u00eda la \u00fanica residencia donde solo pod\u00edan entrar los refa\u00edtas.\n\nMe qued\u00e9 largo rato observando el edificio. El Custodio ya estaba demasiado involucrado para traicionarme. Me estaba ayudando, aunque yo desconociera sus motivos; y yo ten\u00eda que aceptar esa ayuda, aunque solo fuera para socorrer a Liss. Ten\u00eda que aceptarla. Adem\u00e1s, si me met\u00eda en alg\u00fan l\u00edo, siempre podr\u00eda enviarle un mensaje a trav\u00e9s del cord\u00f3n \u00e1ureo. Suponiendo que supiera c\u00f3mo hacerlo. Suponiendo que pudiera soportarlo. Trep\u00e9 al muro, pas\u00e9 una pierna por encima y me dej\u00e9 caer sobre la hierba alta.\n\nComo muchas residencias, aquel complejo estaba construido alrededor de una serie de patios interiores. Mientras cruzaba el primero, fui redactando mentalmente una lista de cosas que necesitar\u00eda para atravesar la Tierra de Nadie. Las armas eran indispensables, dado lo que se ocultaba en el bosque, pero tambi\u00e9n necesitar\u00eda medicinas. Si pon\u00eda el pie donde no deb\u00eda en el campo de minas, necesitar\u00eda un torniquete. Y antis\u00e9pticos. Era un pensamiento estremecedor, pero ten\u00eda que afrontarlo. La adrenalina era muy valiosa: pod\u00eda utilizarla para aumentar mi energ\u00eda y calmar el dolor, pero tambi\u00e9n para resucitarme en caso de que tuviera que abandonar mi cuerpo. Me vendr\u00eda bien asimismo m\u00e1s polen de an\u00e9mona, y cualquier otra sustancia que encontrara: flux, \u00e1ster, sal... Quiz\u00e1 incluso ectoplasma.\n\nPas\u00e9 por delante de varios edificios, pero ninguno se prestaba a que lo registrara: todos ten\u00edan demasiadas habitaciones. Cuando me alej\u00e9 de los patios centrales y fui hacia un extremo de la residencia, encontr\u00e9 un objetivo mejor: un edificio con ventanas grandes y muchos puntos de apoyo para escalar. Pas\u00e9 por debajo de un arco y lo observ\u00e9 desde el otro lado. La fachada estaba recubierta de hiedra roja. Rode\u00e9 el edificio con la esperanza de encontrar una ventana abierta. No vi ninguna. Tendr\u00eda que entrar por otros medios. \u00a1Un momento! S\u00ed, hab\u00eda una: una ventana peque\u00f1a, ligeramente entreabierta, en el primer piso. Trep\u00e9 a un muro bajo, y desde all\u00ed agarr\u00e9 una bajante. La ventana estaba atrancada, pero consegu\u00ed abrirla utilizando un solo brazo. Me met\u00ed en una habitaci\u00f3n min\u00fascula llena de polvo, seguramente un escobero. Abr\u00ed un poco la puerta.\n\nMe encontr\u00e9 en un pasillo de piedra. Vac\u00edo. Mi excursi\u00f3n a la Casa no habr\u00eda podido salir mejor. Examin\u00e9 las puertas en busca de algo que indicara qu\u00e9 pod\u00eda haber detr\u00e1s, y me puse en tensi\u00f3n. Mi sexto sentido se estremeci\u00f3: detect\u00e9 dos auras. Estaban detr\u00e1s de una puerta que ten\u00eda a mi derecha. Me qued\u00e9 quieta.\n\n\u2014\u00a1... s\u00e9 nada! \u00a1Por favor...!\n\nO\u00ed unos ruidos amortiguados. Pegu\u00e9 la oreja contra la puerta.\n\n\u2014La soberana de sangre no escuchar\u00e1 tus s\u00faplicas \u2014dijo una voz de hombre\u2014. Sabemos que los viste juntos.\n\n\u2014\u00a1Los vi una vez, en la pradera! \u00a1Solo una vez! Solo entrenaban. \u00a1No vi nada m\u00e1s, lo juro! \u2014Era una voz aguda, con un deje de p\u00e1nico. La reconoc\u00ed: era Ivy, la palmista. Casi se atragantaba con las palabras\u2014. \u00a1Por favor, otra vez no, no lo soporto!\n\nUn grito espeluznante.\n\n\u2014Cuando nos digas la verdad no habr\u00e1 m\u00e1s dolor. \u2014Ivy sollozaba\u2014. Venga, 24. Seguro que tienes algo para m\u00ed. Solo te pido un poco de informaci\u00f3n. \u00bfLa toc\u00f3?\n\n\u2014\u00c9l... se la llev\u00f3 de la pradera. Estaba cansada. Pero \u00e9l llevaba guantes...\n\n\u2014\u00bfEst\u00e1s segura?\n\nIvy respiraba entrecortadamente.\n\n\u2014No me acuerdo. Lo siento. Por favor, no m\u00e1s...\n\nPasos.\n\n\u2014\u00a1No! \u00a1No!\n\nSus gritos lastimeros me retorcieron el est\u00f3mago. Me habr\u00eda gustado atacar al esp\u00edritu de su torturador, pero el riesgo de que me descubrieran era demasiado grande. Si no consegu\u00eda esas provisiones, no podr\u00eda salvar a nadie. Apret\u00e9 las mand\u00edbulas y segu\u00ed escuchando, temblando de rabia. \u00bfQu\u00e9 le estar\u00edan haciendo a Ivy?\n\nIvy daba unos gritos estremecedores. Cuando par\u00f3, me dio un vuelco el coraz\u00f3n.\n\n\u2014Basta, por favor \u2014dijo entrecortadamente\u2014. \u00a1Es la verdad! \u2014Su torturador permaneci\u00f3 callado\u2014. Pero... le da de comer. S\u00e9 que le da de comer, y ella... ella va limpia. Y... dicen que puede poseer a otros videntes, y \u00e9l debe de... debe de estar ocult\u00e1ndoselo a la soberana de sangre. Si no, ella ya estar\u00eda muerta.\n\nSe produjo un silencio sospechoso. Despu\u00e9s se oy\u00f3 un golpazo, y luego pasos y una puerta que se cerraba.\n\nMe qued\u00e9 paralizada largo rato. Al cabo de un minuto empuj\u00e9 la puerta. Dentro solo hab\u00eda una silla de madera. El asiento estaba manchado de sangre, igual que el suelo.\n\nNot\u00e9 un sudor fr\u00edo. Me pas\u00e9 la manga por el labio. Me agach\u00e9 junto a la pared, con la cabeza entre las manos. Era de m\u00ed de quien estaba hablando Ivy.\n\nNo pod\u00eda entretenerme pensando en eso. Su torturador quiz\u00e1 siguiera en el edificio. Me levant\u00e9 poco a poco y me dirig\u00ed a la siguiente habitaci\u00f3n. La llave estaba puesta en la cerradura de la puerta. Me asom\u00e9 al interior y vi monta\u00f1as de armas: espadas, pu\u00f1ales, una ballesta, una honda con munici\u00f3n de acero. All\u00ed deb\u00eda de ser donde almacenaban las armas que repart\u00edan a los casacas rojas. Agarr\u00e9 un pu\u00f1al. Cerca del mango brillaba un ancla: estaba fabricado en Scion. Weaver enviaba armas a Sheol I mientras sus ministros y \u00e9l se sentaban en el Arconte, lejos de la baliza et\u00e9rea.\n\nJulian ten\u00eda raz\u00f3n. No pod\u00eda marcharme sin m\u00e1s. Quer\u00eda asustar a Frank Weaver. Quer\u00eda que experimentara el miedo de cada uno de los prisioneros videntes a los que hab\u00eda trasladado all\u00ed.\n\nCerr\u00e9 la puerta con llave. Cuando levant\u00e9 la cabeza me encontr\u00e9 ante un gran mapa amarillento. COLONIA PENITENCIARIA DE SHEOL I, dec\u00eda. TERRITORIO OFICIAL DE LA SUZERAN\u00cdA. Lo examin\u00e9. Sheol I estaba construida alrededor de las grandes residencias centrales, y se alargaba hacia la pradera y el bosque. Encontr\u00e9 todos los lugares destacados: Magdalen, la Casa Amaur\u00f3tica, la Residencia del Suzerano, Hawksmoor... y Puerto Pradera. Descolgu\u00e9 el mapa de la pared. Las letras impresas al lado estaban borrosas, pero consegu\u00ed descifrarlas:\n\n\u00abTren\u00bb.\n\nFlexion\u00e9 los dedos y estruj\u00e9 los bordes del mapa. El tren. Ni siquiera se me hab\u00eda ocurrido pensarlo. Si nos hab\u00edan llevado hasta all\u00ed en tren, \u00bfpor qu\u00e9 no \u00edbamos a poder salir en otro tren?\n\nTen\u00eda el cerebro revolucionado. \u00bfC\u00f3mo no se me hab\u00eda ocurrido antes? No necesitaba cruzar la Tierra de Nadie. No necesitaba recorrer kil\u00f3metros a pie, ni pasar por donde estaban los emim, para llegar a la ciudadela. Lo \u00fanico que ten\u00eda que hacer era encontrar el tren. Pod\u00eda llevar a otros conmigo: a Liss, a Julian, a todos. En los trenes de Scion hab\u00eda espacio para casi cuatrocientas personas, para m\u00e1s si iban de pie. Pod\u00eda sacar a todos los prisioneros de aquella ciudad, y todav\u00eda sobrar\u00eda sitio.\n\nAun as\u00ed, \u00edbamos a necesitar armas. Aunque nos escabull\u00e9ramos durante el d\u00eda y fu\u00e9ramos a la pradera en peque\u00f1os grupos, los refa\u00edtas nos perseguir\u00edan. Adem\u00e1s, la entrada deb\u00eda de estar vigilada. Cog\u00ed un pu\u00f1al y me lo guard\u00e9 en la mochila. Luego vi unas cuantas pistolas.\n\nLa minipistola, un modelo parecido al m\u00edo, pod\u00eda resultar \u00fatil: era peque\u00f1a, f\u00e1cil de esconder, y sab\u00eda c\u00f3mo usarla. Apart\u00e9 unos papeles ilegibles de la parte de arriba de una caja met\u00e1lica. En la ciudadela, Nick hab\u00eda intentado dispararle al Custodio, pero sin \u00e9xito. Las balas servir\u00edan para atacar a los casacas rojas leales, pero necesitar\u00edamos algo m\u00e1s que pistolas para disparar contra los refas. Me dispon\u00eda a coger una caja de balas cuando o\u00ed unos pasos.\n\nMe escond\u00ed r\u00e1pidamente detr\u00e1s de una estanter\u00eda. Justo a tiempo: la llave se cay\u00f3 de la cerradura, y entraron dos refas. Deber\u00eda haberlo imaginado. Ten\u00eda la salida bloqueada. Si me arrastraba hasta la ventana, tendr\u00eda que exponerme, y todos conoc\u00edan mi cara. Mir\u00e9 entre los estantes.\n\nThuban.\n\nDijo algo en _gloss_. Me acerqu\u00e9 un poco m\u00e1s a mi mirilla, tratando de identificar a su acompa\u00f1ante. Y entonces Terebell Sheratan entr\u00f3 en mi campo de visi\u00f3n.\n\nNos quedamos quietas las dos, y me pareci\u00f3 que mi coraz\u00f3n dejaba de latir. Esper\u00e9 a que llamara a Thuban, o a que me clavara una navaja en el vientre. Mis dedos buscaron el polen que llevaba escondido en el chaleco, pero me lo pens\u00e9 mejor. Aunque consiguiera matar a Terebell, Thuban me destripar\u00eda.\n\nPero Terebell me sorprendi\u00f3. En lugar de reaccionar a mi presencia, desvi\u00f3 la mirada hacia las pistolas.\n\n\u2014Las armas amaur\u00f3ticas son intrigantes \u2014dijo\u2014. No me extra\u00f1a que se maten entre ellos tan a menudo.\n\n\u2014\u00bfPor qu\u00e9 hablas en la lengua ca\u00edda?\n\n\u2014Gomeisa dice que debemos mantener el dominio del ingl\u00e9s. No me parece mal practicar un poco.\n\nThuban descolg\u00f3 la ballesta de la pared.\n\n\u2014Si quieres que nos ensuciemos la boca con ella, adelante. Podemos rendir homenaje a los d\u00edas en que ten\u00edas poder sobre m\u00ed. \u00a1Cu\u00e1nto tiempo ha pasado! \u2014Desliz\u00f3 los dedos enguantados por encima de la madera torneada\u2014. La So\u00f1adora deber\u00eda haber matado a Jaxon Hall cuando se le present\u00f3 la oportunidad. Habr\u00eda tenido mejor muerte que la que tendr\u00e1 ahora.\n\nSe me cerr\u00f3 la garganta.\n\n\u2014Dudo que lo maten \u2014dijo Terebell\u2014. Adem\u00e1s, la que le interesa a Nashira es Carter.\n\n\u2014Pues tendr\u00e1 que contener a Situla.\n\n\u2014Sin duda. \u2014Pas\u00f3 los dedos por la hoja de una espada\u2014. Refr\u00e9scame la memoria: \u00bfqu\u00e9 se guardaba en esta habitaci\u00f3n antes de que la convirtieran en armer\u00eda?\n\n\u2014Con tu inter\u00e9s blasfemo por el mundo ca\u00eddo, cre\u00eda que sabr\u00edas exactamente d\u00f3nde se guardan todos los recursos.\n\n\u2014Creo que \u00abblasfemo\u00bb es un poco melodram\u00e1tico.\n\n\u2014Pues yo no. \u2014Cogi\u00f3 un pu\u00f1ado de estrellas ninja\u2014. \u00bfQu\u00e9 hab\u00eda antes aqu\u00ed, me preguntas? Suministros m\u00e9dicos. Extractos de plantas. Salvia, \u00e1ster. Otras hojas apestosas.\n\n\u2014\u00bfAd\u00f3nde llevaron todo eso?\n\n\u2014\u00bfQu\u00e9 te pasa, has perdido la memoria de golpe, bellaca? Eres tan est\u00fapida como la concubina.\n\nHab\u00eda que reconocer que, o bien Terebell era inmune a la actitud de Thuban, o disimulaba muy bien sus emociones. Si es que las ten\u00eda.\n\n\u2014Perdona mi curiosidad \u2014dijo.\n\n\u2014En mi familia no perdonamos. Las cicatrices que tienes en la espalda deber\u00edan record\u00e1rtelo todos los d\u00edas. \u2014Ten\u00eda los ojos impregnados del aura de Ivy\u2014. Por eso quieres saberlo. Intentas robar amaranto, \u00bfverdad, Sheratan?\n\n\u00abCicatrices.\u00bb __\n\nTerebell no mud\u00f3 la expresi\u00f3n.\n\n\u2014\u00bfAd\u00f3nde llevaron esos suministros?\n\n\u2014No me gusta ese inter\u00e9s tuyo. Me resulta sospechoso. \u00bfVuelves a conspirar con la concubina?\n\n\u2014De eso hace casi veinte a\u00f1os, Thuban. Es mucho tiempo para los humanos, \u00bfno te parece?\n\n\u2014No me importa c\u00f3mo midan el tiempo los humanos.\n\n\u2014Una cosa es que me guardes rencor por el pasado, pero no creo que a la soberana de sangre le guste tu actitud hacia su consorte. Ni tus cuestionables descripciones del papel que representa.\n\nSu voz se hab\u00eda endurecido. Thuban descolg\u00f3 una espada de la pared y apunt\u00f3 con ella a Terebell. Se detuvo a un cent\u00edmetro de su cuello. Terebell no se inmut\u00f3.\n\n\u2014Una palabra m\u00e1s \u2014susurr\u00f3 Thuban\u2014 y lo llamo. Y esta vez no se moderar\u00e1 tanto.\n\nTerebell guard\u00f3 silencio un momento. Me pareci\u00f3 detectar algo en su cara: dolor, miedo. Deb\u00edan de estar hablando de un Sargas. Gomeisa, quiz\u00e1.\n\n\u2014S\u00ed. Creo que ya recuerdo d\u00f3nde est\u00e1n esos suministros \u2014dijo en voz baja a continuaci\u00f3n\u2014. \u00bfC\u00f3mo habr\u00e9 podido olvidarme de la Torre de Tom?\n\nThuban solt\u00f3 una carcajada. Absorb\u00ed aquella informaci\u00f3n como la sangre absorb\u00eda el flux.\n\n\u2014Nadie podr\u00eda olvidarlo \u2014le susurr\u00f3 al o\u00eddo\u2014. Ni el ta\u00f1ido de su campana. \u00bfNo resuena en tu memoria, Sheratan? \u00bfRecuerdas c\u00f3mo gritabas suplicando piedad?\n\nEmpezaba a dolerme todo, pero no me atrev\u00eda a moverme. Thuban me estaba ayudando sin darse cuenta. La Torre de Tom deb\u00eda de ser la que se alzaba en la entrada, la torre de la campana.\n\n\u2014No gritaba suplicando piedad \u2014dijo Terebell\u2014, sino justicia.\n\nThuban solt\u00f3 un gru\u00f1ido \u00e1spero.\n\n\u2014Idiota \u2014dijo. Levant\u00f3 una mano para pegar a Terebell, pero de pronto se detuvo y olfate\u00f3 el aire\u2014. Noto un aura. \u2014Volvi\u00f3 a olfatear\u2014. Registra la habitaci\u00f3n, Sheratan. Aqu\u00ed huele a humano.\n\n\u2014Yo no noto nada. \u2014Terebell no se movi\u00f3 de donde estaba\u2014. Esta habitaci\u00f3n estaba cerrada con llave cuando hemos llegado.\n\n\u2014Hay otras formas de entrar en una habitaci\u00f3n.\n\n\u2014Pareces paranoico.\n\nPero Thuban no parec\u00eda convencido. Fue hacia mi escondite, con las aletas de la nariz dilatadas y los labios estirados mostrando los dientes. Se me ocurri\u00f3 pensar que pod\u00eda ser un rastreador, que pod\u00eda oler la actividad espiritual. Si me ol\u00eda, estaba perdida.\n\nSus dedos avanzaron hacia la caja que me tapaba. A lo lejos, en otra habitaci\u00f3n, explot\u00f3 algo.\n\nThuban ech\u00f3 a correr por el pasillo al instante. Terebell lo sigui\u00f3, pero antes de salir por la puerta se dio la vuelta.\n\n\u2014Corre \u2014me dijo\u2014. Ve a la torre.\n\nDesapareci\u00f3. No quise tentar a la suerte: me colgu\u00e9 la mochila y salt\u00e9 a la repisa de la ventana. Baj\u00e9 resbalando por la hiedra y me ara\u00f1\u00e9 las manos y los brazos.\n\nMi coraz\u00f3n bombeaba la sangre con fuerza. Cada sombra que ve\u00eda parec\u00eda Thuban. Mientras recorr\u00eda unos soportales, camino del patio central, intent\u00e9 sacar alguna conclusi\u00f3n. Terebell me hab\u00eda ayudado. Me hab\u00eda ocultado. Hasta parec\u00eda ser que alguien hab\u00eda provocado una distracci\u00f3n con el fin de protegerme. Ella sab\u00eda que yo iba a ir all\u00ed, sab\u00eda qu\u00e9 buscaba, y hab\u00eda empezado a hablar en ingl\u00e9s despu\u00e9s de verme. Era uno de ellos. Los marcados. Necesitaba descubrir m\u00e1s informaci\u00f3n sobre su historia, averiguar qu\u00e9 estaba pasando; pero antes ten\u00eda que entrar en la Torre de Tom, hacerme con las medicinas y volver con el Custodio.\n\nLa explosi\u00f3n hab\u00eda atra\u00eddo a un grupo de arrancahuesos que corrieron desde la entrada, alej\u00e1ndose de la torre de la campana. Me par\u00e9 bajo un arco, protegida por su sombra. Justo a tiempo: entraron corriendo en el patio, precisamente por donde yo hab\u00eda estado a punto de salir.\n\n\u201428, 14, proteged el edificio Meadow \u2014dijo uno de ellos\u2014. 6, t\u00fa ven conmigo. Los dem\u00e1s, vigilad los patios. Id a buscar a Kraz y a Mirzam.\n\nNo ten\u00eda mucho tiempo. Me levant\u00e9 y corr\u00ed hacia el patio central.\n\nLa Casa era enorme, unida por una serie de pasillos cubiertos y descubiertos. Me sent\u00eda como una rata en un laberinto. No me atrev\u00eda a parar. Me at\u00e9 las correas de la mochila alrededor del torso. Ten\u00eda que haber una forma de entrar en la Torre de Tom. \u00bfHabr\u00eda una puerta junto a la entrada principal? Ten\u00eda que darme prisa: Kraz y Mirzam eran nombres de refa, y no me interesaba nada que hubiera cuatro refas, tres de los cuales, como m\u00ednimo, eran hostiles, en la Casa y persigui\u00e9ndome. Dudaba que el Custodio tuviera muchos amigos como Terebell.\n\nMe par\u00e9 al borde del patio central. Era muy grande, con un estanque ornamental en medio. En el centro del estanque hab\u00eda una estatua. No ten\u00eda m\u00e1s remedio que dejarme ver. Ten\u00eda que cambiar sigilo por velocidad.\n\nEch\u00e9 a correr por la hierba. Me dol\u00edan las costillas. Al llegar al estanque, me met\u00ed en el agua, poco profunda, y me agach\u00e9 detr\u00e1s de la fuente. El agua me llegaba por la cintura. Levant\u00e9 un momento la cabeza y di un respingo: Nashira me miraba fijamente. Nashira, representada en piedra.\n\nNo hab\u00eda nadie en el patio. Notaba un aura, pero estaba demasiado lejos para representar una amenaza. Sal\u00ed de la fuente y corr\u00ed hacia la torre de la campana. Enseguida vi el arco: deb\u00eda de ser la entrada del pasadizo que conduc\u00eda hasta la campana. Sub\u00ed a toda velocidad los escalones, rezando para que no apareciera ning\u00fan refa; el pasillo era tan estrecho que no podr\u00eda hacer nada. Cuando llegu\u00e9 arriba, me dej\u00f3 pasmada lo que vi.\n\nAquello era un aut\u00e9ntico tesoro. Hab\u00eda tarros de cristal que centelleaban en cientos de anaqueles, ba\u00f1ados por la luz del sol; su brillo y su colorido me recordaron a los caramelos duros. Hab\u00eda l\u00edquidos iridiscentes, polvos de colores intensos, plantas ex\u00f3ticas sumergidas en l\u00edquidos. Todo era hermoso y extra\u00f1o. La habitaci\u00f3n estaba inundada de olores: intensos, desagradables, dulces y fragantes. Registr\u00e9 los estantes en busca de suministros m\u00e9dicos. La mayor\u00eda de las botellas llevaban etiquetas con el s\u00edmbolo de Scion, escritas en ingl\u00e9s, pero algunas ten\u00edan jerogl\u00edficos extra\u00f1os. Tambi\u00e9n hab\u00eda _numa_ , seguramente confiscados. Vi una piedra de adivinaci\u00f3n, varios _sortes_... y una baraja de cartas. Para Liss. Ech\u00e9 un vistazo r\u00e1pido a las ilustraciones. Era una baraja Thoth, de otro dise\u00f1o que la antigua baraja de Liss, pero pod\u00eda utilizarse para la cartomancia.\n\nMe guard\u00e9 la baraja en la mochila. Cog\u00ed sulfadiazina de plata, parafina y antis\u00e9ptico. Vi otra puerta que deb\u00eda de llevar hasta la campana, pero no me met\u00ed por ella. No iba a robar nada m\u00e1s, porque la mochila pesaba tanto que casi no pod\u00eda levantarla. Me colgu\u00e9 las correas de los hombros y me volv\u00ed hacia la escalera. Y di de frente contra un refa.\n\nFue como si se pararan todas mis funciones vitales. Dos ojos amarillos que asomaban por debajo de una capucha me miraban fijamente.\n\n\u2014Vaya, vaya \u2014dijo el refa\u2014. Una traidora en la torre.\n\nAvanz\u00f3 hacia m\u00ed. Solt\u00e9 la mochila y, en un abrir y cerrar de ojos, trep\u00e9 a la estanter\u00eda m\u00e1s cercana.\n\n\u2014T\u00fa debes de ser la onir\u00e1mbula. Yo soy Kraz Sargas, heredero de sangre de los refa\u00edtas. \u2014Fingi\u00f3 una reverencia. Guardaba cierto parecido con Nashira: el cabello rubio, los ojos de p\u00e1rpados gruesos\u2014. \u00bfTe env\u00eda Arcturus?\n\nNo contest\u00e9.\n\n\u2014As\u00ed que deja que su tributo a la soberana de sangre corra suelta por ah\u00ed. Eso no le va a gustar a Nashira. \u2014Me tendi\u00f3 una mano enguantada\u2014. Ven, onir\u00e1mbula. Te acompa\u00f1ar\u00e9 a Magdalen.\n\n\u2014\u00bfY haremos como si esto no hubiera pasado? \u2014No me mov\u00ed del sitio\u2014. Me llevar\u00e1s ante Nashira.\n\nSe le agot\u00f3 la paciencia.\n\n\u2014No me obligues a aplastarte, casaca amarilla.\n\n\u2014Nashira no me quiere muerta.\n\n\u2014Yo no soy Nashira.\n\nEstaba perdida. Si no me mataba, me llevar\u00eda directamente a la Residencia del Suzerano. Mi mirada se pos\u00f3 en un tarro de \u00e1ster blanco. Pod\u00eda borrarle la memoria.\n\nPero no tuve suerte. Con un simple movimiento del brazo, Kraz tir\u00f3 toda la estanter\u00eda, y botellas y viales se estrellaron contra el suelo. Me apart\u00e9 para que no me aplastara, y un trozo de cristal me hizo un tajo en la mejilla. Dej\u00e9 escapar un grito. Me dol\u00edan las costillas, rotas. Mis lesiones me impidieron levantarme suficientemente deprisa.\n\nAll\u00ed no hab\u00eda esp\u00edritus, nada que pudiera utilizar para repelerlo. Kraz me agarr\u00f3 por el chaleco y me lanz\u00f3 contra la pared. Casi perd\u00ed el conocimiento. Notaba como si se me fueran a salir las costillas del pecho. Kraz me agarr\u00f3 por el pelo, me ech\u00f3 la cabeza hacia atr\u00e1s e inhal\u00f3 hondo, como si intentara aspirar algo m\u00e1s que aire. Me di cuenta de lo que estaba pasando cuando se me llenaron los ojos de sangre. Patale\u00e9, di ara\u00f1azos y me retorc\u00ed, intentando asir el \u00e9ter, pero este ya se alejaba de mi alcance.\n\nKraz estaba hambriento. Iba a absorberme el aura.\n\nTen\u00eda el brazo derecho inmovilizado, pero el izquierdo pod\u00eda moverlo. Impulsada por la adrenalina, hice lo que me hab\u00eda ense\u00f1ado mi padre: le clav\u00e9 a Kraz un dedo en el ojo. En cuanto me solt\u00f3 el pelo, saqu\u00e9 el vial que llevaba en el bolsillo. La flor roja.\n\nKraz me agarr\u00f3 por el cuello con una mano, ense\u00f1\u00e1ndome los dientes. Si intentaba atacarle la mente, mi cuerpo quedar\u00eda destrozado. No ten\u00eda alternativa. Le romp\u00ed el vial en la cara.\n\nEl olor a podrido era atroz. Kraz dio un alarido inhumano. El polen se le hab\u00eda metido en los ojos. Se le hab\u00edan oscurecido y goteaban, y la cara se le estaba poniendo de un gris moteado espantoso.\n\n\u2014No \u2014dijo\u2014. \u00a1No!\n\nLas siguientes palabras las dijo en _gloss_. Me fallaba la visi\u00f3n. \u00bfSer\u00eda una reacci\u00f3n al\u00e9rgica? Me dieron arcadas. Busqu\u00e9 a tientas en mi mochila, saqu\u00e9 el rev\u00f3lver y le apunt\u00e9 con \u00e9l a la cabeza. Kraz cay\u00f3 de rodillas.\n\n\u00abM\u00e1talo.\u00bb\n\nTen\u00eda las palmas de las manos sudadas. Me acord\u00e9 del metrovigilante del tren, del crimen por el que estaba all\u00ed; pero no sab\u00eda si ser\u00eda capaz de hacerlo, si podr\u00eda quitar otra vida. Entonces Kraz se quit\u00f3 las manos de la cara, y comprend\u00ed que ya no pod\u00eda salvarse. Ni siquiera pesta\u00f1e\u00e9.\n\nApret\u00e9 el gatillo.\n\nEl viaje de regreso a lugar seguro fue borroso. Corr\u00ed por los tejados, m\u00e1s all\u00e1 de la iglesia antigua y por la larga calle hasta Magdalen. Cuando llegu\u00e9 a la residencia, un brazo sali\u00f3 por una ventana, me agarr\u00f3 y me meti\u00f3 dentro.\n\nEl Custodio. Me estaba esperando. Sin decir palabra, me hizo entrar por una puerta y me llev\u00f3 al patio de levante por pasadizos vac\u00edos. Recorrimos soportales, subimos escaleras. No me atrev\u00eda a hablar. Cuando llegamos a la torre, me dej\u00e9 caer al suelo junto a la chimenea. Mis dedos dejaron polen negro en la alfombrilla. Parec\u00eda holl\u00edn.\n\nEl Custodio ech\u00f3 la llave, apag\u00f3 el gram\u00f3fono y corri\u00f3 las cortinas. Se qued\u00f3 unos minutos mirando por una de las ventanas de la c\u00e1mara, sin separar mucho las cortinas, vigilando la calle. Dej\u00e9 caer mi mochila al suelo. Las correas se me hab\u00edan clavado en los hombros.\n\n\u2014Lo he matado.\n\n\u2014\u00bfA qui\u00e9n?\n\n\u2014A Kraz. Le he disparado. \u2014Temblaba de pies a cabeza\u2014. He matado a un Sargas. Ahora Nashira me matar\u00e1. T\u00fa me matar\u00e1s...\n\n\u2014No.\n\n\u2014Pero \u00bfpor qu\u00e9 no?\n\n\u2014Un Sargas no es una p\u00e9rdida para m\u00ed. \u2014Mir\u00f3 otra vez por la ventana\u2014. \u00bfEst\u00e1s segura de que ha muerto?\n\n\u2014Claro que ha muerto. Le he disparado en la cara.\n\n\u2014Las balas no pueden matarnos. Tendr\u00edas que haber usado el polen.\n\n\u2014S\u00ed. \u2014Intent\u00e9 respirar acompasadamente\u2014. S\u00ed, lo he usado.\n\nSe qued\u00f3 un buen rato callado. Mis pulmones, a punto de explotar, me delataban.\n\n\u2014Si un humano ha matado a un Sargas \u2014dijo por fin\u2014, lo \u00faltimo que querr\u00e1 Nashira ser\u00e1 que se sepa en la ciudad. Nuestra inmortalidad no debe ser cuestionada.\n\n\u2014\u00bfNo es verdad que se\u00e1is inmortales?\n\n\u2014No somos indestructibles. \u2014Se agach\u00f3 delante de m\u00ed y me mir\u00f3 a los ojos\u2014. \u00bfTe ha visto alguien?\n\n\u2014No. Bueno, s\u00ed: Terebell.\n\n\u2014Terebell guardar\u00e1 tu secreto. Si solo te ha visto ella, no hay nada que temer.\n\n\u2014Thuban tambi\u00e9n estaba all\u00ed. Ha habido una explosi\u00f3n. \u2014Lo mir\u00e9\u2014. \u00bfSabes algo de eso?\n\n\u2014Not\u00e9 que estabas en peligro. Ten\u00eda a gente vigilando en la Casa. Han provocado una distracci\u00f3n. Lo \u00fanico que sabr\u00e1 Nashira ser\u00e1 que alguien dej\u00f3 una vela demasiado cerca de un escape de gas.\n\nEsa noticia no consigui\u00f3 reconfortarme mucho. Ya hab\u00eda quitado tres vidas, sin contar las que no hab\u00eda conseguido salvar.\n\n\u2014Est\u00e1s sangrando.\n\nMe mir\u00e9 en el espejo del cuarto de ba\u00f1o, que se ve\u00eda por la puerta abierta. Ten\u00eda un corte en la mejilla, largo pero poco profundo. Lo bastante profundo para sangrar.\n\n\u2014S\u00ed \u2014dije.\n\n\u2014Te ha hecho da\u00f1o.\n\n\u2014Me he cortado con un cristal. \u2014Me toqu\u00e9 la herida\u2014. \u00bfTe enterar\u00e1s de qu\u00e9 ha pasado?\n\nEl Custodio asinti\u00f3 sin dejar de mirarme la mejilla. Sus ojos ten\u00edan algo que me impresion\u00f3: una oscuridad, una tensi\u00f3n. Estaba pensando en otra cosa. No me miraba a los ojos: la herida lo ten\u00eda hipnotizado.\n\n\u2014Si no te curamos la herida, te dejar\u00e1 cicatriz. \u2014Me sujet\u00f3 la barbilla con los dedos enguantados\u2014. Voy a traer algo para limpi\u00e1rtela.\n\n\u2014Y te enterar\u00e1s de qu\u00e9 le ha pasado a Kraz.\n\n\u2014S\u00ed.\n\nNos miramos un instante. Arrugu\u00e9 la frente, y mis labios se fruncieron para formular una pregunta.\n\nPero no pregunt\u00e9 nada.\n\n\u2014Volver\u00e9 en cuanto pueda. \u2014Se levant\u00f3\u2014. Te aconsejo que te laves. Ah\u00ed dentro encontrar\u00e1s ropa limpia.\n\nSe\u00f1al\u00f3 el armario. Me mir\u00e9 el uniforme. Llevaba el chaleco cubierto de polen, lo que era una prueba condenatoria de mis transgresiones.\n\n\u2014De acuerdo \u2014conced\u00ed.\n\n\u2014Y procura que no se te ensucie la herida.\n\nSe march\u00f3 sin que yo hubiera podido decir nada.\n\nMe levant\u00e9 y me acerqu\u00e9 al espejo. El corte era impresionante. \u00bfLe preocupaba al Custodio verme as\u00ed, incluso despu\u00e9s de lo que hab\u00eda hecho Jax? \u00bfPensaba en sus cicatrices cuando me ve\u00eda la cara, las cicatrices que ten\u00eda en la espalda, esas que ocultaba?\n\nTen\u00eda el pelo impregnado de un olor empalagoso. El polen. Cerr\u00e9 la puerta del cuarto de ba\u00f1o, me quit\u00e9 la ropa y llen\u00e9 la ba\u00f1era de agua caliente. Me temblaban las piernas. Me hab\u00eda despellejado una rodilla trepando. Me met\u00ed en la ba\u00f1era y me lav\u00e9 el pelo. Notaba el dolor de las contusiones antiguas, y c\u00f3mo encima se formaban otras nuevas. Esper\u00e9 unos minutos a que el calor relajara mis r\u00edgidos m\u00fasculos; entonces cog\u00ed una pastilla nueva de jab\u00f3n y me frot\u00e9 con fuerza para limpiar el sudor, la sangre y el polen. Mi maltrecha y amarillenta figura no mejor\u00f3 gran cosa con las atenciones que le prodigu\u00e9. No empec\u00e9 a calmarme hasta que la ba\u00f1era se hubo vaciado.\n\n\u00bfDeb\u00eda hablar con \u00e9l de lo del tren? Quiz\u00e1 tratara de detenerme. Me hab\u00eda devuelto a Sheol I cuando habr\u00eda podido dejarme marchar. Por otra parte, necesitaba saber si el tren estaba vigilado o no, en qu\u00e9 lugar de la pradera encontrar\u00eda la entrada. No recordaba haber visto nada durante la sesi\u00f3n de entrenamiento: ninguna trampilla, ninguna puerta. Deb\u00eda de estar bien escondida.\n\nCuando regres\u00e9 a la c\u00e1mara, encontr\u00e9 el uniforme amarillo limpio en el ropero. Hab\u00edan barrido el polen de la alfombra. Me dej\u00e9 caer en el div\u00e1n. Hab\u00eda liquidado a Kraz Sargas, heredero de sangre de los refa\u00edtas, con un solo disparo entre los ojos. Hasta ese momento hab\u00eda cre\u00eddo que era imposible matarlos. Deb\u00eda de haber sido el polen: la bala no hab\u00eda hecho m\u00e1s que liquidar el trabajo. Antes de que yo saliera de la torre, el cad\u00e1ver ya hab\u00eda empezado a pudrirse. Unos pocos granos de polen hab\u00edan bastado para descomponerlo.\n\nSe abri\u00f3 la puerta y me sobresalt\u00e9. El Custodio hab\u00eda vuelto. Tra\u00eda una expresi\u00f3n sombr\u00eda.\n\nSe sent\u00f3 a mi lado. Cogi\u00f3 un pa\u00f1o, lo moj\u00f3 en el l\u00edquido \u00e1mbar de un tarro y me limpi\u00f3 la sangre de la mejilla. Lo mir\u00e9 en silencio, a la espera de su juicio.\n\n\u2014Kraz est\u00e1 muerto \u2014dijo sin revelar emoci\u00f3n alguna. Not\u00e9 una punzada en la mejilla\u2014. Era el heredero al trono de sangre. Si te descubrieran, te torturar\u00edan en p\u00fablico. Saben que faltan provisiones, pero nadie te ha visto. Al portero de d\u00eda le hemos borrado la memoria.\n\n\u2014\u00bfSospecha alguien de m\u00ed?\n\n\u2014Extraoficialmente quiz\u00e1 s\u00ed, pero no tienen pruebas. Por suerte, no utilizaste tu esp\u00edritu para matarlo; si lo hubieras hecho, tu identidad s\u00ed ser\u00eda evidente.\n\nMis temblores se intensificaron. T\u00edpico de m\u00ed: matar a alguien tan importante sin saber siquiera qui\u00e9n era. Si Nashira se enteraba, acabar\u00eda convertida en m\u00e1scara funeraria. Mir\u00e9 al Custodio.\n\n\u2014\u00bfQu\u00e9 le ha hecho el polen a Kraz? Sus ojos... Su cara...\n\n\u2014No somos lo que parecemos, Paige. \u2014Me sostuvo la mirada\u2014. \u00bfCu\u00e1nto rato ha pasado entre la aplicaci\u00f3n del polen y el disparo?\n\nEl disparo. No \u00abel asesinato\u00bb. Hab\u00eda dicho \u00abel disparo\u00bb, como si yo solo hubiera estado all\u00ed por casualidad.\n\n\u2014Unos diez segundos.\n\n\u2014\u00bfQu\u00e9 has visto en esos diez segundos?\n\nIntent\u00e9 recordar. En la habitaci\u00f3n hab\u00eda mucho vaho, y yo hab\u00eda recibido un golpe en la cabeza.\n\n\u2014Fue como si... como si su cara se estuviera... pudriendo. Y ten\u00eda los ojos blancos. Como si hubieran perdido todo el color. Unos ojos muertos.\n\n\u2014Est\u00e1 claro.\n\nNo sab\u00eda a qu\u00e9 se refer\u00eda. \u00abUnos ojos muertos.\u00bb\n\nEl fuego chisporroteaba y calentaba la habitaci\u00f3n en exceso. El Custodio me levant\u00f3 la barbilla para exponer mi corte a la luz.\n\n\u2014Nashira me ver\u00e1 esto \u2014dije\u2014. Y lo sabr\u00e1.\n\n\u2014Eso tiene soluci\u00f3n.\n\n\u2014\u00bfCu\u00e1l?\n\nNo me contest\u00f3. Cada vez que yo preguntaba c\u00f3mo, o por qu\u00e9, parec\u00eda que \u00e9l perdiera inter\u00e9s por la conversaci\u00f3n. Fue hasta su mesa y cogi\u00f3 un cilindro met\u00e1lico lo bastante peque\u00f1o para caber en un bolsillo. La palabra ScionMed estaba impresa a lo largo en letras rojas. Sac\u00f3 tres suturas adhesivas. Me qued\u00e9 quieta mientras me las aplicaba.\n\n\u2014\u00bfTe hago da\u00f1o?\n\n\u2014No.\n\nApart\u00f3 la mano de mi cara. Me toqu\u00e9 los adhesivos.\n\n\u2014En la Casa vi un mapa \u2014dije\u2014. S\u00e9 que hay un tren en Puerto Pradera. Necesito saber d\u00f3nde est\u00e1 la entrada del t\u00fanel.\n\n\u2014Y \u00bfpara qu\u00e9 necesitas saberlo?\n\n\u2014Porque quiero irme de aqu\u00ed. Antes de que Nashira me mate.\n\n\u2014Entiendo. \u2014El Custodio se sent\u00f3 en su butaca\u2014. Y das por hecho que te dejar\u00e9 marchar.\n\n\u2014S\u00ed. \u2014Le mostr\u00e9 su caja de rap\u00e9\u2014. O t\u00fa puedes dar por hecho que esto llegar\u00e1 a manos de Nashira.\n\nLa luz se reflej\u00f3 en el s\u00edmbolo. El Custodio tamborile\u00f3 con los dedos en el brazo de la butaca. No intent\u00f3 negociar conmigo; se limit\u00f3 a mirarme, y sus ojos desped\u00edan un leve resplandor.\n\n\u2014No puedes tomar el tren \u2014dijo.\n\n\u2014Apu\u00e9state algo.\n\n\u2014No me interpretes mal. El tren solo puede activarlo el Arconte de Westminster. Est\u00e1 programado para ir y venir en determinadas fechas, a determinadas horas. Esos horarios no se pueden cambiar.\n\n\u2014Debe de traer la comida.\n\n\u2014El tren solo se utiliza para transporte humano. La comida la traen los mensajeros.\n\n\u2014Entonces, no volver\u00e1 a venir hasta... \u2014cerr\u00e9 los ojos\u2014 la pr\u00f3xima Era de Huesos.\n\nEn 2069. Mi sue\u00f1o de una fuga f\u00e1cil se desvaneci\u00f3. Al final s\u00ed tendr\u00eda que cruzar el campo de minas.\n\n\u2014Qu\u00edtate de la cabeza la idea de cruzar a pie \u2014dijo, como si me hubiera le\u00eddo el pensamiento\u2014. Los emim utilizan el bosque como coto de caza. Ni siquiera t\u00fa, con tu don, durar\u00edas mucho si tuvieras que enfrentarte a una manada.\n\n\u2014No puedo esperar m\u00e1s. \u2014Apret\u00e9 el brazo de la butaca hasta que se me pusieron los nudillos blancos\u2014. Tengo que irme de aqu\u00ed. Ella me matar\u00e1, lo sabes.\n\n\u2014Claro que s\u00ed. Ahora que tu don ha madurado, est\u00e1 impaciente por arrebat\u00e1rtelo. No tardar\u00e1 en actuar.\n\nMe puse en tensi\u00f3n.\n\n\u2014\u00bfQu\u00e9 quiere decir que ha madurado?\n\n\u2014En la ciudadela pose\u00edste a 12. Te vi. Nashira estaba esperando a que desarrollaras todo tu potencial.\n\n\u2014\u00bfSe lo has dicho t\u00fa?\n\n\u2014Se enterar\u00e1, pero no por m\u00ed. Lo que digamos en esta habitaci\u00f3n no saldr\u00e1 de aqu\u00ed.\n\n\u2014\u00bfPor qu\u00e9?\n\n\u2014Consid\u00e9ralo una tentativa de confianza mutua.\n\n\u2014Examinaste mis recuerdos. \u00bfPor qu\u00e9 iba a confiar en ti?\n\n\u2014\u00bfAcaso yo no te ense\u00f1\u00e9 mi onirosaje?\n\n\u2014S\u00ed. Tu onirosaje fr\u00edo y vac\u00edo. No eres m\u00e1s que una c\u00e1scara hueca, \u00bfverdad?\n\nEl Custodio se levant\u00f3 bruscamente, fue hasta la estanter\u00eda y cogi\u00f3 un tomo viejo y enorme. Mis m\u00fasculos se tensaron. Antes de que pudiera decir nada m\u00e1s, sac\u00f3 un folleto de dentro del libro y lo tir\u00f3 encima de la mesa. Me qued\u00e9 de piedra: era _Sobre los m\u00e9ritos de la antinaturalidad_. Mi ejemplar, la prueba irrefutable de que pertenec\u00eda al sindicato. El Custodio lo ten\u00eda guardado.\n\n\u2014Quiz\u00e1 mi onirosaje est\u00e9 privado de su antigua vida, pero yo no veo a las personas ordenadas por categor\u00edas, como el autor de ese panfleto. Ah\u00ed no hay ning\u00fan onirom\u00e1ntico. Ni ning\u00fan refa\u00edta. Yo no veo las cosas as\u00ed. \u2014Me mir\u00f3 a los ojos\u2014. Ya llevo varios meses viviendo contigo. Conozco tu historia, aunque la haya conocido sin tu permiso. No pretend\u00eda invadir tu intimidad, pero quer\u00eda ver c\u00f3mo eras. Quer\u00eda conocerte. No quer\u00eda tratarte como a un vulgar ser humano, inferior y sin valor.\n\nEso me sorprendi\u00f3.\n\n\u2014\u00bfPor qu\u00e9 no? \u2014pregunt\u00e9 sosteni\u00e9ndole la mirada.\n\n\u2014\u00bfQu\u00e9 m\u00e1s te da?\n\n\u2014Me concierne.\n\nCog\u00ed _Sobre los m\u00e9ritos_ y me lo pegu\u00e9 al pecho, como har\u00eda un cr\u00edo con un juguete. Era como si le hubiera salvado la vida a Jaxon. El Custodio me observaba.\n\n\u2014Le tienes un gran aprecio a tu capo \u2014dijo\u2014. Quieres volver a esa vida. Al sindicato.\n\n\u2014Jaxon es algo m\u00e1s que este panfleto.\n\n\u2014Ya me lo imagino.\n\nSe sent\u00f3 a mi lado en el div\u00e1n. Guardamos silencio unos minutos. Una humana y un refa\u00edta, tan diferentes como el d\u00eda y la noche, atrapados en nuestra campana de cristal, como la flor marchita. Cogi\u00f3 la caja de rap\u00e9 y sac\u00f3 un vial de amaranto.\n\n\u2014Te sientes sola. \u2014Lo vaci\u00f3 en un c\u00e1liz\u2014. Lo noto. Noto tu soledad.\n\n\u2014Estoy sola.\n\n\u2014Echas de menos a Nick.\n\n\u2014Es mi mejor amigo. Claro que lo echo de menos.\n\n\u2014\u00c9l era algo m\u00e1s. Tus recuerdos de \u00e9l son extraordinariamente detallados, llenos de color, de vida. Lo adorabas.\n\n\u2014Era joven \u2014dije, cortante, puesto que el Custodio parec\u00eda decidido a pincharme en mi punto m\u00e1s sensible.\n\n\u2014Todav\u00eda eres joven \u2014insisti\u00f3\u2014. No he visto todos tus recuerdos. Falta algo.\n\n\u2014No tiene sentido pensar demasiado en el pasado.\n\n\u2014No estoy de acuerdo.\n\n\u2014Todos tenemos malos recuerdos. \u00bfPor qu\u00e9 te interesan los m\u00edos?\n\n\u2014La memoria es mi cuerda de salvamento. Mi ruta al \u00e9ter, como lo son para ti los onirosajes. \u2014Me toc\u00f3 la frente con un dedo enguantado\u2014. T\u00fa quisiste conocerme a trav\u00e9s de mi onirosaje. Yo, a cambio, te pido tus recuerdos.\n\nMe estremec\u00ed y me apart\u00e9. El Custodio se qued\u00f3 mir\u00e1ndome, evaluando mi reacci\u00f3n; entonces se levant\u00f3 y sacudi\u00f3 el tirador de la campanilla.\n\n\u2014\u00bfQu\u00e9 haces? \u2014pregunt\u00e9.\n\n\u2014Necesitas comer.\n\nEncendi\u00f3 el gram\u00f3fono y se puso a mirar la calle.\n\nMichael lleg\u00f3 en un abrir y cerrar de ojos y escuch\u00f3 las \u00f3rdenes que le dio el Custodio. Diez minutos m\u00e1s tarde volvi\u00f3 con una bandeja, que me puso en el regazo, con suficiente comida para devolverme las fuerzas: una taza de t\u00e9 con leche, un azucarero, sopa de tomate y pan caliente.\n\nLe di las gracias.\n\nMichael me sonri\u00f3 brevemente; luego le hizo una serie de signos complicados al Custodio, que asinti\u00f3 con la cabeza. Tras saludar con una reverencia, sali\u00f3 de la c\u00e1mara. El Custodio me mir\u00f3; quer\u00eda saber si iba a tener que obligarme a comer. Di un sorbo de t\u00e9. Me acord\u00e9 de que mi abuela me daba t\u00e9 cuando yo era muy peque\u00f1a, siempre que me pon\u00eda enferma; mi abuela ten\u00eda una gran fe en el t\u00e9. Com\u00ed un poco de pan. \u00bfMe estar\u00eda leyendo en ese momento, leyendo mis emociones? \u00bfNotar\u00eda que ese recuerdo me tranquilizaba? Intent\u00e9 concentrarme en \u00e9l, utilizar el cord\u00f3n \u00e1ureo, pero no sent\u00ed nada.\n\nCuando hube terminado, el Custodio cogi\u00f3 la bandeja y la dej\u00f3 en la mesita; luego volvi\u00f3 a sentarse a mi lado. Carraspe\u00e9.\n\n\u2014\u00bfQu\u00e9 ha dicho Michael?\n\n\u2014Que Nashira ha citado a los otros Sargas en su residencia.\n\n\u00bbTiene muy buen o\u00eddo \u2014a\u00f1adi\u00f3 con un deje de iron\u00eda\u2014. Me trae mucha informaci\u00f3n de los aposentos de Nashira. Su supuesta amaurosis hace que ella no se fije en sus idas y venidas. \u2014De modo que a Michael no le importaba escuchar a hurtadillas. Lo tendr\u00eda presente\u2014. Va a contarles lo de Kraz.\n\nMe apret\u00e9 las sienes con los dedos.\n\n\u2014Yo no quer\u00eda matarlo. Yo solo...\n\n\u2014Te habr\u00eda matado \u00e9l a ti. Kraz odiaba profundamente a los humanos. Planeaba, cuando llegara el d\u00eda de nuestra revelaci\u00f3n, atraer a los ni\u00f1os humanos a nuestras ciudades de control. Sent\u00eda debilidad por sus huesos, peque\u00f1os y finos. Para practicar cleromancia.\n\nMe dieron n\u00e1useas. La cleromancia empleaba _sortes_ , con los que los esp\u00edritus formaban figuras o que lanzaban en determinada direcci\u00f3n. Hab\u00eda todo tipo de _sortes_ : agujas, dados, llaves. El grupo de los llamados espatulom\u00e1nticos prefer\u00eda los huesos, pero generalmente se manejaban esqueletos muy viejos por respeto hacia los difuntos. Si Kraz hab\u00eda robado los huesos de ni\u00f1os para practicar cleromancia, me alegraba de haberlo matado.\n\n\u2014Me alegro de que est\u00e9 muerto \u2014dijo el Custodio\u2014. Era una plaga terrible para este mundo.\n\nNo dije nada.\n\n\u2014Te sientes culpable \u2014observ\u00f3 el Custodio.\n\n\u2014No. Tengo miedo.\n\n\u2014Miedo \u00bfde qu\u00e9?\n\n\u2014De lo que pueda llegar a hacer. No paro... \u2014Sacud\u00ed la cabeza; estaba agotada\u2014. No paro de matar. No quiero convertirme en un arma.\n\n\u2014Tu don es inestable, pero te mantiene viva. Funciona como un escudo.\n\n\u2014No es ning\u00fan escudo. Es como una pistola. Vivo pendiente de un gatillo. \u2014Clav\u00e9 la vista en el estampado de la alfombra\u2014. Hago da\u00f1o a otros. En eso consiste mi don.\n\n\u2014No lo haces deliberadamente. No siempre has sabido de lo que eras capaz.\n\nSolt\u00e9 una risa amarga.\n\n\u2014Bueno, s\u00ed sab\u00eda que pod\u00eda. No sab\u00eda c\u00f3mo, pero sab\u00eda qui\u00e9n hac\u00eda sangrar a la gente. Sab\u00eda qui\u00e9n hac\u00eda que a la gente le doliera la cabeza. Cuando se burlaban de m\u00ed... Cuando mencionaban las revueltas de Molly... les dol\u00eda algo. Y lo \u00fanico que hab\u00eda hecho era darles un empujoncito mental. Me gustaba, de alguna manera \u2014dije\u2014. Cuando solo ten\u00eda diez a\u00f1os ya me gustaba. Me gustaba vengarme. Era mi secreto. \u2014El Custodio no dejaba de mirarme\u2014. No soy como los sensores, ni como los m\u00e9diums. No me limito a usar esp\u00edritus para tener compa\u00f1\u00eda ni para la defensa personal. Soy uno de ellos, \u00bfme explico? Puedo morir cuando quiera, convertirme en esp\u00edritu cuando quiera. Eso hace que la gente me tema. Y hace que yo tema a la gente.\n\n\u2014S\u00ed, eres diferente. Pero eso no significa que debas tener miedo.\n\n\u2014S\u00ed. Mi esp\u00edritu es peligroso.\n\n\u2014No temes el peligro, Paige. Es m\u00e1s, creo que te gusta. Aceptaste trabajar para Jaxon Hall pese a saber que eso reducir\u00eda considerablemente tu esperanza de vida. Pese a saber que exist\u00eda el riesgo de que te detectaran.\n\n\u2014Necesitaba el dinero.\n\n\u2014Tu padre trabaja para Scion. No necesitabas el dinero. Dudo que lo hayas utilizado jam\u00e1s. El peligro te acerca m\u00e1s al \u00e9ter \u2014continu\u00f3\u2014. Por eso aprovechas cualquier oportunidad que se presenta para experimentarlo.\n\n\u2014No, no era eso. No soy una especie de adicta a la adrenalina. Quer\u00eda estar con otros videntes. \u2014La rabia volv\u00eda a impregnar mi voz\u2014. No quer\u00eda vivir como las colegialas con el cerebro lavado de Scion. Quer\u00eda formar parte de algo. Quer\u00eda importar. \u00bfNo lo entiendes?\n\n\u2014Esas no eran las \u00fanicas razones. Pensabas en una persona en concreto.\n\n\u2014No... \u2014Me temblaban los labios.\n\n\u2014Pensabas en Nick. \u2014Me sostuvo la mirada sin piedad\u2014. Estabas enamorada de \u00e9l. Lo habr\u00edas seguido a cualquier parte.\n\n\u2014No quiero hablar de eso.\n\n\u2014\u00bfPor qu\u00e9 no?\n\n\u2014Porque es m\u00edo. Es privado. \u00bfLos onirom\u00e1nticos no entend\u00e9is el concepto de intimidad?\n\n\u2014Lo has guardado en secreto demasiado tiempo. \u2014No me toc\u00f3, pero su mirada era casi tan \u00edntima como una caricia\u2014. No puedo arrebatarte ese recuerdo mientras est\u00e9s despierta, pero en cuanto te quedes dormida, leer\u00e9 las im\u00e1genes de tu mente, y t\u00fa las so\u00f1ar\u00e1s, como has hecho otras veces. Ese es el don del onirom\u00e1ntico. Crear un sue\u00f1o compartido.\n\n\u2014Supongo que nunca te aburres \u2014dije con desprecio\u2014. Viendo la ropa sucia de la gente.\n\nIgnor\u00f3 la pulla.\n\n\u2014Puedes aprender a apartarme, por supuesto, pero para eso tendr\u00edas que conocer mi esp\u00edritu casi tan bien como conoces el tuyo. Y no es f\u00e1cil conocer un esp\u00edritu tan viejo como el m\u00edo. \u2014Hizo una pausa\u2014. O puedes ahorrarte la molestia y dejar que me asome a tu interior.\n\n\u2014\u00bfQu\u00e9 conseguir\u00e9 con eso?\n\n\u2014Ese recuerdo es una barrera. Lo he notado dentro de ti, enterrado en lo m\u00e1s hondo de tu onirosaje. \u2014Me miraba fijamente\u2014. Sup\u00e9ralo, y te liberar\u00e1s de \u00e9l. Tu esp\u00edritu se liberar\u00e1 de \u00e9l.\n\nInspir\u00e9 hondo. La oferta no deber\u00eda haberme tentado.\n\n\u2014Y \u00bfsolo tengo que dormirme?\n\n\u2014S\u00ed. Puedo ayudarte. \u2014Sac\u00f3 un pu\u00f1ado de hojas secas de la caja de rap\u00e9\u2014. Esto es lo que conten\u00edan las p\u00edldoras. Si preparo una infusi\u00f3n, \u00bfte la tomar\u00e1s?\n\n\u2014\u00bfQu\u00e9 da\u00f1o me har\u00e1 una dosis m\u00e1s? \u2014Hice un gesto con los hombros. El Custodio se qued\u00f3 mir\u00e1ndome en silencio\u2014. Est\u00e1 bien \u2014ced\u00ed.\n\nSali\u00f3 de la c\u00e1mara. Imagin\u00e9 que abajo hab\u00eda una cocina, donde trabajaba Michael.\n\nApoy\u00e9 la cabeza en los cojines. Un temblor lento y fr\u00edo se extendi\u00f3 por mi pecho, llenando los espacios entre mis costillas. Hab\u00eda odiado al Custodio intensamente; lo hab\u00eda odiado por ser lo que era, y porque ten\u00eda la impresi\u00f3n de que me entend\u00eda. Hab\u00eda disfrutado odi\u00e1ndolo, y ahora me dispon\u00eda a mostrarle mi recuerdo m\u00e1s \u00edntimo. Yo cre\u00eda saber cu\u00e1l era, pero no pod\u00eda estar segura. Tendr\u00eda que so\u00f1arlo.\n\nPara cuando regres\u00f3 el Custodio, una certeza desafiante se hab\u00eda apoderado de m\u00ed. Tom\u00e9 el vaso de cristal que me ofreci\u00f3. Estaba lleno hasta el borde de un l\u00edquido ocre y trasl\u00facido que parec\u00eda miel diluida. En la superficie flotaban tres hojas.\n\n\u2014Es amargo \u2014me advirti\u00f3\u2014, pero me permitir\u00e1 ver los recuerdos con mayor claridad.\n\n\u2014\u00bfQu\u00e9 viste las otras veces?\n\n\u2014Fragmentos. Per\u00edodos de silencio. Depende de c\u00f3mo te sintieras en cada momento, de lo intensamente que lo sintieras. De la medida en que esa parte del recuerdo siga preocup\u00e1ndote.\n\nMir\u00e9 la infusi\u00f3n.\n\n\u2014En ese caso, creo que no voy a necesitar esto.\n\n\u2014Ser\u00e1 m\u00e1s f\u00e1cil para ti si te la tomas.\n\nSeguramente ten\u00eda raz\u00f3n. La mera perspectiva de enfrentarme a ese recuerdo ya hac\u00eda que me temblaran las manos. Me llev\u00e9 el vaso a los labios y sent\u00ed como si me dispusiera a renunciar de nuevo a mi vida.\n\n\u2014Espera.\n\nMe detuve.\n\n\u2014Paige, no est\u00e1s obligada a ense\u00f1arme ese recuerdo. Espero que lo hagas, por tu propio bien. Espero que puedas. Pero puedes negarte. Si te niegas, respetar\u00e9 tu intimidad.\n\n\u2014No soy tan cruel \u2014repliqu\u00e9\u2014. No hay nada peor que una historia sin final.\n\nAntes de poder seguir hablando, me beb\u00ed la infusi\u00f3n.\n\nEl Custodio me hab\u00eda mentido: aquello no era amargo sin m\u00e1s. Era el l\u00edquido m\u00e1s repugnante que hab\u00eda probado nunca, semejante a un pu\u00f1ado de fragmentos met\u00e1licos. Decid\u00ed que preferir\u00eda beber lej\u00eda que volver a probar la infusi\u00f3n de salvia. Me dieron arcadas. El Custodio me sujet\u00f3 la cara con ambas manos.\n\n\u2014Aguanta, Paige. \u00a1Aguanta!\n\nLo intent\u00e9. Vomit\u00e9 un poco, pero consegu\u00ed tragarme la mayor parte.\n\n\u2014Y ahora \u00bfqu\u00e9? \u2014dije tosiendo.\n\n\u2014Espera.\n\nNo tuve que esperar mucho. Me encorv\u00e9, estremeci\u00e9ndome y tratando de controlar las n\u00e1useas. El sabor era tan intenso que cre\u00ed que se me quedar\u00eda en la boca para siempre.\n\nY entonces se apagaron las luces. Ca\u00ed sobre los cojines y me qued\u00e9 dormida.\n\nSeis de los Siete Sellos est\u00e1bamos de pie formando un corro, como en una sesi\u00f3n de espiritismo.\n\nNadine iba a matar a alguien. Lo llevaba escrito en la cara. En medio del c\u00edrculo estaba Zeke S\u00e1enz, atado con cintas de terciopelo a una silla; su hermana le sujetaba la cabeza con ambas manos. Llev\u00e1bamos horas atacando la mente de Zeke, pero pese a sus lamentos y sus quejas, Jax no se ablandaba. Si su don pod\u00eda aprenderse, ser\u00eda de gran valor para la banda: la capacidad de resistir a toda influencia externa, ya fuera de esp\u00edritus o de otros videntes. As\u00ed pues, sentado en su butaca fumando un puro, Jax se limitaba a esperar a que alguno de nosotros pudiera con \u00e9l.\n\nJax llevaba mucho tiempo estudiando a Zeke. Del resto de nosotros se hab\u00eda olvidado, y dejaba que nos ocup\u00e1ramos de nuestras actividades delictivas. Pese a la rigurosa investigaci\u00f3n a que lo hab\u00eda sometido, Jax no hab\u00eda previsto que nuestro ilegible fuera a sufrir tanto cuando lo atac\u00e1ramos. Su onirosaje era el\u00e1stico y opaco, impenetrable para los esp\u00edritus. Le hab\u00edamos lanzado una bandada tras otra sin \u00e9xito. Su mente las hac\u00eda rebotar por toda la habitaci\u00f3n, resbalaban por ella como el agua por una canica o un Diamante Negro, su nuevo nombre.\n\n\u2014\u00a1Venga, desgraciados! \u2014grit\u00f3 Jax, y golpe\u00f3 la mesa con un pu\u00f1o\u2014. \u00a1Quiero o\u00edrle gritar el triple de fuerte!\n\nLlevaba todo el d\u00eda poniendo la \u00abDanza Macabra\u00bb y bebiendo vino, y eso nunca era buena se\u00f1al. Eliza, colorada por el esfuerzo de controlar a tantos esp\u00edritus, lo mir\u00f3 con reproche.\n\n\u2014\u00bfTe has levantado con el pie izquierdo, Jaxon?\n\n\u2014Otra vez.\n\n\u2014Est\u00e1 sufriendo \u2014dijo Nadine, furiosa\u2014. \u00a1M\u00edralo! \u00a1No lo soportar\u00e1!\n\n\u2014Yo s\u00ed que sufro, Nadine. Me desespera tu obstinaci\u00f3n. \u2014En voz muy baja, no por ello menos amenazadora\u2014. No me obligu\u00e9is a levantarme, ni\u00f1os. O-tra-vez.\n\nSe produjo un breve silencio. Nadine agarr\u00f3 a su hermano por los hombros; el pelo le tapaba la cara. Ahora lo ten\u00eda casta\u00f1o oscuro, y m\u00e1s corto; llamaba menos la atenci\u00f3n, pero ella lo odiaba. Odiaba la ciudadela. Pero sobre todo nos odiaba a nosotros.\n\nComo nadie hac\u00eda nada, Eliza llam\u00f3 a un esp\u00edritu asesor: JD, una musa del siglo XVII. Cuando salt\u00f3 de su onirosaje al \u00e9ter, las luces parpadearon.\n\n\u2014Probar\u00e9 con JD. \u2014Ten\u00eda la frente arrugada\u2014. Si no funciona con un esp\u00edritu tan antiguo, dudo mucho que funcione con nada.\n\n\u2014\u00bfUn duende, quiz\u00e1? \u2014insinu\u00f3 Jaxon con absoluta seriedad.\n\n\u2014\u00a1No vamos a utilizar un duende!\n\nJax sigui\u00f3 fumando.\n\n\u2014Es una pena.\n\nEn el otro extremo de la habitaci\u00f3n, Nick baj\u00f3 las persianas. Lo que est\u00e1bamos haciendo le horrorizaba, pero no pod\u00eda impedirlo.\n\nZeke no soportaba el suspense. Ten\u00eda los afiebrados ojos clavados en aquel esp\u00edritu.\n\n\u2014\u00bfQu\u00e9 hacen, Di?\n\n\u2014No lo s\u00e9. \u2014Nadine mir\u00f3 fr\u00edamente a Jaxon\u2014. Necesita descansar. Si le lanzas ese esp\u00edritu, yo...\n\n\u2014\u00bfQu\u00e9 har\u00e1s? \u2014De la boca de Jaxon sal\u00edan volutas de humo\u2014. \u00bfMe tocar\u00e1s una melod\u00eda furiosa? Adelante, te lo ruego. Me encanta la m\u00fasica que sale del alma.\n\nNadine hizo pucheros, pero no mordi\u00f3 el anzuelo. Sab\u00eda cu\u00e1l era el castigo por desobedecer a Jaxon. No ten\u00eda ning\u00fan otro sitio adonde ir, ning\u00fan otro sitio adonde llevar a su hermano.\n\nZeke se estremeci\u00f3 en sus brazos. Como si fuera m\u00e1s joven que ella, y no dos a\u00f1os mayor.\n\nEliza mir\u00f3 a Nadine, y luego a Jaxon; emiti\u00f3 una orden silenciosa, y la musa se lanz\u00f3. Yo no lo vi, pero lo sent\u00ed; y a juzgar por el grito de dolor de Zeke, \u00e9l tambi\u00e9n. Ech\u00f3 la cabeza hacia atr\u00e1s, y se le marcaron los m\u00fasculos del cuello.\n\nNadine abrazaba a su hermano y apretaba los labios.\n\n\u2014Lo siento \u2014dijo, y apoy\u00f3 la barbilla en su cabeza\u2014. Lo siento mucho, Zeke.\n\nJD se aplicaba con gran empe\u00f1o en su tarea. Le hab\u00edan dicho que Zeke iba a hacerle da\u00f1o a Eliza, y se hab\u00eda propuesto impedir que eso ocurriera. Las l\u00e1grimas y el sudor hac\u00edan brillar la cara de Zeke. Estaba a punto de asfixiarse.\n\n\u2014Por favor \u2014suplic\u00f3\u2014. Basta...\n\n\u2014Para, Jaxon \u2014le espet\u00e9\u2014. \u00bfNo te parece que ya ha aguantado suficiente?\n\nJaxon arque\u00f3 exageradamente las cejas.\n\n\u2014\u00bfEst\u00e1s cuestionando mis m\u00e9todos, Paige?\n\nMi valor se debilit\u00f3.\n\n\u2014No.\n\n\u2014En el sindicato hay que ganarse el sustento. Soy tu capo. Tu protector. Tu patr\u00f3n. \u00a1El hombre que impide que mueras de hambre como esos desdichados limosneros! \u2014Lanz\u00f3 un fajo de billetes por los aires, y la cara de Frank Weaver, que nos miraba desde cada billete, cay\u00f3 revoloteando por la alfombra\u2014. Ezekiel habr\u00e1 aguantado suficiente cuando yo lo diga, cuando yo decida concederle la libertad por hoy. \u00bfCrees que Hector parar\u00eda? \u00bfCrees que Jimmy o la Abadesa parar\u00edan?\n\n\u2014Nosotros no trabajamos para ellos. \u2014Eliza parec\u00eda angustiada. Le hizo una se\u00f1a al esp\u00edritu\u2014. Vuelve, JD. Ya estoy a salvo.\n\nEl esp\u00edritu se retir\u00f3. Entonces Zeke se sujet\u00f3 la cabeza con las manos.\n\n\u2014Estoy bien \u2014consigui\u00f3 decir\u2014. Solo... solo necesito un minuto.\n\n\u2014No est\u00e1s bien. \u2014Nadine se volvi\u00f3 hacia Jaxon, que estaba encendi\u00e9ndose otro puro\u2014. Te has aprovechado de nosotros. Sab\u00edas lo de la operaci\u00f3n y nos hiciste creer que t\u00fa lo har\u00edas mejor. Dijiste que curar\u00edas a Zeke. \u00a1Prometiste que lo curar\u00edas!\n\n\u2014Dije que lo intentar\u00eda \u2014replic\u00f3 Jaxon, imperturbable\u2014. Que experimentar\u00eda.\n\n\u2014Mientes. Eres igual que...\n\n\u2014Si tan terrible te parece este lugar, querida, vete. La puerta siempre est\u00e1 abierta. \u2014Baj\u00f3 un poco el tono de voz\u2014. La puerta que da a las fr\u00edas y oscuras calles. \u2014Le ech\u00f3 el humo del puro\u2014. Me pregunto cu\u00e1nto tardar\u00e1 la DVN en descubrirte.\n\nNadine temblaba de rabia.\n\n\u2014Me voy a Chat's. \u2014Se puso la chaquetilla de encaje\u2014. Sola.\n\nAgarr\u00f3 sus auriculares y su bolso, sali\u00f3 furiosa y cerr\u00f3 de un portazo.\n\n\u2014\u00a1Di! \u2014la llam\u00f3 Zeke, pero ella no le hizo caso.\n\nLa o\u00ed darle una patada a algo al bajar la escalera. Pieter apareci\u00f3 a trav\u00e9s de la pared, enojado porque lo hab\u00edan molestado y, enfurru\u00f1ado, se qued\u00f3 en un rinc\u00f3n.\n\n\u2014Creo que ya es hora de volver a casa, capit\u00e1n \u2014dijo Eliza con firmeza\u2014. Llevamos horas con esto.\n\n\u2014Espera. \u2014Jax me apunt\u00f3 con un largo dedo\u2014. Todav\u00eda no hemos probado nuestra arma secreta. \u2014Arrugu\u00e9 la frente, y Jax lade\u00f3 la cabeza\u2014. Venga, Paige. No te hagas la loca. Entra en su onirosaje, hazlo por m\u00ed.\n\n\u2014Ya hemos hablado de esto. \u2014Empezaba a dolerme la cabeza\u2014. Yo no entro por la fuerza.\n\n\u2014Ah, \u00bfno? No sab\u00eda que tu contrato de trabajo lo especificara. \u00a1Ah! Espera, ya me acuerdo. No has firmado ning\u00fan contrato. \u2014Apag\u00f3 el puro en el cenicero\u2014. Somos clarividentes. Antinaturales. \u00bfCre\u00edas que ser\u00edamos como tu pap\u00e1, que estar\u00edamos en nuestros despachos de Barbican de nueve a cinco, bebiendo t\u00e9 en vasitos de pl\u00e1stico? \u2014De repente parec\u00eda indignado, como si no soportara pensar en lo amaur\u00f3tica que pod\u00eda llegar a ser la gente\u2014. A nosotros no nos van los vasitos de pl\u00e1stico, Paige. Nos van la plata, el raso, las calles s\u00f3rdidas y los esp\u00edritus.\n\nMe qued\u00e9 mir\u00e1ndolo. Jaxon dio un gran sorbo de vino, con la mirada clavada en la ventana. Eliza sacudi\u00f3 la cabeza.\n\n\u2014Bueno, esto es absurdo. Creo que deber\u00edamos...\n\n\u2014\u00bfQui\u00e9n te paga?\n\nEliza dio un suspiro.\n\n\u2014T\u00fa, Jaxon.\n\n\u2014Correcto. Yo te pago, y t\u00fa obedeces. Y ahora, s\u00e9 buena, sube corriendo y dile a Danica que venga. No quiero que se pierda este espect\u00e1culo de magia.\n\nEliza sali\u00f3 de la habitaci\u00f3n con los labios fruncidos. Zeke me lanz\u00f3 una mirada de agotamiento y desesperaci\u00f3n. Me obligu\u00e9 a insistir:\n\n\u2014Jax, no estoy en condiciones, de verdad. Creo que todos necesitamos descansar un poco.\n\n\u2014Ma\u00f1ana puedes tomarte unas horas libres, coraz\u00f3n \u2014dijo distra\u00eddamente.\n\n\u2014No puedo entrar por la fuerza en un onirosaje. Ya lo sabes.\n\n\u2014Compl\u00e1ceme. Int\u00e9ntalo. \u2014Jaxon se sirvi\u00f3 m\u00e1s vino\u2014. Llevo a\u00f1os esperando esto. Un onir\u00e1mbulo contra un ilegible. El encuentro et\u00e9reo por excelencia. No se me ocurre ninguna coincidencia m\u00e1s peligrosa y audaz.\n\n\u2014\u00bfSabes lo que dices?\n\n\u2014No \u2014dijo Nick, y todas las cabezas se volvieron hacia \u00e9l\u2014. Habla como si se hubiera vuelto loco.\n\nTras un breve silencio, Jaxon alz\u00f3 su copa.\n\n\u2014Un diagn\u00f3stico excelente, doctor. Salud.\n\nBebi\u00f3. Nick mir\u00f3 hacia otro lado.\n\nEn la tensi\u00f3n posterior a ese momento, Eliza regres\u00f3 con una jeringa de adrenalina. Con ella iba Danica Panic, el \u00faltimo miembro de nuestro septeto. Hab\u00eda crecido en la ciudadela Scion de Belgrado, pero la hab\u00edan transferido a Londres, donde trabajaba de ingeniera. La hab\u00eda descubierto Nick; hab\u00eda visto su aura en una recepci\u00f3n celebrada en honor de los reci\u00e9n llegados. Se enorgullec\u00eda mucho de que ninguno de nosotros supiera pronunciar su nombre. Ni su apellido. Estaba dura como la piedra; llevaba el cabello, rizado y rojizo, recogido en un mo\u00f1o bajo; y ten\u00eda los brazos cubiertos de cicatrices y quemaduras. Su \u00fanica debilidad eran los chalecos.\n\n\u2014Danica, querida. \u2014Jaxon le hizo se\u00f1as para que se acercara\u2014. Ven y mira esto, \u00bfquieres?\n\n\u2014\u00bfQu\u00e9 es? \u2014pregunt\u00f3.\n\n\u2014Mi arma.\n\nDani y yo nos miramos. Ella solo llevaba una semana con nosotros, pero ya sab\u00eda c\u00f3mo era Jax.\n\n\u2014Veo que est\u00e1is celebrando una sesi\u00f3n de espiritismo \u2014observ\u00f3.\n\n\u2014Hoy no. \u2014Jax agit\u00f3 una mano\u2014. Empecemos.\n\nTuve que morderme la lengua para no mandar a Jax a la mierda. Siempre halagaba a los nuevos. Dani ten\u00eda un aura brillante, hiperactiva, que \u00e9l no hab\u00eda logrado identificar; pero estaba convencido, como siempre, de que ten\u00eda gran valor.\n\nMe sent\u00e9. Nick me limpi\u00f3 el brazo con un algod\u00f3n y me clav\u00f3 la jeringa.\n\n\u2014Hazlo \u2014me orden\u00f3 Jax\u2014. Lee al ilegible.\n\nEsper\u00e9 un momento a que mi sangre absorbiera la mezcla de f\u00e1rmacos; entonces cerr\u00e9 los ojos y busqu\u00e9 el \u00e9ter. Zeke se prepar\u00f3. Yo no pod\u00eda invadirlo (solo pod\u00eda acariciar su onirosaje, tantear los tenues matices de su superficie), pero su mente era tan sensible, que el m\u00e1s leve empujoncito pod\u00eda hacerle mucho da\u00f1o. Tendr\u00eda que ser muy cuidadosa.\n\nMi esp\u00edritu se desplaz\u00f3. Distingu\u00ed cinco onirosajes que tintineaban y se estremec\u00edan como m\u00f3viles de viento. El de Zeke era diferente. Su ta\u00f1ido era m\u00e1s serio, un acorde menor. Intent\u00e9 ver algo en su interior (un recuerdo, un temor), pero no hab\u00eda nada. Donde normalmente ve\u00eda im\u00e1genes borrosas, que parec\u00edan extra\u00eddas de una pel\u00edcula antigua, solo ve\u00eda negrura. Los recuerdos de Zeke estaban sellados.\n\nSal\u00ed bruscamente del \u00e9ter cuando una mano me agarr\u00f3 por el hombro. Zeke temblaba, tap\u00e1ndose las orejas con las manos.\n\n\u2014\u00a1Basta! \u2014Nick, detr\u00e1s de m\u00ed, me ayudaba a levantarme\u2014. Basta. No deber\u00edas obligar a Paige a hacer esto, Jaxon. No me importa cu\u00e1nto me pagues: me pagas con diamantes manchados de sangre. \u2014Abri\u00f3 la ventana con brusquedad\u2014. Vamos, Paige. Necesitas descansar.\n\nEstaba agotada; y aunque no lo hubiera estado, no habr\u00eda desobedecido a Nick. Los ojos de Jaxon lanzaban dardos que se me clavaban en la espalda. Al d\u00eda siguiente se le habr\u00eda pasado el enfado, despu\u00e9s de haberse bebido todo el vino. Sal\u00ed por la ventana y me agarr\u00e9 a la bajante; ve\u00eda borroso.\n\nNick ech\u00f3 a correr en cuanto pis\u00f3 el tejado. Y corr\u00eda mucho. Por suerte, todav\u00eda ten\u00eda adrenalina en las venas, o no habr\u00eda podido seguirlo.\n\nLo hac\u00edamos a menudo: tom\u00e1bamos un atajo por la ciudad. En teor\u00eda Londres ten\u00eda todo lo que yo odiaba: era enorme, gris y hostil, y llov\u00eda nueve de cada diez d\u00edas. Rug\u00eda, bombeaba y golpeaba como un coraz\u00f3n humano. Pero tras dos a\u00f1os de entrenamiento con Nick, aprendiendo a moverme por los tejados, la ciudadela se hab\u00eda convertido en mi refugio. Pod\u00eda sobrevolar el tr\u00e1fico y las cabezas de la DVN. Pod\u00eda correr como la sangre por el laberinto de calles y callejones. Estaba llena a rebosar, repleta de vida. Por lo menos all\u00ed fuera, era libre.\n\nNick baj\u00f3 a la calle. Seguimos corriendo por la concurrida acera hasta llegar a la esquina de Leicester Square. Sin parar para respirar, Nick empez\u00f3 a trepar por el edificio m\u00e1s cercano, contiguo al Hippodrome Casino. Hab\u00eda muchos sitios donde asirse, repisas y cornisas, pero dudaba que pudiera seguirlo. Ni siquiera la adrenalina pod\u00eda vencer mi fatiga.\n\n\u2014\u00bfQu\u00e9 haces, Nick?\n\n\u2014Necesito despejar la mente. \u2014Sonaba cansado.\n\n\u2014\u00bfEn un casino?\n\n\u2014Arriba. \u2014Me tendi\u00f3 una mano\u2014. Vamos, _s\u00f6tnos_. Pareces a punto de quedarte dormida.\n\n\u2014Ya. Ver\u00e1s, es que no sab\u00eda que hoy les iba a dar una paliza a mi esp\u00edritu y a mi cuerpo. \u2014Dej\u00e9 que me subiera a la primera repisa, y una chica que fumaba un cigarrillo nos mir\u00f3 sorprendida\u2014. \u00bfHasta d\u00f3nde vamos a trepar?\n\n\u2014Hasta el terrado de este edificio. Si aguantas \u2014a\u00f1adi\u00f3.\n\n\u2014\u00bfY si no aguanto?\n\n\u2014Muy bien. Ag\u00e1rrate a m\u00ed. \u2014Se puso mis brazos alrededor del cuello\u2014. A ver, \u00bfcu\u00e1l es la regla de oro?\n\n\u2014No mirar hacia abajo.\n\n\u2014Correcto \u2014dijo imitando a Jax. Re\u00ed.\n\nLlegamos arriba sin problema y sin hacernos da\u00f1o. Nick trepaba a los edificios desde que hab\u00eda dado sus primeros pasos; encontraba puntos de apoyo donde no parec\u00eda haberlos. Volv\u00edamos a movernos por los tejados, y las calles hab\u00edan quedado muy abajo. Pis\u00e9 c\u00e9sped artificial. A mi izquierda vi una fuente peque\u00f1a, sin agua; y a mi derecha, un lecho de flores marchitas.\n\n\u2014\u00bfQu\u00e9 es esto?\n\n\u2014Un jard\u00edn de azotea. Lo encontr\u00e9 hace unas semanas. Nunca he visto que lo usen, y pens\u00e9 que ser\u00eda un buen refugio. \u2014Nick se apoy\u00f3 en la barandilla\u2014. Perd\u00f3name por sacarte de all\u00ed de esta forma, _s\u00f6tnos_. A veces Dials es un poco claustrof\u00f3bico.\n\n\u2014S\u00ed, un poco.\n\nNo hablamos de lo que acababa de pasar. A Nick no le gustaban nada las t\u00e1cticas de Jaxon. Me tir\u00f3 una barrita de cereales. Nos quedamos contemplando el horizonte rosado y crepuscular, casi como si esper\u00e1ramos ver aparecer alg\u00fan barco.\n\n\u2014Paige, \u00bfhas estado enamorada alguna vez?\n\nMe tembl\u00f3 la mano. De pronto no pod\u00eda tragar lo que ten\u00eda en la boca: se me hab\u00eda cerrado la garganta.\n\n\u2014Creo que s\u00ed. \u2014Not\u00e9 escalofr\u00edos en los costados. Apoy\u00e9 la espalda en la barandilla\u2014. No s\u00e9... Puede que s\u00ed. \u00bfPor qu\u00e9 me lo preguntas?\n\n\u2014Porque quiero que me digas qu\u00e9 se siente. Para saber si yo lo estoy o no.\n\nAsent\u00ed con la cabeza y trat\u00e9 de aparentar que estaba tranquila. En realidad mi cuerpo estaba sufriendo una reacci\u00f3n muy extra\u00f1a: ve\u00eda puntitos negros, se me iba la cabeza, ten\u00eda la palma de las manos sudadas y el coraz\u00f3n me lat\u00eda muy deprisa.\n\n\u2014A ver, dime \u2014dije.\n\nNick segu\u00eda contemplando el ocaso.\n\n\u2014Cuando te enamoras de alguien \u2014dijo\u2014, \u00bftienes una actitud protectora para con esa persona?\n\nEra una situaci\u00f3n extra\u00f1a por dos motivos. El primero, que yo estaba enamorada de Nick. Eso lo sab\u00eda desde hac\u00eda mucho tiempo, aunque nunca hubiera hecho nada al respecto. Y el segundo, porque Nick ten\u00eda veintisiete a\u00f1os y yo, dieciocho. Era como si nuestros roles naturales se hubieran invertido.\n\n\u2014S\u00ed. \u2014Agach\u00e9 la cabeza\u2014. Bueno, creo que s\u00ed. Yo ten\u00eda... tengo una actitud protectora hacia \u00e9l.\n\n\u2014Y \u00bfte dan ganas, a veces, de... tocar a esa persona, sin m\u00e1s?\n\n\u2014Continuamente \u2014admit\u00ed con cierta timidez\u2014. O mejor dicho... quiero que \u00e9l me toque. Aunque solo sea...\n\n\u2014Que te abrace.\n\nAsent\u00ed con la cabeza, pero no lo mir\u00e9.\n\n\u2014Porque tengo la impresi\u00f3n de que entiendo a esa persona, y quiero que sea feliz. Pero no s\u00e9 c\u00f3mo hacerla feliz. De hecho, s\u00e9 que, si la amo, la har\u00e9 terriblemente desgraciada. \u2014Se le arrug\u00f3 la frente, como si fuera de papel\u2014. Ni siquiera s\u00e9 si debo arriesgarme a dec\u00edrselo, porque s\u00e9 cu\u00e1nta infelicidad provocar\u00e9. O creo que lo s\u00e9. \u00bfEs importante ser feliz, Paige?\n\n\u2014\u00bfC\u00f3mo puedes pensar que no es importante?\n\n\u2014Porque no s\u00e9 si la sinceridad es mejor que la felicidad. \u00bfSacrificamos la sinceridad para ser felices?\n\n\u2014A veces s\u00ed. Pero creo que es mejor ser sincero. Si no, vives una mentira.\n\nEscog\u00ed minuciosamente mis palabras, anim\u00e1ndolo a hablar y, al mismo tiempo, tratando de ignorar el estruendo que hab\u00eda dentro de mi cabeza.\n\n\u2014Porque hay que confiar.\n\n\u2014S\u00ed.\n\nMe ard\u00edan los ojos. Intent\u00e9 respirar despacio, pero una realidad terrible estaba invadiendo mi pensamiento: Nick no se refer\u00eda a m\u00ed.\n\nClaro, \u00e9l jam\u00e1s hab\u00eda insinuado que sintiera lo mismo que yo. Ni una sola palabra. Pero \u00bfqu\u00e9 hab\u00eda de todos los roces involuntarios, todas las horas de atenci\u00f3n, todas las veces que hab\u00edamos corrido juntos? \u00bfQu\u00e9 hab\u00eda de los dos \u00faltimos a\u00f1os de mi vida, cuando hab\u00eda pasado casi todos los d\u00edas y las noches con \u00e9l?\n\nNick miraba fijamente el cielo.\n\n\u2014\u00a1Mira! \u2014dijo.\n\n\u2014\u00bfQu\u00e9?\n\nSe\u00f1al\u00f3 una estrella.\n\n\u2014Arcturus. Nunca la hab\u00eda visto brillar tanto.\n\nLa estrella ten\u00eda un tono anaranjado, y era enorme y muy brillante. Me sent\u00ed suficientemente peque\u00f1a para desaparecer.\n\n\u2014Bueno \u2014dije aparentando despreocupaci\u00f3n\u2014, \u00bfqui\u00e9n es? \u00bfDe qui\u00e9n crees que est\u00e1s enamorado?\n\nNick se llev\u00f3 una mano a la cabeza.\n\n\u2014De Zeke.\n\nAl principio no estaba segura de haberlo o\u00eddo bien.\n\n\u2014Zeke. \u2014Gir\u00e9 la cabeza y lo mir\u00e9\u2014. \u00bfZeke S\u00e1enz?\n\nNick movi\u00f3 la cabeza afirmativamente.\n\n\u2014\u00bfCrees que tengo alguna posibilidad? \u2014pregunt\u00f3 en voz baja\u2014. \u00bfCrees que podr\u00eda quererme?\n\nMe qued\u00e9 de piedra.\n\n\u2014Nunca me hab\u00edas dicho nada \u2014empec\u00e9. Me costaba respirar\u2014. No sab\u00eda que...\n\n\u2014No pod\u00edas saberlo. \u2014Se pas\u00f3 una mano por la cara\u2014. No puedo evitarlo, Paige. S\u00e9 que podr\u00eda encontrar a otra persona, pero no me apetece buscarla. No sabr\u00eda por d\u00f3nde empezar. Creo que Zeke es la persona m\u00e1s hermosa del mundo. Al principio cre\u00eda que eran imaginaciones m\u00edas, pero ya lleva un a\u00f1o con nosotros... \u2014Cerr\u00f3 los ojos\u2014. No puedo negarlo. Lo quiero de verdad.\n\nNo era yo. Me qued\u00e9 callada; sent\u00eda como si alguien me estuviera inyectando una sustancia sopor\u00edfera en las arterias. No era de m\u00ed de quien estaba enamorado.\n\n\u2014Creo que podr\u00eda ayudarle. \u2014Su voz denotaba verdadera pasi\u00f3n\u2014. Podr\u00eda ayudarle a enfrentarse al pasado. Podr\u00eda ayudarle a recordar cosas. Antes era suspirante. Yo podr\u00eda ayudarle a volver a o\u00edr las voces.\n\nOjal\u00e1 yo pudiera o\u00edr voces. Ojal\u00e1 pudiera o\u00edr a los esp\u00edritus, porque as\u00ed podr\u00eda escucharlos a ellos, y no a Nick. Ten\u00eda que concentrarme en no llorar. Pasara lo que pasase esa noche, no pod\u00eda llorar. No pensaba llorar, ni en broma. Nick ten\u00eda todo el derecho del mundo a amar a otra persona. \u00bfPor qu\u00e9 no? Yo jam\u00e1s le hab\u00eda dicho ni una palabra de lo que sent\u00eda por \u00e9l. Deber\u00eda alegrarme por \u00e9l. Pero una parte de m\u00ed siempre hab\u00eda so\u00f1ado con que \u00e9l sintiera lo mismo, con que Nick hubiera estado esperando el momento adecuado para dec\u00edrmelo. Un momento como aquel.\n\n\u2014\u00bfQu\u00e9 has sacado de su onirosaje? \u2014Nick se qued\u00f3 mir\u00e1ndome, esperando mi respuesta\u2014. \u00bfHas visto algo?\n\n\u2014Solo oscuridad.\n\n\u2014Yo podr\u00eda intentarlo. Quiz\u00e1 consiguiera enviarle una imagen. \u2014Esboz\u00f3 una sonrisa\u2014. O hablar con \u00e9l, como hacen las personas normales.\n\n\u2014\u00c9l te escuchar\u00eda \u2014dije\u2014. Si se lo dijeras. \u00bfC\u00f3mo sabes que \u00e9l no siente lo mismo por ti?\n\n\u2014Creo que ya tiene suficientes problemas. Adem\u00e1s, ya conoces las normas. Nada de relaciones. A Jaxon le dar\u00eda algo si se enterara.\n\n\u2014Que se joda. No es justo que tengas que aguantar esta situaci\u00f3n.\n\n\u2014He aguantado un a\u00f1o, _s\u00f6tnos_. Puedo aguantar m\u00e1s.\n\nNick ten\u00eda raz\u00f3n, claro. Jaxon no nos dejaba tener relaciones serias. No le gustaban las relaciones. Aunque Nick me hubiera querido, no habr\u00edamos podido estar juntos. Pero ahora la verdad me miraba a la cara, y mi sue\u00f1o se hac\u00eda a\u00f1icos; casi no pod\u00eda respirar. Aquel hombre no era m\u00edo. Nunca hab\u00eda sido m\u00edo. Y por mucho que yo lo quisiera, nunca ser\u00eda m\u00edo.\n\n\u2014\u00bfPor qu\u00e9 no me hab\u00edas dicho nada? \u2014Me agarr\u00e9 a la barandilla\u2014. Ya s\u00e9 que no es asunto m\u00edo, pero...\n\n\u2014No quer\u00eda que te preocuparas. T\u00fa ya ten\u00edas tus propios problemas. Sab\u00eda que Jax se interesar\u00eda por ti, pero te ha hecho la vida imposible. Todav\u00eda te trata como si fueras un juguete nuevo. Hace que me arrepienta de haberte metido en esto.\n\n\u2014No digas eso. \u2014Me di la vuelta y le apret\u00e9 la mano, demasiado fuerte\u2014. T\u00fa me salvaste, Nick. Tarde o temprano me habr\u00eda vuelto loca. Ten\u00eda que saberlo, o siempre me habr\u00eda sentido marginada. T\u00fa hiciste que sintiera que formaba parte de algo; parte de muchas cosas, en realidad. Nunca podr\u00e9 agradec\u00e9rtelo lo suficiente.\n\nNick me miraba con gesto de sorpresa.\n\n\u2014\u00bfEst\u00e1s llorando?\n\n\u2014No. \u2014Le solt\u00e9 la mano\u2014. Mira, tengo que irme. He quedado con una persona \u2014le ment\u00ed.\n\n\u2014Espera, Paige. No te vayas. \u2014Me agarr\u00f3 por la mu\u00f1eca y me retuvo\u2014. Te he molestado, \u00bfverdad? \u00bfQu\u00e9 pasa?\n\n\u2014No estoy molesta.\n\n\u2014S\u00ed lo est\u00e1s. Espera un momento, por favor.\n\n\u2014De verdad, tengo que irme, Nick.\n\n\u2014Nunca has tenido que irte cuando te necesitaba.\n\n\u2014Lo siento. \u2014Me ce\u00f1\u00ed el blazer\u2014. Si quieres un consejo, vuelve a la base y dile a Zeke lo que sientes. Si le queda algo de cordura, te dir\u00e1 que s\u00ed. \u2014Lo mir\u00e9 y compuse una sonrisa triste\u2014. Es lo que te dir\u00eda yo.\n\nY entonces lo vi. Primero, confusi\u00f3n; luego, incredulidad; y por \u00faltimo, consternaci\u00f3n. Lo sab\u00eda.\n\n\u2014Paige...\n\n\u2014Es tarde. \u2014Pas\u00e9 una pierna por encima de la barandilla. Me temblaban las manos\u2014. Nos vemos el lunes, \u00bfvale?\n\n\u2014No. Espera, Paige. Espera.\n\n\u2014Por favor, Nick.\n\nNo insisti\u00f3, pero segu\u00eda mir\u00e1ndome con los ojos muy abiertos. Baj\u00e9 por la fachada del edificio y lo dej\u00e9 all\u00ed de pie, bajo la luna. Hasta que llegu\u00e9 abajo no brotaron las primeras y \u00fanicas l\u00e1grimas. Cerr\u00e9 los ojos e inspir\u00e9 el aire nocturno.\n\nNo s\u00e9 muy bien c\u00f3mo llegu\u00e9 al I-5. Quiz\u00e1 tomara el metro. Quiz\u00e1 fuera a pie. Mi padre no hab\u00eda vuelto del trabajo; no me estaba esperando. Me qued\u00e9 un rato de pie en medio del apartamento vac\u00edo, mirando por la ventana del sal\u00f3n. Por primera vez desde que era ni\u00f1a, me habr\u00eda gustado tener una madre, o una hermana, o al menos una amiga. Una amiga que no tuviera nada que ver con los Sellos. Pero no ten\u00eda ninguna de esas cosas. No sab\u00eda qu\u00e9 hacer, ni c\u00f3mo sentirme. \u00bfQu\u00e9 habr\u00eda hecho una chica amaur\u00f3tica en mi situaci\u00f3n? Pasarse una semana metida en la cama, seguramente. Pero yo no era una chica amaur\u00f3tica, y en realidad no hab\u00eda cortado con nadie. Solo con un sue\u00f1o. Un sue\u00f1o infantil.\n\nRecord\u00e9 mi \u00e9poca de colegiala, cuando era la \u00fanica vidente rodeada de amaur\u00f3ticos. Suzette, una de mis pocas amigas, hab\u00eda roto con su novio el \u00faltimo a\u00f1o. Intent\u00e9 recordar qu\u00e9 hab\u00eda hecho: no se hab\u00eda pasado una semana metida en la cama. \u00bfQu\u00e9 hab\u00eda hecho? Un momento... S\u00ed, ya me acordaba. Me hab\u00eda enviado un mensaje para pedirme que la acompa\u00f1ara a un club. \u00abQuiero bailar para olvidar mis problemas\u00bb, me hab\u00eda dicho. Yo me hab\u00eda inventado alguna excusa, como siempre.\n\nAquella ser\u00eda mi noche. Yo tambi\u00e9n bailar\u00eda para olvidar mis problemas. Olvidar\u00eda lo que hab\u00eda pasado. Me librar\u00eda de aquel dolor.\n\nMe quit\u00e9 la ropa, me di una ducha, me sequ\u00e9 y me alis\u00e9 el pelo. Me puse pintalabios, r\u00edmel y kohl. Me puse un poco de perfume en los pulsos. Me pellizqu\u00e9 las mejillas para darles color. Cuando hube terminado, me puse un vestido negro de encaje y unas sandalias de tac\u00f3n y sal\u00ed del apartamento.\n\nEl vigilante me mir\u00f3 extra\u00f1ado cuando pas\u00e9 a su lado.\n\nTom\u00e9 un taxi. En el East End hab\u00eda un tugurio al que sol\u00eda ir Nadine, donde los d\u00edas laborables serv\u00edan _mecks_ barato (y a veces alcohol aut\u00e9ntico, ilegal). Era una zona dura del II-6, considerada uno de los pocos sitios donde los videntes pod\u00edan moverse sin peligro: ni a los _centis_ les gustaba entrar all\u00ed.\n\nUn gorila con traje y sombrero vigilaba la entrada. Me hizo una se\u00f1a para indicarme que pod\u00eda pasar.\n\nDentro estaba oscuro y hac\u00eda calor. El local, peque\u00f1o, estaba abarrotado de cuerpos sudorosos. Una barra discurr\u00eda a lo largo de una pared; en un extremo serv\u00edan ox\u00edgeno y en el otro, _mecks_. A la derecha de la barra hab\u00eda una pista de baile. Casi todos los clientes eran amaur\u00f3ticos, _hapsters_ con pantalones de tweed, sombreritos y pajaritas de colores llamativos. No sab\u00eda qu\u00e9 demonios hac\u00eda all\u00ed, mirando a unos amaur\u00f3ticos brincar al son de una m\u00fasica ensordecedora, pero eso era lo que quer\u00eda: actuar espont\u00e1neamente, olvidar el mundo real.\n\nLlevaba nueve a\u00f1os adorando a Nick. Cortar\u00eda por lo sano. Ni siquiera me parar\u00eda a pensar qu\u00e9 sent\u00eda.\n\nFui a la barra de ox\u00edgeno y me sent\u00e9 en un taburete. El camarero me mir\u00f3 de arriba abajo, pero no me dijo nada. Era vidente (profeta, concretamente); era l\u00f3gico que no quisiera hablar conmigo. Pero no pas\u00f3 mucho rato hasta que alguien m\u00e1s se fij\u00f3 en m\u00ed.\n\nEn el otro extremo de la barra hab\u00eda un grupo de chicos, seguramente alumnos de la USL. Eran todos amaur\u00f3ticos, por supuesto; muy pocos videntes llegaban a la universidad. Me dispon\u00eda a pedir un vaso de Floxy cuando se me acerc\u00f3 uno. Tendr\u00eda diecinueve o veinte a\u00f1os; iba bien afeitado y estaba un poco bronceado. Deb\u00eda de haber pasado su a\u00f1o de intercambio en otra ciudadela. Scion Atenas, quiz\u00e1. Llevaba una gorra de b\u00e9isbol que le tapaba el pelo, casta\u00f1o oscuro.\n\n\u2014Hola \u2014me dijo subiendo la voz para hacerse o\u00edr por encima de la m\u00fasica\u2014. \u00bfHas venido sola?\n\nDije que s\u00ed con la cabeza. El chico se sent\u00f3 a mi lado.\n\n\u2014Me llamo Reuben \u2014se present\u00f3\u2014. \u00bfPuedo invitarte a una copa?\n\n\u2014 _Mecks_ \u2014dije\u2014. Si no te importa.\n\n\u2014Claro que no. \u2014Le hizo se\u00f1as al camarero, al que era evidente que conoc\u00eda\u2014. _Mecks_ sangre, Gresham.\n\nEl camarero me sirvi\u00f3 el _mecks_ sangre con el ce\u00f1o fruncido, pero no dijo nada. Era el sustituto del alcohol m\u00e1s caro, hecho con cerezas, uvas negras y ciruelas. Reuben se inclin\u00f3 para hablarme al o\u00eddo.\n\n\u2014\u00bfA qu\u00e9 has venido?\n\n\u2014A nada especial.\n\n\u2014\u00bfNo tienes novio?\n\n\u2014Puede que s\u00ed.\n\n\u2014Yo acabo de romper con mi novia. Y cuando te he visto entrar, he pensado... Bueno, he pensado cosas que seguramente no deber\u00eda pensar al ver entrar a una chica guapa en un bar. Pero entonces he pensado que una chica tan guapa como t\u00fa deb\u00eda de haber venido con su novio. \u00bfNo es as\u00ed?\n\n\u2014No \u2014dije\u2014. He venido sola.\n\nGresham desliz\u00f3 mi vaso de _mecks_ por la superficie de la barra.\n\n\u2014Ser\u00e1n dos \u2014dijo.\n\nReuben le dio dos monedas de oro.\n\n\u2014Supongo que tienes dieciocho a\u00f1os, \u00bfno, joven?\n\nLe mostr\u00e9 mi documento de identidad, y \u00e9l sigui\u00f3 limpiando vasos, pero sin quitarme los ojos de encima. Me pregunt\u00e9 qu\u00e9 ser\u00eda lo que le preocupaba: \u00bfmi edad, mi aspecto, mi aura? Seguramente las tres cosas.\n\nVolv\u00ed de golpe a la realidad cuando Reuben se acerc\u00f3 m\u00e1s a m\u00ed. Le ol\u00eda el aliento a manzanas.\n\n\u2014\u00bfVas a la universidad? \u2014me pregunt\u00f3.\n\n\u2014No.\n\n\u2014\u00bfQu\u00e9 haces?\n\n\u2014Trabajo en un bar de ox\u00edgeno.\n\nAsinti\u00f3 con la cabeza y dio un trago de su copa.\n\nNo sab\u00eda c\u00f3mo hacerlo. C\u00f3mo dar la se\u00f1al. \u00bfHab\u00eda que dar alguna se\u00f1al? Lo mir\u00e9 a los ojos y le pas\u00e9 la punta del zapato por la pantorrilla. Me pareci\u00f3 que funcionaba, porque mir\u00f3 a sus amigos, que hab\u00edan retomado su juego.\n\n\u2014\u00bfQuieres que vayamos a alg\u00fan sitio? \u2014dijo con voz grave y ronca. \u00abAhora o nunca\u00bb, me dije, y asent\u00ed con la cabeza.\n\nReuben entrelaz\u00f3 los dedos con los m\u00edos y me gui\u00f3 entre el gent\u00edo. Gresham me observaba. Deb\u00eda de pensar que era una fresca.\n\nComprend\u00ed que Reuben no me estaba llevando al rinc\u00f3n oscuro que yo me hab\u00eda imaginado, sino a los lavabos. O eso cre\u00ed hasta que salimos por otra puerta que daba al aparcamiento del personal. Era un espacio rectangular diminuto, donde solo cab\u00edan seis coches. Vale, quer\u00eda intimidad. L\u00f3gico, \u00bfno? Al menos significaba que no buscaba \u00fanicamente fardar delante de sus amigos.\n\nAntes de que pudiera darme cuenta de lo que estaba pasando, Reuben me empuj\u00f3 contra la sucia pared de ladrillo. Ol\u00eda a sudor y a tabaco. Empez\u00f3 a desabrocharse el cintur\u00f3n, y me qued\u00e9 de piedra.\n\n\u2014Espera \u2014dije\u2014. Yo no quer\u00eda...\n\n\u2014Venga, solo es un poco de diversi\u00f3n. Adem\u00e1s... \u2014dej\u00f3 caer el cintur\u00f3n\u2014... no le hacemos da\u00f1o a nadie.\n\nMe bes\u00f3. Ten\u00eda los labios firmes. Una lengua h\u00fameda se introdujo en mi boca, y not\u00e9 un sabor artificial. Nunca me hab\u00edan besado. No estaba segura de si me gustaba.\n\nReuben ten\u00eda raz\u00f3n. Un poco de diversi\u00f3n. Claro. \u00bfQu\u00e9 pod\u00eda pasar? La gente normal lo hac\u00eda, \u00bfno? Beb\u00edan, hac\u00edan estupideces y luego ten\u00edan relaciones sexuales. Eso era precisamente lo que yo necesitaba. Jax no nos lo prohib\u00eda: lo que no quer\u00eda era que nos comprometi\u00e9ramos. Yo no pensaba comprometerme. Nada de ataduras. Eliza lo hac\u00eda.\n\nMi cabeza me dec\u00eda que parara. \u00bfPor qu\u00e9 hac\u00eda aquello? \u00bfC\u00f3mo hab\u00eda acabado all\u00ed, a oscuras, con un desconocido? As\u00ed no iba a demostrar nada. No iba a aliviar mi dolor, sino a empeorarlo. Pero Reuben se hab\u00eda arrodillado y me estaba subiendo el vestido hasta la cintura. Me bes\u00f3 el vientre desnudo.\n\n\u2014Eres preciosa.\n\nYo no me consideraba ni m\u00ednimamente guapa.\n\n\u2014No me has dicho c\u00f3mo te llamas.\n\nResigui\u00f3 el borde de mis bragas. Me estremec\u00ed.\n\n\u2014Eva \u2014ment\u00ed.\n\nLa idea de tener relaciones con \u00e9l me repugnaba. No lo conoc\u00eda de nada. No lo quer\u00eda. Pero razon\u00e9 que eso era porque todav\u00eda estaba enamorada de Nick, y ten\u00eda que borrarlo de mi mente. Agarr\u00e9 a Reuben por el pelo y estamp\u00e9 los labios contra los suyos. \u00c9l dio una especie de gru\u00f1ido, me levant\u00f3 y puso mis piernas alrededor de su cintura.\n\nNot\u00e9 un estremecimiento. No lo hab\u00eda hecho nunca. \u00bfNo dec\u00edan que la primera vez ten\u00eda que ser especial? No, no pod\u00eda parar. Ten\u00eda que continuar.\n\nLa luz de la farola parpadeaba un poco y me deslumbraba. Reuben apoy\u00f3 las manos en la pared de ladrillo. Yo no sab\u00eda qu\u00e9 esperar. Era emocionante.\n\nY de pronto sent\u00ed dolor. Un dolor intenso, apabullante. Como si me hubieran pegado un gancho en el est\u00f3mago.\n\nReuben no ten\u00eda ni idea de qu\u00e9 hab\u00eda pasado. Esper\u00e9 a que se me pasara, pero no se me pas\u00f3. \u00c9l not\u00f3 mi tensi\u00f3n.\n\n\u2014\u00bfEst\u00e1s bien?\n\n\u2014S\u00ed, s\u00ed \u2014dije en voz baja.\n\n\u2014\u00bfEs la primera vez?\n\n\u2014No, qu\u00e9 va.\n\nMe acerc\u00f3 la cabeza al cuello y me bes\u00f3 desde el hombro hasta la oreja. Volv\u00ed a sentir aquel dolor, pero m\u00e1s fuerte: un dolor salvaje, atroz. Reuben se apart\u00f3.\n\n\u2014Es la primera vez \u2014dijo.\n\n\u2014No importa.\n\n\u2014Mira, no creo que...\n\n\u2014Vale. \u2014Lo apart\u00e9 de m\u00ed\u2014. Pues... d\u00e9jame en paz. No quiero saber nada de ti ni de nadie.\n\nMe separ\u00e9 de la pared, me baj\u00e9 el vestido y volv\u00ed al bar. Llegu\u00e9 al lavabo justo a tiempo y vomit\u00e9. El dolor me sacud\u00eda los muslos y el est\u00f3mago. Me dobl\u00e9 por la cintura sobre el v\u00e1ter, tosiendo y sollozando. Jam\u00e1s me hab\u00eda sentido tan est\u00fapida.\n\nPens\u00e9 en Nick. Pens\u00e9 en los a\u00f1os que hab\u00eda pasado so\u00f1ando con \u00e9l, pregunt\u00e1ndome si alg\u00fan d\u00eda volver\u00eda a verlo. Y pens\u00e9 en \u00e9l ahora, imagin\u00e9 su sonrisa, c\u00f3mo se preocupaba por m\u00ed; y era in\u00fatil, porque lo quer\u00eda a \u00e9l. Apoy\u00e9 la cabeza en los brazos y llor\u00e9.\n\nLa intensidad del recuerdo me dej\u00f3 largo rato inconsciente. Hab\u00eda revivido cada detalle de aquella noche, hasta el m\u00e1s d\u00e9bil temblor. Despert\u00e9 en medio de una oscuridad total; no sab\u00eda qu\u00e9 hora era.\n\nEn el gram\u00f3fono sonaba \u00abIt's a Sin to Tell a Lie\u00bb a un volumen muy bajo.\n\nHabr\u00eda podido mostrarle muchos recuerdos (las revueltas de Molly, el duelo de mi padre, la crueldad de mis compa\u00f1eras) y, sin embargo, le hab\u00eda ense\u00f1ado la noche en que el chico del que estaba enamorada me hab\u00eda rechazado. Parec\u00eda algo peque\u00f1o e insignificante, pero era mi \u00fanico recuerdo normal, humano. La \u00fanica vez que me hab\u00eda entregado a un desconocido. La \u00fanica vez que se me hab\u00eda partido el coraz\u00f3n.\n\nYo no cre\u00eda en corazones. Cre\u00eda en onirosajes y esp\u00edritus: eso era lo que importaba. Era con lo que se ganaba dinero. Pero aquel d\u00eda se me hab\u00eda partido el coraz\u00f3n. Por primera vez en la vida me hab\u00eda visto obligada a reconocer que ten\u00eda coraz\u00f3n, y a reconocer su fragilidad. Mi coraz\u00f3n era vulnerable. Y pod\u00eda humillarme.\n\nYa era mayor; quiz\u00e1 hubiera cambiado. Quiz\u00e1 hubiera madurado, quiz\u00e1 fuera m\u00e1s fuerte. Ya no era aquella ni\u00f1a adolescente, desesperada por conectar con alguien, por encontrar a alguien en quien apoyarse. Esa ni\u00f1a hab\u00eda dejado de existir hac\u00eda mucho tiempo. Me hab\u00eda convertido en un arma, un t\u00edtere de las maquinaciones de otros. No sab\u00eda cu\u00e1l de las dos cosas era peor.\n\nUna lengua de fuego segu\u00eda acariciando las brasas de la chimenea y proyectaba su luz sobre la figura junto a la ventana.\n\n\u2014Bienvenida.\n\nNo contest\u00e9. El Custodio lade\u00f3 la cabeza y me mir\u00f3.\n\n\u2014Adelante \u2014dije\u2014. Debes de tener algo que decir.\n\n\u2014No, Paige.\n\nHubo un momento de silencio.\n\n\u2014Crees que fui est\u00fapida. Tienes raz\u00f3n. \u2014Me mir\u00e9 las manos\u2014. Solo quer\u00eda...\n\n\u2014Que te vieran. \u2014Mir\u00f3 hacia el fuego\u2014. Creo que entiendo por qu\u00e9 te afecta tanto ese recuerdo. Es de donde surge tu mayor miedo: que no exista nada m\u00e1s all\u00e1 de tu don. M\u00e1s all\u00e1 de la onir\u00e1mbula. Esa es la parte de ti que consideras verdaderamente valiosa: tu sustento. Lo dem\u00e1s lo perdiste en Irlanda. Ahora dependes de Jaxon Hall, que te trata como una mercanc\u00eda. Para \u00e9l no eres m\u00e1s que carne injertada en un fantasma; un don valios\u00edsimo con envoltorio humano. Pero Nick Nyg\u00e5rd te mostr\u00f3 algo m\u00e1s que eso.\n\n\u00bbEsa noche te abri\u00f3 los ojos. Cuando te enteraste de que Nick amaba a otro, te enfrentaste a tu mayor temor: no ser reconocida como ser humano, como la suma de todas tus partes. No ser m\u00e1s que una curiosidad. No ten\u00edas m\u00e1s remedio que demostrarte a ti misma lo contrario. Buscar al primero que te quisiera, alguien que no supiera nada de la onir\u00e1mbula. Era lo \u00fanico que te quedaba.\n\n\u2014Ni se te ocurra compadecerte de m\u00ed \u2014dije.\n\n\u2014No me compadezco. Pero s\u00e9 lo que se siente cuando te quieren solo por lo que eres.\n\n\u2014No volver\u00e1 a pasar.\n\n\u2014Pero tu soledad no te protegi\u00f3, \u00bfverdad?\n\nDesvi\u00e9 la mirada. Me fastidiaba que \u00e9l lo supiera. Me fastidiaba haber permitido que me descubriera. El Custodio se sent\u00f3 a mi lado en la cama.\n\n\u2014La mente de los amaur\u00f3ticos es como el agua. Anodina, gris, transparente. Suficiente para sostener la vida, pero nada m\u00e1s. En cambio, la mente de los clarividentes se parece m\u00e1s al aceite, es mucho m\u00e1s rica. Y como ocurre con el agua y el aceite, nunca llegan a mezclarse del todo.\n\n\u2014Lo dices porque \u00e9l era amaur\u00f3tico...\n\n\u2014S\u00ed.\n\nAl menos, ahora sab\u00eda que no ten\u00eda ning\u00fan defecto f\u00edsico. Nunca hab\u00eda tenido valor para hablar con un m\u00e9dico sobre lo que me hab\u00eda pasado aquella noche. Los m\u00e9dicos de Scion eran fr\u00edos e implacables en lo relativo a esos asuntos.\n\nEntonces se me ocurri\u00f3 una cosa.\n\n\u2014Si las mentes de los videntes son como el aceite \u2014dije escogiendo con cuidado mis palabras\u2014, \u00bfc\u00f3mo son las vuestras?\n\nPor un instante cre\u00ed que no me iba a contestar. Al final el Custodio dijo:\n\n\u2014Fuego.\n\nEsa \u00fanica palabra, pronunciada en voz baja, me hizo estremecer. Pens\u00e9 en lo que pasaba cuando se un\u00edan el aceite y el fuego: explotaban.\n\nNo. No pod\u00eda pensar en \u00e9l as\u00ed. \u00c9l no era humano. Que me entendiera o no era irrelevante. Segu\u00eda siendo mi guardi\u00e1n, un refa\u00edta, todo lo que hab\u00eda sido desde el principio.\n\nSe volvi\u00f3 y me mir\u00f3.\n\n\u2014Paige \u2014dijo\u2014, he visto otro recuerdo. Antes de que perdieras el conocimiento.\n\n\u2014\u00bfCu\u00e1l?\n\n\u2014Sangre. Mucha sangre.\n\nSacud\u00ed la cabeza; estaba demasiado cansada para pensar.\n\n\u2014Seguramente, cuando se manifest\u00f3 mi clarividencia. En los recuerdos de la duende hab\u00eda mucha sangre.\n\n\u2014No. Ese recuerdo ya lo hab\u00eda visto. Me refiero a otro en el que hab\u00eda mucha m\u00e1s sangre. Por todas partes, y te asfixiaba.\n\n\u2014No s\u00e9 de qu\u00e9 me hablas.\n\nEra la verdad. No sab\u00eda de qu\u00e9 pod\u00eda tratarse.\n\nEl Custodio se qued\u00f3 mir\u00e1ndome un momento.\n\n\u2014Duerme un poco m\u00e1s \u2014dijo por fin\u2014. Ma\u00f1ana, cuando despiertes, piensa en cosas mejores.\n\n\u2014\u00bfPor ejemplo?\n\n\u2014Por ejemplo, en c\u00f3mo escapar de esta ciudad. Cuando llegue el momento, debes estar preparada.\n\n\u2014Entonces \u00bfvas a ayudarme? \u2014Como no me contestaba, perd\u00ed la paciencia\u2014. Te lo he ense\u00f1ado todo: mi vida, mis recuerdos. Y, sin embargo, sigo sin tener ni idea de cu\u00e1les son tus motivos. Dime, \u00bfqu\u00e9 quieres?\n\n\u2014Mientras Nashira nos tenga a los dos dominados, cuanto menos sepas, mejor. As\u00ed, si vuelve a interrogarte, podr\u00e1s decir que no sabes nada del asunto.\n\n\u2014\u00bfDe qu\u00e9 \u00abasunto\u00bb?\n\n\u2014Eres muy insistente.\n\n\u2014\u00bfPor qu\u00e9 crees que sigo viva?\n\n\u2014Porque te has hecho inmune al peligro. \u2014Se dio en las rodillas con la palma de las manos\u2014. No puedo revelarte mis motivos. Pero, si quieres, te explicar\u00e9 una cosa sobre la flor roja.\n\nEsa proposici\u00f3n me pill\u00f3 por sorpresa.\n\n\u2014Adelante.\n\n\u2014\u00bfConoces la historia de Adonis?\n\n\u2014En las escuelas de Scion no ense\u00f1an a los cl\u00e1sicos.\n\n\u2014Claro. Perd\u00f3name.\n\n\u2014Espera. \u2014Pens\u00e9 en los libros robados de Jax. A Jax le encantaba la mitolog\u00eda. Dec\u00eda que era deliciosamente il\u00edcita\u2014. \u00bfEra un dios?\n\n\u2014Era el amante de Afrodita. Era un cazador joven, atractivo y mortal. Afrodita estaba tan prendada de su belleza que prefer\u00eda su compa\u00f1\u00eda a la de los otros dioses. Seg\u00fan la leyenda, el pretendiente de Afrodita, el rey de la guerra Ares, se puso tan celoso que se convirti\u00f3 en jabal\u00ed y mat\u00f3 a Adonis. El joven muri\u00f3 en los brazos de Afrodita, y su sangre manch\u00f3 la tierra.\n\n\u00bbMientras mec\u00eda el cad\u00e1ver de su amado, Afrodita rociaba su sangre con n\u00e9ctar. Y de la sangre de Adonis brot\u00f3 la an\u00e9mona: una flor ef\u00edmera, tan roja como la sangre; y el esp\u00edritu de Adonis acab\u00f3, como todos los esp\u00edritus, languideciendo en el Averno. Zeus oy\u00f3 llorar desconsoladamente a Afrodita, se compadeci\u00f3 de la diosa y accedi\u00f3 a dejar que Adonis pasara la mitad del a\u00f1o vivo y la otra mitad, muerto.\n\nEl Custodio me mir\u00f3.\n\n\u2014Pi\u00e9nsalo, Paige. Quiz\u00e1 no existan los monstruos, pero vuestra mitolog\u00eda encierra algunas verdades.\n\n\u2014No me digas que sois dioses. Creo que no podr\u00eda soportar la idea de que Nashira fuera sagrada.\n\n\u2014Somos muchas cosas, pero \u00absagrados\u00bb no es una de ellas. \u2014Hizo una pausa y agreg\u00f3\u2014: Ya he hablado demasiado. Necesitas descansar.\n\n\u2014No estoy cansada.\n\n\u2014Aun as\u00ed, deber\u00edas dormir. Ma\u00f1ana por la noche quiero ense\u00f1arte una cosa.\n\nMe recost\u00e9 en las almohadas. La verdad era que s\u00ed estaba cansada.\n\n\u2014Esto no significa que conf\u00ede en ti \u2014dije\u2014. Solo significa que lo intento.\n\n\u2014En ese caso, no puedo pedir m\u00e1s. \u2014Dio unas palmaditas en las s\u00e1banas\u2014. Que duermas bien, peque\u00f1a So\u00f1adora.\n\nNo aguantaba m\u00e1s. Me di la vuelta y cerr\u00e9 los ojos; segu\u00ed pensando en flores rojas y dioses.\n\nMe despertaron unos golpes. Detr\u00e1s de la ventana el cielo estaba te\u00f1ido de rosa. Vi al Custodio de pie junto a la chimenea, con una mano en la repisa. Dirigi\u00f3 la mirada hacia la puerta.\n\n\u2014Esc\u00f3ndete, Paige \u2014dijo\u2014. R\u00e1pido.\n\nMe levant\u00e9 de la cama y fui derecha a la puerta detr\u00e1s de las cortinas. La dej\u00e9 un poco entornada, tap\u00e9 la rendija con la cortina de terciopelo rojo y me puse a escuchar. Desde donde estaba ve\u00eda la chimenea.\n\nSe abri\u00f3 la puerta de la c\u00e1mara y entr\u00f3 Nashira; la luz del fuego la ilumin\u00f3. Deb\u00eda de tener una llave de la torre del Custodio. El Custodio se arrodill\u00f3, pero no complet\u00f3 el ritual. Nashira pas\u00f3 una mano por la cama.\n\n\u2014\u00bfD\u00f3nde est\u00e1?\n\n\u2014Durmiendo \u2014contest\u00f3 el Custodio.\n\n\u2014\u00bfEn su habitaci\u00f3n?\n\n\u2014S\u00ed.\n\n\u2014Mentiroso. Duerme aqu\u00ed. Las s\u00e1banas huelen a ella. \u2014Le agarr\u00f3 la barbilla con los dedos desnudos\u2014. \u00bfEst\u00e1s seguro de que quieres ir por ese camino?\n\n\u2014No s\u00e9 qu\u00e9 quieres decir. No pienso en nada ni en nadie m\u00e1s que en ti.\n\n\u2014Puede ser. \u2014Sus dedos se tensaron\u2014. Las cadenas siguen colgadas. No pienses ni por un instante que dudar\u00e9 en enviarte otra vez a la Casa. No pienses ni por un instante que se repetir\u00e1 lo que sucedi\u00f3 en la Era XVIII. Si se repite, no quedar\u00e1 nadie con vida. Ni siquiera t\u00fa. Esta vez, no. \u00bfMe has entendido?\n\nComo \u00e9l no contestaba, le peg\u00f3 en la cara.\n\nDi un respingo.\n\n\u2014\u00a1Cont\u00e9stame!\n\n\u2014He tenido veinte a\u00f1os para reflexionar sobre mi insensatez. Ten\u00edas raz\u00f3n. No se puede confiar en los humanos.\n\nHubo un breve silencio.\n\n\u2014Me alegra o\u00edrlo \u2014dijo Nashira con tono m\u00e1s suave\u2014. Todo ir\u00e1 bien. Pronto tendremos esta torre para nosotros solos. Podr\u00e1s cumplir tu promesa.\n\nEstaba loca. \u00bfC\u00f3mo pod\u00eda pegarle y, a continuaci\u00f3n, hacer una declaraci\u00f3n as\u00ed?\n\n\u2014\u00bfSignifica eso \u2014dijo el Custodio\u2014 que a 40 se le ha agotado el tiempo?\n\nMe qued\u00e9 inm\u00f3vil, escuchando.\n\n\u2014Est\u00e1 preparada. S\u00e9 que posey\u00f3 a 12 en la ciudadela. Me lo dijo tu prima. \u2014Le acarici\u00f3 el ment\u00f3n\u2014. Has cultivado muy bien su don.\n\n\u2014Para ti, mi soberana. \u2014Levant\u00f3 la cabeza y la mir\u00f3\u2014. \u00bfPiensas reclamarla en privado? \u00bfO mostrar\u00e1s a todo Scion tu gran poder?\n\n\u2014Cualquiera de las dos cosas bastar\u00e1. Por fin podr\u00e9 onirambulear. Por fin tendr\u00e9 el poder de invadir, de poseer. Y todo gracias a ti, mi querido Arcturus. \u2014Puso un vial en la repisa de la chimenea, y su tono de voz volvi\u00f3 a enfriarse\u2014. Esta ser\u00e1 tu \u00faltima dosis de amaranto hasta el Bicentenario. Creo que necesitas tiempo para reflexionar sobre tus cicatrices. Para recordar por qu\u00e9 debes mirar hacia el futuro y no hacia el pasado.\n\n\u2014Soportar\u00e9 cualquier cosa que me pidas.\n\n\u2014No tendr\u00e1s que sufrir mucho tiempo. Pronto alcanzaremos la gloria. \u2014Se volvi\u00f3 hacia la puerta\u2014. Cuida de ella, Arcturus.\n\nLa puerta se cerr\u00f3. El Custodio se levant\u00f3. Esper\u00e9: quer\u00eda ver qu\u00e9 hac\u00eda. Entonces dio un pu\u00f1etazo y destroz\u00f3 la urna de cristal de la repisa de la chimenea. Me met\u00ed en la cama y me puse a escuchar el silencio.\n\nNo era mi enemigo, como yo cre\u00eda.\n\nNashira hab\u00eda dicho que volver\u00eda a llevarlo a la Casa. Eso era una prueba de que hab\u00eda participado en la Era de Huesos XVIII. Una prueba de su traici\u00f3n. Eso era lo que hab\u00eda querido decir Thuban cuando hab\u00eda amenazado a Terebell. Hab\u00edan intentado ayudarnos y los hab\u00edan castigado por ello. Hab\u00edan escogido el bando equivocado, el de los perdedores.\n\nMe pas\u00e9 horas dando vueltas en la cama. No pod\u00eda parar de pensar en la conversaci\u00f3n que hab\u00eda o\u00eddo ni en la bofetada que Nashira le hab\u00eda dado al Custodio. En c\u00f3mo \u00e9l se hab\u00eda arrodillado ante ella. En que pronto, muy pronto, Nashira pensaba deshacerse de m\u00ed. Me quit\u00e9 las s\u00e1banas de encima y me qued\u00e9 tumbada en la oscuridad con los ojos abiertos. Hab\u00eda tardado mucho tiempo en entender que el Custodio estaba en el mismo bando que yo.\n\nMe acord\u00e9 de las cicatrices que ten\u00eda Terebell en la espalda, esas que Thuban Sargas hab\u00eda mencionado con un deje de crueldad. Su familia y \u00e9l hab\u00edan marcado a Terebell. El Custodio y ella eran los marcados. Hab\u00eda ocurrido algo terrible en la Casa despu\u00e9s de ese d\u00eda, el Novembertide de 2039. Yo no conoc\u00eda a Terebell, pero ella me hab\u00eda salvado la vida; estaba en deuda con ella. Y estaba en deuda con el Custodio por cuidar de m\u00ed.\n\nSi hab\u00eda algo que no pod\u00eda soportar era estar en deuda con alguien. Pero la pr\u00f3xima vez que el Custodio me hablara, le escuchar\u00eda. Me incorpor\u00e9. No: no la pr\u00f3xima vez, sino en ese mismo momento. Necesitaba hablar con \u00e9l. Confiar en \u00e9l era mi \u00fanica salida. No estaba dispuesta a morir all\u00ed. Necesitaba saber de una vez por todas qu\u00e9 quer\u00eda Arcturus Mesarthim. Necesitaba saber si estaba dispuesto a ayudarme.\n\nMe levant\u00e9 de la cama y baj\u00e9 a la c\u00e1mara. La encontr\u00e9 vac\u00eda. Fuera llov\u00eda a c\u00e1ntaros. El reloj de pared dio las cuatro de la madrugada. Cog\u00ed la nota que hab\u00eda encima del escritorio.\n\nHe ido a la capilla. Volver\u00e9 antes del amanecer.\n\nNo me importaba no dormir. Estaba harta de juegos, de hablar en idiomas diferentes con \u00e9l. Me calc\u00e9 las botas y sal\u00ed de la torre.\n\nEn la calle soplaba un fuerte viento. En el patio hab\u00eda una vigilante. Esper\u00e9 a que hubiera pasado antes de salir corriendo. Los truenos y la oscuridad me cubr\u00edan, y me permitieron escabullirme sin ser vista. Pero por encima de la lluvia se o\u00eda otro sonido: m\u00fasica. La segu\u00ed hasta otro pasillo, donde encontr\u00e9 una gran puerta entornada. Detr\u00e1s hab\u00eda una peque\u00f1a capilla, separada del resto del edificio por una celos\u00eda de piedra muy elaborada. La luz de las velas parpadeaba en la oscuridad. Hab\u00eda alguien tocando un \u00f3rgano. Las notas resonaban en mis o\u00eddos y en mi pecho.\n\nEn la celos\u00eda hab\u00eda una portezuela abierta. Pas\u00e9 por ella y sub\u00ed unos escalones. Arriba estaba el \u00f3rgano. El Custodio estaba sentado en el banco, de espaldas a m\u00ed. La m\u00fasica ascend\u00eda por los tubos hasta el techo, inundaba toda la capilla y segu\u00eda ascendiendo m\u00e1s all\u00e1. Era un sonido de gran intensidad emocional. Nadie pod\u00eda tocar aquello sin cierto grado de sentimiento.\n\nCes\u00f3 la m\u00fasica. El Custodio gir\u00f3 la cabeza. Como no dijo nada, me sent\u00e9 en el banco, a su lado. Permanecimos a oscuras, con la \u00fanica luz de las velas y de sus ojos.\n\n\u2014Deber\u00edas estar durmiendo.\n\n\u2014Ya he dormido suficiente. \u2014Pas\u00e9 los dedos por las teclas\u2014. No sab\u00eda que los refa\u00edtas pudieran tocar instrumentos.\n\n\u2014Con los a\u00f1os hemos llegado a dominar el arte de la imitaci\u00f3n.\n\n\u2014Eso no era una imitaci\u00f3n. Eras t\u00fa.\n\nHubo un largo silencio.\n\n\u2014Has venido a preguntarme por tu libertad \u2014dijo \u00e9l\u2014. Eso es lo que quieres.\n\n\u2014S\u00ed.\n\n\u2014Por supuesto. Quiz\u00e1 no me creas, pero es lo que m\u00e1s deseo. Este lugar me ha provocado unas ansias tremendas de conocer mundo. Anhelo tu fuego, los paisajes que t\u00fa has visto. Y sin embargo aqu\u00ed estoy, doscientos a\u00f1os despu\u00e9s de mi llegada. Sigo siendo un prisionero, por mucho que me disfrace de rey.\n\nCon sus ansias de conocer el mundo, al menos, pod\u00eda empatizar.\n\n\u2014Una vez me traicionaron. La v\u00edspera de Novembertide, antes de que comenzara el levantamiento de la Era de Huesos XVIII, un humano decidi\u00f3 traicionarnos a todos. A cambio de la libertad, el traidor sacrific\u00f3 a todos los habitantes de esta ciudad. \u2014Me mir\u00f3 y continu\u00f3\u2014: Comprender\u00e1s por qu\u00e9 Nashira no se siente amenazada por la perspectiva de una segunda rebeli\u00f3n. Cree que sois todos demasiado ego\u00edstas para uniros y trabajar juntos.\n\nS\u00ed, lo comprend\u00eda. Despu\u00e9s de hacer tantos planes para conseguir la libertad de los humanos, y que luego mordi\u00e9ramos la mano que luchaba por nosotros... No me extra\u00f1aba que el Custodio no hubiera confiado en m\u00ed. No me extra\u00f1aba que se hubiera mostrado tan fr\u00edo.\n\n\u2014Pero t\u00fa, Paige... T\u00fa la amenazas. Sabe que eres uno de los Siete Sellos, que eres la So\u00f1adora P\u00e1lida. T\u00fa tienes poder para trasladar el esp\u00edritu del sindicato a esta ciudad. Y ella teme ese esp\u00edritu.\n\n\u2014No hay nada que temer. Lo forman un hatajo de ladronzuelos y traidores.\n\n\u2014Eso depende de qui\u00e9nes sean sus l\u00edderes. Pero tiene potencial para convertirse en algo mucho mayor.\n\n\u2014El sindicato existe porque existe Scion. Scion existe porque existen los refa\u00edtas \u2014dije\u2014. Hab\u00e9is fabricado a vuestro propio enemigo.\n\n\u2014Ya capto la iron\u00eda. Y Nashira tambi\u00e9n. \u2014Se volvi\u00f3 y me mir\u00f3\u2014. La Era XVIII se rebel\u00f3 porque los prisioneros estaban acostumbrados a estar organizados. Eran fuertes y solidarios. Debemos recuperar esa fuerza. Y esta vez no debemos fallar. \u2014Mir\u00f3 hacia la ventana\u2014. Yo no debo fallar.\n\nNo dije nada. Estuve tentada de tomarle la mano, a solo unos cent\u00edmetros de las m\u00edas, sobre las teclas.\n\nAl final no me arriesgu\u00e9.\n\n\u2014Quiero irme \u2014dije\u2014. Es lo \u00fanico que quiero. Volver a la ciudadela con tanta gente como sea posible.\n\n\u2014Entonces nuestros objetivos son diferentes. Si hemos de ayudarnos el uno al otro, hemos de conciliar esas diferencias.\n\n\u2014T\u00fa \u00bfqu\u00e9 quieres?\n\n\u2014Hacer da\u00f1o a los Sargas. Mostrarles qu\u00e9 significa tener miedo.\n\nPens\u00e9 en Julian. Pens\u00e9 en Finn. Y en Liss, que se deslizaba hacia la amaurosis.\n\n\u2014Y \u00bfc\u00f3mo pretendes conseguirlo?\n\n\u2014Tengo una idea. \u2014Su mirada descendi\u00f3 hasta la m\u00eda\u2014. Me gustar\u00eda ense\u00f1arte una cosa. Si quieres.\n\nFui a contestar, pero en el \u00faltimo momento me call\u00e9. Sus ojos, amarillos, ten\u00edan cada vez m\u00e1s calidez. Yo casi notaba su calor.\n\n\u2014Quiero confiar en ti \u2014dijo.\n\n\u2014Puedes confiar.\n\n\u2014Pues entonces, ven conmigo.\n\n\u2014\u00bfAd\u00f3nde?\n\n\u2014A ver a Michael. \u2014Se levant\u00f3\u2014. Al norte del Gran Patio hay un edificio en desuso. Pero los guardias no deben vernos.\n\nAquello s\u00ed me interesaba. Asent\u00ed con la cabeza.\n\nSalimos de la capilla. El Custodio escudri\u00f1\u00f3 el soportal por si hab\u00eda guardias y no vio a ninguno. Entonces hizo una se\u00f1a con la mano; un fantasma que estaba all\u00ed cerca se dio la vuelta y se alej\u00f3 a toda velocidad por el pasillo apagando las antorchas. Nos quedamos a oscuras, y el Custodio me dio la mano. Tuve que acelerar el paso para seguir el ritmo de sus zancadas. Me gui\u00f3 hasta un sendero de grava.\n\nEl edificio en desuso era tan sobrecogedor como los otros. El arrebol del alba me permiti\u00f3 distinguir una serie de arcos, unas ventanas rectangulares con barrotes y un t\u00edmpano con un c\u00edrculo tallado en su interior. El Custodio me llev\u00f3 por una galer\u00eda, se sac\u00f3 una llave de la manga y abri\u00f3 una puerta medio podrida.\n\n\u2014\u00bfQu\u00e9 es este sitio? \u2014pregunt\u00e9.\n\n\u2014Es lo que vosotros llamar\u00edais un \u00abpiso franco\u00bb.\n\nEntr\u00f3; yo lo segu\u00ed y cerr\u00e9 la puerta.\n\nEl Custodio ech\u00f3 el cerrojo.\n\nDentro estaba oscuro como la boca del lobo. Los ojos del Custodio proyectaban una luz tenue por las paredes.\n\n\u2014Antes esto era una bodega \u2014me explic\u00f3 mientras camin\u00e1bamos\u2014. Pas\u00e9 a\u00f1os vaci\u00e1ndola. Por ser el refa\u00edta de rango m\u00e1s alto de esta residencia, pod\u00eda prohibir la entrada en todos los edificios que quisiera. A esta casa solo pueden acceder unos pocos individuos. Entre ellos, Michael.\n\n\u2014Y \u00bfqui\u00e9n m\u00e1s?\n\n\u2014Ya sabes qui\u00e9n m\u00e1s.\n\n\u00abLos marcados.\u00bb Me estremec\u00ed. Aquello era su piso franco, su lugar de reuni\u00f3n. Abri\u00f3 una cancela que hab\u00eda en la pared. Detr\u00e1s hab\u00eda una abertura por donde se pod\u00eda entrar a gatas.\n\n\u2014Entra \u2014dijo.\n\n\u2014\u00bfQu\u00e9 hay ah\u00ed?\n\n\u2014Alguien que puede ayudarte.\n\n\u2014Cre\u00eda que ibas a ayudarme t\u00fa.\n\n\u2014Los humanos de esta ciudad no se fiar\u00edan de un refa\u00edta para organizarse. Creer\u00edan que les estaba tendiendo una trampa. Tendr\u00e1s que hacerlo t\u00fa.\n\n\u2014Ya guiaste a los humanos una vez.\n\nEl Custodio desvi\u00f3 la mirada.\n\n\u2014Entra \u2014me dijo\u2014. Michael te est\u00e1 esperando.\n\nTen\u00eda una expresi\u00f3n sombr\u00eda. Me pregunt\u00e9 cu\u00e1ntos a\u00f1os de trabajo habr\u00eda desperdiciado.\n\n\u2014Esta vez ser\u00e1 diferente \u2014dije.\n\n\u00c9l no dijo nada m\u00e1s. Ten\u00eda los ojos apagados, y le brillaba la piel. La falta de amaranto ya le estaba pasando factura.\n\nNo ten\u00eda alternativa: me met\u00ed en el t\u00fanel, fr\u00edo y oscuro. El Custodio entr\u00f3 tambi\u00e9n y cerr\u00f3 la cancela.\n\n\u2014Sigue \u2014dijo.\n\nObedec\u00ed. Cuando llegu\u00e9 al final, una mano delgada tom\u00f3 la m\u00eda. Levant\u00e9 la vista y vi a Michael; una vela le iluminaba la cara. El Custodio sali\u00f3 tambi\u00e9n del t\u00fanel.\n\n\u2014Ens\u00e9\u00f1aselo, Michael. Es tu obra.\n\nMichael dio una cabezada anim\u00e1ndome a seguirlo en la oscuridad. Accion\u00f3 un interruptor; se encendi\u00f3 una luz que revel\u00f3 una estancia subterr\u00e1nea. Me qued\u00e9 mirando aquella luz un momento, tratando de discernir por qu\u00e9 parec\u00eda tan rara. Y entonces lo entend\u00ed.\n\n\u2014Electricidad. \u2014No pod\u00eda dejar de mirarla\u2014. Aqu\u00ed no hay electricidad. \u00bfC\u00f3mo hab\u00e9is...?\n\nMichael sonre\u00eda.\n\n\u2014Oficialmente, la corriente solo se puede restablecer en una residencia: Balliol. All\u00ed es donde los casacas rojas se coordinan con el Arconte de Westminster durante las Eras de Huesos \u2014explic\u00f3 el Custodio\u2014. Ese edificio tiene una instalaci\u00f3n el\u00e9ctrica moderna. Por suerte, Magdalen tambi\u00e9n.\n\nMichael me gui\u00f3 hasta el rinc\u00f3n, donde un pa\u00f1o de terciopelo cubr\u00eda un objeto ancho y rectangular. Levant\u00f3 el pa\u00f1o y me mostr\u00f3 el objeto de su orgullo y su alegr\u00eda: un ordenador. Terriblemente anticuado \u2014seguramente de alrededor del a\u00f1o 2030\u2014, pero un ordenador. Una conexi\u00f3n con el mundo exterior.\n\n\u2014Lo rob\u00f3 de Balliol \u2014explic\u00f3 el Custodio, y la sombra de una sonrisa acarici\u00f3 sus labios\u2014. Consigui\u00f3 restablecer la corriente el\u00e9ctrica en este edificio y establecer una conexi\u00f3n con la constelaci\u00f3n sat\u00e9lite de Scion.\n\n\u2014Por lo visto eres un prodigio, Michael. \u2014Me sent\u00e9 ante el ordenador. Michael sonri\u00f3 con timidez\u2014. \u00bfPara qu\u00e9 lo us\u00e1is?\n\n\u2014No podemos conectarlo a menudo, porque corremos un riesgo cada vez que restablecemos la electricidad; pero lo utilizamos para monitorizar el desarrollo de la Era XX, por ejemplo.\n\n\u2014\u00bfPuedo ver algo?\n\nMichael se inclin\u00f3 por encima de mi hombro y abri\u00f3 un archivo con el nombre \u00abMahoney, Paige Eva, 07-Mar-59\u00bb que conten\u00eda secuencias de v\u00eddeo tomadas desde un helic\u00f3ptero. La c\u00e1mara hizo un zoom sobre mi cara. Me vi corriendo por los tejados, saltando desde el borde de un edificio. La distancia parec\u00eda insalvable; contuve la respiraci\u00f3n, pero la chica de la pantalla lo consigui\u00f3. El piloto grit\u00f3: \u00ab\u00a1Lanzadle un flux!\u00bb; me precipit\u00e9 desde una altura de quince metros, y se me enganch\u00f3 la mochila en una cuerda de tender. Qued\u00e9 colgando, inconsciente, como un cad\u00e1ver. El c\u00e1mara de la DVN re\u00eda entrecortadamente. \u00abPor las barbas de Weaver \u2014dec\u00eda\u2014. Nunca hab\u00eda visto a una zorra con tanta suerte.\u00bb Y nada m\u00e1s.\n\n\u2014Qu\u00e9 bonito \u2014dije.\n\nMichael me dio unas palmaditas en el hombro.\n\n\u2014Lamentamos mucho que no lograras evitarlos \u2014dijo el Custodio\u2014, aunque nos alegramos de que sobrevivieras.\n\nArque\u00e9 una ceja y pregunt\u00e9:\n\n\u2014\u00bfInvitaste a tus amigos a ver esto, como quien ve una pel\u00edcula?\n\n\u2014M\u00e1s o menos.\n\nSe levant\u00f3 y empez\u00f3 a pasearse por el s\u00f3tano.\n\n\u2014\u00bfQu\u00e9 quieres que haga? \u2014dije.\n\n\u2014Voy a darte la opci\u00f3n de pedir ayuda. \u2014Me qued\u00e9 mir\u00e1ndolo, y a\u00f1adi\u00f3\u2014: Quiero que llames a los Siete Sellos.\n\n\u2014No. Nashira los localizar\u00eda \u2014dije\u2014. Va detr\u00e1s de Jaxon. No pienso traerlo aqu\u00ed.\n\nEl rostro de Michael se ensombreci\u00f3.\n\n\u2014Al menos hazles saber d\u00f3nde est\u00e1s \u2014propuso el Custodio\u2014. Por si todo sale mal.\n\n\u2014\u00bfPor si todo sale mal?\n\n\u2014Por si se frustra tu fuga.\n\n\u2014Mi fuga.\n\n\u2014S\u00ed. \u2014El Custodio se dio la vuelta y me mir\u00f3\u2014. Me preguntaste por el tren. Pues bien: la noche del Bicentenario, ese tren traer\u00e1 a un nutrido grupo de emisarios de Scion a la colonia. Y luego los devolver\u00e1 a Londres.\n\nTard\u00e9 un momento en asimilar sus palabras.\n\n\u2014Podremos volver a casa \u2014dije\u2014. \u00bfCu\u00e1ndo?\n\n\u2014La v\u00edspera del uno de septiembre. \u2014El Custodio se sent\u00f3 en un barril que serv\u00eda de taburete\u2014. Si no quieres ponerte en contacto con los Siete Sellos, puedes utilizar esta habitaci\u00f3n para dise\u00f1ar tus planes. Tienen que ser mejores que los m\u00edos, Paige. Debes recordar las lecciones del sindicato. \u2014Me mir\u00f3 a los ojos y prosigui\u00f3\u2014: Yo comet\u00ed un error la \u00faltima vez. Planeamos atacar a los Sargas de d\u00eda, mientras el resto de la ciudad dorm\u00eda. Gracias al traidor nos estaban esperando; pero, aunque no nos hubieran traicionado, los Sargas habr\u00edan detectado nuestros movimientos a trav\u00e9s del \u00e9ter. Tenemos que atacar cuando ya haya mucha actividad, cuando los Sargas est\u00e9n distra\u00eddos. Cuando su capacidad de contraatacar est\u00e9 limitada por su necesidad de mantener la apariencia de control. \u00bfQu\u00e9 mejor momento que el Bicentenario?\n\nAsent\u00ed con la cabeza.\n\n\u2014Y de paso, asustaremos a unos cuantos funcionarios de Scion.\n\n\u2014Exactamente. \u2014Me sostuvo la mirada\u2014. A partir de ahora, este ser\u00e1 tu piso franco. En el ordenador hay mapas detallados de Sheol I que puedes usar para planear tu salida del centro de la ciudad. Si logras llegar a la pradera a tiempo, podr\u00e1s tomar el tren a Londres.\n\n\u2014\u00bfA qu\u00e9 hora sale el tren?\n\n\u2014Todav\u00eda no lo s\u00e9. No puedo hacer muchas preguntas, pero Michael tiene todas las antenas desplegadas. Lo averiguaremos.\n\n\u2014Me dijiste que ten\u00edamos objetivos distintos \u2014dije\u2014. Que t\u00fa buscas otra cosa.\n\n\u2014Scion cree que somos tan poderosos que nada puede destruirnos. Que no tenemos debilidades. Quiero que les demuestres que se equivocan.\n\n\u2014\u00bfC\u00f3mo?\n\n\u2014Hace tiempo que sospecho que Nashira intentar\u00e1 matarte durante el Bicentenario para reclamar tu don. Hay una forma muy sencilla de humillarla. \u2014Me puso los dedos bajo la barbilla y me la levant\u00f3 suavemente\u2014. Imp\u00eddeselo.\n\nEscudri\u00f1\u00e9 su cara. Ten\u00eda los ojos apagados.\n\n\u2014Si lo consigo \u2014dije\u2014, quiero que se me conceda un favor.\n\n\u2014Te escucho.\n\n\u2014Liss. No consigo llegar hasta ella. Tengo las cartas, pero quiz\u00e1 no las acepte. Necesito... \u2014Las palabras se atascaron en mi garganta, y tuve que obligarlas a salir\u2014: Necesito tu ayuda.\n\n\u2014Tu amiga lleva mucho tiempo con choque espiritual. Necesitar\u00e1 amaranto para recuperarse.\n\n\u2014Ya lo s\u00e9.\n\n\u2014Sabes que Nashira ha interrumpido mi suministro.\n\nSin desviar la mirada, repliqu\u00e9:\n\n\u2014Tienes la \u00faltima dosis.\n\nEl Custodio se sent\u00f3 a mi lado. Yo sab\u00eda muy bien qu\u00e9 le estaba pidiendo. \u00c9l depend\u00eda del amaranto.\n\n\u2014No quieres hacer venir a tus amigos... \u2014Tamborile\u00f3 con los dedos en una rodilla, con aire pensativo\u2014. Pero si yo te ofreciera la libertad, \u00bfaceptar\u00edas, aunque eso significara dejar a Liss aqu\u00ed?\n\n\u2014\u00bfMe est\u00e1s haciendo una oferta?\n\n\u2014Es posible.\n\nSab\u00eda por qu\u00e9 me lo preguntaba. Estaba poni\u00e9ndome a prueba; quer\u00eda comprobar si yo era lo bastante ego\u00edsta para dejar atr\u00e1s a alguien tan vulnerable.\n\n\u2014Me expongo a un grave peligro \u2014agreg\u00f3\u2014. Si alg\u00fan humano informara a los Sargas, me castigar\u00edan severamente por haber ayudado a un humano. Pero si est\u00e1s dispuesta a quedarte un poco m\u00e1s, a correr un riesgo por m\u00ed y por los tuyos, yo me arriesgar\u00e9 por ella. Ese es el trato que te propongo.\n\nMe lo pens\u00e9. Por un instante, me plante\u00e9 abandonar a Liss y apostar por mi libertad. Volver a Londres, olvidarme de aquel lugar y no mirar atr\u00e1s. Sent\u00ed nacer en m\u00ed la verg\u00fcenza, caliente y veloz. Cerr\u00e9 los ojos.\n\n\u2014No \u2014dije\u2014. Quiero que ayudes a Liss.\n\nNotaba el peso de su mirada sobre m\u00ed.\n\n\u2014En ese caso, la ayudar\u00e9 \u2014dijo.\n\nSe hab\u00eda congregado un grupito de bufones en la choza. Cinco de ellos, entre los que se encontraban Cyril y Julian, estaban api\u00f1ados unos contra otros, cabizbajos y con las manos juntas para protegerse del fr\u00edo. Ca\u00edan gotas de lluvia de la tela que hab\u00edan metido en las rendijas entre los tablones.\n\nLiss llevaba tanto tiempo con choque espiritual que era poco probable que se recuperara. Lo \u00fanico que pod\u00edan hacer era velarla en silencio. Si sobreviv\u00eda, quedar\u00eda reducida a una c\u00e1scara amaur\u00f3tica de su antiguo ser. Si mor\u00eda, uno de ellos recitar\u00eda el treno y la desterrar\u00eda m\u00e1s all\u00e1 del alcance de sus captores. De una forma o de otra, estos perder\u00edan a su actriz m\u00e1s preciada, Liss Rymore, la chica que nunca se ca\u00eda.\n\nAl vernos llegar a Michael y a m\u00ed con el Custodio, todos se apartaron asustados. Cyril se refugi\u00f3 en un rinc\u00f3n, aterrado. Los otros se limitaron a mirar y a murmurar entre ellos. \u00bfQu\u00e9 hac\u00eda all\u00ed el consorte de sangre, la mano derecha de Nashira? \u00bfPor qu\u00e9 interrump\u00eda el velatorio?\n\nJulian fue el \u00fanico que no se movi\u00f3.\n\n\u2014Paige...\n\nMe llev\u00e9 un dedo a los labios.\n\nLiss estaba tumbada sobre sus mantas y tapada con una s\u00e1bana sucia. Llevaba retales de seda entrelazados en el pelo a modo de amuletos. Julian le tom\u00f3 una mano sin quitarle los ojos de encima al intruso.\n\nEl Custodio se arrodill\u00f3 junto a Liss. Ten\u00eda las mand\u00edbulas apretadas, pero no mencion\u00f3 su dolor.\n\n\u2014Dame el amaranto, Paige.\n\nLe di el vial. El \u00faltimo vial. Su \u00faltima dosis.\n\n\u2014Las cartas \u2014dijo entonces. Estaba completamente concentrado en su trabajo. Se las di\u2014. Y el pu\u00f1al.\n\nMichael me dio un pu\u00f1al con el mango negro. Lo saqu\u00e9 de la funda y se lo di al Custodio. M\u00e1s susurros. Julian sujetaba la mano de Liss en el regazo y me miraba fijamente.\n\n\u2014Conf\u00eda en m\u00ed \u2014le dije en voz baja.\n\nTrag\u00f3 saliva.\n\nEl Custodio quit\u00f3 el tap\u00f3n del vial de amaranto. Se puso unas gotas en las yemas de los dedos, sin quitarse los guantes, y moj\u00f3 los labios y el surco del filtrum de Liss con aquel aceite. Julian segu\u00eda sin soltarle la mano, pese a que los fr\u00edos dedos de ella no respond\u00edan. El Custodio le aplic\u00f3 unas gotas de amaranto en las sienes; entonces tap\u00f3 el vial y me lo dio. Cogi\u00f3 el pu\u00f1al por la hoja y se lo ofreci\u00f3 a Julian.\n\n\u2014P\u00ednchale los dedos \u2014dijo.\n\n\u2014\u00bfQu\u00e9?\n\n\u2014Necesito un poco de su sangre.\n\nJulian me mir\u00f3. Asent\u00ed con la cabeza. Con mano firme, Julian asi\u00f3 el mango.\n\n\u2014Perd\u00f3name, Liss \u2014dijo.\n\nLa pinch\u00f3 en cada una de las yemas de los dedos, y aparecieron unas diminutas gotas de sangre. El Custodio asinti\u00f3, satisfecho.\n\n\u2014Paige, Michael: extended las cartas.\n\nObedecimos. Dispusimos la baraja nueva formando un semic\u00edrculo. El Custodio le cogi\u00f3 una mano a Liss y le pas\u00f3 los dedos por las cartas, manchando las ilustraciones con su sangre.\n\nA continuaci\u00f3n el Custodio limpi\u00f3 la hoja del cuchillo con un pa\u00f1o. Se quit\u00f3 el guante de la mano izquierda y lo encerr\u00f3 en el pu\u00f1o. Se oy\u00f3 un grito de asombro. Los refas nunca se quitaban los guantes. \u00bfTen\u00edan manos? S\u00ed. Las del Custodio eran grandes, con cicatrices en los nudillos. Volvi\u00f3 a o\u00edrse otro grito ahogado cuando recorri\u00f3 la palma de su mano con la afilada punta del cuchillo, abriendo en ella un corte del que le brot\u00f3 la sangre. Empec\u00e9 a ver borroso. El Custodio levant\u00f3 el brazo y dej\u00f3 caer unas gotas de ectoplasma en cada carta. Como Afrodita rociando de n\u00e9ctar la sangre de Adonis. Sent\u00ed que empezaban a llegar esp\u00edritus, atra\u00eddos por las cartas, por Liss, por el Custodio. Formaron un tri\u00e1ngulo, una abertura en el \u00e9ter. El Custodio estaba abriendo la puerta. Se puso el guante, recogi\u00f3 las cartas y volvi\u00f3 a juntarlas en una pila. Las puso en el escote de Liss, en contacto con su piel, y le coloc\u00f3 las manos encima.\n\n\u2014Y de la sangre de Adonis \u2014dijo\u2014 surgi\u00f3 la vida.\n\nLiss abri\u00f3 los ojos.\n\nUno de septiembre de 2059. Doscientos a\u00f1os despu\u00e9s de que una tormenta de luces extra\u00f1as cruzara el cielo. Doscientos a\u00f1os despu\u00e9s de que lord Palmerston cerrara su trato con los refa\u00edtas. Doscientos a\u00f1os despu\u00e9s de que comenzara la persecuci\u00f3n de la clarividencia. Y lo m\u00e1s importante: doscientos a\u00f1os desde el establecimiento de Sheol I y la tradici\u00f3n de las Eras de Huesos.\n\nEnfrente de m\u00ed, una chica me observaba desde el espejo de marco dorado. Ten\u00eda las mejillas descarnadas y las mand\u00edbulas apretadas. Todav\u00eda segu\u00eda sorprendi\u00e9ndome que ese semblante duro y fr\u00edo fuera el m\u00edo.\n\nLlevaba un vestido blanco, con mangas tres cuartos y escote cuadrado. La tela, elastizada, se adher\u00eda a la poca figura que me quedaba. El Custodio hac\u00eda todo lo posible por alimentarme, pero no siempre ten\u00eda comida para darme y, si me la daba, se arriesgaba a levantar sospechas. El resto del tiempo, com\u00eda _skilly_ y _toke_ con los bufones.\n\nNashira no me hab\u00eda invitado a ning\u00fan otro banquete.\n\nMe alis\u00e9 el vestido. Me hab\u00edan hecho una concesi\u00f3n especial para que pudiera asistir a la ceremonia vestida de blanco. Seg\u00fan Nashira, era una muestra de buena voluntad. Pero yo no me lo tragaba. Estaba preparada. Oculto bajo el escote llevaba el colgante que me hab\u00eda dado el Custodio. Durante semanas no lo hab\u00eda tocado, pero esa noche quiz\u00e1 me fuera \u00fatil. Dentro de uno de los botines llevaba escondido un cuchillo; casi no pod\u00eda andar con ellos, pero los refa\u00edtas quer\u00edan que pareci\u00e9ramos fuertes, y no d\u00e9biles y maltratados. Esa noche quer\u00edan vernos bien erguidos.\n\nLa c\u00e1mara estaba en silencio, iluminada con una vela. El Custodio hab\u00eda ido a recibir a los emisarios con los otros refa\u00edtas. Me hab\u00eda dejado una nota apoyada en el gram\u00f3fono. Me sent\u00e9 a su escritorio y deslic\u00e9 un dedo sobre las letras:\n\nHa llegado el momento. B\u00fascame en el Consistorio.\n\nLa tir\u00e9 a las brasas de la chimenea. En la penumbra di cuerda al gram\u00f3fono y coloqu\u00e9 la aguja sobre el disco. Ser\u00eda la \u00faltima vez que lo oyera sonar. Pasara lo que pasase esa noche, ya no volver\u00eda a la Torre del Fundador.\n\nUnas voces suaves y resonantes invadieron la c\u00e1mara. Mir\u00e9 el t\u00edtulo del disco. _I'll Be Home_ , volver\u00e9 a casa. S\u00ed, iba a volver a casa. Si todo sal\u00eda como estaba previsto, por la ma\u00f1ana estar\u00eda en mi casa. Ya estaba harta de ver a los bufones en la miseria, y de llamarlos \u00abbufones\u00bb. Estaba harta de ver a Liss comer grasa y pan duro porque no ten\u00eda nada m\u00e1s que llevarse a la boca. Estaba harta de los casacas rojas y de los emim. Estaba harta de que me llamaran 40. Estaba harta de aquella maldita ciudad y de todos sus habitantes. No aguantaba ni una noche m\u00e1s.\n\nUna hoja de papel se desliz\u00f3 por debajo de la puerta. Me arrodill\u00e9 y la recog\u00ed de la alfombra.\n\nLas notas que me dejaba el Custodio me hab\u00edan dado una idea, y hab\u00eda convencido a Julian para que organizara un grupo de recadistas como el que ten\u00eda Jax en la ciudadela, y as\u00ed tener informados a los habitantes de las residencias envi\u00e1ndoles notas que los amaur\u00f3ticos se encargaban de entregar.\n\nOrpheus ya lo ha hecho. Todo listo.\n\nLUCKY\n\nSonre\u00ed para m\u00ed. \u00abLucky\u00bb era Felix: le hab\u00eda pedido que firmara sus mensajes con un nombre falso. Orpheus era Michael.\n\nNo nos hab\u00eda costado convencer a Duckett para que nos prestara su pericia particular. Tras amenazarlo con descubrirle a Nashira su peque\u00f1o negocio de tr\u00e1fico de f\u00e1rmacos (\u00ab\u00a1No, por favor! \u00a1Tened piedad de un pobre anciano!\u00bb), Julian y yo lo hab\u00edamos obligado a prepararles una sorpresa a los casacas rojas. Algo que les hiciera reaccionar con lentitud cuando actu\u00e1ramos contra los refa\u00edtas. Se hab\u00eda resistido un poco, pero al final hab\u00eda cedido (\u00ab\u00a1No lo conseguir\u00e9is, os machacar\u00e1n como hicieron con los anteriores!\u00bb). \u00c1ster morado en polvo mezclado con somn\u00edfero. Perfecto.\n\nUna vez que estuvo hecho, us\u00e9 un pu\u00f1ado de su propio \u00e1ster blanco para borrarle la memoria. No me gustaban los cobardes.\n\nLe hab\u00edamos dado el preparado a Michael, que se hab\u00eda encargado de echarlo en el vino que les hab\u00edan servido a los casacas rojas en su banquete de la v\u00edspera del Bicentenario. Si todo sal\u00eda bien, no quedar\u00eda ni uno solo en forma para defenderse.\n\nMir\u00e9 por la ventana. Los emisarios hab\u00edan llegado a las ocho, con sus mejores galas, escoltados por centinelas armados. Aquellos hombres y mujeres de Scion ven\u00edan a ser testimonios de un nuevo acuerdo, el Gran Tratado Territorial. Ese tratado permitir\u00eda a los refa\u00edtas establecer una ciudad de control en Par\u00eds, la primera fuera de Inglaterra. Sheol II. Scion dejar\u00eda de ser un imperio en fase embrionaria: habr\u00eda nacido, vivir\u00eda.\n\nAquello era solo el principio. Si los refa\u00edtas ten\u00edan a todos los videntes encerrados en colonias penitenciarias, no habr\u00eda forma de que el resto de la humanidad los rechazara. El \u00e9ter era nuestra \u00fanica arma. Si nadie pod\u00eda usarla, ser\u00edamos un blanco seguro. Todos nosotros.\n\nPero nada de eso me importaba aquella noche. Lo \u00fanico que me importaba era volver a Seven Dials. Con el sindicato corrupto. Con mi banda. Con Nick. En ese momento era lo \u00fanico que deseaba.\n\nEl gram\u00f3fono segu\u00eda sonando. Me sent\u00e9 al escritorio y mir\u00e9 la luna por la ventana. No estaba llena, y no se ve\u00edan las estrellas.\n\nLiss, Julian y yo llev\u00e1bamos semanas sembrando la discordia por la ciudad, y utilizando el piso franco como guarida. Suhail y el Capataz no pod\u00edan o\u00edrnos cuando est\u00e1bamos all\u00ed. Liss se hab\u00eda recuperado por completo de su trauma y, con renovadas ganas de sobrevivir, hab\u00eda participado activamente reclutando a bufones. Al principio no estaba convencida, hasta que una noche se derrumb\u00f3. \u00abNo puedo seguir viviendo as\u00ed \u2014dijo\u2014. Y no puedo impedir que os rebel\u00e9is. As\u00ed que adelante.\u00bb\n\nY nos lanzamos.\n\nLa mayor\u00eda de los casacas y los actores nos hab\u00edan ofrecido su apoyo. Los que hab\u00edan visto al Custodio curando a Liss eran los que estaban m\u00e1s seguros, convencidos de que contar\u00edamos con el apoyo de algunos refas. A lo largo de varias semanas hab\u00edamos hecho fondo com\u00fan con nuestras provisiones y las hab\u00edamos guardado en escondites acordados. Unos bufones le hab\u00edan robado a Duckett, al que hab\u00edamos borrado la memoria, cerillas y latas de Sterno. Un par de valientes casacas blancas hab\u00edan intentado entrar en la Casa, pero los refas hab\u00edan reforzado las medidas de seguridad desde que hab\u00edan encontrado muerto a Kraz. Como ya no pod\u00edamos acercarnos por all\u00ed, no hab\u00edamos tenido m\u00e1s remedio que fabricarnos nosotros mismos las armas. No ten\u00edamos muchas, pero en realidad no eran imprescindibles.\n\nJulian, Liss y yo \u00e9ramos los \u00fanicos que sab\u00edamos desde d\u00f3nde acceder\u00edamos al tren. No se lo hab\u00edamos revelado a nadie, pues era demasiado arriesgado. Los dem\u00e1s solo sab\u00edan que tendr\u00edamos una forma de salir de la colonia, y que se\u00f1alar\u00edamos el lugar con una bengala.\n\nBaj\u00e9 las piernas de la cama. La puerta del lavabo estaba abierta, y me vi en el espejo. Parec\u00eda una mu\u00f1eca de porcelana, pero habr\u00eda podido ser peor. Habr\u00eda podido parecerme a Ivy. La \u00faltima vez que la hab\u00eda visto, iba detr\u00e1s de Thuban con otro humano, tan sucia y flaca que me cost\u00f3 reconocerla. Pero no lloraba, solo caminaba en silencio. Me sorprendi\u00f3 que hubiera sobrevivido despu\u00e9s de lo que hab\u00eda pasado en la Casa.\n\nEl Custodio no me hab\u00eda dejado acercarme por all\u00ed. Se hab\u00eda mostrado cada vez m\u00e1s reacio a medida que se acercaba septiembre. Supon\u00eda que deb\u00eda de tener miedo. Miedo de que aquella rebeli\u00f3n fracasara, como la anterior. A veces era algo m\u00e1s que miedo; me dio por pensar que estaba enojado. Porque iba a perderme. Porque iba a perder la batalla contra Nashira.\n\nAhuyent\u00e9 ese pensamiento. Lo \u00fanico que quer\u00eda el Custodio era proteger mi don, como todos.\n\nNo ten\u00eda sentido retrasarlo m\u00e1s. Ten\u00eda que ir al Consistorio. Me levant\u00e9 y volv\u00ed a darle cuerda al gram\u00f3fono. Me consolaba, de alguna manera, que la m\u00fasica siguiera sonando; que, pasara lo que pasase fuera, una canci\u00f3n continuara llenando aquella c\u00e1mara vac\u00eda, al menos durante un rato. Cerr\u00e9 la puerta de la Torre al salir.\n\nLa portera de noche acababa de comenzar su turno. Llevaba el pelo trenzado y recogido en un bonito mo\u00f1o, y los labios pintados de rosa.\n\n\u2014Hola, XX-40 \u2014dijo\u2014. Te esperan en el Consistorio dentro de diez minutos.\n\n\u2014Gracias, ya lo s\u00e9.\n\nEl Capataz me lo hab\u00eda repetido una y otra vez.\n\n\u2014Me han pedido que te recuerde cu\u00e1les son tus instrucciones para esta noche. Tienes prohibido hablar con los embajadores y con los patrocinadores de Scion, a menos que vayas acompa\u00f1ada de un refa\u00edta. El espect\u00e1culo empieza a las once. Sales al escenario cuando acabe la obra.\n\n\u2014\u00bfAl escenario?\n\n\u2014Ay... \u2014Consult\u00f3 su registro\u2014. No, nada. Perdona. Ese mensaje era para otra persona.\n\nIntent\u00e9 mirar lo que hab\u00eda escrito en el libro, pero ella puso la mano.\n\n\u2014\u00bfSeguro?\n\n\u2014Buenas noches.\n\nLevant\u00e9 la cabeza: era David. Llevaba un traje y una corbata roja, e iba reci\u00e9n afeitado. Se me retorci\u00f3 el est\u00f3mago. David no parec\u00eda drogado. Sin embargo, estaba segura de que Michael hab\u00eda cumplido su misi\u00f3n.\n\n\u2014Vengo a escoltarte hasta el Consistorio. \u2014Extendi\u00f3 un brazo\u2014. La soberana de sangre requiere tu presencia all\u00ed.\n\n\u2014No necesito escolta.\n\n\u2014Ellos creen que s\u00ed.\n\nDavid no arrastraba las palabras. No hab\u00eda ni probado la mezcla de Duckett. Pas\u00e9 a su lado ignorando el brazo que me ofrec\u00eda, y enfil\u00e9 la calle. Aquello no era un buen principio.\n\nHab\u00edan trazado un sendero de faroles que recorr\u00eda toda la ciudad. El Consistorio estaba cerca de la Casa, y llevaba el mismo nombre que el cuartel general de la DVN de Londres. Los videntes a los que hab\u00edan invitado al Bicentenario eran los que hab\u00edan conseguido el blus\u00f3n rosa o rojo, o los bufones con alg\u00fan talento especial. Seg\u00fan Nashira, era una recompensa por su buen comportamiento. Se les permitir\u00eda bailar y comer con otros humanos. Ellos, a cambio, ten\u00edan que transmitir el mensaje de que no solo les gustaba pasar el tiempo en compa\u00f1\u00eda de sus guardianes, sino que, adem\u00e1s, les estaban muy agradecidos por su \u00abrehabilitaci\u00f3n\u00bb. De que les gustaba vivir apartados de la sociedad en una colonia penitenciaria repugnante. De que no les importaba que los emim les arrancaran las extremidades.\n\nMuchos de ellos no necesitar\u00edan fingir. Carl estaba contento. Todos los casacas rojas estaban contentos. Hab\u00edan encontrado un lugar en la colonia; yo, en cambio, jam\u00e1s lo encontrar\u00eda. En lo \u00fanico que pensaba era en largarme de all\u00ed.\n\n\u2014Un truco muy h\u00e1bil \u2014dijo David\u2014. Lo del vino.\n\nNo lo mir\u00e9 a la cara.\n\n\u2014El chico se pas\u00f3 un poco. S\u00e9 reconocer el _regal_ con solo olerlo. Pero no te preocupes: funcion\u00f3 con la mayor\u00eda. No ser\u00e9 yo quien estropee la sorpresa.\n\nVi correr a dos bufones cargados con unos rollos de tela y meterse por la calle que discurr\u00eda entre la iglesia y la Residencia del Suzerano. Esa era la ruta que tomar\u00edan para quemar la Sala. Ya deb\u00edan de estar poniendo las cerillas all\u00ed. Cerillas y parafina.\n\nHab\u00eda sido Julian quien hab\u00eda propuesto que peg\u00e1ramos fuego a los edificios del centro de la ciudad; hab\u00eda resultado ser un estratega excelente. Los bufones provocar\u00edan la distracci\u00f3n, lo que dejar\u00eda las otras calles despejadas; as\u00ed nosotros podr\u00edamos ir hacia el norte, hacia la pradera. Lo har\u00edan de madrugada, cuando los emisarios empezaran a estar cansados. \u00abNo se marchar\u00e1n a sus casas mucho m\u00e1s tarde de las dos \u2014hab\u00eda dicho\u2014. Si lo hacemos a medianoche, dispondremos de una hora para poner en marcha la funci\u00f3n. Tendremos la sart\u00e9n por el mango. Y mejor pronto que tarde.\u00bb A m\u00ed me hab\u00eda parecido bien. Todo estaba saliendo seg\u00fan lo previsto, pero el avispado casaca roja que iba a mi lado ten\u00eda el poder de destruirlo todo.\n\n\u2014\u00bfA qui\u00e9n se lo has contado? \u2014pregunt\u00e9 a David.\n\n\u2014Voy a darte algo para pensar \u2014dijo, ignorando la pregunta\u2014. \u00bfCrees que a Scion le gusta recibir \u00f3rdenes de los refa\u00edtas?\n\n\u2014Claro que no.\n\n\u2014Pero crees a Nashira cuando dice que ellos tienen el control. \u00bfNo crees que a alguien, en toda la historia de Scion, deber\u00eda hab\u00e9rsele ocurrido combatirlos?\n\n\u2014\u00bfAd\u00f3nde pretendes llegar?\n\n\u2014Contesta mi pregunta.\n\n\u2014No. Porque les tienen demasiado miedo a los emim.\n\n\u2014Quiz\u00e1 tengas raz\u00f3n. O quiz\u00e1 al Arconte todav\u00eda le quede una pizca de sentido com\u00fan.\n\n\u2014\u00bfQu\u00e9 quieres decir con eso?\n\nComo no me contestaba, me par\u00e9 delante de \u00e9l.\n\n\u2014\u00bfQu\u00e9 demonios tiene que ver el Arconte con todo esto?\n\n\u2014Mucho. \u2014Me apart\u00f3 y sigui\u00f3 caminando\u2014. Sigue adelante con tu fuga masiva, princesa callejera. No te preocupes por m\u00ed.\n\nAntes de que pudiera replicar, \u00e9l ya se hab\u00eda alejado, hab\u00eda entrado en el vest\u00edbulo victoriano y se hab\u00eda perdido entre la multitud. Not\u00e9 un escalofr\u00edo. Lo \u00faltimo que necesitaba era un casaca roja d\u00edscolo, y menos si era tan cr\u00edptico como David. Quiz\u00e1 afirmara odiar a los refa\u00edtas, pero tampoco daba la impresi\u00f3n de que yo le cayera muy bien. Pod\u00eda contarle a Nashira lo del vino, y ella sospechar\u00eda de inmediato.\n\nDentro del Consistorio hab\u00edan encendido miles de velas. Nada m\u00e1s cruzar el umbral, Michael y un casaca blanca me llevaron por una escalera, y dejamos a David buscando a los otros arrancahuesos.\n\nLa tarea que los refas hab\u00edan asignado a Michael consist\u00eda en asegurarse de que nadie tuviera un aspecto desali\u00f1ado o magullado; eso nos proporcion\u00f3 un pretexto perfecto para organizar un \u00faltimo encuentro. Cuando llegamos a la galer\u00eda, me volv\u00ed hacia ellos.\n\n\u2014\u00bfListos?\n\n\u2014\u00a1Listos! \u2014confirm\u00f3 el casaca blanca. Era Charles, un criom\u00e1ntico de Terebell. Apunt\u00f3 hacia el sal\u00f3n, donde los refa\u00edtas se codeaban con los emisarios\u2014. Los arrancahuesos est\u00e1n empezando a venirse abajo. Cuando lo noten los refas, ya ser\u00e1 demasiado tarde.\n\n\u2014Muy bien. \u2014Respir\u00e9 hondo para calmarme\u2014. Bien hecho, Michael.\n\nMichael llevaba un sencillo traje gris. Me sonri\u00f3.\n\n\u2014\u00bfTienes mi mochila?\n\nSe\u00f1al\u00f3 debajo de los bancos de la galer\u00eda, donde estaba mi mochila cargada de f\u00e1rmacos. Todav\u00eda no pod\u00eda cogerla, pero los bufones sab\u00edan d\u00f3nde estaba, por si la necesitaban. Era uno de los escondites de provisiones que ten\u00edamos.\n\n\u2014Paige, \u00bfa qu\u00e9 hora lanzaremos la bengala? \u2014pregunt\u00f3 Charles.\n\n\u2014Todav\u00eda no lo s\u00e9. Disparar\u00e9 una en cuanto encontremos un camino. \u2014Charles asinti\u00f3. Volv\u00ed a mirar hacia el sal\u00f3n. Hab\u00eda mucha gente a punto de poner en peligro su vida. Liss, que tanto miedo ten\u00eda. Julian, que hab\u00eda hecho tanto para ayudarme. Los bufones. Los casacas blancas.\n\nY el Custodio. Ahora entend\u00eda qu\u00e9 significaba para \u00e9l confiar en m\u00ed. Si lo traicionaba, como hab\u00eda hecho el otro humano, no lo marcar\u00edan, sino que lo ejecutar\u00edan. Aquella era su \u00faltima oportunidad.\n\nPero el Custodio ten\u00eda que actuar ahora, mientras entre los refa\u00edtas todav\u00eda hab\u00eda una pizca de compasi\u00f3n. Si perec\u00edan los marcados, esa esperanza se habr\u00eda perdido.\n\nLa puerta de la galer\u00eda se abri\u00f3 de golpe, y Suhail apareci\u00f3 en el umbral. Agarr\u00f3 a Charles por el blus\u00f3n y lo oblig\u00f3 a subir por la escalera.\n\n\u2014A la soberana de sangre no le gusta que la hagan esperar, in\u00fatil \u2014me dijo\u2014. No tienes permiso para estar en la galer\u00eda. Baja inmediatamente.\n\nSe march\u00f3 tan aprisa como hab\u00eda llegado. Michael ech\u00f3 un vistazo a la puerta.\n\n\u2014Es la hora \u2014anunci\u00e9, y le apret\u00e9 la mano\u2014. Buena suerte. Recuerda: procura no llamar la atenci\u00f3n y busca la luz de la bengala.\n\nMichael asinti\u00f3 con la cabeza.\n\n\u2014Vive \u2014se limit\u00f3 a decir.\n\nRecorr\u00ed la planta baja del Consistorio con la cabeza agachada. Nadie me vio entrar.\n\nNueve pa\u00edses europeos aplicaban el sistema de Scion, entre ellos Inglaterra. Sin embargo, a diferencia de Inglaterra, ninguno ten\u00eda un lugar al que enviar a sus clarividentes. Aun as\u00ed, los nueve gobiernos hab\u00edan enviado emisarios a la celebraci\u00f3n del Bicentenario. Incluso Dubl\u00edn, la ciudad de Scion m\u00e1s joven y pol\u00e9mica, hab\u00eda enviado a un delegado: Cathal Bell, un viejo amigo de mi padre. Era un hombre indeciso y nervioso, agobiado por las responsabilidades que implicaba su posici\u00f3n. En un primer momento, al verlo me emocion\u00e9 y pens\u00e9 que quiz\u00e1 \u00e9l podr\u00eda ayudarnos; pero entonces record\u00e9 que no me hab\u00eda visto desde que yo ten\u00eda cinco o seis a\u00f1os. No me reconocer\u00eda, y all\u00ed yo no ten\u00eda nombre. Adem\u00e1s, Bell era d\u00e9bil. Su partido hab\u00eda perdido Dubl\u00edn.\n\nEl Consistorio era espectacular. Del techo, decorado con artesonado, colgaban ara\u00f1as de luces, y el sal\u00f3n principal era muy amplio y despejado. La luz de las velas y la m\u00fasica de Chopin hac\u00edan estremecer la oscuridad. A los delegados se les prodigaban todo tipo de atenciones. Pod\u00edan atiborrarse de toda clase de manjares deliciosos, o charlar entre ellos tom\u00e1ndose una copa de _mecks_. Su amaurosis era un privilegio, un derecho. Los esclavos amaur\u00f3ticos, entre ellos Michael, les serv\u00edan la comida; se supon\u00eda que eran participantes voluntarios en un programa de rehabilitaci\u00f3n. Los otros amaur\u00f3ticos deb\u00edan de estar demasiado desnutridos para aparecer por all\u00ed.\n\nPor encima de un grupo de actores estaba Liss, colgada de sus sedas, haciendo poses como una bailarina voladora. Depend\u00eda \u00fanicamente de su fuerza f\u00edsica para no caer y morir aplastada.\n\nRecorr\u00ed la sala con la mirada tratando de localizar a Weaver, pero no lo vi. Quiz\u00e1 llegara tarde. A otros pa\u00edses pod\u00edan perdonarles que no hubieran enviado a sus inquisidores, pero a Inglaterra no. Identifiqu\u00e9 a otros oficiales de Scion, entre ellos el comandante de los centinelas, Bernard Hock. Era un hombre enorme, calvo y con los m\u00fasculos del cuello excesivamente desarrollados; era muy bueno detectando a videntes: de hecho, yo siempre hab\u00eda sospechado que era rastreador. Vi c\u00f3mo mov\u00eda las aletas de la nariz, y me propuse matarlo si se me presentaba la ocasi\u00f3n.\n\nUn amaur\u00f3tico me ofreci\u00f3 una copa de _mecks_ blanco. La rechac\u00e9. Acababa de ver a Cathal Bell.\n\nBell ten\u00eda una copa en la mano, y no paraba de tocarse la corbata. Intentaba conversar con Radmilo Are\u017eina, el viceministro de Migraci\u00f3n de Serbia. Sonre\u00ed para m\u00ed. Are\u017eina, el muy in\u00fatil, hab\u00eda autorizado el traslado de Dani a Londres. Fui hacia ellos.\n\n\u2014\u00bfSe\u00f1or Bell?\n\nBell dio un respingo y derram\u00f3 un poco de vino.\n\n\u2014\u00bfS\u00ed?\n\nMir\u00e9 a Are\u017eina.\n\n\u2014Perdone que les interrumpa, se\u00f1or ministro, pero \u00bfpuedo hablar con el se\u00f1or Bell en privado?\n\nAre\u017eina me mir\u00f3 de arriba abajo. Arque\u00f3 el labio superior y dijo:\n\n\u2014Perd\u00f3neme, se\u00f1or Bell. Debo volver con mi grupo.\n\nSe alej\u00f3 hacia la seguridad que le ofrec\u00edan sus acompa\u00f1antes. Me qued\u00e9 frente a Bell, que intentaba limpiar la mancha de vino de su chaqueta.\n\n\u2014\u00bfQu\u00e9 quieres, antinatural? \u2014dijo, tartamudeando\u2014. Estaba manteniendo una conversaci\u00f3n muy importante.\n\n\u2014Tranquilo, ahora podr\u00e1 mantener otra. \u2014Le quit\u00e9 la copa de la mano y di un sorbo\u2014. \u00bfSe acuerda de la Incursi\u00f3n, se\u00f1or Bell?\n\nBell se qued\u00f3 paralizado.\n\n\u2014Si te refieres a la Incursi\u00f3n de 2046, s\u00ed. Por supuesto. \u2014Le temblaban los dedos. Ten\u00eda los nudillos morados, hinchados por la artritis\u2014. \u00bfPor qu\u00e9 me lo preguntas? \u00bfQui\u00e9n eres?\n\n\u2014A mi primo lo detuvieron ese d\u00eda. Quiero saber si sigue vivo.\n\n\u2014\u00bfEres irlandesa?\n\n\u2014S\u00ed.\n\nMe mir\u00f3 con los ojos entrecerrados.\n\n\u2014\u00bfC\u00f3mo te llamas?\n\n\u2014Mi nombre no importa. El de mi primo, s\u00ed. Finn McCarthy. Estudiaba en el Trinity College. \u00bfLo conoce?\n\n\u2014S\u00ed. \u2014Contest\u00f3 sin vacilar\u2014. McCarthy estaba en el castillo de Carrickfergus con los otros l\u00edderes estudiantiles. Lo condenaron a la horca.\n\n\u2014\u00bfY lo ahorcaron?\n\n\u2014Yo... yo no tuve conocimiento de los detalles, pero...\n\nSurgi\u00f3 algo oscuro y violento dentro de m\u00ed. Me inclin\u00e9 m\u00e1s hacia \u00e9l y le dije al o\u00eddo:\n\n\u2014Si resulta que ejecutaron a mi primo, se\u00f1or Bell, lo har\u00e9 a usted responsable. Fue su gobierno el que perdi\u00f3 Irlanda. Fue su gobierno el que tir\u00f3 la toalla.\n\n\u2014No fui yo \u2014dijo Bell entrecortadamente. Estaba empezando a sangrarle la nariz\u2014. No me hagas da\u00f1o...\n\n\u2014No solamente usted, se\u00f1or Bell. Usted y otros como usted.\n\n\u2014L\u00e1rgate, antinatural \u2014me espet\u00f3; me perd\u00ed entre la masa de gente y lo dej\u00e9 conteni\u00e9ndose la hemorragia.\n\nMe di cuenta de que estaba temblando. Cog\u00ed otra copa de _mecks_ y me la beb\u00ed de un trago. Siempre hab\u00eda cre\u00eddo que Finn deb\u00eda de haber muerto, pero una peque\u00f1a parte de m\u00ed se aferraba a su recuerdo, a la posibilidad de que siguiera vivo. De todas formas, si mi primo hab\u00eda sobrevivido, no iba a enterarme por Cathal Bell.\n\nVi a Nashira de pie bajo un estrado. El Custodio, a su lado, conversaba con un emisario griego. Despu\u00e9s de sonar la campanada nocturna hab\u00eda recibido su primera dosis de amaranto en varios meses; unas gotas hab\u00edan bastado para transformarlo. Vest\u00eda de negro y dorado, con una joya que imitaba un jacinto en el cuello, y sus ojos brillaban como l\u00e1mparas. Reconoc\u00ed a las personas que estaban m\u00e1s cerca de Nashira: su guardia de \u00e9lite. Una de ellas, la sustituta de Amelia, me vio y por el movimiento de sus labios deduje que hab\u00eda informado a su jefa.\n\nNashira mir\u00f3 por encima de las cabezas de la guardia y dej\u00f3 escapar una risa d\u00e9bil. Al o\u00edrla, el Custodio se dio la vuelta. Sus ojos empezaron a arder inmediatamente.\n\nNashira me llam\u00f3 por se\u00f1as. Le di mi copa vac\u00eda a un amaur\u00f3tico y fui hacia ella.\n\n\u2014Damas y caballeros \u2014dijo a quienes la rodeaban\u2014, quiero presentarles a XX-59-40, una de nuestras clarividentes de m\u00e1s talento.\n\nLos delegados murmuraron, entre intrigados y asqueados.\n\n\u2014Este es Aloys Mynatt, Gran Anecdotista de Francia. Y Birgitta Tj\u00e4der, Jefa de Centinelas de la ciudadela Scion Estocolmo.\n\nMynatt era un hombre de escasa estatura, de postura r\u00edgida, con unas facciones anodinas. Salud\u00f3 con una inclinaci\u00f3n de cabeza.\n\nTj\u00e4der se qued\u00f3 mir\u00e1ndome. Tendr\u00eda alrededor de treinta y cinco a\u00f1os; su cabello era rubio y tupido, y sus ojos de color del aceite de oliva. Nick siempre la hab\u00eda llamado \u00abla Urraca\u00bb; era famosa por la crueldad con que gobernaba Estocolmo. Repar\u00e9 en que no soportaba estar cerca de m\u00ed: ten\u00eda los labios tensos y mostraba los dientes, como si se dispusiera a morder. A m\u00ed tampoco me entusiasmaba su presencia.\n\n\u2014Que no se me acerque \u2014dijo Tj\u00e4der confirmando mis sospechas.\n\n\u2014Pero \u00bfno prefer\u00eds que est\u00e9n aqu\u00ed a que corran libremente por vuestras calles? \u2014dijo Nashira\u2014. Aqu\u00ed no pueden hacer ning\u00fan da\u00f1o, Birgitta. No les dejamos. Cuando se haya creado Sheol III, ya no tendr\u00e1s que volver a mirar a ning\u00fan clarividente.\n\n\u00bfUna tercera colonia penitenciaria? \u00bfTambi\u00e9n ten\u00edan planes para Estocolmo? No quer\u00eda imaginarme un Sheol III con la Urraca como captadora.\n\nTj\u00e4der no me quitaba los ojos de encima. No ten\u00eda aura, pero llevaba escrito en la cara el odio que sent\u00eda.\n\n\u2014Estoy impaciente \u2014dijo.\n\nEntonces el pianista dej\u00f3 de tocar, y le aplaudieron. Las parejas que estaban bailando se separaron. Nashira dirigi\u00f3 la vista hacia un gran reloj de pared.\n\n\u2014Se acerca la hora \u2014dijo en voz baja.\n\n\u2014Disculpadme \u2014dijo Tj\u00e4der; se dio la vuelta y volvi\u00f3 con los suecos, dejando un espacio vac\u00edo entre el Custodio y yo, pero no me atrev\u00ed a mirarlo a los ojos.\n\n\u2014Tengo que dirigirme a los emisarios. \u2014Nashira mir\u00f3 hacia el escenario\u2014. Arcturus, qu\u00e9date con 40. Te avisar\u00e9 cuando la necesite.\n\nDe modo que era cierto que planeaba matarme en p\u00fablico. Evit\u00e9 mirarlos a los dos. El Custodio agach\u00f3 la cabeza y dijo:\n\n\u2014S\u00ed, mi soberana. \u2014Me agarr\u00f3 bruscamente por un brazo y a\u00f1adi\u00f3\u2014: Ven, 40.\n\nAntes de que \u00e9l pudiera apartarme, Nashira gir\u00f3 bruscamente la cabeza. Me agarr\u00f3 por la mu\u00f1eca y tir\u00f3 de m\u00ed hacia ella.\n\n\u2014\u00bfTe has hecho da\u00f1o, 40?\n\nYa no llevaba las suturas adhesivas en la mejilla, pero el corte que me hab\u00eda hecho con el cristal me hab\u00eda dejado una fina cicatriz.\n\n\u2014Tuve que pegarle. \u2014El Custodio me sujetaba el brazo con fuerza\u2014. Me desobedeci\u00f3 y la castigu\u00e9.\n\nCada uno me ten\u00eda agarrado por un brazo; parec\u00eda una mu\u00f1eca de trapo. Se miraron por encima de mi cabeza.\n\n\u2014Estupendo \u2014dijo Nashira\u2014. Has tardado a\u00f1os, pero por fin empiezas a entender qu\u00e9 significa ser mi consorte.\n\nLe dio la espalda, se separ\u00f3 de los emisarios y fue hacia los invitados.\n\nEl m\u00fasico, quienquiera que fuese, empez\u00f3 a tocar unos acordes de piano bien escogidos, acompa\u00f1ados de una letra inquietante. Estaba segura de reconocer aquella voz, pero no consegu\u00eda identificarla.\n\nEl Custodio me llev\u00f3 a un lado de la sala, al pasillo alargado bajo la galer\u00eda, y se inclin\u00f3 para mirarme a los ojos.\n\n\u2014\u00bfEst\u00e1 todo preparado?\n\nDije que s\u00ed con la cabeza.\n\nEl m\u00fasico ten\u00eda una voz francamente bonita, una especie de d\u00e9bil falsete. Volv\u00ed a tener la sensaci\u00f3n de que conoc\u00eda aquella voz.\n\n\u2014Anoche mis compa\u00f1eros y yo hicimos una sesi\u00f3n espiritista \u2014dijo el Custodio con una voz apenas audible\u2014. Tendremos esp\u00edritus a nuestra disposici\u00f3n. Esp\u00edritus humanos, las v\u00edctimas de la Era de Huesos XVIII. Te ayudar\u00e1n a combatir a los refa\u00edtas.\n\n\u2014\u00bfY la DVN? \u00bfLa han tra\u00eddo?\n\n\u2014No pueden entrar en el Consistorio a menos que los llamen. Se hallan estacionados en el puente.\n\n\u2014\u00bfCu\u00e1ntos son?\n\n\u2014Treinta.\n\nVolv\u00ed a asentir. Todos los emisarios ten\u00edan al menos un guardaespaldas, pero todos eran centinelas de la DVD. No quer\u00edan que los protegieran antinaturales. Por suerte para nosotros, la DVD no estaba preparada para el combate espiritista.\n\nEl Custodio mir\u00f3 al techo y vio a Liss trepando por sus sedas.\n\n\u2014Veo que Liss se ha recuperado.\n\n\u2014S\u00ed.\n\n\u2014Entonces estamos en paz. Est\u00e1 todo arreglado.\n\n\u2014S\u00ed, todas las deudas est\u00e1n saldadas \u2014confirm\u00e9.\n\nEl treno. Eso me hizo pensar en lo que todav\u00eda estaba por venir. \u00bfY si Nashira consegu\u00eda matarme?\n\n\u2014Todo saldr\u00e1 como lo hemos planeado, Paige. No debes perder la esperanza. \u2014Mir\u00f3 hacia el escenario y a\u00f1adi\u00f3\u2014: La esperanza es lo \u00fanico que todav\u00eda podr\u00eda salvarnos a todos.\n\nSegu\u00ed la direcci\u00f3n de su mirada. Hab\u00edan colocado la campana de cristal y la flor sin vida en un pedestal cubierto.\n\n\u2014\u00bfQu\u00e9 esperanza?\n\n\u2014La esperanza de un cambio.\n\nCes\u00f3 la m\u00fasica, y volvieron a arrancar aplausos de la pista de baile. Quer\u00eda mirar, averiguar qui\u00e9n era el que tocaba, pero no ve\u00eda m\u00e1s all\u00e1 de las cabezas de los emisarios.\n\nUn casaca roja subi\u00f3 al escenario. 22. Sus andares torcidos indicaban la cantidad de mezcla de Duckett que hab\u00eda ingerido.\n\n\u2014Damas y caballeros \u2014dijo\u2014, la gran... Suzerana, Nashira Sargas, soberana de sangre de la... raza de los refa\u00edtas.\n\nBaj\u00f3 tambale\u00e1ndose. Reprim\u00ed una sonrisa: al menos hab\u00eda un casaca roja del que no tendr\u00edamos que encargarnos.\n\nNashira subi\u00f3 al estrado, y su p\u00fablico la aplaudi\u00f3 con entusiasmo. Nos mir\u00f3. El Custodio la mir\u00f3 tambi\u00e9n.\n\n\u2014Damas y caballeros \u2014dijo Nashira sin desviar la mirada\u2014, bienvenidos a Sheol I, capital de Scion. Quiero darles las gracias por asistir a nuestra celebraci\u00f3n de esta noche.\n\n\u00bbHan transcurrido ya doscientos a\u00f1os desde nuestra llegada a Gran Breta\u00f1a. Ha pasado mucho tiempo desde 1859. Como ver\u00e1n, hemos hecho todo lo posible para convertir nuestra primera ciudad de control en un lugar de belleza, respeto y, por encima de todo, compasi\u00f3n. Nuestro sistema de rehabilitaci\u00f3n permite a los j\u00f3venes clarividentes entrar en nuestra ciudad y recibir la mejor calidad de vida posible. \u2014Como animales en un zoo\u2014. Como todos sabemos, la clarividencia no es culpa de sus v\u00edctimas. Como una enfermedad, se aprovecha de los inocentes. Los hace enfermar de antinaturalidad.\n\n\u00bbHoy Sheol I celebra doscientos a\u00f1os de trabajo bien hecho. Como ver\u00e1n, el proyecto ha tenido un \u00e9xito rotundo, y es la primera de las muchas semillas que queremos plantar. A cambio de su comprensi\u00f3n, no solo les hemos proporcionado un medio humanitario de apartar a los clarividentes de la sociedad, sino que hemos impedido cientos de ataques de emim a la ciudadela. Somos una baliza que los atrae; vienen a nosotros como polillas atra\u00eddas hacia una llama. \u2014Sus ojos tambi\u00e9n parec\u00edan llamas en la penumbra\u2014. Pero el n\u00famero de emim crece d\u00eda a d\u00eda. Esta colonia pronto dejar\u00e1 de ofrecer suficiente protecci\u00f3n. Los emim han sido vistos en Francia, Irlanda y, m\u00e1s recientemente, Suecia.\n\nIrlanda. Por eso estaba all\u00ed Cathal Bell. Por eso estaba tan nervioso y asustado.\n\n\u2014Es imprescindible que establezcamos Sheol II, que prendamos otra llama \u2014prosigui\u00f3 Nashira\u2014. Ya hemos puesto a prueba nuestro m\u00e9todo. Con su ayuda, y la de sus ciudades, confiamos en que la flor de nuestra alianza pueda florecer por fin.\n\nAplausos. El Custodio ten\u00eda las mand\u00edbulas apretadas. La expresi\u00f3n de su cara daba miedo. Enfadado. Cruel. Mort\u00edfero.\n\nEra la primera vez que lo ve\u00eda con aquel semblante.\n\n\u2014Nos quedan unos minutos antes de que d\u00e9 comienzo la obra de teatro escrita por nuestro Capataz. Entretanto, me gustar\u00eda presentarles a mi socio, el segundo soberano de sangre, que quiere hacer un breve anuncio. Damas y caballeros: con ustedes, Gomeisa Sargas.\n\nExtendi\u00f3 una mano. Antes de que me hubiera dado cuenta de que hab\u00eda alguien m\u00e1s all\u00ed, otra mano m\u00e1s grande se la tom\u00f3.\n\nSe me cort\u00f3 la respiraci\u00f3n.\n\nLlevaba una t\u00fanica negra, con un cuello que llegaba a cubrirle las orejas. Era alto y delgado, de cabello rubio y rostro descarnado. Las comisuras de sus labios apuntaban hacia abajo, como si le pesaran las sartas de piedras preciosas del tama\u00f1o de ojos que llevaba alrededor del cuello. Parec\u00eda mayor que los otros refa\u00edtas; se notaba en su porte, y en las sorprendentes dimensiones de su onirosaje. Yo percib\u00eda aquel onirosaje como una pared que chocaba contra mi cr\u00e1neo. Era lo m\u00e1s antiguo y terrible que jam\u00e1s hab\u00eda sentido en el \u00e9ter.\n\n\u2014Buenas noches.\n\nGomeisa nos contempl\u00f3 con la t\u00edpica expresi\u00f3n de los refa\u00edtas: la del observador imperturbable. Su aura semejaba una mano que tapara el sol. No me extra\u00f1aba que Liss le tuviera tanto miedo. La vi envuelta en sus cintas, silenciosa y quieta. Al cabo de un momento, baj\u00f3 a la galer\u00eda.\n\n\u2014Pido perd\u00f3n a los humanos que residen en Sheol I por mis prolongadas ausencias. Soy el principal emisario de los refa\u00edtas ante el Arconte de Westminster. Como tal, paso mucho tiempo en la capital con el Inquisidor, discutiendo sobre cu\u00e1les son los mejores m\u00e9todos para aumentar la eficacia de esta colonia penitenciaria.\n\n\u00bbComo ya ha comentado Nashira, lo que hoy celebramos es un comienzo. Est\u00e1 naciendo una nueva era: la de la colaboraci\u00f3n perfecta entre humanos y refa\u00edtas, dos razas que llevan demasiado tiempo separadas. Celebramos el fin de un viejo mundo, donde reinaban la ignorancia y la oscuridad. Juramos compartir nuestra sabidur\u00eda con vosotros, como vosotros hab\u00e9is compartido vuestro mundo con nosotros. Juramos protegeros, como vosotros nos hab\u00e9is dado cobijo. Y os prometo, amigos, que no permitiremos que nuestro pacto se deshaga. Aqu\u00ed la pureza gobierna con mano de hierro. Y la flor de la transgresi\u00f3n permanecer\u00e1 eternamente marchita.\n\nEch\u00e9 un vistazo a la flor muerta de la campana de cristal. \u00c9l la mir\u00f3 como si mirara una babosa.\n\n\u2014Pero basta de formalidades \u2014dijo\u2014. Que d\u00e9 comienzo la obra.\n\nApareci\u00f3 el Capataz, con un atuendo deslumbrante. Llevaba una larga capa roja, abrochada hasta el cuello, que le tapaba todo el cuerpo. Salud\u00f3 con una reverencia.\n\n\u2014\u00a1Salutaciones, damas y caballeros, y bienvenidos a Sheol I! Soy el Capataz, Beltrame. Me ocupo de la poblaci\u00f3n humana de la ciudad. Quiero dar una bienvenida especialmente calurosa a aquellos de ustedes que han venido desde regiones no convertidas del continente. \u00a1No teman! Despu\u00e9s del espect\u00e1culo tendr\u00e1n la oportunidad de convertir sus ciudades en ciudadelas de Scion, como han hecho otras muchas ciudades. Nuestro programa permite a los gobiernos desenraizar y segregar a los clarividentes cuando todav\u00eda son j\u00f3venes, sin necesidad de recurrir al tremendo gasto que conllevan las ejecuciones en masa.\n\nProcur\u00e9 no escuchar. No todos los pa\u00edses empleaban el NiteKind para ejecutar a los clarividentes. Muchos utilizaban la inyecci\u00f3n letal, o un pelot\u00f3n de fusilamiento, o cosas peores.\n\n\u2014Ya hemos planeado la creaci\u00f3n de Sheol II en asociaci\u00f3n con las ciudadelas Scion de Par\u00eds y Marsella, que se convertir\u00e1n en las primeras ciudadelas sat\u00e9lite francesas. \u2014Aplausos. Mynatt sonri\u00f3\u2014. Esta noche esperamos concretar la futura ubicaci\u00f3n de, por lo menos, dos ciudades de control m\u00e1s en el continente. Pero, antes de todo eso, queremos mostrarles una peque\u00f1a obra de teatro para demostrar que muchos de nuestros clarividentes hacen un buen uso de sus habilidades.\n\n\u00bbEsta obra nos recordar\u00e1 los d\u00edas oscuros anteriores a la llegada de los refa\u00edtas, cuando el Rey Sangriento todav\u00eda ostentaba el poder. El rey que construy\u00f3 su casa con sangre.\n\nEl reloj dio la hora. Los actores, veinte en total, salieron en fila india. Iban a representar la historia de la vida de Eduardo VII, desde su adquisici\u00f3n de una mesa de espiritismo y los cinco asesinatos, hasta el pu\u00f1al hallado en sus dependencias y su huida de Inglaterra con el resto de su familia. Los inicios de la presunta epidemia, y un testimonio de por qu\u00e9 la existencia de Scion era necesaria. Liss estaba de pie al fondo del escenario. La flanqueaban Nell, la chica que la hab\u00eda sustituido cuando hab\u00eda sufrido el choque espiritual, y una profeta que, si no recordaba mal, se llamaba Lotte. Las tres iban disfrazadas de v\u00edctimas del Rey Sangriento.\n\nEn el centro del escenario, el Capataz se desprendi\u00f3 de la capa revelando las vestiduras de un monarca. El p\u00fablico lo abuche\u00f3. Representaba a Eduardo en su \u00e9poca de heredero de la reina Victoria, engalanado con joyas y pieles.\n\nLa primera escena parec\u00eda desarrollarse en su alcoba, donde un ostentoso cal\u00edope interpretaba \u00abDaisy Bell\u00bb. El buf\u00f3n que estaba m\u00e1s cerca del p\u00fablico dijo ser Frederick Ponsonby, primer bar\u00f3n de Sysonby, el secretario personal de Eduardo. La obra iba a presentarse a trav\u00e9s de su mirada.\n\n\u2014Alteza \u2014dijo al Capataz\u2014, \u00bfvamos afuera a dar un paseo?\n\n\u2014\u00bfHas tra\u00eddo tu chaqueta corta, Ponsonby?\n\n\u2014No, solo un frac, alteza.\n\n\u2014Cre\u00eda que todo el mundo sab\u00eda \u2014dijo el Capataz con voz resonante, y con un hilarante acento aristocr\u00e1tico ingl\u00e9s\u2014 que a una exhibici\u00f3n privada celebrada por la ma\u00f1ana siempre hay que ir con chaqueta corta y sombrero de seda. \u00a1Y esos pantalones son los m\u00e1s feos que he visto jam\u00e1s!\n\nAbucheos. Silbidos. Aquella bestia licenciosa se hab\u00eda atrevido a llamarse heredero de Victoria. Ponsonby se volvi\u00f3 hacia el p\u00fablico.\n\n\u2014Tras provocar numerosas aflicciones, con mi frac, por ejemplo, y con mis inadecuados pantalones \u2014risas\u2014, el pr\u00edncipe se cans\u00f3 de sus galas. Esa misma tarde me pidi\u00f3 que lo acompa\u00f1ara a hacer una excursi\u00f3n. \u00a1Ay, amigos m\u00edos! El sufrimiento humano jam\u00e1s ha superado el de la reina al contemplar a su hijo recorriendo el camino del mal.\n\nGir\u00e9 la cabeza para ver la reacci\u00f3n del Custodio, pero ya no estaba all\u00ed.\n\nLa conversaci\u00f3n entre Eduardo y Ponsonby se prolong\u00f3 un poco m\u00e1s. Todas las escenas estaban pensadas para mostrar a Eduardo como un idiota cruel y lujurioso, y un fracaso para su madre. Yo observaba fascinada. Exageraban el papel de Eduardo en la muerte del pr\u00edncipe Alberto hasta lo rid\u00edculo, e incluso introduc\u00edan un duelo. Aparec\u00eda la reina Victoria, que hab\u00eda enviudado, con su coronita de diamantes y su velo. \u00abNo puedo mirarlo sin estremecerme \u2014confes\u00f3 al p\u00fablico\u2014. Para m\u00ed es tan antinatural como si lo hubieran sustituido por otro ni\u00f1o al nacer.\u00bb El p\u00fablico aplaudi\u00f3. Ella era un baluarte de bondad, la \u00faltima monarca impoluta antes de la plaga. Mientras la actriz encandilaba a los emisarios, yo vigilaba el reloj. Hab\u00eda transcurrido casi media hora, y segu\u00eda sin saber a qu\u00e9 hora sal\u00eda el tren. A continuaci\u00f3n ven\u00eda el momento crucial de la obra: la sesi\u00f3n espiritista. Llevaron unos faroles rojos al escenario. Cuando volv\u00ed a mirar, tuve que contener la risa, pues el Capataz estaba muy metido en su papel.\n\n\u2014Los poderes terrenales no son suficiente \u2014dijo, resollando, imbuido de la maldad de su personaje. Hab\u00edan sacado la mesa de espiritismo, y el Capataz describ\u00eda c\u00edrculos con los brazos sobre ella\u2014. Hablan de la era victoriana, pero \u00bfc\u00f3mo ser\u00e1 la de Eduardo? \u00bfQu\u00e9 rey puede gobernar con eficacia, si lo agobian las cadenas de la mortalidad? \u2014Se inclin\u00f3 sobre la mesa, haci\u00e9ndola balancear con las manos\u2014. S\u00ed, levantaos. Salid de las sombras. Atravesad el umbral, esp\u00edritus de los muertos. \u00a1Entrad en m\u00ed y en mis seguidores! \u00a1Reproduc\u00edos en la sangre de Inglaterra!\n\nMientras \u00e9l hablaba, los faroles rojos salieron del escenario llevados por actores vestidos de negro que representaban a los esp\u00edritus antinaturales. Se dispersaron por la sala agarrando a la gente, que chillaba. Eran la plaga de antinaturalidad.\n\nLa m\u00fasica y la risa de los actores acab\u00f3 por aturdirme. Todo me daba vueltas. El Capataz entonaba a gritos sus conjuros. Aprovechando la oscuridad y la confusi\u00f3n, el Custodio me agarr\u00f3 por el brazo.\n\n\u2014R\u00e1pido \u2014me dijo al o\u00eddo\u2014. Ven conmigo.\n\nMe llev\u00f3 al foso, un espacio peque\u00f1o y oscuro bajo el piso del escenario donde se amontonaban los cajones de embalaje. La \u00fanica luz era la que se filtraba entre los tablones. Una luz roja, como los faroles. Unos gruesos cortinajes de terciopelo colgaban a lo largo de uno de los lados de la estancia, ocult\u00e1ndonos del sal\u00f3n de arriba. No era f\u00e1cil pensar, en aquel lugar tan oscuro, en eso a lo que tal vez pronto me enfrentar\u00eda.\n\nAll\u00ed no hab\u00eda tanto ruido. Los actores danzaban por encima de nuestras cabezas, pero las tablas del suelo amortiguaban el estruendo. El Custodio me mir\u00f3.\n\n\u2014T\u00fa ser\u00e1s la \u00faltima escena de la obra. El \u00faltimo acto. \u2014Ten\u00eda los ojos encendidos\u2014. Le o\u00ed coment\u00e1rselo a Gomeisa.\n\nSe me puso la piel de gallina.\n\n\u2014Sab\u00edamos que iba a pasar.\n\n\u2014S\u00ed.\n\nYo sab\u00eda desde el principio que Nashira iba a matarme, pero o\u00edrselo decir a \u00e9l hac\u00eda que pareciera mucho m\u00e1s real. Una parte de m\u00ed hab\u00eda seguido confiando en que Nashira esperar\u00eda unos d\u00edas y me brindar\u00eda la oportunidad de fugarme con los otros en el tren; pero Nashira era cruel. Quer\u00eda hacerlo en p\u00fablico, delante de Scion. No pensaba arriesgarse a mantenerme con vida.\n\nLa luz de los ojos del Custodio oscurec\u00eda a\u00fan m\u00e1s las sombras. Detect\u00e9 algo diferente en ellos: algo salvaje, imprevisible. Me estremec\u00ed, y me sent\u00e9 en un caj\u00f3n.\n\n\u2014No puedo luchar contra ella \u2014dije\u2014. Sus \u00e1ngeles...\n\n\u2014No, Paige. Piensa. Nashira lleva meses esperando que pudieras poseer otro cuerpo. Si no mostrabas esa habilidad, exist\u00eda el peligro de que ella no pudiera obtenerla de ti. Te hizo casaca amarilla para asegurarse de que los emim no volv\u00edan a poner tu vida en peligro. Te puso bajo la protecci\u00f3n de su propio consorte. \u00bfPor qu\u00e9 iba a tomarse tantas molestias para conservarte si t\u00fa no ten\u00edas un don que ella no solo ambicionaba, sino que tambi\u00e9n tem\u00eda?\n\n\u2014T\u00fa me ense\u00f1aste a hacer todo eso. Las sesiones de entrenamiento en la pradera... La mariposa y la cierva... Me ense\u00f1aste a ejercitar mi esp\u00edritu. Me entrenaste para mi propia muerte.\n\n\u2014Ella me escogi\u00f3 para prepararte. Por eso me dej\u00f3 llevarte a Magdalen. Pero no pienso dejar que se salga con la suya. Me he comprometido a desarrollar tu don, pero para ti, Paige. No para ella.\n\nNo dije nada. No hab\u00eda nada que decir.\n\nEl Custodio arranc\u00f3 un pedazo de cortina y, con cuidado, empez\u00f3 a quitarme el maquillaje. Yo le dej\u00e9 hacer. Ten\u00eda los labios entumecidos, la piel congelada. Tal vez estuviera muerta al cabo de unos minutos, flotando alrededor de Nashira en un estado de ciega servidumbre. Cuando hubo terminado, el Custodio me apart\u00f3 el pelo de la cara. Le dej\u00e9 hacer. No pod\u00eda enfocar la mirada.\n\n\u2014Ni se te ocurra \u2014me dijo\u2014. Ni se te ocurra dej\u00e1rselo ver. T\u00fa eres mucho m\u00e1s que eso. Eres mucho m\u00e1s de lo que ella quiere hacerte.\n\n\u2014No tengo miedo.\n\nMe escudri\u00f1\u00f3 la cara.\n\n\u2014Deber\u00edas tenerlo \u2014dijo\u2014. Pero no se lo muestres. Por nada del mundo.\n\n\u2014Le mostrar\u00e9 lo que quiera. No est\u00e1s en posici\u00f3n de darme \u00f3rdenes. \u2014Le apart\u00e9 las manos de mi cabeza\u2014. Debiste dejarme marchar. Debiste dejar que Nick me llevara a Dials. Era lo \u00fanico que ten\u00edas que hacer. Ahora podr\u00eda estar en mi casa.\n\nEl Custodio se agach\u00f3 hasta que nuestras caras quedaron a la misma altura.\n\n\u2014Te traje aqu\u00ed porque sin ti no encontraba la fuerza necesaria para enfrentarme a ella. Pero, por esa misma raz\u00f3n, har\u00e9 todo lo posible para que llegues sana y salva a la ciudadela.\n\nNos quedamos callados. Le sostuve la mirada.\n\n\u2014Tienes que recogerte el pelo \u2014dijo con otra voz, m\u00e1s serena, y me puso un peine ornamentado en la mano.\n\nEl peine estaba fr\u00edo. Me temblaban los dedos.\n\n\u2014Creo que no puedo. \u2014Respir\u00e9 hondo, despacio\u2014. \u00bfQuieres hacerlo t\u00fa?\n\nNo dijo nada, pero cogi\u00f3 el peine. Como si estuviera tocando una telara\u00f1a fin\u00edsima, me puso todo el pelo a un lado del cuello y me lo recogi\u00f3 en un mo\u00f1o. No un mo\u00f1o sencillo, como los que me hac\u00eda yo, sino un mo\u00f1o trenzado muy elaborado, pegado al cogote. Desliz\u00f3 sus callosos dedos por mi cuero cabelludo, dando los \u00faltimos toques al peinado. Me recorri\u00f3 un d\u00e9bil escalofr\u00edo. El Custodio apart\u00f3 las manos, y el mo\u00f1o permaneci\u00f3 en su sitio.\n\nHab\u00eda notado algo raro en su forma de tocarme. Cuando le vi las manos comprend\u00ed por qu\u00e9.\n\nNo llevaba puestos los guantes.\n\nMe toqu\u00e9 la cabeza y pas\u00e9 los dedos por el complicado peinado. Parec\u00eda mentira que unas manos tan grandes como las del Custodio hubieran sido capaces de semejante complejidad.\n\n\u2014El tren sale a la una en punto \u2014me dijo al o\u00eddo\u2014. La entrada est\u00e1 bajo el campo de entrenamiento. Exactamente donde entrenabas conmigo.\n\nLlevaba mucho tiempo esperando o\u00edr esas palabras.\n\n\u2014Si me mata, tienes que dec\u00edrselo a los otros \u2014dije con la voz tomada\u2014. Tienes que guiarlos.\n\nMe acarici\u00f3 un brazo y dijo:\n\n\u2014No har\u00e1 falta que los gu\u00ede.\n\nNot\u00e9 un estremecimiento, pero no de la clase que yo esperaba. Cuando gir\u00e9 la cabeza y lo mir\u00e9, \u00e9l me recogi\u00f3 un rizo rebelde detr\u00e1s de la oreja. Me puso la otra mano en el abdomen y apret\u00f3 mi espalda contra su pecho. Su calor me reconfort\u00f3.\n\nY not\u00e9 sus ansias de m\u00ed. No de mi aura, sino de m\u00ed.\n\nAcerc\u00f3 la cabeza a mi mejilla y me acarici\u00f3 las clav\u00edculas. Su onirosaje estaba muy cerca, y su aura se entrelazaba con la m\u00eda. Mi sexto sentido se acentu\u00f3.\n\n\u2014Tienes la piel fr\u00eda \u2014dijo con voz ronca\u2014. Yo nunca...\n\nSe interrumpi\u00f3. Mis dedos se trabaron con los suyos, desnudos. Mantuve los ojos abiertos.\n\nAcerc\u00f3 los labios a mi ment\u00f3n, y le gui\u00e9 una mano hacia mi cintura. El atractivo de su tacto era insoportable; no pod\u00eda resistirme. No pod\u00eda rechazarlo. Deseaba aquello, antes de que llegara el fin. Quer\u00eda que me tocaran, que me vieran; all\u00ed, en aquella habitaci\u00f3n a oscuras, en aquel silencio rojo. Levant\u00e9 la barbilla, y sus labios se cerraron sobre los m\u00edos.\n\nSiempre hab\u00eda sabido que no exist\u00eda el cielo. Jax me lo hab\u00eda dicho muchas veces. Hasta el Custodio lo hab\u00eda dicho. Solo hab\u00eda luz blanca, la \u00faltima luz: un \u00faltimo descanso al borde de la inconsciencia, el lugar donde todas las cosas encuentran su fin. M\u00e1s all\u00e1, qui\u00e9n sabe. Pero, si exist\u00eda el cielo, deb\u00eda de parecerse a aquello. Tocar el \u00e9ter con las manos desnudas. Jam\u00e1s habr\u00eda podido esperar algo semejante, y menos de \u00e9l. De nadie. Me agarr\u00e9 a su espalda, tir\u00e9 de \u00e9l hacia m\u00ed. Me puso una mano en la nuca. Not\u00e9 los callos de la palma de sus manos.\n\nEl calor de su aliento. Me bes\u00f3 despacio. \u00abNo pares. No pares.\u00bb Solo pod\u00eda pensar en esas palabras: \u00abNo pares\u00bb. Desliz\u00f3 las manos por mis costados, por mi espalda, y me estrech\u00f3 firmemente. Me subi\u00f3 a un caj\u00f3n. Le puse una mano en el cuello y le not\u00e9 el pulso. Su ritmo. Mi ritmo.\n\nMe ard\u00eda la piel. No pod\u00eda parar. Jam\u00e1s hab\u00eda sentido nada parecido, aquel apremio, aquella necesidad de acariciar. Sus labios separaron los m\u00edos. Abr\u00ed los ojos. \u00abPara. Para, Paige.\u00bb Empec\u00e9 a apartar la cabeza. Se me escap\u00f3 una palabra: un \u00abno\u00bb, o quiz\u00e1 un \u00abs\u00ed\u00bb. Quiz\u00e1 su nombre. Me sujet\u00f3 la cara con ambas manos, me acarici\u00f3 los labios, las mejillas. Nuestras frentes se tocaron. Mi onirosaje estaba ardiendo. El Custodio prendi\u00f3 fuego a las amapolas. \u00abNo pares. No pares.\u00bb\n\nSolo fue un momento. Lo mir\u00e9, y \u00e9l me mir\u00f3. Un momento. Una decisi\u00f3n. Mi decisi\u00f3n. La suya. Entonces volvi\u00f3 a besarme, esa vez con m\u00e1s \u00edmpetu. Le dej\u00e9 hacer. Me rode\u00f3 con los brazos y me levant\u00f3. Y yo lo deseaba, con toda el alma. Le as\u00ed la cabeza, me agarr\u00e9 a su cuello. \u00abNo pares.\u00bb Ten\u00eda sus labios en la boca, en los ojos, en los hombros, en el hueco entre las clav\u00edculas. \u00abNo pares.\u00bb Desliz\u00f3 la palma de las manos por mis muslos y los acarici\u00f3 en\u00e9rgicamente, con confianza.\n\nLe abr\u00ed la camisa y le acarici\u00e9 el pecho. Le bes\u00e9 el cuello, y \u00e9l me agarr\u00f3 un mech\u00f3n de pelo. \u00abNo pares.\u00bb Nunca hab\u00eda tocado su piel. La not\u00e9 caliente y suave, y me hizo desear el resto de su cuerpo. Mis manos se deslizaron por debajo de su camisa y buscaron su espalda. Mis dedos encontraron sus cicatrices, unas bregaduras alargadas, crueles. Yo siempre hab\u00eda sabido que las ten\u00eda. Eran las cicatrices de un traidor.\n\nSe puso en tensi\u00f3n.\n\n\u2014Paige \u2014susurr\u00f3, pero no me detuve; emiti\u00f3 un ruido gutural y sus labios se acercaron a los m\u00edos.\n\nYo no iba a traicionarlo. La Era de Huesos XVIII ya era historia, y no se repetir\u00eda.\n\nDoscientos a\u00f1os eran m\u00e1s que suficiente.\n\nMi sexto sentido me sac\u00f3 de mi aturdimiento. Me separ\u00e9 del Custodio. \u00c9l segu\u00eda con las manos en mi cintura, tirando de m\u00ed hacia \u00e9l.\n\nNashira estaba all\u00ed, semioculta entre las sombras. Me dio un vuelco el coraz\u00f3n.\n\n\u00abCorre\u00bb, me dijo mi aturdido cerebro; pero no pod\u00eda correr. Nashira lo hab\u00eda visto todo. Mi piel sudada, mis labios hinchados, mi pelo alborotado. Las manos del Custodio sujet\u00e1ndome las caderas. Su camisa abierta. Mis dedos todav\u00eda tanteando su piel.\n\nMe qued\u00e9 paralizada. Ni siquiera pod\u00eda mover los ojos.\n\nEl Custodio me puso detr\u00e1s de su espalda.\n\n\u2014La he obligado \u2014dijo con voz \u00e1spera.\n\nNashira no dijo nada.\n\nAvanz\u00f3 hacia la d\u00e9bil luz que se colaba entre las cortinas. Vi que ten\u00eda algo en las manos: la campana de cristal. Mir\u00e9 dentro; me zumbaban los o\u00eddos. Dentro hab\u00eda una flor. Una flor fresca, preciosa y extra\u00f1a, con los ocho p\u00e9talos h\u00famedos de n\u00e9ctar. La flor que antes estaba marchita.\n\n\u2014No puede haber piedad \u2014dijo Nashira\u2014 para esto.\n\nEl Custodio mir\u00f3 un momento la flor con ojos resplandecientes. Luego mir\u00f3 a Nashira.\n\nNashira solt\u00f3 la campana de cristal, que se estrell\u00f3 contra el suelo y me sac\u00f3 de mi par\u00e1lisis.\n\nAcababa de destruirlo todo.\n\n\u2014Arcturus Mesarthim, eres mi consorte de sangre. Eres Custodio de los Mesarthim. Pero esto no puede volver a pasar. \u2014Nashira avanz\u00f3 hacia nosotros\u2014. Solo existe una forma de impedir la traici\u00f3n, y consiste en poner a los traidores de ejemplo. Te colgar\u00e9 de los muros de esta ciudad.\n\nEl Custodio no se movi\u00f3.\n\n\u2014Mejor eso que ser utilizado para satisfacer tus placeres.\n\n\u2014Siempre tan audaz. O tan insensato. \u2014Le pas\u00f3 una mano por la cara\u2014. Me encargar\u00e9 de que destruyan a todos tus viejos compa\u00f1eros.\n\n\u2014No. \u2014Sal\u00ed de detr\u00e1s de \u00e9l\u2014. No puedes...\n\nNo me dio tiempo a moverme. El golpe que me asest\u00f3 me tir\u00f3 al suelo. Me di con el canto de un caj\u00f3n, y me hice un corte en la ceja. Par\u00e9 la ca\u00edda apoyando las manos en los cristales rotos. O\u00ed que el Custodio dec\u00eda mi nombre, con rabia; pero entonces aparecieron Thuban y Situla, los fieles servidores de Nashira, dispuestos a no soltarlo. Thuban golpe\u00f3 al Custodio en la cabeza con el mango de su pu\u00f1al, pero no lo derrib\u00f3. Esa vez el Custodio no iba a arrodillarse ante los Sargas.\n\n\u2014Ya me ocupar\u00e9 de tus delitos m\u00e1s tarde, Arcturus. Te despojo de tu posici\u00f3n de consorte de sangre. \u2014Nashira se apart\u00f3 de \u00e9l\u2014. Thuban, Situla: llevadlo a la galer\u00eda.\n\n\u2014S\u00ed, mi soberana \u2014dijo Thuban. Agarr\u00f3 al Custodio por el cuello\u2014. Ha llegado la hora de que pagues lo que debes, traidor.\n\nSitula le hinc\u00f3 los dedos en el hombro, avergonzada de su primo traidor. \u00c9l no dijo nada.\n\n\u00ab\u00a1No!\u00bb Aquello no pod\u00eda acabar as\u00ed, como la Era de Huesos XVIII. El Custodio ya no era consorte de sangre. Estaba perdido. Yo hab\u00eda apagado el \u00faltimo rayo de luz. Busqu\u00e9 su mirada, algo que rescatar; pero el Custodio ten\u00eda los ojos inexpresivos y oscuros, y lo \u00fanico que sent\u00ed fue su silencio. Thuban y Situla se lo llevaron por la fuerza.\n\nNashira camin\u00f3 por encima de los cristales rotos. Me qued\u00e9 inm\u00f3vil y not\u00e9 que se me empa\u00f1aban los ojos. Qu\u00e9 idiota era. \u00bfEn qu\u00e9 estaba pensando? \u00bfQu\u00e9 estaba haciendo?\n\n\u2014Ha llegado tu hora, onir\u00e1mbula.\n\n\u2014Por fin. \u2014Me sangraba la herida de la frente\u2014. Has esperado mucho.\n\n\u2014Deber\u00edas alegrarte. Tengo entendido que los onir\u00e1mbulos ans\u00edan el \u00e9ter. Hoy podr\u00e1s unirte a \u00e9l.\n\n\u2014Nunca dominar\u00e1s este mundo. \u2014Levant\u00e9 la mirada; temblaba de rabia, no de miedo\u2014. Puedes matarme. Puedes reclamarme. Pero no puedes reclamarnos a todos. Los Siete Sellos esperan. Jaxon Hall espera. El sindicato entero te espera. \u2014Alc\u00e9 la barbilla y la mir\u00e9 a los ojos\u2014. Buena suerte.\n\nNashira me levant\u00f3 tir\u00e1ndome del pelo y acerc\u00f3 su cara a la m\u00eda.\n\n\u2014Podr\u00edas haber sido m\u00e1s \u2014dijo\u2014. Mucho m\u00e1s. Pero resulta que pronto no ser\u00e1s nada. Todo cuanto eras ser\u00e1 m\u00edo. \u2014Sacudi\u00f3 el brazo y me lanz\u00f3 contra otro refa\u00edta, que me sujet\u00f3 fuertemente\u2014. Alsafi, lleva este saco de huesos al escenario. Ha llegado la hora de que entregue su esp\u00edritu.\n\nNo me par\u00e9 a pensar: Alsafi me hizo subir la escalera. Me hab\u00eda puesto una bolsa en la cabeza. Ten\u00eda los labios resecos y las mejillas ardiendo. No pod\u00eda respirar, ni pensar con calma.\n\nEl Custodio hab\u00eda desaparecido. Lo hab\u00eda perdido. Era mi \u00fanico aliado refa, y hab\u00eda dejado que lo capturaran. Nashira no se limitar\u00eda a matarlo, despu\u00e9s de que \u00e9l se rebajara hasta el punto de tocar a una humana con las manos desnudas. Eso era algo m\u00e1s que traici\u00f3n. Al besarme, al abrazarme, el consorte de sangre hab\u00eda humillado a toda su familia. Ya no era un candidato digno. Ya no era nada.\n\nAlsafi me sujetaba el brazo con firmeza. Me dispon\u00eda a morir. Dentro de menos de diez minutos me unir\u00eda al \u00e9ter, como un esp\u00edritu cualquiera. Mi cord\u00f3n arg\u00e9nteo se romper\u00eda. Ya no podr\u00eda regresar a mi cuerpo, el cuerpo que hab\u00eda ocupado durante diecinueve a\u00f1os. A partir de ahora tendr\u00eda que servir a Nashira.\n\nMe quitaron la bolsa de la cabeza. Estaba al lado del escenario, donde se representaba el final de la obra de teatro. Ten\u00eda a un refa a cada lado, Alsafi y Terebell. Terebell se agach\u00f3 y me pregunt\u00f3:\n\n\u2014\u00bfD\u00f3nde est\u00e1 Arcturus?\n\n\u2014Se lo han llevado a la galer\u00eda. Thuban y Situla.\n\n\u2014Nos ocuparemos de ellos. \u2014Alsafi me solt\u00f3 el brazo\u2014. Tienes que entretener a la soberana de sangre, onir\u00e1mbula.\n\nYa sab\u00eda que Terebell era una de las colaboradoras del Custodio, pero ignoraba que Alsafi tambi\u00e9n estuviera en su bando. No parec\u00eda un simpatizante, pero el Custodio tampoco.\n\nEl Capataz sali\u00f3 corriendo del escenario, con el traje manchado de sangre artificial y dejando atr\u00e1s su pu\u00f1al. Sus gritos suplicando piedad resonaron por el Consistorio. Los emisarios aplaudieron cuando un grupo de actores lo persigui\u00f3 hasta la calle; llevaban todos uniformes de Scion. La ovaci\u00f3n fue ensordecedora, y se prolong\u00f3 mientras Nashira sub\u00eda de nuevo al escenario.\n\n\u2014Gracias por su bondad, damas y caballeros. Me alegro de que hayan disfrutado con la obra. \u2014No parec\u00eda muy contenta\u2014. Tambi\u00e9n me alegro de poder hacerles una breve demostraci\u00f3n, para poner fin a la velada, de c\u00f3mo funciona el sistema judicial de Sheol I. Una de nuestras clarividentes ha mostrado tal desobediencia que no podemos permitir que siga con vida. Igual que el Rey Sangriento, debe ser desterrada m\u00e1s all\u00e1 del alcance de la poblaci\u00f3n amaur\u00f3tica, donde ya no pueda causar ning\u00fan da\u00f1o.\n\n\u00bbXX-59-40 tiene un largo historial de traici\u00f3n. Proviene de la regi\u00f3n lechera de Tipperary, en el sur de Irlanda, una regi\u00f3n asociada desde hace siglos a la sedici\u00f3n. \u2014Cathal Bell traslad\u00f3 el peso del cuerpo de una pierna a la otra, y algunos emisarios murmuraron\u2014. Nada m\u00e1s venir a Inglaterra, se mezcl\u00f3 con el crimen organizado de Londres. La noche del siete de marzo asesin\u00f3 a dos clarividentes, dos metrovigilantes al servicio de Scion. Los mat\u00f3 a sangre fr\u00eda. Ninguna de las dos v\u00edctimas de 40 tuvo una muerte r\u00e1pida. Esa misma noche la trajeron a Sheol I. \u2014Nashira se paseaba por el escenario\u2014. Confi\u00e1bamos en que podr\u00edamos educarla, ense\u00f1arle a controlar su don. Nos duele perder a clarividentes tan j\u00f3venes. Tambi\u00e9n me duele admitir que nuestros esfuerzos por reformar a 40 han fracasado. Ha correspondido a nuestra compasi\u00f3n con insolencia y crueldad, y por lo tanto, no le queda otra opci\u00f3n que enfrentarse al juicio del Inquisidor.\n\nMir\u00e9 m\u00e1s all\u00e1 de Nashira. En el escenario no hab\u00eda ninguna horca, ninguna camilla ni ning\u00fan tajo. Pero hab\u00eda una espada.\n\nCre\u00ed que mi coraz\u00f3n hab\u00eda dejado de latir. No era una espada normal y corriente. La hoja era de oro, y el pu\u00f1o, negro. Era la _C\u00f3lera del Inquisidor_ , la espada que a tantos traidores pol\u00edticos hab\u00eda decapitado. Solo la utilizaban cuando descubr\u00edan a alg\u00fan esp\u00eda clarividente dentro del Arconte de Westminster. Yo era la hija de un destacado cient\u00edfico de Scion, una traidora en las filas de los naturales. Alsafi y Terebell desaparecieron bajo el escenario. Me qued\u00e9 sola frente a Nashira, que gir\u00f3 la cabeza.\n\n\u2014Ven aqu\u00ed, 40.\n\nNo vacil\u00e9. Se hizo silencio cuando sal\u00ed de detr\u00e1s de las cortinas. \u00ab\u00a1Traidora!\u00bb, grit\u00f3 Cathal Bell, y algunos emisarios lo imitaron. Segu\u00ed sin mirarlos. Ten\u00eda gracia que Bell me llamara traidora.\n\nAvanc\u00e9 con la cabeza bien alta, concentr\u00e1ndome en Nashira y sin mirar a los emisarios. Tampoco mir\u00e9 hacia la galer\u00eda, adonde hab\u00edan llevado al Custodio. Me detuve a cierta distancia de Nashira, que me rode\u00f3 lentamente, y cuando desapareci\u00f3 de mi campo de visi\u00f3n, mantuve la vista al frente.\n\n\u2014Se preguntar\u00e1n c\u00f3mo administramos la justicia en Sheol I. Con la soga, tal vez, o el fuego de tiempos ya lejanos. Aqu\u00ed est\u00e1 la espada del Inquisidor, tra\u00edda desde la ciudadela. \u2014Se\u00f1al\u00f3 la _C\u00f3lera_ \u2014. Pero antes de blandirla, quiero mostrarles otra cosa: el mayor don de los refa\u00edtas.\n\nHubo un murmullo.\n\n\u2014Eduardo VII era un hombre curioso. Sabemos muy bien que jugaba con cosas con las que es mejor no jugar. Intent\u00f3 controlar un poder que est\u00e1 m\u00e1s all\u00e1 del conocimiento humano. Un poder que los refa\u00edtas conocemos muy bien.\n\nBirgitta Tj\u00e4der miraba el escenario muy concentrada y con la frente fruncida. Varios emisarios, entre ellos Bell, miraron a sus guardaespaldas de la DVD.\n\n\u2014Imaginen la energ\u00eda m\u00e1s potente de la Tierra. \u2014Nashira alarg\u00f3 una mano hacia un farol cercano\u2014. La electricidad. Permite que su mundo funcione. Ilumina sus ciudades y sus hogares. Les permite comunicarse. El \u00e9ter (el origen, la fuerza vital de los refa\u00edtas) se parece a la electricidad. Puede alumbrar la oscuridad, y tornar en sabidur\u00eda la ignorancia. \u2014De pronto el farol emiti\u00f3 un intenso destello\u2014. Pero si se usa de forma incorrecta, puede destruir. Puede matar. \u2014Se apag\u00f3 la luz\u2014. Yo tengo un don que en los dos siglos pasados ha resultado muy \u00fatil. Algunos humanos clarividentes exhiben capacidades especialmente err\u00e1ticas. Canalizan el \u00e9ter, el reino de los muertos, de maneras que pueden generar locura y violencia. El Rey Sangriento ten\u00eda esa capacidad, que fue lo que provoc\u00f3 aquella tr\u00e1gica matanza. Yo puedo eliminar esas mutaciones peligrosas del don. \u2014Avanz\u00f3 hacia m\u00ed\u2014. La clarividencia, como la energ\u00eda, no se destruye, solo se transforma. Cuando 40 muera, otro clarividente desarrollar\u00e1 su don. Pero, guard\u00e1ndolo en mi interior, me asegurar\u00e9 de que nadie vuelva a utilizarlo nunca.\n\n\u2014Te gusta inventarte cosas, \u00bfverdad, Nashira?\n\nLo dije antes de que el pensamiento hubiera tomado forma en mi mente. Nashira se volvi\u00f3 y me fulmin\u00f3 con una mirada ardiente.\n\n\u2014No vuelvas a hablar \u2014dijo en voz baja.\n\nMe arriesgu\u00e9 a mirar hacia la galer\u00eda: estaba vac\u00eda. Por debajo de m\u00ed, Michael se llev\u00f3 una mano dentro de la chaqueta, donde ocultaba una pistola.\n\nSe abri\u00f3 una puerta al fondo del Consorcio: eran Terebell, Alsafi y el Custodio. Busqu\u00e9 la mirada del Custodio por encima de las cabezas de los emisarios. El cord\u00f3n \u00e1ureo tembl\u00f3. Vi una imagen del pu\u00f1al que el Capataz hab\u00eda dejado en el suelo del escenario, a escasos palmos de donde estaba Nashira.\n\nCuando se dio la vuelta hacia el p\u00fablico, mi esp\u00edritu se abalanz\u00f3 sobre ella. Con las \u00faltimas fuerzas que me quedaban, irrump\u00ed en su zona hadal. Mi ataque la pill\u00f3 desprevenida. Me vi a m\u00ed misma con una forma on\u00edrica inmensa, un B\u00e9gimo lo bastante grande para derribar cualquier barrera.\n\nEl \u00e9ter retumb\u00f3. Volaron esp\u00edritus por el Consorcio; acud\u00edan junto a Nashira desde todos los \u00e1ngulos. Se unieron a m\u00ed en los bordes de su onirosaje, rompiendo su arcaica coraza. Los cinco \u00e1ngeles intentaban defenderla; pero veinte, cincuenta, doscientos esp\u00edritus hab\u00edan descendido sobre ella, y las paredes empezaban a ceder. No perd\u00ed el tiempo: me lanc\u00e9 hacia el mism\u00edsimo centro de su onirosaje.\n\nVe\u00eda con sus ojos. La habitaci\u00f3n era una mancha borrosa, un torbellino de color y oscuridad, luz y fuego, un espectro de cosas que yo no hab\u00eda visto nunca. \u00bfEra as\u00ed como ve\u00edan los refas? Hab\u00eda auras por todas partes. Ten\u00eda visi\u00f3n, pero en ese momento estaba ciega, y los ojos de Nashira se negaban a ver. No quer\u00edan que yo viera. No eran mis ojos. Los obligu\u00e9 a abrirse y me mir\u00e9 la mano. Demasiado grande, y enguantada. Me costaba mantener los ojos abiertos, porque Nashira estaba luchando. \u00abDate prisa, Paige.\u00bb\n\nEl pu\u00f1al. El pu\u00f1al estaba all\u00ed. \u00abDeprisa.\u00bb Intent\u00e9 cogerlo. Mover la mano era como tratar de levantar unas pesas. \u00abM\u00e1tala.\u00bb En mis o\u00eddos resonaban gritos y otros sonidos extra\u00f1os, y voces, miles de voces. \u00abM\u00e1tala.\u00bb Mis nuevos dedos se enroscaron alrededor del mango.\n\nLo ten\u00eda. Llev\u00e9 el brazo hacia atr\u00e1s y me clav\u00e9 la hoja del pu\u00f1al en el pecho. Los emisarios gritaron. Volv\u00eda a tener visi\u00f3n de t\u00fanel. Todo parpadeaba. Retorc\u00ed el pu\u00f1al con mi nueva mano, hundi\u00e9ndolo en lo que fuera que compon\u00eda el cuerpo de Nashira. No sent\u00ed dolor. Nashira era inmune a las pu\u00f1aladas de un pu\u00f1al amaur\u00f3tico. Volv\u00ed a clavarlo, esa vez en el lado izquierdo, apuntando al sitio donde los humanos ten\u00edamos el coraz\u00f3n. Tampoco sent\u00ed dolor, pero cuando levant\u00e9 el brazo por tercera vez, me vi expulsada del cuerpo de Nashira.\n\nLos esp\u00edritus se dispersaron por la sala y apagaron todas las velas. El caos se apoder\u00f3 del Consorcio. Cuando recuper\u00e9 la visi\u00f3n, no distingu\u00eda nada. Solo o\u00eda gritos.\n\nVolvieron a encenderse las velas. Nashira yac\u00eda sobre las tablas del escenario. No se mov\u00eda. Ten\u00eda el pu\u00f1al clavado en el pecho, hasta el pu\u00f1o.\n\n\u2014\u00a1Soberana de sangre! \u2014grit\u00f3 un refa.\n\nLos emisarios guardaban silencio. Me temblaban las manos cuando me arrastr\u00e9 por el suelo hasta Nashira. Le mir\u00e9 la cara, los ojos carentes de luz. Los esp\u00edritus de la Era de Huesos XVIII segu\u00edan cerni\u00e9ndose sobre ella, como si esperaran a que se reuniera con ellos en el \u00e9ter.\n\nEntonces prendi\u00f3 en sus ojos un d\u00e9bil resplandor. Gir\u00f3 lentamente la cabeza. Me puse a temblar incontroladamente mientras ella se ergu\u00eda cuan alta era.\n\n\u2014Muy lista \u2014dijo\u2014. S\u00ed, muy lista.\n\nSegu\u00ed movi\u00e9ndome y ara\u00f1ando el suelo. Vi a Nashira arrancarse el pu\u00f1al del pecho. El p\u00fablico dio gritos de sorpresa.\n\n\u2014Ens\u00e9\u00f1anos m\u00e1s. \u2014Unas gotas de luz cayeron como l\u00e1grimas\u2014. No tengo objeciones.\n\nDio una sacudida con la mu\u00f1eca y el pu\u00f1al salt\u00f3 por los aires. Se qued\u00f3 un momento suspendido, como si colgara de un hilo invisible, y de pronto sali\u00f3 volando hacia m\u00ed. Me hizo un corte en la mejilla. Las velas chisporrotearon.\n\nUno de los \u00e1ngeles de Nashira era un duende. No era habitual que los duendes pudieran levantar la materia f\u00edsica, pero no era la primera vez que ve\u00eda a uno conseguirlo. \u00abAporte\u00bb, lo llamaba Jaxon. Esp\u00edritus que mov\u00edan objetos. Una pel\u00edcula de sudor fr\u00edo me cubr\u00eda la piel. No deb\u00eda tener miedo: ya me hab\u00eda enfrentado a un duende en otra ocasi\u00f3n. Ahora mi esp\u00edritu hab\u00eda madurado, sab\u00eda defenderme.\n\n\u2014Si insistes \u2014dije.\n\nEsa vez no la pill\u00e9 desprevenida. Levant\u00f3 todas las capas de blindaje con que proteg\u00eda su onirosaje. Como si dos puertas gigantescas se hubieran cerrado ante m\u00ed, me vi lanzada hacia atr\u00e1s, hacia mi cuerpo. Se me estremeci\u00f3 el coraz\u00f3n. La presi\u00f3n de mi cr\u00e1neo se intensific\u00f3. O\u00ed una voz conocida, pero la ahog\u00f3 otro sonido, agudo y prolongado.\n\nTen\u00eda que moverme. Nashira no iba a parar. Nunca parar\u00eda de perseguir mi esp\u00edritu. Me incorpor\u00e9 apoy\u00e1ndome en los codos y busqu\u00e9 el pu\u00f1al. Distingu\u00ed el contorno de Nashira acerc\u00e1ndose a m\u00ed.\n\n\u2014Pareces cansada, Paige. D\u00e9jalo ya. El \u00e9ter te llama.\n\n\u2014No debo de haber o\u00eddo esa llamada \u2014logr\u00e9 articular.\n\nLo que sucedi\u00f3 a continuaci\u00f3n me pill\u00f3 por sorpresa. Sus cinco \u00e1ngeles formaron una bandada y se abalanzaron sobre m\u00ed.\n\nAtravesaron mis defensas como una ola gigantesca y negra. Fuera de mi onirosaje, mi cabeza se golpe\u00f3 contra las tablas del suelo. Dentro, los esp\u00edritus abrieron un camino por el que se esparcieron p\u00e9talos rojos. Las im\u00e1genes pasaban a toda velocidad ante mis ojos. Todos mis pensamientos, todos mis recuerdos estaban rotos. Sangre, fuego, sangre. Un campo moribundo. Sent\u00eda como si una mano me apretara el pecho, inmoviliz\u00e1ndome. En una caja, un ata\u00fad. No pod\u00eda moverme, ni respirar, ni pensar. Los cinco esp\u00edritus me atravesaron como una espada, arranc\u00e1ndome pedazos de mente, de alma. Me puse de lado y me retorc\u00ed como un insecto aplastado.\n\nMe temblaban los m\u00fasculos de los brazos y las piernas. Abr\u00ed los ojos. La luz los abras\u00f3. Solo ve\u00eda a Nashira, con la mano extendida, la luz de las velas reflejada en la hoja del pu\u00f1al. Y entonces desapareci\u00f3. Levant\u00e9 la cabeza del suelo haciendo un gran esfuerzo. Ten\u00eda l\u00e1grimas en los ojos. Michael hab\u00eda saltado sobre la espalda de Nashira para distraerla. Ten\u00eda un pu\u00f1al en la mano. Se lo clav\u00f3 en el cuello, fall\u00f3 por poco. Nashira dio una sacudida con el brazo y lo tir\u00f3 del escenario.\n\nMichael cay\u00f3 sobre un buf\u00f3n, y ambos fueron a parar al suelo. Nashira volver\u00eda a atacarme, y esta vez acabar\u00eda conmigo. Vi su cara por encima de m\u00ed. Se le pusieron los ojos rojos. Sus facciones se desdibujaron. Estaba debilit\u00e1ndome, asegur\u00e1ndose de que no pudiera volver a utilizar mi esp\u00edritu. Alterando mi conexi\u00f3n con el \u00e9ter. Estaba perdida. Nashira se arrodill\u00f3 a mi lado y me coloc\u00f3 la cabeza en el pliegue interno de su codo.\n\n\u2014Gracias, Paige Mahoney. \u2014Me acerc\u00f3 la punta del pu\u00f1al al cuello\u2014. No desperdiciar\u00e9 este don.\n\nYa estaba. Ni siquiera tuve un \u00faltimo pensamiento. Us\u00e9 mi \u00faltima pizca de energ\u00eda para mirarla a los ojos.\n\nY entonces vi al Custodio. La obligaba a retroceder utilizando unas bandadas enormes que hac\u00eda girar para formar escudos, como un tragafuegos con sus antorchas. \u00abSi yo tuviera visi\u00f3n \u2014pens\u00e9 vagamente\u2014, estar\u00eda contemplando un espect\u00e1culo magn\u00edfico.\u00bb Terebell y Alsafi estaban con \u00e9l; y tambi\u00e9n otros. \u00bfNo era esa Pleione? Sus contornos se confund\u00edan. Mi onirosaje enviaba extra\u00f1os espejismos m\u00e1s all\u00e1 de mi l\u00ednea de visi\u00f3n. Entonces alguien me cogi\u00f3 en brazos y me sac\u00f3 del escenario.\n\nVe\u00eda destellos. Hab\u00eda estallado una tormenta en mi onirosaje: los recuerdos se filtraban por unas grietas con forma de rayos, y un fuerte viento destrozaba las flores. Hab\u00edan saqueado mi mente.\n\nNo era plenamente consciente de lo que suced\u00eda en el mundo exterior. El Custodio estaba all\u00ed. Reconoc\u00ed su onirosaje, una presencia conocida contra la m\u00eda. Me llevaba en brazos a la galer\u00eda, lejos de eso que hab\u00eda ocurrido en los pocos minutos que hab\u00eda estado inconsciente. Me dej\u00f3 en el suelo, y not\u00e9 la sangre sec\u00e1ndose en mi cara. Apenas recordaba d\u00f3nde estaba.\n\n\u2014Comb\u00e1telo, Paige. Tienes que combatirlo.\n\nMe acarici\u00f3 el pelo. Observ\u00e9 su cara e intent\u00e9 que las l\u00edneas dejaran de desdibujarse.\n\nApareci\u00f3 otro par de ojos. Me pareci\u00f3 que volv\u00eda a ser Terebell. Perd\u00ed otra vez el conocimiento, y me despert\u00f3 un rugido sordo. El ruido presionaba contra mis sienes. Cuando el dolor me oblig\u00f3 a regresar al mundo de la carne, vi al Custodio mir\u00e1ndome. Est\u00e1bamos en la galer\u00eda, por encima del clamor del sal\u00f3n.\n\n\u2014\u00bfMe oyes, Paige?\n\nParec\u00eda una pregunta. Dije que s\u00ed con la cabeza.\n\n\u2014Nashira \u2014dije con un hilo de voz.\n\n\u2014Est\u00e1 viva. Pero t\u00fa tambi\u00e9n.\n\n\u00abEst\u00e1 viva.\u00bb Nashira segu\u00eda all\u00ed. Sent\u00ed los tenues indicios del p\u00e1nico, pero ten\u00eda el cuerpo demasiado d\u00e9bil para reaccionar. Aquello todav\u00eda no hab\u00eda terminado.\n\nAbajo reson\u00f3 el ruido de un disparo. Salvo los ojos del Custodio, todo estaba oscuro.\n\n\u2014Hab\u00eda... \u2014El Custodio se acerc\u00f3 m\u00e1s a mis labios para o\u00edrme\u2014. Hab\u00eda un duende. Nashira tiene... un duende.\n\n\u2014S\u00ed. Pero t\u00fa estabas preparada. \u2014Me resigui\u00f3 el escote con un dedo\u2014. \u00bfNo te dije que esto podr\u00eda salvarte la vida?\n\nLa luz de sus ojos se reflej\u00f3 en el colgante; el objeto sublimado, dise\u00f1ado para repeler a los duendes. El que me hab\u00eda regalado. El que yo hab\u00eda intentado rechazar, y que hab\u00eda estado a punto de no ponerme esa noche. El Custodio me apoy\u00f3 contra su pecho y me puso una mano detr\u00e1s de la cabeza.\n\n\u2014Va a llegar ayuda \u2014dijo en voz muy baja\u2014. Han venido a buscarte, Paige. Los Sellos han venido por ti.\n\nHubo otro momento de ceguera, y el ruido se intensific\u00f3. Mi onirosaje intentaba curarse. Los da\u00f1os hab\u00edan sido graves; tardar\u00eda d\u00edas en empezar a repararse. Quiz\u00e1 no empezara nunca. No lo sab\u00eda; pero no pod\u00eda moverme, y se estaba agotando el tiempo: ten\u00eda que llegar a la pradera, encontrar la salida. Me marchaba a casa. Ten\u00eda que marcharme a casa.\n\nCuando volv\u00ed a abrir los ojos, una luz muy intensa me los quem\u00f3. No era luz de velas. Intent\u00e9 protegerlos; respiraba entrecortadamente.\n\n\u2014Paige. \u2014Alguien me tom\u00f3 la mano. No era el Custodio, sino alguien m\u00e1s\u2014. Paige, tesoro.\n\nConoc\u00eda esa voz.\n\n\u00c9l no pod\u00eda estar all\u00ed. Deb\u00eda de ser una aparici\u00f3n, una imagen de mi da\u00f1ado onirosaje. Pero cuando me tom\u00f3 la mano supe que era real. Todav\u00eda ten\u00eda la cabeza sobre el regazo del Custodio.\n\n\u2014Nick \u2014consegu\u00ed decir. Llevaba traje negro y corbata roja.\n\n\u2014S\u00ed, _s\u00f6tnos_ , soy yo.\n\nMe mir\u00e9 los dedos. Se me estaban poniendo grises, y las u\u00f1as reposaban sobre lechos de un morado oscuro.\n\n\u2014Paige \u2014dijo Nick con voz grave, y con tono de urgencia\u2014, mant\u00e9n los ojos abiertos. Qu\u00e9date con nosotros, tesoro. Venga.\n\n\u2014Tienes que irte \u2014dije con voz ronca.\n\n\u2014S\u00ed, claro, voy a irme. Y t\u00fa tambi\u00e9n.\n\n\u2014Mu\u00e9vete, Visi\u00f3n. No hay tiempo que perder. \u2014Era otra voz\u2014. Trataremos a nuestra peque\u00f1a So\u00f1adora P\u00e1lida dormida cuando lleguemos a la ciudadela.\n\nEra Jaxon.\n\n\u00ab\u00a1No!\u00bb \u00bfPor qu\u00e9 hab\u00edan venido? Nashira los ver\u00eda.\n\n\u2014Ya ser\u00e1 demasiado tarde. \u2014La misma luz volvi\u00f3 a alumbrarme los ojos\u2014. Las pupilas no responden. Hipoxia cerebral. Si no hacemos esto, morir\u00e1. \u2014Una mano me apart\u00f3 el pelo de la cara, sudada y pegajosa\u2014. \u00bfD\u00f3nde demonios est\u00e1 Danica?\n\nNo entend\u00eda por qu\u00e9 el Custodio no dec\u00eda nada. Estaba all\u00ed: yo lo notaba.\n\nSufr\u00ed otro desmayo. Cuando recuper\u00e9 la visi\u00f3n, ten\u00eda algo sobre la boca y la nariz. Reconoc\u00ed aquel olor a pl\u00e1stico: era un SVP2, la versi\u00f3n port\u00e1til del equipo de soporte vital de Dani. Hab\u00eda otros onirosajes cerca, api\u00f1ados a mi alrededor. Nick me sosten\u00eda la cabeza y sujetaba la mascarilla sobre mi boca. Respir\u00e9 la dosis extra de ox\u00edgeno, adormilada. Jam\u00e1s me hab\u00eda sentido tan agotada.\n\n\u2014No funciona. Su onirosaje se ha fracturado.\n\n\u2014Ese tren no nos esperar\u00e1, Visi\u00f3n \u2014dijo Jaxon con tono cortante\u2014. Ll\u00e9vala. Nos vamos.\n\nEsas palabras se filtraron en mi cerebro. Por primera vez desde hac\u00eda varios minutos o\u00ed hablar al Custodio:\n\n\u2014Yo puedo ayudarla.\n\n\u2014No se acerque \u2014salt\u00f3 Nick.\n\n\u2014No hay tiempo que perder. La DVN est\u00e1 en el puente. Vendr\u00e1n y ver\u00e1n su aura inmediatamente, doctor Nyg\u00e5rd. Perder\u00e1 la reputaci\u00f3n que tiene en Scion. \u2014El Custodio los mir\u00f3\u2014. Si no hacen nada, Paige morir\u00e1. Podemos reparar su onirosaje, pero solo si nos damos prisa. \u00bfQuiere perder a su onir\u00e1mbula, Vinculador Blanco?\n\n\u2014\u00bfC\u00f3mo sabe mi nombre? \u2014salt\u00f3 Jaxon. No lo ve\u00eda en la oscuridad, pero not\u00e9 el cambio repentino en su onirosaje, not\u00e9 que levantaba sus defensas.\n\n\u2014Tenemos formas de saber las cosas.\n\nSus palabras eran como una secuencia de patrones imposible de desentra\u00f1ar. No las entend\u00eda. Nick se inclin\u00f3 sobre m\u00ed; percib\u00ed su aliento c\u00e1lido en la mejilla.\n\n\u2014Paige \u2014me dijo al o\u00eddo\u2014, este hombre dice que puede curarte. \u00bfPuedo confiar en \u00e9l?\n\n\u00abConfiar.\u00bb Reconoc\u00ed esa palabra. Una flor ba\u00f1ada por el sol en el umbral de la percepci\u00f3n, invit\u00e1ndome a entrar en otro mundo. Una vida diferente, anterior al prado de amapolas.\n\n\u2014S\u00ed.\n\nEn cuanto dije eso, el Custodio se me acerc\u00f3. Vi a Pleione detr\u00e1s de \u00e9l.\n\n\u2014Paige, necesito que bajes todas las defensas mentales que puedas \u2014me dijo\u2014. \u00bfPuedes hacerlo?\n\nComo si me quedara alguna defensa.\n\nEl Custodio cogi\u00f3 un vial que Pleione le acerc\u00f3 con una mano enguantada. Un vial de amaranto, casi vac\u00edo. \u00abMarcado.\u00bb Deb\u00edan de hacer acopio de aquellos viales, guardar cada gota de amaranto que pudieran. Me puso un poco bajo la nariz, y otro poco en los labios. Not\u00e9 que un calor se extend\u00eda bajo mi piel. Era como si el \u00e9ter me llamara, como si me pidiera que abriera la mente. Sent\u00ed un calor repentino que cosi\u00f3 los desgarrones de mi onirosaje. El Custodio me acarici\u00f3 la mejilla con un pulgar.\n\n\u2014\u00bfPaige?\n\nParpade\u00e9.\n\n\u2014\u00bfEst\u00e1s bien?\n\n\u2014S\u00ed. Creo que s\u00ed.\n\nMe incorpor\u00e9, y luego intent\u00e9 levantarme. Nick me ayud\u00f3 a sostenerme en pie. No sent\u00eda dolor. Me frot\u00e9 los ojos y parpade\u00e9 tratando de adaptarme a la oscuridad.\n\n\u2014\u00bfC\u00f3mo demonios hab\u00e9is llegado hasta aqu\u00ed? \u2014dije sujet\u00e1ndome a sus brazos. No pod\u00eda dejar de mirarlo. Era real, y estaba all\u00ed, conmigo.\n\n\u2014Con el grupo de Scion. Ya te lo explicar\u00e9. \u2014Me abraz\u00f3 y me apret\u00f3 contra su pecho\u2014. Vamos. Nos largamos de aqu\u00ed.\n\nJaxon, un poco m\u00e1s all\u00e1, sujetaba su bast\u00f3n con ambas manos.\n\nDanica y Zeke estaban cada uno a uno de sus lados. Todos vest\u00edan los colores de Scion. Al otro lado de la galer\u00eda Nadine disparaba al azar contra los emisarios con su pistola. Los refa\u00edtas me miraban.\n\n\u2014Custodio, \u00bfcu\u00e1nto... \u2014inspir\u00e9 hondo\u2014... cu\u00e1nto tiempo nos queda?\n\n\u2014Cincuenta minutos. Ten\u00e9is que iros.\n\nMenos de una hora. Cuanto antes lleg\u00e1ramos al tren, antes podr\u00eda lanzar la bengala para se\u00f1alar el camino a los otros videntes.\n\n\u2014Supongo que todav\u00eda sabes a qui\u00e9n debes lealtad, Paige \u2014dijo Jaxon. Me mir\u00f3 de arriba abajo\u2014. Casi me hiciste ponerlo en duda, dama m\u00eda, con aquel numerito que montaste en Londres.\n\n\u2014Jaxon, aqu\u00ed est\u00e1 muriendo gente. Est\u00e1n muriendo videntes. \u00bfNo podemos dejar eso aparte y concentrarnos en largarnos de aqu\u00ed?\n\nNo tuvo ocasi\u00f3n de contestar. Un grupo de refas irrumpi\u00f3 en la galer\u00eda blandiendo bandadas enormes. El Custodio y Pleione se colocaron delante de nosotros.\n\n\u2014\u00a1Marchaos! \u2014dijo el Custodio.\n\nEstaba en un dilema. Jaxon ya hab\u00eda empezado a bajar la escalera, y los otros lo segu\u00edan.\n\n\u2014\u00a1Vamos, Paige! \u2014me exhort\u00f3 Nick.\n\nPleione par\u00f3 una bandada. El Custodio me mir\u00f3.\n\n\u2014Corre. Ve a Puerto Pradera \u2014dijo\u2014. Nos encontraremos all\u00ed.\n\nNo ten\u00eda alternativa; no pod\u00eda obligarlo a ir conmigo. Solo pod\u00eda obedecerle y confiar en estar haciendo lo que deb\u00eda. Nick me agarr\u00f3 por el brazo, bajamos corriendo la escalera y llegamos al vest\u00edbulo del Consistorio. No hab\u00eda tiempo para detenerse.\n\nLos bufones y los refas hab\u00edan salido del edificio y ocupaban las calles. Los aterrados emisarios y sus guardias de la DVN corr\u00edan por el vest\u00edbulo. Nick los sigui\u00f3. Not\u00e9 que el \u00e9ter temblaba, y me par\u00e9.\n\nMe volv\u00ed hacia el sal\u00f3n. Pasaba algo raro; estaba segura. Sin saber muy bien qu\u00e9 hac\u00eda, volv\u00ed corriendo a la escalera de piedra.\n\n\u2014\u00bfSe puede saber ad\u00f3nde vas? \u2014me grit\u00f3 Jaxon.\n\n\u2014D\u00e9jame, Jaxon. Id al tren.\n\nNo o\u00ed su respuesta. Nick me sigui\u00f3 y me agarr\u00f3 por el brazo.\n\n\u2014\u00bfAd\u00f3nde vas?\n\n\u2014Vete con Jaxon.\n\n\u2014Tenemos que salir de aqu\u00ed. Si la DVN ve mi aura...\n\nSe par\u00f3 cuando llegamos al sal\u00f3n, vac\u00edo.\n\nEstaba muy oscuro. Casi todas las velas se hab\u00edan apagado, pero todav\u00eda hab\u00eda tres faroles rojos encendidos, tirados por el suelo. Las cintas de seda con las que hab\u00eda actuado Liss se hab\u00edan ca\u00eddo y formaban dos montones. Fui hacia ellas, y not\u00e9 el d\u00e9bil parpadeo de un onirosaje. Corr\u00ed por el suelo de m\u00e1rmol y me arrodill\u00e9.\n\n\u2014\u00a1Liss! \u2014le cog\u00ed una mano\u2014. \u00a1Vamos, Liss!\n\n\u00bfPor qu\u00e9 hab\u00eda vuelto adonde estaban sus sedas? Ten\u00eda sangre en el pelo. No pod\u00eda estar muerta, no despu\u00e9s de que le hubi\u00e9ramos salvado la vida. No despu\u00e9s de lo mucho que hab\u00edamos trabajado juntas. No pod\u00eda morir. Seb hab\u00eda muerto; \u00bfpor qu\u00e9 ten\u00eda que seguir ella sus pasos?\n\nLiss entorn\u00f3 los ojos. Todav\u00eda iba vestida de v\u00edctima del rey. Al verme, sus labios esbozaron una sonrisa.\n\n\u2014Hola \u2014dijo. Hac\u00eda ruido al respirar\u2014. Lo siento, he llegado... tarde.\n\n\u2014No. Ni se te ocurra morirte, Liss. Por favor. \u2014Le apret\u00e9 una mano\u2014. Por favor. La otra vez cre\u00edamos que te hab\u00edamos perdido. No nos hagas eso otra vez.\n\n\u2014Me alegro de saber que le importo a alguien. \u2014Se me agolparon las l\u00e1grimas en los ojos; unas l\u00e1grimas fr\u00edas y temblorosas que se resist\u00edan a caer. A Liss le sal\u00eda sangre por la boca. Yo no distingu\u00eda d\u00f3nde terminaba la sangre del escenario y d\u00f3nde empezaba la suya\u2014. Ve... Vete \u2014dijo con voz d\u00e9bil\u2014. Haz lo que... yo no he podido... hacer. Solo... quer\u00eda... ver mi casa.\n\nSe le cay\u00f3 la cabeza hacia un lado. Sus dedos soltaron los m\u00edos, y su esp\u00edritu se desliz\u00f3 hacia el \u00e9ter.\n\nMe qued\u00e9 un momento all\u00ed sentada, mirando el cad\u00e1ver. Nick agach\u00f3 la cabeza y cubri\u00f3 la cara de Liss con una cortina. \u00abLiss se ha ido \u2014me obligu\u00e9 a pensar\u2014. Liss se ha ido, igual que Seb. No los has salvado. Se han ido.\u00bb\n\n\u2014Tienes que recitar el treno \u2014murmur\u00f3 Nick\u2014. Yo no s\u00e9 su nombre, _s\u00f6tnos_.\n\nTen\u00eda raz\u00f3n. A Liss no le gustar\u00eda quedarse all\u00ed, en su prisi\u00f3n.\n\n\u2014Liss Rymore \u2014pronunci\u00e9, y confi\u00e9 en que ese fuera su nombre completo\u2014, vete al \u00e9ter. Est\u00e1 todo arreglado. Todas las deudas est\u00e1n saldadas. Ya no tienes que morar entre los vivos.\n\nSu esp\u00edritu desapareci\u00f3.\n\nNo soportaba ver el cad\u00e1ver. Ya no era Liss, sino un cuerpo, una c\u00e1scara, la sombra que ella hab\u00eda dejado en el mundo.\n\nLa pistola de bengalas estaba debajo de su fr\u00eda mano. Ella era la encargada de dispararla. La cog\u00ed con cuidado.\n\n\u2014A esa chica no le habr\u00eda gustado que abandonaras. \u2014Nick me vio comprobar cu\u00e1ntas bengalas ten\u00eda la pistola\u2014. No le habr\u00eda gustado que murieras por ella.\n\n\u2014Yo creo que s\u00ed.\n\nConoc\u00eda esa voz. No pod\u00eda ver a Gomeisa Sargas, pero su voz reson\u00f3 por toda la sala.\n\n\u2014\u00bfLa has matado t\u00fa, Gomeisa? \u2014Me levant\u00e9\u2014. \u00bfEs lo bastante buena para ti ahora que est\u00e1 muerta?\n\nUn silencio elocuente.\n\nO\u00ed una voz d\u00e9bil detr\u00e1s de m\u00ed.\n\n\u2014No deber\u00edas ocultarte entre las sombras, Gomeisa.\n\nEl Custodio hab\u00eda entrado en el sal\u00f3n y miraba hacia la galer\u00eda.\n\n\u2014A menos que temas a Paige \u2014continu\u00f3\u2014. La ciudad est\u00e1 ardiendo. Vuestra apariencia de poder ya se ha disuelto.\n\nRisas. Me puse en tensi\u00f3n.\n\n\u2014No le temo a Scion. Nos ofrecieron su mundo en bandeja de plata, Arcturus. Ahora vamos a cenar.\n\n\u2014Vete al infierno \u2014dije.\n\n\u2014A ti tampoco te temo, 40. \u00bfQu\u00e9 podemos temer de la muerte, si somos muerte? Adem\u00e1s, ser desterrados de este mundo en descomposici\u00f3n, vuestro peque\u00f1o mundo de flores y carne, ser\u00eda casi... una bendici\u00f3n. Si no quedara tanto trabajo por hacer todav\u00eda. \u2014Se oyeron pasos\u2014. No puedes matar a la muerte. \u00bfQu\u00e9 fuego puede quemar el sol? \u00bfQui\u00e9n puede ahogar el oc\u00e9ano?\n\n\u2014Estoy segura de que se nos ocurrir\u00e1 algo.\n\nLo dije con firmeza pese a que estaba temblando; ya no sab\u00eda si de rabia o de miedo. Detr\u00e1s del Custodio hab\u00eda aparecido otro refa, y a su lado estaba Terebell.\n\n\u2014Me gustar\u00eda que los dos os imaginarais una cosa. Sobre todo t\u00fa, Arcturus. Dado lo que t\u00fa tienes que perder.\n\nEl Custodio no dijo nada. Intent\u00e9 averiguar de d\u00f3nde proven\u00eda la voz. De alg\u00fan sitio por encima de m\u00ed. De la galer\u00eda.\n\n\u2014Me gustar\u00eda que os imaginarais una mariposa. Imagin\u00e1osla: sus alas de colores, iridiscentes. Es hermosa. Muy querida. Y entonces mirad la polilla. Tiene la misma forma, pero \u00a1observad las diferencias! La polilla es p\u00e1lida, d\u00e9bil y fea. Un bicho lastimoso y autodestructivo. No puede dominarse, pues cuando ve un fuego, desea su calor. Y cuando encuentra la llama, se quema. \u2014Su voz resonaba por la estancia, en mis o\u00eddos, en mi cabeza\u2014. As\u00ed es como nosotros vemos vuestro mundo, Paige Mahoney. Una caja llena de polillas que esperan a quemarse.\n\nSu onirosaje estaba muy cerca. Prepar\u00e9 mi esp\u00edritu. No me importaba el da\u00f1o que pudiera causar. \u00c9l hab\u00eda matado a Liss, y yo iba a matarlo a \u00e9l. El Custodio me agarr\u00f3 por la mu\u00f1eca.\n\n\u2014No lo hagas \u2014dijo\u2014. Nosotros nos encargaremos.\n\n\u2014Quiero encargarme yo.\n\n\u2014No puedes vengar a Liss, onir\u00e1mbula. \u2014Pleione no apartaba la vista de su enemigo\u2014. Ve a la pradera. No queda mucho tiempo.\n\n\u2014S\u00ed, ve a la pradera, 40. Toma nuestro tren y ve a nuestra ciudadela. \u2014Gomeisa sali\u00f3 de detr\u00e1s de las columnas. Ten\u00eda los ojos radiantes de aura: era lo \u00faltimo que le quitar\u00eda a Liss Rymore\u2014. \u00bfTan mal lo pasabas aqu\u00ed, 40? Te ofrecimos un refugio, nuestra sabidur\u00eda, un nuevo hogar. Aqu\u00ed no eras antinatural; eras inferior, s\u00ed, pero ten\u00edas tu sitio. Para Scion solo eres un s\u00edntoma de la plaga. Un sarpullido en su piel. \u2014Me tendi\u00f3 una mano enguantada\u2014. All\u00ed ya no tienes un hogar, onir\u00e1mbula. Qu\u00e9date con nosotros. Descubre lo que hay debajo.\n\nYo ten\u00eda los m\u00fasculos tensados al m\u00e1ximo. Gomeisa me miraba fijamente; me miraba a los ojos, miraba en mi onirosaje, en mis partes m\u00e1s oscuras. Sab\u00eda que sus palabras ten\u00edan sentido; conoc\u00eda bien su retorcida l\u00f3gica, pues llevaba dos siglos recurriendo a ella, utiliz\u00e1ndola para tentar a los d\u00e9biles. Antes de que pudiera contestarle, el Custodio me agarr\u00f3 por el brazo y me levant\u00f3 del suelo. Una hoja curvada pas\u00f3 roz\u00e1ndole un hombro, por encima de mi cabeza. Yo no la hab\u00eda visto acercarse. Ca\u00ed al suelo, y el Custodio se abalanz\u00f3 sobre Gomeisa. Terebell y el otro refa salieron tras \u00e9l, ambos reuniendo bandadas y produciendo unos sonidos aterradores. Nick me ayud\u00f3 a levantarme, pero yo no notaba sus manos. Lo \u00fanico que notaba era el \u00e9ter, por donde danzaban los refa\u00edtas.\n\nEl aire que me rodeaba fue volvi\u00e9ndose m\u00e1s ligero, como una gasa plateada. No ve\u00eda a los cuatro refas, pero notaba sus movimientos. Cada flexi\u00f3n de un musculo, cada giro y cada paso lanzaban una onda expansiva por el \u00e9ter. Danzaban al borde de la vida. Un baile de gigantes, una danza macabra.\n\nLos esp\u00edritus de la Era de Huesos XVIII segu\u00edan en el sal\u00f3n. La bandada de Terebell sali\u00f3 volando entre las columnas: treinta esp\u00edritus, serpenteando y elev\u00e1ndose juntos, convergieron en el onirosaje de Gomeisa. Ning\u00fan vidente habr\u00eda sobrevivido a un ataque conjunto de tantos esp\u00edritus. Esper\u00e9 a que se produjera la colisi\u00f3n.\n\nLa risa de Gomeisa se elev\u00f3 hacia el techo. Con una sacudida de la mano deshizo la bandada. Los esp\u00edritus estallaron por todo el sal\u00f3n, como fragmentos de cristal de un espejo roto. El cuerpo inerte de Terebell sali\u00f3 despedido y choc\u00f3 contra una columna. El ruido de huesos contra el m\u00e1rmol invadi\u00f3 la g\u00e9lida atm\u00f3sfera. El otro refa carg\u00f3 contra Gomeisa, que se limit\u00f3 a levantar una mano. El atacante salt\u00f3 por los aires y fue a caer en el escenario. Las tablas del suelo se astillaron bajo su peso, y el refa cay\u00f3 al foso.\n\nRetroced\u00ed; mis botas resbalaban con la sangre del suelo. \u00bfQu\u00e9 era Gomeisa, una especie de duende? Ten\u00eda aporte: pod\u00eda mover cosas sin tocarlas. Al darme cuenta, el coraz\u00f3n empez\u00f3 a latirme con fuerza en el pecho. Gomeisa pod\u00eda estrellarme contra el techo si quer\u00eda.\n\nSolo quedaba el Custodio. Se volvi\u00f3 hacia su enemigo, aterrador en la penumbra.\n\n\u2014Ven, Arcturus \u2014dijo Gomeisa abriendo los brazos\u2014. Paga por tu esplendidez.\n\nEntonces fue cuando explot\u00f3 el escenario.\n\nLa r\u00e1faga de calor me lanz\u00f3 hacia atr\u00e1s y me ensordeci\u00f3. Ca\u00ed sobre el costado derecho y me lastim\u00e9 la cadera. Nick me agarr\u00f3 por la mu\u00f1eca, me levant\u00f3 y me llev\u00f3 hacia el vest\u00edbulo. Nada m\u00e1s llegar junto a la puerta, se encendieron las llamas. Me tir\u00e9 al suelo y me cubr\u00ed la cabeza con los brazos. El fuego hizo estallar los cristales de las ventanas del Consistorio. Me arrastr\u00e9 tan deprisa como pude por el suelo. Todav\u00eda ten\u00eda la pistola de bengalas en la mano.\n\nNing\u00fan buf\u00f3n ten\u00eda la clase de pertrechos necesarios para provocar semejante explosi\u00f3n. Deb\u00eda de haber algo que Julian no me hab\u00eda contado. \u00bfD\u00f3nde habr\u00eda encontrado una mina, o tiempo para plantarla? \u00bfLa habr\u00eda cogido de la Tierra de Nadie? Y \u00bfqu\u00e9 clase de mina era aquella capaz de provocar un incendio que atravesaba un edificio?\n\nEn medio de la nube de humo, Nick me cogi\u00f3 por el codo y me puso de pie. Me cayeron trocitos de cristal del pelo. Me puse a toser. Me escoc\u00edan los ojos.\n\n\u2014Espera. \u2014Me solt\u00e9 de la mano de Nick\u2014. El Custodio...\n\nNo pod\u00eda estar muerto. Nick me estaba gritando algo, pero lo o\u00eda muy lejos. Intent\u00e9 usar el cord\u00f3n \u00e1ureo. Ver, sentir, o\u00edr. Nada.\n\nFuera aullaban las sirenas y un incendio ard\u00eda en la calle de al lado. La Sala arrojaba llamas y nubes negras. Vi arder dos residencias. Una era Balliol, el \u00fanico edificio con electricidad. Los emisarios iban a tener problemas para avisar a la ciudadela. \u00abGracias, Julian \u2014pens\u00e9\u2014. Est\u00e9s donde est\u00e9s, gracias.\u00bb\n\nNick me cogi\u00f3 en brazos.\n\n\u2014Tenemos que irnos \u2014dijo con voz ronca. Mir\u00f3 alrededor con gesto angustiado\u2014. Paige, no conozco esta ciudad. \u00bfD\u00f3nde est\u00e1 el tren?\n\n\u2014Sigue hacia el norte. \u2014Intent\u00e9 bajar al suelo, pero Nick me sujetaba con fuerza\u2014. \u00a1Puedo correr, maldita sea!\n\n\u2014\u00a1Acabas de sobrevivir a una explosi\u00f3n y a un duende! \u2014me grit\u00f3. Estaba colorado de rabia\u2014. No he venido hasta aqu\u00ed para ver c\u00f3mo te matas, Paige. Por una vez en la vida, hazle caso a alguien.\n\nSheol I se hallaba en plena guerra. Los rebeldes hab\u00edan salido del Consorcio en llamas y hab\u00edan invadido las calles, donde luchaban con \u00edmpetu contra los refa\u00edtas. Los emisarios de Scion hu\u00edan en todas direcciones detr\u00e1s de sus guardaespaldas, que disparaban contra los videntes. Los miembros del grupo de Julian, que eran los encargados de provocar el incendio, se hab\u00edan entregado a su misi\u00f3n con entusiasmo, y ya hab\u00edan quemado la mayor parte del Poblado. Quer\u00eda quedarme all\u00ed y pelear, pero ten\u00eda que lanzar la bengala; de ese modo salvar\u00eda m\u00e1s vidas.\n\nNick tom\u00f3 la ruta m\u00e1s segura, manteni\u00e9ndose apartado de los combates. Se meti\u00f3 por una callejuela. Vi otra escaramuza. Los bufones peleaban junto a amaur\u00f3ticos y casacas, asoci\u00e1ndose para vencer a los refas. Hasta Cyril se les hab\u00eda unido.\n\nLleg\u00f3 a mis o\u00eddos un grito desgarrador. Mir\u00e9 por encima del hombro de Nick. Era Nell. Dos refas le sujetaban las manos.\n\n\u2014No ir\u00e1s a ninguna parte, 9. Necesitamos comer.\n\nUno de ellos la agarr\u00f3 por el pelo y le gir\u00f3 la cabeza.\n\n\u2014\u00a1No! \u00a1Qu\u00edtame las manos de encima! \u00a1No volver\u00e1s a alimentarte de m\u00ed, par\u00e1sito!\n\nDej\u00f3 de gritar cuando su guardi\u00e1n le tap\u00f3 la boca con una manaza.\n\n\u2014\u00a1Nick! \u2014grit\u00e9.\n\nNick detect\u00f3 la desesperaci\u00f3n de mi voz. Afloj\u00f3 los brazos; nada m\u00e1s tocar el suelo ech\u00e9 a correr hacia Nell. No iba armada, pero ten\u00eda mi don. Mi don, que ya no era una maldici\u00f3n. Esa noche no iba a usarlo para quitar una vida, sino para salvarla.\n\nLe lanc\u00e9 mi esp\u00edritu al m\u00e1s corpulento de los refas. Empuj\u00e9 contra su onirosaje, me met\u00ed en su zona hadal y volv\u00ed r\u00e1pidamente a mi cuerpo. Llegu\u00e9 justo a tiempo para extender las manos y no parar el golpe con la barbilla. Nell, que no ten\u00eda ni idea de qu\u00e9 hab\u00eda pasado, se solt\u00f3 de los refas y le clav\u00f3 el pu\u00f1al en el costado al que ten\u00eda a la derecha. Al mismo tiempo agarr\u00f3 un esp\u00edritu al vuelo y se lo ech\u00f3 en la cara. El refa dio un gru\u00f1ido espeluznante. Su compinche todav\u00eda se tambaleaba a consecuencia de mi agresi\u00f3n. Nell recogi\u00f3 las cosas que se le hab\u00edan ca\u00eddo y sali\u00f3 corriendo.\n\nLos dos refas estaban heridos, pero segu\u00edan representando una amenaza. El que yo hab\u00eda atacado levant\u00f3 la cabeza, y sus ojos, de color naranja, consiguieron enfocar. Sac\u00f3 un pu\u00f1al de una funda que llevaba atada al brazo.\n\n\u2014Vuelve al \u00e9ter, onir\u00e1mbula.\n\nLanz\u00f3 el pu\u00f1al contra mi cara. No me agach\u00e9 lo bastante deprisa, y me dio en un brazo. Nick dispar\u00f3 varias veces. Una bala le dio al refa en el pecho, pero no sirvi\u00f3 de nada. Lanc\u00e9 mi esp\u00edritu contra su onirosaje. El segundo ataque lo debilit\u00f3. Cog\u00ed el pu\u00f1al que me hab\u00eda lanzado y se lo clav\u00e9 en el cuello.\n\nComet\u00ed el error de olvidarme de su acompa\u00f1ante. El segundo refa se abalanz\u00f3 sobre m\u00ed y me inmoviliz\u00f3 en el suelo; se me cort\u00f3 la respiraci\u00f3n. Con su pu\u00f1o gigantesco me asest\u00f3 un pu\u00f1etazo que esquiv\u00e9 por los pelos.\n\nNick tir\u00f3 su pistola. Cuando el refa levant\u00f3 el pu\u00f1o para intentarlo de nuevo, Nick agarr\u00f3 tres esp\u00edritus y se los lanz\u00f3 en r\u00e1pida sucesi\u00f3n. Not\u00e9 la sacudida del \u00e9ter cuando le envi\u00f3 una v\u00edvida instant\u00e1nea al onirosaje del refa, ceg\u00e1ndolo. El refa me solt\u00f3 solo un segundo para combatir los esp\u00edritus y la visi\u00f3n, y lo aprovech\u00e9 para levantarme y correr hasta donde estaba Nick.\n\nNo hab\u00edamos ido muy lejos cuando not\u00e9 un tir\u00f3n de mi sexto sentido. Gir\u00e9 bruscamente la cabeza, dispuesta a enfrentarme a la amenaza.\n\n\u2014\u00a1Nick!\n\n\u00c9l ya se hab\u00eda dado cuenta. Con un \u00e1gil movimiento, dej\u00f3 la mochila y agarr\u00f3 otra bandada. El objetivo era conocido: Aludra Chertan.\n\n\u2014So\u00f1adora. \u2014Ni siquiera mir\u00f3 a Nick\u2014. Creo que todav\u00eda te debo una por tu peque\u00f1a exhibici\u00f3n en la capilla.\n\n\u2014No te acerques \u2014le previno Nick.\n\n\u2014Es que tienes un aspecto tan reparador... \u2014dijo mientras sus ojos cambiaban de color.\n\nEl rostro de Nick se crisp\u00f3. La sangre se acumul\u00f3 en sus conductos lacrimales, y se le abultaron las venas del cuello.\n\n\u2014Casi tan reparador como la andarina \u2014continu\u00f3 Aludra, avanzando hacia nosotros\u2014. Quiz\u00e1 te salve, or\u00e1culo.\n\nNick hizo un esfuerzo para mantenerse erguido.\n\n\u2014He matado a vuestro heredero \u2014dije\u2014. Estoy dispuesta a hacerte lo mismo a ti, no lo dudes. Vu\u00e9lvete al infierno podrido del que saliste.\n\n\u2014Kraz era un arrogante. Yo no lo soy. S\u00e9 distinguir qu\u00e9 enemigos merecen que les dedique mis valiosos minutos.\n\n\u2014Pues yo soy uno de ellos.\n\n\u2014S\u00ed, ya lo creo.\n\nMe qued\u00e9 quieta. Hab\u00eda algo detr\u00e1s de Aludra: una sombra enorme y torpe. Aludra estaba demasiado excitada para reparar en ella. El gigante podrido. Reconoc\u00ed aquel manch\u00f3n en el \u00e9ter.\n\n\u2014\u00bfCu\u00e1ntos minutos?\n\n\u2014Solo uno. \u2014Levant\u00f3 una mano\u2014. Pero un minuto es tiempo de sobra para morir.\n\nDe pronto la sorpresa se dibuj\u00f3 en su rostro. Hab\u00eda sentido aquella presencia, pero no se volvi\u00f3 con suficiente rapidez. La cosa la hab\u00eda agarrado antes de que ella pudiera moverse. Vi unos ojos blancos, muertos; solo alcanc\u00e9 a ver algunas partes (las l\u00e1mparas de gas se hab\u00edan apagado cuando hab\u00eda aparecido), pero era m\u00e1s que suficiente para que se grabara en mi memoria, para que quedara marcada en el tejido y dejara una cicatriz en la delicada tela de mi onirosaje. Aludra lo ten\u00eda muy dif\u00edcil. Su grito se apag\u00f3 antes incluso de haberse o\u00eddo.\n\n\u2014S\u00ed \u2014dije\u2014. Tiempo de sobra.\n\nNick estaba paralizado. Ten\u00eda los ojos muy abiertos y la boca fuertemente cerrada. Lo agarr\u00e9 por el brazo y echamos a correr.\n\nCorrimos tan aprisa como pudimos. Los emim estaban en la ciudad. Igual que en la Era de Huesos XVIII.\n\n\u2014\u00bfCu\u00e1nto falta? \u2014le grit\u00e9 a Nick.\n\n\u2014No mucho. \u2014Me tom\u00f3 la mano y tir\u00f3 de m\u00ed\u2014. \u00bfQu\u00e9 era ese monstruo? \u00bfQu\u00e9 ha hecho Scion en este sitio?\n\n\u2014Muchas cosas.\n\nTomamos una callejuela, una de tantas que se adentraban en la ciudad fantasma. Vimos que se acercaba alguien desde el extremo opuesto, corriendo y jadeando. Nick y yo reaccionamos al mismo tiempo: Nick le puso la zancadilla al chico, que cay\u00f3 de bruces al suelo, e inmediatamente yo lo agarr\u00e9 por el cuello.\n\n\u2014\u00bfAd\u00f3nde vas, Carl?\n\n\u2014\u00a1Su\u00e9ltame! \u2014Carl estaba empapado de sudor\u2014. \u00a1Ya vienen! \u00a1Los han dejado entrar en la ciudad!\n\n\u2014\u00bfQui\u00e9nes?\n\n\u2014Los zumbadores. \u00a1Los zumbadores! \u2014Me dio un empuj\u00f3n en el pecho; estaba al borde de las l\u00e1grimas\u2014. \u00bfPor qu\u00e9 has tenido que estropearlo? \u00bfPor qu\u00e9 has tenido que intentar cambiarlo todo? Este sitio es lo \u00fanico que tengo, y no voy a dejar que t\u00fa...\n\n\u2014Tienes todo un mundo. \u00bfNo te acuerdas?\n\n\u2014\u00bfTodo un mundo? \u00a1Soy un monstruo! \u00a1Somos todos monstruos, 40! Monstruos que hablan con los muertos. Por eso los necesitamos \u2014dijo apuntando con un dedo hacia el centro de la ciudad\u2014. \u00bfNo lo ves? Este es el \u00fanico sitio seguro para nosotros. Pronto empezar\u00e1n a matarnos...\n\n\u2014\u00bfQui\u00e9nes?\n\n\u2014Los amaur\u00f3ticos. Cuando se den cuenta. Cuando comprendan qu\u00e9 quieren los refa\u00edtas. No pienso volver jam\u00e1s. Puedes quedarte tu precioso mundo. \u00a1Te lo regalo!\n\nLe solt\u00e9 el cuello. Carl se levant\u00f3 y ech\u00f3 a correr. Nick no intent\u00f3 retenerlo.\n\n\u2014Cuando lleguemos a casa tendr\u00e1s que contarme muchas cosas \u2014dijo.\n\nVi a Carl doblar una esquina y desaparecer.\n\nEst\u00e1bamos a solo un kil\u00f3metro de la pradera, pero supon\u00eda que tendr\u00edamos que pelear para llegar. Nashira rondaba por all\u00ed, y cab\u00eda la posibilidad de que no todos los arrancahuesos se hubieran tomado el mejunje de Duckett. Avanzamos por la ciudad fantasma apart\u00e1ndonos del centro de la calle.\n\nO\u00edmos una explosi\u00f3n a lo lejos. Nick no se detuvo. Las ventanas de los edificios vibraron. Yo no pod\u00eda pensar con claridad. \u00bfQu\u00e9 estaba pasando? \u00bfHabr\u00eda intentado alguien atravesar el campo de minas? Deb\u00edan de haberse dejado llevar por el p\u00e1nico; se preguntar\u00edan d\u00f3nde estaba la bengala y tratar\u00edan de encontrar la salida entre los \u00e1rboles. Ten\u00eda que protegerlos. Corrimos hasta llegar al final de la calle y torcimos por el camino que llevaba a Puerto Pradera. Vislumbr\u00e9 las vallas y el letrero. Un grupo de videntes y amaur\u00f3ticos se hab\u00eda congregado fuera. Deb\u00edan de creer que por all\u00ed podr\u00edan salir de la ciudad.\n\nEl Custodio tambi\u00e9n estaba all\u00ed. Estaba sucio, cubierto de tizne, pero vivo. Me abraz\u00f3.\n\n\u2014\u00bfD\u00f3nde demonios estabas? \u2014dije entrecortadamente.\n\n\u2014Perd\u00f3name. Me han entretenido. \u2014Mir\u00f3 hacia la ciudad\u2014. No fuiste t\u00fa la que coloc\u00f3 esa bomba incendiaria debajo del escenario, \u00bfverdad?\n\n\u2014No. \u2014Puse las manos sobre las rodillas e intent\u00e9 recobrar el aliento\u2014. A menos que...\n\n\u2014A menos que \u00bfqu\u00e9?\n\n\u201412. El or\u00e1culo, el casaca roja. Me coment\u00f3 algo de un plan alternativo.\n\n\u2014Tenemos que centrarnos en c\u00f3mo salir de aqu\u00ed. \u2014Nick mir\u00f3 al Custodio, y luego de nuevo a m\u00ed\u2014. \u00bfD\u00f3nde est\u00e1 la entrada del t\u00fanel? Cuando hemos llegado era de d\u00eda.\n\nLa pradera ya estaba completamente oscura, y era muy dif\u00edcil orientarse.\n\n\u2014No est\u00e1 lejos \u2014dijo el Custodio.\n\n\u2014De acuerdo. \u2014Nick mir\u00f3 su viejo reloj Nixie y se enjug\u00f3 el sudor de la cara con una mano temblorosa\u2014. \u00bfY el Vinculador? \u00bfHa conseguido llegar?\n\n\u2014Puedes llamarlo por su nombre, Nick. \u2014Notaba el sudor resbal\u00e1ndome por la nuca\u2014. Lo sabe.\n\n\u2014El se\u00f1or Hall y otros tres de sus acompa\u00f1antes est\u00e1n en la pradera, esper\u00e1ndote \u2014dijo el Custodio sin desviar la mirada de la ciudad\u2014. Paige, te aconsejo que dispares una de esas bengalas. Todav\u00eda tienes tiempo.\n\nNick fue a la verja, donde Jaxon examinaba la valla et\u00e9rea. Me puse al lado del Custodio.\n\n\u2014Siento mucho lo de Liss \u2014me dijo.\n\n\u2014Yo tambi\u00e9n.\n\n\u2014Me asegurar\u00e9 de que Gomeisa no olvide su muerte.\n\n\u2014\u00bfNo lo has matado?\n\n\u2014Nos ha interrumpido la explosi\u00f3n. Gomeisa estaba mucho m\u00e1s fuerte que nosotros, porque hab\u00eda comido, pero hemos podido debilitarlo. Tal vez el incendio del Consistorio haya hecho el resto.\n\nTodav\u00eda llevaba los guantes puestos. Sent\u00ed una punzada, de pena quiz\u00e1. \u00bfC\u00f3mo pod\u00eda haber cre\u00eddo que el Custodio cambiar\u00eda tan f\u00e1cilmente?\n\n\u00c9l no dejaba de mirarme. El cord\u00f3n \u00e1ureo se estremeci\u00f3 d\u00e9bilmente. Yo no sab\u00eda qu\u00e9 intentaba transmitirme el Custodio, pero de pronto me sent\u00ed m\u00e1s centrada, m\u00e1s decidida. As\u00ed el mango de la pistola de bengalas. El Custodio dio un paso atr\u00e1s. Apunt\u00e9 por encima de la pradera, amartill\u00e9 la pistola y gir\u00e9 la cabeza.\n\nLa bengala qued\u00f3 suspendida en el aire, lanzando una se\u00f1al tras otra. La vi arder y echar humo; la luz roja parpadeaba y se reflejaba en los ojos del Custodio.\n\nMir\u00e9 m\u00e1s all\u00e1 de la luz de la bengala, hacia las estrellas. Tal vez fuera la \u00faltima vez que ve\u00eda las estrellas as\u00ed, en una ciudad sin luces y sin contaminaci\u00f3n. O quiz\u00e1 alg\u00fan d\u00eda el mundo entero fuera as\u00ed. El mundo gobernado por Nashira. Una colonia penitenciaria enorme y oscura.\n\nEl Custodio me puso una mano en la espalda.\n\n\u2014Tenemos que irnos.\n\nFui con \u00e9l hasta la verja. Cuando la abri\u00f3, los videntes y los amaur\u00f3ticos (ocho en total) entraron en la pradera. Una vez todos dentro, abri\u00f3 del todo la verja y sac\u00f3 otro vial. Ten\u00eda m\u00e1s viales que un narco.\n\nEl contenido era una sustancia cristalina. Sal. Traz\u00f3 con ella una delgada l\u00ednea en la entrada. Iba a preguntar por los emim cuando Jax me agarr\u00f3 por los brazos y me estamp\u00f3 contra un poste. Not\u00e9 la energ\u00eda de la valla, tan cerca que me chisporrote\u00f3 el pelo.\n\n\u2014Idiota. \u2014Jax me agarr\u00f3 por el vestido\u2014. Acabas de ense\u00f1arles d\u00f3nde estamos, maldita sea.\n\n\u2014Les estoy ense\u00f1ando a todos d\u00f3nde estamos. No voy a dejar morir aqu\u00ed a toda esta gente, Jaxon. Son videntes.\n\nLe temblaban los m\u00fasculos de la cara. Ten\u00eda el rostro transido de ira. Aquel era el Jaxon que yo tem\u00eda, el due\u00f1o de mi vida.\n\n\u2014Acced\u00ed a venir aqu\u00ed a salvar a mi onir\u00e1mbula \u2014dijo en voz baja\u2014. No a salvar a una pandilla de adivinos y de augures.\n\n\u2014Eso no es problema m\u00edo.\n\n\u2014Ya lo creo. Si haces algo m\u00e1s que ponga en peligro este intento (el intento de rescatarte, podr\u00eda a\u00f1adir, golfa desagradecida), me asegurar\u00e9 de que pasas el resto de tus d\u00edas deambulando por las alcantarillas. Te enviar\u00e9 a Jacob's Island, y all\u00ed podr\u00e1s limosnear con los ar\u00faspices y los antropom\u00e1nticos y toda la escoria que va a parar all\u00ed. Ya veremos qu\u00e9 hacen contigo. \u2014Me agarr\u00f3 el cuello con una mano fr\u00eda\u2014. Toda esta gente es prescindible. Nosotros no. Quiz\u00e1 hayas disfrutado de cierta independencia, querida m\u00eda, pero vas a obedecerme. Y todo volver\u00e1 a ser como antes.\n\nEsas palabras retiraron varias capas de mi onirosaje. Volv\u00eda a tener diecis\u00e9is a\u00f1os; volv\u00eda a tenerle miedo al mundo y a todo lo que hab\u00eda dentro de m\u00ed. Entonces levant\u00e9 de nuevo una coraza y me convert\u00ed en otra.\n\n\u2014No \u2014dije\u2014. Lo dejo.\n\nJaxon mud\u00f3 la expresi\u00f3n.\n\n\u2014Los Siete Sellos no \u00abse dejan\u00bb, Paige.\n\n\u2014Yo acabo de hacerlo.\n\n\u2014Tu vida me pertenece. Hicimos un trato. Firmaste un contrato conmigo.\n\n\u2014Me importa un cuerno lo que digan otros capos. Si soy propiedad tuya, Jaxon, no soy otra cosa que una esclava. \u2014Lo empuj\u00e9, tratando de separarlo de m\u00ed\u2014. Ya estoy harta de esa clase de vida.\n\nPronunci\u00e9 esas palabras, pero no parec\u00edan haber salido de mi cabeza. Me estaba quedando atontada.\n\n\u2014Si yo no puedo tenerte, no te tendr\u00e1 nadie \u2014dijo, y cerr\u00f3 los dedos\u2014. No renunciar\u00e9 a una onir\u00e1mbula.\n\nLo dec\u00eda en serio. Despu\u00e9s de lo que hab\u00eda pasado en Trafalgar Square, entend\u00eda su sed de sangre. Su aura lo delataba. Si dejaba de trabajar para \u00e9l, me matar\u00eda.\n\nNick nos hab\u00eda visto.\n\n\u2014\u00bfQu\u00e9 haces, Jaxon?\n\n\u2014Lo dejo \u2014dije. Y lo repet\u00ed\u2014: Lo dejo. \u2014Necesitaba o\u00edrmelo decir\u2014. Cuando volvamos a Londres, no ir\u00e9 al I-4.\n\nNick mir\u00f3 a Jaxon.\n\n\u2014Ya hablaremos \u2014dijo\u2014. Ahora no hay tiempo. Faltan quince minutos.\n\nEse recordatorio me produjo un escalofr\u00edo.\n\n\u2014Hay que subir a todos al tren inmediatamente.\n\nNadine hab\u00eda vuelto.\n\n\u2014\u00bfD\u00f3nde est\u00e1 la entrada? \u2014pregunt\u00f3, sudorosa\u2014. Hemos salido a esta pradera por un pasadizo. \u00bfD\u00f3nde est\u00e1?\n\n\u2014Lo encontraremos. \u2014Mir\u00e9 m\u00e1s all\u00e1 de ella y solo vi a Zeke\u2014. \u00bfD\u00f3nde est\u00e1 Dani?\n\n\u2014No contesta por el transmisor. Podr\u00eda estar en cualquier sitio.\n\n\u2014Ella trabaja para Scion \u2014dijo Nick\u2014. Podr\u00eda librarse diciendo que es una emisaria, pero no es lo ideal.\n\n\u2014\u00bfHa venido Eliza?\n\n\u2014No, la hemos dejado en Dials. Necesit\u00e1bamos que hubiera un Sello en la ciudadela.\n\nJaxon se levant\u00f3 y se sacudi\u00f3 la ropa.\n\n\u2014Bueno, de momento vamos a llevarnos todos bien. Ya discutiremos nuestras diferencias cuando hayamos regresado. \u2014Hizo se\u00f1as con una mano y a\u00f1adi\u00f3\u2014: Diamante, Campana: cubridnos, por favor. Tenemos que tomar un tren.\n\n\u2014\u00bfY Dani? \u2014Zeke parec\u00eda nervioso.\n\n\u2014Tranquilo, lo conseguir\u00e1. Esa chica ser\u00eda capaz de atravesar un campo de minas.\n\nJaxon pas\u00f3 a mi lado y encendi\u00f3 otro puro. \u00bfC\u00f3mo pod\u00eda fumar en un momento as\u00ed? Estaba segura de que solo fing\u00eda despreocupaci\u00f3n. No quer\u00eda perderme. Y yo tampoco estaba segura de querer perderlo a \u00e9l. \u00bfPor qu\u00e9 hab\u00eda dicho todo aquello? Jaxon no era un or\u00e1culo ni un adivino, pero sus palabras hab\u00edan sonado prof\u00e9ticas. Yo no pod\u00eda acabar limosneando (o peor a\u00fan, prostituy\u00e9ndome) en una barriada de videntes como Jacob's Island. Hab\u00eda cosas y lugares mucho peores que el empleo que me ofrec\u00eda Jaxon en la zona segura del I-4.\n\nQuer\u00eda disculparme. Ten\u00eda que disculparme. Era una dama; \u00e9l era mi capo. Pero el orgullo me lo imped\u00eda.\n\nLanc\u00e9 otra bengala. La \u00faltima. Una \u00faltima oportunidad para los \u00faltimos supervivientes. Entonces ech\u00e9 a correr detr\u00e1s de Jaxon. El Custodio me sigui\u00f3.\n\nLa bengala iluminaba el sendero. Unos cuantos humanos m\u00e1s llegaron a la verja y entraron con nosotros en la pradera; algunos iban por parejas y otros, solos. La mayor\u00eda eran videntes. Cuando lleg\u00f3 Michael, me agarr\u00f3 por el brazo. Ten\u00eda un tajo en la cara, desde una ceja hasta la barbilla, pero al menos pod\u00eda andar. Me puso mi mochila en los brazos.\n\n\u2014Gracias, Michael. No hac\u00eda falta que... \u2014Sacudi\u00f3 la cabeza; respiraba con dificultad. Me colgu\u00e9 una correa del hombro\u2014. \u00bfViene alguien m\u00e1s?\n\nMichael hizo tres signos.\n\n\u2014Los emisarios \u2014tradujo el Custodio\u2014. Vienen con sus guardaespaldas. \u00bfCu\u00e1nto falta? \u2014Michael levant\u00f3 dos dedos\u2014. Dos minutos. Tenemos que alejarnos de ellos cuanto podamos.\n\nEra una pesadilla. Gir\u00e9 la cabeza y dije:\n\n\u2014\u00bfPor qu\u00e9 no van a dejarnos marchar?\n\n\u2014Les habr\u00e1n ordenado retener a todos los testigos de este incidente. Quiz\u00e1 tengamos que pelear.\n\n\u2014Pelearemos.\n\nNotaba una punzada en el costado. En el camino encontramos a un herido tirado en la hierba. Respiraba con dificultad. Solo ten\u00eda medio minuto para levantarlo del suelo; si no, tendr\u00edamos que dejarlo all\u00ed.\n\n\u2014Sigue t\u00fa \u2014le dije al Custodio\u2014. Diles que llegar\u00e9 enseguida. \u00bfPuedes abrir el t\u00fanel?\n\n\u2014Sin ti, no. \u2014Mir\u00f3 al herido; yo no sab\u00eda qu\u00e9 estaba pensando\u2014. Date prisa, Paige.\n\nSigui\u00f3 adelante con Michael. Me arrodill\u00e9 junto al herido. Estaba tumbado boca arriba, con los ojos cerrados y las manos juntas sobre el pecho; habr\u00eda parecido una efigie de no ser por el uniforme de Scion: traje negro y corbata roja, manchados de sangre. Le tom\u00e9 el pulso, y \u00e9l abri\u00f3 un ojo. Con repentino apremio, su mano cargada de anillos asi\u00f3 la m\u00eda.\n\n\u2014Eres la chica.\n\nMe qued\u00e9 quieta.\n\n\u2014\u00bfQui\u00e9n eres?\n\n\u2014Mi cartera. Mira ah\u00ed.\n\nLe saqu\u00e9 una cartera de piel del bolsillo interior de la chaqueta. Dentro hab\u00eda un documento de identidad. Era del Starch.\n\n\u2014Trabajas para Weaver \u2014dije\u2014. Cerdo. Eres el responsable de todo esto. \u00bfTe ha enviado \u00e9l a verme morir? \u00bfA vigilar c\u00f3mo nos va en el infierno al que nos ha enviado?\n\nNo era una persona conocida, y no reconoc\u00ed su nombre.\n\n\u2014Lo destruir\u00e1n... todo. \u2014Ten\u00eda sangre en los labios.\n\n\u2014\u00bfQui\u00e9nes?\n\n\u2014Esos... seres. \u2014Inspir\u00f3 trabajosa y ruidosamente\u2014. Busca... a Rackham. B\u00fascalo.\n\nFueron sus \u00faltimas palabras. Me qued\u00e9 con su cartera en las manos, temblando. De pronto hac\u00eda mucho fr\u00edo.\n\n\u2014\u00a1Paige!\n\nNick hab\u00eda venido a buscarme.\n\n\u2014Era de Scion. \u2014Sacud\u00ed la cabeza, agotada\u2014. Ya no entiendo nada.\n\n\u2014Yo tampoco. Est\u00e1n jugando con nosotros, _s\u00f6tnos_. Lo que pasa es que todav\u00eda no sabemos a qu\u00e9 estamos jugando. \u2014Me apret\u00f3 la mano\u2014. Vamos.\n\nDej\u00e9 que me levantara. Nada m\u00e1s ponerme en pie, o\u00ed un disparo a lo lejos. Me puse en tensi\u00f3n. Los emisarios. Deb\u00edan de haber llegado a la verja. Al mismo tiempo el \u00e9ter lanz\u00f3 una se\u00f1al extra\u00f1a. Se nos acercaban cuatro figuras con ojos amarillos.\n\n\u2014Refas \u2014dije, y ech\u00e9 a correr\u2014. \u00a1Corre, Nick, corre!\n\nNick me hizo caso. Nuestras botas golpeaban con fuerza la tierra helada, pero los refas iban pis\u00e1ndonos los talones y eran m\u00e1s r\u00e1pidos que nosotros. Saqu\u00e9 un pu\u00f1al de mi mochila y me volv\u00ed, dispuesta a clav\u00e1rselo a alguno en un ojo, pero Terebell Sheratan me sujet\u00f3 la mano.\n\n\u2014Terebell \u2014dije, jadeando\u2014. \u00bfQu\u00e9 quieres?\n\nTerebell me mir\u00f3 a los ojos. Con ella estaban Pleione, Alsafi y otra refa m\u00e1s joven a la que no reconoc\u00ed. Y detr\u00e1s de ellos, ensangrentada y con la camisa rota, estaba Dani. Al verla sent\u00ed un alivio inmenso.\n\n\u2014Hemos tra\u00eddo a tu amiga \u2014dijo Terebell. En sus ojos hab\u00eda muy poca luz\u2014. Aqu\u00ed no durar\u00eda mucho.\n\nDani pas\u00f3 cojeando a mi lado, ignor\u00e1ndolos a todos, y fue hacia el grupo de rezagados. Estaba destrozada.\n\n\u2014\u00bfQu\u00e9 quer\u00e9is a cambio? \u2014pregunt\u00e9 con recelo\u2014. Supongo que no quer\u00e9is subir al tren.\n\n\u2014Si quisi\u00e9ramos, no podr\u00edas imped\u00edrnoslo. Todos hemos salvado vidas humanas. Te hemos tra\u00eddo a tu amiga, y hemos retrasado a la DVN. Est\u00e1s en deuda con nosotros. \u2014Alsafi me miraba fijamente\u2014. Por suerte para ti, onir\u00e1mbula, no tenemos intenci\u00f3n de ir a la ciudadela. Hemos venido a buscar a Arcturus.\n\n\u2014Vendr\u00e1 cuando pueda.\n\nYo todav\u00eda necesitaba al Custodio.\n\n\u2014Pues transm\u00edtele este mensaje. Tiene que encontrarse con nosotros en el claro en cuanto vosotros os hay\u00e1is marchado. Le estaremos esperando.\n\nSe fueron tan deprisa como hab\u00edan aparecido. Se perdieron en la oscuridad como polvo que se disuelve, huyendo de las inevitables represalias de los Sargas. Me volv\u00ed y fui hacia una plataforma de entrenamiento donde hab\u00eda dos faroles encendidos. Llegar hasta all\u00ed hab\u00eda sido f\u00e1cil. Ahora ten\u00eda que guiar a toda aquella gente por el t\u00fanel hasta el tren.\n\nLos rezagados se hab\u00edan congregado junto a una plataforma de hormig\u00f3n equivocada, pues era rectangular y no ovalada. Nick estaba examin\u00e1ndole la cara a Dani. Ten\u00eda un corte profundo en la ceja, pero ella no le daba importancia. Jaxon, un poco m\u00e1s all\u00e1, contemplaba la ciudad con gesto impasible. No hab\u00eda ni rastro de Julian. Se lo hab\u00eda tragado el fuego, igual que a Finn. Confi\u00e9 en que, al menos, hubieran tenido una muerte r\u00e1pida.\n\n\u2014Tenemos que irnos \u2014dije\u2014. No podemos esperar m\u00e1s.\n\n\u2014No tiene sentido \u2014dijo un chico amaur\u00f3tico agarr\u00e1ndose el pelo con ambas manos\u2014. La DVN viene hacia aqu\u00ed.\n\n\u2014Nosotros hemos llegado primero.\n\nUnas cuantas miradas se avivaron. Saqu\u00e9 una linterna de la mochila y la encend\u00ed.\n\n\u2014Seguidme \u2014dije\u2014. Id tan deprisa como pod\u00e1is. Los que pod\u00e1is, ayudad a los heridos. Tenemos que encontrar otra marca, un \u00f3valo. No nos queda mucho tiempo.\n\n\u2014T\u00fa est\u00e1s con los refas \u2014dijo alguien con rencor\u2014. Yo no voy a ninguna parte con una sanguijuela.\n\nMe volv\u00ed hacia el que acababa de hablar y apunt\u00e9 hacia la ciudad.\n\n\u2014\u00bfPrefieres volver all\u00ed?\n\nNo dijo nada. Pas\u00e9 a su lado ignorando el dolor del costado y ech\u00e9 a correr otra vez.\n\nUna vez que hubimos pasado el estanque, no me cost\u00f3 mucho recordar el lugar exacto. Vi al Custodio de pie en el sitio donde hab\u00edamos entrenado meses atr\u00e1s.\n\n\u2014La entrada est\u00e1 aqu\u00ed \u2014dijo cuando me acerqu\u00e9 a \u00e9l, se\u00f1alando el \u00f3valo de hormig\u00f3n\u2014. A Nashira le gust\u00f3 la idea de poner la estaci\u00f3n bajo el campo de entrenamiento.\n\n\u2014\u00bfCrees que ha muerto?\n\n\u2014Ojal\u00e1.\n\nApart\u00e9 esa idea de mi mente. Ahora no pod\u00eda pensar en Nashira.\n\n\u2014Te est\u00e1n esperando \u2014dije\u2014. En el claro.\n\n\u2014No pienso ir con ellos todav\u00eda.\n\nSent\u00ed alivio. Mir\u00e9 alrededor y dije:\n\n\u2014No hay vigilancia. \u00bfHan dejado la entrada abierta, sin m\u00e1s?\n\n\u2014No, no son tan necios. \u2014El Custodio levant\u00f3 una capa de musgo que revel\u00f3 un candado de plata. Sali\u00f3 un fino haz de luz blanca, como si dentro se hubiera encendido una bombilla\u2014. Este candado contiene una bater\u00eda et\u00e9rea. Dentro hay un duende. Pensaban enviar a un guardia refa\u00edta con los emisarios para abrirlo antes de que se restablezca la corriente; pero si puedes convencer al duende para que salga de ah\u00ed dentro, fallar\u00e1 la bater\u00eda y saltar\u00e1 el cierre.\n\nNot\u00e9 un escozor en la herida de la mano.\n\n\u2014En tu forma on\u00edrica no puede hacerte da\u00f1o, Paige. \u2014\u00c9l lo sab\u00eda\u2014. Eres la mejor preparada para enfrentarse a un quebrajador.\n\n\u2014Jaxon es vinculador.\n\n\u2014Eso no eliminar\u00eda el problema. Hay que convencer al duende, persuadirlo para que abandone el objeto; no se le puede obligar. Hasta que no lo hubieran liberado de sus restricciones f\u00edsicas, tu amigo no podr\u00eda vincularlo.\n\n\u2014\u00bfQu\u00e9 quieres que haga?\n\n\u2014T\u00fa puedes viajar por el \u00e9ter. Puedes comunicarte con el duende sin tocar el candado; nosotros, no.\n\n\u2014No digas \u00abnosotros\u00bb, refa. \u2014Era un augur un poco mayor que yo\u2014. Ap\u00e1rtate de ese candado.\n\nEl Custodio le dej\u00f3 hacer, pero sin dejar de observarlo. El chico iba armado con un trozo de tuber\u00eda.\n\n\u2014\u00bfQu\u00e9 haces? \u2014le pregunt\u00e9.\n\n\u2014Eso de la bater\u00eda et\u00e9rea es un cuento \u2014dijo apretando los dientes\u2014. De esto ya me encargo yo. Quiero largarme de aqu\u00ed.\n\nGolpe\u00f3 el candado con la tuber\u00eda.\n\nEl \u00e9ter se estremeci\u00f3. El augur salt\u00f3 por los aires, gritando, y fue a parar a m\u00e1s de seis metros.\n\n\u2014\u00a1No, por favor! \u00a1No quiero morir! \u00a1Por favor, no quiero ser un esclavo! \u00a1No!\n\nArque\u00f3 la espalda, se estremeci\u00f3 y se qued\u00f3 quieto.\n\nEsas palabras me trajeron recuerdos.\n\n\u2014He cambiado de idea \u2014dije. El Custodio me mir\u00f3\u2014. S\u00ed puedo ocuparme de ese duende.\n\nEl Custodio hizo un gesto afirmativo. Quiz\u00e1 lo hubiera entendido.\n\n\u2014\u00a1Ya vienen!\n\nGir\u00e9 la cabeza.\n\nLa DVN se acercaba por la pradera iluminada por la luna. Armados con escudos y bastones antidisturbios, escoltaban a un grupo de emisarios. Birgitta Tj\u00e4der se hallaba entre ellos, y tambi\u00e9n Cathal Bell. Tj\u00e4der fue la que nos vio primero, y grit\u00f3 de rabia. Nick levant\u00f3 la pistola y le apunt\u00f3 a la cabeza. Contra los amaur\u00f3ticos no pod\u00edamos usar bandadas. Me di la vuelta y mir\u00e9 a los prisioneros. Por primera vez desde mi llegada all\u00ed, necesitaban que los animaran. Necesitaban o\u00edr una voz que les dijera que pod\u00edan hacer lo que estaban haciendo. Que val\u00edan algo.\n\nEsa voz iba a ser la m\u00eda.\n\n\u2014\u00bfVeis a esos centinelas? \u2014dije subiendo la voz y se\u00f1al\u00e1ndolos\u2014. Esos centinelas van a intentar impedirnos salir de aqu\u00ed. Van a matarnos porque no quieren que vayamos a su capital. No quieren que compartamos lo que hemos visto. Quieren vernos morir, aqu\u00ed y ahora. \u2014Ten\u00eda la voz ronca, pero continu\u00e9. Ten\u00eda que aguantar\u2014. Voy a abrir esta trampilla, y saldremos a tiempo de esta ciudad. Os prometo que al amanecer estaremos en Londres. \u00a1Y no habr\u00e1 campanada diurna que nos obligue a volver a nuestras celdas! \u2014Hubo murmullos de aprobaci\u00f3n, de rabia. Michael aplaudi\u00f3\u2014. Pero necesito que defend\u00e1is esta pradera. Necesito que hag\u00e1is esto \u00faltimo para que podamos salir de aqu\u00ed para siempre. Dadme dos minutos, y yo os dar\u00e9 la libertad.\n\nGuardaron silencio. No hubo gritos de guerra, nada. Pero cogieron sus armas, convocaron a cuantos esp\u00edritus pudieron y salieron en tropel hacia la DVN. Nadine y Zeke fueron tras ellos, dispuestos a entrar en la refriega. Los esp\u00edritus de la pradera se unieron a su causa, y se lanzaron contra la DVN con m\u00e1s \u00edmpetu que las balas. Jaxon se qued\u00f3 quieto, evalu\u00e1ndome.\n\n\u2014Un discurso excelente \u2014dijo\u2014 para ser una aficionada.\n\nEra un cumplido. Un halago de un capo a su dama. Pero yo sab\u00eda que no era fruto de una admiraci\u00f3n sincera.\n\nTen\u00eda dos minutos. Esa era mi promesa.\n\n\u2014Dani \u2014dije\u2014, necesito la m\u00e1scara.\n\nMeti\u00f3 una mano en el bolsillo de su abrigo. Ten\u00eda sudor en la frente.\n\n\u2014Toma. \u2014Me la lanz\u00f3\u2014. Le queda poco ox\u00edgeno. Aprov\u00e9chalo bien.\n\nMe coloqu\u00e9 tan cerca como pude del candado y me tumb\u00e9 en la hierba. Nick mir\u00f3 al Custodio.\n\n\u2014No s\u00e9 qui\u00e9n eres, pero espero que sepas lo que haces. Paige no es ning\u00fan juguete.\n\n\u2014No puedo permitir que lleve a esta gente por la Tierra de Nadie. \u2014El Custodio dirigi\u00f3 la mirada hacia el bosque\u2014. A menos que se le ocurra una alternativa, doctor Nyg\u00e5rd, esta es la \u00fanica forma de salir de aqu\u00ed.\n\nMe puse el SVP2 en la cara, tap\u00e1ndome la boca y la nariz. La mascarilla se sell\u00f3 y se ilumin\u00f3, lo que indicaba un flujo constante de ox\u00edgeno.\n\n\u2014No tienes mucho tiempo \u2014me previno Dani\u2014. Te sacudir\u00e9 cuando tengas que volver.\n\nDije que s\u00ed con la cabeza.\n\n\u2014Custodio, \u00bfcu\u00e1l era el segundo nombre de Seb?\n\n\u2014Albert.\n\nCerr\u00e9 los ojos.\n\n\u2014Cronometrar\u00e9 dos minutos \u2014dijo Nick, y eso fue lo \u00faltimo que o\u00ed, al menos en el mundo de la carne.\n\nVislumbraba el diminuto recept\u00e1culo en el \u00e9ter. Me absorb\u00eda como habr\u00eda hecho un onirosaje, como una gotita absorber\u00eda otra. Entonces me di la vuelta y me encontr\u00e9 ante un ni\u00f1o perdido.\n\nNo fui hacia \u00e9l. Me qued\u00e9 quieta. Pero all\u00ed estaba: Sebastian Albert Pearce, el ni\u00f1o al que no hab\u00eda podido salvar. Golpeaba las paredes, sacud\u00eda los barrotes de hierro de la habitaci\u00f3n. Al otro lado de esos barrotes estaba la oscuridad infinita del \u00e9ter. Ten\u00eda la cara manchada de sangre, transida de ira, y el pelo sucio de cenizas.\n\nLa \u00faltima vez que me hab\u00eda tropezado con un duende hab\u00eda sido en mi forma f\u00edsica; aun as\u00ed, Seb pod\u00eda hacerle da\u00f1o a mi esp\u00edritu. Iba a tener que dominarlo.\n\n\u2014Seb \u2014dije con toda la suavidad que pude.\n\n\u00c9l no tard\u00f3 en percatarse de la invasi\u00f3n. Se volvi\u00f3 contra m\u00ed, corri\u00f3 hacia m\u00ed. Lo agarr\u00e9 por las mu\u00f1ecas.\n\n\u2014\u00a1Soy yo, Seb!\n\n\u2014No me salvaste \u2014gru\u00f1\u00f3, furioso\u2014. No me salvaste, y ahora estoy muerto. \u00a1Estoy muerto, Paige! \u00a1Y no puedo... \u2014golpe\u00f3 la pared\u2014... salir... \u2014otra vez\u2014... de esta habitaci\u00f3n!\n\nSu endeble figura temblaba en mis brazos. Se le marcaban los huesos, igual que antes. Me tragu\u00e9 el miedo y le cog\u00ed la sucia cara con las manos. Me estremec\u00ed al ver su cuello roto.\n\nTen\u00eda que hacerlo. Ten\u00eda que aplacar la ira del esp\u00edritu en que se hab\u00eda convertido porque, si no, Seb quedar\u00eda atrapado en ese estado eternamente. Aquel no era Seb. No eran el rencor, el dolor ni el odio de Seb.\n\n\u2014Esc\u00fachame, Seb. Lo siento much\u00edsimo. T\u00fa no te merec\u00edas esto. \u2014Ten\u00eda los ojos negros\u2014. Yo puedo ayudarte. \u00bfQuieres volver a ver a tu madre?\n\n\u2014Mi madre me odia.\n\n\u2014No. Escucha, Seb. No te liber\u00e9, y... lo siento. \u2014Estaba a punto de quebr\u00e1rseme la voz\u2014. Pero ahora podemos liberarnos el uno al otro. Si sales de esta habitaci\u00f3n, yo podr\u00e9 salir de la ciudad.\n\n\u2014Nadie se va de aqu\u00ed. Ella nos avis\u00f3: nadie se va. \u2014Me agarr\u00f3 un brazo, y la cabeza le tembl\u00f3 tan deprisa que vi un borr\u00f3n\u2014. Ni siquiera t\u00fa. Ni siquiera yo.\n\n\u2014Yo puedo hacer que te vayas.\n\n\u2014No quiero irme. \u00bfPara qu\u00e9 voy a irme? Ella me mat\u00f3. \u00a1Yo deber\u00eda haber vivido m\u00e1s!\n\n\u2014Tienes mucha raz\u00f3n. Deber\u00edas haber vivido m\u00e1s. Pero \u00bfseguro que quieres vivir en esta jaula el resto de la eternidad?\n\nSeb empez\u00f3 a temblar otra vez.\n\n\u2014\u00bfEternamente?\n\n\u2014S\u00ed, para siempre. Seguro que no quieres.\n\nPar\u00f3 de temblar.\n\n\u2014Paige \u2014dijo en voz baja\u2014, \u00bftengo que irme para siempre? \u00bfNo puedo volver?\n\nAhora era yo la que temblaba. \u00bfPor qu\u00e9 no lo hab\u00eda salvado? \u00bfPor qu\u00e9 no hab\u00eda parado a Nashira?\n\n\u2014De momento, no. \u2014Despacio, con cuidado, le puse las manos sobre los hombros\u2014. No puedo enviarte hasta la \u00faltima luz. Ya sabes, esa luz blanca que la gente dice que ve al final. No puedo enviarte all\u00ed. Pero puedo enviarte muy lejos, a la oscuridad exterior, para que nadie pueda volver a encerrarte. Y entonces, si de verdad quieres, podr\u00e1s volver.\n\n\u2014Si quiero.\n\n\u2014S\u00ed.\n\nNos quedamos un rato callados, Seb en mis brazos. No ten\u00eda pulso, pero yo sab\u00eda que deb\u00eda de estar asustado. Mi cord\u00f3n arg\u00e9nteo tembl\u00f3.\n\n\u2014No vayas tras ella \u2014dijo Seb aferr\u00e1ndose a mi forma on\u00edrica\u2014. De Nashira. Lo \u00fanico que quieren es sorbernos hasta dejarnos secos. Y hay un secreto.\n\n\u2014\u00bfQu\u00e9 secreto?\n\n\u2014No puedo revelarlo. Lo siento. \u2014Me tom\u00f3 las manos\u2014. Es demasiado tarde para m\u00ed, pero no para ti. T\u00fa todav\u00eda puedes parar esto. Nosotros te ayudaremos. Todos te ayudaremos.\n\nSeb me abraz\u00f3 el cuello. Percib\u00eda su presencia como algo tan real como el ni\u00f1o al que hab\u00eda conocido con vida. As\u00ed era como yo lo recordaba. Recit\u00e9 el treno en voz baja:\n\n\u2014Sebastian Albert Pearce, vete al \u00e9ter. Est\u00e1 todo arreglado. Todas las deudas est\u00e1n saldadas. Ya no tienes que morar entre los vivos. \u2014Cerr\u00e9 los ojos\u2014. Adi\u00f3s.\n\nSeb sonri\u00f3.\n\nY desapareci\u00f3.\n\nEl \u00e9ter contenido en el _numen_ empez\u00f3 a deshacerse. El cord\u00f3n arg\u00e9nteo dio una sacudida, esa vez con m\u00e1s apremio. Tom\u00e9 carrerilla y salt\u00e9, y mi onirosaje me recuper\u00f3.\n\n\u2014Paige. \u00a1Paige!\n\nUna luz me deslumbraba.\n\n\u2014Est\u00e1 bien \u2014dijo Nick\u2014. Nos vamos. Nadine, re\u00fanelos.\n\n\u2014Custodio \u2014murmur\u00e9.\n\nUna mano enguantada apret\u00f3 la m\u00eda, y supe que estaba all\u00ed. Abr\u00ed los ojos. O\u00eda disparos. Y los latidos del coraz\u00f3n del Custodio.\n\nEl Custodio levant\u00f3 la trampilla de acceso: una puerta maciza, cubierta de hormig\u00f3n, que ocultaba una escalera estrecha. El candado vac\u00edo cay\u00f3 repiqueteando. El Custodio me carg\u00f3 sobre el hombro, y lo abrac\u00e9 por el cuello. Los humanos bajaron por la escalera sin dejar de disparar contra la DVN. Tj\u00e4der se hizo con el arma de un centinela muerto; dispar\u00f3 a Cyril en el cuello y lo mat\u00f3. Vi la ciudad por \u00faltima vez (la luz en el cielo, la baliza en la oscuridad), antes de que el Custodio siguiera a los supervivientes. Su cuerpo, s\u00f3lido y c\u00e1lido, era lo \u00fanico en lo que pod\u00eda concentrarme. Iba recuperando la percepci\u00f3n a base de dolorosas sacudidas.\n\nEn el t\u00fanel hac\u00eda fr\u00edo. Lo ol\u00ed: el olor seco y rancio de una habitaci\u00f3n que se usa poco. Los gritos de arriba se confund\u00edan formando una cacofon\u00eda sin sentido, como ladridos de perros. Me agarr\u00e9 fuerte al hombro del Custodio. Necesitaba adrenalina, amaranto, algo.\n\nNo era un t\u00fanel grande, m\u00e1s o menos del tama\u00f1o de los t\u00faneles del metro de Londres, pero en el and\u00e9n, largo y ancho, cab\u00edan al menos cien personas. Al final hab\u00eda unas camillas, apiladas unas sobre otras. Ol\u00eda a desinfectante. Deb\u00edan de haberlas utilizado para trasladar a los videntes a los que hab\u00edan inyectado flux desde all\u00ed hasta la penitenciar\u00eda, o al menos hasta la calle. Pero estaba segura de poder o\u00edr algo en la oscuridad: un zumbido el\u00e9ctrico.\n\nEl Custodio alumbr\u00f3 con su linterna hacia el tren. Al cabo de un momento vi las luces. Entrecerr\u00e9 los ojos.\n\nElectricidad.\n\nEra un tren ligero, de los del metro; no estaba pensado para transportar a muchos pasajeros. En la parte trasera pod\u00eda leerse: SISTEMA DE TRANSPORTE AUTOMATIZADO DE SCION. Los vagones eran blancos, y llevaban la insignia de Scion en las puertas; al abrirse estas, se encendieron las luces del interior.\n\n\u00abBienvenidos a bordo \u2014dijo la voz de Scarlett Burnish\u2014. Este tren partir\u00e1 dentro de tres minutos con destino a la ciudadela Scion Londres.\u00bb\n\nEntre suspiros de alivio, los supervivientes fueron deshaci\u00e9ndose de sus armas rudimentarias y entrando en los vagones. El Custodio se qued\u00f3 quieto en el and\u00e9n.\n\n\u2014Se dar\u00e1n cuenta \u2014dije con voz cansada\u2014. Se dar\u00e1n cuenta de que en el tren no van los que deber\u00edan. Nos estar\u00e1n esperando.\n\n\u2014Y t\u00fa te enfrentar\u00e1s a ellos. Como te enfrentas siempre a todo.\n\nMe dej\u00f3 en el suelo, pero no me solt\u00f3. Me abraz\u00f3 por las caderas. Lo mir\u00e9 y dije:\n\n\u2014Gracias.\n\n\u2014No tienes que agradecerme tu libertad. Tienes derecho a ella.\n\n\u2014Y t\u00fa tambi\u00e9n.\n\n\u2014Ya me has dado la libertad, Paige. Me ha costado veinte a\u00f1os recuperar las fuerzas para reclamarla. Eso tengo que agradec\u00e9rtelo \u00fanicamente a ti.\n\nIba a contestar, pero se me hizo un nudo en la garganta. Unas cuantas personas m\u00e1s subieron al tren, entre ellas Nell y Charles.\n\n\u2014Tenemos que irnos \u2014dije.\n\nEl Custodio no me contest\u00f3. No estaba muy segura de qu\u00e9 era eso que hab\u00eda pasado a lo largo de los seis \u00faltimos meses (no sab\u00eda si era real o no), pero notaba el coraz\u00f3n henchido y la piel caliente, y no ten\u00eda miedo. Ya no. De \u00e9l, no.\n\nSe oy\u00f3 un ruido a lo lejos, una especie de trueno. Otra mina. Otra muerte in\u00fatil. Zeke, Nadine y Jax entraron tambale\u00e1ndose en el t\u00fanel; iban ayudando a Dani, que estaba seminconsciente.\n\n\u2014\u00bfVienes, Paige? \u2014dijo Zeke.\n\n\u2014Id tirando. Voy enseguida.\n\nEntraron en uno de los \u00faltimos vagones. Jaxon asom\u00f3 la cabeza por la puerta y me mir\u00f3.\n\n\u2014Ya hablaremos, So\u00f1adora \u2014dijo\u2014. Cuando volvamos, hablaremos.\n\nPuls\u00f3 el bot\u00f3n del interior del vag\u00f3n, y las puertas se cerraron. Un amaur\u00f3tico y un adivino entraron tambale\u00e1ndose en el siguiente vag\u00f3n; uno llevaba la camisa manchada de sangre.\n\n\u2014Un minuto para la salida. Por favor, p\u00f3nganse c\u00f3modos.\n\nEl Custodio me abraz\u00f3 con fuerza.\n\n\u2014Qu\u00e9 raro que esto sea tan dif\u00edcil \u2014dijo.\n\nEscudri\u00f1\u00e9 su cara. Ten\u00eda los ojos muy apagados.\n\n\u2014No vienes, \u00bfverdad? \u2014pregunt\u00e9.\n\n\u2014No.\n\nLo entend\u00ed poco a poco, con la lentitud con que el polvo se acumula en una estrella. En realidad yo nunca hab\u00eda tenido la certeza de que fuera a venir conmigo; eso solo hab\u00eda sido un deseo impreciso que hab\u00eda surgido en las \u00faltimas horas. Cuando ya era demasiado tarde. Y ahora estaba a punto de irse. O mejor dicho, de quedarse. A partir de ese momento, estar\u00eda sola. Y ser\u00eda libre en esa soledad.\n\nAcerc\u00f3 la nariz a la m\u00eda. Un dolor lento y dulce surgi\u00f3 dentro de m\u00ed, y no supe qu\u00e9 hacer. El Custodio no dejaba de mirarme, pero yo ten\u00eda la cabeza agachada. Miraba nuestras manos; las suyas, m\u00e1s grandes, sobre las m\u00edas, protegidas por los guantes que ocultaban la piel \u00e1spera; y las m\u00edas, p\u00e1lidas, con r\u00edos de venas azuladas. Mis u\u00f1as, todav\u00eda amoratadas.\n\n\u2014Ven con nosotros \u2014dije. Me dol\u00eda la garganta, y me ard\u00edan los labios\u2014. Ven... conmigo. A Londres.\n\nMe hab\u00eda besado. Me hab\u00eda deseado. Quiz\u00e1 todav\u00eda me deseara.\n\nPero no pod\u00eda haber nada entre nosotros. Era imposible. Y supe, por su mirada, que desearme no era suficiente.\n\n\u2014No puedo ir a la ciudadela. \u2014Me pas\u00f3 un pulgar por los labios\u2014. Pero t\u00fa s\u00ed. T\u00fa puedes recuperar tu vida, Paige. Es lo \u00fanico que quiero: que aproveches esa oportunidad.\n\n\u2014Pues yo quiero m\u00e1s cosas.\n\n\u2014\u00bfQu\u00e9 quieres?\n\n\u2014No lo s\u00e9. Quiero tenerte a mi lado.\n\nNunca hab\u00eda dicho esas palabras en voz alta. Ahora que pod\u00eda saborear mi libertad, quer\u00eda que \u00e9l la compartiera conmigo.\n\nPero \u00e9l no pod\u00eda cambiar su vida por m\u00ed. Y yo tampoco pod\u00eda sacrificar mi vida para estar con \u00e9l.\n\n\u2014Ahora debo perseguir a Nashira desde la clandestinidad. \u2014Apoy\u00f3 la frente en mi frente\u2014. Si logro echarla de aqu\u00ed, quiz\u00e1 los dem\u00e1s se marchen tambi\u00e9n. Quiz\u00e1 abandonen. \u2014Abri\u00f3 los ojos y me grab\u00f3 sus palabras a fuego en la mente\u2014. Si no vuelvo, si no vuelves a verme nunca, significar\u00e1 que todo ha salido bien. Que he acabado con ella. Pero si regreso, significar\u00e1 que he fracasado. Que todav\u00eda hay peligro. Y entonces te buscar\u00e9.\n\nLe sostuve la mirada. Recordar\u00eda su promesa.\n\n\u2014\u00bfConf\u00edas en m\u00ed ahora? \u2014me pregunt\u00f3.\n\n\u2014\u00bfDebo confiar?\n\n\u2014Eso no puedo dec\u00edrtelo yo. En eso consiste la confianza, Paige. En no saber si debes confiar en alguien o no.\n\n\u2014Entonces, conf\u00edo en ti.\n\nO\u00ed unos golpes a lo lejos. Pu\u00f1os contra metal, gritos amortiguados. Nick entr\u00f3 corriendo en el t\u00fanel, acompa\u00f1ado por el resto de los supervivientes, que me metieron en tropel en el tren justo antes de que se cerraran las puertas.\n\n\u2014\u00a1Sube, Paige! \u2014me grit\u00f3.\n\nHab\u00eda acabado la cuenta atr\u00e1s. Ya no hab\u00eda tiempo. El Custodio se separ\u00f3 de m\u00ed; el remordimiento ard\u00eda en sus ojos.\n\n\u2014Corre \u2014dijo\u2014. Corre, So\u00f1adora.\n\nEl tren se hab\u00eda puesto en marcha. Nick salt\u00f3 al tren agarr\u00e1ndose a la barandilla del final del \u00faltimo vag\u00f3n, y me tendi\u00f3 una mano.\n\n\u2014\u00a1PAIGE!\n\nVolv\u00ed en m\u00ed. Me dio un vuelco el coraz\u00f3n, y todos mis sentidos me golpearon como un muro de hierro. Me di la vuelta y corr\u00ed por el and\u00e9n. El tren aceler\u00f3; iba muy deprisa. As\u00ed la mano de Nick, salt\u00e9 por encima de la barandilla y me met\u00ed en el vag\u00f3n. Estaba a salvo. Vi volar chispas por la v\u00eda, y sent\u00ed temblar el suelo met\u00e1lico bajo mis pies.\n\nNo cerr\u00e9 los ojos. El Custodio hab\u00eda desaparecido en la oscuridad, como una vela apagada por el viento.\n\nNo volver\u00eda a verlo.\n\nPero mientras ve\u00eda correr las paredes del t\u00fanel, estaba segura de una cosa: confiaba en \u00e9l.\n\nYa solo faltaba que confiara en m\u00ed misma.\n\nEl argot empleado por los clarividentes en _La Era de Huesos_ est\u00e1 inspirado en el l\u00e9xico utilizado por el hampa londinense en el siglo XIX, con algunas modificaciones respecto al uso y el significado. Los t\u00e9rminos espec\u00edficos de la Familia (los humanos residentes en Sheol I) se indican con un asterisco.*\n\n****\n\n**Actor:** [sustantivo] Humano residente en Sheol que ha suspendido los ex\u00e1menes y est\u00e1 a las \u00f3rdenes del Capataz.\n\n**Andar\u00edn:** [sustantivo] Sin\u00f3nimo de \u00abonir\u00e1mbulo\u00bb.\n\n**Amaurosis:** [sustantivo] Carencia de clarividencia.\n\n**Amaur\u00f3tico:** [adjetivo o sustantivo] No clarividente.\n\n**Arrancahuesos:** ***** [sustantivo peyorativo] Casaca roja.\n\n**Bandada:** [sustantivo] Grupo de esp\u00edritus.\n\n**Buf\u00f3n:** ***** [sustantivo] Actor.\n\n**Cantor:** [sustantivo] Pol\u00edglota.\n\n**Carro\u00f1o:** [sustantivo] Amaur\u00f3tico.\n\n**Casaca amarilla:** ***** [sustantivo] El rango m\u00e1s bajo en Sheol I. Lo reciben los humanos que muestran miedo durante un examen. Puede utilizarse como sin\u00f3nimo de \u00abcobarde\u00bb.\n\n**Casaca blanca:** ***** [sustantivo] Primer rango que reciben todos los humanos en Sheol I. Los casacas blancas deben exhibir cierto grado de habilidad en sus respectivos tipos de clarividencia. Si aprueban ese examen, ascienden a casaca rosa; si lo suspenden, son enviados al Poblado.\n\n**Casaca roja:** ***** [sustantivo] El rango m\u00e1s elevado entre los humanos de Sheol I. Los casacas rojas son los encargados de proteger la ciudad de los emim. A cambio de sus servicios, reciben privilegios especiales. Tambi\u00e9n son llamados \u00abarrancahuesos\u00bb.\n\n**Casaca rosa:** ***** [sustantivo] Segunda fase de iniciaci\u00f3n en Sheol I. Los casacas rosa deben enfrentarse a los emim antes de pasar a ser casacas rojas. Si suspenden ese examen, los casacas rosa son degradados a casaca blanca.\n\nCenti **:** [sustantivo] Centinela.\n\n**Cord\u00f3n arg\u00e9nteo:** [sustantivo] Conexi\u00f3n permanente entre el cuerpo y el esp\u00edritu. Permite a una persona permanecer durante a\u00f1os en una misma forma f\u00edsica. Exclusivo de cada individuo. Especialmente importante para los onir\u00e1mbulos, quienes utilizan el cord\u00f3n para abandonar temporalmente su cuerpo. El cord\u00f3n arg\u00e9nteo se desgasta con los a\u00f1os, y una vez roto no puede repararse.\n\n**Cord\u00f3n \u00e1ureo:** [sustantivo] Conexi\u00f3n entre dos esp\u00edritus. Se sabe muy poco sobre \u00e9l.\n\n**Cortesano:** [sustantivo] Adicto al \u00e1ster morado. El nombre viene de St Anne's Court, el callej\u00f3n del Soho donde comenz\u00f3 el comercio de \u00e1ster morado a principios del siglo XXI.\n\n**Dama:** [sustantivo] Clarividente joven asociada con un mimetocapo. A menudo se da por hecho que es [a] la amante del capo y [b] la heredera de su sector.\n\n**Destronado:** [adjetivo] Completamente recuperado de los efectos del \u00e1ster morado.\n\n**Ecto:** [sustantivo] Ectoplasma, o sangre refa\u00edta. De color amarillo verdoso. Luminiscente y ligeramente gelatinoso. Puede emplearse para abrir puntos fr\u00edos.\n\n**Emim, los:** [sustantivo] [singular _emite_ ] Presuntos enemigos de los refa\u00edtas; \u00ablos temidos\u00bb. Nashira Sargas los define como seres carn\u00edvoros y brutales, con debilidad por la carne humana. Su existencia est\u00e1 envuelta en un halo de misterio.\n\n**\u00c9ter:** [sustantivo] El reino de los esp\u00edritus, al que pueden acceder los clarividentes. Tambi\u00e9n llamado el origen.\n\n**Falsante:** [sustantivo] Falsificador de documentos; los contratan los mimetocapos para proporcionar documentos de viaje falsos a sus empleados.\n\n**Familia, la:** ***** [sustantivo] Conjunto de humanos residentes en Sheol I, exceptuando a los arrancahuesos y a otros traidores.\n\n**Fantasma:** [sustantivo] Esp\u00edritu que ha escogido un sitio determinado donde residir, muy frecuentemente el lugar donde muri\u00f3. Sacar a un fantasma de su \u00ablugar predilecto\u00bb puede provocar su enfado.\n\n**Fasc\u00edculos de terror:** [sustantivo] Obras de ficci\u00f3n baratas e ilegales editadas en Grub Street, el centro del mundillo literario de los videntes. Componen series de relatos de terror. Distribuidos entre los clarividentes para compensar la falta de literatura fant\u00e1stica, de la que no se publican obras en Scion. Los fasc\u00edculos de terror abarcan gran variedad de temas sobrenaturales.\n\n**Floxy:** [sustantivo] Ox\u00edgeno aromatizado que se inhala mediante una c\u00e1nula. Es la alternativa de Scion al alcohol. Se sirve en la gran mayor\u00eda de los locales de entretenimiento, incluidos los bares de ox\u00edgeno.\n\n**Flux:** [sustantivo] Fluxion 14, psicof\u00e1rmaco que produce desorientaci\u00f3n y dolor a los clarividentes.\n\n**Lector:** [sustantivo] T\u00e9rmino anticuado, sin\u00f3nimo de \u00abcartom\u00e1ntico\u00bb. Su uso no es frecuente en la ciudadela.\n\n**Limosnear:** [verbo] Clarividencia remunerada. La mayor\u00eda de los limosneros leen el futuro a cambio de dinero. Est\u00e1 prohibido dentro del sindicato de clarividentes.\n\n**Luci\u00e9rnaga:** [sustantivo] Guardaespaldas callejero, contratado para proteger a los ciudadanos de los antinaturales por la noche. Se le identifica por una caracter\u00edstica luz verde.\n\nMecks **:** [sustantivo] Sustituto sin alcohol del vino. Tiene un sabor dulce y una consistencia espesa como el jarabe. Hay tres variedades: blanco, rosado y \u00absangre\u00bb, o tinto.\n\n**Mimetocapo:** [sustantivo] L\u00edder de una banda del sindicato de clarividentes; especialista en mimetodelincuencia. Generalmente dirige a un grupo reducido de entre cinco y diez seguidores, pero tiene el mando de todos los clarividentes de determinado sector de una cohorte. Miembro de la Asamblea Antinatural.\n\n**Mimetodelincuencia:** [sustantivo] Cualquier actividad que implique el uso de, o la comunicaci\u00f3n con, el mundo de los esp\u00edritus, sobre todo si es con fines de lucro. La ley de Scion lo considera alta traici\u00f3n.\n\n**Mundo de la carne:** [sustantivo] El mundo corp\u00f3reo; la Tierra.\n\n**Narco:** [sustantivo] Especialista en drogas et\u00e9reas y sus efectos en el onirosaje.\n\nNumen **:** [sustantivo] [plural _numa_ ] Objetos utilizados por los adivinos y los augures para conectar con el \u00e9ter; por ejemplo espejos, cartas, huesos.\n\n**Onirosaje:** [sustantivo] El interior de la mente, donde se almacenan los recuerdos. Est\u00e1 dividido en cinco \u00abanillos\u00bb de cordura: zona soleada, zona crepuscular, medianoche, baja medianoche y zona hadal. Los clarividentes pueden acceder conscientemente a su propio onirosaje, mientras que los amaur\u00f3ticos solo pueden entreverlo cuando duermen.\n\n**Oscuridad exterior:** [sustantivo] Zona del \u00e9ter muy remota, m\u00e1s all\u00e1 del alcance de los clarividentes.\n\n**Oxista:** [sustantivo] Camarero\/a de un bar de ox\u00edgeno.\n\n**Peste cerebral:** [sustantivo] T\u00e9rmino de argot, sin\u00f3nimo de fantasmagor\u00eda, una fiebre debilitante producida por el Fluxion 14.\n\n**Pirata:** [sustantivo] Taxi ilegal, sin licencia; son los que utilizan, por lo general, los clarividentes.\n\n**Poblado, el:** [sustantivo] Barriada. En Sheol I, barrio de chabolas donde se ven obligados a vivir los actores.\n\n**Pondo:** [sustantivo] Una libra; medida de peso. T\u00e9rmino empleado corrientemente en el \u00e1mbito de las drogas et\u00e9reas.\n\n**Punto fr\u00edo:** [sustantivo] Peque\u00f1a abertura entre el \u00e9ter y el mundo corp\u00f3reo. Se manifiesta como un charco de hielo permanente. Se puede utilizar, con ectoplasma, para abrir un conducto hasta el Inframundo. La materia corp\u00f3rea (por ejemplo, la sangre y la carne) no puede pasar por un punto fr\u00edo.\n\n**Quebrajador:** [sustantivo] Esp\u00edritu capaz de causar un impacto en el mundo corporal debido a su categor\u00eda o su edad. Incluye a los duendes y a los arc\u00e1ngeles.\n\n**Refa\u00edtas, los:** [sustantivo] [singular refa\u00edta] Habitantes humanoides, biol\u00f3gicamente inmortales, del Inframundo; se alimentan del aura de los humanos clarividentes. Su historia y sus or\u00edgenes son inciertos.\n\nRegal **:** [sustantivo] \u00c1ster morado.\n\n**Reinar:** [verbo] Estar bajo los efectos del \u00e1ster morado.\n\n**Sindicato:** [sustantivo] Organizaci\u00f3n criminal de clarividentes, con base en la Ciudadela Scion Londres. Activo desde principios de la d\u00e9cada de 1960. Gobernado por el Subse\u00f1or y la Asamblea Antinatural. Sus miembros se especializan en mimetodelincuencia con fines lucrativos.\n\n**Sindis:** [sustantivo] Miembros del sindicato de clarividentes. T\u00e9rmino empleado mayoritariamente por los centinelas.\n\nSkilly **:** ***** [sustantivo] Gachas ligeras, generalmente hechas con jugo de carne.\n\n**Solicitante:** [sustantivo] Cualquier persona que busca informaci\u00f3n en el \u00e9ter. Pueden formular preguntas u ofrecer una parte de s\u00ed mismos (sangre, palma de la mano) para que les hagan una predicci\u00f3n. Los adivinos y los augures pueden utilizar a un solicitante para concentrarse en determinadas zonas del \u00e9ter y hacer predicciones m\u00e1s f\u00e1cilmente.\n\n**So\u00f1ador:** [sustantivo] Sin\u00f3nimo de \u00abonir\u00e1mbulo\u00bb; lo usan frecuentemente los refa\u00edtas.\n\nSortes **:** [sustantivo] Una de las categor\u00edas de _numa_ empleados por los clarividentes. Incluye las agujas, los dados, las llaves, los huesos y las varillas.\n\n**Sublimaci\u00f3n:** [sustantivo] Proceso por el que un objeto corriente se convierte en un _numen_.\n\n**Subse\u00f1or:** [sustantivo] Jefe de la Asamblea Antinatural y capo supremo del sindicato de clarividentes. Tradicionalmente reside en _Devil's Acre_ (el Acre del Diablo), en el sector 1 de la cohorte I.\n\nSusu **:** [sustantivo] Sin\u00f3nimo de \u00absusurrante\u00bb o \u00abpol\u00edglota\u00bb.\n\n**Tase:** [sustantivo] Abreviatura de \u00abtase\u00f3grafo\u00bb.\n\n**Tincto:** [sustantivo] L\u00e1udano. Narc\u00f3tico ilegal. El nombre proviene de su nombre t\u00e9cnico, tintura de opio.\n\nToke **:** ***** [sustantivo] Pan rancio.\n\n**Treno:** [sustantivo] Serie de palabras utilizadas para desterrar a los esp\u00edritus a la oscuridad exterior.\n\n**\u00daltima luz:** [sustantivo] El centro del \u00e9ter, el lugar del que los esp\u00edritus ya no pueden regresar. Se rumorea que m\u00e1s all\u00e1 de la \u00faltima luz existe otra vida definitiva.\n\n**Vagabundos:** [sustantivo] Esp\u00edritus del \u00e9ter que no han sido desterrados a la oscuridad exterior o \u00faltima luz. Los clarividentes todav\u00eda pueden controlarlos.\n\n**Videncia:** [sustantivo] Clarividencia.\n\n**Vidente:** [sustantivo] Clarividente.\n\n**Whitewash:** [sustantivo] Amnesia duradera provocada por el consumo de \u00e1ster blanco. **Encalar** : [verbo] Emplear \u00e1ster blanco para borrar la memoria de alguien.\n\nZeitgeist **:** [sustantivo] T\u00e9rmino alem\u00e1n que significa \u00abesp\u00edritu del tiempo\u00bb. Los videntes lo utilizan metaf\u00f3ricamente, pero algunos adoran al _zeitgeist_ como a una deidad.\n\n**Zumbadores:** ***** [sustantivo] Emim.\n\nQuiero dar las gracias a la familia Godwin, y muy especialmente a David, por acogerme con tanto cari\u00f1o en el mundo editorial. Gracias tambi\u00e9n a Kirsty McLachlan, Caitlin Ingham y Anna Watkins por su trabajo con los derechos para el cine y para el extranjero. No podr\u00eda haber encontrado una agencia mejor que DGA.\n\nAl equipo de Bloomsbury: antes de conoceros no ten\u00eda ni idea de la pasi\u00f3n y el trabajo de equipo necesarios para crear un libro. Quiero expresar mi gratitud a la inimitable Alexandra Pringle, cuya pasi\u00f3n por esta novela ha sido la mejor inspiraci\u00f3n que pod\u00eda pedir; a Alexa von Hirschberg, mi estupenda editora, que ha ido mucho m\u00e1s all\u00e1 de sus obligaciones y siempre ha estado dispuesta a ayudarme; y a Rachel Mannheimer, Justine Taylor y Sarah Barlow, quienes me han ayudado a perfeccionar muchos aspectos de _La Era de Huesos_. Gracias de coraz\u00f3n a Katie Bond, Jude Drake, Amanda Shipp, Ianthe Cox-Willmott, Eleanor Weil y Oliver Holden-Rea del Reino Unido, y a George Gibson, Cristina Gilbert, Nancy Miller, Marie Coolman y Sara Mercurio de Estados Unidos. Sois todos geniales.\n\nAndy Serkis, Jonathan Cavendish, Chloe Sizer, Will Tennant y el resto del equipo de Imaginarium: es un privilegio trabajar con vosotros. Gracias por vuestra dedicaci\u00f3n a todos los aspectos de este libro, mucho m\u00e1s all\u00e1 de lo visual. En el plano art\u00edstico, muchas gracias a Andr\u00e1s Bereznay por dise\u00f1ar el mapa; a David Mann, por la cubierta; y a Leiana Leatutufu, por ser mi _automatiste_ personal.\n\nDecir que este libro ha sido mi vida estos dos \u00faltimos a\u00f1os es quedarse corto. Sois demasiados para nombraros a todos en tan poco espacio, pero gracias a todos los amigos que han estado a mi lado durante ese tiempo, y mucho antes. Gracias muy especiales a Neil Diamond y a Fran Tracey; a Emma Forward, mi gran profesora de literatura; y a Rian, Jesica y Richard por llevarme con ellos a Irlanda. De no ser por vosotros, no habr\u00eda conocido a Molly Malone.\n\nA mis traductores de todo el mundo, gracias por hacer que este libro pueda leerse en tantos idiomas en los que nunca podr\u00e9 escribir. Muchas gracias a Flo y a Alie por ayudarme con los nombres franceses y serbios, y a Devora de Agam Books por compartir conmigo sus conocimientos de hebreo.\n\nGracias a todas las personas que han seguido mi blog y mis comentarios en Twitter antes de la publicaci\u00f3n del libro, y sobre todo a Susan Hill: tu apoyo me ha dado mucha seguridad. Gracias tambi\u00e9n a los profesores y alumnos del St Anne's College, por ser tan comprensivos conmigo a lo largo de este a\u00f1o ca\u00f3tico.\n\nY, por supuesto, gracias a mi familia, y sobre todo a mam\u00e1, por darme fuerza y energ\u00eda constantemente, y a Mike, mi incre\u00edble padrastro y rey indiscutible de la taseograf\u00eda. Los dos me hab\u00e9is soportado en mis peores momentos, as\u00ed que, seg\u00fan Marilyn Monroe, merec\u00e9is que me acuerde de vosotros en los mejores.\n\nJD, gracias por ser mi musa. Eres mi poeta muerto n\u00famero uno. Y, por \u00faltimo, gracias a Ali Smith por animarme a lanzar _La Era de Huesos_ al mundo.\n\nGracias a todos por apostar por una so\u00f1adora.\n\n### _Samantha Shannon_\n\n# La era de huesos\n\nSamantha Shannon naci\u00f3 en Londres en 1991 y empez\u00f3 a escribir a los quince a\u00f1os. Entre 2010 y 2013 estudi\u00f3 Lengua y Literatura en la Universidad de Oxford. Con s\u00f3lo veintid\u00f3s a\u00f1os salt\u00f3 a la fama con la publicaci\u00f3n de _La era de huesos_ , su primera novela. Esta escal\u00f3 r\u00e1pidamente las listas de m\u00e1s vendidos, cosechando rese\u00f1as espectaculares a ambos lados del Atl\u00e1ntico.\n\n_La era de huesos_ da a conocer a una protagonista valiente y cautivadora, as\u00ed como a una joven autora de imaginaci\u00f3n, ambici\u00f3n y talento desbordantes. Si bien sus m\u00e9ritos le han valido comparaciones con las obras de fantas\u00eda m\u00e1s exitosas de la actualidad y cl\u00e1sicos del siglo XX, es un libro que brilla por su originalidad. En esta primera novela, Samantha Shannon ha creado un universo fascinante y totalmente \u00fanico que est\u00e1 conquistando los corazones y las mentes de lectores en todo el mundo.\nPRIMERA EDICI\u00d3N DIGITAL VINTAGE ESPA\u00d1OL, DICIEMBRE 2014\n\n_Copyright de la traducci\u00f3n \u00a9 2014 por Gemma Rovira Ortega_\n\nTodos los derechos reservados. Publicado en coedici\u00f3n con Penguin Random House Grupo Editorial, S. A., Barcelona, en los Estados Unidos de Am\u00e9rica por Vintage Espa\u00f1ol, una divisi\u00f3n de Random House LLC, Nueva York, y en Canad\u00e1 por Random House of Canada Limited, Toronto, compa\u00f1\u00edas Penguin Random House. Publicado seg\u00fan acuerdo con David Godwin Associates, Londres. Originalmente publicado en ingl\u00e9s en EE.UU. como _The Bone Season_ por Bloomsbury USA en 2013. Copyright \u00a9 2013 por Samantha Shannon-Jones. Copyright de la presente edici\u00f3n en castellano para todo el mundo \u00a9 2014 por Penguin Random House Grupo Editorial, S. A.\n\nVintage es una marca registrada y Vintage Espa\u00f1ol y su colof\u00f3n son marcas de Random House LLC.\n\nEste libro es una obra de ficci\u00f3n. Los nombres, personajes, lugares e incidentes o son producto de la imaginaci\u00f3n de la autora o se usan de forma ficticia. Cualquier parecido con personas, vivas o muertas, eventos o escenarios es puramente casual.\n\nInformaci\u00f3n de catalogaci\u00f3n de publicaciones disponible en la Biblioteca del Congreso de los Estados Unidos.\n\n**Vintage Espa\u00f1ol ISBN en tapa blanda: 978-1-101-87321-2**\n\n**Vintage Espa\u00f1ol eBook ISBN: 978-1-101-87322-9**\n\n_Adaptaci\u00f3n del dise\u00f1o original de David Mann: PRHGE_\n\n_Para venta exclusiva en EE.UU., Canad\u00e1, Puerto Rico y Filipinas._\n\nwww.vintageespanol.com\n\n#### \u00cdndice\n\nSobre _La Era de Huesos_\n\nLa Era de Huesos\n\nLos siete \u00f3rdenes de la clarividencia\n\nMapa\n\n1. La maldici\u00f3n\n\n2. Mentiras\n\n3. Recluida\n\n4. El serm\u00f3n\n\n5. Indiferencia\n\n6. Comunidad\n\n7. El se\u00f1uelo\n\n8. Mi nombre\n\n9. Variedad\n\n10. El mensaje\n\n11. Llanto\n\n12. Fiebre\n\n13. Una imagen extra\u00edda de mi memoria\n\n14. Amanecer\n\n15. La ca\u00edda de un muro\n\n16. La tarea\n\n17. Voluntad\n\n18. Despertar\n\n19. La flor\n\n20. Un mundo peque\u00f1o\n\n21. Una nave quemada\n\n22. Tres veces necia\n\n23. Anticuario\n\n24. El sue\u00f1o\n\n25. La disoluci\u00f3n\n\n26. Cambio\n\n27. El aniversario\n\n28. La prohibici\u00f3n\n\n29. La despedida\n\nGlosario\n\nAgradecimientos\n\nBiograf\u00eda\n\nCr\u00e9ditos\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nCopyright \u00a9 2012 by Anna Anthropy\n\nA Seven Stories Press First Edition\n\nAll rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, including mechanical, electronic, photocopying, recording, or otherwise, without the prior written permission of the publisher.\n\nSeven Stories Press\n\n140 Watts Street\n\nNew York, NY 10013\n\nwww.sevenstories.com\n\nCollege professors may order examination copies of Seven Stories Press titles for a free six-month trial period. To order, visit http:\/\/www.sevenstories.com\/textbook or send a fax on school letterhead to (212) 226-1411.\n\nBook design by Elizabeth DeLong\n\nLibrary of Congress Cataloging-in-Publication Data\n\nAnthropy, Anna.  \nRise of the videogame zinesters : how freaks, normals, amateurs, artists, dreamers, drop-outs, queers, housewives, and people like you are taking back an art form \/ Anna Anthropy.  \np. cm.  \nIncludes bibliographical references and index.  \nISBN 978-1-60980-372-8 (pbk.)  \nEbook ISBN: 978-1-60980-373-5  \n1. Video games. 2. Video games--Social aspects. I. Title.  \nGV1469.3.A53 2012  \n794.8--dc23  \n2011045409\n\nPrinted in the United States\n\n9 8 7 6 5 4 3 2 1\nFor the child I was,\n\nthe book no one could write for me.\nContents\n\nChapter One: The Problem with Videogames\n\nChapter Two: The History of Magic\n\nChapter Three: What Is It Good For?\n\nChapter Four: Changing the Game\n\nChapter Five: The New Videogame\n\nChapter Six: Making the Games\n\nChapter Seven: By Your Bootstraps\n\nChapter Eight: Growing Up\n\nAppendix A: What to Use\n\nAppendix B: Zinester Games\n\nAbout the Author\nChapter One\n\nThe Problem with Videogames\n\nI have a problem with videogames.1\n\nPlenty of people seem to have problems with videogames these days. Newscasters are fond of reporting that videogames are dangerous to children, either because they teach children how to steal cars and kill cops or because they actually connect children electronically to the game-playing predators who are waiting to snatch them away. Religious leaders have wasted no time condemning videogames as a trap for children's souls, and armchair psychologists accuse them of turning kids into antisocial hermits.\n\nPeople condemn videogames because videogames are pervasive in popular culture. They're on our computers and our cell phones, in our homes and purses and pockets. Even if you yourself don't play games, you have a hard time escaping their marketing. When the television isn't telling you to be afraid of videogames, it's telling you to buy them, and to drink World of Warcraft\u2013flavored Mountain Dew while you play.\n\nThese are some problems people have with videogames. What's my problem with videogames?\n\nAs a queer transgendered woman in 2012, in a culture pervaded by videogames\u2014a culture in which, typing on my computer, I am seconds away from a digital game, even if I have not taken the time to buy or install a single game on my computer\u2014I have to strain to find any game that's about a queer woman, to find any game that resembles my own experience.\n\nThis is in spite of the fact that videogames in America and elsewhere are an industry and an institution. I've already brought up World of Warcraft, a game about performing repetitive tasks until numbers increase. So, now that we're in the land of numbers, here are some numbers. The ESA\u2014that's the Entertainment Software Association, who spend half their time assuring the population that videogames aren't worth being mad at, and the other half pursuing litigation against anyone who distributes games that their shareholders have long since stopped distributing or profiting from\u2014claims that, as of 2009, 68 percent of American households play digital games.2 In 2008 alone, people bought 269,100,000 games (the ESA word is units.)3\n\nSo digital games, by the numbers, are here, and they take up a lot of space. And almost none of these games are about me, or anyone like me.\n\nWhat are videogames about?\n\nMostly, videogames are about men shooting men in the face. Sometimes they are about women shooting men in the face. Sometimes the men who are shot in the face are orcs, zombies, or monsters. Most of the other games the ESA is talking about when it mentions \"units\" are abstract games: the story of a blue square who waits for a player to place him in a line with two other blue squares, so he can disappear forever. The few commercial games that involve a woman protagonist in a role other than slaughterer put her in a role of servitude: waiting tables at a diner (or a dress shop, a pet shop, a wedding party). This is not to say that games about head shots are without value, but if one looked solely at videogames, one would think the whole of human experience is shooting men and taking their dinner orders. Surely an artistic form that has as much weight in popular culture as the videogame does now has more to offer than such a narrow view of what it is to be human.\n\nAnd yes, from here on out I'll be talking about videogames as an art form. What I mean by this is that games, digital and otherwise, transmit ideas and culture. This is something they share with poems, novels, music albums, films, sculptures, and paintings. A painting conveys what it's like to experience the subject as an image; a game conveys what it's like to experience the subject as a system of rules. If videogames are compared unfavorably to other art forms such as novels and songs and films\u2014and they are compared unfavorably with these forms, or else this paragraph defending videogames as art wouldn't be necessary\u2014it is likely a result of how limited a perspective videogames have offered up to this point. Imagine a world in which art forms are assigned value by the number of dykes that populate them. This is the world I inhabit; this is the value games have for me. And why not? The number of stories from marginalized cultures\u2014from people who are othered by the mainstream\u2014that a form contains tells us something about that form's maturity. If a form has attracted so many authors, so many voices, that several of them come from experiences outside the social norm and bring those experiences and voices to bear when working in that form, can't that form be said to have reached cultural maturity?\n\nIt should go without saying that novels and films have plenty of dykes in them, as do the media of writing and filmmaking. American comics have been around since 1896\u2014that's over one hundred years\u2014yet comics are still involved in a debate, as videogames are, about their cultural and artistic value. But I can think of many comics about queer women. More important, I can think of plenty of queer women who make comics: to name a few, Diane DiMassa, Alison Bechdel, Jennifer Camper, Kris Dresen, and Colleen Coover, in order of how disappointed I was when they came out in defense of the Michigan Womyn's Music Festival.4 And those are just print comics, in a world where the majority of comics are published on the Internet.\n\nIn Alison Bechdel's Dykes to Watch Out For, Mo (a dyke to watch out for) explains a metric she uses to decide whether she'll watch a movie. This criteria has become known as the Bechdel Test: the movie has to (1) contain at least two women who (2) talk to each other about (3) something other than a man. So why do videogames fail my variant of the Bechdel Test? Why are there no dykes in videogames?\n\nI know at least one of you has been itching, for several pages, to point out games like Fear Effect 2: Retro Helix and Mass Effect, both of which include scenes in which women smooch women, both on and off camera. In Fear Effect 2, women make out for the benefit of the male audience the game's creators expect to buy the game. (The first scene, in fact, is of the protagonist stripping as seen through a hidden camera, which tells us something about her relationship to the player.) And the lady-sex in Mass Effect is just one of many branches on a tree of awkward dialogue, offering nothing resembling the actual lust, desire, and flirtation that women feel for each other. But, aesthetic failures aside, the most important distinction here is that these are stories about queer women that are generally written by white, college-educated men. These are not cases of queer women presenting their own experiences.\n\nWhy are digital games so sparse in the dykes making art department? Why are the experiences that games present, the stories they tell, the voices in which they speak, so limited?\n\nThe limitations of games aren't just thematic. When I criticize games for being mostly about shooting people in the head, that's a design criticism as well. Most games are copies of existing successful games. They play like other games, resemble their contemporaries in shape and structure, have the same buttons that interact with the world in the same way (mouse to aim, left click to shoot), and have the same shortcomings. If there's a vast pool of experiences that contemporary videogames are failing to tap, then there's just as large a pool of aesthetic and design possibilities that are being ignored. I don't believe these are separate issues, either. To tell different stories, we need different ways of interacting with games. Why are games so similar in terms of both content and design?\n\nThe problem with videogames is that they're created by a small, insular group of people. Digital games largely come from within a single culture. When computers were first installed in college campuses and laboratories, only engineers had the access to the machines, the comparative leisure time, and the technical knowledge to teach those computers to play games. It is not surprising that the games they made looked like their own experiences: physics simulations, space adventures drawn from the science fiction they enjoyed, the Dungeons & Dragons tabletop role-playing games they played with their friends. As computers made their way out of labs and into homes, the games that programmers were hacking together became a salable product\u2014and salespeople showed up to profit off of them. And so as businessmen and marketers guided videogames into becoming a billion-dollar industry, publishers installed themselves as the gatekeepers of game creation.\n\nCommercial games have become expensive: according to a presentation at the High Performance Graphics 2009 conference, Gears of War 2\u2014an industry leader in the \"men shooting things\" genre\u2014had a \"development budget\" of 12 million dollars.5 (\"Development\" refers just to the cost of creating the game\u2014it doesn't include all the bucks that were spent marketing, manufacturing, and shipping the game.) If the game cost that much to produce, you can imagine what it would have to earn in sales in order to make any money. Hint: more than 12 million dollars. With that much money at stake, publishers and shareholders are not going to permit a game that is experimental either in terms of its content or in terms of its design. The publisher will do the minimum amount it can get away with in order to differentiate its game from all other games that follow its previously established model and that are being sold to its previously established audience.\n\nNow we have a dangerous cycle: publishers permit only games that follow a previously established model to be marketed to previously established audiences, and only to those audiences. The audiences in question are mostly young adults, and mostly male. And it's these dudes, already entrenched in the existing culture of games, who are eventually driven to enter the videogame industry and to take part in the creation of games. The population who creates games becomes more and more insular and homogeneous: it's the same small group of people who are creating the same games for themselves.\n\nVideogames as they're commonly conceived today both come from and contain exactly one perspective. It should be terrifying that an entire art form can be dominated by a single perspective, that a small and privileged group has a monopoly on the creation of art. Before the adoption of the printing press, the church was responsible for the creation of books, and the books that monks hand-lettered in Latin in monasteries were largely the Bible or books that agreed with the Bible. Not to knock the Bible, but that a single institution can hold power over what works are allowed to exist within any art form should demonstrate the power that institution has over that art form, and therefore over that culture. And so the printing press, which allowed people to print their own versions of the Bible in their own languages\u2014and eventually to print books that had nothing to do with the Bible\u2014had a role to play in the decentralization of religious authority in Europe.\n\nThe printing press is a piece of technology. If digital games, a form that is often (and not entirely correctly) described as being \"technology driven,\" can be compared to books, where then is the printing press for videogames?\n\nWhat Videogames Need\n\nThere's a videogame about a dyke who convinces her girlfriend to stop drinking. Mainstream gamer culture by and large does not know about this game. I know about this game because I made it.\n\nI created Calamity Annie in 2008. I made it by myself: I wrote the dialogue, composed the music, designed the rules, scripted the game, and drew all the characters. It was made in a couple of months. The development costs were the cost of the food that went into my belly. I made the game in a program called Game Maker, which, at the time, cost fifteen dollars.\n\nI am nowhere close to the only person who has used Game Maker, nowhere close to the only person who makes digital games outside of the games industry's publisher model. There are hundreds, if not thousands, of such creators. A few of them have achieved some mainstream recognition, like Jonathan Blow and Jason Rohrer, who was profiled in Esquire magazine. But these rich white dudes were professional programmers before they came to videogames, and so they don't represent the new dynamic that I'm excited about: hobbyists and non-programmers making their first games. There are lots of tools that allow people to make and distribute games without ever having written a line of code and without having to pass through publishers' gates. In years to come, there will be a lot more tools. I hope that there will also be a lot more people.\n\nI once heard the criticism that the phrase \"what videogames need\" can usually be more honestly rephrased as \"what I want from videogames.\" In that case, what I want from videogames is a plurality of voices. I want games to come from a wider set of experiences and present a wider range of perspectives. I can imagine\u2014you are invited to imagine with me\u2014a world in which digital games are not manufactured by publishers for the same small audience, but one in which games are authored by you and me for the benefit of our peers.\n\nThis is something the videogame industry, by its nature, cannot give us. I like to think about zines\u2014self-published, self-distributed magazines and books. Send me a dollar and a self-addressed envelope; I'll send you a stapled book of some stories from my life, or some pictures I took of out-of-the-way nooks of my city, or researched accounts of historical murders, or some jokes about sea life. (What does the merman's waiter bring? He brings the MERMANATEE.6) I like the idea of games as zines: as transmissions of ideas and culture from person to person, as personal artifacts instead of impersonal creations by teams of forty-five artists and fifteen programmers, in the case of Gears of War 2.\n\nThe Internet in particular has made self-publishing and distributing games both possible and easy. Authors are able not only to put their works online, but to find audiences for them. Publishers want to be gatekeepers to the creation of videogames, but the Internet has opened those gates.\n\nCurrently, the only real barrier to game creation is the technical ability to design and create games\u2014and that, too, is a problem that is in the process of being solved.\n\nDigital game creation was once limited to those who knew how to speak with computers: engineers and programmers, people who could code. In the games industry of today, coders are an inescapable fixture of the hierarchy of production, since games that we play with machines need creators capable of negotiating with machines. Game creation is daunting for someone who doesn't code professionally. But more and more game-making tools are being designed with people who aren't professional coders in mind. (I describe several of these tools, and what each is good for, in the appendix.) It's now possible for people with no programming experience\u2014hobbyists, independent game designers, zinesters\u2014to make their own games and to distribute them online.\n\nWhat I want from videogames is for creation to be open to everyone, not just to publishers and programmers. I want games to be personal and meaningful, not just pulp for an established audience. I want game creation to be decentralized. I want open access to the creative act for everyone. I want games as zines.\n\nIt's a tall order, maybe, but the ladder's being built as you read these words.\n\nIs What You Want Really What Games Need?\n\nWhy transform videogames, though? What do I get out of it? What, for that matter, do videogames get out of it?\n\nIn 2005, movie critic Roger Ebert infamously remarked that he does not think games can ever be considered as art. (By whom? By him, apparently.) He argues, mostly by assertion, that he doesn't feel game designers can exercise enough authorial control over the experience of a game. Ebert has gone on to make no attempt to justify or defend his remark or engage in any kind of debate, other than to allow, five years after the original remark, that he should have kept his opinion to himself.7\n\nAs I've made clear above, Ebert is wrong about videogames as a form. But frankly, I don't care whether Ebert is wrong or not. Achieving \"artistic legitimacy\" is not a good reason to transform videogames. Who legitimizes art? To cede the right to decide the value of games to an authority that has nothing to do with games\u2014or to concede the right to decide what is and is not art to any authority outside of the artist\u2014is a dangerous trap. Creation is art. It doesn't need validation beyond that.\n\nWhat it needs is to be free. That an art form exists should be justification enough for people to be able to contribute to it, to work in it. We finally have the means to allow more than just programmers and big game publishers to create games\u2014and the vast majority of people in the world aren't computer engineers, or designers employed by Epic Games. What do we gain from giving so many people the means to create games? We gain a lot more games that explore much wider ground, in terms of both design and subject matter. Many of these games will be mediocre, of course; the majority of work in any form is mediocre. But we'll see many more interesting ideas just by the sheer mathematical virtue of so many people producing so many games without the commercial obligations industry games are beholden to. Remember, I'm talking about hobbyists, people who make games in their spare time with the tools they have on hand. And even if a game isn't original, it's personal, in the way a game designed to appeal to target demographics can't be. And that's a cultural artifact our world is a little bit richer for having.\n\nTo visualize this new world of games, think about network television versus YouTube. The former spends a lot of money and time creating content designed to appeal to the lowest common denominator. Because network shows need to justify themselves monetarily\u2014they need to catch enough viewers to earn advertising dollars\u2014they can rarely afford to be brilliant, daring, or bizarre. (Sometimes a director has enough force of will, and fights the network hard enough, to create a show that is all of these things. But it's certainly not the norm.)\n\nYouTube: millions of videos from millions of authors. Most of them are mediocre: boring, familiar, or unwatchable. That's to be expected in an arena where everyone is allowed to contribute. But others are sublime, brilliant, valuable: Grishno's \"Transgender in New York\" videos,8 wendyvainity's surreal computer animations and music,9 or shaneduarte's Simpsons remixes.10 As long as there's some sort of infrastructure, valuable works\u2014those by both dabbling amateurs and dedicated artists\u2014can reach their audiences. YouTube has its own infrastructure of user ratings and featured videos, but people are just as likely to share the addresses of specific videos with the friends they think those videos will appeal to. And there's far more value in the collective content of YouTube\u2014even given that there are more piles of trash than treasure\u2014than in the collective content of a television network, simply as a function of the number of people contributing and the overwhelming volume of their contributions. YouTube's content is far more diverse, too, since involvement in the television industry isn't a requirement for entry. Network television shows are all made by professionals working in the field, a far smaller set of people than the set of people who own webcams. YouTube's content is made much more quickly and cheaply because it's not (usually) designed with a commercial agenda: videos can be recorded and broadcast, and their value assessed later.\n\nYouTube also gives people the means to make videos of themselves, their friends, their babies, and their puppies\u2014video snapshots\u2014not for the world at large, but for their social circles and themselves. YouTube is a means of transmitting a video directly from the author to an audience\u2014one that can be as small and specific as the author desires. Videos become more specialized, and hence more personalized. A medium that was formerly accessible only to those with money and training can now be used by anyone for personal ends. If Internet television is in the process of reinventing television, imagine how game design tools for nonprogrammers and the free distribution of games online might reinvent videogames.\n\nThe Culture of Alienation\n\nLimiting the creation of games to a small, exclusive group leads not only to creative stagnation, but also to the alienation of anyone outside that group. I've described the round-the-drain cycle the games industry is in: games are designed by a small, male-dominated culture and marketed to a small, male-dominated audience, which in turn produces the next small, male-dominated generation of game designers. It's a bubble, and it largely produces work that has no meaning to those outside that bubble, those not already entrenched in the culture of games.\n\nThere are mechanical consequences as well. Look at how game controllers have changed as their audiences changed. The home game machines of the 1970s and '80s, which marketed themselves to large, general family audiences, had the simplest control pads. The Atari Video Computer System (or the Atari 2600) is a simple joystick with a single button. And here is the design of the Nintendo Entertainment System (NES) controller, released in the United States in 1985:\n\nThe NES controller has a four-way compass rose and two prominent red buttons. (There are also two buttons in the center for secondary functions like pausing the game, but the design of the controller clearly communicates that they're peripheral.) You use the compass to navigate your character or cursor. You use the buttons to perform actions.\n\nAfter over thirty years of catering to an audience that is continuously playing and learning games\u2014an audience that hence requires more and more complicated games to interest it\u2014games and the controllers with which players interact with them have become more and more complex. This is not to say different: layers of complexity have simply been added to the same few models of games and the same few models of controllers. Here's the controller for the Xbox 360, released in the United States in 2005:\n\nThe Xbox 360 controller is the same model as the NES controller: held between two hands, with navigational functions assigned to the left hand and manipulation verbs to the right. But instead of a single navigational pad on the left, two verb buttons on the right, and two option buttons in the center, the Xbox pad has a navigational pad plus a stick on the left, four verb buttons plus another stick on the right, four \"shoulder\" buttons on the top of the controller (two to each side), plus three option buttons in the center. (Additionally, some games call for the player to \"click\" either of the sticks in like a button, adding two more verbs.)\n\nThe means players use to interact with games guides the design of those games. A game for the NES might have a button for jump and a button for shoot, and the compass rose directional pad for moving a character left and right. You can imagine the kinds of games that are designed for eight buttons and four sticks. Imagine introducing someone who had never seen a movie before to Matthew Barney's Cremaster films. The amount of both manual dexterity and game-playing experience required to operate a game designed for the Xbox 360 makes play inaccessible to those who aren't already grounded in the technique of playing games. And to attain that level of familiarity with games requires a huge and continuous investment of time (and money\u2014keeping up with new games costs bucks). This means that older people\u2014people with families and obligations, people trying to raise kids, or any people with a lack of free time to invest\u2014have a harder time gaining access to games. At the same time, as a side effect of this unnatural selection, commercial games become longer and longer, with game covers advertising dozens and occasionally hundreds of hours of gameplay. (Shin Megami Tensei: Persona 3, a PlayStation 2 game from 2008, advertises a \"70+ hour game\" on the back of its box.) Who has that much time to invest in playing a videogame? Answer: the target audience of most of the industry's games, a mostly young and mostly male audience that has few obligations and plenty of disposable income.\n\nThe culture that this audience creates and exists within is one of in-jokes and brand worship, rituals to establish whether the participants are in or out of the tribe. It's an exclusive culture, an alienating environment that speaks only to itself. Its interactions with the outside world are decidedly hostile.\n\nDestructoid, one of the most popular sources for videogame news on the Internet, employs a writer named Jim Sterling who once called my girl a \"feminazi slut\" on Twitter. This isn't some rogue nerd; this is a \"journalist\" whom Destructoid employs to write on such topics as whether the penis is more powerful than the vagina because it can rape,11 or on whether female Mortal Kombat characters have secret cocks.12 And lest you think that such a character couldn't possibly be taken seriously, hundreds of his readers responded publicly to my open letter to Destructoid complaining about Sterling's behavior in an attempt to bully and shame us.13 How is a woman, a trans person, or any rational individual expected to feel safe enough to participate in such a community?\n\nWhat I want from videogames is for videogames to speak to more than just the handful of people already engaged in producing and consuming them. To de-monopolize game creation is to de-monopolize access to games.\n\nBeyond Consumer\n\nIn an era when the Internet makes it easy to transmit and disseminate media, there's no reason for people to accept that their only contribution to the growth of an art form is as a consumer, supporting \"elite\" creators with money.\n\nI've wanted to make videogames since I played Fukio Mitsuji's NES game Bubble Bobble as a kid. I drew characters on construction paper, cut them out, and laid out obstacle courses for them to navigate\u2014Bubble Bobble stages on hardwood floors. But the technical leap to digitize my designs was beyond my reach. Programming was something mystical and arcane. I came into contact with code sometimes: the most basic BASIC examples. But something as simple as making a picture of a character move across a screen required a working knowledge of control loops, writing to video memory buffers, and advanced bit-shifting math\u2014all of which was so inaccessible to me as a kid that I sublimated my childhood desire to make games until well into my adulthood.\n\nIt's not like it was then. There's a commercial product in videogame stores right now\u2014Warioware: D.I.Y., from Nintendo\u2014that allows players to create their own small games.14 What Warioware: D.I.Y. does is to introduce its players to the concept of designing rules, of using art and sound to communicate the state of the game to the player, of scripting the events of a game and of working cleverly within limitations. For kids today, digital game creation doesn't have to be the mystical process it was when I was little.\n\nKids today also have tools like Stencyl,15 a free tool for making games and distributing them online. A website collects kits and resources contributed by the entire community, which are all made available to an individual creator for use in her game. The rules are put together in Scratch,16 a system designed by programmers at MIT for young children to use. It involves snapping simple instructions together like LEGOs.\n\nBut before things like Stencyl and Warioware existed, I made games and digital stories however I could: an old DOS shooting-game creation program that I can no longer remember the name of, the track editor in Nintendo's Excitebike, an editor for creating worlds made out of text called ZZT. People with something to say will always manage to find ways to say it, and there's a history of clever people using whatever means they can find to modify and subvert digital games and to create new ones\u2014to engage with games in a role beyond consumer. Today, this process is easier than ever.\n\nThe Big Crunch\n\nThis same false sense that the knowledge needed to create videogames is unattainable without special institutional training is the same carrot the Big Games Industry uses to entice wannabe game artists into taking jobs within their system\u2014and putting up with insane hours and ridiculous working conditions. There exists within the games industry a phenomenon called \"crunch mode\": working sixteen-hour days, staying at work until the game you're being paid to make is finished. This isn't something you're asked to do\u2014it's expected, a standard condition of the job. And it's likely the reason most people in the games industry, their physical and mental health fizzled, burn out and quit within a few years, forced to retrain and find a new career. According to the International Game Developers Association (IGDA), the closest thing the industry has to an advocacy group for employees, 34 percent of game developers expect to leave the industry within five years, and 51 percent\u2014half of them!\u2014expect not to last a decade.17 That's lunacy.\n\nThe industry gets away with this because it's convinced its employees that these jobs are the only gateway to videogame creation. \"We've graciously allowed you to fulfill your childhood dream of making games. We're even paying you for it! And what's more, we're the only way you'll ever be able to do that.\" Mike Capps, a former member of the board of directors at the IGDA and the president of Epic Games said that Epic expected employees to work more than sixty hours a week and in fact only hired people they expected to be willing to do so.18 The IGDA has no official stance on the hours of unpaid overtime the people it claims to represent are obliged to do by their employers.\n\nSince the industry sees itself as ubiquitous\u2014as the only possible means of creating games\u2014it feels no need to change itself for the benefit of either its employees or its art. Which is another reason why carving new paths to game creation and distribution is valuable. By undermining the industry's claim to being the only route to game creation\u2014especially to making a living from game creation\u2014we force the industry to reconsider its totalitarian attitude toward the people it employs. Publishers need creative people to make games for them. We have one foot in an era when creative people will no longer need publishers to distribute their games.\n\nCreating more and better games will also challenge the industry creatively. Spending millions of dollars to remake the same seventy-hour-long games for the same small audience is no longer feasible when so many people want different experiences out of games and have the means to find them elsewhere. Games from hobbyists have the potential to change the dominant format of the videogame: instead of seventy-hour multimillion dollar games that sell for sixty bucks apiece, digital games can be short and self-contained\u2014less than an hour, short enough to fit comfortably into an adult player's day. The focus of games could shift from features, the ways in which a game is differentiated from similar games\u2014thirty hours of play, twelve unique weapons, advanced four-dimensional graphics acceleration\u2014to ideas. Take Tarn Adams' WWI Medic19 for example: a game not about chain-gunning enemy soldiers but about trying to patch them up as the bullets cut them down. Saving even a single soul\u2014climbing out of the trench, grabbing a fallen body and lugging it back to safety under a senseless hail of bullets\u2014is incredibly difficult. The game takes minutes to play, and communicates an idea about war that may perhaps be more valuable than space marines frotteurizing each other with chainsaws.\n\nSmaller games with smaller budgets and smaller audiences have the luxury of being more experimental or bizarre or interesting than 12 million dollar games that need to play it as safely as possible to ensure a return on investment. Imagine what a videogames industry that wasn't fixated on hits\u2014that wasn't required to make hits\u2014would create.\n\nWhat Are Games Good For?\n\nBut even given all of this, why worry about the accessibility of digital game creation at all when other forms\u2014like the short story or novel\u2014are already established and available for non-professionals to work in?\n\nAnswer: because different forms are suited to different kinds of expression, and some are more effective at communicating in certain ways than others. Broadly, films and photographs are best suited for communicating action and physical detail. Novels are best suited for communicating internal monologue and ambiguity.\n\nWhat are games best suited for? Since games are composed of rules, they're uniquely suited to exploring systems and dynamics. Games are especially good at communicating relationships; digital games are most immediately about the direct relationship between the player's actions or choices and their consequences. Games are a kind of theater in which the audience is an actor and takes on a role\u2014and experiences the circumstances and consequences of that role. It's hard to imagine a more effective way to characterize someone than to allow a player to experience life as that person.\n\nTake, for example, a game called We the Giants.20 Most people who connect to this game's website in order to play it\u2014taking the role of a squat, block-like cyclops\u2014will be unable to reach the game's goal, a star high in the sky. Rather, most players are given the responsibility of voluntarily dying in a position that will allow future players to use their solidified bodies as steps in a staircase leading skyward. Each player guides her cyclops to the position of its sacrifice, presses a button, types a single message to future players of the game, and watches the cyclops's eye close forever. Thereafter, the player is never allowed to play the game again; logging on to the website, she can only watch the ongoing progress of the staircase of which her body is a part.\n\nThat's a pretty compelling way to explore themes of sacrifice in a work: to ask players actually to make a sacrifice, and to show them the meaning of that sacrifice over the course of generations. This is something games are almost uniquely capable of doing, and we haven't even begun to explore the possibilities of this kind of expression.\n\nIt's also the sort of experience\u2014a minutes-long game in which the player is asked to commit voluntary suicide and never allowed to play again afterward\u2014that is unlikely to come out of a commercial publishing system that needs its creations to sell millions in order to justify their having been made. The author of We the Giants, Peter Groeneweg, is a student and created the game as part of a monthly \"experimental gameplay\" challenge.21\n\nThe ability to work in any art form with the digital game's unique capabilities for expression shouldn't be restricted to a privileged (and profit-oriented) few. If everyone is given the means to work in an art form, then we'll invariably see a much more diverse, experimental, and ultimately rich body of work. In a speech at the 2007 Game Developers Conference, Greg Costikyan\u2014a board and videogame designer and critic of the games industry\u2014said: \"I want you to imagine a 21st century in which games are the predominant art form of the age, as film was of the 20th, and the novel of the 19th.\"22\n\nThis is what I want from videogames, and this is what I'm trying to help you imagine. Throughout the rest of this book, I hope to help you imagine how this transformation of games\u2014and the role games will play in the art and culture of the twenty-first century\u2014is not only necessary, but inevitable.\n\nFootnotes\n\n1 I'll use this term interchangeably with \"digital games\" and \"electronic games\" throughout this book.\n\n2 ESA, 2009 Sales, Demographic and Usage Data: Essential Facts about the Computer and Video Game Industry, p. 2, http:\/\/www.theesa.com\/facts\/pdfs\/ESA_EF_2009.pdf.\n\n3 Ibid, p. 10.\n\n4 For insight into the Michfest controversy, see http:\/\/www.auntiepixelante.com\/?p=1247. Also, I don't mean to imply that all of these artists have taken a position on Michfest. I'm really just disappointed in Diane Dimassa.\n\n5 Tim Sweeney, \"End of the GPU Roadmap\" (keynote address, High Performance Graphics 2009, New Orleans, August 3, 2009), http:\/\/www.highperformancegraphics.org\/previous\/www_2009\/presentations\/TimHPG2009.pdf.\n\n6 Sea life joke courtesy Emily Alden Foster.\n\n7 \"I was a fool for mentioning video games in the first place. I would never express an opinion on a movie I hadn't seen. Yet I declared as an axiom that video games can never be Art. I still believe this, but I should never have said so.\" Roger Ebert's Journal, \"Okay, kids, play on my lawn,\" July 1, 2010, http:\/\/blogs.suntimes.com\/ebert\/2010\/07\/okay_kids_play_on_my_lawn.html.\n\n8 http:\/\/www.youtube.com\/user\/grishno.\n\n9 http:\/\/www.youtube.com\/user\/wendyvainity.\n\n10 http:\/\/www.youtube.com\/user\/shaneduarte.\n\n11 Jim Sterling, \"How Aliens are blatantly better than Predators,\" Destructoid, February 9, 2010, http:\/\/www.destructoid.com\/how-aliens-are-blatantly-better-than-predators-162754.phtml.\n\n12 Jim Sterling, \"Why Penis Why?: MK vs. DC's Kitana looks . . . lumpy,\" Destructoid, August 5, 2008, http:\/\/www.destructoid.com\/why-penis-why-mk-vs-dc-s-kitana-looks-lumpy-98300.phtml.\n\n13 Anna Anthropy, \"An Open Letter to Destructoid on Jim Sterling's Misogyny,\" Auntie Pixelante, http:\/\/www.auntiepixelante.com\/?p=912.\n\n14 Because Nintendo controls the infrastructure by which five-second Warioware: D.I.Y. games are distributed, or even allowed to exist, transmission of those games from creator to creator is fairly restricted. What most creators do, in fact, is publish videos of themselves playing their games on YouTube.\n\n15 http:\/\/www.stencyl.com.\n\n16 http:\/\/scratch.mit.edu.\n\n17 Casey O'Donnell, \"Quality of Life in a Global Game Industry,\" International Game Developers Association, http:\/\/www.igda.org\/articles\/codonell_global.\n\n18 Greg Costikyan, \"Mothers, Don't Let Your Children Grow Up to Be Game Developers,\" Play This Thing!, April 3, 2009, http:\/\/playthisthing.com\/mothers-dont-let-your-children-grow-be-game-developers.\n\n19 http:\/\/www.bay12games.com\/ww1medic.\n\n20 Playable online at http:\/\/wethegiants.thegiftedintrovert.com.\n\n21 \"Best of the Net: Art Game,\" Experimental Gameplay Project, January 1, 2010, http:\/\/experimentalgameplay.com\/blog\/2010\/01\/best-of-the-net-art-game.\n\n22 Greg Costikyan, \"Maverick Award Speech,\" Man!festo Games, March 14, 2007, http:\/\/www.manifestogames.com\/node\/3413.\nChapter Two\n\nThe History of Magic\n\nSince digital games have existed, their creation has been dominated by a small part of the population: generally white male engineers. In the 1960s and '70s, universities like MIT and Southern Illinois University contained computers and computer networks that were available for student use. Most of these games existed on the school network and were played and contributed to by only those people on the network. Often they were disguised as other programs, because systems administrators tended to delete games as a waste of time.\n\nIt's beside the point to try to identify the first videogame\u2014as with most inventions, a number of people were working along the same lines simultaneously. But whatever the first game was, it had to have been inspired by something\u2014so what came before it? Answer: an entire history of human civilization in which folk games\u2014Go, Chess, Hide and Seek, Stickball\u2014were important cultural experiences, that's what. But the most immediate predecessors of digital games were carnival games (throwing a ball at a stack of bottles from a set distance), mechanical games (a shooting gallery with moving targets), and pinball machines. Coincidentally, these are the games that typify the shift in the history of games from folk to designed games, or games with identifiable authors. When videogames were first monetized, it's this model that the people making money used: pay-to-play games of skill in public spaces designated for game-playing. But that's getting ahead of ourselves.\n\nSo, to create digital games in the sixties and seventies, one first needed access to a computer. The \"home computer,\" like the Apple Macintosh\u2014a computer designed specifically for non-engineers\u2014wasn't popularized until the eighties. To have access to a computer, then, generally required being connected to an engineering school. But being able to make contact with the computer was only the first barrier: in order to teach computers to play games, one needs to know how to talk to computers.\n\nAt the time, neither computers nor the tools people used to communicate with computers were designed with non-engineers in mind. Most programs were written in the super-technical language Assembly. Here's a sample of game code written in Assembly, from the 1979 Atari 800 game 3-D Tic-Tac-Toe:23\n\nCompletely illegible! By 2010, we have coding languages like Ruby and scripting languages like Lua that are designed to be readable by human beings, and we have tools like Scratch and Twine that minimize, if not obviate, the need for coding entirely. But in 1975, there was no way to make a game on a computer without understanding the computer inside and out.\n\nThe Affairs of Wizards\n\nWhat digital games were being made in the 1970s? And who was creating them?\n\nThe college engineer who programmed games in the mid-seventies had most likely been exposed to the role-playing game Dungeons & Dragons (D&D), published in 1974 by TSR, and possibly to the wargames that preceded it. Dungeons & Dragons is storytelling with rules\u2014a human player, the \"Dungeon Master,\" presents story situations to which the other players must respond. The Dungeon Master keeps the rules and facilitates the adventure of the other players, each of whom plays a role within the game world. It borrows from wargames a complex set of rules and tables for resolving situations, mostly those related to combat: whether a sword hits an opponent, whether it does any damage to that opponent, how much damage it does. And it borrows the fantasy world\u2014the wizards and dragons, orcs and elves\u2014of J. R. R. Tolkien's Lord of the Rings books, popular among student engineers at that time.\n\nThat Dungeons & Dragons was hugely influential on digital game creators of the seventies can be seen in network games like dnd\u2014short for \"Dungeons & Dragons,\" naturally\u2014created by Gary Whisenhunt and Ray Wood at Southern Illinois University in 1974, the year of D&D's release.24 Like TSR's game, dnd involves descending into a dungeon, fighting monsters (dragons included), and collecting treasure. In this version of the game, it's the computer that keeps the rules, taking on the \"Dungeon Master\" role that would formerly have been given to a human participant.\n\nThe Tolkienesque fantasy setting of Dungeons & Dragons is all but ubiquitous in digital games of the time, but what's really interesting is the way designers transformed that setting by transposing it into a digital world. In 1977, at MIT, Tim Anderson, Marc Blank, Bruce Daniels, and Dave Lebling began working on a game initially called Dungeon, later renamed Zork. Zork is a text adventure game: the player is presented with a paragraph of descriptive text, types a sentence explaining the action she wishes to take, and is presented with further text by the game in response. In this way, it resembles both prose fiction and the refereed experience of a game of round-the-table Dungeons & Dragons.\n\nAnderson, Blank, Daniels, and Lebling, along with others, founded Infocom in 1979. The fantasy world that they created in Zork grew into many more games, such as Enchanter and Spellbreaker. But what's interesting to me is the particular way magic is treated within the fantasy world these MIT engineers built. As Jeff Howard writes for the blog The Gameshelf:\n\nInfocom's Spellbreaker trilogy, consisting of Enchanter, Sorcerer, and Spellbreaker, entails a magical grammar, in which spells are verbs that take direct objects, allowing players to type \"frotz stone\" to make a stone glow or \"blorb chest\" to open this locked container. Frotz and Blorb are names for, respectively, an interactive fiction interpreter and a wrapper for multimedia elements. While amusing, this application of the names of in-game spells to the programming and technology outside of and supporting the game also suggests a powerful relationship between programming and the verbal grammars of magic. Simply put, programmers and magicians both master a grammar in order to make things happen. Both hackers and wizards achieve this alteration of reality, whether simulated or real, through an arcane set of words and phrases known as programs or spells. When properly configured, a program causes amazing events to occur (calculates our taxes, launches an anti-missile defense system, summons a longed-for package from Amazon.com to our doorstep), just as magicians can throw fireballs and (when very powerful) grant wishes. However, when the programmer makes the slightest error in the placement of a semicolon or case sensitivity, the program won't compile, much as a spell fizzles.25\n\nIt's not surprising at all that in a fantasy world constructed by programmers, the power to create and change the world would be indistinguishable from programming. For these engineers, technology was their means of making magic happen.\n\nThis theme of magic as technology (or technology as magic) doesn't occur only in Infocom games. The Ultima games were created by role-playing games enthusiast Richard Garriott, who named some of the people in his fantasy world after characters he role-played. The first Ultima game progresses from slaying monsters for Lord British and Shamino to flying a spaceship; the nemesis of the third Ultima game (Ultima III: Exodus) is ultimately revealed to be a computer that the player must reprogram. We can see the trend persisting into the MUDs (multi-user dungeon games, the descendents of games like Dungeon) and MOOs (MUD object oriented games), online games in which game administrators are called \"Wizards\" and have the ability to ban players from the game by \"toading\" them (turning a player into a toad).\n\nThis early in the history of digital game creation, we can still see that games, as with all works of art, contain the values of the people who make them. Which is precisely why more than a single group of people should have access to the means of creating them.\n\nDigital Barkers\n\nThe founders of Atari, NAMCO, and SEGA, three early videogame publishers that still exist (in some form) today, were all involved in either carnival barking or the distribution of pinball and other mechanical games. When these people set out to make digital games that earned them money, they used the system they were already familiar with: installing games in public spaces and calling people over to play them.\n\nThe arcade cabinets that housed these new games were designed to act as their own barkers: they were stylized pieces of wood decorated with artwork, containing video screens that flashed invitations to players and demonstrated their games. For a quarter (sometimes more), a player bought one play of the game. As in pinball, the player's skill extended the game: the better you were, the longer you could play. Arcade cabinets were initially placed in settings like bars and, later\u2014when videogames became popular among kids\u2014pizzerias and malls. Eventually, they came to inhabit dedicated spaces\u2014arcades\u2014that existed solely to house digital games. The arcade cabinet was the way most of mainstream culture first encountered videogames.\n\nHow did the arcade game become more ubiquitous than the carnival and pinball games whose sales model it borrowed? Carnival and mechanical games were huge and required human supervision. Pinball machines, because they contained lots of moving parts that constantly collided with each other, broke frequently and were expensive to maintain. The first digital arcade games were cheaper to maintain than the pinball machine because the parts were all standard pieces of electronics, far easier to replace than a piece of track molded specifically for a single pinball machine. The arcade game was also more compact and self-regulating. Since it didn't take up a lot of space or require constant maintenance by the owner, it could inhabit spaces like the aforementioned bars and pizzerias, spaces not exclusively dedicated to electronics, and thus, spaces not exclusively populated by the engineers and programmers who, up to that point, had been almost exclusively the audience for digital games.\n\nBut naturally, the manufacture and distribution of these arcade cabinets required capital, and here businesspeople gained their foothold (soon to be a stranglehold) on videogames. Engineers, sometimes with the assistance of artists, still designed the games and the hardware that made them possible. Businesspeople handled distribution to bars, malls, and arcades. Venture capitalists were brought in to fund the costs of production and expansion. A need for marketers began to appear, although this was not as important as it would become later, when games weren't sold only to arcade operators but directly to players. But what's important to note is that it was the business folks, not the engineers or artists, who controlled the capital. As long as game creators were hardware manufacturers, this was the case.\n\nThe Invasion of Home\n\nIt wasn't long before the people who manufactured machines for the arcade hit upon the idea of manufacturing machines for the home. This would allow them to market their games not to the middleman arcade operator, but to the players themselves. Atari was among the earliest publishers to have great success in the arcade, and in 1977 it began to publish the Atari Video Computer System\u2014later retroactively renamed the Atari 2600, after the Atari 5200 and 7800 went to market\u2014selling home versions of its most popular arcade cabinets to players. Games for the Video Computer System were distributed on cartridges that plugged into a base machine, rather than on miniature arcade cabinets, which meant that after the initial purchase of the hardware, the actual game software became much cheaper to produce and distribute. The market exploded.\n\nArcade cabinets were more expensive to play than home game cartridges, and now that the player could play digital games in her own home, arcades became less of an attraction. Because home game hardware was fixed and arcade cabinet hardware was not\u2014the home game player buys a single piece of hardware, while most arcade cabinets have hardware specialized to the game that inhabits them\u2014arcade cabinets still managed to offer unique experiences. Arcade games became more and more specialized over time, distributing games that were implausible in the home, either because of the technology or the context. The games that predominate in modern arcades are large ride-on vehicles, dance platforms, or drum sets that make less sense in the home than in a commercial space. Most arcades didn't survive this shift in the market: there aren't many arcades left these days, at least in America.\n\nThe shift in the way people discovered and played games also lead to a shift in game design trends. Arcade games, because they earn money on each play, are designed to be as succinct as possible, and to teach new players how to play quickly. They are also often designed to be hard, because a player, once she loses, will either have to pay again to continue her game or relinquish the machine to a new player. Home games, which players pay for one time in exchange for infinite plays, require publishers to set the price of the game higher than the traditional quarter. Thus home games became longer and longer in an attempt to appear more valuable to potential players. They could have much longer learning curves and be much gentler to play. But this longer game requires more content, and hence bigger teams to design and create that content. Marketing, now that the games were sold directly to the player, became a powerful force, and began to make many of the creative decisions.\n\nTake, for example, this account of a conflict between the marketers and game programmers at Mattel Electronics, publishers of the Intellivision and its software, related by former Mattel Electronics staff:\n\nOn December 6, 1982, all of the programmers and graphic artists were herded into a conference room and shown a series of TV commercials\u2014the new Kool-Aid ad campaign. It was announced that Marketing had made a tie-in deal to release Intellivision and M Network Atari 2600 Kool-Aid Man cartridges. The games were scheduled to be ready in about six months, which meant that programming had to begin immediately. Worse, they wanted game-screen mockups to appear in the 1983 Mattel Electronics catalog at the Consumer Electronics Show\u2014one month away. A two-week contest to come up with the best game concept was announced. Separate ideas were developed for Intellivision and Atari 2600.\n\nThis led to a confrontation with Marketing. The programmers' viewpoint was that the features of a game should be tailored to the system it would be played on, to take full advantage of the system's strengths. Marketing, on the other hand, wanted games designed for multiple systems, with the features being the same on each system. If a game couldn't be ported to other systems, it shouldn't be done on any system.\n\nThe programmers argued that this meant all games would have to be designed for the lowest common denominator\u2014the Atari 2600. Marketing argued that keeping the features the same would make games easier to advertise and make word-of-mouth among customers more favorable.26\n\nOn contemporary home game consoles, most games come from companies other than the manufacturer of the hardware. The hardware manufacturer generally enforces an approval process for games commonly called \"lot check,\" or \"technical requirements,\" which contains a list of requirements the game must meet before it can be printed and distributed. For example: The game must display a message when a game controller is unplugged from the machine. It must support a variety of novelty controllers that have a limited run. There's a lot of room for error, and applying for the process isn't cheap. Rejection means that the fee will have to be paid again after the asked-for changes are made to the game, and there can be many rejections before a game is approved, putting distribution of digital games to early home consoles still well out of the financial reach of almost anyone outside of the growing industry, despite the cheaper manufacturing costs.\n\nGame consoles weren't the only home invaders. In 1984, Apple released the Macintosh computer. Microsoft began distributing its Windows operating system with computers the following year. These machines were conceived and marketed as \"personal computers,\" designed for home use by non-engineers and marketed to the public. Their use of a mouse for navigating between different files and programs visually made these new computers far more approachable to non-engineers than the traditional text prompt, where users typed from a list of hidden commands.\n\nPublishing a game for the home computer was similarly expensive to publishing one for a home game console. While game consoles have identical components (every individual PlayStation 3 has the same pieces and the same capabilities, with a few small deviations), computers aren't homogeneous. Certifying that a game will run on a wide variety of contemporary computers, with hundreds of potential variations in operating system, installed programs, hardware, and input devices, is an extensive and expensive process. Manufacturing the game, getting it on store shelves, providing on-call technical support to players, and marketing it to those players all costs money.\n\nThe Games Publishing Industry Today\n\nGiven the expenses of distributing a game\u2014lot check, compatibility testing, printing, marketing\u2014how does anyone afford to make games?\n\nThe contemporary games industry uses a developer-publisher model. The developer actually designs, programs, and animates the game at the behest of the publisher, who pays the expenses of distributing it. The developer may pitch the game to the publisher, or the publisher may bring the game concept to the developer. The publisher might just own the developer: bigger publishers like EA (formerly Electronic Arts) and Ubisoft have purchased many development studios.\n\nA developer may start a project with her own resources before attempting to find a publisher for the game. Because the publisher controls the distribution of the game, it has control over the content of the game. The publisher's agents will periodically check the progress of the game and demand changes from the developer. Often these changes are for the sake of marketing the game: a publisher will always do what it can to make a game more salable, or what it perceives as being more salable. A publisher may shape a game to better resemble trends in the widely selling games of the day.\n\nWithin a development company, employees are typically divided into three roles: designer, artist, and engineer or programmer. All of these roles have a technical (knowledge) barrier to entry. An artist doesn't just need to be able to draw; she needs to be proficient in the 3-D modeling software the developer prefers. She needs to know how to prepare images in a way that the engineers can use. A designer needs to be familiar with the \"engine\" the game is being developed in, and to be fluent in the scripting language that engine uses in order to create events and interactive elements within the level she designs.\n\nA game is made by at least one team of each of these groups: a team of engineers under a lead engineer, a team of artists under a lead artist, and a team of designers under a lead designer (or \"game designer\"). The engineers\/artists\/designers receive their instructions from the leads. The leads report to a director. The director reports to a producer, who in turn represents the publisher. Within this system, which exists to coordinate teams of increasingly unmanageable numbers of people (numbers needed to produce the huge amount of content Hit Games demand), you can see that the people who exercise the most creative power over the project are the people who are farthest from its creation.\n\nThe expenses of hiring and coordinating all these people mean that a game has to be a hit in the market in order to make a profit. And so the publisher, with its final authority on the content of a game, will almost always make a conservative decision about that content in order to make the game more marketable. If it wants to make a profit, the publisher is obligated to.\n\nPublishers have installed themselves as gatekeepers to videogames publishing. To distribute and sell a game in the contemporary market requires their consent. But for as long as people have had access to computers, there's been a history of game creators who've sought alternative solutions to the problem of game distribution.\n\nRethinking Distribution: Share? Where?\n\nThe personal computer appeared in homes in the eighties. Personal computers are not just for consumption; they are also tools for creation. Anyone with the technical knowledge and the tools can make a game on a computer. And any game I make on my Windows (or Mac or Linux) computer, you can play on your Windows (or Mac or Linux) computer. It's just a matter of getting the game from my computer to yours. Distribution\u2014whether it's intended to make a profit or not\u2014has been the major problem of most small game creators.\n\n\"Shareware\" was a popular concept in small game distribution throughout the eighties and nineties. Shareware relies on the players themselves to distribute a game. If I encounter a game I like, I might duplicate it and give a copy to a friend, who in turn makes more copies. Copying games initially meant floppy disks: the cost of producing digital media containing the game was deferred to the audience. Some authors might include their address in their games and ask for a tip: a donation of any amount, a postcard from somewhere interesting. Some authors, for the cost of a disk and some compensation, might offer an expanded version of the game, a second episode or a sequel.\n\nThis is how Tim Sweeney of Epic MegaGames (now Epic Games, mentioned earlier) and Scott Miller of Apogee Software got their start. They reinvested the money their games earned into creating distribution networks, hiring developers to create more games, marketing their games at first through catalogs and eventually on store shelves. They went the path of the publisher, which unfortunately remains the only viable method for widespread physical distribution.\n\nBut the rise of online networks gave hobbyists and small game developers a new method\u2014and critically, a wholly digital method\u2014for distributing their games. The Bulletin Board System, or BBS, was a public online space that proliferated from the 1970s to the '90s. A home computer user with a modem could dial in to a BBS through the phone line, and would then have access to all the files available on that system: shareware games, for example, that could be downloaded to the user's computer. More important, that user, and countless other hackers, hobbyists, and coders, could upload games to that BBS for other users to download. Games could be passed from computer to computer this way. And they could be passed around without the need for physical copies and the associated costs. This means that non-professionals and non-publishers were able to transmit all sorts of games to players\u2014and in fact, there's a swath of weird, personal, and experimental shareware games around that could never have come from the hit-driven games mainstream.\n\nFor example, I discovered a game when I was young called Evolve! Lite.27 This game simulated life by allowing the player to program a species of digital creature with a set of different reactions to different stimuli (for example: when in the presence of two or more predators, the creature turns and runs in the opposite direction). Individuals of the species who mate pass on these tables of behaviors\u2014this virtual DNA\u2014but not all of it! Some of the behaviors will randomly mutate, as in real sexual reproduction, and individuals with beneficial mutations will survive long enough to pass on their mutated DNA. This shareware game, then, provides a working model of evolution!\n\nThe game was made in 1993 by Matt Bace and Mike Wall, who published under the label \"FunTek.\" This is all I know about them. The game is called Evolve! Lite because there supposedly exists an expanded version of the game called Evolve!, one that allows for a world that's four times larger and populated with twelve competing species, rather than the two of Evolve! Lite. A registration form included with the game offers copies of Evolve! for $19.95 plus shipping. It also encourages me to register on CompuServe (an online network of the time), and it contains an advertisement and phone number for JAB BBS: \"We have one of the largest collection of PD [Public Domain] & Shareware.\"\n\nI actually discovered Evolve! Lite on a CD I bought in a store\u2014a shareware CD containing the noncommercial versions of hundreds of shareware games. This was another solution to the problem of distributing shareware games: a small publisher would offer to distribute shareware authors' games in stores, and the publisher would sell the CD to buyers, promising hundreds of games on a single disc. Many creators were able to infiltrate store shelves this way.\n\nBut it was digital distribution that offered the most potential for the distribution of small games. Side-stepping the cost of printing media entirely, digital distribution promised to ship a game directly from computer to computer, from author to player. The BBS allowed for digital distribution, but was hampered by its bandwidth\u2014stuffing data through phone lines, a BBS could only allow for small, slow downloads\u2014and the isolation of BBS networks. One BBS wasn't connected to another, and a user plugged in to one BBS would only have access to what was available on that BBS. This made widespread distribution more difficult and slow.\n\nAnd so, for a long time, the digital distribution of games was scattershot. But eventually a network would coalesce that would resolve these problems.\n\nI'm Referring to the Internet\n\nToday the Internet is linked by cables, not phone lines. The Internet of BitTorrent gives us a model for file sharing that's fast and decentralized. The Internet of 2012 is different from the BBS systems and early online networks of the eighties and nineties in a few important ways: there's the speed, yes, but more important is the access. The infrastructure of the Internet is different: a user doesn't dial in to an isolated part of it, but rather always has access to any part of it (government censorship aside). Which is to say that if I make a game, I can post it in one location (say, my website), and anyone connected to the Internet can visit that site and download that game.\n\nWhether they can run that game is another question. But there's been a progression toward infrastructure not only in playing games but in running them. Take Flash, for example, an Internet plug-in originally designed, by Macromedia, to allow animators to insert movies into web pages so that visitors could watch them inside their web browsers. Almost immediately creators began to co-opt Flash in order to put playable games into web browsers\u2014obviating the need to download a game before playing it. Look at Newgrounds.com, a Flash \"portal\" whose current slogan is, \"Everything, by everyone.\" Newgrounds (which began as a zine distributed by the thirteen-year-old Tom Fulp28) began accepting visitor submissions in 1999. A decade later, Newgrounds hosts 170,000 Flash movies and games created by over 2.2 million registered users.29 Plenty of those are cartoons about Super Mario, but consider how many creators have found audiences for their creations. Newgrounds has even found ways to earn money for its creators, by selling ads to interested companies and giving creators the option of including those ads in their movies and games. There was a time when I made my living almost exclusively by creating Flash games for Newgrounds.30\n\nSo you can get an impression of how much potential digital distribution has to allow games to proliferate outside the industry. To physically publish games has always been difficult for authors without access to capital: that accounts for the rise of publishers. But the speed and interconnectedness of the contemporary Internet provide authors with a means to distribute their games to players without having to deal with the costs of physical publishing and the marketing these costs engender.\n\nPublishers, incidentally, are aware of the Internet as well. Corporations like Valve, Apple, and Microsoft have set up online infrastuctures (\"Steam,\" the \"App Store,\" and \"XBox Live,\" respectively) to sell games. Users buy games with a credit card, allowing them to digitally download games to their computers. Small game creators have been able to ride the coattails of these online marketplaces, using them to sell and distribute their own creations. The danger is that these markets are maintained and regulated exclusively by the corporations who built them, corporations who will of course police them according to their own interests. Take for example, February 2010, when Apple deleted over 5,000 iPhone games from its digital store overnight for being, in Apple's judgment, too sexual.31\n\nBut digital distribution potentially means the most to the creators of free games\u2014hobbyist game creators. There can be hobbyist game creators because distributing games no longer requires capital. An author can produce a game in her spare time, upload it to the Internet, and watch as an audience finds, downloads, and experiences it.\n\nBut what does she use to produce her game?\n\nNew Tools for Artisans\n\nThe first digital games were created by engineers in university computer labs. They alone had access to computers, and they alone had access to the technical information required to teach those machines to play games. But now personal computers inhabit homes\u2014and, consequently, new game-creating tools have come into being for people who aren't engineers with technical knowledge.\n\nI'm going to discuss many of these tools, and what each is good for, later in this book. But for now, I think a sample of source code might illustrate how far the tools of today have come from the Assembly code at the opening of the chapter. This is a sample of Inform 7 code. Inform is a tool for creating interactive fiction: text adventures. The newest version was created by Graham Nelson to allow authors to write \"natural language\" code\u2014that is, lines of code that look like English sentences. Natural language code isn't necessarily the most efficient or effective way to write a game, but Inform 7 was made with the idea that an interactive story should be as easy to write as a prose story, and that if it was, more people would create games. The following code gives the player a bag of four candies, one of which is poisoned.32\n\nMaybe you don't follow the example totally, but it looks very different from the 1979 Assembly code. If the two biggest barriers to free game creation\u2014and by free, here, I mean creation that's universally accessible\u2014have been the technical knowledge required to teach game logic to computers and the high cost of publishing physical copies of games, then at the time of this writing, both of those barriers have been breached.\n\nRight now, we can imagine a future where creating a game is as easy as writing a story or drawing a picture. We can imagine videogames that are written, like Newgrounds suggests, \"by everyone\" for everyone, rather than by corporations for consumers or by technical wizards for stunned onlookers. This is our time, and games are ours to create.\n\nSo what are games good for?\n\nFootnotes\n\n23 Contributed by Steve Harvey.\n\n24 See \"Pixel Journeys: The Magic of dnd5,\" GameSetWatch, December 19, 2008, http:\/\/www.gamesetwatch.com\/2008\/12\/column_pixel_journeys_dnd.php.\n\n25 Jeff Howard, \"Magick Systems in Theory and Practice, Installment 2: Word and Gesture as Input Methods in Gaming History,\" The Gameshelf, June 30, 2010, http:\/\/gameshelf.jmac.org\/2010\/06\/magick-systems-in-theory-and-p.html.\n\n26 \"1983 Releases,\" Intellivision Lives, http:\/\/www.intellivisionlives.com\/bluesky\/games\/credits\/1983c.html#kool_aid.\n\n27 I host a download at http:\/\/blog.dessgeega.com\/?p=51.\n\n28 \"Newgrounds Wiki: History,\" Newgrounds, http:\/\/www.newgrounds.com\/wiki\/history.\n\n29 I emailed Tom Fulp for the 2.2 million figure.\n\n30 Having done so, by the way, I can assure you that Flash is not an ideal tool for creating games. Since the program was designed as an animation player with interactivity included as an afterthought, it's an unwieldy and confusing game-making tool.\n\n31 Craig Grannell, \"Apple's Stance on 'Adult' Apps Is Indefensible,\" TechRadar, February 23, 2010, [http:\/\/www.techradar.com\/news\/phone-and-communications\/mobile-phones\/apple-s-stance-on-adult-apps-  \nis-indefensible-672642](http:\/\/www.techradar.com\/news\/phone-and-communications\/mobile-phones\/apple-s-stance-on-adult-apps- is-indefensible-672642).\n\n32 From The Inform Recipe Book, http:\/\/inform7.com\/learn\/man\/Rex62.html.\nChapter Three\n\nWhat is it Good For?\n\nSo, for the first time in the history of the videogame form, people who aren't programmers or corporations can easily make and distribute games. But why would they want to? Why make a game\u2014especially when there already exist the means to write stories, play songs, film yourself for YouTube? What can we do with games that we can't do with those forms?\n\nTo begin, let's define what a game is.\n\nYou've played games and you have assumptions about what they are. Maybe when you read game you imagine a videogame; maybe when you imagine a videogame you imagine a big-budget run-jump-shoot game. Maybe you imagine Tetris. Since I'm more interested in games, digital and otherwise, that don't resemble games that already exist, I think a fresh definition is in order. I also think it's worthwhile to have a definition that isn't specific to digital games, because I'm interested in the commonalities between digital and non-digital games, and in connecting videogames to that much older tradition.\n\nSo here's my definition:\n\nThat's pretty broad, huh? I'm interested in as inclusive a definition as possible, though you might argue that mine is too broad: for example, you can use it to describe getting stuck in a traffic jam or paying your taxes. A tax form is nothing but a series of rules you follow to produce a final number, after all. But is it useful to think about your taxes as a game?33 Not really. Do the rules on a tax form really create a strong experience, or are they just a method for producing a number?\n\nA game is an experience, and that experience has a certain character. Maybe a game is a story, or maybe it's the experience of control giving way to panic giving way to relief. Maybe it's about taking something and making it grow bigger and bigger and bigger, or maybe it's about two rivals, equally matched, each trying to out-guess the other's plans. The experience that we identify as a game has character, and we can talk about what that experience is.\n\nAnd if we're discussing an experience, then that implies someone is there to have that experience, someone we refer to as a player. We can't talk about a game without talking about the experience of the player playing that game, even if the playing experience we're talking about is often our own.\n\nThe experience we call a game is created by the interaction between different rules, but the rules themselves aren't the game, the interaction is! A game can't exist without a player or players: someone needs to be engaging with the rules for the experience to happen.\n\nHow does that work? Consider a game of Tag. Rules: One player is IT, and must tag as many of the other players as possible with a touch. Each of those other players is SAFE when she touches this gnarled-up oak tree. You can see the way the interaction between those two rules creates an interesting (and volatile) dynamic. The players who aren't IT want to reach the tree, but the player who is IT wants to stop them.\n\nYou can imagine a situation where the IT player is standing between two other players\u2014one to her left, one to her right\u2014and the SAFEty of the tree. Maybe one of them will make a break for the tree, maybe IT will be forced to pick one of the two to chase while the other gets to make a run at the tree, maybe a fourth player will take advantage of IT's distraction to make a run at the tree from behind. When we talk about a game of Tag, we're talking about this experience. But this situation (and it's a good, tense one) isn't explicitly defined anywhere in the rules. However, notice how these rules guide the creation of that situation. The rules set the players in opposition to each other, give most of the players a goal, and give the other player a reason to intervene, creating a tense dynamic.\n\nWhat if we were to take either of these rules away: the SAFE location or the player who's IT? Without a SAFE location, players have no reason to stay nearby and interact with the other players, especially the IT player. The ideal strategy to avoid IT would be to go as far away as possible, and that breaks the tension and hence the experience of the game. What if there was no IT player? Then it'd just be people running around, and while a bunch of people running around has value, it doesn't have the character or dynamic of a game.\n\nBut there's certainly room to change the details of the rules. Tag, being a folk game, has been played by many people in many places with many, many different versions of the rules. In one version, a player might be done once she's tagged the SAFE tree. As more and more players tag the tree and leave the game, the players who are less fast become greater and greater targets because the IT player can focus less on monitoring the tree and more on pursuing them.\n\nAlternately, what if a player who touches the tree isn't permanently safe\u2014what if players are only allowed to be in contact with the tree for five minutes at a time? That keeps players vulnerable to IT and keeps the game from stagnating. Maybe a player who leaves the tree has temporary immunity to allow her to get safely out of IT's sight, or maybe it becomes a stand-off, where the escaping player has to wait for another player to distract IT's attention before she can make a break for it.\n\nWhat about freeze tag? In this case, a player who's tagged by IT is \"frozen\" and has to wait for another player to come and \"rescue\" her before she can move again. This variation has much more direct interaction between the non-IT players. Instead of just depending on one another as decoys, they have to actively put themselves at risk to aid other players, which only adds to the tension of the game. And it creates a new dynamic between the non-IT players: I rescued you this time, but if I get tagged you're going to have to leave the tree and rescue me.\n\nAnd that's what games are good at: exploring dynamics, relationships, and systems.\n\nThe Story of Tetris\n\nA \"system\" is what we'll call the interaction (or ongoing interactions) between a set of rules. Let's talk about Tetris now.\n\nWhat are the rules of Tetris, essentially? The basic rules that drive Tetris are:\n\nThe game is played with pieces, comprised of every possible combination of four squares. (See the image above.)\n\n  * Pieces fall continuously into a well of a certain volume. The player can guide the pieces' fall to the left and right of the well, and also rotate the pieces both clockwise and counterclockwise.\n  * Pieces are removed from the well when and only when the player organizes them into complete rows.\n  * If there is no room left in the well for a new piece to fall, the player loses.34\n\nYou can see how these rules create a system where the player's mistakes compound on one another to cause further mistakes: Only full rows are eliminated, so incomplete rows stick around and take up space in the well. Clutter in the well then makes it more difficult to position other pieces and to create rows. As the row fills with mistakes, it eventually becomes impossible to fit more pieces, and the game ends.\n\nThese rules function in tandem to give the game a momentum and shape: the player makes errors that cause further errors, until eventually the player is overcome. (And consider how well a commonly added rule, \"the pieces fall faster every time ten lines are made,\" works with these basic rules to help the game escalate.) We could consider this a system.\n\nAll games aren't necessarily simulations of existing systems: it would be difficult to imagine a situation in the world that actually resembled Tetris. But it's easy to imagine simulations that model systems of rules that are far less abstract: urban planning, politics, oil drilling. And there are games whose rules mimic such systems. Will Wright's SimCity is a game in which the player plans a city, Jim Gasperini's Hidden Agenda is a game in which the player governs a post-revolutionary South American nation. Arch D. Robison's Seismic Duck35 models the way drillers use aimed sound waves and seismogram to find oil reservoirs.\n\nYou can begin to see how systems can be translated into game rules: a commercial zone in SimCity, for example, needs people to act both as a work force and as consumers. That means the people need homes to live in, transportation to get them around the city, power to make sure the lights are on. The system teaches concepts about the interdependency of urban forces. To again cite Greg Costikyan's \"Maverick Award Speech\": \"I want you to imagine a world in which the common person is no longer ignorant of economics, physics and the functioning of the environment\u2014things which are themselves interactive systems\u2014because they have interacted with them in the form of games.\"\n\nEvery game of Tetris has the same shape\u2014errors compound on errors until the well is filled and the player is overcome\u2014because the system of rules we've discussed guides the experience in that direction. But the player places all the pieces herself. Every player will place the pieces differently, will play a different game, but experience a similar result. The same holds true for any system of rules, as simple as Tag or Tetris or as complicated as SimCity. Games have a lot of potential for examining the relationships between things\u2014or, rather, for allowing the player to examine the relationships between things, because the player does not merely observe the interactions; she herself engages with the game's systems.\n\nThe Rise of the Designer\n\nTag is an example of a folk game, along with Go, Chess, Poker, Stickball, Hide and Seek, and most of the world's oldest games. Games have been around as long as civilization has; the game is by no means a new form or a recent invention. What is relatively recent is the shift from folk to authored games. Folk games, like folk songs and folk texts such as the Bible, have no single credited author, but rather many untraceable authors over many years. They're artifacts shaped by entire cultures, and generally they can tell us a lot about those cultures.\n\nFor example, compare Chess, a continental European board game of warfare, with Hnefatafl, a Viking board game of warfare. Chess is a game of combat between kings with equal resources. Each player has the same pieces and starts in the same position on opposite sides of the game board. Each player's goal is to capture the other player's king. In Hnefatafl, one player represents a king and his defenders, who start in the center of the game board. The other player represents the attackers, who surround the king's forces on all sides of the board. The king player's goal is to get the king through the attacking hordes to safety, while the other player's goal is to surround and capture the king. The differences between these games' interpretations of combat tell us a lot about the differences between strategic thought between European vassal kings and Viking warrior bands: their priorities, the nature of their battles, and whether they approach warfare as a platonic war between equals. And the games themselves, in turn, shape the strategic thought of those who play them.\n\nOur history is full of folk board games. Authored board games\u2014games created by a single person or small group, and whose authors can be identified\u2014are a more recent phenomenon. For example, I can tell you that the board game Cosmic Encounter was designed in 1977 by Bill Eberle, Bill Norton, Jack Kittredge, and Peter Olotka of Eon Games. (We can date Cribbage, by Sir John Suckling, to the 1630s.) These are games as texts of specific rules, rather than as patterns of rules that are subject to change through mimicry. A game of Tag will always have a chasing player and a safe position, but the actual rules will change from play to play. The majority of contemporary board games are designed by a single author or team, and the same is true of digital games.\n\nCan there be folk videogames? Videogames retain credits better than board, card, and physical games. I think that there are digital games, though, that exist as patterns of similar rules, perpetuated through duplication with small mutations. There are a thousand different versions of Tetris, for example, each coded by one of a thousand different authors, and each version with a slightly different set of rules, a slightly different set of numbers, and often (to avoid litigation) a different name. There's a digital game that's commonly known as \"the snake game,\" which began as an arcade game called Nibbler by Joseph Ulowetz and John Jaugilas. In this game, the player directs a snake to gobble pieces of food. The snake dies whenever it crashes into either a wall or its own body by coiling around itself. Each piece of food causes the snake's tail to grow longer, making it take up more space and making it more difficult for the player to avoid collisions with her own body. So many different authors have remade this game on so many different machines that all of its forms and variants are usually just referred to as \"the snake game.\" Is this how authored games become folk games?\n\nBut what can authored games tell us that's different from folk games? Folk games tell us about the culture that created them; authored games tell us about the author that created them. Authored games have the potential to be more personal, and thus more specific and diverse, than folk games. Two plays of an authored game are likely to be more similar than two plays of a folk game, because the authored game retains the rules set created by its original designer. It's the fact that folk games change with each player that makes them so long-lived, that makes them adapt to suit the culture that adopts them. But in this book, it's authored games, and the diverse set of voices they embody, that I want to focus on.\n\nWhat's Video Good For?\n\nIn a board game, players have to track how much money is left in the bank, which pieces are in play, how high the water level rises. A deck of cards can keep players from knowing in what order pieces will come into play, dice can generate random outcomes to situations, and players have hands of cards that represent information they keep from the other players, but beyond these basic devices, little information can be hidden from the players, because the players must make sure the rules are being observed by tracking most of the information themselves.\n\nIn digital games, the computer keeps the rules. The computer tracks all the numbers. Digital games therefore have much greater control over what information the players have access to, making videogames capable of much greater ambiguity than board or card games.\n\nWhat's ambiguity good for? Telling stories! Digital games have great potential for storytelling. The author has a lot of control over the pace at which information is revealed; therefore the author can pace the telling of a story. This is not to say that videogame stories are being told as well as they could be. But the format of a videogame\u2014which lets rules be changed and introduced over the course of the experience, and which lets the author hide the causes for events and show only the effects\u2014lends itself more easily to an overt, sustained narrative than any physical game format.\n\nBecause the rules are kept by the machine, the rules in digital games tend to be more numerous and more subtle. Think of a game like Shigeru Miyamoto and Takashi Tezuka's Super Mario Bros. Unless you've studied the game in great detail on a technical level, you probably don't know exactly how high Mario can jump relative to the height of the screen, or how fast he accelerates horizontally when he runs. The interactions between these hidden rules in videogames can result in very complex systems without necessarily complicating the game, because the player isn't required to track and compare all the numbers. For example, imagine the designer creating a situation where there's a tiny platform with a long pit on either side. Mario has to run to build up the momentum to clear the pit and land on the platform, but instead of stopping there he needs to immediately jump again in order to make the second pit without losing the momentum that will let him cross it. This is a problem that wouldn't be obvious to someone who had just approached the game.\n\nThrough playing the game, the player develops a sense of the limits and subtleties of these hidden rules. This interaction between the player and the game, dependent on the game's hiding information, gives digital games their special capacity for subtlety and nuance. You could compare it to the use of \"English\" in a physical sport: the difference between hitting a ball and hitting it with a particular force, and in a particular direction.\n\nBecause of this capacity, videogames are often performative: they allow the player room to interact with rich and complex systems with grace and finesse. We usually refer to this as \"skill.\" A system may persist through an entire game, but the game may start very permissive of less graceful playing and require the player to play with more and more finesse as the game goes on. The game gets HARDER, asking that the player become more skillful, but allowing her to learn the game's systems over the course of navigating increasingly difficult situations.\n\nThe systems that the player manipulates in Super Mario Bros. are introduced very early in the game, with the only added rules coming with the periodic introduction of new enemy characters or hazards. But the situations that Mario has to navigate start fairly relaxed and demand more and more skillful playing as the game progresses. In the first stage of the game, obstacles are low enough that a simple jump from a standing position will allow Mario to clear them. In later stages, the height of obstacles will require Mario to run and build momentum before jumping, in order to jump higher. In this way the designer teaches the player the subtleties of the game's complex system through careful use of machine-controlled variables. Digital games are thus good at teaching, and at communicating a sense of the player's progress, which often parallels the progress of the protagonist and the development of a story.\n\nWhat else is handy for telling a story? The ability to generate or play video and audio, either as accompaniments or as central vehicles for information. Digital games can incorporate a variety of media when telling their stories. Consider how the music in Super Mario Bros. speeds up when there's only a hundred ticks left on the time limit to complete a stage, creating a sense of urgency, or how the sound played when Mario jumps on an enemy gets higher and higher pitched, indicating that a reward\u2014in this case, an extra life\u2014will come if the player keeps doing what she's doing. Consider how the player's journey takes her through a changing visual landscape, from a sunlit field to a black-and-blue underground, to treetops, to the mushroom forest, and to Bowser's castle, and the way each of these sights\u2014withheld from the player until her skill develops to give her access to later areas\u2014provides a sense of progression through the Mushroom Kingdom.\n\nI don't mean to imply that non-digital games are incapable of the things I've described, or that digital games are in some absolute sense better or more worthy of interest. There are many different kinds of games, all of them suited to different things. Digital games, because of their ability to withhold and pace the player's access to information, because of the strict narrative control the author is able to have over the player's experience (because the machine enforces the rules), and because of their capacity for generating a wide variety of sights and sounds to enhance or even define the playing-out of the rules, are particularly well suited for the telling of stories. And the telling of stories\u2014games becoming more personal\u2014is what especially interests me about games as a form.\n\nRole-Playing Games\n\nDigital games have certain strengths for telling stories, but the nature of games in general\u2014even without the advantages the machine provides\u2014makes them good for storytelling. And I do think, in the way that film and photography have generally changed the focus of novels and visual art, the mechanical rules keeper that videogames provide has caused similar focus changes in other aspects of games. In the last chapter, I couldn't discuss the earliest mainframe computer games without mentioning role-playing games like Dungeons & Dragons. Role-playing games came out of the tradition of \"miniatures wargaming,\" a set of rules for moving armies of dolls around a tabletop battlefield and pitting them against one another in combat. Later role-playing games kept the rules for combat and situation resolution, gave players the responsibility for a single combatant rather than a larger number of soldiers, and largely got rid of the dolls.36 But the important thing that games like Dungeons & Dragons introduced was the concept of a \"Dungeon Master,\" or \"Game Master.\" This is a player who manages the game for the other players, laying out the scenario and directing the world's responses to the players' actions. The Game Master essentially inhabits the role of storyteller, preparing and guiding the players through a story in which they make decisions.\n\nThis aspect of role-playing games\u2014an overseer who negotiates the player's choices using a set of rules\u2014was eagerly adopted by many of the first digital game authors. Early digital games like Rogue, a graphical game of maze exploration and combat, lifted its rules and probability calculations directly from Dungeons & Dragons. And text adventure games like Zork took the idea of a narrator who relates the world of the game to the player using a consistent voice. Many early digital games are, conceptually, role-playing games in which the computer takes on the role of Game Master.\n\nThe computer's adoption of the responsibilities of rules keeping and number counting has shifted the focus of many tabletop games away from their original focus\u2014providing players with an extremely finely grained simulation of combat and other adventuring situations. Why spend thirty minutes rolling dice and looking up random treasure and critical hit information from lengthy tables when you can play a computer game that resolves everything, with far greater mathematical complexity, in seconds and in color? Instead, tabletop role-playing games have been able to move away from a mathematically dense combat situation and toward collaborative storytelling and improvisation.\n\nTake, for example, Paul Czege's My Life with Master, in which the players invent a Victor Frankenstein\u2013style mad scientist under whom they will serve as Igors. The Game Master, as the titular Master, assigns the servants duties that enable the players to act out the conflict between their duty to obey and their desire to reclaim their humanity. The dice rolls aren't to test the player's ability to penetrate leather armor with her sword, but rather to test the player's \"love\" versus her \"self-loathing\": if a player \"fails\" her roll, her character must either perform the duty assigned her or refuse to perform it, regardless of which outcome the player might prefer, a simulation of the state of being in emotional servitude. In the end, one of the servants will rise up and destroy the Master, a result that the rules of the game make inevitable: the destruction of the Master is the climax of the story, which each player finishes in turn by creating an epilogue for her character.\n\nWhat interests me about My Life with Master isn't just its use of rules as a unique device for telling a story of personal, internal conflict, rather than as a means of resolving physics simulations in a fantasy world. I also find Czege's distribution method interesting. My Life with Master is sold on Czege's website as both a book, sent in the mail, and as a downloadable PDF file that the buyer may print herself if she's interested. Digital distribution! In the past, the rules for tabletop role-playing games were so elaborate that they required hardbound books, distributed through traditional bookstores and novelty stores, which is still the method used for distributing recent editions of Dungeons & Dragons. But authors of small, more experimental role-playing games like My Life with Master are avoiding or mitigating the costs of publishing and distributing by selling their rules online as digital downloads, or in some cases simply posting them for free on the Internet.\n\nRole-playing zinesters! And ones who, through their change of focus from complicated and expensive rule books full of encounter tables to simple rules that create conflicts and guide the players in creating a story, offer useful lessons that the designers of digital games could stand to learn.\n\nGrown-Up Games\n\nGames are useful, I wrote earlier, for exploring and teaching about dynamics and relationships. Gang Rape is a role-playing game made by Tobias Wrigstad in 2007. Outraged by how many rape cases the courts in his native Sweden dismiss without charges, Wrigstad wrote a game that he hoped would allow players to explore and talk about the experience, and the horror, of rape.\n\nMost games are designed to be pleasing and stimulating to play in an immediately rewarding way: they're intended to be fun. This game is not like those games. In fact, the rules\u2014which are only available by directly and personally requesting them from the author37\u2014open with the sentence: \"A scenario about gang rape is not meant to be fun to play.\" The game is intended to be harrowing; its goal is to give players some respect for the severity of its subject.\n\nA game isn't defined by being fun just as comics aren't defined by being funny. A game is defined as an experience created by rules. Wrigstad's Gang Rape is like any other game in this regard.\n\nIn the traditional role-playing games we've discussed, one player takes the role of Game Master. That player then guides the other players' experiences by telling another player, for example, what things her character \"sees.\" Gang Rape has no Game Master\u2014one player is a victim, the others are her attackers\u2014but it gives each player limited, Game Master\u2013like control over the characters the other players are playing. Specifically, the rapists can tell the victim how her body reacts to their actions, but not how she feels about those actions. The victim can tell the rapists how they feel while they perform those actions. Rape is about control: these rules are designed to give the players an impression of the power a rapist has over the person being raped. There are additional rules that allow the rapist players to dictate each other's behavior, and to allow the players to explore the role of peer pressure and \"egging on\" in the dynamic of a gang rape. All the rules are clearly crafted to create a sense of the dynamics at work in a situation where two or more people have power over another, and to give the players the liberty to explore and better understand those dynamics. (Though the scenario the author had in mind is clearly a woman being raped by two or more men, he admits that the characters involved\u2014and their players\u2014can be of any gender, and that the rules can apply to a scenario like bullying and mobbing instead of sexual assault.)\n\nThis game (which, again, is distributed exclusively digitally, and with the special limitation that anyone wishing to play it must identify herself directly to the author) is an example of using the capabilities of games\u2014experiences created by rules\u2014not to indulge an escapist fantasy but rather the direct opposite: to try to educate players about the dynamics at work within a horrible real-life experience, and how those dynamics might come to be as a product of individual choices and responses. The players narrate, through their characters, the events prior to the assault, the events of the assault, and the aftermath.\n\nGames can be topical; they can be relevant to our lives as human beings. They can be relevant without having to be about rape. But Gang Rape is one example of what we gain when people other than commercial publishers author, publish, and distribute games that commercial publishers could never touch.\n\nThe World's a Stage and We are Players\n\nOften, games\u2014particularly digital games, with their use of video and audio\u2014are compared to film, probably because the videogame publishing industry strongly resembles the Hollywood studio system. But I don't think this comparison is particularly constructive, in that it gives us little insight into what the game, as a form, is capable of. Film tells a static story; what's exciting about the game is that it allows the audience to interact with a set of rules. This doesn't mean the game can't tell a story: in the role-playing genre, the players aren't merely watching a story but playing the roles of the characters within the story.\n\nA better comparison than film is theater, which is where a lot of our game vocabulary (\"the player,\" \"stages,\" \"set pieces,\" \"scripting\") comes from. A play defines the roles, events, and scenes of a story. An individual performance of those roles and scenes will always be different: different actors will perform the same role in different ways. Every performance and interpretation of a particular play is different\u2014sometimes in minute ways, sometimes in radical ways\u2014but we consider the play itself and the scene itself to be the same.\n\nCompare this to a game story, particularly a videogame story. Every player will perform the story called Super Mario Bros. differently (and the same player will perform the story differently each play), but the role of Mario and the actions Mario is capable of taking remain the same. There is always a scene called \"World 1-2,\" although each performance of \"World 1-2\" will be different. In a more contemporary videogame such as Half-Life 2, a very clearly cinema-inspired game, each player will always pass through the events the designers have scripted in the order in which they are presented, but each player's (and each play's) performance of Gordon Freeman, the game's protagonist, will be at least subtly different. The player will always get chased across the rooftops by cops, but in one performance she might hesitate, unsure of where to go, in one she might head straight for the escape route, in one she might panic, almost getting Gordon Freeman killed, and in another she might walk a little too close to the edge of the roof, fall, and have to start the scene over.\n\nAs game storytellers, we are not directing static stories take-by-take but rather arranging the scenes that will comprise the shape of our story. We can begin to think of the player as someone performing a role we've written rather than as an audience who experiences our story without any input as to its outcome. We allow room for improvisation, room for the player to make a role her own. The audience of a game can be more usefully compared to the audience for a play than the audience in the movie theater. In videogames, the audience is there, live, with the actors\u2014or as the actors\u2014experiencing a single performance that is unique, despite the story having been performed and continuing to be performed many times.\n\nSome players record videos of their performances, either for documentation or for the purpose of recording a specific achievement, such as reaching the game's conclusion as quickly as possible\u2014what is usually called a \"speed run\" (YouTube has given lots of these videos a means of reaching an audience). That there's an incentive to capture individual performances of a game testifies to the amount of variance there is within a game depending on who's playing it.\n\nGames and Chance\n\nThe board and card game traditions have also given a lot to digital games. What I think digital games have taken the most from board games and card games is the way they manage chance. Both contemporary designed games and older folk games have invented many systems for managing chance. The six-sided die, for example, allows for the random selection of six equally likely outcomes (and can then be further used to access other percentages and ratios; for example, three outcomes, each represented by two sides of the die, or eleven outcomes with different likelihoods represented by two rolls of the die, and so on.)\n\nCard games themselves are designed as a system for managing chance and gradually revealing information. When all cards are in the deck, every card in the game has (as far as the player knows) an equal chance of being in any position. Once a card has come into play and been seen by the players, though, the players then know where it is and can use that information to make guesses about the remaining cards. Cards also allow players to manage the pace at which they reveal information: a player might have a hand of seven cards hidden from the other players, who don't know whether those cards have come into the game yet or not. Poker is a classic game of using limited knowledge of the cards in play to predict the positions of cards not yet in play. This is what makes Poker an elaborate game of bluffing. One player tries to see through the other's \"Poker face\" because the decisions she'll make are based on what she can predict about the information the other player is concealing. Contemporary game designers have contrived even more rules to control the revelation of information.\n\nAside from hiding information, chance is frequently used to break symmetry. Having different starting conditions between players prevents both players from having the same set of ideal moves, and thus having the game become a stalemate. Having different, randomly selected values between one play and another, or having different game events happen at different, impossible-to-predict times (or not at all), means that each game will demand a different strategy, keeping play from becoming stagnant.\n\nFranz-Benno Delonge's and Thomas Ewert's board game Container, for example\u2014a game where players trade and transport commodities\u2014uses chance to ensure that all players do not value the commodities identically. At the start of the game, a number of cards are shuffled and randomly distributed, one to a player. These cards describe how valuable the different commodities are to the players who hold them, and each card values the commodities differently. The cards are also kept hidden until the end of the game, each card seen only by the player that holds it. Because each player is aware of the entire possible set of values on the cards\u2014she knows which cards are in the game, and which card she, and therefore not the others, possesses\u2014she can watch the other players' decisions and make deductions about which players have which cards, and therefore which commodities are valuable to which players.\n\nComputers have an innate capacity for manipulating chance. Though true randomness doesn't exist, computers handle numbers easily and are capable of generating reasonably unpredictable probabilities of any size on the fly. Every computer has access to an infinite number of monkeys rolling an infinite number of dice.\n\nWhy is this useful? Because, as we've discussed, games have a unique capacity for improvisation! Though each scene has the same shape\u2014Link battles a gang of Moblins\u2014each performance is different. So what if, in one performance, one of the Moblins comes from the left instead of the right? Digital games have the capacity to create variations on many subtle details in every play, keeping the experience from becoming stagnant.\n\nThe differences don't have to be subtle, either. In Chris Klimas and Joel Haddock's online game, Where We Remain,38 for example, the player is a boy searching for a girl on an island patrolled by monsters that are intended to evoke characters from Greek mythology. The layout of the island\u2014what tools are hidden in which caves, what areas which monsters patrol, and in which cave the girl is hidden\u2014is different every time, decided by a random number generator. In effect, this randomness makes the characters and events of the game more archetypal because the emphasis is on the shape of the game\u2014the boy's search for the girl while monsters pursue him\u2014rather than on the details like what treasure is hidden where. Games have lots of room for improvisation, for every play of a game or scene to be unique, and digital games in particular have easy access to a great degree of chance.\n\nGames as Culture\n\nI keep bringing up the profound influence role-playing games like Dungeons & Dragons have had on digital games. Both Eastern and Western videogame trends have their roots in Dungeons & Dragons, but both experienced and responded to that influence in different ways, much as Chess and Hnefatafl reflect different experiences of and responses to warfare.\n\nCharacter creation is an important feature of role-playing games: the players literally describe the role they intend to play, both in narrative terms (what is this character's background and personality?) and in mathematical ones (how many times can this character be hit by dragon breath before she collapses?).\n\nThe American game Wizardry, created in 1981 by Andrew Greenberg and Robert Woodhead, was an attempt to bring the dungeon exploring and monster battling of Dungeons & Dragons to the computer. (It has plenty of peers with similar intentions and similar properties, but it's a good example for discussion.) In Wizardry, the player controls a team of up to six adventurers, similar to a team of player characters in D&D. Each of these adventurers is, fundamentally, a set of statistics (Strength, I.Q., Piety, Vitality, Agility, Luck, Age), a Class (the character's job or specialization), and a list of Spells (tactical magical abilities characters can use to help with combat or exploration). Character creation is held over from tabletop role-playing games, but in Wizardry it loses its narrative dimension. Players \"roll\" statistics (as with dice) and assign character classes and ability sets based on those random \"stats,\" and that is the extent of characterization.\n\nCharacter creation is present in many Western digital role-playing games, though implemented in different ways. The Ultima games, which I've mentioned before, ask the player a series of moral choices that determine which character class she plays as. Games like Michael Toy and Glenn Wichman's Rogue, which borrows its combat rules directly from D&D, skip character creation but assign the player's character no properties beyond this set of stats.\n\nYuji Horii's Dragon Quest39 is similar to early Western digital role-playing games: character creation is limited to entering a name, which also determines your stats via a hidden algorithm. Dragon Quest was widely imitated in Japan, in addition to receiving many of its own sequels, but over time Japanese games broke with the mathematical focus of character creation in Western games. The trend Japanese digital role-playing games tend to follow is to have the player character be increasingly designed by the game designer. In the Final Fantasy series, characters wear outfits designed by the authors, answer to names chosen by the authors (though the player is sometimes given the choice of changing these names), and speak scripted dialogue that the player has little say in.\n\nCompare a 2002 Japanese game like Kingdom Hearts, a collaboration between the authors of Final Fantasy and the Disney corporation, to a 2008 American game like Fallout 3, produced by Bethesda Game Studios. In Fallout 3, the player not only names her character, but also designs her face and appearance and decides on her race and gender. In Kingdom Hearts, the player plays a character named Sora, who is given clear motivations by the writers and dialogue by voice actor Haley Joel Osment and whose name cannot be changed. Sora's appearance in Kingdom Hearts is mostly static: he changes appearance to fit some of the worlds he visits, but the only part of him the player is allowed to change is what his weapon looks like when he's swinging it. His outfits, designed by the game's art staff, are law, as is his race and gender: white and male.\n\nWhy has character creation remained such a fixture of American interpretations of digital role-playing games while Japanese role-playing games have phased it out? It could possibly reflect that America is a young country, and a nation that has been capitalist almost since inception. American culture sells the idea of individuality and ego. In Japan, a much older country in which social roles are valued (and connected to uniforms), role-playing might more easily mean playing the role to which you've been assigned. (In Yuji Horii's Dragon Quest, the protagonist's sole characterization is that his ancestor is a hero.) There's an ongoing dialogue between Eastern and Western design these days, so none of these trends are exclusive (and they've never ruled all of design, obviously), but there are clear patterns in games that we can trace to the values of the people who created them.\n\nGames tell stories that communicate the values of their creators in a unique way: not just through their explicit content but through the logic of their design, and the systems they choose to model. And if games communicate the values of their creators in a unique way, then it's absolutely essential that there be more creators passing on more values, more perspectives. Games must become more personal.\n\nFootnotes\n\n33 Sometimes it is: consider the contemporary interest in game theory among non-game designers.\n\n34 This list obviously omits some rules, like how the game selects the pieces to bring into play next, and at what speed the pieces will fall. Digital games tend to have many more of these very specific rules.\n\n35 http:\/\/home.comcast.net\/~arch.robison\/seismic_duck.html.\n\n36 It's worth mentioning, by the way, that labeling traditions of games the predecessors of digital games is not to imply that these traditions are obsolete and gone. I know people with armies of tiny soldiers.\n\n37 You can contact the author to request a copy at http:\/\/jeepen.org\/games\/gr.\n\n38 Playable online at http:\/\/twofoldsecret.com\/games\/whereweremain.\n\n39 The Dragon Quest games appeared in America under the name Dragon Warrior until recently, for legal reasons: Horii borrowed the name Dragon Quest from a specific Dungeons & Dragons supplement.\nChapter Four\n\nChanging the Game\n\nCreating a game from scratch takes time, effort, and tools\u2014tools that haven't started to become widely available until recently. Digital games are capable of presenting video and audio, and creating a game often requires that music be composed, sounds be recorded, art be drawn, or characters be animated. That involves a variety of tools and training and a lot of work. Often it requires a team of people, each with a different specialty, to accomplish a particular design.\n\nAnd while plenty of hobbyist authors work on that level, the truth is that most digital games are not made from scratch. Most tell their stories by borrowing pictures or sounds or music or code. They use kits that other people have made; they use sound effects that other people have recorded and released for their use.\n\nAnd many, many game creators, budding or otherwise, piggyback on existing works, taking advantage of existing infrastructure to make their creations. A commercial game from a large studio, for example, requires thousands of 3-D models, sounds, and assets to be loaded onto the player's computer. Even more important, a commercial game comes with game logic: a means of resolving collision, of determining the effects of gravity, of moving objects around a game world. These things are difficult for someone inexperienced to code, but here a group of paid professionals have already done all the work. The player who's bought this game has all of those resources on her computer: why not use them?\n\nIn this chapter we're going to talk about what are usually called \"hacks\" and \"mods\" (for \"modifications\"): changing existing games to create new stories. There are a variety of ways in which people change games, and a variety of motivations for doing so. Sometimes changes are akin to crude vandalism, such as redrawing Super Mario so he has a dangling naked cock that flops around as the player walks through the Mushroom Kingdom. Sometimes they're akin to clever, subversive vandalism, such as changing the player character into the Princess and the prisoner who waits at the end into the Plumber. And sometimes they're something entirely new, almost indistinguishable from the games that gave birth to them.\n\nSometimes authors want to situate their creations in the context of a particular work. For example, Jesse Petrilla created Quest for Al-Qa'eda: The Hunt for Bin Laden in response to the September 11, 2001, attack on the World Trade Center, which the American media attributed to Osama Bin Laden. This game is a modification of Duke Nukem 3D, a first-person game about ultraviolent American culture in which the stripper-tipping, alien-gunning protagonist appears to the player only as two attributes: a gun that hovers in front of the camera and a series of prerecorded movie quips. The jockish American attitude of military vigilantism totally fits the play of Duke Nukem, a game in which the most logical response to everything the player sees is either to shoot it or to stuff bills between its tits. It makes sense that Quest for Al-Qa'eda and Petrilla's 2003 sequel, Quest for Saddam, should take the structure of that game.\n\nIt further makes sense that the Global Islamic Media Front should, in 2006, reinvent Quest for Saddam as Quest for Bush (or The Night of Bush Capturing), a modification of Quest for Saddam in which the Saddam Husseins the player ceaselessly guns down have been replaced with George W. Bushes, an icon of American militarism. They're structurally the same game, with the faces swapped and the pictures on the wall changed in the simplest possible inversion of the original game's xenophobic aggression\u2014a blunt \"how do you like it?\"40 Hacks and modifications are made to subvert or comment on the original author's intentions, or to simply correct what the modder feels is an oversight on the author's part, or simply because it's an easy existing infrastructure for creating something new. Let's examine the varied motivations for hacking a game by looking at the most-hacked digital game: Super Mario Bros.\n\nAnother Castle\n\nSuper Mario Bros. was published on a cartridge that plugs into the Nintendo Entertainment System. Hardware has existed for a while, however, that allows the contents of a Nintendo cartridge to be \"dumped\" to a computer and distributed digitally as software. The digital file is called a \"ROM,\" and \"emulator\" programs have been written for many different machines that will run that game software as a Nintendo Entertainment System would. In this way, Super Mario Bros. has made the transition from hard media to a purely digital form, available for modification by anyone with the knowledge and tools.\n\nAnd tools have been readily available for a long time. Super Mario Bros. was written in Assembly, the machine language I mentioned in chapter 2, and reprogramming the game requires working knowledge of Assembly. But programmers cracked the code a long time ago, and plenty of tools currently exist for changing the appearance, rules and levels of Super Mario Bros. without having to look at machine code. When I made my 2008 Super Mario Bros. hack, Mario Goes Underground, I used a tool called SMB Utility41 that made it easy to rearrange Super Mario Bros.'s maps.\n\nSuper Mario Bros. has enjoyed popularity with ROM hackers for a variety of reasons. For one, the large number of people who've played the game made it an early target for programmers to crack, and tools have existed for a while. It's also widely played enough that most people have a sense for how the game works: move left, move right, jump, catch a mushroom to grow big. Most players have an idea of what to expect from Super Mario Bros., so they can pick up a hack easily and be surprised effectively when the hack turns out to be different from the original.\n\nSome hacks simply change the game in a superficial way. Silhouette Bros., by Leon Arnott, changes all the characters and architecture to stark black silhouettes on a solid-color background. The hack demonstrates how recognizable the elements of Super Mario Bros. are and the choices of color are well suited to the tones of different parts of the game. Enigmario by Dr. Floppy replaces the soundtrack of the game with covers of songs by the band Enigma. The somber tones of this new soundtrack give Super Mario Bros. a more introspective feel.\n\nMario vs. Airman is an interesting attempt at intertextuality: the \"Airman\" Mario fights is a villain from Capcom's Mega Man 2, and Mario is required to navigate a level from this game to defeat Airman. Mega Man, as a player character, moves very differently than Mario. Momentum does not affect his motion, he can \"brake\" instantly, and his arm gun makes him able to shoot his opponents, rather than requiring him to jump on them to defeat them. Trying to navigate a level designed for Mega Man as Mario is a challenge that emphasizes the importance of the nuances of motion in platform game design.\n\nA hack like Super Mario Forever exists to challenge the player's technical knowledge of Super Mario Bros.'s rules. Mario starts mid-fall into a bottomless pit, and must instantly maneuver to land on a tiny platform. The game then requires the player to generate enough momentum on that tiny platform to make a long jump to the next, which involves jumping off of and back onto the platform in order to build speed.\n\nSome hacks change a single rule of a familiar game in order to give the game a new experience, similar to the concept of \"house rules\" in non-digital games. Normally, the player of Super Mario Bros. must complete each stage within a limited time, which resets at the stage's end. Nanashi's 900 Time Challenge gives the player a single time limit in which to complete the entire game: nine hundred ticks of the game's internal clock. Completing the challenge requires the player to minimize the wasted time in her play, and to exploit shortcuts and warp zones in order to get through the game in as fast a time as possible.\n\nThen there are hacks like Extra Mario Bros. (or my own hack, Mario Goes Underground), which attempt to tell an entirely new story using the familiar framework of Super Mario Bros. Rather than requiring Mario to proceed from the starting point on the left to the castle on the right, Extra Mario Bros. contains a big world for the player to explore in multiple directions, using pipes to travel vertically from area to area. It's essentially Super Mario Bros. through the lens of contemporary game design, which is why it's interesting as a direct modification of Super Mario Bros.: the games that informed Extra Mario Bros. are all descended from Super Mario Bros.\n\nThen there's Super Daisy Land, which changes the star of the Nintendo Game Boy game Super Mario Land from Mario to Daisy, a captured princess who normally waits at the end of the game for rescue. But that's not exactly a hack of Super Mario Bros., and there's a richer subject for gender correction in game mods.\n\nBobs Whose First Name is Betty\n\nAnne-Marie Schleiner, artist and writer, maintains Mutation  \n.fem, a gallery of modifications of first-person shooting games: games where the player's character is personified as the gun she holds in front of her, like the earlier-mentioned Duke Nukem 3D. Each mod changes the gender of the characters in those games. The patches hosted on Mutation.fem42 change male characters to female for a variety of reasons: either for the purposes of gender play, to correct gender bias by the games' authors, or to subvert the aggressive masculinity of this school of game creation.\n\nThe first mod in Mutation.fem that I played was Lynn Forest's FemDOOM. The original DOOM, a 1993 release from id Software, was one of the earliest games to popularize the first-person shooter, and has given tropes to that genre that are still imitated today. The game is viewed from a first person perspective, and the protagonist is given no name or background beyond being a space marine assigned to a tour of duty on Mars. Only the tiny, emoting face of a man in the status bar and the occasional baritone grunts when the player is shot gender the protagonist as male, and reinforce the tacit assumption that the player of a game of aggression must be male.\n\nLynn Forest, a fan and player of DOOM, was frustrated with the implication that she was playing a game for men. So she drew femme faces to replace the male faces that appear in the status bar, and put her patch online for other players to download. A later patch, \"FEMDMSND.WAD,\" credited to \"Amanda, Ivor B. and Rob Lord,\" replaces the masculine sound effects with feminine ones of equal intensity.\n\nGender is even more of an oversight in id Software's later Quake, which puts an even greater emphasis on competitive play. In Quake, one sees other players not as masked figures, but as hulks with a single male face. Players modded the game to introduce feminine faces and bodies. Schleiner notes43 that later versions of Quake acknowledge players' desire for female avatars by including female models. She also hosts \"skins\" worn by PMS Clan, the Psychotic Men Slayers,44 a band of women players who play competitively together in the game Quake 2.\n\nMutation.fem contains patches that paint a mustache on Tomb Raider's protagonist Lara Croft and change Bungie's Marathon from a game about firing guns in dark, angular space stations to a game about fighting with egg flippers and dish towels in a checkerboard-tiled kitchen. Loren Petrich's \"Tina-Bob\" patch replaces the generic jumpsuited men\u2014the \"Bobs\"\u2014that solely populate Marathon's space station with identically jumpsuited women.\n\nThe Hack as Sampling\n\nThe repurposing of commercial game assets can be compared to sampling in music: using part of an existing song as an instrument in your own piece of music. Hip-hop artists often rap their own material over music sampled from another source, using the found music as a background for their own words.\n\nHyperBound was created in 2009 by Michael Iantorno as part of his thesis project at Ryerson University.45 Iantorno's game is a hack of EarthBound, a Super Nintendo Entertainment System game developed by APE and the Japanese writer Shigesato Itoi and released in 1994. EarthBound is a digital role-playing game informed by Yuji Horii's Dragon Quest, but instead of telling the story of a warrior who battles wolfmen and dragons in a fantasy world, EarthBound is set in a world vaguely resembling a Japanese vision of contemporary America. The protagonist is a young boy in a baseball cap who fights aliens and renegade animals using an old baseball bat and his newly discovered psychic abilities.\n\nHyperBound takes its name from \"hypertext,\" text that's arranged in a nonlinear structure. (This book is a text: it's arranged to be read from start to finish, one page to the next. A website, where you might click on a word to \"link\" to a page about that subject, is hypertext.) What better model for the nonlinear exploration of text than the space of a digital game, where the player moves around the world by moving her character across the screen, encountering characters, and listening to what they have to say? That's the part of the design of EarthBound that HyperBound has lifted. What it's rejected is the fighting. The hack is purely about exploring the world and discovering the text, an original script written by Iantorno and his brother.\n\nIn EarthBound, the protagonist typically wears a baseball cap, a striped shirt, and a backpack. There's a brief scene at the beginning of the game, though, where he appears in pajamas with ruffled bed-hair. That means that there were animated sprites of the bed-headed protagonist, seen from all angles, just waiting in the game for Iantorno to use. HyperBound uses those sprites for its protagonist, to give him an identity apart from that of EarthBound's protagonist and to emphasize his confused state: the protagonist is amnesiac, and the information he is trying to recover is knowledge of his own identity.\n\nThroughout the hack, Iantorno repurposes assets from EarthBound to fit his new story. The bearded, sunglasses-wearing criminal the player encounters in EarthBound becomes the radio DJ whose show the protagonist of HyperBound used to call in to before he lost his memory. The boarding school that appears in EarthBound, with its classrooms and lockers, becomes the university that HyperBound's protagonist attended, where he meets former teachers and finds valuable information on his previous life. All of EarthBound's graphics are sampled and given new purposes in the hacked game.\n\nThe other assets Iantorno samples are less obvious, though just as crucial: the ways in which the player presses controller buttons to move the protagonist around a world, to engage characters who wander around of their own accord in conversation, and to advance their dialogue text on the screen. These are all things that are non-trivial to program, and that have been programmed for Iantorno by APE. To present his nonlinear story, he's taken advantage of an existing infrastructure for allowing the player to navigate a world of characters who can be spoken to.\n\nIf you can create a script but you can't create animated characters, scenery, or code to take player input and move a character around a large world that scrolls in eight directions, why not use those existing assets to present your original script? The work has already been done and the resources already exist to be sampled.\n\nChanging the Script\n\nThere's another obvious reason to change the script of a game: to translate it to a new language. Publishers often neglect to translate and publish a game in a new language because of the cost: if they think a game won't sell enough in a part of the world to justify paying people to rewrite the script, to manufacture cartridges for different game-playing hardware, and to market the game in another country, they won't do it. But dumped roms, which are purely digital, don't require the expense of physical publishing, and because they're distributed freely, don't depend so heavily on marketing to make up that expense. And thus, people who read languages neglected by videogame publishers and care enough about games from other parts of the world will invest their own time and effort into the unpaid translation of games.\n\nThe work of translating a game involves more than just translating the script\u2014which is often a lot of work, given that the dynamic nature of games usually means lots of scenes and incidental text to translate. The new script also needs to be inserted into the original game. In formats where space is expensive, such as Nintendo game cartridges for example, programmers use a lot of clever tricks to store text as efficiently as possible. Cracking those codes can be tricky. And if the original game was written in Japanese, the hacker also has to insert a new font. The way the game displays text might not be suited for English letters, so screen space is usually at a premium, too. Japanese is a compact, ideographic language, and a word that takes up two Japanese characters on screen could be an eight-letter word in English, or an idiom that requires translation as a full sentence. How do you translate the word so it fits? All game text tends to be presented in a limited part of the screen (a window that takes up the bottom half of the screen, for example, so the player can see the characters who are speaking on the top half), so all word choice is informed by just how much space is available to display those words\u2014space that was chosen based on the structure of an entirely different language!\n\nThe Japanese Game Boy Advance game Mother 3, a sequel to EarthBound, was translated into English over the course of two years by a two-man team who refer to themselves as \"Jeffman\" and \"Tomato.\"46 The script for the game was written by the Japanese writer and journalist Shigesato Itoi. Nintendo's American branch decided that translating the game would be too much work, and too expensive a project, given the limited audience they expected the game to have. Nine years before, Nintendo of America put a lot of money into translating Mother 2 into EarthBound and manufacturing copies, only to have their marketing department completely mishandle it. Physical distribution is a big investment, after all: Nintendo of America was unwilling to make the investment a second time. American players who had encountered and liked EarthBound, though, were eager to play the sequel and were upset that it wasn't to be published in English. Tomato, who works as a professional Japanese-to-English translator, played the Japanese game and decided that English-reading players deserved to experience the game. He undertook the translation of the game, and Jeffman did most of the coding to get around the memory limits of the original Japanese game: a two-year project, undertaken for free. The English version of Mother 3 was released in July 2009. You can go on the official translation site's forums to read how many players cried during the ending47 (which is, incidentally, Stephen Spielberg's metric for when games will become art,48 not that tears are anywhere close to the only metric for judging the value of an experience).\n\nBut don't be misled into thinking that the majority of translation work is in translating Japanese games to English. As of this writing, people are working to translate Mother 3 into Spanish and Latin American Spanish, Portuguese and Brazilian Portuguese, French, German, Italian, Dutch, and Malay. Whenever I visit the ROM hacking database at romhacking.net, I see translations of English games into German, Polish, Russian, Korean. Hackers are engaged in correcting the oversights of profit-oriented thinking and making the games they care about available in their native tongues.\n\nMachinima\n\nThe product of \"machinima\"\u2014from machine and cinema\u2014isn't games, but the source material is. It's the same kind of sampling I've described: machinima uses the resources and infrastructure of commercial games as the basis for creating animated movies. All the resources are there in most 3-D games: the ability to move a camera through a world full of 3-D objects, and to move and animate those objects. Game hacking provides an immediate avenue to 3-D animation, one that's far more accessible than dedicated 3-D modeling (with a specialized\u2014and costly\u2014modeling program like Maya or 3D Studio Max) because there are already scripts in place to move things around and to operate a camera.\n\nThe origins of the community surrounding machinima can be traced to that of the game Quake, a fully 3-D first-person shooting game that offers players the ability to record, distribute, and play back \"demo\" recordings of their games. Players began to manipulate and position the camera during these playbacks, introducing cinematography to the demos. Soon they were directing story scenes to frame their demos.\n\nMost machinima stars the characters of the games they sample, because that's what's available: usually armored soldiers who crack jokes for the benefit of players familiar with the game. The 1996 Diary of a Camper by Matthew Van Sickler, Heath Brown, and company,49 for example, follows a band of Quake characters as they deal with an opponent who is hiding and picking off their comrades one-by-one. (And who is ultimately revealed to be John Romero, one of the founders of id Software.) Diary of a Camper is widely remembered as the first game demo interested purely in storytelling, rather than presenting a sample of play footage.\n\nBut the subject matter of machinima has become increasingly diverse. The 2003 machinima Anna, produced by Katherine Anna Kang's (wife of Quake programmer John Carmack) Fountainhead Entertainment, follows the life of a single flower on a forest floor.50 Anna bears little resemblance to Quake at all, set in the woods with a cast of plants and animals.\n\nAs more tools become available, machinima looks less and less like videogames. A few years ago I encountered a movie of Sherwood Anderson's short story \"The Dumb Man\" filmed in Second Life,51 a game that gives players the tools to construct anything they like, be it movie sets or actors or other games.\n\nWhile machinima is not games, it's demonstrative of the ways that game tools can be used to create things radically different than the originals: including new games altogether.\n\nSomething Borrowed\n\nFew commercial games, these days, are made entirely of original resources. The creators of the Quake and Unreal \"engines,\" the rules and code that drive the games, license their engines to other developers so that they can save time and work by modifying an existing framework rather than building an entirely new one. The id Tech 3 engine\u2014the engine that runs the third Quake game\u2014was used to create games like American McGee's Alice, Call of Duty, Star Trek: Voyager\u2014Elite Force, and Star Wars Jedi Knight II: Jedi Outcast.\n\nThe publishers of these games paid for licenses allowing them to sell and distribute their modifications, but plenty of players, using their knowledge of how their favorite games work, changed the games into something new and distributed their modifications for free.\n\nBrandon Chung's Citizen Abel: Gravity Bone52 was built in a modified Quake 2 engine, but the finished game bears little resemblance to a game about space marines blowing up aliens. Gravity Bone is a spy pastiche that opens with the player riding a fancy gated elevator into a masked costume party. Once there she must make contact with a fellow agent. This sets forth a rapidly accelerating chain of events that lead to an unexpected outcome.\n\nWhat Gravity Bone takes from Quake is logic for moving a player around a three-dimensional world, looking around with a first-person camera, and collecting and manipulating items. Everything else\u2014the game's appearance and presentation, the locations and scenario, the player's goals and the events they trigger\u2014is original, designed and inserted by Brandon Chung. Quake, here, is just a vehicle for the delivery of Chung's original story.\n\nOr look at Robert Yang's Radiator series,53 which is built in the Half-Life 2 engine. The second episode of the series, Handle With Care, involves the player visiting a marriage counselor. What Half-Life 2 does well is allow the player to manipulate and move objects using a rudimentary physics simulation: in Handle With Care, the protagonist, Jason, withdraws to a \"Repression Facility\" while his husband speaks with the counselor. The Repression Facility looks like the kind of grimy steel structure common to Half-Life 2. In the facility, the player carries crates containing repressed memories (if there's one thing Half-Life 2 does well, it's letting the player carry crates around). If they're put on the facility's shelves, they're locked away forever; if they're broken, Jason revisits the memory he was trying to repress. The therapy session, seen through a monitor, progresses depending on whether Jason confronts or seals away his memories.\n\nBoth of these games look significantly different than the games they're based on. Nevertheless, we can identify rules they've borrowed from their mother games. It's apparent why it's easier to change existing games into new games rather than creating them from scratch: particularly when it comes to 3-D games, solitary creators often don't have the resources that the programmers of Half-Life 2 had. It's unlikely that either game could've existed without such sampling\u2014it's far less likely that Valve would make a game about the roles individual memories and pasts play in negotiating modern queer relationships than a Half-Life 2 level involving building a wall of crates to stop drone planes from killing you.\n\nScripting cameras and collision, as well as creating 3-D architecture and a way of storing it conveniently in computer memory, are all big projects. The Valve Hammer Editor, which edits Half-Life 2 levels, gives a creator immediate access to all the design tools a team of paid programmers put into place. It also gives her access to the resources Half-Life's art team created, which would otherwise be another huge investment of time, effort, and skill. Editors like Hammer allow creators to create without having to build an infrastructure to manage their creations.\n\nNew Worlds\n\nMany game designers have anticipated their players' desires and created editors to accompany their games. The first level editor I ever encountered was for the Nintendo game Excitebike. Excitebike is a stunt bike race. The included EDIT MODE allows players to mix and match the obstacles from the game\u2014ramps, bumps, gravel pits, and boost pads\u2014to create new stunt tracks to race on. When I was little, I spent a lot of time with the instruction book open in my lap to the list of pieces, moving my racer through the track and rearranging the obstacles. The appeal of an editor like this to a child who's creative but lacks experience is that everything\u2014the rules, the code, all of the art and sounds\u2014are provided. The only thing I had to worry about, as a kid, was the track itself.\n\nWhen we were discussing shareware, I mentioned Tim Sweeney, who founded Epic MegaGames\u2014which later became Epic Games, the publisher of the Unreal engine used by many commercial and independent designers to build 3-D worlds. The first game Tim Sweeney sold, as \"Potomac Computer Systems,\" was released in 1991 and is called ZZT. (He picked this name so it would always be the very last game listed on alphabetically ordered shareware CDs.) The shareware part of ZZT was a series of four self-contained adventures, the first of which came with the game and the other three of which had to be purchased. (They were later released for free.) The free part of ZZT, however, was the editor that Sweeney used to make those adventures, and the means to play \"worlds\" created with it.\n\nZZT is a text-mode game: it takes advantage of DOS computers' native ability to display text. In addition to letters and numbers, DOS has a special extended sheet of characters for drawing basic pictures: smiley faces, playing card suits, Greek letters, and a variety of borders, lines, and patterns. ZZT co-opts these: the player is a smiley face, Greek characters such as pi and omega are monsters who roam around, the diamond from the playing card suits is a gem that the player can collect and use to buy items. This set of 255 characters comprises everything that any ZZT world will ever contain.\n\nAnd that's incredibly liberating. It means that ZZT is self-contained: there's no need to go outside the editor to find another tool to draw and animate graphics to import into the game. Authors get 255 characters in sixteen colors. And creative authors can do a lot with those: in that set there are patterns of varying density and lines of varying thickness, and I've seen some pretty impressive environments and portraits drawn with those text characters.\n\nSound and music are similarly self-contained: ZZT uses the PC speaker, which contains a range of simple notes and instruments. ZZT has a simple scripting system for playing PC speaker tunes, which again obviates the need to go outside the world editor for anything.\n\nBut oh, yes: let's talk about scripting. ZZT has a nice library of game pieces, which appear in Tim Sweeney's shareware ZZT worlds. There are pushable boulders, \"rotators\" that move objects around and can form conveyor belts if a series of them is placed in a line, stock monsters, weapons, and power-ups, and several kinds of walls (invisible walls, secret passages, walls that can be shot and destroyed). But the most interesting game piece is the Object, a programmable creature that can take on any appearance. Each Object contains a script written in a language called ZZT-OOP (ZZT Object-Oriented Programming), a simple programming language.\n\nHere's an example of ZZT-OOP:\n\nThat tells an Object to move north (or up) six times, then shoot a bullet east, then display a haughty message on the screen. That's pretty simple, something someone who's never coded before should be able to understand. Though it's simple, ZZT-OOP is also surprisingly robust: each Object has its own script, and can receive messages from the player and from each other. Here's a more complicated example of two Objects interacting:\n\nWhen the vase is shot (the colon indicates a message the Object can receive) it sends a message to the object named Collector (the @ indicates an Object's name), which causes the Collector to display a message and briefly animate (#char is the \"change character command\"; character 1 is a hollow smiley face and character 2 is a filled smiley face. The direction \"i\" means \"idle\"; it tells the object to wait a moment between flipping its character).\n\nAuthors have taken advantage of ZZT-OOP to do things author Tim Sweeney never envisioned. One can create a Super Mario Bros.\u2013style jumping game by creating button Objects that, when touched by the player, send messages causing a Super Mario Object to move left or right or jump, and that falls when there's nothing beneath it. Recently, someone built a machine that graphs sine waves in ZZT. Creative authors have bent the tools they were provided with to entirely different ends than they were intended for.\n\nWhat's interesting about ZZT is not only the robustness and versatility of ZZT-OOP, but the simplicity of the game's overall presentation. Because all of the graphics and sounds that a ZZT world can produce are already inside it, it frees authors from having to worry about asset creation. They can just design with the tools they have in front of them. This makes it an ideal introduction to scripting and design, and I know a handful of game designers who got their start making ZZT worlds.\n\nHouse Rules\n\nThere's a concept in non-digital games called \"house rules.\" Because players keep the rules in board and card and physical games, and because all the players have to agree on what rules they'll play by, it's easy for the players to change the rules or invent new rules to suit themselves. Maybe they feel as though their modification will correct an oversight on the part of the designer: part of the game may not quite work as it's presented. Or perhaps they've thought of a variation on the rules that's more interesting to them as veterans of the game. Maybe they need the game to be easier or harder, or to change it so it better suits the circumstances in which they're playing it: a game of Tag where the goals change to suit the space that's available, for example.\n\nIn digital games, the rules are kept by the computer and thus are somewhat hidden from the player. Adding house rules to a digital game seems more far-fetched. There are plenty of ways, though, to change games: changing the rules, inventing new rules, inventing new scenarios and whole new games out of the same content. Telling new stories!\n\nWhat's really valuable about hacking and modifying games is the realization that there are ways of interacting with games other than just playing them: roles beyond consumer. Inventing rules is, after all, inventing games. The knowledge that games can be changed and remixed is the knowledge that games can be created: it's a small conceptual leap from the one to the other. While game creation tends to be more interdisciplinary\u2014using the Valve Hammer Editor allows the creator access to all of the 3-D models, textures, sounds, characters, and such from Half-Life, letting her focus on design\u2014there are game creation tools now that are as easy to approach as the typical level editor.\n\nI said that the first level editor I encountered was Excitebike's. I've played with plenty through the years: ZZT and its sort-of successor MegaZeux, Dezaemon 3D (a spaceship shooter kit for the Super Famicom), Knytt Stories (a jumping-and-climbing story creator), Tombs of ASCIIroth (a text-mode puzzle game kit), Ragdoll Masters (rubber dolls fighting each other in the air), The Elder Scrolls IV: Oblivion (a fantasy computer role-playing game). I've used whatever tools I could get my hands on to tell stories.\n\nYou can, too. And once you've realized that videogames\u2014even if they're made by big corporations with teams of hundreds and budgets of millions\u2014are mutable, and can be reinvented by a single person, you can start to imagine what's possible when you have the means to create games that are entirely your own.\n\nFootnotes\n\n40 More information on the history of all three games can be found at [http:\/\/www.gameology.org\/reviews\/quest_for_bush_quest_for_saddam_  \ncontent_vs_context](http:\/\/www.gameology.org\/reviews\/quest_for_bush_quest_for_saddam_ content_vs_context).\n\n41 http:\/\/www.zophar.net\/utilities\/neslevel\/smb-utility.html.\n\n42 http:\/\/www.opensorcery.net\/mutation.\n\n43 See \"The Female Skin Pac for Quake\": http:\/\/www.opensorcery.net\/mutation\/patches.html.\n\n44 Newman, James. Videogames, 57. The PMS Clan still exists, but has apparently been co-opted by corporate interests and has been rebranded Pandora's Mighty Soldiers.\n\n45 http:\/\/hyperbound.net.\n\n46 http:\/\/mother3.fobby.net\/or..\n\n47 Though you probably shouldn't. Spoilers.\n\n48 Lev Grossman, \"The Art of the Virtual,\" Time, November 8, 2004, http:\/\/www.time.com\/time\/magazine\/article\/0,9171,995582,00.html.\n\n49 http:\/\/www.youtube.com\/watch?v=uSGZOuD3kCU.\n\n50 http:\/\/www.youtube.com\/watch?v=bKEr5RRKoO4.\n\n51 View it online at http:\/\/vimeo.com\/609147.\n\n52 http:\/\/www.blendogames.com.\n\n53 http:\/\/www.radiator.debacle.us.\nChapter Five\n\nThe New Videogame\n\nThe La La Land series was created by Matt Aldridge under the alias \"The Anemic.\" It's a series of five games, each supposedly based on one of the author's dreams. The games are populated with recurring scrawl-doodle characters whose features are abstract enough to be sinister. The protagonist, Biggt (pronounced like \"bigot\"), has a smile that's Glasgow-wide and a pair of eyes that don't line up. The music is often seemingly incongruous: the climax of La La Land 5 happens with the anti-evolution song \"I'm No Kin to the Monkey\" playing in the background. And the arbitrary goals the games give the player are almost always revealed to be against the protagonist's best interest.\n\nThe first one I stumbled upon was La La Land 2. In this game, Biggt is confronted by a towering snaggletooth fish head, baby pink and jittering, which tells him: \"it's ok to steal from the rich cos they havelots [sic] to spare!\" A counter\u2014which in most games would be nailed to a fixed point on the screen, but in this game jiggers and jumps from point to point\u2014reads \"$0.00.\"\n\nBiggt travels left to a screen where crow-like \"nobles\" with fish heads sit at opposite corners of a long table. A figure that resembles Biggt is on a pedestal, face in his hands. As he cries, some of the tears pass through pipes and fill the nobles' wineglasses. At the far left is a hay-colored, shapeless pile that represents gold. Biggt scoops up an armful, making the pile smaller, and carries it back to the initial fish head, adding money to the counter. The fish head says, \"Oh thanx biggt i want MORE.\"\n\nThe player runs back and forth between the fish head and the nobles, redistributing wealth. When the player has stolen enough money, the fish head rises up to reveal a gold medallion hanging from one of its bones. \"Look at the shini JEWELS I bougt thanks. a lot.\" Returning to the nobles, the player finds them dead at their table. \"without money how were, the nobles to surivve?\" The game then closes.\n\nPersonal and abstract, The Anemic's La La Land games were the first games of their kind that I encountered, and they would never have been made if not for Game Maker.\n\nI'm not sure when La La Land 2 was created, but in 2003, when I was seeking out the other games in the series, I discovered a game called Seiklus by an admirer of the La La Land games. The author, cly5m, had made worlds for Tim Sweeney's ZZT that were as dreamlike as La La Land, if a bit larger in scope. (One, Kudzu, involves making a collect call to heaven at a melting payphone and visiting a hall filled with imprisoned invisible people.) American by birth, cly5m had visited the country of Estonia on a religious mission, then returned to America and discovered Game Maker. Seiklus is the Estonian word for \"adventure.\"\n\nSeiklus begins with a boy and girl watching the stars on a cliff. A meteor hits and causes the boy to fall into a valley below. The game ends when he has found his way back to the girl. To return, the player has to collect colored tokens, which are found in a giant hollow tree, on a quiet, snow-covered landscape, in a dark land full of bones, and inside the stomach of an enormous creature. All of these scenes were hand-drawn by cly5m himself; the thick line art style is as unmistakably his as the collection of pastoral-alien dream scenes.\n\nTo the West\n\nFrom my current perspective, I can say that what I always wanted to do was make videogames, but as a child, making videogames was something mystical that I never thought I'd be capable of it. So I funneled my creativity into other channels. I was better at writing than drawing, so I was a creative writing major in college.\n\nAfter playing Seiklus, I figured that if cly5m could make the transition from making add-ons for ZZT to making self-contained games in Game Maker, then I could do the same.\n\nI had taught myself to program a little, particularly in a language called Blitz Basic, in which I had recreated a few games that I knew. I had made a board game\u2013like version of PAC-MAN where the player and the ghosts took turns moving rather than moving at the same time. Making objects move and react to each other all at once was something that was beyond me, and objects' placement on the screen had to be hard-coded, making it difficult to experiment with actual design. Game Maker was refreshingly similar to ZZT to me: every individual object had its own instructions to tell it what to do and how to react to the other objects. I could draw every object, give it instructions, put it wherever I wanted to on the screen, and let the program play out and resolve interactions based on how I'd told the objects to behave.\n\nMy first Game Maker game was called Jaywalker. In Jaywalker, the player controls a disembodied head named Marjorie with a HotHead Paisan\u2013style crown of orange hair and a stud in her nose (I think I made the game around the time I first got my nose pierced.) Marjorie is fiercely about the rights of pedestrians, and is committed to destroying as many cars as possible. The player moves her back and forth across a busy intersection, causing cars to swerve around her and crash into each other. Cars will crash into cars that have already been wrecked, creating huge pile-ups and causing Marjorie to stick out her tongue triumphantly. If she's ever hit by a car, the game is over.\n\nI dropped out of college some time after creating Jaywalker. When I quit, I was in my fourth year of school with no end in sight. I didn't see myself as a writer, I hated school, and I thought (and think) that \"higher education\" is bullshit.\n\nI spent another year in New York, where I was born. I made some more games during that time. Terrified I'd be stuck in the same spot for the rest of my life, I decided I needed a reason to move somewhere far away and a way to earn rent, a trade that I could learn while I was there. The thought of earning my living by making games was exciting: why not go to a games college in another part of the country, perfect my craft, and find a way to pay my rent by making videogames?\n\nSpecialized videogame schools had been around for a while at this time. I had known of DigiPen, in Seattle, since I was a teen, and had recently heard of Full Sail, a movie school in Florida, but both of those seemed to me like game programs tacked on to visual arts schools. I chose the Guildhall at Southern Methodist University because it was the only school I discovered that had a level design focus. (A Game Maker game I made after quitting creative writing school, Invader, which was about an alien from the game Space Invaders crash-landing on a weird planet and having to find her way off, was part of my portfolio.)\n\nThe Guildhall was in the middle of Plano, Texas. Plano, Texas, is brown and not much else. They have a Frito-Lay factory, parking lots, and a videogame school. At the time, I kept a strict vegan diet and didn't drive. There was nothing to eat and nowhere to go.\n\nBut the latter didn't matter; when you were at the Guildhall you had no life outside the Guildhall. I remember the first day of orientation, sitting in a lecture hall with my future classmates and the spouses they'd brought with them to this wasted brown land. One of the other level design students had his wife and their year-old child with him. \"Give her a kiss and say good-bye,\" the director of the school told him in front of the assembly. \"You're not going to see her for two years.\"\n\nI was in Plano, Texas, for six months.\n\nYou're at school from nine to five. You stay after and do your work with the teams they've assigned you to. Late at night you drag yourself home and do your actual homework. Maybe you get a few hours of sleep. The idea behind the school is that you're always in what the Big Games Industry calls \"crunch time\": unpaid overtime. Your masters want the game done by Christmas, so you don't leave the office until it's done. This is why people in the industry aren't healthy; this is why they burn out and quit games within a few years. This is why you miss the second year of your daughter's life. This is their scheme: you put up with crunch time all the time while you're in school, so when you work for a big publisher\u2014or, rather, a studio contracted by a big publisher\u2014you won't complain about being told you can't see your daughter until the game's done. The Guildhall boasts an over 90 percent employment rate, and it's true: they will get you a job in the games industry. That's because they will make you into exactly the kind of worker the games industry wants. It's that kind of school.\n\nAnd it works; that's the horrifying thing. My classmates were all self-identified gamers and game fans and were willing to put up with anything in order to live their dream of making videogames. That's the carrot the industry dangles, and it's what we take away from the industry when we create a form to which anyone can contribute. As long as the industry is allowed to continue acting as the gatekeeper to game creation, people will continue to accept the ways in which the industry tramples the lives and well-being of the creative people who make games, rather than challenging the insane level of control that publishers ask over developers' lives.\n\nNeedless to say, I was not at the school long. I butted heads with lecturers too many times, I asked too many questions, and my Oblivion mods were too experimental (I didn't include messages that popped up and told the player she'd achieved her goal because I didn't think the player was too stupid to realize that.) Eventually I was pulled into the director's office and asked to leave. \"If your unit is marching to Fort Worth, you don't ask 'what if we go to Austin instead.'\" That's a paraphrase of what the school's director told me.54 I've had a while to think about this, and I've since decided that if you know there's an army waiting to ambush you at Fort Worth, it's fatally irresponsible not to try to dissuade your comrades from marching there blind.\n\nI said that I was in Texas for six months. Each semester at the Guildhall was just over two months long; they canned me the first day of the third semester. I spent the last month or so getting my stuff moved back to New York and making a game called Calamity Annie.\n\nCalamity Strikes\n\nStranded in Plano, Texas, with everyone I knew occupied with school all of the day and most of the night, working on Calamity Annie is what kept me from going stir-crazy until I flew back to New York.\n\nThe protagonist and namesake of Calamity Annie is a brash young dyke who rolls into the Old West (from the New West, naturally) with her breasts bound and a pistol in her holster. She tears across the countryside, dropping hombres in one-on-one shootouts. Along the way she meets a lady named Valentine with a different past but the same breed of loneliness. If the player's sharp enough, Annie and her Valentine ride off into the sunset together at the end.\n\nThe whole game is played with the mouse, by pointing to aim and clicking to shoot. A duel consists of \"holstering\" your cursor at the bottom of the screen, waiting for the call of \"DRAW!\" and quickly aiming and firing before your opponent can do the same. Interactions with Annie's love interest use the same controls. The first time you meet her, Valentine will ask for a light for her cigarette. Aim at the cigarette, click, and PEW! Flirtation has begun.\n\nIt's a hard game\u2014the quick-draw contests get quicker and quicker, and eventually the player has just milliseconds to react. But even when the player gets GAME OVER and Annie spills some blood, Valentine patches her up, and their relationship goes on. What's between them outlasts any individual firefight. Every play of the game isn't self-contained but is\u2014in a small way\u2014fitted into the ongoing tale of love and connection, which the gunfights are just a means to developing.\n\nCalamity Annie is a lot of things to me. It's about being an angry young woman in a hostile land (that just so happens to be Texas), trying to prove your worth. I certainly was trying to do that after they kicked me out of the Guildhall. I was making a game, and I was doing it my way: I built every piece of the game myself. I drew all the pictures; I wrote all the music; I scripted all the events in Game Maker. If the Guildhall taught me anything, it was that videogames needed to be saved from the industry, that a creative form deserves better than an assembly line production process.\n\nBut the game is also about love, and how passion finds love. I had started seeing a loud-ass submissive named Daphny Drucilla Delight David shortly before starting at the Guildhall, and the workload had kept me away from her except for one week of vacation between each semester. The time we got to spend with each other while I was at school was mostly limited to me calling her and crying from the stress of having to split my energy between standing up to teachers, finishing my work, and doing good work, when all I wanted to do was make games.\n\nI'm writing this, three years later, from the apartment where I live with her in California. The theme of Calamity Annie is that being driven will drive you to love, and that passionate people are attracted to passion. Annie was one of the first in a series of games that led to my being able to pay for a home and food by making games, which is what I went to Texas to seek. But it didn't come from the Guildhall. It came from me, from my stubbornness, and the stubborn friends who helped me.\n\nLater games have made me more money\u2014I asked players who wanted to support Calamity Annie for a donation of at least a dollar, in exchange for which they got a bunch of secret bad guys to shoot (I made a few hundred total)\u2014but I still think of Annie as my most important game. It's everything I realized I wanted games to be. It's personal (Annie has my name, after all) and it has a clear, unmuddied mechanical idea that translates perfectly to a storytelling idea. I want games to tell stories, and Calamity Annie tells my story, or at least the story of my stay in Texas.\n\nAuthors\n\nMy friend Lamar Williams is working on a documentary about videogame creators called You Meet the Nicest People Making Videogames. In one of the project's trailers, he points out that the idea of digital games as the product of teams is a myth: many of our most important games, and the ones that have been the most widely duplicated, are almost entirely the projects of impassioned individuals.55 The examples that appear in the trailer are: Another World, a game about finding friendship during a struggle for survival in an alien world, created by Eric Chahi (who even painted the cover art for the game himself!); Karateka, a kung fu game with strong visual storytelling by Prince of Persia creator Jordan Mechner; Defender, a seminal arcade game by Eugene Jarvis and Larry DeMar; and Berzerk, almost as seminal an arcade game by Alan McNeil. He goes on to show contemporary games by individual authors, some of them made in Game Maker.\n\nThere's nothing unnatural about a digital game by an individual creator (or a pair of creators). It is, in fact, much harder to keep the idea behind a game coherent when the designer is managing a team of many people who are each working on one aspect of the game separately. That's part of the reason why contemporary big-budget games have so much clutter and so few strong ideas. The games are all over the place because the creators were all over the place. It's hard to have a strong singular vision when the process of creation is spread so thin.\n\nDigital games contain video, audio, animation, design, and rules. You can parcel out these roles, but the closer they remain to each other, the more cohesive the work you create. If I'm the designer and I'm also drawing the spaceship that appears in the game, I know exactly how I intend to use that spaceship in terms of play, what its place is in the larger story, and what its appearance should express. I have a vision, in other words.\n\nBooks are written by single authors or by author-editor teams. Visual art is typically made by an individual artist. It makes sense for creators to be close to their work and to own their work completely, and that's something that the big teams that big-budget games demand can't have. When an individual or pair is solely responsible for a work you can watch an individual style develop: you can trace themes, both mechanical and otherwise, across a creator's work. (The Anemic's La La Land games, though each game is very different from the rest, have a strong singular style that persists throughout the series.)\n\nAnd being able (or learning to) identify the individual style, and growth, of individual authors leads to better criticism and a critical understanding of games. Not to mention, like I said, more personal games, more relevant games, more games with something to say. I want a world where everyone is capable of sitting down at a computer and making a game by herself. This is not to say that all games need to be made that way, but as a paradigm, I think the individual author has more to offer us than the team, especially at a time when videogames are so seemingly creatively bankrupt.\n\nDeath from Overwork\n\nI first became aware of Jesse Venbrux, a game designer from the Netherlands, in 2008 because of his game Execution. When the player starts this game, she sees, through the sight of a gun, a prisoner tied to a stake. If the player shoots the prisoner, the prisoner dies, and the player is told, \"You lose.\" If the player starts the game a second time to try to avoid losing, she again sees the same prisoner, still riddled with bullet holes.\n\nExecution (a direct inspiration for Calamity Annie) points out something about causality in games: in particular, it points out how, as players, we expect to be allowed to undo and redo our choices at whim. The game takes just a few minutes to make this point: it's the kind of game that could never be a commercial product, but nevertheless has something essential to tell us.\n\nExecution would probably never have been made without Game Maker. Jesse Venbrux might never have dug into game creation were it not for Game Maker. Though he's more recently gotten involved in Flash game creation and released games for the iPhone and its permutations, his website lists Game Maker\u2013created games going as far back as 2004.56 In Venbrux's body of work we can clearly see ongoing themes, both mechanical and contextual. His 2007 Frozzd, which was created for a Game Maker theme competition (entrants were asked to design games with a \"winter\" theme), involves navigating free-floating, irregularly shaped planets that all have their own gravity. That is, the player can walk all the way around the surface of one and arrive where she started, before jumping off and flying to another one. His 2010 game Maru features the same kind of irregular planet navigation, as does his most recent game (as of this writing), They Need To Be Fed for the iPhone.\n\nHis 2008 game Deaths remembers every time any player playing the game anywhere in the world meets death. Players will see the corpses of the last fifty players to have played the game, indicating particularly difficult areas and hidden traps. There is one screen in which players must continually die in order to build a bridge of corpses for future players to reach a high plateau (this is where the earlier-mentioned We the Giants gets its seed). His 2008 and 2009 diptych, You Made It and You Probably Won't Make It, also remember the player's death, painting patches of blood wherever she encounters a deadly spike. In the former game, this is especially important because the blood will paint over the visual information from the previous screen, showing what the terrain actually looks like. (In this game, nothing the player sees is ever erased from the screen. It is only overwritten by new information.)\n\nThe theme of protagonist death as narrative progress is an important one in Venbrux's work. His Karoshi series of games (Karoshi can be translated from Japanese as \"death from overwork\") require the player to guide the protagonist, a suit-and-tied salary man, to his death in a series of increasingly convoluted screens. Sometimes, in a similar vein to Execution, the levels of Karoshi force the player to make use of something she thought was external to the game in order to proceed. For example, there's a stage in Karoshi 2.0 called \"What's On the Menu Today?\" In this stage, the player must quit to the game menu\u2014the screen where the player selects what stage she wants to play. The protagonist follows and falls to his death. Another stage requires the player to put a CD in her computer so that a boom box on the screen can play loudly enough to shunt a safe onto the protagonist.\n\nSure enough, Venbrux's iPhone game They Need To Be Fed requires the player to feed the protagonist to a monster at the end of every stage in order to complete the stage. (His 2008 game Pazzon requires the player to lead characters to the dinner tables of other game inhabitants.) It also features some of Venbrux's recurring \"characters,\" such as the guided missile launchers that appear in his 2009 Focus, which in turn features the protagonist of You Made It.\n\nWe can notice and trace details and themes like these from game to game because each of these games is almost entirely the work of a sole, identifiable author. It makes us better, more literate critics of games to be able to see and discuss the progress of Jesse Venbrux as a designer. And it makes games more richly personal if we can play them in the context of the ongoing work and growth of a knowable author. This is a more useful paradigm than viewing games as the work of nebulous teams of hundreds.\n\nThe Author in the Games marketplace\n\nAt the time that Jesse Venbrux started making games, there was no place for him to market his work. Six years later, he's released a game for sale through the Apple Store. While this book isn't about making games for money\u2014it's about making games for the sake of having more games by more people, and there will always be more hobbyist authors than commercial authors\u2014I think it's worth discussing the changing videogames marketplace in the context of the changing videogames paradigm. Just as digital games are starting to be thought of as the work of an author rather than a corporation, there is a growing place for solo game authors in the marketplace, and people can make a living doing this (though it's not easy).\n\nI've been making my living in recent years by selling Flash games to \"sponsors.\" Flash is an infrastructure for embedding movies and games in websites, and sponsors will pay for the privilege of including good games in their websites because games attract players who look at ads, and then explore the rest of the site to play more games and look at more ads. The more traffic a website gets, the more valuable that site is to advertisers, so traffic from popular games is important to site owners. Flash is much more complicated to use than something like Game Maker (although that's changing in a small way with tools like Stencyl, Flixel, and FlashPunk), and requires the kind of dedication that someone who just wants to make games in her spare time isn't likely to be able to spare. It's an avenue, though, for solitary authors to make the games they want and to find money for them.\n\nThe first Flash game I sold is about a flying pig. I sold it to Newgrounds\u2014the \"Everything, By Everyone\" portal I mentioned earlier\u2014for enough money to help me move from America's East Coast to the West. Later I sold a game called REDDER for twice as much, and while working on this book I sold a game called Lesbian Spider-Queens of Mars to Adult Swim for enough to keep me out of trouble for a while. I like this model because it means that although I get paid for my work, the game remains free to players. That's important, because I want as few barriers between my work and its audience as possible.\n\nLike Jesse Venbrux, I've also released an iPhone game, although mine is totally free. It's also based on a game I made in Game Maker\u2014a friend of mine with greater technical knowledge, Bennett Foddy, programmed the iPhone version. The game's called Chicanery,57 and it's about the interactions that go on between players outside of the digital components of the game (the stuff that's on the screen). Each player's goal is to keep a finger on the screen as long as possible, and to hinder the other players' abilities to do that. Usually the players do this by punching, shoving, or tickling each other. A guy I met who worked at WayForward Technologies later showed me the dent that playing the game had made in his iPhone.\n\nWhen they're not free like Chicanery, iPhone games are usually sold for a dollar or two. And Apple has total control over its marketplace and what games can be sold within it, and is willing to wipe out games without a second word, as they did in 2010 with thousands of games they thought were too \"sexual.\"58 But the fact that authors are selling games there, and sometimes making a profit, shows that the digital games marketplace is making room for a new paradigm. Note that the Apple Store sells games exclusively through digital distribution\u2014there's no publishing or manufacturing cost whatsoever.\n\nSimilar markets are Android phones, Steam for the computer, and the Xbox Live Marketplace. These are all digital distribution stores tied to specific technologies, and each is run by a corporation that exercises total control over what content is available on which device. It's a situation that leaves creators at the mercy of corporations, but it's a sign, at least, that avenues are appearing for solitary authors to make money by creating games. And, hopefully, there will soon be more avenues, and decentralized ones.\n\nSome authors take it upon themselves to sell and distribute their own games. My friend Edmund McMillen sold a CD with a bunch of his games and drawings on it\u2014his attempt to create a game-as-zine. One of the things I used to do was to release a game that was free to play from start to finish\u2014again, free is important\u2014but to offer a secret password to anyone who donated at least a dollar. The password unlocked additional characters that would show up in the game as neat surprises. I got a few people who donated exactly one dollar, but most people gave five or ten: the invitation to donate was all they needed.\n\nBut that's enough about money. What I'm interested in is game creation as a goal in and of itself.\n\nCrap Games\n\nGlorious Trainwrecks is about bringing back the spirit of postcardware, circa 1993. It's about throwing a bunch of random crap into your game and keeping whatever sticks. About bringing back a time when you didn't care so much about \"production values,\" as much as ripping sound samples from your favorite television shows to use in your game, or animating pictures of yourself making goofy faces on your webcam. Where every ridiculous idea you had, you would just sit down and code. When you would make up a \"company name\" to legitimize dorking around on the computer with your friends. . . .\n\nTogether, you and I will bring the true spirit of indie gaming back. Yes, you! For this site is about nothing, if it is not about getting off your ass and creating. Wikipedia claims that they used to stage trainwrecks (with empty trains, of course) for the amusement of the general population. Would the world not be a better place if we brought this tradition back?\n\nSo states Jeremy Penner on his website, Glorious Trainwrecks (at glorioustrainwrecks.com), founded in 2007. The site is interested in a videogame application of the \"Crap Art\" philosophy59: that professionals aren't the only ones who should make art, and that creation is a meaningful goal in and of itself. Games are more wonderful, more creative, and more inventive when they're thrown together around an idea with little regard for production values or painstaking creative choices. There was no place to explore the videogame as an act of raw creation, so Glorious Trainwrecks was built to provide one.\n\nOn the third Saturday of every month, Glorious Trainwrecks holds an event called Klik of the Month Klub, after Klik & Play, the simple, ugly game creation system (designed for children and people who have never made games before) that most of the participants prefer. But there are no enforced rules about what tools people can use, what they should make, or how much time they should take, though it's in the spirit of the thing to finish the game, from start to end, in two hours.\n\nThe experience forces participants to get past their egos and their meticulous plans for future epic games, to stop focusing on details and CREATE. Klik of the Month is about doing, not planning. Having to finish a game in two hours keeps authors from getting hung up on lesser decisions about their games. This is why Klik & Play is so well suited for the exercise: it's full of clip art, stock sound effects, and design shortcuts. All the resources are already there: just get your hands dirty.\n\nThe results are usually sloppy and haphazard. That's what the glory is: having made something raw and improvised. During different Kliks of the Month, I made a version of Pong where touching the ball makes you lose points (Dodgeball), a game about escorting vulvas back to their home planet (Box Pushing Game), a game about pushing sumo wrestlers into basketball nets with your belly (Sumo Dunk), and a pile more. In 2007, Glorious Trainwrecks ran a two-day event (the 100-in-1 Klik & Play Pirate Kart) that produced one hundred games. We did it again in 2010 and produced 529 games by over a hundred people. People have the desire and creativity to make games; they're only stopped by the roadblocks that get put in the way. Tear down the roadblocks and we'll have so many more games\u2014and more important, more authors.\n\nGame Sketches\n\nWhat will help to tear down those roadblocks and allow games to be more creative\u2014in the sense of, \"Let's create something!\"\u2014are tools that allow more spontaneity. Klik & Play comes with a bunch of clip art that the author can easily appropriate toward other ends. Game Maker allows the author to draw sprites right in the program before giving them rules and actions to perform. Stencyl allows instant access to an online database of resources other creators have already made. That's the solution, I think: to put as little distance as possible between the idea and a playable game. Certainly, many authors will spend a lot of time adding details to their games and developing their ideas. But there's value in ideas, especially in a form so young, and those ideas need to be put out there.\n\nThere's no way to sketch in games. In half an hour, I can sit with a pencil and paper and draw some dumb little comic strip. Videogames have no easy equivalent to that. And yet dumb little games are important because they enrich our vocabulary of ideas. The ideal game-making tool, I think, would look like a sketchpad: I could draw a landscape with a character in it, then give that character rules about how she could move across that landscape.\n\nThe editor that Fred Wood made for his game, Love,60 comes close. Love is a simple little game about a stick figure who navigates a world by running and jumping; if she touches anything dangerous, she dies. To create a Love level, all an author has to do is draw it. The game will figure out how the stick figure interacts with the level. What stops the editor from being perfect is the fact that the author has to split the drawing into a few different images: a layer for things the player can touch and a layer for things that will kill the player on touch. But this is the right direction: if level design can be as simple as drawing a sketch, then game making can be as quick as putting an idea onto the screen.\n\nAnd someone that doesn't know how to script or code probably knows how to draw\u2014at least, to sketch. All of the most successful game-making tools for non-professionals I've seen have looked like this. Warioware: D.I.Y. is probably the most mainstream commercial game-making software I've encountered, in that it was developed by a big publisher (Nintendo) for sale on store shelves. And while Nintendo makes it hard for people to actually distribute games they've made with the software (if the tool had a YouTube-like gallery where people could browse and play other peoples' creations, it would have changed a generation), the process of creating a game is brilliant.\n\nWarioware is for the Nintendo DS, which has a touch screen and plastic stylus. Player-authors can draw sprites on the screen like they're drawing with a marker. Then they assign the sprites simple rules to tell them what to do during the game. But there are also a number of constraints that keep the author from being too ambitious: each game is only seconds long, the author only has a handful of objects to work with, and there's an equally limited number of sprites. This is a tool that's designed specifically to create very small games, and that's good, because small games are exactly what we need more of.\n\nThe proliferation of small games also serves another goal: it'll make games more personal. Ian Bogost, writing for Gamasutra, discussed the potential of games by way of talking about snapshots. The introduction of cheap portable cameras like the Kodak Brownie in the 1960s allowed people who were not professional photographers with access to expensive cameras and lenses to document anything, including things that held little or no value for anyone other that the taker:\n\n[Consider] You're Invited to Go to Heaven, a simple quiz game. . . . You're Invited is a rudimentary example of Christian evangelism.\n\nThe game poses just a single question, \"Who is the Lord of your life,\" and offers four answers: Chris Brown, Orlando Bloom, Zac Efron, and Jesus Christ. The \"correct\" choice is obvious, and it's tempting to write off this game as trite, even worthless.\n\nIts single question would seem barely to qualify it as a quiz game, a genre itself on the very fringes of the medium.\n\nBut there is something deliberate and honest about its simplicity: this is not a game meant to inspire conversion or even head-scratching; it's just a little touchstone in someone's day for reinforcing what's really important to the believer.61\n\nIt might be hard to see the value in a thousand versions of You're Invited to Go to Heaven, but the real value of people producing digital games as quickly and easily as photographs is more subtle:\n\nThe Brownie teaches us that snapshots aren't just good pictures created easily thanks to simple tools. They are also good pictures\u2014or games\u2014created for different purposes. The future of video game snapshots will require platform creators to show their potential users how to incorporate games into their individual lives.\n\nThe result could be very important. The snapshot didn't just popularize photography as chaff, it also helped more ordinary people appreciate photography as craft. The successful game creation platform will be the one we can say the same of, someday.62\n\nMaking games more pervasive\u2014not just games, but game CREATION\u2014will help us to better appreciate games and think about the craft and design of them. It'll also, I think, demystify game creation. We won't think of games so narrowly because we'll understand that they're capable of telling many different stories: stories about dyke cowgirls getting their girlfriends off booze, for example, or a record of a dream.\n\nThe ideal game-making tool, the game sketchpad, isn't here yet, but there are many tools that are close\u2014or at least much closer than what engineers had to work with back in the day. But even given that, how do you, with little to no game making experience, get your feet wet and your hands dirty? How do you make a game?\n\nFootnotes\n\n54 Paraphrase only because I don't remember which two actual cities he named.\n\n55 You can watch the trailer online at: http:\/\/www.youtube.com\/watch?v=Cvd0GbVCoYw.\n\n56 http:\/\/venbrux.com.\n\n57 http:\/\/itunes.apple.com\/us\/app\/chicanery\/id395775628?mt=8.\n\n58 Jason Kincaid, \"The New App Store Rules: No Swimsuits, No Skin, and No Innuendo,\" TechCrunch, February 20, 2010, http:\/\/techcrunch.com\/2010\/02\/20\/app-store-rules-sexy.\n\n59 See the principles of Crap Art explicated at: http:\/\/crapart.spacebar.org.\n\n60 http:\/\/www.love-the-game.com.\n\n61 Ian Bogost, \"Persuasive Games: Video Game Snapshots,\" Gamasutra, September 11, 2008, http:\/\/www.gamasutra.com\/view\/feature\/3784\/persuasive_games_video_game_.php.\n\n62 Ibid.\nChapter Six\n\nMaking the Games\n\nHow do you become a game author? By making a game! Get an idea, find some tools, and make something you can play. You can improve your craft later. At the start, just getting some experience putting together a game and making creative decisions about it will be incredibly valuable. I'm going to describe a few of the smaller, simpler games I've made and the process by which I made them.\n\nTitle, Interaction, Resolution: Gay Sniper\n\nI made Gay Sniper in May of 2009 during a Klik of the Month Klub, the two-hour game design contest I described in the last chapter. I think I made it in about an hour. It was inspired by a video made by West Virginia 4 Marriage,63 an organization that absolutely does not want queers to gain the right to legally marry in West Virginia. One scene in the video particularly grabbed me: The ideal heterosexual family\u2014one man, one woman, and two kids\u2014sits on a stoop blowing bubbles. (The man is white and the woman brown, so the filmmakers can position themselves as a anti-racist organization.) The narrator: \"Marriage between a man and a woman is held up as the ideal in all of civilized society because of its profound stabilizing influence on our culture, as well as the important economic benefits of strong and intact families. But today that ideal is under an unrelenting attack, and same-sex marriage in Western Virginia is a closer reality than you may think.\" As he says this, a crosshair appears on the screen with the het couple in its sights, personifying the homo who wants to get married as a family-assassinating sniper.\n\nThe shot of the couple through the rifle sight, at least to me, was reminiscent of a videogame\u2014a sniper game like Silent Scope, for example. I decided it should be a game: about a gay sniper who assassinates long-held American values.\n\nI made Gay Sniper in Game Maker. Here's how the game plays: The game starts with a title card. Then the player sees the ideal family from the video through a crosshair. The crosshair is smaller than the screen, and the player can move the mouse to pan the crosshair over the screen. (The player can only see what's inside the crosshair. Anything else is blacked out.) The voice-over from the video\u2014\"marriage between a man and a woman is held up as the ideal in all of civilized society\"\u2014plays. When the player left-clicks the mouse over the family\u2014to pull the trigger\u2014a gunshot is heard, and the game changes to a screen that displays the text \"America is destroyed\" over a picture of the contiguous United States cracking in half. After a couple of seconds, the game ends.\n\nIt's a simple game, much like You're Invited to Go to Heaven. How exactly did I make it?\n\nI'm going to avoid going into too many details in explaining how I made the game work. This isn't intended to be a guide to working with Game Maker: I'm interested in writing a more universal approach to thinking about game design and in making the decisions that guide an idea into a game. If you want more specific details on how to build a game in Game Maker you ought to be able to find them easily, by asking questions of other Game Maker users or by consulting a tutorial on the Internet.64\n\nFirst, let's consider the shape of the game. It's made of three distinct screens: a Title screen that introduces the game's theme, the Crosshair screen where the player takes action, and a screen that displays the Resolution of the player's action (the destruction of America). This is the basic narrative shape. In Game Maker, I could easily use \"rooms\" to arrange these individual screens.\n\nThe Title and Resolution screens are the easiest. Each of these screens needs to do just a few things: show a picture, wait a few seconds, and advance the game (or end the game, in the case of the Resolution screen). To draw the pictures I used a program called PaintShop Pro, though Game Maker has a built-in drawing program that I could've just as easily used. A free drawing program like GIMP could have done the same thing. Here's a list of the pictures I drew: one that says \"Gay Sniper\" in red text on white, and one that says \"America is destroyed\" over a picture of a cracked United States. For the latter, I downloaded a map from the Internet, dyed it red, and drew a big crack along the middle. Then I used Game Maker's built-in timer system, called an \"alarm clock,\" to make those images stay on the screen for a few seconds so the player can read them before going to the next \"room.\" (Though making a manual timer is just as easy as adding one to a counter on each frame, and then doing something when the counter reaches the right number.)\n\nThat's all these two screens do. The Title screen also plays the same little melody that opens the West Virginia 4 Marriage video (a light twinkly tune that helps us prepare to appreciate heterosexual marriage). I recorded that directly from the video using a free program called Audacity65 and imported it into Game Maker. The Title and Resolution screens are easy, as there's no interaction. They just frame the action that happens in the Crosshair screen. Interaction is the tricky part, and also the most important part.\n\nThe Crosshair screen requires a few things: the narration from the video and the sound of a gunshot, for starters. The former I recorded off the video with Audacity, like I did with the melody on the Title screen, and the latter I made with a free sound effects program called SFXR.66 Next, the family in the background. I took two screenshots of the family, one in which they're about to blow bubbles and one in which bubbles are blown, and I used the timer to make the image switch every few seconds to give them a little bit of life. And then there's the crosshair itself: the interaction. There are two parts to the interaction: the player moves the crosshair by moving the mouse, and the player fires the gun by clicking. For the first part, I just needed a crosshair that moved to wherever the player's mouse cursor might be. I drew a round crosshair in Game Maker's drawing program and made the inside of the circle transparent and the outside black. Now I needed to have it follow the cursor, so I used Game Maker's scripting language to check the mouse's position at every \"frame\" of the game, then set the crosshair's position to mouse_x, mouse_y (x being the horizontal position of the mouse on the screen, y being the vertical).\n\nBut I only wanted what's inside the crosshair to be visible. So every frame, after I moved the crosshair, I drew four big black boxes, large enough to cover the screen: one to the top of the crosshair, one to the left, one to the right, and one below. Since we've told the game to check the crosshair's location at every tick of the game's timer, we can easily use that information to plan out where to put the boxes on the screen. An even easier solution would have been to just make the crosshair image twice the size of the screen, with black all around.\n\nThe second part of the interaction is left-clicking to fire the gun. Game Maker includes predefined \"events\" to check player input, such as pressing a key on the keyboard or clicking the mouse. When the player left-clicks, the simple solution would be to tell the \"left-click\" event to stop the narration, play the gunshot sound effect, and advance the game to the Resolution screen. I wanted something a little more ambitious, though: I wanted the player to have to shoot the family. Clicking away from them should fire the gun, but not destroy America.\n\nWhat I did was take one of the pictures I had made of the family and trace over it in two colors, one of which I set to transparent, indicating where the gun would miss, and the other, which outlined the family, indicating where the gun would hit. (This is called a \"collision mask\" in Game Maker.) When the player left-clicks, the game checks where the hidden outline of the family is on the screen, where the crosshair is, and if the two are touching. If they are, the \"event\" stops the narration, plays the gunshot, and goes on to the Resolution screen. Otherwise, it just plays the gunshot.67\n\nSo that's the game: we have a screen that establishes the scenario (\"Gay Sniper\"), one in which the player takes action (moving the crosshair and firing the gun, inevitably shooting the idealized hetero family), and a final screen that resolves that action (with the inevitable destruction of America). What makes it a game is the Crosshair screen. It's the player who does what the video implies; she fills the role of Gay Sniper and fires the gun. This is why I want the player to miss if her crosshair isn't over the family: I want the player to have to consciously aim and shoot, to make her more complicit in the fantasy that the West Virginia 4 Marriage video suggests, and that I am forcing the player to act out. By forcing the player to inhabit a political ideology\u2014because that's what games, uniquely, can do\u2014I am pointing out how absurd it is. It's not a complicated game or a long one, but one I thought was worthy of bringing into the world.\n\nImperfection: Box Pushing Game\n\nIn November of 2009 I made a game called Box Pushing Game for Klik of the Month. That was a particularly exciting Klik of the Month because some friends of mine were in town and we all got together in the same room to make games, and to play and give criticism of each others' games.\n\nBox Pushing Game was intended as a parody of games like Sokoban where the player's goal is to push and arrange a number of crates in a constrained area. (Sokoban translates from Japanese as \"warehouse manager.\") In Sokoban the boxes move along a grid, and the conflict is that pushing the crates around will put them in the way of other crates, forcing the player to plan and manage them carefully.\n\nIn my game, the boxes aren't crates but vulvas. (Box is old slang for \"vagina\" and I'm not sure anyone knows why.) It was an idea I thought up in less than a minute during a conversation, and it was the perfect level of dumb for Klik of the Month.\n\nSo the ideal version of my game would be one in which the player pushes pictures of vulvas (downloaded from the amateur porn site I Shot Myself) neatly along a grid, managing them carefully to ensure that they all reach their destinations. But in the interest of getting the game done in the two hours of Klik of the Month, I was using Klik & Play to make it. And that meant making a lot of compromises. But I could convey the concept of my game\u2014Sokoban with hoo-hoos\u2014without all the particulars having to be exact.\n\nInstead of pushing the boxes along a grid, whenever the player touches a box it just shunts a bit in the direction the player was moving. Instead of being stopped by walls, it just bounces in the opposite direction with a CLANG when it touches one. Instead of having to fill a number of marked spaces with boxes, each level includes \"Mother Boxes\" (bright pink vulvas with eyestalks), each of which vanishes when a box comes into contact with it, taking the box with it. When all of the Mother Boxes are gone, the player goes on to the next level. And instead of checking whether one box is blocking another\u2014which would take time to figure out how to do properly\u2014when two boxes overlap, they resolve the collision by wiggling in random directions until they aren't touching anymore.\n\nIt's a really clumsy solution to the problem of keeping things from occupying the same space, but it doesn't distract from the idea of the game: push a box with a vulva on it from point A to point B. If anything, it gives the game more character and identity: another Klik of the Month participant described the rule as, \"When my box touches another box it goes crazy.\"\n\nAccidents are creative. At a more recent Klik of the Month that we held on the night of a thunderstorm, my friend Loren Schmidt (hereafter called \"Sparky\") was working on a quick implementation of Atari's old Asteroids game. (The player maneuvers to destroy asteroids by rotating a ship left or right and firing an engine for thrust.) When Sparky called me over to look at it, he hadn't yet implemented a screen refresh\u2014that is, everything that was drawn to the screen stayed there instead of going away. So the player's ship, when it moved across the screen, left a long snake-like trail behind it. Sparky really liked the effect and wanted to keep it.\n\nI told him that he should make the asteroids white, the same color as the background, so that they'd be hidden until they passed over the trails that the player (or her ship's bullets) left on the screen, giving her an incentive to move around and cover as much as the screen as possible (while using bullets to gauge the presence of asteroids in her path). The \"accident\" of not implementing a screen refresh led to the author finding an entirely unique identity for the game.\n\nPerfection isn't a useful goal; if anything, it keeps amateurs from getting their feet wet and authors from finishing their works. It's an ideal that hinders more than it serves. Imperfections, creative accidents, and compromises, on the other hand, give a game personality and individuality. Tolstoy's Anna Karenina begins with the line: \"Happy families are all alike; every unhappy family is unhappy in its own way.\"68 Difference is valuable, and creative accidents and jury-rigs help us achieve it.\n\nImperfection is an invaluable tool when making games, particularly when making your first games. Think about the ways you can approximate an idea with the tools you have and the things you know you're capable of. Don't worry about minutiae: you can express the idea of your game to a player without everything being exactly as you've envisioned it.\n\nUse What's on Hand: Find Shit\n\nMy submissive's first videogame was a Klik of the Month creation called FIND SHIT. She had no prior experience with coding or with game making other than looking over my shoulder. How did she make her game? She was using Klik & Play, so she had access to all of the clip art that comes with it. She put a tiny dancing red bug on the screen. Then she covered the rest of the screen with huge red starbursts. The player's goal is to find the bug hidden in the loud, flashing, abrasive screen before the timer runs out.\n\n69\n\nShe minimized the amount of scripting she had to do (what a lazy pig) by making clever creative decisions about what she had available to her. The only things she had to script were the timer, clicking the bug to advance to the next level, and the message at the beginning of the level that tells the player to find the bug and asks her to click on a button to start. The buttons are a kind of object available in Klik & Play to allow the author to ask yes\/no-type questions of the player. After putting the premade button objects in her game, though, my pig discovered that they're broken: every choice calls the same response. So she made every choice an increasingly apathetic variation on \"YES.\" (For example, the first screen asks the player to \"FIND THE BUG,\" and lets her respond with \"okay\" or \"sure.\")\n\nKlik & Play supports WAV audio files, so she put a bunch in. Playing three of them on top of each other, she found, makes Klik & Play totally freak out and begin playing glitchy machine noises. The shrill nightmare sounds perfectly fit the tone of the game, which is already visually abrasive.\n\nLimitations and Creativity: Shoot! Win!\n\nIn February of 2010, a friend's fianc\u00e9 made a game called Shoot! Win! This game is a shooting gallery of American presidents: images of Abraham Lincoln, William McKinley, James Garfield (represented by a picture of Garfield the cat), and other presidents bounce around the screen, while images of their assassins (John Wilkes Booth is a phone booth) pose along the bottom.\n\nThe player presses keys to make assassins fire at the bouncing presidents, trying to shoot a president using the historically accurate assassin.\n\nWhen a president is shot, a voiceover explains why that assassin did not historically assassinate that president and chides you for your incorrect choice. (\"John Wilkes Booth did NOT shoot James Garfield.\") There's a unique voiceover for every combination of victim and assassin (with five assassins and five victims, that's twenty-five vocal samples). These explanations make up the brunt of the game's content; they're what make the game interesting and entertaining to play.\n\nAll it took to create the vocal samples was a microphone, the historical background, and the willingness to record them. Again, the author keeps scripting to a minimum by making the focus of the game something he can easily create. And because the game is built around components that the author had the means to create, it's again unique, different from anything that an author who had the time, resources, and ability to perfectly mimic any existing game might have produced. Limitations, both self-imposed and otherwise, guide our creativity.\n\nWorking Within Limitations: \"Flimsytown\" and \"Pestilence\"\n\nTim Sweeney's ZZT, created in 1991, is a game-making tool that uses ANSI, the sixteen-color text mode that came preinstalled on PCs of the time, to display all its graphics. This limits ZZT authors to a choice of 256 characters\u2014half of which are the letters of the alphabet written in lower and upper cases and using different combinations of diacritical marks\u2014rendered in sixteen possible colors. (Each character can contain two colors, a foreground and a background color, the former for the character itself and the latter for the negative space that surrounds it. Eight of the sixteen colors can only be chosen as foreground colors.) This may seem constrictive, and it is, but authors took the limitations imposed by the program as a creative challenge.\n\nHow do you animate a spinning turnstile, for example, in sixteen colors and 256 mostly alphabetical characters? You might have a character that alternates between an X and a cross. A twinkling star could be a dot that blinks as a +. A club (the playing card symbol) could be colored green and made to stand for a bush or tree. The symbol for pi could be a table. Three different \"pattern\" characters from the extended ASCII character set are similar to different weights of crosshatching, and when combined with a creative mind and an artful use of color can texture a scene impressively.\n\nIn fact, the scenes are more impressive for being drawn within the limitations of 256 characters and sixteen colors. Working within those limitations teaches the author to be clever and creative with what she's given in order to make an impression. The limitations of ZZT also change the terms of the challenge. An author isn't trying to produce something visually beautiful in comparison to the entire body of contemporary videogames, those produced by small armies of artists and those consumed by millions of mainstream gamers, but rather to produce something visually effective within a small set of limitations and for a much smaller body of peers. This narrowing of the set of possibilities helps the author stay focused and avoid the pitfalls of over-ambition.\n\nThe same applies to scripting. Creating a variant of a game like Lemmings might not be technically impressive in most programming environments because it's already been done many times, but in ZZT-OOP\u2014ZZT's scripting language, which is far less powerful than most programming languages\u2014it's a somewhat astonishing feat.70 (That still doesn't mean creating a ZZT version of Lemmings is necessarily valuable; to do so would be an accomplishment on the order of creating an intricate copy of the statue Rape of the Sabine Women in LEGOs. Limitations force you to be inventive, but you can also fall into the trap of pursuing arbitrary technical goals rather than growing as an author and storyteller.) When a small and isolated community consolidates around a game-making tool, creators start to create more and more for each other (because who else is playing?). Their target audience becomes other creators who possess a detailed mechanical understanding of the game they're working with. As a result of authors creating for one another (and challenging one another), their creations become harder and require players to have more mechanical knowledge in order to play a game at all, until most experiences are too hard for most players.\n\nThis is true of a lot of levels made for Knytt Stories (Nicklas \"Nifflas\" Nygren's expandable game about running, jumping, and climbing.) Many of the levels that circulate on Nifflas's online forums71 are about really tricky, difficult maneuvering. So much of the technical ground of what is possible with the tool has already been explored and so many creators who work with Knytt Stories are familiar with the basic and advanced mechanics of the game that creators are naturally drawn toward designing for this group and taking as their starting point something far beyond what most novices using the tool would be capable of easily appreciating.\n\nBut this tendency to design for a select group can also push authors to further unexplored conceptual ground. Recently, I played a Knytt Stories level called \"Pestilence.\"72 In this story, the author has taken the terrain tiles that come with the tool and arranged them seemingly at random to create a broken-looking world. Power-ups that the player would normally expect to collect are locked inside walls, out of reach. The characters who populate the shattered landscape bemoan the hopelessness of their situation as though they were in glitchy videogame hell. (They are, in fact, trapped\u2014the editor for Knytt Stories doesn't provide characters with any means of moving around the world or leaving the place in which the author set them.) Though the world seems to have little order, a player who explores will eventually be able to find one of several ways out.\n\nThis is the sort of experiment with form that rarely happens in a less restrictive format. With the exception of its music, \"Pestilence\" is made entirely from parts included with Knytt Stories; it's the way in which those parts are arranged that is unique. It's easy to arrange tiles in an unconventional order, but to do so with intention, to build a functioning, solvable world out of them? This is the sort of conceptual ground that, I believe, is usually only reached when possibility-space is finite, and when a lot of that space has already been thoroughly mapped. Think of Los Angeles versus New York City: one city, with a great deal of continental terrain, expands outward; the other, built on an island, expands upward.\n\nZZT, unsurprisingly, has an equivalent level to \"Pestilence.\" It's usually called \"Flimsytown.\" \"Town of ZZT\" is the official level packaged with the ZZT editor and program. \"Flimsytown\" starts identically to \"Town of ZZT,\" but quickly diverges, presenting a jarring and confusing experience that requires the player to actually exploit the bugs and glitches inadvertently built into ZZT in order to progress. For a while, supposedly, the website Z2, the most well-known online archive of ZZT levels,73 included \"Flimsytown\" in its ZZT package in place of \"Town of ZZT.\"\n\nBe Derivative: Mighty Jill Off\n\nHi, I'm Der Rivative. Come to see my work? I've got this medical paraphernalia here, which I've made into little objects. Butt-plug stool\u2014some people say it's like Nayland Blake. No, it's mine, Mr. Der Rivative. I made it. Here you have scaffolded video representing my face in six monitors. Some people say Nam June Paik, some people say Bruce Nauman. No, I made it\u2014Der Rivative. I like to lie in bed, as long as I can, let people in the gallery come and see me. Nothing like the Chris Burden on the shelf thing, it's a Der Rivative extravaganza. I invented it. I made it. People come to the museum and say, \"Hey, there's a Der Rivative.\"74\n\n\u2014Bob Flanagan\n\nNothing is original. You've probably heard the old quote about there only being seven basic stories (or three, or fifteen; the number always changes) a thousand times. Quantification aside, the purpose of the quote is to make the listener realize that what's interesting about the story is not the story itself but the way it's told. There are a lot of novel ways to tell stories with games. We've found very few of them so far.\n\nWe're going to find many of them, probably, by accident. Don't worry about making an original game\u2014just worry about making it yours. Many authors start by trying to replicate games they've already seen: Super Mario Bros., Mega Man, and Final Fantasy are popular. And many of those games end up being very different from their antecedents because the authors didn't have the technical or artistic skill to mimic those antecedents exactly, and instead had either to approximate the parts of the game they couldn't replicate or to improvise something entirely new. Or to incorporate a single new idea that changed something significant about the basic model. Or simply to invest the game with their own personality.\n\nOne of my most successful games (in terms of press, not money; the game was released for free on the Internet) is Mighty Jill Off. It's based on a Nintendo game called Mighty Bomb Jack, published by Tecmo in 1986. Mighty Bomb Jack's predecessor game, Bomb Jack, predated Super Mario Bros. and presents a very different vision of the way a videogame about jumping can work. While the emphasis in Mario is on the planning of a jump\u2014the momentum Mario gains on the runway before the jump, the limited English he can spin on his jump while he's actually in the air\u2014the character Bomb Jack is highly maneuverable while in the air. He can jump the entire height of the screen, reverse a jump on a dime, and flap his cape to keep himself from descending too quickly.\n\nI made Mighty Jill Off during the first semester I spent at the Guildhall because I was bored to shit with their lectures and I needed an actual design task to challenge me. For Jill Off, I lifted the entire vocabulary of Bomb Jack\u2014the high jumps, the reversals, the cape flap\u2014and put it in the service of a story about a submissive masochist climbing a tower to prove herself to her Queen. As a pervert who missed her own submissive (who waited for me in California while I toiled at school in Texas), it was a subject that was both on my mind and relevant to the way I experience games: games about challenge are about the relationship between the player and the designer, the former trying to prove her capacity and will to the latter, the latter trying to continuously challenge the former while maintaining her trust.\n\nBorrow whatever you can. If you were to try to complete a jigsaw puzzle with pieces collected from five different puzzle sets, there'd be blanks that the pieces you've borrowed wouldn't fill in. You'll fill in the blanks with your own personality. You'll smudge the corners in a way that is identifiably yours. And you'll create something new: maybe by design, probably\u2014and preferably\u2014by accident. The modern novel started when Cervantes took the stock idea of \"a knight who has adventures\" and put him into his own contemporary Spain. Modern music started when bluesmen took traditional folk songs and began to bend the notes. Modern sculpture began when Duchamp took a space in the gallery reserved for artwork and put a urinal in it. Setting out to make something utterly new can be a trap to a new author: you'll spend forever planning and no time doing. If you get busy without worrying about being original, you're liable to stumble across many more interesting ideas.\n\nMake Weird Shit: Cactus Block\n\nPut weird shit into your game. Make unusual creative decisions. You're not beholden to anyone, so why be conventional? Whenever you see an opportunity to make something a little more distinctive, take it. These are the parts of the game, after all, that are most uniquely you.\n\nGlorious Trainwrecks user chuchino made a game called Cactus Block75 in February 2010. This is a simple game about a huddled figure trudging through the snow from the left side of the screen to her house on the right side of the screen. She can jump, but her house is located atop ledges and cliffs that are far too high to jump to the top of. Fortunately, the player has the power to create additional platforms\u2014blocks\u2014by moving the mouse and clicking on areas of open space. The goal, then, is to place the blocks appropriately in order to climb over the obstacles.\n\nThis is the point at which the author made a Weird Decision. About half the time, when the player clicks the mouse, a deadly cactus will appear instead of a block that's safe to stand on. (If the player touches the cactus, she must start the screen over.) That's a pretty silly idea, but this is the rule that defines the game.\n\nBecause of the unpredictable way that this introduces chance to the player's actions, the player has to plan carefully to accommodate for potentialities. Maybe I click in a less optimal platform location because there's a fifty-fifty chance of my click producing a cactus, and I don't want to block that prime location. Do I take the risk of trying to move forward quickly, depending on a lucky streak of blocks, or do I work slowly, making sure I always have a back-up plan? Do I try the last resort of jumping into the air and clicking immediately under the protagonist, with an equal chance of producing solid ground or instant death? These are all interesting, unique player choices, and they spring from a single unorthodox creator choice.\n\nAs a Klik of the Month game, Cactus Block was made in two hours. It's not the same case as a big-budget commercial game, where lots of money and hours have been invested in a project, and where a publisher expects a return on its investment. If you do something wacky in a two-hour game and it doesn't quite work out, you've gambled and lost just two hours of your time. And I use the word lost reluctantly, because I don't consider any game a total loss, even if the results don't work very well. A \"loss\" here means that you've made something unique and that you have that much more experience putting a game together.\n\nBeyond the potential for interesting design generated by making something weird, any opportunity to do something weird is also an opportunity to put some of yourself in a game. Most of my games star perverts, even if all the perverts are doing is jumping up a tower. Some hobbyists try to emulate what they think of as \"professional\" game development and divorce all aspects of themselves from their games. That's ridiculous! Your game comes from you; your game should contain you. A game you make should have your signature on it. It should reflect something of your personality, your humor, your values. After all, you're only beholden to yourself. Make something that's your own.\n\nWhat to Make a Game About?\n\nYour dog, your cat, your child, your boyfriend, your girlfriend, your mother, your father, your grandmother, your friends, your imaginary friends, your summer vacation, your winter in the mountains, your childhood home, your current home, your future home, your first job, your worst job, the job you wish you had.\n\nYour first date, your first kiss, your first fuck, your first true love, your second true love, your relationship, your kinks, your deepest secrets, your fantasies, your guilty pleasures, your guiltless pleasures, your break-up, your make-up, your undying love, your dying love.\n\nYour hopes, your dreams, your fears, your secrets, the dream you had last night, the thing you were afraid of when you were little, the thing you're afraid of now, the secret you think will come back and bite you, the secret you were planning to take to your grave, your hope for a better world, your hope for a better you, your hope for a better day.\n\nThe passage of time, the passage of memory, the experience of forgetting, the experience of remembering, the experience of meeting a close friend from long ago on the street and not recognizing her face, the experience of meeting a close friend from long ago and not being recognized, the experience of aging, the experience of becoming more dependent on the people who love you, the experience of becoming less dependent on the people you hate.\n\nThe experience of opening a business, the experience of opening the garage, the experience of opening your heart, the experience of opening someone else's heart via risky surgery, the experience of opening the window, the experience of opening for a famous band at a concert when nobody in the audience knows who you are, the experience of opening your mind, the experience of taking drugs, the experience of your worst trip, the experience of meditation, the experience of learning a language, the experience of writing a book.\n\nA silent moment at a pond, a noisy moment in the heart of a city, a moment that caught you unprepared, a moment you spent a long time preparing for, a moment of revelation, a moment of realization, a moment when you realized the universe was not out to get you, a moment when you realized the universe was out to get you, a moment when you were totally unaware of what was going on, a moment of action, a moment of inaction, a moment of regret, a moment of victory, a slow moment, a long moment, a moment you spent in the branches of a tree.\n\nThe cruelty of children, the brashness of youth, the wisdom of age, the stupidity of age, a fairy tale you heard as a child, a fairy tale you heard as an adult, the lifestyle of an imaginary creature, the lifestyle of yourself, the subtle ways in which we admit authority into our lives, the subtle ways in which we overcome authority, the subtle ways in which we become a little stronger or a little weaker each day.\n\nA trip on a boat, a trip on a plane, a trip down a vanishing path through a forest, waking up in a darkened room, waking up in a friend's room and not knowing how you got there, waking up in a friend's bed and not knowing how you got there, waking up after twenty years of sleep, a sunset, a sunrise, a lingering smile, a heartfelt greeting, a bittersweet goodbye.\n\nYour past lives, your future lives, lies that you've told, lies you plan to tell, lies, truths, grim visions, prophecy, wishes, wants, loves, hates, premonitions, warnings, fables, adages, myths, legends, stories, diary entries.\n\nJumping over a pit, jumping into a pool, jumping into the sky and never coming down.\n\nAnything. Everything.\n\nGet Your Hands Dirty\n\nSo get to it. Make a game. Your first game will be rough and derivative. It may not change the world, but it will be yours. The next chapter will be an attempt to walk you through your first game step by step. It's intended to be as abstract as possible so that you can apply it to whichever game-making tool suits your needs best. You can find out the details of making that tool do what you want through reading the documentation, asking questions of the community surrounding that tool, or best of all, fiddling with the tool to see how it works.\n\nAs for what tool to choose: Appendix A lists a bunch of different game-making tools that are available, describing what they're good for and what the experience of working with them is like. It's by no means a complete overview of all the game-making software out there\u2014there are far more tools than anyone could document\u2014but I've included those that I think are the most useful and approachable. There will also be tools that are created after this book is published. Nothing that I'd consider the ideal game-making tool exists yet, but I'm confident we'll keep getting closer.\n\nIt's possible that your interest in digital game creation is purely academic and doesn't extend to becoming an author. In that case, I hope that what you take away from this book is that the videogame isn't the creation of a corporation, but of an author, that this form is important, and that people are using it to do exciting things.\n\nIn 2008, Nicklas Nygren (a.k.a. Nifflas, author of Knytt Stories) attended a festival called EDGE in Ume\u00e5, Sweden, with a laptop, a printer, and a digital camera. He asked other festivalgoers to contribute to a game he was working on. They drew pictures, photographed themselves, and sang into a microphone. Over the course of the day, he created a game called Det Officiella Edge-Dataspelet, or The Official Edge Videogame.76 Every image and sound in the game was created by someone Nifflas had pulled from the crowd: the protagonist is a photo of someone's head in profile, the stages are drawn in crayon, the music and sound effects were made by strangers' mouths.\n\nThis is what game creation is becoming: small, personal, and made by people's own hands and mouths instead of by corporations with teams, managers, and software renderers. What we call a videogame is not a product. It's the creation of an author and her accomplice, the player; it is handmade by the former and personally distributed to the latter. The videogame is a zine.\n\nFootnotes\n\n63 You can watch the video online at http:\/\/www.youtube.com\/watch?v=5vQm0hJ8Dd0.\n\n64 Although the concepts I'll talk about can be implemented using a wide variety of tools, I'll use Game Maker's names for the concepts where appropriate in this section.\n\n65 http:\/\/audacity.sourceforge.net.\n\n66 http:\/\/www.drpetter.se\/project_sfxr.html.\n\n67 Collision detection, telling whether two things are touching\u2014in this case, the mouse and the family's collision mask\u2014is pretty tricky, because it involves comparing a lot of values, yet it's a basic part of most game interactions. GameMaker lets you accomplish it really easily. Every game-making tool you use will allow you to check (or at least improvise a way of checking) whether two things are touching, but some tools will make it harder than others.\n\n68 I totally stole the idea of repurposing this quote from Jared Diamond.\n\n69 There's a video of FIND SHIT being played at http:\/\/www.youtube.com\/watch?v=zPssxEE_qnE.\n\n70 Lemmings for ZZT exists, naturally. See John D. Moore's Zem! games.\n\n71 http:\/\/nifflas.lpchip.nl\/index.php?PHPSESSID=ebf1b8ac1a8a3fc1f5109d79a1813537&board=39.0.\n\n72 It can be found on Nifflas's forum at http:\/\/nifflas.lpchip.nl\/index.php?topic=1942.0.\n\n73 http:\/\/zzt.org.\n\n74 From Sick: The Life and Death of Bob Flanagan, Supermasochist, DVD, directed by Kirby Dick (Santa Monica, CA: Lions Gate, 1997).\n\n75 http:\/\/www.glorioustrainwrecks.com\/node\/451#comment-3591.\n\n76 http:\/\/nifflas.ni2.se\/?page=Miscellaneous.\nChapter Seven\n\nBy Your Bootstraps\n\nThis chapter is here to help you through your first game. It's not intended as an instruction manual, as an encyclopedic reference, or even as a step-by-step guide. Rather, it's a series of tasks designed to stimulate you creatively, to get you thinking about game design decisions, and to actively involve you in working on a game\u2014which is the most important part. Lots of would-be game creators spend such a long time planning dream games that their ambitions are far too large for them to achieve, or even to know where to start. For others, the whole notion of making a videogame is alien and it's difficult for them to know how to even approach the work from a technical perspective. The steps of this chapter attempt to introduce concepts of game making in a less intimidating way.\n\nThe tasks are purposefully left abstract so that they can be applied to any tool you choose to work with. This means that I'm not providing specific instructions for how to make a character move around in Game Maker, for example. Figuring that out is up to you. Fiddle around, read the documentation, ask questions online. By working it out yourself, you'll start to learn your way around the tool of your choice.\n\nSome tools are very different than others. Twine and Inform, for example, are for making games presented largely through text alone. You'll need to be creative in adapting the tasks below to fit the kind of game you can make with those tools. Remember that these tasks aren't instructions: they're just here to point you in the right direction. Diverge from them as often as possible; do them out of order; ignore the ones that don't fit your vision; leave them behind once you've found your own way. This is just the start of the path. The end is everywhere.\n\nTask #1: Choose a Tool\n\nThe first step is to pick a tool\u2014either one that sounds like it would be useful for the game you have in mind, or one that seems interesting to you and that starts you thinking about what kind of games you might like to make, or that suggests possibilities that your game might explore.\n\nTo make the kind of games most people intuitively think of when they think of videogames, The Games Factory (for Windows) or Game Maker (for Windows or Mac) are strongly recommended for being fairly robust. They're capable of crafting a great variety of games that look and play differently without much difficulty. Game Maker is probably the best-documented tool of its kind, but the current version (rebranded \"GameMaker\" by the soulless corporation that currently controls the intellectual property) makes it difficult and unrewarding to self-publish. But starting with an older version of Game Maker that publishes games for free is a good idea.77\n\nIf Game Maker and The Games Factory are too intimidating, try Scratch. Scratch provides a good introduction to thinking about games in terms of rules, an essential habit for more advanced game making. Every rule in the Scratch editor is essentially a sentence made up of smaller phrases, which helps you understand how rules create the actions in a digital game. There's also a tool called Stencyl that uses the same concept as Scratch, but produces Flash games for online distribution.\n\nI'm also partial to Twine for games that are closer to written fiction. Though you can create projects that include images and other graphical elements if you have knowledge of HTML and CSS (in which Twine is based), Twine is essentially a tool for creating text-only games. It's intended for making branching stories in the vein of Choose Your Own Adventure books, and has a clean visual presentation that shows how passages in the story are connected. Other text game creators like Ren'Py and Inform 7, described more fully later, provide other simple options for creating this kind of game.\n\nIn the next chapter you'll find a fuller list of the tools that I think will make the most sense to people who have no programming or game making experience. I've tried to describe what each tool is good for and what the experience of working with each tool is like. Try more than one. You can always learn to use a new tool. Each one will teach you something different about design.\n\nTask #2: Introduce a Character\n\nA character could be abstract or concrete. It could be a person, an animal, a symbol. It could be represented by a photo\u2014one of yourself or of a loved one or of a stranger that you've grabbed from Google's image search. Mold a character out of clay and take a picture. Open up MS Paint and doodle a cartoon. Find a picture of an American president and draw horns on it. Or just use an image file that comes with whatever tool you're using. Klik & Play includes lots of clip art, and Stencyl lets you grab animated characters from an online database.\n\nTake this image of your character and find a way to start a game and display the character on screen.\n\nOnce you have that, imagine what this character might be capable of doing, and how that will characterize her. It can help to think of this in terms of a question: what job does this character have? Think about what kind of conflict the character might have: what makes it difficult for her to do that job? I once made a game about a robot gardener: her conflict was that coming in contact with water would cause her to short circuit.\n\nLet's say your character starts from a picture of one of your friends, grabbed from Facebook. In the photo your friend is wearing a sombrero. Why would this character you're creating wear a sombrero? Perhaps she's a MATADOR. What forces come into conflict with a matador? Perhaps she'll have to face some RAGING BULLS. A story starts to develop from a single picture that you've taught your program to display on the game screen. (Or the story starts to develop and you change the picture to fit the new story.)\n\nIf you're working with something text heavy like Twine or Inform or Ren'Py, you can, instead of using an image, write an opening passage that characterizes the protagonist, what she's capable of doing, and what her relationship is with the world. In either case\u2014text or image\u2014this is the first piece of information the player will see when she loads your game. So you should choose something that tells her about the game and its world.\n\nTask #3: Teach Your Character to Do Something\n\nSpecifically, teach your character to do something in response to the player. This means teaching the game to TAKE INPUT. The player should be able to press the spacebar, or click the mouse, or type the letter A, or something, and your character should respond to that input somehow. Maybe your character turns invisible while the SHIFT key is held down. Maybe your character flies up the screen when the player hits the UP arrow key. Maybe when the player clicks somewhere on the screen with the mouse, the character teleports there.\n\nAs you do this, think about how what you're teaching the character to do characterizes her. How does being able to teleport anywhere on the screen fit her character? What does that ability tell the player about her? Maybe she's not just any matador, but a MATADOR FROM THE FUTURE. Maybe the raging bulls she comes into conflict with are charging from one side of the screen to the other, and she has to teleport to stay out of their way.\n\nThis is your game's first rule. When the player does X, the protagonist does Q. This is the kind of rule I like to call a \"verb,\" because that's the part of speech this rule would correspond to in a sentence describing your game, one that governs the relationship between the player's character and the other characters in your game. In other words: FUTURE MATADOR (subject) TELEPORTS (verb) to avoid RAGING BULLS (object.)\n\nIf you're working on a text game, write a second passage that describes a location, be it a physical place or a state of mind, so long as it's different from the one the game starts in. Maybe the player starts in a desert, but she can see an oasis in the distance. This second location can be the oasis. Now give the player some means of getting her character to the new location.\n\nTask #4: Introduce a Second Character\n\nThis is the object of the previous sentence. This character should have a relationship to the first character, who we'll start calling the player for short. Maybe the second character represents something the player wants to avoid (raging bulls) or something she wants to acquire (a bag of cash). (Or maybe she wants to avoid the corrupting influence of the bag of cash and touch the confused raging bull, gently, in order to calm it.)\n\nThe relationship between the characters should have something to do with the player's verb. For example, if the player can FLY UPWARD and the object is THE SUN, which the player wants to touch, then writing the game's rules such that the player can fly upward high enough to reach the sun will change the experience of the game. If the player can fly, but never high enough to reach the sun (or perhaps the player can only fly straight upward, and the sun is a little to the left or right), you'll have made a game about unattainable dreams, ones that remain just out of reach. If the protagonist can reach the sun but it burns her wings off, causing her to fall back to the earth and the game to end, you'll have made a cautionary fable. If she can reach the sun, put it in her pocket, and earn one hundred points for it, you'll have made a power fantasy.\n\nRules change not only the meaning but also the feeling of a game. In our future matador game, maybe you'll decide that a raging bull always moves left and right across the screen, or instead, that it always moves toward the protagonist. The latter bull is probably more difficult to avoid than the former. Once you have your rule in mind, experiment with different ways of implementing it to find a feeling and balance that you like.\n\nIt's most likely that you want this character to abide by a rule that's internal to the game\u2014that is, one kept by the computer, not controlled by the player. You could, of course, experiment with giving the player two characters to control in different ways (maybe one of them is a mirror image that moves left when the other moves right), or with making two different characters that are controlled by two different human players.\n\nIf you're writing a text game, introduce an object to the story, one whose relationship with the player introduces a conflict. Maybe when the player reaches into the oasis for a drink, she discovers what appears to be an old, waterlogged map.\n\nTask #5: Make Some Noise\n\nPut a sound effect in your game. Think about which interactions in your game would be clarified by the playing of a corresponding sound. For example, if our matador makes a sci-fi BZZIP sound when she teleports, it would definitely characterize her as a matador that uses future technology. And it would act as punctuation to the teleport, helping the player to understand that clicking to move the character isn't a mistake, but what she should focus on. The bull striking the matador might be accompanied by a drawn-out scream to let the player know that it's an undesirable outcome (or a choir of angels singing if the future matador secretly seeks a glorious death in the ring).\n\nHow do you find sounds to put in your game? Open the sound recording program that comes with your computer (or download a free one, like Audacity78) and record yourself making sounds into your computer's microphone. Try shifting the pitch or the tempo, reversing the sound, adding a thousand phase or distortion effects. Andrew Plotkin made all the sound effects for his 1994 shareware game, System's Twilight, by recording himself making silly noises into a cheap mic. You can even download his sounds if you want; he's made them available.79 You can find plenty of sounds online, at sites like Freesound.org. And many tools, in particular Game Maker and Klik & Play, come with many of sounds for you to use. There are programs that will let you generate your own sounds directly, without a mic: a free program called SFXR80 lets you easily generate Nintendo-like sounds.\n\nIn a typical videogame, the screen communicates a lot of visual information. In a busy game, it can be hard to follow absolutely all of what's happening on the screen. Sound communicates on a different channel, giving you a separate and sometimes more immediate layer of information. Use sound to underline the important relationships in your game.\n\nIf you're making a text game, the tool you're using may not have an easy way to play sound, or playing a sound effect might be jarring in a game that's all read. Your story might not need sound at all. You might try fiddling with some kind of background ambiance, like a loop of birds chirping or water running that only plays at the oasis, to differentiate that location from the desert. Or maybe a quick sound like a SPLOOSH of water or a CHIME of discovery upon pulling the map out of the water.\n\nTask #6: Round Out the Player's Vocabulary\n\nGive the player's character more verbs; give the player more ways to communicate with the game. Try to think of new rules that will interact with the existing ones in interesting ways. For example, what if you make your future matador into a future cowboy? In addition to being able to teleport, she can also drop a ZAPPO LASSO, which will catch bulls that run across it and return them to the future. But maybe being able to get rid of the bulls makes the game feel too easy, so maybe she can only drop the lasso where she's standing. This means that in order to trap a bull, the player will have to use her teleport to put her in the path of the bull, set the trap, and teleport away again. The conflict escalates.\n\nIn the case of the Icarus character who flies UP to reach the sun, maybe she can move LEFT or RIGHT as well. If she moves off the left or right side of the screen, she'll travel around to the other side of the planet, where it's night and where she can land safely on the moon instead. (The transition can be as simple as changing the background from a sky blue to a night black, and the sun to the moon.)\n\nThink about other rules you can add. Rules don't only have to be verbs; rules can also be adverbs. Your future cowboy might be able to teleport SLOWLY. In practice, this means that it takes a few fractions of a second between the player clicking the button and the cowboy arriving at her destination, which forces the player to plan more carefully and react more quickly. Maybe Icarus is affected by gravity, and whenever she's not flying up, she's falling slowly down. You could give her the ability to move up, left, and right when the player presses buttons, but she has to depend on gravity to move down.\n\nTry to ensure that all the rules in the game have relationships with each other. If you think of them as a vocabulary\u2014nouns, verbs, and adverbs\u2014try to construct a vocabulary with which you can tell engaging stories.\n\nIn your text game, give the player a way to interact with the object she's encountered. Maybe if she tries to read it, she's told it's too waterlogged to make sense of. Maybe she has to take it back to the desert to dry it off. The way she interacts with the object should advance the story. Now that the player's returned to the desert, maybe the dried map leads to a treasure hidden somewhere in the sands.\n\nTask #7: Design a Level\n\nI'm using the term level to mean a sequence of events the player has to negotiate using her vocabulary of verbs. In other words, a level is a story: the tension that rises before the climax of the game resolves it.\n\nA story doesn't have to be complicated. Levels for Future Cowboy could just be an escalation. Every time our cowboy catches a bull, it enrages two new bulls who charge onto the screen, causing the bull population to gradually increase with the player's performance, and making the bulls harder for the player to manage. This is rising tension too, and each new bull on the screen is a new chapter, ultimately telling a story about human performance against overwhelming odds (or whatever you like).\n\nHowever, some stories you want to tell might demand more complicated levels. What if the sun our Icarus wants to reach is blocked by storm clouds that will electrocute her if she touches them? The storm clouds make the sun a more valuable goal\u2014even though, in the version of the game where Icarus burns up when she touches the sun, it's a negative goal\u2014because they require Icarus to work harder in order to reach it. They also force the player to recognize Icarus's ability to move left and right, making it more likely that she'll decide to fly Icarus off the edge of the screen, and thus more likely that she'll realize that the moon might also be a valuable goal. You can push this possibility further by leaving clues: plant a trail of storm clouds along the side of the screen, leading to the edge, or\u2014to be even more obvious\u2014make the side of the screen a wall of clouds with an Icarus-sized hole in it.\n\nOnce you've gotten the player on the night side of the planet, you might give her a trickier maze of clouds to navigate, cluing her in that the moon goal is more desirable. Now you're starting to tell a really interactive story: the player can take the easier and more obvious route of flying to the sun and being burned, or she can take the more difficult and oblique route to discover this newer goal. (Notice that I haven't said anything here about telling the story explicitly, having Icarus shout out a text bubble saying, \"I MUST REACH THE GLORIOUS MOON!\"\u2014if done carefully, letting the player discover the moon on her own by accident or by following clues, the player will get the idea, and consider it that much more interesting and personally relevant because she discovered it herself). The most difficult part of the story then becomes the climax: now that the player's goal has changed, she uses all her skill and patience to reach it.\n\nIf navigating the storm cloud maze around the moon is the climax of the game's tension\u2014then maybe on reaching the moon Icarus arrives at a third, easier scene that can serve as the game's denouement. On this screen the player just has to guide Icarus to a Greek-columned house sitting at the bottom of the screen, moving left and right to avoid obstacles (maybe they can be hot stars instead of storm clouds) while gravity pulls her downward. This adds some variety and makes the scene distinct. What literary critics call the \"falling action\" of the story could be Icarus literally falling toward the surface of the moon and the game's conclusion.\n\nNotice how this story was entirely told using our basic verb set: all Icarus is doing is flying, but we've given her interesting goals that cause her to use her verb set in interesting ways. Although the Future Cowboy story is maybe less complex, it's still a story entirely told through the player's two major verbs\u2014TELEPORT and LASSO\u2014and the relationship they put her in with the objects in her world. Plan a story that will develop the player's relationship to the rules of your game, especially her verb set.\n\nIf you're writing a text game, give the player somewhere to go. Maybe the map says something like, \"Go west until you reach a mushroom-shaped rock, then travel north until you find the skull of a giant wizard, then head east to the secret cave.\" Maybe there are things for the player to do at these places, maybe not. But they should be at least interesting to visit: make them distinctive, write them descriptions that are evocative or silly or weird.\n\nTask #8: Finish the Story\n\nThe beginning and resolution of a game do a lot to tell the player how to think about the experience in between. Think about Gay Sniper, which I described in the previous chapter. The opening screen\u2014with the words \"Gay Sniper\" and the opening tinkle of a designed-to-tug-at-your-hetero-heartstrings public service video\u2014and the closing screen\u2014\"America is destroyed\"\u2014do a lot to characterize the absurdity of the political message they bracket.\n\nNot every game needs a title screen, but a title screen is very good at giving the player a context with which to understand the experience that follows. Think about what the first thing the player sees tells her about the game she's about to play. The title Future Cowboy goes a long way toward characterizing the rules of that game: along with whatever art you choose, the title gives a context both to the teleportation and to the bull-trapping in that game. If the title screen of Icarus shows a picture of Icarus on the ground, reaching toward the sun, that effectively conveys to the player that the sun should be Icarus's goal\u2014and also makes the player more excited when she realizes that the title screen is not being entirely straightforward with her about what the game's true goal might be.81\n\nThe way your game ends is the last thing the player will see, and will similarly have a great effect on how she interprets the experience as a whole. The ending could resolve the story or subvert it. Icarus could freeze to death away from the sun's heat, or might live happily ever after soaring in the lower gravity of the moon. If the game program closes, it lets the player know she's reached the resolution of the story and there's no farther to go. But what about the other game-ending scenario in the Icarus game, when Icarus touches a storm cloud and is electrocuted? When this happens, the game could restart rather than close, telling the player she hasn't found the resolution of the story yet. Getting burned by the sun could lead to the game restarting as well, to clue the player in that there's another, more desirable resolution.\n\nIn the text adventure cave, maybe the player finds something valuable or maybe something useless. Maybe, after traveling so far from the oasis, she frees a genie who offers her \"the greatest gift one can give in the desert: water!\" Maybe to enter the darkness of the cave, she has to set the map on fire and use it as a torch, destroying her means of finding her way back. Whatever she discovers in the cave, the text should make it clear to the player that this is the story's resolution: she's found the ending.\n\nThink about how the beginning of your game, what the player experiences during your game, and the ending(s) of your game form a story. What story are you telling, and how will the beginning and ending help provide the context you want for the game?\n\nTask #9: Have Someone Play It, Then Change It\n\nYou should be playing and replaying your game with every change that you make to see if it works in the way you expect. But you should be having other people play it, too. Because of your perfect knowledge of how the game works and what it contains, you can't get an accurate sense of how a player will experience your game from your own plays alone. You'll get the most information about what the player's experience actually is by having people who are unfamiliar with the game play it.\n\nGet as many people as you can to play. Try to have an idea of how experienced these players are with videogames to give you some idea of how players with different backgrounds will react to your game. I usually try to have the widest variety of players as I can, and to have people play the game both while it's being constructed and after I consider it finished. Often I'll have different people play the game at different stages, so that I can ensure that players playing the \"finished\" game are seeing it for the first time.\n\nIf you're still putting the game together, give your player an idea of where you want to take the game so that she can give you ideas for how to take it there. If it's finished, don't tell her anything. Watch her play the game and don't make any comments or assist her unless she asks. Make notes on what she does first, where she gets hung up, whether she reacts as you expected, at what points she gets confused. Also make notes on anything that doesn't work as it should\u2014and if any interesting consequences result when things don't work as they should. Someone less familiar with the game than you is more likely to stumble upon mechanical errors because she doesn't know the range of possibilities in the game. What if the player flies Icarus off the top of the screen and disappears? Nothing prevents the player from doing so because you never expected her to: you knew exactly what the goal was, so in all your plays of the game you've never flown off the top of the screen.\n\nWhen you sit someone down to play the game, keep in mind specific parts of the game that you're uncertain about. Watch how she deals with them, then after she's done, ask her what she thought and whether those parts of the game worked or not. She may have some suggestions of how they or the game as a whole can work better.\n\nNow that you have a better idea of how the player experiences your game, change it! Fix errors, smooth snags, add whole levels if you need to, implement some of the ideas your players suggested, and see if they improve the game. Have them play the game again. Have someone new play the changed game. Change everything, have the original player play again, and see which version they prefer.\n\nTask #10: Distribute Your Game\n\nWhat's the use of a game that no one can play?\n\nBefore broadband Internet, small game authors either had to find a publisher to pay the cost of manufacturing the games and putting them on store shelves, or they had to come up with clever solutions\u2014shareware, which we talked about in chapter 2, being a popular one\u2014for distributing their games. You, however, can take advantage of the speed and accessibility of the Internet to digitally distribute your game.\n\nUpload your finished game to a free file sharing website like MediaFire.com. Some tools, such as Stencyl, have official websites where authors can make their games available. Start a free blog to keep track of all your games; put them on your ancient Myspace page; ask your significant other to upload them to a secret directory of the web server at her job. All you need is some web space for the files and an address you can send people a link to.\n\nOnce you have that, spread the link around! Put it on your Facebook page, on your e-mail signature, on a postcard you staple to local telephone poles, everywhere. E-mail it to IndieGames.com. A good place to start: most tools I talk about in this book have communities with attached forums where you can show your game to peers. Post your link there, then read any feedback or criticism you receive (even though most of it will be dumb). Keep it in mind when you make your next game.\n\nJust because digital distribution makes things easier, however, doesn't mean you shouldn't be creative about passing out your game. A friend of mine who puts out his games under the label \"Amon26\" used the online publishing site Lulu.com to sell CDs containing two games of his that had previously been available on the Internet, along with scans of concept art and MP3 files of the soundtrack.82 Rob Fearon, a British author and father, asks a one-time donation in the amount of the buyer's choosing for a collection of all of his games.83 Burn your game to CD and sneak it onto a store shelf, leave a basket full of free copies at a local coffeehouse.\n\nTask #11: Make Another Game\n\nDo all of this again, using what you've learned from your first game. Make an entirely different game, use an entirely different tool, use completely different verbs than you used before. Or don't: make the same game over again, but slightly better: Icarus 2, with day, night, and twilight levels.\n\nYou're a zinester, after all. Whatever you're doing is right because you're doing it, and that's valuable. Don't worry about being brilliant or original\u2014just make sure you're creative.\n\nFootnotes\n\n77 You can download this older version at http:\/\/www.auntiepixelante.com\/?p=1240.\n\n78 http:\/\/audacity.sourceforge.net.\n\n79 http:\/\/eblong.com\/zarf\/twilight.html.\n\n80 http:\/\/www.drpetter.se\/project_sfxr.html.\n\n81 You might also be clever: if the design and presentation of the game Icarus is exactly as described above, but the title screen shows a picture of Samuel Gompers and bears the title The Story of Organized Labor in America 1886\u20131894, the entire experience essentially changes. A little of this goes a very long way.\n\n82 [http:\/\/www.lulu.com\/product\/cd\/and-you-will-be-fit-for-more-  \nthan-ashes\/6053570](http:\/\/www.lulu.com\/product\/cd\/and-you-will-be-fit-for-more- than-ashes\/6053570).\n\n83 [http:\/\/bagfullofwrong.co.uk\/bagfullofwords\/whats-in-bundle-  \nof-wrong](http:\/\/bagfullofwrong.co.uk\/bagfullofwords\/whats-in-bundle- of-wrong).\nChapter Eight\n\nGrowing Up\n\nWhen I was little I was enchanted by videogames. Kids will doodle and sketch, and like a lot of kids, I sketched videogames. I would draw potential videogame levels, imagine my own versions of Super Mario Bros. where Mario had to collect a magic sea anemone instead of a magic flower to shoot fire underwater. I would draw my own videogame characters, cut them out (or, when I was really young, have my mother cut them out), and then move them through my levels and imagine what my own videogame would be like to play.\n\nThere never seemed to be a way to take it further than that, though. I remember the excitement when my parents bought our first family computer: I knew, somehow, that people used computers to make videogames. I was never able to unlock the secret, though. I learned some QBasic, but I couldn't imagine how to get from writing text on a screen to the games of my imagination.\n\nI shelved my childhood ambition of making videogames. I tried to write fiction, because that was something creative, but I sputtered out, got sick of it, and dropped out of college. Though not before discovering Game Maker.\n\nI was lucky: I left school just as a whole generation of new game-making tools for people like me\u2014people who didn't program, people who hadn't worked in the games industry\u2014were becoming available. But the really lucky ones are the generation who are growing up with access to tools like Scratch and Stencyl. For these kids, game creation won't be the mystical, inaccessible thing it was for me. It'll be something they can actually touch. They'll be able to doodle playable games as easily as doodling a comic or writing a simple story. Imagine what this generation will do with videogames once they've grown up.\n\nWhat videogames need right now is to grow up. The videogame industry has spent millions upon millions of dollars to develop more visually impressive ways for a space marine to kill a monster. What they've invested almost nothing in is finding better ways to tell a story, and in exploring different stories to tell. That's for us to do: the people who don't have to sell thousands of copies of a game to break even, who aren't obliged to fill their games with eighty hours of content, who are beholden to no one, who are free to be silly and weird and creative and personal. Hobbyists and zinesters. You and me.\n\nEvery game that you and I make right now\u2014every five-minute story, every weird experiment, every dinky little game about the experience of putting down your dog84\u2014makes the boundaries of our art form (and it is ours) larger. Every new game is a voice in the darkness.\n\nAnd new voices are important in an art form that has been dominated for so long by a single perspective. Engineering students and venture capitalists have given us valuable pieces of culture, but there's more to the human experience than orcs, elves, and wish-fulfillment power fantasies. If people don't take videogames seriously, it's because, as an art form, they tell us very little about ourselves, so far. But authors outside of the mainstream\u2014those who haven't spent fortunes to bring their works into the mainstream\u2014have revealed much more. They have shown us a new perspective through their unconventionality, their creativity. They have shown us new ways for games to use rules, new ways of giving the players liberty to act and play within those rules, new ways to create experiences that are unique to games.\n\nDigital games\u2014through their ability to keep complex systems of rules, their ability to present audio and visual information, and their reproduceability\u2014have enormous potential for telling stories. So let's tell some stories! I want to hear about your hopes, your dreams, your fears. I want to hear about what it was like to put down your dog or to fall in love or to realize that something you thought controlled you holds no real sway over your life. I want creative people to take a creative form and do amazing things with it.\n\nI want zinesters to find new, inventive ways to distribute games. Fuck Steam and the App Store! Let's invent new networks for letting people sell their games online. Let's bring back CD compilations of games, available at your local co-op or coffee shop or by mail. And let's take advantage of what our predecessors, the shareware authors, never had: a fast, centralized Internet to distribute our games for free.\n\nThere's nothing to stop us from making our voices heard now. And there will be plenty of voices. Among those voices, there will be plenty of mediocrity, and plenty of games that have no meaning to anyone outside the author and maybe her friends. But we'll find new ways to sort that shit. And imagine what we'll gain: real diversity, a plethora of voices and experiences, and a new avenue for human beings to tell their stories and connect with other human beings.\n\nAre you excited? I'm excited just to be alive at this time. You and I are going to see some amazing things in our lifetime. Are you going to be a part of them?\n\nPublishers claimed game creation as their private territory. And for a while, they convinced a lot of people that their claim was legitimate. But that's over now. This is the age of the videogame zinester.\n\nFootnotes\n\n84 See the selectbutton forums at: http:\/\/forums.selectbutton.net\/viewtopic.php?t=33106.\nAppendix A\n\nWhat to Use\n\nThere are many different ways to make digital games, and different people are most comfortable with different tools. Here I've tried to identify the tools I think are the most useful to new authors, based on a few criteria. First, that they require as little prior knowledge of game making as possible. And second, that they're reasonably self-contained. What I mean by this: There are a lot of different tasks required in making a game. For many of my games, I use one program to draw images, another to edit and resize them, another program to record and edit audio, others to generate sound effects, and a final program to put them all together and script the game rules. Here, I've tried to recommend tools that let you produce and edit images and animations within the tools themselves, or tools that don't require you to bring in outside assets at all.\n\nWorking with each of these tools will be a different experience, though there are commonalities between them, and the basic skills you learn in one can often be applied to any game-design project, using any tool. In discussing each of the tools below, I try to describe what the experience of working with the tool is like, as well as what it can teach you about game making that you can take with you to a different tool. Remember, you're not bound to a single tool, and I encourage you to experiment with as many as possible.\n\nYou'll eventually figure out what works best for you. In the meantime, try to figure out what you can make with what you can get your hands on. The more you do, the more you'll be capable of doing.\n\nKlick & Play and the Games Factory\n\nKlik & Play was originally designed as an educational tool for use in schools: a way for kids with little computer experience to make games. It proved popular enough that Clickteam, the creators, have continued to update the program since its initial 1994 release.\n\nI think the original 1994 Klik & Play is almost the ideal place for an inexperienced game creator to start, with its simple, rules-based structure and its vast library of clip art. Unfortunately, it won't run on most contemporary versions of Windows. But in 2010, Clickteam released a free Newgrounds edition of The Games Factory, the third incarnation of Klik & Play. It has less clip art than the original Klik, but it publishes games directly to the Newgrounds website as Flash games\u2014meaning that most people with a web browser will be able to play them (although Newgrounds users, a cantankerous lot, have the power to vote down and remove any game they dislike from the site). It will only run in Windows.\n\nTo use any Klik program, simply drag a character to wherever you want it on the screen. The character image can be one you found on the Internet, one you drew yourself using the program's art functions, or one of the pieces of clip art included with the program. Once your character is in the right place, you can give it some properties\u2014how it moves, whether the player controls its movement, etc. Then, in an \"event editor\" screen, you can set specific rules for that screen: \"every time the character touches a penguin, give the player a hundred points,\" for example, or \"every thirty seconds, add another penguin to the screen.\"\n\nAbove all, Klik & Play\/The Games Factory will help you think about and plan your game in terms of rules, which is, after all, what games are. It will always be an aid to you to be able to envision the experience you want the player to have and then imagine which rules will create that experience, and Klik makes it easy to establish rules and think primarily about their effects and the interactions between them, rather than forcing you to focus on the details of lower-level scripting, displaying art, updating the position of objects on the screen, and other potential obstacles to novice creators. (Construct85 is a newer and more complicated tool designed along the same basic lines as Klik & Play and The Games Factory, and might be a good place to go after experimenting with these older tools and learning some of the basics of this style of design.) http:\/\/www.newgrounds.com\/wiki\/creator-resources\/game-dev-resources\/the-games-factory-2\n\nGamemaker\n\nGame Maker is probably the most popular of these tools because it has the easiest metaphor for managing all the pieces of a game, one that's open enough to support a really diverse collection of graphical games. Every scene in a game is a \"room\"; the characters in those rooms are \"objects\"; the images that you see are \"sprites.\" The concept of \"rooms\" makes it easy to mentally separate your game into a series of scenes and to then assemble them one by one. It's particularly useful for laying out levels: just open up a room and stick objects wherever you want them to appear on the screen.\n\nInitially, you'll script the rules of your games with icons that each mean something: \"move in this direction at this speed,\" \"wait for this long,\" \"play this sound.\" As your games get more complex, though, using the icons will become cluttered and intractable, and you'll want to transition to Game Maker's scripting language, GML. GML looks like a programming language, but it's easy to make the leap to GML from the icon-based vocabulary because most functions will correspond to the icons you've learned to use, and the basic logic of making the game is the same. From there, it's easier to make the leap to a more traditional, restrictive programming language, or to similar scripting languages in other tools.\n\nRegretfully, the corporation that distributes Game Maker, YoYo Games, has decided that in order to make a profit from their product, they have to spin it as a tool for professionals and small studios. Thus, they've hiked up the price to 40 dollars for the \"standard\" edition and 100 dollars for a version that creates web-based games. YoYo Games has always distributed both a \"Pro\" version of the program (costing 25 dollars) and a \"Lite\" version (costing nothing), with the only difference being that the \"Lite\" version was missing only a few of the flashier features of the \"Pro\" version. But the price hike coincided with a new version of the \"Lite\" program: one that stamps a watermark on every single screen of a finished game.\n\nI recommend the Lite version of Game Maker 8.086\u2014the last version before the watermark\u2014to zinesters and newcomers. This version displays a \"Made in Game Maker\" banner on the loading screen for your game, and that's it. You can make self-contained executable files that you can zip up and put on a file-sharing site like MediaFire.com, upload them to your website, anything.\n\nVersion 8.0 of Game Maker and all the earlier versions are made for Windows, but YoYo recently published a version for Mac.87 According to the company's website, though, only the paid version of Game Maker for Mac is able to produce distributable games. http:\/\/www.auntiepixelante.com\/?p=1240\n\nScratch\n\nScratch was developed by the Lifelong Kindergarten Group at MIT Media Lab to allow people of any age to create small games and animations. It's probably the simplest and fastest of the graphical game-making tools: there's almost no time between adding a rule to the game and seeing how it affects the play. Scratch is free for Windows, Mac, and Linux.\n\nScratch uses a very simple metaphor for rule construction: every rule is literally constructed, made up of a collection of jigsaw pieces that each represent part of a full sentence. Snapping a \"when up arrow is pressed\" piece onto a \"move ten steps\" piece, for example, tells a character to move forward when the player holds the up arrow key. Each rule is a single, adjustable object that sits on the \"desktop\" of the game editor and flashes whenever the game applies it. It's really easy to visually keep track of the rules of the game.\n\nIt's also designed to be as quick and dirty as possible, which is great when you're a first-time game maker without ambitious plans. There's a bunch of clip art, an option to pick a piece of clip art at random, and a \"play a musical note\" jigsaw piece that allows you to string together melodies really easily.\n\nYou can also share your games directly to the Scratch website. What's even better, though, is that you can download any game you play on the Scratch site, open it up, and see exactly how it works. See something you want in your game? Peek inside and see exactly how the author did it.\n\nScratch is a good starting place if Game Maker or The Games Factory are too intimidating. It's an excellent introduction to thinking about games as combinations of rules, and to thinking about the ways in which rules interact. It's also a good way just to get some experience making stuff, experimenting with flinging rules together and seeing what results you get.\n\nAfter you've played with Scratch, you might consider trying Stencyl,88 which uses the concept of Scratch's jigsaw pieces as the basis for its own scripting. Stencyl creates Flash games, which you can publish to Stencyl's website or to any other website. It lets you tinker with some fun, silly physics for your games. The program is still in its infancy, though, and I feel like it's a little too spread out right now; doing things that should be quick and simple take too much work. But it's likely that Stencyl will become more concise and usable as its creators update it. http:\/\/scratch.mit.edu\n\nTwine\n\nTwine was created by Chris Klimas to make branching text stories that can be published online. It's a really clean and simple way to make Choose Your Own Adventure\u2013like story games. It doesn't require much from you other than the ability to write, and it runs on Windows, Mac, and Linux.\n\nTwine keeps stories organized by drawing a map of all the passages you've written, with lines showing which scenes connect to which.\n\nPassages are just blocks of text, but you can put brackets around a phrase to create a link to another passage. That's all you need to write a story in Twine, though there are other simple grammatical structures that will let you keep track of different things the player does within the story: which passages the player has seen, for example, or how many magic beans she's carrying around. And if you know CSS, the standard method of designing web pages, you can build features on top of the basic Twine structure to do even wackier things with your finished stories.\n\nThe program outputs stories as web pages\u2014a full story being a single HTML file\u2014meaning it's really easy to upload your text games to the Internet, or to pass them to other players. It is, in fact, the best distribution model I've seen in any tool: you can run an HTML game on any computer that has both the Internet and a contemporary web browser.\n\nWhat you'll take away from Twine is an understanding of how to track the flow of a story with flags and variables: \"Has the player seen this yet?\" or \"How many times has the player done this?\" This is critical for any game involving a long and somewhat specific narrative. You'll also gain, in working with pure text, a sense of how to direct the player's interest with simple cues\u2014if you're describing a gigantic castle hallway filled with suits of armor, tapestries, and ancient candles burning in iron fixtures, how do you get the player to notice that there's a loose brick in the wall?\u2014which is a valuable skill in both text and graphical games.\n\nIf you want to try a similar tool that lets you accompany your prose with pictures and music, fiddle with Ren'Py,89 a tool for creating \"visual novels\"; essentially, Choose Your Own Adventures with pictures. http:\/\/gimcrackd.com\/etc\/src\n\nInform 7\n\nInform, created by the British poet and mathematician Graham Nelson, is another tool for making text-only games, but unlike in Twine\u2014in which the player clicks on links within the text to navigate the game\u2014Inform games require the player to type full sentences that the game interprets or responds to. Naturally this makes Inform (which is for Windows, Mac, and Linux) more complex than Twine, but not as complex to work in as one might think.\n\nInform 7 games are written in what its authors call \"natural language\" code. That means that the code you write (or other people's code that you consult for reference) looks and reads like English sentences. For example, here's a piece of text from the code of Emily Short's Bronze,90 a retelling of Beauty and the Beast, that describes the rules for a helmet that the player can discover, pick up, and wear:\n\nIt's still programming, and it still uses a very particular syntax that you'll have to learn. But it's a very simplified language\u2014compare it, for example, with the Assembly code in chapter 2, or even a random sample of Game Maker script\u2014because the tool itself does the work to reconcile the things you write with the way the system works. The Inform syntax is well documented and, as I've said, example code isn't hard to read. With a printout of the (free) Inform 7 handbook open, I made my first Inform 7 game in a weekend.\n\nInform is intended for people with no programming experience\u2014that's the philosophy behind the \"natural language\" code. But the natural language can, in a way, be deceptive, and there's a lot more trial-and-error than you might expect while you're trying to work out the syntax that Inform expects before it'll do what you want. Because Inform games involve a text parser for player input rather than just linking the player to different predetermined routes, they can allow for a much richer level of player input: for example, the player in a Twine game might click on a link saying \"loose brick\" to connect with a passage of text indicating what happens when she pushes the loose brick, while in an Inform game she might type \"push loose brick,\" \"jiggle loose brick,\" or \"ignore loose brick,\" and these three possibilities might be scripted to have entirely different outcomes. And because the raw materials of Inform are language, it can be an easy way to experiment with managing, controlling, and guiding player input without having to worry as much about actually scripting the outcomes of different verbs\u2014instead of scripting them, you just write them! But the tradeoff is a much steeper learning curve, and so if Inform is too intimidating at first, it might be easier to deal with after getting a handle on Twine.\n\nLike Twine, Inform also lets you output your finished game as an HTML file which you can then easily share online with anyone who has a web browser. http:\/\/inform7.com\n\nWarioware: D.I.Y.\n\nWarioware D.I.Y. is a product for the Nintendo DS (which you'll need to use it), meaning that this is the most strictly commercial of the tools listed here. That's its strength, though: a team of people were paid a salary to come up with the most widely accessible game-making tool they could, and Warioware is one of the simplest tools described here.\n\nThe DS uses a touch screen and stylus, so making the art for a game is just a matter of drawing it on the screen. Everyone knows how to draw, or at least doodle. The game provides a lot of similar shortcuts to creativity that I like: If you can't write music, you can hum into the microphone. Or you can grab a piece of music or a piece of game art from any of the games that come with the package.\n\nThat's another big deal: Warioware comes with ninety games made by the authors of the tool, and you can open each of them in the editor and see exactly how they work. You'll learn more about the scripting system from this than anything else.\n\nYou script objects (up to fifteen per game) by picking rules like \"move a certain way\" or \"play this sound\" from a menu, and applying them to characters on the screen. The rules are simple, but can be combined in surprising and clever ways. Here, though, is what I like the most about Warioware: each game is only seconds long, includes up to fifteen objects with five rules each, contains just a handful of art, and must be controlled entirely by tapping things with the stylus. In other words, working craftily within limitations is an invaluable skill you'll take away from Warioware.\n\nThe big failing of Warioware\u2014and of any creative tool whose distribution infrastructure is entirely owned and operated by a corporation\u2014is that it's very hard to transmit your games to anyone. You need an individual code from every person you want to digitally share a game with (from their copy of Warioware), and you can only make two games available for download at any given time. The only way to get widespread exposure is to enter a game into one of Nintendo's semimonthly themed game making competitions, the prize for which is publication and distribution, or\u2014the avenue most authors take\u2014just to record a video of the game and put it on YouTube. http:\/\/www.wariowarediy.com\n\nKnytt Stories\n\nKnytt Stories was created by Nicklas \"Nifflas\" Nygren as an expansion on his earlier Windows game, Knytt, that allows players to create their own stories using Knytt's built-in verb set: running, jumping, and climbing around a world. (Knytt Stories is also Windows-only.) It's a level-making tool, not a game-making tool, but I include it because there's so little scripting involved and because there's zero variance in the game's basic verb set, making it a good place to start learning about level design.\n\nScripting is mostly limited to placing markers (visible or invisible) that move the player to different scenes, allowing for interesting narrative progression. The bulk of making a Knytt Story is placing tiles on a grid to create a landscape for the player to navigate with the protagonist's running, jumping, and climbing abilities. The seven background, midground, and foreground layers on which tiles are placed can be hard to keep track of, but this is pretty close to pure level design, with minimal barriers between the creator's ideas and their implementation. (For something even simpler, try Love Custom,91 which lets you draw levels as simple images, bypassing the need to arrange tiles.)\n\nThough the rules of Knytt are immutable, you can add your own pictures, tiles, and sounds to your stories. You can also use the music, sounds, and hundreds of background images and tile sets that come with Knytt Stories, which let you make a complete level without ever going outside the program. When you're done, Knytt Stories will compress your level and all the graphics and sound it uses into a package file that you can upload either to a file-sharing site or to Nifflas's own forums for exchanging Knytt Stories.92\n\nBecause scripting exists almost exclusively to handle transitions, what you'll gain from Knytt Stories is a chance to experiment with and gain a sense for level design: how to arrange the elements of a game in order to create an engaging, interesting, fair experience, as well as how to tell a story. http:\/\/nifflas.ni2.se\/?page=Knytt+Stories\n\nZZT\n\nZZT was created by Tim Sweeney in 1991 as a shareware game with a powerful \"World Editor.\" The game is defunct in a lot of ways: it's a DOS game, an operating system modern computers no longer support, meaning you have to use a DOSBox emulator93 to run it (which ironically makes it equally usable on Windows, Mac, and Linux.) I include it because it's an amazing introduction to scripting and because it requires no resources outside of itself\u2014in fact, it allows for no resources outside of itself.\n\nZZT is a text-mode game, meaning that every graphic that appears in the game will be one of 256 text characters in one of sixteen colors. All the sounds in the game are generated directly by the PC speaker\u2014there are commands in ZZT's scripting that will allow you to write PC speaker music for the game to play. Thus, ZZT is entirely self-contained. Not only do the limitations force you to be clever as a game maker, but the self-contained nature of the tool means you'll never have to leave the editor to create your own art or sound. Everything you need is immediately available to you.\n\nThe other wonderful thing about ZZT is its surprisingly robust scripting language, ZZT-OOP (for \"Object-Oriented Programming\"). You'll script objects with commands that have a simple structure like \"#if contact #endgame\" (or \"if this object is touching the player, end the game\"). The objects can send and receive messages, and thus interact with each other, which is where the possibility for complexity arises.\n\nZZT is an old tool\u2014one of the first of its level of sophistication\u2014and thus doesn't have exactly the distribution potential of the above tools, although there is a large community website and archive called Z2.94 But despite its age, ZZT has the potential to teach you a lot about game logic and scripting. It's a really good and deceptively powerful introduction to thinking about how game elements behave in particular ways and communicate with one another, all the more so because its entirely self-contained nature leaves you free to focus on scripting and design. It's a weird old game, but not a bad starting point at all. http:\/\/zzt.belsambar.net\/versions\/\n\nFootnotes\n\n85 http:\/\/www.scirra.com.\n\n86 Again, http:\/\/www.auntiepixelante.com\/?p=1240.\n\n87 http:\/\/www.yoyogames.com\/gamemaker\/mac.\n\n88 http:\/\/www.stencyl.com.\n\n89 http:\/\/www.renpy.org.\n\n90 You can read more about Bronze and download the game at http:\/\/inform7.com\/learn\/eg\/bronze\/index.html.\n\n91 http:\/\/love-the-game.com\/custom.html.\n\n92 Nifflas's forums can be found at: http:\/\/nifflas.lpchip.nl\/index.php?board=39.0.\n\n93 Find out more about DOSBox at: http:\/\/www.dosbox.com.\n\n94 http:\/\/zzt.org.\nAppendix B\n\nZinester Games\n\nEach of the games below was created by just one or two people. Each is valuable, and I encourage you to play them if you can. Many of them, having been made in Game Maker, are playable in Windows only. (Borrow a friend's computer if you need to!) I've avoided including games I mention elsewhere in the book: consider this \"further reading.\" I've tried to pick games that are simple enough that, in playing them, you should be able to see how you could make them or something like them. Each of them also shows off a different idea about the way digital games can be and what they can give us.\n\nStriptease\n\nStephen \"increpare\" Lavelle is a prolific queer game creator from Ireland. Striptease, one of his most affecting works, uses the simple rules of an abstract sliding tile puzzle to achieve a narrative that explores themes of objectification, alienation, and the way we internalize and externalize our bodies. http:\/\/www.increpare.com\/2009\/05\/striptease\n\nReset\n\nBritish author Robin Burkinshaw's Reset is a game set to a single piece of music: \"Rest to Reset\" by Trash80. Every event in the game responds to that music, and the player's fuel supply only lasts as long as the song. The game, which is neither won nor lost, is designed as a way of experiencing the song. It was made in Game Maker. http:\/\/www.roburky.co.uk\/?p=13\n\nAll the Better to See You\n\nBento Smile works in the games industry as an artist and creates small games in her own time. All the Better to See You is a Ren'Py game she created for Valentine's Day of 2010. It uses the simple mechanical conceit of diminishing player options to characterize a relationship that becomes increasingly dominated by one party. http:\/\/bentosmile.com\/mini-games\/all-the-better-to-see-you\/\n\nAll of Our Friends are Dead\n\nAmon26 is a visual artist, musician, and game author. All of Our Friends Are Dead, a running, jumping, and shooting game inspired largely by id Software's 1993 game DOOM, was made in Game Maker using an \"engine\" that someone else had programmed. Amon's lack of complete control over his own creation serves the game's disjointed, hostile tone. http:\/\/amon26.site11.com\/1_5_Games.html\n\nThe Lake\n\nThe author \"agj\" is interested in the expressive capabilities of digital games, and his works tend to have short, simple stories. The Lake, made in Construct, is based on a story fragment from H.P. Lovecraft's journal. The game uses only two buttons\u2014the left and right oars of a small boat\u2014to achieve a subtle and unsettling story. http:\/\/forums.tigsource.com\/index.php?action=printpage;topic=3750.0\n\nSexy Hiking\n\nAuthor Jazzuo likely hit on the rules of Sexy Hiking by accident, then designed an intricate and challenging game around them. Jazzuo uses ugliness of presentation\u2014the game is assembled from crude drawings and animation and stolen songs and clip art\u2014as an avenue to unbounded creativity of design. Sexy Hiking was put together in Game Maker. http:\/\/www.jazzuo.com\/games\/sexyhands\/sexy_hiking\/sexyhiking.htm\n\nThe Strange and Somewhat Sinister Tale of the House at Desert Bridge\n\nVerena and Jonas Kyratzes are a filmmaker and writer couple living in Germany. The House at Desert Bridge, a story about the importance of whimsy as a tool against oppression, is implemented very simply\u2014play consists almost entirely of clicking on crayon illustrations, reading text, and typing information when prompted. http:\/\/www.jonas-kyratzes.net\/games\/the-strange-and-somewhat-sinister-tale-of-the-house-at-desert-bridge\/\n\nBushido Edge\n\nBushido Edge, a Games Factory game by author \"Pizza Time,\" is inspired by the 1997 PlayStation game Bushido Blade. It's a distillation of what's most interesting about Bushido Blade or any two-player fighting game: the importance of reading and anticipating your opponent. It's for two players only. http:\/\/badmspaint.livejournal.com\/195733.html\n\nDigital: A Love Story\n\nChristine Love is a queer Canadian writer of fiction and digital stories. Her storytelling primarily concerns young love, particularly the kind that is mediated by technology. Digital: A Love Story is a \"period piece,\" modeled after the BBS networks of the 1980s and the culture they propagated. It was made in Ren'Py. (See also the semi-sequel, Don't Take It Personally, Babe, It Just Ain't Your Story, which is a meditation on our changing ideas of privacy.95) http:\/\/www.scoutshonour.com\/digital\/\n\nSuper Green Shell Bros.\n\nLeon \"L\" Arnott pays attention to the smaller details of game experiences. Super Green Shell Bros. is a takeoff of Super Mario Bros., or rather a specific part of the experience of Super Mario Bros.: kicking a shell and chasing it as it mows over enemies.\n\nA short game, it has a perfect pace that builds to a brilliant climax. Almost all of the graphics in this game are sampled from other games. http:\/\/l.j-factor.com\/gamemaker\/#\n\nThe Baron\n\nVictor Gijsbers is a Dutch philosophy professor at Leiden University. His text game, The Baron\u2014made in Inform\u2014beckons the player into a difficult moral allegory, then asks her to decide how a heinous situation is resolved, or whether it even can be. The Baron explores themes that a commercial game publisher would be terrified to touch. http:\/\/ifdb.tads.org\/viewgame?id=weac28l51hiqfzxz\n\nSonic 2 XL\n\nSonic 2 XL, by \"Captain Bozo\" and \"Ranger,\" is a hack of Sonic the Hedgehog 2 for the SEGA Genesis, and will require a Genesis emulator to play. In this hack, the gold rings Sonic collects are replaced with fried onion rings that make him fatter and slower. It's interesting both on a thematic level\u2014as commentary on game-related obesity and consumerist collection goals in games\u2014and on a play level: navigating the game without collecting so many rings as to become fat is a challenging variation on the original game. http:\/\/forums.sonicretro.org\/index.php?showtopic=21739\n\nFootnotes\n\n95 http:\/\/scoutshonour.com\/donttakeitpersonallybabeitjustaintyourstory.\n\nAbout the Author\n\nANNA ANTHROPY is a prolific game developer and critic. She is the creator of games such as Calamity Annie, Mighty Jill Off, and Lesbian Spider-Queens of Mars among many others. She lives in the East Bay area, California with her pet girlmonster and two lovely cats. She wants you to stop reading this and GO MAKE A GAME.\nAbout Seven Stories Press\n\nSeven Stories Press is an independent book publisher based in New York City. We publish works of the imagination by such writers as Nelson Algren, Russell Banks, Octavia E. Butler, Ani DiFranco, Assia Djebar, Ariel Dorfman, Coco Fusco, Barry Gifford, Hwang Sok-yong, Lee Stringer, and Kurt Vonnegut, to name a few, together with political titles by voices of conscience, including the Boston Women's Health Collective, Noam Chomsky, Angela Y. Davis, Human Rights Watch, Derrick Jensen, Ralph Nader, Loretta Napoleoni, Gary Null, Project Censored, Barbara Seaman, Alice Walker, Gary Webb, and Howard Zinn, among many others. Seven Stories Press believes publishers have a special responsibility to defend free speech and human rights, and to celebrate the gifts of the human imagination, wherever we can. For additional information, visit www.sevenstories.com.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n## Contact!\nby the same author\n\nHEAVEN'S COMMAND: AN IMPERIAL PROGRESS\n\nPAX BRITANNICA: THE CLIMAX OF AN EMPIRE\n\nFAREWELL THE TRUMPETS: AN IMPERIAL RETREAT\n\nCOAST TO COAST\n\nCORONATION EVEREST\n\nVENICE\n\nCONUNDRUM\n\nTRIESTE AND THE MEANING OF NOWHERE\n\nA WRITER'S WORLD\n\nEUROPE: AN INTIMATE JOURNEY\n\nHAV\n\nFISHER'S FACE\n\nA VENETIAN BESTIARY\n\nSPAIN\n\nAMONG THE CITIES\n\nTHE GREAT PORT\n\nTHE HASHEMITE KINGS\n\nHONG KONG\n\nLINCOLN\n\nMANHATTAN '45\n\nTHE MARKET OF SELEUKIA\n\nSOUTH AFRICAN WINTER\n\nTHE SPECTACLE OF EMPIRE\n\nSYDNEY\n\nTRAVELS\n\nTHE VENETIAN EMPIRE\n\n## Contact!\n\nA Book of Encounters\n\n## JAN MORRIS\n\nW. W. Norton & Company\n\nNew York \u2022 London\nCopyright \u00a9 2009 by Jan Morris\n\nFirst American Edition 2010\n\nFirst published in Great Britain under the title\n\nContact! A Book of Glimpses\n\nAll rights reserved\n\nFor information about permission to reproduce selections from this  \nbook, write to Permissions, W. W. Norton & Company, Inc.,  \n500 Fift h Avenue, New York, NY 10110\n\nManufacturing by Courier Westford  \nProduction manager: Anna Oler\n\nLibrary of Congress Cataloging-in-Publication Data\n\nMorris, Jan, 1926\u2013\n\nContact!: a book of encounters \/ Jan Morris.\n\np. cm.\n\nISBN: 978-0-393-07640-0\n\n1. Morris, Jan, 1926\u2013\u2014Travel. 2. Voyages and travels. I. Title.\n\nG465M658 2009\n\n910.4092\u2014dc22 2009052193\n\nW. W. Norton & Company, Inc.  \n500 Fift h Avenue, New York, N.Y. 10110  \nwww.wwnorton.com\n\nW. W. Norton & Company Ltd.  \nCastle House, 75\/76 Wells Street, London W1T 3QT\nDedicated to the whole lot of them\n\n## Contents\n\nIntroductory\n\nA sighting in Texas\n\nA snatch of sound in Morocco\n\nThe touch of a hand at home\n\n## Introductory\n\n'Contact' is a noun of varied nuances, not all to my taste. Ambitious businessmen make useful contacts at golf clubs or race meetings. Diplomats and journalists are urged to cultivate their contacts. Spies have contacts and so do publicity executives, and people with poor eyesight. On the other hand contacts turn the lights on, start the engine, send the rocket off, launch the movie Spitfires into glory, and it is this meaning of the word that gives me the title of this album.\n\nIn a lifetime of travel and literature I have written relatively little about people. Places, atmospheres, histories have figured far more in my all too often purple prose. But people everywhere, nevertheless, have been sparks of my work, if often only in glimpses\u2013a sighting through a window, a gentle snatch of sound, the touch of a hand\u2013and it is mostly such fugitive moments and observations, scattered across half a century and forty-odd books, that I have here gratefully plucked out of their literary obscurity.\n\nOften I have given them only a few lines, or a paragraph; occasionally the people have known me as James rather than Jan, because until 1972 I wrote in the persona of James Morris; but my fleeting contacts with them have fuelled my travels down the years, generated my motors, excited my laughter and summoned my sympathies. I write of them here more or less as I wrote of them at the time, and I recall them not in any chronological or geographical order, but jumbled. Their locations will generally be self-evident, and I have included dates only when they seem essential to the historical sense of the piece. Otherwise all these encounters simply occurred between Here and There, to Him or Her, after Then and before Now.\n\nRich and poor people are remembered here, young and old, grand and humble, primitive and exquisitely civilized, named and anonymous, in the particular and in the general. Every one of them, of course, deserves more than the handful of words I have resurrected in these pages: but there it is, they are seldom friends or even acquaintances, only contacts.\n\nTrefan Morys, 2009\n\n## Contact!\n## A sighting in Texas\n\nOn my fourth day in the city\n\nI looked through the window\n\nand saw a dreamlike figure sauntering by.\n\nHe had a sack over his arm, and a stick over his\n\nshoulder,\n\nand he wore a high-crowned hat and a cloak, I think,\n\nand he strolled past easy, insolent and amused.\n\nMy heart leapt to see him.\n\n'Who was that?' I cried, rushing to the window,\n\n'that man with the stick, and the high-crowned hat,\n\nand the sack on his arm?'\n\nMy hostess returned me reprovingly to our\n\nconversation.\n\n'I saw nobody,' she sweetly and carefully said.\n\n'But tell me, have you had time to see our new Picasso\n\nin the Fine Arts Museum?\n\nAnd will you have an opportunity to meet with\n\nMrs Oveta Culp Hobby?'\n\n##\n\nManhattan dialogue\n\nI chanced one day, off the joggers' circuit in Central Park, to come across a young black man fast asleep upon a bench below the lake. His overcoat was thrown over him, his books were placed neatly side by side upon the ground. His head upon his clasped hands, as in kindergarten plays, he was breathing regularly and gently, as though bewitched. Even as I watched, a grey squirrel, skipping across the green, leapt across his legs to the back of the bench, where it sat tremulously chewing, and almost at the same time there arose a brisk gust of wind, tangy with salt.\n\nA scatter of leaves and fallen blossoms came with it, flicked and eddied around the bench. The squirrel paused, twitched and vanished. The black man opened his eyes, as the breeze dusted his face, and, seeing me standing there bemused, smiled me a slow sleepy smile. 'Be not afeared,' I said ridiculously, on the spur of the moment, 'the isle is full of noises.'\n\n'Yeah,' the man replied, stretching and scratching mightily in the morning. 'Bugs, too.'\n\nAt the hotel door\n\nI was going out through the door of the Albergo Savoia Excelsior in Trieste when a man simultaneously entered. We bumped into one another, our bags and luggage got mixed up, and we both apologized. He was a theatrical-looking character, with a camel coat slung over his shoulders\u2013perhaps one of the opera singers from the Teatro Verdi. When we had disentangled ourselves he stood there for a moment, motionless.\n\n'Where are you from?' he said.\n\n'Wales.'\n\n'Wales! How wonderful!'\n\nOh you splendid liar, I said to myself, you've never heard of the place. There was a pause. I laughed, and so did he. He shook my hand in both of his. We lingered for a moment and parted. When I think of Trieste, lust and love I often think of him.\n\nSelf-discipline\n\nAt Kanpur, in India, I came across a man with whom I felt an instant affinity. That he was deeply unhappy was obvious, but he numbed his misery by touching things. Day and night he wandered the streets of the city, earnestly and methodically touching windows, doorposts, lamp standards, apparently to strict unwritten rules. Sometimes he appeared to feel that he had neglected his task, and did a street all over again, paying a still more diligent attention to the doorknobs. I spoke to him one morning, but he responded only with an engaging preoccupied smile, as if to say that, although some other time it would be delightful to have a chat, that day he simply hadn't a moment to spare.\n\nHonesty\n\nSometimes Sydney seems to be inhabited chiefly by school-children, children kicking pebbles across bridges, children racing fig leaves down the channels of ornamental fountains, children clambering like invading armies all over the Opera House, or mustered in their thousands in the New South Wales Art Gallery. They seem to me a stalwart crew. 'Now this is a Picasso,' I heard a teacher say in the gallery one day. 'I'm sure you all know who Picasso was.' 'I don't,' piped up a solitary small Australian at the back, and I bowed to him as the only absolutely honest soul in sight.\n\nNot Chopin\n\nWhen communism failed in Poland, materialism took over. There were bright new shops, posh hotels, plenty of cars, all the usual paraphernalia of capitalism. 'Nice car,' I remarked one day to the man who drove me to the airport in his big new Volvo. He shrugged his shoulders and looked at me with a dry smile. I knew what he meant. 'Well, no,' I added in afterthought, 'I suppose it's not Chopin' and he knew what I meant, too.\n\nAfter the ball was over\n\nNot so long ago the chief celebrants of the legendary Venetian carnival were the children of Venice, who bought their funny faces and moustaches from the chain stores and emerged to saunter self-consciously through the city in fancy dress.\n\nOn the final evening of the festivities I was walking home when I saw before me, in a hurried glimpse, three small figures crossing a square from one lane to another. In the middle walked a thin little man, his overcoat rather too long for him and buttoned down the front, his gloves very neat, his hat very precise, his shoes very polished. Clutching his right hand was a tiny Pierrot, his orange pom-pom waggling in the half-light. Clutching his left hand was a minuscule fairy, her legs wobbly in white cotton, her skirt infinitesimal, her wand warped a little with the excitement and labour of the day. Quickly, silently and carefully they crossed the square and disappeared from view; the fairy had to skip a bit to keep up, the Pierrot cherished a sudden determination to walk only on the lines between the paving stones, and the man trod a precarious tightrope between the indulgent and the conventional.\n\nHow small they looked, and respectable, I thought to myself! How carefully their mother had prepared them, all three, to survive the scrutiny of their neighbours! How thin a reflection they offered of Venice's rumbustious carnivals of old, her Doges and masked patriciate, her grand lovers, her tall warships and her princely artists! How touching the little Venetians, tight buttoned in their alleyways!\n\nThe student's request\n\nIn the shadowy underneath of a bridge in Isfahan a student sits, dangling his feet over a sluice and reading a book. His face is dark and meditative, and his air of poetic concentration is all one asks of a Persian student. He has caught sight of us, in a dreamy sort of way, and as he buttons his jacket, gathers his notebooks and moves sidelong in our direction, we recognize one of the more endearing hazards of modern travel, the Student of English. We are too late to escape. 'Sir!' he cries. 'Madam!' fluttering his notes and bearing down on us. 'Allow me please to ask you one question, before you leave the bridge: is it permissible or not, in the English language, to pursue a gerund with a participle? And would you be kind enough to comment on my pronunciation in the following passage, Exercise 12? Sit down, sir; sit down, madam! Be comfortable!'\n\nD\u00e9j\u00e0 entendu!\n\nOn a winter day in Zagreb a man bundled in a greatcoat is playing an instrument of his own invention, consisting of rows of wine and mineral-water bottles strung on a contraption rather like a washing line and tuned by their varying contents of liquid. He is playing with great delicacy a piece I know well, but can't for the life of me place, and around him a smiling crowd has gathered, amused by the instrument, touched by the tune. In the front row of the audience a small child of two or three in a woolly blue and white jumper suit, with hat to match, is performing a shuffly sort of dance to the music. I am curiously affected, partly because of the endearing busker, partly because of the sweetly familiar music\u2013oh, and hang on, I think I remember what that tune is. Isn't it one of those charming Fritz Kreisler fripperies they used to play at palm-court cafes, with a lady violinist in a satin blouse, and the grammar-school music master moonlighting at the piano? 'Sch\u00f6n Rosmarin'\u2013isn't that it?\n\nBreath of the woods\n\nA junior functionary all but monopolized my attention during my visit to the Legislature of the Canadian North West Territories at Yellowknife. She was about fifteen years old, I would guess, indeterminately Caucasian, Indian, Inuit or M\u00e9tis, and tremendous fun. Busy as she was taking perpetual missives from one member to another, bobbing incessantly to the chair en route, she managed to elevate the whole session to a jollier and more sensible level. She laughed to herself and to others, she did her bows with a wonderfully comic jerkiness, she stuck her tongue out at her colleagues, she yawned, she hitched her tights up, she cheerfully swung her legs when she was sitting down and walked in a delightfully insouciant way when she was on her feet. I loved this irrepressible child of the north: the legislators droned on as legislators will, but she brought a breath of the woods inside.\n\nCosta del Sol, 1960s\n\nSometimes I went out on the beach at Fuengirola to watch the fishermen at work. It could be heartbreaking to see. They worked like slaves, wading into the sea with their huge heavy net and laboriously hauling it in, inch by inch, hour by hour up the sands: so much depended on the catch, so much labour and good humour had been expended, so many children were waiting to be fed, so many anxious mothers hoping\u2013and when at last the haul appeared, often enough only a dozen small sardines in the mesh of the net, the fishermen carefully cleared up their tackle and dispersed to their homes in weary silence.\n\nCaravan of martyrs\n\nIn the summer of 1958 the young King Feisal of Iraq was assassinated, leaving his contemporary and close relative King Hussein of Jordan isolated upon his own threatened throne. I was in Jordan when, a few days later, Hussein made a public declaration about the tragedy. His face was lined and tired, and moisture glistened in the corners of his eyes. Ministers, officials, officers and security guards were grouped behind his chair. Clearing his throat huskily, the King said slowly: 'I have now had confirmation of the murder of my cousin, brother and childhood playmate, King Feisal of Iraq, and all his royal family.' He paused, his eyes filling, his lip trembling, a muscle working rhythmically in the side of his jaw, and then he said it again, in identical words, but in a voice that was awkwardly thickening. 'I have now received confirmation of the murder of my cousin, brother and childhood playmate, King Feisal of Iraq, and all his royal family.' And raising his head from his notes, Hussein added in his strange formal English: 'They are only the last in a caravan of martyrs.'\n\nTrial of an alleged spy\n\nThe court sat on a kind of stage against a background of opaque white curtains. In the centre were the military judge and his two assessors, generals all, immaculate in dove grey and sitting in their tall wooden chairs like bulky Buddhas. To the left sat the prosecutor, the procurator-general of the Soviet Union, a heavy and formidable lawyer in a sombre blue uniform. The lights were blinding\u2013chandeliers, strings of bulbs, floodlight, cameramen's flashlights, bathing the whole scene in chill brilliance and giving the members of the court a waxen cosmetic look. Punctually at ten Gary Powers was led in with an escort of two young soldiers in olive-green jackets and blue trousers. He wore a blue Russian suit too large for him, so that he had to hitch up his sleeves now and then, and they put him in the wooden dock, like a big child's playpen beneath the floodlights, and the sentries stood at attention beside it as beside a catafalque. Powers was obviously frightened, and so was I.\n\nThe one left behind\n\nThrough the windows of a lakeside restaurant at M\u00f6lln I watch four German children playing. Their families are lunching inside, and I would judge the children to be between six and ten years old. The two boys are always in the lead, dashing about the lake, the girls follow enthusiastically behind. One is slim, blonde and pretty, and wears a floral dress she likes to flounce about. The other is plain and plump, and wears a blue anorak, with sleeves too long for her, over a short tartan skirt. The plain girl is always last. She can never quite keep up. When they run out to the end of the jetty, she is always left behind. When they rush helter-skelter into the restaurant to speak to their families, the door closes behind the other three and there is a long pause before, panting heavily, the short fat girl opens it again with difficulty and stumbles in. I like her best of the quartet\u2013she tries so hard, laughs so gamely, struggles so constantly to tuck up the sleeves of her anorak. I feel for her, too. However when they all scamper out of the restaurant again, and I offer her a smile as she passes my table, she returns a most malevolent glare.\n\nHaute couture\n\nI was invited to write about one of the Paris summer collections for an American magazine, and sat incongruously in the front row among the condescending New York buyers and unbelievably ugly princesses of American fashion journalism. How awful they looked, draped in their furs, red taloned, emaciated to the point of grotesqueness, while all about them graceful exponents of the art of French allure glided silkily around the room and along the catwalk. The audience otherwise seemed to be composed chiefly of characters from Proust.\n\nDialogues on the Orient Express\n\n'I've always said,' observed one American matron to another, 'I'm not going to be a possessive mother, because his was'\u2013and she jerked her head in the direction of her husband in the next seat. The two ladies eyed him speculatively. 'He'll be no good to us in Venice,' said the other. 'He'll be lost with those gondolier people.'\n\nYoung English wife, on her honeymoon, I guessed: 'Oh, look at the castle. Isn't that a lovely castle?' Young English husband: 'It's a castle. A castle is a castle. You've seen castles before.' She relapsed into thoughtful silence. He returned to his thriller.\n\nSaid an American man, to me: 'You gotta read this book. I've been reading it all the way since London. It's called God Owns My Business. God Owns My Business, that's the title. The guy who wrote it, he's a very low-key man, but he's got a sign above his store, \"Christ Is My Manager\". When do we get to Innsbruck? We might get a hamburger there.'\n\nIf you prick us?\n\nI shared a taxi one day with a lady in a blue silk turban, who was visiting Washington and was about to meet her daughter for lunch at a Hot Shoppe. Down the great thoroughfares we drove, and all the memorials of the American splendour passed us one by one, granite and concrete, obelisk and colonnade. My companion drew my attention now and then to a White House or a Treasury, but it was as we passed the Capitol itself, and were deploring the state of the world in general, that she spoke the words I best remember: 'I sometimes wonder, oh, what kind of a world are we bringing our children into, when you have to pay a quarter for a doughnut?'\n\nTwenty-five cents for a doughnut! Even Americans bleed.\n\nThe sprig of rosemary\n\nI was driving along a road in Portugal when I spied a tray of oranges for sale outside a cottage. It was blazing hot, and I stopped to buy some. Nobody attended the fruit, so I selected three oranges for myself and knocked on the cottage door. Nobody came. There was no sign of life. I peered through the window, I walked around the back, and in the end I opened the front door. It was very dark inside, but when my eyes accustomed themselves I saw that fast asleep in a corner of the room was a small old lady. I coughed and shuffled my feet, and without a start she awoke. Her very first reaction was to smile. Her second was to reach for her straw hat on a nearby chair and put it carefully on her head. She accepted my few coins for the oranges but then, hustling me kindly outside, took two apricots from another box and gave me them as a present. All the time she smiled, and bustled around looking for other kindnesses to perform. When we had said goodbye, and I had returned to the car, she came running out of the garden gate again, smiling still, clutching her hat on her head, to give me a sprig of rosemary.\n\nDifferent responses\n\nMy family and I lived for a time near the Swiss frontier in France. On the French side the gendarmes were jolly, careless, and often had wine on their breaths. On the Swiss side the police were cool, diligent, courteous and unsmiling. My car in those days was a Rolls-Royce, grand, decorative and elderly. The French gendarmes were delightedly amused by this vehicle, and sometimes asked permission to sit at its wheel, or try the squashy grey leather seats behind. To the Swiss border police, on the other hand, a Rolls-Royce was an image of wealth, and a quaint middle-aged example like mine, not old enough to be a valuable antique, certainly not new enough to be a status symbol, seemed to confuse their responses. They habitually greeted us with a mixture of respect and condescension, covering all contingencies.\n\nFlying the flag\n\nThe Aboriginal flag of gold, black and yellow was hoisted above Sydney Town Hall, but was soon pulled down again. By the time I reached the park the Aboriginal Day rally seemed to have fizzled out, too, and all I found was a small huddle of dark-skinned people around an open bonfire, surrounded by litter on the edge of the green. They greeted me with wan concern, offering me beer out of an ice bucket, sidling around me rather and occasionally winking. A small thin boy with cotton wool stuffed in one ear wandered here and there leading a black puppy on a string. Others kicked a football about in the gathering dusk. A strong smell of alcohol hung over us, and the man with the bucket urged me quietly, again and again, to have one for the road. Had the rally been a success? I asked. 'Yeah,' they said, and looked into the fire.\n\nThe Frenchest person\n\nThe Frenchest person I ever met was Yves Saint-Laurent, the couturier. He was utterly French. He told me that the only books he ever read were eleven volumes of Proust's \u00c0 la recherche du temps perdu, over and over again, but that the twelfth and last volume he had never read at all\u2013saving it up, I suppose, for a last splurge of Frenchness on his deathbed chaise longue. Everything that was French seemed to be embodied in him, even a bit of the old gloire, for there was a distinctly grand manner lurking behind his melancholy shyness, and he lived in a grand style too. Saint-Laurent liked to call himself an artisan, and the little world of craftspeople he had built up around him, the dedicated world of cutters, shoemakers, milliners and tailors, seemed to me a true ornament of French civilization, and a vindication of French pride. He told me that all I needed for elegance was one dress, a pair of jeans, some blouses and a raincoat. I asked him if he was consciously contributing to the splendour of France, and he smiled rather distantly. He was, he said, he was.\n\nA fine scoundrel\n\nA celebrated, or notorious, rebel leader in Oman was Suleiman bin Hamyar, who owned the only motorcar in the Green Mountains. I saw him arriving in it one day for a political parley at the village of Nizwa, and the sight of his American convertible careening recklessly out of the road-less mountains was wonderfully inspiriting. The roof of the car was closed, but on the boot there sat a Negro slave, armed with a rifle, with his feet sticking through the back window into the inside of the car; and when it stopped this slave jumped off like lightning, as promptly and neatly as any duke's footman, and opened the door with a flourish.\n\nI walked across to meet the old sheikh, and he greeted me with an expression of unfathomable foxiness, suggesting to me instantly some infinitely clever beast in Aesop, about to hoodwink a goat. Suleiman was a big man with a powerful face, rather Dickensian in concept, and a triangular grey beard. On his head was a twisted blue and white turban. His aba was blue, gold-edged and filmy. In his hand was a cane with a carved end, on his belt a curved dagger of splendid ostentation.\n\nMy own instincts told me that this fine scoundrel should be instantly decapitated, for the public good, but I was rather glad, all the same, when he later reappeared from the parley intact, and was driven away into his mountains with only minor (and I felt sure temporary) modifications of his manner.\n\nEnjoy yourself!\n\nIt took many hours, on a Sunday evening, to drive from Trieste the few miles into Tito's Yugoslavia. I can see now the hundreds of cars lurching and overheating in the gathering dusk, the ad hoc hamburger bars beside the road, the occasional truck bullying its way up the queue by sheer weight and horsepower, pale weary faces at the windows of buses and at last the dim-lit frontier post, and a joyless official with a red star on his cap. A slow flicking through the pages of our passports, a silent gesture of release, and away with us into the communist half of Europe. 'Cheer up,' I said to the frontier official once. 'Enjoy yourself,' the man lugubriously replied.\n\nThe helmsman\n\nOur helmsman, although he is surrounded by talkative friends and relatives of all ages, is ever attentive to his craft and courteously watchful of me: and sometimes, indeed, sailing in such a ramshackle sampan crablike against the current from one island to another, to the laboured chugging of diesel engines and the creaking of timbers\u2013sometimes I feel I would like to be assimilated into Chineseness myself, and sail these waters under Chinese helmsmanship for good.\n\nA literary test\n\nIn Newfoundland in those days it was necessary to find a guarantor before one could cash foreign money orders. Knowing nobody in town, and finding that the public library had a copy of a book of mine about Venice, I introduced myself to the librarian and asked her to endorse a travellers' cheque. How could she confirm, she sensibly demanded, that I was who I said I was? By a simple literary test, I suggested: surely nobody else on earth could recite by heart the last line of my book on Venice, which she had upon her own shelves.\n\nSolemnly she reached for the volume. Nervously I stood at her desk while she turned to the final page, and ran her eye down the paragraph to the end of it. 'Well?' she said. I cleared my throat. The concluding words of my book were not very stately. 'No wonder,' I mumbled then, feeling distinctly disadvantaged, 'No wonder George Eliot's husband fell into the Grand Canal.' Without a flicker that librarian of old St John's closed the book, returned it to the shelf and authorized my money.\n\nJudgement\n\n'What's this?' demanded a Soviet customs official curtly one day, extracting a typescript from my baggage and simultaneously eyeing my then epicene figure. 'It's a psychological novel,' said I. 'Oh, a psychological novel,' he replied in a voice of infinite understanding, as though I had shared a confidence with him; and carefully repacking the script in my suitcase, he waved me through.\n\nByronic\n\nA symbolic figure of the South African tragedy in the years of apartheid was Christopher Gell, one of the most inspiring of the liberal activists, whose name was almost legendary, who was a unique source of guidance and information, and who lived as it happens in an iron lung. From there he cocked a perpetual snook at the Afrikaner Nationalists. A well-trodden path led to his little house, and brought a constant stream of people interested in the African Risorgimento, and anxious to meet this strange Byronic figure.\n\nGell received them in his lung. He was tall, painfully cadaverous, immensely vivacious. He wore glasses and had one arm suspended above him in a kind of sling. Books and elaborate filing cabinets lined the room, and his table was littered with proofs and pamphlets and letters. Often the telephone rang and Gell launched himself into a farrago of opinion, prejudice and argument till the voice of the man at the other end of the line sounded breathless and dispirited, Gell's face was wickedly aglow and the conversation ended in intellectual annihilation.\n\nThen he turned to you. 'Now then, let me put you straight about these bloody Nats.' He presented his case with tremendous energy, witty, outrageous, caustic, irrepressible, pausing sometimes to scribble a name down for you or dash off a note of introduction, swearing, joking, laughing, in a most extraordinary flood of stimulation and conviction. Slowly, though, his damaged physique ran down. His breathing became gasping and spasmodic, his face more strained with effort, and the gusto drained from his body before your eyes, like the symbolism of a Gothic painting. He would still be talking as you left him, though, and his anxious humorous eyes would be looking at you in the little mirror above his head. 'Of course we're intolerant,' he would say as you left him. 'We have to be. We'd never get anywhere with these stiffs if we weren't.'\n\nDoubly damned\n\nAt the Rock Hotel in Gibraltar I overheard two very old-school American matrons commenting upon the grumpy hotel porter who had just dumped their bags unceremoniously on the lobby floor. 'What an unpleasant man!' said one. 'What can you expect?' responded the other. 'He is British, my dear, and male.'\n\nDung and diplomacy\n\nOne day I walked up to the royal palace in Brussels, and just as I arrived a plenipotentiary emerged from its gates in a big black car after a diplomatic presentation to the King of the Belgians. A footling squadron of cavalry awaited him in the ceremonial square outside. Its officers wore romantic white cloaks. Its troopers, in slightly cockeyed bearskins, as in musical comedies, included some sceptical-looking horsemen of the old-sweat school, and at least one rosy-cheeked woman. When they clattered and bounced away with the ambassadorial Cadillac, a municipal road-sweeping truck came trundling around the place where they had mustered, cleaning up the horse shit. Its driver told me he spent his days doing it. There were so many embassies, missions and international institutions in Brussels, he said, that the palace cavalry was always at it\u2013and sure enough, as he spoke, the horsepersons, having disappeared round the corner with their fluttering lances, came ridiculously back again with another couple of limousines.\n\nSteamboat Gothic\n\nLongwood is the oddest of the mansions of Natchez, Mississippi, where Southern myth and prejudice are very powerful. The house was begun shortly before the Civil War, a wild architectural extravaganza, and Northern workmen were brought in to work upon it. Soon after the war began they dropped their tools and left, leaving the house unfinished to this day, with their hammers and wheelbarrows and paintboxes still lying about, ladders propped against walls, scaffolding still in place. Octagonal, domed and balconied, it stands in a wooded garden as a grotesque monument of Steamboat Gothic, its glassless windows gaping. Only the ground floor is inhabitable, and in it there lives all alone Mr John Price, white-haired and nearsighted, who once entertained us to a very enjoyable lunch.\n\nThis was a feat, for he seems usually to live entirely on marshmallows and fig rolls, but a jolly cousin of his came in to do the cooking, and the results were capital. Mr Price had no tablecloth handy, but a sheet on the table served just as well, and we ate Southern fried chicken in enormous quantities, and blancmange and cheese, finishing with either fig rolls or marshmallows, I forget which.\n\nWar story\n\nAn Australian boy once told me that his father had recently taken part in a military parade. 'What kind of a hat did he wear?' I asked for something to say. One of those hats, he replied, which were flat on one side but turned up in the other. 'I know,' I said, 'like they used to wear in the Great War.' There was a silence for a moment, and then the boy spoke. 'I hate the Great War,' he said, and my heart turned.\n\nChinese jingles\n\nEarly in a performance I attended in one of the regime's Children's Palaces, by an orchestra of children under the age of five, the virtuoso lead xylophonist happened to get herself a full tone out of key. She never appeared to notice. Nor did any of the other performers, all dimples, winsome smiles and bobbing heads up there on the stage. On they went in fearful discord, tinkle-tinkle, clang-clang, simpering smugly to the end.\n\nAt El Kharga\n\nEl Kharga is one of the five isolated oases which lie well to the west of the Nile in the Egyptian desert, and it has always been a place of exile. Nestorius was banished there, and Athanasius too, it is said. In our time political prisoners are immured in a detention camp at the oasis, and I once encountered some of them. They were patients in the local hospital, lying on straw palliasses on the floor of a bare ward. A murderous lot they looked, all the more sinister because bandages and plasters covered their eyes and supported their limbs\u2013one and all were enemies of the state, and their interrogations had not been easy. I talked to them warily of this and that, the conditions of their detention and their hopes of release, and they told me that every morning they were given a lecture of indoctrination by a representative of the regime. Something in their eyes, though, told me they were far from brainwashed, and now and then a particularly savage old dissident lying in a corner intervened with a caustic witticism, delivered in the most cultured of English accents and with the bite of an incisive mind. Thus Nestorius might have spoken, I thought, during his exile at El Kharga.\n\nThe European\n\nI met a man so allegorically Dutch that I deliberately engaged him in what I hoped would be allegorically Dutch conversation. He was a tall man with military moustaches, deep-blue eyes and a proper burgher's paunch, but he did not talk about Rembrandt, tulips, dykes, the German occupation, Queen Beatrix, the new season's herrings, Admiral de Ruyter or what was playing at the Concertgebouw that evening. No. He talked about unemployment, too many Asian immigrants, keeping his weight down and his hopes, earlier in life, of being a professional footballer. He was a citizen of the Netherlands, but I have met him all over western Europe, and that's what he always talks about.\n\nHero of the Soviet Union\n\nThe most dramatic as well as the most diligent conductor in the world is to be seen in action at the Theatre of Opera and Ballet in Odessa. He is an elderly man, but passionate. All around him as he works peculiar things are happening. Behind, in the half-empty auditorium, a constant buzz of homely conversation underlies the score, and three ill-shaven Levantines in the second row seem to be in the throes of opium dreams, squirming and sighing in their seats. In front, the stage is alive with minor mishaps\u2013trap-doors mysteriously closing and opening, fans being dropped, iron accessories clattering, while the cast of La Traviata smile resolutely across the footlights with a treasury of gold teeth.\n\nThe conductor is unperturbed. Majestically he sails through the confusions of the evening, impervious to them all, sometimes grunting emotionally, sometimes joining in an aria in a powerful baritone, throwing his fine head back, bending double, conspiratorially withdrawing, pugnaciously advancing, with infinite variations of facial expression and frequent hissed injunctions to the woodwind. Nobody in the socialist bloc fulfils a norm more devotedly.\n\nThe choice\n\n'Are you a man or a woman?' asked the Fijian taxi driver as he drove me from the airport.\n\n'I am a respectable, rich, middle-aged English widow,' I replied.\n\n'Good,' he said, 'just what I want,' and put his hand upon my knee.\n\nA Gypsy kiss\n\nIn the evening the entire population of Tirana seemed to emerge for the twilight passeggiata, strolling up and down the main avenue, sitting on the edges of fountains, milling around funfairs, wandering haphazardly across highways. I loved the louche insouciance of it all, the immense hum over everything, the quirks and surprises. Once I felt a small dry kiss on my arm, and turned to find a Gypsy child irresistibly importuning me for cash.\n\nUnderstanding the truth\n\nWhat would happen, I asked a fundamentalist predicant of the Dutch Reformed Church in South Africa, if an African walked into one of his services?\n\n'I would have him removed. My church is for Europeans, and it would be wrong to allow a native to worship there. God divided the races for His own purposes, and it is not for us to doubt His wisdom.'\n\n'Or if a Chinaman turned up one day, or an Eskimo?'\n\n'No, my church is not for Asiatics. I would send them away. But now you must not misunderstand me,' he added earnestly, tapping his knee with his forefinger. 'I don't say they shouldn't have a service at all. If there was no other church for them to attend I would hold a service myself, not inside my church, of course, but in a field if necessary. I feel this very strongly: that no man, whatever his colour, whatever his race, wheresoever he cometh from, should be deprived of the opportunity of worshipping Him who is the creator of us all.'\n\nAs I left the house the predicant clasped my arm, rather in the Rotarian manner, and pointed across the street outside, where an elderly black woman was hobbling out of a shop, screaming something in a searing treble over her shoulder. She crossed the pavement, closed one nostril with her finger and emptied her nose noisily into the gutter. Then, wiping her nose with her skirt, she turned round, still screeching, and disappeared indoors.\n\n'You see?' said the predicant. 'My dear friend, we are not unkindly, but you must live among them to understand the Truth.'\n\nSocial status in the people's dictatorship\n\nMrs Wang had invited me to lunch at her Shanghai apartment, but it gave me no culture shock. True, we ate eggs in aspic, a kind of pickled small turnip and strips of a glutinous substance which suggested to me jellified seawater, and Mrs Wang evoked for me her hysterectomy by acupuncture ('When they slit me open, oh, it hurt very bad, but after it was very strange feeling, very strange...')\u2013nevertheless her home seemed to me the bourgeois home par excellence. It had the statutory upright piano, a picture of two kittens playing with a ball of wool, a bookshelf of paperbacks and a daily help. It had a daughter who had come over to help cook lunch, and a husband away at the office who sent his regards. 'We are very lucky,' said kind Mrs Wang. 'We have a certain social status.'\n\nBaleful eyes\n\nNo infidel is allowed to enter the most celebrated shrine of Kerbala, the holy city of the Shias in Iraq, but I knocked at the door of a neighbouring house and asked if I might climb to its roof to see into its courtyard. The owner of the house was all smiles, but it turned out to be a simple inn, catering for pilgrims from Iraq, and as I walked up its narrow winding staircase I found myself passing a series of sparsely furnished rooms\u2013a bed and a prayer mat and a hard cold floor. In each of these doorless cells there was a pilgrim, and as I climbed my way up those steep steps each turned his baleful eyes in my direction. I shall never forget the detestation that overcame the faces of those merciless old men when they observed an infidel on the stairs, nor the relief with which at last I escaped the gamut of their loathing and emerged upon the roof, with the golden dome of the mosque in front of me and the wide sunlit courtyard, crowded with robed pilgrims, spread before me like a chessboard.\n\nPassing the nut\n\nI was trekking alone between Namche Bazaar and Thangboche, in the Nepali Himalaya. I was walking fast, in pleasant heathland country, and presently I saw far ahead of me another solitary figure, moving in the same direction. It was a robust Sherpa woman, wearing long aprons and a high embroidered hat. Despite her hampering skirts she, too, was making good time, striding firmly along the track, but gradually I overhauled her until, in a narrow bend of the path, I was able to overtake her.\n\nShe had given no sign that she knew of my presence, never turning round or looking over her shoulder, just ploughing steadily on like a colourful battleship. As I passed her, however, her left hand suddenly shot into mine. For a moment we touched. Neither of us spoke, and I was too surprised to stop, but I felt some small hard object pass from her hand into mine.\n\nI looked down to see what it was, passed so strangely from traveller to traveller, and found it was a small brown nut. When I turned round to thank her for it, she grinned and nodded and waved me on; so I pushed ahead up the hill, cracking its shell between my teeth.\n\nThe master glass-blower\n\nHere stands the master glass-blower of Murano, in the Venetian lagoon. He stands grandly assured beside his furnace, watched by a wondering tour group, with a couple of respectful apprentices to hand him his implements, and his long pipe in his hand like a wand. With a flourish he raises it to his lips, and with a gentle blow produces a small round bubble of glass. A twist, a chip, another delicate breath, and there appears the embryo of an ornament. A twiddle of the pipe follows, a slice with an iron rod, a dollop of molten glass, a swift plunge into the fire, a gulp or two, a flourish in the air, a sudden snap of iron shears\u2013and abruptly the glass-blower lays down his work with a gesture of artistic exhaustion, leaving the apprentice boys around him silent with respect, and the tourists, sweating in the heat, clustered awestruck about a huge glass harlequin, beady eyed and multicoloured, whose long spindly legs, swollen stomach, drunken grin and dissipated attitude breathe a spirit of unsurpassable vulgarity.\n\nAn official of the glass factory shouts through the window to a pair of husbands who have evaded the tour, and are sitting comfortably on the quay outside. 'Gentlemen! Gentlemen!' he calls reprovingly. 'Sirs! Your charming ladies are awaiting you in the vestibule. All the prices are marked!'\n\nAwful, really\n\nA hand touched my shoulder as I stood watching a crowd of masked and black-robed women crowding around a water hole in the Omani village of Ibri. 'You shouldn't stand about here, you know,' a voice said in English, 'you might catch something. The sanitation is ever so bad!' It was an Omani Christian convert, product of a mission school somewhere, whom I had met earlier that day.\n\nHe invited me to visit his nearby house. It stood behind a heavy gate, for there was ever such bad robbers in Ibri, he said, and as we entered I saw, half hidden in a dim and smoky recess, four or five black-shrouded figures almost motionless, and soundless but for a few moaning noises. I did not like to ask what was happening in there, but as we climbed the stairs my companion remarked casually, hitching up his shroud: 'That's my wife. She's got something wrong with her inside, so a few friends came in to look after her.'\n\nWe sat pleasantly in an upstairs room, watching the passers-by from a window and eating some rather stringy pomegranates. Yes, he sighed, Ibri was a funny place. The people was very funny. 'It's awful, really,' concluded the apostate, removing a pip from between his two front teeth.\n\nChanging the guard\n\nEvery other morning they change the guard outside the Presidential Palace in Santiago. An enormous military band plays, and the two guard companies, equipped with high boots, swords and resplendent spurs, march and counter-march with an almost ominous certainty. This is no toy-soldier parade. It feels all too real, as though the participating soldiery, dropping to firing positions by the flick of a command, might easily exterminate each other by numbers. It ends happily, though, for when the ceremony is over the two young subalterns of the guard, marvellously slim and elegant, salute each other with brisk respect, shake hands like brothers and stride off together into the palace. There is a moment or two of silence and then the band strikes up a waltz, and even the undemonstrative Chileans, standing woodenly all about, tap an occasional tight-laced toe.\n\nLooking after the place\n\nI was once driving through the Transvaal when I noticed a small obelisk on a hillock beside the road. I stopped, and found beside it an old Afrikaner farmer, crouched in what seemed to be silent meditation. He wore an unbuttoned waterproof jacket. A linen hat slouched around his ears and a mass of curly hair lay down his neck and oozed over his collar. He turned to look at me, and I found myself gazing into the bluest and clearest and hardest pair of eyes I had ever seen. The face that smiled at me was round and sun-burnt, engraved with innumerable deep lines, but the body was stringy as gristle. 'Who's the memorial to?' I asked him as we shook hands. 'One of our great Boer generals,' he replied, and added simply: 'I'm his son, I look after the place.'\n\nThus I met, almost as in a reincarnation, one of the legendary Boer farmers of tradition. He gave me a packet of biltong, prepared by his wife ('the most beautiful woman in Africa'), and we sat in the back of his car and drank some lukewarm coffee out of a Thermos flask. He suffered from no false modesty ('I'm always giving, it's one of my failings') and he held violent and generally unshakeable opinions. Why, only a few days before he had sent a telegram to the Commonwealth Conference in London, warning the assembled leaders that communism, Catholicism and Jewry were secretly allied in a campaign to overthrow Western civilization. 'But they're blind, you know, blind. Churchill was just the same. I sent him a cable in 1942\u2013it cost me \u00a37\u2013to warn him that Russia was anti-Christ, but he disregarded it. He never answered it at all. I suppose he read it?' said that old Boer, screwing up the Thermos flask. 'What do you think?'\n\nSo we chatted pleasantly, and he told me that if I ever came that way again I was welcome to stay at his farm and eat his biltong and disagree with his arguments for as long as I liked. It was a pleasure, said he, to meet a visitor from Britain, and that reminded him: had I seen the incontrovertible evidence at Bloemfontein concerning the ground glass and the porridge in British concentration camps during the Boer War?\n\nAlien visitors\n\nOne evening I heard music in the street, and looking out of my window I saw two strange figures passing. One was a young man in a tall brown hat, blowing on a shepherd's flute. The other was attached by complex apparatus to a variety of apparently home-made instruments\u2013bagpipes, drums, cymbals, a triangle I think\u2013and in order to beat the biggest drum he had to move in an abrupt but creaky shuffle. Slowly and sporadically these engaging characters pottered down the pavement below me, tootling and drumming as they went. In Trieste that day they were like visitors from another, less inhibited world.\n\nYou're welcome\n\nHere is an exchange I heard during an anti-American protest demonstration at Ottawa:\n\nPolice inspector: Are you a part of this demonstration, which is forbidden as you know to go any closer to the American embassy?\n\nProtestor: No, sir, we are just Canadian citizens exercising our right of free movement.\n\nInspector: Why are you carrying that placard, then?\n\nProtestor: Oh, that's simply an expression of my own personal views, as a Canadian citizen.\n\nInspector: I see. All right, go ahead, then.\n\nProtestor: Thank you, sir.\n\nInspector: You're welcome.\n\nI was in a stingy mood\n\nIn the ill-lit pedestrian tunnel that goes under the Elbe at Dresden I heard ahead of me the strains of a Viennese waltz, played by a pair of Gypsy violinists. I was in a stingy mood, and resolved to give them nothing. As it happened there was nobody else in the tunnel at that moment, and as I passed the musicians, still eloquently playing, I felt their eyes thoughtfully following me. I was decidedly self-conscious, knowing very well that I ought to put something in the open violin case at their feet, and as I walked towards the daylight my resolution wavered. 'You should be ashamed of yourself,' I said to myself, so when I emerged into the open I dug a few coins out of my purse and re-entered the tunnel. Melodies from the Vienna woods were still sounding in its twilight, and the Gypsies were not in the least surprised to see me back. They had read me like a book, and were expecting me. I put my coins in their violin case, and they both bowed courteously, without a smile. I bowed back in admiration.\n\nTwo Kiwis\n\nCarefully and kindly the keeper placed the creature in my arms, and I felt its feathers rustling against my hands so sharp and metallic that they almost felt like scales. The beady little eyes were blind and filmed, the strong wire-like legs scratched and struggled against my chest, and the long tube of a beak, nostrils at the end of it, protruded its way crossly under my arm. It was about the size of a hen. The keeper looked on almost paternally.\n\nTwo Jocks\n\nI watched an elderly man with sparse gingery hair strolling hands in pockets towards a pub on an Edinburgh corner, followed forty or fifty yards behind by his extremely aged collie dog. Sometimes the man looked round with an encouraging smile, and the dog smiled gamely back, and so they progressed in perfect rapport, like figures in a Burns poem, until the pair of them disappeared together into the malty shadows of the pub.\n\nFour Londoners\n\nI had an appointment with the pelicans of St James's Park, to whom each day a grateful Ministry of Works donates a ration of fish. Their keeper was waiting there with his bucket, and punctually at four o'clock a big white pelican waddled staidly out of the water and rubbed his beak ingratiatingly against the man's legs. 'This is Paul,' he said. 'He's a very good-natured bird.' Before long two others turned up, in rather a diffident, squint-eyed, lopsided manner, for they were newcomers to the park. 'They're funny birds, pelicans,' said the keeper. 'Some people like them, some don't.' But when he had fed them their fish, and they waddled away sated, he turned to me again. 'They've had enough, you see. They aren't greedy birds at all. I thought they behaved very well, didn't you? Very well indeed, considering.'\n\nIn academia\n\nI have never forgotten the Christmas parties arranged for us, when I was a child, by the canons of Christ Church, Oxford, in their great canonical houses facing Tom Quad. How tall the candles were! How rich but wholesome the cakes! How twinkling the Regius professors turned out to be, stripped of the awful dignities of office! What thrilling presents we were given\u2013envelopes with penny blacks upon them, magnificent wax seals of bishops or chancellors! How happy the old clergymen's faces looked as, breathlessly piping our gratitude\u2013'Thank you very much indeed, sir!' 'It was jolly nice of you, sir!'\u2013we last saw them nodding their goodbyes, a little exhausted around the eyes, through the narrowing gaps of their front doors!\n\nDefinitely not\n\nI was sitting upon a grassy incline in a park in Adelaide when two small boys, one rather smaller than the other, prepared to ride down the slope on their skateboards. There were a few beer bottles lying around, left over from the night before, and I heard the elder boy say to the younger, in an authoritative voice intended largely for my own ears: 'Please don't hit the lady\u2013I don't mind about the beer bottles, but definitely not the lady.'\n\nThe captain of the Saratoga\n\nThe captain of the USS Saratoga, a tall lean man of ecclesiastical bearing, sits in a raised padded armchair on the port side of his bridge, rather as though he is having his hair cut or is being inducted to his see, and by looking through its tilted windows he can see the big jet bombers on the flight deck below. This ship, the publicity officer at your elbow tells you, has enough paint on her to redecorate 30,000 average American homes. There are 2,000 telephones on board, three escalators, three soda fountains, nine barbers' shops and 3,676 trouser hangers. 'We generate enough electricity to service a city the size of Pittsburgh, an industrial city in the State of Ohio. Our machines peel a thousand potatoes an hour.'\n\nBelow the windows the pilots scramble into their high cockpits. The captain rises from his chair, and a first violent roar of jet engines reverberates though the carrier. ('This ship has seventeen decks,' shouts the publicity officer indefatigably. 'There are more than 7,000 coffee cups on board the giant carrier, which is named from a battleground in the American Revolutionary War.') Then, suddenly, there screams into the corner of your eye a lean silver aircraft, violently projected at breakneck speed down the deck and into the blue, and in a moment there seem to be aircraft everywhere, some careering down the angled deck, some straight towards the bows, flashing and roaring and streaming away into the blue. In a moment or two the whole flight is gone, and is vanishing in the general direction of Turkey. A flash, a blast of jets, a dozen young men hurled brutally into the sky, and a terrible page of history could almost instantly be written. No wonder the captain of the Saratoga, as he returns to the seat of his command, has the air of a thoughtful but authoritative divine.\n\nYoung man with a gun\n\nUp a dingy flight of stairs in Vienna I went to visit Dr Simon Wiesenthal, the Nazi hunter and himself a victim of the camps. Short, balding, in his seventies then, he was surrounded in his cluttered apartment by certificates of merit and scrolls of gratitude. He had devoted his later life to tracking down the last of the Nazi murderers and seeing that they were punished\u2013year after year, decade after decade\u2013while those once-swaggering SS men grew frail and forgetful, and Wiesenthal himself entered old age fired still by his merciless search for justice\u2013or revenge. If he had anything to do with it, he told me, no single Nazi murderer, however old and grey, would ever be allowed to die in peace. I thought his office unforgettably baleful. The files that filled its walls were dreadful registers of death and torture, and Wiesenthal talked disturbingly about the wicked men still alive and flourishing in Europe. There had been an attempt on his life a few weeks before my visit, and a police guard had reluctantly been given him by the Viennese, whose communal conscience about the Jews was less than clear.\n\nThat day's sentry looked up at me as I left Wiesenthal's office. He was a blond long-haired youth with a gun on his lap, lounging there on a bench with his feet upon a chair, chewing something; and as he insolently stared at me, and at the old gentleman saying goodbye to me at the door, I felt an uneasy frisson.\n\nMore organic patriots\n\nBeing myself a sort of self-adopted Swiss patriot, I made a pilgrimage once to the lakeside field of R\u00fctli, which is the traditional birthplace of the Swiss nation. On the Sunday I walked down the track from the heights above, thousands of more organic patriots were making their way to or from the hallowed site, most of them evidently people from the mountain country around. I offered a cheerful good morning to everyone I met, and could not help admiring the utter lack of ingratiation, the courtesy tinged with decidedly suspended and unsmiling judgement, with which most of them responded. I was struck too by the proportion of twisted, stooped or withered old people among them\u2013people of a kind that had almost vanished from the rest of western Europe. They were one generation removed from the goitre, that talismanic affliction of mountain peasantries, and the faces of those crooked ancients\u2013hard hewn, bashed about, gaunt\u2013seemed to speak of centuries of earthy hardship, isolation and suspicion. I could not help remembering, too, that in Switzerland the very last European witch was publicly burnt.\n\nFishing lady\n\nOn the edge of a swamp in Louisiana an old Negro woman in a floppy straw hat was fishing in the oozy water with a home-cut rod. She had already caught a few fish, and they were floundering in the shallows, tied up in a net. She told me she had been dropped there that morning from the train which passed nearby; her husband worked on the railroads, and in the evening, when the train came back again, it would slow down past the swamp and allow her to scramble aboard a freight car. She asked me to drive a little way down the road and fetch her some Coca-Cola. I bought her four bottles, and the last I saw of her she was standing on the boggy bank in her huge hat, with the rod in one hand and a bottle raised to her lips with the other, a portly statuesque figure against a gloomy background of cypress trees.\n\nCoffee time\n\nI was once standing at the entrance to the celebrated whores' alley of Hamburg, beneath the flickering neon sky of the Reeperbahn, when an unexpected figure passed through its portals, weaving a bustling, purposeful, businesslike way among the pallid lechers loitering inside. It was a waiter from a neighbouring cafe, nattily dressed in white and carrying a cup of coffee neatly on his polished tray, with two lumps of sugar hygienically wrapped. He made his way dexterously to one of the brothel windows and, peering into the gloom to pick his customer from the row of ghoul-like prostitutes inside\u2013dim, apparently phosphorescent images of flesh, paint and pink nylon\u2013he handed her the tray with a polite little bow and returned to the world outside.\n\nIn London, 1980s\n\nSomewhere in Oxford Street, towards the end of the afternoon, a sort of hallucination seemed to overcome me, and I found myself in a nightmare limbo. I was aghast. Who were these fearful people, of no particular race, of no particular kind, so crude and elvish of face, so shambling of gait, so shabby of clothes, so degraded and demeaned of bearing? Where were they shoving and sidling their way to? What culture did they represent, what traditions inspired them, what loyalty did they cherish, what God did they worship? I seriously doubt if a less prepossessing citizenry can be found anywhere on earth than the citizenry frequenting such streets of London.\n\nMagnifique!\n\nWhen I was dining one night in a restaurant in the French island of Martinique, an extraordinary girl burst into the dining room and began dancing a kind of ferocious screeching rumba. She wore an enormous tricorn hat and a red swimsuit, and when the management objected to her presence she instantly threw herself into a spectacularly flamboyant tantrum. She screamed, shouted, sang ear-splitting snatches of songs, threw plates about, dropped her hat, made savage faces at the customers, knocked tables over and reduced the whole room to helpless laughter until at last, to crown a splendid entertainment, somebody dialled the wrong number and obtained, instead of the police, the fire brigade, whose clanking red engines skidded to a halt outside our windows and whose helmeted officers, trailing axes and hoses, stared in bewilderment through the open doors at the hilarious chaos inside. 'It was magnificent!' was the general verdict as, wiping our eyes and resuming our victuals, we watched that uninhibited performer withdraw.\n\nWest Pointer\n\nIt was Saturday afternoon at West Point, and many of the cadets were preparing to go out. I saw one emerging from her barracks in what I took to be her semi-dress uniform\u2013a trim grey trouser suit with a shiny peaked cap, very smart and flattering to her lithe figure. I followed her down the path towards the Eisenhower statue\u2013left right, left right, head up, arms swinging, brisk as could be to where her father was waiting to greet her: and then, talk about symbolisms! He was your very image of a kindly homespun countryman, a figure from an old magazine cover, wearing boots and a floppy brown hat, his face shining with pride and happiness. She broke into a run, her cap went askew for a moment, and into his strong American arms she fell.\n\nAn Irish experience\n\nI was in Dublin for the first time in my life, and I took a stroll along the extended breakwater, bleached in sun and sea wind, that protects the mouth of the Liffey from the exuberance of the Irish Sea. Gazing about me pleasurably, presently I saw implanted across the causeway the clubhouse of the Half Moon Swimming Club, and immediately beside the door of the building there was a bench, facing directly down the mole, as though in judgement. Even from a distance I could see that four or five heavy pinkish figure occupied this seat, motionless but glistening in the sun, like Buddhas, and I could feel their eyes steadily focused upon me as I approached them down the causeway until at last, reaching the purlieus of the club, I raised my own eyes modestly to meet those divinities face to face. Five old, fat, gleaming Dubliners looked back at me severely, and they were all entirely nude.\n\nYoung Iceland\n\nChildren play a disproportionate part in Icelandic life, it seems to me. Nothing is more surprising than to hand over one's fare in a country bus and find it accepted by a character apparently not much more than four years old, who grumbles with absolute adult authenticity if you haven't got the right change. And in the Althing, the Icelandic parliament, a common sight is a minuscule page hastening in with a quotation for the Foreign Minister, perhaps, or a statistic for the Minister of Finance: he is likely to be wearing a check shirt, a green jersey and corduroy trousers, and as often as not he interrupts the flow of debate by banging the door behind him. Nobody minds. Drat the boy, one seems to hear them murmuring. And his father was just the same.\n\nAfter Per\u00f3n\n\nGeneral Per\u00f3n's dictatorship of Argentina had ended, but in the plush fin-de-si\u00e8cle cafe I chose for my lunch in Buenos Aires his presence was still palpable. Around me gaggles of elderly women were sipping Cinzano with soda water and nibbling biscuits, nuts and bits of flabby cheese, but in the dimmer recesses of the room various lonely men were deep in the contemplation of La Prensa. When I asked my waiter if there were still many Peronistas about he nodded darkly but wryly, with a flicker of his thumb, towards those several grey solitaries in the corners, who certainly had a brooding conspiratorial look to them but were probably, in fact, looking through the small advertisements for second-hand canoes.\n\nThe proclamation\n\nA stone's throw from the holiday madhouses of Waikiki there stands a row of rickety tables beside the sea, shaded by straw matting, where elderly Honolulu citizens while away their Sunday mornings with chess, chequers and inexplicable card games. I was sitting there in reverie one morning, happily lost in the sun and the salt breezes, when a prickly old gentleman on the benches beside me touched me on the shoulder. 'You look a little melancholy,' he said kindly. 'Aintcha read the proclamation?'\u2013and he pointed to the notice painted on a weatherboard above us. 'This is a Public Park,' it said. 'Have Fun!'\n\nReassurance\n\nI was in the Isle of Man for the first time in my life, to write an essay about it. I had bought a book about Manx folklore and, finding an open-air cafe beside the sea, settled down to read it with a plate of prawns and a Guinness. The sun was lovely, the prawns were excellent, the Guinness went down like a treat, and I congratulated myself upon my choice of profession. Presently a lady came over to my table and handed me a pamphlet. 'Oh, thank you,' I said, 'how kind of you. What's it about?' 'Oh my dear,' she emolliently replied, 'it is only to reassure you that God is always with the lonely.'\n\nNo reply\n\nNowhere on earth is so inexorably improving as Washington, DC. When we came down from the top of the Washington Monument even the elevator operator dismissed us with a parting injunction. 'Let's all work', he said, 'to clean up our country for the two-hundredth anniversary just coming up.'\n\n'Yes sir,' we dutifully replied, 'you're darned right\u2013you hear that, kids?'\n\nHe had not, however, finished yet. 'And I'm talking,' he darkly added, 'about the mental aspects as well as the physical.'\n\nWe had no answer to that.\n\nPossibilities of misfortune\n\nThe Kashmiris are a hospitable people, but not inspiriting. They seem to be considering always the possibilities of misfortune. In the autumn the fall of the leaf seems a personal affliction to them, and the passing of the year depresses them like the fading of their own powers. Then in the chill evenings the women disappear into their private quarters, and the men light their little baskets of charcoal, tuck them under their fustian cloaks and squat morosely in the twilight, their unshaven faces displaying a faint but telling disquiet. There was a touching pathos, I thought, to their style. 'How do you like your life?' I asked one new acquaintance there, when we had progressed into intimacy. 'Excellent,' he replied with a look of inexpressible regret. 'I love every minute of it'\u2013and he withdrew a cold hand from the recesses of his cloak, and waved it listlessly in the air to illustrate his enjoyment.\n\nMy dinner companion\n\nMarvellously lithe and light-footed are the people of Helsinki, big but agile, jovial at smorgasbords or loping across their snowfields like Tibetan holy men. Their children, slithering about with hockey sticks, give the heartening impression that they came into the world on skis. Their wives are neat as pins, and gossip sharply in expensive coffee shops. They are a people that nobody in the world could possibly be sorry for. They are sharp as nails, and twice as spiky. But here's an odd and provoking fact. When I wanted something to read with my dinner some unexpected instinct guided my choice, a kind of reluctant nostalgia, a niggling trace of respect and affection, and when I sat down to my pig's trotters I found myself dining with Turgenev: and all that brave and courteous citizenry, I felt, could not offer me quite such company.\n\nDiplomats and a pianist\n\nI once went to the British embassy in Washington, DC, to see the pianist Vladimir Horowitz presented with the Gold Medal of the Royal Philharmonic Society, brought to him on a cushion by a marvellously suave young secretary and handed over with a graceful ambassadorial speech about violent times and the meaning of art. Mr Horowitz seemed pleased, but instead of replying in kind sat down at the piano and played in a highly vibrant and indeed imperial manner 'God Save the Queen', making full use of the sustaining pedal.\n\nThere was a pause at the end of it, and instantly, as the last notes faded, I clicked the scene in my memory: and so I have held it there like a flash from a dream, the ambassador benignly at attention, the young diplomats rigid all about, the American guests clutching their champagne glasses, the great room aglow with carpets and portraits, the pianist's hand raised in a last grandiloquence\u2013an ornate little vignette of Washington, where life so often shimmers through a gauze curtain, insubstantially.\n\nImpact!\n\nKing Sobhuza of Swaziland, one of the world's last absolute monarchs, offered me a kindly greeting. His subjects fell on their knees, or even on their faces, when he passed, but I looked him Jeffersonianly in the eye, and shall never forget the moment. He had the most remarkable, most twinkling, most mischievous, altogether most entertaining face in the world. He seemed to radiate an amused but resolute complicity, as though he knew what a charade life was but was determined to make the most of it. He was dressed that day in European clothes; when he wore his tribal costume, a stunning assembly of feathers, bright textiles and talismanic brooches, the effect must have been terrific.\n\nStyle\n\nI joined an eminent, kind and cultivated actress in taking a cab to an address on Second Avenue in Manhattan. Said the cab driver: 'Whereabouts is that on Second Avenue, lady?' Without a flicker in her elegant equanimity she replied: 'Don't ask me, bud. You're the fucking cab driver.'\n\nOn an Oxford evening\n\nLoitering around Magdalen College on a classic May evening I saw a company of players making their way through the trees for a performance of A Midsummer Night's Dream. They were moving swiftly in their cowls, ruffs and velvets, all among the elms, and a few shy deer watched them pass between the tree trunks. Their footfalls were silent on the turf, their voices reached me faintly on the warm air, and they disappeared into the shadows merrily, with Puck occasionally practising his jumps, and Titania lifting her crimson skirts, and a few lumpish fairies skirmishing in the flanks. I never caught the spell of the theatre more hauntingly, as I watched them across the fence, and felt like Hamlet when the players came to Elsinore\u2013'You are welcome, masters, welcome all.'\n\nThe moment of victory\n\nAn old woman, horribly crippled, struggles down the last few steps of the Chapel of St Helena, in the Church of the Holy Sepulchre at Jerusalem. It is a faintly illuminated crypt. Her progress is agonizingly slow, but she is determined to reach the altar by herself. Painfully with her two sticks she shuffles down the stone steps, each one a torment. Prayers are mingled with her breathing. When at least she reaches the bottom, though, and I peer into the darkness to watch her, she abruptly leans down and places her sticks beside her on the ground. Then, straightening herself as far as her old crooked frame will allow her, she raises her arms above her in triumph and exuberance, more like some whipcord young athlete at the moment of victory than a poor old woman, distorted and arthritic, who would soon have to face the steps again.\n\nAt Schwab's\n\nHardly a Hollywood memoir is complete without a reference to Schwab's, 'The World's Most Famous Drugstore', and it is still heavy with the old mystique. Elderly widows of \u00e9migr\u00e9 directors reminisce about Prague over their breakfasts. Young men in jerkins and expensive shoes ostentatiously read Variety, or greet each other with stagey endearments. Ever and again one hears exchanges of critiques across the hubbub\u2013'I love her, she's a fine, fine actress, but it just wasn't her...'\u2013'Well, but what can one expect with Philip directing, she needs definite direction'\u2013'True, but shit, it just made me puke, the way she did that last scene...' I took to sharing a table with the divorced wife of a Mexican set designer who shared my enthusiasm for Abyssinian cats.\n\nA royal court\n\nI had an introduction to a Mogul princess, of the dynasty which made Delhi its capital in the seventeenth century and built the very walled city in whose labyrinthine recesses she lives. I found her ensconced in her front sitting room between portraits of her imperial forebears: a short, decisive old lady with a brief mischievous smile and an air of totally liberated self-possession. Her antique mansion is a beguiling shambles in the old Islamic style: a couple of rooms in the Western manner for the convenience of visitors, the rest more or less medieval\u2013wide decrepit courtyard, dusty trellised vine, thickly populated chambers all around. There are granddaughters and sons-in-law and undefined connections; there are skivvies and laundrymen and assorted sweepers; there are children and dogs and unexplained loiterers in doorways. Forty or fifty souls constitute the tumbled court of the Begum Timur Jehan, and through it she moves commandingly in green trousers, issuing instructions, reminiscing about emperors, traitors or ladies of the harem, and frequently consulting her highly organized notebook, all asterisks and cross-references, for addresses or reminders.\n\nPoliticians\n\nI love to watch the politicians ushering their constituents around the Capitol in Washington, DC, benign and avuncular, and to observe the endearing combination of the condescending and the wheedling with which they shake hands with their respectful electors at the end of the tour\u2013'We sure are obliged to you, Congressman'\u2013'We certainly are, sir'\u2013'I shall never forget this day, Congressman'\u2013'Fine, fine, great to have you along...' Meeting a likely looking gent in a Capitol corridor, I tried a gambit myself, as a speculation. 'Morning, Senator,' I said. 'Hullo there, young lady,' he instantly replied. 'Having fun?'\u2013and off he strode to his office, chomping, alas for my purposes, not an actual, but at least a metaphorical cigar.\n\nPerfect understanding\n\nLong after the end of the British Empire, some of its manners balefully survived. In Patna I had occasion to go to the Secretariat to ask permission to take photographs of the city, and found myself before a functionary of such classic insolence, such an unassailable mixture of resentment, patronage, self-satisfaction and effrontery that for a moment I felt like picking up his inkwell and throwing it at him. But I bit my lip and restrained myself, and as I glared back at him there the scales dropped from my eyes: his image blurred and reassembled before me, his colour paled somewhat, and I saw before me his true archetype and inspiration, the lesser English civil servant\u2013now, as in imperial times, the insufferable master of his art. I thanked the man profusely and assured him that I understood perfectly why I would have to make an application in triplicate to the Divisional Officer, who would unfortunately not be on duty until the following Wednesday afternoon.\n\nSnow in Holland\n\nI was once in Amsterdam when the first snow of the winter fell. The men in the central junk market, among their stuffed birds and rusty curios from Surinam, broke up the most hopelessly lopsided of their kitchen chairs and made bon-fires of them. A cold wind whistled in from the North Sea, huddling the more mature housewives in their mutation minks and driving the portly burghers to the felt-covered newspaper tables of the cafes, where they meditated ponderously over their coffees like so many East India merchants considering the price of apes.\n\nJoburgers\n\nI found the poor black inhabitants of the Johannesburg locations, in the cruel days of apartheid, hard to understand. Sometimes they were grave and courteous, and I was reminded of Ethiopian chieftains; sometimes they treated me with such bubbling flippancy that I thought they might be teasing me; sometimes a flash of malice entered their eyes, or something gave them such inexplicable amusement that they burst into a tumult of infectious laughter, or danced little jazzy jigs upon the pavement. When they spoke, they did so with explosive animation, but when they listened the whole of their being supplemented their hearing, they became one great ear, and their white eyes, their tense bodies, their eager fingers and their yellow striped socks all waited upon my words. And once in my hotel I heard tinny twangs of music from the street outside; and there beneath the arcades of President Street a solitary black man was lounging by, in a crumpled brown hat and blue dungarees, plucking away at a guitar as he walked, humming a high-toned melody, and expressing in his every gesture, in the very swing of his shoulders, the spirit of blithe indolence.\n\n'The same again'\n\nKabul in the 1960s is a tense, nervous, shifty capital, and edgiest of all at night, when the streetlights are dimmed, the brilliant Asian stars come up above the hills and only a few shrouded watchmen are left brooding on the doorsteps. Then the whole place feels sleepless and dry-eyed, like some insomniac conspirator. Sometimes a shot rings dead on the night, and sometimes a distant shout, and when a donkey pads softly by you can hear the two men upon its back, nebulous in white robes, murmuring to each other in low sporadic undertones. I once asked an old man of Kabul what would happen if another enemy attacked this capital as the British had catastrophically attacked it in 1845. Would they be exterminated too? He gave an angry tug at his beard and threw me a look of piercing and bloodshot intensity. 'The same,' he hissed through the last of his teeth. 'The same again!'\n\nHome are the hunters!\n\nI first went to Kuwait in the company of a sheikhly hawking party, returning home from a desert sporting expedition. Splendid were the caparisons of those haughty Arabian sportsmen, and their eyes were cold and heavy-lidded. They wore magnificent flowered gowns, and crossed bandoliers, and daggers, and spotless headdresses, and golden swords; and big black lackeys carried their peregrine falcons, hooded upon their pedestals; and a brass band puffed away on the airfield at Kuwait when this gorgeous crew, looking slightly airsick, staggered on to the ancestral soil.\n\nThe quarry clerk\n\nI was only just in time to meet Bob Owen of Croesor, in northern Wales, before he died in 1962, and I am glad I didn't miss him. He had worked as a clerk for a local quarry company for more than thirty years, a small man with a high wrinkled brow, a white moustache and bushy eyebrows, respectably dressed when I met him in jacket, waistcoat and unassertive tie. He was a tremendous talker, a chain smoker and a chapel goer of strong views, and when his quarry work ended he had become a writer and lecturer well known throughout Wales. He took me to the small square house where he and his wife lived and, merciful heavens, the moment he opened the front door for me I found myself hemmed in, towered over, squashed in, squeezed down by an almighty multitude of books. They filled every room of the house\u2013he had amassed more than 40,000 books and pamphlets, many of them rare and valuable. He was born, he told me, in a very small nearby cottage, nicknamed Twll Wenci, and people used to call him Bob Twll Wenci\u2013Bob Weasel's Hole.\n\nPretty children\n\nIn the mountain resort of Flims I saw three small Swiss girls on their way home from school. They looked like modernistic elves, with bright-coloured rucksacks on their backs, and they were burbling brightly to each other as they climbed the hill to their homes above the town. They paused for a bit of gossip and leg-swinging at a bench beside the road, and when they got up to go one of them, meandering off by herself, chanced to leave her sunglasses on the bench. In a trice the other two, laughing and giggling, threw them on the ground and stamped them into pieces before my eyes, alternating kicks in the prettiest way.\n\nLese-majesty\n\n'Yeah,' said a woman loudly and complacently, stepping back from a china cabinet during our guided tour of the White House. 'Just what I thought\u2013chipped!'\n\nA hated man\n\nSoon after World War II a friend was driving me one day along an Oxfordshire lane when we saw a picturesque sight in front of us. A fine four-in-hand was running along at a spanking pace, driven by an elderly gentlemanly looking coachman on his high box. 'D'you see the man driving it?' said my host. 'That's Air Marshal Sir Arthur Harris, the most hated Englishman alive.'\n\n'Bomber' Harris! The man who had unleashed his vast fleets of thudding black aircraft, manned by crews from every corner of the old British Empire, to devastate half Germany and kill scores of thousands of German civilians! I stared rudely at him through our rear window, as we left those trotting horses behind, but he looked a jolly enough old fellow, up there behind the reins.\n\nAmerica, America!\n\nI once came into Pier 86 on the liner United States, the fastest ever built, and I watched the faces of the passengers around me, waiting for the gangplank to open as the ship's band subsided into a last medley of patriotism\u2013'America, America', 'Dixie', 'Star-Spangled Banner' and one or two stirring marches I failed to recognize. Trilly secretarial voices rang, as those grand old tunes reverberated, jewelled spectacles vibrated, stiletto heels tapped the deck; but the expressions on the passengers' faces struck me as sad, as though the hum of the liner's mechanisms, the blaring of those anthems as the vessel docked, were holding the voyagers for a moment in a lost American world\u2013a world encapsulated there still between the decks of the great ship, that would dissipate the moment the gangplank doors were opened, and they returned to 46th Street.\n\nResponses on the road\n\nDriving through Vienna in a rented car, I slowed down uncertainly to decide upon my route. Instantly the driver in the car behind blasted his horn most rudely. I gave him a vulgar two-fingered sign which I would never have dreamed of using had I not recently learnt that it was a gesture devised by Welsh archers to demonstrate to opponents that their shooting fingers were intact. When the other car overtook me, its occupants both looked eagerly in my direction. The stout tight-buttoned horn-rimmed burgher at the wheel shook his jowls at me in affronted astonishment. His wife blew me a kiss.\n\nWidowhood\n\nFor the jollier kind of American widow Los Angeles offers a cheerful refuge, and provides a bedrock, so to speak, upon which they can reconstruct their lives. There is a certain sameness to their appearance, in their bright blouses, leather jerkins, rather too tight slacks and rather too rakish sailor caps. They are bowed often with arthritis but resolutely jaunty of step, and to their attitudes there is a sprightly element of freedom. 'Did you know,' one such lady asked me, supposing me, I imagine, to be a bit lost for social satisfactions, 'did you know that the telephone company offers a free tour every day of the week? My, that's a rewarding way of spending an afternoon!'\n\nMoon men\n\nIn the last years of Rhodesia, before it mutated into Zimbabwe, I sometimes had the feeling that its Europeans were being mutated by history, becoming some sort of new subspecies as they fruitlessly resisted the rise of black power. In Salisbury I sometimes took my lunch beside the hotel pool, and there was generally a group of young officers, on leave from their battle stations in the bush, having a swim or a beer on the terrace. Stripped to their trunks and sun-bleached hair, they seemed to lose all ethnic identity. Some were probably mercenaries: Portuguese, German, Afrikaner; most were doubtless expatriate Britons, subtly changed in posture and physique; all seemed to me specific not simply to the place, but to the time, to the circumstances, the historical prospect. They might have been moon men.\n\nEthel in Egypt\n\nOne evening I went to a salon presided over by one of the younger, richer and more cosmopolitan of the Cairo society ladies. The purpose was to present to the Cairo grand monde a celebrated clairvoyant of indeterminate Levantine origin, named\u2013well, let us say Ethel. Ethel would judge character, tell fortunes, give semi-occult advice and accept confidences. The attendance was soign\u00e9e: a couple of ambassadors' wives, an Indian, a German, a few bangled Egyptian patricians. We were served coffee in very fragile cups, by a Berber in a tarboosh, and we sat on squashy sofas at spindly tables in a room above the Nile.\n\nEthel was closeted on a balcony, and one by one the guests disappeared for consultations, taking their coffee cups with them. In the meantime the rest of us chatted. Did we know that G was almost certainly going to St Tropez with A? Was it really true that B was getting Omani money for his new hotel? Had we tried the chopped liver at the Hilton? What about F selling that awful house of his for a quarter of a million?\n\nFrom time to time another woman left for the confessional, but the returning devotees, I noticed, never seemed dismayed by Ethel's predictions, and returned instantly, without so much as a mention of their brush with the occult, to our distinctly worldly exchange\u2013'Half a million, I heard\u2013and surely it's chopped kidneys?...' But then Ethel, my hostess told me, seldom had unhappy premonitions. 'Well, one would hardly expect her to, would one, actually in one's own drawing room?'\n\nBirth of a taste\n\nI had disliked whisky all my life, but stopping to eat my corned beef sandwich in one of the most famous Scottish distillery areas, I felt it my duty to try once more. I went to a nearby pub and asked for a dram of the local water of life, to drink with my victuals on a bench in the square outside. The barmaid looked at me quizzically. 'I'm not sure the law allows it, but seeing as you're a visitor...' Looking cautiously around her, she poured into my plastic mug a full measure of single malt whisky\u2013one of the very best, she said, from a distillery just down the road. I concealed it in my bag as I left the inn, and turning at the door I saw her winking at me conspiratorially, as if I had poached a salmon. No constable intervened, though. No revenue man expostulated. Unwrapping my sandwich, there on the bench in the square I took a cautious swig of the whisky, and, dear God, I have never looked back.\n\nHospitable cop\n\nI was a guest once at a Buckingham Palace reception for publishers and writers, and at the end of the evening, wishing to leave, I looked around for somebody to thank. Queen, princes, dukes and all seemed to have gone elsewhere, so I left anyway, and at the palace gates I found a policeman. 'I was brought up,' I told him, 'to say thank you for having me when I'd been to a party, so as I can't find the Queen or anybody to say it to, I'll say it to you instead. Thank you very much for having me.'\n\n'Not at all, madam,' he replied. 'Come again.'\n\nA glimpse of power, 1950s\n\nFor a glimpse of power, try the Bolshoi at Moscow, when some gigantic Russian epic is being furiously enacted, with rolls of kettle drums and clashes of armour, a mammoth chorus open-mouthed, a clutch of heroes swelling in the foreground, with a passage and repassage of knights, horses, serfs, a frenzy of conical helmets and chain mail, banners dramatically waving, flames issuing from a backcloth, smoke, flashing beacons, the orchestra in a quivering fortissimo, the conductor wiping his sweating bald head, the enormous audience gripping its seats or craning from the high gilded balconies above the chandelier\u2013then, in the middle of it all, you will glance across your neighbour's shoulder to the great state box in the centre: and there will be sitting the most powerful man on earth, looking bored and rather glazed, a slight sad smile playing around the corners of his mouth, his wife, in a bun and brown sagging dress, demure and attentive at his elbow. You need not wait for the last act. Go home and sleep it off.\n\nAnother time, perhaps\n\nIn the Faroe Islands I repeatedly ran into groups of traditionally dressed folk persons, buckled and aproned, on their way to or from festivals of one kind or another. 'We have been telling rhymes in Klaksvik,' one practitioner told me as we sat together on the deck of a ferry, a celestial scene of mountain and fjord streaming by. 'Long rhymes?' I ventured to ask, thinking I might be fortunate to have missed them. 'Extremely long,' he said with pride.\n\nAt a Patagonian airfield\n\nWhen I was once hanging about an airfield in Patagonia, hoping to arrange a lift to the north, I noticed a small group of people, dressed apparently for apr\u00e8s-ski, who seemed to dominate the waiting room with a kind of steely radiance. They looked very rich and very brassy and very thrusting. Their children were ill mannered but intensely vivacious, their women were gimlet eyed but seductive, their men had a feline Italian elegance to them; and unexpectedly, when I offered a smile in their direction, one and all suddenly, brilliantly, delightfully smiled back. I asked where these magical creatures were making for, and was answered in one short tingling word: 'Rio!'\n\nMakings of a microcosm\n\nI did not want to be rude, but I could not help eyeing my neighbour with interest, for she seemed to me to have the makings of a microcosm. More than most cities, Stockholm projects two images\u2013the one you have been led to expect and the one you discover for yourself\u2013and this plump but not unalluring citizen, wearing a pink linen dress and a white straw hat, her eyes bluish but somehow glazed, her mastication rhythmic and her bosom calmly heaving to the flow of the salad\u2013this lady of Stockholm, evocative partly of Chanel and partly of disinfectant, slipped into my preconceptions like a plug into a socket. She was eating alone, with a half bottle of Niersteiner and what appeared to be the financial page of Svenska Dagbladet. Her lunch was large but looked obscurely colourless, as though it had been bleached in some anti-fattening lotion. Her gaze now and then wandered from her victuals and paraded slowly, resting at last without excitement on somebody else's pudding. Her expression was content without being joyous, and beneath her loose blonde curls, I told myself, all kinds of Swedish neuroses surely festered: anxieties of opulence, spinsterhood or free love, occupational frustrations and suicidal impulses. She seemed to express all that I expected of Stockholm, and when at last I engaged her in conversation, and boldly asked her what she did for a living, I could almost have hugged her in gratitude. 'I am a juvenile social welfare worker,' she replied with a sweet smile, taking a delicate last sip of the hock.\n\nThe only place for him\n\nWhen the US Supreme Court ended racial segregation in American schools, all the simmering discontent of the white Southerners boiled over, and I spent the day in Atlanta listening to angry men and women. The abuse they used was at once so theatrical and so repetitive that I could scarcely believe it had not been plucked wholesale from some common phrasebook of prejudice. I joined a conversation, in a coffee shop, with the manager of the place and a man who told me he was a senior police officer. They spend some time reminiscing about race riots of the past, talking comfortably of niggers bashed and beaten in the streets; and of one especially, hounded by the mob, who had thrown himself into the doorway of that very coffee shop, only to be pushed back on to the pavement. 'The only place for a nigger,' said the manager with finality, 'is at the back door, with his hat in his hand.'\n\nFlowers and the tribesmen\n\nClumps of a rhododendron-like bush brightened the fresh meadows as we drove through the Qara tribal country of south-east Arabia, and I asked the Arab driver to stop while I jumped out of the truck to pick some blossoms.\n\n'What do you want them for?' he asked when I returned with the flowers. 'Are they good for diseases?'\n\n'I don't think so. I just thought they looked nice.'\n\n'So they do, so they do. But the Qara people eat them, for the stomach.'\n\n'Are you sure?'\n\n'Quite sure. The Qara people know everything about flowers and things. They are very strange people, like the animals. By Allah! They are very like the animals.'\n\nAs if to bear him out we saw at that moment three strange fuzzy tribesmen standing on a bank beside the road, leather thongs around their foreheads, dark robes slung over their shoulders, daggers at their belts. My driver shouted them a ribald greeting. Two of them, with long, beautiful faces, did not respond, but simply stood there stiffly, like childish elocutionists waiting to perform; the third, a younger man, ventured to wave his short stick at us, and then, seeing that his companions remained impassive, lowered it shamefacedly as though guilty of some desperate solecism.\n\nWhen we had passed I leant out of the window to look back at them. There they were still, three straight, shy figures, holding their sticks, watching our progress fixedly.\n\nUncle Henry and the planter\n\nThe planter, fresh from a tussle with his tractor, had greasy hands and wore a toupee and an open-necked shirt. But like most Southern gentlemen he had a talent for hospitality, and soon we were sitting on the balustrade of his porch, sipping long cool drinks and looking out through the pines. The plantation had once extended to some 10,000 acres of cotton, tobacco, sweet potatoes and corn, but was now whittled down to about 150 acres. He told me that he ran it with only one full-time employee. His children went to the local state school, his wife did the housework, and 'The Street', the double row of uniform cottages where the slaves used to live, was empty and tumbledown.\n\nWhile we were talking on the porch a great cloud of dust approached us from the drive, and there emerged in stately motion two large mules. They were pulling a kind of sledge, a cross between a bobsled and Cleopatra's barge, and sitting on it, very old and dignified, was a Negro in a straw hat. Round the corner he came in imperial state, the mules panting, the sledge creaking, the dust billowing around us, and as he passed the porch he raised his hat by its crown and called: 'G'd evening, boss, sir; g'd evening, Missus Parker.' 'Good evening, Uncle Henry,' they replied.\n\nHe was an old retainer of the Parkers who lived almost entirely on their kindness. He was given a house and a few acres, firewood and storage space and a loan when he needed one. The planter would not see him in distress for the world. But to suggest that he might invite the old man into his house, or even shake hands with him, would be more than an impertinence, but might well be construed as a deliberate insult. Uncle Henry will always have a home, but, after all, the race must be preserved.\n\nFamily home\n\nAfter a while I felt quite familiar with the social structure of St Andrew's, New Brunswick. Who was this, for instance, smiling at me so kindly from the Wren House on Queen Street? Why, who but Miss Lelia Wren, who lives with her sister Miss Frances in the house their family has occupied for 150 years. Who is at the helm of that white boat out there? Mr Hered Hatt the scallop fisherman, of course\u2013everyone knows that. In no time at all I was acquainted with Mr Ian Mackay, who owns the Shiretown Inn, and with Mrs Bobby Cockburn, whose late husband's pharmacy was one of the town's prime power centres, and very soon the Venerable Nantlais Jones was waving to me from his handsome Buick Park Avenue limousine, which has CLERGY in ecclesiastical lettering on its windscreen. Hardly has one well-known householder introduced me to her stately collection of teddy bears ('That's Boogy, that's Oogy, that's Daddy Bear in the corner') before another is telling me how effective birth control pills have proved in the propagation of her hibiscus plants.\n\nIt was like exploring a rambling old family home, the streets its corridors, the houses its rooms, the citizens its extremely gossipy owners and retainers. One morning I arranged to meet two of the town's many widows and, idly passing the time beforehand by wandering through the town cemetery, I found both those ladies' names already inscribed upon gravestones, below their departed husbands'.\n\nKey West, 1960s\n\nKey West is full of people with nothing much to do, but a talent for lounging gracefully in doorways. If I stood on the waterfront on a sunny morning I would soon find other idlers wandering to my side to stare at the water with me, and sometimes gentlemen would buttonhole me with dark questions. Was I looking for rare fish? Had I spoken to Mr Alvark? Was it right, what the papers were saying about convertibility? Did I realize that the deputation from Ecuador was arriving next day? What did the British government think about labour restrictions in Peru? Most of them had wild gleams in their eyes, and having said their queer bit, shuffled away like disappointed saboteurs. Slow and old is the island city of Key West; also surreptitious, bland and turtle-like.\n\nAll in the family\n\nAt the railway station at Assiut an elderly Copt had come to meet me. We sat in the station cafe for a preliminary cup of coffee, and he undid the buttons of his tight linen jacket and wiped his head fastidiously with a silk handkerchief. 'I come from a family of priests,' he said by way of introduction. 'There have been Christian priests in my family since AD 48, when St Mark paid his visit to Assiut; and before that my family, through unnumbered centuries, provided priestly acolytes for the local god of Assiut, Leci. Come, finish your coffee while I settle this infamous account.'\n\nMarching with us!\n\nOne evening we were driving down a road on the outskirts of Chattanooga when we saw a dirty marquee. From it there came strains of music, with accompanying desultory snatches of women's voices, so we stopped at once and went inside. At the end of the tent a very fat woman was lying on the ground quivering and shaking, sometimes tremulously, like a jelly, sometimes with sharp stabs of impulsive movement. Two fierce women were supporting her head, and standing above them, waving his arms like a Paganini, prancing crazily here and there, a youth was strumming on a guitar. In the background a small girl was banging a hymn tune on an upright piano, and a group of black ladies, respectably dressed, looking a trifle bored, and sometimes pausing to exchange gossip or look out of the tent flap, was half-heartedly singing some sacred words: 'I'll never go hungry or know poverty\/ So long as the good Lord is marching with me.\/ Marching with me! Marching with thee!\/ So long as the good Lord is marching with me.'\n\nPresently the prostrate patient, with heavings and convulsions, tried to gasp a few words, and at this the attendant harpies were galvanized. Seizing the patient by the front of her dress, they yanked her into a sitting position and hissed urgent instructions into her ear. She was still jerking incessantly. 'Take Him in, take Him in!' they hissed, and were soon screaming, 'Take Him in! Roll it! O Jesus, the glory of it!' until the patient herself, jerking and jumping, managed to croak from her constricted throat a few unintelligible syllables.\n\nWhen we left the marquee she was still unhealed. The guitarist still whirled about her. The piano still tinnily clanged. The lady choristers whined their listless hymn. And the convulsed patient, all her draperies loose by now, was still being urged to 'Let Him in, sister! Glory, glory, roll it, roll it!' by the demon women at her side.\n\nEconomic imperative\n\nIn the worst times of the Irish troubles, when Belfast was more or less in a state of war, I once saw a patrol of five or six British infantrymen moving cautiously and watchfully through the city centre in the prescribed mode\u2013guns cocked, helmeted heads constantly turning right and left, lead man well in front, rearguard walking backwards with his finger on the trigger. As they passed an office of the National Westminster Bank one of them peeled away, while the others crouched there covering his back, ready for instant fire. He put his card in the bank's cash dispenser, he tucked his money away in a pocket of his camouflage suit, and they proceeded grimly on their prowl.\n\nSporting pleasures!\n\nMy first floodlit cricket match, in Sydney, was a terrific affair. Australia were playing New Zealand, and passions ran high. If a wicket fell or a catch was missed the crowd burst into magnificent displays of emotion, throwing hats, paper, cups and balloons into the air, shouting, whistling, clapping, booing and cheering. I was exhilarated! In the course of the game I happened to look over the balcony into an open space outside the stadium, and there I saw a succession of young men being hauled in, handcuffed by plainclothes policemen, briskly questioned, photographed there and then and shoved into a windowless van from whose interior emerged muffled thumps of protest. A few yards away, within sight of the police but on the safe side of a high wire-mesh fence, three small boys were getting their own kicks by sniffing aerosol cans.\n\nThe French swimmer\n\nTo my right, as I sat beside the harbour at La Rochelle, there came into my field of view a swimming man. He was in his fifties, I would guess, but stout and muscular, and he was swimming with an absolute rhythmic exactitude. A slow and powerful crawl, one two, one two, deep wallowing in the water went his head, up came his podgy arm, out emerged his face for breath, running with salt water\u2013a slight pause at the top of his stroke, and he was down again half submerged. He never wavered. The pace of his stroke was metronomic, and it suddenly occurred to me that he was on the way somewhere, as one might walk to work, or take a bus. He was the first swimmer I ever saw who was using his crawl as a means of transport. I watched him intently, and once I thought I caught his eye, as he rolled around for breath out there; but if I did it was an entirely dispassionate eye, like the lens of a submarine's periscope. It took him some time to pass me, until he disappeared round the headland to my left, and for some time afterwards I fancied I could hear the regular flop and splashing of his stroke, as one sometimes hears the tread of a ship's engines when it has long sailed out of vision. Since then, whenever I hear such a beat of engines in the night, I think, there goes the French swimmer on his way.\n\nThrashing as they went\n\nOn Ascension Day they beat the bounds of St Michael's Church in Oxford, to establish once again the parish limits and emphasize the old pre-eminence of the Church. Once I followed the course of this antic but moving ceremony. Led by the vicar, a little raggle-taggle group of choirboys and parishioners paraded through the city centre, now and then pausing at immemorially ordained spots to thrash a wall with canes and shout 'Mark! Mark! Mark!' (in the old days they used to thrash the choirboys too, to impress the boundaries on them once and for all). The route they pursued was involved, but the vicar and his crew were not perturbed. Once they scrambled over a high wall, once they marched deadpan through Woolworth's, and once they beat the wall of a banana store in the market. They followed the line to the bitter end, thrashing as they went, as fifteen generations of parsons and giggling choristers had loyally done before them.\n\nForty tailors and a camel driver\n\nIn a little upstairs factory in a Cairo backstreet forty tailors work, year after year, on the vast and splendid carpet, lavishly embroidered with gold thread, that covers the sacred shrine of the Kaaba in Mecca. A new one is woven every year, and is taken to the holy city at the time of the great pilgrimage, escorted by soldiers of the Egyptian army. When I visited the factory the tailors were nearly all Turkish by origin, and nearly all related to one another, and nearly all very old, and some of them represented the third or fourth family generation to work there. They sat at trestle tables in a long rickety room and stitched away there like the tailors of myth. They held their eyes very close to their work, and some of them wore little steel-rimmed spectacles on the ends of their noses. Their director, a portly and paternal official, sauntered up and down the tables with me, and the tailors, working away with their reels of gold thread, threw pleasantries as we passed. I asked one how old he was. As old as the hills, he said, but the director said with pride that he was actually a hundred years old\u2013'And so,' he added, peering round the room and indicating another benevolent ancient, 'so is the one in the corner, the one with the hat on.' All the old men grinned and nodded.\n\nAs we climbed down the staircase to the street, I noticed an elderly Egyptian sitting morosely on a stool outside the door, like a disconsolate watchman, with a white scarf around his head and a string of prayer beads in his lap. Who was he, I wondered. They said that until a few years before the carpet had been taken to Mecca each year on a magnificently caparisoned camel, and the man on the stool had been the camel driver. His unique occupation was gone, and he had never been the same man since. 'Poor fellow,' they said, 'he never leaves the factory'\u2013and when we looked at him, I noticed, he shifted his big feet uneasily.\n\nSingin' in the dawn\n\nOnce very early in Beijing I strayed over a bridge to a leafy path beside a moat. I was led there by a curious cacophony of shouts, singing and twanged instruments, and I found it to be a place of self-fulfilment. Resolutely facing a high stone rampart above the moat, like Jews at the Wailing Wall, all along the path men and women were rehearsing their own particular accomplishments privately in the dawn. As we sing in the evening tub, so the people of Beijing go to that wall. Here was a man, his face a few inches from the masonry, declaiming some heroic soliloquy. Here a woman was practising an astonishing range of arpeggios. A splendid bass was singing a romantic ballad, a poet seemed to be trying out a lyric, an elderly man with a bicycle was plucking the strings of an antique lute. I thought of joining in, so universal did these impulses seem, sending To Be or Not to Be reverberating down that wall, or perhaps reciting some of my own purple passages: but I restrained myself, as a Foreign Guest, and just whistled my way home to breakfast.\n\n'I hope you see it truly'\n\nA young forester walked by, as I picnicked by Loch Ness, and I asked him if he had ever seen the monster. He did not smile at the question. He had lived there always, he said, but he had not seen it yet. For him, though, its existence or nonexistence was not important, because he interpreted it as a didactic figure of faith. 'It teaches us to believe in something we canna see\u2013you understand me?' He thought a great deal about the matter, he told me, and often looked out there on the half chance of glimpsing the creature. I said I seemed to see it every five minutes, but again he did not laugh. 'Well before you go home,' he said meaningfully, looking me straight in the eye, 'I hope you see it truly...'\n\nNot altogether intelligible\n\nThe holy land of the Yezidis is in the Kurdish country of Iraq, and I was taken to meet some in their village north of Mosul. They follow an unusually cloudy religion concerned with the worship or perhaps propitiation of the devil. They seemed to me distinctly vague about it all, but although they were very hospitable I was haunted throughout my visit by the fear of committing some dreadful spiritual solecism. I must never, I had been told, utter the name Satan, for it is anathema to the devil: if somebody does speak it, the really convinced Yezidi must either instantly kill the transgressor, or commit suicide. Lettuces were strictly taboo: it is said that the Evil One once tried to hide inside a lettuce, but found its leaves insufficient to conceal him. Radishes were also unpopular, I was told, and the colour blue was something the Yezidis particularly loathed. They are most welcoming in everyday affairs, though, and if I wandered up the village stream the housewives at their washing would smile at me and make jokes (which, being expressed in a corrupt version of medieval Kurdish, were not altogether intelligible to me).\n\nThe croupier\n\nI remember clearly the appearance of one of the most famous of the Nevada croupiers. He was a tall man who wore a check shirt, open at the neck, narrow trousers sustained by a belt with an ornate buckle, and a black eyeshade. His face was withered and wrinkled like a tortoise's, his nose hooked and slightly crooked, his eye sharp and pale, his mouth thin but humorous, conveying an impression of very calculated bonhomie. His ears were long and protruding and his long thin neck was entwined with a mesh of muscles, like Laoco\u00f6n and the snake. Coldly and knowingly this man presided over the game, taking or paying mechanically with never a flicker of emotion, only the slightest hint of a nod, or the suspicion of a gesture, or the embryo trace of a beckon in the direction of the management. In front of him the piles of big silver dollars (common currency in Nevada then) glittered like stage properties; and once in a decade, I dare say, there passed through his hands a dollar made of gold, withdrawn with heart searchings from beneath some aged indigent's mattress.\n\nMormon faithful\n\nWhenever I think of Salt Lake City, with the pinnacles of the Mormon Temple shining there beneath the mountains, I think of bright clothes and urgent smiles, the voices of the vast Mormon choir ringing across Temple Square on Sunday morning, the unquenchable cheerfulness of the people, the general air of satisfied competence, and the extraordinary blandness of the old lady who told me one fine summer morning that for high religious purposes she had been tracing the course of her ancestry, and had succeeded in establishing it as far back as 64 BC, 'Only a few years,' as she rightly remarked, 'before Caesar went to England, but of course the ancient Americans had been civilized for centuries, as the blessed prophet Moroni told our founder\u2013that's him, that's the prophet Moroni, right up there on the Temple tower\u2013see?'\u2013and I looked up there, shading my eyes against the sun, but could perceive only the vague outline of that antique saint, holding what looked like a trumpet.\n\nHis Highness\n\nOne fine Arabian morning I walked into the palace of the Sultan of Muscat and Oman, on the shores of the Indian Ocean in Dhufar. Through the great gate of the outer courtyard I passed, and the slaves bowed low, into the polished hall of the palace, lined with bearded and begowned retainers, their rifles in their hands, until there approached me from the darkened recesses of the building a small dignified figure in a brown and gold aba, a turban on his head, a sword at his side, a heavy scent of frankincense emanating from his person. 'Good morning,' said His Highness the Sultan Said bin Taimur.\n\nHe was only forty-four, but the voluminous dignity of his robes, his stately bearing and his luxuriant beard all combined to make him look much older. His eyes were large, dark, long-lashed and very serious. His mouth, though kindly and humorous, looked to me capable of an occasional sneer. It was an antique, melancholy face, such you might see in old pictures of the East, and as profoundly enigmatical as the Pyramids. Later I was to encounter him in less autocratic mode, and then his eyes had a soft, thoughtful, almost sleepy look beneath their heavy eyelids, reminding me rather of an elaborately turbaned Cheshire Cat.\n\nFeudalism\n\nIf you go down a gold mine, in the South Africa of the 1950s, you will find that racial feudalism extends even to the face of the reef. A black man brings you your boots, helmet and overalls; and a black man hands you your face rag as you enter the hoist; and a black man blows his whistle and drives you in his trolley along the underground corridor; and a black man helps you off with your jacket when, as you approach the stope, the heat of the pit suddenly blasts you. A white Afrikaner overseer grins you a welcome there, but in front of him, flat on his back in an alcove of the rock, is the African driller, helmeted and bathed in sweat at the very war front of the mine. He holds his big drill with his feet, and he lies there like some hefty freak or prodigy, a handless painter or a three-legged man, his whole body shaking with the vibration of the drill, and the very air about him shuddering with its noise. He pauses in his work as you approach, but the supervisor gives him a flicker of his torch, and he is off again, smiling broadly through his dirt.\n\nMerciless fish\n\nAt sea in the Caribbean an elderly sailor pointed out to me the dark shadow of a shark, loitering beside the hull of our ship, and this is what he told me: 'It's got no marcy, no marcy at all. Big blue fish, so you can't see 'um in de water, he's sly! No marcy, see, not a drop of marcy!'\n\nNanny talk\n\nThe nannies of the London park were there in their battalions, elderly complacent nannies and perky young ones and hard old professionals with starched faces. 'So I said to her, I said, \"No, madam, it is not and never has been my job to make the tea...\"' 'It's never been the same since Lady Sarah passed over but, there, times have changed, haven't they, dear?' 'No, Jeremy darling, keep away from the doggy, dear...' '\"Give him his tea?\" I said, \"I haven't been looking after children for thirty years without knowing when it's teatime,\" I said, and with that I walked out...' 'Try rubbing his back, Mabel, that usually brings it up, doesn't it, dear?'\n\nChief of the Egyptians\n\nGamal Abdel Nasser, the President of Egypt, lived blamelessly with his buxom wife and five children in a modest Cairo house that was plain to the point of ugliness. No rude or ranting orator greeted me there, behind some big officious desk. On the contrary, the Chief of the Egyptians was relaxed and friendly, in shirtsleeves, his vest showing between the buttons, and he gave me coffee and talked pleasantly and intelligently for as long as I liked. Nasser like to call himself the first indigenous ruler of Egypt since the Pharaohs, and he was indeed a genuine through-and-through Egyptian, born of peasant stock on the banks of the Nile. 'What a reasonable sort of man,' I said to myself as we talked across the plain deal table, sipping thick chamomile coffee from cups edged with blue roses and gilt.\n\nI was not deceived, though. For many long years Nasser led an underground revolutionary movement, and I knew he had talents of deception and conspiracy of a very high order. His horizons were limitless, and he liked to talk about circles of power, national destinies, the interventions of fate and that sort of thing. The hours slipped smoothly by as he expounded his theories, the coffee cups came and went, until at last the President rose from the table, his sandals flip-flopping across the linoleum, to see me to the door in his shirtsleeves and wave me goodbye into the night. The sentries saluted obsequiously.\n\nAnglo-Sudanese\n\nGood living is a Sudanese tradition, but it came as a disagreeable surprise to me in a Khartoum bar one evening to meet a young Sudanese, just down from the university, drunk not in the Sudanese but in the British manner; facetious with the sweaty banter of his British companions, not with any African drollery, with his tie loosened precisely as theirs was and a cigarette sticking to his lower lip. His grandfather had charged across the plain at Omdurman, brandishing a spear and screaming, but when this modern Sudanese slurred into the maudlin it was the maudlin of smoky pubs and potato crisps. I was shocked. But the British administrators of the Sudan have a wonderful knack of making you feel slightly ashamed of yourself, and I thought of that unlovely young man when I later read in a pamphlet of theirs: 'A new nation is being born, and in the difficult world of today the new arrival needs all the sincere sympathy and disinterested help you can give or get it.'\n\nI blushed: but it did not matter, for all the electric lights had gone out.\n\nAn exotic\n\nSen Tenzing was a Sherpa who had become well known as a porter with British mountaineering expeditions in the 1930s. He had always been a man of lively tastes, and by the time I met him in Kathmandu, when he was elderly, much respected and semi-retired, his appearance was wonderfully distinctive. On his head he wore a brown balaclava helmet with a peak, like the hats the Red Army used to wear. His grey sports shirt had polished major's crowns upon its epaulettes. Over woollen long johns he wore a voluminous pair of blue shorts, and on his feet were elderly trainers. A confused variety of beads, tokens and Tibetan charms dangled around his neck and a bracelet hung upon his wrist. In one hand he flourished an ice axe, in the other a fly whisk. It was not for nothing that Sen Tenzing, in the old days of gentlemanly climbing, had been affectionately christened by his British employers 'The Foreign Sportsman'.\n\nMr Beebe\n\nVirginia City, the most famous old mining town of Nevada, has been kept boisterously alive by gambling, and by the presence there of Mr Lucius Beebe. Mr Beebe owns and edits a revived newspaper of the Gold Rush days, the Territorial Enterprise, and he lives grandly in a small Victorian mansion, keeping Rolls-Royces and St Bernards. Almost before we had settled in at our hotel he was aware of our presence by bush telegraph, and before long he was showing us the town, wearing a hat with a flat crown and very broad brim, a shirt with a wide and handsome check, an elegant pinstriped suit and a waistcoat embellished with a gold watch-chain. Mr Beebe is a fine sight at any time, but is at his best when he strides into a gambling house with his St Bernard at his heels, pausing for a moment beside a roulette wheel to throw a handful of silver dollars on the table with a satisfying clang, shrugging his shoulders with cheerful nonchalance when he loses the whole lot, bending an ear to a tattered prospector from the hills who has some slight financial worry, raising a negligent hand of greeting to an acquaintance here and there, listening patiently to the report of activities of a man who plans to get even with him for something he published in the paper last week, ushering his guest into the dimness of the bar with a truly Bostonian courtesy before hitching his ample frame on to a bar stool and ordering an enormously large whisky. During our stay in the town Mr Beebe lent us one of his Rolls-Royces, for our convenience.\n\nBattle hardened\n\n'Lucky you got me,' Chicago taxi drivers nearly always seemed to say, if you wanted to visit the tough black neighbourhoods. 'Not many guys would take you. I tell you, I was a Marine for four years, I fought in eight major battles, eight major battles, and believe me if any of these blacks get in my way I'll just run 'em down, just like that. Lock your door now. Like I say, it's lucky you found me. Not many guys would come out this way.'\n\nCelebrating with Breughel\n\nIf ever you attend a rustic wedding in the Orange Free State you will realize how close the Afrikaner can be to the world of the old Dutch masters. The reception is held in the church hall, and the room is packed, and hot with robust gaiety. At the top table sit the bride and groom, flushed and rotund, she in an ornate white headdress, he intolerably corseted in black. Here are the bride's parents, wrinkled and sharp of face, and here also the two small bridesmaids, their plump country figures wrapped in pink and blue, posed self-consciously beside a potted palm. Big black servants scurry about with cold drinks and sweetmeats. 'It's all done to plan,' says your host complacently. 'All the tables are numbered, you see, so that everyone knows just where to sit\u2013no confusion, you see, no pushing or shoving, everyone can have a good time.' And everyone does. Now and then somebody makes a speech, generally disregarded, and the bride and bridegroom sometimes simper at each other at the demand of amateur photographers, and a hubbub of enjoyment and mastication fills the hall. Each trestle table makes a party of its own and eats its pastries with gusto, and shouts cheerfully for the Africans with the drinks; and the whole scene is warm and homely and animated, with the sheen of red velvet dresses, the fizz of bottled pop, smiling weathered faces, white satin, excited little girls and a smell of flowers and scent and sandwiches.\n\nShoeshine\n\nThe waiter at Colombo put down my breakfast and said he hoped I would have an enjoyable day. I told him I was going to make a pilgrimage to the grave of my father-in-law, a planter who had died in Ceylon during the war.\n\n'By God,' he said at once, 'that's good, that's very good\u2013parents is a bigger thing than the Lord Buddha himself,' and picking up my shoes, to clean them for the occasion, he bowed gracefully and withdrew.\n\nWe did not linger\n\nWith an American colleague I once went to a ceremony at Alexandria at which some new Czech weapons were to be handed over to the Egyptian forces. In those days many German specialists and advisers were working for the Egyptian army, and as we waited for the ceremony to begin we noticed a crowd of Egyptian officers milling around a tall figure in a black beret at the corner of the grandstand. We elbowed our way across and found ourselves face to face with as obvious and disagreeable a Nazi officer as ever I saw. His face was congealed with hauteur; his movements were stiff and mechanical, like a robot's; and icy cold were the eyes with which, flicking his cane against his long legs, he turned to look at us. The jostling Egyptians crowded admiringly all around him, but my colleague was a Jew, and we did not linger.\n\nA family outing\n\nIt was a festival day of some kind, and in the evening I asked a taxi driver in Beirut to take me for a run around the neighbouring hills, to observe the village goings-on. He brought along his family for the ride\u2013a plump smiling wife in black, a little boy in jeans and a very small baby girl with enormous brown eyes. The driver had spent some years in America, and his English was sprinkled with rather dated Americanisms\u2013'Say, what you say we stop for a sundae?' or 'How d'ya feel like a Coke, baby?'\u2013as we progressed through the balmy evening. We frequently stopped in villages for some quick refreshment among the celebrations. Candles were burning in many windows, and there was a constant crackling of fire-works and whizzing of rockets. Gangs of young men strolled about the hilly streets, singing and shouting. Innumerable friends and relatives of the taxi driver emerged from houses to impede our progress, and we had so many bottles of pop that the baby was visited by a staccato series of burps. 'What feast day is this?' I asked the driver. 'Christmas, friend,' he replied (it was the middle of July).\n\nWhen we started our journey back the family was fast asleep in the back seat, in a tangle of ungainly abandon, and the driver and I smiled at each other. 'Dig those crazy guys,' said he, as another festive party rollicked by.\n\nTactful parent\n\nDuring my stay in Darjeeling I often saw a young American dressed in the habit of a Buddhist monk. He was studying at a nearby seminary, I was told, and wore the brown cloak, the sandals and the hair bun as to the manner born. Nobody appeared in the least surprised by this anomalous figure, and his father, who was paying him a visit from the States, seemed entirely at home with the turn of family events. 'I'm going to drink, Jimmy,' I heard him saying to his son one day, puffing at his cigar and raising his glass, 'I'm going to drink to all these wonderful, wonderful people of Darjeeling!' (And 'Say,' he tactfully added as he put his glass down, rather hastily I thought, 'is this Indian wine? Delicious!')\n\nA small magnifico\n\nI was strolling through the souk in Qatar when there emerged from a doorway beside me the smallest man I had ever seen. He was about four feet high, stout and prosperous looking, dressed in the resplendent regalia of an Arab gentleman\u2013splendid brown aba, milk-white keffiyeh, black head-band, dagger and beads. With a flourish and a toss of his head this marvellous figure strode down the steps of his house and swaggered away through the souk, as bold and assured as any gigantic African chieftain or Renaissance aristocrat. His proud head bobbed away among the packing cases, and a breath of the incense that perfumed his beard hung in the bazaar behind him.\n\nA large magnifico\n\n'Back again,' said the magnifico at the cafe on the last ridge before Cetinje and the heart of Montenegro. We had met before, you see. He is always there, it seems, summer or winter, like a major-domo of these uplands, or a Chief of Protocol. He wears black breeches, and a wide belt like a cummerbund, and he stands about seven feet tall, and speaks in a basso profundo, and tosses slivovic back like lime juice, and is in all respects the very model of a modern Montenegran.\n\n## A snatch of sound in Morocco\n\n'Go to sleep now,' they said, 'the operation will be later.'\n\nBut when they had gone I got out of bed rather shakily,\n\nfor the drug was beginning to work,\n\nand went to say goodbye to myself in the mirror.\n\nAs I did so a street vendor outside played a delicate\n\narpeggio upon his flute,\n\na very gentle merry sound\n\nwhich he repeated, over and over again,\n\nin sweet diminuendo down the street.\n\nFlights of angels, I said to myself, and so\n\nstaggered to my bed, and oblivion.\n\n##\n\n'Ain't that right?'\n\nIn Montana once I found the road blocked for a mile or more by a mass of sheep. Some were moving very slowly, some were nibbling the sparse grass beside the highway, some were sitting down and one or two seemed to be fast asleep. At the head of this leisurely procession were two cowboys, mounted on fine black horses. The men were very weatherbeaten, dirty and bearded, with their tangled hair escaping from their hats and their fingernails black and broken. They had been rounding up the sheep in the surrounding mountains, to bring them down for shearing and to escape the coming winter storms. 'We been fourteen days in the hills,' said one, 'and seven days on the move. Sheep ain't very fast movers. Boy, will I be glad of a bed!\n\n'As for this horse,' he added affectionately, 'all he wants is a good hot cup of coffee and a place to put his feet up. Ain't that right, boy?'\n\nIn a trance?\n\nWhen I was alone in the Himalaya one day I saw a man. I saw him first in the extreme distance, across an absolutely blank snowfield at about 19,000 feet, to which I had climbed from the glacier below for the sake of the view. At first I could not make out what he was\u2013only a black swaying speck, indescribably alone in the desolation. As he came closer I could see that he could only be human, so I plunged through the loose snow to meet him, and presently, there near the top of the world, thousands of feet and many miles above the trees, we met face to face. It was the strangest encounter of my life.\n\nHe was a holy man, wandering in the mountains, I suppose, for wandering's sake. His brown, crinkled, squashed-up face looked back at me expressionless from beneath a yellow hood, and seemed to find nothing strange in my presence there. He wore a long yellow cloak and hide boots, and from his waist there hung a spoon and a cloth satchel. He carried nothing else, and he wore no gloves. I greeted him as best I could, but he did not answer, only smiling at me distantly and without surprise. Perhaps he was in a trance.\n\nI offered him a piece of chocolate but he did not take it, simply standing there before me, slightly smiling. Presently we parted, and without a word he continued on his unfaltering journey, making, it seemed, for Tibet without visible means of survival and moving with a proud, gliding and effortless motion that seemed inexorable. He did not appear to move fast, but when I looked round he had almost disappeared, and was no more than that small black speck again, inexplicably moving over the snows.\n\nThe truth of it\n\nIn a churchyard in County Monaghan I stood beside the grave of Seamus McElwain, a young IRA man whose whole life had been a succession of bloodshed and imprisonments until he had been killed by British soldiers in a neighbouring meadow. His epitaph was in Irish, and on the cross, together with the relief of a bird escaping through a mesh of barbed wire, was affixed a coloured photograph of him, a good-looking dark-haired boy in a dinner jacket. The tears came into my eye as I stood there (the wind rustling the hedges all around), and a gardener working nearby asked me if perhaps I was a McElwain myself? But I said I was simply crying for them all, whatever side they were on. 'That's the truth of it,' he said, 'that's the truth.'\n\nUrgent inquiries\n\nWide eyed and open handed the Fijians greet me, in their tidy thatched settlements off the highway, or among the mangrove swamps where the women, hitching their skirts up to their waists, scoop about indelicately for shellfish. 'Where are you going? What is your name? Are you married? Where do you live? Have you any children? Would you like a banana? How many people live in London? Do you sleep alone?' Their inquiries are directed urgently at me: the Fijian for curiosity is via kila\u2013knowledge want. I told the maids in my Suva hotel that I was scared of them, because they stared at me so hard, and in the evening I found a little bowl of flowers placed beneath my window in appeasement. 'Aaah,' they said, shaking their big kind heads in remorse, 'we no wish to frighten you.'\n\nTheir forebears used to be cannibals, but I would not mind being eaten in Fiji. The pot would be spiced, the cooking gentle, and the occasion in most ways merry.\n\nOn the waterfront\n\nAfter a day around the docks with the notorious Manhattan longshoremen, still among the more obstreperous of the city's workers, I thought of them with more sympathy. That evening I took a coffee and doughnut in a diner down there, and tried to imagine what it would have been like to be a poor Irish immigrant half a century ago, looking for work on that hard and dirty shore. When the man on the next stool scowled across the napkin stand and said nastily, 'You want all the sugar?' I told myself he was the victim of historical circumstance, and said I was most awfully sorry.\n\nEmpire builders?\n\nI loved to sit on a Hong Kong ferry and contemplate the remaining Britons of Hong Kong. They were trapped for me there, like historical specimens, deep in their tabloids or compiling their shopping lists. What a line they represented, I used to think! What generations of exile culminated in their persons, listlessly looking out across the passing harbour, or doing the crossword puzzle! Their forefathers blazed a way across the world, veld to outback, pioneering in shacks, beachcombing on reefs, disciplining recalcitrant Sioux or bayoneting fuzzy-wuzzies; and there they were beside me, the last of the long parade, indifferent to their origins, unconscious of them, perhaps, unexcited on the slat-seated ferryboat between Kowloon and Victoria!\n\nThe boat shudders; the gangplanks clatter down; the blue-shirted Chinese seamen swing open the iron gates; and in a trice, as the crowd streams off the vessel, those unobtrusive imperialists are so utterly overwhelmed by their Chinese fellow citizens that you would hardly guess there was a Briton left in Hong Kong at all.\n\nIcelandic allegory\n\nBeauty and the Beast is an Icelandic allegory. Everybody who goes to the island is struck by the splendour of the girls. Pick a choice example now\u2013the shop girl, say, who is packing your stuffed puffin in the souvenir shop, or pricing your lava-stone powder box. What a gorgeous strapping girl she is, what a terrific golden girl, with her wide-apart eyes, her hair bleached in Arctic sunshine, her exquisite complexion mixed out of snow and pink blossom!\n\nTake your eyes off her for a moment, though, and observe the young man shambling down the street outside\u2013towards his trawler, perhaps, or off to his fearful shift at the fish factory. He is the original Viking, I suppose. His forehead and his chin symmetrically recede, his cheeks are wolfishly sunken, his eyes blaze, and there is to his loping walk a suggestion of immense loose-limbed power, as though a tap on his shoulder would unleash his doublebladed battleaxe and send him off to Greenland with horns on his head.\n\nWedding pictures\n\nWhen King Hussein of Jordan married it was supposed to be a ceremony of liberated modernity. However, as I joined the society ladies for the occasion I could not help feeling that we were close in spirit to the huddled jealousies and school-girl excitements of the harem. At the head of the stairs were two bold lancers in scarlet tunics and white breeches, as the eunuchs would have stood sentry in an earlier age, and the body of the room was a seething mass of women. They were dressed magnificently, a glitter of satins and brocades and furs, a mosaic of lipsticks and mascaras, a tinkling kaleidoscope of earrings, a flurry of sequinned handbags. Chanel and Dior thickened the air. How often and how brazenly did the women of the court eye each other's coutures! How heavily accentuated were the outlines of their eyes, like eyes seen through diaphanous curtains in forbidden corridors of the Seraglio! How scratchy and talon-like were the fingernails, how pinkly fleshy the figures, and how passive and doll-like those emancipated ladies looked, in serried and perfumed phalanx, as if some lascivious Sultan were about to pass through their ranks, picking a beauty here and a beauty there with a lordly gesture of his forefinger!\n\nBut presently a cameraman in a crumpled jacket suddenly pressed his way past the guards and said (as it were), 'Just one more, ladies, please, give us a nice smile now.'\n\nA holy clock\n\nIn the Church of the Holy Sepulchre at Jerusalem a little chapel stands shrouded in black curtains beside the site of the sepulchre itself. One morning I found myself all alone before this shrine, and as I stood there in the silence I thought I heard a faint ticking noise from the inside. For a moment I stood hesitant, thinking it might be the working of a perpetual censer, or perhaps the swinging of an ornate lantern on its chain. But it was so regular and so insistent that I pulled the heavy curtains aside and looked in, and there on the altar, all alone among the ikons and candlesticks, a red moon-faced kitchen clock ticked away robustly, for all the world as though it were timing the eggs. I laughed with pleasure at this unexpected discovery, and there was an answering chuckle behind my back. Standing among the tall pillars of the rotunda, all but hidden in the shadows, there stood a gigantic Abyssinian priest in an attitude of serene meditation. When I turned to look at him a white gleam in the darkness testified to the smile upon his black bearded face.\n\nSilly question\n\nWildness, freeness, recklessness\u2013not in Vienna! I went to a police court there one day and, noticing one of the accused studying a road map between hearings, asked him if he was planning an escape. 'No,' he said, 'I am deciding the best route to visit my aunt at Graz.'\n\nExpertise\n\nI visited the Pioneer Museum, in Fremont County, Wyoming. During my visit a schoolmistress was taking a group of children round the exhibits, and I heard her drawing their attention to a chair that stood in a corner of the gallery. It was made all of bleached white horn: legs, seat, back and all. 'That is an example', she was saying, 'of the craftsmanship of the very first pioneers to come to Fremont County. Isn't that beautiful, children?' 'Yes, ma'am,' most of them replied, but after the main body had moved on to the Shoshone relics, a section of the museum I preferred to circumvent, I noticed a pair of laggard urchins trailing along behind. They had not heard their teacher's encomium of the chair, but they too paused as they passed it, and inspected it with no less knowledgeable admiration. 'Jeez,' said one to the other. 'Take a look at that elk.'\n\nFirst time lucky!\n\nI crossed to Hoboken on the very last voyage of the very last Manhattan railroad ferry, and the shabby old boat was full for the occasion with reporters, cameramen, roistering office workers, people with banners and leaflets and comic hats and crates of champagne, and a trio of Salvation Army girls singing unaccompanied hymns indefatigably above the hubbub. I sat down on a grubby bench beside an elderly habitu\u00e9 of the ferry. 'You sound kinda British,' he said. 'Funny place for you to be, ain't it? You ride this boat regular?'\n\n'Never before in my life,' said I.\n\n'Well, you don't say. Some people have it easy. I've had to ride this ferry forty years to make the last crossing. You come over and do it first time!'\n\nAdvice\n\nBy and large, it seemed to me, British businessmen in Hong Kong pursued their various careers in a pleasant state of half-speed ahead, eating well, enjoying their friends, gossiping in the club bar, taking the junk out on Sundays\u2013'Whatever you do,' they used to tell me, 'don't go out with Bill [or Simon, or Ted], you'll be drunk before you get out of the harbour.'\n\nRacial tension\n\nSix or seven miles out of Pretoria, on my way to catch an early flight, I saw a black figure running helter-skelter down the road towards the city. A moment later another followed, and then two or three more, and they panted by us, with serious faces and bulging eyes, like participants in some strenuous sunrise celebration.\n\n'What are they doing?' I asked my driver.\n\n'Those Kaffirs? They're on their way to work. They've probably got to start at seven, and they've got a long way to go, so they've got to run fast. It won't do them any harm.'\n\n'You don't like Kaffirs?' I surmised.\n\n'Kaffirs?' he replied with a genial twinkle. 'I love them like they was vermin.'\n\nKurds on the move\n\nHow's this for a glimpse? Once, once only, I encountered a tribe of Kurds on the move. It was spring, the grasslands were thick with flowers, and in the distance the mountain barriers stood blue and purple and formidable, with a fleck of snow on their summits and a cloudless sky above. Against this heavenly background the Kurdish nomads moved triumphantly across my field of vision. Their herds of sheep and floppy-eared goats scrambled and jostled in the sunshine, and behind them the bold horsemen lorded it across the plain, riding their stocky horses like avenging marauders, rifles across their backs, bandoliers across their chests, sheepskin jackets slung about them grandly. The women walked alongside, carrying the baggage; the children scampered or lagged behind; the great herds eddied about and spilled over the landscape; and the effect of the procession, glimpsed in so wide and airy a setting, was that of a community of unusually cheerful brigands crossing a steppe to commit an atrocity.\n\nBefore fame hit him\n\nA couple of days after Edmund Hillary and Tenzing Norkay became the first climbers ever to reach the summit of Everest, I was near the foot of the mountain on my way home. It was a still, oppressive, grey morning, and I saw away up the glacier, coming down from the mountain, a solitary figure moving swiftly and gracefully down the valley, swinging and buoyant, like some unspoilt mountain creature. A wide-brimmed hat! High reindeer boots! A smile that illuminated the glacier! An outstretched hand of greeting! Tenzing!\n\nHe took off his big hat, smiling still, and sat down upon a rock. He was going to rest and wash, he said, and then traverse a neighbouring ridge towards his home village, where his mother lived. We had breakfast together, and he pulled from his wallet a snapshot of himself with a number of little Tibetan terriers. 'Given me by the Dalai Lama,' he explained with pride, and taking a pen from his pocket he slowly wrote his signature (the only word he could write) across the bottom of it and handed it to me with a deprecatory grin. The last I saw of him there, he had stripped his lean lithe body to the waist and was soaping himself with water from a tin basin. It looked a chilly operation.\n\nBy the very next day he would be one of the most famous men on earth.\n\n'Mamma mia!'\n\nA veteran fisherman took me out into the Venetian lagoon to find an island house I had read about, but when we reached the spot we found that nothing remained of it but a pile of rubble. The old man was astonished, but even more affronted. 'Now why should a thing like that happen?' he asked me indignantly. 'Mamma mia! That house was there when I was a child, a fine big house of stone\u2013and now it's gone! Now why should that have happened, eh? Tell me that!'\n\nHe was an urbane man, though, beneath his stubble, and as we moved away from that desolate place, and turned our prow towards San Pietro, I heard a rasping chuckle from the stern of the boat. 'Mamma mia!' the old man said again, shaking his head from side to side: and so we chugged home laughing and drinking wine until, paying insufficient attention to his task, that fisherman ran us aground and broke our forward gear, and we completed the voyage pottering shamefacedly backwards. 'Like a couple of crabs,' he said, unabashed, 'though even the crabs go sideways.'\n\nAn apartheid queue\n\nIn the evening all the poor black workers of Johannesburg, forbidden to live within the precincts of the city, rush for the buses that will take them to their slums and sprawling estates. You can hardly watch such a scene without the stirring of some crusading instinct, some Byronic impulse, or at least a stab of pity. As dusk falls, and as the bitter winter night begins to whistle through the buildings, a vast tattered queue moves in raggety parade towards the bus depot, and thousands of Africans shuffle their slow way in double file towards their shabby buses. There is an air of unutterable degradation to the scene, so heartless and machine-like is the progress of the queue, as the white folk hasten off in their cars to the rich city districts, and the lights glitter in the windows of the department stores, and those poor lost souls are crammed into their buses and packed off to their distant ill-lit townships. Many of them are half starving. Most of them live in fear of robbery or violence when they step off the buses into the dark streets of the locations; half of them spend almost all their leisure hours travelling between the city and the far-flung patches of high veld in which they are obliged to live; they reach their homes long after dark at night, and they start work again when the morning is still only a suggestion.\n\nOld-school flying\n\nIn earlier days of transatlantic flying it was generally necessary for passenger aircraft to refuel en route, at Gander in Newfoundland, Shannon in Ireland, or somewhere or other on the way to San Francisco. I was travelling to America on a British Overseas Airways Corporation aircraft when we were told that because of favourable winds we would not for once have to make an intermediate stop. About halfway across, the aircraft's captain came chattily around the passenger cabins, as was the BOAC custom, in the full glory of his regalia\u2013they ran their planes like ships then, and he was very much the Master.\n\n'Everything all right?' he asked courteously, in a clipped public-school accent, when he came to me. 'Having a comfortable flight?'\n\n'Everything's fine,' I told him, 'but there is one thing: are you quite sure you've got enough fuel to get us over without a stop?'\n\n'Never fear,' he replied in the best old-school British style. 'We're terrible cowards up front.'\n\nFestivities!\n\nOn Princes Street that day, when the Edinburgh Festival was in full fling, half a dozen sideshows were performing. An old-fashioned socialist demagogue was haranguing the crowd from his soapbox. A man in full evening dress was singing 'On the Sunny Side of the Street' from the steps of the National Gallery. Two comedians dressed as ancient Egyptians were doing a comic act, and a tipsy old fellow in a kilt was dancing a reel to the music of a wind-up gramophone. All of a sudden amidst the hubbub two young toughs in shirtsleeves struck up a bit of a fight, punching each other in a tentative way and exchanging high-pitched Scottish insults. Instantly all attention turned to them. The orator found his audience dwindling before his eyes, the ancient Egyptians were soon playing to an empty pavement, and swirling here and there across the pavement went all the audiences, wavering and staggering with each exchange of blows. Through the melee, as it disappeared behind the Scott Memorial, I could see the fierce squabbling heads of the contestants, mouthing festive curses.\n\nAloofly towards the dawn\n\nI once heard a pair of Venetian inebriates passing my window at four o'clock on a May morning, and looking out into the Rio San Trovaso I saw them riding by in a gondola. They were sitting on the floor of the boat, drumming on its floor-boards, banging its seats, singing and shouting incoherently at the tops of their thickened voices; but on the poop of the gondola, rowing with an easy, dry, worldly stroke, an elderly grey-haired gondolier propelled them aloofly towards the dawn.\n\nMother Russia\n\nIn Soviet Russia during the Cold War the foreign writer was generally at the mercy of Intourist, the government tourist organization. It was efficient and courteous enough, though speckled no doubt with agents of counter-intelligence, and its younger employees were often refreshingly undogmatic. Now and then, though, Intourist would send you an interpreter of the old school, a woman of severe bearing and inflexible party loyalty. Polite but unmistakably chill such a lady was likely to be, as though you were an emissary of capitalist encirclement, and a day with such an ideologue could be exhausting. I found there was a solution, though, an exorcism. When my companion was particularly severe about my bourgeois deviations, I would turn to her with an expression of deeply wounded sensibility, allow the warm tears to well into my eyes, sniff a little, blow my nose shakily, and tell her I thought she had been unkind.\n\nThis was a magic word. Instantly there would be released from her bosom a flood of immemorial Russian emotion, dimly lit with ikons and scented with incense. In a trice all thought of norms and Seven-Year Plans would be driven from her kindly mind, and she was likely to be on the telephone half the night, making sure I was warm enough.\n\n'Nobody's used this cup'\n\nOne of the notorieties of the Cape of Good Hope is the 'tot' system, which legally allows a wine farmer to pay his coloured labourers partly in cheap sweet wine. 'You're just in time,' a Huguenot farmer told me when I asked to watch the process. 'We give them six tots a day, you see\u2013one when they start, one at breakfast, one at eleven, one at two, one at four and one when they finish work.'\n\nThere stood the labourers in a quavering crew, seven or eight tattered coloured men, and on the steps of a barn a white overseer was doling out the tots. He had a big bucket of thick red wine before him, and as the workers came shambling up with their old baked-bean tins he scooped them their ration in silence. It was an eerie spectacle, for it was plain to me that those dazed and ragged half-castes were in a state of perpetual dissipation. Quaffing their tots in one experienced and joyless gulp they shuffled away again. It was as though eight elderly machines were being greased or refuelled.\n\n'Yes, we give them six tots a day,' said the farmer chattily, 'that's the law. It comes to a bottle and a quarter a day. They sweat it out very quickly\u2013it gives them kick, you see. It's a good wine\u2013here, taste it.' And with fastidious courtesy the foreman, producing a tin cup from inside the barn, wiped it carefully with his handkerchief and drew me a ration. 'It's quite all right,' he said kindly, 'nobody's used this cup.'\n\nExpatriatism\n\nAlmost anywhere in the world of the 1950s I met expatriate Britons of the upper bourgeoisie, and almost always they liked to tell me their memories, personal or inherited, factual or fictionalized, of an England long extinct: a garden party England it seemed to have been, where nobody talked too loud, and there were parasols on the lawn, and we so often used to visit Sir Henry...Sir Henry...what was his name now?\u2013Never mind, I shall remember it later\u2013Anyway, he had this lovely old house. Oh, the smell of the honeysuckle and such gay tennis parties we used to have. 'Of course I know it's all changed now and I could never go back, it would break my heart to see it all so different, socialism, and strikes, and white girls with black men in the streets, they tell me, and all these death duties and so on. But it will always be home to me, Mr Morris\u2013you may be a little too young to understand just how I feel\u2013that's Lindley Hall there, by the way, above the mantelpiece, painted by Robert...Robert...you know, very famous\u2013but I'll remember later, I always do...'\n\nOn second thoughts\n\nWhen I was writing a book about Oxford I read that a special duty of the High Steward of Oxford University was 'to hear and determine criminal cases of the gravest kind, like treason or felony', if the accused was a resident member of the university. In legal theory it meant that until capital punishment was finally abolished in England, this purely academic official was authorized to hang you.\n\nI once told a proctor, one of the intendants of university discipline, that I proposed to follow him and his officers (popularly called Bulldogs) on their patrol through the streets one night, to see how the undergraduates responded to his authority. He advised me not to follow too closely, in case the Bulldogs took offence at my attentions, and summoned me into the proctorial presence. I bristled a bit at this. They'd better not, I said, I was a free citizen, I knew my rights, I could walk where I liked when I liked, nobody could pull antique usage over my eyes, he and his minions certainly had no authority over me. The proctor smiled darkly. 'Are you quite sure?' he inquired; and by heavens, remembering the bit about the High Steward and the felonies, on second thoughts I wasn't.\n\nAlas, proved right\n\nI was never a very astute political observer, and I really did not know what to make of Jack Kennedy, the 35th President of the United States, when I went to one of his Washington press conferences soon after his inauguration. I was charmed by the look, sound and presence of him, as everyone was. I was impressed by his professionalism and his fluency, but some vague instinct told me that although he was only in his mid-forties he was already in his prime. In a report I wrote for the Guardian I tried to express my feeling that the Kennedy we were seeing then was the definitive Kennedy, that we would never know him greatly changed by time or experience, and, as it sadly happened, for once I was proved right.\n\nA Mikado\n\nSir Charles Dalrymple Belgrave was officially Adviser to the Ruler of Bahrain, but in effect he was prime minister as well, while Lady Belgrave enjoyed the beguiling title of Directoress of Female Education. For thirty years Belgrave had guided the destinies of the island, and his influence was all pervasive. A mere mention of the name Belgrave would instantly bring a price down. There was a street called Belgrave Road, and not a soul in the place, not a sheikh or a tailor or a man picking his teeth on the high curved prow of a dhow, who could not direct you to the house where the Belgraves lived.\n\nThis Mikado viewed his own eminence with a trace of dry amusement, and his home (above his office) was a gay and racy place. Belgrave had a splendid and eclectic library, and he was a man of esoteric tastes, addicted to (for example) roulette, cigars, watercolour painting, swords and pantomimes. Each year he presented a panto of his own in the dining room, with bold backcloths painted by himself and dialogue verging upon the risqu\u00e9. It was curious that his effect upon his bailiwick was almost sanctimonious. 'How good are roads are,' the island seemed to say, 'and how sensible our schools are, and how thriftily we use our oil royalties. Mohammed, stop picking your nose in front of Lady Belgrave.'\n\nBeggars and buskers\n\nThe indigenous beggars and buskers of Venice are treated with indulgence. There is a dear old lady, bundled in shawls, who sits in the evening at the foot of the Accademia bridge, and has many faithful patrons. There is a bent old man who haunts the alleys near Santo Stefano and who is often to be seen pacing from one stand to another, plucking a neat little melody upon his guitar. On Sunday mornings a faun-like couple of countrymen materialize on the quayside of Giudecca with a set of bagpipes and a wooden whistle. A well-known comic figure of the Zattere is a man in a cloth cap and a long blue overcoat who suddenly appears among the tables of the outdoor cafes and, planting himself in an uncompromising posture on the pavement, legs apart, head thrown back, produces a sheet of music from his pocket and throws himself into a loud incomprehensible aria, tuneless and spasmodic, but delivered with such an air of informed authority that there are often a few innocents to be seen following the melodic line with knowledgeable attention. I once asked this man if I could see his music, and discovered it to be a specimen page from a score of Beethoven's 9th, held upside-down and close to the stomach.\n\nAmong the Delhi Spearmen\n\nAmong the officers of the 9th Queen's Royal Lancers there was a powerful sense of family. It was hardly like being in an army at all. Age was disregarded and rank was tacit. Nobody called anybody 'sir'. The Colonel was Colonel Jack, or Colonel Tony. Everyone else was known by his Christian name. Courtesy towards each other was not a deliberate form, it was merely a matter of habit, or convenience. This was a very professional regiment. A sense of heritage accordingly bound us one to another, and made us conscious of lance and plume, saddle-carbine and cuirass. These were the Delhi Spearmen; and though the details of the regimental history were less than vivid to most of us, still there hung always around our mess a general suggestion of glory (not that anyone would have been so insensitive as to mention it, for if there was one attribute the 9th Lancers were not anxious to display, it was keenness).\n\nMixed sensations\n\nIt seems only the other day that communism ended in Romania, but here I am already at the dinner table with a jolly crew of acquaintances, eating pike-perch from the Danube and drinking a happy Moldavian Riesling, to the deafeningly amplified thump of a band in the chandeliered dining room of the Central House of the Army. I have poked my nose into several such unpromising bastions of old Establishment. At the Writers' Union, for instance, which has been for several decades a tribunal of communist orthodoxy, I wandered bemused and unhindered through the accumulated cigar smoke of a thousand ideological debates, amiably nodded at now and then by marvellously literary-looking confr\u00e8res. And at the Military Hotel, strolling in, I was befriended without question by a most formidable captain of the Romanian navy, wearing over his gilded uniform a leather coat like a U-boat commander. It is a queer mixture of sensations. On the one hand nearly everybody is welcoming. On the other hand few seem altogether frank.\n\nSharing the pleasure\n\nIn Wyoming cowboys sometimes walked their horses up to me as I picnicked in the sage, sat sketching in my car or took my morning walk through the scented countryside; and then, after we had exchanged pleasantries and told each other where we came from, and explained what we were doing, sometimes they would slide from their saddles and join me for a few minutes, looking over my shoulder at a sketch, accepting a slice of cheese, or simply sharing the pleasure of the place and the moment\u2013not a talkative presence usually, but one so naturally kind and unembarrassed that a silence was never awkward, and the parting came organically, like the end of a good meal, just before satisfaction moved towards surfeit.\n\nSiren call\n\nSeen from Hong Kong's New Territories in those days, China seemed to me essentially simple, like a world stripped of its complexities and pretensions. I found myself looking towards that silent landscape as though it were calling me home. Home to where? Home to what? As I wandered down the track towards my car, one of the stall sellers spoke to me quietly, without urgency, across his wares. 'Why don't you buy,' he inquired, as though he genuinely, if mildly, wanted to know the answer, 'the thoughts of Chairman Mao?'\u2013and he held up a small red book, bound in plastic.\n\n'Get thee behind me,' I said.\n\nValleys music\n\nI prefer to catch a Welsh male voice choir at practice, when it has not been stiffened up with clean shirts and clasped hands for a concert. Nobody ever tried harder than a choir of the valleys intent on getting an interpretation absolutely right. Just once more, cries the conductor\u2013Unwaith eto, bois!\u2013and the lady accompanist stiffens herself again at the piano, the stocky tenors, the well-paunched basses adjust their spectacles, smooth out the creases in their music sheets and wait in tense taut postures, like tennis players awaiting a service, for the drop of the baton. The rustlings and the coughings stop. Silence falls. The maestro crouches there before his men, half doubled on the dais, a demoniac figure, black of hair, swarthy of face, eyes gleaming. He is irresistible! He raises his baton. The choir takes a breath. The pianist lifts her fingers. Crash, the place reverberates, the whole town surely, perhaps the whole of Wales, with the passion of the opening chord.\n\nR.I.P.\n\nThe foreign news editor, my immediate superior at The Times, was ageing, and week by week I noticed not merely a faltering in the old gentleman himself, but a progressive disregard of his views. People did not listen to him. Decisions were taken without his knowledge. He clearly sensed it too, and for what seems to have been hours at a time, talking in an infinitely slow grating voice that was, I admit, among the heavier of my burdens, he would disclose to me his anxieties or more often his resentments.\n\nHe grew increasingly talkative, bitter and confused, until finally, one winter evening, he gave me a letter. If anything should happen to him, he said, buttoning his thick black overcoat, straightening his homburg and removing his walking stick from its stand behind the desk, I was to hand it to the higher authorities of The Times; and gently chewing\u2013for he generally seemed to have in his mouth, when not a cigar, some kind of lubricant lozenge, perhaps to keep his voice going\u2013he nodded at me in his usual way, said goodnight with his habitual icy trace of a smile, and went home to kill himself with sleeping pills.\n\nLevantine life\n\nI was happy working for the Arab News Agency in Cairo. My friends were mostly in the office, and we were none of us rich. We were boulevardiers, but of a modest rank, frequenting the shabbier of the downtown pavement cafes, murky places with marble-topped tables where the coffee was as thick as porridge and the water glasses were a perpetual dingy grey. There we would sit and talk in the early evening, when the long siesta was nearly over, until we heard the rattle of the heavy steel shutters being raised one by one from the shop fronts, and it was time for us to saunter to the office and start work on the evening bulletin.\n\nThe news that greeted us up there was always full of drama and piquant intelligence\u2013wars and corruptions, desert crime, court conspiracies, religious polemics, family feuds\u2013and we worked in a spirit of Bohemian release. Once we were inside our dim-lit, crowded and untidy rooms we would forget the truth about ourselves, forget the impending misery of the midnight tram, forget the shabby villa off the airport road, forget the swarming children and the skinny black-veiled wife, forget our lost hopes for a career in the law or the Ministry of the Interior, forget that we were indigent Egyptian effendis or struggling Levantines, forget even our sexual ambiguities, and lose ourselves in that strange little world of ours upstairs.\n\nAmerica verbatim\n\nFrom my notebooks:\n\n'I told him, I said, \"Johnnie, if you want me you've just gotta come right down here and get me...\"'\n\n'Crooked? Crooked as a green snake...'\n\n'She said that? She actually said that, right there? She said that to your face...?'\n\n'Listen, Ed, I'll blind you, honest I will, I'll cut your tongue you old son of a gun you...'\n\n'She says to me, \"Leon,\" she says, \"I wantya to know, I'm fond of you, truly I am, but there's this problem of Juan's baby, see?\" \"To hell with Juan's baby,\" I says. \"What's Juan's baby to me?\" And she says, \"Leon, honey,\" she says, \"listen to me...\"'\n\n'If Consolidated Edison could be boiled down into one man I wouldn't have him in my home...'\n\nAn indistinct saint\n\nSt Frideswide was an indistinct medieval divine who is the titular saint of Oxford Cathedral. Every 19 October the bigwigs of the city and the diocese process, begowned, befurred, cassocked, epauletted and even bewigged, to celebrate her memory at her shrine in the building. When I was once at the service I noticed that one of the most venerable canons of the cathedral showed signs of irritable impatience. He scowled, muttered audibly to himself, hitched his hood, twitched his surplice, nudged his companions and occasionally gazed frowardly around the congregation. It was true that 'Jerusalem, My Happy Home' did seem more than usually protracted that day, with so much civic weight to slow it down. I watched that clergyman closely, though, and after a time I reached the conclusion that he was not annoyed by the music, only by the occasion. He didn't dislike the hymn tune. He had doubts about the saint.\n\nMayor Murphy's inducements\n\nIt happened that while I was in town St John's, Newfoundland, was celebrating its centenary as a municipality. The festivities closed with a public party which suggested to me an enormous country wedding\u2013everyone someone else's sister-in-law, everyone ready to talk, with no pretence and no pretension either. Jigs and folk songs sounded from the stage, and when people seemed slow to dance jolly Mayor Murphy took the floor alone, offering free booze coupons to any who would join him. 'You have to get them half tight,' he remarked to me as he handed out these inducements, jigging the while himself.\n\nLater I was walking along a city street when a man launched upon me, without warning, a challenging statement in such advanced Newfoundlandese that I can only reproduce it impressionistically, so to speak. It sounded something like: 'Sish yarkin trapse John Murphy.' He looked at me expectantly for a response, so I simultaneously shook my head and nodded, to be on the safe side.\n\nThings to attend to\n\nThe millionaire, a man of taste as well as power, commanding everything that money can buy\u2013even he seemed restless and impatient; not socially, for he was kindness itself, but temperamentally. The whole luxurious establishment seemed to me somehow disposable: the manservant, the housekeeper, the masterpieces on the walls, the carpets from the East, the gorgeous maps and the calf-bound library, the great silver tray of decanters and silver-topped siphons, the bronze picked up from an unknown but infinitely promising young sculptor in West Africa\u2013none of it seemed destined to last. It was as though he might decide one day to rip it all up and start again. As he saw me off at the door two things happened. An alarm buzzer sounded, announcing that the eldest son of the house had got stuck in the elevator, and the millionaire's wife called through the drawing-room door to say that the White House was on the telephone. 'Excuse me,' he said in his flat velvet voice, shaking hands on the doorstep. 'I have one or two things to attend to. Thanks for coming.'\n\nGive and take\n\nOne still hears the instant give and take in Dublin pubs and parlours. 'Ah, me rheumatism's cured,' says the old lady quick as a flash when the landlord pats her kindly on the knee, 'you should advertise your healing powers.' 'Sure it was only my left hand too,' says the landlord. 'Well and it was only my left knee\u2013try the other one there's a good man.'\n\nThe judge\n\nThe judge at the Court of Session, Scotland's High Court, wore his tight-curled wig as though it had sprouted spontaneously from his pate in childhood. Crouched over his papers at his high dais, he was big nosed, wrinkle eyed, high cheeked, hooded, with eyes that never seemed to blink, and a mouth that expressed, whatever interrogation he supervised, whatever sentence he was decreeing, no flicker of concern, distaste or even particular interest. I could seldom hear what he said, for he spoke in a cracked and high-pitched drone apparently outside my aural range, but I observed that, like an owl peering down from a telephone wire, he missed no nuance or allusion of the proceedings below him. When I left the court I turned at the door for one last fascinated look at him, and discovered that, although his slumped posture had apparently not budged an inch, those pale blue eyes of his were staring fixed and motionless into mine\u2013rather as though, like the owl, he could rotate his head without reference to his body, preparatory to dismembering a mouse.\n\nAcross a chasm\n\nAll the windows in the huge slab of a building are brilliantly lit, and in each a little cameo, separate from all its neighbours, is joylessly displayed. Here four girls sit tense over their sewing machines, silent and unsmiling, motionless but for the quick twist and tug of their fingers. There a solitary shirtsleeved man is hunched over his files and calculators, beneath the dazzling light of his naked bulb, dead to all else and perhaps to himself. Along the way eight or nine families seem to be packed into one room, and one sees only flashes of infant limbs, waves of drapery, buckets, black loose hair, bedclothes and grinning mop faces, as though some perpetual and appalling farce is being played inside.\n\nEvery room there is ablaze, every room full; and across the gloom one hears radios, clicking machines, shouts and children's screams. In another city all that life over there might be a comfort, a reminder that if you happen to be alone that night, all around you is the warmth of community. In Hong Kong it is different. Nobody in that building seems to take the slightest notice of anyone else\u2013let alone of you as, peering out of the night, your wan Western face gazes aghast across the chasm.\n\nDead guys\n\nI stood on the edge of Arlington National Cemetery in Washington, DC\u2013Fame's Eternal Camping Ground, as it says on its triumphal arch\u2013and looked across the massed ranks of the departed, like a vast city of slabs. 'Are all these,' said a child beside me, surveying Fame's Eternal Camping Ground herself, 'are all these dead guys?' 'Dead,' said I, 'as mutton'\u2013but at that moment her grandmother arrived, and throwing me a distinctly accusatory look, as though I were undermining the loyalty of the young, she gave the child's nose a necessary wipe of the Kleenex and hurried her down the hill to catch the Tour-Mobile.\n\nDuke of London\n\nAs it happened the only time I ever saw Winston Churchill was at the very moment of his ultimate triumph\u2013the moment when, on the day of German surrender, he appeared on a balcony in Whitehall to accept the gratitude of London, his battered capital of victory. All around us were grand old monuments of English history, Parliament and Abbey, Nelson on his column up the road, Admiralty and Banqueting Hall and Horse Guards Parade, and it seemed to me then that he was already one of them\u2013so perfectly did his portly smiling presence up there seem to satisfy the setting, the story and the meaning of the day. I always thought he should have been made Duke of London.\n\nMagic in Paris\n\nWandering into Notre Dame on a Sunday night, I found a choir and orchestra celestially performing Bach's Christmas Oratorio. The cathedral was full, a reverent multitude of young people sitting on the floor if they could not get a seat in the nave, or simply milling about like me. It was magical. All Paris seemed to be there, singing its heart out or half lost in the marvel of it all. When I discovered that the choir and orchestra came from Germany, and realized that half the listeners were as foreign as I was, it only seemed more magical still.\n\nInsufficient compassion\n\nScary beggars used to infest the centre of Alexandria. I was walking home in Alex one evening when I felt, rather than actually saw, a legless beggar observing my passage from across the street. He was strapped to a low wooden trolley, which he pushed along with his hands, and made an object at once heart rending and frightening to see: but I had no money with me, neither a pound nor a piastre, so I quickened my step self-consciously and hurried down Zaghloul Street towards my hotel.\n\nBehind me I could hear the whirr of his roller-skate wheels as he pursued me through the town\u2013a thump when he eased himself off the pavement, a clanking when he crossed the tramlines, a change of pitch when he left the tarmac for the flagstones. Faster and faster I walked through the evening crowds, but I could never escape those whirring wheels: over the low wall into Zaghloul Square, across the little garden, and I could still hear them skidding down the path, closer than ever behind my back, so that I could hear the poor man's panting breath, too, until at last, breaking into a run, I threw myself into the revolving door of the Cecil.\n\nThe wheels came to a sudden stop on the sidewalk outside, and a curse bade me goodnight.\n\nDignities\n\nIn the last years of the Iraqi monarchy the real power behind the throne was the Crown Prince Abdul Illah, a sallow but handsome man. I interviewed him once. He talked fluently and amusingly, and our conversation turned to the pictures on the palace wall, a galaxy of chieftainship. 'What a splendid costume it is,' I said of a portrait of the great Feisal I in Arab dress. 'Do you ever wear it yourself?' 'Sometimes,' said the Crown Prince. 'You would be surprised how comfortable it is.' 'And it does make one look so fearfully dignified,' I said lightly. This was not a success. A chill seemed to settle upon our encounter, and the interview soon ended. The Crown Prince Abdul Illah, it seemed, needed no romantic trappings to give him dignity.\n\nIddums-diddums\n\nCome with me, and watch a full-blooded beldame selling fish upon the waterfront of San Sebastian. She is flanked by cronies, sitting brawnily on kitchen chairs like a gangster's bodyguard. She herself stands in the middle with a microphone around her neck, wearing a blue anorak, a pink chiffon scarf, white ankle socks and a suggestion of innumerable underclothes. She looks as though no nuance of life has escaped her. Her face is heavy jowled, her wrists are muscular, and she is built like a boxer, but there is a rich urbanity to her voice as she intones the price of sardines. The crowd is altogether at her command. The seamen stumble in with their fish trays like acolytes in an archiepiscopal presence. The cronies laugh at her every joke. Policemen sheepishly perambulate. Customers never dare to argue. But sometimes that empress of the fish market, pausing to scribble a price upon her pad, notices a baby in somebody's arms, and looks up unexpectedly with the sweetest of grandmotherly smiles, a twiddle of cod-scaled fingers and what I take to be the Basque equivalent of 'Iddums-diddums.'\n\nIn the park\n\nI woke early and walked across Chowringi into the green of Calcutta's Maidan, before the sun rose and the heat haze fell like a web upon us. It was lovely then in the park. Rooks cawed, kites hung, sparrows pecked, smiling pi-dogs padded by. Here and there across the grass white figures moved or loitered, and whenever I paused I was sympathetically accosted. 'What you are seeing is the Theatre, built in honour of our great poet Rabindranath Tagore.' 'If I may say so you would be more comfortable where there not so many ants.' Or: 'Wouldn't you like a game of golf? I am teaching golf, you see. Here are my golf clubs.'\n\nTradition\n\n'Does he, do you think,' tactfully inquired the Bishop of Barrackpore, 'expect a T-I-P?' But no, my guide was not ready for one yet, having high hopes of further services to be performed, so I joined the bishop on his verandah, where during a lull before evensong he was eating peanut butter sandwiches with a kind Anglican lady in blue. He was all an aficionado of the tradition could ask: cassocked, distinguished, fatherly, concerned about that T-I-P. Soon, he told me, he would be retiring. Going home? I wondered, but as the lady replaced the tea cosy with a significant air he answered me in grave italics: 'Staying in India\u2013for ever.'\n\nAt evensong we sang the hymn that says the Lord's throne shall never like earth's proud empires pass away, and as I left the cathedral a Balliol voice called kindly across the transept\u2013'I say! Excuse me! You do know where we are, don't you, if you're coming to the children's dance drama in the parish hall?'\n\nArrival of the tourists\n\nDown in the harbour of Capri I can see the morning vaporetto from the mainland, still hazy about the funnel, and here flooding into the piazza, pouring out of taxis, out of buses, out of horse carriages, out of the steep funicular that runs up from the waterfront\u2013wearing floppy straw hats and rope-soled shoes and pink jeans and multifarious bangles\u2013festooned with cameras, inquiring the price of swimsuits, unfolding maps, touching up their lipsticks beneath the campanile\u2013talking German, English, French and every variety of Italian\u2013young and old, blatant and demure, strait laced and outrageous, earnest and frivolous and thrilled and sick-to-death-of-it-all\u2013here past my cafe table streams the first quota of the morning's tourists.\n\nTeatime in Old Chicago, 1950s\n\nI love to watch the customers at a carriage-trade Chicago restaurant at English Teatime, with Jasmine Tea and Toasted Muffins beside the goldfish pool. Its children behave with almost fictional decorum. Its daughters wear pearls. Its young mums look as though they have come direct from committee meetings of charitable balls. Its husbands look as though they keep fit by riding hunters through parks before breakfast. Its grandmothers, best of all, talk in throaty turtle voices, as though the words are being squeezed out from beneath the carapace and they are heavily loaded with inherited gewgaws, and are inclined to call the waitress 'child', as though expecting pretty curtseys in return. 'Would you care for some more Jasmine Tea, Mrs Windlesham? Do you desire another Toasted English Muffin?' 'Why thank you, child\u2013how pretty you are looking today!' 'Thank you, Mrs Windlesham, it's always a pleasure to serve you and the members of your family.'\n\nA separate sphere?\n\nOne evening at Akureyri, on the northern coast of Iceland, I heard the sound of solemn singing from a restaurant, and peering through its door I saw that a large party was in peculiar progress. I felt as though I were looking in at some utterly separate sphere of existence. There the Icelanders, men and women, sat in ordered ranks, their arms linked around the long tables, and as they sang what seemed to be some kind of sacramental anthem they swayed heavily from side to side in rhythmic motion. The sight of them gave me a queer sense of secret solidarity. Everybody clearly knew the words of the song, and the whole assembly seemed to be in some sort of arcane collusion. I noticed that if ever I caught an eye, as the celebrants sang and swayed there at the table, after a moment's puzzled focusing it abruptly switched away from me, as if to dismiss an illusion.\n\n'Are they?'\n\nEvery evening at the Pera Palace Hotel in Istanbul a string trio plays, attentively listened to by the German package tourists at their communal tables, and gives the place a comfortable, palm-court air. Two elderly gentlemen in Gypsy outfits are on piano and accordion, and they are led by a romantic Gypsy fiddler, adept at waltzes and polkas. I was sitting there one evening when suddenly there burst into the room, driving the trio from its podium and severely disconcerting the hausfraus, a team of ferocious Anatolian folk dancers, accompanied by a young man with a reedy trumpet and an apparently half-crazed drummer. The dancers were fairly crazed themselves. Apparently welded together into a multicoloured phalanx, they shrieked, they roared with laughter, they leapt, they whirled, they waved handkerchiefs\u2013a performance of furious bravura, leaving us all breathless and aghast. They were like so many houris, come to dance over the corpses on a battlefield.\n\nThey withdrew as abruptly as they had arrived, and in the stunned hush that ensued I turned to the Americans at the next table. 'My God,' I said, 'I'm glad they're on our side!' But a knowing look crossed the man's face. 'Ah, but are they?' he replied.\n\nTyrant\n\nA terrifyingly ambitious, inexhaustible girl supervisor works at one of the downtown McDonald's of Manhattan. Over the serving counter one may see the glazed and vacant faces of the cooks, a black man and a couple of Puerto Ricans, who appear to speak no English; in front that small tyrant strides peremptorily up and down, yelling orders, angrily correcting errors and constantly falling back upon an exhortatory slogan of her own: C'mon, guys, today guys, today...The cooks look back in pained incomprehension.\n\nDivine merriment\n\nWe find ourselves lost on the edge of a deserted traffic junction somewhere east of Kranj, Slovenia. Helplessly we consult our map, hopefully we look for somebody to ask the way, and presently there somehow seem to sidle into our company half a dozen Slovene men and a very talkative Slovene woman. Between us we speak five languages, but we are fluent only in our own, and gradually our discussions descend into farce. It's that way, for sure, no it's the other, they haven't been through Preddvor, no, no of course they haven't, they came the Cerklje way\u2013they should go back the way they came then, they should have gone by Duplice\u2013no, no, no, look here, look at the map\u2013and so, as the map gets more and more crumpled, the arguments louder, the languages ever more incomprehensible, we subside into impotent merriment, shake hands with each other, clap one another on the back and, chuckling still, go our various ways. We ourselves are no wiser about our situation, so we leave the car on the grassy verge and go for a drink instead.\n\nGod habitually smiles upon Slovenia, and sometimes he laughs out loud.\n\nVioletta down under\n\nGo to La Traviata at Sydney Opera House and, my, what a robust Australian chorus will be attendant upon Violetta in the opening act, their crinolines and Parisian whiskers delightfully failing to disguise physiques born out of Australian surf and sunshine\u2013while even La Traviata herself, as she subsides to the last curtain, may seem to you the victim of some specifically Australian variety of tuberculosis, since she looks as though immediately after the curtain calls she will be off for a vigorous set of tennis with the conductor, or at least a grilled lobster with orange juice and caramel.\n\nThe Leading Citizen's lesson\n\n'Have your fun, Jan,' said the Leading Citizen. 'Sure thing, this is a Fun Town, but what we especially do not like is these comparisons with Sodom and such. What people forget is that here in Las Vegas we have a thriving civic-minded community. We have 130 church buildings, Jan, in this city of ours. I think I could safely say that you won't find a more lovely home environment anywhere than some of our high-grade home environments here. What I want you to remember, Jan, is this\u2013the Spanish Trail came this way, right over this very spot, before the game of roulette ever entered the Infant Republic\u2013that's what I always tell people like you, who come inquiring\u2013before the game of roulette ever entered the Infant Republic of the United States!'\n\nThe Very Reverend\n\nAlmost at once I met the Dean of Wells, actually in the shadow of Penniless Porch. Eton, Oxford and the Welsh Guards, he was not hard to identify. With a splendid concern his voice rang out, as we sat there watching the citizenry pass by. 'Good morning, good morning! Lovely day! What a success yesterday\u2013what would we have done without you? Morning, Simon! Morning, Bert! Morning, John! (John Harvey, you know, our greatest authority on church architecture...)'In his cathedral, I was later disconcerted to learn, they habitually call him 'Father', but I certainly could not complain about his authenticity qua Dean.\n\nOur Gracie\n\nOn a bus in Capri I chanced to meet, I can't remember how, a man who introduced himself as Boris Alperovici, the third husband of Gracie Fields. She was a famous star of the past, a Dame of the British Empire\u2013'Our Gracie', formerly a household name in her native England but by then somewhat forgotten. She was living in elderly retirement in her villa on the island. Boris took me along to visit her, and she received me graciously, and told me anecdotes of her theatrical life, and had coffee served to me by her seaside swimming pool. It was just as though the old lady were some great Hollywood actress at the height of her career, and she evidently enjoyed it as much as I did. When I got back to Britain I was surprised to meet other people, too, who had chanced to encounter Signor Alperovici on the Capri bus, but couldn't quite remember how, and had sat drinking coffee at the feet of Our Gracie.\n\nThe exchange\n\nWandering around the purlieus of the High Court in Madras, I took out my tape recorder to remind myself of some of its architectural peculiarities. At once I heard an admonitory clapping of hands, and a policeman with a nightstick beckoned me over.\n\n'What have you got there? What is this machine?'\n\n'It's a tape recorder.'\n\n'What are you doing with it here?'\n\n'I am reminding myself of some architectural peculiarities.'\n\n'How do I know it is not a bomb?'\n\n'You can speak into it yourself.'\n\n'What shall I say?'\n\n'Anything.'\n\n'I cannot think of anything to say.'\n\n'Sing a song then.'\n\n'What kind of a song?'\n\n'A Tamil song.'\n\n'Very well, I will sing you a very old Tamil song, a tragic song'\u2013and half closing his eyes, and assuming an unmistakably tragic expression, there in the sunshine outside the court in a high wavering voice he sang several verses of a very, very old Tamil song. I played it back to him.\n\n'Very well,' he said, 'now you have my voice. What will you give me in return?'\n\nBut, bless his heart, I was gone by then.\n\nTemper of the South\n\nThe temper of the South is inescapable in Houston. You can sense the swagger of it in the postures of the cattle people come into town for dinner or convention: hulking rich men in Stetsons and silver belt buckles, paunchy with their generations of beer and prime steaks, lacquered observant women in bangles, talking rather too loud as Texans are apt to, the wives greeting each other with dainty particularities ('Why, hi, Cindy. My you're looking pretty!') the husbands with spacious generics ('Well, boy, what's things like in East Texas?')\n\nAnd you can sense the poignant charm of it in the faded white clapboard houses of the Fifth Ward, stilted above the dust of their unpaved streets. There the black folk still idle away the warm evenings on their splintered porches, as in the old story books; there the vibrant hymns still rise from the pews of the Rose of Sharon Tabernacle Church; there the garbage still blows about the garden lots, and you may still be asked, as I was, if, 'Say, ain't you Miss Mary's daughter from the old store? Bless your heart, I used to be one of Miss Mary's best, best customers...'\n\nMonty\n\nLate one evening during World War II I was walking up Arlington Street towards Piccadilly when there emerged from the door of the Ritz General Sir Bernard Montgomery (not yet a Marshal or a Lord). A policeman saluted as he scuttled down the hotel steps and into his waiting staff car, but I thought there seemed something almost furtive about his movements. I expect he was really in haste to get back to the War Office, or even into battle, but if he had been another kind of general I would have guessed he was hurrying to an assignation down the road in Soho!\n\nConfrontation\n\nThrough the crowd waiting for their luggage at the Toronto airport carousel there staggered ever and again a middle-aged woman in a fur hat and a long coat of faded blue, held together by a leather belt evidently inherited from some earlier ensemble. She was burdened with many packages elaborately stringed, wired and brown-papered, she had a sheaf of travel documents generally in her hands, sometimes between her teeth, and she never stopped moving, talking and gesticulating. If she was not hurling questions at expressionless bystanders in theatrically broken English, she was muttering to herself in unknown tongues, or breaking into sarcastic laughter. Often she dropped things; she got into a terrible mess trying to get a baggage cart out of its stack ('You\u2013must\u2013put\u2013money\u2013in\u2013the\u2013slot.' 'What is slot? How is carriage coming? Slot? What is slot?') and when at last she perceived her travelling accoutrements\u2013awful mounds of canvas and split leather\u2013erupting on to the conveyor, like a tank she forced a passage through the immobile Canadians, toppling them left and right or barging them one into another with virtuoso elbow work.\n\nI lost sight of the lady as she passed through customs (I suspect she was involved in some fracas there, or could not undo the knots on her baggage), but she represented for me the archetypal immigrant, arriving at the emblematic immigrant destination of the late twentieth century, and I watched the confrontation with sympathy for both sides.\n\nThe spy's discomfort\n\nRoller skating was then all the rage around the Lake of Geneva. Whole families skated along the promenade. Dogs rode about in rollered baskets and youths whizzed shatteringly here and there, scattering the crowds with blasts of the whistles that were held between their teeth. I lunched with a spy of my acquaintance. What kind of a spy he is, who he spies for, or against, I have never been able to discover, but he has all the hallmarks of espionage about him, divides his time between Switzerland and the East, wears raincoats and speaks Greek. We ate little grilled fish at the water's edge and discussed the state of the city. Uncomfortable, he thought it, and getting worse. Security getting tougher? I conjectured. Banks turning difficult? Opposition hotter? No, no, he said testily, holding his hands over his ears, nothing like that: only those damned roller skaters.\n\nAdmiral's walk\n\nSplit in Croatia is a naval base, and when I was driving out of town I stopped at the traffic lights near the fleet headquarters. A very senior naval officer started to cross the road. He was loaded with badges, braid and medal ribbons, but wearing as I was a floppy old hat and a less than spotless blue shirt, just for fun I saluted him. His response was Split all over. First he faltered slightly in his steady tread. Then he brought his hand to the peak of his cap in a guarded and cautious way. And then, as the lights changed, I started forward and he scuttled with rather less than an admiral's dignity to the safety of the opposite pavement, he turned round, all rank and propriety discarded, and shared my childish laughter.\n\nTrue gents\n\nAt Three Rivers, stopping for a hamburger, I found that I had locked my car keys in the boot. Small-town Texas swung instantly to my rescue\u2013well, eased itself slowly off its cafe stools, tipped its Stetsons over its eyes, strolled into the car park and stood meditatively eyeing the problem, saying things like Huh or Kindova problem there. In easy stages they approached the task, sniffing it, feeling it, and when in the end they got the hang of it, enlarged the right aperture, unscrewed the right screws, and found that the keys were not in the boot at all, since I had left them on the Dairy Queen counter, they seemed not in the least disconcerted. Deftly reassembling the mechanism, tilting their Stetsons back again, they drifted back into the cafe murmuring, 'You bet, lady, any time.'\n\nThe Low Riders\n\nIn Santa Fe the Spanish culture is relentlessly pressed upon by all the influences and temptations of the American Way. Often in the evenings the cultists called the Low Riders cruise through town. They are the public faces, I suppose, of young Hispanica, and as they drive slowly about the streets in their weirdly low-slung limousines, wearing wide hats and dark glasses, radios booming, unsmiling, proud, stately one really might say, who knows what resentments or aspirations of their race they are trying to declare?\n\nThe call of conscience\n\nOn the Bund in Shanghai one evening a youth with the droopy shadow of a moustache confronted me with a kind of dossier. Would I go through his examination paper for him, and correct his mistakes? But I had been pestered by students all afternoon, and I wanted to go and look at the silks in Department Store No. 10. 'No,' said I. 'I won't.'\n\nAt that a theatrical scowl crossed his face, screwing up his eyes and turning down the corners of his mouth. He looked then, with that suggestion of whiskers around his chin, like a Chinese villain in a bad old movie, with a gong to clash him in. I circumvented him nevertheless and, ah yes, I thought, if the Gang of Four were still around, you would have me up against a wall by now, with a placard around my neck and a mob to jeer me, not to consult me about participles.\n\nBut my conscience pricked me, and I went back and corrected his damned papers after all.\n\nA lesson\n\nI helped a blind lady over a street crossing near the Gare de Lyon. She looked particularly irritable, cross and demanding, but though born and bred in the 12th arrondissement, turned out to be diffidently gentle. It was a lesson to me not to misjudge the hard-mouthed, sharp-eyed, fast-shoving, middle-aged Parisian housewife. I took the lady first to the post office, then to the pharmacy, and when I left her she said: 'Now I give you back your liberty.'\n\nAfter a Mexican dinner\n\nTheatrical characters, it seemed to me, filled the main square of Oaxaca when we strolled down there for a drink after dinner: nut-brown women cloaked in red, and dapper old gents with silvery moustaches, and gaggles of students like opera choruses, and small policemen with nightsticks, and rumble-tumble infants everywhere, and a blind guitar player doing the rounds of the coffees shops, guided by his urchin familiar, and a gringo hippie or two, and barefoot families of peasants loaded with shopping bundles and making, I assumed, for the mountains. The faces were mostly dry and burnt. The movements seemed kind of airy, as though tending towards weightlessness. Among the trees some children were blowing up long sausage balloons and letting them off with a squirt of air into the night sky, where they rotated dizzily off into the darkness like so many flying serpents.\n\nHarry's\n\nIt was in 1946, when the war in Europe was hardly over and Venice was still under the control of the Allied armies, that I first poked my nose through the doors of Harry's Bar in Venice. I was in my twentieth year, and did not know what to expect. The room was smallish and unexpectedly cosy. At the tables were smoky looking, hooded-eyed, tweedy, sometimes hatted, heavily made-up but rather weatherbeaten persons I took to be members of the Italian aristocracy. Sitting at the bar were three or four officers, the British looking disconcertingly suave to me, the Americans dauntingly experienced. The conversation was low but intense, and everyone looked up as I made my entrance. The officers looked up in a cool, officer-like way, holding their glasses. The patricians looked up patricianly, rather disappointedly, as though they had been hoping for better things. But it was the contact I made with the three pairs of eyes behind the counter that I remember best\u2013the eyes of the boss sitting behind his cash till, the eyes of the two busy barmen in their white jackets. The expression in their gazes seemed to me generic to the place. It was at once interested, faintly amused, speculative and all but collusive. It put me simultaneously at my ease and on my guard, made me feel in some way a member of the establishment, and has kept me going back to Harry's from that day to this.\n\nOnly in London\n\nI was sitting over my croissant and the morning paper in a coffee shop in Marylebone High Street when a tall elegant man in late middle age walked stiffly in and ordered a cup of coffee. He wore a long dark coat and a trilby tilted over his brow, and I rather think spectacles were inclined towards the end of his nose. He looked to me as though he had enjoyed perhaps rather too good a dinner the night before, but he emanated an air of unconcerned, if not actually oblivious, composure. I put him down for some mildly eccentric and very likely scholarly earl, of the Irish peerage, perhaps, and thought to myself that only in London could one still see such a genial figure, at once so urbane and so well used, more or less direct from the eighteenth century.\n\n'Know who that was?' said the proprietor, when the man had walked perhaps a little shakily out again. 'That was Peter O'Toole. Remember him in Lawrence of Arabia?'\n\nNo thanks\n\nI went to a place on the Rio Grande which was, I was told, a favourite place for illegal immigrants to cross into the United States. There were a few houses nearby, grazed about by goats, guarded by many dogs, but I found it a chill and spooky spot. It seemed full of secrets, and sure enough one of the neighbours told me that almost every night of the year people from the south clandestinely crossed the river there, and crept damp and dripping through the shrubbery into Texas. 'You see that forest there,' my neighbour said, pointing to a confusion of shrubbery beside the water. 'I'll bet you there's people laying there this very minute, waiting for dark, bad men some of them, from far, far away.' I peered at the bushes through my binoculars, hoping to see glints of weaponry, the smoke of marijuana rising, blackened faces peering back at me through the leaves. All seemed deserted, though. 'Want to go over and see? See if there's men there now?' asked my informant helpfully. 'No, thanks,' I said.\n\nGlaswegians\n\nGeorge Square in Glasgow has a family feel to it. People talk to each other easily on benches. People share gambles, compare prices, take their shoes off to give their poor feet a rest. The five-year-old boy riding his motorized buggy around the benches smiles indiscriminately at us all as he blasts past yet again, and his father proudly tells us how much he paid for the machine. Sitting there among those citizens, looking at the civic statues, cursing the buggy boy, while the big buses slide around the square and the City Chambers look paternalistically down at us, I seem to feel a comforting sense of community. Ay, well, responds a freckled woman sitting beside me, that's all very well, but life's not all statues in George Square\u2013and what's a wee bairn doing with a contraption like that anyway, he'll do himself a damage in the end.\n\nWildlife\n\nWhile searching unsuccessfully for kangaroos in the bush of Mount Ainslie, a wooded hill rising immediately above Canberra, I felt a sudden need to relieve myself. I was just doing so when I heard a padding and a shoving and a rustling through the bushes. Kangaroos at last? Very nearly. Crashing among the branches, as I was in the very act, a few feet away from me there appeared a very large, very bearded, white-shorted and energetically sweating Australian, doing his daily jog, I suppose, during the luncheon break from his duties as Executive Officer Grade Two in the Department of Inter-Administration. 'Ho, ho, ho,' was all he said, as he bounded distinctly roo-like past.\n\nTwo in the morning\n\nAt two in the morning I decided that enough was enough, and clambering upstairs I knocked upon the door of M. le Propri\u00e9taire's private apartment. It sounded as though they were having a football match inside and, sure enough, when the door opened it was the hotelier's three-year-old son, all flushed and tousled with hilarity, who first poked his nose through the crack. 'A million pardons, madame,' came his father after him. 'How can you forgive us? We were having\u2013how do you say it\u2013a little practice match!'\n\nTwo Berlins\n\n'I'm the Boss' was the first T-shirt slogan I saw, on the ample bosom of a housewife dancing a vigorous jig with her decidedly un-henpecked husband. East Berlin was having a public holiday, and at the hotel beside the lake several thousand citizens, great-grandmothers to babes in arms, were enjoying a family feast in the sunshine. How genially they laughed, danced, sang, drank their beer and ate their pickled pork knuckles! With what indefatigable smiles the two bands alternated, one with the old oom-pah-pah, the other exploring the less raucous fringes of rock!\n\nThat same evening, al fresco in the Gr\u00fcnewald woods on the other side of Berlin, I observed two middle-aged ladies, mother and daughter, perhaps, sharing delicate jokes over their asparagus, and balancing their purses carefully on the rims of their glasses to stop the chestnut blossoms falling into their wine.\n\nAlg\u00e9rie Fran\u00e7aise\n\nI stand in the big public forum of Algiers, outside the government buildings, watching the citizenry. The square is packed to suffocation, the crowd spilling away through the pleasant gardens, up and down steps, across neighbouring squares, until it peters away at its fringes into clutches of foot-weary housewives at the tables of deserted cafes. There stand the grim paratroopers, the high priests of mid-century Algiers, dressed in boots and camouflage suits, festooned with tommy guns, grenades and pistols, lounging about in attitudes fearfully tough and jungly, or swapping badinage with the crowd. There are the queer bigwigs of this confused and unhealthy city, hastening up the steps to the Governorate, or briefly appearing upon some flowered balcony: ramrod generals in kepis, greasy double-breasted politicians, wild creatures of the nocturnal right, bearded plotters or fanatic militarists. A sickly cheer greets a token delegation of Muslims: spindly old men with ragged robes and a covey of bewildered white-robed women, with a trilling of high voices and an arabesque of reedy clarinets. Before long that vast crowd, like so many maudlin drunks outside a saloon, is caught up in histrionics, swayed to a man by the querulous, pitiful passions of Algiers\u2013until the whole assembly, with a roll of drums and a sting of hot tears, bursts into the 'Marseillaise', and for a moment all seems clear, all seems honourable.\n\nAllegory in Amsterdam\n\nStanding on a bridge in Amsterdam, I noticed a sleek tourist motor boat, all glass and chrome, gliding down the waterway with a warm hum of diesels. Inside it, snug behind the glass, sat five young Americans in bright open-necked shirts and jeans\u2013servicemen, perhaps, from some air base. They all wore sideburns, and peered through their windows with an air of concentration; and as they passed slowly by, inspecting me, too, as though I were a medieval monument, they emanated a powerful sense of allegory. They were new men in a very old world. Their identity tags flashed at their necks like ritual amulets. They seemed to me like young priests from some distant cloistered seminary, on a mission of dogmatic inquiry.\n\nAll American\n\nFor me the All American has always been the city bus driver. Since I first saw him clicking that little lever above his change machine, to the tinkle of the nickels and dimes sorting themselves out\u2013since I first heard his timeless response: 'Yeah, lady, get out at City Hall'\u2013since I first plucked up courage to ask him if he could manage change for a ten-dollar bill\u2013ever since I first made his acquaintance he has exemplified for me The American. His slumped shirtsleeved posture over the wheel, the weary reach of his arm towards that change machine, the occasional cursing at a cab driver, the unflustered answering of questions as he drives, his eyes always flicking to the mirror\u2013all are the hallmarks of a man who knows the world for what it is, knows his own city to be its epitome, and has no illusions left. 'So it's a big city? Sure it is. So they're tall buildings? So?'\n\nDinkum Aussie\n\nIn Darwin you may meet the Australian male at his most confident, on the edge of the great Outback. He may be of any age, this dinkum Aussie. He may be a humdrum bank clerk, or a prospector driven wildly in from his shack in the wilderness to squander his money on drink and loose living. Whoever he is, he is magnificent to meet: as free a spirit as you can find in the world today, shackled by no inhibition of class or disadvantage, with little sense of thrift and still less of decorum, no agonizing reserve, no contempt, no meanness. It is as though he has been relieved of the burden of the centuries, strengthened and cleansed by the southern sun, and allowed to begin history all over again.\n\nSuddenly there emerges...\n\nSuddenly there emerges from some unexpected alley of Kyoto a vision of the legendary Japan\u2013a geisha in all her plastered glory, moving fast and purposeful towards an assignation. Immensely tall is her mound of hair, jet black and shiny; her face is vivid with white and scarlet, her costume is gorgeous with silks, sashes, the gaudiest of clashing colours and the floridest of patterns; and as she hastens awkwardly down the street, embellished from head to foot with paint and brocade, she seems less like a living woman than some fabulous toy, some last masterpiece by Faberg\u00e9, enamelled like a queenly trinket, animated by ultimate refinements of clockwork.\n\nA queen rides by\n\nThe people around you seem instinct with an air of happy collusion, as though they all know one another, and are linked in one long line of neighbourly acquaintance from Admiralty Arch to the Palace. The soldiers lining the street look fresh faced and rather touching, the policemen are properly genial, and presently you will see, undulating strangely above the crowd, the head of the Queen of England, in a tricorn hat. You can hardly see her horse for the people, but high above the soldiers and the policemen, as she paces grandly by, you may study her pale face\u2013a sad, antique face, it seems to me at such a moment, young but tired, half commanding, half embarrassed, half person, half idea\u2013a face lined with the blood heritage of Alfred the Great, William the Conqueror, Charlemagne, Roderigo the Cid, Barbarossa and her great-grandmother Victoria, Empress of India.\n\nSaturday lunch in Hong Kong\n\nThere seem to be a couple of thousand tables at the restaurant, and at them in uproarious enjoyment sits a vast multitude of Chinese, in families running the gamut from infancy to old age. Nobody is alone. Nobody is silent. The noise is deafening, all that talking and laughter mingling with the clanking of plates, the shouts of waiters from one side of the room to the other, the occasional cries of babies, the sizzling of woks and the Chinese music blaring from hidden loudspeakers. In we go, extremely European, and it is like sitting on the edge of a maelstrom, as we vacantly study the enormous menu (bound in gold and scarlet). But we are offered encouraging nods and incomprehensible explanations from the family at the next table, and we smile ourselves in a baffled and innocuous way across the Chinese mass. In a daze we order, and as by a miracle our food arrives, piping hot and indefinable, and in no time at all we are slurping it happily away, all inhibitions lost, and nodding appreciatively to our neighbours as to the Chinese manner born.\n\nImmigrants\n\nCourtesy of the Department of Immigration, I once stood in the background of an immigration booth at JFK to watch passengers from Europe coming through, and it was revealing to see what emotions passed through their eyes when they noticed me there, looking I suppose like an unusually well-disguised Secret Service agent: suspicion nearly always, ingratiation very often, sometimes a hint of collusion, and occasionally a look I had never encountered before, which I took to be fear. Some of the new arrivals had clearly roistered their way across the Atlantic with champagne and canap\u00e9s. Others, especially the mothers, the squirmy children, the stout beldames with swollen feet, arrived exhausted at that frenzied airport, into the glaring lights, the unremitting noise and movement of the New World; and as they looked wearily from the immigration officer to me, searching I imagine for some warmth of understanding in our faces, I sometimes thought I detected a flicker of regret in theirs. The officer treated everyone exactly the same, down to the badinage: 'Oh, please don't look at my picture there, I look terrible.' 'It's like we always say, ma'am, if you look as sick as you do in your passport, you're not fit to travel.'\n\nAn ugliness\n\nI was once held up on a seashore track by the unloading of live pigs from the Chinese mainland. This is a familiar ugliness of Hong Kong. The pigs are transported in narrow cylindrical cages of wire or wicker, into which they must be jammed so tightly that they lie there grotesquely squashed and distorted, and frequently in pain. That day they were squealing heart-rendingly as they were bumped in barrows at speed towards their slaughter, and I stood helpless and grieving beside the track. At that moment there came in single file from the opposite direction, on their way home from school, a line of small girls in almost exaggeratedly English uniforms, crested blazers, pleated white skirts, small neat knapsacks their backs. Demurely they filed past, their faces exuding school pride and team spirit: and they took not the slightest notice, as they walked daintily by, of the doomed animals screaming in their torture chambers.\n\n'We'd be famous'\n\nOff the top of a building we fell that day, and sidled across the Hudson River, and in few moments the helicopter stopped, shook itself and gingerly descended a couple of hundred feet. Looking out of my side window I found myself hovering, with a disrespectful clatter, close to the nose of the Statue of Liberty. We hung there for a minute, and the sunshine reflected off the water hung about her head. Then, with a last curtsey, we flew away. 'If we hit her we'd be famous,' I said to the pilot as we darted off. 'What a way to go,' he said. 'I'd be the guy who assaulted the Statue of Liberty, and you'd be instant Shakespeare.' Later I climbed up the statue from the ground, and sympathized with the lady who wrote in the visitors' book that it was 'a nice sight but the stairs weren't that wide'.\n\nSubterraneanism\n\nThe station was excavated in the early days of the New York subway system, and suggested to me a particularly cramped and airless cave, or perhaps a sunken submarine. The lady at the booth was elderly and all white, almost albino. Her face was ashen. Her eyes seemed to have no pupils. It was as though she had never in her life emerged into the daylight, but had been born and bred down there. When I asked her how she liked working underground, and whether she did not miss the sun, she was rather affronted. What could I know about it? She had worked in the subway for thirty years, and did not regret a moment of it. She loved the old station, liked to see the trains go by, and had many friends among the passengers; and, sure enough, when a black man walked by he called out unexpectedly to that pallid lady behind her grille: 'Howya doin', ma'am? Keepin' well?' 'I'm fine, Jack, thank you kindly,' she replied. 'Keepin' just fine.'\n\nIn Old Vienna\n\nWatch now\u2013stand back\u2013here come a couple of ministers down the steps from the Council Chamber in the Austrian parliament, portly important men, deep in portly and important matters of state\u2013and swoosh, like a rocket from his office leaps the porter, buttoning his jacket\u2013out of his door, panting a little, urgently smoothing his hair, down the steps two at a go, bitte, bitte!\u2013just in time, my goodness only just in time to open the door for Their Excellencies, who acknowledge his grovel only with slight inclinations of their heads, so as not to interrupt the flow of their discourse, as they lumber out beneath the figures of Minerva and her attendant sages to their waiting limousines.\n\nAltercation on the Zephyr\n\nI had pleasant companions at breakfast on the California Zephyr\u2013a girl from Fresno who had never been on a train before, and two railroad buffs who kept me informed about the state of the track. However, I did have one altercation in the dining car. My ticket, I had been told, entitled me to anything I liked on the menu, but when I asked for cornflakes and scrambled eggs I was told I was entitled to one or the other, but not both. I called for the supervisor to expostulate, but I did not get far. I had got it wrong, the functionary said, not unkindly, and I quote him word for word: 'You're not from this country. You don't understand the lingo.' But the girl from Fresno thought that man had been rather rude, and one of the train buffs offered to share his scrambled eggs with me\u2013only fair, really, because I had already urged upon him some of my Cooper's Oxford Marmalade.\n\nA very Irish lady\n\nA very Irish lady, sitting beside me as the ferry chugged doggedly across Sydney harbour, told me sadly that her car had just been stolen, and deliberately driven over a bluff. Never mind, I said, it was only a thing. Her eyes misted. 'Only a thing! Sure that's the way to look at it. Only a thing! I must look at it that way. God bless you, God bless you for that!' 'God bless you too,' I responded lamely, not knowing, as so often happens in discourse with the Irish, anything better to say.\n\nLiveable city\n\nFollowing the tourist signs towards the Old Town District and the Chinatown of Portland, Oregon, and expecting the usual harmless flummery of restored gas lamps and dragon gates, I crossed Burnside Street and found myself in a corner of hell. Suddenly all around me were the people of Outer America, flat out on the sidewalk, propped against walls, sitting on steps, some apparently drugged, some evidently about to vomit and nearly all of them, it occurred to me, idly wondering whether it was worth while mugging me as I passed. Portland has repeatedly been voted one of the Most Liveable Cities in the United States, but, thought I, you must choose the right part to live in...\n\nA performance in Athens\n\nOn a lovely spring day I climbed the Mouseion hill in Athens, all among the olive trees, to see the celestial view of the Parthenon from its summit. The morning smelt delectably of pines, flowers and dust, and my mind was full of Hellenic glories. Halfway up a Greek sprang from the bushes, opened his mackintosh wide and revealed to me his manly equipment. Well, I supposed, why not? Greek art had been displaying masculine glories for a few thousand years, and for that matter the sentries outside the royal palace, down in the city, were something of a disappointment to one of romantic fancy. Goose-stepping up and down in their full and famous finery, they looked to me less like soldiers of lyrical myth than farm boys in drag\u2013bulging, rather sweaty young men who might easily, in their off hours, mount a performance on the hill of Mouseion.\n\nPositive identification\n\nI was excited when somebody told me that the bronze statue of Enver Hoxha, the late dictator of Albania, still existed in Tirana, preserved in the Monuments Factory where it had been cast. Not so long before it had been the very centrepiece of the capital, dominating Skanderbeg Square until the rebellious populace toppled it\u2013and him. In a flash I was there, accompanied by a young Albanian engineer of my acquaintance. A watchman directed us to a windowless warehouse, apparently sealed off for ever. 'Enver's in there,' he said.\n\nWe circled this gloomy mausoleum until I found a spy-hole between the bricks, and there Enver was, recumbent in the shadows, just his bronze thigh to be glimpsed. It was enough. My engineer positively identified the old monster, and he should know. As a student he had been in the fore-front of the rejoicing crowd when the statue was pulled down in Skanderbeg Square. 'I pissed on it,' he complacently recalled, and you can't get more positive than that.\n\nLast post\n\nEven in the very last days of British Hong Kong one could occasionally see an imperial exhibition of the old kind, bands and sergeant majors shouting, every plume out of its box, judges in wigs and red robes, medals jangling on officers' breasts, swords, white gloves and His Excellency the Governor in full fig. I watched such a parade one Armistice Sunday, from a balcony above Statue Square, and all was as it always was. The commands were barked. The sad old hymns were sung. Trumpets trumpeted. Salutes were saluted.\n\nAround the Cenotaph a handful of Europeans, mostly tourists I suspect, stood watching in twos and threes. Just behind them the Sunday multitude of Filipino women was settling down to its weekly jollities, spreading themselves happily on the ground, chattering, laughing, fussing about with paper bags, and beyond them again the life of the great city proceeded altogether oblivious of the few score imperialists, with guards and musicians, pursuing their rituals at the war memorial.\n\nBreakfast Cokes\n\nAt breakfast in my Lithuanian hotel a long, long table covered with brown velveteen cloth is occupied by twenty young Russian males, while at the end of the dim-lit room there sits alone in silence at her victuals a woman who might be type-cast as a lady commissar: severe, spectacled, muscular, her hair in a bun and her skirts long and heavy. A solitary waiter in shirtsleeves serves us\u2013thick black coffee (they're out of milk), fried eggs with peas, black bread and very good cheese. Halfway through the meal we are each given a bottle of Coca-Cola. Most of the men drink theirs there and then, in tandem with the coffee, but I notice that as the lady commissar leaves the room, wiping her mouth carefully with her paper napkin and studiously not looking anyone on the face, she takes hers with her.\n\nGrand cru\n\nBeing a crude islander, and an iconoclast at that, I decided to cock a snook. I bought for the first and probably the last time in my life a grand cru Montrachet\u2013Marquis de Laguiche, vintage 1993. I got a kindly waitress in a cafe to uncork it for me, and picked up a hefty ham and cheese baguette to eat with it. 'Kindly direct me', I said to a viniculturist who happened to arrive at that moment in his Range Rover, 'to the exact patch of soil that has produced this bottle of wine.' He raised his eyebrows slightly when he saw its label and the napkin-wrapped sandwich in my hand. It was not much of a day for a picnic, he said, but perhaps the wine would help\u2013and with a wonderfully subtle suggestion of disapproval he pointed me the way to Montrachet. 'Bon app\u00e9tit,' he brought himself to say, for your Burgundy wine man is nothing if not gentlemanly.\n\nThrough a hole in the wall\n\nI looked through a big hole just hammered in the Berlin Wall, and saw into the patch of no-man's-land beyond. It was littered with rolls of discarded barbed wire, surrounded by ruined buildings and floored with the dismal mixture of sand, gravel and rubble that had resulted from three decades of herbicide\u2013for nothing was allowed to soften the allegory of the Wall. Three East German soldiers were in there, one tilted back on a kitchen chair with his cap over his eyes, the others kicking an old steel helmet around in the dust.\n\nHomesickness\n\nIn Moscow I made the acquaintance of Guy Burgess, a renegade British diplomat who had been a Soviet agent for some years but was by then sadly nostalgic for England and his mother. I could not help feeling sorry for him, and we agreed to go together one evening to the Bolshoi. We arranged to meet outside the theatre door, and when I got there I saw him waiting for me on the steps. I waved a greeting as I approached him through the crowd, and he waved a response, but by the time I reached the door he had vanished. I never saw him again.\n\nGrecian collusion\n\nI had taken a room in a private house on the outskirts of Monemvasia, and in the evening I walked a mile or so to a taverna for my supper. It was very full and very lively\u2013local people mostly, with some merry Americans. We drank large amounts of furiously resinated retsina out of metal mugs, and I seldom had a happier evening. In the small hours I staggered up the road again to my lodgings, and my landlady, in a flowered housecoat over her nightdress, pulled back the bars and undid the chains of her front door to let me in. I expected her to be tight lipped and disapproving; instead she greeted me with a sly and knowing smile of collusion, as if she had been enjoyably up to no good herself. I went to bed incoherently whistling, and awoke in the morning fresh as a daisy.\n\nNothing to say\n\nIn the bad times of communism a Polish colleague drove me out to a writers' retreat near Zakopane in the southern mountains, and on the way we were stopped by the police. My friend, a man of great charm and intelligence, did not speak when the policeman tapped on his window. He merely took his driving licence from his inside pocket, tucked a banknote into it and handed it out. The policeman did not speak, either. He had no need to. He just took the note, handed the licence back and walked away. My friend drove on without a word to me. There was nothing to say.\n\nWho cared?\n\nIt was midnight, and wartime. Sarajevo, pitted all over with bullet marks, was dark and shuttered, and the airport was closed, but I got a seat on a minibus going down to the coast. There were four other passengers\u2013a Swede, a Finn, a Croat and an Englishman. Behind us a second busload was following us through the night. The snow was deep, every now and then we were stopped at road blocks, sometimes we clattered across a temporary bridge beside a blown-up original. Scattered ruins passed dismally by\u2013house after house gaping in the darkness, with no sign of life but for a single dim light, perhaps, on a ground floor, or a fire burning in a brazier. The awful gorges through the mountains loomed around us, dark and dangerous. At about two in the morning we stopped, and our driver got out and peered rather helplessly into the black emptiness behind him, up the highway banked with snow-drifts. 'What's happening?' said the Englishman in front of me. 'What have we stopped for?' The driver explained that the other bus seemed to be lost: there was no sign of its lights, and he was worried that it might have got into trouble back there. The Englishman stretched, pulled his coat more tightly round his shoulders, and settled down to sleep again. 'Who cares?' he said, but he may have been joking.\n\nToo late\n\nLong ago I came across the ruin of one of the great Anglo-Irish mansions of the old Ascendancy, and paused to imagine all the blithe existence it had once known\u2013hunt balls and elegant dowagers and Etonians larking about with girls in the rose gardens. I remarked to a passing Irishman that it seemed a shame all the festive and colourful life of the house should have come to an end, but he replied, 'Oh, wouldn't you think it was too late for that kind of fandango?' He was right, of course, and years later I came across that ruin again, and found I could no longer hear the hunting horns, or glimpse Lady W's ancestral pearls, still less imagine those young English toffs living it up among the bushes.\n\nCity of Art and Culture\n\nSo enamoured were the Nazis of Weimar that they erected there one of their most celebrated and characteristic monuments. The site they chose was the lovely hill of Ettersberg, just outside the city, which Goethe had long before made famous\u2013he loved to sit and meditate beneath an oak tree there. On my last day in Weimar I paid a visit to this place, now a popular tourist site well publicized in the town. My taxi driver, a gregarious soul, chatted cheerfully to me all the way. Had I enjoyed my stay in Weimar? Did I visit the Goethehaus? What did I think of the food? Did I know that Weimar was to be the European City of Culture in 1999, at the end of the millennium? Congratulations, I said. Recognition once more for the City of Art and Music. 'Exactly,' replied the taxi driver, and just then we turned off the highway up to Buchenwald.\n\nGod bless Swissness!\n\nHalfway through my stay at Weggis I cracked my head open entering the lake for a swim, and had to have it stitched. How glad I was of Swissness then! Calmly and steadily the Herr Doktor worked, assisted by Frau Doktor and by their son the computer specialist, and delicate was his technique, and state-of-the-art his equipment, and whenever I opened my eyes I saw through the spotless windows of his surgery the glistening lake, streaked with leisurely waves and ringed with green hills, like a visual tranquillizer.\n\nNorway, 1950s\n\nNorwegian writers still looked wonderfully writerly then, painters were like painters, middle-aged ladies properly middle aged and cardiganed. I happened in Oslo one night to see some members of a theatre cast assembling for a post-performance supper in a restaurant, and watching their meticulously staged arrivals, their accomplished greetings and their mastery of incidental business was almost as stimulating as seeing the play itself.\n\nWelsh pride\n\nIn Argentinian Patagonia, long ago, Welsh people established a colony where they could speak their own language and live in their own way, far from the intrusive English. They called it simply Y Wladfa, The Colony, and a century and a half later, when Welshness was fading there, in a farm on the outskirts of Trevelin I found a last archetype of its settlers. He was like the smile, as it were, on the face of the Cheshire Cat. Not a soul in his household understood Welsh besides himself, but they all clustered eagerly around us as we talked\u2013a jolly Argentinian wife, diverse unidentified children and grandchildren, dogs and chickens and a horse tied to the fence; and with his cloth cap tilted on his head, his hands in his pockets, that Welshman of South America touched my heart not with melancholy at all, but with grateful pride to be Welsh myself.\n\n'Ai, ai, ai'\n\nSeveral times during my stay in Rome I came across a couple of countrymen who seemed, in their quaint fustian clothes and peculiar shoes, to have stepped more or less out of the Middle Ages. They were like substantial fauns, haunting the city out of its remote rural past. These medieval figured seemed to me wonderfully exotic, until late one night I encountered the pair of them anxiously consulting a bus timetable beneath a streetlight in the Corso. Then I realized that in fact they piquantly illustrated the matter-of-factnesss of the city. Nobody took the slightest notice of them, as they huddled there; they looked up and asked me for advice about the best way to get home, but when I told them I was a foreigner, 'Ai, ai, ai,' they said theatrically, like Italians in old movies.\n\nThe first of the Morgans\n\nOn the land of Mr Harold Childs, a horse breeder of Harolyn Hill in Vermont, is buried the stallion Justin Morgan, the only progenitor of that superb American creature the Morgan horse. Mr Childs kindly allowed me to visit the horse's grave, down the hill below his house, and when I walked back he was waiting for me with a present. It was a short piece of lead piping. 'Now this is true,' he said. 'Just here where we're standing there used to be the stables where Justin Morgan was kept, and when we was digging up there on the hill we found this old lead piping, came straight down the hill here, and a branch of that pipe it came right across the yard here and took the water to the stables. Now that's a fact.\n\n'Now I'm going to give you this bit of that pipe. You can say\u2013and it's true\u2013that Justin Morgan drank from the very water that came through this bit of pipe. You take it away with you, now.' I took it gratefully and I have treasured it ever since. 'I shall mount it on wood,' I said as I started the car to leave, 'and I'll have a card saying \"From this pipe drank Justin Morgan, the first of all the Morgan horses\".' Mr Childs tipped his hat politely, in the old American way. 'Good idea,' he said.\n\nHell's traffic\n\nNobody could be much less Neapolitan than I am, and when at last we reached the hotel, limp with excitement, amusement and exhaustion, and I had paid our driver his exorbitant but entirely justified fare, I told the hotel receptionist that I wanted to go home. 'Don't say that,' he replied. 'Wait till you get up to your room, and everything will seem different.'\n\nSo it did. Dusk was falling by then, the harbour was speckled with small fishing boats, and in the distance Vesuvius loomed hazy in the half-light. The docks were full of white cruise liners, and even as I watched one of them slipped away from the quay towards the open sea. For a long time I could see her lights, fainter and fainter to the west\u2013treading her way, I liked to imagine, towards calm realms of order. But it did not make me in the least homesick. The receptionist was right. I rang for a bottle of wine, and we sat there on our balcony in perfect contentment, while hell's traffic snarled convivially below.\n\nFrenchness like a cloak\n\nNothing had changed in the corner restaurant, the one with the awnings and the menu in the polished brass frame. It remained quintessential France, as we islanders have loved and loathed it for several centuries. Madame remained the epitome of everything false, narrow-minded and unreliable. One waiter seemed, as ever, to be some sort of duke, the other was evidently the village idiot. At the table next to mine sat a prosperous local family out for Sunday dinner, well known to the proprietress and esteemed throughout the community\u2013unsmiling, voluminously napkinned, serious and consistent eaters who sometimes, eyeing me out of the corners of their eyes, exchanged in undertones what were doubtless sly Anglophobics before returning sluggishly to their veal.\n\nI do not doubt the bill was wrong. I am sure Madame disliked me as much as I detested her. The veal was, as a matter of fact, rather stringy. But what a contrary delight it all was! How excellent still the vegetables! How much better the wine in France! How stately that duke! How endearing the idiot! With what real gratitude, evading the final scrutiny of the prefectorial table, and sweetly returning Madame's shifty glittering smile, did I wrap the Frenchness of that cafe around me like a cloak, and return cherished to the autoroute!\n\nThey thought not\n\nIn Beijing the compound called Zhongnanhai is the very heart of the Chinese communist despotism. Its main entrance is to the south, with two great guardian lions. The Red Flag flies from a mast outside, and within the gate an inner wall is inscribed with the cabbalistic text 'Serve the People'. You cannot see past it, though. Two armed sentries stand there, with two more over their shoulders. They look distinctly unwelcoming, as they stare motionless and expressionless into the street: and sure enough, when I asked them if I could take a stroll inside Zhongnanhai, they seemed to think not.\n\nThe Smile Test\n\nThe Smile Test is the system I employ to gauge the responsiveness of cities, and it entails smiling relentlessly, if not unnervingly, at everyone I meet walking along a street. I devised it in Vancouver, which remains a good place to test the system. Pay attention now, as we try it out in Robson Street, one of the raciest of the city's downtown boulevards. Many of our subjects disqualify themselves from the start, so obdurately do they decline eye contact. Others are so shaken that they have no time to register a response before we have passed by. A majority look back with a blank but generally amenable expression, as though they would readily return a smile if they could be sure it was required of them, and were quite certain that the smile was for them and not for somebody else. A few can just summon up the nerve to offer a diffident upturn at the corners of the mouth, but if anybody smiles back instantly, instinctively, joyously, you can be sure it is a visiting American, or an immigrant not yet indoctrinated. Whenever I go to Vancouver people ask me how they're doing in the Smile Test. I respond with a nervous smile myself.\n\nSuburban enchantment\n\nIn the evening I saw Die Fledermaus, staged with a genuine rollicking panache, and so instinct with the magic of the waltz, the whirl of white skirts and the flick of tailcoats, that when I inspected the faces of the women around me, Soviet proletarians every one, I found them glazed with a true suburban enchantment.\n\nGORGE\n\nOne person in particular at Iceland, the Sydney skating rink, seemed to me quintessentially Australian. He was about five years old, I suppose, blond, lively, tough and unsmiling. He could not, it seemed, actually skate, but he was adept at running about the rink on his blades, and his one purpose of the morning was to gather up the slush that fell off other people's boots, and throw it at passing skaters. This task he pursued with skilful and unflagging zeal. Hop, hop, he would abruptly appear upon the rink, and, picking a likely target, staggering his way across the ice, inexorably he would hunt that victim down until slosh! the missile was dispatched\u2013and hobble, hobble, quick as a flash he was out of the rink again, gathering more slush.\n\nHe hardly ever fell over, he seldom missed, and he did everything with a dexterous assiduity. When I asked him his name he spelt out GORGE with his finger on the rail of the rink; when I asked him if he was enjoying himself he just nodded grimly; and in my mind's eye I saw him thirty years from then, exploding into a company meeting perhaps with an irresistible takeover bid, or relentlessly engineering the resignation of a rival undersecretary. I kept my eye firmly on him as I walked out of Iceland, for instinct told me he was assembling slush for me.\n\nSeen from a bus\n\nI sit in a motionless bus near the Sugar Loaf, at Rio de Janeiro, at a place where a small park runs down to the sea. There are military offices nearby, and in constant twos and threes colonels and captains walk by carrying briefcases. My eye is captured, though, by a solitary middle-aged man hanging about at the edge of the park. He bears himself elegantly, slim and erect in a well-cut grey suit, but there is something wrong with him. It seems to be partly physical, partly mental and partly, perhaps, too much coffee. He can never get comfortable. If he sits on a bench, after a moment he gets up again. If he takes a turn around the grass, he abruptly stops. Sometimes he looks up at the hill above, but it seems only to disappoint him, as if he cannot see what he is looking for up there. He inspects the passing officers keenly (was he once a colonel or captain himself?) but he recognizes none. He gazes longingly out to sea, but the sun gets in his eyes. When my bus starts, and we move away from the park, I wave at him through the window, he waves abstractedly back\u2013but not at me, I think, not at me.\n\nVery simple matters\n\n'Certainly,' said the government spokesman, perusing my list of questions. 'By all means, these are very simple matters. We can attend to them for you at once. As I told you, it is our duty! It is what we are paid for! I myself have to attend to an important meeting this afternoon\u2013you will excuse me I hope?\u2013but I will leave all these little matters with our good Mrs Gupta and all will be taken care of. I will telephone with the answers myself without fail\u2013or it not myself, then Mrs Gupta will be sure to telephone you either today or tomorrow morning. Did you sign our register? A duplicate signature here if you would not mind, and the lady at the door will issue you with the requisite application form for a pass\u2013it will make everything easier for you, you see. Have no fear, Mrs Gupta will take care of everything.'\n\nBut neither he nor Mrs Gupta ever did ring.\n\nI smell\n\nI drove direct from the horrible purlieus of San Crist\u00f3bal, one of the worst of the Lima barriadas, to have tea at the Country Club in San Isidro. The odour of the slum went with me, clinging to my clothes and the soles of my shoes like some blasphemous travesty of incense, and as I sat there among the little black dresses and the sticky cakes, the greying distinctions and the foppish playboys, the starched nannies and the exquisite children on the lawn, the chic and the cultivation and the chit-chat of urbanity\u2013as I sat there with the squalor still in my hair I could not help remembering, Pharisaical it seems in retrospect, Dr Johnson's celebrated differentiation, I smell, you stink.\n\nBastille Day\n\nFor hours I had been hanging about the airport at Kharkov, fobbed off by supercilious airline employees through delay after delay in a bitterly cold and comfortless waiting room, until at last the patience of my Soviet passengers expired. They found a boarding ramp, pushed it on to the tarmac, climbed up to the aircraft and, brushing aside the horrified stewardesses, plumped themselves in their seats and called for vodka. I followed in their wake rejoicing, feeling as though we had stormed life's varied Kremlins.\n\nAt the trial of Eichmann in Jerusalem\n\nI looked at Adolf Eichmann to see how he was reacting, half expecting to see some flicker of perverse pride crossing his face. But he was sitting well back in his chair, with his hands in his lap, blinking frequently and moving his lips, and he reminded me irresistibly of some elderly pinched housewife in a flowered pinafore, leaning back on her antimacassar and shifting her false teeth as she listened to the railing gossip of a neighbour. It was only towards the end of the morning, several hours, ten thousand words and an eternity of horrors later, that the old lady in the pinny began to sway and fidget a little in her chair, as though she were pining for a nice cup of tea.\n\nWaiting for Churchill\n\nUp on the mountainside, while the press of the world jostle for scoops and angles in the hostelries below, old Sir Winston lies in bed. He lies there in seclusion, the last of the giants, reading his newspapers and confounding his pleurisies while they wait for him to die. Some of my colleagues depict him demanding brandy, puffing cigars, writing his own health bulletins, calling for splendid enormous meals. For myself, when I was up the mountain one evening I thought I heard a sound from the villa, above the sweetness of the birds and the distant sawing of a woodman. It came from an upstairs window and it sounded to me uncommonly like a rich, quixotic, irrepressible, ageless Harrovian chuckle. 'How come you heard that and nobody else? You got influence some place? Hey, gar\u00e7on, two dries.'\n\nHome thoughts from Barbados\n\nThe parishioners who came to the service were nearly all black people, sugar workers and their families from the island estates, but few of them were really strangers to me. Their white muslins and their wide straw hats once graced the English social fabric, and when they sat down expectantly for the sermon the rustle of their petticoats and the crackling of their starch filtered though to me across the pages of many an Edwardian memoir. I knew what hymns they would sing with gusto, for I had heard the same tentative starts and communal diapasons at many a grumbling British army church parade. The verger in his black cassock I had often met before, pointing out the ravages of death-watch beetle in the shires, and when the piano struck up its preliminary chord I knew from her air of proud command which of those old friends would be the one who always comes in half a beat before the beginning of the verse. 'Amen, amen,' murmured the congregation at the end of the sermon, and it was like the clatter of hobnailed boots on the stone-flagged floor of a dairy.\n\nEnd of a battle\n\nThe Israelis had won. Tanks clattered by. Trucks came and went. Soldiers climbed aboard and waved goodbye to each other. It rained, and I prepared to move on too, but just then a rainbow came. 'Look, a rainbow,' I said to a bearded and taciturn sergeant not long from Romania, and added sentimentally: 'Omen of peace!' 'It is not a reasonable analogy to the present situation,' he replied, shifting his Sten gun on his shoulder. 'God showed Noah the rainbow as a promise for no more floods in the future. When He merely wished to show that Noah could now leave the ark, He dispatched a small bird, carrying a piece of tree in its snout.'\n\nAmong the treasures\n\nIn the vaults of the Central Bank of Persia, before the Iranian Revolution, were kept the priceless and legendary Crown Jewels, in a huge underground strongroom. I was down there one crowded weekday, when it was open to the public, and came across an agreeable case of brooches and little jewelled watches. I stooped to examine them more closely, and as I did so the treasure house suddenly reverberated with the ear-splitting blast of an alarm hooter. Everyone froze. Not a word was spoken. The hooter went on hooting. For a moment nothing else happened, and then a smart young woman in green walked with composure across the room. She avoided the case containing the Gika of Nadir Shah, with its diamond ornaments of bayonets and gun barrels around a monumental emerald. She ignored the sceptre presented to Reza Shah by the people of Azerbaijan, with its gold lions rampant around a jewelled globe. She took no notice of the Sea of Light inherited from the first Mogul Emperor of India. Instead she walked calmly through the room, utterly silent but for the clicking of her heels, directly between the display cases to me. 'May I please ask you,' she said with an amiable smile, 'to remove your elbow from that metal bar around the jewel case?' I moved my arm. The hooter stopped. 'Thank you,' she said, and walked composedly back again.\n\nAffronted\n\n'Those Algerians!' expostulated an elderly politician in Reykjavik, when our conversation turned to politics. He wagged his beard irritably, as elderly Nordic politicians do. 'They're nothing but troublemakers. They were up here, you know, making mischief\u2013Einar Arnarson, I think it was, he put paid to them, he and Jon Olafsson and one or two others'\u2013and suddenly it dawned upon me, as his sharp affronted eyes blazed into mine, that he was talking about the Barbary Pirates.\n\nA lovely dream\n\nThe day I arrived in Harar, Ethiopia, I spent a happy hour in the market, sitting beside a courteous silversmith and watching the rural citizenry at its shopping. There was nothing ugly to be seen there, nothing sham, nothing pretentious. It was like watching an assembly of beautiful lithe-limbed animals, so easily did all those people move, so naturally, so discreetly; and as I sat there on my stone seat, the craftsman tinkling away beside me, or engaged in earnest but desultory haggling with half-naked but otherwise impeccable debutantes, I thought how fortunate were those creatures of nature, those children of the thatched hut and the empty places, those sisters of specious innocence. But alas, even there the dream would soon be over.\n\nFair enough\n\nWould they be casting their votes as Jamaicans, I asked the Kingston Rastafarians, in the forthcoming elections? The idea horrified them. 'Tell your Queen Elizabeth,' they said, 'that the suffering Ethiopians assembled here from the corners of the earth, yea verily from the four corners, it is written, the seventh year of the seventh epoch, yea verily the time has come, Abja!\u2013tell her that we are aliens in this land, and cannot vote in elections imposed upon us by our oppressors. Fair enough?'\n\nProverb in Formosa\n\nThe Vice-President of Nationalist China, in his garden on the island of Formosa, folded his intricate old hands in his lap and projected a Chinese proverb at me. Beneath the trees on the edge of the lawn a stalwart servant waited in attentive silence, and a few soft raindrops were spattering the foliage in a sly, oriental kind of way. 'Among our people we have a saying,' the Vice-President said. '\"It is foolish to judge the character of a man by the complexion of his face.\"' And his pale eyes flickered at me, as an old experienced tiger's eyes might blink in the forests of the night.\n\nOpposite directions\n\nVery early one morning two men met outside my tent on a mountainside in Wales. The younger of the two was a tatterdemalion Welsh shepherd, cloth capped, driving his sheep down to the road with a clatter and scurry, calling to his dog, shouting guttural Welsh encouragements and waving his thick stick like an apparition. The elder, a scholarly looking man in plus fours, was evidently a believer in rhythmic breathing, for as he walked he whistled to himself a monotonous Bach-like melody\u2013two beats to each footstep, round and round, over and over again in an endless classical cadence. The two men passed each other as I gaped at them through my tent flap. The shepherd brandished his stick and grunted casually; the scholar interrupted his fugue to offer a greeting in a reedy academic voice; and so they disappeared into the rain, in opposite directions.\n\nAt the theatre\n\nThe audience at a Tokyo Kabuki theatre consists mostly of women in kimonos, following the drama with an informed avidity I have seen paralleled only among rugby crowds in South Africa. You sit there wedged between the brocades, baffled by the tortuosities of the plot, swathed in the sickly perfumes of Japan, while high above you in the balcony the narrator declaims his lines majestically from a tasselled lectern. The man beside him plucks dreamily or astringently at his ancient instrument, and on the magnificent stage the queer medieval figures sit and strut and gesticulate with falsetto voices and grand flamboyant costumes. All that is grand, awful or ablaze in the old Japan lives on, twitching and quivering, in the theatre of the Kabuki.\n\nSchooner travel\n\nAt Granada the schooner captain kindly signed me on as crew, and to avoid awkward questions stuffed me away in a cubbyhole of his vessel until the heavy footfalls of authority had died away along the quay. When I was released I crept out blinking to find the schooner already scudding gaily out of harbour, and the captain grinning beside the wheel with a tin mug of rum in his hand. African, British, French and Indian ancestors had all contributed to his ship's company, and in the tiny starboard deckhouse, I presently discovered, there resided a seductive mulatto camp follower, immured there silently like a lady about to be sawn in half. We all slept on the open deck, and when the moon came up I heard somebody murmur to this nubile shipmate: 'You got your moonburn lotion, honey?'\n\nThe clock and the nougat\n\n'My brother-in-law,' said the woman I had given a lift to, who was dressed funereally and clutched a posy of lilies in a sanctimonious sort of way\u2013'my brother-in-law has told me that the British are more honest than we Calabrese. Is this so?' I had taken a peculiar dislike to this person, and had noticed that she was eyeing my travelling clock with an interest unmistakably covetous. So when she asked me again, wriggling in her seat in a manner at once obsequious and obscurely arrogant\u2013'Eh, is it true?'\u2013I answered her harshly. 'Perfectly true,' said I. She was unperturbed. When I dropped her she said nastily, 'Haven't you got some small memento to give me, some small gift or souvenir?' 'Only the memory of our meeting,' said I firmly, shoving the travelling clock out of sight beneath the dashboard\u2013and she shamed me then, by pressing into my hand a large and rather nasty bar of nougat.\n\nThe portrait\n\nBrigadier Abdul Karim Kassem has today led a violent coup d'\u00e9tat which has suddenly made him the prime minister of Iraq, and he has invited the gentlemen of the press to meet him at the Ministry of Information. Soldiers stand guard with tommy guns, the Deputy Prime Minister and the Minister of Information (since this morning) are in attendance, and pointedly on the floor of the room is the new leader's camp bed, with a pair of green striped pyjamas folded primly on its pillow. In a corner stands a large, enthusiastic and evidently freshly painted portrait of the brigadier. 'May we ask the Prime Minister', says an American reporter with a courtly air, 'if that is a new portrait of His Excellency, and, if so, who painted it?' The Prime Minister smiles a glittering smile and is silent, so his deputy answers for him. 'Yes, it is a new one. It was painted by the people. It is a present.' 'All the Iraqi people like this government very much,' adds the Prime Minister then. 'Ask the people yourselves. When I go into the street everybody is friendly.' The American clears his throat. The American public would be interested to know, he feels sure, if it is not too personal a question, if any particular one of the Iraqi people painted the portrait, and if so, which? But 'Gentlemen,' intervenes the Minister of Information, 'I think we are all very tired,' so we shake hands with His Excellency and filter through the sentries into the street.\n\n'For heaven's sake'\n\nOne Christmas in Vienna I went for a walk in a park before returning to the hotel where my Christmas dinner was roasting. There was hardly a woman in the park. Everywhere the husbands of Vienna, with their children, aimlessly but expectantly loitered, expelled from under the womanly feet of the city while Gretchen and Helga got on with the job. Christmas is a time when old hierarchies are restored. 'For heaven's sake,' I could hear the housewives of all Europe grumbling that day, 'go out and get yourself some fresh air, and take the children with you.'\n\nWine of the country\n\nThe Colonel's family had lived in the same Scottish Lowlands house for several centuries, but as a retired widower he lived unostentatiously, and I was greeted with a homely plate of scones and raspberry jam, and a pot of tea in a blue flowered teapot beneath a bobbled cosy. He ate and drank his ration with enthusiasm, but very soon afterwards fetched a bottle of whisky from the sideboard and poured a couple of glasses\u2013'Wine of the country\u2013I always say, you should drink the wine of the country.'\n\nHe looked like an Irish deerhound, very tall and elongated, his figure only slightly stooped with age, and he was dressed tweedily, with shoes that looked handmade. At first sight he did not appear to be Scottish at all, but after a while, through that expensively anglicized exterior there began to appear something pricklier, more gingery, more ruthless, perhaps, and I realized in fact that I was talking to a man almost aboriginally Scots. His attitudes were mellow. His subjects of conversation ranged from Bonnie Prince Charlie ('All those Stuarts were a rotten lot') to the price of claret ('I used to get it from a man I know in Bordeaux, but now I just go to the supermarket') to snooker on TV ('They're very sporting fellows, extremely sporting'). He alludes now and then to some duke, marquis or other ('by way of being a relative of mine'), but only to tell a comic tale about him, discredit a medieval anecdote or explain the genesis of a portrait. It was raining when I left him, but he came to the bottom of the garden to wave me away, and as I turned the corner of the drive I glanced in the driving mirror to see that old inheritor of blood feud and cattle raid regain the shelter of his door, as though he were escaping a royal posse, in a single mighty stride.\n\nHoly experiences\n\nConsider this family of Irish people, sitting beneath a canopy in the drizzle of a Marian shrine in County Waterford. The image of the Virgin is not very old, but stands strangely half in shadow on a rock wall in a frond-filled grotto, with a stream running below. During the last few months it has repeatedly been seen to move of its own accord, and to be transfigured. Sometimes its face changes into that of Christ, and sometimes it apparently comes to life\u2013early last night, a bright-eyed lady at the gate tells me, she met the Virgin walking silently by the stream. The family sits there, mother, father, adult son and daughter, in a determined common trance, their eyes fixed immovably upon the statue on the rock\u2013willing it to move, praying for a manifestation, clutching rosaries, lips moving sometimes but bodies still as images themselves. They were like addicts at a gaming table. The rain fell all around.\n\nSacred memories\n\nI went to the 300th-anniversary march of the Prentice Boys in Londonderry (aka Derry), by which the Protestant Orange Order remembers a famous victory over the Catholics, and never did I see such a variety of remarkable faces, pinched, florid, genial or fierce beneath their bowler hats or tam-o'-shanters. Never were pipe-and-drum bands more fervent. Never was I in a crowd so absolutely united in its bigotries. Thirty thousand Orangemen took part in the march, and for five hours an air of perfervid dedication enveloped the city. There were tiny boys of three or four marching with the rest. There were half-crazed bass drummers and clown-like drum majors, juggling their batons, balancing them on the ends of their noses, strutting and gesturing like circus performers. There were ranks of stern elderly men, bowler hatted, some carrying swords, all swathed in the regalia of the Orange Order. Halfway down the procession the hero of the day, a large Presbyterian clergyman, came swaggering by with a cohort of aides, smiling here and there and cheered along the way like a dictator moving among his adoring subjects. Hour after hour the beat of the drums reverberated, and when I left Derry the Orangemen were still streaming across the Craigavon Bridge, banners flying, drummers prancing, strutting infants, determined old men in medals and bowlers marching in steadfast line abreast.\n\nReciprocal ill will\n\nI can see to this day the face of a Benedictine monk I encountered at the Bavarian monastery of Andechs. In his late twenties, I would guess, he looked more like an interrogator than a confessor, far more accusatory than forgiving. Tall, thin, pale, unsmiling, cold eyed, pious as all hell, when I asked him the way to the monastic cemetery he did not at first reply at all, but simply turned his cod-like features upon me with raised eyebrows. When at last he gave me a curt and loveless answer I hardly had time to thank him (not that I was planning to be very fulsome about it) before he turned on his heel with a flounce of his cassock and disappeared inside the church. I hope he choked on his vespers.\n\n'Oes heddwch?'\n\nAssembled on stage at the National Eisteddfod, the great cultural festival of the Welsh nation, are the Bards of the Druidical Orders, a strange conclave of eminent citizens, doctors and philosophers, writers and politicians, dressed in long hooded robes of white and grey. They are presided over by sages and attended by nymphs in green, by matrons with horns of plenty, by harpists and by trumpeters, and they are there to honour the victor of a poetry competition. The winner's identity is a secret, but he is sitting, we know, somewhere in the audience around us. A hum of excitement and speculation accordingly fills the pavilion. Strange preliminaries occur on the stage: harpists pluck arcane strains, elves dance, a gigantic sword is half drawn from its sheath, then majestically slammed home again. 'Oes heddwch?' cries the Archdruid. 'Is there peace?' 'Heddwch!' thunders back the audience, and the trumpets blow their fanfares, and gathering their robes about them a deputation of Druids gravely leaves the stage to summon the victorious poet to his honours. The organ thunders. A spotlight plays at random over the auditorium. The television cameras are poised in their gantries. The audience strains forward in its seats. Presently the light steadies itself, sweeps deliberately along the seats, and falls at last upon the person of the winner\u2013who, blushing with pride and self-consciousness, and pretending hard to be astonished, allows himself with mock reluctance to be led away by the Druids, up through the huge applauding crowd, up through the reverberating organ music, to the throne that is, for those few moments, the very crucible of Wales. Some years ago I was a member of that Druidical delegation, the man who drew the great sword from its sheath was a famous rugby player and my son Twm was the poet.\n\nSmall change\n\nThere is a Sydney street group called the Aussie Small Change Brass Band which might well represent the city at ceremonial functions, so alive is it with the authentic Sydney mixture of fun, fizz and chutzpah. Its players are three very small boys in very large hats, with two trumpets, a tuba and extremely powerful amplifiers, and I can tell you they play 'Puttin' on the Ritz' like nobody's business.\n\nIncidental music\n\nI was driving down the Adriatic coast from Istria to Montenegro, and I was playing a recording by Vladimir Ashkenazy of Mozart's 22nd piano concerto. It seemed to me that the vivacious allegro movement of this work absolutely suited the swashbuckling landscape of karst, sea and island through which I was passing, and I drove down the magnificent coast road playing the tape repeatedly, laughing and singing out loud. In the course of the journey I gave a lift to a frail and elderly Montenegrin traveller, wizard-like with stick and black coat, and when towards the end of the journey, Ashkenazy still playing, me still singing, in the delight of my mood I narrowly escaped head-on collision with a convoy of armoured cars, this delightful old worthy seemed to find it just as funny as I did.\n\nThe Algerian gardener\n\nThe Algerian gardener at our hotel in the Midi was extremely tall and cadaverous, and his eyes blazed deep in their sockets. His luxuriant sideburns, however, gave him a noble scholarly aspect. He was like a professor in some medieval academy of Islam. As he trundled his barrow about, I used to think, surely he was debating within himself subtle mathematical formulae, or composing Sufi couplets? Once I got up at the break of day, when the place was deserted, and I came across the gardener feeding a black and white cat. He stood very erect above the animal, having placed a grisly dish of offal before it, and I heard him murmuring endearments to the creature. They sounded stately endearments\u2013Koranic, perhaps\u2013and he stood there gauntly as the sun rose behind him, looking down at the cat and murmuring. The cat kept circling around his feet, casting glances at the food, rubbing its head against the man's ankles until it felt it had paid its proper respects. Only then did it fall, with snarls and rendings through its purrs, upon the unlovely victuals.\n\nThe matter with me\n\n'Wazzamatterwidyou?' hissed the angry cab driver, as I stumbled bemused across 45th Street. 'Hey, you in the green hat,' shouts the policeman from his horse, 'can't you see that signal?' 'You must wait for the green,' says the passing lady slowly and sympathetically, assuming I speak only Welsh or Lithuanian, and am new to the mysteries of science. But it takes time to readjust, when you return to Manhattan from idler climes.\n\nTo touch the owl\n\nI notice that for mistily religious reasons women in Dijon touch the little figure of an owl in the rue de la Chouette as casually as they might pull on a glove\u2013except that, since it is perched rather high on a wall, small ladies have to jump a bit to reach the bird, and children have to be lifted one after the other, their mothers never interrupting, all the same, the flow of their own conversations.\n\nThe friar's warning\n\nAt the Capuchin catacombs in Palermo the desiccated corpses of generations of citizens are on display, guarded by friars and climaxed by the body of a child labelled BAMBINA\u2013SLEEPING BEAUTY GIRL. 'Be very careful,' one of the friars said in a flat sort of voice when I left this macabre exhibit\u2013'watch out for robbers.' I thought there was a queer, occult look in his eye, and hardly had I left the sacred premises than two thugs on a motorbike snatched my bag and left me destitute.\n\nOne can always tell\n\nIf the hitch-hikers are American I generally stop for them. One can always tell. They try harder for their lifts, holding up well-lettered destination signs and offering ingratiating smiles. They are in the lift-getting business, and they do the job properly. When they are on board they generally work for their keep, too. They tell me all about themselves, they learn all about me, they may give me a brief lecture upon the social customs of my own country, or kindly correct me when I appear to be going the wrong way. They are usually willing to oblige, too. 'Are you going to Scotland?' one young man asked me when I stopped for him just outside London. 'No, I'm going to Wales.' 'OK, make it Wales'\u2013and I drove him all the way to Bala, and left him smoothly chatting up the farmer's wife at a bed-and-breakfast place.\n\nMarvellously goes the elk meat\n\nMy favourite place for a Christmas meal is the Operak\u00e4llaren restaurant in Stockholm. There the restaurant's famous house aquavit is poured most generously by merry waiters of the old school, and everyone soon gives the impression of being acquainted with everyone else. Marvellously goes the elk meat, swiftly pass the herrings, one great salmon succeeds another on the buffet, and very soon I find myself on familiar terms with the Swedes at the next table, complimenting them on their fluent English, admiring little Eva's Christmas frock or little Erik's smart bow tie, exchanging grandmotherly confidences with Mrs Andersson, toasting them one and all with yet more aquavit. Stockholmers are not especially religious people, and I like to think they have been eating those Baltic herrings, downing those fiery liquids, since the days of the pagan kings.\n\nThe red tarboosh\n\nAndrew Holden was one of the very last British officials of the Egyptian government, still a highly respected functionary of the Ministry of Finance when Egypt had long recovered its independence. By the time I knew him he was near retirement, but he still went to work each morning by tram, clinging to the outside like any other Cairene if he could not get a seat. The amiable Egyptians, helping him up the step, would make sure he had a place on the rear platform, where he could hang on to the pole, and there I can see him now as the tram swayed and clanked its way into town, so scholarly looking in his spectacles, so slight, so incongruously at ease\u2013and on his head, tilted at a jaunty but not ostentatious angle, the red tarboosh which was the only badge of his commitment.\n\nDance music\n\nOne evening I came across a dance in a Cretan courtyard. The lights were very bright there. The deafeningly amplified music was a quavery sort of oriental theme. A high gate closed the yard, but along the wall of the road above, from windows and shadowy terraces all around, a crowd of villagers watched. Beneath the lights inside, a long circling line of Cretans, men and women, danced a strange dance. I was bewitched. Gracefully, jauntily, thoughtfully, swankily, the dancers tripped their complex steps, and the music blared through the pergola. Round and round they went, to and fro, and sometimes the man at the head of the line, detaching himself momentarily from the rest, threw himself into a spasm, leaping, kicking his feet together, twirling about in an ecstasy of conceit and accomplishment, before the convulsion left him and he subsided into the music's rhythm. When I tore myself away the half-tone music of the loudspeakers tracked me far into the night.\n\nThe three days\n\nOne of the most demanding of Irish pilgrimages takes the faithful to a grim island in Lough Derg, a remote and dispiriting mountain lake, where they endure a three-day fast, a twenty-four-hour vigil, barefoot peregrinations over stony tracks and the compulsory recitations of 63 Glorias, 124 Creeds, 891 Paternosters and 1,458 Hail Marys. I was once at a wedding at Drogheda, away on the east coast, when I heard a woman ask a worldly young guest with a carnation in his buttonhole and a glass of champagne in his hand where he was going for his holidays that year. I expected Mykonos or Barbados, but no. 'I thought of giving myself', he said, 'the three days at Lough Derg.'\n\nAt a Breton window\n\nMy small daughter and I looked up from the waterfront of Douarnenez, in Brittany, to see an old woman smiling down at us from an open window. She had a shawl around her shoulders, her face was infinitely wrinkled, and her smile was so kind that it seemed to be reaching us from different times altogether\u2013from before the Fall, perhaps. 'I want that lady,' my small daughter said.\n\nDo I know her?\n\nNow and then I chance to see in real life one of those nameless and numberless actresses of television, encountered in the Underground, perhaps, or browsing at a bookshop. At first I think I really know her. Who could she be? Is she a publisher, or a fellow author? Did we meet on an aircraft, or at a literary festival somewhere? Like one of those nagging fragrances one cannot place, or a tune whose words we can never quite remember, her presence tantalizes and disturbs me. But then with a touch of melancholy I realize that I know her only by proxy, through the medium of the TV screen. Some people in these circumstances introduce themselves anyway, and perhaps one should: I sometimes notice that if I chance to catch the woman's eye she will give me one of those closed-lip actress's smiles, turned up a little too resolutely at the corners of the mouth, as if she is dying to be recognized.\n\nSalon life\n\nA Jewish acquaintance of mine in Delhi, being a passionate horsewoman, established a sort of lien upon the social loyalties of a whole covey of equestrian maharajas, polo players to a man but as fascinated by the personality of their hostess as they were by her love of horses. They became a kind of salon. They used to sit in her drawing room, itself a strange and wonderful melange of cultures, or sprawl on the lawn with long cool drinks, hanging upon her every word: dark mustachioed military figures, handsome but rather running to plump, and in their midst that small vivacious woman bestowing a chaff here, a compliment there, like a Jewish maharani herself.\n\nDays of liberty!\n\nI chanced to arrive in Paris when a student rebellion was reaching its climax, and was astonished to find the students surging to and fro between their makeshift barricades, handkerchiefs over their mouths, throwing things now and then and shouting slogans. They were all that old people dreamed themselves to have been when they looked back to their days of liberty, the days when they had causes to throw bricks for, when to be alive was grand enough, but heaven itself was to be young, radical, brandishing a stick and shouting a slogan in Paris!\n\nHarry's Challenge\n\nI had a pre-Christmas luncheon at Harry Ramsden's Fish and Chip Shop at Guiseley, where the menu was dominated by Harry's Challenge, a fish-and-chip dish so gigantic that if you got through it you were given a free pudding and a signed certificate. All the customers were the real thing\u2013not another outsider among them, only celebratory office parties hilarious over Harry's Challenge, and amiably extended families with grandmothers in hats, and burbling children with hand-held video games, and not a few stout parties who would have done better to cut down on the steamed ginger pudding. At one o'clock precisely there arrived outside the front door the Scissett Youth Band of Huddersfield, to serenade us lustily with all the old carols\u2013none of your fancy ecumenicals\u2013setting many a sensibly shod foot tapping to their rhythms and inciting me, as an inveterate whistler, to join in messily over my mushy peas.\n\nIn other circumstances\n\nTrams are essential to the character of Vienna, but there are some places where they run against the flow of the traffic, and are likely to murder you. When I nearly lost my own life to one of them\u2013'Quick! Comes the tram the other way!'\u2013sympathetic onlookers were quick to reassure me that Dr Kurt Waldheim himself had almost died a similar death (although that was of course, they respectfully added, before he became president of our republic...)\n\nBelow the ships\n\nThe longest escalator in Europe plunges beneath the Kiel Canal to take pedestrians to the other side. I stood there one morning looking down this dreadful shaft, which was all empty, dark and rumbling, wondering if anybody ever used it, when a cheerful girl rode up behind me on a bicycle. Without a pause she tucked the bike under her arm, so to speak, and launched herself upon the moving staircase. I stood there watching her go. Down and down into the dark she went, all alone, smaller and smaller, clutching her bicycle, until she disappeared into the hole beneath the Kiel Canal. Above her the ships sailed on.\n\nColombian coffee\n\nI once sat for half an hour over a coffee at a pavement cafe in Buenaventura, Colombia, and never did I see a more piteous and dispiriting citizenry pass by, in the sticky blaze of that tropical afternoon. A mutilated beggar crawled about my feet, silently holding out his hand. A shoeshine boy with a withered arm sat listless at the pavement's edge. A few tattered black men slouched about the surrounding tables. Two small boys played football with an old tin in the street. Sometimes a grey figure in white ducks shambled into the cafe, reaching into his money belt for the price of a brandy, sometimes the beggar scuttled off like a huge black crab towards some new arrival, and sometimes the waitress, with a clang of her bangles, screamed some raucous incantation into the kitchen. All around was filth, heat and degradation, malformation and truncation, stumps of arms and crooked arms and scabbed dry lips.\n\nThrough a glass darkly\n\nThe Yamut Turkoman tribes are the most daunting of the Iranian peoples. On Thursday mornings they hold a horse fair at the village of Pahlevi Dej, and there I went to see them. They converged upon the village in ones and twos, bolt upright on their horses, top heavy in their black fleece hats, in stately lolloping motion across their splendid landscape. Some brought their wives with them, demurely riding pillion and wearing purple or scarlet skirts with brightly flowered shawls. I saluted one formidable tribesman as he rode by; not for a moment did his pace flag, inexorably he continued his progress, kicking up little clouds of dust with each step, and looking distantly down at me from the saddle as through a thick glass plate.\n\nJoking on the coastal route\n\nOnce on the Hurtigruten, the Norwegian coastal shipping service, an entire brass band boarded our vessel, with musicians of all ages down to small boys and girls. They were going to the next port up the coast and earned their passage by playing sombre but rousing marches in the forward lounge. The faces of the instrumentalists were quintessentially Norwegian: pale, long, incurious, handsome faces. One boy asked me where I was from, and when I told him he said, 'I have a grandmother in Wales.' 'You don't mean it!' I exclaimed in delighted surprise. 'No,' he said, 'I was only joking.'\n\nPan's blood\n\nDelight is still the occupation of Corfu, and sweet airs of comfort abound. The peasants of old may have deserted their olives for occupations of easier profit, but the olive trees are still there, and the stony earth beneath them, and the scents of herbs in the evening. On our way back from Kavos we saw, in one of the wayside villages, a pick-up truck run over a cat. In a trice the corpse of the poor animal was removed for burial, and I was struck by the air of true sadness that fell upon the village bystanders. It was the sweet silent sadness, I thought, of the ages. When we drove away the little pool of cat's blood left in the street behind us suggested to me the blood of Adonis, or perhaps of Pan himself.\n\nCanadian arrivals\n\nVery early one morning I went down to Union Station to watch the transcontinental train passengers arrive out of the darkness from Vancouver. I knew exactly what to expect from this experience, but still it stirred me: the hiss and rumble of it, the engineers princely in their high cab, the grey faces peering out of sleeper windows, the proud exhaustion of it all\u2013and then the thick tumble of the disembarking passengers, a blur of boots and lumber jackets and hoods and frosty breaths and bags and bundled children, clattering down the steps to breakfast, Grandma and Toronto.\n\nDestiny in Missouri\n\n'Mr Truman? Certainly, he's expecting you,' said the pleasant secretary in Independence, Missouri, and in a moment there was his familiar figure, sitting at a big polished desk. Beside him there stood a large and splendid globe, in a frame stand, and from time to time during our conversation Harry Truman would reflectively spin it or point to parts of it in a manner that I can only describe as proprietorial. He was, as he reminded me, the president who, in the years after World War II, had decreed an interventionist foreign policy for the United States of America\u2013the Truman Doctrine. When he twirled that globe he was retrospectively reshaping my world, abolishing my empire, and affecting the way I would live for the rest of my life.\n\nA cabman's wink\n\nI was wandering the streets of Alexandria's Arab Quarter\u2013'The best way to see it', E. M. Forster said, 'is to wander aimlessly about'\u2013when I happened to catch the eye of a wrinkled cabby with a towel wrapped round his head, high behind his poor Rosinante on the seat of his gharry. On the impulse of the moment I winked: and instantly there crossed his face an expression of indescribable knowingness and complicity, half comic, half conspiratorial\u2013as though between us, he, the city and I, we had plumbed the depths of human and historical experience, and were still coming up for more.\n\n## The touch of a hand at home\n\nThe baby, we knew, was very near death.\n\nWe lay sleepless in our room overlooking the garden,\n\nand a great moon shone.\n\nTowards midnight a nightingale began to sing.\n\nAll night long it trilled and soared in the moonlight,\n\ninfinitely sad, infinitely beautiful.\n\nWe lay there through it all,\n\neach knowing what the other was thinking,\n\nand the bird sang on, part elegy, part comfort, part\n\nfarewell, until the moon failed\n\nand we fell hand in hand into sleep.\n\nIn the morning the child had gone.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n## SHAZEA QURAISHI\n\n### THE ART OF SCRATCHING\n\nTaking inspiration from sources including historical and medical texts, curator's notes and the Complete Kama Sutra, Shazea Quraishi's poems explore love and loss through a range of voices: an Iraqi mother holds her fragile son; under the guise of ardour, a courtesan searches a client for signs of the woman she loves; a wife is unsettled by her husband's new family...\n\n_The Art of Scratching_ is her first book-length collection, and includes _The Courtesans Reply_ , a sequence written in response to the _Caturbhani_ , four plays written around 300 BC on the life of courtesans in India.\n\n'There is an intriguing collision between the archaeological and the lyrical in Shazea Quraishi's series of poems, _The Courtesans Reply_... The props and rituals bestow on these poems an exotic otherness but the emotions they explore are timeless' \u2013 Stephen Knight, _Ten_.\n\n'Shazea Quraishi is one of a number of younger black and Asian women poets currently gaining ground in UK poetry. In sensual, clear, perfectly measured tones, her poems meet the male gaze with a female voice' \u2013 Katy Evans-Bush, _Poetry International Web_.\n\n'Shazea Quraishi, in _The Courtesans Reply_ , sensitively reconstructs an unfamiliar and vanished culture. Working from historical and literary sources, Quraishi never allows her research to speak louder than the human voices of her characters, a community of courtesans in Ancient India. Their individual feelings and desires emerge through lines which are simultaneously spare and sensuous' \u2013 Richard O'Brien, _Poetry London_.\n\nCover photograph: Jo Tyler\nShazea Quraishi\n\n# THE ART OF  \nSCRATCHING\n\n# ACKNOWLEDGEMENTS\n\nThanks are due to the following publications in which versions of some of these poems appeared: _The Delight-Tree: an anthology of Contemporary International Poetry_ (The United Nations SRC Society of Writers, 2015), _The Financial Times, I am twenty people_ (Enitharmon, 2007), _Images of Women_ (Arrowhead\/Second Light, 2006), _Ink Sweat & Tears, Magma, Modern Poetry in Translation, Ploughshares, PN Review, Poetry International web, Poetry Review, The Rialto, Sentence: a journal of prose poetics, Smiths Knoll,_ and _Ten: new poets from Spread the Word_ (Bloodaxe Books, 2010).\n\n_The Courtesans Reply_ was first published as a pamphlet by flipped eye in 2012\n\nI am indebted to Manomohan Ghosh for his translation of the Caturbh\u0101n\u012b, 'Glimpses of Sexual Life in Nanda-Maurya India', which inspired _The Courtesans Reply_. I am also grateful to Alain Dani\u00e9lou's translation of 'The Complete K\u0101ma S\u016btra', an invaluable source of reference and material.\n\n# CONTENTS\n\n  1. Title Page\n  2. Acknowledgements\n  3.   4. You May Have Heard of Me\n  5. Skyros\n  6. Steps\n  7. Darker than the Pansies\n  8. A Portrait\n  9. Coke and Other Lies\n  10. Mwanza, Malawi\n  11. Story of a Small Block of Flats in Elephant and Castle\n  12. Winter\n  13. The Drowned Sailor\n  14. Singularly Calm, Rather Wise\n  15. True story\n  16. The Wooden Family\n  17. All This Time\n  18. Cape Town\n  19. Fallujah, Basrah\n  20. My Mother's Embroidered Apron\n  21. Gold\n  22. THE COURTESANS REPLY\n  23. Tambulasena\n  24. The Sixty-four Arts\n  25. Vanarajika\n  26. Ramadasi\n  27. Sukumarika\n  28. The Days of Chandragupta Maurya\n  29. Priyangusena\n  30. Ratisena\n  31. Messenger\n  32. Madhavesana\n  33. Pradymnadasi\n  34. Sondasi\n  35. Devadatta\n  36. Carandasi\n  37. Anangadatta\n  38. Epilogue\n  39.   40. The Years\n  41. I Want to Tell You\n  42. At Six Months\n  43. Garden, Night\n  44. The Mummy of Hor\n  45. The Beauty of the Swimming Teacher\n  46. Wild Fennel\n  47. Sweetie Girl\n  48. Turieno\n  49. Roses\n  50. Still Light\n  51. Cousin\n  52. Goodbye, My Loveds\n  53. Notes on Found Poems\n  54.   55. _Additional Notes_\n  56. About the Author\n  57. Copyright\n\n# You May Have Heard of Me\n\nMy father was a bear.\n\nHe carried me through forest, sky\n\nand over frozen sea. At night\n\nI lay along his back\n\nwrapped in fur and heat\n\nand while I slept, he ran,\n\nnever stopping to rest, never\n\nletting me fall.\n\nHe showed me how to be careful as stone\n\nsharp as thorn and quick\n\nas weather. When he hunted alone\n\nhe'd leave me somewhere safe \u2013 high up a tree\n\nor deep within a cave.\n\nAnd then a day went on...\n\nHe didn't come.\n\nI looked and looked for him.\n\nThe seasons changed and changed again.\n\nSleep became my friend. It even brought my father back.\n\nThe dark was like his fur,\n\nthe sea's breathing echoed his breathing.\n\nI left home behind, an empty skin.\n\nAlone, I walked taller, balanced better.\n\nSo I came to the gates of this city\n\n\u2013 tall, black gates with teeth.\n\nHere you find me, keeping my mouth small,\n\nhiding pointed teeth and telling stories,\n\nconcealing their truth as I conceal\n\nthe thick black fur on my back.\n\n# Skyros\n\nI live near the place with the honey.\n\nIt's always spring here. I sing\n\nand cook, bare and brown\n\nor in my purple dress.\n\nIn the day there are birds\n\nwith so many different songs, I can't choose\n\nand at night, the brass shimmering of goat bells\n\nleads me into sleep.\n\nMaybe tomorrow I'll go\n\ndown the road with the purple rock and fig tree at dusk\n\nto find the beekeeper's garden, dripping\n\nwith milk and humming.\n\n# Steps\n\nWhere I come from\n\nwe don't hang portraits of our beloved dead\n\nto comfort us in empty rooms,\n\nwe hang their shoes, wrapped in gauze\n\nsewn with surgical thread.\n\nIn my hall I've hung\n\nmy father's shoes, their soft leather\n\nshaped by his strong\n\nbrown feet shaped by his journey.\n\nI close my eyes and see my father\n\nwalking towards me.\n\nAs the day closes, I press my ear\n\nto the air around me\n\nlistening for his footsteps,\n\nhis key in the door.\n\n# Darker than the Pansies\n\nDarker than the pansies at the cemetery gate-lodge,\n\nbottles of frosted glass filled with thick, pungent\n\nliquids (lavender, pine sap, geranium, grass)\n\nlined the top of the lower window sash, where they caught\n\nthe low light coming in \u2013 caught it and spilled it\n\non the cold, red stone of the kitchen floor,\n\nso there appeared to be pools of liquid around her feet\n\nas she stood with her hands in hot, soapy water,\n\nthinking of the time she jumped into the pond in Hawley's field\n\nand how the cold, green water filled her nose and ears\n\nand closed over her head.\n\n# A Portrait\n\nThe husband makes a house:\n\nthe walls don't meet, there are gaps\n\nin the floor but it houses them \u2013\n\nhusband, wife, daughter, son.\n\nIt is a house of skin.\n\n*\n\nAt night, while the family sleeps\n\nthe husband carries the house on his back.\n\nSometimes the boy wakes on the floor. The girl dreams she is flying.\n\nThe wife sleeps soundly but weeps\n\non waking to disarray.\n\nOne morning before dawn\n\nas he sets the house down\n\nthe husband feels a tearing in his shoulder.\n\n*\n\nHusband and wife stand in a field\n\n\u2013 light pours on the husband, pours out\n\nfrom his skin, his golden hair.\n\nThe wife is darker \u2013 the artist has painted lines on her face\n\nthat won't appear for many years.\n\n# Coke and Other Lies\n\nWhen I look in the mirror I see the wrong face. My sister's hair shines like a summer lake with swans, and when she speaks, it's like those gusts of air you get in the spring, smelling of green. Since he left, Mom's lips have no colour and she reminds me of that albino kid who packs groceries at Doug's Mini Mart. She cries on the foldout couch. I stroke her hair. There, there.\n\nMy sister wraps her hair around Coke cans when it's wet, so when she takes it out, it looks like in the movies. Mom says she's wasting her time.\n\nYou know the man who sweeps up after the show? He said my eyes were the green of the pond at the end of his garden. He said, Come, see.\n\n# Mwanza, Malawi\n\nI am Edith, I am eight.\n\nI have two brothers, Jomo\n\nand David. When we are hungry\n\nmy mother holds us close\n\nwith her body and her eyes.\n\nSome nights I dream the same dream\n\n\u2013 my mother covered in flies.\n\nI jump out of the dream\n\nleaving my mother alone\n\nin the dark and the heat.\n\nThis happens again, again,\n\nuntil dream-me holds her eyes open\n\nputs one foot in front of the other\n\n\u2013 goes like this to my mother\n\nbecause of how much she hates flies \u2013\n\nand reaching her sees\n\na fine, black shawl covers her\n\nwith shiny, black beads\n\nrunning over her like rain...\n\nand my mother is sleeping\n\nin the cool of its shade.\n\n# Story of a Small Block of Flats in Elephant and Castle\n\n# Winter\n\nHe is an excellent swimmer,\n\njumping in the water like a dog, or sliding in backwards.\n\nHe keeps his eyes open and holds\n\nhis breath, staying under for 2 minutes at a time.\n\nWhen he emerges, he shakes\n\nthe water from himself before it freezes.\n\nHis skin is dark under his white fur\n\nwhich stores heat.\n\nHe is solitary, but sometimes\n\nin February, he joins others to feed.\n\nHe is strong, patient and cunning.\n\nHe teaches boys how to be men;\n\nthat to be a man is to be a successful hunter.\n\nHe waits until they come close, or stalks them\n\non ice, spending much of the long winter out of sight.\n\nHe goes for the young ones\n\nbecause they're small and slow.\n\n# The Drowned Sailor\n\n#### I\n\nIn Norse the name means corpse whale\n\nbecause its mottled, greyish\n\nappearance suggests a drowned sailor.\n\nOften feeding on fish\n\nat depths of 1500 metres\n\nunder dense pack ice,\n\nnarwhal is found in Arctic waters.\n\nThe first year of a male's life,\n\nhis left tooth grows out, twisting\n\ninto a helix. This tusk,\n\nten feet in length, is used for jousting\n\nor channelling sonar pulses.\n\nLittle is known about this whale\n\nbut there are stories.\n\n#### II\n\nReturning each year to the bay\n\nwhere he drowned within sight\n\nof his house, he saw his wife pale,\n\nmarry again and brighten.\n\nLater, lonelier still, he saw\n\ntwo boys who weren't his\n\n(such fierce, sweet explosions) grow\n\ninto men, tender as trees.\n\nAgain and again he was drawn,\n\nagain turned away.\n\nHe saw her black hair that fanned\n\non the pillow, dull and grey.\n\nOne moonlit night towards the end\n\nof her life, in a dream\n\nhe went to her and felt her hand\n\non his neck...such warmth.\n\n# Singularly Calm, Rather Wise\n\nThey line up \u2013 good, clean children \u2013\n\nfrom tallest to smallest. The cupboard can't confine\n\ntheir implacable gazes \u2013 they question your\n\nmoving, breathing presence. Coloured lights\n\nfrom the prism in the window veil them in\n\nforget-me-not blue, green-gold, jellied\n\ncherry. They listen. They believe rules\n\nare there for good. The smallest two resemble\n\nthe girls next door. Silence presses \u2013 a silence of graves \u2013\n\nand you want to leave, you want them to be\n\nlike other children, noisy, grass-stained \u2013 not this metaphysical\n\nexistence, this sad patience, this endurance of mules.\n\n# True story\n\nThe woman who was not me sat down in the chair indicated by the doctor. She started to speak and proceeded to flow into the crevices and folds of the rough brown fabric of the chair, stopping just short of ending up at the good doctor's feet.\n\nThe doctor, whose name was Anne, put out her hand to place it on the woman there, and said how she liked to think of all the mothers alone at home all over the world, as hundreds of thousands of little lights shining in the dark.\n\n# The Wooden Family\n\nAlone at home my husband\n\ncalls me, tells me he's painting\n\na wooden family. He says\n\nhe misses us but I can\n\ntell he's captivated by\n\nthe wooden wife: she's mild as\n\na mail-order bride; all day\n\nshe gazes at her wooden\n\nhusband, all night she cannot\n\nclose her adoring brown eyes.\n\nBut the wooden children scare\n\nhim. The girl's too quiet \u2013 she\n\nhas nothing to ask or say.\n\nThe baby doesn't cry or\n\nbabble. He looks and he looks.\n\nI kiss my husband at the\n\nairport, and we are shy as\n\nstrangers, the children like goats\n\nbutting our knees. I say he's\n\npale \u2013 he says it's too much sleep.\n\nWhen we get home, our red house\n\nhas shrunk and filled with sawdust.\n\nEager to meet the family\n\nI'm appalled by the wooden\n\nhusband. His hair is ginger,\n\nhe's wearing a shirt I hate\n\nand he's baring his teeth \u2013 he\n\nlooks insane. His wooden wife\n\nis lovely if you ignore\n\nthe way her head narrows at\n\nthe top. She has my eyebrows,\n\nher red lips shine and she is\n\nwearing a green dress that was\n\nmine. I hold the girl in my\n\nhand. She's serene and pink, her\n\nhair smooth against her head. She's\n\nso still, and she wants nothing.\n\nThe baby is gorgeous with\n\nhis enormous eyes and his\n\nsucculent mouth, but he's too,\n\ntoo tiny \u2013 he will be lost\n\nin this world of sudden holes\n\nin the floor and so many\n\nsharp edges to damage him.\n\nI know what this family means\n\nto my husband, but all I\n\ncan say is, Why's her head so\n\npointy, and what's wrong with him?\n\nIsn't he happy with his\n\nwooden life and his family?\n\n# All This Time\n\nI've been living the wrong life.\n\nI stepped out to bring in the milk\n\n15 years ago\n\nand now I see\n\nI'm in the wrong house.\n\nWho is this man\n\nwith the plaster dust on his hands?\n\nWhat are these children doing in the kitchen?\n\nThe boy is skinny, smells\n\nof goat, mixes Cheerios\n\nand raisins in his cereal bowl. The girl\n\nreminds me of a jug\n\nmy mother had,\n\nthe china so fine, the milk shone\n\na blueish light through it.\n\nWhere are my bright\n\nskirts, my heavy silver rings,\n\nthe red in my hair?\n\n# Cape Town\n\nMy brother Luc at sixteen\n\nis almost a man\n\nand bigger than daddy\n\nbut when he cries in sleep\n\nhe sounds just like\n\nthe skinny boy he used to be.\n\nHe has this dream\n\nof walking on an empty road at night\n\nand looking back\n\nhe sees our mother run out of a house\n\nher hair crazy.\n\nHe can't move\n\ncan't speak\n\ncan only watch\n\nas the dark swallows her\n\nhead-first\n\ndown to the white socks\n\non her shoeless feet.\n\nI hold him then, say\n\n_Hush, Luc,_\n\n_it's just a dream._\n\n# Fallujah, Basrah\n\n_A poem in four voices_\n\n##### _Rahim_\n\nOh my son\n\nLet my love\n\ncushion the weight of your head.\n\nLet me take your hurt\n\nI will hold it\n\nwhile you rest\n\nI will fold it into my body.\n\nSleep, my angel,\n\nmy flower, my brave, sweet boy.\n\n[ _Child with extreme hydrocephalus \u2013 deformity of face, body and ear \u2013 and defects of cerebral nerves._ ]\n\n##### _Sabir_\n\nI dressed him in a long white shirt\n\nwith a blue bird embroidered over the heart\n\nand placed tiny white mittens on his hands.\n\nBorn with thick black hair\n\nlike his father, he was almost\n\nso beautiful, almost perfect\n\nas we had imagined him.\n\n[ _Born without eyes_ ]\n\n##### _Farrah_\n\nWhere is my baby girl,\n\nthe one I dreamed?\n\nI long for sleep\n\nto return her.\n\n[ _Extreme hydrocephalus. The line running down the right side of the head would appear to show that potentially two heads were forming._ ]\n\n##### _Anah_\n\n[ _It isn't clear what has happened to this child._ ]\n\n# My Mother's Embroidered Apron\n\nI am lost in my mother's apron \u2013\n\ngreen parrots drip from the trees,\n\na peacock brushes past me\n\npulling its clockwork tail of children's dreams.\n\nI breathe in the heat of cinnamon,\n\nthe fug of yeast. My mother's voice\n\nfills me like smoke and her stories\n\nlift me \u2013 I rise like a yellow balloon,\n\nmy feet, white ribbons trailing in the long, wet grass.\n\n# Gold\n\nI crossed the land with my small\n\ngold baby. I had only my skin\n\nwhich hung from me in folds, to wrap him in,\n\nonly my hands to cover his miraculous feet.\n\nWe came to a forest where the trees had faces;\n\nthere was a loud ticking and the smell\n\nof waiting \u2013 dry, leafy.\n\nI thought of his soft heart, the blood like finest embroidery\n\nrunning through his body, and I was heavy\n\nwith the knowledge of animals in the forest \u2013\n\nthe claws and eyes, the beaky hunger.\n\nMy baby stirred and the trees leaned closer.\n\nI walked till I came to a wall of grass\n\nreaching over my head. Then I heard\n\na shushing that calmed me, though\n\nit could have been wings beating or knives\n\nslicing the air. The grass\n\nparted, like a sea parting\n\nand my baby's breath on my skin\n\nwas the wind in our sails.\n\n# THE COURTESANS REPLY\n\n_How wonderful is the supreme beauty of Kusumapura!_\n\n_Here between the rows of houses the streets are well-watered,_\n\n_well-cleaned, and are scattered over with flower-offerings great and small..._\n\n_Daughters of courtezans, the beauty of whose lotus-like faces is being drunk_\n\n_by the eyes of all people, are gracefully walking up and down,_\n\n_it seems, to bestow their favour on the thoroughfare._\n\nFROM _The Ubhay \u0101bhis\u0101rika _\n\ntranslated by Manomohan Ghosh\n\n# Tambulasena\n\nIn the beginning\n\nmy whole body was covered with skin\n\nhard as rock. Then he came\n\nand his mouth\n\nrunning over me was a river, cool and quick\n\nwith small silver fish.\n\nNight after night\n\nhe shaped me,\n\nsmoothed me down\n\nto velvet\n\nbones.\n\n*\n\nNow I bathe while he watches,\n\neyes fireflies\n\non my skin.\n\nI bend over,\n\nmy hair between us\n\na curtain of water.\n\nI let him towel me dry,\n\nhis strokes soft...then brisk,\n\na cloth shining a lamp.\n\nWater drips down\n\nmy back. He grasps my hair\n\nand climbs.\n\n# The Sixty-four Arts\n\nOf pleasant disposition,\n\nbeautiful and otherwise attractive,\n\nmaster of sixty-four arts including\n\nmusic, dancing, acting, singing,\n\nthe composition of poetry,\n\nflower-arrangement and garland-making,\n\nthe preparation of perfumes\n\nand cosmetics, dress-making, embroidery, conjuring,\n\nsleight of hand,\n\nlogic, cooking, sorcery, fencing\n\nwith sword and staff,\n\narchery, gymnastics, carpentry,\n\nchemistry, architecture\n\nand mineralogy,\n\nthe composition of riddles, tongue-twisters\n\nand other puzzles, gardening, writing in cipher,\n\nlanguages, making artificial flowers\n\nand clay modelling,\n\ntraining fighting cocks, partridges\n\nand rams, teaching parrots and mynah birds to talk...\n\nSuch a courtesan will be honoured by the King, praised\n\nby the learned, and all will seek her favours\n\nand treat her with consideration.\n\n# Vanarajika\n\n_speaks of the eight varieties of nail marks_\n\nUsing the nail on my middle finger\n\nI mark his neck\n\nwith a half moon, on the place\n\nI like best\n\nto kiss.\n\nA sign of my devotion.\n\nOn his lower belly, I leave\n\na circle.\n\nOften I trace a short, straight line\n\non his chest\n\nhis belly\n\nhis back.\n\nThe dash.\n\nLightly, he touches my cheek\n\ngiving me gooseflesh\n\nthen marks me with his thumb\n\ndeepening the scratch with the other fingers.\n\nA knife stroke.\n\nOn my buttocks\n\na mark resembling a lotus leaf.\n\nThe peacock's claw is for me alone.\n\nThe hare's jump even more.\n\nThe tiger's claw\n\nhe traces under my breast\n\nbinds me to him.\n\n# Ramadasi\n\nReturn\n\nto me, beloved\n\nand take me on your lap.\n\nUndo my braid\n\nstiff\n\nas buffalo horn\n\nand draw your\n\nfingers\n\nthrough my hair.\n\nUntie my belt, open\n\nthe silk cloth\n\ncovering my waist,\n\nlet my oiled limbs, my\n\nperfumed skin\n\nenvelop you\n\nas the rose\n\nswallows\n\nthe bee.\n\n# Sukumarika\n\n_to Ramasena_\n\nMy dearest, my life,\n\nmoon to my night,\n\nremember our happiness?\n\nRecall, if you can,\n\nthe equal kiss, _Sama_\n\nand the pressed kiss, _Pidita_.\n\n_Aschita_ , the devouring kiss\n\nand _Mridu_ , the delicate kiss.\n\nAlso, the inflamer\n\nthe kiss of encouragement\n\nthe awakening kiss\n\nthe vagabond, the joyful\n\nkiss, the vibrant one\n\nthe bowed kiss, the twisted kiss and\n\nthe satisfied kiss.\n\nHave you forgotten\n\nthe taste of my mouth\n\nsweetened with betel?\n\nMy garments, outer and inner,\n\nwhite as milk.\n\nThe sound of my bangles\n\nduring love, their silence\n\nin sleep.\n\nRemember my lips\n\nnibbling\n\npinching\n\nkissing\n\nbrowsing\n\nsucking the mango\n\ndevouring.\n\nRemember\n\nthe way I make you feel\n\n\u2013 like twenty men \u2013\n\nand in your hands\n\nmy painted feet.\n\n# The Days of Chandragupta Maurya\n\nwere split into sixteen hours\n\nof ninety minutes each.\n\nIn the first, he arose\n\nand prepared himself by meditation;\n\nin the second, he studied\n\nthe reports of his agents\n\nand issued secret instructions;\n\nhe met with his councillors in the third hour\n\nand in the fourth, attended to state\n\nfinances and national defence;\n\nin the fifth, he heard the petitions\n\nand suits of his subjects.\n\nIn the sixth hour, he bathed,\n\ndined and read religious literature.\n\nIn the seventh hour, while he made official appointments,\n\nhe received taxes and tribute.\n\nIn the eighth, he met his council again\n\nand heard the reports of his spies and courtesans.\n\nThe ninth hour was devoted to relaxation\n\nand prayer,\n\nthe tenth and eleventh given to military matters\n\nand the twelfth to secret reports.\n\nIn the thirteenth hour, the king indulged\n\nin an evening bath and a meal\n\nand for the next three hours he slept\n\nbut never in the same bed twice.\n\n# Priyangusena\n\n_speaks of the Keeper of the King's Zoo_\n\nHe is not like other men.\n\nHe prefers me unperfumed,\n\nlikes to watch me\n\nremove hair flowers,\n\nundo the _rasana_\n\naround my waist.\n\nIn the morning I am porcupine,\n\nat night, _Dhole_ ,\n\nfour-horned antelope.\n\nHe tells me secrets\n\nof the _Nilgai_ , its fondness\n\nfor almonds,\n\nhow the _Chinkara_ leaps\n\nthe palace walls\n\nand back again.\n\n# Ratisena\n\n_to Chandragupta_\n\nWhile you sleep, I take\n\nyour white shirt\n\nfrom the unpainted chair,\n\nsmoothe it with my hands\n\nthe way I smooth\n\nthe tiredness from your body\n\npressing my self against you.\n\nShh...\n\nLet me take your worries,\n\nyour secrets \u2013 those sharp\n\nsmall stones you carry\n\nwith you always.\n\nI know you have women half my age\n\n\u2013 I see them in the street, swaying\n\nlike long grass, their saris\n\nconcealing slim legs\n\nthat circled your waist.\n\nAre you my King\n\nor the boy I met at the well\n\nso many summers past?\n\nI watch you sleeping;\n\nmy small bed cradles you,\n\nmy only child\n\nmy only man.\n\n# Messenger\n\nMalatika remembers you.\n\nHer passion, hundred-petalled,\n\ngrows and grows, eclipsing all flowers\n\nin her mother's garden: lotus, marigold,\n\n_raat-ki-rani_ , rose.\n\nEach day she strokes her skin\n\nwith perfumed oil,\n\neach day adorns her self\n\nwith pleasing things.\n\nAnd as she walks, thinking of you,\n\nher ankle-bracelets sound,\n\na hundred tiny, silver bells\n\ntrembling.\n\n# Madhavesana\n\nOnce more this\n\npressing of bodies, his desire\n\nbeating against me as the eagle's wings\n\nagainst the air that lifts him up, up.\n\nMy body has learned to soften\n\nand bend, but my heart\n\nlike a child who will not listen, clings\n\nto a soft, worn thing.\n\nAfter I have washed the sweat\n\nand trails of saliva from my skin,\n\nI stand at the open window,\n\nlet the breeze dry my face.\n\n# Pradymnadasi\n\n_on biting_\n\nWhen he gave me the discreet bite on my lower lip\n\nI sighed with disappointment\n\nknowing his mark would fade.\n\nThe coral jewel bite he bestowed on my left breast\n\nand then the right. Around my throat\n\nhe placed a necklace of gems.\n\nI will wear no ornaments today other than kiss\n\nmarks on my ears, filigree\n\nbites on my hot, hot cheeks.\n\nBefore he left, he gave me the bite I like best:\n\nthe nibbling of the wild boar.\n\nAnd so, he knew I would wait.\n\n# Sondasi\n\nI smile slow as honey\n\noffer him\n\nmy pollen-dusted breasts.\n\nI press my nose to his skin\n\nsmell Varunika on him.\n\n_Wait_\n\nthe word a caress\n\nI undress him\n\n\u2013 the first time I have done this.\n\n* *\n\nThe next day\n\nshe is not with him.\n\nI seat him on the low, green chair\n\nmove in his lap\n\nput my mouth to his ear:\n\n_Tell me what you do with her_\n\nHe answers\n\nand I show him\n\nthe flame lit inside me.\n\n* *\n\nVarunika,\n\nqueen of forests.\n\nHer teeth\n\nmarks on his lips,\n\nher nail marks on his back,\n\nher love note to me.\n\n* *\n\nA dark pink flower falls\n\nfrom her hair as she passes\n\n\u2013 I hold it carefully in my hand: five petals,\n\none scattered with small, dark markings.\n\nOpening it, I stroke the velvet\n\ninside \u2013 eleven stamens raise\n\ntheir pollen-tipped nubs\n\nto the tip of my tongue.\n\n# Devadatta\n\nI'm summoned to the jasmine terrace\n\nwhere he waits\n\nreclining on the large, low bed\n\ndraped in blues and reds and oranges.\n\nHe's with Sondasi's servant girl\n\n\u2013 her gaze is lowered, his\n\nrests on her breasts, where a blush blooms\n\nabove her open blouse.\n\nHer waist is a handspan, her hips\n\nhigh and wide.\n\nDuring the love act, he moves my legs\n\nto one side\n\nso she will see as he enters me.\n\nHe doesn't look at me\n\nbut over my shoulder, watches her\n\nsmall, heart-shaped face.\n\n# Carandasi\n\nTell me I am necessary for you like sleep,\n\nnot like opium which carries forgetting,\n\nor pleasant as a breeze\n\nscented with jasmine.\n\nTell me what you see\n\nbehind my art, my bright cloth.\n\nLook into my face and show me.\n\nTell me what you read in books\n\nand hear in coffee houses,\n\nat wedding parties. Teach me.\n\nWhen our tired, gladdened bodies\n\ndrift onto the bed,\n\nkiss me like a husband\n\nand spread over me an endless blue wing...\n\n# Anangadatta\n\n_Dreams I have_\n\nThe peaceful routine of household chores:\n\nsweeping the floors of the house,\n\nsprinkling water on the yellow earth outside the door.\n\nCooking my husband's food,\n\nanticipating his pleasure.\n\nFeeding sweet, milky pudding to my child.\n\nSewing a button on my husband's shirt.\n\nTo spend the whole night dreaming,\n\nmy child pressed against my back,\n\nmy husband's breath in my hair.\n\n# Epilogue:\n\n_in which we explain a few things_\n\n_How can scratching and biting, even if they are painful, create pleasure?_\n\nJust as a whip\n\nwhen used by the charioteer,\n\nmakes horses mindful of speed,\n\nso the use of nails and teeth\n\nduring intercourse\n\nengross the heart in the pleasure of touch.\n\n_Tell more of the art of scratching_\n\nWhen a man sees, even from afar,\n\nnail marks on a girl's breasts,\n\nhe feels interest and desire for her\n\neven without knowing her.\n\nAnd it often happens,\n\nwhen a woman sees nail marks\n\non the various parts of a man's body,\n\nher spirit awakens\n\nand takes her to him.\n\n_How is a courtesan to choose?_\n\nHere is a list of men to be avoided:\n\nThose with tuberculosis,\n\nwith worms in their excrement,\n\nwith bad breath,\n\nin love with their wife,\n\ncoarse in word,\n\nbrutal,\n\ncruel,\n\nabandoned by their parents,\n\ninsensitive to praise and insult,\n\nimmodest,\n\nfrequenting enemies for hope of gain...\n\nAlso one who is a thief,\n\nan idiot\n\nor one who practises magic.\n\n# The Years\n\n##### **10**\n\nClifton Road, Karachi. I pray to God to change me to a boy so I can climb trees and get my clothes dirty.\n\nMy father tells us we're moving to Canada. I can't wait \u2013 it's going to be like Disney movies: houses with porches and grass all around, sidewalks to cycle on, tree-lined streets.\n\n##### **11**\n\nFleetwood Crescent, Bramalea. All the streets around here begin with F. There are hundreds of houses and not many trees, but we have bunk beds, chores and wall-to-wall carpeting.\n\nI have a new friend called Terry; her hands are pale as her white shirt. She has three sisters \u2013 Rita, Shirley, Patsy \u2013 and an endless supply of chips with dip. Her brother Michael says Hi, then stays in his room smoking.\n\nTerry's parents are kind and they're like people on TV.\n\n##### **12**\n\nMy favourite chores are vacuuming and making my bed. There is so much TV, my brothers and I don't talk about our ugly new names.\n\n_You've got a weird accent.  \nDid you live in a hut made of straw?_\n\nMy father gets a new job with Mattel and brings home Slime.\n\n##### **13**\n\nIn the winter, on the way to school we take off our snowsuits, stumbling into drifts, our stiff jeans thawing all morning.\n\nOne day, three boys surround Terry, their words like spit on her face. They don't see me where I've bent down to get a stone out of my shoe.\n\n##### **14**\n\nTim Newman. I follow him unseen, my heart thudding in my flat chest. He is like the songs, his hair is feathered. I buy two copies of the school yearbook to cut out his picture to keep.\n\n_If you didn't hang around so much with Rajini Malhotra, most people wouldn't know you're a Paki._\n\nI know how he'd smell if I got close. Clean, minty.\n\n##### **15**\n\nWe move to the country \u2013 my father's dream. At Erin High everyone smokes weed, my new friends call me Brain and cheat off me. At home I cry to get 98% on my spelling test.\n\nEvery morning the boys line the hall yelling out numbers or barking at the girls. A boy in art class shouts _give face!_ and laughs as I wash out my tray. Next year I'll be sweet sixteen.\n\n##### **16**\n\nEvery Saturday in the summer my father and brothers go fishing. They leave before light for Six Mile Lake and come back smelling of goat. My mother keeps finding fish heads in the freezer.\n\nWalking one morning to babysit, a motorcycle passes me on the dirt road. I hear it slow and then a man is in front of me unzipping his jeans. I run and then I cry.\n\n##### **17**\n\n_I'm desperate for modern languages_ , I tell the guidance counsellor. A yellow bus takes me to my new school in the city. The bus driver's name is Floyd; his unwashed hair and beard reach past his chest. The kids at the back of the bus smoke joints and burn the seats. Sometimes Floyd slows as we pass the chicken factory with the windows down. He can't say my name so he calls me Sophie, for Sophia Loren.\n\n##### **18**\n\nI read 'Are You There God It's Me Margaret', and 'Forever', tingling through English. At lunch with the cool Italian girls we discuss 'Do you swallow or do you spit?' I say I swallow because there's protein in it.\n\nThe stoners on the bus teach me to roll their joints. One afternoon Floyd drops off everyone except me and the girl who goes to a special school. Her straight black hair is always in her eyes. Floyd lets me drive the bus even though I'm scared I'm going to crash it. The road is icy and I keep confusing the brake with the gas. He keeps putting his hand on my bum.\n\nMy father says _Shaz, don't grow up so quickly_. I bury my head in his shoulder.\n\n##### **19**\n\nI start university. Boys notice me.\n\nI discover drinking and indie bands like Change of Heart and The Jellyfish Babies. At midnight I go dancing with my friends at the Bullring which opens when everything else closes. I dance and dance and I am happy.\n\nWeekends, my father takes me home and we talk in the car. He says my skirts are too short. I tell him I'm a communist.\n\n##### **20**\n\nMay. My father is dead.\n\nI'm not yet 20.\n\n# I Want to Tell You\n\nIt was a beautiful evening,\n\nthe light going\n\nthe colour of old gold,\n\ntrees leafing and blossoming,\n\nmy feet meeting the dirt road\n\nas shadows lengthened.\n\nI was thinking of the tree I planted last week\n\nbehind the house,\n\nof dinner warming in the oven\n\nand the trout\n\nmoving like memory\n\nthrough the still water of Six Mile Lake.\n\n# At Six Months\n\nMy daughter doesn't like people\n\nshe doesn't know, especially if they're men,\n\nespecially if they're dark.\n\nThis is embarrassing\n\nconsidering my family,\n\nthe colour of our skin.\n\nMy brother comes to see her.\n\nShe takes one look at him and cries\n\nuntil I take her off to bed.\n\nIt must be colic, I say, thinking\n\nof his three boys, their kisses like bubbles\n\nbursting on my cheek,\n\ntheir little arms necklacing my neck\n\n\u2013 what I wanted for him.\n\nIt makes me sad\n\nmy daughter doesn't see\n\nmy brother is the gentlest man. He\n\nloves God, makes omelettes, silly jokes.\n\nLater, after he goes,\n\nI find his camera on the table\n\nits one eye closed.\n\n# Garden, Night\n\nI watch my father at the bookcase\n\nfingering spines like keys\n\nand guess at what he's taken down to read:\n\nWodehouse? Tennyson? Yeats?\n\nThe standard lamp throws shadows on his face\n\nand he looks young, though it's been twenty years\n\nand memories have taken his place.\n\nShouldn't he be more pale,\n\nhow ghosts appear in books?\n\nI can't move, can only look,\n\nthe glass between us deep as a lake.\n\nMoonlight. He turns.\n\nShadows fall like lace.\n\n# The Mummy of Hor\n\nIn this cave-like room, lamp-lit,\n\nthe Goddess Isis spreads her wings\n\nacross Hor's chest to protect him.\n\nBut that's not all:\n\nthe four sons of Horus guard his entrails\n\nand the human-headed God Imset guards the liver\n\nwhile Ha'py, with a baboon head, guards the lungs.\n\nDuamutef, who has a jackal's head, guards the stomach,\n\nQuebehsenuf, with a hawk's head, guards the intestines\n\nand other Gods watch over his body\n\nwhile sacred symbols protect his soul.\n\nHor's body, wrapped in layers of linen\n\nand bound with black pitch\n\nis here\n\nand you are gone.\n\nI think of you\n\non that country road\n\nwhen your heart stopped\n\nand your breath stopped...\n\nI think of you there alone.\n\n# The Beauty of the Swimming Teacher\n\nThe way he throws rings in the water\n\nfor the children to swim through;\n\nThe way he coaxes them under\n\npulls them along like ducks on a string;\n\nThe way the water shines around him\n\nthe children gleaming like seals;\n\nHow, at lesson end\n\nthey float like stars\n\nthey shine.\n\n# Wild Fennel\n\nOh my sweet girl,\n\nbody outgrowing girl-thoughts\n\nthat bloom and burst...\n\nAt five, your loneliness was hard to bear\n\nbut now you seek its comfort \u2013\n\nbehind the closed door, your thoughts\n\ndrift through the room, mauve smoke\n\ntendrilling through the gap.\n\nOnce you covered yourself in leaves\n\nyour face showing. Now you swim through cold\n\nmountain rivers, brown reed in the water,\n\nyour quietness everywhere\n\nlike the scent of wild fennel.\n\n# Sweetie Girl\n\nI love to watch my grandmother eat\n\n_tarte au citron_ , battenberg,\n\nlemon drizzle cake. Lost\n\nin the feel, the taste,\n\na low moan escapes her.\n\nLater, calling me\n\nby my mother's name, she worries\n\nthey are planning to put her in a home.\n\n_Don't go._\n\nHolding my hand at the door,\n\nshe cradles my cheek,\n\ncalls me sweetie girl.\n\n# Turieno\n\nShe watches her son sleep\n\nbrushing away flies which return\n\nto move again over his clothes\n\nhis perfect skin.\n\nShe draws the thin white sheet over his head.\n\nOutside there are mountains\n\nand the hermitage they walked up to\n\nthat morning. A bee buzzes\n\nat the curtain\n\ntwice the size of bees at home.\n\nThat night it stings her husband\n\nas he sleeps\n\n\u2013 the next morning they find it\n\ndead beside him\n\na small puncture between his eyes.\n\nEight days remain\n\nof churches and mountain roads\n\nwine and coffee\n\nair bright with sound.\n\nIf only life could be\n\nmade of such moments\n\nstrung together.\n\nLove balances on them.\n\nShe tries not to notice\n\nage showing in her hands.\n\n# Roses\n\n_(i.m. Annette)_\n\nYour hair curls and wisps\n\naround your face.\n\nWe hide weak smiles behind gifts of soup, ultra-balm tissues,\n\nOlympic badges, Turkish Delight, garden figs.\n\nI stroke wild rose lotion over your hands\n\n( _how small they are, how tender_ )\n\nand the smell of rose fills the room\n\ndrifting out to the nurses' station.\n\nWalking home through the park I remember\n\nyour daily walk around it, your pleasure.\n\nThe walled garden is closed now\n\nwhile they smooth the cobblestone paths.\n\nNo one will see this season change\n\nthrough its flowers.\n\nThe roses will be going now\n\nor already gone.\n\n# Still Light\n\nYou picture your mother as a tree\n\n\u2013 somehow that makes it easier \u2013\n\na silver birch, undressing\n\nunhurriedly, as though days were years,\n\nwhile a fine rain plays\n\nlike jazz in her hair. She drops\n\nher fine, white leaves\n\none by one. Her branches\n\nare almost bare now. See\n\nhow beautiful she is against the darkening sky.\n\n# Cousin\n\nYou carry them on your back,\n\nyour muffled parents and her\n\nsoft, small children\n\nas you carry your sister's body\n\nwrapped in its white shroud\n\nover the bright, stoney ground.\n\nNow the brown earth pillows\n\nher, holds her small body\n\nin its quiet lap,\n\nrocks her to sleep.\n\n# Goodbye, My Loveds\n\n# Notes on found poems\n\n#### I\n\nThe source material for a found poem can be\n\nany piece of writing whatsoever.\n\nAfter the initial act of selection\n\nmay follow excision,\n\ntrimming,\n\nre-arrangement\n\nor even, to a certain extent,\n\nre-writing.\n\nOr it may be simply\n\na question of selection.\n\nSo be it.\n\n#### II\n\nThe writer of found poems will require the following articles:\n\na. |  | A hammer will be found most generally useful.  \n---|---|---  \nb. |  | Stout and thin paper, and some of a soft kind for wrapping up specimens.  \nc. |  | String, sealing wax.\n\n#### III\n\nMarcel Duchamp never tired of saying:\n\n_The most important element_\n\n_in a picture_\n\n_is its frame._\n\n_in a sculpture,_\n\n_its plinth._\n\n# Additional Notes\n\n'Singularly Calm, Rather Wise' () uses words from Gwendolyn Brooks' poem 'The Children of the Poor' and employs Terrance Hayes' golden shovel form.\n\nIn 'Fallujah, Basrah' (), medical quotes come from Ross B. Mirkarimi (The Arms Control Research Centre, from report _The Environmental and Human Health Impacts of the Gulf Region with special reference to Iraq_ , from May 1992) commenting on photographs of extreme birth deformities experienced in Iraq and Afghanistan following bombing with DU ammunition. All names are fictional.\n\nIn _The Courtesans Reply_ , the first and last lines of 'Carandasi' () are quotes from the poem 'Two stranger birds in our feathers' by Mahmoud Darwish. 'Epilogue' () employs found text from _Glimpses of Sexual Life in Nanda-Maurya India_ , a translation of _The Caturbh \u0101n\u012b_ by Manomohan Ghosh (Manisha Granthalaya Private Ltd, Calcutta, 1975) and from _The Complete K \u0101ma S\u016btra_, translated by Alain Dani\u00e9lou (Park Street Press, 1994).\n\n'The Mummy of Hor' () uses text from the curator's notes for the Mummy of Tem Hor in Swansea Museum.\n\n'Goodbye, My Loveds' () was the painter Arshile Gorky's parting message written in chalk on a crate.\n\n'Notes on Found Poems' () uses text from the following sources: the introduction to Malcom Parr's pamphlet, _Found Poems_ (1972); a footnote from The Metropolitan Police Services' Investigation of Fakes and Forgeries exhibition at the V&A Museum; a quote from Germaine Greer in the _Guardian_ (30 March 2009).\n\n# About the Author\n\n**Shazea Quraishi** was born in Pakistan, emigrated to Canada aged ten, and lived in Madrid before moving to London where she works as a writer, teacher and translator. A selection of her work was included in Bernardine Evaristo and Daljit Nagra's Bloodaxe anthology _Ten: new poets from Spread the Word_ in 2010. Her first pamphlet, _The Courtesans Reply_ , was published by flipped eye in 2012, and is being adapted as a play. _The Art of Scratching_ (Bloodaxe Books, 2015) is her first book-length collection.\n\n# Copyright\n\nCopyright \u00a9 Shazea Quraishi 2015\n\nFirst published 2015 by  \nBloodaxe Books Ltd,  \nEastburn,  \nSouth Park,  \nHexham,  \nNorthumberland NE46 1BS\n\nThis ebook edition first published in 2015.\n\n**www.bloodaxebooks.com**  \nFor further information about Bloodaxe titles  \nplease visit our website or write to  \nthe above address for a catalogue.\n\nCover design: Neil Astley & Pamela Robertson-Pearce.\n\nThe right of Shazea Quraishi to be identified as author of this work has been asserted in accordance with Section 77 of the Copyright, Designs and Patents Act 1988.\n\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 1 78037 249 5 ebook\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"MINEOLA, N.Y. (AP) _ Hall of Fame jockey Angel Cordero is upset with a plea bargain that would give a six-month jail term to the driver who killed his wife in a hit-and-run accident last year.\nRobert Mikols pleaded guilty Tuesday to leaving the scene of the accident that killed Marjorie Clayton Cordero before police arrived. Nassau County Court Judge Ira Wexner indicated he will sentence Mikols to six months in jail and five years probation.\nSentencing is scheduled for Nov. 20.\nMikols, 63, a crane operator from Glen Cove, admitted that on the night of Jan. 22, 2001, his car struck and killed Cordero on Glen Cove Road in Greenvale. Cordero, 41, was struck, apparently from behind, as she walked home.\nMarjorie Cordero was a jockey from 1982 to 1985, winning 71 races from 911 mounts and earning more than $1 million in purses. She turned to training and won 42 races from 400 starters from 1985 to 1999.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Party wear lehenga choli form an essential part of every woman's wardrobe. We bring to you an amazing party wear collection with glamorous styles to help you stand out in a crowd. Party dresses are becoming the choice of every teenager today. Party wear lehenga choli are simply beautiful. Party wear dresses have amazing variety of styles, shades, designs and fabrics. Party wear lehenga choli decorated with heavy embroidery, resham work, stones work, mirror work, patch work and many more. Party wear lehenga choli very comfortable for wearing and washing as well.\nParty wear lehenga choli are highly stylized and bright, attractive to make the right style statement. Party wear lehenga choli can be bought online which makes a person shopping experience simpler. A very special range of latest party wear lehenga choli designs wear, party wear embroidery lehenga choli, a-line lehenga, floor length fully decorated party wear lehenga. Shop online party wear lehenga choli from nikvik.com get good quality of fabric, work and price as well. We as nikvik always deliver your orders before committed time. Nikvik is the largest seller of party wear lehenga choli all over the world. You can choose free ship product from our displayed items as well with the best deal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"May This New Year Illuminate Your Life And Bless You With Prosperity And Joy Today And Always.\nMay This New Year All Your Dreams Turn Into Reality And All Your Efforts Into Great Achievement.\nMay Your New Year Dawn Be\u2026.Peaceful And Bright. Full Of Hopes And Possibility ! Happy New Year SMS.\nVanish Everything That'S Bad, Welcome Everything That'S Good. Wish You A Very Happy New Year 2018.\nHere Wishing You A Happy Prosperous Fun Filled Joyful & Fortunate New Year Ahead. Happy New Year.\nI Wish You In This New Year To Remain Happy And Joyful And To Be Successful In All Phased Of Life.\nThe Object Of New Year Is Not That We Should Have A New Year. It Is That We Should Have A New Soul.\nMay You Have A Heart Of Courage, A Mind Of Will And May Get Whatever You Desire Always At Your Will.\nWishing You Beautiful Moments, Treasured Memories, And All The Blessing A Heart Can Know. Happy New Year 2018.\nAs The New Year Approaches Us With New Hopes, Here Is Wishing You And Your Family A Wonderful Year Ahead.\nMay The Every Day Of The New Year Glow With Good Cheer & Happiness For You & Your Family. Happy New Year.\nEvery New Year Wish I Ever Made Came True When I Met You. Thank You Sweetheart. Happy New Year With Love.\nCherish The Memories Of Yesterday, Celebrate Today & Wishing You. Enjoyable Moments Ahead. Happy New Year.\nHappy New Year. May This Christmas Be Bright And Cheerful And May The New Year Begin On A Prosperous Note.\nThinking Of You Specially And Hoping The New Year Will Hold Many Special Pleasures For You. Happy New Year.\nWishing You A New Year That'S Sparking With Fun, Bursting With Joy ^ Cracking With Laughter. Happy New Year.\nHave Happy New Year'S Day And Then When It Is Through Have A Lot Of Happy Days All Through The New Year, Too.\nSending You My Heartiest New Year Wishes Just To Let You Know You Are Special For Every Beginning In My Life.\nMay The Fortune And Renewed Creativity Help You To Make The Right Decision In Your Professional & Personal Life.\nYear'S End Is Neither And End Nor A Beginning But A Going On, With All The Wisdom That Experience Ca Instil In Us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jeremiah B. Thomson was born in Virginia, April 10, 1886, and moved to Evening Shade, Arkansas when a small boy. He passed away at Roseberg, Oregon in his 87th year.\nHis father was married twice. To his first marriage two sons were born \u2013 James and Malcolm; to the second marriage two children were born -Jeremiah and Narcissa. His brother Malcolm and his parents died in Arkansas. James married and with Jerry and Narcissa came to Oregon. Narcissa married James Vickers and resides at Winlock, Oregon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"7.56am haul myself out of bed, late again. Tomorrow I will put my phone on the other side of the room so I can't hit snooze.\n8.20am wrangle with some hair related angst. When exactly are you too old to do your hair in two plaits? It is a worry. There again, I have the stylistic tendencies of a 9 year old girl in most other respects \u2013 I probably shouldn't fight it.\n8.30am read a tweet on my way out of the door claiming that Sarah Waters redrafts things up to 34 times before sending to an agent or publisher. Nobody should live at that speed. Seriously, I'll be dead by the time I've done that many redrafts.\n6pm get home. Open post, drink some water, consider going for a jog.\n6.40pm head out for a jog. Just call me Haruki Murakami.\n8pm congratulate self on not dying. Open up my shiny new copy of Mslexia for some 'sit down and write' inspiration.\n8.15pm my brother comes round for a visit (and to collect a phone charger. Multitasking). We catch up over some minstrels.\n9.25pm brother heads home. I sit and finish drafting an application for a mentoring project. This involves 20 minutes of editing fiction! Result!\nEagle-eyed readers might have noticed that it's January now, so I suppose I'll have a go at answering that question from the perspective of four days hindsight.\nI still feel a little bit fraudulent.\nYesterday was the busiest I've ever had on the 12 books blog, which is lovely, so I hope people liked it enough to come back. If you are visiting for the first time and would like to know what on earth is going on, you can read a quick overview on IdeasTap or a slightly longer evaluation of the journey so far on Mslexia.\nMeanwhile here is a reading from book seven, which you may remember is aimed at kids and is about a grumpy wizard called Snooky Jim, a slightly batty cat lady called Amelia Trousers, and a young lad called Chris who is well brung up and usually eats his greens. I wrote this yesterday and have just done one take, so it's literally just me reading off a bit of paper. But it's pretty short, so that's a bonus\u2026 And anyway if you don't like it you can't be mean cause it's my birthday, so there.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The SP2 for AutoCAD MAP3D 2016 in now available for you to download from the link below or from the Autodesk Application Manager.\nIn a continuing effort to provide high-quality products, Autodesk has released the AutoCAD Map 3D 2016 Service Pack 2. The Readme contains the latest information regarding the installation and use of this update. It is strongly recommended that you read this entire document before you apply the update to your product. For your reference, you should save the Readme to your hard drive or print a copy.\nThat's right! Another release of InfraWorks 360!\nYesterday, we released a brand new version of InfraWorks 360. For all the existing customers, you just need to go into your Autodesk Application Manager and install the latest release.\nAnd here you can find the Webcast to take you through all the new exciting features of this latest release.\nFinally, If you missed one of the previous webcasts, you can view the recordings by going to the InfraWorks 360 Community site at http:\/\/autodesk.com\/InfraWorks360Community and clicking on the Videos tab.\nAlso, If you're looking for a quick productivity boost, visit this new page to learn some quick InfraWorks 360 tips and tricks InfraTips page.\nFor more info you can also visit InfraWorks 360 Forum, IdeaStation, or InfraWorks 360 Twitter and Facebook pages.\nEnjoy the brand new InfraWorks 360.\nCustomers using Ordnance Survey MasterMap Topography Layer can now access information on the heights of almost 20 million buildings across Great Britain with the alpha release of OS building height attributes (www.ordnancesurvey.co.uk) Released on 17 March, OS MasterMap Topography Layer \u2013 Building Height Attribute is a product enhancement to OS MasterMap Topography Layer, and available to licence holders at no additional cost.\nIf you are one of the lucky customers who can now access this very useful dataset then here you will find how to use it in AutoCAD Map3D\/C3D and Autodesk Infraworks environments.\nFirst of all the OS MasterMap Topography Layer \u2013 Building Height Attribute comes as .cvs file and it is not joint to the topography so in order to reuse it with Autodesk products it is necessary to manipulate first the .csv file then you need to join it to the topography.\nThe Building Height Attribute .csv contains the attribute OS_TOPO_TOID, which is the TOID from Topography Layer. This TOID is the attribute that is used to join the data to your Topography Layer data.\nTOIDs are stored in GML as a character string, with the prefix 'osgb', and can comprise of either a 13-digit number or a 16-digit number after 'osgb' (for example, osgb1000042992516).\nTo make the TOIDs joining process simpler it is reccomend you filter the Building poligons in the OS Mastermap Topography Layer and you save it as sdf file.\na. Command line: MAPIMPORT, Select .gz file and bring it into your Map, > Select \"Topographic Area\" only > Create object data and select Import polygons as polylines.\nYou now need to create a Data Source Name (DSN) .\na. Using ODBC Data Source administrator, create a user DSN connected to the .xlsx file.\nb. Add an ODBC connection;In AutoCAD Map3D or Civil 3D Select, Task Pane > Manage data content > Connect to Data > Add ODBC connection to the user DSN just created.\na. Right click the Topographic Layer and Select \"Create a join\"\nc. In order to Save the joint Map, Right click the topography Layer and select \"Export Layer to SDF\".\nb. Drag and drop a digital terrain of the area.\nc. Drag and drop the .sdf file created earlier and use the AbsH2 coloum for your building height.\naka as the BIM Coordinator released on our Labs site, we have had a good interest in making this available on our newer releases.\nSo for both 2013 and 2014 releases this tool is now available on www.autodesk.com\/subscription under the productivity tools for AutoCAD Civil 3D.\nThe tool exports an xml file from Civil 3D with coordinates and z level\/elevation, rotation and units.\nFrom within Revit you select the same two reference points on your building\/structure and it creates a 'Shared Coordinate System' for your location and can orientate the view to suit and most importantly export the building\/structure correctly to be used in AutoCAD platforms, Navisworks etc.\nTo see this in action I have recorded a new video that covers from the outline planning of a site with InfraWorks through to the project aggregated model in Navisworks.\nAs usual I throw a big supermarket on a public park in the UK close to where I live, of course its all made up!\nThe number of planes is for having different setup that you can come back to when you want.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Quick, informal Halloween survey. Which is your favorite costume: this ingenious amputee idea, this brilliant \"Saved By The Bell\"-inspired couple or my friend Rachel's political effort? Ahem, VOTE FOR MY FRIEND.\nFinally, my dears, rest assured, your pumpkin is inferior to this one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HHNFreak2006 had the most liked content!\nI am into Disney, Universal, and HHN related material (music and soundtracks). I have made HHN CD's since 2004 and have cds all the way up to Current! If u wanna know or want me to share any materials I have, I will do so!\n(WARNING): None of my HHN CD's contain Unreleased Materials!!\nThank you so much! We have a great team and amazing group of friends and one hell of a fan base with this!!\nLots of hard work! LOL!\nFound and added some new tracks for Scary Tales, Slaughter Sinema and Blumhouse! Enjoy!!\nHmmm! I may have missed one in my file, lemme listen to it and message back my AOV friends to see if they know!\nThank you SO MUCH! I just messaged my friends who are the directors of AOV for the tracklisting and they sayed they would post either send me the tracklist or the complete soundtrack to the show at the end of the run! This should complete the listing of AOV! Thanks again!\nI have this really weird feeling that they're using more of the score inside of the house but it's so loud I can't tag anything ! We have an amazing team and amazing fans that help bring life to this thread as well as preserve the memories of HHN's past! I just want to say thank you all for your help and support over the years with this thread!\nI need to figure out was Twisted Traditions is using, sounds like piano and synthesized guitars! Anyone know (any scare-actors in Twisted Traditions) that could help! Thanks!\nSome of the characters from the various zones and such come to gate to greet guests as a deep booming voice welcomes you to HHN and then two chainsaw scareactors stalk the gates for the first hour of the event scaring the daylights out of the guests as they enter the park! Not much, but funny to watch people freakout when they get stalked by chainsaws! LOL!\nI have to work on Vamp one night cause I heard the loop is 4hrs!\nI actually made a live recording of the entire music loop it's almost 4 hours of music ! Also the harvest does not have a separate loop it's just the park wide loop running all the way through as well as the main gate that also plays the park wide loop !\nAdded missing scngs for Saw, Blumhouse and Bill and Ted!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This book will feature the structural scheme of the Family Justice Rules taking into account any relevant judicial interpretations of the rules, academic commentaries, and parallels with the general civil justice regime.\nFamily Procedure in Singapore addresses the key provisions in the Rules of Court for the Family Justice Courts (known as the Family Justice Rules), Family Justice Courts Practice Directions, and Family Justice Courts Circulars. The legislation came into being only in 2015, following sweeping changes to the family justice landscape in 2014. The focus will be on procedural rather than substantive family law, but where necessary, references will be made to the new Family Justice Act 2014 as well. This book addresses the procedural rules by provision for ease of reference for practitioners and follows the structural scheme of the Family Justice Rules.\nThe book also takes into account any relevant judicial interpretations of the rules, academic commentaries, and parallels with the general civil justice regime (from which most of the rules are drawn from).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I posted the cover of Archie's Pal Jughead #103 six days ago in a post entitled \"Ebay Wins: Schwartz, Marek, Bacon,\" at which time I wrote, \"it's not the oddly composed cover of Archie's Pal Jughead #103 that caused me to buy it \u2014 nose meet pocket; pocket, nose \u2014 but the interior art. Can you guess why?\" Well, the reason is that I think it's always cool, when one owns a piece of original comic art, to own a copy of a comic in which the page was printed. Also, by having the comic at hand, I am able to post a decent scan of the printed page along with a digital photo of the art on this blog.\nAlong with the above page, my wife and I also own the original art of a complete story drawn by Samm Schwartz, \"Color Me True Love,\" Jughead #321 (February 1982), created about 19 years after \"In the Swim,\" but still clearly by the same hand, guided by the same minimalist artistic sensibility.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Screen Australia does not itself cashflow the Producer Offset, and it is not able to offer producers advice or recommendations as to third-party cashflow facilities.\nHowever, Screen Australia is aware that the following organisations (listed in alphabetical order) have provided cashflow facilities or brokered finance for Producer Offset applicants.\nProducers should be aware that the above list is not intended as a comprehensive list of providers, either domestic or offshore, and that Screen Australia is not approving or recommending any particular provider. Producers will need to make their own inquiries and seek their own advice as to cashflow providers and cashflow facility terms.\nAny organisations which provide cashflow facilities but which are not listed above may email the Producer Offset and Co-production Unit to request that they be included in the above list.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1.) iPhone 6 Plus Water Jade Shell is made of high quality polycarbonate, ultra-thin minimalist, good toughness, not easy to break, not easily deformed, non-toxic and harmless to the human body, light weight, high strength.\n2.) The ultra-thin fine design, set aside each function key hole, use and demolition feel design perfection.\n3.) Surface the fine matte inkjet processing, workmanship, moderate hardness, smooth feel fine,fingerprint-proof, anti-perspiration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JYP is planning to introduce a new girl group.\nJapanese is planning to debut a Japanese girl group that is just like Twice.\nJYP has always been planning to expand in Japan, and this might be their first step towards getting inside the Japanese local entertainment industry.\nMany do not know this but Twice member's Sana was supposed to be part of the Japanese girl group which JYP created before Sixteen. Later on, Sana was added to Sixteen and successfully became a member of Twice.\nJYP is planning to debut the Japanese girl group next year or early 2020.\nAre you ready for JYP's first ever Japanese girl group?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cannabissearch.com Buzz Rating: This bud is a true old world sativa strain, with an incredible balance between a roaring cerebral effect and a slight, but noticeable body high. Very hard strain to find and the dispensary that had it said it was likely a one-time deal. Needless to say, I'll be back for more.\nVenomberry is not only great at producing quality buds, but due to the incredibly low calyx-to-leaf ratio it is also favored among hash makers. It has lots of frosty sugar leaves. That make some excellent concentrates. These ladies have elongated, fairly dense buds and smell of sweet candy and fruit, after a decent cure. A strong and hearty Durban Uzbek Figwidow male was chosen to pollinate an incredibly resinous Chocolate Thai female. The result is a strain that has become widely popular in the Netherlands. Unfortunately you'll have to find it in the states, if you're not a Dutch resident, as they no longer allow foreigners to frequent their \"coffee\" shops.\nThese ladies will love light. The more lumens you provide, the better they will do. There seems to be no ceiling on the yield, provided enough lumens are used. Venomberry should be vegetated for 2 to 3 weeks, before forcing flowering. Once flowering is forced, these ladies should take anywhere from 10 - 11 weeks to become mature. There is an indica-dominant phenotype that has a flowering time of 8 - 9 weeks, although this pheno is somewhat uncommon. During late flowering, Venomberry often turns a crimson\/lavender color, which makes for an absolutely breathtaking visual display. If you come across clones of this in a dispensary, you should most definitely get them. These ladies are somewhat hard to come across here in North America.\nVenomberry 0 out of 5 based on 0 ratings. 0 user reviews.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let the stimulating begin. Sharp, herbal rosemary pure essential oil can help keep your mind clear and help improve focus. We source our rosemary from Morocco, and you'll find its distinctive herbaceous scent in our rosemary deodorants, shampoos, conditioners, and aromatherapy blends. Rosemary pure essential oil can help refresh both body and mind, anytime. A perfect little respite.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Basically, you get a cash sum that will be paid back, alongside a charge set by the moneylender at http:\/\/www.installment-loans-canada.ca, after accepting your next paycheck. This is a phenomenal strategy for acquiring little sums rapidly and to pay them off similarly as quick. It's an approach to make any day payday when you truly require it to be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MOM IS PREGNANT WITH QUADRUPLETS. WHEN SHE GIVES BIRTH AND SEES THEIR FACES, SHE'S STUNNED!\nMiranda and Josh Crawford used IVF to conceive their first daughter. The couple, who are both nurses, eventually planned on having three or four children. When Miranda wanted to get pregnant a year after the birth of baby Joslyn, they turned to IVF again. Doctors transferred two embryos into her uterus and after six weeks, the couple got the news they were having twins. Then four days later, Miranda and Josh got some more news: they were having quadruplets.\n\"I was shocked, the doctor was shocked,\" Miranda said. \"Never had that happened in his entire career.\" The entire pregnancy was scary for Miranda who believed she might have a miscarriage or go into labor prematurely. Yet, everything went relatively well. At 34 weeks, Miranda gave birth to four babies around four-and-a-half to five-and-a-half pounds each.\nWhat was even more stunning is that Miranda didn't just give birth to quadruplets, she gave birth to two sets of identical twins. Remember that she was implanted with two different embryos; each one split to form two pairs.\n\"Embryo splitting occurs approximately in one out of 100 embryo transfers. The chance of this outcome is approximately one in 10,000. This could also occur in a natural conception, but the chance of that is much rarer,\" Dr. James Grifo said.\nMiranda and Josh welcomed two identical twin girls and two identical twin boys home: Mia, Madison, Jackson and James, each a healthy little bundle of joy.\nPlease SHARE to wish the Crawfords the best of luck!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The aim of this book is to concisely present fundamental ideas, results, and techniques in linear algebra and mainly matrix theory. The book contains eight chapters covering various topics ranging from similarity and special types of matrices to Schur complements and matrix normality. Each chapter focuses on the results, techniques, and methods that are beautiful, interesting, and representative, followed by carefully selected problems. For many theorems several different proofs are given. The book can be used as a text or a supplement for a linear algebra and matrix theory class or seminar for senior or graduate students. The only prerequisites are a decent background in elementary linear algebra and calculus. The book can also serve as a reference for instructors and researchers in the fields of algebra, matrix analysis, operator theory, statistics, computer science, engineering, operations research, economics, and other fields.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bin Harkil is a manufacturing company specializing in Project Management, Design & Engineering, Fabrication and Construction of Quarrying, Crushing and Screening plants. This includes bulk materials handling, storage and mineral processing projects.\nBin Harkil represents various international Original Equipment Manufacturer (OEM) for Mining, Quarrying, Water treatment, Pumps and Earth moving equipments; and supplying the market with after sales parts and consumables.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The RCN welcomes the formation of the new SSP-UL platform. The SSP United Left Statement (June 9th, 2006) makes a number of important points with which we are in agreement. The SSP, since its inception, has been a beacon of hope to the workers' movement in Scotland and internationally. We also very much agree that uniting the left into a working, fighting political party has been a major achievement of the SSP.\nthat representatives may not be sufficiently accountable and note that in recent month, internal debate has been conducted via the mainstream media rather than through the democratic structures of the party.\nThe SSP United Left's specific proposals on Building the SSP, Accountability and Participation, Self-organisation and Education share much of our own thinking on these matters. We particularly value the methods of working outlined in Our Network, and would want to work together to ensure these are entrenched in the practice of the whole SSP. This need is particularly evident after the marked departure from such methods displayed at the May 28th 2006 Emergency National Council meeting.\nSome SSP members have attacked the declaration of the SSP United Left platform. We think that, even where SSP members oppose (some of) its principles, they should support the right to form particular platforms. Of course, it is incumbent on all platforms to conduct themselves in a principled, democratic and comradely manner towards others who do not necessarily share all their views The Role of Platforms in the SSP \u2013 Why have the RCN Platform?.\nThere clearly are political reasons for the formation of the SSP United Left. However, this decision was not taken lightly, partly due to many of the new platform members being former members of the former ISM platform, which had ceased to be viable due to growing political differences amongst its members.\nIt is quite clear that the unanimous agreement, originally shared by all members attending the SSP Emergency EC of November 9th 2004, over the handling of Tommy Sheridan's standing down as SSP Convenor, has long broken down. When the EC maintained a bureaucratic united front against the membership , at the November 27th 2004, NC, the RCN opposed this. It would have been far better for both sides, in this particular dispute, to have brought their grievances to that NC meeting, so they could have been resolved then. Instead these differences have been allowed to fester. As a result some relatively small political differences have become considerably greater, whilst growing personal animosities have not been fully disentangled from political differences.\nWhen Tommy Sheridan launched his Open Letter at the May 28th 2006 National Council meeting, it could be construed as a platform declaration, around which Tommy has asked SSP members to rally. The SW Platform, and quite a number of other SSP members, have answered his call. Therefore, like it or not, the major division in the party at present is over the issue of Tommy's resignation. The formation of the SSP United Left, like Tommy's Open Letter , represents a particular political response to this situation. The formation of particular platforms is the correct response to major differences in politically mature organisations.\nThe RCN also believes that the current crisis has highlighted a particular political weakness, shared by all sections our current leadership. We believe that the EC was correct on November 9th to advise Tommy not to resort to the state's courts to seek redress from the scabby News of the World . The RCN is opposed, in principle, to any resort to the state's courts, when attempting to seek redress and judgement over our internal party matters (and those of other working class organisations), including the conduct of any office bearer The State, the Crisis in the SSP and the Republican Response.\nWe should only seek the opinion and judgement of the working class. Alternative working class methods of seeking justice include:- the use of our own press and website, open letters to the offending media, followed by direct appeals to the trade union members working in these bodies and, if necessary, boycott actions. The state's courts are not our natural arena. We should never initiate actions which seek the state's judgement over the conduct of our own affairs. We might, of course, be forced to attend courts to defend ourselves, as best we can, against the attacks of others. However, we do not do this in the belief that they will deliver real justice. Furthermore, resort to the bourgeois courts for justice is a rich man's game.\nTherefore, the compromise public agreement, whereby the November 9th Emergency EC did not give its backing to Tommy's misguided course of action, but supported Tommy's individual right to pursue court action, caused problems for the party. The state eagerly seized the opportunity provided by Tommy's court case to intervene in the internal affairs of our party. We have now witnessed the jailing of Alan McCombes, raids on our premises and members' houses, a major surveillance operation upon our party, followed by punitive fines. Minutes, or no minutes, the state has been given an opening to find legal excuses to attack our party.\nTommy's case could never have been confined to a personal issue. As far as the News of the World is concerned, Tommy's celebrity status is linked to his political position. Therefore, we welcome those who recently advised Tommy to reconsider and withdraw from his court action, when the damage it was inflicting upon our party became strikingly evident.\nJust as our party has to learn lessons over its relationship with the institutions of the state, the RCN is also concerned that lessons are learned about the off-the-record leaks to the press (including the alleged affadavit given to The Herald ) made by various parties in the current dispute.\nThe SSP currently has no principles set down for dealing with the courts or the press, when dealing with internal matters. Given this weakness, the RCN favours the drawing up of a set of party principles, which would form the basis of our representatives' and members' future conduct over these matters.\nThe SSP United Left has taken no position on the issue of dealing with the state's courts and the bourgeois media. The RCN do not think that this debate can, or should be, avoided. We support the idea of moving the SSP's Annual Conference forward to this Autumn. Furthermore, there are no longer any excuses for not providing SSP members with the Minutes of the November 9th, 14th and 24th ECs, now that they are in the hands of both the state and the News of the World . We need to stop both those who wish to rewrite history (something with a very bad record on the Left!) and those who have speculated on the content of these minutes with malicious intent.\nA Conference, however, which concentrated purely on internal affairs, would represent too obvious a target for the media; and would possibly also discourage many SSP members. The wider political context has also changed considerably since our Conference earlier this year. There is growing evidence that our 2007 electoral campaign for the Scottish Parliament and Local Councils, will take place in a situation where New Labour face the prospect of losing their dominant position. Our Conference needs to address this situation.\nWe also recognise the leading role many SSP United Left members have taken in advancing the party's understanding of women's oppression. Gender Equality forms a significant part of the Statement. Much emphasis is placed upon the hard-won, ground-breaking 50:50 policy . However, the current political divisions in the SSP, over the handling of Tommy's court case, do not reflect the original political divisions in the SSP over the 50:50 decision \u2013 the SW Platform then being prominently in favour.\nDespite reservations, we accept that the 50:50 rule is now in place. But we also recognise that much more needs to be done to ensure that women play a full and active role in the party at all levels. Furthermore, we would wish to deepen our own understanding of women's oppression, feminism and the different political approaches and experiences over this matter. The RCN invites SSP United Left members to give a lead at one of our meetings. We would also welcome an invitation to any SSP United Left or wider party educational on this matter.\nThe RCN will continue its longstanding policy of working with any platform, or individual SSP member\/s, where we can find common agreement. We also accept majority decisions taken by our party (compatible with the wider interests of the working class in Scotland and internationally). We hope for a fruitful working relationship with the SSP United Left.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Lean Tip #346 - Think Strategically about Your Improvement.\nLean Tip #347 \u2013 Leader must ask the right questions.\nLean Tip #348 - Teams are the engines that deliver successful process improvements.\n\u2022 Team members should have good change management skills.\nLean Tip #349 - Learn how to use the word \"WE\".\nLearn how to use the word \"WE\" among your team members because this gives them confidence and encouragement. It shows that their contribution towards the process improvement is recognized and taken into consideration. It will also enable the team to get more information\/ data needed for the process improvement.\nLean Tip #350 - Evaluate your improvement process as a whole.\nOnce you have put your plan into action and have achieved the results from it, you will need to evaluate your improvement process as a whole. Ask yourselves if the process had its desired effect. Was the process successful? Did it fix the problem? Did it eliminate waste? Did you implement the improvements on time and within budget? All of these factors should be taken into consideration.\nLean Tip #351 \u2013 Encourage Your Team Members.\nAlways try and encourage your team members and be sure that any criticism you give is constructive. Encouragement and constructive criticism is a sure way to promote team work.\nLean Tip #352 - Do Not Hide Your Mistakes.\nTo err is completely human, so you should not be afraid of the mistakes you may make and of course, you should never hide them. Each mistake you make saves everyone else from repeating it \u2013 this is the very case, when you should consider team profit higher than yours. Such lessons do improve team knowledgebase and highlight weak points.\nLean Tip #353 \u2013 Teams Require Good Leadership.\nLean Tip #354 - Form Common Skills To Make Your Team More Productive.\nBe sure everyone has a common skill base for communication, conflict resolution, problem solving, giving and receiving peer feedback. I find that teams who have these common skill sets are much more productive than teams that don't. Technical expertise is only half of the success quotient.\nLean Tip #355 - Acknowledge Unique Talents and Contributions.\nEach team member brings value to the team. Point out or showcase various abilities. Take time in a meeting to recognize one or two members. Be sure everyone receives equal recognition.\nLean Tip #356 \u2013 You must believe in your goals to be successful.\nYou must believe that it is at least possible for you to achieve the goal or you will not be motivated to try. More importantly, it is you who must believe, not others. Also, just because you should believe that the goal is possible does not mean that you must expect it to be easy or even probable. Indeed, some argue that completion of only the most difficult goals will have enduring value to you.\nLean Tip #357 \u2013 Make your goals measurable and specific.\nYour goals should be measurable and specific enough for you to know unambiguously whether they have been completed yet or not. Makes sure your actions and their timing will satisfy your goals and if not rethink your plan.\nLean Tip #358 - Prioritize your goals but be flexible.\nDecide which of your goals (and tasks) are most important and assign your due dates accordingly. Be willing to change due dates or even put a goal on hold for a while if necessary.\nLean Tip #359 \u2013 If you get discouraged re-evaluate your expectations.\nIf you feel discouraged, it's probably the result of not meeting one of your own expectations. Ask yourself, \"Was the expectation realistic in the first place?\" If not, you have no reason to feel discouraged. Simply create a new goal (or tasks) that you feel are realistic and keep on going.\nLean Tip #360 \u2013 Don't fear failure, if you don't try, you gain nothing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u00ab The Herouni Antenna \u2013 The Death of the Big Bang!\nDan lawr dygn dyn loer degwch.\nI scan this with some poetic licence as follows, making no attempt to keep the technique intact except for seven syllables per line.\nMoon's beauty has lain buried.\nThis entry was posted on February 5, 2019 at 10:01 am and is filed under asott2.\tYou can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Python OODBProvides a Pythonic Object-Oriented BTree Database that stores pickled Python objects using a variety of Object Picklers ranging from the Python native cPickle to a Safe-Serializer that stores Python objects in a safe manner thus allowing Objects to be stored under Linux and retrieved under Windows and vice-versa.\npoodle also supports json for those who just have to have json.\nThe goal of this project is to take the current oodb and embellish it to allow multi-user access as well as a more powerful Cython based Safe-Serializer.\nThe goal of this project is also to build a SQL front-end for what will become a powerful multi-user database every bit as powerful as ZODB from Zope.\nThis project must support Stackless Python 2.5.2 as well as CPython 2.5.2 on both Windows XP\/Vista and Ubuntu\/CentOS Linux.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your site works on any device with responsive web design.\nThe number of people accessing the internet via mobile devices like smartphones and tablets is growing rapidly. Responsive web design lets you reach this expanding market.\nYour website needs to be ready to accommodate all users, no matter the device or the size of the screen.\nResponsive web design is the process of coding your website so that it responds intelligently to whatever size screen it's being displayed on. This way, no matter how small the screen, the information and functionality of your site is retained and mobile users are able to navigate with ease.\nResponsive design can give you an impressive edge over your competitors. Non-responsive websites can actually alienate mobile users, who are forced to deal with a clunky, slow design.\nGoogle will soon be penalizing sites with bad mobile design. If your site isn't ready to receive mobile users, Google will rank it lower on its search results pages. Sites with good mobile design will sport an icon in mobile search results that indicates to users that the site can be viewed on their device. Read more here.\nResponsive web design also alleviates the scores of issues that come with a mobile version of your website. Going the route of the \"mobile website\" (m.website.com) actually cannibalizes your primary site's traffic as a huge portion of your visitors are now being redirected to a completely new domain. This hurts your site's online presence, reducing traffic and lowering its search ranking.\nDon't get left behind: the number of people who use mobile devices to access the internet will soon overtake the number of people who use desktop computers.\nA huge portion of consumers use their smartphones to look for local products and services. It's fast becoming necessary to offer a well-designed mobile experience. 61% of users have a better opinion of a brand that offers a good mobile experience and 67% of users claim they're more likely to purchase from a mobile-friendly website.\nWest County will keep you ahead of the curve. A website with a responsive design leads to higher conversion rates from mobile customers, who more and more are becoming the #1 source of internet traffic. Our design and development teams have extensive experience creating responsive websites. All of our designs are built from the ground up with responsiveness in mind.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MINI Asia and Eurokars Habitat are jointly launching MINI Connected and the new MINI Lifestyle Collection at the Singapore Motorshow 2019. The Motorshow is held from 10 to 13 January 2019 at the Suntec Singapore Convention Centre. The MINI booth is located at level 4, and the latest range of MINI cars will be on display.\nAll MINI models will soon come with MINI Connected as standard. This new digital feature will allow MINI drivers to access a digital world of personalised services designed to make day-to-day mobility and planning even easier \u2013 and, in so doing, integrate the MINI seamlessly into their digital lives. New services including Real Time Traffic Information and Apple CarPlay will only be available on all MINI Cooper S and John Cooper Works variants as well as the MINI Cooper Countryman.\nThe new MINI Lifestyle Collection 2018-2020 will also be unveiled at the Singapore Motorshow. The new Collection offers a wide range of products designed to ease, enrich and beautify everyday life in all its variety and embodies the essence of the MINI brand in its cars and beyond. The Collection includes more than 100 items, from clothing and bags to luggage, accessories, children's gear and mobility items.\nMINI has carried a passion for motor racing in its genes for 59 years and this is expressed more intensely than ever before in the extreme driving fun offered by the top athlete in the small car segment \u2013 the MINI John Cooper Works 3 Door. The exceptional standing of the MINI John Cooper Works 3 Door is due to the most powerful engine ever installed in one of the brand's serial production models, suspension technology refined with racing expertise and model-specific design features which not only optimise the car's aerodynamic features but also underscore its exclusive charisma. With its irresistible temperament and enthralling performance figures, it outperforms the sportiest variants of the MINI just as strikingly as it does the predecessor model with the John Cooper Works logo on the radiator grille.\nThe MINI Cooper Convertible, the first and still the only premium convertible in the small car segment; combines open-top driving fun on four seats with optimised qualities in terms of sportiness, efficiency, comfort, functionality, safety, connectivity and quality of both materials and workmanship. It takes just 18 seconds to combine the further enhanced driving fun of a MINI with an intensive open-air feeling. For this purpose, the MINI Cooper Convertible is fitted with a fully automatic, fully electrically powered and therefore particularly quiet soft top operation system as well as a rollover protection.\nThe original style icon \u2013 now reinterpreted for modern city driving. Mixing hallmark MINI attitude with more doors, head room and leg room, this reliable 5 Door hatchback is transforming the look of cities, roads and parking spaces the world over. MINI fans and drivers can also enjoy a multitude of customisation options. And the car lets people get more in it \u2013 and out of it \u2013 without sacrificing that agile go-kart handling that MINI is known for.\nMINI drivers are invited on a journey of sophistication with the latest special edition MINI \u2013 The MINI Clubman Savile Row Edition \u2013 now available exclusively in Singapore. In keeping with the gentleman character of the MINI Clubman, the marque's most suave and urbane model, the Savile Row edition takes inspiration from the famed London street that introduced bespoke tailoring to the world.\nThe new generation of the MINI Countryman sees the tradition-steeped British brand continue its advance into the premium compact segment. The MINI Cooper S Countryman is the biggest and most versatile model in the brand's 59-year history. Having been completely newly developed, it now reflects considerable advancements in the areas of space, functionality, athletic flair and premium characteristics. The MINI Cooper S Countryman achieves its unique standing among direct competitors due to its powerfully expressive design, efficiently shaped bodywork and in particular its unmatched driving agility.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"India's nation brand value has in 2015 increased by a whopping 32 per cent to $2.14 trillion, compared with $1.62 trillion last year, shows a report by London-based Brand Finance, a leading independent brand valuation and strategy consultancy. Not only has India's rate of increase been the highest among the top 10 by brand value, it has also helped the country improve its global ranking by a notch to seventh.\nOnly three Asian nations \u2013 China, India and South Korea \u2013 figure among the top 20 most valuable nation brands. Even as China has maintained its second position, it has lost one per cent of its value over a year to $6.3 trillion in 2015. South Korea has improved its ranking to 12th from 17th with a 10 per cent increase in value to $1.1 trillion.\nMeanwhile, in a classic case of how one company's mess can hurt a country, the recent crisis faced by automaker Volkswagen has not only affected Germany's brand value but also cost it its position as the world's strongest nation brand.\nIn addition to a four per cent erosion in brand value to $4.2 trillion, Germany, the third-most-valuable nation brand, has been replaced in the strongest nation brand pecking order by Singapore.\nWith its intolerance for corruption, generous wages for public officials to discourage graft, heavy tax on cars leading to less congestion and good public transport, and a high-quality education system, Singapore, which has a nation brand value of $412 billion, is now the strongest nation brand.\nAccording to David Haigh, chief executive officer, Brand Finance, a nation brand is one of the most important assets for any state in a global marketplace, \"encouraging inward investment, adding value to exports and attracting tourists\".\nIn the case of the UK, there are even stronger negative associations with Iranians, many of whom resent the UK's historical political interference in their country. Germany and France, by contrast, were faster to reach out and had a more established presence in Iran before sanctions were imposed on that country. \"France's Peugeot was the market leader in the Iranian automobile market. However, its perceived abandonment of the country might mean other European firms are better placed to profit,\" says the report.\nBrand Finance measures the strength and value of 100 countries using a method based on the royalty-relief mechanism employed to value large companies. The five-step approach includes preparing a brand strength index on the basis of goods and services, investment and society. The first is sub-divided into governance, market and tourism (for investment, tourism is replaced by people and skills).\nIt emphasises a six-step approach by governments to improve the nation brand through appraisal, macro and micro image, consistent and focused vision, brand strategy, market strategy and execution. India's \"Incredible India\" slogan, used for tourism promotion, has worked well as \"an umbrella brand\", with more targeted and detailed campaigns appealing to the different audiences. \"Who doesn't want to discover something incredible? An overarching slogan or campaign could be used across the board,\" says Courtney Fingar, editor-in-chief of fDi Magazine, which has partnered Brand Finance for this year's Nation Brands study.\nInterestingly, Iran tops the list of best-performing nation brands; the value of its nation brand value has increased 59 per cent over a year ago to $159 billion. Iran is followed by Cameroon, Tanzania, Kenya and Zambia in high rates of increase. The report says Hassan Rouhani's moderate approach is slowly shifting the international perception of Iran's potential.\n\"The conflict on its doorstep and the Sunni-Shia divide will remain an impediment to trade and investment locally but with a market of 77 million people, vast hydrocarbon reserves and a highly educated population, Iran certainly has a receptive audience globally,\" says the report.\nUkraine and Russia, rivals in political arena, are together in the worst-performing nation brand category, at first and third spots, respectively. Russia's brand value, at $810 billion, is more than 60 per cent lower than India's.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rustproof Locks are designed to provide security in alomost any environment, ideal for marine outdoo..\nThe products of this series have passed all rust-proofing test carried out in the IFAM laborotory. T..\nBody made in 316 stainless steel with brass lock, fits most outboards due to clamp slot being adjust..\nA stylish range of hinges fabricated in 316 stainless steel and electro-polished for a long lasting ..\nCorrosion resistant. Ice spray proof and resistant against salt and water. Anti-twist hardened stain..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MADISON - Gov. Scott Walker readied for a rough re-election race with Wednesday's budget proposal, rolling ahead with $593 million in tax and fee cuts and more money for state workers, schools and the University of Wisconsin System.\nIn the two-year, $76.1 billion budget bill, the governor is keeping the brakes on road funding, abandoning Milwaukee's massive I-94 east-west project and bracing for a collision with his fellow Republicans over state highway funding. He also would overhaul state employee insurance, release some nonviolent inmates early and fully repeal special minimum pay rates for workers on public building projects.\nBut elsewhere, Walker made clear his intent to win a third term by driving just as far as he can on a modest state surplus and his own lagging poll numbers. The governor's budget would dole out $203 million in broad income tax cuts; $22 million for a sales tax holiday; and nearly $340 million in property tax cuts.\nOverall, Walker is partly undoing many of the cuts he made in 2011 to close a state budget shortfall, including offering state workers 2% raises for each of the next two years.\n\"We're putting more money into public education than ever before, making college even more affordable, caring for the truly needy, building a stronger infrastructure, rewarding work, and cutting taxes to the lowest point in decades,\" Walker said.\nWhile Walker spoke in the Assembly chambers, protesters filled the Capitol rotunda to demonstrate for abortion rights and health funding for women.\nWalker's budget ramps up some new spending in its second year. That means once the 2018 elections are over, state officials would need to cover a two-year, $735 million shortfall in the following budget, according to one rough calculation.\nWalker's fellow Republicans were skeptical of some of Walker's budget, especially his delays to road projects and borrowing more money to avoid a gas tax increase.\nSenate Republicans such as Majority Leader Scott Fitzgerald of Juneau and Sen. Alberta Darling of River Hills are much more skeptical of a gas tax increase. But Fitzgerald also said he was not enthused by Walker's income tax cuts, adding he wanted to look at other potential tax decreases.\nWalker's two-year budget proposal would drop plans to work on the 3.5-mile east-west portion of I-94 in Milwaukee between the Marquette and Zoo interchanges. That would free $31 million over the next two years for long-delayed work on the section of I-94 south of Milwaukee \u2014 a top priority for Vos \u2014 and avoid for now at least the total costs that could run $852 million or more.\nThat raised concerns for Steve Baas, a lobbyist for the Metropolitan Milwaukee Association of Commerce. He questioned whether it made sense to make massive investments on upgrading the Marquette and Zoo interchanges and then leave a \"choke point\" of aging highway between them.\n\"We'll work with the Legislature to try and make that case and help them find funding for it,\" he said of I-94.\nPatrick Curley, chief of staff to Milwaukee Mayor Tom Barrett, was less critical saying that if it means more funding for local road aid, then nixing the east-west project was \"not necessarily a bad thing.\"\nPeter Skopec, director of WISPIRG, a state public interest advocacy group that opposed the I-94 project as unneeded, praised the move for saving taxpayers money.\nSo far Walker's budget has largely drawn praise from education groups, and left Democrats focusing their attacks on the governor's past actions rather than his current proposals.\n\"He does not deserve praise for making a weak attempt to repair the damage that he himself caused,\" said Rep. Katrina Shankland of Stevens Point, a member of the Legislature's budget committee.\nOK $500 million in new borrowing for roads and roughly $450 million more for other state building projects.\nPut back on track two projects that were expected to be delayed \u2014 Highway 15 in Outagamie County and Highways 18\/151 in Madison. Other projects that are planned but haven't been started would face delays.\nLower the income tax rate by one-tenth of 1 percentage point in the bottom two income brackets and apply those two brackets to more of tax filers' income. This would decrease the taxes on a four-person family with a yearly income of $85,859 by about $70 in both 2017 and 2018, increasing the cumulative income tax cuts approved by Walker and GOP lawmakers to $1,542 between 2013 and 2018.\nEliminate the state's portion of the property tax at a two-year savings of $180 million and give $159 million more in state aid to schools and local governments that they would have to pass through directly to property taxpayers. The state also would give a back-to-school sales tax holiday worth $11 million a year and restore more than $20 million to a tax credit program for the working poor that Walker and lawmakers cut in 2011.\nPut $20 million more into the state's rainy day fund, about enough to run state operations for half a day.\nRoll back some but not all of past education cuts made under Walker by providing $649 million more for K-12 schools and $100 million more for the UW System.\nIncrease taxpayer money to charter and private voucher schools by $217 per pupil in each year of the budget, matching the increases given to public schools. Currently, K-8 voucher pupils bring their private schools $7,323 annually; high school voucher students bring their schools $7,969; and charter schools receive $8,188 per student.\nRequire that, before getting their state aid bumps, school districts implement Walker's signature Act 10 union law and have workers pay at least 12% of their health insurance premiums each year. It's unclear how many districts statewide do not currently meet the 12% threshold, but the Madison School District only requires teachers to pay 3% of their premiums.\nFreeze technical college tuition for Wisconsin students for two years and also keep it flat in the first year for in-state students at UW schools. In the second year, Walker would use $35 million in state tax dollars to lower UW tuition by 5%, or $360 a year for the average student. He also would provide $10 million to technical colleges, $10.2 million for need-based grants for students and require each UW school to publish an online report card starting next year, with the measures on it determined by the Board of Regents.\nSpeed up the release of up to 250 nonviolent inmates. Walker campaigned against such an earned-release program in 2010 and scaled it back in 2011. The governor's bill also would provide $2 million to help fund police overtime costs in cities with high violent crime rates.\nThe governor also put money into the state's juvenile prison complex, Lincoln Hills School for Boys and Copper Lake School for Girls, which is under federal criminal investigation for alleged abuses of youth.\nWalker would approve hiring 20 more workers there \u2014 three to provide mental health services to female inmates, eight to reduce the risk of sexual assaults and nine to distribute medication to inmates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Summer is approaching, and with it, cravings for everyone's favorite campfire treat: S'mores! Walt Disney World has done s'mores in lots of different ways over the years, most recently in latte form at Joffrey's last year. And now The Ganachery in Disney Springs has upped the game on this simple treat.\nBecause The Ganachery is an upscale specialty chocolate and sweets shop, their version of the S'more features high-end components like Vanilla Bean Marshmallow Squares and surprise additions like Salted Caramel Ganache. But for ooey, gooey goodness, it's hitting all the right notes.\nThese pop up from time to time at the Ganachery, and during the holidays, you should look for their gingerbread version!\nThese S'mores are also house-made \u2014 right before your eyes! That's right: you'll get to peek in on the cast members as they craft these squares of dessert heaven.\nCast Members make S'mores on stage!\nYou'll even get to see the cast members playing with fire as they torch your marshmallow to order!\nThe House-Made S'more is sandwiched between two Graham Crackers \u2014 of course! \u2014 with chocolate coating each cracker.\nThe Graham Crackers are super soft. These are not your typical grocery store boxed crackers, y'all. And while the texture is really lovely to bite into, the S'more does fall apart pretty easily.\nStill, it's worth it for the perfect bite of thick Marshmallow! And when we say thick, we mean seriously thick! And because it's torched to order, the hot Marshmallow easily melts the rich Chocolate. Plus there is a thin layer of Salted Caramel Ganache placed directly on top of the hot Marshmallow, marrying all the components in sticky food joy.\nThe S'more is topped with a powdered sugar Mickey, which is a special touch we've come to expect at The Ganachery.\nMickey in powdered sugar form!\nIt's messy, no doubt. But the $8 S'more at The Ganachery is worth the splurge. Every component is top-notch delicious, and it's SO.GOOD.\nWhat are you waiting for? We say head straight for The Ganachery the next time you're in Disney Springs \u2014 just don't forget a napkin (or two)!\nWould you dig into this sweet treat? Tell us in the comments below!\nBlack Velvet Anniversary Cupcakes at Disney Sprinkles Locations \u2014 SEE WHAT'S INSIDE!\nCheck Out These St. Patrick's Day Treats in Disney Springs!\nThe Best Cookie In Disney World. And You Don't Even Have To Pay for a Park Ticket!\n\u00ab Mother's Day Brunch at Walt Disney World!\nAny ideas if we can use a snack credit for the s'more?\nThe location does not yet give DVC or AP discounts, which is a bummer. Very friendly staff though, who told me to keep checking back bc that could change at a moments notice.\nDanielle, that's a good question. You could try calling The Ganachery (1 407-934-3500) to find out!\nOhmigosh! You have got to try another new find here! I just went on Saturday and they had a cold version of their drinking chocolate\u2026 it was like drinking a chocolate bar! So good!!!\nTricia, thanks for the tip! We always love an excuse to eat (or drink!) more chocolate!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Heartland Food Products Group plans to spend $32 million to update its manufacturing and distribution facilities in Marion County\u2014a project that could help it retain 732 local employees and add another 60 workers over the next three years.\nThe Carmel-based company\u2014the maker of Splenda, Java House Cold Brew Coffee and other products\u2014is seeking a personal-property tax abatement from the city of Indianapolis that would save it almost $1.8 million over the next 10 years.\nHeartland has 310,000 square feet of manufacturing space and 468,000 square feet of distribution space in facilities at 4635 W. 84th St. and 4925 W. 86th St. on the city's northwest side, according to a report from the Department of Metropolitan Development.\nCompany officials told Department of Metropolitan Development staff that it needs to expand production capacity to keep up with demand for its products but is out of room to grow physically at its current location. However, the company said it could boost the efficiency of its operations at the existing sites by investing $32 million in new production and logistics equipment.\nHeartland said the project would allow it to keep 732 employees making an average wage of $19.19 per hour and add sixty jobs that pay an average of $17.48 per hour by 2021.\nDMD staff are recommending a 10-year tax abatement to help offset the costs of Heartland's investment. The company would save about 50 percent, or $1.78 million, on its personnel property taxes over abatement period while still paying nearly that amount in taxes on the new equipment.\nThe DMD estimates Heartland would pay nearly $250,000 in taxes annually on the new equipment after the abatement period ends.\nThe Metropolitan Development Commission will hear the abatement request Wednesday afternoon.\nHeartland, which distributes food products globally, has corporate offices in Carmel and additional offices in Horsham, Pennsylvania; Bentonville, Arkansas; Amsterdam; Mexico City; and Bogota, Colombia. It has additional manufacturing facilities in Mexico and Amsterdam.\nThe company has about 1,200 employees worldwide.\nHeartland won an important intellectual property case at the U.S. Supreme Court last year involving patent infringement lawsuits.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you knew everybody was watching, what would you do differently? If the world could see what you were doing right now, how would it change what you were up to?\nIf you're like most of us, you make a bit more of an effort on days when you're going out than on ones you plan to spend on the couch. If you're like most of us, you say things to your family when you're safe inside your house, and say them in a way, that you would never say in public.\nWell, Isaiah says God's hoping for a lot more from you than just good public manners. Isaiah says God's given you\u2014you!\u2014as a light to the nations. You're to be the beacon fire on the mountaintop. The searchlight on the rescue helicopter. The torch in the Statue of Liberty's hand.\nGod doesn't need you to be in the public eye; God needs you to be the light that lets it see. If you knew God needed you not just to not be a public embarrassment, but to enlighten the world by the way you live your life, what would you do differently today?\nGod, this little light of mine doesn't really seem bright enough for the job you have in mind. But it's your call, so I'll do my best to not put it out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Frank was my dad's best friend and confidant. He \"got\" my dad. They had the same politics and outlook on life. Our family did everything with Frank and Dollie and their sons. I have so many fond memories of Sunday dinners at their home. I knew if I had a problem I could talk it through with Frank and he would explain my dad's point of view without the emotion I would get from my dad. I could complain to Frank and he understood my concerns because he understood my dad. I haven't seen Frank in a few years and need to make the effort to drive the 90 minutes it takes to go and see him.\nPat was also one of my dad's dearest friends. He and LaVonne have been my second parents. I love visiting them and even asked Pat to be \"my dad\" when so many of the dad's in my life had died. I try to visit him every year. He also lives a few hours drive from my home. This saying is so true, \"old friends are the best friends\". Can you believe these active seniors are 90 years old? I remember riding my bike to their house to visit them on hot summer nights when I was a young teenager. They thought I was visiting them but I was really trying to get one of their sons to notice me, Ha! Ha! I remember going camping with their family at a Boy Scout Camp (how's that for beating the odds when you are a teenaged girl?). I think since they only had boys I learned a lot from Pat and LaVonne about how to be a mom in an all-boy family. I treasure their love, wisdom and friendship.\nWhat a beautiful post and how generous to not only honor your own father but others as well. Love the photos of your father.\nDi, loved this post. Such a beautiful tribute to all the Dads in your life. Your sons are such a handsome bunch! What a beautiful family you have.\nYou really touched my heart deeply with this devotion of love and admiration. Not only your father but the men in your life who have influenced you to become who you are. That, I love.\nBeautiful post to all the fabulous fathers. Wow! Did you take that photo of Brent's family (the B&W...I see your watermark on it)....that's awesome!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Developer Brad Malsin of Beam Development said this week he's eyeballing a mixed-use wood building of up to 20 stories for the second phase of development of Block 75 at the Burnside Bridgehead. That on top of a timber frame building at 811 S.E. Stark that will offer close to 40,000 square feet of creative office over retail space.\"\nOlder PostWill Portland's Next Skyline Addition Be A 20-story Wooden Tower?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Article featuring quotes from my studio partner\/Yetti bandmate Ben Coll concerning this weekend's festivities. It's the best party around.\nWas afraid this was going to happen after my trip to \"The Rock\" yesterday. Woke up to this in my driveway when I was leaving for work this morning. At least it didn't go flat on a rural part of Route 41 yesterday evening. \u2764 you, Toyota Tacoma.\nThis shot required one heck of a climb up said \"Rock\", straight up a rock face but with the aid of a couple of ropes. Did this in work clothes, but I've been going to this spot since high school. It's one of the best kept secrets of the New River Gorge. As you can see, the very bright sun gave me some problems exposure-wise, but you can at least still see how the New River flows on the left, 180s in the middle and flows out to the right. Also, the Turkey Spur section of Grandview State Park is on the top of the mountain on the left. God I love West Virgina.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published at Friday, April 19th, 2019 - 3:16 PM. jd Wiring Diagram. By Lynette D. Matteson.\nMany thanks for visiting here. Below is a excellent graphic for jd 2040 hydraulics schematic diagram. We have been looking for this image through web and it originated from professional resource. If youre searching for any new plan for your wiring then this jd 2040 hydraulics schematic diagram picture needs to be on the top of guide or you might use it for an alternative idea.\nThis picture has been added by Lynette D. Matteson tagged in category field. And we also believe it could be the most well liked vote in google vote or event in facebook share. Hopefully you like it as we do. Please share this jd 2040 hydraulics schematic diagram image to your friends, family through google plus, facebook, twitter, instagram or some other social networking site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabsdx b/data_all_eng_slimpj/shuffled/split2/finalzzzabsdx
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+{"text":"Four sauces for three meats.\nIn my continuing quest for Buffalo's best BBQ, I made my way to Fat Bob's Smokehouse with a few guys. Unlike Dinosaur, Fat Bob's is a Buffalo native with only one location in Buffalo. Like some other BBQ places, Fat Bob's doesn't seem to have an actual specialty, unless you count their mac and cheese. Fortunately, this one has combo platters to ensure an unvegan like me can sample around.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Over two years ago, the Occupational Safety and Health Administration (\"OSHA\") issued one of the most significant health standards ever promulgated by the Agency. Published on March 25, 2016, the Occupational Exposure to Respirable Crystalline Silica final rule (the \"rule,\" or the \"standard\") is designed to reduce exposures to silica in construction, general industry, and maritime workplaces.\nAfter publication of the rule, numerous organizations challenged it in various federal courts of appeals, citing concerns over OSHA's health effects and risk analyses, technological and economic feasibility analyses, and the reasonableness of several ancillary provisions in the rule such as medical surveillance and the prohibition on dry sweeping and dry brushing. Those challenges were consolidated for review in the U.S. Court of Appeals for the D.C. Circuit, and after the case had been fully briefed, a three-judge panel was selected to decide the petitions.\nOn December 22, 2017, the D.C. Circuit denied several of the challenges to the rule. In a 60-page decision, the court stated that the industry presented plausible alternatives to the final rule, but ultimately the court deferred to the Agency's decision-making.\nFor mason contractors, the decision of the D.C. Circuit means that, for all intents and purposes, the crystalline silica rule is here to stay. Enforcement of the rule in construction has begun, although the pace of enforcement is still slow. The time is now for mason contractors to be fully prepared or risk an OSHA inspection and possible citation.\nThe respirable crystalline silica rule is a comprehensive health standard. It significantly reduces the Permissible Exposure Limit (\"PEL\") for crystalline silica and requires construction employers to implement engineering and work practice controls and follow several other ancillary provisions, such as medical surveillance, a written exposure control plan, and housekeeping practices.\nThe standard sets a PEL of 50 \u03bcg\/m3(from the previous PEL of approximately 250 \u03bcg\/m3) measured as an eight-hour time- weighted average (\"TWA\"). If workers are exposed to crystalline silica above the new PEL, employers must implement engineering controls to reduce exposures to below that level, before using respiratory protection. In other words, if workers are exposed above the PEL, employers must investigate and implement engineering solutions (such as wet methods, vacuums, shrouds, enclosures, or cabs) to reach levels below the PEL. Employers are not permitted to rely on respiratory protection to protect employees.\nDesignate a competent person to implement the written exposure control planby making frequent and regular inspections of job sites, materials, and equipment.\nProhibit dry sweeping and dry brushing where such activities could contribute to employee exposure to respirable crystalline silica unless wet sweeping, HEPA-filtered vacuuming or other methods that minimize the likelihood of exposure are not feasible.\nOffer medical examinations at no charge to employees\u2014including chest X-rays and lung function tests\u2014initially (if not received within the last three years by another employer) and every three years for workers who are required by the standard to wear a respirator for 30 or more days per year.\nIn the standard, OSHA is also restricting the information that employers are permitted to receive regarding an employee's medical condition. Unless the employer receives written authorization from an employee, the employer is only allowed to receive information regarding a medical examination that includes the date of the examination, a statement that the examination complied with the requirements of the standard, and any recommended limitations on the employee's use of respiratory protection.\nOSHA also has implemented a unique, alternative approach to compliance with the revised PEL and exposure monitoring requirements for construction employers. Construction industry employers are exempt from meeting the PEL and performing exposure monitoring to crystalline silica if they comply with the specific engineering controls, work practices and respirator use outlined in \"Table 1\" of the standard. Table 1 identifies 18 construction tasks that could generate exposures to respirable crystalline silica and for each task, specifies engineering controls, work practice controls, and respiratory protection to protect workers.\nOn October 19, 2017, OSHA published a memorandum setting forth interim guidance to compliance officers for enforcing the rule on construction worksites. The memorandum states that compliance officers should be prepared to collect personal breathing zone samples on the first day of the inspection. In addition, compliance officers are directed to review the employer's written exposure control plan and other relevant programs (e.g., respiratory protection program, hazard communication program, etc.) as part of the inspection. If the employer has conducted an exposure assessment, compliance officers should review the assessment to determine what levels might be expected before entering any work area. Compliance officers should also interview affected employees, including the competent person, as part of the overall assessment of the employer's implementation of its silica control plan.\nWhere the construction employer is fully and properly implementing the engineering controls, work practices, and respiratory protections specified in Table 1 for each employee engaged in listed tasks, there is no requirement for the compliance officer to collect personal air samples for those tasks. In this situation, compliance officers will be focused on determining if the controls are fully and properly implemented. Where the construction employer: (1) has employees performing a task or using equipment that is not listed in Table 1; or (2) has not fully and properly implemented the engineering controls, work practices, and respiratory protection listed in Table 1 for all employees engaged in Table 1 tasks, the compliance officer must still collect personal air samples to determine if exposures exceed the PEL.\nThe memorandum gives further guidance regarding the standard's requirements for respiratory protection, housekeeping, written exposure control plans, and medical surveillance. However, the overall emphasis to compliance officers \u2013 and contractors as well \u2013 is ensuring that appropriate engineering and work practice controls are fully and effectively implemented at the worksite.\nWith the rule in effect and enforcement beginning, contractors should be well on their way to full compliance. At this point, contractors should have reviewed their silica-producing tasks and determined whether they fit within Table 1 or will be controlled through alternative means. For those instances where Table 1 cannot be utilized, exposure monitoring should be arranged to identify the extent of exposures with implementation of alternative controls.\nIn addition, all contractors should have developed written exposure control plans for their worksites. As set forth above, compliance officers are instructed to request and review an employer's written exposure control plan when conducting an inspection. Make sure that your plan is developed, up-to-date, and that your supervisors are familiar with its contents.\nAnd finally, continue to keep abreast of new guidance from OSHA on compliance with the rule. OSHA is continually putting out compliance assistance material and new enforcement information that can help keep employers citation free.\nThe time is now! OSHA's silica standard is here to stay \u2013 make sure that you are fully compliant!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FSU Psychology researcher Jessica Ribeiro.\nEvery day, nearly 120 Americans take their lives. This statistic is what attracted a research team led by Jessica Ribeiro from Florida State University (FSU), to create new technology that may help predict a suicide. Ribeiro collaborated with FSU Psychology Assistant Professor Joseph Franklin and Colin Walsh of Vanderbilt University Medical Center on her project: \"Predicting Risk of Suicide Attempts over Time through Machine Learning.\" The technological breakthrough outlined in this paper could help clinicians predict whether a patient will attempt suicide up to two years in advance.\nPrior to Ribeiro's machine learning techniques, this field of psychology remained stagnant in terms of research. There was little to no reliable evidence that would allow physicians to determine whether a patient was at risk or not. In fact, they had as much of a chance in accurately profiling suicide risk as a coin flip would.\n\"I think that we are in an age where it is important to harness available technology and use it as these researchers have. Suicide is a real problem and I am glad to see that something is finally being done to predict and prevent it,\" junior Audrey Aragon said.\nRibeiro's disillusionment with the lack of progress and rise in suicide rates led her to create a predictive model which had an 80% accuracy at two years prior to the attempted suicide, and 92% at one week prior. The project came as a result of the largest research study of its kind ever done, with the research team accessing a massive data repository containing health records of approximately two million Tennessee patients. Ribeiro, Walsh, and Franklin combed through these records and identified those who had committed suicide. They used the data found to create algorithms which are able to use machine-learning to identify which factors will most accurately predict future suicide attempts.\nOnce these findings are published and advanced further, the technology could become available to health care providers around the country. This wide range of access could result in a significant decrease in suicide rates, seeing as physicians would be able to reliably profile suicide risk up to two years in advance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our labor and employment practice, headed by Frank Consolo, an Ohio State Bar Association Certified Specialist in Labor And Employment Law, counsels companies on a variety of labor-management and employment issues, including collective bargaining negotiations, grievance\u2013arbitration, employment discrimination and harassment charges, overtime pay, non-compete agreements and wrongful discharge claims. Our labor and employment attorneys handle matters in State and Federal Courts as well as before the EEOC, OCRC, SERB and NLRB.\nOur workers' compensation practice counsels self-insured and state fund employers, on all workers' compensation issues, including claims allowance, temporary total compensation, medical treatment authorization, permanent total disability, permanent partial disability and violations of specific safety requirements. Over the past 10 years, our workers' compensation attorneys have handled, on average, 3000 Industrial Commission hearings per year, 75 common pleas court cases per year, 7 VSSR matters per year and 3 mandamus matters per year.\nOur business transactions practice, headed by Frank Consolo, evolved out of Frank's years of counseling companies on labor and employment issues. Over the years Frank was called upon to draft and negotiate agreements for the sale and purchase of businesses, real estate transactions, and general corporate matters including incorporation and forming limited liability companies.\nCopyright \u00a9 2012 Consolo Law Firm LTD \/\/ A QuickSite By The Intersoft Group, Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: Three brothers were killed by a driver who never should have been on the road. They were stuck in traffic that had come to a standstill because of road construction. They were unaware that as they sat stopped behind a double-trailer FedEx truck, another tractor-trailer was barreling toward them. Its driver, a man with a driving record littered with violations, had fallen asleep at the wheel. He never even braked.\nThe impact slammed the SUV into the FedEx trailer in front of it, ripping that truck's trailer open and sending its contents flying all over the road. The man driving the tractor-trailer walked away from the crash; the men in the SUV never had a chance.\nThe driver had six recent driving violations and had his commercial driver's license revoked in Tennessee because of multiple traffic citations. Despite his horrible driving record, he was hired anyway and pushed to \"make time.\" Evidence uncovered by our team showed that the trucking company had not only encouraged him to push his limits, they had not installed common technology to record drivers' performance and sound alarms when they drive dangerously.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Explain the implication of maximum price ceiling With example.\nA price ceiling occurs when the government limits how much producers can charge for a good. It is called a ceiling because you cannot charge more than the amount specified by the government. Economic theory tells us this is usually a problem if the price ceiling is set too low as it would create a shortage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"F.W. Barnes and Son Funeral Directors provides details of its past Ballarat and district funeral services so family and friends have an enduring memorial to their loved ones. Please search our listings if can't immediately find the funeral details you're looking for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the normal course of our lives we meet thousands, if not tens of thousands of people. We do things for them, and they do things for us. Or, perhaps, we get in their way, or they get in ours. We may only exchange a few words, but we impact upon each other.\nToday the news is focused on Anne Kirkbride whose sad death has affected so many, and yet the irony is that the vast majority of those distressed by her passing will never have been in the same building with her, let alone exhanged words. Meanwhile faceless people slip away everyday, their passing noticed only by the few with whom they were initmately connected, and yet it is just possible that they will have had a more profound, more lasting impact on a great many more people throughout the their lives. One such person is Gail Swann.\nGail Swann \u2013 General Nursing Assistant. Photo from Phineas' Friends.\nI met Gail twice, I think. The first time she was caring for my critically ill son, Phineas, then three weeks old while he was at the Evelina Children's Hospital. My only recollection of her at that meeting was that she was calm, warm and professional. The second time was when she sat for me for my project Phineas' Friends. Again, she was calm, engaged, polite and professional. I am not entirely certain that she was sure what I was trying to do, but she knew that my intentions were honourable and she was keen to be involved.\nI found out today that Gail died from cancer last month, and I was struck by just how saddened I was to learn of this. Perhaps it is because I could so readily put a face to the name, that I could recall exactly how she influenced my life \u2013 the lives of my wife, my other children, my family and friends \u2013 and most especially Phineas himself. Gail may not directly have been the one who saved his life, but she was utterly instrumental, and as a part of that team, as a part of Phineas' Friends, I am eternally indebted to her.\nIn some respects I am lucky. I know who Gail was and I could say thank you. But there will have been thousands, tens of thousands of people whose lives she will have touched over her career, who would have wanted to say thank you to her but never thought to do so until after she was no longer involved with them. They may not have known her name, or the specific role she played in treating them or their children, but if they had I have no doubt that they too would have been upset to learn of her death. They too would have mourned her and wanted to say good bye and the \"thank you\" that escaped them when it was appropriate. They would have wanted to express their sympathies to Gail's own daughters.\nPerhaps I can do it for all of us: Gail, thank you. You will be missed.\nNext Post What's in a box?\nThis is a very beautiful and caring write up, for an exceptionally beautiful and caring woman. Gail's family and friends will be so touched by your kind words.\nI'm Gail's daughter. I just want to say thank you so much for your beautiful words and memories of our mum. She loved her job and caring for children is what she enjoyed the most in life, and what she was best at!\nI'm so glad to hear that Phineas is doing so well. Thanks again for a very touching and heartfelt tribute to her.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gillette is introducing a new marketing campaign and charitable program dedicated to celebrating the stories of men making a positive impact, and to inspire others in the process. The program-leading short film , entitled We Believe,begins with a compilation of actions commonly associated with \"toxic masculinity.\" More importantly, it then showcases examples of how men can take actions large and small to create meaningful change for themselves, their loved ones, their peers and set the right example for the next generation of men. Because it's only by challenging ourselves to do more that we get closer to our best.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"During a National Day of Prayer breakfast, Rick Perry said God will forgive his \"oops\" moment. We remember a few other things God told him.\nRon Paul debated economic policy with New York Times columnist Paul Krugman on Bloomberg Television, but the throwdown had no teeth.\nThe Republican presidential candidate's campaign stop at the University of Texas had the vibe of an Austin rock festival.\nWith Newt Gingrich dropping out, the governor endorses the presumptive Republican presidential nominee.\nThe governor's infamous debate gaffe topped Politico's list of \"The 50 Craziest Quotes of the 2012 Campaign.\"\nAccording to a new e-book, \"it became an open secret that he was using painkillers in sufficient dosages to keep him standing through the two-hour debates.\"\nThe Comedy Central host calls UT-Austin \"the fightin' only part of Texas that's liberal!\"\nThe Paul campaign says it is \"hoarding cash\" for Texas and California, and still has a \"fighting chance\" to win the nomination. Really?\nOn the day of the Illinois primary, the Paul visits the Tonight Show's Burbank studios where he hones his television skills as the lead guest.\nPaul failed to win even one state on Super Tuesday, but he isn't going anywhere.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..b403bac7306f69185278dfdfb2201f64db09072a
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+{"text":"Ballykinsella is on Logainm.ie: Ballykinsella.\nIt is located at 52\u00b0 11' 47\" N, 7\u00b0 6' 59\" W.\nWe don't know about any subtownlands in Ballykinsella.\nCurious to see who lived in Ballykinsella in the past? Maybe even seeing scans of their handwritten census returns?\nBallykinsella was added to OpenStreetMap on 9 Nov 2015 by Teester.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Outdoor air pollution is one of the major challenges of the 21st century, and about 4 million deaths globally are attributed to it, according to a World Health Organization (WHO) estimate in 2014.\nAnother estimate, from 2016, is that more than 90% of the world's population lives in regions where air pollutant levels are higher than the WHO specified limits. Thus, there is a global drive to tackle this problem, specifically by developing and installing gas and particle sensors.\nFor a long time, the main gas and particle sensor market has been restricted to industrial applications. But nowadays, degradation of the air quality is impacting many different places, homes, offices, hotels, cars and urban areas. Air quality control is therefore becoming a selling argument to promote hotels or restaurants.\nHistorically, gas sensors were used to control air conditioning in buildings to save energy. Today, Yole D\u00e9veloppement (Yole) analysts are observing a shift from comfort to safety as inhabitants want safe indoor air. This trend also extends to other environments, such as personal vehicles, public transportation, and homes.\nAs an example, the market for air purifier systems, which often embed gas and particle sensors, are expected to expand from $16B in 2017 to $33B in 2023 with a 12.5% Compound Annual Growth Rate (CAGR).\nCommercial vehicles have increasingly adopted gas and particle sensors onboard to ensure high quality of air for their customers. Previously sensors were embedded in-cabin to activate the air conditioning system when CO2 and volatile organic compound (VOC) concentrations become critical. Today, additional sensors can be found out of the cabin to sense the outside environment and close the ventilation system if pollution is detected. Metal Oxide Semiconductor (MOS) is the preferred technology because of its low cost, small size and improved sensitivity compared to previous generation. To support this increasing market demand, historical players involved in gas sensors, such as Paragon, and Amphenol, which has acquired SGX, are developing particle sensors. The big sensor leaders like Bosch and ams have identified the air quality market as a very promising one. They are therefore using their know-how to develop new technical solutions for both gas and particle sensing.\nAlthough it had big promise, the consumer segment is taking more time than expected to be a real significant share of the gas sensors market. Therefore some players are currently de-emphasized their efforts to enter this segment (in the short term (example: ams).\nIn this report, Yole analyzes the impact of the global air quality in every market segment. The main impacted segments are Heating, Ventilation and Air Conditioning (HVAC), consumer products and air comfort for transportation. Technical requirements, gases to be measured, market forecasts, technologies, main suppliers and regulations are fully described for each segment. Yole forecasts that the gas sensor market will top $1B in 2022! A chapter is dedicated to environmental combos and electronic noses technology developments as they are considered as the next steps to improve air quality sensing in several applications (but there are still challenges ahead, such as having the right use-case and structuring an industrial food chain for e-noses)..\nMOS has been around for many years but its lack of sensitivity has prevented widespread adoption of this technology. Today, the use of MOS technology has been boosted thanks to two applications: indoor\/outdoor automotive and air purifiers. MOS is the best trade-off technology in cost and size for VOC measurement.\nMOS is the reference technology in automotive air quality sensing, but only for gases. In consumer applications, miniaturized Non-Dispersive Infra-Red (NDIR) sensors compete with MOS. NDIR is more expensive, but its form factor is today close to MOS and its sensitivity is better. The final choice will be application-dependent.\nOn the other hand, optical is the reference technology for particle sensing. Particle monitoring is becoming even more important than gas sensing. Ultra-fine particles, below 1\u03bcm in diameter (PM1) are increasingly being watched as they are the most dangerous species for health. So most players are investigating particle sensors for PM1 and below.\nYole estimates the particle sensor market to be worth about $150M in 2017, growing to more than $200M in 2023. The market is less fragmented than the market for gas sensors as four players today share more than 90% of the market. So there is room for newcomers!\nBecause Asia is very concerned by air quality, our report includes a special chapter on the ecosystem of Asian players. The air purifier market is expected to ship about 20 million units annually by 2020. The Chinese market currently accounts for more than 95% of the global air purifier market, with these products using both dust and VOC sensors.\nAs a consequence, Asian players are developing their own technological portfolio for the local market. They cover all sensing technologies, but with geographical differences. For example, Japan and Taiwan are focusing on MOS and NDIR only, but China has all gas and particle technologies available.\nThe strong interest in gas and particle sensors in Asia and the large diversity in technology and R&D work are reflected by the Intellectual Property (IP) activity. Our report includes a chapter on IP activity in Asia. Patenting activity in China has strongly increased since 2010. On the contrary, patenting activity in other countries is almost stable. 85% of patents in China have been filed as priority patents and only 15% have been filed via an extension of their priority patent. In fact, there is an increasing number of Chinese companies and R&D labs among newcomers in the patent landscape since 2014..\nBundle offer possible for the Gas & Particle Sensors 2018 Report by Yole D\u00e9veloppement, contact us for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nba Finals Game 2 Free - I realize that you're seeking articles on our blog within the headline Nba Finals Game 2 Free of choices of articles that we got. If you are trying to find Nba Finals Game 2 Free article and you fails to dwell right here, you can see from several of the articles that may suit your needs right down below. We are going to help you to get articles Nba Finals Game 2 Free you're looking at some of our content which might be suitable as you are looking for the material.\nNba Finals Game 2 Free - Nba finals - wikipedia, The nba finals is the championship series of the national basketball association (nba). the eastern and western conference champions play a best-of-seven game series to determine the league champion. the winners of the finals are awarded the larry o'brien championship trophy, which replaced the walter a. brown trophy in 1983.. the series was initially known as the baa finals prior to the 1949. 2017 nba finals | warriors . cavaliers - nba., 2017 nba finals: golden state warriors vs. cleveland cavaliers | scores, playoff bracket, schedules, news, stats, analysis and video highlights.. 1977 nba finals - wikipedia, The 1977 nba world championship series was the championship round of the national basketball association (nba)'s 1976\u201377 season, and the culmination of the season's playoffs.the western conference champion portland trail blazers played against the eastern conference champion philadelphia 76ers, with the 76ers holding home-court advantage.their four regular season meetings had been split. Watch nba replays full game online free - nba replay tv, Watch nba replays full game online free, nba all-star ,playoffs , finals . all nba full game replays available for free to watch online.. Heat spurs - june 11, 2013 - game preview - nba., Get exclusive video highlights from the heat at spurs game, as well as game recaps, box scores, in-depth stats, media notes, play-by-play info and expert nba.com analysis.. Nba games today - watch nba game 2018 live stream, Nba finals 2018: schedule, times, dates, matchups, live stream. for the fourth consecutive postseason, the golden state warriors and the cleveland cavaliers will meet in the basketball's biggest for the right to be crowned this season's champion.. Nba live stream, Attention! login required !!! you must create a free account to watch watch nba live stream create my account. 2010 nba finals - boston celtics . los angeles lakers, Summary and statistics for the 2010 nba finals - boston celtics vs. los angeles lakers. Cleveland stuns road lebron james, ties , It's a make or miss league, they tell us. and if you like misses, game 2 was your sort of scene. the warriors had to overcome an 11-point deficit with 3:13 left in the contest in order to force.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"She was born in January 1922 at Launceston, Tasmania, and became a member of the Margaret Street church whilst in her teens and was an active participant in its youth activities and sporting teams. She became secretary of the church's youth committee, working alongside Kenneth Barton who was the youth committee president. During the 1940s they were both also involved in the Tasmanian state youth camps.\nKen attended the College of the Bible in Glen Iris during the war years and on his graduation at the end of 1946 they married and commenced a shared life of service in ministries in South Australia, New South Wales and Victoria - Port Pirie (1947-1951), Taree, Wingham & Comboyne (1951-1954), Warrnambool & Port Fairy (1954-1960) and Camberwell (1960-1963). Later, they also undertook interim ministries at Hartwell and Ashburton during the 1990s. The family worshipped at the Hartwell Church of Christ during Ken's years as a schoolteacher.\nEdith and Ken formed a formidable complementary team and relished working together on all aspects of Christian faith and fellowship - Sunday Schools, youth groups, fellowship groups, representation at State and Federal Conferences, and ecumenical activities. They were particularly delighted to attend the Churches of Christ World Conventions in Auckland, Long Beach, Calgary and Brisbane.\nGood friendships made at the various churches continued through Edith's regular correspondence, which was a great joy to her, with cards marking important events and Christmas. They also extended generous hospitality to many visitors from churches throughout Australia and around the world.\nOn Ken's death in 2001 Edith returned to worship at the Camberwell Vision Church of Christ. She was a member of the Board of Elders who valued her practical and wise advice, help and support. In 2011 Edith stepped down, deciding it was time to do so only at the age of 89. Edith continued to worship during the last three years of her life, attending twice weekly Chapel Services at the Baptcare Hedley Sutton Aged Care Community.\nEdith had a long, interesting and fulfilling life, lived with generosity, kindness, example, humour and intelligence. She wrote that if there was one short statement that could sum up her life it would be found in Psalm 16, verse 6 - \"the lines have fallen unto me in pleasant places, yea I have a goodly heritage\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's not a secret anymore that email is the most effective marketing channel for online stores and that it outperforms all the other social media platforms that you may be using.\nIn order to build a meaningful relationship with your customers and keep them coming back to your site, we recommend using an email strategy, so you can engage them with your brand.\nEducating the customers is a powerful technique you can use to show that you care and to get their attention other than having a massive promotion.\nYou can add value to your relationship with them by sending emails that teach them something about your niche or the products you're selling specifically.\nFor example, if you're selling running shoes, you could send tips for people who run marathons, for those who want to get started as a runner, or content on how to avoid injuries by using proper running shoes.\nYou get the idea. Take the product you're selling and think about what topics can be linked to it.\nOne of the first emails your customers should receive is the one where they get to find out more about your brand, what you stand for, your vision, and what sets you apart from the competition.\nThis is your chance to stand out and turn a visitor or a one-time customer into a fan or loyal buyer and even get them to spread the word out about your online store.\nSome online buyers get frustrated about the fact that it's hard to get in touch with online retailers after they purchased something from them and they require assistance.\nLuckily for you, this represents your chance to stand out.\nA few days after their order should have arrived, you can send them an email asking if they received it and if they're pleased with the products they purchased. Make sure that they can reply directly to the email you sent instead of having to complete a contact form.\nOver time, some of your subscribers and customers will lose interest in your emails. The good news is that you can create a campaign to reactivate them, even though that won't happen with your entire inactive list. At least, in the end, your email list will contain only people who want to stay connected with your brand.\nYou can start by sending an email to remind them that you miss them or that you're wondering if you've done something that might have upset them. Then, you can follow up with a discount to entice them to purchase from you again.\nLastly, if they still don't take any action, it's time to unsubscribe them from your email list by kindly letting them know about it.\nRepeat customers are crucial for online stores, which is why aside from sending regular promotional emails, you should also set up a loyalty program and let them know about it.\nAcquiring new customers is much more expensive than selling to existing ones, and so it's worth spending time crafting a loyalty program so your current customers will spend more money on your online store.\nBefore setting it up, you need to look at the numbers and determine what type of discounts you can offer. Don't try to copy what other online retailers are doing because every company is different and what they might be doing won't work for you as well.\nWe don't know how long email marketing will still be the number one acquisition channel for online stores, but nevertheless, you should take advantage of these tips while you can.\nIf you're not using email to build relationships with your subscribers and customers, you're leaving big money on the table. Take action now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadtgr b/data_all_eng_slimpj/shuffled/split2/finalzzzadtgr
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+{"text":"Promotional Biostick Duo Ballpens are eco-friendly with their recycled paper barrel and recycled corn starch trim and come with a nib at either end. Black and Blue ink refils are standard and the caps contain safety breather holes. 5 working day lead time is based upon a 1 colour print and up to 5000 units, higher units may add extra production time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An overall improvement to the nut which holds the camera.\nI only wish that was true. Even a minor improvement would be good.\nMy vote also goes to \"Better dynamic range\".\nI accidentally clicked on \"Other (Specify)\" in the poll though.\nI miss the Eg-s focus screen on the 5D2. I love the dynamic range, popup flash, and exposure compensation in manual mode on my Nikon.\nWhat do you mean by Exposure compensation in manual mode?\n28 to 30mp and increase buffer size.\nThey will mean Auto-ISO mode, with the mode dial set to \"M\" to allow selection of aperture and shutter speed, with the camera choosing ISO. It is an auto mode like Av or Tv, but some refer to it as \"manual\" because it shares the position on the mode dial with manual mode.\nThere is, of course, no way you could have EC in manual, but it is logical for \"M with auto-ISO\" to have it.\nExactly. When you shoot weddings you just cant go full manual for all changing light situations.\nSo inside I usually go for Manual mode + 1\/160 and I also want my aperture to stay at f 1.4. Autoiso is very fine, BUT when I have strong backlights like windows behind my subject I really need to bump the EC for a correct exposure on my subject. (because Canon doesnt meter for the focus point, unless its the center point - which I never use).\nYep, Nikon's is what everyone should be aiming for IMO. A great system.\nAs a strobist I want ISO 50 for ability to sync at cameras native sync speed at brighter conditions.\nLet's see... My MOST needed feature would be a highlight-weighted metering mode, like the one Nikon has now. And I'd like it to work seamlessly with ETTL.\nA close second would be basic video features, like DPAF & touch screen, zebras & focus peaking.\nPost edited over 4 years ago by I Simonius with reason 'trypoos'.\nI like that idea, not as a most needed but certainly as a faeature. You not talking teh same as highight priority I understand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The inertial heat exchange mechanism (International Patent WO 2013069034 A1) allows the domestic water to be heated even several hours after the end of the sunshine. A double circuit with high thermal inertia is able to guarantee maximum absorption of solar radiation and minimum thermal dispersion even in the most unfavorable environmental conditions, such as in winter.\nThe fluid contained in the vacuum tubes heats up during the day and transmits the heat to the domestic hot water tank. In this phase, the primary circuit is able to accumulate a considerable amount of heat, thus constituting an inertial accumulation, a sort of \"heat battery\", to which it is possible to access in case of need. At night, for example, after emptying the tank completely, the system is able to newly heat the cold water, thanks to the energy previously stored in the primary circuit.\nThe system is available in the lowered version EQUATOR which, having a height below one meter, is practically invisible on flat roofs.\nThe technological peculiarity is the use of a high inertia heat exchange mechanism (international patent WO 2013069034 A1), characterized by reduced temperature ranges and by a large heat storage capacity, allowing water to be heated even hours later the end of the sunshine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THE novels of the quite brilliant Ian McEwan often make for great cinema as evidenced by Enduring Love (2004) and Atonement (2007) amongst many others.\nNow add to the list On Chesil Beach shown on the final day of the BFI London Film Festival at a packed Curzon Mayfair.\nThe focal point for the story, that begins in the early 1960's, is the marriage evening of Edward (Billy Howle) and Florence (Saoirse Ronan). Two beautiful individuals in the prime of life. Both possessing first-class degrees but from contrasting backgrounds. Sitting anxiously in a hotel room close to Chesil Beach in Dorset and nervously waiting to consummate their marriage.\nFlorence, an exceptional violinist and leader of a string quartet, comes from an upper class background, with parents that would frighten the strongest of people. Mother Violet Ponting (Emma Watson) is an arrogant Oxford Don who speaks frequently to Iris Murdoch and possesses a poisonous tongue. At one stage, she warns Florence against 'screeching' despite the fact that in her hands the violin is a thing of musical beauty.\nFather Geoffrey (Samuel West) is a successful businessman who is fiercely competitive as is evidenced when he thrashes Edward at tennis. He will do anything to win. There is also a sinister side to him as we are fleetingly made aware of when he takes Florence sailing \u2013 an episode that might explain Florence's actions (or lack of them) on top of the marital bed.\nIn contrast, Edward's family is more humble, working class and homely. Father Lionel Mayhew (Adrian Scarborough) is a kindly headmaster who cycles to and from school while mother Marjorie (Anne Marie-Duff) is a talented artist (and wonderful cook) who suffers from mental issues following a horrible incident at the railway station when she was hit by an opening door (a truly shocking scene). She spends most of her time in a state of undress.\nEdward, groundsman at the local cricket club, rarely has a book out of his hand and knows the sound of every bird known to man and woman. Like Geoffrey, he is a little hot headed as we see when a gentleman derides a university colleague for being a Jew.\nThrough a series of flashbacks, we learn about how Edward and Florence first met (at a CND meeting in Oxford) and their subsequent courtship. How Florence is welcomed with open arms by Edwards' family ('marry her' urges Lionel). In contrast, Florence's parents believe she is marrying below her station. Even the family vicar questions whether she is doing the right thing in marrying Edward.\nAll the time, the expected consummation edges ever nearer. First the unfinished meal in the hotel room where they drink watered down red wine, then the initial fumblings on the bed, Florence's questioning as to how many lovers Edward has had, the removal of stockings (but not Edwards's socks \u2013 enough to put off any woman) and finally a failed consummation although Edwards does find almost instant relief.\nIt culminates in Florence storming off, fleeing along Chesil Beach with Edward following in her wake. Beside a fishing boat, Florence beseeches him to understand her and to be patient. Will he be? Won't he be? His decision frames the rest of both their lives.\nYears later, when Edward is running a record shop in London, a young school girl asks whether he possesses any Chuck Berry music. It is the trigger which ultimately leads to the film's finale, at Wigmore Hall in London, a venue which during their courtship the couple visited \u2013 and where Edward promised he would sit in the third row when she was successful and performed on stage.\nOn Chesil Beach is a beautiful film that pulls at your heartstrings (it brought a tear to my eye). It speaks of a time when most couples entered marriage as virgins. When class division was rife, anti-semitism still lingered and the youth of the time were slowly (and I mean slowly) being empowered by music, alcohol, education and love.\nThe performances are faultless with Howle (The Sense of an Ending) and Ronan (Brooklyn, Atonement) excelling as the film's leads. Seemingly madly in love (there is one beautiful scene when they are larking around while Edward is rolling the cricket pitch) until the marital bed (and deep lurking gremlins) gets the better of them.\nWatson is waspish as Violet although Anne Marie-Duff delivers the film's most pulsating performance as Marjorie.\nThe cinematography (Sean Bobbitt) is magnificent, especially the scenes on Chesil Beach and in Oxford and its surrounding countryside. As for the music, it is sublime (thank you Esther Yoo).\nOn Chesil Beach does justice to McEwan's book \u2013 helped no doubt by the fact that the author wrote the screenplay. Director Dominic Cooke has put together a film that cannot fail to tear at your heart.\nA must see. Take a handkerchief just in case.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gro Swaddle, comforing wrap for newborn sleep Gro\u2013swaddle is a specially designed shaped blanket for babies from birth to around 3 months. It is the ideal bedding solution to make newborn babies feel cosy and secure. 'Swaddling' is a method of folding, but our swaddle blanket is shaped so parents only use the right amount of fabric. We use a specially designed fabric to make sure it can never be stretched too much which prevents over wrapping.\nOur range of swaddle blankets have been designed to be the easiest to use in the market. Uniquely, our wrap has a simple shape so there is no need for poppers, Velcro or ties. The fabric has a specially developed low elasticity so it can never be wrapped around the baby more than is necessary therefore keeping babies temperature right.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Unemployed? In a rut? Looking to find your \"dream job\"? Xpert Recruiters LLC will find the right employer for your talents. We have successfully placed hundreds of employees each year, with extremely high satisfaction rates amongst both our clients and candidates. Be part of our talent pool and send us a resume today so we can match you with an employer that best suits your skills and experience!\nExpert operates an extremely thorough and efficient selection process to ensure that only the very best candidates are presented to our clients. Our detailed process yields only the most qualified individuals with the right experience & skillset so that our clients can suitably execute their strategy in accordance with project timelines.\n1) Resumes are obtained from a variety of sources that include but not limited to our own extensive database, internet job boards, job fairs, employee referrals, and direct recruitment.\n2) Potential candidates are thoroughly screened in order to confirm their relevant skills, experience, work history, education, licenses, credentials and professional reference checks and availability.\n3) Candidates are signed up in-person and required to sign off on their understanding and agreement to client confidentiality, criminal background checks, drug test (if required by client), health & safety, ethics, sexual harassment, and policies as a condition of their employment contract with Xpert LLC.\nPre Screen Interview: Senior Recruiters invest accurate timing to identify the top talent with the right experience, skillset, attitude and salary.\nInterview: Our recruiters conduct a chronological review of the applicant's education and experience.\nTechnical and Behavioral-Based Assessment: The candidate participates in a comprehensive behavioral-based interview to determine their customer service skills, personality type, performance standards, ethics, reliability, integrity, and fit.\nKnowledge & Skills Assessment: Candidate participates in a technical interview that includes a demonstration of previous abilities, hands on techniques to determine the degree of their technical aptitude and knowledge.\nVerification of Employment Requirements: Applicant confirms their understanding and agreement regarding the job expectations, caliber, duties, salary, understanding of maximum ability, location and if it is suitable for them, available start date, hours or shifts that are required to work, if they have suitable transportation for the job and eligibility to work in the United States complying with both client, Federal and State Laws and the extended offers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the biggest challenges on Thanksgiving day is stovetop space. Every burner is occupied with one task or the other. So why not use your MultiPot pressure cooker to fix the mashed potatoes? Genius! Light, creamy, and fluffy all at once, these may be the best mashed potatoes that you ever eat. Best of all, they are ready in nearly no time, so you don't have to wait for Thanksgiving. You can even make them for weeknight dinners any time of year!\nUsing a potato ricer to 'mash' the potatoes can really improve the texture of the final dish. It keeps them super light and fluffy. Yukon Golds are incredibly versatile medium-size potatoes with a beautiful buttery taste. They are perfect for mashed potatoes.\nPer Serving: 375 calories; 18g fat; 48.9g carbohydrates; 7g protein; 51mg cholesterol; 244mg sodium.\nI'm going five stars even though I ultimately stirred cream cheese into the potatoes because I cannot be stopped in my search for decadent creaminess with my mashed potatoes. So easy and delicious!\nYummy basic mashed potatoes dish! Using a ricer is definitely key for a nice texture. I like to add a little bit of whipped cream cheese to my potatoes for that extra creaminess.\nThese are some of the easiest but most delicious mashed potatoes I've ever made. This will be my go to from now on.\nThis was so easy, never would've thought to use my pressure cooker for this. I added a little extra heavy cream and they were perfection.\nI used this as the base to my pressure cooker beef bourgignon! They were perfect as the base, but would be so yummy alone as a side, too!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The information contained in GentlemensUniversity.com is for general information purposes only. The information is provided by GentlemensUniversity.com and while we endeavor to keep the information up to date and correct, we make no representations or warranties of any kind, express or implied, about the completeness, accuracy, reliability, suitability or availability with respect to GentlemensUniversity.com or the information, products, services, or related graphics contained on GentlemensUniversity.com for any purpose. Any reliance you place on such information is therefore strictly at your own risk.\nIn no event will we be liable for any loss or damage including without limitation, indirect or consequential loss or damage, or any loss or damage whatsoever arising from loss of data or profits arising out of, or in connection with, the use of GentlemensUniversity.com.\nThrough GentlemensUniversity.com you are able to link to other websites which are not under the control of GentlemensUniversity.com. We have no control over the nature, content and availability of those sites. The inclusion of any links does not necessarily imply a recommendation or endorse the views expressed within them.\nEvery effort is made to keep GentlemensUniversity.com up and running smoothly. However, BReal Innovations, Inc. takes no responsibility for, and will not be liable for, GentlemensUniversity.com (the site) being temporarily unavailable due to technical issues beyond our control.\nTarget Focus Training Review \u2013 Is The Survival Pack Good?\nStealth Attraction Review \u2013 Is Richard La Ruina's Program Good?\nLanguage of Lust Review \u2013 Is Lawrence Lanoff's Program Good?\nHOW2TXTHER Review \u2013 Can Christian Hudson Really Help You Win A Girl Through Text?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The EPA approved TDI emissions modification performed by Volkswagen on 3\/1\/2019.\nCarfax certified 2014 Volkswagen Jetta TDI wagon! Low mileage! She is in excellent condition with only 18,736 miles priced at just $12,995 with an estimated monthly payment as low as $220 for qualified buyers. This vehicle comes with just the right features such as a fuel efficient 2.0L turbodiesel engine that returns 39 MPG highway, selectable automatic transmission, AM\/FM\/CD radio with bluetooth, satellite, and backup camera, heated leather seats, air conditioning, cruise control, alloy wheels, power locks and windows, and more! In addition, we have installed 4 brand new tires to give you peace of mind when purchasing a vehicle! We have provided you with a copy of the Carfax report that suggests a retail value of $14,630.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OMF is a not-for-profit retirement project making films to raise funds and\/or awareness for specific charities. OMF is not a company and is not a charity.\nAlthough ostensibly a film about a vehicle owner's club rally, Peak Squadron is not simply a documentary.\nWith its imaginative intro sequence combining elements of 633 Squadron and Star Wars plus a mid-film fantasy sequence inspired by Disney's Beauty & the Beast\u00a9, Peak Squadron re-writes the documentary rule book.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafjci b/data_all_eng_slimpj/shuffled/split2/finalzzzafjci
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index 0000000000000000000000000000000000000000..8eb358bd95e4ed279f72c269db007d03acaa2049
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafjci
@@ -0,0 +1,5 @@
+{"text":"With its forward-thinking vision, capital and strong expertise, Dubai is serving as a strategic partner in driving Africa's digital transformation, according to a panel of fintech experts who spoke on Wednesday at the 4th Global Business Forum on Africa in Dubai.\nTheir comments came during a high-level session entitled, \"Africa-Dubai Dialogue: The Future of Money\", which revealed that African countries have realised the value of fintech and blockchain, and are now aiming to create an enabling environment for these emerging industries and technologies.\nThe session saw the participation of Arif Amiri, Chief Executive Officer of DIFC, UAE; Louis Antoine Muhire, Founder and Chief Executive Officer of Mergims, Rwanda; and Tayo Oviosu, Founder and Chief Executive Officer of Paga, Nigeria, who spoke about the impact of new technologies on Africa's finance industry, as well as Dubai's progress with regards to fostering an ecosystem that drives innovation its fast-growing fintech sector.\n\"When you look at the future of money, innovation is going to be the core. In the Middle East, North Africa and South Asia region, the penetration of [online] financial services is at about 30 per cent. That is a huge upside highlighting inclusive growth and inclusive finance. There is no doubt that everything is going to go digital, and we have global banks, organisations and government collaborating to make it happen,\" said Amiri.\nAccording to Oviosu, the UAE has the potential to be one of the first countries in the world to go completely digital in the near-future. He noted that the shift toward digitalisation is also happening in Africa, but pointed out that more education is needed to expand access to digital services on the continent.\nMuhire reiterated that the future of money will be digital but shared his view that \"people will determine if you go digitally or not.\" He stressed the importance of simplifying the process for making online transactions, adding that financial players in Africa cannot afford to wait until computer literacy reaches the same levels as Western countries.\nUnderlining the fact that more governments in Africa are now receptive of innovative technologies in the financial sector, the panellists shared the view that embracing blockchain technology and fintech services will help reduce the cost of capital, which in turn will drive economic growth on the continent.\nUnder the patronage of H.H. Sheikh Mohammed bin Rashid Al Maktoum, Vice President and Prime Minister of the UAE and Ruler of Dubai, the 4th Global Business Forum on Africa was held November 1 and 2, 2017 at Dubai's Madinat Jumeirah. The event was organised by the Dubai Chamber of Commerce and Industry.\nThe Global Business Forum series, launched by Dubai Chamber in 2013, focuses on Africa, the Commonwealth of Independent States (CIS) and Latin America. To date, the series has hosted 10 heads of state, 74 ministers and dignitaries, and 5,400 executives, as well as a host of influential decision makers from 65 countries around the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Abstract.Fix . For each integer let denote the maximal number of points of contained in a -dimensional linear subspace. Here we compute these integers (for certain ) when arises from Veronese varieties or embeddings of a hyperbolic or an elliptic quadric surface.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"to the Renewal and to the Church.\nThe ICCRS\/CF Office will continue to function as the CHARIS Office, in Palazzo San Calisto, after June 9, 2019.\nThe projects planned by ICCRS will be respected and organized by CHARIS .\nHaving its beginnings in a retreat for college students in 1967, The Catholic Charismatic Renewal has been described as a \"chance for the Church\" (Pope Paul VI), \"a special gift of the Holy Spirit\" (Pope John Paul II), and is called to share the \"current of grace\" which is baptism in the Holy Spirit (Pope Francis). Read about the birth, growth, and objectives of this move of the Holy Spirit and what our recent Popes have said about it as well.\nCalled to be ambassadors of the \"baptism in the Holy Spirit\", making the Catholic faithful aware of their \"charismatic\" dimension of life in order to grow in holiness and to serve the Kingdom of God and all the Church, read how ICCRS operates as a coordination, information and communication centre to serve the worldwide Catholic Charismatic Renewal, as well as providing a link between the Movement and the Holy See.\nThe worldwide Catholic Charismatic Renewal gathered in Rome to celebrate its Golden Jubilee in June 2017. More than merely a nostalgic look backwards, there was an anticipation for the days ahead. There was also a clearly ecumenical focus, initiated by Pope Francis himself! Here you will find links to the audio, video, presentation texts, and other archival information from this historic celebration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Virology. 2018 May;518:14-24. doi: 10.1016\/j.virol.2018.02.005. Epub 2018 Feb 12.\nEvidence for contemporary plant mitoviruses.\nNibert ML1, Vong M2, Fugate KK3, Debat HJ4.\nDepartment of Microbiology & Immunobiology, Harvard Medical School, Boston, MA 02115, USA. Electronic address: mnibert@hms.harvard.edu.\nDepartment of Microbiology & Immunobiology, Harvard Medical School, Boston, MA 02115, USA.\nSugarbeet and Potato Research, United States Department of Agriculture (USDA), Agricultural Research Service (ARS), Red River Valley Agricultural Research Center, Fargo, ND 58102, USA.\nInstituto de Patolog\u00eda Vegetal, Centro de Investigaciones Agropecuarias, Instituto Nacional de Tecnolog\u00eda Agropecuaria (IPAVE-CIAP-INTA), X5020ICA, C\u00f3rdoba, Argentina.\nMitoviruses have small RNA(+) genomes, replicate in mitochondria, and have been shown to infect only fungi to date. For this report, sequences that appear to represent nearly complete plant mitovirus genomes were recovered from publicly available transcriptome data. Twenty of the refined sequences, 2684-2898 nt long and derived from 10 different species of land plants, appear to encompass the complete coding regions of contemporary plant mitoviruses, which furthermore constitute a monophyletic cluster within genus Mitovirus. Complete coding sequences of several of these viruses were recovered from multiple transcriptome (but not genome) studies of the same plant species and also from multiple plant tissues. Crop plants among implicated hosts include beet and hemp. Other new results suggest that such genuine plant mitoviruses were immediate ancestors to endogenized mitovirus elements now widespread in land plant genomes. Whether these mitoviruses are wholly cryptic with regard to plant health remains to be investigated.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The perfect fencing design can dramatically enhance the look and feel of your home and garden. If you have a unique outdoor project in mind that needs a custom fencing solution to match, Everlast can help.\nBuilding a new fence is exciting but it can be a challenge to sort through all of the options and find the perfect combination for you. From modern and contemporary designs that stand out through to traditional and period finishes that add style and elegance, it's important to know exactly what you're looking for to produce the best results.\nWith over 20-years of experience, we can help you with expert advice on defining your fencing needs and navigating the choices. Our experts can even install your new fence for you. Everlast products are all proudly manufactured at our state-of-the-art factory in Hoppers Crossing to Australian standards, to give you the best results for the lowest costs.\nFor your next custom fencing project, come in and visit our fencing specialists to see how we can help you create your ideal fence.\nWhy Choose Everlast For Custom Fencing?\nOur products are Australian made, right here in our modern Hoppers Crossing factory in Melbourne. All materials are built to strict industry specifications to offer durable, tough, and long-lasting fencing you can trust.\nEverlast Services are family owned and operated. We have been helping customers create stunning outdoor settings in Melbourne for over 20 years. We know what it takes to deliver high-quality service, workmanship and value for money.\nOur fencing and landscaping experts are all highly qualified tradesmen who know Melbourne gardens like the back of their hand. They are the perfect advisors to help you create a garden you can truly be proud of.\nEverlast Services offer all of your outdoor needs under one roof. For your fencing, paving, decking and more, look no further than us for a better experience, ever time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafvtq b/data_all_eng_slimpj/shuffled/split2/finalzzzafvtq
new file mode 100644
index 0000000000000000000000000000000000000000..f032223f19fb039ece1cc0c38bb99f89c0794435
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The wheat in south central Kansas is currently into the early stages of grain development but the kernels are not yet watery ripe. Stripe rust was severe at many locations this past week, with incidence near 100% and severity ranging from 20-80% depending on the variety. Severe disease pressure during the early stages of grain development will likely have a significant impact on grain yield. Fields with severe damage to the flag leaves may experience more than 20% yield loss.\nVarieties with the most stripe rust included Armour, Everest, Garrison, LCS Wizard, Ruby Lee, TAM 111, and WB-Redhawk. Interestingly, varieties with a Jagger pedigree, such as 1863, Danby, Fuller, and WB4458, appeared to be moderately resistant. The varieties Oakley CL, Gallagher, WB-Cedar, and WB-Grainfield also appear resistant or moderately resistant at this time. This pattern of susceptibility among varieties indicates that the stripe rust population is similar to what was present in 2012.\nLeaf rust was also present on many of the same locations I visited this week in south central Kansas. In most cases, incidence of leaf rust was around 10-20% with severity less than 1% on the flag leaves. There was more leaf rust at the South Central Experiment Field near Hutchinson than at other locations. At this location, leaf rust severity was approaching 10-15% on the flag leaves of the susceptible variety Overley.\nReports from the central and western regions of the state also indicate that stripe rust has moved to the upper leaves in many fields. There are a few reports of severe disease incidence on the flag leaves in some fields.\nFungicides may be an option for some fields with good yield potential and seed production fields where the value of the grain is greater. Be mindful of the harvest restrictions on the fungicide products, however. Many fungicide products can no longer be applied once the wheat is flowering, while others have a 30-day pre-harvest interval. Take these label restrictions seriously. Failure to follow the label directions can result in quarantines on harvested grain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cone Crusher; Hammer Mill; Mobile Crusher; Machinery is a Manufacturer of Mining Machinery. ... Some Notes to the Maintenance of Wearing Parts of Cone Crusher.\nDiscover all the information about the product Cone crusher \/ mobile \/ diesel engine \/ crawler QH440 - Mining and Rock Technology and find where you can buy it.\nMobile Cone crushers Cone crushers have mobile cone crusher. Cone crusher is used for crushing ores and rocks, and the advantages are : reliable structure, high efficiency, easy-to-adjust and economical.\nWe are professonal mobile crusher manufacturer,our products have been exported to 167 countries and are widely used in global mining and construction industry.\nDesigned and manufactured with German technology, the Plants adopt first level mobile crushing technique, which fully meets the customers' requirement of high mobility, high crushing efficiency and optimizes your business mode.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are some photographs of the piglets that were born on Wednesday night.\nThe photographs were taken on Friday morning, so they are only just over a day old.\nThese newborn piglets are pure bred British Saddleback piglets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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Michael Cimino ( t\u0283 \u026a \u02c8 m i\u02d0 n o\u028a francesco del casino siena February 3, 1939 July francesco del casino siena, 2016) was an American film director, screenwriter, producer, and author.\nBorn in New York City, he graduated from Yale University in \u2026 Nervous-looking lead and supporting actor of the American stage and films, with sandy colored hair, pale complexion and a somewhat nervous. One of several 1978 films dealing with the Vietnam War (including Hal Ashby's Oscar-winning Coming Home), Michael Cimino's epic second feature The Deer Hunter was both francesco del casino siena for its tough portrayal of the war's effect on American working class steel workers and notorious for its ahistorical use of Russian roulette in the Vietnam \u2026 Chatroulette is an online chat website that pairs random users for webcam-based conversations.\nVisitors to the website begin an online chat \u2026 Il cacciatore (The Deer Hunter) 232; un film del 1978 diretto da Michael Cimino. Sebbene cel sia propriamente un film di francesco del casino siena, 232; ritenuto uno dei pi249; celebri in tale genere. franceeco entrato nella storia francesco del casino siena cinema per il suo contrasto francesco del casino siena le inquietanti immagini della guerra e la vita quotidiana dei protagonisti.\nRobert de Niro in The Deer Hunter Illustration: Ronald Grant Archive Casino yak aguascalientes Michael Cimino had but one feature under his belt the spirited caper movie Thunderbolt and Lightfoot before he found himself at the helm of the first epic studio movie directly about the lately concluded Vietnam war that had traumatised his country.\nDeer Hunter (The Deer Hunter) er en amerikansk film fra 1978, skrevet af Deric Washburn og instrueret af Michael Cimino. Filmen fort230;ller om hvordan Vietnamkrigen p229;virker handy spiele poker kostenlos i industribyen Clairton i Pennsylvania syd for Pittsburgh ved Monongahela-floden. 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PIANO VERSION http:gaga. francesco del casino siena LADY GAGA JOANNE http:gaga. lkJoanne FOLLOW LADY GAGA: 2 Michael Jackson Michael Joseph Jackson was an High spirits gambling machine singer, dancer, and songwriter 888 poker anglers tavern on August 29, 1958 in Gary, Indiana and passed away on June 25, 2009. He donated (at least) a remarkable 500,000,000 dollars to charity. Lady Gaga's birthday party drew so many of music's hottest acts it rivaled the Grammys. Gaga, who will ring in\u2026 4 Madonna Madonna Louise Ciccone is an American singer, songwriter, actress, and businesswoman. She influenced a lot of singers such as: Lady Gaga, Jennifer Lopez, Rihanna, Beyonce Knowles and Britney Spears. Lady Gagas performances and out-of-this-world outfits create a strong and memorable impression in all of us, distinguishing her from all other artists. A description of tropes appearing in Lady Gaga. Influences: David Bowie, Madonna, Whitney Houston, Bruce Springsteen, Michael Jackson, Queen, The Beatles, \u2026 Lady Gaga (kunstnernavn for Stefani Joanne Angelina Germanotta, f248;dt 28. marts 1986) er en amerikansk sangerinde, sangskriver, skuespillerinde, pladeproducer og aktivist. Hun er f248;dt og opvokset i New York City, hvor hun prim230;rt studerede p229; Convent of the Sacred Heart, men ogs229; kortvarigt var indskrevet p229; New York Universitys Tisch. Lady Gaga has perfected the poker face and changed celebrity style forever. Get the latest news on Lady Gaga's fashion, outfits and hair-and look back.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here at Avenue Estates we like to be as honest and transparent as we possibly can, so below you can find a breakdown of any and all fees and costs that you will be responsible for paying before, during and after your tenancy.\nReference Fee \u00a3110.00 (inc VAT) Per tenant processing the application and referencing. Anyone over the age of 18 moving into the property will need to pay this fee.\nDeposit Equivalent to a month's rent plus \u00a3100.00 (unless otherwise agreed) This will be protected via MyDeposits and may be returned at the end of the tenancy.\nTenancy Amendment Fee \u00a3264.00 (inc VAT) Contract negotiation, amending terms and updating your tenancy agreement during your tenancy.\nTenancy Renewal Fee \u00a390.00 (inc VAT) Preparation and signing of new tenancy agreements.\nCheck out Fee \u00a354.00 (inc VAT) Full assessment of property condition against inventory, collection of keys and arrangement of deposit return.\nEnd of Tenancy Clean As per the charge of the cleaning company.\nIf you have any questions on our fees please feel free to ask a member of our team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahczy b/data_all_eng_slimpj/shuffled/split2/finalzzzahczy
new file mode 100644
index 0000000000000000000000000000000000000000..842e62bc6cbb452a13e24934d19b6835e101f0bf
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Mobile Legends Bang Bang Online Hack & Cheat Generator Features: Add Unlimited Diamonds. Add Unlimited Battle Points. Mobile Legends Bang Bang Hack Latest Version (Updated Version). Mobile Legends Bang Bang Cheat Tool Undetectable, Safe and Effective (100% Guaranteed). Protected with Anti Ban Security and User-friendly Layout. No Need ROOT or JAILBREAK Mobile Device. No Download Needed. Simple but Worked.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a placeholder page for Wittenbaugh Carol, which means this person is not currently on this site. We do suggest using the tools below to find Wittenbaugh Carol.\nYou are visiting the placeholder page for Wittenbaugh Carol. This page is here because someone used our placeholder utility to look for Wittenbaugh Carol. We created this page automatically in hopes Wittenbaugh Carol would find it. If you are not Wittenbaugh Carol, but are an alumni of New Brighton High School, register on this site for free now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beckett, Lois. \"With its new food blog, WordPress gets into the content-curation game.\" Nieman Journalism Lab. Nieman Foundation for Journalism at Harvard, 22 Nov. 2010. Web. 23 Apr. 2019.\nBeckett, Lois. \"With its new food blog, WordPress gets into the content-curation game.\" Nieman Journalism Lab. Last modified November 22, 2010. Accessed April 23, 2019. https:\/\/www.niemanlab.org\/2010\/11\/with-its-new-food-blog-wordpress-gets-into-the-content-curation-game\/.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Carrie Butler is an award-winning author and graphic designer. Time away from her desk is spent playing with her dog, yelling at the TV during hockey season, and indulging in various forms of geekery--not to mention wielding a thousand-degree pen!\nThe Mark of Nexus series has appeared on Amazon bestselling, top-rated, and hot new release lists in various genres. It has also been mentioned in publications like USA Today and Writing New Adult Fiction--a how-to from Writer's Digest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy November, everyone! This is a big month for me\u2014there's Thanksgiving, plus I'm a Scorpio so my b-day is this month, my husband's birthday is this month (he's not lucky enough to be a Scorpio though, he's on the Sagittarius end of the month), this is the month I was diagnosed with Diabetes all those years ago, AND this is national Diabetes Awareness month. So much going on!!\nOut of all of those, the biggest thing for me is probably Thanksgiving. I love to cook, I love the traditional thanksgiving meal, visiting family and getting ready for Christmas. It's just a fun Holiday. And despite the fact that I have diabetes, I still usually manage to eat everything I want on Thanksgiving. More on that soon (Thanksgiving Menu planning is underway!), but I just wanted to take a sec and give you the november iPhone Download. Grab it here (below), or if you need a little technical help, see the download page with instructions.\nI know with all the food, the family gatherings, and diabetes awareness month, people may be talking more about it. And you're bound to run into people asking you what you can eat (eye roll, eye roll, sigh). Diabetes can feel like more of a burden this month with all the talk of food, and pie, and extra-carb-counting. But I try to remind myself that I'm the one in control here. This is my disease, I can decide what to eat and what to skip (no matter what anyone else says about it), and at the end of the day there are so many other diseases out there that people DON'T have that kind of control over.\nI know there are ups and downs, but in an effort to stay positive I'd ask you to remember to be grateful for the little things. Whether it's for your insulin pump, or for the small amount of carbs in turkey, or for the pie that you're willing to get high blood sugar for.\nWhatever it is, look for the good (even in Diabetes) and you'll find it.\nThat's what this download is all about. Enjoy!!\nI love that quote. Frida's life was full of both physical and emotional pain, and that quote is like a beacon of light for me on tough days. Because let's face it, we are made for more than dwelling on our struggles. So this month's download is a Diabetic take on Frida's words that I hold near and dear to my heart.\nIt's a big file, so for best results download it onto your computer and then email to yourself, open that email on your phone and save the image\u2013from there you can set it as wallpaper from your camera roll or your settings.\nAqu\u00ed lo tienes en espa\u00f1ol! Soy un poquito obsesionada con Frida Khalo, especialmente en octubre con todo los decoraciones del Dia de Los Muertos.\nLa vida de Frida estaba tan llena de dolor f\u00edsico y emocional, que la cita es como un rayo de luz para mi en los dias dif\u00edciles.\nEl archivo es bastante grande, as\u00ed que para obtener los mejores resultados descarga primero en una computadora!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaicxt b/data_all_eng_slimpj/shuffled/split2/finalzzzaicxt
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index 0000000000000000000000000000000000000000..7a81a49da4003a0ebfa2afb42336b73dfa906035
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"It is very frustrating to look for cars because many dealers post their car's price as the down payment. So, say I'm looking for a car under $2000... I get all these cars in my search results that actually cost thousands, but have a down payment of <$2000.\n@LetsMakeADeal New features, one dealing with what type of price to put in and more, for improving the Auto section are coming soon.\nBefore I made my posts, I went through and read some of these posts on this forum and took this into consideration. I completely agree and it doesn't help us either. I believe transparency in critical in this business, I appreciate your feedback and if there's anything I can do to help, please let me know.\nIn the future this app could benefit from subcategories. This way when browsing or searching we can get more relevant search results. The clothing section is another one.\nI actually hate contacting dealers in OfferUp more than CL.\nThey seem to put up lots of generic info (like financing. and perks) but nothing about the vehicle (like mileage, transmission, and vehicle features) and of course some do not post even the price. Just a picture of a vehicle.\nI seen even pictures of multiple vehicles on the same posting.\nThen if you contact them they want you to call them (to a toll free number so they can get your phone number) from someone that is just maning the OfferUp respond line. Just asking what mileage on the convertible will prompt a response: call my manager at 800 1234567 to get info in financing and the vehicle.\nDid they just cut and past this reply on all the responses?\nAnd I could go on but in general I just do not contact dealers in OfferUp, they seem extra sleazy.\nYep each site has them Letgo is bad for that too. It's like if u can't meet and that even scary now days don't even bother not even worth it unless perhaps it's something that u gotta have.\nI can so understand this. Spent 15 minutes muddling through ads that were all down payments not true prices. What made it even worse was the ads were from the same 4 car dealers just different cars. I don't know for sure but are we allowed to spam the feed that way? If I had a car to sell I would not have a chance here cause my ad would be lost in all the car dealers loading their whole lots on the site. I really like using Offer Up for selling my items and purchasing as well. But will have to skip finding my next vehicle here till something is done about the car dealers falsely showing down payments in price category and the spamming of the feed by the same car dealers. I have nothing against car dealers, just the way they are allowed to do business on this site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you find yourself struggling for energy and strength during a workout? If you're looking for that extra push to motivate you in the gym- then following these 6 tips, along with Trec's Energy and Recovery range will help you reach your objectives. We can also help you attain your goals- in less time, with less effort.\nMaintaining a healthy diet will ensure your body is provided with a constant energy supply. Consuming regular five to six meals per day will increase and retain your metabolic level. Skipping meals, on the other hand, will leave you feeling weak as your body tends to lack the essential nutrients for energy. Balanced meals containing: complex carbohydrates, protein, good fats and fibre are necessary to preserve good health. The addition of carbohydrate drinks before, during and after training will also provide you with the energy needed for an intense workout, a prompt recovery and fighting fatigue. Nitrobolon Energizer is a splendid formula that tackles all of these requirements, and helps boost your motivation levels and focus during every visit to the gym.\nTraining makes you sweat, leading you to lose water, salt and minerals from the body. If these elements are not replaced, it can cause dehydration which can have a negative impact on performance. Athletes are advised to consume energy drinks that contain electrolytes to replace those lost while training, as well as carbohydrates to supply energy. The suppleness of Trec Nutrition's Max Carb formula means you can mix your drink to the right solution for before, during and after training.\nIt is essential to optimize recovery by introducing the right nutrients into your body straight after a workout, especially if you're doing a lot of exercise. A ratio of 3:1 of carbohydrates to proteins is proven to help restore lost energy, and assist damaged muscles caused by training to repair and recover faster. By introducing a recovery formula like Hard Mass you will guarantee you are ready to hit the gym for your next session.\nTo ensure damaged muscle tissue has the right fuel for repair and recovery, plenty of protein intake is required. It is recommended to eat around 2 grams of protein per kilogram of bodyweight per day. Products such as Whey Pump Extreme and Perfect Whey Protein deliver high quality Biomax whey protein; a protein that is often called a \"fast\" protein for its capability to offer speedy muscle nourishment.\nBranch Chain Amino Acids (BCAAs) found in some proteins, are proven to prevent imbalances that lead to mental fatigue. Trec's Mass XXL formula contains BCAAs, as well as our whey protein product, Ultimate Protein. For other formulas which promise you a physical and mental boost, you can try high-energy products like Super Energy.\nWhat can be more frustrating than showing huge progress, then getting either ill or injured? Well, the maintenance range, such as Multipack 36 & Mega Mineral Pack sustains immunity by providing your body with the needed nutrients at the right times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Delivery of humanitarian aid to opposition-controlled areas in Aleppo leads to a temporary ceasefire between Syrian regime and Free Syrian Army troops.\nA temporary ceasefire has been declared between the Syrian regime and Free Syrian Army troops due to the delivery of humanitarian aid to opposition-controlled areas in Aleppo by UN Relief and Works Agency for Palestine Refugees in the Near East (UNRWA) and the Syrian Red Crescent.\nMamun Ebu Omar, an editor at Shahba Press Agency in Aleppo, told Anadolu Agency that four truckloads of humanitarian aid was transported to storages in al Basatin and Abu al Kusur by small vehicles and wheelbarrows since the roads in the area were in poor condition.\n\"This is a significant step. It is the first delivery of humanitarian aid to opposition-controlled areas since armed opponents gained control of al-Rashidin six months ago,\" he said.\nFollowing the agreement the UN refugee agency, UNHCR, and Syrian Arab Red Crescent (SARC) carried out a rare and risky operation to deliver humanitarian assistance to the besieged neighborhood of Boustan al Qaser in Eastern Aleppo.\n\"Two trucks packed with blankets, plastic sheeting, hygiene kits and kitchen sets and food were offloaded at the last checkpoint at the outskirts of the city,\" UNCHR noted in a statement on Wednesday.\nThe goods were then transported into the community using 54 pull-carts with 75 workers, making 270 trips back and forth the one and a half kilometers route, for distribution later to a SARC warehouse and then to the needy population.\nUNHCR last accessed the area in June 2013 and no humanitarian aid has reached the population there since then.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Property courtesy of Sandy Jones Realty, LLC.\nHomes for sale in Stewart Ridge are in high demand. With the recent boom in Stewart Ridge Homes for sale you can be assured that you have the best tools available for your search. Check out Stewart Ridge market statistics and the latest listings to find your next Stewart Ridge Home!\nThere are currently 2 Homes for sale in Stewart Ridge. The average price for Stewart Ridge properties is $125,125which is 0% higher than all Homes within a 20 mile radius. The highest priced property in Stewart Ridge is $250,000 while the lowest priced Home is listed at $250. Most of the properties available in Stewart Ridge are Homes . Compared to last month, the average price of real estate in the area has gone up 100%.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for a traditional, classic greeting card to send out for your greeting, invitation or announcement, then look no further than our custom folded cards. Our folding greeting cards can be personalized with photos and text whether you're creating custom wedding invitations, graduation announcements or holiday greeting cards. We offer so many styles and custom options that you're guaranteed to have a card designed that's perfect for any occasion.\nDesigning your own folded photo greeting cards doesn't get simpler than this. Once you have a photo picked out and ideas for your card layout, let your creativity flow and create a stunning card stock folding card that's perfect for any holiday or upcoming event.\nBegin by uploading your photo to Winkflash. Choose a photo from your digital image storage or scan and old photographic to add a unique twist to your greeting.\nAfterwards, select your card's orientation.\nThen, choose your style. Pick from dozens of templates that are tailored to a bevy of occasions.\nOnce you have your style chosen, you can drag and drop your photo onto your card layout in the project builder.\nAfter adding your photo you can enter your own message using our text feature.\nFinally, review your card, add it to the cart and place your order.\nOnce your order arrives in the mail, you can hand write your own message to each of your recipients for a truly heartwarming greeting card. Not only will your cards showcase a favorite photo, but they will allow you send a little love to family and friends throughout the year, no matter the occasion.\nGetting ready for your high school senior's graduation? Make collage graduation announcements that bring their memories to life! You can also include your own text to remind family and friends of the time, date, and location of the ceremony and party afterwards. Our folding photo cards can be custom printed on the front with your text on the inside for a formal graduation announcement that compliments the occasion!\nAlong with a personalized Mother's Day gift, you can create a Mother's Day photo card that features Mom's favorite moments. Just choose a template, upload your photos, and add your own text to complete your card design online. We have so many custom options to choose from, you're sure to the create the perfect greeting that makes Mom's day!\nMake Dad a Happy Father's Day card with his more treasured memories. Maybe you have a couple of pictures from your last hiking trip together. Or, there's some great snapshots on your phone from your last visit. Upload them to Winkflash, pick a template, and arrange your photos on the front of your folding card. Then, choose a text font and size, and add a personalized message on the inside to show him how much you care! Already have a card picked out? We offer photo gifts for Dad that can't be beat! Pick out a gift and personalize it in minutes with pictures from your phone or laptop!\nKeep up with loved ones throughout the year with our folded photo greeting cards. They make ideal personalized stationery since you can choose from a variety of templates that are perfect for a simple \"hello\" or even select from multiple seasonal themed templates to keep up with those closest to you year round.\nIf you are looking for the perfect stationery to use, our custom folding photo cards couldn't be better. Use our full photo template and add your all time favorite photo to the front of your card. Leave the inside blank for your own handwritten message and send out your own stationery, anytime. Personalized greeting cards are a great way to keep up with those you love, no matter the time of year.\nWhether you are planning a birthday party, a baby shower or just a yearly celebration with family and friends, we offer a wide variety of templates that will allow you to build the perfect invitation. Add a photo to give your invitation a unique feel and use our text feature to add details about your upcoming festivities. Our photo invitations will impress your recipients and are a great way to get your guests excited about your fun-filled party.\nIf you're throwing your best friend a baby shower, create for her a custom baby shower invitation that includes a flattering photo printed on our personalized folding cards. Include the time, place and date along with a little message on the interior of the card and send them out to all her closest friends and family members to kick off a baby shower she won't forget.\nOur custom folding cards also make the ultimate birthday invitation. Choose from several birthday invitation templates and include a fun photo of the birthday boy or girl to invite their relatives and friends to their birthday celebration. Customized birthday invitations not only add a personal touch to the occasion but are a great way to make a birthday party special, not matter whose celebration it is.\nDesign your own custom thank you cards that are perfect for showing your gratitude for a special gift or simply thanking a friend or relative for a good deed. No matter the occasion, you can create your own thank you notes to show how much your loved ones are appreciated while incorporating your own style into your card's design. Include a favorite landscape photo from a mountain hike you took at sunset or even scan your own artwork and add it to the front of your card for a personalized look. Whether you're thanking a loved one for a Christmas present or a neighbor for watering your plants during your last vacation, a personalized thank you card is guaranteed to make your recipient feel special.\nAre you in the midst of planning your wedding and have finally decided on a date? Look no further than Winkflash to design the perfect save the date card to share your wedding plans with everyone in your social circle. Our folding save the date cards let you to add a romantic photo with details about your wedding day. We offer a variety of templates that are a great fit for any style and can even compliment your wedding's elegant style and color palette.\nDesign the perfect graduation announcement or thank you graduation cards with your own unique touch. Our 5\u00d77 folded cards are perfect for letting loved ones know about your upcoming graduation plans while including a favorite senior photo. Add text that gives all the details about your graduation celebration and send them out to keep you friends and family in the loop.\nWith a little creativity and a favorite photo, you can design a beautiful custom folded card for any occasion. Whether you want to create your own greeting cards, make your own invitations or design your own announcement cards, Winkflash offers a huge variety of customized options that are perfect for your photo card project. Display a favorite picture and include your own message to give your greeting your own personal touch. Upload your photos to Winkflash and design your own folding photo cards today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaiuzy b/data_all_eng_slimpj/shuffled/split2/finalzzzaiuzy
new file mode 100644
index 0000000000000000000000000000000000000000..ea2d49fd595d85157676f9b9b79c177cf16107db
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaiuzy
@@ -0,0 +1,5 @@
+{"text":"Francis William Aston FRS (1 September 1877 \u2013 20 November 1945) wis an Inglis chemist an pheesicist wha wan the 1922 Nobel Prize in Chemistry for his discovery, bi means o his mass spectrograph, o isotopes, in a lairge nummer o non-radioactive elements, an for his enunciation o the whole nummer rule. He wis a fellae o the Ryal Society an Fellae o Trinity College, Cambridge.\n\u2191 1.0 1.1 Hevesy, G. (1948). \"Francis William Aston. 1877-1945\". Obituary Notices of Fellows of the Royal Society. 5 (16): 634\u2013626. doi:10.1098\/rsbm.1948.0002. JSTOR 768761.\n\u2191 \"The Nobel Prize in Chemistry 1922\". Nobel Foundation. Retrieved 2008-04-14.\n\u2191 Squires, Gordon (1998). \"Francis Aston and the mass spectrograph\". Dalton Transactions (23): 3893\u20133900. doi:10.1039\/a804629h. Retrieved 2007-12-06.\n\u2191 \"Dr. F.W. Aston Dies. Winner of the Nobel Prize in Chemistry in 1922. Noted for His Work With Isotopes. In Tour of This Country 23 Years Ago, He Foresaw the Releasing of New Energy. Developed Isotopes. Headed Atom Committee. Lectured in This Country\". New York Times. November 22, 1945. Retrieved 2010-08-06. The death of Dr. Francis William Aston, fellow of the Royal Society and Fellow of Trinity College, Cambridge, was announced today. He was 68 years old.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"REALSCAPE doesn't just improve businesses. It improves their people, it improves their networks and it improves their communities, too. There's no way to fix just one element without making certain the other components are also in good, strong order. That's why REALSCAPE's clients are so successful, time and again.\nWhat else goes into REALSCAPE's economic development strategies? A large component is accessibility to and integration with universities and community colleges Higher education fosters innovation and technological progress. Additionally, alumni networks and new graduates are valuable sources of professional local talent. . This broadens recruitment networks, connects with technology incubators and helps to secure favorable and appropriate incentive packages. We also put our unique spin on maximizing our clients' Return On Investment (ROI) using cash grants, tax strategies and other financial incentives offered by governments in return for human resources and capital investments. That's most of the secret ingredients in our recipe for success, REALSCAPE's \"special sauce\"\u2013though to learn the entire recipe, you'll need to speak with one of our knowledgeable economic development experts.\nThe convergence of new economic, cultural and human factors Human factors matter! Without the right professionals, there would be no one to establish or monitor processes, let alone make critical day-to-day decisions that measure a business' success. paradigms drives the corporate momentum for operations and delivery centers positioned in non-traditional areas. Migrating from high cost metropolitan centers to new technology centers in lower cost regions provides unparalleled momentum for nearly all up-and-coming businesses.\nREALSCAPE's clients have taken advantage of this opportunity to rationalize existing delivery methods, incorporate new technologies, lower staffing costs and provide the \"clean sheet of paper\" opportunity that Information Technology (IT) executives covet. Once integrated, our training and digital media center provides the necessary on-boarding and instructional materials to make this transition as seamless REALSCAPE is an end-to-end service provider for all your economic strategy, recruitment and training needs. We're happy to help you maximize your potential for success! as possible.\nREALSCAPE has built a template for working in partnership with educational institutions, governmental agencies, community development agencies and private enterprises. This ability to reposition domestic resources and repatriate previously out-sourced jobs to domestic locations with lower costs and strong workforce development partners, is an attractive and ever-popular alternative for our clients.\nAn early adopter of repositioning IT service jobs through the establishment of new workforces in low-cost markets, REALSCAPE uses available public-private partnerships to assist in the funding of new local service businesses. These partnerships include all levels of government, higher education (particularly community colleges), technical and secondary schools, community and economic development foundations and agencies, our own clients and private investors. Early experiences in this market have provided REALSCAPE with significant knowledge in navigating the somewhat complex and often vague rules and guidelines for maximizing all applicable incentives.\nStrategic InsightsRely on REALSCAPE's experience navigating complex incentives, tax guidelines & corporate environments.\nPassionate PeopleFind recently graduated, unemployed and underemployed professionals with innovative recruitment campaigns.\nLeading TechnologyUse leading-edge technology as both a business differentiator AND a source of lucrative economic incentives.\nRobust CommunityBuild up your business and keep it strong by supplementing the surrounding community as well.\nGET IN TOUCH WITH OUR ECONOMIC DEVELOPMENT TEAM TODAY.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2005-07-18 Assigned to GENERAL INSTRUMENT CORPORATION reassignment GENERAL INSTRUMENT CORPORATION ASSIGNMENT OF ASSIGNORS INTEREST (SEE DOCUMENT FOR DETAILS). Assignors: BONTEMPI, RAYMOND C., GARRISON, WILLIAM J., GOFFIN, GLEN P.\nA method (500) for providing access to data at a consumer location (23). The consumer location (23) has a central consumer device (202) operable to access a data recording medium (402), a QAM interface (422) arranged for communication with a first remote consumer device (214), and a packet data interface (424) arranged for communication with a second remote consumer device (220). The method includes: arranging (502) for the central consumer device to receive the data from a broadband communication system; arranging (504) for storage of the received data on the data recording medium; arranging (506) for processing of a request in behalf of the first remote consumer device or the second remote consumer device, for access to the stored data; and arranging (508\/510) for transmission of the stored data from the data recording medium to the requesting device via either the QAM interface or the packet data interface, as appropriate.\nThis application claims the benefit of provisional application No. 60\/605,958, filed Aug. 31, 2004, the entire disclosure of which is incorporated by reference herein.\nAspects of this invention relate generally to data distribution, and more particularly to a method and apparatuses for providing secure access to recorded data, such as audio\/video content received from a broadband communication system, by multiple consumer appliances at a consumer location.\nProgram providers such as multiple system operators (\"MSOs\") deliver data (pre-recorded or live electronic signals representing images and\/or audio, software, or other data, in any format, and services including interactive television, Internet services, telephone services, video-on-demand services, and the like) from a variety of public and private sources to the homes of consumers via broadband communication systems such as cable networks.\nMSOs may employ various types of data communication technologies to deliver data to consumers' homes over cable networks. Quadrature amplitude modulation (\"QAM\") is the traditional interface for delivering data via a hybrid fiber-optic\/coaxial cable (\"HFC\") network. Packet data-based technologies delivered via cable modems and Fiber to the Home (FTTH) such as the Ethernet and Internet Protocol (\"IP\"), however, are increasingly being deployed by MSOs to deliver digital data to consumers.\nA typical consumer is faced with many choices of wired and\/or wireless home consumer appliances for receiving and rendering data, such as set-top boxes, IP audio devices, personal digital assistants, RF communication devices, cable modems, and the like, and generally also must decide what type of physical transmission media in which to invest\u2014coaxial cable, Ethernet wiring, and\/or wireless signal paths, for example.\nConsumers are especially interested in acquiring consumer appliances that allow them to control the recording and home use of digital video and audio data, such as personal video recorders, digital versatile recorders, and audio\/video hard disk devices, and consumer appliances having access to such internal or external hard drive-like storage devices.\nBecause MSOs are concerned with reducing the likelihood of illegal sharing of video and audio data (especially digital data), however, consumers are typically significantly restricted as to how and where they use recorded data, even within their own homes. Historically, recorded digital data has been encrypted in a manner that restricts its use to the consumer appliance that originally received and recorded the data, necessitating a consumer's use of multiple recording devices throughout the home. More recently, whole-home solutions have been proposed\u2014in one example, two TiVO digital video recorders may share recorded content; in another example, a personal computer running Microsoft\u00ae's Windows\u00ae XP Media Center Edition operating system may transmit stored audio\/video programming to certain remote devices using packet data-based communication technologies.\nConsumers having incompatible transmission media or consumer appliances, or consumer appliances that don't support packet data-based communication technologies (such as devices that receive QAM-modulated data over coaxial cables), however, are faced with the inconvenience and expense of replacing existing consumer appliances and\/or adding additional transmission media throughout the home to take advantage of existing whole-home solutions.\nThere are, therefore, needs for methods and apparatuses that provide secure, ubiquitous access to recorded data by multiple home consumer appliances supporting different data communication technologies.\nIn accordance with an aspect of the present invention, a method provides access to data at a consumer location. The consumer location has a central consumer device operable to access a data recording medium, a quadrature amplitude modulation (\"QAM\") interface arranged for communication with a first remote consumer device, and a packet data interface arranged for communication with a second remote consumer device. The method includes: arranging for the central consumer device to receive the data from a broadband communication system; arranging for storage of the received data on the data recording medium; arranging for processing of a request, in behalf of the first remote consumer device or the second remote consumer device, for access to the stored data; when the request for access to the stored data is in behalf of the first remote consumer device, arranging for transmission of the stored data from the data recording medium to the first remote consumer device via the QAM interface; and when the request for access to the stored data is in behalf of the second remote consumer device, arranging for transmission of the stored data from the data recording medium to the second remote consumer device via the packet data interface.\nThe broadband communication system may be a cable television system carrying data such as video programming to remote consumer devices such as set-top boxes over the same or different physical transmission media (for example, coaxial cable, fiber optic cable, twisted pair lines, wireless signals, power lines, and\/or phone lines), the QAM interface may be a QAM modulator, the packet data interface may be an Internet Protocol (\"IP\") adapter, such as a cable modem. The recorded data may be encrypted\/decrypted using a key(s), and\/or formatted according to the signal receiving capability(ies) of the remote consumer devices.\nIn accordance with another aspect of the present invention, a computer-readable medium is encoded with a computer program which, when loaded into a processor (such as a processor associated with the central consumer device), implements the foregoing method.\nIn accordance with a further aspect of the present invention, an apparatus (such as customer premises equipment) for providing access to data at a consumer location, includes: a quadrature amplitude modulation (\"QAM\") interface responsive to a data recording medium; a packet data interface responsive to the data recording medium; a computer-readable storage medium; and a processor responsive to the computer-readable storage medium and to a computer program. When the computer program is loaded into the processor, it is operative to perform a method comprising: arranging for receipt of the data from a broadband communication system (such as a cable television system) arranging for storage of the received data on the (internal or external) data recording medium; arranging for processing of a request, in behalf of one of a first remote consumer device and a second remote consumer device, for access to the stored data; when the request for access to the stored data is in behalf of the first remote consumer device, arranging for transmission of the stored data from the data recording medium to the first remote consumer device via the QAM interface; and when the request for access to the stored data is in behalf of the second remote consumer device, arranging for transmission of the stored data from the data recording medium to the second remote consumer device via the packet data interface.\nFIG. 1 is a block diagram of a typical broadband communication system operated by a multiple system operator.\nFIG. 2 is a block diagram of a consumer location arranged to receive data from the headend of the broadband communication system shown in FIG. 1, in which various aspects of the present invention may be used.\nFIG. 3 is a block diagram of a processing system, illustrating certain components accessible by, or included in, the devices at the consumer location shown in FIG. 2, in accordance with various aspects of the present invention.\nFIG. 4 is a block diagram of the Media Hub shown in FIG. 2, in which various aspects of the present invention may be used.\nFIG. 5 is a flowchart of a method for providing access to data at a consumer location, in accordance with various aspects of the present invention.\nTurning now to the drawings, wherein like numerals designate like components, FIG. 1 is a block diagram of a typical broadband communication system 10, such as a cable network, operated by a multiple system operator (\"MSO\"). Using well-known components, methods and techniques, a headend 22 facilitates delivery of digital content 12 (which may be any pre-recorded or live electronic signals representing images and\/or audio, software, or other data, in any format, and services including interactive television, Internet services, telephone services, video-on-demand services, and the like) over a hybrid fiber-optic\/coaxial cable (\"HFC\") network 25 to consumer appliances (not shown) at a consumer location 23, such as an area in or around a home, business, vehicle, or other local area. HFC network 25 may carry any type of upstream or downstream channels, including but not limited to out-of-band channels, broadcast channels, data carousel channels, or video-on-demand channels.\nAs shown in simplified form, headend 22 includes servers 13 and CMTSs 26, in communication via an IP network 24 and an MSO-specific operational network 34.\nMSO-specific operational network 34 represents a collection of well-known reception and transmission components (not shown) such as servers\/controllers, multiplexers, modulators, and network adapters and\/or interfaces that operate in well-known manners to manage and control the processing of content 12 for delivery to consumer appliances operative to receive Quadrature Amplitude Modulation (\"QAM\")-modulated downstream channels (not shown) (such consumer appliances are hereinafter referred to as \"QAM Terminals\").\nIP network 24 represents a collection of well-known packet data-based reception and transmission components (not shown) such as media gateways, signaling gateways, routers, switches, and firewalls, that operate in well-known manners to manage and control the processing of content 12 for delivery to consumer appliances operative to receive downstream channels (not shown) carrying packet data (such consumer appliances are hereinafter referred to as \"IP Terminals\").\nIt will be understood, however, that system 10 and connections throughout may be any public or private, wired or wireless, content transmission infrastructure or technology for delivery of content 12, including but not limited to a fiber-optic network, a coaxial cable network, a satellite network, a cellular network, a wireless network, the Internet, a television network, a radio network, a copper wire network, or any other existing or future transmission infrastructure or technology, or any combination thereof, operated by any type of program provider, such as a television network or station, a studio, an Internet broadcaster or service provider, a cable operator, or a satellite operator.\nFIG. 2 is a block diagram of an arrangement of consumer location 23 (shown in FIG. 1) in accordance with various aspects of the present invention.\nAs shown, a media terminal 214, which is preferably a QAM Terminal (such as a set-top box\u2014for example, a set-top box from Motorola's DCT series of cable receivers), and a media terminal 220, which is preferably an IP Terminal (such as a set-top box\u2014for example, a QIP700 and\/or Motorola's DCT2500 cable receiver) are arranged for communication with a Media Hub 202 (also discussed further below, in connection with FIG. 4) over physical transmission media 201 and 203, respectively. A processing system 206 (discussed further below, in connection with FIG. 3) is accessible by Media Hub 202, QAM Terminal 214 and IP Terminal 220.\nHereinafter, media terminal 214 will be referred to as \"QAM Terminal 214,\" and media terminal 220 will be referred to as \"IP Terminal 220.\" It will be appreciated that QAM Terminal 214 and IP Terminal 220 may be any device or combination of devices responsive to headend 22 capable of receiving and rendering content 12, including but not limited to home- or office-based personal computer systems, receiving, recording or playback devices (such as IP audio devices), stereo systems, personal computer\/television devices, display devices, personal digital assistants, radiofrequency communication devices, voice-over-IP adapters, and other types of wired or wireless devices, either standing alone or included in other devices.\nGeneral internal arrangements, architectures and principles of operation of QAM Terminal 214 and IP Terminal 220 are well known to those skilled in the art. Aspects of the present invention described herein, however, may be implemented or executed by QAM Terminal 214 and IP Terminal 220 (for example, functionality associated with Data Recording Application 308 (shown in FIG. 3 and discussed further below)), and the operation\/configurations of QAM Terminal 214 and IP Terminal 220 as adapted to implement or execute aspects of the present invention described herein are not deemed to be well-known.\nPhysical transmission medium 201 represents any medium over which QAM-modulated signals may be carried, such as an in-home coaxial cable network. Physical transmission medium 203 represents any medium over which packet data may be carried, such as twisted-pair copper wiring, optical fiber, or coaxial cable. It will be appreciated, however, that physical transmission media 201 and 203 may different media altogether, such as wireless signal paths or power line paths, operating to transmit signals in accordance with any applicable format or protocol. It will also be appreciated that additional consumer appliances and types thereof may be in physical communication with Media Hub 202 via transmission media 201 and 203.\nFIG. 3 is a block diagram of processing system 206. The blocks depicted in processing system 206 are functions\/components generally included in, or accessible by, QAM Terminal 214, IP Terminal 220, and Media Hub 202. As such, processing system 206 may be implemented centrally (by a personal computer, for example), or as separate systems\/components located within, or accessible by, the consumer devices described herein, or any combination thereof.\nA processor 302 is responsive to a computer-readable storage medium 304 and to computer programs 306. Computer-readable storage medium 304 may be any electrical, magnetic, optical, local or remote memory device or system, now known or later developed, capable of recording or storing data, and in particular may be, or may include, a read only memory (\"ROM\"), a flash memory, a random access memory (\"RAM\"), a hard disk drive, any type of compact disk or digital videodisk, and\/or magnetic tape. It will be appreciated that the type of memory used may be dictated by the type data to be stored, and that computer-readable storage medium may include more than one type of memory.\nComputer-readable storage medium 304 operates to store executable instructions, such as computer programs 306, which are loaded into a processor, such as processor 302, and executed.\nComputer programs 306 are generally software components implemented according to well-known software engineering practices for component-based software development. Computer programs 306, however, may be any signal processing methods and\/or stored instructions, in one or more parts, that electronically control or implement the functions discussed herein.\nData Recording Application block 308 represents aspects of the functional arrangement of various computer programs 306 accessible by Media Hub 202 (shown in FIG. 2), QAM Terminal 214 (shown in FIG. 2) and IP Terminal 220 (shown in FIG. 2) that pertain to the provisioning of certain recorded content 12 by Media Hub 202 to QAM Terminal 214 and IP Terminal 220, in accordance with aspects of the present invention. It will be understood that functions represented by block 308 that involve aspects of networking may operate at one or more of the seven vertical layers of the well known abstract model that defines internetworking (the \"OSI Model\"): layer 1, the Physical layer; layer 2, the Data Link Layer; layer 3, the Network Layer; layer 4, the Transport Layer; layer 5, the Session Layer; layer 6, the Presentation layer; and layer 7, the Application layer.\nIn general, Data Recoding Application block 308 includes the following functions: a Discovery and Provisioning function 309 that corresponds to a collection of activities relating to the set-up of media terminals in relation to Media Hub 202; a Media Transmission function 312 corresponding to to a collection of activities relating to the implementation of user interfaces, and the handling of user input therefrom, as those user interfaces relate to the provisioning (e.g., recording and playback) of recorded content by Media Hub 202 to QAM Terminal 214 and IP Terminal 220, as well as to the mechanics of communication between Media Hub 202 and QAM Terminal 214\/IP Terminal 220; QAM Terminal 214\/IP Terminal 220; and a Security function 318 that provides protection against unauthorized use and distribution of recorded content.\nComputer-readable storage medium 304 also stores information usable by Data Recording Application 308 and functions thereof, for example: (1) configuration data associated with QAM Terminal 214 and IP Terminal 220 (user- and system-defined settings, communication settings, network settings, device receiving and\/or display capabilities, device-specific conditions relating to access to certain data from Media Hub 202, device IP\/media access control (\"MAC\") addresses, and other unique device identifiers, such as serial or identification numbers, for example); (2) encryption keys 409 (shown and discussed in connection with FIG. 4); and (3) encoding algorithms\/modules (not shown) usable by Encoder function 407 (shown and discussed in connection with FIG. 4).\nReferring again to FIG. 2, Media Hub 202 functions to receive content 12 over HFC network 25, to accept requests from QAM Terminal 214 and IP Terminal 220 to record content and access selected content 12, and to transmit requested content 12 to QAM Terminal 214 and IP Terminal 220 in a secure manner, using communication technologies compatible with QAM Terminal 214 and IP Terminal 220. Media Hub 202 may act as a proxy for all content received over HFC network 25, or may only receive and process content requested to be recorded by QAM Terminal 214 or IP Terminal 220.\nFIG. 4 is a block diagram of Media Hub 202. As shown, Media Hub 202 communicates with headend 22 over HFC network 25 on behalf of all consumer devices at consumer location 23\u2014filters 403 (for example, notch filters or any other suitable filter) are operable to separate incoming data according to desired criteria such as frequency and destination device.\nMedia Hub 202 has access to processing system 206 (shown in detail in FIG. 3) or a like internal or external processing system, via communication link 310, such as a bus or serial link, which may be selected and implemented by one skilled in the art. It will be understood that certain functions of blocks\/elements shown and discussed in connection with FIG. 4, which relate to aspects of the present invention, may be implemented by, in conjunction with, or in response to computer programs 306 (shown in FIG. 3), such as functional elements of Data Recording Application 308 (also shown in FIG. 3).\nMedia Hub 202 also has access to a recording medium 402, which may be any internal or external electrical, magnetic, optical, local or remote device or system, now known or later developed, capable of recording or storing data, including but not limited to a hard disk device, a personal video recorder, a digital versatile recorder, a VCR, a compact disk, a digital videodisk, a magnetic tape, or a PC\/TV device. Recording medium 402 may be part of, or co-located with, storage medium 304 (shown in FIG. 3), for example. Recording medium 402 is used for recording selected content received by Media Hub 202, in response to requests from QAM Terminal 214 and IP Terminal 220. Recording medium 402 may also store such information as encoded bit rates, video resolution information, Digital Rights Management information, copy control indications, content titles, content information (for example, summary data on the content-actors, running time, ratings, and the like), recording time, access conditions, expiration information, access history, and other such information, created by any network device.\nMedia Hub includes one or more external network interfaces 404 for communication with other devices. External network interfaces 404 support, for example, devices such as modems (using various communication protocols and techniques, for example, SCTE55-1, SCTE 55-2, DOCSIS\u00ae, EuroDOCSIS\u00ae, DSL, or ISDN, among others), streaming media players and other network connection support devices and\/or software that may be coupled through local or wide area networks (not shown) to program providers and providers of other content 12.\nTuner\/demodulator block 406 receives downstream channel signals (not shown) from HFC network 25, and operates to demultiplex, demodulate, decompress (as applicable), and decode the signals. Internal arrangements and principles of operation of tuner\/demodulator block 406 are well known. For example, one or more tuners within tuner\/demodulator block 406 tunes to content selected by a consumer (not shown) via a user interface (not shown). The user interface may be associated with Media Hub 202, QAM Terminal 214, or IP Terminal 220, and may be any type of known or future device or technology (for example, a remote control, a mouse, a microphone, a keyboard, or a display) allowing the consumer to select content 12, such as channels, programming, services, or recorded data, that the consumer wishes to receive. One or more demodulators (for example, a digital demodulator device, such as a QAM demodulator, and a device, such as an NTSC demodulator, responsive to receive analog versions of a channel signal) are responsive to the tuner(s), and output demodulated data in analog or digital form. It will be appreciated that aspects of tuner\/demodulator block 406 may be implemented by, or in conjunction with, elements of processing system 206, such as computer programs 306 (shown in FIG. 3).\nAn Encoder function 407 (for example, an MPEG encoder) is responsive to receive analog or digital video signals from tuner\/demodulator block 406 that are not pre-encoded in formats receivable by QAM Terminal 214 and\/or IP Terminal 220. Encoder function 407 is operative to perform predetermined coding techniques to: (1) arrange video information for transmission to QAM Terminal 214 and\/or IP Terminal 220 (for example, arrange video into one or more formats receivable by QAM Terminal 214 and\/or IP Terminal 220, and\/or displayable by a display device (not shown), which may be in communication with Media Hub 202, QAM Terminal 214, and\/or IP Terminal 220); and\/or (2) arrange video information for storage in recording medium 402 (either directly or indirectly, via security device 408). Internal components of Encoder function 407, and operations thereof, are well known, and may include analog-to-digital converters, one or more storage media and\/or buffers, and general or special-purpose processors or application-specific integrated circuits, arranged to produce multiple outputs in different formats. It will be appreciated that Encoder function 407 may be implemented by, or in conjunction with, elements of processing system 206, such as computer programs 306 (shown in FIG. 3).\nA security device 408 is responsive to receive re-formatted digital video (i.e., information that was digitally encoded by Encoder function 407), and to receive pre-formatted digital video (i.e., digital video in formats receivable by QAM Reciever 14 and\/or IP Terminal 220) from a digital demodulator device, such as a QAM demodulator, associated with tuner\/demodulator block 406. Security device 408 functions to: (1) decrypt digital video information arriving from directly tuner\/demodulator block 406 that was encrypted by headend 22, using well-known methods and techniques; and (2) protect against unauthorized use and distribution of content stored on recording medium 402, in accordance with aspects of the present invention.\nSecurity device 408 provides protection against unauthorized use and distribution of content stored on recording medium 402 by locally encrypting (or re-encrypting, if the received information was previously encrypted by headend 22) content using predetermined encryption techniques and local encryption key(s) 409 (discussed further below) to form encrypted content 410. Encrypted content 410 is decryptable and usable only by those (such as QAM Terminal 214 and IP Terminal 220) having access\u2014authorized by Media Hub 202\u2014to certain local encryption key(s) 409. It will be appreciated that certain functional aspects of security device 408 may be implemented by, or in conjunction with, elements of processing system 206, such as computer programs 306 (shown in FIG. 3).\nWhen Media Hub 202 is set-up for the first time, a locally generated key is created. Each media terminal in communication therewith has a unique X.509 certificate, which it sends to the Media Hub. The X.509 certificate is uniquely associated with an identifier of the media terminal (such as a MAC address, IP address, or a serial number) so that another device cannot masquerade as a valid media terminal. The X.509 certificate includes a public key associated with they media terminal. Media Hub 202 may store the public keys associated with the media terminals. The media terminals stores the keys and are able to decrypt encrypted content as it is transmitted (e.g., streamed). Media terminals may include the same tripleDES decryption engines and Media Hub 202. It will be appreciated that numerous variations are possible. For example, standard public key authentication methods may be used to establish a chain of authenticity back to the certification authority, as is commonly used in PKI systems. Challenge\/response techniques may also be used between Media Hub 202 and media terminals to authenticate the media terminals\u2014in this case, key exchange would likely use a public\/private key-based sharing algorithm.\nRecording medium 402 is responsive to messaging 411 (discussed further below), which includes instructions and information related to the recording and\/or rendering of content 12 to\/from recording medium 402 that is exchanged between QAM Terminal 214 and\/or IP Terminal and Media Hub 202. Messaging 411 is the demodulated data received from quadrature phase shift key (\"QPSK\") demodulator element 416, which operates in a well-known manner using well-known components (in coordination with elements of processing system 206, for example) to demodulate QPSK-modulated control signals 412 received over upstream channel(s) 414 by QPSK demodulator element 416. Control signals 412 are created by QAM Terminal 214 and IP Terminal 220 using upstream transmitters 417 (cable modems, for example) in well-known manners, and transmitted via upstream channel(s) 414 between QAM Terminal 214 and Media Hub 202 and between IP Terminal 220 and Media Hub over physical transmission medium 201 and physical transmission medium 203, respectively, using well-known techniques. Communication protocols utilized on upstream channel(s) 414 may be standard or proprietary protocols operating at various levels of the internetworking model.\nLocal Decryptor 418 receives encrypted content 410 from recording medium 402 for transmission as a digital transport stream 425 to QAM Interface 422 and\/or IP Interface 424 in response to instructions and information received via messaging 411 and\/or elements of processing system 206, such as computer programs 306 (shown in FIG. 3). More specifically, when Media Hub 202 authorizes transmission of recorded encrypted content from recording medium 402 to QAM Terminal 214 or IP Terminal 220, Local Decryptor 418 uses encryption key(s) 409 to manage the decryption of the selected recorded content by Media Hub 202 and\/or QAM Terminal 214 or IP Terminal 220, and forwards digital transport stream 425 to QAM Interface 422 and\/or IP Interface 424. The digital transport stream may also carry a set of authorization rights information (such as access conditions relating to the recorded content-right to copy and limits thereon, right to view and limits thereon, and the like\u2014sent as messages or ancillary information bits) along with the selected recorded content itself. The authorization rights information may be accessible in the same manner (for example, using the same authorization code\/key) as the recorded content. General internal arrangements, architectures and principles of operation of Local Decryptor 418 are well known to those skilled in the art.\nQAM Interface 422 is arranged for receiving digital transport stream 425, and for configuring an output QAM signal 420 (i.e., a carrier onto which digital transport stream 425 is modulated) receivable by QAM Terminal 214 over physical transmission medium 201. Internal arrangements and principles of operation of QAM Interface 422 are well-known\u2014QAM Interface 422 may include, for example, a QAM modulator (configured for 16-, 32-, 64-, 128-, or 256-QAM operation), an upconverter, a multiplexer, a transmitter, and other elements operating in well-known manners. It will be appreciated that certain functional aspects of QAM Interface 422 may be implemented by, or in conjunction with, elements of processing system 206, such as computer programs 306 (shown in FIG. 3).\nIP Interface 424 is arranged for receiving digital transport stream 425, and for configuring an output packet data signal 421 receivable by IP Terminal 220 over physical transmission medium 203. Internal arrangements and principles of operation of IP Interface 424 are well-known\u2014IP Interface 424 may include, for example, a protocol stack, such as a TCP\/IP stack for encapsulating\/framing digital transport stream 425 within data packets, software (such as computer programs 306) for processing the communication protocols that define communication between Media Hub 202 and IP Terminal 220, a virtual or hardware IP network adapter (for example, a cable modem), and an Ethernet port. IP Interface 424 can operate at any level of the OSI Model.\nWith continued reference to FIGS. 1-4, FIG. 5 is a flowchart of a method for providing access to data, such as content 12, at a consumer location, such as consumer location 23, in accordance with certain aspects of the present invention. The method may be implemented when one or more computer programs, such as computer programs 306 (for example, data recording application 308 and functions\/modules thereof associated with Media Hub 202, QAM Terminal 214, and\/or IP Terminal 220), are loaded into a processor, such as processor 302, and executed.\nThe consumer location has a central consumer device, such as Media Hub 202, which is operable to access a data recording medium, such as recording medium 402. The central consumer device has a QAM interface, such as QAM Interface 422, which is arranged for communication (via transmission medium 201, for example) with a first remote consumer device, such as QAM Terminal 214. The central consumer device also has an IP interface, such as IP Interface 424, which is arranged for communication (via transmission medium 203, for example) with a second remote consumer device, such as IP Terminal 220.\nWhen a particular media terminal enters into a local network that includes Media Hub 202, the media terminal sends a discovery message to a well-known IP address and port on which the Media Hub is listening. Media Hub 202 responds, and provides the new media terminal with an IP address and provisioning information. Media Hub 202 may also initiate a key exchange process at this time\u2014if Media Hub 202 is being established for the first time, a locally generated key may be created. Media terminals send their X.509 certificates, which contain the public keys associated with the media terminals, to the Media Hub. The public keys are used to encrypt the Media Hub's local key, and the encrypted public keys are sent to the media terminals.\nThe method begins at oval 500, and continues at block 502, where it is arranged for the central consumer device to receive the data from a broadband communication system, such as a cable television system.\nIn general, Media Hub 202 communicates with headend 22 over HFC network 25 on behalf of QAM Terminal 214 and IP Terminal 220 to receive content 12 from headend 22. More specifically, a consumer may select content for recording using a user interface associated with QAM Terminal, IP Terminal 220 or Media Hub 202. Once the consumer's selection has been communicated to Media Hub 202 (for example, via a recording request made by QAM Terminal 214 or IP Terminal 220 via upstream channel 414), Media Hub 202 may request and receive the selected content from headend 22.\nAt block 504, it is arranged for storage of the received data on the data recording medium.\nMedia Hub 202 may dynamically discover, or access previously stored, configuration data associated with the requesting consumer device and\/or other devices within consumer location 23 that will have access to the stored content, to determine the format(s) in which to store the received content in recording medium 402. For example, device receiving and\/or display capabilities\/preferences, and device-specific conditions relating to access to certain received content (limitations on the type or amount of data that may be received, for example).\nBased on the configuration data, Encoder function 407 may use predetermined encoding algorithms\/modules to re-format (in one or more ways) the content received from headend 22 prior to storing the received content on recording medium 402.\nSecurity device 408 may locally encrypt the received content using predetermined encryption techniques and local encryption key(s) 409 to form encrypted content 410, which is stored on recording medium 402.\nArrangements for processing a request for access to the stored data, in behalf of either the first remote consumer device or the second remote consumer device, are made at block 506.\nA consumer may select recorded content for playback using a user interface such as a menu (the menu may include unique IDs associated with all recorded content) associated with QAM Terminal, IP Terminal 220 or Media Hub 202. Once the consumer's selection has been communicated to Media Hub 202 (for example, via a playback request made by QAM Terminal 214 or IP Terminal 220 via upstream channel 414 that includes the selected unique IDs), Media Hub 202 authorizes or denies the playback request based on security associations established by Security device 408 (for example, the requesting consumer device may have to demonstrate possession of a particular authorization code or key, such as the X.509 certificate and\/or the Media Hub's local key, encrypted with the media terminal's public key).\nWhen the request for access to the stored data is in behalf of the first remote consumer device, arrangements are made, at block 508, for transmission of the stored data from the data recording medium to the first remote consumer device via the QAM interface, and when the request for access to the stored data is in behalf of the second remote consumer device, arrangements are made, at block 510, for transmission of the stored data from the data recording medium to the second remote consumer device via the IP interface.\nLocal Decryptor 418 forwards digital transport stream 425 to QAM Interface 422 and\/or IP Interface 424.\nQAM Interface 422 configures QAM signal 420, receivable by QAM Terminal 214 over physical transmission medium 201. IP Interface 424 configures output packet data signal 421 receivable by IP Terminal 220 over physical transmission medium 203.\nThe receiving media terminal may control the flow of content into its own memory (for example, a buffer) by requesting the content in blocks. The media terminal may track the fullness of its buffer, and request more data to fill the buffer when it is running low, and stopping requests when the buffer fills up. Content may be pulled out of the buffer and displayed, as it is needed by the video decoding system.\nAspects of the present invention described herein address the need for methods and apparatuses for providing secure, ubiquitous access to recorded data by a number of consumer devices within the home, which may function on different transmission media\/networks and\/or support different data communication technologies. A centrally-accessible recording medium may be used to manage all recorded content for the home\u2014allowing consumers more freedom to select content receiving devices\/networks, and obviating the need multiple video recorders associated with receiving\/display devices throughout the home, while maintaining the security of content transmitted over an MSO's broadband communication system.\nIt will be appreciated that aspects of the present invention are not limited to any specific embodiments of computer software or signal processing methods. For example, one or more processors and one or more computer programs packaged together or with other elements of Media Hub 202 or remote consumer devices 14 and 20 may implement functions described herein in a variety of ways. It will be understood, however, that the functions described herein are processes that convey or transform data in a predictable way, and may generally be implemented in hardware, software, firmware, or any combination thereof.\nWhen one element is indicated as being responsive to another element, the elements may be directly or indirectly coupled. Connections depicted herein may be logical or physical in practice to achieve a coupling or communicative interface between elements. Connections may be implemented as inter-process communications among software processes.\nIt will furthermore be apparent that other and further forms of the invention, and embodiments other than the specific embodiments described above, may be devised without departing from the spirit and scope of the appended claims, and it is therefore intended that the scope of this invention will only be governed by the following claims.\nwhen the request for access to the stored data is in behalf of the second remote consumer device, arranging for transmission of the stored data from the data recording medium to the second remote consumer device via the packet data interface.\n2. The method according to claim 1, wherein the broadband communication system comprises a cable television system.\n3. The method according to claim 1, wherein the QAM interface comprises a QAM modulator.\n4. The method according to claim 3, wherein the packet data interface comprises an Internet Protocol (\"IP\") adapter.\n5. The method according to claim 4, wherein the IP adapter comprises a cable modem.\n6. The method according to claim 2, wherein the data comprises video programming.\n7. The method according to claim 6, wherein the first and second remote consumer devices comprise set-top boxes.\n8. The method according to claim 1, wherein the first remote consumer device is arranged for communication with the central consumer device via a first physical transmission medium responsive to the QAM interface, and wherein the second remote consumer device is arranged for communication with the central consumer device via a second physical transmission medium responsive to the packet data interface.\nwhen the request for access to the stored data is in behalf of the second remote consumer device, arranging for placement of the stored data onto the second physical transmission medium using the packet data interface.\n10. The method according to claim 9, wherein the first and second transmission media comprise coaxial cable.\n11. The method according to claim 9, wherein the first and second transmission media are selected from the group consisting of: coaxial cable, fiber optic cable, twisted pair lines, wireless, power lines, and phone lines.\nbased on the record request, acquiring the data from the broadband communication system.\nstoring the encrypted data on the data recording medium.\nwhen the request for access to the stored data is in behalf of the second remote consumer device, determining a signal receiving capability of the second remote consumer device, and authorizing the second remote consumer device to receive the stored data.\n15. A computer-readable medium encoded with a computer program which, when loaded into a processor, implements the method of claim 1.\n16. The computer-readable medium according to claim 15, wherein the processor is associated with the central consumer device.\n18. The apparatus according to claim 17, wherein the data recording medium is external to the apparatus.\n19. The apparatus according to claim 17, wherein the apparatus comprises customer premises equipment, and the broadband communication system comprises a cable television system.\nES200502123A ES2310430A1 (en) 2004-08-31 2005-08-30 Method and apparatus for providing access to data in a user location.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Orla Tinsley is a noted writer and fierce campaigner for the rights of those with Cystic Fibrosis. She faces off against Kathleen Lonsdale, a pioneer for women in STEM and a trailblazing chemist. These two Newbridge women go head to head in this second round match-up for Kildare's All Time Great poll.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"b Calculate both the mean and median to check your prediction. Weight (kg) 6 9 7 024 7 6678 8 00 1 2334 8 55569 9 12 9 8 10 3 8 The stem plot shows the distribution of life expectancies of 23 countries. a From the shape of the distribution, which measure of centre, the mean or the median, do you think would best indicate the typical life expectancy in these countries? b Calculate both the mean how to find out what microsoft payment was for Mew is a Danish alternative rock band, consisting of Jonas Bjerre, Bo Madsen, and Silas Utke Graae J\u00f8rgensen. Bassist Johan Wohlert was also a founding member of the band, but left in 2006. Whilst their music may be classified as indie and on occasion progressive rock, the band's guitarist Bo Madsen refers to the band as \"the world's only indie stadium band\".\nis one of the fictional species of creatures from Nintendo's and Game Freak's Pok\u00e9mon media franchise created by Satoshi Tajiri. Mew is a small, pink, Psychic-type Mythical Pok\u00e9mon.\n29\/03\/2018 \u00b7 This is singular - to mew - it mews; a mew -> many mews. To mew is also a specialised verb to mew a hawk - to tie a hawk to something, or confine a hawk somewhere (usually when moulting.) I would be exceptionally rare to find this verb in any form of normal English.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajkdf b/data_all_eng_slimpj/shuffled/split2/finalzzzajkdf
new file mode 100644
index 0000000000000000000000000000000000000000..78d2ed30440e4de6ad4249e01ad4b1649ee267e8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajkdf
@@ -0,0 +1,5 @@
+{"text":"Instacanvas Gift Cards Are Now Here!\nThe perfect gift for the art or photography lover in your life! Give an Instacanvas gift card, and let the recipient create their own Instacanvas magic.\nChoose your desired dollar amount. Your gift card will be emailed to you within 24 hours of purchase. Simply email or print and deliver to the recipient.\nOver 500 images to choose from!\nPrevious PostFine Art Photography: \"Dollar Sign\"Next PostSay Literally Again!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well, we sat down to do some budgeting, and one of the things that got trimmed was our cell-phone service. The timing was very good: our contract just ended last week. So, we no longer have our cell phones. If you had my (or Erica's) phone number before, just post a comment or send me an email and I'll let you know what the new number is.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When it comes to this economic downturn, everything seems to be worse in California.\nCalifornia, along with Nevada and Florida, have been the three hardest hit states in the real estate downturn.\nDue to a staggering drop in revenues, California has had to fight the hardest to close its budget gap.\nOn top of that, California has one of the highest unemployment rates in the country, coming in at a whopping 10.5% in February.\nBased on the numbers from January of 2009, California was one of four states to have an unemployment rate of over 10%.\nIn January, Michigan led the way with an unemployment rate of 11.6%, while South Carolina (10.4%), Rhode Island (10.3%) and California (10.1%) rounded out the top four. The numbers got worse for California in February, as businesses continued to lay off workers in droves.\nSome counties in California currently possess an unemployment rate of over 20%. Trinity (21.2%), Imperial (24.5%), and Colusa (26.6%) all hold that distinction.\nThis is the highest unemployment rate that California has seen in the past 25 years, and there is a distinct possibility that it will end up topping the 11% unemployment rate that was witnessed in late 1982.\nTo give you an idea of how bad things have gotten in California over the past few years - the official unemployment rate in January of 2006 in California was 4.8%.\nKeep in mind as well that the 10.5% number is the U-3 unemployment rate for California. The U-3 number does not include discouraged workers, marginally affected workers and those who are wanting full-time work but are forced to take part-time jobs for economic reasons. The U-6 number in California, according to some people, could be approaching 20%. (For a detailed explanation of the U-6 number, please click here).\nThe big question is: how much worse will things get in California before they start to get better?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"why can't I get to msn.com?\nFirefox detected a potential security threat and did not continue to www.msn.com because this website requires a secure connection.\nI can get there with Microsoft Edge and Internet Explorer.\nwww.msn.com has a security policy called HTTP Strict Transport Security (HSTS), which means that Firefox can only connect to it securely. You can't add an exception to visit this site.\nHow to access websites that chrome can access?\nI'm trying to access a website that loads perfectly fine in chrome but loads incorrectly in firefox. I think there may be a popup involved, but I've disabled everything I can find to disable, and there is no shield showing by the url. This seems to be happening a lot lately with firefox failing but other browsers accurately showing websites. Please advise!\nCase sensitive search is always on, so I have to switch it off for every single search. How to change this?\nWhen I search (Firefox 52.9.0) for text on a website (ctrl-f) the case-sensitivity button is always on, so I always have to uncheck that. This is very unconvenient especially, when I use quick search (F3).\nHow can I either change the search settings that case-sensitivity is off by default or provide that the last used settings are also used for the next search.\nP.S.: When I uncheck the case-sensitivity button it will remain unchecked for a new search on the same site. But when I open a new site or start a new session it is checked again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published on September 28, 2009 in Site News. 0 Comments Tags: abv, calculator, IBU, php.\nI finally had a chance to sit down and work on the calculators tonight without a baby, cat, or toddler wanting something. Sorry for the delay, it was a pretty simple fix, just needed to replace global variables since the host updated from PHP 4.1 to PHP 5.2. While reading around, I was struck with the temptation to get all AJAXy on them so they do live updating. Given how long the fix took, don't expect anything soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakjpv b/data_all_eng_slimpj/shuffled/split2/finalzzzakjpv
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index 0000000000000000000000000000000000000000..06ed48c19ac376d825dadf59eac5e92b79648b39
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"ESTHER is a app Umito developed, and Shapers designed according to the ideas of a TU Delft IO promovendus as part of his doctorate.\nESTHER was used in experiments where users were motivated to plan daily exercise like walking, and see how they did and reflect on it.\nTechnically there were quite some challenges, main one the stepcounter and the energy usage. Stepcounter was developed based on several public research projects and integrated some low level c++ code for increased battery usage and performance as well as making the detection algorithm more accurate using a Butterworth filter.\nThe circular interface was also quite complex to build, being a circular view and needing to handle lots of different user input and restricting the actions of the user based on time or existing planned activities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"De Villers are an independent form of chartered surveyors providing comprehensive valuation and specialist advice for the commercial and residential sectors. Our team has over 100 years of professional property experience across the UK and a strong reputation for a highly responsive hands on senior level service. Founded in 2007, De Villiers has a proven track record as a trusted business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Aecio Neves, presidential candidate of the Brazilian Social Democracy Party and his wife Leticia Weber, are surrounded by supporters and journalists as they leave a polling station after voting in the presidential runoff election, in Belo Horizonte, Brazil, Sunday, Oct. 26, 2014. Brazilian voters decide Sunday who will next lead the world's fifth-largest country, the left-leaning incumbent Dilma Rousseff or center-right rival Neves.\nRIO DE JANEIRO, BRAZIL (AP) \u2014 President Dilma Rousseff is counting on Brazilians' gratefulness for a decade of progress to overcome concerns about a sluggish economy as the leftist leader seeks re-election on Sunday after a bitter, unpredictable campaign.\nRousseff held a slight lead in one major poll over her center-right opponent, Aecio Neves, but the two were deadlocked in another. The choice between Rousseff and Neves has split Brazilians into two camps \u2014 those who think only the president will continue to protect the poor and advance social inclusion versus those who are certain that only the contender's market-friendly economic policies can see Brazil return to solid growth.\nThe Workers' Party's 12 years in power have seen a profound transformation in Brazil, as it expanded social welfare programs to help lift millions of people from poverty and into the middle class. But four straight years of weak economic growth under Rousseff, with an economy that's now in a technical recession, has some worried those gains are under threat.\n\"Brazilians want it all. They are worried about the economy being sluggish and stagnant but they want to preserve social gains that have been made,\" said Michael Shifter, president of the Washington-based Inter-American Dialogue. \"The question is which candidate is best equipped to deliver both of those.\"\nRousseff and Neves have fought bitterly to convince voters that they can deliver on both growth and social advances. This year's campaign is widely considered the most acrimonious since Brazil's return to democracy in 1985, a battle between the only two parties to have held the presidency since 1995.\nNeves has hammered at Rousseff over a widening kickback scandal at state-run oil company Petrobras, with an informant telling investigators that the Workers' Party directly benefited from the scheme. Rousseff has rejected those allegations and told Brazilians that a vote for Neves would be support for returning Brazil to times of intense economic turbulence, hyperinflation and high unemployment, which the nation encountered when the Social Democrats last held power.\n\"We've worked so hard to better the lives of the people, and we won't let anything in this world, not even in this crisis nor all the pessimism, take away what they've conquered,\" Rousseff said before voting in southern Brazil.\nAfter he voted in his hometown in Minas Gerais state, Neves exuded confidence and said he's ready to lead all Brazilians, rich or poor. \"I'm in a much better position than her,\" he said of his opponent. \"We'll show that we'll maintain the social programs, that we'll make good on all our promises. If I win the election, my first big mission will be to unify the country.\"\nIn Rio de Janeiro, 43-year-old lifeguard Marcelo Barbosa dos Santos voted in the Botafogo neighborhood and said he's been a Rousseff backer from the beginning. \"Many things changed for the better during Dilma's administration,\" he said. \"The poor have seen our lives improved and we want that to continue.\"\nBut Paula Canongia, a 34-year-old hotel owner, said she voted for Neves because of \"the current state of our country.\" \"He's not an ideal candidate, far from it ... but we desperately need change and hopefully he can provide that,\" she said.\nPolls opened at 8 a.m. local (6 a.m. EDT; 10 a.m. GMT). Voting stations in far western Brazil close at 8 p.m. local time (6 p.m. EDT; 10 p.m. GMT), and with the nation's all-electronic voting system, a final result was expected within a few hours.\nOfficials from Brazil's top electoral court said voting went smoothly through late afternoon. However, there was a shooting at a polling location in the northeastern state of Rio Grande do Norte, when a man was shot and killed inside a school where ballots were cast. Police said it appeared to be gang-related.\nAssociated Press writers Jenny Barchfield in Rio de Janeiro and Adriana Gomez in Sao Paulo contributed to this report.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We service the whole state of Oklahoma including the Muldrow area and can help you with and answer your questions about Generac 8000 Watt Standby Generator for residential and commercial use. Service, repair and installation - we do it all!\nGenerators are incredible devices that permit you to continue on with your life when everyone else is waiting on the power to turn back on. This is especially true if you live in our storm-prone state. Electrical power can be out for days or weeks as a result of the damages caused by one of these calamities. Having an extra source of electric power provides peace of mind, safety and security, comfort, and benefits for your family in the house or your staff members and clients at the workplace.\nWe advise speaking to us on the phone for the very best service for your inquiry concerning Generac 8000 Watt Standby Generator. Our service area in Oklahoma is quite extensive and we have installed and serviced generators from one end of the state to the other - even Muldrow!\nWhen you phone us, you can also be directed to our support staff that know about your model and can supply you with the technological specifications or details that you need to keep your system running in tip top condition. From circuitry to batteries to oil changes, you can be assured that when you call us, you can find the most current info and support offered anywhere.\nIf you are considering purchasing a new standby generator and getting it installed at your residence or business property, there are great reasons why you should buy the generator from an accredited Generac dealership like us and not a big box store.\nFirst, a regional dealership will be around to help you with maintenance after the sale. In contrast, you never ever know who you will get when you call a retailer and locating the same contractor might be hard.\nSecondly, dealerships are aware of the current promotions and price cut programs offered. Also crucial, dealing with a regional Generac dealership means that you won't experience any kind of service warranty problems that could develop if a non-authorized installer is contracted to install your generator.\nCall us today for help with your Generac 8000 Watt Standby Generator questions in Muldrow, OK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ryan finally leaves the stage after an 'eventful' gig - The Cribs, Sheffield University, October 6, 2007.\nGary Jarman's obviously the talented member of The Cribs - look, he's playing two guitars! The Cribs, Sheffield University, October 6, 2007.\nRyan Jarman unceremoniously dumped on the stage after being mistaken for a crowd surfer - The Cribs, Sheffield University, October 6, 2007.\n'It's my stage!' Ryan attempts to escape security's clutches - The Cribs, Sheffield University, October 6, 2007.\nSecurity mistake Cribs singer Ryan Jarman for a scruffy Yorkshire urchin - funny that... The Cribs, Sheffield University, October 6, 2007.\nMaybe a case of 'Kiss Me Kate Nash'? The Cribs' Ryan Jarman, Sheffield University, October 6, 2007.\nToo right he looks annoyed, those jeans were brand new. The Cribs' Ryan Jarman, Birmingham Academy, October 5, 2007.\nEither Ryan's got three legs or those jeans aren't quite right... The Cribs, Birmingham Academy, October 5, 2007.\nA security guard heroically carries an unconscious Ryan Jarman back to the stage - The Cribs, Birmingham Academy, October 5, 2007.\nGary Jarman attempts a Spinal Tap moment - The Cribs at Birmingham Academy, October 5, 2007.\nRyan Jarman of The Cribs, perhaps not enjoying the crowd's fondling -Birmingham Academy, October 5, 2007.\nNo, Ryan, don't crowd surf, you never know what might happen... The Cribs, Birmingham Academy, October 5, 2007.\nThe boys in the band get ready for Brum - The Cribs, Birmingham Academy, October 5, 2007.\nRyan Jarman shows the only part of the rider he cares about - The Cribs, Birmingham Academy, October 5, 2007.\nRyan Jarman adrift in the crowd - The Cribs, Sheffield University, October 6, 2007.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalrud b/data_all_eng_slimpj/shuffled/split2/finalzzzalrud
new file mode 100644
index 0000000000000000000000000000000000000000..af5a0c581fbf88e80f5054db0e048c6108c0af5f
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Employers and grad schools want students who can think, and students who can do. ESP gives you the ideas and skills to put the liberal arts to work.\nThis is our bold educational perspective on education. It's a real-world \u2014 even other-worldly \u2014 way to learn that's relevant, rewarding, flexible, and fun.\nHere are the lists of skills and perspectives in the program. Put them together, and you have ESP.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Born in Finnsnes in 1968.\nGrew up in Bod\u00f8 in the north of Norway.\nHe made his debut as a writer in 1993 with the novel Tittell\u00f8s (Without title).\nHis most recent publication was a collection of literary essays in 2009, Livet! Litteraturen! (The Life! The Literature), for which he was named \"Critic of the Year\" by the Norwegian Critics Association.\n\u00d8ver\u00e5s has a degree in comparative literature and worked for several years as the editor of the Norwegian literary review Vinduet.\nToday he divides his time between northern Norway and Berlin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Judith H. Whittlesey, Executive Vice President, has decades of broad-based experience in strategic planning, media relations and spokesperson training, crisis communications strategies and management, institutional positioning for non-profit organizations and institutions, and the organization of issue, cause and political campaigns at the national, state and grassroots levels. Over the years, Ms. Whittlesey has developed the strategies and directed the teams that have launched some of our nation's most recognized institutions.\nMs. Whittlesey has created and administered award-winning communications strategies requiring diverse expertise that have effectively guided clients including government entities, non-profit and professional groups, associations, and national and international corporations through periods of both great opportunities and great challenges. She has developed effective strategic campaigns and initiatives for entities seeking to position an existent or new institution; promote a cause or concept; or launch and manage a groundbreaking national public education campaigns.\nShe has led initiatives and campaigns on behalf of the firm's health and wellness-related clients including Foundation Fighting Blindness; the National Campaign for Hearing Health; LUNGevity, the nation's pre-eminent organization funding and raising awareness for lung cancer research; the Medstar Washington Hospital Center and the Freddie Mac Foundation's annual Adoption Expo.\nMs. Whittlesey has led SDI's teams that created positioning communications campaign and designed and managed the various groundbreakings, dedications and anniversaries for national landmarks including the Women in Military Service for American Memorial, the U.S. Memorial Holocaust Museum, the National Museum of the Marine Corps, the Franklin Delano Roosevelt Memorial, the National Japanese American Memorial, and the Martin Luther King Jr. Memorial.\nIn support of the nation's military services, as well as military families, Ms. Whittlesey led SDI's award-winning work for the Department of Defense Employer Support of the Guard and Reserve Freedom Awards for many years. She leads SDI's team providing communications support and strategic planning for the Elizabeth Dole Foundation's ground-breaking work on behalf of the nation's military and veteran caregivers. And she leads the continuing work for the Marine Corps Heritage Foundation in support of the National Museum of the Marine Corps.\nWith background in the arts and education, Ms. Whittlesey leads SDI's communications initiatives and support for the Institute of Museum and Library Services, one of the U.S. government's primary sources for support for libraries and museums across the nation. She has also led communications and events initiatives for the Phillips Collection and the Museum of Women in the Arts. On behalf of corporate clients, she has led SDI's team in support of exhibitions at the National Gallery of Art, the National Portrait Gallery, the Smithsonian National Museum of Natural History, and in New York, the Metropolitan Museum and the Cooper Hewitt Museum.\nTo meet the multi-faceted needs of firm clients, Ms. Whittlesey has assembled and overseen the work of strong teams of sub-contracting firms, the creation of award-winning materials to support program and project goals, partnership programs to further client objectives, and outreach strategies to carry client messages to broad and diverse audiences.\nA graduate of the University of Oklahoma English Literature, Psychology and Education, Ms. Whittlesey is an enrolled member of the Choctaw Nation of Oklahoma. She was selected to Leadership Greater Washington, the National Capitol Public Relations Society of America Hall of Fame, and PR News inaugural selection of Top Women in PR.\nShe is President and long-serving member of the Board of Directors of the Food Research and Action Center, and a board member of Counterpart International. With a commitment to the greater Washington, DC community, Ms. Whittlesey previously served as a member of the Board of Directors of Decatur House, Suited for Change, the Levine School of Music, and Washington Project for the Arts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Read Katekyo Hitman Reborn 400 Raw online. Katekyo Hitman Reborn 400 Spoiler. You could read the latest and hottest Katekyo Hitman Reborn 400 in manga here. Katekyo Hitman Reborn Chapter 400 is the next chapter to come out on Katekyo Hitman Reborn 400 manga series. This page will soon be updated with Katekyo Hitman Reborn 400 manga scans, raw, spoilers or when Katekyo Hitman Reborn 400 is out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Atmospheric Jig Dyeing Machine is also named Laboratory Jigger Dyeing Machine which widely suitable for dyeing test of cotton, linen, fiber glue and artificial fiber etc., and can simulate the production and dyeing machine. Especially suit for Lab research and University .FYI is best manufacturer and supplier for Jig dyeing machine and supply goods solution for Dyeing machine .\nJigger Dyeing Machine is suitable to test the dyeing of cotton and linen cloth, viscose, man-made fiber, etc., and suitable for dye material, such as direct dyes, acid dyes, basic dyes, insoluble azo dyes, and reactive dyes, etc. This apparatus can achieve bleaching and dyeing with low-liquor ratio, simulates as nearly as possible the mechanism of production jigs.\nDyes Solution Cycle Mode: Auto-stirring, keep dyes solution evenness.\nAtmospheric Jig Dyeing Machine is specially designed to satisfy the requirements of testing dye formula before quantity production of dyeing and finishing enterprise. This apparatus simulates as nearly as possible the mechanism of production jigs' requirements of speed, temperature, pre-tension, etc. of real production process. Compared with production machine, this Laboratory Jig Dyeing Machine's size is smaller, consumes small quantity of dye, and with short testing process, so operator is easier to change dye formula, and finds suitable dye formula in a short time. Therefore this machine is well welcomed by many dyeing and finishing enterprise.\nCompared with production machine, this Laboratory Jig Dyeing Machine's size is smaller, consumes small quantity of dye, and with short testing process, so operator is easier to change dye formula, and finds suitable dye formula in a short time. Therefore this machine is well welcomed by many dyeing and finishing enterprise.\nIf you need more information about Jig Dyeing Machine , Please contact with us .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaluco b/data_all_eng_slimpj/shuffled/split2/finalzzzaluco
new file mode 100644
index 0000000000000000000000000000000000000000..2f968ec620f2e5c131cfae35bebff8a96a066069
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Mount Olive, N.C. --- The 3rd-ranked University of Mount Olive swept the final two game of the series 1-0, 7-3 against North Greenville University in Conference Carolinas baseball on Saturday afternoon in Mount Olive, N.C.\nMount Olive entered game one of the doubleheader in a stalemate as neither team's offenses could get much of anything going for the first three innings. NGU threatened in the top of the third with runners at the corners, but the Trojans struck out the final batter and the game remained scoreless.\nThe game remained deadlocked heading into the final at-bats for the Trojans in the bottom of the seventh.\nZach Mozingo singled with one away to give UMO a baserunner. Jose Lopez then moved Mozingo into scoring position with an infield groundout.\nDrew Ellis then came up with the clutch two-out RBI-single as Mozingo slid in safely at home and Mount Olive walked off with the 1-0 victory.\nAustin Hutchison (8-0) took the win on the mound as he went the distance holding North Greenville to just five hits while striking out seven Crusaders.\nEllis led the team with a 2-for-2 outing at the plate with the key walk-off RBI base hit.\nMount Olive fell behind in the top of the second in game two as North Greenville struck for a run off an RBI-single by Andrew Plunkett.\nThe Crusaders held the Trojans at bay for the next three innings as NGU added a second run off a Clark Scolamiero solo home run in the top of the sixth and led 2-0.\nUMO found the answer in the bottom of the sixth as the Trojans first three batters, Mozingo, Lopez, and Ellis all reached base to start off the inning.\nA wild pitch from the North Greenville pitcher allowed all three runners to advance including Mozingo who put UMO on the scoreboard as the Trojans trailed 2-1.\nDavid Mayo then drew a walk to reload the bases and following a fielder's choice, UMO had Mayo at second, Ellis at third, and Tony Salvaggio at first.\nJustin Carroll flew out to right allowing Ellis to score the tying run. However, on the play an errant throw from the Crusaders' right fielder allowed Mayo to score on the throwing error to give Mount Olive a 3-2 lead.\nThe Trojans went on to add a pair of runs in the bottom of the seventh and eighth innings to expand their lead to 7-3.\nDespite allowing two NGU hits in the top of the ninth Mount Olive shut the door in the final frame to pick up the game three win and a series sweep of the Crusaders.\nBrighton Hudson (5-0) picked up the win on the mound for the Trojans as he went seven and two-thirds, allowing three runs, off five hits, striking out four, and allowing one free pass.\nZach Mozingo (S8) picked up the save as he pitched the final inning and a third giving up two hits and struck out three.\nJose Lopez led the Trojans with two RBI.\nMount Olive will remain at Scarborough Field a single non-conference match-up with Francis Marion Tuesday, March 28. First pitch is set for 5 p.m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Watch Deep Impact (1998). Journalist Jenny Lerner is assigned to look into the background of Secretary Alan Rittenhouse who abruptly resigned from government citing his wife's ill health. She learns from his secretary that Rittenhouse was having an affair with someone named Ellie but when she confronts him, his strange reaction leads her to reconsider her story.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy Final Touch\u00ae Double-Wall Espresso Cup in Stainless Steel (Set of 2) from $19.99 at Bed Bath & Beyond. The Final Touch Double-Wall Espresso Cup makes it possible to take your time enjoying a nice, hot espresso. Hot drinks stay hot longer due to the cup's insulated stainless steel design, while the exterior stays temperature-neutral and easy to handle.The Final Touch Double-Wall Espresso Cup makes it possible to take your time enjoying a nice, hot espresso. Hot drinks stay hot longer due to the cup's insulated stainless steel design, while the exterior stays temperature-neutral and easy to handle.\nNow you can have your coffee and take it with you with these \"Coffee\" Collage To Stay and To Go Mugs by Konitz. Perfect for busy mornings, pour a cup of your favorite brew into the mug at home, then hit the road with the matching travel cup and enjoy.\nBrighten up your morning with the Konitz Poppy Blossom Mugs. Featuring lively poppies set against a rich blue background, these beautifully detailed mugs add a touch of personal style to your coffee or tea time.\nAdd an adorable touch to your coffee or tea time with the Konitz Frog Couple Mugs. Crafted of porcelain, these dishwasher-safe mugs feature a sweet frog couple bordered by charming polka dot trim.\nThe His\/Her Better Half Mug Set from American Atelier is perfect for any couple. Enjoy a morning cup of your favorite drink with these trendy mugs, each featuring a white finish and sloped handle, as well as a special design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"shiny bulbasaur, use tackle again!\nUse tackle to finish him off!!\ntry for one more hit Bulbasaur!! you can do it!\nShiny Bulbasaur inflicted 13 damage!! It was a critical hit!!\nAmanda: I accept your challenge!\nGo bulbasaur! use leech seed!\nShiny Bulbasaur inflicted 14 damage it was super effective!!\nSquirtle HP = 11! Shiny Bulbasaur HP = 32!!\nBulbasaur inflicted 10 damage (from tackle) !!\nBulbasaur inflicted 18 damage! It was a critical hit, It was Super effective!\nSquirtle was Seeded! Squirtle HP=12, Shiny Bulbasaur Hp =32!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dirt is the second studio album by the American grunge band Alice in Chains and was released on September 29, 1992, through Columbia Records.\nPeaking at number six on the Billboard 200, the album was well received by music critics and has since been certified four-times platinum by the RIAA, making Dirt the band's highest selling album to date.\nThe album spawned five singles: \"Would?\", \"Them Bones\", \"Angry Chair\", \"Rooster\", and \"Down in a Hole\". The songs on the album focused on depression, anger, drug use, war, death, and other emotionally heavy topics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamyaj b/data_all_eng_slimpj/shuffled/split2/finalzzzamyaj
new file mode 100644
index 0000000000000000000000000000000000000000..e9d6a3d88b4da8536306135d60bf9e9527198538
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Our detailed needs assessment is based on years of industry experience. We understand that no two individuals are identical and thus they will require unique support and services. Based on use history, co-occurring disorders or related issues, our team will develop a unique program offering that will align your treatment experience for the best possible results.\nMore Than Rehab provides the full continuum of care to its patients. This means that no matter your stage of recovery or need, we can help. We are licensed for all levels of care including detox, full-residential, partial hospitalization (PHP) , and intensive outpatient (IOP). Our goal is to provide a single solution to those struggling with addiction in the Texas area.\nAt More Than Rehab we understand that fads in treatment come and go. We rely on proven modalities that are known to create lasting results. Three types of intense therapy that have proven success records are EMDR, CBT, and DBT. Specialists trained in these modalities are brought in on a case by case basis to provide therapy services for clients requiring this added level of care. We understand that no two cases are the same and never approach our treatment programs with an out of the box solution. Custom catered treatment plans are key to quality success rate More Than Rehab has developed.\nOne of the most impactful and effective modalities in addiction treatment is that of individual therapy. These individual therapy sessions occur between you or your loved one and their individually assigned therapist. During these sessions, clients will explore the deeper underlying issues behind their addiction; self-esteem issues, depression, abuse, PTSD, etc. Our caring therapists have years of experience in providing individual therapy in an addiction treatment environment.\nNever underestimate the power of a group. Group therapy often leads to fellowship and group-wide progress. Learning that we are not alone is key to developing a sense of self-worth. Clients often find peace in processing daily interactions, struggles, or hauntings of the past in a group setting and receiving positive reinforcement from fellow clients. Group therapy can be based on group processes, or a specific topic such as faith, meditation, triggers or families.\nTesting and accountability is key to lasting sobriety. At More Than Rehab we understand that addicts can be sneaky and we are on to it. We offer a safe and secure environment for recovery. One way in which we ensure this is through regular drug and alcohol testing. Our testing is connected to a reputable lab to test levels on over 26 substances. We can monitor medications as well through our testing, ensuring clients are taking what they are prescribed, as prescribed. Often that added level of fear of testing helps to keep clients sober during their first 30 days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever felt like there is another world out there? Maybe a parallel universe that strongly resembles ours but isn't quite the same. Degrees of Separation is a game that stems from the idea of two realms combining forces in an interesting 2D puzzle platformer environment. From the industry veteran and RPG legend, Chris Avellone, comes a title that has the lovely story of a girl and boy those world intertwine and together they must go on a journey although they cannot touch each-other. However love conquers all and how closer the two are in the relationship, how smaller the separation will be and maybe one day they will be together.\n2D Adventure Platforming. Experience beautifully crafted 2D worlds with a deeply immersive art style based on the aspects of contrasting temperatures.\nInnovative Storytelling. Discover the potential of dual protagonists Ember and Rime and follow a complex story by acclaimed video game writer Chris Avellone.\nEnvironmental Puzzles. Navigate distinct worlds and solve complex puzzles to organically alter situations and surroundings.\nCooperative Gameplay. Harness the powers of Ember and Rime in both single-player and cooperative multiplayer to interact with the environment in a variety of ways, both separately and as a team.\nNon-Linear Levels. Traverse through diverse landscapes that each introduce a new game mechanic, weaving together gameplay, narrative, and character progression.\nDegrees of Separation will get released on Valentine's Day 2019 (that's February 14th so don't forget to buy your beloved a present or chocolate) for Xbox one, PlayStation 4, Nintendo Switch and Steam.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Professor Douglas Robertson introduced workshop two by recapitulating the first workshop activities, highlighting what worked and what perhaps wasn't so successful, noting that participant preferences did not always match those of the planning team.\nHe went on to introduce the first activity which explores how notions of home and wellbeing tie into our idea of neighbourhood.\nThe participants were asked to write down three comments and then give away one to each of three people. Professor Deborah Peel asks some of the group to say the word they were given.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today I played tennis, for the first time in ages, with the Meetup group. It was stupidly windy out there (wind must play a huge part in Wellington interclub competition). We didn't play any games, which was probably just as well. One of the guys had never played before, and didn't seem to care that he couldn't keep any kind of rally going, or even hit the ball a lot of the time. I didn't care much either, but I would have felt extremely self-conscious in his situation. He kept talking about \"Sanchez shots\" as if someone called Sanchez was currently a top-ranked male tennis player. We later ascertained that he meant Sampras. As for me, I enjoyed the hit-out and wouldn't mind getting back into it properly if I could find the time.\nWhen I got home after tennis, Ana Ivanovic had just taken the second set in her shock three-set win over Serena Williams. I was happy with that result - I've never been a huge Serena fan, and I remember seeing Ivanovic as a teenager in 2005 and I was mighty impressed with her. There was some drama as Serena served at 2-5 in the final set. She was (rather unfairly) given a time violation, and responded by taking virtually no time in between points. She hung onto her serve, saving a match point, but Ivanovic served out to love, beating the player everyone said was unbeatable. It was a great performance, regardless of any injury Serena might have had, and any of the dozen players left in the tournament has a chance of lifting the trophy.\nI've just updated my CV, but I guess I need several versions. It's tricky - I'm saying, look, this is what I've done for the last ten years, but that's all irrelevant because now I don't want to do anything remotely like it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hand tooling of leather is an ancient art performed by artisans who painstakingly chisel and carve the leather by hand. Each leather hide is selected for quality. Leathers may have natural markings and, since no two are alike, will vary in color.\nThis time honored tradition inspired these hand tooled western leather ottomans with a distinctive Spanish flair. Besides offering a unique design, this leather furniture is created for durability and lasting quality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamzfz b/data_all_eng_slimpj/shuffled/split2/finalzzzamzfz
new file mode 100644
index 0000000000000000000000000000000000000000..5e2313ff849c732ebad730c636af1e31c42a88f6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamzfz
@@ -0,0 +1,5 @@
+{"text":"Unfortunately there was a house break-in yesterday evening in Auchtertool.\nPlease take care to ensure your property is secure and be vigilent for any unusual activity around the village.\nPlease contact the Police if you notice anything suspicious and they'll investigate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NEW ORLEANS - The Cleveland Browns intercepted four passes and returned two for touchdowns Sunday as they took a 30-17 win over the New Orleans Saints.\nCleveland linebacker David Bowens ran back the pair of pick-sixs.\nRookie Colt McCoy threw for just 74 yards but got his first win as an NFL starter.\nCleveland (2-5) scored its only offensive touchdown on a 4-yard run by Peyton Hillis.\nFormer Saints linebacker Scott Fujita had an interception and a sack. Sheldon Brown also picked off a pass from Saints quarterback Drew Brees. The quarterback has not thrown four interceptions since a loss to Tennessee in September 2007.\nNew Orleans (4-3) never led in the game. Cleveland jumped out to a 20-3 halftime lead.\nCleveland punter Reggie Hodges surprised the Saints on a fake run up the middle for 68 yards. The Browns went on to hit a field goal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eastop ERP System is a famous ERP system around the world and highly recommended by our customers since 1993. Eastop ERP System is innovative, mature, stable and flexible, creating tailored results for many industries worldwide.\nEastop ERP System serves companies across a broad spectrum of industries, including trading, manufacturing, wholesale and retail of electronic components, electronic products, electrical appliances, chemicals, cosmetics, food, pharmaceuticals, toys, watches, jewelry, equipment, plastics, printing, lighting, textiles, metalware, apparel, and construction etc., accommodates the unique requirements of each industry without sacrificing ease-of-use.\nBuild-in Function-On-Demand Tool is one of the most important keys why our customers can use Eastop ERP for more than 2 decades. The customizations are compiled and stored in clients' database and seamlessly integrated with our standard EASTOP.EXE, and thus all the clients from more than 20 industries are eligible for standard upgrades.\nKeep searching an ERP System to fulfill your growing business? Then, Eastop ERP System will be the final solution for you.\nAll systems share same database and provide centralized management which greatly simplify our clients' daily operations.\nEastop Office Automation System (EOA) is a fully integrated Trading, Inventory and Accounting System, and also the main components of Eastop ERP System.\nEASTOP Manufacturing Automation System is a production managememnt system designed for various types of factories. It is able to fulfill the concrete requirements issued by the factory. Different modes of production no matter the small batch process or the mass production are applicable.\nEastop POS is an all-rounded retail system for various type of industry. Eastop POS integrates all front-end, back-end, inventory and accounting modules with centralized control at Eastop ERP System. Eastop POS also shows all information real time.\nEastop e-Commerce System manages all homepage, products, pricing, members, e-Coupon, news and promotion in Eastop ERP system to increase efficiency and consistency. Online, in-store and mobile orders are integrated in Eastop ERP system, unifying the data as well as increasing ROI.\nEastop Mobile Apps Store is a moving shop created to satisfy the demands of \"always-on\" consumers. One stop management for all Trading, Wholesale, Shops, Web Shop and Mobile Store. Products, Members, e-Coupon, News, Promotion are all managed in Eastop ERP System efficiently and consistently.\nEastop Warehouse Management System (WMS) designed for trading, wholesale, retail, storage and logistic company, optimized for operation on the Android platform.\nEastop iPOS is POS system which runs on iOS platform, including iPad, iPhone and other iOS devices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are a field service company, you know your customers expect a timely and accurate service. But unfortunately, this does not automatically imply that they pay you in the same timely and accurate manner. Once the urgency gone, the need for a quick follow-up by your customer of your invoice also drops drastically.\nWe give you five tips that may help you to be payed faster.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is there some particular reason to use INSPECT?\nActually I have verified that data will be like 'ABCD123456' or 'ABC1D56789' it won't be as specified.\nOk so the count would be atleast 3 and max of 4?\n03 WS-TEST PIC X(10) VALUE 'ABC1D67892'.\n03 WS-ALPHA-KTR PIC 9(08) COMP.\nOnce the PERFORM completes, WS-ALPHA-KTR will equal 4.\nSo why not adjust the code? You did take the time to understand what the code is doing, didn't you?\n03 ws-input pic x(10) value 'ABC1D67892'.\nWhy do you not use the code tags to present your code in a readable manner with proper indentation as other people have done?\nYou have to bear in mind that we don't know what you want, but you do.\nYou have told us, with some contradictions, how you want to do what you want, but we don't know what you are trying to achieve.\nHave you tested your solution with A1B2C12345?\nWith Bill's solution, why don't you just stop after looking at five bytes or when you find a numeric?\nWith GuyC's, just start from five and wind down.\nIf you are going to attempt to use what you end up with for some calculation, please check it for NUMERIC first.\nRather than ALPHABETIC or SPACE, why not NOT NUMERIC? It'll be less of a strain on the old CPU and more directly describe what you are doing - perhaps, because we don't know.\nIt appears that there may not be an actual requirement. Just some wonder about \"what if I have . . . \".\nSuggest rupeshgullu explain what kind of data this would support.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanahl b/data_all_eng_slimpj/shuffled/split2/finalzzzanahl
new file mode 100644
index 0000000000000000000000000000000000000000..70cd453d1a8d281fef3a084d04da182f8685ad0e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanahl
@@ -0,0 +1,5 @@
+{"text":"When you're in the market for a new or used car, truck or van, a great auto loan can make all the difference.\nWith competitive rates and term options from 60 to 84 months, finding the right loan at ISU Credit Union is easy. Even if the dealer is offering 0% APR on that new car of your dreams, you may actually be better off taking the rebate and financing it with ISU Credit Union. Just plug a few numbers into our easy-to-use online calculator to compare which deal will really save you money!\nIf you're already a member, you can apply for an auto loan online.\nFind a Price, Find a Vehicle, Find a Dealer - Find It Free!\nOrder a CARFAX Vehicle History Report.\nFind new and used car prices, and auto reviews.\nYou don't have to stop by an ISU Credit Union branch to get a great auto loan. At participating dealerships, members can get their ISU Credit Union loan within minutes\u2014right from the dealer.\n*Annual Percentage Rate. Up to 60 months On Approved Credit. Other rates and terms may apply. Offer does not apply to existing ISU Credit Union loans. Rates subject to change without notice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Valves, fluid control systems and component manufacturing and assembly. Aftermarket TPMS, air accessory and tire hardware distribution. Schrader products shipped directly to OEM and Aftermarket customers.\nSales, Product Development, Customer Design, Marketing, Manufacturing and Distribution.\n1898 \u2013 Creation in France of a subsidiary of Dubied (CH), specialty in cycle valves and accessories.\n1998 \u2013 Schrader belongs to the Schrader-Bridgeport International Inc. and Tomkins (UK) group.\n1999 \u2013 Manufacturing of valves for TPMS (Tire Pressure Monitoring System).\n1946 \u2013 Rubber valves manufacturing.\n1984 \u2013 Michelin inflating gauges manufacturing.\n1990 \u2013 Manufacturing of valves and connectors for car air conditioning and injection fuel systems.\nISO 9001 v2008, ISO\/TS 16949 v2009, ISO 14001 v2004, EN\/AS\/JISQ 9100 v2009, Q1 FORD, OSHAS 18001 v2007.\nTake a video tour through this Schrader location.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Office for Sale - $258,500 - 9,993 Sq. Ft.\n918 SF+\/- office\/retail condo located in North Beach Professional Building. Floor Plan includes one (1) private office, reception area\/open work area, conference room and storage\/supply room. Sale subject to existing lease with Edwards Jones. Condo documents available upon request. Perfect opportunity for investor in growing Town of North Beach!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"David A. Winston, a former unpaid adviser to the White House who coordinated the Reagan Administration's strategy on health policy, died of respiratory arrest caused by spinal cord injuries Thursday in San Francisco. He was 44 years old.\nThe injuries came in an assault he suffered the previous Saturday while visiting the city on business. He lived in Alexandria, Va.\nMr. Winston was attacked after he crossed a street, forcing a motorist to brake to avoid a collision. The police said that in an ensuing quarrel, the motorist struck Mr. Winston, causing him to fall, and his head struck the pavement. A 22-year-old sheet metal worker later turned himself in and was charged with manslaughter.\nMr. Winston was corporate senior vice president of Voluntary Hospitals of America, an alliance of major private hospitals.\nHe was previously Republican staff director for the Senate Committee on Labor and Human Resources and headed Ronald Reagan's pre-inauguration team on health care.\nHe was born in Manteca, Calif., and graduated from San Francisco State University. He also held a master's degree from the University of California at Santa Barbara.\nHe is survived by his wife, Susan; a son, Heath; a daughter, Shannon; his mother, Dorothy Winston of Manteca; two brothers, William of Boise, Idaho, and James of Manteca, and a sister, Diane Branaugh of Pittsburg, Calif..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"away from home with family and friends. It's a time in our busy lives when we look forward to strengthening our relationships with those we love. This is a basic need that God has placed inside each and every one of us; a need that cries out to be met. Just like eating, sleeping and working bring satisfaction, so must our relationships be fulfilled.\nWhat if we have experienced rejection? How does it affect our relationships? Maybe a parent abandoned us or gave us up for adoption. It may have begun at conception, while a mother was struggling with the decision of abortion. Was there ever a time when we felt unwanted because we were a boy instead of a girl? Did the peer rejection at school or work leave you feeling as though you did not belong? Was love withheld from you as you experienced abuse sexually, physically or verbally? Did betrayal destroy your value as your spouse was unfaithful in your marriage that led you to a divorce? These deep roots of rejection force us to look for things that will fill our void and sometimes we place unfair demands on our friends and loved ones. Rejection hurts and is painful. We grieve that loss but at the same time we set up defences to ward off possible future hurts.\nOur controlling defence causes us to live in fear, and sometimes, retreat and isolation from others. Why? If I get too close, I might hurt them more. It's a constant striving for acceptance to the point of becoming a people-pleaser, a false identity of who we really are. We begin to focus on no one but ourselves and we respond in anger as rebellion rears its ugly head. We are immersed with self-pity and cease to care.\nThe most important relationship we can work on is our relationship with God. Surrender your whole self, wounds and all. His grace will give you the ability to find healing and restoration. He will set you free from the devastating words or actions that robbed you of knowing His love. We are reminded that we must \"bear with each other and forgive one another; if any of you has a grievance against someone, forgive as the Lord forgave you.\" (Colossians 3:13).\nSurround yourself in healthy relationships that build you up in Christ and let the joy of the Lord be your strength. May your summer vacation be filled with freedom beyond measure!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanhqr b/data_all_eng_slimpj/shuffled/split2/finalzzzanhqr
new file mode 100644
index 0000000000000000000000000000000000000000..821ff9b88fc115048ee0a55da1c4c5f118b7f9a0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanhqr
@@ -0,0 +1,5 @@
+{"text":"Steve is a native of Appleton, WI, and a graduate of the University of Wisconsin-Eau Claire. Steve is a longtime and current resident of Menomonie WI, with over 30 years' experience in the publishing world. He joined Dowell Management in September 2014 as our Vice President and serves as the Executive Director for Momentum West, based in Eau Claire. Since 2007, Steve was vice president\/publisher of Finance & Commerce, Inc. He led the Dolan Company's Twin Cities operations, publishers of Finance & Commerce, Capitol Report\/Politics in Minnesota, and Minnesota Lawyer. His responsibilities included overseeing operations with a staff of 60 employees involved in producing news, data and information. From 1996-2007 he was publisher of the Dunn County News and held several upper management positions at Chippewa Valley Newspapers, a regional group of newspapers in west central Wisconsin.\nWhat are you complimented on most in your work?\nWhat one problem are you best at solving for your clients?\nWhat are you most passionate about \u2013 professionally and personally?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Just love your journals and marvel at one my good friend won a couple of years ago. They are so beautiful. Do you still plan on doing an online class for those of us too far away to take one of your classes around the world? Hope so!\nSo excited you are coming to ESS in April. Looking forward to the class and learning from you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Camelot at Federal Hill!\nWelcome to Camelot at Federal Hill! Here, you will experience a great place to live, work and play with every luxury and convenience you want for your lifestyle. Impressive is the easy access to abundant shopping, the Jersey shore, recreational and cultural advantages, as well as bus and rail transportation to New York City and major New Jersey business centers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who does not know this mask, with his patched outfit and witty tongue? The first Harlequin we hear of is Alberto Naselli, from Bergamo, in 1572.\nHarlequin is a poor character, wearing a colorful patched costume, the patches probably remnants of richer costumes.\nAnarchic kind of behavior, always hungry and without money, in modern times he could have been a hippie.\nHe's hungry, always, but he has all the necessary wit and creativity to get along, survive, take his satisfactions on the world that is constantly abusing him.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The blog suffered a fairly large and extensive hack several weeks ago, just as Lee was starting a busy rotation. I was loathe to ask him to take the time to fix it until yesterday, when he had a somewhat empty day off. Six hours later, we managed to deal with the damage. It actually looks like two different problems, one from several years ago that went undetected, and then the more recent one. Thankfully we backup the content weekly, but it still took quite a lot of time fixing things. I am thankful that my husband was willing to sacrifice the time. Otherwise, I would have simply deleted everything and started over.\nAs it is, our lives are so busy, I'll probably save the appearance issues for later. Lee simply doesn't have the time right now. It will come, in its own time. But content is more important than appearance, right?\nI am hopeful that everything works properly, but if you see something that doesn't look right, would you send me an email and let me know? Thank you. I missed you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanxei b/data_all_eng_slimpj/shuffled/split2/finalzzzanxei
new file mode 100644
index 0000000000000000000000000000000000000000..5d532c59592162cce34238c906859c9cba4560d1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanxei
@@ -0,0 +1,5 @@
+{"text":"The website www.shapeyoursuccesscoaching.com (hereinafter \"Website\") is owned and operated by Shape Your Success Coaching, LLC, a New York Limited Liability Company (hereinafter \"we\" \"us\" \"our\"). Please read the following Disclaimer before using our Website - by visiting and using this Website, you (hereinafter \"visitor\" \"you\" \"your\") are consenting that you have read this Disclaimer to the point that you feel you understand the terms outlined herein, and agree to be bound by them.\nB. I am not a doctor, lawyer, accountant, therapist, counselor, dietician, or other professional; User agrees he or she will consult with his or her personal doctor, lawyer, or other professional prior to or in addition to utilizing information found on this website, and such information will not substitute for any consultation with User's own professional. The information contained on Website or via emails, or available through our products, programs, and services is not intended to be a substitute for any professional, personal advice, nor is anything contained herein designed to provide you with a medical diagnosis, treatment, or other medical services.\nA. You understand and agree that your participation in using our website and\/or adding yourself to our email list is wholly voluntary, and you are solely and personally responsible for your actions, choices, and any results therein. You understand there are sometimes unknown risks and circumstances that may arise during or following use of our Website or products, that cannot be foreseen or anticipated, but may influence or affect your business or you as an individual. You understand and agree that any suggestion or recommendation of a product, service, coach, or otherwise through our Website is purely information \u2013 any decision to act upon these suggestions is to be taken by you, at your own risk, without any liability on the part of Shape Your Success Coaching, LLC. You agree to accept all risks herein.\nB. Your use of this Website constitutes an agreement and acceptance that you will absolve Shape Your Success Coaching as well anyone acting as an agent, consultant, affiliate, guest blogger, joint venture partner, employee, staff, team member, or anyone affiliated with Us in any way of any liability for any loss, damage, injury, or litigation that you or any other person may incur from direct or indirect use of the information, content, or products found on our Website or via materials requested through email.\nA. You agree at all times to defend, fully indemnify and hold harmless Shape Your Success Coaching, LLC and any affiliates, agents, team members or other party associated with it from any causes of action, damages, losses, costs, expenses incurred as a result of your use of our Website or any products or services contained therein, as well as any third party claims of any kind (including attorney's fees) arising from your actions in relation to our Website or any breach by you of any such conditions outlined herein. Should we be required to defend ourselves in any action directly or indirectly involving you, or an action where we decide your participation or assistance would benefit our defense, you agree to participate and provide any evidence, documents, testimony, or other information deemed useful by us, free of charge.\nA. Although we have spent considerable time and effort in creating the products on www.shapeyoursuccesscoaching.com and the content provided herein, you understand and acknowledge that we are not responsible nor liable for any errors, omissions, or liability as a result of any loss or damages incurred as a direct or indirect result of your use of Website content or our products. You also understand there may be inadvertent typographical errors or inaccuracies. By your use of this website, you acknowledge and understand this information, and agree you have chosen\/will choose to utilize our Website and\/or our products voluntarily. You agree that we are not responsible for the accuracy of our Website, or for any errors or omissions that may occur on the site or in our products.\nA. Website may feature testimonials from clients in order to provide readers with additional comments from others' experiences with Website, Shape Your Success Coaching, LLC, and products or services offered. While all information, photos, and quotes used are from actual clients, sharing their real, honest opinions of our website and services, these testimonials are not to be considered as a guarantee that current or future clients will experience the same results, or a guarantee that all clients will have the same experience. You understand and agree that by reading a featured testimonial on our Website, you do not expect the same results, and understand this information is not a guarantee.\nYour use of www.shapeyoursuccesscoaching.com constitutes full and complete acceptance and agreement to this Disclaimer. Please contact crystal@shapeyoursuccesscoaching.com with any questions or concerns.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I hope you had nice thanksgiving. We had a great time with our family and enjoyed some yummy food but after a couple days of splurging, I was glad to get on track today with a green smoothie and some sweet potatoes and a veggie burger. Splurging on food I don't normally eat is fun but I always look forward to going back to healthy eating.\nSo, did you do any Black Friday shopping??? I am normally anti-black Friday because I personally do not have the patience to deal with huge crowds and long lines to get a \"deal\" For the most part, sales are offered all month long and you can buy a lot of gifts online so I usually don't bother on Black Friday. Years ago, I used to show up on Black Friday to find what I came for was already sold out so I have been against it ever since.\nThis year however, something spiked my interest at Target. The $250 giftcard with Iphone. What? I have been wanting a new IPhone for years. I'm not one to get the latest and greatest when it comes to phones or electronics. I couldn't afford to anyway. So as much as I dislike crowds and chaos, my daughter and I went to Target on black Friday which now starts on Thanksgiving at 6pm. It seems to happen earlier every year which seems a little unfair for the employees.\nMy daughter and I showed up 30 minutes before the store opened to find a long line already. When we got in the store, there was already about 8 people lined up for Iphones. It took each person 20-30 minutes to get their phone purchased and activated so I knew it was going to be a while. We waited in line for an hour and then we left for an hour thinking that would be enough time. When we arrived back at the store, there was still a long wait. I was getting a little inpatient and starting to get a headache.\nThis is what we looked like after a few hours. Drained.\nThankfully, we had some nice people around us to talk to while waiting.\nWhile we waiting, we took turns looking around the store. My daughter found a pair of boots and some cute scarves. All apparel and boots were 40% off!\nAfter 4 long hours, I finally had my new Iphone in Rose Gold. Btw, it was the last one in the size and color I wanted.\nAnd my $250 Target giftcard.\nI came home with a headache but I also had a new phone and a giftcard.\nBecause I go the phone I wanted with 0 down (just tax) and was able to buy some things we needed like a new printer.\nAnd of course some gifts, which I purchased online.\nUnless its something big that's going to save a lot of $$$$ I will pass.\nWhat are your thoughts on Black Friday? Do you think it should be on Thanksgiving or is that taking away from the whole meaning of Thanksgiving?\nIf I ever go again, I will not go on Thanksgiving.\nHow far are you willing to go to save big?\nAfter yoga, my husband and I took our dog for a walk and enjoyed this beautiful day.\nHow did you enjoy your thanksgiving?\nEvery year we go to my husband's parents house which I love because they live only a few miles away and my mother in law is the best cook. Every year she does all the cooking and I bring a side, aka my sweet potatoes. This year I also brought a salad.\nThis year I decided to make my mashed sweet potatoes in the crockpot because you know I like convenience when cooking. It was so simple. I peeled and cut the potatoes into chunks, threw in crockpot and 4 hours later they were ready to be mashed. I added cinnamon, butter (vegan), almond milk, and many syrup. Everyone said they turned on it good.\nMy mother in law makes the best broccoli casserole.\nChecking out those \"Black Friday\" deals. Did you or do you plan on partaking in all that deal shopping? I'm normally against it but this year I went for one thing only. More on that later.\nHemphearts recipe, review and giveaway!\nI recently had the opportunity to receive a free sample pack of hemphearts from Manitoba Harvest and write a review. Because I have been using hemphearts for over a year now and love them, I was more than happy to share a recipe and a review.\nI discovered hempseeds during my search for plant-based protein over a year ago when I gave up meat for good and became a vegetarian. Yes, I'm happy to say I have been meatless for over a year and it feels great. It has really helped me to think outside the box and become more creative with my food. And of course, the question I get asked most is, \"where do you get your protein if you don't eat meat?\" My answer is always \"from plants\" where else?\nHempseeds are my favorite way to get protein and Omegas. Did you know that they contain a whopping 10 grams of protein and 10 grams of Omegas per serving? They are lower in carbs than flax and chia seeds, they are raw and NON-GMO. They have a yummy nutty flavor similar to pine nuts or sunflower seeds. I've tried all different plant proteins, rice, soy etc. and hemp is my favorite. For me, it's much easier to digest.\nToday, I'm sharing one of my favorite recipes using hempseeds. Peanut butter chocolate balls. I've shared this peanut butter ball recipe before, you can see it here. This time I added some raw cocao nibs for a more decadent peanut butter experience. Not sure if you know but peanut butter and chocolate are my favorite combo ever. It can be dangerous at times, but thankfully I have learned to create healthier versions of my favorite (but unhealthy) snacks and desserts.\nHere is everything I used. Pretty simple right? I buy my maple syrup and dates in bulk at Costco. Its saves $$$$ and lasts for a while. I also buy my oats in bulk, usually at Sprouts.\n1 cup of dates, 1 cup of oats, 1\/2 cup of peanut butter, 1\/8 cup of maple syrup, 1 Tbs of hemphearts, 2 tbs of raw cocao nibs Place all ingredients into food processor and mix away.\nYou would think you roll them into balls but you actually have to press them into balls. Every time I try to roll them, they fall apart. It gives your hand a nice little workout in the process. Sometimes I have to involve my kids in the process if I'm making a large batch or if I've weight trained that day. Mom's hands get tired sometimes.\nWhen you are done, they should look something like this. They very first time I made them, my husband said they looked like meatballs. Ew! I guess I could see that.\nI ate about 3 of them while photographing. Sorry, I couldn't help myself.\nRusty thought he was going to get one but no can do. Chocolate is not for dogs. I repeat, chocolate is not for dogs. Especially not raw cocao. One time, we had an awful scare with chocolate. I'll share that in another post. Back to hempseeds.\nI want to thank Manitoba Harvest and Sweat Pink for allowing me this opportunity to review hemphearts. I had a lot of fun creating good stuff with them.\nDisclaimer: Although I did receive a sample of hemphearts for free in exchange for a review, all thoughts and opinions are my own.\nNow for the best part, would you like to win a free bag of hemphearts????? Manitoba Harvest will send one lucky reader a 2.5 oz bag of yummy hemphearts. All you have to do is comment below and follow Manitoba Harvest on one or more social media channels. Facebook, Twitter, Instagram Note: This giveaway is open to United States and Canada residents only.\nOne last thing, I usually buy my hemphearts from sprouts or amazon but you can also get them directly from Manitoba Harvest and use promo code HHSP1015 for 20% off your purchase.\nHave you tried hemphearts? Please leave me a comment below for a chance to win a free bag. Good luck!\nHow are you today? I feel kind of like junk. Why? Because I over splurged on Halloween Candy. I always say I'm going to stay strong and resist the candy but who am I trying to fool? I'm human and I'm weak, especially when it comes to chocolate. I often hear, its ok as long as its in moderation but I also hear and believe that if there's a certain food you do not have enough discipline to eat in moderation, you should not have it in your house. This my relationship with peanut butter and chocolate. Its bad. Can anyone relate?\nAnyhow, I threw out all the candy today. Well, I didn't throw it out, I sent it with my husband to work. Not for him, for his co-workers. Maybe that's not setting the best example but I know they will eat it. I have this thing about letting stuff go to waste.\nNow, its time to detox. Here are some of the foods I enjoy after a splurge. It helps me feel clean and normal again.\nFor lunch my son and I shared some quinoa with apples, raisins and pecans. He didn't want it but I made him eat it. Then, he realized its not that bad \ud83d\ude42 Kids.\nFor breakfast, I had a bowl of oatmeal with bananas, almond milk and hempseeds. My favorite. I usually have a smoothie or oatmeal but this morning was so cold and I needed something hot.\nNext I need to make a batch of these green smoothies. I missed them during the time while I was working full time and had time for nothing.\nMango and Banana. That was my favorite this summer. Its making my mouth water right now.\nAnd who can forget the chocolate frosty! This chocolate I never feel guilty about.\nToday is Taco Tuesday at my house, well according to my son. So as they have their turkey meat tacos I will enjoy some veggie tacos and sweet potatoes. Btw, I am vegetarian, my family is not. Although they eat a lot more plant based meals than they used to. I'm working on it.\nSo there you have it. I enjoy eating healthy but I fall short sometimes so this is what I will be eating lately. I have 3 weeks until the next big temptation comes. Thanksgiving! Can you believe its November already? Where has this year gone?\nI'll be back with my hemphearts post later.\nOff to workout and see my clients. Have a good day.\nHow was your Monday? I am not quite used to this time change yet and I'm still not sure why we have a time change anyway. Who enjoys it being dark at 5:00pm?\nThis weekend I made a yummy acai bowl. I recently went to a smoothie bowl shop and ordered one just like this. As I ate it, I thought this would be pretty simple to make myself and probably cost a lot less so I bought a large pack last time I was at Costco.\nDo you ever feel like you're speaking another language when you pronounce Acai? I still have trouble saying it correctly, according to my daughter.\nI think it was around $10. Not bad for 10 individual pouches.\nBtw, I just realized the picture on the front of the package looks a lot like the bowl I made. Trust me, I didn't notice that until now. So I guess acai with bananas and granola is quite common afterall.\nThey have some recipes on the back but I haven't tried any yet.\nI used two of these packets for my smoothie bowl. One packet was not going to cut it.\nI ran under water for 5 seconds like the instructions said, then I threw in my Ninja and blended away.\nNext, I poured into a bowl, added some sliced bananas, granola and hempseeds and enjoyed every minute of it.\nOnce granola was gone, I added more hempseeds. Yum.\nQuite yummy, I must say.\nHow hard is that? It took less than 5 minutes to put together and if it took any longer, I probably wouldn't be writing about it. Lol.\nEnjoy! Now I'm off to watch some Dancing with the Stars. Do you ever watch it? I'm a little obsessed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fashion Gallery London introduces all the women's fashion shows from the latest London fashion week. A wide preview of the best outfits, featuring the top London designers and brands presented in single sections. It is the most complete and quickest trade publication of this kind, containing full-size images in a smart, essential layout. Beside the catwalk sequences, two opening sections show the top trends and key colors of the upcoming season offering creative inputs and ideas for any professional fashion project.\nMagazine covers are used for illustrative purposes only and you may not receive a copy of the particular issue depicted. Your subscription will include the most recent issue once your subscription begins. Magazine covers are the property of the publisher. This site is not officially affiliated with, associated with, or endorsed by FASHION GALLERY LONDON (Digital) or the publisher.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Council has a number of policies and guidelines that provide additional information and guidance to the controls included within the Local Environmental Plan (LEP) and the Waverley Development Control Plan (DCP).\nAs per Section 7.12 of the Environmental Planning and Assessment Act 1979, Waverley Council can levy development contributions towards the cost of public facilities and services to meet the increased demand created by new development.\nA Planning Agreement (PA) is an agreement entered into by Council and a developer. Under the agreement, a developer agrees to provide or fund the likes of public amenities and public services, affordable housing, transport or other infrastructure. Council's Planning Agreement Policy 2014 came into effect on 21 October 2014.\nA list of other current Council policies can be found on the Plans, policies and reports page.\nFor more information about Council's planning controls, policies and agreements, please contact our Duty Planner for advice. This service is available between 8:30am and 5pm, Monday to Friday.\nVisit in person at our Customer Service Centre, 55 Spring Street, Bondi Junction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How do you need to use it?\n1. Snaps on to stirrup 3\/8\"\n4. Used in driveways, walkways, slabs for residential and commercial buildings. columns, bridge construction, swimming pool foundations.\nsnaps on to 3\/8\" rebar, 1\/2\" rebar, 6, 8, and 10 gauge wire mesh and 3\/8\" stirrup.\nsnaps on to 1\/2\" rebar, 5\/8\" rebar and 3\/4\" rebar.\n3\/32\" wall thickness strong enough to stand on.\nFills in with concrete gravel size up to 3\/4\".\nDesigned for 100% filling with PSI 2500, 3000, 4500 concrete.\nThe foundation of a residential structure is affected by many forces acting on it from different directions. Engineering calculations show that concrete can be in compression or tension simultaneously. To avoid the failure of the concrete foundation, steel is added to the foundation to increase the carrying capacity of the concrete, principally in tension. The American Concrete Institute (ACI) in Section 318 of their code specifies that the rebar in a slab-on-grade foundation must be placed at the midpoint of the depth of the foundation. For example, in a four-inch foundation slab, the rebar mat would normally be placed at a two-inch depth. Also, the ACI Concrete Code calls for the steel reinforcement in the concrete beams at all exterior and interior beams to have sufficient cover to avoid exposing the steel to the effects of moisture penetrating corrosion. Once the steel is exposed to the effects of chemical-laden moisture, corrosion starts taking effect.\nMany contractors, because of the lack of a product(s) that places the steel at the correct depth in a uniform and efficient manner, do not install the steel reinforcement correctly. Many times, concrete laborers use stone (in different sizes) or pieces of broken bricks to support the steel mat (or cage in foundation beams) at the perpendicular intersection or rebar piece. Since the pieces of stone or bricks are not uniform in size, the placement of the rebar ends up uneven. Another problem is that while the foundation is being prepared the workers step on the installed steel to go across the foundation and this sometimes leads to the steel rebar falling off the support (stones or bricks). During the placement of the concrete, the rebar might end up not at the mid-depth level as prescribed in specifications. There are concrete foundation supports out in the market that are not strong enough to withstand the weight of the concrete laborers and the weight of the rebar supports. Some of the supports perform well under ideal circumstances. But these supports, either metal or plastic, normally have a single installation purpose.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaoics b/data_all_eng_slimpj/shuffled/split2/finalzzzaoics
new file mode 100644
index 0000000000000000000000000000000000000000..ad3794b1499fd9bc86188876381690728f512481
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaoics
@@ -0,0 +1,5 @@
+{"text":"When family reunions end with comments like, \"Do we really have to wait a whole year to get together again?\" or, \"It's a pity the out-of-towners couldn't make it \u2014 we haven't spoken in ages!\" your family is a perfect candidate for a teleconference. What are family teleconferences? Why do families set them up? And what makes them successful? Members of family teleconferences share the finer details.\nFirst, Bubby's sister spoke about their childhood and teenage years, revealing interesting tidbits most of the family had never been aware of. Afterward, her daughters and daughters-in-law shared memories from their vantage points as children. Then the grandchildren were given a chance to reminisce.\nAside from the fact that family members from all over the globe were able to participate without babysitters or traveling hassles, and no one had to prepare \u2014 or pay for \u2014 a grand yahrtzeit seudah, it turned out to be a lot more meaningful.\nWith the increase of service providers and economical teleconferencing options, many families are finding that there is so much to gain from this conventional and convenient method of keeping in touch with relatives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"India is the largest smartphone user country with over 290+ million users. The number itself is so big that it is really confusing sometimes that who called us? But Don't worry! Today I am going to tell you How you can Check Phone Number Owner Name easily.!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Microsoft Windows Server 2016 Training Courses!\nIf you're ready to learn how your organisation can do more with Windows Server 2016, you're in the right place.\nYou will learn from the experts how to provide more secure access to data and remotely manage your Windows-based servers.\nWe will provide you with the skills to power the next generation of cloud-optimised networks, applications and web services.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Wednesday, the Austin Animal Center posted photos of Police Chief Brian Manley with a dog named Deena and thanked the police department for their work investigating a series of bombs that killed two people and injured five others in Austin and Schertz. Police confirmed the suspect in all the incidents, Mark Conditt, died after a bomb exploded in his car early Wednesday morning.\n\"Their tireless work over the last three weeks put an end to the fear our beautiful city has felt. We appreciate them and their work every day, but are especially thankful for them today,\" the Facebook post reads.\nIn the Facebook post, the center shared that the initial dog the chief posed with named Deena was adopted the night before the \"#APDRescueMe\" campaign launched. The chief redid the photoshoot with another dog named Chin-Chin.\nThe Austin Animal Center's \"#APDRescueMe\" social media campaign has showcased adoptable pets posing alongside Austin police officers since February.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pull on your gumboots, the animals are waiting!\nThere's eggs to collect, lambs to feed and Georgie the Pig is always happy for a treat. For horse lovers there's led pony rides and Clydesdale cart rides around our 'real' working farm located 10 minutes from Bathurst.\nCollecting eggs & feeding the chickens.\nBottle feeding lambs and maybe a calf.\nLed rides on our ponies & Draft Horse.\nTrick lessons with Georgie the Performing Pig, and Eeyore our Donkey.\nDuration: minimum of 1 hour and often longer.\nCost: FREE for guests staying in the units, farmhouse or activity-included camping.\nFor day visitors the cost is $80.00 \u2013 covers up to 3 people taking part in the activity (ie. 3 kids or 2 kids & 1 adult etc). Extra participants are $15.00 each. There is no charge for supervising adults.\nThis farming thing is hard yakka! ?\nPrivate visits can be arranged at a time suitable for you. Please contact us to discuss group sizes, costs and availability.\nSled & sulky rides with our gentle Clydesdale., or our Shetland Ponies.\nTake a farm tour on our trailer, feed the sheep and the cattle, Enjoy the view from the back paddock, learn a little about landmarks and history, maybe spot some wild kangaroos. A maximum of 12 people and the cost is $80.00 per group. The tour will take approximately 1 hour.\nCredit card details are taken at the time of booking. Your card is not debited unless you do not show up and have not let me know within 24hrs of your booked visit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapkji b/data_all_eng_slimpj/shuffled/split2/finalzzzapkji
new file mode 100644
index 0000000000000000000000000000000000000000..d3a0b6c42470e2a9f5070486af5db6ae5604b7c6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzapkji
@@ -0,0 +1,5 @@
+{"text":"It's another wacky episode of the Brian Trust! This week, some exciting news is shared, road trips are discussed, deadlines are mentioned, moving plans are made and jokes are bombed.\nWe're off next week and possibly the week after, but we'll return with all-new episodes as soon as practical! We've got Avengers to endgame!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've all heard of the 7-year itch - that time of potential crisis when you traditionally evaluate your relationship and decide whether it's what you really want or not? It's such a common idiom that a famous namesake film was made back in 1955 starring Marilyn Monroe and Tom Ewell.\nContrary to popular belief, the original seven-year itch wasn't a condition related to relationships. It actually referred to an irritating skin complaint that supposedly lasted for seven years, and so it became a metaphor for all things deemed annoying.\nAs the clock ticked over to 2018, so begins another kind of 7-year itch \u2013 the 2025 SAP S\/4HANA deadline. Assuming you plan to stay with SAP, a move to S\/4HANA is inevitable at some point within the next 7 years. The question is when, how, and where?\nBenjamin Franklin (or Winston Churchill depending on your geography!) advises: If You Fail to Plan, You Plan to Fail.\nMoving to SAP S\/4HANA can be an evolution or a revolution - how you get there will depend on where you are today and your appetite for risk.\nCutting through the hype and the myth can be challenging. There's a lot you need to do before moving. To reduce business risk, prioritize investment, and identify quick wins to pull forward benefits, you need to build a realistic plan of what needs to be done, within what timeframes and levels of investment.\nYou should be advising your business now on all the options so stakeholders can make the right decisions at the right time. Our SAP S\/4HANA Readiness e-book offers advice and thoughts on some of the key considerations to help you get started on planning your digital roadmap.\n> Register for your complimentary eBook to learn more here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Plymouth has never had it easier.\nOur garage storage in Plymouth offers the skill and products to help you take control of the mess in your garage. Our patented Monkey Bars system includes overhead racks, shelving, cabinets, and flooring so that you'll be able to keep your storage high and your stress low.\nWe are here to help you with all of your needs for garage storage in Plymouth. From the free estimate to installation, we will be there through the whole organization process.\nEconomical: Rather than going to a hardware store and buying overpriced shelves, we offer a line of quality products affordable to families. Our professional garage consultant will design a free organization plan around your specific budget.\nDurable: You need storage materials you can trust. Our products are powder coated and made of steel so that your garage can look brand new for years to come. In addition, all of our products are backed by a lifetime warranty so you won't have to worry about replacing worn down parts year after year.\nBest Customer Service: We provide a service that big box stores cannot compete with. We will be there through the whole garage organization process, and we will even help you store your things after our product is installed.\nGarage Shelving: Our shelves hold up to 1,000 pounds for every four feet. With both 16\" and 24\" wide shelves, you can customize shelf space for maximum efficiency.\nGarage Cabinets: Our wall mounted cabinets are great for items that you want to keep out of a child's reach. The cabinets even have the ability to lock.\nOverhead Garage Storage: Garage Storage in Plymouth has been elevated to a higher level thanks to our overhead racks. Get those less used items out of the way by placing them high on your garage overhead rack.\nGarage Flooring: Give your garage a sleek and modern look with our garage tiles or Epoxy floor coating.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Axel Einar Hjorth Archive collect and spread information about the work of the swedish architect AE Hjorth.\nWe will be happy to get and supply you with information available. To get in contact with the AE Hjorth archive please fill in the form below. Your question will be answered asap.\nIf you have a question on a specific furniture produced by Nordiska Kompaniet, please attach the NK-tag number.\nhis life changed dramatically: he was placed as a foster-child with a well off family.\nand left with no heritage, he was forced to break off his studies without examination.\nbut placed Axel Einar Hjorth yet again amongst the bourgeoisie.\nHjorth's major customers and they had models of his in production right up to the forties.\nparticipate in the Paris exhibition that year \u2013 nor was Stockholms stads hantverksf\u00f6rening.\nsociety as a whole and particularly in Swedish industrial art.\nvery distant from the socially oriented ideas characterizing the activities of Svenska Sl\u00f6jdf\u00f6reningen.\nexclusive industrial art of the 20s through the 40s.\nexhibitions at the department store itself every year.\nwas typical for the exhibition.\nAxel Einar Hjorth carried on working, alternating between running a shop and managing a architect office.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you have one site or multi sites we have a service solution for you.\nJapanese Knotweed Specialists has a fantastic reputation in long term partnerships with Facilities Management companies in delivering Invasive Weed solutions. Whether it's a one off job or a long term contract, we can assist you.\nWith our bespoke live reporting system, you can stay in control of your jobs by seeing live photos of the treatment. You can also request and book in additional jobs via our client web portal. Our live reporting system provides you and your client with all of the information you need regarding your treatment plans.\nWhether it's a one-off job or a long term contract, we can assist you.\nTo find out more about our Management plans speak to one of our team today on free phone 0800 122 3326 or send us a message. Download our brochure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapzek b/data_all_eng_slimpj/shuffled/split2/finalzzzapzek
new file mode 100644
index 0000000000000000000000000000000000000000..0360b807a76c95fb7a91798a7dcf6c8383fb0f97
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzapzek
@@ -0,0 +1,5 @@
+{"text":"Arbor Erecta: A Botanical Concept for Masculinity continued Rapoport's exploration of narrative from the male perspective.\nThis artwork interweaves a new story about \"James\", a transexual person who underwent gender reassignment surgery from female to male, with representations of the New Guinean initiation rite of \"tree bonding.\" As James grows from his female body into that of a male, he re-enacts certain New Guinean tribal rituals that purported to purge the female pollutants acquired from the mother from the body of the initiated man.\nThe site is structured by ten botanical plants that are perceived as having gendered attributes, inspired by the symbolic importance in New Guinean culture of the Pandanus tree.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When Newton's 2nd Law is introduced and we begin summing the forces, students often think it something entirely new. It is not. Summing the forces is vector addition.\nA giant thank you to Frances Luke Accord for letting me use their song \"In The Water\" from their album \"Kandote\". They make wonderful music; you should really check them out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It was a mark of Walsall's renaissance under Kevan Broadhurst that they spent more than 70 minutes seem-ingly on course for victory away to a team chasing promotion to the Coca-Cola Championship.\nBut old failings haunted Walsall and, while they deserved their goal, scored by Paul Devlin in the 17th minute, they could only pick up a point when Adebayo Akinfenwa, a substitute, scored with just two minutes remaining. A draw was a fair result but Walsall will be hurt that they could not keep a clean sheet for just another two minutes.\nStill, the evidence of progress is great. The post-Paul Merson era is already looking encouraging, if less of a rollercoaster ride. Broad-hurst's policy of ensuring stability is certainly one that will please his employers and, for the most part, there was much about which visiting supporters could be pleased.\nDevlin's goal was worthy of winning any match. The former Birmingham City winger found himself unmarked inside the penalty area and scored with a low right-footed shot after a neat free kick involving Simon Osborn.\nWalsall should have made the game safe in the 70th minute when Andrew Barrowman shot straight at Gueret and they were left to rue this missed opportunity.\nAndy Robinson struck the ball against the post for Swansea as they chased an equaliser and their pressure finally paid dividends. They made two substitutions in the final 25 minutes and they were rewarded for their changes.\nWith time running out, Akinfenwa, sharp and physical, scored with a shot that took a wicked deflection to beat Andy Oakes in the Walsall goal.\nWalsall will count themselves unfortunate but, with both teams creating enough opportunities to have won the matches, perhaps a draw was t he most appropriate conclusion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bonjour Jolie 4th of July Deal \u2013 20% off Any Subscription!\nBonjour Jolie is having a 4th of July Sale!\nNow through 7\/5, use coupon code MYSUBADDICT4TH to get 20% off any subscription of Bonjour Jolie!\nReading the comments makes me not want to subscribe to them.\nI have been them for years not an issue ever, I see 12 other people have also, don't' let one whiner discourage you!\nI love Bonjour Jolie \u2013 they really do have options for everyone. =) I should also note that any time I've dealt with their staff personally it has been a positive experience. If you are on the fence, you should definitely give it a try \u2013 this deal makes it a no-brainer!\nIve been a subscriber on and off for over a year and always have a good experience. I get the goods only. Just wish they offered the 3 and 6 month price.\nI love this box. Happy to sub again on the sale. I had to cancel since I was saving for a new house but things are good now. We got the house!\nI always had a great experience and their customer service was fast and friendly always. No issues ever.\nI LOVE Bonjour Jolie box! I've been a subscriber for years and any time I've had to contact customer service, they have been super quick to respond and went above and beyond to remedy the situation.\nUnfortunately, too many times these angry commenters are the ones who have tried to scam companies by saying they didn't receive a product(s) when they really did. They are only mad because it didn't work.\nI'm glad that you've had a good experience with them but please understand that is not the case for everyone. No need to imply that I've done something wrong by my having a bad one. My experience was sufficiently bad that I thought others might appreciate the warning.\nI have had nothing but awesome experiences with Bonjour Jolie! I've subbed off and on for over a year and they were always really easy to deal with when I wanted to cancel. I think I'll have to sign up again for this deal! They always include the BEST snacks! Woohoo!\nI agree with \"worst customer service ever\". I rejoined in April because they had some special going on-I I have yet to receive a box from them. And just like Carrie-my numerous emails were ignored, then they charged me again for July for 3 more months, after I sent several emails to cancel. I did receive a refund for that, but not for the 3 months of boxed I paid for and never received! Terrible, Terrible, Terrible!\nI subscribed to this for a while and I had a great experience, and absolutely no issues whatsoever with canceling. Couldn't have been better. I totally recommend them and I might re-sub with this sale!\nAwesome Deal! They rarely have sales, if ever.\nThanks for posting! I used to get them last year, and have been looking for a good deal to get back with them.\nJust a word of warning on this one\u2026worst customer service experience ever. Very hard to unsubscribe from as they completely ignored my emails about a missing box for weeks. It took a threat of a chargeback and a Better Business Bureau complaint before they would cancel.\nI had no issues ever with them, and I was a customer for over 2 years.\nI just signed back up!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lisa Wilson-Wright joined NMR in 2003 and brought with her extensive experience in the use of quantitative and qualitative research techniques to help inform energy-efficiency, clean energy, and environmental policy. This includes conducting multivariate regression analysis, survival analysis, and billing analysis. She also has designed and administered surveys, conducted depth interviews, and analyzed qualitative data. Her research focuses on process, impact, and market analysis evaluations of residential projects in the areas of behavioral change, lighting, appliances, retrofit and weatherization (low-income and non-low-income), and new construction. She also performed an in-depth exploration of the strengths and weaknesses of developing consistent approaches to net-savings in the Northeast United States.\nWilson-Wright holds a Ph.D. in Sociology from the University of Wisconsin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqnwj b/data_all_eng_slimpj/shuffled/split2/finalzzzaqnwj
new file mode 100644
index 0000000000000000000000000000000000000000..ec320fe6a54a45c8f425df6c46a5326be457f76d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaqnwj
@@ -0,0 +1,5 @@
+{"text":"Working in areas of high risk, safety is paramount and compliance is non-negotiable. Major Training Group will help raise the bar with quality education and training solutions.\nWant more than your Dogging, Rigging or Scaffolding licence?\nDelivering the best and setting the highest standards in compliance for industry, Major Training Group bring flexible, reliable and quality education and training solutions that help meet the exact compliance standards of industry.\nFrom a Dogman licence to basic Rigging licence or a Slewing Crane operator licence, to working your way up to Open Class, Major Training Group will make it happen.\nIt's easy to book-in and train with us.\nWe will provide the information you need on courses and funding that may be available.\nIf you want to set the standard for education and training look at a Certificate III in Civil Construction or our new constructions skills program and choose from Certificate III in Dogging, Rigging or Scaffolding to learn major skills for the future.\nContact us today to find out how Major Training Group can help you build your skills, and set the standard for compliance!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Full episodes of TV show The Next Food Network Star (season 1-10, 11, 12, 13, 14, 15) in mp4 avi and mkv download free. All seasons in one place. No torrents and 100% SAFE Downloads. Laptop, PC, mobile and Mac Support! Android and Iphone. Quality formats \u2013 480p, 720p, 1080p, Full HD. Formats: mp4, mkv, avi. Get all episodes in single click.\nFood Network Star (formerly understood under the title The Following Food Network Star) is an American truth tv series that is presently broadcasting on the Food Network in the USA. The competitors series is created by CBS EYEtoo Productions and also exec created by Dave Oliver, Donna MacLetchie, Bob Kirsh, and also Stephanie Steele. Season 1 of Food Network Star premiered on Food Network on June 5,2005 Food Network celebrity is a competitors series included a series of distinct obstacles created to identify that has the individuality, on-air charm and also food preparation abilities to win the best reward\u2013 their very own program on Food Network. Of all previous victors of the program, Person Fieri is the one that has actually come to be a real Food Network Star. He has actually been a host of lots of programs on the network, such as Restaurants, Drive-Ins and also Dives and also Person's Grocery store Gamings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not useful for Full Frame Pro DSLR with battery grip.\nI have Lowepro SlingShot 302 AW but since I can't use it while hiking I wanted a backpack. I looked at official website of Lowepro. And as they say this Flipside 400 AW can hold a PRO-DSLR with 4 to 6 lenses, I went for it. I got good service from Flipkart. Got the bag delivered in time. But when I placed my kit in the bag, I was disappointed.\nthis bag is fulfilling my expectations up to the mark.\ni can easily accomodate my canon 500D with 100-400mm lens attached, 15-55mm, 50mm, 55-250mm, macro lenses kit and still i am left with space for 3 lenses.\nthe main feature what i liked is that its a flipside bag. flip it into ur front with bag opening from backside, take out camera or change the lenses. the bag itself becomes a platform for carrying out such activities.\nBuilt like a tank, light as a feather, brilliantly designed!!\nI read dozens of reviews for several DSLR camera backpacks before deciding to go with the Lowepro Flipside 400 AW Multi Use backpack. My biggest concern was that this bag would be smaller in reality than expected and that the space inside would be cramped. I was thrilled to bits when my bag arrived (within 2 days of purchase, so thank you FlipKart!) and it was a regular sized backpack, with loads of space.\nI love the build quality of the bag. This can easily carry one 70-200mm, 24-70mm, 105mm, 50mm, SB900 with a full frame body. What else do you need. It can also carry a small size tripod. However I do not find it very useful considering the weight of the bag.\nThis is indeed little heavy but that is because it is very well built and full of padding.\nSince the bag opens from the back, no one can access the gears easily, which is good considering the security.\nExcellent product, built like tank. very sturdy. currently carrying Nikon D500 + Nikon 200-500mm Lens without hood + Nikon 105mm Micro Lens + Nikon 16-80mm Lens and few other stuff.\nGreat product.. I brought it for 4600\/- only.. It can easily hold 2bodies and 3 lenses including my nikkor 200 500 5.6 with a laptop. though the tripod attachment is unusable. overall great product.. value for money.\nnice product. just up side down pattern is problematic sometimes.\nQ:Does it hold canon 77d body with sigma 150 600 lens??\nQ:Does it have Space for Laptop ?\nKeep your camera and its accessories protected from extreme weather conditions using this Lowepro Flipside 400 AW Multi Use Backpack.\nThe bag features a padded, adjustable camera compartment that makes storage convenient. For additional security, easy access, and storage, the bag has body-side access, along with a unique back-entry compartment. It has silent zipper pulls for making wildlife photography easy. The front panel lets you store small accessories.\nExtremely comfortable and convenient to carry with you during your shoots, this bag features contoured shoulder straps and a padded waist belt that reduce strain while carrying it for long periods.\nThe bag comes with a Hideaway Tripod Mount\u2122 system to keep your tripod safe. The SlipLock\u2122 attachment loops increase the capacity of the bag. You can take with you a pro DSLR with an attached lens up to 300 mm, along with 4 to 6 additional lenses, a flash unit and other accessories in this bag.\nThis backpack has a built-in All Weather AW Cover that can be used to cover the bag to protect the gear inside from rain water.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As part of the Biblical Gardens mission towards education, we offer a selection of materials available for purchase discussing different topics the garden explores. These range from reference books, to essays, to cookbooks. In addition to our online store, all books are available for purchase in person at the Rodef Shalom Gift Corner, or by phone at 412-621-6566.\nLavishly illustrated with a thorough introduction to the theme. A selection of botanical symbols used by Judaism, Christianity, Islam, Buddhism, Hinduism, Confucianism, and Taoism.\nA complete guide to the plants of the bible, including descriptions of the plant, their growing conditions, and their ancient and modern uses.\nA book of recipes to accompany the special exhibit \"Land of Milk and Honey.\" All recipes contain honey.\nA collection of essays exploring the way Jewish law and ethics intersect with the environment.\nA book of recipes from Ancient Egypt, Greece, and Rome.\nA short collection of recipes, all featuring one of the words most popular beverages, beer.\nA 30-year history of the garden. Exploring its creation and impact.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you pregnant? Desperately needing to get a good nights sleep and want to find the best body pillows for pregnancy?\nCost \u2013 stick with a pillow budget. Some pillows can be quite expensive!\nComfort \u2013 You want the pillow to provide comfort so you can hopefully get a good nights rest!\nQuality \u2013 Sometimes it pays to spend a little bit more money to get a better, long-lasting product.\nShape and Design \u2013 There are many different pregnancy pillow shapes including a C-Shape, U-Shape, Bean Shape, and Wedge.\nDoes it come with extra pregnancy pillow covers?\nHelps to keep you sleeping in one spot that helps give you a peaceful, uninterrupted and comfortable nights rest.\nyou can purchase replacement pregnancy pillow covers.\nLarge and can take up a lot of space.\nCovers can be difficult to get back on pillow due to its shape.\nA favorite pillow for many moms that supports your entire body while you sleep. The Comfort U Total Body Support Pillow has a premium fiber that allows air to flow freely throughout the pillow. The generously oversized U-shape design allows you to stretch out supporting both sides of your body but more specifically your neck, shoulders, back, and hips.\nReplacement pillowcases for the Comfort-U body pillow are also available.\nThe cost of the Pillow is quite expensive.\nThe cost of the replacement covers is expensive.\nHuge and takes up a lot of bed space.\nListed as an Amazon best seller for pregnancy, the Wedge Pillow is a firm and supportive pillow for your back and tummy. Because of its compact design, it makes it easily portable. Expecting moms can place it in the spot that gives her the best comfort. The cotton slipcover is removable and machine washable.\nMay not provide support in enough areas at once.\nreplacement pillow covers that you can purchase for an additional cost.\nThe pillow can help to reduce body aches and pains.\nNot as full and fluffy as other pregnancy pillows.\nLarge and can takes up a lot of bed space.\nPharmedoc's Total Body Pillow is a luxury pillow made out of premium materials to support your body and help get rid of those aches, pains, and sleepless nights. It's made with a 100% cotton cover and a superior interior cushion that helps provide support for sore necks, aching shoulders, and lower back pain. Research states it is the best pillow for side sleepers, back sleepers, and pregnancy sleep support.\nA replacement pillow case is available for additional cost.\nEncourages a good night sleep.\nGood support for upper and lower body.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatlsi b/data_all_eng_slimpj/shuffled/split2/finalzzzatlsi
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+{"text":"Tanjong Ria is a smalll development with full condo facilities and is located at the corner of Tanjong Rhu Road. This is one of the most affordable condo in Tanjong Rhu.\nTanjong Ria is within 5 minutes drive to Suntec City and Raffles Place (Singapore\ufffds financial district). Orchard Road is within a 10 minutes drive away via the East Coast Parkway or the Nicoll Highway. The East Coast Park and beaches are nearby. For water sports enthusiasts, you can head to the nearby Kallang Basin for water sports facilities. It is near the East Coast beach, seafood restaurants, Parkway Parade Mall and eating outlets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are a multi-site retailer then the heart of your business beats in the head office; since every major decision that affects even the remotest store is made there and all feedback is sent back to it, every retail head office tends to be a really busy operational centre that can become difficult to organise and manage. Thankfully, EPoS systems did not only change customers' experience around the world but also the way businesses manage their everyday functions and operations.\nThe RCS Retail Head Office Management System is designed to provide control over a number of retail branch stores as well as over one or more warehouses. Functions such as budget development and implementation, purchasing, accounting, human resources, payroll, facilities management and branch stock control form part of the daily routine in a head office. That is why our Head Office Management System incorporates features that can help you manage all these functions of vital importance to your business \u2013and even more!\nTo begin with, the system gives you the ability to coordinate the product management, the cash management and the purchasing centrally; this way you can always have a detailed image of the functions that determine your company's success. Stock management, which can easily become a headache if you are a multi-site retailer with numerous stores nationwide, becomes easier and more efficient and mistakes are avoided thanks to the \"Branch Stock Control\" function. The system also integrates features like Retail Accounting, Invoice Matching, EPoS Control, Supplier Integration and Web Sites Integration that will help you manage every aspect of the daily routine around the Head Office. The system integrates into Sage, Quick-books, Microsoft Dynamics and SAP, so no matter which software you might already be using it can be easily connected with the RCS system.\nIn addition to the above you can always choose to integrate the Warehouse Management System into the Head Office System. If your business counts not only a number of stores and a head office but also one or more warehouses, then our Warehouse Management System will help you manage your warehouse facilities efficiently \u2013and now you can do that seated behind your desk in your office!\nEffective and efficient head office management determines if a business is successful or not. If you are interested in learning more about the RCS Retail Head Office Management System don't hesitate to contact RCS today or give us a call on 01924 260020! For more information visit our website \u2013 http:\/\/www.rcs-uk.com\/.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As for many African countries, the legally pluralistic architecture of family law in Papua New Guinea (PNG) has become a subject of controversy. My analysis of PNG's village courts addresses scholarly debates about the subject. I use my empirical material to discuss two conflicting positions: 1) that male-dominated village courts contribute to the subjugation of women through the prescribed use of \u00abcustom\u00bb; and 2) that village courts form the only local dispute resolution forum in which women have a chance of their rights being acknowledged.\nIn my PhD project \u00abLegal pluralism and the limits of state sovereignty: an ethnography of the local state in the Markham Valley, Papua New Guinea\u00bb I try to achieve a broad understanding of the various relationships between the forums for dispute resolution existing in the area. I focus on one forum in particular, that of the Nazab Village Court among the Wampar. During the case collection process I concentrated on gender-related disputes rather than on quarrels about damaged gardens, stolen pigs, debts or land rights. My work is situated at the intersection of various debates on issues of legal pluralism, the anthropology of the state and women's access to justice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this weeks episode of The Marlins Catch we will be joined by Phancast and the Amazin Mets podcast as we will be doing our first ever NL preview show! We will also be discussing both the Braves and Nationals and see where they stand and where the competition is! With pitcher and catchers reporting this week and batters we will be breaking down the progress of all the players! Going live at 7:30 Eastern tonight you do not want to miss it!\nMake sure to give our host a follow on Instagram.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An afternoon at the park!\nSign up for my e-mail list to receive monthly updates + promotions!\nI respect your privacy, and will not use your e-mail address for anything besides this e-mail subscription.\nThanks for signing up! Can't wait for you to be a part of the e-mail fam!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatwwl b/data_all_eng_slimpj/shuffled/split2/finalzzzatwwl
new file mode 100644
index 0000000000000000000000000000000000000000..8069d02f6409dd4cd7917438a25602cdf694666d
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+{"text":"While you sleep and dream, your spirit is awake to the messages it needs to hear, understand, and process. Listen to your dreams!\nHas God put big dreams in your heart? How to train for them over the long-run.\nIs your life about fulfillment or frustration? It may be time to re-think things.\nWe can't answer every question you've ever had about your own dreams, but we can shed some light on the act of dreaming itself: how, why and even when you're most likely to dream.\nImagine doing what you loved as a kid for a living. Imagine having your whole family join you.\nA grieving sister is reassured after her brother's death.\nSharing the full story of your life can be downright uplifting\u2013and honest.\nIn the February 2016 issue of Guideposts, cover star Misty Copeland shares her inspiring journey to become the first black principal ballerina in American Ballet Theatre's 75-year history.\nWe don't have to pack a parachute to chase the hopes and dreams He's placed in our hearts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Update: For some time the date of their wedding puzzled me. Family records had it as 1866 which would have meant that they would have married when James was sixteen years old. Those records were in error - and his age at the corrected time of marriage now makes more sense.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CoreDR with Zerto Virtual Replication defies the physical assets that limited older disaster recovery (DR) technologies through its hypervisor-based replication solution. The Zerto Virtual Replication Appliances (Z-VRAs) copy data as it is created and before it leaves the hypervisor, creating continuous replication and allowing for recovery time objectives (RTOs) of minutes and recovery point objectives (RPOs) of seconds.\nFirst, one of Virtacore's engineers will set up a call with you to deploy your new Zerto Virtual Data Replication infrastructure in your environment.\nThen, Virtacore engineers will configure your Virtual Protection Groups (VPGs), which facilitate how your systems failover to Virtacore in the event of a disaster. We'll review the set-up process in this section.\nThird, you will receive a login to your new Zerto Virtual Manager (ZVM).\nThe first step in setting up the Zerto virtual data replication infrastructure is to configure the ZVM. The ZVM manages disaster recovery, business continuity and offsite backup functionality at the site level.\nYou will need to have a VMware vCenter for this step. If you don't have one, you can purchase a vCenter from Virtacore by becoming a Reserve Customer. Your engineer will be happy to get that process started if needed.\nThe ZVM should be installed in a Windows VM with 2vCPU, 4GB of RAM, a static IP and around 100GB of disk space.\nOnce your ZVM is configured, your engineer will walk you through setting up your Z-VRAs. The Z-VRAs replicate the VMs and associated virtual disks; one VRA is installed per ESXi\/Hyper-V host.\nVirtacore then works with you to configure your VPG, which facilitate how your systems failover to Virtacore in the event of a disaster.\nFirst, you'll choose which VMs you want to protect with Zerto and then our engineer will help group them by application\/service dependencies.\nSecond, we will assist you with deciding in what order the VMs should be restored and brought online.\nThird, we'll help determine what SLAs you want for each system in terms of RPO and RTO, which can be set granularly per each VPG in order to obtain different SLAs per different criticality of systems.\nLastly, we'll configure the networking at your environment and Virtacore's environment to allow for replication and seamless re-IP'ing after live failover.\nPractice makes perfect, so once you install Zerto Virtual Replication it's imperative to set up a plan for regular testing, at least on an annual basis. Regular testing ensures that the production systems at your source site are still online and unaffected.\nUsing Zerto's Failover Test Operation, you can create test virtual machines in a sandbox to ensure that they are correctly replicated at the recovery site. The testing is non-disruptive so you can do it any time without worry of your environment remaining protected. The test produces reports that can be used in compliance.\nChoose a checkpoint to which you'll restore your application. This can be existing or you can create a new one before starting the test.\nWithin the remote site, go to the VPG settings to find the network specified for testing and configured to the checkpoint specified for recovery.\nPower on the virtual machines by making them available to the user. This can be accomplished by using the boot order defined in the VPG.\nPower off the virtual machines.\nRemove them from the inventory.\nAdd the following tag to the checkpoint specified for the test: Tested at startDateAndTimeOfTest(OriginalCheckpoint_DateAndTime).\nTo ensure all of our customers are testing their Zerto Virtual Replication, Virtacore offers an annual assisted live disaster recovery exercise, during which your actual production systems go offline and are brought online at Virtacore. We work with you to ensure your Zerto VPG configuration is optimal to ensure the live failover is successful and meets your RPO\/RTO and SLAs.\nYou will be contacted to set up a time when the assisted live recovery will occur. The live recovery will then occur every year on the same date.\nZerto Virtual Replication is fully automated and orchestrated so that getting your data back online in a high pressure situation is as easy as only a few clicks.\nWhen you experience a disaster event, you can log into Virtacore's Zerto Self-Service Portal (ZSSP) to initiate a live failover of your systems to your disaster recovery site at Virtacore. Zerto's built-in automation and orchestration allows these systems to be brought online with RPOs\/RTOs of minutes and seconds.\nIn the event of a disaster you click the red failover button in the lower right hand corner. Once clicked, a pop-up will appear allowing you to select the applications (aka VPGs) that need recovery from a list.\nA pop-up with appear asking you to verify the point in time to which the application need to be restored. The point you choose will need to be before any corruption occurred. Once you make a selection, a button will appear confirming that you want to start the failover. Upon clicking, the failover process begins and VMs will be booted and reconfigured where needed.\nIf you decide after beginning the failover process that you've made a mistake or need to change something, you also have the opportunity to failback. Reverse protection begins synching the additional work that was done at the recovery site to the production site, when the production site is ready for use. After the applications have been updated to the original production site, failback is again just a few clicks away.\nDoes this seem too easy? That's because Zerto subscribes to the philosophy that people make mistakes in high pressure situations. As a result they keep their disaster recovery to only a few clicks to virtually eliminate mistakes.\nIf you're unsure of to what point in time to go back or which applications are corrupted, Virtacore's hands on assistance has you covered. Call Virtacore any time night or day, 365 days a year and Virtacore engineers will help assist you in the failover process. You'll be given our Emergency Phone number upon purchasing Zerto Virtual Replication.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In July of last year I was stranded for a week at a military base in Kuwait, waiting for a flight to Kabul. I was traveling with four members of a new U.S. Army unit called the Human Terrain Team (HTT). One morning the team sat at a long table in the mess hall. It was already 120 degrees outside, and there was no compelling reason to leave the air-conditioning. Our flight wasn't scheduled to leave until that night.\nOrdc Forum\/Blog \u2013 Human quicksand for the U.S. Army, a crash course in cultural studies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Temporary, portable, and bright LED sports field lighting. The Beacon LED Tower from Lind Equipment can be run from external power or a user-supplied small generator (1000W). Units can be linked together easily. Light tower packs up small enough to fit two of them in the back of a pick-up truck. Perfect for lighting sports fields, football, soccer, lacrosse or any other sport.\nWhether for practice, regular season, or the championships, the Beacon LED Tower is ready to light your sports field in any condition at any time. Weatherproof, compact, lightweight, and truly man-portable the Beacon LED Tower can be transported with ease without any compromise in durability and stability.\nIf you are lighting up a completely dark field, or supplementing existing lighting, the Beacon LED Tower is the perfect temporary sports light tower.\nAll this, and at a price point that is less than half of traditional light towers, making it accessible to everyone.\nThere is no reason to compromise. The Beacon LED Tower gives you all the light you need for your team, but in a far superior way.\nThe Beacon LED Tower comes standard with an external fuel tank (inside the cabinet), which supplements the run time of a 2000W generator to about 45 hours. Whether running off external power or a small generator, the Beacon LED Tower is whisper-quiet, perfect for your sporting events.\nTwo Beacon LED Towers will fit in the back of a standard-sized pick-up truck, and it is small enough to fit through a doorway. Finally, a tower light that can move around your fields with ease for the perfect positioning, no matter what the circumstance.\nIf you are ready to adopt the next generation of temporary sports field area lighting, we are excited to welcome you to our community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A Valley man who claims he was shot in the back by Phoenix Police says he's planning to file a lawsuit. The shooting left 35-year-old Edward Brown paralyzed from the chest down.\nPolice say Brown was reaching for an officer's gun when he was shot, but Brown and his attorneys say that's not what happened.\nAttorneys for Brown claim Officer Kenneth Silvia lied in the police report when he said Brown tried to grab his gun.\n\"If Edward had been trying to take the gun from the police officer.. he would've been shot in the front, not in the back,\" said attorney Tom Horne.\nBrown showed reporters where the bullet entered his back. It happened on August 5 near Glenrosa and 21st Avenue. Police were investigating a tip that a group of people in an alley were doing drugs.\nAccording to police records, when Officer Silvia and his partner arrived at the scene, they tried to talk to Brown, but he took off running and officers gave chase. Brown ran through yards, jumped two fences, but couldn't make it over a third. Officer Silvia was not far behind. He says Brown charged at him, made a swiping motion for his gun, so he fired a shot.\nAttorneys claim Brown didn't commit a crime and ran out of fear.\nBrown is a convicted felon who went to prison for unlawful flight from officers and burglary.\nAccording to police, Brown admitted he ran because he had a warrant, but denies charging at the officer and reaching for his gun.\nOfficer Silvia did not have a police body camera. He is back on patrol, and there are internal and criminal investigations underway.\nBrown is facing one charge of aggravated assault on an officer and marijuana possession.\nA protest on Brown's behalf is planned to take place on September 28 in downtown Phoenix.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Leadership Skills I is the recommended first course for first-line, team leads and\/or supervisors. The course concentrates on roles and responsibilities, communication, motivation and teamwork. There is a comprehensive test given after the class. If passed, the exam offers the credential of \"AIDT Certified Leader I\". The class is offered in 2-day format. This class is designed for currently employed Alabama citizens who are in leadership roles in their respective companies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JSM Infotech provides website maintenance and website support services. We can provide you with ongoing, consistent and affordable website solutions, delivered in a well-timed and professional manner. Whether you're a small, home-based business or a larger, corporate or government organization, we can respond to your requirements and provide you with the support you need.\nWe deliver your project on time, on budget and with excellent customer service. No project is too small and there's no such thing as a silly question. Our core values shows who we are and how we work. Read what our clients say about our website maintenance and support services. If you'd like to talk about your requirements in more detail, please don't hesitate to contact us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brighton-based dance artist Amy Bell is an exciting voice in dance, posing questions about how we view and label the female body and offering inventive answers. As a self-identified queer female, her work forges a space in which less 'conventional' bodies and ways of moving can find expression. We caught up with Bell, who recently debuted her piece The Forecast at the Place, to learn more about her work, her inspiration and what the future holds.\nTell us about the inspiration for The Forecast!\nI was thinking about identity, body, sexuality and gender, how people are always in flux. I was thinking how difficult it can be to accept and work with that. Our very human need to know, to put down solid markers in the face of chaos, to predict, convince, to name, are useful and yet ridiculous. I was also thinking about my own experience as a queer female dance artist and wondering how I could draw from all the goofiness, difficulty and sincerity I feel around that, and, importantly, how I could connect it to wider experiences that anyone could relate to.\nYou collaborated on The Forecast with Hetain Patel (animation and dramaturgy), Jamie McCarthy (sound and music) and Lucy Hansom (lighting design). How was working with them?\nI approached Jamie McCarthy and Hetain Patel to work on the project with me with a view to, amongst other things, finding larger metaphors for the ideas that I had been researching for a long time around my own identity. To move the project on, it was important for me to work with artists I had things in common with, but who were also different to me, so could challenge my assumptions and widen the thinking and feeling in the project. I also needed to trust them as artists and people. This collaboration was central to finding the main metaphor of The Forecast. I had been looking at female masculinity in movement and at the lack of queer female presence in dance. For many years, Jamie has been working with long durational drone structures, which often take inspiration from cloud movement and the weather systems. Hetain, who has been working on the formation of cultural identity for many years, proposed some studio tasks, including one that connected a TV weather forecast and gender. Something clicked, and I knew quite quickly this was a direction I wanted to take. Lighting designer Lucy Hansom joined the process later on, but the whole thing has been very collaborative and so a really rich experience. The piece is richer for it too.\nYour work seems both deeply personal, and also touches on universal themes. Do you ever make abstract work? Or do you think something of yourself will always come through?\nI seem to always need to acknowledge what feels to me like the absurdity or, at least, the odd situation of dancing in front of people in the moment of doing it. I can't pretend that that is a normal thing to do. That doesn't necessarily mean the dancing is personal, but perhaps it does make it different to some more abstract work. I love watching abstract work, I often find it very beautiful and indeed personal, but when I am making or performing, I somehow find that I have to acknowledge that we are people here together \u2013 performers and audience. That can be a wonderful material to work with, whether we get personal or not. I think perhaps that is something abstract work doesn't necessarily get so interested in.\nIn terms of working with personal material in The Forecast, I wanted to find ways of making it relatable to people who did not share my experience. I love that struggling with the unpredictability of the weather is something we can all relate to on a very concrete level, and I thought it was a lovely way into thinking about our internal, bodily or psychological gender weather. I hope to share the absurdity and humanity of trying to ride the storms of gender and sexuality as they play out across the body, but you'll have to decide for yourself what you make of it, of course.\nYou read English at Cambridge. Tell us about the relationship between language and movement in your artistic practice.\nYes, this relationship has become very important to me in my work. I enjoy the slippage of meaning that occurs when the body in movement is set against words. The playing out of this tension is happening to us in every moment. Our bodies are often saying something very different to the words that come out of our mouths. There is something so human about it, and I enjoy exploring this interplay further through choreography.\nIn my installation piece TOMBO(Y)LA, for example, I conduct relatively natural conversations with people while, at the same time, I am engaged in improvised movement scores. The piece often happens in gallery or public spaces where people might feel alienated by contemporary dance movement or might start to objectify me. The conversation, since its register is familiar and hopefully unpretentious, draws people together and then allows a playful, subversive, nuanced physical undercurrent to bubble up in our exchanges. In the work I am also asking people to give language to their physical, bodily experience of their gender identities, a place where language often breaks down. I love the way that movement, set against this search for verbal expression, prompts memory, sensation, and a more visceral or grounded searching into their own somatic experiencing and into mine.\nWho and what inspire you to move and create?\nIt's hard to put a response into words. I think movement and creativity are pretty fundamental human urgencies we all share. It's very powerful to tap into that as a way of making a life for yourself in the world. Even though it's not easy to carry on with it and it sometimes feels completely pointless, it's kind of addictive and brings with it glimpses of some kind of truth \u2013 if I can use such a risky word \u2013 that can make you feel 'alive' in a way other things just can't. There are many artists, creative thinkers and activists who inspire me and I think working together, sharing resources, caring for each other and challenging each other is completely at the heart of what we do. No one can do this alone.\nCan you describe your movement style in five words?\nI don't believe in \"style\".\nExcellent use of five words! You're curating Splayed \u2013 a 'festival of disruptive femininities' at the Place this May\/June. Sounds fantastic! Can you give us an idea of what to expect?\nI am also excited about Splayed! You can expect a week of subversive and super engaging work and events! The festival is geared towards disrupting conventional notions of femininity through experimental movement, performances, discussions and a free zine. I wanted to address the fact that there is amazing work going on at the moment rocking what we think we know about desire, identity, power, violence, gender binaries and embodiment. Having these works rub up against each other in a relatively mainstream dance context will hopefully set some sparks flying. I hope that there might be a wider debate about dismantling the norms of physical expression and a celebration of the range and richness of individual experience.\nSplayed! Festival will be at the Place this May\/June and The Forecast will be available for touring in Spring 2019. Learn more about Amy Bell at www.amy-bell.com.\nAlexandra's love of physical theatre first became clear at five years old when she veered off script in the school nativity play. At the entrance of the Angel Gabriel, she cartwheeled across the stage crying 'Yippee, an angel of the lord!' and the Virgin Mary burst into tears. Following this auspicious start, she went on to study dance and theatre and is currently doing her Masters in English Literature. When not in the library or at the theatre, she can be found singing jazz professionally, teaching yoga, and growing broad beans.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In Drive Daniel H Pink, a former political aide-turned author and lecturer, overturns the conventional notions that operate modern companies' dealings with its people and debunks what we think are true regarding motivation and people management.\nMuch like a Malcolm Gladwell, who translates academia and science into layman's terms for wider dissemination, Pink digests behavioural and management science from the last 50 years into a surprisingly effective essay regarding how to motivate human beings, which conventional and accepted ideas are gibberish and which actions are counter-effective. The book is interesting in its audacity and perhaps startling in both its conclusions and how so much that is so tested and known in some circles has not seeped into mainstream business. I say 'perhaps' because much of the contents is actually coherent and often what many of us have experienced and subconsciously identified, but never put quite found in one tome and in this way. Many managers, business owners or human resource professionals should prepare for a jolt judging by how seldom these ideas are practiced and, indeed, how often they are resisted. Apparently, much of the business world is still in the Dark Ages.\nSpeaking of which, Pink begins with definitions of Motivation 1.0 (early\/basic man who seeks survival), moves to Motivation 2.0 (the carrot and stick model for reward and punishment) and his central theme of Motivation 3.0, which address intrinsic motivators for creative and non-routine work. Imagine how a reward could be a negative. Pink does! And when one thinks about the impossibility of that notion Pink pulls the example of volunteer-run success of Wikipedia versus the paid staffers of the deceased Encarta, which was previously a product of Microsoft. Perhaps Motivation 2.0 is increasingly irrelevant in the Western world. While routine and programmatic work can be outsourced, heuristic and non-routine work generally cannot. Since job growth in our circles is mostly from the latter carrots and sticks are a dangerously anachronistic paradigm that need to be severely reconsidered. As well as Pink explains the concept well; imagine the ways we have to go when imagining telling a sales manager to forego assigning a bonus\/commission-based model to his or her salespeople. Unlikely, right? Pink seriously challenges that type of thinking. Adding rewards to pleasurable non-programmatic work has the opposite effect and indeed make the job unpleasant. Taking the extrinsic reward away renders the job pleasant in itself and gives it a sense of purpose and achievement. Pink will explain that financial incentives will make performance worse. Rewards, science shows, narrow the focus when the work calls for thinking and creativity and excitement. To be clear, he hesitates to render the same judgement for algorithmic work.\nHaving said that, Pink clearly elaborates and explains that 'baseline rewards' need to be sufficient for this model to work.\nIn the second part of Drive, Pink explains the three elements of true motivation. These are Autonomy, Mastery and Purpose. He then offers examples, actual case studies and techniques for unleashing these elements. Most managers and leaders prefer and seek Intrinsically motivated people (he calls them Type I) to Extrinsically motivated (type X) individuals. What if there was a template for unleashing the latter and harnessing the ensuing talent and energy? To be clear, Pink insists on fairness and equity being a prerequisite, but not as a form of carrots and stick rule. Going back to the three elements, Drive addresses the dominance of purpose over profit, specific goals over constant reward systems and giving people control over their work environments (including team-mates, work hours and tasks) as essential to motivation and success. Imagine it. Pink and many businesses did and the case studies are enlightening. The examples are there, but whether, as the book jacket proclaims Drive is \"paradigm-shattering\" depends on the dissemination and acceptance of the idea at the top echelon of the business and workforce entities. After all, Daniel Pink is telling sales managers, just to cite one example, to stop the commission-based system and to instead discover that the new hires have longer-term visions of their work. The news is that the message is hard to dismiss.\nThe book is padded with chapter summaries, conversation starters, an index and even a helpful glossary. This makes it a useful handbook on top of its exciting content. Having read it three times over the last several months, useful and exciting are certainly appropriate words here.\nChallenging conventional wisdom could be termed Drive's first and last word were it not for its drawing from actual academic studies and learnings.\nis the fourth non-fiction book by Daniel Pink . The book was published on December 29, 2009 by Riverhead Hardcover . In the text, he argues that human motivation is largely intrinsic , and that the aspects of this motivation can be divided into autonomy , mastery , and purpose.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawfkv b/data_all_eng_slimpj/shuffled/split2/finalzzzawfkv
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+{"text":"They have BOTH blue and green lume as Rolex kind of uses up old stock as they transition models. Mine is green. BLNR is a great style for sure and each has their own proponents. The blue on the BLNR is striking and definitely adds a splash of color. I chose the LN because it doesn't have any color, can go with a suit (bathing or tie) and has a complication. I am just afraid I'll grow tired of the blue in 10 years and find it a bit trendy. I don't scuba dive but do take mine snorkeling. So, it's a sub with an extra complication that IMHO makes the dial more interesting and the PCL's which add a bit of bling.\nThat's ok. Everyone is free to open a new one or to post here.\nHi all, long time reader, first time poster its a fantastic forum and I was hoping someone might have some though on a slightly involved DJ question (apologies if this has been asked before I couldnt find anything using search). Im in my mid twenties and am getting to the point in my life where I can feel like I can justify my first serious watch purchase. Im looking for something thats classic, goes with everything, and will last for a couple of decades or more and to me, thats a Datejust, specifically one in stainless steel with a Jubilee bracelet, acrylic crystal, light coloured dial and baton markers. Im not too keen on the newer models, so Ive narrowed it down to a 1601 or a 16014 the question is, what are the pros and cons or each? I know theyre very similar, but do you think its worth paying extra (say, $5-600) for the 16014? You get the newer 3035 which is closer to the current Rolex movements would that make it more reliable\/more easy to service further on down the line? You also get the quickset which would be nice to have, but that not really that much of a dealbreaker if the 1601 is going to be just as reliable. Obviously, the 16014 would be newer so thats why they often command a higher price than a 1601 from the sixties, but if they were of equivalent ages (say, both late seventies) is there any real difference? Id be really interested to hear from anyone thats owned either (or both!).\nI bought a sub because i love it, not because other people wear them or not. If people do not buy a sub because it's too common and want something no one's wearing. They're more affected by what other people think of what they're wearing and that makes them a lot more common than submariners, imo.\nOh boy, now I can't wait to get home and look at my BLNR and sub. lol I never even thought to check such a thing.\nPicking it up tomorrow, supposedly it was an issue with the rotor being loose. Crossing my fingers they did a good job with it, will update tomorrow.\nYou might be right.. They are just not located near me, which is why I asked if anybody knows a IWC AD near San Francisco..\nvery much looking forward to some photos! i also have a 6 1\/4 inch wrist, though my wrist is more flat than rounded.\nNeed help identifying another 750K Gold IWC Schaffhausen!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gear Up for the 70th Independence Day with India Market Hub \u2013 Celebrate the Spirit of Freedom!\n16th August '16, Chandigarh, India: India Market Hub, a digital and marketing agency, has been bountifully celebrating the glory of freedom on the occasion of 70th Indian Independence Day in-house and across myriad of social media channels. The company, a division of India Softech, organized multiple contests and the participants have been winning amazing prizes including shopping and dinner vouchers, free website, flattering discounts and a lot more! The contest got an overwhelming response online and was open until 14th August, 2016.\nIndia is a country with rich culture, traditions, heritage, history, diverse languages and varied geographies which make it a hub of interesting facts. Well, this contest was to enlighten the audience about the unveiled facts of the country. The top 9 lucky winners got a chance to win exciting gift vouchers worth Rs. 500.\nThis contest was exclusively for the business owners of the Tricity who were asked to exhibit their creative skills by decorating their workplace and sharing the pictures\/videos with us. The business owners with maximum likes on their picture\/video got amazing prizes including a FREE Business website and lavish dinner vouchers.\nThis \"Freedom Sale\" organized by India Market Hub was something not to be missed. The company offered special offers to the customers who registered for a website or digital marketing campaign before 15th August 2016. The company gave flat 50% discount for an advanced e-commerce website to the customers, all they had to pay was INR 10,999\/- for a website. Moreover, it also gave 30% discount for 1 year Digital Marketing Package and the customers had to pay just INR 29,999\/- one time for a yearly plan.\nThe contest gave a chance to the audience to unleash their creativity and share patriotism by participating in the Patriotic Creativity Contest. The contestants just had to get dressed in a tricolor outfit and capture a picture\/dubsmash. The 1st, 2nd and the 3rd winner with maximum like won cash prizes of 3000\/- , 2000\/\u2013 , 1000\/\u2013 respectively.\nIndia Market Hub created a patriotic fervor for the employees by organizing a Kite Flying activity in the company's premises. The kites in tricolor were flown high by the employees donning ethnic wear. The workplace was beautifully decorated and the employees of the company were drenched with the spirit of freedom.\nLet's keep up the fervor and pay tribute to the thousands who laid down their lives for the freedom of the country. It's proud to be a part of such a glorious nation. Vande Mataram!\nCopyright \u00a9 2019 India Market Hub. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The final version of this article with its full bibliographical information is available. To access the article, click here (Vol. 78 (2016), No. 5 pp. 909-912).\nA two-year-old castrated male Pomeranian dog was referred with the chief complaints of coughing and subcutaneous emphysema. On physical examination, the crepitant areas were palpable. When auscultated, the right chest was absent of respiratory sound, while the sound of the opposite side was enhanced. Radiographs presented pneumothorax and pneumomediastinum. On computed tomography, hypoattenuated bulla-like lesion at right middle lung lobe and trapped air in mediastinum were shown. After patient stabilization, surgery for excision of affected lobe was performed. During follow-up period, there were no recurrence and complication on radiographic examination. Based on clinical and pathological findings, the dog was diagnosed as congenital lobar emphysema.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Carden Challenge is almost underway.\nA short e-news to share with you. The Carden Challenge begins at 6pm today! You can follow along on Twitter and Instagram to find out some of the team sightings, see pictures and share in the fun of the Challenge.\nSome challenge participants! Ginny, Tanya, Mark, Ron and Nath.\nTake part and help us to protect the homes of hundreds of species on the alvar!\nwe are able to protect and care for over 8,000 acres of alvar habitat.\nThat love of the alvar is one reason why teams of four go out on the alvar each year in search of as many species as possible in 24 hours and raise funds to help us to protect these precious lands.\nEleven teams are taking part this year. We'll be tweeting and instagramming throughout the Challenge, so follow along in the fun. Scroll to the bottom of this e-news for the links (no account needed).\nAll funds raised during the Challenge support the work we do on the alvar. Our celebrity birder, Nathalie Rockhill's team is SO CLOSE to reaching their fundraising goal. Will you help Nath surpass the goal?\nDonate online through a secure link here.\nWe are currently reaching out to the general public to inform them of our goals and invite them to become more engaged in our work. To that end we are seeking one individual to work with one other Engagement Organizer and volunteers to take our message to the public through an outreach and canvassing project.\nThe Part-time Engagement Organizer position is for up to 16 weeks and the deadline for submissions is Friday, June 10, 2016. Learn more about this job here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you attended Hughes Springs High and want to be added to the alumni directory register now to add your name to the alumni list.\tIf you are an alumn of Hughes Springs you should register now in the alumni directory. We also encourage you to interact with the rest of the alumni on this site. You can find old friends, share photos and stay updated on the latest school news.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Over the years, the members of Kings of Leon have had their fair share of problems. From excessive drinking to simply walking off stage, the four Followill's haven't always gotten along, but now, they all have reason to celebrate because a new member of the family has just been welcomed.\nAccording to US Magazine, lead singer Caleb Followill's wife Lily Aldridge gave birth to the couple's first child on Thursday. The delivery took place at Vanderbilt University & Medical Center and by all accounts, went as swimmingly as he could be expected. The little girl's name is Dixie Pearl, and she's said to be healthy.\nAldridge is best known for her work with Victoria's Secret. She and Followill married a little more than a year ago and have lived happily together ever since, at least when his touring schedule and her appearances don't require one be away. Neither party has announced how much time they might take off in order to spend time with their new little girl, but one would think at least a slight shift in priorities would take place over the next few months.\nPop Blend sends out its sincerest congratulations to Lily, Caleb and the entire Followill crew. Family always comes first, and there are few things more exciting than meeting a new daughter\/ niece. Here's to hoping the bundle of joy has her mother's looks and her father's voice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Location! Location! With Short Distance To Little Poland Shops, Restaurants, Banks, Bus Line, Fastrak, And Highways. Very Well Maintained 6 Family Brick Building With 2 Car Garage. Updated Mechanicals, Fire Alarm System. Each Unit Has 2 Bedrooms And Separate Natural Gas Heating. 1 Year Old Roof. Coin Operated Laundry In The Basement Brings Extra Income. Great Investment Opportunity!\nListing Courtesy of Century 21 Stamm Realty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That's not her real name. Of course it isn't. But Suzie, if you happen to recognise yourself, do get in touch. I cannot imagine that you voted Brexit. If you did, please let me know why. Get in touch either way.\n\u00b7 Taking part in a French exchange.\nEven within the first week at secondary school there were problems. Someone had stolen her brand new trainers whilst she was in the shower, after P.E. They hadn't, of course. The trainers just got moved to the other side of aisle where she'd been getting changed. I'm pretty certain it wasn't deliberate, just feet catching and pushing objects out of the way. Then, next P.E. lesson, one of the staff members had stolen her watch. She'd forgotten to collect it and the P.E. teacher had locked it away safely. She'd then gone on to cover a class in another part of the school. Suzie continued to play the victim and this made her unpopular with her peers and with staff.\nI was Suzie's tutor for four and half years before I found out what was behind all of this and it was whilst we were on the French exchange that it all clarified.\nI had clearance to take children on to a beach. We gave them a half hour of free time. Suddenly Suzie shrieked. Someone had doused her in salt water. She was soaked. The others were reprimanded, they apologised and had the grace to look shame-faced. Between us we managed to find her dry clothes. There was nowhere to change so I fixed a towel across the entrance to some closed toilets. She told me as she got changed: \"It's my own fault. I'm always accusing people of being mean. I don't trust anybody. My dad's friend ran off with all of the money from their business and Dad then disappeared because he was so ashamed.\" Ah.\nWe tried to keep Suzie going.\nAn early highlight was a day trip to France. We went to Boulogne. We took a coach all the way went us and went though the tunnel. She was so nervous that she was sick on the way out. We were in the tunnel and we weren't even moving.\nNext came taking part in Joseph. This was a real professional matter. The students performed with a primary school. There was another pair of schools. Each pair did either Monday, Tuesday and two performances on Wednesday or Thursday, Friday and two performances on Saturday. Normal school staff were not allowed anywhere near them, apart from supervising during the breaks between the performances on Wednesday and Saturdays. They had tea served in the bar and then we'd take them outside to kick a ball or something similar in the local park. Otherwise they were dropped off at the stage door and left with the theatre's chaperone and picked up for the journey home. Every child was given notes after each performance. Serious stuff indeed. Suzie did quite well on the whole though there were a few aggressive notes when she began to lose concentration because she was starving herself. Oh yes, I'll take on anyone who tries to limit the arts in school just as vehemently as I'm challenging Brexit.\nIt was with some trepidation that I took her on an exchange to France. Yet, after the salt water incident and the confession all went well. When I met her on the final morning the French exchange partner's mother came up to me and said \"Elle a \u00e9t\u00e9 charmante. Absolument chramante. Nous l'invitons l'ann\u00e9e prochaien. Et son fer\u00e8 et sa m\u00e8re.\" Wow!\nOkay, so why does this mean we should stay in Europe? The connection might seem tenuous. However, it's what EU programmes such as Erasmus, Socrates and Lapidus make extremely easy. It was about trust and that was what Suzie had found hard. When as a school teacher you arrange an exchange there is a lot of trust. We were on a group passport that was barely looked at. It's quite brave also, operating in a different language in a slightly different culture. Her charm, I believe, came from a willingness now to trust in a process. I believe she'd already begun to see that even before the incident which is why she'd been able to understand it.\nOh and another delight: Suzie worked hard with her exchange partner on swapping vocabulary. She filled a whole thick exercise book with new words and phrases.\nEurope as a super power?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"20.5 micron superwash easy-care Pure Wool.\nWeighing around 200gms pr. makes them substantial socks.\nMachine washable to 30 degree's c.\nNon-restrictive\/ loose tops for comfort and healthy wear.\nColors range from conservative mixes to bright 'out there' combinations.\nWill maintain their shape wash after wash.\nWarm in the cold, cool in the heat, maintain warmth when damp.\nMade on antique sock machines in service since 1928, still going strong. Your feet won't get too hot or too cold no matter what the day brings. Actually you probably won't think of your feet at all. You can wear these pure wool socks for days (if you need to) and still be polite to the nose, more importantly to other peoples noses - less washing has got to be environmentally good! Sustainable and renewal materials Health benefits of the loose top can not be understated, even if tight socks don't annoy you yet, every time you wear them they are undermining your legs vascular health.\nFabulous socks! I live in the UK and have twice bought these socks online for myself and as gifts after picking them up initially in the market in Hobart. Chilly feet are a thing of the past in these lovely hard wearing socks. My daughter used them as slippers for 5 years and darned them when they got holes!! I've bought her a new pair. Thanks for excellent service and product.\nI bought these for my husband who gets very cold aching legs during winter & he loves them, very warm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Descending upon the town, I felt pregnant with expectation, not unlike a certain Lena Grove in Light in August. (Well, maybe that's going too far.) In any case, the real destination for our trip was Faulkner's colonial estate, where he lived with his wife, Estelle, her two children and their daughter together, Jill. The massive two-story plantation is eerily well preserved, as if by a noble ghostly presence. (Jill Faulkner Summers sold the house to the University of Mississippi in 1972, so they're the real custodians.) Elderly cedar trees lead up to the entrance like a collonade, flanked on all sides by 29 acres of severe landscaping and gardens.\nFrom the house you get the feeling that the Faulkners were a family living on the precipice of history. Out back, there's a kitchen where the slaves would prepare daily meals for the family. I also noted wooden stables, and a small cabin that slaves had. A year after purchasing the plantation, Faulkner renamed it \"Rowan Oak\" after a mythical tree believed to bring protection and safety.\nIn 1952 Faulkner added a work room onto the first floor where he later mapped out the outline for his novel on the walls with graphite and red pencil. Every day he scrawled a synopsis of the plot that later became the 1954 novel, A Fable. A trip like this encourages a special attention to relics: In front of the window rests a wooden table given to the author by his mother, and the typewriter he used. He would add a bed with a light bulb attached to the top of the frame for when he worked late nights on his stories. Faulkner's bedroom was furnished with vintage off-white diamond plaid wallpaper and a wooden frame bed with a white, hand-sewn blanket. A nightstand with a couple of books from the author's personal collection, and a clock (although he reportedly used a mockingbird as his alarm). There's more than one story in Oxford, Mississippi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This is a placeholder page for Nancy Capparelli, which means this person is not currently on this site. We do suggest using the tools below to find Nancy Capparelli.\nYou are visiting the placeholder page for Nancy Capparelli. This page is here because someone used our placeholder utility to look for Nancy Capparelli. We created this page automatically in hopes Nancy Capparelli would find it. If you are not Nancy Capparelli, but are an alumni of Beaver Falls High School, register on this site for free now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"coal mining expo june 2012. Home >> Quarry Machine >> coal mining expo june 2012, icie 2012 the 9th xinjiang international coal coal mining expo june 2012.\ncoal mining projects 2012 presentation ppt crusher in India. Crusher machine. Shenbang stone crushing equipment is designed to achieve coal mining expo june 2012.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sustainability is a key feature of the design, including optimum cross-ventilation and abundant natural daylight.\nThe Park View Grand Spa in Gurugram is a large residential development consisting of nine 20-storey towers and one 33-storey tower. Each tower has three lifts and offers three-sided open views. Located in Sector 81 of the city, the scheme benefits from convenient transport links to the rest of the city, and is just five minutes' drive from a proposed Metro station.\nThe buildings are arranged around an eight acre landscaped park full of leisure activities. The development provides an environment in which residents can enjoy themselves both indoors and outdoors \u2013 with a variety of sports areas along with children's recreation facilities. As well as a multi-purpose hall and a high-quality spa amenity with table-tennis and pool tables, Park View provides a school and other facilities for its residents.\nChapman Taylor's New Delhi studio was responsible for masterplanning, architectural design, interior design of the club & the entrance lobbies to the towers.\nSapna has worked at Chapman Taylor since 2007 and is responsible for managing our Indian design studio in New Delhi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's all about the outdoors. The rest is meaningless.\nHodge Army Navy is more than just a local establishment selling great outdoor gear. We help connect people to the outdoors through quality products and an excellent choice of products.\nOur store is located in historic Marietta, Georgia and is home to hundreds of name-brand items for outdoor wear, camping, military garments, and more! We ship world-wide at the lowest prices. Stop by or give us a call today!\nClick on the picture to Order now!\nand can be digested by an elephant.\nTwo Colors in stock now!\nNATO Jerry Cans, In Stock Now!\nGreat Selection of ENO Hammocks, including the ENO CAMO NEST!\nBug Nets, Rain Flys, Camo Lights, and suspension systems!\nThis classic 8\" tactical boot features a side zip for easy on and off and cushioned removable insert for extra comfort.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The queen of Indian Television, Divyanka Tripathi Dahiya is currently entertaining the viewers as Ishita in Star Plus' Yeh Hai Mohabbatein. Divyanka Tripathi entertained the viewers with her expressions and acting on Holi Sequence in Star Plus' Yeh Hai Mohabbatein. A actor of Star Plus' Yeh Hai Mohabbatein left the actress completely shocked with his surprise.\nStar Plus' Yeh Hai Mohabbatein ex-actor Gaurav Wadhwa, who played Sohail in the show recently left Divyanka Tripathi shocked with his unique and beautiful surprise. Gaurav met a fan of Divyanka and did something unexpected. The little fan of Divyanka Tripathi wasn't able to pronounce her name properly and Gaurav teach her how to pronounce Divyanka's name.\nIndeed, it is an adorable surprise as the little fan was finally able to pronounce Divyanka Tripath's name. Indeed, Divyanka Triathi would have been shocked and yet surprised seeing an adorable gift from Star Plus' Yeh Hai Mohabbatein ex-actor Gaurav Wadhwa, who played Sohail in the show. (Also Read: Most SHOCKING Twist Ever going to happen in Star Plus show Yeh Hai Mohabbatein AFTER this\u2026).","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Welcome to Z-Fever, Inc. We are a Nissan and Infiniti performance center located in Tampa, FL. Our phone number is (813) 877-7600.\n(Bing Map Here). Please excuse the mess as we complete the migration.\nWe have been building, repairing, racing, and tuning Nissan vehicles for almost 20 years now.\nIncluding Dyno Services. We are an UpRev Pro Tuner!\nBe sure to browse or download files from our section, which includes numerous documents such as wiring harness applications, installation manuals, and other various technical documents.\nCheckout our Current Hottest Project: Fever Racing S14 Drift Car!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"It literally started a riot when I started playing against my friends\"\n- Grab 3 friends to play against.\n- A Question Card is shown to everyone, e.g. \"Nothing excites a group of Catholic Priests more than ____.\"\n- Each player selects an Answer Card from their hand, e.g. \"middle school crushes.\"\n- First Player to 7 points wins the game, and the respect of all humanity.\n- 4,000+ Answer Cards and 800+ Question Cards! Never gets boring!\n- Various internal bug fixes and improvements.\nI've been enjoying this app for two days now, and I've seen two major issues with it. The first issue I'd address would be the fact that only one of the game modes, that being classic mode, is played by people online. There is little to no traffic in the other two game modes, and therefore there's essentially only one type of game that anyone can play. I'd recommend adding more incentive to the other two game modes, such as larger reward pools for winning and perhaps even participation. The second issue I'd like to address is the online drop rate of players. I've only been able to finish a game averaging maybe 1 in 8 or 9 attempts. People will realize they're losing and will drop out of the game. If you get to below 3 people then the game automatically closes out. This wouldn't be to much of an issue, if we didn't lose our in game premium currency. I hope that this is addressed in some way, whether it be a refund of our premium currency when a game is dropped, or a premium currency deduction from players who quit games. I realize that this has been a mostly negative plug at your product, but I would like to see it survive through these issues, and to do that they must be addressed. For people considering this app, I'll say this: It's fun with friends, and if you can find a group online that won't quit on you, then you could easily spend an hour or two enjoying what this app has to offer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well for me im a little disappointed. I was hoping this would look better on a TV screen, rather than the poorish image we've been used to of it on our computer screens. For me the imitation rain, which of course is vital to make the routine work, does nothing to enhance our viewing pleasure. Still the song is a good one and Flicks interpretation of inclement weather is spot on.\nThe costumes are great. I love the idea of the see through macs, and the red under garments are cool too. Those see through brollie's are so 70s, im sure you dont see them these days. All the girls are having a ball squelching around in the water, dont think them wellies are keeping the water out. Dancer of the day is a toughie, im sure you all agree. No dancer performs a stand out role. Pauline is having lots of fun and i would have given it to her, only her rain hat is to low over her face, it distracts me. So im going to give it to Patti as she catches my eye the most, oh and she looks good from the rear in her see through mac.\nI think a score of 7 out of 10 is about right.\nUpbeat and perky while the costumes add a certain sauciness to proceedings. It splashes and squelches along nicely scoring an 8 from me.\nAfter seeing a glimpse of it on the TOTP documentary I was eagerly awaiting this one but had no idea when it would appear on our screens, so I was a bit of a giddy kipper when it started on what was already shaping up to be a good show with Blondie, The Jam, and those guys in flat caps banging on about Lowry. Well it didn't disappoint, I thought it was brilliant! Great song, the routine was good, the outfits hot, and they used their props (brollies) well, which can't be easy. I always like the heavily themed routines and one of the things that always gives me enjoyment is seeing the girls having fun themselves which they clearly all are and Pauline in particular who looks like she's having an absolute ball, and for that reason she gets my DOTD.\nI like this one too. The song is good and as others have said it's a very apt performance and the cloudy backdrop and huge puddle add to the cheerful storm. The ladies look quite thrilled with it all IMO too and Patti appears to be right in her element, looking divine in her mac and wellies and smiling at me throughout. The only thing missing is a water-fight but we can't have everything in one performance. For me this is 9\/10 material and did I mention that I liked Patti for DotD in this?\nI agree with what all you say Ryan. I was also a little disappointed with the picture for the routine being less crisp than I expected. But when I've finally seen all the Legs' routines - hopefully in 2016 - this could well qualify as one of the very sexiest costumes. The girls having a ball splish sploshing about in see through plastic and red underwear, it could hardly get better. The brollies are skillfully twirled and it all flows. I would also have been minded to give Pauline my DOTD were she not semi-incognito. So I will opt instead for Gill, for her lovely smile.\nBut presumably these ladies could, as they did seem to be enjoying themselves, splashing in the torrential rain, whilest twirling their umbrellas.\nA pity that Lulu seemed to be hidden at the back throughout this routine, I'd have liked to see some dancing in a circle, so that they all got equal amounts of camera exposure. Gill seemed to be particularly enjoying this one, with her beaming smiles at various intervals.\nA shame this is usually picked as an example of \"literal\" routines - tell me how else would any choreographer interpret this song? It's great fun and as already pointed out, the girls have a ball and don't seem to let the damp conditions distract from their performances.\nMy DOTD is Patti as her umbrella spinning is glorious.\n2.33 minutes of heaven.I love this routine,and the song.The opening thunder,the rain,umbrellas and macs,are all great.I first saw this show in 1978 in black and white,and remember it as the time when Kate Bush knocked Abba off number one.It was great to see it in crisp,glorious techno-colour.\nI agree with Col's opening point because the over-literal accusation is a bit of a cynic's cheap-shot : would they prefer the troupe to slide down a polystyrene slope on skis whilst BBC stage-hands shake iberal amounts of fake snow from above for this number?\nAnyway, if they wanted 'literal' , the ladies would have performed with sullen expressions as a result of the soggy elements.\nThe contrast of the red in the clothing works well against the gloomy background, highlighting the dancers, although I do think some of the ground level lighting on their faces is perhaps a touch too weak.\nHowever, the whole thing looks rather 'videoish' to me, almost as if it could have been a specially shot, early 80's band promo.\nThe super-imposed rain works well and the splashing about in the shallow pool doesn't really hinder their movements.\nAll are having a whale of a time, due in no small part to the mini-red boots being a welcome change from the usual heels, I imagine.\nAlthough her Sou'wester's so low that it casts a shadow over her eyes, Pauline's joyous performance only just edges out brolly-twirling Patti in my opinion.\nBesides, those red boots go really well with the colour of her legs.\nA disco classic that I could never warm up to (maybe it's the rain!), it always reminds me of Boney M and that's not a good thing in my book. But I like this routine, it's much better than the song. I like the way the girls are whirling their brollies, while waving them up and down. It's certainly fun to watch and as such this routine would have been more suitable for a song like \"Singing in the Rain\".\nIt's not easy to pick a DotD, but after a bit of consideration, I decided to go for Patti (again!).\nClassic, Ally's Tartan Army was the nations anthem in 1978, sorry my mistake with all those Legs from this show.\nI wasn't disappointed with the new showing of I Can't Stand The Rain. It does help if you like the track and I have always had a fondness for Eruption reworking of Ann Peebles soul number (keeping all thoughts of being a second rate Boney M to the back of my mind). The routine is kept brilliantly simple and devising something that allows six beautiful dancers to splash about while suitably attired for heavy rain works very well. 9.5\/10 Dotd Rosie, LNV would be so proud of me.\nI agree with the points regarding knocking many dance routines for being too literal. I remember a celebrity panel game show in the nineties chaired by Danny Baker. One of the resident captains was Rick Wakeman. I can't recall what the programme was called, but there was a regular round where an excerpt of a Pan's People routine was shown with the sound muted. Then the teams had to try to identify the song danced to. Danny Baker seemed always at pains to remind the contestants what a 'very literal' interpretor of songs Flick Colby was, as if 'Get Down', exemplified most of her choreography. It used to annoy me because I would think most of the Pan's dances were lyrically straightforward and a literal approach was often more or less unavoidable. And it wasn't as if Flick didn't have her fair share of wacky and esoteric interpretations either.\nBoney M were dreadful with things like Rivers Of Babylon and Brown Girl In The Ring, though I do quite like early singles such as Daddy Cool, Sunny and Ma Baker. I imagine Eruption were riding Boney M's coat- tails, Vin.\nI never knew that Far Corporation got their name that way, Rolf Harris's version's quite a hoot, but understandably, is seen as sacrilege by rock music purists worldwide.\nI didn't know Farian produced Eruption Vin, but it's no surprise really. I guess with this Far Corporation he reckoned it was about time he got at least a syllable on the sleeve, seeing his vocals were all over Boney M.\nAs for the abysmal Milli Vanilli - couldn't agree more. Say no more.\nThis one slipped by me! Apologies all.\nI wonder if they get taught props handling at dance school...? Their brolly skills are 2nd to none!\nBut as Ryan says, by the end there must be more water inside their little booties than on the studio floor! Must have been somewhat uncomfortable for them - ever the pros our girls!!\nOverall 8\/10. And DOTD hmmm Rosie or Patti?! I think it has to be Rosie - her smile shines through the inclement weather on this occasion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Designed to improve the competitive performance of collegiate players by providing a better understanding of the relationship between properly fit equipment and lower scores. This program provides access to our entire line of Maxx Fitting products with fees adjusted to a favorable hourly rate. Contact Nate Lammers with questions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GREENWICH, CONN. \u2014 North Castle Partners, a private equity firm with investments in Atkins Nutritionals and Bora Bora Organic Foods, has recapitalized Flatout, Inc. Based in Saline, Mich., Flatout manufactures and markets premium flatbreads, wraps and snack wraps sold under the Flatout brand to the grocer, mass, club store and food service channels.\nStacey Marsh, chief executive officer of Flatout, said the company is excited with the partnership given North Castle's track record of growing strong, niche brands into nationally recognized brands.\nFlatout is the official flatbread of the Best Life Diet, a program created by Bob Greene, Oprah Winfrey's personal trainer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Pinterest, Snapchat or Instagram, when it comes to business marketing what should be your pick?\nMany social networks march in front of us today and it's tough to decide which one is the best.\nHere are some handful of interesting statistics and tips to jumpstart your business promotion on your chosen social networking site.\nInstagram gives its users the platform to share pictures and videos y applying digital filters and caption for more professional and creative look. The app has a vast user base enabling businesses to touch an otherwise unreachable customer base.\nPinterest, the virtual pinboard enables users to curate collections of content. The content shared on this social platform is mainly visual (images and videos). Pinterest pins are 100 times more spreadable compared to a tweet. It's insanely popular for small business marketing.\nSnapchat, a mobile messaging service lets users send photos or videos. These items last only up to 10 seconds before it disappears. Users can discover channels and Live Stories of other not in the friend list into one scrollable row. Snapchat is widely popular among younger audiences.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the greatest ways that you can ensure that your smile is beautiful for years and years to come is by taking care of it with preventative measures. Prevention requires active participation by brushing your teeth at least twice a day, flossing daily, and visiting your dentist on a regular basis for cleanings and checkups. Yet, there is one more way that you can participate in oral care prevention that requires no effort at all \u2013 dental sealants.\nDental sealants are one of the most effective ways of protecting the teeth from cavities and decay, yet they are underused simply because many people do not fully appreciate all that they do. Dental sealants are plastic coatings that are painted onto the molars and premolars in order to \"seal out\" bacteria and food particles that lead to decay. Many parents choose to have dental sealants applied to their young children's teeth, yet they often do not realize that this same prevention is available to them as well.\nThe premolars and molars are the most common teeth affected by decay. The deep grooves, pits, and fissures within these teeth commonly trap bacteria. Brushing these back teeth is often not enough to remove tiny particles trapped within the rough surfaces of these teeth, and floss is not designed to clean these areas. Dental sealants create a barrier on these teeth so that particles do not become trapped.\nDental sealants are painless and quick in application. After the teeth have been thoroughly cleaned, the teeth will be ready for dental sealants. A special coating is painted on the premolars and\/or molars, and the coating is hardened with a special curing light so that the coating will immediately bond with the tooth enamel. This coating is a special plastic material that is designed to seep into the deep fissures and grooves of the back teeth in order to create a blockade against harmful matters that enter the mouth through foods and beverages. As long as the dental sealants remain securely in place, the teeth are fully protected from decay \u2013 without any added work on your part. This added protection is extremely beneficial for children and adolescents who are learning to care for their teeth and whose mouths are continually developing. However, adults can also benefit greatly from dental sealants in order to protect their permanent teeth which are also susceptible to cavities, damage, and decay. Dental sealants typically last from 2-4 years, and they are checked at each dental visit in order to ensure that they are properly secure. Reapplication is as simple as the first application.\nContact Gold Hill Dentistry to learn more about dental sealants and how you can take advantage of this outstanding dental prevention procedure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2013 Originality & Creativity (50%) \u2013 the piece affects the viewer i.e. is powerful, entertaining.\n\u2013 Appropriate use of Theme (50%) \u2013 the piece answers the question and in a way the viewer can understand.\nTo submit your response upload your video to YouTube and provide the link. If you need assistance in uploading or recording your video please contact Cici Tompkins at cici@ceamteam.org or 314-561-8646. Awards in January, date and location TBD.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Put the spatula down and let Schwan's do the cooking for you this Friday night (or any night of the week). Why? Because when the week is almost over the last thing you want to think about is preparing dinner for the family, as you do night in and out!\nThe folks at Schwan's want to give you all a KITCHEN VACATION by providing you a complete VEGETARIAN meal complete with the entrees, sides and dessert that is ready in minutes!\nFIVE CHEESE GARLIC BREAD ~ need I say more!\nBROCCOLI ~ for good measure!\nDESSERT ~ someone once told me dessert has to include CHOCOLATE for it to count as dessert! Schwan's delivers with this decadent, creamy, smooth chocolate ice cream that is only 140 calories, 7 grams of fat and 4 points+ a serving!\n1) Simply leave your FIRST comment telling me why you NEED a \"Kitchen Vacation\".\nThe Fine Print: Thanks to the folks at Schwan's for letting me share in the fun by giving you all the chance to win a complete dinner too!\nThe winner will be randomly drawn and announced on Saturday, March 19, 2011!\n\u00ab How do you \"top\" your ice cream?\nOh my gosh now I'm hungry! Being a SAHM I'm always the one who makes a the meals so a little kitchen vacation would be amazing for a day it would help relax a bit and know that i will still be giving my family a healthy meal. Thk you for the chance!\nI love this giveaway! I need a kitchen vacation giveaway because I've been way too busy lately to cook with my student teaching! This would be perfect. I love the garlic bread.\nOh my gosh, I would love a kitchen vacation especially since I don't have the time to cook like I used to. With kids in activities after school, it leaves little time, PLUS I'd love to try Schwanns!!!\nWAY over retirement age, so a lot of meals have come out of my kitchen. I'm ready for a kitchen vacation. Thanks for a chance Schwans and Danica.\nEveryone needs a kitchen vacation once in awhile.\ni need a kitchen vacay like woah.\nTold a friend about this giveaway.\nI don't cook much, but it would be nice if I won this so I could give my wife a kitchen vacation.\nTold my 3 SIL about this giveaway.\nI need this kitchen vacation because I'm in the process of moving to a new home and I have teenagers that are always on the go. I also love the pointsplus for this meal.\nI don't have a blog but I told three of my friends about your blog and your great giveaway and ideas. Thanks for sharing.\nA kitchen vacation would be amazing. My boyfriend and I are on completely different works schedules and dinner time tends to be a little wonky so this would be so helpful.\nIt is spring braek for me!\nI never can figure out how to do the give-aways but I'm gonna have to figure it out for this one\u2026YUM! For ice cream toppings, if I serve Fr. vanilla I open a can of passion fruit pulp and just pour over\u2026I'm married to an Aussie, and we love passion fruit, lol. Another great post Danika!\nI would love a Kitchen Vacation! On Tuesday nights my daughter has soccer and by the time I get home, I have one hour to have dinner prepared and served and kitchen cleaned before I leave for my Water Aerobics class. This Schwan's dinner would be a huge blessing!\nI need a kitchen vacation!!\nI told my coworkers and fellow WW's about the giveaway!\nWorking 50+ hours per week, bad economy and having to put dinner on the table for 4 every night\u2026.enough said!\nI would love a Kitchen Vacation, because im tired of looking for new and exciting recipes, i would love to just sit down and eat for a change.\nI make breakfast, lunch and dinner EVERYDAY for me, my husband and our 2 kids. I could definitely use a break!\nI run a home daycare. By the end of my day I am about to pull my hair out with having 3 children of my own too:) When daycare is over I barely have the energy to turn around and make a nice big dinner. It would be nice to have a free \"vacation\" from cooking dinner.\nI totally need a kitchen vacation I am a vegetarian so this meal would be awesome, especially since my fianc\u00e9 is going to \"try\" vegetarianism for me.\nI need a kitchen vacation because I teach full-time and now am coaching track and training for a marathon in May! My nights are very limited on time\u2026 I'd be happy to let Schwann's cook for me one night!\nSent notes to coworkers\/friends about this awesome giveaway!!!\nWorking out after work everyday means quick dinners. This prize would be awesome!!\nI have a 3 year old boy\u2026any vacation would be nice!\nSent several we friends an email about this great giveaway!\nI need a kitchen vacation because 1) we are off school this week so I am on \"vacation\" but more importantly 2) I am tired of trying to figure out what to prepare for the six of us for dinner each night.\nMy husband \"can't\" cook, so I'm always the one stuck doing it. I definitely could use vacation in the kitchen!\nI don't have a blog but I will be telling my friend Lauren about this giveaway as soon as she wakes up. The food looks delicious!\nOH my I NEED this! If Mamma don't cook the family don't eat! LOL! I really LOVE to cook, but even I could use a break! With my husband, 2 boys and my MIL living with us I'd love this complete meal it would be such a treat! Thanks!!\nHeck who doesn't need a night off from figuring out what to put together! Great giveaway miss D!\nOMG! I really need this break from cooking. Besides working daily, I take care of my grandkids afterwork 3 days a week and cook for them until their mommy drops in for dinner and finally takes them home. I'm pooped!\nWow! I really need a kitchen vacation. I have a house full of men and I have to cook a full course meal every night. If I even suggest a hotdog they laugh and are like ok nice snack but whats for dinner. WInning this meal would be a great treat for all of us.\nI am in need of a Kitchen Vacation as my husband has been working long hours for the past couple weeks. This requires me to make a supper for the kids at normal supper time and then again when the husband is on his way home. Is he spoiled or what?!\nShared the give away with a friend.\nI need a kitchen vacation, because it is my birthday today! Also, I've been so busy with work, that I haven't had time to cook a real dinner in a few weeks.\nAHH the Schwan's man!!! I remember when I was younger he would come around the neighborhood! This looks delicious!!!\nI just sent an e-mail to two friends letting them know about the give away.\nCould definately use a Kitchen Vacation! We are always so busy so Schwanns meals would be perfect for my family. And they look so yummy!\nI need a kitchen vacation because I'm around kitchens all day at work and the last thing I want to do is cook when I get home. A bowl of cereal is my dinner way too often.\nOh what a fun giveaway! I am a SAHM so I too do all the cooking and meal planning. I absolutely love to cook but sometimes I get tired of trying to figure out what to cook!\nI just told my friend Sandra about the giveaway!\nI need a kitchen vacation because some nights I get home late from school and need an easy meal to make for my family.\nI told my friend about this giveaway!\nI started working full time and really miss being a SAHM. I feel like I am in the kitchen all the time either preparing a meal or prepping for the next one!\nI've been working hard trying to plan and cook healthy meals for my family every week and I definitely need a kitchen vacation!\nI would love love love a kitchen vacation! I live with my fiance and my sister. I work 40+ hours a week, cook dinner every night (thanks to some of your wonderful recipes), and I am in crunch time for planning our wedding (three months away). I get stressed out almost every night and would really appreciate a break from not just the stove but the grocery store as well!\nI need a kitchen \"vacation\" because I feel like I'm in a rut and making the same meals week after week, and spending so much time in the kitchen, and I hate relying on going out to eat as my back-up plan when I have those nights when I don't feel like cooking.\nI need a kitchen vacation because that meal looks AWESOME and if I don't win this, I might have to make it anyways!\nI need a kitchen 'vacation' because I'm tired of cooking out of a mini college fridge while waiting for our new fridge to be delivered. There's nothing like having a general conversation with your husband about a sound you heard coming from the fridge then him jumping up from the couch to open the fridge to see your frozen foods defrosted and the milk warm. On top of that the kid's are hungry and thirsty for anything cold.\nYes I need a vacation!!! A gigantic cooking vacation, I have 5 kids and a hungry husband\u2026.Schwans take me away!!!\nI soooooooo need a kitchen vacation. I am so tired of making the same old thing (frankly, I'm just tired of cooking .. period! lol!). I have a picky 11 yo daughter and an \"always starving\" 24 yo son. Who wouldn't love a break from cooking?!?!?\nI need a kitchen vacation.. Baked Ziti is my son's favorite and he just made the middle school baseball team! I am a busy mom of two, in the Spring we have Lacrosse, Softball, 2 Baseball teams, and 3 Soccer teams.\nFacebooked it and told three of my Biggest Loser Pals @ work !\nI need this kitchen vacation because with the warmer weather, the kids and I are heading to the park and outdoors after school. By the time we get home, we are crunched for time with baths and making supper. This would allow us to enjoy the great weather a little bit longer!\nI need a kitchen vacation because we are in the crazy part of basketball season\u2026March Madness Baby! and I dont want to be prepping alot so this dinner would be easy and healty amidst the games!\nBecause my head hurts from trying to figure out meals that are good AND healthy. It would be great to have a \"personal chef\" fix my dinner, at least for one evening!\ni need a kitchen vacation because I HAVE KIDS and i'd rather play with them instead of slaving over the stove! help a girl out!\nTax season, 'nuff said- I could so use the break from the planning, shopping, preparing & cooking for a night or two! thx!!\nI would love to win this and win a night off from cooking. Count me in.\nI cook every night, I would love a break!!\nWith a newly emptied nest my husband and I are really trying to focus on our health. I've been on Weight Watchers since October I could use a kitchen vacation so I can enjoy prepared foods that look and taste great that fit into my eating plan!\nI don't have a blog, so I will tell my friends with blogs about this opportunity to get a kitchen vacation!\nWhy would I love a kitchen vacation?? Not enough time at the end of the energy!!\nI need a kitchen vacation because sometimes it would be nice to have a meal already planned out that you throw in the oven and you already know the point value. That way you have more time to spend in the evenings with the hubby and kids!\nI need a kitchen vacation because after work, picking up my daughter, going to the gym, and cooking, I get to see my 8 year old about 10 minutes before I have to bustle her off to bed. I miss that little cutie all week.\nI need a kitchen vacation because I run around crazy all week and try to have a healthy dinner ready for the family quickly each night. This would be awesome after a long week and a late evening arriving home from dance practices.\nI would LOVE a kitchen vacation!!! After working 8+ hours a day, I would love to come home and not worry about cooking and spend that time playing with my two little girls!\nBTW- I tweeted your contest too!\nI tweeted about your contest too!\nI emailed two friends about this giveaway.\nI just facebooked about this too!\nI told my co-worker about your blog and contest.\nI need a kitchen vacation! I cook healthy for Dan and I every day. With long hours, commuting, and fitting in workouts, it's exhausting!\nI didn't know that Schwan's carried so much! It would be nice to have a meal decided for me at the end of one of my \"taxi service\" nights!\nFirst of all who doesn't need a kitchen vacation. I for one love to cook but I would welcome to have dinner brought in for me for a change. I would love to have this Italian dinner. Hope I win!\nCause I am sick of making dinner and having no one eat it! After I take all that time to prepare it.\nEveryone one in my family would LOVE this meal! A kitchen vacation would make me soooo happy!\nI need a Kitchen Vacation because I spend way too much time in my closet kitchen in which I can stand in the middle, open the stove, turn on the water, get something from the freezer or fridge, make some toast and get a pan or can from one of my many cabinets\u2026I can do all this without moving from my starting place. Way too small, I need to let someone else cook for me so that I won't become cluastrophobic!\nI just tols My co-worker Kathy, who is also trying to win a Kitchen vacation too!\nHi Danica. I live in Canada, and we don't have Schwan's here, so no need for me to enter, BUT\u2026just wanted to say I love the idea of a kitchen vacation. Actually I love the idea of a vacation period! \ud83d\ude42 Have a great day, good luck to all who enter.\nI need a kitchen vacation because life gets too hectic sometimes!\nI NEED a kitchen vacation! I cook 6 nights out of the week for my family of 5, sometimes 6 or 7 depending if my kids friends come over.\nand I just noted that if #1 wins, we all win, it looks like we are all #1.\nI need a kitchen vacation because I have to cook 7 days a week most of the time.\nWith 5 kids under 8 I could REALLY use a break from cooking!\nBecause everyone needs a kitchen! Although I have to say the commenter in front of me (with 5 kids under 8) probably NEEDS it more!\nAs for all your amazing giveaways \u2013 I told my girlfriends!\nI need a kitchen vacay because this whole wife thing is tough work!\nI need a kitchen vacation, because I cook dinner every night and need a change.\nI need a Kitchen vacation because working tax season is long and exhausting. Which means the last thing I want to do is cook when I get home!\nI dont have a blog but I told my coworkers about your giveaway!\nI don't have a blog, but I can tell all the teachers that I work with and my Mom who absolutely LOVES Schwan's food. For my b-day last yr. she gave me a suprise of a whole freezer food of their products. It was awesome!!\nI need a kitchen vacation because we are currently redoing our kitchen so I don't even have a place to cook!\nI need a kitchen vacation because I am the sole \"chef\" for my house full of boys (and one man) and boy can they eat\u2026alot!!!\nI need a kitchen vacation b\/c I have 3 small kids and am the primary cooker\/baker in the house. I could use a night off.\nI really need a kitchen vacation. It's almost 7 pm when I get home from work each night. It would be great to have a meal to just stick in the oven and go. And, I already am a Schwan's customer and I love their food.\nI need a kitchen vacation because I work full-time supporting my husband who's in school full time and a 3 year old. Hard to get dinner on the table every night, when I don't get home until 6:00.\nI told my friend Thea about your giveaway!\nSounds awesome-would love a kitchen vacation-and what a great way to mix it up tried Schwan's ages ago and it was really good. Would love to try it again!\nI need a Kitchen Vacation since my husband and I gave up eating out for Lent. i could sure use some yummy food that I don't have to cook after a long day at work.\nI told my husband about the contest\u2026he is super excited for something yummy!\nI need a kitchen vacation because I have 5 year-old twins and I'm BEAT by the time I get home from work!!!\nI'm a busy college student trying to juggle several different things at one time so I definitely need a kitchen vacation!\nI need a vacation because I have a two year old who is very good at being two! \ud83d\ude42 I love her to pieces but making dinner while my husband is at work on a weeknight is very tiring. My daughter loves to \"help\" me in the kitchen, so having a meal that is ready to go from Schwans would be wonderful help!\nI told my best friend about this giveaway! Thanks!\nI'm a full-time Administrative Assistant at a busy CPA firm. It's Tax season so we are swamped EVERY day. My husband works night so it's up to me to take care of the day-to-day activities at home plus look after my son and what he has going on. I love to cook but a Kitchen Vacation would be a welcomed and appreciated prize. Thanks, Danica, for the opportunity.\nI need a vacation getaway because it's not the cooking that gets me, it's the CLEANING!\nNot only would I love to win this Kitchen Vacation, but I'm sure my kids and husband would too!! We have had two to many family members in and out of the hospital lately and between that and sports for the kids, we eat out way to much. The kids would love a nice home cooked meal and that is exactly what Schwans food taste like!! I love Schwans!\nI need a Kitchen Vacation because I am starting out on WW and it is SOOOO hard and tedious to count points, look things up, and make food that my Husband will like.\nAwesome awesome giveaway! This would be great for me on late nights when my husband and I have worked all day and are tired but need good food. I also like the idea of having friends over for a delicious dinner without all the dinner-making work!\ni need a vacation b\/c i have to gather 3 different meals all for supper in one night- ugh- 1st my son is 2 and is picky and my daughter is 1 \u2013 she cant eat alot yet and then i cook for me and my hubby and its alot.\nI \"need\" a kitchen vacation because I spend hours in that room and I could use a real change of scenery. It would sure be a treat to win!\nI told my workout buddy Kim about the contest. I know she could use a kitchen vacation.\nI need a kitchen vacation because cooking everyday while on weight watchers is exhausting. especially after chauffeuring children here there and everywhere, getting home late\u2026then having to do baths and homework and get in my own workout. How wonderful to just pop something in the oven and not worry about it ,plus baked pasta is something everyone loves so it is a win, win!\nI told a few people\u2026hubby, good friend also on WW.\nI need it cause I work too many hours!\nI NEED a kitchen vacation because I am the one who is solely responsible for the cooking, and it gets tiring. Same complaint of women everywhere, right? \ud83d\ude42 I love cooking, but sometimes it would be nice to have a break that doesn't include a brown paper bag lol And this just looks really delicious.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fun, traditions and Christian values in a rustic setting - that's what makes us so special, along with over 100 rolling, wooded acres in Tallulah Falls, GA. Our girls camp programs provide opportunities for personal development and spiritual growth with lots of fun - in a safe, protective environment both kids and parents feel good about. The girls camp setting is a lovely setting with a private lake that is complimented by surrounding mountains.. There are opportunities to develop friendships, build self esteem, improve abilities and develop greater strength of character and personality. Of course, the health and safety of each camper is top priority; our 1:6 counselor-to-camper ratio assures close supervision of all activities.\nGirls are placed in a cabin with campers their own ages. This is their home while they are at camp. Cabins house between 8 and 15 campers. Each cabin has a bathroom located in the building and campers are responsible for keeping their living quarters neat and tidy. Twin size bunk beds are used for sleeping and sheets are recommended.We do our best to honor cabinmate request and can usually honor them as long as the campers are within ONE year of age of each other. Be mindful that it is difficult to put four or more boys that know each other in the same cabin due to the negative affects of the cabin dynamics. This will be their home while they are at camp. Each cabin has a bathroom located in the building and campers are responsible for keeping their living quarters neat and tidy. Twin size bunk beds are used for sleeping and sheets are recommended.\nGirls are kept very busy during their time at camp. A typical day consist of cabin cleanup, league games, rest hour, classes, free-time, morning and evening assemblies, evening activities and cabin devotions. Three meals are squeezed into the daily routine.\nActivites: These are some of the fun things there are to do at camp!\nTrips are taken almost everyday by some group of campers. Sliding Rock Gorge trips happen on Tues and Thur for the boys, Fri for the Girls and are limited to 50 people each time by the State of GA DNR. Older campers have priority on this trip.\nremembered but not often experienced. Campers will learn and experience many of the old fashion values and truths that make our camp programs unique.\nPersonal responsibility, maturity and friendship are all life skills that are learned and honed at camp.\nWe offer 8 one-week sessions of summer day camp for ages 7-14. Our sessions start on Mondays and end on Fridays. Campers arrive before breakfast and leave before dinner. Throughout the day, campers are integrated into classes and activities as well as the cabin group of their age and school year for lunch and rest time.\nDay campers enjoy the same historic facilities and exciting programming that make our residential camps so awesome!\nThe Leader In Training (girls ages 15-16) program has been an integral part of both camps from its inception, and has produced many fine leaders over the years. Where do great counselors come from? Great campers, of course. The LIT program is a rigorous 4-week program than focuses on authentic Christian leadership. Participants will learn about themselves and the type of leader that God wants them to be by focusing on five core areas: Values, Relationships, Purpose, Heart and Self-discipline. The program consists of intensive training in areas such as program planning and development, child development, vespers, child abuse recognition, CPR and Lifeguard training.\nThe young girls in this program develop leadership skills such as teaching, public speaking, responsible mentoring and decision making. They also participate as a group in traditional camper activities and overnight trips. Additionally, they take several out of camp trips such as: white water rafting, hiking, caving or even traveling to a pro baseball game or amusement park. A special trip is taken the last week of the program that serves as a bonding experience for the friendships that have formed. Trips from past years include The Outer Banks of NC, Charleston SC, Cumberland Island, GA. and local mountain retreats.\nThe group operates as a close knit unit from which many lifelong friendships are forged. Those participants who have completed the training and have the qualities and skills required of the Camps leadership will be asked to return the following summer as a staff member! The value of the LIT experience is incalculable, and the rewards are many. LITs grow not only as leaders but also as young women. With the process of learning-to-teach, comes greater self-awareness and physical, mental and spiritual maturity. Perhaps the greatest reward is the satisfaction that comes from serving as a positive influence on the lives of others not only in one summer but in many to follow.\nThe Xtremist Program (for girls 12-14) is a beginner level program for campers who are interested in outdoor adventure\/sports. The program is tailored to the age and skill level of the participants. It strives to teach new skills and to continue building upon previously learned skills, all within a framework of fun and safety! Campers will participate in some of the traditional camp programs and reside in a cabin. Classes and league games will be replaced by high and low ropes course activities, zip-line rides, the climbing tower, and rappelling instruction! Participants will also learn team building skills on the high and low ropes courses, have a day of rock climbing and rappelling on \"real rock\" away from camp. X-tremist experiences may include a one to two night trip to NW GA and TN for caving and rafting the Ocoee River. It is not uncommon for the boys and girls adventure program participants to travel together and participate in the off campus activities together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This Fluidigm Assay Design Portal Agreement (\"Portal Agreement\") is a legal agreement that governs the use of the Fluidigm Assay Design Portal.\n3. The entity for which the User works (\"the Customer\").\nIn this Portal Agreement, \"You\" and \"Your\" shall mean the User and the Customer unless specified otherwise. If the User is using the Assay Design Portal on his or her own behalf as an individual, both terms (\"the User\" and \"the Customer\") shall refer to the same natural person.\nThe terms of this Portal Agreement are referenced by and incorporated in the Fluidigm Corporation Assay Design Sales Terms and Conditions. By placing an order through the Assay Design Portal, the User and Customer are agreeing to the terms of this Portal Agreement.\nYou may use the Fluidigm Assay Design Portal only after accepting the terms of this Portal Agreement. By electronically accepting this Portal Agreement on a computer terminal, You agree to be bound by all of the terms and conditions of this Portal Agreement individually. If You do not agree with all of the terms and conditions of this Portal Agreement, You must decline the terms and conditions and You will not be able or permitted to use the Fluidigm Assay Design Portal.\n1.2 \"Assay Design\" shall mean the sequence of primers and probes designed by Fluidigm based on the data You provide.\n1.3 \"Assay Design Portal\" shall mean the internet-based services and features Fluidigm makes available to Customer or User, including through (a) websites at the top-level domain d3.fluidigm.com (b) any other Fluidigm branded or co-branded websites (including sub-domains, international versions, widgets, and mobile versions) and (c) other networks, media, and software (such as mobile applications) now existing or later developed.\n1.4 \"Content\" shall mean any and all web pages, Assay Designs, documents, data, and other information stored in or available for download from the Assay Design Portal.\n1.5 \"Data Intellectual Property\" shall mean all Intellectual Property Rights necessary to upload data provided by User or Customer, including genetic sequence data, to the Assay Design Portal, all Intellectual Property Rights necessary to make or have made any assays ordered using the Assay Design Portal, and all Intellectual Property Rights necessary to process the data uploaded to the Assay Design Portal.\n1.6 \"Fluidigm Copyrights\" shall mean the copyrights, moral rights, and similar rights related to original works of authorship owned or licensed by Fluidigm, including: text; dramatic or non-dramatic literary works; photos; drawings; works of visual arts; works of the performing arts; serials; phone records; sound recordings; software (including source and object code); mask works; and any other kind of copyrightable work.\n1.7 \"Fluidigm Marks\" shall mean the trademarks, service marks, design marks, logos, trade names, tag lines, and other source indicators owned or licensed by Fluidigm, including (a) those listed on Exhibit A to this Portal Agreement and (b) any others designated from time to time by Fluidigm.\n1.8 \"Intellectual Property Rights\" shall mean any patents (including utility models and rights to patent an invention), design patents, designs (whether or not capable of registration), chip topography rights, mask works and other like protection, copyrights, trademarks and any other forms of like protection as well as any applications, divisions, continuations, renewals, re-exams and reissues thereof worldwide as well as any trade secrets and other information or know-how.\n2.1 Notwithstanding anything in this Portal Agreement to the contrary, in the event of any inconsistency between any Assay Design Sales Terms and Conditions and this Portal Agreement, the terms of this Portal Agreement control.\n3.1 Grant of Access. Subject to the terms of this Portal Agreement, Fluidigm grants the User and the Customer access to the Assay Design Portal.\n3.1.1 The grant of access to the Assay Design Portal and the Content is personal to the User and to the Customer and made solely for the purpose of assisting the User and the Customer in developing assays and placing requests for sales quotes for those assays. Any other use by the User or the Customer of Assay Designs or Content is prohibited. By way of example and not limitation, re-sale of Assay Designs or Content to third parties or re-use of Assay Designs or Content for commercial purposes is prohibited.\n3.1.2 Usernames and passwords for the Assay Design Portal are personal and may only be used by the User identified by the credentials. Multiple users of the portal working for the Customer must each have a username and password to access the Assay Design Portal. If one of those employees leaves the employment of the Customer, the Customer shall immediately notify Fluidigm, which will deactivate the username and password for that user.\n3.1.3 The User's and Customer's access to the Assay Design Portal and the Content may be terminated at any time by Fluidigm within its sole discretion without notice to Customer or User. The Content in the Assay Design Portal is subject to change by Fluidigm within its sole discretion without notice to Customer or User.\n3.2.1 Neither Customer nor User shall disrupt or interfere with the security of the Assay Design Portal, or the servers or networks connected to or accessible through the Assay Design Portal. Neither Customer nor User shall attempt to change or alter the Assay Design Portal or the Content except as expressly permitted elsewhere in this Portal Agreement. Neither Customer nor User shall attempt to obtain unauthorized access to the Assay Design Portal or to portions of the Assay Design Portal that are restricted from general access. Neither Customer nor User shall attempt to reverse engineer, deconstruct, disassemble, or decompile any software or technology underlying or provided through the Assay Design Portal.\n3.2.2 Customer and User shall provide accurate, up-to-date, information in the profile section of Portal. Neither Customer nor User shall provide any false information or attempt to create a profile for any other party.\n3.2.3 Neither Customer nor User shall give access to the Assay Design Portal to, or share usernames or passwords with, any unauthorized person, or knowingly take any other action that might jeopardize the security of the Assay Design Portal. Customer and User shall treat Portal usernames and passwords with at least the same degree of care as Customer treats usernames and passwords used with its own systems and networks, but not less than reasonable care.\n3.2.4 Customer and User shall abide by any terms of service that may be posted by Fluidigm or its agents in the Assay Design Portal from time to time.\n3.2.5 You may publish Assay Designs in connection with the non-commercial publication of a scientific article, provided that You acknowledge your use of the Assay Design Portal in any such publication.\n3.3.1 Fluidigm conducts periodic security reviews of the Assay Design Portal to limit security threats and prevent unauthorized access to the portal, and works with commercially reasonable diligence to close any discovered security threats.\n3.3.2 Access to the Assay Design Portal is controlled by username and password, and passwords can be reset via e-mail. Furthermore, usernames, passwords, and other proprietary information may be sent in clear text (without SSL or TLS) over the open internet between a client computer and Fluidigm servers. Data relating to the Assay Design Portal may be stored in unencrypted form on Fluidigm servers.\n3.3.3 Customer and User agree not to submit or upload any information of a sensitivity or confidentiality that requires stronger security measures than those recited above. Furthermore, Customer and User agree that they will not suffer irreparable harm if a third party were to gain access to any Customer or User information from the Assay Design Portal.\n3.3.4 Customer and User agree to keep copies of all data uploaded and submitted to the Fluidigm Assay Design Portal.\n3.3.5 Each nucleic acid sequence or target gene submitted or uploaded by You through the Assay Design Portal that is intended to be detected by use of a custom assay or to be included in primers and probes or other oligonucleotide assays manufactured by Fluidigm and sold to You, and the facts that You placed orders for assays containing or intended to detect such sequence and that You ordered oligonucleotide assays from Fluidigm containing or intending to detect such sequence shall be \"Customer Confidential Information.\" Fluidigm agrees that for five (5) years after the disclosure to Fluidigm of Customer Confidential Information, Fluidigm shall not disclose such Customer Confidential Information to any third party and will use at least the same degree of care as it uses to protect its own confidential information of a like nature, but in no event less than a reasonable degree of care, to prevent the disclosure of such Customer Confidential Information to any third party. This undertaking of confidentiality shall not apply to, and Fluidigm shall have no obligations under this paragraph with respect to (a) any Customer Confidential Information that is or becomes a matter of public knowledge or part of the public domain through no fault of Fluidigm; (b) any Customer Confidential Information that is disclosed by Fluidigm with Customer's prior written approval; or (c) any information shared by You to a person not under a duty of confidentiality to You through. the sharing functionality of the Assay Design Portal or otherwise.. Notwithstanding any provision of this Portal Agreement, Fluidigm may disclose Confidential Information of Customer to the extent required to comply with governmental regulations and other applicable laws or to respond to subpoena or other compulsory legal process.\n3.4.1 Fluidigm does not provide any warranty or guarantee as to the accuracy, completeness, timeliness, or suitability of the Content for any particular purpose.\n3.4.2 IN NO EVENT SHALL FLUIDIGM BE LIABLE TO CUSTOMER OR USER, ANY CUSTOMER OR ANY OTHER THIRD PARTY IN ANY MANNER FOR ANY SPECIAL, NON-COMPENSATORY, CONSEQUENTIAL, INDIRECT, INCIDENTAL, STATUTORY OR PUNITIVE DAMAGES OF ANY KIND, INCLUDING, WITHOUT LIMITATION, FOR LOST PROFITS, LOST SALES, LOST REVENUE OR LOSS OF USE, REGARDLESS OF THE FORM OF ACTION, WHETHER IN CONTRACT, TORT, NEGLIGENCE, STRICT LIABILITY, OR OTHERWISE, EVEN IF FLUIDIGM HAS BEEN INFORMED OF OR IS AWARE OF THE POSSIBILITY OF ANY SUCH DAMAGES IN ADVANCE, ARISING OUT OF OR RELATING TO CUSTOMER OR USER'S USE OF THE ASSAY DESIGN PORTAL OR THE CONTENT. IN NO EVENT SHALL FLUIDIGM'S LIABILITY EXCEED THE AMOUNT PAID BY CUSTOMER TO FLUIDIGM IN THE PRIOR TWELVE (12) MONTHS. IN THE EVENT OF ANY DATA LOSS OR DATA BREACH RESULTING IN DISCLOSURE OF USER INFORMATION, CUSTOMER INFORMATION, OR CUSTOMER CONFIDENTIAL INFORMATION TO THIRD PARTIES, FLUIDIGM SHALL NOT BE LIABLE TO CUSTOMER OR USER FOR MORE THAN THE AGGREGATE PRICE OF ORDERS PLACED THROUGH THE ASSAY DESIGN PORTAL DURING THE PREVIOUS 12 MONTHS, OR IF SUCH AGGREGATE PRICE IS ZERO, THEN NO MORE THAN $5,000 (FIVE THOUSAND UNITED STATES DOLLARS), EVEN WHEN SUCH DATA LOSS OR DATA BREACH IS DUE TO FLUIDIGM'S NEGLIGENT OR RECKLESS CONDUCT.\n3.4.3 The Assay Design Portal or the Content may from time to time include links to other websites. These links are provided for convenience only and do not signify that Fluidigm endorses the website(s) or that the site owner endorses Fluidigm. Fluidigm has no responsibility for the content of the linked website(s).\n4.1 Warranty by Customer and User. Customer and User hereby warrant to Fluidigm that they have all necessary rights, including Intellectual Property Rights, to use and upload any data uploaded to the Assay Design Portal. Furthermore, Customer and User hereby warrant to Fluidigm that they have the necessary rights, including Intellectual Property Rights, to make and have made any assays ordered using the Assay Design Portal, including sufficient rights to grant the rights granted under Article 4.3 below to Fluidigm.\n4.2 Indemnity by Customer and User. Customer and User shall fully indemnify and hold Fluidigm harmless for any damage caused to Fluidigm by a breach of Article 4.1 above, including paying legal fees and any monetary award associated with a patent infringement lawsuit or any other lawsuit.\n4.3 License Grant to Fluidigm. Customer and User hereby grant to Fluidigm a non-exclusive royalty free fully paid up license with the right to sublicense to all Data Intellectual Property to make, use, sell, offer to sell, import, export and otherwise transfer any product and to practice any method or process and use any product involved in the manufacture thereof throughout the world.\n4.4 No License Granted by Fluidigm. The Fluidigm Marks and Fluidigm Copyrights are Fluidigm property, and no license or permission is given to User or Customer to use the Fluidigm Marks or Fluidigm Copyrights other than to the limited extent necessary for User or Customer to exercise the grant of access to the Assay Design Portal set forth in Section 3.1 of this Portal Agreement. Other than use necessarily attendant to use of the Assay Design Portal to place requests for sales quotes for assays, no permission is granted to use the Fluidigm Marks or Fluidigm Copyrights.\n5.1 The Assay Design Portal may be used to design assays and to place requests for sales quotes to Fluidigm. No response generated by or communicated via the Assay Design Portal shall constitute a binding contract, warranty, or statement that Fluidigm can or will make an assay. Furthermore, no response generated by or communicated via the Assay Design Portal constitutes an offer to make or sell any particular assay. Only a formal offer submitted by a Fluidigm sales representative shall comprise an offer to make or sell an assay.\n6.1 Term. This Portal Agreement shall be effective with respect to each party from acceptance in accordance with Article 6.2 below, and remain in force until terminated pursuant to this Portal Agreement.\n6.2 Acceptance. The User, individually, shall be deemed to have irrevocably accepted the terms of this Portal Agreement by indicating agreement on the Assay Design Portal website or by using the Assay Design Portal website. The Customer shall be deemed to have irrevocably accepted the this Portal Agreement when agreeing to this software license by operation of another agreement and\/or when instructing a User to use the Assay Design Portal in any way, or when a user with authority to bind Customer uses the Assay Design Portal in any way.\n6.3 Termination by Fluidigm. Fluidigm may terminate this Portal Agreement with immediate effect without providing notice to either User or Customer. Such termination may include the immediate suspension of access to the Assay Design Portal for User and\/or Customer.\n6.4 Termination by User or Customer. This Portal Agreement shall terminate thirty (30) days after notice of such termination is received by Fluidigm.\n6.5.3 Take such other steps as may be reasonably necessary to stop use of the information and materials provided to Customer or User pursuant to this Portal Agreement.\n6.5.4 The provisions of Articles 1 and 4 and of Sections 3.2.1, 3.2.3, 3.2.4, 3.4, 7.3, and 7.4 shall survive the expiration or termination of this Portal Agreement.\n7.1 Notices. Any notice required or permitted hereunder shall be in English, in writing and shall be deemed to be properly given upon the earlier of (i) actual receipt by the addressee (including facsimile or e-mail) or (ii) five (5) business days after deposit in the mail, postage prepaid, when mailed by registered or certified airmail, return receipt requested, or (iii) two (2) business days after being sent via private industry courier to a party to such address as it may designate in a writing. Notices to Fluidigm shall be to the attention of the Legal Department and sent by registered mail at 7000 Shoreline Court, Suite 100, South San Francisco 94080. An electronic copy of the notice may also be sent via email to legal@fluidigm.com and assay_design_group@fluidigm.com. Notices to You shall be sent to the address You provide when You register to use the Assay Design Portal.\n7.2 Assignment. Except as otherwise expressly provided under this Portal Agreement, neither this Portal Agreement nor any right or obligation hereunder may be assigned or otherwise transferred (whether voluntarily, by operation of law or otherwise) by Customer or User without the prior express written consent of Fluidigm. Fluidigm may freely assign this Portal Agreement or any right or obligation hereunder. Any permitted assignee shall assume all obligations of its assignor under this Portal Agreement. Any purported assignment or transfer in violation of this Article 7.2 shall be void.\n7.3 Arbitration. Fluidigm, Customer and User agree that any dispute or controversy arising out of or in connection with this Portal Agreement shall be finally settled by binding arbitration under the then current rules of the American Arbitration Association. The venue of such arbitration shall be in San Francisco, California. The arbitration shall be conducted by a single independent arbitrator appointed in accordance with such rules. The arbitrator shall determine what discovery will be permitted, based on the principle of limiting the cost and time which the parties must expend on discovery; provided, the arbitrator(s) shall permit such discovery as they deem necessary to achieve an equitable resolution of the dispute. The decision and\/or award rendered by the arbitrator shall be written, final and non-appealable and may be entered in any court of competent jurisdiction. The parties agree that, any provision of applicable law notwithstanding, they will not request, and the arbitrator shall have no authority to award, punitive or exemplary damages against any party. The costs of any arbitration, including administrative fees and fees of the arbitrator, shall be shared equally by the parties, unless otherwise specified by the arbitrator. Each party shall bear the cost of its own attorneys' and expert fees; provided that the arbitrator may in its discretion award to the prevailing party the costs and expenses incurred by the prevailing party in connection with the arbitration proceeding. This Article shall not be construed to prohibit either party from seeking preliminary or permanent injunctive relief, restraining orders, decrees of specific performance, or other equitable remedies, in any court of competent jurisdiction. For avoidance of doubt, any such equitable remedies shall be cumulative and not exclusive and are in addition to any other remedies which either party may have under this Portal Agreement or applicable law.\n7.4 Governing Law and Venue. This Agreement shall be construed and enforced in accordance with the laws of the State of California, USA, without regard to the conflicts of law principles, and shall not be governed by the United Nations Convention on Contracts for the International Sale of Goods. The parties hereby submit to the jurisdiction of, and venue in, the federal courts located in the Northern District of California and the state courts located in San Mateo County, California, USA.\n7.5 Severability. Each party hereby acknowledges that it does not intend to violate any public policy, statutory or common laws, rules, regulations, treaty or decision of any government agency or executive body thereof of any country or community or association of countries. Should one or more provisions of this Portal Agreement be or become invalid, the parties shall substitute, by mutual consent, valid provisions for such invalid provisions which valid provisions in their economic effect are sufficiently similar to the invalid provisions that it can be reasonably assumed that the parties would have entered into this Portal Agreement with such provisions. In case such provisions cannot be agreed upon, the invalidity of one or several provisions of this Portal Agreement shall not affect the validity of this Portal Agreement as a whole, unless the invalid provisions are of such essential importance to this Portal Agreement that it is to be reasonably assumed that the parties would not have entered into this Portal Agreement without the invalid provisions.\n7.6 Entire Agreement; Amendment. This Agreement contains the entire understanding of the parties with respect to the subject matter hereof. All express or implied representations, agreements and understandings with respect to the subject matter hereof, either oral or written, heretofore made are expressly superseded by this Portal Agreement. This Agreement may be amended, or any term hereof modified, only by a written instrument duly executed by both parties.\n7.7 Nature of Relationship. Nothing in this Portal Agreement is intended to, or shall be construed to, create a partnership, agency, joint venture, employment or similar relationship.\n7.8 Waiver. The waiver by a party of any right hereunder, or the failure to perform or of a breach by the other party, shall not constitute a waiver of any other right hereunder or of any other breach or failure by the other party whether of a similar nature or otherwise.\n7.9 Force Majeure. A party shall neither be held liable or responsible to the other party, nor be deemed to have defaulted under or breached this Portal Agreement, for failure or delay in fulfilling or performing any obligation under this Portal Agreement to the extent, and for so long as, such failure or delay is caused by or results from causes beyond the reasonable control of such party including but not limited to fire, floods, embargoes, war, acts of war (whether war be declared or not), insurrections, riots, civil commotions, strikes, lockouts or other labor disturbances, acts of God or acts, supplier delays, omissions or delays in acting by any governmental authority or the other party.\n7.10 Authority. By clicking on the \"I Agree\" button or by otherwise accepting this Portal Agreement or using or accessing the Assay Design Portal, You represent and warrant that You are at least eighteen (18) years of age and have the authority and capacity to enter into this Portal Agreement on Your own behalf. If accepting this Portal Agreement on behalf of the Customer, You represent and warrant if acting on behalf of a corporation or other person or entity that You are authorized and empowered to bind such corporation, person or entity to all of the terms and conditions set forth in this Portal Agreement; and Your use of the Assay Design Portal and Documentation is not governed by any funding, grant or other terns or conditions, whether from the United States federal government or other institution, that would prevent You from complying with all of the terms of this Portal Agreement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Successful investment management requires planning and discipline.\nGareth's focus is on utilizing the independent, value-based investment philosophy of the Odlum Brown Research Department to cut through distracting news headlines to ensure your investment decisions are based on solid fundamentals. Helping clients navigate the risks and rewards of investing and avoiding mistakes is the basis of long-term wealth generation. The success of the research department's hypothetical Odlum Brown Model Portfolio demonstrates the results of discipline over time.\nGareth has a strong focus on the planning process and he works with each client to build a portfolio tailored to complement their financial objectives, needs and risk tolerance. He is dedicated to building long-lasting relationships based upon trust and confidence, allowing him to advise clients through the financial challenges and transitions faced in each stage of their life.\nGareth joined the team at Odlum Brown in 2014. He was born and raised in BC, and has lived and worked within the Victoria community for much of his life. Before joining Odlum Brown, he served Victoria for 10 years as the owner and operator of a successful construction contracting business. On weekends he can be found playing in his woodworking shop, or cycling the streets and trails in and around Victoria.\nIf you would like to receive our monthly newsletter, the Odlum Brown Report, please contact Gareth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mattel will celebrate the 50th anniversary of Star Trek with a trio of Barbie Collector Star Trek 50th Anniversary Dolls inspired by Captain Kirk, Mr. Spock and Uhura. The dolls, for adult collectors and due to ship in the near future, will be sculpted in the images of William Shatner, Leonard Nimoy and Nichelle Nichols, with the individual dolls featuring a fully articulated body and sporting an authentic replication of their respective TOS costumes.\nThe Kirk doll will come with a communicator and phaser, while the Uhura doll will depict her with a communicator in one hand and a tricorder strapped over her shoulder. The Spock doll will make the Vulcan salute with one and hold a communicator in the other.\nEach of the dolls will cost $34.99 and be available in a box that measures 13\u2033H by 3.3\u2033W by 7\u2033L. Go to www.thebarbiecollection.com to pre-order.\nMattel is more than ready for DC's summer blockbuster, \"Batman VS Superman: Dawn of Justice\", and at Toy Fair 2016 showed off toys and dolls from the film.\nAs part of the six-inch line, we will see versions of the titular characters, along with Wonder Woman, Aqua Man, and two other versions of Batman \u2013 armored Batman, and the desert-style Batman. The film will also be getting the Barbie treatment, with Batman, Superman, and Wonder Woman all getting dolls. Lastly, 12-inch figures with limited articulation will also be available of the DC trinity.\nThere was plenty more \"Batman VS Superman: Dawn of Justice\" toys and statues at NY Toy Fair, including the ones seen at DC Direct's booth.\nNot that long ago, Mattel announced that Barbie would be getting a makeover, and not just the kind that involves make-up but an all new look to better match the modern world. At NY Toy Fair 2016, we got to see first hand what the new Barbies would be like, with new body types that represent a more diverse type of woman.\nIn this video, we get a quick overview of the new bodies for Barbie, which come in four sizes along with flat feet and arched feet, plus the new skin tones and hair styles that bring their flagship doll into modernity. We also see the new fashions Barbie will be wearing, as well as a new line of dolls that allows you to dye and change her hair as often as you wish.\nIf you thought there would not be a ton of merchandise for the upcoming \"Batman VS Superman: Dawn of Justice\" movie, you must also believe that diet soda is actually better for you than normal soda. Mattel, who just recently announced new bodies types for their Barbie dolls, will be releasing 12-inch dolls of the DC trilogy characters.\nArriving officially on February 20, the line-up includes Wonder Woman, Batman (with removable mask) and Superman. Check out the photo gallery to see some pictures of the prototypes, as well as their packaging.\nYou can order the dolls on Amazon for $45 as of the date of this article being posted. Head to this link here if you want the Batman \" target=\"_blank\">BatmanWonder Woman , or this one for Superman .\nFans of \"The Hunger Games\" books and now movie series can officially look forward to their first piece of bizarre merchandise that we should have seen coming \u2013 a barbie doll.\nEW gave us the first look at the iconic heroine after being filtered through Mattel's Barbie-making factory, which features the series female protagonist, Katniss Everdeen, ready to fight.\n\"I chose to dress her in the outfit she wears during the games,\" said designer Bill Greening, \"Since this is where all the non-stop action takes place and is instantly recognizable by fans. Of course, she wears her mockingjay pin proudly on her lapel\u2026 The doll's minimalistic style and details \u2014 such as her loosely braided hair and makeup-free look \u2014 also really embody the heroic character Katniss\u2026 I am a huge Hunger Games fan and loved all three books, so it was truly an amazing experience for me to be able to design the Katniss Barbie doll. I think it really pays tribute to Katniss and The Hunger Games.\nHere's the doll in all of her glory, which is available for preorder starting today on Amazon.com , and will be in stores this August.\nAnd here is your way to order it right off of Amazon \u2013 how nice of me!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A dramatic purple, teal and black grand entry to a bat mitzvah party photographed at the Greystone Manor night club in Los Angeles, CA. Large black vases filled with beautiful purple flowers, by Sky Flowers, set on top of tall black box pedestals, make a bold statement to welcome guests to the event planned by Ilana Wenger.\nLocation: 643 N. La Cienega Boulevard, Los Angeles, CA 90069.\nKeywords: Details (1125), Event Photography (384), Flowers (471), Mitzvah (78).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What weapon would you like to get for the next necromancer elite specialization?\nClarification: I added an option for \"other\" weapons, too, so you can tell us what currently non-existant weapon you would like to get as a necromancer.\nI would appreciate if you voted and told us why you'd like to see your chosen weapon for the next necromancer's elite specialization!\nDoesn't matter which weapon Necro gets.\nNecro will be OP in beta, get nerfed.\nNecro will be OP after release, get nerfed to trash tier after one to two weeks.\nNecro will get \"looked at\" in the future after the numbers are crunched.\nNecro will get tooltip fixed so that what was thought a bug was actually intended all along.\nNecro will wait for next xpac.\nI'd like to see mace similar to a holy paladin... Except like literally the opposite and focused on blood or death magic. Conversely I'd love to see an elite spec that has its own variation on staff skills.\nDouble sword or staff to play like Darth Jedi. No rifles or other kitten.\nI picked shield because i want to see a higly defensive necro with tons of evades tons and tons of blocks invulns.Imagine if necro could do what guardian could.\nThis is the part where I'm going to point out that Guardians only ever use Shield for the knockback. It really isn't a defensive set. Focus is the weapon with the condi cleanse and blocks.\nThat does remind me, Necro has Focus too. I had actually forgotten.\nSword as a condi melee weapon and because sword has the best selection of skins (aside from gs which we already have).\nI just can't chose anything else than a meme off-hand axe.\nWhy? Because I believe that I will be worth it to see what they could end up doing with another axe trait.\nThis is a difficult decision for me.\nI had been asking for land spear for years and now thief has some staff animations that might be used with some modifications.\nRifle was another old request for more focused use against single targets rather than staff's AoE.\nNecromancer's off hand weapon selections seem to cover most uses now that torch fills the condi dps role.\nPower Necro has MH\/Dual axe and greatsword, OH horn and focus.\nCorruption Necro has MH sceptre and dagger, OH dagger.\nCondi dps Necro has MH\/Dual sceptre and greatsword, OH torch.\nWhile I wished Scourge brought MH torch to better isolate its condition dps (and builds) from sceptre, that seems way too late to change.\nFor utilities, banners or traps seem to have potential.\nSword would be my first choice, but honestly we have a lot of good offensive options as is.\nShield would give us (presumably) blocking abilities, and it would mark our first active defense abilities (which would be fantastic).\nPerhaps a blocking ability that corrupts boons when attacks are blocked on the \"4\" ability and a reflect that gives us retaliation on the \"5\" ability.\nHow about nothing, joke is if we don't have anything to begin with there's nothing for developers to randomly taken away.\nA real power based necromancer spec would actually be good (unlike reaper).\nI don't see why you're asking for new weapons when almost all of the ones that Necro has right now are trash. If they add any more, it'd be best to just remove what already exists. Or just compact what little good rests in the Necro weapon line-up into 2-3 weapons that aren't hot garbage.\nClose call between Mace and Hammer. Like the idea of a \"pure\" power build with absolutely no damaging condition on any Trait, Weapon-, shroud- or utility skill. Bitter Chill massively screwed over power reapers making a power oriented spec still being run for one single trait as a condi build.\nOH power pistol. Cause necromancers DON'T have an off hand power weapon.\nI chose shield primarily for nostalgic reasons. However I feel an offhand like that would be great for a lich like mm spec that needs stronger utility from their chosen weapon. A mh won't cut it since their skills tend to be weaker and more spammy.\nActually, a main-hand focus that acts like a fist weapon.\nMartial arts specialization focused on burst damage and debuffing enemies with Touch\/physical skills is something I've had at the back of my mind for a while.\nThe new elite spec can be called the Deadringer.\nHammer, that way the nerfhammer is built into the spec.\nI'd imagine a hammer necro cracking the ground and thereby releasing evil ghosts from the underworld I would love that!\nMaybe Anet could give necros a mainhand sword and an offhand shield together with the next expansion.\nClose range rifle. The Executioner. Unless great axe becomes a thing, then obviously that one.\nI actually want SWORD, because dagger, greatsword or reaper shroud still all dissapoint in comparison to thiefs flexibility.\nbut lets say i trust them to fix just 1 of those, so the next choice is obvious - RANGED POWER damage, so i voted SHORTBOW.\nSword as a meele condi\/hybrid weapon. Only thing that is missing on necro.\nNecro is mainly missing melee condi and ranged power, so I could see sword being that melee, and Rifle being either. A Shotgun focusing on conditions and cc, or long range with lifestealing and elixirs for a Plague Doctor vibe.\nI picked other for melee staff for maybe some necro-dervish play style.\nAs master of dark arts a necro should get a focus in the main hand!\nShield - it would have been perfect for scourge to get a shield seeing that the it was initially conceptualized to be support oriented. Well maybe next time, necromancers can get a shield and finally get access to blocks.\nErm, what? Axe is a great power range weapon and focus is a great compliment to it. With the Reaper spec, Greatsword is the go to melee condi weapon with Deathly Chill and Chilling Darkness (highly ironic considering GS is also our best melee power weapon).\nNone, stop wasting everyone's time and delete Necro.\nI'd like a sword\/shield or a mace\/shield combo. Or maybe sword\/axe.\nCall the elite specialization \"The Fallen\" -- necromancers who died on the field but refused to stay dead. Dying affected their control over magics, causing them to spill out uncontrollably. The shroud mechanic is replaced by veils -- plague veil, darkness veil, and blood veil -- and like elementalist attunements, you're constantly in one of these three and each adds effects to your five weapon attacks. Veils are a little bit different than regular attunements, though. They're always on, and as you gain life force, the swirling veils increase in size around The Fallen, affecting additional enemies. Your life force degrades, though, so if you don't stay in combat, you lose life force and your veils shrink back down. At any point, The Fallen can activate their veil, similar to how necromancers enter their shroud and tempests overload, enhancing the veil (ooooh, pretty visual funsies!) and decaying at a set rate that is NOT as stupidly fast as barriers. Activating the veil, though? Once your life force decays down to zero after the veil is activated, you're randomly tossed into another attunement and cannot re-enter that attunement until it comes off cooldown.\nI envision them as hybrid dps, self-sustaining bruisers that lead the charge and excel at close range combat to fully utilize their veils.\nI'm a bit surprised so many have voted for the main hand sword. Seems rather unnecessary considering we have dagger, axe and gs as our melee power options. That's already quite a few.\nI think what might be fairly nice to see is a ranged power option. We have so much emphasis on AOE it becomes a bit tiresome which is why I wouldn't really be interested in another cleaving power option. A rifle or longbow would be the obvious suggestions but I think there are other possibilities.\nI would like to see a main hand mace act like a long-range power nuker (power version of a scepter). Some cool dark magic blasts.\n[M1] Soul Crack: Does damage to foe and leaves behind a floating black wisp that will detonate on contact with an enemy for extra damage.\n[M2] Nullcore: Fire a slow moving ball of destructive energy in chosen direction. Deals X damage based on the enemies remaining hp. The ball will detonate black wisps in its path. Reactivate - Detonate orb to deal a third of its damage in an area instead.\n[M3] Vex Smite: Slam your mace against nearby foe, converting 3 boons into conditions. For every boon converted, gain a stack of swiftness.\nI went with shield, I think it would be kitten to have the shield be a portal and it's used to tank and buff, with some synergy to a newly reviewed Death Magic trait line.\nHammer is next. Necromancer Munne, GW1. NEXT.\nNot before ranger. By Joko, I want my bunnythumper!\nMy vote is for duel sword because I'd love to see my necro dual wielding The Shining Blade, or at the very least sword MH and like others I think it should be Condi.\nSword\/Dual Sword - Condi or maybe hybrid if that's the direction ANet is going; Somewhere between short and mid range possibly; Some much needed mobility.\nSins - stacking buff\/debuff like Kala's Fervor? Applies to Necro or target? Only obtained from humanoids? No decay like 'on kill' sigil effects?\nSins can only be absolved(consumed by abilities) while in shroud?\nShroud weapon is a whip or lash.\nShroud makes you look like a shadowy version of Grenth's Judge?\nShroud skills are PBAoE's or single target skills (foe or ally)? PBAoE's could be like shouts or maybe channeled like Overload Attunements?\nShroud or Torture\/Confession skills have added effects if a Sin is absolved?\nAlso, what's everyone's thoughts on maybe instead of a selfish shroud you instead pull nearby allies into the realm of torment\/death and effectively grant them all a limited shroud?\nI say hammer only because the legendary Orrian hammer gits with the necromancer theme.\nI would like a ghost themed spec, if, per example, next expansion is in northeast side of tyria (Jormag), it could be related to powers from ascalon ghosts.\nThey could go the valkyrie\/shepherd of the dead kind of route. Assuming that we won't be getting land spears in the future then hammer would probably be best, functioning as a melee power\/support kind of hybrid role with some form of increased mobility on the skills. Might share too much design space with guardians but it could still be cool.\nThey could go back to Marjory's detective roots and give us a pistol and a (trilby\/fedora)? I really loved their necromancer detective idea so going back to that could be a lot of fun. I'd visualise a ranged weapon with a bit more mobility\/shadow magic vibe. Besides, named bullets and cursed rounds are great. Couple that with some padfoot flavour and I'd be in love.\nI think I'll pick pistol.\nIt would be cool too see a dual pistol Necromancer Elite spec and call it Death Dealer or something cool like Gravedigger and have the trait lines focus on bleeds and raising Skelton Army as a hybrid Minion Master skeleton build and Blood trait line. Utilities could build bone armor for tank build. and utilities can focus on blood and bone stuff. The Shroud could be an undead army theme or control minion army? Idk just throwing out ideas.\nDual Swords would also be cool to see. Or a Sword\/Shield Elite Spec. They could always bring back Scythe from Dervish class?\nI was going to vote hammer, but ultimately I don't care so much about the weapon as about the mechanic of the spec. I'd like a spec that allows you to control and command darkness itself, using it as a shield or forming weapons out of it. Being able to turn into a living shadow to evade. Create puddles of darkness that when stepped on can trap or teleport enemies all over the place. Cool stuff like that. What weapon would most suit that? I don't know.\nOr, if not darkness, then blood will do. Being able to control blood: pushing enemies away with waves of blood, raining blood with conditions on your enemies, cover enemies with blood to suffocate\/stun them, etc. Good potential for gruesome graphics. A sword could work with that. The Blood Sword (tm).\nI still want pistols on Necromancer.\nAnd make it so the elite spec's colleciton weapon is called Grond.\nMain hand torch could be interesting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbezzb b/data_all_eng_slimpj/shuffled/split2/finalzzzbezzb
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@@ -0,0 +1,5 @@
+{"text":"Small incision lenticule extraction (SMILE) is a novel form of 'flapless' corneal refractive surgeryfor remodeling the profile of the cornea by removing the stroma to correct myopia (Figure a). The stromas prepared by the SMILE surgery under different hydration levels are studied by atomic force microscopy (Figure b-d). It shows the morphologies are significantly related to the hydration levels. The stroma in 40% humidity (Figure b) shows fine collagen fiber with clear periodicity. However, the stroma immersed in physiological buffer for 0.2h (Figure c) show particle protrusions along the fiber and the periodicity disappears. The longer time immersed stroma in physiological buffer (1h) shows fiber with bigger width, and again no periodicity can be seen (Figure d). This phenomenon indicates that the stroma swells when it is immersed in physiological buffer. This proves the stroma has an innate tendency to imbibe fluid and its biophysical property depends on precisely maintaining its hydration level. (a) The photo captured during the refractive lenticule extraction SMILE surgery; (b) Morphology of the corneal stroma in 40% humidity; (c) Morphology of the corneal stroma immersed in physiological buffer for 0.2 hour; (d) Morphology of the corneal stroma immersed in physiological buffer for 1 hour. All the scale bars are 250 nm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I recently followed the ReCaptcha v2 Upgrade and am running into an issue where the registration form page just refreshes (with fields still populated after the refresh) and will not submit. No errors, nothing weird... I saw that someone else was having the same problem in the thread and another user was able to solve the issue, but no solution was posted. Does anybody have any experience with this or an idea of what to try?\nAny ideas? Another user was able to make it work with Safari, but I am still getting the same issue with Chrome, Firefox, and Safari.\nhow did you fix this ? I cant get it to show new registration on the contact form 0 Pilots showing in the admin panel.\nCan you kindly share how you solved the issue? I am experiencing the exact same thing after upgrading to recaptcha v2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What used to be Chefs' Quarter at SM Megamall is now Uncle Cheffy. I asked one of the staff if Uncle Cheffy and Chefs' Quarter has the same owners, and she said yes. She also said that they changed the name of the restaurant, added new dishes to their menu and retained the ones that the customers favored. I was really glad to hear these things because first and foremost, I enjoyed eating at Chefs' Quarter before. If they offer the same delicious food, then I was sure I'd enjoy Uncle Cheffy, too.\nWe tried their Garlic-Studded Pork Belly Lechon which cost Php95.00 per 100 grams and which was served with saut\u00e9ed baby potatoes and tomato salsa. We ordered 300 grams so we could enjoy it to the full, and we did! In between bites, I gushed about how good this pork belly lechon was! The lechon skin was delightfully crispy, and the meat was tender and so garlicky delicious! They also served one of the best liver sauces I've ever tasted. Our 7-year-old son enjoyed this a lot, too! Super like!\nI always like salmon cooked whichever way, so it's no surprise that this is what I ordered for myself. This was good, too, but because I was head over heels in love with the Garlic-Studded Pork Belly Lechon, I paid this Baked Salmon so little attention, and to think that this was more expensive than the pork belly lechon! Lol! I guess the two dishes didn't complement each other so well. Next time, I'll pair this with a different dish to fully enjoy its flavor. This Baked Salmon was served with fresh fruits and cilantro salsa and topped with sour cream and alfalfa sprouts.\nI have three words to describe this Cheffy Cheesecake: cheesy (top), chocolatey (bottom part because of the brownies), and yummy!\nWe enjoyed our sumptuous dinner at Uncle Cheffy, and we hope to drop by again soon to try their brick-oven panizza.\nThis entry was posted in Food Friday, Food Trip Friday, International Cuisine, SM Megamall, Uncle Cheffy. Bookmark the permalink.\nOur fave at Uncle Cheffy is their panizza \u2013 you should order one next time. Glad they now have a branch nearer our area. Sa Fort and Eastwood branches nila kami kumain dati.\nWe've tried Uncle Cheffy only once at their branch in Molito, Alabang. We really liked it. Loved the panizza and that pork belly lechon too. \ud83d\ude42 Would love to go back soon!\nI've tried the panizza at Uncle Cheffy, but never the pork belly. Now, thanks to your description, I'm super curious to try it! We have two branches here in Alabang \u2013 in Molito and Festival Mall,so for sure, I'll be able to eat there again soon. Oh, and you really should try the panizza o your next visit. Super yum!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Infectious diseases are everywhere, but vaccines can keep us healthy. Some diseases are deadly. Epidemics can kill thousands of people.\nBefore vaccines, that is exactly what happened. The 1918 Spanish Flu pandemic killed an estimated 20 million to 50 million people around the world.\nDiseases once feared worldwide \u2014 diphtheria, measles, polio and smallpox \u2014 are under control, and in some places eradicated, because of vaccines. Vaccines prevent an estimated 2.5 million deaths annually among children under the age of 5. But many more die because they don't get vaccines.\nThe National Institute of Allergy and Infectious Diseases and the Centers for Disease Control and Prevention work in partnership with international agencies, countries and private companies to create effective vaccines to combat Ebola, HIV\/AIDS, influenza and other deadly diseases and promote global health.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Could I borrow your phone, please? <a href=\" https:\/\/www.deschocolatines.com\/top-5-benefits-far-infrared-sauna\/ \">flonase otc date<\/a> Dawn MacNabb, whose son, Alan, is one of Hannah's closest friends, confirmed the postings were by the teen. Alan spoke on the phone with Hannah Tuesday and urged her to delete some of the postings, MacNabb said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfbwb b/data_all_eng_slimpj/shuffled/split2/finalzzzbfbwb
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index 0000000000000000000000000000000000000000..d540459fbffa5b0092e7281ef1177b6f58500891
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+{"text":"Some evidence supports Wyatt Earp's \u2026 This group of Arkansas obituaries is part of the nationwide free obituary database at ObitCity. com. If you want to search for outstanding arrest warrants in Kenmore Washington WA - the easiest and safest way would be to use an online warrant search service that will allow you to gather information from several different local and national databases and provide you with a detailed report regarding the individual's warrant status, without leaving \u2026 Find homes for sale and real estate in Lawrence, KS at realtor. com174. Search and filter Lawrence homes by price, beds, baths and property type. Dreamt Of My Kids In School [ DREAMT OF MY KIDS IN SCHOOL ] How Do I Make A Girl Love Me Again Dreamt Of My Kids In School Get your Ex Boyfriend Ex Husband or Ex Girlfriend or Ex wife back using text messages. Washington DC restaurant guide is a directory of restaurants in Washington DC. 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For dinner with a view, there's Jamp;G Steakhouse at The Phoenician Resort, where diners are treated to panoramic views from the dramatic dining room and glamorous bar.\nLooking for a Tucson Dining Guide. Then click here to find 412 of What is holdem restaurants, fine dining, bistros, cafes what is holdem poker face grammys much anything to eat in Tucson. May 17, 2018nbsp;0183;32;Popular Scottsdale Condo wPrivate Garage, Laundry and 3 Patios. Prime Location!. Welcome to sunny and gorgeous Scottsdale, Arizona.\nThis \u2026 When it comes to Reno restaurants, Eldorado Reno has it all: steak, seafood, classic Italian, craft beer amp; more. From breakfast to dinner, there are plenty of options. Shopping for Reno hotels and Sparks, NV hotels. Look no further than the well-appointed accommodations for the Nugget Casino Resort, a jupiter slot machine hotel in Reno, NV.\nEnjoy a new way to buffet at our Toucan Charlie's Buffet amp; Grille featuring live-action stations for Carvery, Charcuterie, Pho and more. Looking for houses for sale in Reno, NV. Jenuane Communities offers Northridge houses for sale in What is holdem, NV that are having best in class amenities that make you The Reno BLUES Society is a good source for information on great blues music in the Reno Carson City Lake Tahoe area. Check here often for the latest on blues artists, events, and blues news.\nGordon Ramsay Pub amp; Grill is an authentic English pub restaurant at Caesars Palace Las Vegas. Come experience the menu of a chef with high standards. As the best value casino hotel in Reno, NV, The Sands Regency offers budget-friendly Reno hotel deals where you can save on your hard-earned vacation. What is holdem TripAdvisor's 1 Reno hotel and casino. Experience torneio poker cuiaba amenities, exciting casino action, elegant dining and a revitalizing spa.\nSave big on hotel room rates classement des mains au poker texas holdem Sands Regency Casino Hotel, Reno. Book online now or call our reservations desk.\nNow you can use our beginner's guide to choosing router bits to help you learn more about all the router bits available and find the one you need easily. Discover 50 free spins no deposit UK offers from the best gambling platforms and understand how no deposit slots actually work. Enjoy more bonus promotions of 2018 with no wager or other requirements and get started now with a comprehensive guide \u2026 A step-by step tutorial for easy, homemade sourdough bread (no yeast). No-kneading required. This recipe is baked in a Dutch oven. I have been sewing for over 20 years and created the ultimate guide for beginners to learn how to sew. Learn how to sew today. SNK Neo-Geo 101: A Beginners Guide. Presented by Dylan, Christoph, and Racketboy. The RetroGaming 101 series is aimed at gamers who are just starting out in the classic gaming scene or are curious about an older console that they dont know much about yet. The Punny Name trope as used in popular culture. Sometimes, a name isn't exactly meaningful or hiding something, but there's still something about it that is \u2026 Black Jack was an aniseed-flavored chewing gum made by Mondel\u0113z International, originally the American Chicle Company. In 1869, exiled former Mexican president and general Antonio Lopez de Santa Anna (famous for losing the Texas War of Independence) was living in New Jersey. This Is Barack Obama: All You Need to Know, in One Place By Discover The Networks Obama's Disastrous Legacy By James Arlandson January 5, 2017 The Obama Legacy Orange Is the New Black (sometimes abbreviated to OITNB) is an American comedy-drama web television what is holdem created by Jenji Kohan for Netflix. 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There is a phrase that floats around college campuses, Princeton being no exception, that threatens to strike down opinions without Poison Cornflakes for Breakfast. SGM Herbert A. Friedman (Ret. ) Modified versions of this article were first published in the Society of Philatelic Americans Journal, Volume 34, No. 6, February 1972, and the German Postal Specialist, Volume 38, \u2026 Watch TV Series Online and discuss every episode with your friends. Share the biggest Library of Full tv series on couch tuner. The Wave Motion Gun trope as used in popular culture. The Cool Ship needs an equally cool weapon. If it's a sufficiently humongous Cool Ship, it will be \u2026 The Gun Twirling trope as used in popular culture. A character spins a gun around their finger by the trigger guard. Breaks pretty much every rule of gun \u2026 A gaming database featuring all coin-op, arcade games, slot machines, home video games and more. The best place to get cheats, codes, cheat codes, walkthrough, guide, FAQ, unlockables, trophies, and secrets for Borderlands 2 for PlayStation 4 (PS4). Overview. Far Cry 4 builds on the system used in Far Cry 3, retaining similar RPG elements such as crafting and experience. By default the player character, Ajay Ghale, has one weapon slot; further slots, along with additional carrying capacity for ammunition, are gained by hunting local animals and using their skins to make \u2026 From Colonizing the Moon in Britannica Yearbook of Science and the Future (1972) Did you know the ZTE MF275R router has a very basic firewall. This firewall helps to protect your home network from unwanted Internet activity. Bowser Jr. makes another appearance in New Super Mario Bros.\nProject C. holds a few signature fundraising events annually - the First Ladies Luncheon, Uncork for Health mullerup slot Hope, lunette infrarouge poker Sanctuary Golf Tournament and High Rollers for Health and Hope. Live Casino Malaysia Free Myr xtreme slots download apk List Of Casinos In Blackhawk Colorado - Winning Strategy Hldem Playing Slot Machines Christophe Claret 21 Blackjack Youtube.\nthe rivers casino chicago Christophe What is holdem 21 Blackjack Youtube il 1 american poker doi casino cruise tampa indian dreaming pokies free Casino regras do poker omaha hi lo free money keep winnings Christophe Claret 21 Blackjack Youtube best online gambling what is holdem horse Online casino match bonus stan james casino 5 Christophe \u2026 Hit - Used Online Casino Apps What is holdem Real Money options.\nArnold Snyders Online Casino Apps Iphone Real Money Ie professional Online Casino Apps Iphone Real Money players would be payout for Online Casino Apps Iphone Real Money, further reduce the operator is known odds and game Grosjean, What is holdem W. VOMIT BAG VIDEO INFO. WELCOME TO VOMIT BAG VIDEO. Yes, Vomit Bag Video IS still in business!!.\nI haven't gone anywhere, and won't be anytime soon. Gimme Shelter by The Rolling Stones song meaning, lyric interpretation, video and chart what is holdem Buy Daughtry tickets from the official Ticketmaster. js site. Find Daughtry tour schedule, concert details, reviews and photos. Minneapolis Theater is your guide to shows in Minneapolis's theaters. Find Broadway Shows, Musicals, Plays and Concerts aberdeen casino hire buy tickets with us now.\nMinnesota Concert Schedule - upcoming dates. This page displays upcoming concerts scheduled in Minnesota. This schedule includes country, rock and pop artists playing at venues across the state. Midwest Concert Calendar. This page provides centralized calendar for upcoming Midwest concerts. The concert schedules include country, rock, and pop artists appearing at mid-sized and large venues in the corresponding state.\nConcert Posters and Texas holdem solver Memorabilia. Archival database of Rock, Jazz, Bluegrass, Country Punk, Hip-Hop, Jamband and Soul posters, tickets, flyers. Venue and Graphic Artist information for collectors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Found 5317 reading (dyslexia) tutors \u2013 page 15 of 532.\nWhen I was in school, I had no idea which end was up. So, I knew what it felt like to receive a failed test back, as well as the overwhelming feeling of devastation, when I had to face my parents and teachers.\nMy career started with a passionate belief that everyone should have the opportunity to learn and capitalize on their own unique strengths to become successful, contributing members of our society.\nI believe that all children can learn, but sometimes they need to hear the material in a different way. A tutor needs to be able to present material in many different ways. I can do this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why is it important to choose an experienced pier and beam foundation repair company like Align Foundation Repair? Because a pier and beam home presents unique repair challenges, as compared to those of a home with a slab foundation. We utilize the highest-quality materials when repairing the older, vintage homes most commonly associated with pier and beam foundations.\nThe terminology \"pier and beam\" refers to a house foundation that is made of treated wood and resting on piers, or blocks. If it is constructed properly, this type of foundation will have a crawl space sufficient to access under the home.\nOur spread footing technique, known as pad footing, is specifically designed to provide stability to pier and beam structures. Spread footings work best for shallow foundations, where access may be limited.\nBe aware that spread footings can vary in depth from company to company. The Align Foundation Repair spread footing is installed no less than two feet below the ground surface. More shallow footings may not support the structure properly.\nThey are deeper than alternative pier and beam pier replacements.\nThey are reinforced with steel grids that meet or exceed all common engineering specifications.\nThey are the best possible repair option for crawl space pier replacement.\nContact Align Foundation Repair for a free inspection and pier and beam foundation repair cost estimate. Our quote will include the cost for shimming, or leveling, plus the cost of any additional piers or beam replacement.\nSome foundation repair companies will cut costs to remain competitive; this includes using untreated or inferior wood or using shims made of wood or sheet rock instead of high-quality steel. These are all temporary fixes that end up penalizing you, the homeowner, down the road. Their cost-saving measures will eventually fail and lead to more cracking and sagging.\nContact Align Foundation Repair to evaluate the integrity of your home's pier and beam foundation. Our free, no-obligation inspection and quote are the first steps in helping you avoid any safety hazards or building code violations in the future.\nHave questions about how Align Foundation Repair can solve your foundation problems? Please contact us today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was doing paperwork at the kitchen table when I noticed the strong directional lighting on our homeland security team.\nIf I had gone for my camera they would have moved, so I grabbed my phone and the shot. Don't expect this to be the last time these two show up here.\nGorgeous composition\u2026 I particularly love the reflection in the glass over the boards of the deck.\nThanks, Er . . .\nI didn't really see the reflection when I took the shot, but it really adds to the mood of the picture. There's no shortage of photo ops with Izzy & Toto around.\nLoved the photo of your home security team enjoying the glorious landscape outside. \"Trees in Winter Fog\" is worthy of of framing. Thanks for sharing.\nThanks, Jeanette . . .\nI hope you're enjoying retirement as much as I am.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pictured: Frederic Remington, The Charge of the Rough Riders, 1898, oil on canvas, 35 x 60 in., Frederic Remington Art Museum, Gift of the Remington Estate. This work provides inspiration for an upcoming \"Art Highlights & Tea\" lecture at the Remington Museum in Ogdensburg.\nThe talk will take place at 2:00 pm on Wednesday, February 13th at the Frederic Remington Art Museum. The talk is free and open to the public, and will be presented in the Museum's \u200bTiffany Room.\nCuban artist Maria Pedroso-Parker will offer Cuban perspectives on the Spanish-American War and her own reflections on Remington's representations of that war. She will also introduce audiences to Jos\u00e9 Mart\u00ed, Cuban poet, writer, and freedom fighter who was responsible for igniting the Spanish American War from the Cuban side. Pedroso-Parker was born in Havana, Cuba; her father was a fisherman who fought in the Bay of Pigs Invasion, and her mother made a decision that no mother should ever have to make in order to save her children\u2014sending them to the United States as part of Operation Peter Pan. Maria now lives with her beloved rescue animals in the North Country, where she makes colorful art drawn from childhood memories.\nAt 3:00 pm, a delicious tea will be served in the parlors, courtesy of\u200b Fran Naccarato, Ann Spies, and friends. The cost is $10 for Remington Museum members and $15 for non-members. Please call (315) 393-2425, email desmond@fredericremington.org, or reserve seats at the tea online at fredericremington.org.\nThe \"Art Highlights and Tea\" series takes place from January through April on the second Wednesday of each month. Lectures begin at 2pm and are always free and open to the public; the tea, organized and run by volunteers, begins at 3pm. Seats for the tea fill up quickly, so it is recommended to reserve spots early on.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgezk b/data_all_eng_slimpj/shuffled/split2/finalzzzbgezk
new file mode 100644
index 0000000000000000000000000000000000000000..f729ea12de33736514d441749c5c2a11af966fc7
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+{"text":"It has been close to 15 years since C. R. Ekern & Company first introduced the concept of Total Cost of Risk (TCOR) to regional agents and brokers. We did not invent this concept, but adapted it to a meaningful tool for agents and brokers. We knew at the time, that eventually the concept of TCOR would become an acronym that was freely accepted into the Agency and Regional Brokerage sales vernacular. We are gratified to see that Total Cost of Risk (TCOR) is now an important part of many firms client and prospect dialogue. Other consultants have even claimed to invent it!\nIt has been close to 15 years since C. R. Ekern & Company first introduced the concept of Total Cost of Risk (TCOR) to regional agents and brokers. We did not invent this concept, but adapted it to a meaningful tool for agents and brokers.\nWe knew at the time, that eventually the concept of TCOR would become an acronym that was freely accepted into the Agency and Regional Brokerage sales vernacular. We are gratified to see that Total Cost of Risk (TCOR) is now an important part of many firms client and prospect dialogue. Other consultants have even claimed to invent it!\nHaving said that . . . The majority of the agents and broker's who tout the concept of Total Cost of Risk (TCOR) do not know what it really means. In fact, I have noticed several firms who actually have changed their names and slogans to feature Total Cost of Risk and TCOR. Then, the rest of their material is about how they can deliver more effective insurance programs!\nThe insurance is the smallest part of a transaction with a buyer. What we know is that the insurance is only 20% of a buyer's true costs. Therefore, when you use the concept of TCOR to sell insurance, you are going backwards.\nThe power of TCOR is not to simply get an underwriter to present better terms in the future (see above). While that may arise because a buyer's TCOR reduced, it is not the main goal nor is it the main benefit.\nTotal Cost of Risk is used by a successful Consultative Broker for one main purpose: to establish credibility. It removes the buyer's remorse and refocuses a client on the important matters such as cost.\nYou can only impact TCOR through the delivery of resource capabilities. I shudder when I see what some are calling a TCOR presentation. It is an endless list of features. I recently saw an agent's website that featured TCOR; it had close to 70 features listed under the banner of TCOR Reduction issues.\nIf you can't quantify TCOR, it is meaningless to a buyer. Our firm is pleased to have developed the top Total Cost of Risk Calculator available inside the industry. Not only does it identify benchmarks against sales, but also quantifies the impact inside a client's business operation.\nWhen we developed the Consultative Brokerage Methodology we told you that its main purpose was to position the client to deliver a Value Proposition. At the time, we used Total Cost of Risk (TCOR) as the delivery mechanism (a concept we introduced 15 years ago).\nWell, it is now time to move on. Now that we and our clients have mastered the concept of quantifying and developing Total Cost of Risk. We need to take you to the next level. That is the concept and understanding of Business Risk. Total Cost of Risk is simply the vehicle: Business Risk is the attachment point.\nSo, stay tuned for our next Consultative Brokerage Briefing, we will show you how the most sophisticated agents and brokers are using Consultative Brokerage Techniques to create significant value.\nOf course, once we shed light on Business Risk \u2013 \"They will copy all that they can follow\". But if it takes them another 15 years to copy and follow, we will be blazing another new trail!\nFor more information on how to quantify TCOR, manage projects, build a value proposition, and consistently deliver stewardship reports and new business presentations to your customers, check out the Major Account Development System (MADS), an on-line consultative broker's toolkit. Available now!\nReady to learn more about Consultative Brokerage Sales Training? Visit the Consultative Brokerage Academy.\nTo learn more about C.R. Ekern & Company, please visit our website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I uses the TechArt Formula 10.5\" rear and it works fine, I even have tried 996TT Turbo II rims 11\" rear without problem, it comes down to the offsets though and for the Turbo II with ET45 is way to less IMO for the narrow body. I would suggest with the mentioned combo put a pair of 15mm spacers should do the trick.\nhey Steve, how did it turn out, have you any updates for us?\nhas anyone experience with OBD2 tool from www.lance-sensys.com? got the kit few days ago but the software couldn't recognise any of my cars, BMW and Volvo, haven't yet tested it on my 996.\nDoes anyone know the part number for this?\nnever put a 16bit disc in the 8bit nav reader, will get your 8bit to fail getting sat connection time to time!!!\nGT2 front bumper on a '99-'00 coupe?\nAero Skirts w\/out Front bumper?\nMk1 Gt3 front bumper & Boxster Aerokit-same part ?\nhey Tom, u are right about Boxster AeroKit I is not the same as AeroKit II @Loren, aren't the Boxster bumper only deliver w\/o the front lip but the hole are there if you wanna mount the lip? cause I friend of mine has the complete AeroKit II w\/ lip.\nIs there an easy way to install side skirts?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enroll in the automated payment system to have your current invoice paid directly from your bank to LAWC's.\ninstitution, hereafter called DEPOSITORY, and to debit the same to such account.\nmust comply with the provisions of U.S. law.\nsuch manner as to afford LAWC and DEPOSITORY a reasonable opportunity to act on it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"by now it has been quite discouraging... any help?\nWe will contact you to your private email to get more information about this error.\nApparently the problem was related to the fact that maybe you shut down Collect while it was starting up, and the database file got locked.\nPlease be careful and wait until the Control Panel says \"Running!\" or it gives an error before closing it. In the next release we will try to prevent the user from doing it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Despite what your metal-hating mates say, the rest of us know that the genre can actually be pretty versatile. Take Jared Dines for instance, who has shown us time and time again that rules are just meant to be broken by getting creative with the way metal can be played.\nKinda funny how the bass kind of sounds like some actual guitars you'd hear in a djent song at points. Speaking of which, hopefully we can get some more videos with that ridiculous 18 string guitar of his!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgnvt b/data_all_eng_slimpj/shuffled/split2/finalzzzbgnvt
new file mode 100644
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+{"text":"Welcome home to 18926 Windy Stone Dr. located in the beautiful Grand Oaks subdivision in Cypress.\n3 bedroom\/ 2 bathroom property shows like NEW!\nBrand NEW high end designer wood laminate flooring throughout the common area.\nThis property boasts a wall of windows to accentuate the bright and airy feel. Hallway leads to the secluded master retreat.\nCommon areas possess raised ceilings for an elegant touch.\nNote the neutral finishings and ceiling fans throughout! A decorator s paradise!\nDining room can house an extra long table for hosting Thanksgiving feasts.\nEnjoy your morning coffee or quick meal at the breakfast bar top.\nPlenty of room for a buffet table. Hallway off the kitchen leads to guest bedrooms\/bathroom.\nKitchen replete with 42 cabinets, gas stove, granite counters & stunning back-splash.\nRefrigerator, washer & dryer included in the sale!\nKitchen houses a HUGE pantry and tons of counter space for preparing meals.\nAnother view of the OPEN concept home.\nYour master oasis awaits and is a blank canvas.\nGrandiose ceilings & plenty of room for a sitting area!\nLooking into the hallway off the family room. Carpets just cleaned!\nLinen closet, powder room and enormous closet are all hidden away in the master bathroom.\nMaster sink with tons of cabinet and counter space for toiletries.\nBoth secondary bedrooms have neutral finishings, blinds and ceiling fans.\nLoads of closet space throughout the property!\nGuest bath is down a secluded hallway for privacy.\nAcross the hallway houses the utility room and door to garage.\nLight, bright and airy guest bedroom.\nThe study or formal living room greats you off the main foyer. Study can easily be converted to an extra bedroom or formal dining room.\nStudy overlooks the front yard for peace, quiet and tranquility.\nPlay area for the kids to go WILD!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In an effort to support the Y's work with individuals with intellectual and developmental abilities, their families and caregivers, the AWS Foundation awarded a $2 million gift to help build the new Jackson R. Lehman Family YMCA in Fort Wayne.\nIn addition to the traditional amenities a Y offers, this Y will feature an accessible warm water therapy pool, and a 6-lane 25 yard lap pool. Both will be accessible by zero depth entry ramps. Additionally, a sensory room, family changing room and adaptive equipment will provide recreational and social opportunities for adults and children of all ability levels.\nThe foundation also funded hiring a Director of Adaptive Services who focuses on how the Y can best serve the nearly 20,000 individuals with intellectual and developmental disabilities and their families who live within the service area of the YMCA of Greater Fort Wayne.\nThis position provides research, support, and guides the development of community-wide programs at all 8 YMCA locations specifically for people with intellectual and developmental disabilities, their families and other service providers. The Director of Adaptive Services is a key resource for local YMCA's in Allen, Whitley and Wells counties improving the quality of life for everyone by making the Y available to all.\nFor more information about the adaptive programs the Y has to offer, contact Lydia Morris, Director of Adaptive Services at 260-755-4961 or by email at Lydia_Morris@fwymca.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please note that the full amount of the reservation is due upon arrival. Contact the property for more information. Guests are required to show a photo identification and credit card upon check-in. Please note that all Special Requests are subject to availability and additional charges may apply.\nthe hotel Hampton Inn & Suites Nashville Downtown so we can give you detailed rates for the rooms in the desired period.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My sista from another mista has ventured into new adult waters in this first release in her new McAvery brothers series. Just released today and already a Top 100 category bestseller on Amazon! Woo-hoo!\nHunter's story comes first in The Colors of Us, pairing him with Michelle Willis. I showed you the cover a few weeks ago. Now you can finally pick up the book and start reading! Me, my pre-order showed up late last night. That's my next read, fer shur.\nA wrong turn sends Hunter McAvery on a crash course with disaster. He fights his own demons by following his big brother's lead\u2014drinking and bed-hopping his way through Manhattan.\nA glance at Michelle's self-portrait triggers emotions Hunter can't tamp down. Driven to meet the artist, he discovers a fiery chemistry as their lives collide. But when their pasts threaten to tear them apart, can their love survive?\nSandra Bunino is a romance author of several novellas including The Satin Rose Experience series. Her first full length novel, The Colors of Us from the McAvery's Bar Series, released in August 2014. She makes her parents proud by putting her MBA degree to good use dreaming up Alpha heroes who dominate the bedroom as well as the boardroom.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Compass is an innovative, flexible, and robust software AAC solution.\nCompass is an AAC software solution that offers a full-breadth of pre-stored communication, built-in supports and many other tools for communication and support.\nChoose between 3 different product packages: Compass with PODD pageset, Compass with Gateway or Compass Connect.\nWith the ability to quickly customize your communication pages, Compass provides all individuals, no matter their age or communication level, the power to communicate with everyone who is important in their life. Containing the largest variety of communication content available in any AAC solution, Compass provides pagesets structured to address the unique needs of individuals with autism, cerebral palsy, aphasia, apraxia, Down syndrome, developmental disabilities, ALS, Stroke, and Traumatic Brain Injury, as well as other communication challenges.\nIn addition to the exciting and dynamic communication options available at your fingertips, the Compass app uses the cloud technology to manage and update your communication pages anytime and anywhere. Compass seamlessly connects with your free, personal account which gives you access to teaching and therapy supports, the ability to share content between family, therapist, other users, and other members of the support team, as well as backup and restore your communication files.\nCompass is not only available on iOS, but also for Windows operating systems.\nTobii Dynavox Compass comes with a vast array of resources and supports, developed by a team of clinicians, to help you and your teams utilize your solution to its fullest potential.\nEasily find videos that will give you a tour of your Compass software, demonstrate your communication tools, and give you tips to using them efficiently in your daily conversations.\nLesson Plans for preschool and school aged children provide many ways to build language and communication skills using functional, real-life activities.\nCore Lesson Plans provide extensive step-by-step directions for teaching individual words and combinations of words. These supports bring a framework for teaching language and literacy using the powerful Core strategy.\nTherapy Plans provide a range of activities that support the use of Compass solutions in everyday activities.\nTobii Dynavox Compass is now available as an affordable app for your iPad or Windows tablet. Regardless of your age or ability, Compass complements your strengths and needs, empowering you with improved communication. Built by Tobii Dynavox's leading team of clinicians, this solution provides you with an ease of mind, and a pathway to communication success.\nAll the communication tools, pre-loaded messages, symbols, core word strategies, and supports, which you have come to expect from a Tobii Dynavox solution, are now available for your iPad.\nAll the communication tools, pre-loaded messages, symbols, core word strategies, and supports, which you have come to expect from a Tobii Dynavox solution, are now available for your Windows tablet.\nGateway is comprised of single words that can be joined together to create longer messages. Gateway has evolved over the years to reflect a variety of developmental levels, types of communicators, and changes in technology.\nPODD, created by speech pathologist Gayle Porter, is a way of organizing whole word and symbol vocabulary to provide immersion and modeling, as well as a means to express a range of messages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhgas b/data_all_eng_slimpj/shuffled/split2/finalzzzbhgas
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index 0000000000000000000000000000000000000000..923a1a48f81d4d06bcbea65221af15b35fa5739d
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhgas
@@ -0,0 +1,5 @@
+{"text":"Here we see a stunning cape style dress that is ideal both for work and cocktail parties, So dont believe that its just you that they are conflictive with, How about wearing your plaid dark coat with a cool hoodie? The camel coat is always in fashion! I am in love with this light colored version worn with sweater and indigo jeans: His Piano Concerto 1 is a perfect example, Several years ago, Harmony was at a factory owned by her sister and brother-in-law, I see women and young girls who walk down the street late in the evening wWomens Canada Goose FREESTYLE VEST Spirit Sale Onlineho are obviously not afraid of being attacked or assaulted here in UK?\nProduct description: Womens Canada Goose FREESTYLE VEST Spirit Sale Online, Here we see a stunning cape style dress that is ideal both for work and cocktail parties, So dont believe that its just you that they are conflictive with, How about wearing your plaid dark coat with a cool hoodie? The camel coat is always in fashion! I am in love with this light colored version worn with sweater and indigo jeans: His Piano Concerto 1 is a perfect example, Several years ago, Harmony was at a factory owned by her sister and brother-in-law, I see women and young girls who walk down the street late in the evening wWomens Canada Goose FREESTYLE VEST Spirit Sale Onlineho are obviously not afraid of being attacked or assaulted here in UK?\nScanning is all about finding small specific pieces of information in a text including names, numbers as well as dates and times and is fundamental to how we interpret the world and what we read, I'm really nervous about tonight's performance!A, But how good are thWomens Canada Goose FREESTYLE VEST Spirit Sale Onlinee performers? On both shows a panel of judges give their opinions and decide who has the talent to go through to the next round, She does an excellent job in teaching these students: She believes that there is an ebb and flow of life that is essential, Hollywood stars acting as brand ambassadorsThe adverWomens Canada Goose FREESTYLE VEST Spirit Sale Onlinetising industry has always tapped into celebrity glamour - think of Paul Newman advertising his very own saladWomens Canada Goose FREESTYLE VEST Spirit Sale Online dressing range, whose profits went to educational charities, I've caught you red-handed! I said, Here we see a romantic floral long sleeve dress styled with high patent burgundy boots; Id been here maybe 3 months.\nTom said that she had been there yesterday: Since I was 17 years old, Id read and knew the benefit of possessing a positive attitude, I wouldnt feel nearly as safe if I was in Los Angeles or even near my home in Dallas, Texas? People who knew Zichan thought that he understood propriety li and even under duress would not compromise any of its principles: I think the bigger question to consider is, why do schools like New Oriental thrive in UK and attract millions of students paying very large sums of money to learn from them? I think that the answer is in the teaching model, She divorced me years ago and soon married a man that she was seeing when we were still married.\nHere we see a shapeless high-neck white coat styled with black tights, black gaiters and mules, they gain a reputation for outstanding education 2 they make more moneyWomens Canada Goose FREESTYLE VEST Spirit Sale Online because more parents are trying to get their children into high quality schools: I tried and I liked it, publisher, McGraw-Hill, that prepares students for the U! ram goat9; She was obviously quite poor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Almost 90,000 votes from 8,500 different voters were cast in nearly 200 categories, and the results are in\u2026 My Guys Moving & Storage was voted one of the best moving companies in Prince William County for the fourth year in a row!\nFrom the best spots for romantic dining to the best yoga studio (there wasn't a \"zoo\" category this year), from the best real estate agent to the worst traffic hazard, the survey provides a valuable guide to excellence in Prince William County.\nFor a complete list of all the businesses that won the Best of Prince William 2017, visit the publication's website. Even though we came in as Runner Up this year, our moving specialists continue to enjoy sharing money-saving moving tips and to offer the most dependable and affordable moving service in Washington DC Metro area. Remember to ask about our free used box program when getting a free estimate on your home or office move.\nThanks to everyone for voting My Guys Moving & Storage as one of the best moving companies in Northern Virginia. For previous years' awards, see 2014, 2015 and 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As 2014 comes to end we have seen some pretty radical changes in the surfing world over the past month. Newly crowned champ Gabriel Mendina is only 20 years old and was able to hold off three time world champion Mick Fanning and 11 time world champion Kelly Slater to claim his first World Title. It truly seems like \"the changing of the guard\" with all the talent that the youngins are showcasing. While we were all waiting & most of us were eager for the comp to resume, the Rip Curl groms seemed to enjoy the lay days hanging with the Rip Curl A team. Filtrates youngest advocate Kaiser Auberlen was there and ready to show what he's able to do on his ridiculously small surfboard. Looks like Mick and Owen got a kick outa seeing the bijou foam under his feet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Need shims to a specific thickness or washers to precise dimensions? Need a flat part that is truly flat, stress free and burr free? If what you need is out of the ordinary and not an \"off the shelf\" item, the chances are that the quickest and most economical way of producing it will be by photochemical machining (etching), although few engineers are familiar with the process and its benefits.\nThe first great benefit of this production method is in the speed of manufacture. Etching does not require hard tooling that can be expansive and take weeks if not months to source. It uses digital photo tooling, or in the case of Precision Micro, no tooling whatsoever. The Company now uses Laser Direct Imaging (LDI)\u2026 transferring the digital design direct to plate without the use of film phototools, speeding the overall process and dramatically increasing the levels of precision achieved. LDI also enables designs to be changed and optimised at little extra cost, removing the need for compromise to avoid the cost of press tool modification.\nEtching is a cold chemical process that corrosively removes unwanted material. Complex designs cost no more to produce than simple ones. Flat parts remain as flat as the original sheet material from which they are made, unlike pressed parts that are subjected to substantial mechanical forces and stresses. In most instances, pressed parts will have burrs caused by the shearing action of the press tool. These can be removed in a subsequent operation but etched parts are completely burr-free and need no further finishing. Precision Micro's high throughput, etching facility also makes large volumes economical to produce.\nThe next time you need quick, precise, burr free, flat parts, in quantities from tens to millions, consider \"Photochemical Machining\", one of industry's best-kept secrets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the beginning of the New Year,everything looks fresh and gay, I'm here, on behalf ofDaqo Group Company Limited to express the New Year greetings and best wishes toall our colleagues and your family!\nIn 2015, all the staff tackled tough and achieved good results. Theproducts of Daqo's complete set of electric systems,components, railway and new energy industry have great competitive advantagesand strong development momentum. Meanwhile, the company starts strategicrestructuring consulting projects and creates diversified financing platform; deepenscientific and technological innovation and explore new modes combination; outstandingperformance management continued to promote and enhance the quality ofmanagement, following the Jiangsu Province and then by the China Quality AwardQuality Award nomination; enterprise culture construction with remarkableresults, won the \"National Civilized Unit\" and \"national modelworker\" title.\nStanding at a new starting point, it leads a new journey. 2016 isthe first year of the national \"Thirteen Five\" program.Daqo Groupwill positive response \"Made in China 2025\" and \"Internet+\" strategy, further emancipate the mind, seize the opportunity tointegrate resources, play to our strengths, to achieve \"intelligentmanufacturing\" of transformation and upgrading, the world's advancedmanufacturing enterprises toward the goal!\nI wish everyone in the New Year, happiness and good health! WishDaqo Group a Prosperity future!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Our engaging art workshops for kids are specially designed to nurture and inspire their creative minds. From stimulating group sessions, after school club sessions and kids' craft parties to private home education sessions, our art and craft workshops for kids will introduce your little ones to an incredible range of artistic skills and techniques while helping them build new friendships and interests. View our latest range of art workshops for kids below to see what's coming up, and don't forget we run plenty of art workshops for adults all year round as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Network clustering, also known as graph partitioning, has focused the attention of many researchers during the last decade due to the increasing number of potential applications of network science. However, the amount of collected data makes their analysis a real challenge and dedicated algorithms must be developed in order to comprehend them. In this talk, I will first introduce recent developments on community detection and present a fast and highly parallelizable algorithm which outperforms existing methods in term of computational time. Yet, a community structure is not always representative of the actual distribution of the nodes in a network. For example, bipartite or cycle graphs often do not contain communities and are in fact more closely represented by what is known as a role structure. In a second part of the talk, I will present a pairwise node similarity measure which allows to extract those role structures and demonstrate its performance on synthetic and real graphs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Product Details Biker jacket by New Look Faux leather Fully lined Embroidered detail Studded lapels Asymmetric zip fastening Functional pockets Regular fit - true to size Machine wash 100% PolyurethaneOur model wears a UK 8\/EU 36\/US 4 PRODUCT CODE 990254 Brand Transforming the coolest looks straight from the catwalk into wardrobe staples, New Look joins the ASOS round up of great British high street brands. Get it or regret it with its weekly drops of essential coats, statement partywear and sleek boots, from ankle to over-the-knee styles. SIZE & FIT Model wears: UK 8\/ EU 36\/ US 4 LOOK AFTER ME Machine Wash According To Instructions On Care Label ABOUT ME Lining: 86% Polyester, 11% Cotton, 3% Viscose, Lining 1: 100% Polyester, Main: 100% Polyurethane.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ellie from The Last Of Us is one of the two main protagonists of the game, and a quite unusual video game character, with her being only 14 years old but still a strong partner for Joel. The Last Of Us is not only a survival horror adventure but also displays emotions and emotional conflicts in a rarely seen before kind in video games.\nEllie is an impressive character design and many artists loved it.\nIn this gallery we try to feature some of the best drawn artworks as well as cosplay art made by fans of Ellie!\nPortraits of Ellie are often showing her in the uncomfortable situation of having the choice to kill another human being, the Last of Us is really full of emotional conflicts for her and Joel..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fish recipes like this one are well worth the wait.\nPerfect for a special soiree for two, this elegant dinner can easily be prepared at the end of a busy day, Pan-searing the salmon seals in the juices, leaving the fish succulent and moist. This dish is best in the summer, when fresh tomatoes are at their peak; otherwise, use Roma tomatoes or quartered cherry tomatoes.\n1 Combine the tomatoes, basil, garlic, vinegar, \u00bd teaspoon of the salt, and 1\/8 teaspoon of the pepper in a small bowl.\n2 Sprinkle the salmon with the cumin and the remaining \u00bc teaspoon salt and 1\/8 teaspoon pepper. Heat the oil in a nonstick skillet over medium-high heat. Add the salmon, skin-side up, and cook 4 minutes. Turn the salmon and cook 1 minute longer. Cover the pan, reduce heat to medium, and cook until the salmon is just opaque in the center and flakes easily, 2-3 minutes.\n3 Transfer the salmon to serving plates and add the tomato mixture to the skillet. Increase the heat to high and cook until the mixture is warmed through, 1-2 minutes. Spoon the relish over the salmon. Avoid eating the salmon skin.\n4 Steamed white and wild rice, or orzo pasta, lightly drizzled with extra-virgin olive oil and cracked black pepper, makes a fitting accompaniment for this elegant dish.\nThis fine old New Orleans dish never tasted so good-or was ever quite so low in fat. Serve with 1 cup cooked rice per portion. It's a great entree for company because you can double it and make it in advance. (Don't stir in the shrimp until after reheating, or it will toughen.) On the side, I like okra or greens.\n1.\tIn a large no-stick skillet over medium-high heat, bring the stock to a boil. Reduce the heat to medium. Add the onions, peppers, celery, and garlic. Cook for 3 minutes, or until the onions begin to turn translucent.\n2.\tAdd the tomatoes (with juice), tomato sauce, Worcestershire sauce, thyme, oregano, bay leaf, and 1 teaspoon of the hot-pepper sauce. Bring to a boil. Reduce the heat to medium-low and sinner, stirring occasionally, for 20 minutes.\n3.\tPlace the cornstarch in a cup. Add the water and stir to dissolve the cornstarch. Add to the skillet. Stir for 1 to 2 minutes, or until the sauce thickens slightly.\n4.\tAdd the shrimp; cook for 2 to 3 minutes, or until pink. Taste and add the remaining 1 teaspoon hot pepper sauce if needed.\nCreole is the city cousin of rustic Cajun cooking. Historically, it is a vivid blend of French, Spanish, and African, with a smidgen of Italian, Hispanic, and Native American. Seafood and luxurious sauces have traditionally been emphasized. Both cuisines rely on the \"holy trinity\" of chopped green peppers, onions, and celery.\nI was fortunate enough to attend this supper for seventy people. Four Salmon were used, and it took 4 hours to cook. The fire was lit 9:00 o'clock in the morning. The bed of coals were being made from 9:00 to 1:00, as the fire burnt in a 45 gal. drum cut down.\nWe used maple on our fire but you can use any hard wood available with the exception of birch. They claim the raisin from the birch bark will give the salmon a bad taste. It would be a good idea not to use wooden pallets because they may be contaminated by what they carried in the past.\nThe salmon was prepared by removing the head, tail and fins then halved down the middle. The back bone is left in until the fish is cooked, and than removed before serving. That way a lot of the bones are removed at the same time.\nThe salmon is placed on the maple planks skin side to the board. The fish is out lined with small nails, and than rapping sneer wire cress cross back and fourth across the fish.\nThe planks of fish are than placed down wind from the fire, standing up close to the coals. The planks are turned end for end each time the fish is basted every 20 minutes. Using a 2-inch brush and basting the fish with melted butter and seasoned to your liking with garlic and any other spice you prefer.\nApproximately 40 lbs. of salmon were done to feed 71 people (about \u00bd lbs. piece of fish each). The fish cooked from 1:00 o'clock until 5:00.\nFish recipes like this one are well worth the wait. I think this is one we might be doing as a annual event.\nWe hope to find more fish recipes from around the country, maybe we might find one you might like.\nLearn how to cook traditional French food in the comfort of your own kitchen and savor the food of France. A list of traditional French food fish recipes from all around France. Regional specialties, using famous foods to create well known dishes from France.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This year, the Bradley Observatory (84 17 38.98 W, 33 45 54.84 N) celebrates its 65th year of operation. Dedicated June 3, 1950, the observatory was named in honor of William C. and Sarah Hall Bradley. It was the brainchild of William Calder, who came to Agnes Scott from Harvard in 1947 under the condition that Agnes Scott would get a telescope that he could use for observations.\nCalder was a student of Harlow Shapley, an American astronomer who helped popularize astronomy. The Lewis H. Beck telescope was bought in 1947 and was the largest telescope in Georgia and the Southeast for 20 years, until the Fernbank Science Center began using its new telescope in 1967. The costs of the telescope and the observatory were $15,000 and $85,000, respectively.\nSARA is a consortium of twelve colleges and universities, which already remotely operates facilities in Arizona and Chile.\nThe member schools are Florida International University, the Florida Institute of Technology, East Tennessee State University, Valdosta State University, Agnes Scott College, Ball State University, the University of Alabama, Butler University, Valparaiso University, Clemson University, Texas A&M University-Commerce and Embry-Riddle Aeronautical University.\nA more in-depth article chronicling the history of the observatory and its impact on Agnes Scott students, as well as the Decatur and Atlanta community, will appear in the upcoming issue of Agnes Scott The Magazine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You will have 7 days Diamonds of Turkey Aegean coastline tours from Istanbul included Sirince Village, Ephesus, Kusadasi, Pamukkale, Didyma, and Bodrum. You will have historical and nature guided tour and enjoy on famous Bodrum beaches. You will fly from Istanbul to Izmir and transfer to Sirince Village. You will stay in the old village. Join to the Pamukkale tour and stay in Kusadasi. After the Pamukkale and Priene tours continue to Bodrum.\nFlight from Istanbul to Izmir. You can choose the flight in the morning, noon, afternoon or late evening. Upon arrival to Izmir Adnan Menderes Airport. You will be picked up from the airport and drive to Sirince Village. It takes 1 hour 10 minutes. Rest of the day and Overnight at the hotel in famous old Sirince Village.\nSirince Village: Sirince is a beautiful hill and local village located about 8 kilometers (5.0 mi) east of the Selcuk town, near Ephesus, in the Aegean hinterland south of Izmir city in Turkey. It is only 30 km away from Kusadasi town. It was settled in the 15th century after Ephesus was abandoned. However, the current buildings date back to the 19th century. It was once Cirkince (ugly). Indeed its habitats gave this name on purpose as they did not want to be bothered by foreigners nor to share the beauty of their village. Still, after years, visitors understood that the village was not ugly at all and called it Sirince (pretty). As the village is located on the top of a mountain, anyone will enjoy the impressive wine yards' and peach trees' views on his way. Today the village is a perfect synthesis of Turk-Greek culture. \u015eirince acquired worldwide fame when tourists flocked to the village in December 2012 to witness the Mayan Apocalypse, as New Age mystics believed its 'positive energy' would aid in weathering the catastrophe, during the 2012 phenomenon.\nPick up from the hotel in Sirince Village and drive to Ephesus Ancient City. You are going to have fully guided Ephesus & Virgin Mary House Tour.\nYou are going to start to visit Celsus library, the Temple Of Hadrian and the Trajan Fountain in the Ancient Ephesus. After walking on the historical marble street the Great Theater where St. Paul preached will meet at the end. Drive Isa Bey Mosque, After visiting Isabey Mosque, stop in a local restaurant for lunch. In the afternoon will visit the Proceed to the shrine of Mother Mary (Virgin Mary House), it is believed where she spent recent years of her life. Then visit Temple Of Artemis (Diana), on the side of one of the seven wonders of ancient World.\nAt the end of the Ephesus tour, Check into hotel in Kusadasi. Overnight in Kusadasi.\nPick up from the hotel in Kusadasi and join to the group. Drive to Pamukkale for fully guided Pamukkale & Hierapolis Ancient City Tour.\nYou will see the Cotton Clifts. Pamukkale is an extraordinary natural wonder. The calceus waters rise from the ground at a temperature of 35 C and tumble down the mountain from a height of 100 meters, forming a myriad of pools. Cream-colored stalactites are formed as the water overflows from the pools, creating a breathtaking sight unequaled in the world. Also visit the ancient ruins of the City of Hierapolis with its basilica, theater, Roman Bath, and Necropolis.\nBreakfast & Check out from the hotel in Kusadasi.\nPick up from the hotel for fully guided Miletus Priene & Didyma Tour. Drive to Priene Ancient City. We will visit the theater, the temple of Athena, the Bouleterium, the Agora, and the Altar of Zeus and Cybele. After visiting Priene, we will have lunch in one of the local restaurants in the area. Afterward, drive to Didyma Ancient City, Visiting Temple Of the Apollon which was the biggest Oracle Center of ancient Turkey. Our last stop will be at Miletos Ancient City, established on the sea shore at that period. Also, it was known Thales's home. The city was destroyed by the Persians in 5th century BC. At the end of the tour, bus transportation to Bodrum. Upon arrival to Bodrum. Check into hotel and Overnight in Bodrum.\nToday is your leisure in Bodrum to explore Bodrum and relax on the famous beaches. Overnight in Bodrum.\nBreakfast and Check out from the hotel in Bodrum. Tour Ends.\nAccording to your travel plan, we can add flight ticket for back to Istanbul, ferry ticket from Bodrum to Kos island or bus ticket to other cities of Turkey. Also, we can combine this program with the Marmaris, Fethiye, Olympos, Antalya and Cappadocia. Please contact us for more details to make your tailor made Turkey tour program.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The global gene therapy market is expanding at an exponential pace due to promising therapeutic outcomes of gene therapy, high prevalence and rise in incidence of cancer, and large number of clinical research pipeline products. In terms of revenue, the market was valued at US$ 17.0 Mn in 2017. It is projected to reach a value of US$ 5164.03 Mn by 2026, expanding at a CAGR of 40.0% from 2018 to 2026. The global gene therapy market is driven by new product approvals and commercialization, increasing demand and number of gene therapy treatment centers, and large number of patient population with unmet medical needs.\nGene therapy is a way of fixing genetic disorders by introducing a normal and healthy gene in place of defective genes in a cell in order to prevent or cure different types of genetic and chronic disorders for which no final cure has been developed. Gene therapy is considered as an important means of treatment, as it helps eliminate the usage of drugs, surgery, or other procedures that can have side-effects on the health of individuals. Moreover, gene therapy results in the formation of beneficial proteins that can help the body perform normal functions to its full potential. Cell-based technologies have been developing at a rapid pace, which in turn has driven the demand for gene therapy. Till date, only five gene therapy products have been approved and are currently in the infancy stage of commercialization. Gendicine was the first commercialized gene therapy product developed by the Shenzhen, China-based SiBiono GeneTech, in 2003. Gendicine was commercialized in China, in 2004.\nThe global market for gene therapy has been expanding due to the intensive research that has offset in the domain of genetics. Demand for gene therapy across the globe has increased significantly due to the rise in awareness about the ability of gene therapy to cure diseases. Gene therapy has considerable potential to eliminate and prevent several genetic disorders and numerous life-threatening disorders, especially cancer, heart diseases, AIDS, cystic fibrosis, and age-related disorders. Gene therapy provides a complete cure to patients affected with genetic disorders, rather than ease symptoms with other therapeutic treatments. Promising therapeutic outcomes and increasing competition among leading biopharmaceutical companies to approve and commercialize gene therapy products in different areas of unmet medical needs, in order to gain the first mover advantage, are projected to boost the gene therapy market during the forecast period. Moreover, high prevalence and rise in incidence rates of different types of cancers for which complete cure has not been developed is a key factor that is expected to drive the gene therapy market.\nLarge number of gene therapy candidates are in late stage clinical studies and are anticipated to commercialize during the forecast period. Significant investments in the field of gene-related research and development by various biopharmaceutical companies, governments, as well as research institutes offers potential opportunity to gain the first mover advantage in the gene therapy market. According to Alliance for Regenerative Medicine, in 2015, more than US$ 10 Bn has been invested by both public as well as private companies in gene therapy-related research. Companies, after initial success rates of gene therapy, are striving to increase the number of treatment centers in order to increase the access to a large number of patient pool requiring gene therapy treatments. Thus, increasing number of gene therapy treatment centers in developed countries offers significant opportunity to global players operating in the gene therapy market.\nThe global gene therapy market has been segmented based on product, application, and region. In terms of product, there are only five gene therapy products that have been approved and commercialized globally. These include Yescarta, Kymriah, Luxturna, Strimvelis, and Gendicine. The Yescarta segment dominated the global market in 2017. It is projected to gain market share by the end of 2026. Yescarta bagged the first mover advantage in 2017 in the U.S, by launching the first CAR T therapy approved by the US FDA for the treatment of certain relapsed or refractory large B-cell lymphoma, including diffuse large B-cell lymphoma (DLBCL). Furthermore, high prevalence of DLBCL and anticipated commercialization of Yescarta in Europe and other developed countries are projected to drive the demand for Yescarta during the forecast period. The Luxturna segment is projected to expand at a significant growth rate during the forecast period.\nBased on application, the global gene therapy market has been divided into ophthalmology, oncology, and adenosine deaminase ?deficient severe combined immunodeficiency (ADA-SCID). The oncology segment accounted for a prominent share of the market in 2017. Of the five gene therapy products approved, three products are directed toward the treatment of certain forms of cancer. Moreover, increased application of gene therapy products in treatment of other forms of cancers is poised to drive the oncology segment during the forecast period.\nBased on region, the global gene therapy market has been segmented into three major regions: U.S., Europe, and Rest of World. Europe is projected to account for a dominant share of more than 40% of the global market by 2026. Increasing number of gene therapy treatment centers in the region and high prevalence and rising incidence rates of non-Hodgkin lymphoma are projected to fuel the gene therapy market in the region during the forecast period. The U.S. is projected to follow Europe, in terms of share of global gene therapy market, by the end of 2026. It is estimated that in the U.S., every year, around 7,500 patients with refractory DLBCL are eligible for CART therapy. The gene therapy market in Rest of World is projected to expand at a notable CAGR during the forecast period. In 2003, Gendicine was approved in China and was commercialized in 2004. Since then, more than 30,000 patients with head and neck cancers have been treated with Gendicine, in China. The anticipated approval and launch of gene therapy products in developed countries, such as Japan, Australia & New Zealand, and GCC is projected to drive the gene therapy market in this region.\nMajority of biopharmaceutical companies have made significant investments in the clinical research and development of gene therapy products. According to a report by the Alliance for Regenerative Medicine, there were around 34 gene therapy candidates in phase III clinical trials as of June 2017. Key players operating in the global market include Gilead Sciences, Inc., Novartis AG, Sibiono GeneTech Co. Ltd., Spark Therapeutics, Inc., CELGENE CORPORATION, and Orchard Therapeutics Limited.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Winter issue of our BECTU sector magazine. Contents include: Head of BECTU Gerry Morrissey, reflects on the achievements of 2017 alongside general secretary, Mike Clancy, who writes on the first year of the merged union. The cover story trails a major new agreement covering big budget film, the first in the UK, which comes into effect in April. Also highlighted are national secretary Sarah Ward's thoughts on the sexual harassment scandal which started in Hollywood but which is now a worldwide concern as well as updates on C&D business and BBC, ITV and West End Theatre pay.\nThe Autumn issue of our BECTU sector magazine. Contents include: Union Week; tax issues for freelancers; the major response to the review on BBC's terms and conditions; ACAS stepping in to help with the West End theatres pay claim; BT pensions; Roger Darlington's column on our changing TV habits.\nProspect, through its BECTU sector, is the union for professionals in the media and entertainment industry.\nProspect's BECTU sector represents people in the IT & telecoms industry as well as media and entertainment.\nMembers in media and entertainment: you can log into the separate BECTU website for full access to your services. IT and telecoms members should continue to use the Prospect site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Currently 605 visitors online at SuggestADoctor.com.\nWhen I had my consultation I felt very comfortable, I asked all of my questions, the day of surgery I laid down on the operating table and I was comfortable! I wasn't scared, I had no reservations at all, before I knew it the surgery was over. You know how you are scared before surgery wondering if it'll come out okay but I did not feel that way at all. It's been a few years since my last procedure and I am looking to have another procedure done soon. I tell all of my friends and relatives to come here because they really do take care of you here, they took care of me for years and now they are taking care of the people I know too, they do great work here.\nI was referred to Dr. Muriel Boreham who is a Urogynecologist and is 2 1\/2 hours away from where I live. I had to have surgery to remove a vaginal cyst. My entire experience with her and her staff was superior. She has a great bedside manner and has a very calming presence. She is attentive, patient, informative, and kind. She spent quality time with me, listening, answering questions, offering solutions, etc. Definitely a hands on, \"real\" doctor that cares for her patient. Absolutely the best experience I have ever had with a physician.\nIt was special, my surgery was very special to me, it changed my life, I am better because of my surgery and where I have been and what I have been through and where I can now go and what I can now do.\nEverybody made me feel really comfortable, if I was nervous at all they helped calm me down, it was definitely a relaxing environment.\nStaff is friendly, thourough and compassionate. I have been extremely happy with Dr. Branch and his staff.\nIs this Review about Sean BRANCH helpful to you?\nAlways friendly and helpful. Never been dissatisfied with my visits.\nI had a great experience with this acne treatment and all of my acne is gone now! Such a great staff and doctor, so friendly and helpful. They were very personably and I would tell all of my friends and family to come here.\nDr Larose is an OBGYN out of Pensacola Florida! Choosing Dr. Larose was the best decision I could have made. He was so great with me in listening to my problems, never questioning the issues I was having but instead taking the next step to find the issue. Dr. Larose is always professional and truly seems to care about his patients well being! He performed my robotic hysterectomy on 11\/15\/17 with zero complications. Truly a great doctor!\nIs this Review about Paul E. LAROSE helpful to you?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..e2a02c5cd9d1cbe2bb6cf8b1ee32f4f82b765d6a
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+{"text":"Teenagers in parts of Caithness and Sutherland are having to make 200 mile-long (322 km) round trips for orthodontic treatment in Inverness.\nThe dental care is not available on the NHS in the areas.\nCampaigners claim 950 orthodontic appointments each year involved local young people, and the trips had an impact on their exam revision time.\nNHS Highland said it, like other health boards, was experiencing a shortage of orthodontic consultants.\nIt said it was working with the North of Scotland Planning Group, which is a collaboration between it and Grampian, Orkney, Shetland, Tayside and Western Isles health boards, on finding a solution.\nCampaign group Caithness Health Action Team (Chat) has begun calling for a consultant to be based in the area because of the impact it says the situation is having on young people.\nCaithness has never had an orthodontist available on the NHS, according to NHS Highland.\nThe nearest available treatment is at Raigmore Hospital in Inverness.\nGetting to the city involves a round trip of more than 200 miles from places such as Wick and Thurso in Caithness and Durness in Sutherland.\nMarie Aitken, of Chat, said the long journeys were affecting young people's schooling, with some having to attend appointments every six weeks.\nShe said: \"They are saying that they are not achieving their best results because they are missing out on exam times and revision times.\nNHS Highland said it recognised that orthodontic treatment was \"one of a number of challenges facing the healthcare system in our more remote and rural areas\".\nA spokeswoman said: \"We have recently employed a locum on a part-time basis based in Inverness but they will not have the capacity to provide a visiting service.\nSniffing lemons key to exam success?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please post all comments or questions in this section about your Silver Wing.\nPlease feel free to post any modifications you have made to your Silver Wing. There are so many available to us all so let's share them with the group. Also, post anything you have about all of the accessories you have installed or know about.\nThis section is for all of us to share information on performing maintenance for the Silver Wing. You can also post pictures here too to help others out too.\nWhat did you do to your SW today.\nMileage on a full tank?\nHow deep is the rider's seat foam on a SW?\nDr. Pulley sliders - which ones do I have?\nWhat is the build history of Silver Wings?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can Bush Be Both Ignorant and a Liar?\nYes. There's no reason for Bush-bashers to choose between the two.\nIs President Bush a liar? The New York Times' David Rosenbaum examined this question with a surfeit of post-Howell-Raines fair-mindedness in the June 22 \"Week in Review\" section. His bottom line: \"[A] review of the president's public statements found little that could lead to a conclusion that the president actually lied\" in two particular instances. The first was when Bush claimed he knew Saddam Hussein to possess large quantities of chemical and biological weapons. The second was when Bush claimed that his tax cut would provide tax relief for everyone who pays income taxes. In both instances, Chatterbox is baffled by Rosenbaum's doubt.\nLet's address Bush's tax claim first. Its falsity is not in dispute. Chatterbox has written elsewhere that Bush lied when he said, \"My jobs and growth plan would reduce tax rates for everyone who pays income tax.\" (The Urban Institute-Brookings Institution Tax Policy Center found 8.1 million people who pay taxesbut will receive no tax cuts.) Rosenbaum recognized that Bush's statement was untrue but expressed doubt that Bush knew it to be untrue. Can a false statement be a lie if the speaker is unaware it is a lie?\nThat leads us immediately to a second question, one that Rosenbaum dared not address: Why is the speaker unaware that his statement is a lie? In Bush's case, the answer is painfully obvious. It's because Bush is a functionally not-bright man. As Chatterbox has explained elsewhere, it's impossible to tell\u2014and, ultimately, of little interest\u2014whether Bush lacks the necessary mental equipment, or whether he's simply incurious. The end result is the same. Even Bush's allies concede that Bush is strikingly ignorant. In the July Vanity Fair, Sam Tanenhaus quoted Richard Perle as saying that when he first met Bush, it was \"clear\" that \"he didn't know very much.\" Perle went on to argue (with what he failed to recognize as condescension) that Bush is an eager pupil. But there isn't much evidence to support even that.\nWe've found the weapons of mass destruction. You know, we found biological laboratories. You remember when Colin Powell stood up in front of the world and he saidIraq has got laboratories, mobile labs to build biological weapons. They're illegal. They're against the United Nations' resolutions and we've so far discovered two. And we'll find more weapons as time goes on.\nIn fact, it has yet to be proved that the two mobile labs were used (or even designed to be used) to build biological weapons. It isn't possible that Bush fails to grasp that. So, why did he say something so obviously untrue? Chatterbox posed the question to The Nation's David Corn, who has written extensively on the question of Bush's veracity. In Corn's view, the key to Bush's lies isn't necessarily that he doesn't know any better, but that he doesn't care. \"He mischaracterizes situations to fit his pattern of thinking,\" Corn explained. \"Does he believe he's lying? I don't know.\" But \"he still should be held accountable, whether he made a mistake of this nature in good faith or in bad faith.\" Amen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A group of children from years four five and six visited Normandy in France for a week long residential.\nDuring the visit they were expected to use their spoken French skills. In addition, they visited: The Normandy beaches, and Museum, The War Memorial Graves, Bayeaux Tapestry, a French market.\nThe staff and children had an amazing week and it was the first of many trips for Valley View.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ali Imsirovic took down the biggest prize yet at this years Poker Masters when he outgunned the field in the second of the $25k NLHE events on the schedule, taking down a career best score of $462,000 for his 5th six-figure score of the year, and his life!\nBy 'work to do', the Canadian meant somehow spin up his 7BB into something playable against his five adversaries from the US, but it wasn't to be \u2013 a cold-deck not allowing him any wiggle room and he eventually hit the rails, although a $99k payout and his second Masters cash was a very good effort.\nYu's $330,000 and a jump to 3rd spot in the overall Purple Jacket standings will soften the blow, while Imsirovic's excellent year makes him a new and dangerous face on the high-stakes streets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"On the pages of this web discuss about Tracky6z. Tracky6z article reviews probably able to provide helpful information to ye.\nPlease see the list of article titles below, pertaining to Tracky6z. You get see the specified information about Tracky6z. May be efficient.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you own a home or a business, some day you're going to have to call a plumber. Problems with soil and waste pipes, water supply and distribution pipes and, for a building, the sanitary and storm drains will occur. Blockages disrupt your daily routine. Albuquerque Plumbing Heating & Cooling can solve your problems. We do high-quality work at affordable prices. We're a Placitas drain cleaning company with the know-how and equipment to complete your Placitas drain cleaning job the right way. Our licensed, bonded and insured plumbers can unclog garbage disposals, dishwasher drains, bathroom sink drains and more. We can even give you advice about how to prevent clogged drains.\nWe are the go-to Placitas drain cleaning professionals and we are always standing by to give you a helping hand. Contact us today.\nOur plumbers are available to handle your problems 24\/7. Our rooter service in Placitas handles such problems as blocked branch mains and clogs in showers and tubs. Our plumbers have the equipment for Placitas drain cleaning. These include powerful mechanical drain rooters. Some of them weigh over 100 pounds and can go as far as 100 feet into a drain. We're Placitas drain cleaners with heavy-duty equipment that can completely unblock clogs anywhere on your property.\nIt's not a good idea for property owners to use drain cleaners to try to move clogs in metal drains because since most of them are caustic, they can damage pipes. Instead, call professional Placitas drain cleaners to take care of your drain cleaning in Placitas. We will clear your drains and sewers without damaging your pipes.\nIn the article \"Clear Any Drain\" by Merle Henkenius that appears on the \"This Old House\" website, the author states that property owners can sometimes successfully clear clogged drains. The author also says that clearing some blockages can only be made by professional plumbers, and the author cautions that putting too much pressure on pipes by using plungers and other equipment can permanently damage pipes them. Hot water poured down a drain can sometimes clear a clog. Caution must be used when pouring hot water PVC pipes. The PVC Fittings Online article \"Can I Use PVC for Hot Water?\" says water over 120 F can damage PVC. So many times it's more cost effective to call our Placitas drain cleaning company for service.\nFor fast, reliable and affordable Placitas drain cleaning service, just pick up the phone and give us a call!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Newmarket Garden Machinery Services is a family business, established in Newmarket for over 45 years. We stock a comprehensive range of petrol, battery and electric garden machinery; from small garden mowers and strimmers to large ride on machines, professional mowers, robotics and Barbecues, plus a large range of accessories and parts.\nWe have recently re-located to our new modern premises with onsite workshop and customer parking. Our experienced technicians are always available to offer advice, guidance and support plus a local delivery and collection service. All new machinery and BBQ's are supplied fully assembled, inspected and can be delivered locally free of charge.\nWe are authorised dealers for Honda, Mountfield, Stiga, Makita, Cub Cadet, Billy Goat and Lawnflight garden machinery. Additionally, we also stock Napoleon barbecues and FloGas LPG products. Visit our showroom, shop online or call us on 01638 600202 for expert advice and guidance to help you choose the right product.\nWe are able to offer repairs and servicing in our fully equipped onsite workshop, with highly trained technicians providing a professional service from start to finish. As main agents for Honda, Stiga, Mountfield, Billy Goat, Cub Cadet, Lawnflite and Makita we can also supply spare parts on request.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When I was a child, the story Peter Rabbit was something I could always relate to. Off limits or not, who could resist a garden of delicious veggies? Old man McGregor wasn't that scary after all; I'd gladly dodge a garden sieve or two to get my furry mouth around a couple of plump radishes. Luckily for me, spring is here, and I don't need to squeeze through a dilapidated wooden fence to peruse Spokane's prime assortment of roughage. I thought it would be unfair if I simply spotlighted the simple garden or Caesar salads you could stumble upon at any restaurant in town. I raised on my haunches, conjured my gluttonous inner bunny and bounced my way to salads that I'd fight a farmer for.\nIf you've ever made fun of someone ordering a salad, this menu item is sweet revenge. Hay J's packs your plate full of langoustine, blackened ahi, avocado, bacon, Gorgonzola and a hard-boiled egg. Mix in some light Gorgonzola vinaigrette and top it with a giant gulf shrimp, and you're quick to learn the joke's on whoever didn't order this one-of-a-kind Spokane masterpiece.\nTwigs Bistro: 4320 S. Regal St.\nIf you're looking for a hearty meal but still want to keep it light, Twigs offers the best of both worlds with tender steak pieces saut\u00e9ed with Worcestershire and garlic over mixed baby greens and romaine, tossed with feta cheese, red onions, tomato, roasted cashews and a balsamic vinaigrette. Perfect eats to accompany Twig's uber famous happy hour martinis.\nThe Blackbird: 905 N. Washington St.\nI've always felt tomatoes are among the world's most underrated foods. Chef Molly buys them locally in bulk and cures them to have fresh flavorful colors on her plates all year long. In a fresh herb salad, they are undeniable. A perfect match with mozzarella curd, arugula and lemon vinaigrette. The special touch is their in-house \"Soil\"\u2014a crumb mixture of cracker and secret spices.\nWiley's Downtown Bistro: 115 N. Washington St.\nIt isn't spring until you start eating berries. If I have a say, filling your salad with pungent tart strawberries is the best way to do so. Wiley's must agree. I got my hands on a delicious version that added grilled chicken, Gorgonzola, candied walnuts, tomato and finished the whole thing with a berry vinaigrette. It's never a bad thing when you get lunch and dessert on the same plate.\nClover: 913 E. Sharp Ave.\nWhen people ask me what my favorite food is, I often answer with vinegar. So when you find a salad that has both stone-ground mustard vinaigrette and cornichons, you start to slowly fall in love. She's beautiful, with crunchy bibb lettuce, avocado, artichoke hearts and chewy flakes of house-smoked salmon. Clover continues to be top of the food chain in just about any category.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The four largest cities in Texas are among those challenging the law.\nA federal judge is hearing arguments Monday from cities, counties and other organizations that want to prevent the implementation of a new Texas law banning sanctuary cities.\nPlaintiffs claim the new sanctuary cities ban violates the U.S. Constitution by threatening free speech and equal protection. The ban allows local law enforcement to ask about the immigration status of people who are detained, even for something as routine as a traffic stop.\n\"It really is a law that would license every individual police officer in this one area of the law, immigration, to be his or her own vigilante deciding on their own what to do, when to do it and who to target,\" Saenz says.\nUnder Senate Bill 4 local law enforcement that refused federal immigration requests to detain immigrants could lose state grant money, or their jobs.\nThe author of the statute, Sen. Charles Perry (R-Lubbock) says he's confident it will pass constitutional muster.\nSo far, Texas' biggest cities: Houston, Dallas, San Antonio and Austin have all joined the lawsuit. Bexar County is also a party.\nThe small border community of El Cenizo with a population of 3,300, and Maverick County, southwest of San Antonio, were the first to file legal challenges in May.\nEl Cenizo Mayor Raul Reyes says 20 percent of his town's residents are undocumented. To protect them, the city passed a \"safe harbor\" ordinance in 1999. It prevents local police and officials from questioning people about their immigration status. That's a policy that won't be permitted under the law scheduled to go into effect September 1.\nReyes says with or without papers people living in his town are considered part of the community.\n\"Who have their homestead in our community, who pay property taxes, why on earth would we try to hinder upon their rights and their livelihoods just because they don't have a piece of paper,\" says Reyes.\nBut on the banks of the Rio Grande River, which divides Texas from Mexico, there's also opposition to the lawsuit and support for the new law. Roy Vasquez didn't want his town of El Cenizo to challenge the law because he's afraid his community will become a target.\nAnd while many urban police chiefs oppose the sanctuary cities ban, the Sheriffs' Association of Texas supports it. Jackson County Sheriff Andy Louderback is the group's legislative director.\nLouderback says the law will free up police officers to report someone to federal immigration officials if they feel the person is a public safety concern without fear of reprisal from local officials.\nOrlando Garcia is the federal judge who is weighing the legal claims. He recently ruled it was unlawful for Bexar County to hold an immigrant for 72 days just because immigration agents had requested his detention. Criminal charges against the immigrant had been dropped.\nWhat plaintiffs want now is for Garcia to put the enforcement of the sanctuary cities ban on hold, while those for and against it continue a legal debate that may last a very long time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Her life is sometimes bright and sometimes dull. She had been through the extreme sorrows and lived her deepest fears..\nShe feels habitual of the darkness as it is the only time where she could be with him in her dreams..\nfor years is back to her..\nSo, she is pushing him away..\nShe still loves him immensely but being with him forever is again a mere dream..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We all want to create products that make people's lives easier and more enjoyable, products that delight people around us and ourselves. Creating great experiences starts with understanding your users, their behavior, motivations and goals.\nRemote user studies are ideal for getting insights from end-users who may be located anywhere in the world. These studies provide access to a larger pool of potential customers, cut out travel time, and can significantly lower the cost of usability testing.\nIn this article I would like to discuss methods and tools for effective remote UX research, as well as how to make sure your research findings are put to good use.\nFlexibility. The remote method allows you to connect with your customers from anywhere in the world. There's also greater user diversity and a quicker set-up time thanks to online implementation.\nEnvironment. User interviews are conducted in the participant's usual working environment with the hardware and software they use on a day-to-day basis, or while they are on the go.\nThere are a lots of reasons you might want to conduct an in-person study instead.\nInformation security.You may want to keep control over confidential test materials.\nInability to use screen sharing.Some studies may require you to talk to users who don't have reliable high-speed internet connections.\nFace and body movements.In some studies, it's importance to be able to see the user's body or face, or track eye movements.\nWhen you need to do a lab study, use reliable equipment, keep the interface 100% secure, and monitor the user's physical movements when you can't use screen sharing tools.\nRemote research can be divided into two categories: moderated and unmoderated research.\nIn moderated research, a moderator speaks directly to the participants. Moderated research allows you to gather in-depth qualitative feedback, as the researcher can make personal observations about the participants.\nSince moderated studies provide lots of context and insight into exactly what users are doing and why, these methods are best for research projects that are looking for new ideas to come from observation.\nUnlike moderated research, automated research does not involve any conversation between the researcher and the participants. Instead, you use online tools and services to collect information automatically.\nWhat percent of users can successfully log in?\nHow long does it take for users to find a product?\nIf all you need is performance data, and not the reasons users behave the way they do, then automated testing is for you.\nWhat. You need to decide what you want to learn in the study. Before starting, everyone on the team should agree on the questions you plan to ask and the assumptions you plan to test.\nWho. You'll need to carefully recruit people based on your goals\u2014do you want to study existing customers, prospective customers, or representative customers? I personally like a collaborative approach here where you work with your team to identify goals for the study, as well list the characteristics of the people you want to interview or exclude from your study. After you've done this, figure out precise criteria you can use to identify those people and write questions to screen them.\nHow. Determine the best research methodology for a particular study that will allow you to reach your goals, and write an interview guide or that keeps you on track during the interviews.\nNext, I would like to focus on \"Who\" and \"How\" steps and discuss what is different when you plan and manage a remote study.\nIt is very important to understand the usage context of an interface\u2014the physical environment where people are using a product. Remote research allows you to instantly recruit people who are right in the middle of performing the task you're interested in, in a process called Time-Aware Research.\nTime-awareness means that users are interested in what they're doing because they're doing it for their own reasons, not because you're asking them to.\n(For more details on Time-aware research, I highly recommend Nate Bolt's amazing book \"Remote Research.\" It also has a lot of practical tips and templates that you can apply in your own remote study).\nAn effective strategy for live recruiting is to use a popup form that will open the recruiting screener, which users can fill out to opt-in to your study. Alternatively, you can place a link to your survey in some noticeable spot on your site. The easiest way to create a survey is to link to a form in created in Google Forms, or use a tool like Ethnio.\nIt will take some trial and error to determine when exactly to show the popup and the right balance between session length and incentive.\nEthnio also works to find participants in real-time from mobile apps.\nFor mobile survey requests, you'll need to place code from Android or iOS libraries into your app, then trigger an Ethnio screener based on actions you define in your app.\nAn alternative approach is to just send a push notification to the selected audience with a request to take part in the study.\nIn addition to the channels listed above, you can also use other channels to recruit someone for a study. Use agencies when you have higher budget\u2014Usertesting is one such agency that screens people very thoroughly. Facebook or Craiglist ads work in some cases too.\nLive recruiting is typically your best option if you have an existing site or an app. But in order to get people to participate, you'll most likely need to provide an incentive.\nThe incentive amount you should offer to the participants depends heavily on whom you are looking to recruit.\nOn the other hand, there are plenty of people who are willing to donate their time for no incentive at all. This is especially true for services with loyal users, who are quite often eager to participate in the study. They may bring up problems they have encountered with your product.\nThere is a great article by Charles Liu, design researcher from KISSmetrics, sharing his experiments with recruiting people via emails with a zero budget.\nIn the article, he shares several email templates that you can use for recruiting people for your own studies. It's definitely worth a look.\nThere are several different tools that can be used for remote research.\nIn the screener form, I'll typically ask the participants what screen sharing service is convenient for them. More often than not, Skype is the first choice, but sometimes they are not familiar with screen sharing. In this case, I suggest what better works for us.\nSkype\u2014you can observe the face of the participant and their screen.\nGoogle Hangout \u2014you can observe either their screen or face.\nGoToMeeting($49\/month) \u2014you can observe their face and the screen, and you can also give keyboard and mouse control to the participant.\nGoToMeeting lets you view your participants' screen, and allows your participants to see and even control your screen. This feature is very useful when you want to test a mobile app launched in an emulator.\nThe first two solutions may be a bit pricey, but they will cover all your screen recording needs. All three have nice video and audio recording options to record portions of the screen or windows.\nThere are a number of automated tools that prompt users to perform a task and then record their behavior. If you want to learn how users interact with some part of your website or app, task elicitation is the way to go.\nDo users have difficulty performing the task?\nHow many people abandon the task, and where?\nWhat is the most common path users take through the interface to complete a task?\nWhere do users go to perform the task and where do they expect to go?\nThere are several different tools you can use for task elicitation.\nLoop11. A browser bar prompts users to perform a task. Automated analysis includes task completion rate, time on task, first click, click stream, most frequent success and fail points.\nUserTesting. Users' spoken are captured and synced with videos of their site behavior as they perform your tasks. Videos are downloadable and can be annotated with comments. You can also see users' demographics and system info. Users are recruited by UserTesting.com from a preselected panel.\nWhen running a mobile study, there are two different tools I recommend.\nLookback. Lookback helps you learn how people are actually using your mobile app. It records the screen of your app, along with the face and voice of the user, as well as all their taps and touches. With recordings like these, it's super simple for anybody to send you feedback or bug reports, straight from within your app.\nLookback for Android records everything on screen, including any native app, mobile website or prototype. (iOS blocks third-party apps from recording this data).\nOnce a record is completed and uploaded, it appears on Lookback's website where you and your collaborators can view, discuss, rename and share them.\nUnicorns.io. Unicorns is an app that streams whatever you're doing on your phone\u2014playing a game, texting, swiping through Tinder, etc. Unicorns allows you to conduct usability tests, showcase pre-release apps or upcoming features, and chat with the users live.\nPlaytestCloud. PlaytestCloud is an online service for crowdsourced testing for mobile games. The video recordings show you the player's screen during their playthrough and highlight their touches on the screen. You can also discuss and share the videos inside your team.\nInvisionapp. Invisionapp allows you to transform your static screens into clickable, interactive prototypes complete with gestures, transitions, and animations. You can share a link with your users to open on their desktop or mobile device, then conduct usability tests with the users live. Invisionapp is great for quickly demonstrating concepts, collecting feedback in one place and making rapid iterations.\nRecently, Invisionapp partnered with Lookback so that you can record a user as they navigate through your Invisionapp prototype on their iOS device. You can capture their screen as well as audio and video reactions.\nUserTesting. I already mentioned this tool before in relation to un-moderated web studies, but it works great for mobile studies as well. You can watch users complete the tasks you specify (including gestures), and listen to them speak their minds as they interact with your app. Their mobile recorder works with whatever is displayed on the mobile device, including wireframes and prototypes.\nRemote research allows a large number of observers to record their thoughts and observations at their computers.\nHaving developers, project managers, marketers and more observe and engage with the research can make them more invested in the outcome. Often, that makes it more likely that the findings will be applied.\nIncluding everyone might sound like a waste, but it's very important. Each team member has a unique perspective and will interpret your findings in different ways, leading to a more complete understanding of how people interact with your app\/product\/service.\nInstruct everyone to capture two kinds of things on paper while watching. Write important insights (good and bad) learned from the user on the different Post-it notes.\nLet's say yellow is for good and pink is for bad. Write any specific implementation ideas on green post-it notes. Then, ask everyone to write each down on their post-it notes using a thick, black marker.\nAfter several remote sessions, gather around a white board and ask everyone to put their Post-its on the wall. The goal of this step is to distill everyone's notes into a common list of insights.\nOnce data is grouped together, work with the team to try and find trends, patterns, and themes that are relevant to the research question. Then, discuss them.\nMake sure you do it all together\u2014get the goals and questions from the team, then do the up-front work conducting a survey or getting participants to the interview. The team should be there during the interviews, usability tests, and debriefing sessions, so they can observe users with their own eyes.\nOne really important aspect of a collaborative process during your research study is that if any member of your team is remote, then everyone should be remote.\nWhy? We at Stanfy did a lot of experiments with this, trying to involve a remote person via Google Hangout or GoToMeeting, but it never worked. Sooner or later, all discussions start to happen between people who are in the same space, ignoring those who are outside.\nThere are many tools that you could use for remote team collaboration. With the design team at Stanfy we tried Murally, Realtimeboard and Boardthing. They have different pricing models but basically do the similar job \u2013 providing you with a shared canvas for online brainstorming and collaboration, which allows you to visually organize your ideas. Murally currently is our favorite space for such collaborative work.\nChoose a research method based on what you want to find out.\nSelect the tool or service that best supports your research goals.\nCreate and post a recruiting screener and confirm valid participants.\nDesign a set of tasks or write an interview guide.\nConduct research session and pay out the incentives (don't forget to include your team).\nSummarize findings together with your team.\nIf you're working with an audience scattered across the globe, all under budget and time constraints, remote user testing can provide the rich data to help make informed design decisions, and to understand how people interact and behave with the interface you've made.\nAnna Iurchenko is a UX Designer and Sketchnoter. She is driven to create products that improve people's day to day life. She is working at Stanfy where she helps clients create meaningful and engaging user experiences for their customers. Anna loves to visualize ideas, and sketching serves her as the great tool for connecting her mind and creativity. Keep up with her on Medium or Twitter.\nI love the illustrations and font, which font has been used?\nGreat article. Speacial thanks for mentioning the tools!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At WasteRatings, we strive to bring you the best service at the best price available. We have a wide selection of dumpster sizes in stock in Upper Jay, NY, with flexible & timley pickup and delivery. Give us a call today for all your dumpster rental and roll off needs.\nThere are many cases where you will need the help of garbage in your home work a few of works that want help of specialist dumpster services contain house demolition, remodeling an integral part of your home, cleaning space for your children books store, cutting garden, packing your property hold stuff in order to make a move to different spot from your home, tearing your wall of your cooking area or bathroom. A professional Dumpster rental vendor will make suggestions the best journey so as to buy the right dumpster that best fits inside your needs.\nYou've got two selections when it comes to pick up. If you aren't completed with the task in advance, but the dumpster regarding rent is entire, you want to call up and ask that they come to get it and change it out with yet another. This way, there is absolutely no reason to quit the work. Just keep performing and keep slinging things away. If you are certain that the job is complete, you can call up to have it eliminated for good.\nDumpers are available with different payload capabilities and throwing heights. You'll be able to choose dumpers depending on the direction of loading as well as unloading. You can pick a dumper with front side, side as well as back loading. Dumpster rental for remodelling is an easy as well as affordable way to take care of all of your dumping needs. Your online 3 Yard Dumpster in Upper Jay, NY helps you inside size, shape and composition of the best dumpster for the project to help make your remodelling as sleek as possible. Regardless of if the container is being conducted the street or your property, your truck dropping away from and obtaining the pot needs area to maneuver thus be sure to select a spot for your dumpster that is straightforward to get to for you personally and for the shipping truck.\nThis resolution obtained should held by necessary activities such as lessen the production of throw away, promote making use of recyclable goods, recycle your generated garbage in eco-friendly method. The first a couple of tasks are in your hands. What about the 3rd task, recycling where possible the trash with eco-friendly manner? Mounting up trash from time to time becomes unavoidable; projects including house remodelling, construction, as well as green backyard clean up as well as industrial jobs are sure to produce huge a ton of trash which should be disposed quickly. This or else will have poor impact on the environment.\nThe clever and easiest way to get out of every one of these troubles is to rent a dumpster from professional 3 Yard Dumpster in Upper Jay, NY vendor who grips the entire task on your behalf. They help you in obtaining the right type of dumpster and they will put it at the explained location at that time specified by you actually. The dumpsters offered by these 3 Yard Dumpster in Upper Jay, NY organizations will be tagged with the a ton of waste which might be handled in order to notify the consumer ahead concerning the tons of throw away that has to be broke up with into the dumpster. This specific precaution can eradicate the possibilities of miscarriage as well as overload.\nBodyweight Limit: Each and every dumpster comes with a bodyweight limit this means you can place your throwaways to a restriction only. The extra weight limit of dumpsters can range involving 2 to 12 tons. The dumpsters usually are weighted in case weight is found above the acceptable limit when compared with you have to pay extra cash. So it is incredibly necessary to get information about your dumpster weight restriction.\nSome junk removal corporations offer work, meaning that they'll not only provide the dumpster and remove the garbage, but they will also allow you to carry products out of the the location of the truck or perhaps dumpster. Consider no matter if you need this particular service based on the degree of the cleanout and your household and friends' chance to do this hard work yourselves. It's going to likely be more pricey, but it could get the job completed quickly. Be aware that when you're buying labor, particularly when you're spending by the hour, you might not be able to spend some time to sift through each box as well as bag seeking sentimental products.\nPromoting trying to recycle in an Eco-friendly way not only inhibits environmental pollution, and also stands for the particular notation that will \"one man's garbage is another person's treasure\". Usage of reused products is especially popularized and plenty of nations, you can get huge income on promoting recycled merchandise in the marketplaces.\nDumpsters usually have as well as end openings as well as a tiny top cracking open for when structure workers want to toss points in previously mentioned. Rental prices on dumpsters and throw offs is usually dependant upon the type of rubbish you have, the length of time you'll need the particular dumpster, and the sized the dumpster anyone rent. Get one that is too small as well as multiple opt for ups may be needed and your prices will rise. A much too big of a dumpster will probably be costlier along with the space defintely won't be needed. Assess which dimension you want very carefully.\nThere are various Md 3 Yard Dumpster in Upper Jays supplied by a vehicle which is designed to have and get rid of bulk resources. These dumpster are generally distinguished through dump vans by arrangement. Large dumpsters may be used in the location of your choice in your business look. Next day delivery options help to make cleanup easy and efficient; it is possible to call and attain a next day delivery of the dumpster. Quick pickup of a full dumpster is also a property to structure 3 Yard Dumpster in Upper Jay. Using a structure 3 Yard Dumpster in Upper Jay makes garbage disposal less difficult and productive as it doesn't require debris to become left in your yard which is your spot of business.\nAnd finally, the use of this particular service would also allow you to stay away from having a front-yard full of trash, throughout remodeling. Have you ever seen the top yard of someone's house filled with settee sets, dried wall, mosaic glass and outdated carpet, amongst other things? Maybe, you might have experienced this specific yourself. While you wait for the rubbish truck to reach you, you're hoping there isn't any nasty weather that will come via and move the debris towards the front yards of many of your neighbors together with into the avenue. This can be aggravating as well as uncomfortable.\nBasically, what it really boils down to could be that the more neighborhood a company is, the more they may care about giving a good experience for each purchaser. But they will also be more prepared to do so. These people probably use a local center, which means you defintely won't be paying to help ship ones trash far, far away. Being a bonus, you'll be putting dollars into your individual, local economy rather than just providing an already huge corporation. If you're still uncomfortable, just give the business a telephone call; if they will take the time to talk to you and soon you are satisfied, chances are that the remainder of the experience will probably be just as pleasant.\nThe first thing that you must do is definitely evaluate your individual needs. Occasionally, big corporations will need extremely large dumpsters that is able to be rolled away by a vehicle. When the vehicle rolls these large dumpsters out, it leaves a new one that is completely bare and ready for that businesses' refuse.\nWithin the flipside, also watch out for businesses who are very big. Sometimes individuals companies have got so many shoppers that each client gets extremely impersonal service. For them, burning off one client just isn't something useful. This is especially true if your company is so large that it is across the country.\nWith spring season arriving, many of our minds use cleaning out your homes as well as selling what exactly we no longer need in the lives. A yard sale is a superb thing on its own, but when you find the whole area in onto it, your capabilities can merge to make a truly fun celebration for your family, your neighbors, as well as anyone who stops by to purchase goods. The following is how you can get your neighborhood together for any community yard sale.\nIf you have a major remodel or even renovation challenge, having just one giant dumpster indicates you only have one means of treatment of waste. Ignore multiple excursions to the dump and the service fees they charge. Have a very 3 Yard Dumpster in Upper Jay, NY company leave a dumpster with your driveway as well as fill it up. It can save you money through cutting back on petrol, unnecessary excursions to the dump, and managing the material. They usually will opt for and dispose of your trash for you. The next time you have a remodelling project, you'll want to call the 3 Yard Dumpster in Upper Jay, NY company to hold your squander to a minimum.\nVery first, take the time to come across customer reviews or even testimonials this company. When anybody had complaints you will find there's decent opportunity they travelled online to vent their frustrations. His or her Better Business Bureau report may be specially worthwhile to look at.\nYou and your family tend to be excited about the modern living room you actually envision. You intend to get rid of the fatigued carpeting, gates, baseboards and many types of molding during your home.\nThere are few things in life which can be certain. You may be pretty certain if your french fries will be cold before you finish your Major Mac. You may be fairly sure when you inadvertently bump into ones water goblet it's going to fall over and spill all over your demonstration notes. And you may guarantee that whenever you get around to starting among the hundreds of do it yourself projects you have been putting off since you moved in, you're going to need to have a 3 Yard Dumpster in Upper Jay, NY to handle the large piles involving construction squander that will be put aside! You can quickly and simply dispose of waste such as aged counter tops as well as sinks, kitchen appliances and cupboards in one cleanse sweep through tossing all of them into a design dumpster. Sometimes a uncomplicated home renovation will take longer than you anticipate and you really need to get rid of ones trash at the earliest opportunity. You can demand a next day shipping of your dumpster and you will have the complete dumpster picked up just like quickly. The garbage and crap can quickly accumulate and overtake you and you can sound stuck not understanding what to do; you are able to clean this mess up insurance agencies the specialist throw just about all excess to the 3 Yard Dumpster in Upper Jay, NY.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Most rivers in Europe have alternative names in different languages. Some rivers have also undergone name changes for political or other reasons. This article attempts to give all known alternative names for all major European rivers. It also includes some lesser rivers that are important because of their location or history.\nThis article does not offer any opinion about what the \"original\", \"official\", \"real\", or \"correct\" name of any river is or was. Rivers are listed alphabetically by their current best-known name in English. The English version is followed by variants in other languages, in alphabetical order by name, and then by any historical variants and former names.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Position for daily support, basic tool configuration changes and user training related to a service catalog tool.\nAssignments include response and troubleshooting to help desk tickets and service requests with escalation to L2 development for complex issues or fixes. Service catalog changes such as adding new offerings and modifying existing offerings. Responding to emails within the support mailbox. Create user documentation and provide regularly scheduled training courses and 1-1 training to requestors, fulfillers and managers.\nExperience handling business partner support including after-hours effort.\nStrong knowledge of MS Excel, MS Word, MS Outlook and MS Powerpoint. Excellent communication skills required.\nITILv3 knowledge or certification is preferred.\nKnowledge of Tivoli Service Request Manager (TSRM) is preferred.","meta":{"redpajama_set_name":"RedPajamaC4"}}